{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta E mphasis" -1 263 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 264 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 128 "Derivation of combined 7 stage, \+ order 4 and 5 Runge-Kutta schemes, each with an additional 8 stage, or der 4 FSAL embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 45 "by Peter St one, Gabriola Island, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Ver sion: 21.12.2012" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "load procedures for \+ constructing Runge-Kutta schemes etc. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 9 "butcher.m" }{TEXT -1 2 ", " }{TEXT 262 7 "roots.m" }{TEXT -1 5 " and " }{TEXT 262 6 "intg.m" }{TEXT -1 33 " are required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read into a Maple session by commands similar to those th at follow, where each file path gives the location of the m-file." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/ butcher.m\";\nread \"C:\\\\Maple/procdrs/roots.m\";\nread \"C:\\\\Mapl e/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 50 "#=================================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Bogacki-Shampine 7 stage, order 5 Runge-Kutta scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 160 "See: An Efficient Runge-Kutta (4,5) Pair by P.Bogacki \+ and L.F.Shampine\n Computers and Mathematics with Applications, Vol. 32, No. 6, 1996, pages 15 to 28" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "See: An Efficient Runge-Kutta (4,5) Pair by P.Bogacki and L.F.Shampine\n Compute rs and Mathematics with Applications, Vol. 32, No. 6, 1996, pages 15 t o 28" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 988 "ee := \{c[2]=1/6,\nc[3]=2/9,\nc[4] =3/7,\nc[5]=2/3,\nc[6]=3/4,\nc[7]=1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=2 /27,\na[3,2]=4/27,\na[4,1]=183/1372,\na[4,2]=-162/343,\na[4,3]=1053/13 72,\na[5,1]=68/297,\na[5,2]=-4/11,\na[5,3]=42/143,\na[5,4]=1960/3861, \na[6,1]=597/22528,\na[6,2]=81/352,\na[6,3]=63099/585728,\na[6,4]=5865 3/366080,\na[6,5]=4617/20480,\na[7,1]=174197/959244,\na[7,2]=-30942/79 937,\na[7,3]=8152137/19744439,\na[7,4]=666106/1039181,\na[7,5]=-29421/ 29068,\na[7,6]=482048/414219,\na[8,1]=587/8064,\na[8,2]=0,\na[8,3]=444 0339/15491840,\na[8,4]=24353/124800,\na[8,5]=387/44800,\na[8,6]=2152/5 985,\na[8,7]=7267/94080,\n\nb[1]=587/8064,\nb[2]=0,\nb[3]=4440339/1549 1840,\nb[4]=24353/124800,\nb[5]=387/44800,\nb[6]=2152/5985,\nb[7]=7267 /94080,\n\n`b^`[1]=6059/80640,\n`b^`[2]=0,\n`b^`[3]=8559189/30983680, \n`b^`[4]=26411/124800,\n`b^`[5]=-927/89600,\n`b^`[6]=443/1197,\n`b^`[ 7]=7267/94080,\n\n`b*`[1]=2479/34992,\n`b*`[2]=0,\n`b*`[3]=123/416,\n` b*`[4]=612941/3411720,\n`b*`[5]=43/1440,\n`b*`[6]=2272/6561,\n`b*`[7]= 79937/1113912,\n`b*`[8]=3293/556956\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i], i=1..7),``],[`b^`,seq(`b^`[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7+#\"\"\"\"\"'F(%! GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$ $=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#F.F6#\"#o\"$(H#!\"%\"#6#\"#U\"$ V\"#\"%g>\"%hQF+F+F+F+7+#F6F3#\"$(f\"&GD##\"#\")\"$_$#\"&*4j\"'Gde#\"& `'e\"'!3m$#\"%u\"\"'W#f*#!&U4$\"&P*z#\"(P@:)\" )RWu>#\"'1hm\"(\"=R5#!&@%H\"&o!H#\"'[?[\"'>UTF+F+7+F)#\"$(e\"%k!)\"\"! #\"(R.W%\")S=\\:#\"&`V#\"'+[7#\"$(Q\"&+[%#\"%_@\"%&)f#\"%ns\"&!3%*F+7+ %\"bGF]pF`pFapFdpFgpFjpF]qF+7+%#b^G#\"%fg\"&S1)F`p#\"(*=f&)\")!o$)4$# \"&6k#Ffp#!$F*\"&+'*)#\"$V%\"%(>\"F]qF+7+%#b*G#\"%zC\"&#*\\$F`p#\"$B\" \"$;%#\"'THh\"(? " 0 "" {MPLTEXT 1 0 372 "RK5_7eqs := [ op(RowSumConditions(7,'expanded')),op(OrderConditions(5,7,'expanded')) ]:\nsubs(ee,RK5_7eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4_7eqs^ ` := subs(b=`b^`,OrderConditions(4,7,'expanded')):\nsubs(ee,`RK4_7eqs^ `):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4_8eqs*` := subs(b=`b*`,Or derConditions(4,8,'expanded')):\nsubs(ee,`RK4_8eqs*`):\nmap(u->lhs(u)- rhs(u),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F $F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that is, the 2-norm \+ of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\nsqrt(add(s ubs(ee,errterms5_7[i])^2,i=1.. nops(errterms5_7))):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zF$p@#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also calculate the 2 norm of the principal error of the order 4 embedded schemes." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7^` := subs(b=`b ^`,PrincipalErrorTerms(4,7,'expanded')):\nsqrt(add(subs(ee,`errterms4_ 7^`[i])^2,i=1.. nops(`errterms4_7^`))):\nevalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+Feaf5!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_8*` := subs(b=`b *`,PrincipalErrorTerms(4,8,'expanded')):\nsqrt(add(subs(ee,`errterms4_ 8*`[i])^2,i=1.. nops(`errterms4_8*`))):\nevalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+y(\\:1\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := StageOrd erConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following r espective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 6) = b[j]*(1-c[j]) " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" \"'*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 5 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "[Sum(b[i]*a[i,1],i=1+1..7 )=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6)];\neval(sub s(Sum=add,%)):\nsubs(ee,%);\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"a G6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6 #F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F -,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FE FjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF- /-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/#\"$(e\"%k!)F%/\"\"!F)/#\"'rL\\\"(?J@#F+/ #\"%zM\"&+7$F//#\"$H\"\"&+[%F3/#\"$Q&\"%&)fF7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "I n particular, the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG 6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 14 "holds (since " }{XPPEDIT 18 0 "b[2]=0" " 6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\" #F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\" 'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1 ),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F +%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Su m(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,%));\ns ubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG \"\"\"-F$6$*&&%\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F *;F5\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"%\"\"\"&% \"aG6$F(\"\"$F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F +6$F4F(F)&F+6$F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FA F(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\" (F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "None of t he principal error conditions are satisfied by the order 5 scheme. " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK5_7err_eqs := PrincipalE rrorConditions(5,7,'expanded'):\nsubs(ee,RK5_7err_eqs):\nmap(u->`if`(l hs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions \+ are satisfied by the 8 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 95 "The largest magnitude of an error term is about 24 t imes the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 147 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms( 4,8,'expanded')):\nsubs(ee,`errterms4_8*`):\nmap(abs,evalf[8](%));\nev alf[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$ \")\\F*3&!#7$\")#3;r(F&$\")9iY5F&$\")C,GNF&$\")(HSf*!#8$\")@$*p:F&$\") f!)=>F&$\")h`qDF&$\")*4!)>$F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%7 C!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions are satisfied by the 7 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 95 "The larges t magnitude of an error term is about 18 times the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterm s4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\nsubs(ee,` errterms4_7^`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(%))/min(op(%)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")M(GG#!#7$\")d`q$)F&$\")c\" >_\"F&$\"),JCMF&$\")$*3&R\"F&F$$\")'y,z#F&F/$\")wH]Y!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"%+=!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 97 "simultaneous construction of the 7 stage, order 5 \+ scheme and the 8 stage, order 4 scheme .. short" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "In this subsection we giv e a " }{TEXT 260 18 "short construction" }{TEXT -1 215 " of the Bogack i-Shampine scheme by first obtaining a 7 stage order 5 scheme with a n umber of parameters and then assigning values to these parameters with out any consideration as to how these values might be chosen." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The foll owing \"simple\" order conditions (given in abreviated form) are relev ant for the construction of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := Simp leOrderConditions(5):\n[seq([i,SO5[i]],i=[1,2,4,5,8,12,13,15,16])]:\nl inalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim ](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7+7%\"\"\"%%~~~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF( #F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@ 6#*&FCF(F2F(F(F(#F(\"#C7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6 #*&FCF(-F@6#*&F8F(FCF(F(F(#F(\"#g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF) /*&F,F(-F@6#*&FMF(FCF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b *`,SimpleOrderConditions(4)):\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nli nalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim] (%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7'7%\"\"\"%%~~~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(# F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF( F(#F(\"#77%\"\")F)/*&F,F()F2F:F(#F(F5Q)pprint306\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We incorporate the simplifying conditions:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\" \"$\"\"(\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i, 2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&% \"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3 ;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 sch eme and use the simple order conditions in the first table above." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We also \+ need to specify that the linking coefficients for the 8th (FSAL) stage are given by the weights of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6 #%\"iG&%\"aG6$\"\")F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"i G\"\"\"" }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "We use the simplifying condition: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i \+ = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+ ;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for t he order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7 ,'expanded'):\n`SO4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expa nded')): \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8 *`[i],i=[1,2,4,7,8])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded') ),op(StageOrderConditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a [i,2],i=3..7)=0,add(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n \+ add(b[i]*c[i]*a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_ eqns := [seq(b[i]=a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op( simp_eqns),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"' !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\" $*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 2/3 ;" "6#/&%\"cG6#\"\"&*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6 $\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\" (\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " \+ are initially parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "e1 := \{c[2]=1/6,c[3]=2/9,c[5]=2/3 ,c[6]=3/4,c[7]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs(e1,cdns):\nn ops(eqns);\nindets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[ 5]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 " " 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "info level[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 : = solve(\{op(eqns)\},indets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*` [3],`b*`[5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3957 "e3 := \{c[ 8] = 1, b[2] = 0, a[8,2] = 0, a[2,1] = 1/6, c[7] = 1, `b*`[2] = 0, c[2 ] = 1/6, a[4,1] = 27/4*c[4]^3-9/2*c[4]^2+c[4], a[7,1] = 1/369360*(2696 3280*b[7]*c[4]^3+2226420*c[4]^3+8208000*b[7]*a[7,5]*c[4]^3-55296000*a[ 6,5]*c[4]^3+516096000*a[6,5]*c[4]^3*b[7]-11344995*c[4]^2-1211105280*a[ 6,5]*c[4]^2*b[7]+128532480*a[6,5]*c[4]^2-17209440*b[7]*a[7,5]*c[4]^2+1 1911860*b[7]*c[4]^2-97878016*a[6,5]*c[4]+11446569*c[4]-44784900*b[7]*c [4]+928972800*a[6,5]*c[4]*b[7]+12011040*b[7]*a[7,5]*c[4]-2790720*b[7]* a[7,5]+24510464*a[6,5]+15513120*b[7]-233963520*b[7]*a[6,5]-3142638)/(4 *c[4]-3)/c[4]/b[7]/(25*c[4]-17), a[4,2] = -9*(3*c[4]-1)*c[4]^2, b[5] = 9/160*(-20*c[4]+420*b[7]*c[4]+27-420*b[7])/(-2+3*c[4]), a[7,6] = 16/4 38615*(51300*c[4]^2-328320*b[7]*a[7,5]*c[4]^2-276480*a[6,5]*c[4]^2+258 0480*a[6,5]*c[4]^2*b[7]-73359*c[4]+465120*b[7]*a[7,5]*c[4]-4300800*a[6 ,5]*c[4]*b[7]+454656*a[6,5]*c[4]+26163-164160*b[7]*a[7,5]+1720320*b[7] *a[6,5]-180224*a[6,5])/b[7]/(4*c[4]-3)^2, b[6] = -64/855*(420*b[7]*c[4 ]-45*c[4]-420*b[7]+44)/(4*c[4]-3), a[3,1] = 2/27, a[6,1] = 1/768*(3840 0*a[6,5]*c[4]^2-4149*c[4]^2-51712*a[6,5]*c[4]+10017*c[4]-4104+17408*a[ 6,5])/c[4]/(25*c[4]-17), `b*`[4] = -1/1458*(532*`b*`[3]-243+108*`b*`[5 ])/c[4]/(3-7*c[4]+4*c[4]^2), a[8,1] = 1/720*(420*b[7]*c[4]-420*b[7]+41 )/c[4], a[8,7] = b[7], a[8,4] = -1/60*(420*b[7]-41)/c[4]/(-12+88*c[4]- 177*c[4]^2+108*c[4]^3), a[6,2] = -27/64*(71*c[4]-27)/(25*c[4]-17), a[5 ,1] = -2/9*(33*c[4]^2-45*c[4]+16)/c[4]/(25*c[4]-17), a[4,3] = 81/4*c[4 ]^3-9/2*c[4]^2, `b*`[8] = 1/1458*(-9396*`b*`[5]*c[4]^2+7047*c[4]^2+422 82*b[7]*c[4]^2-19836*c[4]^2*`b*`[3]-2187*c[4]-55404*b[7]*c[4]+10564*c[ 4]*`b*`[3]+9180*`b*`[5]*c[4]-1944*`b*`[5]-1368*`b*`[3]+13122*b[7])/(26 4*b[7]*c[4]-29*c[4]+9-72*b[7])/(-1+c[4]), a[7,4] = -1/30780*1/c[4]*(-2 7648000*a[6,5]*c[4]^3+3939840*c[4]^3-13789440*b[7]*c[4]^3+258048000*a[ 6,5]*c[4]^3*b[7]+34473600*b[7]*c[4]^2-605552640*a[6,5]*c[4]^2*b[7]-273 6000*b[7]*a[7,5]*c[4]^2+64266240*a[6,5]*c[4]^2-8669700*c[4]^2-48939008 *a[6,5]*c[4]+3912480*b[7]*a[7,5]*c[4]+464486400*a[6,5]*c[4]*b[7]-28440 720*b[7]*c[4]+6381207*c[4]-1395360*b[7]*a[7,5]-116981760*b[7]*a[6,5]-1 571319+7756560*b[7]+12255232*a[6,5])/(144*c[4]^3-248*c[4]^2+129*c[4]-1 8)/b[7]/(25*c[4]-17), a[8,3] = -729/3040*(60*b[7]*c[4]-20*c[4]+9-60*b[ 7])/(-2+9*c[4]), a[8,6] = -64/855*(420*b[7]*c[4]-45*c[4]-420*b[7]+44)/ (4*c[4]-3), a[5,4] = 32/9*(-2+3*c[4])/(-2+9*c[4])/c[4]/(25*c[4]-17), c [3] = 2/9, c[5] = 2/3, c[6] = 3/4, `b*`[1] = -1/4374*(-1215*c[4]+162*` b*`[5]*c[4]+2394*c[4]*`b*`[3]-532*`b*`[3]+243-108*`b*`[5])/c[4], a[6,4 ] = -1/48*1/c[4]*(3200*a[6,5]*c[4]-684*c[4]-2176*a[6,5]+513)/(225*c[4] ^2-203*c[4]+34), a[8,5] = 9/160*(-20*c[4]+420*b[7]*c[4]+27-420*b[7])/( -2+3*c[4]), b[1] = 1/720*(420*b[7]*c[4]-420*b[7]+41)/c[4], `b*`[7] = 1 /243*(-17739*c[4]^2+14256*`b*`[5]*c[4]^2+30096*c[4]^2*`b*`[3]+12150*c[ 4]-14896*c[4]*`b*`[3]-13392*`b*`[5]*c[4]+1824*`b*`[3]-2187+2592*`b*`[5 ])*b[7]/(264*b[7]*c[4]-29*c[4]+9-72*b[7])/(-1+c[4]), b[4] = -1/60*(420 *b[7]-41)/c[4]/(-12+88*c[4]-177*c[4]^2+108*c[4]^3), a[7,3] = -1/28880* (27219780*c[4]^3-212874480*b[7]*c[4]^3+12312000*b[7]*a[7,5]*c[4]^3-552 96000*a[6,5]*c[4]^3+516096000*a[6,5]*c[4]^3*b[7]-42144375*c[4]^2-12111 05280*a[6,5]*c[4]^2*b[7]+128532480*a[6,5]*c[4]^2+303391620*b[7]*c[4]^2 -25814160*b[7]*a[7,5]*c[4]^2-140551740*b[7]*c[4]+18016560*b[7]*a[7,5]* c[4]-97878016*a[6,5]*c[4]+21821139*c[4]+928972800*a[6,5]*c[4]*b[7]+245 10464*a[6,5]+24562440*b[7]-233963520*b[7]*a[6,5]-4142988-4186080*b[7]* a[7,5])/(6-35*c[4]+36*c[4]^2)/b[7]/(25*c[4]-17), a[3,2] = 4/27, b[3] = -729/3040*(60*b[7]*c[4]-20*c[4]+9-60*b[7])/(-2+9*c[4]), `b*`[6] = -16 /2187*(648*`b*`[5]*c[4]+504*c[4]*`b*`[3]-486*c[4]-112*`b*`[3]+243-432* `b*`[5])/(4*c[4]-3), a[5,2] = -4*(-1+c[4])/(25*c[4]-17), a[7,2] = -3/4 *(264*b[7]*c[4]-29*c[4]+9-72*b[7])/b[7]/(25*c[4]-17), a[6,3] = -9/256* (-13851*c[4]^2+19200*a[6,5]*c[4]^2-25856*a[6,5]*c[4]+12597*c[4]+8704*a [6,5]-3078)/(225*c[4]^2-203*c[4]+34), a[5,3] = 4*(18-71*c[4]+63*c[4]^2 )/(-2+9*c[4])/(25*c[4]-17)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "the coefficients in terms of \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] " "6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b *G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6# \"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do\n print(`====== ================================================`);\n temp_eq := c[i i]=subs(e3,c[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print( lhs(temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n e nd if;\n print(``); \n for jj to ii-1 do\n temp_eq := a[ii,j j]=subs(e3,a[ii,jj]);\n if lhs(temp_eq)=rhs(temp_eq) then\n \+ print(lhs(temp_eq)*` is a parameter`);\n else\n pr int(temp_eq);\n end if;\n print(``);\n end do:\nend do: \n`######################################################`;\n``;\nfor \+ ii to 7 do\n temp_eq := b[ii]=subs(e3,b[ii]);\n if lhs(temp_eq)=rh s(temp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else \n print(temp_eq);\n end if;\n print(``);\nend do:\n`######## ##############################################`;\n``;\nfor ii to 8 do \n temp_eq := `b*`[ii]=subs(e3,`b*`[ii]);\n if lhs(temp_eq)=rhs(te mp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n \+ print(temp_eq);\n end if;\n print(``);\nend do:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%W============================================== ========G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"\" '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============================= =========================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"$#\"\"#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"\"#\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" $\"\"##\"\"%\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%W============================================== ========G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\"%0~i s~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",(*&#\"#FF'F(*$)&%\"cG6#F'\" \"$F(F(F(*&#\"\"*\"\"#F(*$)F/F6F(F(!\"\"F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\" \"#,$*(\"\"*\"\"\",&*&\"\"$F,&%\"cG6#F'F,F,F,!\"\"F,)F0F(F,F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"%\"\"$,&*&#\"#\")F'\"\"\"*$)&%\"cG6#F'F(F-F-F-*&#\"\"*\" \"#F-*$)F0F6F-F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W========================================== ============G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"\"# \"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*&#\"\"#\"\"*F(*(,(*&\"#LF()&%\"c G6#\"\"%F,F(F(*&\"#XF(F3F(!\"\"\"#;F(F(F3F9,&*&\"#DF(F3F(F(\"#F.&F%6$F'\"\"&F.F3F.F.*(\"&ce#F.FF(FKF(FF*(\")g=\">\"F(F1F(FKF(F(*(\");!yy*F(FCF(F5F(FF*&\") plW6F(F5F(F(*(\")+\\yWF(F1F(F5F(FF**\"*+G(*G*F(FCF(F5F(F1F(F(**\")S5,7 F(F1F(F>F(F5F(F(*(\"(?2z#F(F1F(F>F(FF*&\")k/^CF(FCF(F(*&\")?J^:F(F1F(F (*(\"*?N'RBF(F1F(FCF(FF\"(QE9$FFF(,&*&F8F(F5F(F(F9FFFFF5FFF1FF,&*&\"#D F(F5F(F(\"#s#F,)&%\"cG6#\"\"%F(F,F,*(\"*![uG@F,&%\"bG6#F'F ,F2F,!\"\"**\")+?J7F,F9F,&F%6$F'\"\"&F,F2F,F,*(\")+gHbF,&F%6$\"\"'FAF, F2F,F<**\"*+g4;&F,FDF,F2F,F9F,F,*&\")vV9UF,)F3\"\"#F,F<**\"+!G06@\"F,F DF,FKF,F9F,F<*(\"*![K&G\"F,FDF,FKF,F,*(\"*?;R.$F,F9F,FKF,F,**\")gT\"e# F,F9F,F?F,FKF,F<*(\"*S8d\"F2*&\"(glv(F,F@F ,F,*&\")K_D7F,F6F,F,F,,**&\"$W\"F,F:F,F,*&\"$[#F,FGF,F2*&\"$H\"F,F/F,F ,\"#=F2F2F@F2,&*&\"#DF,F/F,F,\"#F.F4F.F.**\"(+3I%F.FDF.F4F.F;F.FA* (\"'cYXF.FDF.F4F.F.\"&jh#F.*(\"'gT;F.F;F.F>F.FA*(\"(?.s\"F.F;F.FDF.F.* &\"'C-=F.FDF.FAF.F;FA,&*&F7F.F4F.F.\"\"$FA!\"#F.F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============ ==========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",$*&#F(\"$ ?(F(*&,(*(\"$?%F(&%\"bG6#\"\"(F(&%\"cG6#\"\"%F(F(*&F0F(F1F(!\"\"\"#TF( F(F5F:F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,$*&# \"$H(\"%SI\"\"\"*&,**(\"#gF.&%\"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"#?F.F7 F.!\"\"\"\"*F.*&F2F.F3F.F=F.,&\"\"#F=*&F>F.F7F.F.F=F.F=" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"%,$*&#\"\"\"\"#gF,*(,&*&\"$?%F,&%\"bG6#\"\"(F,F,\"#T!\"\"F,&% \"cG6#F(F7,*\"#7F7*&\"#))F,F8F,F,*&\"$x\"F,)F8\"\"#F,F7*&\"$3\"F,)F8\" \"$F,F,F7F,F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,$*&#\"\"*\"$g\"\"\"\"*&,**&\"#?F .&%\"cG6#\"\"%F.!\"\"*(\"$?%F.&%\"bG6#\"\"(F.F3F.F.\"#FF.*&F9F.F:F.F7F .,&\"\"#F7*&\"\"$F.F3F.F.F7F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',$*&#\"#k\"$b) \"\"\"*&,**(\"$?%F.&%\"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"#XF.F7F.!\"\"*& F2F.F3F.F=\"#WF.F.,&*&F:F.F7F.F.\"\"$F=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"(&%\"bG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W################################################## ####G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"$?(F'*&,(*(\"$?%F'&F%6#\"\"(F'& %\"cG6#\"\"%F'F'*&F/F'F0F'!\"\"\"#TF'F'F3F8F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"$H(\"%SI\"\"\"*&,**(\"#gF-&F%6#\" \"(F-&%\"cG6#\"\"%F-F-*&\"#?F-F5F-!\"\"\"\"*F-*&F1F-F2F-F;F-,&\"\"#F;* &FF+F4F+&F%6#\"\"$F+F:*&\"%(=#F+F5F+F:*(\"&/a&F+ F?F+F5F+F:*(\"&k0\"F+F5F+FEF+F+*(\"%!=*F+F1F+F5F+F+*&\"%W>F+F1F+F:*&\" %o8F+FEF+F:*&\"&AJ\"F+F?F+F+F+,**(\"$k#F+F?F+F5F+F+*&\"#HF+F5F+F:\"\"* F+*&\"#sF+F?F+F:F:,&F+F:F5F+F:F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "RK5 _7eqs := [op(RowSumConditions(7,'expanded')),op(OrderConditions(5,7,'e xpanded'))]:\nsimplify(subs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\n nops(%);\n`RK4_8eqs*` := subs(b=`b*`,OrderConditions(4,8,'expanded')): \nsimplify(subs(e7,`RK4_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The Bogacki-Shampine scheme (excluding the 7 stage, o rder 4 embedded scheme) can be obtained by setting " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=3/7" "6#/&%\"cG6#\"\"%*&\"\"$\" \"\"\"\"(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]=4617/20480" " 6#/&%\"aG6$\"\"'\"\"&*&\"% " 0 "" {MPLTEXT 1 0 144 "e4 := \{c[4 ]=3/7,a[6,5]=4617/20480,a[7,5]=-29421/29068,b[7]=7267/94080,`b*`[3]=12 3/416,`b*`[5]=43/1440\}:\ne5 := `union`(e4,simplify(subs(e4,e3))):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 947 "e5 := \{c[2 ] = 1/6, c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, c[7] = 1, c[8] = 1, b[2] \+ = 0, `b*`[2] = 0, a[2,1] = 1/6, a[5,3] = 42/143, b[5] = 387/44800, a[6 ,3] = 63099/585728, a[8,5] = 387/44800, b[1] = 587/8064, a[8,2] = 0, a [8,7] = 7267/94080, c[4] = 3/7, a[3,2] = 4/27, a[7,4] = 666106/1039181 , `b*`[4] = 612941/3411720, `b*`[6] = 2272/6561, a[4,1] = 183/1372, `b *`[3] = 123/416, `b*`[7] = 79937/1113912, a[5,1] = 68/297, `b*`[8] = 3 293/556956, a[4,2] = -162/343, a[8,4] = 24353/124800, a[6,5] = 4617/20 480, a[7,5] = -29421/29068, b[7] = 7267/94080, `b*`[5] = 43/1440, a[3, 1] = 2/27, a[8,3] = 4440339/15491840, `b*`[1] = 2479/34992, a[7,3] = 8 152137/19744439, a[6,4] = 58653/366080, a[5,4] = 1960/3861, a[8,1] = 5 87/8064, a[7,2] = -30942/79937, a[6,1] = 597/22528, a[5,2] = -4/11, b[ 3] = 4440339/15491840, a[7,6] = 482048/414219, b[6] = 2152/5985, a[4,3 ] = 1053/1372, a[7,1] = 174197/959244, b[4] = 24353/124800, a[6,2] = 8 1/352, a[8,6] = 2152/5985\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(e5,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7 ),``],[`b*`,seq(`b*`[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+# \"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$ V$#\"%`5F:F+F+F+F+F+7+#F.F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"#\"%g>\"%hQF+ F+F+F+7+#F6F3#\"$(f\"&GD##\"#\")\"$_$#\"&*4j\"'Gde#\"&`'e\"'!3m$#\"%u\"\"'W#f*#!&U4$\"&P*z#\"(P@:)\")RWu>#\"'1hm\"( \"=R5#!&@%H\"&o!H#\"'[?[\"'>UTF+F+7+F)#\"$(e\"%k!)\"\"!#\"(R.W%\")S=\\ :#\"&`V#\"'+[7#\"$(Q\"&+[%#\"%_@\"%&)f#\"%ns\"&!3%*F+7+%\"bGF]pF`pFapF dpFgpFjpF]qF+7+%#b*G#\"%zC\"&#*\\$F`p#\"$B\"\"$;%#\"'THh\"(?QL\"F*$!)@.BZF*$\")r#\\n(F*F+F+F+F+F+7+$ \")nmmmF*$\")Bc*G#F*$!)OOOOF*$\")H1PHF*$\")^Sw]F*F+F+F+F+7+$\")+++vF*$ \")b.]EF1$\")k8,BF*$\")[Fx5F*$\")3>-;F*$\")XRaAF*F+F+F+7+$\"\"\"\"\"!$ \")A)f\"=F*$!)$)zqQF*$\")p#)GTF*$\")K\"*4kF*$!)R975!\"($\"):vj6F]oF+F+ 7+FV$\")fEzsF1$FXFX$\")QCmGF*$\")AO^>F*$\")HRQ')!#5$\")el&f$F*$\")sFCx F1F+7+%\"bGFaoFcoFdoFfoFhoF[pF]pF+7+%#b*G$\")lZ%3(F1Fco$\")3tcHF*$\")Z d'z\"F*$\")66')HF1$\")o)GY$F*$\"),CwrF1$\")e\\7fFjoQ(pprint16\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The norm \+ of the principal error terms is as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 120 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded') :\nsqrt(add(evalf(subs(e5,errterms5_7[i]))^2,i=1.. nops(errterms5_7))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zF$p@#!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the we ights " }{XPPEDIT 18 0 "`b^`[i], i=1" "6$&%#b^G6#%\"iG/F&\"\"\"" } {TEXT -1 128 " . . 7, for the auxiliary 7 stage, order 4 embedded sch eme by using the quadrature conditions along with the requirements tha t " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^`[7]=b[7]" " 6#/&%#b^G6#\"\"(&%\"bG6#F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b^`[2]= 0" "6#/&%#b^G6#\"\"#\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b^`[ 6]=443/1197" "6#/&%#b^G6#\"\"'*&\"$V%\"\"\"\"%(>\"!\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "quad_eqns := subs(b=`b^`,QuadratureConditions(4,7,'e xpanded')):\ne6 := \{`b^`[2]=0,`b^`[6]=443/1197\}:\ne7 := `union`(e5,e 6):\ncdns2 := [`b^`[7]=b[7],op(quad_eqns)]:\neqns2 := subs(e7,cdns2): \nnops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%#b^G6#\"\"\"&F%6#\" \"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "e8 := solve(\{op(eqns2)\}):\ne9 := `union`(e7,e8):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "`RK4_7eqs^` := subs(b=`b^ `,OrderConditions(4,7,'expanded')):\nsimplify(subs(e9,`RK4_7eqs^`)):\n map(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The coef ficients for the whole scheme including the 7 stage and 8 stage embedd ed order 4 schemes are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "subs(e9,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b^`,seq(`b^`[i],i=1..7),``] ,[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7,7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1 F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#F. F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"#\"%g>\"%hQF+F+F+F+7+#F6F3#\"$(f\"&GD# #\"#\")\"$_$#\"&*4j\"'Gde#\"&`'e\"'!3m$#\"%u\" \"'W#f*#!&U4$\"&P*z#\"(P@:)\")RWu>#\"'1hm\"(\"=R5#!&@%H\"&o!H#\"'[?[\" '>UTF+F+7+F)#\"$(e\"%k!)\"\"!#\"(R.W%\")S=\\:#\"&`V#\"'+[7#\"$(Q\"&+[% #\"%_@\"%&)f#\"%ns\"&!3%*F+7+%\"bGF]pF`pFapFdpFgpFjpF]qF+7+%#b^G#\"%fg \"&S1)F`p#\"(*=f&)\")!o$)4$#\"&6k#Ffp#!$F*\"&+'*)#\"$V%\"%(>\"F]qF+7+% #b*G#\"%zC\"&#*\\$F`p#\"$B\"\"$;%#\"'THh\"(? " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 98 "simultaneous construction of the 7 stage, order 5 sc heme and the 8 stage, order 4 scheme .. longer" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "In this subsection we giv e a " }{TEXT 260 19 "longer construction" }{TEXT -1 207 " of the Bogac ki-Shampine scheme. As before, we start by obtaining a 7 stage order 5 scheme with a number of parameters and then attempt to discover why t he values that are chosen for the scheme are suitable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following \"sim ple\" order conditions (given in abreviated form) are relevant for the construction of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := SimpleOrderConditi ons(5):\n[seq([i,SO5[i]],i=[1,2,4,5,8,12,13,15,16])]:\nlinalg[augment] (linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg [delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7 %\"\"\"%%~~~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F) /*&F,F()F2F/F(#F(\"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@6#*&FCF(F2F(F( F(#F(\"#C7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6#*&FCF(-F@6#*& F8F(FCF(F(F(#F(\"#g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF)/*&F,F(-F@6#*& FMF(FCF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b*`,SimpleOrderC onditions(4)):\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nlinalg[augment](l inalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7% \"\"\"%%~~~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/ *&F,F()F2F/F(#F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF(F(#F(\"#77%\" \")F)/*&F,F()F2F:F(#F(F5Q)pprint326\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We incorporate the simplifying conditions:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0; " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\" \"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 . . 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum( b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+ F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 scheme a nd use the simple order conditions in the first table above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We also need t o specify that the linking coefficients for the 8th (FSAL) stage are g iven by the weights of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6#%\"iG&% \"aG6$\"\")F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\" " }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"( \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the order 4 \+ embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 158 "simp_eqns := [add(b[i]*a[i,2],i=3..7)=0,add(b [i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i]*c[i]*a[i ,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "simp_eqns2 : = [add(b[i]*a[i,1],i=1+1..7)=b[1],seq(add(b[i]*a[i,j],i=j+1..7)=b[j]*( 1-c[j]),j=2..6)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 538 "SO5_7 := SimpleOrderConditions(5,7,'expande d'):\n`SO4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded')): \+ \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[ 1,2,4,7,8])]:\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(StageO rderConditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3.. 7)=0,add(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i ]*c[i]*a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [se q(b[i]=a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_eqns), op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\" #\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 2/3;" "6#/ &%\"cG6#\"\"&*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " are initial ly parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "e1 := \{c[2]=1/6,c[3]=2/9,c[5]=2/3,c[6]=3/4,c[7 ]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs(e1,cdns):\nnops(eqns);\ni ndets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5]\};\nnops(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 60 "indets(eqn s) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5]\}" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel [solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := sol ve(\{op(eqns)\},indets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],` b*`[5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "inde ts(map(rhs,e3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"bG6#\"\"(&% \"cG6#\"\"%&%\"aG6$\"\"'\"\"&&F-6$F'F0&%#b*G6#\"\"$&F46#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3957 "e3 := \{c[8] = 1, b [2] = 0, a[8,2] = 0, a[2,1] = 1/6, c[7] = 1, `b*`[2] = 0, c[2] = 1/6, \+ a[4,1] = 27/4*c[4]^3-9/2*c[4]^2+c[4], a[7,1] = 1/369360*(26963280*b[7] *c[4]^3+2226420*c[4]^3+8208000*b[7]*a[7,5]*c[4]^3-55296000*a[6,5]*c[4] ^3+516096000*a[6,5]*c[4]^3*b[7]-11344995*c[4]^2-1211105280*a[6,5]*c[4] ^2*b[7]+128532480*a[6,5]*c[4]^2-17209440*b[7]*a[7,5]*c[4]^2+11911860*b [7]*c[4]^2-97878016*a[6,5]*c[4]+11446569*c[4]-44784900*b[7]*c[4]+92897 2800*a[6,5]*c[4]*b[7]+12011040*b[7]*a[7,5]*c[4]-2790720*b[7]*a[7,5]+24 510464*a[6,5]+15513120*b[7]-233963520*b[7]*a[6,5]-3142638)/(4*c[4]-3)/ c[4]/b[7]/(25*c[4]-17), a[4,2] = -9*(3*c[4]-1)*c[4]^2, b[5] = 9/160*(- 20*c[4]+420*b[7]*c[4]+27-420*b[7])/(-2+3*c[4]), a[7,6] = 16/438615*(51 300*c[4]^2-328320*b[7]*a[7,5]*c[4]^2-276480*a[6,5]*c[4]^2+2580480*a[6, 5]*c[4]^2*b[7]-73359*c[4]+465120*b[7]*a[7,5]*c[4]-4300800*a[6,5]*c[4]* b[7]+454656*a[6,5]*c[4]+26163-164160*b[7]*a[7,5]+1720320*b[7]*a[6,5]-1 80224*a[6,5])/b[7]/(4*c[4]-3)^2, b[6] = -64/855*(420*b[7]*c[4]-45*c[4] -420*b[7]+44)/(4*c[4]-3), a[3,1] = 2/27, a[6,1] = 1/768*(38400*a[6,5]* c[4]^2-4149*c[4]^2-51712*a[6,5]*c[4]+10017*c[4]-4104+17408*a[6,5])/c[4 ]/(25*c[4]-17), `b*`[4] = -1/1458*(532*`b*`[3]-243+108*`b*`[5])/c[4]/( 3-7*c[4]+4*c[4]^2), a[8,1] = 1/720*(420*b[7]*c[4]-420*b[7]+41)/c[4], a [8,7] = b[7], a[8,4] = -1/60*(420*b[7]-41)/c[4]/(-12+88*c[4]-177*c[4]^ 2+108*c[4]^3), a[6,2] = -27/64*(71*c[4]-27)/(25*c[4]-17), a[5,1] = -2/ 9*(33*c[4]^2-45*c[4]+16)/c[4]/(25*c[4]-17), a[4,3] = 81/4*c[4]^3-9/2*c [4]^2, `b*`[8] = 1/1458*(-9396*`b*`[5]*c[4]^2+7047*c[4]^2+42282*b[7]*c [4]^2-19836*c[4]^2*`b*`[3]-2187*c[4]-55404*b[7]*c[4]+10564*c[4]*`b*`[3 ]+9180*`b*`[5]*c[4]-1944*`b*`[5]-1368*`b*`[3]+13122*b[7])/(264*b[7]*c[ 4]-29*c[4]+9-72*b[7])/(-1+c[4]), a[7,4] = -1/30780*1/c[4]*(-27648000*a [6,5]*c[4]^3+3939840*c[4]^3-13789440*b[7]*c[4]^3+258048000*a[6,5]*c[4] ^3*b[7]+34473600*b[7]*c[4]^2-605552640*a[6,5]*c[4]^2*b[7]-2736000*b[7] *a[7,5]*c[4]^2+64266240*a[6,5]*c[4]^2-8669700*c[4]^2-48939008*a[6,5]*c [4]+3912480*b[7]*a[7,5]*c[4]+464486400*a[6,5]*c[4]*b[7]-28440720*b[7]* c[4]+6381207*c[4]-1395360*b[7]*a[7,5]-116981760*b[7]*a[6,5]-1571319+77 56560*b[7]+12255232*a[6,5])/(144*c[4]^3-248*c[4]^2+129*c[4]-18)/b[7]/( 25*c[4]-17), a[8,3] = -729/3040*(60*b[7]*c[4]-20*c[4]+9-60*b[7])/(-2+9 *c[4]), a[8,6] = -64/855*(420*b[7]*c[4]-45*c[4]-420*b[7]+44)/(4*c[4]-3 ), a[5,4] = 32/9*(-2+3*c[4])/(-2+9*c[4])/c[4]/(25*c[4]-17), c[3] = 2/9 , c[5] = 2/3, c[6] = 3/4, `b*`[1] = -1/4374*(-1215*c[4]+162*`b*`[5]*c[ 4]+2394*c[4]*`b*`[3]-532*`b*`[3]+243-108*`b*`[5])/c[4], a[6,4] = -1/48 *1/c[4]*(3200*a[6,5]*c[4]-684*c[4]-2176*a[6,5]+513)/(225*c[4]^2-203*c[ 4]+34), a[8,5] = 9/160*(-20*c[4]+420*b[7]*c[4]+27-420*b[7])/(-2+3*c[4] ), b[1] = 1/720*(420*b[7]*c[4]-420*b[7]+41)/c[4], `b*`[7] = 1/243*(-17 739*c[4]^2+14256*`b*`[5]*c[4]^2+30096*c[4]^2*`b*`[3]+12150*c[4]-14896* c[4]*`b*`[3]-13392*`b*`[5]*c[4]+1824*`b*`[3]-2187+2592*`b*`[5])*b[7]/( 264*b[7]*c[4]-29*c[4]+9-72*b[7])/(-1+c[4]), b[4] = -1/60*(420*b[7]-41) /c[4]/(-12+88*c[4]-177*c[4]^2+108*c[4]^3), a[7,3] = -1/28880*(27219780 *c[4]^3-212874480*b[7]*c[4]^3+12312000*b[7]*a[7,5]*c[4]^3-55296000*a[6 ,5]*c[4]^3+516096000*a[6,5]*c[4]^3*b[7]-42144375*c[4]^2-1211105280*a[6 ,5]*c[4]^2*b[7]+128532480*a[6,5]*c[4]^2+303391620*b[7]*c[4]^2-25814160 *b[7]*a[7,5]*c[4]^2-140551740*b[7]*c[4]+18016560*b[7]*a[7,5]*c[4]-9787 8016*a[6,5]*c[4]+21821139*c[4]+928972800*a[6,5]*c[4]*b[7]+24510464*a[6 ,5]+24562440*b[7]-233963520*b[7]*a[6,5]-4142988-4186080*b[7]*a[7,5])/( 6-35*c[4]+36*c[4]^2)/b[7]/(25*c[4]-17), a[3,2] = 4/27, b[3] = -729/304 0*(60*b[7]*c[4]-20*c[4]+9-60*b[7])/(-2+9*c[4]), `b*`[6] = -16/2187*(64 8*`b*`[5]*c[4]+504*c[4]*`b*`[3]-486*c[4]-112*`b*`[3]+243-432*`b*`[5])/ (4*c[4]-3), a[5,2] = -4*(-1+c[4])/(25*c[4]-17), a[7,2] = -3/4*(264*b[7 ]*c[4]-29*c[4]+9-72*b[7])/b[7]/(25*c[4]-17), a[6,3] = -9/256*(-13851*c [4]^2+19200*a[6,5]*c[4]^2-25856*a[6,5]*c[4]+12597*c[4]+8704*a[6,5]-307 8)/(225*c[4]^2-203*c[4]+34), a[5,3] = 4*(18-71*c[4]+63*c[4]^2)/(-2+9*c [4])/(25*c[4]-17)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "the coefficients in terms of " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[ 7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" " 6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6 #\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\" &" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do\n print(`========== ============================================`);\n temp_eq := c[ii]=s ubs(e3,c[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs( temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end i f;\n print(``); \n for jj to ii-1 do\n temp_eq := a[ii,jj]=s ubs(e3,a[ii,jj]);\n if lhs(temp_eq)=rhs(temp_eq) then\n \+ print(lhs(temp_eq)*` is a parameter`);\n else\n print( temp_eq);\n end if;\n print(``);\n end do:\nend do:\n`# #####################################################`;\n``;\nfor ii t o 7 do\n temp_eq := b[ii]=subs(e3,b[ii]);\n if lhs(temp_eq)=rhs(te mp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n \+ print(temp_eq);\n end if;\n print(``);\nend do:\n`############ ##########################################`;\n``;\nfor ii to 8 do\n \+ temp_eq := `b*`[ii]=subs(e3,`b*`[ii]);\n if lhs(temp_eq)=rhs(temp_eq ) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n p rint(temp_eq);\n end if;\n print(``);\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%W=================================================== ===G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"#\"\"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================== ====================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\" $#\"\"#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"\"#\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\" \"##\"\"%\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================================== ====G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\"%0~is~a~ parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",(*&#\"#FF'F(*$)&%\"cG6#F'\"\"$F( F(F(*&#\"\"*\"\"#F(*$)F/F6F(F(!\"\"F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,$*(\" \"*\"\"\",&*&\"\"$F,&%\"cG6#F'F,F,F,!\"\"F,)F0F(F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" %\"\"$,&*&#\"#\")F'\"\"\"*$)&%\"cG6#F'F(F-F-F-*&#\"\"*\"\"#F-*$)F0F6F- F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"\"#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"\",$*&#\"\"#\"\"*F(*(,(*&\"#LF()&%\"cG6#\"\"%F,F(F( *&\"#XF(F3F(!\"\"\"#;F(F(F3F9,&*&\"#DF(F3F(F(\"#F.&F%6$F'\"\"&F.F 3F.F.*(\"&ce#F.F F(FKF(FF*(\")g=\">\"F(F1F(FKF(F(*(\");!yy*F(FCF(F5F(FF*&\")plW6F(F5F(F (*(\")+\\yWF(F1F(F5F(FF**\"*+G(*G*F(FCF(F5F(F1F(F(**\")S5,7F(F1F(F>F(F 5F(F(*(\"(?2z#F(F1F(F>F(FF*&\")k/^CF(FCF(F(*&\")?J^:F(F1F(F(*(\"*?N'RB F(F1F(FCF(FF\"(QE9$FFF(,&*&F8F(F5F(F(F9FFFFF5FFF1FF,&*&\"#DF(F5F(F(\"# s#F,)&%\"cG6#\"\"%F(F,F,*(\"*![uG@F,&%\"bG6#F'F,F2F,!\"\" **\")+?J7F,F9F,&F%6$F'\"\"&F,F2F,F,*(\")+gHbF,&F%6$\"\"'FAF,F2F,F<**\" *+g4;&F,FDF,F2F,F9F,F,*&\")vV9UF,)F3\"\"#F,F<**\"+!G06@\"F,FDF,FKF,F9F ,F<*(\"*![K&G\"F,FDF,FKF,F,*(\"*?;R.$F,F9F,FKF,F,**\")gT\"e#F,F9F,F?F, FKF,F<*(\"*S8d\"F2*&\"(glv(F,F@F,F,*&\")K _D7F,F6F,F,F,,**&\"$W\"F,F:F,F,*&\"$[#F,FGF,F2*&\"$H\"F,F/F,F,\"#=F2F2 F@F2,&*&\"#DF,F/F,F,\"#F.F4F.F.**\"(+3I%F.FDF.F4F.F;F.FA*(\"'cYXF.FDF.F4F.F .\"&jh#F.*(\"'gT;F.F;F.F>F.FA*(\"(?.s\"F.F;F.FDF.F.*&\"'C-=F.FDF.FAF.F ;FA,&*&F7F.F4F.F.\"\"$FA!\"#F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W=============================== =======================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",$*&#F(\"$?(F(*&,(*(\"$?%F(&%\"bG 6#\"\"(F(&%\"cG6#\"\"%F(F(*&F0F(F1F(!\"\"\"#TF(F(F5F:F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,$*&#\"$H(\"%SI\"\"\"*&,**(\" #gF.&%\"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"#?F.F7F.!\"\"\"\"*F.*&F2F.F3F. F=F.,&\"\"#F=*&F>F.F7F.F.F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,$*&#\"\"\"\"#g F,*(,&*&\"$?%F,&%\"bG6#\"\"(F,F,\"#T!\"\"F,&%\"cG6#F(F7,*\"#7F7*&\"#)) F,F8F,F,*&\"$x\"F,)F8\"\"#F,F7*&\"$3\"F,)F8\"\"$F,F,F7F,F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"&,$*&#\"\"*\"$g\"\"\"\"*&,**&\"#?F.&%\"cG6#\"\"%F.!\"\"*( \"$?%F.&%\"bG6#\"\"(F.F3F.F.\"#FF.*&F9F.F:F.F7F.,&\"\"#F7*&\"\"$F.F3F. F.F7F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',$*&#\"#k\"$b)\"\"\"*&,**(\"$?%F.&% \"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"#XF.F7F.!\"\"*&F2F.F3F.F=\"#WF.F.,&* &F:F.F7F.F.\"\"$F=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(&%\"bG6#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%W######################################################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\",$*&#F'\"$?(F'*&,(*(\"$?%F'&F%6#\"\"(F'&%\"cG6#\"\"%F'F'*&F/ F'F0F'!\"\"\"#TF'F'F3F8F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$ ,$*&#\"$H(\"%SI\"\"\"*&,**(\"#gF-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"#?F- F5F-!\"\"\"\"*F-*&F1F-F2F-F;F-,&\"\"#F;*&FF+F4F+&F%6#\"\"$F+F:*&\"%(=#F+F5F+F:*(\"&/a&F+F?F+F5F+F:*(\"&k0\"F +F5F+FEF+F+*(\"%!=*F+F1F+F5F+F+*&\"%W>F+F1F+F:*&\"%o8F+FEF+F:*&\"&AJ\" F+F?F+F+F+,**(\"$k#F+F?F+F5F+F+*&\"#HF+F5F+F:\"\"*F+*&\"#sF+F?F+F:F:,& F+F:F5F+F:F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK5_7eqs := [op(RowSumCondi tions(7,'expanded')),op(OrderConditions(5,7,'expanded'))]:\nsimplify(s ubs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "The stability function of a general 7 stage, order 5 Ru nge-Kutta scheme has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7 ;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F) *&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\" \"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]" "6#/ &%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"&F-&F/6$F1\"\"%F-&F/6$F 4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F-&F+6#\"\"(F-" }{TEXT -1 2 " (" } {XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*a[6,4]*a[4,3]* a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"(\"\"&\"\"\"&F&6$F)\"\"%F*&F&6$F-\" \"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F(\"\"'F*&F&6$F:F-F*&F&6$F-F0F *&F&6$F0F3F*&F56#F3F*F*%!GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[7,6]*a[6, 5]*a[5,3]*a[3,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,2]*c[2]+a[7,6]*a[6,5]*a [5,4]*a[4,3]*c[3]" "6#,(*,&%\"aG6$\"\"(\"\"'\"\"\"&F&6$F)\"\"&F*&F&6$F -\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-\" \"%F*&F&6$F>F3F*&F56#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-F>F*&F&6$F>F 0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "t[7]=b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"'F,&F.6$F0\"\"&F,&F.6 $F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9\"\"#F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 264 "eq7 := b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=269/1379840;\neq6 := b[6 ]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c [2]+\n a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2]*c[2]+a [5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))))=17291/12418560;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6#\"\"(\"\"\"&%\"aG6$F*\"\"'F+&F-6$F/ \"\"&F+&F-6$F2\"\"%F+&F-6$F5\"\"$F+&F-6$F8\"\"#F+&%\"cG6#F;F+#\"$p#\"( S)z8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/,&*.&%\"bG6#\"\"'\"\" \"&%\"aG6$F+\"\"&F,&F.6$F0\"\"%F,&F.6$F3\"\"$F,&F.6$F6\"\"#F,&%\"cG6#F 9F,F,*&&F)6#\"\"(F,,&*,&F.6$F@F0F,F1F,F4F,F7F,F:F,F,*&&F.6$F@F+F,,&**& F.6$F+F3F,F4F,F7F,F:F,F,*&F-F,,&*(&F.6$F0F6F,F7F,F:F,F,*&F1F,,&*&&F.6$ F3F9F,F:F,F,*&F4F,&F;6#F6F,F,F,F,F,F,F,F,F,F,#\"&\"H<\")g&=C\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We shall \+ require the following two equations to be satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve(simplify(subs(e3,\{eq6 \})),a[7,5]));\neqn7 := op(solve(simplify(subs(e3,\{eq7\})),a[7,5])); \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&%\"aG6$\"\"(\"\"&,$*&# \"\"\"\"-?n+JK[F.*,,J**\"1+?FUNA=MF.)&F'6$\"\"'F*\"\"#F.)&%\"cG6#\"\"% F=F.&%\"bG6#F)F.F.*&\"/+%=XCDX\"F.F9F.!\"\"*(\"/++KgV&z'F.F5F.F9F.F.*( \"0+?nO#QiOF.F4F.F9F.FC**\"1?f=2aT1#*F.F4F.)F:\"\"$F.F>F.FC*(\"0+G&=@T p;F.F5F.FJF.FC*(\"0?$)GBIEy*F.F4F.FJF.F.*&\"/7\\Ezb7PF.FJF.F.**\"1![O% G'\\\"[*)F.F4F.)F:F8F.F>F.F.*(\"0/.83*[\\%*F.F4F.FTF.FC*(\"0O6xJ=t]\"F .F5F.FTF.F.*&\"/K')>!>!*[$F.FTF.FC*(\"078978k*QF.F4F.F:F.F.**\"1!3'GIN (oq$F.F4F.F:F.F>F.FC*(\"/w`_RG!*eF.F5F.F:F.FC*&\"/4KT?s>9F.F:F.F.*&\"/ k/ZPfHdF.F4F.FC*(\"0?Nwmd\"paF.F>F.F4F.F.*&\".oJq8wJ)F.F5F.F.\".hr#*e' )3#FCF.,&*&F=F.F:F.F.FKFCFC,**&\"$D#F.FJF.F.*&\"$\"QF.FTF.FC*&\"$3#F.F :F.F.\"#OFCFCF>FCF5FCF.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/ &%\"aG6$\"\"(\"\"&,$*&#\"\"\"\".!)o-CH$>F.*.,B*(\"/;?uAkzAF.&F'6$\"\"' F*F.)&%\"cG6#\"\"%\"\"$F.!\"\"*(\"0CQHhnUN\"F.)F4\"\"#F.F7F.F.**\"1!)[ 6*yOhF\"F.F@F.F7F.&%\"bG6#F)F.F=*(\"/ci*3Q5f\"F.F4F.)F8FAF.F.**\"0![ok p%Hs*F.F@F.FIF.FDF.F.*(\"0s]B\">QC5F.F@F.FIF.F=*(\".39? " 0 "" {MPLTEXT 1 0 38 "factor(simplify(rhs(eqn6)-rh s(eqn7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"%R:\",!G\\OI6\"\" \"*4,&*&\"#DF(&%\"cG6#\"\"%F(F(\"# " 0 "" {MPLTEXT 1 0 143 "denom_factors := select(has,map(op ,select(has,map(denom@rhs,e3),c[4])),c[4]):\nmember(25*c[4]-17,denom_f actors);\nmember(4*c[4]-3,denom_factors);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "This leav es the possibilities " }{XPPEDIT 18 0 "c[4]=21789/78848" "6#/&%\"cG6# \"\"%*&\"&*y@\"\"\"\"&[)y!\"\"" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "c [4]=3/7" "6#/&%\"cG6#\"\"%*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT -1 53 ". Th e Bogacki-Shampine scheme uses the latter value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting for " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 32 " in each of the \+ equations for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 29 " now gives an equation for " }{XPPEDIT 18 0 "a[7,5]" "6 #&%\"aG6$\"\"(\"\"&" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "b[ 7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6 #&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "simplify(subs(c[4]=3/7,eqn6 ));\nsimplify(subs(c[4]=3/7,eqn7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"&,$*&#\"\"\"\".+glQ?T\"F,*(,**(\"/+g,1+t>F,&%\"bG6# F'F,)&F%6$\"\"'F(\"\"#F,F,\",pSq2;&!\"\"*&\"-S#yj)HBF,F6F,F,*&\".+w&3s J?F,F5F,F;F,F2F;F6F;F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"(\"\"&,$*&#\"\"\"\".+glQ?T\"F,*(,**(\"/+g,1+t>F,&%\"bG6#F'F,)&F%6$ \"\"'F(\"\"#F,F,\",pSq2;&!\"\"*&\"-S#yj)HBF,F6F,F,*&\".+w&3sJ?F,F5F,F; F,F2F;F6F;F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 110 "dd := simplify(subs(c[4]=3/7,rhs(eqn6))):\ne4 := \{c[4]=3/7,a[7,5]=dd\}:\ne5 := `union`(e4,simplify(subs(e4,e3))): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1713 " e5 := \{c[8] = 1, b[2] = 0, a[8,2] = 0, a[2,1] = 1/6, c[7] = 1, `b*`[2 ] = 0, c[2] = 1/6, a[4,1] = 183/1372, `b*`[8] = 1/5832*(-540*`b*`[5]-7 29+5832*b[7]+988*`b*`[3])/(12*b[7]-1), a[7,4] = -49/32451674112000*(-3 192704204800*a[6,5]^2+403943915520*a[6,5]-2118057984000*b[7]*a[6,5]+31 004295168000*b[7]*a[6,5]^2-21025360917)/b[7]/a[6,5], a[7,1] = -1/13979 182694400*(-6883688448000*b[7]*a[6,5]+1874481315840*a[6,5]+21703006617 6000*b[7]*a[6,5]^2-63076082751-22348929433600*a[6,5]^2)/b[7]/a[6,5], c [4] = 3/7, a[3,1] = 2/27, `b*`[7] = 1/486*(243+540*`b*`[5]-988*`b*`[3] )*b[7]/(12*b[7]-1), a[8,7] = b[7], a[7,3] = 1/4487145062400*(443380137 9840*a[6,5]-34100733542400*b[7]*a[6,5]+217030066176000*b[7]*a[6,5]^2-8 9634433383-22348929433600*a[6,5]^2)/b[7]/a[6,5], a[7,5] = 1/1412038656 000*(19730006016000*b[7]*a[6,5]^2-51607704069+232986378240*a[6,5]-2031 720857600*a[6,5]^2)/b[7]/a[6,5], a[7,2] = 9/22*(12*b[7]-1)/b[7], a[7,6 ] = 7263/358400*1/(b[7]*a[6,5]), c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, a [4,2] = -162/343, b[5] = -1161/800+189/10*b[7], b[6] = -7168/513*b[7]+ 11072/7695, a[8,1] = -7/9*b[7]+287/2160, `b*`[4] = -45619/39366*`b*`[3 ]+343/648-343/1458*`b*`[5], a[6,1] = -10/9*a[6,5]+195/704, a[6,2] = 81 /352, a[8,4] = -16807/1755*b[7]+98441/105300, a[4,3] = 1053/1372, a[5, 1] = 68/297, a[8,3] = 2187/494*b[7]-2187/39520, a[8,6] = -7168/513*b[7 ]+11072/7695, a[5,4] = 1960/3861, `b*`[1] = 4/27+5/243*`b*`[5]-1729/65 61*`b*`[3], a[6,4] = -392/117*a[6,5]+8379/9152, a[8,5] = -1161/800+189 /10*b[7], b[1] = -7/9*b[7]+287/2160, b[4] = -16807/1755*b[7]+98441/105 300, a[3,2] = 4/27, b[3] = 2187/494*b[7]-2187/39520, a[5,2] = -4/11, ` b*`[6] = -640/729*`b*`[5]+11648/19683*`b*`[3]+16/81, a[5,3] = 42/143, \+ a[6,3] = -1539/2288+45/13*a[6,5]\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We now ha ve a scheme with just the two parameters " }{XPPEDIT 18 0 "b[7]" "6#& %\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6 $\"\"'\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Up to thi s stage the simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/ -%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F )6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1 " "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 23 " ) have not been used. " }}{PARA 0 "" 0 "" {TEXT -1 6 "It is " }{TEXT 260 13 "not necessary" }{TEXT -1 381 " to require that any of t hese conditions be satisfied, but since the particular scheme that we \+ are constructing does satisfy them, we can simplify the discussion reg arding the choice of parameters by insisting that they are satisfied. \+ It turns out that it will be sufficient to pick any one of these equat ions (except the second, which has already been incorporated in the sc heme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "simp_eqns2 := [add(b[i]*a[i,1],i=1+1..7)=b[1],seq(ad d(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6)]:\nnops(simp_eqns2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 61 "We may as well pick the last equation , which is the simplest." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " simp_eqns2[6];\nsubs(e5,%);\nextra_eqn := b[7]=solve(%,b[7]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F(\"\"'F )*&&F&6#F-F),&F)F)&%\"cGF0!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,$*&#\"%js\"'+%e$\"\"\"*&F)F)&%\"aG6$\"\"'\"\"&!\"\"F)F),&*&#\"%#z\"\" $8&F)&%\"bG6#\"\"(F)F0#\"%oF\"%&p(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*extra_eqnG/&%\"bG6#\"\"(,$*&#\"\"\"\"++%en#>F-*&,&\")dx<6!\"\"*& \"*S-T)>F-&%\"aG6$\"\"'\"\"&F-F-F-F5F2F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "e6 := `union`(\{b [7]=(198410240*a[6,5]-11177757)/(1926758400*a[6,5])\},\n simplify(s ubs(b[7]=(198410240*a[6,5]-11177757)/(1926758400*a[6,5]),e5))):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1848 "e6 := \{b[7] = 1/192675840 0*(-11177757+198410240*a[6,5])/a[6,5], `b*`[7] = 1/17496*(243+540*`b*` [5]-988*`b*`[3])*(-11177757+198410240*a[6,5])/(-3725919+12615680*a[6,5 ]), a[7,2] = 162/11*(-3725919+12615680*a[6,5])/(-11177757+198410240*a[ 6,5]), a[8,5] = 9/917504000*(50462720*a[6,5]-11177757)/a[6,5], c[7] = \+ 1, `b*`[2] = 0, c[2] = 1/6, a[4,1] = 183/1372, b[3] = 729/16698572800* (-588303+9175040*a[6,5])/a[6,5], b[2] = 0, a[8,2] = 0, a[2,1] = 1/6, a [8,3] = 729/16698572800*(-588303+9175040*a[6,5])/a[6,5], c[8] = 1, c[4 ] = 3/7, a[3,1] = 2/27, c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, a[4,2] = - 162/343, `b*`[4] = -45619/39366*`b*`[3]+343/648-343/1458*`b*`[5], a[6, 1] = -10/9*a[6,5]+195/704, a[6,2] = 81/352, a[4,3] = 1053/1372, a[5,1] = 68/297, a[5,4] = 1960/3861, `b*`[1] = 4/27+5/243*`b*`[5]-1729/6561* `b*`[3], a[6,4] = -392/117*a[6,5]+8379/9152, a[3,2] = 4/27, a[5,2] = - 4/11, `b*`[6] = -640/729*`b*`[5]+11648/19683*`b*`[3]+16/81, a[5,3] = 4 2/143, a[6,3] = -1539/2288+45/13*a[6,5], a[7,6] = 39045888/(-11177757+ 198410240*a[6,5]), a[8,7] = 1/1926758400*(-11177757+198410240*a[6,5])/ a[6,5], b[1] = 19/825753600*(196101+2293760*a[6,5])/a[6,5], b[4] = -34 3/7667712000*(-1241973+1146880*a[6,5])/a[6,5], a[7,5] = 63/10*(2567168 0*a[6,5]-11177757)/(-11177757+198410240*a[6,5]), b[5] = 9/917504000*(5 0462720*a[6,5]-11177757)/a[6,5], b[6] = 7263/89600*1/a[6,5], a[8,1] = \+ 19/825753600*(196101+2293760*a[6,5])/a[6,5], a[8,4] = -343/7667712000* (-1241973+1146880*a[6,5])/a[6,5], a[8,6] = 7263/89600*1/a[6,5], `b*`[8 ] = -1/8748*(43352064000*`b*`[5]*a[6,5]+10311598080*a[6,5]+2716194951- 79318220800*`b*`[3]*a[6,5])/(-3725919+12615680*a[6,5]), a[7,4] = -4390 4/2145*(848320*a[6,5]-1241973)/(-11177757+198410240*a[6,5]), a[7,1] = \+ 1/132*(421028847+1699962880*a[6,5])/(-11177757+198410240*a[6,5]), a[7, 3] = -189/572*(437719040*a[6,5]-140604417)/(-11177757+198410240*a[6,5] )\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 44 "We now have just 3 parameters in the scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,e6));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%&%#b*G6#\"\"&&F%6#\"\"$&%\"aG6$\"\"'F'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Choosing \+ a value for " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 69 " will determine all the coefficients of the 7 stage, order 5 s cheme." }}{PARA 0 "" 0 "" {TEXT -1 47 "Consider the norm of the princi pal error terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 129 "errterms5_7 := PrincipalErrorTerms(5,7,'expan ded'):\nsm := sqrt(add(simplify(subs(e6,errterms5_7[i]))^2,i=1.. nops( errterms5_7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#smG,$*&#\"\"\"\" /+?z*))*)G#F(*$,&\"3++k[W(*3&G#F(*(\"&%RPF(,&*&\")?FY]F(&%\"aG6$\"\"' \"\"&F(F(\")dx<6!\"\"\"\"#F2!\"#F(#F(F9F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We can find the value for " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 61 " that mak es the norm of the principal error terms a minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "simplify(diff(sm,a[6,5]));\nsolve(%);\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"*()**eo$\",+gC0:A!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(sm,a[6, 5]=0.12..1,font=[HELVETICA,9],color=magenta);" }}{PARA 13 "" 1 "" {GLPLOT2D 511 311 311 {PLOTDATA 2 "6'-%'CURVESG6#7`o7$$\"3%*********** ***>\"!#=$\"3Muid(G?@h$!#@7$$\"3gmm;RP&zC\"F*$\"3LH*y\\8\"H5LF-7$$\"3C LLLyu!fH\"F*$\"3jhPlmy*3.$F-7$$\"3w****\\<7'QM\"F*$\"3=pN%G?a:x#F-7$$ \"3ammmc\\\"=R\"F*$\"3__+8th?IDF-7$$\"3;L$3([#RNV\"F*$\"3[:4nQHYLBF-7$ $\"3w***\\2aj_Z\"F*$\"3%G%\\/t9(z9#F-7$$\"3Qm;zKy)p^\"F*$\"3a,'*yvW#G( >F-7$$\"3GLL$[77(e:F*$\"3-)[$e2p@2=F-7$$\"3$omT]XNcg\"F*$\"3Ed@QTX[J;F -7$$\"35++D&yeDl\"F*$\"33@8]m;-m9F-7$$\"3QL$ea6#[*p\"F*$\"3Ua>#3V_+J\" F-7$$\"3mmmmXaSYHMN>F*$\"3W]ov&prc]'Fio7$$\"3.++DjEOH?F*$\"3'poJhBrYU%Fi o7$$\"3=LL$oS#QB@F*$\"3k!)fO!)4j$y#Fio7$$\"3#)***\\7\\u^9#F*$\"3t?W&*p \"Rz]#Fio7$$\"3smmmvl'p;#F*$\"33;[P#GXEH#Fio7$$\"3OLL3g'e()=#F*$\"3om! e:%4D]@Fio7$$\"3+++]W2b5AF*$\"3m?\"Q\"ygxH#F*$\"3US1:'z#4\"f#Fio7$$\"3;LL$o%o(zQ# F*$\"39WJxP'[rs$Fio7$$\"3!)****\\6YByCF*$\"3O\\^9f$F*$\"3)o^BAe#3Z;F-7$$\"3;+++ 9@3vPF*$\"3%))RQ-oWSx\"F-7$$\"3\"omm,>l=%RF*$\"3U))>Spm=z=F-7$$\"3@LLL *R'=STF*$\"3ww\\uL%oK*>F-7$$\"3\\LLLx/>3VF*$\"3#y>Pb$Qu\"3#F-7$$\"3p** **\\f\">O]%F*$\"3)*f=mp()Rw@F-7$$\"39LLLdmgwYF*$\"3u@m7U(QoCF-7$$\"3-LL$G;r)3aF*$\"3\\*pLu(z#f_#F-7$$ \"3Qnmm6kl&f&F*$\"3+X[N,*4Se#F-7$$\"3Gnm;!zt'*y&F*$\"3U*>OEz!QSEF-7$$ \"3/++])[m&efF*$\"3#*f_%>xoko#F-7$$\"3anmmH_(49'F*$\"3z&z1O[-Mt#F-7$$ \"3Z*****zI@%HjF*$\"3[,]HvJ0zFF-7$$\"3*)*****4.zP^'F*$\"3_TM(\\?h6#GF- 7$$\"3Y++]2Y:#p'F*$\"3SJn-;wpfGF-7$$\"3K+++t-@!*oF*$\"3M1]?\"4^,!HF-7$ $\"3'GLLt4s\"oqF*$\"3)R(4ZE*oX$HF-7$$\"3S+++IV=esF*$\"3Eue7YjXpHF-7$$ \"3Amm;aOOIuF*$\"3K&>4K>I&**HF-7$$\"3R+++M6g=wF*$\"3focL1]&3.$F-7$$\"3 eKL$G$zr&z(F*$\"3-kpl!>\\*eIF-7$$\"3;++]@g&3)zF*$\"35\"GQC&R)p3$F-7$$ \"3gKLLVB'=;)F*$\"3dA'*pIX;8JF-7$$\"3G****\\(=f8N)F*$\"3<%*ej(>e$RJF-7 $$\"3*omm1#[gp[jJF-7$$\"3gmmmVz]?()F*$\"3/L[Bg$=r=$F-7$$ \"3xLL$oB.c!*)F*$\"3/J=.Jld4KF-7$$\"3S+++)e'ov!*F*$\"3ZV-U\">1%HKF-7$$ \"3#RLLBI@1F*F*$\"3'f-@`'*Q7D$F-7$$\"30nmmMK(\\W*F*$\"3or)HQW.+F$F-7$$ \"3w****\\f(o3j*F*$\"3!o:NSQi#*G$F-7$$\"3o++]'\\%z3)*F*$\"3D#>YsL7qI$F -7$$\"\"\"\"\"!$\"3iW!G\"zLQDLF--%%FONTG6$%*HELVETICAG\"\"*-%'COLOURG6 &%$RGBG$\"*++++\"!\")$Ff`lFf`lFbal-%+AXESLABELSG6$Q'a[6,5]6\"Q!Fjal-%% VIEWG6$;$\"#7!\"#Fd`l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Chosing " }{XPPEDIT 18 0 "a[6,5]=11177757 /50462720" "6#/&%\"aG6$\"\"'\"\"&*&\")dx<6\"\"\"\")?FY]!\"\"" }{TEXT -1 79 " gives the following minimum value for the norm of the princip al error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eval(sm, a[6,5]=11177757/50462720);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"/+?z*))*)G#!\"\"\"3++k[W(*3&G##\"\"\"\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BBP)3#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "The corresponding weights are as follows. Note that " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "seq(b[i]=subs(a[6 ,5]=11177757/50462720,subs(e6,b[i])),i=1..7);\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"bG6#\"\"\"#\"#z\"%!3\"/&F%6#\"\"#\"\"!/ &F%6#\"\"$#\"&$o>\"&g\"p/&F%6#\"\"%#\"&2o\"\"&SU)/&F%6#\"\"&F//&F%6#\" \"'#\"%;G\"%&p(/&F%6#\"\"(#\"#V\"$g&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)/&%\"bG6#\"\"\"$\"'\"[J(!\"(/&F%6#\"\"#$\"\"!F0/&F%6#\"\"$$\"',YG!\" '/&F%6#\"\"%$\"'8&*>F7/&F%6#\"\"&F//&F%6#\"\"'$\"'_fOF7/&F%6#\"\"($\"' dywF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The Bogacki-Shampine scheme uses the value " }{XPPEDIT 18 0 "a[6,5]= 4617/20480" "6#/&%\"aG6$\"\"'\"\"&*&\"% " 0 "" {MPLTEXT 1 0 37 "eval(sm,a[6,5]=4617/20480);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"/+?z*))*)G#!\"\"\"3+C#o'R-4vD#\"\"\"\" \"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zF$p@#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " As an aside, both t he numerator and denominator of " }{XPPEDIT 18 0 "4617/20480" "6#*&\" % " 0 "" {MPLTEXT 1 0 47 "[4617/20480,11177757/5046272 0];\nmap(ifactor,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"%\"\"\"-F&6#\"\"#!#7-F&6#F)!\"\"*0F%\"\"(F*F--F&6#\"$p#F-F.!# " 0 "" {MPLTEXT 1 0 68 "seq(b[i]=subs(a[6,5]=4617/20 480,subs(e6,b[i])),i=1..7);\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"bG6#\"\"\"#\"$(e\"%k!)/&F%6#\"\"#\"\"!/&F%6#\"\"$# \"(R.W%\")S=\\:/&F%6#\"\"%#\"&`V#\"'+[7/&F%6#\"\"&#\"$(Q\"&+[%/&F%6#\" \"'#\"%_@\"%&)f/&F%6#\"\"(#\"%ns\"&!3%*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"bG6#\"\"\"$\"'Fzs!\"(/&F%6#\"\"#$\"\"!F0/&F%6#\"\"$$\"'CmG! \"'/&F%6#\"\"%$\"'O^>F7/&F%6#\"\"&$\"'RQ')!\")/&F%6#\"\"'$\"'m&f$F7/&F %6#\"\"($\"'GCxF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Values for the parameters " }{XPPEDIT 18 0 "`b*`[3]" "6# &%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b* G6#\"\"&" }{TEXT -1 110 " can be chosen so as to give a compatible st ability region for the embedded 8 stage, order 4 scheme, namely: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[3] = 123/416;" " 6#/&%#b*G6#\"\"$*&\"$B\"\"\"\"\"$;%!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[5] = 43/1440;" "6#/&%#b*G6#\"\"&*&\"#V\"\"\"\"%S9! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "prm := \{a[6,5]=4617/20480,`b*`[3]=123/4 16,`b*`[5]=43/1440\}:\ne7 := `union`(prm,simplify(subs(prm,e6))):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 947 "e7 := \{c[7 ] = 1, `b*`[2] = 0, c[2] = 1/6, a[4,1] = 183/1372, b[2] = 0, a[8,2] = \+ 0, a[2,1] = 1/6, c[8] = 1, c[4] = 3/7, a[3,1] = 2/27, c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, a[4,2] = -162/343, a[6,2] = 81/352, a[4,3] = 1053/ 1372, a[5,1] = 68/297, a[5,4] = 1960/3861, a[3,2] = 4/27, a[5,2] = -4/ 11, a[5,3] = 42/143, `b*`[3] = 123/416, `b*`[5] = 43/1440, a[6,5] = 46 17/20480, b[7] = 7267/94080, `b*`[7] = 79937/1113912, a[7,2] = -30942/ 79937, a[8,5] = 387/44800, b[3] = 4440339/15491840, a[8,3] = 4440339/1 5491840, a[6,1] = 597/22528, a[6,4] = 58653/366080, a[6,3] = 63099/585 728, a[7,6] = 482048/414219, a[8,7] = 7267/94080, b[1] = 587/8064, b[4 ] = 24353/124800, a[7,5] = -29421/29068, b[5] = 387/44800, b[6] = 2152 /5985, a[8,4] = 24353/124800, a[8,6] = 2152/5985, `b*`[8] = 3293/55695 6, a[7,4] = 666106/1039181, a[7,1] = 174197/959244, a[7,3] = 8152137/1 9744439, a[8,1] = 587/8064, `b*`[1] = 2479/34992, `b*`[6] = 2272/6561, `b*`[4] = 612941/3411720\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(e7,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7 ),``],[`b*`,seq(`b*`[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+# \"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$ V$#\"%`5F:F+F+F+F+F+7+#F.F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"#\"%g>\"%hQF+ F+F+F+7+#F6F3#\"$(f\"&GD##\"#\")\"$_$#\"&*4j\"'Gde#\"&`'e\"'!3m$#\"%u\"\"'W#f*#!&U4$\"&P*z#\"(P@:)\")RWu>#\"'1hm\"( \"=R5#!&@%H\"&o!H#\"'[?[\"'>UTF+F+7+F)#\"$(e\"%k!)\"\"!#\"(R.W%\")S=\\ :#\"&`V#\"'+[7#\"$(Q\"&+[%#\"%_@\"%&)f#\"%ns\"&!3%*F+7+%\"bGF]pF`pFapF dpFgpFjpF]qF+7+%#b*G#\"%zC\"&#*\\$F`p#\"$B\"\"$;%#\"'THh\"(?QL\"F*$!)@.BZF*$\")r#\\n(F*F+F+F+F+F+7+$ \")nmmmF*$\")Bc*G#F*$!)OOOOF*$\")H1PHF*$\")^Sw]F*F+F+F+F+7+$\")+++vF*$ \")b.]EF1$\")k8,BF*$\")[Fx5F*$\")3>-;F*$\")XRaAF*F+F+F+7+$\"\"\"\"\"!$ \")A)f\"=F*$!)$)zqQF*$\")p#)GTF*$\")K\"*4kF*$!)R975!\"($\"):vj6F]oF+F+ 7+FV$\")fEzsF1$FXFX$\")QCmGF*$\")AO^>F*$\")HRQ')!#5$\")el&f$F*$\")sFCx F1F+7+%\"bGFaoFcoFdoFfoFhoF[pF]pF+7+%#b*G$\")lZ%3(F1Fco$\")3tcHF*$\")Z d'z\"F*$\")66')HF1$\")o)GY$F*$\"),CwrF1$\")e\\7fFjoQ(pprint36\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the weights " }{XPPEDIT 18 0 "`b^`[i], i=1" "6$&%#b^G6#%\"iG/F&\"\"\"" }{TEXT -1 127 " . . 7, for \+ an auxiliary 7 stage, order 4 embedded scheme by using the quadrature \+ conditions along with the requirements that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^`[7]=b[7]" "6#/&%#b^G6#\"\"(&%\"bG6# F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b^`[2]=0" "6#/&%#b^G6#\"\"#\"\" !" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b^`[6]=443/1197" "6#/&%#b^G6 #\"\"'*&\"$V%\"\"\"\"%(>\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "quad_eqns : = subs(b=`b^`,QuadratureConditions(4,7,'expanded')):\ne8 := \{`b^`[2]= 0,`b^`[6]=443/1197\}:\ne9 := `union`(e7,e8):\ncdns2 := [`b^`[7]=b[7],o p(quad_eqns)]:\neqns2 := subs(e9,cdns2):\nnops(eqns2);\nindets(eqns2); \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<'&%#b^G6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "e10 := solve (\{op(eqns2)\}):\ne11 := `union`(e9,e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1099 "e11 := \{c[7] = 1, `b*`[2] = 0, c[2] = 1/6, a[4,1] = 183/1372, b[2] = 0, a[8,2] = 0, a[2,1] = 1/ 6, c[8] = 1, c[4] = 3/7, a[3,1] = 2/27, c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, a[4,2] = -162/343, a[6,2] = 81/352, a[4,3] = 1053/1372, a[5,1] = 68/297, a[5,4] = 1960/3861, a[3,2] = 4/27, a[5,2] = -4/11, a[5,3] = 4 2/143, `b*`[3] = 123/416, `b*`[5] = 43/1440, a[6,5] = 4617/20480, b[7] = 7267/94080, `b*`[7] = 79937/1113912, a[7,2] = -30942/79937, a[8,5] \+ = 387/44800, b[3] = 4440339/15491840, a[8,3] = 4440339/15491840, a[6,1 ] = 597/22528, a[6,4] = 58653/366080, a[6,3] = 63099/585728, a[7,6] = \+ 482048/414219, a[8,7] = 7267/94080, b[1] = 587/8064, b[4] = 24353/1248 00, a[7,5] = -29421/29068, b[5] = 387/44800, b[6] = 2152/5985, a[8,4] \+ = 24353/124800, a[8,6] = 2152/5985, `b*`[8] = 3293/556956, a[7,4] = 66 6106/1039181, a[7,1] = 174197/959244, a[7,3] = 8152137/19744439, a[8,1 ] = 587/8064, `b^`[2] = 0, `b^`[6] = 443/1197, `b^`[7] = 7267/94080, ` b*`[1] = 2479/34992, `b*`[6] = 2272/6561, `b*`[4] = 612941/3411720, `b ^`[1] = 6059/80640, `b^`[5] = -927/89600, `b^`[4] = 26411/124800, `b^` [3] = 8559189/30983680\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "`RK4_7eqs^` := subs(b=`b^`,OrderConditions(4,7,'expa nded')):\nsimplify(subs(e11,`RK4_7eqs^`)):\nmap(u->lhs(u)-rhs(u),%);\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 112 "The coefficients for the whole schem e including the 7 stage and 8 stage embedded order 4 schemes are as fo llows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "subs(e11,matrix([ seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1.. 7),``],[`b^`,seq(`b^`[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]]));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7+#\"\"\"\"\"'F(%!GF+F+ F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"% s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#F.F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"# \"%g>\"%hQF+F+F+F+7+#F6F3#\"$(f\"&GD##\"#\")\"$_$#\"&*4j\"'Gde#\"&`'e \"'!3m$#\"%u\"\"'W#f*#!&U4$\"&P*z#\"(P@:)\")RW u>#\"'1hm\"(\"=R5#!&@%H\"&o!H#\"'[?[\"'>UTF+F+7+F)#\"$(e\"%k!)\"\"!#\" (R.W%\")S=\\:#\"&`V#\"'+[7#\"$(Q\"&+[%#\"%_@\"%&)f#\"%ns\"&!3%*F+7+%\" bGF]pF`pFapFdpFgpFjpF]qF+7+%#b^G#\"%fg\"&S1)F`p#\"(*=f&)\")!o$)4$#\"&6 k#Ffp#!$F*\"&+'*)#\"$V%\"%(>\"F]qF+7+%#b*G#\"%zC\"&#*\\$F`p#\"$B\"\"$; %#\"'THh\"(? " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absol ute stability regions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 58 "#-------------------------------------- -------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficien ts of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 988 "ee := \{c[2]=1/6,\nc[3]=2/9,\nc[4] =3/7,\nc[5]=2/3,\nc[6]=3/4,\nc[7]=1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=2 /27,\na[3,2]=4/27,\na[4,1]=183/1372,\na[4,2]=-162/343,\na[4,3]=1053/13 72,\na[5,1]=68/297,\na[5,2]=-4/11,\na[5,3]=42/143,\na[5,4]=1960/3861, \na[6,1]=597/22528,\na[6,2]=81/352,\na[6,3]=63099/585728,\na[6,4]=5865 3/366080,\na[6,5]=4617/20480,\na[7,1]=174197/959244,\na[7,2]=-30942/79 937,\na[7,3]=8152137/19744439,\na[7,4]=666106/1039181,\na[7,5]=-29421/ 29068,\na[7,6]=482048/414219,\na[8,1]=587/8064,\na[8,2]=0,\na[8,3]=444 0339/15491840,\na[8,4]=24353/124800,\na[8,5]=387/44800,\na[8,6]=2152/5 985,\na[8,7]=7267/94080,\n\nb[1]=587/8064,\nb[2]=0,\nb[3]=4440339/1549 1840,\nb[4]=24353/124800,\nb[5]=387/44800,\nb[6]=2152/5985,\nb[7]=7267 /94080,\n\n`b^`[1]=6059/80640,\n`b^`[2]=0,\n`b^`[3]=8559189/30983680, \n`b^`[4]=26411/124800,\n`b^`[5]=-927/89600,\n`b^`[6]=443/1197,\n`b^`[ 7]=7267/94080,\n\n`b*`[1]=2479/34992,\n`b*`[2]=0,\n`b*`[3]=123/416,\n` b*`[4]=612941/3411720,\n`b*`[5]=43/1440,\n`b*`[6]=2272/6561,\n`b*`[7]= 79937/1113912,\n`b*`[8]=3293/556956\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#-------------------------------------------------------- -" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 stage, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 77 "The stability function R for the 7 stage, order 5 \+ scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(5,7,'expa nded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"&\"H<\")g&=C\"F)*$)F'F1F)F)F)*&#\"$p#\"(S)z8F)*$)F'\" \"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability regi on intersects the negative real axis by solving the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"R G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+*>Fz)R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4) :\np1 := plot([R(z),-1],z=-5.19..0.49,color=[red,blue]):\np2 := plot([ [[z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\n p3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplo ts[display]([p1,p2,p3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVE SG6$7W7$$!3Q++++++!>&!#<$!3*RwB]%)fHt(F*7$$!3Ynm;#3'4G^F*$!39PR9e0>`qF *7$$!3kLLLk@>m]F*$!3G]<5GldDkF*7$$!3E+]U'*)HB,&F*$!3d]&>H%3E>fF*7$$!3! pm;&GwYe\\F*$!3wY9xaCqZaF*7$$!3s+](\\(Q*y*[F*$!3%on?sDzj&\\F*7$$!3nLLV @,KP[F*$!3NKW$G,$p.XF*7$$!3'RLLd%[MwZF*$!3Ic'[s4(R%3%F*7$$!3NLL.q&p`r% F*$!3Q0,*=J8\"*p$F*7$$!3;nmJG')*Rf%F*$!3Q?R(3w5U-$F*7$$!3uLLyGAZ\"[%F* $!3Quy23xp&\\#F*7$$!3%3+])fw&\\O%F*$!3I#4N)4)GP.#F*7$$!3$QL$)f7eWC%F*$ !3*[c\\%**RjM;F*7$$!3A++lN]MCTF*$!3JEtl9e\"\\I\"F*7$$!3ummYeRz+SF*$!3h ?u8[;%f-\"F*7$$!3_LLV-,(>*QF*$!3;_G1Y%=5B)!#=7$$!35++S:-YpPF*$!3ke2$3- #[_jFbp7$$!3K+++\"HZkk$F*$!3o)p`-:ay#[Fbp7$$!3;++gW:!z_$F*$!3%4)>B[VRU OFbp7$$!3hLL)*\\1D?MF*$!3!oGFItMcw#Fbp7$$!3'ommSKVAH$F*$!3ABc\"e&ysD>F bp7$$!3/nmEGV!Q=$F*$!3Am:HGcra8Fbp7$$!39++0(*RmdIF*$!3_2hFWla(>)!#>7$$ !39nmEI%3g%HF*$!3=u3FBZk$Q%Ffr7$$!3-++0xX]BGF*$!3o^A-L(>Co*!#?7$$!3*)* **\\\"R>&oq#F*$\"3&G()eOPRer\"Ffr7$$!3gmm;\\r8&e#F*$\"33EVJ&y0l3%Ffr7$ $!3ymmrw\\OtCF*$\"3Qo&)G_c>%*fFfr7$$!3SLL$))e.GN#F*$\"3sLO3_KLpyFfr7$$ !3nLL)**=uvA#F*$\"3ZjwgtMF:(*Ffr7$$!3K++:I;c=@F*$\"3b.6'ps^18\"Fbp7$$! 31LL.z]#3+#F*$\"3FmhG\"\\qpI\"Fbp7$$!3M++?,<>z=F*$\"3%QLOIb=(*\\\"Fbp7 $$!3;++!4<(>gFbp7$ $!3H++q9zA<:F*$\"3#4s\"oumI)=#Fbp7$$!3EnmEY;O-9F*$\"3quXxE=edCFbp7$$!3 #)*****pQ<(z7F*$\"3!=w_%=i%*zFFbp7$$!3)RL$efMeo6F*$\"3')e&e?*y\\2JFbp7 $$!3I****fAZ3Z5F*$\"3l*R)*oym$4NFbp7$$!3xqm;(zQwK*Fbp$\"3<$oV/fiX$RFbp 7$$!3&z***\\)ecE8)Fbp$\"3hmHBz90MWFbp7$$!3'3nmm0VV'pFbp$\"3)\\a8tK&e$) \\Fbp7$$!3P)***\\iqATdFbp$\"3KhJiHm)>j&Fbp7$$!3aFLL*)4AjXFbp$\"3Sl;t.s 4OjFbp7$$!33LLLO'R&eLFbp$\"3%y%)o;xus9(Fbp7$$!3Uim;`O$Q;#Fbp$\"3Gkt@/Z Ea!)Fbp7$$!3?*****>$H-m5Fbp$\"3y4GNQ4$)))*)Fbp7$$\"3v*QLLU?>#>Ffr$\"3q -s.#30%>5F*7$$\"3%ymmY^avJ\"Fbp$\"3\"oXLqHH39\"F*7$$\"3E0+]HcUFz)RF*F[]l-%'SYMBOLG6#%'CIRCLEG-F_\\l6&Fa\\lFf\\lFf\\lFf\\l-% &STYLEG6#%&POINTG-F$6&Fa`l-Ff`l6#%&CROSSGFi`lF[al-F$6&Fa`l-Ff`l6#%(DIA MONDGFi`lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLORG6&Fa\\lFe\\l$\"\"&F\\]lFe \\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z 6\"Q!F_cl-Fgbl6#%(DEFAULTG-%%VIEWG6$;$!$\\%!\"#$\"#\\Fjcl;$!$Z\"Fjcl$ \"$Z\"Fjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1291 "R \+ := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+17291/12418560*z^6+269/ 1379840*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := \+ newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.2,.2,.5)): \np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,.4,.4,1)):\npts := [ ]:\nz0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp (ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]] :\nend do:\np3 := plot(pts,color=COLOR(RGB,.2,.2,.5)):\np4 := plots[po lygonplot]([seq([pts[i-1],pts[i],[1.37,4.19]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.4,.4,1)):\npts := []:\nz0 := 1 .5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I ),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do: \np5 := plot(pts,color=COLOR(RGB,.2,.2,.5)):\np6 := plots[polygonplot] ([seq([pts[i-1],pts[i],[1.37,-4.19]],i=2..nops(pts))],\n styl e=patchnogrid,color=COLOR(RGB,.4,.4,1)):\np7 := plot([[[-4.79,0],[2.19 ,0]],[[0,-4.79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([ p||(1..7)],view=[-4.79..2.19,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 606 615 615 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$$\"3b+++!f<*>f!#G$\"3++++Fjzq:!#=7$$\"3p******4$3Q5&!#E$ \"3(******pi#fTJF07$$\"3=+++(*f-r!)!#D$\"3D+++/%)Q7ZF07$$\"3E+++P,gNg! #C$\"3S+++&yzJG'F07$$\"33+++%p%GlG!#B$\"3S+++g$[R&yF07$$\"3s*****>)evd )*FF$\"3w*****\\jIYU*F07$$\"33+++y<$3i#!#A$\"3'******\\D0&*4\"!#<7$$\" 3I+++6Y&R[&FQ$\"33+++h8[c7FT7$$\"3C+++XY2B')FQ$\"3!******4V6LT\"FT7$$ \"3]*****>Ow'yuFQ$\"3++++iU&)p:FT7$$!3-+++'3\"pH5!#@$\"3!******\\/()es \"FT7$$!3^+++odj+rFao$\"31+++jY4\")=FT7$$!3')*****\\^$>CA!#?$\"3<+++[t 2N?FT7$$!32+++\\v\\4aF\\p$\"3=+++upP(=#FT7$$!3++++O#)>O6!#>$\"36+++QP] PBFT7$$!3'******pt83:#Fgp$\"35+++yV#\\[#FT7$$!3!)*****4:(*Qv$Fgp$\"3/+ ++4w'*GEFT7$$!3<+++3,`ChFgp$\"3!******p0J'oFFT7$$!3Z*****>$4#*=%*Fgp$ \"34+++RsN-HFT7$$!3#******\\$)Q=P\"F0$\"3A+++nS)z-$FT7$$!3)******4&R!y *=F0$\"3\")*****f&48VJFT7$$!3t******[l:-DF0$\"37+++/I(fC$FT7$$!3;+++h( G-;$F0$\"36+++-)[eL$FT7$$!3'******pW;&[QF0$\"3!******pAQKT$FT7$$!35+++ W,T\\XF0$\"3)******z^$HzMFT7$$!3S+++P]p^_F0$\"3<+++QZONNFT7$$!3W+++C6. \\fF0$\"3;+++:@s#e$FT7$$!3a+++.s$FT7$$!3*******p $3=S**F0$\"3!******H[4@t$FT7$$!3/+++rqOd5FT$\"3#)*****zh<&RPFT7$$!3+++ +mQ$*>6FT$\"3@+++\\+xUPFT7$$!3#******fjo<=\"FT$\"38+++2k/UPFT7$$!36+++ V4#HC\"FT$\"34+++l#)[PPFT7$$!3)******fGTMI\"FT$\"3/+++2e?HPFT7$$!31+++ hKQj8FT$\"3&)*****48$GMOFT7$$!3#*******[f*ol\"FT$\"3&*******z'=Zg$ FT7$$!3%*******GS2:FT$\"3#******pq9*>$FT7$$!3/+++\")*\\YF#FT$\"3> +++x.zVJFT7$$!31+++\"y4OM#FT$\"3))*****RU-#)3$FT7$$!3z*****\\0RLT#FT$ \"3.+++:(>G.$FT7$$!33+++8@H$[#FT$\"3'******f76z(HFT7$$!3=+++eY\"Hb#FT$ \"34+++v GFT$\"3@+++$)yl4FFT7$$!31+++Tmj\")GFT$\"3))******ey,cEFT7$$!3++++g(>=% HFT$\"3*******za$)=g#FT7$$!3y******)R-***HFT$\"3++++=-2ZDFT7$$!31+++;N 'e0$FT$\"3))*****p:<9\\#FT7$$!3\")*****p>0(4JFT$\"32+++MoyMCFT7$$!3#** *****RjWhJFT$\"3=+++?H1xBFT7$$!3,+++]$=6@$FT$\"34+++lz9=BFT7$$!3++++rB weKFT$\"3)******\\?hzD#FT7$$!3#)*****\\hFWI$FT$\"3'******ptOk>#FT7$$!3 ?+++j/<[LFT$\"39+++?A_L@FT7$$!3=+++\\S0!R$FT$\"3;+++U\"y\"p?FT7$$!38++ +x![,V$FT$\"3z*****4qxL+#FT7$$!3#******pjG&oMFT$\"3)******RG2h$>FT7$$! 36+++'zx_]$FT$\"33+++6vOn=FT7$$!3*)******=F[SNFT$\"3-+++NZ<(z\"FT7$$!3 /+++!4MUd$FT$\"32+++VGcD\"FT7$$!3')******zM?,QFT$\"3'******fm$=76FT7$$!32+++\"38Z#QFT$\"34 +++K!RC.\"FT7$$!3A+++*yVq%QFT$\"3z*****p8!)\\_*F07$$!3')*****\\)R5oQFT $\"3b*****\\l!yC()F07$$!3=+++/+z()QFT$\"3Z*****RXVZ#zF07$$!3*)*****zb# *f!RFT$\"3)******HE'eDrF07$$!38+++4EgARFT$\"3s*****Hl+yK'F07$$!3-+++/i ^PRFT$\"34+++mKnJbF07$$!3++++u!Q1&RFT$\"3F+++1\\JPZF07$$!3'******H/%)= 'RFT$\"3w*****4X#pWRF07$$!3))******>C=rRFT$\"37+++j)fO:$F07$$!37+++UYZ yRFT$\"3!******R/')RO#F07$$!3++++Gar$)RFT$\"3'*******e&y`d\"F07$$!3%** ****\\mso)RFT$\"3m*****p%>.vyFgp7$$!3/+++*>Fz)RFTF(7$Fjjl$!3m*****p%>. vyFgp7$Fejl$!3'*******e&y`d\"F07$F`jl$!3!******R/')RO#F07$F[jl$!37+++j )fO:$F07$Ffil$!3w*****4X#pWRF07$Fail$!3F+++1\\JPZF07$F\\il$!34+++mKnJb F07$Fghl$!3s*****Hl+yK'F07$Fbhl$!3)******HE'eDrF07$F]hl$!3Z*****RXVZ#z F07$Fhgl$!3b*****\\l!yC()F07$Fcgl$!3z*****p8!)\\_*F07$F^gl$!34+++K!RC. \"FT7$Fifl$!3'******fm$=76FT7$Fdfl$!3'******zS&f\">\"FT7$F_fl$!3))**** *z)R_q7FT7$Fjel$!3'******pB7)[8FT7$Feel$!3/+++QOIE9FT7$F`el$!3******** )e\\G]\"FT7$F[el$!3/+++!\\:$y:FT7$Ffdl$!3/+++w]e_;FT7$Fadl$!32+++VGcD< FT7$F\\dl$!3-+++NZ<(z\"FT7$Fgcl$!33+++6vOn=FT7$Fbcl$!3)******RG2h$>FT7 $F]cl$!3z*****4qxL+#FT7$Fhbl$!3;+++U\"y\"p?FT7$Fcbl$!39+++?A_L@FT7$F^b l$!3'******ptOk>#FT7$Fial$!3)******\\?hzD#FT7$Fdal$!34+++lz9=BFT7$F_al $!3=+++?H1xBFT7$Fj`l$!32+++MoyMCFT7$Fe`l$!3))*****p:<9\\#FT7$F``l$!3++ ++=-2ZDFT7$F[`l$!3*******za$)=g#FT7$Ff_l$!3))******ey,cEFT7$Fa_l$!3@++ +$)yl4FFT7$F\\_l$!35+++P2,jFFT7$Fg^l$!3$******4\")*H;GFT7$Fb^l$!3)**** ***)[^(pGFT7$F]^l$!34+++vG.$FT7$F^]l$!3))*****RU-#)3$FT7$Fi\\l$!3>+++x.zVJFT7$Fd\\l$!3#)****** >q9*>$FT7$F_\\l$!3#)*****Ri:PD$FT7$Fj[l$!38+++G_*oI$FT7$Fe[l$!3A+++\\! =\"eLFT7$F`[l$!3#******pMOFT7$F]y$!3)******fb8-m$FT7$Fhx$!3')*****f:\\Fo$F T7$Fcx$!39+++rEy,PFT7$F^x$!3&)*****48$G.s$FT7$Ffu$!3<+++$=jQq$FT7 $Fau$!3#)*****fJ%Q#o$FT7$F\\u$!3;+++XYVbOFT7$Fgt$!3()*****RD]Ci$FT7$Fb t$!3;+++:@s#e$FT7$F]t$!3<+++QZONNFT7$Fhs$!3)******z^$HzMFT7$Fcs$!3!*** ***pAQKT$FT7$F^s$!36+++-)[eL$FT7$Fir$!37+++/I(fC$FT7$Fdr$!3\")*****f&4 8VJFT7$F_r$!3A+++nS)z-$FT7$Fjq$!34+++RsN-HFT7$Feq$!3!******p0J'oFFT7$F `q$!3/+++4w'*GEFT7$F[q$!35+++yV#\\[#FT7$Fep$!36+++QP]PBFT7$F`p$!3=+++u pP(=#FT7$Fjo$!3<+++[t2N?FT7$Feo$!31+++jY4\")=FT7$F_o$!3!******\\/()es \"FT7$$\"39+++kjnyuFQ$!3++++iU&)p:FT7$Fen$!3!******4V6LT\"FT7$FV$!33++ +h8[c7FT7$FO$!3'******\\D0&*4\"FT7$FJ$!3w*****\\jIYU*F07$FD$!3S+++g$[R &yF07$$\"3^*****f8+c.'F@$!3S+++&yzJG'F07$$\"3s******pd-r!)F:$!3D+++/%) Q7ZF07$$\"3G+++9$3Q5&F4$!3(******pi#fTJF07$$\"3i*****>a4(>fF-$!3++++Fj zq:F0F'-%&COLORG6&%$RGBG$\"\"#!\"\"Fh^n$\"\"&Fj^n-%)POLYGONSG6fw7%F'7$ $\"+!f<*>fF\\p$\"+Fjzq:!#57$$!\"#F)F(7%Fa_n7$$\"+5$3Q5&F0$\"+FEfTJFf_n Fg_n7%F[`n7$$\"+(*f-r!)FT$\"+/%)Q7ZFf_nFg_n7%Fa`n7$$\"+P,gNg!#;$\"+&yz JG'Ff_nFg_n7%Fg`n7$$\"+%p%GlG!#:$\"+g$[R&yFf_nFg_n7%F^an7$$\"+#)evd)*F aan$\"+N1jC%*Ff_nFg_n7%Fean7$$\"+y<$3i#!#9$\"+b_]*4\"!\"*Fg_n7%F[bn7$$ \"+6Y&R[&F^bn$\"+h8[c7FabnFg_n7%Fcbn7$$\"+XY2B')F^bn$\"+J9J89FabnFg_n7 %Fibn7$$\"+ijnyuF^bn$\"+iU&)p:FabnFg_n7%F_cn7$$!+'3\"pH5!#8$\"+Xq)es\" FabnFg_n7%Fecn7$$!+odj+rFhcn$\"+jY4\")=FabnFg_n7%F\\dn7$$!+:N>CA!#7$\" +[t2N?FabnFg_n7%Fbdn7$$!+\\v\\4aFedn$\"+upP(=#FabnFg_n7%Fidn7$$!+O#)>O 6!#6$\"+QP]PBFabnFg_n7%F_en7$$!+PP\"3:#Fben$\"+yV#\\[#FabnFg_n7%Ffen7$ $!+^r*Qv$Fben$\"+4w'*GEFabnFg_n7%F\\fn7$$!+3,`ChFben$\"+d5joFFabnFg_n7 %Fbfn7$$!+K4#*=%*Fben$\"+RsN-HFabnFg_n7%Fhfn7$$!+N)Q=P\"Ff_n$\"+nS)z-$ FabnFg_n7%F^gn7$$!+^R!y*=Ff_n$\"+c48VJFabnFg_n7%Fdgn7$$!+\\l:-DFf_n$\" +/I(fC$FabnFg_n7%Fjgn7$$!+h(G-;$Ff_n$\"+-)[eL$FabnFg_n7%F`hn7$$!+Zk^[Q Ff_n$\"+F#QKT$FabnFg_n7%Ffhn7$$!+W,T\\XFf_n$\"+=NHzMFabnFg_n7%F\\in7$$ !+P]p^_Ff_n$\"+QZONNFabnFg_n7%Fbin7$$!+C6.\\fFf_n$\"+:@s#e$FabnFg_n7%F hin7$$!+.s$FabnFg_n7%Ff[o7$$!+P3=S**Ff _n$\"+$[4@t$FabnFg_n7%F\\\\o7$$!+rqOd5Fabn$\"+=w^RPFabnFg_n7%Fb\\o7$$! +mQ$*>6Fabn$\"+\\+xUPFabnFg_n7%Fh\\o7$$!+O'o<=\"Fabn$\"+2k/UPFabnFg_n7 %F^]o7$$!+V4#HC\"Fabn$\"+l#)[PPFabnFg_n7%Fd]o7$$!+'GTMI\"Fabn$\"+2e?HP FabnFg_n7%Fj]o7$$!+hKQj8Fabn$\"+JJGMOFabnFg_n7%Fh_o 7$$!+\\f*ol\"Fabn$\"+!o=Zg$FabnFg_n7%F^`o7$$!+HS2:F abn$\"+xT!pS$FabnFg_n7%F\\bo7$$!+cmH8?Fabn$\"+\\!=\"eLFabnFg_n7%Fbbo7$ $!+q_?w?Fabn$\"+G_*oI$FabnFg_n7%Fhbo7$$!+p0tS@Fabn$\"+Ccr`KFabnFg_n7%F ^co7$$!+30#p?#Fabn$\"+?q9*>$FabnFg_n7%Fdco7$$!+\")*\\YF#Fabn$\"+x.zVJF abnFg_n7%Fjco7$$!+\"y4OM#Fabn$\"+CC?)3$FabnFg_n7%F`do7$$!+b!RLT#Fabn$ \"+:(>G.$FabnFg_n7%Ffdo7$$!+8@H$[#Fabn$\"+E6\"z(HFabnFg_n7%F\\eo7$$!+e Y\"Hb#Fabn$\"+vGFabn$\"+$)yl4FFabnFg_n7%Fjfo7$$!+Tmj\")GFabn$\"+fy, cEFabnFg_n7%F`go7$$!+g(>=%HFabn$\"+[N)=g#FabnFg_n7%Ffgo7$$!+*R-***HFab n$\"+=-2ZDFabnFg_n7%F\\ho7$$!+;N'e0$Fabn$\"+drT\"\\#FabnFg_n7%Fbho7$$! +(>0(4JFabn$\"+MoyMCFabnFg_n7%Fhho7$$!+SjWhJFabn$\"+?H1xBFabnFg_n7%F^i o7$$!+]$=6@$Fabn$\"+lz9=BFabnFg_n7%Fdio7$$!+rBweKFabn$\"+07'zD#FabnFg_ n7%Fjio7$$!+:wU/LFabn$\"+PnV'>#FabnFg_n7%F`jo7$$!+j/<[LFabn$\"+?A_L@Fa bnFg_n7%Ffjo7$$!+\\S0!R$Fabn$\"+U\"y\"p?FabnFg_n7%F\\[p7$$!+x![,V$Fabn $\"+,xP.?FabnFg_n7%Fb[p7$$!+P'G&oMFabn$\"+%G2h$>FabnFg_n7%Fh[p7$$!+'zx _]$Fabn$\"+6vOn=FabnFg_n7%F^\\p7$$!+>F[SNFabn$\"+NZ<(z\"FabnFg_n7%Fd\\ p7$$!+!4MUd$Fabn$\"+VGcD\"FabnFg_n7%Fd_p7$$!+![.7!QFabn$\"+mO=76FabnFg_n 7%Fj_p7$$!+\"38Z#QFabn$\"+K!RC.\"FabnFg_n7%F``p7$$!+*yVq%QFabn$\"+P,) \\_*Ff_nFg_n7%Ff`p7$$!+&)R5oQFabn$\"+b1yC()Ff_nFg_n7%F\\ap7$$!+/+z()QF abn$\"+aMuCzFf_nFg_n7%Fbap7$$!+eD*f!RFabn$\"+jieDrFf_nFg_n7%Fhap7$$!+4 EgARFabn$\"+`1!yK'Ff_nFg_n7%F^bp7$$!+/i^PRFabn$\"+mKnJbFf_nFg_n7%Fdbp7 $$!+u!Q1&RFabn$\"+1\\JPZFf_nFg_n7%Fjbp7$$!+VS)='RFabn$\"+^CpWRFf_nFg_n 7%F`cp7$$!+?C=rRFabn$\"+j)fO:$Ff_nFg_n7%Ffcp7$$!+UYZyRFabn$\"+Wg)RO#Ff _nFg_n7%F\\dp7$$!+Gar$)RFabn$\"+f&y`d\"Ff_nFg_n7%Fbdp7$$!+lE(o)RFabn$ \"+Z>.vyFbenFg_n7%Fhdp7$$!+*>Fz)RFabnF(Fg_n7%F^ep7$Fidp$!+Z>.vyFbenFg_ n7%Fbep7$Fcdp$!+f&y`d\"Ff_nFg_n7%Ffep7$F]dp$!+Wg)RO#Ff_nFg_n7%Fjep7$Fg cp$!+j)fO:$Ff_nFg_n7%F^fp7$Facp$!+^CpWRFf_nFg_n7%Fbfp7$F[cp$!+1\\JPZFf _nFg_n7%Fffp7$Febp$!+mKnJbFf_nFg_n7%Fjfp7$F_bp$!+`1!yK'Ff_nFg_n7%F^gp7 $Fiap$!+jieDrFf_nFg_n7%Fbgp7$Fcap$!+aMuCzFf_nFg_n7%Ffgp7$F]ap$!+b1yC() Ff_nFg_n7%Fjgp7$Fg`p$!+P,)\\_*Ff_nFg_n7%F^hp7$Fa`p$!+K!RC.\"FabnFg_n7% Fbhp7$F[`p$!+mO=76FabnFg_n7%Ffhp7$Fe_p$!+3af\">\"FabnFg_n7%Fjhp7$F__p$ !+))R_q7FabnFg_n7%F^ip7$Fi^p$!+PA\")[8FabnFg_n7%Fbip7$Fc^p$!+QOIE9Fabn Fg_n7%Ffip7$F]^p$!+*e\\G]\"FabnFg_n7%Fjip7$Fg]p$!+!\\:$y:FabnFg_n7%F^j p7$Fa]p$!+w]e_;FabnFg_n7%Fbjp7$F[]p$!+VGcDFabnFg_n 7%Fb[q7$Fc[p$!+,xP.?FabnFg_n7%Ff[q7$F][p$!+U\"y\"p?FabnFg_n7%Fj[q7$Fgj o$!+?A_L@FabnFg_n7%F^\\q7$Fajo$!+PnV'>#FabnFg_n7%Fb\\q7$F[jo$!+07'zD#F abnFg_n7%Ff\\q7$Feio$!+lz9=BFabnFg_n7%Fj\\q7$F_io$!+?H1xBFabnFg_n7%F^] q7$Fiho$!+MoyMCFabnFg_n7%Fb]q7$Fcho$!+drT\"\\#FabnFg_n7%Ff]q7$F]ho$!+= -2ZDFabnFg_n7%Fj]q7$Fggo$!+[N)=g#FabnFg_n7%F^^q7$Fago$!+fy,cEFabnFg_n7 %Fb^q7$F[go$!+$)yl4FFabnFg_n7%Ff^q7$Fefo$!+P2,jFFabnFg_n7%Fj^q7$F_fo$! +6)*H;GFabnFg_n7%F^_q7$Fieo$!+*[^(pGFabnFg_n7%Fb_q7$Fceo$!+vG.$FabnFg_n7%F^`q7$F ado$!+CC?)3$FabnFg_n7%Fb`q7$F[do$!+x.zVJFabnFg_n7%Ff`q7$Feco$!+?q9*>$F abnFg_n7%Fj`q7$F_co$!+Ccr`KFabnFg_n7%F^aq7$Fibo$!+G_*oI$FabnFg_n7%Fbaq 7$Fcbo$!+\\!=\"eLFabnFg_n7%Ffaq7$F]bo$!+xT!pS$FabnFg_n7%Fjaq7$Fgao$!+D X)GX$FabnFg_n7%F^bq7$Faao$!+%\\(z&\\$FabnFg_n7%Fbbq7$F[ao$!+c2ZNNFabnF g_n7%Ffbq7$Fe`o$!+w#)zrNFabnFg_n7%Fjbq7$F_`o$!+!o=Zg$FabnFg_n7%F^cq7$F i_o$!+&[(>MOFabnFg_n7%Fbcq7$Fc_o$!+cN@gOFabnFg_n7%Ffcq7$F]_o$!+c\"\\Fo $FabnFg_n7%Fjcq7$Fg^o$!+rEy,PFabnFg_n7%F^dq7$Fa^o$!+JJG.s$FabnFg_n7%F^fq7$Fa[o $!+$=jQq$FabnFg_n7%Fbfq7$F[[o$!+;VQ#o$FabnFg_n7%Fffq7$Fejn$!+XYVbOFabn Fg_n7%Fjfq7$F_jn$!+a-XAOFabnFg_n7%F^gq7$Fiin$!+:@s#e$FabnFg_n7%Fbgq7$F cin$!+QZONNFabnFg_n7%Ffgq7$F]in$!+=NHzMFabnFg_n7%Fjgq7$Fghn$!+F#QKT$Fa bnFg_n7%F^hq7$Fahn$!+-)[eL$FabnFg_n7%Fbhq7$F[hn$!+/I(fC$FabnFg_n7%Ffhq 7$Fegn$!+c48VJFabnFg_n7%Fjhq7$F_gn$!+nS)z-$FabnFg_n7%F^iq7$Fifn$!+RsN- HFabnFg_n7%Fbiq7$Fcfn$!+d5joFFabnFg_n7%Ffiq7$F]fn$!+4w'*GEFabnFg_n7%Fj iq7$Fgen$!+yV#\\[#FabnFg_n7%F^jq7$F`en$!+QP]PBFabnFg_n7%Fbjq7$Fjdn$!+u pP(=#FabnFg_n7%Ffjq7$Fcdn$!+[t2N?FabnFg_n7%Fjjq7$F]dn$!+jY4\")=FabnFg_ n7%F^[r7$Ffcn$!+Xq)es\"FabnFg_n7%Fb[r7$$\"+kjnyuF^bn$!+iU&)p:FabnFg_n7 %Ff[r7$Fjbn$!+J9J89FabnFg_n7%F\\\\r7$Fdbn$!+h8[c7FabnFg_n7%F`\\r7$F\\b n$!+b_]*4\"FabnFg_n7%Fd\\r7$Ffan$!+N1jC%*Ff_nFg_n7%Fh\\r7$F_an$!+g$[R& yFf_nFg_n7%F\\]r7$$\"+O,gNgFj`n$!+&yzJG'Ff_nFg_n7%F`]r7$$\"+qd-r!)FT$! +/%)Q7ZFf_nFg_n7%Ff]r7$$\"+9$3Q5&F0$!+FEfTJFf_nFg_n7%F\\^r7$$\"+U&4(>f F\\p$!+Fjzq:Ff_nFg_n7%Fb^rF'Fg_n-Fe^n6&Fg^n$\"\"%Fj^nFj^r$\"\"\"F)-%&S TYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"33+++.>]*[\"FT$\"3#******>&)\\QI%FT7 $$\"3!******ff%R\"[\"FT$\"3()*****HP'f7VFT7$$\"3'******Ru+EZ\"FT$\"3C+ ++&o\\1K%FT7$$\"3/+++&RjJY\"FT$\"3D+++(RozK%FT7$$\"3/+++yi7`9FT$\"3e** ***\\V6XL%FT7$$\"3++++@T`U9FT$\"3F+++kxBSVFT7$$\"3++++9IVJ9FT$\"3i**** **>h5XVFT7$$\"3#******f#3()>9FT$\"3M+++dZ2\\VFT7$$\"3$******Ro(*yS\"FT $\"3e*****p?,@N%FT7$$\"3-+++Glc&R\"FT$\"3=+++^?9aVFT7$$\"33+++xP$HQ\"F T$\"3o*****4r_^N%FT7$$\"3%******R6g+P\"FT$\"37+++os3bVFT7$$\"3!******f U6qN\"FT$\"3/+++A$)*QN%FT7$$\"33+++W*fQM\"FT$\"3y*****p\"p`^VFT7$$\"3, +++CcoI8FT$\"3-+++mD&zM%FT7$$\"3#******4$yd<8FT$\"3k******RM4VVFT7$$\" 3$******\\QPYI\"FT$\"3)******\\z1pL%FT7$$\"3)*******e!z>H\"FT$\"3e**** *zxR$HVFT7$$\"3++++-Ytz7FT$\"3J++++3M?VFT7$$\"3#******pMc!o7FT$\"3V+++ g<')4VFT7$$\"3)*******H97d7FT$\"31+++d:'yH%FT7$$\"3!******H]OrC\"FT$\" 3!******RZ6VG%FT7$$\"32+++7EMQ7FT$\"3e*****HF.#pUFT7$$\"3*******fR>5B \"FT$\"3e*****z/hDD%FT7$$\"3,+++wp[D7FT$\"3v*****p+eWB%FT7$$\"3******* >e-@A\"FT$\"3V+++%)*Q]@%FT7$$\"3.+++`sC@7FT$\"3k*****p=[X>%FT7$$\"3/++ +QlHB7FT$\"3/+++:%eL<%FT7$$\"3)*******4)p&G7FT$\"3)******\\N#*>:%FT7$$ \"3.+++(*)esB\"FT$\"3j*****4&*>68%FT7$$\"3'******>K]$\\7FT$\"3&******* QR^6TFT7$$\"3-+++\\:dk7FT$\"3o*****pBdR4%FT7$$\"3/+++e\\R#G\"FT$\"3G++ +mC7zSFT7$$\"3\"******4_5@I\"FT$\"3))*****\\@ou1%FT7$$\"3$*******z2%HK \"FT$\"3S+++yK>fSFT7$$\"3++++T::W8FT$\"3y*****RkdU0%FT7$$\"3/+++&*R7l8 FT$\"3u*****4JYC0%FT7$$\"3++++P:Q&Q\"FT$\"3=+++%oUM0%FT7$$\"3,+++BPe/9 FT$\"3i*****\\F*)o0%FT7$$\"3)******R-/DU\"FT$\"3#)******>%GC1%FT7$$\"3 #*******RO+R9FT$\"3w*****R5D(pSFT7$$\"3%******>u2SX\"FT$\"3o*****\\=x% ySFT7$$\"3#******fS'[n9FT$\"3#*******yzT)3%FT7$$\"3)*******4.Wz9FT$\"3 A+++eZJ*4%FT7$$\"3'******HW!*)*[\"FT$\"3w*****RUl46%FT7$$\"3%******H)3 ())\\\"FT$\"3R+++wX>BTFT7$$\"3$******H$QU1:FT$\"33+++T'\\e8%FT7$$\"3.+ ++Wif7:FT$\"37+++Wtz[TFT7$$\"3!******fcPu^\"FT$\"3#*******G1#>;%FT7$$ \"30+++g#)*4_\"FT$\"3++++@g6vTFT7$$\"3!******>&)GL_\"FT$\"3G+++<9H)=%F T7$$\"3\"******zLzW_\"FT$\"3:+++=UO,UFT7$$\"3$******4&))\\C:FT$\"3B+++ %yfU@%FT7$$\"3.+++$\\NM_\"FT$\"3j*****H85pA%FT7$$\"3'******4GO8_\"FT$ \"3m******[GDRUFT7$$\"3(******z8_WH%FTFe_rFd^n-F^_n 6jn7%7$$\"+.>]*[\"Fabn$\"+_)\\QI%Fabn7$$\"+'f%R\"[\"Fabn$\"+tjf7VFabn7 $$\"$P\"Fi_n$\"$>%Fi_n7%Fibs7$$\"+W2gs9Fabn$\"+&o\\1K%FabnF^cs7%Fdcs7$ $\"+&RjJY\"Fabn$\"+(RozK%FabnF^cs7%Fjcs7$$\"+yi7`9Fabn$\"+N9^MVFabnF^c s7%F`ds7$$\"+@T`U9Fabn$\"+kxBSVFabnF^cs7%Ffds7$$\"+9IVJ9Fabn$\"+?h5XVF abnF^cs7%F\\es7$$\"+E3()>9Fabn$\"+dZ2\\VFabnF^cs7%Fbes7$$\"+%o(*yS\"Fa bn$\"+275_VFabnF^cs7%Fhes7$$\"+Glc&R\"Fabn$\"+^?9aVFabnF^cs7%F^fs7$$\" +xP$HQ\"Fabn$\"+6F:bVFabnF^cs7%Fdfs7$$\"+9,1q8Fabn$\"+os3bVFabnF^cs7%F jfs7$$\"+E9,d8Fabn$\"+A$)*QN%FabnF^cs7%F`gs7$$\"+W*fQM\"Fabn$\"+H\"Fabn$\"+y(R$HVFabnF^cs7%F^is7$$\"+-Ytz7Fabn$\"++3M?VFabnF^cs7%Fd is7$$\"+Zj0o7Fabn$\"+g<')4VFabnF^cs7%Fjis7$$\"+I97d7Fabn$\"+d:'yH%Fabn F^cs7%F`js7$$\"+.l8Z7Fabn$\"+u9J%G%FabnF^cs7%Ffjs7$$\"+7EMQ7Fabn$\"+tK ?pUFabnF^cs7%F\\[t7$$\"+'R>5B\"Fabn$\"+[5c_UFabnF^cs7%Fb[t7$$\"+wp[D7F abn$\"+2!eWB%FabnF^cs7%Fh[t7$$\"+#e-@A\"Fabn$\"+%)*Q]@%FabnF^cs7%F^\\t 7$$\"+`sC@7Fabn$\"+(=[X>%FabnF^cs7%Fd\\t7$$\"+QlHB7Fabn$\"+:%eL<%FabnF ^cs7%Fj\\t7$$\"+5)p&G7Fabn$\"+bB*>:%FabnF^cs7%F`]t7$$\"+(*)esB\"Fabn$ \"+^*>68%FabnF^cs7%Ff]t7$$\"+A.N\\7Fabn$\"+RR^6TFabnF^cs7%F\\^t7$$\"+ \\:dk7Fabn$\"+Ps&R4%FabnF^cs7%Fb^t7$$\"+e\\R#G\"Fabn$\"+mC7zSFabnF^cs7 %Fh^t7$$\"+@06-8Fabn$\"+:#ou1%FabnF^cs7%F^_t7$$\"+!ySHK\"Fabn$\"+yK>fS FabnF^cs7%Fd_t7$$\"+T::W8Fabn$\"+WwDaSFabnF^cs7%Fj_t7$$\"+&*R7l8Fabn$ \"+6jW_SFabnF^cs7%F``t7$$\"+P:Q&Q\"Fabn$\"+%oUM0%FabnF^cs7%Ff`t7$$\"+B Pe/9Fabn$\"+v#*)o0%FabnF^cs7%F\\at7$$\"+CS]A9Fabn$\"+?%GC1%FabnF^cs7%F bat7$$\"+SO+R9Fabn$\"+/^spSFabnF^cs7%Fhat7$$\"+Ux+a9Fabn$\"+&=x%ySFabn F^cs7%F^bt7$$\"+1k[n9Fabn$\"+zzT)3%FabnF^cs7%Fdbt7$$\"+5.Wz9Fabn$\"+eZ J*4%FabnF^cs7%Fjbt7$$\"+V/*)*[\"Fabn$\"+Ca'46%FabnF^cs7%F`ct7$$\"+$)3( ))\\\"Fabn$\"+wX>BTFabnF^cs7%Ffct7$$\"+LQU1:Fabn$\"+T'\\e8%FabnF^cs7%F \\dt7$$\"+Wif7:Fabn$\"+Wtz[TFabnF^cs7%Fbdt7$$\"+mvV<:Fabn$\"+H1#>;%Fab nF^cs7%Fhdt7$$\"+g#)*4_\"Fabn$\"+@g6vTFabnF^cs7%F^et7$$\"+_)GL_\"Fabn$ \"+<9H)=%FabnF^cs7%Fdet7$$\"+Q$zW_\"Fabn$\"+=UO,UFabnF^cs7%Fjet7$$\"+^ ))\\C:Fabn$\"+%yfU@%FabnF^cs7%F`ft7$$\"+$\\NM_\"Fabn$\"+L,\"pA%FabnF^c s7%Ffft7$$\"+\"GO8_\"Fabn$\"+\\GDRUFabnF^cs7%F\\gt7$$\"+yrC=:Fabn$\"+k ,B^UFabnF^cs7%Fbgt7$$\"+yJ@9:Fabn$\"+!=)yiUFabnF^cs7%Fhgt7$$\"+J%y#4:F abn$\"+.i(QF%FabnF^cs7%F^ht7$$\"+%R'[.:Fabn$\"+mhW%G%FabnF^cs7%Fdht7$$ \"+4+)o\\\"Fabn$\"+K@X%H%FabnF^cs7%FjhtFdbsF^csFh^rF^_r-F$6$7in7$Ff_r$ !3#******>&)\\QI%FT7$F]bs$!3)******>8_WH%FT7$Fhas$!3S+++mhW%G%FT7$Fcas $!3Q+++.i(QF%FT7$F^as$!3E+++!=)yiUFT7$Fi`s$!3))*****R;I7D%FT7$Fd`s$!3m ******[GDRUFT7$F_`s$!3j*****H85pA%FT7$Fj_s$!3B+++%yfU@%FT7$Fe_s$!3:+++ =UO,UFT7$F`_s$!3G+++<9H)=%FT7$F[_s$!3++++@g6vTFT7$Ff^s$!3#*******G1#>; %FT7$Fa^s$!37+++Wtz[TFT7$F\\^s$!33+++T'\\e8%FT7$Fg]s$!3R+++wX>BTFT7$Fb ]s$!3w*****RUl46%FT7$F]]s$!3A+++eZJ*4%FT7$Fh\\s$!3#*******yzT)3%FT7$Fc \\s$!3o*****\\=x%ySFT7$F^\\s$!3w*****R5D(pSFT7$Fi[s$!3#)******>%GC1%FT 7$Fd[s$!3i*****\\F*)o0%FT7$F_[s$!3=+++%oUM0%FT7$Fjjr$!3u*****4JYC0%FT7 $Fejr$!3y*****RkdU0%FT7$F`jr$!3S+++yK>fSFT7$F[jr$!3))*****\\@ou1%FT7$F fir$!3G+++mC7zSFT7$Fair$!3o*****pBdR4%FT7$F\\ir$!3&*******QR^6TFT7$Fgh r$!3j*****4&*>68%FT7$Fbhr$!3)******\\N#*>:%FT7$F]hr$!3/+++:%eL<%FT7$Fh gr$!3k*****p=[X>%FT7$Fcgr$!3V+++%)*Q]@%FT7$F^gr$!3v*****p+eWB%FT7$Fifr $!3e*****z/hDD%FT7$Fdfr$!3e*****HF.#pUFT7$F_fr$!3!******RZ6VG%FT7$Fjer $!31+++d:'yH%FT7$Feer$!3V+++g<')4VFT7$F`er$!3J++++3M?VFT7$F[er$!3e**** *zxR$HVFT7$Ffdr$!3)******\\z1pL%FT7$Fadr$!3k******RM4VVFT7$F\\dr$!3-++ +mD&zM%FT7$Fgcr$!3y*****p\"p`^VFT7$Fbcr$!3/+++A$)*QN%FT7$F]cr$!37+++os 3bVFT7$Fhbr$!3o*****4r_^N%FT7$Fcbr$!3=+++^?9aVFT7$F^br$!3e*****p?,@N%F T7$Fiar$!3M+++dZ2\\VFT7$Fdar$!3i******>h5XVFT7$F_ar$!3F+++kxBSVFT7$Fj` r$!3e*****\\V6XL%FT7$Fe`r$!3D+++(RozK%FT7$F``r$!3C+++&o\\1K%FT7$F[`r$! 3()*****HP'f7VFTFcitFd^n-F^_n6jn7%7$Febs$!+_)\\QI%Fabn7$F[it$!+K@X%H%F abn7$F_cs$!$>%Fi_n7%F]eu7$Feht$!+mhW%G%FabnF`eu7%Fdeu7$F_ht$!+.i(QF%Fa bnF`eu7%Fheu7$Figt$!+!=)yiUFabnF`eu7%F\\fu7$Fcgt$!+k,B^UFabnF`eu7%F`fu 7$F]gt$!+\\GDRUFabnF`eu7%Fdfu7$Fgft$!+L,\"pA%FabnF`eu7%Fhfu7$Faft$!+%y fU@%FabnF`eu7%F\\gu7$F[ft$!+=UO,UFabnF`eu7%F`gu7$Feet$!+<9H)=%FabnF`eu 7%Fdgu7$F_et$!+@g6vTFabnF`eu7%Fhgu7$Fidt$!+H1#>;%FabnF`eu7%F\\hu7$Fcdt $!+Wtz[TFabnF`eu7%F`hu7$F]dt$!+T'\\e8%FabnF`eu7%Fdhu7$Fgct$!+wX>BTFabn F`eu7%Fhhu7$Fact$!+Ca'46%FabnF`eu7%F\\iu7$F[ct$!+eZJ*4%FabnF`eu7%F`iu7 $Febt$!+zzT)3%FabnF`eu7%Fdiu7$F_bt$!+&=x%ySFabnF`eu7%Fhiu7$Fiat$!+/^sp SFabnF`eu7%F\\ju7$Fcat$!+?%GC1%FabnF`eu7%F`ju7$F]at$!+v#*)o0%FabnF`eu7 %Fdju7$Fg`t$!+%oUM0%FabnF`eu7%Fhju7$Fa`t$!+6jW_SFabnF`eu7%F\\[v7$F[`t$ !+WwDaSFabnF`eu7%F`[v7$Fe_t$!+yK>fSFabnF`eu7%Fd[v7$F__t$!+:#ou1%FabnF` eu7%Fh[v7$Fi^t$!+mC7zSFabnF`eu7%F\\\\v7$Fc^t$!+Ps&R4%FabnF`eu7%F`\\v7$ F]^t$!+RR^6TFabnF`eu7%Fd\\v7$Fg]t$!+^*>68%FabnF`eu7%Fh\\v7$Fa]t$!+bB*> :%FabnF`eu7%F\\]v7$F[]t$!+:%eL<%FabnF`eu7%F`]v7$Fe\\t$!+(=[X>%FabnF`eu 7%Fd]v7$F_\\t$!+%)*Q]@%FabnF`eu7%Fh]v7$Fi[t$!+2!eWB%FabnF`eu7%F\\^v7$F c[t$!+[5c_UFabnF`eu7%F`^v7$F][t$!+tK?pUFabnF`eu7%Fd^v7$Fgjs$!+u9J%G%Fa bnF`eu7%Fh^v7$Fajs$!+d:'yH%FabnF`eu7%F\\_v7$F[js$!+g<')4VFabnF`eu7%F`_ v7$Feis$!++3M?VFabnF`eu7%Fd_v7$F_is$!+y(R$HVFabnF`eu7%Fh_v7$Fihs$!+&z1 pL%FabnF`eu7%F\\`v7$Fchs$!+SM4VVFabnF`eu7%F``v7$F]hs$!+mD&zM%FabnF`eu7 %Fd`v7$Fggs$!+#Fi_n;F\\gv$\"$z%Fi _n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or sta bility interval) is the intersection of the stability region with the \+ real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stabili ty interval is (approximately) " }{XPPEDIT 18 0 "[-3.9879, 0];" "6#7$ ,$-%&FloatG6$\"&z)R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The stability radius is h alf the length of the real stability interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(R(z)=-1,z=-4)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++O'R*>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 401 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+17291/12418560*z^6+269/1379840*z^7:\npts := []: \nz0 := 0:\nfor ct from 0 to 360 do\n zz := newton(R(z)=exp(ct*Pi/30 *I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np1 := plot(pts,color=red):\nrd := 1.993963600: \np2 := plot([-rd+rd *cos(t),rd*sin(t),t=0..2*Pi],color=blue):\nplots[display]([p1,p2],scal ing=constrained,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7eal7$$\"\"!F)F(7$F($\"31+++^v> Z5!#=7$$\"3!******>K-Nj$!#F$\"3-+++,^R%4#F-7$$\"3)******>J3Q5&!#E$\"3( ******pi#fTJF-7$$\"3')*****4[#oxN!#D$\"3#******>(**y)=%F-7$$\"31+++qoX \"o\"!#C$\"3d*****zW')fB&F-7$$\"3^*****f8+c.'FC$\"3S+++&yzJG'F-7$$\"3) ******zg#Gv!#A$\"3)** ****4fir/\"!#<7$$\"3n******e@=WMFhn$\"31++++4%=:\"F[o7$$\"3I+++6Y&R[&F hn$\"33+++h8[c7F[o7$$\"3f*****>;kgr(Fhn$\"3/+++z\"e5O\"F[o7$$\"3O+++Ou (f6*Fhn$\"3#*******\\M`l9F[o7$$\"39+++kjnyuFhn$\"3++++iU&)p:F[o7$$!3-+ ++_3%oE\"Fhn$\"31+++L.&Rn\"F[o7$$!3'******fk\\?R#!#@$\"3!******>zKxx\" F[o7$$!3^+++odj+rF]q$\"31+++jY4\")=F[o7$$!3'******>\\*3y:!#?$\"3!***** *4L8R)>F[o7$$!3'******p%4?`IFhq$\"3))******zX0'3#F[o7$$!32+++\\v\\4aFh q$\"3=+++upP(=#F[o7$$!37+++9.q))*)Fhq$\"37+++NVt(G#F[o7$$!3++++&G&*)>9 !#>$\"3#)*****zZtpQ#F[o7$$!3'******pt83:#F]s$\"35+++yV#\\[#F[o7$$!3t** ***R(z-VJF]s$\"3#)*****f)3Q\"e#F[o7$$!39+++?M@]WF]s$\"3%)*****>Usgn#F[ o7$$!3<+++3,`ChF]s$\"3!******p0J'oFF[o7$$!3j+++>P$4@)F]s$\"3;+++CXceGF [o7$$!3++++Fv'R2\"F-$\"3#)*****HO__%HF[o7$$!3#******\\$)Q=P\"F-$\"3A++ +nS)z-$F[o7$$!3-+++:]n7:Q)yJF[o7$$!3t******[l:-DF-$\"37+++/I(fC$F[o7$$!3<+++(e]k$HF-$\" 37+++PLyc$F[o7$$!3[*****pYg(zhF-$\"3=+++O[w'f$F[o7$$ !3a+++n\"G(p$F[o7$$!3O+++vHel))F-$\"3!)*****fO( ))4PF[o7$$!35+++.\\N)H*F-$\"3z*****RP>.s$F[o7$$!3&******4z!>F(*F-$\"38 +++\\!z'GPF[o7$$!35+++sWA:5F[o$\"39+++/N0NPF[o7$$!3/+++rqOd5F[o$\"3#)* ****zh<&RPF[o7$$!3'*******e?;*4\"F[o$\"3,+++1d8UPF[o7$$!3!******RMC19 \"F[o$\"3#******z\"H'Hu$F[o7$$!3#******fjo<=\"F[o$\"38+++2k/UPF[o7$$!3 \"******Rf4EA\"F[o$\"3&******pJE%RPF[o7$$!3/+++]?;j7F[o$\"30+++:m8NPF[ o7$$!3)******fGTMI\"F[o$\"3/+++2e?HPF[o7$$!3-+++^LYV8F[o$\"33+++qvl@PF [o7$$!33+++NaC$Q\"F[o$\"37+++18^7PF[o7$$!31+++xi!GU\"F[o$\"39+++rEy,PF [o7$$!3++++Fm;i9F[o$\"3-+++tS[*o$F[o7$$!3!******\\v\\8]\"F[o$\"3++++(> Dcn$F[o7$$!3.+++/?QS:F[o$\"3)******fb8-m$F[o7$$!3'******fR$Hz:F[o$\"3( )*****H7bKk$F[o7$$!3++++p#=\"=;F[o$\"3A+++t]vCOF[o7$$!3#*******[f*ol\" F[o$\"3&*******z'=Zg$F[o7$$!3!******pRrcp\"F[o$\"3%******>U_Je$F[o7$$! 36+++@e\\MF[o$\"3\")*** **zqdDU$F[o7$$!3%*******3<F[o$\"33+++$*z$4R$F[o7$$!32+++cmH8?F[o$\" 3A+++\\!=\"eLF[o7$$!3#)*****Hsi]0#F[o$\"3?+++%y7UK$F[o7$$!31+++E&Gv4#F [o$\"3#******4\\f$*G$F[o7$$!3.+++p0tS@F[o$\"3#)*****Ri:PD$F[o7$$!3++++ '4wY=#F[o$\"37+++zOX+++x.zVJF[o7$$!3!*******RC^?BF[o$\"34+++Jts1JF[o7$ $!3))*****pK$zmBF[o$\"37+++]G.$ F[o7$$!31+++xV)*fCF[o$\"3!******R1_h*HF[o7$$!3*)*****>mil]#F[o$\"39+++ ]YtfHF[o7$$!3=+++eY\"Hb#F[o$\"34+++v&GF[o7$$!3?+++iyC*o#F[ o$\"3$******4\")*H;GF[o7$$!3!******>MAMt#F[o$\"3++++b&p2y#F[o7$$!3%)** *****3Fox#F[o$\"3))*****zzW_u#F[o7$$!3=+++jgS>GF[o$\"3@+++$)yl4FF[o7$$ !3=+++ho6hGF[o$\"37+++')H%Rn#F[o7$$!3@+++o&G>!HF[o$\"39+++Aw.QEF[o7$$! 3++++g(>=%HF[o$\"3*******za$)=g#F[o7$$!3')*****Rvw2)HF[o$\"31+++(=Fac# F[o7$$!3=+++K@z=IF[o$\"3;+++a(>'GDF[o7$$!31+++;N'e0$F[o$\"3))*****p:<9 \\#F[o7$$!35+++)e#*>4$F[o$\"3/+++&yzPX#F[o7$$!3!)*****>J%=FJF[o$\"34++ +=An:CF[o7$$!3#*******RjWhJF[o$\"3=+++?H1xBF[o7$$!37+++b&)y%>$F[o$\"3$ ******z#R#zL#F[o7$$!3*)*****puAsA$F[o$\"3A+++p0B)H#F[o7$$!3++++rBweKF[ o$\"3)******\\?hzD#F[o7$$!3;+++$RA%*G$F[o$\"3*)*****p-(4#>LF[o$\"3=+++$*=iv@F[o7$$!3?+++j/<[LF[o$\"39+++?A_L@F[o7$$!3'** *****p_HwLF[o$\"3!******p#py!4#F[o7$$!3<+++ORh.MF[o$\"3!******4V2u/#F[ o7$$!38+++x![,V$F[o$\"3z*****4qxL+#F[o7$$!3#)*****>b?fX$F[o$\"33+++cVp e>F[o7$$!39+++7a&4[$F[o$\"3/+++@oN8>F[o7$$!36+++'zx_]$F[o$\"33+++6vOn= F[o7$$!3<+++yP\"*GNF[o$\"3'******41K2#=F[o7$$!39+++W,*=b$F[o$\"3!***** **o&fMx\"F[o7$$!3/+++!4MUd$F[o$\"32+++VGcDyl$F[o$\"3%******f/J\"G:F[o7$$ !3!)******[7RxOF[o$\"3-+++vzWx9F[o7$$!31+++(\\ukp$F[o$\"3/+++QOIE9F[o7 $$!3>+++[R3:PF[o$\"3/+++3)QZP\"F[o7$$!3\")*****4()HKt$F[o$\"33+++gqzA8 F[o7$$!3;+++iv\"4v$F[o$\"3))*****z)R_q7F[o7$$!35+++')p9oPF[o$\"3++++5j 'z@\"F[o7$$!3%)*****>h7\\y$F[o$\"33+++y3S#[(QF[o$\"3M+++0L/e%)F-7$$!3= +++/+z()QF[o$\"3Z*****RXVZ#zF-7$$!3))*****Rm'4+RF[o$\"3C+++0%R=R(F-7$$ !3A+++l9r6RF[o$\"3_*****H@([foF-7$$!38+++4EgARF[o$\"3s*****Hl+yK'F-7$$ !3\")*****4fRF$RF[o$\"3e*****Rqaoz&F-7$$!3?+++VO4URF[o$\"3S+++p&*om_F- 7$$!3++++u!Q1&RF[o$\"3F+++1\\JPZF-7$$!3!)*****H^[$eRF[o$\"3#******>29( 3UF-7$$!3/+++nI?lRF[o$\"3=+++y\"[3o$F-7$$!3))******>C=rRF[o$\"37+++j)f O:$F-7$$!3%)*****\\()pi(RF[o$\"3z*****ppwqi#F-7$$!3!)*****HO^/)RF[o$\" 32+++8U,,@F-7$$!3++++Gar$)RF[o$\"3'*******e&y`d\"F-7$$!3;+++0K0')RF[o$ \"3/+++;)o+0\"F-7$$!39+++'Qeu)RF[o$\"31+++_-z\\_F]s7$$!3/+++*>Fz)RF[oF (7$Fbjm$!31+++_-z\\_F]s7$F]jm$!3/+++;)o+0\"F-7$Fhim$!3'*******e&y`d\"F -7$Fcim$!32+++8U,,@F-7$F^im$!3z*****ppwqi#F-7$Fihm$!37+++j)fO:$F-7$Fdh m$!3=+++y\"[3o$F-7$F_hm$!3#******>29(3UF-7$Fjgm$!3F+++1\\JPZF-7$Fegm$! 3S+++p&*om_F-7$F`gm$!3e*****Rqaoz&F-7$F[gm$!3s*****Hl+yK'F-7$Fffm$!3_* ****H@([foF-7$Fafm$!3C+++0%R=R(F-7$F\\fm$!3Z*****RXVZ#zF-7$Fgem$!3M+++ 0L/e%)F-7$Fbem$!3c*****4)z`\"**)F-7$F]em$!3z*****p8!)\\_*F-7$Fhdm$!3** *****Hh2e+\"F[o7$Fcdm$!3++++*y[!f5F[o7$F^dm$!3'******fm$=76F[o7$Ficm$! 33+++y3F[o7$Fi^m$!33+++cVpe>F[o7$Fd^m$!3z*****4qxL+#F[o7$F_^m$!3! ******4V2u/#F[o7$Fj]m$!3!******p#py!4#F[o7$Fe]m$!39+++?A_L@F[o7$F`]m$! 3=+++$*=iv@F[o7$F[]m$!3*)*****p-(4'GDF[o7$F^jl$!31+++(=Fac#F[o7$Fiil$!3*******za$)=g#F[o7$F dil$!39+++Aw.QEF[o7$F_il$!37+++')H%Rn#F[o7$Fjhl$!3@+++$)yl4FF[o7$Fehl$ !3))*****zzW_u#F[o7$F`hl$!3++++b&p2y#F[o7$F[hl$!3$******4\")*H;GF[o7$F fgl$!3)******\\s->&GF[o7$Fagl$!3#******\\lTw)GF[o7$F\\gl$!34+++vG.$F[o7 $Fhel$!37+++]+++x.zVJF[o7$Fidl$! 3?+++nJv!=$F[o7$Fddl$!37+++zOXU_Je$F[o7$Fc`l$!3&*******z'=Zg$F[o7 $F^`l$!3A+++t]vCOF[o7$Fi_l$!3()*****H7bKk$F[o7$Fd_l$!3)******fb8-m$F[o 7$F__l$!3++++(>Dcn$F[o7$Fj^l$!3-+++tS[*o$F[o7$Fe^l$!39+++rEy,PF[o7$F`^ l$!37+++18^7PF[o7$F[^l$!33+++qvl@PF[o7$Ff]l$!3/+++2e?HPF[o7$Fa]l$!30++ +:m8NPF[o7$F\\]l$!3&******pJE%RPF[o7$Fg\\l$!38+++2k/UPF[o7$Fb\\l$!3#** ****z\"H'Hu$F[o7$F]\\l$!3,+++1d8UPF[o7$Fh[l$!3#)*****zh<&RPF[o7$Fc[l$! 39+++/N0NPF[o7$F^[l$!38+++\\!z'GPF[o7$Fiz$!3z*****RP>.s$F[o7$Fdz$!3!)* ****fO())4PF[o7$F_z$!3!******>n\"G(p$F[o7$Fjy$!3#)*****fJ%Q#o$F[o7$Fey $!35+++gy0lOF[o7$F`y$!3%)*****f4U^k$F[o7$F[y$!3()*****RD]Ci$F[o7$Ffx$! 3=+++O[w'f$F[o7$Fax$!3%******>PLyc$F[o7$F\\x$!3<+++QZONNF[o7$Fgw$!3%)* ****fJF!*\\$F[o7$Fbw$!3/+++C7XeMF[o7$F]w$!3!******pAQKT$F[o7$Fhv$!3!)* ****fL%)HO$F[o7$Fcv$!37+++:Q)yJF[o7$Fdu$!3)*******[h/1JF[o7$F_u$!3A+++nS)z-$F[o7$Fjt$!3#)*****H O__%HF[o7$Fet$!3;+++CXceGF[o7$F`t$!3!******p0J'oFF[o7$F[t$!3%)*****>Us gn#F[o7$Ffs$!3#)*****f)3Q\"e#F[o7$Fas$!35+++yV#\\[#F[o7$F[s$!3#)*****z ZtpQ#F[o7$Ffr$!37+++NVt(G#F[o7$Far$!3=+++upP(=#F[o7$F\\r$!3))******zX0 '3#F[o7$Ffq$!3!******4L8R)>F[o7$Faq$!31+++jY4\")=F[o7$F[q$!3!******>zK xx\"F[o7$Ffp$!31+++L.&Rn\"F[o7$$\"3]*****>Ow'yuFhn$!3++++iU&)p:F[o7$F \\p$!3#*******\\M`l9F[o7$Fgo$!3/+++z\"e5O\"F[o7$Fbo$!33+++h8[c7F[o7$F] o$!31++++4%=:\"F[o7$Ffn$!3)******4fir/\"F[o7$$\"3s*****>)evd)*FN$!3w** ***\\jIYU*F-7$FR$!3A+++cT_x$)F-7$FL$!3]******GQOItF-7$FG$!3S+++&yzJG'F -7$$\"3'******H\"oX\"o\"FC$!3d*****zW')fB&F-7$F;$!3#******>(**y)=%F-7$ $\"3v*****fJ3Q5&F7$!3(******pi#fTJF-7$$\"3$******\\K-Nj$F1$!3-+++,^R%4 #F-7$F($!31+++^v>Z5F-F'F*F.F4F:F@FFFKFQFVFenF\\oFaoFfoF[pF`pFepFjpF`qF eqF[rF`rFerFjrF`sFesFjsF_tFdtFitF^uFcuFhuF]vFbvFgvF\\wFawFfwF[xF`xFexF jxF_yFdyFiyF^zFczFhzF][lFb[lFg[lF\\\\lFa\\lFf\\lF[]lF`]lFe]lFj]lF_^lFd ^l-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$$!3_^*>\"z,5n=F]s$\"3 hPT*)**\\JAFF-7$$!3G'3JMTnU]'F]s$\"3M^TrtfG^]F-7$$!3gF*)p47H)\\\"F-$\" 3]/xaaNE$e(F-7$$!3#f]+(fvs&o#F-$\"3[7\"[6!\\e%***F-7$$!3OO6Ah^%)yTF-$ \"3'ytqa[>9A\"F[o7$$!33vGV84Ct7A)[x:F[o7$$!3UPw!)3*R$[**F-$\"3#>V:9J\"eDF[o7$$!3oy&oQ@[ds \"F[o$\"3]pK9L>%e(>F[o7$$!3=]#y&4U:&*>F[o$\"3;D$RXCjR*>F[o7$$!3Enp*HiP cE#F[o$\"3S&>\"f[$p`(>F[o7$$!3r*=y')o-;_#F[o$\"3T'R[yG&)G#>F[o7$$!39Gx t'y2ju#F[o$\"3'=())oxIeY=F[o7$$!3'*HF`6HQ**HF[o$\"3\\:#y'pT#>s\"F[o7$$ !3+0P%y\"e@)>$F[o$\"3!RD>!e[B*e\"F[o7$$!3K)GLWZZvS$F[o$\"3c\"*puwqH19F [o7$$!3mwQGZJ.qNF[o$\"3)eFv5zJ9A\"F[o7$$!3=P;/Wyf?PF[o$\"37B(=o\"*)3t* *F-7$$!3Su`D#pyX$QF[o$\"3yc8(pL\"HowF-7$$!3!\\c&=_\\&3#RF[o$\"366W:vH; G^F-7$$!3suCGC$*QpRF[o$\"3)**e.1%zj7FF-7$$!3GKh)3hB!y'4m%=F[o7$$!38/qO@?79DF[o$!3B_:L]B# \\#>F[o7$$!3!e.!o8t\\^AF[o$!3%H)pHeDEx>F[o7$$!3'GSp7\"*f++#F[o$!3w$p(f !GaR*>F[o7$$!3cw$Q0F'>PF[o7$$!3OMr,cw\\%[\"F[o$!3PkhI '))zx#>F[o7$$!3J\")eAk$>\"H7F[o$!3')eyVw2WT=F[o7$$!3mIBSrnBj**F-$!31q& fr*HWE 5#**F-7$$!3WFBK*))>`a\"F-$!3ebz4Roj'p(F-7$$!3'3WWWDy`)oF]s$!3]0.iZ &F-7$$!3KS9mWFFb=F]s$!3PArCp#>Pr#F-7$F($\"3S*3zF3vej\"F7-Fcgo6&FegoF(F (Ffgo-%(SCALINGG6#%,CONSTRAINEDG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6%Q!6\"Fhgp-Fagp6#%(DEFAULTG-%%VIEWG6$F\\hpF\\hp" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "We can distort the boundary curve horizontally by taking the 11th root of the real p art of points along the curve. In this way we see that the largest int erval on the nonnegative imaginary axis that contains the origin and l ies inside the stability region is [ 0, 1.67 ] approximately. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+17291/12418560*z ^6+269/1379840*z^7:\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz: \n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,col or=COLOR(RGB,.15,.15,1),thickness=2,font=[HELVETICA,9]);\nDigits := 10 :" }}{PARA 13 "" 1 "" {GLPLOT2D 249 293 293 {PLOTDATA 2 "6(-%'CURVESG6 #7in7$$\"\"!F)F(7$$\"5%Q*GND&*=JJ[!#@$\"5PEs(*e`EfTJF-7$$\"5&*Ga[0Csyh qF-$\"5k8zo<2`=$G'F-7$$\"5cq1v!o)GEJ))F-$\"523![A2'zxC%*F-7$$\"53$3q$3 )*3dO5!#?$\"5c&)4,Shqjc7F=7$$\"5u'**o***RMYv6F=$\"5[bS+iEjzq:F=7$$\"5^ &f7n-hCWI\"F=$\"5F1u$4:fb\\)=F=7$$\"5(e2)Rs+;#F=7$$ \"53>Ml!pYECa\"F=$\"5k;;HqF=$\"5WES*>'z5\"*pPF=7$$\"5#oU[XIvq52#F=$\"5QS/M^d-2%3 %F=7$$\"53$**f(4IRIp@F=$\"5%H*31Lw$H#)R%F=7$$\"5bPB?$>\")ebE#F=$\"5J,% *Q(RS)Q7ZF=7$$\"5BiR:\\]$\\*fBF=$\"5@XTcZ%HZl-&F=7$$\"5xs0!*\\r.c_CF=$ \"5(HG\")RL)fqS`F=7$$\"5IhWe))>ZXVDF=$\"5Ml#*Rs\"Qk[l&F=7$$\"59WNY&fDy Ej#F=$\"5#G=Dr#pB-pfF=7$$\"54C!QqMNj-s#F=$\"5QqG!)Q&yzJG'F=7$$\"5Nk8Sj R8B1GF=$\"5@:,Hv=kL(f'F=7$$\"5M\"y#z3wTf!*GF=$\"5]r;?$[*>\\6pF=7$$\"5' Qv7%z%ebL(HF=$\"5^4'\\q6;YcA(F=7$$\"5m?ZA:d<^aIF=$\"5*[$zdAq%)zRvF=7$$ \"5(p9mw*Q<0MJF=$\"5>84\"4&f$[R&yF=7$$\"5#*oGTC^v&>@$F=$\"5?!)>c!)G^4o \")F=7$$\"5kl**oK#3/#)G$F=$\"5ho;BD9zB#[)F=7$$\"5a[G[do(eFO$F=$\"5Eb*3 n!fcP'z)F=7$$\"5#fuC!Gk5eNMF=$\"5Vn?8u!3206*F=7$$\"5^'yz2uo@m]$F=$\"5M /QT`M1jC%*F=7$$\"58POhN7;#ed$F=$\"5NjV%QJZW(Q(*F=7$$\"5?t#o]Pw5Jk$F=$ \"50%=!GCSYG05!#>7$$\"5ec*HVQM1%3PF=$\"5HquTK$Q$pO5Ffu7$$\"5+*eCayp5;x $F=$\"5jru:#HP+\"o5Ffu7$$\"5S'z]2Fe3E$QF=$\"56+HP2b_]*4\"Ffu7$$\"5lzfU G\\OE\"*QF=$\"5Wb6m^>w!48\"Ffu7$$\"5oAm=UnQTZRF=$\"5if=\\N$*pIi6Ffu7$$ \"52@r3e,b'3+%F=$\"5Cz*R9W$Gq$>\"Ffu7$$\"5E>:w`e]Q^SF=$\"58h%H;V_%4D7F fu7$$\"5EvdMoL$)o)4%F=$\"5/TBcKh8[c7Ffu7$$\"58Ffu7$$\"5!Hh(>$p*)Ht@%F=$\"5 M)Q1dJL/1N\"Ffu7$$\"5F:'H^\"or@ZUF=$\"5$fuKao\"G'>Q\"Ffu7$$\"50vXl7]T& 3F%F=$\"5psBYTJ9J89Ffu7$$\"56-ZE$)R'4pG%F=$\"5^j?__@)[YW\"Ffu7$$\"5H79 5P$zqMH%F=$\"5pJ(z()))\\tfZ\"Ffu7$$\"5JoL?6#fNwG%F=$\"5=#*zy\\OQG2:Ffu 7$$\"5wz&QINtiYE%F=$\"5Y:_Zmj!y&Q:Ffu7$$\"5KqSP#ftFf@%F=$\"5E%)[x\">Ea )p:Ffu7$$\"5hh?L?Ao!G7%F=$\"5ZJyH%>O56g\"Ffu7$$\"5h#**)QwqpIHRF=$\"5z> 3zoTTMK;Ffu7$$\"5])p!>B!\\Tk$GF=$\"52ZH\"fdA`Nm\"Ffu7$$!5&f8c1**pU\"=S F=$\"5Iq'G?i3NZp\"Ffu7$$!5\"yy;.kl!HSVF=$\"5)\\8/s[/()es\"Ffu7$$!5zPFB (Hi;Dc%F=$\"5l\\*fjqF1qv\"Ffu7$$!50aKVyYs'Gu%F=$\"5/S*eXq\")*3)y\"Ffu7 $$!5?'pNP3(\\%)**[F=$\"5Zo+q#[cM\">=Ffu7$$!5ZU6;'Q6zA'fU29t^F=$\"5j(Qc&\\jY4\")=Ffu-%%FONTG6$%*HELVETICAG\" \"*-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"F__l-%&COLORG6&%$RGBG$\"#: !\"#Fe_l$\"\"\"F)-%%VIEWG6$%(DEFAULTGF]`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection poin t of the boundary curve with the imaginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look \+ for points on the boundary curve either side of the intersection point . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 1.7*I:\nfor ct from 52 to 55 do \n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0q`-&>FZM!#>$\"0\"zoTTMK;!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0f$\\`Q>h&*!#@$\"08fdA`Nm\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0WO$H*y$3W!#>$\"0H?i3NZp\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0.\"e&3\"pH5!#=$\"0/s[/()es\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then \+ we apply the bisection method to calculate the parameter value associa ted with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.7*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.52..0.55); \nnewton(R(z)=exp(u0*Pi*I),z=1.7*I);\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0H]M)zW-`!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0r?#))oJk;!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[ 0, 1.6643 ]" "6#7 $\"\"!-%&FloatG6$\"&Vm\"!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "th e stability region for the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R * for the 8 stage, order 4 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFuncti on(4,8,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#\"'s'p#\")4U;KF)*$)F'\"\"&F)F)F)*&#\"(Z!\\B\"+![Car\"F)*$)F'F1F)F)F )*&#\"+.:m48\".gL]\"f;pF)*$)F'\"\"(F)F)F)*&#\"' " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+&y(G)*R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=- 1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.49..0.49,color=[red,blue]):\np _2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],co lor=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RG B,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.49..0.49,-1.47..1.4 7],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 393 267 267 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%!#<$!3A*>r>_hvT#F*7$$!3;+ ]7'3DdV%F*$!3>#*[s1H<0AF*7$$!35++Ds,X\"Q%F*$!3@!GPmjs*3?F*7$$!3(*\\P*) feAMVF*$!3m6j#eJ*f]=F*7$$!3%)*\\Pva,qG%F*$!3jT2fRs+.%y!= %F*$!3L)H^8;+sS\"F*7$$!35+]#oviQ2%F*$!3m'3y*e9da6F*7$$!3A+ve,&\\u'RF*$ !3gh*GXC05U*!#=7$$!3?+v8#z!zoQF*$!3]f*=Wq*p^xFP7$$!3;+v.8ajmPF*$!3+114 OJc'G'FP7$$!3C+v$Q<')4m$F*$!3=v\\Q%esQ,&FP7$$!3G+v3`dnbNF*$!3;#G6hIsZ& RFP7$$!37++0T5NZMF*$!3m<`&[\"\\+]IFP7$$!3?+]sv&Q>N$F*$!37YHx+%*=%Q#FP7 $$!3E++:Qn_WKF*$!37*4'*yunyv\"FP7$$!3#****\\s&QnOJF*$!3Y.hx-SJQ7FP7$$! 3&****\\=iPF.$F*$!3](o-_\"ogE#)!#>7$$!3Q+v$Gc`$QHF*$!3=J%H\")R.L/&Fhp7 $$!3+++l(y@h#GF*$!3Yv8=o*)RZ=Fhp7$$!3=++g_n/JFF*$\"3\"G+ISl@Eh%!#?7$$! 3C+vt%)=X?EF*$\"3%fu6FP7$$ !3$***\\7w!eC+#F*$\"3=C2<5JZ$G\"FP7$$!3%**\\Py(=m#*=F*$\"3sdQ&R^;:Y\"F P7$$!3%)*\\iW'R3(z\"F*$\"3\\KGcw$=gi\"FP7$$!3-+]d*>dQp\"F*$\"3)HRmid$o ;=FP7$$!3)*****p]Q@(e\"F*$\"3^mbO&p*)4.#FP7$$!3I+]FRT)G[\"F*$\"3-F]LPk #3E#FP7$$!3')*\\P*y(R>Q\"F*$\"3.td$[Pc^]#FP7$$!3++]Kx#e)p7F*$\"3@IVEL% 3`!GFP7$$!3=++5j![\"p6F*$\"3o!)*of&*\\U5$FP7$$!3s***\\KT=;1\"F*$\"3Mj' *[=')ydMFP7$$!3'***\\P%o0=k*FP$\"3/Gdu/fG7QFP7$$!3](***\\tEbw&)FP$\"3M P`*>p37C%FP7$$!39-]P2EBuvFP$\"3qNMyrgb)o%FP7$$!3>)*\\(GL>l_'FP$\"3:+&= Dz`l?&FP7$$!3z++]-\")=-bFP$\"3GU/B`U?odFP7$$!3$*)*\\PM#3)HWFP$\"3#H!e& )>]=@kFP7$$!3;(****f'*ypR$FP$\"3c[PuR7&)>rFP7$$!3L)**\\U'=wSBFP$\"3'4+ 8L)[,8zFP7$$!3?$*\\Pt2H$H\"FP$\"3#G#*=zgZoy)FP7$$!33R****pit2LFhp$\"3K n%p36PYn*FP7$$\"3Z^+]FkzBxFhp$\"3!z/S&=\"*H!3\"F*7$$\"35*****z2`!f\"F*7$$\"3l+]i_F06GFP$\"3oSV'47$fC8F*7$$\"3\\.]Pt1&z\"QFP$ \"35W&*fw@\"\\Y\"F*7$$\"3!***************[FP$\"3FS " 0 " " {MPLTEXT 1 0 1374 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+269672/ 32164209*z^5+\n 2349047/1715424480*z^6+1309661503/6916591503360 *z^7+885817/768510167040*z^8:\npts := []:\nz0 := 0:\nfor ct from 0 to \+ 200 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=C OLOR(RGB,0,.1,.2)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[ -2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB, .2,.35,.9)):\npts := []:\nz0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n \+ zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := \+ [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,0 ,.1,.2)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.33,4.23] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.2,.3 5,.9)):\npts := []:\nz0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz : = newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,0,.1,. 2)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.33,-4.23]],i= 2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.2,.35,.9 )):\np_7 := plot([[[-4.79,0],[2.19,0]],[[0,-4.79],[0,4.79]]],color=bla ck,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.79..2.19,-4.79. .4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axe s=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$!33+++*p]c/\"!#E$\"3! *******yizq:!#=7$$!3S*****>1;kY'!#D$\"3(******z>\"fTJF07$$!39+++U%e1'p !#C$\"3y*****RwyBr%F07$$!3#******HXS&>O!#B$\"3y*****>>YJG'F07$$!3-+++k 6(\\D\"!#A$\"3-+++QG(Q&yF07$$!3-+++7\\+jLFF$\"3c*****f`AXU*F07$$!3O*** **HzMne(FF$\"3/+++U:]*4\"!#<7$$!3++++fh'4`\"!#@$\"31+++ki^c7FS7$$!3$** ****4eSf!HFW$\"3%******f:bMT\"FS7$$!3*******>Ff*4aFW$\"35+++u$Q-d\"FS7 $$!3********Gb([,\"!#?$\"3(*******[$Hns\"FS7$$!3-+++56#*H>Fao$\"35+++. 1s#)=FS7$$!3!******f+()yn$Fao$\"3$******p&4$z.#FS7$$!3%******z::1!pFao $\"3#*******3x,#>#FS7$$!3-+++$*)zeD\"!#>$\"3!)*****fQ%fWBFS7$$!31+++%H 1u>#Ffp$\"3;+++-UA&\\#FS7$$!3I+++**HH#o$Ffp$\"3$)*****HdTLk#FS7$$!3.++ +sPC/fFfp$\"37+++N$e!)y#FS7$$!3]*****fJ`d0*Ffp$\"32+++*3sy#HFS7$$!3$** ****fK/sK\"F0$\"3++++)3f/1$FS7$$!3%******HI8c&=F0$\"3A+++rG$H=$FS7$$!3 ++++Xg=uCF0$\"31+++U&eFH$FS7$$!3)******4]0]:$F0$\"3#******4[H()Q$FS7$$ !3=+++wI!)pQF0$\"39+++7G:rMFS7$$!3u*****RH&G(f%F0$\"35+++wtq%*F0$\"3-+++wV6)z$FS7$$!3%******RL`B,\"FS$\"3!)*** **fqB9\"QFS7$$!3-+++C7qw5FS$\"32+++5BG?QFS7$$!3,+++RH=S6FS$\"32+++-E$ \\#QFS7$$!35+++Q*eG?\"FS$\"3@+++;rcDQFS7$$!3/+++.dyk7FS$\"3)******\\6Q B#QFS7$$!3)******Rd+++:( RZ!QFS7$$!3-+++4whY9FS$\"3#******f1K0z$FS7$$!3-+++%f1h]\"FS$\"3/+++kIz sPFS7$$!3#******p\"*[^c\"FS$\"3#)*****p.f:v$FS7$$!3#******z(3$Qi\"FS$ \"32+++[b&os$FS7$$!37+++4lD#o\"FS$\"3/+++#3-()p$FS7$$!3#*******R@bSFS$\"3!)*****\\3k@b$FS7$$!3++ ++-)[f(>FS$\"3;+++(4%R2NFS7$$!3*******4Nvk.#FS$\"3!)*****R\"znfMFS7$$! 31+++1&p\")4#FS$\"3%)*****4?2$4MFS7$$!31+++l]Bh@FS$\"3A+++hUncLFS7$$!3 (******pR%yDAFS$\"38+++0XE-LFS7$$!3-+++RHz\"H#FS$\"3=+++\"*HiYKFS7$$!3 /+++S?2fBFS$\"3)******Ri&H!>$FS7$$!39+++\\aFFCFS$\"3>+++#**eP8$FS7$$!3 #)*******yTf\\#FS$\"37+++]*y`HGFS$\"3=+ ++4Op*z#FS7$$!3%******\\n?=*GFS$\"3&)*****HzdTu#FS7$$!3;+++6v9_HFS$\"3 ++++T'p\")o#FS7$$!3$)******4TV5IFS$\"3))******3JaJEFS7$$!3++++$)RjmIFS $\"3#*******=>6uDFS7$$!3*)*****4zG27$FS$\"3y*****zmId^#FS7$$!3%)****** >6ssJFS$\"3=+++RSFcCFS7$$!3#*******e$HEA$FS$\"37+++j^j&R#FS7$$!31+++^U [qKFS$\"3.+++oPsLBFS7$$!3#******prEjJ$FS$\"3-+++*\\k/F#FS7$$!3!)*****4 g1-O$FS$\"3$******\\i&z0AFS7$$!3:+++Z?=-MFS$\"3@+++s#o'R@FS7$$!3&***** *fA>BW$FS$\"3!******pBY?2#FS7$$!3%)*****p=#p![$FS$\"3*******4V1H+#FS7$ $!33+++fDQFS7$$!3;+++j)yCb$FS$\"3!******zj`+'=FS7$$! 3-+++\\^2'e$FS$\"35+++BAP'y\"FS7$$!3(******Hro#=OFS$\"3&******\\US7r\" FS7$$!3)******>qc\"\\OFS$\"34+++7]sM;FS7$$!33+++d:$)yOFS$\"3#******R5; pb\"FS7$$!3'******>Xvtq$FS$\"35+++3s#zZ\"FS7$$!3-++++U&[t$FS$\"3++++`M *yR\"FS7$$!31+++a8JhPFS$\"3#*******pw'pJ\"FS7$$!3-+++RMw'y$FS$\"3,+++Y VJN7FS7$$!3(******\\@(>6QFS$\"3++++C>5`6FS7$$!3:+++2'pX$QFS$\"3#****** fO%\\q5FS7$$!3'*******>/\"o&QFS$\"3#)******=6Vw)*F07$$!3)******4WFy(QF S$\"3'******H,0o/*F07$$!39+++xI^(*QFS$\"3&)********R:<#)F07$$!35+++9:v :RFS$\"3'******4k7$)Q(F07$$!3$)*****H[DC$RFS$\"3!)*****zLk3c'F07$$!35+ ++!)>UZRFS$\"3*)******H7;NdF07$$!3/+++GmjgRFS$\"3%)*****>ia8\"\\F07$$! 33+++vm(>(RFS$\"37+++%*)H%*3%F07$$!3++++2GO\")RFS$\"3%)*****RJT#pKF07$ $!3y*****p/I())RFS$\"3#******p!Qa]CF07$$!3;+++a!GS*RFS$\"36+++B.-L;F07 $$!3++++z5A(*RFS$\"3E+++^g/j\")Ffp7$$!3/+++&y(G)*RFSF(7$Fijl$!3E+++^g/ j\")Ffp7$Fdjl$!36+++B.-L;F07$F_jl$!3#******p!Qa]CF07$Fjil$!3%)*****RJT #pKF07$Feil$!37+++%*)H%*3%F07$F`il$!3%)*****>ia8\"\\F07$F[il$!3*)***** *H7;NdF07$Ffhl$!3!)*****zLk3c'F07$Fahl$!3'******4k7$)Q(F07$F\\hl$!3&)* *******R:<#)F07$Fggl$!3'******H,0o/*F07$Fbgl$!3#)******=6Vw)*F07$F]gl$ !3#******fO%\\q5FS7$Fhfl$!3++++C>5`6FS7$Fcfl$!3,+++YVJN7FS7$F^fl$!3#** *****pw'pJ\"FS7$Fiel$!3++++`M*yR\"FS7$Fdel$!35+++3s#zZ\"FS7$F_el$!3#** ****R5;pb\"FS7$Fjdl$!34+++7]sM;FS7$Fedl$!3&******\\US7r\"FS7$F`dl$!35+ ++BAP'y\"FS7$F[dl$!3!******zj`+'=FS7$Ffcl$!35+++S*RA$>FS7$Facl$!3***** **4V1H+#FS7$F\\cl$!3!******pBY?2#FS7$Fgbl$!3@+++s#o'R@FS7$Fbbl$!3$**** **\\i&z0AFS7$F]bl$!3-+++*\\k/F#FS7$Fhal$!3.+++oPsLBFS7$Fcal$!37+++j^j& R#FS7$F^al$!3=+++RSFcCFS7$Fi`l$!3y*****zmId^#FS7$Fd`l$!3#*******=>6uDF S7$F_`l$!3))******3JaJEFS7$Fj_l$!3++++T'p\")o#FS7$Fe_l$!3&)*****HzdTu# FS7$F`_l$!3=+++4Op*z#FS7$F[_l$!3-+++\\$y\\&GFS7$Ff^l$!3(******\\O=-\"H FS7$Fa^l$!3++++!)pglHFS7$F\\^l$!36+++#f'H@IFS7$Fg]l$!37+++]+++#**eP8$FS7$F]]l$!3)******Ri&H!>$FS7$Fh\\l$!3=+++\"*HiYKFS7$Fc \\l$!38+++0XE-LFS7$F^\\l$!3A+++hUncLFS7$Fi[l$!3%)*****4?2$4MFS7$Fd[l$! 3!)*****R\"znfMFS7$F_[l$!3;+++(4%R2NFS7$Fjz$!3!)*****\\3k@b$FS7$Fez$!3 ')*****HQ'y$f$FS7$F`z$!3$******4xI@j$FS7$F[z$!3))*****H/=rm$FS7$Ffy$!3 /+++#3-()p$FS7$Fay$!32+++[b&os$FS7$F\\y$!3#)*****p.f:v$FS7$Fgx$!3/+++k IzsPFS7$Fbx$!3#******f1K0z$FS7$F]x$!3>+++:(RZ!QFS7$Fhw$!3@+++M\\O:QFS7 $Fcw$!3)******\\6QB#QFS7$F^w$!3@+++;rcDQFS7$Fiv$!32+++-E$\\#QFS7$Fdv$! 32+++5BG?QFS7$F_v$!3!)*****fqB9\"QFS7$Fju$!3-+++wV6)z$FS7$Feu$!3%)**** *RI\\+y$FS7$F`u$!3#******H#R%ov$FS7$F[u$!37+++.;,GPFS7$Fft$!3#******zA MHp$FS7$Fat$!37+++!QD3l$FS7$F\\t$!3*)*****46*o+OFS7$Fgs$!35+++#FS7$Fjo$!3$******p&4$z.#FS7$Feo$!35+++.1s#)=FS7$ F_o$!3(*******[$Hns\"FS7$Fjn$!35+++u$Q-d\"FS7$Fen$!3%******f:bMT\"FS7$ FU$!31+++ki^c7FS7$FO$!3/+++U:]*4\"FS7$FJ$!3c*****f`AXU*F07$FD$!3-+++QG (Q&yF07$F>$!3y*****>>YJG'F07$$!3++++?&e1'pF:$!3y*****RwyBr%F07$F2$!3(* *****z>\"fTJF07$$!3++++(p]c/\"F-$!3!*******yizq:F0F'-%&COLORG6&%$RGBGF ($\"\"\"!\"\"$\"\"#Fc^n-%)POLYGONSG6fw7%F'7$$!+*p]c/\"F0$\"+zizq:!#57$ $!\"#F)F(7%Fj^n7$$!+igTmkFS$\"+)>\"fTJF__nF`_n7%Fd_n7$$!+U%e1'p!#;$\"+ k(yBr%F__nF`_n7%Fj_n7$$!+`/a>O!#:$\"+#>YJG'F__nF`_n7%Fa`n7$$!+k6(\\D\" !#9$\"+QG(Q&yF__nF`_n7%Fh`n7$$!+7\\+jLF[an$\"+OD_C%*F__nF`_n7%F_an7$$! +$zMne(F[an$\"+U:]*4\"!\"*F`_n7%Fean7$$!+fh'4`\"!#8$\"+ki^c7FjanF`_n7% F\\bn7$$!+\"eSf!HF_bn$\"+c^X89FjanF`_n7%Fcbn7$$!+s#f*4aF_bn$\"+u$Q-d\" FjanF`_n7%Fibn7$$!+Hb([,\"!#7$\"+\\$Hns\"FjanF`_n7%F_cn7$$!+56#*H>Fbcn $\"+.1s#)=FjanF`_n7%Ffcn7$$!+1q)yn$Fbcn$\"+d4$z.#FjanF`_n7%F\\dn7$$!+e ^h+pFbcn$\"+4x,#>#FjanF`_n7%Fbdn7$$!+$*)zeD\"!#6$\"+'Q%fWBFjanF`_n7%Fh dn7$$!+%H1u>#F[en$\"+-UA&\\#FjanF`_n7%F_en7$$!+**HH#o$F[en$\"+t:MVEFja nF`_n7%Feen7$$!+sPC/fF[en$\"+N$e!)y#FjanF`_n7%F[fn7$$!+;Lvb!*F[en$\"+* 3sy#HFjanF`_n7%Fafn7$$!+EV?F8F__n$\"+)3f/1$FjanF`_n7%Fgfn7$$!+.Lhb=F__ n$\"+rG$H=$FjanF`_n7%F]gn7$$!+Xg=uCF__n$\"+U&eFH$FjanF`_n7%Fcgn7$$!+,b +bJF__n$\"+\"[H()Q$FjanF`_n7%Fign7$$!+wI!)pQF__n$\"+7G:rMFjanF`_n7%F_h n7$$!+%H&G(f%F__n$\"+wtq%*F__n$\"+wV6)z$FjanF`_n7%F_[o7$$!+MLN75Fjan$\"+1PU6QFjanF`_n7%Fe [o7$$!+C7qw5Fjan$\"+5BG?QFjanF`_n7%F[\\o7$$!+RH=S6Fjan$\"+-E$\\#QFjanF `_n7%Fa\\o7$$!+Q*eG?\"Fjan$\"+;rcDQFjanF`_n7%Fg\\o7$$!+.dyk7Fjan$\"+: \"QB#QFjanF`_n7%F]]o7$$!+uv,E8Fjan$\"+M\\O:QFjanF`_n7%Fc]o7$$!+r!4mQ\" Fjan$\"+:(RZ!QFjanF`_n7%Fi]o7$$!+4whY9Fjan$\"+m?`!z$FjanF`_n7%F_^o7$$! +%f1h]\"Fjan$\"+kIzsPFjanF`_n7%Fe^o7$$!+<*[^c\"Fjan$\"+P!f:v$FjanF`_n7 %F[_o7$$!+y3$Qi\"Fjan$\"+[b&os$FjanF`_n7%Fa_o7$$!+4lD#o\"Fjan$\"+#3-() p$FjanF`_n7%Fg_o7$$!+S@bSFjan$\"+&3k@b$FjanF`_n7%F_ao7$$!+-)[f(>Fjan$\"+(4%R2NFjanF`_n7%Feao 7$$!+^`ZO?Fjan$\"+9znfMFjanF`_n7%F[bo7$$!+1&p\")4#Fjan$\"+,sI4MFjanF`_ n7%Fabo7$$!+l]Bh@Fjan$\"+hUncLFjanF`_n7%Fgbo7$$!+(R%yDAFjan$\"+0XE-LFj anF`_n7%F]co7$$!+RHz\"H#Fjan$\"+\"*HiYKFjanF`_n7%Fcco7$$!+S?2fBFjan$\" +CcH!>$FjanF`_n7%Fico7$$!+\\aFFCFjan$\"+#**eP8$FjanF`_n7%F_do7$$!+!zTf \\#Fjan$\"+]>6uDFjanF`_n7%F[ho7$$!+\"zG27$Fjan$\"+o1t:DFjanF`_n7%Faho7 $$!+?6ssJFjan$\"+RSFcCFjanF`_n7%Fgho7$$!+f$HEA$Fjan$\"+j^j&R#FjanF`_n7 %F]io7$$!+^U[qKFjan$\"+oPsLBFjanF`_n7%Fcio7$$!+BW$Fjan$\"+Pi/s?FjanF`_n7%F[[p7$$!+(=#p![$Fjan $\"+Jk!H+#FjanF`_n7%Fa[p7$$!+fDQFjanF`_n7%Fg[p7$$!+j)y Cb$Fjan$\"+QO0g=FjanF`_n7%F]\\p7$$!+\\^2'e$Fjan$\"+BAP'y\"FjanF`_n7%Fc \\p7$$!+8(o#=OFjan$\"+D/C66QFjan$\"+C>5`6FjanF`_n7%Fc_p7$$!+2'pX$QFjan$\"+mV\\q5FjanF`_n7%F i_p7$$!+?/\"o&QFjan$\"+>6Vw)*F__nF`_n7%F_`p7$$!+Tu#y(QFjan$\"+8]!o/*F_ _nF`_n7%Fe`p7$$!+xI^(*QFjan$\"++S:<#)F__nF`_n7%F[ap7$$!+9:v:RFjan$\"+T EJ)Q(F__nF`_n7%Faap7$$!+$[DC$RFjan$\"+QV'3c'F__nF`_n7%Fgap7$$!+!)>UZRF jan$\"+I7;NdF__nF`_n7%F]bp7$$!+GmjgRFjan$\"+AYN6\\F__nF`_n7%Fcbp7$$!+v m(>(RFjan$\"+%*)H%*3%F__nF`_n7%Fibp7$$!+2GO\")RFjan$\"+98CpKF__nF`_n7% F_cp7$$!+Z+t))RFjan$\"+2Qa]CF__nF`_n7%Fecp7$$!+a!GS*RFjan$\"+B.-L;F__n F`_n7%F[dp7$$!+z5A(*RFjan$\"+^g/j\")F[enF`_n7%Fadp7$$!+&y(G)*RFjanF(F` _n7%Fgdp7$Fbdp$!+^g/j\")F[enF`_n7%F[ep7$F\\dp$!+B.-L;F__nF`_n7%F_ep7$F fcp$!+2Qa]CF__nF`_n7%Fcep7$F`cp$!+98CpKF__nF`_n7%Fgep7$Fjbp$!+%*)H%*3% F__nF`_n7%F[fp7$Fdbp$!+AYN6\\F__nF`_n7%F_fp7$F^bp$!+I7;NdF__nF`_n7%Fcf p7$Fhap$!+QV'3c'F__nF`_n7%Fgfp7$Fbap$!+TEJ)Q(F__nF`_n7%F[gp7$F\\ap$!++ S:<#)F__nF`_n7%F_gp7$Ff`p$!+8]!o/*F__nF`_n7%Fcgp7$F``p$!+>6Vw)*F__nF`_ n7%Fggp7$Fj_p$!+mV\\q5FjanF`_n7%F[hp7$Fd_p$!+C>5`6FjanF`_n7%F_hp7$F^_p $!+YVJN7FjanF`_n7%Fchp7$Fh^p$!+qw'pJ\"FjanF`_n7%Fghp7$Fb^p$!+`M*yR\"Fj anF`_n7%F[ip7$F\\^p$!+3s#zZ\"FjanF`_n7%F_ip7$Ff]p$!+/h\"pb\"FjanF`_n7% Fcip7$F`]p$!+7]sM;FjanF`_n7%Fgip7$Fj\\p$!+D/C6Fjan F`_n7%Fgjp7$Fb[p$!+Jk!H+#FjanF`_n7%F[[q7$F\\[p$!+Pi/s?FjanF`_n7%F_[q7$ Ffjo$!+s#o'R@FjanF`_n7%Fc[q7$F`jo$!+Dcz0AFjanF`_n7%Fg[q7$Fjio$!+*\\k/F #FjanF`_n7%F[\\q7$Fdio$!+oPsLBFjanF`_n7%F_\\q7$F^io$!+j^j&R#FjanF`_n7% Fc\\q7$Fhho$!+RSFcCFjanF`_n7%Fg\\q7$Fbho$!+o1t:DFjanF`_n7%F[]q7$F\\ho$ !+>>6uDFjanF`_n7%F_]q7$Ffgo$!+4JaJEFjanF`_n7%Fc]q7$F`go$!+T'p\")o#Fjan F`_n7%Fg]q7$Fjfo$!+$zdTu#FjanF`_n7%F[^q7$Fdfo$!+4Op*z#FjanF`_n7%F_^q7$ F^fo$!+\\$y\\&GFjanF`_n7%Fc^q7$Fheo$!+l$=-\"HFjanF`_n7%Fg^q7$Fbeo$!+!) pglHFjanF`_n7%F[_q7$F\\eo$!+#f'H@IFjanF`_n7%F__q7$Ffdo$!+]$FjanF`_n7%F[`q7$Fdc o$!+\"*HiYKFjanF`_n7%F_`q7$F^co$!+0XE-LFjanF`_n7%Fc`q7$Fhbo$!+hUncLFja nF`_n7%Fg`q7$Fbbo$!+,sI4MFjanF`_n7%F[aq7$F\\bo$!+9znfMFjanF`_n7%F_aq7$ Ffao$!+(4%R2NFjanF`_n7%Fcaq7$F`ao$!+&3k@b$FjanF`_n7%Fgaq7$Fj`o$!+$Q'y$ f$FjanF`_n7%F[bq7$Fd`o$!+r28KOFjanF`_n7%F_bq7$F^`o$!+V!=rm$FjanF`_n7%F cbq7$Fh_o$!+#3-()p$FjanF`_n7%Fgbq7$Fb_o$!+[b&os$FjanF`_n7%F[cq7$F\\_o$ !+P!f:v$FjanF`_n7%F_cq7$Ff^o$!+kIzsPFjanF`_n7%Fccq7$F`^o$!+m?`!z$FjanF `_n7%Fgcq7$Fj]o$!+:(RZ!QFjanF`_n7%F[dq7$Fd]o$!+M\\O:QFjanF`_n7%F_dq7$F ^]o$!+:\"QB#QFjanF`_n7%Fcdq7$Fh\\o$!+;rcDQFjanF`_n7%Fgdq7$Fb\\o$!+-E$ \\#QFjanF`_n7%F[eq7$F\\\\o$!+5BG?QFjanF`_n7%F_eq7$Ff[o$!+1PU6QFjanF`_n 7%Fceq7$F`[o$!+wV6)z$FjanF`_n7%Fgeq7$Fjjn$!+/$\\+y$FjanF`_n7%F[fq7$Fdj n$!+BR%ov$FjanF`_n7%F_fq7$F^jn$!+.;,GPFjanF`_n7%Fcfq7$Fhin$!+GU$Hp$Fja nF`_n7%Fgfq7$Fbin$!+!QD3l$FjanF`_n7%F[gq7$F\\in$!+6\"*o+OFjanF`_n7%F_g q7$Ffhn$!+#FjanF`_n7%F_j q7$F]dn$!+d4$z.#FjanF`_n7%Fcjq7$Fgcn$!+.1s#)=FjanF`_n7%Fgjq7$F`cn$!+\\ $Hns\"FjanF`_n7%F[[r7$Fjbn$!+u$Q-d\"FjanF`_n7%F_[r7$Fdbn$!+c^X89FjanF` _n7%Fc[r7$F]bn$!+ki^c7FjanF`_n7%Fg[r7$Ffan$!+U:]*4\"FjanF`_n7%F[\\r7$F `an$!+OD_C%*F__nF`_n7%F_\\r7$Fi`n$!+QG(Q&yF__nF`_n7%Fc\\r7$Fb`n$!+#>YJ G'F__nF`_n7%Fg\\r7$$!+?&e1'pF]`n$!+k(yBr%F__nF`_n7%F[]r7$Fe_n$!+)>\"fT JF__nF`_n7%Fa]r7$$!+(p]c/\"F0$!+zizq:F__nF`_n7%Fe]rF'F`_n-F^^n6&F`^nFd ^n$\"#NFb_n$\"\"*Fc^n-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3%*******G& =^X\"FS$\"3u*****R:u]M%FS7$$\"3'******\\Q^qW\"FS$\"3g*****R8*[aVFS7$$ \"3%******z\"oDQ9FS$\"3Q+++zF@jVFS7$$\"3$********ex(G9FS$\"39+++,K?rVF S7$$\"3'******>'ol=9FS$\"39+++(z=%yVFS7$$\"3.+++1&QzS\"FS$\"3;+++Zz\"[ Q%FS7$$\"33+++cum'R\"FS$\"3?+++A(e.R%FS7$$\"3#*******)4!*[Q\"FS$\"3A++ +\"f)*\\R%FS7$$\"35+++GZls8FS$\"3g*****p9%p)R%FS7$$\"3-+++q?,g8FS$\"3N +++B3S,WFS7$$\"3'******H!f,Z8FS$\"3K+++GE2.WFS7$$\"3*******pzBPL\"FS$ \"3A+++x=m.WFS7$$\"3,+++&*z>?8FS$\"3P+++[*=JS%FS7$$\"35+++$\\1lI\"FS$ \"3#)*****z1#R,WFS7$$\"3/+++bVs#H\"FS$\"33+++`rU)R%FS7$$\"3/+++E`$*y7F S$\"3q*****RvnTR%FS7$$\"3#******H$QBl7FS$\"3')*****p:b&)Q%FS7$$\"3)*** ***4YFM:eA\"FS$\"3')*****>*z)HO%FS7$$\"3*)*****4)=s87FS$\"3%)*****fx`8N% FS7$$\"3#*******e#eC?\"FS$\"3A+++`E2QVFS7$$\"3-+++-'fA>\"FS$\"3\")**** *Rp1JK%FS7$$\"3)******>F/M=\"FS$\"36+++B=W1VFS7$$\"33+++p)>i<\"FS$\"3m *****pP.\")G%FS7$$\"31+++ng3r6FS$\"3%)*****4uy\"oUFS7$$\"3%*******H(H% o6FS$\"37+++*R[oC%FS7$$\"3++++H\\qo6FS$\"37+++nVUCUFS7$$\"3/+++\"H[B< \"FS$\"3\"******4H*Q,UFS7$$\"3'*******\\Cqz6FS$\"3o*****ze>%yTFS7$$\"3 )******4H84>\"FS$\"3O+++!)fOcTFS7$$\"3-+++@<$e?\"FS$\"3%******z=nh8%FS 7$$\"3(*******zB(RA\"FS$\"3)******Ht,(=TFS7$$\"3\"*******R#pXC\"FS$\"3 N+++v\"HY5%FS7$$\"3++++$G9nE\"FS$\"3G+++^CH%4%FS7$$\"37+++W^^*G\"FS$\" 3e*****zb+++T`G+TFS7$$\"30++ +39)*39FS$\"3c******fo84TFS7$$\"3)********H5PU\"FS$\"3m*****f%)*H>TFS7 $$\"35+++oA!oV\"FS$\"3()*****4r808%FS7$$\"3%******4w&G[9FS$\"3(******z $\\bUTFS7$$\"3-+++%*G?e9FS$\"3k*****4vJ_:%FS7$$\"3++++UPgm9FS$\"3)**** **p2z$oTFS7$$\"3/+++#fUNZ\"FS$\"3w******)3a==%FS7$$\"3/+++Ib2z9FS$\"3/ +++3C`&>%FS7$$\"35+++1*eK[\"FS$\"3*)*****z#\\I4UFS7$$\"3/++++%[h[\"FS$ \"3f*****>>vIA%FS7$$\"3!******pZ)z([\"FS$\"3&)*****\\AdnB%FS7$$\"3#*** ***>6i#)[\"FS$\"3g*****put-D%FS7$$\"3)*******y1f([\"FS$\"3q*****RqaNE% FS7$$\"3/+++fR$e[\"FS$\"3i*****p.OlF%FS7$$\"3)******z@SI[\"FS$\"3/+++r 'e\"*G%FS7$$\"3'******RMc#z9FS$\"3N+++\\wO,VFS7$$\"3++++-!GXZ\"FS$\"3B +++4968VFS7$$\"37+++B)**)o9FS$\"3N+++g6MCVFS7$$\"3!******\\g:CY\"FS$\" 3N+++U.,NVFSFh^rF]^n-Fg^n6jn7%7$$\"+H&=^X\"Fjan$\"+aT2XVFjan7$$\"+&Q^q W\"Fjan$\"+M\"*[aVFjan7$$\"$L\"Fb_n$\"$B%Fb_n7%F\\bs7$$\"+=oDQ9Fjan$\" +zF@jVFjanFabs7%Fgbs7$$\"+!fx(G9Fjan$\"+,K?rVFjanFabs7%F]cs7$$\"+iol=9 Fjan$\"+(z=%yVFjanFabs7%Fccs7$$\"+1&QzS\"Fjan$\"+Zz\"[Q%FjanFabs7%Fics 7$$\"+cum'R\"Fjan$\"+A(e.R%FjanFabs7%F_ds7$$\"+*4!*[Q\"Fjan$\"+\"f)*\\ R%FjanFabs7%Feds7$$\"+GZls8Fjan$\"+ZTp)R%FjanFabs7%F[es7$$\"+q?,g8Fjan $\"+B3S,WFjanFabs7%Faes7$$\"+.f,Z8Fjan$\"+GE2.WFjanFabs7%Fges7$$\"+(zB PL\"Fjan$\"+x=m.WFjanFabs7%F]fs7$$\"+&*z>?8Fjan$\"+[*=JS%FjanFabs7%Fcf s7$$\"+$\\1lI\"Fjan$\"+o?R,WFjanFabs7%Fifs7$$\"+bVs#H\"Fjan$\"+`rU)R%F janFabs7%F_gs7$$\"+E`$*y7Fjan$\"+ax;%R%FjanFabs7%Fegs7$$\"+LQBl7Fjan$ \"+d^b)Q%FjanFabs7%F[hs7$$\"+hus^7Fjan$\"+](G:Q%FjanFabs7%Fahs7$$\"+A+ aQ7Fjan$\"+;o-tVFjanFabs7%Fghs7$$\"+U`\"eA\"Fjan$\"+#*z)HO%FjanFabs7%F ]is7$$\"+\")=s87Fjan$\"+wPN^VFjanFabs7%Fcis7$$\"+f#eC?\"Fjan$\"+`E2QVF janFabs7%Fiis7$$\"+-'fA>\"Fjan$\"+%p1JK%FjanFabs7%F_js7$$\"+sUS$=\"Fja n$\"+B=W1VFjanFabs7%Fejs7$$\"+p)>i<\"Fjan$\"+xL5)G%FjanFabs7%F[[t7$$\" +ng3r6Fjan$\"+T(y\"oUFjanFabs7%Fa[t7$$\"+I(H%o6Fjan$\"+*R[oC%FjanFabs7 %Fg[t7$$\"+H\\qo6Fjan$\"+nVUCUFjanFabs7%F]\\t7$$\"+\"H[B<\"Fjan$\"+\"H *Q,UFjanFabs7%Fc\\t7$$\"+]Cqz6Fjan$\"+)e>%yTFjanFabs7%Fi\\t7$$\"+\"H84 >\"Fjan$\"+!)fOcTFjanFabs7%F_]t7$$\"+@<$e?\"Fjan$\"+)=nh8%FjanFabs7%Fe ]t7$$\"+!QsRA\"Fjan$\"+LTFjanFabs7%Fgat7$$\"+oA!oV\"Fjan$\"+6P^ITFjanFabs7%F]bt7$$ \"+hdG[9Fjan$\"+Q\\bUTFjanFabs7%Fcbt7$$\"+%*G?e9Fjan$\"+^%FjanFabs7%F[dt7$$\"+1*eK[\"Fja n$\"+G\\I4UFjanFabs7%Fadt7$$\"++%[h[\"Fjan$\"+#>vIA%FjanFabs7%Fgdt7$$ \"+x%)z([\"Fjan$\"+DsvOUFjanFabs7%F]et7$$\"+7@E)[\"Fjan$\"+ZPF]UFjanFa bs7%Fcet7$$\"+z1f([\"Fjan$\"+/ZbjUFjanFabs7%Fiet7$$\"+fR$e[\"Fjan$\"+P g`wUFjanFabs7%F_ft7$$\"+=-/$[\"Fjan$\"+r'e\"*G%FjanFabs7%Feft7$$\"+WjD z9Fjan$\"+\\wO,VFjanFabs7%F[gt7$$\"+-!GXZ\"Fjan$\"+4968VFjanFabs7%Fagt 7$$\"+B)**)o9Fjan$\"+g6MCVFjanFabs7%Fggt7$$\"+0cTi9Fjan$\"+U.,NVFjanFa bs7%F]htFgasFabsF[^rFa^r-F$6$7in7$Fi^r$!3u*****R:u]M%FS7$F`as$!3N+++U. ,NVFS7$F[as$!3N+++g6MCVFS7$Ff`s$!3B+++4968VFS7$Fa`s$!3N+++\\wO,VFS7$F \\`s$!3/+++r'e\"*G%FS7$Fg_s$!3i*****p.OlF%FS7$Fb_s$!3q*****RqaNE%FS7$F ]_s$!3g*****put-D%FS7$Fh^s$!3&)*****\\AdnB%FS7$Fc^s$!3f*****>>vIA%FS7$ F^^s$!3*)*****z#\\I4UFS7$Fi]s$!3/+++3C`&>%FS7$Fd]s$!3w******)3a==%FS7$ F_]s$!3)******p2z$oTFS7$Fj\\s$!3k*****4vJ_:%FS7$Fe\\s$!3(******z$\\bUT FS7$F`\\s$!3()*****4r808%FS7$F[\\s$!3m*****f%)*H>TFS7$Ff[s$!3c******fo 84TFS7$Fa[s$!3>+++T`G+TFS7$F\\[s$!3a*****\\yZI4%FS7$Fgjr$!3/+++A8x(3%F S7$Fbjr$!3!******4EX[3%FS7$F]jr$!3/+++\\#)o%3%FS7$Fhir$!3e*****zb%yTFS7$Fe gr$!3\"******4H*Q,UFS7$F`gr$!37+++nVUCUFS7$F[gr$!37+++*R[oC%FS7$Fffr$! 3%)*****4uy\"oUFS7$Fafr$!3m*****pP.\")G%FS7$F\\fr$!36+++B=W1VFS7$Fger$ !3\")*****Rp1JK%FS7$Fber$!3A+++`E2QVFS7$F]er$!3%)*****fx`8N%FS7$Fhdr$! 3')*****>*z)HO%FS7$Fcdr$!3K+++;o-tVFS7$F^dr$!3?+++](G:Q%FS7$Ficr$!3')* ****p:b&)Q%FS7$Fdcr$!3q*****RvnTR%FS7$F_cr$!33+++`rU)R%FS7$Fjbr$!3#)** ***z1#R,WFS7$Febr$!3P+++[*=JS%FS7$F`br$!3A+++x=m.WFS7$F[br$!3K+++GE2.W FS7$Ffar$!3N+++B3S,WFS7$Faar$!3g*****p9%p)R%FS7$F\\ar$!3A+++\"f)*\\R%F S7$Fg`r$!3?+++A(e.R%FS7$Fb`r$!3;+++Zz\"[Q%FS7$F]`r$!39+++(z=%yVFS7$Fh_ r$!39+++,K?rVFS7$Fc_r$!3Q+++zF@jVFS7$F^_r$!3g*****R8*[aVFSFfhtF]^n-Fg^ n6jn7%7$Fhas$!+aT2XVFjan7$F^ht$!+U.,NVFjan7$Fbbs$!$B%Fb_n7%F`du7$Fhgt$ !+g6MCVFjanFcdu7%Fgdu7$Fbgt$!+4968VFjanFcdu7%F[eu7$F\\gt$!+\\wO,VFjanF cdu7%F_eu7$Ffft$!+r'e\"*G%FjanFcdu7%Fceu7$F`ft$!+Pg`wUFjanFcdu7%Fgeu7$ Fjet$!+/ZbjUFjanFcdu7%F[fu7$Fdet$!+ZPF]UFjanFcdu7%F_fu7$F^et$!+DsvOUFj anFcdu7%Fcfu7$Fhdt$!+#>vIA%FjanFcdu7%Fgfu7$Fbdt$!+G\\I4UFjanFcdu7%F[gu 7$F\\dt$!+3C`&>%FjanFcdu7%F_gu7$Ffct$!+*3a==%FjanFcdu7%Fcgu7$F`ct$!+x! z$oTFjanFcdu7%Fggu7$Fjbt$!+^TFjanFcdu7%Fghu7$Fbat$! +go84TFjanFcdu7%F[iu7$F\\at$!+T`G+TFjanFcdu7%F_iu7$Ff`t$!+&yZI4%FjanFc du7%Fciu7$F``t$!+A8x(3%FjanFcdu7%Fgiu7$Fj_t$!+h_%[3%FjanFcdu7%F[ju7$Fd _t$!+\\#)o%3%FjanFcdu7%F_ju7$F^_t$!+evr(3%FjanFcdu7%Fcju7$Fh^t$!+^CH%4 %FjanFcdu7%Fgju7$Fb^t$!+v\"HY5%FjanFcdu7%F[[v7$F\\^t$!+L%yTFjanFcdu7%F[\\v7$Fd\\t$!+\"H*Q,UFjanFcdu7%F_\\v7$F^\\t$!+nVUCU FjanFcdu7%Fc\\v7$Fh[t$!+*R[oC%FjanFcdu7%Fg\\v7$Fb[t$!+T(y\"oUFjanFcdu7 %F[]v7$F\\[t$!+xL5)G%FjanFcdu7%F_]v7$Ffjs$!+B=W1VFjanFcdu7%Fc]v7$F`js$ !+%p1JK%FjanFcdu7%Fg]v7$Fjis$!+`E2QVFjanFcdu7%F[^v7$Fdis$!+wPN^VFjanFc du7%F_^v7$F^is$!+#*z)HO%FjanFcdu7%Fc^v7$Fhhs$!+;o-tVFjanFcdu7%Fg^v7$Fb hs$!+](G:Q%FjanFcdu7%F[_v7$F\\hs$!+d^b)Q%FjanFcdu7%F__v7$Ffgs$!+ax;%R% FjanFcdu7%Fc_v7$F`gs$!+`rU)R%FjanFcdu7%Fg_v7$Fjfs$!+o?R,WFjanFcdu7%F[` v7$Fdfs$!+[*=JS%FjanFcdu7%F_`v7$F^fs$!+x=m.WFjanFcdu7%Fc`v7$Fhes$!+GE2 .WFjanFcdu7%Fg`v7$Fbes$!+B3S,WFjanFcdu7%F[av7$F\\es$!+ZTp)R%FjanFcdu7% F_av7$Ffds$!+\"f)*\\R%FjanFcdu7%Fcav7$F`ds$!+A(e.R%FjanFcdu7%Fgav7$Fjc s$!+Zz\"[Q%FjanFcdu7%F[bv7$Fdcs$!+(z=%yVFjanFcdu7%F_bv7$F^cs$!+,K?rVFj anFcdu7%Fcbv7$Fhbs$!+zF@jVFjanFcdu7%Fgbv7$F]bs$!+M\"*[aVFjanFcdu7%F[cv F]duFcduF[^rFa^r-F$6%7$7$$!3/++++++!z%FSF(7$$\"3%**************=#FSF(- %'COLOURG6&F`^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(Fccv7$F($\"3/+++++ +!z%FSFhcvF[dv-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-F\\ ev6$%*HELVETICAGF`^r-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-% %VIEWG6$;$!$z%Fb_n$\"$>#Fb_n;F^fv$\"$z%Fb_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stabil ity interval" }{TEXT -1 37 " for this scheme is (approximately) " } {XPPEDIT 18 0 "[-3.9983, 0];" "6#7$,$-%&FloatG6$\"&$)*R!\"%!\"\"\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the stability region for \+ the 8 stage, order 4 scheme with that of the order 5 scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),p _||(1..6)],view=[-4.79..2.19,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "65-%'CURVESG6$7ew7 $$\"\"!F)F(7$$\"3b+++!f<*>f!#G$\"3++++Fjzq:!#=7$$\"3p******4$3Q5&!#E$ \"3(******pi#fTJF07$$\"3=+++(*f-r!)!#D$\"3D+++/%)Q7ZF07$$\"3E+++P,gNg! #C$\"3S+++&yzJG'F07$$\"33+++%p%GlG!#B$\"3S+++g$[R&yF07$$\"3s*****>)evd )*FF$\"3w*****\\jIYU*F07$$\"33+++y<$3i#!#A$\"3'******\\D0&*4\"!#<7$$\" 3I+++6Y&R[&FQ$\"33+++h8[c7FT7$$\"3C+++XY2B')FQ$\"3!******4V6LT\"FT7$$ \"3]*****>Ow'yuFQ$\"3++++iU&)p:FT7$$!3-+++'3\"pH5!#@$\"3!******\\/()es \"FT7$$!3^+++odj+rFao$\"31+++jY4\")=FT7$$!3')*****\\^$>CA!#?$\"3<+++[t 2N?FT7$$!32+++\\v\\4aF\\p$\"3=+++upP(=#FT7$$!3++++O#)>O6!#>$\"36+++QP] PBFT7$$!3'******pt83:#Fgp$\"35+++yV#\\[#FT7$$!3!)*****4:(*Qv$Fgp$\"3/+ ++4w'*GEFT7$$!3<+++3,`ChFgp$\"3!******p0J'oFFT7$$!3Z*****>$4#*=%*Fgp$ \"34+++RsN-HFT7$$!3#******\\$)Q=P\"F0$\"3A+++nS)z-$FT7$$!3)******4&R!y *=F0$\"3\")*****f&48VJFT7$$!3t******[l:-DF0$\"37+++/I(fC$FT7$$!3;+++h( G-;$F0$\"36+++-)[eL$FT7$$!3'******pW;&[QF0$\"3!******pAQKT$FT7$$!35+++ W,T\\XF0$\"3)******z^$HzMFT7$$!3S+++P]p^_F0$\"3<+++QZONNFT7$$!3W+++C6. \\fF0$\"3;+++:@s#e$FT7$$!3a+++.s$FT7$$!3*******p $3=S**F0$\"3!******H[4@t$FT7$$!3/+++rqOd5FT$\"3#)*****zh<&RPFT7$$!3+++ +mQ$*>6FT$\"3@+++\\+xUPFT7$$!3#******fjo<=\"FT$\"38+++2k/UPFT7$$!36+++ V4#HC\"FT$\"34+++l#)[PPFT7$$!3)******fGTMI\"FT$\"3/+++2e?HPFT7$$!31+++ hKQj8FT$\"3&)*****48$GMOFT7$$!3#*******[f*ol\"FT$\"3&*******z'=Zg$ FT7$$!3%*******GS2:FT$\"3#******pq9*>$FT7$$!3/+++\")*\\YF#FT$\"3> +++x.zVJFT7$$!31+++\"y4OM#FT$\"3))*****RU-#)3$FT7$$!3z*****\\0RLT#FT$ \"3.+++:(>G.$FT7$$!33+++8@H$[#FT$\"3'******f76z(HFT7$$!3=+++eY\"Hb#FT$ \"34+++v GFT$\"3@+++$)yl4FFT7$$!31+++Tmj\")GFT$\"3))******ey,cEFT7$$!3++++g(>=% HFT$\"3*******za$)=g#FT7$$!3y******)R-***HFT$\"3++++=-2ZDFT7$$!31+++;N 'e0$FT$\"3))*****p:<9\\#FT7$$!3\")*****p>0(4JFT$\"32+++MoyMCFT7$$!3#** *****RjWhJFT$\"3=+++?H1xBFT7$$!3,+++]$=6@$FT$\"34+++lz9=BFT7$$!3++++rB weKFT$\"3)******\\?hzD#FT7$$!3#)*****\\hFWI$FT$\"3'******ptOk>#FT7$$!3 ?+++j/<[LFT$\"39+++?A_L@FT7$$!3=+++\\S0!R$FT$\"3;+++U\"y\"p?FT7$$!38++ +x![,V$FT$\"3z*****4qxL+#FT7$$!3#******pjG&oMFT$\"3)******RG2h$>FT7$$! 36+++'zx_]$FT$\"33+++6vOn=FT7$$!3*)******=F[SNFT$\"3-+++NZ<(z\"FT7$$!3 /+++!4MUd$FT$\"32+++VGcD\"FT7$$!3')******zM?,QFT$\"3'******fm$=76FT7$$!32+++\"38Z#QFT$\"34 +++K!RC.\"FT7$$!3A+++*yVq%QFT$\"3z*****p8!)\\_*F07$$!3')*****\\)R5oQFT $\"3b*****\\l!yC()F07$$!3=+++/+z()QFT$\"3Z*****RXVZ#zF07$$!3*)*****zb# *f!RFT$\"3)******HE'eDrF07$$!38+++4EgARFT$\"3s*****Hl+yK'F07$$!3-+++/i ^PRFT$\"34+++mKnJbF07$$!3++++u!Q1&RFT$\"3F+++1\\JPZF07$$!3'******H/%)= 'RFT$\"3w*****4X#pWRF07$$!3))******>C=rRFT$\"37+++j)fO:$F07$$!37+++UYZ yRFT$\"3!******R/')RO#F07$$!3++++Gar$)RFT$\"3'*******e&y`d\"F07$$!3%** ****\\mso)RFT$\"3m*****p%>.vyFgp7$$!3/+++*>Fz)RFTF(7$Fjjl$!3m*****p%>. vyFgp7$Fejl$!3'*******e&y`d\"F07$F`jl$!3!******R/')RO#F07$F[jl$!37+++j )fO:$F07$Ffil$!3w*****4X#pWRF07$Fail$!3F+++1\\JPZF07$F\\il$!34+++mKnJb F07$Fghl$!3s*****Hl+yK'F07$Fbhl$!3)******HE'eDrF07$F]hl$!3Z*****RXVZ#z F07$Fhgl$!3b*****\\l!yC()F07$Fcgl$!3z*****p8!)\\_*F07$F^gl$!34+++K!RC. \"FT7$Fifl$!3'******fm$=76FT7$Fdfl$!3'******zS&f\">\"FT7$F_fl$!3))**** *z)R_q7FT7$Fjel$!3'******pB7)[8FT7$Feel$!3/+++QOIE9FT7$F`el$!3******** )e\\G]\"FT7$F[el$!3/+++!\\:$y:FT7$Ffdl$!3/+++w]e_;FT7$Fadl$!32+++VGcD< FT7$F\\dl$!3-+++NZ<(z\"FT7$Fgcl$!33+++6vOn=FT7$Fbcl$!3)******RG2h$>FT7 $F]cl$!3z*****4qxL+#FT7$Fhbl$!3;+++U\"y\"p?FT7$Fcbl$!39+++?A_L@FT7$F^b l$!3'******ptOk>#FT7$Fial$!3)******\\?hzD#FT7$Fdal$!34+++lz9=BFT7$F_al $!3=+++?H1xBFT7$Fj`l$!32+++MoyMCFT7$Fe`l$!3))*****p:<9\\#FT7$F``l$!3++ ++=-2ZDFT7$F[`l$!3*******za$)=g#FT7$Ff_l$!3))******ey,cEFT7$Fa_l$!3@++ +$)yl4FFT7$F\\_l$!35+++P2,jFFT7$Fg^l$!3$******4\")*H;GFT7$Fb^l$!3)**** ***)[^(pGFT7$F]^l$!34+++vG.$FT7$F^]l$!3))*****RU-#)3$FT7$Fi\\l$!3>+++x.zVJFT7$Fd\\l$!3#)****** >q9*>$FT7$F_\\l$!3#)*****Ri:PD$FT7$Fj[l$!38+++G_*oI$FT7$Fe[l$!3A+++\\! =\"eLFT7$F`[l$!3#******pMOFT7$F]y$!3)******fb8-m$FT7$Fhx$!3')*****f:\\Fo$F T7$Fcx$!39+++rEy,PFT7$F^x$!3&)*****48$G.s$FT7$Ffu$!3<+++$=jQq$FT7 $Fau$!3#)*****fJ%Q#o$FT7$F\\u$!3;+++XYVbOFT7$Fgt$!3()*****RD]Ci$FT7$Fb t$!3;+++:@s#e$FT7$F]t$!3<+++QZONNFT7$Fhs$!3)******z^$HzMFT7$Fcs$!3!*** ***pAQKT$FT7$F^s$!36+++-)[eL$FT7$Fir$!37+++/I(fC$FT7$Fdr$!3\")*****f&4 8VJFT7$F_r$!3A+++nS)z-$FT7$Fjq$!34+++RsN-HFT7$Feq$!3!******p0J'oFFT7$F `q$!3/+++4w'*GEFT7$F[q$!35+++yV#\\[#FT7$Fep$!36+++QP]PBFT7$F`p$!3=+++u pP(=#FT7$Fjo$!3<+++[t2N?FT7$Feo$!31+++jY4\")=FT7$F_o$!3!******\\/()es \"FT7$$\"39+++kjnyuFQ$!3++++iU&)p:FT7$Fen$!3!******4V6LT\"FT7$FV$!33++ +h8[c7FT7$FO$!3'******\\D0&*4\"FT7$FJ$!3w*****\\jIYU*F07$FD$!3S+++g$[R &yF07$$\"3^*****f8+c.'F@$!3S+++&yzJG'F07$$\"3s******pd-r!)F:$!3D+++/%) Q7ZF07$$\"3G+++9$3Q5&F4$!3(******pi#fTJF07$$\"3i*****>a4(>fF-$!3++++Fj zq:F0F'-%&COLORG6&%$RGBG$\"\"#!\"\"Fh^n$\"\"&Fj^n-%)POLYGONSG6fw7%F'7$ $\"+!f<*>fF\\p$\"+Fjzq:!#57$$!\"#F)F(7%Fa_n7$$\"+5$3Q5&F0$\"+FEfTJFf_n Fg_n7%F[`n7$$\"+(*f-r!)FT$\"+/%)Q7ZFf_nFg_n7%Fa`n7$$\"+P,gNg!#;$\"+&yz JG'Ff_nFg_n7%Fg`n7$$\"+%p%GlG!#:$\"+g$[R&yFf_nFg_n7%F^an7$$\"+#)evd)*F aan$\"+N1jC%*Ff_nFg_n7%Fean7$$\"+y<$3i#!#9$\"+b_]*4\"!\"*Fg_n7%F[bn7$$ \"+6Y&R[&F^bn$\"+h8[c7FabnFg_n7%Fcbn7$$\"+XY2B')F^bn$\"+J9J89FabnFg_n7 %Fibn7$$\"+ijnyuF^bn$\"+iU&)p:FabnFg_n7%F_cn7$$!+'3\"pH5!#8$\"+Xq)es\" FabnFg_n7%Fecn7$$!+odj+rFhcn$\"+jY4\")=FabnFg_n7%F\\dn7$$!+:N>CA!#7$\" +[t2N?FabnFg_n7%Fbdn7$$!+\\v\\4aFedn$\"+upP(=#FabnFg_n7%Fidn7$$!+O#)>O 6!#6$\"+QP]PBFabnFg_n7%F_en7$$!+PP\"3:#Fben$\"+yV#\\[#FabnFg_n7%Ffen7$ $!+^r*Qv$Fben$\"+4w'*GEFabnFg_n7%F\\fn7$$!+3,`ChFben$\"+d5joFFabnFg_n7 %Fbfn7$$!+K4#*=%*Fben$\"+RsN-HFabnFg_n7%Fhfn7$$!+N)Q=P\"Ff_n$\"+nS)z-$ FabnFg_n7%F^gn7$$!+^R!y*=Ff_n$\"+c48VJFabnFg_n7%Fdgn7$$!+\\l:-DFf_n$\" +/I(fC$FabnFg_n7%Fjgn7$$!+h(G-;$Ff_n$\"+-)[eL$FabnFg_n7%F`hn7$$!+Zk^[Q Ff_n$\"+F#QKT$FabnFg_n7%Ffhn7$$!+W,T\\XFf_n$\"+=NHzMFabnFg_n7%F\\in7$$ !+P]p^_Ff_n$\"+QZONNFabnFg_n7%Fbin7$$!+C6.\\fFf_n$\"+:@s#e$FabnFg_n7%F hin7$$!+.s$FabnFg_n7%Ff[o7$$!+P3=S**Ff _n$\"+$[4@t$FabnFg_n7%F\\\\o7$$!+rqOd5Fabn$\"+=w^RPFabnFg_n7%Fb\\o7$$! +mQ$*>6Fabn$\"+\\+xUPFabnFg_n7%Fh\\o7$$!+O'o<=\"Fabn$\"+2k/UPFabnFg_n7 %F^]o7$$!+V4#HC\"Fabn$\"+l#)[PPFabnFg_n7%Fd]o7$$!+'GTMI\"Fabn$\"+2e?HP FabnFg_n7%Fj]o7$$!+hKQj8Fabn$\"+JJGMOFabnFg_n7%Fh_o 7$$!+\\f*ol\"Fabn$\"+!o=Zg$FabnFg_n7%F^`o7$$!+HS2:F abn$\"+xT!pS$FabnFg_n7%F\\bo7$$!+cmH8?Fabn$\"+\\!=\"eLFabnFg_n7%Fbbo7$ $!+q_?w?Fabn$\"+G_*oI$FabnFg_n7%Fhbo7$$!+p0tS@Fabn$\"+Ccr`KFabnFg_n7%F ^co7$$!+30#p?#Fabn$\"+?q9*>$FabnFg_n7%Fdco7$$!+\")*\\YF#Fabn$\"+x.zVJF abnFg_n7%Fjco7$$!+\"y4OM#Fabn$\"+CC?)3$FabnFg_n7%F`do7$$!+b!RLT#Fabn$ \"+:(>G.$FabnFg_n7%Ffdo7$$!+8@H$[#Fabn$\"+E6\"z(HFabnFg_n7%F\\eo7$$!+e Y\"Hb#Fabn$\"+vGFabn$\"+$)yl4FFabnFg_n7%Fjfo7$$!+Tmj\")GFabn$\"+fy, cEFabnFg_n7%F`go7$$!+g(>=%HFabn$\"+[N)=g#FabnFg_n7%Ffgo7$$!+*R-***HFab n$\"+=-2ZDFabnFg_n7%F\\ho7$$!+;N'e0$Fabn$\"+drT\"\\#FabnFg_n7%Fbho7$$! +(>0(4JFabn$\"+MoyMCFabnFg_n7%Fhho7$$!+SjWhJFabn$\"+?H1xBFabnFg_n7%F^i o7$$!+]$=6@$Fabn$\"+lz9=BFabnFg_n7%Fdio7$$!+rBweKFabn$\"+07'zD#FabnFg_ n7%Fjio7$$!+:wU/LFabn$\"+PnV'>#FabnFg_n7%F`jo7$$!+j/<[LFabn$\"+?A_L@Fa bnFg_n7%Ffjo7$$!+\\S0!R$Fabn$\"+U\"y\"p?FabnFg_n7%F\\[p7$$!+x![,V$Fabn $\"+,xP.?FabnFg_n7%Fb[p7$$!+P'G&oMFabn$\"+%G2h$>FabnFg_n7%Fh[p7$$!+'zx _]$Fabn$\"+6vOn=FabnFg_n7%F^\\p7$$!+>F[SNFabn$\"+NZ<(z\"FabnFg_n7%Fd\\ p7$$!+!4MUd$Fabn$\"+VGcD\"FabnFg_n7%Fd_p7$$!+![.7!QFabn$\"+mO=76FabnFg_n 7%Fj_p7$$!+\"38Z#QFabn$\"+K!RC.\"FabnFg_n7%F``p7$$!+*yVq%QFabn$\"+P,) \\_*Ff_nFg_n7%Ff`p7$$!+&)R5oQFabn$\"+b1yC()Ff_nFg_n7%F\\ap7$$!+/+z()QF abn$\"+aMuCzFf_nFg_n7%Fbap7$$!+eD*f!RFabn$\"+jieDrFf_nFg_n7%Fhap7$$!+4 EgARFabn$\"+`1!yK'Ff_nFg_n7%F^bp7$$!+/i^PRFabn$\"+mKnJbFf_nFg_n7%Fdbp7 $$!+u!Q1&RFabn$\"+1\\JPZFf_nFg_n7%Fjbp7$$!+VS)='RFabn$\"+^CpWRFf_nFg_n 7%F`cp7$$!+?C=rRFabn$\"+j)fO:$Ff_nFg_n7%Ffcp7$$!+UYZyRFabn$\"+Wg)RO#Ff _nFg_n7%F\\dp7$$!+Gar$)RFabn$\"+f&y`d\"Ff_nFg_n7%Fbdp7$$!+lE(o)RFabn$ \"+Z>.vyFbenFg_n7%Fhdp7$$!+*>Fz)RFabnF(Fg_n7%F^ep7$Fidp$!+Z>.vyFbenFg_ n7%Fbep7$Fcdp$!+f&y`d\"Ff_nFg_n7%Ffep7$F]dp$!+Wg)RO#Ff_nFg_n7%Fjep7$Fg cp$!+j)fO:$Ff_nFg_n7%F^fp7$Facp$!+^CpWRFf_nFg_n7%Fbfp7$F[cp$!+1\\JPZFf _nFg_n7%Fffp7$Febp$!+mKnJbFf_nFg_n7%Fjfp7$F_bp$!+`1!yK'Ff_nFg_n7%F^gp7 $Fiap$!+jieDrFf_nFg_n7%Fbgp7$Fcap$!+aMuCzFf_nFg_n7%Ffgp7$F]ap$!+b1yC() Ff_nFg_n7%Fjgp7$Fg`p$!+P,)\\_*Ff_nFg_n7%F^hp7$Fa`p$!+K!RC.\"FabnFg_n7% Fbhp7$F[`p$!+mO=76FabnFg_n7%Ffhp7$Fe_p$!+3af\">\"FabnFg_n7%Fjhp7$F__p$ !+))R_q7FabnFg_n7%F^ip7$Fi^p$!+PA\")[8FabnFg_n7%Fbip7$Fc^p$!+QOIE9Fabn Fg_n7%Ffip7$F]^p$!+*e\\G]\"FabnFg_n7%Fjip7$Fg]p$!+!\\:$y:FabnFg_n7%F^j p7$Fa]p$!+w]e_;FabnFg_n7%Fbjp7$F[]p$!+VGcDFabnFg_n 7%Fb[q7$Fc[p$!+,xP.?FabnFg_n7%Ff[q7$F][p$!+U\"y\"p?FabnFg_n7%Fj[q7$Fgj o$!+?A_L@FabnFg_n7%F^\\q7$Fajo$!+PnV'>#FabnFg_n7%Fb\\q7$F[jo$!+07'zD#F abnFg_n7%Ff\\q7$Feio$!+lz9=BFabnFg_n7%Fj\\q7$F_io$!+?H1xBFabnFg_n7%F^] q7$Fiho$!+MoyMCFabnFg_n7%Fb]q7$Fcho$!+drT\"\\#FabnFg_n7%Ff]q7$F]ho$!+= -2ZDFabnFg_n7%Fj]q7$Fggo$!+[N)=g#FabnFg_n7%F^^q7$Fago$!+fy,cEFabnFg_n7 %Fb^q7$F[go$!+$)yl4FFabnFg_n7%Ff^q7$Fefo$!+P2,jFFabnFg_n7%Fj^q7$F_fo$! +6)*H;GFabnFg_n7%F^_q7$Fieo$!+*[^(pGFabnFg_n7%Fb_q7$Fceo$!+vG.$FabnFg_n7%F^`q7$F ado$!+CC?)3$FabnFg_n7%Fb`q7$F[do$!+x.zVJFabnFg_n7%Ff`q7$Feco$!+?q9*>$F abnFg_n7%Fj`q7$F_co$!+Ccr`KFabnFg_n7%F^aq7$Fibo$!+G_*oI$FabnFg_n7%Fbaq 7$Fcbo$!+\\!=\"eLFabnFg_n7%Ffaq7$F]bo$!+xT!pS$FabnFg_n7%Fjaq7$Fgao$!+D X)GX$FabnFg_n7%F^bq7$Faao$!+%\\(z&\\$FabnFg_n7%Fbbq7$F[ao$!+c2ZNNFabnF g_n7%Ffbq7$Fe`o$!+w#)zrNFabnFg_n7%Fjbq7$F_`o$!+!o=Zg$FabnFg_n7%F^cq7$F i_o$!+&[(>MOFabnFg_n7%Fbcq7$Fc_o$!+cN@gOFabnFg_n7%Ffcq7$F]_o$!+c\"\\Fo $FabnFg_n7%Fjcq7$Fg^o$!+rEy,PFabnFg_n7%F^dq7$Fa^o$!+JJG.s$FabnFg_n7%F^fq7$Fa[o $!+$=jQq$FabnFg_n7%Fbfq7$F[[o$!+;VQ#o$FabnFg_n7%Fffq7$Fejn$!+XYVbOFabn Fg_n7%Fjfq7$F_jn$!+a-XAOFabnFg_n7%F^gq7$Fiin$!+:@s#e$FabnFg_n7%Fbgq7$F cin$!+QZONNFabnFg_n7%Ffgq7$F]in$!+=NHzMFabnFg_n7%Fjgq7$Fghn$!+F#QKT$Fa bnFg_n7%F^hq7$Fahn$!+-)[eL$FabnFg_n7%Fbhq7$F[hn$!+/I(fC$FabnFg_n7%Ffhq 7$Fegn$!+c48VJFabnFg_n7%Fjhq7$F_gn$!+nS)z-$FabnFg_n7%F^iq7$Fifn$!+RsN- HFabnFg_n7%Fbiq7$Fcfn$!+d5joFFabnFg_n7%Ffiq7$F]fn$!+4w'*GEFabnFg_n7%Fj iq7$Fgen$!+yV#\\[#FabnFg_n7%F^jq7$F`en$!+QP]PBFabnFg_n7%Fbjq7$Fjdn$!+u pP(=#FabnFg_n7%Ffjq7$Fcdn$!+[t2N?FabnFg_n7%Fjjq7$F]dn$!+jY4\")=FabnFg_ n7%F^[r7$Ffcn$!+Xq)es\"FabnFg_n7%Fb[r7$$\"+kjnyuF^bn$!+iU&)p:FabnFg_n7 %Ff[r7$Fjbn$!+J9J89FabnFg_n7%F\\\\r7$Fdbn$!+h8[c7FabnFg_n7%F`\\r7$F\\b n$!+b_]*4\"FabnFg_n7%Fd\\r7$Ffan$!+N1jC%*Ff_nFg_n7%Fh\\r7$F_an$!+g$[R& yFf_nFg_n7%F\\]r7$$\"+O,gNgFj`n$!+&yzJG'Ff_nFg_n7%F`]r7$$\"+qd-r!)FT$! +/%)Q7ZFf_nFg_n7%Ff]r7$$\"+9$3Q5&F0$!+FEfTJFf_nFg_n7%F\\^r7$$\"+U&4(>f F\\p$!+Fjzq:Ff_nFg_n7%Fb^rF'Fg_n-Fe^n6&Fg^n$\"\"%Fj^nFj^r$\"\"\"F)-%&S TYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"33+++.>]*[\"FT$\"3#******>&)\\QI%FT7 $$\"3!******ff%R\"[\"FT$\"3()*****HP'f7VFT7$$\"3'******Ru+EZ\"FT$\"3C+ ++&o\\1K%FT7$$\"3/+++&RjJY\"FT$\"3D+++(RozK%FT7$$\"3/+++yi7`9FT$\"3e** ***\\V6XL%FT7$$\"3++++@T`U9FT$\"3F+++kxBSVFT7$$\"3++++9IVJ9FT$\"3i**** **>h5XVFT7$$\"3#******f#3()>9FT$\"3M+++dZ2\\VFT7$$\"3$******Ro(*yS\"FT $\"3e*****p?,@N%FT7$$\"3-+++Glc&R\"FT$\"3=+++^?9aVFT7$$\"33+++xP$HQ\"F T$\"3o*****4r_^N%FT7$$\"3%******R6g+P\"FT$\"37+++os3bVFT7$$\"3!******f U6qN\"FT$\"3/+++A$)*QN%FT7$$\"33+++W*fQM\"FT$\"3y*****p\"p`^VFT7$$\"3, +++CcoI8FT$\"3-+++mD&zM%FT7$$\"3#******4$yd<8FT$\"3k******RM4VVFT7$$\" 3$******\\QPYI\"FT$\"3)******\\z1pL%FT7$$\"3)*******e!z>H\"FT$\"3e**** *zxR$HVFT7$$\"3++++-Ytz7FT$\"3J++++3M?VFT7$$\"3#******pMc!o7FT$\"3V+++ g<')4VFT7$$\"3)*******H97d7FT$\"31+++d:'yH%FT7$$\"3!******H]OrC\"FT$\" 3!******RZ6VG%FT7$$\"32+++7EMQ7FT$\"3e*****HF.#pUFT7$$\"3*******fR>5B \"FT$\"3e*****z/hDD%FT7$$\"3,+++wp[D7FT$\"3v*****p+eWB%FT7$$\"3******* >e-@A\"FT$\"3V+++%)*Q]@%FT7$$\"3.+++`sC@7FT$\"3k*****p=[X>%FT7$$\"3/++ +QlHB7FT$\"3/+++:%eL<%FT7$$\"3)*******4)p&G7FT$\"3)******\\N#*>:%FT7$$ \"3.+++(*)esB\"FT$\"3j*****4&*>68%FT7$$\"3'******>K]$\\7FT$\"3&******* QR^6TFT7$$\"3-+++\\:dk7FT$\"3o*****pBdR4%FT7$$\"3/+++e\\R#G\"FT$\"3G++ +mC7zSFT7$$\"3\"******4_5@I\"FT$\"3))*****\\@ou1%FT7$$\"3$*******z2%HK \"FT$\"3S+++yK>fSFT7$$\"3++++T::W8FT$\"3y*****RkdU0%FT7$$\"3/+++&*R7l8 FT$\"3u*****4JYC0%FT7$$\"3++++P:Q&Q\"FT$\"3=+++%oUM0%FT7$$\"3,+++BPe/9 FT$\"3i*****\\F*)o0%FT7$$\"3)******R-/DU\"FT$\"3#)******>%GC1%FT7$$\"3 #*******RO+R9FT$\"3w*****R5D(pSFT7$$\"3%******>u2SX\"FT$\"3o*****\\=x% ySFT7$$\"3#******fS'[n9FT$\"3#*******yzT)3%FT7$$\"3)*******4.Wz9FT$\"3 A+++eZJ*4%FT7$$\"3'******HW!*)*[\"FT$\"3w*****RUl46%FT7$$\"3%******H)3 ())\\\"FT$\"3R+++wX>BTFT7$$\"3$******H$QU1:FT$\"33+++T'\\e8%FT7$$\"3.+ ++Wif7:FT$\"37+++Wtz[TFT7$$\"3!******fcPu^\"FT$\"3#*******G1#>;%FT7$$ \"30+++g#)*4_\"FT$\"3++++@g6vTFT7$$\"3!******>&)GL_\"FT$\"3G+++<9H)=%F T7$$\"3\"******zLzW_\"FT$\"3:+++=UO,UFT7$$\"3$******4&))\\C:FT$\"3B+++ %yfU@%FT7$$\"3.+++$\\NM_\"FT$\"3j*****H85pA%FT7$$\"3'******4GO8_\"FT$ \"3m******[GDRUFT7$$\"3(******z8_WH%FTFe_rFd^n-F^_n 6jn7%7$$\"+.>]*[\"Fabn$\"+_)\\QI%Fabn7$$\"+'f%R\"[\"Fabn$\"+tjf7VFabn7 $$\"$P\"Fi_n$\"$>%Fi_n7%Fibs7$$\"+W2gs9Fabn$\"+&o\\1K%FabnF^cs7%Fdcs7$ $\"+&RjJY\"Fabn$\"+(RozK%FabnF^cs7%Fjcs7$$\"+yi7`9Fabn$\"+N9^MVFabnF^c s7%F`ds7$$\"+@T`U9Fabn$\"+kxBSVFabnF^cs7%Ffds7$$\"+9IVJ9Fabn$\"+?h5XVF abnF^cs7%F\\es7$$\"+E3()>9Fabn$\"+dZ2\\VFabnF^cs7%Fbes7$$\"+%o(*yS\"Fa bn$\"+275_VFabnF^cs7%Fhes7$$\"+Glc&R\"Fabn$\"+^?9aVFabnF^cs7%F^fs7$$\" +xP$HQ\"Fabn$\"+6F:bVFabnF^cs7%Fdfs7$$\"+9,1q8Fabn$\"+os3bVFabnF^cs7%F jfs7$$\"+E9,d8Fabn$\"+A$)*QN%FabnF^cs7%F`gs7$$\"+W*fQM\"Fabn$\"+H\"Fabn$\"+y(R$HVFabnF^cs7%F^is7$$\"+-Ytz7Fabn$\"++3M?VFabnF^cs7%Fd is7$$\"+Zj0o7Fabn$\"+g<')4VFabnF^cs7%Fjis7$$\"+I97d7Fabn$\"+d:'yH%Fabn F^cs7%F`js7$$\"+.l8Z7Fabn$\"+u9J%G%FabnF^cs7%Ffjs7$$\"+7EMQ7Fabn$\"+tK ?pUFabnF^cs7%F\\[t7$$\"+'R>5B\"Fabn$\"+[5c_UFabnF^cs7%Fb[t7$$\"+wp[D7F abn$\"+2!eWB%FabnF^cs7%Fh[t7$$\"+#e-@A\"Fabn$\"+%)*Q]@%FabnF^cs7%F^\\t 7$$\"+`sC@7Fabn$\"+(=[X>%FabnF^cs7%Fd\\t7$$\"+QlHB7Fabn$\"+:%eL<%FabnF ^cs7%Fj\\t7$$\"+5)p&G7Fabn$\"+bB*>:%FabnF^cs7%F`]t7$$\"+(*)esB\"Fabn$ \"+^*>68%FabnF^cs7%Ff]t7$$\"+A.N\\7Fabn$\"+RR^6TFabnF^cs7%F\\^t7$$\"+ \\:dk7Fabn$\"+Ps&R4%FabnF^cs7%Fb^t7$$\"+e\\R#G\"Fabn$\"+mC7zSFabnF^cs7 %Fh^t7$$\"+@06-8Fabn$\"+:#ou1%FabnF^cs7%F^_t7$$\"+!ySHK\"Fabn$\"+yK>fS FabnF^cs7%Fd_t7$$\"+T::W8Fabn$\"+WwDaSFabnF^cs7%Fj_t7$$\"+&*R7l8Fabn$ \"+6jW_SFabnF^cs7%F``t7$$\"+P:Q&Q\"Fabn$\"+%oUM0%FabnF^cs7%Ff`t7$$\"+B Pe/9Fabn$\"+v#*)o0%FabnF^cs7%F\\at7$$\"+CS]A9Fabn$\"+?%GC1%FabnF^cs7%F bat7$$\"+SO+R9Fabn$\"+/^spSFabnF^cs7%Fhat7$$\"+Ux+a9Fabn$\"+&=x%ySFabn F^cs7%F^bt7$$\"+1k[n9Fabn$\"+zzT)3%FabnF^cs7%Fdbt7$$\"+5.Wz9Fabn$\"+eZ J*4%FabnF^cs7%Fjbt7$$\"+V/*)*[\"Fabn$\"+Ca'46%FabnF^cs7%F`ct7$$\"+$)3( ))\\\"Fabn$\"+wX>BTFabnF^cs7%Ffct7$$\"+LQU1:Fabn$\"+T'\\e8%FabnF^cs7%F \\dt7$$\"+Wif7:Fabn$\"+Wtz[TFabnF^cs7%Fbdt7$$\"+mvV<:Fabn$\"+H1#>;%Fab nF^cs7%Fhdt7$$\"+g#)*4_\"Fabn$\"+@g6vTFabnF^cs7%F^et7$$\"+_)GL_\"Fabn$ \"+<9H)=%FabnF^cs7%Fdet7$$\"+Q$zW_\"Fabn$\"+=UO,UFabnF^cs7%Fjet7$$\"+^ ))\\C:Fabn$\"+%yfU@%FabnF^cs7%F`ft7$$\"+$\\NM_\"Fabn$\"+L,\"pA%FabnF^c s7%Ffft7$$\"+\"GO8_\"Fabn$\"+\\GDRUFabnF^cs7%F\\gt7$$\"+yrC=:Fabn$\"+k ,B^UFabnF^cs7%Fbgt7$$\"+yJ@9:Fabn$\"+!=)yiUFabnF^cs7%Fhgt7$$\"+J%y#4:F abn$\"+.i(QF%FabnF^cs7%F^ht7$$\"+%R'[.:Fabn$\"+mhW%G%FabnF^cs7%Fdht7$$ \"+4+)o\\\"Fabn$\"+K@X%H%FabnF^cs7%FjhtFdbsF^csFh^rF^_r-F$6$7in7$Ff_r$ !3#******>&)\\QI%FT7$F]bs$!3)******>8_WH%FT7$Fhas$!3S+++mhW%G%FT7$Fcas $!3Q+++.i(QF%FT7$F^as$!3E+++!=)yiUFT7$Fi`s$!3))*****R;I7D%FT7$Fd`s$!3m ******[GDRUFT7$F_`s$!3j*****H85pA%FT7$Fj_s$!3B+++%yfU@%FT7$Fe_s$!3:+++ =UO,UFT7$F`_s$!3G+++<9H)=%FT7$F[_s$!3++++@g6vTFT7$Ff^s$!3#*******G1#>; %FT7$Fa^s$!37+++Wtz[TFT7$F\\^s$!33+++T'\\e8%FT7$Fg]s$!3R+++wX>BTFT7$Fb ]s$!3w*****RUl46%FT7$F]]s$!3A+++eZJ*4%FT7$Fh\\s$!3#*******yzT)3%FT7$Fc \\s$!3o*****\\=x%ySFT7$F^\\s$!3w*****R5D(pSFT7$Fi[s$!3#)******>%GC1%FT 7$Fd[s$!3i*****\\F*)o0%FT7$F_[s$!3=+++%oUM0%FT7$Fjjr$!3u*****4JYC0%FT7 $Fejr$!3y*****RkdU0%FT7$F`jr$!3S+++yK>fSFT7$F[jr$!3))*****\\@ou1%FT7$F fir$!3G+++mC7zSFT7$Fair$!3o*****pBdR4%FT7$F\\ir$!3&*******QR^6TFT7$Fgh r$!3j*****4&*>68%FT7$Fbhr$!3)******\\N#*>:%FT7$F]hr$!3/+++:%eL<%FT7$Fh gr$!3k*****p=[X>%FT7$Fcgr$!3V+++%)*Q]@%FT7$F^gr$!3v*****p+eWB%FT7$Fifr $!3e*****z/hDD%FT7$Fdfr$!3e*****HF.#pUFT7$F_fr$!3!******RZ6VG%FT7$Fjer $!31+++d:'yH%FT7$Feer$!3V+++g<')4VFT7$F`er$!3J++++3M?VFT7$F[er$!3e**** *zxR$HVFT7$Ffdr$!3)******\\z1pL%FT7$Fadr$!3k******RM4VVFT7$F\\dr$!3-++ +mD&zM%FT7$Fgcr$!3y*****p\"p`^VFT7$Fbcr$!3/+++A$)*QN%FT7$F]cr$!37+++os 3bVFT7$Fhbr$!3o*****4r_^N%FT7$Fcbr$!3=+++^?9aVFT7$F^br$!3e*****p?,@N%F T7$Fiar$!3M+++dZ2\\VFT7$Fdar$!3i******>h5XVFT7$F_ar$!3F+++kxBSVFT7$Fj` r$!3e*****\\V6XL%FT7$Fe`r$!3D+++(RozK%FT7$F``r$!3C+++&o\\1K%FT7$F[`r$! 3()*****HP'f7VFTFcitFd^n-F^_n6jn7%7$Febs$!+_)\\QI%Fabn7$F[it$!+K@X%H%F abn7$F_cs$!$>%Fi_n7%F]eu7$Feht$!+mhW%G%FabnF`eu7%Fdeu7$F_ht$!+.i(QF%Fa bnF`eu7%Fheu7$Figt$!+!=)yiUFabnF`eu7%F\\fu7$Fcgt$!+k,B^UFabnF`eu7%F`fu 7$F]gt$!+\\GDRUFabnF`eu7%Fdfu7$Fgft$!+L,\"pA%FabnF`eu7%Fhfu7$Faft$!+%y fU@%FabnF`eu7%F\\gu7$F[ft$!+=UO,UFabnF`eu7%F`gu7$Feet$!+<9H)=%FabnF`eu 7%Fdgu7$F_et$!+@g6vTFabnF`eu7%Fhgu7$Fidt$!+H1#>;%FabnF`eu7%F\\hu7$Fcdt $!+Wtz[TFabnF`eu7%F`hu7$F]dt$!+T'\\e8%FabnF`eu7%Fdhu7$Fgct$!+wX>BTFabn F`eu7%Fhhu7$Fact$!+Ca'46%FabnF`eu7%F\\iu7$F[ct$!+eZJ*4%FabnF`eu7%F`iu7 $Febt$!+zzT)3%FabnF`eu7%Fdiu7$F_bt$!+&=x%ySFabnF`eu7%Fhiu7$Fiat$!+/^sp SFabnF`eu7%F\\ju7$Fcat$!+?%GC1%FabnF`eu7%F`ju7$F]at$!+v#*)o0%FabnF`eu7 %Fdju7$Fg`t$!+%oUM0%FabnF`eu7%Fhju7$Fa`t$!+6jW_SFabnF`eu7%F\\[v7$F[`t$ !+WwDaSFabnF`eu7%F`[v7$Fe_t$!+yK>fSFabnF`eu7%Fd[v7$F__t$!+:#ou1%FabnF` eu7%Fh[v7$Fi^t$!+mC7zSFabnF`eu7%F\\\\v7$Fc^t$!+Ps&R4%FabnF`eu7%F`\\v7$ F]^t$!+RR^6TFabnF`eu7%Fd\\v7$Fg]t$!+^*>68%FabnF`eu7%Fh\\v7$Fa]t$!+bB*> :%FabnF`eu7%F\\]v7$F[]t$!+:%eL<%FabnF`eu7%F`]v7$Fe\\t$!+(=[X>%FabnF`eu 7%Fd]v7$F_\\t$!+%)*Q]@%FabnF`eu7%Fh]v7$Fi[t$!+2!eWB%FabnF`eu7%F\\^v7$F c[t$!+[5c_UFabnF`eu7%F`^v7$F][t$!+tK?pUFabnF`eu7%Fd^v7$Fgjs$!+u9J%G%Fa bnF`eu7%Fh^v7$Fajs$!+d:'yH%FabnF`eu7%F\\_v7$F[js$!+g<')4VFabnF`eu7%F`_ v7$Feis$!++3M?VFabnF`eu7%Fd_v7$F_is$!+y(R$HVFabnF`eu7%Fh_v7$Fihs$!+&z1 pL%FabnF`eu7%F\\`v7$Fchs$!+SM4VVFabnF`eu7%F``v7$F]hs$!+mD&zM%FabnF`eu7 %Fd`v7$Fggs$!+1;kY'F:$\"3(******z>\"fTJF07$$!39+++U%e 1'pF@$\"3y*****RwyBr%F07$$!3#******HXS&>OFF$\"3y*****>>YJG'F07$$!3-+++ k6(\\D\"FQ$\"3-+++QG(Q&yF07$$!3-+++7\\+jLFQ$\"3c*****f`AXU*F07$$!3O*** **HzMne(FQ$\"3/+++U:]*4\"FT7$$!3++++fh'4`\"Fao$\"31+++ki^c7FT7$$!3$*** ***4eSf!HFao$\"3%******f:bMT\"FT7$$!3*******>Ff*4aFao$\"35+++u$Q-d\"FT 7$$!3********Gb([,\"F\\p$\"3(*******[$Hns\"FT7$$!3-+++56#*H>F\\p$\"35+ ++.1s#)=FT7$$!3!******f+()yn$F\\p$\"3$******p&4$z.#FT7$$!3%******z::1! pF\\p$\"3#*******3x,#>#FT7$$!3-+++$*)zeD\"Fgp$\"3!)*****fQ%fWBFT7$$!31 +++%H1u>#Fgp$\"3;+++-UA&\\#FT7$$!3I+++**HH#o$Fgp$\"3$)*****HdTLk#FT7$$ !3.+++sPC/fFgp$\"37+++N$e!)y#FT7$$!3]*****fJ`d0*Fgp$\"32+++*3sy#HFT7$$ !3$******fK/sK\"F0$\"3++++)3f/1$FT7$$!3%******HI8c&=F0$\"3A+++rG$H=$FT 7$$!3++++Xg=uCF0$\"31+++U&eFH$FT7$$!3)******4]0]:$F0$\"3#******4[H()Q$ FT7$$!3=+++wI!)pQF0$\"39+++7G:rMFT7$$!3u*****RH&G(f%F0$\"35+++wtq%*F0$\"3-+++wV6)z$FT7$$!3%******RL`B,\"FT$\"3! )*****fqB9\"QFT7$$!3-+++C7qw5FT$\"32+++5BG?QFT7$$!3,+++RH=S6FT$\"32+++ -E$\\#QFT7$$!35+++Q*eG?\"FT$\"3@+++;rcDQFT7$$!3/+++.dyk7FT$\"3)****** \\6QB#QFT7$$!3)******Rd+ ++:(RZ!QFT7$$!3-+++4whY9FT$\"3#******f1K0z$FT7$$!3-+++%f1h]\"FT$\"3/++ +kIzsPFT7$$!3#******p\"*[^c\"FT$\"3#)*****p.f:v$FT7$$!3#******z(3$Qi\" FT$\"32+++[b&os$FT7$$!37+++4lD#o\"FT$\"3/+++#3-()p$FT7$$!3#*******R@bS FT$\"3!)*****\\3k@b$FT7$$! 3++++-)[f(>FT$\"3;+++(4%R2NFT7$$!3*******4Nvk.#FT$\"3!)*****R\"znfMFT7 $$!31+++1&p\")4#FT$\"3%)*****4?2$4MFT7$$!31+++l]Bh@FT$\"3A+++hUncLFT7$ $!3(******pR%yDAFT$\"38+++0XE-LFT7$$!3-+++RHz\"H#FT$\"3=+++\"*HiYKFT7$ $!3/+++S?2fBFT$\"3)******Ri&H!>$FT7$$!39+++\\aFFCFT$\"3>+++#**eP8$FT7$ $!3#)*******yTf\\#FT$\"37+++]*y`HGFT$ \"3=+++4Op*z#FT7$$!3%******\\n?=*GFT$\"3&)*****HzdTu#FT7$$!3;+++6v9_HF T$\"3++++T'p\")o#FT7$$!3$)******4TV5IFT$\"3))******3JaJEFT7$$!3++++$)R jmIFT$\"3#*******=>6uDFT7$$!3*)*****4zG27$FT$\"3y*****zmId^#FT7$$!3%)* *****>6ssJFT$\"3=+++RSFcCFT7$$!3#*******e$HEA$FT$\"37+++j^j&R#FT7$$!31 +++^U[qKFT$\"3.+++oPsLBFT7$$!3#******prEjJ$FT$\"3-+++*\\k/F#FT7$$!3!)* ****4g1-O$FT$\"3$******\\i&z0AFT7$$!3:+++Z?=-MFT$\"3@+++s#o'R@FT7$$!3& ******fA>BW$FT$\"3!******pBY?2#FT7$$!3%)*****p=#p![$FT$\"3*******4V1H+ #FT7$$!33+++fDQFT7$$!3;+++j)yCb$FT$\"3!******zj`+'=F T7$$!3-+++\\^2'e$FT$\"35+++BAP'y\"FT7$$!3(******Hro#=OFT$\"3&******\\U S7r\"FT7$$!3)******>qc\"\\OFT$\"34+++7]sM;FT7$$!33+++d:$)yOFT$\"3#**** **R5;pb\"FT7$$!3'******>Xvtq$FT$\"35+++3s#zZ\"FT7$$!3-++++U&[t$FT$\"3+ +++`M*yR\"FT7$$!31+++a8JhPFT$\"3#*******pw'pJ\"FT7$$!3-+++RMw'y$FT$\"3 ,+++YVJN7FT7$$!3(******\\@(>6QFT$\"3++++C>5`6FT7$$!3:+++2'pX$QFT$\"3#* *****fO%\\q5FT7$$!3'*******>/\"o&QFT$\"3#)******=6Vw)*F07$$!3)******4W Fy(QFT$\"3'******H,0o/*F07$$!39+++xI^(*QFT$\"3&)********R:<#)F07$$!35+ ++9:v:RFT$\"3'******4k7$)Q(F07$$!3$)*****H[DC$RFT$\"3!)*****zLk3c'F07$ $!35+++!)>UZRFT$\"3*)******H7;NdF07$$!3/+++GmjgRFT$\"3%)*****>ia8\"\\F 07$$!33+++vm(>(RFT$\"37+++%*)H%*3%F07$$!3++++2GO\")RFT$\"3%)*****RJT#p KF07$$!3y*****p/I())RFT$\"3#******p!Qa]CF07$$!3;+++a!GS*RFT$\"36+++B.- L;F07$$!3++++z5A(*RFT$\"3E+++^g/j\")Fgp7$$!3/+++&y(G)*RFTF(7$Fadx$!3E+ ++^g/j\")Fgp7$F\\dx$!36+++B.-L;F07$Fgcx$!3#******p!Qa]CF07$Fbcx$!3%)** ***RJT#pKF07$F]cx$!37+++%*)H%*3%F07$Fhbx$!3%)*****>ia8\"\\F07$Fcbx$!3* )******H7;NdF07$F^bx$!3!)*****zLk3c'F07$Fiax$!3'******4k7$)Q(F07$Fdax$ !3&)********R:<#)F07$F_ax$!3'******H,0o/*F07$Fj`x$!3#)******=6Vw)*F07$ Fe`x$!3#******fO%\\q5FT7$F``x$!3++++C>5`6FT7$F[`x$!3,+++YVJN7FT7$Ff_x$ !3#*******pw'pJ\"FT7$Fa_x$!3++++`M*yR\"FT7$F\\_x$!35+++3s#zZ\"FT7$Fg^x $!3#******R5;pb\"FT7$Fb^x$!34+++7]sM;FT7$F]^x$!3&******\\US7r\"FT7$Fh] x$!35+++BAP'y\"FT7$Fc]x$!3!******zj`+'=FT7$F^]x$!35+++S*RA$>FT7$Fi\\x$ !3*******4V1H+#FT7$Fd\\x$!3!******pBY?2#FT7$F_\\x$!3@+++s#o'R@FT7$Fj[x $!3$******\\i&z0AFT7$Fe[x$!3-+++*\\k/F#FT7$F`[x$!3.+++oPsLBFT7$F[[x$!3 7+++j^j&R#FT7$Ffjw$!3=+++RSFcCFT7$Fajw$!3y*****zmId^#FT7$F\\jw$!3#**** ***=>6uDFT7$Fgiw$!3))******3JaJEFT7$Fbiw$!3++++T'p\")o#FT7$F]iw$!3&)** ***HzdTu#FT7$Fhhw$!3=+++4Op*z#FT7$Fchw$!3-+++\\$y\\&GFT7$F^hw$!3(***** *\\O=-\"HFT7$Figw$!3++++!)pglHFT7$Fdgw$!36+++#f'H@IFT7$F_gw$!37+++]+++#**eP8$FT7$Fefw$!3)******Ri&H!>$FT7$F`fw$!3=+++\"*HiYK FT7$F[fw$!38+++0XE-LFT7$Ffew$!3A+++hUncLFT7$Faew$!3%)*****4?2$4MFT7$F \\ew$!3!)*****R\"znfMFT7$Fgdw$!3;+++(4%R2NFT7$Fbdw$!3!)*****\\3k@b$FT7 $F]dw$!3')*****HQ'y$f$FT7$Fhcw$!3$******4xI@j$FT7$Fccw$!3))*****H/=rm$ FT7$F^cw$!3/+++#3-()p$FT7$Fibw$!32+++[b&os$FT7$Fdbw$!3#)*****p.f:v$FT7 $F_bw$!3/+++kIzsPFT7$Fjaw$!3#******f1K0z$FT7$Feaw$!3>+++:(RZ!QFT7$F`aw $!3@+++M\\O:QFT7$F[aw$!3)******\\6QB#QFT7$Ff`w$!3@+++;rcDQFT7$Fa`w$!32 +++-E$\\#QFT7$F\\`w$!32+++5BG?QFT7$Fg_w$!3!)*****fqB9\"QFT7$Fb_w$!3-++ +wV6)z$FT7$F]_w$!3%)*****RI\\+y$FT7$Fh^w$!3#******H#R%ov$FT7$Fc^w$!37+ ++.;,GPFT7$F^^w$!3#******zAMHp$FT7$Fi]w$!37+++!QD3l$FT7$Fd]w$!3*)***** 46*o+OFT7$F_]w$!35+++#FT7 $Fciv$!3$******p&4$z.#FT7$F^iv$!35+++.1s#)=FT7$Fihv$!3(*******[$Hns\"F T7$Fdhv$!35+++u$Q-d\"FT7$F_hv$!3%******f:bMT\"FT7$Fjgv$!31+++ki^c7FT7$ Fegv$!3/+++U:]*4\"FT7$F`gv$!3c*****f`AXU*F07$F[gv$!3-+++QG(Q&yF07$Fffv $!3y*****>>YJG'F07$$!3++++?&e1'pF@$!3y*****RwyBr%F07$F\\fv$!3(******z> \"fTJF07$$!3++++(p]c/\"F4$!3!*******yizq:F0F'-Fe^n6&Fg^nF($F]_rFj^nFh^ n-F^_n6fw7%F'7$$!+*p]c/\"F0$\"+zizq:Ff_nFg_n7%F[hy7$$!+igTmkFT$\"+)>\" fTJFf_nFg_n7%Fahy7$$!+U%e1'pFj`n$\"+k(yBr%Ff_nFg_n7%Fghy7$$!+`/a>OFaan $\"+#>YJG'Ff_nFg_n7%F]iy7$$!+k6(\\D\"F^bn$\"+QG(Q&yFf_nFg_n7%Fciy7$$!+ 7\\+jLF^bn$\"+OD_C%*Ff_nFg_n7%Fiiy7$$!+$zMne(F^bn$\"+U:]*4\"FabnFg_n7% F_jy7$$!+fh'4`\"Fhcn$\"+ki^c7FabnFg_n7%Fejy7$$!+\"eSf!HFhcn$\"+c^X89Fa bnFg_n7%F[[z7$$!+s#f*4aFhcn$\"+u$Q-d\"FabnFg_n7%Fa[z7$$!+Hb([,\"Fedn$ \"+\\$Hns\"FabnFg_n7%Fg[z7$$!+56#*H>Fedn$\"+.1s#)=FabnFg_n7%F]\\z7$$!+ 1q)yn$Fedn$\"+d4$z.#FabnFg_n7%Fc\\z7$$!+e^h+pFedn$\"+4x,#>#FabnFg_n7%F i\\z7$$!+$*)zeD\"Fben$\"+'Q%fWBFabnFg_n7%F_]z7$$!+%H1u>#Fben$\"+-UA&\\ #FabnFg_n7%Fe]z7$$!+**HH#o$Fben$\"+t:MVEFabnFg_n7%F[^z7$$!+sPC/fFben$ \"+N$e!)y#FabnFg_n7%Fa^z7$$!+;Lvb!*Fben$\"+*3sy#HFabnFg_n7%Fg^z7$$!+EV ?F8Ff_n$\"+)3f/1$FabnFg_n7%F]_z7$$!+.Lhb=Ff_n$\"+rG$H=$FabnFg_n7%Fc_z7 $$!+Xg=uCFf_n$\"+U&eFH$FabnFg_n7%Fi_z7$$!+,b+bJFf_n$\"+\"[H()Q$FabnFg_ n7%F_`z7$$!+wI!)pQFf_n$\"+7G:rMFabnFg_n7%Fe`z7$$!+%H&G(f%Ff_n$\"+wtq%*Ff_n$\"+wV6)z$Fabn Fg_n7%Fecz7$$!+MLN75Fabn$\"+1PU6QFabnFg_n7%F[dz7$$!+C7qw5Fabn$\"+5BG?Q FabnFg_n7%Fadz7$$!+RH=S6Fabn$\"+-E$\\#QFabnFg_n7%Fgdz7$$!+Q*eG?\"Fabn$ \"+;rcDQFabnFg_n7%F]ez7$$!+.dyk7Fabn$\"+:\"QB#QFabnFg_n7%Fcez7$$!+uv,E 8Fabn$\"+M\\O:QFabnFg_n7%Fiez7$$!+r!4mQ\"Fabn$\"+:(RZ!QFabnFg_n7%F_fz7 $$!+4whY9Fabn$\"+m?`!z$FabnFg_n7%Fefz7$$!+%f1h]\"Fabn$\"+kIzsPFabnFg_n 7%F[gz7$$!+<*[^c\"Fabn$\"+P!f:v$FabnFg_n7%Fagz7$$!+y3$Qi\"Fabn$\"+[b&o s$FabnFg_n7%Fggz7$$!+4lD#o\"Fabn$\"+#3-()p$FabnFg_n7%F]hz7$$!+S@bSFabn$\"+&3k@b$FabnFg_n7%Fe iz7$$!+-)[f(>Fabn$\"+(4%R2NFabnFg_n7%F[jz7$$!+^`ZO?Fabn$\"+9znfMFabnFg _n7%Fajz7$$!+1&p\")4#Fabn$\"+,sI4MFabnFg_n7%Fgjz7$$!+l]Bh@Fabn$\"+hUnc LFabnFg_n7%F][[l7$$!+(R%yDAFabn$\"+0XE-LFabnFg_n7%Fc[[l7$$!+RHz\"H#Fab n$\"+\"*HiYKFabnFg_n7%Fi[[l7$$!+S?2fBFabn$\"+CcH!>$FabnFg_n7%F_\\[l7$$ !+\\aFFCFabn$\"+#**eP8$FabnFg_n7%Fe\\[l7$$!+!zTf\\#Fabn$\"+]> 6uDFabnFg_n7%Fa`[l7$$!+\"zG27$Fabn$\"+o1t:DFabnFg_n7%Fg`[l7$$!+?6ssJFa bn$\"+RSFcCFabnFg_n7%F]a[l7$$!+f$HEA$Fabn$\"+j^j&R#FabnFg_n7%Fca[l7$$! +^U[qKFabn$\"+oPsLBFabnFg_n7%Fia[l7$$!+BW$Fabn$\"+Pi/s?FabnFg_n7%Fac[l7$$!+(=#p![$Fabn$\"+ Jk!H+#FabnFg_n7%Fgc[l7$$!+fDQFabnFg_n7%F]d[l7$$!+j)yCb $Fabn$\"+QO0g=FabnFg_n7%Fcd[l7$$!+\\^2'e$Fabn$\"+BAP'y\"FabnFg_n7%Fid[ l7$$!+8(o#=OFabn$\"+D/C66QFabn$\"+C>5`6FabnFg_n7%Fig[l7$$!+2'pX$QFabn$\"+mV\\q5FabnFg _n7%F_h[l7$$!+?/\"o&QFabn$\"+>6Vw)*Ff_nFg_n7%Feh[l7$$!+Tu#y(QFabn$\"+8 ]!o/*Ff_nFg_n7%F[i[l7$$!+xI^(*QFabn$\"++S:<#)Ff_nFg_n7%Fai[l7$$!+9:v:R Fabn$\"+TEJ)Q(Ff_nFg_n7%Fgi[l7$$!+$[DC$RFabn$\"+QV'3c'Ff_nFg_n7%F]j[l7 $$!+!)>UZRFabn$\"+I7;NdFf_nFg_n7%Fcj[l7$$!+GmjgRFabn$\"+AYN6\\Ff_nFg_n 7%Fij[l7$$!+vm(>(RFabn$\"+%*)H%*3%Ff_nFg_n7%F_[\\l7$$!+2GO\")RFabn$\"+ 98CpKFf_nFg_n7%Fe[\\l7$$!+Z+t))RFabn$\"+2Qa]CFf_nFg_n7%F[\\\\l7$$!+a!G S*RFabn$\"+B.-L;Ff_nFg_n7%Fa\\\\l7$$!+z5A(*RFabn$\"+^g/j\")FbenFg_n7%F g\\\\l7$$!+&y(G)*RFabnF(Fg_n7%F]]\\l7$Fh\\\\l$!+^g/j\")FbenFg_n7%Fa]\\ l7$Fb\\\\l$!+B.-L;Ff_nFg_n7%Fe]\\l7$F\\\\\\l$!+2Qa]CFf_nFg_n7%Fi]\\l7$ Ff[\\l$!+98CpKFf_nFg_n7%F]^\\l7$F`[\\l$!+%*)H%*3%Ff_nFg_n7%Fa^\\l7$Fjj [l$!+AYN6\\Ff_nFg_n7%Fe^\\l7$Fdj[l$!+I7;NdFf_nFg_n7%Fi^\\l7$F^j[l$!+QV '3c'Ff_nFg_n7%F]_\\l7$Fhi[l$!+TEJ)Q(Ff_nFg_n7%Fa_\\l7$Fbi[l$!++S:<#)Ff _nFg_n7%Fe_\\l7$F\\i[l$!+8]!o/*Ff_nFg_n7%Fi_\\l7$Ffh[l$!+>6Vw)*Ff_nFg_ n7%F]`\\l7$F`h[l$!+mV\\q5FabnFg_n7%Fa`\\l7$Fjg[l$!+C>5`6FabnFg_n7%Fe` \\l7$Fdg[l$!+YVJN7FabnFg_n7%Fi`\\l7$F^g[l$!+qw'pJ\"FabnFg_n7%F]a\\l7$F hf[l$!+`M*yR\"FabnFg_n7%Faa\\l7$Fbf[l$!+3s#zZ\"FabnFg_n7%Fea\\l7$F\\f[ l$!+/h\"pb\"FabnFg_n7%Fia\\l7$Ffe[l$!+7]sM;FabnFg_n7%F]b\\l7$F`e[l$!+D /C6FabnFg_n7%F]c\\l7$Fhc[l$!+Jk!H+#FabnFg_n 7%Fac\\l7$Fbc[l$!+Pi/s?FabnFg_n7%Fec\\l7$F\\c[l$!+s#o'R@FabnFg_n7%Fic \\l7$Ffb[l$!+Dcz0AFabnFg_n7%F]d\\l7$F`b[l$!+*\\k/F#FabnFg_n7%Fad\\l7$F ja[l$!+oPsLBFabnFg_n7%Fed\\l7$Fda[l$!+j^j&R#FabnFg_n7%Fid\\l7$F^a[l$!+ RSFcCFabnFg_n7%F]e\\l7$Fh`[l$!+o1t:DFabnFg_n7%Fae\\l7$Fb`[l$!+>>6uDFab nFg_n7%Fee\\l7$F\\`[l$!+4JaJEFabnFg_n7%Fie\\l7$Ff_[l$!+T'p\")o#FabnFg_ n7%F]f\\l7$F`_[l$!+$zdTu#FabnFg_n7%Faf\\l7$Fj^[l$!+4Op*z#FabnFg_n7%Fef \\l7$Fd^[l$!+\\$y\\&GFabnFg_n7%Fif\\l7$F^^[l$!+l$=-\"HFabnFg_n7%F]g\\l 7$Fh][l$!+!)pglHFabnFg_n7%Fag\\l7$Fb][l$!+#f'H@IFabnFg_n7%Feg\\l7$F\\] [l$!+]$FabnFg_n7%Fah\\l7$Fj[[l$!+\"*HiYKFabnFg_n7%Feh\\l7$Fd[[l$!+0XE- LFabnFg_n7%Fih\\l7$F^[[l$!+hUncLFabnFg_n7%F]i\\l7$Fhjz$!+,sI4MFabnFg_n 7%Fai\\l7$Fbjz$!+9znfMFabnFg_n7%Fei\\l7$F\\jz$!+(4%R2NFabnFg_n7%Fii\\l 7$Ffiz$!+&3k@b$FabnFg_n7%F]j\\l7$F`iz$!+$Q'y$f$FabnFg_n7%Faj\\l7$Fjhz$ !+r28KOFabnFg_n7%Fej\\l7$Fdhz$!+V!=rm$FabnFg_n7%Fij\\l7$F^hz$!+#3-()p$ FabnFg_n7%F][]l7$Fhgz$!+[b&os$FabnFg_n7%Fa[]l7$Fbgz$!+P!f:v$FabnFg_n7% Fe[]l7$F\\gz$!+kIzsPFabnFg_n7%Fi[]l7$Fffz$!+m?`!z$FabnFg_n7%F]\\]l7$F` fz$!+:(RZ!QFabnFg_n7%Fa\\]l7$Fjez$!+M\\O:QFabnFg_n7%Fe\\]l7$Fdez$!+:\" QB#QFabnFg_n7%Fi\\]l7$F^ez$!+;rcDQFabnFg_n7%F]]]l7$Fhdz$!+-E$\\#QFabnF g_n7%Fa]]l7$Fbdz$!+5BG?QFabnFg_n7%Fe]]l7$F\\dz$!+1PU6QFabnFg_n7%Fi]]l7 $Ffcz$!+wV6)z$FabnFg_n7%F]^]l7$F`cz$!+/$\\+y$FabnFg_n7%Fa^]l7$Fjbz$!+B R%ov$FabnFg_n7%Fe^]l7$Fdbz$!+.;,GPFabnFg_n7%Fi^]l7$F^bz$!+GU$Hp$FabnFg _n7%F]_]l7$Fhaz$!+!QD3l$FabnFg_n7%Fa_]l7$Fbaz$!+6\"*o+OFabnFg_n7%Fe_]l 7$F\\az$!+#F abnFg_n7%Feb]l7$Fd\\z$!+d4$z.#FabnFg_n7%Fib]l7$F^\\z$!+.1s#)=FabnFg_n7 %F]c]l7$Fh[z$!+\\$Hns\"FabnFg_n7%Fac]l7$Fb[z$!+u$Q-d\"FabnFg_n7%Fec]l7 $F\\[z$!+c^X89FabnFg_n7%Fic]l7$Ffjy$!+ki^c7FabnFg_n7%F]d]l7$F`jy$!+U:] *4\"FabnFg_n7%Fad]l7$Fjiy$!+OD_C%*Ff_nFg_n7%Fed]l7$Fdiy$!+QG(Q&yFf_nFg _n7%Fid]l7$F^iy$!+#>YJG'Ff_nFg_n7%F]e]l7$$!+?&e1'pFj`n$!+k(yBr%Ff_nFg_ n7%Fae]l7$Fbhy$!+)>\"fTJFf_nFg_n7%Fge]l7$$!+(p]c/\"F0$!+zizq:Ff_nFg_n7 %F[f]lF'Fg_n-Fe^n6&Fg^nFh^n$\"#NFi_n$\"\"*Fj^nF^_r-F$6$7in7$$\"3%***** **G&=^X\"FT$\"3u*****R:u]M%FT7$$\"3'******\\Q^qW\"FT$\"3g*****R8*[aVFT 7$$\"3%******z\"oDQ9FT$\"3Q+++zF@jVFT7$$\"3$********ex(G9FT$\"39+++,K? rVFT7$$\"3'******>'ol=9FT$\"39+++(z=%yVFT7$$\"3.+++1&QzS\"FT$\"3;+++Zz \"[Q%FT7$$\"33+++cum'R\"FT$\"3?+++A(e.R%FT7$$\"3#*******)4!*[Q\"FT$\"3 A+++\"f)*\\R%FT7$$\"35+++GZls8FT$\"3g*****p9%p)R%FT7$$\"3-+++q?,g8FT$ \"3N+++B3S,WFT7$$\"3'******H!f,Z8FT$\"3K+++GE2.WFT7$$\"3*******pzBPL\" FT$\"3A+++x=m.WFT7$$\"3,+++&*z>?8FT$\"3P+++[*=JS%FT7$$\"35+++$\\1lI\"F T$\"3#)*****z1#R,WFT7$$\"3/+++bVs#H\"FT$\"33+++`rU)R%FT7$$\"3/+++E`$*y 7FT$\"3q*****RvnTR%FT7$$\"3#******H$QBl7FT$\"3')*****p:b&)Q%FT7$$\"3)* *****4YFM:eA\"FT$\"3')*****>*z)HO%FT7$$\"3*)*****4)=s87FT$\"3%)*****fx`8 N%FT7$$\"3#*******e#eC?\"FT$\"3A+++`E2QVFT7$$\"3-+++-'fA>\"FT$\"3\")** ***Rp1JK%FT7$$\"3)******>F/M=\"FT$\"36+++B=W1VFT7$$\"33+++p)>i<\"FT$\" 3m*****pP.\")G%FT7$$\"31+++ng3r6FT$\"3%)*****4uy\"oUFT7$$\"3%*******H( H%o6FT$\"37+++*R[oC%FT7$$\"3++++H\\qo6FT$\"37+++nVUCUFT7$$\"3/+++\"H[B <\"FT$\"3\"******4H*Q,UFT7$$\"3'*******\\Cqz6FT$\"3o*****ze>%yTFT7$$\" 3)******4H84>\"FT$\"3O+++!)fOcTFT7$$\"3-+++@<$e?\"FT$\"3%******z=nh8%F T7$$\"3(*******zB(RA\"FT$\"3)******Ht,(=TFT7$$\"3\"*******R#pXC\"FT$\" 3N+++v\"HY5%FT7$$\"3++++$G9nE\"FT$\"3G+++^CH%4%FT7$$\"37+++W^^*G\"FT$ \"3e*****zb+++T`G+TFT7$$\"30 +++39)*39FT$\"3c******fo84TFT7$$\"3)********H5PU\"FT$\"3m*****f%)*H>TF T7$$\"35+++oA!oV\"FT$\"3()*****4r808%FT7$$\"3%******4w&G[9FT$\"3(***** *z$\\bUTFT7$$\"3-+++%*G?e9FT$\"3k*****4vJ_:%FT7$$\"3++++UPgm9FT$\"3)** ****p2z$oTFT7$$\"3/+++#fUNZ\"FT$\"3w******)3a==%FT7$$\"3/+++Ib2z9FT$\" 3/+++3C`&>%FT7$$\"35+++1*eK[\"FT$\"3*)*****z#\\I4UFT7$$\"3/++++%[h[\"F T$\"3f*****>>vIA%FT7$$\"3!******pZ)z([\"FT$\"3&)*****\\AdnB%FT7$$\"3#* *****>6i#)[\"FT$\"3g*****put-D%FT7$$\"3)*******y1f([\"FT$\"3q*****RqaN E%FT7$$\"3/+++fR$e[\"FT$\"3i*****p.OlF%FT7$$\"3)******z@SI[\"FT$\"3/++ +r'e\"*G%FT7$$\"3'******RMc#z9FT$\"3N+++\\wO,VFT7$$\"3++++-!GXZ\"FT$\" 3B+++4968VFT7$$\"37+++B)**)o9FT$\"3N+++g6MCVFT7$$\"3!******\\g:CY\"FT$ \"3N+++U.,NVFTFjf]lFegy-F^_n6jn7%7$$\"+H&=^X\"Fabn$\"+aT2XVFabn7$$\"+& Q^qW\"Fabn$\"+M\"*[aVFabn7$$\"$L\"Fi_n$\"$B%Fi_n7%F^j^l7$$\"+=oDQ9Fabn $\"+zF@jVFabnFcj^l7%Fij^l7$$\"+!fx(G9Fabn$\"+,K?rVFabnFcj^l7%F_[_l7$$ \"+iol=9Fabn$\"+(z=%yVFabnFcj^l7%Fe[_l7$$\"+1&QzS\"Fabn$\"+Zz\"[Q%Fabn Fcj^l7%F[\\_l7$$\"+cum'R\"Fabn$\"+A(e.R%FabnFcj^l7%Fa\\_l7$$\"+*4!*[Q \"Fabn$\"+\"f)*\\R%FabnFcj^l7%Fg\\_l7$$\"+GZls8Fabn$\"+ZTp)R%FabnFcj^l 7%F]]_l7$$\"+q?,g8Fabn$\"+B3S,WFabnFcj^l7%Fc]_l7$$\"+.f,Z8Fabn$\"+GE2. WFabnFcj^l7%Fi]_l7$$\"+(zBPL\"Fabn$\"+x=m.WFabnFcj^l7%F_^_l7$$\"+&*z>? 8Fabn$\"+[*=JS%FabnFcj^l7%Fe^_l7$$\"+$\\1lI\"Fabn$\"+o?R,WFabnFcj^l7%F [__l7$$\"+bVs#H\"Fabn$\"+`rU)R%FabnFcj^l7%Fa__l7$$\"+E`$*y7Fabn$\"+ax; %R%FabnFcj^l7%Fg__l7$$\"+LQBl7Fabn$\"+d^b)Q%FabnFcj^l7%F]`_l7$$\"+hus^ 7Fabn$\"+](G:Q%FabnFcj^l7%Fc`_l7$$\"+A+aQ7Fabn$\"+;o-tVFabnFcj^l7%Fi`_ l7$$\"+U`\"eA\"Fabn$\"+#*z)HO%FabnFcj^l7%F_a_l7$$\"+\")=s87Fabn$\"+wPN ^VFabnFcj^l7%Fea_l7$$\"+f#eC?\"Fabn$\"+`E2QVFabnFcj^l7%F[b_l7$$\"+-'fA >\"Fabn$\"+%p1JK%FabnFcj^l7%Fab_l7$$\"+sUS$=\"Fabn$\"+B=W1VFabnFcj^l7% Fgb_l7$$\"+p)>i<\"Fabn$\"+xL5)G%FabnFcj^l7%F]c_l7$$\"+ng3r6Fabn$\"+T(y \"oUFabnFcj^l7%Fcc_l7$$\"+I(H%o6Fabn$\"+*R[oC%FabnFcj^l7%Fic_l7$$\"+H \\qo6Fabn$\"+nVUCUFabnFcj^l7%F_d_l7$$\"+\"H[B<\"Fabn$\"+\"H*Q,UFabnFcj ^l7%Fed_l7$$\"+]Cqz6Fabn$\"+)e>%yTFabnFcj^l7%F[e_l7$$\"+\"H84>\"Fabn$ \"+!)fOcTFabnFcj^l7%Fae_l7$$\"+@<$e?\"Fabn$\"+)=nh8%FabnFcj^l7%Fge_l7$ $\"+!QsRA\"Fabn$\"+LTFabnFcj^l7%Fii_l7$$\"+oA!oV\"Fabn$\"+6P^IT FabnFcj^l7%F_j_l7$$\"+hdG[9Fabn$\"+Q\\bUTFabnFcj^l7%Fej_l7$$\"+%*G?e9F abn$\"+^%Fab nFcj^l7%F]\\`l7$$\"+1*eK[\"Fabn$\"+G\\I4UFabnFcj^l7%Fc\\`l7$$\"++%[h[ \"Fabn$\"+#>vIA%FabnFcj^l7%Fi\\`l7$$\"+x%)z([\"Fabn$\"+DsvOUFabnFcj^l7 %F_]`l7$$\"+7@E)[\"Fabn$\"+ZPF]UFabnFcj^l7%Fe]`l7$$\"+z1f([\"Fabn$\"+/ ZbjUFabnFcj^l7%F[^`l7$$\"+fR$e[\"Fabn$\"+Pg`wUFabnFcj^l7%Fa^`l7$$\"+=- /$[\"Fabn$\"+r'e\"*G%FabnFcj^l7%Fg^`l7$$\"+WjDz9Fabn$\"+\\wO,VFabnFcj^ l7%F]_`l7$$\"+-!GXZ\"Fabn$\"+4968VFabnFcj^l7%Fc_`l7$$\"+B)**)o9Fabn$\" +g6MCVFabnFcj^l7%Fi_`l7$$\"+0cTi9Fabn$\"+U.,NVFabnFcj^l7%F_``lFii^lFcj ^lFaf]lF^_r-F$6$7in7$F[g]l$!3u*****R:u]M%FT7$Fbi^l$!3N+++U.,NVFT7$F]i^ l$!3N+++g6MCVFT7$Fhh^l$!3B+++4968VFT7$Fch^l$!3N+++\\wO,VFT7$F^h^l$!3/+ ++r'e\"*G%FT7$Fig^l$!3i*****p.OlF%FT7$Fdg^l$!3q*****RqaNE%FT7$F_g^l$!3 g*****put-D%FT7$Fjf^l$!3&)*****\\AdnB%FT7$Fef^l$!3f*****>>vIA%FT7$F`f^ l$!3*)*****z#\\I4UFT7$F[f^l$!3/+++3C`&>%FT7$Ffe^l$!3w******)3a==%FT7$F ae^l$!3)******p2z$oTFT7$F\\e^l$!3k*****4vJ_:%FT7$Fgd^l$!3(******z$\\bU TFT7$Fbd^l$!3()*****4r808%FT7$F]d^l$!3m*****f%)*H>TFT7$Fhc^l$!3c****** fo84TFT7$Fcc^l$!3>+++T`G+TFT7$F^c^l$!3a*****\\yZI4%FT7$Fib^l$!3/+++A8x (3%FT7$Fdb^l$!3!******4EX[3%FT7$F_b^l$!3/+++\\#)o%3%FT7$Fja^l$!3e***** zb%yTFT7$Fg_^l$!3\"******4H*Q,UFT7$Fb_^l$!37+++nVUCUFT7$F]_^l$!37++ +*R[oC%FT7$Fh^^l$!3%)*****4uy\"oUFT7$Fc^^l$!3m*****pP.\")G%FT7$F^^^l$! 36+++B=W1VFT7$Fi]^l$!3\")*****Rp1JK%FT7$Fd]^l$!3A+++`E2QVFT7$F_]^l$!3% )*****fx`8N%FT7$Fj\\^l$!3')*****>*z)HO%FT7$Fe\\^l$!3K+++;o-tVFT7$F`\\^ l$!3?+++](G:Q%FT7$F[\\^l$!3')*****p:b&)Q%FT7$Ff[^l$!3q*****RvnTR%FT7$F a[^l$!33+++`rU)R%FT7$F\\[^l$!3#)*****z1#R,WFT7$Fgj]l$!3P+++[*=JS%FT7$F bj]l$!3A+++x=m.WFT7$F]j]l$!3K+++GE2.WFT7$Fhi]l$!3N+++B3S,WFT7$Fci]l$!3 g*****p9%p)R%FT7$F^i]l$!3A+++\"f)*\\R%FT7$Fih]l$!3?+++A(e.R%FT7$Fdh]l$ !3;+++Zz\"[Q%FT7$F_h]l$!39+++(z=%yVFT7$Fjg]l$!39+++,K?rVFT7$Feg]l$!3Q+ ++zF@jVFT7$F`g]l$!3g*****R8*[aVFTFh``lFegy-F^_n6jn7%7$Fji^l$!+aT2XVFab n7$F```l$!+U.,NVFabn7$Fdj^l$!$B%Fi_n7%Fb\\al7$Fj_`l$!+g6MCVFabnFe\\al7 %Fi\\al7$Fd_`l$!+4968VFabnFe\\al7%F]]al7$F^_`l$!+\\wO,VFabnFe\\al7%Fa] al7$Fh^`l$!+r'e\"*G%FabnFe\\al7%Fe]al7$Fb^`l$!+Pg`wUFabnFe\\al7%Fi]al7 $F\\^`l$!+/ZbjUFabnFe\\al7%F]^al7$Ff]`l$!+ZPF]UFabnFe\\al7%Fa^al7$F`]` l$!+DsvOUFabnFe\\al7%Fe^al7$Fj\\`l$!+#>vIA%FabnFe\\al7%Fi^al7$Fd\\`l$! +G\\I4UFabnFe\\al7%F]_al7$F^\\`l$!+3C`&>%FabnFe\\al7%Fa_al7$Fh[`l$!+*3 a==%FabnFe\\al7%Fe_al7$Fb[`l$!+x!z$oTFabnFe\\al7%Fi_al7$F\\[`l$!+^TFabnFe\\al7%Fi`al7$Fdi_l$!+go84TFabnFe\\al 7%F]aal7$F^i_l$!+T`G+TFabnFe\\al7%Faaal7$Fhh_l$!+&yZI4%FabnFe\\al7%Fea al7$Fbh_l$!+A8x(3%FabnFe\\al7%Fiaal7$F\\h_l$!+h_%[3%FabnFe\\al7%F]bal7 $Ffg_l$!+\\#)o%3%FabnFe\\al7%Fabal7$F`g_l$!+evr(3%FabnFe\\al7%Febal7$F jf_l$!+^CH%4%FabnFe\\al7%Fibal7$Fdf_l$!+v\"HY5%FabnFe\\al7%F]cal7$F^f_ l$!+L%yTFabnFe\\al7%F]dal7$Ffd_l$!+\"H* Q,UFabnFe\\al7%Fadal7$F`d_l$!+nVUCUFabnFe\\al7%Fedal7$Fjc_l$!+*R[oC%Fa bnFe\\al7%Fidal7$Fdc_l$!+T(y\"oUFabnFe\\al7%F]eal7$F^c_l$!+xL5)G%FabnF e\\al7%Faeal7$Fhb_l$!+B=W1VFabnFe\\al7%Feeal7$Fbb_l$!+%p1JK%FabnFe\\al 7%Fieal7$F\\b_l$!+`E2QVFabnFe\\al7%F]fal7$Ffa_l$!+wPN^VFabnFe\\al7%Faf al7$F`a_l$!+#*z)HO%FabnFe\\al7%Fefal7$Fj`_l$!+;o-tVFabnFe\\al7%Fifal7$ Fd`_l$!+](G:Q%FabnFe\\al7%F]gal7$F^`_l$!+d^b)Q%FabnFe\\al7%Fagal7$Fh__ l$!+ax;%R%FabnFe\\al7%Fegal7$Fb__l$!+`rU)R%FabnFe\\al7%Figal7$F\\__l$! +o?R,WFabnFe\\al7%F]hal7$Ff^_l$!+[*=JS%FabnFe\\al7%Fahal7$F`^_l$!+x=m. WFabnFe\\al7%Fehal7$Fj]_l$!+GE2.WFabnFe\\al7%Fihal7$Fd]_l$!+B3S,WFabnF e\\al7%F]ial7$F^]_l$!+ZTp)R%FabnFe\\al7%Faial7$Fh\\_l$!+\"f)*\\R%FabnF e\\al7%Feial7$Fb\\_l$!+A(e.R%FabnFe\\al7%Fiial7$F\\\\_l$!+Zz\"[Q%FabnF e\\al7%F]jal7$Ff[_l$!+(z=%yVFabnFe\\al7%Fajal7$F`[_l$!+,K?rVFabnFe\\al 7%Fejal7$Fjj^l$!+zF@jVFabnFe\\al7%Fijal7$F_j^l$!+M\"*[aVFabnFe\\al7%F] [blF_\\alFe\\alFaf]lF^_r-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DE FAULTG-Fg[bl6$%*HELVETICAGFff]l-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CON STRAINEDG-%%VIEWG6$;$!$z%Fi_n$\"$>#Fi_n;Fi\\bl$\"$z%Fi_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability regi on for the 7 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "The stability function R^ for the 7 sta ge, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b^`,Sta bilityFunction(4,7,'expanded'))):\n`R^` := unapply(%,z):\n'`R^`(z)'=`R ^`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R^G6#%\"zG,2\"\"\"F)F'F) *&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F' \"\"%F)F)F)*&#\"%H)*\"(?F=\"F)*$)F'\"\"&F)F)F)*&#\"&\\^$\")?r$[#F)*$)F 'F1F)F)F)*&#\"$p#\"(S)z8F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R^`(z) = -1;" "6#/-%#R^G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R ^`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+u6lZS!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "zo0 := newton(`R^`(z)=-1,z=-4):\npo1 := plot([`R^`(z ),-1],z=-4.49..0.49,color=[red,blue]):\npo2 := plot([[[zo0,-1]]$3],sty le=point,symbol=[circle,cross,diamond],color=black):\npo3 := plot([[zo 0,0],[zo0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([ po1,po2,po3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$ $!3A++++++!\\%!#<$!3#=7!*3wdSI#F*7$$!3;+]7'3DdV%F*$!39J9#o0e:4#F*7$$!3 5++Ds,X\"Q%F*$!3S=7NW)Qf*=F*7$$!3(*\\P*)feAMVF*$!3')4*f?zn&Q%y!=%F*$!3Tk8;=D*4I\"F*7$$!35+ ]#oviQ2%F*$!3k*yCR/^T0\"F*7$$!3A+ve,&\\u'RF*$!3!3=TcUCO[)!#=7$$!3?+v8# z!zoQF*$!3ERn1(4t9)oFP7$$!3;+v.8ajmPF*$!3Gj]K/0l)[&FP7$$!3C+v$Q<')4m$F *$!3I**z@zCb\"H%FP7$$!3G+v3`dnbNF*$!3!pCE>i>oI$FP7$$!37++0T5NZMF*$!3i/ MS$\\PgZ#FP7$$!3?+]sv&Q>N$F*$!3!\\/QMziB(=FP7$$!3E++:Qn_WKF*$!317$$!3&****\\=iPF.$F*$!3^R%o $[/$p\"\\Fcp7$$!3Q+v$Gc`$QHF*$!3)>#4`V/Vz@Fcp7$$!3+++l(y@h#GF*$\"3a=kz ,B*\\S&!#?7$$!3=++g_n/JFF*$\"3Mh>R#oo4\\#Fcp7$$!3C+vt%)=X?EF*$\"3+h=#) QwrcWFcp7$$!3D++5(ocD_#F*$\"39k!f!**\\F,gFcp7$$!3#**\\P(R,::CF*$\"3'3i 3!>Y+evFcp7$$!3&)*\\7sqtGJ#F*$\"3]7%pDC)Hl*)Fcp7$$!30+]Pz*eh?#F*$\"31e(4P\\\"FP7$$!3%)*\\iW'R3(z\"F* $\"3W6\"40#Qi];FP7$$!3-+]d*>dQp\"F*$\"3')RffdBwM=FP7$$!3)*****p]Q@(e\" F*$\"3)e6>U]bQ/#FP7$$!3I+]FRT)G[\"F*$\"3'R%=217$)pAFP7$$!3')*\\P*y(R>Q \"F*$\"3$=s3Oqq8^#FP7$$!3++]Kx#e)p7F*$\"3=xE3tvG4GFP7$$!3=++5j![\"p6F* $\"3*4X]n)H#o5$FP7$$!3s***\\KT=;1\"F*$\"3U\\Vzq[LfMFP7$$!3'***\\P%o0=k *FP$\"3^%>V%*H;K\"QFP7$$!3](***\\tEbw&)FP$\"3jQO#)Q1rTUFP7$$!39-]P2EBu vFP$\"3UZx`!4<))o%FP7$$!3>)*\\(GL>l_'FP$\"3e5C+#Htm?&FP7$$!3z++]-\")=- bFP$\"3SMcv*=`#odFP7$$!3$*)*\\PM#3)HWFP$\"3%)Q0P_3?@kFP7$$!3;(****f'*y pR$FP$\"3&)H>8Z_&)>rFP7$$!3L)**\\U'=wSBFP$\"3UT3Zua,8zFP7$$!3?$*\\Pt2H $H\"FP$\"3505qOw%oy)FP7$$!33R****pit2LFcp$\"3ga#**36PYn*FP7$$\"3Z^+]Fk zBxFcp$\"3g9sM=\"*H!3\"F*7$$\"35*****z2`!f\"F*7$$\"3l+ ]i_F06GFP$\"3vZ#**[,$fC8F*7$$\"3\\.]Pt1&z\"QFP$\"3yiP`E<\"\\Y\"F*7$$\" 3!***************[FP$\"3c/6QGeJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\" \"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa \\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l 7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l 7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l 7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\ l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\ l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa \\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3I+++u6lZ SF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#% &POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_` lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINE STYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$% *HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture sho ws the stability region for the 8 stage, order 4 scheme. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1318 "`R^ ` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+9829/1182720*z^5+35149/24837120 *z^6+269/1379840*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R^`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts : = [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo1 := plot(pts,color=COLOR(RGB ,0,0,.2)):\npo2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i= 2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.1,.25,.8 )):\npts := []:\nz0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := ne wton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts), [Re(zz),Im(zz)]]:\nend do:\npo3 := plot(pts,color=COLOR(RGB,0,0,.2)): \npo4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.37,4.17]],i=2..no ps(pts))],\n style=patchnogrid,color=COLOR(RGB,.1,.25,.8)):\n pts := []:\nz0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton( `R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(z z),Im(zz)]]:\nend do:\npo5 := plot(pts,color=COLOR(RGB,0,0,.2)):\npo6 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[1.37,-4.17]],i=2..nops(pt s))],\n style=patchnogrid,color=COLOR(RGB,.1,.25,.8)):\npo7 : = plot([[[-4.79,0],[2.19,0]],[[0,-4.79],[0,4.79]]],color=black,linesty le=3):\nplots[display]([po||(1..7)],view=[-4.79..2.19,-4.79..4.79],fon t=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,sc aling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 586 586 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3=+++%y(>%R(!#F$\"31+++ [jzq:!#=7$$\"37+++(o2lv%!#D$\"3()*****HA$fTJF07$$\"3\\******[o-Va!#C$ \"3#******f*>R7ZF07$$\"37+++[n&R0$!#B$\"3')*****Rh*=$G'F07$$\"3%****** *G*zx9\"!#A$\"3A+++.(eR&yF07$$\"3I+++')e#*)H$FF$\"3#)*****z#4gC%*F07$$ \"3b+++B!emr(FF$\"35+++cf[*4\"!#<7$$\"3,+++N\\0/:!#@$\"33+++I!=kD\"FS7 $$\"3*)*****4F]>T#FW$\"3/+++@A:89FS7$$\"3!)*****>Lf(4HFW$\"35+++'f9&p: FS7$$\"33+++JC%\\Y\"FW$\"34+++cUCD$\"34+++`*\\QL#FS7$$!31+++Y9pmBFfp$\"31+++gL,![#FS7$ $!3,+++H[g2TFfp$\"34+++=_cAEFS7$$!3=+++yk-UmFfp$\"3)******HX[/w#FS7$$! 3/+++)e55,\"F0$\"3'******H'40#*GFS7$$!31+++-1(pX\"F0$\"3/+++4TA:IFS7$$ !35+++>%e`*>F0$\"32+++f%Gx7$FS7$$!3#******HH(e2EF0$\"3%******RLszA$FS7 $$!3u*****f\\@'pKF0$\"3;+++_))\\:LFS7$$!35+++`tQfRF0$\"33+++qA)3R$FS7$ $!3)******RV91m%F0$\"39+++#*3FbMFS7$$!3O+++W)yHO&F0$\"3\"*******fG'*4N FS7$$!3e*****f&*>11'F0$\"3!******f\\mhb$FS7$$!3=+++$GZ0v'F0$\"3!****** \\[6\\f$FS7$$!3$******p:f9V(F0$\"3(******4mRqi$FS7$$!3S+++>*))H5)F0$\" 3!******\\$oA`OFS7$$!3u*****\\,uOFS7$$!3F+++*\\I'=%*F 0$\"3-+++6W#)*o$FS7$$!3\"*******)ecj+\"FS$\"3<+++Uk+,PFS7$$!3'******\\ Cc+2\"FS$\"39+++C6$yq$FS7$$!3$*******z@,L6FS$\"36+++kg^5PFS7$$!33+++@V F&>\"FS$\"3(******f+M#4PFS7$$!3++++PI*oD\"FS$\"3@+++;57/PFS7$$!3(***** *Hq?zJ\"FS$\"3;+++\"*GG&p$FS7$$!3%******>$RTy8FS$\"3y*****zM+Go$FS7$$! 3++++^jVQ9FS$\"3#******4VLnm$FS7$$!3!*******G>1)\\\"FS$\"33+++#fDrk$FS 7$$!35+++U)ytb\"FS$\"3()*****p)y+COFS7$$!3/+++\\Q\\;;FS$\"3y******3QS( f$FS7$$!3++++Xq`v;FS$\"3!)*****4SNtc$FS7$$!33+++BmmMk9>FS$\"3%)*****44`LT$FS7$$!31+++aOLw>FS $\"3#******Rh&*pO$FS7$$!3;+++/]KR?FS$\"3\")*****z3izJ$FS7$$!3)******\\ kVQ5#FS$\"3?+++!RgmE$FS7$$!3?+++UU-q@FS$\"3%)*****zL-O@$FS7$$!3)****** 4lcyB#FS$\"3#******\\Xq$fJFS7$$!34+++rA:2BFS$\"3!******f4bX5$FS7$$!3/+ ++gsaxBFS$\"3))******Rhn\\IFS7$$!31+++.ua[CFS$\"3')*****RL:^*HFS7$$!3* *******e&*f>DFS$\"3*******>]#3THFS7$$!3++++vQ;!f#FS$\"3')******f=i()GF S7$$!39++++hwfEFS$\"33+++/rkMGFS7$$!3#******4:B!GFFS$\"3;+++Xq)>y#FS7$ $!3')*******[YYz#FS$\"39+++,uUHFFS7$$!3#******\\:K%fGFS$\"3%)*****R+Un n#FS7$$!36+++!QXA#HFS$\"3\"******zo7Pi#FS7$$!37+++?[+$)HFS$\"3))*****f 3S,d#FS7$$!3$)*****p')o;/$FS$\"3/+++#=[e^#FS7$$!3=+++<[A)4$FS$\"3;+++, ^ogCFS7$$!3++++93o_JFS$\"3\"******4*=_/CFS7$$!3')*****H9g]?$FS$\"3%)** ***RU]sM#FS7$$!3++++XtRbKFS$\"3?+++86y)G#FS7$$!3z*****4sNPI$FS$\"3>+++ I3/HAFS7$$!3-+++(zD,N$FS$\"3')*****>@rz;#FS7$$!31+++kWi%R$FS$\"3(***** **)[Fb5#FS7$$!3*)*****p\\%HPMFS$\"3;+++XynT?FS7$$!31+++WU?yMFS$\"3'*** ***RM.k(>FS7$$!3*)******\\sUFS7$$!3!)*****RiT]b$FS$ \"3,+++z:dT=FS7$$!3=+++F*G6f$FS$\"37+++!3Z?x\"FS7$$!3*)*****>Tsdi$FS$ \"30+++Ef;,Za!fOFS$\"33+++pt)*G;FS7$$!3!******\\D`5p$FS $\"33+++3(*eb:FS7$$!3!)*****fbQ=s$FS$\"3)******fDq5[\"FS7$$!3y*****pEn 9v$FS$\"3,+++jSa09FS7$$!35+++y)y*zPFS$\"3/+++<69H8FS7$$!3()******e;R2Q FS$\"3%******\\)>+_7FS7$$!3&)*****f;+P$QFS$\"3/+++V@Fu6FS7$$!3)******R fu)eQFS$\"3++++Zb4'4\"FS7$$!3!*******)=iG)QFS$\"3,+++A#3w,\"FS7$$!3'** ****pc!f0RFS$\"3c+++;lK*Q*F07$$!33+++hC(p#RFS$\"3^+++Hht,')F07$$!3#)** ***4?6p%RFS$\"3P+++/y:9yF07$$!3++++?iIlRFS$\"3<+++/6BFqF07$$!3?+++?!e? )RFS$\"3w*****HC(RTiF07$$!3?+++:A2(*RFS$\"3o*****Hj=pX&F07$$!3N+++TDE5 SFS$\"35+++8X!Rn%F07$$!3!*******fGb@SFS$\"3++++:'QB*QF07$$!3i*****RLy3 .%FS$\"3++++>f57JF07$$!3/+++tf=QSFS$\"3(******4q;IL#F07$$!3i*****RxMM/ %FS$\"3))******)3F[b\"F07$$!3w*****4j&fYSFS$\"3M*****4YhDx(Ffp7$$!3I++ +u6lZSFSF(7$Fijl$!3M*****4YhDx(Ffp7$Fdjl$!3))******)3F[b\"F07$F_jl$!3( ******4q;IL#F07$Fjil$!3++++>f57JF07$Feil$!3++++:'QB*QF07$F`il$!35+++8X !Rn%F07$F[il$!3o*****Hj=pX&F07$Ffhl$!3w*****HC(RTiF07$Fahl$!3<+++/6BFq F07$F\\hl$!3P+++/y:9yF07$Fggl$!3^+++Hht,')F07$Fbgl$!3c+++;lK*Q*F07$F]g l$!3,+++A#3w,\"FS7$Fhfl$!3++++Zb4'4\"FS7$Fcfl$!3/+++V@Fu6FS7$F^fl$!3%* *****\\)>+_7FS7$Fiel$!3/+++<69H8FS7$Fdel$!3,+++jSa09FS7$F_el$!3)****** fDq5[\"FS7$Fjdl$!33+++3(*eb:FS7$Fedl$!33+++pt)*G;FS7$F`dl$!30+++Ef;,FS7$F\\c l$!3'******RM.k(>FS7$Fgbl$!3;+++XynT?FS7$Fbbl$!3(*******)[Fb5#FS7$F]bl $!3')*****>@rz;#FS7$Fhal$!3>+++I3/HAFS7$Fcal$!3?+++86y)G#FS7$F^al$!3%) *****RU]sM#FS7$Fi`l$!3\"******4*=_/CFS7$Fd`l$!3;+++,^ogCFS7$F_`l$!3/++ +#=[e^#FS7$Fj_l$!3))*****f3S,d#FS7$Fe_l$!3\"******zo7Pi#FS7$F`_l$!3%)* ****R+Unn#FS7$F[_l$!39+++,uUHFFS7$Ff^l$!3;+++Xq)>y#FS7$Fa^l$!33+++/rkM GFS7$F\\^l$!3')******f=i()GFS7$Fg]l$!3*******>]#3THFS7$Fb]l$!3')*****R L:^*HFS7$F]]l$!3))******Rhn\\IFS7$Fh\\l$!3!******f4bX5$FS7$Fc\\l$!3#** ****\\Xq$fJFS7$F^\\l$!3%)*****zL-O@$FS7$Fi[l$!3?+++!RgmE$FS7$Fd[l$!3\" )*****z3izJ$FS7$F_[l$!3#******Rh&*pO$FS7$Fjz$!3%)*****44`LT$FS7$Fez$!3 -+++!QNnX$FS7$F`z$!3y*****4OQp\\$FS7$F[z$!3!*******49$Q`$FS7$Ffy$!3!)* ****4SNtc$FS7$Fay$!3y******3QS(f$FS7$F\\y$!3()*****p)y+COFS7$Fgx$!33++ +#fDrk$FS7$Fbx$!3#******4VLnm$FS7$F]x$!3y*****zM+Go$FS7$Fhw$!3;+++\"*G G&p$FS7$Fcw$!3@+++;57/PFS7$F^w$!3(******f+M#4PFS7$Fiv$!36+++kg^5PFS7$F dv$!39+++C6$yq$FS7$F_v$!3<+++Uk+,PFS7$Fju$!3-+++6W#)*o$FS7$Feu$!3=+++s >,uOFS7$F`u$!3!******\\$oA`OFS7$F[u$!3(******4mRqi$FS7$Fft$!3!******\\ [6\\f$FS7$Fat$!3!******f\\mhb$FS7$F\\t$!3\"*******fG'*4NFS7$Fgs$!39+++ #*3FbMFS7$Fbs$!33+++qA)3R$FS7$F]s$!3;+++_))\\:LFS7$Fhr$!3%******RLszA$ FS7$Fcr$!32+++f%Gx7$FS7$F^r$!3/+++4TA:IFS7$Fiq$!3'******H'40#*GFS7$Fdq $!3)******HX[/w#FS7$F_q$!34+++=_cAEFS7$Fjp$!31+++gL,![#FS7$Fdp$!34+++` *\\QL#FS7$F_p$!31+++[Mx%=#FS7$Fio$!30+++LEKL?FS7$$!3E+++;G$\\0&FW$!3/+ ++'G*)*z=FS7$F_o$!34+++cUCD$!3')*****Rh*=$G'F07$$\"3C+++Bo-VaF:$ !3#******f*>R7ZF07$F2$!3()*****HA$fTJF07$$\"3W+++wx>%R(F-$!31+++[jzq:F 0F'-%&COLORG6&%$RGBGF(F($\"\"#!\"\"-%)POLYGONSG6fw7%F'7$$\"+%y(>%R(Ffp $\"+[jzq:!#57$$!\"#F)F(7%F\\_n7$$\"+(o2lv%FS$\"+BKfTJFa_nFb_n7%Ff_n7$$ \"+\\o-Va!#;$\"+'*>R7ZFa_nFb_n7%F\\`n7$$\"+[n&R0$!#:$\"+9'*=$G'Fa_nFb_ n7%Fc`n7$$\"+H*zx9\"!#9$\"+.(eR&yFa_nFb_n7%Fj`n7$$\"+')e#*)H$F]an$\"+G 4gC%*Fa_nFb_n7%Faan7$$\"+B!emr(F]an$\"+cf[*4\"!\"*Fb_n7%Fgan7$$\"+N\\0 /:!#8$\"+I!=kD\"F\\bnFb_n7%F^bn7$$\"+r-&>T#Fabn$\"+@A:89F\\bnFb_n7%Feb n7$$\"+K$f(4HFabn$\"+'f9&p:F\\bnFb_n7%F[cn7$$\"+JC%\\Y\"Fabn$\"+cUCD%e`*>Fa_n$\"+f%Gx7$F\\bnFb_n7%F_gn7$$!+$H(e2EF a_n$\"+MB(zA$F\\bnFb_n7%Fegn7$$!+'\\@'pKFa_n$\"+_))\\:LF\\bnFb_n7%F[hn 7$$!+`tQfRFa_n$\"+qA)3R$F\\bnFb_n7%Fahn7$$!+MWhgYFa_n$\"+#*3FbMF\\bnFb _n7%Fghn7$$!+W)yHO&Fa_n$\"+gG'*4NF\\bnFb_n7%F]in7$$!+c*>11'Fa_n$\"+'\\ mhb$F\\bnFb_n7%Fcin7$$!+$GZ0v'Fa_n$\"+&[6\\f$F\\bnFb_n7%Fiin7$$!+d\"f9 V(Fa_n$\"+h'Rqi$F\\bnFb_n7%F_jn7$$!+>*))H5)Fa_n$\"+NoA`OF\\bnFb_n7%Fej n7$$!+v\"f_w)Fa_n$\"+s>,uOF\\bnFb_n7%F[[o7$$!+*\\I'=%*Fa_n$\"+6W#)*o$F \\bnFb_n7%Fa[o7$$!+*ecj+\"F\\bn$\"+Uk+,PF\\bnFb_n7%Fg[o7$$!+Xi0q5F\\bn $\"+C6$yq$F\\bnFb_n7%F]\\o7$$!+!=7I8\"F\\bn$\"+kg^5PF\\bnFb_n7%Fc\\o7$ $!+@VF&>\"F\\bn$\"+1SB4PF\\bnFb_n7%Fi\\o7$$!+PI*oD\"F\\bn$\"+;57/PF\\b nFb_n7%F_]o7$$!+.2#zJ\"F\\bn$\"+\"*GG&p$F\\bnFb_n7%Fe]o7$$!+KRTy8F\\bn $\"+[.!Go$F\\bnFb_n7%F[^o7$$!+^jVQ9F\\bn$\"+JMtmOF\\bnFb_n7%Fa^o7$$!+H >1)\\\"F\\bn$\"+#fDrk$F\\bnFb_n7%Fg^o7$$!+U)ytb\"F\\bn$\"+()y+COF\\bnF b_n7%F]_o7$$!+\\Q\\;;F\\bn$\"+4QS(f$F\\bnFb_n7%Fc_o7$$!+Xq`v;F\\bn$\"+ ,aLnNF\\bnFb_n7%Fi_o7$$!+BmmMk9>F\\bn$\"+\"4`LT$F\\bnFb_n7%Faao7$$!+aOLw>F\\bn$\"+9c*pO$ F\\bnFb_n7%Fgao7$$!+/]KR?F\\bn$\"+)3izJ$F\\bnFb_n7%F]bo7$$!+XO%Q5#F\\b n$\"+!RgmE$F\\bnFb_n7%Fcbo7$$!+UU-q@F\\bn$\"+QBg8KF\\bnFb_n7%Fibo7$$!+ ^m&yB#F\\bn$\"+b/PfJF\\bnFb_n7%F_co7$$!+rA:2BF\\bn$\"+'4bX5$F\\bnFb_n7 %Feco7$$!+gsaxBF\\bn$\"+Shn\\IF\\bnFb_n7%F[do7$$!+.ua[CF\\bn$\"+M`6&*H F\\bnFb_n7%Fado7$$!+f&*f>DF\\bn$\"+-D3THF\\bnFb_n7%Fgdo7$$!+vQ;!f#F\\b n$\"+g=i()GF\\bnFb_n7%F]eo7$$!++hwfEF\\bn$\"+/rkMGF\\bnFb_n7%Fceo7$$!+ ^J-GFF\\bn$\"+Xq)>y#F\\bnFb_n7%Fieo7$$!+!\\YYz#F\\bn$\"+,uUHFF\\bnFb_n 7%F_fo7$$!+b@VfGF\\bn$\"+/?uwEF\\bnFb_n7%Fefo7$$!+!QXA#HF\\bn$\"+)o7Pi #F\\bnFb_n7%F[go7$$!+?[+$)HF\\bn$\"+'3S,d#F\\bnFb_n7%Fago7$$!+n)o;/$F \\bn$\"+#=[e^#F\\bnFb_n7%Fggo7$$!+<[A)4$F\\bn$\"+,^ogCF\\bnFb_n7%F]ho7 $$!+93o_JF\\bn$\"+\"*=_/CF\\bnFb_n7%Fcho7$$!+V,10KF\\bn$\"+C/DZBF\\bnF b_n7%Fiho7$$!+XtRbKF\\bn$\"+86y)G#F\\bnFb_n7%F_io7$$!+@dt.LF\\bn$\"+I3 /HAF\\bnFb_n7%Feio7$$!+(zD,N$F\\bn$\"+77(z;#F\\bnFb_n7%F[jo7$$!+kWi%R$ F\\bn$\"+*[Fb5#F\\bnFb_n7%Fajo7$$!+(\\%HPMF\\bn$\"+XynT?F\\bnFb_n7%Fgj o7$$!+WU?yMF\\bn$\"+WLSw>F\\bnFb_n7%F][p7$$!+]sUF\\bn Fb_n7%Fc[p7$$!+C;/bNF\\bn$\"+z:dT=F\\bnFb_n7%Fi[p7$$!+F*G6f$F\\bn$\"+! 3Z?x\"F\\bnFb_n7%F_\\p7$$!+7CxDOF\\bn$\"+Ef;,+_7F\\bnFb_n7%Fi^p7$$!+m,qLQF\\bn$\"+V@Fu6F\\bnFb_n7%F__p7$$!+ %fu)eQF\\bn$\"+Zb4'4\"F\\bnFb_n7%Fe_p7$$!+*=iG)QF\\bn$\"+A#3w,\"F\\bnF b_n7%F[`p7$$!+n0f0RF\\bn$\"+;lK*Q*Fa_nFb_n7%Fa`p7$$!+hC(p#RF\\bn$\"+Hh t,')Fa_nFb_n7%Fg`p7$$!+,7\"p%RF\\bn$\"+/y:9yFa_nFb_n7%F]ap7$$!+?iIlRF \\bn$\"+/6BFqFa_nFb_n7%Fcap7$$!+?!e?)RF\\bn$\"+VsRTiFa_nFb_n7%Fiap7$$! +:A2(*RF\\bn$\"+L'=pX&Fa_nFb_n7%F_bp7$$!+TDE5SF\\bn$\"+8X!Rn%Fa_nFb_n7 %Febp7$$!+gGb@SF\\bn$\"+:'QB*QFa_nFb_n7%F[cp7$$!+M$y3.%F\\bn$\"+>f57JF a_nFb_n7%Facp7$$!+tf=QSF\\bn$\"+,n,LBFa_nFb_n7%Fgcp7$$!+uZVVSF\\bn$\"+ *3F[b\"Fa_nFb_n7%F]dp7$$!+JcfYSF\\bn$\"+h9csxF]enFb_n7%Fcdp7$$!+u6lZSF \\bnF(Fb_n7%Fidp7$Fddp$!+h9csxF]enFb_n7%F]ep7$F^dp$!+*3F[b\"Fa_nFb_n7% Faep7$Fhcp$!+,n,LBFa_nFb_n7%Feep7$Fbcp$!+>f57JFa_nFb_n7%Fiep7$F\\cp$!+ :'QB*QFa_nFb_n7%F]fp7$Ffbp$!+8X!Rn%Fa_nFb_n7%Fafp7$F`bp$!+L'=pX&Fa_nFb _n7%Fefp7$Fjap$!+VsRTiFa_nFb_n7%Fifp7$Fdap$!+/6BFqFa_nFb_n7%F]gp7$F^ap $!+/y:9yFa_nFb_n7%Fagp7$Fh`p$!+Hht,')Fa_nFb_n7%Fegp7$Fb`p$!+;lK*Q*Fa_n Fb_n7%Figp7$F\\`p$!+A#3w,\"F\\bnFb_n7%F]hp7$Ff_p$!+Zb4'4\"F\\bnFb_n7%F ahp7$F`_p$!+V@Fu6F\\bnFb_n7%Fehp7$Fj^p$!+&)>+_7F\\bnFb_n7%Fihp7$Fd^p$! +<69H8F\\bnFb_n7%F]ip7$F^^p$!+jSa09F\\bnFb_n7%Faip7$Fh]p$!+c-2\"[\"F\\ bnFb_n7%Feip7$Fb]p$!+3(*eb:F\\bnFb_n7%Fiip7$F\\]p$!+pt)*G;F\\bnFb_n7%F ]jp7$Ff\\p$!+Ef;,F\\bnFb_n7%F][q7$F^[p$!+WLSw>F \\bnFb_n7%Fa[q7$Fhjo$!+XynT?F\\bnFb_n7%Fe[q7$Fbjo$!+*[Fb5#F\\bnFb_n7%F i[q7$F\\jo$!+77(z;#F\\bnFb_n7%F]\\q7$Ffio$!+I3/HAF\\bnFb_n7%Fa\\q7$F`i o$!+86y)G#F\\bnFb_n7%Fe\\q7$Fjho$!+C/DZBF\\bnFb_n7%Fi\\q7$Fdho$!+\"*=_ /CF\\bnFb_n7%F]]q7$F^ho$!+,^ogCF\\bnFb_n7%Fa]q7$Fhgo$!+#=[e^#F\\bnFb_n 7%Fe]q7$Fbgo$!+'3S,d#F\\bnFb_n7%Fi]q7$F\\go$!+)o7Pi#F\\bnFb_n7%F]^q7$F ffo$!+/?uwEF\\bnFb_n7%Fa^q7$F`fo$!+,uUHFF\\bnFb_n7%Fe^q7$Fjeo$!+Xq)>y# F\\bnFb_n7%Fi^q7$Fdeo$!+/rkMGF\\bnFb_n7%F]_q7$F^eo$!+g=i()GF\\bnFb_n7% Fa_q7$Fhdo$!+-D3THF\\bnFb_n7%Fe_q7$Fbdo$!+M`6&*HF\\bnFb_n7%Fi_q7$F\\do $!+Shn\\IF\\bnFb_n7%F]`q7$Ffco$!+'4bX5$F\\bnFb_n7%Fa`q7$F`co$!+b/PfJF \\bnFb_n7%Fe`q7$Fjbo$!+QBg8KF\\bnFb_n7%Fi`q7$Fdbo$!+!RgmE$F\\bnFb_n7%F ]aq7$F^bo$!+)3izJ$F\\bnFb_n7%Faaq7$Fhao$!+9c*pO$F\\bnFb_n7%Feaq7$Fbao$ !+\"4`LT$F\\bnFb_n7%Fiaq7$F\\ao$!+!QNnX$F\\bnFb_n7%F]bq7$Ff`o$!+h$Qp\\ $F\\bnFb_n7%Fabq7$F``o$!+59$Q`$F\\bnFb_n7%Febq7$Fj_o$!+,aLnNF\\bnFb_n7 %Fibq7$Fd_o$!+4QS(f$F\\bnFb_n7%F]cq7$F^_o$!+()y+COF\\bnFb_n7%Facq7$Fh^ o$!+#fDrk$F\\bnFb_n7%Fecq7$Fb^o$!+JMtmOF\\bnFb_n7%Ficq7$F\\^o$!+[.!Go$ F\\bnFb_n7%F]dq7$Ff]o$!+\"*GG&p$F\\bnFb_n7%Fadq7$F`]o$!+;57/PF\\bnFb_n 7%Fedq7$Fj\\o$!+1SB4PF\\bnFb_n7%Fidq7$Fd\\o$!+kg^5PF\\bnFb_n7%F]eq7$F^ \\o$!+C6$yq$F\\bnFb_n7%Faeq7$Fh[o$!+Uk+,PF\\bnFb_n7%Feeq7$Fb[o$!+6W#)* o$F\\bnFb_n7%Fieq7$F\\[o$!+s>,uOF\\bnFb_n7%F]fq7$Ffjn$!+NoA`OF\\bnFb_n 7%Fafq7$F`jn$!+h'Rqi$F\\bnFb_n7%Fefq7$Fjin$!+&[6\\f$F\\bnFb_n7%Fifq7$F din$!+'\\mhb$F\\bnFb_n7%F]gq7$F^in$!+gG'*4NF\\bnFb_n7%Fagq7$Fhhn$!+#*3 FbMF\\bnFb_n7%Fegq7$Fbhn$!+qA)3R$F\\bnFb_n7%Figq7$F\\hn$!+_))\\:LF\\bn Fb_n7%F]hq7$Ffgn$!+MB(zA$F\\bnFb_n7%Fahq7$F`gn$!+f%Gx7$F\\bnFb_n7%Fehq 7$Fjfn$!+4TA:IF\\bnFb_n7%Fihq7$Fdfn$!+j40#*GF\\bnFb_n7%F]iq7$F^fn$!+`% [/w#F\\bnFb_n7%Faiq7$Fhen$!+=_cAEF\\bnFb_n7%Feiq7$Fben$!+gL,![#F\\bnFb _n7%Fiiq7$F[en$!+`*\\QL#F\\bnFb_n7%F]jq7$Fedn$!+[Mx%=#F\\bnFb_n7%Fajq7 $F^dn$!+LEKL?F\\bnFb_n7%Fejq7$$!+;G$\\0&Fabn$!+'G*)*z=F\\bnFb_n7%Fijq7 $Fbcn$!+cUCDR7ZFa_n Fb_n7%Fa]r7$Fg_n$!+BKfTJFa_nFb_n7%Fg]r7$$\"+wx>%R(Ffp$!+[jzq:Fa_nFb_n7 %F[^rF'Fb_n-Fb^n6&Fd^n$\"\"\"Fg^n$\"#DFd_n$\"\")Fg^n-%&STYLEG6#%,PATCH NOGRIDG-F$6$7in7$$\"3/+++(R%*)*\\\"FS$\"3-+++2&=ZG%FS7$$\"3/+++\\Zo\" \\\"FS$\"3u*****\\,OKH%FS7$$\"3%*******)4-G[\"FS$\"3C+++.T0,VFS7$$\"3) ******\\n*Gt9FS$\"3()*****4RJ\"3VFS7$$\"3'******\\]\">j9FS$\"3P+++(yEW J%FS7$$\"31+++YFb_9FS$\"3C+++8#**)>VFS7$$\"3++++g+UT9FS$\"3&******pM2X K%FS7$$\"3&*******Q?%)H9FS$\"3;+++'Q4#GVFS7$$\"31+++1(pyT\"FS$\"36+++7 G'4L%FS7$$\"3)******p4dbS\"FS$\"3%)*****H=CFL%FS7$$\"3++++Y>'HR\"FS$\" 3&)*****f%*[ML%FS7$$\"35+++5l9!Q\"FS$\"3)******RG\"4LVFS7$$\"3'******p ayrO\"FS$\"3$)*****R,/;L%FS7$$\"35+++*\\KTN\"FS$\"3F+++i&Q*GVFS7$$\"3) ******\\$44T8FS$\"3e*****p1X]K%FS7$$\"32+++#GY\"G8FS$\"3y*****4ms)>VFS 7$$\"3$******p%HS:8FS$\"3&)*****>/qLJ%FS7$$\"3$*******p)zHI\"FS$\"3N++ +kk[0VFS7$$\"3-+++\"f85H\"FS$\"3C+++tL<'H%FS7$$\"31+++j=mz7FS$\"3#)*** **p*pQ&G%FS7$$\"3'******Rz2\"p7FS$\"3))******HB4tUFS7$$\"3%******\\Kk& f7FS$\"3I+++d\"p#fUFS7$$\"3)******fey7D\"FS$\"3)******>$4#RC%FS7$$\"3% ******z6NXC\"FS$\"3\")*****R`(3FUFS7$$\"31+++cglR7FS$\"3w*****z'G')3UF S7$$\"3#******z1&*pB\"FS$\"3y******>xT*=%FS7$$\"31+++,.#pB\"FS$\"3s*** **zaE!pTFS7$$\"3(******>y!yR7FS$\"3m*****>F$4[TFS7$$\"33+++;E&eC\"FS$ \"3H+++ab;FTFS7$$\"33+++f)o_D\"FS$\"3F+++K=#p5%FS7$$\"35+++!f]zE\"FS$ \"3'******>n;\")3%FS7$$\"3)*******Q#pNG\"FS$\"3M+++RY[rSFS7$$\"3\"**** **4Fn:I\"FS$\"3;+++.#Gw0%FS7$$\"3'*******)fQ7K\"FS$\"3))*****f&>$p/%FS 7$$\"3-+++G\"Q=M\"FS$\"3J+++7[`RSFS7$$\"3/+++F(yEO\"FS$\"3n******>.ONS FS7$$\"3-+++L(*=$Q\"FS$\"3E+++)zzT.%FS7$$\"3\"******fONHS\"FS$\"3B+++* Qzc.%FS7$$\"3!******px0;U\"FS$\"32+++WN^RSFS7$$\"33+++r`**Q9FS$\"3?+++ \">S`/%FS7$$\"35+++$fy\\X\"FS$\"3e******z,%G0%FS7$$\"33+++C$)[p9FS$\"3 7+++')\\#[\"FS$\"35+++7GurSFS7$$\"33+++U<,%\\\"FS$\" 3s*****psoE3%FS7$$\"3%******4!)[S]\"FS$\"3M+++J)4V4%FS7$$\"3-+++VJk7:F S$\"3-+++hu\\1TFS7$$\"3\"******>yN)>:FS$\"3j*****fV%3>TFS7$$\"3%****** RFsc_\"FS$\"3U+++&yT>8%FS7$$\"3,+++?1?I:FS$\"3)******f)e&\\9%FS7$$\"3/ +++l)pM`\"FS$\"3A+++3g-eTFS7$$\"3$******R?Hb`\"FS$\"3y*****HBi5<%FS7$$ \"3\"******\\TFk`\"FS$\"3=+++;P)R=%FS7$$\"35+++RD@O:FS$\"3C+++7sr'>%FS 7$$\"3)******HrJ\\`\"FS$\"3&)*****R&e>4UFS7$$\"32+++=6jK:FS$\"3!)***** 46e8A%FS7$$\"3'******H+c$H:FS$\"3&)*****\\\"p9LUFS7$$\"3\"*******o2:D: FS$\"3>+++=*3XC%FS7$$\"3*)*****R3f+_\"FS$\"3g*****f%QRbUFS7$$\"3++++SS 79:FS$\"3-+++aRvlUFS7$$\"3++++?$)Q2:FS$\"3#)*****f`VbF%FSF`_rFa^n-Fi^n 6jn7%7$$\"+(R%*)*\\\"F\\bn$\"+2&=ZG%F\\bn7$$\"+\\Zo\"\\\"F\\bn$\"+:gB$ H%F\\bn7$$\"$P\"Fd_n$\"$<%Fd_n7%Fdbs7$$\"+*4-G[\"F\\bn$\"+.T0,VF\\bnFi bs7%F_cs7$$\"+v'*Gt9F\\bn$\"+\"RJ\"3VF\\bnFibs7%Fecs7$$\"+0:>j9F\\bn$ \"+(yEWJ%F\\bnFibs7%F[ds7$$\"+YFb_9F\\bn$\"+8#**)>VF\\bnFibs7%Fads7$$ \"+g+UT9F\\bn$\"+Zt]CVF\\bnFibs7%Fgds7$$\"+R?%)H9F\\bn$\"+'Q4#GVF\\bnF ibs7%F]es7$$\"+1(pyT\"F\\bn$\"+7G'4L%F\\bnFibs7%Fces7$$\"+(4dbS\"F\\bn $\"+$=CFL%F\\bnFibs7%Fies7$$\"+Y>'HR\"F\\bn$\"+Y*[ML%F\\bnFibs7%F_fs7$ $\"+5l9!Q\"F\\bn$\"+%G\"4LVF\\bnFibs7%Fefs7$$\"+Z&yrO\"F\\bn$\"+9SgJVF \\bnFibs7%F[gs7$$\"+*\\KTN\"F\\bn$\"+i&Q*GVF\\bnFibs7%Fags7$$\"+N44T8F \\bn$\"+n]/DVF\\bnFibs7%Fggs7$$\"+#GY\"G8F\\bn$\"+hE()>VF\\bnFibs7%F]h s7$$\"+ZHS:8F\\bn$\"+U+P8VF\\bnFibs7%Fchs7$$\"+q)zHI\"F\\bn$\"+kk[0VF \\bnFibs7%Fihs7$$\"+\"f85H\"F\\bn$\"+tL<'H%F\\bnFibs7%F_is7$$\"+j=mz7F \\bn$\"+(*pQ&G%F\\bnFibs7%Feis7$$\"+%z2\"p7F\\bn$\"+IB4tUF\\bnFibs7%F[ js7$$\"+DVcf7F\\bn$\"+d\"p#fUF\\bnFibs7%Fajs7$$\"+'ey7D\"F\\bn$\"+K4#R C%F\\bnFibs7%Fgjs7$$\"+=^`W7F\\bn$\"+Mv3FUF\\bnFibs7%F][t7$$\"+cglR7F \\bn$\"+oG')3UF\\bnFibs7%Fc[t7$$\"+o]*pB\"F\\bn$\"+?xT*=%F\\bnFibs7%Fi [t7$$\"+,.#pB\"F\\bn$\"+[l-pTF\\bnFibs7%F_\\t7$$\"+#y!yR7F\\bn$\"+sK4[ TF\\bnFibs7%Fe\\t7$$\"+;E&eC\"F\\bn$\"+ab;FTF\\bnFibs7%F[]t7$$\"+f)o_D \"F\\bn$\"+K=#p5%F\\bnFibs7%Fa]t7$$\"+!f]zE\"F\\bn$\"+sm6)3%F\\bnFibs7 %Fg]t7$$\"+R#pNG\"F\\bn$\"+RY[rSF\\bnFibs7%F]^t7$$\"+rsc,8F\\bn$\"+.#G w0%F\\bnFibs7%Fc^t7$$\"+*fQ7K\"F\\bn$\"+c>$p/%F\\bnFibs7%Fi^t7$$\"+G\" Q=M\"F\\bn$\"+7[`RSF\\bnFibs7%F__t7$$\"+F(yEO\"F\\bn$\"+?.ONSF\\bnFibs 7%Fe_t7$$\"+L(*=$Q\"F\\bn$\"+)zzT.%F\\bnFibs7%F[`t7$$\"+m`$HS\"F\\bn$ \"+*Qzc.%F\\bnFibs7%Fa`t7$$\"+xdg@9F\\bn$\"+WN^RSF\\bnFibs7%Fg`t7$$\"+ r`**Q9F\\bn$\"+\">S`/%F\\bnFibs7%F]at7$$\"+$fy\\X\"F\\bn$\"+!=SG0%F\\b nFibs7%Fcat7$$\"+C$)[p9F\\bn$\"+')\\#[\"F\\ bn$\"+7GurSF\\bnFibs7%F_bt7$$\"+U<,%\\\"F\\bn$\"+F(oE3%F\\bnFibs7%Febt 7$$\"+,)[S]\"F\\bn$\"+J)4V4%F\\bnFibs7%F[ct7$$\"+VJk7:F\\bn$\"+hu\\1TF \\bnFibs7%Fact7$$\"+#yN)>:F\\bn$\"+OW3>TF\\bnFibs7%Fgct7$$\"+uAnD:F\\b n$\"+&yT>8%F\\bnFibs7%F]dt7$$\"+?1?I:F\\bn$\"+')e&\\9%F\\bnFibs7%Fcdt7 $$\"+l)pM`\"F\\bn$\"+3g-eTF\\bnFibs7%Fidt7$$\"+/#Hb`\"F\\bn$\"+LA1rTF \\bnFibs7%F_et7$$\"+:uUO:F\\bn$\"+;P)R=%F\\bnFibs7%Feet7$$\"+RD@O:F\\b n$\"+7sr'>%F\\bnFibs7%F[ft7$$\"+8<$\\`\"F\\bn$\"+ae>4UF\\bnFibs7%Faft7 $$\"+=6jK:F\\bn$\"+6\"e8A%F\\bnFibs7%Fgft7$$\"+.gNH:F\\bn$\"+:p9LUF\\b nFibs7%F]gt7$$\"+p2:D:F\\bn$\"+=*3XC%F\\bnFibs7%Fcgt7$$\"+%3f+_\"F\\bn $\"+YQRbUF\\bnFibs7%Figt7$$\"+SS79:F\\bn$\"+aRvlUF\\bnFibs7%F_ht7$$\"+ ?$)Q2:F\\bn$\"+ONavUF\\bnFibs7%FehtF_bsFibsFa^rFi^r-F$6$7in7$Fa_r$!3-+ ++2&=ZG%FS7$Fhas$!3#)*****f`VbF%FS7$Fcas$!3-+++aRvlUFS7$F^as$!3g*****f %QRbUFS7$Fi`s$!3>+++=*3XC%FS7$Fd`s$!3&)*****\\\"p9LUFS7$F_`s$!3!)***** 46e8A%FS7$Fj_s$!3&)*****R&e>4UFS7$Fe_s$!3C+++7sr'>%FS7$F`_s$!3=+++;P)R =%FS7$F[_s$!3y*****HBi5<%FS7$Ff^s$!3A+++3g-eTFS7$Fa^s$!3)******f)e&\\9 %FS7$F\\^s$!3U+++&yT>8%FS7$Fg]s$!3j*****fV%3>TFS7$Fb]s$!3-+++hu\\1TFS7 $F]]s$!3M+++J)4V4%FS7$Fh\\s$!3s*****psoE3%FS7$Fc\\s$!35+++7GurSFS7$F^ \\s$!37+++S`/%FS7$F_[s$ !32+++WN^RSFS7$Fjjr$!3B+++*Qzc.%FS7$Fejr$!3E+++)zzT.%FS7$F`jr$!3n***** *>.ONSFS7$F[jr$!3J+++7[`RSFS7$Ffir$!3))*****f&>$p/%FS7$Fair$!3;+++.#Gw 0%FS7$F\\ir$!3M+++RY[rSFS7$Fghr$!3'******>n;\")3%FS7$Fbhr$!3F+++K=#p5% FS7$F]hr$!3H+++ab;FTFS7$Fhgr$!3m*****>F$4[TFS7$Fcgr$!3s*****zaE!pTFS7$ F^gr$!3y******>xT*=%FS7$Fifr$!3w*****z'G')3UFS7$Fdfr$!3\")*****R`(3FUF S7$F_fr$!3)******>$4#RC%FS7$Fjer$!3I+++d\"p#fUFS7$Feer$!3))******HB4tU FS7$F`er$!3#)*****p*pQ&G%FS7$F[er$!3C+++tL<'H%FS7$Ffdr$!3N+++kk[0VFS7$ Fadr$!3&)*****>/qLJ%FS7$F\\dr$!3y*****4ms)>VFS7$Fgcr$!3e*****p1X]K%FS7 $Fbcr$!3F+++i&Q*GVFS7$F]cr$!3$)*****R,/;L%FS7$Fhbr$!3)******RG\"4LVFS7 $Fcbr$!3&)*****f%*[ML%FS7$F^br$!3%)*****H=CFL%FS7$Fiar$!36+++7G'4L%FS7 $Fdar$!3;+++'Q4#GVFS7$F_ar$!3&******pM2XK%FS7$Fj`r$!3C+++8#**)>VFS7$Fe `r$!3P+++(yEWJ%FS7$F``r$!3()*****4RJ\"3VFS7$F[`r$!3C+++.T0,VFS7$Ff_r$! 3u*****\\,OKH%FSF^itFa^n-Fi^n6jn7%7$F`bs$!+2&=ZG%F\\bn7$Ffht$!+ONavUF \\bn7$Fjbs$!$<%Fd_n7%Fhdu7$F`ht$!+aRvlUF\\bnF[eu7%F_eu7$Fjgt$!+YQRbUF \\bnF[eu7%Fceu7$Fdgt$!+=*3XC%F\\bnF[eu7%Fgeu7$F^gt$!+:p9LUF\\bnF[eu7%F [fu7$Fhft$!+6\"e8A%F\\bnF[eu7%F_fu7$Fbft$!+ae>4UF\\bnF[eu7%Fcfu7$F\\ft $!+7sr'>%F\\bnF[eu7%Fgfu7$Ffet$!+;P)R=%F\\bnF[eu7%F[gu7$F`et$!+LA1rTF \\bnF[eu7%F_gu7$Fjdt$!+3g-eTF\\bnF[eu7%Fcgu7$Fddt$!+')e&\\9%F\\bnF[eu7 %Fggu7$F^dt$!+&yT>8%F\\bnF[eu7%F[hu7$Fhct$!+OW3>TF\\bnF[eu7%F_hu7$Fbct $!+hu\\1TF\\bnF[eu7%Fchu7$F\\ct$!+J)4V4%F\\bnF[eu7%Fghu7$Ffbt$!+F(oE3% F\\bnF[eu7%F[iu7$F`bt$!+7GurSF\\bnF[eu7%F_iu7$Fjat$!+S`/%F\\bnF[eu7%F[ju7$Fh`t $!+WN^RSF\\bnF[eu7%F_ju7$Fb`t$!+*Qzc.%F\\bnF[eu7%Fcju7$F\\`t$!+)zzT.%F \\bnF[eu7%Fgju7$Ff_t$!+?.ONSF\\bnF[eu7%F[[v7$F`_t$!+7[`RSF\\bnF[eu7%F_ [v7$Fj^t$!+c>$p/%F\\bnF[eu7%Fc[v7$Fd^t$!+.#Gw0%F\\bnF[eu7%Fg[v7$F^^t$! +RY[rSF\\bnF[eu7%F[\\v7$Fh]t$!+sm6)3%F\\bnF[eu7%F_\\v7$Fb]t$!+K=#p5%F \\bnF[eu7%Fc\\v7$F\\]t$!+ab;FTF\\bnF[eu7%Fg\\v7$Ff\\t$!+sK4[TF\\bnF[eu 7%F[]v7$F`\\t$!+[l-pTF\\bnF[eu7%F_]v7$Fj[t$!+?xT*=%F\\bnF[eu7%Fc]v7$Fd [t$!+oG')3UF\\bnF[eu7%Fg]v7$F^[t$!+Mv3FUF\\bnF[eu7%F[^v7$Fhjs$!+K4#RC% F\\bnF[eu7%F_^v7$Fbjs$!+d\"p#fUF\\bnF[eu7%Fc^v7$F\\js$!+IB4tUF\\bnF[eu 7%Fg^v7$Ffis$!+(*pQ&G%F\\bnF[eu7%F[_v7$F`is$!+tL<'H%F\\bnF[eu7%F__v7$F jhs$!+kk[0VF\\bnF[eu7%Fc_v7$Fdhs$!+U+P8VF\\bnF[eu7%Fg_v7$F^hs$!+hE()>V F\\bnF[eu7%F[`v7$Fhgs$!+n]/DVF\\bnF[eu7%F_`v7$Fbgs$!+i&Q*GVF\\bnF[eu7% Fc`v7$F\\gs$!+9SgJVF\\bnF[eu7%Fg`v7$Fffs$!+%G\"4LVF\\bnF[eu7%F[av7$F`f s$!+Y*[ML%F\\bnF[eu7%F_av7$Fjes$!+$=CFL%F\\bnF[eu7%Fcav7$Fdes$!+7G'4L% F\\bnF[eu7%Fgav7$F^es$!+'Q4#GVF\\bnF[eu7%F[bv7$Fhds$!+Zt]CVF\\bnF[eu7% F_bv7$Fbds$!+8#**)>VF\\bnF[eu7%Fcbv7$F\\ds$!+(yEWJ%F\\bnF[eu7%Fgbv7$Ff cs$!+\"RJ\"3VF\\bnF[eu7%F[cv7$F`cs$!+.T0,VF\\bnF[eu7%F_cv7$Febs$!+:gB$ H%F\\bnF[eu7%FccvFeduF[euFa^rFi^r-F$6%7$7$$!3/++++++!z%FSF(7$$\"3%**** **********=#FSF(-%'COLOURG6&Fd^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F [dv7$F($\"3/++++++!z%FSF`dvFcdv-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONT G6#%(DEFAULTG-Fdev6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6 #%,CONSTRAINEDG-%%VIEWG6$;$!$z%Fd_n$\"$>#Fd_n;Fgfv$\"$z%Fd_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme is (approx imately) " }{XPPEDIT 18 0 "[-4.04765, 0];" "6#7$,$-%&FloatG6$\"'lZS! \"&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the stabi lity region for the 7 stage, order 4 scheme with that of the order 5 s cheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p ||(1..7),po||(1..6)],view=[-4.79..2.19,-4.79..4.79],font=[HELVETICA,9] ,\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 623 575 575 {PLOTDATA 2 "65-%'CURVESG6 $7ew7$$\"\"!F)F(7$$\"3b+++!f<*>f!#G$\"3++++Fjzq:!#=7$$\"3p******4$3Q5& !#E$\"3(******pi#fTJF07$$\"3=+++(*f-r!)!#D$\"3D+++/%)Q7ZF07$$\"3E+++P, gNg!#C$\"3S+++&yzJG'F07$$\"33+++%p%GlG!#B$\"3S+++g$[R&yF07$$\"3s*****> )evd)*FF$\"3w*****\\jIYU*F07$$\"33+++y<$3i#!#A$\"3'******\\D0&*4\"!#<7 $$\"3I+++6Y&R[&FQ$\"33+++h8[c7FT7$$\"3C+++XY2B')FQ$\"3!******4V6LT\"FT 7$$\"3]*****>Ow'yuFQ$\"3++++iU&)p:FT7$$!3-+++'3\"pH5!#@$\"3!******\\/( )es\"FT7$$!3^+++odj+rFao$\"31+++jY4\")=FT7$$!3')*****\\^$>CA!#?$\"3<++ +[t2N?FT7$$!32+++\\v\\4aF\\p$\"3=+++upP(=#FT7$$!3++++O#)>O6!#>$\"36+++ QP]PBFT7$$!3'******pt83:#Fgp$\"35+++yV#\\[#FT7$$!3!)*****4:(*Qv$Fgp$\" 3/+++4w'*GEFT7$$!3<+++3,`ChFgp$\"3!******p0J'oFFT7$$!3Z*****>$4#*=%*Fg p$\"34+++RsN-HFT7$$!3#******\\$)Q=P\"F0$\"3A+++nS)z-$FT7$$!3)******4&R !y*=F0$\"3\")*****f&48VJFT7$$!3t******[l:-DF0$\"37+++/I(fC$FT7$$!3;+++ h(G-;$F0$\"36+++-)[eL$FT7$$!3'******pW;&[QF0$\"3!******pAQKT$FT7$$!35+ ++W,T\\XF0$\"3)******z^$HzMFT7$$!3S+++P]p^_F0$\"3<+++QZONNFT7$$!3W+++C 6.\\fF0$\"3;+++:@s#e$FT7$$!3a+++.s$FT7$$!3****** *p$3=S**F0$\"3!******H[4@t$FT7$$!3/+++rqOd5FT$\"3#)*****zh<&RPFT7$$!3+ +++mQ$*>6FT$\"3@+++\\+xUPFT7$$!3#******fjo<=\"FT$\"38+++2k/UPFT7$$!36+ ++V4#HC\"FT$\"34+++l#)[PPFT7$$!3)******fGTMI\"FT$\"3/+++2e?HPFT7$$!31+ ++hKQj8FT$\"3&)*****48$GMOFT7$$!3#*******[f*ol\"FT$\"3&*******z'=Z g$FT7$$!3%*******GS2:FT$\"3#******pq9*>$FT7$$!3/+++\")*\\YF#FT$\" 3>+++x.zVJFT7$$!31+++\"y4OM#FT$\"3))*****RU-#)3$FT7$$!3z*****\\0RLT#FT $\"3.+++:(>G.$FT7$$!33+++8@H$[#FT$\"3'******f76z(HFT7$$!3=+++eY\"Hb#FT $\"34+++vGFT$\"3@+++$)yl4FFT7$$!31+++Tmj\")GFT$\"3))******ey,cEFT7$$!3++++g(>= %HFT$\"3*******za$)=g#FT7$$!3y******)R-***HFT$\"3++++=-2ZDFT7$$!31+++; N'e0$FT$\"3))*****p:<9\\#FT7$$!3\")*****p>0(4JFT$\"32+++MoyMCFT7$$!3#* ******RjWhJFT$\"3=+++?H1xBFT7$$!3,+++]$=6@$FT$\"34+++lz9=BFT7$$!3++++r BweKFT$\"3)******\\?hzD#FT7$$!3#)*****\\hFWI$FT$\"3'******ptOk>#FT7$$! 3?+++j/<[LFT$\"39+++?A_L@FT7$$!3=+++\\S0!R$FT$\"3;+++U\"y\"p?FT7$$!38+ ++x![,V$FT$\"3z*****4qxL+#FT7$$!3#******pjG&oMFT$\"3)******RG2h$>FT7$$ !36+++'zx_]$FT$\"33+++6vOn=FT7$$!3*)******=F[SNFT$\"3-+++NZ<(z\"FT7$$! 3/+++!4MUd$FT$\"32+++VGcD\"FT7$$!3')******zM?,QFT$\"3'******fm$=76FT7$$!32+++\"38Z#QFT$\"3 4+++K!RC.\"FT7$$!3A+++*yVq%QFT$\"3z*****p8!)\\_*F07$$!3')*****\\)R5oQF T$\"3b*****\\l!yC()F07$$!3=+++/+z()QFT$\"3Z*****RXVZ#zF07$$!3*)*****zb #*f!RFT$\"3)******HE'eDrF07$$!38+++4EgARFT$\"3s*****Hl+yK'F07$$!3-+++/ i^PRFT$\"34+++mKnJbF07$$!3++++u!Q1&RFT$\"3F+++1\\JPZF07$$!3'******H/%) ='RFT$\"3w*****4X#pWRF07$$!3))******>C=rRFT$\"37+++j)fO:$F07$$!37+++UY ZyRFT$\"3!******R/')RO#F07$$!3++++Gar$)RFT$\"3'*******e&y`d\"F07$$!3%* *****\\mso)RFT$\"3m*****p%>.vyFgp7$$!3/+++*>Fz)RFTF(7$Fjjl$!3m*****p%> .vyFgp7$Fejl$!3'*******e&y`d\"F07$F`jl$!3!******R/')RO#F07$F[jl$!37+++ j)fO:$F07$Ffil$!3w*****4X#pWRF07$Fail$!3F+++1\\JPZF07$F\\il$!34+++mKnJ bF07$Fghl$!3s*****Hl+yK'F07$Fbhl$!3)******HE'eDrF07$F]hl$!3Z*****RXVZ# zF07$Fhgl$!3b*****\\l!yC()F07$Fcgl$!3z*****p8!)\\_*F07$F^gl$!34+++K!RC .\"FT7$Fifl$!3'******fm$=76FT7$Fdfl$!3'******zS&f\">\"FT7$F_fl$!3))*** **z)R_q7FT7$Fjel$!3'******pB7)[8FT7$Feel$!3/+++QOIE9FT7$F`el$!3******* *)e\\G]\"FT7$F[el$!3/+++!\\:$y:FT7$Ffdl$!3/+++w]e_;FT7$Fadl$!32+++VGcD FT 7$F]cl$!3z*****4qxL+#FT7$Fhbl$!3;+++U\"y\"p?FT7$Fcbl$!39+++?A_L@FT7$F^ bl$!3'******ptOk>#FT7$Fial$!3)******\\?hzD#FT7$Fdal$!34+++lz9=BFT7$F_a l$!3=+++?H1xBFT7$Fj`l$!32+++MoyMCFT7$Fe`l$!3))*****p:<9\\#FT7$F``l$!3+ +++=-2ZDFT7$F[`l$!3*******za$)=g#FT7$Ff_l$!3))******ey,cEFT7$Fa_l$!3@+ ++$)yl4FFT7$F\\_l$!35+++P2,jFFT7$Fg^l$!3$******4\")*H;GFT7$Fb^l$!3)*** ****)[^(pGFT7$F]^l$!34+++vG.$FT7$F^]l$!3))*****RU-#)3$FT7$Fi\\l$!3>+++x.zVJFT7$Fd\\l$!3#)***** *>q9*>$FT7$F_\\l$!3#)*****Ri:PD$FT7$Fj[l$!38+++G_*oI$FT7$Fe[l$!3A+++\\ !=\"eLFT7$F`[l$!3#******pMOFT7$F]y$!3)******fb8-m$FT7$Fhx$!3')*****f:\\Fo$ FT7$Fcx$!39+++rEy,PFT7$F^x$!3&)*****48$G.s$FT7$Ffu$!3<+++$=jQq$FT 7$Fau$!3#)*****fJ%Q#o$FT7$F\\u$!3;+++XYVbOFT7$Fgt$!3()*****RD]Ci$FT7$F bt$!3;+++:@s#e$FT7$F]t$!3<+++QZONNFT7$Fhs$!3)******z^$HzMFT7$Fcs$!3!** ****pAQKT$FT7$F^s$!36+++-)[eL$FT7$Fir$!37+++/I(fC$FT7$Fdr$!3\")*****f& 48VJFT7$F_r$!3A+++nS)z-$FT7$Fjq$!34+++RsN-HFT7$Feq$!3!******p0J'oFFT7$ F`q$!3/+++4w'*GEFT7$F[q$!35+++yV#\\[#FT7$Fep$!36+++QP]PBFT7$F`p$!3=+++ upP(=#FT7$Fjo$!3<+++[t2N?FT7$Feo$!31+++jY4\")=FT7$F_o$!3!******\\/()es \"FT7$$\"39+++kjnyuFQ$!3++++iU&)p:FT7$Fen$!3!******4V6LT\"FT7$FV$!33++ +h8[c7FT7$FO$!3'******\\D0&*4\"FT7$FJ$!3w*****\\jIYU*F07$FD$!3S+++g$[R &yF07$$\"3^*****f8+c.'F@$!3S+++&yzJG'F07$$\"3s******pd-r!)F:$!3D+++/%) Q7ZF07$$\"3G+++9$3Q5&F4$!3(******pi#fTJF07$$\"3i*****>a4(>fF-$!3++++Fj zq:F0F'-%&COLORG6&%$RGBG$\"\"#!\"\"Fh^n$\"\"&Fj^n-%)POLYGONSG6fw7%F'7$ $\"+!f<*>fF\\p$\"+Fjzq:!#57$$!\"#F)F(7%Fa_n7$$\"+5$3Q5&F0$\"+FEfTJFf_n Fg_n7%F[`n7$$\"+(*f-r!)FT$\"+/%)Q7ZFf_nFg_n7%Fa`n7$$\"+P,gNg!#;$\"+&yz JG'Ff_nFg_n7%Fg`n7$$\"+%p%GlG!#:$\"+g$[R&yFf_nFg_n7%F^an7$$\"+#)evd)*F aan$\"+N1jC%*Ff_nFg_n7%Fean7$$\"+y<$3i#!#9$\"+b_]*4\"!\"*Fg_n7%F[bn7$$ \"+6Y&R[&F^bn$\"+h8[c7FabnFg_n7%Fcbn7$$\"+XY2B')F^bn$\"+J9J89FabnFg_n7 %Fibn7$$\"+ijnyuF^bn$\"+iU&)p:FabnFg_n7%F_cn7$$!+'3\"pH5!#8$\"+Xq)es\" FabnFg_n7%Fecn7$$!+odj+rFhcn$\"+jY4\")=FabnFg_n7%F\\dn7$$!+:N>CA!#7$\" +[t2N?FabnFg_n7%Fbdn7$$!+\\v\\4aFedn$\"+upP(=#FabnFg_n7%Fidn7$$!+O#)>O 6!#6$\"+QP]PBFabnFg_n7%F_en7$$!+PP\"3:#Fben$\"+yV#\\[#FabnFg_n7%Ffen7$ $!+^r*Qv$Fben$\"+4w'*GEFabnFg_n7%F\\fn7$$!+3,`ChFben$\"+d5joFFabnFg_n7 %Fbfn7$$!+K4#*=%*Fben$\"+RsN-HFabnFg_n7%Fhfn7$$!+N)Q=P\"Ff_n$\"+nS)z-$ FabnFg_n7%F^gn7$$!+^R!y*=Ff_n$\"+c48VJFabnFg_n7%Fdgn7$$!+\\l:-DFf_n$\" +/I(fC$FabnFg_n7%Fjgn7$$!+h(G-;$Ff_n$\"+-)[eL$FabnFg_n7%F`hn7$$!+Zk^[Q Ff_n$\"+F#QKT$FabnFg_n7%Ffhn7$$!+W,T\\XFf_n$\"+=NHzMFabnFg_n7%F\\in7$$ !+P]p^_Ff_n$\"+QZONNFabnFg_n7%Fbin7$$!+C6.\\fFf_n$\"+:@s#e$FabnFg_n7%F hin7$$!+.s$FabnFg_n7%Ff[o7$$!+P3=S**Ff _n$\"+$[4@t$FabnFg_n7%F\\\\o7$$!+rqOd5Fabn$\"+=w^RPFabnFg_n7%Fb\\o7$$! +mQ$*>6Fabn$\"+\\+xUPFabnFg_n7%Fh\\o7$$!+O'o<=\"Fabn$\"+2k/UPFabnFg_n7 %F^]o7$$!+V4#HC\"Fabn$\"+l#)[PPFabnFg_n7%Fd]o7$$!+'GTMI\"Fabn$\"+2e?HP FabnFg_n7%Fj]o7$$!+hKQj8Fabn$\"+JJGMOFabnFg_n7%Fh_o 7$$!+\\f*ol\"Fabn$\"+!o=Zg$FabnFg_n7%F^`o7$$!+HS2:F abn$\"+xT!pS$FabnFg_n7%F\\bo7$$!+cmH8?Fabn$\"+\\!=\"eLFabnFg_n7%Fbbo7$ $!+q_?w?Fabn$\"+G_*oI$FabnFg_n7%Fhbo7$$!+p0tS@Fabn$\"+Ccr`KFabnFg_n7%F ^co7$$!+30#p?#Fabn$\"+?q9*>$FabnFg_n7%Fdco7$$!+\")*\\YF#Fabn$\"+x.zVJF abnFg_n7%Fjco7$$!+\"y4OM#Fabn$\"+CC?)3$FabnFg_n7%F`do7$$!+b!RLT#Fabn$ \"+:(>G.$FabnFg_n7%Ffdo7$$!+8@H$[#Fabn$\"+E6\"z(HFabnFg_n7%F\\eo7$$!+e Y\"Hb#Fabn$\"+vGFabn$\"+$)yl4FFabnFg_n7%Fjfo7$$!+Tmj\")GFabn$\"+fy, cEFabnFg_n7%F`go7$$!+g(>=%HFabn$\"+[N)=g#FabnFg_n7%Ffgo7$$!+*R-***HFab n$\"+=-2ZDFabnFg_n7%F\\ho7$$!+;N'e0$Fabn$\"+drT\"\\#FabnFg_n7%Fbho7$$! +(>0(4JFabn$\"+MoyMCFabnFg_n7%Fhho7$$!+SjWhJFabn$\"+?H1xBFabnFg_n7%F^i o7$$!+]$=6@$Fabn$\"+lz9=BFabnFg_n7%Fdio7$$!+rBweKFabn$\"+07'zD#FabnFg_ n7%Fjio7$$!+:wU/LFabn$\"+PnV'>#FabnFg_n7%F`jo7$$!+j/<[LFabn$\"+?A_L@Fa bnFg_n7%Ffjo7$$!+\\S0!R$Fabn$\"+U\"y\"p?FabnFg_n7%F\\[p7$$!+x![,V$Fabn $\"+,xP.?FabnFg_n7%Fb[p7$$!+P'G&oMFabn$\"+%G2h$>FabnFg_n7%Fh[p7$$!+'zx _]$Fabn$\"+6vOn=FabnFg_n7%F^\\p7$$!+>F[SNFabn$\"+NZ<(z\"FabnFg_n7%Fd\\ p7$$!+!4MUd$Fabn$\"+VGcD\"FabnFg_n7%Fd_p7$$!+![.7!QFabn$\"+mO=76FabnFg_n 7%Fj_p7$$!+\"38Z#QFabn$\"+K!RC.\"FabnFg_n7%F``p7$$!+*yVq%QFabn$\"+P,) \\_*Ff_nFg_n7%Ff`p7$$!+&)R5oQFabn$\"+b1yC()Ff_nFg_n7%F\\ap7$$!+/+z()QF abn$\"+aMuCzFf_nFg_n7%Fbap7$$!+eD*f!RFabn$\"+jieDrFf_nFg_n7%Fhap7$$!+4 EgARFabn$\"+`1!yK'Ff_nFg_n7%F^bp7$$!+/i^PRFabn$\"+mKnJbFf_nFg_n7%Fdbp7 $$!+u!Q1&RFabn$\"+1\\JPZFf_nFg_n7%Fjbp7$$!+VS)='RFabn$\"+^CpWRFf_nFg_n 7%F`cp7$$!+?C=rRFabn$\"+j)fO:$Ff_nFg_n7%Ffcp7$$!+UYZyRFabn$\"+Wg)RO#Ff _nFg_n7%F\\dp7$$!+Gar$)RFabn$\"+f&y`d\"Ff_nFg_n7%Fbdp7$$!+lE(o)RFabn$ \"+Z>.vyFbenFg_n7%Fhdp7$$!+*>Fz)RFabnF(Fg_n7%F^ep7$Fidp$!+Z>.vyFbenFg_ n7%Fbep7$Fcdp$!+f&y`d\"Ff_nFg_n7%Ffep7$F]dp$!+Wg)RO#Ff_nFg_n7%Fjep7$Fg cp$!+j)fO:$Ff_nFg_n7%F^fp7$Facp$!+^CpWRFf_nFg_n7%Fbfp7$F[cp$!+1\\JPZFf _nFg_n7%Fffp7$Febp$!+mKnJbFf_nFg_n7%Fjfp7$F_bp$!+`1!yK'Ff_nFg_n7%F^gp7 $Fiap$!+jieDrFf_nFg_n7%Fbgp7$Fcap$!+aMuCzFf_nFg_n7%Ffgp7$F]ap$!+b1yC() Ff_nFg_n7%Fjgp7$Fg`p$!+P,)\\_*Ff_nFg_n7%F^hp7$Fa`p$!+K!RC.\"FabnFg_n7% Fbhp7$F[`p$!+mO=76FabnFg_n7%Ffhp7$Fe_p$!+3af\">\"FabnFg_n7%Fjhp7$F__p$ !+))R_q7FabnFg_n7%F^ip7$Fi^p$!+PA\")[8FabnFg_n7%Fbip7$Fc^p$!+QOIE9Fabn Fg_n7%Ffip7$F]^p$!+*e\\G]\"FabnFg_n7%Fjip7$Fg]p$!+!\\:$y:FabnFg_n7%F^j p7$Fa]p$!+w]e_;FabnFg_n7%Fbjp7$F[]p$!+VGcDFabnFg_n 7%Fb[q7$Fc[p$!+,xP.?FabnFg_n7%Ff[q7$F][p$!+U\"y\"p?FabnFg_n7%Fj[q7$Fgj o$!+?A_L@FabnFg_n7%F^\\q7$Fajo$!+PnV'>#FabnFg_n7%Fb\\q7$F[jo$!+07'zD#F abnFg_n7%Ff\\q7$Feio$!+lz9=BFabnFg_n7%Fj\\q7$F_io$!+?H1xBFabnFg_n7%F^] q7$Fiho$!+MoyMCFabnFg_n7%Fb]q7$Fcho$!+drT\"\\#FabnFg_n7%Ff]q7$F]ho$!+= -2ZDFabnFg_n7%Fj]q7$Fggo$!+[N)=g#FabnFg_n7%F^^q7$Fago$!+fy,cEFabnFg_n7 %Fb^q7$F[go$!+$)yl4FFabnFg_n7%Ff^q7$Fefo$!+P2,jFFabnFg_n7%Fj^q7$F_fo$! +6)*H;GFabnFg_n7%F^_q7$Fieo$!+*[^(pGFabnFg_n7%Fb_q7$Fceo$!+vG.$FabnFg_n7%F^`q7$F ado$!+CC?)3$FabnFg_n7%Fb`q7$F[do$!+x.zVJFabnFg_n7%Ff`q7$Feco$!+?q9*>$F abnFg_n7%Fj`q7$F_co$!+Ccr`KFabnFg_n7%F^aq7$Fibo$!+G_*oI$FabnFg_n7%Fbaq 7$Fcbo$!+\\!=\"eLFabnFg_n7%Ffaq7$F]bo$!+xT!pS$FabnFg_n7%Fjaq7$Fgao$!+D X)GX$FabnFg_n7%F^bq7$Faao$!+%\\(z&\\$FabnFg_n7%Fbbq7$F[ao$!+c2ZNNFabnF g_n7%Ffbq7$Fe`o$!+w#)zrNFabnFg_n7%Fjbq7$F_`o$!+!o=Zg$FabnFg_n7%F^cq7$F i_o$!+&[(>MOFabnFg_n7%Fbcq7$Fc_o$!+cN@gOFabnFg_n7%Ffcq7$F]_o$!+c\"\\Fo $FabnFg_n7%Fjcq7$Fg^o$!+rEy,PFabnFg_n7%F^dq7$Fa^o$!+JJG.s$FabnFg_n7%F^fq7$Fa[o $!+$=jQq$FabnFg_n7%Fbfq7$F[[o$!+;VQ#o$FabnFg_n7%Fffq7$Fejn$!+XYVbOFabn Fg_n7%Fjfq7$F_jn$!+a-XAOFabnFg_n7%F^gq7$Fiin$!+:@s#e$FabnFg_n7%Fbgq7$F cin$!+QZONNFabnFg_n7%Ffgq7$F]in$!+=NHzMFabnFg_n7%Fjgq7$Fghn$!+F#QKT$Fa bnFg_n7%F^hq7$Fahn$!+-)[eL$FabnFg_n7%Fbhq7$F[hn$!+/I(fC$FabnFg_n7%Ffhq 7$Fegn$!+c48VJFabnFg_n7%Fjhq7$F_gn$!+nS)z-$FabnFg_n7%F^iq7$Fifn$!+RsN- HFabnFg_n7%Fbiq7$Fcfn$!+d5joFFabnFg_n7%Ffiq7$F]fn$!+4w'*GEFabnFg_n7%Fj iq7$Fgen$!+yV#\\[#FabnFg_n7%F^jq7$F`en$!+QP]PBFabnFg_n7%Fbjq7$Fjdn$!+u pP(=#FabnFg_n7%Ffjq7$Fcdn$!+[t2N?FabnFg_n7%Fjjq7$F]dn$!+jY4\")=FabnFg_ n7%F^[r7$Ffcn$!+Xq)es\"FabnFg_n7%Fb[r7$$\"+kjnyuF^bn$!+iU&)p:FabnFg_n7 %Ff[r7$Fjbn$!+J9J89FabnFg_n7%F\\\\r7$Fdbn$!+h8[c7FabnFg_n7%F`\\r7$F\\b n$!+b_]*4\"FabnFg_n7%Fd\\r7$Ffan$!+N1jC%*Ff_nFg_n7%Fh\\r7$F_an$!+g$[R& yFf_nFg_n7%F\\]r7$$\"+O,gNgFj`n$!+&yzJG'Ff_nFg_n7%F`]r7$$\"+qd-r!)FT$! +/%)Q7ZFf_nFg_n7%Ff]r7$$\"+9$3Q5&F0$!+FEfTJFf_nFg_n7%F\\^r7$$\"+U&4(>f F\\p$!+Fjzq:Ff_nFg_n7%Fb^rF'Fg_n-Fe^n6&Fg^n$\"\"%Fj^nFj^r$\"\"\"F)-%&S TYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"33+++.>]*[\"FT$\"3#******>&)\\QI%FT7 $$\"3!******ff%R\"[\"FT$\"3()*****HP'f7VFT7$$\"3'******Ru+EZ\"FT$\"3C+ ++&o\\1K%FT7$$\"3/+++&RjJY\"FT$\"3D+++(RozK%FT7$$\"3/+++yi7`9FT$\"3e** ***\\V6XL%FT7$$\"3++++@T`U9FT$\"3F+++kxBSVFT7$$\"3++++9IVJ9FT$\"3i**** **>h5XVFT7$$\"3#******f#3()>9FT$\"3M+++dZ2\\VFT7$$\"3$******Ro(*yS\"FT $\"3e*****p?,@N%FT7$$\"3-+++Glc&R\"FT$\"3=+++^?9aVFT7$$\"33+++xP$HQ\"F T$\"3o*****4r_^N%FT7$$\"3%******R6g+P\"FT$\"37+++os3bVFT7$$\"3!******f U6qN\"FT$\"3/+++A$)*QN%FT7$$\"33+++W*fQM\"FT$\"3y*****p\"p`^VFT7$$\"3, +++CcoI8FT$\"3-+++mD&zM%FT7$$\"3#******4$yd<8FT$\"3k******RM4VVFT7$$\" 3$******\\QPYI\"FT$\"3)******\\z1pL%FT7$$\"3)*******e!z>H\"FT$\"3e**** *zxR$HVFT7$$\"3++++-Ytz7FT$\"3J++++3M?VFT7$$\"3#******pMc!o7FT$\"3V+++ g<')4VFT7$$\"3)*******H97d7FT$\"31+++d:'yH%FT7$$\"3!******H]OrC\"FT$\" 3!******RZ6VG%FT7$$\"32+++7EMQ7FT$\"3e*****HF.#pUFT7$$\"3*******fR>5B \"FT$\"3e*****z/hDD%FT7$$\"3,+++wp[D7FT$\"3v*****p+eWB%FT7$$\"3******* >e-@A\"FT$\"3V+++%)*Q]@%FT7$$\"3.+++`sC@7FT$\"3k*****p=[X>%FT7$$\"3/++ +QlHB7FT$\"3/+++:%eL<%FT7$$\"3)*******4)p&G7FT$\"3)******\\N#*>:%FT7$$ \"3.+++(*)esB\"FT$\"3j*****4&*>68%FT7$$\"3'******>K]$\\7FT$\"3&******* QR^6TFT7$$\"3-+++\\:dk7FT$\"3o*****pBdR4%FT7$$\"3/+++e\\R#G\"FT$\"3G++ +mC7zSFT7$$\"3\"******4_5@I\"FT$\"3))*****\\@ou1%FT7$$\"3$*******z2%HK \"FT$\"3S+++yK>fSFT7$$\"3++++T::W8FT$\"3y*****RkdU0%FT7$$\"3/+++&*R7l8 FT$\"3u*****4JYC0%FT7$$\"3++++P:Q&Q\"FT$\"3=+++%oUM0%FT7$$\"3,+++BPe/9 FT$\"3i*****\\F*)o0%FT7$$\"3)******R-/DU\"FT$\"3#)******>%GC1%FT7$$\"3 #*******RO+R9FT$\"3w*****R5D(pSFT7$$\"3%******>u2SX\"FT$\"3o*****\\=x% ySFT7$$\"3#******fS'[n9FT$\"3#*******yzT)3%FT7$$\"3)*******4.Wz9FT$\"3 A+++eZJ*4%FT7$$\"3'******HW!*)*[\"FT$\"3w*****RUl46%FT7$$\"3%******H)3 ())\\\"FT$\"3R+++wX>BTFT7$$\"3$******H$QU1:FT$\"33+++T'\\e8%FT7$$\"3.+ ++Wif7:FT$\"37+++Wtz[TFT7$$\"3!******fcPu^\"FT$\"3#*******G1#>;%FT7$$ \"30+++g#)*4_\"FT$\"3++++@g6vTFT7$$\"3!******>&)GL_\"FT$\"3G+++<9H)=%F T7$$\"3\"******zLzW_\"FT$\"3:+++=UO,UFT7$$\"3$******4&))\\C:FT$\"3B+++ %yfU@%FT7$$\"3.+++$\\NM_\"FT$\"3j*****H85pA%FT7$$\"3'******4GO8_\"FT$ \"3m******[GDRUFT7$$\"3(******z8_WH%FTFe_rFd^n-F^_n 6jn7%7$$\"+.>]*[\"Fabn$\"+_)\\QI%Fabn7$$\"+'f%R\"[\"Fabn$\"+tjf7VFabn7 $$\"$P\"Fi_n$\"$>%Fi_n7%Fibs7$$\"+W2gs9Fabn$\"+&o\\1K%FabnF^cs7%Fdcs7$ $\"+&RjJY\"Fabn$\"+(RozK%FabnF^cs7%Fjcs7$$\"+yi7`9Fabn$\"+N9^MVFabnF^c s7%F`ds7$$\"+@T`U9Fabn$\"+kxBSVFabnF^cs7%Ffds7$$\"+9IVJ9Fabn$\"+?h5XVF abnF^cs7%F\\es7$$\"+E3()>9Fabn$\"+dZ2\\VFabnF^cs7%Fbes7$$\"+%o(*yS\"Fa bn$\"+275_VFabnF^cs7%Fhes7$$\"+Glc&R\"Fabn$\"+^?9aVFabnF^cs7%F^fs7$$\" +xP$HQ\"Fabn$\"+6F:bVFabnF^cs7%Fdfs7$$\"+9,1q8Fabn$\"+os3bVFabnF^cs7%F jfs7$$\"+E9,d8Fabn$\"+A$)*QN%FabnF^cs7%F`gs7$$\"+W*fQM\"Fabn$\"+H\"Fabn$\"+y(R$HVFabnF^cs7%F^is7$$\"+-Ytz7Fabn$\"++3M?VFabnF^cs7%Fd is7$$\"+Zj0o7Fabn$\"+g<')4VFabnF^cs7%Fjis7$$\"+I97d7Fabn$\"+d:'yH%Fabn F^cs7%F`js7$$\"+.l8Z7Fabn$\"+u9J%G%FabnF^cs7%Ffjs7$$\"+7EMQ7Fabn$\"+tK ?pUFabnF^cs7%F\\[t7$$\"+'R>5B\"Fabn$\"+[5c_UFabnF^cs7%Fb[t7$$\"+wp[D7F abn$\"+2!eWB%FabnF^cs7%Fh[t7$$\"+#e-@A\"Fabn$\"+%)*Q]@%FabnF^cs7%F^\\t 7$$\"+`sC@7Fabn$\"+(=[X>%FabnF^cs7%Fd\\t7$$\"+QlHB7Fabn$\"+:%eL<%FabnF ^cs7%Fj\\t7$$\"+5)p&G7Fabn$\"+bB*>:%FabnF^cs7%F`]t7$$\"+(*)esB\"Fabn$ \"+^*>68%FabnF^cs7%Ff]t7$$\"+A.N\\7Fabn$\"+RR^6TFabnF^cs7%F\\^t7$$\"+ \\:dk7Fabn$\"+Ps&R4%FabnF^cs7%Fb^t7$$\"+e\\R#G\"Fabn$\"+mC7zSFabnF^cs7 %Fh^t7$$\"+@06-8Fabn$\"+:#ou1%FabnF^cs7%F^_t7$$\"+!ySHK\"Fabn$\"+yK>fS FabnF^cs7%Fd_t7$$\"+T::W8Fabn$\"+WwDaSFabnF^cs7%Fj_t7$$\"+&*R7l8Fabn$ \"+6jW_SFabnF^cs7%F``t7$$\"+P:Q&Q\"Fabn$\"+%oUM0%FabnF^cs7%Ff`t7$$\"+B Pe/9Fabn$\"+v#*)o0%FabnF^cs7%F\\at7$$\"+CS]A9Fabn$\"+?%GC1%FabnF^cs7%F bat7$$\"+SO+R9Fabn$\"+/^spSFabnF^cs7%Fhat7$$\"+Ux+a9Fabn$\"+&=x%ySFabn F^cs7%F^bt7$$\"+1k[n9Fabn$\"+zzT)3%FabnF^cs7%Fdbt7$$\"+5.Wz9Fabn$\"+eZ J*4%FabnF^cs7%Fjbt7$$\"+V/*)*[\"Fabn$\"+Ca'46%FabnF^cs7%F`ct7$$\"+$)3( ))\\\"Fabn$\"+wX>BTFabnF^cs7%Ffct7$$\"+LQU1:Fabn$\"+T'\\e8%FabnF^cs7%F \\dt7$$\"+Wif7:Fabn$\"+Wtz[TFabnF^cs7%Fbdt7$$\"+mvV<:Fabn$\"+H1#>;%Fab nF^cs7%Fhdt7$$\"+g#)*4_\"Fabn$\"+@g6vTFabnF^cs7%F^et7$$\"+_)GL_\"Fabn$ \"+<9H)=%FabnF^cs7%Fdet7$$\"+Q$zW_\"Fabn$\"+=UO,UFabnF^cs7%Fjet7$$\"+^ ))\\C:Fabn$\"+%yfU@%FabnF^cs7%F`ft7$$\"+$\\NM_\"Fabn$\"+L,\"pA%FabnF^c s7%Ffft7$$\"+\"GO8_\"Fabn$\"+\\GDRUFabnF^cs7%F\\gt7$$\"+yrC=:Fabn$\"+k ,B^UFabnF^cs7%Fbgt7$$\"+yJ@9:Fabn$\"+!=)yiUFabnF^cs7%Fhgt7$$\"+J%y#4:F abn$\"+.i(QF%FabnF^cs7%F^ht7$$\"+%R'[.:Fabn$\"+mhW%G%FabnF^cs7%Fdht7$$ \"+4+)o\\\"Fabn$\"+K@X%H%FabnF^cs7%FjhtFdbsF^csFh^rF^_r-F$6$7in7$Ff_r$ !3#******>&)\\QI%FT7$F]bs$!3)******>8_WH%FT7$Fhas$!3S+++mhW%G%FT7$Fcas $!3Q+++.i(QF%FT7$F^as$!3E+++!=)yiUFT7$Fi`s$!3))*****R;I7D%FT7$Fd`s$!3m ******[GDRUFT7$F_`s$!3j*****H85pA%FT7$Fj_s$!3B+++%yfU@%FT7$Fe_s$!3:+++ =UO,UFT7$F`_s$!3G+++<9H)=%FT7$F[_s$!3++++@g6vTFT7$Ff^s$!3#*******G1#>; %FT7$Fa^s$!37+++Wtz[TFT7$F\\^s$!33+++T'\\e8%FT7$Fg]s$!3R+++wX>BTFT7$Fb ]s$!3w*****RUl46%FT7$F]]s$!3A+++eZJ*4%FT7$Fh\\s$!3#*******yzT)3%FT7$Fc \\s$!3o*****\\=x%ySFT7$F^\\s$!3w*****R5D(pSFT7$Fi[s$!3#)******>%GC1%FT 7$Fd[s$!3i*****\\F*)o0%FT7$F_[s$!3=+++%oUM0%FT7$Fjjr$!3u*****4JYC0%FT7 $Fejr$!3y*****RkdU0%FT7$F`jr$!3S+++yK>fSFT7$F[jr$!3))*****\\@ou1%FT7$F fir$!3G+++mC7zSFT7$Fair$!3o*****pBdR4%FT7$F\\ir$!3&*******QR^6TFT7$Fgh r$!3j*****4&*>68%FT7$Fbhr$!3)******\\N#*>:%FT7$F]hr$!3/+++:%eL<%FT7$Fh gr$!3k*****p=[X>%FT7$Fcgr$!3V+++%)*Q]@%FT7$F^gr$!3v*****p+eWB%FT7$Fifr $!3e*****z/hDD%FT7$Fdfr$!3e*****HF.#pUFT7$F_fr$!3!******RZ6VG%FT7$Fjer $!31+++d:'yH%FT7$Feer$!3V+++g<')4VFT7$F`er$!3J++++3M?VFT7$F[er$!3e**** *zxR$HVFT7$Ffdr$!3)******\\z1pL%FT7$Fadr$!3k******RM4VVFT7$F\\dr$!3-++ +mD&zM%FT7$Fgcr$!3y*****p\"p`^VFT7$Fbcr$!3/+++A$)*QN%FT7$F]cr$!37+++os 3bVFT7$Fhbr$!3o*****4r_^N%FT7$Fcbr$!3=+++^?9aVFT7$F^br$!3e*****p?,@N%F T7$Fiar$!3M+++dZ2\\VFT7$Fdar$!3i******>h5XVFT7$F_ar$!3F+++kxBSVFT7$Fj` r$!3e*****\\V6XL%FT7$Fe`r$!3D+++(RozK%FT7$F``r$!3C+++&o\\1K%FT7$F[`r$! 3()*****HP'f7VFTFcitFd^n-F^_n6jn7%7$Febs$!+_)\\QI%Fabn7$F[it$!+K@X%H%F abn7$F_cs$!$>%Fi_n7%F]eu7$Feht$!+mhW%G%FabnF`eu7%Fdeu7$F_ht$!+.i(QF%Fa bnF`eu7%Fheu7$Figt$!+!=)yiUFabnF`eu7%F\\fu7$Fcgt$!+k,B^UFabnF`eu7%F`fu 7$F]gt$!+\\GDRUFabnF`eu7%Fdfu7$Fgft$!+L,\"pA%FabnF`eu7%Fhfu7$Faft$!+%y fU@%FabnF`eu7%F\\gu7$F[ft$!+=UO,UFabnF`eu7%F`gu7$Feet$!+<9H)=%FabnF`eu 7%Fdgu7$F_et$!+@g6vTFabnF`eu7%Fhgu7$Fidt$!+H1#>;%FabnF`eu7%F\\hu7$Fcdt $!+Wtz[TFabnF`eu7%F`hu7$F]dt$!+T'\\e8%FabnF`eu7%Fdhu7$Fgct$!+wX>BTFabn F`eu7%Fhhu7$Fact$!+Ca'46%FabnF`eu7%F\\iu7$F[ct$!+eZJ*4%FabnF`eu7%F`iu7 $Febt$!+zzT)3%FabnF`eu7%Fdiu7$F_bt$!+&=x%ySFabnF`eu7%Fhiu7$Fiat$!+/^sp SFabnF`eu7%F\\ju7$Fcat$!+?%GC1%FabnF`eu7%F`ju7$F]at$!+v#*)o0%FabnF`eu7 %Fdju7$Fg`t$!+%oUM0%FabnF`eu7%Fhju7$Fa`t$!+6jW_SFabnF`eu7%F\\[v7$F[`t$ !+WwDaSFabnF`eu7%F`[v7$Fe_t$!+yK>fSFabnF`eu7%Fd[v7$F__t$!+:#ou1%FabnF` eu7%Fh[v7$Fi^t$!+mC7zSFabnF`eu7%F\\\\v7$Fc^t$!+Ps&R4%FabnF`eu7%F`\\v7$ F]^t$!+RR^6TFabnF`eu7%Fd\\v7$Fg]t$!+^*>68%FabnF`eu7%Fh\\v7$Fa]t$!+bB*> :%FabnF`eu7%F\\]v7$F[]t$!+:%eL<%FabnF`eu7%F`]v7$Fe\\t$!+(=[X>%FabnF`eu 7%Fd]v7$F_\\t$!+%)*Q]@%FabnF`eu7%Fh]v7$Fi[t$!+2!eWB%FabnF`eu7%F\\^v7$F c[t$!+[5c_UFabnF`eu7%F`^v7$F][t$!+tK?pUFabnF`eu7%Fd^v7$Fgjs$!+u9J%G%Fa bnF`eu7%Fh^v7$Fajs$!+d:'yH%FabnF`eu7%F\\_v7$F[js$!+g<')4VFabnF`eu7%F`_ v7$Feis$!++3M?VFabnF`eu7%Fd_v7$F_is$!+y(R$HVFabnF`eu7%Fh_v7$Fihs$!+&z1 pL%FabnF`eu7%F\\`v7$Fchs$!+SM4VVFabnF`eu7%F``v7$F]hs$!+mD&zM%FabnF`eu7 %Fd`v7$Fggs$!+%R(! #F$\"31+++[jzq:F07$$\"37+++(o2lv%F:$\"3()*****HA$fTJF07$$\"3\\******[o -VaF@$\"3#******f*>R7ZF07$$\"37+++[n&R0$FF$\"3')*****Rh*=$G'F07$$\"3%* ******G*zx9\"FQ$\"3A+++.(eR&yF07$$\"3I+++')e#*)H$FQ$\"3#)*****z#4gC%*F 07$$\"3b+++B!emr(FQ$\"35+++cf[*4\"FT7$$\"3,+++N\\0/:Fao$\"33+++I!=kD\" FT7$$\"3*)*****4F]>T#Fao$\"3/+++@A:89FT7$$\"3!)*****>Lf(4HFao$\"35+++' f9&p:FT7$$\"33+++JC%\\Y\"Fao$\"34+++cUCD%e`*>F0$\"32+++f%Gx7$FT7$$!3#******HH(e2EF0$\"3%****** RLszA$FT7$$!3u*****f\\@'pKF0$\"3;+++_))\\:LFT7$$!35+++`tQfRF0$\"33+++q A)3R$FT7$$!3)******RV91m%F0$\"39+++#*3FbMFT7$$!3O+++W)yHO&F0$\"3\"**** ***fG'*4NFT7$$!3e*****f&*>11'F0$\"3!******f\\mhb$FT7$$!3=+++$GZ0v'F0$ \"3!******\\[6\\f$FT7$$!3$******p:f9V(F0$\"3(******4mRqi$FT7$$!3S+++>* ))H5)F0$\"3!******\\$oA`OFT7$$!3u*****\\,uOFT7$$!3F++ +*\\I'=%*F0$\"3-+++6W#)*o$FT7$$!3\"*******)ecj+\"FT$\"3<+++Uk+,PFT7$$! 3'******\\Cc+2\"FT$\"39+++C6$yq$FT7$$!3$*******z@,L6FT$\"36+++kg^5PFT7 $$!33+++@VF&>\"FT$\"3(******f+M#4PFT7$$!3++++PI*oD\"FT$\"3@+++;57/PFT7 $$!3(******Hq?zJ\"FT$\"3;+++\"*GG&p$FT7$$!3%******>$RTy8FT$\"3y*****zM +Go$FT7$$!3++++^jVQ9FT$\"3#******4VLnm$FT7$$!3!*******G>1)\\\"FT$\"33+ ++#fDrk$FT7$$!35+++U)ytb\"FT$\"3()*****p)y+COFT7$$!3/+++\\Q\\;;FT$\"3y ******3QS(f$FT7$$!3++++Xq`v;FT$\"3!)*****4SNtc$FT7$$!33+++BmmMk9>FT$\"3%)*****44`LT$FT7$$!31+ ++aOLw>FT$\"3#******Rh&*pO$FT7$$!3;+++/]KR?FT$\"3\")*****z3izJ$FT7$$!3 )******\\kVQ5#FT$\"3?+++!RgmE$FT7$$!3?+++UU-q@FT$\"3%)*****zL-O@$FT7$$ !3)******4lcyB#FT$\"3#******\\Xq$fJFT7$$!34+++rA:2BFT$\"3!******f4bX5$ FT7$$!3/+++gsaxBFT$\"3))******Rhn\\IFT7$$!31+++.ua[CFT$\"3')*****RL:^* HFT7$$!3********e&*f>DFT$\"3*******>]#3THFT7$$!3++++vQ;!f#FT$\"3')**** **f=i()GFT7$$!39++++hwfEFT$\"33+++/rkMGFT7$$!3#******4:B!GFFT$\"3;+++X q)>y#FT7$$!3')*******[YYz#FT$\"39+++,uUHFFT7$$!3#******\\:K%fGFT$\"3%) *****R+Unn#FT7$$!36+++!QXA#HFT$\"3\"******zo7Pi#FT7$$!37+++?[+$)HFT$\" 3))*****f3S,d#FT7$$!3$)*****p')o;/$FT$\"3/+++#=[e^#FT7$$!3=+++<[A)4$FT $\"3;+++,^ogCFT7$$!3++++93o_JFT$\"3\"******4*=_/CFT7$$!3')*****H9g]?$F T$\"3%)*****RU]sM#FT7$$!3++++XtRbKFT$\"3?+++86y)G#FT7$$!3z*****4sNPI$F T$\"3>+++I3/HAFT7$$!3-+++(zD,N$FT$\"3')*****>@rz;#FT7$$!31+++kWi%R$FT$ \"3(*******)[Fb5#FT7$$!3*)*****p\\%HPMFT$\"3;+++XynT?FT7$$!31+++WU?yMF T$\"3'******RM.k(>FT7$$!3*)******\\sUFT7$$!3!)*****R iT]b$FT$\"3,+++z:dT=FT7$$!3=+++F*G6f$FT$\"37+++!3Z?x\"FT7$$!3*)*****>T sdi$FT$\"30+++Ef;,Za!fOFT$\"33+++pt)*G;FT7$$!3!******\\ D`5p$FT$\"33+++3(*eb:FT7$$!3!)*****fbQ=s$FT$\"3)******fDq5[\"FT7$$!3y* ****pEn9v$FT$\"3,+++jSa09FT7$$!35+++y)y*zPFT$\"3/+++<69H8FT7$$!3()**** **e;R2QFT$\"3%******\\)>+_7FT7$$!3&)*****f;+P$QFT$\"3/+++V@Fu6FT7$$!3) ******Rfu)eQFT$\"3++++Zb4'4\"FT7$$!3!*******)=iG)QFT$\"3,+++A#3w,\"FT7 $$!3'******pc!f0RFT$\"3c+++;lK*Q*F07$$!33+++hC(p#RFT$\"3^+++Hht,')F07$ $!3#)*****4?6p%RFT$\"3P+++/y:9yF07$$!3++++?iIlRFT$\"3<+++/6BFqF07$$!3? +++?!e?)RFT$\"3w*****HC(RTiF07$$!3?+++:A2(*RFT$\"3o*****Hj=pX&F07$$!3N +++TDE5SFT$\"35+++8X!Rn%F07$$!3!*******fGb@SFT$\"3++++:'QB*QF07$$!3i** ***RLy3.%FT$\"3++++>f57JF07$$!3/+++tf=QSFT$\"3(******4q;IL#F07$$!3i*** **RxMM/%FT$\"3))******)3F[b\"F07$$!3w*****4j&fYSFT$\"3M*****4YhDx(Fgp7 $$!3I+++u6lZSFTF(7$Fbdx$!3M*****4YhDx(Fgp7$F]dx$!3))******)3F[b\"F07$F hcx$!3(******4q;IL#F07$Fccx$!3++++>f57JF07$F^cx$!3++++:'QB*QF07$Fibx$! 35+++8X!Rn%F07$Fdbx$!3o*****Hj=pX&F07$F_bx$!3w*****HC(RTiF07$Fjax$!3<+ ++/6BFqF07$Feax$!3P+++/y:9yF07$F`ax$!3^+++Hht,')F07$F[ax$!3c+++;lK*Q*F 07$Ff`x$!3,+++A#3w,\"FT7$Fa`x$!3++++Zb4'4\"FT7$F\\`x$!3/+++V@Fu6FT7$Fg _x$!3%******\\)>+_7FT7$Fb_x$!3/+++<69H8FT7$F]_x$!3,+++jSa09FT7$Fh^x$!3 )******fDq5[\"FT7$Fc^x$!33+++3(*eb:FT7$F^^x$!33+++pt)*G;FT7$Fi]x$!30++ +Ef;, FT7$Fe\\x$!3'******RM.k(>FT7$F`\\x$!3;+++XynT?FT7$F[\\x$!3(*******)[Fb 5#FT7$Ff[x$!3')*****>@rz;#FT7$Fa[x$!3>+++I3/HAFT7$F\\[x$!3?+++86y)G#FT 7$Fgjw$!3%)*****RU]sM#FT7$Fbjw$!3\"******4*=_/CFT7$F]jw$!3;+++,^ogCFT7 $Fhiw$!3/+++#=[e^#FT7$Fciw$!3))*****f3S,d#FT7$F^iw$!3\"******zo7Pi#FT7 $Fihw$!3%)*****R+Unn#FT7$Fdhw$!39+++,uUHFFT7$F_hw$!3;+++Xq)>y#FT7$Fjgw $!33+++/rkMGFT7$Fegw$!3')******f=i()GFT7$F`gw$!3*******>]#3THFT7$F[gw$ !3')*****RL:^*HFT7$Fffw$!3))******Rhn\\IFT7$Fafw$!3!******f4bX5$FT7$F \\fw$!3#******\\Xq$fJFT7$Fgew$!3%)*****zL-O@$FT7$Fbew$!3?+++!RgmE$FT7$ F]ew$!3\")*****z3izJ$FT7$Fhdw$!3#******Rh&*pO$FT7$Fcdw$!3%)*****44`LT$ FT7$F^dw$!3-+++!QNnX$FT7$Ficw$!3y*****4OQp\\$FT7$Fdcw$!3!*******49$Q`$ FT7$F_cw$!3!)*****4SNtc$FT7$Fjbw$!3y******3QS(f$FT7$Febw$!3()*****p)y+ COFT7$F`bw$!33+++#fDrk$FT7$F[bw$!3#******4VLnm$FT7$Ffaw$!3y*****zM+Go$ FT7$Faaw$!3;+++\"*GG&p$FT7$F\\aw$!3@+++;57/PFT7$Fg`w$!3(******f+M#4PFT 7$Fb`w$!36+++kg^5PFT7$F]`w$!39+++C6$yq$FT7$Fh_w$!3<+++Uk+,PFT7$Fc_w$!3 -+++6W#)*o$FT7$F^_w$!3=+++s>,uOFT7$Fi^w$!3!******\\$oA`OFT7$Fd^w$!3(** ****4mRqi$FT7$F_^w$!3!******\\[6\\f$FT7$Fj]w$!3!******f\\mhb$FT7$Fe]w$ !3\"*******fG'*4NFT7$F`]w$!39+++#*3FbMFT7$F[]w$!33+++qA)3R$FT7$Ff\\w$! 3;+++_))\\:LFT7$Fa\\w$!3%******RLszA$FT7$F\\\\w$!32+++f%Gx7$FT7$Fg[w$! 3/+++4TA:IFT7$Fb[w$!3'******H'40#*GFT7$F][w$!3)******HX[/w#FT7$Fhjv$!3 4+++=_cAEFT7$Fcjv$!31+++gL,![#FT7$F^jv$!34+++`*\\QL#FT7$Fiiv$!31+++[Mx %=#FT7$Fdiv$!30+++LEKL?FT7$$!3E+++;G$\\0&Fao$!3/+++'G*)*z=FT7$Fjhv$!34 +++cUCDR7ZF07$F]fv$!3()*****HA$fTJF07$$\"3W+++wx>%R(Fiev$!31+++[jzq:F0F'- Fe^n6&Fg^nF(F(Fh^n-F^_n6fw7%F'7$$\"+%y(>%R(Fgp$\"+[jzq:Ff_nFg_n7%F_hy7 $$\"+(o2lv%FT$\"+BKfTJFf_nFg_n7%Fehy7$$\"+\\o-VaFj`n$\"+'*>R7ZFf_nFg_n 7%F[iy7$$\"+[n&R0$Faan$\"+9'*=$G'Ff_nFg_n7%Faiy7$$\"+H*zx9\"F^bn$\"+.( eR&yFf_nFg_n7%Fgiy7$$\"+')e#*)H$F^bn$\"+G4gC%*Ff_nFg_n7%F]jy7$$\"+B!em r(F^bn$\"+cf[*4\"FabnFg_n7%Fcjy7$$\"+N\\0/:Fhcn$\"+I!=kD\"FabnFg_n7%Fi jy7$$\"+r-&>T#Fhcn$\"+@A:89FabnFg_n7%F_[z7$$\"+K$f(4HFhcn$\"+'f9&p:Fab nFg_n7%Fe[z7$$\"+JC%\\Y\"Fhcn$\"+cUCD%e`*>Ff_n$\" +f%Gx7$FabnFg_n7%Fg_z7$$!+$H(e2EFf_n$\"+MB(zA$FabnFg_n7%F]`z7$$!+'\\@' pKFf_n$\"+_))\\:LFabnFg_n7%Fc`z7$$!+`tQfRFf_n$\"+qA)3R$FabnFg_n7%Fi`z7 $$!+MWhgYFf_n$\"+#*3FbMFabnFg_n7%F_az7$$!+W)yHO&Ff_n$\"+gG'*4NFabnFg_n 7%Feaz7$$!+c*>11'Ff_n$\"+'\\mhb$FabnFg_n7%F[bz7$$!+$GZ0v'Ff_n$\"+&[6\\ f$FabnFg_n7%Fabz7$$!+d\"f9V(Ff_n$\"+h'Rqi$FabnFg_n7%Fgbz7$$!+>*))H5)Ff _n$\"+NoA`OFabnFg_n7%F]cz7$$!+v\"f_w)Ff_n$\"+s>,uOFabnFg_n7%Fccz7$$!+* \\I'=%*Ff_n$\"+6W#)*o$FabnFg_n7%Ficz7$$!+*ecj+\"Fabn$\"+Uk+,PFabnFg_n7 %F_dz7$$!+Xi0q5Fabn$\"+C6$yq$FabnFg_n7%Fedz7$$!+!=7I8\"Fabn$\"+kg^5PFa bnFg_n7%F[ez7$$!+@VF&>\"Fabn$\"+1SB4PFabnFg_n7%Faez7$$!+PI*oD\"Fabn$\" +;57/PFabnFg_n7%Fgez7$$!+.2#zJ\"Fabn$\"+\"*GG&p$FabnFg_n7%F]fz7$$!+KRT y8Fabn$\"+[.!Go$FabnFg_n7%Fcfz7$$!+^jVQ9Fabn$\"+JMtmOFabnFg_n7%Fifz7$$ !+H>1)\\\"Fabn$\"+#fDrk$FabnFg_n7%F_gz7$$!+U)ytb\"Fabn$\"+()y+COFabnFg _n7%Fegz7$$!+\\Q\\;;Fabn$\"+4QS(f$FabnFg_n7%F[hz7$$!+Xq`v;Fabn$\"+,aLn NFabnFg_n7%Fahz7$$!+BmmM k9>Fabn$\"+\"4`LT$FabnFg_n7%Fiiz7$$!+aOLw>Fabn$\"+9c*pO$FabnFg_n7%F_jz 7$$!+/]KR?Fabn$\"+)3izJ$FabnFg_n7%Fejz7$$!+XO%Q5#Fabn$\"+!RgmE$FabnFg_ n7%F[[[l7$$!+UU-q@Fabn$\"+QBg8KFabnFg_n7%Fa[[l7$$!+^m&yB#Fabn$\"+b/PfJ FabnFg_n7%Fg[[l7$$!+rA:2BFabn$\"+'4bX5$FabnFg_n7%F]\\[l7$$!+gsaxBFabn$ \"+Shn\\IFabnFg_n7%Fc\\[l7$$!+.ua[CFabn$\"+M`6&*HFabnFg_n7%Fi\\[l7$$!+ f&*f>DFabn$\"+-D3THFabnFg_n7%F_][l7$$!+vQ;!f#Fabn$\"+g=i()GFabnFg_n7%F e][l7$$!++hwfEFabn$\"+/rkMGFabnFg_n7%F[^[l7$$!+^J-GFFabn$\"+Xq)>y#Fabn Fg_n7%Fa^[l7$$!+!\\YYz#Fabn$\"+,uUHFFabnFg_n7%Fg^[l7$$!+b@VfGFabn$\"+/ ?uwEFabnFg_n7%F]_[l7$$!+!QXA#HFabn$\"+)o7Pi#FabnFg_n7%Fc_[l7$$!+?[+$)H Fabn$\"+'3S,d#FabnFg_n7%Fi_[l7$$!+n)o;/$Fabn$\"+#=[e^#FabnFg_n7%F_`[l7 $$!+<[A)4$Fabn$\"+,^ogCFabnFg_n7%Fe`[l7$$!+93o_JFabn$\"+\"*=_/CFabnFg_ n7%F[a[l7$$!+V,10KFabn$\"+C/DZBFabnFg_n7%Faa[l7$$!+XtRbKFabn$\"+86y)G# FabnFg_n7%Fga[l7$$!+@dt.LFabn$\"+I3/HAFabnFg_n7%F]b[l7$$!+(zD,N$Fabn$ \"+77(z;#FabnFg_n7%Fcb[l7$$!+kWi%R$Fabn$\"+*[Fb5#FabnFg_n7%Fib[l7$$!+( \\%HPMFabn$\"+XynT?FabnFg_n7%F_c[l7$$!+WU?yMFabn$\"+WLSw>FabnFg_n7%Fec [l7$$!+]sUFabnFg_n7%F[d[l7$$!+C;/bNFabn$\"+z:dT=FabnFg _n7%Fad[l7$$!+F*G6f$Fabn$\"+!3Z?x\"FabnFg_n7%Fgd[l7$$!+7CxDOFabn$\"+Ef ;,+_7FabnFg_n7%Fag[l7$$!+m,qLQFabn$\"+V@Fu6F abnFg_n7%Fgg[l7$$!+%fu)eQFabn$\"+Zb4'4\"FabnFg_n7%F]h[l7$$!+*=iG)QFabn $\"+A#3w,\"FabnFg_n7%Fch[l7$$!+n0f0RFabn$\"+;lK*Q*Ff_nFg_n7%Fih[l7$$!+ hC(p#RFabn$\"+Hht,')Ff_nFg_n7%F_i[l7$$!+,7\"p%RFabn$\"+/y:9yFf_nFg_n7% Fei[l7$$!+?iIlRFabn$\"+/6BFqFf_nFg_n7%F[j[l7$$!+?!e?)RFabn$\"+VsRTiFf_ nFg_n7%Faj[l7$$!+:A2(*RFabn$\"+L'=pX&Ff_nFg_n7%Fgj[l7$$!+TDE5SFabn$\"+ 8X!Rn%Ff_nFg_n7%F][\\l7$$!+gGb@SFabn$\"+:'QB*QFf_nFg_n7%Fc[\\l7$$!+M$y 3.%Fabn$\"+>f57JFf_nFg_n7%Fi[\\l7$$!+tf=QSFabn$\"+,n,LBFf_nFg_n7%F_\\ \\l7$$!+uZVVSFabn$\"+*3F[b\"Ff_nFg_n7%Fe\\\\l7$$!+JcfYSFabn$\"+h9csxFb enFg_n7%F[]\\l7$$!+u6lZSFabnF(Fg_n7%Fa]\\l7$F\\]\\l$!+h9csxFbenFg_n7%F e]\\l7$Ff\\\\l$!+*3F[b\"Ff_nFg_n7%Fi]\\l7$F`\\\\l$!+,n,LBFf_nFg_n7%F]^ \\l7$Fj[\\l$!+>f57JFf_nFg_n7%Fa^\\l7$Fd[\\l$!+:'QB*QFf_nFg_n7%Fe^\\l7$ F^[\\l$!+8X!Rn%Ff_nFg_n7%Fi^\\l7$Fhj[l$!+L'=pX&Ff_nFg_n7%F]_\\l7$Fbj[l $!+VsRTiFf_nFg_n7%Fa_\\l7$F\\j[l$!+/6BFqFf_nFg_n7%Fe_\\l7$Ffi[l$!+/y:9 yFf_nFg_n7%Fi_\\l7$F`i[l$!+Hht,')Ff_nFg_n7%F]`\\l7$Fjh[l$!+;lK*Q*Ff_nF g_n7%Fa`\\l7$Fdh[l$!+A#3w,\"FabnFg_n7%Fe`\\l7$F^h[l$!+Zb4'4\"FabnFg_n7 %Fi`\\l7$Fhg[l$!+V@Fu6FabnFg_n7%F]a\\l7$Fbg[l$!+&)>+_7FabnFg_n7%Faa\\l 7$F\\g[l$!+<69H8FabnFg_n7%Fea\\l7$Fff[l$!+jSa09FabnFg_n7%Fia\\l7$F`f[l $!+c-2\"[\"FabnFg_n7%F]b\\l7$Fje[l$!+3(*eb:FabnFg_n7%Fab\\l7$Fde[l$!+p t)*G;FabnFg_n7%Feb\\l7$F^e[l$!+Ef;,FabnFg_ n7%Fec\\l7$Ffc[l$!+WLSw>FabnFg_n7%Fic\\l7$F`c[l$!+XynT?FabnFg_n7%F]d\\ l7$Fjb[l$!+*[Fb5#FabnFg_n7%Fad\\l7$Fdb[l$!+77(z;#FabnFg_n7%Fed\\l7$F^b [l$!+I3/HAFabnFg_n7%Fid\\l7$Fha[l$!+86y)G#FabnFg_n7%F]e\\l7$Fba[l$!+C/ DZBFabnFg_n7%Fae\\l7$F\\a[l$!+\"*=_/CFabnFg_n7%Fee\\l7$Ff`[l$!+,^ogCFa bnFg_n7%Fie\\l7$F``[l$!+#=[e^#FabnFg_n7%F]f\\l7$Fj_[l$!+'3S,d#FabnFg_n 7%Faf\\l7$Fd_[l$!+)o7Pi#FabnFg_n7%Fef\\l7$F^_[l$!+/?uwEFabnFg_n7%Fif\\ l7$Fh^[l$!+,uUHFFabnFg_n7%F]g\\l7$Fb^[l$!+Xq)>y#FabnFg_n7%Fag\\l7$F\\^ [l$!+/rkMGFabnFg_n7%Feg\\l7$Ff][l$!+g=i()GFabnFg_n7%Fig\\l7$F`][l$!+-D 3THFabnFg_n7%F]h\\l7$Fj\\[l$!+M`6&*HFabnFg_n7%Fah\\l7$Fd\\[l$!+Shn\\IF abnFg_n7%Feh\\l7$F^\\[l$!+'4bX5$FabnFg_n7%Fih\\l7$Fh[[l$!+b/PfJFabnFg_ n7%F]i\\l7$Fb[[l$!+QBg8KFabnFg_n7%Fai\\l7$F\\[[l$!+!RgmE$FabnFg_n7%Fei \\l7$Ffjz$!+)3izJ$FabnFg_n7%Fii\\l7$F`jz$!+9c*pO$FabnFg_n7%F]j\\l7$Fji z$!+\"4`LT$FabnFg_n7%Faj\\l7$Fdiz$!+!QNnX$FabnFg_n7%Fej\\l7$F^iz$!+h$Q p\\$FabnFg_n7%Fij\\l7$Fhhz$!+59$Q`$FabnFg_n7%F][]l7$Fbhz$!+,aLnNFabnFg _n7%Fa[]l7$F\\hz$!+4QS(f$FabnFg_n7%Fe[]l7$Ffgz$!+()y+COFabnFg_n7%Fi[]l 7$F`gz$!+#fDrk$FabnFg_n7%F]\\]l7$Fjfz$!+JMtmOFabnFg_n7%Fa\\]l7$Fdfz$!+ [.!Go$FabnFg_n7%Fe\\]l7$F^fz$!+\"*GG&p$FabnFg_n7%Fi\\]l7$Fhez$!+;57/PF abnFg_n7%F]]]l7$Fbez$!+1SB4PFabnFg_n7%Fa]]l7$F\\ez$!+kg^5PFabnFg_n7%Fe ]]l7$Ffdz$!+C6$yq$FabnFg_n7%Fi]]l7$F`dz$!+Uk+,PFabnFg_n7%F]^]l7$Fjcz$! +6W#)*o$FabnFg_n7%Fa^]l7$Fdcz$!+s>,uOFabnFg_n7%Fe^]l7$F^cz$!+NoA`OFabn Fg_n7%Fi^]l7$Fhbz$!+h'Rqi$FabnFg_n7%F]_]l7$Fbbz$!+&[6\\f$FabnFg_n7%Fa_ ]l7$F\\bz$!+'\\mhb$FabnFg_n7%Fe_]l7$Ffaz$!+gG'*4NFabnFg_n7%Fi_]l7$F`az $!+#*3FbMFabnFg_n7%F]`]l7$Fj`z$!+qA)3R$FabnFg_n7%Fa`]l7$Fd`z$!+_))\\:L FabnFg_n7%Fe`]l7$F^`z$!+MB(zA$FabnFg_n7%Fi`]l7$Fh_z$!+f%Gx7$FabnFg_n7% F]a]l7$Fb_z$!+4TA:IFabnFg_n7%Faa]l7$F\\_z$!+j40#*GFabnFg_n7%Fea]l7$Ff^ z$!+`%[/w#FabnFg_n7%Fia]l7$F`^z$!+=_cAEFabnFg_n7%F]b]l7$Fj]z$!+gL,![#F abnFg_n7%Fab]l7$Fd]z$!+`*\\QL#FabnFg_n7%Feb]l7$F^]z$!+[Mx%=#FabnFg_n7% Fib]l7$Fh\\z$!+LEKL?FabnFg_n7%F]c]l7$$!+;G$\\0&Fhcn$!+'G*)*z=FabnFg_n7 %Fac]l7$F\\\\z$!+cUCD R7ZFf_nFg_n7%Fie]l7$Ffhy$!+BKfTJFf_nFg_n7%F_f]l7$$\"+wx>%R(Fgp$!+[jzq: Ff_nFg_n7%Fcf]lF'Fg_n-Fe^n6&Fg^n$F]_rFj^n$\"#DFi_n$\"\")Fj^nF^_r-F$6$7 in7$$\"3/+++(R%*)*\\\"FT$\"3-+++2&=ZG%FT7$$\"3/+++\\Zo\"\\\"FT$\"3u*** **\\,OKH%FT7$$\"3%*******)4-G[\"FT$\"3C+++.T0,VFT7$$\"3)******\\n*Gt9F T$\"3()*****4RJ\"3VFT7$$\"3'******\\]\">j9FT$\"3P+++(yEWJ%FT7$$\"31+++ YFb_9FT$\"3C+++8#**)>VFT7$$\"3++++g+UT9FT$\"3&******pM2XK%FT7$$\"3&*** ****Q?%)H9FT$\"3;+++'Q4#GVFT7$$\"31+++1(pyT\"FT$\"36+++7G'4L%FT7$$\"3) ******p4dbS\"FT$\"3%)*****H=CFL%FT7$$\"3++++Y>'HR\"FT$\"3&)*****f%*[ML %FT7$$\"35+++5l9!Q\"FT$\"3)******RG\"4LVFT7$$\"3'******payrO\"FT$\"3$) *****R,/;L%FT7$$\"35+++*\\KTN\"FT$\"3F+++i&Q*GVFT7$$\"3)******\\$44T8F T$\"3e*****p1X]K%FT7$$\"32+++#GY\"G8FT$\"3y*****4ms)>VFT7$$\"3$******p %HS:8FT$\"3&)*****>/qLJ%FT7$$\"3$*******p)zHI\"FT$\"3N+++kk[0VFT7$$\"3 -+++\"f85H\"FT$\"3C+++tL<'H%FT7$$\"31+++j=mz7FT$\"3#)*****p*pQ&G%FT7$$ \"3'******Rz2\"p7FT$\"3))******HB4tUFT7$$\"3%******\\Kk&f7FT$\"3I+++d \"p#fUFT7$$\"3)******fey7D\"FT$\"3)******>$4#RC%FT7$$\"3%******z6NXC\" FT$\"3\")*****R`(3FUFT7$$\"31+++cglR7FT$\"3w*****z'G')3UFT7$$\"3#***** *z1&*pB\"FT$\"3y******>xT*=%FT7$$\"31+++,.#pB\"FT$\"3s*****zaE!pTFT7$$ \"3(******>y!yR7FT$\"3m*****>F$4[TFT7$$\"33+++;E&eC\"FT$\"3H+++ab;FTFT 7$$\"33+++f)o_D\"FT$\"3F+++K=#p5%FT7$$\"35+++!f]zE\"FT$\"3'******>n;\" )3%FT7$$\"3)*******Q#pNG\"FT$\"3M+++RY[rSFT7$$\"3\"******4Fn:I\"FT$\"3 ;+++.#Gw0%FT7$$\"3'*******)fQ7K\"FT$\"3))*****f&>$p/%FT7$$\"3-+++G\"Q= M\"FT$\"3J+++7[`RSFT7$$\"3/+++F(yEO\"FT$\"3n******>.ONSFT7$$\"3-+++L(* =$Q\"FT$\"3E+++)zzT.%FT7$$\"3\"******fONHS\"FT$\"3B+++*Qzc.%FT7$$\"3!* *****px0;U\"FT$\"32+++WN^RSFT7$$\"33+++r`**Q9FT$\"3?+++\">S`/%FT7$$\"3 5+++$fy\\X\"FT$\"3e******z,%G0%FT7$$\"33+++C$)[p9FT$\"37+++')\\#[\"FT$\"35+++7GurSFT7$$\"33+++U<,%\\\"FT$\"3s*****psoE3%F T7$$\"3%******4!)[S]\"FT$\"3M+++J)4V4%FT7$$\"3-+++VJk7:FT$\"3-+++hu\\1 TFT7$$\"3\"******>yN)>:FT$\"3j*****fV%3>TFT7$$\"3%******RFsc_\"FT$\"3U +++&yT>8%FT7$$\"3,+++?1?I:FT$\"3)******f)e&\\9%FT7$$\"3/+++l)pM`\"FT$ \"3A+++3g-eTFT7$$\"3$******R?Hb`\"FT$\"3y*****HBi5<%FT7$$\"3\"******\\ TFk`\"FT$\"3=+++;P)R=%FT7$$\"35+++RD@O:FT$\"3C+++7sr'>%FT7$$\"3)****** HrJ\\`\"FT$\"3&)*****R&e>4UFT7$$\"32+++=6jK:FT$\"3!)*****46e8A%FT7$$\" 3'******H+c$H:FT$\"3&)*****\\\"p9LUFT7$$\"3\"*******o2:D:FT$\"3>+++=*3 XC%FT7$$\"3*)*****R3f+_\"FT$\"3g*****f%QRbUFT7$$\"3++++SS79:FT$\"3-+++ aRvlUFT7$$\"3++++?$)Q2:FT$\"3#)*****f`VbF%FTFcg]lFjgy-F^_n6jn7%7$$\"+( R%*)*\\\"Fabn$\"+2&=ZG%Fabn7$$\"+\\Zo\"\\\"Fabn$\"+:gB$H%Fabn7$F_cs$\" $<%Fi_n7%Fgj^l7$$\"+*4-G[\"Fabn$\"+.T0,VFabnF\\[_l7%F`[_l7$$\"+v'*Gt9F abn$\"+\"RJ\"3VFabnF\\[_l7%Ff[_l7$$\"+0:>j9Fabn$\"+(yEWJ%FabnF\\[_l7%F \\\\_l7$$\"+YFb_9Fabn$\"+8#**)>VFabnF\\[_l7%Fb\\_l7$$\"+g+UT9Fabn$\"+Z t]CVFabnF\\[_l7%Fh\\_l7$$\"+R?%)H9Fabn$\"+'Q4#GVFabnF\\[_l7%F^]_l7$$\" +1(pyT\"Fabn$\"+7G'4L%FabnF\\[_l7%Fd]_l7$$\"+(4dbS\"Fabn$\"+$=CFL%Fabn F\\[_l7%Fj]_l7$$\"+Y>'HR\"Fabn$\"+Y*[ML%FabnF\\[_l7%F`^_l7$$\"+5l9!Q\" Fabn$\"+%G\"4LVFabnF\\[_l7%Ff^_l7$$\"+Z&yrO\"Fabn$\"+9SgJVFabnF\\[_l7% F\\__l7$$\"+*\\KTN\"Fabn$\"+i&Q*GVFabnF\\[_l7%Fb__l7$$\"+N44T8Fabn$\"+ n]/DVFabnF\\[_l7%Fh__l7$$\"+#GY\"G8Fabn$\"+hE()>VFabnF\\[_l7%F^`_l7$$ \"+ZHS:8Fabn$\"+U+P8VFabnF\\[_l7%Fd`_l7$$\"+q)zHI\"Fabn$\"+kk[0VFabnF \\[_l7%Fj`_l7$$\"+\"f85H\"Fabn$\"+tL<'H%FabnF\\[_l7%F`a_l7$$\"+j=mz7Fa bn$\"+(*pQ&G%FabnF\\[_l7%Ffa_l7$$\"+%z2\"p7Fabn$\"+IB4tUFabnF\\[_l7%F \\b_l7$$\"+DVcf7Fabn$\"+d\"p#fUFabnF\\[_l7%Fbb_l7$$\"+'ey7D\"Fabn$\"+K 4#RC%FabnF\\[_l7%Fhb_l7$$\"+=^`W7Fabn$\"+Mv3FUFabnF\\[_l7%F^c_l7$$\"+c glR7Fabn$\"+oG')3UFabnF\\[_l7%Fdc_l7$$\"+o]*pB\"Fabn$\"+?xT*=%FabnF\\[ _l7%Fjc_l7$$\"+,.#pB\"Fabn$\"+[l-pTFabnF\\[_l7%F`d_l7$$\"+#y!yR7Fabn$ \"+sK4[TFabnF\\[_l7%Ffd_l7$$\"+;E&eC\"Fabn$\"+ab;FTFabnF\\[_l7%F\\e_l7 $$\"+f)o_D\"Fabn$\"+K=#p5%FabnF\\[_l7%Fbe_l7$$\"+!f]zE\"Fabn$\"+sm6)3% FabnF\\[_l7%Fhe_l7$$\"+R#pNG\"Fabn$\"+RY[rSFabnF\\[_l7%F^f_l7$$\"+rsc, 8Fabn$\"+.#Gw0%FabnF\\[_l7%Fdf_l7$$\"+*fQ7K\"Fabn$\"+c>$p/%FabnF\\[_l7 %Fjf_l7$$\"+G\"Q=M\"Fabn$\"+7[`RSFabnF\\[_l7%F`g_l7$$\"+F(yEO\"Fabn$\" +?.ONSFabnF\\[_l7%Ffg_l7$$\"+L(*=$Q\"Fabn$\"+)zzT.%FabnF\\[_l7%F\\h_l7 $$\"+m`$HS\"Fabn$\"+*Qzc.%FabnF\\[_l7%Fbh_l7$$\"+xdg@9Fabn$\"+WN^RSFab nF\\[_l7%Fhh_l7$$\"+r`**Q9Fabn$\"+\">S`/%FabnF\\[_l7%F^i_l7$$\"+$fy\\X \"Fabn$\"+!=SG0%FabnF\\[_l7%Fdi_l7$$\"+C$)[p9Fabn$\"+')\\#[\"Fabn$\"+7GurSFabnF\\[_l7%F`j_l7$$\"+U<,%\\\"Fabn$ \"+F(oE3%FabnF\\[_l7%Ffj_l7$$\"+,)[S]\"Fabn$\"+J)4V4%FabnF\\[_l7%F\\[` l7$$\"+VJk7:Fabn$\"+hu\\1TFabnF\\[_l7%Fb[`l7$$\"+#yN)>:Fabn$\"+OW3>TFa bnF\\[_l7%Fh[`l7$$\"+uAnD:Fabn$\"+&yT>8%FabnF\\[_l7%F^\\`l7$$\"+?1?I:F abn$\"+')e&\\9%FabnF\\[_l7%Fd\\`l7$$\"+l)pM`\"Fabn$\"+3g-eTFabnF\\[_l7 %Fj\\`l7$$\"+/#Hb`\"Fabn$\"+LA1rTFabnF\\[_l7%F`]`l7$$\"+:uUO:Fabn$\"+; P)R=%FabnF\\[_l7%Ff]`l7$$\"+RD@O:Fabn$\"+7sr'>%FabnF\\[_l7%F\\^`l7$$\" +8<$\\`\"Fabn$\"+ae>4UFabnF\\[_l7%Fb^`l7$$\"+=6jK:Fabn$\"+6\"e8A%FabnF \\[_l7%Fh^`l7$$\"+.gNH:Fabn$\"+:p9LUFabnF\\[_l7%F^_`l7$$\"+p2:D:Fabn$ \"+=*3XC%FabnF\\[_l7%Fd_`l7$$\"+%3f+_\"Fabn$\"+YQRbUFabnF\\[_l7%Fj_`l7 $$\"+SS79:Fabn$\"+aRvlUFabnF\\[_l7%F```l7$$\"+?$)Q2:Fabn$\"+ONavUFabnF \\[_l7%Ff``lFbj^lF\\[_lFif]lF^_r-F$6$7in7$Fdg]l$!3-+++2&=ZG%FT7$F[j^l$ !3#)*****f`VbF%FT7$Ffi^l$!3-+++aRvlUFT7$Fai^l$!3g*****f%QRbUFT7$F\\i^l $!3>+++=*3XC%FT7$Fgh^l$!3&)*****\\\"p9LUFT7$Fbh^l$!3!)*****46e8A%FT7$F ]h^l$!3&)*****R&e>4UFT7$Fhg^l$!3C+++7sr'>%FT7$Fcg^l$!3=+++;P)R=%FT7$F^ g^l$!3y*****HBi5<%FT7$Fif^l$!3A+++3g-eTFT7$Fdf^l$!3)******f)e&\\9%FT7$ F_f^l$!3U+++&yT>8%FT7$Fje^l$!3j*****fV%3>TFT7$Fee^l$!3-+++hu\\1TFT7$F` e^l$!3M+++J)4V4%FT7$F[e^l$!3s*****psoE3%FT7$Ffd^l$!35+++7GurSFT7$Fad^l $!37+++S`/%FT7$Fbc^l $!32+++WN^RSFT7$F]c^l$!3B+++*Qzc.%FT7$Fhb^l$!3E+++)zzT.%FT7$Fcb^l$!3n* *****>.ONSFT7$F^b^l$!3J+++7[`RSFT7$Fia^l$!3))*****f&>$p/%FT7$Fda^l$!3; +++.#Gw0%FT7$F_a^l$!3M+++RY[rSFT7$Fj`^l$!3'******>n;\")3%FT7$Fe`^l$!3F +++K=#p5%FT7$F``^l$!3H+++ab;FTFT7$F[`^l$!3m*****>F$4[TFT7$Ff_^l$!3s*** **zaE!pTFT7$Fa_^l$!3y******>xT*=%FT7$F\\_^l$!3w*****z'G')3UFT7$Fg^^l$! 3\")*****R`(3FUFT7$Fb^^l$!3)******>$4#RC%FT7$F]^^l$!3I+++d\"p#fUFT7$Fh ]^l$!3))******HB4tUFT7$Fc]^l$!3#)*****p*pQ&G%FT7$F^]^l$!3C+++tL<'H%FT7 $Fi\\^l$!3N+++kk[0VFT7$Fd\\^l$!3&)*****>/qLJ%FT7$F_\\^l$!3y*****4ms)>V FT7$Fj[^l$!3e*****p1X]K%FT7$Fe[^l$!3F+++i&Q*GVFT7$F`[^l$!3$)*****R,/;L %FT7$F[[^l$!3)******RG\"4LVFT7$Ffj]l$!3&)*****f%*[ML%FT7$Faj]l$!3%)*** **H=CFL%FT7$F\\j]l$!36+++7G'4L%FT7$Fgi]l$!3;+++'Q4#GVFT7$Fbi]l$!3&**** **pM2XK%FT7$F]i]l$!3C+++8#**)>VFT7$Fhh]l$!3P+++(yEWJ%FT7$Fch]l$!3()*** **4RJ\"3VFT7$F^h]l$!3C+++.T0,VFT7$Fig]l$!3u*****\\,OKH%FTF_a`lFjgy-F^_ n6jn7%7$Fcj^l$!+2&=ZG%Fabn7$Fg``l$!+ONavUFabn7$F_cs$!$<%Fi_n7%Fi\\al7$ Fa``l$!+aRvlUFabnF\\]al7%F`]al7$F[``l$!+YQRbUFabnF\\]al7%Fd]al7$Fe_`l$ !+=*3XC%FabnF\\]al7%Fh]al7$F__`l$!+:p9LUFabnF\\]al7%F\\^al7$Fi^`l$!+6 \"e8A%FabnF\\]al7%F`^al7$Fc^`l$!+ae>4UFabnF\\]al7%Fd^al7$F]^`l$!+7sr'> %FabnF\\]al7%Fh^al7$Fg]`l$!+;P)R=%FabnF\\]al7%F\\_al7$Fa]`l$!+LA1rTFab nF\\]al7%F`_al7$F[]`l$!+3g-eTFabnF\\]al7%Fd_al7$Fe\\`l$!+')e&\\9%FabnF \\]al7%Fh_al7$F_\\`l$!+&yT>8%FabnF\\]al7%F\\`al7$Fi[`l$!+OW3>TFabnF\\] al7%F``al7$Fc[`l$!+hu\\1TFabnF\\]al7%Fd`al7$F][`l$!+J)4V4%FabnF\\]al7% Fh`al7$Fgj_l$!+F(oE3%FabnF\\]al7%F\\aal7$Faj_l$!+7GurSFabnF\\]al7%F`aa l7$F[j_l$!+S`/%FabnF\\]al7%F\\bal7$Fih_l$!+WN^RSFabnF\\]al7%F`bal7$Fch_l $!+*Qzc.%FabnF\\]al7%Fdbal7$F]h_l$!+)zzT.%FabnF\\]al7%Fhbal7$Fgg_l$!+? .ONSFabnF\\]al7%F\\cal7$Fag_l$!+7[`RSFabnF\\]al7%F`cal7$F[g_l$!+c>$p/% FabnF\\]al7%Fdcal7$Fef_l$!+.#Gw0%FabnF\\]al7%Fhcal7$F_f_l$!+RY[rSFabnF \\]al7%F\\dal7$Fie_l$!+sm6)3%FabnF\\]al7%F`dal7$Fce_l$!+K=#p5%FabnF\\] al7%Fddal7$F]e_l$!+ab;FTFabnF\\]al7%Fhdal7$Fgd_l$!+sK4[TFabnF\\]al7%F \\eal7$Fad_l$!+[l-pTFabnF\\]al7%F`eal7$F[d_l$!+?xT*=%FabnF\\]al7%Fdeal 7$Fec_l$!+oG')3UFabnF\\]al7%Fheal7$F_c_l$!+Mv3FUFabnF\\]al7%F\\fal7$Fi b_l$!+K4#RC%FabnF\\]al7%F`fal7$Fcb_l$!+d\"p#fUFabnF\\]al7%Fdfal7$F]b_l $!+IB4tUFabnF\\]al7%Fhfal7$Fga_l$!+(*pQ&G%FabnF\\]al7%F\\gal7$Faa_l$!+ tL<'H%FabnF\\]al7%F`gal7$F[a_l$!+kk[0VFabnF\\]al7%Fdgal7$Fe`_l$!+U+P8V FabnF\\]al7%Fhgal7$F_`_l$!+hE()>VFabnF\\]al7%F\\hal7$Fi__l$!+n]/DVFabn F\\]al7%F`hal7$Fc__l$!+i&Q*GVFabnF\\]al7%Fdhal7$F]__l$!+9SgJVFabnF\\]a l7%Fhhal7$Fg^_l$!+%G\"4LVFabnF\\]al7%F\\ial7$Fa^_l$!+Y*[ML%FabnF\\]al7 %F`ial7$F[^_l$!+$=CFL%FabnF\\]al7%Fdial7$Fe]_l$!+7G'4L%FabnF\\]al7%Fhi al7$F_]_l$!+'Q4#GVFabnF\\]al7%F\\jal7$Fi\\_l$!+Zt]CVFabnF\\]al7%F`jal7 $Fc\\_l$!+8#**)>VFabnF\\]al7%Fdjal7$F]\\_l$!+(yEWJ%FabnF\\]al7%Fhjal7$ Fg[_l$!+\"RJ\"3VFabnF\\]al7%F\\[bl7$Fa[_l$!+.T0,VFabnF\\]al7%F`[bl7$Fh j^l$!+:gB$H%FabnF\\]al7%Fd[blFf\\alF\\]alFif]lF^_r-%+AXESLABELSG6%%&Re (z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-F^\\bl6$%*HELVETICAG\"\"*-%*AXESSTYLE G6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$z%Fi_n$\"$>#Fi_n;Fa] bl$\"$z%Fi_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#----------------- ----------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "chara cteristics of the embedded schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 988 "ee := \{c[2 ]=1/6,\nc[3]=2/9,\nc[4]=3/7,\nc[5]=2/3,\nc[6]=3/4,\nc[7]=1,\nc[8]=1,\n \na[2,1]=1/6,\na[3,1]=2/27,\na[3,2]=4/27,\na[4,1]=183/1372,\na[4,2]=-1 62/343,\na[4,3]=1053/1372,\na[5,1]=68/297,\na[5,2]=-4/11,\na[5,3]=42/1 43,\na[5,4]=1960/3861,\na[6,1]=597/22528,\na[6,2]=81/352,\na[6,3]=6309 9/585728,\na[6,4]=58653/366080,\na[6,5]=4617/20480,\na[7,1]=174197/959 244,\na[7,2]=-30942/79937,\na[7,3]=8152137/19744439,\na[7,4]=666106/10 39181,\na[7,5]=-29421/29068,\na[7,6]=482048/414219,\na[8,1]=587/8064, \na[8,2]=0,\na[8,3]=4440339/15491840,\na[8,4]=24353/124800,\na[8,5]=38 7/44800,\na[8,6]=2152/5985,\na[8,7]=7267/94080,\n\nb[1]=587/8064,\nb[2 ]=0,\nb[3]=4440339/15491840,\nb[4]=24353/124800,\nb[5]=387/44800,\nb[6 ]=2152/5985,\nb[7]=7267/94080,\n\n`b^`[1]=6059/80640,\n`b^`[2]=0,\n`b^ `[3]=8559189/30983680,\n`b^`[4]=26411/124800,\n`b^`[5]=-927/89600,\n`b ^`[6]=443/1197,\n`b^`[7]=7267/94080,\n\n`b*`[1]=2479/34992,\n`b*`[2]=0 ,\n`b*`[3]=123/416,\n`b*`[4]=612941/3411720,\n`b*`[5]=43/1440,\n`b*`[6 ]=2272/6561,\n`b*`[7]=79937/1113912,\n`b*`[8]=3293/556956\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "T[5,7];" "6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 127 " denote the vector whose components are the principal error terms of \+ the 7 stage, order 5 scheme (the error terms of order 6)." }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,8];" "6#&%#T*G6$\"\"% \"\")" }{TEXT -1 145 " denote the vector whose components are the pri ncipal error terms of the embedded 8 stage, order 4 scheme (the error \+ terms of order 5) and let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\" \"&\"\")" }{TEXT -1 99 " denote the vector whose components are the e rror terms of order 6 of the embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of t hese vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7]));" "6#-%$absG6#-F$ 6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4 ,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\" \")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG 6#-F)6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/abs(abs(`T*`[4, 8]));" "6#/&%\"BG6#\"\"'*&-%$abs G6#-F*6#&%#T*G6$\"\"&\"\")\"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,8]-T[5,7]))/abs(ab s(`T*`[4,8]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\") \"\"\"&%\"TG6$F2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to ensure that \+ " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 73 " is a minimu m, if the embedded scheme is to be used for error control, " } {XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " should be chos en so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magn itude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "From this point of view the Bo gacki-Shampine scheme is near ideal with respect to the 8 stage, order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_7 := PrincipalErrorTerms(5,7,' expanded'):\n`errterms5_8*` :=subs(b=`b*`,PrincipalErrorTerms(5,8,'exp anded')):\n`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expa nded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "snmB := sqrt(add(simplify(subs(ee,`errterms5_8*`[i]) )^2,i=1..nops(`errterms5_8*`))):\nsdnB := sqrt(add(simplify(subs(ee,`e rrterms4_8*`[i]))^2,i=1..nops(`errterms4_8*`))):\nsnmC := sqrt(add((si mplify(subs(ee,`errterms5_8*`[i])-subs(ee,errterms5_7[i])))^2,i=1..nop s(errterms5_7))):\n`B*`[6]= evalf[8](snmB/sdnB);\n`C*`[6]= evalf[8](sn mC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B*G6#\"\"'$\")&fb.\"! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\"\"'$\")/%[/\"!\"(" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The cor responding characterstics for the auxiliary 7 stage, order 4 embedded \+ scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "` errterms5_7^` :=subs(b=`b^`,PrincipalErrorTerms(5,7,'expanded')):\n`er rterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "snmB := sqrt(add(simplify(subs(ee,`errterms5_7^`[i]))^2,i=1..nops (`errterms5_7^`))):\nsdnB := sqrt(add(simplify(subs(ee,`errterms4_7^`[ i]))^2,i=1..nops(`errterms4_7^`))):\nsnmC := sqrt(add((simplify(subs(e e,`errterms5_7^`[i])-subs(ee,errterms5_7[i])))^2,i=1..nops(errterms5_7 ))):\n`B^`[6]= evalf[8](snmB/sdnB);\n`C^`[6]= evalf[8](snmC/sdnB);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B^G6#\"\"'$\")tcn7!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C^G6#\"\"'$\")X!H>\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 42 "#---------------------------------------- -" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#=================================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "A minor modification of the Bogac ki-Shampine scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 42 "#---------------------------------------- -" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the \+ combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1101 "ee := \{c[2]=1/6,\nc[3]=2/9,\nc[4]=3/7,\nc[5 ]=2/3,\nc[6]=3/4,\nc[7]=1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=2/27,\na[3, 2]=4/27,\na[4,1]=183/1372,\na[4,2]=-162/343,\na[4,3]=1053/1372,\na[5,1 ]=68/297,\na[5,2]=-4/11,\na[5,3]=42/143,\na[5,4]=1960/3861,\na[6,1]=32 575/1096128,\na[6,2]=81/352,\na[6,3]=38673/395824,\na[6,4]=2421335/142 49664,\na[6,5]=77/346,\na[7,1]=2218605592257973/11003713657896960,\na[ 7,2]=-63/143,\na[7,3]=1473430904732051/3532056235868160,\na[7,4]=25497 32718751057/3649190753894400,\na[7,5]=-13054508705469277/1222634850877 4400,\na[7,6]=21360483/17937920,\na[8,1]=539/7344,\na[8,2]=0,\na[8,3]= 190269/671840,\na[8,4]=362551/1790100,\na[8,5]=-81/13600,\na[8,6]=4844 8/130815,\na[8,7]=13/170,\n\nb[1]=539/7344,\nb[2]=0,\nb[3]=190269/6718 40,\nb[4]=362551/1790100,\nb[5]=-81/13600,\nb[6]=48448/130815,\nb[7]=1 3/170,\n\n`b^`[1]=137189/1827840,\n`b^`[2]=0,\n`b^`[3]=6386607/2315264 0,\n`b^`[4]=69923/326400,\n`b^`[5]=-118311/6092800,\n`b^`[6]=201/532, \n`b^`[7]=13/170,\n\n`b*`[1]=14222516/196928415,\n`b*`[2]=0,\n`b*`[3]= 13/45,\n`b*`[4]=908468771/4726281960,\n`b*`[5]=33/3335,\n`b*`[6]=21256 5968/590785245,\n`b*`[7]=2066857/29174580,\n`b*`[8]=214645/35009496\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b^`,seq(`b^`[i],i =1..7),``],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\" \"*#F.\"#F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%` 5F:F+F+F+F+F+7+#F.F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"#\"%g>\"%hQF+F+F+F+7 +#F6F3#\"&vD$\"(Gh4\"#\"#\")\"$_$#\"&t'Q\"'CeR#\"(N8U#\")k'\\U\"#\"#x \"$Y$F+F+F+7+F)#\"1tzD#f0'=A\"2gp*yl8P+6#!#jFJ#\"1^?t/4Vt9\"1g\"oeBc?` $#\"1d5v=Ft\\D\"1+W*Qv!>\\O#!2x#paq3X08\"2+Wx3&[jA7#\")$[g8#\")?z$z\"F +F+7+F)#\"$R&\"%Wt\"\"!#\"'p->\"'S=n#\"'^DO\"(+,z\"#!#\")\"&+O\"#\"&[% [\"':38#\"#8\"$q\"F+7+%\"bGF\\pF_pF`pFcpFfpFipF\\qF+7+%#b^G#\"'*=P\"\" (Sy#=F_p#\"(2mQ'\")SE:B#\"&B*p\"'+kK#!'6$=\"\"(+G4'#\"$,#\"$K&F\\qF+7+ %#b*G#\");DA9\"*:%Gp>F_p#F]q\"#X#\"*r(o%3*\"+g>GEZ#\"#L\"%NL#\"*ofc7# \"*X_y!f#\"(do1#\")!eu\"H#\"'XY@\")'\\4]$Q(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "RK5_7eqs := [op(RowSumConditions(7 ,'expanded')),op(OrderConditions(5,7,'expanded'))]:\nsubs(ee,RK5_7eqs) :\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4_7eqs^` := subs(b=`b^`,Orde rConditions(4,7,'expanded')):\nsubs(ee,`RK4_7eqs^`):\nmap(u->lhs(u)-rh s(u),%);\nnops(%);\n`RK4_8eqs*` := subs(b=`b*`,OrderConditions(4,8,'ex panded')):\nsubs(ee,`RK4_8eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%); " }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7* \"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can ca lculate the principal error norm, that is, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_ 7 := PrincipalErrorTerms(5,7,'expanded'):\nsqrt(add(subs(ee,errterms5_ 7[i])^2,i=1.. nops(errterms5_7))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xdk7:!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also calculate the 2 norm of the princip al error of the order 4 embedded schemes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4, 7,'expanded')):\nsqrt(add(subs(ee,`errterms4_7^`[i])^2,i=1.. nops(`err terms4_7^`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)H$3K u!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4, 8,'expanded')):\nsqrt(add(subs(ee,`errterms4_8*`[i])^2,i=1.. nops(`err terms4_8*`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wD\\H u!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " Next we set-up stage-order condtions to check for stage-orders from 2 \+ to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct f rom 2 to 4 do\n so||ct||_7 := StageOrderConditions(ct,7,'expanded'); \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following respective stage-orders. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so ||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimpl ify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In general, the simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$ SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6# F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6 #/%\"jG\"\"\"" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 7 " ) are " }{TEXT 260 13 "not satisfied " }{TEXT -1 15 ", except when " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 ", where the condition reduces to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 152 "[Sum(b[i]*a[i,1],i=1+1..7)=b[1],seq(Sum(b[i]* a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nmap( u->`if`(lhs(u)=rhs(u),0,1),subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F* 6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F- /-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F) F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F, FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F ,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7(\"\"\"\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i ]*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F* ;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\" &%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F )F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = \+ 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&& %\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"! " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Sum(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,% ));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6 #%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\" \"F+/F*;F5\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"%\" \"\"&%\"aG6$F(\"\"$F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F) F)*&&F+6$F4F(F)&F+6$F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&& F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\" (F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "None of t he principal error conditions are satisfied by the order 5 scheme. " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK5_7err_eqs := PrincipalE rrorConditions(5,7,'expanded'):\nsubs(ee,RK5_7err_eqs):\nmap(u->`if`(l hs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions \+ are satisfied by the 8 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 95 "The largest magnitude of an error term is about 32 t imes the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 147 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms( 4,8,'expanded')):\nsubs(ee,`errterms4_8*`):\nmap(abs,evalf[8](%));\nev alf[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$ \")I]aW!#7$\")*z$z\\F&$\")i-d]!#8$\")/3CCF&$\")ugNYF+$\")$Rbe(F+$\")Z@ r#*F+$\")Lzf;F&$\")D?X:F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%BK!\" #" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Non e of the 9 principal error conditions are satisfied by the 7 stage emb edded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 95 "The largest ma gnitude of an error term is about 18 times the lowest magnitude of an \+ error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_7 ^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\nsubs(ee,`errt erms4_7^`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")[/)f\"!#7$\")w\\feF&$\")vAz5F& $\")s1(R#F&$\")%=H*)*!#8$\")7%)=;F&$\")Pey>F&$\")f;`>F&$\")&RwH$F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%x " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 88 "simultaneous construction of the 7 stag e, order 5 scheme and the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "We follow the longer con struction of the Bogacki-Shampine scheme up to the point where the sta bility function is incorporated." }}{PARA 0 "" 0 "" {TEXT -1 123 "The \+ following \"simple\" order conditions (given in abreviated form) are r elevant for the construction of the order 5 scheme." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := Si mpleOrderConditions(5):\n[seq([i,SO5[i]],i=[1,2,4,5,8,12,13,15,16])]: \nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[row dim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7+7%\"\"\"%%~~~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"c GF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF( -F@6#*&FCF(F2F(F(F(#F(\"#C7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(- F@6#*&FCF(-F@6#*&F8F(FCF(F(F(#F(\"#g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%Fj nF)/*&F,F(-F@6#*&FMF(FCF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b =`b*`,SimpleOrderConditions(4)):\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]: \nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[row dim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7'7%\"\"\"%%~~~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"c GF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\" aGF(F(#F(\"#77%\"\")F)/*&F,F()F2F:F(#F(F5Q)pprint346\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We incorporate the simplifying conditio ns:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2 ],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F ,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[ i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6# F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 9 " and " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = \+ 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3 \"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 159 "for t he order 5 scheme. We also need to specify that the linking coefficien ts for the 8th (FSAL) stage are given by the weights of the order 5 sc heme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[i ] = a[8,i];" "6#/&%\"bG6#%\"iG&%\"aG6$\"\")F'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$ \"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the ord er 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7,'expa nded'):\n`SO4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded') ): \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i], i=[1,2,4,7,8])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(S tageOrderConditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2], i=3..7)=0,add(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n ad d(b[i]*c[i]*a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns : = [seq(b[i]=a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_e qns),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "We start in the same way as in the previous section by sp ecifying the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"'!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 2/3;" "6#/&%\"cG6#\"\"&*& \"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 16 "and the weights " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "` b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b *`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " are initially parameters." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "e1 := \{c[2]=1/6,c[3]=2/9,c[5]=2/3,c[6]=3/4,c[7]=1,c[8]=1,b[2]=0, `b*`[2]=0\}:\neqns := subs(e1,cdns):\nnops(eqns);\nindets(eqns) minus \+ \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := solve(\{op(eqns)\},indets(eqns) mi nus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3957 "e3 := \{c[8] = 1, b[2] = 0, a[8,2] = 0, a[2,1] = 1/ 6, c[7] = 1, `b*`[2] = 0, c[2] = 1/6, a[4,1] = 27/4*c[4]^3-9/2*c[4]^2+ c[4], a[7,1] = 1/369360*(26963280*b[7]*c[4]^3+2226420*c[4]^3+8208000*b [7]*a[7,5]*c[4]^3-55296000*a[6,5]*c[4]^3+516096000*a[6,5]*c[4]^3*b[7]- 11344995*c[4]^2-1211105280*a[6,5]*c[4]^2*b[7]+128532480*a[6,5]*c[4]^2- 17209440*b[7]*a[7,5]*c[4]^2+11911860*b[7]*c[4]^2-97878016*a[6,5]*c[4]+ 11446569*c[4]-44784900*b[7]*c[4]+928972800*a[6,5]*c[4]*b[7]+12011040*b [7]*a[7,5]*c[4]-2790720*b[7]*a[7,5]+24510464*a[6,5]+15513120*b[7]-2339 63520*b[7]*a[6,5]-3142638)/(4*c[4]-3)/c[4]/b[7]/(25*c[4]-17), a[4,2] = -9*(3*c[4]-1)*c[4]^2, b[5] = 9/160*(-20*c[4]+420*b[7]*c[4]+27-420*b[7 ])/(-2+3*c[4]), a[7,6] = 16/438615*(51300*c[4]^2-328320*b[7]*a[7,5]*c[ 4]^2-276480*a[6,5]*c[4]^2+2580480*a[6,5]*c[4]^2*b[7]-73359*c[4]+465120 *b[7]*a[7,5]*c[4]-4300800*a[6,5]*c[4]*b[7]+454656*a[6,5]*c[4]+26163-16 4160*b[7]*a[7,5]+1720320*b[7]*a[6,5]-180224*a[6,5])/b[7]/(4*c[4]-3)^2, b[6] = -64/855*(420*b[7]*c[4]-45*c[4]-420*b[7]+44)/(4*c[4]-3), a[3,1] = 2/27, a[6,1] = 1/768*(38400*a[6,5]*c[4]^2-4149*c[4]^2-51712*a[6,5]* c[4]+10017*c[4]-4104+17408*a[6,5])/c[4]/(25*c[4]-17), `b*`[4] = -1/145 8*(532*`b*`[3]-243+108*`b*`[5])/c[4]/(3-7*c[4]+4*c[4]^2), a[8,1] = 1/7 20*(420*b[7]*c[4]-420*b[7]+41)/c[4], a[8,7] = b[7], a[8,4] = -1/60*(42 0*b[7]-41)/c[4]/(-12+88*c[4]-177*c[4]^2+108*c[4]^3), a[6,2] = -27/64*( 71*c[4]-27)/(25*c[4]-17), a[5,1] = -2/9*(33*c[4]^2-45*c[4]+16)/c[4]/(2 5*c[4]-17), a[4,3] = 81/4*c[4]^3-9/2*c[4]^2, `b*`[8] = 1/1458*(-9396*` b*`[5]*c[4]^2+7047*c[4]^2+42282*b[7]*c[4]^2-19836*c[4]^2*`b*`[3]-2187* c[4]-55404*b[7]*c[4]+10564*c[4]*`b*`[3]+9180*`b*`[5]*c[4]-1944*`b*`[5] -1368*`b*`[3]+13122*b[7])/(264*b[7]*c[4]-29*c[4]+9-72*b[7])/(-1+c[4]), a[7,4] = -1/30780*1/c[4]*(-27648000*a[6,5]*c[4]^3+3939840*c[4]^3-1378 9440*b[7]*c[4]^3+258048000*a[6,5]*c[4]^3*b[7]+34473600*b[7]*c[4]^2-605 552640*a[6,5]*c[4]^2*b[7]-2736000*b[7]*a[7,5]*c[4]^2+64266240*a[6,5]*c [4]^2-8669700*c[4]^2-48939008*a[6,5]*c[4]+3912480*b[7]*a[7,5]*c[4]+464 486400*a[6,5]*c[4]*b[7]-28440720*b[7]*c[4]+6381207*c[4]-1395360*b[7]*a [7,5]-116981760*b[7]*a[6,5]-1571319+7756560*b[7]+12255232*a[6,5])/(144 *c[4]^3-248*c[4]^2+129*c[4]-18)/b[7]/(25*c[4]-17), a[8,3] = -729/3040* (60*b[7]*c[4]-20*c[4]+9-60*b[7])/(-2+9*c[4]), a[8,6] = -64/855*(420*b[ 7]*c[4]-45*c[4]-420*b[7]+44)/(4*c[4]-3), a[5,4] = 32/9*(-2+3*c[4])/(-2 +9*c[4])/c[4]/(25*c[4]-17), c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, `b*`[1 ] = -1/4374*(-1215*c[4]+162*`b*`[5]*c[4]+2394*c[4]*`b*`[3]-532*`b*`[3] +243-108*`b*`[5])/c[4], a[6,4] = -1/48*1/c[4]*(3200*a[6,5]*c[4]-684*c[ 4]-2176*a[6,5]+513)/(225*c[4]^2-203*c[4]+34), a[8,5] = 9/160*(-20*c[4] +420*b[7]*c[4]+27-420*b[7])/(-2+3*c[4]), b[1] = 1/720*(420*b[7]*c[4]-4 20*b[7]+41)/c[4], `b*`[7] = 1/243*(-17739*c[4]^2+14256*`b*`[5]*c[4]^2+ 30096*c[4]^2*`b*`[3]+12150*c[4]-14896*c[4]*`b*`[3]-13392*`b*`[5]*c[4]+ 1824*`b*`[3]-2187+2592*`b*`[5])*b[7]/(264*b[7]*c[4]-29*c[4]+9-72*b[7]) /(-1+c[4]), b[4] = -1/60*(420*b[7]-41)/c[4]/(-12+88*c[4]-177*c[4]^2+10 8*c[4]^3), a[7,3] = -1/28880*(27219780*c[4]^3-212874480*b[7]*c[4]^3+12 312000*b[7]*a[7,5]*c[4]^3-55296000*a[6,5]*c[4]^3+516096000*a[6,5]*c[4] ^3*b[7]-42144375*c[4]^2-1211105280*a[6,5]*c[4]^2*b[7]+128532480*a[6,5] *c[4]^2+303391620*b[7]*c[4]^2-25814160*b[7]*a[7,5]*c[4]^2-140551740*b[ 7]*c[4]+18016560*b[7]*a[7,5]*c[4]-97878016*a[6,5]*c[4]+21821139*c[4]+9 28972800*a[6,5]*c[4]*b[7]+24510464*a[6,5]+24562440*b[7]-233963520*b[7] *a[6,5]-4142988-4186080*b[7]*a[7,5])/(6-35*c[4]+36*c[4]^2)/b[7]/(25*c[ 4]-17), a[3,2] = 4/27, b[3] = -729/3040*(60*b[7]*c[4]-20*c[4]+9-60*b[7 ])/(-2+9*c[4]), `b*`[6] = -16/2187*(648*`b*`[5]*c[4]+504*c[4]*`b*`[3]- 486*c[4]-112*`b*`[3]+243-432*`b*`[5])/(4*c[4]-3), a[5,2] = -4*(-1+c[4] )/(25*c[4]-17), a[7,2] = -3/4*(264*b[7]*c[4]-29*c[4]+9-72*b[7])/b[7]/( 25*c[4]-17), a[6,3] = -9/256*(-13851*c[4]^2+19200*a[6,5]*c[4]^2-25856* a[6,5]*c[4]+12597*c[4]+8704*a[6,5]-3078)/(225*c[4]^2-203*c[4]+34), a[5 ,3] = 4*(18-71*c[4]+63*c[4]^2)/(-2+9*c[4])/(25*c[4]-17)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "t he coefficients in terms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do\n print(`======================================== ==============`);\n temp_eq := c[ii]=subs(e3,c[ii]);\n if lhs(temp _eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end if;\n print(``); \n for jj \+ to ii-1 do\n temp_eq := a[ii,jj]=subs(e3,a[ii,jj]);\n if l hs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a par ameter`);\n else\n print(temp_eq);\n end if;\n \+ print(``);\n end do:\nend do:\n`############################### #######################`;\n``;\nfor ii to 7 do\n temp_eq := b[ii]=su bs(e3,b[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(t emp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end if ;\n print(``);\nend do:\n`########################################## ############`;\n``;\nfor ii to 8 do\n temp_eq := `b*`[ii]=subs(e3,`b *`[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_e q)*` is a parameter`);\n else\n print(temp_eq);\n end if;\n \+ print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============ ==========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\" \"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%W=================================================== ===G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"\"#\"\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"$\"\"\"#\"\"#\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\"%\"#F" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%W======================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\"%0~is~a~parameterGF(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"%\"\"\",(*&#\"#FF'F(*$)&%\"cG6#F'\"\"$F(F(F(*&#\"\"*\" \"#F(*$)F/F6F(F(!\"\"F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,$*(\"\"*\"\"\",&*& \"\"$F,&%\"cG6#F'F,F,F,!\"\"F,)F0F(F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,&*&# \"#\")F'\"\"\"*$)&%\"cG6#F'F(F-F-F-*&#\"\"*\"\"#F-*$)F0F6F-F-!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%W======================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"\"#\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"& \"\"\",$*&#\"\"#\"\"*F(*(,(*&\"#LF()&%\"cG6#\"\"%F,F(F(*&\"#XF(F3F(!\" \"\"#;F(F(F3F9,&*&\"#DF(F3F(F(\"#F.&F%6$F'\"\"&F.F3F.F.*(\"& ce#F.FF(FKF(FF*(\")g =\">\"F(F1F(FKF(F(*(\");!yy*F(FCF(F5F(FF*&\")plW6F(F5F(F(*(\")+\\yWF(F 1F(F5F(FF**\"*+G(*G*F(FCF(F5F(F1F(F(**\")S5,7F(F1F(F>F(F5F(F(*(\"(?2z# F(F1F(F>F(FF*&\")k/^CF(FCF(F(*&\")?J^:F(F1F(F(*(\"*?N'RBF(F1F(FCF(FF\" (QE9$FFF(,&*&F8F(F5F(F(F9FFFFF5FFF1FF,&*&\"#DF(F5F(F(\"#s# F,)&%\"cG6#\"\"%F(F,F,*(\"*![uG@F,&%\"bG6#F'F,F2F,!\"\"**\")+?J7F,F9F, &F%6$F'\"\"&F,F2F,F,*(\")+gHbF,&F%6$\"\"'FAF,F2F,F<**\"*+g4;&F,FDF,F2F ,F9F,F,*&\")vV9UF,)F3\"\"#F,F<**\"+!G06@\"F,FDF,FKF,F9F,F<*(\"*![K&G\" F,FDF,FKF,F,*(\"*?;R.$F,F9F,FKF,F,**\")gT\"e#F,F9F,F?F,FKF,F<*(\"*S8d\"F2*&\"(glv(F,F@F,F,*&\")K_D7F,F6F,F,F,,**& \"$W\"F,F:F,F,*&\"$[#F,FGF,F2*&\"$H\"F,F/F,F,\"#=F2F2F@F2,&*&\"#DF,F/F ,F,\"#F.F4F.F.**\"(+3I%F.FDF.F4F.F;F.FA*(\"'cYXF.FDF.F4F.F.\"&jh#F.*(\"'gT; F.F;F.F>F.FA*(\"(?.s\"F.F;F.FDF.F.*&\"'C-=F.FDF.FAF.F;FA,&*&F7F.F4F.F. \"\"$FA!\"#F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================================ ======G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"\",$*&#F(\"$?(F(*&,(*(\"$?%F(&%\"bG6#\"\"(F(&%\"cG6 #\"\"%F(F(*&F0F(F1F(!\"\"\"#TF(F(F5F:F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,$*&#\"$H(\"%SI\"\"\"*&,**(\"#gF.&% \"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"#?F.F7F.!\"\"\"\"*F.*&F2F.F3F.F=F.,& \"\"#F=*&F>F.F7F.F.F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,$*&#\"\"\"\"#gF,*( ,&*&\"$?%F,&%\"bG6#\"\"(F,F,\"#T!\"\"F,&%\"cG6#F(F7,*\"#7F7*&\"#))F,F8 F,F,*&\"$x\"F,)F8\"\"#F,F7*&\"$3\"F,)F8\"\"$F,F,F7F,F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"&,$*&#\"\"*\"$g\"\"\"\"*&,**&\"#?F.&%\"cG6#\"\"%F.!\"\"*(\"$?%F .&%\"bG6#\"\"(F.F3F.F.\"#FF.*&F9F.F:F.F7F.,&\"\"#F7*&\"\"$F.F3F.F.F7F. F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',$*&#\"#k\"$b)\"\"\"*&,**(\"$?%F.&% \"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"#XF.F7F.!\"\"*&F2F.F3F.F=\"#WF.F.,&* &F:F.F7F.F.\"\"$F=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(&%\"bG6#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%W######################################################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\",$*&#F'\"$?(F'*&,(*(\"$?%F'&F%6#\"\"(F'&%\"cG6#\"\"%F'F'*&F/ F'F0F'!\"\"\"#TF'F'F3F8F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$ ,$*&#\"$H(\"%SI\"\"\"*&,**(\"#gF-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"#?F- F5F-!\"\"\"\"*F-*&F1F-F2F-F;F-,&\"\"#F;*&FF+F4F+&F%6#\"\"$F+F:*&\"%(=#F+F5F+F:*(\"&/a&F+F?F+F5F+F:*(\"&k0\"F +F5F+FEF+F+*(\"%!=*F+F1F+F5F+F+*&\"%W>F+F1F+F:*&\"%o8F+FEF+F:*&\"&AJ\" F+F?F+F+F+,**(\"$k#F+F?F+F5F+F+*&\"#HF+F5F+F:\"\"*F+*&\"#sF+F?F+F:F:,& F+F:F5F+F:F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK5_7eqs := [op(RowSumCondi tions(7,'expanded')),op(OrderConditions(5,7,'expanded'))]:\nsimplify(s ubs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "The stability function of a general 7 stage, order 5 Ru nge-Kutta scheme has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7 ;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F) *&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\" \"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]" "6#/ &%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"&F-&F/6$F1\"\"%F-&F/6$F 4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F-&F+6#\"\"(F-" }{TEXT -1 2 " (" } {XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*a[6,4]*a[4,3]* a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"(\"\"&\"\"\"&F&6$F)\"\"%F*&F&6$F-\" \"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F(\"\"'F*&F&6$F:F-F*&F&6$F-F0F *&F&6$F0F3F*&F56#F3F*F*%!GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[7,6]*a[6, 5]*a[5,3]*a[3,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,2]*c[2]+a[7,6]*a[6,5]*a [5,4]*a[4,3]*c[3]" "6#,(*,&%\"aG6$\"\"(\"\"'\"\"\"&F&6$F)\"\"&F*&F&6$F -\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-\" \"%F*&F&6$F>F3F*&F56#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-F>F*&F&6$F>F 0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "t[7]=b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"'F,&F.6$F0\"\"&F,&F.6 $F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9\"\"#F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 264 "eq7 := b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=269/1379840; \neq6 := b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]*(a[7,5]*a[5,4]*a[4 ,3]*a[3,2]*c[2]+\n a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3]*a [3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))))=17291/12418560;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6#\"\"(\"\"\"&%\"aG6$F *\"\"'F+&F-6$F/\"\"&F+&F-6$F2\"\"%F+&F-6$F5\"\"$F+&F-6$F8\"\"#F+&%\"cG 6#F;F+#\"$p#\"(S)z8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/,&*.&% \"bG6#\"\"'\"\"\"&%\"aG6$F+\"\"&F,&F.6$F0\"\"%F,&F.6$F3\"\"$F,&F.6$F6 \"\"#F,&%\"cG6#F9F,F,*&&F)6#\"\"(F,,&*,&F.6$F@F0F,F1F,F4F,F7F,F:F,F,*& &F.6$F@F+F,,&**&F.6$F+F3F,F4F,F7F,F:F,F,*&F-F,,&*(&F.6$F0F6F,F7F,F:F,F ,*&F1F,,&*&&F.6$F3F9F,F:F,F,*&F4F,&F;6#F6F,F,F,F,F,F,F,F,F,F,#\"&\"H< \")g&=C\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We shall require the following two equations to be satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve(simplify (subs(e3,\{eq6\})),a[7,5]));\neqn7 := op(solve(simplify(subs(e3,\{eq7 \})),a[7,5]));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&%\"aG6$ \"\"(\"\"&,$*&#\"\"\"\"-?n+JK[F.*,,J**\"1+?FUNA=MF.)&F'6$\"\"'F*\"\"#F .)&%\"cG6#\"\"%F=F.&%\"bG6#F)F.F.*&\"/+%=XCDX\"F.F9F.!\"\"*(\"/++KgV&z 'F.F5F.F9F.F.*(\"0+?nO#QiOF.F4F.F9F.FC**\"1?f=2aT1#*F.F4F.)F:\"\"$F.F> F.FC*(\"0+G&=@Tp;F.F5F.FJF.FC*(\"0?$)GBIEy*F.F4F.FJF.F.*&\"/7\\Ezb7PF. FJF.F.**\"1![O%G'\\\"[*)F.F4F.)F:F8F.F>F.F.*(\"0/.83*[\\%*F.F4F.FTF.FC *(\"0O6xJ=t]\"F.F5F.FTF.F.*&\"/K')>!>!*[$F.FTF.FC*(\"078978k*QF.F4F.F: F.F.**\"1!3'GIN(oq$F.F4F.F:F.F>F.FC*(\"/w`_RG!*eF.F5F.F:F.FC*&\"/4KT?s >9F.F:F.F.*&\"/k/ZPfHdF.F4F.FC*(\"0?Nwmd\"paF.F>F.F4F.F.*&\".oJq8wJ)F. F5F.F.\".hr#*e')3#FCF.,&*&F=F.F:F.F.FKFCFC,**&\"$D#F.FJF.F.*&\"$\"QF.F TF.FC*&\"$3#F.F:F.F.\"#OFCFCF>FCF5FCF.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/&%\"aG6$\"\"(\"\"&,$*&#\"\"\"\".!)o-CH$>F.*.,B*(\"/;?uA kzAF.&F'6$\"\"'F*F.)&%\"cG6#\"\"%\"\"$F.!\"\"*(\"0CQHhnUN\"F.)F4\"\"#F .F7F.F.**\"1!)[6*yOhF\"F.F@F.F7F.&%\"bG6#F)F.F=*(\"/ci*3Q5f\"F.F4F.)F8 FAF.F.**\"0![okp%Hs*F.F@F.FIF.FDF.F.*(\"0s]B\">QC5F.F@F.FIF.F=*(\".39? " 0 "" {MPLTEXT 1 0 38 "factor(simplify(rh s(eqn6)-rhs(eqn7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"%R:\",! G\\OI6\"\"\"*4,&*&\"#DF(&%\"cG6#\"\"%F(F(\"# " 0 "" {MPLTEXT 1 0 143 "denom_factors := select (has,map(op,select(has,map(denom@rhs,e3),c[4])),c[4]):\nmember(25*c[4] -17,denom_factors);\nmember(4*c[4]-3,denom_factors);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "This l eaves the possibilities " }{XPPEDIT 18 0 "c[4]=21789/78848" "6#/&%\"c G6#\"\"%*&\"&*y@\"\"\"\"&[)y!\"\"" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "c[4]=3/7" "6#/&%\"cG6#\"\"%*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT -1 38 ". As before we take the latter value." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting for " }{XPPEDIT 18 0 "c [4]" "6#&%\"cG6#\"\"%" }{TEXT -1 32 " in each of the equations for \+ " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 29 " now g ives an equation for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"& " }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\" (" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"& " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "simplify(subs(c[4]=3/7,eqn6));\nsimplify(subs( c[4]=3/7,eqn7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\" &,$*&#\"\"\"\".+glQ?T\"F,*(,**(\"/+g,1+t>F,&%\"bG6#F'F,)&F%6$\"\"'F(\" \"#F,F,\",pSq2;&!\"\"*&\"-S#yj)HBF,F6F,F,*&\".+w&3sJ?F,F5F,F;F,F2F;F6F ;F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,$*&#\"\" \"\".+glQ?T\"F,*(,**(\"/+g,1+t>F,&%\"bG6#F'F,)&F%6$\"\"'F(\"\"#F,F,\", pSq2;&!\"\"*&\"-S#yj)HBF,F6F,F,*&\".+w&3sJ?F,F5F,F;F,F2F;F6F;F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "dd := simplify(subs(c[4]=3/7,rhs(eqn6))):\ne4 := \{c[4]=3/7,a[7,5 ]=dd\}:\ne5 := `union`(e4,simplify(subs(e4,e3))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1713 "e5 := \{c[8] = 1, b[2] = 0 , a[8,2] = 0, a[2,1] = 1/6, c[7] = 1, `b*`[2] = 0, c[2] = 1/6, a[4,1] \+ = 183/1372, `b*`[8] = 1/5832*(-540*`b*`[5]-729+5832*b[7]+988*`b*`[3])/ (12*b[7]-1), a[7,4] = -49/32451674112000*(-3192704204800*a[6,5]^2+4039 43915520*a[6,5]-2118057984000*b[7]*a[6,5]+31004295168000*b[7]*a[6,5]^2 -21025360917)/b[7]/a[6,5], a[7,1] = -1/13979182694400*(-6883688448000* b[7]*a[6,5]+1874481315840*a[6,5]+217030066176000*b[7]*a[6,5]^2-6307608 2751-22348929433600*a[6,5]^2)/b[7]/a[6,5], c[4] = 3/7, a[3,1] = 2/27, \+ `b*`[7] = 1/486*(243+540*`b*`[5]-988*`b*`[3])*b[7]/(12*b[7]-1), a[8,7] = b[7], a[7,3] = 1/4487145062400*(4433801379840*a[6,5]-34100733542400 *b[7]*a[6,5]+217030066176000*b[7]*a[6,5]^2-89634433383-22348929433600* a[6,5]^2)/b[7]/a[6,5], a[7,5] = 1/1412038656000*(19730006016000*b[7]*a [6,5]^2-51607704069+232986378240*a[6,5]-2031720857600*a[6,5]^2)/b[7]/a [6,5], a[7,2] = 9/22*(12*b[7]-1)/b[7], a[7,6] = 7263/358400*1/(b[7]*a[ 6,5]), c[3] = 2/9, c[5] = 2/3, c[6] = 3/4, a[4,2] = -162/343, b[5] = - 1161/800+189/10*b[7], b[6] = -7168/513*b[7]+11072/7695, a[8,1] = -7/9* b[7]+287/2160, `b*`[4] = -45619/39366*`b*`[3]+343/648-343/1458*`b*`[5] , a[6,1] = -10/9*a[6,5]+195/704, a[6,2] = 81/352, a[8,4] = -16807/1755 *b[7]+98441/105300, a[4,3] = 1053/1372, a[5,1] = 68/297, a[8,3] = 2187 /494*b[7]-2187/39520, a[8,6] = -7168/513*b[7]+11072/7695, a[5,4] = 196 0/3861, `b*`[1] = 4/27+5/243*`b*`[5]-1729/6561*`b*`[3], a[6,4] = -392/ 117*a[6,5]+8379/9152, a[8,5] = -1161/800+189/10*b[7], b[1] = -7/9*b[7] +287/2160, b[4] = -16807/1755*b[7]+98441/105300, a[3,2] = 4/27, b[3] = 2187/494*b[7]-2187/39520, a[5,2] = -4/11, `b*`[6] = -640/729*`b*`[5]+ 11648/19683*`b*`[3]+16/81, a[5,3] = 42/143, a[6,3] = -1539/2288+45/13* a[6,5]\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "We now have a scheme with just the tw o parameters " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 11 ". Setting " }{XPPEDIT 18 0 "b[7]=7267/94080" "6#/&%\"bG6#\"\"(*&\"%ns \"\"\"\"&!3%*!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]=4617/ 20480" "6#/&%\"aG6$\"\"'\"\"&*&\"% " 0 "" {MPLTEXT 1 0 169 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded' ):\nsm := 0:\nfor ct to nops(errterms5_7) do\n sm := sm+simplify(sub s(e5,errterms5_7[ct]))^2;\nend do:\nsm := simplify(sm):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "sm := 1/223376328345814 268949477457920000*(284971295990003535055945728*a[6,5]^2-1306329072953 2178878955520*a[6,5]+1446793686458671066378281+54178151685970270082826 240000*a[6,5]^3*b[7]-5579059667662414716862464000*a[6,5]^3+45879731972 94995521929216000000*a[6,5]^4*b[7]^2-944904003728612172968755200000*a[ 6,5]^4*b[7]+48651307334836281524879360000*a[6,5]^4-2487212831475813039 231467520*b[7]*a[6,5]^2+16195804484028550022902579200*b[7]^2*a[6,5]^2) /a[6,5]^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indets(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"aG6$\"\"'\"\"&&%\"bG6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "plot3d(sm,b[7]=-0.05 ..0.15,a[6,5]=0.1..0.7,labels=[`b[7]`,`a[6,5]`,``],grid=[40,60],style= patch,linestyle=2,\n lightmodel=light2,color=COLOR(RGB,1,.25,0),axes= boxed,orientation=[-50,62],view=[-0.05..0.15,0.1..0.7,0..4e-6]);" }} {PARA 13 "" 1 "" {GLPLOT3D 509 487 487 {PLOTDATA 3 "6*-%%GRIDG6&;$!\"& !\"#$\"#:F);$\"\"\"!\"\"$\"\"(F/X,%)anythingG6\"6\"[gl'!%\"!!#[at\"I\" gn3ECA9D9AE7F395983ECF906F9D88B5BB3ED2C04CAD0F4FDA3ED630728E1858B73EDA 147DEF6B2DA63EDE69B5AF6651353EE19722D70517CD3EE4307600560DD13EE7006359 3AF08D3EEA069A28DF34363EED42DFDC04C8C13EF05A84AE94D9AD3EF22E7B4D728E75 3EF41D47BB7CAB633EF626E0D1AFF2683EF84B3F8FA777093EFA8A5E90CFCEC93EFCE4 39A793930F3EFF58CD9345B00B3F00F40BE48F94223F02490B258101923F03AB63C424 6F063F051B1522FF837B3F06981EC64DEB603F0822804CF0AA583F09BA396AEE316F3F 0B5F49E523FE363F0D11B18DE2CABD3F0ED1704240000F3F104F42F3FA0AAD3F113C79 35D38F973F12305ADFC9FE803F132AE7ED37F9AC3F142C205B444A8B3F15340428867B 543F16429354BCC3613F1757CDE08FDE453F1873B3CD621B143F1996451D277EBC3F1A BF81D24543063F1BEF69EF77524A3F1D25FD77BAA5B83F1E633C6E3BA1623F1FA726D6 47B5273F2078DE59A0D64F3F21217F044C19AA3F21CD756CDF14D13F227CC195123BFD 3F232F637E9A8EEC3F23E55B2B27A26F3F249EA89C6215443F255B4BD3EA5D8F3F261B 44D357DF373F26DE939C383E903F27A538300EE4373F286F329054A9C53F293C82BE77 A7893F2A0D28BBDB1D833F2AE12489D7723D3F2BB87629BA43853EC839E4563A765B3E CCB6742FEF8D2E3ED111230CAFFF023ED43813888393633ED7CBE008FE3BED3EDBC9CF 6C7E7B2B3EE01806C9825EAB3EE27EAC48F32A9B3EE518673A26874E3EE7E4E6E447E8 FB3EEAE3F0B4193F923EEE1559941890443EF0BC80B96E81FB3EF28768191475FF3EF4 6B59C1FAE6213EF6684EB3BCE5E43EF87E4189C70ACA3EFAAD2E1683EE3D3EFCF51119 467C6A3EFF55E80747B7DC3F00E7D87134D3FB3F0231350D03750F3F03870939956628 3F04E9547B2653B33F065816709741553F07D34ECDEEA0153F095AFD5809ED8D3F0AEF 21E139E3C33F0C8FBC4693EBCC3F0E3CCC6DD07CCB3F0FF6524397AB5A3F10DE26DD12 2B733F11C75F6415E1FE3F12B6D2B3FDC2BD3F13AC80CB6157EA3F14A869A9FED8D93F 15AA8D507F01223F16B2EBC0441FD63F17C184FB4239E73F18D65903DE89183F19F167 DCD4F7C53F1B12B189227F223F1C3A360BF383383F1D67F5689573EA3F1E9BEFA26B1C D03F1FD624BCE329793F208B4A5DB83D1B3F212E9FD0C1FBC13F21D512B943D06B3F22 7EA318ED4FE83F232B50F16718FB3F23DB1C4451A1C13F248E0513444E273F25440B5F CCC27E3F25FD2F2B6E67653F26B97077A216703F2778CF45D5E7F23F283B4B976D1BEC 3F2900E56DC018E73F29C99CCA1C7CB23EC5F5914964592E3ECA00D1A235E3333ECEED BBEABF10E03ED2578E90BDA00A3ED59E4F20C0F0A23ED9486683C7FCBE3EDD540099D3 30963EE0DFEDBC1BEF7C3EE3458A950FE9D93EE5DA861EF0FB993EE89EA5C68114AC3E EB91BE763E3A3F3EEEB3B01CBF953F3EF102314DE3BB7D3EF2C1E1D2A7B1AF3EF498E2 9C46C0B43EF6872E462D7E123EF88CC0A2C6832E3EFAA9967164BC3A3EFCDDAD27412B C13EFF2902C63DD9253F00C5CADEDAB0E73F0202B2695989D63F034B3786D723D53F04 9F59D63482843F05FF190B7816EC3F076A74EB7F5EA43F08E16D489B13B63F0A6401FF E09F343F0BF232F70878433F0D8C001ABAB37D3F0F31695D32304B3F1071375A9219A0 3F114F880E6F279C3F1233A6C8C7CC513F131D93895A3F183F140D4E50CF3B843F1502 D7208910A83F15FE2DFA7BC3773F16FF52E10C8DB23F180645D6F759B93F191306DF39 20BA3F1A2595FCFE46C13F1B3DF333943BB43F1C5C1E865DCB253F1D8017F8C9A0A73F 1EA9DF8E4A9C8B3F1FD9754A51A74A3F20876C982460303F212505A1C788B43F21C585 C3BAEBF23F2268ECFF9F000E3F230F3B570B28EF3F23B870CB8D0AE63F24648D5EA80E 943F25139111D50D8D3F25C57BE68220263F267A4DDE12865B3F273206F9DEA6B83F27 ECA73B341F0B3EC3D0A1C1713E153EC76F87F45BB7CC3ECBE2FACE3BD8443ED08EE3A6 C67EB13ED38BCB36B34BC83ED6E57AF542D5F23EDA9A1EC27589663EDEA874B3A0B8F0 3EE187CD69F7182F3EE3E777D8DA6C113EE672FF133C48093EE92A38039AB14C3EEC0D 02988CD0BF3EEF1B46B3D4F7B63EF12A7903B6550E3EF2DCFB494507783EF4A524C603 289F3EF682F14C5B51E43EF8765D9BA06F823EFA7F67290B83FA3EFC9E0BF67E96B03E FED24A7354450F3F008E10B24BEE863F01BDC7E9605BC53F02F84A7DC86DE53F043D98 238A95EF3F058DB09F84517C3F06E893C4065A953F084E416E261A493F09BEB9839C07 C13F0B39FBF11037953F0CC008A8BD89323F0E50DFA15941253F0FEC80D530F24C3F10 C97620B9D88C3F11A210F2CEF61E3F128010E1808D6A3F136375EE30ED8B3F144C401A D41B6E3F153A6F69CF50D53F162E03DDDE78203F1726FD79FE8A823F18255C415BEC02 3F19292037440C833F1A32495F19B79E3F1B40D7BC4B98E13F1C54CB524C90A23F1D6E 24248D87563F1E8CE236787C743F1FB1058B6C99A53F206D47135D8E2F3F2104BE05D2 78763F219EE79EAC6F903F223BC3DF7917CB3F22DB52C9BBD9C93F237D945EED8F1F3F 242288A07C50213F24CA2F8FCB5CCD3F2574892E331BAD3F2621957D012A8F3EC1CB15 BE6125113EC5029726610AF93EC90202C3D654363ECDBC25953C5EA83ED194544AD54D 5E3ED4A10CC0EF06C63ED802680CEBC7BF3EDBB7244420E3183EDFBE5F71B8249F3EE2 0BBC12043A613EE460FC9A4AD9B03EE6DEC63C2DF56C3EE984F8E644B67D3EEC537C7A 516E373EEF4A3EAA4DA0863EF13498BAB7BA323EF2D82509480A703EF48FC013425A62 3EF65B6697F9963A3EF83B160CA6C0813EFA2ECC732BE09B3EFC36883AE393ED3EFE52 4828D928713F004105A2C1FB823F0162E8675303773F028ECC16261D213F03C4B07418 C6153F050495537BB85E3F064E7A91645D0C3F07A260138B2B483F090045C69837A93F 0A682B9CC661983F0BDA118CCAEDA93F0D55F790F36EB83F0EDBDDA66EF9373F1035E1 E65CFDE83F1102D50292F6D73F11D4C8293BB67D3F12ABBB5C4B41CA3F1387AE9E26D2 813F1468A1F18A53003F154E955972BC793F163988D90C72F53F17297C73A4E65A3F18 1E702C9EE2383F191864076912283F1A1758077656783F1B1B4C3037979E3F1C244085 16D5163F1D3235097338833F1E4529C09DFF613F1F5D1EADD815F23F203D09EA282208 3F20CE049B90E92B3F21617F6CA9C9033F21F77A5EEB9B273F228FF573C477E73F232A F0AC979F423F23C86C0ABD77C53F2468678F839F3D3EBFC9DA80681C3E3EC2B9FF3845 DCBA3EC64AD3CB8E84AD3ECA8A37F88963E73ECF6FD4BA4DEAC93ED27B1BE6CC8F3B3E D58CDC7935EBA43ED8EBEA29B85D723EDC9763037DB0743EE04752CA6E668C3EE2689E 5BACC99D3EE4AF691FF8069F3EE71B9305E746783EE9AD03EF3CDA7C3EEC63A98DF246 933EEF3F75E13DB1C33EF1202F0FFC23883EF2B32CF77B9CA43EF458B1667030683EF6 10B9D212E15B3EF7DB443C45B6E13EF9B84F14634E623EFBA7D91F76F5D63EFDA9E165 F8061C3EFFBE6725A886773F00F2B4E34AAC7F3F020F74693CBC6E3F033571F6FB2D11 3F0464AD699B677D3F059D26A6D5E2D23F06DEDD9B52B3B03F0829D2394CB97F3F097E 04777938CD3F0ADB745025C47D3F0C4221C08170F83F0DB20CC808ACF33F0F2B35680C EF933F1056CDD1A96B7F3F111C9FBEE1368E3F11E7107E1312B53F12B62011FAEA563F 1389CE7D95B6A03F14621BC40FDBA03F153F07E8B6C9373F162092EEED4AFA3F1706BC DA220C7B3F17F185ADC7EE0E3F18E0ED6D4FD8273F19D4F41C23CA463F1ACD99BDA2EE 063F1BCADE551E80E93F1CCCC1E5D76F303F1DD34472FC80B13F1EDE65FFA8FE0F3F1F EE268EE3B8863F20814311CF31A03F210DC2605A97743F219C9134774DBC3F222DAF8F 7DBB023F22C11D72BB7D173EBC3C508DD3F2813EC095C02A0A2D0C3EC3BD6DE56469B4 3EC787FE77740D1D3ECBED1ADB5087B93ED073A866DB6F4C3ED3397C0753F5143ED646 C6646728003ED99AA5893ED3E03EDD34780431E1213EE089E4576217D23EE29C20AEF8 E4E23EE4D0D0F77480AC3EE727DD12973C813EE9A132B25A9C4D3EEC3CC3D5F4C70D3E EEFA85B43EE7CD3EF0ED37F90718AA3EF26E3E07043E063EF40052794FE6853EF5A373 51CC19853EF7579EFFD374703EF91CD44871453D3EFAF312341CE4CF3EFCDA5800985A 633EFED2A515F088173F006DFC7EF034863F017B29AE84B8B03F0290D9F6CB39993F03 AF0D3D7C2E663F04D5C36F3FABB13F0604FC7E5090E53F073CB8616422923F087CF712 C7F3973F09C5B88FAB18613F0B16FCD789FFA13F0C70C3EBB6207E3F0DD30DCEF41921 3F0F3DDA852BF36F3F105895099411723F11167E3F303E233F11D8A8E66778F83F129F 15026626003F1369C29679B5193F1438B1A604F1DC3F150BE2347687DC3F15E3544541 57653F16BF07DBD648F33F179EFCFB9F5C013F188333A7FBBA2D3F196BABE43CA0F73F 1A5865B3A2FCA23F1B4961195D95053F1C3E9E1887B2BE3F1D381CB428270C3F1E35DC EF30A51D3F1F37DECC7D5D943F201F11276A68383F20A453BC73E5623F212BB726A8C4 193EB8ED8DA505CCEB3EBD2BB3F75BF7E13EC159D1115803433EC4B57911FC5A513EC8 A07AF8B271873ECD156482374DFE3ED10846B745E40D3ED3C7B8F42D42BD3ED6C82702 FB8EDF3EDA08EF7207B0DD3EDD899D1AD5889B3EE0A4ECE93090393EE2A4B2BAEC651D 3EE4C407E46094483EE702DA1786A1AB3EE9611B5394B43E3EEBDEC0CF63F7123EEE7B C22FC99CEE3EF09C0C79B5BF1C3EF209E0025DCFFF3EF38759B3BF08853EF51477FD34 06163EF6B139A3C816A73EF85D9DAFF2931C3EFA19A35F7582B43EFBE54A1A5DC7893E FDC0916A665CC03EFFAB78F430B6743F00D30038F3D3623F01D813D77E0DFE3F02E4F7 425F1FA93F03F9AA6BD1E7CB3F05162D4A8BAAF63F063A7FD8D9FC093F0766A213EBEF 703F089A93FB3DF3E03F09D655902180773F0B19E6D55B33663F0C6547CED316903F0D B87881539D6F3F0F1378F2549CCB3F103B2493E803813F10F074940F52153F11A9AC7C EDAA043F1266CC51E5D6E23F1327D4166684463F13ECC3CDE2927B3F14B59B7BCAE9FE 3F15825B23898A473F165302C87D9CF73F1727926DF85F8D3F18000A173ABE493F18DC 69C77381093F19BCB181BDF0643F1AA0E14920DD9E3F1B88F9208DFBE13F1C74F90AE1 7BD13F1D64E10AE1DD6E3F1E58B1233FEDCD3F1F50695696E88B3EB5DD91C5FDAB7D3E B974995A6292D23EBE3FFA9ED2A2B73EC212A7C8224B7D3EC589F51273A8373EC98072 EB1A6CA13ECDF279121771243ED16EC1D90AADAC3ED41FE770B3E1703ED70C0BDE5E3C 4B3EDA32B9FB8D9E1F3EDD939B9D3E11433EE09738504EF3C83EE281846498E1D33EE4 889FBD7656B03EE6AC7C5A1D795D3EE8ED0F716774E53EEB4A50A8297C0E3EEDC4397D 0967433EF02D626D3C9DCB3EF186F7621E83E23EF2EEDA0C8503533EF46509317B6A14 3EF5E983D97911053EF77C49423FFF693EF91D58D3DD17563EFACCB2180B53F63EFC8A 54B36C295C3EFE5640602A69B33F00183A74DB819E3F010C7914B10F973F0207DC01D0 BE313F030A6332EFD1FB3F04140EA25BDDD23F0524DE4D43F6273F063CD2332489A73F 075BEA554F0F7A3F088226B68825CB3F09AF875AB7D67C3F0AE40C46A8950B3F0C1FB5 7FD2363C3F0D62830C2EAC723F0EAC74F216BFBE3F0FFD8B38254FE73F10AAE2F28FFA 0C3F115A927FF201BC3F120DD447AB9F533F12C4A84D2DBB4A3F137F0E93E255193F14 3D071F2896653F14FE91F251BCA93F15C3AF109EB4293F168C5E7D3E44BE3F1758A03B 4BB7013F1828744DCDDC3A3F18FBDAB7B6678C3F19D2D37BE189A23F1AAD5E9D15C274 3F1B8B7C1E03DF1F3F1C6D2C01471B3B3EB30C5CF0BB8E343EB606307D282AE83EBA1F E53F30A8023EBF3F1533CBC1453EC2A98928942BC73EC6287C08603A843ECA18BAF94A E5413ECE77C225FED19D3ED1A1E6D267CB983ED43DCD4935836F3ED70F1F50EC702F3E DA1586BE889C393EDD50C36F38595E3EE06052934025213EE23283A429BB5D3EE41EE6 E98F16643EE625719A4961413EE8461B5B2DCEBA3EEA80DDAAE237383EECD5B373D89F CC3EEF4498B9D517383EF0E6C52DC66C2B3EF23842F18B3F833EF396C4B0B05E8A3EF5 0249A8F7D0803EF67AD1426E777F3EF8005B06CF4EAA3EF992E69ABBCA1C3EFB3273B8 5EBBF93EFCDF022B2912863EFE9891CC6AF6FC3F002F91404D14183F01195A1A90979E 3F0209A36F4D98F13F03006D3BB32C823F03FDB77F3DC0FC3F0501823B3ECD883F060B CD727AF04D3F071C9928DA33343F0833E5632709B73F0951B226D9489A3F0A75FF79EA E2413F0BA0CD62B49EBE3F0CD21BE7D15DD43F0E09EB1006B6B23F0F483AE232007E3F 104685B29C7DEA3F10EC2E4FFEBCD43F1195174CA9BC793F124140ABFCA6763F12F0AA 7148B84C3F13A3549FCEDE3E3F14593F3ABDE0243F15126A453106993F15CED5C22F22 DD3F168E81B4A9E81E3F17516E1F7D87013F18179B05707F833F18E10869339EB93F19 ADB64D62203F3EB079EF253F750F3EB2E0795FACC0253EB6536203CA165D3EBAB8430E 8E33813EBFFE6E7627F86E3EC30D7FDA08B7A53EC68353242624753ECA5E2D4416E842 3ECE9C4A502E9AA93ED19E33B28D86433ED41ECD1AF1FECF3ED6CF9B3640C1523ED9B0 5DE1A81FA13EDCC0E4E0ACBC5E3EE00085CBA0CFB13EE1B85B01E98B553EE387E74A09 BC273EE56F2248D694EF3EE76E057CF5EE133EE9848BD0D9CCA43EEBB2B148463F683E EDF872C1F081303EF02AE6E3F796F53EF1656035987BAB3EF2ABA4939CF5FC3EF3FDB3 6611E8013EF55B8C36B24CE03EF6C52EAA1F98B43EF83A9A7A849D993EF9BBCF735249 DD3EFB48CD6DD8C6B83EFCE1944E8DD2FC3EFE862402DBF7C43F001B3E3FAF2D6A3F00 F94EDF3992833F01DD43DF8999DD3F02C71D41F0BA9F3F03B6DB093392F03F04AC7D39 3A2CBA3F05A803D6CEFB733F06A96EE769D3E53F07B0BE7104A8703F08BDF279F84126 3F09D10B08DF7DCE3F0AEA08247FF5933F0C08E9D3B6FF9B3F0D2DB01D6A5C843F0E58 5B087BDD423F0F88EA9BBF80C83F105FAF6EF9CD2B3F10FDDBEADD52773F119EFAC4CB 3C883F12430BFFF2533A3F12EA0F9F6DDF253F139405A644B18A3F1440EE17687D973F 14F0C8F5B573F33F15A39643F214953F16595604CF2C993F1712083AE7F7993EAC4C90 C712C0283EB0037401F052873EB2DA70EC9EEDCF3EB690D9208BEDB23EBB15FE93E633 0B3EC02F7E6013E4063EC3324192A92EBD3EC690C50C5D9F493ECA4944E384CD493ECE 5A7E34CC89993ED161C3599E49FC3ED3C1D90466808F3ED64D3FF7ED3A5C3ED903C7F7 B71A003EDBE54C67B7275B3EDEF1B14659B0623EE1147080A8859A3EE2C5657123CEAE 3EE48BB0F3448BD63EE6674DF17CC21F3EE858386F9080503EEA5E6D4C3501403EEC79 EA1180E0D03EEEAAACD062D0CE3EF0785A022F6FDC3EF1A5FF3EC768DF3EF2DE45A7B4 4E973EF4212CE197951E3EF56EB4A69C0E953EF6C6DCC232A9443EF829A50DAB8E633E F9970D6D7C7CC63EFB0F15CF0FFD8A3EFC91BE270136703EFE1F066FAE50553EFFB6EE A81028963F00ACBB6964D6BF3F01834F7AB20DB23F025F338BD7C30B3F034067A1A06A 403F0426EBC183D81D3F0512BFF17BFF003F0603E437E1A6FB3F06FA589B4FAFD43F07 F61D228BB0B93F08F731D47300CE3F09FD96B7EB60B33F0B094BD3D6A15A3F0C1A512F 08C1B73F0D30A6D04015093F0E4C4CBE1F16503F0F6D42FF279E123F1049C4CCDB9E00 3F10DF904A0240A83F117803FA0E88413F12131FDFF227FA3F12B0E3FE8950763F1351 50589A81B13F13F464F0D688C33F149A21C9D8A1493EA822D157329E7C3EAADE40C7E5 C41E3EAF6A23F35E5CAC3EB2C8D769C4EFD63EB699C2AA63076D3EBB1CEF35037F483E C0258644D4041C3EC30F897ED2F6B63EC64ABD5ED22F123EC9D5DF017F7E133ECDB004 19E2A36D3ED0EC4028F9D9F23ED32769B207A98C3ED5894E6B9F63293ED811C9B9B40E A23EDAC0BF9AB1F9EE3EDD961A7C4B7B323EE048E4D4157BF73EE1D9E00DCE107D3EE3 7DF9D5C1803A3EE5352E2FB3D9F33EE6FF79FA5A587C3EE8DCDABFCB97BD3EEACD4E90 F6497F3EECD0D3E95E7C3F3EEEE769991DF42F3EF0888759D553D03EF1A6E14123BF60 3EF2CEC23CA50EEA3EF4002A17CA30B53EF53B18ABE34DFD3EF67F8DDD6625913EF7CD 8999BD40903EF9250BD583C4BD3EFA86148B17DAEF3EFBF0A3B972608A3EFD64B96336 43D93EFEE2558DECC12B3F0034BC20B2F62A3F00FD10C39B561E3F01CA28B52755403F 029C03FB50E5EF3F0372A29C70D0393F044E049F21F3E93F052E2A0A29E8253F061312 E466041B3F06FCBF34BC08633F07EB2F020DC5F23F08DE62532F3BBD3F09D6592EDEBD 013F0AD3139BBEC4C23F0BD491A0512B7E3F0CDAD342F380F13F0DE5D889DC56443F0E F5A17B194E043F1005170E46E7433F1091BF39F91C893F1120C94369C6D13F11B2352D 49B3343F124602FA341D4C3EA4769FFADE851E3EA646FD0B68DD7C3EA9C68A55F5AFE1 3EAEC07BD47273DE3EB289BAB99E758F3EB654D712A495073EBABA42754D491E3EBFB4 F536EDDD0A3EC2A0B3C216C0073EC5AE89CB33E9F23EC9031269D62C023ECC9DA147F5 9AEE3ED03EDB0FF76D363ED251783C6597D43ED486838D3855353ED6DDE100DBF3513E D9577B0502C8443EDBF340E35739943EDEB1259924F8163EE0C88F7DA806F63EE24992 88B04C4F3EE3DB98CC6086EC3EE57E9FD2CF52B03EE732A5ACEB61643EE8F7A8D638C1 8C3EEACDA81ED137583EECB4A29A2AB9123EEEAC9791882EEE3EF05AC33C9F9E993EF1 67B77418E0363EF27D28488005853EF39B159E4ACD5D3EF4C17F62E3C0D63EF5F0658A E605B73EF727C810AFC4D83EF867A6F339DB983EF9B002352738443EFB00D9DC011731 3EFC5A2DEF978C283EFDBBFE797F7E1F3EFF264B84A896A43F004C8A8E835D3E3F010A 2DA7A5BCE33F01CC0F14564A093F02922EDB5A9BDB3F035C8D0390097E3F042B2993DC 53923F04FE0493214B0A3F05D51E0832EEDB3F06B075F9CF92433F07900C6E99B0403F 0873E16D1321593F095BF4FB9975423F0A484720633D2C3F0B38D7E17E1B993F0C2DA7 44CD76E83F0D26B55009B05F3F0E240208BFC7F53F0F258D745157D93F1015ABCBFA6B A43EA147FCB216740C3EA2411CCE69F1223EA4CA150103D5413EA8AE1943D197F23EAD CBCD8330FAEA3EB206B4590B09413EB5B224E84220303EB9E330C4930D713EBE96501A A500483EC1E47E91E9CD373EC4BCB1A3172DAB3EC7D314EAD2B6493ECB272823790AA7 3ECEB88AD413700B3ED14379E243FB1B3ED3491578D79C853ED56D029B76F26C3ED7AF 3093CC624E3EDA0F925F239D043EDC8E1DD260ACA73EDF2ACAF50BAEC93EE0F2C9C247 8C893EE25F394A8C11A53EE3DAB2244218833EE56532CAEDAFA13EE6FEBA0EA89B343E E8A74702E8D1873EEA5ED8F0F13ACB3EEC256F4D177B453EEDFB09AE3D6F813EEFDFA7 C70369FB3EF0E9A4B02A74273EF1EAF72A837E5D3EF2F3CB4727F95F3EF4042100760E 0B3EF51BF8556699BF3EF63B51489C8AC73EF7622BDFA11D733EF89088224465973EF9 C6661A1B4A1F3EFB03C5D21574A03EFC48A75626C9DA3EFD950AB300D9ED3EFEE8EFF5 D9646D3F00222B961DCBD53F00D3A031F110FB3F0188D5D54C42453F0241CC871130A5 3F02FE844E13DB113F03BEFD311294CA3F04833736AFD8CD3F054B32656D7F9F3F0616 EEC3A918F53F06E66C579935FF3F07B9AB274B79423F0890AB38A349193F096B6C9159 06CE3F0A49EF36F9B2543F0B2C332EE6E5DC3F0C12387E5718A53E9D2DCEF9B4D6933E 9D994021D1FE393EA074C3F488CCCC3EA35A8721A74BED3EA75C8D84A23E2C3EAC650E 106DB7FC3EB132B3E2868D693EB4A9C5A6957E9B3EB89434810ADECF3EBCEF7AAB424F C33EC0DCE1C5A5A8773EC378DB3A8B05E43EC64B296EADE3CF3EC9536BE91787763ECC 915971AE009B3ED0025D02A4F58E3ED1D6B13FA7F9A93ED3C598B98A721A3ED5CF066D 980FBC3ED7F2F030B4DCA43EDA314E0A68F66D3EDC8A19B81ED2AE3EDEFD4E4E03A942 3EE0C573F6FA6ED53EE21971C77D467F3EE37A9F68A41FC73EE4E8FBEDE4F0FE3EE664 86A082A2513EE7ED3EF4D2D80B3EE9832481B76EB83EEB2636F9D0B8C93EECD676260A 33E33EEE93E1E138F2453EF02F3D0A499FB73EF11B1F5A6AB68C3EF20D97DFF89AFE3E F306A69D963B5B3EF4064B98CCD3F53EF50C86D96C789E3EF61958690A10423EF72CC0 529544753EF846BEA201F9F73EF967536401C0EB3EFA8E7EA5CA58E13EFBBC4074E6F0 2B3EFCF098DF12351B3EFE2B87F217A8EA3EFF6D0DBBBAED7D3F005A9524D200613F01 01EED4A7C49B3F01AC93F4013E693F025A84896046503F030BC09B226C0B3F03C0482F 7E40C03F04781B4C8166FD3F053339F80F45183F05F1A437E03C5B3F06B35A11814CBF 3F07785B8A54106E3F0840A8A78EFEAA3E98C6C0B654D5A93E97D30DA5CC0EB63E998D 2E61092D003E9D8B8ADBE71F9D3EA1C5B57790B4F43EA5B09E40501A673EAA77DEC835 219F3EB008B3DCF5308C3EB33B14B75F1BAA3EB6D08C2CB3F3EA3EBAC745A30338B83E BF1DE86E3D13923EC1E9BA018D65E43EC47393B7D775EA3EC72C3821E2C99C3ECA136F 3C87FCE03ECD290DE32BBC013ED03679549168FF3ED1EF81C48250413ED3BF96164C9D E53ED5A6AE51786F7C3ED7A4C433703AAB3ED9B9D2D06137423EDBE5D64A28C8C93EDE 28CB97CF0C443EE041582CC3C5103EE179C15B1F17753EE2BDA0A03C65873EE40CF570 7153883EE567BF629FBE013EE6CDFE2967F7743EE83FB18DB57D783EE9BCD96A5D624E 3EEB4575A895F1793EECD9863D1B7CB03EEE790B25DFBEB13EF0120234144A063EF0ED 3907843AB83EF1CE2A150FC53F3EF2B4D5664BD0853EF3A13B062806203EF4935B0098 4AD53EF58B35624E2EC13EF688CA387F71723EF78C1990B7413C3EF8952378B04C733E F9A3E7FE36144D3EFAB8672F0C3AB33EFBD2A118DABD933EFCF295C91E435C3EFE1845 4D1BC2213EFF43AFB1D6EADA3F003A6A82056E7F3F00D5DAA8125D713F017428511FE4 CC3F02155383116AE83F02B95C439F510F3F036042985697383F040A068698D7943F04 B6A8139C895A3E955ACE9A0CE5543E932FA228C213D13E937F1D69EE64BF3E95DFA851 6CC7233E9A0E8AB7F8BE803E9FE032837A739A3EA39BAED9FC54C03EA7FFF6CF644681 3EAD15E17B436DBB3EB16C31A82886DE3EB4A1E98D5612BB3EB82ABFC11A83EB3EBC05 B3B82F21D53EC0190240533B6B3EC2578FD52651393EC4BE4A97696E463EC74D076281 3ECF3ECA03A4C9C28DE43ECCE208C7C4BD283ECFE81F064FE0D53ED18AEBCA3A1A063E D33592F683510C3ED4F4001C30CD493ED6C62F5D1FF24D3ED8AC1DB058DD1C3EDAA5C8 B60F16193EDCB32E952E89D83EDED44DE03D08CA3EE08492BFF2EDB23EE1A8DA50F65D 683EE2D6FD55C925F93EE40EFB9756C50D3EE550D4F0744CD73EE69C894A5808E53EE7 F21899BE4AE23EE95182DC98CD933EEABAC8182D6D8A3EEC2DE8578EA3713EEDAAE3AA 5C96F03EEF31BA23C115DA3EF06135ECCDB9AA3EF12E7C71E9BC723EF200B0ADE62368 3EF2D7D2ADF8AE183EF3B3E27FAC8AD73EF494E030BC67F83EF57ACBCEF3C6B13EF665 A5681648E83EF7556D09CBEC903EF84A22C19158193EF943C69CAB818C3EFA4258A81E 19EC3EFB45D8F0A440AC3EFC4E4782AB18203EFD5BA46A4DE5583EFE6DEFB35375093E FF8529692C89C83F0050A8CB7991BE3F00E13423B53B443F017436C27FB8B33E92E9F8 A4DD058C3E8F5DFB55681B033E8D7EAA0782819E3E8F62CD47BF1D073E92427A63CA7A 273E9646FE296A2C5B3E9BA1AFF4C5686B3EA11F388D98AD8D3EA5079127A560BC3EA9 84D63B4011383EAE935E9487BDE23EB2183C6DAE5CDD3EB52D11FC98C9B33EB8876F05 15AFD33EBC26C116F12EF03EC0054C15EE3F5A3EC2194EFD507BD63EC44F47D4F417F3 3EC6A71C977075683EC920B8EE8DA8663ECBBC0CE95BEBFB3ECE790C02B02B953ED0AB D631726B5E3ED22BF326DA5A433ED3BCD9D897FF823ED55E87E6DEE37E3ED710FB789A F2D13ED8D4332051FDDD3EDAA82DC6AF4D273EDC8CEA997699CB3EDE8268FDE888C73E E0445442EE0AA13EE14FD4737DB1E03EE263B4F9D985A73EE37FF5CABDD7AB3EE4A496 E41C62A83EE5D1984B3B03353EE706FA0B2C31F63EE844BC339016963EE98ADED7907E E53EEAD9620D0CBE0F3EEC3045EBEC9D9A3EED8F8A8D933DCA3EEEF7300C6C1DB13EF0 339B41C6CD013EF0EFCF073687AD3EF1B0336450C0173EF274C866D918223EF33D8E1C 778DBD3EF40A8492A8C7623EF4DBABD6B1BB173EF5B103F59619D83EF68A8CFC11031C 3EF76846F68F993E3EF84A31F12D21433EF9304DF7B067E43EFA1A9B158A2FB93EFB09 1955D478A63EFBFBC8C35277123EFCF2A96871196A3E91743ED6C536753E8AA2405743 F7843E869BAA5D0382573E86018BA67EEB4C3E884E73E52D39803E8D2B3EE4DEBE643E 922EB452D080033E96DE59E90F2C3B3E9C9670E7C820A93EA1A6711A34F2453EA57D2D B3993AB23EA9CCBCE7F13CC33EAE931DA0AF86D83EB1E766FCFC97033EB4BF5489B338 A53EB7D0E7702CE0173EBB1BC96732E6083EBE9FB7956EE0673EC12E3EF807C9413EC3 28F9E55292703EC53FFCA1A806E53EC7733AA7DD11D53EC9C2AA204C22F83ECC2E434E B001543ECEB6002118E6F03ED0ACEDEBF6F25A3ED20CE9608369C83ED37AF100B7AA4A 3ED4F703B53EFC523ED68120ABDD19093ED8194749D2A54D3ED9BF7720F69C6A3EDB73 AFE6F322D33EDD35F16E34CF913EDF063BA034461C3EE072473C6A7DEE3EE168750151 55043EE265A729E120F73EE369DDC5BA096F3EE47518E805DC373EE58758A6A3EC833E E6A09D197C03D73EE7C0E659F1426D3EE8E834826F27653EEA1687AE0C0F6E3EEB4BDF F83D57233EEC883D7C9A00F63EEDCBA056A950B33EEF1608A1BB42393EF033BB3C6491 3A3EF0DFF4FB2E6EB83EF18FB19A3EEA993EF242F1265124503EF2F9B3ABD23E323EF3 B3F936DD7D523EF471C1D339AE5E3EF5330D8C5793ED3EF5F7DC6D512DE73EF6C02E80 E9B0B33EF78C03D18E0AC73E90F9A12FC577EB3E882C135717BD5C3E82553BD45FCBAC 3E7F37177E31F28B3E7EF325917499593E81982CBD1417BA3E85BCD59C33E09E3E8BBF 5010C400C23E91C1AEBFFEF9513E967A67DA5F61DB3E9C0280EFC138123EA12A4BA7F2 90CD3EA4B65466D81E0E3EA8A3D8D0B658753EACF1B405257F1E3EB0CF82FBC400E63E B355910AB848BE3EB60AC156192D253EB8EEDFD31571643EBC01C3D53D3E1D3EBF434D 7AB109403EC159B1DC8D54D53EC328F970977F423EC50F75BD31CAEF3EC70D20B06C71 F33EC921F58AAE85273ECB4DF099CFDD9A3ECD910F02DC1C213ECFEB4E9729D1C93ED1 2E56D92038693ED272958F50A18E3ED3C262FFFD1F8C3ED51DBEE0CFD6DE3ED684A904 355E903ED7F72153A65A033ED97527CB063ED03EDAFEBC74E0C5FA3EDC93DF675AE0E2 3EDE3490C1B4DEEF3EDFE0D0AA425B933EE0CC4FA660FEA73EE1ADFE6C81AB8E3EE295 74C0E654B73EE382B2BDFA794C3EE475B87ED475F93EE56E861EE9A7513EE66D1BB9D1 0FC03EE771796B11F3253EE87B9F4DFC4D5B3EE98B8D7D896B453EEAA144144338EE3E EBBCC32C3118683EECDE0ADEC9488E3EEE051B44E60E163EEF31F476BDF3113EF0324B 45EF48703EF0CE80CD94B6673EF16D9ADD69434A3EF20F99803023763EF2B47CC05645 6D3E917A1FAFDDCA0C3E87FB7454E36C503E80AB5E6D975DD33E78619B231A8D8F3E74 0CA2E4785BF43E73A98BB6E8CA823E76C34D9A86E4EA3E7D096DE4A75E9E3E8321B7AF DE965A3E89247AD0C135413E903F2DC2BBC3AA3E94924A36C16C353E9987909756B1B2 3E9F1BFD1CC5BDEE3EA2A6A3031F877C3EA60CD59D43C1FD3EA9BFE9CA623F9F3EADBF 59D7CE2C673EB1055ED7F287953EB350E1FCE39C483EB5C21678DACAA53EB858E34181 E93C3EBB1534A78DD7423EBDF6FB327422E53EC07F155F2AA01E3EC2155CE5FFA8743E C3BE507B91039E3EC579ED44EA52593EC74831136225C33EC9291A427FF0103ECB1CA7 9CC4FB0F3ECD22D845DF3D583ECF3BABA922FFD73ED0B390B5B4B84D3ED1D29CAFD1EB 303ED2FAF9BECC8E203ED42CA7ED305A473ED567A7512240EC3ED6ABF809E291633ED7 F99A3DC4E7473ED9508E1887E8FF3EDAB0D3C9FB298A3EDC1A6B84E4E96C3EDD8D557E 1C26DC3EDF0991EBCD9B273EE047908271FFD23EE10F018046AC993EE1DB1C0AEC179E 3EE2ABE03DB23CD83EE3814E339269333EE45B66071688B23EE53A27D245FD4F3EE61D 93AE97060C3EE705A9B4E3E76E3EE7F269FD632BCD3EE8E3D49FA26C4F3EE9D9E9B283 2E493EEAD4A94C39718F3EEBD413824B9EBC3EECD828699392EF3E92F5BA570E2CB23E 8A106350A704693E819E1228AA387E3E7782A23BB7A7C03E6FD2BF86CB757F3E678428 FFFE9AB43E652F70E33ED6633E6833A1DB3590573E700FE0BE6459443E76518B79B639 DE3E7EC187808860333E84A52508EEB1F13E8AEDE53CA4DD9A3E911A7E65FD83493E95 3AEC1AA914923E99D67BA31A01A93E9EEBD3DA75ACC03EA23CF4ADB137333EA53FF5FD 4DA9D73EA87E9C83307F0E3EABF8A87B9AA37A3EAFADE7D1DFC4843EB1CF1993B4D02A 3EB3E4B5C591A9A83EB617BC5AA6E2D13EB86823D3C29D3A3EBAD5E4CC948B233EBD60 F98F33EE853EC004AEDF26F4973EC167870878AFA63EC2D9040E10728A3EC45925139C 33A03EC5E7E984703BE33EC785510565B96F3EC9315B696DF3183ECAEC08A84FD3BB3E CCB558D722CDB83ECE8D4C2230C3DC3ED039F163FD2A423ED1348E8A935B7E3ED2367D B0999C0F3ED33FBF05DF7F023ED45052BD2B44AC3ED568390B53EB773ED6877226862C AB3ED7ADFE45ACC1713ED8DBDD9FF5AECE3EDA11106C6F7C343EDB4D96E1BA217B3EDC 917135C8386B3EDDDC9F9DAD99213EDF2F224D7907D23EE0447CBC0B810A3EE0F512A7 9E08953EE1A95301AAA4C03EE2613DE1BEEEFE3EE31CD35EBC6C5A3EE3DC138ED71CAC 3EE49EFE879568CE3EE565945DD12D9C3E956C7125569FF43E8E6AE04A6285C43E852D 5705985BEA3E7C9A2CC80941633E72895C2E3CB58D3E66F3E7AAD76F043E5BDBAEF6C5 50C33E52BF24F188FA9E3E51B03C6FC5AD433E580D83C863D7053E62B0EA3FA1A91B3E 6CAB81C837411E3E74EB3B8A8DCE2E3E7D0CD5F450047E3E8358B48F121E0C3E88E8FA 518EF89F3E8F3489F7E957FC3E931CA65BB6F5433E96FAD11CC965B53E9B34228D6C9F BA3E9FC81B26227B593EA25B2B5EC929B53EA4FF434B47D2793EA7D036677854393EAA CDEC6B9B97483EADF852594DDE873EB0A7ACB3F7476D3EB2697B15D615AE3EB4418FF4 F07C4D3EB62FE808555F613EB83480E50753073EBA4F58D26244003EBC806EA70EC405 3EBEC7C1ACBF2F9E3EC092A8C4AB0A1E3EC1CC8F1630A3F83EC31193E007D7A33EC461 B74E7A87B93EC5BCF9A00952EB3EC7235B215B70893EC894DC29EE2D123ECA117D1960 AECD3ECB993E553F76C13ECD2C204738810A3ECECA235BA5409C3ED039A4002D94023E D113C751DB20743ED1F37BD9E9CD1B3ED2D8C1CEF992093ED3C39966FD08CD3ED4B402 D70809963ED5A9FE5329585E3ED6A58C0E4D73183ED7A6AC3A28E2E93ED8AD5F0728BC 353ED9B9A4A46831B03EDACB7D3FAA4DD93EDBE2E9055710243EDCFFE8207B4A413EDE 227ABACAB5E93E98DE441AB723E13E928575A10AF81F3E8B592D0461C7D83E83D41D64 07ACF13E7BEC9824FD4C123E72D129B72E0A123E67C1CEFD7940033E5B2D5B43F704B9 3E4ABFD3C3230E143E35E760AAA9E38C3E282FFD190D93013E36B8D51C06552C3E4A64 AC8937EB373E59622620936DDF3E650C8466F7F5AE3E6F9D7E5C7FA7D43E762C9BE505 F2013E7D9BAAC267B4DC3E830C558CA8A9D43E87D0891162A8C53E8D1970E0658C8B3E 917322538003623E949B33756C89AB3E9804ADF3CE91393E9BAF613F27EA803E9F9B27 5AB11C683EA1E3F15A9477583EA41ABE3B7634793EA671F195580A433EA8E984D657C0 BB3EAB8173287A7F593EAE39B91A7820913EB0892A2CD0E4493EB205A1BCAE17523EB3 9242E2B6373C3EB52F0D8E771A283EB6DC01EA1FA4713EB899204E439B7B3EBA666937 E43C873EBC43DD404FF6023EBE317D168161843EC017A4BCDB65B73EC11EA19A849A85 3EC22DB58DEA4BE83EC344E10365EC823EC4642468CCEE943EC58B802C7B5C1B3EC6BA F4BC8E44563EC7F282864596B73EC93229F58686AE3ECA79EB7478C4A73ECBC9C76B39 D89C3ECD21BE3FA2BE2E3ECE81D0551C88173ECFE9FE0C815EEA3ED0AD23E203D3FF3E D16956EB9ABC1D3ED229984F6D674B3ED2EDE83888A7513ED3B646D0A035E03E9D4B33 372FB84E3E96F8421BE0A1F33E9210CA12833E463E8C56661DE4F8B63E860989D3D3BF 6C3E80E3DB12A322713E7955ABB4C14E883E726C79AB6D503C3E69A745A77A7B823E60 C59586D6679F3E53F29F6DBCA0403E445D7AD65D9D923E2FFB11A2A974693E134514DD 25658B3E1A8EEAAAE2B6473E3504E2DCDA8FF13E47CF3A4B83BD1A3E558B65A0B566BD 3E610FC7B9E105F93E68CCF7C178F1583E70FCA35598A1CB3E7648C99622686A3E7C49 B3ADA7BC393E817F3509C0C1ED3E85331544DB7ECF3E89402E8E95224A3E8DA645C46B 3FAD3E9132969B8BF1C33E93BE60FE020E613E967674E5D387E43E995AC8A70AAE053E 9C6B555F1734173E9FA816689B614C3EA188847530537C3EA35315BDF89D7E3EA533BE EDC23B593EA72A8058EAF5063EA9375AB09A569F3EAB5A4EEED2D9743EAD935E46315A BC3EAFE28A14AB0D673EB123E9EC5E452B3EB2619E944BEA133EB3AA63D57A6F313EB4 FE3A889CB94D3EB65D23895DAD2B3EB7C71FB4755EB73EB93C2FE61DECA83EBABC54F8 D7357F3EBC478FC46BA02D3EBDDDE11D2A8DEE3EBF7F49D34F0E8B3EC095E559458C7B 3EC171B240D869773EC2530C013A46813EC339F2F8A1894F3EC42667829446E43EC518 69F7E27DEE3EC60FFAAEA5D1CB3EC70D19FA45B5FE3EA1599F3D602EBA3E9C8DD595B2 40653E97C34633C33CF43E93E9F848CE42243E90BF33AD8ED3703E8C2ECC8FDA33873E 87AA9C2FDFAFA23E83C998E5D875B83E806F8416FDFFC43E7B105F1C5AB6543E7609EC F7C507C23E71B626FF3D8AB23E6C0A08FCB16B6B3E65D4ECAC7E5EE93E60BAAB0765E1 1E3E595A86871DD0753E5343D874491B823E4E412B0166210F3E49C775F01930123E49 063B7CBF81513E4BED8AB7944D033E51386F3AC16DB73E56432F887D516D3E5D132822 6174833E62D2A808456A813E67FBA3C340EB993E6E039AB8198C3F3E7274EB14DBBCE4 3E7656E536DF79CD3E7AA7873306468C3E7F66AA597963F73E824A198FFBCB683E8518 07788504183E881D1933EB38F43E8B594BF142DAA93E8ECC9F2CFA976E3E913B8A1C44 C9AB3E932C56EC46FC2F3E9538B7FA88D9C13E9760AFAC441B0C3E99A440BD6325733E 9C036E2ADF2B4C3E9E7E3B211C33CC3EA08A5576C510C73EA1E360797A44BD3EA34A40 500381973EA4BEF6B3D0EB893EA641855D56C1D13EA7D1EE0194C13A3EA97032501FB8 CE3EAB1C53F196685C3EACD654866EF1F83EAE9E35A60D436E3EB039FC6F0AD3343EB1 2BCFD8F85EFE3EB22495CD3C0FAC3EB3244F02E20F643EB42AFC2B8A5C643EB5389DF3 683C5D3EB64D35014A97053EA48B32F2B489883EA1A318073FE9B23E9EC40AE5F0DFDC 3E9B265E5F97279A3E982A72542E2E233E95A4B4A99E1C143E9377E35FEDB4C23E918F DFCC7224D03E8FBD15439052B33E8CB3998ABC928A3E89F4A79EE5051F3E87757A1F8A F5E73E852E0CB2789F913E831857C84255503E812FC7DF4915333E7EE1B98A2D35D93E 7BB1C85FEB58673E78CB8EAA65F2D83E762BCBBC893C9E3E73CFF4CF57B9063E71B60B C4D8CB9A3E6FB8FFEF53C6063E6C842CE631DC413E69CBAF5CFDA5ED3E678E02D7C59E B23E65C9F7704E04A63E647EA09C8D15763E63AB479F6B0AA33E634F60CE8C28453E63 6A830BCCF5BC3E63FC60F77FF3E33E6504C37D22B8543E66838571C78F093E6878900B B9EAB33E6AE3D8074B94B83E6DC55B563736433E708E8F9F2CB7073E72759769156C7B 3E7497CCD8E9975A3E76F5398390232A3E798DE858B297E33E7C61E54C35CB943E7F71 3D0FA9C8263E815DFE6C6714E83E8321191968A30D3E8501F56C30B49A3E87009A4A7F E2F23E891D0E962726F93E8B57592325763D3E8DAF80AFCE06353E9012C5EF4DF23E3E 915CC098B3C4F43E92B5B382E05B623E941DA1C91C45E23E95948E7238CC913E971A7C 6FE91A1B3E98AF6E9E5DD5D83E9A5367C416F6663E9C066A91E198303E9DC879A2F959 6F3EA83A54BB94ECB03EA590A8C324AD7E3EA3898C148613953EA1F032A9A6967E3EA0 A3406EE3F7EA3E9F19D8AE6498B93E9D3D2AC529EA513E9B9C58776D11CE3E9A294277 CACC343E98D9D1ABB1A0563E97A6B3AB5F8C1E3E968A85C695353C3E958145D037B91C 3E9487F00091420B3E939C3A96D24D5D3E92BC6579BFBA553E91E717847EEBE73E911B 4551B639803E90581EB620B1063E8F3A01FFFDE7963E8DD2D94FA5B6CE3E8C79FB0887 82C23E8B2EC9B04A45563E89F0C9829A71F83E88BF99607BA7753E879AED505EF3B83E 86828A2FC1E9693E8576424F5E92BF3E8475F2C49E03113E83818147F75DB23E8298DA 817F40593E81BBF0AC15134F3E80EABA7CFD628C3E80253241CB16033E7ED6AA532525 A33E7D7A4561851E0E3E7C353851A4591D3E7B0788ACBF2D073E79F13E42E969053E78 F262A909FAE63E780B00CECA7CCC3E773B24A80FA0703E7682DAE66999523E75E230BF 97ECFE3E755933BEC8F2CD3E74E7F19EAAF1873E748E782ABF38123E744CD526A79199 3E7423163A6204223E741148E294ED053E74177A643661583E7435B7C2F76D723E746C 0DB9F777093E74BA88B658D0FD3E752134D36097773E75A01DD7DB6C213E76374F348B FCEE3E76E6D40372172E3E77AEB707C2352F3E788F02AE6AB1CE3EAC67049801581C3E AA0F9CFE876B9E3EA85836FE8A896F3EA70C0691F829173EA609AFA52E0C4B3EA53B69 2B2047833EA4929223D23C3E3EA4051B39EE80EE3EA38BF4C6C3742A3EA322133D0859 4E3EA2C3CD68DC2F303EA26E71F21F48FA3EA21FFFC24435243EA1D6F4F57008C73EA1 922CAB3B02CA3EA150C6D70792CB3EA11216E66869A43EA0D59726B0B3EB3EA09ADF82 3DF81E3EA0619EA03086D53EA02994BCC94D563E9FE51F86B7F0473E9F78D0AAA48AAB 3E9F0DFF032EA4643E9EA47A00D80D643E9E3C1BA6D94A553E9DD4C663F126EA3E9D6E 63607CA75C3E9D08E1273056633E9CA4329447C29B3E9C404DFBCD3E9A3E9BDD2C7B30 7AF13E9B7AC96D13C34D3E9B1921F84082923E9AB834B440F4FC3E9A58015F4C6EBE3E 99F888A21E84ED3E9999CBDF0649653E993BCD0A08B32B3E98DE8E885F7CCE3E988213 15F50C2F3E98265DAFC28F023E97CB71822C11523E977151DAA1734A3E9718021BEE4C 443E96BF85B4BE2EA63E9667E017F168383E961114B67076643E95BB26FA3AD0013E95 661A427A16073E9511F1E06BC7EF3E94BEB114FC29FD3E946C5B0EF314733E941AF2E9 991D3A3E93CA7BABBF88AE3E937AF8471986863E932C6B97D7BB493E92DED8647A1977 3E9292415DCDC3073E9246A91F0E569F3EB088A143FCE5EF3EAF1FF4B968240B3EADCE 063105D1643EACE6AAE8C04B8D3EAC4886CCF554343EABDDD0D098FFB33EAB97E7F3AE AFC43EAB6CBCE0E0FA9F3EAB553FF17E3EC83EAB4C65983DC0033EAB4E84A13E7CC73E AB58EBB460C8753EAB6999BB1731603EAB7F0CD186CC7F3EAB98201747D9053EABB3F3 7FBCC6B83EABD1DA787846473EABF14F4ECD83A43EAC11E9ED1A06403EAC3358FA7E1E 323EAC555CB33AB89B3EAC77C302A3820A3EAC9A648A2239AB3EACBD2258A1FB133EAC DFE42663AF723EAD0296F50398D33EAD252BFBE11C783EAD4797CF2ABD513EAD69D1B4 3ABFF63EAD8BD3192EECB43EADAD97280C6BC93EADCF1A6F8B167F3EADF05A9CFC1131 3EAE115643C413B43EAE320CAFA839BB3EAE527DBFC42E053EAE72A9C875BB5F3EAE92 917AE46BC53EAEB235D112B9703EAED197FD9D82463EAEF0B95E79F7E03EAF0F9B7224 B31C3EAF2E3FCECFB8823EAF4CA81B32F7B93EAF6AD608B43E933EAF88CB4EA954C53E AFA689A68262D83EAFC412C8B3A39D3EAFE1686A3CD7253EAFFE8C3AB2CAB83EB00DBF F15A1FC33EB01C22815BAB063EB02A6E990FFEB33EB038A4FF493F4F3EB046C675BBBD E63EB054D3B8D446193EB062CD7F9F01B53EB070B47BBBEE523EB07E89595E528A3EB0 8C4CBF55153D3EB31C8745BF23F63EB260D7F9E36B553EB1F57CD5FBF5BF3EB1C00FD6 FF7EEF3EB1AFE2F31CE7C23EB1BA11A3CE3A773EB1D74B691527BF3EB202889846FAD6 3EB238415E0AE3113EB275EFF3BC82253EB2B9BFBF6B57693EB3025815078C733EB34E B8694A68993EB39E1FCA46764E3EB3EFFBC7C7D4FE3EB443DC5B7FBC903EB4996B3C37 859A3EB4F0659098C61A3EB548974DD241503EB5A1D7C7741FCF3EB5FC071B9ED7C23E B6570C3FFC3FA93EB6B2D3854136E23EB70F4D72E34BC43EB76C6DE502F23F3EB7CA2B 5C6E4BC33EB8287E74D50A9D3EB88761784E70CC3EB8E6D00987A0283EB946C6DF8F7C F63EB9A7438F689B2E3EBA0844606EE7AE3EBA69C8294AF4643EBACBCE33AF1D8A3EBB 2E56257DF0C33EBB915FEE45442E3EBBF4EBB833FBC73EBC58F9DBDDDD513EBCBD8AD6 4426583EBD229F40B5448C3EBD8837CA2BD2203EBDEE5531E61B183EBE54F842FD2109 3EBEBC21D0CCDD593EBF23D2B408349B3EBF8C0BC85A0C233EBFF4CDEA7A787B3EC02F 0CFB51CB243EC063F863AAA2F73EC099299A2CF70D3EC0CEA1090077383EC1045F1842 AC543EC13A642DCC91FD3EC170B0AD0739FC3EC1A744F6CCCD943EC1DE216953B2113E C215466021FC383EC24CB43407AB143EC2846B3B1E620C3EC2BC6BC8CD94C03EB5EF34 5147661F3EB57A6756D1C1D93EB55788B7B06BE73EB56C3270DA20163EB5A7B6787DBF 3E3EB5FF30481553893EB66B4F5FA28DC63EB6E70C147AB9373EB76EDE9345051B3EB8 0041155D131C3EB8995F00D262A93EB938DFA9951D5C3EB9DDC2845E8A753EBA87469F 4134713EBB34D98A093ABB3EBBE60B3E67D52E3EBC9A8573265C763EBD52054EEE6682 3EBE0C56C6EEB6A93EBEC9512EB775573EBF88D4A46918F63EC025640ED6BBF63EC087 8BF59CB8BE3EC0EADA4A4149E63EC14F48FAD4A9903EC1B4D347BDE8813EC21B757ED4 DFBB3EC2832CC52530363EC2EBF6EC056AE73EC355D24EFD02053EC3C0BDB88D3FB03E C42CB84C6419F63EC499C175D5DA3B3EC507D8D9BBAE463EC576FE4B06DCEE3EC5E731 C17E51453EC6587352397D7D3EC6CAC32982447B3EC73E2185D9448C3EC7B28EB3E5B5 603EC8280B0B24E3273EC89E96EB35F4303EC91632B9A569963EC98EDEE0213FB23ECA 089BCB02E91F3ECA8369E81FD8503ECAFF49A5D4179D3ECB7C3B723CB53D3ECBFA3FBA 99A1AD3ECC7956EAD00FA23ECCF9816D07AFDE3ECD7ABFA95E0A2D3ECDFD1205AC1AEF 3ECE8078E55AF3A33ECF04F4A944BDD43ECF8A85AF9FDE003ED0089629F935123ED04C 74778630293ED090DDEB83B2373ED0D5D2AFA574E33EB900A86695AC753EB8DCA8737F 157C3EB90D26BDA04B293EB977BD41F009283EBA0BBDF69D30783EBABE445521CB183E BB87FFDD7F8A0D3EBC63E8E50BB85C3EBD4E77986D855A3EBE4526310092C63EBF4620 14D4601F3EC028064BEC8B643EC0B0F59763FF053EC13D7D73D9D03C3EC1CD54A9340E CC3EC26043344B569B3EC2F61DF08453DC3EC38EC37133D1A73EC42A19B0F131A83EC4 C80C598488033EC5688B79FE100F3EC60B8A8D32B4BC3EC6B0FFBB7CE5EF3EC758E348 976ADB3EC8032F22927D203EC8AFDE89D52DDA3EC95EEDCC3756233ECA105A0EC4969C 3ECAC42122D380B13ECB7A4163EB86283ECC32B99C8DF1503ECCED88F068B84E3ECDAA AECAD024353ECE6A2AD09D63133ECF2BFCD4C1BBD63ECFF024CF0419513ED05B516A3D F6ED3ED0BFBB88B98E333ED12550E23521873ED18C119D044CB43ED1F3FDE4E5B5FE3E D25D15E9A8F07A3ED2C759DE143CD23ED332C9F6FE98ED3ED39F666A94BE5C3ED40D2F 6FC166663ED47C253DB295A13ED4EC480B76D3CD3ED55D980FAE18E33ED5D015804AFE 553ED643C092615C9B3ED6B89979FFF7773ED72EA06A134CEC3ED7A5D5944FE6333ED8 1E392922D7413ED897CB57A753913ED9128C4DA064FE3ED98E7C37760B563EDA0B9B40 3517EC3EDA89E9919144A93EBC50E385A9F6E33EBC879B4FEB66443EBD1656E7CB9378 3EBDE2B04A413A303EBEDBF96D7B3B663EBFF74DCAF3A1373EC096AE71560E343EC13C 8F84FCFC263EC1EB8636C232063EC2A24FA35380AD3EC360017DB8A7E13EC423EF6FE9 BC6B3EC4ED99344361CF3EC5BC9E51CEDD6B3EC690B590723EE23EC769A6ED0622343E C8474742EF33A33EC9297525823E353ECA10168F56A3853ECAFB172A3477E63ECBEA67 052BF6623ECCDDF99B120A613ECDD5C5144123583ECED1C1B47408743ECFD1E969BAF3 C23ED06B1BBA3E7B063ED0EF541148F4333ED1759C4C82B5EA3ED1FDF3549708883ED2 88585749A5573ED314CAB9DB31173ED3A34A0E22A72C3ED433D609CA2C123ED4C66E7F 3E56A53ED55B1357F7CA8E3ED5F1C48FD9FD1D3ED68A823170A7713ED7254C52E0BB02 3ED7C223136A87043ED861069961A72D3ED901F71084C1F33ED9A4F4A8A369F63EDA49 FF9483E0073EDAF11808FD22643EDB9A3E3C3BEA2A3EDC4572652AF0E33EDCF2B4BAF8 3B4E3EDDA20574B250E93EDE5364C8F929B23EDF06D2EDBF5F123EDFBC501818C9933E E039EE3E0A16903EE096BC264F03B53EE0F491DDB570DF3EE1536F7C73E8653EE1B355 1A1803513EE21442CD8346FE3EE27638ACE9B34C3EE2D936CDD1B0FA3EE33D3D45151C F23EBFDFE5AE8445883EC03D9FF60B5A1E3EC0B98C9B19226E3EC15685C4E6D9953EC2 0C346E8BF00C3EC2D52654C56AE53EC3ADB33794227A3EC4935741A2BC7A3EC5844E89 44D0773EC67F562B282F653EC78383DA5498DB3EC8902B40C221B23EC9A4CC18CD6D84 3ECAC105E97FC1923ECBE48F7ABF2D873ECD0F30C9644D813ECE40BEB0D3CDA93ECF79 17C46278E13ED05C10FF53D88E3ED0FEE484B5C52B3ED1A4FE79DF1FE03ED24E589C3A 5E643ED2FAEDFFF4B8603ED3AABAC6EB91813ED45DBBE82706C83ED513EF03DAA0873E D5CD5240F24B713ED689E431F3D7C53ED749A3BF8A8E483ED80C90177A27C83ED8D2A8 9F03495C3ED99BECE7FCEE4F3EDA685CA8113B0A3EDB37F7B1ACC6963EDC0ABDEE4834 603EDCE0AF59C6F9E63EDDB9CBFEB4D03B3EDE9613F336A8DF3EDF7587568CD3083EE0 2C13278575283EE09EF88437CA823EE11373D94533383EE189854010CF4E3EE2012CD3 061E123EE27A6AAD3B7C253EE2F53EEA2646383EE371A9A55D7E853EE3EFAAFA686915 3EE46F43049701AC3EE4F071DEE2952E3EE57337A3D50F9E3EE5F7946D75D31D3EE67D 88553B1EA53EE7051373FF368F3EE78E35E1F8A51C3EE818EFB6B5040D3EE8A5410915 D81F3EE93329EF4F21423EE9C2AA7EE748343EEA53C2CCB82A2F3EC1D6D770924C2B3E C25BCB24007FA83EC311B6D46A2FAB3EC3EB67804ABA133EC4E08622B98F483EC5EBA0 7873B4693EC7090E417A01D93EC8364BA8771D413EC97194C3BE9E1D3ECAB9A6AFFE55 6D3ECC0D97203E02E43ECD6CB9BE75BB433ECED68E450222193ED0255A1D763E5E3ED0 E471340D6D783ED1A87064B2EC363ED271421D1910E13ED33ED5A6EA40E33ED4111DFF 722D233ED4E810FB94DFA73ED5C3A6A2DA75FB3ED6A3D8B2AD65E93ED788A23F3AE731 3ED871FF6A605D3A3ED95FED2B25E5403EDA526921BF077B3EDB49717517B0A83EDC45 04B7B5B11F3EDD4521D24451533EDE49C7F2874A6C3EDF52F67DBF416B3EE0305682E1 98B63EE0B975A01E9F7E3EE144D87FCD009A3EE1D27F16A90DB83EE2626962A481813E E2F497690538C13EE3890934DDABDE3EE41FBED5CE02AF3EE4B8B85F000AF23EE553F5 E65317CF3EE5F17783AEF2DB3EE6913D5076BC5E3EE733476715F3413EE7D795E2A2F4 5B3EE87E28DE931C493EE92700767D6D763EE9D21CC5E92BB43EEA7F7DE82652CD3EEB 2F23F82E2FC63EEBE10F108AAEA23EEC953F4B43314E3EED4BB4C1CDF6DE3EEE046F8D 0543E03EEEBF6FC51FA2783EEF7CB581AAAC0B3EF01E206CC3F2E53EF07F08F175A7B9 3EF0E11459ADA8F73EF14442AFD8E5013EC3DD1F8EC577A03EC49E4F31D523C83EC593 AA1FD8F1703EC6AFFD574C3E7E3EC7EAF1D3467B633EC93F155084AD2F3ECAA8BF8F07 AC4C3ECC256CB97A1E523ECDB358E62F9AF03ECF514131D5F2E73ED07F1DA7BA730C3E D15CCD748244853ED2416FDC70BFCD3ED32CD4A30A87793ED41ED72C42A3693ED5175B 768561883ED6164BEF85180C3ED71B97E0EC2C673ED8273248064F793ED93910F9B78B 593EDA512BFD87FD9B3EDB6F7D10E21BDE3EDC93FF47F31DEC3EDDBEAEC498676B3EDE EF887DDA15473EE0134509F5D7F83EE0B1D8D6DC91F13EE1537EEEE420D73EE1F836C6 6228E33EE29FFFF438869C3EE34ADA2B078C8D3EE3F8C533BAB83F3EE4A9C0E9271C01 3EE55DCD348303403EE614EA0A8ABFC93EE6CF1769300C4E3EE78C5555B8C57B3EE84C A3DB3763CC3EE91003094C0F0B3EE9D672F32095063EEA9FF3AE9448C53EEB6C85538E F4113EEC3C27FB73B6F53EED0EDBC0AE10943EEDE4A0BE545DB03EEEBD770FDBFAC83E EF995ED0DBEA6D3EF03C2C0E6DB8253EF0AD31879544313EF11FBFE1613FC23EF193D7 2916A0E03EF209776BB918993EF280A0B603C6563EF2F9531463CC733EF3738E92F46F F13EF3EF533D7C7D8F3EF46CA11F6CB7F13EF4EB7843DF1EE23EF56BD8B596E6D53EF5 EDC27F00FE353EC602CB31DBA5293EC7052C1F89467D3EC83F667D6567C63EC9A44749 EB66F03ECB2B778032B4623ECCCF84DCF855413ECE8CC7203D21DD3ED0305D3A55DFEE 3ED124CD784BE3713ED22312D85783E53ED32AB833FCA12A3ED43B67603745A23ED554 E03A35C30C3ED676F2857CBC213ED7A179A5FF38B23ED8D4599A2986AD3EDA0F7CCFAD FC4E3EDB52D29036FF073EDC9E4DD9103FAC3EDDF1E47F1DC8623EDF4D8E89E7B6A53E E058A2DB6C40163EE10E828D0EAE623EE1C8646AC9D8083EE28646F021CB6C3EE34828 ED304CDF3EE40E09756B52943EE4D7E7D215C4D43EE5A5C3778547D73EE6779BFC9BB8 2C3EE74D7113F968473EE827428689D5DB3EE905102F2212FA3EE9E6D9F6F86B5C3EEA CC9FD2C930803EEBB661C0861D5E3EECA41FC5750E563EED95D9ECA87C463EEE8B9045 C08EA13EEF8542E3E7135A3EF04178EE7DAEC93EF0C24EA4729B623EF14522A083DF99 3EF1C9F4EFE73B0D3EF250C5A027DC083EF2D994BF0070CF3EF364625A3C7AB23EF3F1 2E7F9F9B813EF47FF93CD1D1243EF510C29F4FC2333EF5A38AB45E649E3EF638518901 69743EF6CF1729F3F0153EF767DBA3A31AD43EF8029F022A2ED83EF89F61514FF8CD3E F93E229C853B5A3EF9DEE2EEE3F6233EFA81A2532F5DE13EFB2660D3D460E53EC847DA 59D4D4CC3EC99061ED1CE7BC3ECB14EBED0F92A83ECCC845582833553ECEA217297E3A 423ED04E778EE756413ED15A927A8D313D3ED2741A6D0600E53ED39A2D717B91073ED4 CC2A1644CA0D3ED6099B34E58BD73ED7522AA259E0D03ED8A5983BD01AC93EDA03B3C4 CCDC4C3EDB6C58A1432D453EDCDF6ACF9F5BAF3EDE5CD4BD93BDAC3EDFE485B4CAB8AD 3EE0BB385947FED03EE18945C5E3CB4B3EE25C6723FCD08F3EE334995248496A3EE411 D9DADCD1DF3EE4F426CEA91B8B3EE5DB7EA93134F93EE6C7E03A8EE27C3EE7B94A9638 1A443EE8AFBD056FC4763EE9AB36FC8B857D3EEAABB8126D39C93EEBB13FF9B533EB3E ECBBCE7B4EF1673EEDCB63720F84763EEEDFFEC72D38B83EEFF9A06F645FD43EF08C24 34535A473EF11DFB5C1D09B23EF1B255B4987A973EF249334595C0B63EF2E29418A9C2 F83EF37E7838C42AFA3EF41CDFB1D8DD753EF4BDCA90996A993EF56138E23B91F13EF6 072AB44A81C13EF6AFA01480E8703EF75A9910AA47173EF80815B68A3FC93EF8B81613 C8D0583EF96A9A35E29F2F3EFA1FA22A1CA27A3EFAD72DFD7A8B113EFB913DBCB77887 3EFC4DD174408D173EFD0CE930310DF53EFDCE84FC4FC7A63EFE92A4E40D7CE83EFF59 48F2842D953F001138993B87043F00770ED7298663-%&COLORG6&%$RGBG$F.\"\"!$\" #DF)$F;F;-%*LINESTYLEG6#\"\"#-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%%b[ 7]G%'a[6,5]G%!G-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G-%%VIEWG6% F&F,;F>$\"\"%!\"'-%+PROJECTIONG6%$!#]F;$\"#iF;F." 1 2 0 2 10 0 2 1 4 2 2 1.000000 62.000000 -50.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "evalf[15](mi nimize(evalf[20](sm),b[7]=0..0.1,a[6,5]=0.1..0.7,'location'));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"0FPz2(*zG#!#C<#7$<$/&%\"aG6$\"\"'\" \"&$\"0ua(3eTDA!#:/&%\"bG6#\"\"($\"0![/BhtYw!#;F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use rational appro ximations for the values of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"( " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"& " }{TEXT -1 54 " that mimimize the norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "[.764673612304480e-1,.22 2541580875474];\nmap(convert,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"0![/BhtYw!#;$\"0ua(3eTDA!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"#8\"$q\"#\"#x\"$Y$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We set " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7] = 130/170;" "6#/&%\"bG6#\"\"(*&\"$I\"\" \"\"\"$q\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5] = 77/346;" " 6#/&%\"aG6$\"\"'\"\"&*&\"#x\"\"\"\"$Y$!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3] = 13/45;" "6#/&%#b*G6#\"\"$*&\"#8\"\"\"\"#X!\" \"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5] = 33/3335;" "6#/&%#b *G6#\"\"&*&\"#L\"\"\"\"%NL!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The two weights " } {XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 277 " are chosen \+ so that the ratio of the principal error norm of the order 5 scheme t o that of the order 4 scheme is approximately the same as for the Boga cki-Shampine scheme. In addition, these values give a compatible stabi lity region for the embedded 8 stage, order 4 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "prm := \{b[7]=13/170,a[6,5]=77/346,`b*`[3]=13/45,`b*`[5]=33/3335\}:\ne6 := ` union`(prm,simplify(subs(prm,e5))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1056 "e6 := \{a[7,4] = 2549732718751057/364919075 3894400, a[7,1] = 2218605592257973/11003713657896960, `b*`[2] = 0, a[8 ,2] = 0, c[2] = 1/6, c[8] = 1, b[2] = 0, a[5,4] = 1960/3861, c[7] = 1, a[6,3] = 38673/395824, `b*`[8] = 214645/35009496, c[5] = 2/3, c[4] = \+ 3/7, c[6] = 3/4, `b*`[7] = 2066857/29174580, a[7,3] = 1473430904732051 /3532056235868160, a[7,5] = -13054508705469277/12226348508774400, `b*` [5] = 33/3335, `b*`[4] = 908468771/4726281960, a[7,2] = -63/143, a[7,6 ] = 21360483/17937920, b[5] = -81/13600, a[8,1] = 539/7344, a[6,1] = 3 2575/1096128, a[8,4] = 362551/1790100, b[6] = 48448/130815, a[2,1] = 1 /6, a[3,2] = 4/27, a[5,2] = -4/11, a[5,3] = 42/143, `b*`[1] = 14222516 /196928415, `b*`[6] = 212565968/590785245, b[7] = 13/170, a[6,5] = 77/ 346, a[8,7] = 13/170, a[8,3] = 190269/671840, a[6,4] = 2421335/1424966 4, `b*`[3] = 13/45, a[4,1] = 183/1372, a[8,5] = -81/13600, b[1] = 539/ 7344, b[4] = 362551/1790100, a[8,6] = 48448/130815, b[3] = 190269/6718 40, a[3,1] = 2/27, c[3] = 2/9, a[4,2] = -162/343, a[6,2] = 81/352, a[4 ,3] = 1053/1372, a[5,1] = 68/297\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(e6,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i], i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]]));\n``;\nevalf[8](%%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"\"'F(%!GF+F+F +F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s 8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#F.F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"#\" %g>\"%hQF+F+F+F+7+#F6F3#\"&vD$\"(Gh4\"#\"#\")\"$_$#\"&t'Q\"'CeR#\"(N8U #\")k'\\U\"#\"#x\"$Y$F+F+F+7+F)#\"1tzD#f0'=A\"2gp*yl8P+6#!#jFJ#\"1^?t/ 4Vt9\"1g\"oeBc?`$#\"1d5v=Ft\\D\"1+W*Qv!>\\O#!2x#paq3X08\"2+Wx3&[jA7#\" )$[g8#\")?z$z\"F+F+7+F)#\"$R&\"%Wt\"\"!#\"'p->\"'S=n#\"'^DO\"(+,z\"#!# \")\"&+O\"#\"&[%[\"':38#\"#8\"$q\"F+7+%\"bGF\\pF_pF`pFcpFfpFipF\\qF+7+ %#b*G#\");DA9\"*:%Gp>F_p#F]q\"#X#\"*r(o%3*\"+g>GEZ#\"#L\"%NL#\"*ofc7# \"*X_y!f#\"(do1#\")!eu\"H#\"'XY@\")'\\4]$Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7+7+$\")nmm;!\")F(%!GF+F+F+F+F+F+7+$\")AAAAF*$\")uS2u!\"*$\"):[\"[\"F *F+F+F+F+F+F+7+$\")Vr&G%F*$\")#>QL\"F*$!)@.BZF*$\")r#\\n(F*F+F+F+F+F+7 +$\")nmmmF*$\")Bc*G#F*$!)OOOOF*$\")H1PHF*$\")^Sw]F*F+F+F+F+7+$\")+++vF *$\")X#=(HF1$\")k8,BF*$\")9Dq(*F1$\")DA*p\"F*$\")NVDAF*F+F+F+7+$\"\"\" \"\"!$\")NB;?F*$!)Wf0WF*$\")`frTF*$\")#=r)pF*$!)dtn5!\"($\")/!3>\"F]oF +F+7+FV$\")YKRtF1$FXFX$\")#e?$GF*$\")9JD?F*$!)C)e&f!#5$\")3b.PF*$\"))e qk(F1F+7+%\"bGFaoFcoFdoFfoFhoF[pF]pF+7+%#b*G$\")cF*$\")D0&*)*Fjo$\")V-)f$F*$\")ZW%3(F1$\")30JhFjoQ(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The no rm of the principal error terms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "errterms5_7 := PrincipalErrorTerms(5,7,'expande d'):\nsqrt(add(evalf(subs(e6,errterms5_7[i]))^2,i=1.. nops(errterms5_7 )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xdk7:!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The stability functio n for the order 5 scheme is exactly the same as for the Bogacki-Shampi ne scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(e6,Stabil ityFunction(5,7,'expanded'));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\" \"\"F$%\"zGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F $\"#CF$*$)F%\"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F$*&#\"&\"H<\")g&=C \"F$*$)F%F-F$F$F$*&#\"$p#\"(S)z8F$*$)F%\"\"(F$F$F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the weights " }{XPPEDIT 18 0 "`b^`[i], \+ i=1" "6$&%#b^G6#%\"iG/F&\"\"\"" }{TEXT -1 26 " . . 7, ( initially wit h " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 136 " as a p arameter) for an auxiliary 7 stage, order 4 embedded scheme by using t he quadrature conditions along with the requirements that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^`[7]=b[7]" "6#/&%#b^G6# \"\"(&%\"bG6#F'" }{TEXT -1 7 " qnd " }{XPPEDIT 18 0 "`b^`[2]=0" "6#/ &%#b^G6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "quad_eqns := subs(b=`b^`,Q uadratureConditions(4,7,'expanded')):\ne7 := \{`b^`[2]=0\}:\ne8 := `un ion`(e6,e7):\ncdns2 := [`b^`[7]=b[7],op(quad_eqns)]:\neqns2 := subs(e8 ,cdns2):\nnops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%#b^G6#\"\" $&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e9 := solve(\{op(eqns2)\},\{`b^`[1] ,`b^`[3],`b^`[4],`b^`[5],`b^`[7]\}):\ne10 := `union`(e8,e9):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The coeff icients are expressed in terms of " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^ G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " indets(map(rhs,e10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#&%#b^G6#\" \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e10" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1246 "e10 := \{a[7,4] = 2549732718751057/3649190753894400, a[7,1] = 22 18605592257973/11003713657896960, `b*`[2] = 0, a[8,2] = 0, c[2] = 1/6, c[8] = 1, b[2] = 0, a[5,4] = 1960/3861, c[7] = 1, a[6,3] = 38673/3958 24, `b*`[8] = 214645/35009496, c[5] = 2/3, c[4] = 3/7, c[6] = 3/4, `b* `[7] = 2066857/29174580, a[7,3] = 1473430904732051/3532056235868160, a [7,5] = -13054508705469277/12226348508774400, `b*`[5] = 33/3335, `b^`[ 1] = -37/4080+57/256*`b^`[6], `b^`[4] = -25039/66300+6517/4160*`b^`[6] , `b^`[5] = 9003/13600-4617/2560*`b^`[6], `b^`[3] = 22923/35360-6561/6 656*`b^`[6], `b*`[4] = 908468771/4726281960, a[7,2] = -63/143, a[7,6] \+ = 21360483/17937920, b[5] = -81/13600, a[8,1] = 539/7344, a[6,1] = 325 75/1096128, a[8,4] = 362551/1790100, b[6] = 48448/130815, a[2,1] = 1/6 , a[3,2] = 4/27, a[5,2] = -4/11, a[5,3] = 42/143, `b*`[1] = 14222516/1 96928415, `b*`[6] = 212565968/590785245, `b^`[7] = 13/170, `b^`[2] = 0 , b[7] = 13/170, a[6,5] = 77/346, a[8,7] = 13/170, a[8,3] = 190269/671 840, a[6,4] = 2421335/14249664, `b*`[3] = 13/45, a[4,1] = 183/1372, a[ 8,5] = -81/13600, b[1] = 539/7344, b[4] = 362551/1790100, a[8,6] = 484 48/130815, b[3] = 190269/671840, a[3,1] = 2/27, c[3] = 2/9, a[4,2] = - 162/343, a[6,2] = 81/352, a[4,3] = 1053/1372, a[5,1] = 68/297\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 10 "We choose " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\" \"'" }{TEXT -1 88 " so that the principal error norms of the two embe dded schemes are approximately equal." }}{PARA 0 "" 0 "" {TEXT -1 66 " First we obtain the principal error of the 8 stage order 4 scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "`errterms4_8*` := subs(b=`b *`,PrincipalErrorTerms(4,8,'expanded')):\nprin_err4_8 := sqrt(add(subs (e6,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wD\\Hu!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The principal error norm \+ of the 7 stage order 4 scheme is expressed in terms of " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 151 "`errterms4_7^` := subs(b=`b^`,PrincipalErrorT erms(4,7,'expanded')):\nprin_err4_7 := sqrt(add(subs(e10,`errterms4_7^ `[i])^2,i=1.. nops(`errterms4_7^`)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,prin_err4_7G,$*&#\"\"\"\"3!G`R:9&H+:F(*&\"6e1l*3jLE&RI\"#F(\"\"# *$),&\"&[%[!\"\"*&\"':38F(&%#b^G6#\"\"'F(F(F-F(F,F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "prin_er r4_7=prin_err4_8:\nsolve(%):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+j@$*GO!#5$\"+p%p\"yPF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 145 "We choose a rational approximation for t he larger of the two values since this value gives a larger real stabi lity interval than the other value." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 ".3778169469;\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p%p\"yP!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# #\"$,#\"$K&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "`b^`[6] = 201/532;" "6#/&%#b^G6#\"\"'* &\"$,#\"\"\"\"$K&!\"\"" }{TEXT -1 89 " which gives the following prin cipal error norm for the 7 stage order 4 embedded scheme." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eval( prin_err4_7,`b^`[6]=201/532):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*H$3Ku!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "e11 := `union`(\{`b^`[6]=201/532\},subs(` b^`[6]=201/532,e10)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1212 "e11 := \{a[7,4] = 2549732718751057/3649190753894400 , a[7,1] = 2218605592257973/11003713657896960, `b*`[2] = 0, a[8,2] = 0 , c[2] = 1/6, c[8] = 1, b[2] = 0, `b^`[1] = 137189/1827840, `b^`[4] = \+ 69923/326400, `b^`[5] = -118311/6092800, `b^`[3] = 6386607/23152640, a [5,4] = 1960/3861, c[7] = 1, a[6,3] = 38673/395824, `b*`[8] = 214645/3 5009496, c[5] = 2/3, c[4] = 3/7, c[6] = 3/4, `b*`[7] = 2066857/2917458 0, a[7,3] = 1473430904732051/3532056235868160, a[7,5] = -1305450870546 9277/12226348508774400, `b*`[5] = 33/3335, `b*`[4] = 908468771/4726281 960, a[7,2] = -63/143, a[7,6] = 21360483/17937920, b[5] = -81/13600, a [8,1] = 539/7344, a[6,1] = 32575/1096128, a[8,4] = 362551/1790100, b[6 ] = 48448/130815, a[2,1] = 1/6, a[3,2] = 4/27, a[5,2] = -4/11, a[5,3] \+ = 42/143, `b*`[1] = 14222516/196928415, `b*`[6] = 212565968/590785245, `b^`[7] = 13/170, `b^`[2] = 0, b[7] = 13/170, a[6,5] = 77/346, a[8,7] = 13/170, a[8,3] = 190269/671840, a[6,4] = 2421335/14249664, `b*`[3] \+ = 13/45, a[4,1] = 183/1372, a[8,5] = -81/13600, b[1] = 539/7344, b[4] \+ = 362551/1790100, a[8,6] = 48448/130815, b[3] = 190269/671840, `b^`[6] = 201/532, a[3,1] = 2/27, c[3] = 2/9, a[4,2] = -162/343, a[6,2] = 81/ 352, a[4,3] = 1053/1372, a[5,1] = 68/297\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "`RK4_7eqs^` := subs(b=` b^`,OrderConditions(4,7,'expanded')):\nsimplify(subs(e11,`RK4_7eqs^`)) :\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The coef ficients for the whole scheme including the 7 stage and 8 stage embedd ed order 4 schemes are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "subs(e11,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b^`,seq(`b^`[i],i=1..7),`` ],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7,7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F 1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#F .F6#\"#o\"$(H#!\"%\"#6#\"#U\"$V\"#\"%g>\"%hQF+F+F+F+7+#F6F3#\"&vD$\"(G h4\"#\"#\")\"$_$#\"&t'Q\"'CeR#\"(N8U#\")k'\\U\"#\"#x\"$Y$F+F+F+7+F)#\" 1tzD#f0'=A\"2gp*yl8P+6#!#jFJ#\"1^?t/4Vt9\"1g\"oeBc?`$#\"1d5v=Ft\\D\"1+ W*Qv!>\\O#!2x#paq3X08\"2+Wx3&[jA7#\")$[g8#\")?z$z\"F+F+7+F)#\"$R&\"%Wt \"\"!#\"'p->\"'S=n#\"'^DO\"(+,z\"#!#\")\"&+O\"#\"&[%[\"':38#\"#8\"$q\" F+7+%\"bGF\\pF_pF`pFcpFfpFipF\\qF+7+%#b^G#\"'*=P\"\"(Sy#=F_p#\"(2mQ'\" )SE:B#\"&B*p\"'+kK#!'6$=\"\"(+G4'#\"$,#\"$K&F\\qF+7+%#b*G#\");DA9\"*:% Gp>F_p#F]q\"#X#\"*r(o%3*\"+g>GEZ#\"#L\"%NL#\"*ofc7#\"*X_y!f#\"(do1#\") !eu\"H#\"'XY@\")'\\4]$Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "# ---------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1101 "ee := \{c[2]=1/6,\nc[3]=2/9,\nc[4]=3/7,\nc[5]=2/3,\nc[6]=3/4,\nc [7]=1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=2/27,\na[3,2]=4/27,\na[4,1]=183 /1372,\na[4,2]=-162/343,\na[4,3]=1053/1372,\na[5,1]=68/297,\na[5,2]=-4 /11,\na[5,3]=42/143,\na[5,4]=1960/3861,\na[6,1]=32575/1096128,\na[6,2] =81/352,\na[6,3]=38673/395824,\na[6,4]=2421335/14249664,\na[6,5]=77/34 6,\na[7,1]=2218605592257973/11003713657896960,\na[7,2]=-63/143,\na[7,3 ]=1473430904732051/3532056235868160,\na[7,4]=2549732718751057/36491907 53894400,\na[7,5]=-13054508705469277/12226348508774400,\na[7,6]=213604 83/17937920,\na[8,1]=539/7344,\na[8,2]=0,\na[8,3]=190269/671840,\na[8, 4]=362551/1790100,\na[8,5]=-81/13600,\na[8,6]=48448/130815,\na[8,7]=13 /170,\n\nb[1]=539/7344,\nb[2]=0,\nb[3]=190269/671840,\nb[4]=362551/179 0100,\nb[5]=-81/13600,\nb[6]=48448/130815,\nb[7]=13/170,\n\n`b^`[1]=13 7189/1827840,\n`b^`[2]=0,\n`b^`[3]=6386607/23152640,\n`b^`[4]=69923/32 6400,\n`b^`[5]=-118311/6092800,\n`b^`[6]=201/532,\n`b^`[7]=13/170,\n\n `b*`[1]=14222516/196928415,\n`b*`[2]=0,\n`b*`[3]=13/45,\n`b*`[4]=90846 8771/4726281960,\n`b*`[5]=33/3335,\n`b*`[6]=212565968/590785245,\n`b*` [7]=2066857/29174580,\n`b*`[8]=214645/35009496\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability regi on for the 7 stage, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 108 "The stability region for the 7 stage, or der 5 scheme is exactly the same as for the Bogacki-Shampine scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The sta bility function R is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunctio n(5,7,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#\"&\"H<\")g&=C\"F)*$)F'F1F)F)F)*&#\"$p#\"(S)z8F)*$ )F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-% \"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+*>Fz)R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4) :\np1 := plot([R(z),-1],z=-5.19..0.49,color=[red,blue]):\np2 := plot([ [[z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\n p3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplo ts[display]([p1,p2,p3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVE SG6$7W7$$!3Q++++++!>&!#<$!3*RwB]%)fHt(F*7$$!3Ynm;#3'4G^F*$!39PR9e0>`qF *7$$!3kLLLk@>m]F*$!3G]<5GldDkF*7$$!3E+]U'*)HB,&F*$!3d]&>H%3E>fF*7$$!3! pm;&GwYe\\F*$!3wY9xaCqZaF*7$$!3s+](\\(Q*y*[F*$!3%on?sDzj&\\F*7$$!3nLLV @,KP[F*$!3NKW$G,$p.XF*7$$!3'RLLd%[MwZF*$!3Ic'[s4(R%3%F*7$$!3NLL.q&p`r% F*$!3Q0,*=J8\"*p$F*7$$!3;nmJG')*Rf%F*$!3Q?R(3w5U-$F*7$$!3uLLyGAZ\"[%F* $!3Quy23xp&\\#F*7$$!3%3+])fw&\\O%F*$!3I#4N)4)GP.#F*7$$!3$QL$)f7eWC%F*$ !3*[c\\%**RjM;F*7$$!3A++lN]MCTF*$!3JEtl9e\"\\I\"F*7$$!3ummYeRz+SF*$!3h ?u8[;%f-\"F*7$$!3_LLV-,(>*QF*$!3;_G1Y%=5B)!#=7$$!35++S:-YpPF*$!3ke2$3- #[_jFbp7$$!3K+++\"HZkk$F*$!3o)p`-:ay#[Fbp7$$!3;++gW:!z_$F*$!3%4)>B[VRU OFbp7$$!3hLL)*\\1D?MF*$!3!oGFItMcw#Fbp7$$!3'ommSKVAH$F*$!3ABc\"e&ysD>F bp7$$!3/nmEGV!Q=$F*$!3Am:HGcra8Fbp7$$!39++0(*RmdIF*$!3_2hFWla(>)!#>7$$ !39nmEI%3g%HF*$!3=u3FBZk$Q%Ffr7$$!3-++0xX]BGF*$!3o^A-L(>Co*!#?7$$!3*)* **\\\"R>&oq#F*$\"3&G()eOPRer\"Ffr7$$!3gmm;\\r8&e#F*$\"33EVJ&y0l3%Ffr7$ $!3ymmrw\\OtCF*$\"3Qo&)G_c>%*fFfr7$$!3SLL$))e.GN#F*$\"3sLO3_KLpyFfr7$$ !3nLL)**=uvA#F*$\"3ZjwgtMF:(*Ffr7$$!3K++:I;c=@F*$\"3b.6'ps^18\"Fbp7$$! 31LL.z]#3+#F*$\"3FmhG\"\\qpI\"Fbp7$$!3M++?,<>z=F*$\"3%QLOIb=(*\\\"Fbp7 $$!3;++!4<(>gFbp7$ $!3H++q9zA<:F*$\"3#4s\"oumI)=#Fbp7$$!3EnmEY;O-9F*$\"3quXxE=edCFbp7$$!3 #)*****pQ<(z7F*$\"3!=w_%=i%*zFFbp7$$!3)RL$efMeo6F*$\"3')e&e?*y\\2JFbp7 $$!3I****fAZ3Z5F*$\"3l*R)*oym$4NFbp7$$!3xqm;(zQwK*Fbp$\"3<$oV/fiX$RFbp 7$$!3&z***\\)ecE8)Fbp$\"3hmHBz90MWFbp7$$!3'3nmm0VV'pFbp$\"3)\\a8tK&e$) \\Fbp7$$!3P)***\\iqATdFbp$\"3KhJiHm)>j&Fbp7$$!3aFLL*)4AjXFbp$\"3Sl;t.s 4OjFbp7$$!33LLLO'R&eLFbp$\"3%y%)o;xus9(Fbp7$$!3Uim;`O$Q;#Fbp$\"3Gkt@/Z Ea!)Fbp7$$!3?*****>$H-m5Fbp$\"3y4GNQ4$)))*)Fbp7$$\"3v*QLLU?>#>Ffr$\"3q -s.#30%>5F*7$$\"3%ymmY^avJ\"Fbp$\"3\"oXLqHH39\"F*7$$\"3E0+]HcUFz)RF*F[]l-%'SYMBOLG6#%'CIRCLEG-F_\\l6&Fa\\lFf\\lFf\\lFf\\l-% &STYLEG6#%&POINTG-F$6&Fa`l-Ff`l6#%&CROSSGFi`lF[al-F$6&Fa`l-Ff`l6#%(DIA MONDGFi`lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLORG6&Fa\\lFe\\l$\"\"&F\\]lFe \\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z 6\"Q!F_cl-Fgbl6#%(DEFAULTG-%%VIEWG6$;$!$\\%!\"#$\"#\\Fjcl;$!$Z\"Fjcl$ \"$Z\"Fjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1300 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+ 1/120*z^5+17291/12418560*z^6+269/1379840*z^7:\npts := []:\nz0 := 0:\nf or ct from 0 to 200 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot (pts,color=COLOR(RGB,.4,.18,.4)):\np2 := plots[polygonplot]([seq([pts[ i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,col or=COLOR(RGB,.8,.35,.8)):\npts := []:\nz0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=CO LOR(RGB,.4,.18,.4)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[ 1.37,4.19]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.8,.35,.8)):\npts := []:\nz0 := 1.5-4.3*I:\nfor ct from 0 to 60 d o\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts : = [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB, .4,.18,.4)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.37,-4. 19]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.8 ,.35,.8)):\np7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4.79]]],col or=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-4.79..1.99,- 4.79..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)` ],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 606 615 615 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3b+++!f<*>f!#G$ \"3++++Fjzq:!#=7$$\"3p******4$3Q5&!#E$\"3(******pi#fTJF07$$\"3=+++(*f- r!)!#D$\"3D+++/%)Q7ZF07$$\"3E+++P,gNg!#C$\"3S+++&yzJG'F07$$\"33+++%p%G lG!#B$\"3S+++g$[R&yF07$$\"3s*****>)evd)*FF$\"3w*****\\jIYU*F07$$\"33++ +y<$3i#!#A$\"3'******\\D0&*4\"!#<7$$\"3I+++6Y&R[&FQ$\"33+++h8[c7FT7$$ \"3C+++XY2B')FQ$\"3!******4V6LT\"FT7$$\"3]*****>Ow'yuFQ$\"3++++iU&)p:F T7$$!3-+++'3\"pH5!#@$\"3!******\\/()es\"FT7$$!3^+++odj+rFao$\"31+++jY4 \")=FT7$$!3')*****\\^$>CA!#?$\"3<+++[t2N?FT7$$!32+++\\v\\4aF\\p$\"3=++ +upP(=#FT7$$!3++++O#)>O6!#>$\"36+++QP]PBFT7$$!3'******pt83:#Fgp$\"35++ +yV#\\[#FT7$$!3!)*****4:(*Qv$Fgp$\"3/+++4w'*GEFT7$$!3<+++3,`ChFgp$\"3! ******p0J'oFFT7$$!3Z*****>$4#*=%*Fgp$\"34+++RsN-HFT7$$!3#******\\$)Q=P \"F0$\"3A+++nS)z-$FT7$$!3)******4&R!y*=F0$\"3\")*****f&48VJFT7$$!3t*** ***[l:-DF0$\"37+++/I(fC$FT7$$!3;+++h(G-;$F0$\"36+++-)[eL$FT7$$!3'***** *pW;&[QF0$\"3!******pAQKT$FT7$$!35+++W,T\\XF0$\"3)******z^$HzMFT7$$!3S +++P]p^_F0$\"3<+++QZONNFT7$$!3W+++C6.\\fF0$\"3;+++:@s#e$FT7$$!3a+++.s$FT7$$!3*******p$3=S**F0$\"3!******H[4@t$FT7$$!3/ +++rqOd5FT$\"3#)*****zh<&RPFT7$$!3++++mQ$*>6FT$\"3@+++\\+xUPFT7$$!3#** ****fjo<=\"FT$\"38+++2k/UPFT7$$!36+++V4#HC\"FT$\"34+++l#)[PPFT7$$!3)** ****fGTMI\"FT$\"3/+++2e?HPFT7$$!31+++hKQj8FT$\"3&)*****48$GMOFT7$$ !3#*******[f*ol\"FT$\"3&*******z'=Zg$FT7$$!3%*******GS2:FT$\"3# ******pq9*>$FT7$$!3/+++\")*\\YF#FT$\"3>+++x.zVJFT7$$!31+++\"y4OM#FT$\"3) )*****RU-#)3$FT7$$!3z*****\\0RLT#FT$\"3.+++:(>G.$FT7$$!33+++8@H$[#FT$ \"3'******f76z(HFT7$$!3=+++eY\"Hb#FT$\"34+++vGFT$\"3@+++$)yl4FFT7$$!31+++Tmj\" )GFT$\"3))******ey,cEFT7$$!3++++g(>=%HFT$\"3*******za$)=g#FT7$$!3y**** **)R-***HFT$\"3++++=-2ZDFT7$$!31+++;N'e0$FT$\"3))*****p:<9\\#FT7$$!3\" )*****p>0(4JFT$\"32+++MoyMCFT7$$!3#*******RjWhJFT$\"3=+++?H1xBFT7$$!3, +++]$=6@$FT$\"34+++lz9=BFT7$$!3++++rBweKFT$\"3)******\\?hzD#FT7$$!3#)* ****\\hFWI$FT$\"3'******ptOk>#FT7$$!3?+++j/<[LFT$\"39+++?A_L@FT7$$!3=+ ++\\S0!R$FT$\"3;+++U\"y\"p?FT7$$!38+++x![,V$FT$\"3z*****4qxL+#FT7$$!3# ******pjG&oMFT$\"3)******RG2h$>FT7$$!36+++'zx_]$FT$\"33+++6vOn=FT7$$!3 *)******=F[SNFT$\"3-+++NZ<(z\"FT7$$!3/+++!4MUd$FT$\"32+++VGcD\"FT7$$!3')******zM?,QFT$\"3' ******fm$=76FT7$$!32+++\"38Z#QFT$\"34+++K!RC.\"FT7$$!3A+++*yVq%QFT$\"3 z*****p8!)\\_*F07$$!3')*****\\)R5oQFT$\"3b*****\\l!yC()F07$$!3=+++/+z( )QFT$\"3Z*****RXVZ#zF07$$!3*)*****zb#*f!RFT$\"3)******HE'eDrF07$$!38++ +4EgARFT$\"3s*****Hl+yK'F07$$!3-+++/i^PRFT$\"34+++mKnJbF07$$!3++++u!Q1 &RFT$\"3F+++1\\JPZF07$$!3'******H/%)='RFT$\"3w*****4X#pWRF07$$!3))**** **>C=rRFT$\"37+++j)fO:$F07$$!37+++UYZyRFT$\"3!******R/')RO#F07$$!3++++ Gar$)RFT$\"3'*******e&y`d\"F07$$!3%******\\mso)RFT$\"3m*****p%>.vyFgp7 $$!3/+++*>Fz)RFTF(7$Fjjl$!3m*****p%>.vyFgp7$Fejl$!3'*******e&y`d\"F07$ F`jl$!3!******R/')RO#F07$F[jl$!37+++j)fO:$F07$Ffil$!3w*****4X#pWRF07$F ail$!3F+++1\\JPZF07$F\\il$!34+++mKnJbF07$Fghl$!3s*****Hl+yK'F07$Fbhl$! 3)******HE'eDrF07$F]hl$!3Z*****RXVZ#zF07$Fhgl$!3b*****\\l!yC()F07$Fcgl $!3z*****p8!)\\_*F07$F^gl$!34+++K!RC.\"FT7$Fifl$!3'******fm$=76FT7$Fdf l$!3'******zS&f\">\"FT7$F_fl$!3))*****z)R_q7FT7$Fjel$!3'******pB7)[8FT 7$Feel$!3/+++QOIE9FT7$F`el$!3********)e\\G]\"FT7$F[el$!3/+++!\\:$y:FT7 $Ffdl$!3/+++w]e_;FT7$Fadl$!32+++VGcDFT7$F]cl$!3z*****4qxL+#FT7$Fhbl$!3;+ ++U\"y\"p?FT7$Fcbl$!39+++?A_L@FT7$F^bl$!3'******ptOk>#FT7$Fial$!3)**** **\\?hzD#FT7$Fdal$!34+++lz9=BFT7$F_al$!3=+++?H1xBFT7$Fj`l$!32+++MoyMCF T7$Fe`l$!3))*****p:<9\\#FT7$F``l$!3++++=-2ZDFT7$F[`l$!3*******za$)=g#F T7$Ff_l$!3))******ey,cEFT7$Fa_l$!3@+++$)yl4FFT7$F\\_l$!35+++P2,jFFT7$F g^l$!3$******4\")*H;GFT7$Fb^l$!3)*******)[^(pGFT7$F]^l$!34+++vG.$FT7$F^]l$!3))*****RU-#)3$FT7$ Fi\\l$!3>+++x.zVJFT7$Fd\\l$!3#)******>q9*>$FT7$F_\\l$!3#)*****Ri:PD$FT 7$Fj[l$!38+++G_*oI$FT7$Fe[l$!3A+++\\!=\"eLFT7$F`[l$!3#******pMOFT7$F]y$!3 )******fb8-m$FT7$Fhx$!3')*****f:\\Fo$FT7$Fcx$!39+++rEy,PFT7$F^x$!3&)** ***48$G.s$FT7$Ffu$!3<+++$=jQq$FT7$Fau$!3#)*****fJ%Q#o$FT7$F\\u$!3 ;+++XYVbOFT7$Fgt$!3()*****RD]Ci$FT7$Fbt$!3;+++:@s#e$FT7$F]t$!3<+++QZON NFT7$Fhs$!3)******z^$HzMFT7$Fcs$!3!******pAQKT$FT7$F^s$!36+++-)[eL$FT7 $Fir$!37+++/I(fC$FT7$Fdr$!3\")*****f&48VJFT7$F_r$!3A+++nS)z-$FT7$Fjq$! 34+++RsN-HFT7$Feq$!3!******p0J'oFFT7$F`q$!3/+++4w'*GEFT7$F[q$!35+++yV# \\[#FT7$Fep$!36+++QP]PBFT7$F`p$!3=+++upP(=#FT7$Fjo$!3<+++[t2N?FT7$Feo$ !31+++jY4\")=FT7$F_o$!3!******\\/()es\"FT7$$\"39+++kjnyuFQ$!3++++iU&)p :FT7$Fen$!3!******4V6LT\"FT7$FV$!33+++h8[c7FT7$FO$!3'******\\D0&*4\"FT 7$FJ$!3w*****\\jIYU*F07$FD$!3S+++g$[R&yF07$$\"3^*****f8+c.'F@$!3S+++&y zJG'F07$$\"3s******pd-r!)F:$!3D+++/%)Q7ZF07$$\"3G+++9$3Q5&F4$!3(****** pi#fTJF07$$\"3i*****>a4(>fF-$!3++++Fjzq:F0F'-%&COLORG6&%$RGBG$\"\"%!\" \"$\"#=!\"#Fh^n-%)POLYGONSG6fw7%F'7$$\"+!f<*>fF\\p$\"+Fjzq:!#57$$F]_nF )F(7%Fb_n7$$\"+5$3Q5&F0$\"+FEfTJFg_nFh_n7%F[`n7$$\"+(*f-r!)FT$\"+/%)Q7 ZFg_nFh_n7%Fa`n7$$\"+P,gNg!#;$\"+&yzJG'Fg_nFh_n7%Fg`n7$$\"+%p%GlG!#:$ \"+g$[R&yFg_nFh_n7%F^an7$$\"+#)evd)*Faan$\"+N1jC%*Fg_nFh_n7%Fean7$$\"+ y<$3i#!#9$\"+b_]*4\"!\"*Fh_n7%F[bn7$$\"+6Y&R[&F^bn$\"+h8[c7FabnFh_n7%F cbn7$$\"+XY2B')F^bn$\"+J9J89FabnFh_n7%Fibn7$$\"+ijnyuF^bn$\"+iU&)p:Fab nFh_n7%F_cn7$$!+'3\"pH5!#8$\"+Xq)es\"FabnFh_n7%Fecn7$$!+odj+rFhcn$\"+j Y4\")=FabnFh_n7%F\\dn7$$!+:N>CA!#7$\"+[t2N?FabnFh_n7%Fbdn7$$!+\\v\\4aF edn$\"+upP(=#FabnFh_n7%Fidn7$$!+O#)>O6!#6$\"+QP]PBFabnFh_n7%F_en7$$!+P P\"3:#Fben$\"+yV#\\[#FabnFh_n7%Ffen7$$!+^r*Qv$Fben$\"+4w'*GEFabnFh_n7% F\\fn7$$!+3,`ChFben$\"+d5joFFabnFh_n7%Fbfn7$$!+K4#*=%*Fben$\"+RsN-HFab nFh_n7%Fhfn7$$!+N)Q=P\"Fg_n$\"+nS)z-$FabnFh_n7%F^gn7$$!+^R!y*=Fg_n$\"+ c48VJFabnFh_n7%Fdgn7$$!+\\l:-DFg_n$\"+/I(fC$FabnFh_n7%Fjgn7$$!+h(G-;$F g_n$\"+-)[eL$FabnFh_n7%F`hn7$$!+Zk^[QFg_n$\"+F#QKT$FabnFh_n7%Ffhn7$$!+ W,T\\XFg_n$\"+=NHzMFabnFh_n7%F\\in7$$!+P]p^_Fg_n$\"+QZONNFabnFh_n7%Fbi n7$$!+C6.\\fFg_n$\"+:@s#e$FabnFh_n7%Fhin7$$!+.s$FabnFh_n7%Ff[o7$$!+P3=S**Fg_n$\"+$[4@t$FabnFh_n7%F\\\\o7$$!+ rqOd5Fabn$\"+=w^RPFabnFh_n7%Fb\\o7$$!+mQ$*>6Fabn$\"+\\+xUPFabnFh_n7%Fh \\o7$$!+O'o<=\"Fabn$\"+2k/UPFabnFh_n7%F^]o7$$!+V4#HC\"Fabn$\"+l#)[PPFa bnFh_n7%Fd]o7$$!+'GTMI\"Fabn$\"+2e?HPFabnFh_n7%Fj]o7$$!+hKQj8Fabn$\"+J JGMOFabnFh_n7%Fh_o7$$!+\\f*ol\"Fabn$\"+!o=Zg$FabnFh _n7%F^`o7$$!+HS2:Fabn$\"+xT!pS$FabnFh_n7%F\\bo7$$!+ cmH8?Fabn$\"+\\!=\"eLFabnFh_n7%Fbbo7$$!+q_?w?Fabn$\"+G_*oI$FabnFh_n7%F hbo7$$!+p0tS@Fabn$\"+Ccr`KFabnFh_n7%F^co7$$!+30#p?#Fabn$\"+?q9*>$FabnF h_n7%Fdco7$$!+\")*\\YF#Fabn$\"+x.zVJFabnFh_n7%Fjco7$$!+\"y4OM#Fabn$\"+ CC?)3$FabnFh_n7%F`do7$$!+b!RLT#Fabn$\"+:(>G.$FabnFh_n7%Ffdo7$$!+8@H$[# Fabn$\"+E6\"z(HFabnFh_n7%F\\eo7$$!+eY\"Hb#Fabn$\"+vGFabn$\"+$)yl4FF abnFh_n7%Fjfo7$$!+Tmj\")GFabn$\"+fy,cEFabnFh_n7%F`go7$$!+g(>=%HFabn$\" +[N)=g#FabnFh_n7%Ffgo7$$!+*R-***HFabn$\"+=-2ZDFabnFh_n7%F\\ho7$$!+;N'e 0$Fabn$\"+drT\"\\#FabnFh_n7%Fbho7$$!+(>0(4JFabn$\"+MoyMCFabnFh_n7%Fhho 7$$!+SjWhJFabn$\"+?H1xBFabnFh_n7%F^io7$$!+]$=6@$Fabn$\"+lz9=BFabnFh_n7 %Fdio7$$!+rBweKFabn$\"+07'zD#FabnFh_n7%Fjio7$$!+:wU/LFabn$\"+PnV'>#Fab nFh_n7%F`jo7$$!+j/<[LFabn$\"+?A_L@FabnFh_n7%Ffjo7$$!+\\S0!R$Fabn$\"+U \"y\"p?FabnFh_n7%F\\[p7$$!+x![,V$Fabn$\"+,xP.?FabnFh_n7%Fb[p7$$!+P'G&o MFabn$\"+%G2h$>FabnFh_n7%Fh[p7$$!+'zx_]$Fabn$\"+6vOn=FabnFh_n7%F^\\p7$ $!+>F[SNFabn$\"+NZ<(z\"FabnFh_n7%Fd\\p7$$!+!4MUd$Fabn$\"+VGcD\"FabnFh_n7 %Fd_p7$$!+![.7!QFabn$\"+mO=76FabnFh_n7%Fj_p7$$!+\"38Z#QFabn$\"+K!RC.\" FabnFh_n7%F``p7$$!+*yVq%QFabn$\"+P,)\\_*Fg_nFh_n7%Ff`p7$$!+&)R5oQFabn$ \"+b1yC()Fg_nFh_n7%F\\ap7$$!+/+z()QFabn$\"+aMuCzFg_nFh_n7%Fbap7$$!+eD* f!RFabn$\"+jieDrFg_nFh_n7%Fhap7$$!+4EgARFabn$\"+`1!yK'Fg_nFh_n7%F^bp7$ $!+/i^PRFabn$\"+mKnJbFg_nFh_n7%Fdbp7$$!+u!Q1&RFabn$\"+1\\JPZFg_nFh_n7% Fjbp7$$!+VS)='RFabn$\"+^CpWRFg_nFh_n7%F`cp7$$!+?C=rRFabn$\"+j)fO:$Fg_n Fh_n7%Ffcp7$$!+UYZyRFabn$\"+Wg)RO#Fg_nFh_n7%F\\dp7$$!+Gar$)RFabn$\"+f& y`d\"Fg_nFh_n7%Fbdp7$$!+lE(o)RFabn$\"+Z>.vyFbenFh_n7%Fhdp7$$!+*>Fz)RFa bnF(Fh_n7%F^ep7$Fidp$!+Z>.vyFbenFh_n7%Fbep7$Fcdp$!+f&y`d\"Fg_nFh_n7%Ff ep7$F]dp$!+Wg)RO#Fg_nFh_n7%Fjep7$Fgcp$!+j)fO:$Fg_nFh_n7%F^fp7$Facp$!+^ CpWRFg_nFh_n7%Fbfp7$F[cp$!+1\\JPZFg_nFh_n7%Fffp7$Febp$!+mKnJbFg_nFh_n7 %Fjfp7$F_bp$!+`1!yK'Fg_nFh_n7%F^gp7$Fiap$!+jieDrFg_nFh_n7%Fbgp7$Fcap$! +aMuCzFg_nFh_n7%Ffgp7$F]ap$!+b1yC()Fg_nFh_n7%Fjgp7$Fg`p$!+P,)\\_*Fg_nF h_n7%F^hp7$Fa`p$!+K!RC.\"FabnFh_n7%Fbhp7$F[`p$!+mO=76FabnFh_n7%Ffhp7$F e_p$!+3af\">\"FabnFh_n7%Fjhp7$F__p$!+))R_q7FabnFh_n7%F^ip7$Fi^p$!+PA\" )[8FabnFh_n7%Fbip7$Fc^p$!+QOIE9FabnFh_n7%Ffip7$F]^p$!+*e\\G]\"FabnFh_n 7%Fjip7$Fg]p$!+!\\:$y:FabnFh_n7%F^jp7$Fa]p$!+w]e_;FabnFh_n7%Fbjp7$F[]p $!+VGcDFabnFh_n7%Fb[q7$Fc[p$!+,xP.?FabnFh_n7%Ff[q7 $F][p$!+U\"y\"p?FabnFh_n7%Fj[q7$Fgjo$!+?A_L@FabnFh_n7%F^\\q7$Fajo$!+Pn V'>#FabnFh_n7%Fb\\q7$F[jo$!+07'zD#FabnFh_n7%Ff\\q7$Feio$!+lz9=BFabnFh_ n7%Fj\\q7$F_io$!+?H1xBFabnFh_n7%F^]q7$Fiho$!+MoyMCFabnFh_n7%Fb]q7$Fcho $!+drT\"\\#FabnFh_n7%Ff]q7$F]ho$!+=-2ZDFabnFh_n7%Fj]q7$Fggo$!+[N)=g#Fa bnFh_n7%F^^q7$Fago$!+fy,cEFabnFh_n7%Fb^q7$F[go$!+$)yl4FFabnFh_n7%Ff^q7 $Fefo$!+P2,jFFabnFh_n7%Fj^q7$F_fo$!+6)*H;GFabnFh_n7%F^_q7$Fieo$!+*[^(p GFabnFh_n7%Fb_q7$Fceo$!+vG.$FabnFh_n7%F^`q7$Fado$!+CC?)3$FabnFh_n7%Fb`q7$F[do$!+ x.zVJFabnFh_n7%Ff`q7$Feco$!+?q9*>$FabnFh_n7%Fj`q7$F_co$!+Ccr`KFabnFh_n 7%F^aq7$Fibo$!+G_*oI$FabnFh_n7%Fbaq7$Fcbo$!+\\!=\"eLFabnFh_n7%Ffaq7$F] bo$!+xT!pS$FabnFh_n7%Fjaq7$Fgao$!+DX)GX$FabnFh_n7%F^bq7$Faao$!+%\\(z& \\$FabnFh_n7%Fbbq7$F[ao$!+c2ZNNFabnFh_n7%Ffbq7$Fe`o$!+w#)zrNFabnFh_n7% Fjbq7$F_`o$!+!o=Zg$FabnFh_n7%F^cq7$Fi_o$!+&[(>MOFabnFh_n7%Fbcq7$Fc_o$! +cN@gOFabnFh_n7%Ffcq7$F]_o$!+c\"\\Fo$FabnFh_n7%Fjcq7$Fg^o$!+rEy,PFabnF h_n7%F^dq7$Fa^o$!+JJG.s$FabnFh_n7%F^fq7$Fa[o$!+$=jQq$FabnFh_n7%Fbfq7$F[[o$!+;V Q#o$FabnFh_n7%Fffq7$Fejn$!+XYVbOFabnFh_n7%Fjfq7$F_jn$!+a-XAOFabnFh_n7% F^gq7$Fiin$!+:@s#e$FabnFh_n7%Fbgq7$Fcin$!+QZONNFabnFh_n7%Ffgq7$F]in$!+ =NHzMFabnFh_n7%Fjgq7$Fghn$!+F#QKT$FabnFh_n7%F^hq7$Fahn$!+-)[eL$FabnFh_ n7%Fbhq7$F[hn$!+/I(fC$FabnFh_n7%Ffhq7$Fegn$!+c48VJFabnFh_n7%Fjhq7$F_gn $!+nS)z-$FabnFh_n7%F^iq7$Fifn$!+RsN-HFabnFh_n7%Fbiq7$Fcfn$!+d5joFFabnF h_n7%Ffiq7$F]fn$!+4w'*GEFabnFh_n7%Fjiq7$Fgen$!+yV#\\[#FabnFh_n7%F^jq7$ F`en$!+QP]PBFabnFh_n7%Fbjq7$Fjdn$!+upP(=#FabnFh_n7%Ffjq7$Fcdn$!+[t2N?F abnFh_n7%Fjjq7$F]dn$!+jY4\")=FabnFh_n7%F^[r7$Ffcn$!+Xq)es\"FabnFh_n7%F b[r7$$\"+kjnyuF^bn$!+iU&)p:FabnFh_n7%Ff[r7$Fjbn$!+J9J89FabnFh_n7%F\\\\ r7$Fdbn$!+h8[c7FabnFh_n7%F`\\r7$F\\bn$!+b_]*4\"FabnFh_n7%Fd\\r7$Ffan$! +N1jC%*Fg_nFh_n7%Fh\\r7$F_an$!+g$[R&yFg_nFh_n7%F\\]r7$$\"+O,gNgFj`n$!+ &yzJG'Fg_nFh_n7%F`]r7$$\"+qd-r!)FT$!+/%)Q7ZFg_nFh_n7%Ff]r7$$\"+9$3Q5&F 0$!+FEfTJFg_nFh_n7%F\\^r7$$\"+U&4(>fF\\p$!+Fjzq:Fg_nFh_n7%Fb^rF'Fh_n-F e^n6&Fg^n$\"\")Fj^n$\"#NF]_nFj^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\" 33+++.>]*[\"FT$\"3#******>&)\\QI%FT7$$\"3!******ff%R\"[\"FT$\"3()***** HP'f7VFT7$$\"3'******Ru+EZ\"FT$\"3C+++&o\\1K%FT7$$\"3/+++&RjJY\"FT$\"3 D+++(RozK%FT7$$\"3/+++yi7`9FT$\"3e*****\\V6XL%FT7$$\"3++++@T`U9FT$\"3F +++kxBSVFT7$$\"3++++9IVJ9FT$\"3i******>h5XVFT7$$\"3#******f#3()>9FT$\" 3M+++dZ2\\VFT7$$\"3$******Ro(*yS\"FT$\"3e*****p?,@N%FT7$$\"3-+++Glc&R \"FT$\"3=+++^?9aVFT7$$\"33+++xP$HQ\"FT$\"3o*****4r_^N%FT7$$\"3%******R 6g+P\"FT$\"37+++os3bVFT7$$\"3!******fU6qN\"FT$\"3/+++A$)*QN%FT7$$\"33+ ++W*fQM\"FT$\"3y*****p\"p`^VFT7$$\"3,+++CcoI8FT$\"3-+++mD&zM%FT7$$\"3# ******4$yd<8FT$\"3k******RM4VVFT7$$\"3$******\\QPYI\"FT$\"3)******\\z1 pL%FT7$$\"3)*******e!z>H\"FT$\"3e*****zxR$HVFT7$$\"3++++-Ytz7FT$\"3J++ ++3M?VFT7$$\"3#******pMc!o7FT$\"3V+++g<')4VFT7$$\"3)*******H97d7FT$\"3 1+++d:'yH%FT7$$\"3!******H]OrC\"FT$\"3!******RZ6VG%FT7$$\"32+++7EMQ7FT $\"3e*****HF.#pUFT7$$\"3*******fR>5B\"FT$\"3e*****z/hDD%FT7$$\"3,+++wp [D7FT$\"3v*****p+eWB%FT7$$\"3*******>e-@A\"FT$\"3V+++%)*Q]@%FT7$$\"3.+ ++`sC@7FT$\"3k*****p=[X>%FT7$$\"3/+++QlHB7FT$\"3/+++:%eL<%FT7$$\"3)*** ****4)p&G7FT$\"3)******\\N#*>:%FT7$$\"3.+++(*)esB\"FT$\"3j*****4&*>68% FT7$$\"3'******>K]$\\7FT$\"3&*******QR^6TFT7$$\"3-+++\\:dk7FT$\"3o**** *pBdR4%FT7$$\"3/+++e\\R#G\"FT$\"3G+++mC7zSFT7$$\"3\"******4_5@I\"FT$\" 3))*****\\@ou1%FT7$$\"3$*******z2%HK\"FT$\"3S+++yK>fSFT7$$\"3++++T::W8 FT$\"3y*****RkdU0%FT7$$\"3/+++&*R7l8FT$\"3u*****4JYC0%FT7$$\"3++++P:Q& Q\"FT$\"3=+++%oUM0%FT7$$\"3,+++BPe/9FT$\"3i*****\\F*)o0%FT7$$\"3)***** *R-/DU\"FT$\"3#)******>%GC1%FT7$$\"3#*******RO+R9FT$\"3w*****R5D(pSFT7 $$\"3%******>u2SX\"FT$\"3o*****\\=x%ySFT7$$\"3#******fS'[n9FT$\"3#**** ***yzT)3%FT7$$\"3)*******4.Wz9FT$\"3A+++eZJ*4%FT7$$\"3'******HW!*)*[\" FT$\"3w*****RUl46%FT7$$\"3%******H)3())\\\"FT$\"3R+++wX>BTFT7$$\"3$*** ***H$QU1:FT$\"33+++T'\\e8%FT7$$\"3.+++Wif7:FT$\"37+++Wtz[TFT7$$\"3!*** ***fcPu^\"FT$\"3#*******G1#>;%FT7$$\"30+++g#)*4_\"FT$\"3++++@g6vTFT7$$ \"3!******>&)GL_\"FT$\"3G+++<9H)=%FT7$$\"3\"******zLzW_\"FT$\"3:+++=UO ,UFT7$$\"3$******4&))\\C:FT$\"3B+++%yfU@%FT7$$\"3.+++$\\NM_\"FT$\"3j** ***H85pA%FT7$$\"3'******4GO8_\"FT$\"3m******[GDRUFT7$$\"3(******z8_WH%FTFe_rFd^n-F__n6jn7%7$$\"+.>]*[\"Fabn$\"+_)\\QI%Fab n7$$\"+'f%R\"[\"Fabn$\"+tjf7VFabn7$$\"$P\"F]_n$\"$>%F]_n7%Fibs7$$\"+W2 gs9Fabn$\"+&o\\1K%FabnF^cs7%Fdcs7$$\"+&RjJY\"Fabn$\"+(RozK%FabnF^cs7%F jcs7$$\"+yi7`9Fabn$\"+N9^MVFabnF^cs7%F`ds7$$\"+@T`U9Fabn$\"+kxBSVFabnF ^cs7%Ffds7$$\"+9IVJ9Fabn$\"+?h5XVFabnF^cs7%F\\es7$$\"+E3()>9Fabn$\"+dZ 2\\VFabnF^cs7%Fbes7$$\"+%o(*yS\"Fabn$\"+275_VFabnF^cs7%Fhes7$$\"+Glc&R \"Fabn$\"+^?9aVFabnF^cs7%F^fs7$$\"+xP$HQ\"Fabn$\"+6F:bVFabnF^cs7%Fdfs7 $$\"+9,1q8Fabn$\"+os3bVFabnF^cs7%Fjfs7$$\"+E9,d8Fabn$\"+A$)*QN%FabnF^c s7%F`gs7$$\"+W*fQM\"Fabn$\"+H\"Fabn$\"+y(R$HVFabnF^cs7%F^is7$$ \"+-Ytz7Fabn$\"++3M?VFabnF^cs7%Fdis7$$\"+Zj0o7Fabn$\"+g<')4VFabnF^cs7% Fjis7$$\"+I97d7Fabn$\"+d:'yH%FabnF^cs7%F`js7$$\"+.l8Z7Fabn$\"+u9J%G%Fa bnF^cs7%Ffjs7$$\"+7EMQ7Fabn$\"+tK?pUFabnF^cs7%F\\[t7$$\"+'R>5B\"Fabn$ \"+[5c_UFabnF^cs7%Fb[t7$$\"+wp[D7Fabn$\"+2!eWB%FabnF^cs7%Fh[t7$$\"+#e- @A\"Fabn$\"+%)*Q]@%FabnF^cs7%F^\\t7$$\"+`sC@7Fabn$\"+(=[X>%FabnF^cs7%F d\\t7$$\"+QlHB7Fabn$\"+:%eL<%FabnF^cs7%Fj\\t7$$\"+5)p&G7Fabn$\"+bB*>:% FabnF^cs7%F`]t7$$\"+(*)esB\"Fabn$\"+^*>68%FabnF^cs7%Ff]t7$$\"+A.N\\7Fa bn$\"+RR^6TFabnF^cs7%F\\^t7$$\"+\\:dk7Fabn$\"+Ps&R4%FabnF^cs7%Fb^t7$$ \"+e\\R#G\"Fabn$\"+mC7zSFabnF^cs7%Fh^t7$$\"+@06-8Fabn$\"+:#ou1%FabnF^c s7%F^_t7$$\"+!ySHK\"Fabn$\"+yK>fSFabnF^cs7%Fd_t7$$\"+T::W8Fabn$\"+WwDa SFabnF^cs7%Fj_t7$$\"+&*R7l8Fabn$\"+6jW_SFabnF^cs7%F``t7$$\"+P:Q&Q\"Fab n$\"+%oUM0%FabnF^cs7%Ff`t7$$\"+BPe/9Fabn$\"+v#*)o0%FabnF^cs7%F\\at7$$ \"+CS]A9Fabn$\"+?%GC1%FabnF^cs7%Fbat7$$\"+SO+R9Fabn$\"+/^spSFabnF^cs7% Fhat7$$\"+Ux+a9Fabn$\"+&=x%ySFabnF^cs7%F^bt7$$\"+1k[n9Fabn$\"+zzT)3%Fa bnF^cs7%Fdbt7$$\"+5.Wz9Fabn$\"+eZJ*4%FabnF^cs7%Fjbt7$$\"+V/*)*[\"Fabn$ \"+Ca'46%FabnF^cs7%F`ct7$$\"+$)3())\\\"Fabn$\"+wX>BTFabnF^cs7%Ffct7$$ \"+LQU1:Fabn$\"+T'\\e8%FabnF^cs7%F\\dt7$$\"+Wif7:Fabn$\"+Wtz[TFabnF^cs 7%Fbdt7$$\"+mvV<:Fabn$\"+H1#>;%FabnF^cs7%Fhdt7$$\"+g#)*4_\"Fabn$\"+@g6 vTFabnF^cs7%F^et7$$\"+_)GL_\"Fabn$\"+<9H)=%FabnF^cs7%Fdet7$$\"+Q$zW_\" Fabn$\"+=UO,UFabnF^cs7%Fjet7$$\"+^))\\C:Fabn$\"+%yfU@%FabnF^cs7%F`ft7$ $\"+$\\NM_\"Fabn$\"+L,\"pA%FabnF^cs7%Ffft7$$\"+\"GO8_\"Fabn$\"+\\GDRUF abnF^cs7%F\\gt7$$\"+yrC=:Fabn$\"+k,B^UFabnF^cs7%Fbgt7$$\"+yJ@9:Fabn$\" +!=)yiUFabnF^cs7%Fhgt7$$\"+J%y#4:Fabn$\"+.i(QF%FabnF^cs7%F^ht7$$\"+%R' [.:Fabn$\"+mhW%G%FabnF^cs7%Fdht7$$\"+4+)o\\\"Fabn$\"+K@X%H%FabnF^cs7%F jhtFdbsF^csFh^rF^_r-F$6$7in7$Ff_r$!3#******>&)\\QI%FT7$F]bs$!3)******> 8_WH%FT7$Fhas$!3S+++mhW%G%FT7$Fcas$!3Q+++.i(QF%FT7$F^as$!3E+++!=)yiUFT 7$Fi`s$!3))*****R;I7D%FT7$Fd`s$!3m******[GDRUFT7$F_`s$!3j*****H85pA%FT 7$Fj_s$!3B+++%yfU@%FT7$Fe_s$!3:+++=UO,UFT7$F`_s$!3G+++<9H)=%FT7$F[_s$! 3++++@g6vTFT7$Ff^s$!3#*******G1#>;%FT7$Fa^s$!37+++Wtz[TFT7$F\\^s$!33++ +T'\\e8%FT7$Fg]s$!3R+++wX>BTFT7$Fb]s$!3w*****RUl46%FT7$F]]s$!3A+++eZJ* 4%FT7$Fh\\s$!3#*******yzT)3%FT7$Fc\\s$!3o*****\\=x%ySFT7$F^\\s$!3w**** *R5D(pSFT7$Fi[s$!3#)******>%GC1%FT7$Fd[s$!3i*****\\F*)o0%FT7$F_[s$!3=+ ++%oUM0%FT7$Fjjr$!3u*****4JYC0%FT7$Fejr$!3y*****RkdU0%FT7$F`jr$!3S+++y K>fSFT7$F[jr$!3))*****\\@ou1%FT7$Ffir$!3G+++mC7zSFT7$Fair$!3o*****pBdR 4%FT7$F\\ir$!3&*******QR^6TFT7$Fghr$!3j*****4&*>68%FT7$Fbhr$!3)****** \\N#*>:%FT7$F]hr$!3/+++:%eL<%FT7$Fhgr$!3k*****p=[X>%FT7$Fcgr$!3V+++%)* Q]@%FT7$F^gr$!3v*****p+eWB%FT7$Fifr$!3e*****z/hDD%FT7$Fdfr$!3e*****HF. #pUFT7$F_fr$!3!******RZ6VG%FT7$Fjer$!31+++d:'yH%FT7$Feer$!3V+++g<')4VF T7$F`er$!3J++++3M?VFT7$F[er$!3e*****zxR$HVFT7$Ffdr$!3)******\\z1pL%FT7 $Fadr$!3k******RM4VVFT7$F\\dr$!3-+++mD&zM%FT7$Fgcr$!3y*****p\"p`^VFT7$ Fbcr$!3/+++A$)*QN%FT7$F]cr$!37+++os3bVFT7$Fhbr$!3o*****4r_^N%FT7$Fcbr$ !3=+++^?9aVFT7$F^br$!3e*****p?,@N%FT7$Fiar$!3M+++dZ2\\VFT7$Fdar$!3i*** ***>h5XVFT7$F_ar$!3F+++kxBSVFT7$Fj`r$!3e*****\\V6XL%FT7$Fe`r$!3D+++(Ro zK%FT7$F``r$!3C+++&o\\1K%FT7$F[`r$!3()*****HP'f7VFTFcitFd^n-F__n6jn7%7 $Febs$!+_)\\QI%Fabn7$F[it$!+K@X%H%Fabn7$F_cs$!$>%F]_n7%F]eu7$Feht$!+mh W%G%FabnF`eu7%Fdeu7$F_ht$!+.i(QF%FabnF`eu7%Fheu7$Figt$!+!=)yiUFabnF`eu 7%F\\fu7$Fcgt$!+k,B^UFabnF`eu7%F`fu7$F]gt$!+\\GDRUFabnF`eu7%Fdfu7$Fgft $!+L,\"pA%FabnF`eu7%Fhfu7$Faft$!+%yfU@%FabnF`eu7%F\\gu7$F[ft$!+=UO,UFa bnF`eu7%F`gu7$Feet$!+<9H)=%FabnF`eu7%Fdgu7$F_et$!+@g6vTFabnF`eu7%Fhgu7 $Fidt$!+H1#>;%FabnF`eu7%F\\hu7$Fcdt$!+Wtz[TFabnF`eu7%F`hu7$F]dt$!+T'\\ e8%FabnF`eu7%Fdhu7$Fgct$!+wX>BTFabnF`eu7%Fhhu7$Fact$!+Ca'46%FabnF`eu7% F\\iu7$F[ct$!+eZJ*4%FabnF`eu7%F`iu7$Febt$!+zzT)3%FabnF`eu7%Fdiu7$F_bt$ !+&=x%ySFabnF`eu7%Fhiu7$Fiat$!+/^spSFabnF`eu7%F\\ju7$Fcat$!+?%GC1%Fabn F`eu7%F`ju7$F]at$!+v#*)o0%FabnF`eu7%Fdju7$Fg`t$!+%oUM0%FabnF`eu7%Fhju7 $Fa`t$!+6jW_SFabnF`eu7%F\\[v7$F[`t$!+WwDaSFabnF`eu7%F`[v7$Fe_t$!+yK>fS FabnF`eu7%Fd[v7$F__t$!+:#ou1%FabnF`eu7%Fh[v7$Fi^t$!+mC7zSFabnF`eu7%F\\ \\v7$Fc^t$!+Ps&R4%FabnF`eu7%F`\\v7$F]^t$!+RR^6TFabnF`eu7%Fd\\v7$Fg]t$! +^*>68%FabnF`eu7%Fh\\v7$Fa]t$!+bB*>:%FabnF`eu7%F\\]v7$F[]t$!+:%eL<%Fab nF`eu7%F`]v7$Fe\\t$!+(=[X>%FabnF`eu7%Fd]v7$F_\\t$!+%)*Q]@%FabnF`eu7%Fh ]v7$Fi[t$!+2!eWB%FabnF`eu7%F\\^v7$Fc[t$!+[5c_UFabnF`eu7%F`^v7$F][t$!+t K?pUFabnF`eu7%Fd^v7$Fgjs$!+u9J%G%FabnF`eu7%Fh^v7$Fajs$!+d:'yH%FabnF`eu 7%F\\_v7$F[js$!+g<')4VFabnF`eu7%F`_v7$Feis$!++3M?VFabnF`eu7%Fd_v7$F_is $!+y(R$HVFabnF`eu7%Fh_v7$Fihs$!+&z1pL%FabnF`eu7%F\\`v7$Fchs$!+SM4VVFab nF`eu7%F``v7$F]hs$!+mD&zM%FabnF`eu7%Fd`v7$Fggs$!+FTF(-%'COLOURG6&Fg^ nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F`dv7$F($\"3/++++++!z%FTFedvFhdv -%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAU LTG-%(SCALINGG6#%,CONSTRAINEDG-F]fv6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z% F]_n$\"$*>F]_n;F\\gv$\"$z%F]_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute \+ stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " } {XPPEDIT 18 0 "[-3.9879, 0];" "6#7$,$-%&FloatG6$\"&z)R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The stability radius is half the length of the real stabi lity interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(R(z )=-1,z=-4)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++O'R*>!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 401 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+17291/12418560*z ^6+269/1379840*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 360 do\n \+ zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=red):\nrd := 1. 993963600: \np2 := plot([-rd+rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue ):\nplots[display]([p1,p2],scaling=constrained,font=[HELVETICA,9]);" } }{PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7ea l7$$\"\"!F)F(7$F($\"31+++^v>Z5!#=7$$\"3!******>K-Nj$!#F$\"3-+++,^R%4#F -7$$\"3)******>J3Q5&!#E$\"3(******pi#fTJF-7$$\"3')*****4[#oxN!#D$\"3#* *****>(**y)=%F-7$$\"31+++qoX\"o\"!#C$\"3d*****zW')fB&F-7$$\"3^*****f8+ c.'FC$\"3S+++&yzJG'F-7$$\"3)******zg#Gv!#A$\"3)******4fir/\"!#<7$$\"3n******e@=WMFhn$\"31++ ++4%=:\"F[o7$$\"3I+++6Y&R[&Fhn$\"33+++h8[c7F[o7$$\"3f*****>;kgr(Fhn$\" 3/+++z\"e5O\"F[o7$$\"3O+++Ou(f6*Fhn$\"3#*******\\M`l9F[o7$$\"39+++kjny uFhn$\"3++++iU&)p:F[o7$$!3-+++_3%oE\"Fhn$\"31+++L.&Rn\"F[o7$$!3'****** fk\\?R#!#@$\"3!******>zKxx\"F[o7$$!3^+++odj+rF]q$\"31+++jY4\")=F[o7$$! 3'******>\\*3y:!#?$\"3!******4L8R)>F[o7$$!3'******p%4?`IFhq$\"3))***** *zX0'3#F[o7$$!32+++\\v\\4aFhq$\"3=+++upP(=#F[o7$$!37+++9.q))*)Fhq$\"37 +++NVt(G#F[o7$$!3++++&G&*)>9!#>$\"3#)*****zZtpQ#F[o7$$!3'******pt83:#F ]s$\"35+++yV#\\[#F[o7$$!3t*****R(z-VJF]s$\"3#)*****f)3Q\"e#F[o7$$!39++ +?M@]WF]s$\"3%)*****>Usgn#F[o7$$!3<+++3,`ChF]s$\"3!******p0J'oFF[o7$$! 3j+++>P$4@)F]s$\"3;+++CXceGF[o7$$!3++++Fv'R2\"F-$\"3#)*****HO__%HF[o7$ $!3#******\\$)Q=P\"F-$\"3A+++nS)z-$F[o7$$!3-+++:]n7:Q)yJF[o7$$!3t******[l:-DF-$\"37+++/ I(fC$F[o7$$!3<+++(e]k$HF-$\"37+++PLyc$F[o7$$!3[***** pYg(zhF-$\"3=+++O[w'f$F[o7$$!3a+++n\"G(p$F[o7$$ !3O+++vHel))F-$\"3!)*****fO())4PF[o7$$!35+++.\\N)H*F-$\"3z*****RP>.s$F [o7$$!3&******4z!>F(*F-$\"38+++\\!z'GPF[o7$$!35+++sWA:5F[o$\"39+++/N0N PF[o7$$!3/+++rqOd5F[o$\"3#)*****zh<&RPF[o7$$!3'*******e?;*4\"F[o$\"3,+ ++1d8UPF[o7$$!3!******RMC19\"F[o$\"3#******z\"H'Hu$F[o7$$!3#******fjo< =\"F[o$\"38+++2k/UPF[o7$$!3\"******Rf4EA\"F[o$\"3&******pJE%RPF[o7$$!3 /+++]?;j7F[o$\"30+++:m8NPF[o7$$!3)******fGTMI\"F[o$\"3/+++2e?HPF[o7$$! 3-+++^LYV8F[o$\"33+++qvl@PF[o7$$!33+++NaC$Q\"F[o$\"37+++18^7PF[o7$$!31 +++xi!GU\"F[o$\"39+++rEy,PF[o7$$!3++++Fm;i9F[o$\"3-+++tS[*o$F[o7$$!3!* *****\\v\\8]\"F[o$\"3++++(>Dcn$F[o7$$!3.+++/?QS:F[o$\"3)******fb8-m$F[ o7$$!3'******fR$Hz:F[o$\"3()*****H7bKk$F[o7$$!3++++p#=\"=;F[o$\"3A+++t ]vCOF[o7$$!3#*******[f*ol\"F[o$\"3&*******z'=Zg$F[o7$$!3!******pRrcp\" F[o$\"3%******>U_Je$F[o7$$!36+++@e\\MF[o$\"3\")*****zqdDU$F[o7$$!3%*******3<F[o$\"33+++$*z $4R$F[o7$$!32+++cmH8?F[o$\"3A+++\\!=\"eLF[o7$$!3#)*****Hsi]0#F[o$\"3?+ ++%y7UK$F[o7$$!31+++E&Gv4#F[o$\"3#******4\\f$*G$F[o7$$!3.+++p0tS@F[o$ \"3#)*****Ri:PD$F[o7$$!3++++'4wY=#F[o$\"37+++zOX+++x.zVJF[o7$$!3!***** **RC^?BF[o$\"34+++Jts1JF[o7$$!3))*****pK$zmBF[o$\"37+++]G.$F[o7$$!31+++xV)*fCF[o$\"3!******R1_h*HF[o7 $$!3*)*****>mil]#F[o$\"39+++]YtfHF[o7$$!3=+++eY\"Hb#F[o$\"34+++v&GF[o7$$!3?+++iyC*o#F[o$\"3$******4\")*H;GF[o7$$!3!******>MAMt#F [o$\"3++++b&p2y#F[o7$$!3%)*******3Fox#F[o$\"3))*****zzW_u#F[o7$$!3=+++ jgS>GF[o$\"3@+++$)yl4FF[o7$$!3=+++ho6hGF[o$\"37+++')H%Rn#F[o7$$!3@+++o &G>!HF[o$\"39+++Aw.QEF[o7$$!3++++g(>=%HF[o$\"3*******za$)=g#F[o7$$!3') *****Rvw2)HF[o$\"31+++(=Fac#F[o7$$!3=+++K@z=IF[o$\"3;+++a(>'GDF[o7$$!3 1+++;N'e0$F[o$\"3))*****p:<9\\#F[o7$$!35+++)e#*>4$F[o$\"3/+++&yzPX#F[o 7$$!3!)*****>J%=FJF[o$\"34+++=An:CF[o7$$!3#*******RjWhJF[o$\"3=+++?H1x BF[o7$$!37+++b&)y%>$F[o$\"3$******z#R#zL#F[o7$$!3*)*****puAsA$F[o$\"3A +++p0B)H#F[o7$$!3++++rBweKF[o$\"3)******\\?hzD#F[o7$$!3;+++$RA%*G$F[o$ \"3*)*****p-(4#>LF[o$\"3=+++$*=iv@F[o7$$!3?+++j/<[ LF[o$\"39+++?A_L@F[o7$$!3'*******p_HwLF[o$\"3!******p#py!4#F[o7$$!3<++ +ORh.MF[o$\"3!******4V2u/#F[o7$$!38+++x![,V$F[o$\"3z*****4qxL+#F[o7$$! 3#)*****>b?fX$F[o$\"33+++cVpe>F[o7$$!39+++7a&4[$F[o$\"3/+++@oN8>F[o7$$ !36+++'zx_]$F[o$\"33+++6vOn=F[o7$$!3<+++yP\"*GNF[o$\"3'******41K2#=F[o 7$$!39+++W,*=b$F[o$\"3!*******o&fMx\"F[o7$$!3/+++!4MUd$F[o$\"32+++VGcD yl$ F[o$\"3%******f/J\"G:F[o7$$!3!)******[7RxOF[o$\"3-+++vzWx9F[o7$$!31+++ (\\ukp$F[o$\"3/+++QOIE9F[o7$$!3>+++[R3:PF[o$\"3/+++3)QZP\"F[o7$$!3\")* ****4()HKt$F[o$\"33+++gqzA8F[o7$$!3;+++iv\"4v$F[o$\"3))*****z)R_q7F[o7 $$!35+++')p9oPF[o$\"3++++5j'z@\"F[o7$$!3%)*****>h7\\y$F[o$\"33+++y3S#[ (QF[o$\"3M+++0L/e%)F-7$$!3=+++/+z()QF[o$\"3Z*****RXVZ#zF-7$$!3))*****R m'4+RF[o$\"3C+++0%R=R(F-7$$!3A+++l9r6RF[o$\"3_*****H@([foF-7$$!38+++4E gARF[o$\"3s*****Hl+yK'F-7$$!3\")*****4fRF$RF[o$\"3e*****Rqaoz&F-7$$!3? +++VO4URF[o$\"3S+++p&*om_F-7$$!3++++u!Q1&RF[o$\"3F+++1\\JPZF-7$$!3!)** ***H^[$eRF[o$\"3#******>29(3UF-7$$!3/+++nI?lRF[o$\"3=+++y\"[3o$F-7$$!3 ))******>C=rRF[o$\"37+++j)fO:$F-7$$!3%)*****\\()pi(RF[o$\"3z*****ppwqi #F-7$$!3!)*****HO^/)RF[o$\"32+++8U,,@F-7$$!3++++Gar$)RF[o$\"3'*******e &y`d\"F-7$$!3;+++0K0')RF[o$\"3/+++;)o+0\"F-7$$!39+++'Qeu)RF[o$\"31+++_ -z\\_F]s7$$!3/+++*>Fz)RF[oF(7$Fbjm$!31+++_-z\\_F]s7$F]jm$!3/+++;)o+0\" F-7$Fhim$!3'*******e&y`d\"F-7$Fcim$!32+++8U,,@F-7$F^im$!3z*****ppwqi#F -7$Fihm$!37+++j)fO:$F-7$Fdhm$!3=+++y\"[3o$F-7$F_hm$!3#******>29(3UF-7$ Fjgm$!3F+++1\\JPZF-7$Fegm$!3S+++p&*om_F-7$F`gm$!3e*****Rqaoz&F-7$F[gm$ !3s*****Hl+yK'F-7$Fffm$!3_*****H@([foF-7$Fafm$!3C+++0%R=R(F-7$F\\fm$!3 Z*****RXVZ#zF-7$Fgem$!3M+++0L/e%)F-7$Fbem$!3c*****4)z`\"**)F-7$F]em$!3 z*****p8!)\\_*F-7$Fhdm$!3*******Hh2e+\"F[o7$Fcdm$!3++++*y[!f5F[o7$F^dm $!3'******fm$=76F[o7$Ficm$!33+++y3F[o7$Fi^m$!33+++cVpe>F[o7$Fd^m$ !3z*****4qxL+#F[o7$F_^m$!3!******4V2u/#F[o7$Fj]m$!3!******p#py!4#F[o7$ Fe]m$!39+++?A_L@F[o7$F`]m$!3=+++$*=iv@F[o7$F[]m$!3*)*****p-(4'GDF[o7$F^jl$!31+++(=Fac#F[o7$ Fiil$!3*******za$)=g#F[o7$Fdil$!39+++Aw.QEF[o7$F_il$!37+++')H%Rn#F[o7$ Fjhl$!3@+++$)yl4FF[o7$Fehl$!3))*****zzW_u#F[o7$F`hl$!3++++b&p2y#F[o7$F [hl$!3$******4\")*H;GF[o7$Ffgl$!3)******\\s->&GF[o7$Fagl$!3#******\\lT w)GF[o7$F\\gl$!34+++vG.$F[o7$Fhel$!37+++]+++x.zVJF[o7$Fidl$!3?+++nJv!=$F[o7$Fddl$!37+++zOXU_Je$F[o 7$Fc`l$!3&*******z'=Zg$F[o7$F^`l$!3A+++t]vCOF[o7$Fi_l$!3()*****H7bKk$F [o7$Fd_l$!3)******fb8-m$F[o7$F__l$!3++++(>Dcn$F[o7$Fj^l$!3-+++tS[*o$F[ o7$Fe^l$!39+++rEy,PF[o7$F`^l$!37+++18^7PF[o7$F[^l$!33+++qvl@PF[o7$Ff]l $!3/+++2e?HPF[o7$Fa]l$!30+++:m8NPF[o7$F\\]l$!3&******pJE%RPF[o7$Fg\\l$ !38+++2k/UPF[o7$Fb\\l$!3#******z\"H'Hu$F[o7$F]\\l$!3,+++1d8UPF[o7$Fh[l $!3#)*****zh<&RPF[o7$Fc[l$!39+++/N0NPF[o7$F^[l$!38+++\\!z'GPF[o7$Fiz$! 3z*****RP>.s$F[o7$Fdz$!3!)*****fO())4PF[o7$F_z$!3!******>n\"G(p$F[o7$F jy$!3#)*****fJ%Q#o$F[o7$Fey$!35+++gy0lOF[o7$F`y$!3%)*****f4U^k$F[o7$F[ y$!3()*****RD]Ci$F[o7$Ffx$!3=+++O[w'f$F[o7$Fax$!3%******>PLyc$F[o7$F\\ x$!3<+++QZONNF[o7$Fgw$!3%)*****fJF!*\\$F[o7$Fbw$!3/+++C7XeMF[o7$F]w$!3 !******pAQKT$F[o7$Fhv$!3!)*****fL%)HO$F[o7$Fcv$!37+++:Q)yJF[o7$Fdu$!3)*******[h/1JF[o7$F_u$!3A+ ++nS)z-$F[o7$Fjt$!3#)*****HO__%HF[o7$Fet$!3;+++CXceGF[o7$F`t$!3!****** p0J'oFF[o7$F[t$!3%)*****>Usgn#F[o7$Ffs$!3#)*****f)3Q\"e#F[o7$Fas$!35++ +yV#\\[#F[o7$F[s$!3#)*****zZtpQ#F[o7$Ffr$!37+++NVt(G#F[o7$Far$!3=+++up P(=#F[o7$F\\r$!3))******zX0'3#F[o7$Ffq$!3!******4L8R)>F[o7$Faq$!31+++j Y4\")=F[o7$F[q$!3!******>zKxx\"F[o7$Ffp$!31+++L.&Rn\"F[o7$$\"3]*****>O w'yuFhn$!3++++iU&)p:F[o7$F\\p$!3#*******\\M`l9F[o7$Fgo$!3/+++z\"e5O\"F [o7$Fbo$!33+++h8[c7F[o7$F]o$!31++++4%=:\"F[o7$Ffn$!3)******4fir/\"F[o7 $$\"3s*****>)evd)*FN$!3w*****\\jIYU*F-7$FR$!3A+++cT_x$)F-7$FL$!3]***** *GQOItF-7$FG$!3S+++&yzJG'F-7$$\"3'******H\"oX\"o\"FC$!3d*****zW')fB&F- 7$F;$!3#******>(**y)=%F-7$$\"3v*****fJ3Q5&F7$!3(******pi#fTJF-7$$\"3$* *****\\K-Nj$F1$!3-+++,^R%4#F-7$F($!31+++^v>Z5F-F'F*F.F4F:F@FFFKFQFVFen F\\oFaoFfoF[pF`pFepFjpF`qFeqF[rF`rFerFjrF`sFesFjsF_tFdtFitF^uFcuFhuF]v FbvFgvF\\wFawFfwF[xF`xFexFjxF_yFdyFiyF^zFczFhzF][lFb[lFg[lF\\\\lFa\\lF f\\lF[]lF`]lFe]lFj]lF_^lFd^l-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7 SF'7$$!3_^*>\"z,5n=F]s$\"3hPT*)**\\JAFF-7$$!3G'3JMTnU]'F]s$\"3M^TrtfG^ ]F-7$$!3gF*)p47H)\\\"F-$\"3]/xaaNE$e(F-7$$!3#f]+(fvs&o#F-$\"3[7\"[6!\\ e%***F-7$$!3OO6Ah^%)yTF-$\"3'ytqa[>9A\"F[o7$$!33vGV84Ct7A)[x:F[o7$$!3UPw!)3*R$[**F-$\"3#>V:9J \"eDF[o7$$!3oy&oQ@[ds\"F[o$\"3]pK9L>%e(>F[o7$$!3=]#y&4U:&*>F[o$\"3; D$RXCjR*>F[o7$$!3Enp*HiPcE#F[o$\"3S&>\"f[$p`(>F[o7$$!3r*=y')o-;_#F[o$ \"3T'R[yG&)G#>F[o7$$!39Gxt'y2ju#F[o$\"3'=())oxIeY=F[o7$$!3'*HF`6HQ**HF [o$\"3\\:#y'pT#>s\"F[o7$$!3+0P%y\"e@)>$F[o$\"3!RD>!e[B*e\"F[o7$$!3K)GL WZZvS$F[o$\"3c\"*puwqH19F[o7$$!3mwQGZJ.qNF[o$\"3)eFv5zJ9A\"F[o7$$!3=P; /Wyf?PF[o$\"37B(=o\"*)3t**F-7$$!3Su`D#pyX$QF[o$\"3yc8(pL\"HowF-7$$!3! \\c&=_\\&3#RF[o$\"366W:vH;G^F-7$$!3suCGC$*QpRF[o$\"3)**e.1%zj7FF-7$$!3 GKh)3hB!y'4m%=F[o7 $$!38/qO@?79DF[o$!3B_:L]B#\\#>F[o7$$!3!e.!o8t\\^AF[o$!3%H)pHeDEx>F[o7$ $!3'GSp7\"*f++#F[o$!3w$p(f!GaR*>F[o7$$!3cw$Q0F'>PF[o7 $$!3OMr,cw\\%[\"F[o$!3PkhI'))zx#>F[o7$$!3J\")eAk$>\"H7F[o$!3')eyVw2WT= F[o7$$!3mIBSrnBj**F-$!31q&fr*HWE5#**F-7$$!3WFBK*))>`a\"F-$!3ebz4Roj'p(F-7$$! 3'3WWWDy`)oF]s$!3]0.iZ&F-7$$!3KS9mWFFb=F]s$!3PArCp#>Pr#F-7$F($\"3S *3zF3vej\"F7-Fcgo6&FegoF(F(Ffgo-%(SCALINGG6#%,CONSTRAINEDG-%%FONTG6$%* HELVETICAG\"\"*-%+AXESLABELSG6%Q!6\"Fhgp-Fagp6#%(DEFAULTG-%%VIEWG6$F\\ hpF\\hp" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the largest interval on the nonnegative imaginary axis th at contains the origin and lies inside the stability region is [ 0, 1 .67 ] approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24 *z^4+1/120*z^5+17291/12418560*z^6+269/1379840*z^7:\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct* Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im (zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.8,0,.8),thickness=2,font=[ HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 249 293 293 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)F(7$$\"5%Q*GND&*=JJ[!#@$\" 5PEs(*e`EfTJF-7$$\"5&*Ga[0CsyhqF-$\"5k8zo<2`=$G'F-7$$\"5cq1v!o)GEJ))F- $\"523![A2'zxC%*F-7$$\"53$3q$3)*3dO5!#?$\"5c&)4,Shqjc7F=7$$\"5u'**o*** RMYv6F=$\"5[bS+iEjzq:F=7$$\"5^&f7n-hCWI\"F=$\"5F1u$4:fb\\)=F=7$$\"5(e2 )Rs+;#F=7$$\"53>Ml!pYECa\"F=$\"5k;;HqF=$\"5WES*>'z5\"*pPF=7$$ \"5#oU[XIvq52#F=$\"5QS/M^d-2%3%F=7$$\"53$**f(4IRIp@F=$\"5%H*31Lw$H#)R% F=7$$\"5bPB?$>\")ebE#F=$\"5J,%*Q(RS)Q7ZF=7$$\"5BiR:\\]$\\*fBF=$\"5@XTc Z%HZl-&F=7$$\"5xs0!*\\r.c_CF=$\"5(HG\")RL)fqS`F=7$$\"5IhWe))>ZXVDF=$\" 5Ml#*Rs\"Qk[l&F=7$$\"59WNY&fDyEj#F=$\"5#G=Dr#pB-pfF=7$$\"54C!QqMNj-s#F =$\"5QqG!)Q&yzJG'F=7$$\"5Nk8SjR8B1GF=$\"5@:,Hv=kL(f'F=7$$\"5M\"y#z3wTf !*GF=$\"5]r;?$[*>\\6pF=7$$\"5'Qv7%z%ebL(HF=$\"5^4'\\q6;YcA(F=7$$\"5m?Z A:d<^aIF=$\"5*[$zdAq%)zRvF=7$$\"5(p9mw*Q<0MJF=$\"5>84\"4&f$[R&yF=7$$\" 5#*oGTC^v&>@$F=$\"5?!)>c!)G^4o\")F=7$$\"5kl**oK#3/#)G$F=$\"5ho;BD9zB#[ )F=7$$\"5a[G[do(eFO$F=$\"5Eb*3n!fcP'z)F=7$$\"5#fuC!Gk5eNMF=$\"5Vn?8u!3 206*F=7$$\"5^'yz2uo@m]$F=$\"5M/QT`M1jC%*F=7$$\"58POhN7;#ed$F=$\"5NjV%Q JZW(Q(*F=7$$\"5?t#o]Pw5Jk$F=$\"50%=!GCSYG05!#>7$$\"5ec*HVQM1%3PF=$\"5H quTK$Q$pO5Ffu7$$\"5+*eCayp5;x$F=$\"5jru:#HP+\"o5Ffu7$$\"5S'z]2Fe3E$QF= $\"56+HP2b_]*4\"Ffu7$$\"5lzfUG\\OE\"*QF=$\"5Wb6m^>w!48\"Ffu7$$\"5oAm=U nQTZRF=$\"5if=\\N$*pIi6Ffu7$$\"52@r3e,b'3+%F=$\"5Cz*R9W$Gq$>\"Ffu7$$\" 5E>:w`e]Q^SF=$\"58h%H;V_%4D7Ffu7$$\"5EvdMoL$)o)4%F=$\"5/TBcKh8[c7Ffu7$ $\"5 8Ffu7$$\"5!Hh(>$p*)Ht@%F=$\"5M)Q1dJL/1N\"Ffu7$$\"5F:'H^\"or@ZUF=$\"5$f uKao\"G'>Q\"Ffu7$$\"50vXl7]T&3F%F=$\"5psBYTJ9J89Ffu7$$\"56-ZE$)R'4pG%F =$\"5^j?__@)[YW\"Ffu7$$\"5H795P$zqMH%F=$\"5pJ(z()))\\tfZ\"Ffu7$$\"5JoL ?6#fNwG%F=$\"5=#*zy\\OQG2:Ffu7$$\"5wz&QINtiYE%F=$\"5Y:_Zmj!y&Q:Ffu7$$ \"5KqSP#ftFf@%F=$\"5E%)[x\">Ea)p:Ffu7$$\"5hh?L?Ao!G7%F=$\"5ZJyH%>O56g \"Ffu7$$\"5h#**)QwqpIHRF=$\"5z>3zoTTMK;Ffu7$$\"5])p!>B!\\Tk$GF=$\"52ZH \"fdA`Nm\"Ffu7$$!5&f8c1**pU\"=SF=$\"5Iq'G?i3NZp\"Ffu7$$!5\"yy;.kl!HSVF =$\"5)\\8/s[/()es\"Ffu7$$!5zPFB(Hi;Dc%F=$\"5l\\*fjqF1qv\"Ffu7$$!50aKVy Ys'Gu%F=$\"5/S*eXq\")*3)y\"Ffu7$$!5?'pNP3(\\%)**[F=$\"5Zo+q#[cM\">=Ffu 7$$!5ZU6;'Q6zA'fU29t^F=$\"5j(Qc&\\jY4 \")=Ffu-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"Fj^l-%&COLORG6&%$RGBG$ \"\")!\"\"F(F`_l-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$%(DEFAULTGF[`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The rele vant intersection point of the boundary curve with the imaginary axis \+ can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either sid e of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 1.7*I: \nfor ct from 52 to 55 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0q`-&>FZM! #>$\"0\"zoTTMK;!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0f$\\`Q>h&* !#@$\"08fdA`Nm\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0WO$H*y$3W! #>$\"0H?i3NZp\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0.\"e&3\"pH5 !#=$\"0/s[/()es\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the p arameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton (R(z)=exp(u*Pi*I),z=1.7*I))\nend proc:\nDigits := 15:\nu0 := bisect('r eal_part'(u),u=0.52..0.55);\nnewton(R(z)=exp(u0*Pi*I),z=1.7*I);\nDigit s := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0H]M)zW-`!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0r?#))oJk;!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "la rgest interval on the nonnegative imaginary axis that contains the ori gin and lies inside the stability region" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "[ 0, 1.6643 ]" "6#7$\"\"!-%&FloatG6$\"&Vm\"!\"%" } {TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 8 st age, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 80 "The stability function R* for the 8 stage, order 4 sc heme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(4,8,'expanded'))):\n`R*` := una pply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# R*G6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\" $F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#\"/*315*pfl\"1!3qkb&yHyF)*$)F'\" \"&F)F)F)*&#\"2\"G5NC50G8\"4g&Q6d0eHY'*F)*$)F'F1F)F)F)*&#\",r*\\NB#)\" 0gdkEvwM%F)*$)F'\"\"(F)F)F)*&#\"),za6\".G@f+:m*F)*$)F'\"\")F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " z_0 := newton(`R*`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ z_0G$!+7e)3-%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=-1,z=-4):\np_1 := plo t([`R*`(z),-1],z=-4.49..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1 ]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np_3 := \+ plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[d isplay]([p_1,p_2,p_3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVES G6$7U7$$!3A++++++!\\%!#<$!3GAg:o2'3L#F*7$$!3;+]7'3DdV%F*$!3f]U%oR%oC@F *7$$!35++Ds,X\"Q%F*$!3*3rf[o8V$>F*7$$!3(*\\P*)feAMVF*$!3arB<$o;2y\"F*7 $$!3%)*\\Pva,qG%F*$!3l8kay'Qwj\"F*7$$!3I+](4>%y!=%F*$!3Ski.>!)3^8F*7$$ !35+]#oviQ2%F*$!3D!z1DPGm5\"F*7$$!3A+ve,&\\u'RF*$!3;=x\"zSDE,*!#=7$$!3 ?+v8#z!zoQF*$!3=E$=,a25S(FP7$$!3;+v.8ajmPF*$!3IP%=l;s\"))fFP7$$!3C+v$Q <')4m$F*$!3q[Ag@nYiZFP7$$!3G+v3`dnbNF*$!3Il7Vp7\"Qu$FP7$$!37++0T5NZMF* $!3!G%=,5FwuGFP7$$!3?+]sv&Q>N$F*$!32=bwbi.OAFP7$$!3E++:Qn_WKF*$!3YP??( 4$*ej\"FP7$$!3#****\\s&QnOJF*$!3j&3e;%ReQ6FP7$$!3&****\\=iPF.$F*$!3zmj NK.F5u!#>7$$!3Q+v$Gc`$QHF*$!3+X`\"f9)\\mVFhp7$$!3+++l(y@h#GF*$!3V#Q@'R #Q)48Fhp7$$!3=++g_n/JFF*$\"3Q5kG->P0!*!#?7$$!3C+vt%)=X?EF*$\"3St'y-JO# QJFhp7$$!3D++5(ocD_#F*$\"3o&*3'*Q\\?'*[Fhp7$$!3#**\\P(R,::CF*$\"3#RfGT .?$emFhp7$$!3&)*\\7sqtGJ#F*$\"3\"RRi$\\C^M#)Fhp7$$!30+]Pz*eh?#F*$\"3K^ _/J9dQp\"F*$\"3BOV:3LU>=FP7$$!3)*****p]Q@(e\"F *$\"3Jmi*=oyG.#FP7$$!3I+]FRT)G[\"F*$\"3+F+')4)3@E#FP7$$!3')*\\P*y(R>Q \"F*$\"3v,8@\"*e,1DFP7$$!3++]Kx#e)p7F*$\"33)3q$e8%e!GFP7$$!3=++5j![\"p 6F*$\"3wT\"Qfv%e/JFP7$$!3s***\\KT=;1\"F*$\"3'>00$*o$)zX$FP7$$!3'***\\P %o0=k*FP$\"38b&z\"4+S7QFP7$$!3](***\\tEbw&)FP$\"3u(oS#\\$o7C%FP7$$!39- ]P2EBuvFP$\"3#[%R'4<'e)o%FP7$$!3>)*\\(GL>l_'FP$\"3?N(*pIrc1_FP7$$!3z++ ]-\")=-bFP$\"3)G]:q`4#odFP7$$!3$*)*\\PM#3)HWFP$\"3W/#4ym'=@kFP7$$!3;(* ***f'*ypR$FP$\"3@KCUT;&)>rFP7$$!3L)**\\U'=wSBFP$\"3Ih?9S\\,8zFP7$$!3?$ *\\Pt2H$H\"FP$\"3Wy@c5w%oy)FP7$$!33R****pit2LFhp$\"3Mw?(36PYn*FP7$$\"3 Z^+]FkzBxFhp$\"3QiT_=\"*H!3\"F*7$$\"35*****z2`!f\"F*7$ $\"3l+]i_F06GFP$\"3G:kV8JfC8F*7$$\"3\\.]Pt1&z\"QFP$\"3+$fMx97\\Y\"F*7$ $\"3!***************[FP$\"39hA:krJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\") $\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$F GFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa \\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FdqFa\\l7$FjqFa \\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa \\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vF a\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxF a\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$Ffz Fa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3R+++7e )3-%F*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG 6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDG F_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*L INESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl 6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F` cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture sho ws the stability region for the 8 stage, order 4 scheme. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1430 "`R* ` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+65596991006089/7829785556470080 *z^5+\n 13280510243510281/9646295805571138560*z^6+\n 822335499 71/434767526645760*z^7+11547901/9661500592128*z^8:\npts := []:\nz0 := \+ 0:\nfor ct from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z =z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_ 1 := plot(pts,color=COLOR(RGB,.35,.18,.35)):\np_2 := plots[polygonplot ]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=pat chnogrid,color=COLOR(RGB,.7,.35,.7)):\npts := []:\nz0 := 1.5+4.3*I:\nf or ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := \+ plot(pts,color=COLOR(RGB,.35,.18,.35)):\np_4 := plots[polygonplot]([se q([pts[i-1],pts[i],[1.33,4.23]],i=2..nops(pts))],\n style=pat chnogrid,color=COLOR(RGB,.7,.35,.7)):\npts := []:\nz0 := 1.5-4.3*I:\nf or ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := \+ plot(pts,color=COLOR(RGB,.35,.18,.35)):\np_6 := plots[polygonplot]([se q([pts[i-1],pts[i],[1.33,-4.23]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,.7,.35,.7)):\np_7 := plot([[[-4.79,0],[1.99, 0]],[[0,-4.79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([p _||(1..7)],view=[-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$$!3)******HK[HT)!#F$\"37+++&G'zq:!#=7$$!3++++$=XP>&!#D$\" 30+++g8fTJF07$$!3]+++zXwwb!#C$\"3v*****HqzBr%F07$$!3'******HuMC*G!#B$ \"3K+++$\\[JG'F07$$!33+++*eG:+\"!#A$\"3U+++#ytQ&yF07$$!3!******p$4))*o #FF$\"3R+++!)y]C%*F07$$!3&)*****z2@f7'FF$\"3'******pY%\\*4\"!#<7$$!3-+ ++?`fi7!#@$\"3)*******pZ\\c7FS7$$!3!******>_LP[#FW$\"3'*******4JS89FS7 $$!3;+++#*o)*\\[FW$\"3!******\\()H,d\"FS7$$!3_******Hdjo&*FW$\"36+++ws _E/:&)*=!#?$\"3/+++irP#)=FS7$$!3&******H[R^s$Ffo$\"32+ ++>3RP?FS7$$!3p*****4lTi5(Ffo$\"35+++SIA\">#FS7$$!31+++L**)HI\"!#>$\"3 :+++7#*[VBFS7$$!35+++#p7RG#Ffp$\"3?+++1Uv$\\#FS7$$!3C+++oV&>#QFfp$\"35 +++>iWTEFS7$$!3<+++\\\"y'4hFfp$\"33+++njl&y#FS7$$!3l*****p(oAN$*Ffp$\" 3?+++_*\\[#HFS7$$!3!******H\\^DO\"F0$\"3%)*****R@#ocIFS7$$!33+++t*Gt*= F0$\"3#******>j$HyJFS7$$!3%)*****4(yb?DF0$\"3/+++YHB(G$FS7$$!34+++Q%eV ?$F0$\"31+++iuQ#Q$FS7$$!3&)*****R(4*4#RF0$\"3?+++Lu6kMFS7$$!3C+++<[t\\ YF0$\"3&)*****\\7)oLNFS7$$!3y*****p!f\"yP&F0$\"3y*****fCPEf$FS7$$!3K++ +3UT)4'F0$\"3\")*****zd2Ck$FS7$$!3(********Q'R3oF0$\"3/+++w`@%o$FS7$$! 3`+++M\"zm](F0$\"3,+++>%R!>PFS7$$!3?+++fIB$>)F0$\"3!******pG`wu$FS7$$! 3a+++h\\_o))F0$\"3'******4jl1x$FS7$$!3y******y8AL&*F0$\"3!******p3c&)y $FS7$$!3'******p*Q!)=5FS$\"3*******4,/Ys[X\"FS$\"3=+++Hy\")zPFS7$$!3++++2Ro9:FS$\"3A+++x+%>w$FS7$$!3!****** 4vpSd\"FS$\"3\")*****fin0u$FS7$$!31+++0)=Jj\"FS$\"3z*****pCFdr$FS7$$!3 -+++=z$>p\"FS$\"33+++y/W(o$FS7$$!3#******HUc1v\"FS$\"3++++k)Hdl$FS7$$! 3!******pJI%4=FS$\"3*)******zsi?OFS7$$!3!*******y_Wo=FS$\"3#)******y]= #e$FS7$$!3'*******\\p\"z#>FS$\"31+++q1\\SNFS7$$!31+++Yg3))>FS$\"3))*** **>X%o&\\$FS7$$!3')******3^?\\?FS$\"37+++,'zzW$FS7$$!3>+++&p8:6#FS$\"3 #)*****fkzwR$FS7$$!3!)******\\4?v@FS$\"3#)*****He'=XLFS7$$!3*)*****pmf .C#FS$\"3')*****pC!*4H$FS7$$!3'******H[VpI#FS$\"3))*****z.KcB$FS7$$!3+ +++[TuuBFS$\"3A+++9_kzJFS7$$!3$)*****\\^-MW#FS$\"3@+++In[BJFS7$$!3&)** ***zZ`C^#FS$\"3%)*****>,$[nIFS7$$!3!)******p&*Q\"e#FS$\"3*)*****Hk1=,$ FS7$$!3++++tDAs%GFS$\"3%)*****zr* \\\"z#FS7$$!3*)*****f6p'4HFS$\"3*******p#=@OFFS7$$!3))*****4Ac,(HFS$\" 3%******z)eW!o#FS7$$!32+++'z.'GIFS$\"3++++x(>Si#FS7$$!3/+++z%o\\3$FS$ \"3#*******46xmDFS7$$!3z*****R.M#RJFS$\"3#)******\\\"e&3DFS7$$!37+++;W S\">$FS$\"3#)******p*e#\\CFS7$$!3'******H&))\\TKFS$\"3!******4#)p()Q#F S7$$!39+++M&[&*G$FS$\"3&******HL.qK#FS7$$!3?+++8XfNLFS$\"3/+++))o)QE#F S7$$!3#)*****>\\'ozLFS$\"3)******\\Qh$*>#FS7$$!3()*****f>#)=U$FS$\"34+ ++80QL@FS7$$!3y*****p2ZAY$FS$\"3)******Rf5f1#FS7$$!38+++5T&3]$FS$\"33+ ++D5$p*>FS7$$!3++++WMyPNFS$\"3)******pINk#>FS7$$!3-+++R;7tNFS$\"3'**** ***)eKW&=FS7$$!3')*****>Egpg$FS$\"3%******4f\\4y\"FS7$$!3$******He$RRO FS$\"3&******plKgq\"FS7$$!3%******\\A:0n$FS$\"3!*******o&\\(H;FS7$$!3* ******pr8/q$FS$\"3)******\\l!>_:FS7$$!3!******4;n\"HPFS$\"31+++]'oMZ\" FS7$$!3y*****pUPov$FS$\"3'******R#or$R\"FS7$$!35+++IYY$y$FS$\"3******* *o]388FS7$$!3))*****Hri!4QFS$\"3'******p,M)F07$$!39+++pzUQRFS$\"3;+++*)R'*ptF07$$!3))*****pn.^&RF S$\"3[******fi,XlF07$$!37+++'[$4qRFS$\"38+++lfm@dF07$$!3!********='H$) RFS$\"3/+++V>2+\\F07$$!3;+++v\")>?E$F07$$!3m*****RAZ8,%FS$\"35+++;z?XCF07$$!3u******* )[j;SFS$\"38+++.:]H;F07$$!3@+++`8#)>SFS$\"3[+++%onb9)Ffp7$$!3R+++7e)3- %FSF(7$Fijl$!3[+++%onb9)Ffp7$Fdjl$!38+++.:]H;F07$F_jl$!35+++;z?XCF07$F jil$!3#******>\")>?E$F07$Feil$!3C+++WFB!3%F07$F`il$!3/+++V>2+\\F07$F[i l$!38+++lfm@dF07$Ffhl$!3[******fi,XlF07$Fahl$!3;+++*)R'*ptF07$F\\hl$!3 u******oo:'>)F07$Fggl$!3s*****HN?I-*F07$Fbgl$!3-+++/)Q(\\)*F07$F]gl$!3 ++++l__n5FS7$Fhfl$!3)*******\\w#)\\6FS7$Fcfl$!3'******p,M_:FS7$Fjdl$!3!*******o&\\(H;FS7$Fedl$!3&******plKgq\"F S7$F`dl$!3%******4f\\4y\"FS7$F[dl$!3'*******)eKW&=FS7$Ffcl$!3)******pI Nk#>FS7$Facl$!33+++D5$p*>FS7$F\\cl$!3)******Rf5f1#FS7$Fgbl$!34+++80QL@ FS7$Fbbl$!3)******\\Qh$*>#FS7$F]bl$!3/+++))o)QE#FS7$Fhal$!3&******HL.q K#FS7$Fcal$!3!******4#)p()Q#FS7$F^al$!3#)******p*e#\\CFS7$Fi`l$!3#)*** ***\\\"e&3DFS7$Fd`l$!3#*******46xmDFS7$F_`l$!3++++x(>Si#FS7$Fj_l$!3%** ****z)eW!o#FS7$Fe_l$!3*******p#=@OFFS7$F`_l$!3%)*****zr*\\\"z#FS7$F[_l $!3#******RF2l%GFS7$Ff^l$!39+++-%Q9!HFS7$Fa^l$!39+++,$[nIFS7$Fb]l$!3@+++In[BJFS7$F]]l$!3A++ +9_kzJFS7$Fh\\l$!3))*****z.KcB$FS7$Fc\\l$!3')*****pC!*4H$FS7$F^\\l$!3# )*****He'=XLFS7$Fi[l$!3#)*****fkzwR$FS7$Fd[l$!37+++,'zzW$FS7$F_[l$!3)) *****>X%o&\\$FS7$Fjz$!31+++q1\\SNFS7$Fez$!3#)******y]=#e$FS7$F`z$!3*)* *****zsi?OFS7$F[z$!3++++k)Hdl$FS7$Ffy$!33+++y/W(o$FS7$Fay$!3z*****pCFd r$FS7$F\\y$!3\")*****fin0u$FS7$Fgx$!3A+++x+%>w$FS7$Fbx$!3=+++Hy\")zPFS 7$F]x$!3?+++/S;%z$FS7$Fhw$!3$******p]F\\!QFS7$Fcw$!37+++j\"R?\"QFS7$F^ w$!33+++swS:QFS7$Fiv$!3-+++i\\\"\\\"QFS7$Fdv$!3,+++$\\5/\"QFS7$F_v$!3* ******4,/%R!>PFS7$Fft$!3/+++w`@%o$FS7$Fat$!3\")***** zd2Ck$FS7$F\\t$!3y*****fCPEf$FS7$Fgs$!3&)*****\\7)oLNFS7$Fbs$!3?+++Lu6 kMFS7$F]s$!31+++iuQ#Q$FS7$Fhr$!3/+++YHB(G$FS7$Fcr$!3#******>j$HyJFS7$F ^r$!3%)*****R@#ocIFS7$Fiq$!3?+++_*\\[#HFS7$Fdq$!33+++njl&y#FS7$F_q$!35 +++>iWTEFS7$Fjp$!3?+++1Uv$\\#FS7$Fdp$!3:+++7#*[VBFS7$F_p$!35+++SIA\"># FS7$Fjo$!32+++>3RP?FS7$Fdo$!3/+++irP#)=FS7$F_o$!36+++ws_E$!3K+++$\\[JG'F07$F8$!3v*****HqzBr%F07$F2$!30+++g8fTJF07$$!30++ +H$[HT)F-$!37+++&G'zq:F0F'-%&COLORG6&%$RGBG$\"#N!\"#$\"#=Fc^nFa^n-%)PO LYGONSG6fw7%F'7$$!+B$[HT)Ffp$\"+&G'zq:!#57$$Fc^nF)F(7%Fj^n7$$!+$=XP>&F S$\"+g8fTJF__nF`_n7%Fc_n7$$!+zXwwb!#;$\"+.(zBr%F__nF`_n7%Fi_n7$$!+VZV# *G!#:$\"+$\\[JG'F__nF`_n7%F``n7$$!+*eG:+\"!#9$\"+#ytQ&yF__nF`_n7%Fg`n7 $$!+P4))*o#Fj`n$\"+!)y]C%*F__nF`_n7%F^an7$$!+y5#f7'Fj`n$\"+nW\\*4\"!\" *F`_n7%Fdan7$$!+?`fi7!#8$\"+qZ\\c7FianF`_n7%F[bn7$$!+ANt$[#F^bn$\"+5JS 89FianF`_n7%Fbbn7$$!+#*o)*\\[F^bn$\"+v)H,d\"FianF`_n7%Fhbn7$$!+Idjo&*F ^bn$\"+ws_E3RP?FianF`_n7%F[dn7$$!+^;C1rFgcn$\"+SIA\">#FianF`_n7%Fad n7$$!+L**)HI\"!#6$\"+7#*[VBFianF`_n7%Fgdn7$$!+#p7RG#Fjdn$\"+1Uv$\\#Fia nF`_n7%F^en7$$!+oV&>#QFjdn$\"+>iWTEFianF`_n7%Fden7$$!+\\\"y'4hFjdn$\"+ njl&y#FianF`_n7%Fjen7$$!+xoAN$*Fjdn$\"+_*\\[#HFianF`_n7%F`fn7$$!+$\\^D O\"F__n$\"+9AocIFianF`_n7%Fffn7$$!+t*Gt*=F__n$\"+KOHyJFianF`_n7%F\\gn7 $$!+ryb?DF__n$\"+YHB(G$FianF`_n7%Fbgn7$$!+Q%eV?$F__n$\"+iuQ#Q$FianF`_n 7%Fhgn7$$!+u4*4#RF__n$\"+Lu6kMFianF`_n7%F^hn7$$!+<[t\\YF__n$\"+D\")oLN FianF`_n7%Fdhn7$$!+2f\"yP&F__n$\"+Ysj#f$FianF`_n7%Fjhn7$$!+3UT)4'F__n$ \"+yvSUOFianF`_n7%F`in7$$!+!R'R3oF__n$\"+w`@%o$FianF`_n7%Ffin7$$!+M\"z m](F__n$\"+>%R!>PFianF`_n7%F\\jn7$$!+fIB$>)F__n$\"+(G`wu$FianF`_n7%Fbj n7$$!+h\\_o))F__n$\"+JcmqPFianF`_n7%Fhjn7$$!+z8AL&*F__n$\"+(3c&)y$Fian F`_n7%F^[o7$$!+(*Q!)=5Fian$\"+6Sq,QFianF`_n7%Fd[o7$$!+EvO$3\"Fian$\"+$ \\5/\"QFianF`_n7%Fj[o7$$!+%Gyq9\"Fian$\"+i\\\"\\\"QFianF`_n7%F`\\o7$$! +:n**47Fian$\"+swS:QFianF`_n7%Ff\\o7$$!+@(z@F\"Fian$\"+j\"R?\"QFianF`_ n7%F\\]o7$$!+*R#oL8Fian$\"+2v#\\!QFianF`_n7%Fb]o7$$!+g.c%R\"Fian$\"+/S ;%z$FianF`_n7%Fh]o7$$!+iC([X\"Fian$\"+Hy\")zPFianF`_n7%F^^o7$$!+2Ro9:F ian$\"+x+%>w$FianF`_n7%Fd^o7$$!+^(pSd\"Fian$\"+EwcSPFianF`_n7%Fj^o7$$! +0)=Jj\"Fian$\"+Zss:PFianF`_n7%F`_o7$$!+=z$>p\"Fian$\"+y/W(o$FianF`_n7 %Ff_o7$$!+Bkl]Fian$\"+ q1\\SNFianF`_n7%F^ao7$$!+Yg3))>Fian$\"+_Wo&\\$FianF`_n7%Fdao7$$!+4^?\\ ?Fian$\"+,'zzW$FianF`_n7%Fjao7$$!+&p8:6#Fian$\"+Y'zwR$FianF`_n7%F`bo7$ $!+]4?v@Fian$\"+$e'=XLFianF`_n7%Ffbo7$$!+n'f.C#Fian$\"+Z-*4H$FianF`_n7 %F\\co7$$!+$[VpI#Fian$\"+Q?jNKFianF`_n7%Fbco7$$!+[TuuBFian$\"+9_kzJFia nF`_n7%Fhco7$$!+:DSVCFian$\"+In[BJFianF`_n7%F^do7$$!+yMX7DFian$\"+7I[n IFianF`_n7%Fddo7$$!+q&*Q\"e#Fian$\"+Vm!=,$FianF`_n7%Fjdo7$$!+tSi#FianF`_n7%Fdgo7$$!+z%o\\3$Fian$\"+5 6xmDFianF`_n7%Fjgo7$$!+MSBRJFian$\"+]\"e&3DFianF`_n7%F`ho7$$!+;WS\">$F ian$\"+q*e#\\CFianF`_n7%Ffho7$$!+`))\\TKFian$\"+@)p()Q#FianF`_n7%F\\io 7$$!+M&[&*G$Fian$\"+LL+FBFianF`_n7%Fbio7$$!+8XfNLFian$\"+))o)QE#FianF` _n7%Fhio7$$!+#\\'ozLFian$\"+&Qh$*>#FianF`_n7%F^jo7$$!+'>#)=U$Fian$\"+8 0QL@FianF`_n7%Fdjo7$$!+xqCiMFian$\"+%f5f1#FianF`_n7%Fjjo7$$!+5T&3]$Fia n$\"+D5$p*>FianF`_n7%F`[p7$$!+WMyPNFian$\"+2`VE>FianF`_n7%Ff[p7$$!+R;7 tNFian$\"+*eKW&=FianF`_n7%F\\\\p7$$!+i-'pg$Fian$\"+\"f\\4y\"FianF`_n7% Fb\\p7$$!+$e$RROFian$\"+dE.1_:FianF`_n7%Fd]p7$$!+hr;HPFian$\"+]' oMZ\"FianF`_n7%Fj]p7$$!+Fu$ov$Fian$\"+Cor$R\"FianF`_n7%F`^p7$$!+IYY$y$ Fian$\"+p]388FianF`_n7%Ff^p7$$!+8F14QFian$\"+)F__nF`_n7%Fj`p7$$!+pzUQRFian$\"+*)R' *ptF__nF`_n7%F`ap7$$!+xO5bRFian$\"+gi,XlF__nF`_n7%Ffap7$$!+'[$4qRFian$ \"+lfm@dF__nF`_n7%F\\bp7$$!+!>'H$)RFian$\"+V>2+\\F__nF`_n7%Fbbp7$$!+v< i%*RFian$\"+WFB!3%F__nF`_n7%Fhbp7$$!+%=$*R+%Fian$\"+7)>?E$F__nF`_n7%F^ cp7$$!+CsM6SFian$\"+;z?XCF__nF`_n7%Fdcp7$$!+!*[j;SFian$\"+.:]H;F__nF`_ n7%Fjcp7$$!+`8#)>SFian$\"+%onb9)FjdnF`_n7%F`dp7$$!+7e)3-%FianF(F`_n7%F fdp7$Fadp$!+%onb9)FjdnF`_n7%Fjdp7$F[dp$!+.:]H;F__nF`_n7%F^ep7$Fecp$!+; z?XCF__nF`_n7%Fbep7$F_cp$!+7)>?E$F__nF`_n7%Ffep7$Fibp$!+WFB!3%F__nF`_n 7%Fjep7$Fcbp$!+V>2+\\F__nF`_n7%F^fp7$F]bp$!+lfm@dF__nF`_n7%Fbfp7$Fgap$ !+gi,XlF__nF`_n7%Fffp7$Faap$!+*)R'*ptF__nF`_n7%Fjfp7$F[ap$!+po:'>)F__n F`_n7%F^gp7$Fe`p$!+`.-B!*F__nF`_n7%Fbgp7$F_`p$!+/)Q(\\)*F__nF`_n7%Ffgp 7$Fi_p$!+l__n5FianF`_n7%Fjgp7$Fc_p$!+]w#)\\6FianF`_n7%F^hp7$F]_p$!+_:FianF`_n7%Fbip7$F_]p$! +p&\\(H;FianF`_n7%Ffip7$Fi\\p$!+dE.1FianF`_n7%Ff jp7$Fa[p$!+D5$p*>FianF`_n7%Fjjp7$F[[p$!+%f5f1#FianF`_n7%F^[q7$Fejo$!+8 0QL@FianF`_n7%Fb[q7$F_jo$!+&Qh$*>#FianF`_n7%Ff[q7$Fiio$!+))o)QE#FianF` _n7%Fj[q7$Fcio$!+LL+FBFianF`_n7%F^\\q7$F]io$!+@)p()Q#FianF`_n7%Fb\\q7$ Fgho$!+q*e#\\CFianF`_n7%Ff\\q7$Faho$!+]\"e&3DFianF`_n7%Fj\\q7$F[ho$!+5 6xmDFianF`_n7%F^]q7$Fego$!+x(>Si#FianF`_n7%Fb]q7$F_go$!+))eW!o#FianF`_ n7%Ff]q7$Fifo$!+F=@OFFianF`_n7%Fj]q7$Fcfo$!+=(*\\\"z#FianF`_n7%F^^q7$F ]fo$!+us]YGFianF`_n7%Fb^q7$Fgeo$!+-%Q9!HFianF`_n7%Ff^q7$Faeo$!+w$FianF`_n7%Fbcq7$F_^o$!+Hy\")zPFianF`_n7%Ffcq7 $Fi]o$!+/S;%z$FianF`_n7%Fjcq7$Fc]o$!+2v#\\!QFianF`_n7%F^dq7$F]]o$!+j\" R?\"QFianF`_n7%Fbdq7$Fg\\o$!+swS:QFianF`_n7%Ffdq7$Fa\\o$!+i\\\"\\\"QFi anF`_n7%Fjdq7$F[\\o$!+$\\5/\"QFianF`_n7%F^eq7$Fe[o$!+6Sq,QFianF`_n7%Fb eq7$F_[o$!+(3c&)y$FianF`_n7%Ffeq7$Fijn$!+JcmqPFianF`_n7%Fjeq7$Fcjn$!+( G`wu$FianF`_n7%F^fq7$F]jn$!+>%R!>PFianF`_n7%Fbfq7$Fgin$!+w`@%o$FianF`_ n7%Fffq7$Fain$!+yvSUOFianF`_n7%Fjfq7$F[in$!+Ysj#f$FianF`_n7%F^gq7$Fehn $!+D\")oLNFianF`_n7%Fbgq7$F_hn$!+Lu6kMFianF`_n7%Ffgq7$Fign$!+iuQ#Q$Fia nF`_n7%Fjgq7$Fcgn$!+YHB(G$FianF`_n7%F^hq7$F]gn$!+KOHyJFianF`_n7%Fbhq7$ Fgfn$!+9AocIFianF`_n7%Ffhq7$Fafn$!+_*\\[#HFianF`_n7%Fjhq7$F[fn$!+njl&y #FianF`_n7%F^iq7$Feen$!+>iWTEFianF`_n7%Fbiq7$F_en$!+1Uv$\\#FianF`_n7%F fiq7$Fhdn$!+7#*[VBFianF`_n7%Fjiq7$Fbdn$!+SIA\">#FianF`_n7%F^jq7$F\\dn$ !+>3RP?FianF`_n7%Fbjq7$Fecn$!+irP#)=FianF`_n7%Ffjq7$F_cn$!+ws_ERVFS7$$\"3-+++()y\"*\\9FS$\"3W+++]>a[VFS7$$\"3%*******pU 3T9FS$\"37+++'f$>dVFS7$$\"3\"******>(*p:V\"FS$\"39+++X&4^O%FS7$$\"3#** *****G#=9U\"FS$\"3s*****z9[AP%FS7$$\"3(******4.t1T\"FS$\"3e*****4sn&yV FS7$$\"31+++u%z$*R\"FS$\"3s******fi-%Q%FS7$$\"3'******f:%e(Q\"FS$\"3U+ ++_6e)Q%FS7$$\"3-+++UcLv8FS$\"3e*****p*))=#R%FS7$$\"3)********)\\oi8FS $\"3g*****f$[![R%FS7$$\"3!******fQ'o\\8FS$\"39+++#)GQ'R%FS7$$\"36+++*) yRO8FS$\"3;+++k_(oR%FS7$$\"3-+++PB)GK\"FS$\"3G+++*GKiR%FS7$$\"34+++M%3 #48FS$\"3A+++Q@S%R%FS7$$\"3'******H4_aH\"FS$\"3/+++J2L\"R%FS7$$\"31+++ t!)p\"G\"FS$\"3K+++(ohpQ%FS7$$\"3)*******3?/o7FS$\"3m*****RYO7Q%FS7$$ \"33+++DHfa7FS$\"3;+++\"y%4uVFS7$$\"3,+++QjZT7FS$\"3A+++vaZlVFS7$$\"3+ +++#3Q)G7FS$\"37+++r!=`N%FS7$$\"3/+++a*[o@\"FS$\"36+++KacVVFS7$$\"35++ +U-r07FS$\"3x******)4o,L%FS7$$\"3)******f+gc>\"FS$\"3(******R<\"4:VFS7 $$\"3++++O'zp=\"FS$\"3O+++q\\K)H%FS7$$\"3/+++i%***z6FS$\"3))*****R.,*z UFS7$$\"3++++X35v6FS$\"37+++b_\"*fUFS7$$\"35+++H+rs6FS$\"3++++*>f&QUFS 7$$\"33+++\"HwK<\"FS$\"3\"******>?eh@%FS7$$\"3!******41Ds<\"FS$\"33+++ .)3K>%FS7$$\"3-+++S)z[=\"FS$\"3?+++l#)RqTFS7$$\"3!******fWgj>\"FS$\"33 +++(yz&[TFS7$$\"3.+++D+\\67FS$\"3:+++Y:oGTFS7$$\"3++++$>i(H7FS$\"3i*** ****Ge:6%FS7$$\"3)******fm./D\"FS$\"3K+++)pQy4%FS7$$\"3/+++qQ^s7FS$\"3 I+++**p$y3%FS7$$\"3++++Gv@&H\"FS$\"3C+++g%f:3%FS7$$\"3)*******e8x<8FS$ \"3!)******H1yySFS7$$\"3.+++xogR8FS$\"3o*****zFS\"zSFS7$$\"3!******R*[ Kg8FS$\"3G+++!*\\A#3%FS7$$\"3%*******>kmz8FS$\"33+++;?i(3%FS7$$\"3++++ B#yuR\"FS$\"3O+++n#[\\4%FS7$$\"32+++T9FS$\"3i*****RC/`7%FS7$$\"31++ +Ua`_9FS$\"3v******G_NPTFS7$$\"3++++=/Ki9FS$\"3S+++H-.]TFS7$$\"3++++hw fq9FS$\"3w*****f=mJ;%FS7$$\"3++++X4Ux9FS$\"3E+++j=iwTFS7$$\"33+++%FS7$$\"3)******R5Gp[\"FS$\"3++++#*[,/UFS7$$\"3'***** *z*Hs*[\"FS$\"3l*****RX[x@%FS7$$\"3)******p^%G\"\\\"FS$\"3N+++l%*QJUFS 7$$\"3++++O_m\"\\\"FS$\"3$******f8h[C%FS7$$\"3-+++th\"4\\\"FS$\"3Q+++# z$4eUFS7$$\"35+++-o3*[\"FS$\"3k*****pnB5F%FS7$$\"3!******f7Di[\"FS$\"3 g*****f%>f$G%FS7$$\"33+++3yP#[\"FS$\"3/+++=Qu&H%FS7$$\"35+++H.fx9FS$\" 3y******fyU2VFS7$$\"33+++xr!>Z\"FS$\"3))*****>P&f=VFS7$$\"3-+++R?Pl9FS $\"3N+++_)*>HVFSFf^rF]^n-Fg^n6jn7%7$$\"+D!G!e9Fian$\"+Il>RVFian7$$\"+( )y\"*\\9Fian$\"+]>a[VFian7$$\"$L\"Fc^n$\"$B%Fc^n7%Fjas7$$\"+qU3T9Fian$ \"+'f$>dVFianF_bs7%Febs7$$\"+s*p:V\"Fian$\"+X&4^O%FianF_bs7%F[cs7$$\"+ H#=9U\"Fian$\"+[\"[AP%FianF_bs7%Facs7$$\"+JIn59Fian$\"+@xcyVFianF_bs7% Fgcs7$$\"+u%z$*R\"Fian$\"+gi-%Q%FianF_bs7%F]ds7$$\"+cTe(Q\"Fian$\"+_6e )Q%FianF_bs7%Fcds7$$\"+UcLv8Fian$\"+(*))=#R%FianF_bs7%Fids7$$\"+!*\\oi 8Fian$\"+O[![R%FianF_bs7%F_es7$$\"+'Q'o\\8Fian$\"+#)GQ'R%FianF_bs7%Fee s7$$\"+*)yRO8Fian$\"+k_(oR%FianF_bs7%F[fs7$$\"+PB)GK\"Fian$\"+*GKiR%Fi anF_bs7%Fafs7$$\"+M%3#48Fian$\"+Q@S%R%FianF_bs7%Fgfs7$$\"+$4_aH\"Fian$ \"+J2L\"R%FianF_bs7%F]gs7$$\"+t!)p\"G\"Fian$\"+(ohpQ%FianF_bs7%Fcgs7$$ \"+4?/o7Fian$\"+kkB\"Q%FianF_bs7%Figs7$$\"+DHfa7Fian$\"+\"y%4uVFianF_b s7%F_hs7$$\"+QjZT7Fian$\"+vaZlVFianF_bs7%Fehs7$$\"+#3Q)G7Fian$\"+r!=`N %FianF_bs7%F[is7$$\"+a*[o@\"Fian$\"+KacVVFianF_bs7%Fais7$$\"+U-r07Fian $\"+*4o,L%FianF_bs7%Fgis7$$\"+1+m&>\"Fian$\"+u64:VFianF_bs7%F]js7$$\"+ O'zp=\"Fian$\"+q\\K)H%FianF_bs7%Fcjs7$$\"+i%***z6Fian$\"+M5!*zUFianF_b s7%Fijs7$$\"+X35v6Fian$\"+b_\"*fUFianF_bs7%F_[t7$$\"+H+rs6Fian$\"+*>f& QUFianF_bs7%Fe[t7$$\"+\"HwK<\"Fian$\"+-#eh@%FianF_bs7%F[\\t7$$\"+h]Ax6 Fian$\"+.)3K>%FianF_bs7%Fa\\t7$$\"+S)z[=\"Fian$\"+l#)RqTFianF_bs7%Fg\\ t7$$\"+Y/O'>\"Fian$\"+(yz&[TFianF_bs7%F]]t7$$\"+D+\\67Fian$\"+Y:oGTFia nF_bs7%Fc]t7$$\"+$>i(H7Fian$\"+!He:6%FianF_bs7%Fi]t7$$\"+mOS]7Fian$\"+ )pQy4%FianF_bs7%F_^t7$$\"+qQ^s7Fian$\"+**p$y3%FianF_bs7%Fe^t7$$\"+Gv@& H\"Fian$\"+g%f:3%FianF_bs7%F[_t7$$\"+f8x<8Fian$\"+I1yySFianF_bs7%Fa_t7 $$\"+xogR8Fian$\"+y-9zSFianF_bs7%Fg_t7$$\"+%*[Kg8Fian$\"+!*\\A#3%FianF _bs7%F]`t7$$\"+?kmz8Fian$\"+;?i(3%FianF_bs7%Fc`t7$$\"+B#yuR\"Fian$\"+n #[\\4%FianF_bs7%Fi`t7$$\"+T9Fian$\"+WUIDTFianF_bs7%F[bt7$$ \"+Ua`_9Fian$\"+H_NPTFianF_bs7%Fabt7$$\"+=/Ki9Fian$\"+H-.]TFianF_bs7%F gbt7$$\"+hwfq9Fian$\"+'=mJ;%FianF_bs7%F]ct7$$\"+X4Ux9Fian$\"+j=iwTFian F_bs7%Fcct7$$\"+%FianF_bs7%Fict7$$\"+/\"Gp[\"Fian$ \"+#*[,/UFianF_bs7%F_dt7$$\"+)*Hs*[\"Fian$\"+a%[x@%FianF_bs7%Fedt7$$\" +f$G%FianF_bs7%Fcft7$$\" +3yP#[\"Fian$\"+=Qu&H%FianF_bs7%Fift7$$\"+H.fx9Fian$\"+gyU2VFianF_bs7% F_gt7$$\"+xr!>Z\"Fian$\"+s`f=VFianF_bs7%Fegt7$$\"+R?Pl9Fian$\"+_)*>HVF ianF_bs7%F[htFeasF_bsFj]rF_^r-F$6$7in7$Fg^r$!3s******Hl>RVFS7$F^as$!3N +++_)*>HVFS7$Fi`s$!3))*****>P&f=VFS7$Fd`s$!3y******fyU2VFS7$F_`s$!3/++ +=Qu&H%FS7$Fj_s$!3g*****f%>f$G%FS7$Fe_s$!3k*****pnB5F%FS7$F`_s$!3Q+++# z$4eUFS7$F[_s$!3$******f8h[C%FS7$Ff^s$!3N+++l%*QJUFS7$Fa^s$!3l*****RX[ x@%FS7$F\\^s$!3++++#*[,/UFS7$Fg]s$!3G+++PTF!>%FS7$Fb]s$!3E+++j=iwTFS7$ F]]s$!3w*****f=mJ;%FS7$Fh\\s$!3S+++H-.]TFS7$Fc\\s$!3v******G_NPTFS7$F^ \\s$!3i*****RC/`7%FS7$Fi[s$!3x******)\\mS6%FS7$Fd[s$!32+++3B'Q5%FS7$F_ [s$!3O+++n#[\\4%FS7$Fjjr$!33+++;?i(3%FS7$Fejr$!3G+++!*\\A#3%FS7$F`jr$! 3o*****zFS\"zSFS7$F[jr$!3!)******H1yySFS7$Ffir$!3C+++g%f:3%FS7$Fair$!3 I+++**p$y3%FS7$F\\ir$!3K+++)pQy4%FS7$Fghr$!3i*******Ge:6%FS7$Fbhr$!3:+ ++Y:oGTFS7$F]hr$!33+++(yz&[TFS7$Fhgr$!3?+++l#)RqTFS7$Fcgr$!33+++.)3K>% FS7$F^gr$!3\"******>?eh@%FS7$Fifr$!3++++*>f&QUFS7$Fdfr$!37+++b_\"*fUFS 7$F_fr$!3))*****R.,*zUFS7$Fjer$!3O+++q\\K)H%FS7$Feer$!3(******R<\"4:VF S7$F`er$!3x******)4o,L%FS7$F[er$!36+++KacVVFS7$Ffdr$!37+++r!=`N%FS7$Fa dr$!3A+++vaZlVFS7$F\\dr$!3;+++\"y%4uVFS7$Fgcr$!3m*****RYO7Q%FS7$Fbcr$! 3K+++(ohpQ%FS7$F]cr$!3/+++J2L\"R%FS7$Fhbr$!3A+++Q@S%R%FS7$Fcbr$!3G+++* GKiR%FS7$F^br$!3;+++k_(oR%FS7$Fiar$!39+++#)GQ'R%FS7$Fdar$!3g*****f$[![ R%FS7$F_ar$!3e*****p*))=#R%FS7$Fj`r$!3U+++_6e)Q%FS7$Fe`r$!3s******fi-% Q%FS7$F``r$!3e*****4sn&yVFS7$F[`r$!3s*****z9[AP%FS7$Ff_r$!39+++X&4^O%F S7$Fa_r$!37+++'f$>dVFS7$F\\_r$!3W+++]>a[VFSFdhtF]^n-Fg^n6jn7%7$Ffas$!+ Il>RVFian7$F\\ht$!+_)*>HVFian7$F`bs$!$B%Fc^n7%F^du7$Ffgt$!+s`f=VFianFa du7%Fedu7$F`gt$!+gyU2VFianFadu7%Fidu7$Fjft$!+=Qu&H%FianFadu7%F]eu7$Fdf t$!+Y>f$G%FianFadu7%Faeu7$F^ft$!+xO-rUFianFadu7%Feeu7$Fhet$!+#z$4eUFia nFadu7%Fieu7$Fbet$!+O6'[C%FianFadu7%F]fu7$F\\et$!+l%*QJUFianFadu7%Fafu 7$Ffdt$!+a%[x@%FianFadu7%Fefu7$F`dt$!+#*[,/UFianFadu7%Fifu7$Fjct$!+PTF !>%FianFadu7%F]gu7$Fdct$!+j=iwTFianFadu7%Fagu7$F^ct$!+'=mJ;%FianFadu7% Fegu7$Fhbt$!+H-.]TFianFadu7%Figu7$Fbbt$!+H_NPTFianFadu7%F]hu7$F\\bt$!+ WUIDTFianFadu7%Fahu7$Ffat$!+*\\mS6%FianFadu7%Fehu7$F`at$!+3B'Q5%FianFa du7%Fihu7$Fj`t$!+n#[\\4%FianFadu7%F]iu7$Fd`t$!+;?i(3%FianFadu7%Faiu7$F ^`t$!+!*\\A#3%FianFadu7%Feiu7$Fh_t$!+y-9zSFianFadu7%Fiiu7$Fb_t$!+I1yyS FianFadu7%F]ju7$F\\_t$!+g%f:3%FianFadu7%Faju7$Ff^t$!+**p$y3%FianFadu7% Feju7$F`^t$!+)pQy4%FianFadu7%Fiju7$Fj]t$!+!He:6%FianFadu7%F][v7$Fd]t$! +Y:oGTFianFadu7%Fa[v7$F^]t$!+(yz&[TFianFadu7%Fe[v7$Fh\\t$!+l#)RqTFianF adu7%Fi[v7$Fb\\t$!+.)3K>%FianFadu7%F]\\v7$F\\\\t$!+-#eh@%FianFadu7%Fa \\v7$Ff[t$!+*>f&QUFianFadu7%Fe\\v7$F`[t$!+b_\"*fUFianFadu7%Fi\\v7$Fjjs $!+M5!*zUFianFadu7%F]]v7$Fdjs$!+q\\K)H%FianFadu7%Fa]v7$F^js$!+u64:VFia nFadu7%Fe]v7$Fhis$!+*4o,L%FianFadu7%Fi]v7$Fbis$!+KacVVFianFadu7%F]^v7$ F\\is$!+r!=`N%FianFadu7%Fa^v7$Ffhs$!+vaZlVFianFadu7%Fe^v7$F`hs$!+\"y%4 uVFianFadu7%Fi^v7$Fjgs$!+kkB\"Q%FianFadu7%F]_v7$Fdgs$!+(ohpQ%FianFadu7 %Fa_v7$F^gs$!+J2L\"R%FianFadu7%Fe_v7$Fhfs$!+Q@S%R%FianFadu7%Fi_v7$Fbfs $!+*GKiR%FianFadu7%F]`v7$F\\fs$!+k_(oR%FianFadu7%Fa`v7$Ffes$!+#)GQ'R%F ianFadu7%Fe`v7$F`es$!+O[![R%FianFadu7%Fi`v7$Fjds$!+(*))=#R%FianFadu7%F ]av7$Fdds$!+_6e)Q%FianFadu7%Faav7$F^ds$!+gi-%Q%FianFadu7%Feav7$Fhcs$!+ @xcyVFianFadu7%Fiav7$Fbcs$!+[\"[AP%FianFadu7%F]bv7$F\\cs$!+X&4^O%FianF adu7%Fabv7$Ffbs$!+'f$>dVFianFadu7%Febv7$F[bs$!+]>a[VFianFadu7%FibvF[du FaduFj]rF_^r-F$6%7$7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF(-%'COLOURG6& F`^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(Facv7$F($\"3/++++++!z%FSFfcvF icv-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DE FAULTG-%(SCALINGG6#%,CONSTRAINEDG-F^ev6$%*HELVETICAG\"\"*-%%VIEWG6$;$! $z%Fc^n$\"$*>Fc^n;F]fv$\"$z%Fc^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interv al" }{TEXT -1 37 " for this scheme is (approximately) " }{XPPEDIT 18 0 "[-4.0209, 0];" "6#7$,$-%&FloatG6$\"&4-%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "T he following picture compares the stability region for the 8 stage, or der 4 scheme with that of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),p_||(1..6)],view=[-4. 79..1.99,-4.79..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`, `Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "65-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3 b+++!f<*>f!#G$\"3++++Fjzq:!#=7$$\"3p******4$3Q5&!#E$\"3(******pi#fTJF0 7$$\"3=+++(*f-r!)!#D$\"3D+++/%)Q7ZF07$$\"3E+++P,gNg!#C$\"3S+++&yzJG'F0 7$$\"33+++%p%GlG!#B$\"3S+++g$[R&yF07$$\"3s*****>)evd)*FF$\"3w*****\\jI YU*F07$$\"33+++y<$3i#!#A$\"3'******\\D0&*4\"!#<7$$\"3I+++6Y&R[&FQ$\"33 +++h8[c7FT7$$\"3C+++XY2B')FQ$\"3!******4V6LT\"FT7$$\"3]*****>Ow'yuFQ$ \"3++++iU&)p:FT7$$!3-+++'3\"pH5!#@$\"3!******\\/()es\"FT7$$!3^+++odj+r Fao$\"31+++jY4\")=FT7$$!3')*****\\^$>CA!#?$\"3<+++[t2N?FT7$$!32+++\\v \\4aF\\p$\"3=+++upP(=#FT7$$!3++++O#)>O6!#>$\"36+++QP]PBFT7$$!3'******p t83:#Fgp$\"35+++yV#\\[#FT7$$!3!)*****4:(*Qv$Fgp$\"3/+++4w'*GEFT7$$!3<+ ++3,`ChFgp$\"3!******p0J'oFFT7$$!3Z*****>$4#*=%*Fgp$\"34+++RsN-HFT7$$! 3#******\\$)Q=P\"F0$\"3A+++nS)z-$FT7$$!3)******4&R!y*=F0$\"3\")*****f& 48VJFT7$$!3t******[l:-DF0$\"37+++/I(fC$FT7$$!3;+++h(G-;$F0$\"36+++-)[e L$FT7$$!3'******pW;&[QF0$\"3!******pAQKT$FT7$$!35+++W,T\\XF0$\"3)***** *z^$HzMFT7$$!3S+++P]p^_F0$\"3<+++QZONNFT7$$!3W+++C6.\\fF0$\"3;+++:@s#e $FT7$$!3a+++.s$FT7$$!3*******p$3=S**F0$\"3!***** *H[4@t$FT7$$!3/+++rqOd5FT$\"3#)*****zh<&RPFT7$$!3++++mQ$*>6FT$\"3@+++ \\+xUPFT7$$!3#******fjo<=\"FT$\"38+++2k/UPFT7$$!36+++V4#HC\"FT$\"34+++ l#)[PPFT7$$!3)******fGTMI\"FT$\"3/+++2e?HPFT7$$!31+++hKQj8FT$\"3&)**** *48$GMOFT7$$!3#*******[f*ol\"FT$\"3&*******z'=Zg$FT7$$!3%*******GS 2:FT$\"3#******pq9*>$FT7$$!3/+++\")*\\YF#FT$\"3>+++x.zVJFT7$$!31+ ++\"y4OM#FT$\"3))*****RU-#)3$FT7$$!3z*****\\0RLT#FT$\"3.+++:(>G.$FT7$$ !33+++8@H$[#FT$\"3'******f76z(HFT7$$!3=+++eY\"Hb#FT$\"34+++vGFT$\"3@+++$)yl4FF T7$$!31+++Tmj\")GFT$\"3))******ey,cEFT7$$!3++++g(>=%HFT$\"3*******za$) =g#FT7$$!3y******)R-***HFT$\"3++++=-2ZDFT7$$!31+++;N'e0$FT$\"3))*****p :<9\\#FT7$$!3\")*****p>0(4JFT$\"32+++MoyMCFT7$$!3#*******RjWhJFT$\"3=+ ++?H1xBFT7$$!3,+++]$=6@$FT$\"34+++lz9=BFT7$$!3++++rBweKFT$\"3)******\\ ?hzD#FT7$$!3#)*****\\hFWI$FT$\"3'******ptOk>#FT7$$!3?+++j/<[LFT$\"39++ +?A_L@FT7$$!3=+++\\S0!R$FT$\"3;+++U\"y\"p?FT7$$!38+++x![,V$FT$\"3z**** *4qxL+#FT7$$!3#******pjG&oMFT$\"3)******RG2h$>FT7$$!36+++'zx_]$FT$\"33 +++6vOn=FT7$$!3*)******=F[SNFT$\"3-+++NZ<(z\"FT7$$!3/+++!4MUd$FT$\"32+ ++VGcD\"FT7$$!3')* *****zM?,QFT$\"3'******fm$=76FT7$$!32+++\"38Z#QFT$\"34+++K!RC.\"FT7$$! 3A+++*yVq%QFT$\"3z*****p8!)\\_*F07$$!3')*****\\)R5oQFT$\"3b*****\\l!yC ()F07$$!3=+++/+z()QFT$\"3Z*****RXVZ#zF07$$!3*)*****zb#*f!RFT$\"3)***** *HE'eDrF07$$!38+++4EgARFT$\"3s*****Hl+yK'F07$$!3-+++/i^PRFT$\"34+++mKn JbF07$$!3++++u!Q1&RFT$\"3F+++1\\JPZF07$$!3'******H/%)='RFT$\"3w*****4X #pWRF07$$!3))******>C=rRFT$\"37+++j)fO:$F07$$!37+++UYZyRFT$\"3!******R /')RO#F07$$!3++++Gar$)RFT$\"3'*******e&y`d\"F07$$!3%******\\mso)RFT$\" 3m*****p%>.vyFgp7$$!3/+++*>Fz)RFTF(7$Fjjl$!3m*****p%>.vyFgp7$Fejl$!3'* ******e&y`d\"F07$F`jl$!3!******R/')RO#F07$F[jl$!37+++j)fO:$F07$Ffil$!3 w*****4X#pWRF07$Fail$!3F+++1\\JPZF07$F\\il$!34+++mKnJbF07$Fghl$!3s**** *Hl+yK'F07$Fbhl$!3)******HE'eDrF07$F]hl$!3Z*****RXVZ#zF07$Fhgl$!3b**** *\\l!yC()F07$Fcgl$!3z*****p8!)\\_*F07$F^gl$!34+++K!RC.\"FT7$Fifl$!3'** ****fm$=76FT7$Fdfl$!3'******zS&f\">\"FT7$F_fl$!3))*****z)R_q7FT7$Fjel$ !3'******pB7)[8FT7$Feel$!3/+++QOIE9FT7$F`el$!3********)e\\G]\"FT7$F[el $!3/+++!\\:$y:FT7$Ffdl$!3/+++w]e_;FT7$Fadl$!32+++VGcDFT7$F]cl$!3z*****4q xL+#FT7$Fhbl$!3;+++U\"y\"p?FT7$Fcbl$!39+++?A_L@FT7$F^bl$!3'******ptOk> #FT7$Fial$!3)******\\?hzD#FT7$Fdal$!34+++lz9=BFT7$F_al$!3=+++?H1xBFT7$ Fj`l$!32+++MoyMCFT7$Fe`l$!3))*****p:<9\\#FT7$F``l$!3++++=-2ZDFT7$F[`l$ !3*******za$)=g#FT7$Ff_l$!3))******ey,cEFT7$Fa_l$!3@+++$)yl4FFT7$F\\_l $!35+++P2,jFFT7$Fg^l$!3$******4\")*H;GFT7$Fb^l$!3)*******)[^(pGFT7$F]^ l$!34+++vG.$FT7$F^]l$!3) )*****RU-#)3$FT7$Fi\\l$!3>+++x.zVJFT7$Fd\\l$!3#)******>q9*>$FT7$F_\\l$ !3#)*****Ri:PD$FT7$Fj[l$!38+++G_*oI$FT7$Fe[l$!3A+++\\!=\"eLFT7$F`[l$!3 #******pMOFT7$F]y$!3)******fb8-m$FT7$Fhx$!3')*****f:\\Fo$FT7$Fcx$!39+++rEy ,PFT7$F^x$!3&)*****48$G.s$FT7$Ffu$!3<+++$=jQq$FT7$Fau$!3#)*****fJ %Q#o$FT7$F\\u$!3;+++XYVbOFT7$Fgt$!3()*****RD]Ci$FT7$Fbt$!3;+++:@s#e$FT 7$F]t$!3<+++QZONNFT7$Fhs$!3)******z^$HzMFT7$Fcs$!3!******pAQKT$FT7$F^s $!36+++-)[eL$FT7$Fir$!37+++/I(fC$FT7$Fdr$!3\")*****f&48VJFT7$F_r$!3A++ +nS)z-$FT7$Fjq$!34+++RsN-HFT7$Feq$!3!******p0J'oFFT7$F`q$!3/+++4w'*GEF T7$F[q$!35+++yV#\\[#FT7$Fep$!36+++QP]PBFT7$F`p$!3=+++upP(=#FT7$Fjo$!3< +++[t2N?FT7$Feo$!31+++jY4\")=FT7$F_o$!3!******\\/()es\"FT7$$\"39+++kjn yuFQ$!3++++iU&)p:FT7$Fen$!3!******4V6LT\"FT7$FV$!33+++h8[c7FT7$FO$!3'* *****\\D0&*4\"FT7$FJ$!3w*****\\jIYU*F07$FD$!3S+++g$[R&yF07$$\"3^*****f 8+c.'F@$!3S+++&yzJG'F07$$\"3s******pd-r!)F:$!3D+++/%)Q7ZF07$$\"3G+++9$ 3Q5&F4$!3(******pi#fTJF07$$\"3i*****>a4(>fF-$!3++++Fjzq:F0F'-%&COLORG6 &%$RGBG$\"\"%!\"\"$\"#=!\"#Fh^n-%)POLYGONSG6fw7%F'7$$\"+!f<*>fF\\p$\"+ Fjzq:!#57$$F]_nF)F(7%Fb_n7$$\"+5$3Q5&F0$\"+FEfTJFg_nFh_n7%F[`n7$$\"+(* f-r!)FT$\"+/%)Q7ZFg_nFh_n7%Fa`n7$$\"+P,gNg!#;$\"+&yzJG'Fg_nFh_n7%Fg`n7 $$\"+%p%GlG!#:$\"+g$[R&yFg_nFh_n7%F^an7$$\"+#)evd)*Faan$\"+N1jC%*Fg_nF h_n7%Fean7$$\"+y<$3i#!#9$\"+b_]*4\"!\"*Fh_n7%F[bn7$$\"+6Y&R[&F^bn$\"+h 8[c7FabnFh_n7%Fcbn7$$\"+XY2B')F^bn$\"+J9J89FabnFh_n7%Fibn7$$\"+ijnyuF^ bn$\"+iU&)p:FabnFh_n7%F_cn7$$!+'3\"pH5!#8$\"+Xq)es\"FabnFh_n7%Fecn7$$! +odj+rFhcn$\"+jY4\")=FabnFh_n7%F\\dn7$$!+:N>CA!#7$\"+[t2N?FabnFh_n7%Fb dn7$$!+\\v\\4aFedn$\"+upP(=#FabnFh_n7%Fidn7$$!+O#)>O6!#6$\"+QP]PBFabnF h_n7%F_en7$$!+PP\"3:#Fben$\"+yV#\\[#FabnFh_n7%Ffen7$$!+^r*Qv$Fben$\"+4 w'*GEFabnFh_n7%F\\fn7$$!+3,`ChFben$\"+d5joFFabnFh_n7%Fbfn7$$!+K4#*=%*F ben$\"+RsN-HFabnFh_n7%Fhfn7$$!+N)Q=P\"Fg_n$\"+nS)z-$FabnFh_n7%F^gn7$$! +^R!y*=Fg_n$\"+c48VJFabnFh_n7%Fdgn7$$!+\\l:-DFg_n$\"+/I(fC$FabnFh_n7%F jgn7$$!+h(G-;$Fg_n$\"+-)[eL$FabnFh_n7%F`hn7$$!+Zk^[QFg_n$\"+F#QKT$Fabn Fh_n7%Ffhn7$$!+W,T\\XFg_n$\"+=NHzMFabnFh_n7%F\\in7$$!+P]p^_Fg_n$\"+QZO NNFabnFh_n7%Fbin7$$!+C6.\\fFg_n$\"+:@s#e$FabnFh_n7%Fhin7$$!+.s$FabnFh_n7%Ff[o7$$!+P3=S**Fg_n$\"+$[4@t$FabnFh _n7%F\\\\o7$$!+rqOd5Fabn$\"+=w^RPFabnFh_n7%Fb\\o7$$!+mQ$*>6Fabn$\"+\\+ xUPFabnFh_n7%Fh\\o7$$!+O'o<=\"Fabn$\"+2k/UPFabnFh_n7%F^]o7$$!+V4#HC\"F abn$\"+l#)[PPFabnFh_n7%Fd]o7$$!+'GTMI\"Fabn$\"+2e?HPFabnFh_n7%Fj]o7$$! +hKQj8Fabn$\"+JJGMOFabnFh_n7%Fh_o7$$!+\\f*ol\"Fabn$ \"+!o=Zg$FabnFh_n7%F^`o7$$!+HS2:Fabn$\"+xT!pS$FabnF h_n7%F\\bo7$$!+cmH8?Fabn$\"+\\!=\"eLFabnFh_n7%Fbbo7$$!+q_?w?Fabn$\"+G_ *oI$FabnFh_n7%Fhbo7$$!+p0tS@Fabn$\"+Ccr`KFabnFh_n7%F^co7$$!+30#p?#Fabn $\"+?q9*>$FabnFh_n7%Fdco7$$!+\")*\\YF#Fabn$\"+x.zVJFabnFh_n7%Fjco7$$!+ \"y4OM#Fabn$\"+CC?)3$FabnFh_n7%F`do7$$!+b!RLT#Fabn$\"+:(>G.$FabnFh_n7% Ffdo7$$!+8@H$[#Fabn$\"+E6\"z(HFabnFh_n7%F\\eo7$$!+eY\"Hb#Fabn$\"+v GFabn$\"+$)yl4FFabnFh_n7%Fjfo7$$!+Tmj\")GFabn$\"+fy,cEFabnFh_n7%F`go7$ $!+g(>=%HFabn$\"+[N)=g#FabnFh_n7%Ffgo7$$!+*R-***HFabn$\"+=-2ZDFabnFh_n 7%F\\ho7$$!+;N'e0$Fabn$\"+drT\"\\#FabnFh_n7%Fbho7$$!+(>0(4JFabn$\"+Moy MCFabnFh_n7%Fhho7$$!+SjWhJFabn$\"+?H1xBFabnFh_n7%F^io7$$!+]$=6@$Fabn$ \"+lz9=BFabnFh_n7%Fdio7$$!+rBweKFabn$\"+07'zD#FabnFh_n7%Fjio7$$!+:wU/L Fabn$\"+PnV'>#FabnFh_n7%F`jo7$$!+j/<[LFabn$\"+?A_L@FabnFh_n7%Ffjo7$$!+ \\S0!R$Fabn$\"+U\"y\"p?FabnFh_n7%F\\[p7$$!+x![,V$Fabn$\"+,xP.?FabnFh_n 7%Fb[p7$$!+P'G&oMFabn$\"+%G2h$>FabnFh_n7%Fh[p7$$!+'zx_]$Fabn$\"+6vOn=F abnFh_n7%F^\\p7$$!+>F[SNFabn$\"+NZ<(z\"FabnFh_n7%Fd\\p7$$!+!4MUd$Fabn$ \"+VGcD\"FabnFh_n7%Fd_p7$$!+![.7!QFabn$\"+mO=76FabnFh_n7%Fj_p7$$!+\"38Z# QFabn$\"+K!RC.\"FabnFh_n7%F``p7$$!+*yVq%QFabn$\"+P,)\\_*Fg_nFh_n7%Ff`p 7$$!+&)R5oQFabn$\"+b1yC()Fg_nFh_n7%F\\ap7$$!+/+z()QFabn$\"+aMuCzFg_nFh _n7%Fbap7$$!+eD*f!RFabn$\"+jieDrFg_nFh_n7%Fhap7$$!+4EgARFabn$\"+`1!yK' Fg_nFh_n7%F^bp7$$!+/i^PRFabn$\"+mKnJbFg_nFh_n7%Fdbp7$$!+u!Q1&RFabn$\"+ 1\\JPZFg_nFh_n7%Fjbp7$$!+VS)='RFabn$\"+^CpWRFg_nFh_n7%F`cp7$$!+?C=rRFa bn$\"+j)fO:$Fg_nFh_n7%Ffcp7$$!+UYZyRFabn$\"+Wg)RO#Fg_nFh_n7%F\\dp7$$!+ Gar$)RFabn$\"+f&y`d\"Fg_nFh_n7%Fbdp7$$!+lE(o)RFabn$\"+Z>.vyFbenFh_n7%F hdp7$$!+*>Fz)RFabnF(Fh_n7%F^ep7$Fidp$!+Z>.vyFbenFh_n7%Fbep7$Fcdp$!+f&y `d\"Fg_nFh_n7%Ffep7$F]dp$!+Wg)RO#Fg_nFh_n7%Fjep7$Fgcp$!+j)fO:$Fg_nFh_n 7%F^fp7$Facp$!+^CpWRFg_nFh_n7%Fbfp7$F[cp$!+1\\JPZFg_nFh_n7%Fffp7$Febp$ !+mKnJbFg_nFh_n7%Fjfp7$F_bp$!+`1!yK'Fg_nFh_n7%F^gp7$Fiap$!+jieDrFg_nFh _n7%Fbgp7$Fcap$!+aMuCzFg_nFh_n7%Ffgp7$F]ap$!+b1yC()Fg_nFh_n7%Fjgp7$Fg` p$!+P,)\\_*Fg_nFh_n7%F^hp7$Fa`p$!+K!RC.\"FabnFh_n7%Fbhp7$F[`p$!+mO=76F abnFh_n7%Ffhp7$Fe_p$!+3af\">\"FabnFh_n7%Fjhp7$F__p$!+))R_q7FabnFh_n7%F ^ip7$Fi^p$!+PA\")[8FabnFh_n7%Fbip7$Fc^p$!+QOIE9FabnFh_n7%Ffip7$F]^p$!+ *e\\G]\"FabnFh_n7%Fjip7$Fg]p$!+!\\:$y:FabnFh_n7%F^jp7$Fa]p$!+w]e_;Fabn Fh_n7%Fbjp7$F[]p$!+VGcDFabnFh_n7%Fb[q7$Fc[p$!+,xP. ?FabnFh_n7%Ff[q7$F][p$!+U\"y\"p?FabnFh_n7%Fj[q7$Fgjo$!+?A_L@FabnFh_n7% F^\\q7$Fajo$!+PnV'>#FabnFh_n7%Fb\\q7$F[jo$!+07'zD#FabnFh_n7%Ff\\q7$Fei o$!+lz9=BFabnFh_n7%Fj\\q7$F_io$!+?H1xBFabnFh_n7%F^]q7$Fiho$!+MoyMCFabn Fh_n7%Fb]q7$Fcho$!+drT\"\\#FabnFh_n7%Ff]q7$F]ho$!+=-2ZDFabnFh_n7%Fj]q7 $Fggo$!+[N)=g#FabnFh_n7%F^^q7$Fago$!+fy,cEFabnFh_n7%Fb^q7$F[go$!+$)yl4 FFabnFh_n7%Ff^q7$Fefo$!+P2,jFFabnFh_n7%Fj^q7$F_fo$!+6)*H;GFabnFh_n7%F^ _q7$Fieo$!+*[^(pGFabnFh_n7%Fb_q7$Fceo$!+vG.$FabnFh_n7%F^`q7$Fado$!+CC?)3$FabnFh_ n7%Fb`q7$F[do$!+x.zVJFabnFh_n7%Ff`q7$Feco$!+?q9*>$FabnFh_n7%Fj`q7$F_co $!+Ccr`KFabnFh_n7%F^aq7$Fibo$!+G_*oI$FabnFh_n7%Fbaq7$Fcbo$!+\\!=\"eLFa bnFh_n7%Ffaq7$F]bo$!+xT!pS$FabnFh_n7%Fjaq7$Fgao$!+DX)GX$FabnFh_n7%F^bq 7$Faao$!+%\\(z&\\$FabnFh_n7%Fbbq7$F[ao$!+c2ZNNFabnFh_n7%Ffbq7$Fe`o$!+w #)zrNFabnFh_n7%Fjbq7$F_`o$!+!o=Zg$FabnFh_n7%F^cq7$Fi_o$!+&[(>MOFabnFh_ n7%Fbcq7$Fc_o$!+cN@gOFabnFh_n7%Ffcq7$F]_o$!+c\"\\Fo$FabnFh_n7%Fjcq7$Fg ^o$!+rEy,PFabnFh_n7%F^dq7$Fa^o$!+JJG.s$FabnFh_n7%F^fq7$Fa[o$!+$=jQq$FabnFh_n7% Fbfq7$F[[o$!+;VQ#o$FabnFh_n7%Fffq7$Fejn$!+XYVbOFabnFh_n7%Fjfq7$F_jn$!+ a-XAOFabnFh_n7%F^gq7$Fiin$!+:@s#e$FabnFh_n7%Fbgq7$Fcin$!+QZONNFabnFh_n 7%Ffgq7$F]in$!+=NHzMFabnFh_n7%Fjgq7$Fghn$!+F#QKT$FabnFh_n7%F^hq7$Fahn$ !+-)[eL$FabnFh_n7%Fbhq7$F[hn$!+/I(fC$FabnFh_n7%Ffhq7$Fegn$!+c48VJFabnF h_n7%Fjhq7$F_gn$!+nS)z-$FabnFh_n7%F^iq7$Fifn$!+RsN-HFabnFh_n7%Fbiq7$Fc fn$!+d5joFFabnFh_n7%Ffiq7$F]fn$!+4w'*GEFabnFh_n7%Fjiq7$Fgen$!+yV#\\[#F abnFh_n7%F^jq7$F`en$!+QP]PBFabnFh_n7%Fbjq7$Fjdn$!+upP(=#FabnFh_n7%Ffjq 7$Fcdn$!+[t2N?FabnFh_n7%Fjjq7$F]dn$!+jY4\")=FabnFh_n7%F^[r7$Ffcn$!+Xq) es\"FabnFh_n7%Fb[r7$$\"+kjnyuF^bn$!+iU&)p:FabnFh_n7%Ff[r7$Fjbn$!+J9J89 FabnFh_n7%F\\\\r7$Fdbn$!+h8[c7FabnFh_n7%F`\\r7$F\\bn$!+b_]*4\"FabnFh_n 7%Fd\\r7$Ffan$!+N1jC%*Fg_nFh_n7%Fh\\r7$F_an$!+g$[R&yFg_nFh_n7%F\\]r7$$ \"+O,gNgFj`n$!+&yzJG'Fg_nFh_n7%F`]r7$$\"+qd-r!)FT$!+/%)Q7ZFg_nFh_n7%Ff ]r7$$\"+9$3Q5&F0$!+FEfTJFg_nFh_n7%F\\^r7$$\"+U&4(>fF\\p$!+Fjzq:Fg_nFh_ n7%Fb^rF'Fh_n-Fe^n6&Fg^n$\"\")Fj^n$\"#NF]_nFj^r-%&STYLEG6#%,PATCHNOGRI DG-F$6$7in7$$\"33+++.>]*[\"FT$\"3#******>&)\\QI%FT7$$\"3!******ff%R\"[ \"FT$\"3()*****HP'f7VFT7$$\"3'******Ru+EZ\"FT$\"3C+++&o\\1K%FT7$$\"3/+ ++&RjJY\"FT$\"3D+++(RozK%FT7$$\"3/+++yi7`9FT$\"3e*****\\V6XL%FT7$$\"3+ +++@T`U9FT$\"3F+++kxBSVFT7$$\"3++++9IVJ9FT$\"3i******>h5XVFT7$$\"3#*** ***f#3()>9FT$\"3M+++dZ2\\VFT7$$\"3$******Ro(*yS\"FT$\"3e*****p?,@N%FT7 $$\"3-+++Glc&R\"FT$\"3=+++^?9aVFT7$$\"33+++xP$HQ\"FT$\"3o*****4r_^N%FT 7$$\"3%******R6g+P\"FT$\"37+++os3bVFT7$$\"3!******fU6qN\"FT$\"3/+++A$) *QN%FT7$$\"33+++W*fQM\"FT$\"3y*****p\"p`^VFT7$$\"3,+++CcoI8FT$\"3-+++m D&zM%FT7$$\"3#******4$yd<8FT$\"3k******RM4VVFT7$$\"3$******\\QPYI\"FT$ \"3)******\\z1pL%FT7$$\"3)*******e!z>H\"FT$\"3e*****zxR$HVFT7$$\"3++++ -Ytz7FT$\"3J++++3M?VFT7$$\"3#******pMc!o7FT$\"3V+++g<')4VFT7$$\"3)**** ***H97d7FT$\"31+++d:'yH%FT7$$\"3!******H]OrC\"FT$\"3!******RZ6VG%FT7$$ \"32+++7EMQ7FT$\"3e*****HF.#pUFT7$$\"3*******fR>5B\"FT$\"3e*****z/hDD% FT7$$\"3,+++wp[D7FT$\"3v*****p+eWB%FT7$$\"3*******>e-@A\"FT$\"3V+++%)* Q]@%FT7$$\"3.+++`sC@7FT$\"3k*****p=[X>%FT7$$\"3/+++QlHB7FT$\"3/+++:%eL <%FT7$$\"3)*******4)p&G7FT$\"3)******\\N#*>:%FT7$$\"3.+++(*)esB\"FT$\" 3j*****4&*>68%FT7$$\"3'******>K]$\\7FT$\"3&*******QR^6TFT7$$\"3-+++\\: dk7FT$\"3o*****pBdR4%FT7$$\"3/+++e\\R#G\"FT$\"3G+++mC7zSFT7$$\"3\"**** **4_5@I\"FT$\"3))*****\\@ou1%FT7$$\"3$*******z2%HK\"FT$\"3S+++yK>fSFT7 $$\"3++++T::W8FT$\"3y*****RkdU0%FT7$$\"3/+++&*R7l8FT$\"3u*****4JYC0%FT 7$$\"3++++P:Q&Q\"FT$\"3=+++%oUM0%FT7$$\"3,+++BPe/9FT$\"3i*****\\F*)o0% FT7$$\"3)******R-/DU\"FT$\"3#)******>%GC1%FT7$$\"3#*******RO+R9FT$\"3w *****R5D(pSFT7$$\"3%******>u2SX\"FT$\"3o*****\\=x%ySFT7$$\"3#******fS' [n9FT$\"3#*******yzT)3%FT7$$\"3)*******4.Wz9FT$\"3A+++eZJ*4%FT7$$\"3'* *****HW!*)*[\"FT$\"3w*****RUl46%FT7$$\"3%******H)3())\\\"FT$\"3R+++wX> BTFT7$$\"3$******H$QU1:FT$\"33+++T'\\e8%FT7$$\"3.+++Wif7:FT$\"37+++Wtz [TFT7$$\"3!******fcPu^\"FT$\"3#*******G1#>;%FT7$$\"30+++g#)*4_\"FT$\"3 ++++@g6vTFT7$$\"3!******>&)GL_\"FT$\"3G+++<9H)=%FT7$$\"3\"******zLzW_ \"FT$\"3:+++=UO,UFT7$$\"3$******4&))\\C:FT$\"3B+++%yfU@%FT7$$\"3.+++$ \\NM_\"FT$\"3j*****H85pA%FT7$$\"3'******4GO8_\"FT$\"3m******[GDRUFT7$$ \"3(******z8_WH%FTFe_rFd^n-F__n6jn7%7$$\"+.>]*[\"Fa bn$\"+_)\\QI%Fabn7$$\"+'f%R\"[\"Fabn$\"+tjf7VFabn7$$\"$P\"F]_n$\"$>%F] _n7%Fibs7$$\"+W2gs9Fabn$\"+&o\\1K%FabnF^cs7%Fdcs7$$\"+&RjJY\"Fabn$\"+( RozK%FabnF^cs7%Fjcs7$$\"+yi7`9Fabn$\"+N9^MVFabnF^cs7%F`ds7$$\"+@T`U9Fa bn$\"+kxBSVFabnF^cs7%Ffds7$$\"+9IVJ9Fabn$\"+?h5XVFabnF^cs7%F\\es7$$\"+ E3()>9Fabn$\"+dZ2\\VFabnF^cs7%Fbes7$$\"+%o(*yS\"Fabn$\"+275_VFabnF^cs7 %Fhes7$$\"+Glc&R\"Fabn$\"+^?9aVFabnF^cs7%F^fs7$$\"+xP$HQ\"Fabn$\"+6F:b VFabnF^cs7%Fdfs7$$\"+9,1q8Fabn$\"+os3bVFabnF^cs7%Fjfs7$$\"+E9,d8Fabn$ \"+A$)*QN%FabnF^cs7%F`gs7$$\"+W*fQM\"Fabn$\"+H\"Fabn$\"+y(R$HV FabnF^cs7%F^is7$$\"+-Ytz7Fabn$\"++3M?VFabnF^cs7%Fdis7$$\"+Zj0o7Fabn$\" +g<')4VFabnF^cs7%Fjis7$$\"+I97d7Fabn$\"+d:'yH%FabnF^cs7%F`js7$$\"+.l8Z 7Fabn$\"+u9J%G%FabnF^cs7%Ffjs7$$\"+7EMQ7Fabn$\"+tK?pUFabnF^cs7%F\\[t7$ $\"+'R>5B\"Fabn$\"+[5c_UFabnF^cs7%Fb[t7$$\"+wp[D7Fabn$\"+2!eWB%FabnF^c s7%Fh[t7$$\"+#e-@A\"Fabn$\"+%)*Q]@%FabnF^cs7%F^\\t7$$\"+`sC@7Fabn$\"+( =[X>%FabnF^cs7%Fd\\t7$$\"+QlHB7Fabn$\"+:%eL<%FabnF^cs7%Fj\\t7$$\"+5)p& G7Fabn$\"+bB*>:%FabnF^cs7%F`]t7$$\"+(*)esB\"Fabn$\"+^*>68%FabnF^cs7%Ff ]t7$$\"+A.N\\7Fabn$\"+RR^6TFabnF^cs7%F\\^t7$$\"+\\:dk7Fabn$\"+Ps&R4%Fa bnF^cs7%Fb^t7$$\"+e\\R#G\"Fabn$\"+mC7zSFabnF^cs7%Fh^t7$$\"+@06-8Fabn$ \"+:#ou1%FabnF^cs7%F^_t7$$\"+!ySHK\"Fabn$\"+yK>fSFabnF^cs7%Fd_t7$$\"+T ::W8Fabn$\"+WwDaSFabnF^cs7%Fj_t7$$\"+&*R7l8Fabn$\"+6jW_SFabnF^cs7%F``t 7$$\"+P:Q&Q\"Fabn$\"+%oUM0%FabnF^cs7%Ff`t7$$\"+BPe/9Fabn$\"+v#*)o0%Fab nF^cs7%F\\at7$$\"+CS]A9Fabn$\"+?%GC1%FabnF^cs7%Fbat7$$\"+SO+R9Fabn$\"+ /^spSFabnF^cs7%Fhat7$$\"+Ux+a9Fabn$\"+&=x%ySFabnF^cs7%F^bt7$$\"+1k[n9F abn$\"+zzT)3%FabnF^cs7%Fdbt7$$\"+5.Wz9Fabn$\"+eZJ*4%FabnF^cs7%Fjbt7$$ \"+V/*)*[\"Fabn$\"+Ca'46%FabnF^cs7%F`ct7$$\"+$)3())\\\"Fabn$\"+wX>BTFa bnF^cs7%Ffct7$$\"+LQU1:Fabn$\"+T'\\e8%FabnF^cs7%F\\dt7$$\"+Wif7:Fabn$ \"+Wtz[TFabnF^cs7%Fbdt7$$\"+mvV<:Fabn$\"+H1#>;%FabnF^cs7%Fhdt7$$\"+g#) *4_\"Fabn$\"+@g6vTFabnF^cs7%F^et7$$\"+_)GL_\"Fabn$\"+<9H)=%FabnF^cs7%F det7$$\"+Q$zW_\"Fabn$\"+=UO,UFabnF^cs7%Fjet7$$\"+^))\\C:Fabn$\"+%yfU@% FabnF^cs7%F`ft7$$\"+$\\NM_\"Fabn$\"+L,\"pA%FabnF^cs7%Ffft7$$\"+\"GO8_ \"Fabn$\"+\\GDRUFabnF^cs7%F\\gt7$$\"+yrC=:Fabn$\"+k,B^UFabnF^cs7%Fbgt7 $$\"+yJ@9:Fabn$\"+!=)yiUFabnF^cs7%Fhgt7$$\"+J%y#4:Fabn$\"+.i(QF%FabnF^ cs7%F^ht7$$\"+%R'[.:Fabn$\"+mhW%G%FabnF^cs7%Fdht7$$\"+4+)o\\\"Fabn$\"+ K@X%H%FabnF^cs7%FjhtFdbsF^csFh^rF^_r-F$6$7in7$Ff_r$!3#******>&)\\QI%FT 7$F]bs$!3)******>8_WH%FT7$Fhas$!3S+++mhW%G%FT7$Fcas$!3Q+++.i(QF%FT7$F^ as$!3E+++!=)yiUFT7$Fi`s$!3))*****R;I7D%FT7$Fd`s$!3m******[GDRUFT7$F_`s $!3j*****H85pA%FT7$Fj_s$!3B+++%yfU@%FT7$Fe_s$!3:+++=UO,UFT7$F`_s$!3G++ +<9H)=%FT7$F[_s$!3++++@g6vTFT7$Ff^s$!3#*******G1#>;%FT7$Fa^s$!37+++Wtz [TFT7$F\\^s$!33+++T'\\e8%FT7$Fg]s$!3R+++wX>BTFT7$Fb]s$!3w*****RUl46%FT 7$F]]s$!3A+++eZJ*4%FT7$Fh\\s$!3#*******yzT)3%FT7$Fc\\s$!3o*****\\=x%yS FT7$F^\\s$!3w*****R5D(pSFT7$Fi[s$!3#)******>%GC1%FT7$Fd[s$!3i*****\\F* )o0%FT7$F_[s$!3=+++%oUM0%FT7$Fjjr$!3u*****4JYC0%FT7$Fejr$!3y*****RkdU0 %FT7$F`jr$!3S+++yK>fSFT7$F[jr$!3))*****\\@ou1%FT7$Ffir$!3G+++mC7zSFT7$ Fair$!3o*****pBdR4%FT7$F\\ir$!3&*******QR^6TFT7$Fghr$!3j*****4&*>68%FT 7$Fbhr$!3)******\\N#*>:%FT7$F]hr$!3/+++:%eL<%FT7$Fhgr$!3k*****p=[X>%FT 7$Fcgr$!3V+++%)*Q]@%FT7$F^gr$!3v*****p+eWB%FT7$Fifr$!3e*****z/hDD%FT7$ Fdfr$!3e*****HF.#pUFT7$F_fr$!3!******RZ6VG%FT7$Fjer$!31+++d:'yH%FT7$Fe er$!3V+++g<')4VFT7$F`er$!3J++++3M?VFT7$F[er$!3e*****zxR$HVFT7$Ffdr$!3) ******\\z1pL%FT7$Fadr$!3k******RM4VVFT7$F\\dr$!3-+++mD&zM%FT7$Fgcr$!3y *****p\"p`^VFT7$Fbcr$!3/+++A$)*QN%FT7$F]cr$!37+++os3bVFT7$Fhbr$!3o**** *4r_^N%FT7$Fcbr$!3=+++^?9aVFT7$F^br$!3e*****p?,@N%FT7$Fiar$!3M+++dZ2\\ VFT7$Fdar$!3i******>h5XVFT7$F_ar$!3F+++kxBSVFT7$Fj`r$!3e*****\\V6XL%FT 7$Fe`r$!3D+++(RozK%FT7$F``r$!3C+++&o\\1K%FT7$F[`r$!3()*****HP'f7VFTFci tFd^n-F__n6jn7%7$Febs$!+_)\\QI%Fabn7$F[it$!+K@X%H%Fabn7$F_cs$!$>%F]_n7 %F]eu7$Feht$!+mhW%G%FabnF`eu7%Fdeu7$F_ht$!+.i(QF%FabnF`eu7%Fheu7$Figt$ !+!=)yiUFabnF`eu7%F\\fu7$Fcgt$!+k,B^UFabnF`eu7%F`fu7$F]gt$!+\\GDRUFabn F`eu7%Fdfu7$Fgft$!+L,\"pA%FabnF`eu7%Fhfu7$Faft$!+%yfU@%FabnF`eu7%F\\gu 7$F[ft$!+=UO,UFabnF`eu7%F`gu7$Feet$!+<9H)=%FabnF`eu7%Fdgu7$F_et$!+@g6v TFabnF`eu7%Fhgu7$Fidt$!+H1#>;%FabnF`eu7%F\\hu7$Fcdt$!+Wtz[TFabnF`eu7%F `hu7$F]dt$!+T'\\e8%FabnF`eu7%Fdhu7$Fgct$!+wX>BTFabnF`eu7%Fhhu7$Fact$!+ Ca'46%FabnF`eu7%F\\iu7$F[ct$!+eZJ*4%FabnF`eu7%F`iu7$Febt$!+zzT)3%FabnF `eu7%Fdiu7$F_bt$!+&=x%ySFabnF`eu7%Fhiu7$Fiat$!+/^spSFabnF`eu7%F\\ju7$F cat$!+?%GC1%FabnF`eu7%F`ju7$F]at$!+v#*)o0%FabnF`eu7%Fdju7$Fg`t$!+%oUM0 %FabnF`eu7%Fhju7$Fa`t$!+6jW_SFabnF`eu7%F\\[v7$F[`t$!+WwDaSFabnF`eu7%F` [v7$Fe_t$!+yK>fSFabnF`eu7%Fd[v7$F__t$!+:#ou1%FabnF`eu7%Fh[v7$Fi^t$!+mC 7zSFabnF`eu7%F\\\\v7$Fc^t$!+Ps&R4%FabnF`eu7%F`\\v7$F]^t$!+RR^6TFabnF`e u7%Fd\\v7$Fg]t$!+^*>68%FabnF`eu7%Fh\\v7$Fa]t$!+bB*>:%FabnF`eu7%F\\]v7$ F[]t$!+:%eL<%FabnF`eu7%F`]v7$Fe\\t$!+(=[X>%FabnF`eu7%Fd]v7$F_\\t$!+%)* Q]@%FabnF`eu7%Fh]v7$Fi[t$!+2!eWB%FabnF`eu7%F\\^v7$Fc[t$!+[5c_UFabnF`eu 7%F`^v7$F][t$!+tK?pUFabnF`eu7%Fd^v7$Fgjs$!+u9J%G%FabnF`eu7%Fh^v7$Fajs$ !+d:'yH%FabnF`eu7%F\\_v7$F[js$!+g<')4VFabnF`eu7%F`_v7$Feis$!++3M?VFabn F`eu7%Fd_v7$F_is$!+y(R$HVFabnF`eu7%Fh_v7$Fihs$!+&z1pL%FabnF`eu7%F\\`v7 $Fchs$!+SM4VVFabnF`eu7%F``v7$F]hs$!+mD&zM%FabnF`eu7%Fd`v7$Fggs$!+FTF (-%'COLOURG6&Fg^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F`dv7$F($\"3/+++ +++!z%FTFedvFhdv-F$6$7ewF'7$$!3)******HK[HT)!#F$\"37+++&G'zq:F07$$!3++ ++$=XP>&F:$\"30+++g8fTJF07$$!3]+++zXwwbF@$\"3v*****HqzBr%F07$$!3'***** *HuMC*GFF$\"3K+++$\\[JG'F07$$!33+++*eG:+\"FQ$\"3U+++#ytQ&yF07$$!3!**** **p$4))*o#FQ$\"3R+++!)y]C%*F07$$!3&)*****z2@f7'FQ$\"3'******pY%\\*4\"F T7$$!3-+++?`fi7Fao$\"3)*******pZ\\c7FT7$$!3!******>_LP[#Fao$\"3'****** *4JS89FT7$$!3;+++#*o)*\\[Fao$\"3!******\\()H,d\"FT7$$!3_******Hdjo&*Fa o$\"36+++ws_E/:&)*=F\\p$\"3/+++irP#)=FT7$$!3&******H[R ^s$F\\p$\"32+++>3RP?FT7$$!3p*****4lTi5(F\\p$\"35+++SIA\">#FT7$$!31+++L **)HI\"Fgp$\"3:+++7#*[VBFT7$$!35+++#p7RG#Fgp$\"3?+++1Uv$\\#FT7$$!3C+++ oV&>#QFgp$\"35+++>iWTEFT7$$!3<+++\\\"y'4hFgp$\"33+++njl&y#FT7$$!3l**** *p(oAN$*Fgp$\"3?+++_*\\[#HFT7$$!3!******H\\^DO\"F0$\"3%)*****R@#ocIFT7 $$!33+++t*Gt*=F0$\"3#******>j$HyJFT7$$!3%)*****4(yb?DF0$\"3/+++YHB(G$F T7$$!34+++Q%eV?$F0$\"31+++iuQ#Q$FT7$$!3&)*****R(4*4#RF0$\"3?+++Lu6kMFT 7$$!3C+++<[t\\YF0$\"3&)*****\\7)oLNFT7$$!3y*****p!f\"yP&F0$\"3y*****fC PEf$FT7$$!3K+++3UT)4'F0$\"3\")*****zd2Ck$FT7$$!3(********Q'R3oF0$\"3/+ ++w`@%o$FT7$$!3`+++M\"zm](F0$\"3,+++>%R!>PFT7$$!3?+++fIB$>)F0$\"3!**** **pG`wu$FT7$$!3a+++h\\_o))F0$\"3'******4jl1x$FT7$$!3y******y8AL&*F0$\" 3!******p3c&)y$FT7$$!3'******p*Q!)=5FT$\"3*******4,/Ys[X\"FT$\"3=+++Hy\")zPFT7$$!3++++2Ro9:FT$\"3A+++x+%>w$ FT7$$!3!******4vpSd\"FT$\"3\")*****fin0u$FT7$$!31+++0)=Jj\"FT$\"3z**** *pCFdr$FT7$$!3-+++=z$>p\"FT$\"33+++y/W(o$FT7$$!3#******HUc1v\"FT$\"3++ ++k)Hdl$FT7$$!3!******pJI%4=FT$\"3*)******zsi?OFT7$$!3!*******y_Wo=FT$ \"3#)******y]=#e$FT7$$!3'*******\\p\"z#>FT$\"31+++q1\\SNFT7$$!31+++Yg3 ))>FT$\"3))*****>X%o&\\$FT7$$!3')******3^?\\?FT$\"37+++,'zzW$FT7$$!3>+ ++&p8:6#FT$\"3#)*****fkzwR$FT7$$!3!)******\\4?v@FT$\"3#)*****He'=XLFT7 $$!3*)*****pmf.C#FT$\"3')*****pC!*4H$FT7$$!3'******H[VpI#FT$\"3))***** z.KcB$FT7$$!3++++[TuuBFT$\"3A+++9_kzJFT7$$!3$)*****\\^-MW#FT$\"3@+++In [BJFT7$$!3&)*****zZ`C^#FT$\"3%)*****>,$[nIFT7$$!3!)******p&*Q\"e#FT$\" 3*)*****Hk1=,$FT7$$!3++++tDAs%GF T$\"3%)*****zr*\\\"z#FT7$$!3*)*****f6p'4HFT$\"3*******p#=@OFFT7$$!3))* ****4Ac,(HFT$\"3%******z)eW!o#FT7$$!32+++'z.'GIFT$\"3++++x(>Si#FT7$$!3 /+++z%o\\3$FT$\"3#*******46xmDFT7$$!3z*****R.M#RJFT$\"3#)******\\\"e&3 DFT7$$!37+++;WS\">$FT$\"3#)******p*e#\\CFT7$$!3'******H&))\\TKFT$\"3!* *****4#)p()Q#FT7$$!39+++M&[&*G$FT$\"3&******HL.qK#FT7$$!3?+++8XfNLFT$ \"3/+++))o)QE#FT7$$!3#)*****>\\'ozLFT$\"3)******\\Qh$*>#FT7$$!3()***** f>#)=U$FT$\"34+++80QL@FT7$$!3y*****p2ZAY$FT$\"3)******Rf5f1#FT7$$!38++ +5T&3]$FT$\"33+++D5$p*>FT7$$!3++++WMyPNFT$\"3)******pINk#>FT7$$!3-+++R ;7tNFT$\"3'*******)eKW&=FT7$$!3')*****>Egpg$FT$\"3%******4f\\4y\"FT7$$ !3$******He$RROFT$\"3&******plKgq\"FT7$$!3%******\\A:0n$FT$\"3!******* o&\\(H;FT7$$!3*******pr8/q$FT$\"3)******\\l!>_:FT7$$!3!******4;n\"HPFT $\"31+++]'oMZ\"FT7$$!3y*****pUPov$FT$\"3'******R#or$R\"FT7$$!35+++IYY$ y$FT$\"3********o]388FT7$$!3))*****Hri!4QFT$\"3'******p,M)F07$$!39+++pzUQRFT$\"3;+++*)R'*ptF07$$!3)) *****pn.^&RFT$\"3[******fi,XlF07$$!37+++'[$4qRFT$\"38+++lfm@dF07$$!3!* *******='H$)RFT$\"3/+++V>2+\\F07$$!3;+++v\")>?E$F07$$!3m*****RAZ8,%FT$\"35+++;z?XCF07 $$!3u*******)[j;SFT$\"38+++.:]H;F07$$!3@+++`8#)>SFT$\"3[+++%onb9)Fgp7$ $!3R+++7e)3-%FTF(7$Fbdx$!3[+++%onb9)Fgp7$F]dx$!38+++.:]H;F07$Fhcx$!35+ ++;z?XCF07$Fccx$!3#******>\")>?E$F07$F^cx$!3C+++WFB!3%F07$Fibx$!3/+++V >2+\\F07$Fdbx$!38+++lfm@dF07$F_bx$!3[******fi,XlF07$Fjax$!3;+++*)R'*pt F07$Feax$!3u******oo:'>)F07$F`ax$!3s*****HN?I-*F07$F[ax$!3-+++/)Q(\\)* F07$Ff`x$!3++++l__n5FT7$Fa`x$!3)*******\\w#)\\6FT7$F\\`x$!3'******p,M< B\"FT7$Fg_x$!3********o]388FT7$Fb_x$!3'******R#or$R\"FT7$F]_x$!31+++]' oMZ\"FT7$Fh^x$!3)******\\l!>_:FT7$Fc^x$!3!*******o&\\(H;FT7$F^^x$!3&** ****plKgq\"FT7$Fi]x$!3%******4f\\4y\"FT7$Fd]x$!3'*******)eKW&=FT7$F_]x $!3)******pINk#>FT7$Fj\\x$!33+++D5$p*>FT7$Fe\\x$!3)******Rf5f1#FT7$F` \\x$!34+++80QL@FT7$F[\\x$!3)******\\Qh$*>#FT7$Ff[x$!3/+++))o)QE#FT7$Fa [x$!3&******HL.qK#FT7$F\\[x$!3!******4#)p()Q#FT7$Fgjw$!3#)******p*e#\\ CFT7$Fbjw$!3#)******\\\"e&3DFT7$F]jw$!3#*******46xmDFT7$Fhiw$!3++++x(> Si#FT7$Fciw$!3%******z)eW!o#FT7$F^iw$!3*******p#=@OFFT7$Fihw$!3%)***** zr*\\\"z#FT7$Fdhw$!3#******RF2l%GFT7$F_hw$!39+++-%Q9!HFT7$Fjgw$!39+++< i[cHFT7$Fegw$!3*)*****Hk1=,$FT7$F`gw$!3%)*****>,$[nIFT7$F[gw$!3@+++In[ BJFT7$Fffw$!3A+++9_kzJFT7$Fafw$!3))*****z.KcB$FT7$F\\fw$!3')*****pC!*4 H$FT7$Fgew$!3#)*****He'=XLFT7$Fbew$!3#)*****fkzwR$FT7$F]ew$!37+++,'zzW $FT7$Fhdw$!3))*****>X%o&\\$FT7$Fcdw$!31+++q1\\SNFT7$F^dw$!3#)******y]= #e$FT7$Ficw$!3*)******zsi?OFT7$Fdcw$!3++++k)Hdl$FT7$F_cw$!33+++y/W(o$F T7$Fjbw$!3z*****pCFdr$FT7$Febw$!3\")*****fin0u$FT7$F`bw$!3A+++x+%>w$FT 7$F[bw$!3=+++Hy\")zPFT7$Ffaw$!3?+++/S;%z$FT7$Faaw$!3$******p]F\\!QFT7$ F\\aw$!37+++j\"R?\"QFT7$Fg`w$!33+++swS:QFT7$Fb`w$!3-+++i\\\"\\\"QFT7$F ]`w$!3,+++$\\5/\"QFT7$Fh_w$!3*******4,/%R!>PFT7$ F_^w$!3/+++w`@%o$FT7$Fj]w$!3\")*****zd2Ck$FT7$Fe]w$!3y*****fCPEf$FT7$F `]w$!3&)*****\\7)oLNFT7$F[]w$!3?+++Lu6kMFT7$Ff\\w$!31+++iuQ#Q$FT7$Fa\\ w$!3/+++YHB(G$FT7$F\\\\w$!3#******>j$HyJFT7$Fg[w$!3%)*****R@#ocIFT7$Fb [w$!3?+++_*\\[#HFT7$F][w$!33+++njl&y#FT7$Fhjv$!35+++>iWTEFT7$Fcjv$!3?+ ++1Uv$\\#FT7$F^jv$!3:+++7#*[VBFT7$Fiiv$!35+++SIA\">#FT7$Fdiv$!32+++>3R P?FT7$F_iv$!3/+++irP#)=FT7$Fjhv$!36+++ws_E&FT$\"+g8fTJFg_nFh_n7%Fah y7$$!+zXwwbFj`n$\"+.(zBr%Fg_nFh_n7%Fghy7$$!+VZV#*GFaan$\"+$\\[JG'Fg_nF h_n7%F]iy7$$!+*eG:+\"F^bn$\"+#ytQ&yFg_nFh_n7%Fciy7$$!+P4))*o#F^bn$\"+! )y]C%*Fg_nFh_n7%Fiiy7$$!+y5#f7'F^bn$\"+nW\\*4\"FabnFh_n7%F_jy7$$!+?`fi 7Fhcn$\"+qZ\\c7FabnFh_n7%Fejy7$$!+ANt$[#Fhcn$\"+5JS89FabnFh_n7%F[[z7$$ !+#*o)*\\[Fhcn$\"+v)H,d\"FabnFh_n7%Fa[z7$$!+Idjo&*Fhcn$\"+ws_E3RP ?FabnFh_n7%Fc\\z7$$!+^;C1rFedn$\"+SIA\">#FabnFh_n7%Fi\\z7$$!+L**)HI\"F ben$\"+7#*[VBFabnFh_n7%F_]z7$$!+#p7RG#Fben$\"+1Uv$\\#FabnFh_n7%Fe]z7$$ !+oV&>#QFben$\"+>iWTEFabnFh_n7%F[^z7$$!+\\\"y'4hFben$\"+njl&y#FabnFh_n 7%Fa^z7$$!+xoAN$*Fben$\"+_*\\[#HFabnFh_n7%Fg^z7$$!+$\\^DO\"Fg_n$\"+9Ao cIFabnFh_n7%F]_z7$$!+t*Gt*=Fg_n$\"+KOHyJFabnFh_n7%Fc_z7$$!+ryb?DFg_n$ \"+YHB(G$FabnFh_n7%Fi_z7$$!+Q%eV?$Fg_n$\"+iuQ#Q$FabnFh_n7%F_`z7$$!+u4* 4#RFg_n$\"+Lu6kMFabnFh_n7%Fe`z7$$!+<[t\\YFg_n$\"+D\")oLNFabnFh_n7%F[az 7$$!+2f\"yP&Fg_n$\"+Ysj#f$FabnFh_n7%Faaz7$$!+3UT)4'Fg_n$\"+yvSUOFabnFh _n7%Fgaz7$$!+!R'R3oFg_n$\"+w`@%o$FabnFh_n7%F]bz7$$!+M\"zm](Fg_n$\"+>%R !>PFabnFh_n7%Fcbz7$$!+fIB$>)Fg_n$\"+(G`wu$FabnFh_n7%Fibz7$$!+h\\_o))Fg _n$\"+JcmqPFabnFh_n7%F_cz7$$!+z8AL&*Fg_n$\"+(3c&)y$FabnFh_n7%Fecz7$$!+ (*Q!)=5Fabn$\"+6Sq,QFabnFh_n7%F[dz7$$!+EvO$3\"Fabn$\"+$\\5/\"QFabnFh_n 7%Fadz7$$!+%Gyq9\"Fabn$\"+i\\\"\\\"QFabnFh_n7%Fgdz7$$!+:n**47Fabn$\"+s wS:QFabnFh_n7%F]ez7$$!+@(z@F\"Fabn$\"+j\"R?\"QFabnFh_n7%Fcez7$$!+*R#oL 8Fabn$\"+2v#\\!QFabnFh_n7%Fiez7$$!+g.c%R\"Fabn$\"+/S;%z$FabnFh_n7%F_fz 7$$!+iC([X\"Fabn$\"+Hy\")zPFabnFh_n7%Fefz7$$!+2Ro9:Fabn$\"+x+%>w$FabnF h_n7%F[gz7$$!+^(pSd\"Fabn$\"+EwcSPFabnFh_n7%Fagz7$$!+0)=Jj\"Fabn$\"+Zs s:PFabnFh_n7%Fggz7$$!+=z$>p\"Fabn$\"+y/W(o$FabnFh_n7%F]hz7$$!+Bkl]Fabn$\"+q1\\SNFabnFh_n7%Fei z7$$!+Yg3))>Fabn$\"+_Wo&\\$FabnFh_n7%F[jz7$$!+4^?\\?Fabn$\"+,'zzW$Fabn Fh_n7%Fajz7$$!+&p8:6#Fabn$\"+Y'zwR$FabnFh_n7%Fgjz7$$!+]4?v@Fabn$\"+$e' =XLFabnFh_n7%F][[l7$$!+n'f.C#Fabn$\"+Z-*4H$FabnFh_n7%Fc[[l7$$!+$[VpI#F abn$\"+Q?jNKFabnFh_n7%Fi[[l7$$!+[TuuBFabn$\"+9_kzJFabnFh_n7%F_\\[l7$$! +:DSVCFabn$\"+In[BJFabnFh_n7%Fe\\[l7$$!+yMX7DFabn$\"+7I[nIFabnFh_n7%F[ ][l7$$!+q&*Q\"e#Fabn$\"+Vm!=,$FabnFh_n7%Fa][l7$$!+tSi#FabnFh_n7%F[`[l7$$!+z%o\\3$Fabn$\"+56xmD FabnFh_n7%Fa`[l7$$!+MSBRJFabn$\"+]\"e&3DFabnFh_n7%Fg`[l7$$!+;WS\">$Fab n$\"+q*e#\\CFabnFh_n7%F]a[l7$$!+`))\\TKFabn$\"+@)p()Q#FabnFh_n7%Fca[l7 $$!+M&[&*G$Fabn$\"+LL+FBFabnFh_n7%Fia[l7$$!+8XfNLFabn$\"+))o)QE#FabnFh _n7%F_b[l7$$!+#\\'ozLFabn$\"+&Qh$*>#FabnFh_n7%Feb[l7$$!+'>#)=U$Fabn$\" +80QL@FabnFh_n7%F[c[l7$$!+xqCiMFabn$\"+%f5f1#FabnFh_n7%Fac[l7$$!+5T&3] $Fabn$\"+D5$p*>FabnFh_n7%Fgc[l7$$!+WMyPNFabn$\"+2`VE>FabnFh_n7%F]d[l7$ $!+R;7tNFabn$\"+*eKW&=FabnFh_n7%Fcd[l7$$!+i-'pg$Fabn$\"+\"f\\4y\"FabnF h_n7%Fid[l7$$!+$e$RROFabn$\"+dE.1_:FabnFh_n7%F[f[l7$$!+hr;HPFa bn$\"+]'oMZ\"FabnFh_n7%Faf[l7$$!+Fu$ov$Fabn$\"+Cor$R\"FabnFh_n7%Fgf[l7 $$!+IYY$y$Fabn$\"+p]388FabnFh_n7%F]g[l7$$!+8F14QFabn$\"+)Fg_nFh_n7%Fai[l7$$!+p zUQRFabn$\"+*)R'*ptFg_nFh_n7%Fgi[l7$$!+xO5bRFabn$\"+gi,XlFg_nFh_n7%F]j [l7$$!+'[$4qRFabn$\"+lfm@dFg_nFh_n7%Fcj[l7$$!+!>'H$)RFabn$\"+V>2+\\Fg_ nFh_n7%Fij[l7$$!+v?E$Fg_nFh_n7%Fe[\\l7$$!+CsM6SFabn$\"+;z?XCFg_nFh_n7%F[\\\\l7$$!+ !*[j;SFabn$\"+.:]H;Fg_nFh_n7%Fa\\\\l7$$!+`8#)>SFabn$\"+%onb9)FbenFh_n7 %Fg\\\\l7$$!+7e)3-%FabnF(Fh_n7%F]]\\l7$Fh\\\\l$!+%onb9)FbenFh_n7%Fa]\\ l7$Fb\\\\l$!+.:]H;Fg_nFh_n7%Fe]\\l7$F\\\\\\l$!+;z?XCFg_nFh_n7%Fi]\\l7$ Ff[\\l$!+7)>?E$Fg_nFh_n7%F]^\\l7$F`[\\l$!+WFB!3%Fg_nFh_n7%Fa^\\l7$Fjj[ l$!+V>2+\\Fg_nFh_n7%Fe^\\l7$Fdj[l$!+lfm@dFg_nFh_n7%Fi^\\l7$F^j[l$!+gi, XlFg_nFh_n7%F]_\\l7$Fhi[l$!+*)R'*ptFg_nFh_n7%Fa_\\l7$Fbi[l$!+po:'>)Fg_ nFh_n7%Fe_\\l7$F\\i[l$!+`.-B!*Fg_nFh_n7%Fi_\\l7$Ffh[l$!+/)Q(\\)*Fg_nFh _n7%F]`\\l7$F`h[l$!+l__n5FabnFh_n7%Fa`\\l7$Fjg[l$!+]w#)\\6FabnFh_n7%Fe `\\l7$Fdg[l$!+_:FabnFh_n7%Fia\\l7$Ffe[l$!+p&\\(H;FabnFh_n7%F]b\\l7$F`e[l$!+dE. 1FabnFh_n7%F]c\\l7$Fhc[l$!+D5$p*>FabnFh_n 7%Fac\\l7$Fbc[l$!+%f5f1#FabnFh_n7%Fec\\l7$F\\c[l$!+80QL@FabnFh_n7%Fic \\l7$Ffb[l$!+&Qh$*>#FabnFh_n7%F]d\\l7$F`b[l$!+))o)QE#FabnFh_n7%Fad\\l7 $Fja[l$!+LL+FBFabnFh_n7%Fed\\l7$Fda[l$!+@)p()Q#FabnFh_n7%Fid\\l7$F^a[l $!+q*e#\\CFabnFh_n7%F]e\\l7$Fh`[l$!+]\"e&3DFabnFh_n7%Fae\\l7$Fb`[l$!+5 6xmDFabnFh_n7%Fee\\l7$F\\`[l$!+x(>Si#FabnFh_n7%Fie\\l7$Ff_[l$!+))eW!o# FabnFh_n7%F]f\\l7$F`_[l$!+F=@OFFabnFh_n7%Faf\\l7$Fj^[l$!+=(*\\\"z#Fabn Fh_n7%Fef\\l7$Fd^[l$!+us]YGFabnFh_n7%Fif\\l7$F^^[l$!+-%Q9!HFabnFh_n7%F ]g\\l7$Fh][l$!+w$FabnFh_n7%Fi[]l7$Fffz$!+Hy\")zPFabnFh_n7%F]\\]l7 $F`fz$!+/S;%z$FabnFh_n7%Fa\\]l7$Fjez$!+2v#\\!QFabnFh_n7%Fe\\]l7$Fdez$! +j\"R?\"QFabnFh_n7%Fi\\]l7$F^ez$!+swS:QFabnFh_n7%F]]]l7$Fhdz$!+i\\\"\\ \"QFabnFh_n7%Fa]]l7$Fbdz$!+$\\5/\"QFabnFh_n7%Fe]]l7$F\\dz$!+6Sq,QFabnF h_n7%Fi]]l7$Ffcz$!+(3c&)y$FabnFh_n7%F]^]l7$F`cz$!+JcmqPFabnFh_n7%Fa^]l 7$Fjbz$!+(G`wu$FabnFh_n7%Fe^]l7$Fdbz$!+>%R!>PFabnFh_n7%Fi^]l7$F^bz$!+w `@%o$FabnFh_n7%F]_]l7$Fhaz$!+yvSUOFabnFh_n7%Fa_]l7$Fbaz$!+Ysj#f$FabnFh _n7%Fe_]l7$F\\az$!+D\")oLNFabnFh_n7%Fi_]l7$Ff`z$!+Lu6kMFabnFh_n7%F]`]l 7$F``z$!+iuQ#Q$FabnFh_n7%Fa`]l7$Fj_z$!+YHB(G$FabnFh_n7%Fe`]l7$Fd_z$!+K OHyJFabnFh_n7%Fi`]l7$F^_z$!+9AocIFabnFh_n7%F]a]l7$Fh^z$!+_*\\[#HFabnFh _n7%Faa]l7$Fb^z$!+njl&y#FabnFh_n7%Fea]l7$F\\^z$!+>iWTEFabnFh_n7%Fia]l7 $Ff]z$!+1Uv$\\#FabnFh_n7%F]b]l7$F`]z$!+7#*[VBFabnFh_n7%Fab]l7$Fj\\z$!+ SIA\">#FabnFh_n7%Feb]l7$Fd\\z$!+>3RP?FabnFh_n7%Fib]l7$F^\\z$!+irP#)=Fa bnFh_n7%F]c]l7$Fh[z$!+ws_ERVFT7$$\"3-+++()y\"*\\9FT$\"3W+++]>a[VF T7$$\"3%*******pU3T9FT$\"37+++'f$>dVFT7$$\"3\"******>(*p:V\"FT$\"39+++ X&4^O%FT7$$\"3#*******G#=9U\"FT$\"3s*****z9[AP%FT7$$\"3(******4.t1T\"F T$\"3e*****4sn&yVFT7$$\"31+++u%z$*R\"FT$\"3s******fi-%Q%FT7$$\"3'***** *f:%e(Q\"FT$\"3U+++_6e)Q%FT7$$\"3-+++UcLv8FT$\"3e*****p*))=#R%FT7$$\"3 )********)\\oi8FT$\"3g*****f$[![R%FT7$$\"3!******fQ'o\\8FT$\"39+++#)GQ 'R%FT7$$\"36+++*)yRO8FT$\"3;+++k_(oR%FT7$$\"3-+++PB)GK\"FT$\"3G+++*GKi R%FT7$$\"34+++M%3#48FT$\"3A+++Q@S%R%FT7$$\"3'******H4_aH\"FT$\"3/+++J2 L\"R%FT7$$\"31+++t!)p\"G\"FT$\"3K+++(ohpQ%FT7$$\"3)*******3?/o7FT$\"3m *****RYO7Q%FT7$$\"33+++DHfa7FT$\"3;+++\"y%4uVFT7$$\"3,+++QjZT7FT$\"3A+ ++vaZlVFT7$$\"3++++#3Q)G7FT$\"37+++r!=`N%FT7$$\"3/+++a*[o@\"FT$\"36+++ KacVVFT7$$\"35+++U-r07FT$\"3x******)4o,L%FT7$$\"3)******f+gc>\"FT$\"3( ******R<\"4:VFT7$$\"3++++O'zp=\"FT$\"3O+++q\\K)H%FT7$$\"3/+++i%***z6FT $\"3))*****R.,*zUFT7$$\"3++++X35v6FT$\"37+++b_\"*fUFT7$$\"35+++H+rs6FT $\"3++++*>f&QUFT7$$\"33+++\"HwK<\"FT$\"3\"******>?eh@%FT7$$\"3!******4 1Ds<\"FT$\"33+++.)3K>%FT7$$\"3-+++S)z[=\"FT$\"3?+++l#)RqTFT7$$\"3!**** **fWgj>\"FT$\"33+++(yz&[TFT7$$\"3.+++D+\\67FT$\"3:+++Y:oGTFT7$$\"3++++ $>i(H7FT$\"3i*******Ge:6%FT7$$\"3)******fm./D\"FT$\"3K+++)pQy4%FT7$$\" 3/+++qQ^s7FT$\"3I+++**p$y3%FT7$$\"3++++Gv@&H\"FT$\"3C+++g%f:3%FT7$$\"3 )*******e8x<8FT$\"3!)******H1yySFT7$$\"3.+++xogR8FT$\"3o*****zFS\"zSFT 7$$\"3!******R*[Kg8FT$\"3G+++!*\\A#3%FT7$$\"3%*******>kmz8FT$\"33+++;? i(3%FT7$$\"3++++B#yuR\"FT$\"3O+++n#[\\4%FT7$$\"32+++T9FT$\"3i*****R C/`7%FT7$$\"31+++Ua`_9FT$\"3v******G_NPTFT7$$\"3++++=/Ki9FT$\"3S+++H-. ]TFT7$$\"3++++hwfq9FT$\"3w*****f=mJ;%FT7$$\"3++++X4Ux9FT$\"3E+++j=iwTF T7$$\"33+++%FT7$$\"3)******R5Gp[\"FT$\"3++++#*[, /UFT7$$\"3'******z*Hs*[\"FT$\"3l*****RX[x@%FT7$$\"3)******p^%G\"\\\"FT $\"3N+++l%*QJUFT7$$\"3++++O_m\"\\\"FT$\"3$******f8h[C%FT7$$\"3-+++th\" 4\\\"FT$\"3Q+++#z$4eUFT7$$\"35+++-o3*[\"FT$\"3k*****pnB5F%FT7$$\"3!*** ***f7Di[\"FT$\"3g*****f%>f$G%FT7$$\"33+++3yP#[\"FT$\"3/+++=Qu&H%FT7$$ \"35+++H.fx9FT$\"3y******fyU2VFT7$$\"33+++xr!>Z\"FT$\"3))*****>P&f=VFT 7$$\"3-+++R?Pl9FT$\"3N+++_)*>HVFTFhf]lFfgy-F__n6jn7%7$$\"+D!G!e9Fabn$ \"+Il>RVFabn7$$\"+()y\"*\\9Fabn$\"+]>a[VFabn7$$\"$L\"F]_n$\"$B%F]_n7%F \\j^l7$$\"+qU3T9Fabn$\"+'f$>dVFabnFaj^l7%Fgj^l7$$\"+s*p:V\"Fabn$\"+X&4 ^O%FabnFaj^l7%F][_l7$$\"+H#=9U\"Fabn$\"+[\"[AP%FabnFaj^l7%Fc[_l7$$\"+J In59Fabn$\"+@xcyVFabnFaj^l7%Fi[_l7$$\"+u%z$*R\"Fabn$\"+gi-%Q%FabnFaj^l 7%F_\\_l7$$\"+cTe(Q\"Fabn$\"+_6e)Q%FabnFaj^l7%Fe\\_l7$$\"+UcLv8Fabn$\" +(*))=#R%FabnFaj^l7%F[]_l7$$\"+!*\\oi8Fabn$\"+O[![R%FabnFaj^l7%Fa]_l7$ $\"+'Q'o\\8Fabn$\"+#)GQ'R%FabnFaj^l7%Fg]_l7$$\"+*)yRO8Fabn$\"+k_(oR%Fa bnFaj^l7%F]^_l7$$\"+PB)GK\"Fabn$\"+*GKiR%FabnFaj^l7%Fc^_l7$$\"+M%3#48F abn$\"+Q@S%R%FabnFaj^l7%Fi^_l7$$\"+$4_aH\"Fabn$\"+J2L\"R%FabnFaj^l7%F_ __l7$$\"+t!)p\"G\"Fabn$\"+(ohpQ%FabnFaj^l7%Fe__l7$$\"+4?/o7Fabn$\"+kkB \"Q%FabnFaj^l7%F[`_l7$$\"+DHfa7Fabn$\"+\"y%4uVFabnFaj^l7%Fa`_l7$$\"+Qj ZT7Fabn$\"+vaZlVFabnFaj^l7%Fg`_l7$$\"+#3Q)G7Fabn$\"+r!=`N%FabnFaj^l7%F ]a_l7$$\"+a*[o@\"Fabn$\"+KacVVFabnFaj^l7%Fca_l7$$\"+U-r07Fabn$\"+*4o,L %FabnFaj^l7%Fia_l7$$\"+1+m&>\"Fabn$\"+u64:VFabnFaj^l7%F_b_l7$$\"+O'zp= \"Fabn$\"+q\\K)H%FabnFaj^l7%Feb_l7$$\"+i%***z6Fabn$\"+M5!*zUFabnFaj^l7 %F[c_l7$$\"+X35v6Fabn$\"+b_\"*fUFabnFaj^l7%Fac_l7$$\"+H+rs6Fabn$\"+*>f &QUFabnFaj^l7%Fgc_l7$$\"+\"HwK<\"Fabn$\"+-#eh@%FabnFaj^l7%F]d_l7$$\"+h ]Ax6Fabn$\"+.)3K>%FabnFaj^l7%Fcd_l7$$\"+S)z[=\"Fabn$\"+l#)RqTFabnFaj^l 7%Fid_l7$$\"+Y/O'>\"Fabn$\"+(yz&[TFabnFaj^l7%F_e_l7$$\"+D+\\67Fabn$\"+ Y:oGTFabnFaj^l7%Fee_l7$$\"+$>i(H7Fabn$\"+!He:6%FabnFaj^l7%F[f_l7$$\"+m OS]7Fabn$\"+)pQy4%FabnFaj^l7%Faf_l7$$\"+qQ^s7Fabn$\"+**p$y3%FabnFaj^l7 %Fgf_l7$$\"+Gv@&H\"Fabn$\"+g%f:3%FabnFaj^l7%F]g_l7$$\"+f8x<8Fabn$\"+I1 yySFabnFaj^l7%Fcg_l7$$\"+xogR8Fabn$\"+y-9zSFabnFaj^l7%Fig_l7$$\"+%*[Kg 8Fabn$\"+!*\\A#3%FabnFaj^l7%F_h_l7$$\"+?kmz8Fabn$\"+;?i(3%FabnFaj^l7%F eh_l7$$\"+B#yuR\"Fabn$\"+n#[\\4%FabnFaj^l7%F[i_l7$$\"+ T9Fabn$\"+WUIDTFabnFaj^l7%F]j_l7$$\"+Ua`_9Fabn$\"+H_NPTFabnFaj^l7%Fcj_ l7$$\"+=/Ki9Fabn$\"+H-.]TFabnFaj^l7%Fij_l7$$\"+hwfq9Fabn$\"+'=mJ;%Fabn Faj^l7%F_[`l7$$\"+X4Ux9Fabn$\"+j=iwTFabnFaj^l7%Fe[`l7$$\"+%FabnFaj^l7%F[\\`l7$$\"+/\"Gp[\"Fabn$\"+#*[,/UFabnFaj^l7%Fa\\` l7$$\"+)*Hs*[\"Fabn$\"+a%[x@%FabnFaj^l7%Fg\\`l7$$\"+f$G%FabnFaj^l7%Fe^`l7$$\"+3yP#[\" Fabn$\"+=Qu&H%FabnFaj^l7%F[_`l7$$\"+H.fx9Fabn$\"+gyU2VFabnFaj^l7%Fa_`l 7$$\"+xr!>Z\"Fabn$\"+s`f=VFabnFaj^l7%Fg_`l7$$\"+R?Pl9Fabn$\"+_)*>HVFab nFaj^l7%F]``lFgi^lFaj^lFaf]lF^_r-F$6$7in7$Fif]l$!3s******Hl>RVFT7$F`i^ l$!3N+++_)*>HVFT7$F[i^l$!3))*****>P&f=VFT7$Ffh^l$!3y******fyU2VFT7$Fah ^l$!3/+++=Qu&H%FT7$F\\h^l$!3g*****f%>f$G%FT7$Fgg^l$!3k*****pnB5F%FT7$F bg^l$!3Q+++#z$4eUFT7$F]g^l$!3$******f8h[C%FT7$Fhf^l$!3N+++l%*QJUFT7$Fc f^l$!3l*****RX[x@%FT7$F^f^l$!3++++#*[,/UFT7$Fie^l$!3G+++PTF!>%FT7$Fde^ l$!3E+++j=iwTFT7$F_e^l$!3w*****f=mJ;%FT7$Fjd^l$!3S+++H-.]TFT7$Fed^l$!3 v******G_NPTFT7$F`d^l$!3i*****RC/`7%FT7$F[d^l$!3x******)\\mS6%FT7$Ffc^ l$!32+++3B'Q5%FT7$Fac^l$!3O+++n#[\\4%FT7$F\\c^l$!33+++;?i(3%FT7$Fgb^l$ !3G+++!*\\A#3%FT7$Fbb^l$!3o*****zFS\"zSFT7$F]b^l$!3!)******H1yySFT7$Fh a^l$!3C+++g%f:3%FT7$Fca^l$!3I+++**p$y3%FT7$F^a^l$!3K+++)pQy4%FT7$Fi`^l $!3i*******Ge:6%FT7$Fd`^l$!3:+++Y:oGTFT7$F_`^l$!33+++(yz&[TFT7$Fj_^l$! 3?+++l#)RqTFT7$Fe_^l$!33+++.)3K>%FT7$F`_^l$!3\"******>?eh@%FT7$F[_^l$! 3++++*>f&QUFT7$Ff^^l$!37+++b_\"*fUFT7$Fa^^l$!3))*****R.,*zUFT7$F\\^^l$ !3O+++q\\K)H%FT7$Fg]^l$!3(******R<\"4:VFT7$Fb]^l$!3x******)4o,L%FT7$F] ]^l$!36+++KacVVFT7$Fh\\^l$!37+++r!=`N%FT7$Fc\\^l$!3A+++vaZlVFT7$F^\\^l $!3;+++\"y%4uVFT7$Fi[^l$!3m*****RYO7Q%FT7$Fd[^l$!3K+++(ohpQ%FT7$F_[^l$ !3/+++J2L\"R%FT7$Fjj]l$!3A+++Q@S%R%FT7$Fej]l$!3G+++*GKiR%FT7$F`j]l$!3; +++k_(oR%FT7$F[j]l$!39+++#)GQ'R%FT7$Ffi]l$!3g*****f$[![R%FT7$Fai]l$!3e *****p*))=#R%FT7$F\\i]l$!3U+++_6e)Q%FT7$Fgh]l$!3s******fi-%Q%FT7$Fbh]l $!3e*****4sn&yVFT7$F]h]l$!3s*****z9[AP%FT7$Fhg]l$!39+++X&4^O%FT7$Fcg]l $!37+++'f$>dVFT7$F^g]l$!3W+++]>a[VFTFf``lFfgy-F__n6jn7%7$Fhi^l$!+Il>RV Fabn7$F^``l$!+_)*>HVFabn7$Fbj^l$!$B%F]_n7%F`\\al7$Fh_`l$!+s`f=VFabnFc \\al7%Fg\\al7$Fb_`l$!+gyU2VFabnFc\\al7%F[]al7$F\\_`l$!+=Qu&H%FabnFc\\a l7%F_]al7$Ff^`l$!+Y>f$G%FabnFc\\al7%Fc]al7$F`^`l$!+xO-rUFabnFc\\al7%Fg ]al7$Fj]`l$!+#z$4eUFabnFc\\al7%F[^al7$Fd]`l$!+O6'[C%FabnFc\\al7%F_^al7 $F^]`l$!+l%*QJUFabnFc\\al7%Fc^al7$Fh\\`l$!+a%[x@%FabnFc\\al7%Fg^al7$Fb \\`l$!+#*[,/UFabnFc\\al7%F[_al7$F\\\\`l$!+PTF!>%FabnFc\\al7%F__al7$Ff[ `l$!+j=iwTFabnFc\\al7%Fc_al7$F`[`l$!+'=mJ;%FabnFc\\al7%Fg_al7$Fjj_l$!+ H-.]TFabnFc\\al7%F[`al7$Fdj_l$!+H_NPTFabnFc\\al7%F_`al7$F^j_l$!+WUIDTF abnFc\\al7%Fc`al7$Fhi_l$!+*\\mS6%FabnFc\\al7%Fg`al7$Fbi_l$!+3B'Q5%Fabn Fc\\al7%F[aal7$F\\i_l$!+n#[\\4%FabnFc\\al7%F_aal7$Ffh_l$!+;?i(3%FabnFc \\al7%Fcaal7$F`h_l$!+!*\\A#3%FabnFc\\al7%Fgaal7$Fjg_l$!+y-9zSFabnFc\\a l7%F[bal7$Fdg_l$!+I1yySFabnFc\\al7%F_bal7$F^g_l$!+g%f:3%FabnFc\\al7%Fc bal7$Fhf_l$!+**p$y3%FabnFc\\al7%Fgbal7$Fbf_l$!+)pQy4%FabnFc\\al7%F[cal 7$F\\f_l$!+!He:6%FabnFc\\al7%F_cal7$Ffe_l$!+Y:oGTFabnFc\\al7%Fccal7$F` e_l$!+(yz&[TFabnFc\\al7%Fgcal7$Fjd_l$!+l#)RqTFabnFc\\al7%F[dal7$Fdd_l$ !+.)3K>%FabnFc\\al7%F_dal7$F^d_l$!+-#eh@%FabnFc\\al7%Fcdal7$Fhc_l$!+*> f&QUFabnFc\\al7%Fgdal7$Fbc_l$!+b_\"*fUFabnFc\\al7%F[eal7$F\\c_l$!+M5!* zUFabnFc\\al7%F_eal7$Ffb_l$!+q\\K)H%FabnFc\\al7%Fceal7$F`b_l$!+u64:VFa bnFc\\al7%Fgeal7$Fja_l$!+*4o,L%FabnFc\\al7%F[fal7$Fda_l$!+KacVVFabnFc \\al7%F_fal7$F^a_l$!+r!=`N%FabnFc\\al7%Fcfal7$Fh`_l$!+vaZlVFabnFc\\al7 %Fgfal7$Fb`_l$!+\"y%4uVFabnFc\\al7%F[gal7$F\\`_l$!+kkB\"Q%FabnFc\\al7% F_gal7$Ff__l$!+(ohpQ%FabnFc\\al7%Fcgal7$F`__l$!+J2L\"R%FabnFc\\al7%Fgg al7$Fj^_l$!+Q@S%R%FabnFc\\al7%F[hal7$Fd^_l$!+*GKiR%FabnFc\\al7%F_hal7$ F^^_l$!+k_(oR%FabnFc\\al7%Fchal7$Fh]_l$!+#)GQ'R%FabnFc\\al7%Fghal7$Fb] _l$!+O[![R%FabnFc\\al7%F[ial7$F\\]_l$!+(*))=#R%FabnFc\\al7%F_ial7$Ff\\ _l$!+_6e)Q%FabnFc\\al7%Fcial7$F`\\_l$!+gi-%Q%FabnFc\\al7%Fgial7$Fj[_l$ !+@xcyVFabnFc\\al7%F[jal7$Fd[_l$!+[\"[AP%FabnFc\\al7%F_jal7$F^[_l$!+X& 4^O%FabnFc\\al7%Fcjal7$Fhj^l$!+'f$>dVFabnFc\\al7%Fgjal7$F]j^l$!+]>a[VF abnFc\\al7%F[[blF]\\alFc\\alFaf]lF^_r-%*AXESSTYLEG6#%$BOXG-%+AXESLABEL SG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fi [bl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%F]_n$\"$*>F]_n;Fh\\bl$\"$z%F]_n " 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "th e stability region for the 7 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R ^ for the 7 stage, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b^`,StabilityFunction(4,7,'expanded'))):\n`R^` := unapply(%,z ):\n'`R^`(z)'=`R^`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R^G6#%\" zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)* &#F)\"#CF)*$)F'\"\"%F)F)F)*&#\")V\"z\\\"\"+g2&4!=F)*$)F'\"\"&F)F)F)*&# \",td1*\\7\".!GX`3v))F)*$)F'F1F)F)F)*&#\"$p#\"(S)z8F)*$)F'\"\"(F)F)F) " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We \+ can find the point where the boundary of the stability region intersec ts the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R^`(z) = -1;" "6#/-%#R^G6#%\"zG,$\" \"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R^`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$z_0G$!+wJFHS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "zo0 := newton(`R^`(z)=-1,z=-4):\npo1 := \+ plot([`R^`(z),-1],z=-4.49..0.49,color=[red,blue]):\npo2 := plot([[[zo0 ,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\npo3 \+ := plot([[zo0,0],[zo0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplot s[display]([po1,po2,po3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETIC A,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CUR VESG6$7U7$$!3A++++++!\\%!#<$!3upjYCvmsBF*7$$!3;+]7'3DdV%F*$!33#G/'R=[b @F*7$$!35++Ds,X\"Q%F*$!3g+EX!>Ya&>F*7$$!3(*\\P*)feAMVF*$!3gc;8Z>W%z\"F *7$$!3%)*\\Pva,qG%F*$!3%y!=%F*$!3At5@P+JY8F*7 $$!35+]#oviQ2%F*$!3[F&oK)49$4\"F*7$$!3A+ve,&\\u'RF*$!3E'p,b7H!=))!#=7$ $!3?+v8#z!zoQF*$!3o92O,2YqrFP7$$!3;+v.8ajmPF*$!3b!3/)GG9OdFP7$$!3C+v$Q <')4m$F*$!3*)4_](pq9]%FP7$$!3G+v3`dnbNF*$!3%[RyIgMT[$FP7$$!37++0T5NZMF *$!3)RFQ09FVi#FP7$$!3?+]sv&Q>N$F*$!3mFw#oWr%)*>FP7$$!3E++:Qn_WKF*$!3D& *oX!RpjT\"FP7$$!3#****\\s&QnOJF*$!3q!G31m]XR*!#>7$$!3&****\\=iPF.$F*$! 3MBd8WoXDcFcp7$$!3Q+v$Gc`$QHF*$!3iYn=p#)>qFFcp7$$!3+++l(y@h#GF*$\"3Ol! *R`$R:\"o!#@7$$!3=++g_n/JFF*$\"31e76iiw-@Fcp7$$!3C+vt%)=X?EF*$\"3up'=X >i.:%Fcp7$$!3D++5(ocD_#F*$\"3E![iU#G([v&Fcp7$$!3#**\\P(R,::CF*$\"3Y9:r zF#eO(Fcp7$$!3&)*\\7sqtGJ#F*$\"31WYux49:))Fcp7$$!30+]Pz*eh?#F*$\"3$*)o zTX)[H5FP7$$!3l*\\([Y:;3@F*$\"3uV0^zHkm6FP7$$!3$***\\7w!eC+#F*$\"3!=)p c%z;*>8FP7$$!3%**\\Py(=m#*=F*$\"3!)[bNa')*))[\"FP7$$!3%)*\\iW'R3(z\"F* $\"31-V0LN.Z;FP7$$!3-+]d*>dQp\"F*$\"3WPbDY**=K=FP7$$!3)*****p]Q@(e\"F* $\"3`Rma'pq?/#FP7$$!3I+]FRT)G[\"F*$\"3Ez!\\v+7'oAFP7$$!3')*\\P*y(R>Q\" F*$\"3%pD;+a[0^#FP7$$!3++]Kx#e)p7F*$\"3!*pf2!Hu(3GFP7$$!3=++5j![\"p6F* $\"35L\\!Q[)\\1JFP7$$!3s***\\KT=;1\"F*$\"3/;7&f[W\"fMFP7$$!3'***\\P%o0 =k*FP$\"32)*fY7U58QFP7$$!3](***\\tEbw&)FP$\"3D-rl.;lTUFP7$$!39-]P2EBuv FP$\"3ixGX)3(y)o%FP7$$!3>)*\\(GL>l_'FP$\"3opo%**))fm?&FP7$$!3z++]-\")= -bFP$\"3#f9!3OyCodFP7$$!3$*)*\\PM#3)HWFP$\"3vj=sm\"*>@kFP7$$!3;(****f' *ypR$FP$\"3:3s7K[&)>rFP7$$!3L)**\\U'=wSBFP$\"3![H\"3:a,8zFP7$$!3?$*\\P t2H$H\"FP$\"3mvF!RjZoy)FP7$$!33R****pit2LFcp$\"3]`k*36PYn*FP7$$\"3Z^+] FkzBxFcp$\"3O&ek$=\"*H!3\"F*7$$\"35*****z2`!f\"F*7$$ \"3l+]i_F06GFP$\"3)y!3aBIfC8F*7$$\"3\\.]Pt1&z\"QFP$\"3/^*y3w6\\Y\"F*7$ $\"3!***************[FP$\"3(p8Rq#fJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\" )$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$ FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoF a\\l7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqF a\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tF a\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`v Fa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$Fcx Fa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$Ff zFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3B+++w JFHSF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG 6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDG F_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*L INESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl 6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F` cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture sho ws the stability region for the 8 stage, order 4 scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "subs(ee,subs(b=`b^`,StabilityFunct ion(4,7,'expanded'))):\nnewton(`R^`(z)=-1,z=-4);\n`R^` := unapply(%,z) :\n'`R^`(z)'=`R^`(z);\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R^`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts : = [op(pts),[Re(zz),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.15,0, .15),style=point);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1488 "`R^` := \+ z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+14979143/1800950760*z^5+\n 12499 065773/8875085345280*z^6+269/1379840*z^7:\npts := []: z0 := 0: tt := 0 : \nwhile tt<=201/20 do\n zz := newton(`R^`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13/20<=tt and tt<=6/5) or (87/10<=tt and tt<=93/1 0) then\n hh := 1/40\n else \n hh := 1/20\n end if;\n \+ tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo1 := pl ot(pts,color=COLOR(RGB,.3,.13,.3)):\npo2 := plots[polygonplot]([seq([p ts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid, color=COLOR(RGB,.6,.25,.6)):\npts := []:\nz0 := 1.5+4.3*I:\nfor ct fro m 0 to 60 do\n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo3 := plot(pts,c olor=COLOR(RGB,.3,.13,.3)):\npo4 := plots[polygonplot]([seq([pts[i-1], pts[i],[1.38,4.18]],i=2..nops(pts))],\n style=patchnogrid,col or=COLOR(RGB,.6,.25,.6)):\npts := []:\nz0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz :\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo5 := plot(pts,colo r=COLOR(RGB,.3,.13,.3)):\npo6 := plots[polygonplot]([seq([pts[i-1],pts [i],[1.38,-4.18]],i=2..nops(pts))],\n style=patchnogrid,color =COLOR(RGB,.6,.25,.6)):\npo7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79], [0,4.79]]],color=black,linestyle=3):\nplots[display]([po||(1..7)],view =[-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n labels=[ `Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 586 586 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F(7$$\"3 ;+++s\"HON&!#F$\"3/+++Tjzq:!#=7$$\"3!)******[+t#[$!#D$\"30+++WIfTJF07$ $\"3C+++3+J_S!#C$\"35+++=4R7ZF07$$\"3$)*****>Sm)=B!#B$\"3=+++mm=$G'F07 $$\"3I+++.E;%*))F@$\"3-+++,c&R&yF07$$\"3&)*****4\"G/0E!#A$\"3^+++Y)4YU *F07$$\"3))*****\\kP#)='FK$\"31+++Z<\\*4\"!#<7$$\"3,+++%4%\\<7!#@$\"3% ******4.PkD\"FS7$$\"3))*****4\\#\\Z>FW$\"3#******>(**>89FS7$$\"3/+++8! GAE#FW$\"3'******RT;'p:FS7$$\"3Y+++*RtE=(FK$\"3(*******)yOas\"FS7$$!3k ******>K[ncFW$\"3-+++i+K!)=FS7$$!3'******4.@c@#!#?$\"3-+++-s%Q.#FS7$$! 3;+++!Gj@m$F[p$\"3$)*****fFd*4@FS7$$!3%*******)H'p$p&F[p$\"3')*****\\8 ^b=#FS7$$!3C+++%RJRY)F[p$\"3!******z)pcgAFS7$$!3$******f!\\l97!#>$\"3= +++$*3%\\L#FS7$$!3;+++p*QLp\"F`q$\"3*******RF/'3CFS7$$!36+++r<:.BF`q$ \"3))*****\\yz9[#FS7$$!3!******4V%)e1$F`q$\"3%******z^vMb#FS7$$!3,+++/ .y.SF`q$\"3++++iyZCEFS7$$!3L+++W@kQ^F`q$\"3')*****\\\"QM%p#FS7$$!33+++ ?sh!\\'F`q$\"3#)******=T*Gw#FS7$$!3-+++#yUm2)F`q$\"3)*******=$4*HGFS7$ $!3%******fvO'3**F`q$\"3?+++\"HJ^*GFS7$$!3'******zan\"*>\"F0$\"3'***** *Hcs#eHFS7$$!3(******H;cAV\"F0$\"3(*******pV.>IFS7$$!3/+++2]#*)o\"F0$ \"3%)******y;8xIFS7$$!3)******fxYr'>F0$\"3#*******\\/KKJFS7$$!3')***** f1`UE#F0$\"3;+++q1U%=$FS7$$!3/+++:r>xDF0$\"3()*****HjHLB$FS7$$!3;+++\" QNG!HF0$\"35+++kK-zKFS7$$!3F+++&4p\"QKF0$\"3>+++pqa@LFS7$$!3?+++&460e$ F0$\"3)******pz+5O$FS7$$!3:+++\"*=dFRF0$\"3')*****\\?@vR$FS7$$!3u***** f6zuF%F0$\"36+++LiEJMFS7$$!3s******)\\a-)\\F0$\"3++++G^4\"\\$FS7$$!3s* ****H#*p2o&F0$\"38+++ypxTNFS7$$!3c*****>V%oujF0$\"3'******Hf_We$FS7$$! 3]+++jW$*fqF0$\"37+++\"Qq+i$FS7$$!3]+++S.vNxF0$\"3y******pmR\\OFS7$$!3 v*****pYc?S)F0$\"3%)*****HbVIn$FS7$$!3e******4l7f!*F0$\"3!******\\L)\\ \"p$FS7$$!3I+++%e$R2(*F0$\"3<+++$4\\^q$FS7$$!35+++,^tM5FS$\"3$******\\ A0Vr$FS7$$!3++++@0&z4\"FS$\"3&******4W8#>PFS7$$!3-+++jeVg6FS$\"3=+++[+ 2?PFS7$$!31+++M5CA7FS$\"3,+++%RIqr$FS7$$!35+++anT$G\"FS$\"33+++%=;-r$F S7$$!3*******HY;SM\"FS$\"35+++s7s*p$FS7$$!3!******H!))4/9FS$\"35+++?kh &o$FS7$$!3++++Z1tj9FS$\"3%)*****HZ`zm$FS7$$!36+++e1*H_\"FS$\"3/+++x$pn k$FS7$$!31+++*Qt>e\"FS$\"3()*****R_!4AOFS7$$!3-+++lQzS;FS$\"31+++tz$Rf $FS7$$!33+++OCf*p\"FS$\"39+++]ULiNFS7$$!3)******fRR&eFS$\"3\"******z^@DS$FS7$$!3:+++@2M+?FS$\"3') *****p(>*[N$FS7$$!3!)******oMgj?FS$\"35+++aDt/LFS7$$!3\")*****pVa%G@FS $\"3')*****pT%\\_KFS7$$!3++++Rp(\\>#FS$\"3!******fkB()>$FS7$$!31+++1A4 jAFS$\"3++++A`,WJFS7$$!3-+++R'QDL#FS$\"3?+++[:%*)3$FS7$$!3z*****\\S#*G S#FS$\"3/+++9P(R.$FS7$$!3/+++=gitCFS$\"33+++9iUzHFS7$$!3++++LN=WDFS$\" 3\")******=#Ra#HFS7$$!3$*******om/9EFS$\"3'******zy)*>(GFS7$$!37+++)Rw Fo#FS$\"37+++L&y*=GFS7$$!3-+++Q&G+v#FS$\"3;+++C^=mFFS7$$!3!******pb]b \"GFS$\"37+++WiR8FFS7$$!37+++b&o\"zGFS$\"33+++aqQgEFS7$$!3/+++R7xSHFS$ \"3(******R-Ypg#FS7$$!3*)*****p\"[H+IFS$\"3y******>G)Gb#FS7$$!3%****** HW5x0$FS$\"3'******4/J!)\\#FS7$$!3.+++yX,8JFS$\"3y*****\\*zCUCFS7$$!3/ +++q>AmJFS$\"3=+++%*HT&Q#FS7$$!3++++\"ygt@$FS$\"3********3^UFBFS7$$!3; +++e$pkE$FS$\"3)******4$4?oAFS7$$!32+++.Uf8LFS$\"3$)*****\\isw?#FS7$$! 33+++%z)yeLFS$\"3#)*****pX'yX@FS7$$!3A+++bH6-MFS$\"3'******4&=]#3#FS7$ $!3(*******>DjVMFS$\"33+++85zM[$FS$\"31+++G\"R;&>FS7$$ !3'******p^\\:_$FS$\"35+++u]/%)=FS7$$!3/+++N`5eNFS$\"3%******zeA]\"=FS 7$$!3)******\\?rJf$FS$\"3********G?PFS$\"35+++rI=]9FS7$$!3'******Rm.!\\PFS$\"3-++ +-_(QP\"FS7$$!37+++$=?nx$FS$\"3!******zqQnH\"FS7$$!3(*******y*eL!QFS$ \"3(******fF?*=7FS7$$!32+++D]!*GQFS$\"3!******p9p09\"FS7$$!33+++?$=L&Q FS$\"31+++)HI=1\"FS7$$!3=+++4i`wQFS$\"3v*****Hqz$G)*F07$$!3?+++PvZ)*QF S$\"3$*******RN5P!*F07$$!3-+++E![!>RFS$\"3C+++b)faC)F07$$!3z*****H5Y\" QRFS$\"3k******o0AauF07$$!3!*******o%oc&RFS$\"31+++9]%Rm'F07$$!33+++#* [^rRFS$\"3Y+++VZ*\\(eF07$$!33+++Z>f&)RFS$\"3q*****\\Rdv3&F07$$!3)***** *Rm:y*RFS$\"3\"******R\"\\n,VF07$$!3j*****4>8\"3SFS$\"3u******4BFF0 7$$!3A+++m=!p-%FS$\"3-+++:!y4<\"F07$$!38+++\\%4!HSFS$\"3L+++H^q-RF`q7$ Fe^m$!3L+++H^q-RF`q7$F`^m$!3-+++:!y4<\"F07$F[^m$!3-+++R$z@&>F07$Ff]m$! 3y*****fU%=MFF07$Fa]m$!3u******4BFFS7$Fcfl$!33+++85zG)Gb#FS7$Fa cl$!3(******R-Ypg#FS7$F\\cl$!33+++aqQgEFS7$Fgbl$!37+++WiR8FFS7$Fbbl$!3 ;+++C^=mFFS7$F]bl$!37+++L&y*=GFS7$Fhal$!3'******zy)*>(GFS7$Fcal$!3\")* *****=#Ra#HFS7$F^al$!33+++9iUzHFS7$Fi`l$!3/+++9P(R.$FS7$Fd`l$!3?+++[:% *)3$FS7$F_`l$!3++++A`,WJFS7$Fj_l$!3!******fkB()>$FS7$Fe_l$!3')*****pT% \\_KFS7$F`_l$!35+++aDt/LFS7$F[_l$!3')*****p(>*[N$FS7$Ff^l$!3\"******z^ @DS$FS7$Fa^l$!31+++b,GZMFS7$F\\^l$!3')*****R=H*)[$FS7$Fg]l$!3A+++UDJFN FS7$Fb]l$!39+++]ULiNFS7$F]]l$!31+++tz$Rf$FS7$Fh\\l$!3()*****R_!4AOFS7$ Fc\\l$!3/+++x$pnk$FS7$F^\\l$!3%)*****HZ`zm$FS7$Fi[l$!35+++?kh&o$FS7$Fd [l$!35+++s7s*p$FS7$F_[l$!33+++%=;-r$FS7$Fjz$!3,+++%RIqr$FS7$Fez$!3=+++ [+2?PFS7$F`z$!3&******4W8#>PFS7$F[z$!3$******\\A0Vr$FS7$Ffy$!3<+++$4\\ ^q$FS7$Fay$!3!******\\L)\\\"p$FS7$F\\y$!3%)*****HbVIn$FS7$Fgx$!3y***** *pmR\\OFS7$Fbx$!37+++\"Qq+i$FS7$F]x$!3'******Hf_We$FS7$Fhw$!38+++ypxTN FS7$Fcw$!3++++G^4\"\\$FS7$$!3#)*****>S_(GYF0$!3%********H-CY$FS7$F^w$! 36+++LiEJMFS7$Fiv$!3')*****\\?@vR$FS7$Fdv$!3)******pz+5O$FS7$F_v$!3>++ +pqa@LFS7$Fju$!35+++kK-zKFS7$Feu$!3()*****HjHLB$FS7$F`u$!3;+++q1U%=$FS 7$F[u$!3#*******\\/KKJFS7$Fft$!3%)******y;8xIFS7$Fat$!3(*******pV.>IFS 7$F\\t$!3'******Hcs#eHFS7$Fgs$!3?+++\"HJ^*GFS7$Fbs$!3)*******=$4*HGFS7 $F]s$!3#)******=T*Gw#FS7$Fhr$!3')*****\\\"QM%p#FS7$Fcr$!3++++iyZCEFS7$ F^r$!3%******z^vMb#FS7$Fiq$!3))*****\\yz9[#FS7$Fdq$!3*******RF/'3CFS7$ F^q$!3=+++$*3%\\L#FS7$Fip$!3!******z)pcgAFS7$Fdp$!3')*****\\8^b=#FS7$F _p$!3$)*****fFd*4@FS7$$!31+++6\"y6A\"F[p$!3!*******4;Gd>FS7$$!3******* z,aDg\"FW$!3*)*****fA?I!=FS7$$\"3-+++@_Rk=FW$!3'******RL?wk\"FS7$$\"33 +++WLA>AFW$!3'******z%\\Y\"\\\"FS7$$\"3#******zQsbe\"FW$!3&*******H(\\ [L\"FS7$$\"37+++_c6()))FK$!3%******H!)zz<\"FS7$$\"3p*****\\*4!H6%FK$!3 !******R.$)4-\"FS7$$\"39+++/lqm:FK$!35+++,pIR')F07$$\"3*******R@aYs%F@ $!39+++wxdoqF07$$\"3-+++'Q-A.\"F@$!3a+++j(*y(\\&F07$$\"3*)*****\\GDDM \"F:$!3********G=*p#RF07$$\"3))*****\\QO=9'!#E$!3++++$f%>cBF07$$\"3c** ***pTzsK)!#H$!3I+++R;)R&yF`q7$$\"3S*****HZzsK)Fden$\"3I+++R;)R&yF`q-%& COLORG6&%$RGBG$\"\"$!\"\"$\"#8!\"#F`fn-%)POLYGONSG6^y7%F'7$$\"+s\"HON& F`q$\"+Tjzq:!#57$$FefnF)F(7%Fjfn7$$\"+\\+t#[$FS$\"+WIfTJF_gnF`gn7%Fcgn 7$$\"+3+J_S!#;$\"+=4R7ZF_gnF`gn7%Fign7$$\"+-k')=B!#:$\"+mm=$G'F_gnF`gn 7%F`hn7$$\"+.E;%*))Fchn$\"+,c&R&yF_gnF`gn7%Fghn7$$\"+6G/0E!#9$\"+Y)4YU *F_gnF`gn7%F]in7$$\"+XwB)='F`in$\"+Z<\\*4\"!\"*F`gn7%Fdin7$$\"+%4%\\<7 !#8$\"+JqVc7FiinF`gn7%F[jn7$$\"+\"\\#\\Z>F^jn$\"+s**>89FiinF`gn7%Fbjn7 $$\"+8!GAE#F^jn$\"+9khp:FiinF`gn7%Fhjn7$$\"+*RtE=(F`in$\"+*yOas\"FiinF `gn7%F^[o7$$!+?K[ncF^jn$\"+i+K!)=FiinF`gn7%Fd[o7$$!+J5i:A!#7$\"+-s%Q.# FiinF`gn7%Fj[o7$$!+!Gj@m$F]\\o$\"+ws&*4@FiinF`gn7%Fa\\o7$$!+*H'p$p&F] \\o$\"+N6b&=#FiinF`gn7%Fg\\o7$$!+%RJRY)F]\\o$\"+))pcgAFiinF`gn7%F]]o7$ $!+1\\l97!#6$\"+$*3%\\L#FiinF`gn7%Fc]o7$$!+p*QLp\"Ff]o$\"+uUg3CFiinF`g n7%Fj]o7$$!+r<:.BFf]o$\"+&yz9[#FiinF`gn7%F`^o7$$!+JW)e1$Ff]o$\"+=bZ`DF iinF`gn7%Ff^o7$$!+/.y.SFf]o$\"+iyZCEFiinF`gn7%F\\_o7$$!+W@kQ^Ff]o$\"+: QM%p#FiinF`gn7%Fb_o7$$!+?sh!\\'Ff]o$\"+>T*Gw#FiinF`gn7%Fh_o7$$!+#yUm2) Ff]o$\"+>$4*HGFiinF`gn7%F^`o7$$!+cnj3**Ff]o$\"+\"HJ^*GFiinF`gn7%Fd`o7$ $!+[v;*>\"F_gn$\"+jDFeHFiinF`gn7%Fj`o7$$!+jhDK9F_gn$\"+qV.>IFiinF`gn7% F`ao7$$!+2]#*)o\"F_gn$\"+z;8xIFiinF`gn7%Ffao7$$!+wn9n>F_gn$\"+]/KKJFii nF`gn7%F\\bo7$$!+mIDkAF_gn$\"+q1U%=$FiinF`gn7%Fbbo7$$!+:r>xDF_gn$\"+L' HLB$FiinF`gn7%Fhbo7$$!+\"QNG!HF_gn$\"+kK-zKFiinF`gn7%F^co7$$!+&4p\"QKF _gn$\"+pqa@LFiinF`gn7%Fdco7$$!+&460e$F_gn$\"+(z+5O$FiinF`gn7%Fjco7$$!+ \"*=dFRF_gn$\"+07_(R$FiinF`gn7%F`do7$$!+;\"zuF%F_gn$\"+LiEJMFiinF`gn7% Ffdo7$$!+*\\a-)\\F_gn$\"+G^4\"\\$FiinF`gn7%F\\eo7$$!+B*p2o&F_gn$\"+ypx TNFiinF`gn7%Fbeo7$$!+KWoujF_gn$\"+$f_We$FiinF`gn7%Fheo7$$!+jW$*fqF_gn$ \"+\"Qq+i$FiinF`gn7%F^fo7$$!+S.vNxF_gn$\"+qmR\\OFiinF`gn7%Fdfo7$$!+nk0 -%)F_gn$\"+`N/tOFiinF`gn7%Fjfo7$$!+5l7f!*F_gn$\"+N$)\\\"p$FiinF`gn7%F` go7$$!+%e$R2(*F_gn$\"+$4\\^q$FiinF`gn7%Ffgo7$$!+,^tM5Fiin$\"+D_I9PFiin F`gn7%F\\ho7$$!+@0&z4\"Fiin$\"+TM@>PFiinF`gn7%Fbho7$$!+jeVg6Fiin$\"+[+ 2?PFiinF`gn7%Fhho7$$!+M5CA7Fiin$\"+%RIqr$FiinF`gn7%F^io7$$!+anT$G\"Fii n$\"+%=;-r$FiinF`gn7%Fdio7$$!+jk,W8Fiin$\"+s7s*p$FiinF`gn7%Fjio7$$!+.) )4/9Fiin$\"+?kh&o$FiinF`gn7%F`jo7$$!+Z1tj9Fiin$\"+tM&zm$FiinF`gn7%Ffjo 7$$!+e1*H_\"Fiin$\"+x$pnk$FiinF`gn7%F\\[p7$$!+*Qt>e\"Fiin$\"+C04AOFiin F`gn7%Fb[p7$$!+lQzS;Fiin$\"+tz$Rf$FiinF`gn7%Fh[p7$$!+OCf*p\"Fiin$\"+]U LiNFiinF`gn7%F^\\p7$$!+'RR&eFiin$\"+=:_-MFiinF`gn7%Ff]p7$$!+@2M+?Fiin$\"+x>*[N$FiinF`gn7%F \\^p7$$!+pMgj?Fiin$\"+aDt/LFiinF`gn7%Fb^p7$$!+PWXG@Fiin$\"+#Fiin$\"+YOs)>$FiinF`gn7%F^_p7$$!+1A4jAFiin$\"+A`, WJFiinF`gn7%Fd_p7$$!+R'QDL#Fiin$\"+[:%*)3$FiinF`gn7%Fj_p7$$!+0C*GS#Fii n$\"+9P(R.$FiinF`gn7%F``p7$$!+=gitCFiin$\"+9iUzHFiinF`gn7%Ff`p7$$!+LN= WDFiin$\"+>#Ra#HFiinF`gn7%F\\ap7$$!+pm/9EFiin$\"+)y)*>(GFiinF`gn7%Fbap 7$$!+)RwFo#Fiin$\"+L&y*=GFiinF`gn7%Fhap7$$!+Q&G+v#Fiin$\"+C^=mFFiinF`g n7%F^bp7$$!+d0b:GFiin$\"+WiR8FFiinF`gn7%Fdbp7$$!+b&o\"zGFiin$\"+aqQgEF iinF`gn7%Fjbp7$$!+R7xSHFiin$\"+Cg%pg#FiinF`gn7%F`cp7$$!+<[H+IFiin$\"+? G)Gb#FiinF`gn7%Ffcp7$$!+V/rdIFiin$\"+T5.)\\#FiinF`gn7%F\\dp7$$!+yX,8JF iin$\"+&*zCUCFiinF`gn7%Fbdp7$$!+q>AmJFiin$\"+%*HT&Q#FiinF`gn7%Fhdp7$$! +\"ygt@$Fiin$\"+4^UFBFiinF`gn7%F^ep7$$!+e$pkE$Fiin$\"+J4?oAFiinF`gn7%F dep7$$!+.Uf8LFiin$\"+DEn2AFiinF`gn7%Fjep7$$!+%z)yeLFiin$\"+dkyX@FiinF` gn7%F`fp7$$!+bH6-MFiin$\"+^=]#3#FiinF`gn7%Fffp7$$!+?DjVMFiin$\"+85zM[$Fiin$\"+G\"R;&>FiinF`gn7%Fbgp7$$!+<&\\:_$Fiin $\"+u]/%)=FiinF`gn7%Fhgp7$$!+N`5eNFiin$\"+)eA]\"=FiinF`gn7%F^hp7$$!+07 <$f$Fiin$\"+HH$oi$Fiin$\"+9/$Gn\"FiinF`gn7%Fjh p7$$!+.W?PFiin$\"+rI=]9FiinF`gn7%F\\jp7$$!+kO+\\PFiin$\"+-_(Q P\"FiinF`gn7%Fbjp7$$!+$=?nx$Fiin$\"+3(QnH\"FiinF`gn7%Fhjp7$$!+z*eL!QFi in$\"+w-#*=7FiinF`gn7%F^[q7$$!+D]!*GQFiin$\"+Z\"p09\"FiinF`gn7%Fd[q7$$ !+?$=L&QFiin$\"+)HI=1\"FiinF`gn7%Fj[q7$$!+4i`wQFiin$\"+.(z$G)*F_gnF`gn 7%F`\\q7$$!+PvZ)*QFiin$\"+SN5P!*F_gnF`gn7%Ff\\q7$$!+E![!>RFiin$\"+b)fa C)F_gnF`gn7%F\\]q7$$!+.h9QRFiin$\"+p0AauF_gnF`gn7%Fb]q7$$!+p%oc&RFiin$ \"+9]%Rm'F_gnF`gn7%Fh]q7$$!+#*[^rRFiin$\"+VZ*\\(eF_gnF`gn7%F^^q7$$!+Z> f&)RFiin$\"+&Rdv3&F_gnF`gn7%Fd^q7$$!+kc\"y*RFiin$\"+9\\n,VF_gnF`gn7%Fj ^q7$$!+\">8\"3SFiin$\"+5BFF_gnF`gn7%F\\`q7$$!+m=!p-%Fiin$\"+: !y4<\"F_gnF`gn7%Fb`q7$$!+\\%4!HSFiin$\"+H^q-RFf]oF`gn7%Fh`q7$Fi`q$!+H^ q-RFf]oF`gn7%F^aq7$Fc`q$!+:!y4<\"F_gnF`gn7%Fbaq7$F]`q$!+R$z@&>F_gnF`gn 7%Ffaq7$Fg_q$!+EW=MFF_gnF`gn7%Fjaq7$Fa_q$!+5BFFiinF`gn7%Fbgq7$F]gp$!+85z(GFiinF`gn7%Fb[r7$F]ap$!+>#Ra#HFiinF`gn7%Ff[r7$Fg`p$!+9iUzHFiinF`g n7%Fj[r7$Fa`p$!+9P(R.$FiinF`gn7%F^\\r7$F[`p$!+[:%*)3$FiinF`gn7%Fb\\r7$ Fe_p$!+A`,WJFiinF`gn7%Ff\\r7$F__p$!+YOs)>$FiinF`gn7%Fj\\r7$Fi^p$!+*[N$FiinF`gn7%F f]r7$Fg]p$!+=:_-MFiinF`gn7%Fj]r7$Fa]p$!+b,GZMFiinF`gn7%F^^r7$F[]p$!+%= H*)[$FiinF`gn7%Fb^r7$Fe\\p$!+UDJFNFiinF`gn7%Ff^r7$F_\\p$!+]ULiNFiinF`g n7%Fj^r7$Fi[p$!+tz$Rf$FiinF`gn7%F^_r7$Fc[p$!+C04AOFiinF`gn7%Fb_r7$F][p $!+x$pnk$FiinF`gn7%Ff_r7$Fgjo$!+tM&zm$FiinF`gn7%Fj_r7$Fajo$!+?kh&o$Fii nF`gn7%F^`r7$F[jo$!+s7s*p$FiinF`gn7%Fb`r7$Feio$!+%=;-r$FiinF`gn7%Ff`r7 $F_io$!+%RIqr$FiinF`gn7%Fj`r7$Fiho$!+[+2?PFiinF`gn7%F^ar7$Fcho$!+TM@>P FiinF`gn7%Fbar7$F]ho$!+D_I9PFiinF`gn7%Ffar7$Fggo$!+$4\\^q$FiinF`gn7%Fj ar7$Fago$!+N$)\\\"p$FiinF`gn7%F^br7$F[go$!+`N/tOFiinF`gn7%Fbbr7$Fefo$! +qmR\\OFiinF`gn7%Ffbr7$F_fo$!+\"Qq+i$FiinF`gn7%Fjbr7$Fieo$!+$f_We$Fiin F`gn7%F^cr7$Fceo$!+ypxTNFiinF`gn7%Fbcr7$F]eo$!+G^4\"\\$FiinF`gn7%Ffcr7 $$!+-CvGYF_gn$!++BSiMFiinF`gn7%Fjcr7$Fgdo$!+LiEJMFiinF`gn7%F`dr7$Fado$ !+07_(R$FiinF`gn7%Fddr7$F[do$!+(z+5O$FiinF`gn7%Fhdr7$Feco$!+pqa@LFiinF `gn7%F\\er7$F_co$!+kK-zKFiinF`gn7%F`er7$Fibo$!+L'HLB$FiinF`gn7%Fder7$F cbo$!+q1U%=$FiinF`gn7%Fher7$F]bo$!+]/KKJFiinF`gn7%F\\fr7$Fgao$!+z;8xIF iinF`gn7%F`fr7$Faao$!+qV.>IFiinF`gn7%Fdfr7$F[ao$!+jDFeHFiinF`gn7%Fhfr7 $Fe`o$!+\"HJ^*GFiinF`gn7%F\\gr7$F_`o$!+>$4*HGFiinF`gn7%F`gr7$Fi_o$!+>T *Gw#FiinF`gn7%Fdgr7$Fc_o$!+:QM%p#FiinF`gn7%Fhgr7$F]_o$!+iyZCEFiinF`gn7 %F\\hr7$Fg^o$!+=bZ`DFiinF`gn7%F`hr7$Fa^o$!+&yz9[#FiinF`gn7%Fdhr7$F[^o$ !+uUg3CFiinF`gn7%Fhhr7$Fd]o$!+$*3%\\L#FiinF`gn7%F\\ir7$F^]o$!+))pcgAFi inF`gn7%F`ir7$Fh\\o$!+N6b&=#FiinF`gn7%Fdir7$Fb\\o$!+ws&*4@FiinF`gn7%Fh ir7$$!+6\"y6A\"F]\\o$!+5;Gd>FiinF`gn7%F\\jr7$$!+=Sb-;F^jn$!+E--.=FiinF `gn7%Fbjr7$$\"+@_Rk=F^jn$!+M.iZ;FiinF`gn7%Fhjr7$$\"+WLA>AF^jn$!+[\\Y\" \\\"FiinF`gn7%F^[s7$$\"+)Qsbe\"F^jn$!+I(\\[L\"FiinF`gn7%Fd[s7$$\"+_c6( )))F`in$!+.)zz<\"FiinF`gn7%Fj[s7$$\"+&*4!H6%F`in$!+MI)4-\"FiinF`gn7%F` \\s7$$\"+/lqm:F`in$!+,pIR')F_gnF`gn7%Ff\\s7$$\"+9UlCZFchn$!+wxdoqF_gnF `gn7%F\\]s7$$\"+'Q-A.\"Fchn$!+j(*y(\\&F_gnF`gn7%Fb]s7$$\"+&GDDM\"F\\hn $!+H=*p#RF_gnF`gn7%Fh]s7$$\"+&QO=9'F0$!+$f%>cBF_gnF`gn7%F^^s7$$\"+<%zs K)FW$!+R;)R&yFf]oF`gn7%Fd^s7$$\"+t%zsK)FW$\"+R;)R&yFf]oF`gn-F]fn6&F_fn $\"\"'Fbfn$\"#DFefnFa_s-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3-+++x-\" o\\\"FS$\"3Q+++^=S!H%FS7$$\"3++++2.j)[\"FS$\"3i*****R<()*)H%FS7$$\"3'* *****fJt(z9FS$\"3f*****4!\\(oI%FS7$$\"3/+++xCGq9FS$\"30+++[O-9VFS7$$\" 3!******\\s,-Y\"FS$\"3<+++CBR?VFS7$$\"31+++(3w&\\9FS$\"3T+++k)RfK%FS7$ $\"3,+++`?XQ9FS$\"3m******e\\iIVFS7$$\"31+++(*z(oU\"FS$\"3u******>eSMV FS7$$\"3*)*****Hn/\\T\"FS$\"3u*****z$*RsL%FS7$$\"36+++)y&e-9FS$\"3&*** ***R(Q3RVFS7$$\"3)******\\oy**Q\"FS$\"3s*****41$*)RVFS7$$\"35+++[^9x8F S$\"3;+++W;iRVFS7$$\"31+++wB:k8FS$\"3%******ROA#QVFS7$$\"31+++bT2^8FS$ \"3e*****\\_YcL%FS7$$\"34+++uA*zL\"FS$\"38+++jS%=L%FS7$$\"3-+++^#)*\\K \"FS$\"3&)*****p!QwEVFS7$$\"36+++Ga>78FS$\"3O+++/SN?VFS7$$\"33+++6:q*H \"FS$\"3V+++`Kc7VFS7$$\"3'*******G;l(G\"FS$\"3$)*****\\1ULI%FS7$$\"3#* *****>%>?w7FS$\"3k*****fLXEH%FS7$$\"3/+++;Q`l7FS$\"3I+++)>O/G%FS7$$\"3 %******fPeeD\"FS$\"3i*****f%=pmUFS7$$\"35+++K5UZ7FS$\"3M+++&>79D%FS7$$ \"3.+++)*\\]S7FS$\"3%)******oBjMUFS7$$\"3++++2?VN7FS$\"3<+++$3Sk@%FS7$ $\"35+++[qbK7FS$\"3;+++M%)*p>%FS7$$\"32+++MCDK7FS$\"3#)*****p.tl<%FS7$ $\"3'******zSv[B\"FS$\"3k*****p!*eb:%FS7$$\"3\"*******[^rS7FS$\"3j**** *4(o\\MTFS7$$\"3-+++\"3A*\\7FS$\"3#******RijS6%FS7$$\"3%******p;OCE\"F S$\"3'******H))>]4%FS7$$\"3'******p]WzF\"FS$\"3K+++n\\6ySFS7$$\"3!**** **4P(*eH\"FS$\"3c*****p(>(R1%FS7$$\"3$*******Hne:8FS$\"3)******zu'*H0% FS7$$\"3(******zjdiL\"FS$\"3k*****p/X`/%FS7$$\"3%******>?4sN\"FS$\"3F+ ++f/&4/%FS7$$\"31+++6p&yP\"FS$\"3F+++$)yeRSFS7$$\"3!******RL`xR\"FS$\" 3r*****\\XU4/%FS7$$\"3/+++E*zlT\"FS$\"3x*****43lY/%FS7$$\"3$******4:DT V\"FS$\"3I+++_\"4/0%FS7$$\"3'******\\pf-X\"FS$\"3O+++O4&y0%FS7$$\"3!** ****HD9\\Y\"FS$\"3G+++w&)pmSFS7$$\"3)******Hvh!y9FS$\"3C+++4UpwSFS7$$ \"3%******4,/(*[\"FS$\"3C+++KCh(3%FS7$$\"31+++0@')*\\\"FS$\"3!*******) 4d#*4%FS7$$\"32+++'fp&3:FS$\"3o*****4gd96%FS7$$\"33+++Sy'e^\"FS$\"3=++ +(>lS7%FS7$$\"3!********z-=_\"FS$\"3\"*******p'\\p8%FS7$$\"3.+++#*GUE: FS$\"37+++,k**\\TFS7$$\"3(******>dx(H:FS$\"3u*****4'Q5jTFS7$$\"3++++=k \">`\"FS$\"37+++*\\\"=wTFS7$$\"3)******Hc))G`\"FS$\"3I+++dz9*=%FS7$$\" 36+++jBuK:FS$\"3L+++'eH>?%FS7$$\"3*******HBD:`\"FS$\"3W+++\">fW@%FS7$$ \"3(******ff$GH:FS$\"3v*****R)\\nEUFS7$$\"3-+++2H1E:FS$\"3w*****Hp>&QU FS7$$\"3/+++Px!>_\"FS$\"3i*****p!)R*\\UFS7$$\"3/+++%*='o^\"FS$\"3Q+++H \\)3E%FS7$$\"3!******>do4^\"FS$\"3u******>sIrUFS7$$\"3%******z_qU]\"FS $\"3A+++14;\"G%FSF\\`sF\\fn-Fgfn6jn7%7$$\"+x-\"o\\\"Fiin$\"+^=S!H%Fiin 7$$\"+2.j)[\"Fiin$\"+ur)*)H%Fiin7$$\"$Q\"Fefn$\"$=%Fefn7%F`ct7$$\"+;Lx z9Fiin$\"+,\\(oI%FiinFect7%F[dt7$$\"+xCGq9Fiin$\"+[O-9VFiinFect7%Fadt7 $$\"+D78F iin$\"+/SN?VFiinFect7%F_it7$$\"+6:q*H\"Fiin$\"+`Kc7VFiinFect7%Feit7$$ \"+H;l(G\"Fiin$\"+l?M.VFiinFect7%F[jt7$$\"+U>?w7Fiin$\"+O`k#H%FiinFect 7%Fajt7$$\"+;Q`l7Fiin$\"+)>O/G%FiinFect7%Fgjt7$$\"+w$eeD\"Fiin$\"+Y=pm UFiinFect7%F][u7$$\"+K5UZ7Fiin$\"+&>79D%FiinFect7%Fc[u7$$\"+)*\\]S7Fii n$\"+pBjMUFiinFect7%Fi[u7$$\"+2?VN7Fiin$\"+$3Sk@%FiinFect7%F_\\u7$$\"+ [qbK7Fiin$\"+M%)*p>%FiinFect7%Fe\\u7$$\"+MCDK7Fiin$\"+PIdwTFiinFect7%F []u7$$\"+3a([B\"Fiin$\"+2*eb:%FiinFect7%Fa]u7$$\"+\\^rS7Fiin$\"+ro\\MT FiinFect7%Fg]u7$$\"+\"3A*\\7Fiin$\"+CO19TFiinFect7%F]^u7$$\"+nhVi7Fiin $\"+$))>]4%FiinFect7%Fc^u7$$\"+2X%zF\"Fiin$\"+n\\6ySFiinFect7%Fi^u7$$ \"+rt*eH\"Fiin$\"+x>(R1%FiinFect7%F__u7$$\"+Ine:8Fiin$\"+[n*H0%FiinFec t7%Fe_u7$$\"+QwDO8Fiin$\"+Z]MXSFiinFect7%F[`u7$$\"+-#4sN\"Fiin$\"+f/&4 /%FiinFect7%Fa`u7$$\"+6p&yP\"Fiin$\"+$)yeRSFiinFect7%Fg`u7$$\"+MLv(R\" Fiin$\"+bC%4/%FiinFect7%F]au7$$\"+E*zlT\"Fiin$\"+\"3lY/%FiinFect7%Fcau 7$$\"+^^7M9Fiin$\"+_\"4/0%FiinFect7%Fiau7$$\"+&pf-X\"Fiin$\"+O4&y0%Fii nFect7%F_bu7$$\"+`U\"\\Y\"Fiin$\"+w&)pmSFiinFect7%Febu7$$\"+`<1y9Fiin$ \"+4UpwSFiinFect7%F[cu7$$\"+6Sq*[\"Fiin$\"+KCh(3%FiinFect7%Facu7$$\"+0 @')*\\\"Fiin$\"+*4d#*4%FiinFect7%Fgcu7$$\"+'fp&3:Fiin$\"+,wX6TFiinFect 7%F]du7$$\"+Sy'e^\"Fiin$\"+(>lS7%FiinFect7%Fcdu7$$\"++G!=_\"Fiin$\"+q' \\p8%FiinFect7%Fidu7$$\"+#*GUE:Fiin$\"+,k**\\TFiinFect7%F_eu7$$\"+svxH :Fiin$\"+hQ5jTFiinFect7%Feeu7$$\"+=k\">`\"Fiin$\"+*\\\"=wTFiinFect7%F[ fu7$$\"+j&))G`\"Fiin$\"+dz9*=%FiinFect7%Fafu7$$\"+jBuK:Fiin$\"+'eH>?%F iinFect7%Fgfu7$$\"+L__J:Fiin$\"+\">fW@%FiinFect7%F]gu7$$\"+'f$GH:Fiin$ \"+%)\\nEUFiinFect7%Fcgu7$$\"+2H1E:Fiin$\"+$p>&QUFiinFect7%Figu7$$\"+P x!>_\"Fiin$\"+2)R*\\UFiinFect7%F_hu7$$\"+%*='o^\"Fiin$\"+H\\)3E%FiinFe ct7%Fehu7$$\"+s&o4^\"Fiin$\"+?sIrUFiinFect7%F[iu7$$\"+G0F/:Fiin$\"+14; \"G%FiinFect7%FaiuF[ctFectF__sFe_s-F$6$7in7$F]`s$!3Q+++^=S!H%FS7$Fdbt$ !3A+++14;\"G%FS7$F_bt$!3u******>sIrUFS7$Fjat$!3Q+++H\\)3E%FS7$Feat$!3i *****p!)R*\\UFS7$F`at$!3w*****Hp>&QUFS7$F[at$!3v*****R)\\nEUFS7$Ff`t$! 3W+++\">fW@%FS7$Fa`t$!3L+++'eH>?%FS7$F\\`t$!3I+++dz9*=%FS7$Fg_t$!37+++ *\\\"=wTFS7$Fb_t$!3u*****4'Q5jTFS7$F]_t$!37+++,k**\\TFS7$Fh^t$!3\"**** ***p'\\p8%FS7$Fc^t$!3=+++(>lS7%FS7$F^^t$!3o*****4gd96%FS7$Fi]t$!3!**** ***)4d#*4%FS7$Fd]t$!3C+++KCh(3%FS7$F_]t$!3C+++4UpwSFS7$Fj\\t$!3G+++w&) pmSFS7$Fe\\t$!3O+++O4&y0%FS7$F`\\t$!3I+++_\"4/0%FS7$F[\\t$!3x*****43lY /%FS7$Ff[t$!3r*****\\XU4/%FS7$Fa[t$!3F+++$)yeRSFS7$F\\[t$!3F+++f/&4/%F S7$Fgjs$!3k*****p/X`/%FS7$Fbjs$!3)******zu'*H0%FS7$F]js$!3c*****p(>(R1 %FS7$Fhis$!3K+++n\\6ySFS7$Fcis$!3'******H))>]4%FS7$F^is$!3#******RijS6 %FS7$Fihs$!3j*****4(o\\MTFS7$Fdhs$!3k*****p!*eb:%FS7$F_hs$!3#)*****p.t l<%FS7$Fjgs$!3;+++M%)*p>%FS7$Fegs$!3<+++$3Sk@%FS7$F`gs$!3%)******oBjMU FS7$F[gs$!3M+++&>79D%FS7$Fffs$!3i*****f%=pmUFS7$Fafs$!3I+++)>O/G%FS7$F \\fs$!3k*****fLXEH%FS7$Fges$!3$)*****\\1ULI%FS7$Fbes$!3V+++`Kc7VFS7$F] es$!3O+++/SN?VFS7$Fhds$!3&)*****p!QwEVFS7$Fcds$!38+++jS%=L%FS7$F^ds$!3 e*****\\_YcL%FS7$Fics$!3%******ROA#QVFS7$Fdcs$!3;+++W;iRVFS7$F_cs$!3s* ****41$*)RVFS7$Fjbs$!3&******R(Q3RVFS7$Febs$!3u*****z$*RsL%FS7$F`bs$!3 u******>eSMVFS7$F[bs$!3m******e\\iIVFS7$Ffas$!3T+++k)RfK%FS7$Faas$!3<+ ++CBR?VFS7$F\\as$!30+++[O-9VFS7$Fg`s$!3f*****4!\\(oI%FS7$Fb`s$!3i***** R<()*)H%FSFjiuF\\fn-Fgfn6jn7%7$F\\ct$!+^=S!H%Fiin7$Fbiu$!+14;\"G%Fiin7 $Ffct$!$=%Fefn7%Fdev7$F\\iu$!+?sIrUFiinFgev7%F[fv7$Ffhu$!+H\\)3E%FiinF gev7%F_fv7$F`hu$!+2)R*\\UFiinFgev7%Fcfv7$Fjgu$!+$p>&QUFiinFgev7%Fgfv7$ Fdgu$!+%)\\nEUFiinFgev7%F[gv7$F^gu$!+\">fW@%FiinFgev7%F_gv7$Fhfu$!+'eH >?%FiinFgev7%Fcgv7$Fbfu$!+dz9*=%FiinFgev7%Fggv7$F\\fu$!+*\\\"=wTFiinFg ev7%F[hv7$Ffeu$!+hQ5jTFiinFgev7%F_hv7$F`eu$!+,k**\\TFiinFgev7%Fchv7$Fj du$!+q'\\p8%FiinFgev7%Fghv7$Fddu$!+(>lS7%FiinFgev7%F[iv7$F^du$!+,wX6TF iinFgev7%F_iv7$Fhcu$!+*4d#*4%FiinFgev7%Fciv7$Fbcu$!+KCh(3%FiinFgev7%Fg iv7$F\\cu$!+4UpwSFiinFgev7%F[jv7$Ffbu$!+w&)pmSFiinFgev7%F_jv7$F`bu$!+O 4&y0%FiinFgev7%Fcjv7$Fjau$!+_\"4/0%FiinFgev7%Fgjv7$Fdau$!+\"3lY/%FiinF gev7%F[[w7$F^au$!+bC%4/%FiinFgev7%F_[w7$Fh`u$!+$)yeRSFiinFgev7%Fc[w7$F b`u$!+f/&4/%FiinFgev7%Fg[w7$F\\`u$!+Z]MXSFiinFgev7%F[\\w7$Ff_u$!+[n*H0 %FiinFgev7%F_\\w7$F`_u$!+x>(R1%FiinFgev7%Fc\\w7$Fj^u$!+n\\6ySFiinFgev7 %Fg\\w7$Fd^u$!+$))>]4%FiinFgev7%F[]w7$F^^u$!+CO19TFiinFgev7%F_]w7$Fh]u $!+ro\\MTFiinFgev7%Fc]w7$Fb]u$!+2*eb:%FiinFgev7%Fg]w7$F\\]u$!+PIdwTFii nFgev7%F[^w7$Ff\\u$!+M%)*p>%FiinFgev7%F_^w7$F`\\u$!+$3Sk@%FiinFgev7%Fc ^w7$Fj[u$!+pBjMUFiinFgev7%Fg^w7$Fd[u$!+&>79D%FiinFgev7%F[_w7$F^[u$!+Y= pmUFiinFgev7%F__w7$Fhjt$!+)>O/G%FiinFgev7%Fc_w7$Fbjt$!+O`k#H%FiinFgev7 %Fg_w7$F\\jt$!+l?M.VFiinFgev7%F[`w7$Ffit$!+`Kc7VFiinFgev7%F_`w7$F`it$! +/SN?VFiinFgev7%Fc`w7$Fjht$!+2QwEVFiinFgev7%Fg`w7$Fdht$!+jS%=L%FiinFge v7%F[aw7$F^ht$!+DlkNVFiinFgev7%F_aw7$Fhgt$!+kBAQVFiinFgev7%Fcaw7$Fbgt$ !+W;iRVFiinFgev7%Fgaw7$F\\gt$!+hI*)RVFiinFgev7%F[bw7$Ffft$!+uQ3RVFiinF gev7%F_bw7$F`ft$!+Q*RsL%FiinFgev7%Fcbw7$Fjet$!+?eSMVFiinFgev7%Fgbw7$Fd et$!+f\\iIVFiinFgev7%F[cw7$F^et$!+k)RfK%FiinFgev7%F_cw7$Fhdt$!+CBR?VFi inFgev7%Fccw7$Fbdt$!+[O-9VFiinFgev7%Fgcw7$F\\dt$!+,\\(oI%FiinFgev7%F[d w7$Fact$!+ur)*)H%FiinFgev7%F_dwFaevFgevF__sFe_s-F$6%7$7$$!3/++++++!z%F SF(7$$\"3+++++++!*>FSF(-%'COLOURG6&F_fnF)F)F)-%*LINESTYLEG6#Fafn-F$6%7 $7$F(Fgdw7$F($\"3/++++++!z%FSF\\ewF_ew-%*AXESSTYLEG6#%$BOXG-%+AXESLABE LSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-F cfw6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%Fefn$\"$*>Fefn;Fbgw$\"$z%Fefn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 44.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme \+ is (approximately) " }{XPPEDIT 18 0 "[-4.0293, 0];" "6#7$,$-%&FloatG6 $\"&$HS!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the s tability region for the 7 stage, order 4 scheme with that of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display ]([p||(1..7),po||(1..6)],view=[-4.79..1.99,-4.79..4.79],font=[HELVETIC A,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrai ned);" }}{PARA 13 "" 1 "" {GLPLOT2D 623 575 575 {PLOTDATA 2 "65-%'CURV ESG6$7ew7$$\"\"!F)F(7$$\"3b+++!f<*>f!#G$\"3++++Fjzq:!#=7$$\"3p******4$ 3Q5&!#E$\"3(******pi#fTJF07$$\"3=+++(*f-r!)!#D$\"3D+++/%)Q7ZF07$$\"3E+ ++P,gNg!#C$\"3S+++&yzJG'F07$$\"33+++%p%GlG!#B$\"3S+++g$[R&yF07$$\"3s** ***>)evd)*FF$\"3w*****\\jIYU*F07$$\"33+++y<$3i#!#A$\"3'******\\D0&*4\" !#<7$$\"3I+++6Y&R[&FQ$\"33+++h8[c7FT7$$\"3C+++XY2B')FQ$\"3!******4V6LT \"FT7$$\"3]*****>Ow'yuFQ$\"3++++iU&)p:FT7$$!3-+++'3\"pH5!#@$\"3!****** \\/()es\"FT7$$!3^+++odj+rFao$\"31+++jY4\")=FT7$$!3')*****\\^$>CA!#?$\" 3<+++[t2N?FT7$$!32+++\\v\\4aF\\p$\"3=+++upP(=#FT7$$!3++++O#)>O6!#>$\"3 6+++QP]PBFT7$$!3'******pt83:#Fgp$\"35+++yV#\\[#FT7$$!3!)*****4:(*Qv$Fg p$\"3/+++4w'*GEFT7$$!3<+++3,`ChFgp$\"3!******p0J'oFFT7$$!3Z*****>$4#*= %*Fgp$\"34+++RsN-HFT7$$!3#******\\$)Q=P\"F0$\"3A+++nS)z-$FT7$$!3)***** *4&R!y*=F0$\"3\")*****f&48VJFT7$$!3t******[l:-DF0$\"37+++/I(fC$FT7$$!3 ;+++h(G-;$F0$\"36+++-)[eL$FT7$$!3'******pW;&[QF0$\"3!******pAQKT$FT7$$ !35+++W,T\\XF0$\"3)******z^$HzMFT7$$!3S+++P]p^_F0$\"3<+++QZONNFT7$$!3W +++C6.\\fF0$\"3;+++:@s#e$FT7$$!3a+++.s$FT7$$!3** *****p$3=S**F0$\"3!******H[4@t$FT7$$!3/+++rqOd5FT$\"3#)*****zh<&RPFT7$ $!3++++mQ$*>6FT$\"3@+++\\+xUPFT7$$!3#******fjo<=\"FT$\"38+++2k/UPFT7$$ !36+++V4#HC\"FT$\"34+++l#)[PPFT7$$!3)******fGTMI\"FT$\"3/+++2e?HPFT7$$ !31+++hKQj8FT$\"3&)*****48$GMOFT7$$!3#*******[f*ol\"FT$\"3&******* z'=Zg$FT7$$!3%*******GS2:FT$\"3#******pq9*>$FT7$$!3/+++\")*\\YF#F T$\"3>+++x.zVJFT7$$!31+++\"y4OM#FT$\"3))*****RU-#)3$FT7$$!3z*****\\0RL T#FT$\"3.+++:(>G.$FT7$$!33+++8@H$[#FT$\"3'******f76z(HFT7$$!3=+++eY\"H b#FT$\"34+++vGFT$\"3@+++$)yl4FFT7$$!31+++Tmj\")GFT$\"3))******ey,cEFT7$$!3++++ g(>=%HFT$\"3*******za$)=g#FT7$$!3y******)R-***HFT$\"3++++=-2ZDFT7$$!31 +++;N'e0$FT$\"3))*****p:<9\\#FT7$$!3\")*****p>0(4JFT$\"32+++MoyMCFT7$$ !3#*******RjWhJFT$\"3=+++?H1xBFT7$$!3,+++]$=6@$FT$\"34+++lz9=BFT7$$!3+ +++rBweKFT$\"3)******\\?hzD#FT7$$!3#)*****\\hFWI$FT$\"3'******ptOk>#FT 7$$!3?+++j/<[LFT$\"39+++?A_L@FT7$$!3=+++\\S0!R$FT$\"3;+++U\"y\"p?FT7$$ !38+++x![,V$FT$\"3z*****4qxL+#FT7$$!3#******pjG&oMFT$\"3)******RG2h$>F T7$$!36+++'zx_]$FT$\"33+++6vOn=FT7$$!3*)******=F[SNFT$\"3-+++NZ<(z\"FT 7$$!3/+++!4MUd$FT$\"32+++VGcD\"FT7$$!3')******zM?,QFT$\"3'******fm$=76FT7$$!32+++\"38Z#QFT $\"34+++K!RC.\"FT7$$!3A+++*yVq%QFT$\"3z*****p8!)\\_*F07$$!3')*****\\)R 5oQFT$\"3b*****\\l!yC()F07$$!3=+++/+z()QFT$\"3Z*****RXVZ#zF07$$!3*)*** **zb#*f!RFT$\"3)******HE'eDrF07$$!38+++4EgARFT$\"3s*****Hl+yK'F07$$!3- +++/i^PRFT$\"34+++mKnJbF07$$!3++++u!Q1&RFT$\"3F+++1\\JPZF07$$!3'****** H/%)='RFT$\"3w*****4X#pWRF07$$!3))******>C=rRFT$\"37+++j)fO:$F07$$!37+ ++UYZyRFT$\"3!******R/')RO#F07$$!3++++Gar$)RFT$\"3'*******e&y`d\"F07$$ !3%******\\mso)RFT$\"3m*****p%>.vyFgp7$$!3/+++*>Fz)RFTF(7$Fjjl$!3m**** *p%>.vyFgp7$Fejl$!3'*******e&y`d\"F07$F`jl$!3!******R/')RO#F07$F[jl$!3 7+++j)fO:$F07$Ffil$!3w*****4X#pWRF07$Fail$!3F+++1\\JPZF07$F\\il$!34+++ mKnJbF07$Fghl$!3s*****Hl+yK'F07$Fbhl$!3)******HE'eDrF07$F]hl$!3Z*****R XVZ#zF07$Fhgl$!3b*****\\l!yC()F07$Fcgl$!3z*****p8!)\\_*F07$F^gl$!34+++ K!RC.\"FT7$Fifl$!3'******fm$=76FT7$Fdfl$!3'******zS&f\">\"FT7$F_fl$!3) )*****z)R_q7FT7$Fjel$!3'******pB7)[8FT7$Feel$!3/+++QOIE9FT7$F`el$!3*** *****)e\\G]\"FT7$F[el$!3/+++!\\:$y:FT7$Ffdl$!3/+++w]e_;FT7$Fadl$!32+++ VGcDFT7$F]cl$!3z*****4qxL+#FT7$Fhbl$!3;+++U\"y\"p?FT7$Fcbl$!39+++?A_L@FT 7$F^bl$!3'******ptOk>#FT7$Fial$!3)******\\?hzD#FT7$Fdal$!34+++lz9=BFT7 $F_al$!3=+++?H1xBFT7$Fj`l$!32+++MoyMCFT7$Fe`l$!3))*****p:<9\\#FT7$F``l $!3++++=-2ZDFT7$F[`l$!3*******za$)=g#FT7$Ff_l$!3))******ey,cEFT7$Fa_l$ !3@+++$)yl4FFT7$F\\_l$!35+++P2,jFFT7$Fg^l$!3$******4\")*H;GFT7$Fb^l$!3 )*******)[^(pGFT7$F]^l$!34+++vG.$FT7$F^]l$!3))*****RU-#)3$FT7$Fi\\l$!3>+++x.zVJFT7$Fd\\l$!3#)* *****>q9*>$FT7$F_\\l$!3#)*****Ri:PD$FT7$Fj[l$!38+++G_*oI$FT7$Fe[l$!3A+ ++\\!=\"eLFT7$F`[l$!3#******pMOFT7$F]y$!3)******fb8-m$FT7$Fhx$!3')*****f: \\Fo$FT7$Fcx$!39+++rEy,PFT7$F^x$!3&)*****48$G.s$FT7$Ffu$!3<+++$=j Qq$FT7$Fau$!3#)*****fJ%Q#o$FT7$F\\u$!3;+++XYVbOFT7$Fgt$!3()*****RD]Ci$ FT7$Fbt$!3;+++:@s#e$FT7$F]t$!3<+++QZONNFT7$Fhs$!3)******z^$HzMFT7$Fcs$ !3!******pAQKT$FT7$F^s$!36+++-)[eL$FT7$Fir$!37+++/I(fC$FT7$Fdr$!3\")** ***f&48VJFT7$F_r$!3A+++nS)z-$FT7$Fjq$!34+++RsN-HFT7$Feq$!3!******p0J'o FFT7$F`q$!3/+++4w'*GEFT7$F[q$!35+++yV#\\[#FT7$Fep$!36+++QP]PBFT7$F`p$! 3=+++upP(=#FT7$Fjo$!3<+++[t2N?FT7$Feo$!31+++jY4\")=FT7$F_o$!3!******\\ /()es\"FT7$$\"39+++kjnyuFQ$!3++++iU&)p:FT7$Fen$!3!******4V6LT\"FT7$FV$ !33+++h8[c7FT7$FO$!3'******\\D0&*4\"FT7$FJ$!3w*****\\jIYU*F07$FD$!3S++ +g$[R&yF07$$\"3^*****f8+c.'F@$!3S+++&yzJG'F07$$\"3s******pd-r!)F:$!3D+ ++/%)Q7ZF07$$\"3G+++9$3Q5&F4$!3(******pi#fTJF07$$\"3i*****>a4(>fF-$!3+ +++Fjzq:F0F'-%&COLORG6&%$RGBG$\"\"%!\"\"$\"#=!\"#Fh^n-%)POLYGONSG6fw7% F'7$$\"+!f<*>fF\\p$\"+Fjzq:!#57$$F]_nF)F(7%Fb_n7$$\"+5$3Q5&F0$\"+FEfTJ Fg_nFh_n7%F[`n7$$\"+(*f-r!)FT$\"+/%)Q7ZFg_nFh_n7%Fa`n7$$\"+P,gNg!#;$\" +&yzJG'Fg_nFh_n7%Fg`n7$$\"+%p%GlG!#:$\"+g$[R&yFg_nFh_n7%F^an7$$\"+#)ev d)*Faan$\"+N1jC%*Fg_nFh_n7%Fean7$$\"+y<$3i#!#9$\"+b_]*4\"!\"*Fh_n7%F[b n7$$\"+6Y&R[&F^bn$\"+h8[c7FabnFh_n7%Fcbn7$$\"+XY2B')F^bn$\"+J9J89FabnF h_n7%Fibn7$$\"+ijnyuF^bn$\"+iU&)p:FabnFh_n7%F_cn7$$!+'3\"pH5!#8$\"+Xq) es\"FabnFh_n7%Fecn7$$!+odj+rFhcn$\"+jY4\")=FabnFh_n7%F\\dn7$$!+:N>CA!# 7$\"+[t2N?FabnFh_n7%Fbdn7$$!+\\v\\4aFedn$\"+upP(=#FabnFh_n7%Fidn7$$!+O #)>O6!#6$\"+QP]PBFabnFh_n7%F_en7$$!+PP\"3:#Fben$\"+yV#\\[#FabnFh_n7%Ff en7$$!+^r*Qv$Fben$\"+4w'*GEFabnFh_n7%F\\fn7$$!+3,`ChFben$\"+d5joFFabnF h_n7%Fbfn7$$!+K4#*=%*Fben$\"+RsN-HFabnFh_n7%Fhfn7$$!+N)Q=P\"Fg_n$\"+nS )z-$FabnFh_n7%F^gn7$$!+^R!y*=Fg_n$\"+c48VJFabnFh_n7%Fdgn7$$!+\\l:-DFg_ n$\"+/I(fC$FabnFh_n7%Fjgn7$$!+h(G-;$Fg_n$\"+-)[eL$FabnFh_n7%F`hn7$$!+Z k^[QFg_n$\"+F#QKT$FabnFh_n7%Ffhn7$$!+W,T\\XFg_n$\"+=NHzMFabnFh_n7%F\\i n7$$!+P]p^_Fg_n$\"+QZONNFabnFh_n7%Fbin7$$!+C6.\\fFg_n$\"+:@s#e$FabnFh_ n7%Fhin7$$!+.s$FabnFh_n7%Ff[o7$$!+P3=S **Fg_n$\"+$[4@t$FabnFh_n7%F\\\\o7$$!+rqOd5Fabn$\"+=w^RPFabnFh_n7%Fb\\o 7$$!+mQ$*>6Fabn$\"+\\+xUPFabnFh_n7%Fh\\o7$$!+O'o<=\"Fabn$\"+2k/UPFabnF h_n7%F^]o7$$!+V4#HC\"Fabn$\"+l#)[PPFabnFh_n7%Fd]o7$$!+'GTMI\"Fabn$\"+2 e?HPFabnFh_n7%Fj]o7$$!+hKQj8Fabn$\"+JJGMOFabnFh_n7% Fh_o7$$!+\\f*ol\"Fabn$\"+!o=Zg$FabnFh_n7%F^`o7$$!+HS2:Fabn$\"+xT!pS$FabnFh_n7%F\\bo7$$!+cmH8?Fabn$\"+\\!=\"eLFabnFh_n7%Fb bo7$$!+q_?w?Fabn$\"+G_*oI$FabnFh_n7%Fhbo7$$!+p0tS@Fabn$\"+Ccr`KFabnFh_ n7%F^co7$$!+30#p?#Fabn$\"+?q9*>$FabnFh_n7%Fdco7$$!+\")*\\YF#Fabn$\"+x. zVJFabnFh_n7%Fjco7$$!+\"y4OM#Fabn$\"+CC?)3$FabnFh_n7%F`do7$$!+b!RLT#Fa bn$\"+:(>G.$FabnFh_n7%Ffdo7$$!+8@H$[#Fabn$\"+E6\"z(HFabnFh_n7%F\\eo7$$ !+eY\"Hb#Fabn$\"+vGFabn$\"+$)yl4FFabnFh_n7%Fjfo7$$!+Tmj\")GFabn$\"+ fy,cEFabnFh_n7%F`go7$$!+g(>=%HFabn$\"+[N)=g#FabnFh_n7%Ffgo7$$!+*R-***H Fabn$\"+=-2ZDFabnFh_n7%F\\ho7$$!+;N'e0$Fabn$\"+drT\"\\#FabnFh_n7%Fbho7 $$!+(>0(4JFabn$\"+MoyMCFabnFh_n7%Fhho7$$!+SjWhJFabn$\"+?H1xBFabnFh_n7% F^io7$$!+]$=6@$Fabn$\"+lz9=BFabnFh_n7%Fdio7$$!+rBweKFabn$\"+07'zD#Fabn Fh_n7%Fjio7$$!+:wU/LFabn$\"+PnV'>#FabnFh_n7%F`jo7$$!+j/<[LFabn$\"+?A_L @FabnFh_n7%Ffjo7$$!+\\S0!R$Fabn$\"+U\"y\"p?FabnFh_n7%F\\[p7$$!+x![,V$F abn$\"+,xP.?FabnFh_n7%Fb[p7$$!+P'G&oMFabn$\"+%G2h$>FabnFh_n7%Fh[p7$$!+ 'zx_]$Fabn$\"+6vOn=FabnFh_n7%F^\\p7$$!+>F[SNFabn$\"+NZ<(z\"FabnFh_n7%F d\\p7$$!+!4MUd$Fabn$\"+VGcD\"FabnFh_n7%Fd_p7$$!+![.7!QFabn$\"+mO=76FabnF h_n7%Fj_p7$$!+\"38Z#QFabn$\"+K!RC.\"FabnFh_n7%F``p7$$!+*yVq%QFabn$\"+P ,)\\_*Fg_nFh_n7%Ff`p7$$!+&)R5oQFabn$\"+b1yC()Fg_nFh_n7%F\\ap7$$!+/+z() QFabn$\"+aMuCzFg_nFh_n7%Fbap7$$!+eD*f!RFabn$\"+jieDrFg_nFh_n7%Fhap7$$! +4EgARFabn$\"+`1!yK'Fg_nFh_n7%F^bp7$$!+/i^PRFabn$\"+mKnJbFg_nFh_n7%Fdb p7$$!+u!Q1&RFabn$\"+1\\JPZFg_nFh_n7%Fjbp7$$!+VS)='RFabn$\"+^CpWRFg_nFh _n7%F`cp7$$!+?C=rRFabn$\"+j)fO:$Fg_nFh_n7%Ffcp7$$!+UYZyRFabn$\"+Wg)RO# Fg_nFh_n7%F\\dp7$$!+Gar$)RFabn$\"+f&y`d\"Fg_nFh_n7%Fbdp7$$!+lE(o)RFabn $\"+Z>.vyFbenFh_n7%Fhdp7$$!+*>Fz)RFabnF(Fh_n7%F^ep7$Fidp$!+Z>.vyFbenFh _n7%Fbep7$Fcdp$!+f&y`d\"Fg_nFh_n7%Ffep7$F]dp$!+Wg)RO#Fg_nFh_n7%Fjep7$F gcp$!+j)fO:$Fg_nFh_n7%F^fp7$Facp$!+^CpWRFg_nFh_n7%Fbfp7$F[cp$!+1\\JPZF g_nFh_n7%Fffp7$Febp$!+mKnJbFg_nFh_n7%Fjfp7$F_bp$!+`1!yK'Fg_nFh_n7%F^gp 7$Fiap$!+jieDrFg_nFh_n7%Fbgp7$Fcap$!+aMuCzFg_nFh_n7%Ffgp7$F]ap$!+b1yC( )Fg_nFh_n7%Fjgp7$Fg`p$!+P,)\\_*Fg_nFh_n7%F^hp7$Fa`p$!+K!RC.\"FabnFh_n7 %Fbhp7$F[`p$!+mO=76FabnFh_n7%Ffhp7$Fe_p$!+3af\">\"FabnFh_n7%Fjhp7$F__p $!+))R_q7FabnFh_n7%F^ip7$Fi^p$!+PA\")[8FabnFh_n7%Fbip7$Fc^p$!+QOIE9Fab nFh_n7%Ffip7$F]^p$!+*e\\G]\"FabnFh_n7%Fjip7$Fg]p$!+!\\:$y:FabnFh_n7%F^ jp7$Fa]p$!+w]e_;FabnFh_n7%Fbjp7$F[]p$!+VGcDFabnFh_ n7%Fb[q7$Fc[p$!+,xP.?FabnFh_n7%Ff[q7$F][p$!+U\"y\"p?FabnFh_n7%Fj[q7$Fg jo$!+?A_L@FabnFh_n7%F^\\q7$Fajo$!+PnV'>#FabnFh_n7%Fb\\q7$F[jo$!+07'zD# FabnFh_n7%Ff\\q7$Feio$!+lz9=BFabnFh_n7%Fj\\q7$F_io$!+?H1xBFabnFh_n7%F^ ]q7$Fiho$!+MoyMCFabnFh_n7%Fb]q7$Fcho$!+drT\"\\#FabnFh_n7%Ff]q7$F]ho$!+ =-2ZDFabnFh_n7%Fj]q7$Fggo$!+[N)=g#FabnFh_n7%F^^q7$Fago$!+fy,cEFabnFh_n 7%Fb^q7$F[go$!+$)yl4FFabnFh_n7%Ff^q7$Fefo$!+P2,jFFabnFh_n7%Fj^q7$F_fo$ !+6)*H;GFabnFh_n7%F^_q7$Fieo$!+*[^(pGFabnFh_n7%Fb_q7$Fceo$!+vG.$FabnFh_n7%F^`q7$ Fado$!+CC?)3$FabnFh_n7%Fb`q7$F[do$!+x.zVJFabnFh_n7%Ff`q7$Feco$!+?q9*>$ FabnFh_n7%Fj`q7$F_co$!+Ccr`KFabnFh_n7%F^aq7$Fibo$!+G_*oI$FabnFh_n7%Fba q7$Fcbo$!+\\!=\"eLFabnFh_n7%Ffaq7$F]bo$!+xT!pS$FabnFh_n7%Fjaq7$Fgao$!+ DX)GX$FabnFh_n7%F^bq7$Faao$!+%\\(z&\\$FabnFh_n7%Fbbq7$F[ao$!+c2ZNNFabn Fh_n7%Ffbq7$Fe`o$!+w#)zrNFabnFh_n7%Fjbq7$F_`o$!+!o=Zg$FabnFh_n7%F^cq7$ Fi_o$!+&[(>MOFabnFh_n7%Fbcq7$Fc_o$!+cN@gOFabnFh_n7%Ffcq7$F]_o$!+c\"\\F o$FabnFh_n7%Fjcq7$Fg^o$!+rEy,PFabnFh_n7%F^dq7$Fa^o$!+JJG.s$FabnFh_n7%F^fq7$Fa[ o$!+$=jQq$FabnFh_n7%Fbfq7$F[[o$!+;VQ#o$FabnFh_n7%Fffq7$Fejn$!+XYVbOFab nFh_n7%Fjfq7$F_jn$!+a-XAOFabnFh_n7%F^gq7$Fiin$!+:@s#e$FabnFh_n7%Fbgq7$ Fcin$!+QZONNFabnFh_n7%Ffgq7$F]in$!+=NHzMFabnFh_n7%Fjgq7$Fghn$!+F#QKT$F abnFh_n7%F^hq7$Fahn$!+-)[eL$FabnFh_n7%Fbhq7$F[hn$!+/I(fC$FabnFh_n7%Ffh q7$Fegn$!+c48VJFabnFh_n7%Fjhq7$F_gn$!+nS)z-$FabnFh_n7%F^iq7$Fifn$!+RsN -HFabnFh_n7%Fbiq7$Fcfn$!+d5joFFabnFh_n7%Ffiq7$F]fn$!+4w'*GEFabnFh_n7%F jiq7$Fgen$!+yV#\\[#FabnFh_n7%F^jq7$F`en$!+QP]PBFabnFh_n7%Fbjq7$Fjdn$!+ upP(=#FabnFh_n7%Ffjq7$Fcdn$!+[t2N?FabnFh_n7%Fjjq7$F]dn$!+jY4\")=FabnFh _n7%F^[r7$Ffcn$!+Xq)es\"FabnFh_n7%Fb[r7$$\"+kjnyuF^bn$!+iU&)p:FabnFh_n 7%Ff[r7$Fjbn$!+J9J89FabnFh_n7%F\\\\r7$Fdbn$!+h8[c7FabnFh_n7%F`\\r7$F\\ bn$!+b_]*4\"FabnFh_n7%Fd\\r7$Ffan$!+N1jC%*Fg_nFh_n7%Fh\\r7$F_an$!+g$[R &yFg_nFh_n7%F\\]r7$$\"+O,gNgFj`n$!+&yzJG'Fg_nFh_n7%F`]r7$$\"+qd-r!)FT$ !+/%)Q7ZFg_nFh_n7%Ff]r7$$\"+9$3Q5&F0$!+FEfTJFg_nFh_n7%F\\^r7$$\"+U&4(> fF\\p$!+Fjzq:Fg_nFh_n7%Fb^rF'Fh_n-Fe^n6&Fg^n$\"\")Fj^n$\"#NF]_nFj^r-%& STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"33+++.>]*[\"FT$\"3#******>&)\\QI%FT 7$$\"3!******ff%R\"[\"FT$\"3()*****HP'f7VFT7$$\"3'******Ru+EZ\"FT$\"3C +++&o\\1K%FT7$$\"3/+++&RjJY\"FT$\"3D+++(RozK%FT7$$\"3/+++yi7`9FT$\"3e* ****\\V6XL%FT7$$\"3++++@T`U9FT$\"3F+++kxBSVFT7$$\"3++++9IVJ9FT$\"3i*** ***>h5XVFT7$$\"3#******f#3()>9FT$\"3M+++dZ2\\VFT7$$\"3$******Ro(*yS\"F T$\"3e*****p?,@N%FT7$$\"3-+++Glc&R\"FT$\"3=+++^?9aVFT7$$\"33+++xP$HQ\" FT$\"3o*****4r_^N%FT7$$\"3%******R6g+P\"FT$\"37+++os3bVFT7$$\"3!****** fU6qN\"FT$\"3/+++A$)*QN%FT7$$\"33+++W*fQM\"FT$\"3y*****p\"p`^VFT7$$\"3 ,+++CcoI8FT$\"3-+++mD&zM%FT7$$\"3#******4$yd<8FT$\"3k******RM4VVFT7$$ \"3$******\\QPYI\"FT$\"3)******\\z1pL%FT7$$\"3)*******e!z>H\"FT$\"3e** ***zxR$HVFT7$$\"3++++-Ytz7FT$\"3J++++3M?VFT7$$\"3#******pMc!o7FT$\"3V+ ++g<')4VFT7$$\"3)*******H97d7FT$\"31+++d:'yH%FT7$$\"3!******H]OrC\"FT$ \"3!******RZ6VG%FT7$$\"32+++7EMQ7FT$\"3e*****HF.#pUFT7$$\"3*******fR>5 B\"FT$\"3e*****z/hDD%FT7$$\"3,+++wp[D7FT$\"3v*****p+eWB%FT7$$\"3****** *>e-@A\"FT$\"3V+++%)*Q]@%FT7$$\"3.+++`sC@7FT$\"3k*****p=[X>%FT7$$\"3/+ ++QlHB7FT$\"3/+++:%eL<%FT7$$\"3)*******4)p&G7FT$\"3)******\\N#*>:%FT7$ $\"3.+++(*)esB\"FT$\"3j*****4&*>68%FT7$$\"3'******>K]$\\7FT$\"3&****** *QR^6TFT7$$\"3-+++\\:dk7FT$\"3o*****pBdR4%FT7$$\"3/+++e\\R#G\"FT$\"3G+ ++mC7zSFT7$$\"3\"******4_5@I\"FT$\"3))*****\\@ou1%FT7$$\"3$*******z2%H K\"FT$\"3S+++yK>fSFT7$$\"3++++T::W8FT$\"3y*****RkdU0%FT7$$\"3/+++&*R7l 8FT$\"3u*****4JYC0%FT7$$\"3++++P:Q&Q\"FT$\"3=+++%oUM0%FT7$$\"3,+++BPe/ 9FT$\"3i*****\\F*)o0%FT7$$\"3)******R-/DU\"FT$\"3#)******>%GC1%FT7$$\" 3#*******RO+R9FT$\"3w*****R5D(pSFT7$$\"3%******>u2SX\"FT$\"3o*****\\=x %ySFT7$$\"3#******fS'[n9FT$\"3#*******yzT)3%FT7$$\"3)*******4.Wz9FT$\" 3A+++eZJ*4%FT7$$\"3'******HW!*)*[\"FT$\"3w*****RUl46%FT7$$\"3%******H) 3())\\\"FT$\"3R+++wX>BTFT7$$\"3$******H$QU1:FT$\"33+++T'\\e8%FT7$$\"3. +++Wif7:FT$\"37+++Wtz[TFT7$$\"3!******fcPu^\"FT$\"3#*******G1#>;%FT7$$ \"30+++g#)*4_\"FT$\"3++++@g6vTFT7$$\"3!******>&)GL_\"FT$\"3G+++<9H)=%F T7$$\"3\"******zLzW_\"FT$\"3:+++=UO,UFT7$$\"3$******4&))\\C:FT$\"3B+++ %yfU@%FT7$$\"3.+++$\\NM_\"FT$\"3j*****H85pA%FT7$$\"3'******4GO8_\"FT$ \"3m******[GDRUFT7$$\"3(******z8_WH%FTFe_rFd^n-F__n 6jn7%7$$\"+.>]*[\"Fabn$\"+_)\\QI%Fabn7$$\"+'f%R\"[\"Fabn$\"+tjf7VFabn7 $$\"$P\"F]_n$\"$>%F]_n7%Fibs7$$\"+W2gs9Fabn$\"+&o\\1K%FabnF^cs7%Fdcs7$ $\"+&RjJY\"Fabn$\"+(RozK%FabnF^cs7%Fjcs7$$\"+yi7`9Fabn$\"+N9^MVFabnF^c s7%F`ds7$$\"+@T`U9Fabn$\"+kxBSVFabnF^cs7%Ffds7$$\"+9IVJ9Fabn$\"+?h5XVF abnF^cs7%F\\es7$$\"+E3()>9Fabn$\"+dZ2\\VFabnF^cs7%Fbes7$$\"+%o(*yS\"Fa bn$\"+275_VFabnF^cs7%Fhes7$$\"+Glc&R\"Fabn$\"+^?9aVFabnF^cs7%F^fs7$$\" +xP$HQ\"Fabn$\"+6F:bVFabnF^cs7%Fdfs7$$\"+9,1q8Fabn$\"+os3bVFabnF^cs7%F jfs7$$\"+E9,d8Fabn$\"+A$)*QN%FabnF^cs7%F`gs7$$\"+W*fQM\"Fabn$\"+H\"Fabn$\"+y(R$HVFabnF^cs7%F^is7$$\"+-Ytz7Fabn$\"++3M?VFabnF^cs7%Fd is7$$\"+Zj0o7Fabn$\"+g<')4VFabnF^cs7%Fjis7$$\"+I97d7Fabn$\"+d:'yH%Fabn F^cs7%F`js7$$\"+.l8Z7Fabn$\"+u9J%G%FabnF^cs7%Ffjs7$$\"+7EMQ7Fabn$\"+tK ?pUFabnF^cs7%F\\[t7$$\"+'R>5B\"Fabn$\"+[5c_UFabnF^cs7%Fb[t7$$\"+wp[D7F abn$\"+2!eWB%FabnF^cs7%Fh[t7$$\"+#e-@A\"Fabn$\"+%)*Q]@%FabnF^cs7%F^\\t 7$$\"+`sC@7Fabn$\"+(=[X>%FabnF^cs7%Fd\\t7$$\"+QlHB7Fabn$\"+:%eL<%FabnF ^cs7%Fj\\t7$$\"+5)p&G7Fabn$\"+bB*>:%FabnF^cs7%F`]t7$$\"+(*)esB\"Fabn$ \"+^*>68%FabnF^cs7%Ff]t7$$\"+A.N\\7Fabn$\"+RR^6TFabnF^cs7%F\\^t7$$\"+ \\:dk7Fabn$\"+Ps&R4%FabnF^cs7%Fb^t7$$\"+e\\R#G\"Fabn$\"+mC7zSFabnF^cs7 %Fh^t7$$\"+@06-8Fabn$\"+:#ou1%FabnF^cs7%F^_t7$$\"+!ySHK\"Fabn$\"+yK>fS FabnF^cs7%Fd_t7$$\"+T::W8Fabn$\"+WwDaSFabnF^cs7%Fj_t7$$\"+&*R7l8Fabn$ \"+6jW_SFabnF^cs7%F``t7$$\"+P:Q&Q\"Fabn$\"+%oUM0%FabnF^cs7%Ff`t7$$\"+B Pe/9Fabn$\"+v#*)o0%FabnF^cs7%F\\at7$$\"+CS]A9Fabn$\"+?%GC1%FabnF^cs7%F bat7$$\"+SO+R9Fabn$\"+/^spSFabnF^cs7%Fhat7$$\"+Ux+a9Fabn$\"+&=x%ySFabn F^cs7%F^bt7$$\"+1k[n9Fabn$\"+zzT)3%FabnF^cs7%Fdbt7$$\"+5.Wz9Fabn$\"+eZ J*4%FabnF^cs7%Fjbt7$$\"+V/*)*[\"Fabn$\"+Ca'46%FabnF^cs7%F`ct7$$\"+$)3( ))\\\"Fabn$\"+wX>BTFabnF^cs7%Ffct7$$\"+LQU1:Fabn$\"+T'\\e8%FabnF^cs7%F \\dt7$$\"+Wif7:Fabn$\"+Wtz[TFabnF^cs7%Fbdt7$$\"+mvV<:Fabn$\"+H1#>;%Fab nF^cs7%Fhdt7$$\"+g#)*4_\"Fabn$\"+@g6vTFabnF^cs7%F^et7$$\"+_)GL_\"Fabn$ \"+<9H)=%FabnF^cs7%Fdet7$$\"+Q$zW_\"Fabn$\"+=UO,UFabnF^cs7%Fjet7$$\"+^ ))\\C:Fabn$\"+%yfU@%FabnF^cs7%F`ft7$$\"+$\\NM_\"Fabn$\"+L,\"pA%FabnF^c s7%Ffft7$$\"+\"GO8_\"Fabn$\"+\\GDRUFabnF^cs7%F\\gt7$$\"+yrC=:Fabn$\"+k ,B^UFabnF^cs7%Fbgt7$$\"+yJ@9:Fabn$\"+!=)yiUFabnF^cs7%Fhgt7$$\"+J%y#4:F abn$\"+.i(QF%FabnF^cs7%F^ht7$$\"+%R'[.:Fabn$\"+mhW%G%FabnF^cs7%Fdht7$$ \"+4+)o\\\"Fabn$\"+K@X%H%FabnF^cs7%FjhtFdbsF^csFh^rF^_r-F$6$7in7$Ff_r$ !3#******>&)\\QI%FT7$F]bs$!3)******>8_WH%FT7$Fhas$!3S+++mhW%G%FT7$Fcas $!3Q+++.i(QF%FT7$F^as$!3E+++!=)yiUFT7$Fi`s$!3))*****R;I7D%FT7$Fd`s$!3m ******[GDRUFT7$F_`s$!3j*****H85pA%FT7$Fj_s$!3B+++%yfU@%FT7$Fe_s$!3:+++ =UO,UFT7$F`_s$!3G+++<9H)=%FT7$F[_s$!3++++@g6vTFT7$Ff^s$!3#*******G1#>; %FT7$Fa^s$!37+++Wtz[TFT7$F\\^s$!33+++T'\\e8%FT7$Fg]s$!3R+++wX>BTFT7$Fb ]s$!3w*****RUl46%FT7$F]]s$!3A+++eZJ*4%FT7$Fh\\s$!3#*******yzT)3%FT7$Fc \\s$!3o*****\\=x%ySFT7$F^\\s$!3w*****R5D(pSFT7$Fi[s$!3#)******>%GC1%FT 7$Fd[s$!3i*****\\F*)o0%FT7$F_[s$!3=+++%oUM0%FT7$Fjjr$!3u*****4JYC0%FT7 $Fejr$!3y*****RkdU0%FT7$F`jr$!3S+++yK>fSFT7$F[jr$!3))*****\\@ou1%FT7$F fir$!3G+++mC7zSFT7$Fair$!3o*****pBdR4%FT7$F\\ir$!3&*******QR^6TFT7$Fgh r$!3j*****4&*>68%FT7$Fbhr$!3)******\\N#*>:%FT7$F]hr$!3/+++:%eL<%FT7$Fh gr$!3k*****p=[X>%FT7$Fcgr$!3V+++%)*Q]@%FT7$F^gr$!3v*****p+eWB%FT7$Fifr $!3e*****z/hDD%FT7$Fdfr$!3e*****HF.#pUFT7$F_fr$!3!******RZ6VG%FT7$Fjer $!31+++d:'yH%FT7$Feer$!3V+++g<')4VFT7$F`er$!3J++++3M?VFT7$F[er$!3e**** *zxR$HVFT7$Ffdr$!3)******\\z1pL%FT7$Fadr$!3k******RM4VVFT7$F\\dr$!3-++ +mD&zM%FT7$Fgcr$!3y*****p\"p`^VFT7$Fbcr$!3/+++A$)*QN%FT7$F]cr$!37+++os 3bVFT7$Fhbr$!3o*****4r_^N%FT7$Fcbr$!3=+++^?9aVFT7$F^br$!3e*****p?,@N%F T7$Fiar$!3M+++dZ2\\VFT7$Fdar$!3i******>h5XVFT7$F_ar$!3F+++kxBSVFT7$Fj` r$!3e*****\\V6XL%FT7$Fe`r$!3D+++(RozK%FT7$F``r$!3C+++&o\\1K%FT7$F[`r$! 3()*****HP'f7VFTFcitFd^n-F__n6jn7%7$Febs$!+_)\\QI%Fabn7$F[it$!+K@X%H%F abn7$F_cs$!$>%F]_n7%F]eu7$Feht$!+mhW%G%FabnF`eu7%Fdeu7$F_ht$!+.i(QF%Fa bnF`eu7%Fheu7$Figt$!+!=)yiUFabnF`eu7%F\\fu7$Fcgt$!+k,B^UFabnF`eu7%F`fu 7$F]gt$!+\\GDRUFabnF`eu7%Fdfu7$Fgft$!+L,\"pA%FabnF`eu7%Fhfu7$Faft$!+%y fU@%FabnF`eu7%F\\gu7$F[ft$!+=UO,UFabnF`eu7%F`gu7$Feet$!+<9H)=%FabnF`eu 7%Fdgu7$F_et$!+@g6vTFabnF`eu7%Fhgu7$Fidt$!+H1#>;%FabnF`eu7%F\\hu7$Fcdt $!+Wtz[TFabnF`eu7%F`hu7$F]dt$!+T'\\e8%FabnF`eu7%Fdhu7$Fgct$!+wX>BTFabn F`eu7%Fhhu7$Fact$!+Ca'46%FabnF`eu7%F\\iu7$F[ct$!+eZJ*4%FabnF`eu7%F`iu7 $Febt$!+zzT)3%FabnF`eu7%Fdiu7$F_bt$!+&=x%ySFabnF`eu7%Fhiu7$Fiat$!+/^sp SFabnF`eu7%F\\ju7$Fcat$!+?%GC1%FabnF`eu7%F`ju7$F]at$!+v#*)o0%FabnF`eu7 %Fdju7$Fg`t$!+%oUM0%FabnF`eu7%Fhju7$Fa`t$!+6jW_SFabnF`eu7%F\\[v7$F[`t$ !+WwDaSFabnF`eu7%F`[v7$Fe_t$!+yK>fSFabnF`eu7%Fd[v7$F__t$!+:#ou1%FabnF` eu7%Fh[v7$Fi^t$!+mC7zSFabnF`eu7%F\\\\v7$Fc^t$!+Ps&R4%FabnF`eu7%F`\\v7$ F]^t$!+RR^6TFabnF`eu7%Fd\\v7$Fg]t$!+^*>68%FabnF`eu7%Fh\\v7$Fa]t$!+bB*> :%FabnF`eu7%F\\]v7$F[]t$!+:%eL<%FabnF`eu7%F`]v7$Fe\\t$!+(=[X>%FabnF`eu 7%Fd]v7$F_\\t$!+%)*Q]@%FabnF`eu7%Fh]v7$Fi[t$!+2!eWB%FabnF`eu7%F\\^v7$F c[t$!+[5c_UFabnF`eu7%F`^v7$F][t$!+tK?pUFabnF`eu7%Fd^v7$Fgjs$!+u9J%G%Fa bnF`eu7%Fh^v7$Fajs$!+d:'yH%FabnF`eu7%F\\_v7$F[js$!+g<')4VFabnF`eu7%F`_ v7$Feis$!++3M?VFabnF`eu7%Fd_v7$F_is$!+y(R$HVFabnF`eu7%Fh_v7$Fihs$!+&z1 pL%FabnF`eu7%F\\`v7$Fchs$!+SM4VVFabnF`eu7%F``v7$F]hs$!+mD&zM%FabnF`eu7 %Fd`v7$Fggs$!+FTF(-%'COLOURG6&Fg^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7 $F(F`dv7$F($\"3/++++++!z%FTFedvFhdv-F$6$7]yF'7$$\"3;+++s\"HON&!#F$\"3/ +++Tjzq:F07$$\"3!)******[+t#[$F:$\"30+++WIfTJF07$$\"3C+++3+J_SF@$\"35+ ++=4R7ZF07$$\"3$)*****>Sm)=BFF$\"3=+++mm=$G'F07$$\"3I+++.E;%*))FF$\"3- +++,c&R&yF07$$\"3&)*****4\"G/0EFQ$\"3^+++Y)4YU*F07$$\"3))*****\\kP#)=' FQ$\"31+++Z<\\*4\"FT7$$\"3,+++%4%\\<7Fao$\"3%******4.PkD\"FT7$$\"3))** ***4\\#\\Z>Fao$\"3#******>(**>89FT7$$\"3/+++8!GAE#Fao$\"3'******RT;'p: FT7$$\"3Y+++*RtE=(FQ$\"3(*******)yOas\"FT7$$!3k******>K[ncFao$\"3-+++i +K!)=FT7$$!3'******4.@c@#F\\p$\"3-+++-s%Q.#FT7$$!3;+++!Gj@m$F\\p$\"3$) *****fFd*4@FT7$$!3%*******)H'p$p&F\\p$\"3')*****\\8^b=#FT7$$!3C+++%RJR Y)F\\p$\"3!******z)pcgAFT7$$!3$******f!\\l97Fgp$\"3=+++$*3%\\L#FT7$$!3 ;+++p*QLp\"Fgp$\"3*******RF/'3CFT7$$!36+++r<:.BFgp$\"3))*****\\yz9[#FT 7$$!3!******4V%)e1$Fgp$\"3%******z^vMb#FT7$$!3,+++/.y.SFgp$\"3++++iyZC EFT7$$!3L+++W@kQ^Fgp$\"3')*****\\\"QM%p#FT7$$!33+++?sh!\\'Fgp$\"3#)*** ***=T*Gw#FT7$$!3-+++#yUm2)Fgp$\"3)*******=$4*HGFT7$$!3%******fvO'3**Fg p$\"3?+++\"HJ^*GFT7$$!3'******zan\"*>\"F0$\"3'******Hcs#eHFT7$$!3(**** **H;cAV\"F0$\"3(*******pV.>IFT7$$!3/+++2]#*)o\"F0$\"3%)******y;8xIFT7$ $!3)******fxYr'>F0$\"3#*******\\/KKJFT7$$!3')*****f1`UE#F0$\"3;+++q1U% =$FT7$$!3/+++:r>xDF0$\"3()*****HjHLB$FT7$$!3;+++\"QNG!HF0$\"35+++kK-zK FT7$$!3F+++&4p\"QKF0$\"3>+++pqa@LFT7$$!3?+++&460e$F0$\"3)******pz+5O$F T7$$!3:+++\"*=dFRF0$\"3')*****\\?@vR$FT7$$!3u*****f6zuF%F0$\"36+++LiEJ MFT7$$!3s******)\\a-)\\F0$\"3++++G^4\"\\$FT7$$!3s*****H#*p2o&F0$\"38++ +ypxTNFT7$$!3c*****>V%oujF0$\"3'******Hf_We$FT7$$!3]+++jW$*fqF0$\"37++ +\"Qq+i$FT7$$!3]+++S.vNxF0$\"3y******pmR\\OFT7$$!3v*****pYc?S)F0$\"3%) *****HbVIn$FT7$$!3e******4l7f!*F0$\"3!******\\L)\\\"p$FT7$$!3I+++%e$R2 (*F0$\"3<+++$4\\^q$FT7$$!35+++,^tM5FT$\"3$******\\A0Vr$FT7$$!3++++@0&z 4\"FT$\"3&******4W8#>PFT7$$!3-+++jeVg6FT$\"3=+++[+2?PFT7$$!31+++M5CA7F T$\"3,+++%RIqr$FT7$$!35+++anT$G\"FT$\"33+++%=;-r$FT7$$!3*******HY;SM\" FT$\"35+++s7s*p$FT7$$!3!******H!))4/9FT$\"35+++?kh&o$FT7$$!3++++Z1tj9F T$\"3%)*****HZ`zm$FT7$$!36+++e1*H_\"FT$\"3/+++x$pnk$FT7$$!31+++*Qt>e\" FT$\"3()*****R_!4AOFT7$$!3-+++lQzS;FT$\"31+++tz$Rf$FT7$$!33+++OCf*p\"F T$\"39+++]ULiNFT7$$!3)******fRR&eFT$\"3\"******z^@DS$FT7$$!3:+++@2M+?FT$\"3')*****p(>*[N$FT7$$!3 !)******oMgj?FT$\"35+++aDt/LFT7$$!3\")*****pVa%G@FT$\"3')*****pT%\\_KF T7$$!3++++Rp(\\>#FT$\"3!******fkB()>$FT7$$!31+++1A4jAFT$\"3++++A`,WJFT 7$$!3-+++R'QDL#FT$\"3?+++[:%*)3$FT7$$!3z*****\\S#*GS#FT$\"3/+++9P(R.$F T7$$!3/+++=gitCFT$\"33+++9iUzHFT7$$!3++++LN=WDFT$\"3\")******=#Ra#HFT7 $$!3$*******om/9EFT$\"3'******zy)*>(GFT7$$!37+++)RwFo#FT$\"37+++L&y*=G FT7$$!3-+++Q&G+v#FT$\"3;+++C^=mFFT7$$!3!******pb]b\"GFT$\"37+++WiR8FFT 7$$!37+++b&o\"zGFT$\"33+++aqQgEFT7$$!3/+++R7xSHFT$\"3(******R-Ypg#FT7$ $!3*)*****p\"[H+IFT$\"3y******>G)Gb#FT7$$!3%******HW5x0$FT$\"3'******4 /J!)\\#FT7$$!3.+++yX,8JFT$\"3y*****\\*zCUCFT7$$!3/+++q>AmJFT$\"3=+++%* HT&Q#FT7$$!3++++\"ygt@$FT$\"3********3^UFBFT7$$!3;+++e$pkE$FT$\"3)**** **4$4?oAFT7$$!32+++.Uf8LFT$\"3$)*****\\isw?#FT7$$!33+++%z)yeLFT$\"3#)* ****pX'yX@FT7$$!3A+++bH6-MFT$\"3'******4&=]#3#FT7$$!3(*******>DjVMFT$ \"33+++85zM[$FT$\"31+++G\"R;&>FT7$$!3'******p^\\:_$FT$ \"35+++u]/%)=FT7$$!3/+++N`5eNFT$\"3%******zeA]\"=FT7$$!3)******\\?rJf$ FT$\"3********G?PFT$\"35+++rI=]9FT7$$!3'******Rm.!\\PFT$\"3-+++-_(QP\"FT7$$!37+++ $=?nx$FT$\"3!******zqQnH\"FT7$$!3(*******y*eL!QFT$\"3(******fF?*=7FT7$ $!32+++D]!*GQFT$\"3!******p9p09\"FT7$$!33+++?$=L&QFT$\"31+++)HI=1\"FT7 $$!3=+++4i`wQFT$\"3v*****Hqz$G)*F07$$!3?+++PvZ)*QFT$\"3$*******RN5P!*F 07$$!3-+++E![!>RFT$\"3C+++b)faC)F07$$!3z*****H5Y\"QRFT$\"3k******o0Aau F07$$!3!*******o%oc&RFT$\"31+++9]%Rm'F07$$!33+++#*[^rRFT$\"3Y+++VZ*\\( eF07$$!33+++Z>f&)RFT$\"3q*****\\Rdv3&F07$$!3)******Rm:y*RFT$\"3\"***** *R\"\\n,VF07$$!3j*****4>8\"3SFT$\"3u******4BFF07$$!3A+++m=!p-%FT$\" 3-+++:!y4<\"F07$$!38+++\\%4!HSFT$\"3L+++H^q-RFgp7$F^hx$!3L+++H^q-RFgp7 $Figx$!3-+++:!y4<\"F07$Fdgx$!3-+++R$z@&>F07$F_gx$!3y*****fU%=MFF07$Fjf x$!3u******4BFFT7$F\\`x$!33+++85zG)Gb#FT7$Fj\\x$!3(******R-Ypg#FT7$ Fe\\x$!33+++aqQgEFT7$F`\\x$!37+++WiR8FFT7$F[\\x$!3;+++C^=mFFT7$Ff[x$!3 7+++L&y*=GFT7$Fa[x$!3'******zy)*>(GFT7$F\\[x$!3\")******=#Ra#HFT7$Fgjw $!33+++9iUzHFT7$Fbjw$!3/+++9P(R.$FT7$F]jw$!3?+++[:%*)3$FT7$Fhiw$!3++++ A`,WJFT7$Fciw$!3!******fkB()>$FT7$F^iw$!3')*****pT%\\_KFT7$Fihw$!35+++ aDt/LFT7$Fdhw$!3')*****p(>*[N$FT7$F_hw$!3\"******z^@DS$FT7$Fjgw$!31+++ b,GZMFT7$Fegw$!3')*****R=H*)[$FT7$F`gw$!3A+++UDJFNFT7$F[gw$!39+++]ULiN FT7$Fffw$!31+++tz$Rf$FT7$Fafw$!3()*****R_!4AOFT7$F\\fw$!3/+++x$pnk$FT7 $Fgew$!3%)*****HZ`zm$FT7$Fbew$!35+++?kh&o$FT7$F]ew$!35+++s7s*p$FT7$Fhd w$!33+++%=;-r$FT7$Fcdw$!3,+++%RIqr$FT7$F^dw$!3=+++[+2?PFT7$Ficw$!3&*** ***4W8#>PFT7$Fdcw$!3$******\\A0Vr$FT7$F_cw$!3<+++$4\\^q$FT7$Fjbw$!3!** ****\\L)\\\"p$FT7$Febw$!3%)*****HbVIn$FT7$F`bw$!3y******pmR\\OFT7$F[bw $!37+++\"Qq+i$FT7$Ffaw$!3'******Hf_We$FT7$Faaw$!38+++ypxTNFT7$F\\aw$!3 ++++G^4\"\\$FT7$$!3#)*****>S_(GYF0$!3%********H-CY$FT7$Fg`w$!36+++LiEJ MFT7$Fb`w$!3')*****\\?@vR$FT7$F]`w$!3)******pz+5O$FT7$Fh_w$!3>+++pqa@L FT7$Fc_w$!35+++kK-zKFT7$F^_w$!3()*****HjHLB$FT7$Fi^w$!3;+++q1U%=$FT7$F d^w$!3#*******\\/KKJFT7$F_^w$!3%)******y;8xIFT7$Fj]w$!3(*******pV.>IFT 7$Fe]w$!3'******Hcs#eHFT7$F`]w$!3?+++\"HJ^*GFT7$F[]w$!3)*******=$4*HGF T7$Ff\\w$!3#)******=T*Gw#FT7$Fa\\w$!3')*****\\\"QM%p#FT7$F\\\\w$!3++++ iyZCEFT7$Fg[w$!3%******z^vMb#FT7$Fb[w$!3))*****\\yz9[#FT7$F][w$!3***** **RF/'3CFT7$Fhjv$!3=+++$*3%\\L#FT7$Fcjv$!3!******z)pcgAFT7$F^jv$!3')** ***\\8^b=#FT7$Fiiv$!3$)*****fFd*4@FT7$$!31+++6\"y6A\"F\\p$!3!*******4; Gd>FT7$$!3*******z,aDg\"Fao$!3*)*****fA?I!=FT7$$\"3-+++@_Rk=Fao$!3'*** ***RL?wk\"FT7$$\"33+++WLA>AFao$!3'******z%\\Y\"\\\"FT7$$\"3#******zQsb e\"Fao$!3&*******H(\\[L\"FT7$$\"37+++_c6()))FQ$!3%******H!)zz<\"FT7$$ \"3p*****\\*4!H6%FQ$!3!******R.$)4-\"FT7$$\"39+++/lqm:FQ$!35+++,pIR')F 07$$\"3*******R@aYs%FF$!39+++wxdoqF07$$\"3-+++'Q-A.\"FF$!3a+++j(*y(\\& F07$$\"3*)*****\\GDDM\"F@$!3********G=*p#RF07$$\"3))*****\\QO=9'F4$!3+ +++$f%>cBF07$$\"3c*****pTzsK)!#H$!3I+++R;)R&yFgp7$$\"3S*****HZzsK)F\\_ z$\"3I+++R;)R&yFgp-Fe^n6&Fg^n$F[evFj^n$\"#8F]_nFf_z-F__n6^y7%F'7$$\"+s \"HON&Fgp$\"+Tjzq:Fg_nFh_n7%F\\`z7$$\"+\\+t#[$FT$\"+WIfTJFg_nFh_n7%Fb` z7$$\"+3+J_SFj`n$\"+=4R7ZFg_nFh_n7%Fh`z7$$\"+-k')=BFaan$\"+mm=$G'Fg_nF h_n7%F^az7$$\"+.E;%*))Faan$\"+,c&R&yFg_nFh_n7%Fdaz7$$\"+6G/0EF^bn$\"+Y )4YU*Fg_nFh_n7%Fjaz7$$\"+XwB)='F^bn$\"+Z<\\*4\"FabnFh_n7%F`bz7$$\"+%4% \\<7Fhcn$\"+JqVc7FabnFh_n7%Ffbz7$$\"+\"\\#\\Z>Fhcn$\"+s**>89FabnFh_n7% F\\cz7$$\"+8!GAE#Fhcn$\"+9khp:FabnFh_n7%Fbcz7$$\"+*RtE=(F^bn$\"+*yOas \"FabnFh_n7%Fhcz7$$!+?K[ncFhcn$\"+i+K!)=FabnFh_n7%F^dz7$$!+J5i:AFedn$ \"+-s%Q.#FabnFh_n7%Fddz7$$!+!Gj@m$Fedn$\"+ws&*4@FabnFh_n7%Fjdz7$$!+*H' p$p&Fedn$\"+N6b&=#FabnFh_n7%F`ez7$$!+%RJRY)Fedn$\"+))pcgAFabnFh_n7%Ffe z7$$!+1\\l97Fben$\"+$*3%\\L#FabnFh_n7%F\\fz7$$!+p*QLp\"Fben$\"+uUg3CFa bnFh_n7%Fbfz7$$!+r<:.BFben$\"+&yz9[#FabnFh_n7%Fhfz7$$!+JW)e1$Fben$\"+= bZ`DFabnFh_n7%F^gz7$$!+/.y.SFben$\"+iyZCEFabnFh_n7%Fdgz7$$!+W@kQ^Fben$ \"+:QM%p#FabnFh_n7%Fjgz7$$!+?sh!\\'Fben$\"+>T*Gw#FabnFh_n7%F`hz7$$!+#y Um2)Fben$\"+>$4*HGFabnFh_n7%Ffhz7$$!+cnj3**Fben$\"+\"HJ^*GFabnFh_n7%F \\iz7$$!+[v;*>\"Fg_n$\"+jDFeHFabnFh_n7%Fbiz7$$!+jhDK9Fg_n$\"+qV.>IFabn Fh_n7%Fhiz7$$!+2]#*)o\"Fg_n$\"+z;8xIFabnFh_n7%F^jz7$$!+wn9n>Fg_n$\"+]/ KKJFabnFh_n7%Fdjz7$$!+mIDkAFg_n$\"+q1U%=$FabnFh_n7%Fjjz7$$!+:r>xDFg_n$ \"+L'HLB$FabnFh_n7%F`[[l7$$!+\"QNG!HFg_n$\"+kK-zKFabnFh_n7%Ff[[l7$$!+& 4p\"QKFg_n$\"+pqa@LFabnFh_n7%F\\\\[l7$$!+&460e$Fg_n$\"+(z+5O$FabnFh_n7 %Fb\\[l7$$!+\"*=dFRFg_n$\"+07_(R$FabnFh_n7%Fh\\[l7$$!+;\"zuF%Fg_n$\"+L iEJMFabnFh_n7%F^][l7$$!+*\\a-)\\Fg_n$\"+G^4\"\\$FabnFh_n7%Fd][l7$$!+B* p2o&Fg_n$\"+ypxTNFabnFh_n7%Fj][l7$$!+KWoujFg_n$\"+$f_We$FabnFh_n7%F`^[ l7$$!+jW$*fqFg_n$\"+\"Qq+i$FabnFh_n7%Ff^[l7$$!+S.vNxFg_n$\"+qmR\\OFabn Fh_n7%F\\_[l7$$!+nk0-%)Fg_n$\"+`N/tOFabnFh_n7%Fb_[l7$$!+5l7f!*Fg_n$\"+ N$)\\\"p$FabnFh_n7%Fh_[l7$$!+%e$R2(*Fg_n$\"+$4\\^q$FabnFh_n7%F^`[l7$$! +,^tM5Fabn$\"+D_I9PFabnFh_n7%Fd`[l7$$!+@0&z4\"Fabn$\"+TM@>PFabnFh_n7%F j`[l7$$!+jeVg6Fabn$\"+[+2?PFabnFh_n7%F`a[l7$$!+M5CA7Fabn$\"+%RIqr$Fabn Fh_n7%Ffa[l7$$!+anT$G\"Fabn$\"+%=;-r$FabnFh_n7%F\\b[l7$$!+jk,W8Fabn$\" +s7s*p$FabnFh_n7%Fbb[l7$$!+.))4/9Fabn$\"+?kh&o$FabnFh_n7%Fhb[l7$$!+Z1t j9Fabn$\"+tM&zm$FabnFh_n7%F^c[l7$$!+e1*H_\"Fabn$\"+x$pnk$FabnFh_n7%Fdc [l7$$!+*Qt>e\"Fabn$\"+C04AOFabnFh_n7%Fjc[l7$$!+lQzS;Fabn$\"+tz$Rf$Fabn Fh_n7%F`d[l7$$!+OCf*p\"Fabn$\"+]ULiNFabnFh_n7%Ffd[l7$$!+'RR&eFabn$\"+=:_-MFabnFh_n7%F^f[ l7$$!+@2M+?Fabn$\"+x>*[N$FabnFh_n7%Fdf[l7$$!+pMgj?Fabn$\"+aDt/LFabnFh_ n7%Fjf[l7$$!+PWXG@Fabn$\"+#Fabn$\"+YOs )>$FabnFh_n7%Ffg[l7$$!+1A4jAFabn$\"+A`,WJFabnFh_n7%F\\h[l7$$!+R'QDL#Fa bn$\"+[:%*)3$FabnFh_n7%Fbh[l7$$!+0C*GS#Fabn$\"+9P(R.$FabnFh_n7%Fhh[l7$ $!+=gitCFabn$\"+9iUzHFabnFh_n7%F^i[l7$$!+LN=WDFabn$\"+>#Ra#HFabnFh_n7% Fdi[l7$$!+pm/9EFabn$\"+)y)*>(GFabnFh_n7%Fji[l7$$!+)RwFo#Fabn$\"+L&y*=G FabnFh_n7%F`j[l7$$!+Q&G+v#Fabn$\"+C^=mFFabnFh_n7%Ffj[l7$$!+d0b:GFabn$ \"+WiR8FFabnFh_n7%F\\[\\l7$$!+b&o\"zGFabn$\"+aqQgEFabnFh_n7%Fb[\\l7$$! +R7xSHFabn$\"+Cg%pg#FabnFh_n7%Fh[\\l7$$!+<[H+IFabn$\"+?G)Gb#FabnFh_n7% F^\\\\l7$$!+V/rdIFabn$\"+T5.)\\#FabnFh_n7%Fd\\\\l7$$!+yX,8JFabn$\"+&*z CUCFabnFh_n7%Fj\\\\l7$$!+q>AmJFabn$\"+%*HT&Q#FabnFh_n7%F`]\\l7$$!+\"yg t@$Fabn$\"+4^UFBFabnFh_n7%Ff]\\l7$$!+e$pkE$Fabn$\"+J4?oAFabnFh_n7%F\\^ \\l7$$!+.Uf8LFabn$\"+DEn2AFabnFh_n7%Fb^\\l7$$!+%z)yeLFabn$\"+dkyX@Fabn Fh_n7%Fh^\\l7$$!+bH6-MFabn$\"+^=]#3#FabnFh_n7%F^_\\l7$$!+?DjVMFabn$\"+ 85zM[$Fabn$\"+G\"R;&>FabnFh_n7%Fj_\\l7$$!+<& \\:_$Fabn$\"+u]/%)=FabnFh_n7%F``\\l7$$!+N`5eNFabn$\"+)eA]\"=FabnFh_n7% Ff`\\l7$$!+07<$f$Fabn$\"+HH$oi$Fabn$\"+9/$G n\"FabnFh_n7%Fba\\l7$$!+.W?PFabn$\"+rI=]9FabnFh_n7%Fdb\\l 7$$!+kO+\\PFabn$\"+-_(QP\"FabnFh_n7%Fjb\\l7$$!+$=?nx$Fabn$\"+3(QnH\"Fa bnFh_n7%F`c\\l7$$!+z*eL!QFabn$\"+w-#*=7FabnFh_n7%Ffc\\l7$$!+D]!*GQFabn $\"+Z\"p09\"FabnFh_n7%F\\d\\l7$$!+?$=L&QFabn$\"+)HI=1\"FabnFh_n7%Fbd\\ l7$$!+4i`wQFabn$\"+.(z$G)*Fg_nFh_n7%Fhd\\l7$$!+PvZ)*QFabn$\"+SN5P!*Fg_ nFh_n7%F^e\\l7$$!+E![!>RFabn$\"+b)faC)Fg_nFh_n7%Fde\\l7$$!+.h9QRFabn$ \"+p0AauFg_nFh_n7%Fje\\l7$$!+p%oc&RFabn$\"+9]%Rm'Fg_nFh_n7%F`f\\l7$$!+ #*[^rRFabn$\"+VZ*\\(eFg_nFh_n7%Fff\\l7$$!+Z>f&)RFabn$\"+&Rdv3&Fg_nFh_n 7%F\\g\\l7$$!+kc\"y*RFabn$\"+9\\n,VFg_nFh_n7%Fbg\\l7$$!+\">8\"3SFabn$ \"+5BFFg_nFh_n7%Fdh\\l7$$!+m=!p-%Fabn$\"+:!y4<\"Fg_nFh_n7 %Fjh\\l7$$!+\\%4!HSFabn$\"+H^q-RFbenFh_n7%F`i\\l7$Fai\\l$!+H^q-RFbenFh _n7%Ffi\\l7$F[i\\l$!+:!y4<\"Fg_nFh_n7%Fji\\l7$Feh\\l$!+R$z@&>Fg_nFh_n7 %F^j\\l7$F_h\\l$!+EW=MFFg_nFh_n7%Fbj\\l7$Fig\\l$!+5BFFabnFh_n7%Fj_]l7$Fe_\\l$!+85z< ?FabnFh_n7%F^`]l7$F__\\l$!+^=]#3#FabnFh_n7%Fb`]l7$Fi^\\l$!+dkyX@FabnFh _n7%Ff`]l7$Fc^\\l$!+DEn2AFabnFh_n7%Fj`]l7$F]^\\l$!+J4?oAFabnFh_n7%F^a] l7$Fg]\\l$!+4^UFBFabnFh_n7%Fba]l7$Fa]\\l$!+%*HT&Q#FabnFh_n7%Ffa]l7$F[] \\l$!+&*zCUCFabnFh_n7%Fja]l7$Fe\\\\l$!+T5.)\\#FabnFh_n7%F^b]l7$F_\\\\l $!+?G)Gb#FabnFh_n7%Fbb]l7$Fi[\\l$!+Cg%pg#FabnFh_n7%Ffb]l7$Fc[\\l$!+aqQ gEFabnFh_n7%Fjb]l7$F][\\l$!+WiR8FFabnFh_n7%F^c]l7$Fgj[l$!+C^=mFFabnFh_ n7%Fbc]l7$Faj[l$!+L&y*=GFabnFh_n7%Ffc]l7$F[j[l$!+)y)*>(GFabnFh_n7%Fjc] l7$Fei[l$!+>#Ra#HFabnFh_n7%F^d]l7$F_i[l$!+9iUzHFabnFh_n7%Fbd]l7$Fih[l$ !+9P(R.$FabnFh_n7%Ffd]l7$Fch[l$!+[:%*)3$FabnFh_n7%Fjd]l7$F]h[l$!+A`,WJ FabnFh_n7%F^e]l7$Fgg[l$!+YOs)>$FabnFh_n7%Fbe]l7$Fag[l$!+*[N$FabnFh_n7%F^f]l7$ F_f[l$!+=:_-MFabnFh_n7%Fbf]l7$Fie[l$!+b,GZMFabnFh_n7%Fff]l7$Fce[l$!+%= H*)[$FabnFh_n7%Fjf]l7$F]e[l$!+UDJFNFabnFh_n7%F^g]l7$Fgd[l$!+]ULiNFabnF h_n7%Fbg]l7$Fad[l$!+tz$Rf$FabnFh_n7%Ffg]l7$F[d[l$!+C04AOFabnFh_n7%Fjg] l7$Fec[l$!+x$pnk$FabnFh_n7%F^h]l7$F_c[l$!+tM&zm$FabnFh_n7%Fbh]l7$Fib[l $!+?kh&o$FabnFh_n7%Ffh]l7$Fcb[l$!+s7s*p$FabnFh_n7%Fjh]l7$F]b[l$!+%=;-r $FabnFh_n7%F^i]l7$Fga[l$!+%RIqr$FabnFh_n7%Fbi]l7$Faa[l$!+[+2?PFabnFh_n 7%Ffi]l7$F[a[l$!+TM@>PFabnFh_n7%Fji]l7$Fe`[l$!+D_I9PFabnFh_n7%F^j]l7$F _`[l$!+$4\\^q$FabnFh_n7%Fbj]l7$Fi_[l$!+N$)\\\"p$FabnFh_n7%Ffj]l7$Fc_[l $!+`N/tOFabnFh_n7%Fjj]l7$F]_[l$!+qmR\\OFabnFh_n7%F^[^l7$Fg^[l$!+\"Qq+i $FabnFh_n7%Fb[^l7$Fa^[l$!+$f_We$FabnFh_n7%Ff[^l7$F[^[l$!+ypxTNFabnFh_n 7%Fj[^l7$Fe][l$!+G^4\"\\$FabnFh_n7%F^\\^l7$$!+-CvGYFg_n$!++BSiMFabnFh_ n7%Fb\\^l7$F_][l$!+LiEJMFabnFh_n7%Fh\\^l7$Fi\\[l$!+07_(R$FabnFh_n7%F\\ ]^l7$Fc\\[l$!+(z+5O$FabnFh_n7%F`]^l7$F]\\[l$!+pqa@LFabnFh_n7%Fd]^l7$Fg [[l$!+kK-zKFabnFh_n7%Fh]^l7$Fa[[l$!+L'HLB$FabnFh_n7%F\\^^l7$F[[[l$!+q1 U%=$FabnFh_n7%F`^^l7$Fejz$!+]/KKJFabnFh_n7%Fd^^l7$F_jz$!+z;8xIFabnFh_n 7%Fh^^l7$Fiiz$!+qV.>IFabnFh_n7%F\\_^l7$Fciz$!+jDFeHFabnFh_n7%F`_^l7$F] iz$!+\"HJ^*GFabnFh_n7%Fd_^l7$Fghz$!+>$4*HGFabnFh_n7%Fh_^l7$Fahz$!+>T*G w#FabnFh_n7%F\\`^l7$F[hz$!+:QM%p#FabnFh_n7%F``^l7$Fegz$!+iyZCEFabnFh_n 7%Fd`^l7$F_gz$!+=bZ`DFabnFh_n7%Fh`^l7$Fifz$!+&yz9[#FabnFh_n7%F\\a^l7$F cfz$!+uUg3CFabnFh_n7%F`a^l7$F]fz$!+$*3%\\L#FabnFh_n7%Fda^l7$Fgez$!+))p cgAFabnFh_n7%Fha^l7$Faez$!+N6b&=#FabnFh_n7%F\\b^l7$F[ez$!+ws&*4@FabnFh _n7%F`b^l7$$!+6\"y6A\"Fedn$!+5;Gd>FabnFh_n7%Fdb^l7$$!+=Sb-;Fhcn$!+E--. =FabnFh_n7%Fjb^l7$$\"+@_Rk=Fhcn$!+M.iZ;FabnFh_n7%F`c^l7$$\"+WLA>AFhcn$ !+[\\Y\"\\\"FabnFh_n7%Ffc^l7$$\"+)Qsbe\"Fhcn$!+I(\\[L\"FabnFh_n7%F\\d^ l7$$\"+_c6()))F^bn$!+.)zz<\"FabnFh_n7%Fbd^l7$$\"+&*4!H6%F^bn$!+MI)4-\" FabnFh_n7%Fhd^l7$$\"+/lqm:F^bn$!+,pIR')Fg_nFh_n7%F^e^l7$$\"+9UlCZFaan$ !+wxdoqFg_nFh_n7%Fde^l7$$\"+'Q-A.\"Faan$!+j(*y(\\&Fg_nFh_n7%Fje^l7$$\" +&GDDM\"Fj`n$!+H=*p#RFg_nFh_n7%F`f^l7$$\"+&QO=9'F0$!+$f%>cBFg_nFh_n7%F ff^l7$$\"+<%zsK)Fao$!+R;)R&yFbenFh_n7%F\\g^l7$$\"+t%zsK)Fao$\"+R;)R&yF benFh_n-Fe^n6&Fg^n$\"\"'Fj^n$\"#DF]_nFig^lF^_r-F$6$7in7$$\"3-+++x-\"o \\\"FT$\"3Q+++^=S!H%FT7$$\"3++++2.j)[\"FT$\"3i*****R<()*)H%FT7$$\"3'** ****fJt(z9FT$\"3f*****4!\\(oI%FT7$$\"3/+++xCGq9FT$\"30+++[O-9VFT7$$\"3 !******\\s,-Y\"FT$\"3<+++CBR?VFT7$$\"31+++(3w&\\9FT$\"3T+++k)RfK%FT7$$ \"3,+++`?XQ9FT$\"3m******e\\iIVFT7$$\"31+++(*z(oU\"FT$\"3u******>eSMVF T7$$\"3*)*****Hn/\\T\"FT$\"3u*****z$*RsL%FT7$$\"36+++)y&e-9FT$\"3&**** **R(Q3RVFT7$$\"3)******\\oy**Q\"FT$\"3s*****41$*)RVFT7$$\"35+++[^9x8FT $\"3;+++W;iRVFT7$$\"31+++wB:k8FT$\"3%******ROA#QVFT7$$\"31+++bT2^8FT$ \"3e*****\\_YcL%FT7$$\"34+++uA*zL\"FT$\"38+++jS%=L%FT7$$\"3-+++^#)*\\K \"FT$\"3&)*****p!QwEVFT7$$\"36+++Ga>78FT$\"3O+++/SN?VFT7$$\"33+++6:q*H \"FT$\"3V+++`Kc7VFT7$$\"3'*******G;l(G\"FT$\"3$)*****\\1ULI%FT7$$\"3#* *****>%>?w7FT$\"3k*****fLXEH%FT7$$\"3/+++;Q`l7FT$\"3I+++)>O/G%FT7$$\"3 %******fPeeD\"FT$\"3i*****f%=pmUFT7$$\"35+++K5UZ7FT$\"3M+++&>79D%FT7$$ \"3.+++)*\\]S7FT$\"3%)******oBjMUFT7$$\"3++++2?VN7FT$\"3<+++$3Sk@%FT7$ $\"35+++[qbK7FT$\"3;+++M%)*p>%FT7$$\"32+++MCDK7FT$\"3#)*****p.tl<%FT7$ $\"3'******zSv[B\"FT$\"3k*****p!*eb:%FT7$$\"3\"*******[^rS7FT$\"3j**** *4(o\\MTFT7$$\"3-+++\"3A*\\7FT$\"3#******RijS6%FT7$$\"3%******p;OCE\"F T$\"3'******H))>]4%FT7$$\"3'******p]WzF\"FT$\"3K+++n\\6ySFT7$$\"3!**** **4P(*eH\"FT$\"3c*****p(>(R1%FT7$$\"3$*******Hne:8FT$\"3)******zu'*H0% FT7$$\"3(******zjdiL\"FT$\"3k*****p/X`/%FT7$$\"3%******>?4sN\"FT$\"3F+ ++f/&4/%FT7$$\"31+++6p&yP\"FT$\"3F+++$)yeRSFT7$$\"3!******RL`xR\"FT$\" 3r*****\\XU4/%FT7$$\"3/+++E*zlT\"FT$\"3x*****43lY/%FT7$$\"3$******4:DT V\"FT$\"3I+++_\"4/0%FT7$$\"3'******\\pf-X\"FT$\"3O+++O4&y0%FT7$$\"3!** ****HD9\\Y\"FT$\"3G+++w&)pmSFT7$$\"3)******Hvh!y9FT$\"3C+++4UpwSFT7$$ \"3%******4,/(*[\"FT$\"3C+++KCh(3%FT7$$\"31+++0@')*\\\"FT$\"3!*******) 4d#*4%FT7$$\"32+++'fp&3:FT$\"3o*****4gd96%FT7$$\"33+++Sy'e^\"FT$\"3=++ +(>lS7%FT7$$\"3!********z-=_\"FT$\"3\"*******p'\\p8%FT7$$\"3.+++#*GUE: FT$\"37+++,k**\\TFT7$$\"3(******>dx(H:FT$\"3u*****4'Q5jTFT7$$\"3++++=k \">`\"FT$\"37+++*\\\"=wTFT7$$\"3)******Hc))G`\"FT$\"3I+++dz9*=%FT7$$\" 36+++jBuK:FT$\"3L+++'eH>?%FT7$$\"3*******HBD:`\"FT$\"3W+++\">fW@%FT7$$ \"3(******ff$GH:FT$\"3v*****R)\\nEUFT7$$\"3-+++2H1E:FT$\"3w*****Hp>&QU FT7$$\"3/+++Px!>_\"FT$\"3i*****p!)R*\\UFT7$$\"3/+++%*='o^\"FT$\"3Q+++H \\)3E%FT7$$\"3!******>do4^\"FT$\"3u******>sIrUFT7$$\"3%******z_qU]\"FT $\"3A+++14;\"G%FTF`h^lFd_z-F__n6jn7%7$$\"+x-\"o\\\"Fabn$\"+^=S!H%Fabn7 $$\"+2.j)[\"Fabn$\"+ur)*)H%Fabn7$$\"$Q\"F]_n$\"$=%F]_n7%Fd[`l7$$\"+;Lx z9Fabn$\"+,\\(oI%FabnFi[`l7%F_\\`l7$$\"+xCGq9Fabn$\"+[O-9VFabnFi[`l7%F e\\`l7$$\"+D78Fabn$\"+/SN?VFabnFi[`l7%Fca`l7$$\"+6:q*H \"Fabn$\"+`Kc7VFabnFi[`l7%Fia`l7$$\"+H;l(G\"Fabn$\"+l?M.VFabnFi[`l7%F_ b`l7$$\"+U>?w7Fabn$\"+O`k#H%FabnFi[`l7%Feb`l7$$\"+;Q`l7Fabn$\"+)>O/G%F abnFi[`l7%F[c`l7$$\"+w$eeD\"Fabn$\"+Y=pmUFabnFi[`l7%Fac`l7$$\"+K5UZ7Fa bn$\"+&>79D%FabnFi[`l7%Fgc`l7$$\"+)*\\]S7Fabn$\"+pBjMUFabnFi[`l7%F]d`l 7$$\"+2?VN7Fabn$\"+$3Sk@%FabnFi[`l7%Fcd`l7$$\"+[qbK7Fabn$\"+M%)*p>%Fab nFi[`l7%Fid`l7$$\"+MCDK7Fabn$\"+PIdwTFabnFi[`l7%F_e`l7$$\"+3a([B\"Fabn $\"+2*eb:%FabnFi[`l7%Fee`l7$$\"+\\^rS7Fabn$\"+ro\\MTFabnFi[`l7%F[f`l7$ $\"+\"3A*\\7Fabn$\"+CO19TFabnFi[`l7%Faf`l7$$\"+nhVi7Fabn$\"+$))>]4%Fab nFi[`l7%Fgf`l7$$\"+2X%zF\"Fabn$\"+n\\6ySFabnFi[`l7%F]g`l7$$\"+rt*eH\"F abn$\"+x>(R1%FabnFi[`l7%Fcg`l7$$\"+Ine:8Fabn$\"+[n*H0%FabnFi[`l7%Fig`l 7$$\"+QwDO8Fabn$\"+Z]MXSFabnFi[`l7%F_h`l7$$\"+-#4sN\"Fabn$\"+f/&4/%Fab nFi[`l7%Feh`l7$$\"+6p&yP\"Fabn$\"+$)yeRSFabnFi[`l7%F[i`l7$$\"+MLv(R\"F abn$\"+bC%4/%FabnFi[`l7%Fai`l7$$\"+E*zlT\"Fabn$\"+\"3lY/%FabnFi[`l7%Fg i`l7$$\"+^^7M9Fabn$\"+_\"4/0%FabnFi[`l7%F]j`l7$$\"+&pf-X\"Fabn$\"+O4&y 0%FabnFi[`l7%Fcj`l7$$\"+`U\"\\Y\"Fabn$\"+w&)pmSFabnFi[`l7%Fij`l7$$\"+` <1y9Fabn$\"+4UpwSFabnFi[`l7%F_[al7$$\"+6Sq*[\"Fabn$\"+KCh(3%FabnFi[`l7 %Fe[al7$$\"+0@')*\\\"Fabn$\"+*4d#*4%FabnFi[`l7%F[\\al7$$\"+'fp&3:Fabn$ \"+,wX6TFabnFi[`l7%Fa\\al7$$\"+Sy'e^\"Fabn$\"+(>lS7%FabnFi[`l7%Fg\\al7 $$\"++G!=_\"Fabn$\"+q'\\p8%FabnFi[`l7%F]]al7$$\"+#*GUE:Fabn$\"+,k**\\T FabnFi[`l7%Fc]al7$$\"+svxH:Fabn$\"+hQ5jTFabnFi[`l7%Fi]al7$$\"+=k\">`\" Fabn$\"+*\\\"=wTFabnFi[`l7%F_^al7$$\"+j&))G`\"Fabn$\"+dz9*=%FabnFi[`l7 %Fe^al7$$\"+jBuK:Fabn$\"+'eH>?%FabnFi[`l7%F[_al7$$\"+L__J:Fabn$\"+\">f W@%FabnFi[`l7%Fa_al7$$\"+'f$GH:Fabn$\"+%)\\nEUFabnFi[`l7%Fg_al7$$\"+2H 1E:Fabn$\"+$p>&QUFabnFi[`l7%F]`al7$$\"+Px!>_\"Fabn$\"+2)R*\\UFabnFi[`l 7%Fc`al7$$\"+%*='o^\"Fabn$\"+H\\)3E%FabnFi[`l7%Fi`al7$$\"+s&o4^\"Fabn$ \"+?sIrUFabnFi[`l7%F_aal7$$\"+G0F/:Fabn$\"+14;\"G%FabnFi[`l7%FeaalF_[` lFi[`lFgg^lF^_r-F$6$7in7$Fah^l$!3Q+++^=S!H%FT7$Fhj_l$!3A+++14;\"G%FT7$ Fcj_l$!3u******>sIrUFT7$F^j_l$!3Q+++H\\)3E%FT7$Fii_l$!3i*****p!)R*\\UF T7$Fdi_l$!3w*****Hp>&QUFT7$F_i_l$!3v*****R)\\nEUFT7$Fjh_l$!3W+++\">fW@ %FT7$Feh_l$!3L+++'eH>?%FT7$F`h_l$!3I+++dz9*=%FT7$F[h_l$!37+++*\\\"=wTF T7$Ffg_l$!3u*****4'Q5jTFT7$Fag_l$!37+++,k**\\TFT7$F\\g_l$!3\"*******p' \\p8%FT7$Fgf_l$!3=+++(>lS7%FT7$Fbf_l$!3o*****4gd96%FT7$F]f_l$!3!****** *)4d#*4%FT7$Fhe_l$!3C+++KCh(3%FT7$Fce_l$!3C+++4UpwSFT7$F^e_l$!3G+++w&) pmSFT7$Fid_l$!3O+++O4&y0%FT7$Fdd_l$!3I+++_\"4/0%FT7$F_d_l$!3x*****43lY /%FT7$Fjc_l$!3r*****\\XU4/%FT7$Fec_l$!3F+++$)yeRSFT7$F`c_l$!3F+++f/&4/ %FT7$F[c_l$!3k*****p/X`/%FT7$Ffb_l$!3)******zu'*H0%FT7$Fab_l$!3c*****p (>(R1%FT7$F\\b_l$!3K+++n\\6ySFT7$Fga_l$!3'******H))>]4%FT7$Fba_l$!3#** ****RijS6%FT7$F]a_l$!3j*****4(o\\MTFT7$Fh`_l$!3k*****p!*eb:%FT7$Fc`_l$ !3#)*****p.tl<%FT7$F^`_l$!3;+++M%)*p>%FT7$Fi__l$!3<+++$3Sk@%FT7$Fd__l$ !3%)******oBjMUFT7$F___l$!3M+++&>79D%FT7$Fj^_l$!3i*****f%=pmUFT7$Fe^_l $!3I+++)>O/G%FT7$F`^_l$!3k*****fLXEH%FT7$F[^_l$!3$)*****\\1ULI%FT7$Ff] _l$!3V+++`Kc7VFT7$Fa]_l$!3O+++/SN?VFT7$F\\]_l$!3&)*****p!QwEVFT7$Fg\\_ l$!38+++jS%=L%FT7$Fb\\_l$!3e*****\\_YcL%FT7$F]\\_l$!3%******ROA#QVFT7$ Fh[_l$!3;+++W;iRVFT7$Fc[_l$!3s*****41$*)RVFT7$F^[_l$!3&******R(Q3RVFT7 $Fij^l$!3u*****z$*RsL%FT7$Fdj^l$!3u******>eSMVFT7$F_j^l$!3m******e\\iI VFT7$Fji^l$!3T+++k)RfK%FT7$Fei^l$!3<+++CBR?VFT7$F`i^l$!30+++[O-9VFT7$F [i^l$!3f*****4!\\(oI%FT7$Ffh^l$!3i*****R<()*)H%FTF^balFd_z-F__n6jn7%7$ F`[`l$!+^=S!H%Fabn7$Ffaal$!+14;\"G%Fabn7$Fj[`l$!$=%F]_n7%Fh]bl7$F`aal$ !+?sIrUFabnF[^bl7%F_^bl7$Fj`al$!+H\\)3E%FabnF[^bl7%Fc^bl7$Fd`al$!+2)R* \\UFabnF[^bl7%Fg^bl7$F^`al$!+$p>&QUFabnF[^bl7%F[_bl7$Fh_al$!+%)\\nEUFa bnF[^bl7%F__bl7$Fb_al$!+\">fW@%FabnF[^bl7%Fc_bl7$F\\_al$!+'eH>?%FabnF[ ^bl7%Fg_bl7$Ff^al$!+dz9*=%FabnF[^bl7%F[`bl7$F`^al$!+*\\\"=wTFabnF[^bl7 %F_`bl7$Fj]al$!+hQ5jTFabnF[^bl7%Fc`bl7$Fd]al$!+,k**\\TFabnF[^bl7%Fg`bl 7$F^]al$!+q'\\p8%FabnF[^bl7%F[abl7$Fh\\al$!+(>lS7%FabnF[^bl7%F_abl7$Fb \\al$!+,wX6TFabnF[^bl7%Fcabl7$F\\\\al$!+*4d#*4%FabnF[^bl7%Fgabl7$Ff[al $!+KCh(3%FabnF[^bl7%F[bbl7$F`[al$!+4UpwSFabnF[^bl7%F_bbl7$Fjj`l$!+w&)p mSFabnF[^bl7%Fcbbl7$Fdj`l$!+O4&y0%FabnF[^bl7%Fgbbl7$F^j`l$!+_\"4/0%Fab nF[^bl7%F[cbl7$Fhi`l$!+\"3lY/%FabnF[^bl7%F_cbl7$Fbi`l$!+bC%4/%FabnF[^b l7%Fccbl7$F\\i`l$!+$)yeRSFabnF[^bl7%Fgcbl7$Ffh`l$!+f/&4/%FabnF[^bl7%F[ dbl7$F`h`l$!+Z]MXSFabnF[^bl7%F_dbl7$Fjg`l$!+[n*H0%FabnF[^bl7%Fcdbl7$Fd g`l$!+x>(R1%FabnF[^bl7%Fgdbl7$F^g`l$!+n\\6ySFabnF[^bl7%F[ebl7$Fhf`l$!+ $))>]4%FabnF[^bl7%F_ebl7$Fbf`l$!+CO19TFabnF[^bl7%Fcebl7$F\\f`l$!+ro\\M TFabnF[^bl7%Fgebl7$Ffe`l$!+2*eb:%FabnF[^bl7%F[fbl7$F`e`l$!+PIdwTFabnF[ ^bl7%F_fbl7$Fjd`l$!+M%)*p>%FabnF[^bl7%Fcfbl7$Fdd`l$!+$3Sk@%FabnF[^bl7% Fgfbl7$F^d`l$!+pBjMUFabnF[^bl7%F[gbl7$Fhc`l$!+&>79D%FabnF[^bl7%F_gbl7$ Fbc`l$!+Y=pmUFabnF[^bl7%Fcgbl7$F\\c`l$!+)>O/G%FabnF[^bl7%Fggbl7$Ffb`l$ !+O`k#H%FabnF[^bl7%F[hbl7$F`b`l$!+l?M.VFabnF[^bl7%F_hbl7$Fja`l$!+`Kc7V FabnF[^bl7%Fchbl7$Fda`l$!+/SN?VFabnF[^bl7%Fghbl7$F^a`l$!+2QwEVFabnF[^b l7%F[ibl7$Fh``l$!+jS%=L%FabnF[^bl7%F_ibl7$Fb``l$!+DlkNVFabnF[^bl7%Fcib l7$F\\``l$!+kBAQVFabnF[^bl7%Fgibl7$Ff_`l$!+W;iRVFabnF[^bl7%F[jbl7$F`_` l$!+hI*)RVFabnF[^bl7%F_jbl7$Fj^`l$!+uQ3RVFabnF[^bl7%Fcjbl7$Fd^`l$!+Q*R sL%FabnF[^bl7%Fgjbl7$F^^`l$!+?eSMVFabnF[^bl7%F[[cl7$Fh]`l$!+f\\iIVFabn F[^bl7%F_[cl7$Fb]`l$!+k)RfK%FabnF[^bl7%Fc[cl7$F\\]`l$!+CBR?VFabnF[^bl7 %Fg[cl7$Ff\\`l$!+[O-9VFabnF[^bl7%F[\\cl7$F`\\`l$!+,\\(oI%FabnF[^bl7%F_ \\cl7$Fe[`l$!+ur)*)H%FabnF[^bl7%Fc\\clFe]blF[^blFgg^lF^_r-%*AXESSTYLEG 6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALING G6#%,CONSTRAINEDG-Fa]cl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%F]_n$\"$*>F] _n;F`^cl$\"$z%F]_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------- --------------------------------------------------" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-------- ---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1101 "ee := \{c[2]=1/6,\nc[3]=2/9,\nc[4]=3/7,\nc[5]=2/3,\nc[6]=3/4,\nc [7]=1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=2/27,\na[3,2]=4/27,\na[4,1]=183 /1372,\na[4,2]=-162/343,\na[4,3]=1053/1372,\na[5,1]=68/297,\na[5,2]=-4 /11,\na[5,3]=42/143,\na[5,4]=1960/3861,\na[6,1]=32575/1096128,\na[6,2] =81/352,\na[6,3]=38673/395824,\na[6,4]=2421335/14249664,\na[6,5]=77/34 6,\na[7,1]=2218605592257973/11003713657896960,\na[7,2]=-63/143,\na[7,3 ]=1473430904732051/3532056235868160,\na[7,4]=2549732718751057/36491907 53894400,\na[7,5]=-13054508705469277/12226348508774400,\na[7,6]=213604 83/17937920,\na[8,1]=539/7344,\na[8,2]=0,\na[8,3]=190269/671840,\na[8, 4]=362551/1790100,\na[8,5]=-81/13600,\na[8,6]=48448/130815,\na[8,7]=13 /170,\n\nb[1]=539/7344,\nb[2]=0,\nb[3]=190269/671840,\nb[4]=362551/179 0100,\nb[5]=-81/13600,\nb[6]=48448/130815,\nb[7]=13/170,\n\n`b^`[1]=13 7189/1827840,\n`b^`[2]=0,\n`b^`[3]=6386607/23152640,\n`b^`[4]=69923/32 6400,\n`b^`[5]=-118311/6092800,\n`b^`[6]=201/532,\n`b^`[7]=13/170,\n\n `b*`[1]=14222516/196928415,\n`b*`[2]=0,\n`b*`[3]=13/45,\n`b*`[4]=90846 8771/4726281960,\n`b*`[5]=33/3335,\n`b*`[6]=212565968/590785245,\n`b*` [7]=2066857/29174580,\n`b*`[8]=214645/35009496\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "T[5,7];" "6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 7 st age, order 5 scheme (the error terms of order 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,8];" "6#&%#T*G6$\"\"%\"\")" }{TEXT -1 145 " denote the vector whose components are the principal \+ error terms of the embedded 8 stage, order 4 scheme (the error terms o f order 5) and let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\" )" }{TEXT -1 99 " denote the vector whose components are the error te rms of order 6 of the embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7]));" "6#-%$absG6#-F$6#& %\"TG6$\"\"&\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,8] ));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\" \")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG 6#-F)6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/abs(abs(`T*`[4, 8]));" "6#/&%\"BG6#\"\"'*&-%$abs G6#-F*6#&%#T*G6$\"\"&\"\")\"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,8]-T[5,7]))/abs(ab s(`T*`[4,8]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\") \"\"\"&%\"TG6$F2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to ensure that \+ " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 73 " is a minimu m, if the embedded scheme is to be used for error control, " } {XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " should be chos en so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magn itude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "From this point of view the Bo gacki-Shampine scheme is near ideal with respect to the 8 stage, order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_7 := PrincipalErrorTerms(5,7,' expanded'):\n`errterms5_8*` :=subs(b=`b*`,PrincipalErrorTerms(5,8,'exp anded')):\n`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expa nded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := sqrt(add(simplify(subs(ee,`errterms5_8*`[i]) )^2,i=1..nops(`errterms5_8*`))):\nsdnB := sqrt(add(simplify(subs(ee,`e rrterms4_8*`[i]))^2,i=1..nops(`errterms4_8*`))):\nsnmC := sqrt(add((si mplify(subs(ee,`errterms5_8*`[i])-subs(ee,errterms5_7[i])))^2,i=1..nop s(errterms5_7))):\n'`B*`[6]'= evalf[8](snmB/sdnB);\n'`C*`[6]'= evalf[8 ](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B*G6#\"\"'$\")4)3 6\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\"\"'$\")jH:5!\"( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The \+ corresponding characterstics for the auxiliary 7 stage, order 4 embedd ed scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "`errterms5_7^` :=subs(b=`b^`,PrincipalErrorTerms(5,7,'expanded')) :\n`errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := sqrt(add(simplify(subs(ee,`errterms5_7^`[i]) )^2,i=1..nops(`errterms5_7^`))):\nsdnB := sqrt(add(simplify(subs(ee,`e rrterms4_7^`[i]))^2,i=1..nops(`errterms4_7^`))):\nsnmC := sqrt(add((si mplify(subs(ee,`errterms5_7^`[i])-subs(ee,errterms5_7[i])))^2,i=1..nop s(errterms5_7))):\n'`B^`[6]'= evalf[8](snmB/sdnB);\n'`C^`[6]'= evalf[8 ](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B^G6#\"\"'$\")>Xt 6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C^G6#\"\"'$\")[h$>\"!\"( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#================================== ===============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "An efficient 7 stage, order 5 Runge-Kutta scheme .. A" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#------------------ -----------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checki ng the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1774 "ee := \{c[2]=1/6 ,\nc[3]=16/63,\nc[4]=63/106,\nc[5]=50/69,\nc[6]=63/83,\nc[7]=1,\nc[8]= 1,\n\na[2,1]=1/6,\na[3,1]=80/1323,\na[3,2]=256/1323,\na[4,1]=155816577 /304900096,\na[4,2]=-16967475/9528128,\na[4,3]=568356831/304900096,\na [5,1]=64921911575/789872295744,\na[5,2]=1250725/43533526,\na[5,3]=9953 53041375/3166454547136,\na[5,4]=365139662800/1219687315371,\na[6,1]=-3 0354668306338585/178910020531686528,\na[6,2]=3896053749/4091707772,\na [6,3]=-2492114394031368057/6454959947119420288,\na[6,4]=86139947901867 3064/3177050598972839673,\na[6,5]=15/164,\na[7,1]=22116285273625596605 656418138617/38775599793952140053248982599680,\na[7,2]=-197127/114496, \na[7,3]=24118537135561993104813811090103/1477624772472274508104687555 5840,\na[7,4]=886475909491169318904581727562843/1979638263308617118988 089415360030,\na[7,5]=-185/122,\na[7,6]=408258859796431350937885801600 9/2571428541481836530593320891180,\na[8,1]=16270811903/209372688000,\n a[8,2]=0,\na[8,3]=76925739070593/205684631194240,\na[8,4]=747543949006 532/3259794500756955,\na[8,5]=-35320882628403/140715258838000,\na[8,6] =1551373170541891/3131789077344060,\na[8,7]=82/1099,\n\nb[1]=162708119 03/209372688000,\nb[2]=0,\nb[3]=76925739070593/205684631194240,\nb[4]= 747543949006532/3259794500756955,\nb[5]=-35320882628403/14071525883800 0,\nb[6]=1551373170541891/3131789077344060,\nb[7]=82/1099,\n\n`b^`[1]= 41084480131/527850886920,\n`b^`[2]=0,\n`b^`[3]=207911313213975/5566442 55095896,\n`b^`[4]=6621573612710978/28585355696715237,\n`b^`[5]=-52604 2283593017/2042110550893490,\n`b^`[6]=1/2,\n`b^`[7]=82/1099,\n\n`b*`[1 ]=1131790443129436/14550345379510605,\n`b*`[2]=0,\n`b*`[3]=71/190,\n`b *`[4]=3326466076155851042/14390291580335988345,\n`b*`[5]=-62/241,\n`b* `[6]=63595996553465987137/127170018616922687700,\n`b*`[7]=882970795744 2742/117440240421421575,\n`b*`[8]=-33340565403709/52195662409520700\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 577 "subs(ee,matrix([[c[2],a[2,1],``$2] ,\n [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1],a[4,2],a[4,3]], \n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a[5,4]],\n [c[6],a[6,1] ,a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3 ]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3]],[``,a [8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n [`b`,b[1],b[2],b[3]],[``,b[ 4],b[5],b[6]],[``$3,b[7]],\n [`b^`,`b^`[1],`b^`[2],`b^`[3]],[``,` b^`[4],`b^`[5],`b^`[6]],[``$3,`b^`[7]],\n [`b*`,`b*`[1],`b*`[2],` b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``$2,`b*`[7],`b*`[8]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777&#\"\"\"\"\"'F(%!GF+7&# \"#;\"#j#\"#!)\"%B8#\"$c#F2F+7&#F/\"$1\"#\"*xl\"e:\"*'4+\\I#!)vu'p\"\" (G\"G&*#\"*JoNo&F:7&#\"#]\"#p#\",v:\">#\\'\"-WdHs)*y#\"(D2D\"\")EN`V# \"-v8/``**\".Oraak;$7&F+F+F+#\"-+GmR^O\".r`J(o>77&#F/\"#$)#!2&eQjIoYNI \"3GloJ0-5*y\"#\"+\\P0'*Q\"+sxq\"4%#!4d!o8.%R9@\\#\"4)G?%>r%*f\\X'7&F+ F+#\"3kIn=!z%*Rh)\"4t'RG(*)f]q<$#\"#:\"$k\"7&F)#\"A<'Q\"=kl0mfDOF&G;@# \"A!o*f#)*[K0S@&Rz*fv(Q#!'Fr>\"''\\9\"#\"A.,46Q\"[5$*>cNr`=T#\"ASebvo/ \"3XFsCxCwZ\"7&F+#\"BVGcF#!$&= \"$A\"#\"@4g,e)y$4NJkzf)e#3%\"@!=\"*3K$fIl$=[T&G9d#7&F)#\",.>\"3F;\"-+ !)os$4#\"\"!#\"/$fq!Rd#p(\"0SU>JYo0#7&F+#\"0Kl+\\RaZ(\"1bpv+XzfK#!/.%G E)3KN\"0+!Q)e_rS\"#\"1\"*=aqJP^:\"1gSMx!*yJJ7&F+F+F+#\"##)\"%*4\"7&%\" bGFcpFfpFgpFjpFdq7&%#b^G#\",J,[%3T\"-?p)3&y_Ffp#\"0vR@88\"z?\"0'*e4bUk c&7&F+#\"1y4r7Od@m\"2P_r'pb`eG#!07&F+#\"4U5&e:wgYEL\"5X$))fL! e\"H!R9#!#i\"$T##\"5Pr)fY`l*ffj\"6+xoAph=+_Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "RK5_7eqs := [op(RowSumConditions(7,'expanded')) ,op(OrderConditions(5,7,'expanded'))]:\nsubs(ee,RK5_7eqs):\nmap(u->lhs (u)-rhs(u),%);\nnops(%);\n`RK4_7eqs^` := subs(b=`b^`,OrderConditions(4 ,7,'expanded')):\nsubs(ee,`RK4_7eqs^`):\nmap(u->lhs(u)-rhs(u),%);\nnop s(%);\n`RK4_8eqs*` := subs(b=`b*`,OrderConditions(4,8,'expanded')):\ns ubs(ee,`RK4_8eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F $F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the prin cipal error norm, that is, the 2-norm of the principal error terms." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_7 := PrincipalEr rorTerms(5,7,'expanded'):\nsqrt(add(subs(ee,errterms5_7[i])^2,i=1.. no ps(errterms5_7))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^ m+XX!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also calculate the 2 norm of the principal error of the ord er 4 embedded schemes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "` errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\nsq rt(add(subs(ee,`errterms4_7^`[i])^2,i=1.. nops(`errterms4_7^`))):\neva lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P))yP@!#9" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`err terms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nsqrt( add(subs(ee,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\nevalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+pBuX@!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-or der condtions to check for stage-orders from 2 to 4 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so ||ct||_7 := StageOrderConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 h ave the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j =1..5)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) \+ then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "In general, the simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j] ,i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " \+ . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 7 " ) are " }{TEXT 260 13 "not satisfied" }{TEXT -1 15 ", except when \+ " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 ", where the cond ition reduces to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 "." }}{PARA 256 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "[Sum(b[i ]*a[i,1],i=1+1..7)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j= 2..6)];\neval(subs(Sum=add,%)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),subs(e e,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG \"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\" \"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F 4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FO F-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F -&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF -" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7);\neval(subs (Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6 $*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6# \"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F- F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "T he simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = \+ 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3 \"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Sum(b[i]*Sum(a[i,j ]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F$6$*&& %\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F*;F5\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"%\"\"\"&%\"aG6$F(\"\"$ F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F+6$F4F(F)&F+6 $F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7);\neval(subs(Sum=add,% ));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6 #%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F) F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F) *(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "None of the principal error condit ions are satisfied by the order 5 scheme. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 115 "RK5_7err_eqs := PrincipalErrorConditions(5,7,'expa nded'):\nsubs(ee,RK5_7err_eqs):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions are satisfied by the 8 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 96 "The \+ largest magnitude of an error term is about 104 times the lowest magni tude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "` errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nsu bs(ee,`errterms4_8*`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(%))/min (op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")fl??!#7$\")eYZ`!#8$ \")Sc2SF)$\")(4*Q9F)$\")%Hi!e!#9$\")U\\%*pF0$\")fCh6F)$\")')[#y\"F)$\" ))4a$>F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%W5!\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 prin cipal error conditions are satisfied by the 7 stage embedded order 4 e mbedded." }}{PARA 0 "" 0 "" {TEXT -1 95 "The largest magnitude of an e rror term is about 39 times the lowest magnitude of an error term." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_7^` := subs(b=`b ^`,PrincipalErrorTerms(4,7,'expanded')):\nsubs(ee,`errterms4_7^`):\nma p(abs,evalf[8](%));\nevalf[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\"))[gi\"!#7$\")GRB%*!#8$\")Ta)o)F)$\")1)z$GF)$ \");\")e7F)$\")%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%vQ!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 88 "simultaneous construction of the 7 stage, order 5 \+ scheme and the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 123 "The following \"simple\" order condi tions (given in abreviated form) are relevant for the construction of \+ the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := SimpleOrderConditions(5):\n[seq([i, SO5[i]],i=[1,2,4,5,8,12,13,15,16])]:\nlinalg[augment](linalg[delcols]( %,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%%~~~~G/*& %\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F( \"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@6#*&FCF(F2F(F(F(#F(\"#C7%\"\")F )/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6#*&FCF(-F@6#*&F8F(FCF(F(F(#F(\" #g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF)/*&F,F(-F@6#*&FMF(FCF(F(#F(\"#? 7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b*`,SimpleOrderConditions(4)) :\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nlinalg[augment](linalg[delcols ](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"\"\"%%~~~~G/ *&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(# F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF(F(#F(\"#77%\"\")F)/*&F,F()F2 F:F(#F(F5Q)pprint366\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We i ncorporate the simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$* &&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-% $SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\" (\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[ j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\" \"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6 \"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 scheme and use the simple order conditions in the first table above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We also need to specify that t he linking coefficients for the 8th (FSAL) stage are given by the weig hts of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6#%\"iG&%\"aG6$\"\")F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG 6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the order 4 embedded schem e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7,'expanded'):\n`SO 4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded')): \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[1,2,4,7,8 ])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(StageOrderCon ditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3..7)=0,ad d(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i]*c[i]* a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [seq(b[i]= a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_eqns),op(SO_e qs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 16/63;" "6#/&%\"cG6#\"\"$*&\"#;\"\"\"\"#j !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 50/69;" "6#/&%\"cG6#\" \"&*&\"#]\"\"\"\"#p!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 63/ 83;" "6#/&%\"cG6#\"\"'*&\"#j\"\"\"\"#$)!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " are initial ly parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "e1 := \{c[2]=1/6,c[3]=16/63,c[5]=50/69,c[6]=63/ 83,c[7]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs(e1,cdns):\nnops(eqn s);\nindets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5]\};\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := solve( \{op(eqns)\},indets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*` [5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5973 "e3 := \{c[8] = 1, b[2] = 0, a[8,2] = 0, a[2,1] = 1/6, c[7] = 1, `b*`[2] = 0, c[2] = 1/6, a[5 ,1] = -25/7008192*(3326703*c[4]^2-12145248*c[4]+6274400)/(155*c[4]-109 )/c[4], a[3,1] = 80/1323, b[1] = 1/3024000*(1071600*b[7]*c[4]+118785*c [4]-1071600*b[7]+101507)/c[4], a[7,1] = 1/3473225328564311040*(-180142 1348336362619055*c[4]^3+1231539431628219072480000*a[6,5]*c[4]^3*b[7]-9 4813366096797381000000*a[6,5]*c[4]^3+60837856026891733845720*b[7]*c[4] ^3+3752804078830541750400*b[7]*a[7,5]*c[4]^3-2990009619603397516224000 *a[6,5]*c[4]^2*b[7]-33719129738044628514840*b[7]*c[4]^2-54911081737250 51000445*c[4]^2-8207005859673190859520*b[7]*a[7,5]*c[4]^2+235457488068 437419224000*a[6,5]*c[4]^2-191211581524247163827200*a[6,5]*c[4]+238604 2876369175072544000*a[6,5]*c[4]*b[7]+9149435792179988998392*c[4]+59796 57652874309416320*b[7]*a[7,5]*c[4]-28933595061313475484000*b[7]*c[4]-1 451556892891584864000*b[7]*a[7,5]-2993352216163493828112-6275726883939 96628800000*b[7]*a[6,5]+50997601442095209440000*a[6,5]+147835665344509 54672320*b[7])/(83*c[4]-63)/c[4]/b[7]/(155*c[4]-109), `b*`[8] = 1/2433 85747920*(56771753916906*c[4]^2+296200455218640*b[7]*c[4]^2-1576750621 78368*c[4]^2*`b*`[3]-30640353748200*`b*`[5]*c[4]^2+34892340008400*`b*` [5]*c[4]-418866872170320*b[7]*c[4]+105342924769792*c[4]*`b*`[3]-272135 68939305*c[4]-9195061680000*`b*`[5]+1533330211896+122666416951680*b[7] -16583737024512*`b*`[3])/(11208*b[7]*c[4]-1217*c[4]+504-4032*b[7])/(-1 +c[4]), a[4,1] = 2079/256*c[4]^3-81/16*c[4]^2+c[4], a[6,3] = -441/3871 6841344*(-1599349765797*c[4]^2+2446104786000*a[6,5]*c[4]^2-34927037108 00*a[6,5]*c[4]+1596319479756*c[4]+1246495660000*a[6,5]-399243329352)/( 9765*c[4]^2-9347*c[4]+1744), a[4,3] = 3969/256*c[4]^3-63/16*c[4]^2, b[ 3] = -1750329/576371840*(22800*b[7]*c[4]-10465*c[4]+4269-22800*b[7])/( 63*c[4]-16), a[8,1] = 1/3024000*(1071600*b[7]*c[4]+118785*c[4]-1071600 *b[7]+101507)/c[4], `b*`[4] = -1/12169287396*(96656205376*`b*`[3]-4360 6613169+6932784600*`b*`[5])/c[4]/(63-146*c[4]+83*c[4]^2), b[4] = -1/60 *(1071600*b[7]-101507)/c[4]/(-50400+334402*c[4]-626943*c[4]^2+360801*c [4]^3), `b*`[6] = -571787/15333302118960*(-2028214566*c[4]+2428234200* `b*`[5]*c[4]+2305695168*c[4]*`b*`[3]-1759590000*`b*`[5]+1014107283-585 573376*`b*`[3])/(83*c[4]-63), a[8,5] = 2518569/134354000*(-2735*c[4]+5 6400*b[7]*c[4]-56400*b[7]+3453)/(-50+69*c[4]), b[5] = 2518569/13435400 0*(-2735*c[4]+56400*b[7]*c[4]-56400*b[7]+3453)/(-50+69*c[4]), a[6,1] = 1/116150524032*(33364869281040*a[6,5]*c[4]^2+140931917847*c[4]^2+5326 014664908*c[4]-47640478615312*a[6,5]*c[4]+17002200802400*a[6,5]-306086 5525032)/(155*c[4]-109)/c[4], a[8,4] = -1/60*(1071600*b[7]-101507)/c[4 ]/(-50400+334402*c[4]-626943*c[4]^2+360801*c[4]^3), a[7,3] = -7/256789 790116131840*(23867234227250154897345*c[4]^3+1231539431628219072480000 *a[6,5]*c[4]^3*b[7]-94813366096797381000000*a[6,5]*c[4]^3-190807414811 533765120680*b[7]*c[4]^3+5777777540721508560000*b[7]*a[7,5]*c[4]^3-299 0009619603397516224000*a[6,5]*c[4]^2*b[7]+294996357340902044646120*b[7 ]*c[4]^2-39863789176611824048925*c[4]^2-12635419578675733728000*b[7]*a [7,5]*c[4]^2+235457488068437419224000*a[6,5]*c[4]^2-191211581524247163 827200*a[6,5]*c[4]+9206217794014699248000*b[7]*a[7,5]*c[4]+23860428763 69175072544000*a[6,5]*c[4]*b[7]-149480955957704793297120*b[7]*c[4]+222 14704206919870942632*c[4]-2234801667941589600000*b[7]*a[7,5]+509976014 42095209440000*a[6,5]-4332135862603012221072-627572688393996628800000* b[7]*a[6,5]+26945065022418889778880*b[7])/(1008-5297*c[4]+5229*c[4]^2) /b[7]/(155*c[4]-109), a[5,3] = 59675/778688*(188307*c[4]^2-199216*c[4] +50400)/(63*c[4]-16)/(155*c[4]-109), a[8,7] = b[7], b[6] = -47458321/1 966647060*(53580*b[7]*c[4]-4125*c[4]-53580*b[7]+4354)/(83*c[4]-63), a[ 8,3] = -1750329/576371840*(22800*b[7]*c[4]-10465*c[4]+4269-22800*b[7]) /(63*c[4]-16), a[7,2] = -3/32*(11208*b[7]*c[4]-1217*c[4]+504-4032*b[7] )/b[7]/(155*c[4]-109), c[5] = 50/69, c[6] = 63/83, c[3] = 16/63, a[8,6 ] = -47458321/1966647060*(53580*b[7]*c[4]-4125*c[4]-53580*b[7]+4354)/( 83*c[4]-63), a[7,6] = 571787/35831203487259161940*(5982993740906622000 *a[6,5]*c[4]^2*b[7]-460616819358712500*a[6,5]*c[4]^2+46175433874813845 *c[4]^2-384067881000342000*b[7]*a[7,5]*c[4]^2-10318496451708522000*a[6 ,5]*c[4]*b[7]+819968335758963600*a[6,5]*c[4]+569831462346762000*b[7]*a [7,5]*c[4]-67520580027438636*c[4]+4335502710801900000*b[7]*a[6,5]-2112 47393949900000*b[7]*a[7,5]-352310168025970000*a[6,5]+24647237032129551 )/b[7]/(83*c[4]-63)^2, a[3,2] = 256/1323, a[4,2] = -9/8*(21*c[4]-8)*c[ 4]^2, `b*`[1] = -1/766665105948*(380583808668*c[4]*`b*`[3]-21499074399 6*c[4]+9567242748*`b*`[5]*c[4]-96656205376*`b*`[3]+43606613169-6932784 600*`b*`[5])/c[4], a[5,2] = -25/24334*(102083*c[4]-60200)/(155*c[4]-10 9), a[6,2] = -567/2287148*(696733*c[4]-349272)/(155*c[4]-109), a[5,4] \+ = 34100/4761*(-50+69*c[4])/(63*c[4]-16)/c[4]/(155*c[4]-109), a[7,4] = \+ -1/215353753011180*(-5925835381049836312500*a[6,5]*c[4]^3+443588029343 711686980*c[4]^3-1603741336858034560620*b[7]*c[4]^3+769712144767636920 30000*a[6,5]*c[4]^3*b[7]-186875601225212344764000*a[6,5]*c[4]^2*b[7]+1 4716093004277338701500*a[6,5]*c[4]^2-169963952845586130000*b[7]*a[7,5] *c[4]^2+4038336619317219556260*b[7]*c[4]^2-995683859702860927725*c[4]^ 2-11950723845265447739200*a[6,5]*c[4]+149127679773073442034000*a[6,5]* c[4]*b[7]+248531821292745144000*b[7]*a[7,5]*c[4]+746669485984084400682 *c[4]-3358568190862369662660*b[7]*c[4]+923972908403184667020*b[7]-3922 3293024624789300000*b[7]*a[6,5]-90722305805724054000*b[7]*a[7,5]-18708 4513510218364257+3187350090130950590000*a[6,5])/c[4]/(434007*c[4]^3-76 9078*c[4]^2+417375*c[4]-63504)/b[7]/(155*c[4]-109), a[6,4] = -1/907425 969*1/c[4]*(3022180188500*a[6,5]*c[4]-504071108289*c[4]-2125275100300* a[6,5]+382608190629)/(9765*c[4]^2-9347*c[4]+1744), `b*`[7] = 1/1014107 2830*(-34126738287516*c[4]^2+11757637798200*`b*`[5]*c[4]^2+60504728050 368*c[4]^2*`b*`[3]+26710571726937*c[4]-12749755762800*`b*`[5]*c[4]-371 32434984448*c[4]*`b*`[3]+3065020560000*`b*`[5]-5622210776952+552791234 1504*`b*`[3])*b[7]/(11208*b[7]*c[4]-1217*c[4]+504-4032*b[7])/(-1+c[4]) \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 30 "the coefficients in terms of " }{XPPEDIT 18 0 "c[4]" " 6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$ \"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"( \"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do\n print(`=========================== ===========================`);\n temp_eq := c[ii]=subs(e3,c[ii]);\n \+ if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a p arameter`);\n else\n print(temp_eq);\n end if;\n print(``); \n for jj to ii-1 do\n temp_eq := a[ii,jj]=subs(e3,a[ii,jj]); \n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_e q)*` is a parameter`);\n else\n print(temp_eq);\n \+ end if;\n print(``);\n end do:\nend do:\n`################## ####################################`;\n``;\nfor ii to 7 do\n temp_e q := b[ii]=subs(e3,b[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n \+ print(lhs(temp_eq)*` is a parameter`);\n else\n print(temp_eq );\n end if;\n print(``);\nend do:\n`############################# #########################`;\n``;\nfor ii to 8 do\n temp_eq := `b*`[i i]=subs(e3,`b*`[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n pri nt(lhs(temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n \+ end if;\n print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% W======================================================G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\" \"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%W=================================================== ===G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"#;\"#j" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"$\"\"\"#\"#!)\"%B8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"$c#\"%B8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%W======================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\"%0~is~a~parameterGF(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"%\"\"\",(*&#\"%z?\"$c#F(*$)&%\"cG6#F'\"\"$F(F(F(*&#\"# \")\"#;F(*$)F0\"\"#F(F(!\"\"F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,$*&#\"\"*\"\" )\"\"\"*&,&*&\"#@F.&%\"cG6#F'F.F.F-!\"\"F.)F3F(F.F.F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"%\"\"$,&*&#\"%pR\"$c#\"\"\"*$)&%\"cG6#F'F(F.F.F.*&#\"#j\"#;F.*$)F1\" \"#F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%W=================================================== ===G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#]\"#p" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"\",$*&#\"#D\"(#>3qF(*(,(*&\"(.nK$F()&%\"cG6#\"\"%\" \"#F(F(*&\")[_97F(F3F(!\"\"\"(+WF'F(F(,&*&\"$b\"F(F3F(F(\"$4\"F:F:F3F: F(F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,$*&#\"#D\"&MV#\"\"\"*&,&*&\"'$3-\"F .&%\"cG6#\"\"%F.F.\"&+-'!\"\"F.,&*&\"$b\"F.F3F.F.\"$4\"F8F8F.F8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"$,$*&#\"&v'f\"')oy(\"\"\"*(,(*&\"'2$)=F.)&%\"cG6#\" \"%\"\"#F.F.*&\"';#*>F.F4F.!\"\"\"&+/&F.F.,&*&\"#jF.F4F.F.\"#;F;F;,&*& \"$b\"F.F4F.F.\"$4\"F;F;F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,$*&#\"&+T$\"%hZ\" \"\"**,&\"#]!\"\"*&\"#pF.&%\"cG6#F(F.F.F.,&*&\"#jF.F5F.F.\"#;F2F2F5F2, &*&\"$b\"F.F5F.F.\"$4\"F2F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================ ======================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"'#\"#j\"#$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",$*&#F(\"-KS_]h6F(*(,.*(\"/S5Gp[ OLF(&F%6$F'\"\"&F()&%\"cG6#\"\"%\"\"#F(F(*&\"-Zy\">$49F(F4F(F(*&\".3\\ m9gK&F(F5F(F(*(\"/7`hy/kZF(F1F(F5F(!\"\"*&\"/+C!3?-q\"F(F1F(F(\".K]_l3 1$F@F(,&*&\"$b\"F(F5F(F(\"$4\"F@F@F5F@F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#,$*&#\"$n&\"([rG#\"\"\"*&,&*&\"'LnpF.&%\"cG6#\"\"%F.F.\"'s#\\$!\"\" F.,&*&\"$b\"F.F3F.F.\"$4\"F8F8F.F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\"$T%\" ,W8%orQ\"\"\"*&,.*&\".(zl(\\$*f\"F.)&%\"cG6#\"\"%\"\"#F.!\"\"*(\".+gy/ hW#F.&F%6$F'\"\"&F.F3F.F.*(\".+3r.F\\$F.Fjf\"F.F4F. F.*&\".++m&\\Y7F.F3EQF,F,,(*&\"%l(*F,)F/\"\"#F,F,*&\"%Z$*F,F/F,F2\"% WEOO$[8U,=F()&%\"cG6#\"\"%\"\"$F( !\"\"**\":++[s!>#G;VR:B\"F(&F%6$\"\"'\"\"&F(F1F(&%\"bG6#F'F(F(*(\"8+++ \"Q(z'4mL\"[*F(F:F(F1F(F7*(\"8?d%Qt\"*o-cy$3'F(F>F(F1F(F(**\"7+/vT0$)y S!Gv$F(F>F(&F%6$F'F=F(F1F(F(**\":+SA;vR.'>'4+*HF(F:F()F2\"\"#F(F>F(F7* (\"8S[^GY/Q(H\">P$F(F>F(FKF(F7*&\"7X/+^]st\"36\\&F(FKF(F7**\"7?&f3>t'f e+2#)F(F>F(FGF(FKF(F7*(\"9+SA>uVo!)[daBF(F:F(FKF(F(*(\"9+s#Q;ZU_\"e67> F(F:F(F2F(F7**\":+Sas]F(F(*&\"7#R)**))*z@zN%\\\"* F(F2F(F(**\"7?jT4V(Gld'zfF(F>F(FGF(F2F(F(*(\"8+S[vMJh]fL*GF(F>F(F2F(F7 *(\"7+S'[e\"*G*ob^9F(F>F(FGF(F7\"77\"GQ\\jh@_L*HF7*(\"9++!)Gm*RR)osviF (F>F(F:F(F7*&\"8++W4_4U9g(*4&F(F:F(F(*&\"8?Bna4XMlc$y9F(F>F(F(F(,&*&\" #$)F(F2F(F(\"#jF7F7F2F7F>F7,&*&\"$b\"F(F2F(F(\"$4\"F7F7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"(\"\"#,$*&#\"\"$\"#K\"\"\"*(,**(\"&37\"F.&%\"bG6#F'F.&%\"cG6# \"\"%F.F.*&\"%<7F.F6F.!\"\"\"$/&F.*&\"%KSF.F3F.F#G;VR:B\"F-&F%6$\" \"'\"\"&F-F2F-&%\"bG6#F'F-F-*(\"8+++\"Q(z'4mL\"[*F-F9F-F2F-!\"\"*(\"9! o?^wL:\"[T23>F-F=F-F2F-FB**\"7++c3:sSvxxdF-F=F-&F%6$F'F'4+*HF-F9F-)F3\"\"#F-F=F-FB*(\"9?hkW?!4Mdj*\\HF-F=F-FKF-F-*&\" 8D*[S#=hw\"*yj)RF-FKF-FB**\"8+!GPtv'y&>aj7F-F=F-FGF-FKF-FB*(\"9+SA>uVo !)[daBF-F9F-FKF-F-*(\"9+s#Q;ZU_\"e67>F-F9F-F3F-FB**\"7+![#*p9Szp?/Z@AF-F3F-F-*(\"7++g*eTzm,[B#F-F=F-FGF-FB*&\" 8++W4_4U9g(*4&F-F9F-F-\"7s5A7Igie8KVFB*(\"9++!)Gm*RR)osviF-F=F-F9F-FB* &\"8!))y(*))=C-l]%p#F-F=F-F-F-,(\"%35F-*&\"%(H&F-F3F-FB*&\"%H_F-FKF-F- FBF=FB,&*&\"$b\"F-F3F-F-\"$4\"FBFBF-FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,$*&# \"\"\"\"0!=6Iv``@F,*,,H*(\"7+DJO)\\5QNe#fF,&F%6$\"\"'\"\"&F,)&%\"cG6#F (\"\"$F,!\"\"*&\"6!)po6PMH!)eV%F,F6F,F,*(\"7?1cM!eoLTPg\"F,&%\"bG6#F'F ,F6F,F;**\"8++.#pjnZ97(p(F,F2F,F6F,F@F,F,**\"9+SwWB@D7gvo=F,F2F,)F7\" \"#F,F@F,F;*(\"8+:qQtF/I4;Z\"F,F2F,FGF,F,**\"6++8'eXG&R'*p\"F,F@F,&F%6 $F'F5F,FGF,F;*(\"7gib>sJ>mLQSF,F@F,FGF,F,*&\"6Dx#4'GqfQo&**F,FGF,F;*( \"8+#RxWl_%Qs]>\"F,F2F,F7F,F;**\"9+S.UM2t(zw7\\\"F,F2F,F7F,F@F,F,**\"6 +S9XFH@=`[#F,F@F,FMF,F7F,F,*&\"6#o+W3%)f[pmuF,F7F,F,*(\"7gEmpB'3>o&eLF ,F@F,F7F,F;*&\"6?qm%=.%3H(R#*F,F@F,F,*(\"8++I*yCY-$HB#RF,F@F,F2F,F;*( \"5+S0Cd!eIA2*F,F@F,FMF,F;\"6dUO=-^8X3(=F;*&\"7++f]48!4]t=$F,F2F,F,F,F 7F;,**&\"'2SVF,F6F,F,*&\"'y!p(F,FGF,F;*&\"'vtTF,F7F,F,\"&/N'F;F;F@F;,& *&\"$b\"F,F7F,F,\"$4\"F;F;F,F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\"\"%0~is~a~ parameterGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*&#\"'(yr&\"5S>;fs[.7$e$\"\"\"*( ,:**\"4+?i14u$*H)fF.&F%6$F(\"\"&F.)&%\"cG6#\"\"%\"\"#F.&%\"bG6#F'F.F.* (\"3+Dre$>ohg%F.F3F.F6F.!\"\"*&\"2XQ\"[(QVvh%F.F6F.F.**\"3+?M+5)y1%QF. F )F.F3F.F7F.F.**\"3+?wYBYJ)p&F.F!3r-bL%F.F\"\" \"*&,**(\"&!e`F.&%\"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"%DTF.F7F.!\"\"*&F2 F.F3F.F=\"%aVF.F.,&*&\"#$)F.F7F.F.\"#jF=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"(&%\"bG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W################################################## ####G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"(+S-$F'*&,**(\"(+;2\"F'&F%6#\" \"(F'&%\"cG6#\"\"%F'F'*&\"'&y=\"F'F3F'F'*&F/F'F0F'!\"\"\"'2:5F'F'F3F:F 'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"(H.v\"\"*S =Pw&\"\"\"*&,**(\"&+G#F-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"&l/\"F-F5F-! \"\"\"%pUF-*&F1F-F2F-F;F-,&*&\"#jF-F5F-F-\"#;F;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"%,$*&#\"\"\"\"#gF+*(,&*&\"(+;2\"F+&F%6#\"\"(F+F+\"'2:5!\"\"F+&%\"cGF &F5,*\"&+/&F5*&\"'-WLF+F6F+F+*&\"'VpiF+)F6\"\"#F+F5*&\"',3OF+)F6\"\"$F +F+F5F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"(p&=D\"*+SNM\"\"\"\"*&,**&\"%NFF-& %\"cG6#\"\"%F-!\"\"*(\"&+k&F-&F%6#\"\"(F-F2F-F-*&F8F-F9F-F6\"%`MF-F-,& \"#]F6*&\"#pF-F2F-F-F6F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\")@$eu%\"+gqkm>\" \"\"*&,**(\"&!e`F-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"%DTF-F5F-!\"\"*&F1F -F2F-F;\"%aVF-F-,&*&\"#$)F-F5F-F-\"#jF;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"bG6#\"\"( \"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%W#################################### ##################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\",$*&#F'\"-[f5lmwF'*&,.*(\"-o' 3Qe!QF'&%\"cG6#\"\"%F'&F%6#\"\"$F'F'*&\"-'*Ru!*\\@F'F0F'!\"\"*(\"+[FCn &*F'&F%6#\"\"&F'F0F'F'*&\",w`?cm*F'F4F'F9\",pJh1O%F'*&\"++YyKpF'FF-F6F-,&*&\"#$)F-F2F-F-\"#jF6F6F-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(,$ *&#\"\"\"\",IG2T,\"F+**,4*&\"/;vGQn7MF+)&%\"cG6#\"\"%\"\"#F+!\"\"*(\"/ +#)zPwv6F+&F%6#\"\"&F+F1F+F+*(\"/o.0GZ]gF+F1F+&F%6#\"\"$F+F+*&\"/Ppsr0 rEF+F2F+F+*(\"/+Gwb(\\F\"F+F:F+F2F+F7*(\"/[W)\\VKr$F+F2F+F?F+F7*&\".++ c?]1$F+F:F+F+\"._px5Ai&F7*&\"./:M7z_&F+F?F+F+F+&%\"bGF&F+,**(\"&37\"F+ FMF+F2F+F+*&\"%<7F+F2F+F7\"$/&F+*&\"%KSF+FMF+F7F7,&F+F7F2F+F7F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"),$*&#\"\"\"\"-?zu&QV#F+*(,:*&\"/1p\"Rvrn&F+)&%\"cG6#\" \"%\"\"#F+F+*(\"0S'=_X+iHF+&%\"bG6#\"\"(F+F1F+F+*(\"0o$y@1vw:F+F1F+&F% 6#\"\"$F+!\"\"*(\"/+#[PNS1$F+&F%6#\"\"&F+F1F+FB*(\"/+%3+M#*[$F+FEF+F2F +F+*(\"0?.*F+FEF+FB\".'*=@IL`\"F+*&\"0!o^pTmE7F+F9F+F+*&\"/7X- PPe;F+F?F+FBF+,**(\"&37\"F+F9F+F2F+F+*&\"%<7F+F2F+FB\"$/&F+*&\"%KSF+F9 F+FBFB,&F+FBF2F+FBF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK5_7eqs := [op(R owSumConditions(7,'expanded')),op(OrderConditions(5,7,'expanded'))]:\n simplify(subs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The stability function of a general 7 s tage, order 5 Runge-Kutta scheme has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z^5/120+t [6]*z^6+t[7]*z^7;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*& F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F' F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[5,4]*a[4,3]*a[3, 2]*c[2]+b[7]" "6#/&%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"&F-&F /6$F1\"\"%F-&F/6$F4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F-&F+6#\"\"(F-" } {TEXT -1 2 " (" }{XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7, 6]*a[6,4]*a[4,3]*a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"(\"\"&\"\"\"&F&6$F )\"\"%F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F(\"\"'F*&F&6 $F:F-F*&F&6$F-F0F*&F&6$F0F3F*&F56#F3F*F*%!GF*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[7,6]*a[6,5]*a[5,3]*a[3,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a [4,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,3]*c[3]" "6#,(*,&%\"aG6$\"\"(\"\"' \"\"\"&F&6$F)\"\"&F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F (F)F*&F&6$F)F-F*&F&6$F-\"\"%F*&F&6$F>F3F*&F56#F3F*F**,&F&6$F(F)F*&F&6$ F)F-F*&F&6$F-F>F*&F&6$F>F0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[7]=b[7]*a[7,6]*a[6,5]*a[5,4]*a [4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\"\"\"&%\"aG6$F'\" \"'F,&F.6$F0\"\"&F,&F.6$F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9\"\"#F,&%\"cG6#F " 0 " " {MPLTEXT 1 0 250 "eq7 := b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2 ]=1/5129;\neq6 := b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]*(a[7,5]*a [5,4]*a[4,3]*a[3,2]*c[2]+\n a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]* (a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))))=1/718;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6#\"\"(\"\"\"&%\"aG6$F *\"\"'F+&F-6$F/\"\"&F+&F-6$F2\"\"%F+&F-6$F5\"\"$F+&F-6$F8\"\"#F+&%\"cG 6#F;F+#F+\"%H^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/,&*.&%\"bG6# \"\"'\"\"\"&%\"aG6$F+\"\"&F,&F.6$F0\"\"%F,&F.6$F3\"\"$F,&F.6$F6\"\"#F, &%\"cG6#F9F,F,*&&F)6#\"\"(F,,&*,&F.6$F@F0F,F1F,F4F,F7F,F:F,F,*&&F.6$F@ F+F,,&**&F.6$F+F3F,F4F,F7F,F:F,F,*&F-F,,&*(&F.6$F0F6F,F7F,F:F,F,*&F1F, ,&*&&F.6$F3F9F,F:F,F,*&F4F,&F;6#F6F,F,F,F,F,F,F,F,F,F,#F,\"$=(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We shall \+ require the following two equations to be satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve(simplify(subs(e3,\{eq6 \})),a[7,5]));\neqn7 := op(solve(simplify(subs(e3,\{eq7\})),a[7,5])); \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&%\"aG6$\"\"(\"\"&,$*&# \"\"\"\"<++!Qd=y9#e4DUp%F.*,,J*(\"D++v=JaI\"H-t]**H#*G[MF.)&F'6$\"\"'F *\"\"#F.)&%\"cG6#\"\"%F=F.!\"\"*(\"C+vIA=zis**\"E+++Pkj:!e&R4X#*GU,zWF.F4F.F9F.&%\"bG 6#F)F.F.*&\"CR4([HkF!>[fh*>YjYa\"F.)F:\"\"$F.F.*(\"C+0PC&Q[dbgfl')[4Z' *)F.F5F.FJF.F>**\"F++?P)RcN)HA&=VfvT3z7F.F4F.FJF.FEF.F>*(\"E++&=tnx3g! ***e`r[!)Q+\"F.F4F.FJF.F.*&\"C`Ue>Y2,vAtrg$*)*)R`\"F.)F:F8F.F>*(\"E++E jSJhnNCDe&>cF'f5F.F4F.FTF.F>**\"F++?![EBjd'zCB\\nj;-L\"F.F4F.FTF.FEF.F .*(\"C+6Cm`,>I3xM\"3#Qm!e)F.F5F.FTF.F.*(\"D+++zlg8:_ye?5o#>\"z%F.F4F.F :F.F.*&\"BH'=')fNq7xW)=T%ySumF.F:F.F.**\"E+++!e$*4w[*f,R$GJ%GYfF.F4F.F :F.FEF.F>*(\"C+\"zaC)*>!GER([1?4ed$F.F5F.F:F.F>*&\"B+]WHV%*ykPCC)p)GXV &F.F5F.F.*&\"C+++]Tbbhu(>IU0s\"oxF.F4F.F>*(\"D++++0no9.j_-Osa%f&*F.FEF .F4F.F.\"Bgeu$*o22Us$fLA')Qp5F>F.,&*&\"#$)F.F:F.F.\"#jF>F>,**&\"'bztF. FJF.F.*&\"(V%p8F.FTF.F>*&\"'+A#)F.F:F.F.\"'+v:F>F>FEF>F5F>F.F." }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/&%\"aG6$\"\"(\"\"&,$*&#\"\"\" \";++$pz\\<[_qbzX\"F.*.,B**\"A++>g'=o\"o:]1<>$\\(*)F.)&F'6$\"\"'F*\"\" #F.)&%\"cG6#\"\"%F=F.&%\"bG6#F)F.F.*(\"@+DJ8q9DfV89/#f4pF.F4F.F9F.!\" \"*(\"?Dv+?Rgv\"H*GrclEpF.F5F.F9F.F.*(\"@qMh!3\\TF![%)R2!z9:F.F5F.)F: \"\"$F.FC*(\"A+q%GY*HdDA&z*f!32t\"F.F4F.FHF.F.**\"B++>67$ecFMcHiz?)>#F .F4F.FHF.F>F.FC*(\"A++big.seDoL&*\\!)>9F.F4F.)F:F8F.FC*(\"@&*QgJ(ftmC( zE<#o.6F.F5F.FPF.F.**\"B+++E?;%[$4*Q>Q#))>x\"F.F4F.FPF.F>F.F.*(\"?]xfi bE-XT@T#z\"zEF.F5F.F:F.FC*(\"@++]#>v9JDsPbmkHQF.F4F.F:F.F.**\"A+++v%\\ ^tAeZgnMFr%F.F4F.F:F.F>F.FC*&\">I9D;yCXZoj9O&*)\\F.FHF.FC*&\"?9!>l^![h w'*)RrC$36F.FPF.F.*&\">=mK+EAwI:&H\"=7?)F.F:F.FC\">Ma]=^+UQYmeG:-#F.F. ,&*&\"#$)F.F:F.F.\"#jFCFC,&\"#]FC*&\"#pF.F:F.F.!\"#F>FCF:FCF5FCF.F." } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can \+ eliminate " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 57 " from the two equations by equating the two right sides." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(simplify(rhs(eqn6)-rh s(eqn7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\".\"y(>TrD\"\"7++! eFV\"*QlX1\"\"\"\"*2,&*&\"#$)F(&%\"cG6#\"\"%F(F(\"#j!\"\"F(,&*&\"$b\"F (F-F(F(\"$4\"F2F(,(*&\"*>N0u*F()F-\"\"#F(F(*&\"*g9fm)F(F-F(F2\"*?X)4 " 0 "" {MPLTEXT 1 0 147 "denom_factors := select(has,map(op,select(has,map(de nom@rhs,e3),c[4])),c[4]):\nmember(83*c[4]-63,denom_factors);\nmember(1 55*c[4]-109,denom_factors);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "However, the quadratic factor h as two real zeros." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "974053 519*c[4]^2-866591460*c[4]+170984520;\nsolve(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"*>N0u*\"\"\")&%\"cG6#\"\"%\"\"#F&F&*& \"*g9fm)F&F(F&!\"\"\"*?X)4N0u*\"\"\"*(\"$E\"F'F&!\"\"\".&R6t;N8#F'\"\"#F',&F$F'*(F)F'F &F*F+F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+=B3Vf!#5$\"+U " 0 "" {MPLTEXT 1 0 97 "433295730/974053519+126/974053519*1335167311395^(1/2);\nevalf( %);\nconvert(%,rational,5);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"*IdHL%\"*>N0u*\"\"\"*(\"$E\"F'F&!\"\"\".&R6t;N8#F'\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=B3Vf!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#j\"$1\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+EiRV f!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "W e set " }{XPPEDIT 18 0 "c[4] = 63/106;" "6#/&%\"cG6#\"\"%*&\"#j\"\"\" \"$1\"!\"\"" }{TEXT -1 57 ", which is a rational approximation for the larger root. " }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting for " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 32 " in each of the \+ equations for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 29 " now gives an equation for " }{XPPEDIT 18 0 "a[7,5]" "6 #&%\"aG6$\"\"(\"\"&" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "b[ 7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6 #&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "Si nce these two equations are only approximately the same we pick the fi rst one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(subs(c[4]=63/106,eqn6));\nevalf(%);\nexpand(su bs(c[4]=63/106,eqn7));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\"(\"\"&,**&#\",TnE8f*\"+@>r*\\#\"\"\"&F%6$\"\"'F(F.F.*&#\"8 r*\\#\"\"\"&F%6$ \"\"'F(F.F.*&#\".ZI&=8n8\"-?9%f42%F.*&&%\"bG6#F'!\"\"F/F.F.F:*&#\")*G+ `'\"*+g%*\\'F.*&F.F.F7F:F.F.*&#\"6>YmLF.*&F.F.*& F7F.F/F.F:F.F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&, **&$\"+5G(p$Q!\")\"\"\"&F%6$\"\"'F(F.F.*($\"+'oa#eL!\"*F.&%\"bG6#F'!\" \"F/F.F9*&$\"+)H.Z+\"!#5F.F6F9F.*($\"+A'o\"R:!#6F.F6F9F/F9F9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "cc := 63/106:\ndd := simplify(subs(c[4]=cc,rhs(eqn6))):\ne4 := \{ c[4]=cc,a[7,5]=dd\}:\ne5 := `union`(e4,simplify(subs(e4,e3))):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3023 "e5 := \{c[ 8] = 1, b[2] = 0, a[8,2] = 0, a[2,1] = 1/6, c[7] = 1, `b*`[2] = 0, c[2 ] = 1/6, b[6] = -95913266741/2499711921*b[7]+1367131853047/40709594142 0, a[8,3] = 1128962205/861903416*b[7]+361934780949/1310093192320, a[3, 1] = 80/1323, a[7,5] = 1/617757600859379594942524777765800000*(2370319 1177646860500568422729801800000*b[7]*a[6,5]^2-950943085281642690219253 5673860249-2074587358082811085826506712432530000*a[6,5]^2+620663099212 30588249561973422049700*a[6,5])/b[7]/a[6,5], a[6,3] = -144634480921413 /297614456504192+10506825/9619336*a[6,5], a[4,3] = 568356831/304900096 , b[3] = 1128962205/861903416*b[7]+361934780949/1310093192320, a[8,1] \+ = -38399/158760*b[7]+2606171/27216000, a[7,4] = -2809/9206636238296755 38416502266908378535408250*(-1855718391805074516271810254270898085000* a[6,5]^2+83366462285600734237912342433185118400*a[6,5]-668197427447283 895842949162194087249000*b[7]*a[6,5]+212025045084051167177584541318077 10100000*b[7]*a[6,5]^2-1776599293053525614549588042055152721)/b[7]/a[6 ,5], a[5,3] = 995353041375/3166454547136, a[8,6] = -95913266741/249971 1921*b[7]+1367131853047/407095941420, a[7,1] = -1/54402469913565593874 39866887776182400000*(-1436615624860406441261588414365070732775*a[6,5] +84810018033620466871033816527230840400000*b[7]*a[6,5]^2-7422873567220 298065087241017083592340000*a[6,5]^2+177193624370057606255324148618099 15375000*b[7]*a[6,5]-1541954103404946759797755659142208022)/b[7]/a[6,5 ], a[4,2] = -16967475/9528128, a[5,2] = 1250725/43533526, a[6,2] = 389 6053749/4091707772, a[5,4] = 365139662800/1219687315371, b[1] = -38399 /158760*b[7]+2606171/27216000, a[5,1] = 64921911575/789872295744, a[8, 7] = b[7], `b*`[4] = -2365395888512/981519741027*`b*`[3]+297754/273861 -222898644400/1289509846569*`b*`[5], b[4] = -2254783850560/19774306950 3*b[7]+457679459924/423735148935, a[4,1] = 155816577/304900096, c[5] = 50/69, c[6] = 63/83, c[3] = 16/63, `b*`[8] = 41/290710754460*(-654099 197535+2616396790140*b[7]+1168614653296*`b*`[3]-86933470050*`b*`[5])/( 4424*b[7]-369), a[6,4] = -383147600/227256813*a[6,5]+988954596252/2325 112941439, a[7,2] = 189/57248*(4424*b[7]-369)/b[7], a[3,2] = 256/1323, a[7,6] = 17696378319/1633314298000*1/(b[7]*a[6,5]), `b*`[6] = -571787 /1095444-51766736045/59977202166*`b*`[5]+244338307988/114130202445*`b* `[3], a[8,5] = -487879556697/128039362000+15270083847/320098405*b[7], \+ b[5] = -487879556697/128039362000+15270083847/320098405*b[7], a[6,1] = -324973/799848*a[6,5]-17349293115/130934648704, a[8,4] = -22547838505 60/197743069503*b[7]+457679459924/423735148935, c[4] = 63/106, `b*`[1] = -282140671/992436543*`b*`[3]+4399/23814+3567079/1303852221*`b*`[5], `b*`[7] = 1/93442742505*(68867301226185+13735488267900*`b*`[5]-184641 115220768*`b*`[3])*b[7]/(4424*b[7]-369), a[7,3] = 1/202363429016378345 7648647852114412672000*(2242267641356131706220494701981211412225*a[6,5 ]+84810018033620466871033816527230840400000*b[7]*a[6,5]^2-742287356722 0298065087241017083592340000*a[6,5]^2-25067017954702423593140822379279 381753000*b[7]*a[6,5]-2317310047866048348972848243519660598)/b[7]/a[6, 5]\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "We choose values for " }{XPPEDIT 18 0 "b [7]" "6#&%\"bG6#\"\"(" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 54 " that minimize the norm of the p rincipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "er rterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\nsm := 0:\nfor ct to nops(errterms5_7) do\n sm := sm+simplify(subs(e5,errterms5_7[ct]))^ 2;\nend do:\nsm := simplify(sm):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1081 "sm := 1/157060083404955582067100371961727431 814536547855614950038526392552520814705220059136000000*(-1614862455937 962101429260847313569655057844165243175210206460540756646784182622200* a[6,5]+736554553484832493904559930493603600089341689200443276130660796 11357251938278633+1910024445738622148262307632446983852011928043862033 91740245602646333710466216442600*a[6,5]^2+2572292863112821558021082453 1531594513218123961228879392604223022480321532400416000000*b[7]^2*a[6, 5]^2-38421843023605790862573732472456082902433845973134401463663581573 06160756928917920000*b[7]*a[6,5]^2+65464519310149673129975494670577317 759378376800629111501612698283621151203653600000000*a[6,5]^3*b[7]-5729 6869078067881268550520086578841240921596497608703808209844857912625845 57560000000*a[6,5]^3+2770396874650352980768509312587123197872904010891 4907663118204479805456504208000000000000*a[6,5]^4*b[7]^2-4849499200294 8777157303500470835247942904174665334880381955457304862891382736000000 00000*a[6,5]^4*b[7]+21222268467066455647701532085131246717446382972408 3814989559971398306752442280000000000*a[6,5]^4)/a[6,5]^2:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indets(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"aG6$\"\"'\"\"&&%\"bG6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "plot3d(sm,b[7]=0..0. 13,a[6,5]=0.01..0.4,labels=[`b[7]`,`a[6,5]`,``],grid=[40,60],style=pat ch,linestyle=2,\n lightmodel=light1,color=COLOR(RGB,0,1,0),axes=boxed ,view=[0..0.13,0.01..0.4,0..4e-6],orientation=[-25,60]);" }}{PARA 13 " " 1 "" {GLPLOT3D 451 437 437 {PLOTDATA 3 "6*-%%GRIDG6&;$\"\"!F($\"#8! \"#;$\"\"\"F+$\"\"%!\"\"X,%)anythingG6\"6\"[gl'!%\"!!#[at\"I\"gn3ED37E 9F0E583C7F3EC14FCF1CFD2F583EB993DA37F83D333EB9608EC6214BE23EBD64F76A90 62563EC2275DEF653C053EC6D10C311936563ECC94FDAAB644013ED1B33DA7DD7DAE3E D59F73FA2DAD593EDA0D4CDEC9A9593EDEFBB7730B276D3EE2350742FB782B3EE52BF5 6DC14F623EE862852028AC143EEBD8A0AE4AE9343EEF8E39ABA80ECD3EF1C1A32FDFBA 433EF3DBE01766C0EE3EF615D14A913EB63EF86F753E6C5A263EFAE8CAE6480FE53EFD 81D18D7244CC3F001D445E7F1CB23F01897814CA946D3F030583D35BAD883F0491678E DC254D3F062D2343A8DB443F07D8B6F383E21D3F099422A3EB41953F0B5F665CE55B51 3F0D3A822821F7C83F0F25761058A5C63F1090211069E2983F119573328E21203F12A2 B1745EA9FA3F13B7DBDB8CF4EA3F14D4F26DA792B73F15F9F530103A3A3F1726E427F5 61D53F185BBF5A4E74CB3F199886CBD9F10C3F1ADD3A811CE7CE3F1C29DA7E637CBD3F 1D7E66C7C218E93F1EDADF6117196B3F201FA22706680E3F20D5CAC90DDB863F218FE9 98465FF23F224DFE963DD7133F231009C46CB8113F23D60B243736D73F24A002B6EE63 433F256DF07DD13E103F263FD47A0DC2263F2715AEACC1E1753F27EF7F16FC75193F28 CD45B9BE209B3F29AF0295FA28773F2A94B5AC973C4B3ED32D188317EB113EC09A57D3 69AAF23EB7DE3247275EA73EB73A48802284E23EBAA82DD7DFA0343EC06AC503F1D40B 3EC4A34CA7C8BE1F3EC9E32507C6B22D3ED00ECB8BB5234D3ED3A602D573BCD33ED7B5 63739D1E043EDC3BDC838AFC9E3EE09C646A2110AF3EE355E094DC137C3EE64A41A848 196D3EE97971F87E7D773EECE36318FF47A43EF04405A8A4E7CF3EF233B202CD3E6D3E F440B45920CAFF3EF66B0B20ACB40C3EF8B2B54CC0F6403EFB17B228AB766F3EFD9A01 3D7F75223F001CD11FF316C73F017B4A81AFF3C43F02E86CB8A00ED83F046437C12047 883F05EEAB9CF2B0833F0787C8519551883F092F8DE70E8C3D3F0AE5FC670E29163F0C AB13DC4BB6E13F0E7ED4521195853F10309EE9F4589B3F11292836AE207C3F12290614 E79A283F1330388A2F56663F143EBF9BE70C0D3F15549B4F3D31863F1671CBA929320C 3F179650AE698B913F18C22A63834F4E3F19F558CCC2A0EF3F1B2FDBEE3BE97F3F1C71 B3CBCD861C3F1DBAE06921C8A03F1F0B61C9B12B133F20319BF8624F933F20E13170BB F42B3F2194714F5DFA7D3F224B5B95AC96703F2305F044F8D7E33F23C42F5E81BF933F 248618E37548663F254BACD4F1644D3F2614EB3404EC633F26E1D401B084323F27B267 3EE770363F2886A4EC90600A3ED2DFE62237C95A3EBFDE07D5884B153EB6413FD2AAC4 1B3EB5329F130D4ED03EB811611CCAC2D33EBD8B2DA96817993EC292207C24166E3EC7 53C4B507A5263ECCFDCC72DD4E573ED1C4CD44EC88233ED579C28F7B6DD13ED99CB536 750DF43EDE2D0008DE314F3EE1951DE55639473EE44A1324417B943EE7354A154A3A7E 3EEA56B44BF07E133EEDAE480FB39DFB3EF09DFF626800D63EF27FE9F3CAF7E63EF47C E2310FDAB53EF694E70D86A5EC3EF8C7F7D47D3E633EFB16140F06E4A53EFD7F3B71CC D3DD3F0001B6E827A9F63F01515589F0EFB53F02AE799A9F1AB13F0419231BF43D9A3F 059152136E60303F0717068913E4153F08AA40869491C23F0A4B0016A7F8053F0BF945 449876BD3F0DB5101BEEFA273F0F7E60A836B9443F10AA9B7A6951CB3F119CC9866FED 583F1295BA7D90E6203F13956E64FAB2883F149BE541A4BDD03F15A91F184D85E73F16 BD1BED7A1C073F17D7DBC576A3DB3F18F95EA45786713F1A21A48DFB20E33F1B50AD86 0BC50E3F1C86799001ECFB3F1DC308AF268A523F1F065AE695609E3F2028381C9FAF15 3F20D0A454F675663F217C721DA0131F3F222BA177DB88FD3F22DE3264D6D1E73F2394 24E5AFDFCD3F244D78FB758BCB3F250A2EA72879693F25CA45E9BBED253F268DBEC416 969B3ED29707EBB7D75D3EBE9BA6C6193E443EB4BD02DA826D913EB349927EE1A9AA3E B5A0913951CA303EBA6FA6C357C6933EC09D87AE2B3F483EC4E6DCB2791CED3ECA071B 621412953ECFF7A691301E8B3ED35A6A326498C03ED71E418BC95B753EDB46B423CB20 293EDFD35ABE5F81893EE261F99414D2853EE50C2904AE20473EE7E82D447BB2173EEA F5FC9AFCDF9C3EEE35906C6E10513EF0D3721A8FC56B3EF2A4FA6F95CE1F3EF48F6028 991EEE3EF692A290E79CAD3EF8AEC1319487E83EFAE3BBBF491BCE3EFD31920D85A038 3EFF9844059D8FC63F010BE8D02554BF3F02581D708889623F03B0BFE9766D8B3F0515 D042F562DC3F06874E86B531CD3F08053ABF6D692C3F098F94F86868DB3F0B265D3D2F 1D133F0CC993994CBCD73F0E7938182437AC3F101AA5626957903F10FEE5D50DC8713F 11E95D692DE4DE3F12DA0C23C118193F13D0F20985E00F3F14CE0F1F014DFA3F15D163 687F85853F16DAEEEA14EFBF3F17EAB1A79FE9C13F1900ABA4CAC5683F1A1CDCE50DFC BE3F1B3F456BB2816E3F1C67E53BD416FA3F1D96BC5863ABB53F1ECBCAC429A7703F20 038840E414F53F20A446C9DE9A4E3F214820FE325EA63F21EF16DEFD53F53F2299286D 4E534F3F234655AA2600403F23F69E96779F453F24AA033329DFFA3ED2527DDF981515 3EBD6D8C78862F743EB3517B5EAE5B113EB17F22C39F95763EB355BE2D74B64A3EB782 F555B2B5013EBD8B047BBC71563EC29C6D001B197D3EC73983E50E93543ECC9629C0EC A47C3ED1575A5C589ED03ED4C0818387E51F3ED885E52508EDEB3EDCA71E04D153E73E E091F4F7C21E423EE2FE0EC6AA2ED33EE597CE02A0E3B13EE85F28F32594873EEB5418 FC74A8CA3EEE76999ADE671F3EF0E353DC3E8E4B3EF2A2209DF861433EF477B25DEA91 4B3EF66408A5285EED3EF86723285B055A3EFA8101BB02CC703EFCB1A446732EC73EFE F90AC365EB683F00AB9A9AAF93DC3F01E611D3AD79993F032BEB14B3088F3F047D2667 7009353F05D9C3D69C0A593F0741C36D816BDD3F08B52537A919FC3F0A33E9409E4BB4 3F0BBE0F93C3F08E3F0D53983C372A183F0EF48344BB65FF3F1050685BD6C8883F112C 404F7E40E53F120DC982129A083F12F503F818E5243F13E1EFB5DD45EA3F14D48CBF74 25673F15CCDB18BBE0B43F16CADAC55EC9AF3F17CE8BC8D55A613F18D7EE266884743F 19E701E1340B6D3F1AFBC6FC28DD9A3F1C163D7A0F62D13F1D36655D89BACF3F1E5C3E A915E70B3F1F87C95F0FDD4F3F205C82C0D9C0C33F20F7F9898F42F03F2196490AA918 B73F223771451A86963F22DB7239CA3C293ED21247FDD882883EBC53B8ECCF1EA83EB1 FEA95F2E8C8E3EAFA69FC28E245B3EB130E7F93387263EB4C5196078E2E83EBA142056 7A05303EC074759DED9AD93EC49505FBCCD0973EC96524190EA2143ECEE1261AAF0004 3ED283751DB0AAF33ED5EA930C979A993ED9A5859E01E9AC3EDDB40A9E92BD983EE10A FB5B3E66213EE36596866012E23EE5E9CD182DBCB63EE8979874E3CB153EEB6EF418D2 84A43EEE6FDCEE1436723EF0CD286DA46CEB3EF277273B861C3C3EF435EA69C269B53E F60971AD0290883EF7F1BCD8C6D8923EF9EECBD662BC6A3EFC009E9E8FFB1E3EFE2735 34D2BA0C3F003147D213845B3F015956FE4CD52F3F028BC828C517FB3F03C89B5C33DB 8C3F050FD0A3E37FC43F0661680B5CF0DF3F07BD619E2B65DC3F0923BD67B1CE3F3F0A 947B730D4B993F0C0F9BCB014B9A3F0D951E79EABB0B3F0F250389B8706C3F105FA581 F3B3D03F1131FA78C0E18A3F120980AD8FE50C3F12E6382475276D3F13C820E14F05BF 3F14AF3AE7C7D1E33F159B863B5805E23F168D02DF4893663F1783B0D6B53DF33F187F 90248EF3D83F1980A0CB9E1CEC3F1A86E2CE84D8E83F1B92562FC129483F1CA2FAF1AF 03D53F1DB8D1168A4C763F1ED3D8A070B55C3F1FF4119163859C3F208CBDF5A4A3183F 21220BD7F7AB273ED1D66646791FB23EBB4E2C22F40BD63EB0C48CDC0302123EAC8C33 AFB03FA73EAE641D391C797D3EB23612E3AA50443EB6D662EC8F3A1D3EBCDDED17E142 033EC219A1A64ECA563EC66495999617543ECB4C288AC278AF3ED0671C5A43ACED3ED3 74BDDA7726313ED6CE9189F142D13EDA7455355528433EDE65DD84D58C633EE15186CF B93FA73EE395E90A15582B3EE6000ED5BB77333EE88FF3AEFBE3673EEB45947FF0E9D3 3EEE20EF2F3A83CA3EF0910129BA3D7F3EF224667F62A83F3EF3CAA74D3FBD543EF583 C366D1C49C3EF74FBAB56C38B03EF92E8D31C8D89E3EFB203ADF6BC9C43EFD24C3C951 1BA13EFF3C27FF85917C3F00B333CAB45E203F01D1C15034DCC63F02F9BC9B8EA4913F 042B25B84AA1BE3F0565FCB1F40B4A3F06AA4193EDD0C03F07F7F4695513A63F094F15 3CED41B23F0AAFA4191347AD3F0C19A107B5FC183F0D8D0C12525AD23F0F09E541F286 493F1048164F9762EB3F110FF11917F5CB3F11DC83015958E13F12ADCC0C05DE043F13 83CC3C95FF423F145E839652AE443F153DF21C57AE8F3F162217D195EE713F170AF4B8 D5D5C03F17F888D4B9843A3F18EAD427BEFB533F19E1D6B44230D93F1ADD907C7F08B2 3F1BDE01829335113F1CE329C87FFD083F1DED09502BE9933F1EFBA01B6459EA3ED19E D8B979EC963EBA5CE61AF4F7053EAF464BAA5777303EA9AF014EA57CD13EAAB2642F09 AE2D3EAFABC3BE8DFA2D3EB3D1CC3DFC10203EB917DF944857E93EBF8EADC929012A3E C3947E4283043D3EC7EFBC08EBA79B3ECCD6EE7281D6283ED124658EA790B23ED42241 C89F5F5E3ED764C9B3CB7C843EDAEBD1F85E9E093EDEB73DBD58D4083EE1637CC8DC66 E73EE38D7C1EFBAD253EE5D9985D5A83673EE847CE6E1336B53EEAD81C37C5C0693EED 8A80510DEA2E3EF02F7CE6091A8A3EF1AAC409128BBE3EF33715652390923EF4D470E3 8FA39C3EF682D67D1083EB3EF84246352A56DF3EFA12C016D92BF23EFBF4443229C66E 3EFDE6D29A7BB7403EFFEA6B653E1C0B3F00FF875482DA433F02125E3E722C983F032D BA7BF83C043F04519C1877F80B3F057E031F0E823D3F06B2EF9A7F48463F07F0619527 36F63F09365918F524CD3F0A84D62F660DA63F0BDBD8E18413F43F0D3B6137E77F0D3F 0EA36F3AB9210F3F100A0178DADA173F10C68E3218EE103F11875DCC8F46803F124C70 4B86D50F3F1315C5B21B5D3F3F13E35E033DCD643F14B53941B68D503F158B577027BC C03F1665B8910F5D2B3F17445CA6C9645E3F182743B391B63F3F190E6DB98605023F19 F9DABAA797B73F1AE98AB8DCF7593F1BDD7DB5F383213ED16B9F56DAE9303EB97FE6D4 D1E0303EAD34E8955172423EA70F089F6DDBD43EA74CA4D42EAC5E3EAB490CA69DD2C0 3EB1065C4AC087353EB596C2B11077683EBB3C4B6D3BE6AD3EC0F4DE13D568CF3EC4CB E0952A8CC93EC9210B7550CAC13ECDF3145251B4393ED1A0965A0C3F4B3ED4856819F5 BA5E3ED7A7D4111801343EDB07BD667323EB3EDEA510A905D1D43EE13FE050A46CED3E E34BE223EE64A23EE5768AB87B1D1D3EE7BFD7F4EA8FAE3EEA27C86FD886033EECAE5B 3B6B81313EEF538FC0F5F7913EF10BB2D3BC3C723EF27CEE60CCFD2C3EF3FD7A8066FD 043EF58D57360E61593EF72C848CBF39AA3EF8DB029486493D3EFA98D160C321003EFC 65F106E4DE923EFE42619D8041B43F0017119DD3916E3F01149AFC37F8063F0219CCF5 5044243F0326A79439975B3F043B2AE3B75F573F055756EE2688E83F067B2BBD75EA8A 3F07A6A95B22801B3F08D9CFD0366C103F0A149F2549F5BA3F0B57176285EF3C3F0CA1 388FA712C73F0DF302B40204133F0F4C75D687B73B3F1056C8FEE507C53F110B2B9800 4A073F11C362B98690B03F127F6E6640439A3F133F4EA0CF827D3F1403036BB24ED33F 14CA8CC9449E653F1595EABBC2551C3F16651D4549252D3F17382467DA55A53F180F00 255C6F813F18E9B07F9CD1F63ED13CBA1E9C15833EB8B72E508AC75B3EAB54F078F3F5 5C3EA4AC49A2095CB33EA432DF288B74093EA744007F842A3C3EACE82625B93EBD3EB2 5A966E39A07C3EB73C1C38D645333EBD0B6A1B1A8A103EC1E0962F7F28393EC5AC8FBC F437AF3EC9E85753F604E03ECE931E7C6FC53C3ED1D63067D3E1D13ED499E3CF01B5E9 3ED7948C9AC16EFD3EDAC6175A11BC643EDE2E76D56B6D083EE0E6D102B7871C3EE2D1 C95F289D073EE4D82266A8F1993EE6F9DAAFD44E823EE936F14CD134CE3EEB8F65A6F2 1AE23EEE0337653790793EF049332D24455F3EF19E793BCC43E93EF3016DE217E9383E F472112B0344CB3EF5F063269B19E93EF77C63E83EF97C3EF91613855E01283EFABD72 148CF0AB3EFC727FACDDA07B3EFE353C65667E9F3F0002D42A76B50C3F00F1E1C8D653 053F01E7C7189586E63F02E48424113D803F03E818F5384D4E3F04F2859587B2323F06 03CA0E098E9D3F071BE6675629E13F083ADAA99656203F0960A6DC86CD8D3F0A8D4B07 7C33E13F0BC0C731677D303F0CFB1B60DA8CCD3F0E3C479C0CE9C33F0F844BE8E070AE 3F1069942672F89C3F11146E66B0D5703F11C2B4B7A7D0463F1274671BB3DEEE3F1329 859510E5473F13E21025DC95913F149E06D01836D33F155D6995AA520B3F1620387860 466D3ED1122910BD718E3EB802BC8E1FAC853EA9A663553F00823EA286C45677FF743E A165132C2005393EA39C9F494100A73EA835E12CA0B1323EAEC6B59787A6563EB38E20 2BF81CB73EB88E065F5531D63EBE5BB9AFD2F3E43EC2797B496C1CF03EC62894223C13 5D3ECA3A58EA4492AD3ECEAE453ACBE5B23ED1C201321BBC1E3ED45DAB5A43B5393ED7 2A0DA4DC8D833EDA271ADA5F13E63EDD54C9F36BD5A53EE0598A621BB6783EE220FB8D 00E62C3EE400B7110143A63EE5F8BC00434FF33EE80909C41981723EEA31A0038467E0 3EEC727E912AF86E3EEECBA55E80B1323EF09E8A3946EE773EF1E365F1A54D533EF334 65E868386C3EF4918A30EF40B63EF5FAD2E0A983CC3EF770400E2BC16D3EF8F1D1D087 D2183EFA7F883ED423C83EFC19636FD695843EFDBF6379C96A703EFF718872337DE43F 0097E936E754C13F017D20C03C4D1C3F02686ADE95A3EB3F0359C79AFD7B9B3F045136 FE0C1B813F054EB90FEA55BD3F06524DD854E4813F075BF55EA06B853F086BAFA9BDDE E33F09817CC03F21EC3F0A9D5CA85BBBA53F0BBF4F67F588B23F0CE755049D58B73F0E 156D83976B323F0F4998E9DFC3113F1041EB9E1725F73F10E2143F7D66C63F1185465B 40562F3F122B81F3613B3E3F12D4C709C69EF33F138115A03DE0833ED0EBEC2D3EFD55 3EB762918D908FB03EA829412A3293A83EA09E78BCB9C40F3E9DC681BDD8BFC03EA052 E903D455FB3EA3F5E9AA3765D33EA9621F935E1ED83EB0325746A16D3B3EB47190F45A C8EE3EB967691CD303D83EBF0F9C3570F5083EC2B3CABD23DFAE3EC636DBFD96E6E03E CA107D7557DAF73ECE405874CC27B63ED16319A4F9F6A13ED3D0F38966452A3ED669AC B023CE613ED92D3C11D31F913EDC1B9B82A8D2D63EDF34C6CFE4DAD13EE13C5D935F65 743EE2F3BB55C1D2993EE4C07C186C2B433EE6A29F825EFF1F3EE89A25664143653EEA A70DB586762A3EECC9587736E2383EEF0105C14AA6803EF0A70AD9EDA4CA3EF1D8443A D3F6A73EF3142F18C7667B3EF45ACB8A5CB3FA3EF5AC19A6A5B7A43EF7081984B8DF7F 3EF86ECB3B5C0A8C3EF9E02EE0C97BEA3EFB5C448A880EF43EFCE30C4D519D4E3EFE74 863D03D3EA3F000859364C55443F00DBC87712330C3F01B490E96BCA9A3F0292B29581 EE0E3F03762D831157A13F045F01B96EAB053F054D2F3F8ADC563F0640B61BF7CEE13F 07399654ED09B13F0837CFF04C69643F093B62F3A6BDA63F0A444F644045F03F0B5295 471505313F0C6634A0DCE7013F0D7F2D760FB3253F0E9D7FCAE8CE113F0FC12BA36AC5 D93F10751881B156403F110C47F735A0383ED0CA037420B8CE3EB6D6AD4EDD70D83EA6 DD89F7CEAED93E9DE6CDA99D55053E995AD081E108093E9ACDBB5E7C547E3EA0283F9E 7D5C993EA4876ACFF6AA903EAA518311A46D7E3EB0B609DA2B4F543EB4E43AA5FE805A 3EB9AF1061B2A0C73EBF13F6495AD3A63EC288A7B666C1DC3EC5D3097F4BA36B3EC968 C9CFC17A313ECD49AEF5C8666E3ED0BAC907AEE3623ED2F62C56B99C893ED556F860A4 EBF13ED7DD26F9A56BB93EDA88B3EEFB0E8F3EDD599C6DDD67C83EE027EF4D4CBCC73E E1B5BCA3EA184D3EE35635E1C7562C3EE5095AD98B6B9D3EE6CF2B7CA9D6BA3EE8A7A7 D22AE2413EEA92CFF006AD2A3EEC90A3F656BE093EEEA1240BDA36AB3EF062282DB7A9 303EF17D14891FC8503EF2A1572F3751303EF3CEF03714B1C83EF505DFB77DC9323EF6 4625C6ACDA773EF78FC27A28C1003EF8E2B5E6AB56653EFA3F00201247A93EFBA4A139 578C823EFD1399448F69D63EFE8BE852EA6E573F0006C73A5D1F173F00CC45DCBC26ED 3F01967017E6F25D3F026545F2CE75863F0338C7740493B03F0410F4A1C0D3E53F04ED CD81E512CD3F05CF521A02200D3F06B5826F5C3B1D3F07A05E86EF66F03F088FE66573 8F123F09841A0F607B5A3F0A7CF988F190313F0B7A84D6295BB53F0C7CBBFAD4EFDC3F 0D839EFA8F0B143ED0AC6EE562A4033EB65F0FD20650003EA5C33DBE1352103E9B0B1D 3D6D65AC3E958712A458E35C3E95B0FA96FCFAD73E9999C612E52B093EA036974D5149 6F3EA4E2BDE514F2903EAAB6E2218D8A0C3EB0D22E4B5569593EB4D153179D3D323EB9 564AB1B1638D3EBE5F78296847473EC1F5E958A73F0F3EC4FD5675176FBC3EC845C9B8 04D5F13ECBCF1C3F6CD0533ECF99339C40FCA63ED1D1FEDFE13AD13ED3F7B7292D37AE 3ED63DBE774467A13ED8A411F75E5E003EDB2AAFCDC81CF03EDDD196CD2691243EE04C 6321BD6D133EE1C01EEB0971253EE343FEB3EAD2E43EE4D8028369DD0B3EE67C2A6F7E AE9F3EE830769842CE2B3EE9F4E724755D813EEBC97C3EF497F13EEDAE3614E9FCDC3E EFA314D4793D5C3EF0D40C55E79AA83EF1DEA0E43BD16A3EF2F1482B7386163EF40C02 411594053EF52ECF39DBD4D23EF659AF29A3F57E3EF78CA22367EDBF3EF8C7A8393C02 7F3EFA0AC17C50C26A3EFB55EDFCF7D1AB3EFCA92DCAAAA4C73EFE0480F412831D3EFF 67E7871154DB3F0069B0C86570543F0123778ED71A4A3F01E1481CBF84E93F02A32277 AF7FEA3F036906A4EB4AB13F0432F4A96EE8423F0500EC89F244293F05D2EE4AED2632 3F06A8F9F09AF2CA3F07830F7EFE380F3F08612EF9E407F33F09435864E720F03ED093 2E8104BEEF3EB5FBB9170B2D273EA4DA5C7D007D463E98A9E034E3BA0E3E924B482540 519D3E914F8FB12A9F0F3E93C7A7D62E21343E98DF4A16DBF7013EA0185F0794699C3E A4C38D305A54283EAA628819AF7DCE3EB076645730CA433EB42E92B34B6F253EB85832 30FE18593EBCF23A02D55BB03EC0FDFE64CE084F3EC3BA8390A93BC93EC6AE85A2F9A6 E73EC9D9EA2CB0E78B3ECD3C9F1F10184D3ED06B4C114036A73ED253E668C0E5FE3ED4 581BC341AD853ED677EA450F8F563ED8B350C0CF78323EDB0A4E8482878F3EDD7CE335 76A7DD3EE005875B496AA23EE15A688AF3D2A63EE2BD153FB2AAE63EE42D8D999F7A02 3EE5ABD1BF7961CF3EE737E1DC1E9D933EE8D1BE1CB8ACBB3EEA7966AF6B403C3EEC2E DBC26334323EEDF21D832C468D3EEFC32C1E3AFD923EF0D103DF4E88023EF1C75846E3 18863EF2C49359B8DD5A3EF3C8B52AC9CE3C3EF4D3BDCC2A30063EF5E5AD4F0A91653E F6FE83C3BC969D3EF81E4139B9B4143EF944E5BFAB31313EFA72716372F62D3EFBA6E4 3234C9A03EFCE23E385FB9AA3EFE247F81B77F723EFF6DA8195DBA673F005EDC04ED74 B73F010A57AE9389323F01B9470E5906403F026BAA28B5B3A93F03218101E4F5D83F03 DACB9DE95AEA3F04978A008FF4D73F0557BC2D7382103ED07E42470709933EB5ACA91D EC084D3EA422E63496308E3E96C316900052243E8F4EE2092EA5C33E8B52F55A0A8231 3E8DB4490DAB375F3E926528149981823E97E4CCF245A57C3E9F2429C179F9C43EA402 F7D50B01F33EA93C8840DA8FEC3EAF399C9C51ECD13EB2FB7D838EF6FD3EB6B948F32B DFB23EBAD5833DCA87E53EBF4FB8FF6B2FF93EC213CE3A044AA73EC4AE7C5EC2F9C43E C777D4DF32BFEF3ECA6FCB63BCD1463ECD965786E113613ED075B9D18756573ED2378E 006FD0C73ED410A722CEE5C13ED6010486A5B4B73ED808A5D14228123EDA278AE58B3C 013EDC5DB3D19185F23EDEAB20C14543F83EE087E8FA6CC1923EE1C5E3DCE643953EE3 0F8132ED63423EE464C129ABA0223EE5C5A3EF44AB193EE73229B1E560313EE8AA529F 197D703EEA2E1EE355891E3EEBBD8EA9A739EE3EED58A21B82431F3EEEFF5960A1FEA7 3EF058DA4F7D2DFB3EF137D9FD59F2743EF21CABCB17DB5E3EF3074FC9088CFE3EF3F7 C606A57BB63EF4EE0E9297EEF63EF5EA297AC1CEF73EF6EC16CC46E24F3EF7F3D69396 37173EF90168DC7386803EFA14CDB2006FF63EFB2E051EC572563EFC4D0F2CBA93863E FD71EBE54FAAB93EFE9C9B517447753EFFCD1D799F32B63F0081B932EAC4403F011FCD 0ED8B6793F01C0CA54342E6D3ED06DAA376983F13EB571DFE6A8E1743EA39CDAE4D46B DB3E9556C04EC32DEB3E8B371A84BBCE503E857D771519C2033E85A27849B734E83E89 FD9127B664B33E90E1A873805B983E9644F2656B08653E9D0B57917ABE623EA291E4E6 A56C8E3EA741FB0493F26C3EAC92B44235C67D3EB140FF82520A153EB4874046BA4532 3EB81BA90A53D50F3EBBFDEC091976F13EC016EA3277334D3EC2559F002A6C813EC4BB 08160F9B703EC7471D0EA6A54F3EC9F9D8445EB0F93ECCD335FFD1C2803ECFD333E649 B3823ED17CE849E461783ED32385A97562C73ED4DD71F4BED9CD3ED6AAAD39D15C1E3E D88B37A49D27BB3EDA7F11755549A43EDC863AF978059C3EDEA0B486C1D22E3EE0673F 3BC2D71F3EE187CC94057DE13EE2B2027A55B9543EE3E5E11C3F47933EE52368A636AE D43EE66A994349A5D33EE7BB731CEBDFD23EE915F65AD8B6B43EEA7A2323041A013EEB E7F9999693863EED5F79E0F140823EEEE0A419B769933EF035BC316DFBAF3EF0FFFB6C D8BC6F3EF1CF0FCCFDDF3C3EF2A2F95F012AAA3EF37BB82F51ACD93EF4594C49B31EF3 3EF53BB5B947206A3EF622F4889630163EF70F08C19853D93EF7FFF26DBD62F13EF8F5 B195F4ECE03EF9F04642B5BA9C3EFAEFB07C04E8393EFBF3F0497C99C53EFCFD05B252 4BF83ED06166522C2E053EB54B5D7141B8983EA3483A8DBB2F283E9464DD712C4D813E 884F39BD281CAB3E811EA49382FD783E7EB3BAC10075E43E8158572942133B3E864EA1 25B1EB9D3E8DD2E8990FA88C3E93D547E98D2ADA3E99D9BD3F8455063EA076409F5CEF 923EA483901343BA023EA912BB608FB5A53EAE2267C8D6914E3EB1D8D7420C66CF3EB4 DFFA0525F2D33EB826674F9B28163EBBABFB03EE3C1B3EBF709C72F197133EC1BA1D68 D281E93EC3DB656A7367B53EC61C2286F5817F3EC87C5263A2A88E3ECAFBF39C7D1BCB 3ECD9B057C20AFFF3ED02CC3E42DAEDA3ED19BBD4EA727E63ED31A6F296D011C3ED4A8 D9B4B2478E3ED646FD3DF53F053ED7F4DA1AF25E043ED9B270A5FCA35F3EDB7FC13B5B 71223EDD5CCC37687F463EDF4991F53B49D23EE0A30966DE6EB43EE1A9278B8453AF3E E2B7239203070D3EE3CCFDA215E2DA3EE4EAB5E1B0D68F3EE6104C74FC6BDE3EE73DC1 7E59C0423EE873151E6C1BFE3EE9B047742668133EEAF5589CDB33393EEC4248B44E4D E63EED9717D4C7457A3EEEF3C6172435DC3EF02C29C97648D03EF0E2602F31CBB43EF1 9C86474D22A23EF25A9C1BC0535F3EF31CA1B5FB352C3EF3E2971EED578B3EF4AC7C5F 0D83673EF57A517E60D4F23EF64C168481700F3EF721CB78A4D1BA3ED05976974F07D3 3EB53921BDB68DBC3EA325052F4A7A813E93ED6DF73BB0B33E86973FB273911B3E7C6C FBAA8C69563E75B4F0A40C939C3E75B5445BAC1CA03E7AD715413DCFAD3E82235EEDA0 BB223E88C781649A92BD3E909AEAD86193943E95ACDAD959C7B43E9B931CF48F91823E A124C6FA1AA3E53EA4E6BC2DBB24453EA90E874D29CA993EAD9B8CD05C11B23EB146B1 7D8C38573EB3F1E0C931A9133EB6CF3B9BF0C3773EB9DEB12AC9521A3EBD20362A99A2 8E3EC049E1964ADE973EC21CA9BDA8AA703EC40872276874013EC60D3A1E260FBE3EC8 2B0167AF762F3ECA61C82025D2683ECCB18E9F69A0373ECF1A5565E0F9E93ED0CE0E87 44335F3ED21B73226C69FB3ED37558DEBC1FB03ED4DBC0187AB6503ED64EA92C01E61E 3ED7CE1474692AF03ED95A024A9991723EDAF27304AE86F33EDC9766F58F71EF3EDE48 DE6CB3065E3EE0036CDB00919F3EE0E8AC8CE56E0B3EE1D42E6E6935C33EE2C5F2A02F 31423EE3BDF9412A49563EE4BC426EAD0CF33EE5C0CE447B4C663EE6CB9CDCDC94E83E E7DCAE50AF02A13EE8F402B77A08283EEA119A2780E4043EEB3574B5D494333EEC5F92 766524453EED8FF37C1242DB3EEEC697D8BB0EDB3EF001BFCEA68D1D3EF0A3556CE94E AD3EF1480CCEBFEFFB3EF1EFE5FB5FEDD83ED055DB06D211563EB53B2CCC0760DC3EA3 333AC9824DE53E93F071E0F157B23E860F2C649E2B8C3E798A05B4C6CF1D3E7048923C 92C2553E6A11C8D5C9587E3E6C68355F1EB0A13E72F68F91128FC93E7ADB07AEEC684F 3E82CEA52FC528FA3E898A05B20F3EAD3E90C95EEE83E5333E957043A289BDD73E9AB6 FB784486B33EA04DDC6FDAA0F53EA38EA2646372233EA71D65DD832E3C3EAAF9DEA03E 40283EAF23DB4E3977C73EB1CD9D28BA35BA3EB42FF289D5DAC83EB6B8E65BA3B3633E B96873E8B772D73EBC3E98691596CF3EBF3B5271F5C9023EC12F50C779FD0C3EC2D442 FC293FF23EC48C802EE9295F3EC65808E01FD3523EC836DDAAC9C54E3ECA28FF3A5FEC 4A3ECC2E6E4388464D3ECE472B7ED1970E3ED0399BD277A73E3ED15949B608328C3ED2 829FC30379AA3ED3B59E4F84EA763ED4F245AE73EAED3ED638962F472F4E3ED788901D E6968B3ED8E233C2A333F83EDA4581623F426A3EDBB2793E0152D13EDD291B93CF36D9 3EDEA9689E4E0C1B3EE019B04A82B9CB3EE0E381D64243BE3EE1B2290B43C02B3EE285 A6010EA16B3EE35DF8CDE6265C3EE43B2186DB4CB23EE51D203FDE20353EE603F50BCE 4F4E3EE6EF9FFC8AF9213EE7E0212301AF9D3EE8D5788F3CAAA83EE9CFA6507032943E EACEAA750741073ED05693A0B54A953EB5517E9C3431FE3EA372DB5C62A9493E946DE9 2E4D42623E86B6FFD3A7EBF33E79946745B52C2B3E6CDD3F1526046E3E61571D005BAB 693E5AD14A5961C6333E5ED5194E4A7A8A3E66725CAEB4E8593E70FBD1F81D60A23E78 D3E4F70D93643E8153CC28C8DE2C3E87332F983999F93E8E02B1D02FBAB73E92DF5BD8 56A1243E9732698CC40E223E9BF9AE8CED129E3EA09A4D316D418E3EA3716528EB0153 3EA681FD96EF5CFC3EA9CBFFE536F0303EAD4F5D3629CBBA3EB086060F77AA953EB281 0387C844873EB498A5993E5CC03EB6CCEBCF75E4643EB91DD662B1314F3EBB8B6602B1 3BDD3EBE159BB04237403EC05E3C50AE99D03EC1BFFF167A05173EC33016DE295B6B3E C4AE84604B60FC3EC63B48559385683EC7D6637430C1023EC97FD66DF52CBA3ECB37A1 EF17A3AE3ECCFDC69D66F2AA3ECED24517D080E43ED05A8EFB1206B63ED15328E481DF 723ED252F08CFA051B3ED359E635C108F63ED4680A1CBCBC673ED57D5C7C923D333ED6 99DD8CC92C473ED7BD8D81F0A3DA3ED8E86C8DC4DCBB3EDA1A7ADF54BB5D3EDB53B8A3 26BD123EDC9426035CDDD53EDDDBC327D7356B3EDF2A9036551FE03EE04046A94A6E03 3EE0EEDD4F38C6923EE1A10C1CFFD20D3EE256D321D3D84B3EE310326C1FD3173ED05B A064F8B38A3EB57C172E3D011E3EA3E3E6E7EB8CB83E9565D3DF4F70C83E888EB9FF90 D2523E7C8C205D5780D03E7027188E0D53943E613A85370F30C23E515EF3D66469893E 441711160D19233E454BB613FA41F03E521D08ACFDE6A43E5FCBE078D196CF3E69A593 318978A73E731629C3B9D7EF3E7AB242ABC12FEE3E81CFEF086F663D3E86ED0FD95F40 4B3E8CAEDCD7CA04F73E918A1A8BE0AE3F3E950E298FEC44EE3E98E3584C06F2253E9D 097993A9EB063EA0C037D5D2A8BA3EA32413DF23F8D83EA5B04B55D1E1773EA864DB64 4BB5D63EAB41C321C9579B3EAE4702FECF717C3EB0BA4E2D6FFAFD3EB265489BC9A5CD 3EB42472006DC7D33EB5F7CBB24E3C2E3EB7DF571AB6FDE93EB9DB15AAC595443EBBEB 08D3E0E1CB3EBE0F320266D3A13EC023C94CFDCE983EC14A15F9867A613EC27A7FAC9C 39873EC3B507052C74133EC4F9AC9B06E7F63EC64870FECDBC973EC7A154BA05670C3E C904584F3B09303ECA717C3A3A46453ECBE8C0F04D59F83ECD6A26E08385FE3ECEF5AE 73FAFEC63ED045AC07161A533ED11592069A79093ED1EA896511192E3ED2C4924C9BF5 E93ED3A3ACE51B26DF3ED487D9544E08D63ED57117BDF2DA723ED65F6843E4BEE73ED7 52CB063826C23ED84B402355A2343ED948C7B81324883ED06501539C4C383EB5BAF682 21CE3C3EA4865D6C1CF82D3E96D831F3F7E2E73E8B965AE858DE943E8138987DD6E627 3E7571FD4701B59F3E69BC0179E3E8873E5C796735454B423E4AFB17F4FCDD5A3E32BE 0F05AE3B2D3E0FCD7C784E493B3E1C513664BCC6013E3A70F8FA89570A3E4E83721FD7 0C223E5B8A2C0CDD31B53E65AC5F1AC3BEC33E6F4CFF444CDC8C3E75502D5FB8B3473E 7BD0F6EE3C02043E8193A8AD6B6FF63E85A917FAC2FEE23E8A286F061715133E8F1172 5907F71B3E9231FE24199FAE3E950FFB49DE6EF93E9822AAF2B10CDB3E9B6A0B4D013E 783E9EE61D39D85AC13EA14B71BC1D21313EA33E3105ABE2D63EA54B4EE58A90A63EA7 72CE099CDEBD3EA9B4B1447AC4283EAC10FB785F50803EAE87AF881842F13EB08C6826 30BF393EB1E2304641B26F3EB345317CACB1163EB4B56D150F55953EB632E44D446F4E 3EB7BD9854EB7A0C3EB9558A4D48C3483EBAFABB496537933EBCAD2C4E5B18473EBE6C DE53C1AE543EC01CE92216B72D3EC10A047EDE6EC03EC1FDC1A95632483EC2F82104F6 72763EC3F922EFDCF93A3EC500C7C316C0DB3EC60F0FD2E7C2413EC723FB6F102AB73E C83F8AE30EB4823EC961BE765FDBF83ECA8A966CB9E8513ECBBA130645BB313ECCF034 7FD4745F3ECE2CFB1311F9963ED072B66CA0149E3EB60E1C97E2995F3EA55A3EE8F6EB A73E98C5036C4698CB3E8FCDE28E0010CF3E85A1CC905C08123E7E4F4DB570294E3E75 6DC8E46CEA1E3E6E10523B0239483E649591CE5913B53E5B01B383B524033E50937184 C327523E4276594949CCA93E3246A363D47AED3E22A59626CA3CA43E22C794D8396E5C 3E32063D6E3BE3B23E4176300A3FF8963E4DF0AA2112404D3E57303F1E57AC073E60AE 553192E0DD3E66BCB20F730A4A3E6DC1CCC495F7583E72DE5BBBBE5DBE3E77566DBD4E DBE43E7C48EFE74450A73E80DAE5C68CAD213E83CE7CB44681863E86FF3E7EE253DF3E 8A6D30A467E57E3E8E185B2B38DA293E910063E18ED1B33E93134091D28CE73E9544C9 4B9789743E979503D353E8E73E9A03F5EEA32ADF3E9C91A54EFD11993E9F3E1782F264 043EA104A8F61CFBAD3EA279ACDCA51A833EA3FE19F0C72F2E3EA591F291C22DEF3EA7 3539021EB6553EA8E7EF67EE9ECB3EAAAA17CD6669F63EAC7BB421B4A9AA3EAE5CC63A 0248923EB026A7E946269F3EB126A947E5C74A3EB22E67FFD0E6423EB33DE4CD4316A2 3EB45520625650B83EB5741B67927EC03EB69AD67C7803193EB7C95238044E993EB8FF 8F29305A8C3EBA3D8DD768B5063EBB834EC2FF22803EBCD0D26595E0213EBE26193284 BBDA3ED084BFB0040CBC3EB675896F7F627A3EA65F8B5E79672B3E9B2C48483B92513E 929AA8784334893E8B81AC663B25323E855F84ECAC57133E81264D88FC3C563E7C152A E62875433E774513DC686D5F3E73671A9436763D3E703706640000823E6B171619EE76 4F3E6695B31905A77C3E62C8CA16D84E873E5F355E5982AB7A3E59F9EBB591A04C3E55 CBCB408CCBF03E529DC573430E963E5066D2EE41CA493E4E418FCC4601EA3E4D8EE335 B2E3773E4EAFFA87F105533E50D08F2E822B0D3E532FF9FF6355693E56758BB7BB490E 3E5AA0E9A574A5363E5FB1F6AF8B6CD03E62D460734A94043E6642BA1D67ED403E6A24 286FA446033E6E78D22921842C3E71A071140D6F9F3E743E42CB07F4153E7715F54B58 BE2B3E7A279FAB6DCA8A3E7D7358B1262BD93E807C9B4B627F5F3E825CA77192CC3F3E 8459DB2810CF403E8674405DB305533E88ABE08F7542553E8B00C4C77A12EC3E8D72F5 9E08B11A3E90013D9DF9A1F53E9157AEAEFA047D3E92BCD1AC5751E13E9430AA0E8BA1 683E95B33B1E8898DA3E974487F811B36B3E98E4938C15463D3E9A9360A2FB2F5F3E9C 50F1DEE31CAE3E9E1D49BDCE8EE23E9FF86A9BB34D723EA0F12B5A3AA0563EA1ED8812 E3502F3EA2F14C78773C0D3EA3FC797E3ADF4A3EA50F100ADDD85D3ED09B1D1DC83494 3EB6F13D08F829923EA79642CCA46AAD3E9E0E0087D6CF903E95E65307F5F39D3E916C 1BFFBA1F243E8D6098D95DA1FE3E89BD3BA2CA509A3E8722AF93A3AF243E852721EF29 A29E3E838FF5051B35FB3E823B0C80AF21003E8113C2427E7BB43E800D2EC93C9C203E 7E3E17529C61053E7C87DBD213EAB03E7AF074D0E3DB043E7973060EB807783E780C41 A9C42B6F3E76B9E5CA2B98B63E757A677D14D59B3E744CBA127C8CC73E733028D84799 213E72243CE1B677343E7128AAD80469533E703D463B7940B83E6EC3F0BF1F289E3E6D 2D73FC0CEDDB3E6BB71D360693B23E6A6102652B0A013E692B43A300E1843E681607AE A9F6E03E672179666189953E664DC5F3A718173E659B1B7A2B913A3E6509A826CB0C4E 3E64999985436F773E644B1C0A1B11C43E641E5AC308A8DF3E64137F23BF70FF3E642A B0E79951203E6464160285DF7E3E64BFD29D0ECDD03E653E09185183013E65DEDA1796 50A53E66A2648DC644183E6788C5CD70164B3E6892199A64072C3E69BE7A3C2FCF973E 6B0E0090EF25173E6C80C4201483553E6E16DB2CDEC11F3E6FD05AC84B9E7D3E70D6AB 71337AFD3E71D6F12D6F92EB3E72E9078869985C3E740CF6F8EA86E53E7542C7840E79 B53E768A80C3F97EFB3E77E429EE284C7F3ED0B5CEB5EC8C213EB78137644CEEAE3EA8 FE653377F6383EA0B516158C284A3E99C9F0F61845A23E95D2B7AE03A9733E93957150 65FAC03E923DD75FD058AC3E916631783776073E90D9963B3B44C53E907AB6D0BB4A8A 3E9038850E52E3933E9008ADE44E9B303E8FC97B24103A973E8F9121C36381163E8F62 E4ACEDC5FD3E8F3B28C0B10DE63E8F177FDE833FF53E8EF643157E3D6A3E8ED65179B3 AE063E8EB6E591B6D3E13E8E9779058606833E8E77B17C13B17C3E8E57537F00165B3E 8E363962E8D1403E8E144CE7F1CB893E8DF182B7D856103E8DCDD72F7BF4623E8DA94C 10EF579E3E8D83E6DA3A3C4C3E8D5DAF91BE4B493E8D36AFE743F6223E8D0EF2925A10 443E8CE682DD607ABE3E8CBD6C514368D53E8C93BA7939E4B83E8C6978B8336E6E3E8C 3EB22B5192893E8C1371960204283E8BE7C155318E383E8BBBAB57B6B1A73E8B8F391A 8D46623E8B6273A7D7C99E3E8B356397DD87C83E8B08111370A3CD3E8ADA83D74B5216 3E8AACC3381141913E8A7ED626B6916E3E8A50C3351E900D3E8A22909AD047363E89F4 4439A859AD3E89C5E3A276739E3E899774197A00CE3E8968FA9AB601573E893A7BDE15 FD7A3E890BFC5B61C64F3E88DD804DFDC6C23E88AF0BB8781A713E8880A267E2BD973E 885247F6FC231D3ED0D4D47871136A3EB82578817DB1C93EAA97F292F409D53EA2A065 99000AB23E9E458242AA2AC03E9AF4A93DFA31CA3E995F31217BA8DF3E98B0D36FBFAF 893E9883D7C4BAF82E3E98A354C20D753D3E98F1FB8F96AC363E995EBE02F1F93C3E99 DF490C6BE7983E9A6D28EB5E22FC3E9B043A69FB02623E9BA1C80B6DE3F23E9C44043F B7C92E3E9CE9B7F6C3A7ED3E9D920FB81E6D993E9E3C7B0DD0EFD53E9EE8973B24D02F 3E9F9621141939033EA02275B613B0D23EA07A6C440FB4343EA0D2EAC9280EC63EA12B EBB565BA763EA1856C33380C2E3EA1DF6B59D6E3F33EA239E999C6EDF23EA294E8528A 1BBC3EA2F06985B8D3B93EA34C6F9F4533F13EA3A8FD4D12F26C3EA4061561BA039E3E A463BABF757A3F3EA4C1F04913144A3EA520B8D74EB3D13EA580173170BBFB3EA5E00E 085497423EA6409FF33579FF3EA6A1CF6DC905573EA7039ED74E2EAD3EA76610724D94 BD3EA7C92664D9098D3EA82CE2B925183A3EA891475E604C133EA8F65629B392DB3EA9 5C10D75BF48C3EA9C2790BD2436F3EAA299054F840573EAA91582B4513DE3EAAF9D1F2 ECABE03EAB62FEFCFEE2D83EABCCE0887C75A53EAC3777C3604ABA3EACA2C5CB9C51C4 3EAD0ECBB009AD103EAD7B8A714BDA1D3EADE90302A75F2B3EAE57364ACBD88A3ED0F8 2E6555CA6A3EB8DE00608A72E43EAC62EAEB18A5713EA4C8EECE470EE53EA1AC8376D5 D1633EA068F857CEDBD03EA006C5EFF7EDA43EA01BC9009996623EA0752557AE2F373E A0F8664605B19A3EA196E45F8FE0003EA248188F1A68913EA306D94CCB91233EA3CFEC B8502F7A3EA4A14136C16EE23EA5797C09B4FEA13EA657B6692B5F703EA73B54CD1A0B B13EA823ED79477AD23EA9113832B817CE3EAA03039B10B2D03EAAF92E1C50E1A83EAB F3A120353EEA3EACF24DC925DB3B3EADF52AAC4A1CFE3EAEFC3239AAFFDA3EB003B0CD DBEBA63EB08B5BF4D4423F3EB1151ACA00D9AC3EB1A0EDFD22A9E63EB22ED68F05E73B 3EB2BED5B5A3A0A53EB350ECC7EDB0483EB3E51D2F3011203EB47B685C884BE73EB513 CFC15D409D3EB5AE54CA0CDF2D3EB64AF8DA3B593A3EB6E9BD4A56657E3EB78AA365FB 9D313EB82DAC6B05CFED3EB8D2D9891479F63EB97A2BE16BE9A13EBA23A4871507CB3E BACF447F2A1B1E3EBB7D0CC14269353EBC2CFE37F266F63EBCDF19C15898833EBD9360 2FB16A4E3EBE49D249EDBBCA3EBF0270CC491EF73EBFBD3C68DD8C123EC03D1AE41977 A63EC09CAEC4E4D49B3EC0FD5A224F957C3EC15F1D43D637723EC1C1F86D2F01183EC2 25EBDE82D4763EC28AF7D4A2D0C73EC2F11C893ADCF23ED11FDC7C9AB11E3EB9AACF01 7331FF3EAE5F4E3BE5C9133EA72EB1B56134F73EA4823F7B8E56E53EA3B54701771DC6 3EA3D040C5E149083EA469098A15680E3EA54CC51C0DD4573EA660FECC9A87163EA797 0F2FAB42E33EA8E66F300DB5A43EAA49F545E825CF3EABBE6790E7685A3EAD41B47624 A9333EAED281373597123EB037F3860D59893EB10CA53664BC103EB1E720CE842E773E B2C74230EDE6043EB3ACF0AE744B1D3EB4981B7C17288A3EB588B74EB4CBDB3EB67EBC B7813CF83EB77A2701102FB53EB87AF3636621373EB9812073BABBA03EBA8CADBDA9EF 313EBB9D9B7976120C3EBCB3EA56E01C923EBDCF9B56B4428C3EBEF0AFAEEB92C93EC0 0B945B3BF3E13EC0A183EB529E673EC13A2740488D4C3EC1D57F12D130533EC2738C19 1B7F523EC3144F04F993303EC3B7C882A2479D3EC45DF937E468D53EC506E1C3AD5F28 3EC5B282BDCCE82D3EC660DCB6E52B083EC711F0387A9D033EC7C5BDC51A60763EC87C 45D89018033EC93588E825FFA03EC9F18762EB582C3ECAB041B1FE57C83ECB71B838D7 6DDF3ECC35EB559464B73ECCFCDB614237CB3ECDC688B024DD7A3ECE92F391FC85153E CF621C5247E8A23ED01A019C41C1CC3ED084544432CF7B3ED0F006410B8CB73ED15D17 B1348A963ED1CB88B183BBF23ED14BDEBE3FC78F3EBA8BE46437EF153EB0468E42ADBA 603EA9D1AE4E4E7CEC3EA7A3F52F7EA5E83EA75F409BF5DEAC3EA80C091279E6923EA9 402B54534CF93EAAC8CB2F7C6B903EAC8B73F4C53B513EAE797E381D7EE83EB0453172 2972253EB15C7C38C5D8E93EB281027FBA5E443EB3B1BB799398383EB4EDF9C71C5E0C 3EB6354A0454F0553EB7875E6D380C6C3EB8E40223A8EDB53EBA4B110A29C9C93EBBBC 72718D08F03EBD38158ED275893EBEBDEF16D85D773EC026FBCD6964533EC0F41532AA B5663EC1C642573260A83EC29D82859C3AD13EC379D583B63CAB3EC45B3B6DA190A33E C541B49B3EB2F03EC62D418CF3BDC13EC71DE2DDBD38783EC81399391410793EC90E65 539E43463ECA0E47E5EA947C3ECB1341A8AC74B03ECC1D535212DBF23ECD2C7D93EFE2 853ECE40C11A7A64983ECF5A1E8B812DE63ED03C4B42F8D3A03ED0CE14D0CDC6DD3ED1 626C379084AF3ED1F951BC0347033ED292C59F6C9F3F3ED32EC81FB25F4B3ED3CD5977 79A4933ED46E79DE4A10313ED5122988B2BC223ED5B868A86FE0953ED661376C906335 3ED70C96019AC0F63ED7BA8491B0F5313ED86B0344B318013ED91E12406085AE3ED9D3 B1A8777D583EDA8BE19ED321E73EDB46A24387E38D3EDC03F3B4FE534E3EDCC3D6100C 632D3ED17C352A450DB63EBB814088D8AA263EB1762AE3BCD4393EACB1E4990EE6B83E AB11A492A6BE723EAB66E5274B1E653EACBA1ED5C1C62C3EAEA12E5F5345233EB0749B C8FCFA583EB1BBE2DF42E7113EB31F18BC7349FD3EB499F9D5F4FA0F3EB629F167DB1A 7C3EB7CD6281FC15E33EB9834456E482143EBB4AE9875F36743EBD23DEAF6C73A33EBF 0DD60B06F7073EC0844D5E08FDE73EC18A045287DBC63EC298038B69495B3EC3AE42A3 2D2BC23EC4CCBBF442AA153EC5F36BC743C8263EC7224FC17A5FFF3EC859667EE8AFE5 3EC998AF4A2A8D203ECAE029E90DEFB83ECC2FD677B403023ECD87B54DFD42D63ECEE7 C6EC4FC8EA3ED02805F6D40ED83ED0E0427EBF96923ED19C99683D7ADC3ED25D0B0F95 1D963ED32197D12036C23ED3EA4007F642873ED4B7040D004D683ED587E43659C4923E D65CE0D6EA0EBB3ED735FA3E27DF9E3ED81330B7FB16813ED8F4848CB4C6C23ED9D9F6 01172B093EDAC3855668D4193EDBB132CA8F94053EDCA2FE983088493EDD98E8F6D351 F53EDE92F21B070AD53EDF911A3687EB513EE049B0BC326C993EE0CCE40692287A3EE1 52270F7427653EE1D979EAC874F53EE262DCAB6EBFB33EE2EE4F634626D33EE37BD223 3C3C003EE40B64FB5B37203EE49D07FAD760DD3EE530BB301BB23C3ED1B0DFC0AA8397 3EBC8AE36F55633B3EB2BE7D012032133EAFCF5495A272653EAECB4DA506A0633EAFCC 34A376DD1D3EB0ED4107DC73FB3EB24609558AA8353EB3D70521C337EF3EB592FA14EE 1FBA3EB77294790340F43EB97190C36972A03EBB8D5A3033D7C73EBDC453CF38DAF13E C00ABA698289203EC14007EE31AA933EC281D8C3A9F1E73EC3D00607E8BDC83EC52A75 4BDFAC253EC6911480CFD7913EC803D74F22744D3EC982B5515767773ECB0DA8E1DDD7 FA3ECCA4AE494FCA323ECE47C32CF717343ECFF6E628D5FE563ED0D90B43C42A213ED1 BCAA076E08A43ED2A64F6D7930273ED395FBA1D5DED43ED48BAEE4B721863ED5876983 9B3C0D3ED6892BD43EA4F23ED790F630F45B5B3ED89EC8F603C0E83ED9B2A47FC68D8F 3EDACC8929543DA13EDBEC774B95DD8B3EDD126F3CA6DAA03EDE3E714F6E9BAA3EDF70 7DD363D4103EE0544A8A3731953EE0F35BAD6FAE5F3EE1957275BC7D6D3EE23A8F03C0 E7653EE2E2B1766FD5193EE38DD9EB1BD5473EE43C087D88B8923EE4ED3D47FE0C0D3E E5A1786359EBBD3EE658B9E723CA623EE71301E99EE6763EE7D0507FDC3E113EE890A5 BDCBDC9F3EE95401B64D70A63EEA1A647B4019713EEAE3CE1D91689F3EEBB03EAD4B96 653EEC7FB639A2EB033EED5234D1025FC23ED1E9DE8170292D3EBDA8CD17AE1A4D3EB4 1F849AD7D3EF3EB194FF22048FF53EB16878334F25F03EB24797883C8D623EB3B69960 2FA6013EB5806C1BCCB7793EB78BA1A210EE8C3EB9CAFF9B6447C33EBC373251BEA48D 3EBECBF63A86DBE43EC0C35B48E808513EC232EB33B856DA3EC3B42676FAA47A3EC546 B663145CDE3EC6EA614605A0463EC89F003DCBCD3A3ECA6478DB5881E93ECC3AB90FEC D8883ECE21B483F2054D3ED00CB168F3F6F13ED110DF2A1EDC5F3ED21D61A9C6B5153E D33237BA906DD73ED44F60AA7D26233ED574DC1EDAC8663ED6A2A9FA9050973ED7D8CA 4BAE546D3ED9173D3E2511DC3EDA5E03122796003EDBAD1C1534249A3EDD04889D0733 CE3EDE644903F3C3203EDFCC5DA64133DC3EE09E6370249F373EE15AC28691AFB03EE2 1B4C42DC51133EE2E000D0123A473EE3A8E057A71EEC3EE475EB0156585A3EE54720F3 13D68B3EE61C82510823163EE6F60F3D945B2B3EE7D3C7D95BC7983EE8B5AC435150FA 3EE99BBC98C785F33EEA85F8F582375B3EEB746173C8F2093EEC66F62C79D2963EED5D B7371C4B103EEE58A4A9F39A703EEF57BE9A10BE343EF02D828DB1E14E3EF0B13C2066 2ACB3EF1370C0E14CB4F3EF1BEF25F348B943EF248EF1BCA87EC3EF2D5024B70E4403E F3632BF55D1D5B3ED227316C95FE7F3EBEDAFD81E2CF623EB59941B0E3B9D43EB360F0 D22177A33EB391466BB6E06D3EB4D7EA3728EBA63EB6B91873DA79063EB8FFBF826FD0 443EBB927149E61E223EBE63F372A55F153EC0B6792352BA5B3EC254951DA69AD83EC4 0B034657E29B3EC5D8F52551C6FF3EC7BDE653DA92FC3EC9B9802257B24E3ECBCB88DE C944FD3ECDF3D9A72CA9CB3ED0192C0639BF513ED14378FFEF6F133ED278CD94EBFE24 3ED3B925926F5F663ED5047E25B126063ED65AD572FE542D3ED7BC2A4CFBD69A3ED928 7C01AACCC23EDA9FCA365920FC3EDC2214CDEDCFD83EDDAF5BD6796E4B3EDF479F7BEC 3A973EE0756FFF3CA0E93EE14C8ED5CF63DF3EE2292C6C8CA15A3EE30B48F09DD8EF3E E3F2E49026BB743EE4DFFF795425023EE5D299D9B1D37B3EE6CAB3DDB44D6C3EE7C84D B069495D3EE8CB677B447A7F3EE9D40166013ACF3EEAE21B969379F83EEBF5B63123C1 863EED0ED158132EE23EEE2D6D2C050A6F3EEF5189CBEC3D503EF03D93AA8DAB043EF0 D522F1AB128C3EF16F72C8721BC53EF20C833B51D4EF3EF2AC54560DF7363EF34EE623 C821EF3EF3F438AF08D4023EF49C4C01C813753EF547202575B7423EF5F4B523014F0A 3EF6A50B02E1A3D03EF75821CD1BD17D3EF80DF98949FC613EF8C6923EA1A1123ED268 D8821C03873EC010BA56F9C13A3EB72BB44343E3BC3EB54B7F5B27F0423EB5E0117BBA 7FA53EB797125E8089663EB9F4BE42DCED273EBCC4038973F2B73EBFEB741942C6B23E C1AEEACD58B2E53EC389EA2BDBD8A73EC5849662DE40183EC79DA510697AC13EC9D447 BC68BDF23ECC27FA002254B03ECE98652BFBAA9F3ED092A7C6FA6FF73ED1E7492205A9 C53ED34A096F98516F3ED4BADFA852F4783ED639C5A06A2CBD3ED7C6B7251DED233ED9 61B163A5C8EA3EDB0AB2804EC2333EDCC1B94DBDC5AB3EDE86C519F3F3153EE02CEAC5 1F9A413EE11D7540C3430B3EE2150206ED3EDF3EE313912D95AC793EE41922D4D61287 3EE525B7236D92F23EE6394E443A68EB3EE753E8646613E83EE87585B21444DA3EE99E 265B71D7C73EEACDCA8E0A8AC13EEC04727652E4103EED421E3F589A573EEE86CE128F 60E03EEFD28217B291A03EF0929D3A5B0DF03EF13F7BA6E144EB3EF1EFDC629C7C403E F2A3BF7DDE582B3EF35B2508204D383EF4160D100BA2A23EF4D477A38240C53EF59664 CFA7EE333EF65BD4A0EBB9043EF724C723115A653EF7F13C613A71B33EF8C13465EF7D F33EF994AF3B2884EB3EFA6BACEA555DAC3EFB462D7C65980D3EFC2430F9CFFCEC3EFD 05B76A99A83A3EFDEAC0D65CBE0F3EFED34D444EBB443ED2AED3C20238473EC0BE194D F019C63EB8D6DC51F851A53EB754AABD17F9CF3EB854D9635A03A03EBA850FFE436696 3EBD698ACD3702523EC0669C186C8F553EC24B55081374273EC45C5309C42DDE3EC695 EC427AAD3B3EC8F5FEECEA5DA43ECB7B40A71CD0A93ECE24E2F8FD3B9E3ED07930BDE8 F4CB3ED1F1B2C00023023ED37BDAA9C438B93ED517950A33E51F3ED6C4D4A9C7F7393E D883908120FC463EDA53C264738E523EDC356620FFA0293EDE2878E3FCC5133EE0167C 68DBFFAC3EE121725E6B1DAF3EE2351DF9AC4C793EE3517F0D4681273EE476958AAD39 C93EE5A46178E8C1443EE6DAE2EDF1366B3EE81A1A09E01FE33EE96206F3749F8D3EEA B2A9D58CF0993EEC0C02DD52925D3EED6E1238E9362F3EEED8D8167DB7A43EF0262A51 CDEA923EF0E444065C0A593EF1A6B93E7016A03EF26D8A0EC3E8DF3EF338B68B352E2F 3EF4083EC6BDDE343EF4DC22D3723DE53EF5B462C282DBF93EF690FEA4415C823EF771 F6882734143EF8574A7CDDAA153EF940FA9046A6973EFA2F06CF85F04B3EFB216F470A 955F3EFC183402984EDD3EFD13550D50BC4C3EFE12D271BC5C9F3EFF16AC39D3357D3F 000F7137828F223F0095BA8D20D32B3F011E3221FFCB7E3F01A8D7FA2205F13F0235AC 195494A43F02C4AE833236023ED2F9232C489CC13EC1759BA5D4714A3EBA9AB9DD0103 933EB97C72F7F194493EBAEF9E22956C5A3EBDA1E31671833F3EC08BBF09745C513EC2 8DAEBC4FAA213EC4CA099749416F3EC73A326E9520883EC9DA7F672F38053ECCA8CEBB CAF3823ECFA3D60A71E4603ED165636D87A0173ED30E8E6374A8CD3ED4CD408F32C23F 3ED6A15D17C1FC8A3ED88AD08C21070B3EDA898DB4C8B09A3EDC9D8B8A59868A3EDEC6 C3E10822F93EE08299430A3C4A3EE1AC6A535B0D4E3EE2E0D4339D05B43EE41FD64D22 9BC63EE5697046EC5F423EE6BDA1F3A145593EE81C6B44B4CC363EE985CC412F3E743E EAF9C4FF08BB223EEC78559E5AC8EA3EEE017E45E489903EEF953F2084383C3EF099CC 2DB1AA463EF16E451252C7AA3EF2480A553BE2673EF3271C0D32D9613EF40B7A5071EF CF3EF4F525348001083EF5E41CCE16E6A43EF6D86131144D9B3EF7D1F270722DBF3EF8 D0D09E44CBDA3EF9D4FBCBBCB69C3EFADE74092B92573EFBED39660AD35D3EFD014BF1 03C1133EFE1AABB7F843E43EFF3958C80C22123F002EA996D735073F00C34D7A516A6B 3F015A98140580F33F01F4896937B8043F0291217EE4129C3F03306059C27C623F03D2 45FE4ABCEE3F0476D270B838EB3F051E05B50D7E463F05C7DFCF179EF33F067460C271 599F3ED347C6C0EF30F33EC237415EA6C7D03EBC774CE45DF9833EBBC2D80BB4BFBA3E BDB05FB96CB9D73EC076C5D3856FAD3EC27F4C09F907FE3EC4D739B06349BD3EC771D7 BA42CB433ECA4888FBCB8ADC3ECD57A399F979323ED04E82E7C000E03ED20BB29D345B 073ED3E2F9B14F65BC3ED5D415F0B446703ED7DEDC0395B3173EDA032F10F3BBA43EDC 40FBA7CD0FA03EDE9834909A7DAD3EE0846861FE49B23EE1C9650B13F5593EE31B0E2A 2E3B203EE4796255E8E4C13EE5E460A06A73523EE75C0873055D183EE8E05974BA31E4 3EEA7153782FE6E03EEC0EF66ED9FA2C3EEDB9425FC0B6793EEF703760DC3AC83EF099 EAC92306E13EF1820E8D5EA8853EF270871290200B3EF365546F4C2D243EF46076BAA4 A84A3EF561EE0BAFFF803EF669BA793411E03EF777DC196B225D3EF88C5301DC0C793E F9A71F4740A9BD3EFAC840FD76A71E3EFBEFB83777FC7C3EFD1D850758EEA33EFE51A7 7E4A0C453EFF8C1FAC9CF9913F006676D0E5957D3F010A08B63EF3EF3F01B0C58D4B8C 453F025AAD5C9D41D83F0307C02A6B9B903F03B7FDFC98761A3F046B66D8B4A1303F05 21FAC4045C373F05DBB9C383AA3A3F0698A3DBEA76A73F0758B911B089543F081BF969 1146D23F08E264E60F3D1B3F09ABFB8C777E323F0A78BD5FE4C8793ED39ABE7FF5F4DA 3EC3030A78671D583EBE6C95680F33723EBE27D9F8617C083EC04B8F13EFF6033EC234 04D807BD743EC48F6C682984353EC7433CF4A76E1D3ECA42BF7100118C3ECD8756B167 6CC63ED086AC6D6CB8473ED269521404C41F3ED46AF71B80A2B13ED68B3447D5EEB63E D8C9C765A7CD9F3EDB26851D28F5463EDDA150955975CB3EE01D0B2E9BFF413EE17864 9E9EAF2C3EE2E2B01705114E3EE45BEA81E972A33EE5E411C5EBCC9E3EE77B2479A7E8 D43EE92121AF4448593EEAD608D01361923EEC99D98315C4383EEE6C939AF265AD3EF0 271B848E61D63EF11F61EA4E94903EF21F1D09B5DA903EF3264CF2D0F70B3EF434F1B8 FF7AFA3EF54B0B71B083FB3EF6689A3378D1C03EF78D9E156A3CD33EF8BA172E9B3324 3EF9EE0595D193CD3EFB29696147A1F63EFC6C42A68438C43EFDB6917A4132093EFF08 55F05C3AA33F0030C80DE7A53A3F00E120075753283F019532ED156E683F024D00C74A C90F3F0308899DB41D6D3F03C7CD77A60F233F048ACC5C1192663F05518651888A613F 061BFB5E427E253F06EA2B88214A6D3F07BC16D4B5BEDA3F0891BD49441AD53F096B1E EAC861403F0A483BBDFA7DDF3F0B2913C75238533F0C0DA70B0AF51F3F0CF5F58D2742 663F0DE1FF5174320D3F0ED1C45B8C8286-%&COLORG6&%$RGBGF'$F.F(F'-%+AXESLAB ELSG6%%%b[7]G%'a[6,5]G%!G-%*LINESTYLEG6#\"\"#-%&STYLEG6#%&PATCHG-%*AXE SSTYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_1G-%%VIEWG6%F&F,;F'$F0!\"'-%+PR OJECTIONG6%$!#DF($\"#gF(F." 1 2 0 2 10 0 2 1 3 2 2 1.000000 60.000000 -25.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "evalf[15](minimize(sm,b[7]=0..0.1,a [6,5]=1e-10..2,'location'));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"0O# *z?Oc2#!#D<#7$<$/&%\"aG6$\"\"'\"\"&$\"0bRQD)RY\"*!#;/&%\"bG6#\"\"($\"/ ))ew#p8Y(!#:F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use rational approximations for the values of " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 54 " that mim imize the norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "[.74613692765888e-1,.914639825383955e-1];\nmap(con vert,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"/))ew#p8Y( !#:$\"0bRQD)RY\"*!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"##)\"%*4 \"#\"#:\"$k\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We set " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7] = 82/1099;" "6#/&%\"bG6#\"\"(*&\"##)\"\"\"\"%*4\"!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5] = 15/164;" "6#/&%\"aG6$\" \"'\"\"&*&\"#:\"\"\"\"$k\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The resulting scheme has \+ the following value for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\" \"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(e5,a[7,5]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"E++!ewxZ_U\\fz$f3gdxhF&*(,**(\"G++!=!)HFUo 0]gokx6>.P#F&&%\"bG6#\"\"(F&)&%\"aG6$\"\"'\"\"&\"\"#F&F&\"C\\-'QnND>-p U;G&3V4&*!\"\"*&\"F++`KCr1l#e36G3eteu?F&F0F&F8*&\"D+(\\?Ut>c\\#)eI7#*4 j1iF&F1F&F&F&F,F8F1F8F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "a[7,5]=simplify(subs(\{b[7]=82/1099 ,a[6,5]=15/164\},subs(e5,a[7,5])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"&#!I4&f_7I30z:>(*\\5L+D/AJ\"\"H++]\"f`Gz')>e#eZQ?Ra Pl)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "W e can simplify the coefficients slightly if we replace the rational ex pression for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 65 " by a simpler one which agrees with the value above to 5 digit s." }}{PARA 0 "" 0 "" {TEXT -1 64 "This will not change the characteri stics of the scheme too much." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "-13122042500331049971915790 50830125259509/865375439203847582581986792853591500000.;\nconvert(%,ra tional,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+!*4M;:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$&=\"$A\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "`b*`[3] = 71 /190;" "6#/&%#b*G6#\"\"$*&\"#r\"\"\"\"$!>!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5] = -62/241;" "6#/&%#b*G6#\"\"&,$*&\"#i\"\"\"\"$ T#!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 145 "These value s are chosen so as to give a principal error norm for the order 4 embe dded scheme approximately 4.72 times that of the order 5 scheme." }} {PARA 0 "" 0 "" {TEXT -1 78 "The corresponding ratio for the Bogacki-S hampine scheme is approximately 4.79." }}{PARA 0 "" 0 "" {TEXT -1 16 " The values for " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 110 " also ensure that the stability region for the embedded scheme i s compatible with that of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "e6 := \{c[4 ]=63/106,b[7]=82/1099,a[6,5]=15/164,a[7,5]=-185/122,`b*`[3]=71/190,`b* `[5]=-62/241\};\ne7 := `union`(e6,simplify(subs(e6,e3))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e6G<(/&%\"bG6#\"\"(#\"##)\"%*4\"/&%\"aG6$\" \"'\"\"&#\"#:\"$k\"/&%\"cG6#\"\"%#\"#j\"$1\"/&F06$F*F3#!$&=\"$A\"/&%#b *G6#\"\"$#\"#r\"$!>/&FG6#F3#!#i\"$T#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1667 "e7 := \{a[7,1] = 221162852736255966056564 18138617/38775599793952140053248982599680, `b*`[2] = 0, a[8,2] = 0, c[ 2] = 1/6, c[8] = 1, b[2] = 0, c[7] = 1, a[6,4] = 861399479018673064/31 77050598972839673, b[7] = 82/1099, a[6,5] = 15/164, c[4] = 63/106, a[2 ,1] = 1/6, `b*`[7] = 8829707957442742/117440240421421575, a[8,3] = 769 25739070593/205684631194240, a[7,2] = -197127/114496, a[8,6] = 1551373 170541891/3131789077344060, a[7,6] = 4082588597964313509378858016009/2 571428541481836530593320891180, c[5] = 50/69, c[6] = 63/83, c[3] = 16/ 63, b[5] = -35320882628403/140715258838000, a[7,4] = 88647590949116931 8904581727562843/1979638263308617118988089415360030, `b*`[1] = 1131790 443129436/14550345379510605, b[1] = 16270811903/209372688000, a[7,5] = -185/122, `b*`[3] = 71/190, `b*`[5] = -62/241, `b*`[8] = -33340565403 709/52195662409520700, a[8,7] = 82/1099, a[6,3] = -2492114394031368057 /6454959947119420288, a[8,1] = 16270811903/209372688000, `b*`[4] = 332 6466076155851042/14390291580335988345, b[4] = 747543949006532/32597945 00756955, a[3,2] = 256/1323, b[3] = 76925739070593/205684631194240, `b *`[6] = 63595996553465987137/127170018616922687700, a[8,5] = -35320882 628403/140715258838000, a[6,1] = -30354668306338585/178910020531686528 , a[8,4] = 747543949006532/3259794500756955, b[6] = 1551373170541891/3 131789077344060, a[4,3] = 568356831/304900096, a[7,3] = 24118537135561 993104813811090103/14776247724722745081046875555840, a[3,1] = 80/1323, a[4,1] = 155816577/304900096, a[5,3] = 995353041375/3166454547136, a[ 4,2] = -16967475/9528128, a[5,2] = 1250725/43533526, a[6,2] = 38960537 49/4091707772, a[5,4] = 365139662800/1219687315371, a[5,1] = 649219115 75/789872295744\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "subs(e7,matrix([[ c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1] ,a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a[5,4]],\n \+ [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n [c[7],a[7 ,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8],a[8,1],a[8, 2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n [`b`,b[1],b [2],b[3]],[``,b[4],b[5],b[6]],[``$3,b[7]],\n [`b*`,`b*`[1],`b*`[2 ],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``$2,`b*`[7],`b*`[8]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747&#\"\"\"\"\"'F(%!GF+7&# \"#;\"#j#\"#!)\"%B8#\"$c#F2F+7&#F/\"$1\"#\"*xl\"e:\"*'4+\\I#!)vu'p\"\" (G\"G&*#\"*JoNo&F:7&#\"#]\"#p#\",v:\">#\\'\"-WdHs)*y#\"(D2D\"\")EN`V# \"-v8/``**\".Oraak;$7&F+F+F+#\"-+GmR^O\".r`J(o>77&#F/\"#$)#!2&eQjIoYNI \"3GloJ0-5*y\"#\"+\\P0'*Q\"+sxq\"4%#!4d!o8.%R9@\\#\"4)G?%>r%*f\\X'7&F+ F+#\"3kIn=!z%*Rh)\"4t'RG(*)f]q<$#\"#:\"$k\"7&F)#\"A<'Q\"=kl0mfDOF&G;@# \"A!o*f#)*[K0S@&Rz*fv(Q#!'Fr>\"''\\9\"#\"A.,46Q\"[5$*>cNr`=T#\"ASebvo/ \"3XFsCxCwZ\"7&F+#\"BVGcF#!$&= \"$A\"#\"@4g,e)y$4NJkzf)e#3%\"@!=\"*3K$fIl$=[T&G9d#7&F)#\",.>\"3F;\"-+ !)os$4#\"\"!#\"/$fq!Rd#p(\"0SU>JYo0#7&F+#\"0Kl+\\RaZ(\"1bpv+XzfK#!/.%G E)3KN\"0+!Q)e_rS\"#\"1\"*=aqJP^:\"1gSMx!*yJJ7&F+F+F+#\"##)\"%*4\"7&%\" bGFcpFfpFgpFjpFdq7&%#b*G#\"1O%HJW!zJ6\"201^z`M]X\"Ffp#\"#r\"$!>7&F+#\" 4U5&e:wgYEL\"5X$))fL!e\"H!R9#!#i\"$T##\"5Pr)fY`l*ffj\"6+xoAph=+_Q(pprint96\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8), \n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]])):\nevalf[8](% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+$\")nmm;!\")F(%! GF+F+F+F+F+F+7+$\")DoRDF*$\")K'o/'!\"*$\")i*\\$>F*F+F+F+F+F+F+7+$\")iR VfF*$\")UT5^F*$!)ux!y\"!\"($\")c2k=F;F+F+F+F+F+7+$\")oPYsF*$\")@H>#)F1 $\")c,tGF1$\")2VVJF*$\")`r$*HF*F+F+F+F+7+$\")9O!f(F*$!)Wk'p\"F*$\")!G= _*F*$!)WxgQF*$\")\"=8r#F*$\"):MY\"*F1F+F+F+7+$\"\"\"\"\"!$\")/m.dF*$!) Lp@7x(F1$FY FY$\")[)*RPF*$\")UA$H#F*$!)h45DF*$\")Cj`\\F*$\")&G8Y(F1F+7+%\"bGFaoFco FdoFfoFhoFjoF\\pF+7+%#b*G$\")QWyxF1Fco$\")@%ot$F*$\")Wg6BF*$!)ThsDF*$ \")S'3+&F*$\")*o%=vF1$!)Bh(Q'!#6Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The norm of the principal error terms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\nadd(subs(e7, errterms5_7[i])^2,i=1.. nops(errterms5_7)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^m+XX!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "This is considerably smaller than t he corresponding value for the Bogacki-Shampine scheme. However, since the weight " }{XPPEDIT 18 0 "b[5]" "6#&%\"bG6#\"\"&" }{TEXT -1 43 " \+ is negative, the scheme is not as robust." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The stability function for the \+ order 5 scheme is " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "expand (subs(e7,StabilityFunction(5,7,'expanded')));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$%\"zGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$* $)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F $*&#\"GZxQUwIY@cx5(\\)*>0)3)4#\"J!3?xs3C)HR3=(zAt\"*pwV1:F$*$)F%F-F$F$ F$*&#\"GD*=B\"4l!R%3_8Uc6hw#oJ$\"K7:p^)\\[a:\">MSlYm?@Y'4q\"F$*$)F%\" \"(F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the weights " }{XPPEDIT 18 0 "`b^`[i], i=1" " 6$&%#b^G6#%\"iG/F&\"\"\"" }{TEXT -1 26 " . . 7, ( initially with " } {XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 136 " as a parame ter) for an auxiliary 7 stage, order 4 embedded scheme by using the qu adrature conditions along with the requirements that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^`[7]=b[7]" "6#/&%#b^G6#\"\"(&% \"bG6#F'" }{TEXT -1 7 " qnd " }{XPPEDIT 18 0 "`b^`[2]=0" "6#/&%#b^G6 #\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "quad_eqns := subs(b=`b^`,Qu adratureConditions(4,7,'expanded')):\ne8 := \{`b^`[2]=0\}:\ne9 := `uni on`(e7,e8):\ncdns2 := [`b^`[7]=b[7],op(quad_eqns)]:\neqns2 := subs(e9, cdns2):\nnops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%#b^G6#\"\" $&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 := solve(\{op(eqns2)\},\{`b^`[1 ],`b^`[3],`b^`[4],`b^`[5],`b^`[7]\}):\ne11 := `union`(e9,e10):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The coeff icients are expressed in terms of " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^ G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " indets(map(rhs,e11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#&%#b^G6#\" \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1967 "e11 := \{a[7,1] = 22116285273625596605656418138617/3877559979395 2140053248982599680, `b*`[2] = 0, a[8,2] = 0, c[2] = 1/6, c[8] = 1, b[ 2] = 0, c[7] = 1, a[6,4] = 861399479018673064/3177050598972839673, b[7 ] = 82/1099, a[6,5] = 15/164, c[4] = 63/106, a[2,1] = 1/6, `b*`[7] = 8 829707957442742/117440240421421575, a[8,3] = 76925739070593/2056846311 94240, a[7,2] = -197127/114496, a[8,6] = 1551373170541891/313178907734 4060, a[7,6] = 4082588597964313509378858016009/25714285414818365305933 20891180, c[5] = 50/69, c[6] = 63/83, c[3] = 16/63, b[5] = -3532088262 8403/140715258838000, `b^`[7] = 82/1099, a[7,4] = 88647590949116931890 4581727562843/1979638263308617118988089415360030, `b*`[1] = 1131790443 129436/14550345379510605, b[1] = 16270811903/209372688000, `b^`[5] = 3 2341966557/71429065400-26399965153401/18581533675100*`b^`[6], a[7,5] = -185/122, `b*`[3] = 71/190, `b*`[5] = -62/241, `b*`[8] = -33340565403 709/52195662409520700, a[8,7] = 82/1099, a[6,3] = -2492114394031368057 /6454959947119420288, a[8,1] = 16270811903/209372688000, `b*`[4] = 332 6466076155851042/14390291580335988345, b[4] = 747543949006532/32597945 00756955, `b^`[4] = -925121678/49993014351+619658231432/1238587274003* `b^`[6], a[3,2] = 256/1323, b[3] = 76925739070593/205684631194240, `b* `[6] = 63595996553465987137/127170018616922687700, a[8,5] = -353208826 28403/140715258838000, `b^`[2] = 0, a[6,1] = -30354668306338585/178910 020531686528, a[8,4] = 747543949006532/3259794500756955, b[6] = 155137 3170541891/3131789077344060, `b^`[1] = 2391113/36926400+11966371/45742 9600*`b^`[6], a[4,3] = 568356831/304900096, a[7,3] = 24118537135561993 104813811090103/14776247724722745081046875555840, a[3,1] = 80/1323, `b ^`[3] = 3320501121/7788134464-1498909992111/14182019238112*`b^`[6], a[ 4,1] = 155816577/304900096, a[5,3] = 995353041375/3166454547136, a[4,2 ] = -16967475/9528128, a[5,2] = 1250725/43533526, a[6,2] = 3896053749/ 4091707772, a[5,4] = 365139662800/1219687315371, a[5,1] = 64921911575/ 789872295744\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "We choose " }{XPPEDIT 18 0 "`b^ `[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 88 " so that the principal error no rms of the two embedded schemes are approximately equal." }}{PARA 0 " " 0 "" {TEXT -1 66 "First we obtain the principal error of the 8 stage order 4 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "`errter ms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nprin_err 4_8 := sqrt(add(subs(e7,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*` ))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+pBuX@!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The princ ipal error norm of the 7 stage order 4 scheme is expressed in terms of " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "`errterms4_7^` := subs(b=`b ^`,PrincipalErrorTerms(4,7,'expanded')):\nprin_err4_7 := sqrt(add(subs (e11,`errterms4_7^`[i])^2,i=1.. nops(`errterms4_7^`)));" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%,prin_err4_7G,$*&#\"#<\"PSu2U@$*[\")Q!R " 0 "" {MPLTEXT 1 0 44 "prin_err4 _7=prin_err4_8:\nsolve(%):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+/A42\\!#5$\"+3D<+]F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The larger of the two values gives a larger rea l stability interval than the other value." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "`b^`[6]=1/2" "6#/&%#b^G6#\"\"'*&\"\"\" F)\"\"#!\"\"" }{TEXT -1 89 " which gives the following principal erro r norm for the 7 stage order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eval(prin_er r4_7,`b^`[6]=1/2):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P ))yP@!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e12 := `union`(\{`b^`[6]=1/2\},subs(`b^`[6]=1/2,e11)) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e12" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1885 "e12 := \{b[5] = -35320882628403/140715258838000, b[2] = 0, `b*`[ 2] = 0, `b*`[6] = 63595996553465987137/127170018616922687700, a[5,4] = 365139662800/1219687315371, a[6,4] = 861399479018673064/3177050598972 839673, a[6,2] = 3896053749/4091707772, a[7,2] = -197127/114496, a[5,1 ] = 64921911575/789872295744, a[7,1] = 2211628527362559660565641813861 7/38775599793952140053248982599680, c[3] = 16/63, a[6,5] = 15/164, c[4 ] = 63/106, c[5] = 50/69, c[6] = 63/83, a[6,1] = -30354668306338585/17 8910020531686528, c[7] = 1, a[6,3] = -2492114394031368057/645495994711 9420288, b[3] = 76925739070593/205684631194240, a[8,6] = 1551373170541 891/3131789077344060, a[5,2] = 1250725/43533526, a[8,2] = 0, a[8,7] = \+ 82/1099, a[4,2] = -16967475/9528128, b[7] = 82/1099, a[3,1] = 80/1323, b[1] = 16270811903/209372688000, a[5,3] = 995353041375/3166454547136, c[8] = 1, a[7,6] = 4082588597964313509378858016009/257142854148183653 0593320891180, b[6] = 1551373170541891/3131789077344060, a[8,4] = 7475 43949006532/3259794500756955, a[7,3] = 2411853713556199310481381109010 3/14776247724722745081046875555840, a[2,1] = 1/6, a[4,1] = 155816577/3 04900096, a[3,2] = 256/1323, a[8,5] = -35320882628403/140715258838000, `b^`[2] = 0, b[4] = 747543949006532/3259794500756955, a[8,1] = 162708 11903/209372688000, c[2] = 1/6, a[8,3] = 76925739070593/20568463119424 0, a[7,5] = -185/122, a[7,4] = 886475909491169318904581727562843/19796 38263308617118988089415360030, a[4,3] = 568356831/304900096, `b*`[3] = 71/190, `b^`[7] = 82/1099, `b^`[5] = -526042283593017/204211055089349 0, `b^`[4] = 6621573612710978/28585355696715237, `b^`[1] = 41084480131 /527850886920, `b*`[5] = -62/241, `b^`[6] = 1/2, `b^`[3] = 20791131321 3975/556644255095896, `b*`[8] = -33340565403709/52195662409520700, `b* `[4] = 3326466076155851042/14390291580335988345, `b*`[1] = 11317904431 29436/14550345379510605, `b*`[7] = 8829707957442742/117440240421421575 \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "`RK4_7eqs^` := subs(b=`b^`,OrderConditions(4,7,'expa nded')):\nsimplify(subs(e12,`RK4_7eqs^`)):\nmap(u->lhs(u)-rhs(u),%);\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 112 "The coefficients for the whole schem e including the 7 stage and 8 stage embedded order 4 schemes are as fo llows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 578 "subs(e12,matrix([ [c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1 ],a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a[5,4]],\n \+ [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n [c[7],a[ 7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8],a[8,1],a[8 ,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n [`b`,b[1], b[2],b[3]],[``,b[4],b[5],b[6]],[``$3,b[7]],\n [`b^`,`b^`[1],`b^`[ 2],`b^`[3]],[``,`b^`[4],`b^`[5],`b^`[6]],[``$3,`b^`[7]],\n [`b*`, `b*`[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``$2,`b*`[7],`b *`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777&#\"\"\" \"\"'F(%!GF+7&#\"#;\"#j#\"#!)\"%B8#\"$c#F2F+7&#F/\"$1\"#\"*xl\"e:\"*'4 +\\I#!)vu'p\"\"(G\"G&*#\"*JoNo&F:7&#\"#]\"#p#\",v:\">#\\'\"-WdHs)*y#\" (D2D\"\")EN`V#\"-v8/``**\".Oraak;$7&F+F+F+#\"-+GmR^O\".r`J(o>77&#F/\"# $)#!2&eQjIoYNI\"3GloJ0-5*y\"#\"+\\P0'*Q\"+sxq\"4%#!4d!o8.%R9@\\#\"4)G? %>r%*f\\X'7&F+F+#\"3kIn=!z%*Rh)\"4t'RG(*)f]q<$#\"#:\"$k\"7&F)#\"A<'Q\" =kl0mfDOF&G;@#\"A!o*f#)*[K0S@&Rz*fv(Q#!'Fr>\"''\\9\"#\"A.,46Q\"[5$*>cN r`=T#\"ASebvo/\"3XFsCxCwZ\"7&F+#\"BVGcF#!$&=\"$A\"#\"@4g,e)y$4NJkzf)e#3%\"@!=\"*3K$fIl$=[T&G9d#7&F)# \",.>\"3F;\"-+!)os$4#\"\"!#\"/$fq!Rd#p(\"0SU>JYo0#7&F+#\"0Kl+\\RaZ(\"1 bpv+XzfK#!/.%GE)3KN\"0+!Q)e_rS\"#\"1\"*=aqJP^:\"1gSMx!*yJJ7&F+F+F+#\"# #)\"%*4\"7&%\"bGFcpFfpFgpFjpFdq7&%#b^G#\",J,[%3T\"-?p)3&y_Ffp#\"0vR@88 \"z?\"0'*e4bUkc&7&F+#\"1y4r7Od@m\"2P_r'pb`eG#!07&F+#\"4U5&e:w gYEL\"5X$))fL!e\"H!R9#!#i\"$T##\"5Pr)fY`l*ffj\"6+xoAph=+_Q(pprint46\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 42 "#--------------------------------------- --" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1774 "ee := \+ \{c[2]=1/6,\nc[3]=16/63,\nc[4]=63/106,\nc[5]=50/69,\nc[6]=63/83,\nc[7] =1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=80/1323,\na[3,2]=256/1323,\na[4,1] =155816577/304900096,\na[4,2]=-16967475/9528128,\na[4,3]=568356831/304 900096,\na[5,1]=64921911575/789872295744,\na[5,2]=1250725/43533526,\na [5,3]=995353041375/3166454547136,\na[5,4]=365139662800/1219687315371, \na[6,1]=-30354668306338585/178910020531686528,\na[6,2]=3896053749/409 1707772,\na[6,3]=-2492114394031368057/6454959947119420288,\na[6,4]=861 399479018673064/3177050598972839673,\na[6,5]=15/164,\na[7,1]=221162852 73625596605656418138617/38775599793952140053248982599680,\na[7,2]=-197 127/114496,\na[7,3]=24118537135561993104813811090103/14776247724722745 081046875555840,\na[7,4]=886475909491169318904581727562843/19796382633 08617118988089415360030,\na[7,5]=-185/122,\na[7,6]=4082588597964313509 378858016009/2571428541481836530593320891180,\na[8,1]=16270811903/2093 72688000,\na[8,2]=0,\na[8,3]=76925739070593/205684631194240,\na[8,4]=7 47543949006532/3259794500756955,\na[8,5]=-35320882628403/1407152588380 00,\na[8,6]=1551373170541891/3131789077344060,\na[8,7]=82/1099,\n\nb[1 ]=16270811903/209372688000,\nb[2]=0,\nb[3]=76925739070593/205684631194 240,\nb[4]=747543949006532/3259794500756955,\nb[5]=-35320882628403/140 715258838000,\nb[6]=1551373170541891/3131789077344060,\nb[7]=82/1099, \n\n`b^`[1]=41084480131/527850886920,\n`b^`[2]=0,\n`b^`[3]=20791131321 3975/556644255095896,\n`b^`[4]=6621573612710978/28585355696715237,\n`b ^`[5]=-526042283593017/2042110550893490,\n`b^`[6]=1/2,\n`b^`[7]=82/109 9,\n\n`b*`[1]=1131790443129436/14550345379510605,\n`b*`[2]=0,\n`b*`[3] =71/190,\n`b*`[4]=3326466076155851042/14390291580335988345,\n`b*`[5]=- 62/241,\n`b*`[6]=63595996553465987137/127170018616922687700,\n`b*`[7]= 8829707957442742/117440240421421575,\n`b*`[8]=-33340565403709/52195662 409520700\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 st age, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "The stability function R for the 7 stage, order 5 schem e is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(subs(ee,StabilityFunction(5, 7,'expanded'))):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F) *&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$ )F'\"\"&F)F)F)*&#\"GZxQUwIY@cx5(\\)*>0)3)4#\"J!3?xs3C)HR3=(zAt\"*pwV1: F)*$)F'F1F)F)F)*&#\"GD*=B\"4l!R%3_8Uc6hw#oJ$\"K7:p^)\\[a:\">MSlYm?@Y'4 q\"F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the sta bility region intersects the negative real axis by solving the equatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6 #/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+q6N))R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4) :\np1 := plot([R(z),-1],z=-4.39..0.49,color=[red,blue]):\np2 := plot([ [[z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\n p3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplo ts[display]([p1,p2,p3],view=[-4.39..0.49,-1.47..1.47],font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVE SG6$7U7$$!3o*************Q%!#<$!3F;R*oUS\\7#F*7$$!3lLL$Q%\\\"oL%F*$!3; CeE&=8D$>F*7$$!3ummm())HOG%F*$!3'Hj`:`9^v\"F*7$$!3\")**\\naQNPUF*$!3&= !zjS#)>7;F*7$$!3'GL$o@y2\">%F*$!3o#4^7Sf#z9F*7$$!3MmmwVL*p3%F*$!3wul\" *GFk87F*7$$!3_mmOE*=A)RF*$!3it7j!)GFy)*!#=7$$!3#HL$[$[Uz(QF*$!3Mud\\l0 ;))zFK7$$!3Ammhs[E\"y$F*$!3D`e$e(4M5lFK7$$!3U***\\j!3;\"o$F*$!3%fb'*z7 R'>_FK7$$!3Imm\"y.Lwd$F*$!3+vc.gEN/TFK7$$!3a***\\TGPWZ$F*$!33?>'*=_G\" =$FK7$$!3)GLLrw(GoLF*$!3%=]OaSeqR#FK7$$!3Cmmw97zuKF*$!3\"=*pX_f)G#=FK7 $$!3W****RTi`pJF*$!3])[R*o16%)!# >7$$!3m****f=***>'HF*$!3q&)\\NG0;n[Fcp7$$!3Amm\"=76&pGF*$!37]wW0Fza@Fc p7$$!3'GLLn*H`fFF*$\"3*=q+pg^%Qd!#?7$$!3yKL$*)4njm#F*$\"39()H!)QVubDFc p7$$!3!)***\\:,$*zb#F*$\"3=O>0f/IxXFcp7$$!35LL$4ej?Y#F*$\"3i6&=Om/G='F cp7$$!3g***\\:z8oN#F*$\"3_\"[ojJ8R\"yFcp7$$!3S***\\E5\"fcAF*$\"3)3iK#H \\\"eH*Fcp7$$!3'HLL3C>?:#F*$\"3=\"3IP?=>3\"FK7$$!3sKL)yi*)f0#F*$\"3Z0t siwlB7FK7$$!3?mm;<(3C&>F*$\"3o@Z^SQE#Q\"FK7$$!3\\mm\"=E<[%=F*$\"3kuPH! >=qb\"FK7$$!3m***\\Oee6v\"F*$\"3,92+\\mD?FK7$$!3))****>HF]X:F*$\"3\"y%*yleXB\"F*$ \"3mq*\\)4d&)3HFK7$$!3MLL$p$=$e8\"F*$\"3B;#[]_06@$FK7$$!3r*****pTh/.\" F*$\"37#)o=O%G#oNFK7$$!3qlm;a:!)\\$*FK$\"3cA&)=)e_e#RFK7$$!3r$****f;Rf I)FK$\"3[e[j`W(yN%FK7$$!3$RLL)3guBtFK$\"3X\\/304l2[FK7$$!3o&***\\`62(H 'FK$\"3!Qsyq8wuK&FK7$$!3=HLLB)3LH&FK$\"3K[N$zF*)**)eFK7$$!36(***\\(oiC C%FK$\"3I1^+PuiUlFK7$$!3jcmmiHPIKFK$\"3YV'*\\h&o%RsFK7$$!3_jmmR]O&>#FK $\"3D?oU)[3*G!)FK7$$!3kBL$[]F*o6FK$\"3+V\"=(eg!o*))FK7$$!3%)=****>,QdA Fcp$\"3-UQT13zw(*FK7$$\"3Ozmmcej_&)Fcp$\"35rs4t-H*3\"F*7$$\"3-TLL,V7A= FK$\"3=t)R_0p)*>\"F*7$$\"330+]%[**H&GFK$\"3o#yO%[4;I8F*7$$\"3C4+]r&y'R QFK$\"3mg!))zC)4o9F*7$$\"3!***************[FK$\"38aH1jhJK;F*-%'COLOURG 6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\ l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FMFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[o Fa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F_q Fa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$Fcs Fa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$Ff uFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$Fi wFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F \\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\ lFh[l-F$6&7#7$$!31+++q6N))RF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\ \\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$ 6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF [\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+ AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$R%!\"#$\"#\\F `cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1450 "R := z -> 1+z+1/2*z^2+1/ 6*z^3+1/24*z^4+1/120*z^5+\n 20980880519984971077562146307642387747/1 5064376699173227971808392982408727720080*z^6+\n 33168276611156421352 084390650912318925/170096462120664665403419115544849851691512*z^7:\npt s := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton(R(z)=exp( ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]: \nend do:\np1 := plot(pts,color=COLOR(RGB,.15,.48,.15)):\np2 := plots[ polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.3,.95,.3)):\npts := []:\nz0 := 1. 5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I) ,z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\n p3 := plot(pts,color=COLOR(RGB,.15,.48,.15)):\np4 := plots[polygonplot ]([seq([pts[i-1],pts[i],[1.37,4.19]],i=2..nops(pts))],\n styl e=patchnogrid,color=COLOR(RGB,.3,.95,.3)):\npts := []:\nz0 := 1.5-4.3* I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0) :\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := \+ plot(pts,color=COLOR(RGB,.15,.48,.15)):\np6 := plots[polygonplot]([seq ([pts[i-1],pts[i],[1.37,-4.19]],i=2..nops(pts))],\n style=pat chnogrid,color=COLOR(RGB,.3,.95,.3)):\np7 := plot([[[-4.79,0],[1.99,0] ],[[0,-4.79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([p|| (1..7)],view=[-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 512 477 477 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$$\"3h******4%*y4l!#G$\"3++++Fjzq:!#=7$$\"35+++31vqa!#E$\" 3))*****fi#fTJF07$$\"3M+++Ft*)o%)!#D$\"3()*****fQ)Q7ZF07$$\"3U+++W(3OC '!#C$\"3!)*****plzJG'F07$$\"3&)*****\\*H=PH!#B$\"3y*****>wZR&yF07$$\"3 *******p\\,Y+\"!#A$\"3r*****zcGYU*F07$$\"3/+++GVpgEFL$\"3!******>n/&*4 \"!#<7$$\"39+++#f0Sb&FL$\"3!*******\\*zkD\"FT7$$\"3X+++U45C()FL$\"3%** *****)R3LT\"FT7$$\"3c*****\\S:!)e(FL$\"3)******\\N[)p:FT7$$!3(******4f 7[-\"!#@$\"3.+++dk(es\"FT7$$!3'*******Rr7;rFao$\"3%******R,x5)=FT7$$!3 ')*****pc=-B#!#?$\"30+++[(\\].#FT7$$!3J+++))[eBaF\\p$\"3;+++XhL(=#FT7$ $!30+++lo()Q6!#>$\"3z*****H1YuL#FT7$$!39+++E#*Gb@Fgp$\"3))*****\\uX[[# FT7$$!3!******\\fy1w$Fgp$\"31+++'*H')GEFT7$$!3-+++)HyR8'Fgp$\"3%)***** **4%\\oFFT7$$!3\"******pSt5V*Fgp$\"3.+++I/=-HFT7$$!3))******G7Gt8F0$\" 39+++!Qgx-$FT7$$!37+++\"p'Q**=F0$\"3(******RmcG9$FT7$$!3.+++[yx.DF0$\" 39++++'\\cC$FT7$$!3#******>/1=;$F0$\"3,+++!>#[NLFT7$$!3w*****z!>+]QF0$ \"3%)*****R:OGT$FT7$$!3#)*****RX'y]XF0$\"3++++-M')yMFT7$$!3Q+++QR'HD&F 0$\"3$)*****po7\\`$FT7$$!3C+++MP?]fF0$\"3=+++-GD#e$FT7$$!3#******>f0$R mF0$\"3?+++Sr'>i$FT7$$!3O+++[#**)=tF0$\"3')*****HES\\l$FT7$$!3e*****4^ ;'))zF0$\"3)*******>0)=o$FT7$$!3++++%=0'[')F0$\"33+++L8N.PFT7$$!3U+++_ uD*H*F0$\"3\"******RZ+)>PFT7$$!3>+++?Z1T**F0$\"3!******\\F%eJPFT7$$!3/ +++,XXd5FT$\"3')*****fk')*QPFT7$$!3,+++q6-?6FT$\"32+++8PBUPFT7$$!3!*** ****)ec==\"FT$\"3()*****>*\\]TPFT7$$!3'********G5IC\"FT$\"3\")*****4(> %pt$FT7$$!3!******RvKNI\"FT$\"3,+++mZlGPFT7$$!3/+++&exMO\"FT$\"3>+++Du s;PFT7$$!30+++;U!HU\"FT$\"3%******zMA7q$FT7$$!3#******H8\")=[\"FT$\"3# )*****fC%=#o$FT7$$!32+++X'*[S:FT$\"39+++'3W'fOFT7$$!3++++*=G))f\"FT$\" 34+++0NiLOFT7$$!3/+++6r,d;FT$\"3')*****>HSTg$FT7$$!3%******4l._r\"FT$ \"3%)*****Rn:7d$FT7$$!3!******\\_mNx\"FT$\"3\"*******yU)[`$FT7$$!3!*** ***p')=B$=FT$\"3%)******)p2_\\$FT7$$!3-+++;uq\"*=FT$\"3!)*****pE#H_MFT 7$$!3++++iY+_>FT$\"3%******Hq5jS$FT7$$!3y*****p%)*[8?FT$\"3%******f2Dv N$FT7$$!3')*****4C:k2#FT$\"3#)*****\\'\\I1LFT7$$!3#******z*y&49#FT$\"3 ))*****fqIJD$FT7$$!3.+++C];2AFT$\"3=+++'Hq&)>$FT7$$!3-+++u1\"\\F#FT$\" 3z*****HfCK9$FT7$$!3++++!*[)QM#FT$\"3)******4()\\w3$FT7$$!3$******RLEO T#FT$\"30+++k>GKIFT7$$!3A+++r\"*e$[#FT$\"3%)******[*)QxHFT7$$!3'****** *o#>Kb#FT$\"3!******>BlI#HFT7$$!3>+++$>:?i#FT$\"31+++<,EpGFT7$$!3/+++. Cc*o#FT$\"3!)*****p$G#e\"GFT7$$!3%)******[DabFFT$\"31+++_saiFFT7$$!3)) ******)RE(>GFT$\"3>+++jp?4FFT7$$!3#)*****H)GFT$\"3'******4kybl#FT7$ $!30+++NV9UHFT$\"3@+++q_X,EFT7$$!3(******f-H-+$FT$\"3%******4B_ma#FT7$ $!3++++>B>cIFT$\"3%)*****z#*35\\#FT7$$!3*******fQO+6$FT$\"3')*****>%zQ MCFT7$$!39+++\"=!yhJFT$\"3y*****z/tmP#FT7$$!3<+++`^X6KFT$\"36+++&*ow#F T7$$!3!)******=#=&[LFT$\"3$******49nJ8#FT7$$!3!)******ohS!R$FT$\"3!*** ****p=$)o?FT7$$!3>+++/\\]IMFT$\"3\")*****RWSI+#FT7$$!3;+++!\\!*)oMFT$ \"3*******4Lzd$>FT7$$!39+++g\\k0NFT$\"3%******zA\\q'=FT7$$!3\")*****fU b3a$FT$\"35+++kl'oz\"FT7$$!3;+++0DhuNFT$\"3&******RHl_s\"FT7$$!3!)**** *>%y+2OFT$\"33+++<()H_;FT7$$!3)******RiI\"QOFT$\"3#*******G4/y:FT7$$!3 ')*****47j!oOFT$\"3\"******4X(e-:FT7$$!3!)*****z\\vop$FT$\"3!*******QX 0E9FT7$$!3;+++7/iCPFT$\"3/+++_nd[8FT7$$!3')*****46G8v$FT$\"3#******4i- .F\"FT7$$!39+++]G+xPFT$\"3-+++g&)Q\">\"FT7$$!3:+++T8i,QFT$\"35+++>;*>6 \"FT7$$!39+++RO8DQFT$\"3\"******p$>EK5FT7$$!3;+++7kYZQFT$\"3#)*****f&* eL_*F07$$!3)******f3G&oQFT$\"3?+++H$3Ls)F07$$!3;+++k]@))QFT$\"3L+++S\" =M#zF07$$!3:+++U\"=k!RFT$\"3f*****RV0W7(F07$$!3#)*****RPGI#RFT$\"3`*** **4OhnK'F07$$!38+++**=%z$RFT$\"37+++&Qs2`&F07$$!3%)*****\\`j5&RFT$\"3A +++Y$\\lt%F07$$!3%******z<4B'RFT$\"3%)*****HKfS%RF07$$!3'******>?2;(RF T$\"3y*****4hcJ:$F07$$!3')*****z4**)yRFT$\"3)******z`5OO#F07$$!3++++C' RT)RFT$\"3++++`\"H^d\"F07$$!3')*****Rp'H()RFT$\"3e*****4t(ytyFgp7$$!31 +++q6N))RFTF(7$Fjjl$!3e*****4t(ytyFgp7$Fejl$!3++++`\"H^d\"F07$F`jl$!3) ******z`5OO#F07$F[jl$!3y*****4hcJ:$F07$Ffil$!3%)*****HKfS%RF07$Fail$!3 A+++Y$\\lt%F07$F\\il$!37+++&Qs2`&F07$Fghl$!3`*****4OhnK'F07$Fbhl$!3f** ***RV0W7(F07$F]hl$!3L+++S\"=M#zF07$Fhgl$!3?+++H$3Ls)F07$Fcgl$!3#)***** f&*eL_*F07$F^gl$!3\"******p$>EK5FT7$Fifl$!35+++>;*>6\"FT7$Fdfl$!3-+++g &)Q\">\"FT7$F_fl$!3#******4i-.F\"FT7$Fjel$!3/+++_nd[8FT7$Feel$!3!***** **QX0E9FT7$F`el$!3\"******4X(e-:FT7$F[el$!3#*******G4/y:FT7$Ffdl$!33++ +<()H_;FT7$Fadl$!3&******RHl_s\"FT7$F\\dl$!35+++kl'oz\"FT7$Fgcl$!3%*** ***zA\\q'=FT7$Fbcl$!3*******4Lzd$>FT7$F]cl$!3\")*****RWSI+#FT7$Fhbl$!3 !*******p=$)o?FT7$Fcbl$!3$******49nJ8#FT7$F^bl$!35+++yH2'>#FT7$Fial$!3 7+++5))edAFT7$Fdal$!36+++&*ow%zQMCFT7$Fe`l$!3%)*****z#*35\\#FT7$F``l$!3%******4B_ma#FT7$F[`l$!3@ +++q_X,EFT7$Ff_l$!3'******4kybl#FT7$Fa_l$!3>+++jp?4FFT7$F\\_l$!31+++_s aiFFT7$Fg^l$!3!)*****p$G#e\"GFT7$Fb^l$!31+++<,EpGFT7$F]^l$!3!******>Bl I#HFT7$Fh]l$!3%)******[*)QxHFT7$Fc]l$!30+++k>GKIFT7$F^]l$!3)******4() \\w3$FT7$Fi\\l$!3z*****HfCK9$FT7$Fd\\l$!3=+++'Hq&)>$FT7$F_\\l$!3))**** *fqIJD$FT7$Fj[l$!3#)*****\\'\\I1LFT7$Fe[l$!3%******f2DvN$FT7$F`[l$!3%* *****Hq5jS$FT7$F[[l$!3!)*****pE#H_MFT7$Ffz$!3%)******)p2_\\$FT7$Faz$!3 \"*******yU)[`$FT7$F\\z$!3%)*****Rn:7d$FT7$Fgy$!3')*****>HSTg$FT7$Fby$ !34+++0NiLOFT7$F]y$!39+++'3W'fOFT7$Fhx$!3#)*****fC%=#o$FT7$Fcx$!3%**** **zMA7q$FT7$F^x$!3>+++Dus;PFT7$Fiw$!3,+++mZlGPFT7$Fdw$!3\")*****4(>%pt $FT7$F_w$!3()*****>*\\]TPFT7$Fjv$!32+++8PBUPFT7$Fev$!3')*****fk')*QPFT 7$F`v$!3!******\\F%eJPFT7$F[v$!3\"******RZ+)>PFT7$Ffu$!33+++L8N.PFT7$F au$!3)*******>0)=o$FT7$F\\u$!3')*****HES\\l$FT7$Fgt$!3?+++Sr'>i$FT7$Fb t$!3=+++-GD#e$FT7$F]t$!3$)*****po7\\`$FT7$Fhs$!3++++-M')yMFT7$Fcs$!3%) *****R:OGT$FT7$F^s$!3,+++!>#[NLFT7$Fir$!39++++'\\cC$FT7$Fdr$!3(******R mcG9$FT7$F_r$!39+++!Qgx-$FT7$Fjq$!3.+++I/=-HFT7$Feq$!3%)*******4%\\oFF T7$F`q$!31+++'*H')GEFT7$F[q$!3))*****\\uX[[#FT7$Fep$!3z*****H1YuL#FT7$ F`p$!3;+++XhL(=#FT7$Fjo$!30+++[(\\].#FT7$Feo$!3%******R,x5)=FT7$F_o$!3 .+++dk(es\"FT7$$\"3g+++/a,)e(FL$!3)******\\N[)p:FT7$Fen$!3%*******)R3L T\"FT7$FV$!3!*******\\*zkD\"FT7$FP$!3!******>n/&*4\"FT7$FJ$!3r*****zcG YU*F07$$\"3<+++'*H=PHFF$!3y*****>wZR&yF07$F>$!3!)*****plzJG'F07$F8$!3( )*****fQ)Q7ZF07$$\"3i*****R$*\\2Z&F4$!3))*****fi#fTJF07$$\"3S*****fP*y 4lF-$!3++++Fjzq:F0F'-%&COLORG6&%$RGBG$\"#:!\"#$\"#[Fh^nFf^n-%)POLYGONS G6fw7%F'7$$\"+5%*y4lF\\p$\"+Fjzq:!#57$$Fh^nF)F(7%F__n7$$\"+31vqaF0$\"+ EEfTJFd_nFe_n7%Fh_n7$$\"+Ft*)o%)FT$\"+'Q)Q7ZFd_nFe_n7%F^`n7$$\"+W(3OC' !#;$\"+d'zJG'Fd_nFe_n7%Fd`n7$$\"+&*H=PH!#:$\"+ix%R&yFd_nFe_n7%F[an7$$ \"+(\\,Y+\"!#9$\"+o&GYU*Fd_nFe_n7%Fban7$$\"+GVpgEFean$\"+sY]*4\"!\"*Fe _n7%Fian7$$\"+#f0Sb&Fean$\"+]*zkD\"F^bnFe_n7%F`bn7$$\"+U45C()Fean$\"+* R3LT\"F^bnFe_n7%Ffbn7$$\"+0a,)e(Fean$\"+b$[)p:F^bnFe_n7%F\\cn7$$!+\"f7 [-\"!#8$\"+dk(es\"F^bnFe_n7%Fbcn7$$!+Sr7;rFecn$\"+9q2\")=F^bnFe_n7%Fic n7$$!+n&=-B#!#7$\"+[(\\].#F^bnFe_n7%F_dn7$$!+))[eBaFbdn$\"+XhL(=#F^bnF e_n7%Ffdn7$$!+lo()Q6!#6$\"+jgWPBF^bnFe_n7%F\\en7$$!+E#*Gb@F_en$\"+Xd%[ [#F^bnFe_n7%Fcen7$$!+&fy1w$F_en$\"+'*H')GEF^bnFe_n7%Fien7$$!+)HyR8'F_e n$\"++T\\oFF^bnFe_n7%F_fn7$$!+2M2J%*F_en$\"+I/=-HF^bnFe_n7%Fefn7$$!+H7 Gt8Fd_n$\"+!Qgx-$F^bnFe_n7%F[gn7$$!+\"p'Q**=Fd_n$\"+km&G9$F^bnFe_n7%Fa gn7$$!+[yx.DFd_n$\"++'\\cC$F^bnFe_n7%Fggn7$$!+Ug!=;$Fd_n$\"+!>#[NLF^bn Fe_n7%F]hn7$$!+3>+]QFd_n$\"+ah$GT$F^bnFe_n7%Fchn7$$!+aky]XFd_n$\"+-M') yMF^bnFe_n7%Fihn7$$!+QR'HD&Fd_n$\"+(o7\\`$F^bnFe_n7%F_in7$$!+MP?]fFd_n $\"+-GD#e$F^bnFe_n7%Fein7$$!+#f0$RmFd_n$\"+Sr'>i$F^bnFe_n7%F[jn7$$!+[# **)=tFd_n$\"+j-%\\l$F^bnFe_n7%Fajn7$$!+6lh))zFd_n$\"+?0)=o$F^bnFe_n7%F gjn7$$!+%=0'[')Fd_n$\"+L8N.PF^bnFe_n7%F][o7$$!+_uD*H*Fd_n$\"+u/!)>PF^b nFe_n7%Fc[o7$$!+?Z1T**Fd_n$\"+vUeJPF^bnFe_n7%Fi[o7$$!+,XXd5F^bn$\"+Ym) *QPF^bnFe_n7%F_\\o7$$!+q6-?6F^bn$\"+8PBUPF^bnFe_n7%Fe\\o7$$!+*ec==\"F^ bn$\"+#*\\]TPF^bnFe_n7%F[]o7$$!+!H5IC\"F^bn$\"+r>%pt$F^bnFe_n7%Fa]o7$$ !+aF`.8F^bn$\"+mZlGPF^bnFe_n7%Fg]o7$$!+&exMO\"F^bn$\"+Dus;PF^bnFe_n7%F ]^o7$$!+;U!HU\"F^bn$\"+[BA,PF^bnFe_n7%Fc^o7$$!+L6)=[\"F^bn$\"+YU=#o$F^ bnFe_n7%Fi^o7$$!+X'*[S:F^bn$\"+'3W'fOF^bnFe_n7%F__o7$$!+*=G))f\"F^bn$ \"+0NiLOF^bnFe_n7%Fe_o7$$!+6r,d;F^bn$\"+#HSTg$F^bnFe_n7%F[`o7$$!+^O?:< F^bn$\"+uc@rNF^bnFe_n7%Fa`o7$$!+DlctF^bn$\"+.2J1MF^bnFe_n7%Fiao7$$!+Z)*[8?F^bn$\"+w]_dLF^bnF e_n7%F_bo7$$!+T_Tw?F^bn$\"+l\\I1LF^bnFe_n7%Febo7$$!+)*y&49#F^bn$\"+128 `KF^bnFe_n7%F[co7$$!+C];2AF^bn$\"+'Hq&)>$F^bnFe_n7%Faco7$$!+u1\"\\F#F^ bn$\"+$fCK9$F^bnFe_n7%Fgco7$$!+!*[)QM#F^bn$\"+r)\\w3$F^bnFe_n7%F]do7$$ !+Mji8CF^bn$\"+k>GKIF^bnFe_n7%Fcdo7$$!+r\"*e$[#F^bn$\"+\\*)QxHF^bnFe_n 7%Fido7$$!+p#>Kb#F^bn$\"+K_1BHF^bnFe_n7%F_eo7$$!+$>:?i#F^bn$\"+<,EpGF^ bnFe_n7%Feeo7$$!+.Cc*o#F^bn$\"+PG#e\"GF^bnFe_n7%F[fo7$$!+\\DabFF^bn$\" +_saiFF^bnFe_n7%Fafo7$$!+*RE(>GF^bn$\"+jp?4FF^bnFe_n7%Fgfo7$$!+t\"f>)G F^bn$\"+T'ybl#F^bnFe_n7%F]go7$$!+NV9UHF^bn$\"+q_X,EF^bnFe_n7%Fcgo7$$!+ E!H-+$F^bn$\"+JAlYDF^bnFe_n7%Figo7$$!+>B>cIF^bn$\"+G*35\\#F^bnFe_n7%F_ ho7$$!+'QO+6$F^bn$\"+UzQMCF^bnFe_n7%Feho7$$!+\"=!yhJF^bn$\"+[InwBF^bnF e_n7%F[io7$$!+`^X6KF^bn$\"+&*ow#F^bnFe_n7%F]jo7$$!+>#=&[LF^bn$ \"+Tr;L@F^bnFe_n7%Fcjo7$$!+phS!R$F^bn$\"+q=$)o?F^bnFe_n7%Fijo7$$!+/\\] IMF^bn$\"+W//.?F^bnFe_n7%F_[p7$$!+!\\!*)oMF^bn$\"+J$zd$>F^bnFe_n7%Fe[p 7$$!+g\\k0NF^bn$\"+G#\\q'=F^bnFe_n7%F[\\p7$$!+Ea&3a$F^bn$\"+kl'oz\"F^b nFe_n7%Fa\\p7$$!+0DhuNF^bn$\"+%Hl_s\"F^bnFe_n7%Fg\\p7$$!+Uy+2OF^bn$\"+ <()H_;F^bnFe_n7%F]]p7$$!+C18QOF^bn$\"+H4/y:F^bnFe_n7%Fc]p7$$!+@J1oOF^b n$\"+^ue-:F^bnFe_n7%Fi]p7$$!+)\\vop$F^bn$\"+RX0E9F^bnFe_n7%F_^p7$$!+7/ iCPF^bn$\"+_nd[8F^bnFe_n7%Fe^p7$$!+6\"G8v$F^bn$\"+@EIq7F^bnFe_n7%F[_p7 $$!+]G+xPF^bn$\"+g&)Q\">\"F^bnFe_n7%Fa_p7$$!+T8i,QF^bn$\"+>;*>6\"F^bnF e_n7%Fg_p7$$!+RO8DQF^bn$\"+P>EK5F^bnFe_n7%F]`p7$$!+7kYZQF^bn$\"+c*eL_* Fd_nFe_n7%Fc`p7$$!+'3G&oQF^bn$\"+H$3Ls)Fd_nFe_n7%Fi`p7$$!+k]@))QF^bn$ \"+S\"=M#zFd_nFe_n7%F_ap7$$!+U\"=k!RF^bn$\"+MaSCrFd_nFe_n7%Feap7$$!+u$ GI#RF^bn$\"+h8wEjFd_nFe_n7%F[bp7$$!+**=%z$RF^bn$\"+&Qs2`&Fd_nFe_n7%Fab p7$$!+NN1^RF^bn$\"+Y$\\lt%Fd_nFe_n7%Fgbp7$$!+y\"4B'RF^bn$\"+B$fS%RFd_n Fe_n7%F]cp7$$!+-sgrRF^bn$\"+6m:`JFd_nFe_n7%Fccp7$$!+)4**)yRF^bn$\"+Q0h jBFd_nFe_n7%Ficp7$$!+C'RT)RF^bn$\"+`\"H^d\"Fd_nFe_n7%F_dp7$$!+%p'H()RF ^bn$\"+JxytyF_enFe_n7%Fedp7$$!+q6N))RF^bnF(Fe_n7%F[ep7$Ffdp$!+JxytyF_e nFe_n7%F_ep7$F`dp$!+`\"H^d\"Fd_nFe_n7%Fcep7$Fjcp$!+Q0hjBFd_nFe_n7%Fgep 7$Fdcp$!+6m:`JFd_nFe_n7%F[fp7$F^cp$!+B$fS%RFd_nFe_n7%F_fp7$Fhbp$!+Y$\\ lt%Fd_nFe_n7%Fcfp7$Fbbp$!+&Qs2`&Fd_nFe_n7%Fgfp7$F\\bp$!+h8wEjFd_nFe_n7 %F[gp7$Ffap$!+MaSCrFd_nFe_n7%F_gp7$F`ap$!+S\"=M#zFd_nFe_n7%Fcgp7$Fj`p$ !+H$3Ls)Fd_nFe_n7%Fggp7$Fd`p$!+c*eL_*Fd_nFe_n7%F[hp7$F^`p$!+P>EK5F^bnF e_n7%F_hp7$Fh_p$!+>;*>6\"F^bnFe_n7%Fchp7$Fb_p$!+g&)Q\">\"F^bnFe_n7%Fgh p7$F\\_p$!+@EIq7F^bnFe_n7%F[ip7$Ff^p$!+_nd[8F^bnFe_n7%F_ip7$F`^p$!+RX0 E9F^bnFe_n7%Fcip7$Fj]p$!+^ue-:F^bnFe_n7%Fgip7$Fd]p$!+H4/y:F^bnFe_n7%F[ jp7$F^]p$!+<()H_;F^bnFe_n7%F_jp7$Fh\\p$!+%Hl_s\"F^bnFe_n7%Fcjp7$Fb\\p$ !+kl'oz\"F^bnFe_n7%Fgjp7$F\\\\p$!+G#\\q'=F^bnFe_n7%F[[q7$Ff[p$!+J$zd$> F^bnFe_n7%F_[q7$F`[p$!+W//.?F^bnFe_n7%Fc[q7$Fjjo$!+q=$)o?F^bnFe_n7%Fg[ q7$Fdjo$!+Tr;L@F^bnFe_n7%F[\\q7$F^jo$!+yH2'>#F^bnFe_n7%F_\\q7$Fhio$!+5 ))edAF^bnFe_n7%Fc\\q7$Fbio$!+&*owGKIF^bnFe_n7%F [`q7$F^do$!+r)\\w3$F^bnFe_n7%F_`q7$Fhco$!+$fCK9$F^bnFe_n7%Fc`q7$Fbco$! +'Hq&)>$F^bnFe_n7%Fg`q7$F\\co$!+128`KF^bnFe_n7%F[aq7$Ffbo$!+l\\I1LF^bn Fe_n7%F_aq7$F`bo$!+w]_dLF^bnFe_n7%Fcaq7$Fjao$!+.2J1MF^bnFe_n7%Fgaq7$Fd ao$!+nAH_MF^bnFe_n7%F[bq7$F^ao$!+*p2_\\$F^bnFe_n7%F_bq7$Fh`o$!+zU)[`$F ^bnFe_n7%Fcbq7$Fb`o$!+uc@rNF^bnFe_n7%Fgbq7$F\\`o$!+#HSTg$F^bnFe_n7%F[c q7$Ff_o$!+0NiLOF^bnFe_n7%F_cq7$F`_o$!+'3W'fOF^bnFe_n7%Fccq7$Fj^o$!+YU= #o$F^bnFe_n7%Fgcq7$Fd^o$!+[BA,PF^bnFe_n7%F[dq7$F^^o$!+Dus;PF^bnFe_n7%F _dq7$Fh]o$!+mZlGPF^bnFe_n7%Fcdq7$Fb]o$!+r>%pt$F^bnFe_n7%Fgdq7$F\\]o$!+ #*\\]TPF^bnFe_n7%F[eq7$Ff\\o$!+8PBUPF^bnFe_n7%F_eq7$F`\\o$!+Ym)*QPF^bn Fe_n7%Fceq7$Fj[o$!+vUeJPF^bnFe_n7%Fgeq7$Fd[o$!+u/!)>PF^bnFe_n7%F[fq7$F ^[o$!+L8N.PF^bnFe_n7%F_fq7$Fhjn$!+?0)=o$F^bnFe_n7%Fcfq7$Fbjn$!+j-%\\l$ F^bnFe_n7%Fgfq7$F\\jn$!+Sr'>i$F^bnFe_n7%F[gq7$Ffin$!+-GD#e$F^bnFe_n7%F _gq7$F`in$!+(o7\\`$F^bnFe_n7%Fcgq7$Fjhn$!+-M')yMF^bnFe_n7%Fggq7$Fdhn$! +ah$GT$F^bnFe_n7%F[hq7$F^hn$!+!>#[NLF^bnFe_n7%F_hq7$Fhgn$!++'\\cC$F^bn Fe_n7%Fchq7$Fbgn$!+km&G9$F^bnFe_n7%Fghq7$F\\gn$!+!Qgx-$F^bnFe_n7%F[iq7 $Fffn$!+I/=-HF^bnFe_n7%F_iq7$F`fn$!++T\\oFF^bnFe_n7%Fciq7$Fjen$!+'*H') GEF^bnFe_n7%Fgiq7$Fden$!+Xd%[[#F^bnFe_n7%F[jq7$F]en$!+jgWPBF^bnFe_n7%F _jq7$Fgdn$!+XhL(=#F^bnFe_n7%Fcjq7$F`dn$!+[(\\].#F^bnFe_n7%Fgjq7$Fjcn$! +9q2\")=F^bnFe_n7%F[[r7$Fccn$!+dk(es\"F^bnFe_n7%F_[r7$$\"+/a,)e(Fean$! +b$[)p:F^bnFe_n7%Fc[r7$Fgbn$!+*R3LT\"F^bnFe_n7%Fi[r7$Fabn$!+]*zkD\"F^b nFe_n7%F]\\r7$Fjan$!+sY]*4\"F^bnFe_n7%Fa\\r7$Fcan$!+o&GYU*Fd_nFe_n7%Fe \\r7$$\"+'*H=PHF^an$!+ix%R&yFd_nFe_n7%Fi\\r7$Fe`n$!+d'zJG'Fd_nFe_n7%F_ ]r7$F_`n$!+'Q)Q7ZFd_nFe_n7%Fc]r7$$\"+M*\\2Z&F0$!+EEfTJFd_nFe_n7%Fg]r7$ $\"+w$*y4lF\\p$!+Fjzq:Fd_nFe_n7%F]^rF'Fe_n-Fc^n6&Fe^n$\"\"$!\"\"$\"#&* Fh^nFe^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"35+++\"oG)*[\"FT$\"3w*** **H>]MI%FT7$$\"31+++L,s\"[\"FT$\"3A+++v;>7VFT7$$\"3'******pLDHZ\"FT$\" 3<+++$))R-K%FT7$$\"33+++Ut[j9FT$\"3z*****pU`vK%FT7$$\"3)*******z)\\MX \"FT$\"3d*****>B\"4MVFT7$$\"3(******fodGW\"FT$\"3%******\\E7)RVFT7$$\" 3*******4'ovJ9FT$\"3t******e_nWVFT7$$\"3#******\\G&>?9FT$\"37+++r%Q'[V FT7$$\"31++++JA39FT$\"3y******H%f;N%FT7$$\"3-+++oK*eR\"FT$\"35+++@Zp`V FT7$$\"33+++KAE$Q\"FT$\"3q*****zw*paVFT7$$\"31+++/2Rq8FT$\"3/+++c'GYN% FT7$$\"3)******\\gWtN\"FT$\"3M+++$*RV`VFT7$$\"3++++5i>W8FT$\"3/+++Mo1^ VFT7$$\"35+++Bb-J8FT$\"3A+++2nZZVFT7$$\"36+++m>#zJ\"FT$\"35+++4=hUVFT7 $$\"3++++Fk)\\I\"FT$\"3H+++I%>kL%FT7$$\"31+++ePL#H\"FT$\"3&)*****fwY)G VFT7$$\"31+++)z&4!G\"FT$\"3y*****HHU)>VFT7$$\"35+++y\\Uo7FT$\"3T+++5!e $4VFT7$$\"3%******Ra)\\d7FT$\"3++++WHN(H%FT7$$\"3#*******fK_Z7FT$\"39+ ++g&)z$G%FT7$$\"3*)*****HFS(Q7FT$\"3?+++koooUFT7$$\"3!******zHH9B\"FT$ \"3K+++zA/_UFT7$$\"3)******fY5fA\"FT$\"3U+++K%QRB%FT7$$\"30+++#*3aA7FT $\"3-+++%o?X@%FT7$$\"35+++38q@7FT$\"3')*****\\%Q.%>%FT7$$\"31+++qmwB7F T$\"3G+++>8&G<%FT7$$\"3$******\\Tb!H7FT$\"3k*****\\D'\\^TFT7$$\"35+++B \"exB\"FT$\"3')*****HpQ18%FT7$$\"3#******4&*f)\\7FT$\"3#)*****H*306TFT 7$$\"3!******RB(3l7FT$\"3!)*****Rl9N4%FT7$$\"3!******H\"=\"HG\"FT$\"3S +++I5qySFT7$$\"32+++sQi-8FT$\"3-+++.q1nSFT7$$\"3-+++xnWB8FT$\"3e****** 4,\")eSFT7$$\"30+++ZukW8FT$\"3!******Rj*)Q0%FT7$$\"35+++r\"3cO\"FT$\"3 %******zP!4_SFT7$$\"3/+++UK&eQ\"FT$\"3;+++0f4`SFT7$$\"3.+++jG/09FT$\"3 A+++t!\\l0%FT7$$\"3(*******G4&HU\"FT$\"3K+++=E4iSFT7$$\"36+++())Q%R9FT $\"3e*****H\">RpSFT7$$\"31+++R?Va9FT$\"37+++h^9ySFT7$$\"3#*******30!zY \"FT$\"3)******4'f3)3%FT7$$\"3#******>)\\%)z9FT$\"3u*****4#=)*)4%FT7$$ \"3)******4U'G!\\\"FT$\"3g*****p#3j5TFT7$$\"33+++o)e#*\\\"FT$\"3<+++Cx &G7%FT7$$\"33+++\"[/o]\"FT$\"3I+++d+^NTFT7$$\"3*)*****f;qH^\"FT$\"3%)* ****4ka%[TFT7$$\"31+++I`!y^\"FT$\"3e*******[u:;%FT7$$\"31+++#Rg8_\"FT$ \"3U+++)=mZ<%FT7$$\"3/+++TeoB:FT$\"3g*****RnPz=%FT7$$\"3*******zjJ[_\" FT$\"3V+++'Q15?%FT7$$\"3)*******))o%[_\"FT$\"3N+++5x*Q@%FT7$$\"3*)**** *pmzP_\"FT$\"3Q+++(oVlA%FT7$$\"3'******z'pn@:FT$\"3h*****p\">))QUFT7$$ \"3\"******\\t%e=:FT$\"3k*****fka3D%FT7$$\"3)******f%za9:FT$\"3m*****z )zSiUFT7$$\"34+++N2h4:FT$\"3W+++`7\\tUFT7$$\"3*)*****4b;Q]\"FT$\"3K+++ %QcSG%FT7$$\"31+++F$3s\\\"FT$\"3'******\\WdSH%FTFa_rFb^n-F\\_n6jn7%7$$ \"+\"oG)*[\"F^bn$\"+$>]MI%F^bn7$$\"+L,s\"[\"F^bn$\"+v;>7VF^bn7$$\"$P\" Fh^n$\"$>%Fh^n7%Febs7$$\"+P`#HZ\"F^bn$\"+$))R-K%F^bnFjbs7%F`cs7$$\"+Ut [j9F^bn$\"+FMbFVF^bnFjbs7%Ffcs7$$\"+!))\\MX\"F^bn$\"+K74MVF^bnFjbs7%F \\ds7$$\"+'odGW\"F^bn$\"+lA\")RVF^bnFjbs7%Fbds7$$\"+hovJ9F^bn$\"+f_nWV F^bnFjbs7%Fhds7$$\"+&G&>?9F^bn$\"+r%Q'[VF^bnFjbs7%F^es7$$\"++JA39F^bn$ \"+I%f;N%F^bnFjbs7%Fdes7$$\"+oK*eR\"F^bn$\"+@Zp`VF^bnFjbs7%Fjes7$$\"+K AE$Q\"F^bn$\"+o(*paVF^bnFjbs7%F`fs7$$\"+/2Rq8F^bn$\"+c'GYN%F^bnFjbs7%F ffs7$$\"+0YMd8F^bn$\"+$*RV`VF^bnFjbs7%F\\gs7$$\"+5i>W8F^bn$\"+Mo1^VF^b nFjbs7%Fbgs7$$\"+Bb-J8F^bn$\"+2nZZVF^bnFjbs7%Fhgs7$$\"+m>#zJ\"F^bn$\"+ 4=hUVF^bnFjbs7%F^hs7$$\"+Fk)\\I\"F^bn$\"+I%>kL%F^bnFjbs7%Fdhs7$$\"+ePL #H\"F^bn$\"+mn%)GVF^bnFjbs7%Fjhs7$$\"+)z&4!G\"F^bn$\"+$HU)>VF^bnFjbs7% F`is7$$\"+y\\Uo7F^bn$\"+5!e$4VF^bnFjbs7%Ffis7$$\"+W&)\\d7F^bn$\"+WHN(H %F^bnFjbs7%F\\js7$$\"+gK_Z7F^bn$\"+g&)z$G%F^bnFjbs7%Fbjs7$$\"+t-uQ7F^b n$\"+koooUF^bnFjbs7%Fhjs7$$\"+)HH9B\"F^bn$\"+zA/_UF^bnFjbs7%F^[t7$$\"+ m/\"fA\"F^bn$\"+K%QRB%F^bnFjbs7%Fd[t7$$\"+#*3aA7F^bn$\"+%o?X@%F^bnFjbs 7%Fj[t7$$\"+38q@7F^bn$\"+XQ.%>%F^bnFjbs7%F`\\t7$$\"+qmwB7F^bn$\"+>8&G< %F^bnFjbs7%Ff\\t7$$\"+:a0H7F^bn$\"+bi\\^TF^bnFjbs7%F\\]t7$$\"+B\"exB\" F^bn$\"+$pQ18%F^bnFjbs7%Fb]t7$$\"+^*f)\\7F^bn$\"+$*306TF^bnFjbs7%Fh]t7 $$\"+Ms3l7F^bn$\"+aY^$4%F^bnFjbs7%F^^t7$$\"+8=\"HG\"F^bn$\"+I5qySF^bnF jbs7%Fd^t7$$\"+sQi-8F^bn$\"+.q1nSF^bnFjbs7%Fj^t7$$\"+xnWB8F^bn$\"+5,\" )eSF^bnFjbs7%F`_t7$$\"+ZukW8F^bn$\"+M'*)Q0%F^bnFjbs7%Ff_t7$$\"+r\"3cO \"F^bn$\"+y.4_SF^bnFjbs7%F\\`t7$$\"+UK&eQ\"F^bn$\"+0f4`SF^bnFjbs7%Fb`t 7$$\"+jG/09F^bn$\"+t!\\l0%F^bnFjbs7%Fh`t7$$\"+H4&HU\"F^bn$\"+=E4iSF^bn Fjbs7%F^at7$$\"+())Q%R9F^bn$\"+8>RpSF^bnFjbs7%Fdat7$$\"+R?Va9F^bn$\"+h ^9ySF^bnFjbs7%Fjat7$$\"+40!zY\"F^bn$\"+hf3)3%F^bnFjbs7%F`bt7$$\"+#)\\% )z9F^bn$\"+@=)*)4%F^bnFjbs7%Ffbt7$$\"+@kG!\\\"F^bn$\"+F3j5TF^bnFjbs7%F \\ct7$$\"+o)e#*\\\"F^bn$\"+Cx&G7%F^bnFjbs7%Fbct7$$\"+\"[/o]\"F^bn$\"+d +^NTF^bnFjbs7%Fhct7$$\"+m,(H^\"F^bn$\"+TYX[TF^bnFjbs7%F^dt7$$\"+I`!y^ \"F^bn$\"+!\\u:;%F^bnFjbs7%Fddt7$$\"+#Rg8_\"F^bn$\"+)=mZ<%F^bnFjbs7%Fj dt7$$\"+TeoB:F^bn$\"+uw$z=%F^bnFjbs7%F`et7$$\"+Q;$[_\"F^bn$\"+'Q15?%F^ bnFjbs7%Ffet7$$\"+*)o%[_\"F^bn$\"+5x*Q@%F^bnFjbs7%F\\ft7$$\"+n'zP_\"F^ bn$\"+(oVlA%F^bnFjbs7%Fbft7$$\"+opn@:F^bn$\"+<>))QUF^bnFjbs7%Fhft7$$\" +NZe=:F^bn$\"+YY&3D%F^bnFjbs7%F^gt7$$\"+Yza9:F^bn$\"+))zSiUF^bnFjbs7%F dgt7$$\"+N2h4:F^bn$\"+`7\\tUF^bnFjbs7%Fjgt7$$\"+^l\"Q]\"F^bn$\"+%QcSG% F^bnFjbs7%F`ht7$$\"+F$3s\\\"F^bn$\"+Xu0%H%F^bnFjbs7%FfhtF`bsFjbsFc^rFj ^r-F$6$7in7$Fb_r$!3w*****H>]MI%FT7$Fias$!3'******\\WdSH%FT7$Fdas$!3K++ +%QcSG%FT7$F_as$!3W+++`7\\tUFT7$Fj`s$!3m*****z)zSiUFT7$Fe`s$!3k*****fk a3D%FT7$F``s$!3h*****p\">))QUFT7$F[`s$!3Q+++(oVlA%FT7$Ff_s$!3N+++5x*Q@ %FT7$Fa_s$!3V+++'Q15?%FT7$F\\_s$!3g*****RnPz=%FT7$Fg^s$!3U+++)=mZ<%FT7 $Fb^s$!3e*******[u:;%FT7$F]^s$!3%)*****4ka%[TFT7$Fh]s$!3I+++d+^NTFT7$F c]s$!3<+++Cx&G7%FT7$F^]s$!3g*****p#3j5TFT7$Fi\\s$!3u*****4#=)*)4%FT7$F d\\s$!3)******4'f3)3%FT7$F_\\s$!37+++h^9ySFT7$Fj[s$!3e*****H\">RpSFT7$ Fe[s$!3K+++=E4iSFT7$F`[s$!3A+++t!\\l0%FT7$F[[s$!3;+++0f4`SFT7$Ffjr$!3% ******zP!4_SFT7$Fajr$!3!******Rj*)Q0%FT7$F\\jr$!3e******4,\")eSFT7$Fgi r$!3-+++.q1nSFT7$Fbir$!3S+++I5qySFT7$F]ir$!3!)*****Rl9N4%FT7$Fhhr$!3#) *****H*306TFT7$Fchr$!3')*****HpQ18%FT7$F^hr$!3k*****\\D'\\^TFT7$Figr$! 3G+++>8&G<%FT7$Fdgr$!3')*****\\%Q.%>%FT7$F_gr$!3-+++%o?X@%FT7$Fjfr$!3U +++K%QRB%FT7$Fefr$!3K+++zA/_UFT7$F`fr$!3?+++koooUFT7$F[fr$!39+++g&)z$G %FT7$Ffer$!3++++WHN(H%FT7$Faer$!3T+++5!e$4VFT7$F\\er$!3y*****HHU)>VFT7 $Fgdr$!3&)*****fwY)GVFT7$Fbdr$!3H+++I%>kL%FT7$F]dr$!35+++4=hUVFT7$Fhcr $!3A+++2nZZVFT7$Fccr$!3/+++Mo1^VFT7$F^cr$!3M+++$*RV`VFT7$Fibr$!3/+++c' GYN%FT7$Fdbr$!3q*****zw*paVFT7$F_br$!35+++@Zp`VFT7$Fjar$!3y******H%f;N %FT7$Fear$!37+++r%Q'[VFT7$F`ar$!3t******e_nWVFT7$F[ar$!3%******\\E7)RV FT7$Ff`r$!3d*****>B\"4MVFT7$Fa`r$!3z*****pU`vK%FT7$F\\`r$!3<+++$))R-K% FT7$Fg_r$!3A+++v;>7VFTF_itFb^n-F\\_n6jn7%7$Fabs$!+$>]MI%F^bn7$Fght$!+X u0%H%F^bn7$F[cs$!$>%Fh^n7%Fidu7$Faht$!+%QcSG%F^bnF\\eu7%F`eu7$F[ht$!+` 7\\tUF^bnF\\eu7%Fdeu7$Fegt$!+))zSiUF^bnF\\eu7%Fheu7$F_gt$!+YY&3D%F^bnF \\eu7%F\\fu7$Fift$!+<>))QUF^bnF\\eu7%F`fu7$Fcft$!+(oVlA%F^bnF\\eu7%Fdf u7$F]ft$!+5x*Q@%F^bnF\\eu7%Fhfu7$Fget$!+'Q15?%F^bnF\\eu7%F\\gu7$Faet$! +uw$z=%F^bnF\\eu7%F`gu7$F[et$!+)=mZ<%F^bnF\\eu7%Fdgu7$Fedt$!+!\\u:;%F^ bnF\\eu7%Fhgu7$F_dt$!+TYX[TF^bnF\\eu7%F\\hu7$Fict$!+d+^NTF^bnF\\eu7%F` hu7$Fcct$!+Cx&G7%F^bnF\\eu7%Fdhu7$F]ct$!+F3j5TF^bnF\\eu7%Fhhu7$Fgbt$!+ @=)*)4%F^bnF\\eu7%F\\iu7$Fabt$!+hf3)3%F^bnF\\eu7%F`iu7$F[bt$!+h^9ySF^b nF\\eu7%Fdiu7$Feat$!+8>RpSF^bnF\\eu7%Fhiu7$F_at$!+=E4iSF^bnF\\eu7%F\\j u7$Fi`t$!+t!\\l0%F^bnF\\eu7%F`ju7$Fc`t$!+0f4`SF^bnF\\eu7%Fdju7$F]`t$!+ y.4_SF^bnF\\eu7%Fhju7$Fg_t$!+M'*)Q0%F^bnF\\eu7%F\\[v7$Fa_t$!+5,\")eSF^ bnF\\eu7%F`[v7$F[_t$!+.q1nSF^bnF\\eu7%Fd[v7$Fe^t$!+I5qySF^bnF\\eu7%Fh[ v7$F_^t$!+aY^$4%F^bnF\\eu7%F\\\\v7$Fi]t$!+$*306TF^bnF\\eu7%F`\\v7$Fc]t $!+$pQ18%F^bnF\\eu7%Fd\\v7$F]]t$!+bi\\^TF^bnF\\eu7%Fh\\v7$Fg\\t$!+>8&G <%F^bnF\\eu7%F\\]v7$Fa\\t$!+XQ.%>%F^bnF\\eu7%F`]v7$F[\\t$!+%o?X@%F^bnF \\eu7%Fd]v7$Fe[t$!+K%QRB%F^bnF\\eu7%Fh]v7$F_[t$!+zA/_UF^bnF\\eu7%F\\^v 7$Fijs$!+koooUF^bnF\\eu7%F`^v7$Fcjs$!+g&)z$G%F^bnF\\eu7%Fd^v7$F]js$!+W HN(H%F^bnF\\eu7%Fh^v7$Fgis$!+5!e$4VF^bnF\\eu7%F\\_v7$Fais$!+$HU)>VF^bn F\\eu7%F`_v7$F[is$!+mn%)GVF^bnF\\eu7%Fd_v7$Fehs$!+I%>kL%F^bnF\\eu7%Fh_ v7$F_hs$!+4=hUVF^bnF\\eu7%F\\`v7$Figs$!+2nZZVF^bnF\\eu7%F``v7$Fcgs$!+M o1^VF^bnF\\eu7%Fd`v7$F]gs$!+$*RV`VF^bnF\\eu7%Fh`v7$Fgfs$!+c'GYN%F^bnF \\eu7%F\\av7$Fafs$!+o(*paVF^bnF\\eu7%F`av7$F[fs$!+@Zp`VF^bnF\\eu7%Fdav 7$Fees$!+I%f;N%F^bnF\\eu7%Fhav7$F_es$!+r%Q'[VF^bnF\\eu7%F\\bv7$Fids$!+ f_nWVF^bnF\\eu7%F`bv7$Fcds$!+lA\")RVF^bnF\\eu7%Fdbv7$F]ds$!+K74MVF^bnF \\eu7%Fhbv7$Fgcs$!+FMbFVF^bnF\\eu7%F\\cv7$Facs$!+$))R-K%F^bnF\\eu7%F`c v7$Ffbs$!+v;>7VF^bnF\\eu7%FdcvFfduF\\euFc^rFj^r-F$6%7$7$$!3/++++++!z%F TF(7$$\"3+++++++!*>FTF(-%'COLOURG6&Fe^nF)F)F)-%*LINESTYLEG6#Ff^r-F$6%7 $7$F(F\\dv7$F($\"3/++++++!z%FTFadvFddv-%+AXESLABELSG6%%&Re(z)G%&Im(z)G -%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fdev6$%*HELVETICAG\"\" *-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%Fh^n$\"$*>Fh^n;Fgfv$\"$z%Fh^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stab ility interval) is the intersection of the stability region with the r eal line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stabilit y interval is (approximately) " }{XPPEDIT 18 0 "[-3.98835, 0];" "6#7$ ,$-%&FloatG6$\"'N))R!\"&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The stability radius is half the length of the real stability interval." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "newton(R(z)=-1,z=-4)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+&evT*>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 545 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+\n 20980880519984971077562146307642387747/1506 4376699173227971808392982408727720080*z^6+\n 33168276611156421352084 390650912318925/170096462120664665403419115544849851691512*z^7:\npts : = []:\nz0 := 0:\nfor ct from 0 to 360 do\n zz := newton(R(z)=exp(ct* Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\ne nd do:\np1 := plot(pts,color=red):\nrd := 1.994175585: \np2 := plot([- rd+rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue):\nplots[display]([p1,p2] ,scaling=constrained,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7eal7$$\"\"!F)F(7$F($ \"31+++^v>Z5!#=7$$\"3'******>c-E'R!#F$\"3-+++,^R%4#F-7$$\"3K+++/1vqa!# E$\"3))*****fi#fTJF-7$$\"3')*****\\Fpwx$!#D$\"3#)*****R'**y)=%F-7$$\"3 %******pjUZv\"!#C$\"3%)*****4T')fB&F-7$$\"3U+++W(3OC'FC$\"3!)*****plzJ G'F-7$$\"3.+++1$eX#=!#B$\"3\")*****pXj.L(F-7$$\"3#)*****RHZgc%FN$\"3/+ ++GK_x$)F-7$$\"3*******p\\,Y+\"!#A$\"3r*****zcGYU*F-7$$\"3%******\\/8X (>FY$\"33+++r@;Z5!#<7$$\"3)******p2xK\\$FY$\"3++++0,%=:\"F[o7$$\"39+++ #f0Sb&FY$\"3!*******\\*zkD\"F[o7$$\"3P******)))Gx!yFY$\"3++++,e0h8F[o7 $$\"3O*****zqOSA*FY$\"3)******RiHbY\"F[o7$$\"3g+++/a,)e(FY$\"3)****** \\N[)p:F[o7$$!31+++\")[B'=\"FY$\"35+++U:%Rn\"F[o7$$!39+++K2#>R#!#@$\"3 %******>9?xx\"F[o7$$!3'*******Rr7;rF]q$\"3%******R,x5)=F[o7$$!3)****** \\X\"G#e\"!#?$\"31+++&R*)Q)>F[o7$$!39+++*Ge91$Fhq$\"3#)******\\H-'3#F[ o7$$!3J+++))[eBaFhq$\"3;+++XhL(=#F[o7$$!3d+++Coo5!*Fhq$\"3/+++,Fo(G#F[ o7$$!3$********48JU\"!#>$\"3)*******3$4pQ#F[o7$$!39+++E#*Gb@F]s$\"3))* ****\\uX[[#F[o7$$!3++++u+**[JF]s$\"3?+++obG\"e#F[o7$$!31+++/e&yX%F]s$ \"33+++2y&fn#F[o7$$!3-+++)HyR8'F]s$\"3%)*******4%\\oFF[o7$$!3]*****H(y ?A#)F]s$\"3))*****\\&=SeGF[o7$$!31+++\"yl_2\"F-$\"3#******zgg]%HF[o7$$ !3))******G7Gt8F-$\"39+++!Qgx-$F[o7$$!3%******\\KAUr\"F-$\"3?+++H))y0J F[o7$$!37+++c%4L4#F-$\"3!)*****>1Z&yJF[o7$$!3.+++[yx.DF-$\"39++++'\\cC $F[o7$$!3s*****Hi\\!QHF-$\"3%)*****>QnpI$F[o7$$!3C+++`4'))Q$F-$\"3y*** **f-0EO$F[o7$$!3w*****z!>+]QF-$\"3%)*****R:OGT$F[o7$$!3*)*****f)>g;VF- $\"3))*****RqH!eMF[o7$$!3')*****>k&3&y%F-$\"3#)*****HG*e)\\$F[o7$$!3Q+ ++QR'HD&F-$\"3$)*****po7\\`$F[o7$$!3w*****>c%e=dF-$\"3=+++v$ptc$F[o7$$ !3s*****H:/4='F-$\"3;+++!f!H'f$F[o7$$!3#******>f0$RmF-$\"3?+++Sr'>i$F[ o7$$!3U+++q8Y$4(F-$\"3*******\\B^Yk$F[o7$$!3J+++CdBVvF-$\"3!******>;gX m$F[o7$$!3e*****4^;'))zF-$\"3)*******>0)=o$F[o7$$!3))*****zuo'H%)F-$\" 31+++!Qsnp$F[o7$$!3K++++N]m))F-$\"3>+++pIP4PF[o7$$!3U+++_uD*H*F-$\"3\" ******RZ+)>PF[o7$$!3B+++T)z!G(*F-$\"3/+++ue:GPF[o7$$!34+++WCJ:5F[o$\"3 z*****>KEXt$F[o7$$!3/+++,XXd5F[o$\"3')*****fk')*QPF[o7$$!33+++;$\\#*4 \"F[o$\"3)******47,;u$F[o7$$!3&*******z*\\]TPF[o7$$!3-+++%R)pA7F[o$\"3,+++K;))QPF[ o7$$!3%******H-_KE\"F[o$\"39+++E()eMPF[o7$$!3!******RvKNI\"F[o$\"3,+++ mZlGPF[o7$$!33+++SmbV8F[o$\"3;+++7M5@PF[o7$$!3/+++y3M$Q\"F[o$\"3*)**** *z0a>r$F[o7$$!30+++;U!HU\"F[o$\"3%******zMA7q$F[o7$$!3-+++>uEi9F[o$\"3 9+++*o?*)o$F[o7$$!3%******Hx`9]\"F[o$\"3++++j(e]n$F[o7$$!32+++X'*[S:F[ o$\"39+++'3W'fOF[o7$$!3)******430%z:F[o$\"39+++QEoUOF[o7$$!3'******4XM #=;F[o$\"3.+++4'zTi$F[o7$$!3/+++6r,d;F[o$\"3')*****>HSTg$F[o7$$!31+++h !)z&p\"F[o$\"3))*****\\>rDe$F[o7$$!3-+++Z&GYt\"F[o$\"35+++09[fNF[o7$$! 3!******\\_mNx\"F[o$\"3\"*******yU)[`$F[o7$$!3/+++#*on7=F[o$\"3;+++$\\ *z3NF[o7$$!3%******zIJ?&=F[o$\"3))*****p[b7[$F[o7$$!3-+++;uq\"*=F[o$\" 3!)*****pE#H_MF[o7$$!3++++8syJ>F[o$\"31+++tW'>U$F[o7$$!3)*******3VNs>F [o$\"37+++rWM!R$F[o7$$!3y*****p%)*[8?F[o$\"3%******f2DvN$F[o7$$!3y**** *f+n_0#F[o$\"37+++V8iBLF[o7$$!35+++`Uu(4#F[o$\"3!)*****4qq()G$F[o7$$!3 #******z*y&49#F[o$\"3))*****fqIJD$F[o7$$!3#)*****H&\\\"\\=#F[o$\"3++++ 1R(o@$F[o7$$!3%******fg(eHAF[o$\"3))*******zz,=$F[o7$$!3-+++u1\"\\F#F[ o$\"3z*****HfCK9$F[o7$$!3$)*****f(Hy?BF[o$\"3'******R:qh5$F[o7$$!3')** ***pvsqO#F[o$\"3#)*****f*R:pIF[o7$$!3$******RLEOT#F[o$\"30+++k>GKIF[o7 $$!3%)*****fVy-Y#F[o$\"3-+++dYi&*HF[o7$$!3\")*****4Zio]#F[o$\"3;+++3x@ fHF[o7$$!3'*******o#>Kb#F[o$\"3!******>BlI#HF[o7$$!3?+++tM?*f#F[o$\"3) )*****4IXr)GF[o7$$!35+++etoWEF[o$\"3/+++[iT^GF[o7$$!3/+++.Cc*o#F[o$\"3 !)*****p$G#e\"GF[o7$$!3(******R7RPt#F[o$\"3?+++x;I!y#F[o7$$!3!******>x Xrx#F[o$\"3!)*****4g&yWFF[o7$$!3))******)RE(>GF[o$\"3>+++jp?4FF[o7$$!3 y*****zoQ9'GF[o$\"3?+++d**\\tEF[o7$$!3A+++!z^A!HF[o$\"3'******48-wj#F[ o7$$!30+++NV9UHF[o$\"3@+++q_X,EF[o7$$!3++++)o-6)HF[o$\"3-+++Ne+lDF[o7$ $!3<+++l%>\">IF[o$\"33+++t]?GDF[o7$$!3++++>B>cIF[o$\"3%)*****z#*35\\#F [o7$$!37+++aHK#4$F[o$\"3/+++?yP`CF[o7$$!3')*****HN;v7$F[o$\"3=+++ijF:C F[o7$$!39+++\"=!yhJF[o$\"3y*****z/tmP#F[o7$$!3%******RLC^>$F[o$\"3\")* ****>%*RvL#F[o7$$!3\"******4hgvA$F[o$\"3-++++C&yH#F[o7$$!3%)*****4Z-\" fKF[o$\"37+++5))edAF[o7$$!3')*****\\)[w*G$F[o$\"3A+++)QIn@#F[o7$$!3&** ****zAk&>LF[o$\"3/+++=5Ev@F[o7$$!3!)******=#=&[LF[o$\"3$******49nJ8#F[ o7$$!3')*****f*ekwLF[o$\"3%)*******pP/4#F[o7$$!3*******zenRS$F[o$\"3++ ++PT1Z?F[o7$$!3>+++/\\]IMF[o$\"3\")*****RWSI+#F[o7$$!3'******pp!GcMF[o $\"3-+++lKOe>F[o7$$!3?+++2!>8[$F[o$\"3'******p/KI\">F[o7$$!39+++g\\k0N F[o$\"3%******zA\\q'=F[o7$$!3*******fh%GHNF[o$\"32+++m/U?=F[o7$$!3/+++ SZE_NF[o$\"3-+++z[:tvh$F[o$\"3(******HJ&oF; F[o7$$!3)******RiI\"QOF[o$\"3#*******G4/y:F[o7$$!3#******fB8#eOF[o$\"3 (******fpky_\"F[o7$$!35+++$p)yxOF[o$\"31+++;,>x9F[o7$$!3!)*****z\\vop$ F[o$\"3!*******QX0E9F[o7$$!3>+++R$)[:PF[o$\"37+++E()\\u8F[o7$$!3A+++bu jLPF[o$\"32+++IicA8F[o7$$!3')*****46G8v$F[o$\"3#******4i-.F\"F[o7$$!3# ******pCg&oPF[o$\"3++++&eax@\"F[o7$$!3-+++9$G`y$F[o$\"3)*******Q*o\\; \"F[o7$$!3:+++T8i,QF[o$\"35+++>;*>6\"F[o7$$!3,+++7KU;+**)F-7$$!3'****** H\\[_(QF[o$\"3;+++F-ic%)F-7$$!3;+++k]@))QF[o$\"3L+++S\"=M#zF-7$$!3z*** **\\6A0!RF[o$\"3))*****ps51R(F-7$$!3=+++Ur87RF[o$\"3u*****\\!RNeoF-7$$ !3#)*****RPGI#RF[o$\"3`*****4OhnK'F-7$$!3#)*****HLlJ$RF[o$\"3O+++O!3fz &F-7$$!3*******\\F>D%RF[o$\"3;+++KT$eE&F-7$$!3%)*****\\`j5&RF[o$\"3A++ +Y$\\lt%F-7$$!3y*****4wt(eRF[o$\"39+++-r.3UF-7$$!3++++%3Gc'RF[o$\"3%)* ****pge-o$F-7$$!3'******>?2;(RF[o$\"3y*****4hcJ:$F-7$$!37+++MWpwRF[o$ \"3%)*****f\")emi#F-7$$!3/+++Bd(3)RF[o$\"3))*****p*3o+@F-7$$!3++++C'RT )RF[o$\"3++++`\"H^d\"F-7$$!3=+++ysZ')RF[o$\"31+++AG!*\\5F-7$$!35+++$Q# )y)RF[o$\"3o*****H7h*[_F]s7$$!31+++q6N))RF[oF(7$Fbjm$!3o*****H7h*[_F]s 7$F]jm$!31+++AG!*\\5F-7$Fhim$!3++++`\"H^d\"F-7$Fcim$!3))*****p*3o+@F-7 $F^im$!3%)*****f\")emi#F-7$Fihm$!3y*****4hcJ:$F-7$Fdhm$!3%)*****pge-o$ F-7$F_hm$!39+++-r.3UF-7$Fjgm$!3A+++Y$\\lt%F-7$Fegm$!3;+++KT$eE&F-7$F`g m$!3O+++O!3fz&F-7$F[gm$!3`*****4OhnK'F-7$Fffm$!3u*****\\!RNeoF-7$Fafm$ !3))*****ps51R(F-7$F\\fm$!3L+++S\"=M#zF-7$Fgem$!3;+++F-ic%)F-7$Fbem$!3 Y*******>;+**)F-7$F]em$!3#)*****f&*eL_*F-7$Fhdm$!3&******R^Nc+\"F[o7$F cdm$!3#******4qm)e5F[o7$F^dm$!35+++>;*>6\"F[o7$Ficm$!3)*******Q*o\\;\" F[o7$Fdcm$!3++++&eax@\"F[o7$F_cm$!3#******4i-.F\"F[o7$Fjbm$!32+++IicA8 F[o7$Febm$!37+++E()\\u8F[o7$F`bm$!3!*******QX0E9F[o7$F[bm$!31+++;,>x9F [o7$Ffam$!3(******fpky_\"F[o7$Faam$!3#*******G4/y:F[o7$F\\am$!3(****** HJ&oF;F[o7$Fg`m$!3!******fZonn\"F[o7$Fb`m$!3&******RHl_s\"F[o7$F]`m$!3 -+++z[:tF[o7$Fi^m$!3-+++lKOe>F[o7$Fd^m$!3\")*****RWSI+#F[o7$F_^m$! 3++++PT1Z?F[o7$Fj]m$!3%)*******pP/4#F[o7$Fe]m$!3$******49nJ8#F[o7$F`]m $!3/+++=5Ev@F[o7$F[]m$!3A+++)QIn@#F[o7$Ff\\m$!37+++5))edAF[o7$Fa\\m$!3 -++++C&yH#F[o7$F\\\\m$!3\")*****>%*RvL#F[o7$Fg[m$!3y*****z/tmP#F[o7$Fb [m$!3=+++ijF:CF[o7$F][m$!3/+++?yP`CF[o7$Fhjl$!3%)*****z#*35\\#F[o7$Fcj l$!33+++t]?GDF[o7$F^jl$!3-+++Ne+lDF[o7$Fiil$!3@+++q_X,EF[o7$Fdil$!3'** ****48-wj#F[o7$F_il$!3?+++d**\\tEF[o7$Fjhl$!3>+++jp?4FF[o7$Fehl$!3!)** ***4g&yWFF[o7$F`hl$!3?+++x;I!y#F[o7$F[hl$!3!)*****p$G#e\"GF[o7$Ffgl$!3 /+++[iT^GF[o7$Fagl$!3))*****4IXr)GF[o7$F\\gl$!3!******>BlI#HF[o7$Fgfl$ !3;+++3x@fHF[o7$Fbfl$!3-+++dYi&*HF[o7$F]fl$!30+++k>GKIF[o7$Fhel$!3#)** ***f*R:pIF[o7$Fcel$!3'******R:qh5$F[o7$F^el$!3z*****HfCK9$F[o7$Fidl$!3 ))*******zz,=$F[o7$Fddl$!3++++1R(o@$F[o7$F_dl$!3))*****fqIJD$F[o7$Fjcl $!3!)*****4qq()G$F[o7$Fecl$!37+++V8iBLF[o7$F`cl$!3%******f2DvN$F[o7$F[ cl$!37+++rWM!R$F[o7$Ffbl$!31+++tW'>U$F[o7$Fabl$!3!)*****pE#H_MF[o7$F\\ bl$!3))*****p[b7[$F[o7$Fgal$!3;+++$\\*z3NF[o7$Fbal$!3\"*******yU)[`$F[ o7$F]al$!35+++09[fNF[o7$Fh`l$!3))*****\\>rDe$F[o7$Fc`l$!3')*****>HSTg$ F[o7$F^`l$!3.+++4'zTi$F[o7$Fi_l$!39+++QEoUOF[o7$Fd_l$!39+++'3W'fOF[o7$ F__l$!3++++j(e]n$F[o7$Fj^l$!39+++*o?*)o$F[o7$Fe^l$!3%******zMA7q$F[o7$ F`^l$!3*)*****z0a>r$F[o7$F[^l$!3;+++7M5@PF[o7$Ff]l$!3,+++mZlGPF[o7$Fa] l$!39+++E()eMPF[o7$F\\]l$!3,+++K;))QPF[o7$Fg\\l$!3()*****>*\\]TPF[o7$F b\\l$!3-+++i[UUPF[o7$F]\\l$!3)******47,;u$F[o7$Fh[l$!3')*****fk')*QPF[ o7$Fc[l$!3z*****>KEXt$F[o7$F^[l$!3/+++ue:GPF[o7$Fiz$!3\"******RZ+)>PF[ o7$Fdz$!3>+++pIP4PF[o7$F_z$!31+++!Qsnp$F[o7$Fjy$!3)*******>0)=o$F[o7$F ey$!3!******>;gXm$F[o7$F`y$!3*******\\B^Yk$F[o7$F[y$!3?+++Sr'>i$F[o7$F fx$!3;+++!f!H'f$F[o7$Fax$!3=+++v$ptc$F[o7$F\\x$!3$)*****po7\\`$F[o7$Fg w$!3#)*****HG*e)\\$F[o7$Fbw$!3))*****RqH!eMF[o7$F]w$!3%)*****R:OGT$F[o 7$Fhv$!3y*****f-0EO$F[o7$Fcv$!3%)*****>QnpI$F[o7$F^v$!39++++'\\cC$F[o7 $Fiu$!3!)*****>1Z&yJF[o7$Fdu$!3?+++H))y0JF[o7$F_u$!39+++!Qgx-$F[o7$Fjt $!3#******zgg]%HF[o7$Fet$!3))*****\\&=SeGF[o7$F`t$!3%)*******4%\\oFF[o 7$F[t$!33+++2y&fn#F[o7$$!33+++v+**[JF]s$!3?+++obG\"e#F[o7$Fas$!3))**** *\\uX[[#F[o7$F[s$!3)*******3$4pQ#F[o7$Ffr$!3/+++,Fo(G#F[o7$Far$!3;+++X hL(=#F[o7$F\\r$!3#)******\\H-'3#F[o7$Ffq$!31+++&R*)Q)>F[o7$Faq$!3%**** **R,x5)=F[o7$F[q$!3%******>9?xx\"F[o7$Ffp$!35+++U:%Rn\"F[o7$$\"3c***** \\S:!)e(FY$!3)******\\N[)p:F[o7$F\\p$!3)******RiHbY\"F[o7$Fgo$!3++++,e 0h8F[o7$Fbo$!3!*******\\*zkD\"F[o7$F]o$!3++++0,%=:\"F[o7$Fgn$!33+++r@; Z5F[o7$FW$!3r*****zcGYU*F-7$FR$!3/+++GK_x$)F-7$FL$!3\")*****pXj.L(F-7$ FG$!3!)*****plzJG'F-7$$\"3++++oEuaZ5F-F'F*F.F4F:F@FFFKFQFVFfnF\\ oFaoFfoF[pF`pFepFjpF`qFeqF[rF`rFerFjrF`sFesFjsF_tFdtFitF^uFcuFhuF]vFbv FgvF\\wFawFfwF[xF`xFexFjxF_yFdyFiyF^zFczFhzF][lFb[lFg[lF\\\\lFa\\lFf\\ lF[]lF`]lFe]lFj]lF_^lFd^l-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF' 7$$!33+:%on)Hn=F]s$\"3%pm*Q=WgAFF-7$$!3.%*=0/*e\\]'F]s$\"3'3DuH*H#=0&F -7$$!3%f?%4)\\]%)\\\"F-$\"3e>**)ovpSe(F-7$$!3w#yh%)38go#F-$\"3ai4?fuk& ***F-7$$!3CZRNG%*GzTF-$\"3'Qc)pQ$\\:A\"F[o7$$!3#p0\\\"F[o$\"3kFiu **z`H>F[o7$$!3nWc)RoJfs\"F[o$\"3%[qn:*>0w>F[o7$$!3osf:@jO&*>F[o$\"3#\\ `,&H_<%*>F[o7$$!3sf#z0\\yeE#F[o$\"3+sNxc$zb(>F[o7$$!3)R]q(o2(=_#F[o$\" 3=SJF;(*3B>F[o7$$!39.u)fv*fYFF[o$\"3UJ\"oTRzn%=F[o7$$!39eI0'y,(**HF[o$ \"3+P#[KB2@s\"F[o7$$!3up8'4$eb)>$F[o$\"3!GsS\\\"QS*e\"F[o7$$!3&['eIU(4 zS$F[o$\"3A8b$\\eYkS\"F[o7$$!3IGnb*o7/d$F[o$\"3F_8hX;c@7F[o7$$!3,hbN$R $*4s$F[o$\"3AI4c*=\\T(**F-7$$!3%4'zDfj)\\$QF[o$\"3U#4/*yl5pwF-7$$!3*)G &3:zr7#RF[o$\"3u&*zYn\"3(G^F-7$$!3)4MeMK6)pRF[o$\"3*eG)HIj#Hr#F-7$$!3/ \"om)z:M))RF[o$\"3y-Hyfbb%='Fhq7$$!3kGBC.@4qRF[o$!3WhH;%)**R#p#F-7$$!3 3l)[75aL#RF[o$!3+QpH*)p\")\\]F-7$$!3Py$o#>FVTQF[o$!3iKPO*>0D^(F-7$$!3W 'fPaggRs$F[o$!3>I6$=q+E#**F-7$$!3!)yTi>5wyNF[o$!3nGq=4Fr57F[o7$$!3o;!G PGZ@T$F[o$!3\\hpk8U<-9F[o7$$!3aJ.FB*p.?$F[o$!3w]Q^Wv-)e\"F[o7$$!3q$[-Y [1%*)HF[o$!3=&RCOYx!GF[o7$$!3y?$eznOF[o7$$!3&*G))QW DF+?F[o$!3q;pgki;%*>F[o7$$!3J%)ptd4QPF[o7$$!3MnJ(zZbY [\"F[o$!3a*=#4N[)z#>F[o7$$!3B0Q0O+DH7F[o$!3-<C\"fPgVEF-$! 3XSwA6W2A**F-7$$!3!*f_\\xT[X:F-$!3I*oOX4bup(F-7$$!3W:]Ci-6')oF]s$!3Cfl ^2S>&>&F-7$$!3WF&o\\)*pa&=F]s$!3:;:!Rx2Sr#F-7$F($\"3uk-=)**[gj\"F7-Feg o6&FggoF(F(Fhgo-%(SCALINGG6#%,CONSTRAINEDG-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6%Q!6\"Fjgp-Fcgp6#%(DEFAULTG-%%VIEWG6$F^hpF^hp" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of th e real part of points along the curve. In this way we see that the lar gest interval on the nonnegative imaginary axis that contains the orig in and lies inside the stability region is " }{XPPEDIT 18 0 "[ 0, 1.6 7 ]" "6#7$\"\"!-%&FloatG6$\"$n\"!\"#" }{TEXT -1 18 " approximately. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 478 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 2098088 0519984971077562146307642387747/15064376699173227971808392982408727720 080*z^6+\n 33168276611156421352084390650912318925/170096462120664665 403419115544849851691512*z^7:\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplo t(pts,color=COLOR(RGB,0,.95,0),thickness=2,font=[HELVETICA,9]);\nDigit s := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 249 293 293 {PLOTDATA 2 "6(-%'C URVESG6#7in7$$\"\"!F)F(7$$\"5rrj`u#3+*y[!#@$\"5t?r(*e`EfTJF-7$$\"5#HQ* yk@l=IrF-$\"5\"fRuwrI&=$G'F-7$$\"5vez5()G5\\9*)F-$\"5H%=%)f/\"!#?$\"5%*3#Q)Rhqjc7F=7$$\"5?V)or=Dhc=\"F=$\"5\"o3\"= hEjzq:F=7$$\"5_,`9]Nh::8F=$\"5MvC*z9fb\\)=F=7$$\"5,(3$R7*3IsV\"F=$\"5j gi!oa&[6*>#F=7$$\"5Rfs['[aVOb\"F=$\"54-TI[F=$\"53[P%**e26*pPF=7$$\"5f%H!y A1Of\"3#F=$\"5'Hq&y,^-2%3%F=7$$\"54@F7x+*R&z@F=$\"5dDNVXl$H#)R%F=7$$\" 5:B#*fs\\7\\vAF=$\"5O5ktS'Q)Q7ZF=7$$\"5C?4;z\"))p&pBF=$\"5r*pT\")pEZl- &F=7$$\"5k[`Z!*zo'=Y#F=$\"5`-)z![TfqS`F=7$$\"5(Qzr)*3t\\Cb#F=$\"5-pC&Q &>V'[l&F=7$$\"5cAqP![mn8k#F=$\"57TZ&*))yA-pfF=7$$\"5Py9i]OalGFF=$\"5$f eb%ec'zJG'F=7$$\"5AN::T&*fL9GF=$\"58!pOH%QiL(f'F=7$$\"5H'>*Qp\\BU)*GF= $\"5)HY&eUY<\\6pF=7$$\"5X\"z7zm$)=4)HF=$\"5B42S)R#ekDsF=7$$\"5,1\"=;\\ %>#=1$F=$\"5!y$G9n=!)zRvF=7$$\"5G=e3(oy?69$F=$\"5d-9pEix%R&yF=7$$\"5ke XT\\(G(z=KF=$\"57A\"4*)yM%4o\")F=7$$\"5X74,zGh#[H$F=$\"5c#\\-!)R!pB#[) F=7$$\"5v*\\aM3Zu\"pLF=$\"5e]]w$\\Oujz)F=7$$\"5?PIs<`9!=W$F=$\"5FN:/@Q a]5\"*F=7$$\"5m@MrNlul7NF=$\"5/#Q6$pn&GYU*F=7$$\"5(H)*yA(fJo\"e$F=$\"5 q*p4&H$*=uQ(*F=7$$\"5>X`A=i\"3)[OF=$\"5*QM8f1K%G05!#>7$$\"5\")R_7!\\T \\Rr$F=$\"5h\\n8D!*HpO5Ffu7$$\"5/#HWmm04qx$F=$\"5#3IX4F*)*4o5Ffu7$$\"5 O+$QCyqry$QF=$\"5f$Q%Q8sY]*4\"Ffu7$$\"5Wh\\rjr4S'*QF=$\"56U#)o8;p!48\" Ffu7$$\"5uZ;+^$)[V_RF=$\"5SM$y2&\\hIi6Ffu7$$\"5`.t*[\">*zd+%F=$\"5\\#G jfv#=q$>\"Ffu7$$\"5-\\43q()H?cSF=$\"51Mg)*>HL4D7Ffu7$$\"5LrZC%Qa?M5%F= $\"5pJZhx\\*zkD\"Ffu7$$\"5)G$z.txM3ZTF=$\"53Z+4w$*3'yG\"Ffu7$$\"5nLeL% Gkbn=%F=$\"5!4)y!\\tEN#>8Ffu7$$\"5P!\\dfg[#)=A%F=$\"5!GA6c52-1N\"Ffu7$ $\"5&=nQ&f&pVg>Q\"Ffu7$$\"538n>7(ey,,Ze+$F=$\"5+-@)eVwWNm\"Ffu7$$!5!)[G=zcyL7SF=$\"5Q3p$pqgDZp \"Ffu7$$!56mBhR1sTQVF=$\"5yxP$[qXwes\"Ffu7$$!5bPn]Sbd*>c%F=$\"5T:(o(3& [%*pv\"Ffu7$$!5@XB3!H@JIu%F=$\"5m\">R(*4sw!)y\"Ffu7$$!51\"*Q)GU/=/!\\F =$\"5oi=?\\g+7>=Ffu7$$!5J.$H59g>E/&F=$\"5Bk,p[t77]=Ffu7$$!57:\\iwUd;u^ F=$\"5C;w7l8q2\")=Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\"\"#-% +AXESLABELSG6$Q!6\"F__l-%&COLORG6&%$RGBGF($\"#&*!\"#F(-%%VIEWG6$%(DEFA ULTGF[`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection point of the boundary curve with the im aginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve ei ther side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 1. 7*I:\nfor ct from 52 to 55 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0); \nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0B4b( y:WN!#>$\"0@e$\"0PpqgDZp\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0d%)3 f7[-\"!#=$\"0M[qXwes\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate \+ the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton (R(z)=exp(u*Pi*I),z=1.7*I))\nend proc:\nDigits := 15:\nu0 := bisect('r eal_part'(u),u=0.52..0.55);\nnewton(R(z)=exp(u0*Pi*I),z=1.7*I);\nDigit s := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0(oktHg/`!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"03:*H1)\\m\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "l argest interval on the nonnegative imaginary axis that contains the or igin and lies inside the stability region" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "[0, 1.6650];" "6#7$\"\"!-%&FloatG6$\"&]m\"!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 " #---------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 8 stage, ord er 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 stage, order 4 scheme is \+ given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b*`,StabilityFunction(4,8,'expanded'))):\n`R*` := unapply(%,z ):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\" zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)* &#F)\"#CF)*$)F'\"\"%F)F)F)*&#\"In>%)f#fet )>z*\"en+S?dcI-40)Gq85xc!oRTEfYB]mj;'*pF)*$)F'F1F)F)F)*&#\"fnB>[w_`7X8 -gYq$Hfs4ZhXnEfa%Q.b5#\"jn+SuS<%[(G=^>D$o\")y9_7`!QuY50]r:Ik2\"F)*$)F' \"\"(F)F)F)*&#\"S8dVfTN[(p3V%y3izYp)GIFQ'RBW\"ZO>8F>A^K#zj\"HwOiT9sOI` hhb+>8b$F)*$)F'\"\")F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R* `(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+B " 0 "" {MPLTEXT 1 0 322 "z_0 := ne wton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.49..0.49,color=[ red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cro ss,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3, color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.49..0 .49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%!#<$!3y)3D4IK 3[#F*7$$!3;+]7'3DdV%F*$!3m!=eOs=_D#F*7$$!35++Ds,X\"Q%F*$!3)*G*)*pHEt/# F*7$$!3(*\\P*)feAMVF*$!3G/2o))[#*z=F*7$$!3%)*\\Pva,qG%F*$!3E(>goU&GC%y!=%F*$!3>)R?rJvNT\"F*7$$!35+]#oviQ2%F*$!3k%HcXu!y\\6F*7$ $!3A+ve,&\\u'RF*$!3W06r-_J$H*!#=7$$!3?+v8#z!zoQF*$!3#y>/^j1Gd(FP7$$!3; +v.8ajmPF*$!3qeg<_(*GtgFP7$$!3C+v$Q<')4m$F*$!3c1d4GT#4y%FP7$$!3G+v3`dn bNF*$!3[BQFbkq9PFP7$$!37++0T5NZMF*$!3.Y'*G<&RC\"GFP7$$!3?+]sv&Q>N$F*$! 3y1;_=w!\\:#FP7$$!3E++:Qn_WKF*$!3=!pu<]0Fa\"FP7$$!3#****\\s&QnOJF*$!3D Z9F$*foS5FP7$$!3&****\\=iPF.$F*$!3#\\Qq>E\"RPk!#>7$$!3Q+v$Gc`$QHF*$!3M Y,u[y[IMFhp7$$!3+++l(y@h#GF*$!3_FR&eMI&QW!#?7$$!3=++g_n/JFF*$\"3e#)e(z c$G$p\"Fhp7$$!3C+vt%)=X?EF*$\"3sK?lx\"ew$QFhp7$$!3D++5(ocD_#F*$\"3oRzz V3!4^&Fhp7$$!3#**\\P(R,::CF*$\"3U&e5\"GZ3#=(Fhp7$$!3&)*\\7sqtGJ#F*$\"3 AgEj%*4aw')Fhp7$$!30+]Pz*eh?#F*$\"3LJA%fJ+$>5FP7$$!3l*\\([Y:;3@F*$\"3' 4Yj#pi1f6FP7$$!3$***\\7w!eC+#F*$\"3H*[F\")e*\\98FP7$$!3%**\\Py(=m#*=F* $\"3_f(*ee+:&[\"FP7$$!3%)*\\iW'R3(z\"F*$\"3Knge#QiVk\"FP7$$!3-+]d*>dQp \"F*$\"3;,-w1sPI=FP7$$!3)*****p]Q@(e\"F*$\"3#pO-oV))3/#FP7$$!3I+]FRT)G [\"F*$\"3E4s!Q;dyE#FP7$$!3')*\\P*y(R>Q\"F*$\"3T8&)3Uc25DFP7$$!3++]Kx#e )p7F*$\"3-PX8Kb]3GFP7$$!3=++5j![\"p6F*$\"3[2\\WgXM1JFP7$$!3s***\\KT=;1 \"F*$\"3kr,iy[1fMFP7$$!3'***\\P%o0=k*FP$\"3mU?WtM18QFP7$$!3](***\\tEbw &)FP$\"3)3)oehRjTUFP7$$!39-]P2EBuvFP$\"3K**=@6,y)o%FP7$$!3>)*\\(GL>l_' FP$\"3MIi<$zdm?&FP7$$!3z++]-\")=-bFP$\"3Y7*>0WZ#odFP7$$!3$*)*\\PM#3)HW FP$\"326Bx0#*>@kFP7$$!3;(****f'*ypR$FP$\"3)fwDe)[&)>rFP7$$!3L)**\\U'=w SBFP$\"3r)3*=Ia,8zFP7$$!3?$*\\Pt2H$H\"FP$\"3#=>?]jZoy)FP7$$!33R****pit 2LFhp$\"3E7!)*36PYn*FP7$$\"3Z^+]FkzBxFhp$\"3YH6N=\"*H!3\"F*7$$\"35**** *z2`!f\"F*7$$\"3l+]i_F06GFP$\"3!375>,$fC8F*7$$\"3\\.]P t1&z\"QFP$\"3WoyK+<\"\\Y\"F*7$$\"3!***************[FP$\"3[r/4\"p:Bj\"F *-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\ \\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$Ff nFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$Fj pFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^ sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$F auFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$F dwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$F gyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[ lF[\\lF[\\lFh[l-F$6&7#7$$!3%******Hs^O+%F*Fa\\l-%'SYMBOLG6#%'CIRCLEG-F e[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSS GF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&C OLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q \"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$ \\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture shows the stability region for the 8 stage, order 4 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1771 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+\n 2642583634938982724957017735859259841967/\n \+ 317880829535588751008411727805166380812000*z^5+\n 979198196685797 3252888651898756643431399476766248808593/\n 6996166366502346592641 396805677101370288050902305657204000*z^6+\n 2105503384545926674561 4709725929370466002134512535276481923/\n 1076430157150051046743805 31252147881683251951182874841740744000*z^7-\n 44233963827302886946 796208784430869748354159435713/\n 35513190055616153303672144162367 6291637923251221927131936*z^8:\npts := []: z0 := 0:\nfor ct from 0 to \+ 200 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=C OLOR(RGB,.1,.41,.1)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i] ,[-2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RG B,.2,.83,.2)):\npts := []: z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n \+ zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB, .1,.41,.1)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.38,4. 19]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.2 ,.83,.2)):\npts := []: z0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.1,. 41,.1)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.38,-4.19] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.2,.8 3,.2)):\np_7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4.79]]],color =black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.79..1.99,-4 .79..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`] ,axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 594 594 594 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3#)*****\\G5jp% !#F$\"3!******fM'zq:!#=7$$\"3!)*****f!)3J4$!#D$\"3y*****\\>$fTJF07$$\" 3#******zwvhm$!#C$\"3D+++bAR7ZF07$$\"3#)*****\\9od9#!#B$\"3M+++]L>$G'F 07$$\"3m*****poh$R%)F@$\"3k******3(zR&yF07$$\"3#******ze++a#!#A$\"3Z++ +$fzYU*F07$$\"3v*****zT_.A'FK$\"3/+++r)3&*4\"!#<7$$\"3++++cEuq7!#@$\"3 %******H#RZc7FS7$$\"3!******zYu+:#FW$\"31+++37F89FS7$$\"3v*****\\6Ad\" GFW$\"3)******ROT(p:FS7$$\"33+++*e5z(>FW$\"3$*******evjD$\"3&)*****zn^bL#FS7$$!3++++[ eoH@Ffp$\"3?+++-f;#[#FS7$$!3)******prp=w$Ffp$\"3:+++\"oF_i#FS7$$!31+++ \"e5x;'Ffp$\"3y*****HmDPw#FS7$$!3)*******4;<&\\*Ffp$\"3?+++\"z4h*GFS7$ $!3/+++yz^\"Q\"F0$\"3>+++Y&p--$FS7$$!3/+++w^_2>F0$\"3.+++K=$R8$FS7$$!3 x*****HP.*4DF0$\"3-+++4zRNKFS7$$!3%*******))>mkJF0$\"3#******Hw*3CLFS7 $$!3y******>+9\\QF0$\"3$)*****HK/0S$FS7$$!31+++dAMYXF0$\"39+++/mwlMFS7 $$!3E+++znOX_F0$\"3')*****4b&=@NFS7$$!3;+++G^(*RfF0$\"39+++bN*zc$FS7$$ !3/+++%*[&pi'F0$\"3\"******p\"eC2OFS7$$!3%)******f=)[I(F0$\"3'******fZ 1)ROFS7$$!3W+++$4CL(zF0$\"3;+++%Rpjm$FS7$$!3k*****4()pBj)F0$\"3y***** \\o([(o$FS7$$!3Y+++&3hBG*F0$\"3%******\\l,Oq$FS7$$!37+++:\"\\P#**F0$\" 3#)*****z3i]r$FS7$$!33+++r@qb5FS$\"3!*******R'[@s$FS7$$!3.+++uqE=6FS$ \"37+++dS3DPFS7$$!33+++u$=,=\"FS$\"3!******f2XSs$FS7$$!32+++i\\IT7FS$ \"3********>3<>PFS7$$!3!******pOx=I\"FS$\"3'******HMp0r$FS7$$!3-+++A** )=O\"FS$\"31+++pGK)p$FS7$$!3$******f\\.9U\"FS$\"3/+++!>#\\#o$FS7$$!3/+ ++\"y)[![\"FS$\"3%******Hm?Jm$FS7$$!35+++X,BR:FS$\"3/+++T#Q-k$FS7$$!3+ +++]+t(f\"FS$\"3*******f0mQh$FS7$$!3%******4$R6c;FS$\"37+++y@-%e$FS7$$ !3!******>DNXr\"FS$\"3A+++>$H2b$FS7$$!3!******R.!=tFS$\"3')******\\dQ%Q$FS7$$!3z*******Hb[,#FS$\"3-+++ .\"3`L$FS7$$!33+++RURy?FS$\"31+++F7'QG$FS7$$!3(******>W2O9#FS$\"3***** ***z\"[0B$FS7$$!3')*****48>0@#FS$\"3-+++I_&f<$FS7$$!3!)*****fqs*yAFS$ \"3-+++s8p?JFS7$$!39+++9gi[BFS$\"3!*******>uIlIFS7$$!3\")*****HJ\"**=C FS$\"3'******p=@-,$FS7$$!3!)*****f@2&*[#FS$\"3++++C`nbHFS7$$!3;+++Vdhf DFS$\"33+++;zt,HFS7$$!3.+++B2#)GEFS$\"3!)*****H'fL[GFS7$$!3'******\\5= np#FS$\"3#)*****zd-`z#FS7$$!3=+++b=+jFFS$\"32+++5;UUFFS7$$!3y******zYX FGFS$\"3%******4bh%*o#FS7$$!3#)*****HGK**)GFS$\"38+++3u>OEFS7$$!33+++E oM]HFS$\"3#)*****HpACe#FS7$$!3()*****zW_'3IFS$\"3;++++Y&z_#FS7$$!3/+++ hP$[1$FS$\"3!)*****f.NEZ#FS7$$!3-+++&H(*)=JFS$\"3++++A%HjT#FS7$$!3%*** ****Qa'3<$FS$\"3'*******>X#*eBFS7$$!3.+++ZBx?KFS$\"3-+++EgK+BFS7$$!3++ ++38moKFS$\"3&)*****pMc/C#FS7$$!3;+++\"=$e9LFS$\"39+++-GDz@FS7$$!3))** ***z`&feLFS$\"3%)*****pLmm6#FS7$$!3,+++\\Aw+MFS$\"3')*****4$3m_?FS7$$! 3:+++mK:TMFS$\"3%******p)H@()>FS7$$!3'******fPW)zMFS$\"3&******\\\"GJ? >FS7$$!34+++(p;p^$FS$\"37+++AY'>&=FS7$$!3;+++)pbCb$FS$\"3'******HV)=#y \"FS7$$!3#******z_\\le$FS$\"3,+++S;-6OFS$\"30+++5 1_Q;FS7$$!31+++*\\^2l$FS$\"3++++u=wk:FS7$$!3/+++]=-\"o$FS$\"3#******>Y U)*[\"FS7$$!3')*****H8i,r$FS$\"3&******H#*yQT\"FS7$$!3')*****4D>#QPFS$ \"3++++yY+P8FS7$$!39+++az@lPFS$\"3)********fl$f7FS7$$!3)*******\\x:\"z $FS$\"3%*******GS6\"=\"FS7$$!3&)*****>z6g\"QFS$\"3'******\\!=S-6FS7$$! 30+++q!G(RQFS$\"36+++sJPB5FS7$$!3y*****\\yKA'QFS$\"3u******)\\%eT%*F07 $$!3?++++aV$)QFS$\"3Q+++\"e#p[')F07$$!3-+++.YB.RFS$\"3a*****H\"p&f&yF0 7$$!3#)******fV_@RFS$\"3o*****p'\\0kqF07$$!3@+++0%*>QRFS$\"3g*****R`\\ MF'F07$$!3#*******3(fJ&RFS$\"3u*****Rt4W[&F07$$!3')******3QJmRFS$\"3w* ****f(*Qqp%F07$$!3#******f*4exRFS$\"3w*****HD08\"RF07$$!3/+++&f#*o)RFS $\"3>+++5/2FJF07$$!3-+++KD>%*RFS$\"3'******>N;TM#F07$$!39+++fuV**RFS$ \"3-+++Ct;i:F07$$!3l*****Rj'f-SFS$\"3h+++!*G34yFfp7$$!3%******Hs^O+%FS F(7$Fijl$!3h+++!*G34yFfp7$Fdjl$!3-+++Ct;i:F07$F_jl$!3'******>N;TM#F07$ Fjil$!3>+++5/2FJF07$Feil$!3w*****HD08\"RF07$F`il$!3w*****f(*Qqp%F07$F[ il$!3u*****Rt4W[&F07$Ffhl$!3g*****R`\\MF'F07$Fahl$!3o*****p'\\0kqF07$F \\hl$!3a*****H\"p&f&yF07$Fggl$!3Q+++\"e#p[')F07$Fbgl$!3u******)\\%eT%* F07$F]gl$!36+++sJPB5FS7$Fhfl$!3'******\\!=S-6FS7$Fcfl$!3%*******GS6\"= \"FS7$F^fl$!3)********fl$f7FS7$Fiel$!3++++yY+P8FS7$Fdel$!3&******H#*yQ T\"FS7$F_el$!3#******>YU)*[\"FS7$Fjdl$!3++++u=wk:FS7$Fedl$!30+++51_Q;F S7$F`dl$!3,+++S;-6&=FS7$ Facl$!3&******\\\"GJ?>FS7$F\\cl$!3%******p)H@()>FS7$Fgbl$!3')*****4$3m _?FS7$Fbbl$!3%)*****pLmm6#FS7$F]bl$!39+++-GDz@FS7$Fhal$!3&)*****pMc/C# FS7$Fcal$!3-+++EgK+BFS7$F^al$!3'*******>X#*eBFS7$Fi`l$!3++++A%HjT#FS7$ Fd`l$!3!)*****f.NEZ#FS7$F_`l$!3;++++Y&z_#FS7$Fj_l$!3#)*****HpACe#FS7$F e_l$!38+++3u>OEFS7$F`_l$!3%******4bh%*o#FS7$F[_l$!32+++5;UUFFS7$Ff^l$! 3#)*****zd-`z#FS7$Fa^l$!3!)*****H'fL[GFS7$F\\^l$!33+++;zt,HFS7$Fg]l$!3 ++++C`nbHFS7$Fb]l$!3'******p=@-,$FS7$F]]l$!3!*******>uIlIFS7$Fh\\l$!3- +++s8p?JFS7$Fc\\l$!3-+++I_&f<$FS7$F^\\l$!3********z\"[0B$FS7$Fi[l$!31+ ++F7'QG$FS7$Fd[l$!3-+++.\"3`L$FS7$F_[l$!3')******\\dQ%Q$FS7$Fjz$!3=+++ 0DqIMFS7$Fez$!3++++zy(RZ$FS7$F`z$!3++++&RDS^$FS7$F[z$!3A+++>$H2b$FS7$F fy$!37+++y@-%e$FS7$Fay$!3*******f0mQh$FS7$F\\y$!3/+++T#Q-k$FS7$Fgx$!3% ******Hm?Jm$FS7$Fbx$!3/+++!>#\\#o$FS7$F]x$!31+++pGK)p$FS7$Fhw$!3'***** *HMp0r$FS7$Fcw$!3********>3<>PFS7$F^w$!3!******f2XSs$FS7$Fiv$!37+++dS3 DPFS7$Fdv$!3!*******R'[@s$FS7$F_v$!3#)*****z3i]r$FS7$Fju$!3%******\\l, Oq$FS7$Feu$!3y*****\\o([(o$FS7$F`u$!3;+++%Rpjm$FS7$F[u$!3'******fZ1)RO FS7$Fft$!3\"******p\"eC2OFS7$Fat$!39+++bN*zc$FS7$F\\t$!3')*****4b&=@NF S7$Fgs$!39+++/mwlMFS7$Fbs$!3$)*****HK/0S$FS7$F]s$!3#******Hw*3CLFS7$Fh r$!3-+++4zRNKFS7$Fcr$!3.+++K=$R8$FS7$F^r$!3>+++Y&p--$FS7$Fiq$!3?+++\"z 4h*GFS7$Fdq$!3y*****HmDPw#FS7$F_q$!3:+++\"oF_i#FS7$Fjp$!3?+++-f;#[#FS7 $Fdp$!3&)*****zn^bL#FS7$F_p$!37+++Zj1'=#FS7$Fio$!3?+++=LDM?FS7$Fdo$!3' *******)4<1)=FS7$F_o$!3$*******evjD$!3M+++]L>$G'F07$F8$!3D+++bAR7ZF07$F2$!3y* ****\\>$fTJF07$$\"3)********H5jp%F-$!3!******fM'zq:F0F'-%&COLORG6&%$RG BG$\"\"\"!\"\"$\"#T!\"#F_^n-%)POLYGONSG6fw7%F'7$$\"+&G5jp%Ffp$\"+Yjzq: !#57$$Fd^nF)F(7%Fi^n7$$\"+1)3J4$FS$\"+&>$fTJF^_nF__n7%Fb_n7$$\"+od$G'F^_nF__n7%F_`n7$$\" +(oh$R%)Fb`n$\"+4(zR&yF^_nF__n7%Ff`n7$$\"+)e++a#!#9$\"+$fzYU*F^_nF__n7 %F\\an7$$\"+=CN?iF_an$\"+r)3&*4\"!\"*F__n7%Fcan7$$\"+cEuq7!#8$\"+BRZc7 FhanF__n7%Fjan7$$\"+oW2]@F]bn$\"+37F89FhanF__n7%Fabn7$$\"+:@s:GF]bn$\" +k8up:FhanF__n7%Fgbn7$$\"+*e5z(>F]bn$\"+fvjDF^_n$\"+K=$R8$FhanF__n7%F[gn7$$!+tL!*4DF^_n$\"+4zRNKFhanF__n7%Fagn7$ $!+*)>mkJF^_n$\"+j(*3CLFhanF__n7%Fggn7$$!+?+9\\QF^_n$\"+BV]+MFhanF__n7 %F]hn7$$!+dAMYXF^_n$\"+/mwlMFhanF__n7%Fchn7$$!+znOX_F^_n$\"+^b=@NFhanF __n7%Fihn7$$!+G^(*RfF^_n$\"+bN*zc$FhanF__n7%F_in7$$!+%*[&pi'F^_n$\"+PFhanF__n7%F[]o7$$!+nt(=I\"Fhan$\"+V$p0r$FhanF__n7%Fa]o7$$!+A**)=O \"Fhan$\"+pGK)p$FhanF__n7%Fg]o7$$!+'\\.9U\"Fhan$\"+!>#\\#o$FhanF__n7%F ]^o7$$!+\"y)[![\"Fhan$\"+j17jOFhanF__n7%Fc^o7$$!+X,BR:Fhan$\"+T#Q-k$Fh anF__n7%Fi^o7$$!+]+t(f\"Fhan$\"+cg'Qh$FhanF__n7%F__o7$$!+JR6c;Fhan$\"+ y@-%e$FhanF__n7%Fe_o7$$!+__`9$H2b$FhanF__n7%F[`o7$$!+M+=tFhan$\"+]dQ%Q$FhanF__n7%F cao7$$!++`&[,#Fhan$\"+.\"3`L$FhanF__n7%Fiao7$$!+RURy?Fhan$\"+F7'QG$Fha nF__n7%F_bo7$$!+UugV@Fhan$\"+!=[0B$FhanF__n7%Febo7$$!+J\">0@#Fhan$\"+I _&f<$FhanF__n7%F[co7$$!+1F(*yAFhan$\"+s8p?JFhanF__n7%Faco7$$!+9gi[BFha n$\"+?uIlIFhanF__n7%Fgco7$$!+88**=CFhan$\"+(=@-,$FhanF__n7%F]do7$$!+;s ]*[#Fhan$\"+C`nbHFhanF__n7%Fcdo7$$!+VdhfDFhan$\"+;zt,HFhanF__n7%Fido7$ $!+B2#)GEFhan$\"+jfL[GFhanF__n7%F_eo7$$!+0\"=np#Fhan$\"+yDI&z#FhanF__n 7%Feeo7$$!+b=+jFFhan$\"+5;UUFFhanF__n7%F[fo7$$!+!oau#GFhan$\"+^:Y*o#Fh anF__n7%Fafo7$$!+$GK**)GFhan$\"+3u>OEFhanF__n7%Fgfo7$$!+EoM]HFhan$\"+$ pACe#FhanF__n7%F]go7$$!+[Cl3IFhan$\"++Y&z_#FhanF__n7%Fcgo7$$!+hP$[1$Fh an$\"+O]jsCFhanF__n7%Figo7$$!+&H(*)=JFhan$\"+A%HjT#FhanF__n7%F_ho7$$!+ Ra'3<$Fhan$\"+?X#*eBFhanF__n7%Feho7$$!+ZBx?KFhan$\"+EgK+BFhanF__n7%F[i o7$$!+38moKFhan$\"+ZjXSAFhanF__n7%Faio7$$!+\"=$e9LFhan$\"+-GDz@FhanF__ n7%Fgio7$$!+QbfeLFhan$\"+Pjm;@FhanF__n7%F]jo7$$!+\\Aw+MFhan$\"+J3m_?Fh anF__n7%Fcjo7$$!+mK:TMFhan$\"+()H@()>FhanF__n7%Fijo7$$!+wV%)zMFhan$\"+ :GJ?>FhanF__n7%F_[p7$$!+(p;p^$Fhan$\"+AY'>&=FhanF__n7%Fe[p7$$!+)pbCb$F han$\"+L%)=#y\"FhanF__n7%F[\\p7$$!+G&\\le$Fhan$\"+S;-6OFhan$\"+51_Q;FhanF__n7%Fg\\p7$$!+*\\^2l$Fhan$\"+u=wk:FhanF__ n7%F]]p7$$!+]=-\"o$Fhan$\"+iC%)*[\"FhanF__n7%Fc]p7$$!+L@;5PFhan$\"+B*y QT\"FhanF__n7%Fi]p7$$!+^#>#QPFhan$\"+yY+P8FhanF__n7%F_^p7$$!+az@lPFhan $\"++cOf7FhanF__n7%Fe^p7$$!+]x:\"z$Fhan$\"+HS6\"=\"FhanF__n7%F[_p7$$!+ #z6g\"QFhan$\"+0=S-6FhanF__n7%Fa_p7$$!+q!G(RQFhan$\"+sJPB5FhanF__n7%Fg _p7$$!+&yKA'QFhan$\"+*\\%eT%*F^_nF__n7%F]`p7$$!++aV$)QFhan$\"+\"e#p[') F^_nF__n7%Fc`p7$$!+.YB.RFhan$\"+8p&f&yF^_nF__n7%Fi`p7$$!+gV_@RFhan$\"+ n\\0kqF^_nF__n7%F_ap7$$!+0%*>QRFhan$\"+M&\\MF'F^_nF__n7%Feap7$$!+4(fJ& RFhan$\"+M(4W[&F^_nF__n7%F[bp7$$!+4QJmRFhan$\"+w*Qqp%F^_nF__n7%Fabp7$$ !+'*4exRFhan$\"+`_I6RF^_nF__n7%Fgbp7$$!+&f#*o)RFhan$\"+5/2FJF^_nF__n7% F]cp7$$!+KD>%*RFhan$\"+_j6WBF^_nF__n7%Fccp7$$!+fuV**RFhan$\"+Ct;i:F^_n F__n7%Ficp7$$!+Mmf-SFhan$\"+!*G34yFidnF__n7%F_dp7$$!+B&=FhanF__ n7%Fejp7$F`[p$!+:GJ?>FhanF__n7%Fijp7$Fjjo$!+()H@()>FhanF__n7%F][q7$Fdj o$!+J3m_?FhanF__n7%Fa[q7$F^jo$!+Pjm;@FhanF__n7%Fe[q7$Fhio$!+-GDz@FhanF __n7%Fi[q7$Fbio$!+ZjXSAFhanF__n7%F]\\q7$F\\io$!+EgK+BFhanF__n7%Fa\\q7$ Ffho$!+?X#*eBFhanF__n7%Fe\\q7$F`ho$!+A%HjT#FhanF__n7%Fi\\q7$Fjgo$!+O]j sCFhanF__n7%F]]q7$Fdgo$!++Y&z_#FhanF__n7%Fa]q7$F^go$!+$pACe#FhanF__n7% Fe]q7$Fhfo$!+3u>OEFhanF__n7%Fi]q7$Fbfo$!+^:Y*o#FhanF__n7%F]^q7$F\\fo$! +5;UUFFhanF__n7%Fa^q7$Ffeo$!+yDI&z#FhanF__n7%Fe^q7$F`eo$!+jfL[GFhanF__ n7%Fi^q7$Fjdo$!+;zt,HFhanF__n7%F]_q7$Fddo$!+C`nbHFhanF__n7%Fa_q7$F^do$ !+(=@-,$FhanF__n7%Fe_q7$Fhco$!+?uIlIFhanF__n7%Fi_q7$Fbco$!+s8p?JFhanF_ _n7%F]`q7$F\\co$!+I_&f<$FhanF__n7%Fa`q7$Ffbo$!+!=[0B$FhanF__n7%Fe`q7$F `bo$!+F7'QG$FhanF__n7%Fi`q7$Fjao$!+.\"3`L$FhanF__n7%F]aq7$Fdao$!+]dQ%Q $FhanF__n7%Faaq7$F^ao$!+0DqIMFhanF__n7%Feaq7$Fh`o$!+zy(RZ$FhanF__n7%Fi aq7$Fb`o$!+&RDS^$FhanF__n7%F]bq7$F\\`o$!+>$H2b$FhanF__n7%Fabq7$Ff_o$!+ y@-%e$FhanF__n7%Febq7$F`_o$!+cg'Qh$FhanF__n7%Fibq7$Fj^o$!+T#Q-k$FhanF_ _n7%F]cq7$Fd^o$!+j17jOFhanF__n7%Facq7$F^^o$!+!>#\\#o$FhanF__n7%Fecq7$F h]o$!+pGK)p$FhanF__n7%Ficq7$Fb]o$!+V$p0r$FhanF__n7%F]dq7$F\\]o$!+?3<>P FhanF__n7%Fadq7$Ff\\o$!+w]/CPFhanF__n7%Fedq7$F`\\o$!+dS3DPFhanF__n7%Fi dq7$Fj[o$!+S'[@s$FhanF__n7%F]eq7$Fd[o$!+)3i]r$FhanF__n7%Faeq7$F^[o$!+b ;g.PFhanF__n7%Feeq7$Fhjn$!+&o([(o$FhanF__n7%Fieq7$Fbjn$!+%Rpjm$FhanF__ n7%F]fq7$F\\jn$!+wk!)ROFhanF__n7%Fafq7$Ffin$!+$G'F^_nF__n7%Fe\\r7$Fi_n$!+bAR 7ZF^_nF__n7%Fi\\r7$Fc_n$!+&>$fTJF^_nF__n7%F]]r7$$\"++.J'p%Ffp$!+Yjzq:F ^_nF__n7%Fa]rF'F__n-F\\^n6&F^^n$\"\"#Fa^n$\"#$)Fd^nFi]r-%&STYLEG6#%,PA TCHNOGRIDG-F$6$7in7$$\"3%******p>lr\\\"FS$\"3g*****H'pw'H%FS7$$\"35+++ qq0*[\"FS$\"3%******fn$Q0VFS7$$\"3%******RGq-[\"FS$\"3T+++!32LJ%FS7$$ \"3'******z'z%3Z\"FS$\"3\"******f'e\\?VFS7$$\"33+++))R$3Y\"FS$\"3U+++_ !4pK%FS7$$\"31+++qKF]9FS$\"3w*****Hr0DL%FS7$$\"32+++o@@R9FS$\"3\"***** *prWsL%FS7$$\"3*)******z))pF9FS$\"30+++zW3TVFS7$$\"3-+++MRy:9FS$\"3)** ****\\t#)RM%FS7$$\"3$******>v?NS\"FS$\"3e******>j*eM%FS7$$\"3#******>K m4R\"FS$\"3'******z(4yYVFS7$$\"3++++*)>=y8FS$\"3o******)>\"fYVFS7$$\"3 ++++5WBl8FS$\"3=+++,,GXVFS7$$\"31+++*p'>_8FS$\"3i******)Q*zUVFS7$$\"3/ +++?)\\\"R8FS$\"3n*****\\U*4RVFS7$$\"3(******HF%=E8FS$\"3i******o%HTL% FS7$$\"36+++W@S88FS$\"3P+++y\"QyK%FS7$$\"3,+++%f>4I\"FS$\"3O+++7X@&yfUFS7$$\"3&******Hoa9C \"FS$\"3v*****f6vJC%FS7$$\"36+++,lCO7FS$\"3\"******>^]^A%FS7$$\"3++++> ()>L7FS$\"3%******45ne?%FS7$$\"3$******Rw!oK7FS$\"3++++?,e&=%FS7$$\"3) *******H.0N7FS$\"3C+++Z7nkTFS7$$\"33+++aDgS7FS$\"3'******z?mO9%FS7$$\" 3#******H0,&\\7FS$\"3N+++8xABTFS7$$\"33+++mfqh7FS$\"3f*****4&z5/TFS7$$ \"3)*******f\"GpF\"FS$\"3I+++6s0(3%FS7$$\"3%******zRRYH\"FS$\"3E+++?-r sSFS7$$\"3\"********RXTJ\"FS$\"3W+++)4#\\hSFS7$$\"35+++e]pM8FS$\"3A+++ z*zN0%FS7$$\"3#*******4Ieb8FS$\"3q******pe#*[SFS7$$\"3++++(\\:iP\"FS$ \"3!******fi0%FS7$$\"31+++;iiK9FS$\"3!)*****p-Rv0%FS7$$\"3-+++%eV)[ 9FS$\"3g******)oG[1%FS7$$\"35+++/$*ej9FS$\"3-+++QaatSFS7$$\"3!******f* G$oZ\"FS$\"3m*****p6HM3%FS7$$\"3'******Ryt&)[\"FS$\"3#******\\3_U4%FS7 $$\"3++++(RJ))\\\"FS$\"3k*****p9;e5%FS7$$\"3(*******y\"Qw]\"FS$\"3v*** **f))[z6%FS7$$\"3/+++_Z.::FS$\"3i**********\\ITFS7$$\"3!******\\gm5_\" FS$\"3-+++ZzLVTFS7$$\"3,+++p=yD:FS$\"3'******f)pMcTFS7$$\"3,+++Q)H#H:F S$\"3++++_YUpTFS7$$\"3&******\\,g9`\"FS$\"3o******4'zC=%FS7$$\"3++++:: _K:FS$\"3;+++_)Ha>%FS7$$\"3#******HriC`\"FS$\"3#******R>,#3UFS7$$\"30+ ++l5LJ:FS$\"39+++-gs?UFS7$$\"3$******f/t\"H:FS$\"3!)*****>8UHB%FS7$$\" 36+++pT.E:FS$\"3;+++I?zWUFS7$$\"3++++h!f>_\"FS$\"3q*****>'>AcUFS7$$\"3 #*******y:*p^\"FS$\"3)******fO\"=nUFS7$$\"3'*******f\\<6:FS$\"3s*****f GAwF%FS7$$\"34+++*)>b/:FS$\"3u*****R())\\(G%FSFd^rF[^n-Ff^n6jn7%7$$\"+ (>lr\\\"Fhan$\"+jpw'H%Fhan7$$\"+qq0*[\"Fhan$\"+wOQ0VFhan7$$\"$Q\"Fd^n$ \"$>%Fd^n7%Fhas7$$\"+%Gq-[\"Fhan$\"+!32LJ%FhanF]bs7%Fcbs7$$\"+oz%3Z\"F han$\"+me\\?VFhanF]bs7%Fibs7$$\"+))R$3Y\"Fhan$\"+_!4pK%FhanF]bs7%F_cs7 $$\"+qKF]9Fhan$\"+8d]KVFhanF]bs7%Fecs7$$\"+o@@R9Fhan$\"+=y8Fhan$\"+*>\"fYVFhanF]bs7%Fies7$$\"+ 5WBl8Fhan$\"+,,GXVFhanF]bs7%F_fs7$$\"+*p'>_8Fhan$\"+*Q*zUVFhanF]bs7%Fe fs7$$\"+?)\\\"R8Fhan$\"+D%*4RVFhanF]bs7%F[gs7$$\"+tU=E8Fhan$\"+p%HTL%F hanF]bs7%Fags7$$\"+W@S88Fhan$\"+y\"QyK%FhanF]bs7%Fggs7$$\"+%f>4I\"Fhan $\"+7X()>L7Fhan$\"+,r'e?%FhanF]bs7%F][t7$$ \"+k2oK7Fhan$\"+?,e&=%FhanF]bs7%Fc[t7$$\"+I.0N7Fhan$\"+Z7nkTFhanF]bs7% Fi[t7$$\"+aDgS7Fhan$\"+3imVTFhanF]bs7%F_\\t7$$\"+`5]\\7Fhan$\"+8xABTFh anF]bs7%Fe\\t7$$\"+mfqh7Fhan$\"+^z5/TFhanF]bs7%F[]t7$$\"+g\"GpF\"Fhan$ \"+6s0(3%FhanF]bs7%Fa]t7$$\"+)RRYH\"Fhan$\"+?-rsSFhanF]bs7%Fg]t7$$\"++ a998Fhan$\"+)4#\\hSFhanF]bs7%F]^t7$$\"+e]pM8Fhan$\"+z*zN0%FhanF]bs7%Fc ^t7$$\"+5Ieb8Fhan$\"+qe#*[SFhanF]bs7%Fi^t7$$\"+(\\:iP\"Fhan$\"+EwJZSFh anF]bs7%F__t7$$\"+6W8'R\"Fhan$\"+&zZ%[SFhanF]bs7%Fe_t7$$\"+_7,:9Fhan$ \"+w.(>0%FhanF]bs7%F[`t7$$\"+;iiK9Fhan$\"+F!Rv0%FhanF]bs7%Fa`t7$$\"+%e V)[9Fhan$\"+*oG[1%FhanF]bs7%Fg`t7$$\"+/$*ej9Fhan$\"+QaatSFhanF]bs7%F]a t7$$\"+'*G$oZ\"Fhan$\"+<\"HM3%FhanF]bs7%Fcat7$$\"+%yt&)[\"Fhan$\"+&3_U 4%FhanF]bs7%Fiat7$$\"+(RJ))\\\"Fhan$\"+Zh\"e5%FhanF]bs7%F_bt7$$\"+z\"Q w]\"Fhan$\"+'))[z6%FhanF]bs7%Febt7$$\"+_Z.::Fhan$\"+++]ITFhanF]bs7%F[c t7$$\"+0m1@:Fhan$\"+ZzLVTFhanF]bs7%Fact7$$\"+p=yD:Fhan$\"+')pMcTFhanF] bs7%Fgct7$$\"+Q)H#H:Fhan$\"+_YUpTFhanF]bs7%F]dt7$$\"+:+YJ:Fhan$\"+5'zC =%FhanF]bs7%Fcdt7$$\"+::_K:Fhan$\"+_)Ha>%FhanF]bs7%Fidt7$$\"+8FYK:Fhan $\"+%>,#3UFhanF]bs7%F_et7$$\"+l5LJ:Fhan$\"+-gs?UFhanF]bs7%Feet7$$\"+YI _\"Fhan$\"+i>AcUFhanF]bs7%Fgft7$$\"+z:*p^\"Fhan$\"+m8=nUFhanF ]bs7%F]gt7$$\"+g\\<6:Fhan$\"+'GAwF%FhanF]bs7%Fcgt7$$\"+*)>b/:Fhan$\"+u ))\\(G%FhanF]bs7%FigtFcasF]bsFg]rF]^r-F$6$7in7$Fe^r$!3g*****H'pw'H%FS7 $F\\as$!3u*****R())\\(G%FS7$Fg`s$!3s*****fGAwF%FS7$Fb`s$!3)******fO\"= nUFS7$F]`s$!3q*****>'>AcUFS7$Fh_s$!3;+++I?zWUFS7$Fc_s$!3!)*****>8UHB%F S7$F^_s$!39+++-gs?UFS7$Fi^s$!3#******R>,#3UFS7$Fd^s$!3;+++_)Ha>%FS7$F_ ^s$!3o******4'zC=%FS7$Fj]s$!3++++_YUpTFS7$Fe]s$!3'******f)pMcTFS7$F`]s $!3-+++ZzLVTFS7$F[]s$!3i**********\\ITFS7$Ff\\s$!3v*****f))[z6%FS7$Fa \\s$!3k*****p9;e5%FS7$F\\\\s$!3#******\\3_U4%FS7$Fg[s$!3m*****p6HM3%FS 7$Fb[s$!3-+++QaatSFS7$F][s$!3g******)oG[1%FS7$Fhjr$!3!)*****p-Rv0%FS7$ Fcjr$!3))*****fPq>0%FS7$F^jr$!3S+++&zZ%[SFS7$Fiir$!3!******fi^]^A%FS7$Fher$!3v*****f6 vJC%FS7$Fcer$!3s*****>@&yfUFS7$F^er$!31+++(*z*[F%FS7$Fidr$!3!******fR \"[)G%FS7$Fddr$!37+++Xs`+VFS7$F_dr$!3v*****HA*36VFS7$Fjcr$!3O+++7X\"fYVF S7$Fgar$!3'******z(4yYVFS7$Fbar$!3e******>j*eM%FS7$F]ar$!3)******\\t#) RM%FS7$Fh`r$!30+++zW3TVFS7$Fc`r$!3\"******prWsL%FS7$F^`r$!3w*****Hr0DL %FS7$Fi_r$!3U+++_!4pK%FS7$Fd_r$!3\"******f'e\\?VFS7$F__r$!3T+++!32LJ%F S7$Fj^r$!3%******fn$Q0VFSFbhtF[^n-Ff^n6jn7%7$Fdas$!+jpw'H%Fhan7$Fjgt$! +u))\\(G%Fhan7$F^bs$!$>%Fd^n7%F\\du7$Fdgt$!+'GAwF%FhanF_du7%Fcdu7$F^gt $!+m8=nUFhanF_du7%Fgdu7$Fhft$!+i>AcUFhanF_du7%F[eu7$Fbft$!+I?zWUFhanF_ du7%F_eu7$F\\ft$!+K@%HB%FhanF_du7%Fceu7$Ffet$!+-gs?UFhanF_du7%Fgeu7$F` et$!+%>,#3UFhanF_du7%F[fu7$Fjdt$!+_)Ha>%FhanF_du7%F_fu7$Fddt$!+5'zC=%F hanF_du7%Fcfu7$F^dt$!+_YUpTFhanF_du7%Fgfu7$Fhct$!+')pMcTFhanF_du7%F[gu 7$Fbct$!+ZzLVTFhanF_du7%F_gu7$F\\ct$!+++]ITFhanF_du7%Fcgu7$Ffbt$!+'))[ z6%FhanF_du7%Fggu7$F`bt$!+Zh\"e5%FhanF_du7%F[hu7$Fjat$!+&3_U4%FhanF_du 7%F_hu7$Fdat$!+<\"HM3%FhanF_du7%Fchu7$F^at$!+QaatSFhanF_du7%Fghu7$Fh`t $!+*oG[1%FhanF_du7%F[iu7$Fb`t$!+F!Rv0%FhanF_du7%F_iu7$F\\`t$!+w.(>0%Fh anF_du7%Fciu7$Ff_t$!+&zZ%[SFhanF_du7%Fgiu7$F`_t$!+EwJZSFhanF_du7%F[ju7 $Fj^t$!+qe#*[SFhanF_du7%F_ju7$Fd^t$!+z*zN0%FhanF_du7%Fcju7$F^^t$!+)4# \\hSFhanF_du7%Fgju7$Fh]t$!+?-rsSFhanF_du7%F[[v7$Fb]t$!+6s0(3%FhanF_du7 %F_[v7$F\\]t$!+^z5/TFhanF_du7%Fc[v7$Ff\\t$!+8xABTFhanF_du7%Fg[v7$F`\\t $!+3imVTFhanF_du7%F[\\v7$Fj[t$!+Z7nkTFhanF_du7%F_\\v7$Fd[t$!+?,e&=%Fha nF_du7%Fc\\v7$F^[t$!+,r'e?%FhanF_du7%Fg\\v7$Fhjs$!+70:DUFhanF_du7%F[]v 7$Fbjs$!+;^\"fYVFhanF_du7%F_`v7$Fdes$!+y4yYVFhanF_du7%Fc`v7$F^es$!+?j*e M%FhanF_du7%Fg`v7$Fhds$!+NF)RM%FhanF_du7%F[av7$Fbds$!+zW3TVFhanF_du7%F _av7$F\\ds$!+FSF(-%'COLOURG6&F^^nF)F)F)-%*LINESTYLEG6#\"\"$- F$6%7$7$F(F_cv7$F($\"3/++++++!z%FSFdcvFgcv-%+AXESLABELSG6%%&Re(z)G%&Im (z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fhdv6$%*HELVETICAG \"\"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%Fd^n$\"$*>Fd^n;F[fv$\"$z%Fd^ n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme \+ is (approximately) " }{XPPEDIT 18 0 "[-4.0037, 0];" "6#7$,$-%&FloatG6 $\"&P+%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the s tability region for the 8 stage, order 4 scheme with that of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 61 "There is little difference \+ between the two stability regions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),p _||(1..6)],view=[-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 761 1041 1041 {PLOTDATA 2 "65-%'CURVESG6$7e w7$$\"\"!F)F(7$$\"3h******4%*y4l!#G$\"3++++Fjzq:!#=7$$\"35+++31vqa!#E$ \"3))*****fi#fTJF07$$\"3M+++Ft*)o%)!#D$\"3()*****fQ)Q7ZF07$$\"3U+++W(3 OC'!#C$\"3!)*****plzJG'F07$$\"3&)*****\\*H=PH!#B$\"3y*****>wZR&yF07$$ \"3*******p\\,Y+\"!#A$\"3r*****zcGYU*F07$$\"3/+++GVpgEFL$\"3!******>n/ &*4\"!#<7$$\"39+++#f0Sb&FL$\"3!*******\\*zkD\"FT7$$\"3X+++U45C()FL$\"3 %*******)R3LT\"FT7$$\"3c*****\\S:!)e(FL$\"3)******\\N[)p:FT7$$!3(***** *4f7[-\"!#@$\"3.+++dk(es\"FT7$$!3'*******Rr7;rFao$\"3%******R,x5)=FT7$ $!3')*****pc=-B#!#?$\"30+++[(\\].#FT7$$!3J+++))[eBaF\\p$\"3;+++XhL(=#F T7$$!30+++lo()Q6!#>$\"3z*****H1YuL#FT7$$!39+++E#*Gb@Fgp$\"3))*****\\uX [[#FT7$$!3!******\\fy1w$Fgp$\"31+++'*H')GEFT7$$!3-+++)HyR8'Fgp$\"3%)** *****4%\\oFFT7$$!3\"******pSt5V*Fgp$\"3.+++I/=-HFT7$$!3))******G7Gt8F0 $\"39+++!Qgx-$FT7$$!37+++\"p'Q**=F0$\"3(******RmcG9$FT7$$!3.+++[yx.DF0 $\"39++++'\\cC$FT7$$!3#******>/1=;$F0$\"3,+++!>#[NLFT7$$!3w*****z!>+]Q F0$\"3%)*****R:OGT$FT7$$!3#)*****RX'y]XF0$\"3++++-M')yMFT7$$!3Q+++QR'H D&F0$\"3$)*****po7\\`$FT7$$!3C+++MP?]fF0$\"3=+++-GD#e$FT7$$!3#******>f 0$RmF0$\"3?+++Sr'>i$FT7$$!3O+++[#**)=tF0$\"3')*****HES\\l$FT7$$!3e**** *4^;'))zF0$\"3)*******>0)=o$FT7$$!3++++%=0'[')F0$\"33+++L8N.PFT7$$!3U+ ++_uD*H*F0$\"3\"******RZ+)>PFT7$$!3>+++?Z1T**F0$\"3!******\\F%eJPFT7$$ !3/+++,XXd5FT$\"3')*****fk')*QPFT7$$!3,+++q6-?6FT$\"32+++8PBUPFT7$$!3! *******)ec==\"FT$\"3()*****>*\\]TPFT7$$!3'********G5IC\"FT$\"3\")***** 4(>%pt$FT7$$!3!******RvKNI\"FT$\"3,+++mZlGPFT7$$!3/+++&exMO\"FT$\"3>++ +Dus;PFT7$$!30+++;U!HU\"FT$\"3%******zMA7q$FT7$$!3#******H8\")=[\"FT$ \"3#)*****fC%=#o$FT7$$!32+++X'*[S:FT$\"39+++'3W'fOFT7$$!3++++*=G))f\"F T$\"34+++0NiLOFT7$$!3/+++6r,d;FT$\"3')*****>HSTg$FT7$$!3%******4l._r\" FT$\"3%)*****Rn:7d$FT7$$!3!******\\_mNx\"FT$\"3\"*******yU)[`$FT7$$!3! ******p')=B$=FT$\"3%)******)p2_\\$FT7$$!3-+++;uq\"*=FT$\"3!)*****pE#H_ MFT7$$!3++++iY+_>FT$\"3%******Hq5jS$FT7$$!3y*****p%)*[8?FT$\"3%******f 2DvN$FT7$$!3')*****4C:k2#FT$\"3#)*****\\'\\I1LFT7$$!3#******z*y&49#FT$ \"3))*****fqIJD$FT7$$!3.+++C];2AFT$\"3=+++'Hq&)>$FT7$$!3-+++u1\"\\F#FT $\"3z*****HfCK9$FT7$$!3++++!*[)QM#FT$\"3)******4()\\w3$FT7$$!3$******R LEOT#FT$\"30+++k>GKIFT7$$!3A+++r\"*e$[#FT$\"3%)******[*)QxHFT7$$!3'*** ****o#>Kb#FT$\"3!******>BlI#HFT7$$!3>+++$>:?i#FT$\"31+++<,EpGFT7$$!3/+ ++.Cc*o#FT$\"3!)*****p$G#e\"GFT7$$!3%)******[DabFFT$\"31+++_saiFFT7$$! 3))******)RE(>GFT$\"3>+++jp?4FFT7$$!3#)*****H)GFT$\"3'******4kybl#F T7$$!30+++NV9UHFT$\"3@+++q_X,EFT7$$!3(******f-H-+$FT$\"3%******4B_ma#F T7$$!3++++>B>cIFT$\"3%)*****z#*35\\#FT7$$!3*******fQO+6$FT$\"3')*****> %zQMCFT7$$!39+++\"=!yhJFT$\"3y*****z/tmP#FT7$$!3<+++`^X6KFT$\"36+++&*o w#FT7$$!3!)******=#=&[LFT$\"3$******49nJ8#FT7$$!3!)******ohS!R$FT$\"3! *******p=$)o?FT7$$!3>+++/\\]IMFT$\"3\")*****RWSI+#FT7$$!3;+++!\\!*)oMF T$\"3*******4Lzd$>FT7$$!39+++g\\k0NFT$\"3%******zA\\q'=FT7$$!3\")***** fUb3a$FT$\"35+++kl'oz\"FT7$$!3;+++0DhuNFT$\"3&******RHl_s\"FT7$$!3!)** ***>%y+2OFT$\"33+++<()H_;FT7$$!3)******RiI\"QOFT$\"3#*******G4/y:FT7$$ !3')*****47j!oOFT$\"3\"******4X(e-:FT7$$!3!)*****z\\vop$FT$\"3!******* QX0E9FT7$$!3;+++7/iCPFT$\"3/+++_nd[8FT7$$!3')*****46G8v$FT$\"3#******4 i-.F\"FT7$$!39+++]G+xPFT$\"3-+++g&)Q\">\"FT7$$!3:+++T8i,QFT$\"35+++>;* >6\"FT7$$!39+++RO8DQFT$\"3\"******p$>EK5FT7$$!3;+++7kYZQFT$\"3#)*****f &*eL_*F07$$!3)******f3G&oQFT$\"3?+++H$3Ls)F07$$!3;+++k]@))QFT$\"3L+++S \"=M#zF07$$!3:+++U\"=k!RFT$\"3f*****RV0W7(F07$$!3#)*****RPGI#RFT$\"3`* ****4OhnK'F07$$!38+++**=%z$RFT$\"37+++&Qs2`&F07$$!3%)*****\\`j5&RFT$\" 3A+++Y$\\lt%F07$$!3%******z<4B'RFT$\"3%)*****HKfS%RF07$$!3'******>?2;( RFT$\"3y*****4hcJ:$F07$$!3')*****z4**)yRFT$\"3)******z`5OO#F07$$!3++++ C'RT)RFT$\"3++++`\"H^d\"F07$$!3')*****Rp'H()RFT$\"3e*****4t(ytyFgp7$$! 31+++q6N))RFTF(7$Fjjl$!3e*****4t(ytyFgp7$Fejl$!3++++`\"H^d\"F07$F`jl$! 3)******z`5OO#F07$F[jl$!3y*****4hcJ:$F07$Ffil$!3%)*****HKfS%RF07$Fail$ !3A+++Y$\\lt%F07$F\\il$!37+++&Qs2`&F07$Fghl$!3`*****4OhnK'F07$Fbhl$!3f *****RV0W7(F07$F]hl$!3L+++S\"=M#zF07$Fhgl$!3?+++H$3Ls)F07$Fcgl$!3#)*** **f&*eL_*F07$F^gl$!3\"******p$>EK5FT7$Fifl$!35+++>;*>6\"FT7$Fdfl$!3-++ +g&)Q\">\"FT7$F_fl$!3#******4i-.F\"FT7$Fjel$!3/+++_nd[8FT7$Feel$!3!*** ****QX0E9FT7$F`el$!3\"******4X(e-:FT7$F[el$!3#*******G4/y:FT7$Ffdl$!33 +++<()H_;FT7$Fadl$!3&******RHl_s\"FT7$F\\dl$!35+++kl'oz\"FT7$Fgcl$!3%* *****zA\\q'=FT7$Fbcl$!3*******4Lzd$>FT7$F]cl$!3\")*****RWSI+#FT7$Fhbl$ !3!*******p=$)o?FT7$Fcbl$!3$******49nJ8#FT7$F^bl$!35+++yH2'>#FT7$Fial$ !37+++5))edAFT7$Fdal$!36+++&*ow%zQMCFT7$Fe`l$!3%)*****z#*35\\#FT7$F``l$!3%******4B_ma#FT7$F[`l$! 3@+++q_X,EFT7$Ff_l$!3'******4kybl#FT7$Fa_l$!3>+++jp?4FFT7$F\\_l$!31+++ _saiFFT7$Fg^l$!3!)*****p$G#e\"GFT7$Fb^l$!31+++<,EpGFT7$F]^l$!3!******> BlI#HFT7$Fh]l$!3%)******[*)QxHFT7$Fc]l$!30+++k>GKIFT7$F^]l$!3)******4( )\\w3$FT7$Fi\\l$!3z*****HfCK9$FT7$Fd\\l$!3=+++'Hq&)>$FT7$F_\\l$!3))*** **fqIJD$FT7$Fj[l$!3#)*****\\'\\I1LFT7$Fe[l$!3%******f2DvN$FT7$F`[l$!3% ******Hq5jS$FT7$F[[l$!3!)*****pE#H_MFT7$Ffz$!3%)******)p2_\\$FT7$Faz$! 3\"*******yU)[`$FT7$F\\z$!3%)*****Rn:7d$FT7$Fgy$!3')*****>HSTg$FT7$Fby $!34+++0NiLOFT7$F]y$!39+++'3W'fOFT7$Fhx$!3#)*****fC%=#o$FT7$Fcx$!3%*** ***zMA7q$FT7$F^x$!3>+++Dus;PFT7$Fiw$!3,+++mZlGPFT7$Fdw$!3\")*****4(>%p t$FT7$F_w$!3()*****>*\\]TPFT7$Fjv$!32+++8PBUPFT7$Fev$!3')*****fk')*QPF T7$F`v$!3!******\\F%eJPFT7$F[v$!3\"******RZ+)>PFT7$Ffu$!33+++L8N.PFT7$ Fau$!3)*******>0)=o$FT7$F\\u$!3')*****HES\\l$FT7$Fgt$!3?+++Sr'>i$FT7$F bt$!3=+++-GD#e$FT7$F]t$!3$)*****po7\\`$FT7$Fhs$!3++++-M')yMFT7$Fcs$!3% )*****R:OGT$FT7$F^s$!3,+++!>#[NLFT7$Fir$!39++++'\\cC$FT7$Fdr$!3(****** RmcG9$FT7$F_r$!39+++!Qgx-$FT7$Fjq$!3.+++I/=-HFT7$Feq$!3%)*******4%\\oF FT7$F`q$!31+++'*H')GEFT7$F[q$!3))*****\\uX[[#FT7$Fep$!3z*****H1YuL#FT7 $F`p$!3;+++XhL(=#FT7$Fjo$!30+++[(\\].#FT7$Feo$!3%******R,x5)=FT7$F_o$! 3.+++dk(es\"FT7$$\"3g+++/a,)e(FL$!3)******\\N[)p:FT7$Fen$!3%*******)R3 LT\"FT7$FV$!3!*******\\*zkD\"FT7$FP$!3!******>n/&*4\"FT7$FJ$!3r*****zc GYU*F07$$\"3<+++'*H=PHFF$!3y*****>wZR&yF07$F>$!3!)*****plzJG'F07$F8$!3 ()*****fQ)Q7ZF07$$\"3i*****R$*\\2Z&F4$!3))*****fi#fTJF07$$\"3S*****fP* y4lF-$!3++++Fjzq:F0F'-%&COLORG6&%$RGBG$\"#:!\"#$\"#[Fh^nFf^n-%)POLYGON SG6fw7%F'7$$\"+5%*y4lF\\p$\"+Fjzq:!#57$$Fh^nF)F(7%F__n7$$\"+31vqaF0$\" +EEfTJFd_nFe_n7%Fh_n7$$\"+Ft*)o%)FT$\"+'Q)Q7ZFd_nFe_n7%F^`n7$$\"+W(3OC '!#;$\"+d'zJG'Fd_nFe_n7%Fd`n7$$\"+&*H=PH!#:$\"+ix%R&yFd_nFe_n7%F[an7$$ \"+(\\,Y+\"!#9$\"+o&GYU*Fd_nFe_n7%Fban7$$\"+GVpgEFean$\"+sY]*4\"!\"*Fe _n7%Fian7$$\"+#f0Sb&Fean$\"+]*zkD\"F^bnFe_n7%F`bn7$$\"+U45C()Fean$\"+* R3LT\"F^bnFe_n7%Ffbn7$$\"+0a,)e(Fean$\"+b$[)p:F^bnFe_n7%F\\cn7$$!+\"f7 [-\"!#8$\"+dk(es\"F^bnFe_n7%Fbcn7$$!+Sr7;rFecn$\"+9q2\")=F^bnFe_n7%Fic n7$$!+n&=-B#!#7$\"+[(\\].#F^bnFe_n7%F_dn7$$!+))[eBaFbdn$\"+XhL(=#F^bnF e_n7%Ffdn7$$!+lo()Q6!#6$\"+jgWPBF^bnFe_n7%F\\en7$$!+E#*Gb@F_en$\"+Xd%[ [#F^bnFe_n7%Fcen7$$!+&fy1w$F_en$\"+'*H')GEF^bnFe_n7%Fien7$$!+)HyR8'F_e n$\"++T\\oFF^bnFe_n7%F_fn7$$!+2M2J%*F_en$\"+I/=-HF^bnFe_n7%Fefn7$$!+H7 Gt8Fd_n$\"+!Qgx-$F^bnFe_n7%F[gn7$$!+\"p'Q**=Fd_n$\"+km&G9$F^bnFe_n7%Fa gn7$$!+[yx.DFd_n$\"++'\\cC$F^bnFe_n7%Fggn7$$!+Ug!=;$Fd_n$\"+!>#[NLF^bn Fe_n7%F]hn7$$!+3>+]QFd_n$\"+ah$GT$F^bnFe_n7%Fchn7$$!+aky]XFd_n$\"+-M') yMF^bnFe_n7%Fihn7$$!+QR'HD&Fd_n$\"+(o7\\`$F^bnFe_n7%F_in7$$!+MP?]fFd_n $\"+-GD#e$F^bnFe_n7%Fein7$$!+#f0$RmFd_n$\"+Sr'>i$F^bnFe_n7%F[jn7$$!+[# **)=tFd_n$\"+j-%\\l$F^bnFe_n7%Fajn7$$!+6lh))zFd_n$\"+?0)=o$F^bnFe_n7%F gjn7$$!+%=0'[')Fd_n$\"+L8N.PF^bnFe_n7%F][o7$$!+_uD*H*Fd_n$\"+u/!)>PF^b nFe_n7%Fc[o7$$!+?Z1T**Fd_n$\"+vUeJPF^bnFe_n7%Fi[o7$$!+,XXd5F^bn$\"+Ym) *QPF^bnFe_n7%F_\\o7$$!+q6-?6F^bn$\"+8PBUPF^bnFe_n7%Fe\\o7$$!+*ec==\"F^ bn$\"+#*\\]TPF^bnFe_n7%F[]o7$$!+!H5IC\"F^bn$\"+r>%pt$F^bnFe_n7%Fa]o7$$ !+aF`.8F^bn$\"+mZlGPF^bnFe_n7%Fg]o7$$!+&exMO\"F^bn$\"+Dus;PF^bnFe_n7%F ]^o7$$!+;U!HU\"F^bn$\"+[BA,PF^bnFe_n7%Fc^o7$$!+L6)=[\"F^bn$\"+YU=#o$F^ bnFe_n7%Fi^o7$$!+X'*[S:F^bn$\"+'3W'fOF^bnFe_n7%F__o7$$!+*=G))f\"F^bn$ \"+0NiLOF^bnFe_n7%Fe_o7$$!+6r,d;F^bn$\"+#HSTg$F^bnFe_n7%F[`o7$$!+^O?:< F^bn$\"+uc@rNF^bnFe_n7%Fa`o7$$!+DlctF^bn$\"+.2J1MF^bnFe_n7%Fiao7$$!+Z)*[8?F^bn$\"+w]_dLF^bnF e_n7%F_bo7$$!+T_Tw?F^bn$\"+l\\I1LF^bnFe_n7%Febo7$$!+)*y&49#F^bn$\"+128 `KF^bnFe_n7%F[co7$$!+C];2AF^bn$\"+'Hq&)>$F^bnFe_n7%Faco7$$!+u1\"\\F#F^ bn$\"+$fCK9$F^bnFe_n7%Fgco7$$!+!*[)QM#F^bn$\"+r)\\w3$F^bnFe_n7%F]do7$$ !+Mji8CF^bn$\"+k>GKIF^bnFe_n7%Fcdo7$$!+r\"*e$[#F^bn$\"+\\*)QxHF^bnFe_n 7%Fido7$$!+p#>Kb#F^bn$\"+K_1BHF^bnFe_n7%F_eo7$$!+$>:?i#F^bn$\"+<,EpGF^ bnFe_n7%Feeo7$$!+.Cc*o#F^bn$\"+PG#e\"GF^bnFe_n7%F[fo7$$!+\\DabFF^bn$\" +_saiFF^bnFe_n7%Fafo7$$!+*RE(>GF^bn$\"+jp?4FF^bnFe_n7%Fgfo7$$!+t\"f>)G F^bn$\"+T'ybl#F^bnFe_n7%F]go7$$!+NV9UHF^bn$\"+q_X,EF^bnFe_n7%Fcgo7$$!+ E!H-+$F^bn$\"+JAlYDF^bnFe_n7%Figo7$$!+>B>cIF^bn$\"+G*35\\#F^bnFe_n7%F_ ho7$$!+'QO+6$F^bn$\"+UzQMCF^bnFe_n7%Feho7$$!+\"=!yhJF^bn$\"+[InwBF^bnF e_n7%F[io7$$!+`^X6KF^bn$\"+&*ow#F^bnFe_n7%F]jo7$$!+>#=&[LF^bn$ \"+Tr;L@F^bnFe_n7%Fcjo7$$!+phS!R$F^bn$\"+q=$)o?F^bnFe_n7%Fijo7$$!+/\\] IMF^bn$\"+W//.?F^bnFe_n7%F_[p7$$!+!\\!*)oMF^bn$\"+J$zd$>F^bnFe_n7%Fe[p 7$$!+g\\k0NF^bn$\"+G#\\q'=F^bnFe_n7%F[\\p7$$!+Ea&3a$F^bn$\"+kl'oz\"F^b nFe_n7%Fa\\p7$$!+0DhuNF^bn$\"+%Hl_s\"F^bnFe_n7%Fg\\p7$$!+Uy+2OF^bn$\"+ <()H_;F^bnFe_n7%F]]p7$$!+C18QOF^bn$\"+H4/y:F^bnFe_n7%Fc]p7$$!+@J1oOF^b n$\"+^ue-:F^bnFe_n7%Fi]p7$$!+)\\vop$F^bn$\"+RX0E9F^bnFe_n7%F_^p7$$!+7/ iCPF^bn$\"+_nd[8F^bnFe_n7%Fe^p7$$!+6\"G8v$F^bn$\"+@EIq7F^bnFe_n7%F[_p7 $$!+]G+xPF^bn$\"+g&)Q\">\"F^bnFe_n7%Fa_p7$$!+T8i,QF^bn$\"+>;*>6\"F^bnF e_n7%Fg_p7$$!+RO8DQF^bn$\"+P>EK5F^bnFe_n7%F]`p7$$!+7kYZQF^bn$\"+c*eL_* Fd_nFe_n7%Fc`p7$$!+'3G&oQF^bn$\"+H$3Ls)Fd_nFe_n7%Fi`p7$$!+k]@))QF^bn$ \"+S\"=M#zFd_nFe_n7%F_ap7$$!+U\"=k!RF^bn$\"+MaSCrFd_nFe_n7%Feap7$$!+u$ GI#RF^bn$\"+h8wEjFd_nFe_n7%F[bp7$$!+**=%z$RF^bn$\"+&Qs2`&Fd_nFe_n7%Fab p7$$!+NN1^RF^bn$\"+Y$\\lt%Fd_nFe_n7%Fgbp7$$!+y\"4B'RF^bn$\"+B$fS%RFd_n Fe_n7%F]cp7$$!+-sgrRF^bn$\"+6m:`JFd_nFe_n7%Fccp7$$!+)4**)yRF^bn$\"+Q0h jBFd_nFe_n7%Ficp7$$!+C'RT)RF^bn$\"+`\"H^d\"Fd_nFe_n7%F_dp7$$!+%p'H()RF ^bn$\"+JxytyF_enFe_n7%Fedp7$$!+q6N))RF^bnF(Fe_n7%F[ep7$Ffdp$!+JxytyF_e nFe_n7%F_ep7$F`dp$!+`\"H^d\"Fd_nFe_n7%Fcep7$Fjcp$!+Q0hjBFd_nFe_n7%Fgep 7$Fdcp$!+6m:`JFd_nFe_n7%F[fp7$F^cp$!+B$fS%RFd_nFe_n7%F_fp7$Fhbp$!+Y$\\ lt%Fd_nFe_n7%Fcfp7$Fbbp$!+&Qs2`&Fd_nFe_n7%Fgfp7$F\\bp$!+h8wEjFd_nFe_n7 %F[gp7$Ffap$!+MaSCrFd_nFe_n7%F_gp7$F`ap$!+S\"=M#zFd_nFe_n7%Fcgp7$Fj`p$ !+H$3Ls)Fd_nFe_n7%Fggp7$Fd`p$!+c*eL_*Fd_nFe_n7%F[hp7$F^`p$!+P>EK5F^bnF e_n7%F_hp7$Fh_p$!+>;*>6\"F^bnFe_n7%Fchp7$Fb_p$!+g&)Q\">\"F^bnFe_n7%Fgh p7$F\\_p$!+@EIq7F^bnFe_n7%F[ip7$Ff^p$!+_nd[8F^bnFe_n7%F_ip7$F`^p$!+RX0 E9F^bnFe_n7%Fcip7$Fj]p$!+^ue-:F^bnFe_n7%Fgip7$Fd]p$!+H4/y:F^bnFe_n7%F[ jp7$F^]p$!+<()H_;F^bnFe_n7%F_jp7$Fh\\p$!+%Hl_s\"F^bnFe_n7%Fcjp7$Fb\\p$ !+kl'oz\"F^bnFe_n7%Fgjp7$F\\\\p$!+G#\\q'=F^bnFe_n7%F[[q7$Ff[p$!+J$zd$> F^bnFe_n7%F_[q7$F`[p$!+W//.?F^bnFe_n7%Fc[q7$Fjjo$!+q=$)o?F^bnFe_n7%Fg[ q7$Fdjo$!+Tr;L@F^bnFe_n7%F[\\q7$F^jo$!+yH2'>#F^bnFe_n7%F_\\q7$Fhio$!+5 ))edAF^bnFe_n7%Fc\\q7$Fbio$!+&*owGKIF^bnFe_n7%F [`q7$F^do$!+r)\\w3$F^bnFe_n7%F_`q7$Fhco$!+$fCK9$F^bnFe_n7%Fc`q7$Fbco$! +'Hq&)>$F^bnFe_n7%Fg`q7$F\\co$!+128`KF^bnFe_n7%F[aq7$Ffbo$!+l\\I1LF^bn Fe_n7%F_aq7$F`bo$!+w]_dLF^bnFe_n7%Fcaq7$Fjao$!+.2J1MF^bnFe_n7%Fgaq7$Fd ao$!+nAH_MF^bnFe_n7%F[bq7$F^ao$!+*p2_\\$F^bnFe_n7%F_bq7$Fh`o$!+zU)[`$F ^bnFe_n7%Fcbq7$Fb`o$!+uc@rNF^bnFe_n7%Fgbq7$F\\`o$!+#HSTg$F^bnFe_n7%F[c q7$Ff_o$!+0NiLOF^bnFe_n7%F_cq7$F`_o$!+'3W'fOF^bnFe_n7%Fccq7$Fj^o$!+YU= #o$F^bnFe_n7%Fgcq7$Fd^o$!+[BA,PF^bnFe_n7%F[dq7$F^^o$!+Dus;PF^bnFe_n7%F _dq7$Fh]o$!+mZlGPF^bnFe_n7%Fcdq7$Fb]o$!+r>%pt$F^bnFe_n7%Fgdq7$F\\]o$!+ #*\\]TPF^bnFe_n7%F[eq7$Ff\\o$!+8PBUPF^bnFe_n7%F_eq7$F`\\o$!+Ym)*QPF^bn Fe_n7%Fceq7$Fj[o$!+vUeJPF^bnFe_n7%Fgeq7$Fd[o$!+u/!)>PF^bnFe_n7%F[fq7$F ^[o$!+L8N.PF^bnFe_n7%F_fq7$Fhjn$!+?0)=o$F^bnFe_n7%Fcfq7$Fbjn$!+j-%\\l$ F^bnFe_n7%Fgfq7$F\\jn$!+Sr'>i$F^bnFe_n7%F[gq7$Ffin$!+-GD#e$F^bnFe_n7%F _gq7$F`in$!+(o7\\`$F^bnFe_n7%Fcgq7$Fjhn$!+-M')yMF^bnFe_n7%Fggq7$Fdhn$! +ah$GT$F^bnFe_n7%F[hq7$F^hn$!+!>#[NLF^bnFe_n7%F_hq7$Fhgn$!++'\\cC$F^bn Fe_n7%Fchq7$Fbgn$!+km&G9$F^bnFe_n7%Fghq7$F\\gn$!+!Qgx-$F^bnFe_n7%F[iq7 $Fffn$!+I/=-HF^bnFe_n7%F_iq7$F`fn$!++T\\oFF^bnFe_n7%Fciq7$Fjen$!+'*H') GEF^bnFe_n7%Fgiq7$Fden$!+Xd%[[#F^bnFe_n7%F[jq7$F]en$!+jgWPBF^bnFe_n7%F _jq7$Fgdn$!+XhL(=#F^bnFe_n7%Fcjq7$F`dn$!+[(\\].#F^bnFe_n7%Fgjq7$Fjcn$! +9q2\")=F^bnFe_n7%F[[r7$Fccn$!+dk(es\"F^bnFe_n7%F_[r7$$\"+/a,)e(Fean$! +b$[)p:F^bnFe_n7%Fc[r7$Fgbn$!+*R3LT\"F^bnFe_n7%Fi[r7$Fabn$!+]*zkD\"F^b nFe_n7%F]\\r7$Fjan$!+sY]*4\"F^bnFe_n7%Fa\\r7$Fcan$!+o&GYU*Fd_nFe_n7%Fe \\r7$$\"+'*H=PHF^an$!+ix%R&yFd_nFe_n7%Fi\\r7$Fe`n$!+d'zJG'Fd_nFe_n7%F_ ]r7$F_`n$!+'Q)Q7ZFd_nFe_n7%Fc]r7$$\"+M*\\2Z&F0$!+EEfTJFd_nFe_n7%Fg]r7$ $\"+w$*y4lF\\p$!+Fjzq:Fd_nFe_n7%F]^rF'Fe_n-Fc^n6&Fe^n$\"\"$!\"\"$\"#&* Fh^nFe^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"35+++\"oG)*[\"FT$\"3w*** **H>]MI%FT7$$\"31+++L,s\"[\"FT$\"3A+++v;>7VFT7$$\"3'******pLDHZ\"FT$\" 3<+++$))R-K%FT7$$\"33+++Ut[j9FT$\"3z*****pU`vK%FT7$$\"3)*******z)\\MX \"FT$\"3d*****>B\"4MVFT7$$\"3(******fodGW\"FT$\"3%******\\E7)RVFT7$$\" 3*******4'ovJ9FT$\"3t******e_nWVFT7$$\"3#******\\G&>?9FT$\"37+++r%Q'[V FT7$$\"31++++JA39FT$\"3y******H%f;N%FT7$$\"3-+++oK*eR\"FT$\"35+++@Zp`V FT7$$\"33+++KAE$Q\"FT$\"3q*****zw*paVFT7$$\"31+++/2Rq8FT$\"3/+++c'GYN% FT7$$\"3)******\\gWtN\"FT$\"3M+++$*RV`VFT7$$\"3++++5i>W8FT$\"3/+++Mo1^ VFT7$$\"35+++Bb-J8FT$\"3A+++2nZZVFT7$$\"36+++m>#zJ\"FT$\"35+++4=hUVFT7 $$\"3++++Fk)\\I\"FT$\"3H+++I%>kL%FT7$$\"31+++ePL#H\"FT$\"3&)*****fwY)G VFT7$$\"31+++)z&4!G\"FT$\"3y*****HHU)>VFT7$$\"35+++y\\Uo7FT$\"3T+++5!e $4VFT7$$\"3%******Ra)\\d7FT$\"3++++WHN(H%FT7$$\"3#*******fK_Z7FT$\"39+ ++g&)z$G%FT7$$\"3*)*****HFS(Q7FT$\"3?+++koooUFT7$$\"3!******zHH9B\"FT$ \"3K+++zA/_UFT7$$\"3)******fY5fA\"FT$\"3U+++K%QRB%FT7$$\"30+++#*3aA7FT $\"3-+++%o?X@%FT7$$\"35+++38q@7FT$\"3')*****\\%Q.%>%FT7$$\"31+++qmwB7F T$\"3G+++>8&G<%FT7$$\"3$******\\Tb!H7FT$\"3k*****\\D'\\^TFT7$$\"35+++B \"exB\"FT$\"3')*****HpQ18%FT7$$\"3#******4&*f)\\7FT$\"3#)*****H*306TFT 7$$\"3!******RB(3l7FT$\"3!)*****Rl9N4%FT7$$\"3!******H\"=\"HG\"FT$\"3S +++I5qySFT7$$\"32+++sQi-8FT$\"3-+++.q1nSFT7$$\"3-+++xnWB8FT$\"3e****** 4,\")eSFT7$$\"30+++ZukW8FT$\"3!******Rj*)Q0%FT7$$\"35+++r\"3cO\"FT$\"3 %******zP!4_SFT7$$\"3/+++UK&eQ\"FT$\"3;+++0f4`SFT7$$\"3.+++jG/09FT$\"3 A+++t!\\l0%FT7$$\"3(*******G4&HU\"FT$\"3K+++=E4iSFT7$$\"36+++())Q%R9FT $\"3e*****H\">RpSFT7$$\"31+++R?Va9FT$\"37+++h^9ySFT7$$\"3#*******30!zY \"FT$\"3)******4'f3)3%FT7$$\"3#******>)\\%)z9FT$\"3u*****4#=)*)4%FT7$$ \"3)******4U'G!\\\"FT$\"3g*****p#3j5TFT7$$\"33+++o)e#*\\\"FT$\"3<+++Cx &G7%FT7$$\"33+++\"[/o]\"FT$\"3I+++d+^NTFT7$$\"3*)*****f;qH^\"FT$\"3%)* ****4ka%[TFT7$$\"31+++I`!y^\"FT$\"3e*******[u:;%FT7$$\"31+++#Rg8_\"FT$ \"3U+++)=mZ<%FT7$$\"3/+++TeoB:FT$\"3g*****RnPz=%FT7$$\"3*******zjJ[_\" FT$\"3V+++'Q15?%FT7$$\"3)*******))o%[_\"FT$\"3N+++5x*Q@%FT7$$\"3*)**** *pmzP_\"FT$\"3Q+++(oVlA%FT7$$\"3'******z'pn@:FT$\"3h*****p\">))QUFT7$$ \"3\"******\\t%e=:FT$\"3k*****fka3D%FT7$$\"3)******f%za9:FT$\"3m*****z )zSiUFT7$$\"34+++N2h4:FT$\"3W+++`7\\tUFT7$$\"3*)*****4b;Q]\"FT$\"3K+++ %QcSG%FT7$$\"31+++F$3s\\\"FT$\"3'******\\WdSH%FTFa_rFb^n-F\\_n6jn7%7$$ \"+\"oG)*[\"F^bn$\"+$>]MI%F^bn7$$\"+L,s\"[\"F^bn$\"+v;>7VF^bn7$$\"$P\" Fh^n$\"$>%Fh^n7%Febs7$$\"+P`#HZ\"F^bn$\"+$))R-K%F^bnFjbs7%F`cs7$$\"+Ut [j9F^bn$\"+FMbFVF^bnFjbs7%Ffcs7$$\"+!))\\MX\"F^bn$\"+K74MVF^bnFjbs7%F \\ds7$$\"+'odGW\"F^bn$\"+lA\")RVF^bnFjbs7%Fbds7$$\"+hovJ9F^bn$\"+f_nWV F^bnFjbs7%Fhds7$$\"+&G&>?9F^bn$\"+r%Q'[VF^bnFjbs7%F^es7$$\"++JA39F^bn$ \"+I%f;N%F^bnFjbs7%Fdes7$$\"+oK*eR\"F^bn$\"+@Zp`VF^bnFjbs7%Fjes7$$\"+K AE$Q\"F^bn$\"+o(*paVF^bnFjbs7%F`fs7$$\"+/2Rq8F^bn$\"+c'GYN%F^bnFjbs7%F ffs7$$\"+0YMd8F^bn$\"+$*RV`VF^bnFjbs7%F\\gs7$$\"+5i>W8F^bn$\"+Mo1^VF^b nFjbs7%Fbgs7$$\"+Bb-J8F^bn$\"+2nZZVF^bnFjbs7%Fhgs7$$\"+m>#zJ\"F^bn$\"+ 4=hUVF^bnFjbs7%F^hs7$$\"+Fk)\\I\"F^bn$\"+I%>kL%F^bnFjbs7%Fdhs7$$\"+ePL #H\"F^bn$\"+mn%)GVF^bnFjbs7%Fjhs7$$\"+)z&4!G\"F^bn$\"+$HU)>VF^bnFjbs7% F`is7$$\"+y\\Uo7F^bn$\"+5!e$4VF^bnFjbs7%Ffis7$$\"+W&)\\d7F^bn$\"+WHN(H %F^bnFjbs7%F\\js7$$\"+gK_Z7F^bn$\"+g&)z$G%F^bnFjbs7%Fbjs7$$\"+t-uQ7F^b n$\"+koooUF^bnFjbs7%Fhjs7$$\"+)HH9B\"F^bn$\"+zA/_UF^bnFjbs7%F^[t7$$\"+ m/\"fA\"F^bn$\"+K%QRB%F^bnFjbs7%Fd[t7$$\"+#*3aA7F^bn$\"+%o?X@%F^bnFjbs 7%Fj[t7$$\"+38q@7F^bn$\"+XQ.%>%F^bnFjbs7%F`\\t7$$\"+qmwB7F^bn$\"+>8&G< %F^bnFjbs7%Ff\\t7$$\"+:a0H7F^bn$\"+bi\\^TF^bnFjbs7%F\\]t7$$\"+B\"exB\" F^bn$\"+$pQ18%F^bnFjbs7%Fb]t7$$\"+^*f)\\7F^bn$\"+$*306TF^bnFjbs7%Fh]t7 $$\"+Ms3l7F^bn$\"+aY^$4%F^bnFjbs7%F^^t7$$\"+8=\"HG\"F^bn$\"+I5qySF^bnF jbs7%Fd^t7$$\"+sQi-8F^bn$\"+.q1nSF^bnFjbs7%Fj^t7$$\"+xnWB8F^bn$\"+5,\" )eSF^bnFjbs7%F`_t7$$\"+ZukW8F^bn$\"+M'*)Q0%F^bnFjbs7%Ff_t7$$\"+r\"3cO \"F^bn$\"+y.4_SF^bnFjbs7%F\\`t7$$\"+UK&eQ\"F^bn$\"+0f4`SF^bnFjbs7%Fb`t 7$$\"+jG/09F^bn$\"+t!\\l0%F^bnFjbs7%Fh`t7$$\"+H4&HU\"F^bn$\"+=E4iSF^bn Fjbs7%F^at7$$\"+())Q%R9F^bn$\"+8>RpSF^bnFjbs7%Fdat7$$\"+R?Va9F^bn$\"+h ^9ySF^bnFjbs7%Fjat7$$\"+40!zY\"F^bn$\"+hf3)3%F^bnFjbs7%F`bt7$$\"+#)\\% )z9F^bn$\"+@=)*)4%F^bnFjbs7%Ffbt7$$\"+@kG!\\\"F^bn$\"+F3j5TF^bnFjbs7%F \\ct7$$\"+o)e#*\\\"F^bn$\"+Cx&G7%F^bnFjbs7%Fbct7$$\"+\"[/o]\"F^bn$\"+d +^NTF^bnFjbs7%Fhct7$$\"+m,(H^\"F^bn$\"+TYX[TF^bnFjbs7%F^dt7$$\"+I`!y^ \"F^bn$\"+!\\u:;%F^bnFjbs7%Fddt7$$\"+#Rg8_\"F^bn$\"+)=mZ<%F^bnFjbs7%Fj dt7$$\"+TeoB:F^bn$\"+uw$z=%F^bnFjbs7%F`et7$$\"+Q;$[_\"F^bn$\"+'Q15?%F^ bnFjbs7%Ffet7$$\"+*)o%[_\"F^bn$\"+5x*Q@%F^bnFjbs7%F\\ft7$$\"+n'zP_\"F^ bn$\"+(oVlA%F^bnFjbs7%Fbft7$$\"+opn@:F^bn$\"+<>))QUF^bnFjbs7%Fhft7$$\" +NZe=:F^bn$\"+YY&3D%F^bnFjbs7%F^gt7$$\"+Yza9:F^bn$\"+))zSiUF^bnFjbs7%F dgt7$$\"+N2h4:F^bn$\"+`7\\tUF^bnFjbs7%Fjgt7$$\"+^l\"Q]\"F^bn$\"+%QcSG% F^bnFjbs7%F`ht7$$\"+F$3s\\\"F^bn$\"+Xu0%H%F^bnFjbs7%FfhtF`bsFjbsFc^rFj ^r-F$6$7in7$Fb_r$!3w*****H>]MI%FT7$Fias$!3'******\\WdSH%FT7$Fdas$!3K++ +%QcSG%FT7$F_as$!3W+++`7\\tUFT7$Fj`s$!3m*****z)zSiUFT7$Fe`s$!3k*****fk a3D%FT7$F``s$!3h*****p\">))QUFT7$F[`s$!3Q+++(oVlA%FT7$Ff_s$!3N+++5x*Q@ %FT7$Fa_s$!3V+++'Q15?%FT7$F\\_s$!3g*****RnPz=%FT7$Fg^s$!3U+++)=mZ<%FT7 $Fb^s$!3e*******[u:;%FT7$F]^s$!3%)*****4ka%[TFT7$Fh]s$!3I+++d+^NTFT7$F c]s$!3<+++Cx&G7%FT7$F^]s$!3g*****p#3j5TFT7$Fi\\s$!3u*****4#=)*)4%FT7$F d\\s$!3)******4'f3)3%FT7$F_\\s$!37+++h^9ySFT7$Fj[s$!3e*****H\">RpSFT7$ Fe[s$!3K+++=E4iSFT7$F`[s$!3A+++t!\\l0%FT7$F[[s$!3;+++0f4`SFT7$Ffjr$!3% ******zP!4_SFT7$Fajr$!3!******Rj*)Q0%FT7$F\\jr$!3e******4,\")eSFT7$Fgi r$!3-+++.q1nSFT7$Fbir$!3S+++I5qySFT7$F]ir$!3!)*****Rl9N4%FT7$Fhhr$!3#) *****H*306TFT7$Fchr$!3')*****HpQ18%FT7$F^hr$!3k*****\\D'\\^TFT7$Figr$! 3G+++>8&G<%FT7$Fdgr$!3')*****\\%Q.%>%FT7$F_gr$!3-+++%o?X@%FT7$Fjfr$!3U +++K%QRB%FT7$Fefr$!3K+++zA/_UFT7$F`fr$!3?+++koooUFT7$F[fr$!39+++g&)z$G %FT7$Ffer$!3++++WHN(H%FT7$Faer$!3T+++5!e$4VFT7$F\\er$!3y*****HHU)>VFT7 $Fgdr$!3&)*****fwY)GVFT7$Fbdr$!3H+++I%>kL%FT7$F]dr$!35+++4=hUVFT7$Fhcr $!3A+++2nZZVFT7$Fccr$!3/+++Mo1^VFT7$F^cr$!3M+++$*RV`VFT7$Fibr$!3/+++c' GYN%FT7$Fdbr$!3q*****zw*paVFT7$F_br$!35+++@Zp`VFT7$Fjar$!3y******H%f;N %FT7$Fear$!37+++r%Q'[VFT7$F`ar$!3t******e_nWVFT7$F[ar$!3%******\\E7)RV FT7$Ff`r$!3d*****>B\"4MVFT7$Fa`r$!3z*****pU`vK%FT7$F\\`r$!3<+++$))R-K% FT7$Fg_r$!3A+++v;>7VFTF_itFb^n-F\\_n6jn7%7$Fabs$!+$>]MI%F^bn7$Fght$!+X u0%H%F^bn7$F[cs$!$>%Fh^n7%Fidu7$Faht$!+%QcSG%F^bnF\\eu7%F`eu7$F[ht$!+` 7\\tUF^bnF\\eu7%Fdeu7$Fegt$!+))zSiUF^bnF\\eu7%Fheu7$F_gt$!+YY&3D%F^bnF \\eu7%F\\fu7$Fift$!+<>))QUF^bnF\\eu7%F`fu7$Fcft$!+(oVlA%F^bnF\\eu7%Fdf u7$F]ft$!+5x*Q@%F^bnF\\eu7%Fhfu7$Fget$!+'Q15?%F^bnF\\eu7%F\\gu7$Faet$! +uw$z=%F^bnF\\eu7%F`gu7$F[et$!+)=mZ<%F^bnF\\eu7%Fdgu7$Fedt$!+!\\u:;%F^ bnF\\eu7%Fhgu7$F_dt$!+TYX[TF^bnF\\eu7%F\\hu7$Fict$!+d+^NTF^bnF\\eu7%F` hu7$Fcct$!+Cx&G7%F^bnF\\eu7%Fdhu7$F]ct$!+F3j5TF^bnF\\eu7%Fhhu7$Fgbt$!+ @=)*)4%F^bnF\\eu7%F\\iu7$Fabt$!+hf3)3%F^bnF\\eu7%F`iu7$F[bt$!+h^9ySF^b nF\\eu7%Fdiu7$Feat$!+8>RpSF^bnF\\eu7%Fhiu7$F_at$!+=E4iSF^bnF\\eu7%F\\j u7$Fi`t$!+t!\\l0%F^bnF\\eu7%F`ju7$Fc`t$!+0f4`SF^bnF\\eu7%Fdju7$F]`t$!+ y.4_SF^bnF\\eu7%Fhju7$Fg_t$!+M'*)Q0%F^bnF\\eu7%F\\[v7$Fa_t$!+5,\")eSF^ bnF\\eu7%F`[v7$F[_t$!+.q1nSF^bnF\\eu7%Fd[v7$Fe^t$!+I5qySF^bnF\\eu7%Fh[ v7$F_^t$!+aY^$4%F^bnF\\eu7%F\\\\v7$Fi]t$!+$*306TF^bnF\\eu7%F`\\v7$Fc]t $!+$pQ18%F^bnF\\eu7%Fd\\v7$F]]t$!+bi\\^TF^bnF\\eu7%Fh\\v7$Fg\\t$!+>8&G <%F^bnF\\eu7%F\\]v7$Fa\\t$!+XQ.%>%F^bnF\\eu7%F`]v7$F[\\t$!+%o?X@%F^bnF \\eu7%Fd]v7$Fe[t$!+K%QRB%F^bnF\\eu7%Fh]v7$F_[t$!+zA/_UF^bnF\\eu7%F\\^v 7$Fijs$!+koooUF^bnF\\eu7%F`^v7$Fcjs$!+g&)z$G%F^bnF\\eu7%Fd^v7$F]js$!+W HN(H%F^bnF\\eu7%Fh^v7$Fgis$!+5!e$4VF^bnF\\eu7%F\\_v7$Fais$!+$HU)>VF^bn F\\eu7%F`_v7$F[is$!+mn%)GVF^bnF\\eu7%Fd_v7$Fehs$!+I%>kL%F^bnF\\eu7%Fh_ v7$F_hs$!+4=hUVF^bnF\\eu7%F\\`v7$Figs$!+2nZZVF^bnF\\eu7%F``v7$Fcgs$!+M o1^VF^bnF\\eu7%Fd`v7$F]gs$!+$*RV`VF^bnF\\eu7%Fh`v7$Fgfs$!+c'GYN%F^bnF \\eu7%F\\av7$Fafs$!+o(*paVF^bnF\\eu7%F`av7$F[fs$!+@Zp`VF^bnF\\eu7%Fdav 7$Fees$!+I%f;N%F^bnF\\eu7%Fhav7$F_es$!+r%Q'[VF^bnF\\eu7%F\\bv7$Fids$!+ f_nWVF^bnF\\eu7%F`bv7$Fcds$!+lA\")RVF^bnF\\eu7%Fdbv7$F]ds$!+K74MVF^bnF \\eu7%Fhbv7$Fgcs$!+FMbFVF^bnF\\eu7%F\\cv7$Facs$!+$))R-K%F^bnF\\eu7%F`c v7$Ffbs$!+v;>7VF^bnF\\eu7%FdcvFfduF\\euFc^rFj^r-F$6%7$7$$!3/++++++!z%F TF(7$$\"3+++++++!*>FTF(-%'COLOURG6&Fe^nF)F)F)-%*LINESTYLEG6#Ff^r-F$6%7 $7$F(F\\dv7$F($\"3/++++++!z%FTFadvFddv-F$6$7ewF'7$$\"3#)*****\\G5jp%!# F$\"3!******fM'zq:F07$$\"3!)*****f!)3J4$F:$\"3y*****\\>$fTJF07$$\"3#** ****zwvhm$F@$\"3D+++bAR7ZF07$$\"3#)*****\\9od9#FF$\"3M+++]L>$G'F07$$\" 3m*****poh$R%)FF$\"3k******3(zR&yF07$$\"3#******ze++a#FL$\"3Z+++$fzYU* F07$$\"3v*****zT_.A'FL$\"3/+++r)3&*4\"FT7$$\"3++++cEuq7Fao$\"3%******H #RZc7FT7$$\"3!******zYu+:#Fao$\"31+++37F89FT7$$\"3v*****\\6Ad\"GFao$\" 3)******ROT(p:FT7$$\"33+++*e5z(>Fao$\"3$*******evjD+++Y&p--$FT7$$!3/+++w^_2>F0$\"3.+++K=$R8$FT7$$!3x* ****HP.*4DF0$\"3-+++4zRNKFT7$$!3%*******))>mkJF0$\"3#******Hw*3CLFT7$$ !3y******>+9\\QF0$\"3$)*****HK/0S$FT7$$!31+++dAMYXF0$\"39+++/mwlMFT7$$ !3E+++znOX_F0$\"3')*****4b&=@NFT7$$!3;+++G^(*RfF0$\"39+++bN*zc$FT7$$!3 /+++%*[&pi'F0$\"3\"******p\"eC2OFT7$$!3%)******f=)[I(F0$\"3'******fZ1) ROFT7$$!3W+++$4CL(zF0$\"3;+++%Rpjm$FT7$$!3k*****4()pBj)F0$\"3y*****\\o ([(o$FT7$$!3Y+++&3hBG*F0$\"3%******\\l,Oq$FT7$$!37+++:\"\\P#**F0$\"3#) *****z3i]r$FT7$$!33+++r@qb5FT$\"3!*******R'[@s$FT7$$!3.+++uqE=6FT$\"37 +++dS3DPFT7$$!33+++u$=,=\"FT$\"3!******f2XSs$FT7$$!32+++i\\IT7FT$\"3** ******>3<>PFT7$$!3!******pOx=I\"FT$\"3'******HMp0r$FT7$$!3-+++A**)=O\" FT$\"31+++pGK)p$FT7$$!3$******f\\.9U\"FT$\"3/+++!>#\\#o$FT7$$!3/+++\"y )[![\"FT$\"3%******Hm?Jm$FT7$$!35+++X,BR:FT$\"3/+++T#Q-k$FT7$$!3++++]+ t(f\"FT$\"3*******f0mQh$FT7$$!3%******4$R6c;FT$\"37+++y@-%e$FT7$$!3!** ****>DNXr\"FT$\"3A+++>$H2b$FT7$$!3!******R.!=tFT$\"3')******\\dQ%Q$FT7$$!3z*******Hb[,#FT$\"3-+++.\"3` L$FT7$$!33+++RURy?FT$\"31+++F7'QG$FT7$$!3(******>W2O9#FT$\"3********z \"[0B$FT7$$!3')*****48>0@#FT$\"3-+++I_&f<$FT7$$!3!)*****fqs*yAFT$\"3-+ ++s8p?JFT7$$!39+++9gi[BFT$\"3!*******>uIlIFT7$$!3\")*****HJ\"**=CFT$\" 3'******p=@-,$FT7$$!3!)*****f@2&*[#FT$\"3++++C`nbHFT7$$!3;+++VdhfDFT$ \"33+++;zt,HFT7$$!3.+++B2#)GEFT$\"3!)*****H'fL[GFT7$$!3'******\\5=np#F T$\"3#)*****zd-`z#FT7$$!3=+++b=+jFFT$\"32+++5;UUFFT7$$!3y******zYXFGFT $\"3%******4bh%*o#FT7$$!3#)*****HGK**)GFT$\"38+++3u>OEFT7$$!33+++EoM]H FT$\"3#)*****HpACe#FT7$$!3()*****zW_'3IFT$\"3;++++Y&z_#FT7$$!3/+++hP$[ 1$FT$\"3!)*****f.NEZ#FT7$$!3-+++&H(*)=JFT$\"3++++A%HjT#FT7$$!3%******* Qa'3<$FT$\"3'*******>X#*eBFT7$$!3.+++ZBx?KFT$\"3-+++EgK+BFT7$$!3++++38 moKFT$\"3&)*****pMc/C#FT7$$!3;+++\"=$e9LFT$\"39+++-GDz@FT7$$!3))*****z `&feLFT$\"3%)*****pLmm6#FT7$$!3,+++\\Aw+MFT$\"3')*****4$3m_?FT7$$!3:++ +mK:TMFT$\"3%******p)H@()>FT7$$!3'******fPW)zMFT$\"3&******\\\"GJ?>FT7 $$!34+++(p;p^$FT$\"37+++AY'>&=FT7$$!3;+++)pbCb$FT$\"3'******HV)=#y\"FT 7$$!3#******z_\\le$FT$\"3,+++S;-6OFT$\"30+++51_Q; FT7$$!31+++*\\^2l$FT$\"3++++u=wk:FT7$$!3/+++]=-\"o$FT$\"3#******>YU)*[ \"FT7$$!3')*****H8i,r$FT$\"3&******H#*yQT\"FT7$$!3')*****4D>#QPFT$\"3+ +++yY+P8FT7$$!39+++az@lPFT$\"3)********fl$f7FT7$$!3)*******\\x:\"z$FT$ \"3%*******GS6\"=\"FT7$$!3&)*****>z6g\"QFT$\"3'******\\!=S-6FT7$$!30++ +q!G(RQFT$\"36+++sJPB5FT7$$!3y*****\\yKA'QFT$\"3u******)\\%eT%*F07$$!3 ?++++aV$)QFT$\"3Q+++\"e#p[')F07$$!3-+++.YB.RFT$\"3a*****H\"p&f&yF07$$! 3#)******fV_@RFT$\"3o*****p'\\0kqF07$$!3@+++0%*>QRFT$\"3g*****R`\\MF'F 07$$!3#*******3(fJ&RFT$\"3u*****Rt4W[&F07$$!3')******3QJmRFT$\"3w***** f(*Qqp%F07$$!3#******f*4exRFT$\"3w*****HD08\"RF07$$!3/+++&f#*o)RFT$\"3 >+++5/2FJF07$$!3-+++KD>%*RFT$\"3'******>N;TM#F07$$!39+++fuV**RFT$\"3-+ ++Ct;i:F07$$!3l*****Rj'f-SFT$\"3h+++!*G34yFgp7$$!3%******Hs^O+%FTF(7$F ]dx$!3h+++!*G34yFgp7$Fhcx$!3-+++Ct;i:F07$Fccx$!3'******>N;TM#F07$F^cx$ !3>+++5/2FJF07$Fibx$!3w*****HD08\"RF07$Fdbx$!3w*****f(*Qqp%F07$F_bx$!3 u*****Rt4W[&F07$Fjax$!3g*****R`\\MF'F07$Feax$!3o*****p'\\0kqF07$F`ax$! 3a*****H\"p&f&yF07$F[ax$!3Q+++\"e#p[')F07$Ff`x$!3u******)\\%eT%*F07$Fa `x$!36+++sJPB5FT7$F\\`x$!3'******\\!=S-6FT7$Fg_x$!3%*******GS6\"=\"FT7 $Fb_x$!3)********fl$f7FT7$F]_x$!3++++yY+P8FT7$Fh^x$!3&******H#*yQT\"FT 7$Fc^x$!3#******>YU)*[\"FT7$F^^x$!3++++u=wk:FT7$Fi]x$!30+++51_Q;FT7$Fd ]x$!3,+++S;-6&=FT7$Fe\\ x$!3&******\\\"GJ?>FT7$F`\\x$!3%******p)H@()>FT7$F[\\x$!3')*****4$3m_? FT7$Ff[x$!3%)*****pLmm6#FT7$Fa[x$!39+++-GDz@FT7$F\\[x$!3&)*****pMc/C#F T7$Fgjw$!3-+++EgK+BFT7$Fbjw$!3'*******>X#*eBFT7$F]jw$!3++++A%HjT#FT7$F hiw$!3!)*****f.NEZ#FT7$Fciw$!3;++++Y&z_#FT7$F^iw$!3#)*****HpACe#FT7$Fi hw$!38+++3u>OEFT7$Fdhw$!3%******4bh%*o#FT7$F_hw$!32+++5;UUFFT7$Fjgw$!3 #)*****zd-`z#FT7$Fegw$!3!)*****H'fL[GFT7$F`gw$!33+++;zt,HFT7$F[gw$!3++ ++C`nbHFT7$Fffw$!3'******p=@-,$FT7$Fafw$!3!*******>uIlIFT7$F\\fw$!3-++ +s8p?JFT7$Fgew$!3-+++I_&f<$FT7$Fbew$!3********z\"[0B$FT7$F]ew$!31+++F7 'QG$FT7$Fhdw$!3-+++.\"3`L$FT7$Fcdw$!3')******\\dQ%Q$FT7$F^dw$!3=+++0Dq IMFT7$Ficw$!3++++zy(RZ$FT7$Fdcw$!3++++&RDS^$FT7$F_cw$!3A+++>$H2b$FT7$F jbw$!37+++y@-%e$FT7$Febw$!3*******f0mQh$FT7$F`bw$!3/+++T#Q-k$FT7$F[bw$ !3%******Hm?Jm$FT7$Ffaw$!3/+++!>#\\#o$FT7$Faaw$!31+++pGK)p$FT7$F\\aw$! 3'******HMp0r$FT7$Fg`w$!3********>3<>PFT7$Fb`w$!3!******f2XSs$FT7$F]`w $!37+++dS3DPFT7$Fh_w$!3!*******R'[@s$FT7$Fc_w$!3#)*****z3i]r$FT7$F^_w$ !3%******\\l,Oq$FT7$Fi^w$!3y*****\\o([(o$FT7$Fd^w$!3;+++%Rpjm$FT7$F_^w $!3'******fZ1)ROFT7$Fj]w$!3\"******p\"eC2OFT7$Fe]w$!39+++bN*zc$FT7$F`] w$!3')*****4b&=@NFT7$F[]w$!39+++/mwlMFT7$Ff\\w$!3$)*****HK/0S$FT7$Fa\\ w$!3#******Hw*3CLFT7$F\\\\w$!3-+++4zRNKFT7$Fg[w$!3.+++K=$R8$FT7$Fb[w$! 3>+++Y&p--$FT7$F][w$!3?+++\"z4h*GFT7$Fhjv$!3y*****HmDPw#FT7$Fcjv$!3:++ +\"oF_i#FT7$F^jv$!3?+++-f;#[#FT7$Fiiv$!3&)*****zn^bL#FT7$Fdiv$!37+++Zj 1'=#FT7$F_iv$!3?+++=LDM?FT7$Fjhv$!3'*******)4<1)=FT7$Fehv$!3$*******ev jD$G'F07$F]fv$!3D+++bAR7ZF07$Fhev$!3y*****\\>$fTJF07$$ \"3)********H5jp%Fdev$!3!******fM'zq:F0F'-Fc^n6&Fe^n$\"\"\"Fg^r$\"#TFh ^nFagy-F\\_n6fw7%F'7$$\"+&G5jp%Fgp$\"+Yjzq:Fd_nFe_n7%Fhgy7$$\"+1)3J4$F T$\"+&>$fTJFd_nFe_n7%F^hy7$$\"+od$G'Fd_nFe_n7%Fjhy7$$\"+(oh$R%)F^an$\"+4(zR&yFd_nFe_n7 %F`iy7$$\"+)e++a#Fean$\"+$fzYU*Fd_nFe_n7%Ffiy7$$\"+=CN?iFean$\"+r)3&*4 \"F^bnFe_n7%F\\jy7$$\"+cEuq7Fecn$\"+BRZc7F^bnFe_n7%Fbjy7$$\"+oW2]@Fecn $\"+37F89F^bnFe_n7%Fhjy7$$\"+:@s:GFecn$\"+k8up:F^bnFe_n7%F^[z7$$\"+*e5 z(>Fecn$\"+fvjDFd_n$\"+K=$R8$F^bnFe_n7% F`_z7$$!+tL!*4DFd_n$\"+4zRNKF^bnFe_n7%Ff_z7$$!+*)>mkJFd_n$\"+j(*3CLF^b nFe_n7%F\\`z7$$!+?+9\\QFd_n$\"+BV]+MF^bnFe_n7%Fb`z7$$!+dAMYXFd_n$\"+/m wlMF^bnFe_n7%Fh`z7$$!+znOX_Fd_n$\"+^b=@NF^bnFe_n7%F^az7$$!+G^(*RfFd_n$ \"+bN*zc$F^bnFe_n7%Fdaz7$$!+%*[&pi'Fd_n$\"+PF^bnFe_n7%F`ez7$$!+nt (=I\"F^bn$\"+V$p0r$F^bnFe_n7%Ffez7$$!+A**)=O\"F^bn$\"+pGK)p$F^bnFe_n7% F\\fz7$$!+'\\.9U\"F^bn$\"+!>#\\#o$F^bnFe_n7%Fbfz7$$!+\"y)[![\"F^bn$\"+ j17jOF^bnFe_n7%Fhfz7$$!+X,BR:F^bn$\"+T#Q-k$F^bnFe_n7%F^gz7$$!+]+t(f\"F ^bn$\"+cg'Qh$F^bnFe_n7%Fdgz7$$!+JR6c;F^bn$\"+y@-%e$F^bnFe_n7%Fjgz7$$!+ __`9$H2b$F^bnFe_n7%F`hz7$$!+M+=tF^bn$\"+]dQ%Q$F^bnFe_n7%Fhiz7$$!++`&[,#F^bn$\"+. \"3`L$F^bnFe_n7%F^jz7$$!+RURy?F^bn$\"+F7'QG$F^bnFe_n7%Fdjz7$$!+UugV@F^ bn$\"+!=[0B$F^bnFe_n7%Fjjz7$$!+J\">0@#F^bn$\"+I_&f<$F^bnFe_n7%F`[[l7$$ !+1F(*yAF^bn$\"+s8p?JF^bnFe_n7%Ff[[l7$$!+9gi[BF^bn$\"+?uIlIF^bnFe_n7%F \\\\[l7$$!+88**=CF^bn$\"+(=@-,$F^bnFe_n7%Fb\\[l7$$!+;s]*[#F^bn$\"+C`nb HF^bnFe_n7%Fh\\[l7$$!+VdhfDF^bn$\"+;zt,HF^bnFe_n7%F^][l7$$!+B2#)GEF^bn $\"+jfL[GF^bnFe_n7%Fd][l7$$!+0\"=np#F^bn$\"+yDI&z#F^bnFe_n7%Fj][l7$$!+ b=+jFF^bn$\"+5;UUFF^bnFe_n7%F`^[l7$$!+!oau#GF^bn$\"+^:Y*o#F^bnFe_n7%Ff ^[l7$$!+$GK**)GF^bn$\"+3u>OEF^bnFe_n7%F\\_[l7$$!+EoM]HF^bn$\"+$pACe#F^ bnFe_n7%Fb_[l7$$!+[Cl3IF^bn$\"++Y&z_#F^bnFe_n7%Fh_[l7$$!+hP$[1$F^bn$\" +O]jsCF^bnFe_n7%F^`[l7$$!+&H(*)=JF^bn$\"+A%HjT#F^bnFe_n7%Fd`[l7$$!+Ra' 3<$F^bn$\"+?X#*eBF^bnFe_n7%Fj`[l7$$!+ZBx?KF^bn$\"+EgK+BF^bnFe_n7%F`a[l 7$$!+38moKF^bn$\"+ZjXSAF^bnFe_n7%Ffa[l7$$!+\"=$e9LF^bn$\"+-GDz@F^bnFe_ n7%F\\b[l7$$!+QbfeLF^bn$\"+Pjm;@F^bnFe_n7%Fbb[l7$$!+\\Aw+MF^bn$\"+J3m_ ?F^bnFe_n7%Fhb[l7$$!+mK:TMF^bn$\"+()H@()>F^bnFe_n7%F^c[l7$$!+wV%)zMF^b n$\"+:GJ?>F^bnFe_n7%Fdc[l7$$!+(p;p^$F^bn$\"+AY'>&=F^bnFe_n7%Fjc[l7$$!+ )pbCb$F^bn$\"+L%)=#y\"F^bnFe_n7%F`d[l7$$!+G&\\le$F^bn$\"+S;-6OF^bn$\"+51_Q;F^bnFe_n7%F\\e[l7$$!+*\\^2l$F^bn$\"+u=wk :F^bnFe_n7%Fbe[l7$$!+]=-\"o$F^bn$\"+iC%)*[\"F^bnFe_n7%Fhe[l7$$!+L@;5PF ^bn$\"+B*yQT\"F^bnFe_n7%F^f[l7$$!+^#>#QPF^bn$\"+yY+P8F^bnFe_n7%Fdf[l7$ $!+az@lPF^bn$\"++cOf7F^bnFe_n7%Fjf[l7$$!+]x:\"z$F^bn$\"+HS6\"=\"F^bnFe _n7%F`g[l7$$!+#z6g\"QF^bn$\"+0=S-6F^bnFe_n7%Ffg[l7$$!+q!G(RQF^bn$\"+sJ PB5F^bnFe_n7%F\\h[l7$$!+&yKA'QF^bn$\"+*\\%eT%*Fd_nFe_n7%Fbh[l7$$!++aV$ )QF^bn$\"+\"e#p[')Fd_nFe_n7%Fhh[l7$$!+.YB.RF^bn$\"+8p&f&yFd_nFe_n7%F^i [l7$$!+gV_@RF^bn$\"+n\\0kqFd_nFe_n7%Fdi[l7$$!+0%*>QRF^bn$\"+M&\\MF'Fd_ nFe_n7%Fji[l7$$!+4(fJ&RF^bn$\"+M(4W[&Fd_nFe_n7%F`j[l7$$!+4QJmRF^bn$\"+ w*Qqp%Fd_nFe_n7%Ffj[l7$$!+'*4exRF^bn$\"+`_I6RFd_nFe_n7%F\\[\\l7$$!+&f# *o)RF^bn$\"+5/2FJFd_nFe_n7%Fb[\\l7$$!+KD>%*RF^bn$\"+_j6WBFd_nFe_n7%Fh[ \\l7$$!+fuV**RF^bn$\"+Ct;i:Fd_nFe_n7%F^\\\\l7$$!+Mmf-SF^bn$\"+!*G34yF_ enFe_n7%Fd\\\\l7$$!+B&=F^bnFe_n7%Fjb\\l7$Fec[l$!+:GJ?>F^bnF e_n7%F^c\\l7$F_c[l$!+()H@()>F^bnFe_n7%Fbc\\l7$Fib[l$!+J3m_?F^bnFe_n7%F fc\\l7$Fcb[l$!+Pjm;@F^bnFe_n7%Fjc\\l7$F]b[l$!+-GDz@F^bnFe_n7%F^d\\l7$F ga[l$!+ZjXSAF^bnFe_n7%Fbd\\l7$Faa[l$!+EgK+BF^bnFe_n7%Ffd\\l7$F[a[l$!+? X#*eBF^bnFe_n7%Fjd\\l7$Fe`[l$!+A%HjT#F^bnFe_n7%F^e\\l7$F_`[l$!+O]jsCF^ bnFe_n7%Fbe\\l7$Fi_[l$!++Y&z_#F^bnFe_n7%Ffe\\l7$Fc_[l$!+$pACe#F^bnFe_n 7%Fje\\l7$F]_[l$!+3u>OEF^bnFe_n7%F^f\\l7$Fg^[l$!+^:Y*o#F^bnFe_n7%Fbf\\ l7$Fa^[l$!+5;UUFF^bnFe_n7%Fff\\l7$F[^[l$!+yDI&z#F^bnFe_n7%Fjf\\l7$Fe][ l$!+jfL[GF^bnFe_n7%F^g\\l7$F_][l$!+;zt,HF^bnFe_n7%Fbg\\l7$Fi\\[l$!+C`n bHF^bnFe_n7%Ffg\\l7$Fc\\[l$!+(=@-,$F^bnFe_n7%Fjg\\l7$F]\\[l$!+?uIlIF^b nFe_n7%F^h\\l7$Fg[[l$!+s8p?JF^bnFe_n7%Fbh\\l7$Fa[[l$!+I_&f<$F^bnFe_n7% Ffh\\l7$F[[[l$!+!=[0B$F^bnFe_n7%Fjh\\l7$Fejz$!+F7'QG$F^bnFe_n7%F^i\\l7 $F_jz$!+.\"3`L$F^bnFe_n7%Fbi\\l7$Fiiz$!+]dQ%Q$F^bnFe_n7%Ffi\\l7$Fciz$! +0DqIMF^bnFe_n7%Fji\\l7$F]iz$!+zy(RZ$F^bnFe_n7%F^j\\l7$Fghz$!+&RDS^$F^ bnFe_n7%Fbj\\l7$Fahz$!+>$H2b$F^bnFe_n7%Ffj\\l7$F[hz$!+y@-%e$F^bnFe_n7% Fjj\\l7$Fegz$!+cg'Qh$F^bnFe_n7%F^[]l7$F_gz$!+T#Q-k$F^bnFe_n7%Fb[]l7$Fi fz$!+j17jOF^bnFe_n7%Ff[]l7$Fcfz$!+!>#\\#o$F^bnFe_n7%Fj[]l7$F]fz$!+pGK) p$F^bnFe_n7%F^\\]l7$Fgez$!+V$p0r$F^bnFe_n7%Fb\\]l7$Faez$!+?3<>PF^bnFe_ n7%Ff\\]l7$F[ez$!+w]/CPF^bnFe_n7%Fj\\]l7$Fedz$!+dS3DPF^bnFe_n7%F^]]l7$ F_dz$!+S'[@s$F^bnFe_n7%Fb]]l7$Ficz$!+)3i]r$F^bnFe_n7%Ff]]l7$Fccz$!+b;g .PF^bnFe_n7%Fj]]l7$F]cz$!+&o([(o$F^bnFe_n7%F^^]l7$Fgbz$!+%Rpjm$F^bnFe_ n7%Fb^]l7$Fabz$!+wk!)ROF^bnFe_n7%Ff^]l7$F[bz$!+$G'Fd _nFe_n7%Fjd]l7$Fehy$!+bAR7ZFd_nFe_n7%F^e]l7$F_hy$!+&>$fTJFd_nFe_n7%Fbe ]l7$$\"++.J'p%Fgp$!+Yjzq:Fd_nFe_n7%Ffe]lF'Fe_n-Fc^n6&Fe^n$\"\"#Fg^r$\" #$)Fh^nF^f]lFj^r-F$6$7in7$$\"3%******p>lr\\\"FT$\"3g*****H'pw'H%FT7$$ \"35+++qq0*[\"FT$\"3%******fn$Q0VFT7$$\"3%******RGq-[\"FT$\"3T+++!32LJ %FT7$$\"3'******z'z%3Z\"FT$\"3\"******f'e\\?VFT7$$\"33+++))R$3Y\"FT$\" 3U+++_!4pK%FT7$$\"31+++qKF]9FT$\"3w*****Hr0DL%FT7$$\"32+++o@@R9FT$\"3 \"******prWsL%FT7$$\"3*)******z))pF9FT$\"30+++zW3TVFT7$$\"3-+++MRy:9FT $\"3)******\\t#)RM%FT7$$\"3$******>v?NS\"FT$\"3e******>j*eM%FT7$$\"3#* *****>Km4R\"FT$\"3'******z(4yYVFT7$$\"3++++*)>=y8FT$\"3o******)>\"fYVF T7$$\"3++++5WBl8FT$\"3=+++,,GXVFT7$$\"31+++*p'>_8FT$\"3i******)Q*zUVFT 7$$\"3/+++?)\\\"R8FT$\"3n*****\\U*4RVFT7$$\"3(******HF%=E8FT$\"3i***** *o%HTL%FT7$$\"36+++W@S88FT$\"3P+++y\"QyK%FT7$$\"3,+++%f>4I\"FT$\"3O+++ 7X@&yfUFT7$$\"3&**** **Hoa9C\"FT$\"3v*****f6vJC%FT7$$\"36+++,lCO7FT$\"3\"******>^]^A%FT7$$ \"3++++>()>L7FT$\"3%******45ne?%FT7$$\"3$******Rw!oK7FT$\"3++++?,e&=%F T7$$\"3)*******H.0N7FT$\"3C+++Z7nkTFT7$$\"33+++aDgS7FT$\"3'******z?mO9 %FT7$$\"3#******H0,&\\7FT$\"3N+++8xABTFT7$$\"33+++mfqh7FT$\"3f*****4&z 5/TFT7$$\"3)*******f\"GpF\"FT$\"3I+++6s0(3%FT7$$\"3%******zRRYH\"FT$\" 3E+++?-rsSFT7$$\"3\"********RXTJ\"FT$\"3W+++)4#\\hSFT7$$\"35+++e]pM8FT $\"3A+++z*zN0%FT7$$\"3#*******4Ieb8FT$\"3q******pe#*[SFT7$$\"3++++(\\: iP\"FT$\"3!******fi0%FT7$$\"31+++;iiK9FT$\"3!)*****p-Rv0%FT7$$\"3-+ ++%eV)[9FT$\"3g******)oG[1%FT7$$\"35+++/$*ej9FT$\"3-+++QaatSFT7$$\"3!* *****f*G$oZ\"FT$\"3m*****p6HM3%FT7$$\"3'******Ryt&)[\"FT$\"3#******\\3 _U4%FT7$$\"3++++(RJ))\\\"FT$\"3k*****p9;e5%FT7$$\"3(*******y\"Qw]\"FT$ \"3v*****f))[z6%FT7$$\"3/+++_Z.::FT$\"3i**********\\ITFT7$$\"3!****** \\gm5_\"FT$\"3-+++ZzLVTFT7$$\"3,+++p=yD:FT$\"3'******f)pMcTFT7$$\"3,++ +Q)H#H:FT$\"3++++_YUpTFT7$$\"3&******\\,g9`\"FT$\"3o******4'zC=%FT7$$ \"3++++::_K:FT$\"3;+++_)Ha>%FT7$$\"3#******HriC`\"FT$\"3#******R>,#3UF T7$$\"30+++l5LJ:FT$\"39+++-gs?UFT7$$\"3$******f/t\"H:FT$\"3!)*****>8UH B%FT7$$\"36+++pT.E:FT$\"3;+++I?zWUFT7$$\"3++++h!f>_\"FT$\"3q*****>'>Ac UFT7$$\"3#*******y:*p^\"FT$\"3)******fO\"=nUFT7$$\"3'*******f\\<6:FT$ \"3s*****fGAwF%FT7$$\"34+++*)>b/:FT$\"3u*****R())\\(G%FTFef]lF_gy-F\\_ n6jn7%7$$\"+(>lr\\\"F^bn$\"+jpw'H%F^bn7$$\"+qq0*[\"F^bn$\"+wOQ0VF^bn7$ $\"$Q\"Fh^nF]cs7%Fii^l7$$\"+%Gq-[\"F^bn$\"+!32LJ%F^bnF^j^l7%Fbj^l7$$\" +oz%3Z\"F^bn$\"+me\\?VF^bnF^j^l7%Fhj^l7$$\"+))R$3Y\"F^bn$\"+_!4pK%F^bn F^j^l7%F^[_l7$$\"+qKF]9F^bn$\"+8d]KVF^bnF^j^l7%Fd[_l7$$\"+o@@R9F^bn$\" +=y8F^bn$\" +*>\"fYVF^bnF^j^l7%Fh]_l7$$\"+5WBl8F^bn$\"+,,GXVF^bnF^j^l7%F^^_l7$$\"+ *p'>_8F^bn$\"+*Q*zUVF^bnF^j^l7%Fd^_l7$$\"+?)\\\"R8F^bn$\"+D%*4RVF^bnF^ j^l7%Fj^_l7$$\"+tU=E8F^bn$\"+p%HTL%F^bnF^j^l7%F`__l7$$\"+W@S88F^bn$\"+ y\"QyK%F^bnF^j^l7%Ff__l7$$\"+%f>4I\"F^bn$\"+7X()>L7F^bn$\"+,r'e?%F^bnF^j^l7%F\\c_l7$$\"+k2oK7F^ bn$\"+?,e&=%F^bnF^j^l7%Fbc_l7$$\"+I.0N7F^bn$\"+Z7nkTF^bnF^j^l7%Fhc_l7$ $\"+aDgS7F^bn$\"+3imVTF^bnF^j^l7%F^d_l7$$\"+`5]\\7F^bn$\"+8xABTF^bnF^j ^l7%Fdd_l7$$\"+mfqh7F^bn$\"+^z5/TF^bnF^j^l7%Fjd_l7$$\"+g\"GpF\"F^bn$\" +6s0(3%F^bnF^j^l7%F`e_l7$$\"+)RRYH\"F^bn$\"+?-rsSF^bnF^j^l7%Ffe_l7$$\" ++a998F^bn$\"+)4#\\hSF^bnF^j^l7%F\\f_l7$$\"+e]pM8F^bn$\"+z*zN0%F^bnF^j ^l7%Fbf_l7$$\"+5Ieb8F^bn$\"+qe#*[SF^bnF^j^l7%Fhf_l7$$\"+(\\:iP\"F^bn$ \"+EwJZSF^bnF^j^l7%F^g_l7$$\"+6W8'R\"F^bn$\"+&zZ%[SF^bnF^j^l7%Fdg_l7$$ \"+_7,:9F^bn$\"+w.(>0%F^bnF^j^l7%Fjg_l7$$\"+;iiK9F^bn$\"+F!Rv0%F^bnF^j ^l7%F`h_l7$$\"+%eV)[9F^bn$\"+*oG[1%F^bnF^j^l7%Ffh_l7$$\"+/$*ej9F^bn$\" +QaatSF^bnF^j^l7%F\\i_l7$$\"+'*G$oZ\"F^bn$\"+<\"HM3%F^bnF^j^l7%Fbi_l7$ $\"+%yt&)[\"F^bn$\"+&3_U4%F^bnF^j^l7%Fhi_l7$$\"+(RJ))\\\"F^bn$\"+Zh\"e 5%F^bnF^j^l7%F^j_l7$$\"+z\"Qw]\"F^bn$\"+'))[z6%F^bnF^j^l7%Fdj_l7$$\"+_ Z.::F^bn$\"+++]ITF^bnF^j^l7%Fjj_l7$$\"+0m1@:F^bn$\"+ZzLVTF^bnF^j^l7%F` [`l7$$\"+p=yD:F^bn$\"+')pMcTF^bnF^j^l7%Ff[`l7$$\"+Q)H#H:F^bn$\"+_YUpTF ^bnF^j^l7%F\\\\`l7$$\"+:+YJ:F^bn$\"+5'zC=%F^bnF^j^l7%Fb\\`l7$$\"+::_K: F^bn$\"+_)Ha>%F^bnF^j^l7%Fh\\`l7$$\"+8FYK:F^bn$\"+%>,#3UF^bnF^j^l7%F^] `l7$$\"+l5LJ:F^bn$\"+-gs?UF^bnF^j^l7%Fd]`l7$$\"+YI_\"F^bn $\"+i>AcUF^bnF^j^l7%Ff^`l7$$\"+z:*p^\"F^bn$\"+m8=nUF^bnF^j^l7%F\\_`l7$ $\"+g\\<6:F^bn$\"+'GAwF%F^bnF^j^l7%Fb_`l7$$\"+*)>b/:F^bn$\"+u))\\(G%F^ bnF^j^l7%Fh_`lFdi^lF^j^lF\\f]lFj^r-F$6$7in7$Fff]l$!3g*****H'pw'H%FT7$F ]i^l$!3u*****R())\\(G%FT7$Fhh^l$!3s*****fGAwF%FT7$Fch^l$!3)******fO\"= nUFT7$F^h^l$!3q*****>'>AcUFT7$Fig^l$!3;+++I?zWUFT7$Fdg^l$!3!)*****>8UH B%FT7$F_g^l$!39+++-gs?UFT7$Fjf^l$!3#******R>,#3UFT7$Fef^l$!3;+++_)Ha>% FT7$F`f^l$!3o******4'zC=%FT7$F[f^l$!3++++_YUpTFT7$Ffe^l$!3'******f)pMc TFT7$Fae^l$!3-+++ZzLVTFT7$F\\e^l$!3i**********\\ITFT7$Fgd^l$!3v*****f) )[z6%FT7$Fbd^l$!3k*****p9;e5%FT7$F]d^l$!3#******\\3_U4%FT7$Fhc^l$!3m** ***p6HM3%FT7$Fcc^l$!3-+++QaatSFT7$F^c^l$!3g******)oG[1%FT7$Fib^l$!3!)* ****p-Rv0%FT7$Fdb^l$!3))*****fPq>0%FT7$F_b^l$!3S+++&zZ%[SFT7$Fja^l$!3! ******fi^]^A%FT7$Fi]^l$!3v*****f6vJC%FT7$Fd]^l$!3s*****>@&yfUFT7$F_]^l$!3 1+++(*z*[F%FT7$Fj\\^l$!3!******fR\"[)G%FT7$Fe\\^l$!37+++Xs`+VFT7$F`\\^ l$!3v*****HA*36VFT7$F[\\^l$!3O+++7X\"fYVFT7$Fhi]l$!3'******z(4yYVF T7$Fci]l$!3e******>j*eM%FT7$F^i]l$!3)******\\t#)RM%FT7$Fih]l$!30+++zW3 TVFT7$Fdh]l$!3\"******prWsL%FT7$F_h]l$!3w*****Hr0DL%FT7$Fjg]l$!3U+++_! 4pK%FT7$Feg]l$!3\"******f'e\\?VFT7$F`g]l$!3T+++!32LJ%FT7$F[g]l$!3%**** **fn$Q0VFTFa``lF_gy-F\\_n6jn7%7$Fei^l$!+jpw'H%F^bn7$Fi_`l$!+u))\\(G%F^ bn7$F_j^lF]eu7%F[\\al7$Fc_`l$!+'GAwF%F^bnF^\\al7%F`\\al7$F]_`l$!+m8=nU F^bnF^\\al7%Fd\\al7$Fg^`l$!+i>AcUF^bnF^\\al7%Fh\\al7$Fa^`l$!+I?zWUF^bn F^\\al7%F\\]al7$F[^`l$!+K@%HB%F^bnF^\\al7%F`]al7$Fe]`l$!+-gs?UF^bnF^\\ al7%Fd]al7$F_]`l$!+%>,#3UF^bnF^\\al7%Fh]al7$Fi\\`l$!+_)Ha>%F^bnF^\\al7 %F\\^al7$Fc\\`l$!+5'zC=%F^bnF^\\al7%F`^al7$F]\\`l$!+_YUpTF^bnF^\\al7%F d^al7$Fg[`l$!+')pMcTF^bnF^\\al7%Fh^al7$Fa[`l$!+ZzLVTF^bnF^\\al7%F\\_al 7$F[[`l$!+++]ITF^bnF^\\al7%F`_al7$Fej_l$!+'))[z6%F^bnF^\\al7%Fd_al7$F_ j_l$!+Zh\"e5%F^bnF^\\al7%Fh_al7$Fii_l$!+&3_U4%F^bnF^\\al7%F\\`al7$Fci_ l$!+<\"HM3%F^bnF^\\al7%F``al7$F]i_l$!+QaatSF^bnF^\\al7%Fd`al7$Fgh_l$!+ *oG[1%F^bnF^\\al7%Fh`al7$Fah_l$!+F!Rv0%F^bnF^\\al7%F\\aal7$F[h_l$!+w.( >0%F^bnF^\\al7%F`aal7$Feg_l$!+&zZ%[SF^bnF^\\al7%Fdaal7$F_g_l$!+EwJZSF^ bnF^\\al7%Fhaal7$Fif_l$!+qe#*[SF^bnF^\\al7%F\\bal7$Fcf_l$!+z*zN0%F^bnF ^\\al7%F`bal7$F]f_l$!+)4#\\hSF^bnF^\\al7%Fdbal7$Fge_l$!+?-rsSF^bnF^\\a l7%Fhbal7$Fae_l$!+6s0(3%F^bnF^\\al7%F\\cal7$F[e_l$!+^z5/TF^bnF^\\al7%F `cal7$Fed_l$!+8xABTF^bnF^\\al7%Fdcal7$F_d_l$!+3imVTF^bnF^\\al7%Fhcal7$ Fic_l$!+Z7nkTF^bnF^\\al7%F\\dal7$Fcc_l$!+?,e&=%F^bnF^\\al7%F`dal7$F]c_ l$!+,r'e?%F^bnF^\\al7%Fddal7$Fgb_l$!+70:DUF^bnF^\\al7%Fhdal7$Fab_l$!+; ^\"fYVF^bnF^\\al7%F\\h al7$Fc]_l$!+y4yYVF^bnF^\\al7%F`hal7$F]]_l$!+?j*eM%F^bnF^\\al7%Fdhal7$F g\\_l$!+NF)RM%F^bnF^\\al7%Fhhal7$Fa\\_l$!+zW3TVF^bnF^\\al7%F\\ial7$F[ \\_l$!+Fh^n;Fa\\bl$\"$z%Fh^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 st age, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 80 "The stability function R^ for the 7 stage, order 4 sc heme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b^`,StabilityFuncti on(4,7,'expanded'))):\n`R^` := unapply(%,z):\n'`R^`(z)'=`R^`(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R^G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#\"9Jy!QPq?'*o$=MSlYm?@Y'4q\"F)*$)F'\"\"(F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R^`(z) = -1;" "6#/-%#R^G6#%\"zG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " z_0 := newton(`R^`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ z_0G$!+qnI7S!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "zo0 := newton(`R^`(z)=-1,z=-4):\npo1 := plot ([`R^`(z),-1],z=-4.49..0.49,color=[red,blue]):\npo2 := plot([[[zo0,-1] ]$3],style=point,symbol=[circle,cross,diamond],color=black):\npo3 := p lot([[zo0,0],[zo0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[di splay]([po1,po2,po3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9] );" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG 6$7U7$$!3A++++++!\\%!#<$!3m!\\^6sE!RCF*7$$!3;+]7'3DdV%F*$!3Q<\">p9Yr@# F*7$$!35++Ds,X\"Q%F*$!38HJDshp7?F*7$$!3(*\\P*)feAMVF*$!3WCv9+(o![=F*7$ $!3%)*\\Pva,qG%F*$!3ys2:;*4]p\"F*7$$!3I+](4>%y!=%F*$!3CG*oTgc%*Q\"F*7$ $!35+]#oviQ2%F*$!3X!oC+6X+8\"F*7$$!3A+ve,&\\u'RF*$!3T>IzgUhK\"*!#=7$$! 3?+v8#z!zoQF*$!3vH'og4<2W(FP7$$!3;+v.8ajmPF*$!3])>A^->h'fFP7$$!3C+v$Q< ')4m$F*$!3i(ovXt(=&p%FP7$$!3G+v3`dnbNF*$!3QyY=bieYOFP7$$!37++0T5NZMF*$ !3C5C!HLH\"fFFP7$$!3?+]sv&Q>N$F*$!39'='Q*)\\G7@FP7$$!3E++:Qn_WKF*$!3#> I5(R$*))4:FP7$$!3#****\\s&QnOJF*$!3AEv0!QHd,\"FP7$$!3&****\\=iPF.$F*$! 3uB&3c;gzC'!#>7$$!3Q+v$Gc`$QHF*$!3c)3fq\"Rm%G$Fhp7$$!3+++l(y@h#GF*$!31 XO6tz*oQ$!#?7$$!3=++g_n/JFF*$\"3W0i=Dyzr60M1#)\\bFhp7$$!3#**\\P(R,::CF*$\"3&p& o@`M33sFhp7$$!3&)*\\7sqtGJ#F*$\"3G]i5GdQp\"F*$\"3\"*Ru\\eALI=FP7$$!3)*****p]Q@ (e\"F*$\"3d359@d\"3/#FP7$$!3I+]FRT)G[\"F*$\"3;ifxp#zxE#FP7$$!3')*\\P*y (R>Q\"F*$\"3Sd+?BV+5DFP7$$!3++]Kx#e)p7F*$\"3X*\\h3kZ%3GFP7$$!3=++5j![ \"p6F*$\"3#[$>]G,I1JFP7$$!3s***\\KT=;1\"F*$\"3O,qWAP.fMFP7$$!3'***\\P% o0=k*FP$\"3u@Hg*HUI\"QFP7$$!3](***\\tEbw&)FP$\"3:li](4@;C%FP7$$!39-]P2 EBuvFP$\"3sX*3kqs()o%FP7$$!3>)*\\(GL>l_'FP$\"3Yb/\"G0am?&FP7$$!3z++]- \")=-bFP$\"3=_O!RwX#odFP7$$!3$*)*\\PM#3)HWFP$\"33KH47')>@kFP7$$!3;(*** *f'*ypR$FP$\"3^A,PAZ&)>rFP7$$!3L)**\\U'=wSBFP$\"3e3=&RS:I\"zFP7$$!3?$* \\Pt2H$H\"FP$\"3#puJOjZoy)FP7$$!33R****pit2LFhp$\"3;gk*36PYn*FP7$$\"3Z ^+]FkzBxFhp$\"3+G@O=\"*H!3\"F*7$$\"35*****z2`!f\"F*7$ $\"3l+]i_F06GFP$\"3fvB8>IfC8F*7$$\"3\\.]Pt1&z\"QFP$\"3!R`MSt6\\Y\"F*7$ $\"3!***************[FP$\"3HtwQ4eJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\") $\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$F GFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa \\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa \\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa \\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vF a\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxF a\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$Ffz Fa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3O+++qn I7SF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6 #%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF _`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LI NESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl6 $%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`c l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture sho ws the stability region for the 8 stage, order 4 scheme. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1525 "`R^ ` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+\n 301718368962070373807831/3 6276990063928779775644720*z^5+\n 55904026309238947888070173528838300 1733/399205982528090541252922414033831284582120*z^6+\n 3316827661115 6421352084390650912318925/170096462120664665403419115544849851691512*z ^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R ^`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\npo1 := plot(pts,color=COLOR(RGB,.08,.35,.08)):\np o2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts)) ],\n style=patchnogrid,color=COLOR(RGB,.15,.7,.15)):\npts := \+ []:\nz0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R^`(z) =exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\npo3 := plot(pts,color=COLOR(RGB,.08,.35,.08)):\npo4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.38,4.19]],i=2..nops(pts)) ],\n style=patchnogrid,color=COLOR(RGB,.15,.7,.15)):\npts := \+ []:\nz0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R^`(z) =exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\npo5 := plot(pts,color=COLOR(RGB,.08,.35,.08)):\npo6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.38,-4.19]],i=2..nops(pts) )],\n style=patchnogrid,color=COLOR(RGB,.15,.7,.15)):\npo7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4.79]]],color=black,linestyl e=3):\nplots[display]([po||(1..7)],view=[-4.79..1.99,-4.79..4.79],font =[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,sca ling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 586 586 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3$)*****H'='f@%!#F$\"37 +++Ujzq:!#=7$$\"3;+++F4i#y#!#D$\"3#******f2$fTJF07$$\"3\"******\\Q4rI$ !#C$\"33+++u8R7ZF07$$\"3%******pRa-%>!#B$\"35+++y(*=$G'F07$$\"3x+++%=% )*RwF@$\"3A+++^%pR&yF07$$\"3;+++&3XqH#!#A$\"3-+++MjlC%*F07$$\"39+++\"e =(*f&FK$\"3-+++2X]*4\"!#<7$$\"3)*******zfwJ6!#@$\"3!******z4nkD\"FS7$$ \"3%*******pk=q=FW$\"34+++/FE89FS7$$\"3#*******ozd+BFW$\"3!******4oM(p :FS7$$\"3(*******=(pD5\"FW$\"35+++R5kD!#?$\"3$******fcLV.#FS7$$!3:+++m$\"36+++V1&eL#FS7$$!39+++5[Uw@Ffp$\"3)**** **>S]E[#FS7$$!32+++pm;>QFfp$\"3=+++ma&fi#FS7$$!30+++7'fuB'Ffp$\"3'**** **H;`Zw#FS7$$!3%)*******fx3e*Ffp$\"35+++QB[(*GFS7$$!3)******zY\">#R\"F 0$\"3'******f!y+AIFS7$$!3*******H&z$3#>F0$\"3\")********=-OJFS7$$!3+++ +wABEDF0$\"3)******zt&zPKFS7$$!3?+++,N7%=$F0$\"3\"******\\qQnK$FS7$$!3 ++++^8jrQF0$\"3\"*******\\@N.MFS7$$!3=+++\"QX;d%F0$\"3')*****RWMKF&F0$\"3%*******GGKCNFS7$$!3/+++]r%f5OFS7$$!39+++0>?RtF0$\"3y*****4RXKk$ FS7$$!3e*****zu(\\4!)F0$\"31+++3E*)pOFS7$$!3q*****>A*Hq')F0$\"3)****** *404\"p$FS7$$!3@+++$3r>K*F0$\"3)******zR!G2PFS7$$!3')******yJ)\\'**F0$ \"3z*****Hs8)=PFS7$$!3-+++7Q)*f5FS$\"3()******G1(fs$FS7$$!33+++vRqA6FS $\"3*******>Du*GPFS7$$!3/+++?&3Z=\"FS$\"3$******>o,!GPFS7$$!3'******z5 ZgC\"FS$\"3@+++*R#>BPFS7$$!30+++t3x18FS$\"3,+++W[l9PFS7$$!3+++++Y$pO\" FS$\"3#)*****RorCq$FS7$$!3&******pY*fE9FS$\"3%******HB/no$FS7$$!3/+++= j$e[\"FS$\"3%******\\Y'RnOFS7$$!3!******p[HZa\"FS$\"3z******e!zXk$FS7$ $!36+++r6Q.;FS$\"35+++'*RF=OFS7$$!3)******>C;>m\"FS$\"3#******>I+&)e$F S7$$!3*******z?([?FS$\"3#)*****p_#H*Q$FS7$$!3\"******4Ad9-#FS$\"3$**** **fw;.M$FS7$$!39+++BI3&3#FS$\"35+++SA(*)G$FS7$$!3))*****f%3O]@FS$\"3)* *****\\McdB$FS7$$!3<+++GXJ!G2hHFS7$$!3!)*****zs_ kc#FS$\"3')*****>(z42HFS7$$!3y*****Rgvcj#FS$\"3;++++'RO&GFS7$$!3(***** *R2-Oq#FS$\"3y*****pNN0!GFS7$$!39+++ig#*pFFS$\"3=+++mIdZFFS7$$!3A+++H$ HW$GFS$\"3?+++ZV_%p#FS7$$!3)******R>mp*GFS$\"3&******p\\m6k#FS7$$!33++ +0tWdHFS$\"3%)*****4i%H(e#FS7$$!3$)*****4LDe,$FS$\"3&)*****>(psKDFS7$$ !3#******\\a$3sIFS$\"3%******z71tZ#FS7$$!3>+++\"HFi7$FS$\"31+++Bz*3U#F S7$$!39+++IzFyJFS$\"39+++f$*QjBFS7$$!3)*******>(o#GKFS$\"3z*****HE'o/B FS7$$!3!******>E ,M=#FS7$$!3%******4]WjO$FS$\"35+++owq?@FS7$$!3(*******3Of3MFS$\"3!**** **H@%fc?FS7$$!3#)******o^1\\MFS$\"33+++#fP5*>FS7$$!3++++?W$y[$FS$\"35+ ++$yFS#>FS7$$!34+++O>)\\_$FS$\"3/+++N!pb&=FS7$$!31+++PEfgNFS$\"3!***** **o7o&y\"FS7$$!3=+++JTv%f$FS$\"3++++0=avi$FS$\"3,+ ++akyT;FS7$$!3#******HEx!fOFS$\"3%*******)p6zc\"FS7$$!3')******G-S*o$F S$\"3%******\\,uG\\\"FS7$$!3#)*****>S(e=PFS$\"3#******fS*y;9FS7$$!3!)* ****zb&oYPFS$\"3.+++(o!zR8FS7$$!37+++3&>Px$FS$\"35+++2J-i7FS7$$!3-+++= !*o*z$FS$\"3/+++D%QO=\"FS7$$!36+++xwcCQFS$\"3$******f+)y/6FS7$$!37+++` SI[QFS$\"34+++sdhD5FS7$$!3#******\\+D3(QFS$\"3G+++?\">DY*F07$$!3'***** *RnS?*QFS$\"3!)*****\\4\"3o')F07$$!3/+++*Q]=\"RFS$\"3s*****>,ZP(yF07$$ !3:+++t'[,$RFS$\"3e******)>)>!3(F07$$!3'*******42$o%RFS$\"3))*****zH-z G'F07$$!3)*******3ozhRFS$\"3k*****f%Q8(\\&F07$$!3!)*****ztb\\(RFS$\"3$ ******z&=+3ZF07$$!3?+++LpA')RFS$\"3'******4&)z/#RF07$$!37+++$zTb*RFS$ \"3?+++P\\VMJF07$$!3<+++vU%G+%FS$\"3)******p%Rl\\BF07$$!3>+++J543SFS$ \"39+++/d'ec\"F07$$!3l*****4J^7,%FS$\"3)******p^%fFyFfp7$$!3O+++qnI7SF SF(7$Fijl$!3)******p^%fFyFfp7$Fdjl$!39+++/d'ec\"F07$F_jl$!3)******p%Rl \\BF07$Fjil$!3?+++P\\VMJF07$Feil$!3'******4&)z/#RF07$F`il$!3$******z&= +3ZF07$F[il$!3k*****f%Q8(\\&F07$Ffhl$!3))*****zH-zG'F07$Fahl$!3e****** )>)>!3(F07$F\\hl$!3s*****>,ZP(yF07$Fggl$!3!)*****\\4\"3o')F07$Fbgl$!3G +++?\">DY*F07$F]gl$!34+++sdhD5FS7$Fhfl$!3$******f+)y/6FS7$Fcfl$!3/+++D %QO=\"FS7$F^fl$!35+++2J-i7FS7$Fiel$!3.+++(o!zR8FS7$Fdel$!3#******fS*y; 9FS7$F_el$!3%******\\,uG\\\"FS7$Fjdl$!3%*******)p6zc\"FS7$Fedl$!3,+++a kyT;FS7$F`dl$!3++++0FS7$F\\cl$!33+++#fP5*>FS7$Fgbl$!3!******H@%fc?FS 7$Fbbl$!35+++owq?@FS7$F]bl$!3%)*****>E,M=#FS7$Fhal$!3z*****R66ZC#FS7$F cal$!3z*****HE'o/BFS7$F^al$!39+++f$*QjBFS7$Fi`l$!31+++Bz*3U#FS7$Fd`l$! 3%******z71tZ#FS7$F_`l$!3&)*****>(psKDFS7$Fj_l$!3%)*****4i%H(e#FS7$Fe_ l$!3&******p\\m6k#FS7$F`_l$!3?+++ZV_%p#FS7$F[_l$!3=+++mIdZFFS7$Ff^l$!3 y*****pNN0!GFS7$Fa^l$!3;++++'RO&GFS7$F\\^l$!3')*****>(z42HFS7$Fg]l$!3! ******>!G2hHFS7$Fb]l$!3z******=Ej:IFS7$F]]l$!3/+++rgqqIFS7$Fh\\l$!3)** ****zr\\g7$FS7$Fc\\l$!3%*******R#[7=$FS7$F^\\l$!3)******\\McdB$FS7$Fi[ l$!35+++SA(*)G$FS7$Fd[l$!3$******fw;.M$FS7$F_[l$!3#)*****p_#H*Q$FS7$Fj z$!3:+++G:^NMFS7$Fez$!35+++-^pyMFS7$F`z$!3*)*****p@d'=NFS7$F[z$!3?+++G I+&)e$FS7$Fay$!35+++'*RF=OFS7$F\\y$!3z******e!zX k$FS7$Fgx$!3%******\\Y'RnOFS7$Fbx$!3%******HB/no$FS7$F]x$!3#)*****RorC q$FS7$Fhw$!3,+++W[l9PFS7$Fcw$!3@+++*R#>BPFS7$F^w$!3$******>o,!GPFS7$Fi v$!3*******>Du*GPFS7$Fdv$!3()******G1(fs$FS7$F_v$!3z*****Hs8)=PFS7$Fju $!3)******zR!G2PFS7$Feu$!3)*******404\"p$FS7$F`u$!31+++3E*)pOFS7$F[u$! 3y*****4RXKk$FS7$Fft$!3)********>%f5OFS7$Fat$!3/+++1FCrNFS7$F\\t$!3%** *****GGKCNFS7$Fgs$!3')*****RS]E[#FS7$Fdp$!36+++V1&eL#FS7$F_p$!35+++sJB'=# FS7$Fio$!3$******fcLV.#FS7$Fdo$!3!******4SY1)=FS7$F_o$!35+++R5kD$ !35+++y(*=$G'F07$F8$!33+++u8R7ZF07$$\"39+++a2i#y#F4$!3#******f2$fTJF07 $$\"3-+++l='f@%F-$!37+++Ujzq:F0F'-%&COLORG6&%$RGBG$\"\")!\"#$\"#NFe^nF c^n-%)POLYGONSG6fw7%F'7$$\"+j='f@%Ffp$\"+Ujzq:!#57$$Fe^nF)F(7%F\\_n7$$ \"+F4i#y#FS$\"+wIfTJFa_nFb_n7%Fe_n7$$\"+&Q4rI$!#;$\"+u8R7ZFa_nFb_n7%F[ `n7$$\"+(Ra-%>!#:$\"+y(*=$G'Fa_nFb_n7%Fb`n7$$\"+%=%)*RwFe`n$\"+^%pR&yF a_nFb_n7%Fi`n7$$\"+&3XqH#!#9$\"+MjlC%*Fa_nFb_n7%F_an7$$\"+\"e=(*f&Fban $\"+2X]*4\"!\"*Fb_n7%Ffan7$$\"+!)fwJ6!#8$\"+)4nkD\"F[bnFb_n7%F]bn7$$\" +qk=q=F`bn$\"+/FE89F[bnFb_n7%Fdbn7$$\"+pzd+BF`bn$\"+\"oM(p:F[bnFb_n7%F jbn7$$\"+>(pD5\"F`bn$\"+R5kD!#7$\"+mNLM?F[bnFb_n7%F\\dn7$$!+mQF\\en$\"+ma&fi#F[bnFb_n7%Ffen7$$!+7 'fuB'F\\en$\"+jJvkFF[bnFb_n7%F\\fn7$$!++w(3e*F\\en$\"+QB[(*GF[bnFb_n7% Fbfn7$$!+o9>#R\"Fa_n$\"+1y+AIF[bnFb_n7%Fhfn7$$!+`z$3#>Fa_n$\"++>-OJF[b nFb_n7%F^gn7$$!+wABEDFa_n$\"+QdzPKF[bnFb_n7%Fdgn7$$!+,N7%=$Fa_n$\"+0(Q nK$F[bnFb_n7%Fjgn7$$!+^8jrQFa_n$\"+]@N.MF[bnFb_n7%F`hn7$$!+\"QX;d%Fa_n $\"+uJxoMF[bnFb_n7%Ffhn7$$!+UWBt_Fa_n$\"+HGKCNF[bnFb_n7%F\\in7$$!+]r?RtFa_n$\"+\"RXKk$F[bnFb_n7%F^jn7$$!+[x\\4!)Fa_n$\"+3E*)pOF[bnFb_n7% Fdjn7$$!+A#*Hq')Fa_n$\"+504\"p$F[bnFb_n7%Fjjn7$$!+$3r>K*Fa_n$\"+)R!G2P F[bnFb_n7%F`[o7$$!+zJ)\\'**Fa_n$\"+BP\")=PF[bnFb_n7%Ff[o7$$!+7Q)*f5F[b n$\"+H1(fs$F[bnFb_n7%F\\\\o7$$!+vRqA6F[bn$\"+_U(*GPF[bnFb_n7%Fb\\o7$$! +?&3Z=\"F[bn$\"+#o,!GPF[bnFb_n7%Fh\\o7$$!+3r/Y7F[bn$\"+*R#>BPF[bnFb_n7 %F^]o7$$!+t3x18F[bn$\"+W[l9PF[bnFb_n7%Fd]o7$$!++Y$pO\"F[bn$\"+%orCq$F[ bnFb_n7%Fj]o7$$!+n%*fE9F[bn$\"+LUq'o$F[bnFb_n7%F`^o7$$!+=j$e[\"F[bn$\" +lkRnOF[bnFb_n7%Ff^o7$$!+([HZa\"F[bn$\"+f!zXk$F[bnFb_n7%F\\_o7$$!+r6Q. ;F[bn$\"+'*RF=OF[bnFb_n7%Fb_o7$$!+Ui\">m\"F[bn$\"+-.])e$F[bnFb_n7%Fh_o 7$$!+3s[?F[bn$\"+FDH*Q$F[bnFb_n7%Ffao7$$!+@sX@?F[bn$\"+mn JSLF[bnFb_n7%F\\bo7$$!+BI3&3#F[bn$\"+SA(*)G$F[bnFb_n7%Fbbo7$$!+Y3O]@F[ bn$\"+XjvNKF[bnFb_n7%Fhbo7$$!+GXJEj:IF[bnFb_n7%F`do7$$!+TgL'\\#F[bn$\"+-G2hHF[bnFb _n7%Ffdo7$$!+GFXmDF[bn$\"+sz42HF[bnFb_n7%F\\eo7$$!+/cnNEF[bn$\"++'RO&G F[bnFb_n7%Fbeo7$$!+u?g.FF[bn$\"+d``+GF[bnFb_n7%Fheo7$$!+ig#*pFF[bn$\"+ mIdZFF[bnFb_n7%F^fo7$$!+H$HW$GF[bn$\"+ZV_%p#F[bnFb_n7%Fdfo7$$!+%>mp*GF [bn$\"+(\\m6k#F[bnFb_n7%Fjfo7$$!+0tWdHF[bn$\"+@YH(e#F[bnFb_n7%F`go7$$! +J`#e,$F[bn$\"+spsKDF[bnFb_n7%Ffgo7$$!+XN3sIF[bn$\"+GhIxCF[bnFb_n7%F\\ ho7$$!+\"HFi7$F[bn$\"+Bz*3U#F[bnFb_n7%Fbho7$$!+IzFyJF[bn$\"+f$*QjBF[bn Fb_n7%Fhho7$$!+?(o#GKF[bn$\"+jio/BF[bnFb_n7%F^io7$$!+s@CwKF[bn$\"+96rW AF[bnFb_n7%Fdio7$$!+^%[AK$F[bn$\"+i7S$=#F[bnFb_n7%Fjio7$$!+,XMmLF[bn$ \"+owq?@F[bnFb_n7%F`jo7$$!+4Of3MF[bn$\"+8Ufc?F[bnFb_n7%Ffjo7$$!+p^1\\M F[bn$\"+#fP5*>F[bnFb_n7%F\\[p7$$!+?W$y[$F[bn$\"+$yFS#>F[bnFb_n7%Fb[p7$ $!+O>)\\_$F[bn$\"+N!pb&=F[bnFb_n7%Fh[p7$$!+PEfgNF[bn$\"+p7o&y\"F[bnFb_ n7%F^\\p7$$!+JTv%f$F[bn$\"+0Px$F[bn$\"+2J-i7F[bnFb_n7%Fh^ p7$$!+=!*o*z$F[bn$\"+D%QO=\"F[bnFb_n7%F^_p7$$!+xwcCQF[bn$\"+1!)y/6F[bn Fb_n7%Fd_p7$$!+`SI[QF[bn$\"+sdhD5F[bnFb_n7%Fj_p7$$!+0]#3(QF[bn$\"+?\"> DY*Fa_nFb_n7%F``p7$$!+u1/#*QF[bn$\"+&4\"3o')Fa_nFb_n7%Ff`p7$$!+*Q]=\"R F[bn$\"+7qutyFa_nFb_n7%F\\ap7$$!+t'[,$RF[bn$\"+*>)>!3(Fa_nFb_n7%Fbap7$ $!+52$o%RF[bn$\"+)H-zG'Fa_nFb_n7%Fhap7$$!+4ozhRF[bn$\"+YQ8(\\&Fa_nFb_n 7%F^bp7$$!+Qd&\\(RF[bn$\"+e=+3ZFa_nFb_n7%Fdbp7$$!+LpA')RF[bn$\"+^)z/#R Fa_nFb_n7%Fjbp7$$!+$zTb*RF[bn$\"+P\\VMJFa_nFb_n7%F`cp7$$!+vU%G+%F[bn$ \"+ZRl\\BFa_nFb_n7%Ffcp7$$!+J543SF[bn$\"+/d'ec\"Fa_nFb_n7%F\\dp7$$!+68 D6SF[bn$\"+)>!3(Fa_nFb_n7%F\\gp7$F]ap$!+7qutyFa_nFb_n7%F`g p7$Fg`p$!+&4\"3o')Fa_nFb_n7%Fdgp7$Fa`p$!+?\">DY*Fa_nFb_n7%Fhgp7$F[`p$! +sdhD5F[bnFb_n7%F\\hp7$Fe_p$!+1!)y/6F[bnFb_n7%F`hp7$F__p$!+D%QO=\"F[bn Fb_n7%Fdhp7$Fi^p$!+2J-i7F[bnFb_n7%Fhhp7$Fc^p$!+(o!zR8F[bnFb_n7%F\\ip7$ F]^p$!+1%*y;9F[bnFb_n7%F`ip7$Fg]p$!+:S(G\\\"F[bnFb_n7%Fdip7$Fa]p$!+*p6 zc\"F[bnFb_n7%Fhip7$F[]p$!+akyT;F[bnFb_n7%F\\jp7$Fe\\p$!+0F[bnFb_n7%F\\[q7$F][p$!+#fP5*>F[bnFb_n7%F`[q7$Fgjo$!+8Ufc?F [bnFb_n7%Fd[q7$Fajo$!+owq?@F[bnFb_n7%Fh[q7$F[jo$!+i7S$=#F[bnFb_n7%F\\ \\q7$Feio$!+96rWAF[bnFb_n7%F`\\q7$F_io$!+jio/BF[bnFb_n7%Fd\\q7$Fiho$!+ f$*QjBF[bnFb_n7%Fh\\q7$Fcho$!+Bz*3U#F[bnFb_n7%F\\]q7$F]ho$!+GhIxCF[bnF b_n7%F`]q7$Fggo$!+spsKDF[bnFb_n7%Fd]q7$Fago$!+@YH(e#F[bnFb_n7%Fh]q7$F[ go$!+(\\m6k#F[bnFb_n7%F\\^q7$Fefo$!+ZV_%p#F[bnFb_n7%F`^q7$F_fo$!+mIdZF F[bnFb_n7%Fd^q7$Fieo$!+d``+GF[bnFb_n7%Fh^q7$Fceo$!++'RO&GF[bnFb_n7%F\\ _q7$F]eo$!+sz42HF[bnFb_n7%F`_q7$Fgdo$!+-G2hHF[bnFb_n7%Fd_q7$Fado$!+>Ej :IF[bnFb_n7%Fh_q7$F[do$!+rgqqIF[bnFb_n7%F\\`q7$Feco$!+=(\\g7$F[bnFb_n7 %F``q7$F_co$!+S#[7=$F[bnFb_n7%Fd`q7$Fibo$!+XjvNKF[bnFb_n7%Fh`q7$Fcbo$! +SA(*)G$F[bnFb_n7%F\\aq7$F]bo$!+mnJSLF[bnFb_n7%F`aq7$Fgao$!+FDH*Q$F[bn Fb_n7%Fdaq7$Faao$!+G:^NMF[bnFb_n7%Fhaq7$F[ao$!+-^pyMF[bnFb_n7%F\\bq7$F e`o$!+BPF[bnFb_n7 %Fddq7$Fi\\o$!+#o,!GPF[bnFb_n7%Fhdq7$Fc\\o$!+_U(*GPF[bnFb_n7%F\\eq7$F] \\o$!+H1(fs$F[bnFb_n7%F`eq7$Fg[o$!+BP\")=PF[bnFb_n7%Fdeq7$Fa[o$!+)R!G2 PF[bnFb_n7%Fheq7$F[[o$!+504\"p$F[bnFb_n7%F\\fq7$Fejn$!+3E*)pOF[bnFb_n7 %F`fq7$F_jn$!+\"RXKk$F[bnFb_n7%Fdfq7$Fiin$!++Uf5OF[bnFb_n7%Fhfq7$Fcin$ !+1FCrNF[bnFb_n7%F\\gq7$F]in$!+HGKCNF[bnFb_n7%F`gq7$Fghn$!+uJxoMF[bnFb _n7%Fdgq7$Fahn$!+]@N.MF[bnFb_n7%Fhgq7$F[hn$!+0(QnK$F[bnFb_n7%F\\hq7$Fe gn$!+QdzPKF[bnFb_n7%F`hq7$F_gn$!++>-OJF[bnFb_n7%Fdhq7$Fifn$!+1y+AIF[bn Fb_n7%Fhhq7$Fcfn$!+QB[(*GF[bnFb_n7%F\\iq7$F]fn$!+jJvkFF[bnFb_n7%F`iq7$ Fgen$!+ma&fi#F[bnFb_n7%Fdiq7$Faen$!+-/l#[#F[bnFb_n7%Fhiq7$Fjdn$!+V1&eL #F[bnFb_n7%F\\jq7$Fddn$!+sJB'=#F[bnFb_n7%F`jq7$F]dn$!+mNLM?F[bnFb_n7%F djq7$Fgcn$!+,kk!)=F[bnFb_n7%Fhjq7$Facn$!+R5kDK%FS7$$\"3!****** **y$\\e9FS$\"3#******\\/r$GVFS7$$\"33+++Q&3zW\"FS$\"3r******[*)*RL%FS7 $$\"30++++0#oV\"FS$\"3.+++ezwQVFS7$$\"30+++ByFD9FS$\"3Q+++tjjUVFS7$$\" 3%******4(4L89FS$\"3A+++(zhbM%FS7$$\"3/+++'HL5S\"FS$\"3q*****p!4]ZVFS7 $$\"3'******\\qT%)Q\"FS$\"3#******\\E4%[VFS7$$\"3/+++`uhv8FS$\"3;+++W6 C[VFS7$$\"33++++rii8FS$\"3;+++0%\\pM%FS7$$\"31+++fOa\\8FS$\"3j******\\ a[WVFS7$$\"3#********)zWO8FS$\"3i******y#*zSVFS7$$\"3/+++$\\IMK\"FS$\" 3P+++8(ReL%FS7$$\"33+++uJf58FS$\"3u******))[bHVFS7$$\"3/+++N@0)H\"FS$ \"3e*****>8$*=K%FS7$$\"3'******pfSfG\"FS$\"3\"*******HW!GJ%FS7$$\"3%** ****HZ7WF\"FS$\"3d*****HzUAI%FS7$$\"31+++5lkj7FS$\"3++++F*p,H%FS7$$\"3 1+++m3&QD\"FS$\"3c*****H$3cwUFS7$$\"33+++CxEX7FS$\"3y*****p879E%FS7$$ \"3#*******[qz(*GB\"FS$\"3W+++sz mEUFS7$$\"3*)*****p\\$yH7FS$\"3,+++YvI2UFS7$$\"3&******Rr2#H7FS$\"3o** ***R3Fp=%FS7$$\"30+++9M`J7FS$\"3-+++%[3f;%FS7$$\"3#******>thqB\"FS$\"3 .+++>)zZ9%FS7$$\"3++++9>'fC\"FS$\"3U+++y*3U7%FS7$$\"31+++&o(>e7FS$\"34 +++Ch&\\5%FS7$$\"35+++A)zMF\"FS$\"3)*******y5y(3%FS7$$\"3'*******[dF\" H\"FS$\"3!******pRFL2%FS7$$\"3)******pl$)3J\"FS$\"33+++/h-iSFS7$$\"3-+ ++oQaJ8FS$\"31+++Vj0aSFS7$$\"3/+++KMa_8FS$\"33+++Q(o$\\SFS7$$\"35+++wI Gt8FS$\"3')*****\\rZx/%FS7$$\"33+++q8I$R\"FS$\"3'*******y;))[SFS7$$\"3 4+++Z#oAT\"FS$\"3G+++i1U_SFS7$$\"35+++=N'*H9FS$\"3u*****>&\\,eSFS7$$\" 3)******fx^iW\"FS$\"3C+++EoLlSFS7$$\"31+++I'f5Y\"FS$\"3i******Q.4uSFS7 $$\"33+++\"RdVZ\"FS$\"3m*****z%Q,%3%FS7$$\"31+++!HXh[\"FS$\"3=+++I'y[4 %FS7$$\"3%******p]Vk\\\"FS$\"3!******fr&[1TFS7$$\"3,+++k^G0:FS$\"3k*** ***R@m=TFS7$$\"31+++u9r7:FS$\"3*******p?d78%FS7$$\"31+++M%o(=:FS$\"38+ ++)4RT9%FS7$$\"3$******fg/N_\"FS$\"3#)*****p(=>dTFS7$$\"3$******4lpp_ \"FS$\"3s*****4%HJqTFS7$$\"3$******4Q8#H:FS$\"3)******\\&3T$=%FS7$$\"3 &******p;&GI:FS$\"3s*****Ha.k>%FS7$$\"3++++*fL-`\"FS$\"3-+++nn@4UFS7$$ \"30+++6j5H:FS$\"3v*****R&Gy@UFS7$$\"3!******f#*\\p_\"FS$\"3n*****ziRS B%FS7$$\"35+++v+\"Q_\"FS$\"3C+++/&HfC%FS7$$\"3++++\"[J(>:FS$\"3%)***** \\s)RdUFS7$$\"31+++y!eZ^\"FS$\"3=+++*p'RoUFS7$$\"3\"*******eJ$*3:FS$\" 3/+++Qa()yUFS7$$\"3/+++]&*H-:FS$\"3I+++j!*y)G%FSF\\_rF_^n-Fi^n6jn7%7$$ \"+5)**[\\\"F[bn$\"+lL4)H%F[bn7$$\"+Pkx'[\"F[bn$\"+Bau1VF[bn7$$\"$Q\"F e^n$\"$>%Fe^n7%F`bs7$$\"+5@(zZ\"F[bn$\"+]Kq9VF[bnFebs7%F[cs7$$\"+Y*H&o 9F[bn$\"+va#>K%F[bnFebs7%Facs7$$\"+!z$\\e9F[bn$\"+X5PGVF[bnFebs7%Fgcs7 $$\"+Q&3zW\"F[bn$\"+\\*)*RL%F[bnFebs7%F]ds7$$\"++0#oV\"F[bn$\"+ezwQVF[ bnFebs7%Fcds7$$\"+ByFD9F[bn$\"+tjjUVF[bnFebs7%Fids7$$\"+r4L89F[bn$\"+( zhbM%F[bnFebs7%F_es7$$\"+'HL5S\"F[bn$\"+24]ZVF[bnFebs7%Fees7$$\"+0)zZ9%F[bnFebs7%Fg\\t7$$\"+9>'fC\"F[bn$\"+ y*3U7%F[bnFebs7%F]]t7$$\"+&o(>e7F[bn$\"+Ch&\\5%F[bnFebs7%Fc]t7$$\"+A)z MF\"F[bn$\"+z5y(3%F[bnFebs7%Fi]t7$$\"+\\dF\"H\"F[bn$\"+(RFL2%F[bnFebs7 %F_^t7$$\"+dO)3J\"F[bn$\"+/h-iSF[bnFebs7%Fe^t7$$\"+oQaJ8F[bn$\"+Vj0aSF [bnFebs7%F[_t7$$\"+KMa_8F[bn$\"+Q(o$\\SF[bnFebs7%Fa_t7$$\"+wIGt8F[bn$ \"+:xuZSF[bnFebs7%Fg_t7$$\"+q8I$R\"F[bn$\"+z;))[SF[bnFebs7%F]`t7$$\"+Z #oAT\"F[bn$\"+i1U_SF[bnFebs7%Fc`t7$$\"+=N'*H9F[bn$\"+_\\,eSF[bnFebs7%F i`t7$$\"+wdTF[bnF ebs7%F_dt7$$\"+^'pp_\"F[bn$\"+THJqTF[bnFebs7%Fedt7$$\"+\"Q8#H:F[bn$\"+ b3T$=%F[bnFebs7%F[et7$$\"+n^GI:F[bn$\"+VNS'>%F[bnFebs7%Faet7$$\"+*fL-` \"F[bn$\"+nn@4UF[bnFebs7%Fget7$$\"+6j5H:F[bn$\"+aGy@UF[bnFebs7%F]ft7$$ \"+E*\\p_\"F[bn$\"+G'RSB%F[bnFebs7%Fcft7$$\"+v+\"Q_\"F[bn$\"+/&HfC%F[b nFebs7%Fift7$$\"+\"[J(>:F[bn$\"+D()RdUF[bnFebs7%F_gt7$$\"+y!eZ^\"F[bn$ \"+*p'RoUF[bnFebs7%Fegt7$$\"+fJ$*3:F[bn$\"+Qa()yUF[bnFebs7%F[ht7$$\"+] &*H-:F[bn$\"+j!*y)G%F[bnFebs7%FahtF[bsFebsF^^rFe^r-F$6$7in7$F]_r$!3s** ***\\O$4)H%FS7$Fdas$!3I+++j!*y)G%FS7$F_as$!3/+++Qa()yUFS7$Fj`s$!3=+++* p'RoUFS7$Fe`s$!3%)*****\\s)RdUFS7$F``s$!3C+++/&HfC%FS7$F[`s$!3n*****zi RSB%FS7$Ff_s$!3v*****R&Gy@UFS7$Fa_s$!3-+++nn@4UFS7$F\\_s$!3s*****Ha.k> %FS7$Fg^s$!3)******\\&3T$=%FS7$Fb^s$!3s*****4%HJqTFS7$F]^s$!3#)*****p( =>dTFS7$Fh]s$!38+++)4RT9%FS7$Fc]s$!3*******p?d78%FS7$F^]s$!3k******R@m =TFS7$Fi\\s$!3!******fr&[1TFS7$Fd\\s$!3=+++I'y[4%FS7$F_\\s$!3m*****z%Q ,%3%FS7$Fj[s$!3i******Q.4uSFS7$Fe[s$!3C+++EoLlSFS7$F`[s$!3u*****>&\\,e SFS7$F[[s$!3G+++i1U_SFS7$Ffjr$!3'*******y;))[SFS7$Fajr$!3')*****\\rZx/ %FS7$F\\jr$!33+++Q(o$\\SFS7$Fgir$!31+++Vj0aSFS7$Fbir$!33+++/h-iSFS7$F] ir$!3!******pRFL2%FS7$Fhhr$!3)*******y5y(3%FS7$Fchr$!34+++Ch&\\5%FS7$F ^hr$!3U+++y*3U7%FS7$Figr$!3.+++>)zZ9%FS7$Fdgr$!3-+++%[3f;%FS7$F_gr$!3o *****R3Fp=%FS7$Fjfr$!3,+++YvI2UFS7$Fefr$!3W+++szmEUFS7$F`fr$!3o*****\\ =aZC%FS7$F[fr$!3y*****p879E%FS7$Ffer$!3c*****H$3cwUFS7$Faer$!3++++F*p, H%FS7$F\\er$!3d*****HzUAI%FS7$Fgdr$!3\"*******HW!GJ%FS7$Fbdr$!3e*****> 8$*=K%FS7$F]dr$!3u******))[bHVFS7$Fhcr$!3P+++8(ReL%FS7$Fccr$!3i******y #*zSVFS7$F^cr$!3j******\\a[WVFS7$Fibr$!3;+++0%\\pM%FS7$Fdbr$!3;+++W6C[ VFS7$F_br$!3#******\\E4%[VFS7$Fjar$!3q*****p!4]ZVFS7$Fear$!3A+++(zhbM% FS7$F`ar$!3Q+++tjjUVFS7$F[ar$!3.+++ezwQVFS7$Ff`r$!3r******[*)*RL%FS7$F a`r$!3#******\\/r$GVFS7$F\\`r$!3d*****\\ZD>K%FS7$Fg_r$!3********\\Kq9V FS7$Fb_r$!3u*****HUXnI%FSFjhtF_^n-Fi^n6jn7%7$F\\bs$!+lL4)H%F[bn7$Fbht$ !+j!*y)G%F[bn7$Ffbs$!$>%Fe^n7%Fddu7$F\\ht$!+Qa()yUF[bnFgdu7%F[eu7$Ffgt $!+*p'RoUF[bnFgdu7%F_eu7$F`gt$!+D()RdUF[bnFgdu7%Fceu7$Fjft$!+/&HfC%F[b nFgdu7%Fgeu7$Fdft$!+G'RSB%F[bnFgdu7%F[fu7$F^ft$!+aGy@UF[bnFgdu7%F_fu7$ Fhet$!+nn@4UF[bnFgdu7%Fcfu7$Fbet$!+VNS'>%F[bnFgdu7%Fgfu7$F\\et$!+b3T$= %F[bnFgdu7%F[gu7$Ffdt$!+THJqTF[bnFgdu7%F_gu7$F`dt$!+x=>dTF[bnFgdu7%Fcg u7$Fjct$!+)4RT9%F[bnFgdu7%Fggu7$Fdct$!+2sDJTF[bnFgdu7%F[hu7$F^ct$!+S@m =TF[bnFgdu7%F_hu7$Fhbt$!+;d[1TF[bnFgdu7%Fchu7$Fbbt$!+I'y[4%F[bnFgdu7%F ghu7$F\\bt$!+[Q,%3%F[bnFgdu7%F[iu7$Ffat$!+R.4uSF[bnFgdu7%F_iu7$F`at$!+ EoLlSF[bnFgdu7%Fciu7$Fj`t$!+_\\,eSF[bnFgdu7%Fgiu7$Fd`t$!+i1U_SF[bnFgdu 7%F[ju7$F^`t$!+z;))[SF[bnFgdu7%F_ju7$Fh_t$!+:xuZSF[bnFgdu7%Fcju7$Fb_t$ !+Q(o$\\SF[bnFgdu7%Fgju7$F\\_t$!+Vj0aSF[bnFgdu7%F[[v7$Ff^t$!+/h-iSF[bn Fgdu7%F_[v7$F`^t$!+(RFL2%F[bnFgdu7%Fc[v7$Fj]t$!+z5y(3%F[bnFgdu7%Fg[v7$ Fd]t$!+Ch&\\5%F[bnFgdu7%F[\\v7$F^]t$!+y*3U7%F[bnFgdu7%F_\\v7$Fh\\t$!+> )zZ9%F[bnFgdu7%Fc\\v7$Fb\\t$!+%[3f;%F[bnFgdu7%Fg\\v7$F\\\\t$!+%3Fp=%F[ bnFgdu7%F[]v7$Ff[t$!+YvI2UF[bnFgdu7%F_]v7$F`[t$!+szmEUF[bnFgdu7%Fc]v7$ Fjjs$!+&=aZC%F[bnFgdu7%Fg]v7$Fdjs$!+P@ThUF[bnFgdu7%F[^v7$F^js$!+L3cwUF [bnFgdu7%F_^v7$Fhis$!+F*p,H%F[bnFgdu7%Fc^v7$Fbis$!+$zUAI%F[bnFgdu7%Fg^ v7$F\\is$!+IW!GJ%F[bnFgdu7%F[_v7$Ffhs$!+KJ*=K%F[bnFgdu7%F__v7$F`hs$!+* )[bHVF[bnFgdu7%Fc_v7$Fjgs$!+8(ReL%F[bnFgdu7%Fg_v7$Fdgs$!+z#*zSVF[bnFgd u7%F[`v7$F^gs$!+]a[WVF[bnFgdu7%F_`v7$Fhfs$!+0%\\pM%F[bnFgdu7%Fc`v7$Fbf s$!+W6C[VF[bnFgdu7%Fg`v7$F\\fs$!+l#4%[VF[bnFgdu7%F[av7$Ffes$!+24]ZVF[b nFgdu7%F_av7$F`es$!+(zhbM%F[bnFgdu7%Fcav7$Fjds$!+tjjUVF[bnFgdu7%Fgav7$ Fdds$!+ezwQVF[bnFgdu7%F[bv7$F^ds$!+\\*)*RL%F[bnFgdu7%F_bv7$Fhcs$!+X5PG VF[bnFgdu7%Fcbv7$Fbcs$!+va#>K%F[bnFgdu7%Fgbv7$F\\cs$!+]Kq9VF[bnFgdu7%F [cv7$Fabs$!+Bau1VF[bnFgdu7%F_cvFaduFgduF^^rFe^r-F$6%7$7$$!3/++++++!z%F SF(7$$\"3+++++++!*>FSF(-%'COLOURG6&Fb^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6% 7$7$F(Fgcv7$F($\"3/++++++!z%FSF\\dvF_dv-%+AXESLABELSG6%%&Re(z)G%&Im(z) G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-F`ev6$%*HELVETICAG\" \"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%Fe^n$\"$*>Fe^n;Fcfv$\"$z%Fe^n " 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme \+ is (approximately) " }{XPPEDIT 18 0 "[-4.0123, 0];" "6#7$,$-%&FloatG6 $\"&B,%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the s tability region for the 7 stage, order 4 scheme with that of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display ]([p||(1..7),po||(1..6)],view=[-4.79..1.99,-4.79..4.79],font=[HELVETIC A,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrai ned);" }}{PARA 13 "" 1 "" {GLPLOT2D 1003 919 919 {PLOTDATA 2 "65-%'CUR VESG6$7ew7$$\"\"!F)F(7$$\"3h******4%*y4l!#G$\"3++++Fjzq:!#=7$$\"35+++3 1vqa!#E$\"3))*****fi#fTJF07$$\"3M+++Ft*)o%)!#D$\"3()*****fQ)Q7ZF07$$\" 3U+++W(3OC'!#C$\"3!)*****plzJG'F07$$\"3&)*****\\*H=PH!#B$\"3y*****>wZR &yF07$$\"3*******p\\,Y+\"!#A$\"3r*****zcGYU*F07$$\"3/+++GVpgEFL$\"3!** ****>n/&*4\"!#<7$$\"39+++#f0Sb&FL$\"3!*******\\*zkD\"FT7$$\"3X+++U45C( )FL$\"3%*******)R3LT\"FT7$$\"3c*****\\S:!)e(FL$\"3)******\\N[)p:FT7$$! 3(******4f7[-\"!#@$\"3.+++dk(es\"FT7$$!3'*******Rr7;rFao$\"3%******R,x 5)=FT7$$!3')*****pc=-B#!#?$\"30+++[(\\].#FT7$$!3J+++))[eBaF\\p$\"3;+++ XhL(=#FT7$$!30+++lo()Q6!#>$\"3z*****H1YuL#FT7$$!39+++E#*Gb@Fgp$\"3))** ***\\uX[[#FT7$$!3!******\\fy1w$Fgp$\"31+++'*H')GEFT7$$!3-+++)HyR8'Fgp$ \"3%)*******4%\\oFFT7$$!3\"******pSt5V*Fgp$\"3.+++I/=-HFT7$$!3))****** G7Gt8F0$\"39+++!Qgx-$FT7$$!37+++\"p'Q**=F0$\"3(******RmcG9$FT7$$!3.+++ [yx.DF0$\"39++++'\\cC$FT7$$!3#******>/1=;$F0$\"3,+++!>#[NLFT7$$!3w**** *z!>+]QF0$\"3%)*****R:OGT$FT7$$!3#)*****RX'y]XF0$\"3++++-M')yMFT7$$!3Q +++QR'HD&F0$\"3$)*****po7\\`$FT7$$!3C+++MP?]fF0$\"3=+++-GD#e$FT7$$!3#* *****>f0$RmF0$\"3?+++Sr'>i$FT7$$!3O+++[#**)=tF0$\"3')*****HES\\l$FT7$$ !3e*****4^;'))zF0$\"3)*******>0)=o$FT7$$!3++++%=0'[')F0$\"33+++L8N.PFT 7$$!3U+++_uD*H*F0$\"3\"******RZ+)>PFT7$$!3>+++?Z1T**F0$\"3!******\\F%e JPFT7$$!3/+++,XXd5FT$\"3')*****fk')*QPFT7$$!3,+++q6-?6FT$\"32+++8PBUPF T7$$!3!*******)ec==\"FT$\"3()*****>*\\]TPFT7$$!3'********G5IC\"FT$\"3 \")*****4(>%pt$FT7$$!3!******RvKNI\"FT$\"3,+++mZlGPFT7$$!3/+++&exMO\"F T$\"3>+++Dus;PFT7$$!30+++;U!HU\"FT$\"3%******zMA7q$FT7$$!3#******H8\") =[\"FT$\"3#)*****fC%=#o$FT7$$!32+++X'*[S:FT$\"39+++'3W'fOFT7$$!3++++*= G))f\"FT$\"34+++0NiLOFT7$$!3/+++6r,d;FT$\"3')*****>HSTg$FT7$$!3%****** 4l._r\"FT$\"3%)*****Rn:7d$FT7$$!3!******\\_mNx\"FT$\"3\"*******yU)[`$F T7$$!3!******p')=B$=FT$\"3%)******)p2_\\$FT7$$!3-+++;uq\"*=FT$\"3!)*** **pE#H_MFT7$$!3++++iY+_>FT$\"3%******Hq5jS$FT7$$!3y*****p%)*[8?FT$\"3% ******f2DvN$FT7$$!3')*****4C:k2#FT$\"3#)*****\\'\\I1LFT7$$!3#******z*y &49#FT$\"3))*****fqIJD$FT7$$!3.+++C];2AFT$\"3=+++'Hq&)>$FT7$$!3-+++u1 \"\\F#FT$\"3z*****HfCK9$FT7$$!3++++!*[)QM#FT$\"3)******4()\\w3$FT7$$!3 $******RLEOT#FT$\"30+++k>GKIFT7$$!3A+++r\"*e$[#FT$\"3%)******[*)QxHFT7 $$!3'*******o#>Kb#FT$\"3!******>BlI#HFT7$$!3>+++$>:?i#FT$\"31+++<,EpGF T7$$!3/+++.Cc*o#FT$\"3!)*****p$G#e\"GFT7$$!3%)******[DabFFT$\"31+++_sa iFFT7$$!3))******)RE(>GFT$\"3>+++jp?4FFT7$$!3#)*****H)GFT$\"3'***** *4kybl#FT7$$!30+++NV9UHFT$\"3@+++q_X,EFT7$$!3(******f-H-+$FT$\"3%***** *4B_ma#FT7$$!3++++>B>cIFT$\"3%)*****z#*35\\#FT7$$!3*******fQO+6$FT$\"3 ')*****>%zQMCFT7$$!39+++\"=!yhJFT$\"3y*****z/tmP#FT7$$!3<+++`^X6KFT$\" 36+++&*ow#FT7$$!3!)******=#=&[LFT$\"3$******49nJ8#FT7$$!3!)******ohS!R $FT$\"3!*******p=$)o?FT7$$!3>+++/\\]IMFT$\"3\")*****RWSI+#FT7$$!3;+++! \\!*)oMFT$\"3*******4Lzd$>FT7$$!39+++g\\k0NFT$\"3%******zA\\q'=FT7$$!3 \")*****fUb3a$FT$\"35+++kl'oz\"FT7$$!3;+++0DhuNFT$\"3&******RHl_s\"FT7 $$!3!)*****>%y+2OFT$\"33+++<()H_;FT7$$!3)******RiI\"QOFT$\"3#*******G4 /y:FT7$$!3')*****47j!oOFT$\"3\"******4X(e-:FT7$$!3!)*****z\\vop$FT$\"3 !*******QX0E9FT7$$!3;+++7/iCPFT$\"3/+++_nd[8FT7$$!3')*****46G8v$FT$\"3 #******4i-.F\"FT7$$!39+++]G+xPFT$\"3-+++g&)Q\">\"FT7$$!3:+++T8i,QFT$\" 35+++>;*>6\"FT7$$!39+++RO8DQFT$\"3\"******p$>EK5FT7$$!3;+++7kYZQFT$\"3 #)*****f&*eL_*F07$$!3)******f3G&oQFT$\"3?+++H$3Ls)F07$$!3;+++k]@))QFT$ \"3L+++S\"=M#zF07$$!3:+++U\"=k!RFT$\"3f*****RV0W7(F07$$!3#)*****RPGI#R FT$\"3`*****4OhnK'F07$$!38+++**=%z$RFT$\"37+++&Qs2`&F07$$!3%)*****\\`j 5&RFT$\"3A+++Y$\\lt%F07$$!3%******z<4B'RFT$\"3%)*****HKfS%RF07$$!3'*** ***>?2;(RFT$\"3y*****4hcJ:$F07$$!3')*****z4**)yRFT$\"3)******z`5OO#F07 $$!3++++C'RT)RFT$\"3++++`\"H^d\"F07$$!3')*****Rp'H()RFT$\"3e*****4t(yt yFgp7$$!31+++q6N))RFTF(7$Fjjl$!3e*****4t(ytyFgp7$Fejl$!3++++`\"H^d\"F0 7$F`jl$!3)******z`5OO#F07$F[jl$!3y*****4hcJ:$F07$Ffil$!3%)*****HKfS%RF 07$Fail$!3A+++Y$\\lt%F07$F\\il$!37+++&Qs2`&F07$Fghl$!3`*****4OhnK'F07$ Fbhl$!3f*****RV0W7(F07$F]hl$!3L+++S\"=M#zF07$Fhgl$!3?+++H$3Ls)F07$Fcgl $!3#)*****f&*eL_*F07$F^gl$!3\"******p$>EK5FT7$Fifl$!35+++>;*>6\"FT7$Fd fl$!3-+++g&)Q\">\"FT7$F_fl$!3#******4i-.F\"FT7$Fjel$!3/+++_nd[8FT7$Fee l$!3!*******QX0E9FT7$F`el$!3\"******4X(e-:FT7$F[el$!3#*******G4/y:FT7$ Ffdl$!33+++<()H_;FT7$Fadl$!3&******RHl_s\"FT7$F\\dl$!35+++kl'oz\"FT7$F gcl$!3%******zA\\q'=FT7$Fbcl$!3*******4Lzd$>FT7$F]cl$!3\")*****RWSI+#F T7$Fhbl$!3!*******p=$)o?FT7$Fcbl$!3$******49nJ8#FT7$F^bl$!35+++yH2'>#F T7$Fial$!37+++5))edAFT7$Fdal$!36+++&*ow%zQMCFT7$Fe`l$!3%)*****z#*35\\#FT7$F``l$!3%******4B_ma#FT 7$F[`l$!3@+++q_X,EFT7$Ff_l$!3'******4kybl#FT7$Fa_l$!3>+++jp?4FFT7$F\\_ l$!31+++_saiFFT7$Fg^l$!3!)*****p$G#e\"GFT7$Fb^l$!31+++<,EpGFT7$F]^l$!3 !******>BlI#HFT7$Fh]l$!3%)******[*)QxHFT7$Fc]l$!30+++k>GKIFT7$F^]l$!3) ******4()\\w3$FT7$Fi\\l$!3z*****HfCK9$FT7$Fd\\l$!3=+++'Hq&)>$FT7$F_\\l $!3))*****fqIJD$FT7$Fj[l$!3#)*****\\'\\I1LFT7$Fe[l$!3%******f2DvN$FT7$ F`[l$!3%******Hq5jS$FT7$F[[l$!3!)*****pE#H_MFT7$Ffz$!3%)******)p2_\\$F T7$Faz$!3\"*******yU)[`$FT7$F\\z$!3%)*****Rn:7d$FT7$Fgy$!3')*****>HSTg $FT7$Fby$!34+++0NiLOFT7$F]y$!39+++'3W'fOFT7$Fhx$!3#)*****fC%=#o$FT7$Fc x$!3%******zMA7q$FT7$F^x$!3>+++Dus;PFT7$Fiw$!3,+++mZlGPFT7$Fdw$!3\")** ***4(>%pt$FT7$F_w$!3()*****>*\\]TPFT7$Fjv$!32+++8PBUPFT7$Fev$!3')***** fk')*QPFT7$F`v$!3!******\\F%eJPFT7$F[v$!3\"******RZ+)>PFT7$Ffu$!33+++L 8N.PFT7$Fau$!3)*******>0)=o$FT7$F\\u$!3')*****HES\\l$FT7$Fgt$!3?+++Sr' >i$FT7$Fbt$!3=+++-GD#e$FT7$F]t$!3$)*****po7\\`$FT7$Fhs$!3++++-M')yMFT7 $Fcs$!3%)*****R:OGT$FT7$F^s$!3,+++!>#[NLFT7$Fir$!39++++'\\cC$FT7$Fdr$! 3(******RmcG9$FT7$F_r$!39+++!Qgx-$FT7$Fjq$!3.+++I/=-HFT7$Feq$!3%)***** **4%\\oFFT7$F`q$!31+++'*H')GEFT7$F[q$!3))*****\\uX[[#FT7$Fep$!3z*****H 1YuL#FT7$F`p$!3;+++XhL(=#FT7$Fjo$!30+++[(\\].#FT7$Feo$!3%******R,x5)=F T7$F_o$!3.+++dk(es\"FT7$$\"3g+++/a,)e(FL$!3)******\\N[)p:FT7$Fen$!3%** *****)R3LT\"FT7$FV$!3!*******\\*zkD\"FT7$FP$!3!******>n/&*4\"FT7$FJ$!3 r*****zcGYU*F07$$\"3<+++'*H=PHFF$!3y*****>wZR&yF07$F>$!3!)*****plzJG'F 07$F8$!3()*****fQ)Q7ZF07$$\"3i*****R$*\\2Z&F4$!3))*****fi#fTJF07$$\"3S *****fP*y4lF-$!3++++Fjzq:F0F'-%&COLORG6&%$RGBG$\"#:!\"#$\"#[Fh^nFf^n-% )POLYGONSG6fw7%F'7$$\"+5%*y4lF\\p$\"+Fjzq:!#57$$Fh^nF)F(7%F__n7$$\"+31 vqaF0$\"+EEfTJFd_nFe_n7%Fh_n7$$\"+Ft*)o%)FT$\"+'Q)Q7ZFd_nFe_n7%F^`n7$$ \"+W(3OC'!#;$\"+d'zJG'Fd_nFe_n7%Fd`n7$$\"+&*H=PH!#:$\"+ix%R&yFd_nFe_n7 %F[an7$$\"+(\\,Y+\"!#9$\"+o&GYU*Fd_nFe_n7%Fban7$$\"+GVpgEFean$\"+sY]*4 \"!\"*Fe_n7%Fian7$$\"+#f0Sb&Fean$\"+]*zkD\"F^bnFe_n7%F`bn7$$\"+U45C()F ean$\"+*R3LT\"F^bnFe_n7%Ffbn7$$\"+0a,)e(Fean$\"+b$[)p:F^bnFe_n7%F\\cn7 $$!+\"f7[-\"!#8$\"+dk(es\"F^bnFe_n7%Fbcn7$$!+Sr7;rFecn$\"+9q2\")=F^bnF e_n7%Ficn7$$!+n&=-B#!#7$\"+[(\\].#F^bnFe_n7%F_dn7$$!+))[eBaFbdn$\"+XhL (=#F^bnFe_n7%Ffdn7$$!+lo()Q6!#6$\"+jgWPBF^bnFe_n7%F\\en7$$!+E#*Gb@F_en $\"+Xd%[[#F^bnFe_n7%Fcen7$$!+&fy1w$F_en$\"+'*H')GEF^bnFe_n7%Fien7$$!+) HyR8'F_en$\"++T\\oFF^bnFe_n7%F_fn7$$!+2M2J%*F_en$\"+I/=-HF^bnFe_n7%Fef n7$$!+H7Gt8Fd_n$\"+!Qgx-$F^bnFe_n7%F[gn7$$!+\"p'Q**=Fd_n$\"+km&G9$F^bn Fe_n7%Fagn7$$!+[yx.DFd_n$\"++'\\cC$F^bnFe_n7%Fggn7$$!+Ug!=;$Fd_n$\"+!> #[NLF^bnFe_n7%F]hn7$$!+3>+]QFd_n$\"+ah$GT$F^bnFe_n7%Fchn7$$!+aky]XFd_n $\"+-M')yMF^bnFe_n7%Fihn7$$!+QR'HD&Fd_n$\"+(o7\\`$F^bnFe_n7%F_in7$$!+M P?]fFd_n$\"+-GD#e$F^bnFe_n7%Fein7$$!+#f0$RmFd_n$\"+Sr'>i$F^bnFe_n7%F[j n7$$!+[#**)=tFd_n$\"+j-%\\l$F^bnFe_n7%Fajn7$$!+6lh))zFd_n$\"+?0)=o$F^b nFe_n7%Fgjn7$$!+%=0'[')Fd_n$\"+L8N.PF^bnFe_n7%F][o7$$!+_uD*H*Fd_n$\"+u /!)>PF^bnFe_n7%Fc[o7$$!+?Z1T**Fd_n$\"+vUeJPF^bnFe_n7%Fi[o7$$!+,XXd5F^b n$\"+Ym)*QPF^bnFe_n7%F_\\o7$$!+q6-?6F^bn$\"+8PBUPF^bnFe_n7%Fe\\o7$$!+* ec==\"F^bn$\"+#*\\]TPF^bnFe_n7%F[]o7$$!+!H5IC\"F^bn$\"+r>%pt$F^bnFe_n7 %Fa]o7$$!+aF`.8F^bn$\"+mZlGPF^bnFe_n7%Fg]o7$$!+&exMO\"F^bn$\"+Dus;PF^b nFe_n7%F]^o7$$!+;U!HU\"F^bn$\"+[BA,PF^bnFe_n7%Fc^o7$$!+L6)=[\"F^bn$\"+ YU=#o$F^bnFe_n7%Fi^o7$$!+X'*[S:F^bn$\"+'3W'fOF^bnFe_n7%F__o7$$!+*=G))f \"F^bn$\"+0NiLOF^bnFe_n7%Fe_o7$$!+6r,d;F^bn$\"+#HSTg$F^bnFe_n7%F[`o7$$ !+^O?:F^bn$\"+.2J1MF^bnFe_n7%Fiao7$$!+Z)*[8?F^bn$\"+w]_ dLF^bnFe_n7%F_bo7$$!+T_Tw?F^bn$\"+l\\I1LF^bnFe_n7%Febo7$$!+)*y&49#F^bn $\"+128`KF^bnFe_n7%F[co7$$!+C];2AF^bn$\"+'Hq&)>$F^bnFe_n7%Faco7$$!+u1 \"\\F#F^bn$\"+$fCK9$F^bnFe_n7%Fgco7$$!+!*[)QM#F^bn$\"+r)\\w3$F^bnFe_n7 %F]do7$$!+Mji8CF^bn$\"+k>GKIF^bnFe_n7%Fcdo7$$!+r\"*e$[#F^bn$\"+\\*)QxH F^bnFe_n7%Fido7$$!+p#>Kb#F^bn$\"+K_1BHF^bnFe_n7%F_eo7$$!+$>:?i#F^bn$\" +<,EpGF^bnFe_n7%Feeo7$$!+.Cc*o#F^bn$\"+PG#e\"GF^bnFe_n7%F[fo7$$!+\\Dab FF^bn$\"+_saiFF^bnFe_n7%Fafo7$$!+*RE(>GF^bn$\"+jp?4FF^bnFe_n7%Fgfo7$$! +t\"f>)GF^bn$\"+T'ybl#F^bnFe_n7%F]go7$$!+NV9UHF^bn$\"+q_X,EF^bnFe_n7%F cgo7$$!+E!H-+$F^bn$\"+JAlYDF^bnFe_n7%Figo7$$!+>B>cIF^bn$\"+G*35\\#F^bn Fe_n7%F_ho7$$!+'QO+6$F^bn$\"+UzQMCF^bnFe_n7%Feho7$$!+\"=!yhJF^bn$\"+[I nwBF^bnFe_n7%F[io7$$!+`^X6KF^bn$\"+&*ow#F^bnFe_n7%F]jo7$$!+>#=& [LF^bn$\"+Tr;L@F^bnFe_n7%Fcjo7$$!+phS!R$F^bn$\"+q=$)o?F^bnFe_n7%Fijo7$ $!+/\\]IMF^bn$\"+W//.?F^bnFe_n7%F_[p7$$!+!\\!*)oMF^bn$\"+J$zd$>F^bnFe_ n7%Fe[p7$$!+g\\k0NF^bn$\"+G#\\q'=F^bnFe_n7%F[\\p7$$!+Ea&3a$F^bn$\"+kl' oz\"F^bnFe_n7%Fa\\p7$$!+0DhuNF^bn$\"+%Hl_s\"F^bnFe_n7%Fg\\p7$$!+Uy+2OF ^bn$\"+<()H_;F^bnFe_n7%F]]p7$$!+C18QOF^bn$\"+H4/y:F^bnFe_n7%Fc]p7$$!+@ J1oOF^bn$\"+^ue-:F^bnFe_n7%Fi]p7$$!+)\\vop$F^bn$\"+RX0E9F^bnFe_n7%F_^p 7$$!+7/iCPF^bn$\"+_nd[8F^bnFe_n7%Fe^p7$$!+6\"G8v$F^bn$\"+@EIq7F^bnFe_n 7%F[_p7$$!+]G+xPF^bn$\"+g&)Q\">\"F^bnFe_n7%Fa_p7$$!+T8i,QF^bn$\"+>;*>6 \"F^bnFe_n7%Fg_p7$$!+RO8DQF^bn$\"+P>EK5F^bnFe_n7%F]`p7$$!+7kYZQF^bn$\" +c*eL_*Fd_nFe_n7%Fc`p7$$!+'3G&oQF^bn$\"+H$3Ls)Fd_nFe_n7%Fi`p7$$!+k]@)) QF^bn$\"+S\"=M#zFd_nFe_n7%F_ap7$$!+U\"=k!RF^bn$\"+MaSCrFd_nFe_n7%Feap7 $$!+u$GI#RF^bn$\"+h8wEjFd_nFe_n7%F[bp7$$!+**=%z$RF^bn$\"+&Qs2`&Fd_nFe_ n7%Fabp7$$!+NN1^RF^bn$\"+Y$\\lt%Fd_nFe_n7%Fgbp7$$!+y\"4B'RF^bn$\"+B$fS %RFd_nFe_n7%F]cp7$$!+-sgrRF^bn$\"+6m:`JFd_nFe_n7%Fccp7$$!+)4**)yRF^bn$ \"+Q0hjBFd_nFe_n7%Ficp7$$!+C'RT)RF^bn$\"+`\"H^d\"Fd_nFe_n7%F_dp7$$!+%p 'H()RF^bn$\"+JxytyF_enFe_n7%Fedp7$$!+q6N))RF^bnF(Fe_n7%F[ep7$Ffdp$!+Jx ytyF_enFe_n7%F_ep7$F`dp$!+`\"H^d\"Fd_nFe_n7%Fcep7$Fjcp$!+Q0hjBFd_nFe_n 7%Fgep7$Fdcp$!+6m:`JFd_nFe_n7%F[fp7$F^cp$!+B$fS%RFd_nFe_n7%F_fp7$Fhbp$ !+Y$\\lt%Fd_nFe_n7%Fcfp7$Fbbp$!+&Qs2`&Fd_nFe_n7%Fgfp7$F\\bp$!+h8wEjFd_ nFe_n7%F[gp7$Ffap$!+MaSCrFd_nFe_n7%F_gp7$F`ap$!+S\"=M#zFd_nFe_n7%Fcgp7 $Fj`p$!+H$3Ls)Fd_nFe_n7%Fggp7$Fd`p$!+c*eL_*Fd_nFe_n7%F[hp7$F^`p$!+P>EK 5F^bnFe_n7%F_hp7$Fh_p$!+>;*>6\"F^bnFe_n7%Fchp7$Fb_p$!+g&)Q\">\"F^bnFe_ n7%Fghp7$F\\_p$!+@EIq7F^bnFe_n7%F[ip7$Ff^p$!+_nd[8F^bnFe_n7%F_ip7$F`^p $!+RX0E9F^bnFe_n7%Fcip7$Fj]p$!+^ue-:F^bnFe_n7%Fgip7$Fd]p$!+H4/y:F^bnFe _n7%F[jp7$F^]p$!+<()H_;F^bnFe_n7%F_jp7$Fh\\p$!+%Hl_s\"F^bnFe_n7%Fcjp7$ Fb\\p$!+kl'oz\"F^bnFe_n7%Fgjp7$F\\\\p$!+G#\\q'=F^bnFe_n7%F[[q7$Ff[p$!+ J$zd$>F^bnFe_n7%F_[q7$F`[p$!+W//.?F^bnFe_n7%Fc[q7$Fjjo$!+q=$)o?F^bnFe_ n7%Fg[q7$Fdjo$!+Tr;L@F^bnFe_n7%F[\\q7$F^jo$!+yH2'>#F^bnFe_n7%F_\\q7$Fh io$!+5))edAF^bnFe_n7%Fc\\q7$Fbio$!+&*owGKIF^bnF e_n7%F[`q7$F^do$!+r)\\w3$F^bnFe_n7%F_`q7$Fhco$!+$fCK9$F^bnFe_n7%Fc`q7$ Fbco$!+'Hq&)>$F^bnFe_n7%Fg`q7$F\\co$!+128`KF^bnFe_n7%F[aq7$Ffbo$!+l\\I 1LF^bnFe_n7%F_aq7$F`bo$!+w]_dLF^bnFe_n7%Fcaq7$Fjao$!+.2J1MF^bnFe_n7%Fg aq7$Fdao$!+nAH_MF^bnFe_n7%F[bq7$F^ao$!+*p2_\\$F^bnFe_n7%F_bq7$Fh`o$!+z U)[`$F^bnFe_n7%Fcbq7$Fb`o$!+uc@rNF^bnFe_n7%Fgbq7$F\\`o$!+#HSTg$F^bnFe_ n7%F[cq7$Ff_o$!+0NiLOF^bnFe_n7%F_cq7$F`_o$!+'3W'fOF^bnFe_n7%Fccq7$Fj^o $!+YU=#o$F^bnFe_n7%Fgcq7$Fd^o$!+[BA,PF^bnFe_n7%F[dq7$F^^o$!+Dus;PF^bnF e_n7%F_dq7$Fh]o$!+mZlGPF^bnFe_n7%Fcdq7$Fb]o$!+r>%pt$F^bnFe_n7%Fgdq7$F \\]o$!+#*\\]TPF^bnFe_n7%F[eq7$Ff\\o$!+8PBUPF^bnFe_n7%F_eq7$F`\\o$!+Ym) *QPF^bnFe_n7%Fceq7$Fj[o$!+vUeJPF^bnFe_n7%Fgeq7$Fd[o$!+u/!)>PF^bnFe_n7% F[fq7$F^[o$!+L8N.PF^bnFe_n7%F_fq7$Fhjn$!+?0)=o$F^bnFe_n7%Fcfq7$Fbjn$!+ j-%\\l$F^bnFe_n7%Fgfq7$F\\jn$!+Sr'>i$F^bnFe_n7%F[gq7$Ffin$!+-GD#e$F^bn Fe_n7%F_gq7$F`in$!+(o7\\`$F^bnFe_n7%Fcgq7$Fjhn$!+-M')yMF^bnFe_n7%Fggq7 $Fdhn$!+ah$GT$F^bnFe_n7%F[hq7$F^hn$!+!>#[NLF^bnFe_n7%F_hq7$Fhgn$!++'\\ cC$F^bnFe_n7%Fchq7$Fbgn$!+km&G9$F^bnFe_n7%Fghq7$F\\gn$!+!Qgx-$F^bnFe_n 7%F[iq7$Fffn$!+I/=-HF^bnFe_n7%F_iq7$F`fn$!++T\\oFF^bnFe_n7%Fciq7$Fjen$ !+'*H')GEF^bnFe_n7%Fgiq7$Fden$!+Xd%[[#F^bnFe_n7%F[jq7$F]en$!+jgWPBF^bn Fe_n7%F_jq7$Fgdn$!+XhL(=#F^bnFe_n7%Fcjq7$F`dn$!+[(\\].#F^bnFe_n7%Fgjq7 $Fjcn$!+9q2\")=F^bnFe_n7%F[[r7$Fccn$!+dk(es\"F^bnFe_n7%F_[r7$$\"+/a,)e (Fean$!+b$[)p:F^bnFe_n7%Fc[r7$Fgbn$!+*R3LT\"F^bnFe_n7%Fi[r7$Fabn$!+]*z kD\"F^bnFe_n7%F]\\r7$Fjan$!+sY]*4\"F^bnFe_n7%Fa\\r7$Fcan$!+o&GYU*Fd_nF e_n7%Fe\\r7$$\"+'*H=PHF^an$!+ix%R&yFd_nFe_n7%Fi\\r7$Fe`n$!+d'zJG'Fd_nF e_n7%F_]r7$F_`n$!+'Q)Q7ZFd_nFe_n7%Fc]r7$$\"+M*\\2Z&F0$!+EEfTJFd_nFe_n7 %Fg]r7$$\"+w$*y4lF\\p$!+Fjzq:Fd_nFe_n7%F]^rF'Fe_n-Fc^n6&Fe^n$\"\"$!\" \"$\"#&*Fh^nFe^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"35+++\"oG)*[\"FT $\"3w*****H>]MI%FT7$$\"31+++L,s\"[\"FT$\"3A+++v;>7VFT7$$\"3'******pLDH Z\"FT$\"3<+++$))R-K%FT7$$\"33+++Ut[j9FT$\"3z*****pU`vK%FT7$$\"3)****** *z)\\MX\"FT$\"3d*****>B\"4MVFT7$$\"3(******fodGW\"FT$\"3%******\\E7)RV FT7$$\"3*******4'ovJ9FT$\"3t******e_nWVFT7$$\"3#******\\G&>?9FT$\"37++ +r%Q'[VFT7$$\"31++++JA39FT$\"3y******H%f;N%FT7$$\"3-+++oK*eR\"FT$\"35+ ++@Zp`VFT7$$\"33+++KAE$Q\"FT$\"3q*****zw*paVFT7$$\"31+++/2Rq8FT$\"3/++ +c'GYN%FT7$$\"3)******\\gWtN\"FT$\"3M+++$*RV`VFT7$$\"3++++5i>W8FT$\"3/ +++Mo1^VFT7$$\"35+++Bb-J8FT$\"3A+++2nZZVFT7$$\"36+++m>#zJ\"FT$\"35+++4 =hUVFT7$$\"3++++Fk)\\I\"FT$\"3H+++I%>kL%FT7$$\"31+++ePL#H\"FT$\"3&)*** **fwY)GVFT7$$\"31+++)z&4!G\"FT$\"3y*****HHU)>VFT7$$\"35+++y\\Uo7FT$\"3 T+++5!e$4VFT7$$\"3%******Ra)\\d7FT$\"3++++WHN(H%FT7$$\"3#*******fK_Z7F T$\"39+++g&)z$G%FT7$$\"3*)*****HFS(Q7FT$\"3?+++koooUFT7$$\"3!******zHH 9B\"FT$\"3K+++zA/_UFT7$$\"3)******fY5fA\"FT$\"3U+++K%QRB%FT7$$\"30+++# *3aA7FT$\"3-+++%o?X@%FT7$$\"35+++38q@7FT$\"3')*****\\%Q.%>%FT7$$\"31++ +qmwB7FT$\"3G+++>8&G<%FT7$$\"3$******\\Tb!H7FT$\"3k*****\\D'\\^TFT7$$ \"35+++B\"exB\"FT$\"3')*****HpQ18%FT7$$\"3#******4&*f)\\7FT$\"3#)***** H*306TFT7$$\"3!******RB(3l7FT$\"3!)*****Rl9N4%FT7$$\"3!******H\"=\"HG \"FT$\"3S+++I5qySFT7$$\"32+++sQi-8FT$\"3-+++.q1nSFT7$$\"3-+++xnWB8FT$ \"3e******4,\")eSFT7$$\"30+++ZukW8FT$\"3!******Rj*)Q0%FT7$$\"35+++r\"3 cO\"FT$\"3%******zP!4_SFT7$$\"3/+++UK&eQ\"FT$\"3;+++0f4`SFT7$$\"3.+++j G/09FT$\"3A+++t!\\l0%FT7$$\"3(*******G4&HU\"FT$\"3K+++=E4iSFT7$$\"36++ +())Q%R9FT$\"3e*****H\">RpSFT7$$\"31+++R?Va9FT$\"37+++h^9ySFT7$$\"3#** *****30!zY\"FT$\"3)******4'f3)3%FT7$$\"3#******>)\\%)z9FT$\"3u*****4#= )*)4%FT7$$\"3)******4U'G!\\\"FT$\"3g*****p#3j5TFT7$$\"33+++o)e#*\\\"FT $\"3<+++Cx&G7%FT7$$\"33+++\"[/o]\"FT$\"3I+++d+^NTFT7$$\"3*)*****f;qH^ \"FT$\"3%)*****4ka%[TFT7$$\"31+++I`!y^\"FT$\"3e*******[u:;%FT7$$\"31++ +#Rg8_\"FT$\"3U+++)=mZ<%FT7$$\"3/+++TeoB:FT$\"3g*****RnPz=%FT7$$\"3*** ****zjJ[_\"FT$\"3V+++'Q15?%FT7$$\"3)*******))o%[_\"FT$\"3N+++5x*Q@%FT7 $$\"3*)*****pmzP_\"FT$\"3Q+++(oVlA%FT7$$\"3'******z'pn@:FT$\"3h*****p \">))QUFT7$$\"3\"******\\t%e=:FT$\"3k*****fka3D%FT7$$\"3)******f%za9:F T$\"3m*****z)zSiUFT7$$\"34+++N2h4:FT$\"3W+++`7\\tUFT7$$\"3*)*****4b;Q] \"FT$\"3K+++%QcSG%FT7$$\"31+++F$3s\\\"FT$\"3'******\\WdSH%FTFa_rFb^n-F \\_n6jn7%7$$\"+\"oG)*[\"F^bn$\"+$>]MI%F^bn7$$\"+L,s\"[\"F^bn$\"+v;>7VF ^bn7$$\"$P\"Fh^n$\"$>%Fh^n7%Febs7$$\"+P`#HZ\"F^bn$\"+$))R-K%F^bnFjbs7% F`cs7$$\"+Ut[j9F^bn$\"+FMbFVF^bnFjbs7%Ffcs7$$\"+!))\\MX\"F^bn$\"+K74MV F^bnFjbs7%F\\ds7$$\"+'odGW\"F^bn$\"+lA\")RVF^bnFjbs7%Fbds7$$\"+hovJ9F^ bn$\"+f_nWVF^bnFjbs7%Fhds7$$\"+&G&>?9F^bn$\"+r%Q'[VF^bnFjbs7%F^es7$$\" ++JA39F^bn$\"+I%f;N%F^bnFjbs7%Fdes7$$\"+oK*eR\"F^bn$\"+@Zp`VF^bnFjbs7% Fjes7$$\"+KAE$Q\"F^bn$\"+o(*paVF^bnFjbs7%F`fs7$$\"+/2Rq8F^bn$\"+c'GYN% F^bnFjbs7%Fffs7$$\"+0YMd8F^bn$\"+$*RV`VF^bnFjbs7%F\\gs7$$\"+5i>W8F^bn$ \"+Mo1^VF^bnFjbs7%Fbgs7$$\"+Bb-J8F^bn$\"+2nZZVF^bnFjbs7%Fhgs7$$\"+m>#z J\"F^bn$\"+4=hUVF^bnFjbs7%F^hs7$$\"+Fk)\\I\"F^bn$\"+I%>kL%F^bnFjbs7%Fd hs7$$\"+ePL#H\"F^bn$\"+mn%)GVF^bnFjbs7%Fjhs7$$\"+)z&4!G\"F^bn$\"+$HU)> VF^bnFjbs7%F`is7$$\"+y\\Uo7F^bn$\"+5!e$4VF^bnFjbs7%Ffis7$$\"+W&)\\d7F^ bn$\"+WHN(H%F^bnFjbs7%F\\js7$$\"+gK_Z7F^bn$\"+g&)z$G%F^bnFjbs7%Fbjs7$$ \"+t-uQ7F^bn$\"+koooUF^bnFjbs7%Fhjs7$$\"+)HH9B\"F^bn$\"+zA/_UF^bnFjbs7 %F^[t7$$\"+m/\"fA\"F^bn$\"+K%QRB%F^bnFjbs7%Fd[t7$$\"+#*3aA7F^bn$\"+%o? X@%F^bnFjbs7%Fj[t7$$\"+38q@7F^bn$\"+XQ.%>%F^bnFjbs7%F`\\t7$$\"+qmwB7F^ bn$\"+>8&G<%F^bnFjbs7%Ff\\t7$$\"+:a0H7F^bn$\"+bi\\^TF^bnFjbs7%F\\]t7$$ \"+B\"exB\"F^bn$\"+$pQ18%F^bnFjbs7%Fb]t7$$\"+^*f)\\7F^bn$\"+$*306TF^bn Fjbs7%Fh]t7$$\"+Ms3l7F^bn$\"+aY^$4%F^bnFjbs7%F^^t7$$\"+8=\"HG\"F^bn$\" +I5qySF^bnFjbs7%Fd^t7$$\"+sQi-8F^bn$\"+.q1nSF^bnFjbs7%Fj^t7$$\"+xnWB8F ^bn$\"+5,\")eSF^bnFjbs7%F`_t7$$\"+ZukW8F^bn$\"+M'*)Q0%F^bnFjbs7%Ff_t7$ $\"+r\"3cO\"F^bn$\"+y.4_SF^bnFjbs7%F\\`t7$$\"+UK&eQ\"F^bn$\"+0f4`SF^bn Fjbs7%Fb`t7$$\"+jG/09F^bn$\"+t!\\l0%F^bnFjbs7%Fh`t7$$\"+H4&HU\"F^bn$\" +=E4iSF^bnFjbs7%F^at7$$\"+())Q%R9F^bn$\"+8>RpSF^bnFjbs7%Fdat7$$\"+R?Va 9F^bn$\"+h^9ySF^bnFjbs7%Fjat7$$\"+40!zY\"F^bn$\"+hf3)3%F^bnFjbs7%F`bt7 $$\"+#)\\%)z9F^bn$\"+@=)*)4%F^bnFjbs7%Ffbt7$$\"+@kG!\\\"F^bn$\"+F3j5TF ^bnFjbs7%F\\ct7$$\"+o)e#*\\\"F^bn$\"+Cx&G7%F^bnFjbs7%Fbct7$$\"+\"[/o] \"F^bn$\"+d+^NTF^bnFjbs7%Fhct7$$\"+m,(H^\"F^bn$\"+TYX[TF^bnFjbs7%F^dt7 $$\"+I`!y^\"F^bn$\"+!\\u:;%F^bnFjbs7%Fddt7$$\"+#Rg8_\"F^bn$\"+)=mZ<%F^ bnFjbs7%Fjdt7$$\"+TeoB:F^bn$\"+uw$z=%F^bnFjbs7%F`et7$$\"+Q;$[_\"F^bn$ \"+'Q15?%F^bnFjbs7%Ffet7$$\"+*)o%[_\"F^bn$\"+5x*Q@%F^bnFjbs7%F\\ft7$$ \"+n'zP_\"F^bn$\"+(oVlA%F^bnFjbs7%Fbft7$$\"+opn@:F^bn$\"+<>))QUF^bnFjb s7%Fhft7$$\"+NZe=:F^bn$\"+YY&3D%F^bnFjbs7%F^gt7$$\"+Yza9:F^bn$\"+))zSi UF^bnFjbs7%Fdgt7$$\"+N2h4:F^bn$\"+`7\\tUF^bnFjbs7%Fjgt7$$\"+^l\"Q]\"F^ bn$\"+%QcSG%F^bnFjbs7%F`ht7$$\"+F$3s\\\"F^bn$\"+Xu0%H%F^bnFjbs7%FfhtF` bsFjbsFc^rFj^r-F$6$7in7$Fb_r$!3w*****H>]MI%FT7$Fias$!3'******\\WdSH%FT 7$Fdas$!3K+++%QcSG%FT7$F_as$!3W+++`7\\tUFT7$Fj`s$!3m*****z)zSiUFT7$Fe` s$!3k*****fka3D%FT7$F``s$!3h*****p\">))QUFT7$F[`s$!3Q+++(oVlA%FT7$Ff_s $!3N+++5x*Q@%FT7$Fa_s$!3V+++'Q15?%FT7$F\\_s$!3g*****RnPz=%FT7$Fg^s$!3U +++)=mZ<%FT7$Fb^s$!3e*******[u:;%FT7$F]^s$!3%)*****4ka%[TFT7$Fh]s$!3I+ ++d+^NTFT7$Fc]s$!3<+++Cx&G7%FT7$F^]s$!3g*****p#3j5TFT7$Fi\\s$!3u*****4 #=)*)4%FT7$Fd\\s$!3)******4'f3)3%FT7$F_\\s$!37+++h^9ySFT7$Fj[s$!3e**** *H\">RpSFT7$Fe[s$!3K+++=E4iSFT7$F`[s$!3A+++t!\\l0%FT7$F[[s$!3;+++0f4`S FT7$Ffjr$!3%******zP!4_SFT7$Fajr$!3!******Rj*)Q0%FT7$F\\jr$!3e******4, \")eSFT7$Fgir$!3-+++.q1nSFT7$Fbir$!3S+++I5qySFT7$F]ir$!3!)*****Rl9N4%F T7$Fhhr$!3#)*****H*306TFT7$Fchr$!3')*****HpQ18%FT7$F^hr$!3k*****\\D'\\ ^TFT7$Figr$!3G+++>8&G<%FT7$Fdgr$!3')*****\\%Q.%>%FT7$F_gr$!3-+++%o?X@% FT7$Fjfr$!3U+++K%QRB%FT7$Fefr$!3K+++zA/_UFT7$F`fr$!3?+++koooUFT7$F[fr$ !39+++g&)z$G%FT7$Ffer$!3++++WHN(H%FT7$Faer$!3T+++5!e$4VFT7$F\\er$!3y** ***HHU)>VFT7$Fgdr$!3&)*****fwY)GVFT7$Fbdr$!3H+++I%>kL%FT7$F]dr$!35+++4 =hUVFT7$Fhcr$!3A+++2nZZVFT7$Fccr$!3/+++Mo1^VFT7$F^cr$!3M+++$*RV`VFT7$F ibr$!3/+++c'GYN%FT7$Fdbr$!3q*****zw*paVFT7$F_br$!35+++@Zp`VFT7$Fjar$!3 y******H%f;N%FT7$Fear$!37+++r%Q'[VFT7$F`ar$!3t******e_nWVFT7$F[ar$!3%* *****\\E7)RVFT7$Ff`r$!3d*****>B\"4MVFT7$Fa`r$!3z*****pU`vK%FT7$F\\`r$! 3<+++$))R-K%FT7$Fg_r$!3A+++v;>7VFTF_itFb^n-F\\_n6jn7%7$Fabs$!+$>]MI%F^ bn7$Fght$!+Xu0%H%F^bn7$F[cs$!$>%Fh^n7%Fidu7$Faht$!+%QcSG%F^bnF\\eu7%F` eu7$F[ht$!+`7\\tUF^bnF\\eu7%Fdeu7$Fegt$!+))zSiUF^bnF\\eu7%Fheu7$F_gt$! +YY&3D%F^bnF\\eu7%F\\fu7$Fift$!+<>))QUF^bnF\\eu7%F`fu7$Fcft$!+(oVlA%F^ bnF\\eu7%Fdfu7$F]ft$!+5x*Q@%F^bnF\\eu7%Fhfu7$Fget$!+'Q15?%F^bnF\\eu7%F \\gu7$Faet$!+uw$z=%F^bnF\\eu7%F`gu7$F[et$!+)=mZ<%F^bnF\\eu7%Fdgu7$Fedt $!+!\\u:;%F^bnF\\eu7%Fhgu7$F_dt$!+TYX[TF^bnF\\eu7%F\\hu7$Fict$!+d+^NTF ^bnF\\eu7%F`hu7$Fcct$!+Cx&G7%F^bnF\\eu7%Fdhu7$F]ct$!+F3j5TF^bnF\\eu7%F hhu7$Fgbt$!+@=)*)4%F^bnF\\eu7%F\\iu7$Fabt$!+hf3)3%F^bnF\\eu7%F`iu7$F[b t$!+h^9ySF^bnF\\eu7%Fdiu7$Feat$!+8>RpSF^bnF\\eu7%Fhiu7$F_at$!+=E4iSF^b nF\\eu7%F\\ju7$Fi`t$!+t!\\l0%F^bnF\\eu7%F`ju7$Fc`t$!+0f4`SF^bnF\\eu7%F dju7$F]`t$!+y.4_SF^bnF\\eu7%Fhju7$Fg_t$!+M'*)Q0%F^bnF\\eu7%F\\[v7$Fa_t $!+5,\")eSF^bnF\\eu7%F`[v7$F[_t$!+.q1nSF^bnF\\eu7%Fd[v7$Fe^t$!+I5qySF^ bnF\\eu7%Fh[v7$F_^t$!+aY^$4%F^bnF\\eu7%F\\\\v7$Fi]t$!+$*306TF^bnF\\eu7 %F`\\v7$Fc]t$!+$pQ18%F^bnF\\eu7%Fd\\v7$F]]t$!+bi\\^TF^bnF\\eu7%Fh\\v7$ Fg\\t$!+>8&G<%F^bnF\\eu7%F\\]v7$Fa\\t$!+XQ.%>%F^bnF\\eu7%F`]v7$F[\\t$! +%o?X@%F^bnF\\eu7%Fd]v7$Fe[t$!+K%QRB%F^bnF\\eu7%Fh]v7$F_[t$!+zA/_UF^bn F\\eu7%F\\^v7$Fijs$!+koooUF^bnF\\eu7%F`^v7$Fcjs$!+g&)z$G%F^bnF\\eu7%Fd ^v7$F]js$!+WHN(H%F^bnF\\eu7%Fh^v7$Fgis$!+5!e$4VF^bnF\\eu7%F\\_v7$Fais$ !+$HU)>VF^bnF\\eu7%F`_v7$F[is$!+mn%)GVF^bnF\\eu7%Fd_v7$Fehs$!+I%>kL%F^ bnF\\eu7%Fh_v7$F_hs$!+4=hUVF^bnF\\eu7%F\\`v7$Figs$!+2nZZVF^bnF\\eu7%F` `v7$Fcgs$!+Mo1^VF^bnF\\eu7%Fd`v7$F]gs$!+$*RV`VF^bnF\\eu7%Fh`v7$Fgfs$!+ c'GYN%F^bnF\\eu7%F\\av7$Fafs$!+o(*paVF^bnF\\eu7%F`av7$F[fs$!+@Zp`VF^bn F\\eu7%Fdav7$Fees$!+I%f;N%F^bnF\\eu7%Fhav7$F_es$!+r%Q'[VF^bnF\\eu7%F\\ bv7$Fids$!+f_nWVF^bnF\\eu7%F`bv7$Fcds$!+lA\")RVF^bnF\\eu7%Fdbv7$F]ds$! +K74MVF^bnF\\eu7%Fhbv7$Fgcs$!+FMbFVF^bnF\\eu7%F\\cv7$Facs$!+$))R-K%F^b nF\\eu7%F`cv7$Ffbs$!+v;>7VF^bnF\\eu7%FdcvFfduF\\euFc^rFj^r-F$6%7$7$$!3 /++++++!z%FTF(7$$\"3+++++++!*>FTF(-%'COLOURG6&Fe^nF)F)F)-%*LINESTYLEG6 #Ff^r-F$6%7$7$F(F\\dv7$F($\"3/++++++!z%FTFadvFddv-F$6$7ewF'7$$\"3$)*** **H'='f@%!#F$\"37+++Ujzq:F07$$\"3;+++F4i#y#F:$\"3#******f2$fTJF07$$\"3 \"******\\Q4rI$F@$\"33+++u8R7ZF07$$\"3%******pRa-%>FF$\"35+++y(*=$G'F0 7$$\"3x+++%=%)*RwFF$\"3A+++^%pR&yF07$$\"3;+++&3XqH#FL$\"3-+++MjlC%*F07 $$\"39+++\"e=(*f&FL$\"3-+++2X]*4\"FT7$$\"3)*******zfwJ6Fao$\"3!******z 4nkD\"FT7$$\"3%*******pk=q=Fao$\"34+++/FE89FT7$$\"3#*******ozd+BFao$\" 3!******4oM(p:FT7$$\"3(*******=(pD5\"Fao$\"35+++R5kDF\\p$\"3$******fcLV.#FT7$$!3: +++mS]E[#FT7$$!32+++pm;>QFgp$\"3=+++ma&fi#FT7$$!30+++7 'fuB'Fgp$\"3'******H;`Zw#FT7$$!3%)*******fx3e*Fgp$\"35+++QB[(*GFT7$$!3 )******zY\">#R\"F0$\"3'******f!y+AIFT7$$!3*******H&z$3#>F0$\"3\")***** ***=-OJFT7$$!3++++wABEDF0$\"3)******zt&zPKFT7$$!3?+++,N7%=$F0$\"3\"*** ***\\qQnK$FT7$$!3++++^8jrQF0$\"3\"*******\\@N.MFT7$$!3=+++\"QX;d%F0$\" 3')*****RWMKF&F0$\"3%*******GGKCNFT7$$!3/+++]r%f5OFT7$$!39+++0>?RtF 0$\"3y*****4RXKk$FT7$$!3e*****zu(\\4!)F0$\"31+++3E*)pOFT7$$!3q*****>A* Hq')F0$\"3)*******404\"p$FT7$$!3@+++$3r>K*F0$\"3)******zR!G2PFT7$$!3') ******yJ)\\'**F0$\"3z*****Hs8)=PFT7$$!3-+++7Q)*f5FT$\"3()******G1(fs$F T7$$!33+++vRqA6FT$\"3*******>Du*GPFT7$$!3/+++?&3Z=\"FT$\"3$******>o,!G PFT7$$!3'******z5ZgC\"FT$\"3@+++*R#>BPFT7$$!30+++t3x18FT$\"3,+++W[l9PF T7$$!3+++++Y$pO\"FT$\"3#)*****RorCq$FT7$$!3&******pY*fE9FT$\"3%******H B/no$FT7$$!3/+++=j$e[\"FT$\"3%******\\Y'RnOFT7$$!3!******p[HZa\"FT$\"3 z******e!zXk$FT7$$!36+++r6Q.;FT$\"35+++'*RF=OFT7$$!3)******>C;>m\"FT$ \"3#******>I+&)e$FT7$$!3*******z?([?FT$\"3#)*****p_#H*Q$FT7$$!3\"***** *4Ad9-#FT$\"3$******fw;.M$FT7$$!39+++BI3&3#FT$\"35+++SA(*)G$FT7$$!3))* ****f%3O]@FT$\"3)******\\McdB$FT7$$!3<+++GXJ!G2h HFT7$$!3!)*****zs_kc#FT$\"3')*****>(z42HFT7$$!3y*****Rgvcj#FT$\"3;++++ 'RO&GFT7$$!3(******R2-Oq#FT$\"3y*****pNN0!GFT7$$!39+++ig#*pFFT$\"3=+++ mIdZFFT7$$!3A+++H$HW$GFT$\"3?+++ZV_%p#FT7$$!3)******R>mp*GFT$\"3&***** *p\\m6k#FT7$$!33+++0tWdHFT$\"3%)*****4i%H(e#FT7$$!3$)*****4LDe,$FT$\"3 &)*****>(psKDFT7$$!3#******\\a$3sIFT$\"3%******z71tZ#FT7$$!3>+++\"HFi7 $FT$\"31+++Bz*3U#FT7$$!39+++IzFyJFT$\"39+++f$*QjBFT7$$!3)*******>(o#GK FT$\"3z*****HE'o/BFT7$$!3!******>E,M=#FT7$$!3%******4]WjO$FT$\"35+++owq?@FT7$$!3(***** **3Of3MFT$\"3!******H@%fc?FT7$$!3#)******o^1\\MFT$\"33+++#fP5*>FT7$$!3 ++++?W$y[$FT$\"35+++$yFS#>FT7$$!34+++O>)\\_$FT$\"3/+++N!pb&=FT7$$!31++ +PEfgNFT$\"3!*******o7o&y\"FT7$$!3=+++JTv%f$FT$\"3++++0=avi$FT$\"3,+++akyT;FT7$$!3#******HEx!fOFT$\"3%*******)p6zc\"FT7$ $!3')******G-S*o$FT$\"3%******\\,uG\\\"FT7$$!3#)*****>S(e=PFT$\"3#**** **fS*y;9FT7$$!3!)*****zb&oYPFT$\"3.+++(o!zR8FT7$$!37+++3&>Px$FT$\"35++ +2J-i7FT7$$!3-+++=!*o*z$FT$\"3/+++D%QO=\"FT7$$!36+++xwcCQFT$\"3$****** f+)y/6FT7$$!37+++`SI[QFT$\"34+++sdhD5FT7$$!3#******\\+D3(QFT$\"3G+++? \">DY*F07$$!3'******RnS?*QFT$\"3!)*****\\4\"3o')F07$$!3/+++*Q]=\"RFT$ \"3s*****>,ZP(yF07$$!3:+++t'[,$RFT$\"3e******)>)>!3(F07$$!3'*******42$ o%RFT$\"3))*****zH-zG'F07$$!3)*******3ozhRFT$\"3k*****f%Q8(\\&F07$$!3! )*****ztb\\(RFT$\"3$******z&=+3ZF07$$!3?+++LpA')RFT$\"3'******4&)z/#RF 07$$!37+++$zTb*RFT$\"3?+++P\\VMJF07$$!3<+++vU%G+%FT$\"3)******p%Rl\\BF 07$$!3>+++J543SFT$\"39+++/d'ec\"F07$$!3l*****4J^7,%FT$\"3)******p^%fFy Fgp7$$!3O+++qnI7SFTF(7$F]dx$!3)******p^%fFyFgp7$Fhcx$!39+++/d'ec\"F07$ Fccx$!3)******p%Rl\\BF07$F^cx$!3?+++P\\VMJF07$Fibx$!3'******4&)z/#RF07 $Fdbx$!3$******z&=+3ZF07$F_bx$!3k*****f%Q8(\\&F07$Fjax$!3))*****zH-zG' F07$Feax$!3e******)>)>!3(F07$F`ax$!3s*****>,ZP(yF07$F[ax$!3!)*****\\4 \"3o')F07$Ff`x$!3G+++?\">DY*F07$Fa`x$!34+++sdhD5FT7$F\\`x$!3$******f+) y/6FT7$Fg_x$!3/+++D%QO=\"FT7$Fb_x$!35+++2J-i7FT7$F]_x$!3.+++(o!zR8FT7$ Fh^x$!3#******fS*y;9FT7$Fc^x$!3%******\\,uG\\\"FT7$F^^x$!3%*******)p6z c\"FT7$Fi]x$!3,+++akyT;FT7$Fd]x$!3++++0FT7$F`\\x$!33+++#fP5*>FT7$F [\\x$!3!******H@%fc?FT7$Ff[x$!35+++owq?@FT7$Fa[x$!3%)*****>E,M=#FT7$F \\[x$!3z*****R66ZC#FT7$Fgjw$!3z*****HE'o/BFT7$Fbjw$!39+++f$*QjBFT7$F]j w$!31+++Bz*3U#FT7$Fhiw$!3%******z71tZ#FT7$Fciw$!3&)*****>(psKDFT7$F^iw $!3%)*****4i%H(e#FT7$Fihw$!3&******p\\m6k#FT7$Fdhw$!3?+++ZV_%p#FT7$F_h w$!3=+++mIdZFFT7$Fjgw$!3y*****pNN0!GFT7$Fegw$!3;++++'RO&GFT7$F`gw$!3') *****>(z42HFT7$F[gw$!3!******>!G2hHFT7$Fffw$!3z******=Ej:IFT7$Fafw$!3/ +++rgqqIFT7$F\\fw$!3)******zr\\g7$FT7$Fgew$!3%*******R#[7=$FT7$Fbew$!3 )******\\McdB$FT7$F]ew$!35+++SA(*)G$FT7$Fhdw$!3$******fw;.M$FT7$Fcdw$! 3#)*****p_#H*Q$FT7$F^dw$!3:+++G:^NMFT7$Ficw$!35+++-^pyMFT7$Fdcw$!3*)** ***p@d'=NFT7$F_cw$!3?+++GI+&)e$FT7$Febw$!35+++' *RF=OFT7$F`bw$!3z******e!zXk$FT7$F[bw$!3%******\\Y'RnOFT7$Ffaw$!3%**** **HB/no$FT7$Faaw$!3#)*****RorCq$FT7$F\\aw$!3,+++W[l9PFT7$Fg`w$!3@+++*R #>BPFT7$Fb`w$!3$******>o,!GPFT7$F]`w$!3*******>Du*GPFT7$Fh_w$!3()***** *G1(fs$FT7$Fc_w$!3z*****Hs8)=PFT7$F^_w$!3)******zR!G2PFT7$Fi^w$!3)**** ***404\"p$FT7$Fd^w$!31+++3E*)pOFT7$F_^w$!3y*****4RXKk$FT7$Fj]w$!3)**** ****>%f5OFT7$Fe]w$!3/+++1FCrNFT7$F`]w$!3%*******GGKCNFT7$F[]w$!3')**** *RS]E[#FT7$Fiiv$!36+++V1&eL#FT7$Fdiv$!35+++sJB'=#FT7$F_iv$!3 $******fcLV.#FT7$Fjhv$!3!******4SY1)=FT7$Fehv$!35+++R5kD(pD5\"Fecn$\"+R5kDFbdn$\"+mNLM?F^bnFe_n7%Fd\\z7$$!+mQF_en$\"+ma&fi#F^bnFe_n7%F\\^z7$$!+7'f uB'F_en$\"+jJvkFF^bnFe_n7%Fb^z7$$!++w(3e*F_en$\"+QB[(*GF^bnFe_n7%Fh^z7 $$!+o9>#R\"Fd_n$\"+1y+AIF^bnFe_n7%F^_z7$$!+`z$3#>Fd_n$\"++>-OJF^bnFe_n 7%Fd_z7$$!+wABEDFd_n$\"+QdzPKF^bnFe_n7%Fj_z7$$!+,N7%=$Fd_n$\"+0(QnK$F^ bnFe_n7%F``z7$$!+^8jrQFd_n$\"+]@N.MF^bnFe_n7%Ff`z7$$!+\"QX;d%Fd_n$\"+u JxoMF^bnFe_n7%F\\az7$$!+UWBt_Fd_n$\"+HGKCNF^bnFe_n7%Fbaz7$$!+]r?Rt Fd_n$\"+\"RXKk$F^bnFe_n7%Fdbz7$$!+[x\\4!)Fd_n$\"+3E*)pOF^bnFe_n7%Fjbz7 $$!+A#*Hq')Fd_n$\"+504\"p$F^bnFe_n7%F`cz7$$!+$3r>K*Fd_n$\"+)R!G2PF^bnF e_n7%Ffcz7$$!+zJ)\\'**Fd_n$\"+BP\")=PF^bnFe_n7%F\\dz7$$!+7Q)*f5F^bn$\" +H1(fs$F^bnFe_n7%Fbdz7$$!+vRqA6F^bn$\"+_U(*GPF^bnFe_n7%Fhdz7$$!+?&3Z= \"F^bn$\"+#o,!GPF^bnFe_n7%F^ez7$$!+3r/Y7F^bn$\"+*R#>BPF^bnFe_n7%Fdez7$ $!+t3x18F^bn$\"+W[l9PF^bnFe_n7%Fjez7$$!++Y$pO\"F^bn$\"+%orCq$F^bnFe_n7 %F`fz7$$!+n%*fE9F^bn$\"+LUq'o$F^bnFe_n7%Fffz7$$!+=j$e[\"F^bn$\"+lkRnOF ^bnFe_n7%F\\gz7$$!+([HZa\"F^bn$\"+f!zXk$F^bnFe_n7%Fbgz7$$!+r6Q.;F^bn$ \"+'*RF=OF^bnFe_n7%Fhgz7$$!+Ui\">m\"F^bn$\"+-.])e$F^bnFe_n7%F^hz7$$!+3 s[?F^bn$\"+FDH*Q$F^bnFe_n7%F\\jz7$$!+@sX@?F^bn$\"+mnJSLF^ bnFe_n7%Fbjz7$$!+BI3&3#F^bn$\"+SA(*)G$F^bnFe_n7%Fhjz7$$!+Y3O]@F^bn$\"+ XjvNKF^bnFe_n7%F^[[l7$$!+GXJEj:IF^bnFe_n7%Ff\\[l7$$!+TgL'\\#F^bn$\"+-G2hHF^bnF e_n7%F\\][l7$$!+GFXmDF^bn$\"+sz42HF^bnFe_n7%Fb][l7$$!+/cnNEF^bn$\"++'R O&GF^bnFe_n7%Fh][l7$$!+u?g.FF^bn$\"+d``+GF^bnFe_n7%F^^[l7$$!+ig#*pFF^b n$\"+mIdZFF^bnFe_n7%Fd^[l7$$!+H$HW$GF^bn$\"+ZV_%p#F^bnFe_n7%Fj^[l7$$!+ %>mp*GF^bn$\"+(\\m6k#F^bnFe_n7%F`_[l7$$!+0tWdHF^bn$\"+@YH(e#F^bnFe_n7% Ff_[l7$$!+J`#e,$F^bn$\"+spsKDF^bnFe_n7%F\\`[l7$$!+XN3sIF^bn$\"+GhIxCF^ bnFe_n7%Fb`[l7$$!+\"HFi7$F^bn$\"+Bz*3U#F^bnFe_n7%Fh`[l7$$!+IzFyJF^bn$ \"+f$*QjBF^bnFe_n7%F^a[l7$$!+?(o#GKF^bn$\"+jio/BF^bnFe_n7%Fda[l7$$!+s@ CwKF^bn$\"+96rWAF^bnFe_n7%Fja[l7$$!+^%[AK$F^bn$\"+i7S$=#F^bnFe_n7%F`b[ l7$$!+,XMmLF^bn$\"+owq?@F^bnFe_n7%Ffb[l7$$!+4Of3MF^bn$\"+8Ufc?F^bnFe_n 7%F\\c[l7$$!+p^1\\MF^bn$\"+#fP5*>F^bnFe_n7%Fbc[l7$$!+?W$y[$F^bn$\"+$yF S#>F^bnFe_n7%Fhc[l7$$!+O>)\\_$F^bn$\"+N!pb&=F^bnFe_n7%F^d[l7$$!+PEfgNF ^bn$\"+p7o&y\"F^bnFe_n7%Fdd[l7$$!+JTv%f$F^bn$\"+0Px$F ^bn$\"+2J-i7F^bnFe_n7%F^g[l7$$!+=!*o*z$F^bn$\"+D%QO=\"F^bnFe_n7%Fdg[l7 $$!+xwcCQF^bn$\"+1!)y/6F^bnFe_n7%Fjg[l7$$!+`SI[QF^bn$\"+sdhD5F^bnFe_n7 %F`h[l7$$!+0]#3(QF^bn$\"+?\">DY*Fd_nFe_n7%Ffh[l7$$!+u1/#*QF^bn$\"+&4\" 3o')Fd_nFe_n7%F\\i[l7$$!+*Q]=\"RF^bn$\"+7qutyFd_nFe_n7%Fbi[l7$$!+t'[,$ RF^bn$\"+*>)>!3(Fd_nFe_n7%Fhi[l7$$!+52$o%RF^bn$\"+)H-zG'Fd_nFe_n7%F^j[ l7$$!+4ozhRF^bn$\"+YQ8(\\&Fd_nFe_n7%Fdj[l7$$!+Qd&\\(RF^bn$\"+e=+3ZFd_n Fe_n7%Fjj[l7$$!+LpA')RF^bn$\"+^)z/#RFd_nFe_n7%F`[\\l7$$!+$zTb*RF^bn$\" +P\\VMJFd_nFe_n7%Ff[\\l7$$!+vU%G+%F^bn$\"+ZRl\\BFd_nFe_n7%F\\\\\\l7$$! +J543SF^bn$\"+/d'ec\"Fd_nFe_n7%Fb\\\\l7$$!+68D6SF^bn$\"+)>!3(Fd_nFe_n7%Fb_\\l7$Fci[l$!+7quty Fd_nFe_n7%Ff_\\l7$F]i[l$!+&4\"3o')Fd_nFe_n7%Fj_\\l7$Fgh[l$!+?\">DY*Fd_ nFe_n7%F^`\\l7$Fah[l$!+sdhD5F^bnFe_n7%Fb`\\l7$F[h[l$!+1!)y/6F^bnFe_n7% Ff`\\l7$Feg[l$!+D%QO=\"F^bnFe_n7%Fj`\\l7$F_g[l$!+2J-i7F^bnFe_n7%F^a\\l 7$Fif[l$!+(o!zR8F^bnFe_n7%Fba\\l7$Fcf[l$!+1%*y;9F^bnFe_n7%Ffa\\l7$F]f[ l$!+:S(G\\\"F^bnFe_n7%Fja\\l7$Fge[l$!+*p6zc\"F^bnFe_n7%F^b\\l7$Fae[l$! +akyT;F^bnFe_n7%Fbb\\l7$F[e[l$!+0F^bnFe _n7%Fbc\\l7$Fcc[l$!+#fP5*>F^bnFe_n7%Ffc\\l7$F]c[l$!+8Ufc?F^bnFe_n7%Fjc \\l7$Fgb[l$!+owq?@F^bnFe_n7%F^d\\l7$Fab[l$!+i7S$=#F^bnFe_n7%Fbd\\l7$F[ b[l$!+96rWAF^bnFe_n7%Ffd\\l7$Fea[l$!+jio/BF^bnFe_n7%Fjd\\l7$F_a[l$!+f$ *QjBF^bnFe_n7%F^e\\l7$Fi`[l$!+Bz*3U#F^bnFe_n7%Fbe\\l7$Fc`[l$!+GhIxCF^b nFe_n7%Ffe\\l7$F]`[l$!+spsKDF^bnFe_n7%Fje\\l7$Fg_[l$!+@YH(e#F^bnFe_n7% F^f\\l7$Fa_[l$!+(\\m6k#F^bnFe_n7%Fbf\\l7$F[_[l$!+ZV_%p#F^bnFe_n7%Fff\\ l7$Fe^[l$!+mIdZFF^bnFe_n7%Fjf\\l7$F_^[l$!+d``+GF^bnFe_n7%F^g\\l7$Fi][l $!++'RO&GF^bnFe_n7%Fbg\\l7$Fc][l$!+sz42HF^bnFe_n7%Ffg\\l7$F]][l$!+-G2h HF^bnFe_n7%Fjg\\l7$Fg\\[l$!+>Ej:IF^bnFe_n7%F^h\\l7$Fa\\[l$!+rgqqIF^bnF e_n7%Fbh\\l7$F[\\[l$!+=(\\g7$F^bnFe_n7%Ffh\\l7$Fe[[l$!+S#[7=$F^bnFe_n7 %Fjh\\l7$F_[[l$!+XjvNKF^bnFe_n7%F^i\\l7$Fijz$!+SA(*)G$F^bnFe_n7%Fbi\\l 7$Fcjz$!+mnJSLF^bnFe_n7%Ffi\\l7$F]jz$!+FDH*Q$F^bnFe_n7%Fji\\l7$Fgiz$!+ G:^NMF^bnFe_n7%F^j\\l7$Faiz$!+-^pyMF^bnFe_n7%Fbj\\l7$F[iz$!+BPF^bnFe_n7%Fj \\]l7$F_ez$!+#o,!GPF^bnFe_n7%F^]]l7$Fidz$!+_U(*GPF^bnFe_n7%Fb]]l7$Fcdz $!+H1(fs$F^bnFe_n7%Ff]]l7$F]dz$!+BP\")=PF^bnFe_n7%Fj]]l7$Fgcz$!+)R!G2P F^bnFe_n7%F^^]l7$Facz$!+504\"p$F^bnFe_n7%Fb^]l7$F[cz$!+3E*)pOF^bnFe_n7 %Ff^]l7$Febz$!+\"RXKk$F^bnFe_n7%Fj^]l7$F_bz$!++Uf5OF^bnFe_n7%F^_]l7$Fi az$!+1FCrNF^bnFe_n7%Fb_]l7$Fcaz$!+HGKCNF^bnFe_n7%Ff_]l7$F]az$!+uJxoMF^ bnFe_n7%Fj_]l7$Fg`z$!+]@N.MF^bnFe_n7%F^`]l7$Fa`z$!+0(QnK$F^bnFe_n7%Fb` ]l7$F[`z$!+QdzPKF^bnFe_n7%Ff`]l7$Fe_z$!++>-OJF^bnFe_n7%Fj`]l7$F__z$!+1 y+AIF^bnFe_n7%F^a]l7$Fi^z$!+QB[(*GF^bnFe_n7%Fba]l7$Fc^z$!+jJvkFF^bnFe_ n7%Ffa]l7$F]^z$!+ma&fi#F^bnFe_n7%Fja]l7$Fg]z$!+-/l#[#F^bnFe_n7%F^b]l7$ Fa]z$!+V1&eL#F^bnFe_n7%Fbb]l7$F[]z$!+sJB'=#F^bnFe_n7%Ffb]l7$Fe\\z$!+mN LM?F^bnFe_n7%Fjb]l7$F_\\z$!+,kk!)=F^bnFe_n7%F^c]l7$Fi[z$!+R5kDK%FT7$$\"3!********y$\\ e9FT$\"3#******\\/r$GVFT7$$\"33+++Q&3zW\"FT$\"3r******[*)*RL%FT7$$\"30 ++++0#oV\"FT$\"3.+++ezwQVFT7$$\"30+++ByFD9FT$\"3Q+++tjjUVFT7$$\"3%**** **4(4L89FT$\"3A+++(zhbM%FT7$$\"3/+++'HL5S\"FT$\"3q*****p!4]ZVFT7$$\"3' ******\\qT%)Q\"FT$\"3#******\\E4%[VFT7$$\"3/+++`uhv8FT$\"3;+++W6C[VFT7 $$\"33++++rii8FT$\"3;+++0%\\pM%FT7$$\"31+++fOa\\8FT$\"3j******\\a[WVFT 7$$\"3#********)zWO8FT$\"3i******y#*zSVFT7$$\"3/+++$\\IMK\"FT$\"3P+++8 (ReL%FT7$$\"33+++uJf58FT$\"3u******))[bHVFT7$$\"3/+++N@0)H\"FT$\"3e*** **>8$*=K%FT7$$\"3'******pfSfG\"FT$\"3\"*******HW!GJ%FT7$$\"3%******HZ7 WF\"FT$\"3d*****HzUAI%FT7$$\"31+++5lkj7FT$\"3++++F*p,H%FT7$$\"31+++m3& QD\"FT$\"3c*****H$3cwUFT7$$\"33+++CxEX7FT$\"3y*****p879E%FT7$$\"3#**** ***[qz(*GB\"FT$\"3W+++szmEUFT7$$ \"3*)*****p\\$yH7FT$\"3,+++YvI2UFT7$$\"3&******Rr2#H7FT$\"3o*****R3Fp= %FT7$$\"30+++9M`J7FT$\"3-+++%[3f;%FT7$$\"3#******>thqB\"FT$\"3.+++>)zZ 9%FT7$$\"3++++9>'fC\"FT$\"3U+++y*3U7%FT7$$\"31+++&o(>e7FT$\"34+++Ch&\\ 5%FT7$$\"35+++A)zMF\"FT$\"3)*******y5y(3%FT7$$\"3'*******[dF\"H\"FT$\" 3!******pRFL2%FT7$$\"3)******pl$)3J\"FT$\"33+++/h-iSFT7$$\"3-+++oQaJ8F T$\"31+++Vj0aSFT7$$\"3/+++KMa_8FT$\"33+++Q(o$\\SFT7$$\"35+++wIGt8FT$\" 3')*****\\rZx/%FT7$$\"33+++q8I$R\"FT$\"3'*******y;))[SFT7$$\"34+++Z#oA T\"FT$\"3G+++i1U_SFT7$$\"35+++=N'*H9FT$\"3u*****>&\\,eSFT7$$\"3)****** fx^iW\"FT$\"3C+++EoLlSFT7$$\"31+++I'f5Y\"FT$\"3i******Q.4uSFT7$$\"33++ +\"RdVZ\"FT$\"3m*****z%Q,%3%FT7$$\"31+++!HXh[\"FT$\"3=+++I'y[4%FT7$$\" 3%******p]Vk\\\"FT$\"3!******fr&[1TFT7$$\"3,+++k^G0:FT$\"3k******R@m=T FT7$$\"31+++u9r7:FT$\"3*******p?d78%FT7$$\"31+++M%o(=:FT$\"38+++)4RT9% FT7$$\"3$******fg/N_\"FT$\"3#)*****p(=>dTFT7$$\"3$******4lpp_\"FT$\"3s *****4%HJqTFT7$$\"3$******4Q8#H:FT$\"3)******\\&3T$=%FT7$$\"3&******p; &GI:FT$\"3s*****Ha.k>%FT7$$\"3++++*fL-`\"FT$\"3-+++nn@4UFT7$$\"30+++6j 5H:FT$\"3v*****R&Gy@UFT7$$\"3!******f#*\\p_\"FT$\"3n*****ziRSB%FT7$$\" 35+++v+\"Q_\"FT$\"3C+++/&HfC%FT7$$\"3++++\"[J(>:FT$\"3%)*****\\s)RdUFT 7$$\"31+++y!eZ^\"FT$\"3=+++*p'RoUFT7$$\"3\"*******eJ$*3:FT$\"3/+++Qa() yUFT7$$\"3/+++]&*H-:FT$\"3I+++j!*y)G%FTF[g]lFcgy-F\\_n6jn7%7$$\"+5)**[ \\\"F^bn$\"+lL4)H%F^bn7$$\"+Pkx'[\"F^bn$\"+Bau1VF^bn7$$\"$Q\"Fh^nF]cs7 %F_j^l7$$\"+5@(zZ\"F^bn$\"+]Kq9VF^bnFdj^l7%Fhj^l7$$\"+Y*H&o9F^bn$\"+va #>K%F^bnFdj^l7%F^[_l7$$\"+!z$\\e9F^bn$\"+X5PGVF^bnFdj^l7%Fd[_l7$$\"+Q& 3zW\"F^bn$\"+\\*)*RL%F^bnFdj^l7%Fj[_l7$$\"++0#oV\"F^bn$\"+ezwQVF^bnFdj ^l7%F`\\_l7$$\"+ByFD9F^bn$\"+tjjUVF^bnFdj^l7%Ff\\_l7$$\"+r4L89F^bn$\"+ (zhbM%F^bnFdj^l7%F\\]_l7$$\"+'HL5S\"F^bn$\"+24]ZVF^bnFdj^l7%Fb]_l7$$\" +0)zZ9%F^bnFdj^l7%Fdd_l7$$\"+9>'fC\"F^bn$\"+y*3U7%F^bnFdj^l7%Fjd_l7$$\" +&o(>e7F^bn$\"+Ch&\\5%F^bnFdj^l7%F`e_l7$$\"+A)zMF\"F^bn$\"+z5y(3%F^bnF dj^l7%Ffe_l7$$\"+\\dF\"H\"F^bn$\"+(RFL2%F^bnFdj^l7%F\\f_l7$$\"+dO)3J\" F^bn$\"+/h-iSF^bnFdj^l7%Fbf_l7$$\"+oQaJ8F^bn$\"+Vj0aSF^bnFdj^l7%Fhf_l7 $$\"+KMa_8F^bn$\"+Q(o$\\SF^bnFdj^l7%F^g_l7$$\"+wIGt8F^bn$\"+:xuZSF^bnF dj^l7%Fdg_l7$$\"+q8I$R\"F^bn$\"+z;))[SF^bnFdj^l7%Fjg_l7$$\"+Z#oAT\"F^b n$\"+i1U_SF^bnFdj^l7%F`h_l7$$\"+=N'*H9F^bn$\"+_\\,eSF^bnFdj^l7%Ffh_l7$ $\"+wdTF^bnFdj^l7%F\\\\`l7$$\"+^'pp_\"F^bn$\"+THJqTF^bnFdj^l7%Fb\\`l7 $$\"+\"Q8#H:F^bn$\"+b3T$=%F^bnFdj^l7%Fh\\`l7$$\"+n^GI:F^bn$\"+VNS'>%F^ bnFdj^l7%F^]`l7$$\"+*fL-`\"F^bn$\"+nn@4UF^bnFdj^l7%Fd]`l7$$\"+6j5H:F^b n$\"+aGy@UF^bnFdj^l7%Fj]`l7$$\"+E*\\p_\"F^bn$\"+G'RSB%F^bnFdj^l7%F`^`l 7$$\"+v+\"Q_\"F^bn$\"+/&HfC%F^bnFdj^l7%Ff^`l7$$\"+\"[J(>:F^bn$\"+D()Rd UF^bnFdj^l7%F\\_`l7$$\"+y!eZ^\"F^bn$\"+*p'RoUF^bnFdj^l7%Fb_`l7$$\"+fJ$ *3:F^bn$\"+Qa()yUF^bnFdj^l7%Fh_`l7$$\"+]&*H-:F^bn$\"+j!*y)G%F^bnFdj^l7 %F^``lFji^lFdj^lFdf]lFj^r-F$6$7in7$F\\g]l$!3s*****\\O$4)H%FT7$Fci^l$!3 I+++j!*y)G%FT7$F^i^l$!3/+++Qa()yUFT7$Fih^l$!3=+++*p'RoUFT7$Fdh^l$!3%)* ****\\s)RdUFT7$F_h^l$!3C+++/&HfC%FT7$Fjg^l$!3n*****ziRSB%FT7$Feg^l$!3v *****R&Gy@UFT7$F`g^l$!3-+++nn@4UFT7$F[g^l$!3s*****Ha.k>%FT7$Fff^l$!3)* *****\\&3T$=%FT7$Faf^l$!3s*****4%HJqTFT7$F\\f^l$!3#)*****p(=>dTFT7$Fge ^l$!38+++)4RT9%FT7$Fbe^l$!3*******p?d78%FT7$F]e^l$!3k******R@m=TFT7$Fh d^l$!3!******fr&[1TFT7$Fcd^l$!3=+++I'y[4%FT7$F^d^l$!3m*****z%Q,%3%FT7$ Fic^l$!3i******Q.4uSFT7$Fdc^l$!3C+++EoLlSFT7$F_c^l$!3u*****>&\\,eSFT7$ Fjb^l$!3G+++i1U_SFT7$Feb^l$!3'*******y;))[SFT7$F`b^l$!3')*****\\rZx/%F T7$F[b^l$!33+++Q(o$\\SFT7$Ffa^l$!31+++Vj0aSFT7$Faa^l$!33+++/h-iSFT7$F \\a^l$!3!******pRFL2%FT7$Fg`^l$!3)*******y5y(3%FT7$Fb`^l$!34+++Ch&\\5% FT7$F]`^l$!3U+++y*3U7%FT7$Fh_^l$!3.+++>)zZ9%FT7$Fc_^l$!3-+++%[3f;%FT7$ F^_^l$!3o*****R3Fp=%FT7$Fi^^l$!3,+++YvI2UFT7$Fd^^l$!3W+++szmEUFT7$F_^^ l$!3o*****\\=aZC%FT7$Fj]^l$!3y*****p879E%FT7$Fe]^l$!3c*****H$3cwUFT7$F `]^l$!3++++F*p,H%FT7$F[]^l$!3d*****HzUAI%FT7$Ff\\^l$!3\"*******HW!GJ%F T7$Fa\\^l$!3e*****>8$*=K%FT7$F\\\\^l$!3u******))[bHVFT7$Fg[^l$!3P+++8( ReL%FT7$Fb[^l$!3i******y#*zSVFT7$F][^l$!3j******\\a[WVFT7$Fhj]l$!3;+++ 0%\\pM%FT7$Fcj]l$!3;+++W6C[VFT7$F^j]l$!3#******\\E4%[VFT7$Fii]l$!3q*** **p!4]ZVFT7$Fdi]l$!3A+++(zhbM%FT7$F_i]l$!3Q+++tjjUVFT7$Fjh]l$!3.+++ezw QVFT7$Feh]l$!3r******[*)*RL%FT7$F`h]l$!3#******\\/r$GVFT7$F[h]l$!3d*** **\\ZD>K%FT7$Ffg]l$!3********\\Kq9VFT7$Fag]l$!3u*****HUXnI%FTFg``lFcgy -F\\_n6jn7%7$F[j^l$!+lL4)H%F^bn7$F_``l$!+j!*y)G%F^bn7$Fej^lF]eu7%Fa\\a l7$Fi_`l$!+Qa()yUF^bnFd\\al7%Ff\\al7$Fc_`l$!+*p'RoUF^bnFd\\al7%Fj\\al7 $F]_`l$!+D()RdUF^bnFd\\al7%F^]al7$Fg^`l$!+/&HfC%F^bnFd\\al7%Fb]al7$Fa^ `l$!+G'RSB%F^bnFd\\al7%Ff]al7$F[^`l$!+aGy@UF^bnFd\\al7%Fj]al7$Fe]`l$!+ nn@4UF^bnFd\\al7%F^^al7$F_]`l$!+VNS'>%F^bnFd\\al7%Fb^al7$Fi\\`l$!+b3T$ =%F^bnFd\\al7%Ff^al7$Fc\\`l$!+THJqTF^bnFd\\al7%Fj^al7$F]\\`l$!+x=>dTF^ bnFd\\al7%F^_al7$Fg[`l$!+)4RT9%F^bnFd\\al7%Fb_al7$Fa[`l$!+2sDJTF^bnFd \\al7%Ff_al7$F[[`l$!+S@m=TF^bnFd\\al7%Fj_al7$Fej_l$!+;d[1TF^bnFd\\al7% F^`al7$F_j_l$!+I'y[4%F^bnFd\\al7%Fb`al7$Fii_l$!+[Q,%3%F^bnFd\\al7%Ff`a l7$Fci_l$!+R.4uSF^bnFd\\al7%Fj`al7$F]i_l$!+EoLlSF^bnFd\\al7%F^aal7$Fgh _l$!+_\\,eSF^bnFd\\al7%Fbaal7$Fah_l$!+i1U_SF^bnFd\\al7%Ffaal7$F[h_l$!+ z;))[SF^bnFd\\al7%Fjaal7$Feg_l$!+:xuZSF^bnFd\\al7%F^bal7$F_g_l$!+Q(o$ \\SF^bnFd\\al7%Fbbal7$Fif_l$!+Vj0aSF^bnFd\\al7%Ffbal7$Fcf_l$!+/h-iSF^b nFd\\al7%Fjbal7$F]f_l$!+(RFL2%F^bnFd\\al7%F^cal7$Fge_l$!+z5y(3%F^bnFd \\al7%Fbcal7$Fae_l$!+Ch&\\5%F^bnFd\\al7%Ffcal7$F[e_l$!+y*3U7%F^bnFd\\a l7%Fjcal7$Fed_l$!+>)zZ9%F^bnFd\\al7%F^dal7$F_d_l$!+%[3f;%F^bnFd\\al7%F bdal7$Fic_l$!+%3Fp=%F^bnFd\\al7%Ffdal7$Fcc_l$!+YvI2UF^bnFd\\al7%Fjdal7 $F]c_l$!+szmEUF^bnFd\\al7%F^eal7$Fgb_l$!+&=aZC%F^bnFd\\al7%Fbeal7$Fab_ l$!+P@ThUF^bnFd\\al7%Ffeal7$F[b_l$!+L3cwUF^bnFd\\al7%Fjeal7$Fea_l$!+F* p,H%F^bnFd\\al7%F^fal7$F_a_l$!+$zUAI%F^bnFd\\al7%Fbfal7$Fi`_l$!+IW!GJ% F^bnFd\\al7%Fffal7$Fc`_l$!+KJ*=K%F^bnFd\\al7%Fjfal7$F]`_l$!+*)[bHVF^bn Fd\\al7%F^gal7$Fg__l$!+8(ReL%F^bnFd\\al7%Fbgal7$Fa__l$!+z#*zSVF^bnFd\\ al7%Ffgal7$F[__l$!+]a[WVF^bnFd\\al7%Fjgal7$Fe^_l$!+0%\\pM%F^bnFd\\al7% F^hal7$F_^_l$!+W6C[VF^bnFd\\al7%Fbhal7$Fi]_l$!+l#4%[VF^bnFd\\al7%Ffhal 7$Fc]_l$!+24]ZVF^bnFd\\al7%Fjhal7$F]]_l$!+(zhbM%F^bnFd\\al7%F^ial7$Fg \\_l$!+tjjUVF^bnFd\\al7%Fbial7$Fa\\_l$!+ezwQVF^bnFd\\al7%Ffial7$F[\\_l $!+\\*)*RL%F^bnFd\\al7%Fjial7$Fe[_l$!+X5PGVF^bnFd\\al7%F^jal7$F_[_l$!+ va#>K%F^bnFd\\al7%Fbjal7$Fij^l$!+]Kq9VF^bnFd\\al7%Ffjal7$F`j^l$!+Bau1V F^bnFd\\al7%FjjalF^\\alFd\\alFdf]lFj^r-%+AXESLABELSG6%%&Re(z)G%&Im(z)G -%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fd[bl6$%*HELVETICAG\" \"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%Fh^n$\"$*>Fh^n;Fg\\bl$\"$z%Fh^ n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" " Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------------------------ ---------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#------------------------------ -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1774 "ee := \{c[2]=1/6,\nc[3]= 16/63,\nc[4]=63/106,\nc[5]=50/69,\nc[6]=63/83,\nc[7]=1,\nc[8]=1,\n\na[ 2,1]=1/6,\na[3,1]=80/1323,\na[3,2]=256/1323,\na[4,1]=155816577/3049000 96,\na[4,2]=-16967475/9528128,\na[4,3]=568356831/304900096,\na[5,1]=64 921911575/789872295744,\na[5,2]=1250725/43533526,\na[5,3]=995353041375 /3166454547136,\na[5,4]=365139662800/1219687315371,\na[6,1]=-303546683 06338585/178910020531686528,\na[6,2]=3896053749/4091707772,\na[6,3]=-2 492114394031368057/6454959947119420288,\na[6,4]=861399479018673064/317 7050598972839673,\na[6,5]=15/164,\na[7,1]=2211628527362559660565641813 8617/38775599793952140053248982599680,\na[7,2]=-197127/114496,\na[7,3] =24118537135561993104813811090103/14776247724722745081046875555840,\na [7,4]=886475909491169318904581727562843/197963826330861711898808941536 0030,\na[7,5]=-185/122,\na[7,6]=4082588597964313509378858016009/257142 8541481836530593320891180,\na[8,1]=16270811903/209372688000,\na[8,2]=0 ,\na[8,3]=76925739070593/205684631194240,\na[8,4]=747543949006532/3259 794500756955,\na[8,5]=-35320882628403/140715258838000,\na[8,6]=1551373 170541891/3131789077344060,\na[8,7]=82/1099,\n\nb[1]=16270811903/20937 2688000,\nb[2]=0,\nb[3]=76925739070593/205684631194240,\nb[4]=74754394 9006532/3259794500756955,\nb[5]=-35320882628403/140715258838000,\nb[6] =1551373170541891/3131789077344060,\nb[7]=82/1099,\n\n`b^`[1]=41084480 131/527850886920,\n`b^`[2]=0,\n`b^`[3]=207911313213975/556644255095896 ,\n`b^`[4]=6621573612710978/28585355696715237,\n`b^`[5]=-5260422835930 17/2042110550893490,\n`b^`[6]=1/2,\n`b^`[7]=82/1099,\n\n`b*`[1]=113179 0443129436/14550345379510605,\n`b*`[2]=0,\n`b*`[3]=71/190,\n`b*`[4]=33 26466076155851042/14390291580335988345,\n`b*`[5]=-62/241,\n`b*`[6]=635 95996553465987137/127170018616922687700,\n`b*`[7]=8829707957442742/117 440240421421575,\n`b*`[8]=-33340565403709/52195662409520700\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5,7];" "6#&%\"TG6$\"\"&\" \"(" }{TEXT -1 127 " denote the vector whose components are the princ ipal error terms of the 7 stage, order 5 scheme (the error terms of or der 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,8 ];" "6#&%#T*G6$\"\"%\"\")" }{TEXT -1 145 " denote the vector whose co mponents are the principal error terms of the embedded 8 stage, order \+ 4 scheme (the error terms of order 5) and let " }{XPPEDIT 18 0 "`T*`[ 5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 99 " denote the vector whose \+ components are the error terms of order 6 of the embedded order 4 sche me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "De note the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7] ));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\")" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6 #-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/abs(abs(`T*`[4, 8]));" " 6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\")\"\"\"-F*6#-F*6#&F/6 $\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T* `[5,8]-T[5,7]))/abs(abs(`T*`[4,8]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6 #,&&%#T*G6$\"\"&\"\")\"\"\"&%\"TG6$F2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F 3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "Dormand and Prince have suggested that as well as attem pting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" } {TEXT -1 73 " is a minimum, if the embedded scheme is to be used for \+ error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "er rterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\n`errterms5_8*` :=su bs(b=`b*`,PrincipalErrorTerms(5,8,'expanded')):\n`errterms4_8*` := sub s(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := sqr t(add(simplify(subs(ee,`errterms5_8*`[i]))^2,i=1..nops(`errterms5_8*`) )):\nsdnB := sqrt(add(simplify(subs(ee,`errterms4_8*`[i]))^2,i=1..nops (`errterms4_8*`))):\nsnmC := sqrt(add((simplify(subs(ee,`errterms5_8*` [i])-subs(ee,errterms5_7[i])))^2,i=1..nops(errterms5_7))):\n'`B*`[6]'= evalf[8](snmB/sdnB);\n'`C*`[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B*G6#\"\"'$\")2>k5!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\"\"'$\")_!G+\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The corresponding characterstic s for the auxiliary 7 stage, order 4 embedded scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "`errterms5_7^` :=subs(b=` b^`,PrincipalErrorTerms(5,7,'expanded')):\n`errterms4_7^` := subs(b=`b ^`,PrincipalErrorTerms(4,7,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := sqrt(add(sim plify(subs(ee,`errterms5_7^`[i]))^2,i=1..nops(`errterms5_7^`))):\nsdnB := sqrt(add(simplify(subs(ee,`errterms4_7^`[i]))^2,i=1..nops(`errterm s4_7^`))):\nsnmC := sqrt(add((simplify(subs(ee,`errterms5_7^`[i])-subs (ee,errterms5_7[i])))^2,i=1..nops(errterms5_7))):\n'`B^`[6]'= evalf[8] (snmB/sdnB);\n'`C^`[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B^G6#\"\"'$\")V0@5!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C^G6#\"\"'$\")ln!*)*!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#=================================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "An efficient 7 stage, order 5 Runge-Kutta scheme .. B" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1640 "ee := \{c[2]=1/6,\nc[3]=155/594,\nc[4]=12/19,\nc[ 5]=66/85,\nc[6]=27/34,\nc[7]=1,\nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=6665/1 17612,\na[3,2]=24025/117612,\na[4,1]=109269372/164787475,\na[4,2]=-234 1872/1063145,\na[4,3]=357797088/164787475,\na[5,1]=-67885276899/135740 048750,\na[5,2]=37863936/17514845,\na[5,3]=-383179645721664/2839003119 60625,\na[5,4]=54884775429/118168704250,\na[6,1]=-22877286938601/28657 439092100,\na[6,2]=2786718411/910771940,\na[6,3]=-57606687522659088/29 968516930563575,\na[6,4]=3984899849779/12473888420630,\na[6,5]=55/406, \na[7,1]=357804108795684026067391/386149007682840983490000,\na[7,2]=-7 556229/2688010,\na[7,3]=1811005935036153778848695517/74559043506132986 2990895000,\na[7,4]=258902691791384553326881/1307299004967194685810000 ,\na[7,5]=-154/529,\na[7,6]=138208410082586660479/25192626558677203500 0,\na[8,1]=51964327/654617700,\na[8,2]=0,\na[8,3]=6965638145689311/179 45236957333900,\na[8,4]=14557376984/60894963675,\na[8,5]=2059161125/95 272074612,\na[8,6]=1793996689/9043706700,\na[8,7]=29/395,\n\nb[1]=5196 4327/654617700,\nb[2]=0,\nb[3]=6965638145689311/17945236957333900,\nb[ 4]=14557376984/60894963675,\nb[5]=2059161125/95272074612,\nb[6]=179399 6689/9043706700,\nb[7]=29/395,\n\n`b^`[1]=126880675943/1597558129200, \n`b^`[2]=0,\n`b^`[3]=28404964428638589/73207500841545650,\n`b^`[4]=19 29236304929/8045094629520,\n`b^`[5]=63707991275/3620338835256,\n`b^`[6 ]=23/114,\n`b^`[7]=29/395,\n\n`b*`[1]=4457141939387131/561142682134155 00,\n`b*`[2]=0,\n`b*`[3]=71/183,\n`b*`[4]=366435142393882979/152755507 9142977500,\n`b*`[5]=1/57,\n`b*`[6]=2399113615956319/11892729854919375 ,\n`b*`[7]=3306969852589408717937/45125218883586784548375,\n`b*`[8]=81 82415847667565831/54150262660304141458050\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "s ubs(ee,matrix([[c[2],a[2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],a[4,1],a[4,2],a[4,3],``],\n [c[5],a[5,1],a[5,2],a[5,3 ],a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3],a[6,4]],[``$4,a[6,5]],\n \+ [c[7],a[7,1],a[7,2],a[7,3],a[7,4]],[``$3,a[7,5],a[7,6]],\n [ c[8],a[8,1],a[8,2],a[8,3],a[8,4]],[``$2,a[8,5],a[8,6],a[8,7]],\n \+ [`b`,b[1],b[2],b[3],b[4]],[``$2,b[5],b[6],b[7]],\n [`b^`,`b^`[1], `b^`[2],`b^`[3],`b^`[4]],[``$2,`b^`[5],`b^`[6],`b^`[7]],\n [`b*`, `b*`[1],`b*`[2],`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[6],`b*`[7],`b*`[8]]] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#727'#\"\"\"\"\"'F(% !GF+F+7'#\"$b\"\"$%f#\"%lm\"'7w6#\"&DS#F2F+F+7'#\"#7\"#>#\"*s$p#4\"\"* vuyk\"#!(s=M#\"(XJ1\"#\"*)3(zd$F;F+7'#\"#m\"#&)#!,**oF&)y'\"-]([+uN\"# \")OR'y$\")X[^<#!0k;sX'zJQ\"0D1'>J+RG#\",Hax%)[&\"-]Uqo\"=\"7'#\"#F\"# M#!/,'QpGxG#\"/+@4RulG#\"+6%=ny#\"*S>x5*#!2)3fE_(o1w&\"2vNcIp^o*H#\".z (\\)**[)R\"/I1U))QZ77'F+F+F+F+#\"#b\"$1%7'F)#\"9\"Rng-%o&z3T!yN\"9++\\ $)4%Go2!\\hQ#!(Hib(\"(5!)o##\"=n\\+*H287'F+F+F+#!$a\"\"$H&#\"6z/m'e#35%3 #Q\"\"6+].sneli#>D7'F)#\")FV'>&\"*+xha'\"\"!#\"16$*oX\"Qc'p\"2+RLdpBXz \"#\",%)pPdX\"\",vO'\\*3'7'F+F+#\"+D6;f?\",7Y2s_*#\"+*o'*Rz\"\"++nqV!* #\"#H\"$&R7'%\"bGFdpFgpFhpF[qF^q7'%#b^G#\"-Vfn!)o7\".+#H\"evf\"Fgp#\"2 *eQ'GW'\\SG\"2]caT3]2K(#\".H\\IO#H>\".?&HY4X!)7'F+F+#\",v7*zqj\".c_$)Q .i$#\"#B\"$9\"Feq7'%#b*G#\"1JrQR>9dW\"2+bT8#oU6cFgp#\"#r\"$$=#\"3zH)QR U^Vm$\"4+v(H9z]bF:7'F+#F)\"#d#\"1>j&fh8\"*R#\"2v$>\\&)HF*=\"#\"7Pzr3%* e_)ppI$\"8v$[Xy'e$))=_7X#\"4Jecnw%eT#=)\"8]!e99/.mi-:aQ(pprint46\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "RK5_7eqs := [op(RowSumCon ditions(7,'expanded')),op(OrderConditions(5,7,'expanded'))]:\nsubs(ee, RK5_7eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4_7eqs^` := subs(b= `b^`,OrderConditions(4,7,'expanded')):\nsubs(ee,`RK4_7eqs^`):\nmap(u-> lhs(u)-rhs(u),%);\nnops(%);\n`RK4_8eqs*` := subs(b=`b*`,OrderCondition s(4,8,'expanded')):\nsubs(ee,`RK4_8eqs*`):\nmap(u->lhs(u)-rhs(u),%);\n nops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that is, the 2-norm of the princi pal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterm s5_7 := PrincipalErrorTerms(5,7,'expanded'):\nsqrt(add(subs(ee,errterm s5_7[i])^2,i=1.. nops(errterms5_7))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+R0kG$)!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 88 "We can also calculate the 2 norm of the p rincipal error of the order 4 embedded schemes." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 145 "`errterms4_7^` := subs(b=`b^`,PrincipalErrorT erms(4,7,'expanded')):\nsqrt(add(subs(ee,`errterms4_7^`[i])^2,i=1.. no ps(`errterms4_7^`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+o,'o'R!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTer ms(4,8,'expanded')):\nsqrt(add(subs(ee,`errterms4_8*`[i])^2,i=1.. nops (`errterms4_8*`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ #**zL(R!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-orders \+ from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f or ct from 2 to 4 do\n so||ct||_7 := StageOrderConditions(ct,7,'expa nded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs( ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc(L) local i; for i to no ps(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\n simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In genera l, the simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$ SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6# F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6 #/%\"jG\"\"\"" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 7 " ) are " }{TEXT 260 13 "not satisfied " }{TEXT -1 15 ", except when " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 ", where the condition reduces to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 152 "[Sum(b[i]*a[i,1],i=1+1..7)=b[1],seq(Sum(b[i]* a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nmap( u->`if`(lhs(u)=rhs(u),0,1),subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F* 6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F- /-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F) F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F, FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F ,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7(\"\"\"\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i ]*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F* ;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\" &%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F )F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = \+ 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&& %\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"! " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Sum(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,% ));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6 #%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\" \"F+/F*;F5\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"%\" \"\"&%\"aG6$F(\"\"$F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F) F)*&&F+6$F4F(F)&F+6$F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&& F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\" (F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "None of t he principal error conditions are satisfied by the order 5 scheme. " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK5_7err_eqs := PrincipalE rrorConditions(5,7,'expanded'):\nsubs(ee,RK5_7err_eqs):\nmap(u->`if`(l hs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions \+ are satisfied by the 8 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 96 "The largest magnitude of an error term is about 167 \+ times the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms (4,8,'expanded')):\nsubs(ee,`errterms4_8*`):\nmap(abs,evalf[8](%));\ne valf[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$ \")4D%e$!#7$\")`JW9F&$\")3'y3(!#8$\")xT4JF+$\")&paW'!#9$\"))Q7E&F0$\") R4*G\"F+$\")UQ9[F+$\"))*[[@F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%p ;!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions are satisfied by the 7 st age embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 96 "The lar gest magnitude of an error term is about 213 times the lowest magnitud e of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`err terms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\nsubs( ee,`errterms4_7^`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(%))/min(op (%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")vdbO!#7$\")_qH8F&$\")N W]c!#8$\")Q\\8FF+$\")\"H$Q^!#9$\")-E%>%F0$\")emF5F+$\")2NKWF+$\")kx7 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 88 "simultaneous construction of the \+ 7 stage, order 5 scheme and the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following \"sim ple\" order conditions (given in abreviated form) are relevant for the construction of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := SimpleOrderConditi ons(5):\n[seq([i,SO5[i]],i=[1,2,4,5,8,12,13,15,16])]:\nlinalg[augment] (linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg [delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7 %\"\"\"%%~~~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F) /*&F,F()F2F/F(#F(\"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@6#*&FCF(F2F(F( F(#F(\"#C7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6#*&FCF(-F@6#*& F8F(FCF(F(F(#F(\"#g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF)/*&F,F(-F@6#*& FMF(FCF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b*`,SimpleOrderC onditions(4)):\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nlinalg[augment](l inalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7% \"\"\"%%~~~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/ *&F,F()F2F/F(#F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF(F(#F(\"#77%\" \")F)/*&F,F()F2F:F(#F(F5Q)pprint386\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We incorporate the simplifying conditions:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0; " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\" \"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 . . 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum( b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+ F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 scheme a nd use the simple order conditions in the first table above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We also need t o specify that the linking coefficients for the 8th (FSAL) stage are g iven by the weights of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6#%\"iG&% \"aG6$\"\")F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\" " }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"( \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the order 4 \+ embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7,'expanded' ):\n`SO4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded')): \n cdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[1, 2,4,7,8])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(StageO rderConditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3.. 7)=0,add(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i ]*c[i]*a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [se q(b[i]=a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_eqns), op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 12/19;" "6#/&%\"cG6#\"\"%*&\" #7\"\"\"\"#>!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 66/85;" "6 #/&%\"cG6#\"\"&*&\"#m\"\"\"\"#&)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 27/34;" "6#/&%\"cG6#\"\"'*&\"#F\"\"\"\"#M!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$ " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " are initial ly parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "This situation is " }{TEXT 260 18 "slightly different" } {TEXT -1 46 " from that in the preceding examples in that " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 2 " " }{TEXT 260 26 "is a parameter rather than" }{TEXT -1 2 " " }{XPPEDIT 18 0 "c[4]" "6 #&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "e1 := \{c[2]=1/6,c[4]=12/19 ,c[5]=66/85,c[6]=27/34,c[7]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs (e1,cdns):\nnops(eqns);\nindets(eqns) minus \{c[3],a[6,5],a[7,5],b[7], `b*`[3],`b*`[5]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{c[3],a [6,5],a[7,5],b[7],`b*`[3],`b*`[5]\}):\ninfolevel[solve] := 0:\ne3 := ` union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5414 "e3 := \{c[2] = 1/6, c[4] = 12/19, c[6] = 27/34, c[5] = 66/85, \+ a[8,5] = 614125/185328*(-330*c[3]+980*b[7]*c[3]+139-980*b[7])/(85*c[3] -66), a[3,2] = 3*c[3]^2, a[4,3] = -144/6859*(19*c[3]-12)/c[3]^2, a[5,4 ] = -464607/1228250*(425*c[3]^2-500*c[3]+132)/(19*c[3]-12)/(25*c[3]-3) , b[4] = -130321/1179360*(532*b[7]*c[3]-402*c[3]+149-532*b[7])/(19*c[3 ]-12), a[5,3] = -396/614125*(759050*c[3]^3-1347495*c[3]^2+741654*c[3]- 118800)/(19*c[3]-12)/(25*c[3]-3)/c[3]^2, c[7] = 1, c[8] = 1, b[2] = 0, `b*`[2] = 0, b[6] = -19652/8505*(532*b[7]*c[3]-167*c[3]-532*b[7]+74)/ (34*c[3]-27), a[2,1] = 1/6, `b*`[6] = 746776/19845*`b*`[3]*c[3]^3+1572 16/6615*`b*`[3]*c[3]-86944/91875*`b*`[5]-1218424/19845*`b*`[3]*c[3]^2- 9826/11907, a[8,2] = 0, a[8,7] = b[7], b[5] = 614125/185328*(-330*c[3] +980*b[7]*c[3]+139-980*b[7])/(85*c[3]-66), a[6,2] = 729/39304*(13600*c [3]^2-12073*c[3]+2376)/(25*c[3]-3)/c[3], a[7,3] = -1/45517500*(-999746 3700000*c[3]^5+59984782200000*c[3]^5*b[7]+34382402250000*c[3]^4-221185 372590000*c[3]^4*b[7]+101663097536000*a[6,5]*c[3]^4*b[7]-3191304001600 0*a[6,5]*c[3]^4-192801074385920*b[7]*a[6,5]*c[3]^3+42750209307520*a[6, 5]*c[3]^3+318895787070000*c[3]^3*b[7]-47073310908750*c[3]^3-4962157200 0*b[7]*a[7,5]*c[3]^3+45359954640*b[7]*a[7,5]*c[3]^2+31388155651125*c[3 ]^2+100610585467392*a[6,5]*c[3]^2*b[7]-223340762767500*b[7]*c[3]^2-156 50631439872*a[6,5]*c[3]^2-9472608617472*a[6,5]*c[3]*b[7]+7520205624750 0*b[7]*c[3]-9987122124000*c[3]+1317618491904*a[6,5]*c[3]-4728643920*b[ 7]*a[7,5]*c[3]+1194561270000-9556490160000*b[7])/(646*c[3]^2-921*c[3]+ 324)/b[7]/(34*c[3]-27)/(25*c[3]-3)/c[3]^2, a[7,2] = 3/4*(720*b[7]*c[3] ^2-824*b[7]*c[3]+192*b[7]-120*c[3]^2+115*c[3]-24)/b[7]/c[3]/(25*c[3]-3 ), b[3] = -1/4*(3724*b[7]+1993)/c[3]/(88326*c[3]-120921*c[3]^2+54910*c [3]^3-21384), a[7,4] = -361/114704100000*(684947340000*b[7]*a[7,5]*c[3 ]^3-2721187875000*c[3]^3+203326195072000*b[7]*a[6,5]*c[3]^3+5687411625 000*c[3]^3*b[7]-63826080032000*a[6,5]*c[3]^3+3981301931250*c[3]^2-1247 8850212500*b[7]*c[3]^2-1157963914800*b[7]*a[7,5]*c[3]^2+85500418615040 *a[6,5]*c[3]^2-385602148771840*a[6,5]*c[3]^2*b[7]+8598028162500*b[7]*c [3]-1786359322125*c[3]+551437120080*b[7]*a[7,5]*c[3]+201221170934784*a [6,5]*c[3]*b[7]-31301262879744*a[6,5]*c[3]+270619744500-1806589575000* b[7]-18945217234944*b[7]*a[6,5]-50681363040*b[7]*a[7,5]+2635236983808* a[6,5])/(646*c[3]^2-921*c[3]+324)/b[7]/(25*c[3]-3), a[7,6] = 2/1116281 25*(2033261950720*a[6,5]*c[3]^2*b[7]-56237781600*b[7]*a[7,5]*c[3]^2-63 8260800320*a[6,5]*c[3]^2-10962131250*c[3]^2+88326398160*b[7]*a[7,5]*c[ 3]+10020677625*c[3]+778412890112*a[6,5]*c[3]-3612030053632*a[6,5]*c[3] *b[7]+1578768102912*b[7]*a[6,5]-1044626625-219603081984*a[6,5]-3467672 2080*b[7]*a[7,5])/(1156*c[3]^2-1836*c[3]+729)/b[7], a[7,1] = 1/2949534 0000*(81220364925000*c[3]^4-488676335175000*c[3]^4*b[7]+20332619507200 0*a[6,5]*c[3]^4*b[7]+127813140000*b[7]*a[7,5]*c[3]^4-63826080032000*a[ 6,5]*c[3]^4-158164777968750*c[3]^3-216079390800*b[7]*a[7,5]*c[3]^3-385 602148771840*b[7]*a[6,5]*c[3]^3+1043632386247500*c[3]^3*b[7]+855004186 15040*a[6,5]*c[3]^3-755679213607500*b[7]*c[3]^2+105245401912875*c[3]^2 -31301262879744*a[6,5]*c[3]^2+201221170934784*a[6,5]*c[3]^2*b[7]+10289 9749680*b[7]*a[7,5]*c[3]^2-9457287840*b[7]*a[7,5]*c[3]+210751578135000 *b[7]*c[3]-18945217234944*a[6,5]*c[3]*b[7]+2635236983808*a[6,5]*c[3]-2 7517677453000*c[3]+2389122540000-19112980320000*b[7])/b[7]/(34*c[3]-27 )/(25*c[3]-3)/c[3]^2, a[8,3] = -1/4*(3724*b[7]+1993)/c[3]/(88326*c[3]- 120921*c[3]^2+54910*c[3]^3-21384), a[6,4] = -361/7860800*(1184832*a[6, 5]-11399520*a[6,5]*c[3]+12716000*a[6,5]*c[3]^2+1275750-4795875*c[3]+40 16250*c[3]^2)/(475*c[3]^2-357*c[3]+36), b[1] = 1/85536*(15202*c[3]+372 4*b[7]*c[3]-3724*b[7]-1993)/c[3], `b*`[8] = 1/3540250*(10656360*`b*`[5 ]*c[3]+84966000*b[7]+137094375*c[3]-2223936*`b*`[5]+561816000*`b*`[3]* c[3]-4289049000*`b*`[3]*c[3]^2+11581602750*`b*`[3]*c[3]^3-143055000*c[ 3]^2+5600820000*`b*`[3]*c[3]^5-13352522500*`b*`[3]*c[3]^4-407128750*b[ 7]*c[3]+424830000*b[7]*c[3]^2-11119680*c[3]^2*`b*`[5]-28611000)/(720*b [7]*c[3]^2-824*b[7]*c[3]+192*b[7]-120*c[3]^2+115*c[3]-24), a[8,6] = -1 9652/8505*(532*b[7]*c[3]-167*c[3]-532*b[7]+74)/(34*c[3]-27), a[4,2] = \+ 1296/6859*(19*c[3]-8)/c[3], a[5,2] = 3564/122825*(7990*c[3]^2-6881*c[3 ]+1320)/(25*c[3]-3)/c[3], a[6,1] = 3/7860800*(-583503750*c[3]^3+150280 00*a[6,5]*c[3]^3+639636525*c[3]^2-13472160*a[6,5]*c[3]^2-202706550*c[3 ]+1400256*a[6,5]*c[3]+19245600)/(25*c[3]-3)/c[3]^2, `b*`[4] = -116603/ 4410*`b*`[3]*c[3]^3+418399/8820*`b*`[3]*c[3]^2-20577/980*`b*`[3]*c[3]+ 6859/5292-1433531/17701250*`b*`[5], a[3,1] = -3*c[3]^2+c[3], a[8,1] = \+ 1/85536*(15202*c[3]+3724*b[7]*c[3]-3724*b[7]-1993)/c[3], a[6,3] = -27/ 982600*(19380000*c[3]^3+1944800*a[6,5]*c[3]^2-35007150*c[3]^2+19617525 *c[3]-233376*a[6,5]*c[3]-3207600)/c[3]^2/(475*c[3]^2-357*c[3]+36), a[8 ,4] = -130321/1179360*(532*b[7]*c[3]-402*c[3]+149-532*b[7])/(19*c[3]-1 2), a[4,1] = -12/6859*(1691*c[3]^2-1092*c[3]+144)/c[3]^2, a[5,1] = -33 /1228250*(7591775*c[3]^3-8131980*c[3]^2+2532708*c[3]-237600)/(25*c[3]- 3)/c[3]^2, `b*`[1] = 323/162*`b*`[3]*c[3]^3-1567/324*`b*`[3]*c[3]^2+41 5/108*`b*`[3]*c[3]+187/972+247/216750*`b*`[5]-`b*`[3], `b*`[7] = -1/17 70125*(16802460000*`b*`[3]*c[3]^5-43184692000*`b*`[3]*c[3]^4+403233030 00*`b*`[3]*c[3]^3-16032564000*`b*`[3]*c[3]^2-216750000*c[3]^2-33359040 *c[3]^2*`b*`[5]+38177568*`b*`[5]*c[3]+287591125*c[3]+2247264000*`b*`[3 ]*c[3]-8895744*`b*`[5]-71961000)*b[7]/(720*b[7]*c[3]^2-824*b[7]*c[3]+1 92*b[7]-120*c[3]^2+115*c[3]-24)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "the coefficients in terms of \+ " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b *`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do \n print(`======================================================`); \n temp_eq := c[ii]=subs(e3,c[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n pr int(temp_eq);\n end if;\n print(``); \n for jj to ii-1 do\n \+ temp_eq := a[ii,jj]=subs(e3,a[ii,jj]);\n if lhs(temp_eq)=rhs(t emp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n \+ else\n print(temp_eq);\n end if;\n print(``);\n \+ end do:\nend do:\n`################################################ ######`;\n``;\nfor ii to 7 do\n temp_eq := b[ii]=subs(e3,b[ii]);\n \+ if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a pa rameter`);\n else\n print(temp_eq);\n end if;\n print(``); \nend do:\n`######################################################`;\n ``;\nfor ii to 8 do\n temp_eq := `b*`[ii]=subs(e3,`b*`[ii]);\n if \+ lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a parame ter`);\n else\n print(temp_eq);\n end if;\n print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============================ ==========================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG 6#\"\"##\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W========= =============================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"#;\"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"#! )\"%B8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"$c#\"%B8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============== ========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\"%0~is~a~parameterGF(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"%\"\"\",(*&#\"%z?\"$c#F(*$)&%\"cG6#F'\"\"$F(F(F(*&#\"#\")\"#;F(*$) F0\"\"#F(F(!\"\"F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,$*&#\"\"*\"\")\"\"\"*&,& *&\"#@F.&%\"cG6#F'F.F.F-!\"\"F.)F3F(F.F.F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\" \"$,&*&#\"%pR\"$c#\"\"\"*$)&%\"cG6#F'F(F.F.F.*&#\"#j\"#;F.*$)F1\"\"#F. F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#]\"#p" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"\",$*&#\"#D\"(#>3qF(*(,(*&\"(.nK$F()&%\"cG6#\"\"%\" \"#F(F(*&\")[_97F(F3F(!\"\"\"(+WF'F(F(,&*&\"$b\"F(F3F(F(\"$4\"F:F:F3F: F(F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,$*&#\"#D\"&MV#\"\"\"*&,&*&\"'$3-\"F .&%\"cG6#\"\"%F.F.\"&+-'!\"\"F.,&*&\"$b\"F.F3F.F.\"$4\"F8F8F.F8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"$,$*&#\"&v'f\"')oy(\"\"\"*(,(*&\"'2$)=F.)&%\"cG6#\" \"%\"\"#F.F.*&\"';#*>F.F4F.!\"\"\"&+/&F.F.,&*&\"#jF.F4F.F.\"#;F;F;,&*& \"$b\"F.F4F.F.\"$4\"F;F;F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,$*&#\"&+T$\"%hZ\" \"\"**,&\"#]!\"\"*&\"#pF.&%\"cG6#F(F.F.F.,&*&\"#jF.F5F.F.\"#;F2F2F5F2, &*&\"$b\"F.F5F.F.\"$4\"F2F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================ ======================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"'#\"#j\"#$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",$*&#F(\"-KS_]h6F(*(,.*(\"/S5Gp[ OLF(&F%6$F'\"\"&F()&%\"cG6#\"\"%\"\"#F(F(*&\"-Zy\">$49F(F4F(F(*&\".3\\ m9gK&F(F5F(F(*(\"/7`hy/kZF(F1F(F5F(!\"\"*&\"/+C!3?-q\"F(F1F(F(\".K]_l3 1$F@F(,&*&\"$b\"F(F5F(F(\"$4\"F@F@F5F@F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#,$*&#\"$n&\"([rG#\"\"\"*&,&*&\"'LnpF.&%\"cG6#\"\"%F.F.\"'s#\\$!\"\" F.,&*&\"$b\"F.F3F.F.\"$4\"F8F8F.F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\"$T%\" ,W8%orQ\"\"\"*&,.*&\".(zl(\\$*f\"F.)&%\"cG6#\"\"%\"\"#F.!\"\"*(\".+gy/ hW#F.&F%6$F'\"\"&F.F3F.F.*(\".+3r.F\\$F.Fjf\"F.F4F. F.*&\".++m&\\Y7F.F3EQF,F,,(*&\"%l(*F,)F/\"\"#F,F,*&\"%Z$*F,F/F,F2\"% WEOO$[8U,=F()&%\"cG6#\"\"%\"\"$F( !\"\"**\":++[s!>#G;VR:B\"F(&F%6$\"\"'\"\"&F(F1F(&%\"bG6#F'F(F(*(\"8+++ \"Q(z'4mL\"[*F(F:F(F1F(F7*(\"8?d%Qt\"*o-cy$3'F(F>F(F1F(F(**\"7+/vT0$)y S!Gv$F(F>F(&F%6$F'F=F(F1F(F(**\":+SA;vR.'>'4+*HF(F:F()F2\"\"#F(F>F(F7* (\"8S[^GY/Q(H\">P$F(F>F(FKF(F7*&\"7X/+^]st\"36\\&F(FKF(F7**\"7?&f3>t'f e+2#)F(F>F(FGF(FKF(F7*(\"9+SA>uVo!)[daBF(F:F(FKF(F(*(\"9+s#Q;ZU_\"e67> F(F:F(F2F(F7**\":+Sas]F(F(*&\"7#R)**))*z@zN%\\\"* F(F2F(F(**\"7?jT4V(Gld'zfF(F>F(FGF(F2F(F(*(\"8+S[vMJh]fL*GF(F>F(F2F(F7 *(\"7+S'[e\"*G*ob^9F(F>F(FGF(F7\"77\"GQ\\jh@_L*HF7*(\"9++!)Gm*RR)osviF (F>F(F:F(F7*&\"8++W4_4U9g(*4&F(F:F(F(*&\"8?Bna4XMlc$y9F(F>F(F(F(,&*&\" #$)F(F2F(F(\"#jF7F7F2F7F>F7,&*&\"$b\"F(F2F(F(\"$4\"F7F7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"(\"\"#,$*&#\"\"$\"#K\"\"\"*(,**(\"&37\"F.&%\"bG6#F'F.&%\"cG6# \"\"%F.F.*&\"%<7F.F6F.!\"\"\"$/&F.*&\"%KSF.F3F.F#G;VR:B\"F-&F%6$\" \"'\"\"&F-F2F-&%\"bG6#F'F-F-*(\"8+++\"Q(z'4mL\"[*F-F9F-F2F-!\"\"*(\"9! o?^wL:\"[T23>F-F=F-F2F-FB**\"7++c3:sSvxxdF-F=F-&F%6$F'F'4+*HF-F9F-)F3\"\"#F-F=F-FB*(\"9?hkW?!4Mdj*\\HF-F=F-FKF-F-*&\" 8D*[S#=hw\"*yj)RF-FKF-FB**\"8+!GPtv'y&>aj7F-F=F-FGF-FKF-FB*(\"9+SA>uVo !)[daBF-F9F-FKF-F-*(\"9+s#Q;ZU_\"e67>F-F9F-F3F-FB**\"7+![#*p9Szp?/Z@AF-F3F-F-*(\"7++g*eTzm,[B#F-F=F-FGF-FB*&\" 8++W4_4U9g(*4&F-F9F-F-\"7s5A7Igie8KVFB*(\"9++!)Gm*RR)osviF-F=F-F9F-FB* &\"8!))y(*))=C-l]%p#F-F=F-F-F-,(\"%35F-*&\"%(H&F-F3F-FB*&\"%H_F-FKF-F- FBF=FB,&*&\"$b\"F-F3F-F-\"$4\"FBFBF-FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,$*&# \"\"\"\"0!=6Iv``@F,*,,H*(\"7+DJO)\\5QNe#fF,&F%6$\"\"'\"\"&F,)&%\"cG6#F (\"\"$F,!\"\"*&\"6!)po6PMH!)eV%F,F6F,F,*(\"7?1cM!eoLTPg\"F,&%\"bG6#F'F ,F6F,F;**\"8++.#pjnZ97(p(F,F2F,F6F,F@F,F,**\"9+SwWB@D7gvo=F,F2F,)F7\" \"#F,F@F,F;*(\"8+:qQtF/I4;Z\"F,F2F,FGF,F,**\"6++8'eXG&R'*p\"F,F@F,&F%6 $F'F5F,FGF,F;*(\"7gib>sJ>mLQSF,F@F,FGF,F,*&\"6Dx#4'GqfQo&**F,FGF,F;*( \"8+#RxWl_%Qs]>\"F,F2F,F7F,F;**\"9+S.UM2t(zw7\\\"F,F2F,F7F,F@F,F,**\"6 +S9XFH@=`[#F,F@F,FMF,F7F,F,*&\"6#o+W3%)f[pmuF,F7F,F,*(\"7gEmpB'3>o&eLF ,F@F,F7F,F;*&\"6?qm%=.%3H(R#*F,F@F,F,*(\"8++I*yCY-$HB#RF,F@F,F2F,F;*( \"5+S0Cd!eIA2*F,F@F,FMF,F;\"6dUO=-^8X3(=F;*&\"7++f]48!4]t=$F,F2F,F,F,F 7F;,**&\"'2SVF,F6F,F,*&\"'y!p(F,FGF,F;*&\"'vtTF,F7F,F,\"&/N'F;F;F@F;,& *&\"$b\"F,F7F,F,\"$4\"F;F;F,F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\"\"%0~is~a~ parameterGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*&#\"'(yr&\"5S>;fs[.7$e$\"\"\"*( ,:**\"4+?i14u$*H)fF.&F%6$F(\"\"&F.)&%\"cG6#\"\"%\"\"#F.&%\"bG6#F'F.F.* (\"3+Dre$>ohg%F.F3F.F6F.!\"\"*&\"2XQ\"[(QVvh%F.F6F.F.**\"3+?M+5)y1%QF. F )F.F3F.F7F.F.**\"3+?wYBYJ)p&F.F!3r-bL%F.F\"\" \"*&,**(\"&!e`F.&%\"bG6#\"\"(F.&%\"cG6#\"\"%F.F.*&\"%DTF.F7F.!\"\"*&F2 F.F3F.F=\"%aVF.F.,&*&\"#$)F.F7F.F.\"#jF=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"(&%\"bG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W################################################## ####G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"(+S-$F'*&,**(\"(+;2\"F'&F%6#\" \"(F'&%\"cG6#\"\"%F'F'*&\"'&y=\"F'F3F'F'*&F/F'F0F'!\"\"\"'2:5F'F'F3F:F 'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"(H.v\"\"*S =Pw&\"\"\"*&,**(\"&+G#F-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"&l/\"F-F5F-! \"\"\"%pUF-*&F1F-F2F-F;F-,&*&\"#jF-F5F-F-\"#;F;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"%,$*&#\"\"\"\"#gF+*(,&*&\"(+;2\"F+&F%6#\"\"(F+F+\"'2:5!\"\"F+&%\"cGF &F5,*\"&+/&F5*&\"'-WLF+F6F+F+*&\"'VpiF+)F6\"\"#F+F5*&\"',3OF+)F6\"\"$F +F+F5F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"(p&=D\"*+SNM\"\"\"\"*&,**&\"%NFF-& %\"cG6#\"\"%F-!\"\"*(\"&+k&F-&F%6#\"\"(F-F2F-F-*&F8F-F9F-F6\"%`MF-F-,& \"#]F6*&\"#pF-F2F-F-F6F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\")@$eu%\"+gqkm>\" \"\"*&,**(\"&!e`F-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"%DTF-F5F-!\"\"*&F1F -F2F-F;\"%aVF-F-,&*&\"#$)F-F5F-F-\"#jF;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"bG6#\"\"( \"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%W#################################### ##################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\",$*&#F'\"-[f5lmwF'*&,.*(\"-o' 3Qe!QF'&%\"cG6#\"\"%F'&F%6#\"\"$F'F'*&\"-'*Ru!*\\@F'F0F'!\"\"*(\"+[FCn &*F'&F%6#\"\"&F'F0F'F'*&\",w`?cm*F'F4F'F9\",pJh1O%F'*&\"++YyKpF'FF-F6F-,&*&\"#$)F-F2F-F-\"#jF6F6F-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(,$ *&#\"\"\"\",IG2T,\"F+**,4*&\"/;vGQn7MF+)&%\"cG6#\"\"%\"\"#F+!\"\"*(\"/ +#)zPwv6F+&F%6#\"\"&F+F1F+F+*(\"/o.0GZ]gF+F1F+&F%6#\"\"$F+F+*&\"/Ppsr0 rEF+F2F+F+*(\"/+Gwb(\\F\"F+F:F+F2F+F7*(\"/[W)\\VKr$F+F2F+F?F+F7*&\".++ c?]1$F+F:F+F+\"._px5Ai&F7*&\"./:M7z_&F+F?F+F+F+&%\"bGF&F+,**(\"&37\"F+ FMF+F2F+F+*&\"%<7F+F2F+F7\"$/&F+*&\"%KSF+FMF+F7F7,&F+F7F2F+F7F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"),$*&#\"\"\"\"-?zu&QV#F+*(,:*&\"/1p\"Rvrn&F+)&%\"cG6#\" \"%\"\"#F+F+*(\"0S'=_X+iHF+&%\"bG6#\"\"(F+F1F+F+*(\"0o$y@1vw:F+F1F+&F% 6#\"\"$F+!\"\"*(\"/+#[PNS1$F+&F%6#\"\"&F+F1F+FB*(\"/+%3+M#*[$F+FEF+F2F +F+*(\"0?.*F+FEF+FB\".'*=@IL`\"F+*&\"0!o^pTmE7F+F9F+F+*&\"/7X- PPe;F+F?F+FBF+,**(\"&37\"F+F9F+F2F+F+*&\"%<7F+F2F+FB\"$/&F+*&\"%KSF+F9 F+FBFB,&F+FBF2F+FBF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "RK5_7eqs := [op(R owSumConditions(7,'expanded')),op(OrderConditions(5,7,'expanded'))]:\n simplify(subs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4 _8eqs*` := subs(b=`b*`,OrderConditions(4,8,'expanded')):\nsimplify(sub s(e3,`RK4_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The stability functi on of a general 7 stage, order 5 Runge-Kutta scheme has the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/ 6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*& F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F) *&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 7 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[ 5,4]*a[4,3]*a[3,2]*c[2]+b[7]" "6#/&%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&% \"aG6$F'\"\"&F-&F/6$F1\"\"%F-&F/6$F4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F -&F+6#\"\"(F-" }{TEXT -1 2 " (" }{XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]* a[3,2]*c[2]+a[7,6]*a[6,4]*a[4,3]*a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"( \"\"&\"\"\"&F&6$F)\"\"%F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,& F&6$F(\"\"'F*&F&6$F:F-F*&F&6$F-F0F*&F&6$F0F3F*&F56#F3F*F*%!GF*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[7,6]*a[6,5]*a[5,3]*a[3,2]*c[2]+a[7,6] *a[6,5]*a[5,4]*a[4,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,3]*c[3]" "6#,(*,&% \"aG6$\"\"(\"\"'\"\"\"&F&6$F)\"\"&F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG 6#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-\"\"%F*&F&6$F>F3F*&F56#F3F*F**, &F&6$F(F)F*&F&6$F)F-F*&F&6$F-F>F*&F&6$F>F0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[7]=b[7]*a[7,6]* a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\" \"\"&%\"aG6$F'\"\"'F,&F.6$F0\"\"&F,&F.6$F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9 \"\"#F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 250 "eq7 \+ := b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=1/5338;\neq6 := b[6]*a [6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2] +\n a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5, 4]*(a[4,2]*c[2]+a[4,3]*c[3]))))=1/717;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6#\"\"(\"\"\"&%\"aG6$F*\"\"'F+&F-6$F/\"\"&F+&F-6 $F2\"\"%F+&F-6$F5\"\"$F+&F-6$F8\"\"#F+&%\"cG6#F;F+#F+\"%Q`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/,&*.&%\"bG6#\"\"'\"\"\"&%\"aG6$F+\" \"&F,&F.6$F0\"\"%F,&F.6$F3\"\"$F,&F.6$F6\"\"#F,&%\"cG6#F9F,F,*&&F)6#\" \"(F,,&*,&F.6$F@F0F,F1F,F4F,F7F,F:F,F,*&&F.6$F@F+F,,&**&F.6$F+F3F,F4F, F7F,F:F,F,*&F-F,,&*(&F.6$F0F6F,F7F,F:F,F,*&F1F,,&*&&F.6$F3F9F,F:F,F,*& F4F,&F;6#F6F,F,F,F,F,F,F,F,F,F,#F,\"$<(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We shall require the following two equations to be satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve(simplify(subs(e3,\{eq6\})),a[7,5]));\neqn7 := op (solve(simplify(subs(e3,\{eq7\})),a[7,5]));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&%\"aG6$\"\"(\"\"&,$*&#\"%8\\\"/!3%\\W5k5\"\" \"*,,J*(\"2+++!y7_W7F0&F'6$\"\"'F*F0)&%\"cG6#\"\"$\"\"%F0!\"\"*&\"1+DJ &pn&pNF0F8F0F0*&\"1++D\"QCr9(F0)F9F*&\"1DJXHm&>8\"F0F9F0F>*&\"0+S sOqi9$F0F5F0F>*&\"1DcE7!)4CZF0)F9\"\"#F0F0*(\"2++5yY%=HDF0F5F0FCF0F0*( \"1+!y#)3]TM%F0F5F0F9F0F0*(\"2++\\g7'z?*&\"2ciH2&*RTh'F0) F5FKF0F>*(\"33up&HMI]v%F0&%\"bG6#F)F0FTF0F0\"/]7`HyTzF0*(\"4?\"42+(3v' H9F0FTF0FJF0F>**\"4+g@\")QqW$3BF0FTF0F8F0FWF0F0**\"4?&z\"f9s/**)pF0FTF 0FJF0FWF0F0**\"4+?\"*=Xo_*(\"3+g4;`'>hC(F0FTF0F8F0F >*(\"4+!)GOU0TRp\"F0FTF0FCF0F0*(\"3)[f`,XQ@8&F0FTF0F9F0F0**\"4G4=<^x** >4$F0FTF0F9F0FWF0F>F0F5F>FWF>,**&\"'+q')F0FCF0F0*&\"(DEk\"F0FJF0F>*&\" 'SF)*F0F9F0F0\"''fy\"F>F>,&*&\"#MF0F9F0F0\"#FF>F>F0F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/&%\"aG6$\"\"(\"\"&,$*&#\"$*G\".S!*>U,*p\" \"\"*.,H*&\"0+!)o4HRr\"F0&F'6$\"\"'F*F0!\"\"*(\"3+S)>[**o;P$F0)F5\"\"# F0)&%\"cG6#\"\"$\"\"%F0F8*(\"1++]dD%3z&F0F5F0F=F0F8*(\"2++:BtE1@\"F0F5 F0)F>FAF0F0*(\"1+g*G#zJ$H#F0F5F0F>F0F0*&\"2s54X>\\Ig$F0F;F0F8*(\"1++vT 71y&)F0F5F0)F>Fk#F0F;F0F>F0F0*(\"3'*o\"GF0FQF0F8** \"4g,)R@!R0CT$F0F;F0FNF0FQF0F0**\"4+%e.w`.srJF0F;F0FGF0FQF0F8*&\"/+voa 5f%*F0FGF0F8*&\"0+]7yPeh\"F0FNF0F0*&\"/vV[@#zw(F0F>F0F8\".Dc^`\"erF0F0 ,&*&\"#&)F0F>F0F0\"#mF8F8FQF8F5F8,(*&\"$D%F0FNF0F0*&\"$+&F0F>F0F8\"$K \"F0F8,&*&\"#MF0F>F0F0\"#FF8F8F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 18 "We can eliminate " }{XPPEDIT 18 0 "a[7, 5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 57 " from the two equations by e quating the two right sides." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(simplify(rhs(eqn6)-rhs(eqn7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\")v$4;$\".C-qJOI&\"\"\"*2,&*&\"#DF(&%\"cG6#\"\"$F (F(F0!\"\"F(,&*&\"#MF(F-F(F(\"#FF1F(,(*&\")D=%y%F()F-\"\"#F(F(*&\")gxZ IF(F-F(F1\"(Z`p%F(F(,&*&\"\"&F(F-F(F(F:F1F1,&*&\"#&)F(F-F(F(\"#mF1!\"# ,&*&\"$?\"F(F-F(F(\"#TF1F1&%\"bG6#\"\"(F1&%\"aG6$\"\"'F@F1F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The facto rs " }{XPPEDIT 18 0 "``(25*c[3]-3);" "6#-%!G6#,&*&\"#D\"\"\"&%\"cG6#\" \"$F)F)F-!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "``(34*c[3]-27);" "6#-%!G6#,&*&\"#M\"\"\"&%\"cG6#\"\"$F)F)\"#F!\"\"" }{TEXT -1 141 " in the numerator of the last expression occur as factors in the denomina tors in the expressions that we have obtained for the coefficients." } }{PARA 0 "" 0 "" {TEXT -1 23 "This rules out taking " }{XPPEDIT 18 0 "c[3] = 3/25;" "6#/&%\"cG6#\"\"$*&F'\"\"\"\"#D!\"\"" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "c[3] = 27/34;" "6#/&%\"cG6#\"\"$*&\"#F\"\"\"\"#M! \"\"" }{TEXT -1 39 " to make the the last expression zero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "denom_factors := select(has,map(op ,select(has,map(denom@rhs,e3),c[3])),c[3]):\nmember(25*c[3]-3,denom_fa ctors);\nmember(34*c[3]-27,denom_factors);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "However, \+ the quadratic factor has two real zeros." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "47841825*c[3]^2-30477760*c[3]+4695347;\nsolve(%);\nev alf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\")D=%y%\"\"\")&%\"cG6# \"\"$\"\"#F&F&*&\")gxZIF&F(F&!\"\"\"(Z`p%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"(wx/$\"(l$o&*\"\"\"*&F&!\"\"\"-XmwzNI#F'\"\"#F',&F $F'*&F&F)F*F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+D*)4hP!#5$\"+$G F%4EF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "3047776/9568365-1/9568365*303579766645^(1/2);\nevalf( %);\nconvert(%,rational,5);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"(wx/$\"(l$o&*\"\"\"*&F&!\"\"\"-XmwzNI#F'\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$GF%4E!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$b\"\"$%f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4wU4E!#5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set \+ " }{XPPEDIT 18 0 "c[3] = 155/594;" "6#/&%\"cG6#\"\"$*&\"$b\"\"\"\"\"$% f!\"\"" }{TEXT -1 56 ", which is a rational approximation for the lowe r root. " }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting for " } {XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$" }{TEXT -1 32 " in each of the equations for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 29 " now gives an equation for " }{XPPEDIT 18 0 "a[7,5]" "6 #&%\"aG6$\"\"(\"\"&" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "b[ 7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6 #&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "Si nce these two equations are only approximately the same we pick the fi rst one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "expand(subs(c[3]=155/594,eqn6));\nevalf(%);\nexpand(s ubs(c[3]=155/594,eqn7));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"&,**&#\")D[x>\"*C%p5P\"\"\"*&F.F.&%\"bG6#F'!\"\"F.F .*&#\"3v=#>cyM>&>\"5O\"RaPYu?\\,$F.*&F.F.*&F0F.&F%6$\"\"'F(F.F3F.F3*&# \")BGy))\")ga*G#F.*&F0F3F:F.F.F3*&#\")L$z4%\"'&p<)F.F:F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,**&$\"+%GW\"H`!#6\"\"\"& %\"bG6#F'!\"\"F.*($\"+!3\\UZ'!#7F.F/F2&F%6$\"\"'F(F2F2*($\"+ZzuxQ!\"*F .F/F2F7F.F2*&$\"+Fqc6]!\")F.F7F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\"(\"\"&,**&#\")D[x>\"*C%p5P\"\"\"*&F.F.&%\"bG6#F'!\"\"F.F. *&#\")BGy))\")ga*G#F.*&F0F3&F%6$\"\"'F(F.F.F3*&#\")L$z4%\"'&p<)F.F9F.F .*&#\"/vV)Ru\"[Y\"1_`%pYu%zrF.*&F.F.*&F0F.F9F.F3F.F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,**&$\"+%GW\"H`!#6\"\"\"&%\"bG6#F '!\"\"F.*($\"+ZzuxQ!\"*F.F/F2&F%6$\"\"'F(F.F2*&$\"+Fqc6]!\")F.F7F.F.*( $\"+wVDuk!#7F.F/F2F7F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "cc := 155/594:\ndd := simplify(sub s(c[3]=cc,rhs(eqn6))):\ne4 := \{c[3]=cc,a[7,5]=dd\}:\ne5 := `union`(e4 ,simplify(subs(e4,e3))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2674 "e5 := \{c[2] = 1/6, c[4] = 12/19, c[6] = 27/34, c[5 ] = 66/85, a[7,2] = 9/92690*(218496*b[7]-18167)/b[7], `b*`[7] = -1/154 57954656375*(2296120703150240000*`b*`[3]-856497190816587375-8840414569 9364352*`b*`[5])*b[7]/(218496*b[7]-18167), a[7,1] = -1/452286903338690 070170520000*(-207142695931137038298199050*a[6,5]+24803366751627778152 28013400*b[7]*a[6,5]+10273303077003790986041847968*b[7]*a[6,5]^2-29492 626262961343015625-794906656895094399903926536*a[6,5]^2)/b[7]/a[6,5], \+ c[3] = 155/594, a[8,5] = -73080875/18272358+66052214375/1205975628*b[7 ], b[3] = 7243950571011/11357744909705*b[7]+15507189568233/45430979638 820, a[7,6] = 66293688325/12149926150032*1/(b[7]*a[6,5]), c[7] = 1, c[ 8] = 1, b[2] = 0, `b*`[2] = 0, a[2,1] = 1/6, `b*`[8] = 18167/927477279 38250*(-2427618236472*`b*`[5]+92747727938250*b[7]-31231377775125+63052 523702500*`b*`[3])/(218496*b[7]-18167), a[8,2] = 0, a[7,5] = 4913/6391 481232594361219636320*(69328568449208318250*a[6,5]-8422560856449140625 -5044688220031949635432*a[6,5]^2+65197102784130472771616*b[7]*a[6,5]^2 )/b[7]/a[6,5], a[4,3] = 357797088/164787475, a[8,7] = b[7], a[3,2] = 2 4025/117612, b[5] = -73080875/18272358+66052214375/1205975628*b[7], a[ 5,4] = 54884775429/118168704250, a[7,3] = 297/125611377922386902173113 9410000*(6145514721658248294189036800*a[6,5]-7152650098089814957437251 0400*b[7]*a[6,5]+246559273848090983665004351232*b[7]*a[6,5]^2-19077759 765482265597694236864*a[6,5]^2-1054240876683597368390625)/b[7]/a[6,5], b[4] = -1087007461/176187960*b[7]+284490743/411105240, a[7,4] = -361/ 107383376217902217208776000*(59602727579887679360250*a[6,5]-3575497405 86060946875-1972473094032492307453912*a[6,5]^2+25492067188595014853701 856*b[7]*a[6,5]^2-648606552346214121429000*b[7]*a[6,5])/b[7]/a[6,5], b [1] = 26647/301320-408709/3314520*b[7], a[4,2] = -2341872/1063145, a[8 ,3] = 7243950571011/11357744909705*b[7]+15507189568233/45430979638820, a[6,4] = -103361159/60444350*a[6,5]+13545844515/24579090484, a[5,3] = -383179645721664/283900311960625, a[8,6] = -40979333/817695*b[7]+8878 2823/22895460, a[5,2] = 37863936/17514845, a[6,1] = -26007090309/35292 412675-1015131/2239750*a[6,5], b[6] = -40979333/817695*b[7]+88782823/2 2895460, a[3,1] = 6665/117612, a[8,1] = 26647/301320-408709/3314520*b[ 7], `b*`[4] = -62820600163/23106700386*`b*`[3]+6859/5292-1433531/17701 250*`b*`[5], a[4,1] = 109269372/164787475, a[5,1] = -67885276899/13574 0048750, a[6,3] = -307040611671288/147628162219525+5449199184/46844371 25*a[6,5], a[8,4] = -1087007461/176187960*b[7]+284490743/411105240, `b *`[6] = 279679634111/103980151737*`b*`[3]-86944/91875*`b*`[5]-9826/119 07, `b*`[1] = -1235854801/4244087826*`b*`[3]+187/972+247/216750*`b*`[5 ], a[6,2] = 2786718411/910771940\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We choose values for " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 8 " \+ and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 54 " \+ that minimize the norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "errterms5_7 := PrincipalErrorTerms(5,7,' expanded'):\nsm := 0:\nfor ct to nops(errterms5_7) do\n sm := sm+sim plify(subs(e5,errterms5_7[ct]))^2;\nend do:\nsm := simplify(sm):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 743 "sm := 1/169 75260428050085377707710308712327041511281304271360000000*(-21874115612 098228022565933837856138495577915625000*a[6,5]+17881058077077074726264 444552360515745171049476117760*a[6,5]^2-380793107098914951688425263793 455901434518608690950400*b[7]*a[6,5]^2+2592360127591731334897760084405 329999213710004075104000*b[7]^2*a[6,5]^2-10997805667640123439737835942 79304466482252514982385280*a[6,5]^3+7925846441909044327403986676209971 1484094447306868642432*a[6,5]^4+14213466427237095617862277177618449423 828073176310448640*a[6,5]^3*b[7]+1323834171640723754845830585961135284 8039307910466136107008*a[6,5]^4*b[7]^2-2048658718183801100743230900620 315800751180341942842975232*a[6,5]^4*b[7]+1475674685100778935877498229 036090164239111328125)/a[6,5]^2:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "in dets(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"bG6#\"\"(&%\"aG6$\" \"'\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "plot3d(sm,b[7]=0..0.13,a[6,5]=0.01..0.4,labels=[`b[7 ]`,`a[6,5]`,``],grid=[40,60],style=patch,linestyle=2,\n lightmodel=li ght3,color=COLOR(RGB,1,.2,.2),axes=boxed,view=[0..0.13,0.01..0.4,0..4e -6],orientation=[-25,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 451 437 437 {PLOTDATA 3 "6*-%%GRIDG6&;$\"\"!F($\"#8!\"#;$\"\"\"F+$\"\"%!\"\"X,%)an ythingG6\"6\"[gl'!%\"!!#[at\"I\"gn3EBB0F1BE74862693EB936FE744F54433EC2 38BF555BC2503ECBEF2A1E5BE9A33ED4A5BAD79516623EDD16146D2CFFFF3EE3A1337E B2B5AD3EE993FCEF3E92603EF0315E20094FAF3EF4068B77D2EF4F3EF8496CD612A7A1 3EFCF9F2E9252DF83F010C0A13CADD893F03D1E535DD4E233F06CE88D09EE7103F0A01 F380D837263F0D6C244FB66C713F10868D4746F3E83F12726ADFE4D7B73F1479AAC3EF F4653F169C4CD1A518C93F18DA50EFDA2C293F1B33B70BA2A34F3F1DA87F16A77F313F 201C5482FC98003F21721A689AC5DD3F22D59138F44D013F2446B8F18B8BCF3F25C591 9073752F3F27521B142EA7B03F28EC557B967BC93F2A9440C5C7F76D3F2C49DCF21528 9C3F2E0D29FFF9D0773F2FDE27EF12919B3F30DE6B5F8B05343F31D49B37E7B46B3F32 D1A3808D11713F33D584396F49183F34E03D6287A0D53F35F1CEFBD363643F370A3905 5304943F38297B7F0970C83F394F9668FB7F533F3A7C89C32F80813F3BB0558DACE1FF 3F3CEAF9C87BE5193F3E2C7673A563603F3F74CB8F329F023F4061FC8D968E433F410C FF8BCF42113F41BB6EC24844B33F426D4A31066BE73F432291D80E81C73F43DB45B765 3ED23F449765CF0F45523F4556F21F111DEA3F4619EAA76F34F63F46E04F682DD8BC3F 47AA20615138273EB9A40F5431DF803EB6F1D07F2D7FD53EC05F394A826AB93EC914D6 A3F760133ED2933B32FBD64B3EDA394FB18B624E3EE1BB36FD8D06DF3EE723EEB2F352 763EED5624B0D11D2D3EF228BF3ACEB4823EF60AE457C29A8B3EFA51725DA0F6033EFE FC5FC2F4FFAE3F0205D3310A82703F04BFA1135581113F07AB9825240A8E3F0AC9B76F A34CFA3F0E19FE4427A9333F10CE36128E12AE3F12A8805B7B737D3F149BDDDB18C5DC 3F16A84E783BF1163F18CDD21FF869EF3F1B0C68C3F7315F3F1D64125948B7943F1FD4 CED78AD2923F212F4F1C24C4FD3F2280C03B44CD483F23DEBAC73875133F25493EBE82 5AEB3F26C04C1FFBD74A3F2843E2EAC1F0223F29D4031E26B3713F2B70ACB9A5E2593F 2D19DFBCDC1F7B3F2ECF9C278009303F3048F0FCAE66BB3F3130589926F1243F321E04 E91DD0A33F3311F5EC8C4AB03F340C2BA36FAA063F350CA60DC862733F3613652B9960 5C3F372068FCE77B113F3833B181B902E23F394D3EBA1565763F3A6D10A604E41F3F3B 93274590586F3F3CBF8298C1048A3F3DF2229FA06D013F3F2B075A383A0C3F40351864 490F973F40D7CF755BE3F73F417DA8E059643F3F4226A4A54648EC3F42D2C2C427344A 3F4382033D00AEFB3F4434660FD7255C3F44E9EB3CAEE5B43F45A292C38C1EE83EB84F 07961EE69B3EB4CFBF115F49A33EBD421BE585FCDC3EC662EAE264F0073ED09D1638CB C3213ED782ACB9771AB93EDFDBFE20A438743EE4D328E248F4553EEA70298A789EF03E F0625343AC0F013EF3E836438A38583EF7C9AE733380CF3EFC06B2493321233F004F9D CFEFB3563F02C9A3317126FA3F057167E5E67A533F0846EAF67CDB723F0B4A2BB488AB 313F0E7B29A274EFBD3F10ECF2325BE7493F12B32DDBE572B13F149047B4AD015D3F16 843FA9C608133F188F15ACD987CE3F1AB0C9B2F7F0BD3F1CE95BB3BF18DB3F1F38CBA8 BB07CF3F20CF8CC6783D2F3F220E22AE42AB2C3F2358278A3F6C6C3F24AD9B5947D968 3F260E7E1A78F8133F277ACFCD24D66D3F28F29070C735953F2A75C004FCB82B3F2C04 5E897BFC8C3F2D9E6BFE1030B23F2F43E86294CA993F307A69DB7911F23F315896FD8D 63C13F323C7B9784A5723F332617A95F4AD63F34156B332040503F350A7634CC5D323F 360538AE69F1CA3F3705B2A0006BC13F380BE409980C673F3917CCEB39AD4D3F3A296D 44EE909E3F3B40C516C03AE43F3C5DD460B854553F3D809B22E0907C3F3EA9195D429A DB3F3FD74F0FE807A43F40859E1D6D23E93F4122706F1150003F41C21E7CE510B53F42 64A846ECD2693F430A0DCD2CE3613F43B24F0FA972863EB71004AD0F77BC3EB2D0CA2A E4B1AF3EB9FC7A9C3AFADE3EC3D966D9A499863ECD8697D209B9C33ED4F22B84F0293F 3EDC73176E03EB823EE2A1AB7D3F78013EE7B0CACD0924A03EED668F24D5FD9B3EF1E1 62996981083EF562A729DCCE513EF9370BBA501F6C3EFD5E8A2519C1A83F00EC8F2AF1 D8CD3F035362C31F86723F05E3BEE44317D23F089DA2DFB0EDD03F0B810E37D40E8E3F 0E8E0091229F943F10E23CD40B1F433F12923CA52D5CFC3F1456FFA90B7DC03F163085 D14E82833F181ECF1306DB713F1A21DB65D25E8F3F1C39AAC33D137F3F1E663D264BB7 053F2053C94592177B3F217ED57765DC323F22B443277A82243F23F41254ED0F443F25 3E42FF0F918F3F2692D5255DCA263F27F1C8C7745BAB3F295B1DE509E4783F2ACED47D E9928B3F2C4CEC91EEDBD83F2DD56621021A0C3F2F68412B15D80D3F3082BED81255A9 3F31568DD817BDBE3F322F8D959E10A73F330DBE10AA25B83F33F11F49424D3C3F34D9 B13F6DF4E23F35C773F3355DF23F36BA6764A162033F37B28B93BB43423F38AFE0808C 862C3F39B2662B1ED3063F3ABA1C937BDD543F3BC703B9AD50973F3CD91B9DBCC1063F 3DF0643FB39F973F3F0CDD9F9B30E93F401743DEBE431B3F40AAB14CB03C1F3F4140B7 19A7D1C63F41D95545A932F93EB5E706990392E33EB0F4F1CBBDB7F93EB6ED8EB923CF 713EC1784A89B65C8B3ECA0BB8874E471E3ED287CC13F68DE03ED93BB9E33926E03EE0 8F7683D6DD763EE518087882AE3E3EEA37384E1707CC3EEFECD2B2C0E9343EF31C5C81 9CDE933EF68D6C164BFA8C3EFA4991F1C415DD3EFE50C9FFAF2D143F015188BCCF2EEB 3F03A03338F602213F061463C5A0710D3F08AE19E53981D43F0B6D553C3759F93F0E52 15871397273F10AE2D49BD03F83F1246121DC8CAF23F13F0B9315621783F15AE227975 77B73F177E4DEDC4A3B03F19613B87CFAD0C3F1B56EB429B50873F1D5F5D1A4D73F83F 1F7A910BEB54783F20D4438A93D17E3F21F49F9A1E35B03F231E5CB3E6E4D73F24517A D769A00F3F258DFA044309FC3F26D3DA3A29C0F63F28231B78E8F2FC3F297BBDC05C16 063F2ADDC1106B83B83F2C49256909C7023F2DBDEACA3175543F2F3C1133E3764C3F30 61CC5312D15F3F312A409080D4A13F31F765524215383F32C93A985E00D43F339FC062 DCD8BF3F347AF6B1C7768B3F355ADD85271C683F363F74DD054EDA3F3728BCB96BB619 3F3816B51A6405B13F39095DFFF7E9223F3A00B76A30F49F3F3AFCC1591899253F3BFD 7BCCB81B4B3F3D02E6C5188C533F3E0D024242C4F83F3F1BCE443F61C83F4017A5658B 60473EB4D40D59FB380F3EAE786BE7D4B8FA3EB415583C407A9F3EBE7F2BE534723C3E C6C98E91652E513ED0438E668A489F3ED635E58043EA983EDD3913EC1E496E3EE2A5E2 8CE53BD03EE736A2031B3C933EEC4E9506DE26213EF0F6CE7A73B18D3EF409D35D26B2 833EF7605305DE63443EFAFA496044C0603EFED7B3A5EAE77A3F017C47F4959A5C3F03 AE6E665734E93F06024CAAA5498C3F0877E265F5FDC93F0B0F2F54C2EF463F0DC83344 B7EC9A3F10517707FDEFAA3F11CFAFCCF064AE3F135EC3E643C58B3F14FEB34B95E840 3F16AF7DF672D4733F187123E1DF46E53F1A43A50A0125673F1C27016BDDAD163F1E1B 3905278EE53F201025EA0C6B5B3F211B1CEBAAD0463F222E8186EAB74E3F234A53BB68 C31B3F246E9388DB92063F259B40EF0E520C3F26D05BEDDC79213F280DE4852E60EE3F 2953DAB4F694633F2AA23E7D2FA8ED3F2BF90FDDDA86303F2D584ED6FD04F63F2EBFFB 68A0D3D93F30180AC96949BF3F30D44EAAD08F9B3F3194C9588E7CCE3F32597AD2ABEA EB3F33226319321C1C3F33EF822C2A94ED3F34C0D80B9EFD903F359664B79909953F36 7028302264793F374E227544A2743F3830538709347B3F3916BB65795F2A3F3A015A10 9E33C53F3AF02F88808AFE3F3BE33BCD29016F3F3CDA7EDE9FF4E03EB3D718EFF66740 3EAB4D2D46D53E803EB173D72590FC633EBA5E9228A05E723EC3C019F04E6F5D3ECC4A E4F956B2EC3ED3619A452436A43ED991CBA7D09B833EE05A590A30CD503EE464CC43E2 9BF33EE8E80C2F2AB8D03EEDE3FA28C28E883EF1AC418EE0474F3EF4A2CD6168A9E03E F7D59C77A46B7F3EFB44AC0B24A9D43EFEEFFA2E43C1043F016BC2C1D539623F037DA6 881765B63F05ADA80E5E8AFC3F07FBC7112446E33F0A68035E1467FA3F0CF25CCF55D7 D13F0F9AD3483A98543F1130B35971C4F13F12A30B7F462C3F3F1424720F2689B73F15 B4E704179A1C3F17546A5A3F43423F1902FC0EA2C2423F1AC09C1EF4C80A3F1C8D4A89 6F60843F1E69074CB6A7B13F2029E933E10FE53F2126D5ECE5870D3F222B49D11F57A6 3F233744E059AFB43F244AC71A7005283F2565D07F4AB1AC3F2688610EDC402E3F27B2 78C91F46183F28E417AE14AB163F2A1D3DBDC247EC3F2B5DEAF831CB393F2CA61F5D6F D59C3F2DF5DAED8B42693F2F4D1DA894943E3F3055F3C74EBF1D3F31091C4FDC425E3F 31C0086DFC58643F327AB821B8A96D3F33392B6B1AE8FF3F33FB624A2CC2A13F34C15C BEF7CA7F3F358B1AC98571983F36589C69DEFC873F3729E1A00D7C873F37FEEA6C19CA 583F38D7B6CE0C82843F39B446C5EE02DF3EB2F0295AF520783EA86827B47D007E3EAE 1216EA2AA9803EB68EC7DDB07DB73EC0EF5AA40A0A423EC85AF0ACB380DB3ED0BED831 DA0B003ED629143AC4B1323EDC6AD7E0CAC57E3EE1C1B7106D25ED3EE5B9382BA6A14B 3EEA1BD09ECB3F6F3EEEE96D56F171E93EF211010462E9AD3EF4E2C345CE2E703EF7E9 FAA94BA4E03EFB26A54135A9273EFE98C1B034FCF43F0120277D8FD6523F030EA63571 019B3F0517DCBC379DFD3F073BCADF8F7A0B3F097A70799F7F593F0BD3CD6DB9AFD03F 0E47E1A5FEEBCC3F106B5688D56FAA3F11C017D1EACCD53F132234A9444A2E3F1491AD 0B07CD833F160E80F43A93F33F1798B0628F4E693F19303B544008D13F1AD521C7F0DF 283F1C8763BC9953A83F1E47013172AB9A3F2009FD12F511DB3F20F7274CCB0BF93F21 EAFF4616BA1E3F22E584FEC075F13F23E6B876BACA5F3F24EE99AE004CDD3F25FD28A4 91E5063F2712655A756BAA3F282E4FCFB48F5F3F2950E8045BF0CE3F2A7A2DF87A6B44 3F2BAA21AC20815F3F2CE0C31F5FE64B3F2E1E12524B1E4B3F2F620F44F532833F3056 5CFBB8B9AC3F30FF0934E9A3F43F31AB0C4E1703933F325A66474A6CC63F330D17208D 50803F33C31ED9E8F35E3F347C7D7366669D3F353932ED0E82FC3F35F93F46E9E50B3F 36BCA28100EA963EB21F3E9AF763B53EA5C95B30CBFEF63EA9A9EA559B075C3EB30FCD 0464D00F3EBCAEA1592FFDF83EC4B73FE72AFB013ECC9B3E8CCACF6B3ED2FEEDA4FA8A 703ED86E367F05F8353EDE9AC4D175B4FF3EE2C218FC51DF853EE69520570175CD3EEA C66565E00EDC3EEF55DBDD9A455D3EF221BDCAC209363EF4C79F805FD8A23EF79C9122 00ED203EFAA091524E08653EFDD39F161D36C43F009ADCDB2D619F3F02637055FCF494 3F044389C92922D43F063B290ED8D5F13F084A4E0A5E0FCF3F0A70F8A5D9B0D93F0CAF 28D08765073F0F04DE7D7F3BA23F10B90CD165571A3F11FB6D1C5AC42F3F1349901CA5 222C3F14A375CFF722043F16091E348ACF953F177A89490446E83F18F7B70C5B0A353F 1A80A77DC85EBD3F1C155A9CB9813F3F1DB5D068C4CDB43F1F6208E1A130033F208D02 038FADC03F216EE0EC9233013F2256A12BD2BD303F234442C15233FD3F2437C5AD1670 263F253129EF29204F3F26306F8796E5133F273596766E99C93F28409EBBC0C1063F29 5188579F0DF93F2A68534A1C04F93F2B84FF934AAF0B3F2CA78D333E5CA33F2DCFFC2A 0A74D63F2EFE4C77C24EAE3F30193F0E3C89443F30B6488C20D12F3F315642B59743B0 3F31F92D8AA8F2053F329F090B5EB4F53F3347D537C128FF3F33F3920FD8ABFD3EB164 58AFFD30F53EA370C7BBC239EA3EA5AF288D73126C3EAFC343397AAAEE3EB7EFF813F0 9B1B3EC15FD2A8BD215A3EC81BDF058C997B3ED01357E67227493ED4BECDEF1332D13E DA0F9C999773543EE002AEA12C738A3EE34FE9516531A13EE6EF6B4A8C657F3EEAE128 414EA9CA3EEF25180CFFF7923EF1DD9A906145173EF451BDD0A58CEF3EF6EEF469F595 133EF9B53D612769C93EFCA497FF2756123EFFBD03BCE895533F017F401AE162513F03 34868F01DB923F04FE551E27B8573F06DCABB273D9043F08CF8A3B21E9933F0AD6F0AB 49F9503F0CF2DEF8F581C33F0F23551C704E8B3F10B42987E26CF23F11E0EC672C42D3 3F1317F32A4FB4D23F14593DCFF0DEF43F15A4CC5707436C3F16FA9EBECC27863F185A B506ACC7EF3F19C50F2E3F80AC3F1B39AD353B3DA73F1CB88F1B70B2273F1E41B4E0C4 F40F3F1FD51E852D2E413F20B966045597FC3F218D5EB5A555683F226679568F7E063F 2344B5E720B26B3F2428146767CDF23F251094D77553323F25FE37375AF55A3F26F0FB 872B38B53F27E8E1C6F926563F28E5E9F6D80EAE3F29E81416DB58D53F2AEF6027165B CA3F2BFBCE279C3FF43F2D0D5E187FE74C3F2E240FF9D3DAFC3F2F3FE3CBAA3D8D3F30 306CC70A60393F30C378A0924E653F315915727547173EB0BF779A06883D3EA15E6D55 5FB1543EA221D191B2CAA63EAA088B4D741BEC3EB3A2B97855EBF53EBCA951E2D3E7DE 3EC3FF91CDF9742B3ECACCA5FE570F6C3ED15C9E30F275503ED5E1F5793F86E03EDAF5 F2346CBAA13EE04C2B8DF672F43EE3647F04F675D23EE6C3E733E3009B3EEA6A5BF210 0C6D3EEE57D7B29FD4863EF1462B4D2388983EF383EAF72BA2F83EF5E529DC3E45B23E F869E74547BBB73EFB1222AB3B40773EFDDDDBA97071043F006688FA1A90443F01EFE2 A916A9683F038AFACBCD64513F0537D1517A6D003F06F6662D35D2B73F08C6B955090F 063F0AA8CAC13FED8E3F0C9C9A6BE4E8833F0EA228505D61C13F105CBA358EB88F3F11 713F5CB6A74E3F128EA39C9DFF4D3F13B4E6F47E05EB3F14E40963C3F7C63F161C0AEA 0630DC3F175CEB86FB9D223F18A6AB3A74EFE73F19F94A04573EFB3F1B54C7E497B547 3F1CB924DB38220D3F1E2660E84436E33F1F9C7C0BCF51043F208DBB22F958D93F2151 A7CB6607C23F221A03FF3E37E43F22E6CFBE939C603F23B80B0978B9873F248DB5E000 98743F2567D0423E89843F26465A3045F3E43F272953AA2A2E863F2810BCAFFE61D53F 28FC9541D56FCC3F29ECDD5FC1E1923F2AE19509D5D9AE3F2BDABC402309A03F2CD853 02BAAA6C3F2DDA5951AD77C83EB0309B591369893E9F2497FB48CA783E9E03CAC4B460 1D3EA4EF7244B5F30F3EAF8DCB0CBFE0EC3EB72B858262E5673EC04656E6115F8D3EC5 EFBDDE4D57663ECC8F4E89477F4B3ED211CF706DEF9B3ED655F0CEDF39BE3EDB13CE19 6A73713EE025A0951E3FD33EE2FE18B55749D13EE6134744B450E43EE96526B6579041 3EECF3B32EF5C0363EF05F74F9F032203EF263648761CA7F3EF485A788BBF42F3EF6C6 3D76F1EA943EF92525ED5B4AD03EFBA260A045E7FC3EFE3DED5655C5F63F007BE5F1E6 52BF3F01E7FE1390EF423F03633F0342C7D73F04EDA8B70556013F06873B2724655D3F 082FF64DAA70323F09E7DA25FCD84F3F0BAEE6AC8FB5873F0D851BDEAB3FE73F0F6A79 BA3E7BCD3F10AF801EDDF9F13F11B157B3FF10BA3F12BAC39C18DE423F13CBC3D6E24E 7E3F14E458642C14AC3F16048143DB46B83F172C3E75E50F6D3F185B8FFA4B3E293F19 9275D119847D3F1AD0EFFA633F903F1C16FE7641B0A73F1D64A144D28E7F3F1EB9D866 36DE2F3F200B51ED4903103F20BD81D10487763F21737BDE6105573F222D401571CD2A 3F22EACE764A45FC3F23AC2700FDC6DE3F247149B59F782B3F253A3694423BDA3F2606 ED9CF89B1D3F26D76ECFD4B87B3F27ABBA2CE8455F3F2883CFB4447AEE3F295FAF65FA 14C93EAF6F87DA47A9B33E9C18C76920AB2F3E989EC7FED2853E3EA077F81F4030533E A8B8F87C1D51553EB2464030273B753EB9E05C9BA8B7363EC18FF76CC7269C3EC6FFD2 544E23C63ECD3E54FE455B003ED2255911B064673ED612379B430BDE3EDA659FF60786 F03EDF1F798B570AEA3EE21FDA04ECC5083EE4E3222BE9BD653EE7D9915F5726EA3EEB 0324E48684FF3EEE5FDAC523F0583EF0F7D8C983FF733EF2D954200C93AE3EF4D45F01 8351F93EF6E8F922360D863EF9172248C8CA253EFB5EDA497D488D3EFDC02102CAE0BD 3F001D7B2D70D8A63F0167AD1EEA56AB3F02BEA64E1DB6083F042266B51570F73F0592 EE4F36E9523F07103D18F636EC3F089A530F9B91C43F0A31303115FE593F0BD4D47BD8 07253F0D853FEEBC25A43F0F427288EF11C43F10863624EF51AF3F117196989620873F 12635A9F510B473F135B8239153CCC3F145A0D65E484643F155EFC25CA93983F166A4E 78DAC79D3F177C045F2E61CA3F18941DD8E318BB3F19B29AE619F1A43F1AD77B86F652 CA3F1C02BFBB9D44E43F1D34678434DA083F1E6C72E0E3B31F3F1FAAE1D1D09E593F20 77DA2B9124D33F211D75387F82F33F21C6420FC64B7E3F227240B17807A03F2321711D A6D9E63F23D3D3546473C63F24896755C20E503F25422D21D065303EAEA9E2AC6F9463 3E999968F44704E13E94149AD1C004C33E994439BA25A77A3EA2C6FB3EC429263EABF3 03D841D3D93EB3FA300A84D0943EBB5AA55388F9F13EC20AC7C2F8D7E73EC7140D4ABB 812F3ECCC855F9C075453ED19393A176AF483ED5181A6D4E019A3ED8F1A6C9BF67003E DD20286572D19A3EE0D1CA13565BF53EE33DF12B6B45553EE5D486C049A8483EE89588 DB9D9D923EEB80F60F3FBB153EEE96CD4D1677783EF0EB86E5E8869B3EF2A0DB7A0591 383EF46B6429865F673EF64B20C8ACB1DE3EF8401135EFE0BA3EFA4A3557800A743EFC 698D197022283EFE9E186C57BEFA3F0073EBA225EACC3F01A364CC0B94C83F02DD77B0 50F5403F0422244C3E443F3F05716A9DC286353F06CB4AA35045BE3F082FC45BC1FC24 3F099ED7C644616C3F0B1884E2454D7C3F0C9CCBAF6626DE3F0E2BAC2D71195E3F0FC5 265C507AA73F10B49D1E03F4A73F118BF3E65764353F1268978735E9483F134A8800B8 C50D3F1431C552FDAE4C3F151E4F7E25AA1A3F16102682541ECE3F17074A5FAE150C3F 1803BB16599EFF3F190578A67D5D823F1A0C8310401ECC3F1B18DA53C890BF3F1C2A7E 713D044D3F1D416F68C33D563F1E5DAD3A804E2E3F1F7F37E6987BF73F205307B69794 C63F20E919E733648C3F2181D2853068F43EAE1047289E931B3E97A67C9CBBD78A3E90 65433D7CDEAB3E92DBC0FC5BBA8C3E9B6FA6A968D0993EA48A956C9FE1B03EADB45031 6E168E3EB48F9F2A8AB5053EBB605DAA8F37BA3EC1A4C7C63E51BC3EC624CD20DD78C0 3ECB2FC4580ABB583ED062B0900FEFAC3ED372B925E7A7DC3ED6C7EB9C34786A3EDA62 3CD93AD7C03EDE41A5266436C23EE1330F8729CE023EE367D352309C833EE5BF1C861F 1CD63EE838EA14DBC58E3EEAD53B3515D1453EED940F4F68E5EC3EF03AB2F88E85AD3E F1BC9F615AE17A3EF34FCCC090DE5E3EF4F43AFC609AFC3EF6A9EA00DD0A4F3EF870D9 BE9DC3A63EFA490A29B7BB673EFC327B38F5B5633EFE2D2CE53FE11E3F001C8F949357 533F012B290044137B3F024262B424AF7C3F03623CAF0FA4ED3F048AB6F031AB823F05 BBD176F89B4B3F06F58C4305DAD53F0837E7542395C83F0982E2AA3C21FD3F0AD67E45 531DF53F0C32BA257FECC83F0D97964AE949073F0F0512B5C1B50C3F103D97B3224F2E 3F10FCF62E5A07903F11C0A4CCAB6A1D3F1288A38E3B7F623F1354F2733059893F1425 917BB099783F14FA80A7E30D443F15D3BFF7EE631D3F16B14F6BF8EBA63F17932F0428 6ADB3F18795EC0A1F3253F1963DEA189C9783F1A52AEA70350CB3F1B45CED130FB593F 1C3D3F203440503EADA2B54ED4A5E03E964002627F23213E8B2182841225B13E8B6D0C 0A4533D63E9317017BDC1DB33E9CA66A3AD140983EA500898C7ECEB23EAD7DB8BD26FC AB3EB3E00F1674DF003EB9E108E19B99833EC0601798B7D35A3EC43E4675DE2DC43EC8 8AC4BC9AA22E3ECD45613F9F9B0E3ED136FDAE1E7E873ED4023E6F7DDA973ED7046B2D 5752463EDA3D7E724DEC743EDDAD7451B9DA8C3EE0AA24F7A624363EE298FE976911A9 3EE4A3463ED4F0253EE6C8FB568565A83EE90A1D6BC279F33EEB66AC270FAEA73EEDDE A7455BB3813EF03907498362F63EF19070F41B65ED3EF2F590930D79F83EF468661A6E 93673EF5E8F1810976193EF77732BFC651B43EF91329D13596003EFABCD6B1354C1B3E FC74395CAA891D3EFE3951D14A3FED3F00061006B6F00C3F00F65207F88CCF3F01ED6E EC0B5CE53F02EB66B2B98BEC3F03F0395BED6FB03F04FBE6E7AAA6BD3F060E6F560894 303F0727D2A72DF27F3F084810DB4D44403F096F29F2A1F6B33F0A9D1DED6E14383F0B D1ECCBF8697B3F0D0D968E8B07BB3F0E501B3572134B3F0F997AC0FACDF63F1074DA98 B969F13F1120654393C0AE3F11CF5D6132BBDC3F1281C2F1BC1F7A3F133795F554FDEF 3F13F0D66C219C423F14AD8456455D4E3F156D9FB3E2B33A3F163128851B157D3EAD61 2D1F11CCB03E9565FA4590E7B43E872E29BEC942E73E83A911A8F48B2E3E89080DE9C4 73373E9299C5D5381FD33E9BB2184C7792D93EA3D0B9DF46AC013EAB294793454BC83E B1EE8694D3E4333EB6F46AC29F0A033EBCA55B38CF6C653EC1805FB00C4B983EC5031A 6EEFB0033EC8DABD3661C8053ECD0731D2EB803F3ED0C4346BAFDD353ED32F2BAC8241 E93ED5C47ABF451FBA3ED8841EC7A9A2663EDB6E15A97CB76A3EDE825DD21CD39F3EE0 E07B0960A1933EE294EEC2FD0AB13EE45E89BE6726BB3EE63D4BB88DA5D13EE831347D D0C4AC3EEA3A43E65669D73EEC5879D34DC3A63EEE8BD62CE0BB033EF06A2C70526BC6 3EF19900F0353C013EF2D26890A93E9D3EF416634D8C7C073EF564F123C409C33EF6BE 121104DA6B3EF821C613A85E083EF9900D2A8A44353EFB08E754ED5A193EFC8C549265 F76A3EFE1A54E2C8C76B3EFFB2E8461D074E3F00AB075E48BEA63F0181E42339CF393F 025E0A721437BC3F033F7A4B1365AB3F042633AE7963403F0512369C8CFD5E3F060383 1598450C3F06FA1919E75EC23F07F5F8A9C78C093F08F721C58669943F09FD946D7152 E23F0B0950A1D4EA053F0C1A5662FCB6963F0D30A5B132DD5B3F0E4C3E8CBFE89B3F0F 6D20F5EA9E0F3F1049A6767BF0F03F10DF613915516C3EAD4BAE995607873E95186445 F125393E84F07C2B1F14903E7CD725A98AF63E3E7EDADC51EC90773E85DE7B50E8C12C 3E907BAFFD4A0D7D3E98308376E8F0543EA0FC378831189F3EA6E4114C4B07863EADCC 99EA9237563EB2D9F3974EA8C23EB74C63F4E9B7B43EBC3D3BB37F1D583EC0D61CC6D7 51533EC3CC988E4511173EC70201C2DBAECD3ECA764D79E138363ECE2973DA0610953E D10DB7955647723ED3261D99B7479B3ED55DEA67847C663ED7B51CCFF5333E3EDA2BB3 ED997B2B3EDCC1AF117857143EDF770DB56F26C33EE125E7B91F32643EE29FF9FC18EE AC3EE429BD84D3F6E13EE5C3323B7A33B13EE76C580DA12C8D3EE9252EED1940CA3EEA EDB6CF03509C3EECC5EFAB1B6D733EEEADD97B2BC1B63EF052BA1D4F199F3EF1565FF3 12D0AF3EF261DE3DBCB6163EF37534FC8F96D93EF490642F1FCA1F3EF5B36BD541FC1A 3EF6DE4BEEFD69653EF811047C80D85F3EF94B957E19BDF93EFA8DFEF42D1F063EFBD8 40DF31D6873EFD2A5B3FABF8EB3EFE844E16291F903EFFE619633D6D853F00A7DE93C0 95503F01609CB1C76D343F021D470C00DB333F02DDDDA2BA3AE43F03A26076402DED3F 046ACF86DE3C823F05372AD4DE8AEA3F0607726089A34A3F06DBA62A264BDF3F07B3C6 31F968AD3F088FD27845E8503EAD6239BDA1566D3E955740639FDBAB3E846879C9139B 413E77691F1B7006B53E72BC476A9B75B83E769B46D2180D123E80BBB456EA1B393E89 514146C183AD3E9271DC175A47F03E99AE3C5641DFE43EA12BAB92DBB69F3EA634AC0E 74227E3EABF0EE6F514ABB3EB12FD7001E6DC23EB4C0341B3B33003EB8A9622210CEAA 3EBCEB423A4503D83EC0C2DF4731F7223EC33C63B5E729033EC5E228B0A46FC23EC8B4 29FF03A6CE3ECBB2647BC1B64A3ECEDCD5C94D37E03ED119BE0E6324EE3ED2DB2B0C1D 79413ED4B2B157B2F8CA3ED6A05089E41E2B3ED8A40852FCB5033EDABDD8755B1AA73E DCEDC0C15321213EDF33C1120FD6F53EE0C7ECA598FD3E3EE20104AB7DA6623EE34528 9271F82B3EE4945854401D1D3EE5EE93EC51FAAF3EE753DB575A70AD3EE8C42E9310DE DE3EEA3F8D9DFADF6A3EEBC5F8774120E23EED576F1E8CFADE3EEEF3F193ECE77E3EF0 4DBFEBDF39663EF1270CF54E43B33EF205DFE6A897263EF2EA38C0650EC03EF3D41783 07BF543EF4C37C2F1E421B3EF5B866C53CB9893EF6B2D745FB6E393EF7B2CDB1F4E357 3EF8B84A09C4523E3EF9C34C4E0472973EFAD3D47F4E87943EFBE9E29E39A0C13EFD05 76AB5A08193EFE2690A740D16E3EFF4D30927B88B53F003CAB36C9FB153F00D5811C87 FB9B3EADA4CE8BF3B9593E96228E9E9D0B0B3E85962298A6D6903E77080FA79848F93E 6B68BA3B2B2C653E66040FCB683BD73E6AC4B6B7C494F83E74272D0FD333F53E7EFB58 EDCB747C3E86D8CF18A6073A3E900C14FC3524BA3E959A0A224694EC3E9C13E4A15583 8F3EA1BC0D877AA3203EA5E2D42AD280983EAA7DED6CD60DD73EAF8D1B98D0CF1F3EB2 881981ED7EDF3EB5838A245B117A3EB8B8D44275CECD3EBC27EF6B716C9E3EBFD0D554 C7BD583EC1D9C0A1AAFA043EC3E7F7D0AE25C33EC6130ED97B69613EC85B04AFCD8366 3ECABFD8852569F43ECD4189BA1AB4153ECFE017D36A1BAD3ED14DC138DDA38A3ED2B9 E4A5B228213ED4347613D35B813ED5BD756D82FD1C3ED754E2A23930A73ED8FABDA58A 4AA03EDAAF066E4A16B53EDC71BCF5DE53973EDE42E137B5C0A23EE01139986DC9813E E108396FD093F53EE2067021AC96F73EE30BDDAE104C933EE41882155941503EE52C5D 58225F3E3EE6476F7735AFF43EE769B87380E9333EE893384E0C33F13EE9C3EF07F2C3 2E3EEAFBDCA25CDCB23EEC3B011E7B0FFF3EED815C7D8264ED3EEECEEEC0A94D633EF0 11DBF4929C5D3EF0BFDBFC1456853EF171777770E5963EF226AE673C92813EF2DF80CC 08719A3EF39BEEA6620E3B3EF45BF7F6D331923EF51F9CBDE1B8D93EAE136D044D3055 3E977A4EF6E8B3673E88797699D8C6663E7BB3F74E03BC3C3E6F863AD5B5E2B43E61E2 73B94B6CE53E56BE5FFD49CF653E5540358A5111A03E5DF5D7CABF06BC3E681521D64B D69F3E72DDB5627656893E7BE791B55650623E838F4E377F4C7C3E8A3E57F2DE42643E 90FF6DF0E32B353E9567BB666328E13E9A5798F08624B83E9FCEAF397C8F1B3EA2E65F BB64BDA93EA628CDF1BFD4C03EA9AE915E36F77F3EAD779D6BBDCAEC3EB0C1F4520759 CD3EB2E9B5EDD514F73EB53310CD6423853EB79E02D82A06C13EBA2A8A7128A58A3EBC D8A6598F325A3EBFA8559AD7217B3EC14CCBBB27DC1C3EC2D635AC529A493EC4706868 2D47C23EC61B63C33B647C3EC7D7279C6D37AE3EC9A3B3DAEB6AE43ECB81086C5D9437 3ECD6F25438F31683ECF6E0A575DC2CB3ED0BEDBD0EFD89F3ED1CF168FDBDBDF3ED2E7 B567C599503ED408B858CA04F43ED5321F63A438033ED663EA898A080A3ED79E19CC0F 860A3ED8E0AD2D101D643EDA2BA4AE9C22713EDB7F0052E9FA453EDCDAC01C4A2DB53E DE3EE40D1DE0A53EDFAB6C27CF1F1C3EE0902C376566BC3EE14ED4723DE5D93EE211AE C5A3931D3EE2D8BB32C48DA83EE3A3F9BAC969173EE4736A5ED44D413EE5470D200052 7F3EE61EE1FF6107763EE6FAE8FE022A7F3EAEAE1526ADBB553E995E816C82D4B03E8D 1275CCA96B1B3E82B66B0759303E3E78E888526B88453E7068DCB6ECD5193E64BE3D30 49676A3E58748E1718E66E3E4A6E3B60B03ACD3E3C3C9869514EB53E36580EBC2FDDAD 3E411CDFC2A842FB3E4FA489BC1FC7303E5B48EBFE7491BD3E657109F53A47B93E6F49 5173430A4A3E7594B8970679EA3E7C87573B8DC9833E823DC4BEA1D6373E86B84C2439 91F23E8BB2FE482BD3633E9096D46B238F8B3E939412FABA82563E96D12C7BDBBAF43E 9A4E15FEF763BD3E9E0AC721E389163EA1039CB853DDE63EA321B3F8006A723EA55FA7 62EA45DD3EA7BD757BB00E783EAA3B1D1BAA32263EACD89D5FDE9EE63EAF95F59A5B61 583EB13992A370C87C3EB2B8160109E76E3EB44684C1483E473EB5E4DEC9C2C7413EB7 93240834818C3EB9515470CA3D4F3EBB1F6FFCC972B63EBCFD76A97CDAA43EBEEB6877 5844E83EC074A2B4A40CA13EC17B86C211D1913EC28A60671D99E23EC3A12FA77E3AE1 3EC4BFF487545BDA3EC5E6AF0B0CC4BF3EC7155F37488D093EC84C0510C9F5A73EC98A A09C651A4F3ECAD131DEF3C5283ECC1FB8DD4BB3A33ECD76359C36FE723ECED4A8206E 23DD3ED01D883749B9963ED0D4B74597C4923ED18FE13D5760D93ED24F0620AFB45D3E D31225F1B832D43EAF74C6F3155A643E9BCF25FF6B6EFB3E91B090188C62363E8A1955 F4D21B103E84924EEA23ACA83E80B4783A44BE1B3E7B70EF265907B73E768E6F796915 623E726C4D59FE3FA33E6DBA90073311C73E678F6D197501C53E62388EA42CF08A3E5B 45C37B98CF483E53843625DBE9FB3E4A3847DE2B72043E4004E63778F9F43E30BDB455 0464433E19D9C7EAA2359A3DF2452C0A5518143DE77CAD751329823E14D2E4BDA29362 3E2D11070B70316C3E3CB2894BC8DA413E47D79A60ECA3333E51DEF280FC8BDB3E5905 8C21E5F5EB3E60AF9969D5D80A3E6575CB710D9BF53E6AD53D55C85FCC3E7066EBA0F7 6A723E73AFC365848B3B3E77451EE0BAECFE3E7B26F8A30AB8923E7F554C8A7CC3933E 81E80BBE1308CA3E844BAB967CE2DA3E86D584FB33D17F3E8985975A0F910F3E8C5BE2 5476C38D3E8F5865B4955F6B3E913D90B260694B3E92E20AB44BF22B3E9499A0EBBECF AA3E9664536C163EC73E9842224E937AEE3E9A330DB0ED385D3E9C3715B428964E3E9E 4E3A7BABF9113EA03C3E164006823EA15AED765B6DC33EA2832B7178AC813EA3B4F81A FECAD73EA4F0538644D8BB3EA6353DC67339803EA783B6EE6BDC893EA8DBBF10B7F72D 3EAA3D563F7A15623EABA87C8C63A4293EAD1D3208ADB4D73EAE9B76C513F0643EB033 C134C206BF3E9ECC3CAFA282283E95B2BAE39369383E92015E37B64F323E901DD77C1B 99883E8DF8BA8ABDD51C3E8C7284D4F7B91C3E8B51F89EBE6B693E8A6C83DE343A643E 89AB863119A6003E89022B22AE76CC3E8868C9E14D56CA3E87DA9DB2929F133E875493 C1A500F83E86D4A1E382CA663E865964E0BABAD93E85E1E5E67AF19E3E856D7649244C A63E84FB9873BCEB4D3E848BF0DC6A54EE3E841E3BFCCA33493E83B24780AB21703E83 47ED8EF6A05A3E82DF1179AE9D913E82779D61B37A293E82118082B147C23E81ACADF4 B732B53E81491BC13EDF3B3E80E6C23413BF963E80859B57B04F183E8025A291763DA9 3E7F8DA8AAFAB9A33E7ED25BD7FB990F3E7E195A87F7C5803E7D62A1A004479D3E7CAE 2ED514770C3E7BFC00809789C43E7B4C157E3D8C433E7A9E6D10D3C0533E79F306CCB1 88BA3E7949E286826C213E78A300458360BB3E77FE60387ADC7C3E775C02ACDDD4C63E 76BBE807B10DF83E761E10BFCF4C993E75827D594D72633E74E92E61C4932D3E745224 6D575D033E73BD60144E8A0E3E732AE1F12D49D53E729AAA9F2CA39B3E720CBAB9040F B03E718112D7F72D093E70F7B39315E2023E70709D7EA95B6F3E6FD7A2578A68363E6E D29E4FECEA7B3E6DD22FFA09A1563E6CD658619B53193EB0C323C4FCEA4F3EA12AE2BE 9407223E9A8FBB4769CA943E98394F3B945D1A3E97B83229B82B213E98009AA670D700 3E98AEDB88B9FC383E99976A34805F423E9AA4FDA4D527DF3E9BCC45A898965F3E9D06 D8056A8E803E9E50E0527865033E9FA7FC329048DE3EA085511DA2C89A3EA13BE6AB8B B0303EA1F765F5268F203EA2B79145BEEAF73EA37C3CF26CFABA3EA4454995F248343E A512A04DD8345B3EA5E430388628963EA6B9ECC0EFCA023EA793CC70CF7F883EA871C8 1CA5BDC23EA953DA4CAAA3A13EAA39FECFC929CD3EAB24326C052A053EAC1272A3BD02 063EAD04BD89E397073EADFB11A1177F993EAEF56DC2B12C303EAFF3D10BB691A43EB0 7B1D671AD8E13EB0FE5542F7D7133EB1838FE7C38FE23EB20ACD30CD36023EB2940D03 ABC2853EB31F4F4E1A36E03EB3AC94044561433EB43BDB1F72BB253EB4CD249CECFB09 3EB560707D27F3C23EB5F5BEC3100D8E3EB68D0F737C90373EB7266294BDCC3B3EB7C1 B82E4173A13EB85F104848C6CD3EB8FE6AEBAD66B13EB99FC821B178A93EBA4327F3D9 808B3EBAE88A6BCDB9453EBB8FEF9341AF873EBC395773E0E22D3EBCE4C2173F81EE3E BD922F86CE8C523EBE419FCBD29F853EBEF312EF5CF6383EBFA688FA4651F63EC02E00 FA95970C3EC089BEF434B6F43EB1688B2A3B57E73EA335E033FE09BA3EA023C8A207C3 2A3E9FB47E060337413EA08C1BBEF3C5E73EA1B37A202C13133EA31E839228509C3EA4 B772C6DEE0C53EA673A33BD858543EA84D6CD884D62A3EAA419A7FB44D833EAC4E41FC FD67263EAE723221C73C1D3EB056531C6DBAF03EB17E8DDBBCD4193EB2B19CE7E2C2CF 3EB3EF6166854A3B3EB537C58230850F3EB68AB9884539093EB7E832078815683EB950 268F2CD1273EBAC290D4AD3D2D3EBC3F6C1CE68C803EBDC6B4D218FAD53EBF58683860 96923EC07A421B996E923EC14D8398C8DCB03EC225F7F4562D743EC3039EB2FE1C6A3E C3E67775684B0F3EC4CE81F1EAD8DD3EC5BBBDEFC7423C3EC6AE2B438105AE3EC7A5C9 CC086E343EC8A299708620D23EC9A49A1EA1B7FE3ECAABCBC926AEF73ECBB82E66F285 843ECCC9C1F21BA4C23ECDE0866744C8923ECEFC7BC5134E083ED00ED105E08E013ED0 A1FC9E6232933ED137C0AD44690B3ED1D01D341B44B63ED26B1234C22FB73ED3089FB1 497F793ED3A8C5ABE796963ED44B8426ED01383ED4F0DB24BADED33ED598CAA7BB410E 3ED64352B25B07A23ED6F073470513FD3ED7A02C681E72863ED8527E180363B53ED907 6859050B8D3ED9BEEB2D67BA563EDA790697619F8E3EDB35BA9919E0973EDBF50734A7 EED83EB223F7647D4F863EA58716B80F48BA3EA36D1E6CC24E2B3EA439754B816EC73E A61EF3BC54DD713EA897B5098A6B513EAB71E29E9FE57A3EAE97B72D8BAEF63EB0FF47 19EB2DE43EB2D05FCC7805AB3EB4BD8B1FD457DE3EB6C5D3F9CB79D03EB8E8A1C310F7 0A3EBB25922109A54A3EBD7C63CE554DDC3EBFECEA6405C9D43EC13B8383E06E393EC2 8D51F20950563EC3EBD8A32DEDF03EC55711DEB09F673EC6CEF96C2B3F113EC8538C26 5AB65B3EC9E4C7AFAD9DD73ECB82AA3D4414373ECD2D32712C1F953ECEE45F3F2080DA 3ED05417EC51158E3ED13C51CF245BB13ED22ADD0A686FAC3ED31FB96E7124103ED41A E6D66987303ED51C6525F257233ED62434474D55313ED7325429F2A3883ED846C4C176 97B23ED9618604ACFC8E3EDA8297ECFB91F53EDBA9FA75D116053EDCD7AD9C38BDE13E DE0BB15E83E7ED3EDF4605BC0541D93EE043555A6CDFC53EE0E6D024DE3D9F3EE18D73 3DF1A3AE3EE2373EA6711C553EE2E4325F4A5CC63EE3944E69858E9E3EE44792C63DE4 123EE4FDFF769BA3503EE5B7947BCF5BF23EE67451D70E15E23EE73437898E42853EE7 F745948551493EE8BD7BF925C9C43EE986DAB89DCAFE3EEA5361D415E6903EEB23114C B046383EEBF5E92388009A3EECCBE959B0A8083EEDA511F035F6103EB2F56873C2D128 3EA81E864AC7C43B3EA723DF03E486653EA93A4A7749A8423EAC94A10CFF5C963EB056 7F07A9BA433EB2A8C574E1DEB93EB5364127234CDF3EB7F99FDD324EF63EBAF00D8AC6 F4C13EBE17EFA14937763EC0B828834235C53EC27C4C91960EDA3EC45832CE5291843E C64BBA96C7A2243EC856CDB77E2DF33ECA795CC34916123ECCB35CCF6E684A3ECF04C6 029E872C3ED0B6C9521DE5483ED1F6DF3DF006CE3ED342A332241C813ED49A13FFF173 343ED5FD30C1E8197A3ED76BF8C90F134D3ED8E66B8F43BFE33EDA6C88AD469A8B3EDB FE4FD3636DDB3EDD9BC0C3F8982E3EDF44DB4F59EB8C3EE07CCFA8593AE53EE15D0655 D17AC23EE24311A53695EE3EE32EF18E449D4E3EE420A60AC5BC7E3EE5182F1623D795 3EE6158CAD11CFA83EE718BECD4703393EE821C575490D113EE930A0A4409C883EEA45 5059D707A13EEB5FD4961AC95A3EEC802D59696D923EEDA65AA45DDEF73EEED25C77C2 27773EF002196A41FE1B3EF09DEEDDD5C2833EF13CAE9729FBCF3EF1DE5896D2CBCE3E F282ECDD687A6A3EF32A6B6B858C033EF3D4D441C539073EF4822760C239A13EF53264 C915D0873EF5E58C7B570DF53EF69B9E781A3B7C3EF7549ABFF06CF73EF8108153672D 6E3EF8CF52330846CA3EF9910D5F5995FC3EB3DCDE580BDCCD3EAAFC2EEC277C323EAB 480A676E6BB53EAEDCBE865A47B53EB1F691D879A1663EB4F9AA98C397323EB85EBE39 C9EBC13EBC1AEA1487D0AC3EC0146DF3E0C7BD3EC242DFD3979C3E3EC497FD621C6264 3EC7134C1254C4B33EC9B47F989A88C33ECC7B66C7A019F03ECF67E0FCB55BF53ED13C EC02319E603ED2D89E38BE4E6F3ED48701ACA3C4E83ED64812713A33823ED81BCDAA32 C6173EDA02313B5A69D33EDBFB3B920F93683EDE06EB7F898F813EE012A00F2C36853E E12B1C5FC197AF3EE24CEA6E739B373EE3780A07A27E623EE4AC7B037426EE3EE5EA3D 4317C3B73EE73150AEB73D913EE881B533E81BB73EE9DB6AC47ABDA03EEB3E71559003 AA3EECAAC8DEE3FE653EEE20715A40D9BA3EEF9F6AC3107A1E3EF093DA8B02DFE23EF1 5CA8286C052F3EF22A1E390679C83EF2FC3CBC66959A3EF3D303B26103CF3EF4AE731A FD01963EF58E8AF66955453EF6734B44F373533EF75CB4070061E03EF84AC53D06FA7B 3EF93D7EE78B526A3EFA34E1071B02843EFB30EB9C4A2E1E3EFC319EA7B11D0A3EFD36 FA29EA53473EFE40FE239109353EFF4FAA953FF7083F00317FBFC82FF43F00BD7E718C A9AC3F014BD160378DBF3F01DC788C1165C13F026F73F560F7F33F0304C39C6B28613F 039C678172E5DA3EB4DA591158727A3EAE20109C2E70A53EAFD9A0975FFE533EB29068 BC59A6B13EB6143DD418484E3EBA355D3812CC9C3EBEDADB9E081A543EC1FCEB2F79B1 633EC4C67D9CCC77D93EC7C8BB10D868823ECB02D64451802B3ECE7454AA1D692A3ED1 0E74FB4AF4C13ED2FE327E36B5CB3ED509528C79EA433ED72FCA0C59086A3ED9719147 3D80E03EDBCEA2C7CA5ABF3EDE46FAA157C6E73EE06D4AFBCB78FD3EE1C4B9572A645D 3EE329C79977DFF23EE49C752B4F92253EE61CC199F982653EE7AAAC8DF9324C3EE946 35C43D51D63EEAEF5D09261E7A3EECA62234D97DB23EEE6A8528869E8F3EF01E42E62B B4C73EF10E1206F0B32A3EF204AFEF7A7A363EF3021C9A59798B3EF40658036BBA843E F51162279651FB3EF6233B048E32F13EF73BE298ACCCF03EF85B58E2CDCFF43EF9819D E23409653EFAAEB19673D04F3EFBE293FF61D04A3EFD1D451D05469C3EFE5EC4EF8CF9 923EFFA71377465DC53F007B185A4B3C5D3F01260E53FA121F3F01D46BA8F1BB0A3F02 86305979036B3F033B5C65D9FCB83F03F3EFCE60CB3A3F04AFEA935AB05F3F056F4CB5 1548CE3F06321633DDF09D3F06F847100149403F07C1DF49CADAA93F088EDEE184C952 3F095F45D7779F883F0A33142BEA233B3F0B0A49DF2138D03F0BE4E6F15FCE3D3EB5ED D89FA8922B3EB0C515AD6E50CC3EB26C50C9DC9F093EB60341A72A5C9D3EBAA354795B A2E03EC004CB72CBAD473EC30E8ED0CE350C3EC66983033301793ECA12FEE95C37B83E CE09987D25DF773ED126413BA1FA983ED36DA1254E11CB3ED5DAC5D5C418543ED86D96 B65D43B43EDB260375C185D93EDE0400FA3555563EE083C3C69111733EE21848DC957A C73EE3BF8DC8D3FF263EE579911D25348B3EE74651CB6F91E73EE925CF0A62510B3EEB 1808429918CA3EED1CFD015BF05D3EEF34ACEF2E597E3EF0AF8BE47F82173EF1CE1EAD 9716AE3EF2F60EBEF0DDC63EF4275C09246E4F3EF562068046BD033EF6A60E1B228DB3 3EF7F372D2A00F723EF94A34A14FB2CD3EFAAA53830F80803EFC13CF74C48DB43EFD86 A87423CD3D3EFF02DE7F86AEE63F004438CAE4713B3F010BB0DB169A983F01D7D77023 FF323F02A8AC89F6F4F13F037E3028931A193F0458624C0FD21E3F053742F493D7343F 061AD22251ABD13F07030FD584BDB13F07EFFC0E6F16DE3F08E196CD5784273F09D7E0 128816C83F0AD2D7DE4CF2D13F0BD27E30F35A653F0CD6D30AC8E9803F0DDFD66C1AFC 603F0EED88553634823F0FFFE8C66619C33F108B7BDFFA68913F11195AA11572113F11 A990A6A78CAB3F123C1DF0D32A463F12D1027FB9C23A3EB7175D02FC3BE23EB29D3F94 19077F3EB52286AE3D157F3EB9C6EA039F45933EBFA3D5C843B1303EC33B2BD0A8A05D 3EC712C222436D983ECB533C856FD8E13ECFF9F1D99007233ED282BC0C4000433ED53A 80FD79E0993ED8240A79E879FC3EDB3F325BB8AF753EDE8BE00C43B64E3EE105019D18 C0663EE2DCC865E342923EE4CD4085361A253EE6D6674062CA943EE8F83AA1156C7E3E EB32B9392695CB3EED85E1FA7CBD2B3EEFF1B41BC71D0E3EF13B1782D0ADD63EF289A9 22A9C0373EF3E48EC1B086953EF54BC83E5C58E53EF6BF557EDD55863EF83F366F466F EA3EF9CB6B002F3EF33EFB63F325ACB7683EFD08CED6899CAA3EFEB9FE0BAE1EA63F00 3BC05FD556A23F0120AB772EA8A13F020BC04A5590053F02FCFED82486133F03F4671F C842E23F04F1F920AE9E2C3F05F5B4DA78FE973F06FF9A4CF190023F080FA97802A7C0 3F0925E25BAFE4F23F0A4244F810AA493F0B64D14D4BB27D3F0C8D875B937FD13F0DBC 6723237FF63F0EF170A43DBCEE3F101651EF9481BC3F10B7006A17B2833F115AC3C1CE 82DA3F12019BF6DF93A83F12AB890971B3283F13588AF9AB8F2D3F1408A1C7B378C63F 14BBCD73AF33AA3F15720DFDC3D2433F162B63661599C83F16E7CDACC7ECE73F17A74C D1FD3DF33F1869E0D5D704183EB856E63B536F9D3EB4988602175C6A3EB80F71F8D162 7A3EBDDB61D1B861A23EC28AE0E068398C3EC6BDCFB5A03FBF3ECB7A07C363B6C63ED0 5D0BDB181BA83ED33DAB36B3F32E3ED65E2CF17365F53ED9BE2A67B072283EDD5D6652 DDED003EE09DDD46945CE43EE2AC873FF506C33EE4DAA8ECE4ED5B3EE7283CC0864BCA 3EE9953EDF8DDA863EEC21AC8F4D1CEB3EEECD83D9702B943EF0CC61A7E7CE403EF241 B4F228F3433EF3C6BB66D321DD3EF55B74BA342D483EF6FFE0B2F0193E3EF8B3FF2548 A68F3EFA77CFEFB52D3A3EFC4B52F865CBF13EFE2E882B6D75833F0010B7BCB1E0AD3F 0112046B2ED1F33F021B2A1C92F0653F032C28CD5253C83F0445007AB534493F0565B1 22AA96563F068E3AC3A504543F07BE9D5C7EF7BD3F08F6D8EC6528A03F0A36ED72C56E B73F0B7EDAEF4130AE3F0CCEA161A29A423F0E2640C9D401093F0F85B927D903C73F10 76853DE482C93F112E1A62E5605E3F11E99C03085C0C3F12A90A1E6B2C563F136C64B5 2ED6703F1433ABC776C0AF3F14FEDF5567F3D73F15CDFF5F2881CA3F16A10BE4DF0BB8 3F177804E6B25F853F1852EA64C92B743F1931BC5F49C0193F1A147AD659E16D3F1AFB 25CA1EA1FE3F1BE5BD3ABC46823F1CD441285632503F1DC6B1930ED7323F1EBD0E7B07 ACA83EB9AC7448AE2D5E3EB6B6E8F7694F933EBB3312A99986183EC1205488BAD85F3E C57C8C3180F4553ECA8CB721B28B633ED0222FDA17885D3ED34F0A4ABA0E893ED6CB96 5271EABD3EDA971EEE2D20E23EDEB13D7A45AF473EE18CDA581735933EE3E82F350A1B C93EE66A9108A824DF3EE913F7AA454A113EEBE45D8D03C6833EEEDBBED59852A53EF0 FD0C64AA38CD3EF29FB4B8F21E3E3EF455D7B09024873EF61F74C47786E03EF7FC8B8F FFE2A83EF9ED1BC7770AEA3EFBF1253181035E3EFE08A7A25F8D063F0019D17C44FFC9 3F01390B8D183CE13F026201F9B2F7553F0394B4BA60FAA53F04D123C92CC1403F0617 4F217BACD63F076736BFC1D5693F08C0DAA14772B53F0A243AC3FB897C3F0B91572650 A3F23F0D082FC7213BC33F0E88C4A59A08F83F10098AE09471903F10D3918CB798793F 11A276571B8F243F1276393FB5805C3F134EDA46873B853F142C596B9C71FB3F150EB6 AF08813F3F15F5F210E4AA563F16E20B914EA41A3F17D303306773783F18C8D8EE527F 0A3F19C38CCB34CF4F3F1AC31EC734766F3F1BC78EE27815543F1CD0DD1D267A2E3F1D DF09776652D43F1EF213F15DF0683F2004FE45998B403F209361A2856BD63F2124340F 84BC683F21B7758CA92E883F224D261A03FB1F3F22E545B7A5DE193EBB18072B0C7523 3EB8F868740EE0FF3EBE8D68C09580563EC37B5FE16B99713EC8A6ECD76C09023ECEA7 E214DF83253ED2B8E4FA52BDA33ED67F99919DC4EE3EDAA6BA4001EA133EDF2D92026D 30F03EE209DD1A9CCBF23EE4AC7AC8ECFA0F3EE77E8EF93D94483EEA800D603B35693E EDB0EDD539D65B3EF0889565ADD9503EF2506033AAC12B3EF42FD5F73C647B3EF626F5 B538CF903EF835BEB68C4D803EFA5C30742A19873EFC9A4A8969D0D83EFEF00CAA9946 9B3F00AEBB4F2E3F463F01F1441C7A9CBE3F033FA0AC6D67AD3F0499D0F21EAF713F05 FFD4E397ED8E3F0771AC7924ED7E3F08EF57ACD029A13F0A78D679FF03DB3F0C0E28DD 2593D53F0DAF4ED38C11AD3F0F5C485B2181CD3F108A8AB92C376E3F116CDB0C05A90B 3F12551525B371C33F13433905EC7D863F143746AC817FA73F15313E19578DFE3F1631 1F4C63D3663F1736EA45A81F1A3F18429F053022BD3F19543D8B0F3B9A3F1A6BC5D75E AAC43F1B8937EA3C27083F1CAC93C3C8B57F3F1DD5D96427BCAF3F1F0508CB7E44EF3F 201D10FCF9304E3F20BA9277D558453F215B08D667016C3F21FE7418C182CC3F22A4D4 3EF804B53F234E29491D69473F23FA73374439873F24A9B2097E982D3F255BE5BFDE36 6B3F26110E5A744D943F26C92BD951992D3EBC999EE26E46EF3EBB5D04780810AE3EC1 0F3A1EE2A8903EC5FED2F2EE74163ECC0A02D22977883ED187A8479393903ED5812342 637B3A3ED9EEB9AFC33EFC3EDECF16FF63F1593EE210C31719CB273EE4F2D04C461611 3EE80D947BF044083EEB60FC932EC6983EEEECFC46AE38423EF158C5B6E149303EF357 523DC708053EF57221CA62B4DB3EF7A932FF5D11693EF9FC84E18C29CC3EFC6C16B9DC 494F3EFEF7E80140AB273F00CFFC2988763F3F023223B1CD702E3F03A26A7CC145543F 0520D07484D61D3F06AD558853CE923F0847F9AB463DCB3F09F0BCD3659D763F0BA79E F8FDB9253F0D6CA016190B203F0F3FC0261CF5433F10907F92BEC79E3F11882E88C188 9F3F1286ECF40E3FCF3F138CBAD3DE32703F149998279E9D503F15AD84EEE5DC563F16 C881296ADB763F17EA8CD6FE4DEC3F1913A7F78549A73F1A43D28AF4F9903F1B7B0C91 4F2CCA3F1CB9560A9F95043F1DFEAEF6F98F833F1F4B1756765D573F204F479499DAB0 3F20FC8B37A94E533F21AD56947B3CCF3F2261A9AB222A603F2319847BB120383F23D4 E7063B6EAB3F2493D14AD47CDC3F255643498FA1BA3F261C3D028005CC3F26E5BE75B8 8AEB3F27B2C7A34BBA5D3F2883588B4BB68A3F2957712DCA30F93F2A2F118AD862D33F 2B0A39A28707AC3EBE313B6ED3A2C23EBDE4BD0354DE893EC2F31A90947C413EC8AAAD BD4368353ECFA5CE21B93FDF3ED3E1814844BBB53ED87AEAB249C1213EDD9C6AA52A7C 883EE1A256484C00393EE4B97DB8C028623EE81378521EB5E63EEBB0277121136B3EEF 8F7802DDB2773EF1D8AEDE0096C83EF40AE839EFBF0D3EF65E654ECD6F7C3EF8D3242E F404713EFB69237D0C3F6F3EFE20623DEC2CFC3F007C6FDD400BF83F01F94DB5DD9DD7 3F0386CA767A9ABC3F0524E5F93DEC263F06D3A0217994053F0892F8D94E72A83F0A62 F00FF8342F3F0C4385B88EE7CC3F0E34B9C91C07113F101B461CF5AED23F11247E8283 B2D23F12360612EAC0863F134FDCCC64E3D83F147202AD9638B53F159C77B57641613F 16CF3BE33E432C3F180A4F365B7ABB3F194DB1AE64441E3F1A99634B0F8B263F1BED64 0C2E033C3F1D49B3F1A4C21D3F1EAE52FB68F2CF3F200DA094BE324A3F20C83F3DF564 693F2187057963BE743F2249F34715E1003F231108A71AA77F3F23DC45998294653F24 ABAA1E5F5AEB3F257F3635C37F5B3F2656E9DFC20AE83F2732C51C6E4DD33F2812C7EB DBAF3F3F28F6F24E1D864E3F29DF444346FB3E3F2ACBBDCB6AF0213F2BBC5EE69BEDF6 3F2CB12794EC179C3F2DAA17D66D1E3B3F2EA72FAB303B043F2FA86F134629973EBFDE DCD03C889C3EC047C90AFAA5563EC4F255B5603B3E3ECB7EF0406A75E93ED1BD27630D B1033ED6617C0C8339F63EDBA63B4A058F6B3EE0C45638E9BEDE3EE403BD79CF0BB93E E790F8E629B0443EEB6BD52C26AB8B3EEF9433A87F68443EF20500A4252BFD3EF46698 E01A0AB03EF6EEDE73C84CA43EF99DCE98C10FA33EFC7367615EAFC93EFF6FA77049EE CD3F014946E52C6C803F02EE0CDC3BDCB33F04A62559CCE5A53F0671902B8B55F43F08 504D2B9E17263F0A425C3D572B273F0C47BD4AD772583F0E6070435A98B43F10463A8C FC56C63F1165E5E25D953E3F128F3A1DF6ED743F13C2373CCD9CA73F14FEDD3C9453AC 3F16452C1B851E7F3F179523D844190E3F18EEC471C8C59E3F1A520DE74C69753F1BBF 00383C414E3F1D359B642EA90C3F1EB5DF6ADA8CBC3F201FE626084FC83F20E9B103DA FBBB3F21B8504EDFDFAA3F228BC40717E3253F23640C2C88DF133F244128BF3C81F53F 252319BF3F6C773F2609DF2CA079F33F26F57907702CF13F27E5E74FC038B33F28DB2A 05A321783F29D541292BF0503F2AD42CBA6DF5B93F2BD7ECB97C98C83F2CE081266B30 7B3F2DEDEA014CE5213F2F00274A3498DF3F300B9C809A6A5A3F30998F932FDDA23F31 29ECDCE37F003F31BCB45DBDEAFA3F3251E615C77F6D3EC0D14183547C3B3EC1AEC1D7 F4AA843EC70CEB8D45E5883ECE7B9A7C639D263ED3C3C25FA7EF083ED90798944F0E4C 3EDF03150996E6023EE2D9BF8ADF21423EE68BC1143B1B2D3EEA97349F5662B03EEEFB E6DA5DF7073EF1DCDC9105A1573EF4684C31BA5BAE3EF7203C29A377B03EFA04A8646A F2203EFD158E1BA1E88C3F002975B0D15B8E3F01DE5F6C8B0FBB3F03A983C36916FD3F 058AE259E196D13F07827AEC6E2CF43F09904D48BAA7E73F0BB45948EDF1363F0DEE9E D05A0AD63F101F8EE48FEA903F1152EB113D7E113F129163E7C1C78B3F13DAF9632183 C63F152FAB7F82988B3F168F7A39EA42FE3F17FA65900B340B3F19706D801F779B3F1A F19208CB29BE3F1C7DD32905CC933F1E1530E008A56A3F1FB7AB2D40F1083F20B2A108 2285A43F218EFAC465F0203F2270E2CB53118D3F2358591CDC74D73F24455DB8FCAF79 3F2537F09FB4A91B3F263011D10A3A7F3F272DC14D0712443F2830FF13B7D1093F2939 CB252B52133F2A4825817218113F2B5C0E289DD6293F2C75851AC110A73F2D948A57EE D0993F2EB91DE03A667A3F2FE33FB3B739603F308977E93C50213F3124171E48E1C73F 31C17D790AC2933F3261AAF98B37313F33049F9FD350DA3F33AA5B6BEBE83D3F3452DE 5DDD99F33F34FE2875B0C3D73EC1BE17090C792A3EC32748E8987ECD3EC942DC18457B 193ED0D05638976EF53ED5E6B806AB59F63EDBD3D6DFA838B13EE148BBF87EE2773EE5 0E714875656B3EE93A6117902E833EEDCC30E4463FB73EF161D6AE624C193EF4105BEE E2513E3EF6F19EAA2E68153EFA0598BA9CDDF13EFD4C460BD7AF6F3F0062D1EBB7FD0A 3F0238D817E00D123F042834DBB8689C3F0630E7B9AC15DD3F0852F056313A573F0A8E 4E6DC173B13F0CE301CE0890733F0F510A512D7A3F3F10EC33ED41197C3F123C8D2A13 CD813F139990D6ACAF243F15033EEC97C61E3F16799766D9CF203F17FC9A4198AFFB3F 198C4779D9A5E03F1B289F0D4F61A53F1CD1A0FA33EF473F1E874D3F2B6AA83F2024D1 ED96AB1A3F210C5266B97B763F21FA280AB4C4EF3F22EE52D953B54F3F23E8D2D271C2 BD3F24E9A7F5F746BC3F25F0D243D6CC523F26FE51BC0AE8DB3F2812265E9484083F29 2C502B7976AF3F2A4CCF22C36F493F2B73A3447F0E8B3F2CA0CC90BB2FCA3F2DD44B07 8855A43F2F0E1EA8F8332D3F302723BA8EA66A3F30CA62B60555D03F3170CC46E9CFFD 3F321A606D45C8753F32C71F2922EAD03F3377087A8ACB1F3F342A1C6186DA7F3F34E0 5ADE205D753F3599C3F060654E3F365657984FCAC13F371615D5F72A513F37D8FEA95E E1DA-%&COLORG6&%$RGBG$F.F($\"\"#F1F;-%*LINESTYLEG6#F<-%&STYLEG6#%&PATC HG-%+LIGHTMODELG6#%(LIGHT_3G-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%%b[7 ]G%'a[6,5]G%!G-%%VIEWG6%F&F,;F'$F0!\"'-%+PROJECTIONG6%$!#DF($\"#gF(F. " 1 2 0 2 10 0 2 1 5 2 2 1.000000 60.000000 -25.000000 1 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "evalf[15](minimize(sm,b[7]=0..0.1,a[6,5]=1e-10..2,'location')) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"08m)e`66p!#D<#7$<$/&%\"bG6#\" \"($\".jd@e8M(!#9/&%\"aG6$\"\"'\"\"&$\"0x,u<-ZN\"!#:F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use rational \+ approximations for the values of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6# \"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"' \"\"&" }{TEXT -1 54 " that mimimize the norm of the principal error t erms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "[.7341358215763e-1, .135470217740177];\nmap(convert,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\".jd@e8M(!#9$\"0x,u<-ZN\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"#H\"$&R#\"#b\"$1%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "We set " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7] = 29/395;" "6#/&%\"bG6#\"\"(*&\"#H\"\"\"\"$ &R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5] = 55/406;" "6#/& %\"aG6$\"\"'\"\"&*&\"#b\"\"\"\"$1%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "The resulting scheme has the following value for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "a[7,5]=simplify(subs(\{b[7]=29/395,a[6,5]=55/406\},subs(e5,a[7,5]) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#!=JwEbT9vuh ^0`Cg\">7(e%**\\Z_c&*3vl%p?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "c_2 := 1/6: c_4 := 12/19: c_ 5 := 66/85: c_6 := 27/34: bs_3 := 118/311: bs_5 := 11/150:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We can simplify the coefficients slightly if we replace the rational expression for \+ " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 65 " by a \+ simpler one which agrees with the value above to 5 digits." }}{PARA 0 "" 0 "" {TEXT -1 64 "This will not change the characteristics of the s cheme too much." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "-6024530551617475144155267631/2069465750895565 2474999458712.;\nconvert(%,rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+wD:6H!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$a\"\"$H&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "`b*`[3] = 71/183;" "6#/&%#b*G6#\"\"$*&\"#r\"\"\"\"$$=! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5] = 1/57;" "6#/&%#b* G6#\"\"&*&\"\"\"F)\"#d!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 139 "These values are chosen so as to give a principal error norm f or the order 4 embedded scheme roughly 4.77 times that of the order 5 \+ scheme." }}{PARA 0 "" 0 "" {TEXT -1 78 "The corresponding ratio for th e Bogacki-Shampine scheme is approximately 4.79." }}{PARA 0 "" 0 "" {TEXT -1 16 "The values for " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\" \"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 110 " also ensure that the stability region for the embedde d scheme is compatible with that of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "e6 := \+ \{c[3]=155/594,b[7]=29/395,a[6,5]=55/406,a[7,5]=-154/529,`b*`[3]=71/18 3,`b*`[5]=1/57\};\ne7 := `union`(e6,simplify(subs(e6,e3))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e6G<(/&%\"bG6#\"\"(#\"#H\"$&R/&%\"cG6#\" \"$#\"$b\"\"$%f/&%#b*G6#\"\"&#\"\"\"\"#d/&%\"aG6$F*F:#!$a\"\"$H&/&F@6$ \"\"'F:#\"#b\"$1%/&F8F1#\"#r\"$$=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1540 "e7 := \{a[7,4] = 258902691791384553326881/13 07299004967194685810000, a[4,1] = 109269372/164787475, `b*`[2] = 0, a[ 8,7] = 29/395, a[8,2] = 0, b[7] = 29/395, c[5] = 66/85, c[2] = 1/6, c[ 3] = 155/594, c[8] = 1, b[2] = 0, a[6,3] = -57606687522659088/29968516 930563575, a[6,1] = -22877286938601/28657439092100, c[4] = 12/19, c[6] = 27/34, `b*`[4] = 366435142393882979/1527555079142977500, b[5] = 205 9161125/95272074612, a[4,3] = 357797088/164787475, a[8,1] = 51964327/6 54617700, c[7] = 1, `b*`[1] = 4457141939387131/56114268213415500, a[5, 4] = 54884775429/118168704250, a[2,1] = 1/6, `b*`[8] = 818241584766756 5831/54150262660304141458050, b[6] = 1793996689/9043706700, `b*`[5] = \+ 1/57, `b*`[6] = 2399113615956319/11892729854919375, a[6,2] = 278671841 1/910771940, a[5,2] = 37863936/17514845, b[4] = 14557376984/6089496367 5, a[5,3] = -383179645721664/283900311960625, a[3,2] = 24025/117612, a [4,2] = -2341872/1063145, `b*`[7] = 3306969852589408717937/45125218883 586784548375, b[1] = 51964327/654617700, b[3] = 6965638145689311/17945 236957333900, a[7,2] = -7556229/2688010, a[5,1] = -67885276899/1357400 48750, a[3,1] = 6665/117612, a[8,6] = 1793996689/9043706700, a[8,5] = \+ 2059161125/95272074612, a[8,4] = 14557376984/60894963675, a[8,3] = 696 5638145689311/17945236957333900, a[7,6] = 138208410082586660479/251926 265586772035000, a[7,3] = 1811005935036153778848695517/745590435061329 862990895000, a[7,1] = 357804108795684026067391/3861490076828409834900 00, a[7,5] = -154/529, a[6,5] = 55/406, a[6,4] = 3984899849779/1247388 8420630, `b*`[3] = 71/183\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "subs(e 7,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [ c[4],a[4,1],a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a [5,4]],\n [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n \+ [c[7],a[7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8], a[8,1],a[8,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n \+ [`b`,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],[``$3,b[7]],\n [`b*`,`b* `[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``$2,`b*`[7],`b*`[ 8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747&#\"\"\"\"\" 'F(%!GF+7&#\"$b\"\"$%f#\"%lm\"'7w6#\"&DS#F2F+7&#\"#7\"#>#\"*s$p#4\"\"* vuyk\"#!(s=M#\"(XJ1\"#\"*)3(zd$F;7&#\"#m\"#&)#!,**oF&)y'\"-]([+uN\"#\" )OR'y$\")X[^<#!0k;sX'zJQ\"0D1'>J+RG7&F+F+F+#\",Hax%)[&\"-]Uqo\"=\"7&# \"#F\"#M#!/,'QpGxG#\"/+@4RulG#\"+6%=ny#\"*S>x5*#!2)3fE_(o1w&\"2vNcIp^o *H7&F+F+#\".z(\\)**[)R\"/I1U))QZ7#\"#b\"$1%7&F)#\"9\"Rng-%o&z3T!yN\"9+ +\\$)4%Go2!\\hQ#!(Hib(\"(5!)o##\"=n\\+*H28#!$a\"\"$H&#\"6z/m'e#35%3#Q \"\"6+].sneli#>D7&F)#\")FV'>&\"*+xha'\"\"!#\"16$*oX\"Qc'p\"2+RLdpBXz\" 7&F+#\",%)pPdX\"\",vO'\\*3'#\"+D6;f?\",7Y2s_*#\"+*o'*Rz\"\"++nqV!*7&F+ F+F+#\"#H\"$&R7&%\"bGFepFhpFipF\\qFfq7&%#b*G#\"1JrQR>9dW\"2+bT8#oU6cFh p#\"#r\"$$=7&F+#\"3zH)QRU^Vm$\"4+v(H9z]bF:#F)\"#d#\"1>j&fh8\"*R#\"2v$> \\&)HF*=\"7&F+F+#\"7Pzr3%*e_)ppI$\"8v$[Xy'e$))=_7X#\"4Jecnw%eT#=)\"8]! e99/.mi-:aQ(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "subs(e7,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(` b*`[i],i=1..8)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7+7+$\")nmm;!\")F(%!GF+F+F+F+F+F+7+$\")wU4EF*$\")(Qpm&!\"*$ \")PtU?F*F+F+F+F+F+F+7+$\")&*y:jF*$\")q#4j'F*$!)wx-A!\"($\")REr@F;F+F+ F+F+F+7+$\")fqkxF*$!)P7,]F*$\")'>=;#F;$!)zp\\8F;$\")=hWYF*F+F+F+F+7+$ \")lF;$\")Jf%>$F*$\"))zYN\"F*F+F+F+7+$ \"\"\"\"\"!$\")'ffE*F*$!)m36GF;$\")a&*GCF;$\"))R/)>F*$!)J:6HF*$\")f1'[ &F*F+F+7+FW$\")#=\"QzF1$FYFY$\")$3;)QF*$\");d!R#F*$\")![8;#F1$\")ip$)> F*$\")AxTtF1F+7+%\"bGFaoFcoFdoFfoFhoFjoF\\pF+7+%#b*G$\")t%p@)F1Fco$\") )yyy$F*$\")1^.EF*$\")LLLtF1$\")![@C\"F*$\")w46:F1$\")$[Kg'F1Q)pprint20 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "T he norm of the principal error terms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "errterms5_7 := PrincipalErrorTerms(5,7,'ex panded'):\nsqrt(add(evalf(subs(e7,errterms5_7[i]))^2,i=1.. nops(errter ms5_7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P0kG$)!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "This is somewh at smaller than the corresponding value for the Bogacki-Shampine schem e. Since all the weights are positive, the scheme should be fairly rob ust." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "T he stability function for the order 5 scheme is " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "expand(subs(e7,StabilityFunction(5,7,'expanded ')));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$%\"zGF$*&#F$\"\"#F$ *$)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*& #F$\"$?\"F$*$)F%\"\"&F$F$F$*&#\">4X.=oy%G-E`d-g&*\"A+v$4(Q#=e;kfMkQX&o F$*$)F%F-F$F$F$*&#\"=f7_fCnJ(>A$*HZL#\"A+]i&))eI$[)HcLvziC\"F$*$)F%\" \"(F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the weights " }{XPPEDIT 18 0 "`b^`[i], i=1" " 6$&%#b^G6#%\"iG/F&\"\"\"" }{TEXT -1 26 " . . 7, ( initially with " } {XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 136 " as a parame ter) for an auxiliary 7 stage, order 4 embedded scheme by using the qu adrature conditions along with the requirements that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^`[7]=b[7]" "6#/&%#b^G6#\"\"(&% \"bG6#F'" }{TEXT -1 7 " qnd " }{XPPEDIT 18 0 "`b^`[2]=0" "6#/&%#b^G6 #\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "quad_eqns := subs(b=`b^`,Qu adratureConditions(4,7,'expanded')):\ne8 := \{`b^`[2]=0\}:\ne9 := `uni on`(e7,e8):\ncdns2 := [`b^`[7]=b[7],op(quad_eqns)]:\neqns2 := subs(e9, cdns2):\nnops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%#b^G6#\"\" $&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 := solve(\{op(eqns2)\},\{`b^`[1 ],`b^`[3],`b^`[4],`b^`[5],`b^`[7]\}):\ne11 := `union`(e9,e10):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The coeff icients are expressed in terms of " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^ G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " indets(map(rhs,e11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#&%#b^G6#\" \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1814 "e11 := \{a[7,4] = 258902691791384553326881/130729900496719468581 0000, a[4,1] = 109269372/164787475, `b^`[5] = 48968239475/190544149224 -8833125/7444294*`b^`[6], `b*`[2] = 0, a[8,7] = 29/395, a[8,2] = 0, b[ 7] = 29/395, c[5] = 66/85, c[2] = 1/6, c[3] = 155/594, c[8] = 1, b[2] \+ = 0, a[6,3] = -57606687522659088/29968516930563575, a[6,1] = -22877286 938601/28657439092100, c[4] = 12/19, c[6] = 27/34, `b*`[4] = 366435142 393882979/1527555079142977500, b[5] = 2059161125/95272074612, a[4,3] = 357797088/164787475, a[8,1] = 51964327/654617700, c[7] = 1, `b*`[1] = 4457141939387131/56114268213415500, `b^`[4] = 5438288471/27837697680+ 13848321/62862124*`b^`[6], `b^`[1] = 22405523/290941200+4711/394196*`b ^`[6], a[5,4] = 54884775429/118168704250, a[2,1] = 1/6, `b*`[8] = 8182 415847667565831/54150262660304141458050, b[6] = 1793996689/9043706700, `b*`[5] = 1/57, `b*`[6] = 2399113615956319/11892729854919375, a[6,2] \+ = 2786718411/910771940, a[5,2] = 37863936/17514845, b[4] = 14557376984 /60894963675, a[5,3] = -383179645721664/283900311960625, `b^`[3] = 264 7944030402/6666135571075-44562922173/975449711413*`b^`[6], a[3,2] = 24 025/117612, a[4,2] = -2341872/1063145, `b*`[7] = 330696985258940871793 7/45125218883586784548375, b[1] = 51964327/654617700, b[3] = 696563814 5689311/17945236957333900, a[7,2] = -7556229/2688010, a[5,1] = -678852 76899/135740048750, a[3,1] = 6665/117612, a[8,6] = 1793996689/90437067 00, a[8,5] = 2059161125/95272074612, a[8,4] = 14557376984/60894963675, a[8,3] = 6965638145689311/17945236957333900, a[7,6] = 138208410082586 660479/251926265586772035000, a[7,3] = 1811005935036153778848695517/74 5590435061329862990895000, a[7,1] = 357804108795684026067391/386149007 682840983490000, a[7,5] = -154/529, a[6,5] = 55/406, a[6,4] = 39848998 49779/12473888420630, `b^`[2] = 0, `b^`[7] = 29/395, `b*`[3] = 71/183 \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "We choose " }{XPPEDIT 18 0 "`b^`[6]" "6#& %#b^G6#\"\"'" }{TEXT -1 88 " so that the principal error norms of the two embedded schemes are approximately equal." }}{PARA 0 "" 0 "" {TEXT -1 66 "First we obtain the principal error of the 8 stage order \+ 4 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nprin_err4_8 := \+ sqrt(add(subs(e7,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\ne valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#**zL(R!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The principal e rror norm of the 7 stage order 4 scheme is expressed in terms of " } {XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "`errterms4_7^` := subs(b=`b^`,Prin cipalErrorTerms(4,7,'expanded')):\nprin_err4_7 := sqrt(add(subs(e11,`e rrterms4_7^`[i])^2,i=1.. nops(`errterms4_7^`)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,prin_err4_7G,$*&#\"\"\"\"?++;B,:sh1tK%3*)R#F(*&\"Dq: 5_cAJ`$[gg!GPGVm*#F(\"\"#*$),&\"+*o'*Rz\"!\"\"*&\"++nqV!*F(&%#b^G6#\" \"'F(F(F-F(F,F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "prin_err4_7=prin_err4_8:\nsolve(%):\nevalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+!\\*fF%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "We choos e a rational approximation for the larger of the two values since this value gives a larger real stability interval than the other value." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 ".2017599490;\nconvert(%,rat ional,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!\\*f " 0 "" {MPLTEXT 1 0 43 "eval(prin_err4_7,`b^`[6]=23/114):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m,'o'R!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "e12 := `unio n`(\{`b^`[6]=23/114\},subs(`b^`[6]=23/114,e11)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1751 "e12 := \{a[7,4] = 25890269 1791384553326881/1307299004967194685810000, a[4,1] = 109269372/1647874 75, `b*`[2] = 0, a[8,7] = 29/395, a[8,2] = 0, b[7] = 29/395, c[5] = 66 /85, c[2] = 1/6, c[3] = 155/594, c[8] = 1, b[2] = 0, a[6,3] = -5760668 7522659088/29968516930563575, a[6,1] = -22877286938601/28657439092100, c[4] = 12/19, c[6] = 27/34, `b*`[4] = 366435142393882979/152755507914 2977500, b[5] = 2059161125/95272074612, a[4,3] = 357797088/164787475, \+ a[8,1] = 51964327/654617700, c[7] = 1, `b*`[1] = 4457141939387131/5611 4268213415500, `b^`[3] = 28404964428638589/73207500841545650, a[5,4] = 54884775429/118168704250, a[2,1] = 1/6, `b*`[8] = 8182415847667565831 /54150262660304141458050, b[6] = 1793996689/9043706700, `b*`[5] = 1/57 , `b^`[6] = 23/114, `b*`[6] = 2399113615956319/11892729854919375, a[6, 2] = 2786718411/910771940, a[5,2] = 37863936/17514845, b[4] = 14557376 984/60894963675, a[5,3] = -383179645721664/283900311960625, a[3,2] = 2 4025/117612, a[4,2] = -2341872/1063145, `b^`[5] = 63707991275/36203388 35256, `b^`[4] = 1929236304929/8045094629520, `b^`[1] = 126880675943/1 597558129200, `b*`[7] = 3306969852589408717937/45125218883586784548375 , b[1] = 51964327/654617700, b[3] = 6965638145689311/17945236957333900 , a[7,2] = -7556229/2688010, a[5,1] = -67885276899/135740048750, a[3,1 ] = 6665/117612, a[8,6] = 1793996689/9043706700, a[8,5] = 2059161125/9 5272074612, a[8,4] = 14557376984/60894963675, a[8,3] = 696563814568931 1/17945236957333900, a[7,6] = 138208410082586660479/251926265586772035 000, a[7,3] = 1811005935036153778848695517/745590435061329862990895000 , a[7,1] = 357804108795684026067391/386149007682840983490000, a[7,5] = -154/529, a[6,5] = 55/406, a[6,4] = 3984899849779/12473888420630, `b^ `[2] = 0, `b^`[7] = 29/395, `b*`[3] = 71/183\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "`RK4_7eqs^` := su bs(b=`b^`,OrderConditions(4,7,'expanded')):\nsimplify(subs(e12,`RK4_7e qs^`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The coefficients for the whole scheme including the 7 stage and 8 stage embedded order 4 schemes are as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 556 "subs(e12,matrix([[c[2],a[2,1],``$3],\n [ c[3],a[3,1],a[3,2],``$2],\n [c[4],a[4,1],a[4,2],a[4,3],``],\n \+ [c[5],a[5,1],a[5,2],a[5,3],a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3 ],a[6,4]],[``$4,a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3],a[7,4]],[`` $3,a[7,5],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3],a[8,4]],[``$2,a[8 ,5],a[8,6],a[8,7]],\n [`b`,b[1],b[2],b[3],b[4]],[``$2,b[5],b[6],b [7]],\n [`b^`,`b^`[1],`b^`[2],`b^`[3],`b^`[4]],[``$2,`b^`[5],`b^` [6],`b^`[7]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3],`b*`[4]],[``,`b*`[5 ],`b*`[6],`b*`[7],`b*`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#727'#\"\"\"\"\"'F(%!GF+F+7'#\"$b\"\"$%f#\"%lm\"'7w6#\"&DS#F2F+ F+7'#\"#7\"#>#\"*s$p#4\"\"*vuyk\"#!(s=M#\"(XJ1\"#\"*)3(zd$F;F+7'#\"#m \"#&)#!,**oF&)y'\"-]([+uN\"#\")OR'y$\")X[^<#!0k;sX'zJQ\"0D1'>J+RG#\",H ax%)[&\"-]Uqo\"=\"7'#\"#F\"#M#!/,'QpGxG#\"/+@4RulG#\"+6%=ny#\"*S>x5*#! 2)3fE_(o1w&\"2vNcIp^o*H#\".z(\\)**[)R\"/I1U))QZ77'F+F+F+F+#\"#b\"$1%7' F)#\"9\"Rng-%o&z3T!yN\"9++\\$)4%Go2!\\hQ#!(Hib(\"(5!)o##\"=n\\+*H287'F+F+ F+#!$a\"\"$H&#\"6z/m'e#35%3#Q\"\"6+].sneli#>D7'F)#\")FV'>&\"*+xha'\"\" !#\"16$*oX\"Qc'p\"2+RLdpBXz\"#\",%)pPdX\"\",vO'\\*3'7'F+F+#\"+D6;f?\", 7Y2s_*#\"+*o'*Rz\"\"++nqV!*#\"#H\"$&R7'%\"bGFdpFgpFhpF[qF^q7'%#b^G#\"- Vfn!)o7\".+#H\"evf\"Fgp#\"2*eQ'GW'\\SG\"2]caT3]2K(#\".H\\IO#H>\".?&HY4 X!)7'F+F+#\",v7*zqj\".c_$)Q.i$#\"#B\"$9\"Feq7'%#b*G#\"1JrQR>9dW\"2+bT8 #oU6cFgp#\"#r\"$$=#\"3zH)QRU^Vm$\"4+v(H9z]bF:7'F+#F)\"#d#\"1>j&fh8\"*R #\"2v$>\\&)HF*=\"#\"7Pzr3%*e_)ppI$\"8v$[Xy'e$))=_7X#\"4Jecnw%eT#=)\"8] !e99/.mi-:aQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 " #-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1640 "ee := \{c[2]=1/6,\nc[3]=155/594, \nc[4]=12/19,\nc[5]=66/85,\nc[6]=27/34,\nc[7]=1,\nc[8]=1,\n\na[2,1]=1/ 6,\na[3,1]=6665/117612,\na[3,2]=24025/117612,\na[4,1]=109269372/164787 475,\na[4,2]=-2341872/1063145,\na[4,3]=357797088/164787475,\na[5,1]=-6 7885276899/135740048750,\na[5,2]=37863936/17514845,\na[5,3]=-383179645 721664/283900311960625,\na[5,4]=54884775429/118168704250,\na[6,1]=-228 77286938601/28657439092100,\na[6,2]=2786718411/910771940,\na[6,3]=-576 06687522659088/29968516930563575,\na[6,4]=3984899849779/12473888420630 ,\na[6,5]=55/406,\na[7,1]=357804108795684026067391/3861490076828409834 90000,\na[7,2]=-7556229/2688010,\na[7,3]=1811005935036153778848695517/ 745590435061329862990895000,\na[7,4]=258902691791384553326881/13072990 04967194685810000,\na[7,5]=-154/529,\na[7,6]=138208410082586660479/251 926265586772035000,\na[8,1]=51964327/654617700,\na[8,2]=0,\na[8,3]=696 5638145689311/17945236957333900,\na[8,4]=14557376984/60894963675,\na[8 ,5]=2059161125/95272074612,\na[8,6]=1793996689/9043706700,\na[8,7]=29/ 395,\n\nb[1]=51964327/654617700,\nb[2]=0,\nb[3]=6965638145689311/17945 236957333900,\nb[4]=14557376984/60894963675,\nb[5]=2059161125/95272074 612,\nb[6]=1793996689/9043706700,\nb[7]=29/395,\n\n`b^`[1]=12688067594 3/1597558129200,\n`b^`[2]=0,\n`b^`[3]=28404964428638589/73207500841545 650,\n`b^`[4]=1929236304929/8045094629520,\n`b^`[5]=63707991275/362033 8835256,\n`b^`[6]=23/114,\n`b^`[7]=29/395,\n\n`b*`[1]=4457141939387131 /56114268213415500,\n`b*`[2]=0,\n`b*`[3]=71/183,\n`b*`[4]=366435142393 882979/1527555079142977500,\n`b*`[5]=1/57,\n`b*`[6]=2399113615956319/1 1892729854919375,\n`b*`[7]=3306969852589408717937/45125218883586784548 375,\n`b*`[8]=8182415847667565831/54150262660304141458050\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "th e stability region for the 7 stage, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability function R \+ for the 7 stage, order 5 scheme is given as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(sub s(ee,StabilityFunction(5,7,'expanded'))):\nR := unapply(%,z):\n'R(z)'= R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)* &#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\" \"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\">4X.=oy%G-E`d-g&*\"A+v$4(Q# =e;kfMkQX&oF)*$)F'F1F)F)F)*&#\"=f7_fCnJ(>A$*HZL#\"A+]i&))eI$[)HcLvziC \"F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabi lity region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/ -%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+?L:eS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4): \np1 := plot([R(z),-1],z=-4.49..0.49,color=[red,blue]):\np2 := plot([[ [z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np 3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplot s[display]([p1,p2,p3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVES G6$7U7$$!3A++++++!\\%!#<$!3F8-ss7SLAF*7$$!3;+]7'3DdV%F*$!3NIi!)Ga\\H?F *7$$!35++Ds,X\"Q%F*$!3qQ,1X,nT=F*7$$!3(*\\P*)feAMVF*$!3OH;Z'[v/p\"F*7$ $!3%)*\\Pva,qG%F*$!3ym+&o%\\%*\\:F*7$$!3I+](4>%y!=%F*$!3%o3tcRC&p7F*7$ $!35+]#oviQ2%F*$!3C@Q\"p!**eJ5F*7$$!3A+ve,&\\u'RF*$!31Si&H&QgG$)!#=7$$ !3?+v8#z!zoQF*$!3u#Q)fD+IynFP7$$!3;+v.8ajmPF*$!3&4HERakvU&FP7$$!3C+v$Q <')4m$F*$!3[7jWv5jjUFP7$$!3G+v3`dnbNF*$!3o<+(o:?LI$FP7$$!37++0T5NZMF*$ !3UT*R1Bh.\\#FP7$$!3?+]sv&Q>N$F*$!39eE)R.Wu*=FP7$$!3E++:Qn_WKF*$!3RIt \"f6dYM\"FP7$$!3#****\\s&QnOJF*$!3`n=M?fU.*)!#>7$$!3&****\\=iPF.$F*$!3 aphxLVR*H&Fcp7$$!3Q+v$Gc`$QHF*$!3Aj1)\\!QidDFcp7$$!3+++l(y@h#GF*$\"3gr PMTU^==!#?7$$!3=++g_n/JFF*$\"3))[XmKc=d@Fcp7$$!3C+vt%)=X?EF*$\"3c-qPKf cdTFcp7$$!3D++5(ocD_#F*$\"3>)Q\"yN-v yC5FP7$$!3l*\\([Y:;3@F*$\"33P$R798@;\"FP7$$!3$***\\7w!eC+#F*$\"3f%*pX( *H\"eJ\"FP7$$!3%**\\Py(=m#*=F*$\"3#\\H_txv`[\"FP7$$!3%)*\\iW'R3(z\"F*$ \"35(*oXkc0W;FP7$$!3-+]d*>dQp\"F*$\"3R))*4Z=)yH=FP7$$!3)*****p]Q@(e\"F *$\"3CGj8jO@S?FP7$$!3I+]FRT)G[\"F*$\"35hs>ss@nAFP7$$!3')*\\P*y(R>Q\"F* $\"3t=5u=l_4DFP7$$!3++]Kx#e)p7F*$\"3%ezz?X!33GFP7$$!3=++5j![\"p6F*$\"3 z\"p&y*fGg5$FP7$$!3s***\\KT=;1\"F*$\"3#yBZZc\\)eMFP7$$!3'***\\P%o0=k*F P$\"3s:mi60#H\"QFP7$$!3](***\\tEbw&)FP$\"3PQrD$=\\:C%FP7$$!39-]P2EBuvF P$\"3%G$QV-Ct)o%FP7$$!3>)*\\(GL>l_'FP$\"3;^e+sUj1_FP7$$!3z++]-\")=-bFP $\"3)ye>u:P#odFP7$$!3$*)*\\PM#3)HWFP$\"3!*H(3cm&>@kFP7$$!3;(****f'*ypR $FP$\"33n'[&RR&)>rFP7$$!3L)**\\U'=wSBFP$\"3h\"4\"H$G:I\"zFP7$$!3?$*\\P t2H$H\"FP$\"3U(4HviZoy)FP7$$!33R****pit2LFcp$\"3#4&**)36PYn*FP7$$\"3Z^ +]FkzBxFcp$\"3k%\\0%=\"*H!3\"F*7$$\"35*****z2`!f\"F*7 $$\"3l+]i_F06GFP$\"3`:'3Q/$fC8F*7$$\"3\\.]Pt1&z\"QFP$\"3ykPHR=\"\\Y\"F *7$$\"3!***************[FP$\"34XW7QhJK;F*-%'COLOURG6&%$RGBG$\"*++++\"! \")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l 7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$Fe oFa\\l7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$Fj qFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F] tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F `vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$F cxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$ FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3u** ****>L:eSF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&S TYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIA MONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\ l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6 \"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\" $Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1418 "R : = z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 9560025753260228478 6818034509/68545386434596416581823870937500*z^6+\n 23347299322197316 72459521259/12462797533562984833058885625000*z^7:\npts := []: z0 := 0: \nfor ct from 0 to 200 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := p lot(pts,color=COLOR(RGB,.48,.13,.13)):\np2 := plots[polygonplot]([seq( [pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogri d,color=COLOR(RGB,.95,.25,.25)):\npts := []: z0 := 1.5+4.3*I:\nfor ct \+ from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,co lor=COLOR(RGB,.48,.13,.13)):\np4 := plots[polygonplot]([seq([pts[i-1], pts[i],[1.33,4.18]],i=2..nops(pts))],\n style=patchnogrid,col or=COLOR(RGB,.95,.25,.25)):\npts := []: z0 := 1.5-4.3*I:\nfor ct from \+ 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz: \n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color= COLOR(RGB,.48,.13,.13)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[ i],[1.33,-4.18]],i=2..nops(pts))],\n style=patchnogrid,color= COLOR(RGB,.95,.25,.25)):\np7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79], [0,4.79]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],view= [-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n labels=[` Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 512 477 477 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3 ))******HiHB\"*!#G$\"3#******fK'zq:!#=7$$\"3S+++KjhSl!#E$\"3z*****zf#f TJF07$$\"3e+++E^!)y&)!#D$\"3#******p$zQ7ZF07$$\"3Y+++M4Yua!#C$\"3g**** *\\bwJG'F07$$\"3?+++qhbiA!#B$\"3I+++QX$R&yF07$$\"3S+++t*>nv'FF$\"3D+++ qxeC%*F07$$\"3'*******p&z:\\\"!#A$\"3!*******\\[\\*4\"!#<7$$\"3/+++\"* 4A#>#FQ$\"3'******>(4Yc7FT7$$\"3/+++>S'yk%FF$\"3-+++='zKT\"FT7$$!3-+++ 'eLB.\"!#@$\"31+++=\">)p:FT7$$!3B+++ET,EXF\\o$\"3\"******>uyes\"FT7$$! 31+++D8]P8!#?$\"33+++FM=\")=FT7$$!3%)*****\\X\"4lKFgo$\"3')*****>A%RN? FT7$$!3l*****49&GBqFgo$\"33+++>=8)=#FT7$$!3)******HIeGP\"!#>$\"3@+++KO **QBFT7$$!3/+++&[=Q[#Fgp$\"35++++++sP[QIFT7$$!36+++![oF/# F0$\"3#******4oYa:$FT7$$!3-+++gd!Ho#F0$\"3;+++2cZfKFT7$$!3%******pdidP $F0$\"3;+++cv*)\\LFT7$$!3!)******[Ni&4%F0$\"3&******>p\\tU$FT7$$!3y*** **>muT#[F0$\"3=+++wQ<$\\$FT7$$!3k*****4!*G/b&F0$\"3/+++md')[NFT7$$!3]* ****4#fioiF0$\"3#*******Qey&f$FT7$$!3!******\\7yh(pF0$\"3&)*****pvn]j$ FT7$$!3;+++n&\\An(F0$\"3=+++D3inOFT7$$!3w*******)G!pN)F0$\"3%********* Q;%p$FT7$$!3f*****z:B1.*F0$\"3>+++*\\j_r$FT7$$!3#******pxkSp*F0$\"3:++ +/pOJPFT7$$!33+++2CzM5FT$\"3$******fVGGu$FT7$$!3#******f5)G*4\"FT$\"3= +++*eH*\\PFT7$$!3#******>#z&H;\"FT$\"3#******\\A%*Gv$FT7$$!3.+++'fieA \"FT$\"3?+++s+!>v$FT7$$!3%******R]g!)G\"FT$\"37+++tu3ZPFT7$$!3'******f h4'\\8FT$\"3*)******ehcQPFT7$$!3/+++^*p0T\"FT$\"3A+++i1UEPFT7$$!35+++l k+r9FT$\"3z*****>%[r5PFT7$$!3&******RM#*4`\"FT$\"3')******))f\\\"p$FT7 $$!33+++;Fh!f\"FT$\"35+++F*)zoOFT7$$!3/+++k(o*\\;FT$\"3,+++51lUOFT7$$! 30+++F@=4FT$\"33+++haThMFT7$$!3*)*****\\nR#4?FT$\"3*)*****z=!e+AFT$\"3%)*****RC(pjKFT7$$!37+++*p4tE#FT$\"3\"**** **H>G'4KFT7$$!3,+++MbTNBFT$\"3y*****Ri[Z:$FT7$$!3()*****z\"4g/CFT$\"3% )*****H*ec*4$FT7$$!33+++QnVuCFT$\"3#)*****>AxW/$FT7$$!3=+++Z*>Wa#FT$\" 3/+++W*H(*)HFT7$$!30+++(QRSh#FT$\"3\"******p[;a$HFT7$$!3A+++z!HGo#FT$ \"37+++C%)\\\")GFT7$$!3!******\\?*R]FFT$\"36+++-]%y#GFT7$$!3()*****>$e W;GFT$\"31+++WAFuFFT7$$!3%******H4Y2)GFT$\"37++++Xd?FFT7$$!3-+++Mh9VHF T$\"3!******R7Ylm#FT7$$!3'*******4ra.IFT$\"3-+++UV*>h#FT7$$!3/+++,A*=1 $FT$\"3A+++QcucDFT7$$!3@+++!)e:=JFT$\"3()*****\\%zk+DFT7$$!3A+++kcLsJF T$\"33+++1/dVCFT7$$!3!******44YWA$FT$\"39+++P>S&Q#FT7$$!39+++MX^uKFT$ \"3&)*****RM\\gK#FT7$$!3!******pTyDK$FT$\"37+++yaVlAFT7$$!3++++uMooLFT $\"3&)*****Rn(\\.AFT7$$!3y*****\\s#)GT$FT$\"3=+++:m=S@FT7$$!31+++2fBbM FT$\"3))******=dYv?FT7$$!3/+++I\"4e\\$FT$\"3'******\\06$4?FT7$$!3z**** *\\_uY`$FT$\"33+++&y6<%>FT7$$!37+++v'4>d$FT$\"3'*******46ns=FT7$$!37++ +8mf2OFT$\"3.+++%e2A!=FT7$$!3*******p=?=k$FT$\"3!******pmc.t\"FT7$$!3% )******>cmuOFT$\"3)******RAsrl\"FT7$$!3;+++#>:iq$FT$\"35+++cws#e\"FT7$ $!3()*****>5Wlt$FT$\"34+++Di62:FT7$$!39+++idrlPFT$\"3********o,XI9FT7$ $!3=+++wpx$z$FT$\"3#******fEeGN\"FT7$$!3<+++[Ov?QFT$\"3++++L<[u7FT7$$! 3%******RrZm%QFT$\"3/+++H'oa>\"FT7$$!35+++)*eVrQFT$\"36+++#Hnf6\"FT7$$ !3'******4hq]*QFT$\"30+++c&>h.\"FT7$$!39+++MH[L:eSFTF(7$Fjjl$!3p+++a\"*)R\"zFg p7$Fejl$!31+++H$GJe\"F07$F`jl$!37+++9k]vBF07$F[jl$!3C+++Xb#)oJF07$Ffil $!37+++6mLjRF07$Fail$!3#)*****zEI#fZF07$F\\il$!3))*****fk9mb&F07$Fghl$ !3]*****zi*[bjF07$Fbhl$!3W+++#p=d:(F07$F]hl$!3I+++C/+dzF07$Fhgl$!3]*** **p-V)e()F07$Fcgl$!39+++Yiag&*F07$F^gl$!30+++c&>h.\"FT7$Fifl$!36+++#Hn f6\"FT7$Fdfl$!3/+++H'oa>\"FT7$F_fl$!3++++L<[u7FT7$Fjel$!3#******fEeGN \"FT7$Feel$!3********o,XI9FT7$F`el$!34+++Di62:FT7$F[el$!35+++cws#e\"FT 7$Ffdl$!3)******RAsrl\"FT7$Fadl$!3!******pmc.t\"FT7$F\\dl$!3.+++%e2A!= FT7$Fgcl$!3'*******46ns=FT7$Fbcl$!33+++&y6<%>FT7$F]cl$!3'******\\06$4? FT7$Fhbl$!3))******=dYv?FT7$Fcbl$!3=+++:m=S@FT7$F^bl$!3&)*****Rn(\\.AF T7$Fial$!37+++yaVlAFT7$Fdal$!3&)*****RM\\gK#FT7$F_al$!39+++P>S&Q#FT7$F j`l$!33+++1/dVCFT7$Fe`l$!3()*****\\%zk+DFT7$F``l$!3A+++QcucDFT7$F[`l$! 3-+++UV*>h#FT7$Ff_l$!3!******R7Ylm#FT7$Fa_l$!37++++Xd?FFT7$F\\_l$!31++ +WAFuFFT7$Fg^l$!36+++-]%y#GFT7$Fb^l$!37+++C%)\\\")GFT7$F]^l$!3\"****** p[;a$HFT7$Fh]l$!3/+++W*H(*)HFT7$Fc]l$!3#)*****>AxW/$FT7$F^]l$!3%)***** H*ec*4$FT7$Fi\\l$!3y*****Ri[Z:$FT7$Fd\\l$!3\"******H>G'4KFT7$F_\\l$!3% )*****RC(pjKFT7$Fj[l$!3=+++,GS;LFT7$Fe[l$!3'******R%R@nLFT7$F`[l$!3*)* ****z%[r5PFT7$F^x $!3A+++i1UEPFT7$Fiw$!3*)******ehcQPFT7$Fdw$!37+++tu3ZPFT7$F_w$!3?+++s+ !>v$FT7$Fjv$!3#******\\A%*Gv$FT7$Fev$!3=+++*eH*\\PFT7$F`v$!3$******fVG Gu$FT7$F[v$!3:+++/pOJPFT7$Ffu$!3>+++*\\j_r$FT7$Fau$!3%*********Q;%p$FT 7$F\\u$!3=+++D3inOFT7$Fgt$!3&)*****pvn]j$FT7$Fbt$!3#*******Qey&f$FT7$F ]t$!3/+++md')[NFT7$Fhs$!3=+++wQ<$\\$FT7$Fcs$!3&******>p\\tU$FT7$F^s$!3 ;+++cv*)\\LFT7$Fir$!3;+++2cZfKFT7$Fdr$!3#******4oYa:$FT7$F_r$!3>+++sP[ QIFT7$Fjq$!32+++'\\f1\"HFT7$Feq$!37+++!==8)=#FT7$F[p$!3')* ****>A%RN?FT7$Feo$!33+++FM=\")=FT7$F`o$!3\"******>uyes\"FT7$Fjn$!31+++ =\">)p:FT7$Fen$!3-+++='zKT\"FT7$$\"3-+++#*4A#>#FQ$!3'******>(4Yc7FT7$F O$!3!*******\\[\\*4\"FT7$FJ$!3D+++qxeC%*F07$FD$!3I+++QX$R&yF07$$\"3q** ***H$4YuaF@$!3g*****\\bwJG'F07$F8$!3#******p$zQ7ZF07$$\"3()*****RL;1a' F4$!3z*****zf#fTJF07$$\"3m*****H,'HB\"*F-$!3#******fK'zq:F0F'-%&COLORG 6&%$RGBG$\"#[!\"#$\"#8Fh^nFi^n-%)POLYGONSG6fw7%F'7$$\"+IiHB\"*Fgo$\"+E jzq:!#57$$Fh^nF)F(7%F__n7$$\"+KjhSlF0$\"+)f#fTJFd_nFe_n7%Fh_n7$$\"+E^! )y&)FT$\"+PzQ7ZFd_nFe_n7%F^`n7$$\"+M4Yua!#;$\"+bl<$G'Fd_nFe_n7%Fd`n7$$ \"+qhbiA!#:$\"+QX$R&yFd_nFe_n7%F[an7$$\"+t*>nv'F^an$\"+qxeC%*Fd_nFe_n7 %Fban7$$\"+q&z:\\\"!#9$\"+][\\*4\"!\"*Fe_n7%Fhan7$$\"+\"*4A#>#F[bn$\"+ s4Yc7F^bnFe_n7%F`bn7$$\"+>S'yk%F^an$\"+='zKT\"F^bnFe_n7%Ffbn7$$!+'eLB. \"!#8$\"+=\">)p:F^bnFe_n7%F\\cn7$$!+ET,EXF_cn$\"+U(yes\"F^bnFe_n7%Fccn 7$$!+D8]P8!#7$\"+FM=\")=F^bnFe_n7%Ficn7$$!+b94lKF\\dn$\"+AURN?F^bnFe_n 7%F`dn7$$!+T^GBqF\\dn$\"+>=8)=#F^bnFe_n7%Ffdn7$$!+.$eGP\"!#6$\"+KO**QB F^bnFe_n7%F\\en7$$!+&[=Q[#F_en$\"+v$F^bnFe_n7% F[]o7$$!+/01)G\"F^bn$\"+tu3ZPF^bnFe_n7%Fa]o7$$!+;'4'\\8F^bn$\"+fhcQPF^ bnFe_n7%Fg]o7$$!+^*p0T\"F^bn$\"+i1UEPF^bnFe_n7%F]^o7$$!+lk+r9F^bn$\"+U [r5PF^bnFe_n7%Fc^o7$$!+WB*4`\"F^bn$\"+*)f\\\"p$F^bnFe_n7%Fi^o7$$!+;Fh! f\"F^bn$\"+F*)zoOF^bnFe_n7%F__o7$$!+k(o*\\;F^bn$\"+51lUOF^bnFe_n7%Fe_o 7$$!+F@=4F^bn$\"+haThMF^bnFe_n7%Fcao7$$!+v'R#4?F^bn$\"+ y@n:MF^bnFe_n7%Fiao7$$!+n!H;2#F^bn$\"+WR@nLF^bnFe_n7%F_bo7$$!+G4ON@F^b n$\"+,GS;LF^bnFe_n7%Febo7$$!+#=!e+AF^bn$\"+WspjKF^bnFe_n7%F[co7$$!+*p4 tE#F^bn$\"+$>G'4KF^bnFe_n7%Faco7$$!+MbTNBF^bn$\"+C'[Z:$F^bnFe_n7%Fgco7 $$!+=4g/CF^bn$\"+$*ec*4$F^bnFe_n7%F]do7$$!+QnVuCF^bn$\"+AsZWIF^bnFe_n7 %Fcdo7$$!+Z*>Wa#F^bn$\"+W*H(*)HF^bnFe_n7%Fido7$$!+(QRSh#F^bn$\"+([;a$H F^bnFe_n7%F_eo7$$!+z!HGo#F^bn$\"+C%)\\\")GF^bnFe_n7%Feeo7$$!+0#*R]FF^b n$\"+-]%y#GF^bnFe_n7%F[fo7$$!+KeW;GF^bn$\"+WAFuFF^bnFe_n7%Fafo7$$!+$4Y 2)GF^bn$\"++Xd?FF^bnFe_n7%Fgfo7$$!+Mh9VHF^bn$\"+ChamEF^bnFe_n7%F]go7$$ !+5ra.IF^bn$\"+UV*>h#F^bnFe_n7%Fcgo7$$!+,A*=1$F^bn$\"+QcucDF^bnFe_n7%F igo7$$!+!)e:=JF^bn$\"+Xzk+DF^bnFe_n7%F_ho7$$!+kcLsJF^bn$\"+1/dVCF^bnFe _n7%Feho7$$!+\"4YWA$F^bn$\"+P>S&Q#F^bnFe_n7%F[io7$$!+MX^uKF^bn$\"+W$\\ gK#F^bnFe_n7%Faio7$$!+<%yDK$F^bn$\"+yaVlAF^bnFe_n7%Fgio7$$!+uMooLF^bn$ \"+uw\\.AF^bnFe_n7%F]jo7$$!+DF)GT$F^bn$\"+:m=S@F^bnFe_n7%Fcjo7$$!+2fBb MF^bn$\"+>dYv?F^bnFe_n7%Fijo7$$!+I\"4e\\$F^bn$\"+b5J4?F^bnFe_n7%F_[p7$ $!+DXnMNF^bn$\"+&y6<%>F^bnFe_n7%Fe[p7$$!+v'4>d$F^bn$\"+56ns=F^bnFe_n7% F[\\p7$$!+8mf2OF^bn$\"+%e2A!=F^bnFe_n7%Fa\\p7$$!+(=?=k$F^bn$\"+nmNI:iq$F^bn$\"+c ws#e\"F^bnFe_n7%Fc]p7$$!+-TaOPF^bn$\"+Di62:F^bnFe_n7%Fi]p7$$!+idrlPF^b n$\"+p,XI9F^bnFe_n7%F_^p7$$!+wpx$z$F^bn$\"+m#eGN\"F^bnFe_n7%Fe^p7$$!+[ Ov?QF^bn$\"+L<[u7F^bnFe_n7%F[_p7$$!+9xkYQF^bn$\"+H'oa>\"F^bnFe_n7%Fa_p 7$$!+)*eVrQF^bn$\"+#Hnf6\"F^bnFe_n7%Fg_p7$$!+612&*QF^bn$\"+c&>h.\"F^bn Fe_n7%F]`p7$$!+MH[h.\"F^bnFe_n7%F_hp7$Fh_p$!+#Hnf6\"F^bnFe_n7%Fchp7$Fb_ p$!+H'oa>\"F^bnFe_n7%Fghp7$F\\_p$!+L<[u7F^bnFe_n7%F[ip7$Ff^p$!+m#eGN\" F^bnFe_n7%F_ip7$F`^p$!+p,XI9F^bnFe_n7%Fcip7$Fj]p$!+Di62:F^bnFe_n7%Fgip 7$Fd]p$!+cws#e\"F^bnFe_n7%F[jp7$F^]p$!+CAF^bnFe_n7%F_[q7$F`[p$!+b5J4?F^bnFe_n7%Fc[q7$Fjjo $!+>dYv?F^bnFe_n7%Fg[q7$Fdjo$!+:m=S@F^bnFe_n7%F[\\q7$F^jo$!+uw\\.AF^bn Fe_n7%F_\\q7$Fhio$!+yaVlAF^bnFe_n7%Fc\\q7$Fbio$!+W$\\gK#F^bnFe_n7%Fg\\ q7$F\\io$!+P>S&Q#F^bnFe_n7%F[]q7$Ffho$!+1/dVCF^bnFe_n7%F_]q7$F`ho$!+Xz k+DF^bnFe_n7%Fc]q7$Fjgo$!+QcucDF^bnFe_n7%Fg]q7$Fdgo$!+UV*>h#F^bnFe_n7% F[^q7$F^go$!+ChamEF^bnFe_n7%F_^q7$Fhfo$!++Xd?FF^bnFe_n7%Fc^q7$Fbfo$!+W AFuFF^bnFe_n7%Fg^q7$F\\fo$!+-]%y#GF^bnFe_n7%F[_q7$Ffeo$!+C%)\\\")GF^bn Fe_n7%F__q7$F`eo$!+([;a$HF^bnFe_n7%Fc_q7$Fjdo$!+W*H(*)HF^bnFe_n7%Fg_q7 $Fddo$!+AsZWIF^bnFe_n7%F[`q7$F^do$!+$*ec*4$F^bnFe_n7%F_`q7$Fhco$!+C'[Z :$F^bnFe_n7%Fc`q7$Fbco$!+$>G'4KF^bnFe_n7%Fg`q7$F\\co$!+WspjKF^bnFe_n7% F[aq7$Ffbo$!+,GS;LF^bnFe_n7%F_aq7$F`bo$!+WR@nLF^bnFe_n7%Fcaq7$Fjao$!+y @n:MF^bnFe_n7%Fgaq7$Fdao$!+haThMF^bnFe_n7%F[bq7$F^ao$!+-*yT]$F^bnFe_n7 %F_bq7$Fh`o$!+85yVNF^bnFe_n7%Fcbq7$Fb`o$!+vU5!e$F^bnFe_n7%Fgbq7$F\\`o$ !+nd28OF^bnFe_n7%F[cq7$Ff_o$!+51lUOF^bnFe_n7%F_cq7$F`_o$!+F*)zoOF^bnFe _n7%Fccq7$Fj^o$!+*)f\\\"p$F^bnFe_n7%Fgcq7$Fd^o$!+U[r5PF^bnFe_n7%F[dq7$ F^^o$!+i1UEPF^bnFe_n7%F_dq7$Fh]o$!+fhcQPF^bnFe_n7%Fcdq7$Fb]o$!+tu3ZPF^ bnFe_n7%Fgdq7$F\\]o$!+s+!>v$F^bnFe_n7%F[eq7$Ff\\o$!+DU*Gv$F^bnFe_n7%F_ eq7$F`\\o$!+*eH*\\PF^bnFe_n7%Fceq7$Fj[o$!+O%GGu$F^bnFe_n7%Fgeq7$Fd[o$! +/pOJPF^bnFe_n7%F[fq7$F^[o$!+*\\j_r$F^bnFe_n7%F_fq7$Fhjn$!++R;%p$F^bnF e_n7%Fcfq7$Fbjn$!+D3inOF^bnFe_n7%Fgfq7$F\\jn$!+dx1NOF^bnFe_n7%F[gq7$Ff in$!+Rey&f$F^bnFe_n7%F_gq7$F`in$!+md')[NF^bnFe_n7%Fcgq7$Fjhn$!+wQ<$\\$ F^bnFe_n7%Fggq7$Fdhn$!+#p\\tU$F^bnFe_n7%F[hq7$F^hn$!+cv*)\\LF^bnFe_n7% F_hq7$Fhgn$!+2cZfKF^bnFe_n7%Fchq7$Fbgn$!+\"oYa:$F^bnFe_n7%Fghq7$F\\gn$ !+sP[QIF^bnFe_n7%F[iq7$Fffn$!+'\\f1\"HF^bnFe_n7%F_iq7$F`fn$!+!==8)=#F^bnFe_n7%Fcjq7$Fadn$!+AURN? F^bnFe_n7%Fgjq7$Fjcn$!+FM=\")=F^bnFe_n7%F[[r7$Fdcn$!+U(yes\"F^bnFe_n7% F_[r7$F]cn$!+=\">)p:F^bnFe_n7%Fc[r7$Fgbn$!+='zKT\"F^bnFe_n7%Fg[r7$$\"+ #*4A#>#F[bn$!+s4Yc7F^bnFe_n7%F[\\r7$Fian$!+][\\*4\"F^bnFe_n7%Fa\\r7$Fc an$!+qxeC%*Fd_nFe_n7%Fe\\r7$F\\an$!+QX$R&yFd_nFe_n7%Fi\\r7$$\"+L4YuaFg `n$!+bl<$G'Fd_nFe_n7%F]]r7$F_`n$!+PzQ7ZFd_nFe_n7%Fc]r7$$\"+MjhSlF0$!+) f#fTJFd_nFe_n7%Fg]r7$$\"+8gHB\"*Fgo$!+Ejzq:Fd_nFe_n7%F]^rF'Fe_n-Fc^n6& Fe^n$\"#&*Fh^n$\"#DFh^nFg^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3-+++ ZVwa9FT$\"38+++]F`-VFT7$$\"3-+++J)ojW\"FT$\"3,+++ot]6VFT7$$\"3.+++^MFP 9FT$\"3$)*****zuq(>VFT7$$\"31+++N9_F9FT$\"3!)*****pI\"GFVFT7$$\"3!**** **pgcrT\"FT$\"3U+++[w*RL%FT7$$\"35+++2PA19FT$\"3J+++w#y)RVFT7$$\"35+++ m(oZR\"FT$\"3s*****zI\")[M%FT7$$\"3!******H_RGQ\"FT$\"33+++*>k*[VFT7$$ \"3))*****p\"f[q8FT$\"3G+++#[$3_VFT7$$\"3++++a1wd8FT$\"3G+++,X>aVFT7$$ \"3)*******z)>ZM\"FT$\"3K+++l6DbVFT7$$\"3(******p)RUJ8FT$\"39+++8d?bVF T7$$\"3%******4eQzJ\"FT$\"3I+++:&3SN%FT7$$\"3&******>mNVI\"FT$\"3w**** *\\\"zg^VFT7$$\"3))*****R1&p!H\"FT$\"3S+++m,&zM%FT7$$\"3%******4A1rF\" FT$\"30+++)\\zHM%FT7$$\"3%*******\\.nj7FT$\"3')******3%QmL%FT7$$\"3/++ +TJ]]7FT$\"3)******4Io)GVFT7$$\"3#******z0QxB\"FT$\"3,+++x1h>VFT7$$\"3 ,+++t.`D7FT$\"3(*******p\"4)3VFT7$$\"3&*******))>197FT$\"3c*****\\*GT' H%FT7$$\"3,+++Oqa.7FT$\"33+++u\"FT$\"3k******3[p mUFT7$$\"3))*****HRJk=\"FT$\"3Q+++5EO\\UFT7$$\"3!*******e>Z!=\"FT$\"3U +++^dWIUFT7$$\"3-+++$)4vw6FT$\"3T+++k.35UFT7$$\"35+++*H%pv6FT$\"3C+++? 6^)=%FT7$$\"3!******>NJx<\"FT$\"3q*****>,Gh;%FT7$$\"31+++BvB$=\"FT$\"3 0+++&='\\VTFT7$$\"3'********R[C>\"FT$\"3,+++U%e87%FT7$$\"33+++aCO07FT$ \"3W+++`ce+TFT7$$\"3*)******\\=n@7FT$\"3g*****4@k?3%FT7$$\"34+++!\\n2C \"FT$\"3g*****zVYl1%FT7$$\"3-+++$*G$=E\"FT$\"3k*****fSCX0%FT7$$\"3++++ F*))RG\"FT$\"3%******4V\"=YSFT7$$\"3)******p:EkI\"FT$\"3()******z#G9/% FT7$$\"30+++DK[G8FT$\"37+++5x)*RSFT7$$\"3#******RTo'\\8FT$\"3n*****pl$ [TSFT7$$\"3#******\\CW'p8FT$\"3)******zq4b/%FT7$$\"3+++++o>)Q\"FT$\"3Q +++L:n^SFT7$$\"3!******fo._S\"FT$\"3A+++E$3'fSFT7$$\"3*)********og?9FT $\"3#******Hj+!pSFT7$$\"32+++!3#RM9FT$\"3!******phq&zSFT7$$\"3%******z atlW\"FT$\"3C+++u'y54%FT7$$\"3%******\\!R=d9FT$\"3,+++6(=L5%FT7$$\"3-+ ++WpEm9FT$\"3K+++DK6;TFT7$$\"3%******p#H(QZ\"FT$\"3A+++l(3$HTFT7$$\"3% ******>Yb+[\"FT$\"3r*****R%>xUTFT7$$\"33+++%[p[[\"FT$\"3Q+++agQcTFT7$$ \"3!******H)*p$)[\"FT$\"3g*****pB[+<%FT7$$\"31+++A7h!\\\"FT$\"39+++Lrm $=%FT7$$\"3)******>HY;\\\"FT$\"3#)******R4;(>%FT7$$\"3-+++()o_\"\\\"FT $\"3x*****p!eX5UFT7$$\"3)******RB..\\\"FT$\"3O+++)\\%[BUFT7$$\"3/+++8T -)[\"FT$\"3%*******z_=OUFT7$$\"3'******f%pt%[\"FT$\"3w*****f*4][UFT7$$ \"3)******\\$z[![\"FT$\"3G+++p#y.E%FT7$$\"3%*******=AKv9FT$\"3I+++Oowr UFT7$$\"3++++kSGp9FT$\"3')*****>(*=EG%FT7$$\"35+++iqTi9FT$\"3w******y* ))GH%FTF`_rFb^n-F\\_n6jn7%7$$\"+ZVwa9F^bn$\"+]F`-VF^bn7$$\"+J)ojW\"F^b n$\"+ot]6VF^bn7$$\"$L\"Fh^n$\"$=%Fh^n7%Fdbs7$$\"+^MFP9F^bn$\"+[2x>VF^b nFibs7%F_cs7$$\"+N9_F9F^bn$\"+28GFVF^bnFibs7%Fecs7$$\"+2m:<9F^bn$\"+[w *RL%F^bnFibs7%F[ds7$$\"+2PA19F^bn$\"+w#y)RVF^bnFibs7%Fads7$$\"+m(oZR\" F^bn$\"+38)[M%F^bnFibs7%Fgds7$$\"+B&RGQ\"F^bn$\"+*>k*[VF^bnFibs7%F]es7 $$\"+aVF^bnFibs 7%Fies7$$\"+!))>ZM\"F^bn$\"+l6DbVF^bnFibs7%F_fs7$$\"+()RUJ8F^bn$\"+8d? bVF^bnFibs7%Fefs7$$\"+\"eQzJ\"F^bn$\"+:&3SN%F^bnFibs7%F[gs7$$\"+icL/8F ^bn$\"+:zg^VF^bnFibs7%Fags7$$\"+k]p!H\"F^bn$\"+m,&zM%F^bnFibs7%Fggs7$$ \"+@i5x7F^bn$\"+)\\zHM%F^bnFibs7%F]hs7$$\"+].nj7F^bn$\"+4%QmL%F^bnFibs 7%Fchs7$$\"+TJ]]7F^bn$\"+,$o)GVF^bnFibs7%Fihs7$$\"+e!QxB\"F^bn$\"+x1h> VF^bnFibs7%F_is7$$\"+t.`D7F^bn$\"+q\"4)3VF^bnFibs7%Feis7$$\"+*)>197F^b n$\"+&*GT'H%F^bnFibs7%F[js7$$\"+Oqa.7F^bn$\"+u\"F^bn$\"+4[pmUF^bnFibs7%Fgjs7$$\"+$RJk=\"F^bn$\"+5EO\\UF^bnFibs 7%F][t7$$\"+f>Z!=\"F^bn$\"+^dWIUF^bnFibs7%Fc[t7$$\"+$)4vw6F^bn$\"+k.35 UF^bnFibs7%Fi[t7$$\"+*H%pv6F^bn$\"+?6^)=%F^bnFibs7%F_\\t7$$\"+_8tx6F^b n$\"+7!Gh;%F^bnFibs7%Fe\\t7$$\"+BvB$=\"F^bn$\"+&='\\VTF^bnFibs7%F[]t7$ $\"++%[C>\"F^bn$\"+U%e87%F^bnFibs7%Fa]t7$$\"+aCO07F^bn$\"+`ce+TF^bnFib s7%Fg]t7$$\"+]=n@7F^bn$\"+6U1#3%F^bnFibs7%F]^t7$$\"+!\\n2C\"F^bn$\"+Qk amSF^bnFibs7%Fc^t7$$\"+$*G$=E\"F^bn$\"+1W_aSF^bnFibs7%Fi^t7$$\"+F*))RG \"F^bn$\"+J9=YSF^bnFibs7%F__t7$$\"+dhU18F^bn$\"+!GG9/%F^bnFibs7%Fe_t7$ $\"+DK[G8F^bn$\"+5x)*RSF^bnFibs7%F[`t7$$\"+9%o'\\8F^bn$\"+dO[TSF^bnFib s7%Fa`t7$$\"+XUkp8F^bn$\"+3(4b/%F^bnFibs7%Fg`t7$$\"++o>)Q\"F^bn$\"+L:n ^SF^bnFibs7%F]at7$$\"+'o._S\"F^bn$\"+E$3'fSF^bnFibs7%Fcat7$$\"++pg?9F^ bn$\"+L1+pSF^bnFibs7%Fiat7$$\"+!3#RM9F^bn$\"+<1dzSF^bnFibs7%F_bt7$$\"+ [NdY9F^bn$\"+u'y54%F^bnFibs7%Febt7$$\"+0R=d9F^bn$\"+6(=L5%F^bnFibs7%F[ ct7$$\"+WpEm9F^bn$\"+DK6;TF^bnFibs7%Fact7$$\"+FH(QZ\"F^bn$\"+l(3$HTF^b nFibs7%Fgct7$$\"+ia0![\"F^bn$\"+W>xUTF^bnFibs7%F]dt7$$\"+%[p[[\"F^bn$ \"+agQcTF^bnFibs7%Fcdt7$$\"+$)*p$)[\"F^bn$\"+P#[+<%F^bnFibs7%Fidt7$$\" +A7h!\\\"F^bn$\"+Lrm$=%F^bnFibs7%F_et7$$\"+#HY;\\\"F^bn$\"+S4;(>%F^bnF ibs7%Feet7$$\"+()o_\"\\\"F^bn$\"+2eX5UF^bnFibs7%F[ft7$$\"+MKI!\\\"F^bn $\"+)\\%[BUF^bnFibs7%Faft7$$\"+8T-)[\"F^bn$\"+!G&=OUF^bnFibs7%Fgft7$$ \"+Ypt%[\"F^bn$\"+'*4][UF^bnFibs7%F]gt7$$\"+Nz[![\"F^bn$\"+p#y.E%F^bnF ibs7%Fcgt7$$\"+>AKv9F^bn$\"+OowrUF^bnFibs7%Figt7$$\"+kSGp9F^bn$\"+s*=E G%F^bnFibs7%F_ht7$$\"+iqTi9F^bn$\"+z*))GH%F^bnFibs7%FehtF_bsFibsFc^rFi ^r-F$6$7in7$Fa_r$!38+++]F`-VFT7$Fhas$!3w******y*))GH%FT7$Fcas$!3')**** *>(*=EG%FT7$F^as$!3I+++OowrUFT7$Fi`s$!3G+++p#y.E%FT7$Fd`s$!3w*****f*4] [UFT7$F_`s$!3%*******z_=OUFT7$Fj_s$!3O+++)\\%[BUFT7$Fe_s$!3x*****p!eX5 UFT7$F`_s$!3#)******R4;(>%FT7$F[_s$!39+++Lrm$=%FT7$Ff^s$!3g*****pB[+<% FT7$Fa^s$!3Q+++agQcTFT7$F\\^s$!3r*****R%>xUTFT7$Fg]s$!3A+++l(3$HTFT7$F b]s$!3K+++DK6;TFT7$F]]s$!3,+++6(=L5%FT7$Fh\\s$!3C+++u'y54%FT7$Fc\\s$!3 !******phq&zSFT7$F^\\s$!3#******Hj+!pSFT7$Fi[s$!3A+++E$3'fSFT7$Fd[s$!3 Q+++L:n^SFT7$F_[s$!3)******zq4b/%FT7$Fjjr$!3n*****pl$[TSFT7$Fejr$!37++ +5x)*RSFT7$F`jr$!3()******z#G9/%FT7$F[jr$!3%******4V\"=YSFT7$Ffir$!3k* ****fSCX0%FT7$Fair$!3g*****zVYl1%FT7$F\\ir$!3g*****4@k?3%FT7$Fghr$!3W+ ++`ce+TFT7$Fbhr$!3,+++U%e87%FT7$F]hr$!30+++&='\\VTFT7$Fhgr$!3q*****>,G h;%FT7$Fcgr$!3C+++?6^)=%FT7$F^gr$!3T+++k.35UFT7$Fifr$!3U+++^dWIUFT7$Fd fr$!3Q+++5EO\\UFT7$F_fr$!3k******3[pmUFT7$Fjer$!33+++uVFT7$Ffdr$! 3)******4Io)GVFT7$Fadr$!3')******3%QmL%FT7$F\\dr$!30+++)\\zHM%FT7$Fgcr $!3S+++m,&zM%FT7$Fbcr$!3w*****\\\"zg^VFT7$F]cr$!3I+++:&3SN%FT7$Fhbr$!3 9+++8d?bVFT7$Fcbr$!3K+++l6DbVFT7$F^br$!3G+++,X>aVFT7$Fiar$!3G+++#[$3_V FT7$Fdar$!33+++*>k*[VFT7$F_ar$!3s*****zI\")[M%FT7$Fj`r$!3J+++w#y)RVFT7 $Fe`r$!3U+++[w*RL%FT7$F``r$!3!)*****pI\"GFVFT7$F[`r$!3$)*****zuq(>VFT7 $Ff_r$!3,+++ot]6VFTF^itFb^n-F\\_n6jn7%7$F`bs$!+]F`-VF^bn7$Ffht$!+z*))G H%F^bn7$Fjbs$!$=%Fh^n7%Fhdu7$F`ht$!+s*=EG%F^bnF[eu7%F_eu7$Fjgt$!+OowrU F^bnF[eu7%Fceu7$Fdgt$!+p#y.E%F^bnF[eu7%Fgeu7$F^gt$!+'*4][UF^bnF[eu7%F[ fu7$Fhft$!+!G&=OUF^bnF[eu7%F_fu7$Fbft$!+)\\%[BUF^bnF[eu7%Fcfu7$F\\ft$! +2eX5UF^bnF[eu7%Fgfu7$Ffet$!+S4;(>%F^bnF[eu7%F[gu7$F`et$!+Lrm$=%F^bnF[ eu7%F_gu7$Fjdt$!+P#[+<%F^bnF[eu7%Fcgu7$Fddt$!+agQcTF^bnF[eu7%Fggu7$F^d t$!+W>xUTF^bnF[eu7%F[hu7$Fhct$!+l(3$HTF^bnF[eu7%F_hu7$Fbct$!+DK6;TF^bn F[eu7%Fchu7$F\\ct$!+6(=L5%F^bnF[eu7%Fghu7$Ffbt$!+u'y54%F^bnF[eu7%F[iu7 $F`bt$!+<1dzSF^bnF[eu7%F_iu7$Fjat$!+L1+pSF^bnF[eu7%Fciu7$Fdat$!+E$3'fS F^bnF[eu7%Fgiu7$F^at$!+L:n^SF^bnF[eu7%F[ju7$Fh`t$!+3(4b/%F^bnF[eu7%F_j u7$Fb`t$!+dO[TSF^bnF[eu7%Fcju7$F\\`t$!+5x)*RSF^bnF[eu7%Fgju7$Ff_t$!+!G G9/%F^bnF[eu7%F[[v7$F`_t$!+J9=YSF^bnF[eu7%F_[v7$Fj^t$!+1W_aSF^bnF[eu7% Fc[v7$Fd^t$!+QkamSF^bnF[eu7%Fg[v7$F^^t$!+6U1#3%F^bnF[eu7%F[\\v7$Fh]t$! +`ce+TF^bnF[eu7%F_\\v7$Fb]t$!+U%e87%F^bnF[eu7%Fc\\v7$F\\]t$!+&='\\VTF^ bnF[eu7%Fg\\v7$Ff\\t$!+7!Gh;%F^bnF[eu7%F[]v7$F`\\t$!+?6^)=%F^bnF[eu7%F _]v7$Fj[t$!+k.35UF^bnF[eu7%Fc]v7$Fd[t$!+^dWIUF^bnF[eu7%Fg]v7$F^[t$!+5E O\\UF^bnF[eu7%F[^v7$Fhjs$!+4[pmUF^bnF[eu7%F_^v7$Fbjs$!+uVF^bnF[eu7%F__v7$Fjhs$!+,$o)GVF^bnF[eu7%Fc_v7$Fdhs$!+4%QmL%F^ bnF[eu7%Fg_v7$F^hs$!+)\\zHM%F^bnF[eu7%F[`v7$Fhgs$!+m,&zM%F^bnF[eu7%F_` v7$Fbgs$!+:zg^VF^bnF[eu7%Fc`v7$F\\gs$!+:&3SN%F^bnF[eu7%Fg`v7$Fffs$!+8d ?bVF^bnF[eu7%F[av7$F`fs$!+l6DbVF^bnF[eu7%F_av7$Fjes$!+,X>aVF^bnF[eu7%F cav7$Fdes$!+#[$3_VF^bnF[eu7%Fgav7$F^es$!+*>k*[VF^bnF[eu7%F[bv7$Fhds$!+ 38)[M%F^bnF[eu7%F_bv7$Fbds$!+w#y)RVF^bnF[eu7%Fcbv7$F\\ds$!+[w*RL%F^bnF [eu7%Fgbv7$Ffcs$!+28GFVF^bnF[eu7%F[cv7$F`cs$!+[2x>VF^bnF[eu7%F_cv7$Feb s$!+ot]6VF^bnF[eu7%FccvFeduF[euFc^rFi^r-F$6%7$7$$!3/++++++!z%FTF(7$$\" 3+++++++!*>FTF(-%'COLOURG6&Fe^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F[ dv7$F($\"3/++++++!z%FTF`dvFcdv-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&R e(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fhev6$%*H ELVETICAG\"\"*-%%VIEWG6$;$!$z%Fh^n$\"$*>Fh^n;Fgfv$\"$z%Fh^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability inte rval) is the intersection of the stability region with the real line. " }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interva l is (approximately) " }{XPPEDIT 18 0 "[-4.05815, 0];" "6#7$,$-%&Floa tG6$\"':eS!\"&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The stability radius is half the l ength of the real stability interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(R(z)=-1,z=-4)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+gm2H?!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 511 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/1 20*z^5+\n 95600257532602284786818034509/6854538643459641658182387093 7500*z^6+\n 2334729932219731672459521259/124627975335629848330588856 25000*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 360 do\n zz := ne wton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re (zz),Im(zz)]]:\nend do:\npp1 := plot(pts,color=red):\nrd := 2.02907666 0: \npp2 := plot([-rd+rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue):\nplo ts[display]([pp1,pp2],scaling=constrained,font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7eal 7$$\"\"!F)F(7$F($\"31+++^v>Z5!#=7$$\"3W+++c)[EH&!#F$\"3')******)4&R%4# F-7$$\"3s*****HL;1a'!#E$\"3z*****zf#fTJF-7$$\"3F+++1?$f.%!#D$\"3%)**** *Hw*y)=%F-7$$\"3#******f[>)*o\"!#C$\"3Y*****R\\&)fB&F-7$$\"3q*****H$4Y uaFC$\"3g*****\\bwJG'F-7$$\"33+++9ohm9!#B$\"3J+++c\\NItF-7$$\"3#****** ftv(oLFN$\"3/+++pL]x$)F-7$$\"3S+++t*>nv'FN$\"3D+++qxeC%*F-7$$\"3#***** *HA!H(=\"!#A$\"35+++dY:Z5!#<7$$\"37+++oq\"oz\"Fhn$\"35+++rv#=:\"F[o7$$ \"3-+++#*4A#>#Fhn$\"3'******>(4Yc7F[o7$$\"39+++HyA$f\"Fhn$\"3-+++Z*H5O \"F[o7$$!3/+++*GnFg\"Fhn$\"3++++^))\\l9F[o7$$!3-+++'eLB.\"!#@$\"31+++= \">)p:F[o7$$!3++++Ul^[HFcp$\"3++++xw#Rn\"F[o7$$!39+++0+*4o'Fcp$\"31+++ nkuxF[o7$$!3s*****4\\XjE%Fcq$\"3))*****\\')*['3#F[o7$$!3l*****49&GBqF cq$\"33+++>=8)=#F[o7$$!31+++59H36!#>$\"33+++64%*)G#F[o7$$!33+++)*p5'o \"Fhr$\"3#)*****R!))y)Q#F[o7$$!3/+++&[=Q[#Fhr$\"35+++i&\\Fhr$\"31+++&e>3o#F [o7$$!3g*****pd-Tu'Fhr$\"37+++!=@m6F-$\"31+++CrIaHF[o7$$!3%*******))*QM[\"F-$\"3>++ +sP[QIF[o7$$!3'******Rpsg%=F-$\"3&)*****\\r=y6$F[o7$$!31+++0p][AF-$\"3 3+++C\\i\">$F[o7$$!3-+++gd!Ho#F-$\"3;+++2cZfKF[o7$$!3\")*****4Gr19$F-$ \"3-+++c/D@LF[o7$$!3G+++![(y8OF-$\"3')*****f\"z4xLF[o7$$!3!)******[Ni& 4%F-$\"3&******>p\\tU$F[o7$$!3-+++3p;\"e%F-$\"3;+++j5VsMF[o7$$!3[+++6X )o1&F-$\"3*)*****4?#z7NF[o7$$!3k*****4!*G/b&F-$\"3/+++md')[NF[o7$$!3n* *****p(G..'F-$\"3)******>iX5e$F[o7$$!3))*****\\^Ed]'F-$\"3z*****fCz'4O F[o7$$!3!******\\7yh(pF-$\"3&)*****pvn]j$F[o7$$!3)******He4:W(F-$\"3#) *******epul$F[o7$$!3h*****>i<v$F[o7$$!3/+++OGSn7F[o$\"3)******\\g3\"\\PF[o7$$!3(** ****p1Y'38F[o$\"3z*****zYaYu$F[o7$$!3'******fh4'\\8F[o$\"3*)******ehcQ PF[o7$$!3%******\\Q6.R\"F[o$\"3-+++Fv'3t$F[o7$$!35+++Z-xI9F[o$\"3$**** **>My:s$F[o7$$!35+++lk+r9F[o$\"3z*****>%[r5PF[o7$$!3.+++c@/6:F[o$\"3'* *****4D!H)p$F[o7$$!3++++)y,4b\"F[o$\"3-+++D`J%o$F[o7$$!33+++;Fh!f\"F[o $\"35+++F*)zoOF[o7$$!3/+++Xd?I;F[o$\"33+++)o[fn=F[o$\"3)********yPx^$ F[o7$$!33+++oni2>F[o$\"3y******G#o-\\$F[o7$$!3#******4Zsz%>F[o$\"33+++ haThMF[o7$$!3#*******\\Zq))>F[o$\"39+++!fO7V$F[o7$$!34+++R&**)H?F[o$\" 3?+++!H0)*R$F[o7$$!3%******p1H;2#F[o$\"3'******R%R@nLF[o7$$!3')*****zy cR6#F[o$\"38+++<[dLLF[o7$$!3))*****4NJp:#F[o$\"37+++o'>!*H$F[o7$$!3%** ****>=!e+AF[o$\"3%)*****RC(pjKF[o7$$!3y*****4a.\\C#F[o$\"3>+++A$oxA$F[ o7$$!3!)*****H;q)*G#F[o$\"3#)*****\\Q)R\">$F[o7$$!3,+++MbTNBF[o$\"3y** ***Ri[Z:$F[o7$$!3/++++NW\"Q#F[o$\"3++++ck'z6$F[o7$$!3,+++A7$yU#F[o$\"3 %******\\0x63$F[o7$$!33+++QnVuCF[o$\"3#)*****>AxW/$F[o7$$!3&)*****>*y5 @DF[o$\"3=+++!QKz+$F[o7$$!3!)*****f=\"pnDF[o$\"3'******>!odrHF[o7$$!30 +++(QRSh#F[o$\"3\"******p[;a$HF[o7$$!3/+++!><+m#F[o$\"3))*****>!QV**GF [o7$$!30+++PZ]0FF[o$\"37+++!)GfjGF[o7$$!3!******\\?*R]FF[o$\"36+++-]%y #GF[o7$$!3))*****p([h%z#F[o$\"3-+++&RL@z#F[o7$$!3)******p<#3QGF[o$\"3# ******\\>(RcFF[o7$$!3%******H4Y2)GF[o$\"37++++Xd?FF[o7$$!3,+++RVcAHF[o $\"3!******fd/Yo#F[o7$$!39+++oa]jHF[o$\"3'******>NH%[EF[o7$$!3'******* 4ra.IF[o$\"3-+++UV*>h#F[o7$$!3,+++)QuE/$F[o$\"3-+++T\"\\_d#F[o7$$!3%)* *****e&y33$F[o$\"3&)*****piZ\"QDF[o7$$!3@+++!)e:=JF[o$\"3()*****\\%zk+ DF[o7$$!3!*******zm]aJF[o$\"3/+++oBriCF[o7$$!3')*****\\hM**=$F[o$\"3(* *****H32VU#F[o7$$!3!******44YWA$F[o$\"39+++P>S&Q#F[o7$$!3/+++9)\\!eKF[ o$\"3-+++%=qfM#F[o7$$!33+++dkv!H$F[o$\"3$)*****HC))fI#F[o7$$!3!******p TyDK$F[o$\"37+++yaVlAF[o7$$!3++++R'HNN$F[o$\"37+++BSHCAF[o7$$!3y*****z \\DOQ$F[o$\"3#******fj[D=#F[o7$$!3y*****\\s#)GT$F[o$\"3=+++:m=S@F[o7$$ !39+++X$>8W$F[o$\"3#*******Rx>(4#F[o7$$!3#)*****fka*oMF[o$\"3?+++%GuN0 #F[o7$$!3/+++I\"4e\\$F[o$\"3'******\\06$4?F[o7$$!3:+++nW!>_$F[o$\"3'** ****4\\0W'>F[o7$$!31+++3MEZNF[o$\"3.+++(zd)=>F[o7$$!37+++v'4>d$F[o$\"3 '*******46ns=F[o7$$!31+++)ynef$F[o$\"30+++(p^e#=F[o7$$!3$)*****z$G;>OF [o$\"3-+++l\"4%y:iq$F[o$\"35+++cws#e\"F[o7$$!3A+++%)ecEPF[o$\"35+++IJWK:F[o7$$ !3!)*****R(QRYPF[o$\"3/+++-?n\"[\"F[o7$$!39+++idrlPF[o$\"3********o,XI 9F[o7$$!3%)*****>bWXy$F[o$\"31+++)e;)y8F[o7$$!3\"******f4*)G!QF[o$\"3/ +++0F\"oK\"F[o7$$!3<+++[Ov?QF[o$\"3++++L<[u7F[o7$$!3(******pcP\"QQF[o$ \"3)******R$y'=A\"F[o7$$!3++++)=N]&QF[o$\"31+++I_,p6F[o7$$!35+++)*eVrQ F[o$\"36+++#Hnf6\"F[o7$$!3)******pRCt)QF[o$\"3))*****>mlF1\"F[o7$$!30+ ++U6o-RF[o$\"3%******HY\\%45F[o7$$!39+++MH[#)RF[o$\"33+++H,%)))oF-7$$!3=+++rJ1$*RF[o$\"3]*****zi *[bjF-7$$!3\")*******ewJ+%F[o$\"3;+++TVuAeF-7$$!39+++HU]7SF[o$\"3]+++' p^1H&F-7$$!3v*****f'3-@SF[o$\"3#)*****zEI#fZF-7$$!3:+++3MqGSF[o$\"3$)* ****Rgt%GUF-7$$!3.+++d5`NSF[o$\"3!******R\"RN)p$F-7$$!3W+++;a[TSF[o$\" 3C+++Xb#)oJF-7$$!3++++\\0bYSF[o$\"3B+++b\"G)REF-7$$!3o*****f*Hr]SF[o$ \"3&******>d*G6@F-7$$!35+++$yhR0%F[o$\"31+++H$GJe\"F-7$$!3o*******R)Gc SF[o$\"3-+++5fDb5F-7$$!3#)*****zy'odSF[o$\"3?+++Kuyv_Fhr7$$!3u******>L :eSF[oF(7$Fbjm$!3?+++Kuyv_Fhr7$F]jm$!3-+++5fDb5F-7$Fhim$!31+++H$GJe\"F -7$Fcim$!3&******>d*G6@F-7$F^im$!3B+++b\"G)REF-7$Fihm$!3C+++Xb#)oJF-7$ Fdhm$!3!******R\"RN)p$F-7$F_hm$!3$)*****Rgt%GUF-7$Fjgm$!3#)*****zEI#fZ F-7$Fegm$!3]+++'p^1H&F-7$F`gm$!3;+++TVuAeF-7$F[gm$!3]*****zi*[bjF-7$Ff fm$!33+++H,%)))oF-7$Fafm$!3$******Rz:FU(F-7$F\\fm$!3I+++C/+dzF-7$Fgem$ !31+++]'Q:\\)F-7$Fbem$!3a*****zpLh-*F-7$F]em$!39+++Yiag&*F-7$Fhdm$!3%* *****HY\\%45F[o7$Fcdm$!3))*****>mlF1\"F[o7$F^dm$!36+++#Hnf6\"F[o7$Ficm $!31+++I_,p6F[o7$Fdcm$!3)******R$y'=A\"F[o7$F_cm$!3++++L<[u7F[o7$Fjbm$ !3/+++0F\"oK\"F[o7$Febm$!31+++)e;)y8F[o7$F`bm$!3********o,XI9F[o7$F[bm $!3/+++-?n\"[\"F[o7$Ffam$!35+++IJWK:F[o7$Faam$!35+++cws#e\"F[o7$F\\am$ !3-+++$=$\\K;F[o7$Fg`m$!3/+++(36F[o7$Fi^m$!3'******4\\0W'>F[o7$Fd^m$!3'******\\06$4?F[o7 $F_^m$!3?+++%GuN0#F[o7$Fj]m$!3#*******Rx>(4#F[o7$Fe]m$!3=+++:m=S@F[o7$ F`]m$!3#******fj[D=#F[o7$F[]m$!37+++BSHCAF[o7$Ff\\m$!37+++yaVlAF[o7$Fa \\m$!3$)*****HC))fI#F[o7$F\\\\m$!3-+++%=qfM#F[o7$Fg[m$!39+++P>S&Q#F[o7 $Fb[m$!3(******H32VU#F[o7$F][m$!3/+++oBriCF[o7$Fhjl$!3()*****\\%zk+DF[ o7$Fcjl$!3&)*****piZ\"QDF[o7$F^jl$!3-+++T\"\\_d#F[o7$Fiil$!3-+++UV*>h# F[o7$Fdil$!3'******>NH%[EF[o7$F_il$!3!******fd/Yo#F[o7$Fjhl$!37++++Xd? FF[o7$Fehl$!3#******\\>(RcFF[o7$F`hl$!3-+++&RL@z#F[o7$F[hl$!36+++-]%y# GF[o7$Ffgl$!37+++!)GfjGF[o7$Fagl$!3))*****>!QV**GF[o7$F\\gl$!3\"****** p[;a$HF[o7$Fgfl$!3'******>!odrHF[o7$Fbfl$!3=+++!QKz+$F[o7$F]fl$!3#)*** **>AxW/$F[o7$Fhel$!3%******\\0x63$F[o7$Fcel$!3++++ck'z6$F[o7$F^el$!3y* ****Ri[Z:$F[o7$Fidl$!3#)*****\\Q)R\">$F[o7$Fddl$!3>+++A$oxA$F[o7$F_dl$ !3%)*****RC(pjKF[o7$Fjcl$!37+++o'>!*H$F[o7$Fecl$!38+++<[dLLF[o7$F`cl$! 3'******R%R@nLF[o7$F[cl$!3?+++!H0)*R$F[o7$Ffbl$!39+++!fO7V$F[o7$Fabl$! 33+++haThMF[o7$F\\bl$!3y******G#o-\\$F[o7$Fgal$!3)********yPx^$F[o7$Fb al$!3\")*****H,\"yVNF[o7$F]al$!3y******fgOoNF[o7$Fh`l$!37+++W)p9f$F[o7 $Fc`l$!3A+++nd28OF[o7$F^`l$!3*)*****pwrJj$F[o7$Fi_l$!33+++)o[%[r5PF[o7$F`^l$!3$******>My:s$F[o7$F[^l$!3-+++Fv'3t$F[o7$ Ff]l$!3*)******ehcQPF[o7$Fa]l$!3z*****zYaYu$F[o7$F\\]l$!3)******\\g3\" \\PF[o7$Fg\\l$!3?+++s+!>v$F[o7$Fb\\l$!35+++y]*Hv$F[o7$F]\\l$!3))*****4 \"QN_PF[o7$Fh[l$!3=+++*eH*\\PF[o7$Fc[l$!3%******p&ymXPF[o7$F^[l$!3A+++ **\\]RPF[o7$Fiz$!3:+++/pOJPF[o7$Fdz$!3&)*****4Cn6s$F[o7$F_z$!3:+++%Q0) 3PF[o7$Fjy$!3%*********Q;%p$F[o7$Fey$!3')******Gb5xOF[o7$F`y$!3#)***** **epul$F[o7$F[y$!3&)*****pvn]j$F[o7$Ffx$!3z*****fCz'4OF[o7$Fax$!3)**** **>iX5e$F[o7$F\\x$!3/+++md')[NF[o7$Fgw$!3*)*****4?#z7NF[o7$Fbw$!3;+++j 5VsMF[o7$F]w$!3&******>p\\tU$F[o7$Fhv$!3')*****f\"z4xLF[o7$Fcv$!3-+++c /D@LF[o7$F^v$!3;+++2cZfKF[o7$Fiu$!33+++C\\i\">$F[o7$Fdu$!3&)*****\\r=y 6$F[o7$F_u$!3>+++sP[QIF[o7$Fjt$!31+++CrIaHF[o7$Fet$!3-+++!)H6mGF[o7$F` t$!37+++!=3o#F[o7$Ffs$!3/+++))*f\\e#F[o7$Fas$!35 +++=8)=#F[o7$F\\r$!3))*****\\')*['3#F[o7$Fgq$!33+++O]8%)>F[o7$Faq$!33 +++FM=\")=F[o7$F\\q$!31+++nkux)p:F[o7$F\\p$!3++++^))\\l9F[o7$Fgo$!3-+++Z*H5O\"F[o7$Fbo$!3'******> (4Yc7F[o7$F]o$!35+++rv#=:\"F[o7$Ffn$!35+++dY:Z5F[o7$FW$!3D+++qxeC%*F-7 $FR$!3/+++pL]x$)F-7$FL$!3J+++c\\NItF-7$FG$!3g*****\\bwJG'F-7$$\"3!**** **z]>)*o\"FC$!3Y*****R\\&)fB&F-7$F;$!3%)*****Hw*y)=%F-7$$\"35+++IjhSlF 7$!3z*****zf#fTJF-7$$\"3[+++f)[EH&F1$!3')******)4&R%4#F-7$F($!31+++^v> Z5F-F'F*F.F4F:F@FFFKFQFVFenF\\oFaoFfoF[pF`pFfpF[qF`qFfqF[rF`rFerF[sF`s FesFjsF_tFdtFitF^uFcuFhuF]vFbvFgvF\\wFawFfwF[xF`xFexFjxF_yFdyFiyF^zFcz FhzF][lFb[lFg[lF\\\\lFa\\lFf\\lF[]lF`]lFe]lFj]lF_^lFd^l-%'COLOURG6&%$R GBG$\"*++++\"!\")F(F(-F$6$7SF'7$$!3C*eL\\@z***=Fhr$\"3uv\\j%4a-x#F-7$$ !3qq+>ya!)=mFhr$\"3uW(\\4]P-9&F-7$$!3S\"\\^xjvY_\"F-$\"36/5d&R-or(F-7$ $!3![`W8OAIt#F-$\"3%*RS$Qjeq,\"F[o7$$!3yscL&GLCD%F-$\"3M.u=\"HGHC\"F[o 7$$!3j#*\\ph1o>fF-$\"3L0aYK`VK9F[o7$$!3=n0A.!4+)yF-$\"3$zo>LGn_g\"F[o7 $$!31s/l)p_B,\"F[o$\"3))[1KX#ofv\"F[o7$$!3ktd!4/TRD\"F[o$\"3O)GN!)[%=v =F[o7$$!3'3ZV\\Ucm^\"F[o$\"3-#R\"p>yIj>F[o7$$!3;4D(o3Qhv\"F[o$\"3Z'G(R wej5?F[o7$$!33u8$eB)GI?F[o$\"3Jw(z#)Hw!H?F[o7$$!3&e')zaxMbI#F[o$\"3D(p #yp\\:5?F[o7$$!3YJ)RRI2gc#F[o$\"3-M&\\:oYn&>F[o7$$!3%z-mnQpYz#F[o$\"3Y 08#R\"35z=F[o7$$!3A#*oS$3,A0$F[o$\"3#oU*[-nC_*f<:1J9F[o7$$!3Feg$pG+Hj$F[o$\"3 x$p*[83%HC\"F[o7$$!3)G(Hvzj6'y$F[o$\"3`o:m\"=r[,\"F[o7$$!31co<%R/@!RF[ o$\"3<*o%***[FL!yF-7$$!3GveR^***)*)RF[o$\"3Qh*[Q^F-7$$!39^D>vNm3RF[ o$!3P\\la]`)Rk(F-7$$!3uGO^Ab8*y$F[o$!3(fNu_6E'45F[o7$$!3$3B[`s%RTOF[o$ !3K,ZM/?!>B\"F[o7$$!3G.VB)*\\'=Z$F[o$!32p4QtVrE9F[o7$$!3w$[M^@\"QcKF[o $!3KAJM\"o)fD)e>F[o7$$!3'3MZ)fa9\"H# F[o$!3%ep/$=:37?F[o7$$!3%4?d`K!GN?F[o$!3K0g[wr1H?F[o7$$!3/e::SxynF[o7$$!3_\"))RBwj2D\"F[o $!3w!>$Rez'Q(=F[o7$$!3)zd2yhoQ,\"F[o$!3!3gu]5Xov\"F[o7$$!3C;nj'HEW*yF- $!3._3w*)3Q1;F[o7$$!3e.'*[$)=u&)eF-$!3YP2g&HA!H9F[o7$$!3/$oi9,*pgUF-$! 3q8U:TT*RC\"F[o7$$!3CLc*[!3()*o#F-$!32 " 0 "" {MPLTEXT 1 0 440 "R := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 95600257532602284786818034509/685453864 34596416581823870937500*z^6+\n 2334729932219731672459521259/12462797 533562984833058885625000*z^7:\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 50 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplo t(pts,color=COLOR(RGB,.95,0,0),thickness=2,font=[HELVETICA,9]);\nDigit s := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 249 293 293 {PLOTDATA 2 "6(-%'C URVESG6#7U7$$\"\"!F)F(7$$\"5:&[cRzIWC1&!#@$\"5b=U(*e`EfTJF-7$$\"5okx** [+.@#R(F-$\"5(GR.trI&=$G'F-7$$\"5$=r:j/tA&H#*F-$\"5&=y\"plgzxC%*F-7$$ \"55BrcI/U(43\"!#?$\"5>(o8^81PmD\"F=7$$\"5>n>n*))*fgA7F=$\"54J'R(QEjzq :F=7$$\"5)*)3H1bU&p_8F=$\"5(pHgz1fb\\)=F=7$$\"5_N7k$=;XTZ\"F=$\"5%[6xE J&[6*>#F=7$$\"5a%31o6k[))e\"F=$\"5#3Gcf:6uK^#F=7$$\"5a$=\\:`]7\")p\"F= $\"57SJH1iLVFGF=7$$\"5\\I*oJu#Q(G!=F=$\"5JZuU-)f#fTJF=7$$\"5!*)*Gbr'zA Q!>F=$\"5'4KnP)3=vbMF=7$$\"5#z!\\u;HrZ,?F=$\"5m;rQ=y4\"*pPF=7$$\"5*fVe \"o%yHi4#F=$\"5zNjSz\"3qS3%F=7$$\"5#f?0(>T4Q)=#F=$\"5z8rNf%3H#)R%F=7$$ \"5%*e]Zrq,;yAF=$\"5&\\(H$4r$zQ7ZF=7$$\"57?!)RNIAulBF=$\"5.v&485dYl-&F =7$$\"5-uz9mB&e7X#F=$\"5;(4pgO*[qS`F=7$$\"5+>AK=@b![`#F=$\"5(zavw9yi[l &F=7$$\"5oIYR/&3>uFF=$\"5e2AZ7f>L(f'F=7$$\"5CXn!ykC7.&GF=$\"5]d9SI Vf[6pF=7$$\"5G!>>Ky!)*eCHF=$\"5ve/(Hw2QcA(F=7$$\"5fL7*)z\"H%*p*HF=$\"5 JlrQGGyyRvF=7$$\"58Gi#z)[,[nIF=$\"5#zbnEw`MR&yF=7$$\"5'y%p(G_\"\\)f8$F =$\"5W`*HZ@Sx!o\")F=7$$\"5_KHz/**yU-KF=$\"5]Zjz)F=7$$\"5KahLgJ4qGLF=$\"5'\\,8Gn+x(eC%*F=7$$\"5xL1R^I?=XMF=$\"5rf4b.eBpQ(*F=7$$\"5 5Vbvo%>c#*\\$F=$\"52ty*fEOy_+\"!#>7$$\"5g5f)z#*4&>]NF=$\"5AMCTG*)eoO5F fu7$$\"5H%>^ic6Zwf$F=$\"5**=DZi/:4o5Ffu7$$\"57@nj%Q)o;TOF=$\"5Mb)[%** \\[\\*4\"Ffu7$$\"5UwG*>7))z,o$F=$\"5_J8p#R^&*38\"Ffu7$$\"5M\"Hp&R)pFRr $F=$\"5=OCUzEIHi6Ffu7$$\"5R1%QGf\")z8u$F=$\"5\"=b'=Aboo$>\"Ffu7$$\"54* 49#***f$3hPF=$\"5ZkbLM'Rw]A\"Ffu7$$\"5(pfA!GW9*3x$F=$\"5i=D\"[>(4Yc7Ff u7$$\"5^E)pVGg=uw$F=$\"5B'\\3%eA)RyG\"Ffu7$$\"5DwXW<8Ffu7$$\"5/&)>x/\\mc\"p$F=$\"5[*ej4)pod]8Ffu7$$\"54uSi.U/gyNF=$\"5 yS'fh!*3L>Q\"Ffu7$$\"5&Q6Cb?P^\\F$F=$\"5\\yg2#yhzKT\"Ffu7$$!5\"))\\iH) >TArLF=$\"5_(=S7%=%F=$\"5(p.G#y1@aQ:Ffu7$$ !5Hld\"=(o?ITVF=$\"5G6I)f\"=\">)p:Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THI CKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"F]\\l-%&COLORG6&%$RGBG$\"#&*!\"#F(F (-%%VIEWG6$%(DEFAULTGFi\\l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 123 "The relevant intersection point of the boundary c urve with the imaginary axis can be determined more accurately as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the b oundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits : = 15:\nz0 := 1.43*I:\nfor ct from 44 to 47 do\n newton(R(z)=exp(ct*P i/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0U=WtlEB\"!#>$\"0gh!*3L>Q\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02D)=S'yk%!#?$\"0w?yhzKT\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0h&[!3*R#R'!#?$\"02de?:YW\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0_f<;WJ;#!#>$\"0$o&**[QfZ\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisect ion method to calculate the parameter value associated with the inters ection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.43*I))\nend proc:\nDigit s := 15:\nu0 := bisect('real_part'(u),u=0.44..0.47);\nnewton(R(z)=exp( u0*Pi*I),z=1.43*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#u0G$\"0!Q(o2:ja%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0#*['fT zF9!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imaginary a xis that contains the origin and lies inside the stability region" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.4278];" "6#7$\"\"!-%&FloatG6$ \"&yU\"!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability regi on for the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 sta ge, order 4 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(4,8,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#\"F\"[^Z!fj;e \\(Qucb&>F,e\"\"H++vICF3!)ot:/If#3MM/>F)*$)F'\"\"&F)F)F)*&#\"S$))>ia]I '\\NDe]'y2-ie1aE&[z=W\"V++v=sf.3%*f(p)eD6NK%p%[-X0zA8$F)*$)F'F1F)F)F)* &#\"T " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+42O=T!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := ne wton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.59..0.49,color=[ red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cro ss,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3, color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.59..0 .49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3')*************e%!#<$!3# fV(\\G$o5U#F*7$$!3mmmTG_jMXF*$!3#)[X&\\O%p&>#F*7$$!3[LL$oXq#zWF*$!318_ O7OR))>F*7$$!39+D6ly4JWF*$!3+6)\\ybe<#=F*7$$!3#om\"Rt_#HQ%F*$!3w[&emx! 4n;F*7$$!3SLL=Q]duUF*$!3%)3Wgxy=f8F*7$$!3yKLG(e1b;%F*$!3;7k===)))4\"F* 7$$!3/n;p>l&p0%F*$!3()or%*RtZB))!#=7$$!3KL$e;r;j&RF*$!3(oX`j&*[;9(FP7$ $!3++]s>+6_QF*$!3S/f(388Lo&FP7$$!3IL$e)4$RVu$F*$!3fy%H$*p\"4MFP7$$!3Ymm'\\J9k_$F*$!3)*QxGvBF[DFP7$$!3GLLoO f3HMF*$!36zRdKHTD>FP7$$!3t*****[B<&>LF*$!39a31#4<*\\8FP7$$!3(*****\\L' )\\4KF*$!30^_'))[A/#))!#>7$$!3z****4D`Z.JF*$!3SB2@l;7`^Fcp7$$!3>L$eQ+' >2IF*$!3a#*e$H@_vR#Fcp7$$!3ommcy0r#*GF*$\"3-uGIY3'H=$!#?7$$!3ommE1ks&z #F*$\"3=;q>7Z,\\AFcp7$$!3y**\\#zv5Ho#F*$\"3/2\"ywWi#yTFcp7$$!3]mmE$z\\ Ie#F*$\"3mo&Fcp7$$!3#)**\\#z['[tCF*$\"3%R>\")zicv=(Fcp7$$!3') **\\x6j:pBF*$\"3$)>Pia=NU&)Fcp7$$!3Emm\"zr)HgAF*$\"3(oJ[wXI'H**Fcp7$$! 37m;4lMLg@F*$\"3$QF=`so@7\"FP7$$!3ALL3Nu]_?F*$\"3gzjg)4=uE\"FP7$$!3SL$ eQ\\10%>F*$\"3!y5wG/>'G9FP7$$!3/+]FX$4I%=F*$\"3OM@pL]R!e\"FP7$$!3ELLy` (4xt\"F*$\"3]^[2'fq!fFP7$$!39++l'G+ D_\"F*$\"3%)QpH[=L$=#FP7$$!3*)**\\7J*G&>9F*$\"3(R[>e.8*>CFP7$$!3z***\\ pz'>08F*$\"3HDYk!RLDr#FP7$$!3/nmE*GkC?\"F*$\"3,*oDv!Ra0IFP7$$!3u****\\ 4ax#4\"F*$\"3E#eaI&o``LFP7$$!3BMLe9)4Q$**FP$\"3P$Hz;UxOq$FP7$$!3'o**** 4=mr%))FP$\"3\"HSs8Ut>FP7$$!3Oqm\"f?>Z#yFP$\"3c/*3]0')Gd%FP7$$!3E'** \\A^nfv'FP$\"3u;y(H8&f)3&FP7$$!3%zmm;Qn5r&FP$\"3e0V+.z-\\cFP7$$!3J'** \\7y`rh%FP$\"3I/awT\"FP$\"3#em6\"H#[#y')FP7$$!3Mf ****>C4eVFcp$\"3C%e\"ox2bt&*FP7$$\"3wMLL)*p&\\*oFcp$\"3Am7\"z<#Qr5F*7$ $\"31mmma=)fp\"FP$\"3S`kqp'G[=\"F*7$$\"370+v?g5pFFP$\"3h1ix%R[!>8F*7$$ \"3g1+DvFA'z$FP$\"3%eXq[6K " 0 "" {MPLTEXT 1 0 1733 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+\n 1580127195555674387495816635904751481/190434340 825930041573688008272430750000*z^5+\n 4418794852654065862020778650 5825354963050546219883/\n 3132279054502484694323511255886975994 0803597218750000*z^6+\n 173715716040994083884354152868720974884441 669417517/\n 92793766989636109069334020955651663824630656760546 8750*z^7+\n 1736702836128959477289403551290677408738772839/\n \+ 61351250902238749797906790714480438892317789593750000*z^8:\npts := \+ []: z0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct* Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\ne nd do:\np_1 := plot(pts,color=COLOR(RGB,.44,.1,.1)):\np_2 := plots[pol ygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.87,.2,.2)):\npts := []: z0 := 1.5+4. 3*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I), z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np _3 := plot(pts,color=COLOR(RGB,.44,.1,.1)):\np_4 := plots[polygonplot] ([seq([pts[i-1],pts[i],[1.35,4.18]],i=2..nops(pts))],\n style =patchnogrid,color=COLOR(RGB,.87,.2,.2)):\npts := []: z0 := 1.5-4.3*I: \nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 : = plot(pts,color=COLOR(RGB,.44,.1,.1)):\np_6 := plots[polygonplot]([se q([pts[i-1],pts[i],[1.35,-4.18]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,.87,.2,.2)):\np_7 := plot([[[-4.79,0],[1.99, 0]],[[0,-4.79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([p _||(1..7)],view=[-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 594 594 594 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$$\"3'******4A()=l)!#F$\"3'*******fjzq:!#=7$$\"37+++#=*e2b !#D$\"3'******Hf$fTJF07$$\"3O++++_g*>'!#C$\"3B+++cXR7ZF07$$\"3*)*****R N\\\\S$!#B$\"3-+++;\"*>$G'F07$$\"3++++u(*pZ7!#A$\"3++++CN)R&yF07$$\"3- +++Huu$[$FF$\"3(******H6`YU*F07$$\"3))*****R]7.)yFF$\"3%******pn&\\*4 \"!#<7$$\"3!******zADJZ\"!#@$\"33+++0cVc7FS7$$\"3&******4fp.A#FW$\"3++ ++wa=89FS7$$\"3*******p$eEOBFW$\"3)******f(=ep:FS7$$\"3++++!)R#4[\"FF$ \"3)******Rh%QD66 N\"!#>$\"31+++6\"*fNBFS7$$!34+++EKk8DFfp$\"35+++N\"GG[#FS7$$!3-+++F=l8 VFfp$\"3)******fjToi#FS7$$!3(******>1;Y$pFfp$\"3)******fG$fmFFS7$$!3++ ++QNJ`5F0$\"3;+++/KQ+HFS7$$!3#******fjs!=:F0$\"3)******zGMe-$FS7$$!3++ ++r@)33#F0$\"3')*****Rg%QSJFS7$$!3&******p\"*)4@FF0$\"32+++`?=UKFS7$$! 35+++%RM=T$F0$\"31+++K_pILFS7$$!3u*****\\'\\#)GTF0$\"3#)*****pL$e1MFS7 $$!3?+++cNwa[F0$\"3=+++Hg8rMFS7$$!3M+++LR9zbF0$\"36+++=uxDNFS7$$!3o*** ***QIM'H'F0$\"3%******p([!=d$FS7$$!3j*****zZVP+(F0$\"3z******4UI5OFS7$ $!36+++@zT+xF0$\"3<+++5/:UOFS7$$!3o*****\\*pK'Q)F0$\"3))******R!Q!oOFS 7$$!39+++vS'=1*F0$\"3A+++Si^)o$FS7$$!3@+++Q!4ws*F0$\"3!******4d>Sq$FS7 $$!3#******>4-As$FS7$$!3!******RCaJH\"FS$\"33+++hgy;PFS7$$!3%******Hmg^N\"FS$\"3 0+++Onk2PFS7$$!3(******>8=mT\"FS$\"3=+++\\b'[p$FS7$$!3/+++:cfx9FS$\"3y *****H#Q]yOFS7$$!3,+++O<1&e;FS$\"3%*******3tL3OFS7$$!3'******>c1& =!yNFS7$$!3-+++G%)fyFS$\"3>+++X#eLU$FS7$$!3\"******>nyS-#FS$\"3!)*****\\X]pP$FS7$$!31+ ++ol&z3#FS$\"31+++(>BzK$FS7$$!3!******4&oH`@FS$\"3!)*****pFymF$FS7$$!3 @+++fI@?AFS$\"30+++f6rBKFS7$$!37+++n0o)G#FS$\"39+++.ydpJFS7$$!3')***** fb3&eBFS$\"3,+++ce$[6$FS7$$!31+++JbMHCFS$\"3-+++%ot*fIFS7$$!3-+++I9s+D FS$\"3?+++'>]`+$FS7$$!3?+++!e7@d#FS$\"39+++I^;^HFS7$$!3++++Rq+VEFS$\"3 !******H$[Y(*GFS7$$!3++++HC&Hr#FS$\"3!)*****4ktT%GFS7$$!3%)*******3!e \"y#FS$\"35+++!3N6z#FS7$$!3')*****pA4'[GFS$\"3-+++r,:QFFS7$$!33+++q'QQ \"HFS$\"3!********f1]o#FS7$$!3=+++YK8xHFS$\"3/+++zx\\JEFS7$$!30+++B%4% QIFS$\"3')*****pdLud#FS7$$!3!)*****\\fAw4$FS$\"3#)*****z=XE_#FS7$$!3#) *****p%ovaJFS$\"39+++dj)pY#FS7$$!36+++Ds\")4KFS$\"32+++.EL5CFS7$$!3?++ +cT#GE$FS$\"3!)*****p\\zDN#FS7$$!3$)******)f4QJ$FS$\"3?+++'pSOH#FS7$$! 31+++wW\"GO$FS$\"3#******p3YMB#FS7$$!3.+++]q))4MFS$\"3++++#GS><#FS7$$! 3')*****z\">3bMFS$\"3\")*****za\"34@FS7$$!3=+++y$f%)\\$FS$\"31+++Q7%[/ #FS7$$!3!******f%\\3SNFS$\"3++++;P?z>FS7$$!3!******R!)G+e$FS$\"3)***** **Rn;7>FS7$$!3.+++(*\\O=OFS$\"30+++`AuV=FS7$$!3%******>>q^l$FS$\"31+++ Au&Rx\"FS7$$!3))*****>%=_!p$FS$\"35+++#ybGq\"FS7$$!3=+++hc\\CPFS$\"3\" ******R=)\\I;FS7$$!3!********Rirv$FS$\"3-+++qK'pb\"FS7$$!3'******40%e) y$FS$\"3++++,rM#[\"FS7$$!3-+++p(4)=QFS$\"3,+++P;w19FS7$$!3y*****H'>(y% QFS$\"33+++#)=LI8FS7$$!3')*****z4$yvQFS$\"3%******\\p\">`7FS7$$!3'**** **fvLD!RFS$\"3%******H`ya<\"FS7$$!3()******eB4GRFS$\"35+++YwK(4\"FS7$$ !3=+++!o1C&RFS$\"3)******4Rm)=5FS7$$!3!******p*pSvRFS$\"3*)******pJ4-% *F07$$!39+++P/,(*RFS$\"3=+++=ca9')F07$$!3L+++0f7#4%FS$\"3))*****R^!z+RF07$$!3#)*****R2Y:5%FS$\"3.+++x.4>J F07$$!3s******HO()3TFS$\"3))*****R['QQBF07$$!30+++'4NT6%FS$\"3-+++l*p% e:F07$$!3\"******z$HIJF07$Feil$!3))*****R^!z+RF07$F`il$!3C+++kWk$o%F07$F[il$!3_*****pcR xY&F07$Ffhl$!33+++a52`iF07$Fahl$!3Y*****4\"z^RqF07$F\\hl$!3?+++&3=o#yF 07$Fggl$!3=+++=ca9')F07$Fbgl$!3*)******pJ4-%*F07$F]gl$!3)******4Rm)=5F S7$Fhfl$!35+++YwK(4\"FS7$Fcfl$!3%******H`ya<\"FS7$F^fl$!3%******\\p\"> `7FS7$Fiel$!33+++#)=LI8FS7$Fdel$!3,+++P;w19FS7$F_el$!3++++,rM#[\"FS7$F jdl$!3-+++qK'pb\"FS7$Fedl$!3\"******R=)\\I;FS7$F`dl$!35+++#ybGq\"FS7$F [dl$!31+++Au&Rx\"FS7$Ffcl$!30+++`AuV=FS7$Facl$!3)*******Rn;7>FS7$F\\cl $!3++++;P?z>FS7$Fgbl$!31+++Q7%[/#FS7$Fbbl$!3\")*****za\"34@FS7$F]bl$!3 ++++#GS><#FS7$Fhal$!3#******p3YMB#FS7$Fcal$!3?+++'pSOH#FS7$F^al$!3!)** ***p\\zDN#FS7$Fi`l$!32+++.EL5CFS7$Fd`l$!39+++dj)pY#FS7$F_`l$!3#)*****z =XE_#FS7$Fj_l$!3')*****pdLud#FS7$Fe_l$!3/+++zx\\JEFS7$F`_l$!3!******** f1]o#FS7$F[_l$!3-+++r,:QFFS7$Ff^l$!35+++!3N6z#FS7$Fa^l$!3!)*****4ktT%G FS7$F\\^l$!3!******H$[Y(*GFS7$Fg]l$!39+++I^;^HFS7$Fb]l$!3?+++'>]`+$FS7 $F]]l$!3-+++%ot*fIFS7$Fh\\l$!3,+++ce$[6$FS7$Fc\\l$!39+++.ydpJFS7$F^\\l $!30+++f6rBKFS7$Fi[l$!3!)*****pFymF$FS7$Fd[l$!31+++(>BzK$FS7$F_[l$!3!) *****\\X]pP$FS7$Fjz$!3>+++X#eLU$FS7$Fez$!3?+++)pXoY$FS7$F`z$!3$)*****H N-s]$FS7$F[z$!3;+++8$*GWNFS7$Ffy$!3#******p,>!yNFS7$Fay$!3%*******3tL3 OFS7$F\\y$!3()******G!4_j$FS7$Fgx$!3++++1qgeOFS7$Fbx$!3y*****H#Q]yOFS7 $F]x$!3=+++\\b'[p$FS7$Fhw$!30+++Onk2PFS7$Fcw$!33+++hgy;PFS7$F^w$!3&)** ***>4-As$FS7$Fiv$!3@+++M()yBPFS7$Fdv$!31+++w)49s$FS7$F_v$!3&******Rx#* [r$FS7$Fju$!3!******4d>Sq$FS7$Feu$!3A+++Si^)o$FS7$F`u$!3))******R!Q!oO FS7$F[u$!3<+++5/:UOFS7$Fft$!3z******4UI5OFS7$Fat$!3%******p([!=d$FS7$F \\t$!36+++=uxDNFS7$Fgs$!3=+++Hg8rMFS7$Fbs$!3#)*****pL$e1MFS7$F]s$!31++ +K_pILFS7$Fhr$!32+++`?=UKFS7$Fcr$!3')*****Rg%QSJFS7$F^r$!3)******zGMe- $FS7$Fiq$!3;+++/KQ+HFS7$Fdq$!3)******fG$fmFFS7$F_q$!3)******fjToi#FS7$ Fjp$!35+++N\"GG[#FS7$Fdp$!31+++6\"*fNBFS7$F_p$!3-+++&Q&z&=#FS7$Fio$!3; +++[?)Q.#FS7$Fdo$!3!*******omF!)=FS7$F_o$!3)******Rh%QD$!3-+++;\"*>$G'F07$F8 $!3B+++cXR7ZF07$F2$!3'******Hf$fTJF07$$\"3_******HT'=l)F-$!3'*******fj zq:F0F'-%&COLORG6&%$RGBG$\"#W!\"#$\"\"\"!\"\"Fb^n-%)POLYGONSG6fw7%F'7$ $\"+@s)=l)Ffp$\"+gjzq:!#57$$Fa^nF)F(7%Fi^n7$$\"+#=*e2bFS$\"+$f$fTJF^_n F__n7%Fb_n7$$\"++_g*>'!#;$\"+cXR7ZF^_nF__n7%Fh_n7$$\"+a$\\\\S$!#:$\"+; \"*>$G'F^_nF__n7%F_`n7$$\"+u(*pZ7!#9$\"+CN)R&yF^_nF__n7%Ff`n7$$\"+Huu$ [$Fi`n$\"+8JlC%*F^_nF__n7%F]an7$$\"+/DJ!)yFi`n$\"+xc\\*4\"!\"*F__n7%Fc an7$$\"+G_7t9!#8$\"+0cVc7FhanF__n7%Fjan7$$\"+\"fp.A#F]bn$\"+wa=89FhanF __n7%Fabn7$$\"+PeEOBF]bn$\"+w=ep:FhanF__n7%Fgbn7$$\"+!)R#4[\"Fi`n$\"+9 YQD66N\"!#6$\"+6\"*fNBFhanF__n7%Ffdn7$$!+EKk8DFidn$\"+N\"GG[#FhanF__n7 %F]en7$$!+F=l8VFidn$\"+O;%oi#FhanF__n7%Fcen7$$!+ighMpFidn$\"+'G$fmFFha nF__n7%Fien7$$!+QNJ`5F^_n$\"+/KQ+HFhanF__n7%F_fn7$$!+OE2=:F^_n$\"+)GMe -$FhanF__n7%Fefn7$$!+r@)33#F^_n$\"+/YQSJFhanF__n7%F[gn7$$!+<*)4@FF^_n$ \"+`?=UKFhanF__n7%Fagn7$$!+%RM=T$F^_n$\"+K_pILFhanF__n7%Fggn7$$!+l\\#) GTF^_n$\"+PLe1MFhanF__n7%F]hn7$$!+cNwa[F^_n$\"+Hg8rMFhanF__n7%Fchn7$$! +LR9zbF^_n$\"+=uxDNFhanF__n7%Fihn7$$!+RIM'H'F^_n$\"+x[!=d$FhanF__n7%F_ in7$$!+yMu.qF^_n$\"+5UI5OFhanF__n7%Fein7$$!+@zT+xF^_n$\"+5/:UOFhanF__n 7%F[jn7$$!+&*pK'Q)F^_n$\"+S!Q!oOFhanF__n7%Fajn7$$!+vS'=1*F^_n$\"+Si^)o $FhanF__n7%Fgjn7$$!+Q!4ws*F^_n$\"+r&>Sq$FhanF__n7%F][o7$$!+s,UQ5Fhan$ \"+uF*[r$FhanF__n7%Fc[o7$$!+[#GK5\"Fhan$\"+w)49s$FhanF__n7%Fi[o7$$!+Dq Cn6Fhan$\"+M()yBPFhanF__n7%F_\\o7$$!+mh`I7Fhan$\"+#4-As$FhanF__n7%Fe\\ o7$$!+WU:$H\"Fhan$\"+hgy;PFhanF__n7%F[]o7$$!+j1;b8Fhan$\"+Onk2PFhanF__ n7%Fa]o7$$!+K\"=mT\"Fhan$\"+\\b'[p$FhanF__n7%Fg]o7$$!+:cfx9Fhan$\"+BQ] yOFhanF__n7%F]^o7$$!+O<1&e;Fhan$\"+4tL3OFhanF__n7%F__o7$$!+il ]=!yNFhanF__n7%Fe_o7$$!+G%)fyFhan$\"+X#eLU$FhanF__n7%F]ao7$$!+s'yS-#Fhan$\"+b/&p P$FhanF__n7%Fcao7$$!+ol&z3#Fhan$\"+(>BzK$FhanF__n7%Fiao7$$!+^oH`@Fhan$ \"+x#ymF$FhanF__n7%F_bo7$$!+fI@?AFhan$\"+f6rBKFhanF__n7%Febo7$$!+n0o)G #Fhan$\"+.ydpJFhanF__n7%F[co7$$!+c&3&eBFhan$\"+ce$[6$FhanF__n7%Faco7$$ !+JbMHCFhan$\"+%ot*fIFhanF__n7%Fgco7$$!+I9s+DFhan$\"+'>]`+$FhanF__n7%F ]do7$$!+!e7@d#Fhan$\"+I^;^HFhanF__n7%Fcdo7$$!+Rq+VEFhan$\"+L[Y(*GFhanF __n7%Fido7$$!+HC&Hr#Fhan$\"+TO<#FhanF__n7%Fgio7$$!+=>3bMFhan$\"+[:34@FhanF__n7%F]jo7$$!+y$ f%)\\$Fhan$\"+Q7%[/#FhanF__n7%Fcjo7$$!+Y\\3SNFhan$\"+;P?z>FhanF__n7%Fi jo7$$!+/)G+e$Fhan$\"+Sn;7>FhanF__n7%F_[p7$$!+(*\\O=OFhan$\"+`AuV=FhanF __n7%Fe[p7$$!+#>q^l$Fhan$\"+Au&Rx\"FhanF__n7%F[\\p7$$!+U=_!p$Fhan$\"+# ybGq\"FhanF__n7%Fa\\p7$$!+hc\\CPFhan$\"+%=)\\I;FhanF__n7%Fg\\p7$$!++C; dPFhan$\"+qK'pb\"FhanF__n7%F]]p7$$!+^Se)y$Fhan$\"+,rM#[\"FhanF__n7%Fc] p7$$!+p(4)=QFhan$\"+P;w19FhanF__n7%Fi]p7$$!+j>(y%QFhan$\"+#)=LI8FhanF_ _n7%F_^p7$$!+)4$yvQFhan$\"+&p\">`7FhanF__n7%Fe^p7$$!+cP`-RFhan$\"+L&ya <\"FhanF__n7%F[_p7$$!+fB4GRFhan$\"+YwK(4\"FhanF__n7%Fa_p7$$!+!o1C&RFha n$\"+\"Rm)=5FhanF__n7%Fg_p7$$!+(*pSvRFhan$\"+qJ4-%*F^_nF__n7%F]`p7$$!+ P/,(*RFhan$\"+=ca9')F^_nF__n7%Fc`p7$$!+0f7#4%Fhan$\"+90z+RF^_nF__n7%Fgbp7$$!+uga,TFhan$\" +x.4>JF^_nF__n7%F]cp7$$!+IO()3TFhan$\"+%['QQBF^_nF__n7%Fccp7$$!+'4NT6% Fhan$\"+l*p%e:F^_nF__n7%Ficp7$$!+QHIJF^_nF__n7%Feep7$F hbp$!+90z+RF^_nF__n7%Fiep7$Fbbp$!+kWk$o%F^_nF__n7%F]fp7$F\\bp$!+n&RxY& F^_nF__n7%Fafp7$Ffap$!+a52`iF^_nF__n7%Fefp7$F`ap$!+6z^RqF^_nF__n7%Fifp 7$Fj`p$!+&3=o#yF^_nF__n7%F]gp7$Fd`p$!+=ca9')F^_nF__n7%Fagp7$F^`p$!+qJ4 -%*F^_nF__n7%Fegp7$Fh_p$!+\"Rm)=5FhanF__n7%Figp7$Fb_p$!+YwK(4\"FhanF__ n7%F]hp7$F\\_p$!+L&ya<\"FhanF__n7%Fahp7$Ff^p$!+&p\">`7FhanF__n7%Fehp7$ F`^p$!+#)=LI8FhanF__n7%Fihp7$Fj]p$!+P;w19FhanF__n7%F]ip7$Fd]p$!+,rM#[ \"FhanF__n7%Faip7$F^]p$!+qK'pb\"FhanF__n7%Feip7$Fh\\p$!+%=)\\I;FhanF__ n7%Fiip7$Fb\\p$!+#ybGq\"FhanF__n7%F]jp7$F\\\\p$!+Au&Rx\"FhanF__n7%Fajp 7$Ff[p$!+`AuV=FhanF__n7%Fejp7$F`[p$!+Sn;7>FhanF__n7%Fijp7$Fjjo$!+;P?z> FhanF__n7%F][q7$Fdjo$!+Q7%[/#FhanF__n7%Fa[q7$F^jo$!+[:34@FhanF__n7%Fe[ q7$Fhio$!+#GS><#FhanF__n7%Fi[q7$Fbio$!+(3YMB#FhanF__n7%F]\\q7$F\\io$!+ 'pSOH#FhanF__n7%Fa\\q7$Ffho$!+(\\zDN#FhanF__n7%Fe\\q7$F`ho$!+.EL5CFhan F__n7%Fi\\q7$Fjgo$!+dj)pY#FhanF__n7%F]]q7$Fdgo$!+)=XE_#FhanF__n7%Fa]q7 $F^go$!+xNVxDFhanF__n7%Fe]q7$Fhfo$!+zx\\JEFhanF__n7%Fi]q7$Fbfo$!++m+&o #FhanF__n7%F]^q7$F\\fo$!+r,:QFFhanF__n7%Fa^q7$Ffeo$!+!3N6z#FhanF__n7%F e^q7$F`eo$!+TO]`+$FhanF__n7%Fe_q7$Fhco$!+%ot*fIFhanF__n 7%Fi_q7$Fbco$!+ce$[6$FhanF__n7%F]`q7$F\\co$!+.ydpJFhanF__n7%Fa`q7$Ffbo $!+f6rBKFhanF__n7%Fe`q7$F`bo$!+x#ymF$FhanF__n7%Fi`q7$Fjao$!+(>BzK$Fhan F__n7%F]aq7$Fdao$!+b/&pP$FhanF__n7%Faaq7$F^ao$!+X#eLU$FhanF__n7%Feaq7$ Fh`o$!+)pXoY$FhanF__n7%Fiaq7$Fb`o$!+`B?2NFhanF__n7%F]bq7$F\\`o$!+8$*GW NFhanF__n7%Fabq7$Ff_o$!+!yNFhanF__n7%Febq7$F`_o$!+4tL3OFhanF__n7%Fi bq7$Fj^o$!+H!4_j$FhanF__n7%F]cq7$Fd^o$!+1qgeOFhanF__n7%Facq7$F^^o$!+BQ ]yOFhanF__n7%Fecq7$Fh]o$!+\\b'[p$FhanF__n7%Ficq7$Fb]o$!+Onk2PFhanF__n7 %F]dq7$F\\]o$!+hgy;PFhanF__n7%Fadq7$Ff\\o$!+#4-As$FhanF__n7%Fedq7$F`\\ o$!+M()yBPFhanF__n7%Fidq7$Fj[o$!+w)49s$FhanF__n7%F]eq7$Fd[o$!+uF*[r$Fh anF__n7%Faeq7$F^[o$!+r&>Sq$FhanF__n7%Feeq7$Fhjn$!+Si^)o$FhanF__n7%Fieq 7$Fbjn$!+S!Q!oOFhanF__n7%F]fq7$F\\jn$!+5/:UOFhanF__n7%Fafq7$Ffin$!+5UI 5OFhanF__n7%Fefq7$F`in$!+x[!=d$FhanF__n7%Fifq7$Fjhn$!+=uxDNFhanF__n7%F ]gq7$Fdhn$!+Hg8rMFhanF__n7%Fagq7$F^hn$!+PLe1MFhanF__n7%Fegq7$Fhgn$!+K_ pILFhanF__n7%Figq7$Fbgn$!+`?=UKFhanF__n7%F]hq7$F\\gn$!+/YQSJFhanF__n7% Fahq7$Fffn$!+)GMe-$FhanF__n7%Fehq7$F`fn$!+/KQ+HFhanF__n7%Fihq7$Fjen$!+ 'G$fmFFhanF__n7%F]iq7$Fden$!+O;%oi#FhanF__n7%Faiq7$F^en$!+N\"GG[#FhanF __n7%Feiq7$Fgdn$!+6\"*fNBFhanF__n7%Fiiq7$Fadn$!+&Q&z&=#FhanF__n7%F]jq7 $Fjcn$!+[?)Q.#FhanF__n7%Fajq7$Fdcn$!+pmF!)=FhanF__n7%Fejq7$F^cn$!+9YQD $G'F^_ nF__n7%Fe\\r7$Fi_n$!+cXR7ZF^_nF__n7%Fi\\r7$Fc_n$!+$f$fTJF^_nF__n7%F]]r 7$$\"+IT'=l)Ffp$!+gjzq:F^_nF__n7%Fa]rF'F__n-F\\^n6&F^^n$\"#()Fa^n$\"\" #Fd^nF[^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3++++a#pfY\"FS$\"35+++O HX#H%FS7$$\"31+++\\ubd9FS$\"3B+++zlB,VFS7$$\"3/+++toX[9FS$\"3$******fv 2$4VFS7$$\"3/+++64rQ9FS$\"3N+++&*[i;VFS7$$\"32+++ROOG9FS$\"3%******ziY JK%FS7$$\"31+++\\,Y<9FS$\"3k******)\\J)GVFS7$$\"3#******R$p/19FS$\"3,+ ++$pPOL%FS7$$\"3)*******3B<%R\"FS$\"3s*****4xAvL%FS7$$\"3-+++;p)=Q\"FS $\"3!)*****4UV/M%FS7$$\"3'******RIW#p8FS$\"35+++6_NUVFS7$$\"3/+++=;Ic8 FS$\"3'******4O7KM%FS7$$\"34+++C/7V8FS$\"3S+++Wv'HM%FS7$$\"33+++(pn(H8 FS$\"3))*****prr:M%FS7$$\"31+++GqJ;8FS$\"37+++>S(*QVFS7$$\"30+++$3]GI \"FS$\"3o******o<7NVFS7$$\"3+++++%e%*G\"FS$\"3$******Rig*HVFS7$$\"3%** ****feXiF\"FS$\"3\")*****z)\\VBVFS7$$\"3)******\\,ILE\"FS$\"3!******> \"))[:VFS7$$\"3/+++6\"[3D\"FS$\"3()*****>*p11VFS7$$\"32+++/$e*Q7FS$\"3 c******)o<^H%FS7$$\"32+++.d%yA\"FS$\"3;+++Ogf#G%FS7$$\"3.+++ots<7FS$\" 3%)*****\\\"*p%oUFS7$$\"3\"******z$y&)37FS$\"3')*****pvGFD%FS7$$\"30++ +%*R`,7FS$\"3$******Hf'RNUFS7$$\"3#******zP(4'>\"FS$\"3u******R2b;UFS7 $$\"31+++P0$H>\"FS$\"3k*****>8Zj>%FS7$$\"3))*****R*=W#>\"FS$\"3!****** *><0vTFS7$$\"3#******f*=.&>\"FS$\"3#******fRtI:%FS7$$\"3++++%>M5?\"FS$ \"3#******f)p)48%FS7$$\"33+++5Nj57FS$\"3j******))o_4TFS7$$\"3,+++Z*yPA \"FS$\"3o******>1`*3%FS7$$\"3'******zSI,C\"FS$\"3c*****ptF=2%FS7$$\"3# *******eh2f7FS$\"3%)*****H;.r0%FS7$$\"3'*******Qu#)z7FS$\"35+++f)*yXSF S7$$\"3'******pn^:I\"FS$\"3E+++6U.QSFS7$$\"3/+++-$*[B8FS$\"39+++s[tLSF S7$$\"3$******f&4-X8FS$\"3x*****H>@E.%FS7$$\"3-+++G[ol8FS$\"3<+++IPLMS FS7$$\"3!******R'>;&Q\"FS$\"3j*****Rz&[QSFS7$$\"36+++7$[KS\"FS$\"3#*** ***p#=qWSFS7$$\"3!*******ob#)>9FS$\"3C+++\"4PE0%FS7$$\"3.+++1%FS7$$\"33+++T[E[9FS$\"3k*****z@yC2%FS7$$\"3#******>;B,Y\" FS$\"3!)*****Hc%)Q3%FS7$$\"3++++;7Wq9FS$\"3K+++kR+'4%FS7$$\"3%******45 g#z9FS$\"3B+++zLm3TFS7$$\"31+++Dvi'[\"FS$\"3(******Hr7<7%FS7$$\"3)**** **\\w%f#\\\"FS$\"3?+++46-NTFS7$$\"3!*******zY@(\\\"FS$\"3S+++mPZ[TFS7$ $\"3)******>VS0]\"FS$\"39+++f#p>;%FS7$$\"3)******zpCE]\"FS$\"3E+++5tTv TFS7$$\"3.+++,#>N]\"FS$\"37+++Ipt)=%FS7$$\"3&*******oWF.:FS$\"3>+++s[& =?%FS7$$\"3++++R(R>]\"FS$\"3/+++_Vq9UFS7$$\"3%******>'Hc*\\\"FS$\"3+++ +sRAFUFS7$$\"3++++w3>'\\\"FS$\"3f*****>\"oNRUFS7$$\"35+++E\"p=\\\"FS$ \"3c******p'\\5D%FS7$$\"33+++9Ck'[\"FS$\"3m*****>S_AE%FS7$$\"3-+++!oa0 [\"FS$\"3-+++ft\"HF%FS7$$\"33+++\"H\\OZ\"FS$\"3i*****f*))*HG%FSFd^rF[^ n-Ff^n6jn7%7$$\"+a#pfY\"Fhan$\"+OHX#H%Fhan7$$\"+\\ubd9Fhan$\"+zlB,VFha n7$$\"$N\"Fa^n$\"$=%Fa^n7%Fhas7$$\"+toX[9Fhan$\"+cxI4VFhanF]bs7%Fcbs7$ $\"+64rQ9Fhan$\"+&*[i;VFhanF]bs7%Fibs7$$\"+ROOG9Fhan$\"+Gm9BVFhanF]bs7 %F_cs7$$\"+\\,Y<9Fhan$\"+*\\J)GVFhanF]bs7%Fecs7$$\"+Mp/19Fhan$\"+$pPOL %FhanF]bs7%F[ds7$$\"+4B<%R\"Fhan$\"+rF_PVFhanF]bs7%Fads7$$\"+;p)=Q\"Fh an$\"+@MWSVFhanF]bs7%Fgds7$$\"+/VCp8Fhan$\"+6_NUVFhanF]bs7%F]es7$$\"+= ;Ic8Fhan$\"+hB@VVFhanF]bs7%Fces7$$\"+C/7V8Fhan$\"+Wv'HM%FhanF]bs7%Fies 7$$\"+(pn(H8Fhan$\"+<S(*QVFhanF] bs7%Fefs7$$\"+$3]GI\"Fhan$\"+p<7NVFhanF]bs7%F[gs7$$\"++%e%*G\"Fhan$\"+ C1'*HVFhanF]bs7%Fags7$$\"+'eXiF\"Fhan$\"+))\\VBVFhanF]bs7%Fggs7$$\"+:+ Lj7Fhan$\"+7))[:VFhanF]bs7%F]hs7$$\"+6\"[3D\"Fhan$\"+#*p11VFhanF]bs7%F chs7$$\"+/$e*Q7Fhan$\"+*o<^H%FhanF]bs7%Fihs7$$\"+.d%yA\"Fhan$\"+Ogf#G% FhanF]bs7%F_is7$$\"+ots<7Fhan$\"+:*p%oUFhanF]bs7%Feis7$$\"+Qy&)37Fhan$ \"+d(GFD%FhanF]bs7%F[js7$$\"+%*R`,7Fhan$\"+$f'RNUFhanF]bs7%Fajs7$$\"+y t4'>\"Fhan$\"+S2b;UFhanF]bs7%Fgjs7$$\"+P0$H>\"Fhan$\"+KrM'>%FhanF]bs7% F][t7$$\"+%*=W#>\"Fhan$\"+?<0vTFhanF]bs7%Fc[t7$$\"+'*=.&>\"Fhan$\"+'Rt I:%FhanF]bs7%Fi[t7$$\"+%>M5?\"Fhan$\"+')p)48%FhanF]bs7%F_\\t7$$\"+5Nj5 7Fhan$\"+*)o_4TFhanF]bs7%Fe\\t7$$\"+Z*yPA\"Fhan$\"+?1`*3%FhanF]bs7%F[] t7$$\"+3/8S7Fhan$\"+Px#=2%FhanF]bs7%Fa]t7$$\"+fh2f7Fhan$\"+jJ5dSFhanF] bs7%Fg]t7$$\"+Ru#)z7Fhan$\"+f)*yXSFhanF]bs7%F]^t7$$\"+x;b,8Fhan$\"+6U. QSFhanF]bs7%Fc^t7$$\"+-$*[B8Fhan$\"+s[tLSFhanF]bs7%Fi^t7$$\"+c4-X8Fhan $\"+$>@E.%FhanF]bs7%F__t7$$\"+G[ol8Fhan$\"+IPLMSFhanF]bs7%Fe_t7$$\"+k> ;&Q\"Fhan$\"+%z&[QSFhanF]bs7%F[`t7$$\"+7$[KS\"Fhan$\"+F=qWSFhanF]bs7%F a`t7$$\"+pb#)>9Fhan$\"+\"4PE0%FhanF]bs7%Fg`t7$$\"+1 %FhanF]bs7%F]at7$$\"+T[E[9Fhan$\"+=#yC2%FhanF]bs7%Fcat7$$\"+iJ7g9Fhan$ \"+jX)Q3%FhanF]bs7%Fiat7$$\"+;7Wq9Fhan$\"+kR+'4%FhanF]bs7%F_bt7$$\"+,, Ez9Fhan$\"+zLm3TFhanF]bs7%Febt7$$\"+Dvi'[\"Fhan$\"+8Fr@TFhanF]bs7%F[ct 7$$\"+lZf#\\\"Fhan$\"+46-NTFhanF]bs7%Fact7$$\"+!o9s\\\"Fhan$\"+mPZ[TFh anF]bs7%Fgct7$$\"+K/a+:Fhan$\"+f#p>;%FhanF]bs7%F]dt7$$\"+)pCE]\"Fhan$ \"+5tTvTFhanF]bs7%Fcdt7$$\"+,#>N]\"Fhan$\"+Ipt)=%FhanF]bs7%Fidt7$$\"+p WF.:Fhan$\"+s[&=?%FhanF]bs7%F_et7$$\"+R(R>]\"Fhan$\"+_Vq9UFhanF]bs7%Fe et7$$\"+iHc*\\\"Fhan$\"+sRAFUFhanF]bs7%F[ft7$$\"+w3>'\\\"Fhan$\"+7oNRU FhanF]bs7%Faft7$$\"+E\"p=\\\"Fhan$\"+q'\\5D%FhanF]bs7%Fgft7$$\"+9Ck'[ \"Fhan$\"+-CDiUFhanF]bs7%F]gt7$$\"+!oa0[\"Fhan$\"+ft\"HF%FhanF]bs7%Fcg t7$$\"+\"H\\OZ\"Fhan$\"+'*))*HG%FhanF]bs7%FigtFcasF]bsFg]rF]^r-F$6$7in 7$Fe^r$!35+++OHX#H%FS7$F\\as$!3i*****f*))*HG%FS7$Fg`s$!3-+++ft\"HF%FS7 $Fb`s$!3m*****>S_AE%FS7$F]`s$!3c******p'\\5D%FS7$Fh_s$!3f*****>\"oNRUF S7$Fc_s$!3++++sRAFUFS7$F^_s$!3/+++_Vq9UFS7$Fi^s$!3>+++s[&=?%FS7$Fd^s$! 37+++Ipt)=%FS7$F_^s$!3E+++5tTvTFS7$Fj]s$!39+++f#p>;%FS7$Fe]s$!3S+++mPZ [TFS7$F`]s$!3?+++46-NTFS7$F[]s$!3(******Hr7<7%FS7$Ff\\s$!3B+++zLm3TFS7 $Fa\\s$!3K+++kR+'4%FS7$F\\\\s$!3!)*****Hc%)Q3%FS7$Fg[s$!3k*****z@yC2%F S7$Fb[s$!37+++?^)>1%FS7$F][s$!3C+++\"4PE0%FS7$Fhjr$!3#******p#=qWSFS7$ Fcjr$!3j*****Rz&[QSFS7$F^jr$!3<+++IPLMSFS7$Fiir$!3x*****H>@E.%FS7$Fdir $!39+++s[tLSFS7$F_ir$!3E+++6U.QSFS7$Fjhr$!35+++f)*yXSFS7$Fehr$!3%)**** *H;.r0%FS7$F`hr$!3c*****ptF=2%FS7$F[hr$!3o******>1`*3%FS7$Ffgr$!3j**** **))o_4TFS7$Fagr$!3#******f)p)48%FS7$F\\gr$!3#******fRtI:%FS7$Fgfr$!3! *******><0vTFS7$Fbfr$!3k*****>8Zj>%FS7$F]fr$!3u******R2b;UFS7$Fher$!3$ ******Hf'RNUFS7$Fcer$!3')*****pvGFD%FS7$F^er$!3%)*****\\\"*p%oUFS7$Fid r$!3;+++Ogf#G%FS7$Fddr$!3c******)o<^H%FS7$F_dr$!3()*****>*p11VFS7$Fjcr $!3!******>\"))[:VFS7$Fecr$!3\")*****z)\\VBVFS7$F`cr$!3$******Rig*HVFS 7$F[cr$!3o******o<7NVFS7$Ffbr$!37+++>S(*QVFS7$Fabr$!3))*****prr:M%FS7$ F\\br$!3S+++Wv'HM%FS7$Fgar$!3'******4O7KM%FS7$Fbar$!35+++6_NUVFS7$F]ar $!3!)*****4UV/M%FS7$Fh`r$!3s*****4xAvL%FS7$Fc`r$!3,+++$pPOL%FS7$F^`r$! 3k******)\\J)GVFS7$Fi_r$!3%******ziYJK%FS7$Fd_r$!3N+++&*[i;VFS7$F__r$! 3$******fv2$4VFS7$Fj^r$!3B+++zlB,VFSFbhtF[^n-Ff^n6jn7%7$Fdas$!+OHX#H%F han7$Fjgt$!+'*))*HG%Fhan7$F^bs$!$=%Fa^n7%F\\du7$Fdgt$!+ft\"HF%FhanF_du 7%Fcdu7$F^gt$!+-CDiUFhanF_du7%Fgdu7$Fhft$!+q'\\5D%FhanF_du7%F[eu7$Fbft $!+7oNRUFhanF_du7%F_eu7$F\\ft$!+sRAFUFhanF_du7%Fceu7$Ffet$!+_Vq9UFhanF _du7%Fgeu7$F`et$!+s[&=?%FhanF_du7%F[fu7$Fjdt$!+Ipt)=%FhanF_du7%F_fu7$F ddt$!+5tTvTFhanF_du7%Fcfu7$F^dt$!+f#p>;%FhanF_du7%Fgfu7$Fhct$!+mPZ[TFh anF_du7%F[gu7$Fbct$!+46-NTFhanF_du7%F_gu7$F\\ct$!+8Fr@TFhanF_du7%Fcgu7 $Ffbt$!+zLm3TFhanF_du7%Fggu7$F`bt$!+kR+'4%FhanF_du7%F[hu7$Fjat$!+jX)Q3 %FhanF_du7%F_hu7$Fdat$!+=#yC2%FhanF_du7%Fchu7$F^at$!+?^)>1%FhanF_du7%F ghu7$Fh`t$!+\"4PE0%FhanF_du7%F[iu7$Fb`t$!+F=qWSFhanF_du7%F_iu7$F\\`t$! +%z&[QSFhanF_du7%Fciu7$Ff_t$!+IPLMSFhanF_du7%Fgiu7$F`_t$!+$>@E.%FhanF_ du7%F[ju7$Fj^t$!+s[tLSFhanF_du7%F_ju7$Fd^t$!+6U.QSFhanF_du7%Fcju7$F^^t $!+f)*yXSFhanF_du7%Fgju7$Fh]t$!+jJ5dSFhanF_du7%F[[v7$Fb]t$!+Px#=2%Fhan F_du7%F_[v7$F\\]t$!+?1`*3%FhanF_du7%Fc[v7$Ff\\t$!+*)o_4TFhanF_du7%Fg[v 7$F`\\t$!+')p)48%FhanF_du7%F[\\v7$Fj[t$!+'RtI:%FhanF_du7%F_\\v7$Fd[t$! +?<0vTFhanF_du7%Fc\\v7$F^[t$!+KrM'>%FhanF_du7%Fg\\v7$Fhjs$!+S2b;UFhanF _du7%F[]v7$Fbjs$!+$f'RNUFhanF_du7%F_]v7$F\\js$!+d(GFD%FhanF_du7%Fc]v7$ Ffis$!+:*p%oUFhanF_du7%Fg]v7$F`is$!+Ogf#G%FhanF_du7%F[^v7$Fjhs$!+*o<^H %FhanF_du7%F_^v7$Fdhs$!+#*p11VFhanF_du7%Fc^v7$F^hs$!+7))[:VFhanF_du7%F g^v7$Fhgs$!+))\\VBVFhanF_du7%F[_v7$Fbgs$!+C1'*HVFhanF_du7%F__v7$F\\gs$ !+p<7NVFhanF_du7%Fc_v7$Fffs$!+>S(*QVFhanF_du7%Fg_v7$F`fs$!+<FSF(-%'COLOURG6&F^^nF)F)F)-%*LINESTYLEG6 #\"\"$-F$6%7$7$F(F_cv7$F($\"3/++++++!z%FSFdcvFgcv-%*AXESSTYLEG6#%$BOXG -%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CON STRAINEDG-F\\ev6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%Fa^n$\"$*>Fa^n;F[fv$ \"$z%Fa^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme is (approximately) " }{XPPEDIT 18 0 "[-4.1184, 0];" "6#7$,$-% &FloatG6$\"&%=T!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture co mpares the stability region for the 8 stage, order 4 scheme with that \+ of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "p lots[display]([p||(1..7),p_||(1..6)],view=[-4.79..1.99,-4.79..4.79],fo nt=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scal ing=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "65-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3))******HiHB\"*!#G$\"3#******fK' zq:!#=7$$\"3S+++KjhSl!#E$\"3z*****zf#fTJF07$$\"3e+++E^!)y&)!#D$\"3#*** ***p$zQ7ZF07$$\"3Y+++M4Yua!#C$\"3g*****\\bwJG'F07$$\"3?+++qhbiA!#B$\"3 I+++QX$R&yF07$$\"3S+++t*>nv'FF$\"3D+++qxeC%*F07$$\"3'*******p&z:\\\"!# A$\"3!*******\\[\\*4\"!#<7$$\"3/+++\"*4A#>#FQ$\"3'******>(4Yc7FT7$$\"3 /+++>S'yk%FF$\"3-+++='zKT\"FT7$$!3-+++'eLB.\"!#@$\"31+++=\">)p:FT7$$!3 B+++ET,EXF\\o$\"3\"******>uyes\"FT7$$!31+++D8]P8!#?$\"33+++FM=\")=FT7$ $!3%)*****\\X\"4lKFgo$\"3')*****>A%RN?FT7$$!3l*****49&GBqFgo$\"33+++>= 8)=#FT7$$!3)******HIeGP\"!#>$\"3@+++KO**QBFT7$$!3/+++&[=Q[#Fgp$\"35+++ +++sP[QIFT7$$!36+++![oF/#F0$\"3#******4oYa:$FT7$$!3-+++gd! Ho#F0$\"3;+++2cZfKFT7$$!3%******pdidP$F0$\"3;+++cv*)\\LFT7$$!3!)****** [Ni&4%F0$\"3&******>p\\tU$FT7$$!3y*****>muT#[F0$\"3=+++wQ<$\\$FT7$$!3k *****4!*G/b&F0$\"3/+++md')[NFT7$$!3]*****4#fioiF0$\"3#*******Qey&f$FT7 $$!3!******\\7yh(pF0$\"3&)*****pvn]j$FT7$$!3;+++n&\\An(F0$\"3=+++D3inO FT7$$!3w*******)G!pN)F0$\"3%*********Q;%p$FT7$$!3f*****z:B1.*F0$\"3>++ +*\\j_r$FT7$$!3#******pxkSp*F0$\"3:+++/pOJPFT7$$!33+++2CzM5FT$\"3$**** **fVGGu$FT7$$!3#******f5)G*4\"FT$\"3=+++*eH*\\PFT7$$!3#******>#z&H;\"F T$\"3#******\\A%*Gv$FT7$$!3.+++'fieA\"FT$\"3?+++s+!>v$FT7$$!3%******R] g!)G\"FT$\"37+++tu3ZPFT7$$!3'******fh4'\\8FT$\"3*)******ehcQPFT7$$!3/+ ++^*p0T\"FT$\"3A+++i1UEPFT7$$!35+++lk+r9FT$\"3z*****>%[r5PFT7$$!3&**** **RM#*4`\"FT$\"3')******))f\\\"p$FT7$$!33+++;Fh!f\"FT$\"35+++F*)zoOFT7 $$!3/+++k(o*\\;FT$\"3,+++51lUOFT7$$!30+++F@=4FT$\"33+++haThMFT7$$! 3*)*****\\nR#4?FT$\"3*)*****z=!e+AFT$\"3%)**** *RC(pjKFT7$$!37+++*p4tE#FT$\"3\"******H>G'4KFT7$$!3,+++MbTNBFT$\"3y*** **Ri[Z:$FT7$$!3()*****z\"4g/CFT$\"3%)*****H*ec*4$FT7$$!33+++QnVuCFT$\" 3#)*****>AxW/$FT7$$!3=+++Z*>Wa#FT$\"3/+++W*H(*)HFT7$$!30+++(QRSh#FT$\" 3\"******p[;a$HFT7$$!3A+++z!HGo#FT$\"37+++C%)\\\")GFT7$$!3!******\\?*R ]FFT$\"36+++-]%y#GFT7$$!3()*****>$eW;GFT$\"31+++WAFuFFT7$$!3%******H4Y 2)GFT$\"37++++Xd?FFT7$$!3-+++Mh9VHFT$\"3!******R7Ylm#FT7$$!3'*******4r a.IFT$\"3-+++UV*>h#FT7$$!3/+++,A*=1$FT$\"3A+++QcucDFT7$$!3@+++!)e:=JFT $\"3()*****\\%zk+DFT7$$!3A+++kcLsJFT$\"33+++1/dVCFT7$$!3!******44YWA$F T$\"39+++P>S&Q#FT7$$!39+++MX^uKFT$\"3&)*****RM\\gK#FT7$$!3!******pTyDK $FT$\"37+++yaVlAFT7$$!3++++uMooLFT$\"3&)*****Rn(\\.AFT7$$!3y*****\\s#) GT$FT$\"3=+++:m=S@FT7$$!31+++2fBbMFT$\"3))******=dYv?FT7$$!3/+++I\"4e \\$FT$\"3'******\\06$4?FT7$$!3z*****\\_uY`$FT$\"33+++&y6<%>FT7$$!37+++ v'4>d$FT$\"3'*******46ns=FT7$$!37+++8mf2OFT$\"3.+++%e2A!=FT7$$!3****** *p=?=k$FT$\"3!******pmc.t\"FT7$$!3%)******>cmuOFT$\"3)******RAsrl\"FT7 $$!3;+++#>:iq$FT$\"35+++cws#e\"FT7$$!3()*****>5Wlt$FT$\"34+++Di62:FT7$ $!39+++idrlPFT$\"3********o,XI9FT7$$!3=+++wpx$z$FT$\"3#******fEeGN\"FT 7$$!3<+++[Ov?QFT$\"3++++L<[u7FT7$$!3%******RrZm%QFT$\"3/+++H'oa>\"FT7$ $!35+++)*eVrQFT$\"36+++#Hnf6\"FT7$$!3'******4hq]*QFT$\"30+++c&>h.\"FT7 $$!39+++MH[L:eSFTF(7$Fjjl$!3p+++a\"*)R\"zFgp7$Fejl$!31+++H$GJe\"F07$F`jl$!37++ +9k]vBF07$F[jl$!3C+++Xb#)oJF07$Ffil$!37+++6mLjRF07$Fail$!3#)*****zEI#f ZF07$F\\il$!3))*****fk9mb&F07$Fghl$!3]*****zi*[bjF07$Fbhl$!3W+++#p=d:( F07$F]hl$!3I+++C/+dzF07$Fhgl$!3]*****p-V)e()F07$Fcgl$!39+++Yiag&*F07$F ^gl$!30+++c&>h.\"FT7$Fifl$!36+++#Hnf6\"FT7$Fdfl$!3/+++H'oa>\"FT7$F_fl$ !3++++L<[u7FT7$Fjel$!3#******fEeGN\"FT7$Feel$!3********o,XI9FT7$F`el$! 34+++Di62:FT7$F[el$!35+++cws#e\"FT7$Ffdl$!3)******RAsrl\"FT7$Fadl$!3!* *****pmc.t\"FT7$F\\dl$!3.+++%e2A!=FT7$Fgcl$!3'*******46ns=FT7$Fbcl$!33 +++&y6<%>FT7$F]cl$!3'******\\06$4?FT7$Fhbl$!3))******=dYv?FT7$Fcbl$!3= +++:m=S@FT7$F^bl$!3&)*****Rn(\\.AFT7$Fial$!37+++yaVlAFT7$Fdal$!3&)**** *RM\\gK#FT7$F_al$!39+++P>S&Q#FT7$Fj`l$!33+++1/dVCFT7$Fe`l$!3()*****\\% zk+DFT7$F``l$!3A+++QcucDFT7$F[`l$!3-+++UV*>h#FT7$Ff_l$!3!******R7Ylm#F T7$Fa_l$!37++++Xd?FFT7$F\\_l$!31+++WAFuFFT7$Fg^l$!36+++-]%y#GFT7$Fb^l$ !37+++C%)\\\")GFT7$F]^l$!3\"******p[;a$HFT7$Fh]l$!3/+++W*H(*)HFT7$Fc]l $!3#)*****>AxW/$FT7$F^]l$!3%)*****H*ec*4$FT7$Fi\\l$!3y*****Ri[Z:$FT7$F d\\l$!3\"******H>G'4KFT7$F_\\l$!3%)*****RC(pjKFT7$Fj[l$!3=+++,GS;LFT7$ Fe[l$!3'******R%R@nLFT7$F`[l$!3*)*****z%[r5PFT7$F^x$!3A+++i1UEPFT7$Fiw$!3*)******ehcQP FT7$Fdw$!37+++tu3ZPFT7$F_w$!3?+++s+!>v$FT7$Fjv$!3#******\\A%*Gv$FT7$Fe v$!3=+++*eH*\\PFT7$F`v$!3$******fVGGu$FT7$F[v$!3:+++/pOJPFT7$Ffu$!3>++ +*\\j_r$FT7$Fau$!3%*********Q;%p$FT7$F\\u$!3=+++D3inOFT7$Fgt$!3&)***** pvn]j$FT7$Fbt$!3#*******Qey&f$FT7$F]t$!3/+++md')[NFT7$Fhs$!3=+++wQ<$\\ $FT7$Fcs$!3&******>p\\tU$FT7$F^s$!3;+++cv*)\\LFT7$Fir$!3;+++2cZfKFT7$F dr$!3#******4oYa:$FT7$F_r$!3>+++sP[QIFT7$Fjq$!32+++'\\f1\"HFT7$Feq$!37 +++!==8)=#FT7$F[p$!3')*****>A%RN?FT7$Feo$!33+++FM=\")=FT7$ F`o$!3\"******>uyes\"FT7$Fjn$!31+++=\">)p:FT7$Fen$!3-+++='zKT\"FT7$$\" 3-+++#*4A#>#FQ$!3'******>(4Yc7FT7$FO$!3!*******\\[\\*4\"FT7$FJ$!3D+++q xeC%*F07$FD$!3I+++QX$R&yF07$$\"3q*****H$4YuaF@$!3g*****\\bwJG'F07$F8$! 3#******p$zQ7ZF07$$\"3()*****RL;1a'F4$!3z*****zf#fTJF07$$\"3m*****H,'H B\"*F-$!3#******fK'zq:F0F'-%&COLORG6&%$RGBG$\"#[!\"#$\"#8Fh^nFi^n-%)PO LYGONSG6fw7%F'7$$\"+IiHB\"*Fgo$\"+Ejzq:!#57$$Fh^nF)F(7%F__n7$$\"+KjhSl F0$\"+)f#fTJFd_nFe_n7%Fh_n7$$\"+E^!)y&)FT$\"+PzQ7ZFd_nFe_n7%F^`n7$$\"+ M4Yua!#;$\"+bl<$G'Fd_nFe_n7%Fd`n7$$\"+qhbiA!#:$\"+QX$R&yFd_nFe_n7%F[an 7$$\"+t*>nv'F^an$\"+qxeC%*Fd_nFe_n7%Fban7$$\"+q&z:\\\"!#9$\"+][\\*4\"! \"*Fe_n7%Fhan7$$\"+\"*4A#>#F[bn$\"+s4Yc7F^bnFe_n7%F`bn7$$\"+>S'yk%F^an $\"+='zKT\"F^bnFe_n7%Ffbn7$$!+'eLB.\"!#8$\"+=\">)p:F^bnFe_n7%F\\cn7$$! +ET,EXF_cn$\"+U(yes\"F^bnFe_n7%Fccn7$$!+D8]P8!#7$\"+FM=\")=F^bnFe_n7%F icn7$$!+b94lKF\\dn$\"+AURN?F^bnFe_n7%F`dn7$$!+T^GBqF\\dn$\"+>=8)=#F^bn Fe_n7%Ffdn7$$!+.$eGP\"!#6$\"+KO**QBF^bnFe_n7%F\\en7$$!+&[=Q[#F_en$\"+< i_([#F^bnFe_n7%Fcen7$$!+pQp5UF_en$\"+uv5LEF^bnFe_n7%Fien7$$!+xD5WnF_en $\"+!=v$F^bnFe_n7%F[]o7$$!+/01)G\"F^bn$\"+tu3ZPF^bnFe_n 7%Fa]o7$$!+;'4'\\8F^bn$\"+fhcQPF^bnFe_n7%Fg]o7$$!+^*p0T\"F^bn$\"+i1UEP F^bnFe_n7%F]^o7$$!+lk+r9F^bn$\"+U[r5PF^bnFe_n7%Fc^o7$$!+WB*4`\"F^bn$\" +*)f\\\"p$F^bnFe_n7%Fi^o7$$!+;Fh!f\"F^bn$\"+F*)zoOF^bnFe_n7%F__o7$$!+k (o*\\;F^bn$\"+51lUOF^bnFe_n7%Fe_o7$$!+F@=4F^bn$\"+haThM F^bnFe_n7%Fcao7$$!+v'R#4?F^bn$\"+y@n:MF^bnFe_n7%Fiao7$$!+n!H;2#F^bn$\" +WR@nLF^bnFe_n7%F_bo7$$!+G4ON@F^bn$\"+,GS;LF^bnFe_n7%Febo7$$!+#=!e+AF^ bn$\"+WspjKF^bnFe_n7%F[co7$$!+*p4tE#F^bn$\"+$>G'4KF^bnFe_n7%Faco7$$!+M bTNBF^bn$\"+C'[Z:$F^bnFe_n7%Fgco7$$!+=4g/CF^bn$\"+$*ec*4$F^bnFe_n7%F]d o7$$!+QnVuCF^bn$\"+AsZWIF^bnFe_n7%Fcdo7$$!+Z*>Wa#F^bn$\"+W*H(*)HF^bnFe _n7%Fido7$$!+(QRSh#F^bn$\"+([;a$HF^bnFe_n7%F_eo7$$!+z!HGo#F^bn$\"+C%) \\\")GF^bnFe_n7%Feeo7$$!+0#*R]FF^bn$\"+-]%y#GF^bnFe_n7%F[fo7$$!+KeW;GF ^bn$\"+WAFuFF^bnFe_n7%Fafo7$$!+$4Y2)GF^bn$\"++Xd?FF^bnFe_n7%Fgfo7$$!+M h9VHF^bn$\"+ChamEF^bnFe_n7%F]go7$$!+5ra.IF^bn$\"+UV*>h#F^bnFe_n7%Fcgo7 $$!+,A*=1$F^bn$\"+QcucDF^bnFe_n7%Figo7$$!+!)e:=JF^bn$\"+Xzk+DF^bnFe_n7 %F_ho7$$!+kcLsJF^bn$\"+1/dVCF^bnFe_n7%Feho7$$!+\"4YWA$F^bn$\"+P>S&Q#F^ bnFe_n7%F[io7$$!+MX^uKF^bn$\"+W$\\gK#F^bnFe_n7%Faio7$$!+<%yDK$F^bn$\"+ yaVlAF^bnFe_n7%Fgio7$$!+uMooLF^bn$\"+uw\\.AF^bnFe_n7%F]jo7$$!+DF)GT$F^ bn$\"+:m=S@F^bnFe_n7%Fcjo7$$!+2fBbMF^bn$\"+>dYv?F^bnFe_n7%Fijo7$$!+I\" 4e\\$F^bn$\"+b5J4?F^bnFe_n7%F_[p7$$!+DXnMNF^bn$\"+&y6<%>F^bnFe_n7%Fe[p 7$$!+v'4>d$F^bn$\"+56ns=F^bnFe_n7%F[\\p7$$!+8mf2OF^bn$\"+%e2A!=F^bnFe_ n7%Fa\\p7$$!+(=?=k$F^bn$\"+nmNI:iq$F^bn$\"+cws#e\"F^bnFe_n7%Fc]p7$$!+-TaOPF^bn$ \"+Di62:F^bnFe_n7%Fi]p7$$!+idrlPF^bn$\"+p,XI9F^bnFe_n7%F_^p7$$!+wpx$z$ F^bn$\"+m#eGN\"F^bnFe_n7%Fe^p7$$!+[Ov?QF^bn$\"+L<[u7F^bnFe_n7%F[_p7$$! +9xkYQF^bn$\"+H'oa>\"F^bnFe_n7%Fa_p7$$!+)*eVrQF^bn$\"+#Hnf6\"F^bnFe_n7 %Fg_p7$$!+612&*QF^bn$\"+c&>h.\"F^bnFe_n7%F]`p7$$!+MH[h.\"F^bnFe_n7%F_hp 7$Fh_p$!+#Hnf6\"F^bnFe_n7%Fchp7$Fb_p$!+H'oa>\"F^bnFe_n7%Fghp7$F\\_p$!+ L<[u7F^bnFe_n7%F[ip7$Ff^p$!+m#eGN\"F^bnFe_n7%F_ip7$F`^p$!+p,XI9F^bnFe_ n7%Fcip7$Fj]p$!+Di62:F^bnFe_n7%Fgip7$Fd]p$!+cws#e\"F^bnFe_n7%F[jp7$F^] p$!+CAF^bnFe_n7%F_[ q7$F`[p$!+b5J4?F^bnFe_n7%Fc[q7$Fjjo$!+>dYv?F^bnFe_n7%Fg[q7$Fdjo$!+:m=S @F^bnFe_n7%F[\\q7$F^jo$!+uw\\.AF^bnFe_n7%F_\\q7$Fhio$!+yaVlAF^bnFe_n7% Fc\\q7$Fbio$!+W$\\gK#F^bnFe_n7%Fg\\q7$F\\io$!+P>S&Q#F^bnFe_n7%F[]q7$Ff ho$!+1/dVCF^bnFe_n7%F_]q7$F`ho$!+Xzk+DF^bnFe_n7%Fc]q7$Fjgo$!+QcucDF^bn Fe_n7%Fg]q7$Fdgo$!+UV*>h#F^bnFe_n7%F[^q7$F^go$!+ChamEF^bnFe_n7%F_^q7$F hfo$!++Xd?FF^bnFe_n7%Fc^q7$Fbfo$!+WAFuFF^bnFe_n7%Fg^q7$F\\fo$!+-]%y#GF ^bnFe_n7%F[_q7$Ffeo$!+C%)\\\")GF^bnFe_n7%F__q7$F`eo$!+([;a$HF^bnFe_n7% Fc_q7$Fjdo$!+W*H(*)HF^bnFe_n7%Fg_q7$Fddo$!+AsZWIF^bnFe_n7%F[`q7$F^do$! +$*ec*4$F^bnFe_n7%F_`q7$Fhco$!+C'[Z:$F^bnFe_n7%Fc`q7$Fbco$!+$>G'4KF^bn Fe_n7%Fg`q7$F\\co$!+WspjKF^bnFe_n7%F[aq7$Ffbo$!+,GS;LF^bnFe_n7%F_aq7$F `bo$!+WR@nLF^bnFe_n7%Fcaq7$Fjao$!+y@n:MF^bnFe_n7%Fgaq7$Fdao$!+haThMF^b nFe_n7%F[bq7$F^ao$!+-*yT]$F^bnFe_n7%F_bq7$Fh`o$!+85yVNF^bnFe_n7%Fcbq7$ Fb`o$!+vU5!e$F^bnFe_n7%Fgbq7$F\\`o$!+nd28OF^bnFe_n7%F[cq7$Ff_o$!+51lUO F^bnFe_n7%F_cq7$F`_o$!+F*)zoOF^bnFe_n7%Fccq7$Fj^o$!+*)f\\\"p$F^bnFe_n7 %Fgcq7$Fd^o$!+U[r5PF^bnFe_n7%F[dq7$F^^o$!+i1UEPF^bnFe_n7%F_dq7$Fh]o$!+ fhcQPF^bnFe_n7%Fcdq7$Fb]o$!+tu3ZPF^bnFe_n7%Fgdq7$F\\]o$!+s+!>v$F^bnFe_ n7%F[eq7$Ff\\o$!+DU*Gv$F^bnFe_n7%F_eq7$F`\\o$!+*eH*\\PF^bnFe_n7%Fceq7$ Fj[o$!+O%GGu$F^bnFe_n7%Fgeq7$Fd[o$!+/pOJPF^bnFe_n7%F[fq7$F^[o$!+*\\j_r $F^bnFe_n7%F_fq7$Fhjn$!++R;%p$F^bnFe_n7%Fcfq7$Fbjn$!+D3inOF^bnFe_n7%Fg fq7$F\\jn$!+dx1NOF^bnFe_n7%F[gq7$Ffin$!+Rey&f$F^bnFe_n7%F_gq7$F`in$!+m d')[NF^bnFe_n7%Fcgq7$Fjhn$!+wQ<$\\$F^bnFe_n7%Fggq7$Fdhn$!+#p\\tU$F^bnF e_n7%F[hq7$F^hn$!+cv*)\\LF^bnFe_n7%F_hq7$Fhgn$!+2cZfKF^bnFe_n7%Fchq7$F bgn$!+\"oYa:$F^bnFe_n7%Fghq7$F\\gn$!+sP[QIF^bnFe_n7%F[iq7$Fffn$!+'\\f1 \"HF^bnFe_n7%F_iq7$F`fn$!+!==8)=#F^bnFe_n7%Fcjq7$Fadn$!+AURN?F^bnFe_n7%Fgjq7$Fjcn$!+FM=\")=F^bnF e_n7%F[[r7$Fdcn$!+U(yes\"F^bnFe_n7%F_[r7$F]cn$!+=\">)p:F^bnFe_n7%Fc[r7 $Fgbn$!+='zKT\"F^bnFe_n7%Fg[r7$$\"+#*4A#>#F[bn$!+s4Yc7F^bnFe_n7%F[\\r7 $Fian$!+][\\*4\"F^bnFe_n7%Fa\\r7$Fcan$!+qxeC%*Fd_nFe_n7%Fe\\r7$F\\an$! +QX$R&yFd_nFe_n7%Fi\\r7$$\"+L4YuaFg`n$!+bl<$G'Fd_nFe_n7%F]]r7$F_`n$!+P zQ7ZFd_nFe_n7%Fc]r7$$\"+MjhSlF0$!+)f#fTJFd_nFe_n7%Fg]r7$$\"+8gHB\"*Fgo $!+Ejzq:Fd_nFe_n7%F]^rF'Fe_n-Fc^n6&Fe^n$\"#&*Fh^n$\"#DFh^nFg^r-%&STYLE G6#%,PATCHNOGRIDG-F$6$7in7$$\"3-+++ZVwa9FT$\"38+++]F`-VFT7$$\"3-+++J)o jW\"FT$\"3,+++ot]6VFT7$$\"3.+++^MFP9FT$\"3$)*****zuq(>VFT7$$\"31+++N9_ F9FT$\"3!)*****pI\"GFVFT7$$\"3!******pgcrT\"FT$\"3U+++[w*RL%FT7$$\"35+ ++2PA19FT$\"3J+++w#y)RVFT7$$\"35+++m(oZR\"FT$\"3s*****zI\")[M%FT7$$\"3 !******H_RGQ\"FT$\"33+++*>k*[VFT7$$\"3))*****p\"f[q8FT$\"3G+++#[$3_VFT 7$$\"3++++a1wd8FT$\"3G+++,X>aVFT7$$\"3)*******z)>ZM\"FT$\"3K+++l6DbVFT 7$$\"3(******p)RUJ8FT$\"39+++8d?bVFT7$$\"3%******4eQzJ\"FT$\"3I+++:&3S N%FT7$$\"3&******>mNVI\"FT$\"3w*****\\\"zg^VFT7$$\"3))*****R1&p!H\"FT$ \"3S+++m,&zM%FT7$$\"3%******4A1rF\"FT$\"30+++)\\zHM%FT7$$\"3%*******\\ .nj7FT$\"3')******3%QmL%FT7$$\"3/+++TJ]]7FT$\"3)******4Io)GVFT7$$\"3#* *****z0QxB\"FT$\"3,+++x1h>VFT7$$\"3,+++t.`D7FT$\"3(*******p\"4)3VFT7$$ \"3&*******))>197FT$\"3c*****\\*GT'H%FT7$$\"3,+++Oqa.7FT$\"33+++u\"FT$\"3k******3[pmUFT7$$\"3))*****HRJk=\"FT$\"3Q+++5 EO\\UFT7$$\"3!*******e>Z!=\"FT$\"3U+++^dWIUFT7$$\"3-+++$)4vw6FT$\"3T++ +k.35UFT7$$\"35+++*H%pv6FT$\"3C+++?6^)=%FT7$$\"3!******>NJx<\"FT$\"3q* ****>,Gh;%FT7$$\"31+++BvB$=\"FT$\"30+++&='\\VTFT7$$\"3'********R[C>\"F T$\"3,+++U%e87%FT7$$\"33+++aCO07FT$\"3W+++`ce+TFT7$$\"3*)******\\=n@7F T$\"3g*****4@k?3%FT7$$\"34+++!\\n2C\"FT$\"3g*****zVYl1%FT7$$\"3-+++$*G $=E\"FT$\"3k*****fSCX0%FT7$$\"3++++F*))RG\"FT$\"3%******4V\"=YSFT7$$\" 3)******p:EkI\"FT$\"3()******z#G9/%FT7$$\"30+++DK[G8FT$\"37+++5x)*RSFT 7$$\"3#******RTo'\\8FT$\"3n*****pl$[TSFT7$$\"3#******\\CW'p8FT$\"3)*** ***zq4b/%FT7$$\"3+++++o>)Q\"FT$\"3Q+++L:n^SFT7$$\"3!******fo._S\"FT$\" 3A+++E$3'fSFT7$$\"3*)********og?9FT$\"3#******Hj+!pSFT7$$\"32+++!3#RM9 FT$\"3!******phq&zSFT7$$\"3%******zatlW\"FT$\"3C+++u'y54%FT7$$\"3%**** **\\!R=d9FT$\"3,+++6(=L5%FT7$$\"3-+++WpEm9FT$\"3K+++DK6;TFT7$$\"3%**** **p#H(QZ\"FT$\"3A+++l(3$HTFT7$$\"3%******>Yb+[\"FT$\"3r*****R%>xUTFT7$ $\"33+++%[p[[\"FT$\"3Q+++agQcTFT7$$\"3!******H)*p$)[\"FT$\"3g*****pB[+ <%FT7$$\"31+++A7h!\\\"FT$\"39+++Lrm$=%FT7$$\"3)******>HY;\\\"FT$\"3#)* *****R4;(>%FT7$$\"3-+++()o_\"\\\"FT$\"3x*****p!eX5UFT7$$\"3)******RB.. \\\"FT$\"3O+++)\\%[BUFT7$$\"3/+++8T-)[\"FT$\"3%*******z_=OUFT7$$\"3'** ****f%pt%[\"FT$\"3w*****f*4][UFT7$$\"3)******\\$z[![\"FT$\"3G+++p#y.E% FT7$$\"3%*******=AKv9FT$\"3I+++OowrUFT7$$\"3++++kSGp9FT$\"3')*****>(*= EG%FT7$$\"35+++iqTi9FT$\"3w******y*))GH%FTF`_rFb^n-F\\_n6jn7%7$$\"+ZVw a9F^bn$\"+]F`-VF^bn7$$\"+J)ojW\"F^bn$\"+ot]6VF^bn7$$\"$L\"Fh^n$\"$=%Fh ^n7%Fdbs7$$\"+^MFP9F^bn$\"+[2x>VF^bnFibs7%F_cs7$$\"+N9_F9F^bn$\"+28GFV F^bnFibs7%Fecs7$$\"+2m:<9F^bn$\"+[w*RL%F^bnFibs7%F[ds7$$\"+2PA19F^bn$ \"+w#y)RVF^bnFibs7%Fads7$$\"+m(oZR\"F^bn$\"+38)[M%F^bnFibs7%Fgds7$$\"+ B&RGQ\"F^bn$\"+*>k*[VF^bnFibs7%F]es7$$\"+aVF^bnFibs7%Fies7$$\"+!))>ZM\"F^bn$\"+l6DbVF ^bnFibs7%F_fs7$$\"+()RUJ8F^bn$\"+8d?bVF^bnFibs7%Fefs7$$\"+\"eQzJ\"F^bn $\"+:&3SN%F^bnFibs7%F[gs7$$\"+icL/8F^bn$\"+:zg^VF^bnFibs7%Fags7$$\"+k] p!H\"F^bn$\"+m,&zM%F^bnFibs7%Fggs7$$\"+@i5x7F^bn$\"+)\\zHM%F^bnFibs7%F ]hs7$$\"+].nj7F^bn$\"+4%QmL%F^bnFibs7%Fchs7$$\"+TJ]]7F^bn$\"+,$o)GVF^b nFibs7%Fihs7$$\"+e!QxB\"F^bn$\"+x1h>VF^bnFibs7%F_is7$$\"+t.`D7F^bn$\"+ q\"4)3VF^bnFibs7%Feis7$$\"+*)>197F^bn$\"+&*GT'H%F^bnFibs7%F[js7$$\"+Oq a.7F^bn$\"+u\"F^bn$\"+4[pmUF^bnFibs7%Fgj s7$$\"+$RJk=\"F^bn$\"+5EO\\UF^bnFibs7%F][t7$$\"+f>Z!=\"F^bn$\"+^dWIUF^ bnFibs7%Fc[t7$$\"+$)4vw6F^bn$\"+k.35UF^bnFibs7%Fi[t7$$\"+*H%pv6F^bn$\" +?6^)=%F^bnFibs7%F_\\t7$$\"+_8tx6F^bn$\"+7!Gh;%F^bnFibs7%Fe\\t7$$\"+Bv B$=\"F^bn$\"+&='\\VTF^bnFibs7%F[]t7$$\"++%[C>\"F^bn$\"+U%e87%F^bnFibs7 %Fa]t7$$\"+aCO07F^bn$\"+`ce+TF^bnFibs7%Fg]t7$$\"+]=n@7F^bn$\"+6U1#3%F^ bnFibs7%F]^t7$$\"+!\\n2C\"F^bn$\"+QkamSF^bnFibs7%Fc^t7$$\"+$*G$=E\"F^b n$\"+1W_aSF^bnFibs7%Fi^t7$$\"+F*))RG\"F^bn$\"+J9=YSF^bnFibs7%F__t7$$\" +dhU18F^bn$\"+!GG9/%F^bnFibs7%Fe_t7$$\"+DK[G8F^bn$\"+5x)*RSF^bnFibs7%F [`t7$$\"+9%o'\\8F^bn$\"+dO[TSF^bnFibs7%Fa`t7$$\"+XUkp8F^bn$\"+3(4b/%F^ bnFibs7%Fg`t7$$\"++o>)Q\"F^bn$\"+L:n^SF^bnFibs7%F]at7$$\"+'o._S\"F^bn$ \"+E$3'fSF^bnFibs7%Fcat7$$\"++pg?9F^bn$\"+L1+pSF^bnFibs7%Fiat7$$\"+!3# RM9F^bn$\"+<1dzSF^bnFibs7%F_bt7$$\"+[NdY9F^bn$\"+u'y54%F^bnFibs7%Febt7 $$\"+0R=d9F^bn$\"+6(=L5%F^bnFibs7%F[ct7$$\"+WpEm9F^bn$\"+DK6;TF^bnFibs 7%Fact7$$\"+FH(QZ\"F^bn$\"+l(3$HTF^bnFibs7%Fgct7$$\"+ia0![\"F^bn$\"+W> xUTF^bnFibs7%F]dt7$$\"+%[p[[\"F^bn$\"+agQcTF^bnFibs7%Fcdt7$$\"+$)*p$)[ \"F^bn$\"+P#[+<%F^bnFibs7%Fidt7$$\"+A7h!\\\"F^bn$\"+Lrm$=%F^bnFibs7%F_ et7$$\"+#HY;\\\"F^bn$\"+S4;(>%F^bnFibs7%Feet7$$\"+()o_\"\\\"F^bn$\"+2e X5UF^bnFibs7%F[ft7$$\"+MKI!\\\"F^bn$\"+)\\%[BUF^bnFibs7%Faft7$$\"+8T-) [\"F^bn$\"+!G&=OUF^bnFibs7%Fgft7$$\"+Ypt%[\"F^bn$\"+'*4][UF^bnFibs7%F] gt7$$\"+Nz[![\"F^bn$\"+p#y.E%F^bnFibs7%Fcgt7$$\"+>AKv9F^bn$\"+OowrUF^b nFibs7%Figt7$$\"+kSGp9F^bn$\"+s*=EG%F^bnFibs7%F_ht7$$\"+iqTi9F^bn$\"+z *))GH%F^bnFibs7%FehtF_bsFibsFc^rFi^r-F$6$7in7$Fa_r$!38+++]F`-VFT7$Fhas $!3w******y*))GH%FT7$Fcas$!3')*****>(*=EG%FT7$F^as$!3I+++OowrUFT7$Fi`s $!3G+++p#y.E%FT7$Fd`s$!3w*****f*4][UFT7$F_`s$!3%*******z_=OUFT7$Fj_s$! 3O+++)\\%[BUFT7$Fe_s$!3x*****p!eX5UFT7$F`_s$!3#)******R4;(>%FT7$F[_s$! 39+++Lrm$=%FT7$Ff^s$!3g*****pB[+<%FT7$Fa^s$!3Q+++agQcTFT7$F\\^s$!3r*** **R%>xUTFT7$Fg]s$!3A+++l(3$HTFT7$Fb]s$!3K+++DK6;TFT7$F]]s$!3,+++6(=L5% FT7$Fh\\s$!3C+++u'y54%FT7$Fc\\s$!3!******phq&zSFT7$F^\\s$!3#******Hj+! pSFT7$Fi[s$!3A+++E$3'fSFT7$Fd[s$!3Q+++L:n^SFT7$F_[s$!3)******zq4b/%FT7 $Fjjr$!3n*****pl$[TSFT7$Fejr$!37+++5x)*RSFT7$F`jr$!3()******z#G9/%FT7$ F[jr$!3%******4V\"=YSFT7$Ffir$!3k*****fSCX0%FT7$Fair$!3g*****zVYl1%FT7 $F\\ir$!3g*****4@k?3%FT7$Fghr$!3W+++`ce+TFT7$Fbhr$!3,+++U%e87%FT7$F]hr $!30+++&='\\VTFT7$Fhgr$!3q*****>,Gh;%FT7$Fcgr$!3C+++?6^)=%FT7$F^gr$!3T +++k.35UFT7$Fifr$!3U+++^dWIUFT7$Fdfr$!3Q+++5EO\\UFT7$F_fr$!3k******3[p mUFT7$Fjer$!33+++uVFT7$Ffdr$!3)******4Io)GVFT7$Fadr$!3')******3%Q mL%FT7$F\\dr$!30+++)\\zHM%FT7$Fgcr$!3S+++m,&zM%FT7$Fbcr$!3w*****\\\"zg ^VFT7$F]cr$!3I+++:&3SN%FT7$Fhbr$!39+++8d?bVFT7$Fcbr$!3K+++l6DbVFT7$F^b r$!3G+++,X>aVFT7$Fiar$!3G+++#[$3_VFT7$Fdar$!33+++*>k*[VFT7$F_ar$!3s*** **zI\")[M%FT7$Fj`r$!3J+++w#y)RVFT7$Fe`r$!3U+++[w*RL%FT7$F``r$!3!)***** pI\"GFVFT7$F[`r$!3$)*****zuq(>VFT7$Ff_r$!3,+++ot]6VFTF^itFb^n-F\\_n6jn 7%7$F`bs$!+]F`-VF^bn7$Ffht$!+z*))GH%F^bn7$Fjbs$!$=%Fh^n7%Fhdu7$F`ht$!+ s*=EG%F^bnF[eu7%F_eu7$Fjgt$!+OowrUF^bnF[eu7%Fceu7$Fdgt$!+p#y.E%F^bnF[e u7%Fgeu7$F^gt$!+'*4][UF^bnF[eu7%F[fu7$Fhft$!+!G&=OUF^bnF[eu7%F_fu7$Fbf t$!+)\\%[BUF^bnF[eu7%Fcfu7$F\\ft$!+2eX5UF^bnF[eu7%Fgfu7$Ffet$!+S4;(>%F ^bnF[eu7%F[gu7$F`et$!+Lrm$=%F^bnF[eu7%F_gu7$Fjdt$!+P#[+<%F^bnF[eu7%Fcg u7$Fddt$!+agQcTF^bnF[eu7%Fggu7$F^dt$!+W>xUTF^bnF[eu7%F[hu7$Fhct$!+l(3$ HTF^bnF[eu7%F_hu7$Fbct$!+DK6;TF^bnF[eu7%Fchu7$F\\ct$!+6(=L5%F^bnF[eu7% Fghu7$Ffbt$!+u'y54%F^bnF[eu7%F[iu7$F`bt$!+<1dzSF^bnF[eu7%F_iu7$Fjat$!+ L1+pSF^bnF[eu7%Fciu7$Fdat$!+E$3'fSF^bnF[eu7%Fgiu7$F^at$!+L:n^SF^bnF[eu 7%F[ju7$Fh`t$!+3(4b/%F^bnF[eu7%F_ju7$Fb`t$!+dO[TSF^bnF[eu7%Fcju7$F\\`t $!+5x)*RSF^bnF[eu7%Fgju7$Ff_t$!+!GG9/%F^bnF[eu7%F[[v7$F`_t$!+J9=YSF^bn F[eu7%F_[v7$Fj^t$!+1W_aSF^bnF[eu7%Fc[v7$Fd^t$!+QkamSF^bnF[eu7%Fg[v7$F^ ^t$!+6U1#3%F^bnF[eu7%F[\\v7$Fh]t$!+`ce+TF^bnF[eu7%F_\\v7$Fb]t$!+U%e87% F^bnF[eu7%Fc\\v7$F\\]t$!+&='\\VTF^bnF[eu7%Fg\\v7$Ff\\t$!+7!Gh;%F^bnF[e u7%F[]v7$F`\\t$!+?6^)=%F^bnF[eu7%F_]v7$Fj[t$!+k.35UF^bnF[eu7%Fc]v7$Fd[ t$!+^dWIUF^bnF[eu7%Fg]v7$F^[t$!+5EO\\UF^bnF[eu7%F[^v7$Fhjs$!+4[pmUF^bn F[eu7%F_^v7$Fbjs$!+uVF^bnF[eu7%F__v7$Fjhs$!+,$o )GVF^bnF[eu7%Fc_v7$Fdhs$!+4%QmL%F^bnF[eu7%Fg_v7$F^hs$!+)\\zHM%F^bnF[eu 7%F[`v7$Fhgs$!+m,&zM%F^bnF[eu7%F_`v7$Fbgs$!+:zg^VF^bnF[eu7%Fc`v7$F\\gs $!+:&3SN%F^bnF[eu7%Fg`v7$Fffs$!+8d?bVF^bnF[eu7%F[av7$F`fs$!+l6DbVF^bnF [eu7%F_av7$Fjes$!+,X>aVF^bnF[eu7%Fcav7$Fdes$!+#[$3_VF^bnF[eu7%Fgav7$F^ es$!+*>k*[VF^bnF[eu7%F[bv7$Fhds$!+38)[M%F^bnF[eu7%F_bv7$Fbds$!+w#y)RVF ^bnF[eu7%Fcbv7$F\\ds$!+[w*RL%F^bnF[eu7%Fgbv7$Ffcs$!+28GFVF^bnF[eu7%F[c v7$F`cs$!+[2x>VF^bnF[eu7%F_cv7$Febs$!+ot]6VF^bnF[eu7%FccvFeduF[euFc^rF i^r-F$6%7$7$$!3/++++++!z%FTF(7$$\"3+++++++!*>FTF(-%'COLOURG6&Fe^nF)F)F )-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F[dv7$F($\"3/++++++!z%FTF`dvFcdv-F$6$7 ewF'7$$\"3'******4A()=l)!#F$\"3'*******fjzq:F07$$\"37+++#=*e2bF:$\"3'* *****Hf$fTJF07$$\"3O++++_g*>'F@$\"3B+++cXR7ZF07$$\"3*)*****RN\\\\S$FF$ \"3-+++;\"*>$G'F07$$\"3++++u(*pZ7FQ$\"3++++CN)R&yF07$$\"3-+++Huu$[$FQ$ \"3(******H6`YU*F07$$\"3))*****R]7.)yFQ$\"3%******pn&\\*4\"FT7$$\"3!** ****zADJZ\"F\\o$\"33+++0cVc7FT7$$\"3&******4fp.A#F\\o$\"3++++wa=89FT7$ $\"3*******p$eEOBF\\o$\"3)******f(=ep:FT7$$\"3++++!)R#4[\"FQ$\"3)***** *Rh%QD66N\"Fgp$ \"31+++6\"*fNBFT7$$!34+++EKk8DFgp$\"35+++N\"GG[#FT7$$!3-+++F=l8VFgp$\" 3)******fjToi#FT7$$!3(******>1;Y$pFgp$\"3)******fG$fmFFT7$$!3++++QNJ`5 F0$\"3;+++/KQ+HFT7$$!3#******fjs!=:F0$\"3)******zGMe-$FT7$$!3++++r@)33 #F0$\"3')*****Rg%QSJFT7$$!3&******p\"*)4@FF0$\"32+++`?=UKFT7$$!35+++%R M=T$F0$\"31+++K_pILFT7$$!3u*****\\'\\#)GTF0$\"3#)*****pL$e1MFT7$$!3?++ +cNwa[F0$\"3=+++Hg8rMFT7$$!3M+++LR9zbF0$\"36+++=uxDNFT7$$!3o******QIM' H'F0$\"3%******p([!=d$FT7$$!3j*****zZVP+(F0$\"3z******4UI5OFT7$$!36+++ @zT+xF0$\"3<+++5/:UOFT7$$!3o*****\\*pK'Q)F0$\"3))******R!Q!oOFT7$$!39+ ++vS'=1*F0$\"3A+++Si^)o$FT7$$!3@+++Q!4ws*F0$\"3!******4d>Sq$FT7$$!3#** ****>4-As$ FT7$$!3!******RCaJH\"FT$\"33+++hgy;PFT7$$!3%******Hmg^N\"FT$\"30+++Onk 2PFT7$$!3(******>8=mT\"FT$\"3=+++\\b'[p$FT7$$!3/+++:cfx9FT$\"3y*****H# Q]yOFT7$$!3,+++O<1&e;FT$\"3%*******3tL3OFT7$$!3'******>c1&=!yNFT7$$!3-+++G%)fyFT$ \"3>+++X#eLU$FT7$$!3\"******>nyS-#FT$\"3!)*****\\X]pP$FT7$$!31+++ol&z3 #FT$\"31+++(>BzK$FT7$$!3!******4&oH`@FT$\"3!)*****pFymF$FT7$$!3@+++fI@ ?AFT$\"30+++f6rBKFT7$$!37+++n0o)G#FT$\"39+++.ydpJFT7$$!3')*****fb3&eBF T$\"3,+++ce$[6$FT7$$!31+++JbMHCFT$\"3-+++%ot*fIFT7$$!3-+++I9s+DFT$\"3? +++'>]`+$FT7$$!3?+++!e7@d#FT$\"39+++I^;^HFT7$$!3++++Rq+VEFT$\"3!****** H$[Y(*GFT7$$!3++++HC&Hr#FT$\"3!)*****4ktT%GFT7$$!3%)*******3!e\"y#FT$ \"35+++!3N6z#FT7$$!3')*****pA4'[GFT$\"3-+++r,:QFFT7$$!33+++q'QQ\"HFT$ \"3!********f1]o#FT7$$!3=+++YK8xHFT$\"3/+++zx\\JEFT7$$!30+++B%4%QIFT$ \"3')*****pdLud#FT7$$!3!)*****\\fAw4$FT$\"3#)*****z=XE_#FT7$$!3#)***** p%ovaJFT$\"39+++dj)pY#FT7$$!36+++Ds\")4KFT$\"32+++.EL5CFT7$$!3?+++cT#G E$FT$\"3!)*****p\\zDN#FT7$$!3$)******)f4QJ$FT$\"3?+++'pSOH#FT7$$!31+++ wW\"GO$FT$\"3#******p3YMB#FT7$$!3.+++]q))4MFT$\"3++++#GS><#FT7$$!3')** ***z\">3bMFT$\"3\")*****za\"34@FT7$$!3=+++y$f%)\\$FT$\"31+++Q7%[/#FT7$ $!3!******f%\\3SNFT$\"3++++;P?z>FT7$$!3!******R!)G+e$FT$\"3)*******Rn; 7>FT7$$!3.+++(*\\O=OFT$\"30+++`AuV=FT7$$!3%******>>q^l$FT$\"31+++Au&Rx \"FT7$$!3))*****>%=_!p$FT$\"35+++#ybGq\"FT7$$!3=+++hc\\CPFT$\"3\"***** *R=)\\I;FT7$$!3!********Rirv$FT$\"3-+++qK'pb\"FT7$$!3'******40%e)y$FT$ \"3++++,rM#[\"FT7$$!3-+++p(4)=QFT$\"3,+++P;w19FT7$$!3y*****H'>(y%QFT$ \"33+++#)=LI8FT7$$!3')*****z4$yvQFT$\"3%******\\p\">`7FT7$$!3'******fv LD!RFT$\"3%******H`ya<\"FT7$$!3()******eB4GRFT$\"35+++YwK(4\"FT7$$!3=+ ++!o1C&RFT$\"3)******4Rm)=5FT7$$!3!******p*pSvRFT$\"3*)******pJ4-%*F07 $$!39+++P/,(*RFT$\"3=+++=ca9')F07$$!3L+++0f7#4%FT$\"3))*****R^!z+RF07$$!3#)*****R2Y:5%FT$\"3.+++x.4>JF07$ $!3s******HO()3TFT$\"3))*****R['QQBF07$$!30+++'4NT6%FT$\"3-+++l*p%e:F0 7$$!3\"******z$HI JF07$Fibx$!3))*****R^!z+RF07$Fdbx$!3C+++kWk$o%F07$F_bx$!3_*****pcRxY&F 07$Fjax$!33+++a52`iF07$Feax$!3Y*****4\"z^RqF07$F`ax$!3?+++&3=o#yF07$F[ ax$!3=+++=ca9')F07$Ff`x$!3*)******pJ4-%*F07$Fa`x$!3)******4Rm)=5FT7$F \\`x$!35+++YwK(4\"FT7$Fg_x$!3%******H`ya<\"FT7$Fb_x$!3%******\\p\">`7F T7$F]_x$!33+++#)=LI8FT7$Fh^x$!3,+++P;w19FT7$Fc^x$!3++++,rM#[\"FT7$F^^x $!3-+++qK'pb\"FT7$Fi]x$!3\"******R=)\\I;FT7$Fd]x$!35+++#ybGq\"FT7$F_]x $!31+++Au&Rx\"FT7$Fj\\x$!30+++`AuV=FT7$Fe\\x$!3)*******Rn;7>FT7$F`\\x$ !3++++;P?z>FT7$F[\\x$!31+++Q7%[/#FT7$Ff[x$!3\")*****za\"34@FT7$Fa[x$!3 ++++#GS><#FT7$F\\[x$!3#******p3YMB#FT7$Fgjw$!3?+++'pSOH#FT7$Fbjw$!3!)* ****p\\zDN#FT7$F]jw$!32+++.EL5CFT7$Fhiw$!39+++dj)pY#FT7$Fciw$!3#)***** z=XE_#FT7$F^iw$!3')*****pdLud#FT7$Fihw$!3/+++zx\\JEFT7$Fdhw$!3!******* *f1]o#FT7$F_hw$!3-+++r,:QFFT7$Fjgw$!35+++!3N6z#FT7$Fegw$!3!)*****4ktT% GFT7$F`gw$!3!******H$[Y(*GFT7$F[gw$!39+++I^;^HFT7$Fffw$!3?+++'>]`+$FT7 $Fafw$!3-+++%ot*fIFT7$F\\fw$!3,+++ce$[6$FT7$Fgew$!39+++.ydpJFT7$Fbew$! 30+++f6rBKFT7$F]ew$!3!)*****pFymF$FT7$Fhdw$!31+++(>BzK$FT7$Fcdw$!3!)** ***\\X]pP$FT7$F^dw$!3>+++X#eLU$FT7$Ficw$!3?+++)pXoY$FT7$Fdcw$!3$)***** HN-s]$FT7$F_cw$!3;+++8$*GWNFT7$Fjbw$!3#******p,>!yNFT7$Febw$!3%******* 3tL3OFT7$F`bw$!3()******G!4_j$FT7$F[bw$!3++++1qgeOFT7$Ffaw$!3y*****H#Q ]yOFT7$Faaw$!3=+++\\b'[p$FT7$F\\aw$!30+++Onk2PFT7$Fg`w$!33+++hgy;PFT7$ Fb`w$!3&)*****>4-As$FT7$F]`w$!3@+++M()yBPFT7$Fh_w$!31+++w)49s$FT7$Fc_w $!3&******Rx#*[r$FT7$F^_w$!3!******4d>Sq$FT7$Fi^w$!3A+++Si^)o$FT7$Fd^w $!3))******R!Q!oOFT7$F_^w$!3<+++5/:UOFT7$Fj]w$!3z******4UI5OFT7$Fe]w$! 3%******p([!=d$FT7$F`]w$!36+++=uxDNFT7$F[]w$!3=+++Hg8rMFT7$Ff\\w$!3#)* ****pL$e1MFT7$Fa\\w$!31+++K_pILFT7$F\\\\w$!32+++`?=UKFT7$Fg[w$!3')**** *Rg%QSJFT7$Fb[w$!3)******zGMe-$FT7$F][w$!3;+++/KQ+HFT7$Fhjv$!3)******f G$fmFFT7$Fcjv$!3)******fjToi#FT7$F^jv$!35+++N\"GG[#FT7$Fiiv$!31+++6\"* fNBFT7$Fdiv$!3-+++&Q&z&=#FT7$F_iv$!3;+++[?)Q.#FT7$Fjhv$!3!*******omF!) =FT7$Fehv$!3)******Rh%QD$G'F07$F]fv$!3B+++cXR7ZF07$F hev$!3'******Hf$fTJF07$$\"3_******HT'=l)Fdev$!3'*******fjzq:F0F'-Fc^n6 &Fe^n$\"#WFh^n$\"\"\"!\"\"Fcgy-F\\_n6fw7%F'7$$\"+@s)=l)Fgp$\"+gjzq:Fd_ nFe_n7%Figy7$$\"+#=*e2bFT$\"+$f$fTJFd_nFe_n7%F_hy7$$\"++_g*>'Fg`n$\"+c XR7ZFd_nFe_n7%Fehy7$$\"+a$\\\\S$F^an$\"+;\"*>$G'Fd_nFe_n7%F[iy7$$\"+u( *pZ7F[bn$\"+CN)R&yFd_nFe_n7%Faiy7$$\"+Huu$[$F[bn$\"+8JlC%*Fd_nFe_n7%Fg iy7$$\"+/DJ!)yF[bn$\"+xc\\*4\"F^bnFe_n7%F]jy7$$\"+G_7t9F_cn$\"+0cVc7F^ bnFe_n7%Fcjy7$$\"+\"fp.A#F_cn$\"+wa=89F^bnFe_n7%Fijy7$$\"+PeEOBF_cn$\" +w=ep:F^bnFe_n7%F_[z7$$\"+!)R#4[\"F[bn$\"+9YQD66N\"F_en$\"+6\"*f NBF^bnFe_n7%F]]z7$$!+EKk8DF_en$\"+N\"GG[#F^bnFe_n7%Fc]z7$$!+F=l8VF_en$ \"+O;%oi#F^bnFe_n7%Fi]z7$$!+ighMpF_en$\"+'G$fmFF^bnFe_n7%F_^z7$$!+QNJ` 5Fd_n$\"+/KQ+HF^bnFe_n7%Fe^z7$$!+OE2=:Fd_n$\"+)GMe-$F^bnFe_n7%F[_z7$$! +r@)33#Fd_n$\"+/YQSJF^bnFe_n7%Fa_z7$$!+<*)4@FFd_n$\"+`?=UKF^bnFe_n7%Fg _z7$$!+%RM=T$Fd_n$\"+K_pILF^bnFe_n7%F]`z7$$!+l\\#)GTFd_n$\"+PLe1MF^bnF e_n7%Fc`z7$$!+cNwa[Fd_n$\"+Hg8rMF^bnFe_n7%Fi`z7$$!+LR9zbFd_n$\"+=uxDNF ^bnFe_n7%F_az7$$!+RIM'H'Fd_n$\"+x[!=d$F^bnFe_n7%Feaz7$$!+yMu.qFd_n$\"+ 5UI5OF^bnFe_n7%F[bz7$$!+@zT+xFd_n$\"+5/:UOF^bnFe_n7%Fabz7$$!+&*pK'Q)Fd _n$\"+S!Q!oOF^bnFe_n7%Fgbz7$$!+vS'=1*Fd_n$\"+Si^)o$F^bnFe_n7%F]cz7$$!+ Q!4ws*Fd_n$\"+r&>Sq$F^bnFe_n7%Fccz7$$!+s,UQ5F^bn$\"+uF*[r$F^bnFe_n7%Fi cz7$$!+[#GK5\"F^bn$\"+w)49s$F^bnFe_n7%F_dz7$$!+DqCn6F^bn$\"+M()yBPF^bn Fe_n7%Fedz7$$!+mh`I7F^bn$\"+#4-As$F^bnFe_n7%F[ez7$$!+WU:$H\"F^bn$\"+hg y;PF^bnFe_n7%Faez7$$!+j1;b8F^bn$\"+Onk2PF^bnFe_n7%Fgez7$$!+K\"=mT\"F^b n$\"+\\b'[p$F^bnFe_n7%F]fz7$$!+:cfx9F^bn$\"+BQ]yOF^bnFe_n7%Fcfz7$$!+O< 1&e;F^bn$\"+4tL3OF^bnFe_n7%Fegz7$$!+il]=!yNF^bnFe _n7%F[hz7$$!+G%)fyF^bn $\"+X#eLU$F^bnFe_n7%Fciz7$$!+s'yS-#F^bn$\"+b/&pP$F^bnFe_n7%Fiiz7$$!+ol &z3#F^bn$\"+(>BzK$F^bnFe_n7%F_jz7$$!+^oH`@F^bn$\"+x#ymF$F^bnFe_n7%Fejz 7$$!+fI@?AF^bn$\"+f6rBKF^bnFe_n7%F[[[l7$$!+n0o)G#F^bn$\"+.ydpJF^bnFe_n 7%Fa[[l7$$!+c&3&eBF^bn$\"+ce$[6$F^bnFe_n7%Fg[[l7$$!+JbMHCF^bn$\"+%ot*f IF^bnFe_n7%F]\\[l7$$!+I9s+DF^bn$\"+'>]`+$F^bnFe_n7%Fc\\[l7$$!+!e7@d#F^ bn$\"+I^;^HF^bnFe_n7%Fi\\[l7$$!+Rq+VEF^bn$\"+L[Y(*GF^bnFe_n7%F_][l7$$! +HC&Hr#F^bn$\"+TO<#F^bnFe_n7%F]b[l7$$!+=>3bMF^bn$\"+[:34@F^bnFe_n7%Fcb[l7$$!+y $f%)\\$F^bn$\"+Q7%[/#F^bnFe_n7%Fib[l7$$!+Y\\3SNF^bn$\"+;P?z>F^bnFe_n7% F_c[l7$$!+/)G+e$F^bn$\"+Sn;7>F^bnFe_n7%Fec[l7$$!+(*\\O=OF^bn$\"+`AuV=F ^bnFe_n7%F[d[l7$$!+#>q^l$F^bn$\"+Au&Rx\"F^bnFe_n7%Fad[l7$$!+U=_!p$F^bn $\"+#ybGq\"F^bnFe_n7%Fgd[l7$$!+hc\\CPF^bn$\"+%=)\\I;F^bnFe_n7%F]e[l7$$ !++C;dPF^bn$\"+qK'pb\"F^bnFe_n7%Fce[l7$$!+^Se)y$F^bn$\"+,rM#[\"F^bnFe_ n7%Fie[l7$$!+p(4)=QF^bn$\"+P;w19F^bnFe_n7%F_f[l7$$!+j>(y%QF^bn$\"+#)=L I8F^bnFe_n7%Fef[l7$$!+)4$yvQF^bn$\"+&p\">`7F^bnFe_n7%F[g[l7$$!+cP`-RF^ bn$\"+L&ya<\"F^bnFe_n7%Fag[l7$$!+fB4GRF^bn$\"+YwK(4\"F^bnFe_n7%Fgg[l7$ $!+!o1C&RF^bn$\"+\"Rm)=5F^bnFe_n7%F]h[l7$$!+(*pSvRF^bn$\"+qJ4-%*Fd_nFe _n7%Fch[l7$$!+P/,(*RF^bn$\"+=ca9')Fd_nFe_n7%Fih[l7$$!+0f7#4%F^bn$\"+90z+RFd_nFe_n7%F ][\\l7$$!+uga,TF^bn$\"+x.4>JFd_nFe_n7%Fc[\\l7$$!+IO()3TF^bn$\"+%['QQBF d_nFe_n7%Fi[\\l7$$!+'4NT6%F^bn$\"+l*p%e:Fd_nFe_n7%F_\\\\l7$$!+QHIJFd_nFe_n7%F[^\\l7$F^[\\l$!+90z+RFd_nF e_n7%F_^\\l7$Fhj[l$!+kWk$o%Fd_nFe_n7%Fc^\\l7$Fbj[l$!+n&RxY&Fd_nFe_n7%F g^\\l7$F\\j[l$!+a52`iFd_nFe_n7%F[_\\l7$Ffi[l$!+6z^RqFd_nFe_n7%F__\\l7$ F`i[l$!+&3=o#yFd_nFe_n7%Fc_\\l7$Fjh[l$!+=ca9')Fd_nFe_n7%Fg_\\l7$Fdh[l$ !+qJ4-%*Fd_nFe_n7%F[`\\l7$F^h[l$!+\"Rm)=5F^bnFe_n7%F_`\\l7$Fhg[l$!+YwK (4\"F^bnFe_n7%Fc`\\l7$Fbg[l$!+L&ya<\"F^bnFe_n7%Fg`\\l7$F\\g[l$!+&p\">` 7F^bnFe_n7%F[a\\l7$Fff[l$!+#)=LI8F^bnFe_n7%F_a\\l7$F`f[l$!+P;w19F^bnFe _n7%Fca\\l7$Fje[l$!+,rM#[\"F^bnFe_n7%Fga\\l7$Fde[l$!+qK'pb\"F^bnFe_n7% F[b\\l7$F^e[l$!+%=)\\I;F^bnFe_n7%F_b\\l7$Fhd[l$!+#ybGq\"F^bnFe_n7%Fcb \\l7$Fbd[l$!+Au&Rx\"F^bnFe_n7%Fgb\\l7$F\\d[l$!+`AuV=F^bnFe_n7%F[c\\l7$ Ffc[l$!+Sn;7>F^bnFe_n7%F_c\\l7$F`c[l$!+;P?z>F^bnFe_n7%Fcc\\l7$Fjb[l$!+ Q7%[/#F^bnFe_n7%Fgc\\l7$Fdb[l$!+[:34@F^bnFe_n7%F[d\\l7$F^b[l$!+#GS><#F ^bnFe_n7%F_d\\l7$Fha[l$!+(3YMB#F^bnFe_n7%Fcd\\l7$Fba[l$!+'pSOH#F^bnFe_ n7%Fgd\\l7$F\\a[l$!+(\\zDN#F^bnFe_n7%F[e\\l7$Ff`[l$!+.EL5CF^bnFe_n7%F_ e\\l7$F``[l$!+dj)pY#F^bnFe_n7%Fce\\l7$Fj_[l$!+)=XE_#F^bnFe_n7%Fge\\l7$ Fd_[l$!+xNVxDF^bnFe_n7%F[f\\l7$F^_[l$!+zx\\JEF^bnFe_n7%F_f\\l7$Fh^[l$! ++m+&o#F^bnFe_n7%Fcf\\l7$Fb^[l$!+r,:QFF^bnFe_n7%Fgf\\l7$F\\^[l$!+!3N6z #F^bnFe_n7%F[g\\l7$Ff][l$!+TO]`+$F^bnFe_n7%F [h\\l7$F^\\[l$!+%ot*fIF^bnFe_n7%F_h\\l7$Fh[[l$!+ce$[6$F^bnFe_n7%Fch\\l 7$Fb[[l$!+.ydpJF^bnFe_n7%Fgh\\l7$F\\[[l$!+f6rBKF^bnFe_n7%F[i\\l7$Ffjz$ !+x#ymF$F^bnFe_n7%F_i\\l7$F`jz$!+(>BzK$F^bnFe_n7%Fci\\l7$Fjiz$!+b/&pP$ F^bnFe_n7%Fgi\\l7$Fdiz$!+X#eLU$F^bnFe_n7%F[j\\l7$F^iz$!+)pXoY$F^bnFe_n 7%F_j\\l7$Fhhz$!+`B?2NF^bnFe_n7%Fcj\\l7$Fbhz$!+8$*GWNF^bnFe_n7%Fgj\\l7 $F\\hz$!+!yNF^bnFe_n7%F[[]l7$Ffgz$!+4tL3OF^bnFe_n7%F_[]l7$F`gz$!+H! 4_j$F^bnFe_n7%Fc[]l7$Fjfz$!+1qgeOF^bnFe_n7%Fg[]l7$Fdfz$!+BQ]yOF^bnFe_n 7%F[\\]l7$F^fz$!+\\b'[p$F^bnFe_n7%F_\\]l7$Fhez$!+Onk2PF^bnFe_n7%Fc\\]l 7$Fbez$!+hgy;PF^bnFe_n7%Fg\\]l7$F\\ez$!+#4-As$F^bnFe_n7%F[]]l7$Ffdz$!+ M()yBPF^bnFe_n7%F_]]l7$F`dz$!+w)49s$F^bnFe_n7%Fc]]l7$Fjcz$!+uF*[r$F^bn Fe_n7%Fg]]l7$Fdcz$!+r&>Sq$F^bnFe_n7%F[^]l7$F^cz$!+Si^)o$F^bnFe_n7%F_^] l7$Fhbz$!+S!Q!oOF^bnFe_n7%Fc^]l7$Fbbz$!+5/:UOF^bnFe_n7%Fg^]l7$F\\bz$!+ 5UI5OF^bnFe_n7%F[_]l7$Ffaz$!+x[!=d$F^bnFe_n7%F__]l7$F`az$!+=uxDNF^bnFe _n7%Fc_]l7$Fj`z$!+Hg8rMF^bnFe_n7%Fg_]l7$Fd`z$!+PLe1MF^bnFe_n7%F[`]l7$F ^`z$!+K_pILF^bnFe_n7%F_`]l7$Fh_z$!+`?=UKF^bnFe_n7%Fc`]l7$Fb_z$!+/YQSJF ^bnFe_n7%Fg`]l7$F\\_z$!+)GMe-$F^bnFe_n7%F[a]l7$Ff^z$!+/KQ+HF^bnFe_n7%F _a]l7$F`^z$!+'G$fmFF^bnFe_n7%Fca]l7$Fj]z$!+O;%oi#F^bnFe_n7%Fga]l7$Fd]z $!+N\"GG[#F^bnFe_n7%F[b]l7$F^]z$!+6\"*fNBF^bnFe_n7%F_b]l7$Fh\\z$!+&Q&z &=#F^bnFe_n7%Fcb]l7$Fb\\z$!+[?)Q.#F^bnFe_n7%Fgb]l7$F\\\\z$!+pmF!)=F^bn Fe_n7%F[c]l7$Ff[z$!+9YQD$G'Fd_nFe_n7%F[e]l7$Ffhy$!+cXR7ZFd_nFe_n7%F_e] l7$F`hy$!+$f$fTJFd_nFe_n7%Fce]l7$$\"+IT'=l)Fgp$!+gjzq:Fd_nFe_n7%Fge]lF 'Fe_n-Fc^n6&Fe^n$\"#()Fh^n$\"\"#FegyFaf]lFi^r-F$6$7in7$$\"3++++a#pfY\" FT$\"35+++OHX#H%FT7$$\"31+++\\ubd9FT$\"3B+++zlB,VFT7$$\"3/+++toX[9FT$ \"3$******fv2$4VFT7$$\"3/+++64rQ9FT$\"3N+++&*[i;VFT7$$\"32+++ROOG9FT$ \"3%******ziYJK%FT7$$\"31+++\\,Y<9FT$\"3k******)\\J)GVFT7$$\"3#******R $p/19FT$\"3,+++$pPOL%FT7$$\"3)*******3B<%R\"FT$\"3s*****4xAvL%FT7$$\"3 -+++;p)=Q\"FT$\"3!)*****4UV/M%FT7$$\"3'******RIW#p8FT$\"35+++6_NUVFT7$ $\"3/+++=;Ic8FT$\"3'******4O7KM%FT7$$\"34+++C/7V8FT$\"3S+++Wv'HM%FT7$$ \"33+++(pn(H8FT$\"3))*****prr:M%FT7$$\"31+++GqJ;8FT$\"37+++>S(*QVFT7$$ \"30+++$3]GI\"FT$\"3o******o<7NVFT7$$\"3+++++%e%*G\"FT$\"3$******Rig*H VFT7$$\"3%******feXiF\"FT$\"3\")*****z)\\VBVFT7$$\"3)******\\,ILE\"FT$ \"3!******>\"))[:VFT7$$\"3/+++6\"[3D\"FT$\"3()*****>*p11VFT7$$\"32+++/ $e*Q7FT$\"3c******)o<^H%FT7$$\"32+++.d%yA\"FT$\"3;+++Ogf#G%FT7$$\"3.++ +ots<7FT$\"3%)*****\\\"*p%oUFT7$$\"3\"******z$y&)37FT$\"3')*****pvGFD% FT7$$\"30+++%*R`,7FT$\"3$******Hf'RNUFT7$$\"3#******zP(4'>\"FT$\"3u*** ***R2b;UFT7$$\"31+++P0$H>\"FT$\"3k*****>8Zj>%FT7$$\"3))*****R*=W#>\"FT $\"3!*******><0vTFT7$$\"3#******f*=.&>\"FT$\"3#******fRtI:%FT7$$\"3+++ +%>M5?\"FT$\"3#******f)p)48%FT7$$\"33+++5Nj57FT$\"3j******))o_4TFT7$$ \"3,+++Z*yPA\"FT$\"3o******>1`*3%FT7$$\"3'******zSI,C\"FT$\"3c*****ptF =2%FT7$$\"3#*******eh2f7FT$\"3%)*****H;.r0%FT7$$\"3'*******Qu#)z7FT$\" 35+++f)*yXSFT7$$\"3'******pn^:I\"FT$\"3E+++6U.QSFT7$$\"3/+++-$*[B8FT$ \"39+++s[tLSFT7$$\"3$******f&4-X8FT$\"3x*****H>@E.%FT7$$\"3-+++G[ol8FT $\"3<+++IPLMSFT7$$\"3!******R'>;&Q\"FT$\"3j*****Rz&[QSFT7$$\"36+++7$[K S\"FT$\"3#******p#=qWSFT7$$\"3!*******ob#)>9FT$\"3C+++\"4PE0%FT7$$\"3. +++1%FT7$$\"33+++T[E[9FT$\"3k*****z@yC2%FT7$$\"3# ******>;B,Y\"FT$\"3!)*****Hc%)Q3%FT7$$\"3++++;7Wq9FT$\"3K+++kR+'4%FT7$ $\"3%******45g#z9FT$\"3B+++zLm3TFT7$$\"31+++Dvi'[\"FT$\"3(******Hr7<7% FT7$$\"3)******\\w%f#\\\"FT$\"3?+++46-NTFT7$$\"3!*******zY@(\\\"FT$\"3 S+++mPZ[TFT7$$\"3)******>VS0]\"FT$\"39+++f#p>;%FT7$$\"3)******zpCE]\"F T$\"3E+++5tTvTFT7$$\"3.+++,#>N]\"FT$\"37+++Ipt)=%FT7$$\"3&*******oWF.: FT$\"3>+++s[&=?%FT7$$\"3++++R(R>]\"FT$\"3/+++_Vq9UFT7$$\"3%******>'Hc* \\\"FT$\"3++++sRAFUFT7$$\"3++++w3>'\\\"FT$\"3f*****>\"oNRUFT7$$\"35+++ E\"p=\\\"FT$\"3c******p'\\5D%FT7$$\"33+++9Ck'[\"FT$\"3m*****>S_AE%FT7$ $\"3-+++!oa0[\"FT$\"3-+++ft\"HF%FT7$$\"33+++\"H\\OZ\"FT$\"3i*****f*))* HG%FTFff]lF_gy-F\\_n6jn7%7$$\"+a#pfY\"F^bn$\"+OHX#H%F^bn7$$\"+\\ubd9F^ bn$\"+zlB,VF^bn7$$\"$N\"Fh^nF\\cs7%Fji^l7$$\"+toX[9F^bn$\"+cxI4VF^bnF_ j^l7%Fcj^l7$$\"+64rQ9F^bn$\"+&*[i;VF^bnF_j^l7%Fij^l7$$\"+ROOG9F^bn$\"+ Gm9BVF^bnF_j^l7%F_[_l7$$\"+\\,Y<9F^bn$\"+*\\J)GVF^bnF_j^l7%Fe[_l7$$\"+ Mp/19F^bn$\"+$pPOL%F^bnF_j^l7%F[\\_l7$$\"+4B<%R\"F^bn$\"+rF_PVF^bnF_j^ l7%Fa\\_l7$$\"+;p)=Q\"F^bn$\"+@MWSVF^bnF_j^l7%Fg\\_l7$$\"+/VCp8F^bn$\" +6_NUVF^bnF_j^l7%F]]_l7$$\"+=;Ic8F^bn$\"+hB@VVF^bnF_j^l7%Fc]_l7$$\"+C/ 7V8F^bn$\"+Wv'HM%F^bnF_j^l7%Fi]_l7$$\"+(pn(H8F^bn$\"+<S(*QVF^bnF_j^l7%Fe^_l7$$\"+$3]GI\"F^bn$\"+p<7N VF^bnF_j^l7%F[__l7$$\"++%e%*G\"F^bn$\"+C1'*HVF^bnF_j^l7%Fa__l7$$\"+'eX iF\"F^bn$\"+))\\VBVF^bnF_j^l7%Fg__l7$$\"+:+Lj7F^bn$\"+7))[:VF^bnF_j^l7 %F]`_l7$$\"+6\"[3D\"F^bn$\"+#*p11VF^bnF_j^l7%Fc`_l7$$\"+/$e*Q7F^bn$\"+ *o<^H%F^bnF_j^l7%Fi`_l7$$\"+.d%yA\"F^bn$\"+Ogf#G%F^bnF_j^l7%F_a_l7$$\" +ots<7F^bn$\"+:*p%oUF^bnF_j^l7%Fea_l7$$\"+Qy&)37F^bn$\"+d(GFD%F^bnF_j^ l7%F[b_l7$$\"+%*R`,7F^bn$\"+$f'RNUF^bnF_j^l7%Fab_l7$$\"+yt4'>\"F^bn$\" +S2b;UF^bnF_j^l7%Fgb_l7$$\"+P0$H>\"F^bn$\"+KrM'>%F^bnF_j^l7%F]c_l7$$\" +%*=W#>\"F^bn$\"+?<0vTF^bnF_j^l7%Fcc_l7$$\"+'*=.&>\"F^bn$\"+'RtI:%F^bn F_j^l7%Fic_l7$$\"+%>M5?\"F^bn$\"+')p)48%F^bnF_j^l7%F_d_l7$$\"+5Nj57F^b n$\"+*)o_4TF^bnF_j^l7%Fed_l7$$\"+Z*yPA\"F^bn$\"+?1`*3%F^bnF_j^l7%F[e_l 7$$\"+3/8S7F^bn$\"+Px#=2%F^bnF_j^l7%Fae_l7$$\"+fh2f7F^bn$\"+jJ5dSF^bnF _j^l7%Fge_l7$$\"+Ru#)z7F^bn$\"+f)*yXSF^bnF_j^l7%F]f_l7$$\"+x;b,8F^bn$ \"+6U.QSF^bnF_j^l7%Fcf_l7$$\"+-$*[B8F^bn$\"+s[tLSF^bnF_j^l7%Fif_l7$$\" +c4-X8F^bn$\"+$>@E.%F^bnF_j^l7%F_g_l7$$\"+G[ol8F^bn$\"+IPLMSF^bnF_j^l7 %Feg_l7$$\"+k>;&Q\"F^bn$\"+%z&[QSF^bnF_j^l7%F[h_l7$$\"+7$[KS\"F^bn$\"+ F=qWSF^bnF_j^l7%Fah_l7$$\"+pb#)>9F^bn$\"+\"4PE0%F^bnF_j^l7%Fgh_l7$$\"+ 1%F^bnF_j^l7%F]i_l7$$\"+T[E[9F^bn$\"+=#yC2%F^bnF_j^ l7%Fci_l7$$\"+iJ7g9F^bn$\"+jX)Q3%F^bnF_j^l7%Fii_l7$$\"+;7Wq9F^bn$\"+kR +'4%F^bnF_j^l7%F_j_l7$$\"+,,Ez9F^bn$\"+zLm3TF^bnF_j^l7%Fej_l7$$\"+Dvi' [\"F^bn$\"+8Fr@TF^bnF_j^l7%F[[`l7$$\"+lZf#\\\"F^bn$\"+46-NTF^bnF_j^l7% Fa[`l7$$\"+!o9s\\\"F^bn$\"+mPZ[TF^bnF_j^l7%Fg[`l7$$\"+K/a+:F^bn$\"+f#p >;%F^bnF_j^l7%F]\\`l7$$\"+)pCE]\"F^bn$\"+5tTvTF^bnF_j^l7%Fc\\`l7$$\"+, #>N]\"F^bn$\"+Ipt)=%F^bnF_j^l7%Fi\\`l7$$\"+pWF.:F^bn$\"+s[&=?%F^bnF_j^ l7%F_]`l7$$\"+R(R>]\"F^bn$\"+_Vq9UF^bnF_j^l7%Fe]`l7$$\"+iHc*\\\"F^bn$ \"+sRAFUF^bnF_j^l7%F[^`l7$$\"+w3>'\\\"F^bn$\"+7oNRUF^bnF_j^l7%Fa^`l7$$ \"+E\"p=\\\"F^bn$\"+q'\\5D%F^bnF_j^l7%Fg^`l7$$\"+9Ck'[\"F^bn$\"+-CDiUF ^bnF_j^l7%F]_`l7$$\"+!oa0[\"F^bn$\"+ft\"HF%F^bnF_j^l7%Fc_`l7$$\"+\"H\\ OZ\"F^bn$\"+'*))*HG%F^bnF_j^l7%Fi_`lFei^lF_j^lF]f]lFi^r-F$6$7in7$Fgf]l $!35+++OHX#H%FT7$F^i^l$!3i*****f*))*HG%FT7$Fih^l$!3-+++ft\"HF%FT7$Fdh^ l$!3m*****>S_AE%FT7$F_h^l$!3c******p'\\5D%FT7$Fjg^l$!3f*****>\"oNRUFT7 $Feg^l$!3++++sRAFUFT7$F`g^l$!3/+++_Vq9UFT7$F[g^l$!3>+++s[&=?%FT7$Fff^l $!37+++Ipt)=%FT7$Faf^l$!3E+++5tTvTFT7$F\\f^l$!39+++f#p>;%FT7$Fge^l$!3S +++mPZ[TFT7$Fbe^l$!3?+++46-NTFT7$F]e^l$!3(******Hr7<7%FT7$Fhd^l$!3B+++ zLm3TFT7$Fcd^l$!3K+++kR+'4%FT7$F^d^l$!3!)*****Hc%)Q3%FT7$Fic^l$!3k**** *z@yC2%FT7$Fdc^l$!37+++?^)>1%FT7$F_c^l$!3C+++\"4PE0%FT7$Fjb^l$!3#***** *p#=qWSFT7$Feb^l$!3j*****Rz&[QSFT7$F`b^l$!3<+++IPLMSFT7$F[b^l$!3x***** H>@E.%FT7$Ffa^l$!39+++s[tLSFT7$Faa^l$!3E+++6U.QSFT7$F\\a^l$!35+++f)*yX SFT7$Fg`^l$!3%)*****H;.r0%FT7$Fb`^l$!3c*****ptF=2%FT7$F]`^l$!3o******> 1`*3%FT7$Fh_^l$!3j******))o_4TFT7$Fc_^l$!3#******f)p)48%FT7$F^_^l$!3#* *****fRtI:%FT7$Fi^^l$!3!*******><0vTFT7$Fd^^l$!3k*****>8Zj>%FT7$F_^^l$ !3u******R2b;UFT7$Fj]^l$!3$******Hf'RNUFT7$Fe]^l$!3')*****pvGFD%FT7$F` ]^l$!3%)*****\\\"*p%oUFT7$F[]^l$!3;+++Ogf#G%FT7$Ff\\^l$!3c******)o<^H% FT7$Fa\\^l$!3()*****>*p11VFT7$F\\\\^l$!3!******>\"))[:VFT7$Fg[^l$!3\") *****z)\\VBVFT7$Fb[^l$!3$******Rig*HVFT7$F][^l$!3o******o<7NVFT7$Fhj]l $!37+++>S(*QVFT7$Fcj]l$!3))*****prr:M%FT7$F^j]l$!3S+++Wv'HM%FT7$Fii]l$ !3'******4O7KM%FT7$Fdi]l$!35+++6_NUVFT7$F_i]l$!3!)*****4UV/M%FT7$Fjh]l $!3s*****4xAvL%FT7$Feh]l$!3,+++$pPOL%FT7$F`h]l$!3k******)\\J)GVFT7$F[h ]l$!3%******ziYJK%FT7$Ffg]l$!3N+++&*[i;VFT7$Fag]l$!3$******fv2$4VFT7$F \\g]l$!3B+++zlB,VFTFb``lF_gy-F\\_n6jn7%7$Ffi^l$!+OHX#H%F^bn7$Fj_`l$!+' *))*HG%F^bn7$F`j^lF\\eu7%F\\\\al7$Fd_`l$!+ft\"HF%F^bnF_\\al7%Fa\\al7$F ^_`l$!+-CDiUF^bnF_\\al7%Fe\\al7$Fh^`l$!+q'\\5D%F^bnF_\\al7%Fi\\al7$Fb^ `l$!+7oNRUF^bnF_\\al7%F]]al7$F\\^`l$!+sRAFUF^bnF_\\al7%Fa]al7$Ff]`l$!+ _Vq9UF^bnF_\\al7%Fe]al7$F`]`l$!+s[&=?%F^bnF_\\al7%Fi]al7$Fj\\`l$!+Ipt) =%F^bnF_\\al7%F]^al7$Fd\\`l$!+5tTvTF^bnF_\\al7%Fa^al7$F^\\`l$!+f#p>;%F ^bnF_\\al7%Fe^al7$Fh[`l$!+mPZ[TF^bnF_\\al7%Fi^al7$Fb[`l$!+46-NTF^bnF_ \\al7%F]_al7$F\\[`l$!+8Fr@TF^bnF_\\al7%Fa_al7$Ffj_l$!+zLm3TF^bnF_\\al7 %Fe_al7$F`j_l$!+kR+'4%F^bnF_\\al7%Fi_al7$Fji_l$!+jX)Q3%F^bnF_\\al7%F]` al7$Fdi_l$!+=#yC2%F^bnF_\\al7%Fa`al7$F^i_l$!+?^)>1%F^bnF_\\al7%Fe`al7$ Fhh_l$!+\"4PE0%F^bnF_\\al7%Fi`al7$Fbh_l$!+F=qWSF^bnF_\\al7%F]aal7$F\\h _l$!+%z&[QSF^bnF_\\al7%Faaal7$Ffg_l$!+IPLMSF^bnF_\\al7%Feaal7$F`g_l$!+ $>@E.%F^bnF_\\al7%Fiaal7$Fjf_l$!+s[tLSF^bnF_\\al7%F]bal7$Fdf_l$!+6U.QS F^bnF_\\al7%Fabal7$F^f_l$!+f)*yXSF^bnF_\\al7%Febal7$Fhe_l$!+jJ5dSF^bnF _\\al7%Fibal7$Fbe_l$!+Px#=2%F^bnF_\\al7%F]cal7$F\\e_l$!+?1`*3%F^bnF_\\ al7%Facal7$Ffd_l$!+*)o_4TF^bnF_\\al7%Fecal7$F`d_l$!+')p)48%F^bnF_\\al7 %Fical7$Fjc_l$!+'RtI:%F^bnF_\\al7%F]dal7$Fdc_l$!+?<0vTF^bnF_\\al7%Fada l7$F^c_l$!+KrM'>%F^bnF_\\al7%Fedal7$Fhb_l$!+S2b;UF^bnF_\\al7%Fidal7$Fb b_l$!+$f'RNUF^bnF_\\al7%F]eal7$F\\b_l$!+d(GFD%F^bnF_\\al7%Faeal7$Ffa_l $!+:*p%oUF^bnF_\\al7%Feeal7$F`a_l$!+Ogf#G%F^bnF_\\al7%Fieal7$Fj`_l$!+* o<^H%F^bnF_\\al7%F]fal7$Fd`_l$!+#*p11VF^bnF_\\al7%Fafal7$F^`_l$!+7))[: VF^bnF_\\al7%Fefal7$Fh__l$!+))\\VBVF^bnF_\\al7%Fifal7$Fb__l$!+C1'*HVF^ bnF_\\al7%F]gal7$F\\__l$!+p<7NVF^bnF_\\al7%Fagal7$Ff^_l$!+>S(*QVF^bnF_ \\al7%Fegal7$F`^_l$!+<Fh^n;Fb\\bl$\"$z%Fh^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#-------------------------------------------------------- -" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 80 "The stability function R^ for the 7 stage, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b^`,Stabilit yFunction(4,7,'expanded'))):\n`R^` := unapply(%,z):\n'`R^`(z)'=`R^`(z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R^G6#%\"zG,2\"\"\"F)F'F)*&#F) \"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F )F)F)*&#\"5@(*G')*)fsdN5\"7+]1?]\">/o\"[7F)*$)F'\"\"&F)F)F)*&#\"?BlIi] #Qp=`Z!*RX=*\"B]i!RxEt_dfmG6<\"=^'F)*$)F'F1F)F)F)*&#\"=f7_fCnJ(>A$*HZL #\"A+]i&))eI$[)HcLvziC\"F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R^`(z) = -1;" "6#/-%#R^G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R ^`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+DBk:T!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "zo0 := newton(`R^`(z)=-1,z=-4):\npo1 := plot([`R^`(z ),-1],z=-4.49..0.49,color=[red,blue]):\npo2 := plot([[[zo0,-1]]$3],sty le=point,symbol=[circle,cross,diamond],color=black):\npo3 := plot([[zo 0,0],[zo0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([ po1,po2,po3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$ $!3A++++++!\\%!#<$!3S$yMZ'*)pP?F*7$$!3;+]7'3DdV%F*$!3Q0b\\qg!o%=F*7$$! 35++Ds,X\"Q%F*$!3CZ[IxqEr;F*7$$!3(*\\P*)feAMVF*$!3b#R%pQp>I:F*7$$!3%)* \\Pva,qG%F*$!3y4KB%y!=%F*$!3+2&3n_j(Q6F*7$$!35+]#o viQ2%F*$!39I1A7*[g=*!#=7$$!3A+ve,&\\u'RF*$!3mIK>=`FbtFK7$$!3?+v8#z!zoQ F*$!32Pu%[>\"eLfFK7$$!3;+v.8ajmPF*$!3dq-rMn#3q%FK7$$!3C+v$Q<')4m$F*$!3 k[CHgB>WOFK7$$!3G+v3`dnbNF*$!3E))y8TjTxFFK7$$!37++0T5NZMF*$!3e'41H*\\8 [?FK7$$!3?+]sv&Q>N$F*$!3%HIc!GhZ>:FK7$$!3E++:Qn_WKF*$!3t9nI;'p&H5FK7$$ !3#****\\s&QnOJF*$!3[K3Ug`7%H'!#>7$$!3&****\\=iPF.$F*$!3Vpc*o&HmOJFcp7 $$!3Q+v$Gc`$QHF*$!3ti*z_*GtOu!#?7$$!3+++l(y@h#GF*$\"3xlTh.T()*=^Fcp7$$! 3D++5(ocD_#F*$\"3AKZK!zAV^'Fcp7$$!3#**\\P(R,::CF*$\"3vf[_z4uTzFcp7$$!3 &)*\\7sqtGJ#F*$\"3'puJ'yk@`#*Fcp7$$!30+]Pz*eh?#F*$\"3o6a%f^V?1\"FK7$$! 3l*\\([Y:;3@F*$\"3U/<=e`:\">\"FK7$$!3$***\\7w!eC+#F*$\"3'p=JX&HtP8FK7$ $!3%**\\Py(=m#*=F*$\"3*yH=?N!\\,:FK7$$!3%)*\\iW'R3(z\"F*$\"3])=?R15il \"FK7$$!3-+]d*>dQp\"F*$\"3=[jW9QgQ=FK7$$!3)*****p]Q@(e\"F*$\"3GMc?!=8k /#FK7$$!3I+]FRT)G[\"F*$\"3;f9PGB^rAFK7$$!3')*\\P*y(R>Q\"F*$\"3#Rv'zFbY 7DFK7$$!3++]Kx#e)p7F*$\"3<;LfNw%*4GFK7$$!3=++5j![\"p6F*$\"3U;+6D#Hs5$F K7$$!3s***\\KT=;1\"F*$\"31D6\\!)ycfMFK7$$!3'***\\P%o0=k*FK$\"3'o6V)3;N 8QFK7$$!3](***\\tEbw&)FK$\"3*zqShZ\"yTUFK7$$!39-]P2EBuvFK$\"33*[+ZD`)) o%FK7$$!3>)*\\(GL>l_'FK$\"36=F\\E(*o1_FK7$$!3z++]-\")=-bFK$\"3mB7rFK7$$!3L)**\\U'=wSBFK$\"39.S2mb,8zFK7$$!3?$*\\Pt2H$H\"FK$\"3C3?\\T w%oy)FK7$$!33R****pit2LFcp$\"3SJY!46PYn*FK7$$\"3Z^+]FkzBxFcp$\"37]$3$= \"*H!3\"F*7$$\"35*****z2`!f\"F*7$$\"3l+]i_F06GFK$\"3d l5T()HfC8F*7$$\"3\\.]Pt1&z\"QFK$\"3- " 0 "" {MPLTEXT 1 0 1625 "`R^` := z \+ -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+\n 10355772598986289721/1248168041 915020065000*z^5+\n 918453990475318693825062306523/651181171128665 957527326773906250*z^6+\n 2334729932219731672459521259/12462797533 562984833058885625000*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=2 01/20 do\n zz := newton(`R^`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n \+ if (13/20<=tt and tt<=6/5) or (87/10<=tt and tt<=93/10) then\n \+ hh := 1/40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo1 := plot(pts,color=CO LOR(RGB,.42,.08,.08)):\npo2 := plots[polygonplot]([seq([pts[i-1],pts[i ],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(R GB,.83,.17,.17)):\npts := []: z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do \n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo3 := plot(pts,color=COLOR(R GB,.42,.08,.08)):\npo4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1. 34,4.19]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(R GB,.83,.17,.17)):\npts := []: z0 := 1.5-4.3*I:\nfor ct from 0 to 60 do \n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo5 := plot(pts,color=COLOR(R GB,.42,.08,.08)):\npo6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1. 34,-4.19]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.83,.17,.17)):\npo7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4. 79]]],color=black,linestyle=3):\nplots[display]([po||(1..7)],view=[-4. 79..1.99,-4.79..4.79],font=[HELVETICA,9],\n labels=[`Re(z )`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 586 586 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F(7$$\"3 E+++q*Hor)!#F$\"3/+++hjzq:!#=7$$\"3?+++S\\U]b!#D$\"36+++:OfTJF07$$\"3M +++\"yx1D'!#C$\"3<+++BZR7ZF07$$\"3&)*****>@9`V$!#B$\"3_*****4#)*>$G'F0 7$$\"31+++]@-g7!#A$\"33+++Xc)R&yF07$$\"3x*******f')G_$FF$\"3[*****f?eY U*F07$$\"3A+++)*)pZ)zFF$\"3/+++1n\\*4\"!#<7$$\"3-+++Vi_(\\\"!#@$\"3%** ****zQPkD\"FS7$$\"3-+++Ms_rAFW$\"3(*******=\")=89FS7$$\"3-+++JN3MCFW$ \"3++++v]ep:FS7$$\"3/+++R/*\\?$FF$\"3-+++RsQD&Gq7E!#?$\"31+++KU(Q.#FS7$$!3\"******>$o\"=A%F[ p$\"3!*******z@15@FS7$$!3')*****\\O2BX'F[p$\"3-+++&otd=#FS7$$!3O+++&\\ I7Y*F[p$\"3A+++sh&4E#FS7$$!3/+++_$\"3y*****f([bNBFS7$$!3;+++>lE a=F`q$\"31+++2)4&4CFS7$$!36+++4.W-DF`q$\"3!)*****R$4v#[#FS7$$!3B+++w?p 4LF`q$\"3)*******\\:>bDFS7$$!3q*****4B)H*H%F`q$\"3')******\\+sEEFS7$$! 37+++MJM%\\&F`q$\"3%)*****4\"4>(p#FS7$$!3d*****R(oV;pF`q$\"3#******>L; kw#FS7$$!3S+++nR*Re)F`q$\"3&******>&4;MGFS7$$!3)******Hu950\"F0$\"3))* ****>uU,!HFS7$$!3-+++is0q7F0$\"35+++lJ/kHFS7$$!3()*****Rkr^^\"F0$\"3\" ******z0Gb-$FS7$$!3')*****zY.]y\"F0$\"3!)*****fOxU3$FS7$$!33+++!pos2#F 0$\"3<+++Oy,SJFS7$$!3/+++d/(*)Q#F0$\"3=+++H/b#>$FS7$$!3&******pA8nr#F0 $\"3#)******pawTKFS7$$!33+++ZT.dIF0$\"3-+++$)Hk(G$FS7$$!34+++U&ymS$F0$ \"32+++o1CILFS7$$!35+++o+yiPF0$\"3#)*****H#\\npLFS7$$!3w*****z>VH7%F0$ \"3)******4W+hS$FS7$$!3;+++^%\\_[%F0$\"3>+++p@pRMFS7$$!3++++8At5_F0$\" 39+++Gh4*\\$FS7$$!3v*****p**=9$fF0$\"3/+++?ZE\\NFS7$$!3))******y=QVmF0 $\"3%******>t\"R\"f$FS7$$!3P+++Zu'\\M(F0$\"3*)*****>-aki$FS7$$!3M+++[( ed.)F0$\"3!******4cJ_l$FS7$$!3(******p9nfr)F0$\"3))*****>zT$yOFS7$$!3w *****>[&4'Q*F0$\"3++++$*HF'p$FS7$$!3\"******>*fn/5FS$\"3/+++b?T4PFS7$$ !3++++e,')p5FS$\"37+++mi1=PFS7$$!3/+++R[AM6FS$\"3@+++\")*zCs$FS7$$!3#* *****4\\Iy>\"FS$\"3?+++%GZGs$FS7$$!3)******\\$etg7FS$\"3%)******[;K>PF S7$$!3$******>T**HK\"FS$\"39+++1M-7PFS7$$!35+++&z\"o%Q\"FS$\"3$)*****4 lX5q$FS7$$!3%******\\I[eW\"FS$\"3#)*****49fko$FS7$$!34+++'=sl]\"FS$\"3 #)*****pi;$oOFS7$$!3!******4CQpc\"FS$\"3-+++$3dmk$FS7$$!33++++p/F;FS$ \"3$******HJ5:i$FS7$$!3/+++N'=qo\"FS$\"3')*****p\\-Hf$FS7$$!3!******4W )*pu\"FS$\"3)*******\\L'3c$FS7$$!32+++,)fr!=FS$\"3?+++[cVDNFS7$$!31+++ )*pqn=FS$\"35+++**yo'[$FS7$$!34+++%*Q()G>FS$\"3.+++)fIZW$FS7$$!3)***** **Qj\"4*>FS$\"31+++fct*R$FS7$$!35+++tZ4a?FS$\"3%)*****>qc>N$FS7$$!3@++ +'3W'=@FS$\"3*******p%\\u,LFS7$$!3!)*****f+JZ=#FS$\"35+++t@b\\KFS7$$!3 %******Rs0CD#FS$\"39+++x6\"f>$FS7$$!3A+++]Bc@BFS$\"30+++@))QTJFS7$$!3- +++Kh#>R#FS$\"3#******RM=l3$FS7$$!3%******\\1yIY#FS$\"3++++X/tJIFS7$$! 3=+++j7^MDFS$\"35+++4kIxHFS7$$!31+++P#*p0EFS$\"3A+++qlOBHFS7$$!31+++qU :wEFS$\"3/+++**3*)pGFS7$$!3!)******GNYXFFS$\"3.+++Tuv;GFS7$$!3&******> O-L\"GFS$\"3#******zV$yjFFS7$$!3#******z1FPg#FS7$ $!3-+++;\\\"G!QCFS7$$!3()*****\\v\"RMKFS$\"3#** ****\\Yc3Q#FS7$$!3++++`TN'G$FS$\"3A+++p/aABFS7$$!3&******z39jL$FS$\"3A +++A?+jAFS7$$!3;+++dkJ%Q$FS$\"34+++Z%y@?#FS7$$!3++++BHTIMFS$\"3'****** R?@+9#FS7$$!3%******f8hYZ$FS$\"3%******R1&\\w?FS7$$!3?+++-U7qyb$FS$\"3/+++x-EX>FS7$$!31+++<:(pf$FS$\"3++++.sa x=FS7$$!3$)*****H)Q]MOFS$\"3(******\\\"zX3=FS7$$!3'******\\\"[aqOFS$\" 32+++'eF!Q:FS7$$!3*)*****Rfm7!QF S$\"33+++I@?W9FS7$$!39+++9p)3$QFS$\"3!******fU'>o8FS7$$!3#)*****p_a$fQ FS$\"3'******Rs79H\"FS7$$!37+++w6n')QFS$\"31++++w)R@\"FS7$$!3$)*****\\ H;G\"RFS$\"3'******pOeg8\"FS7$$!3<+++EwuPRFS$\"33+++:svd5FS7$$!3%***** *Hg.9'RFS$\"3%)*****R=e?z*F07$$!37+++!=2P)RFS$\"3))*****\\35^+*F07$$!3 D+++f0d/SFS$\"3q*****HJ>w@)F07$$!3*)*****4A+R-%FS$\"3%)******QzGIuF07$ $!3c*****4n,;/%FS$\"3E+++$oIOk'F07$$!3#******H_$edSFS$\"3A+++!H))z&eF0 7$$!3E+++(og<2%FS$\"3Y*****fF[N2&F07$$!3!*******fm0%3%FS$\"3C+++Q#p.H% F07$$!3s*****\\)\\S%4%FS$\"37+++fcS3NF07$$!3\")*****z0]F5%FS$\"37+++>6 `FFF07$$!3J+++su/4TFS$\"3))*****fyev%>F07$$!3L+++oSE8TFS$\"3++++O)e#o6 F07$$!3M+++YyP:TFS$\"3')*******=IP*QF`q7$Fe^m$!3')*******=IP*QF`q7$F`^ m$!3++++O)e#o6F07$F[^m$!3))*****fyev%>F07$Ff]m$!37+++>6`FFF07$Fa]m$!37 +++fcS3NF07$F\\]m$!3C+++Q#p.H%F07$Fg\\m$!3Y*****fF[N2&F07$Fb\\m$!3A+++ !H))z&eF07$F]\\m$!3E+++$oIOk'F07$Fh[m$!3%)******QzGIuF07$Fc[m$!3q***** HJ>w@)F07$F^[m$!3))*****\\35^+*F07$Fijl$!3%)*****R=e?z*F07$Fdjl$!33+++ :svd5FS7$F_jl$!3'******pOeg8\"FS7$Fjil$!31++++w)R@\"FS7$Feil$!3'****** Rs79H\"FS7$F`il$!3!******fU'>o8FS7$F[il$!33+++I@?W9FS7$Ffhl$!30+++8'*H >:FS7$Fahl$!3))*****zzpLf\"FS7$F\\hl$!3'******fH3jm\"FS7$Fggl$!32+++'e F!Q FS7$Fcfl$!31+++&yx:,#FS7$F^fl$!3%******R1&\\w?FS7$Fiel$!3'******R?@+9# FS7$Fdel$!34+++Z%y@?#FS7$F_el$!3A+++A?+jAFS7$Fjdl$!3A+++p/aABFS7$Fedl$ !3#******\\Yc3Q#FS7$F`dl$!3!)******>\"G!QCFS7$F[dl$!3++++%p]T\\#FS7$Ff cl$!3)******faQ$\\DFS7$Facl$!3$******>1FPg#FS7$F\\cl$!3;+++ASZdEFS7$Fg bl$!3\")*****Hdf2r#FS7$Fbbl$!3#******zV$yjFFS7$F]bl$!3.+++Tuv;GFS7$Fha l$!3/+++**3*)pGFS7$Fcal$!3A+++qlOBHFS7$F^al$!35+++4kIxHFS7$Fi`l$!3++++ X/tJIFS7$Fd`l$!3#******RM=l3$FS7$F_`l$!30+++@))QTJFS7$Fj_l$!39+++x6\"f >$FS7$Fe_l$!35+++t@b\\KFS7$F`_l$!3*******p%\\u,LFS7$F[_l$!3%)*****>qc> N$FS7$Ff^l$!31+++fct*R$FS7$Fa^l$!3.+++)fIZW$FS7$F\\^l$!35+++**yo'[$FS7 $Fg]l$!3?+++[cVDNFS7$Fb]l$!3)*******\\L'3c$FS7$F]]l$!3')*****p\\-Hf$FS 7$Fh\\l$!3$******HJ5:i$FS7$Fc\\l$!3-+++$3dmk$FS7$F^\\l$!3#)*****pi;$oO FS7$Fi[l$!3#)*****49fko$FS7$Fd[l$!3$)*****4lX5q$FS7$F_[l$!39+++1M-7PFS 7$Fjz$!3%)******[;K>PFS7$Fez$!3?+++%GZGs$FS7$F`z$!3@+++\")*zCs$FS7$F[z $!37+++mi1=PFS7$Ffy$!3/+++b?T4PFS7$Fay$!3++++$*HF'p$FS7$F\\y$!3))***** >zT$yOFS7$Fgx$!3!******4cJ_l$FS7$Fbx$!3*)*****>-aki$FS7$F]x$!3%******> t\"R\"f$FS7$Fhw$!3/+++?ZE\\NFS7$Fcw$!39+++Gh4*\\$FS7$$!3!******p2;#[[F 0$!3?+++%RJ1Z$FS7$F^w$!3>+++p@pRMFS7$Fiv$!3)******4W+hS$FS7$Fdv$!3#)** ***H#\\npLFS7$F_v$!32+++o1CILFS7$Fju$!3-+++$)Hk(G$FS7$Feu$!3#)******pa wTKFS7$F`u$!3=+++H/b#>$FS7$F[u$!3<+++Oy,SJFS7$Fft$!3!)*****fOxU3$FS7$F at$!3\"******z0Gb-$FS7$F\\t$!35+++lJ/kHFS7$Fgs$!3))*****>uU,!HFS7$Fbs$ !3&******>&4;MGFS7$F]s$!3#******>L;kw#FS7$Fhr$!3%)*****4\"4>(p#FS7$Fcr $!3')******\\+sEEFS7$F^r$!3)*******\\:>bDFS7$Fiq$!3!)*****R$4v#[#FS7$F dq$!31+++2)4&4CFS7$F^q$!3y*****f([bNBFS7$Fip$!3A+++sh&4E#FS7$Fdp$!3-++ +&otd=#FS7$F_p$!3!*******z@15@FS7$$!3/+++CFS7 $$!3@+++s\"[-`#FW$!3#******\\VoH!=FS7$$\"37+++#e^/#=FW$!3)******4ryvk \"FS7$$\"3F+++G4j.DFW$!3))*****z)RW\"\\\"FS7$$\"32+++wB^,>FW$!3,+++P]% [L\"FS7$$\"3)******4'G()>6FW$!31+++hK)z<\"FS7$$\"3&******z3iJV&FF$!3,+ ++9$))4-\"FS7$$\"35+++\"3k(o@FF$!3_+++enMR')F07$$\"3s*****\\`1r%oF@$!3 k*****zc)foqF07$$\"3'******Hm2:c\"F@$!33+++&G(z(\\&F07$$\"3(******\\lS u5#F:$!3,+++tM*p#RF07$$\"3c*****ptK&4**!#E$!35+++NZ>cBF07$$\"3'******H hiMO\"!#G$!3S*****\\k\")R&yF`q7$$\"3-+++1EYj8Fden$\"3S*****\\k\")R&yF` q-%&COLORG6&%$RGBG$\"#U!\"#$\"\")FbfnFcfn-%)POLYGONSG6^y7%F'7$$\"+q*Ho r)F`q$\"+hjzq:!#57$$FbfnF)F(7%Fifn7$$\"+S\\U]bFS$\"+:OfTJF^gnF_gn7%Fbg n7$$\"+\"yx1D'!#;$\"+BZR7ZF^gnF_gn7%Fhgn7$$\"+7UJNM!#:$\"+@)*>$G'F^gnF _gn7%F_hn7$$\"+]@-g7!#9$\"+Xc)R&yF^gnF_gn7%Ffhn7$$\"++m)G_$Fihn$\"+1#e YU*F^gnF_gn7%F]in7$$\"+)*)pZ)zFihn$\"+1n\\*4\"!\"*F_gn7%Fcin7$$\"+Vi_( \\\"!#8$\"+)QPkD\"FhinF_gn7%Fjin7$$\"+Ms_rAF]jn$\"+>\")=89FhinF_gn7%Fa jn7$$\"+JN3MCF]jn$\"+v]ep:FhinF_gn7%Fgjn7$$\"+R/*\\?$Fihn$\"+RsQDlEa=Fe]o$\"+2)4&4CFhinF_ gn7%Fi]o7$$!+4.W-DFe]o$\"+M4v#[#FhinF_gn7%F_^o7$$!+w?p4LFe]o$\"+]:>bDF hinF_gn7%Fe^o7$$!+J#)H*H%Fe]o$\"+]+sEEFhinF_gn7%F[_o7$$!+MJM%\\&Fe]o$ \"+64>(p#FhinF_gn7%Fa_o7$$!+uoV;pFe]o$\"+KjTmFFhinF_gn7%Fg_o7$$!+nR*Re )Fe]o$\"+_4;MGFhinF_gn7%F]`o7$$!+VZ,^5F^gn$\"+UF9+HFhinF_gn7%Fc`o7$$!+ is0q7F^gn$\"+lJ/kHFhinF_gn7%Fi`o7$$!+W;<::F^gn$\"+e!Gb-$FhinF_gn7%F_ao 7$$!+oM+&y\"F^gn$\"+mtF%3$FhinF_gn7%Feao7$$!+!pos2#F^gn$\"+Oy,SJFhinF_ gn7%F[bo7$$!+d/(*)Q#F^gn$\"+H/b#>$FhinF_gn7%Fabo7$$!+FKr;FF^gn$\"+qawT KFhinF_gn7%Fgbo7$$!+ZT.dIF^gn$\"+$)Hk(G$FhinF_gn7%F]co7$$!+U&ymS$F^gn$ \"+o1CILFhinF_gn7%Fcco7$$!+o+yiPF^gn$\"+B\\npLFhinF_gn7%Fico7$$!+)>VH7 %F^gn$\"+T/51MFhinF_gn7%F_do7$$!+^%\\_[%F^gn$\"+p@pRMFhinF_gn7%Fedo7$$ !+8At5_F^gn$\"+Gh4*\\$FhinF_gn7%F[eo7$$!+(**=9$fF^gn$\"+?ZE\\NFhinF_gn 7%Faeo7$$!+z=QVmF^gn$\"+K\"Fhin$\"+%GZGs$Fhin F_gn7%Fgho7$$!+Netg7Fhin$\"+\\;K>PFhinF_gn7%F]io7$$!+7%**HK\"Fhin$\"+1 M-7PFhinF_gn7%Fcio7$$!+&z\"o%Q\"Fhin$\"+^c/,PFhinF_gn7%Fiio7$$!+0$[eW \"Fhin$\"+T\"fko$FhinF_gn7%F_jo7$$!+'=sl]\"Fhin$\"+FmJoOFhinF_gn7%Fejo 7$$!+T#Qpc\"Fhin$\"+$3dmk$FhinF_gn7%F[[p7$$!++p/F;Fhin$\"+8.^@OFhinF_g n7%Fa[p7$$!+N'=qo\"Fhin$\"+(\\-Hf$FhinF_gn7%Fg[p7$$!+T%)*pu\"Fhin$\"+] L'3c$FhinF_gn7%F]\\p7$$!+,)fr!=Fhin$\"+[cVDNFhinF_gn7%Fc\\p7$$!+)*pqn= Fhin$\"+**yo'[$FhinF_gn7%Fi\\p7$$!+%*Q()G>Fhin$\"+)fIZW$FhinF_gn7%F_]p 7$$!+Rj\"4*>Fhin$\"+fct*R$FhinF_gn7%Fe]p7$$!+tZ4a?Fhin$\"+-n&>N$FhinF_ gn7%F[^p7$$!+'3W'=@Fhin$\"+Z\\u,LFhinF_gn7%Fa^p7$$!+15t%=#Fhin$\"+t@b \\KFhinF_gn7%Fg^p7$$!+CdS_AFhin$\"+x6\"f>$FhinF_gn7%F]_p7$$!+]Bc@BFhin $\"+@))QTJFhinF_gn7%Fc_p7$$!+Kh#>R#Fhin$\"+W$=l3$FhinF_gn7%Fi_p7$$!+l! yIY#Fhin$\"+X/tJIFhinF_gn7%F_`p7$$!+j7^MDFhin$\"+4kIxHFhinF_gn7%Fe`p7$ $!+P#*p0EFhin$\"+qlOBHFhinF_gn7%F[ap7$$!+qU:wEFhin$\"+**3*)pGFhinF_gn7 %Faap7$$!+HNYXFFhin$\"+Tuv;GFhinF_gn7%Fgap7$$!+iBI8GFhin$\"+QMyjFFhinF _gn7%F]bp7$$!+y6VzGFhin$\"+t&f2r#FhinF_gn7%Fcbp7$$!+(o$oVHFhin$\"+ASZd EFhinF_gn7%Fibp7$$!+6C&f+$Fhin$\"+iqs.EFhinF_gn7%F_cp7$$!+;\\qyb$Fhin$\"+x-EX>FhinF_gn7%Fagp7$$!+<:(pf$Fh in$\"+.sax=FhinF_gn7%Fggp7$$!+$)Q]MOFhin$\"+:zX3=FhinF_gn7%F]hp7$$!+:[ aqOFhin$\"+'eF!Q:FhinF_g n7%Feip7$$!+%fm7!QFhin$\"+I@?W9FhinF_gn7%F[jp7$$!+9p)3$QFhin$\"+Ek>o8F hinF_gn7%Fajp7$$!+FXNfQFhin$\"+CFT\"H\"FhinF_gn7%Fgjp7$$!+w6n')QFhin$ \"++w)R@\"FhinF_gn7%F][q7$$!+&H;G\"RFhin$\"+n$eg8\"FhinF_gn7%Fc[q7$$!+ EwuPRFhin$\"+:svd5FhinF_gn7%Fi[q7$$!+.OShRFhin$\"+%=e?z*F^gnF_gn7%F_\\ q7$$!+!=2P)RFhin$\"+&35^+*F^gnF_gn7%Fe\\q7$$!+f0d/SFhin$\"+8$>w@)F^gnF _gn7%F[]q7$$!+@-!R-%Fhin$\"+RzGIuF^gnF_gn7%Fa]q7$$!+r;gTSFhin$\"+$oIOk 'F^gnF_gn7%Fg]q7$$!+BNedSFhin$\"+!H))z&eF^gnF_gn7%F]^q7$$!+(og<2%Fhin$ \"+w#[N2&F^gnF_gn7%Fc^q7$$!+gm0%3%Fhin$\"+Q#p.H%F^gnF_gn7%Fi^q7$$!+&) \\S%4%Fhin$\"+fcS3NF^gnF_gn7%F__q7$$!+e+v-TFhin$\"+>6`FFF^gnF_gn7%Fe_q 7$$!+su/4TFhin$\"+'yev%>F^gnF_gn7%F[`q7$$!+oSE8TFhin$\"+O)e#o6F^gnF_gn 7%Fa`q7$$!+YyP:TFhin$\"+!>IP*QFe]oF_gn7%Fg`q7$Fh`q$!+!>IP*QFe]oF_gn7%F ]aq7$Fb`q$!+O)e#o6F^gnF_gn7%Faaq7$F\\`q$!+'yev%>F^gnF_gn7%Feaq7$Ff_q$! +>6`FFF^gnF_gn7%Fiaq7$F`_q$!+fcS3NF^gnF_gn7%F]bq7$Fj^q$!+Q#p.H%F^gnF_g n7%Fabq7$Fd^q$!+w#[N2&F^gnF_gn7%Febq7$F^^q$!+!H))z&eF^gnF_gn7%Fibq7$Fh ]q$!+$oIOk'F^gnF_gn7%F]cq7$Fb]q$!+RzGIuF^gnF_gn7%Facq7$F\\]q$!+8$>w@)F ^gnF_gn7%Fecq7$Ff\\q$!+&35^+*F^gnF_gn7%Ficq7$F`\\q$!+%=e?z*F^gnF_gn7%F ]dq7$Fj[q$!+:svd5FhinF_gn7%Fadq7$Fd[q$!+n$eg8\"FhinF_gn7%Fedq7$F^[q$!+ +w)R@\"FhinF_gn7%Fidq7$Fhjp$!+CFT\"H\"FhinF_gn7%F]eq7$Fbjp$!+Ek>o8Fhin F_gn7%Faeq7$F\\jp$!+I@?W9FhinF_gn7%Feeq7$Ffip$!+8'*H>:FhinF_gn7%Fieq7$ F`ip$!+)zpLf\"FhinF_gn7%F]fq7$Fjhp$!+'H3jm\"FhinF_gn7%Fafq7$Fdhp$!+'eF !QFhinF_gn7%Fagq7$F\\gp$!+&yx:,#FhinF_gn7%Fegq7$Fffp$!+ k]\\w?FhinF_gn7%Figq7$F`fp$!+/7-S@FhinF_gn7%F]hq7$Fjep$!+Z%y@?#FhinF_g n7%Fahq7$Fdep$!+A?+jAFhinF_gn7%Fehq7$F^ep$!+p/aABFhinF_gn7%Fihq7$Fhdp$ !+lk&3Q#FhinF_gn7%F]iq7$Fbdp$!+?\"G!QCFhinF_gn7%Faiq7$F\\dp$!+%p]T\\#F hinF_gn7%Feiq7$Ffcp$!+Y&Q$\\DFhinF_gn7%Fiiq7$F`cp$!+iqs.EFhinF_gn7%F]j q7$Fjbp$!+ASZdEFhinF_gn7%Fajq7$Fdbp$!+t&f2r#FhinF_gn7%Fejq7$F^bp$!+QMy jFFhinF_gn7%Fijq7$Fhap$!+Tuv;GFhinF_gn7%F][r7$Fbap$!+**3*)pGFhinF_gn7% Fa[r7$F\\ap$!+qlOBHFhinF_gn7%Fe[r7$Ff`p$!+4kIxHFhinF_gn7%Fi[r7$F``p$!+ X/tJIFhinF_gn7%F]\\r7$Fj_p$!+W$=l3$FhinF_gn7%Fa\\r7$Fd_p$!+@))QTJFhinF _gn7%Fe\\r7$F^_p$!+x6\"f>$FhinF_gn7%Fi\\r7$Fh^p$!+t@b\\KFhinF_gn7%F]]r 7$Fb^p$!+Z\\u,LFhinF_gn7%Fa]r7$F\\^p$!+-n&>N$FhinF_gn7%Fe]r7$Ff]p$!+fc t*R$FhinF_gn7%Fi]r7$F`]p$!+)fIZW$FhinF_gn7%F]^r7$Fj\\p$!+**yo'[$FhinF_ gn7%Fa^r7$Fd\\p$!+[cVDNFhinF_gn7%Fe^r7$F^\\p$!+]L'3c$FhinF_gn7%Fi^r7$F h[p$!+(\\-Hf$FhinF_gn7%F]_r7$Fb[p$!+8.^@OFhinF_gn7%Fa_r7$F\\[p$!+$3dmk $FhinF_gn7%Fe_r7$Ffjo$!+FmJoOFhinF_gn7%Fi_r7$F`jo$!+T\"fko$FhinF_gn7%F ]`r7$Fjio$!+^c/,PFhinF_gn7%Fa`r7$Fdio$!+1M-7PFhinF_gn7%Fe`r7$F^io$!+\\ ;K>PFhinF_gn7%Fi`r7$Fhho$!+%GZGs$FhinF_gn7%F]ar7$Fbho$!+\")*zCs$FhinF_ gn7%Faar7$F\\ho$!+mi1=PFhinF_gn7%Fear7$Ffgo$!+b?T4PFhinF_gn7%Fiar7$F`g o$!+$*HF'p$FhinF_gn7%F]br7$Fjfo$!+#zT$yOFhinF_gn7%Fabr7$Fdfo$!+h:BbOFh inF_gn7%Febr7$F^fo$!+ASXEOFhinF_gn7%Fibr7$Fheo$!+K$FhinF_gn7%Fger7$F\\bo$!+Oy,SJFhinF_gn7%F[fr7$Ffao$!+mtF%3$FhinF_gn7 %F_fr7$F`ao$!+e!Gb-$FhinF_gn7%Fcfr7$Fj`o$!+lJ/kHFhinF_gn7%Fgfr7$Fd`o$! +UF9+HFhinF_gn7%F[gr7$F^`o$!+_4;MGFhinF_gn7%F_gr7$Fh_o$!+KjTmFFhinF_gn 7%Fcgr7$Fb_o$!+64>(p#FhinF_gn7%Fggr7$F\\_o$!+]+sEEFhinF_gn7%F[hr7$Ff^o $!+]:>bDFhinF_gn7%F_hr7$F`^o$!+M4v#[#FhinF_gn7%Fchr7$Fj]o$!+2)4&4CFhin F_gn7%Fghr7$Fc]o$!+w[bNBFhinF_gn7%F[ir7$F]]o$!+sh&4E#FhinF_gn7%F_ir7$F g\\o$!+&otd=#FhinF_gn7%Fcir7$Fa\\o$!+!=i+6#FhinF_gn7%Fgir7$$!+CFhinF_gn7%F[jr7$$!+s\"[-`#F]jn$!+N%oH!=FhinF_gn7%Fajr7$$ \"+#e^/#=F]jn$!+6(yvk\"FhinF_gn7%Fgjr7$$\"+G4j.DF]jn$!+))RW\"\\\"FhinF _gn7%F][s7$$\"+wB^,>F]jn$!+P]%[L\"FhinF_gn7%Fc[s7$$\"+hG()>6F]jn$!+hK) z<\"FhinF_gn7%Fi[s7$$\"+)3iJV&Fihn$!+9$))4-\"FhinF_gn7%F_\\s7$$\"+\"3k (o@Fihn$!+enMR')F^gnF_gn7%Fe\\s7$$\"+Nl5ZoFbhn$!+o&)foqF^gnF_gn7%F[]s7 $$\"+jw]h:Fbhn$!+&G(z(\\&F^gnF_gn7%Fa]s7$$\"+b1W2@F[hn$!+tM*p#RF^gnF_g n7%Fg]s7$$\"+PF`4**F0$!+NZ>cBF^gnF_gn7%F]^s7$$\"+8EYj8F[p$!+X;)R&yFe]o F_gn7%Fc^s7$$\"+1EYj8F[p$\"+X;)R&yFe]oF_gn-F]fn6&F_fn$\"#$)Fbfn$\"#Eg\"R9FS$\"3m******GVT;VFS7$$\"3\"******4\\ =)G9FS$\"3%)*****RHIHK%FS7$$\"3!******z%4#zT\"FS$\"34+++(p4'GVFS7$$\"3 1+++HT^19FS$\"3.+++W2TLVFS7$$\"3++++fjk%R\"FS$\"3&)*****\\-\"HPVFS7$$ \"3'*******)GoBQ\"FS$\"3K+++ds?SVFS7$$\"3-+++xMtp8FS$\"3k*****>/:@M%FS 7$$\"3/+++!3*zc8FS$\"35+++Y'oHM%FS7$$\"3$******HnEOM\"FS$\"3F+++)y?FM% FS7$$\"34+++WKGI8FS$\"3\")*****>[A8M%FS7$$\"3.+++%RUoJ\"FS$\"3v*****R$ HsQVFS7$$\"3,+++(z&Q.8FS$\"3&)*****>ap[L%FS7$$\"3'******f*\\+!H\"FS$\" 3e*****z03(HVFS7$$\"3/++++O!oF\"FS$\"3*)*****f*H=BVFS7$$\"3!*******o** *QE\"FS$\"3!*******R%Q_J%FS7$$\"3)******4]I9D\"FS$\"3-+++X%>eI%FS7$$\" 3%*******zNbR7FS$\"3m******fV([H%FS7$$\"3)******fAa%G7FS$\"3))*****>ae BG%FS7$$\"3%*******f$\\$=7FS$\"3y*****48S#oUFS7$$\"32+++0L\\47FS$\"39+ ++`)3DD%FS7$$\"32+++4E=-7FS$\"3l******e!*=NUFS7$$\"3#******RKen>\"FS$ \"3r*****pPej@%FS7$$\"3%*******4Bg$>\"FS$\"39+++#*H<'>%FS7$$\"3%****** *o?7$>\"FS$\"3U+++J*)*[<%FS7$$\"3#******p\"pr&>\"FS$\"3!*******Q[%H:%F S7$$\"3$********H>J@I\"FS$\"3&******H\"[/QSFS7$$\"30+++F_/C8FS $\"3y*****fEZP.%FS7$$\"3++++1_bX8FS$\"3G+++::jKSFS7$$\"31+++M(*>m8FS$ \"3*)*****f7RV.%FS7$$\"33+++Q*fcQ\"FS$\"3!******HF%[QSFS7$$\"3++++a;t. 9FS$\"3O+++k?pWSFS7$$\"3.+++wjH?9FS$\"3y*****pD=E0%FS7$$\"33+++`kHN9FS $\"3E+++Hn&>1%FS7$$\"3/+++[fr[9FS$\"3w*****f-SC2%FS7$$\"35+++xmcg9FS$ \"37+++([OQ3%FS7$$\"3!******pRy3Z\"FS$\"35+++Fg%f4%FS7$$\"31+++d?pz9FS $\"36+++vcf3TFS7$$\"3'******zBbq[\"FS$\"3$******4QN;7%FS7$$\"3/+++7\"> I\\\"FS$\"3#******pJM\\8%FS7$$\"3'******zWOw\\\"FS$\"3%)*****fpx$[TFS7 $$\"31+++Q.'4]\"FS$\"3M+++1T'=;%FS7$$\"31+++*RVI]\"FS$\"3w*****4F.`<%F S7$$\"33+++1t$R]\"FS$\"3O+++0Uh)=%FS7$$\"3\"******fa#p.:FS$\"36+++jOs, UFS7$$\"3-+++C$eB]\"FS$\"3e*****>'[c9UFS7$$\"35+++jD)**\\\"FS$\"3z**** *>SwqA%FS7$$\"3'*******z>h'\\\"FS$\"3s*****pO,#RUFS7$$\"3'******R?#H# \\\"FS$\"3?+++cl)3D%FS7$$\"33+++Dz1([\"FS$\"3G+++F=3iUFS7$$\"3/+++rI)4 [\"FS$\"3s*****z`RFF%FS7$$\"3++++.53u9FS$\"3u******[S\"GG%FSF[`sF\\fn- Fffn6jn7%7$$\"+CZSm9Fhin$\"+!HhAH%Fhin7$$\"+6r*zX\"Fhin$\"+t$Q5I%Fhin7 $$\"$M\"Fbfn$\"$>%Fbfn7%F_ct7$$\"+h6!*[9Fhin$\"+QK54VFhinFdct7%Fjct7$$ \"+i-;R9Fhin$\"+HVT;VFhinFdct7%F`dt7$$\"+\"\\=)G9Fhin$\"+%HIHK%FhinFdc t7%Ffdt7$$\"+[4#zT\"Fhin$\"+(p4'GVFhinFdct7%F\\et7$$\"+HT^19Fhin$\"+W2 TLVFhinFdct7%Fbet7$$\"+fjk%R\"Fhin$\"+D5HPVFhinFdct7%Fhet7$$\"+*GoBQ\" Fhin$\"+ds?SVFhinFdct7%F^ft7$$\"+xMtp8Fhin$\"+U]6UVFhinFdct7%Fdft7$$\" +!3*zc8Fhin$\"+Y'oHM%FhinFdct7%Fjft7$$\"+tmiV8Fhin$\"+)y?FM%FhinFdct7% F`gt7$$\"+WKGI8Fhin$\"+#[A8M%FhinFdct7%Ffgt7$$\"+%RUoJ\"Fhin$\"+MHsQVF hinFdct7%F\\ht7$$\"+(z&Q.8Fhin$\"+U&p[L%FhinFdct7%Fbht7$$\"+'*\\+!H\"F hin$\"+e!3(HVFhinFdct7%Fhht7$$\"++O!oF\"Fhin$\"+'*H=BVFhinFdct7%F^it7$ $\"+p***QE\"Fhin$\"+S%Q_J%FhinFdct7%Fdit7$$\"+,0V^7Fhin$\"+X%>eI%FhinF dct7%Fjit7$$\"+!e`&R7Fhin$\"+gV([H%FhinFdct7%F`jt7$$\"+EUXG7Fhin$\"+U& eBG%FhinFdct7%Ffjt7$$\"+g$\\$=7Fhin$\"+J,CoUFhinFdct7%F\\[u7$$\"+0L\\4 7Fhin$\"+`)3DD%FhinFdct7%Fb[u7$$\"+4E=-7Fhin$\"+f!*=NUFhinFdct7%Fh[u7$ $\"+C$en>\"Fhin$\"+x$ej@%FhinFdct7%F^\\u7$$\"+5Bg$>\"Fhin$\"+#*H<'>%Fh inFdct7%Fd\\u7$$\"+p?7$>\"Fhin$\"+J*)*[<%FhinFdct7%Fj\\u7$$\"+\"F hin$\"+R[%H:%FhinFdct7%F`]u7$$\"++$>J@I\"Fh in$\"+8[/QSFhinFdct7%Fj_u7$$\"+F_/C8Fhin$\"+msuLSFhinFdct7%F``u7$$\"+1 _bX8Fhin$\"+::jKSFhinFdct7%Ff`u7$$\"+M(*>m8Fhin$\"+E\"RV.%FhinFdct7%F \\au7$$\"+Q*fcQ\"Fhin$\"+tU[QSFhinFdct7%Fbau7$$\"+a;t.9Fhin$\"+k?pWSFh inFdct7%Fhau7$$\"+wjH?9Fhin$\"+d#=E0%FhinFdct7%F^bu7$$\"+`kHN9Fhin$\"+ Hn&>1%FhinFdct7%Fdbu7$$\"+[fr[9Fhin$\"+E+WsSFhinFdct7%Fjbu7$$\"+xmcg9F hin$\"+([OQ3%FhinFdct7%F`cu7$$\"+(Ry3Z\"Fhin$\"+Fg%f4%FhinFdct7%Ffcu7$ $\"+d?pz9Fhin$\"+vcf3TFhinFdct7%F\\du7$$\"+Q_0([\"Fhin$\"+\"QN;7%FhinF dct7%Fbdu7$$\"+7\">I\\\"Fhin$\"+h' \\\"Fhin$\"+n8?RUFhinFdct7%Fhgu7$$\"+/AH#\\\"Fhin$\"+cl)3D%FhinFdct7%F ^hu7$$\"+Dz1([\"Fhin$\"+F=3iUFhinFdct7%Fdhu7$$\"+rI)4[\"Fhin$\"+Q&RFF% FhinFdct7%Fjhu7$$\"+.53u9Fhin$\"+\\S\"GG%FhinFdct7%F`iuFjbtFdctF^_sFd_ s-F$6$7in7$F\\`s$!3!********GhAH%FS7$Fcbt$!3u******[S\"GG%FS7$F^bt$!3s *****z`RFF%FS7$Fiat$!3G+++F=3iUFS7$Fdat$!3?+++cl)3D%FS7$F_at$!3s*****p O,#RUFS7$Fj`t$!3z*****>SwqA%FS7$Fe`t$!3e*****>'[c9UFS7$F``t$!36+++jOs, UFS7$F[`t$!3O+++0Uh)=%FS7$Ff_t$!3w*****4F.`<%FS7$Fa_t$!3M+++1T'=;%FS7$ F\\_t$!3%)*****fpx$[TFS7$Fg^t$!3#******pJM\\8%FS7$Fb^t$!3$******4QN;7% FS7$F]^t$!36+++vcf3TFS7$Fh]t$!35+++Fg%f4%FS7$Fc]t$!37+++([OQ3%FS7$F^]t $!3w*****f-SC2%FS7$Fi\\t$!3E+++Hn&>1%FS7$Fd\\t$!3y*****pD=E0%FS7$F_\\t $!3O+++k?pWSFS7$Fj[t$!3!******HF%[QSFS7$Fe[t$!3*)*****f7RV.%FS7$F`[t$! 3G+++::jKSFS7$F[[t$!3y*****fEZP.%FS7$Ffjs$!3&******H\"[/QSFS7$Fajs$!3) )*****fz$zXSFS7$F\\js$!3-+++N[4dSFS7$Fgis$!31+++l9!=2%FS7$Fbis$!3O+++] :[*3%FS7$F]is$!3#******f#=X4TFS7$Fhhs$!3(******Rz%)38%FS7$Fchs$!3!**** ***Q[%H:%FS7$F^hs$!3U+++J*)*[<%FS7$Figs$!39+++#*H<'>%FS7$Fdgs$!3r***** pPej@%FS7$F_gs$!3l******e!*=NUFS7$Fjfs$!39+++`)3DD%FS7$Fefs$!3y*****48 S#oUFS7$F`fs$!3))*****>aeBG%FS7$F[fs$!3m******fV([H%FS7$Ffes$!3-+++X%> eI%FS7$Faes$!3!*******R%Q_J%FS7$F\\es$!3*)*****f*H=BVFS7$Fgds$!3e***** z03(HVFS7$Fbds$!3&)*****>ap[L%FS7$F]ds$!3v*****R$HsQVFS7$Fhcs$!3\")*** **>[A8M%FS7$Fccs$!3F+++)y?FM%FS7$F^cs$!35+++Y'oHM%FS7$Fibs$!3k*****>/: @M%FS7$Fdbs$!3K+++ds?SVFS7$F_bs$!3&)*****\\-\"HPVFS7$Fjas$!3.+++W2TLVF S7$Feas$!34+++(p4'GVFS7$F`as$!3%)*****RHIHK%FS7$F[as$!3m******GVT;VFS7 $Ff`s$!3h*****zB.\"4VFS7$Fa`s$!3\"******HPQ5I%FSFiiuF\\fn-Fffn6jn7%7$F [ct$!+!HhAH%Fhin7$Faiu$!+\\S\"GG%Fhin7$Fect$!$>%Fbfn7%Fcev7$F[iu$!+Q&R FF%FhinFfev7%Fjev7$Fehu$!+F=3iUFhinFfev7%F^fv7$F_hu$!+cl)3D%FhinFfev7% Fbfv7$Figu$!+n8?RUFhinFfev7%Fffv7$Fcgu$!+-k2FUFhinFfev7%Fjfv7$F]gu$!+i [c9UFhinFfev7%F^gv7$Fgfu$!+jOs,UFhinFfev7%Fbgv7$Fafu$!+0Uh)=%FhinFfev7 %Ffgv7$F[fu$!+rKIvTFhinFfev7%Fjgv7$Feeu$!+1T'=;%FhinFfev7%F^hv7$F_eu$! +'px$[TFhinFfev7%Fbhv7$Fidu$!+1% FhinFfev7%F^jv7$F_bu$!+d#=E0%FhinFfev7%Fbjv7$Fiau$!+k?pWSFhinFfev7%Ffj v7$Fcau$!+tU[QSFhinFfev7%Fjjv7$F]au$!+E\"RV.%FhinFfev7%F^[w7$Fg`u$!+:: jKSFhinFfev7%Fb[w7$Fa`u$!+msuLSFhinFfev7%Ff[w7$F[`u$!+8[/QSFhinFfev7%F j[w7$Fe_u$!+'z$zXSFhinFfev7%F^\\w7$F__u$!+N[4dSFhinFfev7%Fb\\w7$Fi^u$! +l9!=2%FhinFfev7%Ff\\w7$Fc^u$!+]:[*3%FhinFfev7%Fj\\w7$F]^u$!+E=X4TFhin Ffev7%F^]w7$Fg]u$!+%z%)38%FhinFfev7%Fb]w7$Fa]u$!+R[%H:%FhinFfev7%Ff]w7 $F[]u$!+J*)*[<%FhinFfev7%Fj]w7$Fe\\u$!+#*H<'>%FhinFfev7%F^^w7$F_\\u$!+ x$ej@%FhinFfev7%Fb^w7$Fi[u$!+f!*=NUFhinFfev7%Ff^w7$Fc[u$!+`)3DD%FhinFf ev7%Fj^w7$F][u$!+J,CoUFhinFfev7%F^_w7$Fgjt$!+U&eBG%FhinFfev7%Fb_w7$Faj t$!+gV([H%FhinFfev7%Ff_w7$F[jt$!+X%>eI%FhinFfev7%Fj_w7$Feit$!+S%Q_J%Fh inFfev7%F^`w7$F_it$!+'*H=BVFhinFfev7%Fb`w7$Fiht$!+e!3(HVFhinFfev7%Ff`w 7$Fcht$!+U&p[L%FhinFfev7%Fj`w7$F]ht$!+MHsQVFhinFfev7%F^aw7$Fggt$!+#[A8 M%FhinFfev7%Fbaw7$Fagt$!+)y?FM%FhinFfev7%Ffaw7$F[gt$!+Y'oHM%FhinFfev7% Fjaw7$Feft$!+U]6UVFhinFfev7%F^bw7$F_ft$!+ds?SVFhinFfev7%Fbbw7$Fiet$!+D 5HPVFhinFfev7%Ffbw7$Fcet$!+W2TLVFhinFfev7%Fjbw7$F]et$!+(p4'GVFhinFfev7 %F^cw7$Fgdt$!+%HIHK%FhinFfev7%Fbcw7$Fadt$!+HVT;VFhinFfev7%Ffcw7$F[dt$! +QK54VFhinFfev7%Fjcw7$F`ct$!+t$Q5I%FhinFfev7%F^dwF`evFfevF^_sFd_s-F$6% 7$7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF(-%'COLOURG6&F_fnF)F)F)-%*LINE STYLEG6#\"\"$-F$6%7$7$F(Ffdw7$F($\"3/++++++!z%FSF[ewF^ew-%*AXESSTYLEG6 #%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG 6#%,CONSTRAINEDG-Fcfw6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%Fbfn$\"$*>Fbfn ;Fbgw$\"$z%Fbfn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme is (approximately) " }{XPPEDIT 18 0 "[-4.1156, 0 ];" "6#7$,$-%&FloatG6$\"&c6%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The following \+ picture compares the stability region for the 7 stage, order 4 scheme \+ with that of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),po||(1..6)],view=[-4.79..1.99,-4.79 ..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes= boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 623 575 575 {PLOTDATA 2 "65-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3))******HiHB\"*!#G$\"3 #******fK'zq:!#=7$$\"3S+++KjhSl!#E$\"3z*****zf#fTJF07$$\"3e+++E^!)y&)! #D$\"3#******p$zQ7ZF07$$\"3Y+++M4Yua!#C$\"3g*****\\bwJG'F07$$\"3?+++qh biA!#B$\"3I+++QX$R&yF07$$\"3S+++t*>nv'FF$\"3D+++qxeC%*F07$$\"3'******* p&z:\\\"!#A$\"3!*******\\[\\*4\"!#<7$$\"3/+++\"*4A#>#FQ$\"3'******>(4Y c7FT7$$\"3/+++>S'yk%FF$\"3-+++='zKT\"FT7$$!3-+++'eLB.\"!#@$\"31+++=\"> )p:FT7$$!3B+++ET,EXF\\o$\"3\"******>uyes\"FT7$$!31+++D8]P8!#?$\"33+++F M=\")=FT7$$!3%)*****\\X\"4lKFgo$\"3')*****>A%RN?FT7$$!3l*****49&GBqFgo $\"33+++>=8)=#FT7$$!3)******HIeGP\"!#>$\"3@+++KO**QBFT7$$!3/+++&[=Q[#F gp$\"35++++++sP[QIFT7$$!36+++![oF/#F0$\"3#******4oYa:$FT7$ $!3-+++gd!Ho#F0$\"3;+++2cZfKFT7$$!3%******pdidP$F0$\"3;+++cv*)\\LFT7$$ !3!)******[Ni&4%F0$\"3&******>p\\tU$FT7$$!3y*****>muT#[F0$\"3=+++wQ<$ \\$FT7$$!3k*****4!*G/b&F0$\"3/+++md')[NFT7$$!3]*****4#fioiF0$\"3#***** **Qey&f$FT7$$!3!******\\7yh(pF0$\"3&)*****pvn]j$FT7$$!3;+++n&\\An(F0$ \"3=+++D3inOFT7$$!3w*******)G!pN)F0$\"3%*********Q;%p$FT7$$!3f*****z:B 1.*F0$\"3>+++*\\j_r$FT7$$!3#******pxkSp*F0$\"3:+++/pOJPFT7$$!33+++2CzM 5FT$\"3$******fVGGu$FT7$$!3#******f5)G*4\"FT$\"3=+++*eH*\\PFT7$$!3#*** ***>#z&H;\"FT$\"3#******\\A%*Gv$FT7$$!3.+++'fieA\"FT$\"3?+++s+!>v$FT7$ $!3%******R]g!)G\"FT$\"37+++tu3ZPFT7$$!3'******fh4'\\8FT$\"3*)******eh cQPFT7$$!3/+++^*p0T\"FT$\"3A+++i1UEPFT7$$!35+++lk+r9FT$\"3z*****>%[r5P FT7$$!3&******RM#*4`\"FT$\"3')******))f\\\"p$FT7$$!33+++;Fh!f\"FT$\"35 +++F*)zoOFT7$$!3/+++k(o*\\;FT$\"3,+++51lUOFT7$$!30+++F@=4FT$\"33++ +haThMFT7$$!3*)*****\\nR#4?FT$\"3*)*****z=!e+ AFT$\"3%)*****RC(pjKFT7$$!37+++*p4tE#FT$\"3\"******H>G'4KFT7$$!3,+++Mb TNBFT$\"3y*****Ri[Z:$FT7$$!3()*****z\"4g/CFT$\"3%)*****H*ec*4$FT7$$!33 +++QnVuCFT$\"3#)*****>AxW/$FT7$$!3=+++Z*>Wa#FT$\"3/+++W*H(*)HFT7$$!30+ ++(QRSh#FT$\"3\"******p[;a$HFT7$$!3A+++z!HGo#FT$\"37+++C%)\\\")GFT7$$! 3!******\\?*R]FFT$\"36+++-]%y#GFT7$$!3()*****>$eW;GFT$\"31+++WAFuFFT7$ $!3%******H4Y2)GFT$\"37++++Xd?FFT7$$!3-+++Mh9VHFT$\"3!******R7Ylm#FT7$ $!3'*******4ra.IFT$\"3-+++UV*>h#FT7$$!3/+++,A*=1$FT$\"3A+++QcucDFT7$$! 3@+++!)e:=JFT$\"3()*****\\%zk+DFT7$$!3A+++kcLsJFT$\"33+++1/dVCFT7$$!3! ******44YWA$FT$\"39+++P>S&Q#FT7$$!39+++MX^uKFT$\"3&)*****RM\\gK#FT7$$! 3!******pTyDK$FT$\"37+++yaVlAFT7$$!3++++uMooLFT$\"3&)*****Rn(\\.AFT7$$ !3y*****\\s#)GT$FT$\"3=+++:m=S@FT7$$!31+++2fBbMFT$\"3))******=dYv?FT7$ $!3/+++I\"4e\\$FT$\"3'******\\06$4?FT7$$!3z*****\\_uY`$FT$\"33+++&y6<% >FT7$$!37+++v'4>d$FT$\"3'*******46ns=FT7$$!37+++8mf2OFT$\"3.+++%e2A!=F T7$$!3*******p=?=k$FT$\"3!******pmc.t\"FT7$$!3%)******>cmuOFT$\"3)**** **RAsrl\"FT7$$!3;+++#>:iq$FT$\"35+++cws#e\"FT7$$!3()*****>5Wlt$FT$\"34 +++Di62:FT7$$!39+++idrlPFT$\"3********o,XI9FT7$$!3=+++wpx$z$FT$\"3#*** ***fEeGN\"FT7$$!3<+++[Ov?QFT$\"3++++L<[u7FT7$$!3%******RrZm%QFT$\"3/++ +H'oa>\"FT7$$!35+++)*eVrQFT$\"36+++#Hnf6\"FT7$$!3'******4hq]*QFT$\"30+ ++c&>h.\"FT7$$!39+++MH[L:eSFTF(7$Fjjl$!3p+++a\"*)R\"zFgp7$Fejl$!31+++H$GJe\"F0 7$F`jl$!37+++9k]vBF07$F[jl$!3C+++Xb#)oJF07$Ffil$!37+++6mLjRF07$Fail$!3 #)*****zEI#fZF07$F\\il$!3))*****fk9mb&F07$Fghl$!3]*****zi*[bjF07$Fbhl$ !3W+++#p=d:(F07$F]hl$!3I+++C/+dzF07$Fhgl$!3]*****p-V)e()F07$Fcgl$!39++ +Yiag&*F07$F^gl$!30+++c&>h.\"FT7$Fifl$!36+++#Hnf6\"FT7$Fdfl$!3/+++H'oa >\"FT7$F_fl$!3++++L<[u7FT7$Fjel$!3#******fEeGN\"FT7$Feel$!3********o,X I9FT7$F`el$!34+++Di62:FT7$F[el$!35+++cws#e\"FT7$Ffdl$!3)******RAsrl\"F T7$Fadl$!3!******pmc.t\"FT7$F\\dl$!3.+++%e2A!=FT7$Fgcl$!3'*******46ns= FT7$Fbcl$!33+++&y6<%>FT7$F]cl$!3'******\\06$4?FT7$Fhbl$!3))******=dYv? FT7$Fcbl$!3=+++:m=S@FT7$F^bl$!3&)*****Rn(\\.AFT7$Fial$!37+++yaVlAFT7$F dal$!3&)*****RM\\gK#FT7$F_al$!39+++P>S&Q#FT7$Fj`l$!33+++1/dVCFT7$Fe`l$ !3()*****\\%zk+DFT7$F``l$!3A+++QcucDFT7$F[`l$!3-+++UV*>h#FT7$Ff_l$!3!* *****R7Ylm#FT7$Fa_l$!37++++Xd?FFT7$F\\_l$!31+++WAFuFFT7$Fg^l$!36+++-]% y#GFT7$Fb^l$!37+++C%)\\\")GFT7$F]^l$!3\"******p[;a$HFT7$Fh]l$!3/+++W*H (*)HFT7$Fc]l$!3#)*****>AxW/$FT7$F^]l$!3%)*****H*ec*4$FT7$Fi\\l$!3y**** *Ri[Z:$FT7$Fd\\l$!3\"******H>G'4KFT7$F_\\l$!3%)*****RC(pjKFT7$Fj[l$!3= +++,GS;LFT7$Fe[l$!3'******R%R@nLFT7$F`[l$!3*)*****z%[r5PFT7$F^x$!3A+++i1UEPFT7$Fiw$!3*) ******ehcQPFT7$Fdw$!37+++tu3ZPFT7$F_w$!3?+++s+!>v$FT7$Fjv$!3#******\\A %*Gv$FT7$Fev$!3=+++*eH*\\PFT7$F`v$!3$******fVGGu$FT7$F[v$!3:+++/pOJPFT 7$Ffu$!3>+++*\\j_r$FT7$Fau$!3%*********Q;%p$FT7$F\\u$!3=+++D3inOFT7$Fg t$!3&)*****pvn]j$FT7$Fbt$!3#*******Qey&f$FT7$F]t$!3/+++md')[NFT7$Fhs$! 3=+++wQ<$\\$FT7$Fcs$!3&******>p\\tU$FT7$F^s$!3;+++cv*)\\LFT7$Fir$!3;++ +2cZfKFT7$Fdr$!3#******4oYa:$FT7$F_r$!3>+++sP[QIFT7$Fjq$!32+++'\\f1\"H FT7$Feq$!37+++!==8)=#FT7$F[p$!3')*****>A%RN?FT7$Feo$!33+++ FM=\")=FT7$F`o$!3\"******>uyes\"FT7$Fjn$!31+++=\">)p:FT7$Fen$!3-+++='z KT\"FT7$$\"3-+++#*4A#>#FQ$!3'******>(4Yc7FT7$FO$!3!*******\\[\\*4\"FT7 $FJ$!3D+++qxeC%*F07$FD$!3I+++QX$R&yF07$$\"3q*****H$4YuaF@$!3g*****\\bw JG'F07$F8$!3#******p$zQ7ZF07$$\"3()*****RL;1a'F4$!3z*****zf#fTJF07$$\" 3m*****H,'HB\"*F-$!3#******fK'zq:F0F'-%&COLORG6&%$RGBG$\"#[!\"#$\"#8Fh ^nFi^n-%)POLYGONSG6fw7%F'7$$\"+IiHB\"*Fgo$\"+Ejzq:!#57$$Fh^nF)F(7%F__n 7$$\"+KjhSlF0$\"+)f#fTJFd_nFe_n7%Fh_n7$$\"+E^!)y&)FT$\"+PzQ7ZFd_nFe_n7 %F^`n7$$\"+M4Yua!#;$\"+bl<$G'Fd_nFe_n7%Fd`n7$$\"+qhbiA!#:$\"+QX$R&yFd_ nFe_n7%F[an7$$\"+t*>nv'F^an$\"+qxeC%*Fd_nFe_n7%Fban7$$\"+q&z:\\\"!#9$ \"+][\\*4\"!\"*Fe_n7%Fhan7$$\"+\"*4A#>#F[bn$\"+s4Yc7F^bnFe_n7%F`bn7$$ \"+>S'yk%F^an$\"+='zKT\"F^bnFe_n7%Ffbn7$$!+'eLB.\"!#8$\"+=\">)p:F^bnFe _n7%F\\cn7$$!+ET,EXF_cn$\"+U(yes\"F^bnFe_n7%Fccn7$$!+D8]P8!#7$\"+FM=\" )=F^bnFe_n7%Ficn7$$!+b94lKF\\dn$\"+AURN?F^bnFe_n7%F`dn7$$!+T^GBqF\\dn$ \"+>=8)=#F^bnFe_n7%Ffdn7$$!+.$eGP\"!#6$\"+KO**QBF^bnFe_n7%F\\en7$$!+&[ =Q[#F_en$\"+v$F^bnFe_n7%F[]o7$$!+/01)G\"F^bn$\"+ tu3ZPF^bnFe_n7%Fa]o7$$!+;'4'\\8F^bn$\"+fhcQPF^bnFe_n7%Fg]o7$$!+^*p0T\" F^bn$\"+i1UEPF^bnFe_n7%F]^o7$$!+lk+r9F^bn$\"+U[r5PF^bnFe_n7%Fc^o7$$!+W B*4`\"F^bn$\"+*)f\\\"p$F^bnFe_n7%Fi^o7$$!+;Fh!f\"F^bn$\"+F*)zoOF^bnFe_ n7%F__o7$$!+k(o*\\;F^bn$\"+51lUOF^bnFe_n7%Fe_o7$$!+F@=4 F^bn$\"+haThMF^bnFe_n7%Fcao7$$!+v'R#4?F^bn$\"+y@n:MF^bnFe_n7%Fiao7$$!+ n!H;2#F^bn$\"+WR@nLF^bnFe_n7%F_bo7$$!+G4ON@F^bn$\"+,GS;LF^bnFe_n7%Febo 7$$!+#=!e+AF^bn$\"+WspjKF^bnFe_n7%F[co7$$!+*p4tE#F^bn$\"+$>G'4KF^bnFe_ n7%Faco7$$!+MbTNBF^bn$\"+C'[Z:$F^bnFe_n7%Fgco7$$!+=4g/CF^bn$\"+$*ec*4$ F^bnFe_n7%F]do7$$!+QnVuCF^bn$\"+AsZWIF^bnFe_n7%Fcdo7$$!+Z*>Wa#F^bn$\"+ W*H(*)HF^bnFe_n7%Fido7$$!+(QRSh#F^bn$\"+([;a$HF^bnFe_n7%F_eo7$$!+z!HGo #F^bn$\"+C%)\\\")GF^bnFe_n7%Feeo7$$!+0#*R]FF^bn$\"+-]%y#GF^bnFe_n7%F[f o7$$!+KeW;GF^bn$\"+WAFuFF^bnFe_n7%Fafo7$$!+$4Y2)GF^bn$\"++Xd?FF^bnFe_n 7%Fgfo7$$!+Mh9VHF^bn$\"+ChamEF^bnFe_n7%F]go7$$!+5ra.IF^bn$\"+UV*>h#F^b nFe_n7%Fcgo7$$!+,A*=1$F^bn$\"+QcucDF^bnFe_n7%Figo7$$!+!)e:=JF^bn$\"+Xz k+DF^bnFe_n7%F_ho7$$!+kcLsJF^bn$\"+1/dVCF^bnFe_n7%Feho7$$!+\"4YWA$F^bn $\"+P>S&Q#F^bnFe_n7%F[io7$$!+MX^uKF^bn$\"+W$\\gK#F^bnFe_n7%Faio7$$!+<% yDK$F^bn$\"+yaVlAF^bnFe_n7%Fgio7$$!+uMooLF^bn$\"+uw\\.AF^bnFe_n7%F]jo7 $$!+DF)GT$F^bn$\"+:m=S@F^bnFe_n7%Fcjo7$$!+2fBbMF^bn$\"+>dYv?F^bnFe_n7% Fijo7$$!+I\"4e\\$F^bn$\"+b5J4?F^bnFe_n7%F_[p7$$!+DXnMNF^bn$\"+&y6<%>F^ bnFe_n7%Fe[p7$$!+v'4>d$F^bn$\"+56ns=F^bnFe_n7%F[\\p7$$!+8mf2OF^bn$\"+% e2A!=F^bnFe_n7%Fa\\p7$$!+(=?=k$F^bn$\"+nmNI:iq$F^bn$\"+cws#e\"F^bnFe_n7%Fc]p7$$! +-TaOPF^bn$\"+Di62:F^bnFe_n7%Fi]p7$$!+idrlPF^bn$\"+p,XI9F^bnFe_n7%F_^p 7$$!+wpx$z$F^bn$\"+m#eGN\"F^bnFe_n7%Fe^p7$$!+[Ov?QF^bn$\"+L<[u7F^bnFe_ n7%F[_p7$$!+9xkYQF^bn$\"+H'oa>\"F^bnFe_n7%Fa_p7$$!+)*eVrQF^bn$\"+#Hnf6 \"F^bnFe_n7%Fg_p7$$!+612&*QF^bn$\"+c&>h.\"F^bnFe_n7%F]`p7$$!+MH[h.\"F^b nFe_n7%F_hp7$Fh_p$!+#Hnf6\"F^bnFe_n7%Fchp7$Fb_p$!+H'oa>\"F^bnFe_n7%Fgh p7$F\\_p$!+L<[u7F^bnFe_n7%F[ip7$Ff^p$!+m#eGN\"F^bnFe_n7%F_ip7$F`^p$!+p ,XI9F^bnFe_n7%Fcip7$Fj]p$!+Di62:F^bnFe_n7%Fgip7$Fd]p$!+cws#e\"F^bnFe_n 7%F[jp7$F^]p$!+CAF^ bnFe_n7%F_[q7$F`[p$!+b5J4?F^bnFe_n7%Fc[q7$Fjjo$!+>dYv?F^bnFe_n7%Fg[q7$ Fdjo$!+:m=S@F^bnFe_n7%F[\\q7$F^jo$!+uw\\.AF^bnFe_n7%F_\\q7$Fhio$!+yaVl AF^bnFe_n7%Fc\\q7$Fbio$!+W$\\gK#F^bnFe_n7%Fg\\q7$F\\io$!+P>S&Q#F^bnFe_ n7%F[]q7$Ffho$!+1/dVCF^bnFe_n7%F_]q7$F`ho$!+Xzk+DF^bnFe_n7%Fc]q7$Fjgo$ !+QcucDF^bnFe_n7%Fg]q7$Fdgo$!+UV*>h#F^bnFe_n7%F[^q7$F^go$!+ChamEF^bnFe _n7%F_^q7$Fhfo$!++Xd?FF^bnFe_n7%Fc^q7$Fbfo$!+WAFuFF^bnFe_n7%Fg^q7$F\\f o$!+-]%y#GF^bnFe_n7%F[_q7$Ffeo$!+C%)\\\")GF^bnFe_n7%F__q7$F`eo$!+([;a$ HF^bnFe_n7%Fc_q7$Fjdo$!+W*H(*)HF^bnFe_n7%Fg_q7$Fddo$!+AsZWIF^bnFe_n7%F [`q7$F^do$!+$*ec*4$F^bnFe_n7%F_`q7$Fhco$!+C'[Z:$F^bnFe_n7%Fc`q7$Fbco$! +$>G'4KF^bnFe_n7%Fg`q7$F\\co$!+WspjKF^bnFe_n7%F[aq7$Ffbo$!+,GS;LF^bnFe _n7%F_aq7$F`bo$!+WR@nLF^bnFe_n7%Fcaq7$Fjao$!+y@n:MF^bnFe_n7%Fgaq7$Fdao $!+haThMF^bnFe_n7%F[bq7$F^ao$!+-*yT]$F^bnFe_n7%F_bq7$Fh`o$!+85yVNF^bnF e_n7%Fcbq7$Fb`o$!+vU5!e$F^bnFe_n7%Fgbq7$F\\`o$!+nd28OF^bnFe_n7%F[cq7$F f_o$!+51lUOF^bnFe_n7%F_cq7$F`_o$!+F*)zoOF^bnFe_n7%Fccq7$Fj^o$!+*)f\\\" p$F^bnFe_n7%Fgcq7$Fd^o$!+U[r5PF^bnFe_n7%F[dq7$F^^o$!+i1UEPF^bnFe_n7%F_ dq7$Fh]o$!+fhcQPF^bnFe_n7%Fcdq7$Fb]o$!+tu3ZPF^bnFe_n7%Fgdq7$F\\]o$!+s+ !>v$F^bnFe_n7%F[eq7$Ff\\o$!+DU*Gv$F^bnFe_n7%F_eq7$F`\\o$!+*eH*\\PF^bnF e_n7%Fceq7$Fj[o$!+O%GGu$F^bnFe_n7%Fgeq7$Fd[o$!+/pOJPF^bnFe_n7%F[fq7$F^ [o$!+*\\j_r$F^bnFe_n7%F_fq7$Fhjn$!++R;%p$F^bnFe_n7%Fcfq7$Fbjn$!+D3inOF ^bnFe_n7%Fgfq7$F\\jn$!+dx1NOF^bnFe_n7%F[gq7$Ffin$!+Rey&f$F^bnFe_n7%F_g q7$F`in$!+md')[NF^bnFe_n7%Fcgq7$Fjhn$!+wQ<$\\$F^bnFe_n7%Fggq7$Fdhn$!+# p\\tU$F^bnFe_n7%F[hq7$F^hn$!+cv*)\\LF^bnFe_n7%F_hq7$Fhgn$!+2cZfKF^bnFe _n7%Fchq7$Fbgn$!+\"oYa:$F^bnFe_n7%Fghq7$F\\gn$!+sP[QIF^bnFe_n7%F[iq7$F ffn$!+'\\f1\"HF^bnFe_n7%F_iq7$F`fn$!+!==8)=#F^bnFe_n7%Fcjq7$Fadn$!+AURN?F^bnFe_n7%Fgjq7$Fjcn$!+F M=\")=F^bnFe_n7%F[[r7$Fdcn$!+U(yes\"F^bnFe_n7%F_[r7$F]cn$!+=\">)p:F^bn Fe_n7%Fc[r7$Fgbn$!+='zKT\"F^bnFe_n7%Fg[r7$$\"+#*4A#>#F[bn$!+s4Yc7F^bnF e_n7%F[\\r7$Fian$!+][\\*4\"F^bnFe_n7%Fa\\r7$Fcan$!+qxeC%*Fd_nFe_n7%Fe \\r7$F\\an$!+QX$R&yFd_nFe_n7%Fi\\r7$$\"+L4YuaFg`n$!+bl<$G'Fd_nFe_n7%F] ]r7$F_`n$!+PzQ7ZFd_nFe_n7%Fc]r7$$\"+MjhSlF0$!+)f#fTJFd_nFe_n7%Fg]r7$$ \"+8gHB\"*Fgo$!+Ejzq:Fd_nFe_n7%F]^rF'Fe_n-Fc^n6&Fe^n$\"#&*Fh^n$\"#DFh^ nFg^r-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3-+++ZVwa9FT$\"38+++]F`-VFT 7$$\"3-+++J)ojW\"FT$\"3,+++ot]6VFT7$$\"3.+++^MFP9FT$\"3$)*****zuq(>VFT 7$$\"31+++N9_F9FT$\"3!)*****pI\"GFVFT7$$\"3!******pgcrT\"FT$\"3U+++[w* RL%FT7$$\"35+++2PA19FT$\"3J+++w#y)RVFT7$$\"35+++m(oZR\"FT$\"3s*****zI \")[M%FT7$$\"3!******H_RGQ\"FT$\"33+++*>k*[VFT7$$\"3))*****p\"f[q8FT$ \"3G+++#[$3_VFT7$$\"3++++a1wd8FT$\"3G+++,X>aVFT7$$\"3)*******z)>ZM\"FT $\"3K+++l6DbVFT7$$\"3(******p)RUJ8FT$\"39+++8d?bVFT7$$\"3%******4eQzJ \"FT$\"3I+++:&3SN%FT7$$\"3&******>mNVI\"FT$\"3w*****\\\"zg^VFT7$$\"3)) *****R1&p!H\"FT$\"3S+++m,&zM%FT7$$\"3%******4A1rF\"FT$\"30+++)\\zHM%FT 7$$\"3%*******\\.nj7FT$\"3')******3%QmL%FT7$$\"3/+++TJ]]7FT$\"3)****** 4Io)GVFT7$$\"3#******z0QxB\"FT$\"3,+++x1h>VFT7$$\"3,+++t.`D7FT$\"3(*** ****p\"4)3VFT7$$\"3&*******))>197FT$\"3c*****\\*GT'H%FT7$$\"3,+++Oqa.7 FT$\"33+++u\"FT$\"3k******3[pmUFT7$$\"3))*****HR Jk=\"FT$\"3Q+++5EO\\UFT7$$\"3!*******e>Z!=\"FT$\"3U+++^dWIUFT7$$\"3-++ +$)4vw6FT$\"3T+++k.35UFT7$$\"35+++*H%pv6FT$\"3C+++?6^)=%FT7$$\"3!***** *>NJx<\"FT$\"3q*****>,Gh;%FT7$$\"31+++BvB$=\"FT$\"30+++&='\\VTFT7$$\"3 '********R[C>\"FT$\"3,+++U%e87%FT7$$\"33+++aCO07FT$\"3W+++`ce+TFT7$$\" 3*)******\\=n@7FT$\"3g*****4@k?3%FT7$$\"34+++!\\n2C\"FT$\"3g*****zVYl1 %FT7$$\"3-+++$*G$=E\"FT$\"3k*****fSCX0%FT7$$\"3++++F*))RG\"FT$\"3%**** **4V\"=YSFT7$$\"3)******p:EkI\"FT$\"3()******z#G9/%FT7$$\"30+++DK[G8FT $\"37+++5x)*RSFT7$$\"3#******RTo'\\8FT$\"3n*****pl$[TSFT7$$\"3#****** \\CW'p8FT$\"3)******zq4b/%FT7$$\"3+++++o>)Q\"FT$\"3Q+++L:n^SFT7$$\"3!* *****fo._S\"FT$\"3A+++E$3'fSFT7$$\"3*)********og?9FT$\"3#******Hj+!pSF T7$$\"32+++!3#RM9FT$\"3!******phq&zSFT7$$\"3%******zatlW\"FT$\"3C+++u' y54%FT7$$\"3%******\\!R=d9FT$\"3,+++6(=L5%FT7$$\"3-+++WpEm9FT$\"3K+++D K6;TFT7$$\"3%******p#H(QZ\"FT$\"3A+++l(3$HTFT7$$\"3%******>Yb+[\"FT$\" 3r*****R%>xUTFT7$$\"33+++%[p[[\"FT$\"3Q+++agQcTFT7$$\"3!******H)*p$)[ \"FT$\"3g*****pB[+<%FT7$$\"31+++A7h!\\\"FT$\"39+++Lrm$=%FT7$$\"3)***** *>HY;\\\"FT$\"3#)******R4;(>%FT7$$\"3-+++()o_\"\\\"FT$\"3x*****p!eX5UF T7$$\"3)******RB..\\\"FT$\"3O+++)\\%[BUFT7$$\"3/+++8T-)[\"FT$\"3%***** **z_=OUFT7$$\"3'******f%pt%[\"FT$\"3w*****f*4][UFT7$$\"3)******\\$z[![ \"FT$\"3G+++p#y.E%FT7$$\"3%*******=AKv9FT$\"3I+++OowrUFT7$$\"3++++kSGp 9FT$\"3')*****>(*=EG%FT7$$\"35+++iqTi9FT$\"3w******y*))GH%FTF`_rFb^n-F \\_n6jn7%7$$\"+ZVwa9F^bn$\"+]F`-VF^bn7$$\"+J)ojW\"F^bn$\"+ot]6VF^bn7$$ \"$L\"Fh^n$\"$=%Fh^n7%Fdbs7$$\"+^MFP9F^bn$\"+[2x>VF^bnFibs7%F_cs7$$\"+ N9_F9F^bn$\"+28GFVF^bnFibs7%Fecs7$$\"+2m:<9F^bn$\"+[w*RL%F^bnFibs7%F[d s7$$\"+2PA19F^bn$\"+w#y)RVF^bnFibs7%Fads7$$\"+m(oZR\"F^bn$\"+38)[M%F^b nFibs7%Fgds7$$\"+B&RGQ\"F^bn$\"+*>k*[VF^bnFibs7%F]es7$$\"+aVF^bnFibs7%Fies7$$\"+!))>Z M\"F^bn$\"+l6DbVF^bnFibs7%F_fs7$$\"+()RUJ8F^bn$\"+8d?bVF^bnFibs7%Fefs7 $$\"+\"eQzJ\"F^bn$\"+:&3SN%F^bnFibs7%F[gs7$$\"+icL/8F^bn$\"+:zg^VF^bnF ibs7%Fags7$$\"+k]p!H\"F^bn$\"+m,&zM%F^bnFibs7%Fggs7$$\"+@i5x7F^bn$\"+) \\zHM%F^bnFibs7%F]hs7$$\"+].nj7F^bn$\"+4%QmL%F^bnFibs7%Fchs7$$\"+TJ]]7 F^bn$\"+,$o)GVF^bnFibs7%Fihs7$$\"+e!QxB\"F^bn$\"+x1h>VF^bnFibs7%F_is7$ $\"+t.`D7F^bn$\"+q\"4)3VF^bnFibs7%Feis7$$\"+*)>197F^bn$\"+&*GT'H%F^bnF ibs7%F[js7$$\"+Oqa.7F^bn$\"+u\"F^bn$\"+4 [pmUF^bnFibs7%Fgjs7$$\"+$RJk=\"F^bn$\"+5EO\\UF^bnFibs7%F][t7$$\"+f>Z!= \"F^bn$\"+^dWIUF^bnFibs7%Fc[t7$$\"+$)4vw6F^bn$\"+k.35UF^bnFibs7%Fi[t7$ $\"+*H%pv6F^bn$\"+?6^)=%F^bnFibs7%F_\\t7$$\"+_8tx6F^bn$\"+7!Gh;%F^bnFi bs7%Fe\\t7$$\"+BvB$=\"F^bn$\"+&='\\VTF^bnFibs7%F[]t7$$\"++%[C>\"F^bn$ \"+U%e87%F^bnFibs7%Fa]t7$$\"+aCO07F^bn$\"+`ce+TF^bnFibs7%Fg]t7$$\"+]=n @7F^bn$\"+6U1#3%F^bnFibs7%F]^t7$$\"+!\\n2C\"F^bn$\"+QkamSF^bnFibs7%Fc^ t7$$\"+$*G$=E\"F^bn$\"+1W_aSF^bnFibs7%Fi^t7$$\"+F*))RG\"F^bn$\"+J9=YSF ^bnFibs7%F__t7$$\"+dhU18F^bn$\"+!GG9/%F^bnFibs7%Fe_t7$$\"+DK[G8F^bn$\" +5x)*RSF^bnFibs7%F[`t7$$\"+9%o'\\8F^bn$\"+dO[TSF^bnFibs7%Fa`t7$$\"+XUk p8F^bn$\"+3(4b/%F^bnFibs7%Fg`t7$$\"++o>)Q\"F^bn$\"+L:n^SF^bnFibs7%F]at 7$$\"+'o._S\"F^bn$\"+E$3'fSF^bnFibs7%Fcat7$$\"++pg?9F^bn$\"+L1+pSF^bnF ibs7%Fiat7$$\"+!3#RM9F^bn$\"+<1dzSF^bnFibs7%F_bt7$$\"+[NdY9F^bn$\"+u'y 54%F^bnFibs7%Febt7$$\"+0R=d9F^bn$\"+6(=L5%F^bnFibs7%F[ct7$$\"+WpEm9F^b n$\"+DK6;TF^bnFibs7%Fact7$$\"+FH(QZ\"F^bn$\"+l(3$HTF^bnFibs7%Fgct7$$\" +ia0![\"F^bn$\"+W>xUTF^bnFibs7%F]dt7$$\"+%[p[[\"F^bn$\"+agQcTF^bnFibs7 %Fcdt7$$\"+$)*p$)[\"F^bn$\"+P#[+<%F^bnFibs7%Fidt7$$\"+A7h!\\\"F^bn$\"+ Lrm$=%F^bnFibs7%F_et7$$\"+#HY;\\\"F^bn$\"+S4;(>%F^bnFibs7%Feet7$$\"+() o_\"\\\"F^bn$\"+2eX5UF^bnFibs7%F[ft7$$\"+MKI!\\\"F^bn$\"+)\\%[BUF^bnFi bs7%Faft7$$\"+8T-)[\"F^bn$\"+!G&=OUF^bnFibs7%Fgft7$$\"+Ypt%[\"F^bn$\"+ '*4][UF^bnFibs7%F]gt7$$\"+Nz[![\"F^bn$\"+p#y.E%F^bnFibs7%Fcgt7$$\"+>AK v9F^bn$\"+OowrUF^bnFibs7%Figt7$$\"+kSGp9F^bn$\"+s*=EG%F^bnFibs7%F_ht7$ $\"+iqTi9F^bn$\"+z*))GH%F^bnFibs7%FehtF_bsFibsFc^rFi^r-F$6$7in7$Fa_r$! 38+++]F`-VFT7$Fhas$!3w******y*))GH%FT7$Fcas$!3')*****>(*=EG%FT7$F^as$! 3I+++OowrUFT7$Fi`s$!3G+++p#y.E%FT7$Fd`s$!3w*****f*4][UFT7$F_`s$!3%**** ***z_=OUFT7$Fj_s$!3O+++)\\%[BUFT7$Fe_s$!3x*****p!eX5UFT7$F`_s$!3#)**** **R4;(>%FT7$F[_s$!39+++Lrm$=%FT7$Ff^s$!3g*****pB[+<%FT7$Fa^s$!3Q+++agQ cTFT7$F\\^s$!3r*****R%>xUTFT7$Fg]s$!3A+++l(3$HTFT7$Fb]s$!3K+++DK6;TFT7 $F]]s$!3,+++6(=L5%FT7$Fh\\s$!3C+++u'y54%FT7$Fc\\s$!3!******phq&zSFT7$F ^\\s$!3#******Hj+!pSFT7$Fi[s$!3A+++E$3'fSFT7$Fd[s$!3Q+++L:n^SFT7$F_[s$ !3)******zq4b/%FT7$Fjjr$!3n*****pl$[TSFT7$Fejr$!37+++5x)*RSFT7$F`jr$!3 ()******z#G9/%FT7$F[jr$!3%******4V\"=YSFT7$Ffir$!3k*****fSCX0%FT7$Fair $!3g*****zVYl1%FT7$F\\ir$!3g*****4@k?3%FT7$Fghr$!3W+++`ce+TFT7$Fbhr$!3 ,+++U%e87%FT7$F]hr$!30+++&='\\VTFT7$Fhgr$!3q*****>,Gh;%FT7$Fcgr$!3C+++ ?6^)=%FT7$F^gr$!3T+++k.35UFT7$Fifr$!3U+++^dWIUFT7$Fdfr$!3Q+++5EO\\UFT7 $F_fr$!3k******3[pmUFT7$Fjer$!33+++uVFT7$Ffdr$!3)******4Io)GVFT7$ Fadr$!3')******3%QmL%FT7$F\\dr$!30+++)\\zHM%FT7$Fgcr$!3S+++m,&zM%FT7$F bcr$!3w*****\\\"zg^VFT7$F]cr$!3I+++:&3SN%FT7$Fhbr$!39+++8d?bVFT7$Fcbr$ !3K+++l6DbVFT7$F^br$!3G+++,X>aVFT7$Fiar$!3G+++#[$3_VFT7$Fdar$!33+++*>k *[VFT7$F_ar$!3s*****zI\")[M%FT7$Fj`r$!3J+++w#y)RVFT7$Fe`r$!3U+++[w*RL% FT7$F``r$!3!)*****pI\"GFVFT7$F[`r$!3$)*****zuq(>VFT7$Ff_r$!3,+++ot]6VF TF^itFb^n-F\\_n6jn7%7$F`bs$!+]F`-VF^bn7$Ffht$!+z*))GH%F^bn7$Fjbs$!$=%F h^n7%Fhdu7$F`ht$!+s*=EG%F^bnF[eu7%F_eu7$Fjgt$!+OowrUF^bnF[eu7%Fceu7$Fd gt$!+p#y.E%F^bnF[eu7%Fgeu7$F^gt$!+'*4][UF^bnF[eu7%F[fu7$Fhft$!+!G&=OUF ^bnF[eu7%F_fu7$Fbft$!+)\\%[BUF^bnF[eu7%Fcfu7$F\\ft$!+2eX5UF^bnF[eu7%Fg fu7$Ffet$!+S4;(>%F^bnF[eu7%F[gu7$F`et$!+Lrm$=%F^bnF[eu7%F_gu7$Fjdt$!+P #[+<%F^bnF[eu7%Fcgu7$Fddt$!+agQcTF^bnF[eu7%Fggu7$F^dt$!+W>xUTF^bnF[eu7 %F[hu7$Fhct$!+l(3$HTF^bnF[eu7%F_hu7$Fbct$!+DK6;TF^bnF[eu7%Fchu7$F\\ct$ !+6(=L5%F^bnF[eu7%Fghu7$Ffbt$!+u'y54%F^bnF[eu7%F[iu7$F`bt$!+<1dzSF^bnF [eu7%F_iu7$Fjat$!+L1+pSF^bnF[eu7%Fciu7$Fdat$!+E$3'fSF^bnF[eu7%Fgiu7$F^ at$!+L:n^SF^bnF[eu7%F[ju7$Fh`t$!+3(4b/%F^bnF[eu7%F_ju7$Fb`t$!+dO[TSF^b nF[eu7%Fcju7$F\\`t$!+5x)*RSF^bnF[eu7%Fgju7$Ff_t$!+!GG9/%F^bnF[eu7%F[[v 7$F`_t$!+J9=YSF^bnF[eu7%F_[v7$Fj^t$!+1W_aSF^bnF[eu7%Fc[v7$Fd^t$!+QkamS F^bnF[eu7%Fg[v7$F^^t$!+6U1#3%F^bnF[eu7%F[\\v7$Fh]t$!+`ce+TF^bnF[eu7%F_ \\v7$Fb]t$!+U%e87%F^bnF[eu7%Fc\\v7$F\\]t$!+&='\\VTF^bnF[eu7%Fg\\v7$Ff \\t$!+7!Gh;%F^bnF[eu7%F[]v7$F`\\t$!+?6^)=%F^bnF[eu7%F_]v7$Fj[t$!+k.35U F^bnF[eu7%Fc]v7$Fd[t$!+^dWIUF^bnF[eu7%Fg]v7$F^[t$!+5EO\\UF^bnF[eu7%F[^ v7$Fhjs$!+4[pmUF^bnF[eu7%F_^v7$Fbjs$!+uVF^bnF[e u7%F__v7$Fjhs$!+,$o)GVF^bnF[eu7%Fc_v7$Fdhs$!+4%QmL%F^bnF[eu7%Fg_v7$F^h s$!+)\\zHM%F^bnF[eu7%F[`v7$Fhgs$!+m,&zM%F^bnF[eu7%F_`v7$Fbgs$!+:zg^VF^ bnF[eu7%Fc`v7$F\\gs$!+:&3SN%F^bnF[eu7%Fg`v7$Fffs$!+8d?bVF^bnF[eu7%F[av 7$F`fs$!+l6DbVF^bnF[eu7%F_av7$Fjes$!+,X>aVF^bnF[eu7%Fcav7$Fdes$!+#[$3_ VF^bnF[eu7%Fgav7$F^es$!+*>k*[VF^bnF[eu7%F[bv7$Fhds$!+38)[M%F^bnF[eu7%F _bv7$Fbds$!+w#y)RVF^bnF[eu7%Fcbv7$F\\ds$!+[w*RL%F^bnF[eu7%Fgbv7$Ffcs$! +28GFVF^bnF[eu7%F[cv7$F`cs$!+[2x>VF^bnF[eu7%F_cv7$Febs$!+ot]6VF^bnF[eu 7%FccvFeduF[euFc^rFi^r-F$6%7$7$$!3/++++++!z%FTF(7$$\"3+++++++!*>FTF(-% 'COLOURG6&Fe^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F[dv7$F($\"3/++++++ !z%FTF`dvFcdv-F$6$7]yF'7$$\"3E+++q*Hor)!#F$\"3/+++hjzq:F07$$\"3?+++S\\ U]bF:$\"36+++:OfTJF07$$\"3M+++\"yx1D'F@$\"3<+++BZR7ZF07$$\"3&)*****>@9 `V$FF$\"3_*****4#)*>$G'F07$$\"31+++]@-g7FQ$\"33+++Xc)R&yF07$$\"3x***** **f')G_$FQ$\"3[*****f?eYU*F07$$\"3A+++)*)pZ)zFQ$\"3/+++1n\\*4\"FT7$$\" 3-+++Vi_(\\\"F\\o$\"3%******zQPkD\"FT7$$\"3-+++Ms_rAF\\o$\"3(*******= \")=89FT7$$\"3-+++JN3MCF\\o$\"3++++v]ep:FT7$$\"3/+++R/*\\?$FQ$\"3-+++R sQD&Gq7EFgo$\"31+++K U(Q.#FT7$$!3\"******>$o\"=A%Fgo$\"3!*******z@15@FT7$$!3')*****\\O2BX'F go$\"3-+++&otd=#FT7$$!3O+++&\\I7Y*Fgo$\"3A+++sh&4E#FT7$$!3/+++_lEa=Fgp$\"31+++2)4&4CFT7$$!36+++4.W-DFgp$ \"3!)*****R$4v#[#FT7$$!3B+++w?p4LFgp$\"3)*******\\:>bDFT7$$!3q*****4B) H*H%Fgp$\"3')******\\+sEEFT7$$!37+++MJM%\\&Fgp$\"3%)*****4\"4>(p#FT7$$ !3d*****R(oV;pFgp$\"3#******>L;kw#FT7$$!3S+++nR*Re)Fgp$\"3&******>&4;M GFT7$$!3)******Hu950\"F0$\"3))*****>uU,!HFT7$$!3-+++is0q7F0$\"35+++lJ/ kHFT7$$!3()*****Rkr^^\"F0$\"3\"******z0Gb-$FT7$$!3')*****zY.]y\"F0$\"3 !)*****fOxU3$FT7$$!33+++!pos2#F0$\"3<+++Oy,SJFT7$$!3/+++d/(*)Q#F0$\"3= +++H/b#>$FT7$$!3&******pA8nr#F0$\"3#)******pawTKFT7$$!33+++ZT.dIF0$\"3 -+++$)Hk(G$FT7$$!34+++U&ymS$F0$\"32+++o1CILFT7$$!35+++o+yiPF0$\"3#)*** **H#\\npLFT7$$!3w*****z>VH7%F0$\"3)******4W+hS$FT7$$!3;+++^%\\_[%F0$\" 3>+++p@pRMFT7$$!3++++8At5_F0$\"39+++Gh4*\\$FT7$$!3v*****p**=9$fF0$\"3/ +++?ZE\\NFT7$$!3))******y=QVmF0$\"3%******>t\"R\"f$FT7$$!3P+++Zu'\\M(F 0$\"3*)*****>-aki$FT7$$!3M+++[(ed.)F0$\"3!******4cJ_l$FT7$$!3(******p9 nfr)F0$\"3))*****>zT$yOFT7$$!3w*****>[&4'Q*F0$\"3++++$*HF'p$FT7$$!3\"* *****>*fn/5FT$\"3/+++b?T4PFT7$$!3++++e,')p5FT$\"37+++mi1=PFT7$$!3/+++R [AM6FT$\"3@+++\")*zCs$FT7$$!3#******4\\Iy>\"FT$\"3?+++%GZGs$FT7$$!3)** ****\\$etg7FT$\"3%)******[;K>PFT7$$!3$******>T**HK\"FT$\"39+++1M-7PFT7 $$!35+++&z\"o%Q\"FT$\"3$)*****4lX5q$FT7$$!3%******\\I[eW\"FT$\"3#)**** *49fko$FT7$$!34+++'=sl]\"FT$\"3#)*****pi;$oOFT7$$!3!******4CQpc\"FT$\" 3-+++$3dmk$FT7$$!33++++p/F;FT$\"3$******HJ5:i$FT7$$!3/+++N'=qo\"FT$\"3 ')*****p\\-Hf$FT7$$!3!******4W)*pu\"FT$\"3)*******\\L'3c$FT7$$!32+++,) fr!=FT$\"3?+++[cVDNFT7$$!31+++)*pqn=FT$\"35+++**yo'[$FT7$$!34+++%*Q()G >FT$\"3.+++)fIZW$FT7$$!3)*******Qj\"4*>FT$\"31+++fct*R$FT7$$!35+++tZ4a ?FT$\"3%)*****>qc>N$FT7$$!3@+++'3W'=@FT$\"3*******p%\\u,LFT7$$!3!)**** *f+JZ=#FT$\"35+++t@b\\KFT7$$!3%******Rs0CD#FT$\"39+++x6\"f>$FT7$$!3A++ +]Bc@BFT$\"30+++@))QTJFT7$$!3-+++Kh#>R#FT$\"3#******RM=l3$FT7$$!3%**** **\\1yIY#FT$\"3++++X/tJIFT7$$!3=+++j7^MDFT$\"35+++4kIxHFT7$$!31+++P#*p 0EFT$\"3A+++qlOBHFT7$$!31+++qU:wEFT$\"3/+++**3*)pGFT7$$!3!)******GNYXF FT$\"3.+++Tuv;GFT7$$!3&******>O-L\"GFT$\"3#******zV$yjFFT7$$!3#******z 1FPg#FT7$$!3-+++;\\\"G!QCF T7$$!3()*****\\v\"RMKFT$\"3#******\\Yc3Q#FT7$$!3++++`TN'G$FT$\"3A+++p/ aABFT7$$!3&******z39jL$FT$\"3A+++A?+jAFT7$$!3;+++dkJ%Q$FT$\"34+++Z%y@? #FT7$$!3++++BHTIMFT$\"3'******R?@+9#FT7$$!3%******f8hYZ$FT$\"3%******R 1&\\w?FT7$$!3?+++-U7qyb$FT$\"3/+++x-EX> FT7$$!31+++<:(pf$FT$\"3++++.sax=FT7$$!3$)*****H)Q]MOFT$\"3(******\\\"z X3=FT7$$!3'******\\\"[aqOFT$\"32+++'eF!Q:FT7$$!3*)*****Rfm7!QFT$\"33+++I@?W9FT7$$!39+++9p)3$QFT$\"3!** ****fU'>o8FT7$$!3#)*****p_a$fQFT$\"3'******Rs79H\"FT7$$!37+++w6n')QFT$ \"31++++w)R@\"FT7$$!3$)*****\\H;G\"RFT$\"3'******pOeg8\"FT7$$!3<+++Ewu PRFT$\"33+++:svd5FT7$$!3%******Hg.9'RFT$\"3%)*****R=e?z*F07$$!37+++!=2 P)RFT$\"3))*****\\35^+*F07$$!3D+++f0d/SFT$\"3q*****HJ>w@)F07$$!3*)**** *4A+R-%FT$\"3%)******QzGIuF07$$!3c*****4n,;/%FT$\"3E+++$oIOk'F07$$!3#* *****H_$edSFT$\"3A+++!H))z&eF07$$!3E+++(og<2%FT$\"3Y*****fF[N2&F07$$!3 !*******fm0%3%FT$\"3C+++Q#p.H%F07$$!3s*****\\)\\S%4%FT$\"37+++fcS3NF07 $$!3\")*****z0]F5%FT$\"37+++>6`FFF07$$!3J+++su/4TFT$\"3))*****fyev%>F0 7$$!3L+++oSE8TFT$\"3++++O)e#o6F07$$!3M+++YyP:TFT$\"3')*******=IP*QFgp7 $Figx$!3')*******=IP*QFgp7$Fdgx$!3++++O)e#o6F07$F_gx$!3))*****fyev%>F0 7$Fjfx$!37+++>6`FFF07$Fefx$!37+++fcS3NF07$F`fx$!3C+++Q#p.H%F07$F[fx$!3 Y*****fF[N2&F07$Ffex$!3A+++!H))z&eF07$Faex$!3E+++$oIOk'F07$F\\ex$!3%)* *****QzGIuF07$Fgdx$!3q*****HJ>w@)F07$Fbdx$!3))*****\\35^+*F07$F]dx$!3% )*****R=e?z*F07$Fhcx$!33+++:svd5FT7$Fccx$!3'******pOeg8\"FT7$F^cx$!31+ +++w)R@\"FT7$Fibx$!3'******Rs79H\"FT7$Fdbx$!3!******fU'>o8FT7$F_bx$!33 +++I@?W9FT7$Fjax$!30+++8'*H>:FT7$Feax$!3))*****zzpLf\"FT7$F`ax$!3'**** **fH3jm\"FT7$F[ax$!32+++'eF!QFT7$Fg_x$!31+++&yx:,#FT7$Fb_x$!3%******R1& \\w?FT7$F]_x$!3'******R?@+9#FT7$Fh^x$!34+++Z%y@?#FT7$Fc^x$!3A+++A?+jAF T7$F^^x$!3A+++p/aABFT7$Fi]x$!3#******\\Yc3Q#FT7$Fd]x$!3!)******>\"G!QC FT7$F_]x$!3++++%p]T\\#FT7$Fj\\x$!3)******faQ$\\DFT7$Fe\\x$!3$******>1F Pg#FT7$F`\\x$!3;+++ASZdEFT7$F[\\x$!3\")*****Hdf2r#FT7$Ff[x$!3#******zV $yjFFT7$Fa[x$!3.+++Tuv;GFT7$F\\[x$!3/+++**3*)pGFT7$Fgjw$!3A+++qlOBHFT7 $Fbjw$!35+++4kIxHFT7$F]jw$!3++++X/tJIFT7$Fhiw$!3#******RM=l3$FT7$Fciw$ !30+++@))QTJFT7$F^iw$!39+++x6\"f>$FT7$Fihw$!35+++t@b\\KFT7$Fdhw$!3**** ***p%\\u,LFT7$F_hw$!3%)*****>qc>N$FT7$Fjgw$!31+++fct*R$FT7$Fegw$!3.+++ )fIZW$FT7$F`gw$!35+++**yo'[$FT7$F[gw$!3?+++[cVDNFT7$Fffw$!3)*******\\L '3c$FT7$Fafw$!3')*****p\\-Hf$FT7$F\\fw$!3$******HJ5:i$FT7$Fgew$!3-+++$ 3dmk$FT7$Fbew$!3#)*****pi;$oOFT7$F]ew$!3#)*****49fko$FT7$Fhdw$!3$)**** *4lX5q$FT7$Fcdw$!39+++1M-7PFT7$F^dw$!3%)******[;K>PFT7$Ficw$!3?+++%GZG s$FT7$Fdcw$!3@+++\")*zCs$FT7$F_cw$!37+++mi1=PFT7$Fjbw$!3/+++b?T4PFT7$F ebw$!3++++$*HF'p$FT7$F`bw$!3))*****>zT$yOFT7$F[bw$!3!******4cJ_l$FT7$F faw$!3*)*****>-aki$FT7$Faaw$!3%******>t\"R\"f$FT7$F\\aw$!3/+++?ZE\\NFT 7$Fg`w$!39+++Gh4*\\$FT7$$!3!******p2;#[[F0$!3?+++%RJ1Z$FT7$Fb`w$!3>+++ p@pRMFT7$F]`w$!3)******4W+hS$FT7$Fh_w$!3#)*****H#\\npLFT7$Fc_w$!32+++o 1CILFT7$F^_w$!3-+++$)Hk(G$FT7$Fi^w$!3#)******pawTKFT7$Fd^w$!3=+++H/b#> $FT7$F_^w$!3<+++Oy,SJFT7$Fj]w$!3!)*****fOxU3$FT7$Fe]w$!3\"******z0Gb-$ FT7$F`]w$!35+++lJ/kHFT7$F[]w$!3))*****>uU,!HFT7$Ff\\w$!3&******>&4;MGF T7$Fa\\w$!3#******>L;kw#FT7$F\\\\w$!3%)*****4\"4>(p#FT7$Fg[w$!3')***** *\\+sEEFT7$Fb[w$!3)*******\\:>bDFT7$F][w$!3!)*****R$4v#[#FT7$Fhjv$!31+ ++2)4&4CFT7$Fcjv$!3y*****f([bNBFT7$F^jv$!3A+++sh&4E#FT7$Fiiv$!3-+++&ot d=#FT7$Fdiv$!3!*******z@15@FT7$$!3/+++CFT7$$! 3@+++s\"[-`#F\\o$!3#******\\VoH!=FT7$$\"37+++#e^/#=F\\o$!3)******4ryvk \"FT7$$\"3F+++G4j.DF\\o$!3))*****z)RW\"\\\"FT7$$\"32+++wB^,>F\\o$!3,++ +P]%[L\"FT7$$\"3)******4'G()>6F\\o$!31+++hK)z<\"FT7$$\"3&******z3iJV&F Q$!3,+++9$))4-\"FT7$$\"35+++\"3k(o@FQ$!3_+++enMR')F07$$\"3s*****\\`1r% oFF$!3k*****zc)foqF07$$\"3'******Hm2:c\"FF$!33+++&G(z(\\&F07$$\"3(**** **\\lSu5#F@$!3,+++tM*p#RF07$$\"3c*****ptK&4**F4$!35+++NZ>cBF07$$\"3'** ****HhiMO\"F-$!3S*****\\k\")R&yFgp7$$\"3-+++1EYj8F-$\"3S*****\\k\")R&y Fgp-Fc^n6&Fe^n$\"#UFh^n$\"\")Fh^nFb_z-F\\_n6^y7%F'7$$\"+q*Hor)Fgp$\"+h jzq:Fd_nFe_n7%Fg_z7$$\"+S\\U]bFT$\"+:OfTJFd_nFe_n7%F]`z7$$\"+\"yx1D'Fg `n$\"+BZR7ZFd_nFe_n7%Fc`z7$$\"+7UJNMF^an$\"+@)*>$G'Fd_nFe_n7%Fi`z7$$\" +]@-g7F[bn$\"+Xc)R&yFd_nFe_n7%F_az7$$\"++m)G_$F[bn$\"+1#eYU*Fd_nFe_n7% Feaz7$$\"+)*)pZ)zF[bn$\"+1n\\*4\"F^bnFe_n7%F[bz7$$\"+Vi_(\\\"F_cn$\"+) QPkD\"F^bnFe_n7%Fabz7$$\"+Ms_rAF_cn$\"+>\")=89F^bnFe_n7%Fgbz7$$\"+JN3M CF_cn$\"+v]ep:F^bnFe_n7%F]cz7$$\"+R/*\\?$F[bn$\"+RsQDlEa=F_en$\"+2)4&4CF^bnFe_n7%F]fz7$$!+4.W- DF_en$\"+M4v#[#F^bnFe_n7%Fcfz7$$!+w?p4LF_en$\"+]:>bDF^bnFe_n7%Fifz7$$! +J#)H*H%F_en$\"+]+sEEF^bnFe_n7%F_gz7$$!+MJM%\\&F_en$\"+64>(p#F^bnFe_n7 %Fegz7$$!+uoV;pF_en$\"+KjTmFF^bnFe_n7%F[hz7$$!+nR*Re)F_en$\"+_4;MGF^bn Fe_n7%Fahz7$$!+VZ,^5Fd_n$\"+UF9+HF^bnFe_n7%Fghz7$$!+is0q7Fd_n$\"+lJ/kH F^bnFe_n7%F]iz7$$!+W;<::Fd_n$\"+e!Gb-$F^bnFe_n7%Fciz7$$!+oM+&y\"Fd_n$ \"+mtF%3$F^bnFe_n7%Fiiz7$$!+!pos2#Fd_n$\"+Oy,SJF^bnFe_n7%F_jz7$$!+d/(* )Q#Fd_n$\"+H/b#>$F^bnFe_n7%Fejz7$$!+FKr;FFd_n$\"+qawTKF^bnFe_n7%F[[[l7 $$!+ZT.dIFd_n$\"+$)Hk(G$F^bnFe_n7%Fa[[l7$$!+U&ymS$Fd_n$\"+o1CILF^bnFe_ n7%Fg[[l7$$!+o+yiPFd_n$\"+B\\npLF^bnFe_n7%F]\\[l7$$!+)>VH7%Fd_n$\"+T/5 1MF^bnFe_n7%Fc\\[l7$$!+^%\\_[%Fd_n$\"+p@pRMF^bnFe_n7%Fi\\[l7$$!+8At5_F d_n$\"+Gh4*\\$F^bnFe_n7%F_][l7$$!+(**=9$fFd_n$\"+?ZE\\NF^bnFe_n7%Fe][l 7$$!+z=QVmFd_n$\"+K\"F^bn$\"+%GZGs$F^ bnFe_n7%F[a[l7$$!+Netg7F^bn$\"+\\;K>PF^bnFe_n7%Faa[l7$$!+7%**HK\"F^bn$ \"+1M-7PF^bnFe_n7%Fga[l7$$!+&z\"o%Q\"F^bn$\"+^c/,PF^bnFe_n7%F]b[l7$$!+ 0$[eW\"F^bn$\"+T\"fko$F^bnFe_n7%Fcb[l7$$!+'=sl]\"F^bn$\"+FmJoOF^bnFe_n 7%Fib[l7$$!+T#Qpc\"F^bn$\"+$3dmk$F^bnFe_n7%F_c[l7$$!++p/F;F^bn$\"+8.^@ OF^bnFe_n7%Fec[l7$$!+N'=qo\"F^bn$\"+(\\-Hf$F^bnFe_n7%F[d[l7$$!+T%)*pu \"F^bn$\"+]L'3c$F^bnFe_n7%Fad[l7$$!+,)fr!=F^bn$\"+[cVDNF^bnFe_n7%Fgd[l 7$$!+)*pqn=F^bn$\"+**yo'[$F^bnFe_n7%F]e[l7$$!+%*Q()G>F^bn$\"+)fIZW$F^b nFe_n7%Fce[l7$$!+Rj\"4*>F^bn$\"+fct*R$F^bnFe_n7%Fie[l7$$!+tZ4a?F^bn$\" +-n&>N$F^bnFe_n7%F_f[l7$$!+'3W'=@F^bn$\"+Z\\u,LF^bnFe_n7%Fef[l7$$!+15t %=#F^bn$\"+t@b\\KF^bnFe_n7%F[g[l7$$!+CdS_AF^bn$\"+x6\"f>$F^bnFe_n7%Fag [l7$$!+]Bc@BF^bn$\"+@))QTJF^bnFe_n7%Fgg[l7$$!+Kh#>R#F^bn$\"+W$=l3$F^bn Fe_n7%F]h[l7$$!+l!yIY#F^bn$\"+X/tJIF^bnFe_n7%Fch[l7$$!+j7^MDF^bn$\"+4k IxHF^bnFe_n7%Fih[l7$$!+P#*p0EF^bn$\"+qlOBHF^bnFe_n7%F_i[l7$$!+qU:wEF^b n$\"+**3*)pGF^bnFe_n7%Fei[l7$$!+HNYXFF^bn$\"+Tuv;GF^bnFe_n7%F[j[l7$$!+ iBI8GF^bn$\"+QMyjFF^bnFe_n7%Faj[l7$$!+y6VzGF^bn$\"+t&f2r#F^bnFe_n7%Fgj [l7$$!+(o$oVHF^bn$\"+ASZdEF^bnFe_n7%F][\\l7$$!+6C&f+$F^bn$\"+iqs.EF^bn Fe_n7%Fc[\\l7$$!+;\\qyb$F^bn$\"+x-EX>F^bnFe_n7%Fe_\\l7$$!+<:(pf$F^bn$\"+.sax=F^bnFe _n7%F[`\\l7$$!+$)Q]MOF^bn$\"+:zX3=F^bnFe_n7%Fa`\\l7$$!+:[aqOF^bn$\"+'e F!Q:F^bnFe_n7%Fia \\l7$$!+%fm7!QF^bn$\"+I@?W9F^bnFe_n7%F_b\\l7$$!+9p)3$QF^bn$\"+Ek>o8F^b nFe_n7%Feb\\l7$$!+FXNfQF^bn$\"+CFT\"H\"F^bnFe_n7%F[c\\l7$$!+w6n')QF^bn $\"++w)R@\"F^bnFe_n7%Fac\\l7$$!+&H;G\"RF^bn$\"+n$eg8\"F^bnFe_n7%Fgc\\l 7$$!+EwuPRF^bn$\"+:svd5F^bnFe_n7%F]d\\l7$$!+.OShRF^bn$\"+%=e?z*Fd_nFe_ n7%Fcd\\l7$$!+!=2P)RF^bn$\"+&35^+*Fd_nFe_n7%Fid\\l7$$!+f0d/SF^bn$\"+8$ >w@)Fd_nFe_n7%F_e\\l7$$!+@-!R-%F^bn$\"+RzGIuFd_nFe_n7%Fee\\l7$$!+r;gTS F^bn$\"+$oIOk'Fd_nFe_n7%F[f\\l7$$!+BNedSF^bn$\"+!H))z&eFd_nFe_n7%Faf\\ l7$$!+(og<2%F^bn$\"+w#[N2&Fd_nFe_n7%Fgf\\l7$$!+gm0%3%F^bn$\"+Q#p.H%Fd_ nFe_n7%F]g\\l7$$!+&)\\S%4%F^bn$\"+fcS3NFd_nFe_n7%Fcg\\l7$$!+e+v-TF^bn$ \"+>6`FFFd_nFe_n7%Fig\\l7$$!+su/4TF^bn$\"+'yev%>Fd_nFe_n7%F_h\\l7$$!+o SE8TF^bn$\"+O)e#o6Fd_nFe_n7%Feh\\l7$$!+YyP:TF^bn$\"+!>IP*QF_enFe_n7%F[ i\\l7$F\\i\\l$!+!>IP*QF_enFe_n7%Fai\\l7$Ffh\\l$!+O)e#o6Fd_nFe_n7%Fei\\ l7$F`h\\l$!+'yev%>Fd_nFe_n7%Fii\\l7$Fjg\\l$!+>6`FFFd_nFe_n7%F]j\\l7$Fd g\\l$!+fcS3NFd_nFe_n7%Faj\\l7$F^g\\l$!+Q#p.H%Fd_nFe_n7%Fej\\l7$Fhf\\l$ !+w#[N2&Fd_nFe_n7%Fij\\l7$Fbf\\l$!+!H))z&eFd_nFe_n7%F][]l7$F\\f\\l$!+$ oIOk'Fd_nFe_n7%Fa[]l7$Ffe\\l$!+RzGIuFd_nFe_n7%Fe[]l7$F`e\\l$!+8$>w@)Fd _nFe_n7%Fi[]l7$Fjd\\l$!+&35^+*Fd_nFe_n7%F]\\]l7$Fdd\\l$!+%=e?z*Fd_nFe_ n7%Fa\\]l7$F^d\\l$!+:svd5F^bnFe_n7%Fe\\]l7$Fhc\\l$!+n$eg8\"F^bnFe_n7%F i\\]l7$Fbc\\l$!++w)R@\"F^bnFe_n7%F]]]l7$F\\c\\l$!+CFT\"H\"F^bnFe_n7%Fa ]]l7$Ffb\\l$!+Ek>o8F^bnFe_n7%Fe]]l7$F`b\\l$!+I@?W9F^bnFe_n7%Fi]]l7$Fja \\l$!+8'*H>:F^bnFe_n7%F]^]l7$Fda\\l$!+)zpLf\"F^bnFe_n7%Fa^]l7$F^a\\l$! +'H3jm\"F^bnFe_n7%Fe^]l7$Fh`\\l$!+'eF!QF^bnFe _n7%Fe_]l7$F`_\\l$!+&yx:,#F^bnFe_n7%Fi_]l7$Fj^\\l$!+k]\\w?F^bnFe_n7%F] `]l7$Fd^\\l$!+/7-S@F^bnFe_n7%Fa`]l7$F^^\\l$!+Z%y@?#F^bnFe_n7%Fe`]l7$Fh ]\\l$!+A?+jAF^bnFe_n7%Fi`]l7$Fb]\\l$!+p/aABF^bnFe_n7%F]a]l7$F\\]\\l$!+ lk&3Q#F^bnFe_n7%Faa]l7$Ff\\\\l$!+?\"G!QCF^bnFe_n7%Fea]l7$F`\\\\l$!+%p] T\\#F^bnFe_n7%Fia]l7$Fj[\\l$!+Y&Q$\\DF^bnFe_n7%F]b]l7$Fd[\\l$!+iqs.EF^ bnFe_n7%Fab]l7$F^[\\l$!+ASZdEF^bnFe_n7%Feb]l7$Fhj[l$!+t&f2r#F^bnFe_n7% Fib]l7$Fbj[l$!+QMyjFF^bnFe_n7%F]c]l7$F\\j[l$!+Tuv;GF^bnFe_n7%Fac]l7$Ff i[l$!+**3*)pGF^bnFe_n7%Fec]l7$F`i[l$!+qlOBHF^bnFe_n7%Fic]l7$Fjh[l$!+4k IxHF^bnFe_n7%F]d]l7$Fdh[l$!+X/tJIF^bnFe_n7%Fad]l7$F^h[l$!+W$=l3$F^bnFe _n7%Fed]l7$Fhg[l$!+@))QTJF^bnFe_n7%Fid]l7$Fbg[l$!+x6\"f>$F^bnFe_n7%F]e ]l7$F\\g[l$!+t@b\\KF^bnFe_n7%Fae]l7$Fff[l$!+Z\\u,LF^bnFe_n7%Fee]l7$F`f [l$!+-n&>N$F^bnFe_n7%Fie]l7$Fje[l$!+fct*R$F^bnFe_n7%F]f]l7$Fde[l$!+)fI ZW$F^bnFe_n7%Faf]l7$F^e[l$!+**yo'[$F^bnFe_n7%Fef]l7$Fhd[l$!+[cVDNF^bnF e_n7%Fif]l7$Fbd[l$!+]L'3c$F^bnFe_n7%F]g]l7$F\\d[l$!+(\\-Hf$F^bnFe_n7%F ag]l7$Ffc[l$!+8.^@OF^bnFe_n7%Feg]l7$F`c[l$!+$3dmk$F^bnFe_n7%Fig]l7$Fjb [l$!+FmJoOF^bnFe_n7%F]h]l7$Fdb[l$!+T\"fko$F^bnFe_n7%Fah]l7$F^b[l$!+^c/ ,PF^bnFe_n7%Feh]l7$Fha[l$!+1M-7PF^bnFe_n7%Fih]l7$Fba[l$!+\\;K>PF^bnFe_ n7%F]i]l7$F\\a[l$!+%GZGs$F^bnFe_n7%Fai]l7$Ff`[l$!+\")*zCs$F^bnFe_n7%Fe i]l7$F``[l$!+mi1=PF^bnFe_n7%Fii]l7$Fj_[l$!+b?T4PF^bnFe_n7%F]j]l7$Fd_[l $!+$*HF'p$F^bnFe_n7%Faj]l7$F^_[l$!+#zT$yOF^bnFe_n7%Fej]l7$Fh^[l$!+h:Bb OF^bnFe_n7%Fij]l7$Fb^[l$!+ASXEOF^bnFe_n7%F][^l7$F\\^[l$!+K$F^bnFe_n7%F[^^l7$F`jz$!+Oy,SJF^bnF e_n7%F_^^l7$Fjiz$!+mtF%3$F^bnFe_n7%Fc^^l7$Fdiz$!+e!Gb-$F^bnFe_n7%Fg^^l 7$F^iz$!+lJ/kHF^bnFe_n7%F[_^l7$Fhhz$!+UF9+HF^bnFe_n7%F__^l7$Fbhz$!+_4; MGF^bnFe_n7%Fc_^l7$F\\hz$!+KjTmFF^bnFe_n7%Fg_^l7$Ffgz$!+64>(p#F^bnFe_n 7%F[`^l7$F`gz$!+]+sEEF^bnFe_n7%F_`^l7$Fjfz$!+]:>bDF^bnFe_n7%Fc`^l7$Fdf z$!+M4v#[#F^bnFe_n7%Fg`^l7$F^fz$!+2)4&4CF^bnFe_n7%F[a^l7$Fhez$!+w[bNBF ^bnFe_n7%F_a^l7$Fbez$!+sh&4E#F^bnFe_n7%Fca^l7$F\\ez$!+&otd=#F^bnFe_n7% Fga^l7$Ffdz$!+!=i+6#F^bnFe_n7%F[b^l7$$!+CF^bnFe_n7% F_b^l7$$!+s\"[-`#F_cn$!+N%oH!=F^bnFe_n7%Feb^l7$$\"+#e^/#=F_cn$!+6(yvk \"F^bnFe_n7%F[c^l7$$\"+G4j.DF_cn$!+))RW\"\\\"F^bnFe_n7%Fac^l7$$\"+wB^, >F_cn$!+P]%[L\"F^bnFe_n7%Fgc^l7$$\"+hG()>6F_cn$!+hK)z<\"F^bnFe_n7%F]d^ l7$$\"+)3iJV&F[bn$!+9$))4-\"F^bnFe_n7%Fcd^l7$$\"+\"3k(o@F[bn$!+enMR')F d_nFe_n7%Fid^l7$$\"+Nl5ZoF^an$!+o&)foqFd_nFe_n7%F_e^l7$$\"+jw]h:F^an$! +&G(z(\\&Fd_nFe_n7%Fee^l7$$\"+b1W2@Fg`n$!+tM*p#RFd_nFe_n7%F[f^l7$$\"+P F`4**F0$!+NZ>cBFd_nFe_n7%Faf^l7$$\"+8EYj8Fgo$!+X;)R&yF_enFe_n7%Fgf^l7$ $\"+1EYj8Fgo$\"+X;)R&yF_enFe_n-Fc^n6&Fe^n$\"#$)Fh^n$\"#Eg\"R 9FT$\"3m******GVT;VFT7$$\"3\"******4\\=)G9FT$\"3%)*****RHIHK%FT7$$\"3! ******z%4#zT\"FT$\"34+++(p4'GVFT7$$\"31+++HT^19FT$\"3.+++W2TLVFT7$$\"3 ++++fjk%R\"FT$\"3&)*****\\-\"HPVFT7$$\"3'*******)GoBQ\"FT$\"3K+++ds?SV FT7$$\"3-+++xMtp8FT$\"3k*****>/:@M%FT7$$\"3/+++!3*zc8FT$\"35+++Y'oHM%F T7$$\"3$******HnEOM\"FT$\"3F+++)y?FM%FT7$$\"34+++WKGI8FT$\"3\")*****>[ A8M%FT7$$\"3.+++%RUoJ\"FT$\"3v*****R$HsQVFT7$$\"3,+++(z&Q.8FT$\"3&)*** **>ap[L%FT7$$\"3'******f*\\+!H\"FT$\"3e*****z03(HVFT7$$\"3/++++O!oF\"F T$\"3*)*****f*H=BVFT7$$\"3!*******o***QE\"FT$\"3!*******R%Q_J%FT7$$\"3 )******4]I9D\"FT$\"3-+++X%>eI%FT7$$\"3%*******zNbR7FT$\"3m******fV([H% FT7$$\"3)******fAa%G7FT$\"3))*****>aeBG%FT7$$\"3%*******f$\\$=7FT$\"3y *****48S#oUFT7$$\"32+++0L\\47FT$\"39+++`)3DD%FT7$$\"32+++4E=-7FT$\"3l* *****e!*=NUFT7$$\"3#******RKen>\"FT$\"3r*****pPej@%FT7$$\"3%*******4Bg $>\"FT$\"39+++#*H<'>%FT7$$\"3%*******o?7$>\"FT$\"3U+++J*)*[<%FT7$$\"3# ******p\"pr&>\"FT$\"3!*******Q[%H:%FT7$$\"3$********H>J@I\"FT$ \"3&******H\"[/QSFT7$$\"30+++F_/C8FT$\"3y*****fEZP.%FT7$$\"3++++1_bX8F T$\"3G+++::jKSFT7$$\"31+++M(*>m8FT$\"3*)*****f7RV.%FT7$$\"33+++Q*fcQ\" FT$\"3!******HF%[QSFT7$$\"3++++a;t.9FT$\"3O+++k?pWSFT7$$\"3.+++wjH?9FT $\"3y*****pD=E0%FT7$$\"33+++`kHN9FT$\"3E+++Hn&>1%FT7$$\"3/+++[fr[9FT$ \"3w*****f-SC2%FT7$$\"35+++xmcg9FT$\"37+++([OQ3%FT7$$\"3!******pRy3Z\" FT$\"35+++Fg%f4%FT7$$\"31+++d?pz9FT$\"36+++vcf3TFT7$$\"3'******zBbq[\" FT$\"3$******4QN;7%FT7$$\"3/+++7\">I\\\"FT$\"3#******pJM\\8%FT7$$\"3'* *****zWOw\\\"FT$\"3%)*****fpx$[TFT7$$\"31+++Q.'4]\"FT$\"3M+++1T'=;%FT7 $$\"31+++*RVI]\"FT$\"3w*****4F.`<%FT7$$\"33+++1t$R]\"FT$\"3O+++0Uh)=%F T7$$\"3\"******fa#p.:FT$\"36+++jOs,UFT7$$\"3-+++C$eB]\"FT$\"3e*****>'[ c9UFT7$$\"35+++jD)**\\\"FT$\"3z*****>SwqA%FT7$$\"3'*******z>h'\\\"FT$ \"3s*****pO,#RUFT7$$\"3'******R?#H#\\\"FT$\"3?+++cl)3D%FT7$$\"33+++Dz1 ([\"FT$\"3G+++F=3iUFT7$$\"3/+++rI)4[\"FT$\"3s*****z`RFF%FT7$$\"3++++.5 3u9FT$\"3u******[S\"GG%FTF[h^lF^_z-F\\_n6jn7%7$$\"+CZSm9F^bn$\"+!HhAH% F^bn7$$\"+6r*zX\"F^bn$\"+t$Q5I%F^bn7$$\"$M\"Fh^n$\"$>%Fh^n7%F_[`l7$$\" +h6!*[9F^bn$\"+QK54VF^bnFd[`l7%Fj[`l7$$\"+i-;R9F^bn$\"+HVT;VF^bnFd[`l7 %F`\\`l7$$\"+\"\\=)G9F^bn$\"+%HIHK%F^bnFd[`l7%Ff\\`l7$$\"+[4#zT\"F^bn$ \"+(p4'GVF^bnFd[`l7%F\\]`l7$$\"+HT^19F^bn$\"+W2TLVF^bnFd[`l7%Fb]`l7$$ \"+fjk%R\"F^bn$\"+D5HPVF^bnFd[`l7%Fh]`l7$$\"+*GoBQ\"F^bn$\"+ds?SVF^bnF d[`l7%F^^`l7$$\"+xMtp8F^bn$\"+U]6UVF^bnFd[`l7%Fd^`l7$$\"+!3*zc8F^bn$\" +Y'oHM%F^bnFd[`l7%Fj^`l7$$\"+tmiV8F^bn$\"+)y?FM%F^bnFd[`l7%F`_`l7$$\"+ WKGI8F^bn$\"+#[A8M%F^bnFd[`l7%Ff_`l7$$\"+%RUoJ\"F^bn$\"+MHsQVF^bnFd[`l 7%F\\``l7$$\"+(z&Q.8F^bn$\"+U&p[L%F^bnFd[`l7%Fb``l7$$\"+'*\\+!H\"F^bn$ \"+e!3(HVF^bnFd[`l7%Fh``l7$$\"++O!oF\"F^bn$\"+'*H=BVF^bnFd[`l7%F^a`l7$ $\"+p***QE\"F^bn$\"+S%Q_J%F^bnFd[`l7%Fda`l7$$\"+,0V^7F^bn$\"+X%>eI%F^b nFd[`l7%Fja`l7$$\"+!e`&R7F^bn$\"+gV([H%F^bnFd[`l7%F`b`l7$$\"+EUXG7F^bn $\"+U&eBG%F^bnFd[`l7%Ffb`l7$$\"+g$\\$=7F^bn$\"+J,CoUF^bnFd[`l7%F\\c`l7 $$\"+0L\\47F^bn$\"+`)3DD%F^bnFd[`l7%Fbc`l7$$\"+4E=-7F^bn$\"+f!*=NUF^bn Fd[`l7%Fhc`l7$$\"+C$en>\"F^bn$\"+x$ej@%F^bnFd[`l7%F^d`l7$$\"+5Bg$>\"F^ bn$\"+#*H<'>%F^bnFd[`l7%Fdd`l7$$\"+p?7$>\"F^bn$\"+J*)*[<%F^bnFd[`l7%Fj d`l7$$\"+\"F^bn$\"+R[%H:%F^bnFd[`l7%F`e`l7$$\"++$>J@I\"F^bn$\"+8[/QSF^bnFd[`l7%Fjg`l7$$\"+F_/C8F ^bn$\"+msuLSF^bnFd[`l7%F`h`l7$$\"+1_bX8F^bn$\"+::jKSF^bnFd[`l7%Ffh`l7$ $\"+M(*>m8F^bn$\"+E\"RV.%F^bnFd[`l7%F\\i`l7$$\"+Q*fcQ\"F^bn$\"+tU[QSF^ bnFd[`l7%Fbi`l7$$\"+a;t.9F^bn$\"+k?pWSF^bnFd[`l7%Fhi`l7$$\"+wjH?9F^bn$ \"+d#=E0%F^bnFd[`l7%F^j`l7$$\"+`kHN9F^bn$\"+Hn&>1%F^bnFd[`l7%Fdj`l7$$ \"+[fr[9F^bn$\"+E+WsSF^bnFd[`l7%Fjj`l7$$\"+xmcg9F^bn$\"+([OQ3%F^bnFd[` l7%F`[al7$$\"+(Ry3Z\"F^bn$\"+Fg%f4%F^bnFd[`l7%Ff[al7$$\"+d?pz9F^bn$\"+ vcf3TF^bnFd[`l7%F\\\\al7$$\"+Q_0([\"F^bn$\"+\"QN;7%F^bnFd[`l7%Fb\\al7$ $\"+7\">I\\\"F^bn$\"+h'\\\"F^bn$\"+n8?RUF^bnFd[`l7%Fh_al7$$\"+/AH#\\\"F^bn$\"+cl)3D%F^bn Fd[`l7%F^`al7$$\"+Dz1([\"F^bn$\"+F=3iUF^bnFd[`l7%Fd`al7$$\"+rI)4[\"F^b n$\"+Q&RFF%F^bnFd[`l7%Fj`al7$$\"+.53u9F^bn$\"+\\S\"GG%F^bnFd[`l7%F`aal Fjj_lFd[`lFbg^lFi^r-F$6$7in7$F\\h^l$!3!********GhAH%FT7$Fcj_l$!3u***** *[S\"GG%FT7$F^j_l$!3s*****z`RFF%FT7$Fii_l$!3G+++F=3iUFT7$Fdi_l$!3?+++c l)3D%FT7$F_i_l$!3s*****pO,#RUFT7$Fjh_l$!3z*****>SwqA%FT7$Feh_l$!3e**** *>'[c9UFT7$F`h_l$!36+++jOs,UFT7$F[h_l$!3O+++0Uh)=%FT7$Ffg_l$!3w*****4F .`<%FT7$Fag_l$!3M+++1T'=;%FT7$F\\g_l$!3%)*****fpx$[TFT7$Fgf_l$!3#***** *pJM\\8%FT7$Fbf_l$!3$******4QN;7%FT7$F]f_l$!36+++vcf3TFT7$Fhe_l$!35+++ Fg%f4%FT7$Fce_l$!37+++([OQ3%FT7$F^e_l$!3w*****f-SC2%FT7$Fid_l$!3E+++Hn &>1%FT7$Fdd_l$!3y*****pD=E0%FT7$F_d_l$!3O+++k?pWSFT7$Fjc_l$!3!******HF %[QSFT7$Fec_l$!3*)*****f7RV.%FT7$F`c_l$!3G+++::jKSFT7$F[c_l$!3y*****fE ZP.%FT7$Ffb_l$!3&******H\"[/QSFT7$Fab_l$!3))*****fz$zXSFT7$F\\b_l$!3-+ ++N[4dSFT7$Fga_l$!31+++l9!=2%FT7$Fba_l$!3O+++]:[*3%FT7$F]a_l$!3#****** f#=X4TFT7$Fh`_l$!3(******Rz%)38%FT7$Fc`_l$!3!*******Q[%H:%FT7$F^`_l$!3 U+++J*)*[<%FT7$Fi__l$!39+++#*H<'>%FT7$Fd__l$!3r*****pPej@%FT7$F___l$!3 l******e!*=NUFT7$Fj^_l$!39+++`)3DD%FT7$Fe^_l$!3y*****48S#oUFT7$F`^_l$! 3))*****>aeBG%FT7$F[^_l$!3m******fV([H%FT7$Ff]_l$!3-+++X%>eI%FT7$Fa]_l $!3!*******R%Q_J%FT7$F\\]_l$!3*)*****f*H=BVFT7$Fg\\_l$!3e*****z03(HVFT 7$Fb\\_l$!3&)*****>ap[L%FT7$F]\\_l$!3v*****R$HsQVFT7$Fh[_l$!3\")*****> [A8M%FT7$Fc[_l$!3F+++)y?FM%FT7$F^[_l$!35+++Y'oHM%FT7$Fij^l$!3k*****>/: @M%FT7$Fdj^l$!3K+++ds?SVFT7$F_j^l$!3&)*****\\-\"HPVFT7$Fji^l$!3.+++W2T LVFT7$Fei^l$!34+++(p4'GVFT7$F`i^l$!3%)*****RHIHK%FT7$F[i^l$!3m******GV T;VFT7$Ffh^l$!3h*****zB.\"4VFT7$Fah^l$!3\"******HPQ5I%FTFiaalF^_z-F\\_ n6jn7%7$F[[`l$!+!HhAH%F^bn7$Faaal$!+\\S\"GG%F^bn7$Fe[`l$!$>%Fh^n7%Fc]b l7$F[aal$!+Q&RFF%F^bnFf]bl7%Fj]bl7$Fe`al$!+F=3iUF^bnFf]bl7%F^^bl7$F_`a l$!+cl)3D%F^bnFf]bl7%Fb^bl7$Fi_al$!+n8?RUF^bnFf]bl7%Ff^bl7$Fc_al$!+-k2 FUF^bnFf]bl7%Fj^bl7$F]_al$!+i[c9UF^bnFf]bl7%F^_bl7$Fg^al$!+jOs,UF^bnFf ]bl7%Fb_bl7$Fa^al$!+0Uh)=%F^bnFf]bl7%Ff_bl7$F[^al$!+rKIvTF^bnFf]bl7%Fj _bl7$Fe]al$!+1T'=;%F^bnFf]bl7%F^`bl7$F_]al$!+'px$[TF^bnFf]bl7%Fb`bl7$F i\\al$!+1%F ^bnFf]bl7%F^bbl7$F_j`l$!+d#=E0%F^bnFf]bl7%Fbbbl7$Fii`l$!+k?pWSF^bnFf]b l7%Ffbbl7$Fci`l$!+tU[QSF^bnFf]bl7%Fjbbl7$F]i`l$!+E\"RV.%F^bnFf]bl7%F^c bl7$Fgh`l$!+::jKSF^bnFf]bl7%Fbcbl7$Fah`l$!+msuLSF^bnFf]bl7%Ffcbl7$F[h` l$!+8[/QSF^bnFf]bl7%Fjcbl7$Feg`l$!+'z$zXSF^bnFf]bl7%F^dbl7$F_g`l$!+N[4 dSF^bnFf]bl7%Fbdbl7$Fif`l$!+l9!=2%F^bnFf]bl7%Ffdbl7$Fcf`l$!+]:[*3%F^bn Ff]bl7%Fjdbl7$F]f`l$!+E=X4TF^bnFf]bl7%F^ebl7$Fge`l$!+%z%)38%F^bnFf]bl7 %Fbebl7$Fae`l$!+R[%H:%F^bnFf]bl7%Ffebl7$F[e`l$!+J*)*[<%F^bnFf]bl7%Fjeb l7$Fed`l$!+#*H<'>%F^bnFf]bl7%F^fbl7$F_d`l$!+x$ej@%F^bnFf]bl7%Fbfbl7$Fi c`l$!+f!*=NUF^bnFf]bl7%Fffbl7$Fcc`l$!+`)3DD%F^bnFf]bl7%Fjfbl7$F]c`l$!+ J,CoUF^bnFf]bl7%F^gbl7$Fgb`l$!+U&eBG%F^bnFf]bl7%Fbgbl7$Fab`l$!+gV([H%F ^bnFf]bl7%Ffgbl7$F[b`l$!+X%>eI%F^bnFf]bl7%Fjgbl7$Fea`l$!+S%Q_J%F^bnFf] bl7%F^hbl7$F_a`l$!+'*H=BVF^bnFf]bl7%Fbhbl7$Fi``l$!+e!3(HVF^bnFf]bl7%Ff hbl7$Fc``l$!+U&p[L%F^bnFf]bl7%Fjhbl7$F]``l$!+MHsQVF^bnFf]bl7%F^ibl7$Fg _`l$!+#[A8M%F^bnFf]bl7%Fbibl7$Fa_`l$!+)y?FM%F^bnFf]bl7%Ffibl7$F[_`l$!+ Y'oHM%F^bnFf]bl7%Fjibl7$Fe^`l$!+U]6UVF^bnFf]bl7%F^jbl7$F_^`l$!+ds?SVF^ bnFf]bl7%Fbjbl7$Fi]`l$!+D5HPVF^bnFf]bl7%Ffjbl7$Fc]`l$!+W2TLVF^bnFf]bl7 %Fjjbl7$F]]`l$!+(p4'GVF^bnFf]bl7%F^[cl7$Fg\\`l$!+%HIHK%F^bnFf]bl7%Fb[c l7$Fa\\`l$!+HVT;VF^bnFf]bl7%Ff[cl7$F[\\`l$!+QK54VF^bnFf]bl7%Fj[cl7$F`[ `l$!+t$Q5I%F^bnFf]bl7%F^\\clF`]blFf]blFbg^lFi^r-%*AXESSTYLEG6#%$BOXG-% +AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONST RAINEDG-F\\]cl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%Fh^n$\"$*>Fh^n;F[^cl$ \"$z%Fh^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#----------------- ----------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "chara cteristics of the embedded schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1640 "ee := \{c[ 2]=1/6,\nc[3]=155/594,\nc[4]=12/19,\nc[5]=66/85,\nc[6]=27/34,\nc[7]=1, \nc[8]=1,\n\na[2,1]=1/6,\na[3,1]=6665/117612,\na[3,2]=24025/117612,\na [4,1]=109269372/164787475,\na[4,2]=-2341872/1063145,\na[4,3]=357797088 /164787475,\na[5,1]=-67885276899/135740048750,\na[5,2]=37863936/175148 45,\na[5,3]=-383179645721664/283900311960625,\na[5,4]=54884775429/1181 68704250,\na[6,1]=-22877286938601/28657439092100,\na[6,2]=2786718411/9 10771940,\na[6,3]=-57606687522659088/29968516930563575,\na[6,4]=398489 9849779/12473888420630,\na[6,5]=55/406,\na[7,1]=3578041087956840260673 91/386149007682840983490000,\na[7,2]=-7556229/2688010,\na[7,3]=1811005 935036153778848695517/745590435061329862990895000,\na[7,4]=25890269179 1384553326881/1307299004967194685810000,\na[7,5]=-154/529,\na[7,6]=138 208410082586660479/251926265586772035000,\na[8,1]=51964327/654617700, \na[8,2]=0,\na[8,3]=6965638145689311/17945236957333900,\na[8,4]=145573 76984/60894963675,\na[8,5]=2059161125/95272074612,\na[8,6]=1793996689/ 9043706700,\na[8,7]=29/395,\n\nb[1]=51964327/654617700,\nb[2]=0,\nb[3] =6965638145689311/17945236957333900,\nb[4]=14557376984/60894963675,\nb [5]=2059161125/95272074612,\nb[6]=1793996689/9043706700,\nb[7]=29/395, \n\n`b^`[1]=126880675943/1597558129200,\n`b^`[2]=0,\n`b^`[3]=284049644 28638589/73207500841545650,\n`b^`[4]=1929236304929/8045094629520,\n`b^ `[5]=63707991275/3620338835256,\n`b^`[6]=23/114,\n`b^`[7]=29/395,\n\n` b*`[1]=4457141939387131/56114268213415500,\n`b*`[2]=0,\n`b*`[3]=71/183 ,\n`b*`[4]=366435142393882979/1527555079142977500,\n`b*`[5]=1/57,\n`b* `[6]=2399113615956319/11892729854919375,\n`b*`[7]=33069698525894087179 37/45125218883586784548375,\n`b*`[8]=8182415847667565831/5415026266030 4141458050\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5,7];" "6#&%\"TG6$\"\"&\" \"(" }{TEXT -1 127 " denote the vector whose components are the princ ipal error terms of the 7 stage, order 5 scheme (the error terms of or der 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,8 ];" "6#&%#T*G6$\"\"%\"\")" }{TEXT -1 145 " denote the vector whose co mponents are the principal error terms of the embedded 8 stage, order \+ 4 scheme (the error terms of order 5) and let " }{XPPEDIT 18 0 "`T*`[ 5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 99 " denote the vector whose \+ components are the error terms of order 6 of the embedded order 4 sche me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "De note the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7] ));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\")" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6 #-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/abs(abs(`T*`[4, 8]));" " 6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\")\"\"\"-F*6#-F*6#&F/6 $\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T* `[5,8]-T[5,7]))/abs(abs(`T*`[4,8]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6 #,&&%#T*G6$\"\"&\"\")\"\"\"&%\"TG6$F2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F 3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "Dormand and Prince have suggested that as well as attem pting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" } {TEXT -1 73 " is a minimum, if the embedded scheme is to be used for \+ error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "From this point o f view the Bogacki-Shampine scheme is near ideal with respect to the 8 stage, order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_7 := PrincipalErr orTerms(5,7,'expanded'):\n`errterms5_8*` :=subs(b=`b*`,PrincipalErrorT erms(5,8,'expanded')):\n`errterms4_8*` := subs(b=`b*`,PrincipalErrorTe rms(4,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := sqrt(add(simplify(subs(ee, `errterms5_8*`[i]))^2,i=1..nops(`errterms5_8*`))):\nsdnB := sqrt(add(s implify(subs(ee,`errterms4_8*`[i]))^2,i=1..nops(`errterms4_8*`))):\nsn mC := sqrt(add((simplify(subs(ee,`errterms5_8*`[i])-subs(ee,errterms5_ 7[i])))^2,i=1..nops(errterms5_7))):\n'`B*`[6]'= evalf[8](snmB/sdnB);\n '`C*`[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% #B*G6#\"\"'$\")rGZ6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\" \"'$\")7-q5!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The corresponding characterstics for the auxiliary 7 stag e, order 4 embedded scheme are as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 134 "`errterms5_7^` :=subs(b=`b^`,PrincipalErrorTerms(5 ,7,'expanded')):\n`errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4, 7,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 341 "snmB := sqrt(add(simplify(subs(ee,`errterms5_ 7^`[i]))^2,i=1..nops(`errterms5_7^`))):\nsdnB := sqrt(add(simplify(sub s(ee,`errterms4_7^`[i]))^2,i=1..nops(`errterms4_7^`))):\nsnmC := sqrt( add((simplify(subs(ee,`errterms5_7^`[i])-subs(ee,errterms5_7[i])))^2,i =1..nops(errterms5_7))):\n'`B^`[6]'= evalf[8](snmB/sdnB);\n'`C^`[6]'= \+ evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B^G6#\"\"' $\")bVV6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C^G6#\"\"'$\")mWq5 !\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#================================== ===============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "An efficient 7 stage, order 5 Runge-Kutta scheme .. C" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#------------------ -----------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checki ng the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1861 "ee := \{c[2]=9/5 3,\nc[3]=83/311,\nc[4]=167/250,\nc[5]=34/43,\nc[6]=22/27,\nc[7]=1,\nc[ 8]=1,\n\na[2,1]=9/53,\na[3,1]=99517/1740978,\na[3,2]=365117/1740978,\n a[4,1]=30637228987/38750625000,\na[4,2]=-7690642751/2917968750,\na[4,3 ]=270499789573/107640625000,\na[5,1]=-361825370105678/277428090660453, \na[5,2]=495002920909/100075063365,\na[5,3]=-98837524323878781/2878293 3677403685,\na[5,4]=81285337012500/139548553465711,\na[6,1]=-756520667 0807304418/3682741770828399897,\na[6,2]=898568957363/123874468425,\na[ 6,3]=-728964729183934695599/146954383037547095025,\na[6,4]=40187634332 27500/15832894287040767,\na[6,5]=28/87,\na[7,1]=9308989791751701487421 63494117/783561862284207348259733379696,\na[7,2]=-4034678053/110765160 0,\na[7,3]=90799375237487744698230149845605767/30002662701192241345485 814174532400,\na[7,4]=397667103608613164068909964375/15665400669015897 62118166511052,\na[7,5]=30/1003,\na[7,6]=8695756950351830559524295/602 28935845641756095491952,\na[8,1]=70352057/870914352,\na[8,2]=0,\na[8,3 ]=186846453120293/467197050726588,\na[8,4]=3348317773437500/1186604274 7794963,\na[8,5]=1671793689/16419376288,\na[8,6]=4684298121/7864382556 8,\na[8,7]=44/581,\n\nb[1]=70352057/870914352,\nb[2]=0,\nb[3]=18684645 3120293/467197050726588,\nb[4]=3348317773437500/11866042747794963,\nb[ 5]=1671793689/16419376288,\nb[6]=4684298121/78643825568,\nb[7]=44/581, \n\n`b^`[1]=49834625381/616859728443,\n`b^`[2]=0,\n`b^`[3]=37965116260 159111/94936400477927190,\n`b^`[4]=421475563121093750/1492642808170910 487,\n`b^`[5]=83111238435887/822384252596970,\n`b^`[6]=8/133,\n`b^`[7] =44/581,\n\n`b*`[1]=259574608264943289575/3212741792256328977546,\n`b* `[2]=0,\n`b*`[3]=635/1588,\n`b*`[4]=3392901624133516714843750/12011711 392657264881367779,\n`b*`[5]=31/307,\n`b*`[6]=955366029466507585764/15 887360858912285512715,\n`b*`[7]=2086094673191355153536794882/276259857 72827856064347644535,\n`b*`[8]=321531195708483055925831/13313728083290 53304305910580\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "subs(ee,matrix([[c[2],a[2,1 ],``$3],\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],a[4,1],a[4,2], a[4,3],``],\n [c[5],a[5,1],a[5,2],a[5,3],a[5,4]],\n [c[6],a[ 6,1],a[6,2],a[6,3],a[6,4]],[``$4,a[6,5]],\n [c[7],a[7,1],a[7,2],a [7,3],a[7,4]],[``$3,a[7,5],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3], a[8,4]],[``$2,a[8,5],a[8,6],a[8,7]],\n [`b`,b[1],b[2],b[3],b[4]], [``$2,b[5],b[6],b[7]],\n [`b^`,`b^`[1],`b^`[2],`b^`[3],`b^`[4]],[ ``$2,`b^`[5],`b^`[6],`b^`[7]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3],`b *`[4]],[``,`b*`[5],`b*`[6],`b*`[7],`b*`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#727'#\"\"*\"#`F(%!GF+F+7'#\"#$)\"$6$#\"&<&* *\"(y4u\"#\"'<^OF2F+F+7'#\"$n\"\"$]##\",()*GsjI\",+]i](Q#!+^Fk!p(\"+]( oz\"H#\"-t&*y*\\q#\"-+]iSw5F+7'#\"#M\"#V#!0yc5q`#=O\"0`/m!4GuF#\"-44#H +&\\\"-lL1v+5#!2\"yyQKCv$))*\"2&o.unLHyG#\"/+D,P`G\")\"06dY`&[&R\"7'# \"#A\"#F#!4=WI23n1_c(\"4(*)*RG3xTFo$#\"-jt&*o&)*)\"-D%oW(Q7#!6*f&pMR=H Z'*G(\"6D]4Zv.$Qap9#\"1+vALMw=S\"2n2/(G%*G$e\"7'F+F+F+F+#\"#G\"#()7'\" \"\"#\"?,Fm-+$#\"?vV'*4*oS;8'3O5nwR\"@_5^ m\"=@w*e,p1Slc\"7'F+F+F+#\"#I\"%.5#\":&HC&f0$=N]pv&p)\";_>\\&4cn%#\"1+vVtxJ[L\"2j \\zZF/m=\"7'F+F+#\"+*o$zr;\",)Gw$>k\"#\"+@\")H%o%\",ob#Qky#\"#W\"$\"e7 '%\"bGFfpFipFjpF]qF`q7'%#b^G#\",\"QDY$)\\\"-V%G(fohFip#\"26\"f,E;^'z$ \"2!>FzZ+k$\\*#\"3]P4@Jcv9U\"4([54<3Gk#\\\"7'F+F+#\"/()eVQ76$)\"0qpf_U QA)#\"\")\"$L\"Fgq7'%#b*G#\"6v&*GV\\E3Ydf#\"7Yv(*GjD#zTF@$Fip#\"$N'\"% )e\"#\":]P%[r;N8C;!HR$\";zxO\")[EdER6<,77'F+#\"#J\"$2$#\"6kde2lYHgOb* \"8:F^&G7*e3O()e\"#\"=#)[zON:b8>tY4'3#\">NXkZV1cy#Gx&)fiF#\"9Je#f0$[3d >J:K\"=!e5fI/L0H$3GPJ8Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "RK5_7eqs := [op(RowSumConditions(7,'expanded')),op(O rderConditions(5,7,'expanded'))]:\nsubs(ee,RK5_7eqs):\nmap(u->lhs(u)-r hs(u),%);\nnops(%);\n`RK4_7eqs^` := subs(b=`b^`,OrderConditions(4,7,'e xpanded')):\nsubs(ee,`RK4_7eqs^`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%); \n`RK4_8eqs*` := subs(b=`b*`,OrderConditions(4,8,'expanded')):\nsubs(e e,`RK4_8eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F $F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the prin cipal error norm, that is, the 2-norm of the principal error terms." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_7 := PrincipalEr rorTerms(5,7,'expanded'):\nsqrt(add(subs(ee,errterms5_7[i])^2,i=1.. no ps(errterms5_7))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+- %>#[R!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also calculate the 2 norm of the principal error of the ord er 4 embedded schemes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "` errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\nsq rt(add(subs(ee,`errterms4_7^`[i])^2,i=1.. nops(`errterms4_7^`))):\neva lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+fODJD!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errte rms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nsqrt(ad d(subs(ee,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\nevalf(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gc1YD!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-orde r condtions to check for stage-orders from 2 to 4 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so ||ct||_7 := StageOrderConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 h ave the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j =1..5)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) \+ then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "In general, the simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j] ,i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " \+ . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 7 " ) are " }{TEXT 260 13 "not satisfied" }{TEXT -1 15 ", except when \+ " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 ", where the cond ition reduces to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 "." }}{PARA 256 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "[Sum(b[i ]*a[i,1],i=1+1..7)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j= 2..6)];\neval(subs(Sum=add,%)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),subs(e e,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG \"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\" \"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F 4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FO F-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F -&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF -" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7);\neval(subs (Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6 $*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6# \"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F- F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "T he simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = \+ 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3 \"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Sum(b[i]*Sum(a[i,j ]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F$6$*&& %\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F*;F5\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"%\"\"\"&%\"aG6$F(\"\"$ F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F+6$F4F(F)&F+6 $F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7);\neval(subs(Sum=add,% ));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6 #%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F) F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F) *(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "None of the principal error condit ions are satisfied by the order 5 scheme. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 115 "RK5_7err_eqs := PrincipalErrorConditions(5,7,'expa nded'):\nsubs(ee,RK5_7err_eqs):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions are satisfied by the 8 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 96 "The \+ largest magnitude of an error term is about 283 times the lowest magni tude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "` errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nsu bs(ee,`errterms4_8*`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(%))/min (op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")9\\%Q#!#7$\")FF!y'! #8$\")!Ri?&F)$\")HEq5F)$\")$>5`#!#9$\").^z%)!#:$\")&Q?1&F0$\")44gAF)$ \")4tO%)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%EG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 pri ncipal error conditions are satisfied by the 7 stage embedded order 4 \+ embedded." }}{PARA 0 "" 0 "" {TEXT -1 96 "The largest magnitude of an \+ error term is about 638 times the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_7^` := subs(b= `b^`,PrincipalErrorTerms(4,7,'expanded')):\nsubs(ee,`errterms4_7^`):\n map(abs,evalf[8](%));\nevalf[4](max(op(%))/min(op(%)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7+$\")v?oC!#7$\")!)o*z%!#8$\").h&Q#F)$\")7-?S!#9 $\")twf6F.$\")L\\&)Q!#:$\")X`>BF.$\")g*)*f\"F)$\")3*e'QF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%%Q'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 88 "simultaneous construction of the 7 stage, order 5 scheme and the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following \"simple\" order conditions (given in abreviated form) are relevant for the cons truction of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := SimpleOrderCondition s(5):\n[seq([i,SO5[i]],i=[1,2,4,5,8,12,13,15,16])]:\nlinalg[augment](l inalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7% \"\"\"%%~~~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/ *&F,F()F2F/F(#F(\"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@6#*&FCF(F2F(F(F (#F(\"#C7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6#*&FCF(-F@6#*&F 8F(FCF(F(F(#F(\"#g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF)/*&F,F(-F@6#*&F MF(FCF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b*`,SimpleOrderC onditions(4)):\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nlinalg[augment](l inalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7% \"\"\"%%~~~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/ *&F,F()F2F/F(#F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF(F(#F(\"#77%\" \")F)/*&F,F()F2F:F(#F(F5Q)pprint386\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We incorporate the simplifying conditions:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0; " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\" \"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 . . 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum( b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+ F,F,!\"\"F,/F+;F6\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 scheme a nd use the simple order conditions in the first table above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We also need t o specify that the linking coefficients for the 8th (FSAL) stage are g iven by the weights of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6#%\"iG&% \"aG6$\"\")F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\" " }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"( \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the order 4 \+ embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7,'expanded' ):\n`SO4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded')): \n cdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[1, 2,4,7,8])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(StageO rderConditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3.. 7)=0,add(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i ]*c[i]*a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [se q(b[i]=a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_eqns), op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 9/53;" "6#/&%\"cG6#\"\"#*&\"\"*\"\"\"\"#`!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 167/250;" "6#/&%\"cG6#\" \"%*&\"$n\"\"\"\"\"$]#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = \+ 34/43;" "6#/&%\"cG6#\"\"&*&\"#M\"\"\"\"#V!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 22/27;" "6#/&%\"cG6#\"\"'*&\"#A\"\"\"\"#F!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]; " "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"a G6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\" \"(\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " \+ are initially parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "As in the last example, " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 2 " " }{TEXT 260 26 "is a parameter rathe r than" }{TEXT -1 2 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "e1 := \{c[2]=9/53,c[4]=167/250,c[5]=34/43,c[6]= 22/27,c[7]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs(e1,cdns):\nnops( eqns);\nindets(eqns) minus \{c[3],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5]\} ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#&F,6$F;F'&F,6$F;FE&F,6$F.F'&F,6$F 1F*&F,6$F.FE&F,6$F.F>&F,6$F1F>&F,6$F1F'&F,6$F1FE&F,6$F*F>&F,6$F*F'&F,6 $F'F>&F,6$F*FE&F,6$F>FE&F,6$F'FE&F%6#FE&%#b*GF)&FgoF3&Fgo6#F;&Fgo6#F6& FgoFeo" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := solve( \{op(eqns)\},indets(eqns) minus \{c[3],a[6,5],a[7,5],b[7],`b*`[3],`b*` [5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6310 "e3 := \{b[4] = -195312 500/654871629*(2700*b[7]*c[3]-2575*c[3]-2700*b[7]+932)/(250*c[3]-167), c[2] = 9/53, c[5] = 34/43, `b*`[2] = 0, b[5] = 3418801/15068256*(-225 8*c[3]+4980*b[7]*c[3]+869-4980*b[7])/(43*c[3]-34), `b*`[4] = -42187500 0/13736251*`b*`[3]*c[3]^3+765625000/13736251*`b*`[3]*c[3]^2-343750000/ 13736251*`b*`[3]*c[3]+66406250/41208753-133875000000/1092128108257*`b* `[5], a[4,2] = 1478117/140625000*(375*c[3]-167)/c[3], c[6] = 22/27, c[ 4] = 167/250, a[3,1] = -53/18*c[3]^2+c[3], a[6,1] = 1/273499910505*(-1 82144739210885*c[3]^3+4219843487580*a[6,5]*c[3]^3+204525952614423*c[3] ^2-4190636375955*a[6,5]*c[3]^2+675268420770*a[6,5]*c[3]-66785580291892 *c[3]+6514119516564)/(84*c[3]-17)/c[3]^2, a[7,1] = 1/16048756759986720 *(-575895269331311363280*b[7]*c[3]^4+197884240684542720*b[7]*a[7,5]*c[ 3]^4-62651231369211570120*a[6,5]*c[3]^4+95635797389475207480*c[3]^4+15 6262002780637872720*a[6,5]*c[3]^4*b[7]-311442457922982293940*b[7]*a[6, 5]*c[3]^3-357753620932424640*b[7]*a[7,5]*c[3]^3-187862162743263311072* c[3]^3+87825461609364444990*a[6,5]*c[3]^3+1238727253056922708464*b[7]* c[3]^3+180185836539689019900*a[6,5]*c[3]^2*b[7]+126184002367121929076* c[3]^2-905964123426332290320*b[7]*c[3]^2+191788876886745600*b[7]*a[7,5 ]*c[3]^2-35456205861909911025*a[6,5]*c[3]^2+256263410801261634848*b[7] *c[3]-25005381397344598680*a[6,5]*c[3]*b[7]+4097825220069078030*a[6,5] *c[3]-33445264194370310128*c[3]-25801812933684480*b[7]*a[7,5]*c[3]+294 8156616809560464-23585252934476483712*b[7])/b[7]/(84*c[3]-17)/c[3]^2/( 27*c[3]-22), a[8,3] = -5/4*(2988*b[7]+2627)/c[3]/(498288*c[3]-659887*c [3]^2+290250*c[3]^3-124916), a[3,2] = 53/18*c[3]^2, b[6] = -177147/244 1824*(8964*b[7]*c[3]-3594*c[3]-8964*b[7]+1469)/(27*c[3]-22), c[8] = 1, c[7] = 1, b[2] = 0, a[6,2] = 583/885735*(1149795*c[3]^2-1045602*c[3]+ 213092)/(84*c[3]-17)/c[3], a[2,1] = 9/53, a[5,3] = -17/397535*(4488125 00*c[3]^3-820892445*c[3]^2+466815524*c[3]-77754532)/(250*c[3]-167)/c[3 ]^2/(84*c[3]-17), a[4,1] = -167/140625000*(2756625*c[3]^2-1853867*c[3] +251001)/c[3]^2, a[8,2] = 0, `b*`[6] = 492075/10901*`b*`[3]*c[3]^3+328 7061/109010*`b*`[3]*c[3]-3972167181/4333529035*`b*`[5]-8207811/109010* `b*`[3]*c[3]^2-137781/109010, a[7,2] = 53/180*(5010*b[7]*c[3]^2-835*c[ 3]^2-5676*b[7]*c[3]+793*c[3]+1336*b[7]-167)/c[3]/b[7]/(84*c[3]-17), a[ 8,4] = -195312500/654871629*(2700*b[7]*c[3]-2575*c[3]-2700*b[7]+932)/( 250*c[3]-167), `b*`[1] = 3375/1837*`b*`[3]*c[3]^3-16759/3674*`b*`[3]*c [3]^2+13683/3674*`b*`[3]*c[3]+1097/5511+166194/146054359*`b*`[5]-`b*`[ 3], b[3] = -5/4*(2988*b[7]+2627)/c[3]/(498288*c[3]-659887*c[3]^2+29025 0*c[3]^3-124916), a[8,7] = b[7], a[8,5] = 3418801/15068256*(-2258*c[3] +4980*b[7]*c[3]+869-4980*b[7])/(43*c[3]-34), a[6,3] = -1/181969335*1/c [3]^2*(3897613417500*c[3]^3+370735603560*a[6,5]*c[3]^2-7291523466095*c [3]^2+4241814463656*c[3]-75029824530*a[6,5]*c[3]-723791057396)/(84*c[3 ]-17)/(250*c[3]-167), a[8,6] = -177147/2441824*(8964*b[7]*c[3]-3594*c[ 3]-8964*b[7]+1469)/(27*c[3]-22), b[1] = 1/1498992*(44820*b[7]*c[3]+278 062*c[3]-39405-44820*b[7])/c[3], a[8,1] = 1/1498992*(44820*b[7]*c[3]+2 78062*c[3]-39405-44820*b[7])/c[3], a[7,4] = -3125/12048725719050636*(- 1166338155695314140*c[3]^3-20883743789737190040*a[6,5]*c[3]^3+52087334 260212624240*b[7]*a[6,5]*c[3]^3+425073552140160000*b[7]*a[7,5]*c[3]^3+ 1616837984215118280*b[7]*c[3]^3+29275153869788148330*a[6,5]*c[3]^2-358 0996720594965672*b[7]*c[3]^2-768487687117920000*b[7]*a[7,5]*c[3]^2-103 814152640994097980*a[6,5]*c[3]^2*b[7]+1717900620458670956*c[3]^2+60061 945513229673300*a[6,5]*c[3]*b[7]-11818735287303303675*a[6,5]*c[3]+2491 128153457367424*b[7]*c[3]+411980149996800000*b[7]*a[7,5]*c[3]-75941424 9725765480*c[3]+109187070608729344-526969417077520032*b[7]+13659417400 23026010*a[6,5]-55424667661440000*b[7]*a[7,5]-8335127132448199560*b[7] *a[6,5])/(6750*c[3]^2-10009*c[3]+3674)/b[7]/(84*c[3]-17), a[5,2] = 901 /3577815*(2692875*c[3]^2-2356842*c[3]+465596)/(84*c[3]-17)/c[3], `b*`[ 8] = 1/197972430*(-156992136990*b[7]*c[3]+165306979050*b[7]*c[3]^2+711 82378579*c[3]+33061395810*b[7]+292692935436*`b*`[3]*c[3]-1258199376*`b *`[5]-2187230414370*`b*`[3]*c[3]^2+5787557980434*`b*`[3]*c[3]^3-749524 41505*c[3]^2+2688727972500*`b*`[3]*c[3]^5-6540372236130*`b*`[3]*c[3]^4 +5974563504*c[3]*`b*`[5]-6290996880*c[3]^2*`b*`[5]-14990488301)/(5010* b[7]*c[3]^2-835*c[3]^2-5676*b[7]*c[3]+793*c[3]+1336*b[7]-167), `b*`[7] = -1/98986215*(8066183917500*`b*`[3]*c[3]^5-21099112084890*`b*`[3]*c[ 3]^4+20092012976304*`b*`[3]*c[3]^3-142203834990*c[3]^2-18872990640*c[3 ]^2*`b*`[5]-8163535786608*`b*`[3]*c[3]^2+176252469219*c[3]+11707717417 44*`b*`[3]*c[3]+21381855264*c[3]*`b*`[5]-43431255299-5032797504*`b*`[5 ])*b[7]/(5010*b[7]*c[3]^2-835*c[3]^2-5676*b[7]*c[3]+793*c[3]+1336*b[7] -167), a[4,3] = -27889/15625000*(250*c[3]-167)/c[3]^2, a[5,4] = -28028 7500/13277669*(215*c[3]^2-256*c[3]+68)/(84*c[3]-17)/(250*c[3]-167), a[ 7,3] = -1/1456065755760*(-7386075554312070000*c[3]^5+44316453325872420 000*b[7]*c[3]^5-166817253615750710280*b[7]*c[3]^4+52087334260212624240 *a[6,5]*c[3]^4*b[7]+25827830153802604620*c[3]^4-20883743789737190040*a [6,5]*c[3]^4+29275153869788148330*a[6,5]*c[3]^3-103814152640994097980* b[7]*a[6,5]*c[3]^3+245704928244352025064*b[7]*c[3]^3-36140174210592873 652*c[3]^3-52155536304453120*b[7]*a[7,5]*c[3]^3+60061945513229673300*a [6,5]*c[3]^2*b[7]-11818735287303303675*a[6,5]*c[3]^2-17588413678233885 3792*b[7]*c[3]^2+53052390764714880*b[7]*a[7,5]*c[3]^2+2468079365430117 0232*c[3]^2-8335127132448199560*a[6,5]*c[3]*b[7]+1365941740023026010*a [6,5]*c[3]-8600604311228160*b[7]*a[7,5]*c[3]-8038499257157195288*c[3]+ 60541759806023946912*b[7]*c[3]+982718872269853488-7861750978158827904* b[7])/(6750*c[3]^2-10009*c[3]+3674)/b[7]/(84*c[3]-17)/c[3]^2/(27*c[3]- 22), a[5,1] = -17/597495105*(20862170895*c[3]^3-22752715557*c[3]^2+727 4738912*c[3]-699790788)/(84*c[3]-17)/c[3]^2, a[7,6] = 6561/21163430404 71968*(-55039285567728*c[3]^2-1841599981458306*a[6,5]*c[3]^2+459323935 2752436*a[6,5]*c[3]^2*b[7]-190704767099616*b[7]*a[7,5]*c[3]^2+22088784 75515109*a[6,5]*c[3]+55985728308972*c[3]-8225103027021804*a[6,5]*c[3]* b[7]+306178885334784*b[7]*a[7,5]*c[3]-9076143210904-595181586066678*a[ 6,5]-122865775874688*b[7]*a[7,5]+3631863674269368*b[7]*a[6,5])/(729*c[ 3]^2-1188*c[3]+484)/b[7], a[6,4] = -12500/6077775789*(2721053115*c[3]^ 2+12086149320*a[6,5]*c[3]^2-3305575636*c[3]+886861756-12002496570*a[6, 5]*c[3]+1934051580*a[6,5])/(21000*c[3]^2-18278*c[3]+2839)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "t he coefficients in terms of " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$ " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do\n print(`======================================== ==============`);\n temp_eq := c[ii]=subs(e3,c[ii]);\n if lhs(temp _eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end if;\n print(``); \n for jj \+ to ii-1 do\n temp_eq := a[ii,jj]=subs(e3,a[ii,jj]);\n if l hs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a par ameter`);\n else\n print(temp_eq);\n end if;\n \+ print(``);\n end do:\nend do:\n`############################### #######################`;\n``;\nfor ii to 7 do\n temp_eq := b[ii]=su bs(e3,b[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(t emp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end if ;\n print(``);\nend do:\n`########################################## ############`;\n``;\nfor ii to 8 do\n temp_eq := `b*`[ii]=subs(e3,`b *`[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_e q)*` is a parameter`);\n else\n print(temp_eq);\n end if;\n \+ print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============ ==========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"*\"#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#\"\" *\"#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"$\"\"\"%0~is~a~par ameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",&*&#\"#`\"#=F(*$)&%\"cG6#F'\"\"#F( F(!\"\"F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,$*&#\"#`\"#=\"\"\"*$)&%\"cG6#F'F( F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"$n\"\"$]#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"%\"\"\",$*&#\"$n\"\"*+]iS\"F(*&,(*&\"(Dmv#F()&%\"cG6#\" \"$\"\"#F(F(*&\"(nQ&=F(F3F(!\"\"\"',5DF(F(F3!\"#F(F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" %\"\"#,$*&#\"(<\"y9\"*+]iS\"\"\"\"*&,&*&\"$v$F.&%\"cG6#\"\"$F.F.\"$n\" !\"\"F.F3F8F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,$*&#\"&*)y#\")+]i:\"\"\"*&,&*& \"$]#F.&%\"cG6#F(F.F.\"$n\"!\"\"F.F3!\"#F.F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============== ========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#M\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*&# \"#<\"*0^\\(fF(*(,**&\",&*3UF(&F%6$F'\"\"&F( F1F(F(*&\"0BWh_f_/#F()F2\"\"#F(F(*(\".bfPO1>%F(F9F(F>F(F6*(\"-q2Uo_nF( F9F(F2F(F(*&\"/#*=H!e&ymF(F2F(F6\".kl^>T^'F(F(,&*&\"#%)F(F2F(F(\"#=F,**&% \"cG6#F(!\"#,.*&\".+vT8w*QF,)F/F(F,F,*(\"-gNgN2PF,&F%6$F'\"\"&F,)F/\" \"#F,F,*&\".&4mM_\"H(F,FF.F;F.F. F.,(*&\"&+5#F.F3F.F.*&\"&y#=F.F4F.F@\"%RGF.F@F.F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"' \"\"&\"\"\"%0~is~a~parameterGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================ ======================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",$*&#F(\"2?n)*fnv[g\"F(*,,N*(\"6!Gj 8JJ$p_*edF(&%\"bG6#F'F()&%\"cG6#\"\"$\"\"%F(!\"\"**\"3?Fa%oSU)y>F(F1F( &F%6$F'\"\"&F(F4F(F(*(\"5?,d6#p8B^E'F(&F%6$\"\"'F?F(F4F(F:*&\"5![2_Z*Q (zNc*F(F4F(F(**\"6?F(yj!y-?Ec\"F(FBF(F4F(F1F(F(**\"6SRH#)H#zXU9JF(F1F( FBF()F5F8F(F:**\"3SYUK4i`xNF(F1F(F=F(FKF(F:*&\"6s5JjKui@'y=F(FKF(F:*( \"5!*\\Wk$4;YDy)F(FBF(FKF(F(*(\"7k%3F#p0`ssQ7F(F1F(FKF(F(**\"6+*>!*oRl $e=!=F(FBF()F5\"\"#F(F1F(F(*&\"6w!H>7nB+%=E\"F(FVF(F(*(\"6?.HKjUBT'f!* F(F1F(FVF(F:**\"3+cu')o())y\">F(F1F(F=F(FVF(F(*(\"5D5\"*4>'e?ca$F(FBF( FVF(F:*(\"6[[jh7!3TjiDF(F1F(F5F(F(**\"5!o)fWtR\"Q0]#F(FBF(F5F(F1F(F:*( \"4I!y!p+ADy4%F(FBF(F5F(F(*&\"5G,JqV>k_WLF(F5F(F:**\"2![%oLH\"=!e#F(F1 F(F=F(F5F(F:\"4k/c4ohc\"[HF(*&\"57P[wW$HD&eBF(F1F(F:F(F1F:,&*&\"#%)F(F 5F(F(\"#P(*yVP)3#F,FCF,F?F,F7*(\"5I$[\")ypQ:v#HF,FCF,)F3F(F,F,**\"6!)z4%*4 k_T\"Q5F,F:F,FCF,FLF,F7*(\"6k]-_VCG\\qX#F,F:F,FLF,F,*&\"5_O(Gf5U1gF,FCF,)F3\" \"#F,F:F,F,*(\"5vOI.tGN(==\"F,FCF,FYF,F7*(\"6#z`)QByOT)e[C8F^L) F,FCF,F3F,F:F,F7*(\"45g-B+uTfO\"F,FCF,F3F,F,**\"1g\"G7J/1g)F,F:F,FUF,F 3F,F7*&\"4)G&>drD*\\Q!)F,F3F,F7*(\"57p%R-1)fds[?\"\"\"\"**,J*&\"4STJ&pb \"Qj;\"F.)&%\"cG6#\"\"$F7F.!\"\"*(\"5S+>P(*yVP)3#F.&F%6$\"\"'\"\"&F.F3 F.F8**\"5SUi7-EMt3_F.&%\"bG6#F'F.F;F.F3F.F.**\"3++;S@bt]UF.FAF.&F%6$F' F>F.F3F.F.*(\"4!G=^@%)z$oh\"F.FAF.F3F.F.*(\"5I$[\")ypQ:v#HF.F;F.)F4\" \"#F.F.*(\"4sc'\\f?n*4e$F.FAF.FLF.F8**\"3++#z6(o([o(F.FAF.FFF.FLF.F8** \"6!)z4%*4k_T\"Q5F.F;F.FLF.FAF.F8*&\"4c4ne/i+zr\"F.FLF.F.**\"5+LnHK^X> 1gF.F;F.F4F.FAF.F.*(\"5vOI.tGN(==\"F.F;F.F4F.F8*(\"4CuOdM:G6\\#F.FAF.F 4F.F.**\"3++!o**\\,)>TF.FAF.FFF.F4F.F.*&\"3![lds\\UTf(F.F4F.F8\"3W$H(3 12(=4\"F.*&\"3K+_xqTpp_F.FAF.F8*&\"45g-B+uTfO\"F.F;F.F.*(\"2++WhwmCa&F .FAF.FFF.F8*(\"4g&*>[C8F^L)F.FAF.F;F.F8F.,(*&\"%]nF.FLF.F.*&\"&4+\"F.F 4F.F8\"%uOF.F8FAF8,&*&\"#%)F.F4F.F.\"# ZSIM;@\"\"\"*(,:*&\"/Gxc&GR]&F.)&%\"cG6#\"\"$\"\"#F.!\"\"*(\"11$e9)**f T=F.&F%6$F(\"\"&F.F3F.F9**\"1OCv_$RKf%F.FF.FAF.&F%6$F'F>F.F3F.F9*(\"14^^v%y)3AF.F\"*H;([l\"\"\"*&,**(\"%+FF.&%\"bG6#\"\"(F.&%\"cG6#\"\" $F.F.*&\"%vDF.F7F.!\"\"*&F2F.F3F.F=\"$K*F.F.,&*&\"$]#F.F7F.F.\"$n\"F=F =F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,$*&#\"(,)=M\")c#o]\"\"\"\"*&,**&\"% eAF.&%\"cG6#\"\"$F.!\"\"*(\"%!)\\F.&%\"bG6#\"\"(F.F3F.F.\"$p)F.*&F9F.F :F.F7F.,&*&\"#VF.F3F.F.\"#MF7F7F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',$*&#\"'Zr< \"(C=W#\"\"\"*&,**(\"%k*)F.&%\"bG6#\"\"(F.&%\"cG6#\"\"$F.F.*&\"%%f$F.F 7F.!\"\"*&F2F.F3F.F=\"%p9F.F.,&*&\"#FF.F7F.F.\"#AF=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"(&%\"bG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W########################################## ############G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"(#**)\\\"F'*&,**(\"&?[%F'&F% 6#\"\"(F'&%\"cG6#\"\"$F'F'*&\"'i!y#F'F3F'F'\"&0%R!\"\"*&F/F'F0F'F:F'F3 F:F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"\"&\"\"%\" \"\"*(,&*&\"%))HF-&F%6#\"\"(F-F-\"%FEF-F-&%\"cGF&!\"\",**&\"')G)\\F-F6 F-F-*&\"'())f'F-)F6\"\"#F-F8*&\"']-HF-)F6F'F-F-\"';\\7F8F8F-F8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"%,$*&#\"*+DJ&>\"*H;([l\"\"\"*&,**(\"%+FF-&F%6#\"\"(F-&% \"cG6#\"\"$F-F-*&\"%vDF-F5F-!\"\"*&F1F-F2F-F;\"$K*F-F-,&*&\"$]#F-F5F-F -\"$n\"F;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"(,)=M\")c#o]\"\"\"\"*&,**&\"% eAF-&%\"cG6#\"\"$F-!\"\"*(\"%!)\\F-&F%6#\"\"(F-F2F-F-\"$p)F-*&F8F-F9F- F6F-,&*&\"#VF-F2F-F-\"#MF6F6F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\"'Zr<\"(C=W# \"\"\"*&,**(\"%k*)F-&F%6#\"\"(F-&%\"cG6#\"\"$F-F-*&\"%%f$F-F5F-!\"\"*& F1F-F2F-F;\"%p9F-F-,&*&\"#FF-F5F-F-\"#AF;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"bG6#\"\"( \"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%W#################################### ##################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\",.*&#\"%vL\"%P=F'*&&F%6#\"\"$ F')&%\"cGF/F0F'F'F'*&#\"&fn\"\"%uOF'*&F.F')F2\"\"#F'F'!\"\"*&#\"&$o8F7 F'*&F.F'F2F'F'F'#\"%(4\"\"%6bF'*&#\"'%>m\"\"*fV0Y\"F'&F%6#\"\"&F'F'F.F ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#b*G6#\"\"$\"\"\"%0~is~a~parameter GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%,,*&#\"*+](=U\")^it8\"\"\"*&&F%6#\"\"$F-) &%\"cGF0F1F-F-!\"\"*&#\"*+]il(F,F-*&F/F-)F3\"\"#F-F-F-*&#\"*++vV$F,F-* &F/F-F3F-F-F5#\")]iSm\")`(37%F-*&#\"-+++vQ8\".d#3\"G@4\"F-&F%6#\"\"&F- F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#b*G6#\"\"&\"\"\"%0~is~a~parameterGF(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"',,*&#\"'v?\\\"&,4\"\"\"\"*&&F%6#\"\"$F-)&%\"cGF0F1F-F-F-*&#\"(hqG $\"'5!4\"F-*&F/F-F3F-F-F-*&#\"+\"=n@(R\"+N!HNL%F-&F%6#\"\"&F-!\"\"*&# \"(6y?)F8F-*&F/F-)F3\"\"#F-F-FA#\"'\"yP\"F8FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(,$ *&#\"\"\"\"):i)*)*F+*(,8*(\".+v\"R=m!)F+&F%6#\"\"$F+)&%\"cGF2\"\"&F+F+ *(\"/!*[37\"*4@F+F1F+)F5\"\"%F+!\"\"*(\"//j(H,#4?F+F1F+)F5F3F+F+*&\"-! *\\$Q?U\"F+)F5\"\"#F+F<*(\",S1*H()=F+FBF+&F%6#F7F+F<*(\".3myNN;)F+F1F+ FBF+F<*&\"->#pCDw\"F+F5F+F+*(\".WF+*&,>*(\"-!*p8#*p:F+& %\"bG6#\"\"(F+&%\"cG6#\"\"$F+!\"\"*(\"-]!zpIl\"F+F1F+)F5\"\"#F+F+*&\", z&yB=rF+F5F+F+*&\",5eRhI$F+F1F+F+*(\"-Oa$Hp#HF+&F%F7F+F5F+F+*&\"+w$*>e 7F+&F%6#\"\"&F+F9*(\".qVTIs=#F+FDF+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "RK5_7eqs := [op(RowSumConditions(7,'expanded')),op(OrderConditions(5,7,'expanded' ))]:\nsimplify(subs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%); \n`RK4_8eqs*` := subs(b=`b*`,OrderConditions(4,8,'expanded')):\nsimpli fy(subs(e3,`RK4_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The sta bility function of a general 7 stage, order 5 Runge-Kutta scheme has t he form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1 +z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7;" "6#/-%\"RG6#%\"zG,2 \"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F' \"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[ 6]= b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]" "6#/&%\"tG6#\"\"',&*.& %\"bG6#F'\"\"\"&%\"aG6$F'\"\"&F-&F/6$F1\"\"%F-&F/6$F4\"\"$F-&F/6$F7\" \"#F-&%\"cG6#F:F-F-&F+6#\"\"(F-" }{TEXT -1 2 " (" }{XPPEDIT 18 0 " a[7 ,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*a[6,4]*a[4,3]*a[3,2]*c[2]+``" "6# ,(*,&%\"aG6$\"\"(\"\"&\"\"\"&F&6$F)\"\"%F*&F&6$F-\"\"$F*&F&6$F0\"\"#F* &%\"cG6#F3F*F**,&F&6$F(\"\"'F*&F&6$F:F-F*&F&6$F-F0F*&F&6$F0F3F*&F56#F3 F*F*%!GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[7,6]*a[6,5]*a[5,3]*a[3,2] *c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,3]*c[3 ]" "6#,(*,&%\"aG6$\"\"(\"\"'\"\"\"&F&6$F)\"\"&F*&F&6$F-\"\"$F*&F&6$F0 \"\"#F*&%\"cG6#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-\"\"%F*&F&6$F>F3F* &F56#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-F>F*&F&6$F>F0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[ 7]=b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"( *2&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"'F,&F.6$F0\"\"&F,&F.6$F3\"\"%F,&F.6$F6 \"\"$F,&F.6$F9\"\"#F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 250 "eq7 := b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]= 1/5450;\neq6 := b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]*(a[7,5]*a[5 ,4]*a[4,3]*a[3,2]*c[2]+\n a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a [5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))))=1/718;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6#\"\"(\"\"\"&%\"aG6$F*\"\"' F+&F-6$F/\"\"&F+&F-6$F2\"\"%F+&F-6$F5\"\"$F+&F-6$F8\"\"#F+&%\"cG6#F;F+ #F+\"%]a" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/,&*.&%\"bG6#\"\"' \"\"\"&%\"aG6$F+\"\"&F,&F.6$F0\"\"%F,&F.6$F3\"\"$F,&F.6$F6\"\"#F,&%\"c G6#F9F,F,*&&F)6#\"\"(F,,&*,&F.6$F@F0F,F1F,F4F,F7F,F:F,F,*&&F.6$F@F+F,, &**&F.6$F+F3F,F4F,F7F,F:F,F,*&F-F,,&*(&F.6$F0F6F,F7F,F:F,F,*&F1F,,&*&& F.6$F3F9F,F:F,F,*&F4F,&F;6#F6F,F,F,F,F,F,F,F,F,F,#F,\"$=(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We shall require \+ the following two equations to be satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve(simplify(subs(e3,\{eq6\})),a[7,5 ]));\neqn7 := op(solve(simplify(subs(e3,\{eq7\})),a[7,5]));\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&%\"aG6$\"\"(\"\"&,$*&#\"\"\" \"9!ov3f@&\\\"3/8I\"F.*,,J*(\"A?6nd)=!Hdop0XPdW6F.&%\"bG6#F)F.)&F'6$\" \"'F*\"\"#F.F.*&\">g`1jguT_F5JE.'GF.F8F.!\"\"*&\"?EFw\"f34.(HbaWf:TF.) &%\"cG6#\"\"$FEF.F>*(\"??z$fYJ'=N(p9]iE%HF.F8F.FBF.F.*(\"@!Q]vXgA,>t#H .UN+\"F.F8F.)FBF;F.F>*&\">_8!)4$)G'G#3Vn)RZ$)F.FBF.F>*&\"?SNOs>&3+tvt+ `!zHF.FJF.F.*&\"@?SJO,2De$[o2:qv=F.F7F.F>*(\"Agw#)yw9C!42qf+l`4#F.F7F. )FB\"\"%F.F>*(\"A+9[(zF^*p-mHM$eS!\\F.F7F.FAF.F.*(\"AS,1Y'[4?RQkG(3!)o 9F.F7F.FBF.F.*(\"@+I;33`#=[)e]QN:N\"F.F8F.FAF.F.*(\"?!3#37=N()zFcPwMii F.F8F.FSF.F>*&\"?SkT[$4\"zITH(z_/$>F.FSF.F.**\"AS)*z`$f1$\\A)f'*o(32tF .F7F.FBF.F4F.F>**\"BgE!fBL7SlYD#[u(yD;F.F7F.FJF.F4F.F.**\"Ag>lA9Ac;rnx Y\"phA&F.F7F.FSF.F4F.F.**\"B+fUiT\"e&y$Q*R]E`@`\"F.F7F.FAF.F4F.F>*(\"A &)z9U]K1L^d)Hk&=DTF.F7F.FJF.F>\"=[!GvkJ8k-Sw[&zxF.F.F8F>F4F>,**&\"'mvh F.FAF.F.*&\"(\\H>\"F.FJF.F>*&\"'7#G(F.FBF.F.\"'__8F>F>,&*&\"#FF.FBF.F. \"#AF>F>F.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/&%\"aG6$\"\"( \"\"&,$*&#\"\"\"\"9/>N>x()R1'=))R%F.*.,H*&\"?OH_:X_BG)*)*p)RQ3\"F.)&F' 6$\"\"'F*\"\"#F.!\"\"*(\">K\"[@(=y%)>!>jbu'*pF.F5F.)&%\"cG6#\"\"$F@F.F .*(\"??G/Q;@rU6aW(eF5)F.F4F.F=F.F.*(\"?;#\\PcT$)y)[c6/r8mF.&%\"bG6#F)F .F4F.F.*(\"@Q$p2T.FG9$RPRnB>#F.F4F.)F=F8F.F9*(\">!o$43\"f+#p4=HRkU!\\k,67yZ _$>IO&F.F5F.FJF.F9*(\"@RW9C$*)4Jrba:bKMDF.F4F.F C'G.1\"F.F4F.FOF.F9*(\">!oL'))z\"p=PBg)o*oJF.F5F.FOF.F9**\"@%GdJhU+3Ca t3cn%)yF.F4F.F$)ojh\\Nty_;F.F5F.F9**\"@?z[G%f?\"=r%Q O$yw)RF.F4F.F=F.FEF.F9**\"@GBAg*H\")\\pNm&)F.F4F.FJF.FEF.F.*&\"<+o :#[8kby)y4>:&F.F1D`$Q%*F.FJF.F.*&\"<+wX%4i`2D%3.'>^ F.F=F.F9\";+k%= " 0 "" {MPLTEXT 1 0 38 "factor(simplify(rhs(eqn6)-rhs(eqn7)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"/\\rfy\"4*o\"8gLBk(\\iru7_:\" \"\"*2,&*&\"#FF(&%\"cG6#\"\"$F(F(\"#A!\"\"F(,&*&\"#%)F(F-F(F(\"# " 0 "" {MPLTEXT 1 0 145 "denom_factors := select(has,map(op,select(has,map(de nom@rhs,e3),c[3])),c[3]):\nmember(27*c[3]-22,denom_factors);\nmember(8 4*c[3]-17,denom_factors);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "However, the quadratic factor h as two real zeros." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "272831 11475*c[3]^2-17657167180*c[3]+2769081836;\nsolve(%);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\",v96$GF\"\"\")&%\"cG6#\"\"$\"\"# F&F&*&\",!=nrlZR##F'\"\"# F',&F$F'*(F)F'F*F+F,F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+8;6.Q! #5$\"+V)=(oEF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 96 "10573154/32674385-4/1091324459*sqrt(2394719790 63162);\nevalf(%);\nconvert(%,rational,5);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\")aJd5\")&QuE$\"\"\"*(\"\"%F'\"+fWK\"4\"!\"\" \"0iJ1z>ZR##F'\"\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V)=(oE!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#$)\"$6$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*G5)oE!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[3] = 83/311;" "6#/&%\"cG6#\"\"$*&\"#$)\"\"\"\"$6$!\"\"" }{TEXT -1 56 ", which is a r ational approximation for the lower root. " }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting for " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$" } {TEXT -1 32 " in each of the equations for " }{XPPEDIT 18 0 "a[7,5] " "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 29 " now gives an equation for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 15 " in term s of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 80 "Since these two equations are only approx imately the same we pick the first one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(subs(c[3]=83/311 ,eqn6));\nevalf(%);\nexpand(subs(c[3]=83/311,eqn7));\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,**&#\",RjAGE#\"*9q <-(\"\"\"&F%6$\"\"'F(F.F.*&#\")fQzE\"*_4&RvF.*&F.F.&%\"bG6#F'!\"\"F.F. *&#\"+!e%pOn/%>TKv+tF.*&F.F.*&F0F.F4F.F3F.F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,**($\"+gE2+D!\"*\"\"\"&%\"bG6#F'!\" \"&F%6$\"\"'F(F.F2*&$\"+xLz`N!#6F.F/F2F.*&$\"+q9eAK!\")F.F3F.F.*($\"+Z A&*RX!#7F.F/F2F3F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "cc := 83/311:\ndd := simplify(subs(c[3]= cc,rhs(eqn6))):\ne4 := \{c[3]=cc,a[7,5]=dd\}:\ne5 := `union`(e4,simpli fy(subs(e4,e3))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3255 "e5 := \{a[7,3] = 311/506637692378289123137537214986225735440 08798000*(387860298386477201988761709696604481520970680*a[6,5]-4589922 757751295364627079811837636463154174880*b[7]*a[6,5]+475769831753638718 6038585021129500011507022000*b[7]*a[6,5]^2-369101343059184566265412490 701221128825496375*a[6,5]^2-29494402458926539687647220803693556311464) /b[7]/a[6,5], c[2] = 9/53, c[5] = 34/43, `b*`[2] = 0, c[6] = 22/27, c[ 4] = 167/250, `b*`[8] = 1435013/5955056426031330*(5955056426031330*b[7 ]-2700103897586093+5681467713692082*`b*`[3]-226628310623568*`b*`[5])/( 17218558*b[7]-1435013), a[5,2] = 495002920909/100075063365, a[8,3] = 2 8064855523/67010477729*b[7]+24575458486307/66742435818084, a[3,2] = 36 5117/1740978, c[8] = 1, c[7] = 1, b[2] = 0, a[2,1] = 9/53, a[8,2] = 0, a[7,4] = -625/2312567613416808121956616952195892290248572*(8335020259 83893749074405079880133925944*a[6,5]-142990486599459406603421721884794 92070875*a[6,5]^2+184314040116855351413573975172568086294000*b[7]*a[6, 5]^2-4264413593253329012107626148877282000-103225678801712554642387449 19666680206304*b[7]*a[6,5])/b[7]/a[6,5], a[6,4] = 2883941307500/383856 3729351-14885625000/9630015421*a[6,5], `b*`[7] = -2/2977528213015665*( 24456670338333627014439*`b*`[3]-9486578044880926339151-975553177728480 443736*`b*`[5])*b[7]/(17218558*b[7]-1435013), c[3] = 83/311, a[8,7] = \+ b[7], a[7,5] = 1/16971671319245245753214864441438412000*(5469259350648 52674817726929295454262000*b[7]*a[6,5]^2+60313813135446654510224409736 9366500*a[6,5]-77032410533829635096626652429888774-4243041145384552124 7306148927238848875*a[6,5]^2)/b[7]/a[6,5], a[4,3] = 270499789573/10764 0625000, a[5,4] = 81285337012500/139548553465711, b[6] = -22628226339/ 702177014*b[7]+4012556697/1604976032, a[6,2] = 898568957363/1238744684 25, a[5,3] = -98837524323878781/28782933677403685, a[7,1] = -1/1316773 704358516684233529882812447556322400*(12990893741929983519988809851255 484152595600*b[7]*a[6,5]+15298065329698994167326639939323151162402000* b[7]*a[6,5]^2-1092993808352353981855435364343431378372100*a[6,5]-63774 692961066558405979740658836922972-118682103877551307480840029164379784 1882625*a[6,5]^2)/b[7]/a[6,5], b[4] = -40078125000000/6807827164541*b[ 7]+14868554687500/20423481493623, a[7,6] = 27510782285263/781780538223 2400*1/(b[7]*a[6,5]), `b*`[1] = -16358018118/55257384347*`b*`[3]+1097/ 5511+166194/146054359*`b*`[5], a[5,1] = -361825370105678/2774280906604 53, b[5] = -899144663/335089312+5391449177/146601574*b[7], a[4,1] = 30 637228987/38750625000, a[8,4] = -40078125000000/6807827164541*b[7]+148 68554687500/20423481493623, `b*`[6] = 5808283908102/1639522990655*`b*` [3]-3972167181/4333529035*`b*`[5]-137781/109010, b[3] = 28064855523/67 010477729*b[7]+24575458486307/66742435818084, a[8,5] = -899144663/3350 89312+5391449177/146601574*b[7], a[7,2] = 53/25173900*(17218558*b[7]-1 435013)/b[7], `b*`[4] = -16391062500000/4978187989807*`b*`[3]+66406250 /41208753-133875000000/1092128108257*`b*`[5], a[6,3] = -18712358234446 8847/35627962542438825+4337549966/4786175329*a[6,5], a[8,6] = -2262822 6339/702177014*b[7]+4012556697/1604976032, b[1] = -2565/31229*b[7]+108 24191/124416336, a[8,1] = -2565/31229*b[7]+10824191/124416336, a[4,2] \+ = -7690642751/2917968750, a[3,1] = 99517/1740978, a[6,1] = -1331177716 63118/68680960273557-9239595/25628989*a[6,5]\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We choose values for " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 54 " that minimize the norm of the principal error terms." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "errterms5_7 := PrincipalEr rorTerms(5,7,'expanded'):\nsm := 0:\nfor ct to nops(errterms5_7) do\n \+ sm := sm+simplify(subs(e5,errterms5_7[ct]))^2;\nend do:\nsm := simpl ify(sm):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1101 " sm := 1/95440983847194118618916864650690959900240751353162132630562694 32550267008751554260000000000*(-42836864653886887673765833300252705378 3496987326722435381846934780770118042443856010400*b[7]*a[6,5]^2-123176 3919042772151272037015571083986120070210294259488247911162159831712589 6107000*a[6,5]+1976707195791378643407707941485540381401263623362970719 615543315422051146597668060+183808540764628228616950797389227873608384 79766522150700336541612153395881589966673059*a[6,5]^2+7193333337404194 4427770645977772690637051722409454382347030533258524310965463549429000 00*a[6,5]^3*b[7]-93853091510879485548435200194731857978658168301616394 1986992075459412209249886875175000000*a[6,5]^4*b[7]+604880887016036521 6546966538102745810084629978793026278630014782971874461446813437400000 000*a[6,5]^4*b[7]^2+28277859660156109533154109928868035180108505693245 68134663601877584988449478371470560000*b[7]^2*a[6,5]^2-558057451041396 0218375465954606782152654971714379867668431092920351894504798435460562 50*a[6,5]^3+3640552617558567258826544001854420412038530141863602219541 7024459587380139963928146093750*a[6,5]^4)/a[6,5]^2:" }{TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indets(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\" aG6$\"\"'\"\"&&%\"bG6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "plot3d(sm,b[7]=0..0.15,a[6, 5]=0.01..0.5,labels=[`b[7]`,`a[6,5]`,``],grid=[40,60],style=patch,line style=2,\n lightmodel=light4,color=COLOR(RGB,.8,.3,.3),axes=boxed,vie w=[0..0.15,0.01..0.5,0..3e-6],orientation=[-57,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 453 447 447 {PLOTDATA 3 "6*-%%GRIDG6&;$\"\"!F($\"#:!\"#;$ \"\"\"F+$\"\"&!\"\"X,%)anythingG6\"6\"[gl'!%\"!!#[at\"I\"gn3ECEBDDB626 753AA3EC67E0A58B007663ECBCED307E5BBCE3ED3938E9396F69A3EDBB435BF37F87B3 EE311BF220D137F3EE969FE86863AB23EF06FFFB44121733EF4B931408514683EF9903 73601C96B3EFEF4DD1605ECF73F027381B844B29B3F05B34B48E5F45F3F0939C4D3C03 2943F0D06EA0A5EDCF63F108D5BF9E8942D3F12BA963B35D8EB3F150B230736B3183F1 77F01CF8FE9E23F1A16322A0CD6403F1CD0B3C64EDF913F1FAE8666BCBE4F3F2157D4E DCA3AD53F22EA0EFFB5FF4E3F248DF15A5637E03F26437BF1DD8FD43F280AAEBCCEBB5 C3F29E389B37A3D983F2BCE0CCF9BF39B3F2DCA380C0FE7D93F2FD80B649865C03F30F BC36AD87B873F3214552E35915E3F3335BAFB294DFA3F345FF4D0AB94303F359302ADD D7A063F36CEE492022CCE3F38139A7C7930A33F3961246CB9B2773F3AB782624EB4B23 F3C16B45CD3EA7E3F3D7EBA5BF31F953F3EEF945F6211E63F4034A1337053B43F40F5E 2391BB7AA3F41BB8D409B309E3F4285A249D9E7753F43542154C5C0313F44270A614EF 9623F44FE5D6F67DA0E3F45DA1A7F046BE63F46BA41901A3F763F479ED2A2A0390E3F4 887CDB68E64DE3F497532CBDDD1153F4A6701E2886D503F4B5D3AFA88EE673F4C57DE1 3DAB6013F4D56EB2E79BD5F3F4E5A624A6282DC3ECD4399494468563EC47D48A3985C5 83EC8DA4D7AC451913ED168C7C2F6E22B3ED8A248C67E04FE3EE0F9E4829DEA5B3EE6A 7EEB4FFA24F3EED586955DEFB643EF284FA8F2279293EF6DDEC550B96C03EFBB6D57DF 982F03F0087CB4CF211F23F03740DFA85981A3F06A02C5E90CABD3F0A0C222A9F19943 F0DB7EC65BFB9733F10D1C47B23A3BC3F12E77B2C3A582F3F151D19B7C8C1463F17729 FB39A37F33F19E80CCF50239A3F1C7D60CD513CAF3F1F329B7DDB86683F2103DE5DC27 CDD3F227E62345B56A33F2408D936EAFBCC3F25A3435BF4208A3F274DA09BC748003F2 907F0F0204F3F3F2AD23453DB40BB3F2CAC6AC2BA67E13F2E969439394E713F3048585 A34BDB23F314D6018EB50F03F325A6157B843CB3F336F5C15BCAC473F348C50523BB7B 63F35B13E0C94EA313F36DE25443F70AE3F381305F8C64D933F394FE029C5340A3F3A9 4B3D6E5EFCC3F3BE180FFDE3ECC3F3D3647A46E06FC3F3E9307C45DD7953F3FF7C15F7 DA8303F40B23A3AD1E5473F416C908356035C3F422AE3893B6CE73F42ED334C7468ED3 F43B37FCCF501213F447DC90AB2C60C3F454C0F05A49C003F461E51BDC28F2C3F46F49 13305ADBF3F47CECD6567E7583F48AD0654E3F0CE3F498F3C01752BC83F4A756E6B179 1843F4B5F9D91C7A0603ECBE0311B7C99E33EC29CD8178F98BC3EC6151E59F25CDC3EC EBFEB99CA62953ED5BF614F7F7FA13EDE02F0F20C5CE33EE40E9CCFAD5E2D3EEA04440 73020103EF070876C4BB8673EF45221EFE47B663EF8A6BD0FB105833EFD6E395BA93CE 73F01544190066C133F042AC6C547626F3F073AA7FF22F13F3F0A83E244A84D5A3F0E0 6737C2C13D93F10E12D10F698BE3F12DBCA8C9A9A533F14F311C6CD679F3F1727026F3 068053F19779C482A53FB3F1BE4DF21F92EB53F1E6ECAD732F1283F208AAFA52058FA3 F21EC4E31C3E3D43F235C410806CD543F24DA882039989A3F266723741822BB3F28021 2FE7E762A3F29AB56BB2EDE513F2B62EEA6A4E4F43F2D28DABDF2110A3F2EFD1AFEA21 5BC3F306FD7B3525C5A3F31684BFA1E12FF3F3267EA52F75C203F336EB2BD3DBBD63F3 47CA538685F133F3591C1C40248443F36AE085FA72A8C3F37D1790B00D1A93F38FC13C 5C4FB8C3F3A2DD88FB38E263F3B66C7689518B33F3CA6E0503992C73F3DEE2346774E2 93F3F3C904B2A12DF3F404913AF192F033F40F7743FBA5ED83F41A969D76CA29C3F425 EF476258ADD3F4318141BDBFBED3F43D4C8C88801F73F4495127C22AB2B3F4558F136A 5E72B3F462064F80C6ACA3F46EB6DC05197B33F47BA0B8F7167213F488C3E656857743 ECA93A2D90FE8533EC0DCB8B495BC943EC37F45A56FDDAE3ECAF23162C3C7F43ED30B7 F5A3C68613EDA50F60A8BBF7F3EE19E08D68F6E4D3EE6E38F7C75B0E73EECF7AFB001A 4413EF1ECD8068C77613EF5C493CB2C74B13EFA02EBB5D8B03B3EFEA7CC12D0E0923F0 1D99407E3F9A93F04927B87EA63F83F077E99908AE40B3F0A9DEC081A16FA3F0DF0716 AD6E9873F10BB144E05750A3F12978863A665453F148D94A5EFACD63F169D38D748043 23F18C674C7ED6E913F1B09485275E4E53F1D65B3594A7DCD3F1FDBB5C4D08FDC3F213 5A7C106C1B83F228A4040D12F693F23EBA45B836E113F2559D40BF988273F26D4CF4DF 5C9133F285C961DF3BA983F29F1287904E3AE3F2B92865CB4F7803F2D40AFC6F3BBB43 F2EFBA4B6035C5C3F3061B294351A0C3F314BF88E73A58D3F323CA449347DA83F3333B 5C402A4C33F34312CFE79CE043F353509F843C52B3F363F4CB11648263F374FF528B13 CE73F3867035EDD32AB3F398477536A21053F3AA851062E59BC3F3BD2907705A4D73F3 D0335A5D07F703F3E3A409273779C3F3F77B13CD6A0B63F405DC3D2728DEC3F4102E1E 54658D43F41AB32D6DEBD3D3F4256B6A734C9583F43056D56426CC63F43B756E4025C5 C3F446C73506FF9C33F4524C29B873E363F45E044C544A8173EC95DEE81FE539F3EBE7 9D4F5558FB03EC118C35D3CD4053EC76860E0D9F4753ED086A2E6B4BF433ED6DDD84EB 9FC8C3EDEAC65934BA5613EE3F64BB5AFADEC3EE94DD7A4838CB03EEF5C1D320715603 EF31059B06BD0793EF6CDADA8727DE13EFAE5F6CD5749783EFF59284CCD20DB3F02139 CC4F571C03F04A81249677D873F0769F29A1150DC3F0A593C3331D8823F0D75EDF812A 2D63F1060038A2530E33F121BC3738DF20A3F13EE367AAA4D553F15D75C6FB845FC3F1 7D7352D4DD4EF3F19EDC095D410D13F1C1AFE91B050303F1E5EEF0DE7FB703F205CC8F D8E0C693F219573A66231423F22D9777C4C76B03F2428D47B0F28223F25838A9F25CF5 C3F26E999E5A1F3533F285B024C0F472F3F29D7C3D05D909E3F2B5FDE70CEFBAA3F2CF 3522BE9E2F63F2E921F006D4EB83F301E2276A3CC6B3F30F8E1F8C763113F31D94E063 D1E753F32BF669EAECA513F33AB2BC1D2249B3F349C9D6F67133F3F3593BBA736257D3 F369086690F52E73F3792FDB4C8ED453F389B218A3EBC9D3F39A8F1E9513E0A3F3ABC6 ED1E4FF9E3F3BD59843E214B83F3CF46E3F339E6F3F3E18F0C3C765693F3F431FD18D8 1FF3F40397DB43C08463F40D441C43D782D3F4171DC18C5C5843F42124CB1D051F83F4 2B5938F5916C43F435BB0B15C92473EC83F141647DBCC3EBB7ADAD39D75213EBDC32F0 2B27FC53EC4227A140CE8153ECC6197E9D108863ED3A997BE97140B3EDA6E3551E116A B3EE13C78B2DE171E3EE5E386B61D2A163EEB2B8B4E936AA23EF08A0EBF6F18DC3EF3C E7F3376A5D83EF763034FA012D93EFB478E419E4D793EFF7C176C88352C3F02004C6F3 E19CF3F046A873211C17F3F06FCBA7AFDFE693F09B6E52D4C5EEF3F0C9906749394F93 F0FA31DB0166F483F116A9532512F653F1317961959B4F53F14D89167BAC14A3F16AD8 6FFDD6B013F189676CA2708A83F1A9360B39D02A83F1CA444ACE05F3C3F1EC922A968D 8963F2080FD4F7741C93F21A766427AFB843F22D7CC2A3B23413F24122F03C93FF53F2 5568ECCB104CA3F26A4EB82E2376C3F27FD45249F03E63F295F9BB06DC4D93F2ACBEF2 50D827C3F2C423F816C96BB3F2DC28CC4A1065C3F2F4CD6EDE237B63F30708EFE41E11 F3F313FB0F7F890EA3F3213D163D5112F3F32ECF0419FF12B3F33CB0D912928703F34A E29524708C73F35964384D55A343F36835C28B499D23F3775733DC955B83F386C88C3F BA13E3F39689CBB369D813F3A69AF236813223F3B6FBFFC801A7C3F3C7ACF4670CFED3 F3D8ADD012E12B53F3E9FE92CAD4C833F3FB9F3C8E540A73F406C7E6AE6F0CA3F40FE8 229B016063EC7371395EC80DB3EB8BC83040329743EB9B384238A42903EC1207CFC5CA 2D83EC813F509AF6EC33ED0B4345A2305FA3ED68180E8DF30793EDD6C2CE801D8FC3EE 2B8BCE4CE7C763EE747FA62BDEE8B3EEC6365F06C9BAF3EF1056056E528223EF41EF19 9AB3CB33EF77E59EE3B79293EFB2390B7ACBCF63EFF0E90041D71C43F019FA9D01B68E 33F03DAEC423B5B3E3F06390E3BB81E5A3F08BA0EE82864193F0B5DEDA6CEFB433F0E2 4A9FC7954C03F108721C4D1BB7E3F120D5D01BCA9F33F13A50697668C593F154E1E6E3 4B9433F1708A4732C99133F18D49896EF320D3F1AB1FACCF43E5C3F1CA0CB0AF3D2DB3 F1EA1094872866C3F20595ABF33B6463F216AE7D37AC9963F22852BDE9A30523F23A82 6DE81B0223F24D3D8D17375103F260841B5F5D9C43F2745618AC7E66B3F288B384ED7F 4F93F29D9C6013C0A363F2B310AA12B8C6C3F2C91062DFA13203F2DF9B8A7131A263F2 F6B220BF66D6E3F3072A12E1A95B33F31340CCBB7A19E3F31F9D3DEA8AC3E3F32C3F66 6C97D9A3F33927463FA92CB3F34654DD62079E63F353C82BD2346483F36181318EE190 A3F36F7FEE96EBAD03F37DC462E9543F13F38C4E8E853D0CC3F39B1E7169E40A73F3AA 340B969FD263F3B98F5D0ADC9A33F3C93065C6198903F3D91725C7E66AC3EC645ED00E C42C93EB63ECD8686ACA63EB602861D00F0673EBCC4D3339249723EC4245D2D04B1413 ECBFB5C42BBA4B03ED2E6485845F2CA3ED8C649F2305C133EDF9AF4612F07A23EE3B16 A6E86A11A3EE80E8CB582DEE03EECE4A2257C097E3EF119C1AB78C7083EF3FD8B52A4A 3EE3EF71DA56B587ADC3EFA7A0A03B2B57E3EFE12B4E85C8E0B3F00F3D188E9EF033F0 2FC692355E11A3F0523206F08CF2D3F0767F6CB4588053F09CAEBBCD97C8D3F0C4BFEE 440B3293F0EEB2FF6A71DD23F10D43F5C6F74DD3F1241F57DD962013F13BEBA4C96BEB 83F154A8DB94891483F16E56FB76693D33F188F603CA8DB443F1A485F4093FE6E3F1C1 06CBC1F10D83F1DE788A96D20703F1FCDB3039593893F20E175E33BFABC3F21E399774 C4F293F22ED443C8221B33F23FE76319C7A863F25172F5589B3963F26376FA75FD1AB3 F275F3726563B123F288E85D1C0874A3F29C55BA90A32303F2B03B8ABB307A93F2C499 CD94C262C3F2D970831757CE03F2EEBFAB3DBAF593F30243A301B26CF3F30D63A9B232 8F33F318BFE9AEA6C2F3F3245862F5903D53F3302D1585A110D3F33C3E015DB5C733F3 488B267CCFE5D3F3551484E2113323F361DA1C8CB7A2A3F36EDBED7C19CF83F37C19F7 AFA3EEC3F389943B26D527A3F3974AB7E13D4643EC56BA0574721983EB401BA5B27FEB D3EB2B034EF1689473EB7D07FD8A4DB753EC092D053D0D0003EC70C0A288EF2523ECF3 917402ABB473ED487488447B78C3EDA437D32F080473EE067DB71ED824E3EE41591CE2 0FB453EE82AA2CE0277583EECA6E70A1163B13EF0C5226ED9CDC43EF36A55878B6EDF3 EF64306DD3BFED63EF94F323C94B7D53EFC8ED49E13736D3F0000F5E425A72B3F01D43 B0934D6333F03C1391D7A158F3F05C7EFA5C2D6313F07E85E428B1E7B3F0A2284A8FEE 05D3F0C76629DF049163F0EE3F7F22A05C23F10B5A23FDB73933F12062413EC7CE33F1 3638168BFD8FD3F14CDBA340D9CC43F1644CE6D5A5F0D3F17C8BE0D9D33683F1959890 EF927B03F1AF72F6C85A2443F1CA1B122222E783F1E590E2C5324603F200EA344129CA 43F20F72D198B3ECA3F21E6249581D28F3F22DB89B70C5CBE3F23D75C7D6243A73F24D 99CE7D71EBF3F25E24AF5D669EE3F26F166A6DFF1123F2806EFFA84D2A63F2922E6F06 4FDD43F2A454B882D16243F2B6E1DC194ABAB3F2C9D5D9C5CB8953F2DD30B184E59103 F2F0F263539B3D33F3028D7797A85883F30CD52A8ADF8083F317504A82749C33F321FE D77D897143F32CE0D17B5BF3E3F337F6387B42BF33F3433F0C7CAA07E3F34EBB4D7F10 F573F35A6AFB820753C3EC4A82D98FD1D473EB2054981E71FB53EAF792133961A743EB 363FFE7F0FBB93EBABE9CFC2795F63EC29A7265BFF4D53EC94895809AE1FF3ED0AF289 E47EB5B3ED56B143EE162E13EDAD69AD9E5245C3EE078753A46F0DE3EE3DCC2A75D99D B3EE7980E4CB5FA433EEBAA3E85B5ED5D3EF009A10C4598FC3EF2698690B94DBB3EF4F 4CB9CDF4F213EF7AB6D293576B13EFA8D68FC4EE1293EFD9ABD6D58F2563F0069B49D6 CA3E53F021BB5B73561A93F03E361A482B8EA3F05C0B81A809B893F07B3B8DE0136C73 F09BC63BFCF37CF3F0BDAB899F56F463F0E0EB74DB5E9D03F102C2FE1000DDF3F115BD 8F12217663F129656CEC5A8533F13DBA972E1D4313F152BD0D799A8F03F1686CCF8048 CD83F17EC9DD0020B463F195D435C167C4F3F1AD8BD994E953D3F1C5F0C852866783F1 DF0301D80A3C93F1F8C28608356E33F20997AA64FA6263F21724B703DD9823F2250868 D77C15F3F23342BFD7D29AE3F241D3BBFDF30DC3F250BB5D43DC6153F25FF9A3A458CE 13F26F8E8F1AE15583F27F7A1FA3859A43F28FBC553AD75FB3F2A0552FDDD91003F2B1 44AF89EECFD3F2C28AD43CD1B2D3F2D4279DF484C453F2E61B0CAF4B8FC3F2F865206B A1FDE3F30582EC941AA1D3F30EFE9B71EEE5C3F318A59CCECCF253F32277F0AA449313 EC3FB94C60E35D63EB0497AFAC40F8F3EAA4F323A3CF86A3EAEFEA6C2ED547D3EB513A F579B44653EBD4D29F49D58723EC3FB0B71DC59C33ECA7BD48061EF013ED111B98501A F6B3ED57780BF2BA15D3EDA6E6DF3E97F553EDFF603631AE1FE3EE306F91EDF51CD3EE 65B039D503D593EE9F70FF30DF2703EEDDB123C55447B3EF10381093C53F13EF33D6CB 339E7CC3EF59B49E2B67AA33EF81D16EEDF702B3EFAC2D2963A65FF3EFD8C7BE2623DF B3F003D090A2782713F01C5CA4B2C4F553F0360817911B2CF3F050D2E64A25A0E3F06C BD0E7E915D83F089C68E427F2A93F0A7EF6404E64DC3F0C7378E7CC96343F0E79F0C9A BB46F3F10492EEBECF33E3F115E60034EA4263F127C8BA612533C3F13A3B1D0178BD63 F14D3D27DE2A6163F160CEDAC80573C3F174F03596EAFA33F189A13828A63303F19EE1 E25FF7B793F1B4B23423CC5223F1CB122D5E96F173F1E201CDFDC71123F1F98115F156 2ED3F208C80295B40C33F215174DCFFD5993F221AE6CA25138B3F22E8D5F0828AA53F2 3BB424FD935153F24922BE7F22F0B3F256D92B89D9F383F264D76C1B1C7D63F2731D80 30A3A2E3F281AB67C8726E93F2908122E0CC6C73F29F9EB1782D84E3F2AF04138D42ED E3F2BEB1491EE510C3F2CEA6522C123BF3F2DEE32EB3EA08B3EC365D5DE7A6B483EAD9 C9D8B7D9C993EA5E29CF221AC7D3EA844F48A6BCE0E3EB024D7B9FCAB593EB660E3CC7 630F53EBEA0F227DE451A3EC4671AD405B8033ECA6EDA0AA2CBCB3ED0B26893AE7BB53 ED4A3AE1A54CF583ED90ABFD923F9943EDDE74F011AD4903EE19C9424828B783EE4801 49E9F1F273EE79E1D0B1FF8AE3EEAF6A503578C7C3EEE89A678418D903EF12B8E7B821 ABB3EF32F8296FD25E03EF550AE4D1785CD3EF78F10A76C1C503EF9EAA8E6F2F6393EF C63767603F76A3EFEF978DE437A533F00D657E0A36C9B3F023E8D6991DACD3F03B55CE B2F144B3F053AC64C6A8D6B3F06CEC978B46FE53F0871665F15E9733F0A229CF17D211 F3F0BE26D243032C23F0DB0D6ED5DEAF03F0F8DDA44C560753F10BCBB91B7A1C33F11B 9D6C1BA7F463F12BE3EAFE959293F13C9F35A20E3543F14DCF4BE8D275B3F15F742DB9 CF1D23F1718DDAFF771BC3F1841C53A739ECE3F1971F97A1110D33F1AA97A6DF204C63 F1BE848155658D53F1D2E626F9754473F1E7BC97C2417223F1FD07D3A7E95CA3F20963 ED51C84503F2147E56579DE7C3F21FD324E2D9BAA3F22B6258F134D1E3F2372BF280B2 3833F2432FF18F957A03F24F6E561C5A7F43F25BE72025AE7D63F2689A4FAA69DEB3F2 7587E4A98AF1C3F282AFDF22314F23EC2E6F0E241BD973EAB2789C5AEB7D83EA233615 B4436AC3EA29AE9265D64163EA7E42C4697957A3EB07012530A73543EB691BCCDA678B D3EBE40486EF6637B3EC3B85D7FA10C9F3EC90EA4AEDB66AD3ECF2155CF9FA38C3ED2F 7BAB0D67A683ED6BC32E380877A3EDADDE03699AF763EDF5CA0367D70223EE21C2D8DD 2B80E3EE4B8800C5B4C1B3EE7834187D4273B3EEA7C6D8D6383093EEDA400CB64270B3 EF07CFC5F70A7253EF23F29BD885E4E3EF41887C0F145C93EF60915D40341733EF810D 38062ABAE3EFA2FC067A4DEE23EFC65DC3DDF7C723EFEB326C5969D7A3F008BCFE6549 55B3F01C9A394FBBBC23F03130E5DC9EFB33F04681032AD58333F05C8A907EC11203F0 734D8D3B4E7253F08AC9F8DC6F0E23F0A2FFD2F2ADE883F0BBEF1B1FA1AAA3F0D597D1 130C5FA3F0EFF9F4888485E3F1058AC2A2C5A713F11375418BFD2613F121BC76EA5BAF 23F1306062AB50BEF3F13F6104BED5D193F14EBE5D170EB673F15E786BA7F95103F16E 8F30672A1523F17F02AB4B930383F18FD2DC4D536283F1A0FFC3658EB733F1B289608E 49D893F1C46FB3C24D0CE3F1D6B2BCFD0A7F63F1E9527C3A884233F1FC4EF1774D6F73 F207D40E5828EBB3F211AEFEF177F253F21BB84A866C36A3F225EFF1160405D3F23055 F29F5EF8A3EC27EE5D1642CC93EA933BAA41B70D93E9E82FEEB492DE93E9C01092D842 D363EA0F6D52711455A3EA6F56B10B43E963EAF90EDAA229CEB3EB54DE15584B8763EB BFFFAD1FC42DD3EC1EC7C14D290813EC66AC8B8B50D403ECB79E7D464C91C3ED08C9DE 4EFBC593ED3A42F035E445E3ED70384D1D97A683EDAAA8788DB05213EDE99265B07CD7 43EE167AA952B9CA53EE3A685888ADDE93EE60920B5FC72043EE88F799A8B94293EEB3 98E496E08033EEE075D4493E7213EF07C72B0567CB03EF20712D880F9BF3EF3A38EBC5 EC51D3EF551E6100580173EF7121895F944013EF8E4261C18F9613EFAC80E7944F3B83 EFCBDD18B8F8E873EFEC56F36D517B83F006F73B1D0E3963F01851CFF299B013F02A3B 37B33C8DF3F03CB371EF81D493F04FBA7E49002E13F063505C6F79A633F077750C1E84 B883F08C288D1BA29853F0A16ADF34ACD9B3F0B73C023E8958A3F0CD9BF614170EC3F0 E48ABA9548F8A3F0FC084FA666AEF3F10A0A5A97B5FE53F11657F58DC0E3C3F122ECF8 AC4608B3F12FC963EB64B353F13CED3744406AE3F14A5872B0DE02E3F1580B162C05C3 43F1660521B129D713F17446953C505123F182CF70CA004B93F1919FB4573194D3F1A0 B75FE13E9AD3F1B0167365D82F33F1BFBCEEE2FAEF43F1CFAAD256E60BE3EC22DB4ABE 1B8D83EA7C13026C3C7A33E9A19EE82859AB03E94EB8DB733CB6E3E9703542ACCCC843 E9E0335B194D48B3EA48A407C3D8C5A3EABEE00B76CDD653EB28B738D74143C3EB7FCA EB284E9F83EBE4769DFD3B7AC3EC2B4D70A6F70023EC6B1200AD0E5FF3ECB18295EA5A AFE3ECFE9AE1EA4BA5D3ED292BF5DE0B09A3ED565BCCDA2B8AF3ED86DC3408FDB7A3ED BAAC9D0F48C303EDF1CC9DB8659C43EE161DF2D2DDD3E3EE34FD1B9F01ACA3EE5583B1 5F9FF2B3EE77B1A15FB524E3EE9B86DCD3CC8F53EEC10357EE917853EEE8270932B815 63EF0878F47861C1F3EF1DB1F872486DE3EF33BE8DF51545E3EF4A9EB2E1EE0093EF62 526576977F83EF7AD9A43BD53E83EF94346DF780D143EFAE62C1A17D0DA3EFC9649E5A DFF973EFE53A0366D6E5A3F000F1781272F603F00FAFB2061D02E3F01ED7AF53EACB73 F02E696FC9B5E083F03E64F33C6424D3F04ECA3986D48F73F05F994288FA17C3F070D2 0E271C5DF3F082749C493728D3F09480ECDA736083F0A6F6FFC8B50423F0B9D6D5043A A333F0CD206C7F4B4693F0E0D3C62DF09C93F0F4F0E205BB3323F104BBDFFEC7AD33F1 0F343006BC9F13F119E16117438963F124C3732D543843F12FDA66464919A3F13B263A 5FD30343F146A6EF77EE8EF3F1525C858CDAEE53EC1F35D71BA61C93EA6CFEA4DA7BC3 53E972B917C3DB39E3E8FEABFD393452A3E8E12AC465BE0183E920967FCAED32A3E981 EE4233B80603EA0774B03170CBD3EA6264A63531ADD3EAD10D9F0AC50243EB29834EA7 C1E6C3EB740860799D2DC3EBC802513AAAE6B3EC12B2274EEC6013EC4612DDDD030B13 EC7E205356CE4AF3ECBAD86E10EB3FC3ECFC39AA4A46A523ED212175A63772D3ED4679 AE570AECD3ED6E252ED90528C3ED9823B992DE9CB3EDC4751EC29A78D3EDF31938FE31 B813EE1207F5575D7BD3EE2BAC8E1CC66343EE467A5D5207BD53EE62715B47A4B913EE 7F9181B97A64A3EE9DDACB839329F3EEBD4D34683A80A3EEDDE8B8E2FA2A93EEFFAD56 055911D3EF114D84AD8537D3EF2358E869E9F213EF35F8D558F0BE83EF492D4ABDC6F2 73EF5CF647F7F0A393EF7153CC7E9AC803EF8645D7DCC7C663EF9BCC69AE2AC7A3EFB1 E7819C8F88C3EFC8971F5DB3F783EFDFDB42B17E0753EFF7B3EB607CF553F008108C9D 5429B3F01491660B28F4A3F0215C81E7A0DDF3F02E71E022FA75A3F03BD180AF7CC473 F0497B63813175C3F0576F888DA918C3F065ADEFCBC7FDD3F0743699339A55A3F08309 84BE2E49B3F09226B265737E73F0A18E22241EC8A3F0B13FD3F591D323F0C13BC7D5C5 D6D3F0D181FDC1393243EC1CFE022EE279B3EA65FE918C74E813E95B7E7D87178B33E8 A3CFF8A8B653C3E83FF606E8BEB463E83FADA056CF0D53E888209A90CE04D3E906F431 C0FFDEA3E965C3EFA72B8EF3E9DF0CBEC636DB43EA38FE522C69D4D3EA8F0AF01F4D57 F3EAF184D1BB331983EB30292932FB2F33EB6DB11C26AAFD33EBB1544FD712A213EBFB 0E90ED6C5553EC256E6BF3C8CFD3EC505E75AC4F94D3EC7E569136FC7373ECAF561DD6 9E2103ECE35CA058A59813ED0D34DC9C8C9A53ED2A3E8ED1F574D3ED48CB4935A3F613 ED68DAF42CEEB733ED8A6D7CBD3E1283EDAD82D387984213EDD21AEC03B78683EDF835 BBE9B0B3E3EE0FE99D5F0D52A3EE2479B0BC047543EE39CB151D44A963EE4FDDC80E81 2923EE66B1C77F7F94F3EE7E4711B26BFC43EE969DA52DA10343EEAFB580B026D743EE C98EA328576F93EEE4290BAC379123EEFF84B9732D8CE3EF0DD0D5E86869F3EF1C3FF1 184EAC63EF2B0FAE092F7573EF3A400C839A0743EF49D10C5785F573EF59C2AD5B4507 E3EF6A14EF6A9F42B3EF7AC7D26611BE93EF8BDB56322A9753EF9D4F7AB6FD22A3EFAF 243FDFAA77F3EFC159A599FB4013EFD3EFABD607D3B3EFE6E65285EC8873EFFA3D999D 88AAF3F006FAC08922F3E3F01107046D73C5E3F01B439877B003C3F025B07CA45611A3 EC1C33CBF7D0A4C3EA6712C88227E9A3E95BEF19720EA153E88CDDA934FF7813E7F998 99C53743F3E777D130EAF13B73E75EB8C316FA01C3E797822C1FDA05D3E80B999A26BB 76E3E86C005D6681E663E8EB516A44A43AA3E94448A04029F5F3E9A191210BA2ADF3EA 06A777283968A3EA43AFCA350BFAD3EA87D5A5E91C8A83EAD310A396DF5AE3EB12AD5A 1D03B8E3EB3F57B27C674023EB6F85B216FDF573EBA3361772657573EBDA67EC297F90 D3EC0A8D389A2C90B3EC29A688756AFF83EC4A7FAA78A84783EC6D186F6E919713EC91 70B161F7AAB3ECB7885194E6FA13ECDF5F36F6D40293ED047AA67BF430D3ED1A254150 A8EB03ED30AF64F09BB283ED48190B7EDA7C33ED60623012CB95F3ED798ACE8BFA5623 ED9392E36EBED343EDAE7A6BC7CE4C43EDCA416515393713EDE6E7CD33B358A3EE0236 D127A33183EE1169716B5F6FB3EE210BC6B8A5E873EE311DD07AF27123EE419F8E3210 FA43EE5290FF6F1FF263EE63F223D20E8143EE75C2FB07803C73EE880384C70531D3EE 9AB3C0D1978A63EEADD3AEF0537A83EEC1634EF35FAE93EED562A0B0FE6A13EEE9D1A4 04C0F303EEFEB058CED7D613EF09FF5F79BE0263EF14DE6B2D35F303EF1FF54F773D68 03EF2B440C4E97B9D3EF36CAA1AB375B33EF42890F86190A73EC1CD73476709DF3EA70 3B49BB94C703E9740AEB84C07BB3E8B9D50EDE0FBE33E817AD965354D183E76BBBCFE3 AE18C3E6E63F655DF41F03E65D7361BE2C0203E6271F2577CF51B3E637AC6EE5435183 E6888B67DC187593E70AE707AC8A2513E76E7EF6A9840AA3E7EE405C13D4BC03E844D0 B4ACB906C3E8A02180307BCC53E904787AEF81FF73E93F936CD38EEE73E98158B010D7 D713E9C9C1A15CEBB8E3EA0C649D67207A93EA3735D1682F4FB3EA6552EEA1BC50A3EA 96BAC9C38EDD03EACB6C76165BDE03EB01B39B626C2963EB1F5539FD2335D3EB3E9AD9 5F6C84F3EB5F84476830AC83EB82115AF7C30F03EBA641F2207F4A13EBCC15F0BEB32A D3EBF38D3F5E96AC43EC0E53E52EE16A13EC23B2C09D329033EC39E32B02EF0B33EC50 E51E79813FB3EC68B895FD0A1C53EC815D8D46F12833EC9AD400AF49CE53ECB51BED13 AD20A3ECD0354FC278E963ECEC2026699D07B3ED046E3784247313ED133513F17B0833 ED2264A7BCB0BCA3ED31FCF23F0CBBD3ED41FDF2E7AEFA23ED5267A9388BED73ED633A 14C3DBF523ED74753529EC5573ED86190A1741AC13ED98259342FE8513EDAA9AD06D84 21C3EDBD78C15F4404C3EDD0BF65E7BB2CB3EDE46EBDDC905F53EDF886C918D0DF53EE 0683C3BE232DF3EE10F87C75720A63EC1EE83BAAC26503EA81781538BB8123E9A3D1F3 BF2D1623E9155B14D1F393C3E89099E33AEA3F53E82D8D8ECBEA5E43E7CD7603E99C3A 43E761775541B354D3E70C24D9E7FFED43E68F66E213691413E621B65F2332C5B3E596 9453F845F5F3E51357952A166633E46D03DD2E7BB7C3E3F7B097DE60D103E3A0670B96 44D173E3CFF7BEBED9C5A3E441AD9798CCDA73E4DC2BF805ADA2A3E55B514BABF34FF3 E5E8385C2F22EFF3E64A46CA226F6933E6B01098F82B7623E712B372CAA0C733E7551D 72427EE883E79F40643957D613E7F1178A0940D303E8254F85F9E4DA13E855E9E3F3C1 2DA3E88A5995F4CEDEF3E8C29D8C8FEB1233E8FEB4E6A7842B13E91F4F744EACEA93E9 412D7AAF1AF7F3E964F4446DC0ECB3E98AA399D020F6A3E9B23B4BA18FCB03E9DBBB31 C312FCD3EA039194FFF44E63EA1A398B8C54BE43EA31D56FFD104123EA4A65378B9DE7 13EA63E8D8EFBC4913EA7E604C2629C0F3EA99CB8A40CCFDD3EAB62A8D3E987E63EAD3 7D4FE9C69333EAF1C3CDBB56C523EB087F01617625C3EB1895F5C9252763EB2926C295 EDCD63EB3A316679E1E6E3EB4BB5E0518BDF33EB5DB42F1E712AF3EB702C5202576C33 EB831E483B3319B3EB968A111F9EDAE3EBAA6FAC1BCF72C3EBBECF18AEED3923EBD3A8 5668C282E3EC2266E194C5FA33EA9AC92AF99C1743E9EB443221547693E96FC07CC342 D333E933C899CCADF5B3E9117BC681D94733E8F6DDDB637035D3E8D7DD8CA3D90C43E8 C07984E8DFDD73E8ADCC33C91E4723E89E31A3B9F840C3E890AE547A9F0563E884A4A8 08C71283E879AE0BFBDD8CF3E86F85990ADDB4E3E865FBC0291538B3E85CEEE6A0559A 43E85446D05F6ABEF3E84BF1B1FF3AFEE3E843E244EAE439C3E83C0E7D4F53AC33E834 6EA803975EB3E82CFCCCC8160983E825B43E5BEF45C3E81E9149E1F556B3E81790FC04 FB0113E810B0F5720D4923E809EF4A49F274A3E8034A6983F29E23E7F98214454F3753 E7ECA43971DDB5F3E7DFF980526CD833E7D38071AA7B21B3E7C737D3539C0A93E7BB1E 9D30A58F93E7AF33F0579FBD13E7A3770FF5D8B1F3E797E75B8F680863E78C844A514A 60D3E7814D673DE3EA23E776424E085815A3E76B62A87C6A5B53E760AE2C57FFDC13E7 562499811A6E83E74BC5B886E949C3E741915960FEFAA3E7378752612F2323E72DA77F 4F7BD3B3E723F1C0A8F28813E71A65FAFBBA5153E71104165B1F8E63E707CBFDE84875 43E6FD7B3ED7B09A33E6EBB1D5FC352773E6DA3BA576EAA323E6C9189506583D93E6B8 488F7DDAB923E6A7CB8264143633E697A15D9DCD4DD3E687CA1324C47FC3EC27532634 7B5D43EABC2E8AFE3689E3EA2530D3559B4D03E9EC1ABF42F5A843E9CE49C3B754ECB3 E9CB0DF14C97D0E3E9D4FC32D1B1A8E3E9E6613F3C4492E3E9FC7E889E49E473EA0AF0 A361873A63EA18DBBF72A22053EA27C1B87241F703EA377B26DFD3D043EA47EE662133 0B93EA590A3C641E6C93EA6AC2C5E5694C73EA7D0F9BF78818D3EA8FEAAF962DD983EA A34F8DE7940713EAB73AE5468265C3EACBAA32BE33CBF3EAE09B89846C18A3EAF60D6B 8944F6A3EB05FF56BE42F8C3EB1137315371F333EB1CADE71989F5C3EB28632C262C22 C3EB3456C2FD71BE03EB4088797E437F43EB4CF82688F4A733EB59A5A82FE0F4E3EB66 90E24F561B23EB73B9BD738C5613EB812025FA1C65B3EB8EC40B6133F193EB9CA55FBA 331813EBAAC41737EF36A3EBB92027D2A9D173EBC7B988FD323DC3EBD6903367AF2BE3 EBE5A420CD5237F3EBF4F54BCAD842A3EC0241D7DE0FCD73EC0A27A44FC3DD03EC122C 097F020CB3EC1A4F05EB870203EC22909847E115E3EC2AF0BF725066E3EC336F7A6EBD C533EC3C0CC86199E1D3EC44C8A88B7F8343EC4DA31A45777F63EC569C1CFDC6F6A3EC 5FB3B03530BDB3EC68E9D37C97B643EC723E8672F604F3EC7BB1C8C398C633EC85439A 249B4AA3EC8EF3FA558FF0D3EC98C2E91E5DB6C3EC2DAD0989E28EC3EAE5A835468AD8 63EA609528AE69BFA3EA4534EE28860613EA4BE837AEB50093EA61BEA3E3186583EA80 02A82D29E2D3EAA3DAA007BA7AF3EACBE7447BC6EE43EAF76F316EF59283EB1304B4BF 4FB623EB2BBB864D486473EB45C859A11B68D3EB611E5C6DB66873EB7DB4F1C9F8CBD3 EB9B8627D44C3C63EBBA8DCB35EAE7B3EBDAC8D46CAE5853EBFC350A0BB35F83EC0F68 61A02ED8F3EC214D5F48257FB3EC33C902CB2E7AF3EC46DAECA55036B3EC5A82D2027C A393EC6EC077B4E10AD3EC8393AE872A7683EC98FC508429B8E3ECAEFA3EEDEB51C3EC C58D60B3E39A13ECDCB5A1463C6063ECF472EFAE2337A3ED06629EECDFAA83ED12D640 079638D3ED1F94563E9B8643ED2C9CDD717B1453ED39EFD2248CF2D3ED478D31648658 33ED5574F8AF779743ED63A725E214AB83ED7223B728672B13ED80EAAAF13B94C3ED8F FBFFE3C11553ED9F57B4D6EF31B3EDAEFDC8CA5DDFD3EDBEEE3AE049E193EDCF290A58 919F13EDDFAE368C7C3DA3EDF07DBEEB29DFB3EE00CBD17B476453EE097DF12071B063 EE12553E353A8723EE1B51B88FE41C03EE2473600BFFBD53EE2DBA3489BE20C3EE3726 35ED57F643EE40B7641E8BC2B3EE4A6DBF082CE3A3EE54494697C20C73EE5E49FABD30 0973EE686FDB6A706063EC35748B94FB8DD3EB0B9B14E94C8193EAA7CF191B1593F3EA A556E9F6C30043EAC82E4E5F769893EAFD64E4F7521C93EB1F229192E795C3EB43FAB2 32DF6D33EB6C8B30E00EEC93EB98775B8D4943D3EBC78AB3893AF0A3EBF9A5C8CBCA1C D3EC175A73CA56B463EC33559ECB693193EC50C016F48C3EF3EC6F96E354F4AEE3EC8F D7EA413FA543ECB181A7F859D453ECD492FFC3D1B5B3ECF90B1D3A5918E3ED0F74AFCF 5F6473ED2316A5DF39DC03ED376B410ABF25C3ED4C725BE7E83A43ED622BD8DFCBAE93 ED7897A057C02383ED8FB59F552DE9B3EDA785C6791B4783EDC008093B427243EDD93C 5D53B84283EDF322BA4713A293EE06DD8C8633FD53EE1482B9E361A863EE2280E2C6C5 EDD3EE30D80520268303EE3F881F31AF7B93EE4E912F81BB6AC3EE5DF334CF5286A3EE 6DAE2E08CEC0D3EE7DC21A4434E743EE8E2EF8B8EB48D3EE9EF4C8BA894DD3EEB01389 B48AF6E3EEC18B3B26BC08A3EED35BDCA23B0C43EEE5856DC6F71F83EEF807EE4193DC 73EF0571AEE4D0A663EF0F0BDE100CCFB3EF18D284870C8463EF22C5A231E31463EF2C E536FBB1A2B3EF373142D240D9F3EF41A9C5A5A8FA13EF4C4EBF68067843EF5720300D 389A23EF621E178AA8FC33EF6D4875D71AEF73EF789F4AEA80FA83EF842296BDD7C0B3 EC3EA9AC55C65B13EB286C3451308533EAFADEA49B9ECA93EB0B39A98618E083EB2DFB 92F6F7A083EB5C3CDDF17C85F3EB92A2C529645FA3EBCFC0765C17B283EC097327070A 6DD3EC2DDBCD191788C3EC54FFEE0B8963C3EC7ECFD1DA5318D3ECAB419EA54FEE83EC DA4EEB45F6B263ED05F9B4A3DB9323ED20160FDCE306F3ED3B7B7A786048A3ED582932 95C9B523ED761EA9D9E68383ED955B76100FA043EDB5DF46D9AC6C73EDD7A9DE9D7566 83EDFABB0D996DBCC3EE0F895731C735E3EE225851B2770373EE35CA6A81719583EE49 DF98216B1303EE5E97D2E2E6AAA3EE73F31481C16C73EE89F157DA05C303EEA09298AD FF2253EEB7D6D379374793EECFBE054D352EB3EEE8482BB5AC04F3EF00BAA25130B593 EF0DA2A72AC0B5F3EF1ADC23AAB35943EF286717308CB583EF3643813379C583EF4471 613E7CEF53EF52F0B6ED4951B3EF61C181E9AAB193EF70E3C1E95F01B3EF805776AC4C 2A33EF901C9FFB0170E3EFA0333DA57663B3EFB09B4F81FCCE53EFC154D56C5CAAD3EF D25FCF451321E3EFE3BC3CF0AE45B3EFF56A1E57416E93F003B4B9B1F6DA63F00CDD1E 023DE0C3F0162E3C1481F2E3F01FA813E1D14233F0294AA5641B6DA3F03315F0961553 83F03D09F5732097F3F04726B3F7366A73F0516C2C1ED53CA3EC494C6BCC42F653EB49 4778DAF17733EB2CE1E59802B1A3EB4C4514B4692503EB83A15F2D0F7E13EBC97E9453 069A53EC0D40F76D05A4C3EC3A9F4E3FC30983EC6C827CE3FAA893ECA2BC2A53161C63 ECDD3217346B9E33ED0DEA45C1F72473ED2F4CD6B0BCB423ED52BD89D341E203ED783A 37EF3E8C33ED9FC1648C4D1F43EDC95202D4BC1DE3EDF4EB50E801FA53EE1146603572 EAF3EE291AF394605753EE41F33A617CF033EE5BCF15D024EF43EE76AE6DFF59F533EE 9291303A187713EEAF774DB4EBC4D3EECD60BAA27F28D3EEEC4D6D8586D323EF061EAF 5742F193EF169843ECACBC03EF279371F105F633EF391037457463C3EF4B0E9227BCDD 03EF5D8E8120A2DEB3EF709002F6000383EF8413169FB714E3EF9817BB3EDE2513EFAC 9DF016A27F83EFC1A5B4868841B3EFD72F0805BC5A13EFED39EA1F413713F001E32D37 650023F00D6A2C4F1131D3F01931F2318432F3F0253A7EBEB0F733F03183D1DADF1AD3 F03E0DEB6E0A6D83F04AD8CB635BCD93F057E471A8B742A3F06530DE2E5B55B3F072BE 10E68F11C3F0808C09C55C2453F08E9AC8C0531DB3F09CEA4DCE584EF3F0AB7A98E777 E553F0BA4BAA04C004B3F0C95D8120206113F0D8B01E344DC1A3F0E843813CA9D703F0 F817AA352DD483F104164C8D2C1253EC555CC9F8715FE3EB6E2CE2868F5713EB623F46 6C24AE23EB95CDB686524E33EBE5088BD202E283EC233BCAD023A423EC5B6007526E29 23EC9A3A924E954323ECDF739A06D81D53ED156E62BA502AE3ED3E2DEA9FA21353ED69 F088B15B5303ED98B14FFF698FC3EDCA6D0BF68830E3EDFF21955C6F0EE3EE1B66B7DC 5C4C73EE38B7C71AD9ADC3EE578397796AD983EE77C9E1CA736C53EE998A70F39FD9E3 EEBCC51CC5A3D783EEE179C673DBA2E3EF03D42B0EA46813EF17A85C8673EF03EF2C39 703BA20DB3EF41876047846793EF579227EB7515F3EF6E59C34FB5A893EF85DE2F5234 A993EF9E1F6960F753D3EFB71D6F5D235DE3EFD0D83F847DA453EFEB4FD85FC9A603F0 03421C5A707C13F0113AAFBDD33773F01F91A5EA9805B3F02E46FE82559643F03D5AB9 34CDEFD3F04CCCD5BD978CA3F05C9D53E233A573F06CCC33707BCBB3F07D59743D55DD A3F08E451623A0CF03F09F8F19034FA223F0B1377CC0A9EDF3F0C33E4143AB7973F0D5 A366777D2FC3F0E866EC4A031633F0FB88D2AB7BAB33F107848CC7170473F11173E073 11EC43F11B92645476CFD3F125E01866B7DE03F1305CFCA5DB27B3F13B09110E67C753 F145E4559D5589E3F150EECA4FFEE4B3F15C286F24149E23F167914417935DC3F17329 4928BA4C03EC62DAC6DA519733EB971C71540A2583EB9D8774CA355BB3EBE7D38EFBD4 5C23EC29188C72E8E813EC6993F0EC9F4A93ECB3AE9244EBBF33ED035903AD4143C3ED 31233F37D16A03ED631ECF3EDB1CB3ED993E9416097C63EDD37AB1BB5615A3EE08E71D 9F574553EE2A1AFE06672073EE4D57E675B68C63EE729D18B112BD93EE99EA0E4DA2F9 93EEC33E665B227263EEEE99D9ABF4BF73EF0DFE1992E32D13EF25B2A54CA57B83EF3E 6A809CEF6573EF58259F9341B953EF72E3F8D51AB813EF8EA584FCC00463EFAB6A3E23 874323EFC9321F8ACA9023EFE7FD255ACB7603F003E5A638BC43E3F0144E491E6B7CC3 F025387A4F063F13F036A438E96EFCE3F048918432077543F05B005B8BBC7123F06DF0 BE727F5B53F08162AC76DB3623F09556253A66AC33F0A9CB286CE3C893F0BEC1B5C9E9 0703F0D439CD16F79AA3F0EA336E21E91613F100574C5FD1ACA3F10BD5A66582ADE3F1 1794C512010C83F12394A8577095E3F12FD55029CF2DB3F13C56BC7EB148A3F14918ED 4D08E813F1561BE28CF55013F1635F9C379A0853F170E41A46FBE523F17EA95CB5E330 23F18CAF637FC20F93F19AF62EA09E9B93F1A97DBE14FFEA83F1B84611D9DDCCB3F1C7 4F29EC92AE53F1D699064ACF64B3F1E623A6F2908913F1F5EF0BE2154FB3EC71C66271 E39C53EBC416254361E173EBDEBA70B234BA83EC212B4F0A77A673EC658D83343E2223 ECB7C7BC7EF63E63ED0B164C223F3223ED4002D6B1C56BB3ED7A7D950F2D63D3EDBA6F 5AB72BE423EDFFCB07FD6C0E13EE2544606FF3B593EE4D51CC3745D033EE780C2D0B0A 0B83EEA5726F4E1C5473EED583D4C449DF53EF041FEB815CFA83EF1ED30A8C93E2F3EF 3ADB23ECFB74D3EF58381D1569EBD3EF76E9E1EE392233EF96F063111734E3EFB84B94 8D84EED3EFDAFB6D09009153EFFEFFE51DCFC393F0122C7B72A41383F025834ED061EB 53F039846ABC4230E3F04E2FCDA53C29C3F063857642527153F07985638417E373F090 2F94896EF073F0A7840896B95783F0BF82BF0EE3E0F3F0D82BB76DDFF983F0F17EF144 38A1C3F105BE3619C241B3F1131213F5C2D583F120BB1213E94AD3F12EB93056F63BA3 F13D0C6EA5D66E33F14BB4CCEAFCD433F15AB24B13D8EA63F16A04E910642DF3F179AC A6D2C26A33F189A9844EF186C3F199FB817A85F5E3F1AAA29E4C71B693F1BB9EDABCD4 1523F1CCF036C4D08EF3F1DE96B25E6BFF73F1F0924D846EAD93F201718419255FC3F2 0AC47132032663F21441EE0B143803F21DE9FAA2D47973F227BC96F7F122A3F231B9C3 09429E43F23BE17ED5C73903F24633CA5C9E8D53EC821F9CBF276FB3EBF519FE54968B D3EC12EC1D121164C3EC52AB71E8D190F3ECA7E32A2D012023ED06EB96C3944053ED41 6D0CA8930D93ED831AC234D71A13EDCBC8CE897FFCA3EE0DB00291A14073EE38E9A32A E4E2D3EE678D5B0F87AB03EE9998B3C7806C33EECF0A13092CD983EF03F0329BB47013 EF220D760B00DAF3EF41DC909D0F5133EF635D51D9BD6A83EF868F962A3CF2C3EFAB73 43016427D3EFD20844478CE183EFFA4E8A96653CD3F0122304FEB704D3F027F75C9112 C143F03EA4484F68A6E3F05629C546638B03F06E87D116B07613F087BE69D4703163F0 A1CD8DEE9A0443F0BCB53C1C308973F0D875734DC69B33F0F50E32A23EAD73F1093FBC AEFD3D33F11864A371F36693F127F5CD57FA87C3F137F33A295823E3F1485CE9B6D78E 13F15932DBD859CC83F16A75106BAA1D33F17C238753891B93F18E3E4076E38DD3F1A0 C53BC02C6663F1B3B8791CD31DD3F1C717F87CD134F3F1DAE3B9D24A76D3F1EF1BBD11 3CCA13F201E001179E5113F20C6844919DFEA3F21726A8F2AD12E3F2221B2E375D4E63 F22D45D45CB11DF3F238A69B600CA3C3F2443D833F28F683F2500A8BF8091EF3F25C0D B588F0A9D3F26846FFF05B7E73F274B66B2CF6D573F2815BF73D9B13C3F28E37A42146 8C03F29B4971D718D5A3EC93E675C21D1123EC1513FE43D41263EC3970688FFFC4F3EC 886A3018F7ED53ECF019815D31E283ED35E122029B0883ED7CDB8AB5717143EDCCA0C6 36764DD3EE128275D3649A33EE42F867418F7833EE77ABA78F9151F3EEB09848BC66EB 13EEEDBBD450071613EF178A580067C473EF3A50E418CE17B3EF5F312F531D1F43EF86 2AF679E87FB3EFAF3E09150DCF33EFDA6A438DBEDCD3F003D7C5AB68F123F01B86E62C 505D53F034427B966CC133F04E0A7FF17E8353F068DEEE9044C273F084BFC3BFE252B3 F0A1ACFC8D020593F0BFA6969850E623F0DEAC8FF5EFC003F0FEBEE714D5D663F10FEE CD5602E293F1210454D60933E3F132A00999EF1873F144C1EB43E56D93F15769F98562 9963F16A98341C5F5053F17E4C9AD1211463F192872D74733C33F1A747EBDE36CA53F1 BC8ED5EC36FCD3F1D25BEB81346FF3F1E8AF2C841BEC03F1FF8898DF606023F20B7418 4038A3C3F21766F9ABA410A3F2239CF0AB045D33F23015FD38587B83F23CD21F4E6AA5 D3F249D156E8B3DB13F25713A40343C083F26499069AAC17F3F272617EABEF4CB3F280 6D0C34718163F28EBBAF31EBC9C3F29D4D67A26136F3F2AC223584154BF3F2BB3A18D5 83F693F2CA9511955A6A23F2DA331FC2711323F2EA14435BC63D13F2FA387C60798253 ECA71AED7AC48063EC31A00FEE4B5593EC62EA1AD2E57D33ECC267899AEABBE3ED1F18 44626833F3ED68C47FFC8F77B3EDBD61C648DA5D63EE0E4A715B5183A3EE4318F63384 8573EE7D10DB6B609B23EEBC2B912CBB5323EF003294BB48BAA3EF24DD96E86CE8A3EF 4C1601F8F90D53EF75DB4C1E5B2113EFA22D163A79BD93EFD10B1D17E88653F0013A98 1F428A13F01B35960BC09813F036767B0AD98E23F052FD3D10BA5643F070C9D46A3A01 23F08FDC3B1F18F373F0B0346C82164CF3F0D1D264E054E223F0F4B621467F8413F10C 6FCFAAA19F33F11F276E90606D73F13281EC8BF7CEB3F1467F48F8E91243F15B1F834F 7DA513F170629B1F26BE33F18648900A153E03F19CD161C1BF89A3F1B3FD10041E5593 F1CBCB9A997723A3F1E43D01529549B3F1FD51440759CAA3F20B84314AC7F363F218B1 2E6FFC1BC3F2262F9966BFC3B3F233FF72244C61E3F24220B89F59B4A3F250936CCFE4 7A03F25F578EAEFE99D3F26E6D1E36A70603F27DD41B61A7F953F28D8C862B7A7263F2 9D965E902E1443F2ADF1A48C54A3E3F2BE9E581CF08B83F2CF9C793F65EDB3F2E0EC07 F16DDEB3F2F28D04310B6B03F3023FB6FE410F63F30B61A2A926F033F314AC45188DF2 83F31E1F9E4BE24D23F327BBAE42A325B3F3318074FC604913ECBBBD03E91DBDC3EC50 313429B11003EC8F5933DAC28DD3ED0051BF3754FE33ED49142031EE58F3ED9F95B0B1 718DA3EE017FDFB166E8C3EE397B8BDAAEA283EE77A7E8651FC0A3EEBBF95F0F14A863 EF0334B001317493EF2B79FE9F03A483EF56CB6024FC4D73EF8528076E4A4803EFB68F 6AAC5B8C63EFEB012AC116B513F0113E823B87B293F02E8163AB119A33F04D4927E3C1 F733F06D95C19F03E993F08F6726D1045C03F0B2BD4FC69A5E03F0D79836878655A3F0 FDF7D666876213F112EE15D8602BB3F127A299B96E0353F13D1975A6C3BEB3F15352A8 AA71C173F16A4E31FBF3AE33F1820C10F6CAD353F19A8C4513409E73F1B3CECDE0C649 03F1CDD3AB018CAE63F1E89ADC270A37B3F2021230879BCB93F210381CC12D2853F21E BF32A89EDB83F22DA77229E16913F23CF0DB33DA6EE3F24C9B6DB7EA3A43F25CA729A9 873033F26D140EFDEAC7C3F27DE21DABCCC013F28F1155AB29D4F3F2A0A1B6F513F093 F2B29341838A05B3F2C4E5F551564B23F2D799D259F1C253F2EAAED8996C0E43F2FE25 080C56F303F308FE3057DA6D83F3131A714074F483F31D6746BED798B3F327E4B0D203 E0A3F33292AF791B9073F33D7142B35C9DD3F348806A801DA363F353C026DECAD5F3F3 5F3077CEE35C53F36AD15D4FF702B3ECD1CCB90D28C973EC70C76AF6054183ECBEBDB3 A796F713ED218F074A1AD7A3ED7600541D2B5FB3EDDA54B421414AE3EE26DABB01A5E6 E3EE67E3B2995284B3EEB02F4C68364B73EEFFB1F22CAB9F83EF2B328A08440923EF5A 226209643413EF8CA745DDB1B933EFC2C068605B73D3EFFC6D3FC2CF5A23F01CD6B673 7A0333F03D40564BAE8DF3F05F73672DF419D3F0836FD74EE38B83F0A9359967E80393 F0D0C4A36D2E6DC3F0FA1CEDAB8DD843F1129F39156354D3F12914961ECC00F3F1406E 8C18925923F158AD19890BC713F171D03D408ED173F18BD7F6492BDB63F1A6C443DA5E 88B3F1C295254FA82713F1DF4A9A21522303F1FCE4A1DECDB6B3F20DB19E1525DFA3F2 1D63345AA150F3F22D87139ED58AF3F23E1D3BC5E54DE3F24F25ACB83642E3F260A066 61B8EF63F2728D68B152EF93F284ECB398649273F297BE470A643B63F2AB0222FC8B62 E3F2BEB8476591C6E3F2D2E0B43D742493F2E77B697D446493F2FC88671F017A13F309 03D68EBACE53F313FC9DBA0CE673F31F2E890F7ED6D3F32A99988D5982B3F3363DCC32 1E1D43F3421B23FC7EB993F34E319FEB57E1A3F35A813FFDAB1A43F3670A04329A2AC3 F373CBEC896305D3F380C6F9015C46E3F38DFB2999F22343F39B687E52A3C1E3F3A90E F72B00EE33ECE94A0CE6E5A293EC9362B45347E9F3ECF1179A3962B883ED44EB9D05C6 E983EDA5DCE0241F4843EE0C80C525FF5773EE4EC175152A2933EE9982E5973D2A03EE ECAF223CC82563EF241D4A6212C083EF560F56AB9566C3EF8C2BBEFA6A67C3EFC67148 128D2D13F0026F9267964803F023BA65B0DB43D3F04718EE5608D743F06C8B0ABC68D4 B3F09410A2A7EA0763F0BDA9A44D2555C3F0E956026585DDB3F10B8AD9729C4633F123 74570C8A3823F13C6777046CF783F156643703A414B3F1716A9530C0F893F18D7A9012 190CF3F1AA94267802D743F1C8B7576C8EBBB3F1E7E42227385E03F2040D4301C085C3 F214AD413CD91873F225D20B8C9E83C3F2377BA1C2293703F249AA03B6346543F25C5D 3147BC6833F26F952A5AE40193F28351EED810DB33F297937EAB337773F2AC59D9C331 7643F2C1A500116B2363F2D774F18956E4E3F2EDC9AE202E3253F302519AE65463E3F3 0E00C44361B2F3F319F25323C7F9B3F32626478486B103F3329CA16302F5E3F33F5560 BCF94843F34C508590717A03F3598E0FDBB46C23F3670DFF9D435553F374D054D3D046 13F382D50F7E37C933F3911C2F9B7B6363F39FA5B52ABCDC33F3AE71A02B3A25D3F3BD 7FF09C49DC73F3CCD0A67D5834E3F3DC63C1CDE455C3F3EC39428D7E0B83ED011A7FBB 2A2513ECB8031041790993ED133373C812E923ED6A67806A5934E3EDD8A9C446CA1303 EE2DCE1998D4DD43EE79340DEBF3AF93EECE5924D46E91C3EF1693B4F16AA783EF4AC9 A36AC126E3EF83CB15EB258EA3EFC1961572164163F00214B361C75473F025C21E5D5E CED3F04BD306C4888C13F074473D0837D953F09F1E9F8DB68833F0CC591618F36433F0 FBF68EDE875423F116FB7E4BEEBB73F1312D2A9C915BA3F14C904887971313F16924D5 10E01403F186EACDE1CBEE33F1A5E23120EC0B53F1C60AFD5495D4F3F1E765314D1FD1 43F204F8660A4D31F3F216D6E6714097B3F2294E19892AC0F3F23C5DFF0E306473F250 0696C80A1F23F26447E087D05F53F27921DC263E5843F28E9489825064A3F2A49FE880 294503F2BB43F9082EA343F2D280BB06510343F2EA562E69760313F3016229917EF783 F30DE594932F9713F31AB55834699B73F327D1747088E2D3F33539E9438BCD93F342EE B6A9FB4F33F350EFDCA0D6E013F35F3D5B25839D63F36DD73235BE0763F37CBD61CF8D F053F38BEFE9F13C3883F39B6ECA994A15F3F3AB3A03C6699AC3F3BB5195777750B3F3 CBB57FAB74BBE3F3DC65C26183A5B3F3ED625D98E1FFC3F3FEAB5150E66503F408204E C47E85D3F411112120528C73F41A281FBBB72E13ED0E46C85DBA5FC3ECDEA87EC0989F D3ED2F55CDD5F02243ED9202B177D1B933EE0733804295DFB3EE51125769213603EEA6 3285860279F3EF03333828735E13EF38CC11AD2EE433EF73DE0430610203EFB465C7C6 F47F93EFFA61657067BF63F022E7D0F85B1583F04B57D811874E23F07680831C6F84B3 F0A461A28A070773F0D4FB14BF97A773F1042660C08817F3F11F2B4B8184C433F13B8C 43FF776713F159494534767753F178624B46ED7BF3F198D7533ABCA883F1BAA85AB943 8C43F1DDD55FE9139063F2012F30A8410F73F21421AEDFF2DEC3F227C22A20A76863F2 3C10A2061C7D13F2510D163E12C4A3F266B78684AEF5A3F27D0FF2A1A9ABE3F294165A 661B56B3F2ABCABDAABF29C3F2C42D1C4E917EC3F2DD3D7635B51603F2F6FBCB488F9A 93F308B40DB988C7D3F316413352104A43F3242556668E7A03F3326076F0BE8A83F340 F294EB3D3763F34FDBB051665ED3F35F1BC91F386083F36EB2DF513C3173F37EA0F2E4 7148E3F38EE603D63CC3F3F39F8212245B2373F3B0751DCCD438C3F3C1BF26CDF0E513 F3D3602D26325E23F3E55830D44AB6C3F3F7A731D7167663F405269816CB9243F40EA5 15EB7743A3F4184F12692D4913F422248D8F98EFB3F42C25875E7052B3F43651FFD573 05E3F440A9F6F468EDC3ED1C29E05B238183ED03A97FE85356C3ED4CF2DB46490793ED BBBD302E307633EE238A4A6FA226F3EE764D7E96E462A3EED5BCDBE3568873EF20D564 0652D4A3EF5D00A751B0C863EF9F5A6CB2F25213EFE7DF6C3F023A83F01B46D77AAF71 63F045B1FCCD01DAC3F07330BF840FC663F0A3C2DAB8902EC3F0D7681EDB766253F107 103529061933F123F5D270203543F14264DE5D55F8D3F1625D524D5CF2E3F183DF293A 4B9933F1A6EA5F4A8D7433F1CB7EF18202B623F1F19CDD8A0AEF03F20CA210C49BC403 F2213A5E02EEF583F2369755E82A4EE3F24CB8F7F9560263F2639F43D22FE093F27B4A 3920789163F293B9D7A054CBA3F2ACEE1F197D2A83F2C6E70F5D0A1E93F2E1A4A843B6 DA13F2FD26E9AC7FB763F30CB6E9BDC3BB83F31B3CB2CC99E1F3F32A24CFF8BA9043F3 396F4139989603F3491C0687E419D3F3592B1FDD584DE3F3699C8D3491ED93F37A704E 88ECD8E3F38BA663D6676E43F39D3ECD198AA193F3AF398A4F55EC73F3C1969B752E6A 73F3D4560088D09CC3F3E777B9884454A3F3FAFBC671D273B3F4077113A1FE17D3F411 956DFEB9CC93F41BEAF24E8A9E33F42671A090F14E33F4312978C57EC0C3F43C127AEB D1EF03F4472CA703962703F45277FD0C8182F3F45DF47D06539643F469A226F0D44763 ED2AC3C7B36589F3ED190149B8D19903ED6C0A9C191D9913EDE796FC8D756C23EE4159 40AA89DF03EE9D7F8F221E6293EF03E98881F7ED93EF40131603D5ACC3EF833175DEF0 5423EFCD3EDCF27517A3F00F1C01A9A75F73F03B0D79007DD5C3F06A7336DFBBA3F3F0 9D4CD4B4F836E3F0D39A0D98EA89C3F106AD58FE42F503F1254750228A1483F1459AE0 1B420A43F167A80002B747D3F18B6EA9359F5DB3F1B0EED6AE10C143F1D828849276FB 33F2008DD7F3591E43F215E42B2A110B33F22C173B00ABF8E3F2432706BA549C83F25B 138DBF363863F273DCCF9458FEF3F28D82CBD57AC1A3F2A80582305C27B3F2C364F261 21E693F2DFA11C2F849BF3F2FCB9FF6C9CB683F30D57CDF892B423F31CC0F8CE0D8683 F32C988028D03313F33CDE63FD0D8E13F34D92A440BDD983F35EB540EB53E4E3F37046 39F57FD623F382458F58FCE053F394B3411067BE73F3A78F4F171C5063F3BAD9B96918 F483F3CE928002E69F73F3E2B9A2E184C9E3F3F74F2202588FD3F406297EB18F3983F4 10E29A80EF2193F41BD2E46E707133F426FA5C7953C413F4325902A0F22233F43DEED6 E4B9D053F449BBD9442B9213F455C009BED84973F461FB68545EF173F46E6DF5046AD7 23F47B17AFCEB214A3F487F898B2F43E33F49510AFB0F93993ED3A147E668079C3ED2F 5B9CD1C71713ED8C9D104E6DD6C3EE0AC80B4AD04DB3EE60A062F34D0843EEC6A8890A CF3663EF1E3A9273E73903EF60EC4E20FBE623EFAB5E7D54ED8883EFFD8B54EEE952F3 F02BB7C681ED06A3F05C8497499F0CC3F0912B7F308870F3F0C9AC17A4409FE3F10303 0DDEBF4AA3F1231CADF69ACF53F14522DB4C57C6B3F1691589C1ED96A3F18EF4B071A8 B1E3F1B6C048B83EA8A3F1E0784D8FC5F0A3F2060E5D8F550833F21CD6C734659E53F2 3495628BC481A3F24D4A2EA8BA67E3F266F52ACE7204E3F281965665169C03F29D2DB0 F1B05F73F2B9BB3A0FFD2063F2D73EF16DBD85C3F2F5B8D6C71646A3F30A9474F1DFFF 73F31AC7954A698EF3F32B75CC59856B13F33C9F1A0EB1C1A3F34E437E5BFFF283F360 62F935A2D323F372FD8A9192A493F3861332674236C3F399A3F0AF61AFF3F3ADAFC563 AC4223F3C236B07EBEA9F3F3D738B1FBF4C5C3F3ECB5C9D74CF4E3F40156FC06A81593 F40C909E4D7EF0D3F41807CBBEDD9AA3F423BC8459A252C3F42FAEC81CD10273F43BDE 97078E1A13F4484BF1191A3503F454F6D650CE5B43F461DF46AE18D263F46F0542307A 5D43F47C68C8D783DEE3F48A09DAA2D450D3F497E877921700E3F4A6049FA5020943F4 B45E52DB54FDB3F4C2F59134D7C4D-%&COLORG6&%$RGBG$\"\")F1$\"\"$F1F<-%*LIN ESTYLEG6#\"\"#-%+AXESLABELSG6%%%b[7]G%'a[6,5]G%!G-%*AXESSTYLEG6#%$BOXG -%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_4G-%%VIEWG6%F&F,;F'$F=!\"'- %+PROJECTIONG6%$!#dF($\"#gF(F." 1 2 0 2 10 0 2 1 6 2 2 1.000000 60.000000 -57.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "evalf[15](minimize(sm,b[7]= 0..0.1,a[6,5]=1e-10..0.5,'location'));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"0t$)3j,Qb\"!#D<#7$<$/&%\"bG6#\"\"($\".*f!fXKd(!#9/&%\"aG6$\" \"'\"\"&$\"0=V%pBh=K!#:F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use rational approximations for the values of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 54 " that mim imize the norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "[.7573245590599e-1,.321861236944318];\nmap(convert ,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\".*f!fXKd(!#9$ \"0=V%pBh=K!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"#W\"$\"e#\"#G\" #()" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We set " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7] = 44/58 1;" "6#/&%\"bG6#\"\"(*&\"#W\"\"\"\"$\"e!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,5] = 28/87;" "6#/&%\"aG6$\"\"'\"\"&*&\"#G\"\"\"\"#( )!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "The resulting s cheme has the following value for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG 6$\"\"(\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "a[7,5]=simplify(subs(\{b[7]= 44/581,a[6,5]=28/87\},subs(e5,a[7,5])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#\"GxJd'Rex$\\Fx\"G&[G@.$)oh\"I+Sa'*GpS#\\t0T .m+F:iC1#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We can simplify the coefficients slightly if we replace the rat ional expression for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"& " }{TEXT -1 65 " by a simpler one which agrees with the value above t o 5 digits." }}{PARA 0 "" 0 "" {TEXT -1 64 "This will not change the c haracteristics of the scheme too much." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "61688303212848528177274 937758396573177/2062462152700660341057349240692896544000.;\nconvert(%, rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?H+\"*H!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"#I\"%.5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "`b*`[3 ] = 635/1588;" "6#/&%#b*G6#\"\"$*&\"$N'\"\"\"\"%)e\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5] = 31/307;" "6#/&%#b*G6#\"\"&*&\"#J \"\"\"\"$2$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 139 "Thes e values are chosen so as to give a principal error norm for the order 4 embedded scheme roughly 6.45 times that of the order 5 scheme." }} {PARA 0 "" 0 "" {TEXT -1 16 "The values for " }{XPPEDIT 18 0 "`b*`[3] " "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6# &%#b*G6#\"\"&" }{TEXT -1 110 " also ensure that the stability region \+ for the embedded scheme is compatible with that of the order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "e6 := \{c[3]=83/311,b[7]=44/581,a[6,5]=28/87,a[7,5]=30/1003,` b*`[3]=635/1588,`b*`[5]=31/307\};\ne7 := `union`(e6,simplify(subs(e6,e 3))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e6G<(/&%\"aG6$\"\"'\"\"&# \"#G\"#()/&%#b*G6#F+#\"#J\"$2$/&%\"cG6#\"\"$#\"#$)\"$6$/&F(6$\"\"(F+# \"#I\"%.5/&%\"bG6#FA#\"#W\"$\"e/&F1F9#\"$N'\"%)e\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1540 "e7 := \{a[7,4] = 25890 2691791384553326881/1307299004967194685810000, a[4,1] = 109269372/1647 87475, `b*`[2] = 0, a[8,7] = 29/395, a[8,2] = 0, b[7] = 29/395, c[5] = 66/85, c[2] = 1/6, c[3] = 155/594, c[8] = 1, b[2] = 0, a[6,3] = -5760 6687522659088/29968516930563575, a[6,1] = -22877286938601/286574390921 00, c[4] = 12/19, c[6] = 27/34, `b*`[4] = 366435142393882979/152755507 9142977500, b[5] = 2059161125/95272074612, a[4,3] = 357797088/16478747 5, a[8,1] = 51964327/654617700, c[7] = 1, `b*`[1] = 4457141939387131/5 6114268213415500, a[5,4] = 54884775429/118168704250, a[2,1] = 1/6, `b* `[8] = 8182415847667565831/54150262660304141458050, b[6] = 1793996689/ 9043706700, `b*`[5] = 1/57, `b*`[6] = 2399113615956319/118927298549193 75, a[6,2] = 2786718411/910771940, a[5,2] = 37863936/17514845, b[4] = \+ 14557376984/60894963675, a[5,3] = -383179645721664/283900311960625, a[ 3,2] = 24025/117612, a[4,2] = -2341872/1063145, `b*`[7] = 330696985258 9408717937/45125218883586784548375, b[1] = 51964327/654617700, b[3] = \+ 6965638145689311/17945236957333900, a[7,2] = -7556229/2688010, a[5,1] \+ = -67885276899/135740048750, a[3,1] = 6665/117612, a[8,6] = 1793996689 /9043706700, a[8,5] = 2059161125/95272074612, a[8,4] = 14557376984/608 94963675, a[8,3] = 6965638145689311/17945236957333900, a[7,6] = 138208 410082586660479/251926265586772035000, a[7,3] = 1811005935036153778848 695517/745590435061329862990895000, a[7,1] = 357804108795684026067391/ 386149007682840983490000, a[7,5] = -154/529, a[6,5] = 55/406, a[6,4] = 3984899849779/12473888420630, `b*`[3] = 71/183\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "subs(e7,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3 ,1],a[3,2],``],\n [c[4],a[4,1],a[4,2],a[4,3]],\n [c[5],a[5,1 ],a[5,2],a[5,3]],[``$3,a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3]],[`` $2,a[6,4],a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5 ],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6] ],[``$3,a[8,7]],\n [`b`,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],[``$3 ,b[7]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*` [6]],[``$2,`b*`[7],`b*`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#747&#\"\"*\"#`F(%!GF+7&#\"#$)\"$6$#\"&<&**\"(y4u\"#\"'<^OF2F+ 7&#\"$n\"\"$]##\",()*GsjI\",+]i](Q#!+^Fk!p(\"+](oz\"H#\"-t&*y*\\q#\"-+ ]iSw57&#\"#M\"#V#!0yc5q`#=O\"0`/m!4GuF#\"-44#H+&\\\"-lL1v+5#!2\"yyQKCv $))*\"2&o.unLHyG7&F+F+F+#\"/+D,P`G\")\"06dY`&[&R\"7&#\"#A\"#F#!4=WI23n 1_c(\"4(*)*RG3xTFo$#\"-jt&*o&)*)\"-D%oW(Q7#!6*f&pMR=HZ'*G(\"6D]4Zv.$Qa p97&F+F+#\"1+vALMw=S\"2n2/(G%*G$e\"#\"#G\"#()7&\"\"\"#\"?,Fm-+$7&F+#\"?vV'*4*oS;8'3O5nwR\"@_5^m\"=@w*e,p1Slc\" #\"#I\"%.5#\":&HC&f0$=N]pv&p)\";_>\\&4cn%7&F+#\"1+vVtxJ[L\"2j\\zZF/m=\"#\"+*o$zr; \",)Gw$>k\"#\"+@\")H%o%\",ob#Qky7&F+F+F+#\"#W\"$\"e7&%\"bGFgpFjpF[qF^q Fhq7&%#b*G#\"6v&*GV\\E3Ydf#\"7Yv(*GjD#zTF@$Fjp#\"$N'\"%)e\"7&F+#\":]P% [r;N8C;!HR$\";zxO\")[EdER6<,7#\"#J\"$2$#\"6kde2lYHgOb*\"8:F^&G7*e3O()e \"7&F+F+#\"=#)[zON:b8>tY4'3#\">NXkZV1cy#Gx&)fiF#\"9Je#f0$[3d>J:K\"=!e5 fI/L0H$3GPJ8Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "subs(e7,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(` b*`[i],i=1..8)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7+7+$\")K6)p\"!\")F(%!GF+F+F+F+F+F+7+$\").\")oEF*$\")\\:;d! \"*$\")[>(4#F*F+F+F+F+F+F+7+$\")++!o'F*$\")UD1zF*$!)_hNE!\"($\"))*)H^# F;F+F+F+F+F+7+$\")n(p!zF*$!)K@/8F;$\")jJY\\F;$!)L*QV$F;$\")&y[#eF*F+F+ F+F+7+$\")\"[\"[\")F*$!)ABa?F;$\")t'QD(F;$!)J[g\\F;$\")oBQDF*$\")3R=KF *F+F+F+7+$\"\"\"\"\"!$\")^.)=\"F;$!);bUOF;$\")sPEIF;$\")e]QDF*$\")p-\" *HF1$\")RyV9F*F+F+7+FW$\")O&z2)F1$FYFY$\")sI**RF*$\")Xw@GF*$\")M==5F*$ \")dMcfF1$\")(\\Jd(F1F+7+%\"bGFaoFcoFdoFfoFhoFjoF\\pF+7+%#b*G$\")``z!) F1Fco$\")1u)*RF*$\")8mCGF*$\")?x45F*$\"):P8gF1$\")X?^vF1$\")].:C!#6Q(p print36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The norm of the principal error terms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "errterms5_7 := PrincipalErrorTerms (5,7,'expanded'):\nsqrt(add(evalf(subs(e7,errterms5_7[i]))^2,i=1.. nop s(errterms5_7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-%>#[R!#:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The stab ility function for the order 5 scheme is " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "expand(subs(e7,StabilityFunction(5,7,'expanded')));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$%\"zGF$*&#F$\"\"#F$*$)F%F( F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*&#F$\"$? \"F$*$)F%\"\"&F$F$F$*&#\"HBxsh\\*QJ0\">\"K++AMg@+8LmfY#eND$* ))eFP\"F$*$)F%F-F$F$F$*&#\"E6\"G/e\\<0t/Q,$R0jc&H'\"I+b&3S0DGe\"\\;c*Q 8LA(*=V$F$*$)F%\"\"(F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the weights " }{XPPEDIT 18 0 "`b^`[i], i=1" "6$&%#b^G6#%\"iG/F&\"\"\"" }{TEXT -1 26 " . . 7, ( ini tially with " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 136 " as a parameter) for an auxiliary 7 stage, order 4 embedded sche me by using the quadrature conditions along with the requirements that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^`[7]=b[7]" "6 #/&%#b^G6#\"\"(&%\"bG6#F'" }{TEXT -1 7 " qnd " }{XPPEDIT 18 0 "`b^`[ 2]=0" "6#/&%#b^G6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "quad_eqns : = subs(b=`b^`,QuadratureConditions(4,7,'expanded')):\ne8 := \{`b^`[2]= 0\}:\ne9 := `union`(e7,e8):\ncdns2 := [`b^`[7]=b[7],op(quad_eqns)]:\ne qns2 := subs(e9,cdns2):\nnops(eqns2);\nindets(eqns2);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%#b^G6#\"\"\"&F%6#\"\"%&F%6#\"\"$&F%6#\"\"'&F%6#\"\"(&F%6#\"\"& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 := solve(\{op( eqns2)\},\{`b^`[1],`b^`[3],`b^`[4],`b^`[5],`b^`[7]\}):\ne11 := `union` (e9,e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The coefficients are expressed in terms of " }{XPPEDIT 18 0 "` b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#&%#b^G6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2056 "e11 := \{c[2] = 9/53, c[5] = 34/43, `b*`[2] = 0, b[ 4] = 3348317773437500/11866042747794963, a[6,5] = 28/87, c[6] = 22/27, c[4] = 167/250, b[5] = 1671793689/16419376288, a[5,2] = 495002920909/ 100075063365, `b^`[2] = 0, a[8,3] = 186846453120293/467197050726588, ` b^`[7] = 44/581, a[3,2] = 365117/1740978, c[8] = 1, c[7] = 1, b[2] = 0 , a[2,1] = 9/53, `b*`[4] = 3392901624133516714843750/12011711392657264 881367779, b[6] = 4684298121/78643825568, `b*`[5] = 31/307, `b*`[6] = \+ 955366029466507585764/15887360858912285512715, a[8,2] = 0, a[8,5] = 16 71793689/16419376288, a[7,2] = -4034678053/1107651600, `b^`[5] = 13094 8029/733007870-3987713418007/3091670122545*`b^`[6], `b*`[8] = 32153119 5708483055925831/1331372808329053304305910580, c[3] = 83/311, `b^`[4] \+ = 3145144531250/11973807010893+44284625000000/135215400685833*`b^`[6], a[8,7] = 44/581, `b^`[3] = 102302865631/253856254470-18362657495336/3 56903761195215*`b^`[6], a[7,5] = 30/1003, a[4,3] = 270499789573/107640 625000, a[5,4] = 81285337012500/139548553465711, `b*`[7] = 20860946731 91355153536794882/27625985772827856064347644535, a[6,2] = 898568957363 /123874468425, a[5,3] = -98837524323878781/28782933677403685, a[5,1] = -361825370105678/277428090660453, a[8,4] = 3348317773437500/118660427 47794963, `b^`[1] = 63834274/4638043071*`b^`[6]+395671/4948377, b[7] = 44/581, a[6,4] = 4018763433227500/15832894287040767, `b*`[1] = 259574 608264943289575/3212741792256328977546, a[4,1] = 30637228987/387506250 00, a[7,1] = 930898979175170148742163494117/78356186228420734825973337 9696, a[4,2] = -7690642751/2917968750, a[3,1] = 99517/1740978, b[3] = \+ 186846453120293/467197050726588, a[6,3] = -728964729183934695599/14695 4383037547095025, a[8,6] = 4684298121/78643825568, b[1] = 70352057/870 914352, a[8,1] = 70352057/870914352, a[6,1] = -7565206670807304418/368 2741770828399897, a[7,4] = 397667103608613164068909964375/156654006690 1589762118166511052, a[7,3] = 90799375237487744698230149845605767/3000 2662701192241345485814174532400, a[7,6] = 8695756950351830559524295/60 228935845641756095491952, `b*`[3] = 635/1588\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "We choose " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 88 " so that the principal error norms of the two embedded schemes are ap proximately equal." }}{PARA 0 "" 0 "" {TEXT -1 66 "First we obtain the principal error of the 8 stage order 4 scheme." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 160 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorT erms(4,8,'expanded')):\nprin_err4_8 := sqrt(add(subs(e7,`errterms4_8*` [i])^2,i=1.. nops(`errterms4_8*`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gc1YD!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The principal error norm of the 7 stage order 4 scheme is expressed in terms of " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G 6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 " `errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\np rin_err4_7 := sqrt(add(subs(e11,`errterms4_7^`[i])^2,i=1.. nops(`errte rms4_7^`)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,prin_err4_7G,$*&#\" \"\"\"I+?N^\\p!3?=\\b8Oz\")R;rY\"F(*&\"W!>%*e'H$>y&Qo)H!o3(4l`Y!4uZ%4R JZ'#F(\"\"#*$),&\"+@\")H%o%!\"\"*&\",ob#QkyF(&%#b^G6#\"\"'F(F(F-F(F,F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "prin_err4_7=prin_err4_8:\nsolve(%):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+f.J(*e!#6$\"+P5Q:gF%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "We choose a ratio nal approximation for the larger of the two values since this value gi ves a larger real stability interval than the other value." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 ".6015381037e-1;\nconvert(%,rational ,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P5Q:g!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\")\"$L\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "`b^`[6] = 8/133; " "6#/&%#b^G6#\"\"'*&\"\")\"\"\"\"$L\"!\"\"" }{TEXT -1 89 " which giv es the following principal error norm for the 7 stage order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eval(prin_err4_7,`b^`[6]=8/133):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+eODJD!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "e12 := `union`(\{`b^`[6 ]=8/133\},subs(`b^`[6]=8/133,e11)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1972 "e12 := \{`b^`[4] = 421475563121093750/1492 642808170910487, c[2] = 9/53, c[5] = 34/43, `b*`[2] = 0, `b^`[5] = 831 11238435887/822384252596970, `b^`[1] = 49834625381/616859728443, b[4] \+ = 3348317773437500/11866042747794963, a[6,5] = 28/87, `b^`[6] = 8/133, c[6] = 22/27, c[4] = 167/250, b[5] = 1671793689/16419376288, `b^`[3] \+ = 37965116260159111/94936400477927190, a[5,2] = 495002920909/100075063 365, `b^`[2] = 0, a[8,3] = 186846453120293/467197050726588, `b^`[7] = \+ 44/581, a[3,2] = 365117/1740978, c[8] = 1, c[7] = 1, b[2] = 0, a[2,1] \+ = 9/53, `b*`[4] = 3392901624133516714843750/12011711392657264881367779 , b[6] = 4684298121/78643825568, `b*`[5] = 31/307, `b*`[6] = 955366029 466507585764/15887360858912285512715, a[8,2] = 0, a[8,5] = 1671793689/ 16419376288, a[7,2] = -4034678053/1107651600, `b*`[8] = 32153119570848 3055925831/1331372808329053304305910580, c[3] = 83/311, a[8,7] = 44/58 1, a[7,5] = 30/1003, a[4,3] = 270499789573/107640625000, a[5,4] = 8128 5337012500/139548553465711, `b*`[7] = 2086094673191355153536794882/276 25985772827856064347644535, a[6,2] = 898568957363/123874468425, a[5,3] = -98837524323878781/28782933677403685, a[5,1] = -361825370105678/277 428090660453, a[8,4] = 3348317773437500/11866042747794963, b[7] = 44/5 81, a[6,4] = 4018763433227500/15832894287040767, `b*`[1] = 25957460826 4943289575/3212741792256328977546, a[4,1] = 30637228987/38750625000, a [7,1] = 930898979175170148742163494117/783561862284207348259733379696, a[4,2] = -7690642751/2917968750, a[3,1] = 99517/1740978, b[3] = 18684 6453120293/467197050726588, a[6,3] = -728964729183934695599/1469543830 37547095025, a[8,6] = 4684298121/78643825568, b[1] = 70352057/87091435 2, a[8,1] = 70352057/870914352, a[6,1] = -7565206670807304418/36827417 70828399897, a[7,4] = 397667103608613164068909964375/15665400669015897 62118166511052, a[7,3] = 90799375237487744698230149845605767/300026627 01192241345485814174532400, a[7,6] = 8695756950351830559524295/6022893 5845641756095491952, `b*`[3] = 635/1588\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "`RK4_7eqs^` := subs(b=`b^ `,OrderConditions(4,7,'expanded')):\nsimplify(subs(e12,`RK4_7eqs^`)): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The coef ficients for the whole scheme including the 7 stage and 8 stage embedd ed order 4 schemes are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 556 "subs(e12,matrix([[c[2],a[2,1],``$3],\n [c[3],a[ 3,1],a[3,2],``$2],\n [c[4],a[4,1],a[4,2],a[4,3],``],\n [c[5] ,a[5,1],a[5,2],a[5,3],a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3],a[6,4 ]],[``$4,a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3],a[7,4]],[``$3,a[7, 5],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3],a[8,4]],[``$2,a[8,5],a[8 ,6],a[8,7]],\n [`b`,b[1],b[2],b[3],b[4]],[``$2,b[5],b[6],b[7]],\n [`b^`,`b^`[1],`b^`[2],`b^`[3],`b^`[4]],[``$2,`b^`[5],`b^`[6],`b^ `[7]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[ 6],`b*`[7],`b*`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #727'#\"\"*\"#`F(%!GF+F+7'#\"#$)\"$6$#\"&<&**\"(y4u\"#\"'<^OF2F+F+7'# \"$n\"\"$]##\",()*GsjI\",+]i](Q#!+^Fk!p(\"+](oz\"H#\"-t&*y*\\q#\"-+]iS w5F+7'#\"#M\"#V#!0yc5q`#=O\"0`/m!4GuF#\"-44#H+&\\\"-lL1v+5#!2\"yyQKCv$ ))*\"2&o.unLHyG#\"/+D,P`G\")\"06dY`&[&R\"7'#\"#A\"#F#!4=WI23n1_c(\"4(* )*RG3xTFo$#\"-jt&*o&)*)\"-D%oW(Q7#!6*f&pMR=HZ'*G(\"6D]4Zv.$Qap9#\"1+vA LMw=S\"2n2/(G%*G$e\"7'F+F+F+F+#\"#G\"#()7'\"\"\"#\"?,Fm-+$#\"?vV'*4*oS;8'3O5nwR\"@_5^m\"=@w*e,p1Slc\"7'F+F+F+# \"#I\"%.5#\":&HC&f0$=N]pv&p)\";_>\\&4cn%#\"1+vVtxJ[L\"2j\\zZF/m=\"7'F+F+#\"+*o$zr ;\",)Gw$>k\"#\"+@\")H%o%\",ob#Qky#\"#W\"$\"e7'%\"bGFfpFipFjpF]qF`q7'%# b^G#\",\"QDY$)\\\"-V%G(fohFip#\"26\"f,E;^'z$\"2!>FzZ+k$\\*#\"3]P4@Jcv9 U\"4([54<3Gk#\\\"7'F+F+#\"/()eVQ76$)\"0qpf_UQA)#\"\")\"$L\"Fgq7'%#b*G# \"6v&*GV\\E3Ydf#\"7Yv(*GjD#zTF@$Fip#\"$N'\"%)e\"#\":]P%[r;N8C;!HR$\";z xO\")[EdER6<,77'F+#\"#J\"$2$#\"6kde2lYHgOb*\"8:F^&G7*e3O()e\"#\"=#)[zO N:b8>tY4'3#\">NXkZV1cy#Gx&)fiF#\"9Je#f0$[3d>J:K\"=!e5fI/L0H$3GPJ8Q(ppr int46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absol ute stability regions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 58 "#-------------------------------------- -------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficien ts of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1861 "ee := \{c[2]=9/53,\nc[3]=83/311, \nc[4]=167/250,\nc[5]=34/43,\nc[6]=22/27,\nc[7]=1,\nc[8]=1,\n\na[2,1]= 9/53,\na[3,1]=99517/1740978,\na[3,2]=365117/1740978,\na[4,1]=306372289 87/38750625000,\na[4,2]=-7690642751/2917968750,\na[4,3]=270499789573/1 07640625000,\na[5,1]=-361825370105678/277428090660453,\na[5,2]=4950029 20909/100075063365,\na[5,3]=-98837524323878781/28782933677403685,\na[5 ,4]=81285337012500/139548553465711,\na[6,1]=-7565206670807304418/36827 41770828399897,\na[6,2]=898568957363/123874468425,\na[6,3]=-7289647291 83934695599/146954383037547095025,\na[6,4]=4018763433227500/1583289428 7040767,\na[6,5]=28/87,\na[7,1]=930898979175170148742163494117/7835618 62284207348259733379696,\na[7,2]=-4034678053/1107651600,\na[7,3]=90799 375237487744698230149845605767/30002662701192241345485814174532400,\na [7,4]=397667103608613164068909964375/1566540066901589762118166511052, \na[7,5]=30/1003,\na[7,6]=8695756950351830559524295/602289358456417560 95491952,\na[8,1]=70352057/870914352,\na[8,2]=0,\na[8,3]=1868464531202 93/467197050726588,\na[8,4]=3348317773437500/11866042747794963,\na[8,5 ]=1671793689/16419376288,\na[8,6]=4684298121/78643825568,\na[8,7]=44/5 81,\n\nb[1]=70352057/870914352,\nb[2]=0,\nb[3]=186846453120293/4671970 50726588,\nb[4]=3348317773437500/11866042747794963,\nb[5]=1671793689/1 6419376288,\nb[6]=4684298121/78643825568,\nb[7]=44/581,\n\n`b^`[1]=498 34625381/616859728443,\n`b^`[2]=0,\n`b^`[3]=37965116260159111/94936400 477927190,\n`b^`[4]=421475563121093750/1492642808170910487,\n`b^`[5]=8 3111238435887/822384252596970,\n`b^`[6]=8/133,\n`b^`[7]=44/581,\n\n`b* `[1]=259574608264943289575/3212741792256328977546,\n`b*`[2]=0,\n`b*`[3 ]=635/1588,\n`b*`[4]=3392901624133516714843750/12011711392657264881367 779,\n`b*`[5]=31/307,\n`b*`[6]=955366029466507585764/15887360858912285 512715,\n`b*`[7]=2086094673191355153536794882/276259857728278560643476 44535,\n`b*`[8]=321531195708483055925831/1331372808329053304305910580 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#----------------- ----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 stage, order 5 scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stabi lity function R for the 7 stage, order 5 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(subs(ee,StabilityFunction(5,7,'expanded'))):\nR := unappl y(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"z G,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*& #F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"HBxsh\\*QJ0 \">\"K++AMg@+8LmfY#eND$*))eFP\"F)*$)F'F1F)F)F)*&#\"E6\"G/e\\ <0t/Q,$R0jc&H'\"I+b&3S0DGe\"\\;c*Q8LA(*=V$F)*$)F'\"\"(F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"! \"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 \+ := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+b hF!4%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1], z=-4.59..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],style=point ,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,- 1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],v iew=[-4.59..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3')********** ***e%!#<$!3,)[pr6uh]#F*7$$!3mmmTG_jMXF*$!3ut?u:*RtF#F*7$$!3[LL$oXq#zWF *$!3'>=_#*o&fm?F*7$$!39+D6ly4JWF*$!3-UXv\\q)p*=F*7$$!3#om\"Rt_#HQ%F*$! 3M^\"f3!RPRl&p0%F*$!3kJ^F0aie$*!#=7$$!3KL$e;r;j&RF*$!3VM;ZZB>C wFP7$$!3++]s>+6_QF*$!3U7Yo,Z^9hFP7$$!3IL$e)4$RVu$F*$!3'pC:\\7a_\"[FP7$ $!39+]-AU\"pj$F*$!3%*Q%z8r4]u$FP7$$!3Ymm'\\J9k_$F*$!3:r+ea4'3%GFP7$$!3 GLLoOf3HMF*$!3WP)4l!e5$=#FP7$$!3t*****[B<&>LF*$!3#o$\\$ftT=d\"FP7$$!3( *****\\L')\\4KF*$!3%=>WGQ$or5FP7$$!3z****4D`Z.JF*$!3Cm$\\Qh49x'!#>7$$! 3>L$eQ+'>2IF*$!37>>:]X%**y$Fhp7$$!3ommcy0r#*GF*$!3'zRIfXDeO)!#?7$$!3om mE1ks&z#F*$\"3['RFQ#Qiq7Fhp7$$!3y**\\#zv5Ho#F*$\"3sVIen,#)yLFhp7$$!3]m mE$z\\Ie#F*$\"3hHSK:n8>]Fhp7$$!3#)**\\#z['[tCF*$\"3+fRtfpQ`mFhp7$$!3') **\\x6j:pBF*$\"3a,aQ%3:>6)Fhp7$$!3Emm\"zr)HgAF*$\"3wm(Qj>Y+f*Fhp7$$!37 m;4lMLg@F*$\"3W1;Pcj;&4\"FP7$$!3ALL3Nu]_?F*$\"3[17i`BhY7FP7$$!3SL$eQ\\ 10%>F*$\"3*\\AckS/IT\"FP7$$!3/+]FX$4I%=F*$\"3%y[7x#=To:FP7$$!3ELLy`(4x t\"F*$\"3)ohlet?-v\"FP7$$!35++?s\\#*G;F*$\"3M55$*=uHc>FP7$$!39++l'G+D_ \"F*$\"3ifZv:q')y@FP7$$!3*)**\\7J*G&>9F*$\"3%)z/61#4oT#FP7$$!3z***\\pz '>08F*$\"3\"fs+A$y_5FFP7$$!3/nmE*GkC?\"F*$\"3tc&=@.PFP7$$!3'o****4=m r%))FP$\"3G#*34K)3$GTFP7$$!3Oqm\"f?>Z#yFP$\"3]jk5'RYFd%FP7$$!3E'**\\A^ nfv'FP$\"3)=\\gg:A+(FP7$$!39LL$))o eh[#FP$\"3'*HCuK\\z)z(FP7$$!3yim\">/awT\"FP$\"3a*ysr?[#y')FP7$$!3Mf*** *>C4eVFhp$\"3'e,Cwx]Nd*FP7$$\"3wMLL)*p&\\*oFhp$\"3+i^'z<#Qr5F*7$$\"31m mma=)fp\"FP$\"3IpGHu'G[=\"F*7$$\"370+v?g5pFFP$\"3)H!4cW%[!>8F*7$$\"3g1 +DvFA'z$FP$\"3oDsQRBth9F*7$$\"3!***************[FP$\"3kL@%[3;Bj\"F*-%' COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7 $F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa \\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa \\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa \\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauF a\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwF a\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyF a\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[ \\lF[\\lFh[l-F$6&7#7$$!3u*****\\:w-4%F*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l 6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_ `lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLO RG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICA G\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$f%! \"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1597 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z ^5+\n 191192127213679368741705313894961727723/13727588893253558246 5966331300216034220000*z^6+\n 629556630539301380473051749580428111 /3431897223313389561649158282505400855500*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=201/20 do\n zz := newton(`R`(z)=exp(tt*Pi*I),z=z0 ):\n z0 := zz:\n if (13/20<=tt and tt<=6/5) or (87/10<=tt and tt<= 93/10) then\n hh := 1/40\n else \n hh := 1/20\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 : = plot(pts,color=COLOR(RGB,.38,.18,.18)):\np2 := plots[polygonplot]([s eq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchno grid,color=COLOR(RGB,.75,.35,.35)):\npts := []:\nz0 := 1.4+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z 0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pt s,color=COLOR(RGB,.38,.18,.18)):\np4 := plots[polygonplot]([seq([pts[i -1],pts[i],[1.31,4.19]],i=2..nops(pts))],\n style=patchnogrid ,color=COLOR(RGB,.75,.35,.35)):\npts := []:\nz0 := 1.4-4.3*I:\nfor ct \+ from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,co lor=COLOR(RGB,.38,.18,.18)):\np6 := plots[polygonplot]([seq([pts[i-1], pts[i],[1.31,-4.19]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.75,.35,.35)):\np7 := plot([[[-4.79,0],[1.99,0]],[[0,-4. 79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],v iew=[-4.79..1.99,-4.79..4.79],font=[HELVETICA,9],\n label s=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 512 477 477 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F(7$$ \"3s******z7^)4'!#G$\"3#******fK'zq:!#=7$$\"3#)*****p$Gv,W!#E$\"3$**** **Hf#fTJF07$$\"3G+++.0+sd!#D$\"3z*****\\&yQ7ZF07$$\"3z*******=-Ki$!#C$ \"3?+++Rg<$G'F07$$\"3.+++GR&GV\"!#B$\"3G+++'oKR&yF07$$\"3o********p5xQ FF$\"3Z*****zm$eC%*F07$$\"3w******G'ePi'FF$\"35+++uW\\*4\"!#<7$$\"3/++ +**4')y7FF$\"3)********GikD\"FS7$$!3#******RJBu4%!#A$\"35++++!)G89FS7$ $!3,+++-oRV>!#@$\"3)******f&y%)p:FS7$$!3U+++>BT!>'F\\o$\"3#******4cbfs \"FS7$$!3++++n)\\#=;!#?$\"33+++o!f8)=FS7$$!3$)******\\:&fq$Fgo$\"38+++ y=vN?FS7$$!3o*****>$3(HR&Fgo$\"3!)*****\\7kC6#FS7$$!3E+++QL!Qn(Fgo$\"3 =+++k`z)=#FS7$$!3%******4e<*p5!#>$\"35+++I1qkAFS7$$!3)******4#=?k9F\\q $\"3(******H]K,M#FS7$$!3#******4tA(p>F\\q$\"3#)*****HAQ]T#FS7$$!35+++a .03EF\\q$\"3))*****p$\\N*[#FS7$$!3F+++B&zDS$F\\q$\"3?+++-G+jDFS7$$!3q* *****Hm0yVF\\q$\"3()*****4)f(ej#FS7$$!3'******Rlu(fbF\\q$\"37+++??$yq# FS7$$!3i*****z=+B(pF\\q$\"3?+++o4oyFFS7$$!3Q+++54iP')F\\q$\"3/+++*yw\" [GFS7$$!3)******p&>Fd5F0$\"3A+++()[,;HFS7$$!3/+++9Boy7F0$\"3A+++1-%=)H FS7$$!3&*******R-*y_\"F0$\"3?+++\"4p_/$FS7$$!32+++FSi.=F0$\"3(******\\ 8Cf5$FS7$$!35+++S*zM5#F0$\"3A+++%RvM;$FS7$$!34+++*[eTU#F0$\"3!******40 xw@$FS7$$!3y******yT%=w#F0$\"3A+++5\"*QoKFS7$$!3x*****zQ@E6$F0$\"3-+++ !fzbJ$FS7$$!3!******4fAGZ$F0$\"39+++J+JfLFS7$$!3))*****>!>DRQF0$\"3#)* *****e;r*R$FS7$$!3D+++&y!G4UF0$\"3!******H%)fpV$FS7$$!3-+++$eY3e%F0$\" 37+++h>DrMFS7$$!3a+++`*)yA`F0$\"3z*****fgt<`$FS7$$!3y*****4I-s0'F0$\"3 ))*****H:'z#e$FS7$$!3i******\\CX!y'F0$\"31+++#*>hDOFS7$$!3k*****4Bt7\\ (F0$\"3%******>$>EhOFS7$$!3q*****4J9'*=)F0$\"3#******>[m0p$FS7$$!3-+++ yk*f())F0$\"3&******f))oTr$FS7$$!3a+++W0<^&*F0$\"3=++++UdKPFS7$$!32+++ 1Yf@5FS$\"35+++J1=YPFS7$$!33+++>76(3\"FS$\"3)******>[.`v$FS7$$!3'***** *>!3u^6FS$\"3'******H9$>gPFS7$$!3/+++:>b:7FS$\"3/+++i&[5w$FS7$$!3-+++q )3'y7FS$\"33+++1u-ePFS7$$!3%******\\it4M\"FS$\"3'******R#QD^PFS7$$!3-+ ++)>3FS\"FS$\"37+++RX#3u$FS7$$!3#******RXxQY\"FS$\"3<+++3Q\"os$FS7$$!3 %******zD_X_\"FS$\"3-+++RxF4PFS7$$!35+++uH\"[e\"FS$\"3-+++*Qe#)o$FS7$$ !3'******zK`Zk\"FS$\"3!******H5)yjOFS7$$!3*******fa%[/.KNFS7$$!3!******>PsV%>FS$ \"3#)*****pk,4\\$FS7$$!3&)*****f3\"e0?FS$\"3#)*****ptCnW$FS7$$!3y***** *)*Rvn?FS$\"3%)*****fUA(*R$FS7$$!3=+++R&>68#FS$\"3=+++qK?]LFS7$$!3')** ***H^`e>#FS$\"3%******HVp&)H$FS7$$!3/+++'fN?E#FS$\"39+++[ZIXKFS7$$!3\" )*****H\")4'HBFS$\"3#******4pQ4>$FS7$$!3#)*****\\nk$)R#FS$\"3\")*****p /*)f8$FS7$$!39+++F^%zY#FS$\"33+++yv*33$FS7$$!38+++Qf*y`#FS$\"3/+++b$zf -$FS7$$!3>+++(>Axg#FS$\"3y*****fA,9(HFS7$$!3')*****f1Zpn#FS$\"3%)***** 4J\">\"GFS$\"36+ ++ZY[4GFS7$$!3#******4AEs(GFS$\"35+++@#\\cv#FS7$$!3A+++z\"\\1%HFS$\"39 +++jVc,FFS7$$!39+++>89-IFS$\"3;+++:p.ZEFS7$$!3%******4(RihIFS$\"3#)*** **zz))=f#FS7$$!3$******\\[a!>JFS$\"3$)*****\\Ihf`#FS7$$!31+++qqTuJFS$ \"3y*****z?;\"zCFS7$$!3!******\\(erFKFS$\"3$******>3N7U#FS7$$!3?+++&** p*yKFS$\"3')*****f!y@iBFS7$$!3=+++T+@GLFS$\"3(******z$3)>I#FS7$$!31+++ *)eZvLFS$\"3:+++EcXSAFS7$$!38+++o_\"3U$FS$\"3%)*****pL(ex@FS7$$!3=+++V HGkMFS$\"38+++WSL8@FS7$$!3#******\\?Sf]$FS$\"3)******pRmw/#FS7$$!37+++ \"[aea$FS$\"3%*******4yc!)>FS7$$!3y******R))4%e$FS$\"36+++J^.7>FS7$$!3 #)*****p<^2i$FS$\"3(******>tz?%=FS7$$!3#)*****p)G*el$FS$\"3)*******y)G 2x\"FS7$$!35+++]og*o$FS$\"3/+++*=F!)p\"FS7$$!38+++/Y(>s$FS$\"3\"****** *oy.C;FS7$$!37+++0F2`PFS$\"35+++)QV)[:FS7$$!3A+++G%oHy$FS$\"3++++h^as9 FS7$$!3.+++N^r6QFS$\"3'******H!>E&R\"FS7$$!3$)*****>(yMRQFS$\"3******* 4UErJ\"FS7$$!3!)*****>!)ze'QFS$\"33+++')3GQ7FS7$$!3%******48+8*QFS$\"3 '******4\"4()e6FS7$$!3;+++bRd:RFS$\"3++++m!R!z5FS7$$!3<+++_TkQRFS$\"3= +++!*f=*)**F07$$!31+++#3N/'RFS$\"3k*****\\[\"G'=*F07$$!3%)*****pad3)RF S$\"39+++R)zEQ)F07$$!3'******HI9)**RFS$\"3'******4R!=zvF07$$!31+++ma?< SFS$\"3g*****zdwjx'F07$$!3x*****pELH.%FS$\"3g*****>MnY(fF07$$!3'****** >(e!p/%FS$\"3O+++*4yU<&F07$$!3))*****fF07$$!3!)*****>,@z3%FS$\"3'******>#) )=\">\"F07$$!3o*****4A9+4%FS$\"3')*****>./,(RF\\q7$Ff^m$!3')*****>./,( RF\\q7$Fa^m$!3'******>#))=\">\"F07$F\\^m$!3))******QR$e)>F07$Fg]m$!3w* *****zeB\"y#F07$Fb]m$!3!)*****HP^wd$F07$F]]m$!37+++,oGvVF07$Fh\\m$!3O+ ++*4yU<&F07$Fc\\m$!3g*****>MnY(fF07$F^\\m$!3g*****zdwjx'F07$Fi[m$!3'** ****4R!=zvF07$Fd[m$!39+++R)zEQ)F07$F_[m$!3k*****\\[\"G'=*F07$Fjjl$!3=+ ++!*f=*)**F07$Fejl$!3++++m!R!z5FS7$F`jl$!3'******4\"4()e6FS7$F[jl$!33+ ++')3GQ7FS7$Ffil$!3*******4UErJ\"FS7$Fail$!3'******H!>E&R\"FS7$F\\il$! 3++++h^as9FS7$Fghl$!35+++)QV)[:FS7$Fbhl$!3\"*******oy.C;FS7$F]hl$!3/++ +*=F!)p\"FS7$Fhgl$!3)*******y)G2x\"FS7$Fcgl$!3(******>tz?%=FS7$F^gl$!3 6+++J^.7>FS7$Fifl$!3%*******4yc!)>FS7$Fdfl$!3)******pRmw/#FS7$F_fl$!38 +++WSL8@FS7$Fjel$!3%)*****pL(ex@FS7$Feel$!3:+++EcXSAFS7$F`el$!3(****** z$3)>I#FS7$F[el$!3')*****f!y@iBFS7$Ffdl$!3$******>3N7U#FS7$Fadl$!3y*** **z?;\"zCFS7$F\\dl$!3$)*****\\Ihf`#FS7$Fgcl$!3#)*****zz))=f#FS7$Fbcl$! 3;+++:p.ZEFS7$F]cl$!39+++jVc,FFS7$Fhbl$!35+++@#\\cv#FS7$Fcbl$!36+++ZY[ 4GFS7$F^bl$!3?+++R%pK'GFS7$Fial$!3%)*****4J\">$FS7$F[`l$!39+++[ZIXKFS7$Ff_l$!3%******HVp&)H$FS 7$Fa_l$!3=+++qK?]LFS7$F\\_l$!3%)*****fUA(*R$FS7$Fg^l$!3#)*****ptCnW$FS 7$Fb^l$!3#)*****pk,4\\$FS7$F]^l$!3\")*****p*>.KNFS7$Fh]l$!3!******R9m* pNFS7$Fc]l$!3%)*****pO2Yg$FS7$F^]l$!3!******\\f%*ej$FS7$Fi\\l$!3!***** *H5)yjOFS7$Fd\\l$!3-+++*Qe#)o$FS7$F_\\l$!3-+++RxF4PFS7$Fj[l$!3<+++3Q\" os$FS7$Fe[l$!37+++RX#3u$FS7$F`[l$!3'******R#QD^PFS7$F[[l$!33+++1u-ePFS 7$Ffz$!3/+++i&[5w$FS7$Faz$!3'******H9$>gPFS7$F\\z$!3)******>[.`v$FS7$F gy$!35+++J1=YPFS7$Fby$!3=++++UdKPFS7$F]y$!3&******f))oTr$FS7$Fhx$!3#** ****>[m0p$FS7$Fcx$!3%******>$>EhOFS7$F^x$!31+++#*>hDOFS7$Fiw$!3))***** H:'z#e$FS7$Fdw$!3z*****fgt<`$FS7$$!3A+++tMR_\\F0$!33+++;3z-NFS7$F_w$!3 7+++h>DrMFS7$Fjv$!3!******H%)fpV$FS7$Fev$!3#)******e;r*R$FS7$F`v$!39++ +J+JfLFS7$F[v$!3-+++!fzbJ$FS7$Ffu$!3A+++5\"*QoKFS7$Fau$!3!******40xw@$ FS7$F\\u$!3A+++%RvM;$FS7$Fgt$!3(******\\8Cf5$FS7$Fbt$!3?+++\"4p_/$FS7$ F]t$!3A+++1-%=)HFS7$Fhs$!3A+++()[,;HFS7$Fcs$!3/+++*yw\"[GFS7$F^s$!3?++ +o4oyFFS7$Fir$!37+++??$yq#FS7$Fdr$!3()*****4)f(ej#FS7$F_r$!3?+++-G+jDF S7$Fjq$!3))*****p$\\N*[#FS7$Feq$!3#)*****HAQ]T#FS7$F`q$!3(******H]K,M# FS7$Fjp$!35+++I1qkAFS7$Fep$!3=+++k`z)=#FS7$F`p$!3!)*****\\7kC6#FS7$$!3 ++++(RyP[#Fgo$!3#*******)\\-(e>FS7$$!35+++`QZ?5Fgo$!3/+++aMw.=FS7$$!33 +++E[SzNF\\o$!3'******f\\szk\"FS7$$!3X*****pi,jj*Ffn$!33+++P7h\"\\\"FS 7$$!32+++YZK07Ffn$!3/+++&H**[L\"FS7$$\"3m*****pYk:1'FF$!3%******Ha!*z< \"FS7$$\"3#******zB$))eaFF$!3,+++o=)4-\"FS7$$\"3;+++;z3rCFF$!3&)*****R Oz#R')F07$$\"3i*****>809d(F@$!3W+++N7coqF07$$\"3++++/Fa[:F@$!3!******p n$y(\\&F07$$\"3++++ND1.=F:$!3:+++;0*p#RF07$$\"3)*******G6uKt!#F$!3)*** ***>[%>cBF07$F($!3))*****Rj\")R&yF\\q7$F($\"3))*****Rj\")R&yF\\q-%&COL ORG6&%$RGBG$\"#Q!\"#$\"#=F^fnF_fn-%)POLYGONSG6^y7%F'7$$\"+!G6&)4'Fgo$ \"+Ejzq:!#57$$F^fnF)F(7%Fefn7$$\"+PGv,WF0$\"+$f#fTJFjfnF[gn7%F^gn7$$\" +.0+sdFS$\"+byQ7ZFjfnF[gn7%Fdgn7$$\"+!>-Ki$!#;$\"+Rg<$G'FjfnF[gn7%Fjgn 7$$\"+GR&GV\"!#:$\"+'oKR&yFjfnF[gn7%Fahn7$$\"++q5xQFdhn$\"+oOeC%*FjfnF [gn7%Fhhn7$$\"+H'ePi'Fdhn$\"+uW\\*4\"!\"*F[gn7%F^in7$$\"+**4')y7Fdhn$ \"+!HikD\"FcinF[gn7%Fein7$$!+9LU(4%!#9$\"++!)G89FcinF[gn7%F[jn7$$!+-oR V>!#8$\"+cy%)p:FcinF[gn7%Fbjn7$$!+>BT!>'Fejn$\"+hb&fs\"FcinF[gn7%Fijn7 $$!+n)\\#=;!#7$\"+o!f8)=FcinF[gn7%F_[o7$$!+]:&fq$Fb[o$\"+y=vN?FcinF[gn 7%Ff[o7$$!+K3(HR&Fb[o$\"+DTY7@FcinF[gn7%F\\\\o7$$!+QL!Qn(Fb[o$\"+k`z)= #FcinF[gn7%Fb\\o7$$!+\"e<*p5!#6$\"+I1qkAFcinF[gn7%Fh\\o7$$!+@=?k9F[]o$ \"+.D8SBFcinF[gn7%F_]o7$$!+JFsp>F[]o$\"+B#Q]T#FcinF[gn7%Fe]o7$$!+a.03E F[]o$\"+P\\N*[#FcinF[gn7%F[^o7$$!+B&zDS$F[]o$\"+-G+jDFcinF[gn7%Fa^o7$$ !+Im0yVF[]o$\"+\")f(ej#FcinF[gn7%Fg^o7$$!+aYxfbF[]o$\"+??$yq#FcinF[gn7 %F]_o7$$!+)=+B(pF[]o$\"+o4oyFFcinF[gn7%Fc_o7$$!+54iP')F[]o$\"+*yw\"[GF cinF[gn7%Fi_o7$$!+d>Fd5Fjfn$\"+()[,;HFcinF[gn7%F_`o7$$!+9Boy7Fjfn$\"+1 -%=)HFcinF[gn7%Fe`o7$$!+S-*y_\"Fjfn$\"+\"4p_/$FcinF[gn7%F[ao7$$!+FSi.= Fjfn$\"+NT#f5$FcinF[gn7%Faao7$$!+S*zM5#Fjfn$\"+%RvM;$FcinF[gn7%Fgao7$$ !+*[eTU#Fjfn$\"+^qnDRQFjfn$\"+f;r*R$FcinF[gn7%Feco7$$!+&y!G4UFjfn$\"+ V)fpV$FcinF[gn7%F[do7$$!+$eY3e%Fjfn$\"+h>DrMFcinF[gn7%Fado7$$!+`*)yA`F jfn$\"+1OxJNFcinF[gn7%Fgdo7$$!+,B?dgFjfn$\"+`hz#e$FcinF[gn7%F]eo7$$!+] CX!y'Fjfn$\"+#*>hDOFcinF[gn7%Fceo7$$!+JKF\"\\(Fjfn$\"+K>EhOFcinF[gn7%F ieo7$$!+6Vh*=)Fjfn$\"+#[m0p$FcinF[gn7%F_fo7$$!+yk*f())Fjfn$\"+'))oTr$F cinF[gn7%Fefo7$$!+W0<^&*Fjfn$\"++UdKPFcinF[gn7%F[go7$$!+1Yf@5Fcin$\"+J 1=YPFcinF[gn7%Fago7$$!+>76(3\"Fcin$\"+#[.`v$FcinF[gn7%Fggo7$$!+-3u^6Fc in$\"+VJ>gPFcinF[gn7%F]ho7$$!+:>b:7Fcin$\"+i&[5w$FcinF[gn7%Fcho7$$!+q) 3'y7Fcin$\"+1u-ePFcinF[gn7%Fiho7$$!+DO(4M\"Fcin$\"+CQD^PFcinF[gn7%F_io 7$$!+)>3FS\"Fcin$\"+RX#3u$FcinF[gn7%Feio7$$!+au(QY\"Fcin$\"+3Q\"os$Fci nF[gn7%F[jo7$$!+eAbC:Fcin$\"+RxF4PFcinF[gn7%Fajo7$$!+uH\"[e\"Fcin$\"+* Qe#)o$FcinF[gn7%Fgjo7$$!+GLvW;Fcin$\"+.\")yjOFcinF[gn7%F][p7$$!+YX[/.KNFcinF[gn 7%Fe\\p7$$!+sBPW>Fcin$\"+Z;!4\\$FcinF[gn7%F[]p7$$!+'3\"e0?Fcin$\"+PZsY MFcinF[gn7%Fa]p7$$!+**Rvn?Fcin$\"+ECs*R$FcinF[gn7%Fg]p7$$!+R&>68#Fcin$ \"+qK?]LFcinF[gn7%F]^p7$$!+8N&e>#Fcin$\"+L%p&)H$FcinF[gn7%Fc^p7$$!+'fN ?E#Fcin$\"+[ZIXKFcinF[gn7%Fi^p7$$!+8)4'HBFcin$\"+\"pQ4>$FcinF[gn7%F__p 7$$!+vYO)R#Fcin$\"+Z!*)f8$FcinF[gn7%Fe_p7$$!+F^%zY#Fcin$\"+yv*33$FcinF [gn7%F[`p7$$!+Qf*y`#Fcin$\"+b$zf-$FcinF[gn7%Fa`p7$$!+(>Axg#Fcin$\"+E7S rHFcinF[gn7%Fg`p7$$!+mq%pn#Fcin$\"+68>\"GFcin$\"+ZY[4GFcinF[gn7%Fiap7$$!+@i AxGFcin$\"+@#\\cv#FcinF[gn7%F_bp7$$!+z\"\\1%HFcin$\"+jVc,FFcinF[gn7%Fe bp7$$!+>89-IFcin$\"+:p.ZEFcinF[gn7%F[cp7$$!+rRihIFcin$\"+)z))=f#FcinF[ gn7%Facp7$$!+&[a!>JFcin$\"+08'f`#FcinF[gn7%Fgcp7$$!+qqTuJFcin$\"+3i6zC FcinF[gn7%F]dp7$$!+verFKFcin$\"+#3N7U#FcinF[gn7%Fcdp7$$!+&**p*yKFcin$ \"+1y@iBFcinF[gn7%Fidp7$$!+T+@GLFcin$\"+Q3)>I#FcinF[gn7%F_ep7$$!+*)eZv LFcin$\"+EcXSAFcinF[gn7%Feep7$$!+o_\"3U$Fcin$\"+Ptex@FcinF[gn7%F[fp7$$ !+VHGkMFcin$\"+WSL8@FcinF[gn7%Fafp7$$!+0-%f]$Fcin$\"+(Rmw/#FcinF[gn7%F gfp7$$!+\"[aea$Fcin$\"+5yc!)>FcinF[gn7%F]gp7$$!+S))4%e$Fcin$\"+J^.7>Fc inF[gn7%Fcgp7$$!+x6v?OFcin$\"+K(z?%=FcinF[gn7%Figp7$$!+()G*el$Fcin$\"+ z)G2x\"FcinF[gn7%F_hp7$$!+]og*o$Fcin$\"+*=F!)p\"FcinF[gn7%Fehp7$$!+/Y( >s$Fcin$\"+py.C;FcinF[gn7%F[ip7$$!+0F2`PFcin$\"+)QV)[:FcinF[gn7%Faip7$ $!+G%oHy$Fcin$\"+h^as9FcinF[gn7%Fgip7$$!+N^r6QFcin$\"+.>E&R\"FcinF[gn7 %F]jp7$$!+syMRQFcin$\"+@k7<8FcinF[gn7%Fcjp7$$!+-)ze'QFcin$\"+')3GQ7Fci nF[gn7%Fijp7$$!+J,I\"*QFcin$\"+64()e6FcinF[gn7%F_[q7$$!+bRd:RFcin$\"+m !R!z5FcinF[gn7%Fe[q7$$!+_TkQRFcin$\"+!*f=*)**FjfnF[gn7%F[\\q7$$!+#3N/' RFcin$\"+&[\"G'=*FjfnF[gn7%Fa\\q7$$!+Zv&3)RFcin$\"+R)zEQ)FjfnF[gn7%Fg \\q7$$!+.V\")**RFcin$\"+\"R!=zvFjfnF[gn7%F]]q7$$!+ma?FjfnF[gn7%Fg_q7$$!+75#z3%Fcin$\"+A))=\">\"FjfnF[gn 7%F]`q7$$!+@U,!4%Fcin$\"+KS5qRF[]oF[gn7%Fc`q7$Fd`q$!+KS5qRF[]oF[gn7%Fi `q7$F^`q$!+A))=\">\"FjfnF[gn7%F]aq7$Fh_q$!+RR$e)>FjfnF[gn7%Faaq7$Fb_q$ !+!)eB\"y#FjfnF[gn7%Feaq7$F\\_q$!+t8lxNFjfnF[gn7%Fiaq7$Ff^q$!+,oGvVFjf nF[gn7%F]bq7$F`^q$!+*4yU<&FjfnF[gn7%Fabq7$Fj]q$!+UtmufFjfnF[gn7%Febq7$ Fd]q$!+ylPwnFjfnF[gn7%Fibq7$F^]q$!+\"R!=zvFjfnF[gn7%F]cq7$Fh\\q$!+R)zE Q)FjfnF[gn7%Facq7$Fb\\q$!+&[\"G'=*FjfnF[gn7%Fecq7$F\\\\q$!+!*f=*)**Fjf nF[gn7%Ficq7$Ff[q$!+m!R!z5FcinF[gn7%F]dq7$F`[q$!+64()e6FcinF[gn7%Fadq7 $Fjjp$!+')3GQ7FcinF[gn7%Fedq7$Fdjp$!+@k7<8FcinF[gn7%Fidq7$F^jp$!+.>E&R \"FcinF[gn7%F]eq7$Fhip$!+h^as9FcinF[gn7%Faeq7$Fbip$!+)QV)[:FcinF[gn7%F eeq7$F\\ip$!+py.C;FcinF[gn7%Fieq7$Ffhp$!+*=F!)p\"FcinF[gn7%F]fq7$F`hp$ !+z)G2x\"FcinF[gn7%Fafq7$Fjgp$!+K(z?%=FcinF[gn7%Fefq7$Fdgp$!+J^.7>Fcin F[gn7%Fifq7$F^gp$!+5yc!)>FcinF[gn7%F]gq7$Fhfp$!+(Rmw/#FcinF[gn7%Fagq7$ Fbfp$!+WSL8@FcinF[gn7%Fegq7$F\\fp$!+Ptex@FcinF[gn7%Figq7$Ffep$!+EcXSAF cinF[gn7%F]hq7$F`ep$!+Q3)>I#FcinF[gn7%Fahq7$Fjdp$!+1y@iBFcinF[gn7%Fehq 7$Fddp$!+#3N7U#FcinF[gn7%Fihq7$F^dp$!+3i6zCFcinF[gn7%F]iq7$Fhcp$!+08'f `#FcinF[gn7%Faiq7$Fbcp$!+)z))=f#FcinF[gn7%Feiq7$F\\cp$!+:p.ZEFcinF[gn7 %Fiiq7$Ffbp$!+jVc,FFcinF[gn7%F]jq7$F`bp$!+@#\\cv#FcinF[gn7%Fajq7$Fjap$ !+ZY[4GFcinF[gn7%Fejq7$Fdap$!+R%pK'GFcinF[gn7%Fijq7$F^ap$!+68>$FcinF[gn7%Fa\\r7$Fj^p$!+[ZIXKFcinF[gn7%Fe\\r7$Fd^p$!+L%p&)H$FcinF[gn 7%Fi\\r7$F^^p$!+qK?]LFcinF[gn7%F]]r7$Fh]p$!+ECs*R$FcinF[gn7%Fa]r7$Fb]p $!+PZsYMFcinF[gn7%Fe]r7$F\\]p$!+Z;!4\\$FcinF[gn7%Fi]r7$Ff\\p$!+(*>.KNF cinF[gn7%F]^r7$F`\\p$!+Wh'*pNFcinF[gn7%Fa^r7$Fj[p$!+ntg/OFcinF[gn7%Fe^ r7$Fd[p$!+&f%*ej$FcinF[gn7%Fi^r7$F^[p$!+.\")yjOFcinF[gn7%F]_r7$Fhjo$!+ *Qe#)o$FcinF[gn7%Fa_r7$Fbjo$!+RxF4PFcinF[gn7%Fe_r7$F\\jo$!+3Q\"os$Fcin F[gn7%Fi_r7$Ffio$!+RX#3u$FcinF[gn7%F]`r7$F`io$!+CQD^PFcinF[gn7%Fa`r7$F jho$!+1u-ePFcinF[gn7%Fe`r7$Fdho$!+i&[5w$FcinF[gn7%Fi`r7$F^ho$!+VJ>gPFc inF[gn7%F]ar7$Fhgo$!+#[.`v$FcinF[gn7%Faar7$Fbgo$!+J1=YPFcinF[gn7%Fear7 $F\\go$!++UdKPFcinF[gn7%Fiar7$Fffo$!+'))oTr$FcinF[gn7%F]br7$F`fo$!+#[m 0p$FcinF[gn7%Fabr7$Fjeo$!+K>EhOFcinF[gn7%Febr7$Fdeo$!+#*>hDOFcinF[gn7% Fibr7$F^eo$!+`hz#e$FcinF[gn7%F]cr7$Fhdo$!+1OxJNFcinF[gn7%Facr7$$!+tMR_ \\Fjfn$!+;3z-NFcinF[gn7%Fecr7$Fbdo$!+h>DrMFcinF[gn7%F[dr7$F\\do$!+V)fp V$FcinF[gn7%F_dr7$Ffco$!+f;r*R$FcinF[gn7%Fcdr7$F`co$!+J+JfLFcinF[gn7%F gdr7$Fjbo$!+!fzbJ$FcinF[gn7%F[er7$Fdbo$!+5\"*QoKFcinF[gn7%F_er7$F^bo$! +^qnFcinF[gn7%Fgir7$$!+`QZ?5Fb[o$!+aMw.=FcinF[gn7% F]jr7$$!+E[SzNFejn$!+'\\szk\"FcinF[gn7%Fcjr7$$!+F;IO'*F^jn$!+P7h\"\\\" FcinF[gn7%Fijr7$$!+YZK07F^jn$!+&H**[L\"FcinF[gn7%F_[s7$$\"+nWchgFdhn$! +V0*z<\"FcinF[gn7%Fe[s7$$\"+QK))eaFdhn$!+o=)4-\"FcinF[gn7%F[\\s7$$\"+; z3rCFdhn$!+k$z#R')FjfnF[gn7%Fa\\s7$$\"+K^SrvF]hn$!+N7coqFjfnF[gn7%Fg\\ s7$$\"+/Fa[:F]hn$!+xOy(\\&FjfnF[gn7%F]]s7$$\"+ND1.=FS$!+;0*p#RFjfnF[gn 7%Fc]s7$$\"+H6uKtF\\q$!+#[%>cBFjfnF[gn7%Fi]s7$F($!+M;)R&yF[]oF[gn7%F_^ s7$F($\"+M;)R&yF[]oF[gn-Fien6&F[fn$\"#vF^fn$\"#NF^fnFj^s-%&STYLEG6#%,P ATCHNOGRIDG-F$6$7in7$$\"3,+++4OyN9FS$\"3z*****p`oTI%FS7$$\"3/+++0*\\sU \"FS$\"3.+++xqG8VFS7$$\"3-+++W!3!=9FS$\"3')*****zz&o@VFS7$$\"35+++?853 9FS$\"3#)*****\\.B$HVFS7$$\"3-+++fNd(R\"FS$\"3Q+++^s:OVFS7$$\"3-+++\\% pkQ\"FS$\"3?+++Pn9UVFS7$$\"3!*******4\\$[P\"FS$\"3U+++?$\\sM%FS7$$\"3' ******z^*p;]8FS$\"37+++d5iaVFS7 $$\"32+++pdBP8FS$\"3g*****f*4!oN%FS7$$\"3.+++r'zRK\"FS$\"3)******40:zN %FS7$$\"3*******Hse/J\"FS$\"3++++OX\"zN%FS7$$\"3!******H7QnH\"FS$\"3#* ******z'[nN%FS7$$\"3))*****RV*)GG\"FS$\"3S+++IWOaVFS7$$\"3)******R/#** o7FS$\"3u*****>K12N%FS7$$\"3'******f'\\8b7FS$\"3')*****pZ;dM%FS7$$\"3/ +++v!>9C\"FS$\"3T+++*zM$RVFS7$$\"3*)*****f()fzA\"FS$\"3<+++'\\*\\JVFS7 $$\"3(******R%4*[@\"FS$\"3y*****H8[@K%FS7$$\"3-+++g\"oB?\"FS$\"3#***** **4&>7J%FS7$$\"31+++a[d!>\"FS$\"3k******Hol)H%FS7$$\"3-+++\")ysz6FS$\" 3!)*****4(GT%G%FS7$$\"3#******\\V%3q6FS$\"3g*******=e%oUFS7$$\"3)***** *p!*[>;\"FS$\"3!)*****Rz$z]UFS7$$\"3'******>9yc:\"FS$\"32+++5,ZJUFS7$$ \"3)******Rr\"o^6FS$\"3?+++#391@%FS7$$\"33+++77T]6FS$\"3E+++p]Y)=%FS7$ $\"31+++9&HB:\"FS$\"3q*****4V;a;%FS7$$\"3!******f2]y:\"FS$\"3s*****Rg^ ?9%FS7$$\"3#******H&HCn6FS$\"3L+++o=:>TFS7$$\"3++++MV_!=\"FS$\"3/+++&3 Yw4%FS7$$\"3!******4$\\P(>\"FS$\"3K+++Lr[ySFS7$$\"3(*******QE9<7FS$\"3 A+++GJ[iSFS7$$\"3\"*******p$[*Q7FS$\"3w*****\\ba,0%FS7$$\"3/+++d,&=E\" FS$\"3U+++O=oTSFS7$$\"3/+++)[!*\\G\"FS$\"36+++P-&p.%FS7$$\"3.+++$fzwI \"FS$\"3!)*****H^Zc.%FS7$$\"31+++#=9%H8FS$\"3W+++TKOPSFS7$$\"3'****** \\Vc)\\8FS$\"35+++#Ri;/%FS7$$\"3'******z=)zo8FS$\"3s*****>)*G\"[SFS7$$ \"3%******4ODhQ\"FS$\"3'******fz'QcSFS7$$\"3/+++t()y,9FS$\"3/+++xe5mSF S7$$\"3*******HN#y:9FS$\"3-+++]9+xSFS7$$\"32+++sx7G9FS$\"3;+++z%H))3%F S7$$\"32+++CR')Q9FS$\"3H+++L6Q,TFS7$$\"3)******f!*R![9FS$\"33+++8tZ9TF S7$$\"3!******zR5dX\"FS$\"3o*****>rjz7%FS7$$\"3)******fnK>Y\"FS$\"3\") *****fd1<9%FS7$$\"3#******frknY\"FS$\"3S+++M\"*ebTFS7$$\"35+++DSEq9FS$ \"3?+++_'3&pTFS7$$\"3++++`p[s9FS$\"3O+++3SP$=%FS7$$\"3-+++E$)[t9FS$\"3 S+++*o.r>%FS7$$\"3-+++y7Kt9FS$\"3)*******eTi5UFS7$$\"33+++Ms.s9FS$\"3U +++%\\oQA%FS7$$\"33+++ofop9FS$\"3-+++h_xOUFS7$$\"35+++GdJm9FS$\"33+++s vG\\UFS7$$\"31+++$Rt>Y\"FS$\"3o*****4H_8E%FS7$$\"31+++hYqc9FS$\"3m**** *>O>HF%FS7$$\"3'******\\Ca0X\"FS$\"3#*******=7%RG%FS7$$\"3(******>4mNW \"FS$\"3#******zGsVH%FSFc_sFhen-Fbfn6jn7%7$$\"+4OyN9Fcin$\"+P&oTI%Fcin 7$$\"+0*\\sU\"Fcin$\"+xqG8VFcin7$$\"$J\"F^fn$\"$>%F^fn7%Fgbt7$$\"+W!3! =9Fcin$\"+)z&o@VFcinF\\ct7%Fbct7$$\"+?8539Fcin$\"+NIKHVFcinF\\ct7%Fhct 7$$\"+fNd(R\"Fcin$\"+^s:OVFcinF\\ct7%F^dt7$$\"+\\%pkQ\"Fcin$\"+Pn9UVFc inF\\ct7%Fddt7$$\"+5\\$[P\"Fcin$\"+?$\\sM%FcinF\\ct7%Fjdt7$$\"+=vri8Fc in$\"+x?U^VFcinF\\ct7%F`et7$$\"+#*p;]8Fcin$\"+d5iaVFcinF\\ct7%Ffet7$$ \"+pdBP8Fcin$\"+'*4!oN%FcinF\\ct7%F\\ft7$$\"+r'zRK\"Fcin$\"+^]\"zN%Fci nF\\ct7%Fbft7$$\"+B(e/J\"Fcin$\"+OX\"zN%FcinF\\ct7%Fhft7$$\"+B\"QnH\"F cin$\"+!o[nN%FcinF\\ct7%F^gt7$$\"+M%*)GG\"Fcin$\"+IWOaVFcinF\\ct7%Fdgt 7$$\"+W?**o7Fcin$\"+Ajq]VFcinF\\ct7%Fjgt7$$\"+m\\8b7Fcin$\"+xkrXVFcinF \\ct7%F`ht7$$\"+v!>9C\"Fcin$\"+*zM$RVFcinF\\ct7%Ffht7$$\"+w)fzA\"Fcin$ \"+'\\*\\JVFcinF\\ct7%F\\it7$$\"+W4*[@\"Fcin$\"+L\"[@K%FcinF\\ct7%Fbit 7$$\"+g\"oB?\"Fcin$\"+5&>7J%FcinF\\ct7%Fhit7$$\"+a[d!>\"Fcin$\"+Iol)H% FcinF\\ct7%F^jt7$$\"+\")ysz6Fcin$\"+rGT%G%FcinF\\ct7%Fdjt7$$\"+NW3q6Fc in$\"+!>e%oUFcinF\\ct7%Fjjt7$$\"+2*[>;\"Fcin$\"+%z$z]UFcinF\\ct7%F`[u7 $$\"+U\"yc:\"Fcin$\"+5,ZJUFcinF\\ct7%Ff[u7$$\"+9TFcinF\\ct7%Fd]u7$$\"+MV_!=\"Fcin$\"+&3Yw4%FcinF\\ ct7%Fj]u7$$\"+J\\P(>\"Fcin$\"+Lr[ySFcinF\\ct7%F`^u7$$\"+RE9<7Fcin$\"+G J[iSFcinF\\ct7%Ff^u7$$\"+q$[*Q7Fcin$\"+bX:]SFcinF\\ct7%F\\_u7$$\"+d,&= E\"Fcin$\"+O=oTSFcinF\\ct7%Fb_u7$$\"+)[!*\\G\"Fcin$\"+P-&p.%FcinF\\ct7 %Fh_u7$$\"+$fzwI\"Fcin$\"+8vkNSFcinF\\ct7%F^`u7$$\"+#=9%H8Fcin$\"+TKOP SFcinF\\ct7%Fd`u7$$\"+Nk&)\\8Fcin$\"+#Ri;/%FcinF\\ct7%Fj`u7$$\"+)=)zo8 Fcin$\"+#)*G\"[SFcinF\\ct7%F`au7$$\"+h`7'Q\"Fcin$\"+'z'QcSFcinF\\ct7%F fau7$$\"+t()y,9Fcin$\"+xe5mSFcinF\\ct7%F\\bu7$$\"+`By:9Fcin$\"+]9+xSFc inF\\ct7%Fbbu7$$\"+sx7G9Fcin$\"+z%H))3%FcinF\\ct7%Fhbu7$$\"+CR')Q9Fcin $\"+L6Q,TFcinF\\ct7%F^cu7$$\"+1*R![9Fcin$\"+8tZ9TFcinF\\ct7%Fdcu7$$\"+ )R5dX\"Fcin$\"+7P'z7%FcinF\\ct7%Fjcu7$$\"+wE$>Y\"Fcin$\"+wlqTTFcinF\\c t7%F`du7$$\"+;Zwm9Fcin$\"+M\"*ebTFcinF\\ct7%Ffdu7$$\"+DSEq9Fcin$\"+_'3 &pTFcinF\\ct7%F\\eu7$$\"+`p[s9Fcin$\"+3SP$=%FcinF\\ct7%Fbeu7$$\"+E$)[t 9Fcin$\"+*o.r>%FcinF\\ct7%Fheu7$$\"+y7Kt9Fcin$\"+fTi5UFcinF\\ct7%F^fu7 $$\"+Ms.s9Fcin$\"+%\\oQA%FcinF\\ct7%Fdfu7$$\"+ofop9Fcin$\"+h_xOUFcinF \\ct7%Fjfu7$$\"+GdJm9Fcin$\"+svG\\UFcinF\\ct7%F`gu7$$\"+$Rt>Y\"Fcin$\" +\"H_8E%FcinF\\ct7%Ffgu7$$\"+hYqc9Fcin$\"+i$>HF%FcinF\\ct7%F\\hu7$$\"+ XUb]9Fcin$\"+>7%RG%FcinF\\ct7%Fbhu7$$\"+#4mNW\"Fcin$\"+)GsVH%FcinF\\ct 7%FhhuFbbtF\\ctFf^sF\\_s-F$6$7in7$Fd_s$!3z*****p`oTI%FS7$F[bt$!3#***** *zGsVH%FS7$Ffat$!3#*******=7%RG%FS7$Faat$!3m*****>O>HF%FS7$F\\at$!3o** ***4H_8E%FS7$Fg`t$!33+++svG\\UFS7$Fb`t$!3-+++h_xOUFS7$F]`t$!3U+++%\\oQ A%FS7$Fh_t$!3)*******eTi5UFS7$Fc_t$!3S+++*o.r>%FS7$F^_t$!3O+++3SP$=%FS 7$Fi^t$!3?+++_'3&pTFS7$Fd^t$!3S+++M\"*ebTFS7$F_^t$!3\")*****fd1<9%FS7$ Fj]t$!3o*****>rjz7%FS7$Fe]t$!33+++8tZ9TFS7$F`]t$!3H+++L6Q,TFS7$F[]t$!3 ;+++z%H))3%FS7$Ff\\t$!3-+++]9+xSFS7$Fa\\t$!3/+++xe5mSFS7$F\\\\t$!3'*** ***fz'QcSFS7$Fg[t$!3s*****>)*G\"[SFS7$Fb[t$!35+++#Ri;/%FS7$F][t$!3W+++ TKOPSFS7$Fhjs$!3!)*****H^Zc.%FS7$Fcjs$!36+++P-&p.%FS7$F^js$!3U+++O=oTS FS7$Fiis$!3w*****\\ba,0%FS7$Fdis$!3A+++GJ[iSFS7$F_is$!3K+++Lr[ySFS7$Fj hs$!3/+++&3Yw4%FS7$Fehs$!3L+++o=:>TFS7$F`hs$!3s*****Rg^?9%FS7$F[hs$!3q *****4V;a;%FS7$Ffgs$!3E+++p]Y)=%FS7$Fags$!3?+++#391@%FS7$F\\gs$!32+++5 ,ZJUFS7$Fgfs$!3!)*****Rz$z]UFS7$Fbfs$!3g*******=e%oUFS7$F]fs$!3!)***** 4(GT%G%FS7$Fhes$!3k******Hol)H%FS7$Fces$!3#*******4&>7J%FS7$F^es$!3y** ***H8[@K%FS7$Fids$!3<+++'\\*\\JVFS7$Fdds$!3T+++*zM$RVFS7$F_ds$!3')**** *pZ;dM%FS7$Fjcs$!3u*****>K12N%FS7$Fecs$!3S+++IWOaVFS7$F`cs$!3#*******z '[nN%FS7$F[cs$!3++++OX\"zN%FS7$Ffbs$!3)******40:zN%FS7$Fabs$!3g*****f* 4!oN%FS7$F\\bs$!37+++d5iaVFS7$Fgas$!3g*****p2A9N%FS7$Fbas$!3U+++?$\\sM %FS7$F]as$!3?+++Pn9UVFS7$Fh`s$!3Q+++^s:OVFS7$Fc`s$!3#)*****\\.B$HVFS7$ F^`s$!3')*****zz&o@VFS7$Fi_s$!3.+++xqG8VFSFaiuFhen-Fbfn6jn7%7$Fcbt$!+P &oTI%Fcin7$Fihu$!+)GsVH%Fcin7$F]ct$!$>%F^fn7%F[ev7$Fchu$!+>7%RG%FcinF^ ev7%Fbev7$F]hu$!+i$>HF%FcinF^ev7%Ffev7$Fggu$!+\"H_8E%FcinF^ev7%Fjev7$F agu$!+svG\\UFcinF^ev7%F^fv7$F[gu$!+h_xOUFcinF^ev7%Fbfv7$Fefu$!+%\\oQA% FcinF^ev7%Fffv7$F_fu$!+fTi5UFcinF^ev7%Fjfv7$Fieu$!+*o.r>%FcinF^ev7%F^g v7$Fceu$!+3SP$=%FcinF^ev7%Fbgv7$F]eu$!+_'3&pTFcinF^ev7%Ffgv7$Fgdu$!+M \"*ebTFcinF^ev7%Fjgv7$Fadu$!+wlqTTFcinF^ev7%F^hv7$F[du$!+7P'z7%FcinF^e v7%Fbhv7$Fecu$!+8tZ9TFcinF^ev7%Ffhv7$F_cu$!+L6Q,TFcinF^ev7%Fjhv7$Fibu$ !+z%H))3%FcinF^ev7%F^iv7$Fcbu$!+]9+xSFcinF^ev7%Fbiv7$F]bu$!+xe5mSFcinF ^ev7%Ffiv7$Fgau$!+'z'QcSFcinF^ev7%Fjiv7$Faau$!+#)*G\"[SFcinF^ev7%F^jv7 $F[au$!+#Ri;/%FcinF^ev7%Fbjv7$Fe`u$!+TKOPSFcinF^ev7%Ffjv7$F_`u$!+8vkNS FcinF^ev7%Fjjv7$Fi_u$!+P-&p.%FcinF^ev7%F^[w7$Fc_u$!+O=oTSFcinF^ev7%Fb[ w7$F]_u$!+bX:]SFcinF^ev7%Ff[w7$Fg^u$!+GJ[iSFcinF^ev7%Fj[w7$Fa^u$!+Lr[y SFcinF^ev7%F^\\w7$F[^u$!+&3Yw4%FcinF^ev7%Fb\\w7$Fe]u$!+o=:>TFcinF^ev7% Ff\\w7$F_]u$!+/;0UTFcinF^ev7%Fj\\w7$Fi\\u$!+JkTlTFcinF^ev7%F^]w7$Fc\\u $!+p]Y)=%FcinF^ev7%Fb]w7$F]\\u$!+#391@%FcinF^ev7%Ff]w7$Fg[u$!+5,ZJUFci nF^ev7%Fj]w7$Fa[u$!+%z$z]UFcinF^ev7%F^^w7$F[[u$!+!>e%oUFcinF^ev7%Fb^w7 $Fejt$!+rGT%G%FcinF^ev7%Ff^w7$F_jt$!+Iol)H%FcinF^ev7%Fj^w7$Fiit$!+5&>7 J%FcinF^ev7%F^_w7$Fcit$!+L\"[@K%FcinF^ev7%Fb_w7$F]it$!+'\\*\\JVFcinF^e v7%Ff_w7$Fght$!+*zM$RVFcinF^ev7%Fj_w7$Faht$!+xkrXVFcinF^ev7%F^`w7$F[ht $!+Ajq]VFcinF^ev7%Fb`w7$Fegt$!+IWOaVFcinF^ev7%Ff`w7$F_gt$!+!o[nN%FcinF ^ev7%Fj`w7$Fift$!+OX\"zN%FcinF^ev7%F^aw7$Fcft$!+^]\"zN%FcinF^ev7%Fbaw7 $F]ft$!+'*4!oN%FcinF^ev7%Ffaw7$Fget$!+d5iaVFcinF^ev7%Fjaw7$Faet$!+x?U^ VFcinF^ev7%F^bw7$F[et$!+?$\\sM%FcinF^ev7%Fbbw7$Fedt$!+Pn9UVFcinF^ev7%F fbw7$F_dt$!+^s:OVFcinF^ev7%Fjbw7$Fict$!+NIKHVFcinF^ev7%F^cw7$Fcct$!+)z &o@VFcinF^ev7%Fbcw7$Fhbt$!+xqG8VFcinF^ev7%FfcwFhdvF^evFf^sF\\_s-F$6%7$ 7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF(-%'COLOURG6&F[fnF)F)F)-%*LINEST YLEG6#\"\"$-F$6%7$7$F(F^dw7$F($\"3/++++++!z%FSFcdwFfdw-%%FONTG6$%*HELV ETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G -Fbew6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%F^fn$\"$*>F^fn;F jfw$\"$z%F^fn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability reg ion with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this schem e the stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.090 3, 0];" "6#7$,$-%&FloatG6$\"&.4%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The stabi lity radius is half the length of the real stability interval." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(R(z)=-1,z=-4)/2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+y!Q^/#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 690 "R := z -> 1 +z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 1911921272136793687417053 13894961727723/137275888932535582465966331300216034220000*z^6+\n 6 29556630539301380473051749580428111/3431897223313389561649158282505400 855500*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=201/20 do\n zz := newton(`R`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13/20<=tt \+ and tt<=6/5) or (87/10<=tt and tt<=93/10) then\n hh := 1/40\n e lse \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts ),[Re(zz),Im(zz)]]:\nend do:\npp1 := plot(pts,color=red):\nrd := 2.045 138078: \npp2 := plot([-rd+rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue): \nplots[display]([pp1,pp2],scaling=constrained,font=[HELVETICA,9]);" } }{PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7]y 7$$\"\"!F)F(7$$\"3s******z7^)4'!#G$\"3#******fK'zq:!#=7$$\"3#)*****p$G v,W!#E$\"3$******Hf#fTJF07$$\"3G+++.0+sd!#D$\"3z*****\\&yQ7ZF07$$\"3z* ******=-Ki$!#C$\"3?+++Rg<$G'F07$$\"3.+++GR&GV\"!#B$\"3G+++'oKR&yF07$$ \"3o********p5xQFF$\"3Z*****zm$eC%*F07$$\"3w******G'ePi'FF$\"35+++uW\\ *4\"!#<7$$\"3/+++**4')y7FF$\"3)********GikD\"FS7$$!3#******RJBu4%!#A$ \"35++++!)G89FS7$$!3,+++-oRV>!#@$\"3)******f&y%)p:FS7$$!3U+++>BT!>'F\\ o$\"3#******4cbfs\"FS7$$!3++++n)\\#=;!#?$\"33+++o!f8)=FS7$$!3$)****** \\:&fq$Fgo$\"38+++y=vN?FS7$$!3o*****>$3(HR&Fgo$\"3!)*****\\7kC6#FS7$$! 3E+++QL!Qn(Fgo$\"3=+++k`z)=#FS7$$!3%******4e<*p5!#>$\"35+++I1qkAFS7$$! 3)******4#=?k9F\\q$\"3(******H]K,M#FS7$$!3#******4tA(p>F\\q$\"3#)***** HAQ]T#FS7$$!35+++a.03EF\\q$\"3))*****p$\\N*[#FS7$$!3F+++B&zDS$F\\q$\"3 ?+++-G+jDFS7$$!3q******Hm0yVF\\q$\"3()*****4)f(ej#FS7$$!3'******Rlu(fb F\\q$\"37+++??$yq#FS7$$!3i*****z=+B(pF\\q$\"3?+++o4oyFFS7$$!3Q+++54iP' )F\\q$\"3/+++*yw\"[GFS7$$!3)******p&>Fd5F0$\"3A+++()[,;HFS7$$!3/+++9Bo y7F0$\"3A+++1-%=)HFS7$$!3&*******R-*y_\"F0$\"3?+++\"4p_/$FS7$$!32+++FS i.=F0$\"3(******\\8Cf5$FS7$$!35+++S*zM5#F0$\"3A+++%RvM;$FS7$$!34+++*[e TU#F0$\"3!******40xw@$FS7$$!3y******yT%=w#F0$\"3A+++5\"*QoKFS7$$!3x*** **zQ@E6$F0$\"3-+++!fzbJ$FS7$$!3!******4fAGZ$F0$\"39+++J+JfLFS7$$!3))** ***>!>DRQF0$\"3#)******e;r*R$FS7$$!3D+++&y!G4UF0$\"3!******H%)fpV$FS7$ $!3-+++$eY3e%F0$\"37+++h>DrMFS7$$!3a+++`*)yA`F0$\"3z*****fgt<`$FS7$$!3 y*****4I-s0'F0$\"3))*****H:'z#e$FS7$$!3i******\\CX!y'F0$\"31+++#*>hDOF S7$$!3k*****4Bt7\\(F0$\"3%******>$>EhOFS7$$!3q*****4J9'*=)F0$\"3#***** *>[m0p$FS7$$!3-+++yk*f())F0$\"3&******f))oTr$FS7$$!3a+++W0<^&*F0$\"3=+ +++UdKPFS7$$!32+++1Yf@5FS$\"35+++J1=YPFS7$$!33+++>76(3\"FS$\"3)******> [.`v$FS7$$!3'******>!3u^6FS$\"3'******H9$>gPFS7$$!3/+++:>b:7FS$\"3/+++ i&[5w$FS7$$!3-+++q)3'y7FS$\"33+++1u-ePFS7$$!3%******\\it4M\"FS$\"3'*** ***R#QD^PFS7$$!3-+++)>3FS\"FS$\"37+++RX#3u$FS7$$!3#******RXxQY\"FS$\"3 <+++3Q\"os$FS7$$!3%******zD_X_\"FS$\"3-+++RxF4PFS7$$!35+++uH\"[e\"FS$ \"3-+++*Qe#)o$FS7$$!3'******zK`Zk\"FS$\"3!******H5)yjOFS7$$!3*******fa %[/.KNFS7$$ !3!******>PsV%>FS$\"3#)*****pk,4\\$FS7$$!3&)*****f3\"e0?FS$\"3#)*****p tCnW$FS7$$!3y******)*Rvn?FS$\"3%)*****fUA(*R$FS7$$!3=+++R&>68#FS$\"3=+ ++qK?]LFS7$$!3')*****H^`e>#FS$\"3%******HVp&)H$FS7$$!3/+++'fN?E#FS$\"3 9+++[ZIXKFS7$$!3\")*****H\")4'HBFS$\"3#******4pQ4>$FS7$$!3#)*****\\nk$ )R#FS$\"3\")*****p/*)f8$FS7$$!39+++F^%zY#FS$\"33+++yv*33$FS7$$!38+++Qf *y`#FS$\"3/+++b$zf-$FS7$$!3>+++(>Axg#FS$\"3y*****fA,9(HFS7$$!3')*****f 1Zpn#FS$\"3%)*****4J\">\"GFS$\"36+++ZY[4GFS7$$!3#******4AEs(GFS$\"35+++@#\\cv#FS7$$!3A +++z\"\\1%HFS$\"39+++jVc,FFS7$$!39+++>89-IFS$\"3;+++:p.ZEFS7$$!3%***** *4(RihIFS$\"3#)*****zz))=f#FS7$$!3$******\\[a!>JFS$\"3$)*****\\Ihf`#FS 7$$!31+++qqTuJFS$\"3y*****z?;\"zCFS7$$!3!******\\(erFKFS$\"3$******>3N 7U#FS7$$!3?+++&**p*yKFS$\"3')*****f!y@iBFS7$$!3=+++T+@GLFS$\"3(******z $3)>I#FS7$$!31+++*)eZvLFS$\"3:+++EcXSAFS7$$!38+++o_\"3U$FS$\"3%)*****p L(ex@FS7$$!3=+++VHGkMFS$\"38+++WSL8@FS7$$!3#******\\?Sf]$FS$\"3)****** pRmw/#FS7$$!37+++\"[aea$FS$\"3%*******4yc!)>FS7$$!3y******R))4%e$FS$\" 36+++J^.7>FS7$$!3#)*****p<^2i$FS$\"3(******>tz?%=FS7$$!3#)*****p)G*el$ FS$\"3)*******y)G2x\"FS7$$!35+++]og*o$FS$\"3/+++*=F!)p\"FS7$$!38+++/Y( >s$FS$\"3\"*******oy.C;FS7$$!37+++0F2`PFS$\"35+++)QV)[:FS7$$!3A+++G%oH y$FS$\"3++++h^as9FS7$$!3.+++N^r6QFS$\"3'******H!>E&R\"FS7$$!3$)*****>( yMRQFS$\"3*******4UErJ\"FS7$$!3!)*****>!)ze'QFS$\"33+++')3GQ7FS7$$!3%* *****48+8*QFS$\"3'******4\"4()e6FS7$$!3;+++bRd:RFS$\"3++++m!R!z5FS7$$! 3<+++_TkQRFS$\"3=+++!*f=*)**F07$$!31+++#3N/'RFS$\"3k*****\\[\"G'=*F07$ $!3%)*****pad3)RFS$\"39+++R)zEQ)F07$$!3'******HI9)**RFS$\"3'******4R!= zvF07$$!31+++ma?Mn Y(fF07$$!3'******>(e!p/%FS$\"3O+++*4yU<&F07$$!3))*****fF07$$!3!)*****>,@z3 %FS$\"3'******>#))=\">\"F07$$!3o*****4A9+4%FS$\"3')*****>./,(RF\\q7$Ff ^m$!3')*****>./,(RF\\q7$Fa^m$!3'******>#))=\">\"F07$F\\^m$!3))******QR $e)>F07$Fg]m$!3w******zeB\"y#F07$Fb]m$!3!)*****HP^wd$F07$F]]m$!37+++,o GvVF07$Fh\\m$!3O+++*4yU<&F07$Fc\\m$!3g*****>MnY(fF07$F^\\m$!3g*****zdw jx'F07$Fi[m$!3'******4R!=zvF07$Fd[m$!39+++R)zEQ)F07$F_[m$!3k*****\\[\" G'=*F07$Fjjl$!3=+++!*f=*)**F07$Fejl$!3++++m!R!z5FS7$F`jl$!3'******4\"4 ()e6FS7$F[jl$!33+++')3GQ7FS7$Ffil$!3*******4UErJ\"FS7$Fail$!3'******H! >E&R\"FS7$F\\il$!3++++h^as9FS7$Fghl$!35+++)QV)[:FS7$Fbhl$!3\"*******oy .C;FS7$F]hl$!3/+++*=F!)p\"FS7$Fhgl$!3)*******y)G2x\"FS7$Fcgl$!3(****** >tz?%=FS7$F^gl$!36+++J^.7>FS7$Fifl$!3%*******4yc!)>FS7$Fdfl$!3)******p Rmw/#FS7$F_fl$!38+++WSL8@FS7$Fjel$!3%)*****pL(ex@FS7$Feel$!3:+++EcXSAF S7$F`el$!3(******z$3)>I#FS7$F[el$!3')*****f!y@iBFS7$Ffdl$!3$******>3N7 U#FS7$Fadl$!3y*****z?;\"zCFS7$F\\dl$!3$)*****\\Ihf`#FS7$Fgcl$!3#)***** zz))=f#FS7$Fbcl$!3;+++:p.ZEFS7$F]cl$!39+++jVc,FFS7$Fhbl$!35+++@#\\cv#F S7$Fcbl$!36+++ZY[4GFS7$F^bl$!3?+++R%pK'GFS7$Fial$!3%)*****4J\">$FS7$F[`l$!39+++[ZIXKFS7$Ff_l$! 3%******HVp&)H$FS7$Fa_l$!3=+++qK?]LFS7$F\\_l$!3%)*****fUA(*R$FS7$Fg^l$ !3#)*****ptCnW$FS7$Fb^l$!3#)*****pk,4\\$FS7$F]^l$!3\")*****p*>.KNFS7$F h]l$!3!******R9m*pNFS7$Fc]l$!3%)*****pO2Yg$FS7$F^]l$!3!******\\f%*ej$F S7$Fi\\l$!3!******H5)yjOFS7$Fd\\l$!3-+++*Qe#)o$FS7$F_\\l$!3-+++RxF4PFS 7$Fj[l$!3<+++3Q\"os$FS7$Fe[l$!37+++RX#3u$FS7$F`[l$!3'******R#QD^PFS7$F [[l$!33+++1u-ePFS7$Ffz$!3/+++i&[5w$FS7$Faz$!3'******H9$>gPFS7$F\\z$!3) ******>[.`v$FS7$Fgy$!35+++J1=YPFS7$Fby$!3=++++UdKPFS7$F]y$!3&******f)) oTr$FS7$Fhx$!3#******>[m0p$FS7$Fcx$!3%******>$>EhOFS7$F^x$!31+++#*>hDO FS7$Fiw$!3))*****H:'z#e$FS7$Fdw$!3z*****fgt<`$FS7$$!3A+++tMR_\\F0$!33+ ++;3z-NFS7$F_w$!37+++h>DrMFS7$Fjv$!3!******H%)fpV$FS7$Fev$!3#)******e; r*R$FS7$F`v$!39+++J+JfLFS7$F[v$!3-+++!fzbJ$FS7$Ffu$!3A+++5\"*QoKFS7$Fa u$!3!******40xw@$FS7$F\\u$!3A+++%RvM;$FS7$Fgt$!3(******\\8Cf5$FS7$Fbt$ !3?+++\"4p_/$FS7$F]t$!3A+++1-%=)HFS7$Fhs$!3A+++()[,;HFS7$Fcs$!3/+++*yw \"[GFS7$F^s$!3?+++o4oyFFS7$Fir$!37+++??$yq#FS7$Fdr$!3()*****4)f(ej#FS7 $F_r$!3?+++-G+jDFS7$Fjq$!3))*****p$\\N*[#FS7$Feq$!3#)*****HAQ]T#FS7$F` q$!3(******H]K,M#FS7$Fjp$!35+++I1qkAFS7$Fep$!3=+++k`z)=#FS7$F`p$!3!)** ***\\7kC6#FS7$$!3++++(RyP[#Fgo$!3#*******)\\-(e>FS7$$!35+++`QZ?5Fgo$!3 /+++aMw.=FS7$$!33+++E[SzNF\\o$!3'******f\\szk\"FS7$$!3X*****pi,jj*Ffn$ !33+++P7h\"\\\"FS7$$!32+++YZK07Ffn$!3/+++&H**[L\"FS7$$\"3m*****pYk:1'F F$!3%******Ha!*z<\"FS7$$\"3#******zB$))eaFF$!3,+++o=)4-\"FS7$$\"3;+++; z3rCFF$!3&)*****ROz#R')F07$$\"3i*****>809d(F@$!3W+++N7coqF07$$\"3++++/ Fa[:F@$!3!******pn$y(\\&F07$$\"3++++ND1.=F:$!3:+++;0*p#RF07$$\"3)***** **G6uKt!#F$!3)******>[%>cBF07$F($!3))*****Rj\")R&yF\\q7$F($\"3))*****R j\")R&yF\\q-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$$!3HUpXX(=] \">F\\q$\"3u98@)R#=#z#F07$$!35=`g'[(>rmF\\q$\"37%eWQrD4=&F07$$!3la`S/W uO:F0$\"3)yj)>Sd)yx(F07$$!3A&4MB&flaFF0$\"3X\"4CEH4^-\"FS7$$!3c:$e%pR4 'G%F0$\"3Q3:Q]ow_7FS7$$!3sskXd'Ql'fF0$\"3b$>'3fRxV9FS7$$!3@K/!3B%QUzF0 $\"3Gg#QN)R(zh\"FS7$$!3hNO+2hO?5FS$\"3u=/WQy')pFS7$$!3'z)\\RC!R+x\"FS$ \"3*3I'*RK^l-#FS7$$!3#*Gh@W#fj/#FS$\"3QLlT8x8X?FS7$$!3I1)[L`%yBBFS$\"3 dzLq5m1E?FS7$$!3/M>ql)=je#FS$\"3gNkFibBs>FS7$$!3(Q;OF&4z;GFS$\"3]6(o(y ](R*=FS7$$!3sNmtu6OwIFS$\"3[MpDPo6m%4A+j \"FS7$$!3_Sbxe6+&\\$FS$\"30@7hp#*QU9FS7$$!3eKjqxplhOFS$\"3](p0=ZzFD\"F S7$$!3H/oEse3;QFS$\"3vC=RDX!H-\"FS7$$!3$pFyZ+#*H$RFS$\"3-0n;Nd4lyF07$$ !3t\"z`!>D[@SFS$\"39R*y/uv(f_F07$$!3Qu)f7^i72%FS$\"3mv_4!)pD#y#F07$$!3 m*R%y?jE!4%FS$\"3=r4B?ggUjFgo7$$!3%*Q$flY]:2%FS$!3#\\eL%pg?hFF07$$!395 XhV!=O-%FS$!3C[%*RX%o)y^F07$$!3Dks$e5.'RRFS$!3\")*ziCI#\\/xF07$$!3&>PT M\"*G\">QFS$!3-[^+Tzh<5FS7$$!3[;8%\\l=-n$FS$!3f+Ka$G`;C\"FS7$$!33AvS$* pM*\\$FS$!3aqA,:x+Q9FS7$$!3Y33j?v:#G$FS$!38N^D_1hG;FS7$$!3qH0'*pF!e1$F S$!3s$*)p3kRAx\"FS7$$!3*f+C[)Rs;GFS$!39VEzhB+%*=FS7$$!37DpI4ikyDFS$!3] 1vN0\\Ku>FS7$$!3#oaKsJ\"G4BFS$!3Iy'o!4%3!G?FS7$$!3YY/F\\3R^?FS$!3KF!=% >&G^/#FS7$$!3Cnq+#*3y\"y\"FS$!33zyqN16G?FS7$$!3`_OIVofA:FS$!3k7i8zdDx> FS7$$!3Ykk@WVmg7FS$!3oix*=!3q))=FS7$$!3hVVhDS*=-\"FS$!35X34R;vq;FS7$$!3s()\\ZK7LKfF0$!3*3.))e!RLS9FS7$$!3'* [l+C^U%H%F0$!3s(>@s8TQD\"FS7$$!3*)*puT#G;6FF0$!3%\\!R(p'Rc<5FS7$$!3%G+ #*41!)\\e\"F0$!3E\\t5;(oT*yF07$$!35Tb4B%*3iqF\\q$!3e6(42ngzK&F07$$!3:l *QMw()G!>F\\q$!3=gUrggO$y#F07$F($\"3'*=-ME$fyn\"F4-Fien6&F[fnF(F(F\\fn -%%FONTG6$%*HELVETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6% Q!6\"F^fo-Fceo6#%(DEFAULTG-%%VIEWG6$FbfoFbfo" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the bo undary curve horizontally by taking the 11th root of the real part of \+ points along the curve. In this way we see that the largest interval o n the nonnegative imaginary axis that contains the origin and lies ins ide the stability region is " }{XPPEDIT 18 0 "[ 0, 1.43 ]" "6#7$\"\"! -%&FloatG6$\"$V\"!\"#" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 19119212721367936874 1705313894961727723/137275888932535582465966331300216034220000*z^6+\n \+ 629556630539301380473051749580428111/34318972233133895616491582825 05400855500*z^7:\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 45 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n \+ pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=C OLOR(RGB,.75,.2,.2),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" } }{PARA 13 "" 1 "" {GLPLOT2D 249 293 293 {PLOTDATA 2 "6(-%'CURVESG6#7P7 $$\"\"!F)F(7$$\"5!)yW\"GM4o(y[!#@$\"5nHO(*e`EfTJF-7$$\"5]LOm-a/MCrF-$ \"5n,#GsrI&=$G'F-7$$\"59`,YO\"QXd*))F-$\"5!QN7W1'zxC%*F-7$$\"5lzU.@i`) >/\"!#?$\"5iM2;Mhqjc7F=7$$\"5\"Q(Ra\"GhZ'y6F=$\"5)>\\HUjK'zq:F=7$$\"5/ N%)*)*[lGUI\"F=$\"5R@y%>0fb\\)=F=7$$\"5X-vd[=s^@9F=$\"5a0T3m_[6*>#F=7$ $\"5$HLz)o1eKK:F=$\"5$)fQ\")Q5TF8DF=7$$\"5z[v5Jw$)*yj\"F=$\"5:d+,VfLVF GF=7$$\"5;5S!H7A?\"Ros[%\\_#4$>F=$\"5^un.`f4\"*pPF=7$$\"5p$*e,xs #F=$\"59,07kayQ7ZF=7$$\"5d$eu!QB*y;G#F=$\"5lsnZIXkaE]F=7$$\"5ccM\\SEhj jBF=$\"5D!pfF=7$$\"5.%)R)H)[i%pf#F=$\"5=IFSDRg<$G'F=7$$\"5a[ PwL_[aqEF=$\"59FIh6n7L(f'F=7$$\"5sFgalFF+UFF=$\"5%pgTwE.&[6pF=7$$\"5M< KWoa*\\7\"GF=$\"5hp@AF-pjDsF=7$$\"5P_G@#))>!>yGF=$\"5Q'za2%RjyRvF=7$$ \"5c&eXF)evpUHF=$\"5*H:N#3'oKR&yF=7$$\"5S&=u#QV;h/IF=$\"5*f$y-yT^2o\") F=7$$\"5T>>x&3KIP1$F=$\"5@;#)[)*RF@#[)F=7$$\"5&px>U3C*z>JF=$\"5j`&)=6A VM'z)F=7$$\"5j4\"zdZm'\\sJF=$\"5,y:sw3&o/6*F=7$$\"5o4oSJ'=69A$F=$\"56n >xEoOeC%*F=7$$\"5NIQOg6#3gE$F=$\"5x0URR#)yoQ(*F=7$$\"5MoG/$R%yd0LF=$\" 5F0T#Q4*yF05!#>7$$\"5h$*QCViid54o5Ffu7$$\"5\"eq5mP&z8#Q$F=$\"5E$)oqXuW\\*4\"Ffu7$$\"5\"4v $4J)QkiQ$F=$\"5Mio)y!p_*38\"Ffu7$$\"5Dfc)z$H\\2sLF=$\"5DOtU-\"*HHi6Ffu 7$$\"5$p#y)>ZD)=GLF=$\"5jrb$Q,8(o$>\"Ffu7$$\"5OT0+Z2&G]A$F=$\"5)3\"Q%3 96x]A\"Ffu7$$\"5#zB&[/dyV7HF=$\"5\"3@$)H(*GikD\"Ffu7$$!5Pf/O[$=Tx6$F=$ \"5lbvrmW>%yG\"Ffu7$$!5'eg@_YiPkX$F=$\"5+*[Fz_F:#>8Ffu7$$!5&Q-H$Qm^xqO F=$\"50:9M3&R\"e]8Ffu7$$!5G&*Q&fj*Q@UQF=$\"5e=&pUsKR>Q\"Ffu7$$!5+rh_Wm 6^\"*RF=$\"5&*R7o1+!)G89Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q!6\"Fdz-%&COLORG6&%$RGBG$\"#v!\"#$F`z!\"\"F][l-% %VIEWG6$%(DEFAULTGFb[l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 123 "The relevant intersection point of the boundary c urve with the imaginary axis can be determined more accurately as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the b oundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits : = 15:\nz0 := 1.25*I:\nfor ct from 39 to 42 do\n newton(R(z)=exp(ct*P i/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0e8ASm_#R!#?$\"0W396x]A\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0z'4**4')y7!#?$\"0$)H(*GikD\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0\"\\)f8daq#!#?$\"0=nY%>%yG\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0F\\OYBBT)!#?$\"0Fz_F:#>8!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisect ion method to calculate the parameter value associated with the inters ection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.25*I))\nend proc:\nDigit s := 15:\nu0 := bisect('real_part'(u),u=0.39..0.42);\nnewton(R(z)=exp( u0*Pi*I),z=1.25*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#u0G$\"0Y(zddXOS!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0;.4m-zE \"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 260 112 "largest interval on the nonnegative imaginary ax is that contains the origin and lies inside the stability region" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.2679];" "6#7$\"\"!-%&FloatG6$ \"&zE\"!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability regi on for the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 sta ge, order 4 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(4,8,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#\"N\">*f8MctT7 q+B]#)\\b\"R!Gg8)z\"P+DK*)GTL'R.FcBg&f!)eYF)o5g\"^oDc'[4'G8$**f0v,0FC%o61KSZn\"^o++tLGhLvd'o'=[L8_Gam?5$4 ??)eZm8)[/B:F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R* `(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+(*[^LT!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := n ewton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.59..0.49,color= [red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cr oss,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.59.. 0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3')*************e%!#<$!3: o^1D(fGM#F*7$$!3mmmTG_jMXF*$!3mHz\\28BD@F*7$$!3[LL$oXq#zWF*$!3#*ps*=@O ]#>F*7$$!39+D6ly4JWF*$!3kl&p0%F*$!3mNW5Z;Fj&)!#=7$$!3KL$e;r;j&RF*$!3>L(f:41l$pFP7$$!3++]s> +6_QF*$!3vnm4.I6DbFP7$$!3IL$e)4$RVu$F*$!30-NIVHz9VFP7$$!39+]-AU\"pj$F* $!3`f\"R)=ap@LFP7$$!3Ymm'\\J9k_$F*$!3+c&3)=YA'[#FP7$$!3GLLoOf3HMF*$!3_ L()z#)p'4)=FP7$$!3t*****[B<&>LF*$!3%zI^hrY3K\"FP7$$!3(*****\\L')\\4KF* $!3+lkV/aMX')!#>7$$!3z****4D`Z.JF*$!3kr[E97()f]Fcp7$$!3>L$eQ+'>2IF*$!3 -1&QCW/%eBFcp7$$!3ommcy0r#*GF*$\"3rH3]F1sGJ!#?7$$!3ommE1ks&z#F*$\"3UfK \"pLq*=AFcp7$$!3y**\\#zv5Ho#F*$\"3d!*RBPF/JTFcp7$$!3]mmE$z\\Ie#F*$\"3w 2/f\"H!=FcFcp7$$!3#)**\\#z['[tCF*$\"38z4%fl208(Fcp7$$!3')**\\x6j:pBF*$ \"3!z-e;pxq[)Fcp7$$!3Emm\"zr)HgAF*$\"3C-8$)=:%)y)*Fcp7$$!37m;4lMLg@F*$ \"3*>dYu!zj<6FP7$$!3ALL3Nu]_?F*$\"3)*Qe'f'Haj7FP7$$!3SL$eQ\\10%>F*$\"3 *Qj<$o4VD9FP7$$!3/+]FX$4I%=F*$\"3K'32#eLxx:FP7$$!3ELLy`(4xt\"F*$\"3_*p c>g.qv\"FP7$$!35++?s\\#*G;F*$\"3kY;b=P1h>FP7$$!39++l'G+D_\"F*$\"3Kypdo x;#=#FP7$$!3*)**\\7J*G&>9F*$\"3)\\7KHzn!>CFP7$$!3z***\\pz'>08F*$\"3u(= >)fM'>r#FP7$$!3/nmE*GkC?\"F*$\"38j6Q^#f^+$FP7$$!3u****\\4ax#4\"F*$\"3o 5Q\\_cH`LFP7$$!3BMLe9)4Q$**FP$\"3jg:KVp_.PFP7$$!3'o****4=mr%))FP$\"3)e z?b8*[GTFP7$$!3Oqm\"f?>Z#yFP$\"3#4n%=72%Gd%FP7$$!3E'**\\A^nfv'FP$\"3!o *fVOOd)3&FP7$$!3%zmm;Qn5r&FP$\"3)H@6sy=!\\cFP7$$!3J'**\\7y`rh%FP$\"30t a#zTB?I'FP7$$!3!)GLLp\\ejNFP$\"3!z27<>/awT\"FP$\"3sahn@#[#y')FP7$$!3Mf****>C4eVFcp$\"3_< Amx2bt&*FP7$$\"3wMLL)*p&\\*oFcp$\"3%R7Hz<#Qr5F*7$$\"31mmma=)fp\"FP$\"3 mOZ>r'G[=\"F*7$$\"370+v?g5pFFP$\"3M%\\@/T[!>8F*7$$\"3g1+DvFA'z$FP$\"3y gEV#=K " 0 " " {MPLTEXT 1 0 1994 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+\n \+ 798136028039155498250230070124173563413599191/\n 9605116316897 8395061620922693212917412889322500*z^5+\n 601068827465880595602356 2703396331998093266859440860269485936177/\n 428356372884321341 8181511849562151771625041775055999313286094865625*z^6+\n 111637679 37251435974964428506891134973160059549773322251369498591/\n 60 921795254659035280803724082661714085333927467463101344513349200000*z^7 +\n 67474032061168424270501192917177594747335098551652147811747/\n 1523044881366475882020093102066542852133348186686577533612833 730000*z^8:\npts := []: z0 := 0: tt := 0: \nwhile tt<=201/20 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13/20<=tt and tt<=6/5) or (87/10<=tt and tt<=93/10) then\n hh := 1/40\n \+ else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pt s),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.32,.08 ,.08)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2.. nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.63,.15,.15) ):\npts := []: z0 := 1.4+4.3*I:\nfor ct from 0 to 60 do\n zz := newt on(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[R e(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.32,.08,.08) ):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.32,4.19]],i=2.. nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.63,.15,.15) ):\npts := []: z0 := 1.4-4.3*I:\nfor ct from 0 to 60 do\n zz := newt on(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[R e(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.32,.08,.08) ):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.32,-4.19]],i=2. .nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.63,.15,.15 )):\np_7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4.79]]],color=bla ck,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.79..1.99,-4.79. .4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axe s=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 594 594 594 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F(7$$\"3B+++\"R)4Dd!#F$\"3 ))******[jzq:!#=7$$\"31+++4'4uk$!#D$\"3v*****\\D$fTJF07$$\"3#)*****f\" >(f5%!#C$\"3*)*****RE#R7ZF07$$\"3-+++'e3#\\A!#B$\"3]******)4\">$G'F07$ $\"3?+++_5;u\")F@$\"39+++Kb'R&yF07$$\"3!******R)4%)RA!#A$\"3')*****p7G YU*F07$$\"37+++p][t[FK$\"3++++P`\\*4\"!#<7$$\"3e+++&H@\")Q)FK$\"35+++i iWc7FS7$$\"31+++Os#=-\"!#@$\"3\"******RSFKT\"FS7$$\"35+++1kW_FFK$\"3'* *****\\R%pp:FS7$$!3C+++l)3'3JFfn$\"3.+++9`jDMItFfo$\"3%)*****HKws=#FS7$$!31+++\\n6V 5!#>$\"3')******[d&GE#FS7$$!3-+++CGV[9F[q$\"3\"******>;AzL#FS7$$!37+++ S$4(o>F[q$\"3!)*****fl@CT#FS7$$!3)********>$eDEF[q$\"3))*****\\&)*G'[# FS7$$!3x*****4'**HUMF[q$\"3?+++m[WfDFS7$$!3L+++RA>VWF[q$\"3=+++1$y0XJq#FS7$$!3W+++$G1Z4(F[q$\"3'****** *oJNtFFS7$$!3w******R%4%*y)F[q$\"39+++'fc@%GFS7$$!31+++,dAv5F0$\"3>+++ %Ga#4HFS7$$!31+++&>t!*H\"F0$\"3'******Rw*HuHFS7$$!3!******HID-b\"F0$\" 35+++EO#p.$FS7$$!35+++\"z/t#=F0$\"31+++Fuw'4$FS7$$!3#******4guy7#F0$\" 3A+++7B_`JFS7$$!35+++#fy'[CF0$\"32+++U@'p?$FS7$$!3u*****Rg0gy#F0$\"3%) *****H4ipD$FS7$$!3w*****H>xg8$F0$\"3,+++E#*\\.LFS7$$!3#******>-n`\\$F0 $\"3'*******z!QmM$FS7$$!3%******zh.3'QF0$\"31+++=\"3lQ$FS7$$!31+++Xf&) HUF0$\"3$)*****R))zKU$FS7$$!3-+++//`+YF0$\"3?+++0W9dMFS7$$!3`*****\\x- 6M&F0$\"39+++)eIp^$FS7$$!3')*****\\3&zugF0$\"3;+++?vLnNFS7$$!3%******* \\]'zz'F0$\"3A+++X+i4OFS7$$!3!******fox#4vF0$\"34+++/BzWOFS7$$!3\"**** **RW,'3#)F0$\"35+++)GcOn$FS7$$!3[+++\"=$Q'*))F0$\"3%******pGUop$FS7$$! 3++++4LJt&*F0$\"3-+++&3Y[r$FS7$$!3'******R_:S-\"FS$\"30+++Q!f!GPFS7$$! 3-+++5lw*3\"FS$\"3%)*****\\?\"zOPFS7$$!3%******z.cY:\"FS$\"3*******4\" ))GTPFS7$$!33+++G:v=7FS$\"3()*****HQZLDeq$FS7$$!3.+++`?cH:FS$ \"31+++GF$yo$FS7$$!33+++z!z-f\"FS$\"3#******zbSjm$FS7$$!31+++%><2l\"FS $\"3()*****f5\"QTOFS7$$!3/+++4T*4r\"FS$\"3()******RJ)Hh$FS7$$!3%****** *4/Dr)y6e$FS7$$!3*******HA`;$=FS$\"3))*****4z7ga$FS7$$ !3/+++u#)R#*=FS$\"3:+++QTb2NFS7$$!31+++='3P&>FS$\"3#)*****fe6fY$FS7$$! 31+++'[Fe,#FS$\"3@+++?:D@MFS7$$!3'******\\:-!z?FS$\"3=+++4n\"QP$FS7$$! 3(******HFcM9#FS$\"39+++75%RK$FS7$$!3-+++D3N4AFS$\"33+++jC0sKFS7$$!3#) *****zfRnF#FS$\"3%)*****H`XBFS$\"31+++-CHkJFS7$$!3<+ ++:)yaT#FS$\"3%)*****f2v%4JFS7$$!35+++#f#>'[#FS$\"3#******>bFY0$FS7$$! 37+++@m>dDFS$\"3%******RgO++$FS7$$!3>+++*)z)zi#FS$\"37+++\\y$e%HFS7$$! 3%******4S$4)p#FS$\"3#******>PJ?*GFS7$$!3')******)z0rw#FS$\"36+++,c^QG FS7$$!3,+++:)*pMGFS$\"3'******>_D^y#FS7$$!3!)*****4:J1!HFS$\"32+++OdmJ FFS7$$!3;+++poskHFS$\"3-+++*\\Mzn#FS7$$!3!)*****zHso-$FS$\"3/+++)*)QPi #FS7$$!3!)*****H1)*p3$FS$\"3?+++lI!*oDFS7$$!3'******4$*o]9$FS$\"3$)*** **pkrK^#FS7$$!3;+++r\\2,KFS$\"3-+++!R5nX#FS7$$!3/+++Q\\-bKFS$\"33+++$= 0\"*R#FS7$$!3!******R`TpI$FS$\"3$)*****>Ug.M#FS7$$!31+++j\"eoN$FS$\"3' *******[sR!G#FS7$$!3=+++xn\"[S$FS$\"3%)*****H'>:>AFS7$$!3=+++Bl'3X$FS$ \"3/+++&zul:#FS7$$!3y*****\\*G1&\\$FS$\"3)******p6HE4#FS7$$!3;+++krYPN FS$\"3%******R8\"HF?FS7$$!3++++oe9yNFS$\"3'******\\/]0'>FS7$$!3\"***** *z;qrh$FS$\"3++++V'3C*=FS7$$!37+++w[haOFS$\"3$******>D%)G#=FS7$$!3%*** ***p(pb!p$FS$\"3/+++g)4?v\"FS7$$!39+++;N2DPFS$\"3%******>DN)z;FS7$$!3= +++q(Q#ePFS$\"3&******z#yU1;FS7$$!3++++j27!z$FS$\"3&******4\"G(=`\"FS7 $$!3$)*****>Hx2#QFS$\"3#******RZsiX\"FS7$$!35+++K=D]QFS$\"3-+++?Tuz8FS 7$$!31+++?)p&yQFS$\"3.+++1nT-8FS7$$!39+++het0RFS$\"3++++&=EWA\"FS7$$!3 9+++!QK<$RFS$\"3++++w*4f9\"FS7$$!3A+++@+_cRFS$\"3#******z5,q1\"FS7$$!3 !)*****>\")R+)RFS$\"3=+++>vAy)*F07$$!3)*******Go@-SFS$\"3a*****RaP[3*F 07$$!3>+++H['H-%FS$\"3]*****z,c2H)F07$$!3r*****\\C\">USFS$\"3-+++3(4n \\(F07$$!3++++&)>!)fSFS$\"3x*****z`PKq'F07$$!3!)*****pf0d2%FS$\"3%**** **Rc-2\"fF07$$!3#*******zl\")*3%FS$\"38+++5=J>^F07$$!3M+++&zd?5%FS$\"3 2+++>,9HVF07$$!3A+++M?O7TFS$\"35+++'*[:SNF07$$!3#)*****H$Hn?TFS$\"3E++ +W&RAv#F07$$!3#)*****RRXp7%FS$\"3'******pP9_'>F07$$!3B+++*oX68%FS$\"3/ +++kX&)y6F07$$!3o******>9DLTFS$\"3?+++AH0HRF[q7$Fe^m$!3?+++AH0HRF[q7$F `^m$!3/+++kX&)y6F07$F[^m$!3'******pP9_'>F07$Ff]m$!3E+++W&RAv#F07$Fa]m$ !35+++'*[:SNF07$F\\]m$!32+++>,9HVF07$Fg\\m$!38+++5=J>^F07$Fb\\m$!3%*** ***Rc-2\"fF07$F]\\m$!3x*****z`PKq'F07$Fh[m$!3-+++3(4n\\(F07$Fc[m$!3]** ***z,c2H)F07$F^[m$!3a*****RaP[3*F07$Fijl$!3=+++>vAy)*F07$Fdjl$!3#***** *z5,q1\"FS7$F_jl$!3++++w*4f9\"FS7$Fjil$!3++++&=EWA\"FS7$Feil$!3.+++1nT -8FS7$F`il$!3-+++?Tuz8FS7$F[il$!3#******RZsiX\"FS7$Ffhl$!3&******4\"G( =`\"FS7$Fahl$!3&******z#yU1;FS7$F\\hl$!3%******>DN)z;FS7$Fggl$!3/+++g) 4?v\"FS7$Fbgl$!3$******>D%)G#=FS7$F]gl$!3++++V'3C*=FS7$Fhfl$!3'****** \\/]0'>FS7$Fcfl$!3%******R8\"HF?FS7$F^fl$!3)******p6HE4#FS7$Fiel$!3/++ +&zul:#FS7$Fdel$!3%)*****H'>:>AFS7$F_el$!3'*******[sR!G#FS7$Fjdl$!3$)* ****>Ug.M#FS7$Fedl$!33+++$=0\"*R#FS7$F`dl$!3-+++!R5nX#FS7$F[dl$!3$)*** **pkrK^#FS7$Ffcl$!3?+++lI!*oDFS7$Facl$!3/+++)*)QPi#FS7$F\\cl$!3-+++*\\ Mzn#FS7$Fgbl$!32+++OdmJFFS7$Fbbl$!3'******>_D^y#FS7$F]bl$!36+++,c^QGFS 7$Fhal$!3#******>PJ?*GFS7$Fcal$!37+++\\y$e%HFS7$F^al$!3%******RgO++$FS 7$Fi`l$!3#******>bFY0$FS7$Fd`l$!3%)*****f2v%4JFS7$F_`l$!31+++-CHkJFS7$ Fj_l$!3%)*****H)y6e$FS7$F]]l$!3() ******RJ)Hh$FS7$Fh\\l$!3()*****f5\"QTOFS7$Fc\\l$!3#******zbSjm$FS7$F^ \\l$!31+++GF$yo$FS7$Fi[l$!3')*****>LDeq$FS7$Fd[l$!3()******RrF?PFS7$F_ [l$!3%)******=M8JPFS7$Fjz$!3')*****fp@$QPFS7$Fez$!3()*****HQZ+++%Ga# 4HFS7$Fbs$!39+++'fc@%GFS7$F]s$!3'*******oJNtFFS7$Fhr$!3*******>0XJq#FS 7$Fcr$!3=+++1$y;AzL#FS7$Fip$!3')******[d&GE#FS7$Fd p$!3%)*****HKws=#FS7$F_p$!3#)*****zKK76#FS7$$!31+++=]s#3#Ffo$!33+++\\V $z&>FS7$$!3`+++5vPEnFfn$!33+++#\\BL!=FS7$$!3W*****>R['y#*FK$!3++++-muZ ;FS7$$\"3v*****zVJ&H%)FK$!3!******zv6:\\\"FS7$$\"3B+++TOpF)*FK$!3$**** ****el[L\"FS7$$\"3d+++f=,.mFK$!3#******RP&)z<\"FS7$$\"3w*****f(4f'R$FK $!3)******z_&)4-\"FS7$$\"3%******z^r`R\"FK$!39+++@0KR')F07$$\"37+++lG% oY%F@$!3=+++gYeoqF07$$\"3%******fR!eC5F@$!3%*******eBz(\\&F07$$\"3)*** ***fY<\\Q\"F:$!3#)*****RU#*p#RF07$$\"3')*****RCt.^'!#E$!3'******fk%>cB F07$$\"3y*****41AI&*)!#H$!3Y+++T;)R&yF[q7$$\"3u+++L@-`*)Fden$\"3Y+++T; )R&yF[q-%&COLORG6&%$RGBG$\"#K!\"#$\"\")FbfnFcfn-%)POLYGONSG6^y7%F'7$$ \"+\"R)4DdF[q$\"+\\jzq:!#57$$FbfnF)F(7%Fifn7$$\"+4'4uk$FS$\"+bKfTJF^gn F_gn7%Fbgn7$$\"+;>(f5%!#;$\"+kAR7ZF^gnF_gn7%Fhgn7$$\"+'e3#\\A!#:$\"+*4 \">$G'F^gnF_gn7%F_hn7$$\"+_5;u\")Fbhn$\"+Kb'R&yF^gnF_gn7%Ffhn7$$\"+%)4 %)RA!#9$\"+F\"GYU*F^gnF_gn7%F\\in7$$\"+p][t[F_in$\"+P`\\*4\"!\"*F_gn7% Fcin7$$\"+&H@\")Q)F_in$\"+iiWc7FhinF_gn7%Fjin7$$\"+Os#=-\"!#8$\"+/uA89 FhinF_gn7%F`jn7$$\"+1kW_FF_in$\"+&R%pp:FhinF_gn7%Fgjn7$$!+l)3'3JFcjn$ \"+9`jDMItFf[o$\"+BjF(=#FhinF_gn7%Ff\\o7$$!+\\n6V5!#6$\"+\\d&GE#FhinF_gn7 %F\\]o7$$!+CGV[9F_]o$\"+i@#zL#FhinF_gn7%Fc]o7$$!+S$4(o>F_]o$\"+c;U7CFh inF_gn7%Fi]o7$$!++KeDEF_]o$\"+b)*G'[#FhinF_gn7%F_^o7$$!+h**HUMF_]o$\"+ m[WfDFhinF_gn7%Fe^o7$$!+RA>VWF_]o$\"+1$yt!*H\"F^gn$\"+k(*HuHFhinF_gn7%Fi`o7$$!+.`A]:F^gn$\"+EO#p.$Fhi nF_gn7%F_ao7$$!+\"z/t#=F^gn$\"+Fuw'4$FhinF_gn7%Feao7$$!+,Y(y7#F^gn$\"+ 7B_`JFhinF_gn7%F[bo7$$!+#fy'[CF^gn$\"+U@'p?$FhinF_gn7%Fabo7$$!+/c+'y#F ^gn$\"+$4ipD$FhinF_gn7%Fgbo7$$!+$>xg8$F^gn$\"+E#*\\.LFhinF_gn7%F]co7$$ !+AqO&\\$F^gn$\"+!3QmM$FhinF_gn7%Fcco7$$!+=O!3'QF^gn$\"+=\"3lQ$FhinF_g n7%Fico7$$!+Xf&)HUF^gn$\"+%))zKU$FhinF_gn7%F_do7$$!+//`+YF^gn$\"+0W9dM FhinF_gn7%Fedo7$$!+vF5T`F^gn$\"+)eIp^$FhinF_gn7%F[eo7$$!+&3&zugF^gn$\" +?vLnNFhinF_gn7%Faeo7$$!+]]'zz'F^gn$\"+X+i4OFhinF_gn7%Fgeo7$$!+'ox#4vF ^gn$\"+/BzWOFhinF_gn7%F]fo7$$!+W9g3#)F^gn$\"+)GcOn$FhinF_gn7%Fcfo7$$!+ \"=$Q'*))F^gn$\"+(GUop$FhinF_gn7%Fifo7$$!+4LJt&*F^gn$\"+&3Y[r$FhinF_gn 7%F_go7$$!+Cb,C5Fhin$\"+Q!f!GPFhinF_gn7%Fego7$$!+5lw*3\"Fhin$\"+07zOPF hinF_gn7%F[ho7$$!+Qgla6Fhin$\"+6))GTPFhinF_gn7%Faho7$$!+G:v=7Fhin$\"+$ QZM8JPFhinF_gn7%Fcio7$$!+%[4pS\"Fhin$\"+SrF?PFhinF_gn7%Fiio7$ $!+eqYo9Fhin$\"+K`#eq$FhinF_gn7%F_jo7$$!+`?cH:Fhin$\"+GF$yo$FhinF_gn7% Fejo7$$!+z!z-f\"Fhin$\"+e0MmOFhinF_gn7%F[[p7$$!+%><2l\"Fhin$\"+16QTOFh inF_gn7%Fa[p7$$!+4T*4r\"Fhin$\"+SJ)Hh$FhinF_gn7%Fg[p7$$!+5/DrFhin$\"+'e6fY$FhinF_gn7%F_]p 7$$!+'[Fe,#Fhin$\"+?:D@MFhinF_gn7%Fe]p7$$!+b@+z?Fhin$\"+4n\"QP$FhinF_g n7%F[^p7$$!+tiXV@Fhin$\"+75%RK$FhinF_gn7%Fa^p7$$!+D3N4AFhin$\"+jC0sKFh inF_gn7%Fg^p7$$!+)fRnF#Fhin$\"+thl=KFhinF_gn7%F]_p7$$!+L>`XBFhin$\"+-C HkJFhinF_gn7%Fc_p7$$!+:)yaT#Fhin$\"+w]Z4JFhinF_gn7%Fi_p7$$!+#f#>'[#Fhi n$\"+_viaIFhinF_gn7%F_`p7$$!+@m>dDFhin$\"+/m.+IFhinF_gn7%Fe`p7$$!+*)z) zi#Fhin$\"+\\y$e%HFhinF_gn7%F[ap7$$!+,M4)p#Fhin$\"+s8.#*GFhinF_gn7%Faa p7$$!+*z0rw#Fhin$\"+,c^QGFhinF_gn7%Fgap7$$!+:)*pMGFhin$\"+Ab7&y#FhinF_ gn7%F]bp7$$!+^6j+HFhin$\"+OdmJFFhinF_gn7%Fcbp7$$!+poskHFhin$\"+*\\Mzn# FhinF_gn7%Fibp7$$!+)Hso-$Fhin$\"+)*)QPi#FhinF_gn7%F_cp7$$!+j!)*p3$Fhin $\"+lI!*oDFhinF_gn7%Fecp7$$!+J*o]9$Fhin$\"+Z;F8DFhinF_gn7%F[dp7$$!+r\\ 2,KFhin$\"+!R5nX#FhinF_gn7%Fadp7$$!+Q\\-bKFhin$\"+$=0\"*R#FhinF_gn7%Fg dp7$$!+M:%pI$Fhin$\"+A/OSBFhinF_gn7%F]ep7$$!+j\"eoN$Fhin$\"+\\sR!G#Fhi nF_gn7%Fcep7$$!+xn\"[S$Fhin$\"+j>:>AFhinF_gn7%Fiep7$$!+Bl'3X$Fhin$\"+& zul:#FhinF_gn7%F_fp7$$!+&*G1&\\$Fhin$\"+<\"HE4#FhinF_gn7%Fefp7$$!+krYP NFhin$\"+M6HF?FhinF_gn7%F[gp7$$!+oe9yNFhin$\"+X+bg>FhinF_gn7%Fagp7$$!+ o,<vAy)*F^gnF_gn7 %F_\\q7$$!+Ho@-SFhin$\"+Wv$[3*F^gnF_gn7%Fe\\q7$$!+H['H-%Fhin$\"+=gv!H) F^gnF_gn7%F[]q7$$!+X7>USFhin$\"+3(4n\\(F^gnF_gn7%Fa]q7$$!+&)>!)fSFhin$ \"+QvB.nF^gnF_gn7%Fg]q7$$!+(f0d2%Fhin$\"+kDq5fF^gnF_gn7%F]^q7$$!+!e;)* 3%Fhin$\"+5=J>^F^gnF_gn7%Fc^q7$$!+&zd?5%Fhin$\"+>,9HVF^gnF_gn7%Fi^q7$$ !+M?O7TFhin$\"+'*[:SNF^gnF_gn7%F__q7$$!+LHn?TFhin$\"+W&RAv#F^gnF_gn7%F e_q7$$!+%RXp7%Fhin$\"+xV@l>F^gnF_gn7%F[`q7$$!+*oX68%Fhin$\"+kX&)y6F^gn F_gn7%Fa`q7$$!+?9DLTFhin$\"+AH0HRF_]oF_gn7%Fg`q7$Fh`q$!+AH0HRF_]oF_gn7 %F]aq7$Fb`q$!+kX&)y6F^gnF_gn7%Faaq7$F\\`q$!+xV@l>F^gnF_gn7%Feaq7$Ff_q$ !+W&RAv#F^gnF_gn7%Fiaq7$F`_q$!+'*[:SNF^gnF_gn7%F]bq7$Fj^q$!+>,9HVF^gnF _gn7%Fabq7$Fd^q$!+5=J>^F^gnF_gn7%Febq7$F^^q$!+kDq5fF^gnF_gn7%Fibq7$Fh] q$!+QvB.nF^gnF_gn7%F]cq7$Fb]q$!+3(4n\\(F^gnF_gn7%Facq7$F\\]q$!+=gv!H)F ^gnF_gn7%Fecq7$Ff\\q$!+Wv$[3*F^gnF_gn7%Ficq7$F`\\q$!+>vAy)*F^gnF_gn7%F ]dq7$Fj[q$!+36+n5FhinF_gn7%Fadq7$Fd[q$!+w*4f9\"FhinF_gn7%Fedq7$F^[q$!+ &=EWA\"FhinF_gn7%Fidq7$Fhjp$!+1nT-8FhinF_gn7%F]eq7$Fbjp$!+?Tuz8FhinF_g n7%Faeq7$F\\jp$!+uCFc9FhinF_gn7%Feeq7$Ffip$!+6G(=`\"FhinF_gn7%Fieq7$F` ip$!+GyU1;FhinF_gn7%F]fq7$Fjhp$!+__$)z;FhinF_gn7%Fafq7$Fdhp$!+g)4?v\"F hinF_gn7%Fefq7$F^hp$!+_U)G#=FhinF_gn7%Fifq7$Fhgp$!+V'3C*=FhinF_gn7%F]g q7$Fbgp$!+X+bg>FhinF_gn7%Fagq7$F\\gp$!+M6HF?FhinF_gn7%Fegq7$Fffp$!+<\" HE4#FhinF_gn7%Figq7$F`fp$!+&zul:#FhinF_gn7%F]hq7$Fjep$!+j>:>AFhinF_gn7 %Fahq7$Fdep$!+\\sR!G#FhinF_gn7%Fehq7$F^ep$!+A/OSBFhinF_gn7%Fihq7$Fhdp$ !+$=0\"*R#FhinF_gn7%F]iq7$Fbdp$!+!R5nX#FhinF_gn7%Faiq7$F\\dp$!+Z;F8DFh inF_gn7%Feiq7$Ffcp$!+lI!*oDFhinF_gn7%Fiiq7$F`cp$!+)*)QPi#FhinF_gn7%F]j q7$Fjbp$!+*\\Mzn#FhinF_gn7%Fajq7$Fdbp$!+OdmJFFhinF_gn7%Fejq7$F^bp$!+Ab 7&y#FhinF_gn7%Fijq7$Fhap$!+,c^QGFhinF_gn7%F][r7$Fbap$!+s8.#*GFhinF_gn7 %Fa[r7$F\\ap$!+\\y$e%HFhinF_gn7%Fe[r7$Ff`p$!+/m.+IFhinF_gn7%Fi[r7$F``p $!+_viaIFhinF_gn7%F]\\r7$Fj_p$!+w]Z4JFhinF_gn7%Fa\\r7$Fd_p$!+-CHkJFhin F_gn7%Fe\\r7$F^_p$!+thl=KFhinF_gn7%Fi\\r7$Fh^p$!+jC0sKFhinF_gn7%F]]r7$ Fb^p$!+75%RK$FhinF_gn7%Fa]r7$F\\^p$!+4n\"QP$FhinF_gn7%Fe]r7$Ff]p$!+?:D @MFhinF_gn7%Fi]r7$F`]p$!+'e6fY$FhinF_gn7%F]^r7$Fj\\p$!+QTb2NFhinF_gn7% Fa^r7$Fd\\p$!+\"z7ga$FhinF_gn7%Fe^r7$F^\\p$!+?)y6e$FhinF_gn7%Fi^r7$Fh[ p$!+SJ)Hh$FhinF_gn7%F]_r7$Fb[p$!+16QTOFhinF_gn7%Fa_r7$F\\[p$!+e0MmOFhi nF_gn7%Fe_r7$Ffjo$!+GF$yo$FhinF_gn7%Fi_r7$F`jo$!+K`#eq$FhinF_gn7%F]`r7 $Fjio$!+SrF?PFhinF_gn7%Fa`r7$Fdio$!+>M8JPFhinF_gn7%Fe`r7$F^io$!+'p@$QP FhinF_gn7%Fi`r7$Fhho$!+$QZFhinF_gn7%F[jr7$$!+5vPEnFcjn$!+#\\BL!=FhinF_gn7%Fajr7$$!+ #R['y#*F_in$!+-muZ;FhinF_gn7%Fgjr7$$\"+Q9`H%)F_in$!+e<^\"\\\"FhinF_gn7 %F][s7$$\"+TOpF)*F_in$!+!fl[L\"FhinF_gn7%Fc[s7$$\"+f=,.mF_in$!+u`)z<\" FhinF_gn7%Fi[s7$$\"+w4f'R$F_in$!+Gb)4-\"FhinF_gn7%F_\\s7$$\"+=:P&R\"F_ in$!+@0KR')F^gnF_gn7%Fe\\s7$$\"+lG%oY%Fbhn$!+gYeoqF^gnF_gn7%F[]s7$$\"+ 'R!eC5Fbhn$!+fBz(\\&F^gnF_gn7%Fa]s7$$\"+mu\"\\Q\"F[hn$!+CC*p#RF^gnF_gn 7%Fg]s7$$\"+WKP5lF0$!+YY>cBF^gnF_gn7%F]^s7$$\"+h?-`*)Ffn$!+T;)R&yF_]oF _gn7%Fc^s7$$\"+L@-`*)Ffn$\"+T;)R&yF_]oF_gn-F]fn6&F_fn$\"#jFbfn$\"#:Fbf nFb_s-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3%******>%))4V9FS$\"3%***** *R>s#)H%FS7$$\"3,+++5&eXV\"FS$\"3J+++\"\\osI%FS7$$\"3&******>*pJD9FS$ \"3C+++yVa:VFS7$$\"3#******poLJKE\"FS$\"3M+++*H_zL% FS7$$\"3'******>8h'\\7FS$\"32+++fRXJVFS7$$\"3'******H1mjB\"FS$\"3L+++j u]BVFS7$$\"3++++l;[B7FS$\"3)******fc_SJ%FS7$$\"3\"******\\$f;67FS$\"39 +++$*4..VFS7$$\"33+++qVg*>\"FS$\"3K+++g)*Q!H%FS7$$\"3!******H&e,*=\"FS $\"31+++Yr3wUFS7$$\"3%******H4f'z6FS$\"3q******p-5gUFS7$$\"3#******f&) Q=<\"FS$\"3G+++U\"RCC%FS7$$\"3!******z;5f;\"FS$\"3U+++:a;BUFS7$$\"3#** ****\\,xA;\"FS$\"3)******R[>C?%FS7$$\"33+++<&z8;\"FS$\"3K+++)Rb/=%FS7$ $\"31+++w4mj6FS$\"3%)*****z&)zw:%FS7$$\"33+++?h]p6FS$\"3#******>Y$oMTF S7$$\"35+++O%\\\"z6FS$\"3B+++Z>C7TFS7$$\"3%******RNsD>\"FS$\"3%******> kg74%FS7$$\"3#******HFL%47FS$\"3k*****\\z^E2%FS7$$\"3(******\\fz!H7FS$ \"3/+++;b#fvR8FS$\"3s*****\\z(\\LSFS7$$\"31+ ++ht')f8FS$\"3&*******o/'y.%FS7$$\"33+++(Q.&y8FS$\"3&******H)[MWSFS7$$ \"3-+++U;b&R\"FS$\"3O+++Nee_SFS7$$\"31+++5D'4T\"FS$\"39+++G?EiSFS7$$\" 31+++)RGZU\"FS$\"33+++3`4tSFS7$$\"3-+++#)*ooV\"FS$\"3++++/m%[3%FS7$$\" 3!******p0@uW\"FS$\"3y******)y5t4%FS7$$\"3-+++&pJkX\"FS$\"3B+++/:J5TFS 7$$\"3!*******>P&RY\"FS$\"3?+++xkpBTFS7$$\"3%******puU+Z\"FS$\"3-+++_M LPTFS7$$\"3+++++`vu9FS$\"3++++on5^TFS7$$\"3++++Cw9y9FS$\"3=+++7X\"\\;% FS7$$\"3))*****\\)\\F![\"FS$\"3o******fhmyTFS7$$\"3-+++$G\">\"[\"FS$\" 3/+++U1G#>%FS7$$\"3/+++o)[4[\"FS$\"3M+++PZo0UFS7$$\"31+++.&)fz9FS$\"3p *****>v6)=UFS7$$\"3!******4T*=x9FS$\"3M+++4/gJUFS7$$\"3-+++)QpPZ\"FS$ \"33+++GR*RC%FS7$$\"3++++Z\\Qp9FS$\"35+++T#RfD%FS7$$\"3'*******)\\\"3k 9FS$\"3u*****zK'QnUFS7$$\"35+++WN!zX\"FS$\"3W+++UwGyUFS7$$\"3#******H) [*3X\"FS$\"39+++6wf)G%FSF[`sF\\fn-Fffn6jn7%7$$\"+U))4V9Fhin$\"+%>s#)H% Fhin7$$\"+5&eXV\"Fhin$\"+\"\\osI%Fhin7$$\"$K\"Fbfn$\"$>%Fbfn7%F_ct7$$ \"+#*pJD9Fhin$\"+yVa:VFhinFdct7%Fjct7$$\"+(o\"Fhin$\"+g)*Q!H%FhinFdct7%Ffjt7$$\"+`e,* =\"Fhin$\"+Yr3wUFhinFdct7%F\\[u7$$\"+$4f'z6Fhin$\"+q-5gUFhinFdct7%Fb[u 7$$\"+c)Q=<\"Fhin$\"+U\"RCC%FhinFdct7%Fh[u7$$\"+o,\"f;\"Fhin$\"+:a;BUF hinFdct7%F^\\u7$$\"+:qFi6Fhin$\"+%[>C?%FhinFdct7%Fd\\u7$$\"+<&z8;\"Fhi n$\"+)Rb/=%FhinFdct7%Fj\\u7$$\"+w4mj6Fhin$\"+e)zw:%FhinFdct7%F`]u7$$\" +?h]p6Fhin$\"+iMoMTFhinFdct7%Ff]u7$$\"+O%\\\"z6Fhin$\"+Z>C7TFhinFdct7% F\\^u7$$\"+aBd#>\"Fhin$\"+U1E\"4%FhinFdct7%Fb^u7$$\"+tKV47Fhin$\"+&z^E 2%FhinFdct7%Fh^u7$$\"+&fz!H7Fhin$\"+;b\"[\"Fhin$\"+U1G#>%FhinFdct7%F`fu7$$\"+o)[4[\"Fhin$ \"+PZo0UFhinFdct7%Fffu7$$\"+.&)fz9Fhin$\"+_<\")=UFhinFdct7%F\\gu7$$\"+ 6%*=x9Fhin$\"+4/gJUFhinFdct7%Fbgu7$$\"+)QpPZ\"Fhin$\"+GR*RC%FhinFdct7% Fhgu7$$\"+Z\\Qp9Fhin$\"+T#RfD%FhinFdct7%F^hu7$$\"+*\\\"3k9Fhin$\"+GjQn UFhinFdct7%Fdhu7$$\"+WN!zX\"Fhin$\"+UwGyUFhinFdct7%Fjhu7$$\"+$)[*3X\"F hin$\"+6wf)G%FhinFdct7%F`iuFjbtFdctF^_sFd_s-F$6$7in7$F\\`s$!3%******R> s#)H%FS7$Fcbt$!39+++6wf)G%FS7$F^bt$!3W+++UwGyUFS7$Fiat$!3u*****zK'QnUF S7$Fdat$!35+++T#RfD%FS7$F_at$!33+++GR*RC%FS7$Fj`t$!3M+++4/gJUFS7$Fe`t$ !3p*****>v6)=UFS7$F``t$!3M+++PZo0UFS7$F[`t$!3/+++U1G#>%FS7$Ff_t$!3o*** ***fhmyTFS7$Fa_t$!3=+++7X\"\\;%FS7$F\\_t$!3++++on5^TFS7$Fg^t$!3-+++_ML PTFS7$Fb^t$!3?+++xkpBTFS7$F]^t$!3B+++/:J5TFS7$Fh]t$!3y******)y5t4%FS7$ Fc]t$!3++++/m%[3%FS7$F^]t$!33+++3`4tSFS7$Fi\\t$!39+++G?EiSFS7$Fd\\t$!3 O+++Nee_SFS7$F_\\t$!3&******H)[MWSFS7$Fj[t$!3&*******o/'y.%FS7$Fe[t$!3 s*****\\z(\\LSFS7$F`[t$!3f******f&f;.%FS7$F[[t$!32+++?iwKSFS7$Ffjs$!3 \")******y\\@PSFS7$Fajs$!3/+++j6JXSFS7$F\\js$!3/+++;bkg74%FS7$F]is$!3B+++Z>C7TFS7$Fhhs$!3#**** **>Y$oMTFS7$Fchs$!3%)*****z&)zw:%FS7$F^hs$!3K+++)Rb/=%FS7$Figs$!3)**** **R[>C?%FS7$Fdgs$!3U+++:a;BUFS7$F_gs$!3G+++U\"RCC%FS7$Fjfs$!3q******p- 5gUFS7$Fefs$!31+++Yr3wUFS7$F`fs$!3K+++g)*Q!H%FS7$F[fs$!39+++$*4..VFS7$ Ffes$!3)******fc_SJ%FS7$Faes$!3L+++ju]BVFS7$F\\es$!32+++fRXJVFS7$Fgds$ !3M+++*H_zL%FS7$Fbds$!3I+++SE1VVFS7$F]ds$!3!)******eQ%oM%FS7$Fhcs$!3!* *****f#HN\\VFS7$Fccs$!3K+++FYk]VFS7$F^cs$!3++++;:x]VFS7$Fibs$!3?+++'*R y\\VFS7$Fdbs$!3w*****p[IxM%FS7$F_bs$!3w*****RkdYM%FS7$Fjas$!3K+++'f51M %FS7$Feas$!3@+++6KjNVFS7$F`as$!3&)*****pNo(HVFS7$F[as$!3J+++!=eIK%FS7$ Ff`s$!3C+++yVa:VFS7$Fa`s$!3J+++\"\\osI%FSFiiuF\\fn-Fffn6jn7%7$F[ct$!+% >s#)H%Fhin7$Faiu$!+6wf)G%Fhin7$Fect$!$>%Fbfn7%Fcev7$F[iu$!+UwGyUFhinFf ev7%Fjev7$Fehu$!+GjQnUFhinFfev7%F^fv7$F_hu$!+T#RfD%FhinFfev7%Fbfv7$Fig u$!+GR*RC%FhinFfev7%Fffv7$Fcgu$!+4/gJUFhinFfev7%Fjfv7$F]gu$!+_<\")=UFh inFfev7%F^gv7$Fgfu$!+PZo0UFhinFfev7%Fbgv7$Fafu$!+U1G#>%FhinFfev7%Ffgv7 $F[fu$!+ghmyTFhinFfev7%Fjgv7$Feeu$!+7X\"\\;%FhinFfev7%F^hv7$F_eu$!+on5 ^TFhinFfev7%Fbhv7$Fidu$!+_MLPTFhinFfev7%Ffhv7$Fcdu$!+xkpBTFhinFfev7%Fj hv7$F]du$!+/:J5TFhinFfev7%F^iv7$Fgcu$!+*y5t4%FhinFfev7%Fbiv7$Facu$!+/m %[3%FhinFfev7%Ffiv7$F[cu$!+3`4tSFhinFfev7%Fjiv7$Febu$!+G?EiSFhinFfev7% F^jv7$F_bu$!+Nee_SFhinFfev7%Fbjv7$Fiau$!+$)[MWSFhinFfev7%Ffjv7$Fcau$!+ p/'y.%FhinFfev7%Fjjv7$F]au$!+&z(\\LSFhinFfev7%F^[w7$Fg`u$!+g&f;.%FhinF fev7%Fb[w7$Fa`u$!+?iwKSFhinFfev7%Ff[w7$F[`u$!+z\\@PSFhinFfev7%Fj[w7$Fe _u$!+j6JXSFhinFfev7%F^\\w7$F__u$!+;bC7TFhinFfev7%F ^]w7$Fg]u$!+iMoMTFhinFfev7%Fb]w7$Fa]u$!+e)zw:%FhinFfev7%Ff]w7$F[]u$!+) Rb/=%FhinFfev7%Fj]w7$Fe\\u$!+%[>C?%FhinFfev7%F^^w7$F_\\u$!+:a;BUFhinFf ev7%Fb^w7$Fi[u$!+U\"RCC%FhinFfev7%Ff^w7$Fc[u$!+q-5gUFhinFfev7%Fj^w7$F] [u$!+Yr3wUFhinFfev7%F^_w7$Fgjt$!+g)*Q!H%FhinFfev7%Fb_w7$Fajt$!+$*4..VF hinFfev7%Ff_w7$F[jt$!+mD09VFhinFfev7%Fj_w7$Feit$!+ju]BVFhinFfev7%F^`w7 $F_it$!+fRXJVFhinFfev7%Fb`w7$Fiht$!+*H_zL%FhinFfev7%Ff`w7$Fcht$!+SE1VV FhinFfev7%Fj`w7$F]ht$!+fQ%oM%FhinFfev7%F^aw7$Fggt$!+EHN\\VFhinFfev7%Fb aw7$Fagt$!+FYk]VFhinFfev7%Ffaw7$F[gt$!+;:x]VFhinFfev7%Fjaw7$Feft$!+'*R y\\VFhinFfev7%F^bw7$F_ft$!+([IxM%FhinFfev7%Fbbw7$Fiet$!+WwlWVFhinFfev7 %Ffbw7$Fcet$!+'f51M%FhinFfev7%Fjbw7$F]et$!+6KjNVFhinFfev7%F^cw7$Fgdt$! +d$o(HVFhinFfev7%Fbcw7$Fadt$!+!=eIK%FhinFfev7%Ffcw7$F[dt$!+yVa:VFhinFf ev7%Fjcw7$F`ct$!+\"\\osI%FhinFfev7%F^dwF`evFfevF^_sFd_s-F$6%7$7$$!3/++ ++++!z%FSF(7$$\"3+++++++!*>FSF(-%'COLOURG6&F_fnF)F)F)-%*LINESTYLEG6#\" \"$-F$6%7$7$F(Ffdw7$F($\"3/++++++!z%FSF[ewF^ew-%%FONTG6$%*HELVETICAG\" \"*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fjew6#% (DEFAULTG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%Fbfn$\"$*>Fbfn;Fbgw$\"$z %Fbfn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+ 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this sch eme is (approximately) " }{XPPEDIT 18 0 "[-4.1335, 0];" "6#7$,$-%&Flo atG6$\"&N8%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the stability region for the 8 stage, order 4 scheme with that of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[d isplay]([p||(1..7),p_||(1..6)],view=[-4.79..1.99,-4.79..4.79],font=[HE LVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=co nstrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "65- %'CURVESG6$7]y7$$\"\"!F)F(7$$\"3s******z7^)4'!#G$\"3#******fK'zq:!#=7$ $\"3#)*****p$Gv,W!#E$\"3$******Hf#fTJF07$$\"3G+++.0+sd!#D$\"3z*****\\& yQ7ZF07$$\"3z*******=-Ki$!#C$\"3?+++Rg<$G'F07$$\"3.+++GR&GV\"!#B$\"3G+ ++'oKR&yF07$$\"3o********p5xQFF$\"3Z*****zm$eC%*F07$$\"3w******G'ePi'F F$\"35+++uW\\*4\"!#<7$$\"3/+++**4')y7FF$\"3)********GikD\"FS7$$!3#**** **RJBu4%!#A$\"35++++!)G89FS7$$!3,+++-oRV>!#@$\"3)******f&y%)p:FS7$$!3U +++>BT!>'F\\o$\"3#******4cbfs\"FS7$$!3++++n)\\#=;!#?$\"33+++o!f8)=FS7$ $!3$)******\\:&fq$Fgo$\"38+++y=vN?FS7$$!3o*****>$3(HR&Fgo$\"3!)*****\\ 7kC6#FS7$$!3E+++QL!Qn(Fgo$\"3=+++k`z)=#FS7$$!3%******4e<*p5!#>$\"35+++ I1qkAFS7$$!3)******4#=?k9F\\q$\"3(******H]K,M#FS7$$!3#******4tA(p>F\\q $\"3#)*****HAQ]T#FS7$$!35+++a.03EF\\q$\"3))*****p$\\N*[#FS7$$!3F+++B&z DS$F\\q$\"3?+++-G+jDFS7$$!3q******Hm0yVF\\q$\"3()*****4)f(ej#FS7$$!3'* *****Rlu(fbF\\q$\"37+++??$yq#FS7$$!3i*****z=+B(pF\\q$\"3?+++o4oyFFS7$$ !3Q+++54iP')F\\q$\"3/+++*yw\"[GFS7$$!3)******p&>Fd5F0$\"3A+++()[,;HFS7 $$!3/+++9Boy7F0$\"3A+++1-%=)HFS7$$!3&*******R-*y_\"F0$\"3?+++\"4p_/$FS 7$$!32+++FSi.=F0$\"3(******\\8Cf5$FS7$$!35+++S*zM5#F0$\"3A+++%RvM;$FS7 $$!34+++*[eTU#F0$\"3!******40xw@$FS7$$!3y******yT%=w#F0$\"3A+++5\"*QoK FS7$$!3x*****zQ@E6$F0$\"3-+++!fzbJ$FS7$$!3!******4fAGZ$F0$\"39+++J+JfL FS7$$!3))*****>!>DRQF0$\"3#)******e;r*R$FS7$$!3D+++&y!G4UF0$\"3!****** H%)fpV$FS7$$!3-+++$eY3e%F0$\"37+++h>DrMFS7$$!3a+++`*)yA`F0$\"3z*****fg t<`$FS7$$!3y*****4I-s0'F0$\"3))*****H:'z#e$FS7$$!3i******\\CX!y'F0$\"3 1+++#*>hDOFS7$$!3k*****4Bt7\\(F0$\"3%******>$>EhOFS7$$!3q*****4J9'*=)F 0$\"3#******>[m0p$FS7$$!3-+++yk*f())F0$\"3&******f))oTr$FS7$$!3a+++W0< ^&*F0$\"3=++++UdKPFS7$$!32+++1Yf@5FS$\"35+++J1=YPFS7$$!33+++>76(3\"FS$ \"3)******>[.`v$FS7$$!3'******>!3u^6FS$\"3'******H9$>gPFS7$$!3/+++:>b: 7FS$\"3/+++i&[5w$FS7$$!3-+++q)3'y7FS$\"33+++1u-ePFS7$$!3%******\\it4M \"FS$\"3'******R#QD^PFS7$$!3-+++)>3FS\"FS$\"37+++RX#3u$FS7$$!3#******R XxQY\"FS$\"3<+++3Q\"os$FS7$$!3%******zD_X_\"FS$\"3-+++RxF4PFS7$$!35+++ uH\"[e\"FS$\"3-+++*Qe#)o$FS7$$!3'******zK`Zk\"FS$\"3!******H5)yjOFS7$$ !3*******fa%[/.KNFS7$$!3!******>PsV%>FS$\"3#)*****pk,4\\$FS7$$!3&)*****f3\"e0?FS$ \"3#)*****ptCnW$FS7$$!3y******)*Rvn?FS$\"3%)*****fUA(*R$FS7$$!3=+++R&> 68#FS$\"3=+++qK?]LFS7$$!3')*****H^`e>#FS$\"3%******HVp&)H$FS7$$!3/+++' fN?E#FS$\"39+++[ZIXKFS7$$!3\")*****H\")4'HBFS$\"3#******4pQ4>$FS7$$!3# )*****\\nk$)R#FS$\"3\")*****p/*)f8$FS7$$!39+++F^%zY#FS$\"33+++yv*33$FS 7$$!38+++Qf*y`#FS$\"3/+++b$zf-$FS7$$!3>+++(>Axg#FS$\"3y*****fA,9(HFS7$ $!3')*****f1Zpn#FS$\"3%)*****4J\">\"GFS$\"36+++ZY[4GFS7$$!3#******4AEs(GFS$\"35+++@#\\ cv#FS7$$!3A+++z\"\\1%HFS$\"39+++jVc,FFS7$$!39+++>89-IFS$\"3;+++:p.ZEFS 7$$!3%******4(RihIFS$\"3#)*****zz))=f#FS7$$!3$******\\[a!>JFS$\"3$)*** **\\Ihf`#FS7$$!31+++qqTuJFS$\"3y*****z?;\"zCFS7$$!3!******\\(erFKFS$\" 3$******>3N7U#FS7$$!3?+++&**p*yKFS$\"3')*****f!y@iBFS7$$!3=+++T+@GLFS$ \"3(******z$3)>I#FS7$$!31+++*)eZvLFS$\"3:+++EcXSAFS7$$!38+++o_\"3U$FS$ \"3%)*****pL(ex@FS7$$!3=+++VHGkMFS$\"38+++WSL8@FS7$$!3#******\\?Sf]$FS $\"3)******pRmw/#FS7$$!37+++\"[aea$FS$\"3%*******4yc!)>FS7$$!3y******R ))4%e$FS$\"36+++J^.7>FS7$$!3#)*****p<^2i$FS$\"3(******>tz?%=FS7$$!3#)* ****p)G*el$FS$\"3)*******y)G2x\"FS7$$!35+++]og*o$FS$\"3/+++*=F!)p\"FS7 $$!38+++/Y(>s$FS$\"3\"*******oy.C;FS7$$!37+++0F2`PFS$\"35+++)QV)[:FS7$ $!3A+++G%oHy$FS$\"3++++h^as9FS7$$!3.+++N^r6QFS$\"3'******H!>E&R\"FS7$$ !3$)*****>(yMRQFS$\"3*******4UErJ\"FS7$$!3!)*****>!)ze'QFS$\"33+++')3G Q7FS7$$!3%******48+8*QFS$\"3'******4\"4()e6FS7$$!3;+++bRd:RFS$\"3++++m !R!z5FS7$$!3<+++_TkQRFS$\"3=+++!*f=*)**F07$$!31+++#3N/'RFS$\"3k*****\\ [\"G'=*F07$$!3%)*****pad3)RFS$\"39+++R)zEQ)F07$$!3'******HI9)**RFS$\"3 '******4R!=zvF07$$!31+++ma?MnY(fF07$$!3'******>(e!p/%FS$\"3O+++*4yU<&F07$$!3))*****fF07$$!3 !)*****>,@z3%FS$\"3'******>#))=\">\"F07$$!3o*****4A9+4%FS$\"3')*****>. /,(RF\\q7$Ff^m$!3')*****>./,(RF\\q7$Fa^m$!3'******>#))=\">\"F07$F\\^m$ !3))******QR$e)>F07$Fg]m$!3w******zeB\"y#F07$Fb]m$!3!)*****HP^wd$F07$F ]]m$!37+++,oGvVF07$Fh\\m$!3O+++*4yU<&F07$Fc\\m$!3g*****>MnY(fF07$F^\\m $!3g*****zdwjx'F07$Fi[m$!3'******4R!=zvF07$Fd[m$!39+++R)zEQ)F07$F_[m$! 3k*****\\[\"G'=*F07$Fjjl$!3=+++!*f=*)**F07$Fejl$!3++++m!R!z5FS7$F`jl$! 3'******4\"4()e6FS7$F[jl$!33+++')3GQ7FS7$Ffil$!3*******4UErJ\"FS7$Fail $!3'******H!>E&R\"FS7$F\\il$!3++++h^as9FS7$Fghl$!35+++)QV)[:FS7$Fbhl$! 3\"*******oy.C;FS7$F]hl$!3/+++*=F!)p\"FS7$Fhgl$!3)*******y)G2x\"FS7$Fc gl$!3(******>tz?%=FS7$F^gl$!36+++J^.7>FS7$Fifl$!3%*******4yc!)>FS7$Fdf l$!3)******pRmw/#FS7$F_fl$!38+++WSL8@FS7$Fjel$!3%)*****pL(ex@FS7$Feel$ !3:+++EcXSAFS7$F`el$!3(******z$3)>I#FS7$F[el$!3')*****f!y@iBFS7$Ffdl$! 3$******>3N7U#FS7$Fadl$!3y*****z?;\"zCFS7$F\\dl$!3$)*****\\Ihf`#FS7$Fg cl$!3#)*****zz))=f#FS7$Fbcl$!3;+++:p.ZEFS7$F]cl$!39+++jVc,FFS7$Fhbl$!3 5+++@#\\cv#FS7$Fcbl$!36+++ZY[4GFS7$F^bl$!3?+++R%pK'GFS7$Fial$!3%)***** 4J\">$FS7$F[`l$!39+++[ZI XKFS7$Ff_l$!3%******HVp&)H$FS7$Fa_l$!3=+++qK?]LFS7$F\\_l$!3%)*****fUA( *R$FS7$Fg^l$!3#)*****ptCnW$FS7$Fb^l$!3#)*****pk,4\\$FS7$F]^l$!3\")**** *p*>.KNFS7$Fh]l$!3!******R9m*pNFS7$Fc]l$!3%)*****pO2Yg$FS7$F^]l$!3!*** ***\\f%*ej$FS7$Fi\\l$!3!******H5)yjOFS7$Fd\\l$!3-+++*Qe#)o$FS7$F_\\l$! 3-+++RxF4PFS7$Fj[l$!3<+++3Q\"os$FS7$Fe[l$!37+++RX#3u$FS7$F`[l$!3'***** *R#QD^PFS7$F[[l$!33+++1u-ePFS7$Ffz$!3/+++i&[5w$FS7$Faz$!3'******H9$>gP FS7$F\\z$!3)******>[.`v$FS7$Fgy$!35+++J1=YPFS7$Fby$!3=++++UdKPFS7$F]y$ !3&******f))oTr$FS7$Fhx$!3#******>[m0p$FS7$Fcx$!3%******>$>EhOFS7$F^x$ !31+++#*>hDOFS7$Fiw$!3))*****H:'z#e$FS7$Fdw$!3z*****fgt<`$FS7$$!3A+++t MR_\\F0$!33+++;3z-NFS7$F_w$!37+++h>DrMFS7$Fjv$!3!******H%)fpV$FS7$Fev$ !3#)******e;r*R$FS7$F`v$!39+++J+JfLFS7$F[v$!3-+++!fzbJ$FS7$Ffu$!3A+++5 \"*QoKFS7$Fau$!3!******40xw@$FS7$F\\u$!3A+++%RvM;$FS7$Fgt$!3(******\\8 Cf5$FS7$Fbt$!3?+++\"4p_/$FS7$F]t$!3A+++1-%=)HFS7$Fhs$!3A+++()[,;HFS7$F cs$!3/+++*yw\"[GFS7$F^s$!3?+++o4oyFFS7$Fir$!37+++??$yq#FS7$Fdr$!3()*** **4)f(ej#FS7$F_r$!3?+++-G+jDFS7$Fjq$!3))*****p$\\N*[#FS7$Feq$!3#)***** HAQ]T#FS7$F`q$!3(******H]K,M#FS7$Fjp$!35+++I1qkAFS7$Fep$!3=+++k`z)=#FS 7$F`p$!3!)*****\\7kC6#FS7$$!3++++(RyP[#Fgo$!3#*******)\\-(e>FS7$$!35++ +`QZ?5Fgo$!3/+++aMw.=FS7$$!33+++E[SzNF\\o$!3'******f\\szk\"FS7$$!3X*** **pi,jj*Ffn$!33+++P7h\"\\\"FS7$$!32+++YZK07Ffn$!3/+++&H**[L\"FS7$$\"3m *****pYk:1'FF$!3%******Ha!*z<\"FS7$$\"3#******zB$))eaFF$!3,+++o=)4-\"F S7$$\"3;+++;z3rCFF$!3&)*****ROz#R')F07$$\"3i*****>809d(F@$!3W+++N7coqF 07$$\"3++++/Fa[:F@$!3!******pn$y(\\&F07$$\"3++++ND1.=F:$!3:+++;0*p#RF0 7$$\"3)*******G6uKt!#F$!3)******>[%>cBF07$F($!3))*****Rj\")R&yF\\q7$F( $\"3))*****Rj\")R&yF\\q-%&COLORG6&%$RGBG$\"#Q!\"#$\"#=F^fnF_fn-%)POLYG ONSG6^y7%F'7$$\"+!G6&)4'Fgo$\"+Ejzq:!#57$$F^fnF)F(7%Fefn7$$\"+PGv,WF0$ \"+$f#fTJFjfnF[gn7%F^gn7$$\"+.0+sdFS$\"+byQ7ZFjfnF[gn7%Fdgn7$$\"+!>-Ki $!#;$\"+Rg<$G'FjfnF[gn7%Fjgn7$$\"+GR&GV\"!#:$\"+'oKR&yFjfnF[gn7%Fahn7$ $\"++q5xQFdhn$\"+oOeC%*FjfnF[gn7%Fhhn7$$\"+H'ePi'Fdhn$\"+uW\\*4\"!\"*F [gn7%F^in7$$\"+**4')y7Fdhn$\"+!HikD\"FcinF[gn7%Fein7$$!+9LU(4%!#9$\"++ !)G89FcinF[gn7%F[jn7$$!+-oRV>!#8$\"+cy%)p:FcinF[gn7%Fbjn7$$!+>BT!>'Fej n$\"+hb&fs\"FcinF[gn7%Fijn7$$!+n)\\#=;!#7$\"+o!f8)=FcinF[gn7%F_[o7$$!+ ]:&fq$Fb[o$\"+y=vN?FcinF[gn7%Ff[o7$$!+K3(HR&Fb[o$\"+DTY7@FcinF[gn7%F\\ \\o7$$!+QL!Qn(Fb[o$\"+k`z)=#FcinF[gn7%Fb\\o7$$!+\"e<*p5!#6$\"+I1qkAFci nF[gn7%Fh\\o7$$!+@=?k9F[]o$\"+.D8SBFcinF[gn7%F_]o7$$!+JFsp>F[]o$\"+B#Q ]T#FcinF[gn7%Fe]o7$$!+a.03EF[]o$\"+P\\N*[#FcinF[gn7%F[^o7$$!+B&zDS$F[] o$\"+-G+jDFcinF[gn7%Fa^o7$$!+Im0yVF[]o$\"+\")f(ej#FcinF[gn7%Fg^o7$$!+a YxfbF[]o$\"+??$yq#FcinF[gn7%F]_o7$$!+)=+B(pF[]o$\"+o4oyFFcinF[gn7%Fc_o 7$$!+54iP')F[]o$\"+*yw\"[GFcinF[gn7%Fi_o7$$!+d>Fd5Fjfn$\"+()[,;HFcinF[ gn7%F_`o7$$!+9Boy7Fjfn$\"+1-%=)HFcinF[gn7%Fe`o7$$!+S-*y_\"Fjfn$\"+\"4p _/$FcinF[gn7%F[ao7$$!+FSi.=Fjfn$\"+NT#f5$FcinF[gn7%Faao7$$!+S*zM5#Fjfn $\"+%RvM;$FcinF[gn7%Fgao7$$!+*[eTU#Fjfn$\"+^qnDRQFjfn$\"+f;r*R$FcinF[ gn7%Feco7$$!+&y!G4UFjfn$\"+V)fpV$FcinF[gn7%F[do7$$!+$eY3e%Fjfn$\"+h>Dr MFcinF[gn7%Fado7$$!+`*)yA`Fjfn$\"+1OxJNFcinF[gn7%Fgdo7$$!+,B?dgFjfn$\" +`hz#e$FcinF[gn7%F]eo7$$!+]CX!y'Fjfn$\"+#*>hDOFcinF[gn7%Fceo7$$!+JKF\" \\(Fjfn$\"+K>EhOFcinF[gn7%Fieo7$$!+6Vh*=)Fjfn$\"+#[m0p$FcinF[gn7%F_fo7 $$!+yk*f())Fjfn$\"+'))oTr$FcinF[gn7%Fefo7$$!+W0<^&*Fjfn$\"++UdKPFcinF[ gn7%F[go7$$!+1Yf@5Fcin$\"+J1=YPFcinF[gn7%Fago7$$!+>76(3\"Fcin$\"+#[.`v $FcinF[gn7%Fggo7$$!+-3u^6Fcin$\"+VJ>gPFcinF[gn7%F]ho7$$!+:>b:7Fcin$\"+ i&[5w$FcinF[gn7%Fcho7$$!+q)3'y7Fcin$\"+1u-ePFcinF[gn7%Fiho7$$!+DO(4M\" Fcin$\"+CQD^PFcinF[gn7%F_io7$$!+)>3FS\"Fcin$\"+RX#3u$FcinF[gn7%Feio7$$ !+au(QY\"Fcin$\"+3Q\"os$FcinF[gn7%F[jo7$$!+eAbC:Fcin$\"+RxF4PFcinF[gn7 %Fajo7$$!+uH\"[e\"Fcin$\"+*Qe#)o$FcinF[gn7%Fgjo7$$!+GLvW;Fcin$\"+.\")y jOFcinF[gn7%F][p7$$!+YX[/.KNFcinF[gn7%Fe\\p7$$!+sBPW>Fcin$\"+Z;!4\\$FcinF[gn7%F []p7$$!+'3\"e0?Fcin$\"+PZsYMFcinF[gn7%Fa]p7$$!+**Rvn?Fcin$\"+ECs*R$Fci nF[gn7%Fg]p7$$!+R&>68#Fcin$\"+qK?]LFcinF[gn7%F]^p7$$!+8N&e>#Fcin$\"+L% p&)H$FcinF[gn7%Fc^p7$$!+'fN?E#Fcin$\"+[ZIXKFcinF[gn7%Fi^p7$$!+8)4'HBFc in$\"+\"pQ4>$FcinF[gn7%F__p7$$!+vYO)R#Fcin$\"+Z!*)f8$FcinF[gn7%Fe_p7$$ !+F^%zY#Fcin$\"+yv*33$FcinF[gn7%F[`p7$$!+Qf*y`#Fcin$\"+b$zf-$FcinF[gn7 %Fa`p7$$!+(>Axg#Fcin$\"+E7SrHFcinF[gn7%Fg`p7$$!+mq%pn#Fcin$\"+68>\"GFcin$\"+ ZY[4GFcinF[gn7%Fiap7$$!+@iAxGFcin$\"+@#\\cv#FcinF[gn7%F_bp7$$!+z\"\\1% HFcin$\"+jVc,FFcinF[gn7%Febp7$$!+>89-IFcin$\"+:p.ZEFcinF[gn7%F[cp7$$!+ rRihIFcin$\"+)z))=f#FcinF[gn7%Facp7$$!+&[a!>JFcin$\"+08'f`#FcinF[gn7%F gcp7$$!+qqTuJFcin$\"+3i6zCFcinF[gn7%F]dp7$$!+verFKFcin$\"+#3N7U#FcinF[ gn7%Fcdp7$$!+&**p*yKFcin$\"+1y@iBFcinF[gn7%Fidp7$$!+T+@GLFcin$\"+Q3)>I #FcinF[gn7%F_ep7$$!+*)eZvLFcin$\"+EcXSAFcinF[gn7%Feep7$$!+o_\"3U$Fcin$ \"+Ptex@FcinF[gn7%F[fp7$$!+VHGkMFcin$\"+WSL8@FcinF[gn7%Fafp7$$!+0-%f]$ Fcin$\"+(Rmw/#FcinF[gn7%Fgfp7$$!+\"[aea$Fcin$\"+5yc!)>FcinF[gn7%F]gp7$ $!+S))4%e$Fcin$\"+J^.7>FcinF[gn7%Fcgp7$$!+x6v?OFcin$\"+K(z?%=FcinF[gn7 %Figp7$$!+()G*el$Fcin$\"+z)G2x\"FcinF[gn7%F_hp7$$!+]og*o$Fcin$\"+*=F!) p\"FcinF[gn7%Fehp7$$!+/Y(>s$Fcin$\"+py.C;FcinF[gn7%F[ip7$$!+0F2`PFcin$ \"+)QV)[:FcinF[gn7%Faip7$$!+G%oHy$Fcin$\"+h^as9FcinF[gn7%Fgip7$$!+N^r6 QFcin$\"+.>E&R\"FcinF[gn7%F]jp7$$!+syMRQFcin$\"+@k7<8FcinF[gn7%Fcjp7$$ !+-)ze'QFcin$\"+')3GQ7FcinF[gn7%Fijp7$$!+J,I\"*QFcin$\"+64()e6FcinF[gn 7%F_[q7$$!+bRd:RFcin$\"+m!R!z5FcinF[gn7%Fe[q7$$!+_TkQRFcin$\"+!*f=*)** FjfnF[gn7%F[\\q7$$!+#3N/'RFcin$\"+&[\"G'=*FjfnF[gn7%Fa\\q7$$!+Zv&3)RFc in$\"+R)zEQ)FjfnF[gn7%Fg\\q7$$!+.V\")**RFcin$\"+\"R!=zvFjfnF[gn7%F]]q7 $$!+ma?FjfnF[gn7%Fg_q7$$!+75#z3%F cin$\"+A))=\">\"FjfnF[gn7%F]`q7$$!+@U,!4%Fcin$\"+KS5qRF[]oF[gn7%Fc`q7$ Fd`q$!+KS5qRF[]oF[gn7%Fi`q7$F^`q$!+A))=\">\"FjfnF[gn7%F]aq7$Fh_q$!+RR$ e)>FjfnF[gn7%Faaq7$Fb_q$!+!)eB\"y#FjfnF[gn7%Feaq7$F\\_q$!+t8lxNFjfnF[g n7%Fiaq7$Ff^q$!+,oGvVFjfnF[gn7%F]bq7$F`^q$!+*4yU<&FjfnF[gn7%Fabq7$Fj]q $!+UtmufFjfnF[gn7%Febq7$Fd]q$!+ylPwnFjfnF[gn7%Fibq7$F^]q$!+\"R!=zvFjfn F[gn7%F]cq7$Fh\\q$!+R)zEQ)FjfnF[gn7%Facq7$Fb\\q$!+&[\"G'=*FjfnF[gn7%Fe cq7$F\\\\q$!+!*f=*)**FjfnF[gn7%Ficq7$Ff[q$!+m!R!z5FcinF[gn7%F]dq7$F`[q $!+64()e6FcinF[gn7%Fadq7$Fjjp$!+')3GQ7FcinF[gn7%Fedq7$Fdjp$!+@k7<8Fcin F[gn7%Fidq7$F^jp$!+.>E&R\"FcinF[gn7%F]eq7$Fhip$!+h^as9FcinF[gn7%Faeq7$ Fbip$!+)QV)[:FcinF[gn7%Feeq7$F\\ip$!+py.C;FcinF[gn7%Fieq7$Ffhp$!+*=F!) p\"FcinF[gn7%F]fq7$F`hp$!+z)G2x\"FcinF[gn7%Fafq7$Fjgp$!+K(z?%=FcinF[gn 7%Fefq7$Fdgp$!+J^.7>FcinF[gn7%Fifq7$F^gp$!+5yc!)>FcinF[gn7%F]gq7$Fhfp$ !+(Rmw/#FcinF[gn7%Fagq7$Fbfp$!+WSL8@FcinF[gn7%Fegq7$F\\fp$!+Ptex@FcinF [gn7%Figq7$Ffep$!+EcXSAFcinF[gn7%F]hq7$F`ep$!+Q3)>I#FcinF[gn7%Fahq7$Fj dp$!+1y@iBFcinF[gn7%Fehq7$Fddp$!+#3N7U#FcinF[gn7%Fihq7$F^dp$!+3i6zCFci nF[gn7%F]iq7$Fhcp$!+08'f`#FcinF[gn7%Faiq7$Fbcp$!+)z))=f#FcinF[gn7%Feiq 7$F\\cp$!+:p.ZEFcinF[gn7%Fiiq7$Ffbp$!+jVc,FFcinF[gn7%F]jq7$F`bp$!+@#\\ cv#FcinF[gn7%Fajq7$Fjap$!+ZY[4GFcinF[gn7%Fejq7$Fdap$!+R%pK'GFcinF[gn7% Fijq7$F^ap$!+68>$FcinF[gn7%Fa\\r7$Fj^p$!+[ZIXKFcinF[gn7%Fe\\r 7$Fd^p$!+L%p&)H$FcinF[gn7%Fi\\r7$F^^p$!+qK?]LFcinF[gn7%F]]r7$Fh]p$!+EC s*R$FcinF[gn7%Fa]r7$Fb]p$!+PZsYMFcinF[gn7%Fe]r7$F\\]p$!+Z;!4\\$FcinF[g n7%Fi]r7$Ff\\p$!+(*>.KNFcinF[gn7%F]^r7$F`\\p$!+Wh'*pNFcinF[gn7%Fa^r7$F j[p$!+ntg/OFcinF[gn7%Fe^r7$Fd[p$!+&f%*ej$FcinF[gn7%Fi^r7$F^[p$!+.\")yj OFcinF[gn7%F]_r7$Fhjo$!+*Qe#)o$FcinF[gn7%Fa_r7$Fbjo$!+RxF4PFcinF[gn7%F e_r7$F\\jo$!+3Q\"os$FcinF[gn7%Fi_r7$Ffio$!+RX#3u$FcinF[gn7%F]`r7$F`io$ !+CQD^PFcinF[gn7%Fa`r7$Fjho$!+1u-ePFcinF[gn7%Fe`r7$Fdho$!+i&[5w$FcinF[ gn7%Fi`r7$F^ho$!+VJ>gPFcinF[gn7%F]ar7$Fhgo$!+#[.`v$FcinF[gn7%Faar7$Fbg o$!+J1=YPFcinF[gn7%Fear7$F\\go$!++UdKPFcinF[gn7%Fiar7$Fffo$!+'))oTr$Fc inF[gn7%F]br7$F`fo$!+#[m0p$FcinF[gn7%Fabr7$Fjeo$!+K>EhOFcinF[gn7%Febr7 $Fdeo$!+#*>hDOFcinF[gn7%Fibr7$F^eo$!+`hz#e$FcinF[gn7%F]cr7$Fhdo$!+1OxJ NFcinF[gn7%Facr7$$!+tMR_\\Fjfn$!+;3z-NFcinF[gn7%Fecr7$Fbdo$!+h>DrMFcin F[gn7%F[dr7$F\\do$!+V)fpV$FcinF[gn7%F_dr7$Ffco$!+f;r*R$FcinF[gn7%Fcdr7 $F`co$!+J+JfLFcinF[gn7%Fgdr7$Fjbo$!+!fzbJ$FcinF[gn7%F[er7$Fdbo$!+5\"*Q oKFcinF[gn7%F_er7$F^bo$!+^qnFcinF[gn7%Fgir7$$!+`QZ ?5Fb[o$!+aMw.=FcinF[gn7%F]jr7$$!+E[SzNFejn$!+'\\szk\"FcinF[gn7%Fcjr7$$ !+F;IO'*F^jn$!+P7h\"\\\"FcinF[gn7%Fijr7$$!+YZK07F^jn$!+&H**[L\"FcinF[g n7%F_[s7$$\"+nWchgFdhn$!+V0*z<\"FcinF[gn7%Fe[s7$$\"+QK))eaFdhn$!+o=)4- \"FcinF[gn7%F[\\s7$$\"+;z3rCFdhn$!+k$z#R')FjfnF[gn7%Fa\\s7$$\"+K^SrvF] hn$!+N7coqFjfnF[gn7%Fg\\s7$$\"+/Fa[:F]hn$!+xOy(\\&FjfnF[gn7%F]]s7$$\"+ ND1.=FS$!+;0*p#RFjfnF[gn7%Fc]s7$$\"+H6uKtF\\q$!+#[%>cBFjfnF[gn7%Fi]s7$ F($!+M;)R&yF[]oF[gn7%F_^s7$F($\"+M;)R&yF[]oF[gn-Fien6&F[fn$\"#vF^fn$\" #NF^fnFj^s-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3,+++4OyN9FS$\"3z***** p`oTI%FS7$$\"3/+++0*\\sU\"FS$\"3.+++xqG8VFS7$$\"3-+++W!3!=9FS$\"3')*** **zz&o@VFS7$$\"35+++?8539FS$\"3#)*****\\.B$HVFS7$$\"3-+++fNd(R\"FS$\"3 Q+++^s:OVFS7$$\"3-+++\\%pkQ\"FS$\"3?+++Pn9UVFS7$$\"3!*******4\\$[P\"FS $\"3U+++?$\\sM%FS7$$\"3'******z^*p;]8FS$\"37+++d5iaVFS7$$\"32+++pdBP8FS$\"3g*****f*4!oN%FS7$$\"3.+++r 'zRK\"FS$\"3)******40:zN%FS7$$\"3*******Hse/J\"FS$\"3++++OX\"zN%FS7$$ \"3!******H7QnH\"FS$\"3#*******z'[nN%FS7$$\"3))*****RV*)GG\"FS$\"3S+++ IWOaVFS7$$\"3)******R/#**o7FS$\"3u*****>K12N%FS7$$\"3'******f'\\8b7FS$ \"3')*****pZ;dM%FS7$$\"3/+++v!>9C\"FS$\"3T+++*zM$RVFS7$$\"3*)*****f()f zA\"FS$\"3<+++'\\*\\JVFS7$$\"3(******R%4*[@\"FS$\"3y*****H8[@K%FS7$$\" 3-+++g\"oB?\"FS$\"3#*******4&>7J%FS7$$\"31+++a[d!>\"FS$\"3k******Hol)H %FS7$$\"3-+++\")ysz6FS$\"3!)*****4(GT%G%FS7$$\"3#******\\V%3q6FS$\"3g* ******=e%oUFS7$$\"3)******p!*[>;\"FS$\"3!)*****Rz$z]UFS7$$\"3'******>9 yc:\"FS$\"32+++5,ZJUFS7$$\"3)******Rr\"o^6FS$\"3?+++#391@%FS7$$\"33+++ 77T]6FS$\"3E+++p]Y)=%FS7$$\"31+++9&HB:\"FS$\"3q*****4V;a;%FS7$$\"3!*** ***f2]y:\"FS$\"3s*****Rg^?9%FS7$$\"3#******H&HCn6FS$\"3L+++o=:>TFS7$$ \"3++++MV_!=\"FS$\"3/+++&3Yw4%FS7$$\"3!******4$\\P(>\"FS$\"3K+++Lr[ySF S7$$\"3(*******QE9<7FS$\"3A+++GJ[iSFS7$$\"3\"*******p$[*Q7FS$\"3w***** \\ba,0%FS7$$\"3/+++d,&=E\"FS$\"3U+++O=oTSFS7$$\"3/+++)[!*\\G\"FS$\"36+ ++P-&p.%FS7$$\"3.+++$fzwI\"FS$\"3!)*****H^Zc.%FS7$$\"31+++#=9%H8FS$\"3 W+++TKOPSFS7$$\"3'******\\Vc)\\8FS$\"35+++#Ri;/%FS7$$\"3'******z=)zo8F S$\"3s*****>)*G\"[SFS7$$\"3%******4ODhQ\"FS$\"3'******fz'QcSFS7$$\"3/+ ++t()y,9FS$\"3/+++xe5mSFS7$$\"3*******HN#y:9FS$\"3-+++]9+xSFS7$$\"32++ +sx7G9FS$\"3;+++z%H))3%FS7$$\"32+++CR')Q9FS$\"3H+++L6Q,TFS7$$\"3)***** *f!*R![9FS$\"33+++8tZ9TFS7$$\"3!******zR5dX\"FS$\"3o*****>rjz7%FS7$$\" 3)******fnK>Y\"FS$\"3\")*****fd1<9%FS7$$\"3#******frknY\"FS$\"3S+++M\" *ebTFS7$$\"35+++DSEq9FS$\"3?+++_'3&pTFS7$$\"3++++`p[s9FS$\"3O+++3SP$=% FS7$$\"3-+++E$)[t9FS$\"3S+++*o.r>%FS7$$\"3-+++y7Kt9FS$\"3)*******eTi5U FS7$$\"33+++Ms.s9FS$\"3U+++%\\oQA%FS7$$\"33+++ofop9FS$\"3-+++h_xOUFS7$ $\"35+++GdJm9FS$\"33+++svG\\UFS7$$\"31+++$Rt>Y\"FS$\"3o*****4H_8E%FS7$ $\"31+++hYqc9FS$\"3m*****>O>HF%FS7$$\"3'******\\Ca0X\"FS$\"3#*******=7 %RG%FS7$$\"3(******>4mNW\"FS$\"3#******zGsVH%FSFc_sFhen-Fbfn6jn7%7$$\" +4OyN9Fcin$\"+P&oTI%Fcin7$$\"+0*\\sU\"Fcin$\"+xqG8VFcin7$$\"$J\"F^fn$ \"$>%F^fn7%Fgbt7$$\"+W!3!=9Fcin$\"+)z&o@VFcinF\\ct7%Fbct7$$\"+?8539Fci n$\"+NIKHVFcinF\\ct7%Fhct7$$\"+fNd(R\"Fcin$\"+^s:OVFcinF\\ct7%F^dt7$$ \"+\\%pkQ\"Fcin$\"+Pn9UVFcinF\\ct7%Fddt7$$\"+5\\$[P\"Fcin$\"+?$\\sM%Fc inF\\ct7%Fjdt7$$\"+=vri8Fcin$\"+x?U^VFcinF\\ct7%F`et7$$\"+#*p;]8Fcin$ \"+d5iaVFcinF\\ct7%Ffet7$$\"+pdBP8Fcin$\"+'*4!oN%FcinF\\ct7%F\\ft7$$\" +r'zRK\"Fcin$\"+^]\"zN%FcinF\\ct7%Fbft7$$\"+B(e/J\"Fcin$\"+OX\"zN%Fcin F\\ct7%Fhft7$$\"+B\"QnH\"Fcin$\"+!o[nN%FcinF\\ct7%F^gt7$$\"+M%*)GG\"Fc in$\"+IWOaVFcinF\\ct7%Fdgt7$$\"+W?**o7Fcin$\"+Ajq]VFcinF\\ct7%Fjgt7$$ \"+m\\8b7Fcin$\"+xkrXVFcinF\\ct7%F`ht7$$\"+v!>9C\"Fcin$\"+*zM$RVFcinF \\ct7%Ffht7$$\"+w)fzA\"Fcin$\"+'\\*\\JVFcinF\\ct7%F\\it7$$\"+W4*[@\"Fc in$\"+L\"[@K%FcinF\\ct7%Fbit7$$\"+g\"oB?\"Fcin$\"+5&>7J%FcinF\\ct7%Fhi t7$$\"+a[d!>\"Fcin$\"+Iol)H%FcinF\\ct7%F^jt7$$\"+\")ysz6Fcin$\"+rGT%G% FcinF\\ct7%Fdjt7$$\"+NW3q6Fcin$\"+!>e%oUFcinF\\ct7%Fjjt7$$\"+2*[>;\"Fc in$\"+%z$z]UFcinF\\ct7%F`[u7$$\"+U\"yc:\"Fcin$\"+5,ZJUFcinF\\ct7%Ff[u7 $$\"+9TFcinF\\ct7%Fd]u7$$\"+ MV_!=\"Fcin$\"+&3Yw4%FcinF\\ct7%Fj]u7$$\"+J\\P(>\"Fcin$\"+Lr[ySFcinF\\ ct7%F`^u7$$\"+RE9<7Fcin$\"+GJ[iSFcinF\\ct7%Ff^u7$$\"+q$[*Q7Fcin$\"+bX: ]SFcinF\\ct7%F\\_u7$$\"+d,&=E\"Fcin$\"+O=oTSFcinF\\ct7%Fb_u7$$\"+)[!* \\G\"Fcin$\"+P-&p.%FcinF\\ct7%Fh_u7$$\"+$fzwI\"Fcin$\"+8vkNSFcinF\\ct7 %F^`u7$$\"+#=9%H8Fcin$\"+TKOPSFcinF\\ct7%Fd`u7$$\"+Nk&)\\8Fcin$\"+#Ri; /%FcinF\\ct7%Fj`u7$$\"+)=)zo8Fcin$\"+#)*G\"[SFcinF\\ct7%F`au7$$\"+h`7' Q\"Fcin$\"+'z'QcSFcinF\\ct7%Ffau7$$\"+t()y,9Fcin$\"+xe5mSFcinF\\ct7%F \\bu7$$\"+`By:9Fcin$\"+]9+xSFcinF\\ct7%Fbbu7$$\"+sx7G9Fcin$\"+z%H))3%F cinF\\ct7%Fhbu7$$\"+CR')Q9Fcin$\"+L6Q,TFcinF\\ct7%F^cu7$$\"+1*R![9Fcin $\"+8tZ9TFcinF\\ct7%Fdcu7$$\"+)R5dX\"Fcin$\"+7P'z7%FcinF\\ct7%Fjcu7$$ \"+wE$>Y\"Fcin$\"+wlqTTFcinF\\ct7%F`du7$$\"+;Zwm9Fcin$\"+M\"*ebTFcinF \\ct7%Ffdu7$$\"+DSEq9Fcin$\"+_'3&pTFcinF\\ct7%F\\eu7$$\"+`p[s9Fcin$\"+ 3SP$=%FcinF\\ct7%Fbeu7$$\"+E$)[t9Fcin$\"+*o.r>%FcinF\\ct7%Fheu7$$\"+y7 Kt9Fcin$\"+fTi5UFcinF\\ct7%F^fu7$$\"+Ms.s9Fcin$\"+%\\oQA%FcinF\\ct7%Fd fu7$$\"+ofop9Fcin$\"+h_xOUFcinF\\ct7%Fjfu7$$\"+GdJm9Fcin$\"+svG\\UFcin F\\ct7%F`gu7$$\"+$Rt>Y\"Fcin$\"+\"H_8E%FcinF\\ct7%Ffgu7$$\"+hYqc9Fcin$ \"+i$>HF%FcinF\\ct7%F\\hu7$$\"+XUb]9Fcin$\"+>7%RG%FcinF\\ct7%Fbhu7$$\" +#4mNW\"Fcin$\"+)GsVH%FcinF\\ct7%FhhuFbbtF\\ctFf^sF\\_s-F$6$7in7$Fd_s$ !3z*****p`oTI%FS7$F[bt$!3#******zGsVH%FS7$Ffat$!3#*******=7%RG%FS7$Faa t$!3m*****>O>HF%FS7$F\\at$!3o*****4H_8E%FS7$Fg`t$!33+++svG\\UFS7$Fb`t$ !3-+++h_xOUFS7$F]`t$!3U+++%\\oQA%FS7$Fh_t$!3)*******eTi5UFS7$Fc_t$!3S+ ++*o.r>%FS7$F^_t$!3O+++3SP$=%FS7$Fi^t$!3?+++_'3&pTFS7$Fd^t$!3S+++M\"*e bTFS7$F_^t$!3\")*****fd1<9%FS7$Fj]t$!3o*****>rjz7%FS7$Fe]t$!33+++8tZ9T FS7$F`]t$!3H+++L6Q,TFS7$F[]t$!3;+++z%H))3%FS7$Ff\\t$!3-+++]9+xSFS7$Fa \\t$!3/+++xe5mSFS7$F\\\\t$!3'******fz'QcSFS7$Fg[t$!3s*****>)*G\"[SFS7$ Fb[t$!35+++#Ri;/%FS7$F][t$!3W+++TKOPSFS7$Fhjs$!3!)*****H^Zc.%FS7$Fcjs$ !36+++P-&p.%FS7$F^js$!3U+++O=oTSFS7$Fiis$!3w*****\\ba,0%FS7$Fdis$!3A++ +GJ[iSFS7$F_is$!3K+++Lr[ySFS7$Fjhs$!3/+++&3Yw4%FS7$Fehs$!3L+++o=:>TFS7 $F`hs$!3s*****Rg^?9%FS7$F[hs$!3q*****4V;a;%FS7$Ffgs$!3E+++p]Y)=%FS7$Fa gs$!3?+++#391@%FS7$F\\gs$!32+++5,ZJUFS7$Fgfs$!3!)*****Rz$z]UFS7$Fbfs$! 3g*******=e%oUFS7$F]fs$!3!)*****4(GT%G%FS7$Fhes$!3k******Hol)H%FS7$Fce s$!3#*******4&>7J%FS7$F^es$!3y*****H8[@K%FS7$Fids$!3<+++'\\*\\JVFS7$Fd ds$!3T+++*zM$RVFS7$F_ds$!3')*****pZ;dM%FS7$Fjcs$!3u*****>K12N%FS7$Fecs $!3S+++IWOaVFS7$F`cs$!3#*******z'[nN%FS7$F[cs$!3++++OX\"zN%FS7$Ffbs$!3 )******40:zN%FS7$Fabs$!3g*****f*4!oN%FS7$F\\bs$!37+++d5iaVFS7$Fgas$!3g *****p2A9N%FS7$Fbas$!3U+++?$\\sM%FS7$F]as$!3?+++Pn9UVFS7$Fh`s$!3Q+++^s :OVFS7$Fc`s$!3#)*****\\.B$HVFS7$F^`s$!3')*****zz&o@VFS7$Fi_s$!3.+++xqG 8VFSFaiuFhen-Fbfn6jn7%7$Fcbt$!+P&oTI%Fcin7$Fihu$!+)GsVH%Fcin7$F]ct$!$> %F^fn7%F[ev7$Fchu$!+>7%RG%FcinF^ev7%Fbev7$F]hu$!+i$>HF%FcinF^ev7%Ffev7 $Fggu$!+\"H_8E%FcinF^ev7%Fjev7$Fagu$!+svG\\UFcinF^ev7%F^fv7$F[gu$!+h_x OUFcinF^ev7%Fbfv7$Fefu$!+%\\oQA%FcinF^ev7%Fffv7$F_fu$!+fTi5UFcinF^ev7% Fjfv7$Fieu$!+*o.r>%FcinF^ev7%F^gv7$Fceu$!+3SP$=%FcinF^ev7%Fbgv7$F]eu$! +_'3&pTFcinF^ev7%Ffgv7$Fgdu$!+M\"*ebTFcinF^ev7%Fjgv7$Fadu$!+wlqTTFcinF ^ev7%F^hv7$F[du$!+7P'z7%FcinF^ev7%Fbhv7$Fecu$!+8tZ9TFcinF^ev7%Ffhv7$F_ cu$!+L6Q,TFcinF^ev7%Fjhv7$Fibu$!+z%H))3%FcinF^ev7%F^iv7$Fcbu$!+]9+xSFc inF^ev7%Fbiv7$F]bu$!+xe5mSFcinF^ev7%Ffiv7$Fgau$!+'z'QcSFcinF^ev7%Fjiv7 $Faau$!+#)*G\"[SFcinF^ev7%F^jv7$F[au$!+#Ri;/%FcinF^ev7%Fbjv7$Fe`u$!+TK OPSFcinF^ev7%Ffjv7$F_`u$!+8vkNSFcinF^ev7%Fjjv7$Fi_u$!+P-&p.%FcinF^ev7% F^[w7$Fc_u$!+O=oTSFcinF^ev7%Fb[w7$F]_u$!+bX:]SFcinF^ev7%Ff[w7$Fg^u$!+G J[iSFcinF^ev7%Fj[w7$Fa^u$!+Lr[ySFcinF^ev7%F^\\w7$F[^u$!+&3Yw4%FcinF^ev 7%Fb\\w7$Fe]u$!+o=:>TFcinF^ev7%Ff\\w7$F_]u$!+/;0UTFcinF^ev7%Fj\\w7$Fi \\u$!+JkTlTFcinF^ev7%F^]w7$Fc\\u$!+p]Y)=%FcinF^ev7%Fb]w7$F]\\u$!+#391@ %FcinF^ev7%Ff]w7$Fg[u$!+5,ZJUFcinF^ev7%Fj]w7$Fa[u$!+%z$z]UFcinF^ev7%F^ ^w7$F[[u$!+!>e%oUFcinF^ev7%Fb^w7$Fejt$!+rGT%G%FcinF^ev7%Ff^w7$F_jt$!+I ol)H%FcinF^ev7%Fj^w7$Fiit$!+5&>7J%FcinF^ev7%F^_w7$Fcit$!+L\"[@K%FcinF^ ev7%Fb_w7$F]it$!+'\\*\\JVFcinF^ev7%Ff_w7$Fght$!+*zM$RVFcinF^ev7%Fj_w7$ Faht$!+xkrXVFcinF^ev7%F^`w7$F[ht$!+Ajq]VFcinF^ev7%Fb`w7$Fegt$!+IWOaVFc inF^ev7%Ff`w7$F_gt$!+!o[nN%FcinF^ev7%Fj`w7$Fift$!+OX\"zN%FcinF^ev7%F^a w7$Fcft$!+^]\"zN%FcinF^ev7%Fbaw7$F]ft$!+'*4!oN%FcinF^ev7%Ffaw7$Fget$!+ d5iaVFcinF^ev7%Fjaw7$Faet$!+x?U^VFcinF^ev7%F^bw7$F[et$!+?$\\sM%FcinF^e v7%Fbbw7$Fedt$!+Pn9UVFcinF^ev7%Ffbw7$F_dt$!+^s:OVFcinF^ev7%Fjbw7$Fict$ !+NIKHVFcinF^ev7%F^cw7$Fcct$!+)z&o@VFcinF^ev7%Fbcw7$Fhbt$!+xqG8VFcinF^ ev7%FfcwFhdvF^evFf^sF\\_s-F$6%7$7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF (-%'COLOURG6&F[fnF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F^dw7$F($\"3/+++ +++!z%FSFcdwFfdw-F$6$7]yF'7$$\"3B+++\"R)4DdF_en$\"3))******[jzq:F07$$ \"31+++4'4uk$F:$\"3v*****\\D$fTJF07$$\"3#)*****f\">(f5%F@$\"3*)*****RE #R7ZF07$$\"3-+++'e3#\\AFF$\"3]******)4\">$G'F07$$\"3?+++_5;u\")FF$\"39 +++Kb'R&yF07$$\"3!******R)4%)RAFfn$\"3')*****p7GYU*F07$$\"37+++p][t[Ff n$\"3++++P`\\*4\"FS7$$\"3e+++&H@\")Q)Ffn$\"35+++iiWc7FS7$$\"31+++Os#=- \"F\\o$\"3\"******RSFKT\"FS7$$\"35+++1kW_FFfn$\"3'******\\R%pp:FS7$$!3 C+++l)3'3JF\\o$\"3.+++9`jDMItFgo$\"3%)*****HKws=#FS7$$!31+++\\n6V5F\\q$\"3')******[ d&GE#FS7$$!3-+++CGV[9F\\q$\"3\"******>;AzL#FS7$$!37+++S$4(o>F\\q$\"3!) *****fl@CT#FS7$$!3)********>$eDEF\\q$\"3))*****\\&)*G'[#FS7$$!3x*****4 '**HUMF\\q$\"3?+++m[WfDFS7$$!3L+++RA>VWF\\q$\"3=+++1$y0XJq#FS7$$!3W+++$G1Z4(F\\q$\"3'*******oJNtFFS7$ $!3w******R%4%*y)F\\q$\"39+++'fc@%GFS7$$!31+++,dAv5F0$\"3>+++%Ga#4HFS7 $$!31+++&>t!*H\"F0$\"3'******Rw*HuHFS7$$!3!******HID-b\"F0$\"35+++EO#p .$FS7$$!35+++\"z/t#=F0$\"31+++Fuw'4$FS7$$!3#******4guy7#F0$\"3A+++7B_` JFS7$$!35+++#fy'[CF0$\"32+++U@'p?$FS7$$!3u*****Rg0gy#F0$\"3%)*****H4ip D$FS7$$!3w*****H>xg8$F0$\"3,+++E#*\\.LFS7$$!3#******>-n`\\$F0$\"3'**** ***z!QmM$FS7$$!3%******zh.3'QF0$\"31+++=\"3lQ$FS7$$!31+++Xf&)HUF0$\"3$ )*****R))zKU$FS7$$!3-+++//`+YF0$\"3?+++0W9dMFS7$$!3`*****\\x-6M&F0$\"3 9+++)eIp^$FS7$$!3')*****\\3&zugF0$\"3;+++?vLnNFS7$$!3%*******\\]'zz'F0 $\"3A+++X+i4OFS7$$!3!******fox#4vF0$\"34+++/BzWOFS7$$!3\"******RW,'3#) F0$\"35+++)GcOn$FS7$$!3[+++\"=$Q'*))F0$\"3%******pGUop$FS7$$!3++++4LJt &*F0$\"3-+++&3Y[r$FS7$$!3'******R_:S-\"FS$\"30+++Q!f!GPFS7$$!3-+++5lw* 3\"FS$\"3%)*****\\?\"zOPFS7$$!3%******z.cY:\"FS$\"3*******4\"))GTPFS7$ $!33+++G:v=7FS$\"3()*****HQZLDeq$FS7$$!3.+++`?cH:FS$\"31+++GF $yo$FS7$$!33+++z!z-f\"FS$\"3#******zbSjm$FS7$$!31+++%><2l\"FS$\"3()*** **f5\"QTOFS7$$!3/+++4T*4r\"FS$\"3()******RJ)Hh$FS7$$!3%*******4/Dr)y6e$FS7$$!3*******HA`;$=FS$\"3))*****4z7ga$FS7$$!3/+++u#) R#*=FS$\"3:+++QTb2NFS7$$!31+++='3P&>FS$\"3#)*****fe6fY$FS7$$!31+++'[Fe ,#FS$\"3@+++?:D@MFS7$$!3'******\\:-!z?FS$\"3=+++4n\"QP$FS7$$!3(******H FcM9#FS$\"39+++75%RK$FS7$$!3-+++D3N4AFS$\"33+++jC0sKFS7$$!3#)*****zfRn F#FS$\"3%)*****H`XBFS$\"31+++-CHkJFS7$$!3<+++:)yaT#F S$\"3%)*****f2v%4JFS7$$!35+++#f#>'[#FS$\"3#******>bFY0$FS7$$!37+++@m>d DFS$\"3%******RgO++$FS7$$!3>+++*)z)zi#FS$\"37+++\\y$e%HFS7$$!3%******4 S$4)p#FS$\"3#******>PJ?*GFS7$$!3')******)z0rw#FS$\"36+++,c^QGFS7$$!3,+ ++:)*pMGFS$\"3'******>_D^y#FS7$$!3!)*****4:J1!HFS$\"32+++OdmJFFS7$$!3; +++poskHFS$\"3-+++*\\Mzn#FS7$$!3!)*****zHso-$FS$\"3/+++)*)QPi#FS7$$!3! )*****H1)*p3$FS$\"3?+++lI!*oDFS7$$!3'******4$*o]9$FS$\"3$)*****pkrK^#F S7$$!3;+++r\\2,KFS$\"3-+++!R5nX#FS7$$!3/+++Q\\-bKFS$\"33+++$=0\"*R#FS7 $$!3!******R`TpI$FS$\"3$)*****>Ug.M#FS7$$!31+++j\"eoN$FS$\"3'*******[s R!G#FS7$$!3=+++xn\"[S$FS$\"3%)*****H'>:>AFS7$$!3=+++Bl'3X$FS$\"3/+++&z ul:#FS7$$!3y*****\\*G1&\\$FS$\"3)******p6HE4#FS7$$!3;+++krYPNFS$\"3%** ****R8\"HF?FS7$$!3++++oe9yNFS$\"3'******\\/]0'>FS7$$!3\"******z;qrh$FS $\"3++++V'3C*=FS7$$!37+++w[haOFS$\"3$******>D%)G#=FS7$$!3%******p(pb!p $FS$\"3/+++g)4?v\"FS7$$!39+++;N2DPFS$\"3%******>DN)z;FS7$$!3=+++q(Q#eP FS$\"3&******z#yU1;FS7$$!3++++j27!z$FS$\"3&******4\"G(=`\"FS7$$!3$)*** **>Hx2#QFS$\"3#******RZsiX\"FS7$$!35+++K=D]QFS$\"3-+++?Tuz8FS7$$!31+++ ?)p&yQFS$\"3.+++1nT-8FS7$$!39+++het0RFS$\"3++++&=EWA\"FS7$$!39+++!QK<$ RFS$\"3++++w*4f9\"FS7$$!3A+++@+_cRFS$\"3#******z5,q1\"FS7$$!3!)*****> \")R+)RFS$\"3=+++>vAy)*F07$$!3)*******Go@-SFS$\"3a*****RaP[3*F07$$!3>+ ++H['H-%FS$\"3]*****z,c2H)F07$$!3r*****\\C\">USFS$\"3-+++3(4n\\(F07$$! 3++++&)>!)fSFS$\"3x*****z`PKq'F07$$!3!)*****pf0d2%FS$\"3%******Rc-2\"f F07$$!3#*******zl\")*3%FS$\"38+++5=J>^F07$$!3M+++&zd?5%FS$\"32+++>,9HV F07$$!3A+++M?O7TFS$\"35+++'*[:SNF07$$!3#)*****H$Hn?TFS$\"3E+++W&RAv#F0 7$$!3#)*****RRXp7%FS$\"3'******pP9_'>F07$$!3B+++*oX68%FS$\"3/+++kX&)y6 F07$$!3o******>9DLTFS$\"3?+++AH0HRF\\q7$F[hy$!3?+++AH0HRF\\q7$Ffgy$!3/ +++kX&)y6F07$Fagy$!3'******pP9_'>F07$F\\gy$!3E+++W&RAv#F07$Fgfy$!35+++ '*[:SNF07$Fbfy$!32+++>,9HVF07$F]fy$!38+++5=J>^F07$Fhey$!3%******Rc-2\" fF07$Fcey$!3x*****z`PKq'F07$F^ey$!3-+++3(4n\\(F07$Fidy$!3]*****z,c2H)F 07$Fddy$!3a*****RaP[3*F07$F_dy$!3=+++>vAy)*F07$Fjcy$!3#******z5,q1\"FS 7$Fecy$!3++++w*4f9\"FS7$F`cy$!3++++&=EWA\"FS7$F[cy$!3.+++1nT-8FS7$Ffby $!3-+++?Tuz8FS7$Faby$!3#******RZsiX\"FS7$F\\by$!3&******4\"G(=`\"FS7$F gay$!3&******z#yU1;FS7$Fbay$!3%******>DN)z;FS7$F]ay$!3/+++g)4?v\"FS7$F h`y$!3$******>D%)G#=FS7$Fc`y$!3++++V'3C*=FS7$F^`y$!3'******\\/]0'>FS7$ Fi_y$!3%******R8\"HF?FS7$Fd_y$!3)******p6HE4#FS7$F__y$!3/+++&zul:#FS7$ Fj^y$!3%)*****H'>:>AFS7$Fe^y$!3'*******[sR!G#FS7$F`^y$!3$)*****>Ug.M#F S7$F[^y$!33+++$=0\"*R#FS7$Ff]y$!3-+++!R5nX#FS7$Fa]y$!3$)*****pkrK^#FS7 $F\\]y$!3?+++lI!*oDFS7$Fg\\y$!3/+++)*)QPi#FS7$Fb\\y$!3-+++*\\Mzn#FS7$F ]\\y$!32+++OdmJFFS7$Fh[y$!3'******>_D^y#FS7$Fc[y$!36+++,c^QGFS7$F^[y$! 3#******>PJ?*GFS7$Fijx$!37+++\\y$e%HFS7$Fdjx$!3%******RgO++$FS7$F_jx$! 3#******>bFY0$FS7$Fjix$!3%)*****f2v%4JFS7$Feix$!31+++-CHkJFS7$F`ix$!3% )*****H)y6e$FS7$Fcfx$!3()******RJ )Hh$FS7$F^fx$!3()*****f5\"QTOFS7$Fiex$!3#******zbSjm$FS7$Fdex$!31+++GF $yo$FS7$F_ex$!3')*****>LDeq$FS7$Fjdx$!3()******RrF?PFS7$Fedx$!3%)***** *=M8JPFS7$F`dx$!3')*****fp@$QPFS7$F[dx$!3()*****HQZ+++%Ga#4HFS7$Fh\\x$!39+++'fc@%GFS7$Fc\\x$!3'*******oJNtFFS7$F^\\x $!3*******>0XJq#FS7$Fi[x$!3=+++1$y;AzL#FS7$F`j w$!3')******[d&GE#FS7$F[jw$!3%)*****HKws=#FS7$Ffiw$!3#)*****zKK76#FS7$ $!31+++=]s#3#Fgo$!33+++\\V$z&>FS7$$!3`+++5vPEnF\\o$!33+++#\\BL!=FS7$$! 3W*****>R['y#*Ffn$!3++++-muZ;FS7$$\"3v*****zVJ&H%)Ffn$!3!******zv6:\\ \"FS7$$\"3B+++TOpF)*Ffn$!3$********el[L\"FS7$$\"3d+++f=,.mFfn$!3#***** *RP&)z<\"FS7$$\"3w*****f(4f'R$Ffn$!3)******z_&)4-\"FS7$$\"3%******z^r` R\"Ffn$!39+++@0KR')F07$$\"37+++lG%oY%FF$!3=+++gYeoqF07$$\"3%******fR!e C5FF$!3%*******eBz(\\&F07$$\"3)******fY<\\Q\"F@$!3#)*****RU#*p#RF07$$ \"3')*****RCt.^'F4$!3'******fk%>cBF07$$\"3y*****41AI&*)!#H$!3Y+++T;)R& yF\\q7$$\"3u+++L@-`*)Fi^[l$\"3Y+++T;)R&yF\\q-Fien6&F[fn$\"#KF^fn$\"\") F^fnFe_[l-Fbfn6^y7%F'7$$\"+\"R)4DdF\\q$\"+\\jzq:FjfnF[gn7%Fj_[l7$$\"+4 '4uk$FS$\"+bKfTJFjfnF[gn7%F``[l7$$\"+;>(f5%F]hn$\"+kAR7ZFjfnF[gn7%Ff`[ l7$$\"+'e3#\\AFdhn$\"+*4\">$G'FjfnF[gn7%F\\a[l7$$\"+_5;u\")Fdhn$\"+Kb' R&yFjfnF[gn7%Fba[l7$$\"+%)4%)RAF^jn$\"+F\"GYU*FjfnF[gn7%Fha[l7$$\"+p][ t[F^jn$\"+P`\\*4\"FcinF[gn7%F^b[l7$$\"+&H@\")Q)F^jn$\"+iiWc7FcinF[gn7% Fdb[l7$$\"+Os#=-\"Fejn$\"+/uA89FcinF[gn7%Fjb[l7$$\"+1kW_FF^jn$\"+&R%pp :FcinF[gn7%F`c[l7$$!+l)3'3JFejn$\"+9`jDMItFb[o$\"+BjF(=#FcinF[gn7%F^ e[l7$$!+\\n6V5F[]o$\"+\\d&GE#FcinF[gn7%Fde[l7$$!+CGV[9F[]o$\"+i@#zL#Fc inF[gn7%Fje[l7$$!+S$4(o>F[]o$\"+c;U7CFcinF[gn7%F`f[l7$$!++KeDEF[]o$\"+ b)*G'[#FcinF[gn7%Fff[l7$$!+h**HUMF[]o$\"+m[WfDFcinF[gn7%F\\g[l7$$!+RA> VWF[]o$\"+1$yt!*H\"Fjfn$\" +k(*HuHFcinF[gn7%F`i[l7$$!+.`A]:Fjfn$\"+EO#p.$FcinF[gn7%Ffi[l7$$!+\"z/ t#=Fjfn$\"+Fuw'4$FcinF[gn7%F\\j[l7$$!+,Y(y7#Fjfn$\"+7B_`JFcinF[gn7%Fbj [l7$$!+#fy'[CFjfn$\"+U@'p?$FcinF[gn7%Fhj[l7$$!+/c+'y#Fjfn$\"+$4ipD$Fci nF[gn7%F^[\\l7$$!+$>xg8$Fjfn$\"+E#*\\.LFcinF[gn7%Fd[\\l7$$!+AqO&\\$Fjf n$\"+!3QmM$FcinF[gn7%Fj[\\l7$$!+=O!3'QFjfn$\"+=\"3lQ$FcinF[gn7%F`\\\\l 7$$!+Xf&)HUFjfn$\"+%))zKU$FcinF[gn7%Ff\\\\l7$$!+//`+YFjfn$\"+0W9dMFcin F[gn7%F\\]\\l7$$!+vF5T`Fjfn$\"+)eIp^$FcinF[gn7%Fb]\\l7$$!+&3&zugFjfn$ \"+?vLnNFcinF[gn7%Fh]\\l7$$!+]]'zz'Fjfn$\"+X+i4OFcinF[gn7%F^^\\l7$$!+' ox#4vFjfn$\"+/BzWOFcinF[gn7%Fd^\\l7$$!+W9g3#)Fjfn$\"+)GcOn$FcinF[gn7%F j^\\l7$$!+\"=$Q'*))Fjfn$\"+(GUop$FcinF[gn7%F`_\\l7$$!+4LJt&*Fjfn$\"+&3 Y[r$FcinF[gn7%Ff_\\l7$$!+Cb,C5Fcin$\"+Q!f!GPFcinF[gn7%F\\`\\l7$$!+5lw* 3\"Fcin$\"+07zOPFcinF[gn7%Fb`\\l7$$!+Qgla6Fcin$\"+6))GTPFcinF[gn7%Fh` \\l7$$!+G:v=7Fcin$\"+$QZM8JPFcinF[gn7%Fja\\l7$$!+%[4pS\"F cin$\"+SrF?PFcinF[gn7%F`b\\l7$$!+eqYo9Fcin$\"+K`#eq$FcinF[gn7%Ffb\\l7$ $!+`?cH:Fcin$\"+GF$yo$FcinF[gn7%F\\c\\l7$$!+z!z-f\"Fcin$\"+e0MmOFcinF[ gn7%Fbc\\l7$$!+%><2l\"Fcin$\"+16QTOFcinF[gn7%Fhc\\l7$$!+4T*4r\"Fcin$\" +SJ)Hh$FcinF[gn7%F^d\\l7$$!+5/Dr`XBFcin$\"+-CHkJFcinF[gn7% Fjg\\l7$$!+:)yaT#Fcin$\"+w]Z4JFcinF[gn7%F`h\\l7$$!+#f#>'[#Fcin$\"+_via IFcinF[gn7%Ffh\\l7$$!+@m>dDFcin$\"+/m.+IFcinF[gn7%F\\i\\l7$$!+*)z)zi#F cin$\"+\\y$e%HFcinF[gn7%Fbi\\l7$$!+,M4)p#Fcin$\"+s8.#*GFcinF[gn7%Fhi\\ l7$$!+*z0rw#Fcin$\"+,c^QGFcinF[gn7%F^j\\l7$$!+:)*pMGFcin$\"+Ab7&y#Fcin F[gn7%Fdj\\l7$$!+^6j+HFcin$\"+OdmJFFcinF[gn7%Fjj\\l7$$!+poskHFcin$\"+* \\Mzn#FcinF[gn7%F`[]l7$$!+)Hso-$Fcin$\"+)*)QPi#FcinF[gn7%Ff[]l7$$!+j!) *p3$Fcin$\"+lI!*oDFcinF[gn7%F\\\\]l7$$!+J*o]9$Fcin$\"+Z;F8DFcinF[gn7%F b\\]l7$$!+r\\2,KFcin$\"+!R5nX#FcinF[gn7%Fh\\]l7$$!+Q\\-bKFcin$\"+$=0\" *R#FcinF[gn7%F^]]l7$$!+M:%pI$Fcin$\"+A/OSBFcinF[gn7%Fd]]l7$$!+j\"eoN$F cin$\"+\\sR!G#FcinF[gn7%Fj]]l7$$!+xn\"[S$Fcin$\"+j>:>AFcinF[gn7%F`^]l7 $$!+Bl'3X$Fcin$\"+&zul:#FcinF[gn7%Ff^]l7$$!+&*G1&\\$Fcin$\"+<\"HE4#Fci nF[gn7%F\\_]l7$$!+krYPNFcin$\"+M6HF?FcinF[gn7%Fb_]l7$$!+oe9yNFcin$\"+X +bg>FcinF[gn7%Fh_]l7$$!+o,<vAy)*FjfnF[gn7%Ffd]l7$$!+Ho@-SFcin$\"+Wv$[3*Fjf nF[gn7%F\\e]l7$$!+H['H-%Fcin$\"+=gv!H)FjfnF[gn7%Fbe]l7$$!+X7>USFcin$\" +3(4n\\(FjfnF[gn7%Fhe]l7$$!+&)>!)fSFcin$\"+QvB.nFjfnF[gn7%F^f]l7$$!+(f 0d2%Fcin$\"+kDq5fFjfnF[gn7%Fdf]l7$$!+!e;)*3%Fcin$\"+5=J>^FjfnF[gn7%Fjf ]l7$$!+&zd?5%Fcin$\"+>,9HVFjfnF[gn7%F`g]l7$$!+M?O7TFcin$\"+'*[:SNFjfnF [gn7%Ffg]l7$$!+LHn?TFcin$\"+W&RAv#FjfnF[gn7%F\\h]l7$$!+%RXp7%Fcin$\"+x V@l>FjfnF[gn7%Fbh]l7$$!+*oX68%Fcin$\"+kX&)y6FjfnF[gn7%Fhh]l7$$!+?9DLTF cin$\"+AH0HRF[]oF[gn7%F^i]l7$F_i]l$!+AH0HRF[]oF[gn7%Fdi]l7$Fih]l$!+kX& )y6FjfnF[gn7%Fhi]l7$Fch]l$!+xV@l>FjfnF[gn7%F\\j]l7$F]h]l$!+W&RAv#FjfnF [gn7%F`j]l7$Fgg]l$!+'*[:SNFjfnF[gn7%Fdj]l7$Fag]l$!+>,9HVFjfnF[gn7%Fhj] l7$F[g]l$!+5=J>^FjfnF[gn7%F\\[^l7$Fef]l$!+kDq5fFjfnF[gn7%F`[^l7$F_f]l$ !+QvB.nFjfnF[gn7%Fd[^l7$Fie]l$!+3(4n\\(FjfnF[gn7%Fh[^l7$Fce]l$!+=gv!H) FjfnF[gn7%F\\\\^l7$F]e]l$!+Wv$[3*FjfnF[gn7%F`\\^l7$Fgd]l$!+>vAy)*FjfnF [gn7%Fd\\^l7$Fad]l$!+36+n5FcinF[gn7%Fh\\^l7$F[d]l$!+w*4f9\"FcinF[gn7%F \\]^l7$Fec]l$!+&=EWA\"FcinF[gn7%F`]^l7$F_c]l$!+1nT-8FcinF[gn7%Fd]^l7$F ib]l$!+?Tuz8FcinF[gn7%Fh]^l7$Fcb]l$!+uCFc9FcinF[gn7%F\\^^l7$F]b]l$!+6G (=`\"FcinF[gn7%F`^^l7$Fga]l$!+GyU1;FcinF[gn7%Fd^^l7$Faa]l$!+__$)z;Fcin F[gn7%Fh^^l7$F[a]l$!+g)4?v\"FcinF[gn7%F\\_^l7$Fe`]l$!+_U)G#=FcinF[gn7% F`_^l7$F_`]l$!+V'3C*=FcinF[gn7%Fd_^l7$Fi_]l$!+X+bg>FcinF[gn7%Fh_^l7$Fc _]l$!+M6HF?FcinF[gn7%F\\`^l7$F]_]l$!+<\"HE4#FcinF[gn7%F``^l7$Fg^]l$!+& zul:#FcinF[gn7%Fd`^l7$Fa^]l$!+j>:>AFcinF[gn7%Fh`^l7$F[^]l$!+\\sR!G#Fci nF[gn7%F\\a^l7$Fe]]l$!+A/OSBFcinF[gn7%F`a^l7$F_]]l$!+$=0\"*R#FcinF[gn7 %Fda^l7$Fi\\]l$!+!R5nX#FcinF[gn7%Fha^l7$Fc\\]l$!+Z;F8DFcinF[gn7%F\\b^l 7$F]\\]l$!+lI!*oDFcinF[gn7%F`b^l7$Fg[]l$!+)*)QPi#FcinF[gn7%Fdb^l7$Fa[] l$!+*\\Mzn#FcinF[gn7%Fhb^l7$F[[]l$!+OdmJFFcinF[gn7%F\\c^l7$Fej\\l$!+Ab 7&y#FcinF[gn7%F`c^l7$F_j\\l$!+,c^QGFcinF[gn7%Fdc^l7$Fii\\l$!+s8.#*GFci nF[gn7%Fhc^l7$Fci\\l$!+\\y$e%HFcinF[gn7%F\\d^l7$F]i\\l$!+/m.+IFcinF[gn 7%F`d^l7$Fgh\\l$!+_viaIFcinF[gn7%Fdd^l7$Fah\\l$!+w]Z4JFcinF[gn7%Fhd^l7 $F[h\\l$!+-CHkJFcinF[gn7%F\\e^l7$Feg\\l$!+thl=KFcinF[gn7%F`e^l7$F_g\\l $!+jC0sKFcinF[gn7%Fde^l7$Fif\\l$!+75%RK$FcinF[gn7%Fhe^l7$Fcf\\l$!+4n\" QP$FcinF[gn7%F\\f^l7$F]f\\l$!+?:D@MFcinF[gn7%F`f^l7$Fge\\l$!+'e6fY$Fci nF[gn7%Fdf^l7$Fae\\l$!+QTb2NFcinF[gn7%Fhf^l7$F[e\\l$!+\"z7ga$FcinF[gn7 %F\\g^l7$Fed\\l$!+?)y6e$FcinF[gn7%F`g^l7$F_d\\l$!+SJ)Hh$FcinF[gn7%Fdg^ l7$Fic\\l$!+16QTOFcinF[gn7%Fhg^l7$Fcc\\l$!+e0MmOFcinF[gn7%F\\h^l7$F]c \\l$!+GF$yo$FcinF[gn7%F`h^l7$Fgb\\l$!+K`#eq$FcinF[gn7%Fdh^l7$Fab\\l$!+ SrF?PFcinF[gn7%Fhh^l7$F[b\\l$!+>M8JPFcinF[gn7%F\\i^l7$Fea\\l$!+'p@$QPF cinF[gn7%F`i^l7$F_a\\l$!+$QZFcinF[gn7%Fbb_l7$$!+5vPEnFejn$!+#\\BL !=FcinF[gn7%Fhb_l7$$!+#R['y#*F^jn$!+-muZ;FcinF[gn7%F^c_l7$$\"+Q9`H%)F^ jn$!+e<^\"\\\"FcinF[gn7%Fdc_l7$$\"+TOpF)*F^jn$!+!fl[L\"FcinF[gn7%Fjc_l 7$$\"+f=,.mF^jn$!+u`)z<\"FcinF[gn7%F`d_l7$$\"+w4f'R$F^jn$!+Gb)4-\"Fcin F[gn7%Ffd_l7$$\"+=:P&R\"F^jn$!+@0KR')FjfnF[gn7%F\\e_l7$$\"+lG%oY%Fdhn$ !+gYeoqFjfnF[gn7%Fbe_l7$$\"+'R!eC5Fdhn$!+fBz(\\&FjfnF[gn7%Fhe_l7$$\"+m u\"\\Q\"F]hn$!+CC*p#RFjfnF[gn7%F^f_l7$$\"+WKP5lF0$!+YY>cBFjfnF[gn7%Fdf _l7$$\"+h?-`*)F\\o$!+T;)R&yF[]oF[gn7%Fjf_l7$$\"+L@-`*)F\\o$\"+T;)R&yF[ ]oF[gn-Fien6&F[fn$\"#jF^fn$\"#:F^fnFig_lF\\_s-F$6$7in7$$\"3%******>%)) 4V9FS$\"3%******R>s#)H%FS7$$\"3,+++5&eXV\"FS$\"3J+++\"\\osI%FS7$$\"3&* *****>*pJD9FS$\"3C+++yVa:VFS7$$\"3#******poLJKE\"F S$\"3M+++*H_zL%FS7$$\"3'******>8h'\\7FS$\"32+++fRXJVFS7$$\"3'******H1m jB\"FS$\"3L+++ju]BVFS7$$\"3++++l;[B7FS$\"3)******fc_SJ%FS7$$\"3\"***** *\\$f;67FS$\"39+++$*4..VFS7$$\"33+++qVg*>\"FS$\"3K+++g)*Q!H%FS7$$\"3!* *****H&e,*=\"FS$\"31+++Yr3wUFS7$$\"3%******H4f'z6FS$\"3q******p-5gUFS7 $$\"3#******f&)Q=<\"FS$\"3G+++U\"RCC%FS7$$\"3!******z;5f;\"FS$\"3U+++: a;BUFS7$$\"3#******\\,xA;\"FS$\"3)******R[>C?%FS7$$\"33+++<&z8;\"FS$\" 3K+++)Rb/=%FS7$$\"31+++w4mj6FS$\"3%)*****z&)zw:%FS7$$\"33+++?h]p6FS$\" 3#******>Y$oMTFS7$$\"35+++O%\\\"z6FS$\"3B+++Z>C7TFS7$$\"3%******RNsD> \"FS$\"3%******>kg74%FS7$$\"3#******HFL%47FS$\"3k*****\\z^E2%FS7$$\"3( ******\\fz!H7FS$\"3/+++;b#fvR8FS$\"3s*****\\ z(\\LSFS7$$\"31+++ht')f8FS$\"3&*******o/'y.%FS7$$\"33+++(Q.&y8FS$\"3&* *****H)[MWSFS7$$\"3-+++U;b&R\"FS$\"3O+++Nee_SFS7$$\"31+++5D'4T\"FS$\"3 9+++G?EiSFS7$$\"31+++)RGZU\"FS$\"33+++3`4tSFS7$$\"3-+++#)*ooV\"FS$\"3+ +++/m%[3%FS7$$\"3!******p0@uW\"FS$\"3y******)y5t4%FS7$$\"3-+++&pJkX\"F S$\"3B+++/:J5TFS7$$\"3!*******>P&RY\"FS$\"3?+++xkpBTFS7$$\"3%******puU +Z\"FS$\"3-+++_MLPTFS7$$\"3+++++`vu9FS$\"3++++on5^TFS7$$\"3++++Cw9y9FS $\"3=+++7X\"\\;%FS7$$\"3))*****\\)\\F![\"FS$\"3o******fhmyTFS7$$\"3-++ +$G\">\"[\"FS$\"3/+++U1G#>%FS7$$\"3/+++o)[4[\"FS$\"3M+++PZo0UFS7$$\"31 +++.&)fz9FS$\"3p*****>v6)=UFS7$$\"3!******4T*=x9FS$\"3M+++4/gJUFS7$$\" 3-+++)QpPZ\"FS$\"33+++GR*RC%FS7$$\"3++++Z\\Qp9FS$\"35+++T#RfD%FS7$$\"3 '*******)\\\"3k9FS$\"3u*****zK'QnUFS7$$\"35+++WN!zX\"FS$\"3W+++UwGyUFS 7$$\"3#******H)[*3X\"FS$\"39+++6wf)G%FSF^h_lFa_[l-Fbfn6jn7%7$$\"+U))4V 9Fcin$\"+%>s#)H%Fcin7$$\"+5&eXV\"Fcin$\"+\"\\osI%Fcin7$$\"$K\"F^fnF_ct 7%Fb[al7$$\"+#*pJD9Fcin$\"+yVa:VFcinFg[al7%F[\\al7$$\"+(o\"Fcin$\"+g)*Q!H%FcinFg[al7%Fgbal7$$\"+`e,*=\"Fcin$\"+Yr3wUFcinF g[al7%F]cal7$$\"+$4f'z6Fcin$\"+q-5gUFcinFg[al7%Fccal7$$\"+c)Q=<\"Fcin$ \"+U\"RCC%FcinFg[al7%Fical7$$\"+o,\"f;\"Fcin$\"+:a;BUFcinFg[al7%F_dal7 $$\"+:qFi6Fcin$\"+%[>C?%FcinFg[al7%Fedal7$$\"+<&z8;\"Fcin$\"+)Rb/=%Fci nFg[al7%F[eal7$$\"+w4mj6Fcin$\"+e)zw:%FcinFg[al7%Faeal7$$\"+?h]p6Fcin$ \"+iMoMTFcinFg[al7%Fgeal7$$\"+O%\\\"z6Fcin$\"+Z>C7TFcinFg[al7%F]fal7$$ \"+aBd#>\"Fcin$\"+U1E\"4%FcinFg[al7%Fcfal7$$\"+tKV47Fcin$\"+&z^E2%Fcin Fg[al7%Fifal7$$\"+&fz!H7Fcin$\"+;b\"[\"Fcin$\"+U1G# >%FcinFg[al7%Fa^bl7$$\"+o)[4[\"Fcin$\"+PZo0UFcinFg[al7%Fg^bl7$$\"+.&)f z9Fcin$\"+_<\")=UFcinFg[al7%F]_bl7$$\"+6%*=x9Fcin$\"+4/gJUFcinFg[al7%F c_bl7$$\"+)QpPZ\"Fcin$\"+GR*RC%FcinFg[al7%Fi_bl7$$\"+Z\\Qp9Fcin$\"+T#R fD%FcinFg[al7%F_`bl7$$\"+*\\\"3k9Fcin$\"+GjQnUFcinFg[al7%Fe`bl7$$\"+WN !zX\"Fcin$\"+UwGyUFcinFg[al7%F[abl7$$\"+$)[*3X\"Fcin$\"+6wf)G%FcinFg[a l7%FaablF][alFg[alFeg_lF\\_s-F$6$7in7$F_h_l$!3%******R>s#)H%FS7$Ffj`l$ !39+++6wf)G%FS7$Faj`l$!3W+++UwGyUFS7$F\\j`l$!3u*****zK'QnUFS7$Fgi`l$!3 5+++T#RfD%FS7$Fbi`l$!33+++GR*RC%FS7$F]i`l$!3M+++4/gJUFS7$Fhh`l$!3p**** *>v6)=UFS7$Fch`l$!3M+++PZo0UFS7$F^h`l$!3/+++U1G#>%FS7$Fig`l$!3o******f hmyTFS7$Fdg`l$!3=+++7X\"\\;%FS7$F_g`l$!3++++on5^TFS7$Fjf`l$!3-+++_MLPT FS7$Fef`l$!3?+++xkpBTFS7$F`f`l$!3B+++/:J5TFS7$F[f`l$!3y******)y5t4%FS7 $Ffe`l$!3++++/m%[3%FS7$Fae`l$!33+++3`4tSFS7$F\\e`l$!39+++G?EiSFS7$Fgd` l$!3O+++Nee_SFS7$Fbd`l$!3&******H)[MWSFS7$F]d`l$!3&*******o/'y.%FS7$Fh c`l$!3s*****\\z(\\LSFS7$Fcc`l$!3f******f&f;.%FS7$F^c`l$!32+++?iwKSFS7$ Fib`l$!3\")******y\\@PSFS7$Fdb`l$!3/+++j6JXSFS7$F_b`l$!3/+++;bkg74%FS7$F`a`l$!3B+++Z>C7TFS7$ F[a`l$!3#******>Y$oMTFS7$Ff``l$!3%)*****z&)zw:%FS7$Fa``l$!3K+++)Rb/=%F S7$F\\``l$!3)******R[>C?%FS7$Fg_`l$!3U+++:a;BUFS7$Fb_`l$!3G+++U\"RCC%F S7$F]_`l$!3q******p-5gUFS7$Fh^`l$!31+++Yr3wUFS7$Fc^`l$!3K+++g)*Q!H%FS7 $F^^`l$!39+++$*4..VFS7$Fi]`l$!3)******fc_SJ%FS7$Fd]`l$!3L+++ju]BVFS7$F _]`l$!32+++fRXJVFS7$Fj\\`l$!3M+++*H_zL%FS7$Fe\\`l$!3I+++SE1VVFS7$F`\\` l$!3!)******eQ%oM%FS7$F[\\`l$!3!******f#HN\\VFS7$Ff[`l$!3K+++FYk]VFS7$ Fa[`l$!3++++;:x]VFS7$F\\[`l$!3?+++'*Ry\\VFS7$Fgj_l$!3w*****p[IxM%FS7$F bj_l$!3w*****RkdYM%FS7$F]j_l$!3K+++'f51M%FS7$Fhi_l$!3@+++6KjNVFS7$Fci_ l$!3&)*****pNo(HVFS7$F^i_l$!3J+++!=eIK%FS7$Fih_l$!3C+++yVa:VFS7$Fdh_l$ !3J+++\"\\osI%FSFjablFa_[l-Fbfn6jn7%7$F^[al$!+%>s#)H%Fcin7$Fbabl$!+6wf )G%Fcin7$Fh[alF_ev7%Fd]cl7$F\\abl$!+UwGyUFcinFg]cl7%Fi]cl7$Ff`bl$!+GjQ nUFcinFg]cl7%F]^cl7$F``bl$!+T#RfD%FcinFg]cl7%Fa^cl7$Fj_bl$!+GR*RC%Fcin Fg]cl7%Fe^cl7$Fd_bl$!+4/gJUFcinFg]cl7%Fi^cl7$F^_bl$!+_<\")=UFcinFg]cl7 %F]_cl7$Fh^bl$!+PZo0UFcinFg]cl7%Fa_cl7$Fb^bl$!+U1G#>%FcinFg]cl7%Fe_cl7 $F\\^bl$!+ghmyTFcinFg]cl7%Fi_cl7$Ff]bl$!+7X\"\\;%FcinFg]cl7%F]`cl7$F`] bl$!+on5^TFcinFg]cl7%Fa`cl7$Fj\\bl$!+_MLPTFcinFg]cl7%Fe`cl7$Fd\\bl$!+x kpBTFcinFg]cl7%Fi`cl7$F^\\bl$!+/:J5TFcinFg]cl7%F]acl7$Fh[bl$!+*y5t4%Fc inFg]cl7%Faacl7$Fb[bl$!+/m%[3%FcinFg]cl7%Feacl7$F\\[bl$!+3`4tSFcinFg]c l7%Fiacl7$Ffjal$!+G?EiSFcinFg]cl7%F]bcl7$F`jal$!+Nee_SFcinFg]cl7%Fabcl 7$Fjial$!+$)[MWSFcinFg]cl7%Febcl7$Fdial$!+p/'y.%FcinFg]cl7%Fibcl7$F^ia l$!+&z(\\LSFcinFg]cl7%F]ccl7$Fhhal$!+g&f;.%FcinFg]cl7%Faccl7$Fbhal$!+? iwKSFcinFg]cl7%Feccl7$F\\hal$!+z\\@PSFcinFg]cl7%Ficcl7$Ffgal$!+j6JXSFc inFg]cl7%F]dcl7$F`gal$!+;bC7TFcinFg]cl7%F]ec l7$Fheal$!+iMoMTFcinFg]cl7%Faecl7$Fbeal$!+e)zw:%FcinFg]cl7%Feecl7$F\\e al$!+)Rb/=%FcinFg]cl7%Fiecl7$Ffdal$!+%[>C?%FcinFg]cl7%F]fcl7$F`dal$!+: a;BUFcinFg]cl7%Fafcl7$Fjcal$!+U\"RCC%FcinFg]cl7%Fefcl7$Fdcal$!+q-5gUFc inFg]cl7%Fifcl7$F^cal$!+Yr3wUFcinFg]cl7%F]gcl7$Fhbal$!+g)*Q!H%FcinFg]c l7%Fagcl7$Fbbal$!+$*4..VFcinFg]cl7%Fegcl7$F\\bal$!+mD09VFcinFg]cl7%Fig cl7$Ffaal$!+ju]BVFcinFg]cl7%F]hcl7$F`aal$!+fRXJVFcinFg]cl7%Fahcl7$Fj`a l$!+*H_zL%FcinFg]cl7%Fehcl7$Fd`al$!+SE1VVFcinFg]cl7%Fihcl7$F^`al$!+fQ% oM%FcinFg]cl7%F]icl7$Fh_al$!+EHN\\VFcinFg]cl7%Faicl7$Fb_al$!+FYk]VFcin Fg]cl7%Feicl7$F\\_al$!+;:x]VFcinFg]cl7%Fiicl7$Ff^al$!+'*Ry\\VFcinFg]cl 7%F]jcl7$F`^al$!+([IxM%FcinFg]cl7%Fajcl7$Fj]al$!+WwlWVFcinFg]cl7%Fejcl 7$Fd]al$!+'f51M%FcinFg]cl7%Fijcl7$F^]al$!+6KjNVFcinFg]cl7%F][dl7$Fh\\a l$!+d$o(HVFcinFg]cl7%Fa[dl7$Fb\\al$!+!=eIK%FcinFg]cl7%Fe[dl7$F\\\\al$! +yVa:VFcinFg]cl7%Fi[dl7$Fc[al$!+\"\\osI%FcinFg]cl7%F]\\dlFa]clFg]clFeg _lF\\_s-%%FONTG6$%*HELVETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLA BELSG6%%&Re(z)G%&Im(z)G-Fb\\dl6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%%VIEW G6$;$!$z%F^fn$\"$*>F^fn;Fj]dl$\"$z%F^fn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#-------------------------------------------------------- -" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 80 "The stability function R^ for the 7 stage, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b^`,Stabilit yFunction(4,7,'expanded'))):\n`R^` := unapply(%,z):\n'`R^`(z)'=`R^`(z) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R^G6#%\"zG,2\"\"\"F)F'F)*&#F) \"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F )F)F)*&#\"8VqBR5dH<@sg\"\":+;VzWe#)Gz%RM>F)*$)F'\"\"&F)F)F)*&#\"J$zg12 ?[ON?$>KY?!4QnT!G\"\"L++jviOk9.hn)RB;O,9m%)G\"*F)*$)F'F1F)F)F)*&#\"E6 \"G/e\\<0t/Q,$R0jc&H'\"I+b&3S0DGe\"\\;c*Q8LA(*=V$F)*$)F'\"\"(F)F)F)" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects \+ the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R^`(z) = -1;" "6#/-%#R^G6#%\"zG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " z_0 := newton(`R^`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ z_0G$!+X2mGT!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "zo0 := newton(`R^`(z)=-1,z=-4):\npo1 := plot ([`R^`(z),-1],z=-4.49..0.49,color=[red,blue]):\npo2 := plot([[[zo0,-1] ]$3],style=point,symbol=[circle,cross,diamond],color=black):\npo3 := p lot([[zo0,0],[zo0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[di splay]([po1,po2,po3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9] );" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG 6$7U7$$!3A++++++!\\%!#<$!3sJ'3oF*7$$!3;+]7'3DdV%F*$!3'f5;yJd`z\"F *7$$!35++Ds,X\"Q%F*$!3f=$)4l7@D;F*7$$!3(*\\P*)feAMVF*$!3CP[v!*\\X)[\"F *7$$!3%)*\\Pva,qG%F*$!3[A*ebmG:O\"F*7$$!3I+](4>%y!=%F*$!3Y\">N>Wm)36F* 7$$!35+]#oviQ2%F*$!3ELk2k`b_*)!#=7$$!3A+ve,&\\u'RF*$!33a)yQOu_<(FK7$$! 3?+v8#z!zoQF*$!3oju!f&*fUz&FK7$$!3;+v.8ajmPF*$!3_,N$F*$!3[oK>grz&\\\"FK7$$!3E++:Qn_WKF*$!3; T#Ha7$$!3&****\\=iPF.$F *$!3_)GEhYlC8$Fcp7$$!3Q+v$Gc`$QHF*$!3Fu&ewb^%>x!#?7$$!3+++l(y@h#GF*$\" 3!Q#48pxP>O,LFcp7$$!3C+vt%)=X?EF*$\"3 wyX8z\"=)[]Fcp7$$!3D++5(ocD_#F*$\"3?I0hNK8WkFcp7$$!3#**\\P(R,::CF*$\"3 EkAj'=E_(yFcp7$$!3&)*\\7sqtGJ#F*$\"3;M+*yh([#>*Fcp7$$!30+]Pz*eh?#F*$\" 3wL1>'4/n0\"FK7$$!3l*\\([Y:;3@F*$\"3g8_Asa`'=\"FK7$$!3$***\\7w!eC+#F*$ \"3*e;`'3H)QL\"FK7$$!3%**\\Py(=m#*=F*$\"3sBh&)o3R)\\\"FK7$$!3%)*\\iW'R 3(z\"F*$\"3]H*e76,Pl\"FK7$$!3-+]d*>dQp\"F*$\"3hP(R$yWlO=FK7$$!3)*****p ]Q@(e\"F*$\"37^(4X!>&\\/#FK7$$!3I+]FRT)G[\"F*$\"3-zL:59WqAFK7$$!3')*\\ P*y(R>Q\"F*$\"3OE1t$)ep6DFK7$$!3++]Kx#e)p7F*$\"3QW(*G9UV4GFK7$$!3=++5j ![\"p6F*$\"3!QRvx\"f)o5$FK7$$!3s***\\KT=;1\"F*$\"3YR$y'HZNfMFK7$$!3'** *\\P%o0=k*FK$\"36sOq`)>K\"QFK7$$!3](***\\tEbw&)FK$\"3MbS&4Z3)*\\(GL>l_'FK$\"3%f6WCdrm?&FK7$$!3z ++]-\")=-bFK$\"3?6(=?R_#odFK7$$!3$*)*\\PM#3)HWFK$\"3Wn4\\.1?@kFK7$$!3; (****f'*ypR$FK$\"3p/b!y>b)>rFK7$$!3L)**\\U'=wSBFK$\"3]x1Cra,8zFK7$$!3? $*\\Pt2H$H\"FK$\"3o$y_ojZoy)FK7$$!33R****pit2LFcp$\"3eG)**36PYn*FK7$$ \"3Z^+]FkzBxFcp$\"3Sh#R$=\"*H!3\"F*7$$\"35*****z2`!f;cX^K#>\" F*7$$\"3l+]i_F06GFK$\"3V)HXQ+$fC8F*7$$\"3\\.]Pt1&z\"QFK$\"3O3B%*e;\"\\ Y\"F*7$$\"3!***************[FK$\"3m$\\GR_:Bj\"F*-%'COLOURG6&%$RGBG$\"* ++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$ FBFa\\l7$FGFa\\l7$FMFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa \\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F`qFa\\l7$FeqFa \\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa \\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vF a\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xF a\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$Faz Fa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7 $$!3O+++X2mGTF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l -%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#% (DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\l F[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q \"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`c l$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following p icture shows the stability region for the 8 stage, order 4 scheme. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1662 "`R^` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+\n 1607221172957 1039237043/1934394792882584479431600*z^5+\n 1280416738090204632193 2035364820070660793/9128846614013616233986761031464366275630000*z^6+\n 629556630539301380473051749580428111/3431897223313389561649158282 505400855500*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=201/20 do \n zz := newton(`R^`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13 /20<=tt and tt<=6/5) or (87/10<=tt and tt<=93/10) then\n hh := 1/ 40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo1 := plot(pts,color=COLOR(RGB, .29,.05,.05)):\npo2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.57,. 1,.1)):\npts := []: z0 := 1.4+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pt s),[Re(zz),Im(zz)]]:\nend do:\npo3 := plot(pts,color=COLOR(RGB,.29,.05 ,.05)):\npo4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.32,4.19]], i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.57,.1, .1)):\npts := []: z0 := 1.4-4.3*I:\nfor ct from 0 to 60 do\n zz := n ewton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts) ,[Re(zz),Im(zz)]]:\nend do:\npo5 := plot(pts,color=COLOR(RGB,.29,.05,. 05)):\npo6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.32,-4.19]],i =2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.57,.1,. 1)):\npo7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4.79]]],color=bl ack,linestyle=3):\nplots[display]([po||(1..7)],view=[-4.79..1.99,-4.79 ..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],ax es=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 586 586 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F(7$$\"3u*****R2bKw&!#F$\" 3))******[jzq:!#=7$$\"3K+++n#)euO!#D$\"3w*****>G$fTJF07$$\"3(******f]y >9%!#C$\"3A+++wCR7ZF07$$\"3%******\\.$>tA!#B$\"33+++K?>$G'F07$$\"3M*** **zSzEG)F@$\"3')*****R\\oR&yF07$$\"3')******[`%yF#!#A$\"3u******zcjC%* F07$$\"3%******4F#*Q)\\FK$\"3'******z'p\\*4\"!#<7$$\"3g+++=RZl')FK$\"3 1+++@$\\kD\"FS7$$\"3/+++DAr$3\"!#@$\"3&*******QCB89FS7$$\"39+++-xH,SFK $\"3-+++/;qp:FS7$$!35+++c'yx(GFfn$\"34+++kRkD\"!#?$\" 3/+++8%y2)=FS7$$!33+++`$4zB$Ffo$\"3-+++#[rZ.#FS7$$!3$)*****zE$*y#\\Ffo $\"3$)*****\\/G76#FS7$$!3S+++IwROsFfo$\"3)******Rnjs=#FS7$$!3)******Re j=.\"!#>$\"3!******H$=$GE#FS7$$!3/+++)f:^V\"F[q$\"3:+++(o$)yL#FS7$$!39 +++oq6`>F[q$\"3&)******))\\O7CFS7$$!3<+++9N\\2EF[q$\"31+++)46i[#FS7$$! 3!******RVq9U$F[q$\"3!)*****z&*R$fDFS7$$!3E+++/&\\$>WF[q$\"3<+++pJkJEF S7$$!33+++;llDcF[q$\"3))*****zxvHq#FS7$$!3I+++C(QQ1(F[q$\"3#******fEYJ x#FS7$$!3))*****HTDWv)F[q$\"3#******fG4>%GFS7$$!3-+++g\"p72\"F0$\"3\") *****H#['*3HFS7$$!3!*******4+h%H\"F0$\"3/+++Ix'R(HFS7$$!33+++@)3_a\"F0 $\"3'*******y+bOIFS7$$!34+++_[p@=F0$\"31+++n[N'4$FS7$$!30+++YUF0$\" 3')*****HD1FU$FS7$$!3/+++)ps+f%F0$\"3#)*****\\*)flX$FS7$$!37+++Q9kH`F0 $\"3@+++&)zK;NFS7$$!3!)*****\\'fTigF0$\"3')*****Hd>nc$FS7$$!3#******R! ft%y'F0$\"3@+++B!))*3OFS7$$!3_+++Z+D&\\(F0$\"3%******pyYTk$FS7$$!3c*** **4g8Q>)F0$\"3-+++!f(*Hn$FS7$$!3_+++`H'3)))F0$\"3/+++s1<'p$FS7$$!3Y+++ (G!3d&*F0$\"34+++i=;9PFS7$$!3/+++[AKA5FS$\"31+++$fits$FS7$$!33+++RS+)3 \"FS$\"3))*****f+$3OPFS7$$!3#*******>Z#G:\"FS$\"36+++!Rp0u$FS7$$!3*)** ***fM^o@\"FS$\"3%)*****pJ<5u$FS7$$!3/+++$oZ,G\"FS$\"3#)*****pd\"ePPFS7 $$!3\"******\\/wFM\"FS$\"3&)*****>#QQIPFS7$$!31+++q'*z/9FS$\"32+++#e=& >PFS7$$!3#******Ha&Gm9FS$\"37+++I#e]q$FS7$$!30+++'e2t_\"FS$\"3#)****** [t0(o$FS7$$!3!******RD]ze\"FS$\"3/+++]pblOFS7$$!35+++([7$[;FS$\"34+++N !*eSOFS7$$!3)******H%>^3FS$\"3/+++))y0lMFS7$$!3*)*******oTH, #FS$\"3y*****4nx.U$FS7$$!3#*******pq.w?FS$\"35+++G!>HP$FS7$$!3))*****R O:/9#FS$\"3%******>'f,BLFS7$$!3:+++[#Ri?#FS$\"3%******zH(4rKFS7$$!3?++ +bRctAFS$\"3\")*****fopw@$FS7$$!3;+++B'*HUBFS$\"3;+++QbFjJFS7$$!30+++# 3(>7CFS$\"3\"******f;J%3JFS7$$!3%******4cnG[#FS$\"3%)*****4-iN0$FS7$$! 3/+++IE$Qb#FS$\"3;+++_i&*)*HFS7$$!3*)*****\\Q(eCEFS$\"3))*****>(*\\Z%H FS7$$!3)******RucYp#FS$\"3%)*****H+V4*GFS7$$!3!******4IKOw#FS$\"3%**** **p,Lu$GFS7$$!3;+++>y=JGFS$\"33+++IS0%y#FS7$$!3!)******\\&yq*GFS$\"3-+ ++d'41t#FS7$$!3'******zSJ6'HFS$\"35+++ms*on#FS7$$!3$******>xJK-$FS$\"3 <+++*>BFi#FS7$$!3))*****zU5L3$FS$\"3)******H)4\"zc#FS7$$!32+++jBLTJFS$ \"32+++'y/B^#FS7$$!3#)*****4)yG(>$FS$\"3A+++t*pdX#FS7$$!37+++tf=^KFS$ \"37+++@@>)R#FS7$$!3$******>d\\II$FS$\"33+++p`ZRBFS7$$!3'******fH7HN$F S$\"3!******ziS&zAFS7$$!3,+++3j\"3S$FS$\"3-+++!*RK=AFS7$$!3=+++h4\"oW$ FS$\"3')*****R[vd:#FS7$$!3-+++i>&4\\$FS$\"3#)*****\\Ge=4#FS7$$!3=+++13 ILNFS$\"3(******>S[l-#FS7$$!3/+++\"GCRd$FS$\"3-+++U[$)f>FS7$$!3<+++FQ* Gh$FS$\"3#******H@?<*=FS7$$!3!******>d%G]OFS$\"3/+++t;AA=FS7$$!33+++eQ <'o$FS$\"3++++w?P^I\"FS7$$!3>+++o'\\5!RFS$\"3*)*****49NRA\"FS7$$! 3#)*****fc;q#RFS$\"3%********)*Qa9\"FS7$$!33+++9%y<&RFS$\"33+++f6bm5FS 7$$!3/+++'3w_(RFS$\"33+++RA&R()*F07$$!3A+++)RMu*RFS$\"3I+++tN!33*F07$$ !3T+++zn;=SFS$\"3m*****zw#)pG)F07$$!3&******HI!QPSFS$\"3E+++^y@$\\(F07 $$!3!)*****p]!)\\0%FS$\"3U+++(*z/+nF07$$!33+++\"fv32%FS$\"3G+++Kc$y!fF 07$$!3!)*****>uz\\3%FS$\"3b*****fU'y;^F07$$!3e******\\b@(4%FS$\"3.+++N L(pK%F07$$!38+++*e:v5%FS$\"3*******z\\h$QNF07$$!3i******>L#e6%FS$\"3y* ****H\"=$3v#F07$$!35+++6N4ATFS$\"35+++U>?k>F07$$!3D+++VBHETFS$\"3)**** **RLW#y6F07$$!3!)*****fN(RGTFS$\"3!******f?9q#RF[q7$Fe^m$!3!******f?9q #RF[q7$F`^m$!3)******RLW#y6F07$F[^m$!35+++U>?k>F07$Ff]m$!3y*****H\"=$3 v#F07$Fa]m$!3*******z\\h$QNF07$F\\]m$!3.+++NL(pK%F07$Fg\\m$!3b*****fU' y;^F07$Fb\\m$!3G+++Kc$y!fF07$F]\\m$!3U+++(*z/+nF07$Fh[m$!3E+++^y@$\\(F 07$Fc[m$!3m*****zw#)pG)F07$F^[m$!3I+++tN!33*F07$Fijl$!33+++RA&R()*F07$ Fdjl$!33+++f6bm5FS7$F_jl$!3%********)*Qa9\"FS7$Fjil$!3*)*****49NRA\"FS 7$Feil$!3-+++Ph!>I\"FS7$F`il$!33+++4T@z8FS7$F[il$!35+++'oAdX\"FS7$Ffhl $!3++++nDIJ:FS7$Fahl$!3\"*******)=Oeg\"FS7$F\\hl$!33+++@6Az;FS7$Fggl$! 3++++w?P^FS7$Fcfl$!3(******>S[l-#FS7$F^fl$!3#)*****\\Ge=4#FS7$Fiel$!3')*** **R[vd:#FS7$Fdel$!3-+++!*RK=AFS7$F_el$!3!******ziS&zAFS7$Fjdl$!33+++p` ZRBFS7$Fedl$!37+++@@>)R#FS7$F`dl$!3A+++t*pdX#FS7$F[dl$!32+++'y/B^#FS7$ Ffcl$!3)******H)4\"zc#FS7$Facl$!3<+++*>BFi#FS7$F\\cl$!35+++ms*on#FS7$F gbl$!3-+++d'41t#FS7$Fbbl$!33+++IS0%y#FS7$F]bl$!3%******p,Lu$GFS7$Fhal$ !3%)*****H+V4*GFS7$Fcal$!3))*****>(*\\Z%HFS7$F^al$!3;+++_i&*)*HFS7$Fi` l$!3%)*****4-iN0$FS7$Fd`l$!3\"******f;J%3JFS7$F_`l$!3;+++QbFjJFS7$Fj_l $!3\")*****fopw@$FS7$Fe_l$!3%******zH(4rKFS7$F`_l$!3%******>'f,BLFS7$F [_l$!35+++G!>HP$FS7$Ff^l$!3y*****4nx.U$FS7$Fa^l$!3/+++))y0lMFS7$F\\^l$ !3%******H@PFS7$F_[l$!3&)** ***>#QQIPFS7$Fjz$!3#)*****pd\"ePPFS7$Fez$!3%)*****pJ<5u$FS7$F`z$!36+++ !Rp0u$FS7$F[z$!3))*****f+$3OPFS7$Ffy$!31+++$fits$FS7$Fay$!34+++i=;9PFS 7$F\\y$!3/+++s1<'p$FS7$Fgx$!3-+++!f(*Hn$FS7$Fbx$!3%******pyYTk$FS7$F]x $!3@+++B!))*3OFS7$Fhw$!3')*****Hd>nc$FS7$Fcw$!3@+++&)zK;NFS7$$!3/+++NR Mg\\F0$!3?+++#p.x[$FS7$F^w$!3#)*****\\*)flX$FS7$Fiv$!3')*****HD1FU$FS7 $Fdv$!3#******f@ZfQ$FS7$F_v$!35+++gB4YLFS7$Fju$!3,+++7=(HI$FS7$Feu$!3? +++vnXcKFS7$F`u$!34+++VK[1KFS7$F[u$!3A+++)RuI:$FS7$Fft$!31+++n[N'4$FS7 $Fat$!3'*******y+bOIFS7$F\\t$!3/+++Ix'R(HFS7$Fgs$!3\")*****H#['*3HFS7$ Fbs$!3#******fG4>%GFS7$F]s$!3#******fEYJx#FS7$Fhr$!3))*****zxvHq#FS7$F cr$!3<+++pJkJEFS7$F^r$!3!)*****z&*R$fDFS7$Fiq$!31+++)46i[#FS7$Fdq$!3&) ******))\\O7CFS7$F_q$!3:+++(o$)yL#FS7$Fip$!3!******H$=$GE#FS7$Fdp$!3)* *****Rnjs=#FS7$F_p$!3$)*****\\/G76#FS7$$!3A+++;8`K?Ffo$!33+++'yRz&>FS7 $$!3%******f?4=U'Ffn$!3++++%3KL!=FS7$$!3u******HW9jvFK$!3%******4raxk \"FS7$$\"3h+++C%p*=$*FK$!3/+++(*y^\"\\\"FS7$$\"3,+++xIwC5Ffn$!3*)***** *z&p[L\"FS7$$\"3C+++ls'3y'FK$!3'******\\k()z<\"FS7$$\"3s*****>)opiMFK$ !3*******zl')4-\"FS7$$\"3.+++s7B;9FK$!3!******>OD$R')F07$$\"36+++R0%)> XF@$!31+++wjeoqF07$$\"3/+++_gXM5F@$!3#)*****p#Gz(\\&F07$$\"3-+++%zbgR \"F:$!3-+++2D*p#RF07$$\"3;+++'3&*eb'!#E$!3)******Hl%>cBF07$$\"3l+++L(H 4,*!#H$!3Y+++T;)R&yF[q7$$\"3i+++&yH4,*Fden$\"3Y+++T;)R&yF[q-%&COLORG6& %$RGBG$\"#H!\"#$\"\"&FbfnFcfn-%)POLYGONSG6^y7%F'7$$\"+u]DjdF[q$\"+\\jz q:!#57$$FbfnF)F(7%Fifn7$$\"+n#)euOFS$\"+#G$fTJF^gnF_gn7%Fbgn7$$\"+1&y> 9%!#;$\"+wCR7ZF^gnF_gn7%Fhgn7$$\"+NI>tA!#:$\"+K?>$G'F^gnF_gn7%F_hn7$$ \"+3%zEG)Fbhn$\"+%\\oR&yF^gnF_gn7%Ffhn7$$\"+\\`%yF#!#9$\"+!oNYU*F^gnF_ gn7%F\\in7$$\"+rA*Q)\\F_in$\"+op\\*4\"!\"*F_gn7%Fcin7$$\"+=RZl')F_in$ \"+@$\\kD\"FhinF_gn7%Fjin7$$\"+DAr$3\"!#8$\"+RCB89FhinF_gn7%F`jn7$$\"+ -xH,SF_in$\"+/;qp:FhinF_gn7%Fgjn7$$!+c'yx(GFcjn$\"+kRkD\"!#7$\"+8%y2)=FhinF_gn7%Fc[o7$$!+`$4zB$Ff[o$\"+#[rZ.#FhinF_ gn7%Fj[o7$$!+oK*y#\\Ff[o$\"+X!G76#FhinF_gn7%F`\\o7$$!+IwROsFf[o$\"+uOE (=#FhinF_gn7%Ff\\o7$$!+%ej=.\"!#6$\"+L=$GE#FhinF_gn7%F\\]o7$$!+)f:^V\" F_]o$\"+(o$)yL#FhinF_gn7%Fc]o7$$!+oq6`>F_]o$\"+*)\\O7CFhinF_gn7%Fi]o7$ $!+9N\\2EF_]o$\"+)46i[#FhinF_gn7%F_^o7$$!+M/Z@MF_]o$\"+e*R$fDFhinF_gn7 %Fe^o7$$!+/&\\$>WF_]o$\"+pJkJEFhinF_gn7%F[_o7$$!+;llDcF_]o$\"+yd(Hq#Fh inF_gn7%Fa_o7$$!+C(QQ1(F_]o$\"+mi9tFFhinF_gn7%Fg_o7$$!+8aUa()F_]o$\"+' G4>%GFhinF_gn7%F]`o7$$!+g\"p72\"F^gn$\"+B['*3HFhinF_gn7%Fc`o7$$!+5+h%H \"F^gn$\"+Ix'R(HFhinF_gn7%Fi`o7$$!+@)3_a\"F^gn$\"+z+bOIFhinF_gn7%F_ao7 $$!+_[p@=F^gn$\"+n[N'4$FhinF_gn7%Feao7$$!+YUF^g n$\"+`iqAMFhinF_gn7%F_do7$$!+)ps+f%F^gn$\"+&*)flX$FhinF_gn7%Fedo7$$!+Q 9kH`F^gn$\"+&)zK;NFhinF_gn7%F[eo7$$!+lfTigF^gn$\"+t&>nc$FhinF_gn7%Faeo 7$$!+/ft%y'F^gn$\"+B!))*3OFhinF_gn7%Fgeo7$$!+Z+D&\\(F^gn$\"+(yYTk$Fhin F_gn7%F]fo7$$!+,O\"Q>)F^gn$\"+!f(*Hn$FhinF_gn7%Fcfo7$$!+`H'3)))F^gn$\" +s1<'p$FhinF_gn7%Fifo7$$!+(G!3d&*F^gn$\"+i=;9PFhinF_gn7%F_go7$$!+[AKA5 Fhin$\"+$fits$FhinF_gn7%Fego7$$!+RS+)3\"Fhin$\"+1I3OPFhinF_gn7%F[ho7$$ !+?Z#G:\"Fhin$\"+!Rp0u$FhinF_gn7%Faho7$$!+Y8&o@\"Fhin$\"+PFhinF_gn7%Fiio7$$!+VbGm9Fhin$\"+I #e]q$FhinF_gn7%F_jo7$$!+'e2t_\"Fhin$\"+\\t0(o$FhinF_gn7%Fejo7$$!+a-&ze \"Fhin$\"+]pblOFhinF_gn7%F[[p7$$!+([7$[;Fhin$\"+N!*eSOFhinF_gn7%Fa[p7$ $!+V>^3Fhin$\"+))y0lMFhinF_gn7%F_]p7$$!+!pTH,#Fhin $\"+rwP?MFhinF_gn7%Fe]p7$$!+qq.w?Fhin$\"+G!>HP$FhinF_gn7%F[^p7$$!+k`TS @Fhin$\"+if,BLFhinF_gn7%Fa^p7$$!+[#Ri?#Fhin$\"+)H(4rKFhinF_gn7%Fg^p7$$ !+bRctAFhin$\"+'opw@$FhinF_gn7%F]_p7$$!+B'*HUBFhin$\"+QbFjJFhinF_gn7%F c_p7$$!+#3(>7CFhin$\"+m6V3JFhinF_gn7%Fi_p7$$!+hv'G[#Fhin$\"+@?c`IFhinF _gn7%F_`p7$$!+IE$Qb#Fhin$\"+_i&*)*HFhinF_gn7%Fe`p7$$!+&Q(eCEFhin$\"+s* \\Z%HFhinF_gn7%F[ap7$$!+Wnl%p#Fhin$\"+.I%4*GFhinF_gn7%Faap7$$!+,BjjFFh in$\"+y=JGFhin$\"+IS0%y#FhinF_gn7%F]bp7$$!+]& yq*GFhin$\"+d'41t#FhinF_gn7%Fcbp7$$!+398hHFhin$\"+ms*on#FhinF_gn7%Fibp 7$$!+sBFi#FhinF_gn7%F_cp7$$!+G/J$3$Fhin$\"+$)4\"zc#FhinF _gn7%Fecp7$$!+jBLTJFhin$\"+'y/B^#FhinF_gn7%F[dp7$$!+\")yG(>$Fhin$\"+t* pdX#FhinF_gn7%Fadp7$$!+tf=^KFhin$\"+@@>)R#FhinF_gn7%Fgdp7$$!+s&\\II$Fh in$\"+p`ZRBFhinF_gn7%F]ep7$$!+'H7HN$Fhin$\"+G1azAFhinF_gn7%Fcep7$$!+3j \"3S$Fhin$\"+!*RK=AFhinF_gn7%Fiep7$$!+h4\"oW$Fhin$\"+%[vd:#FhinF_gn7%F _fp7$$!+i>&4\\$Fhin$\"+&Ge=4#FhinF_gn7%Fefp7$$!+13ILNFhin$\"+-%[l-#Fhi nF_gn7%F[gp7$$!+\"GCRd$Fhin$\"+U[$)f>FhinF_gn7%Fagp7$$!+FQ*Gh$Fhin$\"+ 8-s\"*=FhinF_gn7%Fggp7$$!+sXG]OFhin$\"+t;AA=FhinF_gn7%F]hp7$$!+eQ<'o$F hin$\"+w?P^I\"FhinF_gn7%Fgjp7$$!+o'\\5!RFhin$\"+ T^$RA\"FhinF_gn7%F][q7$$!+ml,FRFhin$\"+!**Qa9\"FhinF_gn7%Fc[q7$$!+9%y< &RFhin$\"+f6bm5FhinF_gn7%Fi[q7$$!+'3w_(RFhin$\"+RA&R()*F^gnF_gn7%F_\\q 7$$!+)RMu*RFhin$\"+tN!33*F^gnF_gn7%Fe\\q7$$!+zn;=SFhin$\"+oF)pG)F^gnF_ gn7%F[]q7$$!+..QPSFhin$\"+^y@$\\(F^gnF_gn7%Fa]q7$$!+20)\\0%Fhin$\"+(*z /+nF^gnF_gn7%Fg]q7$$!+\"fv32%Fhin$\"+Kc$y!fF^gnF_gn7%F]^q7$$!+U(z\\3%F hin$\"+Eky;^F^gnF_gn7%Fc^q7$$!+]b@(4%Fhin$\"+NL(pK%F^gnF_gn7%Fi^q7$$!+ *e:v5%Fhin$\"+)\\h$QNF^gnF_gn7%F__q7$$!+?L#e6%Fhin$\"+8=$3v#F^gnF_gn7% Fe_q7$$!+6N4ATFhin$\"+U>?k>F^gnF_gn7%F[`q7$$!+VBHETFhin$\"+MVCy6F^gnF_ gn7%Fa`q7$$!+ctRGTFhin$\"+1U,FRF_]oF_gn7%Fg`q7$Fh`q$!+1U,FRF_]oF_gn7%F ]aq7$Fb`q$!+MVCy6F^gnF_gn7%Faaq7$F\\`q$!+U>?k>F^gnF_gn7%Feaq7$Ff_q$!+8 =$3v#F^gnF_gn7%Fiaq7$F`_q$!+)\\h$QNF^gnF_gn7%F]bq7$Fj^q$!+NL(pK%F^gnF_ gn7%Fabq7$Fd^q$!+Eky;^F^gnF_gn7%Febq7$F^^q$!+Kc$y!fF^gnF_gn7%Fibq7$Fh] q$!+(*z/+nF^gnF_gn7%F]cq7$Fb]q$!+^y@$\\(F^gnF_gn7%Facq7$F\\]q$!+oF)pG) F^gnF_gn7%Fecq7$Ff\\q$!+tN!33*F^gnF_gn7%Ficq7$F`\\q$!+RA&R()*F^gnF_gn7 %F]dq7$Fj[q$!+f6bm5FhinF_gn7%Fadq7$Fd[q$!+!**Qa9\"FhinF_gn7%Fedq7$F^[q $!+T^$RA\"FhinF_gn7%Fidq7$Fhjp$!+Ph!>I\"FhinF_gn7%F]eq7$Fbjp$!+4T@z8Fh inF_gn7%Faeq7$F\\jp$!+'oAdX\"FhinF_gn7%Feeq7$Ffip$!+nDIJ:FhinF_gn7%Fie q7$F`ip$!+*=Oeg\"FhinF_gn7%F]fq7$Fjhp$!+@6Az;FhinF_gn7%Fafq7$Fdhp$!+w? P^FhinF_gn7%Fagq7$F\\gp$!+-%[l-#FhinF_gn7%Fegq7$Fffp $!+&Ge=4#FhinF_gn7%Figq7$F`fp$!+%[vd:#FhinF_gn7%F]hq7$Fjep$!+!*RK=AFhi nF_gn7%Fahq7$Fdep$!+G1azAFhinF_gn7%Fehq7$F^ep$!+p`ZRBFhinF_gn7%Fihq7$F hdp$!+@@>)R#FhinF_gn7%F]iq7$Fbdp$!+t*pdX#FhinF_gn7%Faiq7$F\\dp$!+'y/B^ #FhinF_gn7%Feiq7$Ffcp$!+$)4\"zc#FhinF_gn7%Fiiq7$F`cp$!+*>BFi#FhinF_gn7 %F]jq7$Fjbp$!+ms*on#FhinF_gn7%Fajq7$Fdbp$!+d'41t#FhinF_gn7%Fejq7$F^bp$ !+IS0%y#FhinF_gn7%Fijq7$Fhap$!+HP$FhinF_gn7%Fe]r7$Ff]p $!+rwP?MFhinF_gn7%Fi]r7$F`]p$!+))y0lMFhinF_gn7%F]^r7$Fj\\p$!+8sr1NFhin F_gn7%Fa^r7$Fd\\p$!+.)*=XNFhinF_gn7%Fe^r7$F^\\p$!+evO!e$FhinF_gn7%Fi^r 7$Fh[p$!+.?=7OFhinF_gn7%F]_r7$Fb[p$!+N!*eSOFhinF_gn7%Fa_r7$F\\[p$!+]pb lOFhinF_gn7%Fe_r7$Ffjo$!+\\t0(o$FhinF_gn7%Fi_r7$F`jo$!+I#e]q$FhinF_gn7 %F]`r7$Fjio$!+#e=&>PFhinF_gn7%Fa`r7$Fdio$!+AQQIPFhinF_gn7%Fe`r7$F^io$! +x:ePPFhinF_gn7%Fi`r7$Fhho$!+nc$FhinF_gn7%Facr7$F\\eo$!+&)zK;NFhinF_gn7%Fecr7$$!+NRMg \\F^gn$!+#p.x[$FhinF_gn7%Ficr7$Ffdo$!+&*)flX$FhinF_gn7%F_dr7$F`do$!+`i qAMFhinF_gn7%Fcdr7$Fjco$!+;s%fQ$FhinF_gn7%Fgdr7$Fdco$!+gB4YLFhinF_gn7% F[er7$F^co$!+7=(HI$FhinF_gn7%F_er7$Fhbo$!+vnXcKFhinF_gn7%Fcer7$Fbbo$!+ VK[1KFhinF_gn7%Fger7$F\\bo$!+)RuI:$FhinF_gn7%F[fr7$Ffao$!+n[N'4$FhinF_ gn7%F_fr7$F`ao$!+z+bOIFhinF_gn7%Fcfr7$Fj`o$!+Ix'R(HFhinF_gn7%Fgfr7$Fd` o$!+B['*3HFhinF_gn7%F[gr7$F^`o$!+'G4>%GFhinF_gn7%F_gr7$Fh_o$!+mi9tFFhi nF_gn7%Fcgr7$Fb_o$!+yd(Hq#FhinF_gn7%Fggr7$F\\_o$!+pJkJEFhinF_gn7%F[hr7 $Ff^o$!+e*R$fDFhinF_gn7%F_hr7$F`^o$!+)46i[#FhinF_gn7%Fchr7$Fj]o$!+*)\\ O7CFhinF_gn7%Fghr7$Fd]o$!+(o$)yL#FhinF_gn7%F[ir7$F]]o$!+L=$GE#FhinF_gn 7%F_ir7$Fg\\o$!+uOE(=#FhinF_gn7%Fcir7$Fa\\o$!+X!G76#FhinF_gn7%Fgir7$$! +;8`K?Ff[o$!+'yRz&>FhinF_gn7%F[jr7$$!+1#4=U'Fcjn$!+%3KL!=FhinF_gn7%Faj r7$$!+IW9jvF_in$!+6ZvZ;FhinF_gn7%Fgjr7$$\"+C%p*=$*F_in$!+(*y^\"\\\"Fhi nF_gn7%F][s7$$\"+xIwC5Fcjn$!+!ep[L\"FhinF_gn7%Fc[s7$$\"+ls'3y'F_in$!+X w)z<\"FhinF_gn7%Fi[s7$$\"+#)opiMF_in$!+em)4-\"FhinF_gn7%F_\\s7$$\"+s7B ;9F_in$!+i`KR')F^gnF_gn7%Fe\\s7$$\"+R0%)>XFbhn$!+wjeoqF^gnF_gn7%F[]s7$ $\"+_gXM5Fbhn$!+FGz(\\&F^gnF_gn7%Fa]s7$$\"+%zbgR\"F[hn$!+2D*p#RF^gnF_g n7%Fg]s7$$\"+'3&*eb'F0$!+`Y>cBF^gnF_gn7%F]^s7$$\"+L(H4,*Ffn$!+T;)R&yF_ ]oF_gn7%Fc^s7$$\"+&yH4,*Ffn$\"+T;)R&yF_]oF_gn-F]fn6&F_fn$\"#dFbfn$\"\" \"!\"\"Fb_s-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"3!******4*HnV9FS$\"35 +++%>*3)H%FS7$$\"3*)*****\\xR^V\"FS$\"3#)******[p22VFS7$$\"3)*******\\ f!fU\"FS$\"3<+++:ZM:VFS7$$\"34+++:\\,;9FS$\"3%)*****p$3&GK%FS7$$\"33++ +m1^09FS$\"3S+++-QbHVFS7$$\"3)******z7QWR\"FS$\"3F+++T>TNVFS7$$\"31+++ HN%GQ\"FS$\"3m*****\\9$QSVFS7$$\"3#*******G[xq8FS$\"3h*****\\eCWM%FS7$ $\"35+++;AGe8FS$\"3C+++WC\\ZVFS7$$\"37+++$o=aM\"FS$\"3C+++V;a\\VFS7$$ \"3'******\\rSAL\"FS$\"3o*****\\fD0N%FS7$$\"3!*******4\"4)=8FS$\"3C+++ ]fR]VFS7$$\"3,+++n**=08FS$\"3!******HR-\"\\VFS7$$\"33+++*)eX\"H\"FS$\" 3/+++qCfYVFS7$$\"3++++kuox7FS$\"3w******4:\"GM%FS7$$\"3)******R2vRE\"F S$\"3n*****4o-xL%FS7$$\"33+++V6U]7FS$\"3#******pE27L%FS7$$\"3********4 H9P7FS$\"3E+++-`EBVFS7$$\"3++++[eFC7FS$\"33+++in\"QJ%FS7$$\"3++++Zy(>@ \"FS$\"3S+++RO!GI%FS7$$\"3)******RCM/?\"FS$\"39+++`L/0!=\"FS$\"3&)*****pIuA6%FS7$$\"3/+++o0X$>\"FS$\"3u*****R^ G84%FS7$$\"3)*******>4G57FS$\"3C+++V\"\\F2%FS7$$\"3,+++M9*)H7FS$\"3c** ***H2%HdSFS7$$\"3)******HvF9D\"FS$\"3))*****f)=WXSFS7$$\"3-+++S_)RF\"F S$\"3T+++z%\\t.%FS7$$\"3-+++cRu'H\"FS$\"3++++Uw*G.%FS7$$\"3'******f'R/ >8FS$\"3)*******RGyJSFS7$$\"3,+++Q=SS8FS$\"3$)******)[4O.%FS7$$\"3-+++ s4\\g8FS$\"3A+++9%ez.%FS7$$\"31+++'H3\"z8FS$\"3;+++UyUWSFS7$$\"3'***** *R+ThR\"FS$\"3Q+++0Kl_SFS7$$\"32+++[!R:T\"FS$\"3)******pi8B1%FS7$$\"3- +++XWHD9FS$\"3++++978tSFS7$$\"3(******z`EuV\"FS$\"3#)*****>1n[3%FS7$$ \"35+++X=(zW\"FS$\"3k*****H8;t4%FS7$$\"3%*******3s(pX\"FS$\"3R+++C@I5T FS7$$\"3%*******o_\\k9FS$\"3/+++%ysO7%FS7$$\"3#******4\\\"eq9FS$\"3Q++ +keHPTFS7$$\"3%******\\F#Hv9FS$\"3')*****>qb5:%FS7$$\"3-+++oPoy9FS$\"3 S+++w.&[;%FS7$$\"3#******f:63[\"FS$\"3O+++_$*eyTFS7$$\"35+++t#G<[\"FS$ \"3q*****>b\">#>%FS7$$\"3/+++>u[\"[\"FS$\"37+++WPe0UFS7$$\"3-+++8$R,[ \"FS$\"36+++F#*p=UFS7$$\"3#******p9LxZ\"FS$\"3q*****zrw9B%FS7$$\"3!*** ****)o;VZ\"FS$\"3\"******HVfQC%FS7$$\"3/+++Jk$*p9FS$\"39+++/VzbUFS7$$ \"35+++pxjk9FS$\"3#******HJJsE%FS7$$\"3-+++#>l%e9FS$\"3A+++=H7yUFS7$$ \"35+++#\\i9X\"FS$\"3i*****\\bB%)G%FSF\\`sF\\fn-Fffn6jn7%7$$\"+\"*HnV9 Fhin$\"+%>*3)H%Fhin7$$\"+v(R^V\"Fhin$\"+\\p22VFhin7$$\"$K\"Fbfn$\"$>%F bfn7%F`ct7$$\"+]f!fU\"Fhin$\"+:ZM:VFhinFect7%F[dt7$$\"+:\\,;9Fhin$\"+P 3&GK%FhinFect7%Fadt7$$\"+m1^09Fhin$\"+-QbHVFhinFect7%Fgdt7$$\"+G\"QWR \"Fhin$\"+T>TNVFhinFect7%F]et7$$\"+HN%GQ\"Fhin$\"+XJQSVFhinFect7%Fcet7 $$\"+H[xq8Fhin$\"+&eCWM%FhinFect7%Fiet7$$\"+;AGe8Fhin$\"+WC\\ZVFhinFec t7%F_ft7$$\"+$o=aM\"Fhin$\"+V;a\\VFhinFect7%Feft7$$\"+:2CK8Fhin$\"+&fD 0N%FhinFect7%F[gt7$$\"+5\"4)=8Fhin$\"+]fR]VFhinFect7%Fagt7$$\"+n**=08F hin$\"+$R-\"\\VFhinFect7%Fggt7$$\"+*)eX\"H\"Fhin$\"+qCfYVFhinFect7%F]h t7$$\"+kuox7Fhin$\"+5:\"GM%FhinFect7%Fcht7$$\"+u](RE\"Fhin$\"+\"o-xL%F hinFect7%Fiht7$$\"+V6U]7Fhin$\"+ns?JVFhinFect7%F_it7$$\"+5H9P7Fhin$\"+ -`EBVFhinFect7%Feit7$$\"+[eFC7Fhin$\"+in\"QJ%FhinFect7%F[jt7$$\"+Zy(>@ \"Fhin$\"+RO!GI%FhinFect7%Fajt7$$\"+WUV+7Fhin$\"+`L\"Fhin$\"+9&G84%FhinFect7%Fc^u7$$\"+?4G57Fhin$\" +V\"\\F2%FhinFect7%Fi^u7$$\"+M9*)H7Fhin$\"+tSHdSFhinFect7%F__u7$$\"+`x U^7Fhin$\"+')=WXSFhinFect7%Fe_u7$$\"+S_)RF\"Fhin$\"+z%\\t.%FhinFect7%F [`u7$$\"+cRu'H\"Fhin$\"+Uw*G.%FhinFect7%Fa`u7$$\"+mR/>8Fhin$\"+SGyJSFh inFect7%Fg`u7$$\"+Q=SS8Fhin$\"+*[4O.%FhinFect7%F]au7$$\"+s4\\g8Fhin$\" +9%ez.%FhinFect7%Fcau7$$\"+'H3\"z8Fhin$\"+UyUWSFhinFect7%Fiau7$$\"+/59 'R\"Fhin$\"+0Kl_SFhinFect7%F_bu7$$\"+[!R:T\"Fhin$\"+FOJiSFhinFect7%Feb u7$$\"+XWHD9Fhin$\"+978tSFhinFect7%F[cu7$$\"+QlUP9Fhin$\"+iq'[3%FhinFe ct7%Facu7$$\"+X=(zW\"Fhin$\"+LhJ(4%FhinFect7%Fgcu7$$\"+4s(pX\"Fhin$\"+ C@I5TFhinFect7%F]du7$$\"+p_\\k9Fhin$\"+%ysO7%FhinFect7%Fcdu7$$\"+\"\\ \"eq9Fhin$\"+keHPTFhinFect7%Fidu7$$\"+vAHv9Fhin$\"+-d0^TFhinFect7%F_eu 7$$\"+oPoy9Fhin$\"+w.&[;%FhinFect7%Feeu7$$\"+c6\"3[\"Fhin$\"+_$*eyTFhi nFect7%F[fu7$$\"+t#G<[\"Fhin$\"+_:>#>%FhinFect7%Fafu7$$\"+>u[\"[\"Fhin $\"+WPe0UFhinFect7%Fgfu7$$\"+8$R,[\"Fhin$\"+F#*p=UFhinFect7%F]gu7$$\"+ ZJtx9Fhin$\"+=nZJUFhinFect7%Fcgu7$$\"+*o;VZ\"Fhin$\"+L%fQC%FhinFect7%F igu7$$\"+Jk$*p9Fhin$\"+/VzbUFhinFect7%F_hu7$$\"+pxjk9Fhin$\"+88BnUFhin Fect7%Fehu7$$\"+#>l%e9Fhin$\"+=H7yUFhinFect7%F[iu7$$\"+#\\i9X\"Fhin$\" +bNU)G%FhinFect7%FaiuF[ctFectF^_sFe_s-F$6$7in7$F]`s$!35+++%>*3)H%FS7$F dbt$!3i*****\\bB%)G%FS7$F_bt$!3A+++=H7yUFS7$Fjat$!3#******HJJsE%FS7$Fe at$!39+++/VzbUFS7$F`at$!3\"******HVfQC%FS7$F[at$!3q*****zrw9B%FS7$Ff`t $!36+++F#*p=UFS7$Fa`t$!37+++WPe0UFS7$F\\`t$!3q*****>b\">#>%FS7$Fg_t$!3 O+++_$*eyTFS7$Fb_t$!3S+++w.&[;%FS7$F]_t$!3')*****>qb5:%FS7$Fh^t$!3Q+++ keHPTFS7$Fc^t$!3/+++%ysO7%FS7$F^^t$!3R+++C@I5TFS7$Fi]t$!3k*****H8;t4%F S7$Fd]t$!3#)*****>1n[3%FS7$F_]t$!3++++978tSFS7$Fj\\t$!3)******pi8B1%FS 7$Fe\\t$!3Q+++0Kl_SFS7$F`\\t$!3;+++UyUWSFS7$F[\\t$!3A+++9%ez.%FS7$Ff[t $!3$)******)[4O.%FS7$Fa[t$!3)*******RGyJSFS7$F\\[t$!3++++Uw*G.%FS7$Fgj s$!3T+++z%\\t.%FS7$Fbjs$!3))*****f)=WXSFS7$F]js$!3c*****H2%HdSFS7$Fhis $!3C+++V\"\\F2%FS7$Fcis$!3u*****R^G84%FS7$F^is$!3&)*****pIuA6%FS7$Fihs $!39+++YtnMTFS7$Fdhs$!3g*****fHNw:%FS7$F_hs$!3?+++tYP!=%FS7$Fjgs$!3=++ +0iI-UFS7$Fegs$!3O+++SP-BUFS7$F`gs$!3e*****HDtAC%FS7$F[gs$!3;+++'49*fU FS7$Fffs$!3c*****fB%)eF%FS7$Fafs$!39+++`LTNVFS7$Faas$!3S+++-QbHVFS7$F\\as$!3%)***** p$3&GK%FS7$Fg`s$!3<+++:ZM:VFS7$Fb`s$!3#)******[p22VFSFjiuF\\fn-Fffn6jn 7%7$F\\ct$!+%>*3)H%Fhin7$Fbiu$!+bNU)G%Fhin7$Ffct$!$>%Fbfn7%Fdev7$F\\iu $!+=H7yUFhinFgev7%F[fv7$Ffhu$!+88BnUFhinFgev7%F_fv7$F`hu$!+/VzbUFhinFg ev7%Fcfv7$Fjgu$!+L%fQC%FhinFgev7%Fgfv7$Fdgu$!+=nZJUFhinFgev7%F[gv7$F^g u$!+F#*p=UFhinFgev7%F_gv7$Fhfu$!+WPe0UFhinFgev7%Fcgv7$Fbfu$!+_:>#>%Fhi nFgev7%Fggv7$F\\fu$!+_$*eyTFhinFgev7%F[hv7$Ffeu$!+w.&[;%FhinFgev7%F_hv 7$F`eu$!+-d0^TFhinFgev7%Fchv7$Fjdu$!+keHPTFhinFgev7%Fghv7$Fddu$!+%ysO7 %FhinFgev7%F[iv7$F^du$!+C@I5TFhinFgev7%F_iv7$Fhcu$!+LhJ(4%FhinFgev7%Fc iv7$Fbcu$!+iq'[3%FhinFgev7%Fgiv7$F\\cu$!+978tSFhinFgev7%F[jv7$Ffbu$!+F OJiSFhinFgev7%F_jv7$F`bu$!+0Kl_SFhinFgev7%Fcjv7$Fjau$!+UyUWSFhinFgev7% Fgjv7$Fdau$!+9%ez.%FhinFgev7%F[[w7$F^au$!+*[4O.%FhinFgev7%F_[w7$Fh`u$! +SGyJSFhinFgev7%Fc[w7$Fb`u$!+Uw*G.%FhinFgev7%Fg[w7$F\\`u$!+z%\\t.%Fhin Fgev7%F[\\w7$Ff_u$!+')=WXSFhinFgev7%F_\\w7$F`_u$!+tSHdSFhinFgev7%Fc\\w 7$Fj^u$!+V\"\\F2%FhinFgev7%Fg\\w7$Fd^u$!+9&G84%FhinFgev7%F[]w7$F^^u$!+ 2VF7TFhinFgev7%F_]w7$Fh]u$!+YtnMTFhinFgev7%Fc]w7$Fb]u$!+'HNw:%FhinFgev 7%Fg]w7$F\\]u$!+tYP!=%FhinFgev7%F[^w7$Ff\\u$!+0iI-UFhinFgev7%F_^w7$F` \\u$!+SP-BUFhinFgev7%Fc^w7$Fj[u$!+`KFUUFhinFgev7%Fg^w7$Fd[u$!+'49*fUFh inFgev7%F[_w7$F^[u$!+OU)eF%FhinFgev7%F__w7$Fhjt$!+`LT NVFhinFgev7%F_cw7$Fhdt$!+-QbHVFhinFgev7%Fccw7$Fbdt$!+P3&GK%FhinFgev7%F gcw7$F\\dt$!+:ZM:VFhinFgev7%F[dw7$Fact$!+\\p22VFhinFgev7%F_dwFaevFgevF ^_sFe_s-F$6%7$7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF(-%'COLOURG6&F_fnF )F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(Fgdw7$F($\"3/++++++!z%FSF\\ewF_ew- %%FONTG6$%*HELVETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%% &Re(z)G%&Im(z)G-F[fw6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%F bfn$\"$*>Fbfn;Fcgw$\"$z%Fbfn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interv al" }{TEXT -1 37 " for this scheme is (approximately) " }{XPPEDIT 18 0 "[-4.1287, 0];" "6#7$,$-%&FloatG6$\"&(GT!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "T he following picture compares the stability region for the 7 stage, or der 4 scheme with that of the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),po||(1..6)],view=[-4. 79..1.99,-4.79..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`, `Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 623 575 575 {PLOTDATA 2 "65-%'CURVESG6$7]y7$$\"\"!F)F(7$$\"3 s******z7^)4'!#G$\"3#******fK'zq:!#=7$$\"3#)*****p$Gv,W!#E$\"3$******H f#fTJF07$$\"3G+++.0+sd!#D$\"3z*****\\&yQ7ZF07$$\"3z*******=-Ki$!#C$\"3 ?+++Rg<$G'F07$$\"3.+++GR&GV\"!#B$\"3G+++'oKR&yF07$$\"3o********p5xQFF$ \"3Z*****zm$eC%*F07$$\"3w******G'ePi'FF$\"35+++uW\\*4\"!#<7$$\"3/+++** 4')y7FF$\"3)********GikD\"FS7$$!3#******RJBu4%!#A$\"35++++!)G89FS7$$!3 ,+++-oRV>!#@$\"3)******f&y%)p:FS7$$!3U+++>BT!>'F\\o$\"3#******4cbfs\"F S7$$!3++++n)\\#=;!#?$\"33+++o!f8)=FS7$$!3$)******\\:&fq$Fgo$\"38+++y=v N?FS7$$!3o*****>$3(HR&Fgo$\"3!)*****\\7kC6#FS7$$!3E+++QL!Qn(Fgo$\"3=++ +k`z)=#FS7$$!3%******4e<*p5!#>$\"35+++I1qkAFS7$$!3)******4#=?k9F\\q$\" 3(******H]K,M#FS7$$!3#******4tA(p>F\\q$\"3#)*****HAQ]T#FS7$$!35+++a.03 EF\\q$\"3))*****p$\\N*[#FS7$$!3F+++B&zDS$F\\q$\"3?+++-G+jDFS7$$!3q**** **Hm0yVF\\q$\"3()*****4)f(ej#FS7$$!3'******Rlu(fbF\\q$\"37+++??$yq#FS7 $$!3i*****z=+B(pF\\q$\"3?+++o4oyFFS7$$!3Q+++54iP')F\\q$\"3/+++*yw\"[GF S7$$!3)******p&>Fd5F0$\"3A+++()[,;HFS7$$!3/+++9Boy7F0$\"3A+++1-%=)HFS7 $$!3&*******R-*y_\"F0$\"3?+++\"4p_/$FS7$$!32+++FSi.=F0$\"3(******\\8Cf 5$FS7$$!35+++S*zM5#F0$\"3A+++%RvM;$FS7$$!34+++*[eTU#F0$\"3!******40xw@ $FS7$$!3y******yT%=w#F0$\"3A+++5\"*QoKFS7$$!3x*****zQ@E6$F0$\"3-+++!fz bJ$FS7$$!3!******4fAGZ$F0$\"39+++J+JfLFS7$$!3))*****>!>DRQF0$\"3#)**** **e;r*R$FS7$$!3D+++&y!G4UF0$\"3!******H%)fpV$FS7$$!3-+++$eY3e%F0$\"37+ ++h>DrMFS7$$!3a+++`*)yA`F0$\"3z*****fgt<`$FS7$$!3y*****4I-s0'F0$\"3))* ****H:'z#e$FS7$$!3i******\\CX!y'F0$\"31+++#*>hDOFS7$$!3k*****4Bt7\\(F0 $\"3%******>$>EhOFS7$$!3q*****4J9'*=)F0$\"3#******>[m0p$FS7$$!3-+++yk* f())F0$\"3&******f))oTr$FS7$$!3a+++W0<^&*F0$\"3=++++UdKPFS7$$!32+++1Yf @5FS$\"35+++J1=YPFS7$$!33+++>76(3\"FS$\"3)******>[.`v$FS7$$!3'******>! 3u^6FS$\"3'******H9$>gPFS7$$!3/+++:>b:7FS$\"3/+++i&[5w$FS7$$!3-+++q)3' y7FS$\"33+++1u-ePFS7$$!3%******\\it4M\"FS$\"3'******R#QD^PFS7$$!3-+++) >3FS\"FS$\"37+++RX#3u$FS7$$!3#******RXxQY\"FS$\"3<+++3Q\"os$FS7$$!3%** ****zD_X_\"FS$\"3-+++RxF4PFS7$$!35+++uH\"[e\"FS$\"3-+++*Qe#)o$FS7$$!3' ******zK`Zk\"FS$\"3!******H5)yjOFS7$$!3*******fa%[/.KNFS7$$!3!******>PsV%>FS$\"3# )*****pk,4\\$FS7$$!3&)*****f3\"e0?FS$\"3#)*****ptCnW$FS7$$!3y******)*R vn?FS$\"3%)*****fUA(*R$FS7$$!3=+++R&>68#FS$\"3=+++qK?]LFS7$$!3')*****H ^`e>#FS$\"3%******HVp&)H$FS7$$!3/+++'fN?E#FS$\"39+++[ZIXKFS7$$!3\")*** **H\")4'HBFS$\"3#******4pQ4>$FS7$$!3#)*****\\nk$)R#FS$\"3\")*****p/*)f 8$FS7$$!39+++F^%zY#FS$\"33+++yv*33$FS7$$!38+++Qf*y`#FS$\"3/+++b$zf-$FS 7$$!3>+++(>Axg#FS$\"3y*****fA,9(HFS7$$!3')*****f1Zpn#FS$\"3%)*****4J\" >\"GFS$\"36+++ZY [4GFS7$$!3#******4AEs(GFS$\"35+++@#\\cv#FS7$$!3A+++z\"\\1%HFS$\"39+++j Vc,FFS7$$!39+++>89-IFS$\"3;+++:p.ZEFS7$$!3%******4(RihIFS$\"3#)*****zz ))=f#FS7$$!3$******\\[a!>JFS$\"3$)*****\\Ihf`#FS7$$!31+++qqTuJFS$\"3y* ****z?;\"zCFS7$$!3!******\\(erFKFS$\"3$******>3N7U#FS7$$!3?+++&**p*yKF S$\"3')*****f!y@iBFS7$$!3=+++T+@GLFS$\"3(******z$3)>I#FS7$$!31+++*)eZv LFS$\"3:+++EcXSAFS7$$!38+++o_\"3U$FS$\"3%)*****pL(ex@FS7$$!3=+++VHGkMF S$\"38+++WSL8@FS7$$!3#******\\?Sf]$FS$\"3)******pRmw/#FS7$$!37+++\"[ae a$FS$\"3%*******4yc!)>FS7$$!3y******R))4%e$FS$\"36+++J^.7>FS7$$!3#)*** **p<^2i$FS$\"3(******>tz?%=FS7$$!3#)*****p)G*el$FS$\"3)*******y)G2x\"F S7$$!35+++]og*o$FS$\"3/+++*=F!)p\"FS7$$!38+++/Y(>s$FS$\"3\"*******oy.C ;FS7$$!37+++0F2`PFS$\"35+++)QV)[:FS7$$!3A+++G%oHy$FS$\"3++++h^as9FS7$$ !3.+++N^r6QFS$\"3'******H!>E&R\"FS7$$!3$)*****>(yMRQFS$\"3*******4UErJ \"FS7$$!3!)*****>!)ze'QFS$\"33+++')3GQ7FS7$$!3%******48+8*QFS$\"3'**** **4\"4()e6FS7$$!3;+++bRd:RFS$\"3++++m!R!z5FS7$$!3<+++_TkQRFS$\"3=+++!* f=*)**F07$$!31+++#3N/'RFS$\"3k*****\\[\"G'=*F07$$!3%)*****pad3)RFS$\"3 9+++R)zEQ)F07$$!3'******HI9)**RFS$\"3'******4R!=zvF07$$!31+++ma?MnY(fF07$$!3'******>(e! p/%FS$\"3O+++*4yU<&F07$$!3))*****fF07$$!3!)*****>,@z3%FS$\"3'******>#))=\" >\"F07$$!3o*****4A9+4%FS$\"3')*****>./,(RF\\q7$Ff^m$!3')*****>./,(RF\\ q7$Fa^m$!3'******>#))=\">\"F07$F\\^m$!3))******QR$e)>F07$Fg]m$!3w***** *zeB\"y#F07$Fb]m$!3!)*****HP^wd$F07$F]]m$!37+++,oGvVF07$Fh\\m$!3O+++*4 yU<&F07$Fc\\m$!3g*****>MnY(fF07$F^\\m$!3g*****zdwjx'F07$Fi[m$!3'****** 4R!=zvF07$Fd[m$!39+++R)zEQ)F07$F_[m$!3k*****\\[\"G'=*F07$Fjjl$!3=+++!* f=*)**F07$Fejl$!3++++m!R!z5FS7$F`jl$!3'******4\"4()e6FS7$F[jl$!33+++') 3GQ7FS7$Ffil$!3*******4UErJ\"FS7$Fail$!3'******H!>E&R\"FS7$F\\il$!3+++ +h^as9FS7$Fghl$!35+++)QV)[:FS7$Fbhl$!3\"*******oy.C;FS7$F]hl$!3/+++*=F !)p\"FS7$Fhgl$!3)*******y)G2x\"FS7$Fcgl$!3(******>tz?%=FS7$F^gl$!36+++ J^.7>FS7$Fifl$!3%*******4yc!)>FS7$Fdfl$!3)******pRmw/#FS7$F_fl$!38+++W SL8@FS7$Fjel$!3%)*****pL(ex@FS7$Feel$!3:+++EcXSAFS7$F`el$!3(******z$3) >I#FS7$F[el$!3')*****f!y@iBFS7$Ffdl$!3$******>3N7U#FS7$Fadl$!3y*****z? ;\"zCFS7$F\\dl$!3$)*****\\Ihf`#FS7$Fgcl$!3#)*****zz))=f#FS7$Fbcl$!3;++ +:p.ZEFS7$F]cl$!39+++jVc,FFS7$Fhbl$!35+++@#\\cv#FS7$Fcbl$!36+++ZY[4GFS 7$F^bl$!3?+++R%pK'GFS7$Fial$!3%)*****4J\">$FS7$F[`l$!39+++[ZIXKFS7$Ff_l$!3%******HVp&)H$FS7$Fa _l$!3=+++qK?]LFS7$F\\_l$!3%)*****fUA(*R$FS7$Fg^l$!3#)*****ptCnW$FS7$Fb ^l$!3#)*****pk,4\\$FS7$F]^l$!3\")*****p*>.KNFS7$Fh]l$!3!******R9m*pNFS 7$Fc]l$!3%)*****pO2Yg$FS7$F^]l$!3!******\\f%*ej$FS7$Fi\\l$!3!******H5) yjOFS7$Fd\\l$!3-+++*Qe#)o$FS7$F_\\l$!3-+++RxF4PFS7$Fj[l$!3<+++3Q\"os$F S7$Fe[l$!37+++RX#3u$FS7$F`[l$!3'******R#QD^PFS7$F[[l$!33+++1u-ePFS7$Ff z$!3/+++i&[5w$FS7$Faz$!3'******H9$>gPFS7$F\\z$!3)******>[.`v$FS7$Fgy$! 35+++J1=YPFS7$Fby$!3=++++UdKPFS7$F]y$!3&******f))oTr$FS7$Fhx$!3#****** >[m0p$FS7$Fcx$!3%******>$>EhOFS7$F^x$!31+++#*>hDOFS7$Fiw$!3))*****H:'z #e$FS7$Fdw$!3z*****fgt<`$FS7$$!3A+++tMR_\\F0$!33+++;3z-NFS7$F_w$!37+++ h>DrMFS7$Fjv$!3!******H%)fpV$FS7$Fev$!3#)******e;r*R$FS7$F`v$!39+++J+J fLFS7$F[v$!3-+++!fzbJ$FS7$Ffu$!3A+++5\"*QoKFS7$Fau$!3!******40xw@$FS7$ F\\u$!3A+++%RvM;$FS7$Fgt$!3(******\\8Cf5$FS7$Fbt$!3?+++\"4p_/$FS7$F]t$ !3A+++1-%=)HFS7$Fhs$!3A+++()[,;HFS7$Fcs$!3/+++*yw\"[GFS7$F^s$!3?+++o4o yFFS7$Fir$!37+++??$yq#FS7$Fdr$!3()*****4)f(ej#FS7$F_r$!3?+++-G+jDFS7$F jq$!3))*****p$\\N*[#FS7$Feq$!3#)*****HAQ]T#FS7$F`q$!3(******H]K,M#FS7$ Fjp$!35+++I1qkAFS7$Fep$!3=+++k`z)=#FS7$F`p$!3!)*****\\7kC6#FS7$$!3++++ (RyP[#Fgo$!3#*******)\\-(e>FS7$$!35+++`QZ?5Fgo$!3/+++aMw.=FS7$$!33+++E [SzNF\\o$!3'******f\\szk\"FS7$$!3X*****pi,jj*Ffn$!33+++P7h\"\\\"FS7$$! 32+++YZK07Ffn$!3/+++&H**[L\"FS7$$\"3m*****pYk:1'FF$!3%******Ha!*z<\"FS 7$$\"3#******zB$))eaFF$!3,+++o=)4-\"FS7$$\"3;+++;z3rCFF$!3&)*****ROz#R ')F07$$\"3i*****>809d(F@$!3W+++N7coqF07$$\"3++++/Fa[:F@$!3!******pn$y( \\&F07$$\"3++++ND1.=F:$!3:+++;0*p#RF07$$\"3)*******G6uKt!#F$!3)******> [%>cBF07$F($!3))*****Rj\")R&yF\\q7$F($\"3))*****Rj\")R&yF\\q-%&COLORG6 &%$RGBG$\"#Q!\"#$\"#=F^fnF_fn-%)POLYGONSG6^y7%F'7$$\"+!G6&)4'Fgo$\"+Ej zq:!#57$$F^fnF)F(7%Fefn7$$\"+PGv,WF0$\"+$f#fTJFjfnF[gn7%F^gn7$$\"+.0+s dFS$\"+byQ7ZFjfnF[gn7%Fdgn7$$\"+!>-Ki$!#;$\"+Rg<$G'FjfnF[gn7%Fjgn7$$\" +GR&GV\"!#:$\"+'oKR&yFjfnF[gn7%Fahn7$$\"++q5xQFdhn$\"+oOeC%*FjfnF[gn7% Fhhn7$$\"+H'ePi'Fdhn$\"+uW\\*4\"!\"*F[gn7%F^in7$$\"+**4')y7Fdhn$\"+!Hi kD\"FcinF[gn7%Fein7$$!+9LU(4%!#9$\"++!)G89FcinF[gn7%F[jn7$$!+-oRV>!#8$ \"+cy%)p:FcinF[gn7%Fbjn7$$!+>BT!>'Fejn$\"+hb&fs\"FcinF[gn7%Fijn7$$!+n) \\#=;!#7$\"+o!f8)=FcinF[gn7%F_[o7$$!+]:&fq$Fb[o$\"+y=vN?FcinF[gn7%Ff[o 7$$!+K3(HR&Fb[o$\"+DTY7@FcinF[gn7%F\\\\o7$$!+QL!Qn(Fb[o$\"+k`z)=#FcinF [gn7%Fb\\o7$$!+\"e<*p5!#6$\"+I1qkAFcinF[gn7%Fh\\o7$$!+@=?k9F[]o$\"+.D8 SBFcinF[gn7%F_]o7$$!+JFsp>F[]o$\"+B#Q]T#FcinF[gn7%Fe]o7$$!+a.03EF[]o$ \"+P\\N*[#FcinF[gn7%F[^o7$$!+B&zDS$F[]o$\"+-G+jDFcinF[gn7%Fa^o7$$!+Im0 yVF[]o$\"+\")f(ej#FcinF[gn7%Fg^o7$$!+aYxfbF[]o$\"+??$yq#FcinF[gn7%F]_o 7$$!+)=+B(pF[]o$\"+o4oyFFcinF[gn7%Fc_o7$$!+54iP')F[]o$\"+*yw\"[GFcinF[ gn7%Fi_o7$$!+d>Fd5Fjfn$\"+()[,;HFcinF[gn7%F_`o7$$!+9Boy7Fjfn$\"+1-%=)H FcinF[gn7%Fe`o7$$!+S-*y_\"Fjfn$\"+\"4p_/$FcinF[gn7%F[ao7$$!+FSi.=Fjfn$ \"+NT#f5$FcinF[gn7%Faao7$$!+S*zM5#Fjfn$\"+%RvM;$FcinF[gn7%Fgao7$$!+*[e TU#Fjfn$\"+^qnDRQFjfn$\"+f;r*R$FcinF[gn7%Feco7$$!+&y!G4UFjfn$\"+V)fpV $FcinF[gn7%F[do7$$!+$eY3e%Fjfn$\"+h>DrMFcinF[gn7%Fado7$$!+`*)yA`Fjfn$ \"+1OxJNFcinF[gn7%Fgdo7$$!+,B?dgFjfn$\"+`hz#e$FcinF[gn7%F]eo7$$!+]CX!y 'Fjfn$\"+#*>hDOFcinF[gn7%Fceo7$$!+JKF\"\\(Fjfn$\"+K>EhOFcinF[gn7%Fieo7 $$!+6Vh*=)Fjfn$\"+#[m0p$FcinF[gn7%F_fo7$$!+yk*f())Fjfn$\"+'))oTr$FcinF [gn7%Fefo7$$!+W0<^&*Fjfn$\"++UdKPFcinF[gn7%F[go7$$!+1Yf@5Fcin$\"+J1=YP FcinF[gn7%Fago7$$!+>76(3\"Fcin$\"+#[.`v$FcinF[gn7%Fggo7$$!+-3u^6Fcin$ \"+VJ>gPFcinF[gn7%F]ho7$$!+:>b:7Fcin$\"+i&[5w$FcinF[gn7%Fcho7$$!+q)3'y 7Fcin$\"+1u-ePFcinF[gn7%Fiho7$$!+DO(4M\"Fcin$\"+CQD^PFcinF[gn7%F_io7$$ !+)>3FS\"Fcin$\"+RX#3u$FcinF[gn7%Feio7$$!+au(QY\"Fcin$\"+3Q\"os$FcinF[ gn7%F[jo7$$!+eAbC:Fcin$\"+RxF4PFcinF[gn7%Fajo7$$!+uH\"[e\"Fcin$\"+*Qe# )o$FcinF[gn7%Fgjo7$$!+GLvW;Fcin$\"+.\")yjOFcinF[gn7%F][p7$$!+YX[/.KNFcinF[gn7%F e\\p7$$!+sBPW>Fcin$\"+Z;!4\\$FcinF[gn7%F[]p7$$!+'3\"e0?Fcin$\"+PZsYMFc inF[gn7%Fa]p7$$!+**Rvn?Fcin$\"+ECs*R$FcinF[gn7%Fg]p7$$!+R&>68#Fcin$\"+ qK?]LFcinF[gn7%F]^p7$$!+8N&e>#Fcin$\"+L%p&)H$FcinF[gn7%Fc^p7$$!+'fN?E# Fcin$\"+[ZIXKFcinF[gn7%Fi^p7$$!+8)4'HBFcin$\"+\"pQ4>$FcinF[gn7%F__p7$$ !+vYO)R#Fcin$\"+Z!*)f8$FcinF[gn7%Fe_p7$$!+F^%zY#Fcin$\"+yv*33$FcinF[gn 7%F[`p7$$!+Qf*y`#Fcin$\"+b$zf-$FcinF[gn7%Fa`p7$$!+(>Axg#Fcin$\"+E7SrHF cinF[gn7%Fg`p7$$!+mq%pn#Fcin$\"+68>\"GFcin$\"+ZY[4GFcinF[gn7%Fiap7$$!+@iAxGF cin$\"+@#\\cv#FcinF[gn7%F_bp7$$!+z\"\\1%HFcin$\"+jVc,FFcinF[gn7%Febp7$ $!+>89-IFcin$\"+:p.ZEFcinF[gn7%F[cp7$$!+rRihIFcin$\"+)z))=f#FcinF[gn7% Facp7$$!+&[a!>JFcin$\"+08'f`#FcinF[gn7%Fgcp7$$!+qqTuJFcin$\"+3i6zCFcin F[gn7%F]dp7$$!+verFKFcin$\"+#3N7U#FcinF[gn7%Fcdp7$$!+&**p*yKFcin$\"+1y @iBFcinF[gn7%Fidp7$$!+T+@GLFcin$\"+Q3)>I#FcinF[gn7%F_ep7$$!+*)eZvLFcin $\"+EcXSAFcinF[gn7%Feep7$$!+o_\"3U$Fcin$\"+Ptex@FcinF[gn7%F[fp7$$!+VHG kMFcin$\"+WSL8@FcinF[gn7%Fafp7$$!+0-%f]$Fcin$\"+(Rmw/#FcinF[gn7%Fgfp7$ $!+\"[aea$Fcin$\"+5yc!)>FcinF[gn7%F]gp7$$!+S))4%e$Fcin$\"+J^.7>FcinF[g n7%Fcgp7$$!+x6v?OFcin$\"+K(z?%=FcinF[gn7%Figp7$$!+()G*el$Fcin$\"+z)G2x \"FcinF[gn7%F_hp7$$!+]og*o$Fcin$\"+*=F!)p\"FcinF[gn7%Fehp7$$!+/Y(>s$Fc in$\"+py.C;FcinF[gn7%F[ip7$$!+0F2`PFcin$\"+)QV)[:FcinF[gn7%Faip7$$!+G% oHy$Fcin$\"+h^as9FcinF[gn7%Fgip7$$!+N^r6QFcin$\"+.>E&R\"FcinF[gn7%F]jp 7$$!+syMRQFcin$\"+@k7<8FcinF[gn7%Fcjp7$$!+-)ze'QFcin$\"+')3GQ7FcinF[gn 7%Fijp7$$!+J,I\"*QFcin$\"+64()e6FcinF[gn7%F_[q7$$!+bRd:RFcin$\"+m!R!z5 FcinF[gn7%Fe[q7$$!+_TkQRFcin$\"+!*f=*)**FjfnF[gn7%F[\\q7$$!+#3N/'RFcin $\"+&[\"G'=*FjfnF[gn7%Fa\\q7$$!+Zv&3)RFcin$\"+R)zEQ)FjfnF[gn7%Fg\\q7$$ !+.V\")**RFcin$\"+\"R!=zvFjfnF[gn7%F]]q7$$!+ma?FjfnF[gn7%Fg_q7$$!+75#z3%Fcin$\"+A))=\">\"FjfnF[gn7%F]`q 7$$!+@U,!4%Fcin$\"+KS5qRF[]oF[gn7%Fc`q7$Fd`q$!+KS5qRF[]oF[gn7%Fi`q7$F^ `q$!+A))=\">\"FjfnF[gn7%F]aq7$Fh_q$!+RR$e)>FjfnF[gn7%Faaq7$Fb_q$!+!)eB \"y#FjfnF[gn7%Feaq7$F\\_q$!+t8lxNFjfnF[gn7%Fiaq7$Ff^q$!+,oGvVFjfnF[gn7 %F]bq7$F`^q$!+*4yU<&FjfnF[gn7%Fabq7$Fj]q$!+UtmufFjfnF[gn7%Febq7$Fd]q$! +ylPwnFjfnF[gn7%Fibq7$F^]q$!+\"R!=zvFjfnF[gn7%F]cq7$Fh\\q$!+R)zEQ)Fjfn F[gn7%Facq7$Fb\\q$!+&[\"G'=*FjfnF[gn7%Fecq7$F\\\\q$!+!*f=*)**FjfnF[gn7 %Ficq7$Ff[q$!+m!R!z5FcinF[gn7%F]dq7$F`[q$!+64()e6FcinF[gn7%Fadq7$Fjjp$ !+')3GQ7FcinF[gn7%Fedq7$Fdjp$!+@k7<8FcinF[gn7%Fidq7$F^jp$!+.>E&R\"Fcin F[gn7%F]eq7$Fhip$!+h^as9FcinF[gn7%Faeq7$Fbip$!+)QV)[:FcinF[gn7%Feeq7$F \\ip$!+py.C;FcinF[gn7%Fieq7$Ffhp$!+*=F!)p\"FcinF[gn7%F]fq7$F`hp$!+z)G2 x\"FcinF[gn7%Fafq7$Fjgp$!+K(z?%=FcinF[gn7%Fefq7$Fdgp$!+J^.7>FcinF[gn7% Fifq7$F^gp$!+5yc!)>FcinF[gn7%F]gq7$Fhfp$!+(Rmw/#FcinF[gn7%Fagq7$Fbfp$! +WSL8@FcinF[gn7%Fegq7$F\\fp$!+Ptex@FcinF[gn7%Figq7$Ffep$!+EcXSAFcinF[g n7%F]hq7$F`ep$!+Q3)>I#FcinF[gn7%Fahq7$Fjdp$!+1y@iBFcinF[gn7%Fehq7$Fddp $!+#3N7U#FcinF[gn7%Fihq7$F^dp$!+3i6zCFcinF[gn7%F]iq7$Fhcp$!+08'f`#Fcin F[gn7%Faiq7$Fbcp$!+)z))=f#FcinF[gn7%Feiq7$F\\cp$!+:p.ZEFcinF[gn7%Fiiq7 $Ffbp$!+jVc,FFcinF[gn7%F]jq7$F`bp$!+@#\\cv#FcinF[gn7%Fajq7$Fjap$!+ZY[4 GFcinF[gn7%Fejq7$Fdap$!+R%pK'GFcinF[gn7%Fijq7$F^ap$!+68>$Fcin F[gn7%Fa\\r7$Fj^p$!+[ZIXKFcinF[gn7%Fe\\r7$Fd^p$!+L%p&)H$FcinF[gn7%Fi\\ r7$F^^p$!+qK?]LFcinF[gn7%F]]r7$Fh]p$!+ECs*R$FcinF[gn7%Fa]r7$Fb]p$!+PZs YMFcinF[gn7%Fe]r7$F\\]p$!+Z;!4\\$FcinF[gn7%Fi]r7$Ff\\p$!+(*>.KNFcinF[g n7%F]^r7$F`\\p$!+Wh'*pNFcinF[gn7%Fa^r7$Fj[p$!+ntg/OFcinF[gn7%Fe^r7$Fd[ p$!+&f%*ej$FcinF[gn7%Fi^r7$F^[p$!+.\")yjOFcinF[gn7%F]_r7$Fhjo$!+*Qe#)o $FcinF[gn7%Fa_r7$Fbjo$!+RxF4PFcinF[gn7%Fe_r7$F\\jo$!+3Q\"os$FcinF[gn7% Fi_r7$Ffio$!+RX#3u$FcinF[gn7%F]`r7$F`io$!+CQD^PFcinF[gn7%Fa`r7$Fjho$!+ 1u-ePFcinF[gn7%Fe`r7$Fdho$!+i&[5w$FcinF[gn7%Fi`r7$F^ho$!+VJ>gPFcinF[gn 7%F]ar7$Fhgo$!+#[.`v$FcinF[gn7%Faar7$Fbgo$!+J1=YPFcinF[gn7%Fear7$F\\go $!++UdKPFcinF[gn7%Fiar7$Fffo$!+'))oTr$FcinF[gn7%F]br7$F`fo$!+#[m0p$Fci nF[gn7%Fabr7$Fjeo$!+K>EhOFcinF[gn7%Febr7$Fdeo$!+#*>hDOFcinF[gn7%Fibr7$ F^eo$!+`hz#e$FcinF[gn7%F]cr7$Fhdo$!+1OxJNFcinF[gn7%Facr7$$!+tMR_\\Fjfn $!+;3z-NFcinF[gn7%Fecr7$Fbdo$!+h>DrMFcinF[gn7%F[dr7$F\\do$!+V)fpV$Fcin F[gn7%F_dr7$Ffco$!+f;r*R$FcinF[gn7%Fcdr7$F`co$!+J+JfLFcinF[gn7%Fgdr7$F jbo$!+!fzbJ$FcinF[gn7%F[er7$Fdbo$!+5\"*QoKFcinF[gn7%F_er7$F^bo$!+^qnFcinF[gn7%Fgir7$$!+`QZ?5Fb[o$!+aMw.=FcinF[gn7%F]jr7$ $!+E[SzNFejn$!+'\\szk\"FcinF[gn7%Fcjr7$$!+F;IO'*F^jn$!+P7h\"\\\"FcinF[ gn7%Fijr7$$!+YZK07F^jn$!+&H**[L\"FcinF[gn7%F_[s7$$\"+nWchgFdhn$!+V0*z< \"FcinF[gn7%Fe[s7$$\"+QK))eaFdhn$!+o=)4-\"FcinF[gn7%F[\\s7$$\"+;z3rCFd hn$!+k$z#R')FjfnF[gn7%Fa\\s7$$\"+K^SrvF]hn$!+N7coqFjfnF[gn7%Fg\\s7$$\" +/Fa[:F]hn$!+xOy(\\&FjfnF[gn7%F]]s7$$\"+ND1.=FS$!+;0*p#RFjfnF[gn7%Fc]s 7$$\"+H6uKtF\\q$!+#[%>cBFjfnF[gn7%Fi]s7$F($!+M;)R&yF[]oF[gn7%F_^s7$F($ \"+M;)R&yF[]oF[gn-Fien6&F[fn$\"#vF^fn$\"#NF^fnFj^s-%&STYLEG6#%,PATCHNO GRIDG-F$6$7in7$$\"3,+++4OyN9FS$\"3z*****p`oTI%FS7$$\"3/+++0*\\sU\"FS$ \"3.+++xqG8VFS7$$\"3-+++W!3!=9FS$\"3')*****zz&o@VFS7$$\"35+++?8539FS$ \"3#)*****\\.B$HVFS7$$\"3-+++fNd(R\"FS$\"3Q+++^s:OVFS7$$\"3-+++\\%pkQ \"FS$\"3?+++Pn9UVFS7$$\"3!*******4\\$[P\"FS$\"3U+++?$\\sM%FS7$$\"3'*** ***z^*p;]8FS$\"37+++d5iaVFS7$$ \"32+++pdBP8FS$\"3g*****f*4!oN%FS7$$\"3.+++r'zRK\"FS$\"3)******40:zN%F S7$$\"3*******Hse/J\"FS$\"3++++OX\"zN%FS7$$\"3!******H7QnH\"FS$\"3#*** ****z'[nN%FS7$$\"3))*****RV*)GG\"FS$\"3S+++IWOaVFS7$$\"3)******R/#**o7 FS$\"3u*****>K12N%FS7$$\"3'******f'\\8b7FS$\"3')*****pZ;dM%FS7$$\"3/++ +v!>9C\"FS$\"3T+++*zM$RVFS7$$\"3*)*****f()fzA\"FS$\"3<+++'\\*\\JVFS7$$ \"3(******R%4*[@\"FS$\"3y*****H8[@K%FS7$$\"3-+++g\"oB?\"FS$\"3#******* 4&>7J%FS7$$\"31+++a[d!>\"FS$\"3k******Hol)H%FS7$$\"3-+++\")ysz6FS$\"3! )*****4(GT%G%FS7$$\"3#******\\V%3q6FS$\"3g*******=e%oUFS7$$\"3)******p !*[>;\"FS$\"3!)*****Rz$z]UFS7$$\"3'******>9yc:\"FS$\"32+++5,ZJUFS7$$\" 3)******Rr\"o^6FS$\"3?+++#391@%FS7$$\"33+++77T]6FS$\"3E+++p]Y)=%FS7$$ \"31+++9&HB:\"FS$\"3q*****4V;a;%FS7$$\"3!******f2]y:\"FS$\"3s*****Rg^? 9%FS7$$\"3#******H&HCn6FS$\"3L+++o=:>TFS7$$\"3++++MV_!=\"FS$\"3/+++&3Y w4%FS7$$\"3!******4$\\P(>\"FS$\"3K+++Lr[ySFS7$$\"3(*******QE9<7FS$\"3A +++GJ[iSFS7$$\"3\"*******p$[*Q7FS$\"3w*****\\ba,0%FS7$$\"3/+++d,&=E\"F S$\"3U+++O=oTSFS7$$\"3/+++)[!*\\G\"FS$\"36+++P-&p.%FS7$$\"3.+++$fzwI\" FS$\"3!)*****H^Zc.%FS7$$\"31+++#=9%H8FS$\"3W+++TKOPSFS7$$\"3'******\\V c)\\8FS$\"35+++#Ri;/%FS7$$\"3'******z=)zo8FS$\"3s*****>)*G\"[SFS7$$\"3 %******4ODhQ\"FS$\"3'******fz'QcSFS7$$\"3/+++t()y,9FS$\"3/+++xe5mSFS7$ $\"3*******HN#y:9FS$\"3-+++]9+xSFS7$$\"32+++sx7G9FS$\"3;+++z%H))3%FS7$ $\"32+++CR')Q9FS$\"3H+++L6Q,TFS7$$\"3)******f!*R![9FS$\"33+++8tZ9TFS7$ $\"3!******zR5dX\"FS$\"3o*****>rjz7%FS7$$\"3)******fnK>Y\"FS$\"3\")*** **fd1<9%FS7$$\"3#******frknY\"FS$\"3S+++M\"*ebTFS7$$\"35+++DSEq9FS$\"3 ?+++_'3&pTFS7$$\"3++++`p[s9FS$\"3O+++3SP$=%FS7$$\"3-+++E$)[t9FS$\"3S++ +*o.r>%FS7$$\"3-+++y7Kt9FS$\"3)*******eTi5UFS7$$\"33+++Ms.s9FS$\"3U+++ %\\oQA%FS7$$\"33+++ofop9FS$\"3-+++h_xOUFS7$$\"35+++GdJm9FS$\"33+++svG \\UFS7$$\"31+++$Rt>Y\"FS$\"3o*****4H_8E%FS7$$\"31+++hYqc9FS$\"3m*****> O>HF%FS7$$\"3'******\\Ca0X\"FS$\"3#*******=7%RG%FS7$$\"3(******>4mNW\" FS$\"3#******zGsVH%FSFc_sFhen-Fbfn6jn7%7$$\"+4OyN9Fcin$\"+P&oTI%Fcin7$ $\"+0*\\sU\"Fcin$\"+xqG8VFcin7$$\"$J\"F^fn$\"$>%F^fn7%Fgbt7$$\"+W!3!=9 Fcin$\"+)z&o@VFcinF\\ct7%Fbct7$$\"+?8539Fcin$\"+NIKHVFcinF\\ct7%Fhct7$ $\"+fNd(R\"Fcin$\"+^s:OVFcinF\\ct7%F^dt7$$\"+\\%pkQ\"Fcin$\"+Pn9UVFcin F\\ct7%Fddt7$$\"+5\\$[P\"Fcin$\"+?$\\sM%FcinF\\ct7%Fjdt7$$\"+=vri8Fcin $\"+x?U^VFcinF\\ct7%F`et7$$\"+#*p;]8Fcin$\"+d5iaVFcinF\\ct7%Ffet7$$\"+ pdBP8Fcin$\"+'*4!oN%FcinF\\ct7%F\\ft7$$\"+r'zRK\"Fcin$\"+^]\"zN%FcinF \\ct7%Fbft7$$\"+B(e/J\"Fcin$\"+OX\"zN%FcinF\\ct7%Fhft7$$\"+B\"QnH\"Fci n$\"+!o[nN%FcinF\\ct7%F^gt7$$\"+M%*)GG\"Fcin$\"+IWOaVFcinF\\ct7%Fdgt7$ $\"+W?**o7Fcin$\"+Ajq]VFcinF\\ct7%Fjgt7$$\"+m\\8b7Fcin$\"+xkrXVFcinF\\ ct7%F`ht7$$\"+v!>9C\"Fcin$\"+*zM$RVFcinF\\ct7%Ffht7$$\"+w)fzA\"Fcin$\" +'\\*\\JVFcinF\\ct7%F\\it7$$\"+W4*[@\"Fcin$\"+L\"[@K%FcinF\\ct7%Fbit7$ $\"+g\"oB?\"Fcin$\"+5&>7J%FcinF\\ct7%Fhit7$$\"+a[d!>\"Fcin$\"+Iol)H%Fc inF\\ct7%F^jt7$$\"+\")ysz6Fcin$\"+rGT%G%FcinF\\ct7%Fdjt7$$\"+NW3q6Fcin $\"+!>e%oUFcinF\\ct7%Fjjt7$$\"+2*[>;\"Fcin$\"+%z$z]UFcinF\\ct7%F`[u7$$ \"+U\"yc:\"Fcin$\"+5,ZJUFcinF\\ct7%Ff[u7$$\"+9TFcinF\\ct7%Fd]u7$$\"+MV_!=\"Fcin$\"+&3Yw4%FcinF\\ ct7%Fj]u7$$\"+J\\P(>\"Fcin$\"+Lr[ySFcinF\\ct7%F`^u7$$\"+RE9<7Fcin$\"+G J[iSFcinF\\ct7%Ff^u7$$\"+q$[*Q7Fcin$\"+bX:]SFcinF\\ct7%F\\_u7$$\"+d,&= E\"Fcin$\"+O=oTSFcinF\\ct7%Fb_u7$$\"+)[!*\\G\"Fcin$\"+P-&p.%FcinF\\ct7 %Fh_u7$$\"+$fzwI\"Fcin$\"+8vkNSFcinF\\ct7%F^`u7$$\"+#=9%H8Fcin$\"+TKOP SFcinF\\ct7%Fd`u7$$\"+Nk&)\\8Fcin$\"+#Ri;/%FcinF\\ct7%Fj`u7$$\"+)=)zo8 Fcin$\"+#)*G\"[SFcinF\\ct7%F`au7$$\"+h`7'Q\"Fcin$\"+'z'QcSFcinF\\ct7%F fau7$$\"+t()y,9Fcin$\"+xe5mSFcinF\\ct7%F\\bu7$$\"+`By:9Fcin$\"+]9+xSFc inF\\ct7%Fbbu7$$\"+sx7G9Fcin$\"+z%H))3%FcinF\\ct7%Fhbu7$$\"+CR')Q9Fcin $\"+L6Q,TFcinF\\ct7%F^cu7$$\"+1*R![9Fcin$\"+8tZ9TFcinF\\ct7%Fdcu7$$\"+ )R5dX\"Fcin$\"+7P'z7%FcinF\\ct7%Fjcu7$$\"+wE$>Y\"Fcin$\"+wlqTTFcinF\\c t7%F`du7$$\"+;Zwm9Fcin$\"+M\"*ebTFcinF\\ct7%Ffdu7$$\"+DSEq9Fcin$\"+_'3 &pTFcinF\\ct7%F\\eu7$$\"+`p[s9Fcin$\"+3SP$=%FcinF\\ct7%Fbeu7$$\"+E$)[t 9Fcin$\"+*o.r>%FcinF\\ct7%Fheu7$$\"+y7Kt9Fcin$\"+fTi5UFcinF\\ct7%F^fu7 $$\"+Ms.s9Fcin$\"+%\\oQA%FcinF\\ct7%Fdfu7$$\"+ofop9Fcin$\"+h_xOUFcinF \\ct7%Fjfu7$$\"+GdJm9Fcin$\"+svG\\UFcinF\\ct7%F`gu7$$\"+$Rt>Y\"Fcin$\" +\"H_8E%FcinF\\ct7%Ffgu7$$\"+hYqc9Fcin$\"+i$>HF%FcinF\\ct7%F\\hu7$$\"+ XUb]9Fcin$\"+>7%RG%FcinF\\ct7%Fbhu7$$\"+#4mNW\"Fcin$\"+)GsVH%FcinF\\ct 7%FhhuFbbtF\\ctFf^sF\\_s-F$6$7in7$Fd_s$!3z*****p`oTI%FS7$F[bt$!3#***** *zGsVH%FS7$Ffat$!3#*******=7%RG%FS7$Faat$!3m*****>O>HF%FS7$F\\at$!3o** ***4H_8E%FS7$Fg`t$!33+++svG\\UFS7$Fb`t$!3-+++h_xOUFS7$F]`t$!3U+++%\\oQ A%FS7$Fh_t$!3)*******eTi5UFS7$Fc_t$!3S+++*o.r>%FS7$F^_t$!3O+++3SP$=%FS 7$Fi^t$!3?+++_'3&pTFS7$Fd^t$!3S+++M\"*ebTFS7$F_^t$!3\")*****fd1<9%FS7$ Fj]t$!3o*****>rjz7%FS7$Fe]t$!33+++8tZ9TFS7$F`]t$!3H+++L6Q,TFS7$F[]t$!3 ;+++z%H))3%FS7$Ff\\t$!3-+++]9+xSFS7$Fa\\t$!3/+++xe5mSFS7$F\\\\t$!3'*** ***fz'QcSFS7$Fg[t$!3s*****>)*G\"[SFS7$Fb[t$!35+++#Ri;/%FS7$F][t$!3W+++ TKOPSFS7$Fhjs$!3!)*****H^Zc.%FS7$Fcjs$!36+++P-&p.%FS7$F^js$!3U+++O=oTS FS7$Fiis$!3w*****\\ba,0%FS7$Fdis$!3A+++GJ[iSFS7$F_is$!3K+++Lr[ySFS7$Fj hs$!3/+++&3Yw4%FS7$Fehs$!3L+++o=:>TFS7$F`hs$!3s*****Rg^?9%FS7$F[hs$!3q *****4V;a;%FS7$Ffgs$!3E+++p]Y)=%FS7$Fags$!3?+++#391@%FS7$F\\gs$!32+++5 ,ZJUFS7$Fgfs$!3!)*****Rz$z]UFS7$Fbfs$!3g*******=e%oUFS7$F]fs$!3!)***** 4(GT%G%FS7$Fhes$!3k******Hol)H%FS7$Fces$!3#*******4&>7J%FS7$F^es$!3y** ***H8[@K%FS7$Fids$!3<+++'\\*\\JVFS7$Fdds$!3T+++*zM$RVFS7$F_ds$!3')**** *pZ;dM%FS7$Fjcs$!3u*****>K12N%FS7$Fecs$!3S+++IWOaVFS7$F`cs$!3#*******z '[nN%FS7$F[cs$!3++++OX\"zN%FS7$Ffbs$!3)******40:zN%FS7$Fabs$!3g*****f* 4!oN%FS7$F\\bs$!37+++d5iaVFS7$Fgas$!3g*****p2A9N%FS7$Fbas$!3U+++?$\\sM %FS7$F]as$!3?+++Pn9UVFS7$Fh`s$!3Q+++^s:OVFS7$Fc`s$!3#)*****\\.B$HVFS7$ F^`s$!3')*****zz&o@VFS7$Fi_s$!3.+++xqG8VFSFaiuFhen-Fbfn6jn7%7$Fcbt$!+P &oTI%Fcin7$Fihu$!+)GsVH%Fcin7$F]ct$!$>%F^fn7%F[ev7$Fchu$!+>7%RG%FcinF^ ev7%Fbev7$F]hu$!+i$>HF%FcinF^ev7%Ffev7$Fggu$!+\"H_8E%FcinF^ev7%Fjev7$F agu$!+svG\\UFcinF^ev7%F^fv7$F[gu$!+h_xOUFcinF^ev7%Fbfv7$Fefu$!+%\\oQA% FcinF^ev7%Fffv7$F_fu$!+fTi5UFcinF^ev7%Fjfv7$Fieu$!+*o.r>%FcinF^ev7%F^g v7$Fceu$!+3SP$=%FcinF^ev7%Fbgv7$F]eu$!+_'3&pTFcinF^ev7%Ffgv7$Fgdu$!+M \"*ebTFcinF^ev7%Fjgv7$Fadu$!+wlqTTFcinF^ev7%F^hv7$F[du$!+7P'z7%FcinF^e v7%Fbhv7$Fecu$!+8tZ9TFcinF^ev7%Ffhv7$F_cu$!+L6Q,TFcinF^ev7%Fjhv7$Fibu$ !+z%H))3%FcinF^ev7%F^iv7$Fcbu$!+]9+xSFcinF^ev7%Fbiv7$F]bu$!+xe5mSFcinF ^ev7%Ffiv7$Fgau$!+'z'QcSFcinF^ev7%Fjiv7$Faau$!+#)*G\"[SFcinF^ev7%F^jv7 $F[au$!+#Ri;/%FcinF^ev7%Fbjv7$Fe`u$!+TKOPSFcinF^ev7%Ffjv7$F_`u$!+8vkNS FcinF^ev7%Fjjv7$Fi_u$!+P-&p.%FcinF^ev7%F^[w7$Fc_u$!+O=oTSFcinF^ev7%Fb[ w7$F]_u$!+bX:]SFcinF^ev7%Ff[w7$Fg^u$!+GJ[iSFcinF^ev7%Fj[w7$Fa^u$!+Lr[y SFcinF^ev7%F^\\w7$F[^u$!+&3Yw4%FcinF^ev7%Fb\\w7$Fe]u$!+o=:>TFcinF^ev7% Ff\\w7$F_]u$!+/;0UTFcinF^ev7%Fj\\w7$Fi\\u$!+JkTlTFcinF^ev7%F^]w7$Fc\\u $!+p]Y)=%FcinF^ev7%Fb]w7$F]\\u$!+#391@%FcinF^ev7%Ff]w7$Fg[u$!+5,ZJUFci nF^ev7%Fj]w7$Fa[u$!+%z$z]UFcinF^ev7%F^^w7$F[[u$!+!>e%oUFcinF^ev7%Fb^w7 $Fejt$!+rGT%G%FcinF^ev7%Ff^w7$F_jt$!+Iol)H%FcinF^ev7%Fj^w7$Fiit$!+5&>7 J%FcinF^ev7%F^_w7$Fcit$!+L\"[@K%FcinF^ev7%Fb_w7$F]it$!+'\\*\\JVFcinF^e v7%Ff_w7$Fght$!+*zM$RVFcinF^ev7%Fj_w7$Faht$!+xkrXVFcinF^ev7%F^`w7$F[ht $!+Ajq]VFcinF^ev7%Fb`w7$Fegt$!+IWOaVFcinF^ev7%Ff`w7$F_gt$!+!o[nN%FcinF ^ev7%Fj`w7$Fift$!+OX\"zN%FcinF^ev7%F^aw7$Fcft$!+^]\"zN%FcinF^ev7%Fbaw7 $F]ft$!+'*4!oN%FcinF^ev7%Ffaw7$Fget$!+d5iaVFcinF^ev7%Fjaw7$Faet$!+x?U^ VFcinF^ev7%F^bw7$F[et$!+?$\\sM%FcinF^ev7%Fbbw7$Fedt$!+Pn9UVFcinF^ev7%F fbw7$F_dt$!+^s:OVFcinF^ev7%Fjbw7$Fict$!+NIKHVFcinF^ev7%F^cw7$Fcct$!+)z &o@VFcinF^ev7%Fbcw7$Fhbt$!+xqG8VFcinF^ev7%FfcwFhdvF^evFf^sF\\_s-F$6%7$ 7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF(-%'COLOURG6&F[fnF)F)F)-%*LINEST YLEG6#\"\"$-F$6%7$7$F(F^dw7$F($\"3/++++++!z%FSFcdwFfdw-F$6$7]yF'7$$\"3 u*****R2bKw&F_en$\"3))******[jzq:F07$$\"3K+++n#)euOF:$\"3w*****>G$fTJF 07$$\"3(******f]y>9%F@$\"3A+++wCR7ZF07$$\"3%******\\.$>tAFF$\"33+++K?> $G'F07$$\"3M*****zSzEG)FF$\"3')*****R\\oR&yF07$$\"3')******[`%yF#Ffn$ \"3u******zcjC%*F07$$\"3%******4F#*Q)\\Ffn$\"3'******z'p\\*4\"FS7$$\"3 g+++=RZl')Ffn$\"31+++@$\\kD\"FS7$$\"3/+++DAr$3\"F\\o$\"3&*******QCB89F S7$$\"39+++-xH,SFfn$\"3-+++/;qp:FS7$$!35+++c'yx(GF\\o$\"34+++kRkD\"Fgo$\"3/+++8%y2)=FS7$$!33+++`$4zB$Fgo$\"3-+++#[rZ.#FS7$$ !3$)*****zE$*y#\\Fgo$\"3$)*****\\/G76#FS7$$!3S+++IwROsFgo$\"3)******Rn js=#FS7$$!3)******Rej=.\"F\\q$\"3!******H$=$GE#FS7$$!3/+++)f:^V\"F\\q$ \"3:+++(o$)yL#FS7$$!39+++oq6`>F\\q$\"3&)******))\\O7CFS7$$!3<+++9N\\2E F\\q$\"31+++)46i[#FS7$$!3!******RVq9U$F\\q$\"3!)*****z&*R$fDFS7$$!3E++ +/&\\$>WF\\q$\"3<+++pJkJEFS7$$!33+++;llDcF\\q$\"3))*****zxvHq#FS7$$!3I +++C(QQ1(F\\q$\"3#******fEYJx#FS7$$!3))*****HTDWv)F\\q$\"3#******fG4>% GFS7$$!3-+++g\"p72\"F0$\"3\")*****H#['*3HFS7$$!3!*******4+h%H\"F0$\"3/ +++Ix'R(HFS7$$!33+++@)3_a\"F0$\"3'*******y+bOIFS7$$!34+++_[p@=F0$\"31+ ++n[N'4$FS7$$!30+++YUF0$\"3')*****HD1FU$FS7$$!3/+++)ps+f%F0$\"3#)** ***\\*)flX$FS7$$!37+++Q9kH`F0$\"3@+++&)zK;NFS7$$!3!)*****\\'fTigF0$\"3 ')*****Hd>nc$FS7$$!3#******R!ft%y'F0$\"3@+++B!))*3OFS7$$!3_+++Z+D&\\(F 0$\"3%******pyYTk$FS7$$!3c*****4g8Q>)F0$\"3-+++!f(*Hn$FS7$$!3_+++`H'3) ))F0$\"3/+++s1<'p$FS7$$!3Y+++(G!3d&*F0$\"34+++i=;9PFS7$$!3/+++[AKA5FS$ \"31+++$fits$FS7$$!33+++RS+)3\"FS$\"3))*****f+$3OPFS7$$!3#*******>Z#G: \"FS$\"36+++!Rp0u$FS7$$!3*)*****fM^o@\"FS$\"3%)*****pJ<5u$FS7$$!3/+++$ oZ,G\"FS$\"3#)*****pd\"ePPFS7$$!3\"******\\/wFM\"FS$\"3&)*****>#QQIPFS 7$$!31+++q'*z/9FS$\"32+++#e=&>PFS7$$!3#******Ha&Gm9FS$\"37+++I#e]q$FS7 $$!30+++'e2t_\"FS$\"3#)******[t0(o$FS7$$!3!******RD]ze\"FS$\"3/+++]pbl OFS7$$!35+++([7$[;FS$\"34+++N!*eSOFS7$$!3)******H%>^3FS$\"3/ +++))y0lMFS7$$!3*)*******oTH,#FS$\"3y*****4nx.U$FS7$$!3#*******pq.w?FS $\"35+++G!>HP$FS7$$!3))*****RO:/9#FS$\"3%******>'f,BLFS7$$!3:+++[#Ri?# FS$\"3%******zH(4rKFS7$$!3?+++bRctAFS$\"3\")*****fopw@$FS7$$!3;+++B'*H UBFS$\"3;+++QbFjJFS7$$!30+++#3(>7CFS$\"3\"******f;J%3JFS7$$!3%******4c nG[#FS$\"3%)*****4-iN0$FS7$$!3/+++IE$Qb#FS$\"3;+++_i&*)*HFS7$$!3*)**** *\\Q(eCEFS$\"3))*****>(*\\Z%HFS7$$!3)******RucYp#FS$\"3%)*****H+V4*GFS 7$$!3!******4IKOw#FS$\"3%******p,Lu$GFS7$$!3;+++>y=JGFS$\"33+++IS0%y#F S7$$!3!)******\\&yq*GFS$\"3-+++d'41t#FS7$$!3'******zSJ6'HFS$\"35+++ms* on#FS7$$!3$******>xJK-$FS$\"3<+++*>BFi#FS7$$!3))*****zU5L3$FS$\"3)**** **H)4\"zc#FS7$$!32+++jBLTJFS$\"32+++'y/B^#FS7$$!3#)*****4)yG(>$FS$\"3A +++t*pdX#FS7$$!37+++tf=^KFS$\"37+++@@>)R#FS7$$!3$******>d\\II$FS$\"33+ ++p`ZRBFS7$$!3'******fH7HN$FS$\"3!******ziS&zAFS7$$!3,+++3j\"3S$FS$\"3 -+++!*RK=AFS7$$!3=+++h4\"oW$FS$\"3')*****R[vd:#FS7$$!3-+++i>&4\\$FS$\" 3#)*****\\Ge=4#FS7$$!3=+++13ILNFS$\"3(******>S[l-#FS7$$!3/+++\"GCRd$FS $\"3-+++U[$)f>FS7$$!3<+++FQ*Gh$FS$\"3#******H@?<*=FS7$$!3!******>d%G]O FS$\"3/+++t;AA=FS7$$!33+++eQ<'o$FS$\"3++++w?P^I\"FS7$$!3>+++o'\\5 !RFS$\"3*)*****49NRA\"FS7$$!3#)*****fc;q#RFS$\"3%********)*Qa9\"FS7$$! 33+++9%y<&RFS$\"33+++f6bm5FS7$$!3/+++'3w_(RFS$\"33+++RA&R()*F07$$!3A++ +)RMu*RFS$\"3I+++tN!33*F07$$!3T+++zn;=SFS$\"3m*****zw#)pG)F07$$!3&**** **HI!QPSFS$\"3E+++^y@$\\(F07$$!3!)*****p]!)\\0%FS$\"3U+++(*z/+nF07$$!3 3+++\"fv32%FS$\"3G+++Kc$y!fF07$$!3!)*****>uz\\3%FS$\"3b*****fU'y;^F07$ $!3e******\\b@(4%FS$\"3.+++NL(pK%F07$$!38+++*e:v5%FS$\"3*******z\\h$QN F07$$!3i******>L#e6%FS$\"3y*****H\"=$3v#F07$$!35+++6N4ATFS$\"35+++U>?k >F07$$!3D+++VBHETFS$\"3)******RLW#y6F07$$!3!)*****fN(RGTFS$\"3!******f ?9q#RF\\q7$F[hy$!3!******f?9q#RF\\q7$Ffgy$!3)******RLW#y6F07$Fagy$!35+ ++U>?k>F07$F\\gy$!3y*****H\"=$3v#F07$Fgfy$!3*******z\\h$QNF07$Fbfy$!3. +++NL(pK%F07$F]fy$!3b*****fU'y;^F07$Fhey$!3G+++Kc$y!fF07$Fcey$!3U+++(* z/+nF07$F^ey$!3E+++^y@$\\(F07$Fidy$!3m*****zw#)pG)F07$Fddy$!3I+++tN!33 *F07$F_dy$!33+++RA&R()*F07$Fjcy$!33+++f6bm5FS7$Fecy$!3%********)*Qa9\" FS7$F`cy$!3*)*****49NRA\"FS7$F[cy$!3-+++Ph!>I\"FS7$Ffby$!33+++4T@z8FS7 $Faby$!35+++'oAdX\"FS7$F\\by$!3++++nDIJ:FS7$Fgay$!3\"*******)=Oeg\"FS7 $Fbay$!33+++@6Az;FS7$F]ay$!3++++w?P^FS7$Fi_y$!3(******>S[l-#FS7$Fd_y$!3#)* ****\\Ge=4#FS7$F__y$!3')*****R[vd:#FS7$Fj^y$!3-+++!*RK=AFS7$Fe^y$!3!** ****ziS&zAFS7$F`^y$!33+++p`ZRBFS7$F[^y$!37+++@@>)R#FS7$Ff]y$!3A+++t*pd X#FS7$Fa]y$!32+++'y/B^#FS7$F\\]y$!3)******H)4\"zc#FS7$Fg\\y$!3<+++*>BF i#FS7$Fb\\y$!35+++ms*on#FS7$F]\\y$!3-+++d'41t#FS7$Fh[y$!33+++IS0%y#FS7 $Fc[y$!3%******p,Lu$GFS7$F^[y$!3%)*****H+V4*GFS7$Fijx$!3))*****>(*\\Z% HFS7$Fdjx$!3;+++_i&*)*HFS7$F_jx$!3%)*****4-iN0$FS7$Fjix$!3\"******f;J% 3JFS7$Feix$!3;+++QbFjJFS7$F`ix$!3\")*****fopw@$FS7$F[ix$!3%******zH(4r KFS7$Ffhx$!3%******>'f,BLFS7$Fahx$!35+++G!>HP$FS7$F\\hx$!3y*****4nx.U$ FS7$Fggx$!3/+++))y0lMFS7$Fbgx$!3%******H@PFS7$Fedx$!3&)*****>#QQIPFS7$F`dx$!3#)*****pd\"ePPFS7$F[dx$ !3%)*****pJ<5u$FS7$Ffcx$!36+++!Rp0u$FS7$Facx$!3))*****f+$3OPFS7$F\\cx$ !31+++$fits$FS7$Fgbx$!34+++i=;9PFS7$Fbbx$!3/+++s1<'p$FS7$F]bx$!3-+++!f (*Hn$FS7$Fhax$!3%******pyYTk$FS7$Fcax$!3@+++B!))*3OFS7$F^ax$!3')*****H d>nc$FS7$Fi`x$!3@+++&)zK;NFS7$$!3/+++NRMg\\F0$!3?+++#p.x[$FS7$Fd`x$!3# )*****\\*)flX$FS7$F_`x$!3')*****HD1FU$FS7$Fj_x$!3#******f@ZfQ$FS7$Fe_x $!35+++gB4YLFS7$F`_x$!3,+++7=(HI$FS7$F[_x$!3?+++vnXcKFS7$Ff^x$!34+++VK [1KFS7$Fa^x$!3A+++)RuI:$FS7$F\\^x$!31+++n[N'4$FS7$Fg]x$!3'*******y+bOI FS7$Fb]x$!3/+++Ix'R(HFS7$F]]x$!3\")*****H#['*3HFS7$Fh\\x$!3#******fG4> %GFS7$Fc\\x$!3#******fEYJx#FS7$F^\\x$!3))*****zxvHq#FS7$Fi[x$!3<+++pJk JEFS7$Fd[x$!3!)*****z&*R$fDFS7$F_[x$!31+++)46i[#FS7$Fjjw$!3&)******)) \\O7CFS7$Fejw$!3:+++(o$)yL#FS7$F`jw$!3!******H$=$GE#FS7$F[jw$!3)****** Rnjs=#FS7$Ffiw$!3$)*****\\/G76#FS7$$!3A+++;8`K?Fgo$!33+++'yRz&>FS7$$!3 %******f?4=U'F\\o$!3++++%3KL!=FS7$$!3u******HW9jvFfn$!3%******4raxk\"F S7$$\"3h+++C%p*=$*Ffn$!3/+++(*y^\"\\\"FS7$$\"3,+++xIwC5F\\o$!3*)****** z&p[L\"FS7$$\"3C+++ls'3y'Ffn$!3'******\\k()z<\"FS7$$\"3s*****>)opiMFfn $!3*******zl')4-\"FS7$$\"3.+++s7B;9Ffn$!3!******>OD$R')F07$$\"36+++R0% )>XFF$!31+++wjeoqF07$$\"3/+++_gXM5FF$!3#)*****p#Gz(\\&F07$$\"3-+++%zbg R\"F@$!3-+++2D*p#RF07$$\"3;+++'3&*eb'F4$!3)******Hl%>cBF07$$\"3l+++L(H 4,*!#H$!3Y+++T;)R&yF\\q7$$\"3i+++&yH4,*Fi^[l$\"3Y+++T;)R&yF\\q-Fien6&F [fn$\"#HF^fn$\"\"&F^fnFe_[l-Fbfn6^y7%F'7$$\"+u]DjdF\\q$\"+\\jzq:FjfnF[ gn7%Fj_[l7$$\"+n#)euOFS$\"+#G$fTJFjfnF[gn7%F``[l7$$\"+1&y>9%F]hn$\"+wC R7ZFjfnF[gn7%Ff`[l7$$\"+NI>tAFdhn$\"+K?>$G'FjfnF[gn7%F\\a[l7$$\"+3%zEG )Fdhn$\"+%\\oR&yFjfnF[gn7%Fba[l7$$\"+\\`%yF#F^jn$\"+!oNYU*FjfnF[gn7%Fh a[l7$$\"+rA*Q)\\F^jn$\"+op\\*4\"FcinF[gn7%F^b[l7$$\"+=RZl')F^jn$\"+@$ \\kD\"FcinF[gn7%Fdb[l7$$\"+DAr$3\"Fejn$\"+RCB89FcinF[gn7%Fjb[l7$$\"+-x H,SF^jn$\"+/;qp:FcinF[gn7%F`c[l7$$!+c'yx(GFejn$\"+kRkD\"Fb[o$\"+8%y2)=FcinF[gn7%F\\d[l7$$!+`$4zB$Fb[o$\"+#[rZ.#Fci nF[gn7%Fbd[l7$$!+oK*y#\\Fb[o$\"+X!G76#FcinF[gn7%Fhd[l7$$!+IwROsFb[o$\" +uOE(=#FcinF[gn7%F^e[l7$$!+%ej=.\"F[]o$\"+L=$GE#FcinF[gn7%Fde[l7$$!+)f :^V\"F[]o$\"+(o$)yL#FcinF[gn7%Fje[l7$$!+oq6`>F[]o$\"+*)\\O7CFcinF[gn7% F`f[l7$$!+9N\\2EF[]o$\"+)46i[#FcinF[gn7%Fff[l7$$!+M/Z@MF[]o$\"+e*R$fDF cinF[gn7%F\\g[l7$$!+/&\\$>WF[]o$\"+pJkJEFcinF[gn7%Fbg[l7$$!+;llDcF[]o$ \"+yd(Hq#FcinF[gn7%Fhg[l7$$!+C(QQ1(F[]o$\"+mi9tFFcinF[gn7%F^h[l7$$!+8a Ua()F[]o$\"+'G4>%GFcinF[gn7%Fdh[l7$$!+g\"p72\"Fjfn$\"+B['*3HFcinF[gn7% Fjh[l7$$!+5+h%H\"Fjfn$\"+Ix'R(HFcinF[gn7%F`i[l7$$!+@)3_a\"Fjfn$\"+z+bO IFcinF[gn7%Ffi[l7$$!+_[p@=Fjfn$\"+n[N'4$FcinF[gn7%F\\j[l7$$!+YUFjfn$\"+`iqAMFcinF[gn7%Ff\\\\l7$$!+)ps+f%Fj fn$\"+&*)flX$FcinF[gn7%F\\]\\l7$$!+Q9kH`Fjfn$\"+&)zK;NFcinF[gn7%Fb]\\l 7$$!+lfTigFjfn$\"+t&>nc$FcinF[gn7%Fh]\\l7$$!+/ft%y'Fjfn$\"+B!))*3OFcin F[gn7%F^^\\l7$$!+Z+D&\\(Fjfn$\"+(yYTk$FcinF[gn7%Fd^\\l7$$!+,O\"Q>)Fjfn $\"+!f(*Hn$FcinF[gn7%Fj^\\l7$$!+`H'3)))Fjfn$\"+s1<'p$FcinF[gn7%F`_\\l7 $$!+(G!3d&*Fjfn$\"+i=;9PFcinF[gn7%Ff_\\l7$$!+[AKA5Fcin$\"+$fits$FcinF[ gn7%F\\`\\l7$$!+RS+)3\"Fcin$\"+1I3OPFcinF[gn7%Fb`\\l7$$!+?Z#G:\"Fcin$ \"+!Rp0u$FcinF[gn7%Fh`\\l7$$!+Y8&o@\"Fcin$\"+PFcinF[gn7%F`b\\l7$$!+VbGm9Fcin$\"+I#e]q $FcinF[gn7%Ffb\\l7$$!+'e2t_\"Fcin$\"+\\t0(o$FcinF[gn7%F\\c\\l7$$!+a-&z e\"Fcin$\"+]pblOFcinF[gn7%Fbc\\l7$$!+([7$[;Fcin$\"+N!*eSOFcinF[gn7%Fhc \\l7$$!+V>^3Fcin$\"+))y0lMFcinF[gn7%Ffe\\l7$$ !+!pTH,#Fcin$\"+rwP?MFcinF[gn7%F\\f\\l7$$!+qq.w?Fcin$\"+G!>HP$FcinF[gn 7%Fbf\\l7$$!+k`TS@Fcin$\"+if,BLFcinF[gn7%Fhf\\l7$$!+[#Ri?#Fcin$\"+)H(4 rKFcinF[gn7%F^g\\l7$$!+bRctAFcin$\"+'opw@$FcinF[gn7%Fdg\\l7$$!+B'*HUBF cin$\"+QbFjJFcinF[gn7%Fjg\\l7$$!+#3(>7CFcin$\"+m6V3JFcinF[gn7%F`h\\l7$ $!+hv'G[#Fcin$\"+@?c`IFcinF[gn7%Ffh\\l7$$!+IE$Qb#Fcin$\"+_i&*)*HFcinF[ gn7%F\\i\\l7$$!+&Q(eCEFcin$\"+s*\\Z%HFcinF[gn7%Fbi\\l7$$!+Wnl%p#Fcin$ \"+.I%4*GFcinF[gn7%Fhi\\l7$$!+,BjjFFcin$\"+ y=JGFcin$\"+IS0%y#FcinF[gn7%Fdj\\l7$$!+]&yq*GFcin$\"+d'41t#FcinF[gn7%F jj\\l7$$!+398hHFcin$\"+ms*on#FcinF[gn7%F`[]l7$$!+sBFi#Fc inF[gn7%Ff[]l7$$!+G/J$3$Fcin$\"+$)4\"zc#FcinF[gn7%F\\\\]l7$$!+jBLTJFci n$\"+'y/B^#FcinF[gn7%Fb\\]l7$$!+\")yG(>$Fcin$\"+t*pdX#FcinF[gn7%Fh\\]l 7$$!+tf=^KFcin$\"+@@>)R#FcinF[gn7%F^]]l7$$!+s&\\II$Fcin$\"+p`ZRBFcinF[ gn7%Fd]]l7$$!+'H7HN$Fcin$\"+G1azAFcinF[gn7%Fj]]l7$$!+3j\"3S$Fcin$\"+!* RK=AFcinF[gn7%F`^]l7$$!+h4\"oW$Fcin$\"+%[vd:#FcinF[gn7%Ff^]l7$$!+i>&4 \\$Fcin$\"+&Ge=4#FcinF[gn7%F\\_]l7$$!+13ILNFcin$\"+-%[l-#FcinF[gn7%Fb_ ]l7$$!+\"GCRd$Fcin$\"+U[$)f>FcinF[gn7%Fh_]l7$$!+FQ*Gh$Fcin$\"+8-s\"*=F cinF[gn7%F^`]l7$$!+sXG]OFcin$\"+t;AA=FcinF[gn7%Fd`]l7$$!+eQ<'o$Fcin$\" +w?P^I\"FcinF[gn7%F^c]l7$$!+o'\\5!RFcin$ \"+T^$RA\"FcinF[gn7%Fdc]l7$$!+ml,FRFcin$\"+!**Qa9\"FcinF[gn7%Fjc]l7$$! +9%y<&RFcin$\"+f6bm5FcinF[gn7%F`d]l7$$!+'3w_(RFcin$\"+RA&R()*FjfnF[gn7 %Ffd]l7$$!+)RMu*RFcin$\"+tN!33*FjfnF[gn7%F\\e]l7$$!+zn;=SFcin$\"+oF)pG )FjfnF[gn7%Fbe]l7$$!+..QPSFcin$\"+^y@$\\(FjfnF[gn7%Fhe]l7$$!+20)\\0%Fc in$\"+(*z/+nFjfnF[gn7%F^f]l7$$!+\"fv32%Fcin$\"+Kc$y!fFjfnF[gn7%Fdf]l7$ $!+U(z\\3%Fcin$\"+Eky;^FjfnF[gn7%Fjf]l7$$!+]b@(4%Fcin$\"+NL(pK%FjfnF[g n7%F`g]l7$$!+*e:v5%Fcin$\"+)\\h$QNFjfnF[gn7%Ffg]l7$$!+?L#e6%Fcin$\"+8= $3v#FjfnF[gn7%F\\h]l7$$!+6N4ATFcin$\"+U>?k>FjfnF[gn7%Fbh]l7$$!+VBHETFc in$\"+MVCy6FjfnF[gn7%Fhh]l7$$!+ctRGTFcin$\"+1U,FRF[]oF[gn7%F^i]l7$F_i] l$!+1U,FRF[]oF[gn7%Fdi]l7$Fih]l$!+MVCy6FjfnF[gn7%Fhi]l7$Fch]l$!+U>?k>F jfnF[gn7%F\\j]l7$F]h]l$!+8=$3v#FjfnF[gn7%F`j]l7$Fgg]l$!+)\\h$QNFjfnF[g n7%Fdj]l7$Fag]l$!+NL(pK%FjfnF[gn7%Fhj]l7$F[g]l$!+Eky;^FjfnF[gn7%F\\[^l 7$Fef]l$!+Kc$y!fFjfnF[gn7%F`[^l7$F_f]l$!+(*z/+nFjfnF[gn7%Fd[^l7$Fie]l$ !+^y@$\\(FjfnF[gn7%Fh[^l7$Fce]l$!+oF)pG)FjfnF[gn7%F\\\\^l7$F]e]l$!+tN! 33*FjfnF[gn7%F`\\^l7$Fgd]l$!+RA&R()*FjfnF[gn7%Fd\\^l7$Fad]l$!+f6bm5Fci nF[gn7%Fh\\^l7$F[d]l$!+!**Qa9\"FcinF[gn7%F\\]^l7$Fec]l$!+T^$RA\"FcinF[ gn7%F`]^l7$F_c]l$!+Ph!>I\"FcinF[gn7%Fd]^l7$Fib]l$!+4T@z8FcinF[gn7%Fh]^ l7$Fcb]l$!+'oAdX\"FcinF[gn7%F\\^^l7$F]b]l$!+nDIJ:FcinF[gn7%F`^^l7$Fga] l$!+*=Oeg\"FcinF[gn7%Fd^^l7$Faa]l$!+@6Az;FcinF[gn7%Fh^^l7$F[a]l$!+w?P^ FcinF[gn7%Fh_^l7$Fc_]l$!+-%[l-#FcinF[gn7%F\\` ^l7$F]_]l$!+&Ge=4#FcinF[gn7%F``^l7$Fg^]l$!+%[vd:#FcinF[gn7%Fd`^l7$Fa^] l$!+!*RK=AFcinF[gn7%Fh`^l7$F[^]l$!+G1azAFcinF[gn7%F\\a^l7$Fe]]l$!+p`ZR BFcinF[gn7%F`a^l7$F_]]l$!+@@>)R#FcinF[gn7%Fda^l7$Fi\\]l$!+t*pdX#FcinF[ gn7%Fha^l7$Fc\\]l$!+'y/B^#FcinF[gn7%F\\b^l7$F]\\]l$!+$)4\"zc#FcinF[gn7 %F`b^l7$Fg[]l$!+*>BFi#FcinF[gn7%Fdb^l7$Fa[]l$!+ms*on#FcinF[gn7%Fhb^l7$ F[[]l$!+d'41t#FcinF[gn7%F\\c^l7$Fej\\l$!+IS0%y#FcinF[gn7%F`c^l7$F_j\\l $!+HP$FcinF[gn7%F\\f^l7$F]f \\l$!+rwP?MFcinF[gn7%F`f^l7$Fge\\l$!+))y0lMFcinF[gn7%Fdf^l7$Fae\\l$!+8 sr1NFcinF[gn7%Fhf^l7$F[e\\l$!+.)*=XNFcinF[gn7%F\\g^l7$Fed\\l$!+evO!e$F cinF[gn7%F`g^l7$F_d\\l$!+.?=7OFcinF[gn7%Fdg^l7$Fic\\l$!+N!*eSOFcinF[gn 7%Fhg^l7$Fcc\\l$!+]pblOFcinF[gn7%F\\h^l7$F]c\\l$!+\\t0(o$FcinF[gn7%F`h ^l7$Fgb\\l$!+I#e]q$FcinF[gn7%Fdh^l7$Fab\\l$!+#e=&>PFcinF[gn7%Fhh^l7$F[ b\\l$!+AQQIPFcinF[gn7%F\\i^l7$Fea\\l$!+x:ePPFcinF[gn7%F`i^l7$F_a\\l$!+ nc$FcinF[gn7%Fh[_l7$Fc]\\l$!+&)zK;NFcinF[gn7%F\\\\_l7$$!+N RMg\\Fjfn$!+#p.x[$FcinF[gn7%F`\\_l7$F]]\\l$!+&*)flX$FcinF[gn7%Ff\\_l7$ Fg\\\\l$!+`iqAMFcinF[gn7%Fj\\_l7$Fa\\\\l$!+;s%fQ$FcinF[gn7%F^]_l7$F[\\ \\l$!+gB4YLFcinF[gn7%Fb]_l7$Fe[\\l$!+7=(HI$FcinF[gn7%Ff]_l7$F_[\\l$!+v nXcKFcinF[gn7%Fj]_l7$Fij[l$!+VK[1KFcinF[gn7%F^^_l7$Fcj[l$!+)RuI:$FcinF [gn7%Fb^_l7$F]j[l$!+n[N'4$FcinF[gn7%Ff^_l7$Fgi[l$!+z+bOIFcinF[gn7%Fj^_ l7$Fai[l$!+Ix'R(HFcinF[gn7%F^__l7$F[i[l$!+B['*3HFcinF[gn7%Fb__l7$Feh[l $!+'G4>%GFcinF[gn7%Ff__l7$F_h[l$!+mi9tFFcinF[gn7%Fj__l7$Fig[l$!+yd(Hq# FcinF[gn7%F^`_l7$Fcg[l$!+pJkJEFcinF[gn7%Fb`_l7$F]g[l$!+e*R$fDFcinF[gn7 %Ff`_l7$Fgf[l$!+)46i[#FcinF[gn7%Fj`_l7$Faf[l$!+*)\\O7CFcinF[gn7%F^a_l7 $F[f[l$!+(o$)yL#FcinF[gn7%Fba_l7$Fee[l$!+L=$GE#FcinF[gn7%Ffa_l7$F_e[l$ !+uOE(=#FcinF[gn7%Fja_l7$Fid[l$!+X!G76#FcinF[gn7%F^b_l7$$!+;8`K?Fb[o$! +'yRz&>FcinF[gn7%Fbb_l7$$!+1#4=U'Fejn$!+%3KL!=FcinF[gn7%Fhb_l7$$!+IW9j vF^jn$!+6ZvZ;FcinF[gn7%F^c_l7$$\"+C%p*=$*F^jn$!+(*y^\"\\\"FcinF[gn7%Fd c_l7$$\"+xIwC5Fejn$!+!ep[L\"FcinF[gn7%Fjc_l7$$\"+ls'3y'F^jn$!+Xw)z<\"F cinF[gn7%F`d_l7$$\"+#)opiMF^jn$!+em)4-\"FcinF[gn7%Ffd_l7$$\"+s7B;9F^jn $!+i`KR')FjfnF[gn7%F\\e_l7$$\"+R0%)>XFdhn$!+wjeoqFjfnF[gn7%Fbe_l7$$\"+ _gXM5Fdhn$!+FGz(\\&FjfnF[gn7%Fhe_l7$$\"+%zbgR\"F]hn$!+2D*p#RFjfnF[gn7% F^f_l7$$\"+'3&*eb'F0$!+`Y>cBFjfnF[gn7%Fdf_l7$$\"+L(H4,*F\\o$!+T;)R&yF[ ]oF[gn7%Fjf_l7$$\"+&yH4,*F\\o$\"+T;)R&yF[]oF[gn-Fien6&F[fn$\"#dF^fn$\" \"\"!\"\"Fig_lF\\_s-F$6$7in7$$\"3!******4*HnV9FS$\"35+++%>*3)H%FS7$$\" 3*)*****\\xR^V\"FS$\"3#)******[p22VFS7$$\"3)*******\\f!fU\"FS$\"3<+++: ZM:VFS7$$\"34+++:\\,;9FS$\"3%)*****p$3&GK%FS7$$\"33+++m1^09FS$\"3S+++- QbHVFS7$$\"3)******z7QWR\"FS$\"3F+++T>TNVFS7$$\"31+++HN%GQ\"FS$\"3m*** **\\9$QSVFS7$$\"3#*******G[xq8FS$\"3h*****\\eCWM%FS7$$\"35+++;AGe8FS$ \"3C+++WC\\ZVFS7$$\"37+++$o=aM\"FS$\"3C+++V;a\\VFS7$$\"3'******\\rSAL \"FS$\"3o*****\\fD0N%FS7$$\"3!*******4\"4)=8FS$\"3C+++]fR]VFS7$$\"3,++ +n**=08FS$\"3!******HR-\"\\VFS7$$\"33+++*)eX\"H\"FS$\"3/+++qCfYVFS7$$ \"3++++kuox7FS$\"3w******4:\"GM%FS7$$\"3)******R2vRE\"FS$\"3n*****4o-x L%FS7$$\"33+++V6U]7FS$\"3#******pE27L%FS7$$\"3********4H9P7FS$\"3E+++- `EBVFS7$$\"3++++[eFC7FS$\"33+++in\"QJ%FS7$$\"3++++Zy(>@\"FS$\"3S+++RO! GI%FS7$$\"3)******RCM/?\"FS$\"39+++`L/0!= \"FS$\"3&)*****pIuA6%FS7$$\"3/+++o0X$>\"FS$\"3u*****R^G84%FS7$$\"3)*** ****>4G57FS$\"3C+++V\"\\F2%FS7$$\"3,+++M9*)H7FS$\"3c*****H2%HdSFS7$$\" 3)******HvF9D\"FS$\"3))*****f)=WXSFS7$$\"3-+++S_)RF\"FS$\"3T+++z%\\t.% FS7$$\"3-+++cRu'H\"FS$\"3++++Uw*G.%FS7$$\"3'******f'R/>8FS$\"3)******* RGyJSFS7$$\"3,+++Q=SS8FS$\"3$)******)[4O.%FS7$$\"3-+++s4\\g8FS$\"3A+++ 9%ez.%FS7$$\"31+++'H3\"z8FS$\"3;+++UyUWSFS7$$\"3'******R+ThR\"FS$\"3Q+ ++0Kl_SFS7$$\"32+++[!R:T\"FS$\"3)******pi8B1%FS7$$\"3-+++XWHD9FS$\"3++ ++978tSFS7$$\"3(******z`EuV\"FS$\"3#)*****>1n[3%FS7$$\"35+++X=(zW\"FS$ \"3k*****H8;t4%FS7$$\"3%*******3s(pX\"FS$\"3R+++C@I5TFS7$$\"3%*******o _\\k9FS$\"3/+++%ysO7%FS7$$\"3#******4\\\"eq9FS$\"3Q+++keHPTFS7$$\"3%** ****\\F#Hv9FS$\"3')*****>qb5:%FS7$$\"3-+++oPoy9FS$\"3S+++w.&[;%FS7$$\" 3#******f:63[\"FS$\"3O+++_$*eyTFS7$$\"35+++t#G<[\"FS$\"3q*****>b\">#>% FS7$$\"3/+++>u[\"[\"FS$\"37+++WPe0UFS7$$\"3-+++8$R,[\"FS$\"36+++F#*p=U FS7$$\"3#******p9LxZ\"FS$\"3q*****zrw9B%FS7$$\"3!*******)o;VZ\"FS$\"3 \"******HVfQC%FS7$$\"3/+++Jk$*p9FS$\"39+++/VzbUFS7$$\"35+++pxjk9FS$\"3 #******HJJsE%FS7$$\"3-+++#>l%e9FS$\"3A+++=H7yUFS7$$\"35+++#\\i9X\"FS$ \"3i*****\\bB%)G%FSF_h_lFa_[l-Fbfn6jn7%7$$\"+\"*HnV9Fcin$\"+%>*3)H%Fci n7$$\"+v(R^V\"Fcin$\"+\\p22VFcin7$$\"$K\"F^fnF_ct7%Fc[al7$$\"+]f!fU\"F cin$\"+:ZM:VFcinFh[al7%F\\\\al7$$\"+:\\,;9Fcin$\"+P3&GK%FcinFh[al7%Fb \\al7$$\"+m1^09Fcin$\"+-QbHVFcinFh[al7%Fh\\al7$$\"+G\"QWR\"Fcin$\"+T>T NVFcinFh[al7%F^]al7$$\"+HN%GQ\"Fcin$\"+XJQSVFcinFh[al7%Fd]al7$$\"+H[xq 8Fcin$\"+&eCWM%FcinFh[al7%Fj]al7$$\"+;AGe8Fcin$\"+WC\\ZVFcinFh[al7%F`^ al7$$\"+$o=aM\"Fcin$\"+V;a\\VFcinFh[al7%Ff^al7$$\"+:2CK8Fcin$\"+&fD0N% FcinFh[al7%F\\_al7$$\"+5\"4)=8Fcin$\"+]fR]VFcinFh[al7%Fb_al7$$\"+n**=0 8Fcin$\"+$R-\"\\VFcinFh[al7%Fh_al7$$\"+*)eX\"H\"Fcin$\"+qCfYVFcinFh[al 7%F^`al7$$\"+kuox7Fcin$\"+5:\"GM%FcinFh[al7%Fd`al7$$\"+u](RE\"Fcin$\"+ \"o-xL%FcinFh[al7%Fj`al7$$\"+V6U]7Fcin$\"+ns?JVFcinFh[al7%F`aal7$$\"+5 H9P7Fcin$\"+-`EBVFcinFh[al7%Ffaal7$$\"+[eFC7Fcin$\"+in\"QJ%FcinFh[al7% F\\bal7$$\"+Zy(>@\"Fcin$\"+RO!GI%FcinFh[al7%Fbbal7$$\"+WUV+7Fcin$\"+`L \"Fcin$\"+ 9&G84%FcinFh[al7%Fdfal7$$\"+?4G57Fcin$\"+V\"\\F2%FcinFh[al7%Fjfal7$$\" +M9*)H7Fcin$\"+tSHdSFcinFh[al7%F`gal7$$\"+`xU^7Fcin$\"+')=WXSFcinFh[al 7%Ffgal7$$\"+S_)RF\"Fcin$\"+z%\\t.%FcinFh[al7%F\\hal7$$\"+cRu'H\"Fcin$ \"+Uw*G.%FcinFh[al7%Fbhal7$$\"+mR/>8Fcin$\"+SGyJSFcinFh[al7%Fhhal7$$\" +Q=SS8Fcin$\"+*[4O.%FcinFh[al7%F^ial7$$\"+s4\\g8Fcin$\"+9%ez.%FcinFh[a l7%Fdial7$$\"+'H3\"z8Fcin$\"+UyUWSFcinFh[al7%Fjial7$$\"+/59'R\"Fcin$\" +0Kl_SFcinFh[al7%F`jal7$$\"+[!R:T\"Fcin$\"+FOJiSFcinFh[al7%Ffjal7$$\"+ XWHD9Fcin$\"+978tSFcinFh[al7%F\\[bl7$$\"+QlUP9Fcin$\"+iq'[3%FcinFh[al7 %Fb[bl7$$\"+X=(zW\"Fcin$\"+LhJ(4%FcinFh[al7%Fh[bl7$$\"+4s(pX\"Fcin$\"+ C@I5TFcinFh[al7%F^\\bl7$$\"+p_\\k9Fcin$\"+%ysO7%FcinFh[al7%Fd\\bl7$$\" +\"\\\"eq9Fcin$\"+keHPTFcinFh[al7%Fj\\bl7$$\"+vAHv9Fcin$\"+-d0^TFcinFh [al7%F`]bl7$$\"+oPoy9Fcin$\"+w.&[;%FcinFh[al7%Ff]bl7$$\"+c6\"3[\"Fcin$ \"+_$*eyTFcinFh[al7%F\\^bl7$$\"+t#G<[\"Fcin$\"+_:>#>%FcinFh[al7%Fb^bl7 $$\"+>u[\"[\"Fcin$\"+WPe0UFcinFh[al7%Fh^bl7$$\"+8$R,[\"Fcin$\"+F#*p=UF cinFh[al7%F^_bl7$$\"+ZJtx9Fcin$\"+=nZJUFcinFh[al7%Fd_bl7$$\"+*o;VZ\"Fc in$\"+L%fQC%FcinFh[al7%Fj_bl7$$\"+Jk$*p9Fcin$\"+/VzbUFcinFh[al7%F``bl7 $$\"+pxjk9Fcin$\"+88BnUFcinFh[al7%Ff`bl7$$\"+#>l%e9Fcin$\"+=H7yUFcinFh [al7%F\\abl7$$\"+#\\i9X\"Fcin$\"+bNU)G%FcinFh[al7%FbablF^[alFh[alFeg_l F\\_s-F$6$7in7$F`h_l$!35+++%>*3)H%FS7$Fgj`l$!3i*****\\bB%)G%FS7$Fbj`l$ !3A+++=H7yUFS7$F]j`l$!3#******HJJsE%FS7$Fhi`l$!39+++/VzbUFS7$Fci`l$!3 \"******HVfQC%FS7$F^i`l$!3q*****zrw9B%FS7$Fih`l$!36+++F#*p=UFS7$Fdh`l$ !37+++WPe0UFS7$F_h`l$!3q*****>b\">#>%FS7$Fjg`l$!3O+++_$*eyTFS7$Feg`l$! 3S+++w.&[;%FS7$F`g`l$!3')*****>qb5:%FS7$F[g`l$!3Q+++keHPTFS7$Fff`l$!3/ +++%ysO7%FS7$Faf`l$!3R+++C@I5TFS7$F\\f`l$!3k*****H8;t4%FS7$Fge`l$!3#)* ****>1n[3%FS7$Fbe`l$!3++++978tSFS7$F]e`l$!3)******pi8B1%FS7$Fhd`l$!3Q+ ++0Kl_SFS7$Fcd`l$!3;+++UyUWSFS7$F^d`l$!3A+++9%ez.%FS7$Fic`l$!3$)****** )[4O.%FS7$Fdc`l$!3)*******RGyJSFS7$F_c`l$!3++++Uw*G.%FS7$Fjb`l$!3T+++z %\\t.%FS7$Feb`l$!3))*****f)=WXSFS7$F`b`l$!3c*****H2%HdSFS7$F[b`l$!3C++ +V\"\\F2%FS7$Ffa`l$!3u*****R^G84%FS7$Faa`l$!3&)*****pIuA6%FS7$F\\a`l$! 39+++YtnMTFS7$Fg``l$!3g*****fHNw:%FS7$Fb``l$!3?+++tYP!=%FS7$F]``l$!3=+ ++0iI-UFS7$Fh_`l$!3O+++SP-BUFS7$Fc_`l$!3e*****HDtAC%FS7$F^_`l$!3;+++'4 9*fUFS7$Fi^`l$!3c*****fB%)eF%FS7$Fd^`l$!39+++`LTNVFS7$Fdi_l$!3S++ +-QbHVFS7$F_i_l$!3%)*****p$3&GK%FS7$Fjh_l$!3<+++:ZM:VFS7$Feh_l$!3#)*** ***[p22VFSF[bblFa_[l-Fbfn6jn7%7$F_[al$!+%>*3)H%Fcin7$Fcabl$!+bNU)G%Fci n7$Fi[alF_ev7%Fe]cl7$F]abl$!+=H7yUFcinFh]cl7%Fj]cl7$Fg`bl$!+88BnUFcinF h]cl7%F^^cl7$Fa`bl$!+/VzbUFcinFh]cl7%Fb^cl7$F[`bl$!+L%fQC%FcinFh]cl7%F f^cl7$Fe_bl$!+=nZJUFcinFh]cl7%Fj^cl7$F__bl$!+F#*p=UFcinFh]cl7%F^_cl7$F i^bl$!+WPe0UFcinFh]cl7%Fb_cl7$Fc^bl$!+_:>#>%FcinFh]cl7%Ff_cl7$F]^bl$!+ _$*eyTFcinFh]cl7%Fj_cl7$Fg]bl$!+w.&[;%FcinFh]cl7%F^`cl7$Fa]bl$!+-d0^TF cinFh]cl7%Fb`cl7$F[]bl$!+keHPTFcinFh]cl7%Ff`cl7$Fe\\bl$!+%ysO7%FcinFh] cl7%Fj`cl7$F_\\bl$!+C@I5TFcinFh]cl7%F^acl7$Fi[bl$!+LhJ(4%FcinFh]cl7%Fb acl7$Fc[bl$!+iq'[3%FcinFh]cl7%Ffacl7$F][bl$!+978tSFcinFh]cl7%Fjacl7$Fg jal$!+FOJiSFcinFh]cl7%F^bcl7$Fajal$!+0Kl_SFcinFh]cl7%Fbbcl7$F[jal$!+Uy UWSFcinFh]cl7%Ffbcl7$Feial$!+9%ez.%FcinFh]cl7%Fjbcl7$F_ial$!+*[4O.%Fci nFh]cl7%F^ccl7$Fihal$!+SGyJSFcinFh]cl7%Fbccl7$Fchal$!+Uw*G.%FcinFh]cl7 %Ffccl7$F]hal$!+z%\\t.%FcinFh]cl7%Fjccl7$Fggal$!+')=WXSFcinFh]cl7%F^dc l7$Fagal$!+tSHdSFcinFh]cl7%Fbdcl7$F[gal$!+V\"\\F2%FcinFh]cl7%Ffdcl7$Fe fal$!+9&G84%FcinFh]cl7%Fjdcl7$F_fal$!+2VF7TFcinFh]cl7%F^ecl7$Fieal$!+Y tnMTFcinFh]cl7%Fbecl7$Fceal$!+'HNw:%FcinFh]cl7%Ffecl7$F]eal$!+tYP!=%Fc inFh]cl7%Fjecl7$Fgdal$!+0iI-UFcinFh]cl7%F^fcl7$Fadal$!+SP-BUFcinFh]cl7 %Fbfcl7$F[dal$!+`KFUUFcinFh]cl7%Fffcl7$Fecal$!+'49*fUFcinFh]cl7%Fjfcl7 $F_cal$!+OU)eF%FcinFh]cl7%F^gcl7$Fibal$!+`LTNVFcinFh]cl7%F^[dl7$Fi\\al$!+-QbHVFc inFh]cl7%Fb[dl7$Fc\\al$!+P3&GK%FcinFh]cl7%Ff[dl7$F]\\al$!+:ZM:VFcinFh] cl7%Fj[dl7$Fd[al$!+\\p22VFcinFh]cl7%F^\\dlFb]clFh]clFeg_lF\\_s-%%FONTG 6$%*HELVETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G %&Im(z)G-Fc\\dl6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%F^fn$ \"$*>F^fn;F[^dl$\"$z%F^fn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------- --------------------------------------------------" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-------- ---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1861 "ee := \{c[2]=9/53,\nc[3]=83/311,\nc[4]=167/250,\nc[5]=34/43,\nc[ 6]=22/27,\nc[7]=1,\nc[8]=1,\n\na[2,1]=9/53,\na[3,1]=99517/1740978,\na[ 3,2]=365117/1740978,\na[4,1]=30637228987/38750625000,\na[4,2]=-7690642 751/2917968750,\na[4,3]=270499789573/107640625000,\na[5,1]=-3618253701 05678/277428090660453,\na[5,2]=495002920909/100075063365,\na[5,3]=-988 37524323878781/28782933677403685,\na[5,4]=81285337012500/1395485534657 11,\na[6,1]=-7565206670807304418/3682741770828399897,\na[6,2]=89856895 7363/123874468425,\na[6,3]=-728964729183934695599/14695438303754709502 5,\na[6,4]=4018763433227500/15832894287040767,\na[6,5]=28/87,\na[7,1]= 930898979175170148742163494117/783561862284207348259733379696,\na[7,2] =-4034678053/1107651600,\na[7,3]=90799375237487744698230149845605767/3 0002662701192241345485814174532400,\na[7,4]=39766710360861316406890996 4375/1566540066901589762118166511052,\na[7,5]=30/1003,\na[7,6]=8695756 950351830559524295/60228935845641756095491952,\na[8,1]=70352057/870914 352,\na[8,2]=0,\na[8,3]=186846453120293/467197050726588,\na[8,4]=33483 17773437500/11866042747794963,\na[8,5]=1671793689/16419376288,\na[8,6] =4684298121/78643825568,\na[8,7]=44/581,\n\nb[1]=70352057/870914352,\n b[2]=0,\nb[3]=186846453120293/467197050726588,\nb[4]=3348317773437500/ 11866042747794963,\nb[5]=1671793689/16419376288,\nb[6]=4684298121/7864 3825568,\nb[7]=44/581,\n\n`b^`[1]=49834625381/616859728443,\n`b^`[2]=0 ,\n`b^`[3]=37965116260159111/94936400477927190,\n`b^`[4]=4214755631210 93750/1492642808170910487,\n`b^`[5]=83111238435887/822384252596970,\n` b^`[6]=8/133,\n`b^`[7]=44/581,\n\n`b*`[1]=259574608264943289575/321274 1792256328977546,\n`b*`[2]=0,\n`b*`[3]=635/1588,\n`b*`[4]=339290162413 3516714843750/12011711392657264881367779,\n`b*`[5]=31/307,\n`b*`[6]=95 5366029466507585764/15887360858912285512715,\n`b*`[7]=2086094673191355 153536794882/27625985772827856064347644535,\n`b*`[8]=32153119570848305 5925831/1331372808329053304305910580\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T [5,7];" "6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 127 " denote the vector whos e components are the principal error terms of the 7 stage, order 5 sch eme (the error terms of order 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "`T*`[4,8];" "6#&%#T*G6$\"\"%\"\")" }{TEXT -1 145 " \+ denote the vector whose components are the principal error terms of th e embedded 8 stage, order 4 scheme (the error terms of order 5) and le t " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 99 " denote the vector whose components are the error terms of order 6 of the embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\" (" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,8]));" "6#-%$absG6 #-F$6#&%#T*G6$\"\"%\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(ab s(`T*`[5,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 15 " r espectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6 ] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&\" \"(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/ abs(abs(`T*`[4, 8]));" "6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"& \"\")\"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,8]-T[5,7]))/abs(abs(`T*`[4,8])); " "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\")\"\"\"&%\"TG6$F 2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince hav e suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 73 " is a minimum, if the embed ded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\" CG6#\"\"'" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 " " {TEXT -1 68 "they are \"similar in magnitude\" and also not differ t oo much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "From this point of view the Bogacki-Shampine scheme is n ear ideal with respect to the 8 stage, order 4 embedded scheme." }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\n`errterms5 _8*` :=subs(b=`b*`,PrincipalErrorTerms(5,8,'expanded')):\n`errterms4_8 *` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := sqrt(add(simplify(subs(ee,`errterms5_8*`[i]))^2,i=1..nops(`errterm s5_8*`))):\nsdnB := sqrt(add(simplify(subs(ee,`errterms4_8*`[i]))^2,i= 1..nops(`errterms4_8*`))):\nsnmC := sqrt(add((simplify(subs(ee,`errter ms5_8*`[i])-subs(ee,errterms5_7[i])))^2,i=1..nops(errterms5_7))):\n'`B *`[6]'= evalf[8](snmB/sdnB);\n'`C*`[6]'= evalf[8](snmC/sdnB);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B*G6#\"\"'$\")4T]6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\"\"'$\")[?*4\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The corresponding ch aracterstics for the auxiliary 7 stage, order 4 embedded scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "`errterms5_7^` :=subs(b=`b^`,PrincipalErrorTerms(5,7,'expanded')):\n`errterms4_7^` : = subs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "snmB := s qrt(add(simplify(subs(ee,`errterms5_7^`[i]))^2,i=1..nops(`errterms5_7^ `))):\nsdnB := sqrt(add(simplify(subs(ee,`errterms4_7^`[i]))^2,i=1..no ps(`errterms4_7^`))):\nsnmC := sqrt(add((simplify(subs(ee,`errterms5_7 ^`[i])-subs(ee,errterms5_7[i])))^2,i=1..nops(errterms5_7))):\n'`B^`[6] '= evalf[8](snmB/sdnB);\n'`C^`[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#B^G6#\"\"'$\"),'y;\"!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#C^G6#\"\"'$\")xV=6!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#=================================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "A 7 stage, order 5 Runge-Kutta scheme wit h maximal stability radius " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#------------------------------- ----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combin ed scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6072 "ee := \{c[2]=5/33,\nc[3]=29/115, \nc[4]=814716465/3253796668-23/3253796668*105151417455945^(1/2),\nc[5] =317/476,\nc[6]=14/19,\nc[7]=1,\nc[8]=1,\n\na[2,1]=5/33,\na[3,1]=5597/ 132250,\na[3,2]=27753/132250,\na[4,1]=379368932539978890648812440551/7 242812371317610711938898070128-\n 2455501968740121696101/7242 812371317610711938898070128*105151417455945^(1/2),\na[4,2]=98990961703 68469289460332229/62438037683772506137404293708+\n 4520702423 4619537323/62438037683772506137404293708*105151417455945^(1/2),\na[4,3 ]=285859955732027554698452564025/7242812371317610711938898070128-\n \+ 53985530215629267813075/7242812371317610711938898070128*1051514 17455945^(1/2),\na[5,1]=123752083413065843138789879/209127510371847411 054672640+\n 1608851500638397121047/9097046701175362380878259 8400*105151417455945^(1/2),\na[5,2]=-2000041633670013681/5324522345395 547392+\n 15875554484889/3860278700411771859200*1051514174559 45^(1/2),\na[5,3]=18902912452932794771976769428125/1744461728257485982 2209614750208+\n 100662884268346078346663939/5058939011946709 34844078827756032*105151417455945^(1/2),\na[5,4]=-51631824529882111195 9896996980826931/814696711715230147396851820173032960-\n 1079 20362352247138873225158999/488818027029138088438111092103819776*105151 417455945^(1/2),\na[6,1]=-49225589022301233744183735779/63282599719433 884959882336960-\n 3510687788120584753272949/8258379263386121 9872646449732800*105151417455945^(1/2),\na[6,2]=-245632221609852/42328 2790552285-16879401/3230316033162175*105151417455945^(1/2),\na[6,3]=86 276540353665039847710695810845/833493552221848817093515269462144-\n \+ 216788956050328897851037816063/45925494727423869821852691347364 1344*105151417455945^(1/2),\na[6,4]=3598731589064308024912379859285060 42059/246529621131827805043897351525007341440+\n 230654465754 417653852094653348797/443753318037290049079015232745013214592*10515141 7455945^(1/2),\na[6,5]=67/126,\na[7,1]=1038263320304911199846238164482 685063317425776107411696073055841920084894548437/\n 347736854 1762557310220792568694054276803350656737359370378854857516086788658304 0+\n 67568031989344520349682615047488926280454856996955945138 286279519534217777/\n 302531063133342485989208953476382722081 8915071361502652229603726038995506132724480*105151417455945^(1/2),\na[ 7,2]=59734302131229/44926355375720-1441341/1302864305895880*1051514174 55945^(1/2),\na[7,3]=1524979679765223198896979253792114247685185443478 5659517885170619034922780974618277305/\n 20726650173902839432 4952685445159640092572437165623318950343912141403953196522628760064+\n 318954894142076600206191264254918082660742098807666981116708 49008793214933062155/\n 2862251690681820683535060894242680744 13552413228717916645713021528605459176150296859136*\n 1051514 17455945^(1/2),\na[7,4]=-169366012381783914299345101325740304220565539 747522073844711653458806174655894526676245216204127995/\n 155 1626650438147832342495416245937016875264170869570046445892922517413954 15822637890959585918820864-\n 2918940970688744346217763899407 61653039224034151992442739658375212058942004402007722166107207/\n \+ 23274399756572217485137431243689055253128962563043550696688393837 76120931237339568364393788782312960*\n 105151417455945^(1/2), \na[7,5]=4615655708906632101974370650179465855337591303167331051987992 414499309155/\n 130189169814299310919242778592368588246338984 24165122087931906513654014256-\n 2010359019055948882034781883 36475409356265197258304205097572636855/\n 1301891698142993109 1924277859236858824633898424165122087931906513654014256*10515141745594 5^(1/2),\na[7,6]=9714101019226554178088904960/319579088073501979101332 49679+\n 261835787666411012880/31957908807350197910133249679* 105151417455945^(1/2),\na[8,1]=9053807089231/136378733497104-501797/20 45681002456560*105151417455945^(1/2),\na[8,2]=0,\na[8,3]=1195011372178 00311358375/307441845354011196253392-\n 2018586005231875/3074 41845354011196253392*105151417455945^(1/2),\na[8,4]=-37638654822534247 872916940674662423328915/9844674333579068498057764513689180633044624+ \n 3121650305280233144636524202663962531/44301034501105808241 2599403116013128487008080*105151417455945^(1/2),\na[8,5]=1190204022066 367942642960/4085963763691414586732367-\n 2178285514342448/40 85963763691414586732367*105151417455945^(1/2),\na[8,6]=830592632727775 794785/5052174577198328191248+1504077797251/5052174577198328191248*105 151417455945^(1/2),\na[8,7]=91/978,\n\nb[1]=9053807089231/136378733497 104-501797/2045681002456560*105151417455945^(1/2),\nb[2]=0,\nb[3]=1195 01137217800311358375/307441845354011196253392-\n 201858600523 1875/307441845354011196253392*105151417455945^(1/2),\nb[4]=-3763865482 2534247872916940674662423328915/98446743335790684980577645136891806330 44624+\n 3121650305280233144636524202663962531/44301034501105 8082412599403116013128487008080*\n 105151417455945^(1/2),\nb[ 5]=1190204022066367942642960/4085963763691414586732367-\n 217 8285514342448/4085963763691414586732367*105151417455945^(1/2),\nb[6]=8 30592632727775794785/5052174577198328191248+1504077797251/505217457719 8328191248*105151417455945^(1/2),\nb[7]=91/978,\n\n`b*`[1]=14475354021 635991174701/197802622822953213408000-\n 478269920776171/3296 71038038255355680000*105151417455945^(1/2),\n`b*`[2]=0,\n`b*`[3]=15/53 ,\n`b*`[4]=644068307267606241927194247557046508737/6369286362605991319 749019126566156224000+\n 111760904420113865733188135964353/22 2925022691209696191215669429815467840000*105151417455945^(1/2),\n`b*`[ 5]=39/125,\n`b*`[6]=65572753075427021134812949889/49617744328423854037 3654560000+\n 9841360159811270667/719097743890200783150224000 0*105151417455945^(1/2),\n`b*`[7]=16859428709040130745055637936223921/ 188074229500551422433322061497920000+\n 244403905585142824289 185521/2194199344173099928388757384142400000*105151417455945^(1/2),\n` b*`[8]=113093754206156431292803151127249/12734192622433169227256181247 255000-\n 52967678767390756540012887153/998360701598760467416 88460978479200000*105151417455945^(1/2)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 484 "for \+ ii from 2 to 8 do\n print(`========================================= =============`);\n print(c[ii]=subs(ee,c[ii]));print(``); \n for j j to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]));print(``);\n \+ end do:\nend do:\n`################################################### ###`;\n``;\nfor ii to 7 do\n print(b[ii]=subs(ee,b[ii]));print(``); \nend do:\n`######################################################`;\n ``;\nfor ii to 8 do\n print(`b*`[ii]=subs(ee,`b*`[ii]));print(``);\n end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W========================= =============================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"##\"\"&\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#\"\"&\"#L" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======= ===============================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"#H\"$:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"%( f&\"']A8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"&`x#\"']A8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============== ========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,&#\"*lkr9)\"+omz`K\"\"\"*(\"#BF,F+!\"\" \"0XfX<9:0\"#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",&#\"?^0W7)[1*)y*R D$*o$z$\"@G,2)*)Q>r5wJrB\"GC(F(*(\"7,hp@,uo>]bCF(F,!\"\"\"0XfX<9:0\"#F (\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,&#\"=HALg%*Gp%o.<'4**)*\">3PH/u81Dx $oP!QC'\"\"\"*(\"5Bt`>YBCq?XF-F,!\"\"\"0XfX<9:0\"#F-F(F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"%\"\"$,&#\"?DSc_%)pav-Kd&*feG\"@G,2)*)Q>r5wJrB\"GC(\"\"\"*(\"8vI\" yEHc@Ib)R&F-F,!\"\"\"0XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============== ========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$<$\"$w%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\" +%)f#y3Qi`<,n /(4*!\"\"\"0XfX<9:0\"#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,&#\"4\"o8+nL;/ +?\"4#RZbRXB_C`!\"\"*(\"/*)[[ab(e\"\"\"\"\"7+#f=x6/qy-'QF-\"0XfX<9:0\" #F0F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"AD\"G%pn(>x%zKHX7H!*=\"A3-v9'4A #)f[d#G,R*e]!\"\"\"0 XfX<9:0\"#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"EJp#3)p*p*)f>6@))HX# =j^\"EgH.t,#=&oRZ,B:Q5#46 \"Q%)3Q\"Hq-=))[F-\"0XfX<9:0\"#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======================= ===============================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"cG6#\"\"'#\"#9\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",&#\">zdt$=WPB,B-*eD#\\ \">gpL#))f\\)QV>(*f#Gj!\"\"*(\":\\HF`Ze?\")y(o5NF(\"A+Gt\\kks)>7'Qj#z$ e#)F-\"0XfX<9:0\"#F(\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#,&#\"0_)4;AKcC\"0&G _0z#GB%!\"\"*(\"),%zo\"\"\"\"\"1v@;LgJIKF-\"0XfX<9:0\"#F0F(F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"'\"\"$,&#\"AX3\"ep5x%)R]m`.awi)\"BW@Yp_^$4<)[=A_N\\L)\" \"\"*(\"?jg\"y.^y*)G.0c*)y;#F-\"EW8ktM\"p_=#)pQUFZ\\Df%!\"\"\"0XfX<9:0 \"#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,&#\"Hf?/1&Gf)zB\"\\-3V1*eJ()f$\"H S9M2]_^t*Q/0y#=8@'HlC\"\"\"*(\"B(z[Ll%4_Ql%e0tgp6u5wdU6\\I?LEQ5\"[pSIe')y '3;v&[&)y.Pfttc1N.oFaSpoDz?-JdDwT&otZ$F(*(\"eoxx@M&>&ziGQ^%f&p*p&[X!GE *)[Z]h#o\\._W$*)>.ov'F(\"]p![CF81b**Qgs.'HAl-:Or]\"*=3AFQwM&*3#*)f[UL8 j5`-$!\"\"\"0XfX<9:0\"#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#,&#\"/H78-Vtf \"/?dPbj#\\%\"\"\"*(\"(T8W\"F-\"1!)e*eIkGI\"!\"\"\"0XfX<9:0\"#F-F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,&#\"ap0tF=Y(4yA\\.>1<&)y^fcyMW&=&oZU6#z`#zp*))> B_wz'z\\_\"\"bpk+wGE_'>`RST@\"RM]*=LilrVsD4S'f^W&o_\\K%RG!R<]ms?\"\"\" *(\"[pb@1L\\@$z3!\\3n6\")pm2))4U2m#3=\\Dk7>1-gw?9%*[&*=$F-\"bpO\"foH]h :MSARIlh2%**Qw0Jt; .8fP`&eYz,lqV(>5Km!*3dl:Y\"eocU,aO^1>$z3A^;C%)*QjC)eoBfyFC>4J*H9)p\"*= I\"\"\"\"*(\"]obojsv40UIes>liN4aZO$)=yM?))[f0>!f.,#F-F,!\"\"\"0XfX<9:0 \"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"=g\\!*)3yTbE#>559(*\">z'\\K85 z>]t!)3z&>$\"\"\"*(\"6!)G,6km(yN=EF-F,!\"\"\"0XfX<9:0\"#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%W======================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"\",&#\".J#*32Q0*\"0/r\\L(yj8F(*(\"'(z,&F(\"1glX-5oX?!\"\"\"0XfX<9:0 \"#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"$,&#\"9v$e8J+y@P6]>\"\"9#R`i>6SNX=W2$\"\"\"*(\"1v=B0ge=?F-F,!\"\"\"0 XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\"J:*GLUiYnSp\"H(yCMD# [lQw$\"LCY/L1=*o8Xwd!)\\o!zNLuY%)*!\"\"*(\"FJD'Rm-U_OY9L-G0.l@J\"\"\" \"N!33q[GJ,;JS*f7C3e5,X.,V%F-\"0XfX<9:0\"#F0\"\"#F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"&,&#\":gHkUzOm?-/->\"\":nBt'e99pjP'f3%\"\"\"*(\"1[CM9bGy@F-F,!\"\" \"0XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"6&y%zvxsKEfI) \"7[7>G$)>xX<_]\"\"\"*(\".^szxS]\"F-F,!\"\"\"0XfX<9:0\"#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"(#\"#\"*\"$y*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W############################### #######################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",&#\".J#*32Q0*\"0/r\\L( yj8F'*(\"'(z,&F'\"1glX-5oX?!\"\"\"0XfX<9:0\"#F'\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,&#\"9v$e8J+y@P6]>\"\"9#R`i>6SNX=W2$\"\" \"*(\"1v=B0ge=?F,F+!\"\"\"0XfX<9:0\"#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%,& #\"J:*GLUiYnSp\"H(yCMD#[lQw$\"LCY/L1=*o8Xwd!)\\o!zNLuY%)*!\"\"*(\"FJD' Rm-U_OY9L-G0.l@J\"\"\"\"N!33q[GJ,;JS*f7C3e5,X.,V%F,\"0XfX<9:0\"#F/\"\" #F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,&#\":gHkUzOm?-/->\"\":nBt'e99pjP'f3%\"\" \"*(\"1[CM9bGy@F,F+!\"\"\"0XfX<9:0\"#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',& #\"6&y%zvxsKEfI)\"7[7>G$)>xX<_]\"\"\"*(\".^szxS]\"F,F+!\"\"\"0XfX<9:0 \"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"#\"*\"$y*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W############## ########################################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\", &#\"8,Z<\"*fj@SNvW\"\"9+!3M@`H#Gi-y>F'*(\"0rhx?*p#y%F'\"9++ob`DQ!Q5nH$ !\"\"\"0XfX<9:0\"#F'\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$ #\"#:\"#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%,&#\"HP(3l/dvC%>F>C1wE2$oSk\"I+SAchcE\">! \\(>8*fgijGpj\"\"\"*(\"B`V'f8)=Ld'Q6?W!4w6\"F,\"K++%yY:)H%pc@\">'p47pA ]#HA!\"\"\"0XfX<9:0\"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&#\"#R\"$D\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"',&#\">*))\\H\"[8@qUvIvsb'\"?++caOPS&QUGVu<'\\\"\"\"*(\" 4n1F6)f,OT)*F,\"=++C-:$y+-*Qu(4>(!\"\"\"0XfX<9:0\"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b *G6#\"\"(,&#\"D@RAOzjb]uI,/4(G%fo\"\"E++#z\\h?KLCU^0]HU2)=\"\"\"*(\"<@ b=*GCG9&e0RSW#F,\"F++SUTQd()QG**4tTM*>%>#!\"\"\"0XfX<9:0\"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"),&#\"B\\s7^J!GHJk:1Uv$48\"\"D+]DZ7=csApJVAE>MF\"\"\" \"*(\">`r)G,Slv!Rn(yw'H&F,\"G++?z%y4Y)oTn/w)f,2O)**!\"\"\"0XfX<9:0\"#F ,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "RK5_7eqs := [op(RowSumCondi tions(7,'expanded')),op(OrderConditions(5,7,'expanded'))]:\nexpand(sub s(ee,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4_8eqs*` := s ubs(b=`b*`,OrderConditions(4,8,'expanded')):\nexpand(subs(ee,`RK4_8eqs *`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that is, the 2-norm of \+ the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\nsqrt(add(sub s(ee,errterms5_7[i])^2,i=1.. nops(errterms5_7))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:q=-c!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We can also calculate the 2 norm of \+ the principal error of the order 4 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_8*` := subs(b=`b*`,PrincipalErr orTerms(4,8,'expanded')):\nsqrt(add(subs(ee,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+$omb'y!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-o rders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := StageOrderConditions(ct, 7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 58 "Stages 3 to 7 have the following respective stage- orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expan d(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc(L) local i; for \+ i to nops(L) do if not evalb(L[i]) then break end if end do; i end pro c,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$F $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$ SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6# F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6 #/%\"jG\"\"\"" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 7 " ) are " }{TEXT 260 13 "not satisfied " }{TEXT -1 15 ", except when " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 ", where the condition reduces to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 160 "[Sum(b[i]*a[i,1],i=1+1..7)=b[1],seq(Sum(b[i]* a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nmap( u->`if`(lhs(u)=rhs(u),0,1),expand(subs(ee,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\" #\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cG FB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/ -F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F) F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$ F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sum(b[i]*a[i,2],i=3..7); \neval(subs(Sum=add,%));\nexpand(subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\" $\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"a G6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*& &F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1 ),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F +%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Su m(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,%));\ne xpand(subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"b G6#%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+! \"\"F+/F*;F5\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"% \"\"\"&%\"aG6$F(\"\"$F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F. F)F)*&&F+6$F4F(F)&F+6$F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)* &&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 72 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nexpand(subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"# F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$ \"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)* (&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(& F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "N one of the principal error conditions are satisfied by the order 5 sch eme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK5_7err_eqs := Pr incipalErrorConditions(5,7,'expanded'):\nexpand(subs(ee,RK5_7err_eqs)) :\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal e rror conditions are satisfied by the 8 stage embedded order 4 embedded ." }}{PARA 0 "" 0 "" {TEXT -1 96 "The largest magnitude of an error te rm is about 267 times the lowest magnitude of an error term." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_8*` := subs(b=`b *`,PrincipalErrorTerms(4,8,'expanded')):\nsubs(ee,`errterms4_8*`):\nma p(abs,evalf[8](%));\nevalf[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")t^\")H!#6$\")+gelF&$\"(#Ho7F&$\")%QZE#F&$\"' qftF&$\"(&**45F&$\"(SAZ\"F&$\")$)>'=#F&$\")\"[QX#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%tE!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 88 "simultaneous construction of the 7 stage, order 5 \+ scheme and the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 290 "From a theoretical point of view, it is interesting to observe that it is possible to construct a 7 stage, order 5 Runge-Kutta scheme which has the stability function that lead s to the scheme having maximal stability radius (along with a compatib le embedded 8 stage, order 4 [FSAL] scheme.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following \"simple\" order conditions (given in abreviated form) are relevant for the constructi on of the order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := SimpleOrderConditions(5):\n [seq([i,SO5[i]],i=[1,2,4,5,8,12,13,15,16])]:\nlinalg[augment](linalg[d elcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols] (%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%% ~~~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F 2F/F(#F(\"\"$7%\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@6#*&FCF(F2F(F(F(#F(\"#C 7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6#*&FCF(-F@6#*&F8F(FCF(F (F(#F(\"#g7%\"#8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF)/*&F,F(-F@6#*&FMF(FCF(F (#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b*`,SimpleOrderConditions (4)):\n[seq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nlinalg[augment](linalg[del cols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](% ,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"\"\"%%~~ ~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F /F(#F(\"\"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF(F(#F(\"#77%\"\")F)/*&F,F ()F2F:F(#F(F5Q)pprint406\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " We incorporate the simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" " 6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\" $\"\"(\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i, j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG \"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F +;F6\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 scheme and use the simple order conditions in the first table above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We also need to specify that t he linking coefficients for the 8th (FSAL) stage are given by the weig hts of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6#%\"iG&%\"aG6$\"\")F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG 6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the order 4 embedded schem e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7,'expanded'):\n`SO 4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded')): \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[1,2,4,7,8 ])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(StageOrderCon ditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3..7)=0,ad d(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i]*c[i]* a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [seq(b[i]= a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_eqns),op(SO_e qs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 5/33;" "6#/&%\"cG6#\"\"#*&\"\"&\"\"\"\"#L!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 29/115;" "6#/&%\"cG6#\"\"$*&\"#H\"\" \"\"$:\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 317/476;" "6#/ &%\"cG6#\"\"&*&\"$<$\"\"\"\"$w%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 14/19;" "6#/&%\"cG6#\"\"'*&\"#9\"\"\"\"#>!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " are initial ly parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "e1 := \{c[2]=5/33,c[3]=29/115,c[5]=317/476,c[6] =14/19,c[7]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs(e1,cdns):\nnops (eqns);\nindets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*`[5] \};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F,6$F;F'&F,6$F;FE&F,6$F.F'&F, 6$F1F*&F,6$F.FE&F,6$F.F>&F,6$F1F>&F,6$F1F'&F,6$F1FE&F,6$F*F>&F,6$F*F'& F,6$F'F>&F,6$F*FE&F,6$F>FE&F,6$F'FE&F%6#FE&%#b*GF)&FgoF3&Fgo6#F;&Fgo6# F6&FgoFeo" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := solve( \{op(eqns)\},indets(eqns) minus \{c[4],a[6,5],a[7,5],b[7],`b*`[3],`b*` [5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6340 "e3 := \{a[4,3] = 13225 /841*c[4]^3-115/29*c[4]^2, `b*`[2] = 0, c[2] = 5/33, a[2,1] = 5/33, a[ 6,1] = 1/6534926516720*(9368548281120*c[4]^2+611220615055260*a[6,5]*c[ 4]^2+93904762837424*c[4]-835979131803245*a[6,5]*c[4]-57963268214880+28 5650796067400*a[6,5])/c[4]/(57*c[4]-40), a[8,4] = -1/12*(820440*b[7]-7 3261)/c[4]/(-128702+878293*c[4]-1721281*c[4]^2+1040060*c[4]^3), a[7,2] = -33/1160*(15060*b[7]*c[4]-1636*c[4]+667-5336*b[7])/b[7]/(57*c[4]-40 ), a[4,1] = 8786/841*c[4]^3-1721/290*c[4]^2+c[4], c[8] = 1, c[7] = 1, \+ b[2] = 0, `b*`[7] = 1/615099886590000*(4302282180915028800*c[4]^2*`b*` [3]+1661513983767697500*c[4]^2*`b*`[5]-2241423986733960000*c[4]^2-1695 213567391729125*c[4]*`b*`[5]-2609291042894813760*c[4]*`b*`[3]+16998080 33261912000*c[4]-355568741108126000+384405770427573888*`b*`[3]+3920551 46599995750*`b*`[5])*b[7]/(15060*b[7]*c[4]-1636*c[4]+667-5336*b[7])/(c [4]-1), c[3] = 29/115, a[8,1] = 1/1544424*(820440*b[7]*c[4]+39610*c[4] -820440*b[7]+73261)/c[4], a[8,2] = 0, a[8,7] = b[7], a[8,3] = -3498012 5/8347618332*(47700*b[7]*c[4]-15060*c[4]+6983-47700*b[7])/(-29+115*c[4 ]), c[6] = 14/19, b[3] = -34980125/8347618332*(47700*b[7]*c[4]-15060*c [4]+6983-47700*b[7])/(-29+115*c[4]), a[6,4] = -1/1554084784*1/c[4]*(28 07257823421*a[6,5]*c[4]-542513347824*c[4]-1970005490120*a[6,5]+3997466 77344)/(6555*c[4]^2-6253*c[4]+1160), b[1] = 1/1544424*(820440*b[7]*c[4 ]+39610*c[4]-820440*b[7]+73261)/c[4], b[4] = -1/12*(820440*b[7]-73261) /c[4]/(-128702+878293*c[4]-1721281*c[4]^2+1040060*c[4]^3), a[8,5] = 12 834170944/13807845741*(-347*c[4]+5160*b[7]*c[4]+385-5160*b[7])/(-317+4 76*c[4]), a[4,2] = -33/290*(230*c[4]-87)*c[4]^2, c[5] = 317/476, a[8,6 ] = -130321/114041592*(164088*b[7]*c[4]-18342*c[4]-164088*b[7]+17371)/ (19*c[4]-14), a[5,1] = -317/907019980160*(27447372528*c[4]^2-383212793 54*c[4]+17699604655)/c[4]/(57*c[4]-40), `b*`[1] = -1/6889118729808000* (221348652430500*c[4]*`b*`[5]-1886306318876000*c[4]+3388709690513280*c [4]*`b*`[3]-147410762227875*`b*`[5]+369059931954000-854544182825088*`b *`[3])/c[4], a[5,3] = 825742205/181403996032*(-1635861*c[4]+1364130*c[ 4]^2+422878)/(-29+115*c[4])/(57*c[4]-40), b[6] = -130321/114041592*(16 4088*b[7]*c[4]-18342*c[4]-164088*b[7]+17371)/(19*c[4]-14), a[7,3] = -1 15/10954084466153684878848*(9829205643990223318665600*c[4]^3+263978839 513731950142806880*a[6,5]*c[4]^3*b[7]-77097542662426368794181120*b[7]* c[4]^3+4581520678135818371656800*b[7]*a[7,5]*c[4]^3-295079461896108882 39964920*a[6,5]*c[4]^3+68304467467554751232103750*a[6,5]*c[4]^2-625028 181822263534170870440*a[6,5]*c[4]^2*b[7]-16220765997350591847799104*c[ 4]^2-9642098342998428602387400*b[7]*a[7,5]*c[4]^2+11698630070549059447 0848000*b[7]*c[4]^2+8988780193093079872265472*c[4]+6758380226118063225 891600*b[7]*a[7,5]*c[4]-52012487274079409780644445*a[6,5]*c[4]+4844184 88889473686630554760*a[6,5]*c[4]*b[7]-58537116478923063898097664*b[7]* c[4]+10460017484025097988450304*b[7]-1748724713350050214265088-1233691 46580942102602491200*b[7]*a[6,5]+13060342287416174639875400*a[6,5]-157 7689484513665117392000*b[7]*a[7,5])/(406-2161*c[4]+2185*c[4]^2)/b[7]/( 57*c[4]-40), `b*`[4] = -3/164026636424000*(94949353647232*`b*`[3]-4100 6659106000+16378973580875*`b*`[5])/c[4]/(14-33*c[4]+19*c[4]^2), a[3,2] = 27753/132250, a[7,6] = 6859/182350989924080366592*(7308778838108146 7616*a[6,5]*c[4]^2*b[7]-7405774215706643616*b[7]*a[7,5]*c[4]^2+1110021 370608947328*c[4]^2-8169861382220493144*a[6,5]*c[4]^2+1038888295827843 7368*b[7]*a[7,5]*c[4]+13178213333272059070*a[6,5]*c[4]-159687284894620 4928*c[4]-121761798710499167688*a[6,5]*c[4]*b[7]-5152821860418280849*a [6,5]+573972269197608960+48674010329417700072*b[7]*a[6,5]-363410282721 2083632*b[7]*a[7,5])/b[7]/(19*c[4]-14)^2, `b*`[8] = 1/2460399546360000 *(536695154379328000*c[4]^2-1869464448333861120*c[4]^2*`b*`[3]-7219752 66253374000*c[4]^2*`b*`[5]+4025213657844960000*b[7]*c[4]^2-15172463869 2200000*c[4]+1233614020390137792*c[4]*`b*`[3]-5666300155267080000*b[7] *c[4]+775161810954561000*c[4]*`b*`[5]-196027573299997875*`b*`[5]-27351 441623702000-192202885213786944*`b*`[3]+1641086497422120000*b[7])/(150 60*b[7]*c[4]-1636*c[4]+667-5336*b[7])/(c[4]-1), a[3,1] = 5597/132250, \+ a[5,4] = 7180367/12688256*(-317+476*c[4])/(-29+115*c[4])/c[4]/(57*c[4] -40), a[7,1] = 1/724066017592783703040*(11477340848423128267078560*a[6 ,5]*c[4]^3*b[7]-81265190341877638121088*c[4]^3+12376905998515671129552 0*b[7]*a[7,5]*c[4]^3+1582043352916052061504000*b[7]*c[4]^3-12829541821 56995140868040*a[6,5]*c[4]^3-1111730791220279338364928*b[7]*c[4]^2-271 75138340098414529168280*a[6,5]*c[4]^2*b[7]-48232300112536537890816*c[4 ]^2+2969759455111076140526250*a[6,5]*c[4]^2-260479769062820041710360*b [7]*a[7,5]*c[4]^2-386556375778448132662272*b[7]*c[4]+15231902760839858 5167744*c[4]+182576578034623091160240*b[7]*a[7,5]*c[4]-226141249017736 5642636715*a[6,5]*c[4]+21061673429977116810024120*a[6,5]*c[4]*b[7]-426 21033094665881788800*b[7]*a[7,5]-5363875938301830547934400*b[7]*a[6,5] -53537191662873326042880+567840969018094549559800*a[6,5]+2555204008291 22359203840*b[7])/(19*c[4]-14)/c[4]/b[7]/(57*c[4]-40), a[6,2] = -231/9 94555*(224295*c[4]-121394)/(57*c[4]-40), `b*`[6] = -6859/3444559364904 0000*(-82013318212000*c[4]+92797526935680*c[4]*`b*`[3]+109465714228500 *c[4]*`b*`[5]-23401115488128*`b*`[3]-72900486156375*`b*`[5]+4100665910 6000)/(19*c[4]-14), a[7,4] = -1/12299405768520192*1/c[4]*(-32457024876 60561987072*b[7]*c[4]^3-8847959876944794074952*a[6,5]*c[4]^3+791540748 16711229428128*a[6,5]*c[4]^3*b[7]+899487317316783651456*c[4]^3+8028842 995791916494336*b[7]*c[4]^2+20481099690421214762250*a[6,5]*c[4]^2-5684 55404452255541592*b[7]*a[7,5]*c[4]^2-187414747173092513994264*a[6,5]*c [4]^2*b[7]-2003598310978716672192*c[4]^2+145252920206738736620856*a[6, 5]*c[4]*b[7]+1489058307880318344960*c[4]-6545350169021853536256*b[7]*c [4]+817777950264648322992*b[7]*a[7,5]*c[4]-15595948208119763052667*a[6 ,5]*c[4]-293938159273557805440*b[7]*a[7,5]-36992247850357452054720*b[7 ]*a[6,5]+1762209660890499028992*b[7]-369222011468091903744+39161446139 17893445240*a[6,5])/(41515*c[4]^3-71649*c[4]^2+37968*c[4]-5684)/b[7]/( 57*c[4]-40), b[5] = 12834170944/13807845741*(-347*c[4]+5160*b[7]*c[4]+ 385-5160*b[7])/(-317+476*c[4]), a[5,2] = -10461/31276551040*(19261686* c[4]-33195923)/(57*c[4]-40), a[6,3] = -115/1306985303344*(-12110812676 4240*c[4]^2+196742285301660*a[6,5]*c[4]^2-269088510440045*a[6,5]*c[4]+ 124694089605120*c[4]+91946490403400*a[6,5]-31368356916288)/(6555*c[4]^ 2-6253*c[4]+1160)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "the coefficients in terms of " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[ 7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" " 6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6 #\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\" &" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "for ii from 2 to 8 do\n print(`========== ============================================`);\n temp_eq := c[ii]=s ubs(e3,c[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs( temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end i f;\n print(``); \n for jj to ii-1 do\n temp_eq := a[ii,jj]=s ubs(e3,a[ii,jj]);\n if lhs(temp_eq)=rhs(temp_eq) then\n \+ print(lhs(temp_eq)*` is a parameter`);\n else\n print( temp_eq);\n end if;\n print(``);\n end do:\nend do:\n`# #####################################################`;\n``;\nfor ii t o 7 do\n temp_eq := b[ii]=subs(e3,b[ii]);\n if lhs(temp_eq)=rhs(te mp_eq) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n \+ print(temp_eq);\n end if;\n print(``);\nend do:\n`############ ##########################################`;\n``;\nfor ii to 8 do\n \+ temp_eq := `b*`[ii]=subs(e3,`b*`[ii]);\n if lhs(temp_eq)=rhs(temp_eq ) then\n print(lhs(temp_eq)*` is a parameter`);\n else\n p rint(temp_eq);\n end if;\n print(``);\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%W=================================================== ===G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"&\"#L" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"#\"\"\"#\"\"&\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================ ======================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"$#\"#H\"$:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"%(f&\"']A8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"$\"\"##\"&`x#\"']A8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W========================================== ============G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\" %0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",(*&#\"%'y)\"$T)F(*$)&% \"cG6#F'\"\"$F(F(F(*&#\"%@<\"$!HF(*$)F0\"\"#F(F(!\"\"F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"%\"\"#,$*&#\"#L\"$!H\"\"\"*&,&*&\"$I#F.&%\"cG6#F'F.F.\"#()!\"\"F. )F3F(F.F.F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,&*&#\"&DK\"\"$T)\"\"\"*$)&%\"cG6# F'F(F.F.F.*&#\"$:\"\"#HF.*$)F1\"\"#F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============== ========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$<$\"$w%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*&# \"$<$\"-g,)*>q!*F(*(,(*&\",GDPZu#F()&%\"cG6#\"\"%\"\"#F(F(*&\",a$z7KQF (F3F(!\"\"\",bYg*pF.&%\"cG6#\"\"%F.F. \")Bf>L!\"\"F.,&*&\"#dF.F3F.F.\"#SF8F8F.F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\" \"$,$*&#\"*0AuD)\"-Kg*RS\"=\"\"\"*(,(*&\"(hej\"F.&%\"cG6#\"\"%F.!\"\"* &\"(ITO\"F.)F3\"\"#F.F.\"'yGUF.F.,&\"#HF7*&\"$:\"F.F3F.F.F7,&*&\"#dF.F 3F.F.\"#SF7F7F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,$*&#\"(n.=(\")c#)o7\"\"\"**, &\"$<$!\"\"*&\"$w%F.&%\"cG6#F(F.F.F.,&\"#HF2*&\"$:\"F.F5F.F.F2F5F2,&*& \"#dF.F5F.F.\"#SF2F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%W==================================== ==================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'# \"#9\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",$*&#F(\".?n^E\\`'F(*(,.*&\".?6G[&o $*F()&%\"cG6#\"\"%\"\"#F(F(*(\"0g_0:1A6'F(&F%6$F'\"\"&F(F1F(F(*&\"/Cu$ Gw/R*F(F2F(F(*(\"0XK!=8zf$)F(F9F(F2F(!\"\"\"/!)[@oK'z&F@*&\"0+u1'z]cGF (F9F(F(F(F2F@,&*&\"#dF(F2F(F(\"#SF@F@F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#,$*&#\"$J#\"'bX**\"\"\"*&,&*&\"'&HC#F.&%\"cG6#\"\"%F.F.\"'%R@\"!\" \"F.,&*&\"#dF.F3F.F.\"#SF8F8F.F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\"$:\"\". WLI&)pI\"\"\"\"*&,.*&\"0SUwE\"367F.)&%\"cG6#\"\"%\"\"#F.!\"\"*(\"0g;I& GUn>F.&F%6$F'\"\"&F.F3F.F.*(\"0X+W5&)3p#F.F*F.FF,F6F,F2\"-WtnY(*RF,F,,(*&\"%blF,)F/\"\"#F,F,*&\"%`iF,F/F,F2\" %g6F,F2F,F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&&%\"aG6$\"\"'\"\"&\"\"\"%0~is~a~parameterGF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%W======================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"\",$*&#F(\"6SIq$y#fl7)F(F5F(!\"\"** \"9?bH6n:&)*f!pP7F(F;F(&F%6$F'F4F(F5F(F(*(\":+S]h?0;HNV?e\"F(F;F(F5F(F (*(\":S!o39&*p:#=aHG\"F(F1F(F5F(F@*(\":G\\OQ$z-A\"zI<6\"F(F;F()F6\"\"# F(F@**\";!Go\"HXT)4S$Q^B:F(F6F(F(**\"9S-;\"4BY.yld#=F(F ;F(FCF(F6F(F(*(\"::njUcOx,\\79E#F(F1F(F6F(F@**\";?T-5o6x*HMnh5#F(F1F(F 6F(F;F(F(*(\"8+))y\")em%4L5iUF(F;F(FCF(F@*(\":+W$zaI=IQf(QO&F(F;F(F1F( F@\"8!)G/EL(Gm\">PN&F@*&\"9+)f&\\X4=!p4%ycF(F1F(F(*&\"9SQ?fB7H3S?bDF(F ;F(F(F(,&*&\"#>F(F6F(F(\"#9F@F@F6F@F;F@,&*&\"#dF(F6F(F(\"#SF@F@F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"(\"\"#,$*&#\"#L\"%g6\"\"\"*(,**(\"&g]\"F.&%\"bG6#F'F.&% \"cG6#\"\"%F.F.*&\"%O;F.F6F.!\"\"\"$n'F.*&\"%O`F.F3F.Ft8 &R)yREF.&F%6$\"\"'\"\"&F.F3F.&%\"bG6#F'F.F.*(\";?6=%zojUiEa(4xF.F>F.F3 F.!\"\"**\":+olr$=e8y1_\"e%F.F>F.&F%6$F'F=F.F3F.F.*(\";?\\'*R#))3h*=Yz ]HF.F:F.F3F.FC*(\";]P5K7vavYnWIoF.F:F.)F4\"\"#F.F.**\"F.FC*&\";/\"*zZ=f]t*fw?i\"F.FLF.FC**\":+uQ-'G%)*HM)4U'*F.F> F.FFF.FLF.FC*(\"<+![3Z%f!\\02I')p6F.F>F.FLF.F.*&\":saEs)zI4$>!y))*)F.F 4F.F.**\":+;*eAj!=hA!QenF.F>F.FFF.F4F.F.*(\";XWk!y4%zSF([7?&F.F:F.F4F. FC**\"F.F.*(\";kw4)*Q1B*yk6P&eF.F>F.F4F.F C*&\";/.X))z4DS[<+Y5F.F>F.F.\":)3lU@]+N8Zs[F.F:F.FC*&\";+a()RY<;uGU.18F.F:F.F.*(\":+?R<^m8X[*ox:F.F>F.FFF.FCF.,( \"$1%F.*&\"%h@F.F4F.FC*&\"%&=#F.FLF.F.FCF>FC,&*&\"#dF.F4F.F.\"#SFCFCF. FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,$*&#\"\"\"\"2#>?&odS*H7F,*,&%\"cG6# F(!\"\",H*(\"7sq)>cgw[-dC$F,&%\"bG6#F'F,)F/\"\"$F,F2*(\"7_\\2%zWp()fz% ))F,&F%6$\"\"'\"\"&F,F9F,F2**\"8G\"G%H7r;[2a\"zF,F=F,F9F,F6F,F,*&\"6c9 l$y;tJ([**)F,F9F,F,*(\"7OV\\;>z&*H%)G!)F,F6F,)F/\"\"#F,F,*(\"8]Aw97U!p *4\"[?F,F=F,FGF,F,**\"6#fTbD_WSb%o&F,F6F,&F%6$F'F@F,FGF,F2**\"9kU*R^#4 tru9u=F,F=F,FGF,F6F,F2*&\"7#>smry4J)f.?F,FGF,F2**\"9c3iO(Qn??HDX\"F,F= F,F/F,F6F,F,*&\"7g\\M=.)yIe!*[\"F,F/F,F,*(\"7ci``=-p,NXlF,F6F,F/F,F2** \"6#*HK[YE]zx<)F,F6F,FMF,F/F,F,*(\"8nE0j(>\"3#[ff:F,F=F,F/F,F2*(\"6Sa! ybt#f\"QRHF,F6F,FMF,F2*(\"8?Z0_uN]yC#*p$F,F6F,F=F,F2*&\"7#**G!*\\!*3m4 Aw\"F,F6F,F,\"6WP!>4o9,A#p$F2*&\"7S_W$*y\"RhWh\"RF,F=F,F,F,,**&\"&::%F ,F9F,F,*&\"&\\;(F,FGF,F2*&\"&oz$F,F/F,F,\"%%o&F2F2F6F2,&*&\"#dF,F/F,F, \"#SF2F2F,F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\"\"%0~is~a~parameterGF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"',$*&#\"%fo\"6#fm.3C**)4N#=\"\"\"*(,:**\"5;wY\"3\"Q )y(3tF.&F%6$F(\"\"&F.)&%\"cG6#\"\"%\"\"#F.&%\"bG6#F'F.F.**\"4;Ok1d@udS (F.FpA(RdF.*(\"5s+q<%H.,u'[F.FF.F7F.F.\"#9FC!\"#F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W====== ================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",$*&#F(\"( CWa\"F(*&,**(\"'S/#)F(&%\"bG6#\"\"(F(&%\"cG6#\"\"%F(F(*&\"&5'RF(F5F(F( *&F0F(F1F(!\"\"\"&hK(F(F(F5FF.F7F.F.\"#9F=F=F.F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"(&%\"bG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%W#################################### ##################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"(CWa\"F'*&,**(\"'S/# )F'&F%6#\"\"(F'&%\"cG6#\"\"%F'F'*&\"&5'RF'F3F'F'*&F/F'F0F'!\"\"\"&hK(F 'F'F3F:F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\")D,)\\$ \"+K$=wM)\"\"\"*&,**(\"&+x%F-&F%6#\"\"(F-&%\"cG6#\"\"%F-F-*&\"&g]\"F-F 5F-!\"\"\"%$)pF-*&F1F-F2F-F;F-,&\"#HF;*&\"$:\"F-F5F-F-F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"%,$*&#\"\"\"\"#7F+*(,&*&\"'S/#)F+&F%6#\"\"(F+F+\"&hK(!\"\"F+&% \"cGF&F5,*\"'-(G\"F5*&\"'$Hy)F+F6F+F+*&\"(\"G@F-F5F-F-\"#9F;F;F-F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"bG6#\"\"(\"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W############## ########################################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\", $*&#F'\"1+!3)H(=\"*)oF'*&,.*(\"0+0V_'[8AF'&%\"cG6#\"\"%F'&F%6#\"\"&F'F '*&\"1+g()=jI')=F'F0F'!\"\"*(\"1!G80p4()Q$F'F0F'&F%6#\"\"$F'F'*&\"0vyA i2TZ\"F'F4F'F9\"0+S&>$*f!p$F'*&\"0)3DG=WX&)F'FF-)F;\"\"#F-F-F5F-F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#b*G6#\"\"& \"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"',$*&#\"%fo\"2++/\\OfXW$ \"\"\"*&,.*&\"/+?@=L,#)F-&%\"cG6#\"\"%F-!\"\"*(\"/!oNp_(z#*F-F2F-&F%6# \"\"$F-F-*(\"0+&GUrl%4\"F-F2F-&F%6#\"\"&F-F-*&\"/G\")[:6SBF-F9F-F6*&\" /vj:'[+H(F-F>F-F6\"/+g5fm+TF-F-,&*&\"#>F-F2F-F-\"#9F6F6F-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b *G6#\"\"(,$*&#\"\"\"\"0++f'))*4:'F+**,4*(\"4+)G]\"4=#G-VF+)&%\"cG6#\" \"%\"\"#F+&F%6#\"\"$F+F+*(\"4+vpnP)R^h;F+F1F+&F%6#\"\"&F+F+*&\"4++'Rt' )RUTAF+F1F+!\"\"*(\"4D\"HEL!3)*p\"F+F2F+F+\"3+g736uobNFA*&\"3))QdF/x0WQF+F7F+F+ *&\"3]d***fY^0#RF+FsF+F1F+&F%6#\"\"&F+F< *(\"4++'\\%yl8_-%F+&%\"bG6#\"\"(F+F1F+F+*&\"3++?#pQYs^\"F+F2F+F<*(\"4# zP,R?ShL7F+F2F+F9F+F+*(\"4++3n_:+jm&F+FDF+F2F+F<*(\"3+5ca4\"=;v(F+F2F+ F?F+F+*&\"3vy***Htv-'>F+F?F+F<\"2+?qB;W^t#F<*&\"3Wpy8_)G?#>F+F9F+F<*& \"4++7Au\\'3T;F+FDF+F+F+,**(\"&g]\"F+FDF+F2F+F+*&\"%O;F+F2F+F<\"$n'F+* &\"%O`F+FDF+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK5 _7eqs := [op(RowSumConditions(7,'expanded')),op(OrderConditions(5,7,'e xpanded'))]:\nsimplify(subs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The stability function of a gen eral 7 stage, order 5 Runge-Kutta scheme has the form: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z^ 5/120+t[6]*z^6+t[7]*z^7;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&F'\"\"#F+! \"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F)*&&%\"tG6# F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[5,4]*a[4,3] *a[3,2]*c[2]+b[7]" "6#/&%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&%\"aG6$F'\"\" &F-&F/6$F1\"\"%F-&F/6$F4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F-&F+6#\"\"(F -" }{TEXT -1 2 " (" }{XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+ a[7,6]*a[6,4]*a[4,3]*a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"(\"\"&\"\"\"&F &6$F)\"\"%F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F(\"\"'F* &F&6$F:F-F*&F&6$F-F0F*&F&6$F0F3F*&F56#F3F*F*%!GF*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[7,6]*a[6,5]*a[5,3]*a[3,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a [4,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,3]*c[3]" "6#,(*,&%\"aG6$\"\"(\"\"' \"\"\"&F&6$F)\"\"&F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F&6$F (F)F*&F&6$F)F-F*&F&6$F-\"\"%F*&F&6$F>F3F*&F56#F3F*F**,&F&6$F(F)F*&F&6$ F)F-F*&F&6$F-F>F*&F&6$F>F0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[7]=b[7]*a[7,6]*a[6,5]*a[5,4]*a [4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\"\"\"&%\"aG6$F'\" \"'F,&F.6$F0\"\"&F,&F.6$F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9\"\"#F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 58 "[0.1103840e-2,0.6537663796e- 4]:\nmap(convert,%,rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$# \"#8\"&x<\"#\"\"\"\"&'H:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "eq7 := b[7]*a[7,6]*a[6,5]*a[5,4]*a [4,3]*a[3,2]*c[2]=1/15296;\neq6 := b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[ 2]+b[7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+\n a[7,6]*(a[6,4]*a[4,3]*a[ 3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3])) ))=13/11777;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6#\"\" (\"\"\"&%\"aG6$F*\"\"'F+&F-6$F/\"\"&F+&F-6$F2\"\"%F+&F-6$F5\"\"$F+&F-6 $F8\"\"#F+&%\"cG6#F;F+#F+\"&'H:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ eq6G/,&*.&%\"bG6#\"\"'\"\"\"&%\"aG6$F+\"\"&F,&F.6$F0\"\"%F,&F.6$F3\"\" $F,&F.6$F6\"\"#F,&%\"cG6#F9F,F,*&&F)6#\"\"(F,,&*,&F.6$F@F0F,F1F,F4F,F7 F,F:F,F,*&&F.6$F@F+F,,&**&F.6$F+F3F,F4F,F7F,F:F,F,*&F-F,,&*(&F.6$F0F6F ,F7F,F:F,F,*&F1F,,&*&&F.6$F3F9F,F:F,F,*&F4F,&F;6#F6F,F,F,F,F,F,F,F,F,F ,#\"#8\"&x<\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We shall require the following two equations to be satisf ied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve(si mplify(subs(e3,\{eq6\})),a[7,5]));\neqn7 := op(solve(simplify(subs(e3, \{eq7\})),a[7,5]));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&%\" aG6$\"\"(\"\"&,$*&#\"\"\"\"?%Q*)3^#e$**34fHX&\\ZF.*,,J*&\"FW(>b,'*=u?x #[DK#y>&eP$F.)&%\"cG6#\"\"%F8F.!\"\"*(\"GwH]p5Hl*Qd$f*G)>5'[ou\"F.&F'6 $\"\"'F*F.F4F.F.**\"IsU8GbH![&GWXv#H\"fzx=]6F.)F<\"\"#F.F4F.&%\"bG6#F) F.F.*(\"H[;bl;J0gw4,VJQ&epp&G\"F.FAF.F4F.F9*(\"GK3#H)4^H(3?05/ybZ@7Q%F .F,*Rnf%ezB\"pOXU\"F.FAF.F5F.F.*(\"G3)*zn?oj,( =R1g=y2B8k\"F.F$F.F5F.F.*(\"H7J:t[$='G)\\)*H;pgnPQe?F.FAF.FCF.F. *&\"Fgh([D.S1w\\s?!o`S\"GFCF.FF.F5F.F.\"#9F9F9,**&\")K[\"H\"F.FJF. F.*&\")OHTAF.FTF.F9*&\")0*oE\"F.F5F.F.\"(Z7J#F9F9FCF9F%%eqn7G/&%\"aG6$\"\"(\"\"&,$*&#\"\"\" \"=)GBzF6UJP9!*G'Q'*F.*.,B**\"E?fQd3(4sq3`5wqNN$\\&*F.)&F'6$\"\"'F*\" \"#F.)&%\"cG6#\"\"%\"\"$F.&%\"bG6#F)F.!\"\"*(\"DQ;Hmqy)G@f&zW(oxI/yF.F 4F.)F:F8F.FB*(\"DonlwFY(HFxC,wF.F4F.F:F.F.*(\"CS%*y\\MBARm#R_V i%oT@F.F5F.F:F.FB*(\"C70[k1D7m^]7)ycxV<*F.F5F.FEF.F.**\"EcGF9F@-)QYlqI d(p\\qsF.F4F.FEF.F?F.F.**\"E33j'Hm>;=F.F4F.F:F.F?F.FB**\"Es QuRW_gOk=s'osHI]4%F.F4F.)F:F=F.F?F.F.*(\"Cw(>A]z/Q49'oER0L>iF.F5F.FVF. F.*(\"D[!)z9(o!yu$*y4)Rgj$[\\*)=lF.F.,&*&\"#>F.F:F.F.\"#9FBFB,&\"$<$FB*&\"$w%F. F:F.F.!\"#F?FBF:FBF5FBF.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can eliminate " }{XPPEDIT 18 0 "a[7,5]" "6# &%\"aG6$\"\"(\"\"&" }{TEXT -1 57 " from the two equations by equating the two right sides." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fac tor(simplify(rhs(eqn6)-rhs(eqn7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&#\"3Oj3(Q2qL6#\"9`_uF(F-F(F(\"#9F2F(,(*&\"+M$)*oi\"F()F-\"\"#F(F(*&\"* lkr9)F(F-F(F2\")&\\]M*F(F(F-F2,&\"$<$F2*&\"$w%F(F-F(F(!\"#&%\"aG6$\"\" '\"\"&F2&%\"bG6#\"\"(F2,&*&F,F(F-F(F(\"#BF2F2F(F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The factors " }{XPPEDIT 18 0 "``(57*c[4]-40);" "6#-%!G6#,&*&\"#d\"\"\"&%\"cG6#\"\"%F)F)\"#S!\" \"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "``(19*c[4]-14);" "6#-%!G6#,&* &\"#>\"\"\"&%\"cG6#\"\"%F)F)\"#9!\"\"" }{TEXT -1 141 " in the numerat or of the last expression occur as factors in the denominators in the \+ expressions that we have obtained for the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 23 "This rules out taking " }{XPPEDIT 18 0 "c[4] = 40/5 7;" "6#/&%\"cG6#\"\"%*&\"#S\"\"\"\"#d!\"\"" }{TEXT -1 6 " or " } {XPPEDIT 18 0 "c[4] = 14/19;" "6#/&%\"cG6#\"\"%*&\"#9\"\"\"\"#>!\"\"" }{TEXT -1 39 " to make the the last expression zero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "denom_factors := select(has,map(op,selec t(has,map(denom@rhs,e3),c[4])),c[4]):\nmember(57*c[4]-40,denom_factors );\nmember(19*c[4]-14,denom_factors);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "However, the quadrat ic factor has two real zeros." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "1626898334*c[4]^2-814716465*c[4]+93450495;\nsolve(%);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"+M$)*oi\"\"\"\")&%\"cG6#\"\" %\"\"#F&F&*&\"*lkr9)F&F(F&!\"\"\")&\\]M*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"*lkr9)\"+omz`K\"\"\"*(\"#BF'F&!\"\"\"0XfX<9:0\"#F' \"\"#F',&F$F'*(F)F'F&F*F+F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+p &R(GK!#5$\"+*3]!z " 0 "" {MPLTEXT 1 0 187 "n ormal(expand(rationalize(subs(c[4]=(814716465-23*sqrt(105151417455945) )/3253796668,eqn6))));\nnormal(expand(rationalize(subs(c[4]=(814716465 -23*sqrt(105151417455945))/3253796668,eqn7))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,$*&#\"\"\"\"`o)3JGT)GnX&\\c^B0nV%\\>$y'*3o,\"F,*(,2*&\"_ogDo%p$45jmT3MH39ld))ps.r1?!4()o&eNbOp3i)H;>p(fIYCI#F,F2F,\"0X fX<9:0\"#F,\"\"#F,*(\"ao[I(G$*f$RfIs[n%f5/VF,F8F 9F,*&\"`o\"4e0#oK#))4'eo'3LxR8$G9?6 W'yj<]LcEc8BF,*&\"fn?F-HE\\rT_P!RO$H$Q>o8E>Y$pY,'*)Q)o&F,F8F9F,F,F>F5F 2F5F,F5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,$*&#\" \"\"\"`o)3JGT)GnX&\\c^B0nV%\\>$y'*3o,\"F,*(,2*&\"_ogDo%p$45 jmT3MH39ld))ps.r1?!4()o& eNbOp3i)H;>p(fIYCI#F,F2F,\"0XfX<9:0\"#F,\"\"#F,*(\"ao[I(G$*f$RfIs[n%f5 /VF,F8F9F,*&\"`o\"4e0#oK#))4'eo'3LxR8$G9?6W'yj<]LcEc8BF,*&\"fn?F-HE\\rT_P!RO$H$Q>o8 E>Y$pY,'*)Q)o&F,F8F9F,F,F>F5F2F5F,F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "cc := (814716465-23*sqr t(105151417455945))/3253796668:\ndd := normal(expand(rationalize(subs( c[4]=cc,rhs(eqn6))))):\ne4 := \{c[4]=cc,a[7,5]=dd\}:\ne5 := `union`(e4 ,normal(expand(rationalize(subs(e4,e3))))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11427 "e5 := \{a[6,2] = -2456322 21609852/423282790552285-16879401/3230316033162175*105151417455945^(1/ 2), `b*`[2] = 0, c[2] = 5/33, a[2,1] = 5/33, b[6] = -94270962821304652 9/61497889000856074*b[7]+34155179285/430485223005992518*b[7]*105151417 455945^(1/2)+49304693440419039647/30994936056431461296-219591275963/30 994936056431461296*105151417455945^(1/2), c[8] = 1, c[7] = 1, b[2] = 0 , a[7,5] = -1/10168089678319494436705231971412059658250270175156495456 7288412831088*(-165941769450856874172006710372698857651408293408416663 10093694682560*a[6,5]+230244630597691916298620869365535585688709019506 46220655168*a[6,5]*105151417455945^(1/2)-15586805004236896477908668785 43446817430410594674872305939359932873048*a[6,5]^2*b[7]+80674761266805 63444238292170417168907714293645180629833239560*a[6,5]^2*b[7]*10515141 7455945^(1/2)+16174724269521588742690899020631728313397733086685860988 2326820558091*a[6,5]^2-72038341440781136766654663680090221265181715511 1376971153239*a[6,5]^2*105151417455945^(1/2)+2313562656335017637864411 201419020669794351429508810097080563435520+568838896014669346192613681 93832933639037524171492629022720*105151417455945^(1/2))/b[7]/a[6,5], a [5,1] = 123752083413065843138789879/209127510371847411054672640+160885 1500638397121047/90970467011753623808782598400*105151417455945^(1/2), \+ c[3] = 29/115, a[7,1] = -1/1181420461012607629002588913659265376734426 40461800208230027403252145688595200*(648883927333024704209053394737695 24965965710737591149626775354824700130842320*a[6,5]+249150459074527320 6185452372526029816705979610933464067105201815624688*a[6,5]*1051514174 55945^(1/2)-5750093640621793578579555787021100718559979425024393350169 61551835619382473525*a[6,5]^2-2265911057700864580787109842943083179127 9300043627889472150351440381415*a[6,5]^2*105151417455945^(1/2)+5522597 6157319565158476679103071876596894084021537686090515782360348416459028 00*a[6,5]^2*b[7]+21427151798974483861935789783681272550354794281414822 9430147795760874320*a[6,5]^2*b[7]*105151417455945^(1/2)-33331451915361 7433137111894686686601891973636310504541971250386186393871784320*b[7]* a[6,5]-127569839678256739243883334219138036625291181512978637647681336 15517184*b[7]*a[6,5]*105151417455945^(1/2)-546230131599244647649189927 98985645436462487989830998079414521907200*105151417455945^(1/2)-897300 658910440767378731575558400814051083306878698637980418127326434432000) /b[7]/a[6,5], a[8,2] = 0, a[5,4] = -5163182452988211119598969969808269 31/814696711715230147396851820173032960-107920362352247138873225158999 /488818027029138088438111092103819776*105151417455945^(1/2), a[5,2] = \+ -2000041633670013681/5324522345395547392+15875554484889/38602787004117 71859200*105151417455945^(1/2), a[8,7] = b[7], a[5,3] = 18902912452932 794771976769428125/17444617282574859822209614750208+100662884268346078 346663939/505893901194670934844078827756032*105151417455945^(1/2), a[4 ,3] = 285859955732027554698452564025/7242812371317610711938898070128-5 3985530215629267813075/7242812371317610711938898070128*105151417455945 ^(1/2), c[6] = 14/19, a[7,4] = 1/1817787026919735504981952970132778435 3842219485807657705732244458837062187003284740851424110080*(1012390365 8706486054719431536572196036358995996530816289726040724801606312991616 407833495167824*a[6,5]+43486432679596195570313715698059552898423711382 61329203691250111931156023102397049471600*a[6,5]*105151417455945^(1/2) +980386726753211552011711254341978383080237925081330634171844525677412 879447185066955752868217440*a[6,5]^2*b[7]+4410644965447847043248629659 89837705483240077192341260994272992444532395102556101403610528*a[6,5]^ 2*b[7]*105151417455945^(1/2)-10526171330992583339791073696683404550249 0378093143236370811003073348818551903045223550615118975*a[6,5]^2-45861 8095042339149806902853709790635336445435451902790338586591819231917166 34667307798581*a[6,5]^2*105151417455945^(1/2)-410319352779014513556760 6650296194983483414856214451560619712237654017412713723697017995531827 2*b[7]*a[6,5]-19478707067090555874056054244619379023815206360342476175 948199802618875101477336255525376*b[7]*a[6,5]*105151417455945^(1/2)-36 5154285033900928657809635875551543832711888056762427265842282810430956 121971048142607590400-979572657095865317049947349390458600501014448341 97052003980977065715012731111534382080*105151417455945^(1/2))/b[7]/a[6 ,5], `b*`[4] = -592788841464939612701317111677/55713963039240793196233 7666500*`b*`[3]-102862718604078808697213/19499887063734277618681818327 500*`b*`[3]*105151417455945^(1/2)+1346662792889767425279747/2930626805 713331769999968+233677468653559043/102571938199966611949998880*1051514 17455945^(1/2)-14502805494921569790272143811427/7901715419662515923522 2017198592*`b*`[5]-2516575711693853553604963/2765600396881880573232770 601950720*`b*`[5]*105151417455945^(1/2), a[4,1] = 37936893253997889064 8812440551/7242812371317610711938898070128-2455501968740121696101/7242 812371317610711938898070128*105151417455945^(1/2), a[8,4] = -769241794 0571585303632737642692173975725/75495968815790402592467519276757520192 06*b[7]+14177528055635349228127387485120817/75495968815790402592467519 27675752019206*b[7]*105151417455945^(1/2)+5495141443559211261561484008 008106456395/60396775052632322073974015421406016153648-455752471647170 390435208659301200603/2717854877368454493328830693963270726914160*1051 51417455945^(1/2), a[7,3] = -529/4694524668987732792414401991541218212 4578056950749207256964576271708292467795685888*(3917034701315472897295 7070350948385011283431781384966016111230193187591659325040*a[6,5]+2009 7078654691321987797993029827868216730286674972673922383846632251494992 *a[6,5]*105151417455945^(1/2)-2231260212254834301558022262974984576063 89858264082805369352436854326513361562975*a[6,5]^2-2062455653214721653 31658825946139154679405642456991279466685174941372229525*a[6,5]^2*1051 51417455945^(1/2)+1998209359622004382916281781432498455464101028933017 051924851842704302494705652800*a[6,5]^2*b[7]+1985260141974865651545469 135652124749827673116936575648997354603509133574400*a[6,5]^2*b[7]*1051 51417455945^(1/2)-1786776152603381568702752495655692065531712768142748 83027724199957265829051272960*b[7]*a[6,5]-9376517557680313754049069360 3184957933123761182387838103611856741158832896*b[7]*a[6,5]*10515141745 5945^(1/2)-45086332994312425594834561769323268083154050593082912130404 9841834880000*105151417455945^(1/2)-1793192856871112825722326865701293 130459836240143343483054394021235362721920000)/b[7]/a[6,5], a[7,6] = 3 3703180/110073164663663138611251147*(49118108088841104+1323939137*1051 51417455945^(1/2))/b[7]/a[6,5], `b*`[7] = 1/85700671444405169040000*b[ 7]*(-31865308889721975513068750408448*`b*`[3]*b[7]+4301028367493240347 1647488*`b*`[3]*b[7]*105151417455945^(1/2)-265008029849303663512957559 75424*`b*`[3]-9110315207567081157963264*`b*`[3]*105151417455945^(1/2)+ 962363906869137700229025025401750*b[7]*`b*`[5]+74193690947594350118872 50*`b*`[5]*b[7]*105151417455945^(1/2)-21488051949361734384719261949412 5*`b*`[5]-8724670643651603785758375*`b*`[5]*105151417455945^(1/2)-1449 486210235141568393961451604000*b[7]+53499014059415963995236000*b[7]*10 5151417455945^(1/2)+273386682544172631265829189038000-9122981896844064 57926000*105151417455945^(1/2))/(113792409224*b[7]^2-49452519266*b[7]+ 4719263081), c[5] = 317/476, a[4,2] = 9899096170368469289460332229/624 38037683772506137404293708+45207024234619537323/6243803768377250613740 4293708*105151417455945^(1/2), c[4] = 814716465/3253796668-23/32537966 68*105151417455945^(1/2), a[6,1] = 5869152856890694106603/664999190006 1903433630+27960789874590937/964248825508975997876350*105151417455945^ (1/2)-11115420061407/3559626326368*a[6,5]-7180367/53394394895520*a[6,5 ]*105151417455945^(1/2), `b*`[8] = -1/4456434915109068790080000*(29347 364414144425182241814222564000*b[7]+68719876794408145507312069655424*` b*`[3]-2075408380326842389377516590088375*`b*`[5]-11019096674855314941 6375254840640000*b[7]^2-2098148045712163160330575312595712*`b*`[3]*b[7 ]-467211203954657318404724000*b[7]*105151417455945^(1/2)+4005889096557 8340976302000*105151417455945^(1/2)+1057413658189103890047606013416825 0*b[7]*`b*`[5]+498228950497047264233254656*`b*`[3]*b[7]*10515141745594 5^(1/2)-286016701610819525905421250*`b*`[5]*b[7]*105151417455945^(1/2) -16000413862253835321406125*`b*`[5]*105151417455945^(1/2)-927548327010 64297816450944*`b*`[3]*105151417455945^(1/2)-1443978653065216832071854 740758000+3747861763606726852457280000*b[7]^2*105151417455945^(1/2))/( 113792409224*b[7]^2-49452519266*b[7]+4719263081), a[6,3] = 38255781995 508908579388045/29195626157390435764414844+4990324590494804162637/1608 6790012722130106192579044*105151417455945^(1/2)-3369463217005108075/14 84652086711472448*a[6,5]-2184088132225/1484652086711472448*a[6,5]*1051 51417455945^(1/2), a[3,2] = 27753/132250, b[4] = -76924179405715853036 32737642692173975725/7549596881579040259246751927675752019206*b[7]+141 77528055635349228127387485120817/7549596881579040259246751927675752019 206*b[7]*105151417455945^(1/2)+549514144355921126156148400800810645639 5/60396775052632322073974015421406016153648-45575247164717039043520865 9301200603/2717854877368454493328830693963270726914160*105151417455945 ^(1/2), `b*`[6] = 98520735705503/228300445747920+473271/76100148582640 *105151417455945^(1/2)+266706498861868711/14467351684452826250*`b*`[3] -9057810207/629015290628383750*`b*`[3]*105151417455945^(1/2)-200319736 8175497265953/2051853604565968636160*`b*`[5]-5096864111511/20518536045 65968636160*`b*`[5]*105151417455945^(1/2), a[3,1] = 5597/132250, a[7,2 ] = -759/143171901746800*(297562833440*b[7]+17544*b[7]*105151417455945 ^(1/2)-51024131635-1613*105151417455945^(1/2))/b[7], a[8,6] = -9427096 28213046529/61497889000856074*b[7]+34155179285/430485223005992518*b[7] *105151417455945^(1/2)+49304693440419039647/30994936056431461296-21959 1275963/30994936056431461296*105151417455945^(1/2), a[8,3] = -33960912 645734178125/26196476257158418222*b[7]-45838830303125/2619647625715841 8222*b[7]*105151417455945^(1/2)+960653090962834563625/1886146290515406 111984+294708077826875/1886146290515406111984*105151417455945^(1/2), b [5] = -35682296032836675364240/25067262353935058814309+318023772693424 /25067262353935058814309*105151417455945^(1/2)+15398721484436804454976 0/8355754117978352938103*b[7]-1187168361832320/8355754117978352938103* b[7]*105151417455945^(1/2), `b*`[1] = 50014237660083/817977580652288*` b*`[5]+10769441/4089887903261440*`b*`[5]*105151417455945^(1/2)+3666659 /91012656-1/151687760*105151417455945^(1/2)+281559050571/5767453049750 *`b*`[3]+440191/28837265248750*`b*`[3]*105151417455945^(1/2), a[6,4] = 60907122103756303924071/13867213633780112188352*a[6,5]+33397568407445 6563/208008204506701682825280*a[6,5]*105151417455945^(1/2)-22687761240 911572992214558187343/25906331138739929274640250167820-865231957572881 3734795829/25906331138739929274640250167820*105151417455945^(1/2), b[1 ] = -8886903525/4980234206*b[7]-2279/34861639442*b[7]*105151417455945^ (1/2)+194464032337/836679346608+73261/12550190199120*105151417455945^( 1/2), b[3] = -33960912645734178125/26196476257158418222*b[7]-458388303 03125/26196476257158418222*b[7]*105151417455945^(1/2)+9606530909628345 63625/1886146290515406111984+294708077826875/1886146290515406111984*10 5151417455945^(1/2), a[8,5] = -35682296032836675364240/250672623539350 58814309+318023772693424/25067262353935058814309*105151417455945^(1/2) +153987214844368044549760/8355754117978352938103*b[7]-1187168361832320 /8355754117978352938103*b[7]*105151417455945^(1/2), a[8,1] = -88869035 25/4980234206*b[7]-2279/34861639442*b[7]*105151417455945^(1/2)+1944640 32337/836679346608+73261/12550190199120*105151417455945^(1/2)\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 22 "We choose values for " }{XPPEDIT 18 0 "b[7]" "6#& %\"bG6#\"\"(" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG 6$\"\"'\"\"&" }{TEXT -1 54 " that minimize the norm of the principal \+ error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\nsm := 0:\nfor ct to nops(err terms5_7) do\n sm := sm+expand(rationalize(subs(e5,errterms5_7[ct])) )^2;\nend do:\nsm := rationalize(sm):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3537 "sm := 1/681176131781726524633294910466023 6924849138305799172715919808662682552031155852018791236946036421220973 49269206965871163626170273839183031367237632000000*(-22646834719201002 8359011287886850566304097012812374116401197247069035917712743861912193 81176297849758357703318536186598436244702934977782727680000*a[6,5]+195 6294198876608267220058506438086172029366311695957947735008450290424851 8840796917983181001275663921416086567888650518977518805188179534080000 *a[6,5]^3*b[7]*105151417455945^(1/2)+580822307218493401506212034445205 9232943927205956293804516349277812536696926241131611866713783685822557 98270180408899447020296030296292469691014400000*a[6,5]^3*b[7]-10213876 3903029049913355452682947278139206605151065193363176818643033158577970 2167734953394665264054192652874722242804310797667905061185270615996780 0000*a[6,5]^4*b[7]+490862467687264713494647297672031106034357344755032 5248391195511330345389587786917976885892668004290956629299738458424555 3036911952002808991427699200000*a[6,5]^4*b[7]^2+8801572364254424528786 9013788207400314360048678303415860063438750593569544769050338444031272 007273498839469298427271895436487830167003190615505920000*b[7]^2*a[6,5 ]^2+287829529280780093709085333170504745800035973465966420890544799969 7814379443586982839346338687311560215454860416942309114655332680712293 422955281152*a[6,5]^2-167851107657269238299658166813984094291183668943 1264457467783151020399172078087180089070669332766695062704068929411682 5515933497765712133254405785600*a[6,5]^2*b[7]+531352986513944379162993 0086093507619508174523756889358616824123254988008017044366450910109446 66931842345950989651977754127834581399751684291064565625*a[6,5]^4-6045 1364396134543845698397585999175090391743751808835122461551837001867523 8178214541006327785316006531920651805599067750976489852184987549013280 00000*a[6,5]^3-3005631901427359348634776688076904744104737173313964482 7074123267180863073005560700355663529435200378655583830195585037393238 4483729649693800000*a[6,5]^4*b[7]*105151417455945^(1/2)+15984903630480 3151186463556335927810096506927462532999061629525027367994075729400971 17096852809231643705509715416832812794431854299773558021875*a[6,5]^4*1 05151417455945^(1/2)+3158681766025450569464029966132052051662229243079 8651416829469704686820074062455129123671432045884552543477092850907330 25995446168545111040000*b[7]^2*a[6,5]^2*105151417455945^(1/2)+44276987 4509468539843578847950669291839437149148343352876957396990342247210193 4623349174872581929149241540892487563689430774876354191238348800000+33 0718327384028752060965800517112824270919186962223562459278394969779170 824349654864392049077087024508866685074668480271853374443929600000*105 151417455945^(1/2)+871105994085594239451077871260795275488162135943860 4559809459549067321258959400051609067221221016884168040890618047949825 7956781065063914240*a[6,5]^2*105151417455945^(1/2)-1152447275982401223 7040079501550862445212890234032411707948074254564682702145734087027338 81797866345652904768289226490301067096529024000000*a[6,5]*105151417455 945^(1/2)-628547020523256687570001298317950679116106467696835862190241 6121558448814625898391893471620924614534080719709910967181294927891827 61656985600*a[6,5]^2*b[7]*105151417455945^(1/2)+1412090889973284366271 5354494952132843127797996437838962528101454378160758887349560458092304 20322935867659681125772623022384577005379096736000000*a[6,5]^4*b[7]^2* 105151417455945^(1/2)-207452849192841153310521211276118745332196466802 8773013932929240631459893517226652263586210169146265803179175558244542 001287417486753804160000*a[6,5]^3*105151417455945^(1/2))/a[6,5]^2:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indets(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\" aG6$\"\"'\"\"&&%\"bG6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "plot3d(sm,b[7]=0.01..0.105, a[6,5]=0.1..0.7,labels=[`b[7]`,`a[6,5]`,``],grid=[40,60],style=patch,l inestyle=2,\n lightmodel=light2,color=COLOR(RGB,1,.45,0),axes=boxed,o rientation=[-62,59],view=[0.01..0.105,0.1..0.7,0..4e-6]);" }}{PARA 13 "" 1 "" {GLPLOT3D 509 487 487 {PLOTDATA 3 "6*-%%GRIDG6&;$\"\"\"!\"#$\" $0\"!\"$;$F(!\"\"$\"\"(F/X,%)anythingG6\"6\"[gl'!%\"!!#[at\"I\"gn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b[7]G%'a[6,5]G%!G-%%VIEWG6%F&F-;F> $\"\"%!\"'-%+PROJECTIONG6%$!#iF;$\"#fF;F(" 1 2 0 2 10 0 2 1 4 2 2 1.000000 59.000000 -62.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "evalf[15](minimize (evalf[20](sm),b[7]=0..0.1,a[6,5]=0.1..0.7,'location'));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$$\"0nV8-]%QJ!#@<#7$<$/&%\"aG6$\"\"'\"\"&$\"0LI[N yqJ&!#:/&%\"bG6#\"\"($\"0*o:)=CYI*!#;F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use rational approximations for the values of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 54 " that mimimize the norm of the principal error terms." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "[.930462418815689e-1,.531707 835483033];\nmap(convert,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"0*o:)=CYI*!#;$\"0LI[NyqJ&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"#\"*\"$y*#\"#n\"$E\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "We set " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[7] = 91/978;" "6#/&%\"bG6#\"\"(*&\"#\"*\"\"\"\"$y* !\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5] = 67/126;" "6#/&% \"aG6$\"\"'\"\"&*&\"#n\"\"\"\"$E\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Values for the param eters " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 110 " can be chosen so as to give a compatible stability region for the embedded 8 stage, order 4 scheme, namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`b*`[3] = 15/53;" "6#/&%#b*G6#\"\"$*&\"#:\"\"\"\"#`!\" \"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5] = 39/125;" "6#/&%#b* G6#\"\"&*&\"#R\"\"\"\"$D\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "prm := \{b[ 7]=91/978,a[6,5]=67/126,`b*`[3]=15/53,`b*`[5]=39/125\}:\ne6 := `union` (prm,simplify(subs(prm,e5))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5762 "e6 := \{`b*`[2] = 0, a[8,7] = 91/978, b[7] = 91/978, a[6,5] = 67/126, c[8] = 1, c[7] = 1, b[2] = 0, a[7,6] = 97141 01019226554178088904960/31957908807350197910133249679+2618357876664110 12880/31957908807350197910133249679*105151417455945^(1/2), `b*`[5] = 3 9/125, `b*`[3] = 15/53, `b*`[7] = 16859428709040130745055637936223921/ 188074229500551422433322061497920000+244403905585142824289185521/21941 99344173099928388757384142400000*105151417455945^(1/2), `b*`[8] = 1130 93754206156431292803151127249/12734192622433169227256181247255000-5296 7678767390756540012887153/99836070159876046741688460978479200000*10515 1417455945^(1/2), a[8,2] = 0, `b*`[6] = 65572753075427021134812949889/ 496177443284238540373654560000+9841360159811270667/7190977438902007831 502240000*105151417455945^(1/2), `b*`[1] = 14475354021635991174701/197 802622822953213408000-478269920776171/329671038038255355680000*1051514 17455945^(1/2), a[6,1] = -49225589022301233744183735779/63282599719433 884959882336960-3510687788120584753272949/8258379263386121987264644973 2800*105151417455945^(1/2), `b*`[4] = 64406830726760624192719424755704 6508737/6369286362605991319749019126566156224000+111760904420113865733 188135964353/222925022691209696191215669429815467840000*10515141745594 5^(1/2), b[4] = -37638654822534247872916940674662423328915/98446743335 79068498057764513689180633044624+3121650305280233144636524202663962531 /443010345011058082412599403116013128487008080*105151417455945^(1/2), \+ a[6,3] = 86276540353665039847710695810845/8334935522218488170935152694 62144-216788956050328897851037816063/459254947274238698218526913473641 344*105151417455945^(1/2), b[5] = 1190204022066367942642960/4085963763 691414586732367-2178285514342448/4085963763691414586732367*10515141745 5945^(1/2), b[1] = 9053807089231/136378733497104-501797/20456810024565 60*105151417455945^(1/2), a[8,3] = 119501137217800311358375/3074418453 54011196253392-2018586005231875/307441845354011196253392*1051514174559 45^(1/2), a[8,6] = 830592632727775794785/5052174577198328191248+150407 7797251/5052174577198328191248*105151417455945^(1/2), a[6,4] = 3598731 58906430802491237985928506042059/2465296211318278050438973515250073414 40+230654465754417653852094653348797/443753318037290049079015232745013 214592*105151417455945^(1/2), a[7,2] = 59734302131229/44926355375720-1 441341/1302864305895880*105151417455945^(1/2), a[8,5] = 11902040220663 67942642960/4085963763691414586732367-2178285514342448/408596376369141 4586732367*105151417455945^(1/2), b[3] = 119501137217800311358375/3074 41845354011196253392-2018586005231875/307441845354011196253392*1051514 17455945^(1/2), a[8,1] = 9053807089231/136378733497104-501797/20456810 02456560*105151417455945^(1/2), c[5] = 317/476, a[4,2] = 9899096170368 469289460332229/62438037683772506137404293708+45207024234619537323/624 38037683772506137404293708*105151417455945^(1/2), c[4] = 814716465/325 3796668-23/3253796668*105151417455945^(1/2), a[7,3] = 1524979679765223 1988969792537921142476851854434785659517885170619034922780974618277305 /207266501739028394324952685445159640092572437165623318950343912141403 953196522628760064+318954894142076600206191264254918082660742098807666 98111670849008793214933062155/2862251690681820683535060894242680744135 52413228717916645713021528605459176150296859136*105151417455945^(1/2), a[3,2] = 27753/132250, a[7,1] = 1038263320304911199846238164482685063 317425776107411696073055841920084894548437/347736854176255731022079256 86940542768033506567373593703788548575160867886583040+6756803198934452 0349682615047488926280454856996955945138286279519534217777/30253106313 3342485989208953476382722081891507136150265222960372603899550613272448 0*105151417455945^(1/2), a[3,1] = 5597/132250, a[7,5] = 46156557089066 32101974370650179465855337591303167331051987992414499309155/1301891698 1429931091924277859236858824633898424165122087931906513654014256-20103 5901905594888203478188336475409356265197258304205097572636855/13018916 981429931091924277859236858824633898424165122087931906513654014256*105 151417455945^(1/2), a[8,4] = -3763865482253424787291694067466242332891 5/9844674333579068498057764513689180633044624+312165030528023314463652 4202663962531/443010345011058082412599403116013128487008080*1051514174 55945^(1/2), a[5,1] = 123752083413065843138789879/20912751037184741105 4672640+1608851500638397121047/90970467011753623808782598400*105151417 455945^(1/2), c[3] = 29/115, a[5,2] = -2000041633670013681/53245223453 95547392+15875554484889/3860278700411771859200*105151417455945^(1/2), \+ a[5,4] = -516318245298821111959896996980826931/81469671171523014739685 1820173032960-107920362352247138873225158999/4888180270291380884381110 92103819776*105151417455945^(1/2), a[5,3] = 18902912452932794771976769 428125/17444617282574859822209614750208+100662884268346078346663939/50 5893901194670934844078827756032*105151417455945^(1/2), a[4,3] = 285859 955732027554698452564025/7242812371317610711938898070128-5398553021562 9267813075/7242812371317610711938898070128*105151417455945^(1/2), a[7, 4] = -1693660123817839142993451013257403042205655397475220738447116534 58806174655894526676245216204127995/1551626650438147832342495416245937 01687526417086957004644589292251741395415822637890959585918820864-2918 9409706887443462177638994076165303922403415199244273965837521205894200 4402007722166107207/23274399756572217485137431243689055253128962563043 55069668839383776120931237339568364393788782312960*105151417455945^(1/ 2), b[6] = 830592632727775794785/5052174577198328191248+1504077797251/ 5052174577198328191248*105151417455945^(1/2), c[6] = 14/19, a[2,1] = 5 /33, a[6,2] = -245632221609852/423282790552285-16879401/32303160331621 75*105151417455945^(1/2), c[2] = 5/33, a[4,1] = 3793689325399788906488 12440551/7242812371317610711938898070128-2455501968740121696101/724281 2371317610711938898070128*105151417455945^(1/2)\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "printed coe fficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 484 "for ii from 2 to 8 do\n print(`= =====================================================`);\n print(c[i i]=subs(e6,c[ii]));print(``); \n for jj to ii-1 do\n print(a[ii ,jj]=subs(e6,a[ii,jj]));print(``);\n end do:\nend do:\n`########### ###########################################`;\n``;\nfor ii to 7 do\n \+ print(b[ii]=subs(e6,b[ii]));print(``);\nend do:\n`################### ###################################`;\n``;\nfor ii to 8 do\n print(` b*`[ii]=subs(e6,`b*`[ii]));print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"&\"#L" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"#\"\"\"#\"\"&\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W================================ ======================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"$#\"#H\"$:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"%(f&\"']A8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"$\"\"##\"&`x#\"']A8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W========================================== ============G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,&#\"*l kr9)\"+omz`K\"\"\"*(\"#BF,F+!\"\"\"0XfX<9:0\"#F,\"\"#F/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"%\"\"\",&#\"?^0W7)[1*)y*RD$*o$z$\"@G,2)*)Q>r5wJrB\"GC(F(*(\"7,hp@, uo>]bCF(F,!\"\"\"0XfX<9:0\"#F(\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,&#\"=HAL g%*Gp%o.<'4**)*\">3PH/u81Dx$oP!QC'\"\"\"*(\"5Bt`>YBCq?XF-F,!\"\"\"0XfX <9:0\"#F-F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,&#\"?DSc_%)pav-Kd&*feG\"@G,2)*)Q >r5wJrB\"GC(\"\"\"*(\"8vI\"yEHc@Ib)R&F-F,!\"\"\"0XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======================================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$<$\"$w%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"& \"\"\",&#\"+% )f#y3Qi`<,n/(4*!\"\"\"0XfX<9:0\"#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\" \"#,&#\"4\"o8+nL;/+?\"4#RZbRXB_C`!\"\"*(\"/*)[[ab(e\"\"\"\"\"7+#f=x6/q y-'QF-\"0XfX<9:0\"#F0F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"AD\"G%pn(>x%zK HX7H!*=\"A3-v9'4A#)f[d#G,R*e]!\"\"\"0XfX<9:0\"#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"EJp#3 )p*p*)f>6@))HX#=j^\"EgH.t,#=&oRZ,B:Q5#46\"Q%)3Q\"Hq-=))[F-\"0XfX<9:0\"#F0\"\"#F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======= ===============================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#9\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",&#\" >zdt$=WPB,B-*eD#\\\">gpL#))f\\)QV>(*f#Gj!\"\"*(\":\\HF`Ze?\")y(o5NF(\" A+Gt\\kks)>7'Qj#z$e#)F-\"0XfX<9:0\"#F(\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#,&#\"0_)4;AKcC\"0&G_0z#GB%!\"\"*(\"),%zo\"\"\"\"\"1v@;LgJIKF-\"0XfX <9:0\"#F0F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,&#\"AX3\"ep5x%)R]m`.awi)\"BW@Yp_ ^$4<)[=A_N\\L)\"\"\"*(\"?jg\"y.^y*)G.0c*)y;#F-\"EW8ktM\"p_=#)pQUFZ\\Df %!\"\"\"0XfX<9:0\"#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,&#\"Hf?/1&Gf)zB\" \\-3V1*eJ()f$\"HS9M2]_^t*Q/0y#=8@'HlC\"\"\"*(\"B(z[Ll%4_Ql%e0tgp6u5wdU6\\I? LEQ5\"[pSIe')y'3;v&[&)y.Pfttc1N.oFaSpoDz?-JdDwT&otZ$F(*(\"eoxx@M&>&ziG Q^%f&p*p&[X!GE*)[Z]h#o\\._W$*)>.ov'F(\"]p![CF81b**Qgs.'HAl-:Or]\"*=3AF QwM&*3#*)f[UL8j5`-$!\"\"\"0XfX<9:0\"#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"#,&#\"/H78-Vtf\"/?dPbj#\\%\"\"\"*(\"(T8W\"F-\"1!)e*eIkGI\"!\"\"\"0Xf X<9:0\"#F-F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,&#\"ap0tF=Y(4yA\\.>1<&)y^fcyMW &=&oZU6#z`#zp*))>B_wz'z\\_\"\"bpk+wGE_'>`RST@\"RM]*=LilrVsD4S'f^W&o_\\ K%RG!R<]ms?\"\"\"*(\"[pb@1L\\@$z3!\\3n6\")pm2))4U2m#3=\\Dk7>1-gw?9%*[& *=$F-\"bpO\"foH]h:MSARIlh2%**Q w0Jt;.8fP`&eYz,lqV(>5Km!*3dl:Y\"eocU,aO^1>$z3A^;C%)*QjC)eoBfy FC>4J*H9)p\"*=I\"\"\"\"*(\"]obojsv40UIes>liN4aZO$)=yM?))[f0>!f.,#F-F,! \"\"\"0XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"=g\\!*)3yTbE#> 559(*\">z'\\K85z>]t!)3z&>$\"\"\"*(\"6!)G,6km(yN=EF-F,!\"\"\"0XfX<9:0\" #F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"\",&#\".J#*32Q0*\"0/r\\L(yj8F(*(\"'(z,&F(\"1glX-5oX?!\"\" \"0XfX<9:0\"#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"$,&#\"9v$e8J+y@P6]>\"\"9#R`i>6SNX=W2$\"\"\"*(\"1v=B0ge=?F-F,! \"\"\"0XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\"J:*GLUiYnSp\"H (yCMD#[lQw$\"LCY/L1=*o8Xwd!)\\o!zNLuY%)*!\"\"*(\"FJD'Rm-U_OY9L-G0.l@J \"\"\"\"N!33q[GJ,;JS*f7C3e5,X.,V%F-\"0XfX<9:0\"#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"&,&#\":gHkUzOm?-/->\"\":nBt'e99pjP'f3%\"\"\"*(\"1[CM9bGy@F-F ,!\"\"\"0XfX<9:0\"#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"6&y%zvxsKEfI) \"7[7>G$)>xX<_]\"\"\"*(\".^szxS]\"F-F,!\"\"\"0XfX<9:0\"#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"(#\"#\"*\"$y*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W############################### #######################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",&#\".J#*32Q0*\"0/r\\L( yj8F'*(\"'(z,&F'\"1glX-5oX?!\"\"\"0XfX<9:0\"#F'\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,&#\"9v$e8J+y@P6]>\"\"9#R`i>6SNX=W2$\"\" \"*(\"1v=B0ge=?F,F+!\"\"\"0XfX<9:0\"#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%,& #\"J:*GLUiYnSp\"H(yCMD#[lQw$\"LCY/L1=*o8Xwd!)\\o!zNLuY%)*!\"\"*(\"FJD' Rm-U_OY9L-G0.l@J\"\"\"\"N!33q[GJ,;JS*f7C3e5,X.,V%F,\"0XfX<9:0\"#F/\"\" #F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,&#\":gHkUzOm?-/->\"\":nBt'e99pjP'f3%\"\" \"*(\"1[CM9bGy@F,F+!\"\"\"0XfX<9:0\"#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',& #\"6&y%zvxsKEfI)\"7[7>G$)>xX<_]\"\"\"*(\".^szxS]\"F,F+!\"\"\"0XfX<9:0 \"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"#\"*\"$y*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W############## ########################################G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\", &#\"8,Z<\"*fj@SNvW\"\"9+!3M@`H#Gi-y>F'*(\"0rhx?*p#y%F'\"9++ob`DQ!Q5nH$ !\"\"\"0XfX<9:0\"#F'\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$ #\"#:\"#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%,&#\"HP(3l/dvC%>F>C1wE2$oSk\"I+SAchcE\">! \\(>8*fgijGpj\"\"\"*(\"B`V'f8)=Ld'Q6?W!4w6\"F,\"K++%yY:)H%pc@\">'p47pA ]#HA!\"\"\"0XfX<9:0\"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&#\"#R\"$D\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"',&#\">*))\\H\"[8@qUvIvsb'\"?++caOPS&QUGVu<'\\\"\"\"*(\" 4n1F6)f,OT)*F,\"=++C-:$y+-*Qu(4>(!\"\"\"0XfX<9:0\"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b *G6#\"\"(,&#\"D@RAOzjb]uI,/4(G%fo\"\"E++#z\\h?KLCU^0]HU2)=\"\"\"*(\"<@ b=*GCG9&e0RSW#F,\"F++SUTQd()QG**4tTM*>%>#!\"\"\"0XfX<9:0\"#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"),&#\"B\\s7^J!GHJk:1Uv$48\"\"D+]DZ7=csApJVAE>MF\"\"\" \"*(\">`r)G,Slv!Rn(yw'H&F,\"G++?z%y4Y)oTn/w)f,2O)**!\"\"\"0XfX<9:0\"#F ,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "s ubs(e6,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b` ,seq(b[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]])):\nevalf[8](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+$\")::::!\")F(%!GF+F+F +F+F+F+7+$\")\"R<_#F*$\")h8KU!\"*$\")b_)4#F*F+F+F+F+F+F+7+$\")+0z-*[F1$\") " 0 "" {MPLTEXT 1 0 144 "errterms5_7 := PrincipalErrorTerms(5,7,'expanded'):\nsqrt(exp and(rationalize(add(subs(e6,errterms5_7[i])^2,i=1.. nops(errterms5_7)) )));\nevalf(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&\"^p+?V!4z\\u-G \\:%\\![eh!G5]KXs*)zvx>'Hlk17@(3ra$**y]!\"\",&\"[u9@=!yG@'e3Hmu;_+o4** 3)R`H*eI#QLeAA067daS0em\"*>w7-(HxL)RPTtBG7e$4w%pJI$*Q9v$)3cS_P4D#o8iY7 \"\"\"*&\"ctq\\%p^[J_Ly:`JBkrYkF,`cu.#>)Q3wUf)*\\Q\"G/dYI>=E%F)\"0XfX<9:0\"#F)\"\"#F&F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&4(=-c!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "expand(subs(e6,Stabil ityFunction(5,7,'expanded')));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\" \"\"F$%\"zGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F $\"#CF$*$)F%\"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F$*&#\"#8\"&x<\"F$*$ )F%F-F$F$F$*&#F$\"&'H:F$*$)F%\"\"(F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "# ---------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6072 "ee := \{c[2]=5/33,\nc[3]=29/115,\nc[4]=814716465/3253796668-23/3 253796668*105151417455945^(1/2),\nc[5]=317/476,\nc[6]=14/19,\nc[7]=1, \nc[8]=1,\n\na[2,1]=5/33,\na[3,1]=5597/132250,\na[3,2]=27753/132250,\n a[4,1]=379368932539978890648812440551/7242812371317610711938898070128- \n 2455501968740121696101/7242812371317610711938898070128*105 151417455945^(1/2),\na[4,2]=9899096170368469289460332229/6243803768377 2506137404293708+\n 45207024234619537323/62438037683772506137 404293708*105151417455945^(1/2),\na[4,3]=28585995573202755469845256402 5/7242812371317610711938898070128-\n 53985530215629267813075/ 7242812371317610711938898070128*105151417455945^(1/2),\na[5,1]=1237520 83413065843138789879/209127510371847411054672640+\n 160885150 0638397121047/90970467011753623808782598400*105151417455945^(1/2),\na[ 5,2]=-2000041633670013681/5324522345395547392+\n 158755544848 89/3860278700411771859200*105151417455945^(1/2),\na[5,3]=1890291245293 2794771976769428125/17444617282574859822209614750208+\n 10066 2884268346078346663939/505893901194670934844078827756032*1051514174559 45^(1/2),\na[5,4]=-516318245298821111959896996980826931/81469671171523 0147396851820173032960-\n 107920362352247138873225158999/4888 18027029138088438111092103819776*105151417455945^(1/2),\na[6,1]=-49225 589022301233744183735779/63282599719433884959882336960-\n 351 0687788120584753272949/82583792633861219872646449732800*10515141745594 5^(1/2),\na[6,2]=-245632221609852/423282790552285-16879401/32303160331 62175*105151417455945^(1/2),\na[6,3]=86276540353665039847710695810845/ 833493552221848817093515269462144-\n 216788956050328897851037 816063/459254947274238698218526913473641344*105151417455945^(1/2),\na[ 6,4]=359873158906430802491237985928506042059/2465296211318278050438973 51525007341440+\n 230654465754417653852094653348797/443753318 037290049079015232745013214592*105151417455945^(1/2),\na[6,5]=67/126, \na[7,1]=1038263320304911199846238164482685063317425776107411696073055 841920084894548437/\n 347736854176255731022079256869405427680 33506567373593703788548575160867886583040+\n 6756803198934452 0349682615047488926280454856996955945138286279519534217777/\n \+ 302531063133342485989208953476382722081891507136150265222960372603899 5506132724480*105151417455945^(1/2),\na[7,2]=59734302131229/4492635537 5720-1441341/1302864305895880*105151417455945^(1/2),\na[7,3]=152497967 9765223198896979253792114247685185443478565951788517061903492278097461 8277305/\n 20726650173902839432495268544515964009257243716562 3318950343912141403953196522628760064+\n 31895489414207660020 619126425491808266074209880766698111670849008793214933062155/\n \+ 2862251690681820683535060894242680744135524132287179166457130215286 05459176150296859136*\n 105151417455945^(1/2),\na[7,4]=-16936 6012381783914299345101325740304220565539747522073844711653458806174655 894526676245216204127995/\n 155162665043814783234249541624593 701687526417086957004644589292251741395415822637890959585918820864-\n \+ 2918940970688744346217763899407616530392240341519924427396583 75212058942004402007722166107207/\n 2327439975657221748513743 1243689055253128962563043550696688393837761209312373395683643937887823 12960*\n 105151417455945^(1/2),\na[7,5]=461565570890663210197 4370650179465855337591303167331051987992414499309155/\n 13018 916981429931091924277859236858824633898424165122087931906513654014256- \n 2010359019055948882034781883364754093562651972583042050975 72636855/\n 1301891698142993109192427785923685882463389842416 5122087931906513654014256*105151417455945^(1/2),\na[7,6]=9714101019226 554178088904960/31957908807350197910133249679+\n 261835787666 411012880/31957908807350197910133249679*105151417455945^(1/2),\na[8,1] =9053807089231/136378733497104-501797/2045681002456560*105151417455945 ^(1/2),\na[8,2]=0,\na[8,3]=119501137217800311358375/307441845354011196 253392-\n 2018586005231875/307441845354011196253392*105151417 455945^(1/2),\na[8,4]=-37638654822534247872916940674662423328915/98446 74333579068498057764513689180633044624+\n 3121650305280233144 636524202663962531/443010345011058082412599403116013128487008080*10515 1417455945^(1/2), a[8,5]=1190204022066367942642960/4085963763691414586 732367-\n 2178285514342448/4085963763691414586732367*10515141 7455945^(1/2),\na[8,6]=830592632727775794785/5052174577198328191248+15 04077797251/5052174577198328191248*105151417455945^(1/2), a[8,7]=91/97 8,\n\nb[1]=9053807089231/136378733497104-501797/2045681002456560*10515 1417455945^(1/2),\nb[2]=0,\nb[3]=119501137217800311358375/307441845354 011196253392-\n 2018586005231875/307441845354011196253392*105 151417455945^(1/2),\nb[4]=-37638654822534247872916940674662423328915/9 844674333579068498057764513689180633044624+\n 312165030528023 3144636524202663962531/443010345011058082412599403116013128487008080* \n 105151417455945^(1/2),\nb[5]=1190204022066367942642960/408 5963763691414586732367-\n 2178285514342448/408596376369141458 6732367*105151417455945^(1/2), b[6]=830592632727775794785/505217457719 8328191248+1504077797251/5052174577198328191248*105151417455945^(1/2), \nb[7]=91/978,\n\n`b*`[1]=14475354021635991174701/19780262282295321340 8000-\n 478269920776171/329671038038255355680000*105151417455 945^(1/2),\n`b*`[2]=0,\n`b*`[3]=15/53,\n`b*`[4]=6440683072676062419271 94247557046508737/6369286362605991319749019126566156224000+\n \+ 111760904420113865733188135964353/22292502269120969619121566942981546 7840000*105151417455945^(1/2),\n`b*`[5]=39/125,\n`b*`[6]=6557275307542 7021134812949889/496177443284238540373654560000+\n 9841360159 811270667/7190977438902007831502240000*105151417455945^(1/2), `b*`[7]= 16859428709040130745055637936223921/1880742295005514224333220614979200 00+\n 244403905585142824289185521/219419934417309992838875738 4142400000*105151417455945^(1/2), `b*`[8]=1130937542061564312928031511 27249/12734192622433169227256181247255000-\n 5296767876739075 6540012887153/99836070159876046741688460978479200000*105151417455945^( 1/2)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#----------------- ----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability region for the 7 stage, order 5 scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stabi lity function R for the 7 stage, order 5 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(subs(ee,StabilityFunction(5,7,'expanded'))):\nR := unappl y(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"z G,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*& #F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"#8\"&x<\"F) *$)F'F1F)F)F)*&#F)\"&'H:F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=- 1,z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+s7/[j!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-6):\np1 := plot([R(z),-1],z=-6.89..0.49,c olor=[red,blue]):\np2 := plot([[[z0,-1]]$3],style=point,symbol=[circle ,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,-1]],linestyle=3 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-6.89..0.4 9,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3o*************)o!#<$!3-tC?8; '*eAF*7$$!3y**\\7,&o&4oF*$!3:*HQqAER+#F*7$$!3*)***\\A+P\"HnF*$!3%fKb#> gSw5\\MA%*) !#=7$$!3P*\\(3Ozh:hF*$!313Lr,'3o3(FK7$$!35+vjgGTpfF*$!3uX/Uz*G%[dFK7$$ !3&**\\PN(f-=eF*$!3-[6&R&)o5n%FK7$$!3q*\\P$Q9YhcF*$!3Zx+!p2`o\"QFK7$$! 3B*\\(e2!*R0bF*$!31Uk_LV^oJFK7$$!3Y***\\]hp[M&F*$!3K0H'\\Jq8m#FK7$$!3% )**\\sQ_Z._F*$!3*QcOK`u[J#FK7$$!3&****\\,m)HW]F*$!3IHot3Wj1?FK7$$!3Q++ D([oW)[F*$!3yqTF*$!3DY#e0'e'pT*!#>7$$!39 ++5N7RuRF*$!3u4(4j#\\1u!)F\\r7$$!3V*\\Pi\\A_\"QF*$!3YG*)yA-%3g'F\\r7$$ !3_*\\7n@cOm$F*$!3'Gq,+F=&*=&F\\r7$$!3i**\\P/F^0NF*$!3bsn@EOz.PF\\r7$$ !3c*\\()Hf(GgLF*$!3-6(4U=PaK#F\\r7$$!3y**\\7\"pUO?$F*$!38XE\"*p]&y?)!# ?7$$!3q*\\PBmK4/$F*$\"3o$Q$*RgEpn(F[t7$$!31+D'R5$H**GF*$\"3=w!R]Mh*z@F \\r7$$!3s**\\d+&=ju#F*$\"3=!GR@s+$\\PF\\r7$$!3i****pm2G)e#F*$\"3bB?)H1 H(QaF\\r7$$!3#***\\Fw;nLCF*$\"3#4X_I)3J&=(F\\r7$$!39+vVJ%zSG#F*$\"3%)* y55E0^**)F\\r7$$!3y**\\K[F)z6#F*$\"3/$yN_w,)=6FK7$$!3k++5\"\\P(o>F*$\" 3o\"Gnb)H4P8FK7$$!3/++DBjQ4=F*$\"3G)3y,9Q$)f\"FK7$$!3-+v$4[!*\\m\"F*$ \"3'o*f\\j(eg'=FK7$$!3_***\\`oFr]\"F*$\"3M?()=9Q!**>#FK7$$!3Q+vB(4\"fe 8F*$\"35/CU'3r6c#FK7$$!30+vyjvK.7F*$\"37FV^EB=(*HFK7$$!3#****\\-3H:0\" F*$\"3WuRKmpt\"\\$FK7$$!3F)*\\Pf85E*)FK$\"392fk#)3#\\4%FK7$$!35#****f/ BbR(FK$\"3I1q+l(\\Hx%FK7$$!37)**\\Ul&GIeFK$\"3O)4*yGy'>e&FK7$$!3#=*\\P =#4!yUFK$\"3_,byTwR>lFK7$$!35)****pQG;&GFK$\"3ubhSLn\"*=vFK7$$!3]***\\ s)\\$o@\"FK$\"3/'[Xbi&Ga))FK7$$\"31i++!=PMX#F\\r$\"3\"e?Ag\"y$[-\"F*7$ $\"3A,]i(=JV!=FK$\"3yQC/-gt(>\"F*7$$\"3c2]P=6['H$FK$\"3_xOrB$y/R\"F*7$ $\"3!***************[FK$\"3d.$z^G6Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"! \")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l 7$FGFa\\l7$FMFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$Fe oFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FipFa\\l7$F^qFa\\l7$FcqFa\\l7$Fh qFa\\l7$F^rFa\\l7$FcrFa\\l7$FhrFa\\l7$F]sFa\\l7$FbsFa\\l7$FgsFa\\l7$F] tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F `vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$F cxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$ FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3o** ***>FT![jF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&S TYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIA MONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\ l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6 \"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$*o!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$ Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1152 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/12 0*z^5+13/11777*z^6+1/15296*z^7:\npts := []: z0 := 0: tt := 0:\nwhile t t<=841/60 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (17/207$$!3) *******Q/9aP!#F$\"31+++^v>Z5!#=7$$!3%)******)R6\"pU!#E$\"3'*******Hjzq :F77$$!3!******RX4JR#!#D$\"37+++H^R%4#F77$$!3^*****H%3s,\"*FA$\"35+++1 S*zh#F77$$!3>+++ZP!yq#!#C$\"38+++7JfTJF77$$!3_*****4j)Q)z'FL$\"3E+++wF >lOF77$$!3!******4our]\"!#B$\"3A+++KOz)=%F77$$!3.+++e;$z.$FW$\"31+++)y 'R7ZF77$$!3C+++lo]zcFW$\"3N+++iS+O_F77$$!3o++++kN*)**FW$\"3]*****fE='f dF77$$!3%)******[oQq;!#A$\"3[+++wNC$G'F77$$!37+++r'fnn#Ffo$\"3/+++jf)o !oF77$$!3u*****pQ4i8%Ffo$\"30+++;PbItF77$$!3#******zyaL>'Ffo$\"3K+++]! eU&yF77$$!3_*****zYR=-*Ffo$\"3y*****f!Q,y$)F77$$!3#******\\sgEG\"!#@$ \"3X+++]-%=!*)F77$$!33+++$)3e%y\"F`q$\"3u*****H(>wD%*F77$$!3&******z*e ANCF`q$\"3Q+++5*4)\\**F77$$!3%)*****fvOaE$F`q$\"35+++YASZ5!#<7$$!31+++ &GZ'4VF`q$\"3)******HnW)*4\"Fbr7$$!3v*****p<^eg&F`q$\"3#******H*RJ_6Fb r7$$!3=+++Go\\&>(F`q$\"3)******z3<[?\"Fbr7$$!3)******\\s/L7*F`q$\"37++ +s@Od7Fbr7$$!3&******pD+P9\"!#?$\"33+++t*e*48Fbr7$$!35+++etp=9Fjs$\"3' ******H%*=EO\"Fbr7$$!31+++58eU)eDFjs$\"35+++$GB6_\"Fbr7$$!3A+++u#eA1$ Fjs$\"37+++QO>u:Fbr7$$!3\")******3**eOOFjs$\"3-++++&>ui\"Fbr7$$!3S+++X :9(G%Fjs$\"35+++x(G3o\"Fbr7$$!3J+++*\\_*=]Fjs$\"31+++:OXM& >Fbr7$$!33+++N&3K+\"F0$\"3\")*****4uGt+#Fbr7$$!3%******>zK=8\"F0$\"3!) *****HXjK1#Fbr7$$!31+++7Fcp7F0$\"3!******>#o$)>@Fbr7$$!3#******RL8dT\" F0$\"3=+++v\"er<#Fbr7$$!33+++#Gi\"p:F0$\"3&******Rve`B#Fbr7$$!3:+++L\\ =G+++0,]:EFbr7$$!3\" ******>bjk^#F0$\"37+++X*Gyo#Fbr7$$!3=+++sTL&[#F0$\"3++++#=R\\w#Fbr7$$! 35+++tv.%G#F0$\"3=+++j#o([GFbr7$$!3'******H@$GDz87I\"F0$\"3#*******fBi%) HFbr7$$!3&******H5'z95F0$\"3;+++!ojn+$Fbr7$$!3U+++9k;)e'Fjs$\"30+++Pu' )HIFbr7$$!3++++]uP+@Fjs$\"33+++M*fS0$Fbr7$$\"3)*******yi\\eOFjs$\"3!)* ****RHl%zIFbr7$$\"31+++1gI?6F0$\"3\"******H]gh5$Fbr7$$\"3()*****zL1%H@ F0$\"3:+++=M0MJFbr7$$\"37+++f)QX\\$F0$\"3-+++&[+E;$Fbr7$$\"3J+++_!)*oI &F0$\"3)******4'fP!>$Fbr7$$\"3k*****>quga(F0$\"3%)*****>UD^@$Fbr7$$\"3 1+++G[;-5F7$\"3>+++8?=NKFbr7$$\"3!******pw31D\"F7$\"3#)******pak]KFbr7 $$\"3/+++$>Hp[\"F7$\"3%******z0EEE$Fbr7$$\"3))*****H<>rq\"F7$\"3;+++Ez @sKFbr7$$\"38+++\")4O6>F7$\"3;+++-x>!G$Fbr7$$\"30+++ni=,@F7$\"3++++'Hq qG$Fbr7$$\"3))*****z:^$yAF7$\"37+++/9;$H$Fbr7$$\"3#******>&4ZWCF7$\"3# ******fA&o)H$Fbr7$$\"3%*******R&=3:$F7$\"32+++I\"*f@LFbr7$$\"3B+++DEy; PF7$\"3(******fi<4M$Fbr7$$\"37+++*emC>%F7$\"3'******f-;)eLFbr7$$\"3E++ ++t!Rg%F7$\"3%)******p2.wLFbr7$$\"33+++sM]m\\F7$\"3=+++i-(GR$Fbr7$$\"3 [+++8*>-H&F7$\"3=+++K1[4MFbr7$$\"3%)******3[$>e&F7$\"3%)******=j$fU$Fb r7$$\"3I+++!GIm%eF7$\"32+++1mFUMFbr7$$\"3y******>A0)3'F7$\"3#*******4A _eMFbr7$$\"3[+++b>54jF7$\"3%)*****\\\\$ouMFbr7$$\"3Q+++@<37lF7$\"3))** ****>[w!\\$Fbr7$$\"3M+++ra&))p'F7$\"35+++kpw1NFbr7$$\"3'******H9e4(oF7 $\"3/+++E&)oANFbr7$$\"3!)*****RAs'HqF7$\"3/+++yn_QNFbr7$$\"3s*****\\#G 3wrF7$\"3%)******z\"yUb$Fbr7$$\"3[*****4@=6J(F7$\"3A+++0(Q*pNFbr7$$\"3 9+++5-eNuF7$\"3-+++^S]&e$Fbr7$$\"3y*****>Wn,b(F7$\"3/+++n(p4g$Fbr7$$\" 3'******HK$\\bwF7$\"3.+++Z8L;OFbr7$$\"3[+++E-5_xF7$\"31+++yUeJOFbr7$$ \"3a+++_2ZSyF7$\"35+++ySsYOFbr7$$\"38+++\"zO5#zF7$\"3!******fJY\")F7$\"3;+++HV11PFbr7$$\"3a*****H@(\\s\")F7$\"3!****** zLt0s$Fbr7$$\"34+++xP>>#)F7$\"3#)*****HkV\\t$Fbr7$$\"3s*****p^Y*f#)F7$ \"39+++;7<\\PFbr7$$\"3[*****\\'4)\\H)F7$\"3%********3_Kw$Fbr7$$\"3^*** **Rp0XK)F7$\"33+++_B=xPFbr7$$\"3s*****f;9([$)F7$\"3y******Q\"e4z$Fbr7$ $\"3W+++Ojyn$)F7$\"32+++Jcd/QFbr7$$\"3-+++X+*=Q)F7$\"3@+++Y5.=QFbr7$$ \"3W+++HA=\"R)F7$\"30+++O1KJQFbr7$$\"3%)*****>)*4eR)F7$\"3$)*****RoSW% QFbr7$$\"3'******>\\6fR)F7$\"3)******>\\(QdQFbr7$$\"3i******foh\"R)F7$ \"3++++$Qd,(QFbr7$$\"3i*****>$)[IQ)F7$\"3++++#pYF)QFbr7$$\"3q*****\\WB .P)F7$\"3/+++l<:&*QFbr7$$\"3a+++-1b`$)F7$\"3()******[*ot!RFbr7$$\"3/++ +!fMGL)F7$\"3=+++#f%R>RFbr7$$\"3/+++QXF3$)F7$\"3#******p.D7$RFbr7$$\"3 %)*****R$['*z#)F7$\"3\")*****fhcG%RFbr7$$\"3C+++(\\&*zC)F7$\"3%)*****p k&GaRFbr7$$\"3V+++9DX7#)F7$\"3')******H%3b'RFbr7$$\"3o*****H9=M<)F7$\" 3A+++R7_wRFbr7$$\"3^+++#>r48)F7$\"3)******>K?t)RFbr7$$\"3;+++ms=&3)F7$ \"3y*****>*=!z*RFbr7$$\"3#**********Qh.)F7$\"3!******\\7i#3SFbr7$$\"3q *****4E'*Q)zF7$\"3s*****\\:(R=SFbr7$$\"3')*****\\ME&GzF7$\"3;+++kIIGSF br7$$\"3W+++VT4qyF7$\"3#)*****\\)e(z.%Fbr7$$\"3_+++2Bm3yF7$\"3?+++*e6u /%Fbr7$$\"38+++$R\"HWxF7$\"39+++$31m0%Fbr7$$\"3')*****>**Rqn(F7$\"3s** ***H?bb1%Fbr7$$\"3]******f['pg(F7$\"3q*****pqaU2%Fbr7$$\"3A+++E57MvF7$ \"3I+++s-q#3%Fbr7$$\"39+++)*=ceuF7$\"3/+++zu)34%Fbr7$$\"3')*****zNR.Q( F7$\"3=+++8=\"))4%Fbr7$$\"3G+++cQ]*H(F7$\"3Q+++a'ok5%Fbr7$$\"3w*****>[ /h@(F7$\"3?+++jK&Q6%Fbr7$$\"3*)*****40*=IrF7$\"3!)******y2'47%Fbr7$$\" 3y*****p;/=/(F7$\"3w*****p?'yFTFbr7$$\"3M+++)G&*4&pF7$\"3E+++2WKMTFbr7 $$\"3E+++'y1y&oF7$\"32+++$3q09%Fbr7$$\"3C+++'*>GinF7$\"33+++rx^YTFbr7$ $\"3u*****HFjWm'F7$\"3S+++E=;_TFbr7$$\"3G+++I?RklF7$\"3++++2k\\dTFbr7$ $\"3))******z(3@Y'F7$\"3++++ia^iTFbr7$$\"3u*****R<`wN'F7$\"3U+++7F@nTF br7$$\"3-+++LS1^iF7$\"3Y+++J;erTFbr7$$\"3l*****4PzB9'F7$\"3;+++DahvTFb r7$$\"36+++GkjJgF7$\"35+++;qIzTFbr7$$\"3!)*****>or)=fF7$\"3=+++2!\\E=% Fbr7$$\"3'******p'37/eF7$\"3G+++nOj&=%Fbr7$$\"35+++&)*=uo&F7$\"3e***** H*GD)=%Fbr7$$\"3_+++5.!)obF7$\"3y*****>=)\\!>%Fbr7$$\"3Y+++\"R)H[aF7$ \"3!******4fgB>%Fbr7$$\"3d*****\\/YfK&F7$\"3q*****HqIQ>%Fbr7$$\"3_**** *zMv%Fbr7$$\"3!******Hh%Fbr7$ $\"3?+++nL5[\\F7$\"3M+++?\\y&>%Fbr7$$\"3s*****fJi'=[F7$\"3A+++:/e&>%Fb r7$$\"3%******HHBvo%F7$\"37+++$eF\\>%Fbr7$$\"3-+++0UraXF7$\"3#******H) H\"Q>%Fbr7$$\"30+++c>E?WF7$\"3M+++RAA#>%Fbr7$$\"3'******\\N#>%G%F7$\"3 k*****p$*R,>%Fbr7$$\"3))*****f+Il9%F7$\"3U+++-&\\v=%Fbr7$$\"36+++m\")H 2SF7$\"3;+++jIV%=%Fbr7$$\"3)******zd=l'QF7$\"3=+++#Hr2=%Fbr7$$\"3;+++j 7@CPF7$\"3O+++BKawTFbr7$$\"3%*******fUR!e$F7$\"3!******f.E<<%Fbr7$$\"3 >+++>L3NMF7$\"35+++*z%HmTFbr7$$\"3%)*****H]\"H)G$F7$\"3'******4=A-;%Fb r7$$\"3;+++5(G+9$F7$\"3'******z3yM:%Fbr7$$\"3')*****H1,.*HF7$\"3y***** *\\#Hg9%Fbr7$$\"3G+++[,6RGF7$\"3)******RqQy8%Fbr7$$\"37+++X?X'o#F7$\"3 W+++&=l)GTFbr7$$\"3$)*****\\3;B`#F7$\"3c*****\\Oi!>TFbr7$$\"3)******zB $owBF7$\"3&)*****>#zP3TFbr7$$\"3-+++6S_>AF7$\"3;+++tBv'4%Fbr7$$\"3%*** ****Rczg?F7$\"3K+++0w6%3%Fbr7$$\"3&*******H$Q/!>F7$\"3K+++))\\RqSFbr7$ $\"3()*******3q$Q*))p@SFbr7$$\"3'*******eD dS7F7$\"3x******3R_-SFbr7$$\"3'******zYn+2\"F7$\"3))*****RB*e\")RFbr7$ $\"3G+++0.0n*)F0$\"3=+++Q'e'eRFbr7$$\"3/+++$)y?*>(F0$\"3B+++v7VLRFbr7$ $\"3=+++S_%))Q&F0$\"39+++;S^0RFbr7$$\"3!)******3(fK_$F0$\"39+++*ewV(QF br7$$\"3)******>Nc>e\"F0$\"3;+++oWFRQFbr7$$!3.+++RTP1ZFjs$\"3=+++]j5*z $Fbr7$$!3))******ze2.FF0$\"3-+++AZ7_PFbr7$$!3)******>z\"Qq_F0$\"3%**** **f%GF&p$Fbr7$$!3')*****4_rG&eF0$\"3=+++&yrAo$Fbr7$$!3E+++&e9,Z'F0$\"3 #)*****Hi#eoOFbr7$$!3o+++*HE08(F0$\"3#******f;@Tl$Fbr7$$!3/+++L`hXyF0$ \"3#******4n)yQOFbr7$$!3a*****>s?:j)F0$\"3!)*****>1rCi$Fbr7$$!33+++z;[ 6&*F0$\"33+++V?/0OFbr7$$!3/+++w3)>0\"F7$\"3))*****f\\xje$Fbr7$$!3-+++( R'zq6F7$\"3')*****4W+kc$Fbr7$$!3/+++pB?:8F7$\"3!******p-._a$Fbr7$$!3-+ ++cxL&\\\"F7$\"3\")*****f\")GL_$Fbr7$$!3(******49A.s\"F7$\"3$)*****Rb. A]$Fbr7$$!35+++OZ6))>F7$\"3-+++u;3%[$Fbr7$$!3%******\\ht'zAF7$\"3?+++O +jqMFbr7$$!3=+++\"zXAd#F7$\"3#******4\"*[,3=X$Fbr7$$!3#)*****>w01g$F7$\"3%*******GtU^MFbr7$$ !3@+++3U`AQF7$\"37+++Yb!=X$Fbr7$$!3x*****H%\\aLSF7$\"3%******fYGFX$Fbr 7$$!33+++[8Em\\F7$\"31+++!y4xF7$\"3;+++uDR7NFbr7$$!3a+++aZi&G)F7$\"3$****** *H!o[_$Fbr7$$!3<+++[4*H#))F7$\"3>+++M'Qp`$Fbr7$$!3P+++=3cO$*F7$\"3;+++ 'yl&[NFbr7$$!3r*****\\dy+$)*F7$\"33+++*RH(fNFbr7$$!3!******pCW1.\"Fbr$ \"3/+++u'>/d$Fbr7$$!3/+++O]zw5Fbr$\"33+++_Fj!e$Fbr7$$!3\"******p1Y;7\" Fbr$\"3-+++#Go.f$Fbr7$$!3-+++o,Nl6Fbr$\"3?+++=!G'*f$Fbr7$$!34+++F[.37F br$\"3(******H'\\T3OFbr7$$!3#******fq2)\\7Fbr$\"3!)*****H≫OFbr7$$!3 '******4qg2H\"Fbr$\"3&)*****z#eeCOFbr7$$!3%******p*H(4L\"Fbr$\"3.+++C# y>j$Fbr7$$!3!******\\R80P\"Fbr$\"3&******pV9*QOFbr7$$!32+++E@W49Fbr$\" 3#*******R))RXOFbr7$$!3/+++tA\"yW\"Fbr$\"3'******psN9l$Fbr7$$!3-+++/4n &[\"Fbr$\"3#******pFHql$Fbr7$$!34+++#**fI_\"Fbr$\"3%*******QN=iOFbr7$$ !35+++dr,g:Fbr$\"3-+++:C!pm$Fbr7$$!3/+++xid'f\"Fbr$\"35+++h'*=rOFbr7$$ !33+++,!oFj\"Fbr$\"3=+++%))[]n$Fbr7$$!3/+++n,io;Fbr$\"3-+++VN[yOFbr7$$ !3.+++m\"eTq\"Fbr$\"3')*****z&p\\\"o$Fbr7$$!30+++T_SR\">Fbr$\"31+++a6!4p$Fbr7$$!33+++5 w'e%>Fbr$\"31+++P%[5p$Fbr7$$!31+++#)\\fz>Fbr$\"3;+++Poz!p$Fbr7$$!3#*** ***>![;8?Fbr$\"3))*****Hv[,p$Fbr7$$!3')******)H*eY?Fbr$\"3=+++Jl5*o$Fb r7$$!3#)*****R/!))z?Fbr$\"3!******\\Yswo$Fbr7$$!3$)******=![I6#Fbr$\"3 -+++;)[eo$Fbr7$$!3%******\\o.h9#Fbr$\"31+++;yj$o$Fbr7$$!39+++Jq0z@Fbr$ \"3?+++%oT5o$Fbr7$$!3>+++Mw\">@#Fbr$\"3/+++KE1yOFbr7$$!3#)*****fp%pWAF br$\"3!)******yGquOFbr7$$!35+++\"4(RxAFbr$\"3#)*****>mk4n$Fbr7$$!3=+++ -M.5BFbr$\"3)*******\\-&om$Fbr7$$!35+++a>hUBFbr$\"33+++a>OiOFbr7$$!3y* ****\\%39vBFbr$\"3))*****H9-vl$Fbr7$$!3&******R(zi2CFbr$\"3))*****\\Dt Al$Fbr7$$!3&)*****f1\"3SCFbr$\"3!)*****R\"ynYOFbr7$$!3?+++#p2DZ#Fbr$\" 3#******HW=2k$Fbr7$$!3*******\\H:\\]#Fbr$\"3')******zyRMOFbr7$$!32+++, 7JPDFbr$\"36+++(**=xi$Fbr7$$!3)******Rj-(pDFbr$\"3++++8[o?OFbr7$$!3A++ +Ln4-EFbr$\"3;+++8&)H8OFbr7$$!3%******>c+Xj#Fbr$\"3!)******oMc0OFbr7$$ !3=+++66#pm#Fbr$\"3/+++`K[(f$Fbr7$$!3?+++6`O*p#Fbr$\"3%)*****Hmh!*e$Fb r7$$!3:+++G+%=t#Fbr$\"3!)*****pt-.e$Fbr7$$!3))*****>2_Vw#Fbr$\"37+++x2 @rNFbr7$$!3=+++)=3pz#Fbr$\"3/+++o.zhNFbr7$$!3\")*****R0:&HGFbr$\"3'*** ****)RY?b$Fbr7$$!3!******\\Fz@'GFbr$\"3#******>3%)>a$Fbr7$$!3')******p t!\\*GFbr$\"3()*****p(*3;`$Fbr7$$!35+++edqFHFbr$\"3*)*****\\+F4_$Fbr7$ $!3%)*****pu!egHFbr$\"3,+++tW%*4NFbr7$$!3;+++1&QN*HFbr$\"39+++*3o')\\$ Fbr7$$!33+++^]eEIFbr$\"3'******>)\\5([$Fbr7$$!3!)*****H@E(fIFbr$\"3y** ***Hri_Z$Fbr7$$!3'******RhnH4$Fbr$\"3')*****HH\\JY$Fbr7$$!32+++JYJEJFb r$\"3/+++!>t2X$Fbr7$$!3)******pOs(fJFbr$\"3))******RL9QMFbr7$$!3!***** *pgXL>$Fbr$\"35+++]\"p_U$Fbr7$$!3@+++)yQqA$Fbr$\"3()*****>]g@T$Fbr7$$! 3?+++\\f&3E$Fbr$\"3!)*****fuF))R$Fbr7$$!3#******>p+[H$Fbr$\"3;++++V$Fbr$\"3*** ****\\P'>HLFbr7$$!3++++h!GlY$Fbr$\"3&)*****HEhZJ$Fbr7$$!3()*****4!RG,N Fbr$\"3A+++,l=+LFbr7$$!3#******Hgwh`$Fbr$\"3')*****fa&[&G$Fbr7$$!3%)** ***Rr/7d$Fbr$\"3#******>3s1F$Fbr7$$!3?+++wfO1OFbr$\"3!******>(*fdD$Fbr 7$$!3&)*****\\Id;k$Fbr$\"3()******)>j2C$Fbr7$$!33+++hZ2xOFbr$\"3A+++3e pDKFbr7$$!3>+++PNh7PFbr$\"3&******R%=d5KFbr7$$!3#******f%zE[PFbr$\"3;+ ++'G0a>$Fbr7$$!3=+++>9.%y$Fbr$\"3))******z*4-=$Fbr7$$!3.++++l*)>QFbr$ \"3')*****Rd**\\;$Fbr7$$!3#*******e[&e&QFbr$\"3#)******\\uy\\JFbr7$$!3 #)*****>I(*=*QFbr$\"3&******Hl'eMJFbr7$$!39+++\"z8!GRFbr$\"39+++F)4%>J Fbr7$$!3')*****pY$>kRFbr$\"3?+++[\"pU5$Fbr7$$!3f*****\\oC/+%Fbr$\"3y** ***REw\"*3$Fbr7$$!3,+++W]pOSFbr$\"3#******zAUT2$Fbr7$$!3;+++J9*H2%Fbr$ \"3?+++Uu+IFbr7$$!3s*****>7hVD%Fbr$\"33+++pSq&)HFbr7$$!3 i*****Hu/0H%Fbr$\"3%)*****p?I8(HFbr7$$!3')*****R0nlK%Fbr$\"31+++gf2dHF br7$$!3I+++a7`iVFbr$\"3()*****H$[%H%HFbr7$$!3/+++$R!Q)R%Fbr$\"3z*****z KR*GHFbr7$$!3;+++[u4MWFbr$\"3-+++H41:HFbr7$$!3=+++5ampWFbr$\"3<+++)458 !HFbr7$$!30+++ht10XFbr$\"33+++vio()GFbr7$$!3s*****\\c'GSXFbr$\"3/+++\" *y=uGFbr7$$!37+++XlIvXFbr$\"3-+++&Q73'GFbr7$$!3M+++n665YFbr$\"3=+++Gib ZGFbr7$$!3))******>ZoWYFbr$\"3.+++k\\TMGFbr7$$!3I+++\"*>,zYFbr$\"3:+++ \\KQ@GFbr7$$!3q*****fLyIr%Fbr$\"3(******R!\\X3GFbr7$$!3i*****Rupou%Fbr $\"3'******H(Hi&z#Fbr7$$!3U+++)*GP!y%Fbr$\"3))******)yzGy#Fbr7$$!3))** ***>AvN\"[Fbr$\"3;+++Vq@qFFbr7$$!3=+++C\\YY[Fbr$\"3#)*****>'eidFFbr7$$ !3;+++I5.z[Fbr$\"3;+++&)o4XFFbr7$$!3')*****pSj7\"\\Fbr$\"3()*****RM?Et #Fbr7$$!38+++!y_J%\\Fbr$\"3++++Xh=?FFbr7$$!3++++K2pu\\Fbr$\"3*)*****z% Ry2FFbr7$$!3O+++1(pe+&Fbr$\"3w*****pC.ap#Fbr7$$!3++++*)HoO]Fbr$\"3y*** **HWLIo#Fbr7$$!3w*****zpCr1&Fbr$\"3))******=RmqEFbr7$$!3#)*****pu*=(4& Fbr$\"3)******f5%GeEFbr7$$!3A+++@Q(o7&Fbr$\"31+++VN)ek#Fbr7$$!3U+++RL< c^Fbr$\"3\"******H$>XLEFbr7$$!3')*****pS&3&=&Fbr$\"3'******f>z4i#Fbr7$ $!3h*****Hy2O@&Fbr$\"3.+++3bX3EFbr7$$!3m*****\\#)QCN.N&Fbr$\"37+ ++>7vWDFbr7$$!3)******zv-lP&Fbr$\"3z******)*QtJDFbr7$$!3p*****H5zAS&Fb r$\"3-+++bkg=DFbr7$$!3v*****HblwU&Fbr$\"3********z>O0DFbr7$$!3u*****pi jEX&Fbr$\"3#******4'R*>\\#Fbr7$$!3y*****\\0vsZ&Fbr$\"33+++xi\\yCFbr7$$ !3R+++V<],bFbr$\"3!*******)=j[Y#Fbr7$$!3f******QdMDbFbr$\"3%)******H$* 3^CFbr7$$!3-+++6#4)[bFbr$\"3(******HpprV#Fbr7$$!3!)*****>T%*=d&Fbr$\"3 1+++A'*4BCFbr7$$!3C+++mOg%f&Fbr$\"3))*****zyu)3CFbr7$$!3)*******Q$Rph& Fbr$\"37+++&=\"\\%R#Fbr7$$!3()*****\\#Q!*QcFbr$\"3;+++2^%*zBFbr7$$!3!* *****z_*\\gcFbr$\"31+++OJBlBFbr7$$!3Q+++Y)G7_.N#Fbr7 $$!3Q+++hTf-dFbr$\"3()******p\"*HNBFbr7$$!3(******\\#y4BdFbr$\"3$)**** *fir+K#Fbr7$$!3m*****>:UKu&Fbr$\"3$)*****R0nYI#Fbr7$$!3y*****4THIw&Fbr $\"39+++IK3*G#Fbr7$$!3s*****p\"=Y#y&Fbr$\"33+++<\"=LF#Fbr7$$!3O+++F:a, eFbr$\"3#******4')ptD#Fbr7$$!30+++O1F?eFbr$\"3$)*****zwO7C#Fbr7$$!3h** ***><^'QeFbr$\"3$******RH<\\A#Fbr7$$!3c*****\\4&oceFbr$\"3$)*****H.5%3 AFbr7$$!3))*****RHuV(eFbr$\"33+++/Pr\">#Fbr7$$!33+++)e?<*eFbr$\"3/+++M r#[<#Fbr7$$!3/+++Dds3fFbr$\"39+++b#\\x:#Fbr7$$!3o******y8RDfFbr$\"3&** ****\\3z/9#Fbr7$$!3O+++b\"><%fFbr$\"3*)*****>s:I7#Fbr7$$!3m*****4f5x&f Fbr$\"39+++J$e`5#Fbr7$$!3'******R:nL(fFbr$\"35+++Ph](3#Fbr7$$!33+++Z-p ))fFbr$\"3++++8%e%p?Fbr7$$!3'******4@\"o.gFbr$\"3%)*****>Z970#Fbr7$$!3 #)*****fKT$=gFbr$\"3!******pltF.#Fbr7$$!3g*****\\\"=nKgFbr$\"3$)*****f LNT,#Fbr7$$!35+++ZQnYgFbr$\"3-+++\"*))H&*>Fbr7$$!3%)*****>a[.1'Fbr$\"3 5+++6PEw>Fbr7$$!3m*****H(pptgFbr$\"3/+++%=Hq&>Fbr7$$!3y*****HFbr7$$!3w*****\\8>%*4'Fbr$\"32+++(ffz\">Fbr7$$!3C+++A[z6h Fbr$\"33+++[K7)*=Fbr7$$!3f******p\"[Q7'Fbr$\"3'******>'\\3y=Fbr7$$!3Q+ ++!4!eNhFbr$\"31+++>S%y&=Fbr7$$!3q*****4[\"*p9'Fbr$\"3,+++j'*RP=Fbr7$$ !3#)******GK3ehFbr$\"3/+++*3^n\"=Fbr7$$!33+++DH:Fbr7$$!3!)*** **p&*R.G'Fbr$\"3\"******>N@c]\"Fbr7$$!3q*****zufnG'Fbr$\"3!******f]l<[ \"Fbr7$$!3o*****RFzGH'Fbr$\"3#******zA#od9Fbr7$$!3c*****\\h+()H'Fbr$\" 3)******zCpLV\"Fbr7$$!37+++NgA/jFbr$\"31+++$>C)39Fbr7$$!3K+++,!e%4jFbr $\"3'******>fWSQ\"Fbr7$$!3j*****4?*R9jFbr$\"3/+++\")y-f8Fbr7$$!3z***** \\c_!>jFbr$\"3%******RSrPL\"Fbr7$$!3t******y7UBjFbr$\"3$*******HCF38Fb r7$$!3w*****R\")3vK'Fbr$\"3'******4:GDG\"Fbr7$$!31+++S*=8L'Fbr$\"3/+++ /d`c7Fbr7$$!34+++^d&[L'Fbr$\"3$******p<#HI7Fbr7$$!39+++%oB\"QjFbr$\"3) *******HYz.7Fbr7$$!31+++Yv7TjFbr$\"33+++5,/x6Fbr7$$!3U+++HD(QM'Fbr$\"3 #******pnD+:\"Fbr7$$!3;+++MUOYjFbr$\"3-+++E%[F7\"Fbr7$$!3C+++&p3'[jFbr $\"35+++9b?&4\"Fbr7$$!3A+++!R71N'Fbr$\"35+++&>%Rn5Fbr7$$!3I+++nAQ_jFbr $\"3!******4&=JR5Fbr7$$!3o*****\\vDRN'Fbr$\"3.+++Lg&4,\"Fbr7$$!3=+++x2 DbjFbr$\"3s*****z,XK#)*F77$$!3E+++idOcjFbr$\"32+++]F:M&*F77$$!3k****** ['zsN'Fbr$\"3t*****pumAC*F77$$!3q*******)=+ejFbr$\"3w*****\\!QdZ*)F77$ $!3K+++XCaejFbr$\"3#*******\\[1]')F77$$!3;+++v<\"*ejFbr$\"3W+++6bt\\$) F77$$!3#******p#37fjFbr$\"3k******>neY!)F77$$!35+++25=fjFbr$\"3')***** >KD1u(F77$$!3)******p:/\"fjFbr$\"3R+++'ok=V(F77$$!3I+++8D!*ejFbr$\"3[* ****4KD.7(F77$$!3m******f')eejFbr$\"31+++na.1oF77$$!3'******4[v\"ejFbr $\"3#******>mJ!*['F77$$!33+++\"4wwN'Fbr$\"3]*****zFf$phF77$$!3;+++qP5d jFbr$\"3C+++)*H2ZeF77$$!37+++#)=ZcjFbr$\"3;+++KsBAbF77$$!39+++$z$zbjFb r$\"3=+++ik#\\>&F77$$!37+++\"y#3bjFbr$\"3?+++'\\D_'[F77$$!3K+++M>NajFb r$\"3#)*****\\iHK`%F77$$!3q*****p09ON'Fbr$\"31+++xY/*>%F77$$!3%)*****p c\")GN'Fbr$\"3#)*****R&pyiQF77$$!3Q+++'Qm@N'Fbr$\"3A+++cIeCNF77$$!3#)* ****>%)z9N'Fbr$\"37+++!epX=$F77$$!3K+++nD$3N'Fbr$\"3A+++)p#*G%GF77$$!3 s*******RM-N'Fbr$\"3!******Ri2(*\\#F77$$!3w*****HH%p\\jFbr$\"3!******R &z)[jFbr$ \"3/+++'\\wFY\"F77$$!3#******z%o\\[jFbr$\"32+++4,]jFbr$!3!******4o Z?O#F77$$!3C+++&)oe]jFbr$!31+++q1z0FF77$$!3K+++]c@^jFbr$!3++++S**3[IF7 7$$!3o******>x)=N'Fbr$!3y******Qoy)Q$F77$$!31+++oFf_jFbr$!3++++2:tFPF7 7$$!3O+++N(>LN'Fbr$!3#)******eMykSF77$$!3#)*****>#p0ajFbr$!3!)*****f07 )*R%F77$$!3!)*****p4#zajFbr$!3=+++YopKZF77$$!3/+++,E^bjFbr$!3I+++*zFL1 &F77$$!3U+++Wa?cjFbr$!3%)******3ag\"R&F77$$!3=+++8u&oN'Fbr$!3_+++(oSur &F77$$!3M+++n^XdjFbr$!3^+++[]vSgF77$$!3)*******H`)zN'Fbr$!31+++*3![hjF 77$$!3y*****zdM%ejFbr$!3M+++\"Gd&zmF77$$!3%******4s*yejFbr$!3_+++%eP\\ *pF77$$!3;+++xx.fjFbr$!3h*****f)4e2tF77$$!3C+++Mg;fjFbr$!3t*****R*fX75(ejFbr$!3b*****Rjr-`)F77$$!3#)******Q$R#ejFbr$!3@+++-&3* G))F77$$!3'******p*4fdjFbr$!3(*******4qsC\"*F77$$!3A+++J\\vcjFbr$!3L++ +AWt<%*F77$$!3G+++([@dN'Fbr$!3C+++fB%zq*F77$$!39+++>:[ajFbr$!3Q+++LiO& ***F77$$!3u*****fREIN'Fbr$!3(******4Z-!G5Fbr7$$!3;+++%*zM^jFbr$!31+++A R>c5Fbr7$$!31+++'oQ%\\jFbr$!3++++XL6%3\"Fbr7$$!3U+++N8HZjFbr$!3)****** pHj<6\"Fbr7$$!3e*****fF**[M'Fbr$!33+++wk9R6Fbr7$$!3%)*******HcAM'Fbr$! 3!********oli;\"Fbr7$$!3w*****fjc$RjFbr$!3%******R#Q7$>\"Fbr7$$!3O+++H \\>OjFbr$!3(******f!Qs>7Fbr7$$!3(******4AmFL'Fbr$!3!******peogC\"Fbr7$ $!3y*****>$f1HjFbr$!31+++:6;s7Fbr7$$!3e******H)*3DjFbr$!3%******RJ/!)H \"Fbr7$$!3R+++?S$3K'Fbr$!30+++q5gB8Fbr7$$!3H+++:\\H;jFbr$!31+++8U&*[8F br7$$!3!)*****\\@p9J'Fbr$!33+++3l1u8Fbr7$$!3d*******)QN1jFbr$!3%****** *R1%*)R\"Fbr7$$!3j******fh%4I'Fbr$!32+++6#zNU\"Fbr7$$!3%*******pMC&H'F br$!3-+++JZ)zW\"Fbr7$$!3I+++xMC*G'Fbr$!3-+++7'f@Z\"Fbr7$$!3W+++JS%HG'F br$!3'******R;1h\\\"Fbr7$$!3=+++bJMwiFbr$!33+++)fE)>:Fbr7$$!3G+++L!R%p iFbr$!3*******f,BLa\"Fbr7$$!3!)*******)*HAE'Fbr$!3/+++>ufm:Fbr7$$!3))* ****H[9ZD'Fbr$!3-+++*p^'*e\"Fbr7$$!3C+++\"3\"*oC'Fbr$!3\"******4l([7;F br7$$!3K+++d%e(QiFbr$!3\"******H'p5N;Fbr7$$!3u*****>P:.B'Fbr$!3%****** zA6vl\"Fbr7$$!3S+++o1c@iFbr$!3))*****>%>qz;Fbr7$$!3W+++\\K\\7iFbr$!3$* *****z]!o,C[Ms\"Fbr7$$!37+++thT$>'F br$!3#******RP1]u\"Fbr7$$!3y*****Hd/M='Fbr$!31+++agNmFbr7$$!3++++6'Q=4'Fbr$!3%******4wk(H>Fbr 7$$!3#******RAX*ygFbr$!31+++<%z#\\>Fbr7$$!3k*****z+Fd1'Fbr$!3/+++cQfo> Fbr7$$!3?+++XH=_gFbr$!3%******>q3x)>Fbr7$$!3Q+++z>JQgFbr$!3*)*****zcCm +#Fbr7$$!3x******4I6CgFbr$!3;+++k?MD?Fbr7$$!35+++*)[e4gFbr$!3')*****p! ='Q/#Fbr7$$!3q*****\\TEZ*fFbr$!3!*******>W=i?Fbr7$$!3E+++Jj`zfFbr$!3=+ ++^0J!3#Fbr7$$!3Q+++@L,kfFbr$!3')*****>(3C)4#Fbr7$$!3o*****p+c\"[fFbr$ !34+++%4wf6#Fbr7$$!3r*****p%H'>$fFbr$!3:+++rp^L@Fbr7$$!3++++LEV:fFbr$! 37+++9V'3:#Fbr7$$!3m*****p[j&)*eFbr$!3))*****p**=!o@Fbr7$$!3-+++hQN\") eFbr$!37+++n>)\\=#Fbr7$$!3A+++Q?!Q'eFbr$!3!)*****zDa+++E9tMAFbr7$$!3#****** z7v!4eFbr$!3#******R*)Q4D#Fbr7$$!3s*****f!f8!z&Fbr$!3++++!*3'pE#Fbr7$$ !3/+++M[%3x&Fbr$!39+++Z!*z#G#Fbr7$$!3C+++z(*>^dFbr$!3#******H:b%)H#Fbr 7$$!31+++[&)>JdFbr$!3/+++q6$RJ#Fbr7$$!36+++%*)Q3r&Fbr$!31+++X#H#HBFbr7 $$!3Q+++?&=,p&Fbr$!3$)*****ps^VM#Fbr7$$!39+++(3N!pcFbr$!3#******H=,$fB Fbr7$$!3S+++DieZcFbr$!3?+++3/3uBFbr7$$!3Q+++T&pdi&Fbr$!35+++\\Cp)Q#Fbr 7$$!34+++JEe.cFbr$!3:+++819.CFbr7$$!3-+++'4B5e&Fbr$!3!******4\\GuT#Fbr 7$$!3K+++b&)3ebFbr$!35+++n*f:V#Fbr7$$!3'******fmwZ`&Fbr$!3?+++P#RbW#Fb r7$$!3-+++X^36bFbr$!3y*****f#3PfCFbr7$$!39+++(y6q[&Fbr$!3%*******)ffIZ #Fbr7$$!3')*****H\\aDY&Fbr$!35+++x2h'[#Fbr7$$!3s*****\\H6xV&Fbr$!3(*** ****[*H+]#Fbr7$$!3\")*******Q![7aFbr$!3))*****>3BL^#Fbr7$$!3O+++r,'oQ& Fbr$!3\"******>`'\\EDFbr7$$!33+++f#\\3O&Fbr$!3=+++ZqbRDFbr7$$!3o*****f aYWL&Fbr$!3y*****fx6Db#Fbr7$$!3W+++K7l2`Fbr$!3))*****HEo`c#Fbr7$$!3/++ +qGY!G&Fbr$!3%******4XM\"yDFbr7$$!3G+++29)GD&Fbr$!3!*******o'=3f#Fbr7$ $!3r*****HB2\\A&Fbr$!3<+++A'HMg#Fbr7$$!3))*****fATl>&Fbr$!3))*****fPwf h#Fbr7$$!3s*******z%yn^Fbr$!3-+++I$o%GEFbr7$$!37+++_*R'Q^Fbr$!3>+++%>: 4k#Fbr7$$!3,+++5$4\"4^Fbr$!3-+++_pK`EFbr7$$!3')*****\\<'>z]Fbr$!3'**** **z\"QrlEFbr7$$!3))*****Rc/*[]Fbr$!3%******H>'3yEFbr7$$!3r******\\#R#= ]Fbr$!3!******>ha/p#Fbr7$$!3I+++%z0s)\\Fbr$!3z******z'HGq#Fbr7$$!3-+++ u0\"e&\\Fbr$!3#)*****4,A_r#Fbr7$$!37+++331C\\Fbr$!34+++e@kFFFbr7$$!3e* ****pck>*[Fbr$!3#******f`+,u#Fbr7$$!3Q+++%yI&f[Fbr$!3-+++Ytg_FFbr7$$!3 !*******\\#pn#[Fbr$!3-+++(\\s^w#Fbr7$$!3=+++11p$z%Fbr$!31+++Bb!yx#Fbr7 $$!3g*****zJ1.w%Fbr$!3y*****4]:0z#Fbr7$$!3o*****HlGms%Fbr$!37+++w4J.GF br7$$!39+++R1n#p%Fbr$!35+++!))*>;GFbr7$$!3++++:gWeYFbr$!3#)******f%*=H GFbr7$$!3d******z\"pRi%Fbr$!36+++:iGUGFbr7$$!3I+++E^D*e%Fbr$!35+++He\\ bGFbr7$$!3e*****fP>Vb%Fbr$!31+++?J#)oGFbr7$$!3-+++-z<>XFbr$!3++++!*>F# )GFbr7$$!3%******R0ZQ[%Fbr$!35+++$RXe*GFbr7$$!3k*****RZV$[WFbr$!3?+++( HX&4HFbr7$$!3H+++>So7WFbr$!3?+++nEPBHFbr7$$!31+++qc)oP%Fbr$!3<+++]uKPH Fbr7$$!3++++aa'4M%Fbr$!31+++s&39&HFbr7$$!3))*****4OS\\I%Fbr$!3-+++ORhl HFbr7$$!3?+++hs#)oUFbr$!39+++O/%*zHFbr7$$!3G+++FGkKUFbr$!39+++tRQ%*HFb r7$$!3/+++hMS'>%Fbr$!3')*****HZR*3IFbr7$$!3I+++?_7gTFbr$!38+++C4gBIFbr 7$$!3K+++ZP#Q7%Fbr$!3/+++-9OQIFbr7$$!31+++4U^(3%Fbr$!39+++:J@`IFbr7$$! 33+++L7@^SFbr$!31+++Wu9oIFbr7$$!3W+++])G\\,%Fbr$!39+++!*\\:$3$Fbr7$$!3 !******pW!oyRFbr$!31+++EcA)4$Fbr7$$!3%******RryC%RFbr$!3!)*****Hb[L6$F br7$$!3'******>gNj!RFbr$!33+++`B^GJFbr7$$!3))*******Gi-(QFbr$!3y*****R 00P9$Fbr7$$!3,+++Y\"pU$QFbr$!37+++#>9*eJFbr7$$!39+++0dO)z$Fbr$!31+++to 7uJFbr7$$!3?+++X1ciPFbr$!3))*****H%)H$*=$Fbr7$$!3\"******>xhos$Fbr$!3% )*****Hb4X?$Fbr7$$!3y*****4,w7p$Fbr$!3,+++EAl>KFbr7$$!3/+++-%4el$Fbr$! 30+++IRuMKFbr7$$!3-+++,rY?OFbr$!31+++O1x\\KFbr7$$!3)******RR`_e$Fbr$!3 9+++%H=ZE$Fbr7$$!3)******fqr,b$Fbr$!39+++))GdzKFbr7$$!3&)*****>iC_^$Fb r$!31+++,0K%H$Fbr7$$!3/+++8RT![$Fbr$!3>+++pt%*3LFbr7$$!3++++m0uXMFbr$! 3$******z$*RMK$Fbr7$$!31+++:[?6MFbr$!3@+++9\\yPLFbr7$$!3!)*****H=1oP$F br$!3#********Hp>N$Fbr7$$!3-+++:NaULFbr$!3%)*****RW!)fO$Fbr7$$!3%)**** **G]T3LFbr$!3!)******og!)zLFbr7$$!3#*******\\$=WF$Fbr$!3=+++-VV$R$Fbr7 $$!3A+++j0bSKFbr$!3&)*****pp`oS$Fbr7$$!33+++_#3o?$Fbr$!35+++cK0?MFbr7$ $!3')*****fa(=tJFbr$!3&******>WAIV$Fbr7$$!3<+++kToRJFbr$!31+++$=^dW$Fb r7$$!3z*****RX$H1JFbr$!33+++%))H#eMFbr7$$!3%)*****zV5I2$Fbr$!3#******* =%\\/Z$Fbr7$$!31+++U)H)RIFbr$!3%******\\8,C[$Fbr7$$!35+++Rhu1IFbr$!3'* ******Ro2%\\$Fbr7$$!35+++xNvtHFbr$!3$)*****>zoa]$Fbr7$$!3$******H@Y3%H Fbr$!36+++$ppl^$Fbr7$$!33+++Mz,3HFbr$!3$******zmst_$Fbr7$$!3++++'[i_(G Fbr$!3))*****4Dry`$Fbr7$$!3<+++$\\tD%GFbr$!3#)******p$f![NFbr7$$!3$)** ***HZW*4GFbr$!3/+++B8$zb$Fbr7$$!37+++a)otx#Fbr$!3;+++g<[nNFbr7$$!3@+++ ()*R[u#Fbr$!39+++ncqwNFbr7$$!3=+++`6N7FFbr$!31+++Q$)f&e$Fbr7$$!3)***** *pc&*)zEFbr$!3))*****4ObTf$Fbr7$$!31+++'Qmuk#Fbr$!3$******ffsBg$Fbr7$$ !37+++/n0:EFbr$!3'*******[hC5OFbr7$$!3')*****fbfEe#Fbr$!3>+++iBx#Fbr$!31+++!*e*)zOFbr7$$!3!******pv'Hf@Fbr$!3#******fXXEo$Fbr7$$!3\" ******H>$GE@Fbr$!3?+++u2,&o$Fbr7$$!3!)******p8;$4#Fbr$!3y******H'*)po$ Fbr7$$!3++++?5#*f?Fbr$!37+++)zz&)o$Fbr7$$!3=+++y8bE?Fbr$!37+++K!z(*o$F br7$$!3))*****f6TI*>Fbr$!3/+++\\]e!p$Fbr7$$!3))*****pFy$f>Fbr$!35+++Fb *4p$Fbr7$$!3$******>?]b#>Fbr$!3')******)335p$Fbr7$$!3#******\\WV:*=Fbr $!39+++(H?1p$Fbr7$$!3-+++xOMd=Fbr$!37+++S'H)*o$Fbr7$$!33+++\"eNH#=Fbr$ !35+++CNj)o$Fbr7$$!3#******4t-$)y\"Fbr$!3')*****zEHqo$Fbr7$$!37+++auU` 7BTm$Fbr7$$!3,+++(o!o+:Fbr$!3'******RZV\"fOF br7$$!3%******4l9IY\"Fbr$!3;+++OFbr7$$!3#*******Hz%[A\"Fbr$!33+++!=)z6OFbr7$$!3#***** *4%*RD=\"Fbr$!39+++M$*>.OFbr7$$!32+++*pd#R6Fbr$!3!)*****\\7HTf$Fbr7$$! 3-+++;A)[4\"Fbr$!3!)*****p?%e%e$Fbr7$$!3-+++z:F\\5Fbr$!3))*****zCiXd$F br7$$!3,+++]UD-5Fbr$!37+++EE1kNFbr7$$!3.+++mg>O&*F7$!3!)*****fd(3`NFbr 7$$!3d*****H:U5.*F7$!3')*****\\8W;a$Fbr7$$!3(******H*=r.&)F7$!3\")**** *pcZ(HNFbr7$$!3k*****f!)y(\\zF7$!3%******fpFu^$Fbr7$$!3W+++>qAjtF7$!3( ******\\JTZ]$Fbr7$$!3#*******)p>bt'F7$!3w*****4_)z\"\\$Fbr7$$!3]*****p hJQ0'F7$!3))*****\\mC)yMFbr7$$!3Y*****zdDtH&F7$!31+++J*GjY$Fbr7$$!3s** ***>4T\"GWF7$!3'******RdtcX$Fbr7$Fh[o$!3')*****>,3=X$Fbr7$Fc[o$!3)**** **4GtKX$Fbr7$F^[o$!3#)******R!GjX$Fbr7$Fijn$!3#******4\"*[1rCi$ Fbr7$Fggn$!3#******4n)yQOFbr7$Fbgn$!3#******f;@Tl$Fbr7$F]gn$!3#)*****H i#eoOFbr7$Fhfn$!3=+++&yrAo$Fbr7$Fcfn$!3%******f%GF&p$Fbr7$$!3(*******o )\\jr%F0$!30+++Ztl2PFbr7$$!3!)*****p!)of=%F0$!3@+++.i[>PFbr7$$!35+++?4 ]vOF0$!3;+++;:\"3t$Fbr7$$!3')*****\\d#*>=$F0$!3z*****f4y;u$Fbr7$$!3!3+ +IFte**)Fjs$!3/+++oxK!z$Fbr7$$\"3-+++/W)==\"F0$!3!)*****f7!pJQFbr7$$\" 3p*****>M5>9$F0$!35+++6AqnQFbr7$$\"3D+++/'f1-&F0$!3/+++RUc**QFbr7$$\"3 J*****z?l3%oF0$!3.+++Ww2GRFbr7$$\"3n+++Z'Qkh)F0$!3#)******Qz!Q&RFbr7$$ \"31+++M(Qc.\"F7$!3#******\\Mrr(RFbr7$$\"31+++e#zm?\"F7$!3++++VX[)*RFb r7$$\"3'*******z!*=v8F7$!34+++7]*z,%Fbr7$$\"33+++#Hq9a\"F7$!3r*****4Z+ f.%Fbr7$$\"3&******z%>u0+++J%Fbr7$$\" 3!******fOaz_%F7$!3)******HSLN>%Fbr7$$\"3#)*****\\?%4hYF7$!3K+++]?u%>% Fbr7$$\"3!)*****>WpDz%F7$!33+++Kh[&>%Fbr7$$\"3:+++mBNA\\F7$!3'*******o #zd>%Fbr7$$\"3u*****>K9/0&F7$!3q*****4AMc>%Fbr7$$\"3_*****\\xDn<&F7$!3 A+++oH1&>%Fbr7$$\"3k*****4Oc7I&F7$!3-+++&ywS>%Fbr7$$\"3t******=\\(RU&F 7$!3%)*****zI'o#>%Fbr7$$\"3h*****\\_\\[a&F7$!3++++)e,4>%Fbr7$$\"3c**** **3v%Qm&F7$!3M+++X@t)=%Fbr7$$\"3K+++pa$4y&F7$!3i*****H,(='=%Fbr7$$\"33 +++u#zg*eF7$!3k******zZF$=%Fbr7$$\"3T+++pSC4gF7$!3C+++CO+!=%Fbr7$$\"3+ +++nUR?hF7$!3#******>N\"QwTFbr7$$\"3Z******QN\\HiF7$!3E+++AaTsTFbr7$$ \"3C+++,[]OjF7$!3u*****f(H6oTFbr7$$\"3[*****H>!RTkF7$!37+++i3[jTFbr7$$ \"3l******f56WlF7$!3A+++dc_eTFbr7$$\"3++++(F7$!3T+++\"4(H:T Fbr7$$\"39+++Uv-$G(F7$!3x*****z]nz5%Fbr7$$\"3S+++0(zVO(F7$!3)******4tk .5%Fbr7$$\"3k*****z#)GJW(F7$!3#)*****>_$\\#4%Fbr7$$\"3C+++UZA>vF7$!3%) *****\\_eV3%Fbr7$$\"3E+++eeh#f(F7$!3m*****fFkf2%Fbr7$$\"3i******y!\\Km (F7$!3I+++5_JnSFbr7$$\"3K+++5(p5t(F7$!3o*****4n:%eSFbr7$$\"3c*****zM@g z(F7$!33+++;*p#\\SFbr7$$\"3K+++!yX!eyF7$!35+++D@))RSFbr7$$\"3?+++^G3z'R?SFbr7$$\"3.+++@Q%f-)F7 $!3P+++ssI5SFbr7$$\"3i*****f[Oc2)F7$!3?+++p<****RFbr7$$\"3u******)**y? 7)F7$!37+++fTX*)RFbr7$$\"3k*****\\H*>l\")F7$!3=+++v#)pyRFbr7$$\"3\\+++ \"QA\\?)F7$!3;+++4zsnRFbr7$$\"3%******z5q6C)F7$!3/+++BoacRFbr7$$\"3!** *****H4'QF)F7$!3++++W(e^%RFbr7$$\"3K+++G'4HI)F7$!3&*******ztcLRFbr7$$ \"3[+++;qAG$)F7$!3*******RTw<#RFbr7$$\"3S+++5'>(\\$)F7$!3>+++=&*y4RFbr 7$$\"3C+++e#*Gn$)F7$!3!)******[.h(*QFbr7$$\"3]*****>uK3Q)F7$!3=+++iDC& )QFbr7$$\"3K+++?8C!R)F7$!31+++3)*osQFbr7$$\"3a+++F-S&R)F7$!3#)*****4ub *fQFbr7$$\"37+++4\")='R)F7$!3'******>-Wq%QFbr7$$\"3!)******)QwCR)F7$!3 -+++G$eR$QFbr7$$\"3[+++h>Q)F7$!3$)******[Bq?QFbr7$$\"3e*****HQ**4P)F 7$!3))*******zzs!QFbr7$$\"3C+++nR$HN)F7$!3#)*****HU%p$z$Fbr7$$\"39+++X nwH$)F7$!31+++\"**\\*zPFbr7$$\"3Y*****4H?8I)F7$!3%)*****fJ]gw$Fbr7$$\" 3m*****R'RSn#)F7$!31+++_#**>v$Fbr7$$\"3C+++`B\"yA)F7$!34+++/2!yt$Fbr7$ $\"3)*******zMK#=)F7$!38+++D'eMs$Fbr7$$\"3;+++1npI\")F7$!3&******>.x*3 PFbr7$$\"3K+++\"HqE2)F7$!3;+++,+O%p$Fbr7$$\"3h*****\\_ez+)F7$!3%****** fn6'zOFbr7$$\"3()*****\\M[i$zF7$!3?+++ritkOFbr7$$\"3g*****>I&>dyF7$!3! )*****42Q(\\OFbr7$$\"3Q+++U(=/x(F7$!3z*****4V@Yj$Fbr7$$\"33+++\")f\\vw F7$!3z******o2R>OFbr7$$\"3/+++E_&>d(F7$!3'******zb]Sg$Fbr7$$\"3e*****4 )oEfuF7$!3++++3`g)e$Fbr7$$\"3)******pwKoL(F7$!35+++I&fId$Fbr7$$\"3q*** **\\gsR?(F7$!3')*******od2iR$Fbr7$$\"3!******H)o&*zY F7$!3'******\\]A%zLFbr7$$\"3))*****>J%HzUF7$!3-+++1-IiLFbr7$$\"3++++?q 3=QF7$!33+++brdWLFbr7$$\"3!)*****zY,JF$F7$!3=+++^FkDLFbr7$$\"34+++zJ$4 g#F7$!31+++?ly.LFbr7$F_bl$!3#******fA&o)H$Fbr7$Fjal$!37+++/9;$H$Fbr7$F eal$!3++++'HqqG$Fbr7$F`al$!3;+++-x>!G$Fbr7$F[al$!3;+++Ez@sKFbr7$Ff`l$! 3%******z0EEE$Fbr7$Fa`l$!3#)******pak]KFbr7$F\\`l$!3>+++8?=NKFbr7$Fg_l $!3%)*****>UD^@$Fbr7$Fb_l$!3)******4'fP!>$Fbr7$F]_l$!3-+++&[+E;$Fbr7$F h^l$!3:+++=M0MJFbr7$Fc^l$!3\"******H]gh5$Fbr7$F^^l$!3!)*****RHl%zIFbr7 $Fi]l$!33+++M*fS0$Fbr7$Fd]l$!30+++Pu')HIFbr7$F_]l$!3;+++!ojn+$Fbr7$Fj \\l$!3#*******fBi%)HFbr7$Fe\\l$!3))*****pRHL'HFbr7$$!31+++K`Y5AF0$!3'* *****Rb-m'GFbr7$$!39+++d]QiCF0$!33+++gh3\"y#Fbr7$$!3;+++6_E>DF0$!3**** ***pF0$!3))*****\\OOtO#Fbr7$$!3-+++Fbr7$$!3_*****H!pDczFjs$!3-+++=5(z!>Fbr7$$!3H+++sDwOpFjs$!3# ******4-yL&=Fbr7$$!3D+++EO$3,'Fjs$!31+++.09*z\"Fbr7$$!39+++X1[v^Fjs$!3 ++++:w?X v(4aEFjs$!30+++DhsJ:Fbr7$$!3#)*****p]@M?#Fjs$!3))*****>ei(y9Fbr7$$!3)* *****z\")*p8=Fjs$!3,+++\"Fjs$!3.+++9b[?8Fbr7$$!3O+++Wlf`&*F`q$!3/+++ bq(yE\"Fbr7$$!3w******z%>Db(F`q$!3%******zXA`@\"Fbr7$$!33++++'>*)*eF`q $!3%******pm6G;\"Fbr7$$!39+++s5XZXF`q$!3.+++$=O.6\"Fbr7$$!31+++2R*fX$F `q$!3)*******p)))y0\"Fbr7$$!39+++_E&ee#F`q$!3\"******zyja+\"Fbr7$$!3.+ ++Sm(=!>F`q$!3C+++].cI&*F77$$!3'******4&p]s8F`q$!3Z******zgh1!*F77$$!3 G+++-%evp*Ffo$!3[+++OFx#[)F77$$!3q*****\\Q]7p'Ffo$!3!)******RX+fzF77$$ !3I+++\\&*p%\\%Ffo$!3L+++R7HNuF77$$!3,+++vO<<;\"pF77$$ !31+++pfWT=Ffo$!3s******e/(zQ'F77$$!33+++hUG66Ffo$!3')*****zFUV'eF77$$ !3=+++)=7sQ'FW$!3r*****pBE2M&F77$$!3-+++,U)=Y$FW$!3w******Ry6<[F77$$!3 1+++l![fu\"FW$!3))*****H.9NH%F77$$!3R+++$RKE/)FL$!3C+++DG\"*pPF77$$!31 +++N?'QH$FL$!3)******\\)HJYKF77$$!3)******>\"e(3:\"FL$!3;+++,QrAFF77$$ !3y*****>N]_?$FA$!3-+++&*[6*>#F77$$!3O+++07Q&G'F;$!3-+++)3;bn\"F77$$!3 K+++`:x[mF4$!3'******pI<>:\"F77$$!3-+++FU8`Z5F07$F($\"3?+++0-z)=%F0-%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"$Fa b\\lF(-%&STYLEG6#%%LINEG-%)POLYGONSG6jcn7%F'7$$!++-GveF`q$\"+cx)fB&!#6 7$$!\"%F)F(7%F\\c\\l7$$!+R/9aPF0$\"+^v>Z5!#5Fbc\\l7%Ffc\\l7$$!+*R6\"pU F7$\"+Ijzq:F[d\\lFbc\\l7%F]d\\l7$$!+a%4JR#Fbr$\"+H^R%4#F[d\\lFbc\\l7%F cd\\l7$$!+V3s,\"*Fbr$\"+1S*zh#F[d\\lFbc\\l7%Fid\\l7$$!+ZP!yq#!#;$\"+7J fTJF[d\\lFbc\\l7%F_e\\l7$$!+J')Q)z'Fbe\\l$\"+wF>lOF[d\\lFbc\\l7%Ffe\\l 7$$!+\"our]\"!#:$\"+KOz)=%F[d\\lFbc\\l7%F\\f\\l7$$!+e;$z.$F_f\\l$\"+)y 'R7ZF[d\\lFbc\\l7%Fcf\\l7$$!+lo]zcF_f\\l$\"+iS+O_F[d\\lFbc\\l7%Fif\\l7 $$!++kN*)**F_f\\l$\"+m#='fdF[d\\lFbc\\l7%F_g\\l7$$!+\\oQq;!#9$\"+wNC$G 'F[d\\lFbc\\l7%Feg\\l7$$!+r'fnn#Fhg\\l$\"+jf)o!oF[d\\lFbc\\l7%F\\h\\l7 $$!+(Q4i8%Fhg\\l$\"+;PbItF[d\\lFbc\\l7%Fbh\\l7$$!+)yaL>'Fhg\\l$\"+]!eU &yF[d\\lFbc\\l7%Fhh\\l7$$!+o%R=-*Fhg\\l$\"+1Q,y$)F[d\\lFbc\\l7%F^i\\l7 $$!+D2m#G\"!#8$\"+]-%=!*)F[d\\lFbc\\l7%Fdi\\l7$$!+$)3e%y\"Fgi\\l$\"+t> wD%*F[d\\lFbc\\l7%F[j\\l7$$!+)*eANCFgi\\l$\"+5*4)\\**F[d\\lFbc\\l7%Faj \\l7$$!+cnVlKFgi\\l$\"+YASZ5!\"*Fbc\\l7%Fgj\\l7$$!+&GZ'4VFgi\\l$\"+tY% )*4\"F\\[]lFbc\\l7%F^[]l7$$!+x6&eg&Fgi\\l$\"+$*RJ_6F\\[]lFbc\\l7%Fd[]l 7$$!+Go\\&>(Fgi\\l$\"+)3<[?\"F\\[]lFbc\\l7%Fj[]l7$$!+DZIB\"*Fgi\\l$\"+ s@Od7F\\[]lFbc\\l7%F`\\]l7$$!+d-qV6!#7$\"+t*e*48F\\[]lFbc\\l7%Ff\\]l7$ $!+etp=9Fi\\]l$\"+V*=EO\"F\\[]lFbc\\l7%F]]]l7$$!+58eU)eDFi\\]l$\"+$GB6_\"F\\[]lFbc\\l7%F_^]l7$$!+u#eA1$Fi\\]l$\"+QO>u: F\\[]lFbc\\l7%Fe^]l7$$!+4**eOOFi\\]l$\"++&>ui\"F\\[]lFbc\\l7%F[_]l7$$! +X:9(G%Fi\\]l$\"+x(G3o\"F\\[]lFbc\\l7%Fa_]l7$$!+*\\_*=]Fi\\]l$\"+:OXM< F\\[]lFbc\\l7%Fg_]l7$$!+$RHm$eFi\\]l$\"+a5L)y\"F\\[]lFbc\\l7%F]`]l7$$! +6F&>F\\[]lFbc\\l7%F_a]l7$$!+N&3K +\"Fac\\l$\"+T(Gt+#F\\[]lFbc\\l7%Fea]l7$$!+#zK=8\"Fac\\l$\"+`MEj?F\\[] lFbc\\l7%F[b]l7$$!+7Fcp7Fac\\l$\"+Ao$)>@F\\[]lFbc\\l7%Fab]l7$$!+MLr:9F ac\\l$\"+v\"er<#F\\[]lFbc\\l7%Fgb]l7$$!+#Gi\"p:Fac\\l$\"+a(e`B#F\\[]lF bc\\l7%F]c]l7$$!+L\\=G $F\\[]lFbc\\l7%Fej]l7$$!+T@K?F[d\\l$\"+u;3%[$F\\[]lFbc\\l7%Fa[^l7$$!+:OnzAF[d\\l$\"+O+jqMF\\[] lFbc\\l7%Fg[^l7$$!+\"zXAd#F[d\\l$\"+6*[xF[d\\l$\"+uDR7NF\\[]lFbc\\l7%F_ `^l7$$!+aZi&G)F[d\\l$\"+I!o[_$F\\[]lFbc\\l7%Fe`^l7$$!+[4*H#))F[d\\l$\" +M'Qp`$F\\[]lFbc\\l7%F[a^l7$$!+=3cO$*F[d\\l$\"+'yl&[NF\\[]lFbc\\l7%Faa ^l7$$!+v&y+$)*F[d\\l$\"+*RH(fNF\\[]lFbc\\l7%Fga^l7$$!+ZUkI5F\\[]l$\"+u '>/d$F\\[]lFbc\\l7%F]b^l7$$!+O]zw5F\\[]l$\"+_Fj!e$F\\[]lFbc\\l7%Fcb^l7 $$!+ngk@6F\\[]l$\"+#Go.f$F\\[]lFbc\\l7%Fib^l7$$!+o,Nl6F\\[]l$\"+=!G'*f $F\\[]lFbc\\l7%F_c^l7$$!+F[.37F\\[]l$\"+j\\T3OF\\[]lFbc\\l7%Fec^l7$$!+ 1x!)\\7F\\[]l$\"+`Gt;OF\\[]lFbc\\l7%F[d^l7$$!+,2w!H\"F\\[]l$\"+GeeCOF \\[]lFbc\\l7%Fad^l7$$!+(*H(4L\"F\\[]l$\"+C#y>j$F\\[]lFbc\\l7%Fgd^l7$$! +&R80P\"F\\[]l$\"+PW\"*QOF\\[]lFbc\\l7%F]e^l7$$!+E@W49F\\[]l$\"+S))RXO F\\[]lFbc\\l7%Fce^l7$$!+tA\"yW\"F\\[]l$\"+FdV^OF\\[]lFbc\\l7%Fie^l7$$! +/4n&[\"F\\[]l$\"+x#Hql$F\\[]lFbc\\l7%F_f^l7$$!+#**fI_\"F\\[]l$\"+RN=i OF\\[]lFbc\\l7%Fef^l7$$!+dr,g:F\\[]l$\"+:C!pm$F\\[]lFbc\\l7%F[g^l7$$!+ xid'f\"F\\[]l$\"+h'*=rOF\\[]lFbc\\l7%Fag^l7$$!+,!oFj\"F\\[]l$\"+%))[]n $F\\[]lFbc\\l7%Fgg^l7$$!+n,io;F\\[]l$\"+VN[yOF\\[]lFbc\\l7%F]h^l7$$!+m \"eTq\"F\\[]l$\"+ep\\\"o$F\\[]lFbc\\l7%Fch^l7$$!+T_SR\">F\\[]l$\"+a6!4p$F\\[]lFbc\\l7%Fgj^l7$$!+5w'e%>F\\[]l$\"+P%[5p$F\\[ ]lFbc\\l7%F][_l7$$!+#)\\fz>F\\[]l$\"+Poz!p$F\\[]lFbc\\l7%Fc[_l7$$!+-[; 8?F\\[]l$\"+`([,p$F\\[]lFbc\\l7%Fi[_l7$$!+*H*eY?F\\[]l$\"+Jl5*o$F\\[]l Fbc\\l7%F_\\_l7$$!+W+))z?F\\[]l$\"+lCn(o$F\\[]lFbc\\l7%Fe\\_l7$$!+>![I 6#F\\[]l$\"+;)[eo$F\\[]lFbc\\l7%F[]_l7$$!+&o.h9#F\\[]l$\"+;yj$o$F\\[]l Fbc\\l7%Fa]_l7$$!+Jq0z@F\\[]l$\"+%oT5o$F\\[]lFbc\\l7%Fg]_l7$$!+Mw\">@# F\\[]l$\"+KE1yOF\\[]lFbc\\l7%F]^_l7$$!+'p%pWAF\\[]l$\"+zGquOF\\[]lFbc \\l7%Fc^_l7$$!+\"4(RxAF\\[]l$\"+iY'4n$F\\[]lFbc\\l7%Fi^_l7$$!+-M.5BF\\ []l$\"+]-&om$F\\[]lFbc\\l7%F___l7$$!+a>hUBF\\[]l$\"+a>OiOF\\[]lFbc\\l7 %Fe__l7$$!+X39vBF\\[]l$\"+V@]dOF\\[]lFbc\\l7%F[`_l7$$!+uzi2CF\\[]l$\"+ bKF_OF\\[]lFbc\\l7%Fa`_l7$$!+m53SCF\\[]l$\"+9ynYOF\\[]lFbc\\l7%Fg`_l7$ $!+#p2DZ#F\\[]l$\"+V%=2k$F\\[]lFbc\\l7%F]a_l7$$!+&H:\\]#F\\[]l$\"+!)yR MOF\\[]lFbc\\l7%Fca_l7$$!+,7JPDF\\[]l$\"+(**=xi$F\\[]lFbc\\l7%Fia_l7$$ !+MEqpDF\\[]l$\"+8[o?OF\\[]lFbc\\l7%F_b_l7$$!+Ln4-EF\\[]l$\"+8&)H8OF\\ []lFbc\\l7%Feb_l7$$!+i0]MEF\\[]l$\"+pMc0OF\\[]lFbc\\l7%F[c_l7$$!+66#pm #F\\[]l$\"+`K[(f$F\\[]lFbc\\l7%Fac_l7$$!+6`O*p#F\\[]l$\"+j;1*e$F\\[]lF bc\\l7%Fgc_l7$$!+G+%=t#F\\[]l$\"+PFI!e$F\\[]lFbc\\l7%F]d_l7$$!+s?NkFF \\[]l$\"+x2@rNF\\[]lFbc\\l7%Fcd_l7$$!+)=3pz#F\\[]l$\"+o.zhNF\\[]lFbc\\ l7%Fid_l7$$!+a]^HGF\\[]l$\"+*RY?b$F\\[]lFbc\\l7%F_e_l7$$!+v#z@'GF\\[]l $\"+#3%)>a$F\\[]lFbc\\l7%Fee_l7$$!+qt!\\*GF\\[]l$\"+x*3;`$F\\[]lFbc\\l 7%F[f_l7$$!+edqFHF\\[]l$\"+0q#4_$F\\[]lFbc\\l7%Faf_l7$$!+Z2egHF\\[]l$ \"+tW%*4NF\\[]lFbc\\l7%Fgf_l7$$!+1&QN*HF\\[]l$\"+*3o')\\$F\\[]lFbc\\l7 %F]g_l7$$!+^]eEIF\\[]l$\"+#)\\5([$F\\[]lFbc\\l7%Fcg_l7$$!+8isfIF\\[]l$ \"+8FEvMF\\[]lFbc\\l7%Fig_l7$$!+9w'H4$F\\[]l$\"+$H\\JY$F\\[]lFbc\\l7%F _h_l7$$!+JYJEJF\\[]l$\"+!>t2X$F\\[]lFbc\\l7%Feh_l7$$!+nBxfJF\\[]l$\"+S L9QMF\\[]lFbc\\l7%F[i_l7$$!+2cM$>$F\\[]l$\"+]\"p_U$F\\[]lFbc\\l7%Fai_l 7$$!+)yQqA$F\\[]l$\"+-0;7MF\\[]lFbc\\l7%Fgi_l7$$!+\\f&3E$F\\[]l$\"+Yx# ))R$F\\[]lFbc\\l7%F]j_l7$$!+#p+[H$F\\[]l$\"++V$F\\[]l$\"+vj>HLF\\[]lFbc\\l7%F[\\`l7$$!+h!GlY$F\\[]l$\"+j7w9LF\\ []lFbc\\l7%Fa\\`l7$$!+,RG,NF\\[]l$\"+,l=+LF\\[]lFbc\\l7%Fg\\`l7$$!+.m< ONF\\[]l$\"+Yb[&G$F\\[]lFbc\\l7%F]]`l7$$!+9Z?rNF\\[]l$\"+#3s1F$F\\[]lF bc\\l7%Fc]`l7$$!+wfO1OF\\[]l$\"+s*fdD$F\\[]lFbc\\l7%Fi]`l7$$!+0tlTOF\\ []l$\"+*>j2C$F\\[]lFbc\\l7%F_^`l7$$!+hZ2xOF\\[]l$\"+3epDKF\\[]lFbc\\l7 %Fe^`l7$$!+PNh7PF\\[]l$\"+W=d5KF\\[]lFbc\\l7%F[_`l7$$!+YzE[PF\\[]l$\"+ 'G0a>$F\\[]lFbc\\l7%Fa_`l7$$!+>9.%y$F\\[]l$\"+!)*4-=$F\\[]lFbc\\l7%Fg_ `l7$$!++l*)>QF\\[]l$\"+u&**\\;$F\\[]lFbc\\l7%F]``l7$$!+f[&e&QF\\[]l$\" +]uy\\JF\\[]lFbc\\l7%Fc``l7$$!+-t*=*QF\\[]l$\"+`meMJF\\[]lFbc\\l7%Fi`` l7$$!+\"z8!GRF\\[]l$\"+F)4%>JF\\[]lFbc\\l7%F_a`l7$$!+nM>kRF\\[]l$\"+[ \"pU5$F\\[]lFbc\\l7%Fea`l7$$!+&oC/+%F\\[]l$\"+ki<*3$F\\[]lFbc\\l7%F[b` l7$$!+W]pOSF\\[]l$\"+GA9uIF\\[]lFbc\\l7%Fab`l7$$!+J9*H2%F\\[]l$\"+Uu+IF\\[]lFbc\\l7%F_d`l7$$!+A6OaUF \\[]l$\"+pSq&)HF\\[]lFbc\\l7%Fed`l7$$!+VZ]!H%F\\[]l$\"+2-LrHF\\[]lFbc \\l7%F[e`l7$$!+aqcEVF\\[]l$\"+gf2dHF\\[]lFbc\\l7%Fae`l7$$!+a7`iVF\\[]l $\"+L[%H%HF\\[]lFbc\\l7%Fge`l7$$!+$R!Q)R%F\\[]l$\"+G$R*GHF\\[]lFbc\\l7 %F]f`l7$$!+[u4MWF\\[]l$\"+H41:HF\\[]lFbc\\l7%Fcf`l7$$!+5ampWF\\[]l$\"+ )458!HF\\[]lFbc\\l7%Fif`l7$$!+ht10XF\\[]l$\"+vio()GF\\[]lFbc\\l7%F_g`l 7$$!+llGSXF\\[]l$\"+\"*y=uGF\\[]lFbc\\l7%Feg`l7$$!+XlIvXF\\[]l$\"+&Q73 'GF\\[]lFbc\\l7%F[h`l7$$!+n665YF\\[]l$\"+GibZGF\\[]lFbc\\l7%Fah`l7$$!+ ?ZoWYF\\[]l$\"+k\\TMGF\\[]lFbc\\l7%Fgh`l7$$!+\"*>,zYF\\[]l$\"+\\KQ@GF \\[]lFbc\\l7%F]i`l7$$!+O$yIr%F\\[]l$\"+/\\X3GF\\[]lFbc\\l7%Fci`l7$$!+W (pou%F\\[]l$\"+tHi&z#F\\[]lFbc\\l7%Fii`l7$$!+)*GP!y%F\\[]l$\"+*yzGy#F \\[]lFbc\\l7%F_j`l7$$!+A_d8[F\\[]l$\"+Vq@qFF\\[]lFbc\\l7%Fej`l7$$!+C\\ YY[F\\[]l$\"+ieidFF\\[]lFbc\\l7%F[[al7$$!+I5.z[F\\[]l$\"+&)o4XFF\\[]lF bc\\l7%Fa[al7$$!+2ME6\\F\\[]l$\"+W.iKFF\\[]lFbc\\l7%Fg[al7$$!+!y_J%\\F \\[]l$\"+Xh=?FF\\[]lFbc\\l7%F]\\al7$$!+K2pu\\F\\[]l$\"+[Ry2FF\\[]lFbc \\l7%Fc\\al7$$!+1(pe+&F\\[]l$\"+ZKS&p#F\\[]lFbc\\l7%Fi\\al7$$!+*)HoO]F \\[]l$\"+VM.$o#F\\[]lFbc\\l7%F_]al7$$!+)pCr1&F\\[]l$\"+>RmqEF\\[]lFbc \\l7%Fe]al7$$!+Z(*=(4&F\\[]l$\"+1TGeEF\\[]lFbc\\l7%F[^al7$$!+@Q(o7&F\\ []l$\"+VN)ek#F\\[]lFbc\\l7%Fa^al7$$!+RLXLEF\\[]lFbc\\l7 %Fg^al7$$!+2a3&=&F\\[]l$\"+'>z4i#F\\[]lFbc\\l7%F]_al7$$!+$y2O@&F\\[]l$ \"+3bX3EF\\[]lFbc\\l7%Fc_al7$$!+D)Q7vWDF\\[]lFbc\\l7%Faaal7$$!+eF]w`F\\[]l$\"+**QtJDF \\[]lFbc\\l7%Fgaal7$$!+.\"zAS&F\\[]l$\"+bkg=DF\\[]lFbc\\l7%F]bal7$$!+` bmFaF\\[]l$\"+!)>O0DF\\[]lFbc\\l7%Fcbal7$$!+FOm_aF\\[]l$\"+hR*>\\#F\\[ ]lFbc\\l7%Fibal7$$!+b]FxaF\\[]l$\"+xi\\yCF\\[]lFbc\\l7%F_cal7$$!+V<],b F\\[]l$\"+*=j[Y#F\\[]lFbc\\l7%Fecal7$$!+RdMDbF\\[]l$\"+I$*3^CF\\[]lFbc \\l7%F[dal7$$!+6#4)[bF\\[]l$\"+$pprV#F\\[]lFbc\\l7%Fadal7$$!+7W*=d&F\\ []l$\"+A'*4BCF\\[]lFbc\\l7%Fgdal7$$!+mOg%f&F\\[]l$\"+)yu)3CF\\[]lFbc\\ l7%F]eal7$$!+R$Rph&F\\[]l$\"+&=\"\\%R#F\\[]lFbc\\l7%Fceal7$$!+DQ!*QcF \\[]l$\"+2^%*zBF\\[]lFbc\\l7%Fieal7$$!+G&*\\gcF\\[]l$\"+OJBlBF\\[]lFbc \\l7%F_fal7$$!+Y)G# F\\[]lFbc\\l7%Fajal7$$!+)e?<*eF\\[]l$\"+Mr#[<#F\\[]lFbc\\l7%Fgjal7$$!+ Dds3fF\\[]l$\"+b#\\x:#F\\[]lFbc\\l7%F][bl7$$!+z8RDfF\\[]l$\"+&3z/9#F\\ []lFbc\\l7%Fc[bl7$$!+b\"><%fF\\[]l$\"+Ad,B@F\\[]lFbc\\l7%Fi[bl7$$!+\"f 5x&fF\\[]l$\"+J$e`5#F\\[]lFbc\\l7%F_\\bl7$$!+arOtfF\\[]l$\"+Ph](3#F\\[ ]lFbc\\l7%Fe\\bl7$$!+Z-p))fF\\[]l$\"+8%e%p?F\\[]lFbc\\l7%F[]bl7$$!+67o .gF\\[]l$\"+sW@^?F\\[]lFbc\\l7%Fa]bl7$$!+E8M=gF\\[]l$\"+dOxK?F\\[]lFbc \\l7%Fg]bl7$$!+:=nKgF\\[]l$\"+O`89?F\\[]lFbc\\l7%F]^bl7$$!+ZQnYgF\\[]l $\"+\"*))H&*>F\\[]lFbc\\l7%Fc^bl7$$!+U&[.1'F\\[]l$\"+6PEw>F\\[]lFbc\\l 7%Fi^bl7$$!+tpptgF\\[]l$\"+%=Hq&>F\\[]lFbc\\l7%F__bl7$$!+t,s'3'F\\[]l$ \"+!p%fP>F\\[]lFbc\\l7%Fe_bl7$$!+N\">%*4'F\\[]l$\"+(ffz\">F\\[]lFbc\\l 7%F[`bl7$$!+A[z6hF\\[]l$\"+[K7)*=F\\[]lFbc\\l7%Fa`bl7$$!+q\"[Q7'F\\[]l $\"+i\\3y=F\\[]lFbc\\l7%Fg`bl7$$!+!4!eNhF\\[]l$\"+>S%y&=F\\[]lFbc\\l7% F]abl7$$!+\"[\"*p9'F\\[]l$\"+j'*RP=F\\[]lFbc\\l7%Fcabl7$$!+HK3ehF\\[]l $\"+*3^n\"=F\\[]lFbc\\l7%Fiabl7$$!+DH:F\\[]lFbc\\l7 %Fgfbl7$$!+d*R.G'F\\[]l$\"+_8i0:F\\[]lFbc\\l7%F]gbl7$$!+[(fnG'F\\[]l$ \"+1bw\"[\"F\\[]lFbc\\l7%Fcgbl7$$!+u#zGH'F\\[]l$\"+GAod9F\\[]lFbc\\l7% Figbl7$$!+:1q)H'F\\[]l$\"+[#pLV\"F\\[]lFbc\\l7%F_hbl7$$!+NgA/jF\\[]l$ \"+$>C)39F\\[]lFbc\\l7%Fehbl7$$!+,!e%4jF\\[]l$\"+#fWSQ\"F\\[]lFbc\\l7% F[ibl7$$!+,#*R9jF\\[]l$\"+\")y-f8F\\[]lFbc\\l7%Faibl7$$!+lD0>jF\\[]l$ \"+/9xL8F\\[]lFbc\\l7%Fgibl7$$!+z7UBjF\\[]l$\"+ICF38F\\[]lFbc\\l7%F]jb l7$$!+9)3vK'F\\[]l$\"+^\"GDG\"F\\[]lFbc\\l7%Fcjbl7$$!+S*=8L'F\\[]l$\"+ /d`c7F\\[]lFbc\\l7%Fijbl7$$!+^d&[L'F\\[]l$\"+x@HI7F\\[]lFbc\\l7%F_[cl7 $$!+%oB\"QjF\\[]l$\"+IYz.7F\\[]lFbc\\l7%Fe[cl7$$!+Yv7TjF\\[]l$\"+5,/x6 F\\[]lFbc\\l7%F[\\cl7$$!+HD(QM'F\\[]l$\"+xc-]6F\\[]lFbc\\l7%Fa\\cl7$$! +MUOYjF\\[]l$\"+E%[F7\"F\\[]lFbc\\l7%Fg\\cl7$$!+&p3'[jF\\[]l$\"+9b?&4 \"F\\[]lFbc\\l7%F]]cl7$$!+!R71N'F\\[]l$\"+&>%Rn5F\\[]lFbc\\l7%Fc]cl7$$ !+nAQ_jF\\[]l$\"+^=JR5F\\[]lFbc\\l7%Fi]cl7$$!+bd#RN'F\\[]l$\"+Lg&4,\"F \\[]lFbc\\l7%F_^cl7$$!+x2DbjF\\[]l$\"+=]CB)*F[d\\lFbc\\l7%Fe^cl7$$!+id OcjF\\[]l$\"+]F:M&*F[d\\lFbc\\l7%F[_cl7$$!+\\'zsN'F\\[]l$\"+ZnEU#*F[d \\lFbc\\l7%Fa_cl7$$!+!*=+ejF\\[]l$\"+0QdZ*)F[d\\lFbc\\l7%Fg_cl7$$!+XCa ejF\\[]l$\"+][1]')F[d\\lFbc\\l7%F]`cl7$$!+v<\"*ejF\\[]l$\"+6bt\\$)F[d \\lFbc\\l7%Fc`cl7$$!+F37fjF\\[]l$\"+?neY!)F[d\\lFbc\\l7%Fi`cl7$$!+25=f jF\\[]l$\"+A`iSxF[d\\lFbc\\l7%F_acl7$$!+dT5fjF\\[]l$\"+'ok=V(F[d\\lFbc \\l7%Feacl7$$!+8D!*ejF\\[]l$\"+@`K?rF[d\\lFbc\\l7%F[bcl7$$!+g')eejF\\[ ]l$\"+na.1oF[d\\lFbc\\l7%Fabcl7$$!+\"[v\"ejF\\[]l$\"+i;.*['F[d\\lFbc\\ l7%Fgbcl7$$!+\"4wwN'F\\[]l$\"+y#f$phF[d\\lFbc\\l7%F]ccl7$$!+qP5djF\\[] l$\"+)*H2ZeF[d\\lFbc\\l7%Fcccl7$$!+#)=ZcjF\\[]l$\"+KsBAbF[d\\lFbc\\l7% Ficcl7$$!+$z$zbjF\\[]l$\"+ik#\\>&F[d\\lFbc\\l7%F_dcl7$$!+\"y#3bjF\\[]l $\"+'\\D_'[F[d\\lFbc\\l7%Fedcl7$$!+M>NajF\\[]l$\"+D'HK`%F[d\\lFbc\\l7% F[ecl7$$!+dSh`jF\\[]l$\"+xY/*>%F[d\\lFbc\\l7%Faecl7$$!+n:)GN'F\\[]l$\" +apyiQF[d\\lFbc\\l7%Fgecl7$$!+'Qm@N'F\\[]l$\"+cIeCNF[d\\lFbc\\l7%F]fcl 7$$!+U)z9N'F\\[]l$\"+!epX=$F[d\\lFbc\\l7%Fcfcl7$$!+nD$3N'F\\[]l$\"+)p# *G%GF[d\\lFbc\\l7%Fifcl7$$!++WB]jF\\[]l$\"+Cwq*\\#F[d\\lFbc\\l7%F_gcl7 $$!+$H%p\\jF\\[]l$\"+az)[jF\\[]l$\"+'\\wFY\"F[d\\lFbc\\l7%Fahcl 7$$!+[o\\[jF\\[]l$\"+LN'F\\[]l$!+fM ykSF[d\\lFbc\\l7%Fa^dl7$$!+Ap0ajF\\[]l$!+c?\")*R%F[d\\lFbc\\l7%Fg^dl7$ $!+(4#zajF\\[]l$!+YopKZF[d\\lFbc\\l7%F]_dl7$$!+,E^bjF\\[]l$!+*zFL1&F[d \\lFbc\\l7%Fc_dl7$$!+Wa?cjF\\[]l$!+4ag\"R&F[d\\lFbc\\l7%Fi_dl7$$!+8u&o N'F\\[]l$!+(oSur&F[d\\lFbc\\l7%F_`dl7$$!+n^XdjF\\[]l$!+[]vSgF[d\\lFbc \\l7%Fe`dl7$$!+I`)zN'F\\[]l$!+*3![hjF[d\\lFbc\\l7%F[adl7$$!+yXVejF\\[] l$!+\"Gd&zmF[d\\lFbc\\l7%Faadl7$$!+@(*yejF\\[]l$!+%eP\\*pF[d\\lFbc\\l7 %Fgadl7$$!+xx.fjF\\[]l$!+')4e2tF[d\\lFbc\\l7%F]bdl7$$!+Mg;fjF\\[]l$!+% *fX:[ajF \\[]l$!+LiO&***F[d\\lFbc\\l7%Fcedl7$$!+'REIN'F\\[]l$!+rC+G5F\\[]lFbc\\ l7%Fiedl7$$!+%*zM^jF\\[]l$!+AR>c5F\\[]lFbc\\l7%F_fdl7$$!+'oQ%\\jF\\[]l $!+XL6%3\"F\\[]lFbc\\l7%Fefdl7$$!+N8HZjF\\[]l$!+(Hj<6\"F\\[]lFbc\\l7%F [gdl7$$!+w#**[M'F\\[]l$!+wk9R6F\\[]lFbc\\l7%Fagdl7$$!++jDUjF\\[]l$!+!p li;\"F\\[]lFbc\\l7%Fggdl7$$!+OmNRjF\\[]l$!+CQ7$>\"F\\[]lFbc\\l7%F]hdl7 $$!+H\\>OjF\\[]l$!+1Qs>7F\\[]lFbc\\l7%Fchdl7$$!+@iwKjF\\[]l$!+(eogC\"F \\[]lFbc\\l7%Fihdl7$$!+Kf1HjF\\[]l$!+:6;s7F\\[]lFbc\\l7%F_idl7$$!+I)*3 DjF\\[]l$!+9V+)H\"F\\[]lFbc\\l7%Feidl7$$!+?S$3K'F\\[]l$!+q5gB8F\\[]lFb c\\l7%F[jdl7$$!+:\\H;jF\\[]l$!+8U&*[8F\\[]lFbc\\l7%Fajdl7$$!+:#p9J'F\\ []l$!+3l1u8F\\[]lFbc\\l7%Fgjdl7$$!+!*QN1jF\\[]l$!+S1%*)R\"F\\[]lFbc\\l 7%F][el7$$!+gh%4I'F\\[]l$!+6#zNU\"F\\[]lFbc\\l7%Fc[el7$$!+qMC&H'F\\[]l $!+JZ)zW\"F\\[]lFbc\\l7%Fi[el7$$!+xMC*G'F\\[]l$!+7'f@Z\"F\\[]lFbc\\l7% F_\\el7$$!+JS%HG'F\\[]l$!+kh5'\\\"F\\[]lFbc\\l7%Fe\\el7$$!+bJMwiF\\[]l $!+)fE)>:F\\[]lFbc\\l7%F[]el7$$!+L!R%piF\\[]l$!+;IKV:F\\[]lFbc\\l7%Fa] el7$$!+!**HAE'F\\[]l$!+>ufm:F\\[]lFbc\\l7%Fg]el7$$!+$[9ZD'F\\[]l$!+*p^ '*e\"F\\[]lFbc\\l7%F]^el7$$!+\"3\"*oC'F\\[]l$!+^w[7;F\\[]lFbc\\l7%Fc^e l7$$!+d%e(QiF\\[]l$!+jp5N;F\\[]lFbc\\l7%Fi^el7$$!+s`JIiF\\[]l$!+G7^d;F \\[]lFbc\\l7%F__el7$$!+o1c@iF\\[]l$!+U>qz;F\\[]lFbc\\l7%Fe_el7$$!+\\K \\7iF\\[]l$!+30o,'F\\[]l$!+uj+XF\\[]lFbc\\l7%Fgce l7$$!+6'Q=4'F\\[]l$!+hZwH>F\\[]lFbc\\l7%F]del7$$!+C_%*ygF\\[]l$!+<%z# \\>F\\[]lFbc\\l7%Fcdel7$$!+3qslgF\\[]l$!+cQfo>F\\[]lFbc\\l7%Fidel7$$!+ XH=_gF\\[]l$!+-(3x)>F\\[]lFbc\\l7%F_eel7$$!+z>JQgF\\[]l$!+oXi1?F\\[]lF bc\\l7%Feeel7$$!+5I6CgF\\[]l$!+k?MD?F\\[]lFbc\\l7%F[fel7$$!+*)[e4gF\\[ ]l$!+2='Q/#F\\[]lFbc\\l7%Fafel7$$!+:ks%*fF\\[]l$!+?W=i?F\\[]lFbc\\l7%F gfel7$$!+Jj`zfF\\[]l$!+^0J!3#F\\[]lFbc\\l7%F]gel7$$!+@L,kfF\\[]l$!+s3C )4#F\\[]lFbc\\l7%Fcgel7$$!+2g:[fF\\[]l$!+%4wf6#F\\[]lFbc\\l7%Figel7$$! +ZH'>$fF\\[]l$!+rp^L@F\\[]lFbc\\l7%F_hel7$$!+LEV:fF\\[]l$!+9V'3:#F\\[] lFbc\\l7%Fehel7$$!+([j&)*eF\\[]l$!+(**=!o@F\\[]lFbc\\l7%F[iel7$$!+hQN \")eF\\[]l$!+n>)\\=#F\\[]lFbc\\l7%Faiel7$$!+Q?!Q'eF\\[]l$!+eUv,AF\\[]l Fbc\\l7%Fgiel7$$!+Fi!f%eF\\[]l$!+**pL=AF\\[]lFbc\\l7%F]jel7$$!+oXmFeF \\[]l$!+E9tMAF\\[]lFbc\\l7%Fcjel7$$!+G^24eF\\[]l$!+%*)Q4D#F\\[]lFbc\\l 7%Fijel7$$!+1f8!z&F\\[]l$!+!*3'pE#F\\[]lFbc\\l7%F_[fl7$$!+M[%3x&F\\[]l $!+Z!*z#G#F\\[]lFbc\\l7%Fe[fl7$$!+z(*>^dF\\[]l$!+`^X)H#F\\[]lFbc\\l7%F [\\fl7$$!+[&)>JdF\\[]l$!+q6$RJ#F\\[]lFbc\\l7%Fa\\fl7$$!+%*)Q3r&F\\[]l$ !+X#H#HBF\\[]lFbc\\l7%Fg\\fl7$$!+?&=,p&F\\[]l$!+F&F\\[]l$!+wj(fh#F\\[]lFbc\\l7%Fedfl7$$!++[yn^F \\[]l$!+I$o%GEF\\[]lFbc\\l7%F[efl7$$!+_*R'Q^F\\[]l$!+%>:4k#F\\[]lFbc\\ l7%Faefl7$$!+5$4\"4^F\\[]l$!+_pK`EF\\[]lFbc\\l7%Fgefl7$$!+vh>z]F\\[]l$ !+=QrlEF\\[]lFbc\\l7%F]ffl7$$!+kX!*[]F\\[]l$!+$>'3yEF\\[]lFbc\\l7%Fcff l7$$!+]#R#=]F\\[]l$!+7YX!p#F\\[]lFbc\\l7%Fiffl7$$!+%z0s)\\F\\[]l$!+!oH Gq#F\\[]lFbc\\l7%F_gfl7$$!+u0\"e&\\F\\[]l$!+6?A:FF\\[]lFbc\\l7%Fegfl7$ $!+331C\\F\\[]l$!+e@kFFF\\[]lFbc\\l7%F[hfl7$$!+nX'>*[F\\[]l$!+O05SFF\\ []lFbc\\l7%Fahfl7$$!+%yI&f[F\\[]l$!+Ytg_FF\\[]lFbc\\l7%Fghfl7$$!+]#pn# [F\\[]l$!+(\\s^w#F\\[]lFbc\\l7%F]ifl7$$!+11p$z%F\\[]l$!+Bb!yx#F\\[]lFb c\\l7%Fcifl7$$!+=jIgZF\\[]l$!+,b^!z#F\\[]lFbc\\l7%Fiifl7$$!+`'Gms%F\\[ ]l$!+w4J.GF\\[]lFbc\\l7%F_jfl7$$!+R1n#p%F\\[]l$!+!))*>;GF\\[]lFbc\\l7% Fejfl7$$!+:gWeYF\\[]l$!+g%*=HGF\\[]lFbc\\l7%F[[gl7$$!+!=pRi%F\\[]l$!+: iGUGF\\[]lFbc\\l7%Fa[gl7$$!+E^D*e%F\\[]l$!+He\\bGF\\[]lFbc\\l7%Fg[gl7$ $!+w$>Vb%F\\[]l$!+?J#)oGF\\[]lFbc\\l7%F]\\gl7$$!+-z<>XF\\[]l$!+!*>F#)G F\\[]lFbc\\l7%Fc\\gl7$$!+aq%Q[%F\\[]l$!+$RXe*GF\\[]lFbc\\l7%Fi\\gl7$$! +uMM[WF\\[]l$!+(HX&4HF\\[]lFbc\\l7%F_]gl7$$!+>So7WF\\[]l$!+nEPBHF\\[]l Fbc\\l7%Fe]gl7$$!+qc)oP%F\\[]l$!+]uKPHF\\[]lFbc\\l7%F[^gl7$$!+aa'4M%F \\[]l$!+s&39&HF\\[]lFbc\\l7%Fa^gl7$$!+h.%\\I%F\\[]l$!+ORhlHF\\[]lFbc\\ l7%Fg^gl7$$!+hs#)oUF\\[]l$!+O/%*zHF\\[]lFbc\\l7%F]_gl7$$!+FGkKUF\\[]l$ !+tRQ%*HF\\[]lFbc\\l7%Fc_gl7$$!+hMS'>%F\\[]l$!+t%R*3IF\\[]lFbc\\l7%Fi_ gl7$$!+?_7gTF\\[]l$!+C4gBIF\\[]lFbc\\l7%F_`gl7$$!+ZP#Q7%F\\[]l$!+-9OQI F\\[]lFbc\\l7%Fe`gl7$$!+4U^(3%F\\[]l$!+:J@`IF\\[]lFbc\\l7%F[agl7$$!+L7 @^SF\\[]l$!+Wu9oIF\\[]lFbc\\l7%Faagl7$$!+])G\\,%F\\[]l$!+!*\\:$3$F\\[] lFbc\\l7%Fgagl7$$!+Z/oyRF\\[]l$!+EcA)4$F\\[]lFbc\\l7%F]bgl7$$!+9(yC%RF \\[]l$!+`&[L6$F\\[]lFbc\\l7%Fcbgl7$$!+-cL1RF\\[]l$!+`B^GJF\\[]lFbc\\l7 %Fibgl7$$!+!Hi-(QF\\[]l$!+a]qVJF\\[]lFbc\\l7%F_cgl7$$!+Y\"pU$QF\\[]l$! +#>9*eJF\\[]lFbc\\l7%Fecgl7$$!+0dO)z$F\\[]l$!+to7uJF\\[]lFbc\\l7%F[dgl 7$$!+X1ciPF\\[]l$!+V)H$*=$F\\[]lFbc\\l7%Fadgl7$$!+s<'os$F\\[]l$!+`&4X? $F\\[]lFbc\\l7%Fgdgl7$$!+6gF\"p$F\\[]l$!+EAl>KF\\[]lFbc\\l7%F]egl7$$!+ -%4el$F\\[]l$!+IRuMKF\\[]lFbc\\l7%Fcegl7$$!+,rY?OF\\[]l$!+O1x\\KF\\[]l Fbc\\l7%Fiegl7$$!+%R`_e$F\\[]l$!+%H=ZE$F\\[]lFbc\\l7%F_fgl7$$!+1<<]NF \\[]l$!+))GdzKF\\[]lFbc\\l7%Fefgl7$$!+AYA:NF\\[]l$!+,0K%H$F\\[]lFbc\\l 7%F[ggl7$$!+8RT![$F\\[]l$!+pt%*3LF\\[]lFbc\\l7%Faggl7$$!+m0uXMF\\[]l$! +Q*RMK$F\\[]lFbc\\l7%Fgggl7$$!+:[?6MF\\[]l$!+9\\yPLF\\[]lFbc\\l7%F]hgl 7$$!+$=1oP$F\\[]l$!++$p>N$F\\[]lFbc\\l7%Fchgl7$$!+:NaULF\\[]l$!+W/)fO$ F\\[]lFbc\\l7%Fihgl7$$!+H]T3LF\\[]l$!+pg!)zLF\\[]lFbc\\l7%F_igl7$$!+]$ =WF$F\\[]l$!+-VV$R$F\\[]lFbc\\l7%Feigl7$$!+j0bSKF\\[]l$!+(p`oS$F\\[]lF bc\\l7%F[jgl7$$!+_#3o?$F\\[]l$!+cK0?MF\\[]lFbc\\l7%Fajgl7$$!+Yv=tJF\\[ ]l$!+UC-LMF\\[]lFbc\\l7%Fgjgl7$$!+kToRJF\\[]l$!+$=^dW$F\\[]lFbc\\l7%F] [hl7$$!+aMH1JF\\[]l$!+%))H#eMF\\[]lFbc\\l7%Fc[hl7$$!+Q/,tIF\\[]l$!+>% \\/Z$F\\[]lFbc\\l7%Fi[hl7$$!+U)H)RIF\\[]l$!+N6S#[$F\\[]lFbc\\l7%F_\\hl 7$$!+Rhu1IF\\[]l$!+So2%\\$F\\[]lFbc\\l7%Fe\\hl7$$!+xNvtHF\\[]l$!+#zoa] $F\\[]lFbc\\l7%F[]hl7$$!+8i%3%HF\\[]l$!+$ppl^$F\\[]lFbc\\l7%Fa]hl7$$!+ Mz,3HF\\[]l$!+oEPFNF\\[]lFbc\\l7%Fg]hl7$$!+'[i_(GF\\[]l$!+^7(y`$F\\[]l Fbc\\l7%F]^hl7$$!+$\\tD%GF\\[]l$!+q$f![NF\\[]lFbc\\l7%Fc^hl7$$!+tW%*4G F\\[]l$!+B8$zb$F\\[]lFbc\\l7%Fi^hl7$$!+a)otx#F\\[]l$!+g<[nNF\\[]lFbc\\ l7%F__hl7$$!+()*R[u#F\\[]l$!+ncqwNF\\[]lFbc\\l7%Fe_hl7$$!+`6N7FF\\[]l$ !+Q$)f&e$F\\[]lFbc\\l7%F[`hl7$$!+nb*)zEF\\[]l$!+h`:%f$F\\[]lFbc\\l7%Fa `hl7$$!+'Qmuk#F\\[]l$!+'fsBg$F\\[]lFbc\\l7%Fg`hl7$$!+/n0:EF\\[]l$!+\\h C5OF\\[]lFbc\\l7%F]ahl7$$!+c&fEe#F\\[]l$!+iBx#F\\[]l$!+!*e*)zOF\\[]lFbc\\l7%F[fhl7$$!+dnHf@F\\[]l$!+cak#o$F\\ []lFbc\\l7%Fafhl7$$!+$>$GE@F\\[]l$!+u2,&o$F\\[]lFbc\\l7%Fgfhl7$$!+q8;$ 4#F\\[]l$!+I'*)po$F\\[]lFbc\\l7%F]ghl7$$!+?5#*f?F\\[]l$!+)zz&)o$F\\[]l Fbc\\l7%Fcghl7$$!+y8bE?F\\[]l$!+K!z(*o$F\\[]lFbc\\l7%Fighl7$$!+;6/$*>F \\[]l$!+\\]e!p$F\\[]lFbc\\l7%F_hhl7$$!+x#y$f>F\\[]l$!+Fb*4p$F\\[]lFbc \\l7%Fehhl7$$!+--bD>F\\[]l$!+*335p$F\\[]lFbc\\l7%F[ihl7$$!+XMa\"*=F\\[ ]l$!+(H?1p$F\\[]lFbc\\l7%Faihl7$$!+xOMd=F\\[]l$!+S'H)*o$F\\[]lFbc\\l7% Fgihl7$$!+\"eNH#=F\\[]l$!+CNj)o$F\\[]lFbc\\l7%F]jhl7$$!+JFI)y\"F\\[]l$ !+o#Hqo$F\\[]lFbc\\l7%Fcjhl7$$!+auU`OF\\[]lFbc\\l7%Fg_il7$$!+Iz%[A\"F\\[]l$!+!=)z6OF\\[]lFbc\\l7%F ]`il7$$!+T*RD=\"F\\[]l$!+M$*>.OF\\[]lFbc\\l7%Fc`il7$$!+*pd#R6F\\[]l$!+ D\"HTf$F\\[]lFbc\\l7%Fi`il7$$!+;A)[4\"F\\[]l$!+2Ue%e$F\\[]lFbc\\l7%F_a il7$$!+z:F\\5F\\[]l$!+[AcuNF\\[]lFbc\\l7%Feail7$$!+]UD-5F\\[]l$!+EE1kN F\\[]lFbc\\l7%F[bil7$$!+mg>O&*F[d\\l$!+wv3`NF\\[]lFbc\\l7%Fabil7$$!+`@ /J!*F[d\\l$!+NTkTNF\\[]lFbc\\l7%Fgbil7$$!+$*=r.&)F[d\\l$!+nvuHNF\\[]lF bc\\l7%F]cil7$$!+1)y(\\zF[d\\l$!+'pFu^$F\\[]lFbc\\l7%Fccil7$$!+>qAjtF[ d\\l$!+:8u/NF\\[]lFbc\\l7%Ficil7$$!+*p>bt'F[d\\l$!+@&)z\"\\$F\\[]lFbc \\l7%F_dil7$$!+<;$Q0'F[d\\l$!+lY#)yMF\\[]lFbc\\l7%Fedil7$$!+ybK(H&F[d \\l$!+J*GjY$F\\[]lFbc\\l7%F[eil7$$!+#4T\"GWF[d\\l$!+uNnbMF\\[]lFbc\\l7 %Faeil7$F`]^l$!+7!3=X$F\\[]lFbc\\l7%Fgeil7$Fj\\^l$!+\"GtKX$F\\[]lFbc\\ l7%F[fil7$Fd\\^l$!+S!GjX$F\\[]lFbc\\l7%F_fil7$F^\\^l$!+6*[$F\\[]lFbc\\l7%F_hil7$F`j]l$!+&[+E;$F\\[]lFbc\\l7%Fchil7$Fj i]l$!+=M0MJF\\[]lFbc\\l7%Fghil7$Fdi]l$!+.0;1JF\\[]lFbc\\l7%F[iil7$F^i] l$!+%Hl%zIF\\[]lFbc\\l7%F_iil7$Fhh]l$!+M*fS0$F\\[]lFbc\\l7%Fciil7$Fbh] l$!+Pu')HIF\\[]lFbc\\l7%Fgiil7$F\\h]l$!+!ojn+$F\\[]lFbc\\l7%F[jil7$Ffg ]l$!+gBi%)HF\\[]lFbc\\l7%F_jil7$F`g]l$!+(RHL'HF\\[]lFbc\\l7%Fcjil7$$!+ K`Y5AFac\\l$!+aDgmGF\\[]lFbc\\l7%Fgjil7$$!+d]QiCFac\\l$!+gh3\"y#F\\[]l Fbc\\l7%F][jl7$$!+6_E>DFac\\l$!+x,#Gq#F\\[]lFbc\\l7%Fc[jl7$$!+zRjtCFac \\l$!+\\\"['HEF\\[]lFbc\\l7%Fi[jl7$$!+nqvqBFac\\l$!+&*yDgDF\\[]lFbc\\l 7%F_\\jl7$$!+jfgNAFac\\l$!+.Zy$\\#F\\[]lFbc\\l7%Fe\\jl7$$!+`D5$3#Fac\\ l$!+AXiHCF\\[]lFbc\\l7%F[]jl7$$!+HImA>Fac\\l$!+ljLnBF\\[]lFbc\\l7%Fa]j l7$$!+F\\[]lFbc\\l7%F[`jl7$$!+.pDczFi\\]l$!+=5(z!>F\\[]lFbc\\l 7%Fa`jl7$$!+sDwOpFi\\]l$!+@!yL&=F\\[]lFbc\\l7%Fg`jl7$$!+EO$3,'Fi\\]l$! +.09*z\"F\\[]lFbc\\l7%F]ajl7$$!+X1[v^Fi\\]l$!+:w?X\"Fi\\]l$!+9b[?8F\\[]lFbc \\l7%Fcdjl7$$!+Wlf`&*Fgi\\l$!+bq(yE\"F\\[]lFbc\\l7%Fidjl7$$!+![>Db(Fgi \\l$!+eCK:7F\\[]lFbc\\l7%F_ejl7$$!++'>*)*eFgi\\l$!+n;\"G;\"F\\[]lFbc\\ l7%Feejl7$$!+s5XZXFgi\\l$!+$=O.6\"F\\[]lFbc\\l7%F[fjl7$$!+2R*fX$Fgi\\l $!+q)))y0\"F\\[]lFbc\\l7%Fafjl7$$!+_E&ee#Fgi\\l$!+)yja+\"F\\[]lFbc\\l7 %Fgfjl7$$!+Sm(=!>Fgi\\l$!+].cI&*F[d\\lFbc\\l7%F]gjl7$$!+^p]s8Fgi\\l$!+ !3;m+*F[d\\lFbc\\l7%Fcgjl7$$!+-%evp*Fhg\\l$!+OFx#[)F[d\\lFbc\\l7%Figjl 7$$!+&Q]7p'Fhg\\l$!+SX+fzF[d\\lFbc\\l7%F_hjl7$$!+\\&*p%\\%Fhg\\l$!+R7H NuF[d\\lFbc\\l7%Fehjl7$$!+vO#F[d\\lFbc\\l7%Fa\\[m7$$!+07Q&G'F7$!+)3;bn\" F[d\\lFbc\\l7%Fg\\[m7$$!+`:x[mF0$!+2t\">:\"F[d\\lFbc\\l7%F]][m7$$!+FU8 `Z5Fac\\lFbc\\l7%Fi][m7$F($ \"+0-z)=%Fac\\lFbc\\l7%Fej]l7$Fjgil$\"+Aa7:KF\\[]l7$$\"#O!\"#$\"$u$Fg^ [m7%Fa^[m7$$\"+G[;-5F[d\\l$\"+8?=NKF\\[]lFd^[m7%F[_[m7$$\"+n(31D\"F[d \\l$\"+qak]KF\\[]lFd^[m7%Fa_[m7$$\"+$>Hp[\"F[d\\l$\"+egiiKF\\[]lFd^[m7 %Fg_[m7$$\"+t\">rq\"F[d\\l$\"+Ez@sKF\\[]lFd^[m7%F]`[m7$$\"+\")4O6>F[d \\l$\"+-x>!G$F\\[]lFd^[m7%Fc`[m7$$\"+ni=,@F[d\\l$\"+'HqqG$F\\[]lFd^[m7 %Fi`[m7$$\"+e6NyAF[d\\l$\"+/9;$H$F\\[]lFd^[m7%F_a[m7$$\"+_4ZWCF[d\\l$ \"+E_o)H$F\\[]lFd^[m7%Fea[m7$$\"+S&=3:$F[d\\l$\"+I\"*f@LF\\[]lFd^[m7%F [b[m7$$\"+DEy;PF[d\\l$\"+Ew\"4M$F\\[]lFd^[m7%Fab[m7$$\"+*emC>%F[d\\l$ \"+Eg\")eLF\\[]lFd^[m7%Fgb[m7$$\"++t!Rg%F[d\\l$\"+q2.wLF\\[]lFd^[m7%F] c[m7$$\"+sM]m\\F[d\\l$\"+i-(GR$F\\[]lFd^[m7%Fcc[m7$$\"+8*>-H&F[d\\l$\" +K1[4MF\\[]lFd^[m7%Fic[m7$$\"+4[$>e&F[d\\l$\"+>j$fU$F\\[]lFd^[m7%F_d[m 7$$\"+!GIm%eF[d\\l$\"+1mFUMF\\[]lFd^[m7%Fed[m7$$\"+?A0)3'F[d\\l$\"+5A_ eMF\\[]lFd^[m7%F[e[m7$$\"+b>54jF[d\\l$\"+&\\$ouMF\\[]lFd^[m7%Fae[m7$$ \"+@<37lF[d\\l$\"+?[w!\\$F\\[]lFd^[m7%Fge[m7$$\"+ra&))p'F[d\\l$\"+kpw1 NF\\[]lFd^[m7%F]f[m7$$\"+V\"e4(oF[d\\l$\"+E&)oANF\\[]lFd^[m7%Fcf[m7$$ \"+CAnHqF[d\\l$\"+yn_QNF\\[]lFd^[m7%Fif[m7$$\"+DG3wrF[d\\l$\"+!=yUb$F \\[]lFd^[m7%F_g[m7$$\"+6#=6J(F[d\\l$\"+0(Q*pNF\\[]lFd^[m7%Feg[m7$$\"+5 -eNuF[d\\l$\"+^S]&e$F\\[]lFd^[m7%F[h[m7$$\"+Uu;]vF[d\\l$\"+n(p4g$F\\[] lFd^[m7%Fah[m7$$\"+BL\\bwF[d\\l$\"+Z8L;OF\\[]lFd^[m7%Fgh[m7$$\"+E-5_xF [d\\l$\"+yUeJOF\\[]lFd^[m7%F]i[m7$$\"+_2ZSyF[d\\l$\"+ySsYOF\\[]lFd^[m7 %Fci[m7$$\"+\"zO5#zF[d\\l$\"+;juhOF\\[]lFd^[m7%Fii[m7$$\"+0p=%*zF[d\\l $\"+DmkwOF\\[]lFd^[m7%F_j[m7$$\"+mBFg!)F[d\\l$\"+02U\"p$F\\[]lFd^[m7%F ej[m7$$\"+s@h>\")F[d\\l$\"+HV11PF\\[]lFd^[m7%F[[\\m7$$\"+8s\\s\")F[d\\ l$\"+QLd?PF\\[]lFd^[m7%Fa[\\m7$$\"+xP>>#)F[d\\l$\"+VO%\\t$F\\[]lFd^[m7 %Fg[\\m7$$\"+RF\\[]lFd^[m7%F[a\\m7$$\"+QXF3$)F[d\\l$\"+P]AJ RF\\[]lFd^[m7%Faa\\m7$$\"+M['*z#)F[d\\l$\"+;m&G%RF\\[]lFd^[m7%Fga\\m7$ $\"+(\\&*zC)F[d\\l$\"+ZcGaRF\\[]lFd^[m7%F]b\\m7$$\"+9DX7#)F[d\\l$\"+I% 3b'RF\\[]lFd^[m7%Fcb\\m7$$\"+V\"=M<)F[d\\l$\"+R7_wRF\\[]lFd^[m7%Fib\\m 7$$\"+#>r48)F[d\\l$\"+A.K()RF\\[]lFd^[m7%F_c\\m7$$\"+ms=&3)F[d\\l$\"+# *=!z*RF\\[]lFd^[m7%Fec\\m7$$\"++!Rh.)F[d\\l$\"+D@E3SF\\[]lFd^[m7%F[d\\ m7$$\"+hi*Q)zF[d\\l$\"+brR=SF\\[]lFd^[m7%Fad\\m7$$\"+Xj_GzF[d\\l$\"+kI IGSF\\[]lFd^[m7%Fgd\\m7$$\"+VT4qyF[d\\l$\"+&)e(z.%F\\[]lFd^[m7%F]e\\m7 $$\"+2Bm3yF[d\\l$\"+*e6u/%F\\[]lFd^[m7%Fce\\m7$$\"+$R\"HWxF[d\\l$\"+$3 1m0%F\\[]lFd^[m7%Fie\\m7$$\"+#**Rqn(F[d\\l$\"+._blSF\\[]lFd^[m7%F_f\\m 7$$\"+g['pg(F[d\\l$\"+2ZDuSF\\[]lFd^[m7%Fef\\m7$$\"+E57MvF[d\\l$\"+s-q #3%F\\[]lFd^[m7%F[g\\m7$$\"+)*=ceuF[d\\l$\"+zu)34%F\\[]lFd^[m7%Fag\\m7 $$\"+e$R.Q(F[d\\l$\"+8=\"))4%F\\[]lFd^[m7%Fgg\\m7$$\"+cQ]*H(F[d\\l$\"+ a'ok5%F\\[]lFd^[m7%F]h\\m7$$\"+#[/h@(F[d\\l$\"+jK&Q6%F\\[]lFd^[m7%Fch \\m7$$\"+^!*=IrF[d\\l$\"+z2'47%F\\[]lFd^[m7%Fih\\m7$$\"+nT!=/(F[d\\l$ \"+2iyFTF\\[]lFd^[m7%F_i\\m7$$\"+)G&*4&pF[d\\l$\"+2WKMTF\\[]lFd^[m7%Fe i\\m7$$\"+'y1y&oF[d\\l$\"+$3q09%F\\[]lFd^[m7%F[j\\m7$$\"+'*>GinF[d\\l$ \"+rx^YTF\\[]lFd^[m7%Faj\\m7$$\"+tKYkmF[d\\l$\"+E=;_TF\\[]lFd^[m7%Fgj \\m7$$\"+I?RklF[d\\l$\"+2k\\dTF\\[]lFd^[m7%F][]m7$$\"+!y3@Y'F[d\\l$\"+ ia^iTF\\[]lFd^[m7%Fc[]m7$$\"+uJldjF[d\\l$\"+7F@nTF\\[]lFd^[m7%Fi[]m7$$ \"+LS1^iF[d\\l$\"+J;erTF\\[]lFd^[m7%F_\\]m7$$\"+r$zB9'F[d\\l$\"+DahvTF \\[]lFd^[m7%Fe\\]m7$$\"+GkjJgF[d\\l$\"+;qIzTF\\[]lFd^[m7%F[]]m7$$\"+#o r)=fF[d\\l$\"+2!\\E=%F\\[]lFd^[m7%Fa]]m7$$\"+n37/eF[d\\l$\"+nOj&=%F\\[ ]lFd^[m7%Fg]]m7$$\"+&)*=uo&F[d\\l$\"+$*GD)=%F\\[]lFd^[m7%F]^]m7$$\"+5. !)obF[d\\l$\"+#=)\\!>%F\\[]lFd^[m7%Fc^]m7$$\"+\"R)H[aF[d\\l$\"+\"fgB>% F\\[]lFd^[m7%Fi^]m7$$\"+Xg%fK&F[d\\l$\"+.2$Q>%F\\[]lFd^[m7%F__]m7$$\"+ [`x,_F[d\\l$\"+w&)*[>%F\\[]lFd^[m7%Fe_]m7$$\"+8w\"e2&F[d\\l$\"+'p`b>%F \\[]lFd^[m7%F[`]m7$$\"+nL5[\\F[d\\l$\"+?\\y&>%F\\[]lFd^[m7%Fa`]m7$$\"+ ;Bm=[F[d\\l$\"+:/e&>%F\\[]lFd^[m7%Fg`]m7$$\"+$HBvo%F[d\\l$\"+$eF\\>%F \\[]lFd^[m7%F]a]m7$$\"+0UraXF[d\\l$\"+$)H\"Q>%F\\[]lFd^[m7%Fca]m7$$\"+ c>E?WF[d\\l$\"+RAA#>%F\\[]lFd^[m7%Fia]m7$$\"+bB>%G%F[d\\l$\"+P*R,>%F\\ []lFd^[m7%F_b]m7$$\"+1+`YTF[d\\l$\"+-&\\v=%F\\[]lFd^[m7%Feb]m7$$\"+m\" )H2SF[d\\l$\"+jIV%=%F\\[]lFd^[m7%F[c]m7$$\"+y&=l'QF[d\\l$\"+#Hr2=%F\\[ ]lFd^[m7%Fac]m7$$\"+j7@CPF[d\\l$\"+BKawTF\\[]lFd^[m7%Fgc]m7$$\"+gUR!e$ F[d\\l$\"+OgsrTF\\[]lFd^[m7%F]d]m7$$\"+>L3NMF[d\\l$\"+*z%HmTF\\[]lFd^[ m7%Fcd]m7$$\"+.:H)G$F[d\\l$\"+\"=A-;%F\\[]lFd^[m7%Fid]m7$$\"+5(G+9$F[d \\l$\"+)3yM:%F\\[]lFd^[m7%F_e]m7$$\"+j5I!*HF[d\\l$\"+]#Hg9%F\\[]lFd^[m 7%Fee]m7$$\"+[,6RGF[d\\l$\"+/(Qy8%F\\[]lFd^[m7%F[f]m7$$\"+X?X'o#F[d\\l $\"+&=l)GTF\\[]lFd^[m7%Faf]m7$$\"+&3;B`#F[d\\l$\"+lB1>TF\\[]lFd^[m7%Fg f]m7$$\"+QKowBF[d\\l$\"+AzP3TF\\[]lFd^[m7%F]g]m7$$\"+6S_>AF[d\\l$\"+tB v'4%F\\[]lFd^[m7%Fcg]m7$$\"+Sczg?F[d\\l$\"+0w6%3%F\\[]lFd^[m7%Fig]m7$$ \"+I$Q/!>F[d\\l$\"+))\\RqSF\\[]lFd^[m7%F_h]m7$$\"+!4q$Q(Fac\\l$\"+v7VLRF \\[]lFd^[m7%Fij]m7$$\"+S_%))Q&Fac\\l$\"+;S^0RF\\[]lFd^[m7%F_[^m7$$\"+4 (fK_$Fac\\l$\"+*ewV(QF\\[]lFd^[m7%Fe[^m7$$\"+_j&>e\"Fac\\l$\"+oWFRQF\\ []lFd^[m7%F[\\^m7$$!+RTP1ZFi\\]l$\"+]j5*z$F\\[]lFd^[m7%Fa\\^m7$$!+!)e2 .FFac\\l$\"+AZ7_PF\\[]lFd^[m7%Fg\\^m7$$!+#z\"Qq_Fac\\l$\"+YGF&p$F\\[]l Fd^[m7%F]]^m7$$!+@:(G&eFac\\l$\"+&yrAo$F\\[]lFd^[m7%Fc]^m7$$!+&e9,Z'Fa c\\l$\"+BEeoOF\\[]lFd^[m7%Fi]^m7$$!+*HE08(Fac\\l$\"+m67aOF\\[]lFd^[m7% F_^^m7$$!+L`hXyFac\\l$\"+r')yQOF\\[]lFd^[m7%Fe^^m7$$!+A2_J')Fac\\l$\"+ i5ZAOF\\[]lFd^[m7%F[_^m7$$!+z;[6&*Fac\\l$\"+V?/0OF\\[]lFd^[m7%Fa_^m7$$ !+w3)>0\"F[d\\l$\"+'\\xje$F\\[]lFd^[m7%Fg_^m7$$!+(R'zq6F[d\\l$\"+T/SmN F\\[]lFd^[m7%F]`^m7$$!+pB?:8F[d\\l$\"+FI?XNF\\[]lFd^[m7%Fc`^m7$Fdgil$ \"+;)GL_$F\\[]lFd^[m7%Fi`^mF[[^lFd^[m7%Fej]lF[[^lFd^[m7%Fcgil7$Fd`^m$! +FI?XNF\\[]l7$Fe^[m$!$u$Fg^[m7%F_a^m7$F^`^m$!+T/SmNF\\[]lFba^m7%Ffa^m7 $Fh_^m$!+'\\xje$F\\[]lFba^m7%Fja^m7$Fb_^m$!+V?/0OF\\[]lFba^m7%F^b^m7$F \\_^m$!+i5ZAOF\\[]lFba^m7%Fbb^m7$Ff^^m$!+r')yQOF\\[]lFba^m7%Ffb^m7$F`^ ^m$!+m67aOF\\[]lFba^m7%Fjb^m7$Fj]^m$!+BEeoOF\\[]lFba^m7%F^c^m7$Fd]^m$! +&yrAo$F\\[]lFba^m7%Fbc^m7$F^]^m$!+YGF&p$F\\[]lFba^m7%Ffc^m7$$!+p)\\jr %Fac\\l$!+Ztl2PF\\[]lFba^m7%Fjc^m7$$!+2)of=%Fac\\l$!+.i[>PF\\[]lFba^m7 %F`d^m7$$!+?4]vOFac\\l$!+;:\"3t$F\\[]lFba^m7%Ffd^m7$$!+vD*>=$Fac\\l$!+ '4y;u$F\\[]lFba^m7%F\\e^m7$$!+tK(e**)Fi\\]l$!+oxK!z$F\\[]lFba^m7%Fbe^m 7$$\"+/W)==\"Fac\\l$!+E,pJQF\\[]lFba^m7%Fhe^m7$$\"+U.\">9$Fac\\l$!+6Aq nQF\\[]lFba^m7%F^f^m7$$\"+/'f1-&Fac\\l$!+RUc**QF\\[]lFba^m7%Fdf^m7$$\" +3_'3%oFac\\l$!+Ww2GRF\\[]lFba^m7%Fjf^m7$$\"+Z'Qkh)Fac\\l$!+Rz!Q&RF\\[ ]lFba^m7%F`g^m7$$\"+M(Qc.\"F[d\\l$!+X8u0%F\\[]lFba^m7%Fj__m 7$$\"+mV&z_%F[d\\l$!+.M`$>%F\\[]lFba^m7%F``_m7$$\"+0U4hYF[d\\l$!+]?u%> %F\\[]lFba^m7%Ff`_m7$$\"+U%pDz%F[d\\l$!+Kh[&>%F\\[]lFba^m7%F\\a_m7$$\" +mBNA\\F[d\\l$!+p#zd>%F\\[]lFba^m7%Fba_m7$$\"+AVT]]F[d\\l$!+@Uj&>%F\\[ ]lFba^m7%Fha_m7$$\"+vdsw^F[d\\l$!+oH1&>%F\\[]lFba^m7%F^b_m7$$\"+hjD,`F [d\\l$!+&ywS>%F\\[]lFba^m7%Fdb_m7$$\"+>\\(RU&F[d\\l$!+3jo#>%F\\[]lFba^ m7%Fjb_m7$$\"+D&\\[a&F[d\\l$!+)e,4>%F\\[]lFba^m7%F`c_m7$$\"+4v%Qm&F[d \\l$!+X@t)=%F\\[]lFba^m7%Ffc_m7$$\"+pa$4y&F[d\\l$!+8q='=%F\\[]lFba^m7% F\\d_m7$$\"+u#zg*eF[d\\l$!+!yuK=%F\\[]lFba^m7%Fbd_m7$$\"+pSC4gF[d\\l$! +CO+!=%F\\[]lFba^m7%Fhd_m7$$\"+nUR?hF[d\\l$!+_8QwTF\\[]lFba^m7%F^e_m7$ $\"+RN\\HiF[d\\l$!+AaTsTF\\[]lFba^m7%Fde_m7$$\"+,[]OjF[d\\l$!+wH6oTF\\ []lFba^m7%Fje_m7$$\"+$>!RTkF[d\\l$!+i3[jTF\\[]lFba^m7%F`f_m7$$\"+g56Wl F[d\\l$!+dc_eTF\\[]lFba^m7%Fff_m7$$\"+(F[d\\l$!+\"4(H:TF\\[]lFba^m7%F`i_m7$$ \"+Uv-$G(F[d\\l$!+3v'z5%F\\[]lFba^m7%Ffi_m7$$\"+0(zVO(F[d\\l$!+JZO+TF \\[]lFba^m7%F\\j_m7$$\"+G)GJW(F[d\\l$!+AN\\#4%F\\[]lFba^m7%Fbj_m7$$\"+ UZA>vF[d\\l$!+D&eV3%F\\[]lFba^m7%Fhj_m7$$\"+eeh#f(F[d\\l$!+wU'f2%F\\[] lFba^m7%F^[`m7$$\"+z!\\Km(F[d\\l$!+5_JnSF\\[]lFba^m7%Fd[`m7$$\"+5(p5t( F[d\\l$!+rcTeSF\\[]lFba^m7%Fj[`m7$$\"+[8-'z(F[d\\l$!+;*p#\\SF\\[]lFba^ m7%F`\\`m7$$\"+!yX!eyF[d\\l$!+D@))RSF\\[]lFba^m7%Ff\\`m7$$\"+^G3l\")F[d\\l$!+v#)pyRF\\[]lFba^m7%Fj^`m7$$\"+\"QA\\?)F[d\\l$!+ 4zsnRF\\[]lFba^m7%F`_`m7$$\"+3,(\\$)F[d\\l$!+=&*y4RF\\[]lFba^m7%F^a`m7$$\"+e#*Gn$)F[d\\l$!+\\.h(*QF \\[]lFba^m7%Fda`m7$$\"+UF$3Q)F[d\\l$!+iDC&)QF\\[]lFba^m7%Fja`m7$$\"+?8 C!R)F[d\\l$!+3)*osQF\\[]lFba^m7%F`b`m7$$\"+F-S&R)F[d\\l$!+Td&*fQF\\[]l Fba^m7%Ffb`m7$$\"+4\")='R)F[d\\l$!+AS/ZQF\\[]lFba^m7%F\\c`m7$$\"+*QwCR )F[d\\l$!+G$eR$QF\\[]lFba^m7%Fbc`m7$$\"+h>Q)F[d\\l$!+\\Bq?QF\\[]lFba ^m7%Fhc`m7$$\"+$Q**4P)F[d\\l$!++)zs!QF\\[]lFba^m7%F^d`m7$$\"+nR$HN)F[d \\l$!+BWp$z$F\\[]lFba^m7%Fdd`m7$$\"+XnwH$)F[d\\l$!+\"**\\*zPF\\[]lFba^ m7%Fjd`m7$$\"+\"H?8I)F[d\\l$!+;.0mPF\\[]lFba^m7%F`e`m7$$\"+kRSn#)F[d\\ l$!+_#**>v$F\\[]lFba^m7%Ffe`m7$$\"+`B\"yA)F[d\\l$!+/2!yt$F\\[]lFba^m7% F\\f`m7$$\"+![BB=)F[d\\l$!+D'eMs$F\\[]lFba^m7%Fbf`m7$$\"+1npI\")F[d\\l $!+Kq(*3PF\\[]lFba^m7%Fhf`m7$$\"+\"HqE2)F[d\\l$!+,+O%p$F\\[]lFba^m7%F^ g`m7$$\"+D&ez+)F[d\\l$!+w;hzOF\\[]lFba^m7%Fdg`m7$$\"+X$[i$zF[d\\l$!+ri tkOF\\[]lFba^m7%Fjg`m7$$\"+-`>dyF[d\\l$!+r!Q(\\OF\\[]lFba^m7%F`h`m7$$ \"+U(=/x(F[d\\l$!+J9iMOF\\[]lFba^m7%Ffh`m7$$\"+\")f\\vwF[d\\l$!+p2R>OF \\[]lFba^m7%F\\i`m7$$\"+E_&>d(F[d\\l$!+e00/OF\\[]lFba^m7%Fbi`m7$$\"+\" )oEfuF[d\\l$!+3`g)e$F\\[]lFba^m7%Fhi`m7$$\"+nF$oL(F[d\\l$!+I&fId$F\\[] lFba^m7%F^j`m7$$\"+0E(R?(F[d\\l$!+!p!G$F\\[]lFba^m7%F`aam7$F^`[m$!+Ez@sKF\\[]lFba^m7%Fdaam7$Fh_[m$!+egii KF\\[]lFba^m7%Fhaam7$Fb_[m$!+qak]KF\\[]lFba^m7%F\\bam7$F\\_[m$!+8?=NKF \\[]lFba^m7%F`bamFigilFba^m7%FcgilFigilFba^m-F\\b\\l6&F^b\\l$\"\"\"F)$ \"#bFg^[m$\"\"#Fab\\l-Feb\\l6#%,PATCHNOGRIDG-F$6%7$7$$!3Q++++++!>(FbrF (7$$\"3!**************Q\"FbrF(-%'COLOURG6&F^b\\lF)F)F)-%*LINESTYLEG6#F cb\\l-F$6%7$7$F($!3/++++++!z%Fbr7$F($\"3/++++++!z%FbrFicamF\\dam-%(SCA LINGG6#%,CONSTRAINEDG-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG- %+AXESLABELSG6%%&Re(z)G%&Im(z)G-F]eam6#%(DEFAULTG-%%VIEWG6$;$!$>(Fg^[m $\"$R\"Fg^[m;$!$z%Fg^[m$\"$z%Fg^[m" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stab ility interval) is the intersection of the stability region with the r eal line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stabilit y interval is (approximately) " }{XPPEDIT 18 0 "[-6.34804, 0];" "6#7$ ,$-%&FloatG6$\"'/[j!\"&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The stability radius is h alf the length of the real stability interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(R(z)=-1,z=-6)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+O1-uJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+13/11777*z^6+1/15296*z^7:\npts := []:\nz0 := 0: \nfor ct from 0 to 420 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := p lot(pts,color=red):\nrd := 3.174020636: \np2 := plot([-rd+rd*cos(t),rd *sin(t),t=0..2*Pi],color=blue):\nplots[display]([p1,p2],scaling=constr ained,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7ael7$$\"\"!F)F(7$$!3)******z`Z5!#=7$$!31+++1&4JR#!#D$\"37+++H^R%4#F07$$!34+++\\P!yq#!#C$\"38++ +7JfTJF07$$!3!******4our]\"!#B$\"3A+++KOz)=%F07$$!3%)******oo]zcF@$\"3 N+++iS+O_F07$$!3;+++]oQq;!#A$\"3[+++wNC$G'F07$$!3u*****pQ4i8%FK$\"30++ +;PbItF07$$!3_*****zYR=-*FK$\"3y*****f!Q,y$)F07$$!33+++$)3e%y\"!#@$\"3 u*****H(>wD%*F07$$!3%)*****fvOaE$Fen$\"35+++YASZ5!#<7$$!3v*****p<^eg&F en$\"3#******H*RJ_6F]o7$$!3)******\\s/L7*Fen$\"37+++s@Od7F]o7$$!35+++e tp=9!#?$\"3'******H%*=EO\"F]o7$$!35+++(zy27#F[p$\"3/+++CU=o9F]o7$$!3A+ ++u#eA1$F[p$\"37+++QO>u:F]o7$$!3S+++X:9(G%F[p$\"35+++x(G3o\"F]o7$$!3.+ ++$RHm$eF[p$\"3#******R0J$)y\"F]o7$$!3#)*****p5lZu(F[p$\"3))*****pF@q* =F]o7$$!33+++N&3K+\"!#>$\"3\")*****4uGt+#F]o7$$!31+++7Fcp7Fjq$\"3!**** **>#o$)>@F]o7$$!33+++#Gi\"p:Fjq$\"3&******Rve`B#F]o7$$!3/+++Nl?!*=Fjq$ \"3=+++%es]N#F]o7$$!33+++&\\Jh?#Fjq$\"31+++[Oy![#F]o7$$!3=+++;!QmX#Fjq $\"3>+++0,]:EF]o7$$!3=+++sTL&[#Fjq$\"3++++#=R\\w#F]o7$$!3'******H@$GD< Fjq$\"3=+++TgyUHF]o7$$\"3J+++_!)*oI&Fjq$\"3)******4'fP!>$F]o7$$\"34+++ zJ$4g#F0$\"31+++?ly.LF]o7$$\"3++++?q3=QF0$\"33+++brdWLF]o7$$\"3!****** H)o&*zYF0$\"3'******\\]A%zLF]o7$$\"3%)*****faT4N&F0$\"35+++ADy7MF]o7$$ \"3o*****\\Utm*eF0$\"3#)*****f&G`XMF]o7$$\"3!******4Oy5N'F0$\"3#****** 441zZ$F]o7$$\"39+++=aSMnF0$\"30+++&zd*4NF]o7$$\"3_*****p(o!*fqF0$\"3:+ ++IToTNF]o7$$\"3)******pwKoL(F0$\"35+++I&fId$F]o7$$\"3/+++E_&>d(F0$\"3 '******zb]Sg$F]o7$$\"3Q+++U(=/x(F0$\"3z*****4V@Yj$F]o7$$\"3()*****\\M[ i$zF0$\"3?+++ritkOF]o7$$\"3K+++\"HqE2)F0$\"3;+++,+O%p$F]o7$$\"3)****** *zMK#=)F0$\"38+++D'eMs$F]o7$$\"3m*****R'RSn#)F0$\"31+++_#**>v$F]o7$$\" 39+++XnwH$)F0$\"31+++\"**\\*zPF]o7$$\"3e*****HQ**4P)F0$\"3))*******zzs !QF]o7$$\"3!)******)QwCR)F0$\"3-+++G$eR$QF]o7$$\"3a+++F-S&R)F0$\"3#)** ***4ub*fQF]o7$$\"3]*****>uK3Q)F0$\"3=+++iDC&)QF]o7$$\"3S+++5'>(\\$)F0$ \"3>+++=&*y4RF]o7$$\"3K+++G'4HI)F0$\"3&*******ztcLRF]o7$$\"3%******z5q 6C)F0$\"3/+++BoacRF]o7$$\"3k*****\\H*>l\")F0$\"3=+++v#)pyRF]o7$$\"3i** ***f[Oc2)F0$\"3?+++p<****RF]o7$$\"3U+++U.2tzF0$\"3m*****>z'R?SF]o7$$\" 3K+++!yX!eyF0$\"35+++D@))RSF]o7$$\"3K+++5(p5t(F0$\"3o*****4n:%eSF]o7$$ \"3E+++eeh#f(F0$\"3m*****fFkf2%F]o7$$\"3k*****z#)GJW(F0$\"3#)*****>_$ \\#4%F]o7$$\"39+++Uv-$G(F0$\"3x*****z]nz5%F]o7$$\"3s******f\"3F6(F0$\" 3)******Rf[B7%F]o7$$\"3++++dlaKpF0$\"3m*****p6(fNTF]o7$$\"3W+++G/!Hu'F 0$\"3;+++d5nZTF]o7$$\"3l******f56WlF0$\"3A+++dc_eTF]o7$$\"3C+++,[]OjF0 $\"3u*****f(H6oTF]o7$$\"3++++nUR?hF0$\"3#******>N\"QwTF]o7$$\"33+++u#z g*eF0$\"3k******zZF$=%F]o7$$\"3c******3v%Qm&F0$\"3M+++X@t)=%F]o7$$\"3t ******=\\(RU&F0$\"3%)*****zI'o#>%F]o7$$\"3_*****\\xDn<&F0$\"3A+++oH1&> %F]o7$$\"3:+++mBNA\\F0$\"3'*******o#zd>%F]o7$$\"3#)*****\\?%4hYF0$\"3K +++]?u%>%F]o7$$\"3!)*****zhwJR%F0$\"3C+++%F]o7$$\"30+++y%3)=TF0$ \"3C+++\"*)op=%F]o7$$\"3)******fsy\"QQF0$\"3%)*****Ror*zTF]o7$$\"3'*** ***z7^9b$F0$\"3=+++](*oqTF]o7$$\"3!)*****zOc(eKF0$\"3O+++'4G*eTF]o7$$ \"3/+++!)*z,'HF0$\"3>+++Ju0M5>9$Fjq$\"35+++6AqnQF]o7$$!3!3++IFte **)F[p$\"3/+++oxK!z$F]o7$$!3')*****4_rG&eFjq$\"3=+++&yrAo$F]o7$$!3(*** ***49A.s\"F0$\"3$)*****Rb.A]$F]o7$$!3')*****\\fD]B%F0$\"3?+++Q80aMF]o7 $$!3`*****Ra%R4fF0$\"3/+++HAEwMF]o7$$!3m*****p;68C(F0$\"39+++6$o@]$F]o 7$$!3-+++n(=_R)F0$\"3@+++QhJFNF]o7$$!3++++$\\lnV*F0$\"3/+++=g$3b$F]o7$ $!32+++an)*R5F]o$\"3%)*****f^+Dd$F]o7$$!30+++PUZI6F]o$\"3')*****4@eAf$ F]o7$$!32++++#fk@\"F]o$\"3A+++Pf65OF]o7$$!3#******z2g))H\"F]o$\"3***** **H,,hi$F]o7$$!3++++\"GY$y8F]o$\"3<+++CtCSOF]o7$$!31+++4NUb9F]o$\"3!)* ****>x*e_OF]o7$$!3$******\\%[[I:F]o$\"3%)*****>-iJm$F]o7$$!31+++&3VQg \"F]o$\"3-+++Nc*>n$F]o7$$!31+++G@vv;F]o$\"33+++A)>\"zOF]o7$$!3-+++b;UY TX$)o$F]o7$$!3' ******pS>Z)=F]o$\"35+++xW\\!p$F]o7$$!31+++*HEE&>F]o$\"3))******\\*H5p$ F]o7$$!3#)*****z%4')>?F]o$\"3?+++m<(**o$F]o7$$!3?+++>J_'3#F]o$\"3%)*** ***H)Qto$F]o7$$!3!)*****pA-F:#F]o$\"3++++I$\\Jo$F]o7$$!3%******4Wz%=AF ]o$\"3'******f1@un$F]o7$$!33+++q$HRG#F]o$\"3/+++-=9m$F]o7$$!3%)*****42@TT#F]o$\"3;+++4M=^OF]o7$$!3-+++*\\ !**yCF]o$\"3&******f8$[ROF]o7$$!36+++:'*yVDF]o$\"31+++&[Sji$F]o7$$!3(* *****f\\w&3EF]o$\"36+++$Hz(oGF]o$\"3')********Q$*RNF]o7$$!3z******GWFMHF]o $\"3/+++MVv=NF]o7$$!3y*****H_S,+$F]o$\"3%)******p\"yj\\$F]o7$$!3;+++)H mj1$F]o$\"3/+++d9'GZ$F]o7$$!3?+++jr*H8$F]o$\"3%******\\An#[MF]o7$$!37+ ++8X2+KF]o$\"3<+++_emAMF]o7$$!31+++%eMwE$F]o$\"3/+++Y_8'R$F]o7$$!3\")* ****4:2dL$F]o$\"3?+++(zg(oLF]o7$$!3$******f99VS$F]o$\"3@+++[]jSLF]o7$$ !3++++O#oMZ$F]o$\"3-+++cq&=J$F]o7$$!3!)*****HXrJa$F]o$\"3)******>TJDG$ F]o7$$!32+++^RT8OF]o$\"3)******4%pw_KF]o7$$!3'*******yH<%o$F]o$\"3))** ***H4vEA$F]o7$$!3'******>97av$F]o$\"3#******R3oB>$F]o7$$!3'******>*43F QF]o$\"3)******R'o&>;$F]o7$$!3?+++7]6**QF]o$\"38+++t*[:8$F]o7$$!3#)*** **R(fVrRF]o$\"30+++&QY75$F]o7$$!3b*****4n_R/%F]o$\"3!*******4M9rIF]o7$ $!3\"*******3?c;TF]o$\"3/+++5ZKTIF]o7$$!3O+++;/:*=%F]o$\"39+++'\\j=,$F ]o7$$!3=+++_bfhUF]o$\"3;+++H*>G)HF]o7$$!3e*****RGoPL%F]o$\"3#********y RU&HF]o7$$!3p*****f\"\\`0WF]o$\"3:+++]M:EHF]o7$$!33+++>'fnZ%F]o$\"3+++ +3_d)*GF]o7$$!3'******4-2ta%F]o$\"39+++fI]rGF]o7$$!3y*****p&[/fYeo%F]o$\"3))*****>_*y=GF]o7$$!3g******[Q f`ZF]o$\"37+++$\\nIz#F]o7$$!3:+++N&y,#[F]o$\"3!******\\P$pnFF]o7$$!3)) *****H^/b)[F]o$\"3A+++,xfUFF]o7$$!3M+++Q()[\\\\F]o$\"3&*******zMq&F]o$\"3?+ ++W)y%=EF]o7$$!3%******z'zJZ_F]o$\"3-+++OkM$f#F]o7$$!37+++6]C-`F]o$\"3 '******RYGzc#F]o7$$!3E+++!4+cN&F]o$\"3&)******>i:UDF]o7$$!38+++RvQ2aF] o$\"33+++Pr'f^#F]o7$$!3++++'y;wX&F]o$\"39+++()\\I*[#F]o7$$!3$*******)3 ,j]&F]o$\"31+++W)>@Y#F]o7$$!3k******QkX`bF]o$\"3!)*****p&yOMCF]o7$$!3) )*****4h+\"*f&F]o$\"3&)*****>*3,1CF]o7$$!3()*****4R_Kk&F]o$\"31+++*3;q P#F]o7$$!3!)*****R&4$fo&F]o$\"3!)*****\\PbtM#F]o7$$!3p*****RObrs&F]o$ \"37+++[\\+'f37fF]o$\"3y*****Rj5V:#F]o7$$!3o *****>IW\\%fF]o$\"30+++#z*\\>@F]o7$$!3!******fSek(fF]o$\"3!)*****zD7R3 #F]o7$$!3V+++i'Rm+'F]o$\"3%)******y?aZ?F]o7$$!3i*****>Y)\\NgF]o$\"3'** ****p\"RQ5?F]o7$$!3;+++'HWI1'F]o$\"3)******HxKC(>F]o7$$!3o*****H(eG*3' F]o$\"3)*******RPoL>F]o7$$!3P+++\\7B9hF]o$\"3!*******o<8%*=F]o7$$!3j** ***>)z)y8'F]o$\"35+++Y9x`=F]o7$$!3O+++*Hj-;'F]o$\"32+++%z'f7=F]o7$$!39 +++)Gk8='F]o$\"31+++s5gq,iF]o$\"3-+++*fwxs\"F]o7 $$!3w*****\\Bs(>iF]o$\"3-+++%e9To\"F]o7$$!3d*****\\r%4PiF]o$\"3-+++r\\ gR;F]o7$$!3/+++yW<`iF]o$\"3-+++,jB%f\"F]o7$$!3O+++p;-oiF]o$\"3/+++'f&* za\"F]o7$$!3Q+++I![;G'F]o$\"3&******>Fo3]\"F]o7$$!3g*****pKnSH'F]o$\"3 1+++$3QGX\"F]o7$$!3s*****>#eH0jF]o$\"3-+++UX(QjF]o$\"3#******zPk%)>\"F]o7$$!3y*** **\\%4RWjF]o$\"3/+++X8fW6F]o7$$!3A+++z%G!\\jF]o$\"31+++4[m*3\"F]o7$$!3 ++++H)3FN'F]o$\"3(*******REmL5F]o7$$!3S+++*H!\\bjF]o$\"3Q+++0$\\cw*F07 $$!3G+++=\"RuN'F]o$\"3%)*****p1`N=*F07$$!3o*****fmH'ejF]o$\"3;+++kYA!f )F07$$!3#********RW\"fjF]o$\"3K+++>\">c)zF07$$!3/+++9M2fjF]o$\"3C+++B$ y(ptF07$$!3'******Rb8&ejF]o$\"3_++++0&Gu'F07$$!3w*****\\.nvN'F]o$\"3q* ****\\@4^5'F07$$!3))*****H`RjN'F]o$\"3A+++90(pX&F07$$!3c*****HyP\\N'F] o$\"3u******zm+*z%F07$$!3#*******QnY`jF]o$\"3>+++]]&>8%F07$$!3'****** \\VE?N'F]o$\"3!******\\e?nX$F07$$!3u*****\\q32N'F]o$\"3!)******>7PuFF0 7$$!3!*******eRf\\jF]o$\"3!******>hDh3#F07$$!3G+++7#[([jF]o$\"39+++_ML $R\"F07$$!3#******Hc?#[jF]o$\"3S+++)H\"\\upFjq7$$!3o*****>FT![jF]oF(7$ F_]o$!3S+++)H\"\\upFjq7$Fj\\o$!39+++_ML$R\"F07$Fe\\o$!3!******>hDh3#F0 7$F`\\o$!3!)******>7PuFF07$F[\\o$!3!******\\e?nX$F07$Ff[o$!3>+++]]&>8% F07$Fa[o$!3u******zm+*z%F07$F\\[o$!3A+++90(pX&F07$Fgjn$!3q*****\\@4^5' F07$Fbjn$!3_++++0&Gu'F07$F]jn$!3C+++B$y(ptF07$Fhin$!3K+++>\">c)zF07$Fc in$!3;+++kYA!f)F07$F^in$!3%)*****p1`N=*F07$Fihn$!3Q+++0$\\cw*F07$Fdhn$ !3(*******REmL5F]o7$F_hn$!31+++4[m*3\"F]o7$Fjgn$!3/+++X8fW6F]o7$Fegn$! 3#******zPk%)>\"F]o7$F`gn$!35+++zrI^7F]o7$F[gn$!34+++*GVJI\"F]o7$Fffn$ !3-+++`e*RN\"F]o7$Fafn$!3-+++Fo3]\"F]o7$Fben$!3/+++'f&*za\"F]o7$F]en$!3-+++,jB%f\"F]o7$Fh dn$!3-+++r\\gR;F]o7$Fcdn$!3-+++%e9To\"F]o7$F^dn$!3-+++*fwxs\"F]o7$Ficn $!31+++s5gqF]o7$F`bn$!3)******HxKC(>F]o7$F[bn $!3'******p\"RQ5?F]o7$Ffan$!3%)******y?aZ?F]o7$Faan$!3!)*****zD7R3#F]o 7$F\\an$!30+++#z*\\>@F]o7$Fg`n$!3y*****Rj5V:#F]o7$Fb`n$!3++++X;N)=#F]o 7$F]`n$!3&)*****z!4j@AF]o7$Fh_n$!3#******43eTD#F]o7$Fc_n$!37+++sZ%fG#F ]o7$F^_n$!37+++[\\+*3,1CF]o7$Fj]n$!3!)*****p&yOMCF]o7$Fe]n$!31+++W)>@Y #F]o7$F`]n$!39+++()\\I*[#F]o7$F[]n$!33+++Pr'f^#F]o7$Ff\\n$!3&)******>i :UDF]o7$Fa\\n$!3'******RYGzc#F]o7$F\\\\n$!3-+++OkM$f#F]o7$Fg[n$!3?+++W )y%=EF]o7$Fb[n$!3,+++I+SVEF]o7$F][n$!3#)*****p6*=oEF]o7$Fhjm$!3'****** RvGHp#F]o7$Fcjm$!3&*******zMq_*y=GF]o7$Fjhm$!31+ ++U*=\\%GF]o7$Fehm$!39+++fI]rGF]o7$F`hm$!3++++3_d)*GF]o7$F[hm$!3:+++]M :EHF]o7$Ffgm$!3#********yRU&HF]o7$Fagm$!3;+++H*>G)HF]o7$F\\gm$!39+++' \\j=,$F]o7$Fgfm$!3/+++5ZKTIF]o7$Fbfm$!3!*******4M9rIF]o7$F]fm$!30+++&Q Y75$F]o7$Fhem$!38+++t*[:8$F]o7$Fcem$!3)******R'o&>;$F]o7$F^em$!3#***** *R3oB>$F]o7$Fidm$!3))*****H4vEA$F]o7$Fddm$!3)******4%pw_KF]o7$F_dm$!3) ******>TJDG$F]o7$Fjcm$!3-+++cq&=J$F]o7$Fecm$!3@+++[]jSLF]o7$F`cm$!3?++ +(zg(oLF]o7$F[cm$!3/+++Y_8'R$F]o7$Ffbm$!3<+++_emAMF]o7$Fabm$!3%****** \\An#[MF]o7$F\\bm$!3/+++d9'GZ$F]o7$Fgam$!3%)******p\"yj\\$F]o7$Fbam$!3 /+++MVv=NF]o7$F]am$!3')********Q$*RNF]o7$Fh`m$!3#)*****pHn)fNF]o7$Fc`m $!3#)*****\\%3^yNF]o7$F^`m$!3#)******=h#ef$F]o7$Fi_m$!36+++$Hz9m$F]o7$F`^m$!3/+++-=TX$ )o$F]o7$Fh[m$!3-+++*[hXo$F]o7$Fc[m$!33+++A)>\"zOF]o7$F^[m$!3-+++Nc*>n$ F]o7$Fijl$!3%)*****>-iJm$F]o7$Fdjl$!3!)*****>x*e_OF]o7$F_jl$!3<+++CtCS OF]o7$Fjil$!3*******H,,hi$F]o7$Feil$!3A+++Pf65OF]o7$F`il$!3')*****4@eA f$F]o7$F[il$!3%)*****f^+Dd$F]o7$Ffhl$!3/+++=g$3b$F]o7$Fahl$!3@+++QhJFN F]o7$F\\hl$!39+++6$o@]$F]o7$Fggl$!3/+++HAEwMF]o7$Fbgl$!3?+++Q80aMF]o7$ F]gl$!3$)*****Rb.A]$F]o7$Fhfl$!3=+++&yrAo$F]o7$Fcfl$!3/+++oxK!z$F]o7$F ^fl$!35+++6AqnQF]o7$Fiel$!3.+++Ww2GRF]o7$Fdel$!3#******\\Mrr(RF]o7$F_e l$!34+++7]*z,%F]o7$Fjdl$!3s******H=O_SF]o7$Fedl$!3/+++OGY\"3%F]o7$F`dl $!3%)*****\\9Ih5%F]o7$F[dl$!3F+++ZD(p7%F]o7$Ffcl$!3>+++J%F]o7$Fhal$!3K+++]?u%>%F]o7$Fcal$!3 '*******o#zd>%F]o7$F^al$!3A+++oH1&>%F]o7$Fi`l$!3%)*****zI'o#>%F]o7$Fd` l$!3M+++X@t)=%F]o7$F_`l$!3k******zZF$=%F]o7$Fj_l$!3#******>N\"QwTF]o7$ Fe_l$!3u*****f(H6oTF]o7$F`_l$!3A+++dc_eTF]o7$F[_l$!3;+++d5nZTF]o7$Ff^l $!3m*****p6(fNTF]o7$Fa^l$!3)******Rf[B7%F]o7$F\\^l$!3x*****z]nz5%F]o7$ Fg]l$!3#)*****>_$\\#4%F]o7$Fb]l$!3m*****fFkf2%F]o7$F]]l$!3o*****4n:%eS F]o7$Fh\\l$!35+++D@))RSF]o7$Fc\\l$!3m*****>z'R?SF]o7$F^\\l$!3?+++p<*** *RF]o7$Fi[l$!3=+++v#)pyRF]o7$Fd[l$!3/+++BoacRF]o7$F_[l$!3&*******ztcLR F]o7$Fjz$!3>+++=&*y4RF]o7$Fez$!3=+++iDC&)QF]o7$F`z$!3#)*****4ub*fQF]o7 $F[z$!3-+++G$eR$QF]o7$Ffy$!3))*******zzs!QF]o7$Fay$!31+++\"**\\*zPF]o7 $F\\y$!31+++_#**>v$F]o7$Fgx$!38+++D'eMs$F]o7$Fbx$!3;+++,+O%p$F]o7$F]x$ !3?+++ritkOF]o7$Fhw$!3z*****4V@Yj$F]o7$Fcw$!3'******zb]Sg$F]o7$F^w$!35 +++I&fId$F]o7$Fiv$!3:+++IToTNF]o7$Fdv$!30+++&zd*4NF]o7$F_v$!3#******44 1zZ$F]o7$Fju$!3#)*****f&G`XMF]o7$Feu$!35+++ADy7MF]o7$F`u$!3'******\\]A %zLF]o7$F[u$!33+++brdWLF]o7$Fft$!31+++?ly.LF]o7$Fat$!3)******4'fP!>$F] o7$F\\t$!3=+++TgyUHF]o7$Fgs$!3++++#=R\\w#F]o7$Fbs$!3>+++0,]:EF]o7$F]s$ !31+++[Oy![#F]o7$Fhr$!3=+++%es]N#F]o7$Fcr$!3&******Rve`B#F]o7$F^r$!3!* *****>#o$)>@F]o7$Fhq$!3\")*****4uGt+#F]o7$Fcq$!3))*****pF@q*=F]o7$F^q$ !3#******R0J$)y\"F]o7$Fip$!35+++x(G3o\"F]o7$Fdp$!37+++QO>u:F]o7$F_p$!3 /+++CU=o9F]o7$Fio$!3'******H%*=EO\"F]o7$Fdo$!37+++s@Od7F]o7$F_o$!3#*** ***H*RJ_6F]o7$Fin$!35+++YASZ5F]o7$FY$!3u*****H(>wD%*F07$FT$!3y*****f!Q ,y$)F07$FO$!30+++;PbItF07$FI$!3[+++wNC$G'F07$$!3;+++qo]zcF@$!3N+++iS+O _F07$F>$!3A+++KOz)=%F07$F8$!38+++7JfTJF07$F2$!37+++H^R%4#F07$$!35+++Mv 6aPF-$!31+++^v>Z5F0F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$$! 3cOa`tv2sHFjq$\"3p,'=`7@ML%F07$$!35D`vp)e`.\"F0$\"31h_qM9rS!)F07$$!3?& \\kA--]Q#F0$\"3gmy?l]627F]o7$$!3q%Gn*)p!=vUF0$\"3/'R9)\\F&4f\"F]o7$$!3 sB'4bHZ>l'F0$\"3ztQUKNFW>F]o7$$!3ce\"QFzp*f#*F0$\"3^`bL@MrSAF]o7$$!3zm ,?g\\kK7F]o$\"3'Q-;@**o5^#F]o7$$!3kQ)*z.9f$e\"F]o$\"3_\"o$*[v0ou#F]o7$ $!3m48aH/]h>F]o$\"3+V]5k>HLHF]o7$$!3s'=S0#yXsBF]o$\"3)R(e+809rIF]o7$$! 3Y7wnK92ZFF]o$\"3]4oq*Hu^9$F]o7$$!3_!=oe'e\"f<$F]o$\"3ygC8;P%F]o$\"3>LV>\\\"=%RHF]o7$$!3Cz&p\"4>YuZF]o$\"3&z([ziU)4u#F]o7 $$!31(*e))eq'44&F]o$\"3y$Ga:Yn(HDF]o7$$!3I%=wRF%=CaF]o$\"3w'3hgSk&QAF] o7$$!3/)=e>OJGo&F]o$\"3Tykm?JHW>F]o7$$!3Yr6ckC]AfF]o$\"3a?$et)4`(e\"F] o7$$!3q-O0e)QR5'F]o$\"3jKu)G\"*\\1A\"F]o7$$!3Wt#Q(zYFTiF]o$\"3iVUCy`3j \")F07$$!3A'45$3C`=jF]o$\"3q)el(Rq,=VF07$$!3yms!z+E![jF]o$\"3s^y/],iV) *F[p7$$!3s\"R&>5$z*=jF]o$!3M/p*GPY`G%F07$$!3_vxqy*)eWiF]o$!3)37Rze=v.) F07$$!3VDY3y\"*>9hF]o$!3a%)*pW_Cd>\"F]o7$$!3[%\\yV@Ds#fF]o$!3aACS&HE$z :F]o7$$!3)**p4T%*=hp&F]o$!3F9(4'fa-F>F]o7$$!3JamzP*G4V&F]o$!3)zzRaIkeEKv'[]FF ]o7$$!3;%)*4QE4:P%F]o$!3'>U)*>\\g%RHF]o7$$!3rDXuuk,-SF]o$!3#QIxvv>T1$F ]o7$$!3kd(QdfmRe$F]o$!3a%)ovrcVZJF]o7$$!3#z(p4M[s$=$F]o$!3//\"Q7!e+uJF ]o7$$!3uB9&zh%HlFF]o$!3%)p$=eK%fZJF]o7$$!3'3D!HS_/jBF]o$!3o>CWD%o'oIF] o7$$!3/z.(GJIl&>F]o$!3'\\F+$eDBJHF]o7$$!3)*[bL3F'fe\"F]o$!3u7m&*)Rx\"[ FF]o7$$!3AXp%>@+\\B\"F]o$!3b&Qc:(4\"G^#F]o7$$!3s]!*=K/)o?*F0$!3Mc?R:XP NAF]o7$$!3C'[X-6x[m'F0$!31F]o7$$!37L+#*z+o2UF0$!3E&H%fGDCz: F]o7$$!39uKPnH')fCF0$!3w=!y;*>;D7F]o7$$!3'o6F\\lCg4\"F0$!3_aBqns!*o#)F 07$$!3u0\"3\"*[]K&HFjq$!3BD=j9$Q(>VF07$F($\"3oZ!)[e+,/E!#E-F^eq6&F`eqF (F(Faeq-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"F_er-%%FONTG6#% (DEFAULTG-Fber6$%*HELVETICAG\"\"*-%%VIEWG6$FderFder" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real p art of points along the curve. In this way we see that there is " } {TEXT 260 53 "no largest interval on the nonnegative imaginary axis" } {TEXT -1 65 " that contains the origin and lies inside the stability r egion. " }}{PARA 0 "" 0 "" {TEXT -1 120 "However, the stability regio n intersects the nonnegative imaginary axis in an interval that does n ot contain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24 *z^4+1/120*z^5+13/11777*z^6+1/15296*z^7:\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 285 do\n zz := newton(R(z)=exp(ct*Pi/100*I) ,z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\n end do:\nplot(pts,color=COLOR(RGB,.9,.45,0),thickness=2,font=[HELVETIC A,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 189 270 270 {PLOTDATA 2 "6(-%'CURVESG6#7j\\l7$$\"\"!F)F(7$$!575!\\y@B32@(!#@$\"5HL _-f`EfTJF-7$$!5s=kC6OWN_5!#?$\"5P4Z$QsI&=$G'F-7$$!5;EA)fM8pFJ\"F3$\"5] ugKxhzxC%*F-7$$!5GqBnc%=\"pN:F3$\"5%zX(y=iqjc7F37$$!5;l?.[%\\*GMF3$\"5=V^m*\\gb\\)=F37$$!5#G`hb$)3AJ3#F3 $\"51qG^G&*[6*>#F37$$!5dt'Q(*z[V,C#F3$\"5r(G!)H!>UF8DF37$$!5u@))z2%>V$ )Q#F3$\"5>VABY2OVFGF37$$!5PjjX9C`4HDF3$\"5XM5.x6JfTJF37$$!5cf@iv[6WjEF 3$\"5Tw2mU6GvbMF37$$!5t(H?G54#>#z#F3$\"5K^&*obCG\"*pPF37$$!5W)yjRJ(\\* f\"HF3$\"5>F(**f0KtS3%F37$$!5O6h>Tw^PNIF3$\"5VN$*4>NXB)R%F37$$!5>%eBI \"Giw]JF3$\"5L;%e7$)y'R7ZF37$$!5Y18b#)*pIDE$F3$\"5(fH&\\^/0cE]F37$$!5! *>Jp_\\`(4P$F3$\"5bzB()zOisS`F37$$!5B^wDAp<*yNF3$\"5qY1lqon1pfF37$$!5!\\3dr`\"y$)yOF3$\"5K0Q)RhdVKG'F37$ $!5uiM*oga%HwPF3$\"5k9+r+([Euf'F37$$!5`q;kFl\"R9(QF3$\"557_,lrrh6pF37$ $!5J\"\\$G]&f0W'RF3$\"5CY0&\\`kP%Q]!eU&yF37$$!5y#yK^mol8B%F3$\"5Fro\"z%4U]o \")F37$$!5N*p[Rxk#p;VF3$\"5Ny<[yNFx#[)F37$$!5+ZS%fw_E.S%F3$\"5*pv,v*4# oqz)F37$$!5-$>JF'))yL#[%F3$\"5#y;JBn#fR6\"*F37$$!5r%yoXB!4ziXF3$\"5\" \\+T$Ht>wD%*F37$$!5&4*)*z)f3WZF3 $\"5C-OGn(yja+\"!#>7$$!5PIUs(yXg`z%F3$\"5hGL9Plk\"p.\"Ffu7$$!5K8ghBkw6 q[F3$\"5_A>OozjPo5Ffu7$$!5yF3'eb&RcV\\F3$\"5pQ/$oFnW)*4\"Ffu7$$!5jk'e# *)*[Pd,&F3$\"55DNJ2DEKJ6Ffu7$$!5NBB,'3\\tm3&F3$\"5\"3^'3Sn;\"G;\"Ffu7$ $!5SA7]d>XSc^F3$\"5Up5Cr$R8V>\"Ffu7$$!5o:'4%z72'\\A&F3$\"5,dM*>ZdHeA\" Ffu7$$!5'zh?)\\\"**pBH&F3$\"5u()*3LB[\\QU&F3$\"5w#o'HE9b[?8Ffu7$$!535Q(*)[)f'z[& F3$\"575*oP>G\"3_8Ffu7$$!5jmaR@8%G5b&F3$\"5=&[R(>\\Nq$Q\"Ffu7$$!5&pNdt F\"f08cF3$\"5Z7Gsa'[b`T\"Ffu7$$!5Zh4M_*>mSn&F3$\"5#*3KNbl0/Z9Ffu7$$!5' pPrC\"Hc2MdF3$\"5$\\w,&H#ei(y9Ffu7$$!5N)QOj:D*4$z&F3$\"5\\hy?!GoD0^\"F fu7$$!5yho5/^3:^eF3$\"5(>xtd2RMBa\"Ffu7$$!5]![gL6(HC3fF3$\"5QuT6yPO>u: Ffu7$$!5xM#)G\"\\#pQkfF3$\"5)p(30n'z3hg\"Ffu7$$!5K;ulS*y#f>gF3$\"5W([x [!=d3Q;Ffu7$$!5e:UeS$QpQ2'F3$\"5A0fT$4xI,n\"Ffu7$$!5i*))36sACs7'F3$\"5 Zr?B&y)3D-JiF3$ \"5\"\\'>+()prum7e+UmU'=Ffu7$$ !59r.!QxsL#GkF3$\"5(3o`JsF@q*=Ffu7$$!581@f%=p=_Z'F3$\"5%eOB:%f1#*H>Ffu 7$$!53i8UR?BG@lF3$\"5EpW$4s\\yH'>Ffu7$$!5L*HW#R`STmlF3$\"5a')o:&*y+@'* >Ffu7$$!58tmXnW(*f5mF3$\"5s8U]X#Ffu7$$!5=&HY mnsh&yqF3$\"5(4!R!QEq%y$\\#Ffu7$$!57Pb/o7\"GB5(F3$\"5q`KUZ*G7(F3$\"58b^&e/\\zQd#Ffu7$$!56$\\RrF3$\"5P3L_(\\5+bh#Ff u7$$!58I*z&[z=.^rF3$\"5m&)\\\"*QYDSeEFfu7$$!5\"f:^o`c]e:(F3$\"5+%4WFq< ?Gq#Ffu7$$!5Q*GhQXp;8:(F3$\"5x')3c6;l0\\FFfu7$$!5)***3Eg-5&H8(F3$\"5Et ,(>#fz^(z#Ffu7$$!5L^s\\GeqO#4(F3$\"5'z'GUUj#o([GFfu7$$!5a\\=0)Guc9,(F3 $\"5!f1#[#z+CO!HFfu7$$!5O)>u!RJ@QSoF3$\"5j*GdprRHL'HFfu7$$!5UZa#*p6cSM jF3$\"5WOw!4uVn)HIFfu7$$\"5^*)4.$f*>jZmF3$\"5Ge5&oE]gh5$Ffu7$$\"5[rwF^ eNJdwF3$\"5!e6n,7'fP!>$Ffu7$$\"5W&*p9#o>**yF)F3$\"5G'z=X.ZX1D$Ffu7$$\" 5Ry#p>Ie[Lg)F3$\"5f(Ghs?q(>!G$Ffu7$$\"5V^=Mk9)Gzz)F3$\"5PYW8`D_o)H$Ffu 7$$\"5HieYrZ[LFfu7$$\"5%H%3Q1D2:S#*F3$\"5J*R0rh-;)eLFfu7$$\"5aDc9D?[e*G*F3$ \"5>#[8e9)>?pLFfu7$$\"5NK\"pQz[:IL*F3$\"5?Eh+V0DUzLFfu7$$\"58_reS$3@;P *F3$\"5/&Hj)p'*R_*Q$Ffu7$$\"5K-V$G))zjiS*F3$\"5*=t)ycgm`*R$Ffu7$$\"5mQ lodwIfP%*F3$\"5)R_km@j![4MFfu7$$\"5MFz#[/05hY*F3$\"5oyQWC_&p$>MFfu7$$ \"5\\s$>&fk$4A\\*F3$\"5-Hux=YF@HMFfu7$$\"5\"zuv,dD2i^*F3$\"5K39V***f;! RMFfu7$$\"5.H@^Op4OQ&*F3$\"5)oK)otbay[MFfu7$$\"5/H!>$oFD))e&*F3$\"59!> \"e_4A_eMFfu7$$\"55Bdce3([zd*F3$\"5:.A1x>(G#oMFfu7$$\"5Z+,_m?%3df*F3$ \"5rGQuL\"41zZ$Ffu7$$\"5OiUIhZ#)G7'*F3$\"5cq%eom([b([$Ffu7$$\"5Bwl.#4o 'zF'*F3$\"5ZHirA?_<(\\$Ffu7$$\"5c#=AkfRFBk*F3$\"50c+='fl*F3$\"5K=I([EsHj^$Ffu7$$\"5Ui?&QR;r(o'*F3$\"5[j(p-_%H'e_$Ffu7$ $\"5#Qi7+!)R(*F3$\"5<#o;.Va-owp4g$Ffu7$$\"5-SxW`(\\\"Gb(*F3$\"5&Qy<+-]*>5OFfu7$$ \"5@\")>^MLqOi(*F3$\"5Vz'oj'o2R>OFfu7$$\"5w(pB#o@a/p(*F3$\"5&=Tc1!*fU& GOFfu7$$\"5k.pxJIlLv(*F3$\"5p%G\\`F-awj$Ffu7$$\"5j0CI!Rde7y*F3$\"5z>F^ uxSsYOFfu7$$\"5D*p$\\-(GGoy*F3$\"5B4A?G4=vbOFfu7$$\"53'Q(G0v51#z*F3$\" 5`>[&>4FOZm$Ffu7$$\"5qWPJ5Y6(pz*F3$\"5!GC,))\\_wOn$Ffu7$$\"5uUZ5)oir:! )*F3$\"50Qpk*GkrDo$Ffu7$$\"5YqL5qkY(e!)*F3$\"5+=akE02U\"p$Ffu7$$\"5^Gu 5h<:*)4)*F3$\"5!ovFK>!GA+PFfu7$$\"5e36;CMEj8)*F3$\"5y\"o)p$=.x*3PFfu7$ $\"5.\"GBZZs2r\")*F3$\"5zI[r%G]#otUO%\\t$Ffu7$$\"5B*=HOX9Cg#)*F3$\"5 e%RI(\\pv\\VPFfu7$$\"5Ek^b@Q&>&G)*F3$\"5CMXzX_#**>v$Ffu7$$\"5j'*H,_r$) yI)*F3$\"57%>(*z&RyWgPFfu7$$\"5n$3ht8/PG$)*F3$\"5Y'e-Yd[U)oPFfu7$$\"5% R'z-KK:nM)*F3$\"5IMfON_B=xPFfu7$$\"5-``P\\-EQFfu7$$\"5!*yT^j0Z%>% )*F3$\"5R<9+)yKeR$QFfu7$$\"5-!*)3>YWSA%)*F3$\"5SS'\\QFfu7$$\"5FP'=xR&3JU)*F3$\"5\"[D7G<\\(QdQFfu 7$$\"5vl^.8(e\"4U)*F3$\"56-X\"e5*32lQFfu7$$\"5>Bib-'R1<%)*F3$\"5x5e*z! 3)*osQFfu7$$\"5sl24\"*\\y:T)*F3$\"5aD*psAXV-)QFfu7$$\"5wc6M5t()QFfu7$$\"51zLr&49!eR)*F3$\"5GTRCsk<:&*QFfu7$$\"5rH$fjEJb& Q)*F3$\"5*f1uxa&[]-RFfu7$$\"5))y\\tS8ePP)*F3$\"5*zE#pn<&*y4RFfu7$$\"5* oSJhMXVg$)*F3$\"5iDds;i\\+`f\"\\^[p6)*F3$\"5J[SK&[Ez:*RFfu7$$\"5_'*[Pk0`\"e7%e)z*F3$\"5R.'3eAN)Q;SFfu7$$\"5ZCViXA( e\\z*F3$\"56KSNoLsQASFfu7$$\"5>H3j+OX>\"z*F3$\"5J^fH)R1.$GSFfu7$$\"5^G (H)[v:H(y*F3$\"5ma&>$[$)\\8MSFfu7$$\"5F8$QM\"p(\\Ky*F3$\"5hWrV&[7#))RS Ffu7$$\"5PA#e%)\\'*o!z(*F3$\"5LR$pjEhVb/%Ffu7$$\"53Xm.JJ*[Zx*F3$\"55^B s\"Hc=60%Ffu7$$\"5^:)fmoN*Gq(*F3$\"5C!3p)e#31m0%Ffu7$$\"5&p4;u4&)*ol(* F3$\"54bwV8p_+iSFfu7$$\"5`ex[VV*\\4w*F3$\"5PslI**4_JnSFfu7$$\"5`lS&zU3 pgv*F3$\"53C2+2#)\\`sSFfu7$$\"5rJH\"\\LkY5v*F3$\"5b4tP?^OmxSFfu7$$\"57 BN&3(4>)eu*F3$\"5y!*[[hr-q#3%Ffu7$$\"5Rxs8*34u0u*F3$\"5w?EZL&)Qk(3%Ffu 7$$\"58G#pt>JA^t*F3$\"5`?*>9;_$\\#4%Ffu7$$\"5d=ZOZ9c_H(*F3$\"54e&\\4j> [s4%Ffu7$$\"5o_zL2bHyB(*F3$\"5]X\"zW7\"p!>5%Ffu7$$\"5z`x&eT?$*yr*F3$\" 55pu&\\Nlok5%Ffu7$$\"5uO<.TFfu7$$\"5N[8] !\\dPGp*F3$\"51nK(=`3DP7%Ffu7$$\"5*[gi$QvA>'o*F3$\"5y,#)3>2iyFTFfu7$$ \"5es(3I\")=\"Rz'*F3$\"5i'zU6*HRuJTFfu7$$\"5XD$f$)=_KCn*F3$\"5+2G5r;rf NTFfu7$$\"5%[lP&**zVJl'*F3$\"59)Q3+(4YMRTFfu7$$\"5%=X(y'pvM!e'*F3$\"5` O'*F3$\"5#H%4#*GQAafTFfu7$$\"5 /PCl@$))Q3h*F3$\"5wtA%o>Y:D;%Ffu7$$\"5FJ\"=(*)GFO-'*F3$\"5B`/9ZSGPlTFf u7$$\"5b?(y<%Ffu7$$\"5=`(G5L'pcZ&*F3$ \"5rDfPGCO+!=%Ffu7$$\"5ni(*zIDSuP&*F3$\"5M6VR(H%*3?=%Ffu7$$\"5o2_2YbIr F&*F3$\"5y;2$[i>')Q=%Ffu7$$\"5t[:PAL&pu^*F3$\"5wFgwDnOj&=%Ffu7$$\"53P+ ;l0(3q]*F3$\"59%za^(*e\\s=%Ffu7$$\"5Tkng+ubK'\\*F3$\"5&p8F)\\X@t)=%Ffu 7$$\"5cg*om#z[T&[*F3$\"5<^l*R:Yz+>%Ffu7$$\"5cJ,3W'3rUZ*F3$\"5Wgs%psg*G \">%Ffu7$$\"5'[#z`bn$))GY*F3$\"5Ur![98fgB>%Ffu7$$\"5CUk-?$eg7X*F3$\"5% Gv)H@e.H$>%Ffu7$$\"5SQQjZi7QR%*F3$\"5i(*zD'[ywS>%Ffu7$$\"5J#yO%=\"eVsU *F3$\"5]%Ffu7$$\"5-*y!)p!Q.%[T*F3$\"5\"*fj'fPw5_>%Ffu7$$\"52/ $[:*GR;-%*F3$\"5o`t!ygp`b>%Ffu7$$\"5i%*esC=j?*Q*F3$\"5'p[on#RSu&>%Ffu7 $$\"56a**pQ3!ffP*F3$\"5<.$=#Qp#zd>%Ffu7$$\"5?Wy:b0ITi$*F3$\"5'**e5^vwc c>%Ffu7$$\"5^xdcl#ye&[$*F3$\"5]k2(yT\"QP&>%Ffu7$$\"5tXp$G%QiQM$*F3$\"5 #ek;#p#eF\\>%Ffu7$$\"50qU()R_Y))>$*F3$\"514+bzK^J%>%Ffu7$$\"5hRy-CNE/0 $*F3$\"5S^&[!=.M`$>%Ffu7$$\"5#HIt*)Q2[)*G*F3$\"5:??,k.#zD>%Ffu7$$\"5Gv @owq!)Gu#*F3$\"5,!RgBl?\\9>%Ffu7$$\"50L$f%Ffu 7$$\"5id%\\\")R\"e,U#*F3$\"5i\\MQIlxk)=%Ffu7$$\"5y&*>n`'>t_A*F3$\"5aZP Z7\"*)op=%Ffu7$$\"5$3()\\0EB/\"3#*F3$\"5=**)e%HO$*4&=%Ffu7$$\"5zl*\\%* =#4\\!>*F3$\"5'QaG4%H\\.$=%Ffu7$$\"5$H=.)\\PRTs\"*F3$\"5+eP)4?Hr2=%Ffu 7$$\"5x]sPH$\\_Q:*F3$\"5)*eA(GH#QIyTFfu7$$\"5JI'[+T>%yM\"*F3$\"5qrs<5 \"oFc<%Ffu7$$\"5#y!R&He'[=:\"*F3$\"5'Qc'ylmxtsTFfu7$$\"5IVGxKi$G]4*F3$ \"5&3h'42+(G'pTFfu7$$\"5['\\N`\\M'Gu!*F3$\"59;Fl7*z%HmTFfu7$$\"5\"3`j= ,.GH0*F3$\"5n&y\"fT`+tiTFfu7$$\"5x;!*z-**)>4.*F3$\"5nU/s-'4G*eTFfu7$$ \"5#G*>!*ei`A3!*F3$\"50:!zY?<#)[:%Ffu7$$\"5#GnE**GP/[)*)F3$\"5<6uAU.^e ]TFfu7$$\"5&>l`@2(Hhg*)F3$\"5JY>,m\\#Hg9%Ffu7$$\"50]n_#ev-c$*)F3$\"5aE R%RPY179%Ffu7$$\"5SJSt\"QE?(4*)F3$\"5!*Hl<^U!3h8%Ffu7$$\"5g<5^()Gi!H)) )F3$\"5!f3V@2nC28%Ffu7$$\"5Fg,*)H1Z4b))F3$\"5/Daihdj/DTFfu7$$\"53%)3@c @?@E))F3$\"5'3V%f\"\\Oi!>TFfu7$$\"5M@qj!H]vhz)F3$\"5[C2ilC6w7TFfu7$$\" 5\"He!*z:%>*[w)F3$\"5vkhq6X,81TFfu7$$\"5X_8Y?1dDK()F3$\"5LCtul-f:*4%Ff u7$$\"5M^&)Q]8k9)p)F3$\"5Y%Qs#z;P#=4%Ffu7$$\"5&y4)>xBgUi')F3$\"5y3FgB0 w6%3%Ffu7$$\"5IbrC_\"=N\\i)F3$\"5zNx9%[6?g2%Ffu7$$\"5Qkkdv_&)[&e)F3$\" 5&eUdwL77v1%Ffu7$$\"5!zw+G#4)oQa)F3$\"5I[%zVdgs&eSFfu7$$\"5$R>q\">f(=) *\\)F3$\"5L[qpbc$y\"\\SFfu7$$\"5e;m)>Q*3.`%)F3$\"5FE'z6(eOIRSFfu7$$\"5 96H,neK8.%)F3$\"5E`%**fX))>*GSFfu7$$\"5[n#\\#y\")*p'\\$)F3$\"5#['e#*>7 ]*z,%Ffu7$$\"5q*3JCO$Q2#H)F3$\"5Tj6*zP.$\\1SFfu7$$\"5-!4\"zxaliH#)F3$ \"5?_y(R\\AtV*RFfu7$$\"5DsBp#=w*Rh\")F3$\"5<,sR]L#*e\")RFfu7$$\"5Whk6e pP;'3)F3$\"5)*yP5()Hz3oRFfu7$$\"5^J]P/V3C-!)F3$\"5NRO,gQz!Q&RFfu7$$\"5 .C!)=u%[cs!zF3$\"5nO@[mOxnQRFfu7$$\"57kk'*o(z(F3$\"5dx9.7_JhARFfu7$ $\"5*)p**3Pm!))zm(F3$\"5[:/DY;S^0RFfu7$$\"5Y*RLxN4<'3vF3$\"55bu`N&ffs) QFfu7$$\"5f:Jik\"f\")4I(F3$\"5xb3,i5AqnQFfu7$$\"5KM;q&py5,+(F3$\"5Q!oS tc5em%QFfu7$$\"5TO]U(Hr%=IkF3$\"58!QeG'oR*Q#QFfu7$$!5s\"*)[[=)RkVhF3$ \"5_&z6'[]j5*z$Ffu7$$!5Dq\"yt.12S$pF3$\"5%4&*pmQ%*))=x$Ffu7$$!5(e]6)\\ ?-S4tF3$\"5H,;61'4y;u$Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\" \"#-%+AXESLABELSG6$Q!6\"Fdep-%&COLORG6&%$RGBG$F\\ep!\"\"$\"#X!\"#F(-%% VIEWG6$%(DEFAULTGFbfp" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 108 "The relevant intersection points of the boundary \+ curve with the imaginary axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundary curve \+ either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "Digits := 15:\nz0 := 3.1*I:\nfor ct from 87 to 90 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0 );\nend do;``;\nz0 := 3.8*I:\nfor ct from 281 to 284 do\n newton(R(z )=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0yd&\\r?M:!#;$\"0dprRHL'H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0A$f8k;)e'!#<$\"034uVn)HI!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0a#)f+1.7\"!#;$\"0^oE]gh5$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0K\")=0)*oI&!#;$\"0n,7'fP!>$!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0@[0,rw(> !#;$\"0Ttc5em%Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"01,U\\g1x(! #<$\"0eG'oR*Q#Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0:-&QTP1Z!#< $\"07'[]j5*z$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0e*=5#p;y\"!#; $\"0qmQ%*))=x$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to calculate the para meter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part := proc(u)\n \+ Re(newton(R(z)=exp(u*Pi*I),z=3.1*I))\nend proc:\nu0 := bisect('real_pa rt'(u),u=0.87..0.90);\nnewton(R(z)=exp(u0*Pi*I),z=3.1*I);``;\nreal_par t := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.8*I))\nend proc:\nu0 : = bisect('real_part'(u),u=2.81..2.84);\nnewton(R(z)=exp(u0*Pi*I),z=3.8 *I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0`h%Q) 3l%))!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0k/pF'oO$*!#H$\"0\"** **32&R1$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0^7M2zi#G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0H['45!)*H\"!#G$\"0Fs40*e3Q!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability regi on intersects the nonegative imaginary axis in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[3.06395, 3.8086];" "6#7$-%&FloatG6$\"'&R1$ !\"&-F%6$\"&'3Q!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------- -----------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stabilit y region for the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for th e 8 stage, order 4 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "taylor(expand(subs(ee,subs(b=`b*`,StabilityFunc tion(4,8,'expanded')))),z,9):\nconvert(%,polynom,z):\n`R*` := unapply( %,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6# %\"zG,4\"\"\"F)F'F)*&\"\"#!\"\"F'F+F)*&\"\"'F,F'\"\"$F)*&\"#CF,F'\"\"% F)*&,&#\"jo.(z%fps<7J0FJ:'4iBxL9g_S>5'ph6e9-ocuF%QvM\")\"]p++!oF$[%))z bqjVwFMPiJ\"G[8#zOp1-LtxwFhd%e$oAL+\"F)*(\"co()ykQJ4(G$))zc=d=K6X1P?T* 4h9t$Q*o+lLJhY(\\F)\"^p+++w$HQ\">f!Rfa]V*RhO@pzV\\ad&oW6LTuV*G.#4&yeK- (F,\"0XfX<9:0\"#F)F+F,F))F'\"\"&F)F)*&,&#\"io8i$o!>(GI8OJS;ZzJ^&=T*GH* R*)[$p@XIS?B_n6v:6#\"\\p++![%[9SH%=%R]p1fnD>&R`%fYHv5i$z*=\\j8Zuv'>(=) >F)*(\"co*oem$z!*GHX'*R4>s\\*R?mN9:\"\\v#>IG)3K[dh#Q)\\F)\"_p+++3]j5`t C^nV![&>\"H4Pv.bf.Y[d\"\\1`*\\:jit!Gqg=cF,F;FqF)\"O++!GT=&zN*)3!zF&Gu?ywi$GMQ\"F,F;FF)*(\">`r)G,Slv!Rn(yw'H&F) \"L++?Vy\"o7*pm3'4,ka;HD4F:F,F;F " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R *`(z)=-1,z=-7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+\"3f@!o! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=-1,z=-7):\np_1 := plot([`R*`(z ),-1],z=-7.39..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],sty le=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_ 0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([ p_1,p_2,p_3],view=[-7.39..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7W7$ $!3o*************Q(!#<$!3.v%3ycol]$F*7$$!3>LLe7#>TI(F*$!3eGAIA)H\")*HF *7$$!3pmm;D%Q#=sF*$!3#RS<\\=ida#F*7$$!3_*\\P7,9N9(F*$!3)QyW_(3'R>#F*7$ $!3AL$3tf*yoqF*$!3%R/Igo#)y(=F*7$$!3g*\\i'yTv%)pF*$!3%[*=D^sdh:F*7$$!3 &fm;+w=2!pF*$!3?2\"H-6/JG\"F*7$$!3Jmm1+j7;oF*$!3W;M\"zc[u.\"F*7$$!3nmm 6SQ`JnF*$!3NX$e_\\>EB)!#=7$$!35L$3m-`Jc'F*$!3=%o5GJgky%FU7$$!3_m;CeC/2 kF*$!3jgU8%)f#)4CFU7$$!31+](p+*RXiF*$!3AP*4,,Ba<'!#>7$$!3Im;W=rAygF*$ \"31&za?9@cm'F_o7$$!3))**\\F_8f6fF*$\"3]hkFU7$$!3cmm^V?@*e&F*$\"3ZtU/hJ%H@#FU7$$!3^*****Q9^#>aF* $\"3Qri#o5RaF#FU7$$!3y****\\oBf[_F*$\"3k1,Fz4S/AFU7$$!3u****4;58%3&F*$ \"3)[/O]/LZ0#FU7$$!3em;W_VyM\\F*$\"3kGKU%z*py=FU7$$!3GLLBCn>dZF*$\"3_A G50S@Z;FU7$$!3ALL$*3nv1YF*$\"3o(H`Fy_\"\\9FU7$$!3F+]<2#f5J7FU7$$!3?LL$fwcoF%F*$\"3anbOsscd5FU7$$!3v**\\T*GZh(F_o7$$!3')**\\P#>D?Z#F*$\"3,4+:3v.K() F_o7$$!3/++XY`n%H#F*$\"3M;P'ezku-\"FU7$$!3*RLL>i=`8#F*$\"3%fsdPPJf>\"F U7$$!37++]/8F*$\"3*)y$=Wn/6T\"FU7$$!3rm;/YD*4\"=F*$\"3\"\\?,flF@k\" FU7$$!3k****4RWVU;F*$\"3H=88UCBS>FU7$$!3gL$3lRMQ[\"F*$\"3=W*R\"3(R8F#F U7$$!3(***\\Z`;0=8F*$\"3+m!>)=f.zEFU7$$!3%RLL)>(of:\"F*$\"33i;)pL%4\\J FU7$$!3%H+]P4HG')*FU$\"3uK%p#GL[IPFU7$$!3?fmmiIbG#)FU$\"3?u\\A!p%>#R%F U7$$!3Ejm;x(psb'FU$\"3%R_E*3x\"3>&FU7$$!3yIL3hb#)**[FU$\"3o'\\:2h>k7'F U7$$!3M*****>Y1oP$FU$\"3:vH8/*RU8(FU7$$!3@JL$=qa7j\"FU$\"3aiHXci%[\\)F U7$$!3L'ymm')R7+(!#?$\"35t4;<@BI**FU7$$\"3!p**\\#Gvf%f\"FU$\"3'[.A+-xG <\"F*7$$\"3e'**\\xiTy=$FU$\"3[pbjgIXv8F*7$$\"3!***************[FU$\"3U D$>KZ-Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Ff\\lFe\\l-F$6$7S7$F ($!\"\"Ff\\l7$F3F[]l7$F=F[]l7$FGF[]l7$FQF[]l7$FWF[]l7$FfnF[]l7$F[oF[]l 7$FaoF[]l7$FfoF[]l7$F[pF[]l7$F`pF[]l7$FepF[]l7$FjpF[]l7$F_qF[]l7$FdqF[ ]l7$FiqF[]l7$F^rF[]l7$FcrF[]l7$FhrF[]l7$F]sF[]l7$FbsF[]l7$FgsF[]l7$F\\ tF[]l7$FatF[]l7$FftF[]l7$F[uF[]l7$F`uF[]l7$FeuF[]l7$FjuF[]l7$F_vF[]l7$ FdvF[]l7$FivF[]l7$F^wF[]l7$FcwF[]l7$FhwF[]l7$F]xF[]l7$FbxF[]l7$FgxF[]l 7$F\\yF[]l7$FayF[]l7$FfyF[]l7$F[zF[]l7$F`zF[]l7$FezF[]l7$FjzF[]l7$F`[l F[]l7$Fe[lF[]l7$Fj[lF[]l-F_\\l6&Fa\\lFe\\lFe\\lFb\\l-F$6&7#7$$!36+++\" 3f@!oF*F[]l-%'SYMBOLG6#%'CIRCLEG-F_\\l6&Fa\\lFf\\lFf\\lFf\\l-%&STYLEG6 #%&POINTG-F$6&Fa`l-Ff`l6#%&CROSSGFi`lF[al-F$6&Fa`l-Ff`l6#%(DIAMONDGFi` lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLORG6&Fa\\lFe\\l$\"\"&F\\]lFe\\l-%*LIN ESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F_cl -Fgbl6#%(DEFAULTG-%%VIEWG6$;$!$R(!\"#$\"#\\Fjcl;$!$Z\"Fjcl$\"$Z\"Fjcl " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture sho ws the stability region for the 8 stage, order 4 scheme. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2196 "`R* ` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+(813475384277456680214581161696 1019405260143377236209615312705311217726959479703/10033226835845761277 67773330206693679213482813162373427764363705579884483276800000-4974661 31336500689383731461099412037064511321857185679883287093138647887/7023 2587850920328943744133114468557544943796921366139943505459390591913829 376000000*105151417455945^(1/2))*z^5+(21115751167522320403045216934889 3992928941185513179471640313613302871906836213/19818719675744713634918 9793621075294659453395192567590669503941842940144844800000+49838261574 8320882830192754911514356620399497219093996452928907936658689/56186070 2807362631549953064915748460359550375370929119548043675124735310635008 000000*105151417455945^(1/2))*z^6+(28770489040067702993855489479129890 01663/39526524650509164081507971682673700526080000-7019235678659109333 15369596990787/1383428362767820742852779008893579518412800000*10515141 7455945^(1/2))*z^7+(113093754206156431292803151127249/1947822103527377 56500110548358012480000-52967678767390756540012887153/1527092529165464 010960866699126817843200000*105151417455945^(1/2))*z^8:\ntt := 0: z0 : = 0: pts := []:\nwhile tt<=281/20 do\n zz := newton(`R*`(z)=exp(tt*P i*I),z=z0):\n z0 := zz: \n if (7/10+++bIg4WFP$\"3!******f>s xD\"FS7$$!3W******3'*R%4(FP$\"3!*******Hj(eT\"FS7$$!3i*****f()\\6u)FP$ \"3!******f>&ou:FS7$$!3F+++?QE9[FP$\"3&******4(ReMg?FS7$$\"3%)*****f4p( p@!#?$\"34+++Oz,GAFS7$$\"3l******f[o-aF`p$\"3*)*****H+e:S#FS7$$\"3y*** **48Qkp'F`p$\"3;+++A!ohW#FS7$$\"3Y*****p@/JF)F`p$\"3'******z>(Q\"\\#FS 7$$\"3$*******obr>5!#>$\"3:+++\\(*GPDFS7$$\"33+++%*\\0b7Feq$\"3#)***** pCgRe#FS7$$\"3%******p%e(Qa\"Feq$\"3$)*****f%[\\JEFS7$$\"3%)*****45P** *=Feq$\"3++++-@+!o#FS7$$\"3))*****R\\`9M#Feq$\"3*******zg-'HFFS7$$\"3) *******>\"4H*GFeq$\"3')*****zuD/y#FS7$$\"3/+++wfp(e$Feq$\"3@+++I**fKGF S7$$\"3?+++wE&=Z%Feq$\"3')*****z]Gi)GFS7$$\"3,+++/^\"*3cFeq$\"3>+++$QN 8%HFS7$$\"3Y*****RTvX3(Feq$\"3')*******H]x*HFS7$$\"3)******z^Gb+*Feq$ \"3++++*)G*[0$FS7$$\"3'******4LTx9\"F0$\"3;+++:HY6JFS7$$\"3!******>v2U X\"F0$\"3;+++)*)*GlJFS7$$\"3++++4Nx4=F0$\"3/+++.D#R@$FS7$$\"3!******H$ )f1>#F0$\"3/+++\"\\CfD$FS7$$\"3;+++O!))=d#F0$\"37+++vk[\"H$FS7$$\"3E++ +0o@PHF0$\"31+++(zw&zKF0$\"39+++`-5[LFS7$$\"3%)* *****>_=(f$F0$\"3y******p1_rLFS7$$\"3C+++()>9\"*QF0$\"3%******H58GR$FS 7$$\"3')*****plYK;%F0$\"3,+++fB_7MFS7$$\"3\")*****4j^bT%F0$\"3$******R =I5V$FS7$$\"3!*******H'4+l%F0$\"3#******z!pg[MFS7$$\"37+++cPPo[F0$\"3% ******z+ZaY$FS7$$\"3Y*****znz@2&F0$\"3#******>\"Gp\"[$FS7$$\"3o*****zj fFE&F0$\"35+++Y-X(\\$FS7$$\"3`+++imETaF0$\"3********)Q*z7NFS7$$\"3i*** **\\1+(3cF0$\"3'******>l,y_$FS7$$\"3)******H%)Gfw&F0$\"32+++NZ]UNFS7$$ \"3\"******p%>r8fF0$\"3%******z6Ypb$FS7$$\"31+++ejfAkF0$\"3A++++$3Eh$F S7$$\"3b+++mM#>#oF0$\"3/+++#*[llOFS7$$\"3W+++!RpE8(F0$\"3++++K`i;PFS7$ $\"3y*****4tW#ptF0$\"3y*****>%*odw$FS7$$\"3(******\\]e?a(F0$\"3&)***** \\W&>8QFS7$$\"3U+++$RP*ewF0$\"3?+++(4T*eQFS7$$\"3j*****>;ifs(F0$\"3,++ +e0+.RFS7$$\"3g*****fD()zu(F0$\"3))*****zKV`%RFS7$$\"3')******G!y*GxF0 $\"3)******\\lAf)RFS7$$\"3'*******pxBswF0$\"3y******z/oCSFS7$$\"3G+++' Hl0e(F0$\"3!)*****p?]:1%FS7$$\"3E+++'=njX(F0$\"3#)*****p:ek4%FS7$$\"3o *****pSRH)*y%F0$\"3%) *****\\i@iJ%FS7$$\"3#)*****>`NkS%F0$\"3:+++)4iBK%FS7$$\"3>+++FZ$\\+%F0 $\"3j*****pXf[K%FS7$$\"3C+++iwl&e$F0$\"3<+++&f:MK%FS7$$\"31+++`0q[JF0$ \"3:+++7&pwJ%FS7$$\"3C+++4k#Qp#F0$\"3%*******)purI%FS7$$\"3,+++OtH?AF0 $\"37+++]_O\"H%FS7$$\"3'******R7xls\"F0$\"3#)*****p<.&pUFS7$$\"3/+++CC x47F0$\"3?+++YGfSUFS7$$\"3^*****Ri!*\\k'Feq$\"3&)*****f'[B.UFS7$$\"3^* ****z%GaL!)F`p$\"3))******\\BObTFS7$$!3=+++lQo#R(F`p$\"3C+++J!499%FS7$ $!3$******R$Q#RK#Feq$\"3G+++@+aETFS7$$!39+++.b>dRFeq$\"3u*****\\-!o5TF S7$$!3?+++KKTZcFeq$\"3m******zMu$4%FS7$$!3H+++JdD0uFeq$\"33+++DLjvSFS7 $$!33+++%e5YC*Feq$\"3?+++w,CcSFS7$$!3-+++zeAm!RFS7$$!3++++-))G9FF0$\"3;+++%o/k(Q FS7$$!3'******zAS;7$F0$\"3A+++5\\WYQFS7$$!3;+++!y$z*e$F0$\"3>+++)[8*=Q FS7$$!3:+++'4z&3TF0$\"3#)*****>bhiz$FS7$$!35+++$4DUl%F0$\"3)******pgk*F0$\"35+++)=8**z$FS7$$!3,+++L\"4/+\"FS $\"3<+++*3Zk!QFS7$$!32+++>XnN5FS$\"36+++JI,8QFS7$$!3%******>Fy+2\"FS$ \"33+++.Mc>QFS7$$!3\"******\\B'p.6FS$\"3()*****R\"31EQFS7$$!3/+++4\\fO 6FS$\"3()*****RgvC$QFS7$$!3-+++#)G$)o6FS$\"3/+++LSyQQFS7$$!3++++4?Y+7F S$\"38+++`p'\\%QFS7$$!3(********RG:B\"FS$\"39+++:)35&QFS7$$!35+++n&H4N \"FS$\"3++++xKatQFS7$$!3)*******RU(RY\"FS$\"37+++Ag7$*QFS7$$!3%******z 9s?d\"FS$\"3!)*****\\#\\]4RFS7$$!33+++\"HEin\"FS$\"3)******\\giD#RFS7$ $!3-+++UkFS$\"3#******\\6z9%RFS7$$!3))*****z(3$e1#FS$\"3\")*****H =W5%RFS7$$!3))*****>4)\\e@FS$\"3&)*****R+rs$RFS7$$!3#*******31#)\\AFS$ \"34+++z`=IRFS7$$!3&******R!R**RBFS$\"3z*****>=:)>RFS7$$!3:+++Qd=HCFS$ \"3))*****p0!>1RFS7$$!3y******)*Ha0 EFS$\"3/+++3ZIpQFS7$$!3/+++\"*oE#p#FS$\"3?+++![QFS7$$!31+++6-4lGFS$\"3-+++'Ru/z$FS7$$!35+++&\\W 5&HFS$\"3!)*****pfG\"ePFS7$$!3*)*****p'y\"o.$FS$\"3:+++$>cGs$FS7$$!35+ ++6^\\AJFS$\"3%******>'*QZo$FS7$$!34+++#**elMFS$\"3))******pP#f]$FS7$$!3!******fvp;b$FS$\"3#)**** *4Io_X$FS7$$!3\")*****R\\))yj$FS$\"35+++,q^-MFS7$$!3\")*****z$\\@CPFS$ \"3;+++[h\"yM$FS7$$!32+++^_f5QFS$\"33+++gbI\"H$FS7$$!3')*****p^bp*QFS$ \"3'******\\18JB$FS7$$!3-++++g?$)RFS$\"3%)*****z')\\L<$FS7$$!35+++\\mC pSFS$\"3%)*****\\Q2@6$FS7$$!35+++PK(\\:%FS$\"3%******p%eX\\IFS7$$!3q** ***4Z$GSUFS$\"3!)*****\\sWa)HFS7$$!3\"******H^#3DVFS$\"3!******\\r/,#H FS7$$!3R+++UvG4WFS$\"3;+++94X`GFS7$$!3'******f?JG\\%FS$\"3-+++]n[&y#FS 7$$!3++++rRmvXFS$\"3.+++!G3ir#FS7$$!3)******\\\\bxl%FS$\"3!)*****\\d3c k#FS7$$!32+++R_4RZFS$\"3/+++N?otDFS7$$!3))*****4X#p>[FS$\"3&******feG/ ]#FS7$$!3c*****pov&**[FS$\"39+++@y&eU#FS7$$!3I+++.>zy\\FS$\"3%******\\ :$**\\BFS7$$!3C+++(30u0&FS$\"39+++sh(GF#FS7$$!3*******\\H%\\N^FS$\"39+ ++n7d%>#FS7$$!3e*****R\"3:8_FS$\"3'******>pq^6#FS7$$!3c*****H1u/H&FS$ \"3;+++;+![.#FS7$$!3-+++*plvO&FS$\"31+++MQi`>FS7$$!3C+++s7_WaFS$\"3%** ****fI^=(=FS7$$!3*******Ru=9_&FS$\"3-+++V1u*y\"FS7$$!3s******)R.$)f&FS $\"3,+++66g2v&FS$\" 3%*******zYqW:FS7$$!3(******>CR%GeFS$\"3++++k)eZY\"FS7$$!3I+++'FS$\"3'******R\"Qn$4\"FS7$$!3u******3xJgiFS$\"3!******4-#3E5F S7$$!3g******zsEBjFS$\"3?+++#G?Wg*F07$$!3e*****p`_DQ'FS$\"3o******f:wk *)F07$$!3C+++rB'zV'FS$\"3[+++XL')Q$)F07$$!3=+++@HP*['FS$\"3y*****p&\\$ Qs(F07$$!3U+++R&Hn`'FS$\"3()*****z#e1mFS$\"3K+++U#p(>fF07$$!3W+++u_eamFS$\"3r******GA'fK &F07$$!35+++WA&fo'FS$\"3y*****>+)zLZF07$$!3W+++*RfMr'FS$\"3++++P,ZUTF0 7$$!31+++5m;PnFS$\"3%)*****HR=9b$F07$$!37+++4)Grv'FS$\"33+++ADGgHF07$$ !39+++-TRtnFS$\"37+++y7')oBF07$$!3?+++[E+'y'FS$\"37+++To2x+)zLZF07$Fhdn$!3r* *****GA'fK&F07$Fcdn$!3K+++U#p(>fF07$F^dn$!37+++e8K;lF07$Ficn$!3()***** z#e1FS7$F_^n$!3;+++;+![ .#FS7$Fj]n$!3'******>pq^6#FS7$Fe]n$!39+++n7d%>#FS7$F`]n$!39+++sh(GF#FS 7$F[]n$!3%******\\:$**\\BFS7$Ff\\n$!39+++@y&eU#FS7$Fa\\n$!3&******feG/ ]#FS7$F\\\\n$!3/+++N?otDFS7$Fg[n$!3!)*****\\d3ck#FS7$Fb[n$!3.+++!G3ir# FS7$F][n$!3-+++]n[&y#FS7$Fhjm$!3;+++94X`GFS7$Fcjm$!3!******\\r/,#HFS7$ F^jm$!3!)*****\\sWa)HFS7$Fiim$!3%******p%eX\\IFS7$Fdim$!3%)*****\\Q2@6 $FS7$F_im$!3%)*****z')\\L<$FS7$Fjhm$!3'******\\18JB$FS7$Fehm$!33+++gbI \"H$FS7$F`hm$!3;+++[h\"yM$FS7$F[hm$!35+++,q^-MFS7$Ffgm$!3#)*****4Io_X$ FS7$Fagm$!3))******pP#f]$FS7$F\\gm$!3')******)ROVb$FS7$Fgfm$!37+++'oj. g$FS7$Fbfm$!3)******p)=(Qk$FS7$F]fm$!3%******>'*QZo$FS7$Fhem$!3:+++$>c Gs$FS7$Fcem$!3!)*****pfG\"ePFS7$F^em$!3-+++'Ru/z$FS7$Fidm$!3$*******4Q #)>QFS7$Fddm$!3?+++![1RFS7$F`cm$!3z*****>=:)>RFS7$F[cm$!34+++z`=IRFS7$Ffb m$!3&)*****R+rs$RFS7$Fabm$!3\")*****H=W5%RFS7$F\\bm$!3#******\\6z9%RFS 7$Fgam$!3?+++ZKbQRFS7$Fbam$!3$******H!)\\A$RFS7$F]am$!3)******\\giD#RF S7$Fh`m$!3!)*****\\#\\]4RFS7$Fc`m$!37+++Ag7$*QFS7$F^`m$!3++++xKatQFS7$ Fi_m$!39+++:)35&QFS7$Fe^m$!3()*****R\"31EQFS7$F`^m$!33+++.Mc>QFS7$F[^m $!36+++JI,8QFS7$Ff]m$!3<+++*3Zk!QFS7$Fa]m$!35+++)=8**z$FS7$F\\]m$!37++ +YEZ$z$FS7$Fg\\m$!39+++ca?(y$FS7$Fb\\m$!3y*****4Y<7y$FS7$F]\\m$!39+++7 4lvPFS7$Fh[m$!3;+++?-qqPFS7$Fc[m$!3?+++jcjmPFS7$F^[m$!3!)******\\*QQw$ FS7$Fijl$!3))******o[&Gw$FS7$Fdjl$!3!)*****pZeWw$FS7$F_jl$!3/+++IOrpPF S7$Fjil$!3)******pbhiz$FS7$F`il$!3>+++)[8*=QFS 7$F[il$!3A+++5\\WYQFS7$Ffhl$!3;+++%o/k(QFS7$Fahl$!3z*****>eAm!RFS7$F\\ hl$!30+++q')zNRFS7$Fggl$!37+++IHNjRFS7$Fbgl$!3%)******=b6*)RFS7$F]gl$! 3h******3368SFS7$Fhfl$!3++++!pUa.%FS7$Fcfl$!3?+++w,CcSFS7$F^fl$!33+++D LjvSFS7$Fiel$!3m******zMu$4%FS7$Fdel$!3u*****\\-!o5TFS7$F_el$!3G+++@+a ETFS7$Fjdl$!3C+++J!499%FS7$Fedl$!3))******\\BObTFS7$$\"3%******zb4#4BF eq$!3)******\\Hm%oTFS7$$\"3++++gQb#y$Feq$!3=+++=%z2=%FS7$$\"3&******z0 hoA&Feq$!3M+++wNN#>%FS7$F`dl$!3&)*****f'[B.UFS7$F[dl$!3?+++YGfSUFS7$Ff cl$!3#)*****p<.&pUFS7$Facl$!37+++]_O\"H%FS7$F\\cl$!3%*******)purI%FS7$ Fgbl$!3:+++7&pwJ%FS7$Fbbl$!3<+++&f:MK%FS7$$\"3G+++GZ$\\+%F0$!3j*****pX f[K%FS7$Fhal$!3:+++)4iBK%FS7$Fcal$!3%)*****\\i@iJ%FS7$F^al$!3F+++P))o1 VFS7$Fi`l$!3K+++]#yRH%FS7$Fd`l$!3u*****H3w#yUFS7$F_`l$!3s******fhufUFS 7$Fj_l$!3$******pLM&QUFS7$Fe_l$!3y******o=x9UFS7$F`_l$!3m******))zd)=% FS7$F[_l$!3W+++))=1gTFS7$Ff^l$!39+++6UKHTFS7$Fa^l$!3#)*****p:ek4%FS7$F \\^l$!3!)*****p?]:1%FS7$Fg]l$!3y******z/oCSFS7$Fb]l$!3)******\\lAf)RFS 7$F]]l$!3))*****zKV`%RFS7$Fh\\l$!3,+++e0+.RFS7$Fc\\l$!3?+++(4T*eQFS7$F ^\\l$!3&)*****\\W&>8QFS7$Fi[l$!3y*****>%*odw$FS7$Fd[l$!3++++K`i;PFS7$F _[l$!3/+++#*[llOFS7$Fjz$!3A++++$3Eh$FS7$Fez$!3%******z6Ypb$FS7$Fay$!35 +++Y-X(\\$FS7$F\\y$!3#******>\"Gp\"[$FS7$Fgx$!3%******z+ZaY$FS7$Fbx$!3 #******z!pg[MFS7$F]x$!3$******R=I5V$FS7$Fhw$!3,+++fB_7MFS7$Fcw$!3%**** **H58GR$FS7$F^w$!3y******p1_rLFS7$Fiv$!39+++`-5[LFS7$Fdv$!31+++(zw+++$QN8%HFS7$Fgs$!3')*****z]Gi)GFS7$Fbs$!3@+++I**fK GFS7$F]s$!3')*****zuD/y#FS7$Fhr$!3*******zg-'HFFS7$Fcr$!3++++-@+!o#FS7 $F^r$!3$)*****f%[\\JEFS7$Fiq$!3#)*****pCgRe#FS7$Fcq$!3:+++\\(*GPDFS7$F ^q$!3'******z>(Q\"\\#FS7$Fip$!3;+++A!ohW#FS7$Fdp$!3*)*****H+e:S#FS7$$ \"3')******z(o0M%F`p$!3y*****\\)>\\dBFS7$$\"37+++q^\"*oMF`p$!3.+++pE\" RJ#FS7$$\"3))*******=\"QaFF`p$!3')******R&p2F#FS7$F^p$!34+++Oz,GAFS7$F io$!3;+++9r>g?FS7$Fco$!33+++!ySh*=FS7$F^o$!3&******4(ReM&ou:FS7$FZ$!3!*******Hj(eT\"FS7$FU$!3!******f>sxD\"FS7$FN$!3,+++ wF6+6FS7$FI$!3a+++*yXsU*F07$FC$!3c*****fnS\\&yF07$F>$!3C+++07\\$G'F07$ $!35+++v.\"ot\"F:$!3@+++!4gCr%F07$F2$!3w*****H)=gTJF07$$!3.+++@8HGCF-$ !31+++8mzq:F0F'F*-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"#Fi\\qF(-%&STYLEG6#% %LINEG-%)POLYGONSG6acl7%F'7$$!+<8HGCFeq$\"+8mzq:!#57$$!#MFi\\qF(7%Fd]q 7$$!+!3@Ua\"FS$\"+$)=gTJFi]qFj]q7%F^^q7$$!+]/\"ot\"!#;$\"+!4gCr%Fi]qFj ]q7%Fd^q7$$!+$=4[a*Fg^q$\"+07\\$G'Fi]qFj]q7%F[_q7$$!+p?&3^$!#:$\"+w1% \\&yFi]qFj]q7%Fa_q7$$!+z&pY*)*Fd_q$\"+*yXsU*Fi]qFj]q7%Fh_q7$$!+6q()zA! #9$\"+wF6+6!\"*Fj]q7%F^`q7$$!+bIg4WFa`q$\"+'>sxD\"Fd`qFj]q7%Ff`q7$$!+4 '*R%4(Fa`q$\"+Ij(eT\"Fd`qFj]q7%F\\aq7$$!+w)\\6u)Fa`q$\"+'>&ou:Fd`qFj]q 7%Fbaq7$$!+?QE9[Fa`q$\"+rReMg?Fd`qFj]q7%Febq7$$\"+'4p(p@!#7$\" +Oz,GAFd`qFj]q7%F[cq7$$\"+g[o-aF^cq$\"+.!e:S#Fd`qFj]q7%Fbcq7$$\"+J\"Qk p'F^cq$\"+A!ohW#Fd`qFj]q7%Fhcq7$$\"+(Q\"\\#Fd`qFj]q7%F ^dq7$$\"+pbr>5!#6$\"+\\(*GPDFd`qFj]q7%Fddq7$$\"+%*\\0b7Fgdq$\"+Z-'Re#F d`qFj]q7%F[eq7$$\"+Ze(Qa\"Fgdq$\"+Y[\\JEFd`qFj]q7%Faeq7$$\"+,r$***=Fgd q$\"+-@+!o#Fd`qFj]q7%Fgeq7$$\"+%\\`9M#Fgdq$\"+3EgHFFd`qFj]q7%F]fq7$$\" +?\"4H*GFgdq$\"+[dU!y#Fd`qFj]q7%Fcfq7$$\"+wfp(e$Fgdq$\"+I**fKGFd`qFj]q 7%Fifq7$$\"+wE&=Z%Fgdq$\"+3&Gi)GFd`qFj]q7%F_gq7$$\"+/^\"*3cFgdq$\"+$QN 8%HFd`qFj]q7%Fegq7$$\"+9ad%3(Fgdq$\"++.v(*HFd`qFj]q7%F[hq7$$\"+=&Gb+*F gdq$\"+*)G*[0$Fd`qFj]q7%Fahq7$$\"+J8uZ6Fi]q$\"+:HY6JFd`qFj]q7%Fghq7$$! +G-k@JFi]q$\"+5\\WYQFd`qFj]q7%F]iq7$$!+!y$z*e$Fi]q$\"+)[8*=QFd`qFj]q7% Fciq7$$!+'4z&3TFi]q$\"+_:E'z$Fd`qFj]q7%Fiiq7$$!+$4DUl%Fi]q$\"+x^#*zPFd `qFj]q7%F_jq7$$!+&>gk*Fi]q$ \"+)=8**z$Fd`qFj]q7%Fa^r7$$!+L\"4/+\"Fd`q$\"+*3Zk!QFd`qFj]q7%Fg^r7$$!+ >XnN5Fd`q$\"+JI,8QFd`qFj]q7%F]_r7$$!+s#y+2\"Fd`q$\"+.Mc>QFd`qFj]q7%Fc_ r7$$!+Nip.6Fd`q$\"+931EQFd`qFj]q7%Fi_r7$$!+4\\fO6Fd`q$\"+/cZKQFd`qFj]q 7%F_`r7$$!+#)G$)o6Fd`q$\"+LSyQQFd`qFj]q7%Fe`r7$$!+4?Y+7Fd`q$\"+`p'\\%Q Fd`qFj]q7%F[ar7$$!++%G:B\"Fd`q$\"+:)35&QFd`qFj]q7%Faar7$$!+n&H4N\"Fd`q $\"+xKatQFd`qFj]q7%Fgar7$$!+SU(RY\"Fd`q$\"+Ag7$*QFd`qFj]q7%F]br7$$!+[@ 2s:Fd`q$\"+D\\]4RFd`qFj]q7%Fcbr7$$!+\"HEin\"Fd`q$\"+0EcARFd`qFj]q7%Fib r7$$!+UkFd`q$\"+:\"z9%RFd`qFj]q7%F[dr7$$!+y3$e1#Fd`q$\"+$=W5% RFd`qFj]q7%Fadr7$$!+#4)\\e@Fd`q$\"+/5FPRFd`qFj]q7%Fgdr7$$!+41#)\\AFd`q $\"+z`=IRFd`qFj]q7%F]er7$$!+/R**RBFd`q$\"+#=:)>RFd`qFj]q7%Fcer7$$!+Qd= HCFd`q$\"+d+>1RFd`qFj]q7%Fier7$$!+**Ha0EFd`q$\"+3ZIpQFd`qFj]q7%Fefr7$$!+\"*oE#p#Fd`q$\"+![QFd`qFj]q7%Fagr7$$!+6-4lGFd`q$\"+'Ru/z$ Fd`qFj]q7%Fggr7$$!+&\\W5&HFd`q$\"+(fG\"ePFd`qFj]q7%F]hr7$$!+ny\"o.$Fd` q$\"+$>cGs$Fd`qFj]q7%Fchr7$$!+6^\\AJFd`q$\"+i*QZo$Fd`qFj]q7%Fihr7$$!+< E:3KFd`q$\"+()=(Qk$Fd`qFj]q7%F_ir7$$!+Zf&QH$Fd`q$\"+'oj.g$Fd`qFj]q7%Fe ir7$$!+KolzLFd`q$\"+*ROVb$Fd`qFj]q7%F[jr7$$!+A**elMFd`q$\"+qP#f]$Fd`qF j]q7%Fajr7$$!+c(p;b$Fd`q$\"+,$o_X$Fd`qFj]q7%Fgjr7$$!+%\\))yj$Fd`q$\"+, q^-MFd`qFj]q7%F][s7$$!+Q\\@CPFd`q$\"+[h\"yM$Fd`qFj]q7%Fc[s7$$!+^_f5QFd `q$\"+gbI\"H$Fd`qFj]q7%Fi[s7$$!+[Fd`q$\"+'eG/]#Fd`qFj]q7%Fa` s7$$!+(ov&**[Fd`q$\"+@y&eU#Fd`qFj]q7%Fg`s7$$!+.>zy\\Fd`q$\"+bJ**\\BFd` qFj]q7%F]as7$$!+(30u0&Fd`q$\"+sh(GF#Fd`qFj]q7%Fcas7$$!+&H%\\N^Fd`q$\"+ n7d%>#Fd`qFj]q7%Fias7$$!+93:8_Fd`q$\"+#pq^6#Fd`qFj]q7%F_bs7$$!+jSZ!H&F d`q$\"+;+![.#Fd`qFj]q7%Febs7$$!+*plvO&Fd`q$\"+MQi`>Fd`qFj]q7%F[cs7$$!+ s7_WaFd`q$\"+18&=(=Fd`qFj]q7%Facs7$$!+W(=9_&Fd`q$\"+V1u*y\"Fd`qFj]q7%F gcs7$$!+*R.$)f&Fd`q$\"+66g2v&Fd`q$\"+!o/Za\"Fd`qFj]q7%Fids7$$!+U#R%GeFd`q$\"+ k)eZY\"Fd`qFj]q7%F_es7$$!+'Fd`q$\"+9Qn$4\"Fd`qFj ]q7%F]gs7$$!+4xJgiFd`q$\"+@?3E5Fd`qFj]q7%Fcgs7$$!+!GnKK'Fd`q$\"+#G?Wg* Fi]qFj]q7%Figs7$$!+PDb#Q'Fd`q$\"+g:wk*)Fi]qFj]q7%F_hs7$$!+rB'zV'Fd`q$ \"+XL')Q$)Fi]qFj]q7%Fehs7$$!+@HP*['Fd`q$\"+d\\$Qs(Fi]qFj]q7%F[is7$$!+R &Hn`'Fd`q$\"+Ge1y-!e'Fd`q$\"+e8K;lFi]qFj]q7%Fgis 7$$!+_nH>mFd`q$\"+U#p(>fFi]qFj]q7%F]js7$$!+u_eamFd`q$\"+HA'fK&Fi]qFj]q 7%Fcjs7$$!+WA&fo'Fd`q$\"+-!)zLZFi]qFj]q7%Fijs7$$!+*RfMr'Fd`q$\"+P,ZUTF i]qFj]q7%F_[t7$$!+5m;PnFd`q$\"+$R=9b$Fi]qFj]q7%Fe[t7$$!+4)Grv'Fd`q$\"+ ADGgHFi]qFj]q7%F[\\t7$$!+-TRtnFd`q$\"+y7')oBFi]qFj]q7%Fa\\t7$$!+[E+'y' Fd`q$\"+To2xT%\\=\"Fi]qFj]q7%F]]t 7$$!+fpO+oFd`q$\"+\\1WDfFgdqFj]q7%Fc]t7$$!+\"3f@!oFd`qF(Fj]q7%Fi]t7$Fd ]t$!+\\1WDfFgdqFj]q7%F]^t7$F^]t$!+>T%\\=\"Fi]qFj]q7%Fa^t7$Fh\\t$!+To2x fFi]qF j]q7%Fa`t7$Fhis$!+e8K;lFi]qFj]q7%Fe`t7$Fbis$!+Ge1Fd`qFj ]q7%F]et7$Ffbs$!+;+![.#Fd`qFj]q7%Faet7$F`bs$!+#pq^6#Fd`qFj]q7%Feet7$Fj as$!+n7d%>#Fd`qFj]q7%Fiet7$Fdas$!+sh(GF#Fd`qFj]q7%F]ft7$F^as$!+bJ**\\B Fd`qFj]q7%Faft7$Fh`s$!+@y&eU#Fd`qFj]q7%Feft7$Fb`s$!+'eG/]#Fd`qFj]q7%Fi ft7$F\\`s$!+N?otDFd`qFj]q7%F]gt7$Ff_s$!+v&3ck#Fd`qFj]q7%Fagt7$F`_s$!+! G3ir#Fd`qFj]q7%Fegt7$Fj^s$!+]n[&y#Fd`qFj]q7%Figt7$Fd^s$!+94X`GFd`qFj]q 7%F]ht7$F^^s$!+:Z5?HFd`qFj]q7%Faht7$Fh]s$!+DZW&)HFd`qFj]q7%Feht7$Fb]s$ !+ZeX\\IFd`qFj]q7%Fiht7$F\\]s$!+&Q2@6$Fd`qFj]q7%F]it7$Ff\\s$!+o)\\L<$F d`qFj]q7%Fait7$F`\\s$!+lI6LKFd`qFj]q7%Feit7$Fj[s$!+gbI\"H$Fd`qFj]q7%Fi it7$Fd[s$!+[h\"yM$Fd`qFj]q7%F]jt7$F^[s$!+,q^-MFd`qFj]q7%Fajt7$Fhjr$!+, $o_X$Fd`qFj]q7%Fejt7$Fbjr$!+qP#f]$Fd`qFj]q7%Fijt7$F\\jr$!+*ROVb$Fd`qFj ]q7%F][u7$Ffir$!+'oj.g$Fd`qFj]q7%Fa[u7$F`ir$!+()=(Qk$Fd`qFj]q7%Fe[u7$F jhr$!+i*QZo$Fd`qFj]q7%Fi[u7$Fdhr$!+$>cGs$Fd`qFj]q7%F]\\u7$F^hr$!+(fG\" ePFd`qFj]q7%Fa\\u7$Fhgr$!+'Ru/z$Fd`qFj]q7%Fe\\u7$Fbgr$!+5Q#)>QFd`qFj]q 7%Fi\\u7$F\\gr$!+![1RFd`qFj]q7%Fi]u7$Fder$!+#=:)>RFd` qFj]q7%F]^u7$F^er$!+z`=IRFd`qFj]q7%Fa^u7$Fhdr$!+/5FPRFd`qFj]q7%Fe^u7$F bdr$!+$=W5%RFd`qFj]q7%Fi^u7$F\\dr$!+:\"z9%RFd`qFj]q7%F]_u7$Ffcr$!+ZKbQ RFd`qFj]q7%Fa_u7$F`cr$!+.)\\A$RFd`qFj]q7%Fe_u7$Fjbr$!+0EcARFd`qFj]q7%F i_u7$Fdbr$!+D\\]4RFd`qFj]q7%F]`u7$F^br$!+Ag7$*QFd`qFj]q7%Fa`u7$Fhar$!+ xKatQFd`qFj]q7%Fe`u7$Fbar$!+:)35&QFd`qFj]q7%Fi`u7$Fj_r$!+931EQFd`qFj]q 7%F]au7$Fd_r$!+.Mc>QFd`qFj]q7%Faau7$F^_r$!+JI,8QFd`qFj]q7%Feau7$Fh^r$! +*3Zk!QFd`qFj]q7%Fiau7$Fb^r$!+)=8**z$Fd`qFj]q7%F]bu7$F\\^r$!+YEZ$z$Fd` qFj]q7%Fabu7$Ff]r$!+ca?(y$Fd`qFj]q7%Febu7$F`]r$!+hu@\"y$Fd`qFj]q7%Fibu 7$Fj\\r$!+74lvPFd`qFj]q7%F]cu7$Fd\\r$!+?-qqPFd`qFj]q7%Facu7$F^\\r$!+jc jmPFd`qFj]q7%Fecu7$Fh[r$!+]*QQw$Fd`qFj]q7%Ficu7$Fb[r$!+p[&Gw$Fd`qFj]q7 %F]du7$F\\[r$!+x%eWw$Fd`qFj]q7%Fadu7$Ffjq$!+IOrpPFd`qFj]q7%Fedu7$F`jq$ !+x^#*zPFd`qFj]q7%Fidu7$Fjiq$!+_:E'z$Fd`qFj]q7%F]eu7$Fdiq$!+)[8*=QFd`q Fj]q7%Faeu7$Fhhq$!+:HY6JFd`qFj]q7%Feeu7$Fbhq$!+*)G*[0$Fd`qFj]q7%Fieu7$ F\\hq$!++.v(*HFd`qFj]q7%F]fu7$Ffgq$!+$QN8%HFd`qFj]q7%Fafu7$F`gq$!+3&Gi )GFd`qFj]q7%Fefu7$Fjfq$!+I**fKGFd`qFj]q7%Fifu7$Fdfq$!+[dU!y#Fd`qFj]q7% F]gu7$F^fq$!+3EgHFFd`qFj]q7%Fagu7$Fheq$!+-@+!o#Fd`qFj]q7%Fegu7$Fbeq$!+ Y[\\JEFd`qFj]q7%Figu7$F\\eq$!+Z-'Re#Fd`qFj]q7%F]hu7$Fedq$!+\\(*GPDFd`q Fj]q7%Fahu7$F_dq$!+)>(Q\"\\#Fd`qFj]q7%Fehu7$Ficq$!+A!ohW#Fd`qFj]q7%Fih u7$Fccq$!+.!e:S#Fd`qFj]q7%F]iu7$$\"+!yo0M%F^cq$!+&)>\\dBFd`qFj]q7%Faiu 7$$\"+q^\"*oMF^cq$!+pE\"RJ#Fd`qFj]q7%Fgiu7$$\"+!>\"QaFF^cq$!+S&p2F#Fd` qFj]q7%F]ju7$F\\cq$!+Oz,GAFd`qFj]q7%Fcju7$Ffbq$!+9r>g?Fd`qFj]q7%Fgju7$ F_bq$!+!ySh*=Fd`qFj]q7%F[[v7$Fiaq$!+rReM&ou: Fd`qFj]q7%Fc[v7$F]aq$!+Ij(eT\"Fd`qFj]q7%Fg[v7$Fg`q$!+'>sxD\"Fd`qFj]q7% F[\\v7$F_`q$!+wF6+6Fd`qFj]q7%F_\\v7$Fi_q$!+*yXsU*Fi]qFj]q7%Fc\\v7$Fb_q $!+w1%\\&yFi]qFj]q7%Fg\\v7$F\\_q$!+07\\$G'Fi]qFj]q7%F[]v7$$!+v.\"ot\"F g^q$!+!4gCr%Fi]qFj]q7%F_]v7$F_^q$!+$)=gTJFi]qFj]q7%Fe]v7$$!+@8HGCFeq$! +8mzq:Fi]qFj]q7%Fi]vF'Fj]qFc]q7%Fghq7$$\"+_x?a9Fi]q$\"+)*)*GlJFd`q7$$ \"\"(!\"#$\"$d$Fh^v7%F`^v7$$\"+4Nx4=Fi]q$\"+.D#R@$Fd`qFe^v7%F\\_v7$$\" +L)f1>#Fi]q$\"+\"\\CfD$Fd`qFe^v7%Fb_v7$$\"+O!))=d#Fi]q$\"+vk[\"H$Fd`qF e^v7%Fh_v7$$\"+0o@PHFi]q$\"+(zw&zKFi]q$\"+`- 5[LFd`qFe^v7%Fd`v7$$\"+?_=(f$Fi]q$\"+q1_rLFd`qFe^v7%Fj`v7$$\"+()>9\"*Q Fi]q$\"+.J\"GR$Fd`qFe^v7%F`av7$$\"+dmCjTFi]q$\"+fB_7MFd`qFe^v7%Ffav7$$ \"+J;b:WFi]q$\"+%=I5V$Fd`qFe^v7%F\\bv7$$\"+I'4+l%Fi]q$\"+3pg[MFd`qFe^v 7%Fbbv7$$\"+cPPo[Fi]q$\"+3qWlMFd`qFe^v7%Fhbv7$$\"+y'z@2&Fi]q$\"+7Gp\"[ $Fd`qFe^v7%F^cv7$$\"+Q'fFE&Fi]q$\"+Y-X(\\$Fd`qFe^v7%Fdcv7$$\"+imETaFi] q$\"+*Q*z7NFd`qFe^v7%Fjcv7$$\"+l+q3cFi]q$\"+_;!y_$Fd`qFe^v7%F`dv7$$\"+ V)Gfw&Fi]q$\"+NZ]UNFd`qFe^v7%Ffdv7$$\"+Z>r8fFi]q$\"+=h%pb$Fd`qFe^v7%F \\ev7$$\"+ejfAkFi]q$\"++$3Eh$Fd`qFe^v7%Fbev7$$\"+mM#>#oFi]q$\"+#*[llOF d`qFe^v7%Fhev7$$\"+!RpE8(Fi]q$\"+K`i;PFd`qFe^v7%F^fv7$$\"+JZCptFi]q$\" +U*odw$Fd`qFe^v7%Fdfv7$$\"+0&e?a(Fi]q$\"+Xa>8QFd`qFe^v7%Fjfv7$$\"+$RP* ewFi]q$\"+(4T*eQFd`qFe^v7%F`gv7$$\"+i@'fs(Fi]q$\"+e0+.RFd`qFe^v7%Ffgv7 $$\"+cs)zu(Fi]q$\"+GLMXRFd`qFe^v7%F\\hv7$$\"+H!y*GxFi]q$\"+bE#f)RFd`qF e^v7%Fbhv7$$\"+qxBswFi]q$\"+![!oCSFd`qFe^v7%Fhhv7$$\"+'Hl0e(Fi]q$\"+2- bhSFd`qFe^v7%F^iv7$$\"+'=njX(Fi]q$\"+d\"ek4%Fd`qFe^v7%Fdiv7$$\"+2%RH)*y%Fi]q$\"+D;A;VFd`qFe^v7%F`]w7$$\"+KbV1WFi]q$\"+)4iBK%Fd`qFe^v 7%Ff]w7$$\"+FZ$\\+%Fi]q$\"+d%f[K%Fd`qFe^v7%F\\^w7$$\"+iwl&e$Fi]q$\"+&f :MK%Fd`qFe^v7%Fb^w7$$\"+`0q[JFi]q$\"+7&pwJ%Fd`qFe^v7%Fh^w7$$\"+4k#Qp#F i]q$\"+*purI%Fd`qFe^v7%F^_w7$$\"+OtH?AFi]q$\"+]_O\"H%Fd`qFe^v7%Fd_w7$$ \"+CrdEdRFgdq$\"+D+o5TFd`qFe^v7%F^bw7$$!+KKTZcFgd q$\"+![VP4%Fd`qFe^v7%Fdbw7$$!+JdD0uFgdq$\"+DLjvSFd`qFe^v7%Fjbw7$$!+%e5 YC*Fgdq$\"+w,CcSFd`qFe^v7%F`cw7$$!+zb6*)RFd`qF e^v7%Fbdw7$$!+'y'[*y\"Fi]q$\"+IHNjRFd`qFe^v7%Fhdw7$$!+t$Q)e?Fi]q$\"+q' )zNRFd`qFe^v7%F^ew7$$!+EfajBFi]q$\"+#eAm!RFd`qFe^v7%Fdew7$$!+-))G9FFi] q$\"+%o/k(QFd`qFe^v7%FjewF]iqFe^v7%FghqF]iqFe^v7%Faeu7$F^iq$!+5\\WYQFd `q7$Ff^v$!$d$Fh^v7%Fbfw7$F[fw$!+%o/k(QFd`qFefw7%Fifw7$Feew$!+#eAm!RFd` qFefw7%F]gw7$F_ew$!+q')zNRFd`qFefw7%Fagw7$Fidw$!+IHNjRFd`qFefw7%Fegw7$ Fcdw$!+>b6*)RFd`qFefw7%Figw7$F]dw$!+4368SFd`qFefw7%F]hw7$Fgcw$!+!pUa.% Fd`qFefw7%Fahw7$Facw$!+w,CcSFd`qFefw7%Fehw7$F[cw$!+DLjvSFd`qFefw7%Fihw 7$Febw$!+![VP4%Fd`qFefw7%F]iw7$F_bw$!+D+o5TFd`qFefw7%Faiw7$Fiaw$!+@+aE TFd`qFefw7%Feiw7$Fcaw$!+J!499%Fd`qFefw7%Fiiw7$F]aw$!+]BObTFd`qFefw7%F] jw7$$\"+e&4#4BFgdq$!+&Hm%oTFd`qFefw7%Fajw7$$\"+gQb#y$Fgdq$!+=%z2=%Fd`q Fefw7%Fgjw7$$\"+e5'oA&Fgdq$!+wNN#>%Fd`qFefw7%F][x7$Fg`w$!+m[B.UFd`qFef w7%Fc[x7$Fa`w$!+YGfSUFd`qFefw7%Fg[x7$F[`w$!+xJ]pUFd`qFefw7%F[\\x7$Fe_w $!+]_O\"H%Fd`qFefw7%F_\\x7$F__w$!+*purI%Fd`qFefw7%Fc\\x7$Fi^w$!+7&pwJ% Fd`qFefw7%Fg\\x7$Fc^w$!+&f:MK%Fd`qFefw7%F[]x7$$\"+GZ$\\+%Fi]q$!+d%f[K% Fd`qFefw7%F_]x7$Fg]w$!+)4iBK%Fd`qFefw7%Fe]x7$Fa]w$!+D;A;VFd`qFefw7%Fi] x7$F[]w$!+P))o1VFd`qFefw7%F]^x7$Fe\\w$!+]#yRH%Fd`qFefw7%Fa^x7$F_\\w$!+ $3w#yUFd`qFefw7%Fe^x7$Fi[w$!+ghufUFd`qFefw7%Fi^x7$Fc[w$!+PV`QUFd`qFefw 7%F]_x7$F][w$!+p=x9UFd`qFefw7%Fa_x7$Fgjv$!+*)zd)=%Fd`qFefw7%Fe_x7$Fajv $!+))=1gTFd`qFefw7%Fi_x7$F[jv$!+6UKHTFd`qFefw7%F]`x7$Feiv$!+d\"ek4%Fd` qFefw7%Fa`x7$F_iv$!+2-bhSFd`qFefw7%Fe`x7$Fihv$!+![!oCSFd`qFefw7%Fi`x7$ Fchv$!+bE#f)RFd`qFefw7%F]ax7$F]hv$!+GLMXRFd`qFefw7%Faax7$Fggv$!+e0+.RF d`qFefw7%Feax7$Fagv$!+(4T*eQFd`qFefw7%Fiax7$F[gv$!+Xa>8QFd`qFefw7%F]bx 7$Fefv$!+U*odw$Fd`qFefw7%Fabx7$F_fv$!+K`i;PFd`qFefw7%Febx7$Fiev$!+#*[l lOFd`qFefw7%Fibx7$Fcev$!++$3Eh$Fd`qFefw7%F]cx7$F]ev$!+=h%pb$Fd`qFefw7% Facx7$Fecv$!+Y-X(\\$Fd`qFefw7%Fecx7$F_cv$!+7Gp\"[$Fd`qFefw7%Ficx7$Fibv $!+3qWlMFd`qFefw7%F]dx7$Fcbv$!+3pg[MFd`qFefw7%Fadx7$F]bv$!+%=I5V$Fd`qF efw7%Fedx7$Fgav$!+fB_7MFd`qFefw7%Fidx7$Faav$!+.J\"GR$Fd`qFefw7%F]ex7$F [av$!+q1_rLFd`qFefw7%Faex7$Fe`v$!+`-5[LFd`qFefw7%Feex7$F_`v$!+(zw " 0 "" {MPLTEXT 1 0 157 "plots[display]([p_||(1..3),p||(1..2)],view=[-7.39..1.39,-4.79 ..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes= boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6--%'CURVESG6%7\\cl7$$\"\"!F)F(7$$!3;+++<8HGC!#F$\"31+++8 mzq:!#=7$$!38+++!3@Ua\"!#D$\"3w*****H)=gTJF07$$!35+++]/\"ot\"!#C$\"3@+ ++!4gCr%F07$$!3i*****H=4[a*F:$\"3C+++07\\$G'F07$$!3!)******o?&3^$!#B$ \"3c*****fnS\\&yF07$$!3=+++z&pY*)*FE$\"3a+++*yXsU*F07$$!3))*****4,x)zA !#A$\"3,+++wF6+6!#<7$$!3>+++bIg4WFP$\"3!******f>sxD\"FS7$$!3W******3'* R%4(FP$\"3!*******Hj(eT\"FS7$$!3i*****f()\\6u)FP$\"3!******f>&ou:FS7$$ !3F+++?QE9[FP$\"3&******4(ReMg?FS7$$\"3%)*****f4p(p@!#?$\"34+++Oz,GAFS7 $$\"3l******f[o-aF`p$\"3*)*****H+e:S#FS7$$\"3y*****48Qkp'F`p$\"3;+++A! ohW#FS7$$\"3Y*****p@/JF)F`p$\"3'******z>(Q\"\\#FS7$$\"3$*******obr>5!# >$\"3:+++\\(*GPDFS7$$\"33+++%*\\0b7Feq$\"3#)*****pCgRe#FS7$$\"3%****** p%e(Qa\"Feq$\"3$)*****f%[\\JEFS7$$\"3%)*****45P***=Feq$\"3++++-@+!o#FS 7$$\"3))*****R\\`9M#Feq$\"3*******zg-'HFFS7$$\"3)*******>\"4H*GFeq$\"3 ')*****zuD/y#FS7$$\"3/+++wfp(e$Feq$\"3@+++I**fKGFS7$$\"3?+++wE&=Z%Feq$ \"3')*****z]Gi)GFS7$$\"3,+++/^\"*3cFeq$\"3>+++$QN8%HFS7$$\"3Y*****RTvX 3(Feq$\"3')*******H]x*HFS7$$\"3)******z^Gb+*Feq$\"3++++*)G*[0$FS7$$\"3 '******4LTx9\"F0$\"3;+++:HY6JFS7$$\"3!******>v2UX\"F0$\"3;+++)*)*GlJFS 7$$\"3++++4Nx4=F0$\"3/+++.D#R@$FS7$$\"3!******H$)f1>#F0$\"3/+++\"\\CfD $FS7$$\"3;+++O!))=d#F0$\"37+++vk[\"H$FS7$$\"3E+++0o@PHF0$\"31+++(zw&zKF0$\"39+++`-5[LFS7$$\"3%)******>_=(f$F0$\"3y**** **p1_rLFS7$$\"3C+++()>9\"*QF0$\"3%******H58GR$FS7$$\"3')*****plYK;%F0$ \"3,+++fB_7MFS7$$\"3\")*****4j^bT%F0$\"3$******R=I5V$FS7$$\"3!*******H '4+l%F0$\"3#******z!pg[MFS7$$\"37+++cPPo[F0$\"3%******z+ZaY$FS7$$\"3Y* ****znz@2&F0$\"3#******>\"Gp\"[$FS7$$\"3o*****zjfFE&F0$\"35+++Y-X(\\$F S7$$\"3`+++imETaF0$\"3********)Q*z7NFS7$$\"3i*****\\1+(3cF0$\"3'****** >l,y_$FS7$$\"3)******H%)Gfw&F0$\"32+++NZ]UNFS7$$\"3\"******p%>r8fF0$\" 3%******z6Ypb$FS7$$\"31+++ejfAkF0$\"3A++++$3Eh$FS7$$\"3b+++mM#>#oF0$\" 3/+++#*[llOFS7$$\"3W+++!RpE8(F0$\"3++++K`i;PFS7$$\"3y*****4tW#ptF0$\"3 y*****>%*odw$FS7$$\"3(******\\]e?a(F0$\"3&)*****\\W&>8QFS7$$\"3U+++$RP *ewF0$\"3?+++(4T*eQFS7$$\"3j*****>;ifs(F0$\"3,+++e0+.RFS7$$\"3g*****fD ()zu(F0$\"3))*****zKV`%RFS7$$\"3')******G!y*GxF0$\"3)******\\lAf)RFS7$ $\"3'*******pxBswF0$\"3y******z/oCSFS7$$\"3G+++'Hl0e(F0$\"3!)*****p?]: 1%FS7$$\"3E+++'=njX(F0$\"3#)*****p:ek4%FS7$$\"3o*****pSRH)*y%F0$\"3%)*****\\i@iJ%FS7$$\"3#) *****>`NkS%F0$\"3:+++)4iBK%FS7$$\"3>+++FZ$\\+%F0$\"3j*****pXf[K%FS7$$ \"3C+++iwl&e$F0$\"3<+++&f:MK%FS7$$\"31+++`0q[JF0$\"3:+++7&pwJ%FS7$$\"3 C+++4k#Qp#F0$\"3%*******)purI%FS7$$\"3,+++OtH?AF0$\"37+++]_O\"H%FS7$$ \"3'******R7xls\"F0$\"3#)*****p<.&pUFS7$$\"3/+++CCx47F0$\"3?+++YGfSUFS 7$$\"3^*****Ri!*\\k'Feq$\"3&)*****f'[B.UFS7$$\"3^*****z%GaL!)F`p$\"3)) ******\\BObTFS7$$!3=+++lQo#R(F`p$\"3C+++J!499%FS7$$!3$******R$Q#RK#Feq $\"3G+++@+aETFS7$$!39+++.b>dRFeq$\"3u*****\\-!o5TFS7$$!3?+++KKTZcFeq$ \"3m******zMu$4%FS7$$!3H+++JdD0uFeq$\"33+++DLjvSFS7$$!33+++%e5YC*Feq$ \"3?+++w,CcSFS7$$!3-+++zeAm!RFS7$$!3++++-))G9FF0$\"3;+++%o/k(QFS7$$!3'******zAS; 7$F0$\"3A+++5\\WYQFS7$$!3;+++!y$z*e$F0$\"3>+++)[8*=QFS7$$!3:+++'4z&3TF 0$\"3#)*****>bhiz$FS7$$!35+++$4DUl%F0$\"3)******pgk*F0$\"35+++)=8**z$FS7$$!3,+++L\"4/+\"FS$\"3<+++*3Zk!QFS7$ $!32+++>XnN5FS$\"36+++JI,8QFS7$$!3%******>Fy+2\"FS$\"33+++.Mc>QFS7$$!3 \"******\\B'p.6FS$\"3()*****R\"31EQFS7$$!3/+++4\\fO6FS$\"3()*****RgvC$ QFS7$$!3-+++#)G$)o6FS$\"3/+++LSyQQFS7$$!3++++4?Y+7FS$\"38+++`p'\\%QFS7 $$!3(********RG:B\"FS$\"39+++:)35&QFS7$$!35+++n&H4N\"FS$\"3++++xKatQFS 7$$!3)*******RU(RY\"FS$\"37+++Ag7$*QFS7$$!3%******z9s?d\"FS$\"3!)***** \\#\\]4RFS7$$!33+++\"HEin\"FS$\"3)******\\giD#RFS7$$!3-+++UkFS$\" 3#******\\6z9%RFS7$$!3))*****z(3$e1#FS$\"3\")*****H=W5%RFS7$$!3))***** >4)\\e@FS$\"3&)*****R+rs$RFS7$$!3#*******31#)\\AFS$\"34+++z`=IRFS7$$!3 &******R!R**RBFS$\"3z*****>=:)>RFS7$$!3:+++Qd=HCFS$\"3))*****p0!>1RFS7 $$!3y******)*Ha0EFS$\"3/+++3ZIpQFS7 $$!3/+++\"*oE#p#FS$\"3?+++![QFS7$$!31+++6-4lGFS$\"3-+++'Ru/z$FS7$$!35+++&\\W5&HFS$\"3!)*****pfG \"ePFS7$$!3*)*****p'y\"o.$FS$\"3:+++$>cGs$FS7$$!35+++6^\\AJFS$\"3%**** **>'*QZo$FS7$$!34+++#**elM FS$\"3))******pP#f]$FS7$$!3!******fvp;b$FS$\"3#)*****4Io_X$FS7$$!3\")* ****R\\))yj$FS$\"35+++,q^-MFS7$$!3\")*****z$\\@CPFS$\"3;+++[h\"yM$FS7$ $!32+++^_f5QFS$\"33+++gbI\"H$FS7$$!3')*****p^bp*QFS$\"3'******\\18JB$F S7$$!3-++++g?$)RFS$\"3%)*****z')\\L<$FS7$$!35+++\\mCpSFS$\"3%)*****\\Q 2@6$FS7$$!35+++PK(\\:%FS$\"3%******p%eX\\IFS7$$!3q*****4Z$GSUFS$\"3!)* ****\\sWa)HFS7$$!3\"******H^#3DVFS$\"3!******\\r/,#HFS7$$!3R+++UvG4WFS $\"3;+++94X`GFS7$$!3'******f?JG\\%FS$\"3-+++]n[&y#FS7$$!3++++rRmvXFS$ \"3.+++!G3ir#FS7$$!3)******\\\\bxl%FS$\"3!)*****\\d3ck#FS7$$!32+++R_4R ZFS$\"3/+++N?otDFS7$$!3))*****4X#p>[FS$\"3&******feG/]#FS7$$!3c*****po v&**[FS$\"39+++@y&eU#FS7$$!3I+++.>zy\\FS$\"3%******\\:$**\\BFS7$$!3C++ +(30u0&FS$\"39+++sh(GF#FS7$$!3*******\\H%\\N^FS$\"39+++n7d%>#FS7$$!3e* ****R\"3:8_FS$\"3'******>pq^6#FS7$$!3c*****H1u/H&FS$\"3;+++;+![.#FS7$$ !3-+++*plvO&FS$\"31+++MQi`>FS7$$!3C+++s7_WaFS$\"3%******fI^=(=FS7$$!3* ******Ru=9_&FS$\"3-+++V1u*y\"FS7$$!3s******)R.$)f&FS$\"3,+++66g2v&FS$\"3%*******zYqW:FS7$ $!3(******>CR%GeFS$\"3++++k)eZY\"FS7$$!3I+++'FS$\"3'** ****R\"Qn$4\"FS7$$!3u******3xJgiFS$\"3!******4-#3E5FS7$$!3g******zsEBj FS$\"3?+++#G?Wg*F07$$!3e*****p`_DQ'FS$\"3o******f:wk*)F07$$!3C+++rB'zV 'FS$\"3[+++XL')Q$)F07$$!3=+++@HP*['FS$\"3y*****p&\\$Qs(F07$$!3U+++R&Hn `'FS$\"3()*****z#e1mFS$\"3K+++U#p(>fF07$$!3W+++u_eamFS$\"3r******GA'fK&F07$$!35+++WA&fo' FS$\"3y*****>+)zLZF07$$!3W+++*RfMr'FS$\"3++++P,ZUTF07$$!31+++5m;PnFS$ \"3%)*****HR=9b$F07$$!37+++4)Grv'FS$\"33+++ADGgHF07$$!39+++-TRtnFS$\"3 7+++y7')oBF07$$!3?+++[E+'y'FS$\"37+++To2x+)zLZF07$Fhdn$!3r******GA'fK&F07$Fc dn$!3K+++U#p(>fF07$F^dn$!37+++e8K;lF07$Ficn$!3()*****z#e1FS7$F_^n$!3;+++;+![.#FS7$Fj]n$!3'*** ***>pq^6#FS7$Fe]n$!39+++n7d%>#FS7$F`]n$!39+++sh(GF#FS7$F[]n$!3%****** \\:$**\\BFS7$Ff\\n$!39+++@y&eU#FS7$Fa\\n$!3&******feG/]#FS7$F\\\\n$!3/ +++N?otDFS7$Fg[n$!3!)*****\\d3ck#FS7$Fb[n$!3.+++!G3ir#FS7$F][n$!3-+++] n[&y#FS7$Fhjm$!3;+++94X`GFS7$Fcjm$!3!******\\r/,#HFS7$F^jm$!3!)*****\\ sWa)HFS7$Fiim$!3%******p%eX\\IFS7$Fdim$!3%)*****\\Q2@6$FS7$F_im$!3%)** ***z')\\L<$FS7$Fjhm$!3'******\\18JB$FS7$Fehm$!33+++gbI\"H$FS7$F`hm$!3; +++[h\"yM$FS7$F[hm$!35+++,q^-MFS7$Ffgm$!3#)*****4Io_X$FS7$Fagm$!3))*** ***pP#f]$FS7$F\\gm$!3')******)ROVb$FS7$Fgfm$!37+++'oj.g$FS7$Fbfm$!3)** ****p)=(Qk$FS7$F]fm$!3%******>'*QZo$FS7$Fhem$!3:+++$>cGs$FS7$Fcem$!3!) *****pfG\"ePFS7$F^em$!3-+++'Ru/z$FS7$Fidm$!3$*******4Q#)>QFS7$Fddm$!3? +++![1RFS7$F`cm$!3z*****>=:)>RFS7$F[cm$!34+++z`=IRFS7$Ffbm$!3&)*****R+rs$ RFS7$Fabm$!3\")*****H=W5%RFS7$F\\bm$!3#******\\6z9%RFS7$Fgam$!3?+++ZKb QRFS7$Fbam$!3$******H!)\\A$RFS7$F]am$!3)******\\giD#RFS7$Fh`m$!3!)**** *\\#\\]4RFS7$Fc`m$!37+++Ag7$*QFS7$F^`m$!3++++xKatQFS7$Fi_m$!39+++:)35& QFS7$Fe^m$!3()*****R\"31EQFS7$F`^m$!33+++.Mc>QFS7$F[^m$!36+++JI,8QFS7$ Ff]m$!3<+++*3Zk!QFS7$Fa]m$!35+++)=8**z$FS7$F\\]m$!37+++YEZ$z$FS7$Fg\\m $!39+++ca?(y$FS7$Fb\\m$!3y*****4Y<7y$FS7$F]\\m$!39+++74lvPFS7$Fh[m$!3; +++?-qqPFS7$Fc[m$!3?+++jcjmPFS7$F^[m$!3!)******\\*QQw$FS7$Fijl$!3))*** ***o[&Gw$FS7$Fdjl$!3!)*****pZeWw$FS7$F_jl$!3/+++IOrpPFS7$Fjil$!3)***** *pbhiz$FS7$F`il$!3>+++)[8*=QFS7$F[il$!3A+++5\\ WYQFS7$Ffhl$!3;+++%o/k(QFS7$Fahl$!3z*****>eAm!RFS7$F\\hl$!30+++q')zNRF S7$Fggl$!37+++IHNjRFS7$Fbgl$!3%)******=b6*)RFS7$F]gl$!3h******3368SFS7 $Fhfl$!3++++!pUa.%FS7$Fcfl$!3?+++w,CcSFS7$F^fl$!33+++DLjvSFS7$Fiel$!3m ******zMu$4%FS7$Fdel$!3u*****\\-!o5TFS7$F_el$!3G+++@+aETFS7$Fjdl$!3C++ +J!499%FS7$Fedl$!3))******\\BObTFS7$$\"3%******zb4#4BFeq$!3)******\\Hm %oTFS7$$\"3++++gQb#y$Feq$!3=+++=%z2=%FS7$$\"3&******z0hoA&Feq$!3M+++wN N#>%FS7$F`dl$!3&)*****f'[B.UFS7$F[dl$!3?+++YGfSUFS7$Ffcl$!3#)*****p<.& pUFS7$Facl$!37+++]_O\"H%FS7$F\\cl$!3%*******)purI%FS7$Fgbl$!3:+++7&pwJ %FS7$Fbbl$!3<+++&f:MK%FS7$$\"3G+++GZ$\\+%F0$!3j*****pXf[K%FS7$Fhal$!3: +++)4iBK%FS7$Fcal$!3%)*****\\i@iJ%FS7$F^al$!3F+++P))o1VFS7$Fi`l$!3K+++ ]#yRH%FS7$Fd`l$!3u*****H3w#yUFS7$F_`l$!3s******fhufUFS7$Fj_l$!3$****** pLM&QUFS7$Fe_l$!3y******o=x9UFS7$F`_l$!3m******))zd)=%FS7$F[_l$!3W+++) )=1gTFS7$Ff^l$!39+++6UKHTFS7$Fa^l$!3#)*****p:ek4%FS7$F\\^l$!3!)*****p? ]:1%FS7$Fg]l$!3y******z/oCSFS7$Fb]l$!3)******\\lAf)RFS7$F]]l$!3))***** zKV`%RFS7$Fh\\l$!3,+++e0+.RFS7$Fc\\l$!3?+++(4T*eQFS7$F^\\l$!3&)*****\\ W&>8QFS7$Fi[l$!3y*****>%*odw$FS7$Fd[l$!3++++K`i;PFS7$F_[l$!3/+++#*[llO FS7$Fjz$!3A++++$3Eh$FS7$Fez$!3%******z6Ypb$FS7$Fay$!35+++Y-X(\\$FS7$F \\y$!3#******>\"Gp\"[$FS7$Fgx$!3%******z+ZaY$FS7$Fbx$!3#******z!pg[MFS 7$F]x$!3$******R=I5V$FS7$Fhw$!3,+++fB_7MFS7$Fcw$!3%******H58GR$FS7$F^w $!3y******p1_rLFS7$Fiv$!39+++`-5[LFS7$Fdv$!31+++(zw+++$QN8%HFS7$Fgs$!3')*****z]Gi)GFS7$Fbs$!3@+++I**fKGFS7$F]s$!3')** ***zuD/y#FS7$Fhr$!3*******zg-'HFFS7$Fcr$!3++++-@+!o#FS7$F^r$!3$)*****f %[\\JEFS7$Fiq$!3#)*****pCgRe#FS7$Fcq$!3:+++\\(*GPDFS7$F^q$!3'******z>( Q\"\\#FS7$Fip$!3;+++A!ohW#FS7$Fdp$!3*)*****H+e:S#FS7$$\"3')******z(o0M %F`p$!3y*****\\)>\\dBFS7$$\"37+++q^\"*oMF`p$!3.+++pE\"RJ#FS7$$\"3))*** ****=\"QaFF`p$!3')******R&p2F#FS7$F^p$!34+++Oz,GAFS7$Fio$!3;+++9r>g?FS 7$Fco$!33+++!ySh*=FS7$F^o$!3&******4(ReM&ou:FS7$FZ $!3!*******Hj(eT\"FS7$FU$!3!******f>sxD\"FS7$FN$!3,+++wF6+6FS7$FI$!3a+ ++*yXsU*F07$FC$!3c*****fnS\\&yF07$F>$!3C+++07\\$G'F07$$!35+++v.\"ot\"F :$!3@+++!4gCr%F07$F2$!3w*****H)=gTJF07$$!3.+++@8HGCF-$!31+++8mzq:F0F'F *-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"#Fi\\qF(-%&STYLEG6#%%LINEG-%)POLYGON SG6acl7%F'7$$!+<8HGCFeq$\"+8mzq:!#57$$!#MFi\\qF(7%Fd]q7$$!+!3@Ua\"FS$ \"+$)=gTJFi]qFj]q7%F^^q7$$!+]/\"ot\"!#;$\"+!4gCr%Fi]qFj]q7%Fd^q7$$!+$= 4[a*Fg^q$\"+07\\$G'Fi]qFj]q7%F[_q7$$!+p?&3^$!#:$\"+w1%\\&yFi]qFj]q7%Fa _q7$$!+z&pY*)*Fd_q$\"+*yXsU*Fi]qFj]q7%Fh_q7$$!+6q()zA!#9$\"+wF6+6!\"*F j]q7%F^`q7$$!+bIg4WFa`q$\"+'>sxD\"Fd`qFj]q7%Ff`q7$$!+4'*R%4(Fa`q$\"+Ij (eT\"Fd`qFj]q7%F\\aq7$$!+w)\\6u)Fa`q$\"+'>&ou:Fd`qFj]q7%Fbaq7$$!+?QE9[ Fa`q$\"+rReMg?Fd`qFj]q7%Febq7$$\"+'4p(p@!#7$\"+Oz,GAFd`qFj]q7% F[cq7$$\"+g[o-aF^cq$\"+.!e:S#Fd`qFj]q7%Fbcq7$$\"+J\"Qkp'F^cq$\"+A!ohW# Fd`qFj]q7%Fhcq7$$\"+(Q\"\\#Fd`qFj]q7%F^dq7$$\"+pbr>5!# 6$\"+\\(*GPDFd`qFj]q7%Fddq7$$\"+%*\\0b7Fgdq$\"+Z-'Re#Fd`qFj]q7%F[eq7$$ \"+Ze(Qa\"Fgdq$\"+Y[\\JEFd`qFj]q7%Faeq7$$\"+,r$***=Fgdq$\"+-@+!o#Fd`qF j]q7%Fgeq7$$\"+%\\`9M#Fgdq$\"+3EgHFFd`qFj]q7%F]fq7$$\"+?\"4H*GFgdq$\"+ [dU!y#Fd`qFj]q7%Fcfq7$$\"+wfp(e$Fgdq$\"+I**fKGFd`qFj]q7%Fifq7$$\"+wE&= Z%Fgdq$\"+3&Gi)GFd`qFj]q7%F_gq7$$\"+/^\"*3cFgdq$\"+$QN8%HFd`qFj]q7%Feg q7$$\"+9ad%3(Fgdq$\"++.v(*HFd`qFj]q7%F[hq7$$\"+=&Gb+*Fgdq$\"+*)G*[0$Fd `qFj]q7%Fahq7$$\"+J8uZ6Fi]q$\"+:HY6JFd`qFj]q7%Fghq7$$!+G-k@JFi]q$\"+5 \\WYQFd`qFj]q7%F]iq7$$!+!y$z*e$Fi]q$\"+)[8*=QFd`qFj]q7%Fciq7$$!+'4z&3T Fi]q$\"+_:E'z$Fd`qFj]q7%Fiiq7$$!+$4DUl%Fi]q$\"+x^#*zPFd`qFj]q7%F_jq7$$ !+&>gk*Fi]q$\"+)=8**z$Fd`qF j]q7%Fa^r7$$!+L\"4/+\"Fd`q$\"+*3Zk!QFd`qFj]q7%Fg^r7$$!+>XnN5Fd`q$\"+JI ,8QFd`qFj]q7%F]_r7$$!+s#y+2\"Fd`q$\"+.Mc>QFd`qFj]q7%Fc_r7$$!+Nip.6Fd`q $\"+931EQFd`qFj]q7%Fi_r7$$!+4\\fO6Fd`q$\"+/cZKQFd`qFj]q7%F_`r7$$!+#)G$ )o6Fd`q$\"+LSyQQFd`qFj]q7%Fe`r7$$!+4?Y+7Fd`q$\"+`p'\\%QFd`qFj]q7%F[ar7 $$!++%G:B\"Fd`q$\"+:)35&QFd`qFj]q7%Faar7$$!+n&H4N\"Fd`q$\"+xKatQFd`qFj ]q7%Fgar7$$!+SU(RY\"Fd`q$\"+Ag7$*QFd`qFj]q7%F]br7$$!+[@2s:Fd`q$\"+D\\] 4RFd`qFj]q7%Fcbr7$$!+\"HEin\"Fd`q$\"+0EcARFd`qFj]q7%Fibr7$$!+UkFd`q$\"+:\"z9%RFd`qFj]q7%F[dr7$$!+y3$e1#Fd`q$\"+$=W5%RFd`qFj]q7%Fadr 7$$!+#4)\\e@Fd`q$\"+/5FPRFd`qFj]q7%Fgdr7$$!+41#)\\AFd`q$\"+z`=IRFd`qFj ]q7%F]er7$$!+/R**RBFd`q$\"+#=:)>RFd`qFj]q7%Fcer7$$!+Qd=HCFd`q$\"+d+>1R Fd`qFj]q7%Fier7$$!+**Ha0EFd`q$\" +3ZIpQFd`qFj]q7%Fefr7$$!+\"*oE#p#Fd`q$\"+![QFd`qFj]q7%Fagr7$$!+6-4lGFd`q$\"+'Ru/z$Fd`qFj]q7%Fggr7 $$!+&\\W5&HFd`q$\"+(fG\"ePFd`qFj]q7%F]hr7$$!+ny\"o.$Fd`q$\"+$>cGs$Fd`q Fj]q7%Fchr7$$!+6^\\AJFd`q$\"+i*QZo$Fd`qFj]q7%Fihr7$$!+[Fd`q$\"+'eG/]#Fd`qFj]q7%Fa`s7$$!+(ov&**[F d`q$\"+@y&eU#Fd`qFj]q7%Fg`s7$$!+.>zy\\Fd`q$\"+bJ**\\BFd`qFj]q7%F]as7$$ !+(30u0&Fd`q$\"+sh(GF#Fd`qFj]q7%Fcas7$$!+&H%\\N^Fd`q$\"+n7d%>#Fd`qFj]q 7%Fias7$$!+93:8_Fd`q$\"+#pq^6#Fd`qFj]q7%F_bs7$$!+jSZ!H&Fd`q$\"+;+![.#F d`qFj]q7%Febs7$$!+*plvO&Fd`q$\"+MQi`>Fd`qFj]q7%F[cs7$$!+s7_WaFd`q$\"+1 8&=(=Fd`qFj]q7%Facs7$$!+W(=9_&Fd`q$\"+V1u*y\"Fd`qFj]q7%Fgcs7$$!+*R.$)f &Fd`q$\"+66g2v&Fd`q$\"+!o/Za\"Fd`qFj]q7%Fids7$$!+U#R%GeFd`q$\"+k)eZY\"Fd`qFj] q7%F_es7$$!+'Fd`q$\"+9Qn$4\"Fd`qFj]q7%F]gs7$$!+4 xJgiFd`q$\"+@?3E5Fd`qFj]q7%Fcgs7$$!+!GnKK'Fd`q$\"+#G?Wg*Fi]qFj]q7%Figs 7$$!+PDb#Q'Fd`q$\"+g:wk*)Fi]qFj]q7%F_hs7$$!+rB'zV'Fd`q$\"+XL')Q$)Fi]qF j]q7%Fehs7$$!+@HP*['Fd`q$\"+d\\$Qs(Fi]qFj]q7%F[is7$$!+R&Hn`'Fd`q$\"+Ge 1y-!e'Fd`q$\"+e8K;lFi]qFj]q7%Fgis7$$!+_nH>mFd`q$ \"+U#p(>fFi]qFj]q7%F]js7$$!+u_eamFd`q$\"+HA'fK&Fi]qFj]q7%Fcjs7$$!+WA&f o'Fd`q$\"+-!)zLZFi]qFj]q7%Fijs7$$!+*RfMr'Fd`q$\"+P,ZUTFi]qFj]q7%F_[t7$ $!+5m;PnFd`q$\"+$R=9b$Fi]qFj]q7%Fe[t7$$!+4)Grv'Fd`q$\"+ADGgHFi]qFj]q7% F[\\t7$$!+-TRtnFd`q$\"+y7')oBFi]qFj]q7%Fa\\t7$$!+[E+'y'Fd`q$\"+To2xT%\\=\"Fi]qFj]q7%F]]t7$$!+fpO+oFd`q$ \"+\\1WDfFgdqFj]q7%Fc]t7$$!+\"3f@!oFd`qF(Fj]q7%Fi]t7$Fd]t$!+\\1WDfFgdq Fj]q7%F]^t7$F^]t$!+>T%\\=\"Fi]qFj]q7%Fa^t7$Fh\\t$!+To2xfFi]qFj]q7%Fa`t7$Fhis $!+e8K;lFi]qFj]q7%Fe`t7$Fbis$!+Ge1Fd`qFj]q7%F]et7$Ffbs$ !+;+![.#Fd`qFj]q7%Faet7$F`bs$!+#pq^6#Fd`qFj]q7%Feet7$Fjas$!+n7d%>#Fd`q Fj]q7%Fiet7$Fdas$!+sh(GF#Fd`qFj]q7%F]ft7$F^as$!+bJ**\\BFd`qFj]q7%Faft7 $Fh`s$!+@y&eU#Fd`qFj]q7%Feft7$Fb`s$!+'eG/]#Fd`qFj]q7%Fift7$F\\`s$!+N?o tDFd`qFj]q7%F]gt7$Ff_s$!+v&3ck#Fd`qFj]q7%Fagt7$F`_s$!+!G3ir#Fd`qFj]q7% Fegt7$Fj^s$!+]n[&y#Fd`qFj]q7%Figt7$Fd^s$!+94X`GFd`qFj]q7%F]ht7$F^^s$!+ :Z5?HFd`qFj]q7%Faht7$Fh]s$!+DZW&)HFd`qFj]q7%Feht7$Fb]s$!+ZeX\\IFd`qFj] q7%Fiht7$F\\]s$!+&Q2@6$Fd`qFj]q7%F]it7$Ff\\s$!+o)\\L<$Fd`qFj]q7%Fait7$ F`\\s$!+lI6LKFd`qFj]q7%Feit7$Fj[s$!+gbI\"H$Fd`qFj]q7%Fiit7$Fd[s$!+[h\" yM$Fd`qFj]q7%F]jt7$F^[s$!+,q^-MFd`qFj]q7%Fajt7$Fhjr$!+,$o_X$Fd`qFj]q7% Fejt7$Fbjr$!+qP#f]$Fd`qFj]q7%Fijt7$F\\jr$!+*ROVb$Fd`qFj]q7%F][u7$Ffir$ !+'oj.g$Fd`qFj]q7%Fa[u7$F`ir$!+()=(Qk$Fd`qFj]q7%Fe[u7$Fjhr$!+i*QZo$Fd` qFj]q7%Fi[u7$Fdhr$!+$>cGs$Fd`qFj]q7%F]\\u7$F^hr$!+(fG\"ePFd`qFj]q7%Fa \\u7$Fhgr$!+'Ru/z$Fd`qFj]q7%Fe\\u7$Fbgr$!+5Q#)>QFd`qFj]q7%Fi\\u7$F\\gr $!+![1RFd`qFj]q7%Fi]u7$Fder$!+#=:)>RFd`qFj]q7%F]^u7$F ^er$!+z`=IRFd`qFj]q7%Fa^u7$Fhdr$!+/5FPRFd`qFj]q7%Fe^u7$Fbdr$!+$=W5%RFd `qFj]q7%Fi^u7$F\\dr$!+:\"z9%RFd`qFj]q7%F]_u7$Ffcr$!+ZKbQRFd`qFj]q7%Fa_ u7$F`cr$!+.)\\A$RFd`qFj]q7%Fe_u7$Fjbr$!+0EcARFd`qFj]q7%Fi_u7$Fdbr$!+D \\]4RFd`qFj]q7%F]`u7$F^br$!+Ag7$*QFd`qFj]q7%Fa`u7$Fhar$!+xKatQFd`qFj]q 7%Fe`u7$Fbar$!+:)35&QFd`qFj]q7%Fi`u7$Fj_r$!+931EQFd`qFj]q7%F]au7$Fd_r$ !+.Mc>QFd`qFj]q7%Faau7$F^_r$!+JI,8QFd`qFj]q7%Feau7$Fh^r$!+*3Zk!QFd`qFj ]q7%Fiau7$Fb^r$!+)=8**z$Fd`qFj]q7%F]bu7$F\\^r$!+YEZ$z$Fd`qFj]q7%Fabu7$ Ff]r$!+ca?(y$Fd`qFj]q7%Febu7$F`]r$!+hu@\"y$Fd`qFj]q7%Fibu7$Fj\\r$!+74l vPFd`qFj]q7%F]cu7$Fd\\r$!+?-qqPFd`qFj]q7%Facu7$F^\\r$!+jcjmPFd`qFj]q7% Fecu7$Fh[r$!+]*QQw$Fd`qFj]q7%Ficu7$Fb[r$!+p[&Gw$Fd`qFj]q7%F]du7$F\\[r$ !+x%eWw$Fd`qFj]q7%Fadu7$Ffjq$!+IOrpPFd`qFj]q7%Fedu7$F`jq$!+x^#*zPFd`qF j]q7%Fidu7$Fjiq$!+_:E'z$Fd`qFj]q7%F]eu7$Fdiq$!+)[8*=QFd`qFj]q7%Faeu7$F hhq$!+:HY6JFd`qFj]q7%Feeu7$Fbhq$!+*)G*[0$Fd`qFj]q7%Fieu7$F\\hq$!++.v(* HFd`qFj]q7%F]fu7$Ffgq$!+$QN8%HFd`qFj]q7%Fafu7$F`gq$!+3&Gi)GFd`qFj]q7%F efu7$Fjfq$!+I**fKGFd`qFj]q7%Fifu7$Fdfq$!+[dU!y#Fd`qFj]q7%F]gu7$F^fq$!+ 3EgHFFd`qFj]q7%Fagu7$Fheq$!+-@+!o#Fd`qFj]q7%Fegu7$Fbeq$!+Y[\\JEFd`qFj] q7%Figu7$F\\eq$!+Z-'Re#Fd`qFj]q7%F]hu7$Fedq$!+\\(*GPDFd`qFj]q7%Fahu7$F _dq$!+)>(Q\"\\#Fd`qFj]q7%Fehu7$Ficq$!+A!ohW#Fd`qFj]q7%Fihu7$Fccq$!+.!e :S#Fd`qFj]q7%F]iu7$$\"+!yo0M%F^cq$!+&)>\\dBFd`qFj]q7%Faiu7$$\"+q^\"*oM F^cq$!+pE\"RJ#Fd`qFj]q7%Fgiu7$$\"+!>\"QaFF^cq$!+S&p2F#Fd`qFj]q7%F]ju7$ F\\cq$!+Oz,GAFd`qFj]q7%Fcju7$Ffbq$!+9r>g?Fd`qFj]q7%Fgju7$F_bq$!+!ySh*= Fd`qFj]q7%F[[v7$Fiaq$!+rReM&ou:Fd`qFj]q7%Fc[ v7$F]aq$!+Ij(eT\"Fd`qFj]q7%Fg[v7$Fg`q$!+'>sxD\"Fd`qFj]q7%F[\\v7$F_`q$! +wF6+6Fd`qFj]q7%F_\\v7$Fi_q$!+*yXsU*Fi]qFj]q7%Fc\\v7$Fb_q$!+w1%\\&yFi] qFj]q7%Fg\\v7$F\\_q$!+07\\$G'Fi]qFj]q7%F[]v7$$!+v.\"ot\"Fg^q$!+!4gCr%F i]qFj]q7%F_]v7$F_^q$!+$)=gTJFi]qFj]q7%Fe]v7$$!+@8HGCFeq$!+8mzq:Fi]qFj] q7%Fi]vF'Fj]qFc]q7%Fghq7$$\"+_x?a9Fi]q$\"+)*)*GlJFd`q7$$\"\"(!\"#$\"$d $Fh^v7%F`^v7$$\"+4Nx4=Fi]q$\"+.D#R@$Fd`qFe^v7%F\\_v7$$\"+L)f1>#Fi]q$\" +\"\\CfD$Fd`qFe^v7%Fb_v7$$\"+O!))=d#Fi]q$\"+vk[\"H$Fd`qFe^v7%Fh_v7$$\" +0o@PHFi]q$\"+(zw&zKFi]q$\"+`-5[LFd`qFe^v7%F d`v7$$\"+?_=(f$Fi]q$\"+q1_rLFd`qFe^v7%Fj`v7$$\"+()>9\"*QFi]q$\"+.J\"GR $Fd`qFe^v7%F`av7$$\"+dmCjTFi]q$\"+fB_7MFd`qFe^v7%Ffav7$$\"+J;b:WFi]q$ \"+%=I5V$Fd`qFe^v7%F\\bv7$$\"+I'4+l%Fi]q$\"+3pg[MFd`qFe^v7%Fbbv7$$\"+c PPo[Fi]q$\"+3qWlMFd`qFe^v7%Fhbv7$$\"+y'z@2&Fi]q$\"+7Gp\"[$Fd`qFe^v7%F^ cv7$$\"+Q'fFE&Fi]q$\"+Y-X(\\$Fd`qFe^v7%Fdcv7$$\"+imETaFi]q$\"+*Q*z7NFd `qFe^v7%Fjcv7$$\"+l+q3cFi]q$\"+_;!y_$Fd`qFe^v7%F`dv7$$\"+V)Gfw&Fi]q$\" +NZ]UNFd`qFe^v7%Ffdv7$$\"+Z>r8fFi]q$\"+=h%pb$Fd`qFe^v7%F\\ev7$$\"+ejfA kFi]q$\"++$3Eh$Fd`qFe^v7%Fbev7$$\"+mM#>#oFi]q$\"+#*[llOFd`qFe^v7%Fhev7 $$\"+!RpE8(Fi]q$\"+K`i;PFd`qFe^v7%F^fv7$$\"+JZCptFi]q$\"+U*odw$Fd`qFe^ v7%Fdfv7$$\"+0&e?a(Fi]q$\"+Xa>8QFd`qFe^v7%Fjfv7$$\"+$RP*ewFi]q$\"+(4T* eQFd`qFe^v7%F`gv7$$\"+i@'fs(Fi]q$\"+e0+.RFd`qFe^v7%Ffgv7$$\"+cs)zu(Fi] q$\"+GLMXRFd`qFe^v7%F\\hv7$$\"+H!y*GxFi]q$\"+bE#f)RFd`qFe^v7%Fbhv7$$\" +qxBswFi]q$\"+![!oCSFd`qFe^v7%Fhhv7$$\"+'Hl0e(Fi]q$\"+2-bhSFd`qFe^v7%F ^iv7$$\"+'=njX(Fi]q$\"+d\"ek4%Fd`qFe^v7%Fdiv7$$\"+2%RH)*y%Fi]q$ \"+D;A;VFd`qFe^v7%F`]w7$$\"+KbV1WFi]q$\"+)4iBK%Fd`qFe^v7%Ff]w7$$\"+FZ$ \\+%Fi]q$\"+d%f[K%Fd`qFe^v7%F\\^w7$$\"+iwl&e$Fi]q$\"+&f:MK%Fd`qFe^v7%F b^w7$$\"+`0q[JFi]q$\"+7&pwJ%Fd`qFe^v7%Fh^w7$$\"+4k#Qp#Fi]q$\"+*purI%Fd `qFe^v7%F^_w7$$\"+OtH?AFi]q$\"+]_O\"H%Fd`qFe^v7%Fd_w7$$\"+CrdEdRFgdq$\"+D+o5TFd`qFe^v7%F^bw7$$!+KKTZcFgdq$\"+![VP4%Fd`q Fe^v7%Fdbw7$$!+JdD0uFgdq$\"+DLjvSFd`qFe^v7%Fjbw7$$!+%e5YC*Fgdq$\"+w,Cc SFd`qFe^v7%F`cw7$$!+zb6*)RFd`qFe^v7%Fbdw7$$!+' y'[*y\"Fi]q$\"+IHNjRFd`qFe^v7%Fhdw7$$!+t$Q)e?Fi]q$\"+q')zNRFd`qFe^v7%F ^ew7$$!+EfajBFi]q$\"+#eAm!RFd`qFe^v7%Fdew7$$!+-))G9FFi]q$\"+%o/k(QFd`q Fe^v7%FjewF]iqFe^v7%FghqF]iqFe^v7%Faeu7$F^iq$!+5\\WYQFd`q7$Ff^v$!$d$Fh ^v7%Fbfw7$F[fw$!+%o/k(QFd`qFefw7%Fifw7$Feew$!+#eAm!RFd`qFefw7%F]gw7$F_ ew$!+q')zNRFd`qFefw7%Fagw7$Fidw$!+IHNjRFd`qFefw7%Fegw7$Fcdw$!+>b6*)RFd `qFefw7%Figw7$F]dw$!+4368SFd`qFefw7%F]hw7$Fgcw$!+!pUa.%Fd`qFefw7%Fahw7 $Facw$!+w,CcSFd`qFefw7%Fehw7$F[cw$!+DLjvSFd`qFefw7%Fihw7$Febw$!+![VP4% Fd`qFefw7%F]iw7$F_bw$!+D+o5TFd`qFefw7%Faiw7$Fiaw$!+@+aETFd`qFefw7%Feiw 7$Fcaw$!+J!499%Fd`qFefw7%Fiiw7$F]aw$!+]BObTFd`qFefw7%F]jw7$$\"+e&4#4BF gdq$!+&Hm%oTFd`qFefw7%Fajw7$$\"+gQb#y$Fgdq$!+=%z2=%Fd`qFefw7%Fgjw7$$\" +e5'oA&Fgdq$!+wNN#>%Fd`qFefw7%F][x7$Fg`w$!+m[B.UFd`qFefw7%Fc[x7$Fa`w$! +YGfSUFd`qFefw7%Fg[x7$F[`w$!+xJ]pUFd`qFefw7%F[\\x7$Fe_w$!+]_O\"H%Fd`qF efw7%F_\\x7$F__w$!+*purI%Fd`qFefw7%Fc\\x7$Fi^w$!+7&pwJ%Fd`qFefw7%Fg\\x 7$Fc^w$!+&f:MK%Fd`qFefw7%F[]x7$$\"+GZ$\\+%Fi]q$!+d%f[K%Fd`qFefw7%F_]x7 $Fg]w$!+)4iBK%Fd`qFefw7%Fe]x7$Fa]w$!+D;A;VFd`qFefw7%Fi]x7$F[]w$!+P))o1 VFd`qFefw7%F]^x7$Fe\\w$!+]#yRH%Fd`qFefw7%Fa^x7$F_\\w$!+$3w#yUFd`qFefw7 %Fe^x7$Fi[w$!+ghufUFd`qFefw7%Fi^x7$Fc[w$!+PV`QUFd`qFefw7%F]_x7$F][w$!+ p=x9UFd`qFefw7%Fa_x7$Fgjv$!+*)zd)=%Fd`qFefw7%Fe_x7$Fajv$!+))=1gTFd`qFe fw7%Fi_x7$F[jv$!+6UKHTFd`qFefw7%F]`x7$Feiv$!+d\"ek4%Fd`qFefw7%Fa`x7$F_ iv$!+2-bhSFd`qFefw7%Fe`x7$Fihv$!+![!oCSFd`qFefw7%Fi`x7$Fchv$!+bE#f)RFd `qFefw7%F]ax7$F]hv$!+GLMXRFd`qFefw7%Faax7$Fggv$!+e0+.RFd`qFefw7%Feax7$ Fagv$!+(4T*eQFd`qFefw7%Fiax7$F[gv$!+Xa>8QFd`qFefw7%F]bx7$Fefv$!+U*odw$ Fd`qFefw7%Fabx7$F_fv$!+K`i;PFd`qFefw7%Febx7$Fiev$!+#*[llOFd`qFefw7%Fib x7$Fcev$!++$3Eh$Fd`qFefw7%F]cx7$F]ev$!+=h%pb$Fd`qFefw7%Facx7$Fecv$!+Y- X(\\$Fd`qFefw7%Fecx7$F_cv$!+7Gp\"[$Fd`qFefw7%Ficx7$Fibv$!+3qWlMFd`qFef w7%F]dx7$Fcbv$!+3pg[MFd`qFefw7%Fadx7$F]bv$!+%=I5V$Fd`qFefw7%Fedx7$Fgav $!+fB_7MFd`qFefw7%Fidx7$Faav$!+.J\"GR$Fd`qFefw7%F]ex7$F[av$!+q1_rLFd`q Fefw7%Faex7$Fe`v$!+`-5[LFd`qFefw7%Feex7$F_`v$!+(zwZ5F07$$!3%)******)R6\"pU!#E$\"3'*******Hjzq:F07$$!3! ******RX4JR#F4$\"37+++H^R%4#F07$$!3^*****H%3s,\"*F4$\"35+++1S*zh#F07$$ !3>+++ZP!yq#F:$\"38+++7JfTJF07$$!3_*****4j)Q)z'F:$\"3E+++wF>lOF07$$!3! ******4our]\"FE$\"3A+++KOz)=%F07$$!3.+++e;$z.$FE$\"31+++)y'R7ZF07$$!3C +++lo]zcFE$\"3N+++iS+O_F07$$!3o++++kN*)**FE$\"3]*****fE='fdF07$$!3%)** ****[oQq;FP$\"3[+++wNC$G'F07$$!37+++r'fnn#FP$\"3/+++jf)o!oF07$$!3u**** *pQ4i8%FP$\"30+++;PbItF07$$!3#******zyaL>'FP$\"3K+++]!eU&yF07$$!3_**** *zYR=-*FP$\"3y*****f!Q,y$)F07$$!3#******\\sgEG\"Feo$\"3X+++]-%=!*)F07$ $!33+++$)3e%y\"Feo$\"3u*****H(>wD%*F07$$!3&******z*eANCFeo$\"3Q+++5*4) \\**F07$$!3%)*****fvOaE$Feo$\"35+++YASZ5FS7$$!31+++&GZ'4VFeo$\"3)***** *HnW)*4\"FS7$$!3v*****p<^eg&Feo$\"3#******H*RJ_6FS7$$!3=+++Go\\&>(Feo$ \"3)******z3<[?\"FS7$$!3)******\\s/L7*Feo$\"37+++s@Od7FS7$$!3&******pD +P9\"F`p$\"33+++t*e*48FS7$$!35+++etp=9F`p$\"3'******H%*=EO\"FS7$$!31++ +58eU)eDF`p$\"35+++$GB6_\"FS7$$!3A+++u#eA1$F`p$\"37+++QO>u:FS7$$!3\")** ****3**eOOF`p$\"3-++++&>ui\"FS7$$!3S+++X:9(G%F`p$\"35+++x(G3o\"FS7$$!3 J+++*\\_*=]F`p$\"31+++:OXM&>FS7$$!33+++N&3K+\"Feq$\"3\")*****4 uGt+#FS7$$!3%******>zK=8\"Feq$\"3!)*****HXjK1#FS7$$!31+++7Fcp7Feq$\"3! ******>#o$)>@FS7$$!3#******RL8dT\"Feq$\"3=+++v\"er<#FS7$$!33+++#Gi\"p: Feq$\"3&******Rve`B#FS7$$!3:+++L\\=G+++0,]:EFS7$$!3\"******>bjk^#Feq$\"37+++X*Gyo#FS7$$!3 =+++sTL&[#Feq$\"3++++#=R\\w#FS7$$!35+++tv.%G#Feq$\"3=+++j#o([GFS7$$!3' ******H@$GDz87I\"Feq$\"3#*******fBi%)HFS7$$!3&******H5'z95Feq$\"3;+++!oj n+$FS7$$!3U+++9k;)e'F`p$\"30+++Pu')HIFS7$$!3++++]uP+@F`p$\"33+++M*fS0$ FS7$$\"3)*******yi\\eOF`p$\"3!)*****RHl%zIFS7$$\"31+++1gI?6Feq$\"3\"** ****H]gh5$FS7$$\"3()*****zL1%H@Feq$\"3:+++=M0MJFS7$$\"37+++f)QX\\$Feq$ \"3-+++&[+E;$FS7$$\"3J+++_!)*oI&Feq$\"3)******4'fP!>$FS7$$\"3k*****>qu ga(Feq$\"3%)*****>UD^@$FS7$$\"31+++G[;-5F0$\"3>+++8?=NKFS7$$\"3!****** pw31D\"F0$\"3#)******pak]KFS7$$\"3/+++$>Hp[\"F0$\"3%******z0EEE$FS7$$ \"3))*****H<>rq\"F0$\"3;+++Ez@sKFS7$$\"38+++\")4O6>F0$\"3;+++-x>!G$FS7 $$\"30+++ni=,@F0$\"3++++'HqqG$FS7$$\"3))*****z:^$yAF0$\"37+++/9;$H$FS7 $$\"3#******>&4ZWCF0$\"3#******fA&o)H$FS7$$\"3%*******R&=3:$F0$\"32+++ I\"*f@LFS7$$\"3B+++DEy;PF0$\"3(******fi<4M$FS7$$\"37+++*emC>%F0$\"3'** ****f-;)eLFS7$$\"3E++++t!Rg%F0$\"3%)******p2.wLFS7$$\"33+++sM]m\\F0$\" 3=+++i-(GR$FS7$$\"3[+++8*>-H&F0$\"3=+++K1[4MFS7$$\"3%)******3[$>e&F0$ \"3%)******=j$fU$FS7$$\"3I+++!GIm%eF0$\"32+++1mFUMFS7$$\"3y******>A0)3 'F0$\"3#*******4A_eMFS7$$\"3[+++b>54jF0$\"3%)*****\\\\$ouMFS7$$\"3Q+++ @<37lF0$\"3))******>[w!\\$FS7$$\"3M+++ra&))p'F0$\"35+++kpw1NFS7$$\"3'* *****H9e4(oF0$\"3/+++E&)oANFS7$$\"3!)*****RAs'HqF0$\"3/+++yn_QNFS7$$\" 3s*****\\#G3wrF0$\"3%)******z\"yUb$FS7$$\"3[*****4@=6J(F0$\"3A+++0(Q*p NFS7$$\"39+++5-eNuF0$\"3-+++^S]&e$FS7$$\"3y*****>Wn,b(F0$\"3/+++n(p4g$ FS7$$\"3'******HK$\\bwF0$\"3.+++Z8L;OFS7$$\"3[+++E-5_xF0$\"31+++yUeJOF S7$$\"3a+++_2ZSyF0$\"35+++ySsYOFS7$$\"38+++\"zO5#zF0$\"3!******fJY\")F0$\"3;+++HV11PFS7$$\"3a*****H@(\\s\")F0$\"3!******z Lt0s$FS7$$\"34+++xP>>#)F0$\"3#)*****HkV\\t$FS7$$\"3s*****p^Y*f#)F0$\"3 9+++;7<\\PFS7$$\"3[*****\\'4)\\H)F0$\"3%********3_Kw$FS7$$\"3^*****Rp0 XK)F0$\"33+++_B=xPFS7$$\"3s*****f;9([$)F0$\"3y******Q\"e4z$FS7$$\"3W++ +Ojyn$)F0$\"32+++Jcd/QFS7$$\"3-+++X+*=Q)F0$\"3@+++Y5.=QFS7$$\"3W+++HA= \"R)F0$\"30+++O1KJQFS7$$\"3%)*****>)*4eR)F0$\"3$)*****RoSW%QFS7$$\"3'* *****>\\6fR)F0$\"3)******>\\(QdQFS7$$\"3i******foh\"R)F0$\"3++++$Qd,(Q FS7$$\"3i*****>$)[IQ)F0$\"3++++#pYF)QFS7$$\"3q*****\\WB.P)F0$\"3/+++l< :&*QFS7$$\"3a+++-1b`$)F0$\"3()******[*ot!RFS7$$\"3/+++!fMGL)F0$\"3=+++ #f%R>RFS7$$\"3/+++QXF3$)F0$\"3#******p.D7$RFS7$$\"3%)*****R$['*z#)F0$ \"3\")*****fhcG%RFS7$$\"3C+++(\\&*zC)F0$\"3%)*****pk&GaRFS7$$\"3V+++9D X7#)F0$\"3')******H%3b'RFS7$$\"3o*****H9=M<)F0$\"3A+++R7_wRFS7$$\"3^++ +#>r48)F0$\"3)******>K?t)RFS7$$\"3;+++ms=&3)F0$\"3y*****>*=!z*RFS7$$\" 3#**********Qh.)F0$\"3!******\\7i#3SFS7$$\"3q*****4E'*Q)zF0$\"3s***** \\:(R=SFS7$$\"3')*****\\ME&GzF0$\"3;+++kIIGSFS7$$\"3W+++VT4qyF0$\"3#)* ****\\)e(z.%FS7$$\"3_+++2Bm3yF0$\"3?+++*e6u/%FS7$$\"38+++$R\"HWxF0$\"3 9+++$31m0%FS7$$\"3')*****>**Rqn(F0$\"3s*****H?bb1%FS7$$\"3]******f['pg (F0$\"3q*****pqaU2%FS7$$\"3A+++E57MvF0$\"3I+++s-q#3%FS7$$\"39+++)*=ceu F0$\"3/+++zu)34%FS7$$\"3')*****zNR.Q(F0$\"3=+++8=\"))4%FS7$$\"3G+++cQ] *H(F0$\"3Q+++a'ok5%FS7$$\"3w*****>[/h@(F0$\"3?+++jK&Q6%FS7$$\"3*)***** 40*=IrF0$\"3!)******y2'47%FS7$$\"3y*****p;/=/(F0$\"3w*****p?'yFTFS7$$ \"3M+++)G&*4&pF0$\"3E+++2WKMTFS7$$\"3E+++'y1y&oF0$\"32+++$3q09%FS7$$\" 3C+++'*>GinF0$\"33+++rx^YTFS7$$\"3u*****HFjWm'F0$\"3S+++E=;_TFS7$$\"3G +++I?RklF0$\"3++++2k\\dTFS7$$\"3))******z(3@Y'F0$\"3++++ia^iTFS7$$\"3u *****R<`wN'F0$\"3U+++7F@nTFS7$$\"3-+++LS1^iF0$\"3Y+++J;erTFS7$$\"3l*** **4PzB9'F0$\"3;+++DahvTFS7$$\"36+++GkjJgF0$\"35+++;qIzTFS7$$\"3!)***** >or)=fF0$\"3=+++2!\\E=%FS7$$\"3'******p'37/eF0$\"3G+++nOj&=%FS7$$\"35+ ++&)*=uo&F0$\"3e*****H*GD)=%FS7$$\"3_+++5.!)obF0$\"3y*****>=)\\!>%FS7$ $\"3Y+++\"R)H[aF0$\"3!******4fgB>%FS7$$\"3d*****\\/YfK&F0$\"3q*****HqI Q>%FS7$$\"3_*****zMv%FS7$$\"3!******Hh%FS7$$\"3?+++nL5[\\F0$\"3M+++?\\y&>%FS7$$\"3s*****fJi'=[F0$\" 3A+++:/e&>%FS7$$\"3%******HHBvo%F0$\"37+++$eF\\>%FS7$$\"3-+++0UraXF0$ \"3#******H)H\"Q>%FS7$$\"30+++c>E?WF0$\"3M+++RAA#>%FS7$$\"3'******\\N# >%G%F0$\"3k*****p$*R,>%FS7$$\"3))*****f+Il9%F0$\"3U+++-&\\v=%FS7$$\"36 +++m\")H2SF0$\"3;+++jIV%=%FS7$$\"3)******zd=l'QF0$\"3=+++#Hr2=%FS7$$\" 3;+++j7@CPF0$\"3O+++BKawTFS7$$\"3%*******fUR!e$F0$\"3!******f.E<<%FS7$ $\"3>+++>L3NMF0$\"35+++*z%HmTFS7$$\"3%)*****H]\"H)G$F0$\"3'******4=A-; %FS7$$\"3;+++5(G+9$F0$\"3'******z3yM:%FS7$$\"3')*****H1,.*HF0$\"3y**** **\\#Hg9%FS7$$\"3G+++[,6RGF0$\"3)******RqQy8%FS7$$\"37+++X?X'o#F0$\"3W +++&=l)GTFS7$$\"3$)*****\\3;B`#F0$\"3c*****\\Oi!>TFS7$$\"3)******zB$ow BF0$\"3&)*****>#zP3TFS7$$\"3-+++6S_>AF0$\"3;+++tBv'4%FS7$$\"3%*******R czg?F0$\"3K+++0w6%3%FS7$$\"3&*******H$Q/!>F0$\"3K+++))\\RqSFS7$$\"3()* ******3q$Q*))p@SFS7$$\"3'*******eDdS7F0$\"3x ******3R_-SFS7$$\"3'******zYn+2\"F0$\"3))*****RB*e\")RFS7$$\"3G+++0.0n *)Feq$\"3=+++Q'e'eRFS7$$\"3/+++$)y?*>(Feq$\"3B+++v7VLRFS7$$\"3=+++S_%) )Q&Feq$\"39+++;S^0RFS7$$\"3!)******3(fK_$Feq$\"39+++*ewV(QFS7$$\"3)*** ***>Nc>e\"Feq$\"3;+++oWFRQFS7$$!3.+++RTP1ZF`p$\"3=+++]j5*z$FS7$$!3))** ****ze2.FFeq$\"3-+++AZ7_PFS7$$!3)******>z\"Qq_Feq$\"3%******f%GF&p$FS7 $$!3')*****4_rG&eFeq$\"3=+++&yrAo$FS7$$!3E+++&e9,Z'Feq$\"3#)*****Hi#eo OFS7$$!3o+++*HE08(Feq$\"3#******f;@Tl$FS7$$!3/+++L`hXyFeq$\"3#******4n )yQOFS7$$!3a*****>s?:j)Feq$\"3!)*****>1rCi$FS7$$!33+++z;[6&*Feq$\"33++ +V?/0OFS7$$!3/+++w3)>0\"F0$\"3))*****f\\xje$FS7$$!3-+++(R'zq6F0$\"3')* ****4W+kc$FS7$$!3/+++pB?:8F0$\"3!******p-._a$FS7$$!3-+++cxL&\\\"F0$\"3 \")*****f\")GL_$FS7$$!3(******49A.s\"F0$\"3$)*****Rb.A]$FS7$$!35+++OZ6 ))>F0$\"3-+++u;3%[$FS7$$!3%******\\ht'zAF0$\"3?+++O+jqMFS7$$!3=+++\"zX Ad#F0$\"3#******4\"*[,3=X $FS7$$!3#)*****>w01g$F0$\"3%*******GtU^MFS7$$!3@+++3U`AQF0$\"37+++Yb!= X$FS7$$!3x*****H%\\aLSF0$\"3%******fYGFX$FS7$$!33+++[8Em\\F0$\"31+++!y 4xF0$\"3; +++uDR7NFS7$$!3a+++aZi&G)F0$\"3$*******H!o[_$FS7$$!3<+++[4*H#))F0$\"3> +++M'Qp`$FS7$$!3P+++=3cO$*F0$\"3;+++'yl&[NFS7$$!3r*****\\dy+$)*F0$\"33 +++*RH(fNFS7$$!3!******pCW1.\"FS$\"3/+++u'>/d$FS7$$!3/+++O]zw5FS$\"33+ ++_Fj!e$FS7$$!3\"******p1Y;7\"FS$\"3-+++#Go.f$FS7$$!3-+++o,Nl6FS$\"3?+ ++=!G'*f$FS7$$!34+++F[.37FS$\"3(******H'\\T3OFS7$$!3#******fq2)\\7FS$ \"3!)*****H≫OFS7$$!3'******4qg2H\"FS$\"3&)*****z#eeCOFS7$$!3%****** p*H(4L\"FS$\"3.+++C#y>j$FS7$$!3!******\\R80P\"FS$\"3&******pV9*QOFS7$$ !32+++E@W49FS$\"3#*******R))RXOFS7$$!3/+++tA\"yW\"FS$\"3'******psN9l$F S7$$!3-+++/4n&[\"FS$\"3#******pFHql$FS7$$!34+++#**fI_\"FS$\"3%*******Q N=iOFS7$$!35+++dr,g:FS$\"3-+++:C!pm$FS7$$!3/+++xid'f\"FS$\"35+++h'*=rO FS7$$!33+++,!oFj\"FS$\"3=+++%))[]n$FS7$$!3/+++n,io;FS$\"3-+++VN[yOFS7$ $!3.+++m\"eTq\"FS$\"3')*****z&p\\\"o$FS7$$!30+++T_SR\">FS$\"31+++a6!4p$FS7$$!33+++5w'e%>FS$\"31+ ++P%[5p$FS7$$!31+++#)\\fz>FS$\"3;+++Poz!p$FS7$$!3#******>![;8?FS$\"3)) *****Hv[,p$FS7$$!3')******)H*eY?FS$\"3=+++Jl5*o$FS7$$!3#)*****R/!))z?F S$\"3!******\\Yswo$FS7$$!3$)******=![I6#FS$\"3-+++;)[eo$FS7$$!3%****** \\o.h9#FS$\"31+++;yj$o$FS7$$!39+++Jq0z@FS$\"3?+++%oT5o$FS7$$!3>+++Mw\" >@#FS$\"3/+++KE1yOFS7$$!3#)*****fp%pWAFS$\"3!)******yGquOFS7$$!35+++\" 4(RxAFS$\"3#)*****>mk4n$FS7$$!3=+++-M.5BFS$\"3)*******\\-&om$FS7$$!35+ ++a>hUBFS$\"33+++a>OiOFS7$$!3y*****\\%39vBFS$\"3))*****H9-vl$FS7$$!3&* *****R(zi2CFS$\"3))*****\\DtAl$FS7$$!3&)*****f1\"3SCFS$\"3!)*****R\"yn YOFS7$$!3?+++#p2DZ#FS$\"3#******HW=2k$FS7$$!3*******\\H:\\]#FS$\"3')** ****zyRMOFS7$$!32+++,7JPDFS$\"36+++(**=xi$FS7$$!3)******Rj-(pDFS$\"3++ ++8[o?OFS7$$!3A+++Ln4-EFS$\"3;+++8&)H8OFS7$$!3%******>c+Xj#FS$\"3!)*** ***oMc0OFS7$$!3=+++66#pm#FS$\"3/+++`K[(f$FS7$$!3?+++6`O*p#FS$\"3%)**** *Hmh!*e$FS7$$!3:+++G+%=t#FS$\"3!)*****pt-.e$FS7$$!3))*****>2_Vw#FS$\"3 7+++x2@rNFS7$$!3=+++)=3pz#FS$\"3/+++o.zhNFS7$$!3\")*****R0:&HGFS$\"3'* ******)RY?b$FS7$$!3!******\\Fz@'GFS$\"3#******>3%)>a$FS7$$!3')******pt !\\*GFS$\"3()*****p(*3;`$FS7$$!35+++edqFHFS$\"3*)*****\\+F4_$FS7$$!3%) *****pu!egHFS$\"3,+++tW%*4NFS7$$!3;+++1&QN*HFS$\"39+++*3o')\\$FS7$$!33 +++^]eEIFS$\"3'******>)\\5([$FS7$$!3!)*****H@E(fIFS$\"3y*****Hri_Z$FS7 $$!3'******RhnH4$FS$\"3')*****HH\\JY$FS7$$!32+++JYJEJFS$\"3/+++!>t2X$F S7$$!3)******pOs(fJFS$\"3))******RL9QMFS7$$!3!******pgXL>$FS$\"35+++] \"p_U$FS7$$!3@+++)yQqA$FS$\"3()*****>]g@T$FS7$$!3?+++\\f&3E$FS$\"3!)** ***fuF))R$FS7$$!3#******>p+[H$FS$\"3;++++V$FS$\"3*******\\P'>HLFS7$$!3++++h!GlY$FS$\" 3&)*****HEhZJ$FS7$$!3()*****4!RG,NFS$\"3A+++,l=+LFS7$$!3#******Hgwh`$F S$\"3')*****fa&[&G$FS7$$!3%)*****Rr/7d$FS$\"3#******>3s1F$FS7$$!3?+++w fO1OFS$\"3!******>(*fdD$FS7$$!3&)*****\\Id;k$FS$\"3()******)>j2C$FS7$$ !33+++hZ2xOFS$\"3A+++3epDKFS7$$!3>+++PNh7PFS$\"3&******R%=d5KFS7$$!3#* *****f%zE[PFS$\"3;+++'G0a>$FS7$$!3=+++>9.%y$FS$\"3))******z*4-=$FS7$$! 3.++++l*)>QFS$\"3')*****Rd**\\;$FS7$$!3#*******e[&e&QFS$\"3#)******\\u y\\JFS7$$!3#)*****>I(*=*QFS$\"3&******Hl'eMJFS7$$!39+++\"z8!GRFS$\"39+ ++F)4%>JFS7$$!3')*****pY$>kRFS$\"3?+++[\"pU5$FS7$$!3f*****\\oC/+%FS$\" 3y*****REw\"*3$FS7$$!3,+++W]pOSFS$\"3#******zAUT2$FS7$$!3;+++J9*H2%FS$ \"3?+++Uu+IFS7$$!3s*****>7hVD%FS$\"33+++pSq&)HFS7$$!3i*****Hu/0 H%FS$\"3%)*****p?I8(HFS7$$!3')*****R0nlK%FS$\"31+++gf2dHFS7$$!3I+++a7` iVFS$\"3()*****H$[%H%HFS7$$!3/+++$R!Q)R%FS$\"3z*****zKR*GHFS7$$!3;+++[ u4MWFS$\"3-+++H41:HFS7$$!3=+++5ampWFS$\"3<+++)458!HFS7$$!30+++ht10XFS$ \"33+++vio()GFS7$$!3s*****\\c'GSXFS$\"3/+++\"*y=uGFS7$$!37+++XlIvXFS$ \"3-+++&Q73'GFS7$$!3M+++n665YFS$\"3=+++GibZGFS7$$!3))******>ZoWYFS$\"3 .+++k\\TMGFS7$$!3I+++\"*>,zYFS$\"3:+++\\KQ@GFS7$$!3q*****fLyIr%FS$\"3( ******R!\\X3GFS7$$!3i*****Rupou%FS$\"3'******H(Hi&z#FS7$$!3U+++)*GP!y% FS$\"3))******)yzGy#FS7$$!3))*****>AvN\"[FS$\"3;+++Vq@qFFS7$$!3=+++C\\ YY[FS$\"3#)*****>'eidFFS7$$!3;+++I5.z[FS$\"3;+++&)o4XFFS7$$!3')*****pS j7\"\\FS$\"3()*****RM?Et#FS7$$!38+++!y_J%\\FS$\"3++++Xh=?FFS7$$!3++++K 2pu\\FS$\"3*)*****z%Ry2FFS7$$!3O+++1(pe+&FS$\"3w*****pC.ap#FS7$$!3++++ *)HoO]FS$\"3y*****HWLIo#FS7$$!3w*****zpCr1&FS$\"3))******=RmqEFS7$$!3# )*****pu*=(4&FS$\"3)******f5%GeEFS7$$!3A+++@Q(o7&FS$\"31+++VN)ek#FS7$$ !3U+++RLXLEFS7$$!3')*****pS&3&=&FS$\"3'******f>z4i #FS7$$!3h*****Hy2O@&FS$\"3.+++3bX3EFS7$$!3m*****\\#)QCN.N&FS$\"37+++>7vW DFS7$$!3)******zv-lP&FS$\"3z******)*QtJDFS7$$!3p*****H5zAS&FS$\"3-+++b kg=DFS7$$!3v*****HblwU&FS$\"3********z>O0DFS7$$!3u*****pijEX&FS$\"3#** ****4'R*>\\#FS7$$!3y*****\\0vsZ&FS$\"33+++xi\\yCFS7$$!3R+++V<],bFS$\"3 !*******)=j[Y#FS7$$!3f******QdMDbFS$\"3%)******H$*3^CFS7$$!3-+++6#4)[b FS$\"3(******HpprV#FS7$$!3!)*****>T%*=d&FS$\"31+++A'*4BCFS7$$!3C+++mOg %f&FS$\"3))*****zyu)3CFS7$$!3)*******Q$Rph&FS$\"37+++&=\"\\%R#FS7$$!3( )*****\\#Q!*QcFS$\"3;+++2^%*zBFS7$$!3!******z_*\\gcFS$\"31+++OJBlBFS7$ $!3Q+++Y)G7_.N#FS7$$!3Q+++hTf-dFS$\"3()******p\"*HNBF S7$$!3(******\\#y4BdFS$\"3$)*****fir+K#FS7$$!3m*****>:UKu&FS$\"3$)**** *R0nYI#FS7$$!3y*****4THIw&FS$\"39+++IK3*G#FS7$$!3s*****p\"=Y#y&FS$\"33 +++<\"=LF#FS7$$!3O+++F:a,eFS$\"3#******4')ptD#FS7$$!30+++O1F?eFS$\"3$) *****zwO7C#FS7$$!3h*****><^'QeFS$\"3$******RH<\\A#FS7$$!3c*****\\4&oce FS$\"3$)*****H.5%3AFS7$$!3))*****RHuV(eFS$\"33+++/Pr\">#FS7$$!33+++)e? <*eFS$\"3/+++Mr#[<#FS7$$!3/+++Dds3fFS$\"39+++b#\\x:#FS7$$!3o******y8RD fFS$\"3&******\\3z/9#FS7$$!3O+++b\"><%fFS$\"3*)*****>s:I7#FS7$$!3m**** *4f5x&fFS$\"39+++J$e`5#FS7$$!3'******R:nL(fFS$\"35+++Ph](3#FS7$$!33+++ Z-p))fFS$\"3++++8%e%p?FS7$$!3'******4@\"o.gFS$\"3%)*****>Z970#FS7$$!3# )*****fKT$=gFS$\"3!******pltF.#FS7$$!3g*****\\\"=nKgFS$\"3$)*****fLNT, #FS7$$!35+++ZQnYgFS$\"3-+++\"*))H&*>FS7$$!3%)*****>a[.1'FS$\"35+++6PEw >FS7$$!3m*****H(pptgFS$\"3/+++%=Hq&>FS7$$!3y*****HFS7$$!3w*****\\8>%*4'FS$\"32+++(ffz\">FS7$$!3C+++A[z6hFS$\"33+++[K7)* =FS7$$!3f******p\"[Q7'FS$\"3'******>'\\3y=FS7$$!3Q+++!4!eNhFS$\"31+++> S%y&=FS7$$!3q*****4[\"*p9'FS$\"3,+++j'*RP=FS7$$!3#)******GK3ehFS$\"3/+ ++*3^n\"=FS7$$!33+++DH:FS7$$!3!)*****p&*R.G'FS$\"3\"******>N@c]\"FS7$$!3q*****zufnG'F S$\"3!******f]l<[\"FS7$$!3o*****RFzGH'FS$\"3#******zA#od9FS7$$!3c***** \\h+()H'FS$\"3)******zCpLV\"FS7$$!37+++NgA/jFS$\"31+++$>C)39FS7$$!3K++ +,!e%4jFS$\"3'******>fWSQ\"FS7$$!3j*****4?*R9jFS$\"3/+++\")y-f8FS7$$!3 z*****\\c_!>jFS$\"3%******RSrPL\"FS7$$!3t******y7UBjFS$\"3$*******HCF3 8FS7$$!3w*****R\")3vK'FS$\"3'******4:GDG\"FS7$$!31+++S*=8L'FS$\"3/+++/ d`c7FS7$$!34+++^d&[L'FS$\"3$******p<#HI7FS7$$!39+++%oB\"QjFS$\"3)***** **HYz.7FS7$$!31+++Yv7TjFS$\"33+++5,/x6FS7$$!3U+++HD(QM'FS$\"3#******pn D+:\"FS7$$!3;+++MUOYjFS$\"3-+++E%[F7\"FS7$$!3C+++&p3'[jFS$\"35+++9b?&4 \"FS7$$!3A+++!R71N'FS$\"35+++&>%Rn5FS7$$!3I+++nAQ_jFS$\"3!******4&=JR5 FS7$$!3o*****\\vDRN'FS$\"3.+++Lg&4,\"FS7$$!3=+++x2DbjFS$\"3s*****z,XK# )*F07$$!3E+++idOcjFS$\"32+++]F:M&*F07$$!3k******['zsN'FS$\"3t*****pumA C*F07$$!3q*******)=+ejFS$\"3w*****\\!QdZ*)F07$$!3K+++XCaejFS$\"3#***** **\\[1]')F07$$!3;+++v<\"*ejFS$\"3W+++6bt\\$)F07$$!3#******p#37fjFS$\"3 k******>neY!)F07$$!35+++25=fjFS$\"3')*****>KD1u(F07$$!3)******p:/\"fjF S$\"3R+++'ok=V(F07$$!3I+++8D!*ejFS$\"3[*****4KD.7(F07$$!3m******f')eej FS$\"31+++na.1oF07$$!3'******4[v\"ejFS$\"3#******>mJ!*['F07$$!33+++\"4 wwN'FS$\"3]*****zFf$phF07$$!3;+++qP5djFS$\"3C+++)*H2ZeF07$$!37+++#)=Zc jFS$\"3;+++KsBAbF07$$!39+++$z$zbjFS$\"3=+++ik#\\>&F07$$!37+++\"y#3bjFS $\"3?+++'\\D_'[F07$$!3K+++M>NajFS$\"3#)*****\\iHK`%F07$$!3q*****p09ON' FS$\"31+++xY/*>%F07$$!3%)*****pc\")GN'FS$\"3#)*****R&pyiQF07$$!3Q+++'Q m@N'FS$\"3A+++cIeCNF07$$!3#)*****>%)z9N'FS$\"37+++!epX=$F07$$!3K+++nD$ 3N'FS$\"3A+++)p#*G%GF07$$!3s*******RM-N'FS$\"3!******Ri2(*\\#F07$$!3w* ****HH%p\\jFS$\"3!******R&z)[jFS$\"3/+++'\\wFY\"F07$$!3#******z%o\\[jFS$\"32+++< lE:6F07$$!3;+++(*zD[jFS$\"3_+++YfLrwFeq7$$!3')*****R,1\"[jFS$\"31+++l1 q&=%Feq7$$!3c*****R2V![jFS$\"3K+++(p&3xpF`p7$$!3y*****R2q![jFS$!37+++$ )on!z#Feq7$$!3u******Gm=[jFS$!3?+++e)*[xiFeq7$$!3K+++v5R[jFS$!3y*****f ,u2w*Feq7$$!3q*****4[!o[jFS$!3#******\\UfQK\"F07$$!3%)*****Ro]!\\jFS$! 3'******z\"[!4n\"F07$$!3%*******Gj\\\\jFS$!30+++4 ,]jFS$!3!******4oZ?O#F07$$!3C+++&)oe]jFS$!31+++q1z0FF07$$!3K+++]c@^jFS $!3++++S**3[IF07$$!3o******>x)=N'FS$!3y******Qoy)Q$F07$$!31+++oFf_jFS$ !3++++2:tFPF07$$!3O+++N(>LN'FS$!3#)******eMykSF07$$!3#)*****>#p0ajFS$! 3!)*****f07)*R%F07$$!3!)*****p4#zajFS$!3=+++YopKZF07$$!3/+++,E^bjFS$!3 I+++*zFL1&F07$$!3U+++Wa?cjFS$!3%)******3ag\"R&F07$$!3=+++8u&oN'FS$!3_+ ++(oSur&F07$$!3M+++n^XdjFS$!3^+++[]vSgF07$$!3)*******H`)zN'FS$!31+++*3 ![hjF07$$!3y*****zdM%ejFS$!3M+++\"Gd&zmF07$$!3%******4s*yejFS$!3_+++%e P\\*pF07$$!3;+++xx.fjFS$!3h*****f)4e2tF07$$!3C+++Mg;fjFS$!3t*****R*fX< wF07$$!3w******4@;fjFS$!3C+++y!RX#zF07$$!3C+++,S,fjFS$!3A+++&39)G#)F07 $$!3u*****>75(ejFS$!3b*****Rjr-`)F07$$!3#)******Q$R#ejFS$!3@+++-&3*G)) F07$$!3'******p*4fdjFS$!3(*******4qsC\"*F07$$!3A+++J\\vcjFS$!3L+++AWt< %*F07$$!3G+++([@dN'FS$!3C+++fB%zq*F07$$!39+++>:[ajFS$!3Q+++LiO&***F07$ $!3u*****fREIN'FS$!3(******4Z-!G5FS7$$!3;+++%*zM^jFS$!31+++AR>c5FS7$$! 31+++'oQ%\\jFS$!3++++XL6%3\"FS7$$!3U+++N8HZjFS$!3)******pHj<6\"FS7$$!3 e*****fF**[M'FS$!33+++wk9R6FS7$$!3%)*******HcAM'FS$!3!********oli;\"FS 7$$!3w*****fjc$RjFS$!3%******R#Q7$>\"FS7$$!3O+++H\\>OjFS$!3(******f!Qs >7FS7$$!3(******4AmFL'FS$!3!******peogC\"FS7$$!3y*****>$f1HjFS$!31+++: 6;s7FS7$$!3e******H)*3DjFS$!3%******RJ/!)H\"FS7$$!3R+++?S$3K'FS$!30+++ q5gB8FS7$$!3H+++:\\H;jFS$!31+++8U&*[8FS7$$!3!)*****\\@p9J'FS$!33+++3l1 u8FS7$$!3d*******)QN1jFS$!3%*******R1%*)R\"FS7$$!3j******fh%4I'FS$!32+ ++6#zNU\"FS7$$!3%*******pMC&H'FS$!3-+++JZ)zW\"FS7$$!3I+++xMC*G'FS$!3-+ ++7'f@Z\"FS7$$!3W+++JS%HG'FS$!3'******R;1h\\\"FS7$$!3=+++bJMwiFS$!33++ +)fE)>:FS7$$!3G+++L!R%piFS$!3*******f,BLa\"FS7$$!3!)*******)*HAE'FS$!3 /+++>ufm:FS7$$!3))*****H[9ZD'FS$!3-+++*p^'*e\"FS7$$!3C+++\"3\"*oC'FS$! 3\"******4l([7;FS7$$!3K+++d%e(QiFS$!3\"******H'p5N;FS7$$!3u*****>P:.B' FS$!3%******zA6vl\"FS7$$!3S+++o1c@iFS$!3))*****>%>qz;FS7$$!3W+++\\K\\7 iFS$!3$******z]!o,C[Ms\"FS7$$!37+++th T$>'FS$!3#******RP1]u\"FS7$$!3y*****Hd/M='FS$!31+++agNmFS7$$!3++++6'Q=4'FS$!3%******4wk(H>FS7$$!3#******RAX *ygFS$!31+++<%z#\\>FS7$$!3k*****z+Fd1'FS$!3/+++cQfo>FS7$$!3?+++XH=_gFS $!3%******>q3x)>FS7$$!3Q+++z>JQgFS$!3*)*****zcCm+#FS7$$!3x******4I6CgF S$!3;+++k?MD?FS7$$!35+++*)[e4gFS$!3')*****p!='Q/#FS7$$!3q*****\\TEZ*fF S$!3!*******>W=i?FS7$$!3E+++Jj`zfFS$!3=+++^0J!3#FS7$$!3Q+++@L,kfFS$!3' )*****>(3C)4#FS7$$!3o*****p+c\"[fFS$!34+++%4wf6#FS7$$!3r*****p%H'>$fFS $!3:+++rp^L@FS7$$!3++++LEV:fFS$!37+++9V'3:#FS7$$!3m*****p[j&)*eFS$!3)) *****p**=!o@FS7$$!3-+++hQN\")eFS$!37+++n>)\\=#FS7$$!3A+++Q?!Q'eFS$!3!) *****zDa+ ++E9tMAFS7$$!3#******z7v!4eFS$!3#******R*)Q4D#FS7$$!3s*****f!f8!z&FS$! 3++++!*3'pE#FS7$$!3/+++M[%3x&FS$!39+++Z!*z#G#FS7$$!3C+++z(*>^dFS$!3#** ****H:b%)H#FS7$$!31+++[&)>JdFS$!3/+++q6$RJ#FS7$$!36+++%*)Q3r&FS$!31+++ X#H#HBFS7$$!3Q+++?&=,p&FS$!3$)*****ps^VM#FS7$$!39+++(3N!pcFS$!3#****** H=,$fBFS7$$!3S+++DieZcFS$!3?+++3/3uBFS7$$!3Q+++T&pdi&FS$!35+++\\Cp)Q#F S7$$!34+++JEe.cFS$!3:+++819.CFS7$$!3-+++'4B5e&FS$!3!******4\\GuT#FS7$$ !3K+++b&)3ebFS$!35+++n*f:V#FS7$$!3'******fmwZ`&FS$!3?+++P#RbW#FS7$$!3- +++X^36bFS$!3y*****f#3PfCFS7$$!39+++(y6q[&FS$!3%*******)ffIZ#FS7$$!3') *****H\\aDY&FS$!35+++x2h'[#FS7$$!3s*****\\H6xV&FS$!3(*******[*H+]#FS7$ $!3\")*******Q![7aFS$!3))*****>3BL^#FS7$$!3O+++r,'oQ&FS$!3\"******>`' \\EDFS7$$!33+++f#\\3O&FS$!3=+++ZqbRDFS7$$!3o*****faYWL&FS$!3y*****fx6D b#FS7$$!3W+++K7l2`FS$!3))*****HEo`c#FS7$$!3/+++qGY!G&FS$!3%******4XM\" yDFS7$$!3G+++29)GD&FS$!3!*******o'=3f#FS7$$!3r*****HB2\\A&FS$!3<+++A'H Mg#FS7$$!3))*****fATl>&FS$!3))*****fPwfh#FS7$$!3s*******z%yn^FS$!3-+++ I$o%GEFS7$$!37+++_*R'Q^FS$!3>+++%>:4k#FS7$$!3,+++5$4\"4^FS$!3-+++_pK`E FS7$$!3')*****\\<'>z]FS$!3'******z\"QrlEFS7$$!3))*****Rc/*[]FS$!3%**** **H>'3yEFS7$$!3r******\\#R#=]FS$!3!******>ha/p#FS7$$!3I+++%z0s)\\FS$!3 z******z'HGq#FS7$$!3-+++u0\"e&\\FS$!3#)*****4,A_r#FS7$$!37+++331C\\FS$ !34+++e@kFFFS7$$!3e*****pck>*[FS$!3#******f`+,u#FS7$$!3Q+++%yI&f[FS$!3 -+++Ytg_FFS7$$!3!*******\\#pn#[FS$!3-+++(\\s^w#FS7$$!3=+++11p$z%FS$!31 +++Bb!yx#FS7$$!3g*****zJ1.w%FS$!3y*****4]:0z#FS7$$!3o*****HlGms%FS$!37 +++w4J.GFS7$$!39+++R1n#p%FS$!35+++!))*>;GFS7$$!3++++:gWeYFS$!3#)****** f%*=HGFS7$$!3d******z\"pRi%FS$!36+++:iGUGFS7$$!3I+++E^D*e%FS$!35+++He \\bGFS7$$!3e*****fP>Vb%FS$!31+++?J#)oGFS7$$!3-+++-z<>XFS$!3++++!*>F#)G FS7$$!3%******R0ZQ[%FS$!35+++$RXe*GFS7$$!3k*****RZV$[WFS$!3?+++(HX&4HF S7$$!3H+++>So7WFS$!3?+++nEPBHFS7$$!31+++qc)oP%FS$!3<+++]uKPHFS7$$!3+++ +aa'4M%FS$!31+++s&39&HFS7$$!3))*****4OS\\I%FS$!3-+++ORhlHFS7$$!3?+++hs #)oUFS$!39+++O/%*zHFS7$$!3G+++FGkKUFS$!39+++tRQ%*HFS7$$!3/+++hMS'>%FS$ !3')*****HZR*3IFS7$$!3I+++?_7gTFS$!38+++C4gBIFS7$$!3K+++ZP#Q7%FS$!3/++ +-9OQIFS7$$!31+++4U^(3%FS$!39+++:J@`IFS7$$!33+++L7@^SFS$!31+++Wu9oIFS7 $$!3W+++])G\\,%FS$!39+++!*\\:$3$FS7$$!3!******pW!oyRFS$!31+++EcA)4$FS7 $$!3%******RryC%RFS$!3!)*****Hb[L6$FS7$$!3'******>gNj!RFS$!33+++`B^GJF S7$$!3))*******Gi-(QFS$!3y*****R00P9$FS7$$!3,+++Y\"pU$QFS$!37+++#>9*eJ FS7$$!39+++0dO)z$FS$!31+++to7uJFS7$$!3?+++X1ciPFS$!3))*****H%)H$*=$FS7 $$!3\"******>xhos$FS$!3%)*****Hb4X?$FS7$$!3y*****4,w7p$FS$!3,+++EAl>KF S7$$!3/+++-%4el$FS$!30+++IRuMKFS7$$!3-+++,rY?OFS$!31+++O1x\\KFS7$$!3)* *****RR`_e$FS$!39+++%H=ZE$FS7$$!3)******fqr,b$FS$!39+++))GdzKFS7$$!3&) *****>iC_^$FS$!31+++,0K%H$FS7$$!3/+++8RT![$FS$!3>+++pt%*3LFS7$$!3++++m 0uXMFS$!3$******z$*RMK$FS7$$!31+++:[?6MFS$!3@+++9\\yPLFS7$$!3!)*****H= 1oP$FS$!3#********Hp>N$FS7$$!3-+++:NaULFS$!3%)*****RW!)fO$FS7$$!3%)*** ***G]T3LFS$!3!)******og!)zLFS7$$!3#*******\\$=WF$FS$!3=+++-VV$R$FS7$$! 3A+++j0bSKFS$!3&)*****pp`oS$FS7$$!33+++_#3o?$FS$!35+++cK0?MFS7$$!3')** ***fa(=tJFS$!3&******>WAIV$FS7$$!3<+++kToRJFS$!31+++$=^dW$FS7$$!3z**** *RX$H1JFS$!33+++%))H#eMFS7$$!3%)*****zV5I2$FS$!3#*******=%\\/Z$FS7$$!3 1+++U)H)RIFS$!3%******\\8,C[$FS7$$!35+++Rhu1IFS$!3'*******Ro2%\\$FS7$$ !35+++xNvtHFS$!3$)*****>zoa]$FS7$$!3$******H@Y3%HFS$!36+++$ppl^$FS7$$! 33+++Mz,3HFS$!3$******zmst_$FS7$$!3++++'[i_(GFS$!3))*****4Dry`$FS7$$!3 <+++$\\tD%GFS$!3#)******p$f![NFS7$$!3$)*****HZW*4GFS$!3/+++B8$zb$FS7$$ !37+++a)otx#FS$!3;+++g<[nNFS7$$!3@+++()*R[u#FS$!39+++ncqwNFS7$$!3=+++` 6N7FFS$!31+++Q$)f&e$FS7$$!3)******pc&*)zEFS$!3))*****4ObTf$FS7$$!31+++ 'Qmuk#FS$!3$******ffsBg$FS7$$!37+++/n0:EFS$!3'*******[hC5OFS7$$!3')*** **fbfEe#FS$!3>+++iBx#FS$!31+++!*e*)zOFS7$$!3!******pv'Hf@FS$!3#******fXXEo$FS7$$!3\"* *****H>$GE@FS$!3?+++u2,&o$FS7$$!3!)******p8;$4#FS$!3y******H'*)po$FS7$ $!3++++?5#*f?FS$!37+++)zz&)o$FS7$$!3=+++y8bE?FS$!37+++K!z(*o$FS7$$!3)) *****f6TI*>FS$!3/+++\\]e!p$FS7$$!3))*****pFy$f>FS$!35+++Fb*4p$FS7$$!3$ ******>?]b#>FS$!3')******)335p$FS7$$!3#******\\WV:*=FS$!39+++(H?1p$FS7 $$!3-+++xOMd=FS$!37+++S'H)*o$FS7$$!33+++\"eNH#=FS$!35+++CNj)o$FS7$$!3# ******4t-$)y\"FS$!3')*****zEHqo$FS7$$!37+++auU`7BTm$FS7$$ !3,+++(o!o+:FS$!3'******RZV\"fOFS7$$!3%******4l9IY\"FS$!3;+++OFS7$$!3#*******Hz%[A\"FS$!33+++!= )z6OFS7$$!3#******4%*RD=\"FS$!39+++M$*>.OFS7$$!32+++*pd#R6FS$!3!)***** \\7HTf$FS7$$!3-+++;A)[4\"FS$!3!)*****p?%e%e$FS7$$!3-+++z:F\\5FS$!3))** ***zCiXd$FS7$$!3,+++]UD-5FS$!37+++EE1kNFS7$$!3.+++mg>O&*F0$!3!)*****fd (3`NFS7$$!3d*****H:U5.*F0$!3')*****\\8W;a$FS7$$!3(******H*=r.&)F0$!3\" )*****pcZ(HNFS7$$!3k*****f!)y(\\zF0$!3%******fpFu^$FS7$$!3W+++>qAjtF0$ !3(******\\JTZ]$FS7$$!3#*******)p>bt'F0$!3w*****4_)z\"\\$FS7$$!3]***** phJQ0'F0$!3))*****\\mC)yMFS7$$!3Y*****zdDtH&F0$!31+++J*GjY$FS7$$!3s*** **>4T\"GWF0$!3'******RdtcX$FS7$F]i\\l$!3')*****>,3=X$FS7$Fhh\\l$!3)*** ***4GtKX$FS7$Fch\\l$!3#)******R!GjX$FS7$F^h\\l$!3#******4\"*[1rCi$FS7$F\\e\\l$!3#******4n)yQOFS7$Fgd\\l$!3#******f;@Tl$FS7$Fb d\\l$!3#)*****Hi#eoOFS7$F]d\\l$!3=+++&yrAo$FS7$Fhc\\l$!3%******f%GF&p$ FS7$$!3(*******o)\\jr%Feq$!30+++Ztl2PFS7$$!3!)*****p!)of=%Feq$!3@+++.i [>PFS7$$!35+++?4]vOFeq$!3;+++;:\"3t$FS7$$!3')*****\\d#*>=$Feq$!3z***** f4y;u$FS7$$!3!3++IFte**)F`p$!3/+++oxK!z$FS7$$\"3-+++/W)==\"Feq$!3!)*** **f7!pJQFS7$$\"3p*****>M5>9$Feq$!35+++6AqnQFS7$$\"3D+++/'f1-&Feq$!3/++ +RUc**QFS7$$\"3J*****z?l3%oFeq$!3.+++Ww2GRFS7$$\"3n+++Z'Qkh)Feq$!3#)** ****Qz!Q&RFS7$$\"31+++M(Qc.\"F0$!3#******\\Mrr(RFS7$$\"31+++e#zm?\"F0$ !3++++VX[)*RFS7$$\"3'*******z!*=v8F0$!34+++7]*z,%FS7$$\"33+++#Hq9a\"F0 $!3r*****4Z+f.%FS7$$\"3&******z%>u0+++J%FS7$$\"3!****** fOaz_%F0$!3)******HSLN>%FS7$$\"3#)*****\\?%4hYF0$!3K+++]?u%>%FS7$$\"3! )*****>WpDz%F0$!33+++Kh[&>%FS7$$\"3:+++mBNA\\F0$!3'*******o#zd>%FS7$$ \"3u*****>K9/0&F0$!3q*****4AMc>%FS7$$\"3_*****\\xDn<&F0$!3A+++oH1&>%FS 7$$\"3k*****4Oc7I&F0$!3-+++&ywS>%FS7$$\"3t******=\\(RU&F0$!3%)*****zI' o#>%FS7$$\"3h*****\\_\\[a&F0$!3++++)e,4>%FS7$$\"3c******3v%Qm&F0$!3M++ +X@t)=%FS7$$\"3K+++pa$4y&F0$!3i*****H,(='=%FS7$$\"33+++u#zg*eF0$!3k*** ***zZF$=%FS7$$\"3T+++pSC4gF0$!3C+++CO+!=%FS7$$\"3++++nUR?hF0$!3#****** >N\"QwTFS7$$\"3Z******QN\\HiF0$!3E+++AaTsTFS7$$\"3C+++,[]OjF0$!3u***** f(H6oTFS7$$\"3[*****H>!RTkF0$!37+++i3[jTFS7$$\"3l******f56WlF0$!3A+++d c_eTFS7$$\"3++++(F0$!3T+++\"4(H:TFS7$$\"39+++Uv-$G(F0$!3x*****z]nz5 %FS7$$\"3S+++0(zVO(F0$!3)******4tk.5%FS7$$\"3k*****z#)GJW(F0$!3#)***** >_$\\#4%FS7$$\"3C+++UZA>vF0$!3%)*****\\_eV3%FS7$$\"3E+++eeh#f(F0$!3m** ***fFkf2%FS7$$\"3i******y!\\Km(F0$!3I+++5_JnSFS7$$\"3K+++5(p5t(F0$!3o* ****4n:%eSFS7$$\"3c*****zM@gz(F0$!33+++;*p#\\SFS7$$\"3K+++!yX!eyF0$!35 +++D@))RSFS7$$\"3?+++^G3z 'R?SFS7$$\"3.+++@Q%f-)F0$!3P+++ssI5SFS7$$\"3i*****f[Oc2)F0$!3?+++p<*** *RFS7$$\"3u******)**y?7)F0$!37+++fTX*)RFS7$$\"3k*****\\H*>l\")F0$!3=++ +v#)pyRFS7$$\"3\\+++\"QA\\?)F0$!3;+++4zsnRFS7$$\"3%******z5q6C)F0$!3/+ ++BoacRFS7$$\"3!*******H4'QF)F0$!3++++W(e^%RFS7$$\"3K+++G'4HI)F0$!3&** *****ztcLRFS7$$\"3[+++;qAG$)F0$!3*******RTw<#RFS7$$\"3S+++5'>(\\$)F0$! 3>+++=&*y4RFS7$$\"3C+++e#*Gn$)F0$!3!)******[.h(*QFS7$$\"3]*****>uK3Q)F 0$!3=+++iDC&)QFS7$$\"3K+++?8C!R)F0$!31+++3)*osQFS7$$\"3a+++F-S&R)F0$!3 #)*****4ub*fQFS7$$\"37+++4\")='R)F0$!3'******>-Wq%QFS7$$\"3!)******)Qw CR)F0$!3-+++G$eR$QFS7$$\"3[+++h>Q)F0$!3$)******[Bq?QFS7$$\"3e*****HQ **4P)F0$!3))*******zzs!QFS7$$\"3C+++nR$HN)F0$!3#)*****HU%p$z$FS7$$\"39 +++XnwH$)F0$!31+++\"**\\*zPFS7$$\"3Y*****4H?8I)F0$!3%)*****fJ]gw$FS7$$ \"3m*****R'RSn#)F0$!31+++_#**>v$FS7$$\"3C+++`B\"yA)F0$!34+++/2!yt$FS7$ $\"3)*******zMK#=)F0$!38+++D'eMs$FS7$$\"3;+++1npI\")F0$!3&******>.x*3P FS7$$\"3K+++\"HqE2)F0$!3;+++,+O%p$FS7$$\"3h*****\\_ez+)F0$!3%******fn6 'zOFS7$$\"3()*****\\M[i$zF0$!3?+++ritkOFS7$$\"3g*****>I&>dyF0$!3!)**** *42Q(\\OFS7$$\"3Q+++U(=/x(F0$!3z*****4V@Yj$FS7$$\"33+++\")f\\vwF0$!3z* *****o2R>OFS7$$\"3/+++E_&>d(F0$!3'******zb]Sg$FS7$$\"3e*****4)oEfuF0$! 3++++3`g)e$FS7$$\"3)******pwKoL(F0$!35+++I&fId$FS7$$\"3q*****\\gsR?(F0 $!3')*******od2iR$FS7$$\"3!******H)o&*zYF0$!3'******\\]A%zLFS7 $$\"3))*****>J%HzUF0$!3-+++1-IiLFS7$$\"3++++?q3=QF0$!33+++brdWLFS7$$\" 3!)*****zY,JF$F0$!3=+++^FkDLFS7$$\"34+++zJ$4g#F0$!31+++?ly.LFS7$Fd_z$! 3#******fA&o)H$FS7$F__z$!37+++/9;$H$FS7$Fj^z$!3++++'HqqG$FS7$Fe^z$!3;+ ++-x>!G$FS7$F`^z$!3;+++Ez@sKFS7$F[^z$!3%******z0EEE$FS7$Ff]z$!3#)***** *pak]KFS7$Fa]z$!3>+++8?=NKFS7$F\\]z$!3%)*****>UD^@$FS7$Fg\\z$!3)****** 4'fP!>$FS7$Fb\\z$!3-+++&[+E;$FS7$F]\\z$!3:+++=M0MJFS7$Fh[z$!3\"******H ]gh5$FS7$Fc[z$!3!)*****RHl%zIFS7$F^[z$!33+++M*fS0$FS7$Fijy$!30+++Pu')H IFS7$Fdjy$!3;+++!ojn+$FS7$F_jy$!3#*******fBi%)HFS7$Fjiy$!3))*****pRHL' HFS7$$!31+++K`Y5AFeq$!3'******Rb-m'GFS7$$!39+++d]QiCFeq$!33+++gh3\"y#F S7$$!3;+++6_E>DFeq$!3*******pFeq$!3))***** \\OOtO#FS7$$!3-+++FS7$$!3_*****H!pDczF`p$!3-+++=5(z! >FS7$$!3H+++sDwOpF`p$!3#******4-yL&=FS7$$!3D+++EO$3,'F`p$!31+++.09*z\" FS7$$!39+++X1[v^F`p$!3++++:w?Xv(4aEF`p$!30+++DhsJ:FS7$$!3#)*****p]@M?#F`p$!3))**** *>ei(y9FS7$$!3)******z\")*p8=F`p$!3,+++\"F`p$!3.+++9b[?8FS7$$!3O+++Wlf `&*Feo$!3/+++bq(yE\"FS7$$!3w******z%>Db(Feo$!3%******zXA`@\"FS7$$!33++ ++'>*)*eFeo$!3%******pm6G;\"FS7$$!39+++s5XZXFeo$!3.+++$=O.6\"FS7$$!31+ ++2R*fX$Feo$!3)*******p)))y0\"FS7$$!39+++_E&ee#Feo$!3\"******zyja+\"FS 7$$!3.+++Sm(=!>Feo$!3C+++].cI&*F07$$!3'******4&p]s8Feo$!3Z******zgh1!* F07$$!3G+++-%evp*FP$!3[+++OFx#[)F07$$!3q*****\\Q]7p'FP$!3!)******RX+fz F07$$!3I+++\\&*p%\\%FP$!3L+++R7HNuF07$$!3,+++vO<<;\"pF0 7$$!31+++pfWT=FP$!3s******e/(zQ'F07$$!33+++hUG66FP$!3')*****zFUV'eF07$ $!3=+++)=7sQ'FE$!3r*****pBE2M&F07$$!3-+++,U)=Y$FE$!3w******Ry6<[F07$$! 31+++l![fu\"FE$!3))*****H.9NH%F07$$!3R+++$RKE/)F:$!3C+++DG\"*pPF07$$!3 1+++N?'QH$F:$!3)******\\)HJYKF07$$!3)******>\"e(3:\"F:$!3;+++,QrAFF07$ $!3y*****>N]_?$F4$!3-+++&*[6*>#F07$$!3O+++07Q&G'Fajx$!3-+++)3;bn\"F07$ $!3K+++`:x[mF-$!3'******pI<>:\"F07$$!3-+++FU8`Z5Feq7$F($\"3?+++0-z)=%Feq-Fd\\q6&Ff\\q$\"\"&Fi\\qFbgxF( F\\]q-Fa]q6jcn7%F'7$$!++-GveFeo$\"+cx)fB&Fgdq7$$!\"%F)F(7%Fg_jl7$$!+R/ 9aPFeq$\"+^v>Z5Fi]qF\\`jl7%F``jl7$$!+*R6\"pUF0$\"+Ijzq:Fi]qF\\`jl7%Ff` jl7$$!+a%4JR#FS$\"+H^R%4#Fi]qF\\`jl7%F\\ajl7$$!+V3s,\"*FS$\"+1S*zh#Fi] qF\\`jl7%Fbajl7$$!+ZP!yq#Fg^q$\"+7JfTJFi]qF\\`jl7%Fhajl7$$!+J')Q)z'Fg^ q$\"+wF>lOFi]qF\\`jl7%F^bjl7$$!+\"our]\"Fd_q$\"+KOz)=%Fi]qF\\`jl7%Fdbj l7$$!+e;$z.$Fd_q$\"+)y'R7ZFi]qF\\`jl7%Fjbjl7$$!+lo]zcFd_q$\"+iS+O_Fi]q F\\`jl7%F`cjl7$$!++kN*)**Fd_q$\"+m#='fdFi]qF\\`jl7%Ffcjl7$$!+\\oQq;Fa` q$\"+wNC$G'Fi]qF\\`jl7%F\\djl7$$!+r'fnn#Fa`q$\"+jf)o!oFi]qF\\`jl7%Fbdj l7$$!+(Q4i8%Fa`q$\"+;PbItFi]qF\\`jl7%Fhdjl7$$!+)yaL>'Fa`q$\"+]!eU&yFi] qF\\`jl7%F^ejl7$$!+o%R=-*Fa`q$\"+1Q,y$)Fi]qF\\`jl7%Fdejl7$$!+D2m#G\"Fa bq$\"+]-%=!*)Fi]qF\\`jl7%Fjejl7$$!+$)3e%y\"Fabq$\"+t>wD%*Fi]qF\\`jl7%F `fjl7$$!+)*eANCFabq$\"+5*4)\\**Fi]qF\\`jl7%Fffjl7$$!+cnVlKFabq$\"+YASZ 5Fd`qF\\`jl7%F\\gjl7$$!+&GZ'4VFabq$\"+tY%)*4\"Fd`qF\\`jl7%Fbgjl7$$!+x6 &eg&Fabq$\"+$*RJ_6Fd`qF\\`jl7%Fhgjl7$$!+Go\\&>(Fabq$\"+)3<[?\"Fd`qF\\` jl7%F^hjl7$$!+DZIB\"*Fabq$\"+s@Od7Fd`qF\\`jl7%Fdhjl7$$!+d-qV6F^cq$\"+t *e*48Fd`qF\\`jl7%Fjhjl7$$!+etp=9F^cq$\"+V*=EO\"Fd`qF\\`jl7%F`ijl7$$!+5 8eU)eDF^cq$\"+$GB6_\"Fd`qF\\`jl7%Fbjjl7$$!+u#eA1$F^cq$\" +QO>u:Fd`qF\\`jl7%Fhjjl7$$!+4**eOOF^cq$\"++&>ui\"Fd`qF\\`jl7%F^[[m7$$! +X:9(G%F^cq$\"+x(G3o\"Fd`qF\\`jl7%Fd[[m7$$!+*\\_*=]F^cq$\"+:OXM&>Fd`qF\\`jl7%Fb][m7$$!+N&3K+\"Fgdq$\"+T(Gt+#Fd` qF\\`jl7%Fh][m7$$!+#zK=8\"Fgdq$\"+`MEj?Fd`qF\\`jl7%F^^[m7$$!+7Fcp7Fgdq $\"+Ao$)>@Fd`qF\\`jl7%Fd^[m7$$!+MLr:9Fgdq$\"+v\"er<#Fd`qF\\`jl7%Fj^[m7 $$!+#Gi\"p:Fgdq$\"+a(e`B#Fd`qF\\`jl7%F`_[m7$$!+L\\=G$Fd`qF\\`jl7%Fhf[m7$$!+T@K?Fi]q$\"+u;3%[$Fd`qF\\`jl7%Fdg[m7$$!+:OnzAFi]q $\"+O+jqMFd`qF\\`jl7%Fjg[m7$$!+\"zXAd#Fi]q$\"+6*[xFi]q$\"+uDR7NFd`qF\\`jl7%Fb\\\\m7$$!+aZi&G)Fi]q$\"+I!o[_$Fd` qF\\`jl7%Fh\\\\m7$$!+[4*H#))Fi]q$\"+M'Qp`$Fd`qF\\`jl7%F^]\\m7$$!+=3cO$ *Fi]q$\"+'yl&[NFd`qF\\`jl7%Fd]\\m7$$!+v&y+$)*Fi]q$\"+*RH(fNFd`qF\\`jl7 %Fj]\\m7$$!+ZUkI5Fd`q$\"+u'>/d$Fd`qF\\`jl7%F`^\\m7$$!+O]zw5Fd`q$\"+_Fj !e$Fd`qF\\`jl7%Ff^\\m7$$!+ngk@6Fd`q$\"+#Go.f$Fd`qF\\`jl7%F\\_\\m7$$!+o ,Nl6Fd`q$\"+=!G'*f$Fd`qF\\`jl7%Fb_\\m7$$!+F[.37Fd`q$\"+j\\T3OFd`qF\\`j l7%Fh_\\m7$$!+1x!)\\7Fd`q$\"+`Gt;OFd`qF\\`jl7%F^`\\m7$$!+,2w!H\"Fd`q$ \"+GeeCOFd`qF\\`jl7%Fd`\\m7$$!+(*H(4L\"Fd`q$\"+C#y>j$Fd`qF\\`jl7%Fj`\\ m7$$!+&R80P\"Fd`q$\"+PW\"*QOFd`qF\\`jl7%F`a\\m7$$!+E@W49Fd`q$\"+S))RXO Fd`qF\\`jl7%Ffa\\m7$$!+tA\"yW\"Fd`q$\"+FdV^OFd`qF\\`jl7%F\\b\\m7$$!+/4 n&[\"Fd`q$\"+x#Hql$Fd`qF\\`jl7%Fbb\\m7$$!+#**fI_\"Fd`q$\"+RN=iOFd`qF\\ `jl7%Fhb\\m7$$!+dr,g:Fd`q$\"+:C!pm$Fd`qF\\`jl7%F^c\\m7$$!+xid'f\"Fd`q$ \"+h'*=rOFd`qF\\`jl7%Fdc\\m7$$!+,!oFj\"Fd`q$\"+%))[]n$Fd`qF\\`jl7%Fjc \\m7$$!+n,io;Fd`q$\"+VN[yOFd`qF\\`jl7%F`d\\m7$$!+m\"eTq\"Fd`q$\"+ep\\ \"o$Fd`qF\\`jl7%Ffd\\m7$$!+T_SR\">Fd`q$\"+a6!4p$Fd`qF\\`jl7%Fjf\\ m7$$!+5w'e%>Fd`q$\"+P%[5p$Fd`qF\\`jl7%F`g\\m7$$!+#)\\fz>Fd`q$\"+Poz!p$ Fd`qF\\`jl7%Ffg\\m7$$!+-[;8?Fd`q$\"+`([,p$Fd`qF\\`jl7%F\\h\\m7$$!+*H*e Y?Fd`q$\"+Jl5*o$Fd`qF\\`jl7%Fbh\\m7$$!+W+))z?Fd`q$\"+lCn(o$Fd`qF\\`jl7 %Fhh\\m7$$!+>![I6#Fd`q$\"+;)[eo$Fd`qF\\`jl7%F^i\\m7$$!+&o.h9#Fd`q$\"+; yj$o$Fd`qF\\`jl7%Fdi\\m7$$!+Jq0z@Fd`q$\"+%oT5o$Fd`qF\\`jl7%Fji\\m7$$!+ Mw\">@#Fd`q$\"+KE1yOFd`qF\\`jl7%F`j\\m7$$!+'p%pWAFd`q$\"+zGquOFd`qF\\` jl7%Ffj\\m7$$!+\"4(RxAFd`q$\"+iY'4n$Fd`qF\\`jl7%F\\[]m7$$!+-M.5BFd`q$ \"+]-&om$Fd`qF\\`jl7%Fb[]m7$$!+a>hUBFd`q$\"+a>OiOFd`qF\\`jl7%Fh[]m7$$! +X39vBFd`q$\"+V@]dOFd`qF\\`jl7%F^\\]m7$$!+uzi2CFd`q$\"+bKF_OFd`qF\\`jl 7%Fd\\]m7$$!+m53SCFd`q$\"+9ynYOFd`qF\\`jl7%Fj\\]m7$$!+#p2DZ#Fd`q$\"+V% =2k$Fd`qF\\`jl7%F`]]m7$$!+&H:\\]#Fd`q$\"+!)yRMOFd`qF\\`jl7%Ff]]m7$$!+, 7JPDFd`q$\"+(**=xi$Fd`qF\\`jl7%F\\^]m7$$!+MEqpDFd`q$\"+8[o?OFd`qF\\`jl 7%Fb^]m7$$!+Ln4-EFd`q$\"+8&)H8OFd`qF\\`jl7%Fh^]m7$$!+i0]MEFd`q$\"+pMc0 OFd`qF\\`jl7%F^_]m7$$!+66#pm#Fd`q$\"+`K[(f$Fd`qF\\`jl7%Fd_]m7$$!+6`O*p #Fd`q$\"+j;1*e$Fd`qF\\`jl7%Fj_]m7$$!+G+%=t#Fd`q$\"+PFI!e$Fd`qF\\`jl7%F ``]m7$$!+s?NkFFd`q$\"+x2@rNFd`qF\\`jl7%Ff`]m7$$!+)=3pz#Fd`q$\"+o.zhNFd `qF\\`jl7%F\\a]m7$$!+a]^HGFd`q$\"+*RY?b$Fd`qF\\`jl7%Fba]m7$$!+v#z@'GFd `q$\"+#3%)>a$Fd`qF\\`jl7%Fha]m7$$!+qt!\\*GFd`q$\"+x*3;`$Fd`qF\\`jl7%F^ b]m7$$!+edqFHFd`q$\"+0q#4_$Fd`qF\\`jl7%Fdb]m7$$!+Z2egHFd`q$\"+tW%*4NFd `qF\\`jl7%Fjb]m7$$!+1&QN*HFd`q$\"+*3o')\\$Fd`qF\\`jl7%F`c]m7$$!+^]eEIF d`q$\"+#)\\5([$Fd`qF\\`jl7%Ffc]m7$$!+8isfIFd`q$\"+8FEvMFd`qF\\`jl7%F\\ d]m7$$!+9w'H4$Fd`q$\"+$H\\JY$Fd`qF\\`jl7%Fbd]m7$$!+JYJEJFd`q$\"+!>t2X$ Fd`qF\\`jl7%Fhd]m7$$!+nBxfJFd`q$\"+SL9QMFd`qF\\`jl7%F^e]m7$$!+2cM$>$Fd `q$\"+]\"p_U$Fd`qF\\`jl7%Fde]m7$$!+)yQqA$Fd`q$\"+-0;7MFd`qF\\`jl7%Fje] m7$$!+\\f&3E$Fd`q$\"+Yx#))R$Fd`qF\\`jl7%F`f]m7$$!+#p+[H$Fd`q$\"++V$Fd`q$\"+vj>HLFd`qF\\`jl7%F^h]m7$$!+h!GlY$Fd`q$\"+j7w9LFd` qF\\`jl7%Fdh]m7$$!+,RG,NFd`q$\"+,l=+LFd`qF\\`jl7%Fjh]m7$$!+.mj2C$Fd`qF\\ `jl7%Fbj]m7$$!+hZ2xOFd`q$\"+3epDKFd`qF\\`jl7%Fhj]m7$$!+PNh7PFd`q$\"+W= d5KFd`qF\\`jl7%F^[^m7$$!+YzE[PFd`q$\"+'G0a>$Fd`qF\\`jl7%Fd[^m7$$!+>9.% y$Fd`q$\"+!)*4-=$Fd`qF\\`jl7%Fj[^m7$$!++l*)>QFd`q$\"+u&**\\;$Fd`qF\\`j l7%F`\\^m7$$!+f[&e&QFd`q$\"+]uy\\JFd`qF\\`jl7%Ff\\^m7$$!+-t*=*QFd`q$\" +`meMJFd`qF\\`jl7%F\\]^m7$$!+\"z8!GRFd`q$\"+F)4%>JFd`qF\\`jl7%Fb]^m7$$ !+nM>kRFd`q$\"+[\"pU5$Fd`qF\\`jl7%Fh]^m7$$!+&oC/+%Fd`q$\"+ki<*3$Fd`qF \\`jl7%F^^^m7$$!+W]pOSFd`q$\"+GA9uIFd`qF\\`jl7%Fd^^m7$$!+J9*H2%Fd`q$\" +Uu+IFd`qF\\`jl7%Fb`^m7$$!+A6OaUFd`q$\"+pSq&) HFd`qF\\`jl7%Fh`^m7$$!+VZ]!H%Fd`q$\"+2-LrHFd`qF\\`jl7%F^a^m7$$!+aqcEVF d`q$\"+gf2dHFd`qF\\`jl7%Fda^m7$$!+a7`iVFd`q$\"+L[%H%HFd`qF\\`jl7%Fja^m 7$$!+$R!Q)R%Fd`q$\"+G$R*GHFd`qF\\`jl7%F`b^m7$$!+[u4MWFd`q$\"+H41:HFd`q F\\`jl7%Ffb^m7$$!+5ampWFd`q$\"+)458!HFd`qF\\`jl7%F\\c^m7$$!+ht10XFd`q$ \"+vio()GFd`qF\\`jl7%Fbc^m7$$!+llGSXFd`q$\"+\"*y=uGFd`qF\\`jl7%Fhc^m7$ $!+XlIvXFd`q$\"+&Q73'GFd`qF\\`jl7%F^d^m7$$!+n665YFd`q$\"+GibZGFd`qF\\` jl7%Fdd^m7$$!+?ZoWYFd`q$\"+k\\TMGFd`qF\\`jl7%Fjd^m7$$!+\"*>,zYFd`q$\"+ \\KQ@GFd`qF\\`jl7%F`e^m7$$!+O$yIr%Fd`q$\"+/\\X3GFd`qF\\`jl7%Ffe^m7$$!+ W(pou%Fd`q$\"+tHi&z#Fd`qF\\`jl7%F\\f^m7$$!+)*GP!y%Fd`q$\"+*yzGy#Fd`qF \\`jl7%Fbf^m7$$!+A_d8[Fd`q$\"+Vq@qFFd`qF\\`jl7%Fhf^m7$$!+C\\YY[Fd`q$\" +ieidFFd`qF\\`jl7%F^g^m7$$!+I5.z[Fd`q$\"+&)o4XFFd`qF\\`jl7%Fdg^m7$$!+2 ME6\\Fd`q$\"+W.iKFFd`qF\\`jl7%Fjg^m7$$!+!y_J%\\Fd`q$\"+Xh=?FFd`qF\\`jl 7%F`h^m7$$!+K2pu\\Fd`q$\"+[Ry2FFd`qF\\`jl7%Ffh^m7$$!+1(pe+&Fd`q$\"+ZKS &p#Fd`qF\\`jl7%F\\i^m7$$!+*)HoO]Fd`q$\"+VM.$o#Fd`qF\\`jl7%Fbi^m7$$!+)p Cr1&Fd`q$\"+>RmqEFd`qF\\`jl7%Fhi^m7$$!+Z(*=(4&Fd`q$\"+1TGeEFd`qF\\`jl7 %F^j^m7$$!+@Q(o7&Fd`q$\"+VN)ek#Fd`qF\\`jl7%Fdj^m7$$!+RLXL EFd`qF\\`jl7%Fjj^m7$$!+2a3&=&Fd`q$\"+'>z4i#Fd`qF\\`jl7%F`[_m7$$!+$y2O@ &Fd`q$\"+3bX3EFd`qF\\`jl7%Ff[_m7$$!+D)Q7vWDFd`qF\\`jl7%Fd]_m7$$!+eF]w`Fd`q$\"+**QtJDFd`qF\\`jl7%Fj] _m7$$!+.\"zAS&Fd`q$\"+bkg=DFd`qF\\`jl7%F`^_m7$$!+`bmFaFd`q$\"+!)>O0DFd `qF\\`jl7%Ff^_m7$$!+FOm_aFd`q$\"+hR*>\\#Fd`qF\\`jl7%F\\__m7$$!+b]FxaFd `q$\"+xi\\yCFd`qF\\`jl7%Fb__m7$$!+V<],bFd`q$\"+*=j[Y#Fd`qF\\`jl7%Fh__m 7$$!+RdMDbFd`q$\"+I$*3^CFd`qF\\`jl7%F^`_m7$$!+6#4)[bFd`q$\"+$pprV#Fd`q F\\`jl7%Fd`_m7$$!+7W*=d&Fd`q$\"+A'*4BCFd`qF\\`jl7%Fj`_m7$$!+mOg%f&Fd`q $\"+)yu)3CFd`qF\\`jl7%F`a_m7$$!+R$Rph&Fd`q$\"+&=\"\\%R#Fd`qF\\`jl7%Ffa _m7$$!+DQ!*QcFd`q$\"+2^%*zBFd`qF\\`jl7%F\\b_m7$$!+G&*\\gcFd`q$\"+OJBlB Fd`qF\\`jl7%Fbb_m7$$!+Y)G#Fd`qF\\`jl7%Fdf_m7$$!+)e?<*eFd`q$ \"+Mr#[<#Fd`qF\\`jl7%Fjf_m7$$!+Dds3fFd`q$\"+b#\\x:#Fd`qF\\`jl7%F`g_m7$ $!+z8RDfFd`q$\"+&3z/9#Fd`qF\\`jl7%Ffg_m7$$!+b\"><%fFd`q$\"+Ad,B@Fd`qF \\`jl7%F\\h_m7$$!+\"f5x&fFd`q$\"+J$e`5#Fd`qF\\`jl7%Fbh_m7$$!+arOtfFd`q $\"+Ph](3#Fd`qF\\`jl7%Fhh_m7$$!+Z-p))fFd`q$\"+8%e%p?Fd`qF\\`jl7%F^i_m7 $$!+67o.gFd`q$\"+sW@^?Fd`qF\\`jl7%Fdi_m7$$!+E8M=gFd`q$\"+dOxK?Fd`qF\\` jl7%Fji_m7$$!+:=nKgFd`q$\"+O`89?Fd`qF\\`jl7%F`j_m7$$!+ZQnYgFd`q$\"+\"* ))H&*>Fd`qF\\`jl7%Ffj_m7$$!+U&[.1'Fd`q$\"+6PEw>Fd`qF\\`jl7%F\\[`m7$$!+ tpptgFd`q$\"+%=Hq&>Fd`qF\\`jl7%Fb[`m7$$!+t,s'3'Fd`q$\"+!p%fP>Fd`qF\\`j l7%Fh[`m7$$!+N\">%*4'Fd`q$\"+(ffz\">Fd`qF\\`jl7%F^\\`m7$$!+A[z6hFd`q$ \"+[K7)*=Fd`qF\\`jl7%Fd\\`m7$$!+q\"[Q7'Fd`q$\"+i\\3y=Fd`qF\\`jl7%Fj\\` m7$$!+!4!eNhFd`q$\"+>S%y&=Fd`qF\\`jl7%F`]`m7$$!+\"[\"*p9'Fd`q$\"+j'*RP =Fd`qF\\`jl7%Ff]`m7$$!+HK3ehFd`q$\"+*3^n\"=Fd`qF\\`jl7%F\\^`m7$$!+DH:Fd`qF\\`jl7%Fjb`m7$$!+d*R.G'Fd`q $\"+_8i0:Fd`qF\\`jl7%F`c`m7$$!+[(fnG'Fd`q$\"+1bw\"[\"Fd`qF\\`jl7%Ffc`m 7$$!+u#zGH'Fd`q$\"+GAod9Fd`qF\\`jl7%F\\d`m7$$!+:1q)H'Fd`q$\"+[#pLV\"Fd `qF\\`jl7%Fbd`m7$$!+NgA/jFd`q$\"+$>C)39Fd`qF\\`jl7%Fhd`m7$$!+,!e%4jFd` q$\"+#fWSQ\"Fd`qF\\`jl7%F^e`m7$$!+,#*R9jFd`q$\"+\")y-f8Fd`qF\\`jl7%Fde `m7$$!+lD0>jFd`q$\"+/9xL8Fd`qF\\`jl7%Fje`m7$$!+z7UBjFd`q$\"+ICF38Fd`qF \\`jl7%F`f`m7$$!+9)3vK'Fd`q$\"+^\"GDG\"Fd`qF\\`jl7%Fff`m7$$!+S*=8L'Fd` q$\"+/d`c7Fd`qF\\`jl7%F\\g`m7$$!+^d&[L'Fd`q$\"+x@HI7Fd`qF\\`jl7%Fbg`m7 $$!+%oB\"QjFd`q$\"+IYz.7Fd`qF\\`jl7%Fhg`m7$$!+Yv7TjFd`q$\"+5,/x6Fd`qF \\`jl7%F^h`m7$$!+HD(QM'Fd`q$\"+xc-]6Fd`qF\\`jl7%Fdh`m7$$!+MUOYjFd`q$\" +E%[F7\"Fd`qF\\`jl7%Fjh`m7$$!+&p3'[jFd`q$\"+9b?&4\"Fd`qF\\`jl7%F`i`m7$ $!+!R71N'Fd`q$\"+&>%Rn5Fd`qF\\`jl7%Ffi`m7$$!+nAQ_jFd`q$\"+^=JR5Fd`qF\\ `jl7%F\\j`m7$$!+bd#RN'Fd`q$\"+Lg&4,\"Fd`qF\\`jl7%Fbj`m7$$!+x2DbjFd`q$ \"+=]CB)*Fi]qF\\`jl7%Fhj`m7$$!+idOcjFd`q$\"+]F:M&*Fi]qF\\`jl7%F^[am7$$ !+\\'zsN'Fd`q$\"+ZnEU#*Fi]qF\\`jl7%Fd[am7$$!+!*=+ejFd`q$\"+0QdZ*)Fi]qF \\`jl7%Fj[am7$$!+XCaejFd`q$\"+][1]')Fi]qF\\`jl7%F`\\am7$$!+v<\"*ejFd`q $\"+6bt\\$)Fi]qF\\`jl7%Ff\\am7$$!+F37fjFd`q$\"+?neY!)Fi]qF\\`jl7%F\\]a m7$$!+25=fjFd`q$\"+A`iSxFi]qF\\`jl7%Fb]am7$$!+dT5fjFd`q$\"+'ok=V(Fi]qF \\`jl7%Fh]am7$$!+8D!*ejFd`q$\"+@`K?rFi]qF\\`jl7%F^^am7$$!+g')eejFd`q$ \"+na.1oFi]qF\\`jl7%Fd^am7$$!+\"[v\"ejFd`q$\"+i;.*['Fi]qF\\`jl7%Fj^am7 $$!+\"4wwN'Fd`q$\"+y#f$phFi]qF\\`jl7%F`_am7$$!+qP5djFd`q$\"+)*H2ZeFi]q F\\`jl7%Ff_am7$$!+#)=ZcjFd`q$\"+KsBAbFi]qF\\`jl7%F\\`am7$$!+$z$zbjFd`q $\"+ik#\\>&Fi]qF\\`jl7%Fb`am7$$!+\"y#3bjFd`q$\"+'\\D_'[Fi]qF\\`jl7%Fh` am7$$!+M>NajFd`q$\"+D'HK`%Fi]qF\\`jl7%F^aam7$$!+dSh`jFd`q$\"+xY/*>%Fi] qF\\`jl7%Fdaam7$$!+n:)GN'Fd`q$\"+apyiQFi]qF\\`jl7%Fjaam7$$!+'Qm@N'Fd`q $\"+cIeCNFi]qF\\`jl7%F`bam7$$!+U)z9N'Fd`q$\"+!epX=$Fi]qF\\`jl7%Ffbam7$ $!+nD$3N'Fd`q$\"+)p#*G%GFi]qF\\`jl7%F\\cam7$$!++WB]jFd`q$\"+Cwq*\\#Fi] qF\\`jl7%Fbcam7$$!+$H%p\\jFd`q$\"+az)[jFd`q$\"+'\\wFY\"Fi]qF\\`jl7%Fd dam7$$!+[o\\[jFd`q$\"+LN'F d`q$!+fMykSFi]qF\\`jl7%Fdjam7$$!+Ap0ajFd`q$!+c?\")*R%Fi]qF\\`jl7%Fjjam 7$$!+(4#zajFd`q$!+YopKZFi]qF\\`jl7%F`[bm7$$!+,E^bjFd`q$!+*zFL1&Fi]qF\\ `jl7%Ff[bm7$$!+Wa?cjFd`q$!+4ag\"R&Fi]qF\\`jl7%F\\\\bm7$$!+8u&oN'Fd`q$! +(oSur&Fi]qF\\`jl7%Fb\\bm7$$!+n^XdjFd`q$!+[]vSgFi]qF\\`jl7%Fh\\bm7$$!+ I`)zN'Fd`q$!+*3![hjFi]qF\\`jl7%F^]bm7$$!+yXVejFd`q$!+\"Gd&zmFi]qF\\`jl 7%Fd]bm7$$!+@(*yejFd`q$!+%eP\\*pFi]qF\\`jl7%Fj]bm7$$!+xx.fjFd`q$!+')4e 2tFi]qF\\`jl7%F`^bm7$$!+Mg;fjFd`q$!+%*fX:[ajFd`q$!+LiO&***Fi]qF\\`jl7%Ffabm7$$!+'REIN'Fd`q$!+rC+G5Fd`qF\\`j l7%F\\bbm7$$!+%*zM^jFd`q$!+AR>c5Fd`qF\\`jl7%Fbbbm7$$!+'oQ%\\jFd`q$!+XL 6%3\"Fd`qF\\`jl7%Fhbbm7$$!+N8HZjFd`q$!+(Hj<6\"Fd`qF\\`jl7%F^cbm7$$!+w# **[M'Fd`q$!+wk9R6Fd`qF\\`jl7%Fdcbm7$$!++jDUjFd`q$!+!pli;\"Fd`qF\\`jl7% Fjcbm7$$!+OmNRjFd`q$!+CQ7$>\"Fd`qF\\`jl7%F`dbm7$$!+H\\>OjFd`q$!+1Qs>7F d`qF\\`jl7%Ffdbm7$$!+@iwKjFd`q$!+(eogC\"Fd`qF\\`jl7%F\\ebm7$$!+Kf1HjFd `q$!+:6;s7Fd`qF\\`jl7%Fbebm7$$!+I)*3DjFd`q$!+9V+)H\"Fd`qF\\`jl7%Fhebm7 $$!+?S$3K'Fd`q$!+q5gB8Fd`qF\\`jl7%F^fbm7$$!+:\\H;jFd`q$!+8U&*[8Fd`qF\\ `jl7%Fdfbm7$$!+:#p9J'Fd`q$!+3l1u8Fd`qF\\`jl7%Fjfbm7$$!+!*QN1jFd`q$!+S1 %*)R\"Fd`qF\\`jl7%F`gbm7$$!+gh%4I'Fd`q$!+6#zNU\"Fd`qF\\`jl7%Ffgbm7$$!+ qMC&H'Fd`q$!+JZ)zW\"Fd`qF\\`jl7%F\\hbm7$$!+xMC*G'Fd`q$!+7'f@Z\"Fd`qF\\ `jl7%Fbhbm7$$!+JS%HG'Fd`q$!+kh5'\\\"Fd`qF\\`jl7%Fhhbm7$$!+bJMwiFd`q$!+ )fE)>:Fd`qF\\`jl7%F^ibm7$$!+L!R%piFd`q$!+;IKV:Fd`qF\\`jl7%Fdibm7$$!+!* *HAE'Fd`q$!+>ufm:Fd`qF\\`jl7%Fjibm7$$!+$[9ZD'Fd`q$!+*p^'*e\"Fd`qF\\`jl 7%F`jbm7$$!+\"3\"*oC'Fd`q$!+^w[7;Fd`qF\\`jl7%Ffjbm7$$!+d%e(QiFd`q$!+jp 5N;Fd`qF\\`jl7%F\\[cm7$$!+s`JIiFd`q$!+G7^d;Fd`qF\\`jl7%Fb[cm7$$!+o1c@i Fd`q$!+U>qz;Fd`qF\\`jl7%Fh[cm7$$!+\\K\\7iFd`q$!+30o,'Fd`q$!+uj+XFd`qF\\`jl7%Fj_cm7$$ !+6'Q=4'Fd`q$!+hZwH>Fd`qF\\`jl7%F``cm7$$!+C_%*ygFd`q$!+<%z#\\>Fd`qF\\` jl7%Ff`cm7$$!+3qslgFd`q$!+cQfo>Fd`qF\\`jl7%F\\acm7$$!+XH=_gFd`q$!+-(3x )>Fd`qF\\`jl7%Fbacm7$$!+z>JQgFd`q$!+oXi1?Fd`qF\\`jl7%Fhacm7$$!+5I6CgFd `q$!+k?MD?Fd`qF\\`jl7%F^bcm7$$!+*)[e4gFd`q$!+2='Q/#Fd`qF\\`jl7%Fdbcm7$ $!+:ks%*fFd`q$!+?W=i?Fd`qF\\`jl7%Fjbcm7$$!+Jj`zfFd`q$!+^0J!3#Fd`qF\\`j l7%F`ccm7$$!+@L,kfFd`q$!+s3C)4#Fd`qF\\`jl7%Ffccm7$$!+2g:[fFd`q$!+%4wf6 #Fd`qF\\`jl7%F\\dcm7$$!+ZH'>$fFd`q$!+rp^L@Fd`qF\\`jl7%Fbdcm7$$!+LEV:fF d`q$!+9V'3:#Fd`qF\\`jl7%Fhdcm7$$!+([j&)*eFd`q$!+(**=!o@Fd`qF\\`jl7%F^e cm7$$!+hQN\")eFd`q$!+n>)\\=#Fd`qF\\`jl7%Fdecm7$$!+Q?!Q'eFd`q$!+eUv,AFd `qF\\`jl7%Fjecm7$$!+Fi!f%eFd`q$!+**pL=AFd`qF\\`jl7%F`fcm7$$!+oXmFeFd`q $!+E9tMAFd`qF\\`jl7%Fffcm7$$!+G^24eFd`q$!+%*)Q4D#Fd`qF\\`jl7%F\\gcm7$$ !+1f8!z&Fd`q$!+!*3'pE#Fd`qF\\`jl7%Fbgcm7$$!+M[%3x&Fd`q$!+Z!*z#G#Fd`qF \\`jl7%Fhgcm7$$!+z(*>^dFd`q$!+`^X)H#Fd`qF\\`jl7%F^hcm7$$!+[&)>JdFd`q$! +q6$RJ#Fd`qF\\`jl7%Fdhcm7$$!+%*)Q3r&Fd`q$!+X#H#HBFd`qF\\`jl7%Fjhcm7$$! +?&=,p&Fd`q$!+F&Fd`q$!+wj(fh #Fd`qF\\`jl7%Fh`dm7$$!++[yn^Fd`q$!+I$o%GEFd`qF\\`jl7%F^adm7$$!+_*R'Q^F d`q$!+%>:4k#Fd`qF\\`jl7%Fdadm7$$!+5$4\"4^Fd`q$!+_pK`EFd`qF\\`jl7%Fjadm 7$$!+vh>z]Fd`q$!+=QrlEFd`qF\\`jl7%F`bdm7$$!+kX!*[]Fd`q$!+$>'3yEFd`qF\\ `jl7%Ffbdm7$$!+]#R#=]Fd`q$!+7YX!p#Fd`qF\\`jl7%F\\cdm7$$!+%z0s)\\Fd`q$! +!oHGq#Fd`qF\\`jl7%Fbcdm7$$!+u0\"e&\\Fd`q$!+6?A:FFd`qF\\`jl7%Fhcdm7$$! +331C\\Fd`q$!+e@kFFFd`qF\\`jl7%F^ddm7$$!+nX'>*[Fd`q$!+O05SFFd`qF\\`jl7 %Fdddm7$$!+%yI&f[Fd`q$!+Ytg_FFd`qF\\`jl7%Fjddm7$$!+]#pn#[Fd`q$!+(\\s^w #Fd`qF\\`jl7%F`edm7$$!+11p$z%Fd`q$!+Bb!yx#Fd`qF\\`jl7%Ffedm7$$!+=jIgZF d`q$!+,b^!z#Fd`qF\\`jl7%F\\fdm7$$!+`'Gms%Fd`q$!+w4J.GFd`qF\\`jl7%Fbfdm 7$$!+R1n#p%Fd`q$!+!))*>;GFd`qF\\`jl7%Fhfdm7$$!+:gWeYFd`q$!+g%*=HGFd`qF \\`jl7%F^gdm7$$!+!=pRi%Fd`q$!+:iGUGFd`qF\\`jl7%Fdgdm7$$!+E^D*e%Fd`q$!+ He\\bGFd`qF\\`jl7%Fjgdm7$$!+w$>Vb%Fd`q$!+?J#)oGFd`qF\\`jl7%F`hdm7$$!+- z<>XFd`q$!+!*>F#)GFd`qF\\`jl7%Ffhdm7$$!+aq%Q[%Fd`q$!+$RXe*GFd`qF\\`jl7 %F\\idm7$$!+uMM[WFd`q$!+(HX&4HFd`qF\\`jl7%Fbidm7$$!+>So7WFd`q$!+nEPBHF d`qF\\`jl7%Fhidm7$$!+qc)oP%Fd`q$!+]uKPHFd`qF\\`jl7%F^jdm7$$!+aa'4M%Fd` q$!+s&39&HFd`qF\\`jl7%Fdjdm7$$!+h.%\\I%Fd`q$!+ORhlHFd`qF\\`jl7%Fjjdm7$ $!+hs#)oUFd`q$!+O/%*zHFd`qF\\`jl7%F`[em7$$!+FGkKUFd`q$!+tRQ%*HFd`qF\\` jl7%Ff[em7$$!+hMS'>%Fd`q$!+t%R*3IFd`qF\\`jl7%F\\\\em7$$!+?_7gTFd`q$!+C 4gBIFd`qF\\`jl7%Fb\\em7$$!+ZP#Q7%Fd`q$!+-9OQIFd`qF\\`jl7%Fh\\em7$$!+4U ^(3%Fd`q$!+:J@`IFd`qF\\`jl7%F^]em7$$!+L7@^SFd`q$!+Wu9oIFd`qF\\`jl7%Fd] em7$$!+])G\\,%Fd`q$!+!*\\:$3$Fd`qF\\`jl7%Fj]em7$$!+Z/oyRFd`q$!+EcA)4$F d`qF\\`jl7%F`^em7$$!+9(yC%RFd`q$!+`&[L6$Fd`qF\\`jl7%Ff^em7$$!+-cL1RFd` q$!+`B^GJFd`qF\\`jl7%F\\_em7$$!+!Hi-(QFd`q$!+a]qVJFd`qF\\`jl7%Fb_em7$$ !+Y\"pU$QFd`q$!+#>9*eJFd`qF\\`jl7%Fh_em7$$!+0dO)z$Fd`q$!+to7uJFd`qF\\` jl7%F^`em7$$!+X1ciPFd`q$!+V)H$*=$Fd`qF\\`jl7%Fd`em7$$!+s<'os$Fd`q$!+`& 4X?$Fd`qF\\`jl7%Fj`em7$$!+6gF\"p$Fd`q$!+EAl>KFd`qF\\`jl7%F`aem7$$!+-%4 el$Fd`q$!+IRuMKFd`qF\\`jl7%Ffaem7$$!+,rY?OFd`q$!+O1x\\KFd`qF\\`jl7%F\\ bem7$$!+%R`_e$Fd`q$!+%H=ZE$Fd`qF\\`jl7%Fbbem7$$!+1<<]NFd`q$!+))GdzKFd` qF\\`jl7%Fhbem7$$!+AYA:NFd`q$!+,0K%H$Fd`qF\\`jl7%F^cem7$$!+8RT![$Fd`q$ !+pt%*3LFd`qF\\`jl7%Fdcem7$$!+m0uXMFd`q$!+Q*RMK$Fd`qF\\`jl7%Fjcem7$$!+ :[?6MFd`q$!+9\\yPLFd`qF\\`jl7%F`dem7$$!+$=1oP$Fd`q$!++$p>N$Fd`qF\\`jl7 %Ffdem7$$!+:NaULFd`q$!+W/)fO$Fd`qF\\`jl7%F\\eem7$$!+H]T3LFd`q$!+pg!)zL Fd`qF\\`jl7%Fbeem7$$!+]$=WF$Fd`q$!+-VV$R$Fd`qF\\`jl7%Fheem7$$!+j0bSKFd `q$!+(p`oS$Fd`qF\\`jl7%F^fem7$$!+_#3o?$Fd`q$!+cK0?MFd`qF\\`jl7%Fdfem7$ $!+Yv=tJFd`q$!+UC-LMFd`qF\\`jl7%Fjfem7$$!+kToRJFd`q$!+$=^dW$Fd`qF\\`jl 7%F`gem7$$!+aMH1JFd`q$!+%))H#eMFd`qF\\`jl7%Ffgem7$$!+Q/,tIFd`q$!+>%\\/ Z$Fd`qF\\`jl7%F\\hem7$$!+U)H)RIFd`q$!+N6S#[$Fd`qF\\`jl7%Fbhem7$$!+Rhu1 IFd`q$!+So2%\\$Fd`qF\\`jl7%Fhhem7$$!+xNvtHFd`q$!+#zoa]$Fd`qF\\`jl7%F^i em7$$!+8i%3%HFd`q$!+$ppl^$Fd`qF\\`jl7%Fdiem7$$!+Mz,3HFd`q$!+oEPFNFd`qF \\`jl7%Fjiem7$$!+'[i_(GFd`q$!+^7(y`$Fd`qF\\`jl7%F`jem7$$!+$\\tD%GFd`q$ !+q$f![NFd`qF\\`jl7%Ffjem7$$!+tW%*4GFd`q$!+B8$zb$Fd`qF\\`jl7%F\\[fm7$$ !+a)otx#Fd`q$!+g<[nNFd`qF\\`jl7%Fb[fm7$$!+()*R[u#Fd`q$!+ncqwNFd`qF\\`j l7%Fh[fm7$$!+`6N7FFd`q$!+Q$)f&e$Fd`qF\\`jl7%F^\\fm7$$!+nb*)zEFd`q$!+h` :%f$Fd`qF\\`jl7%Fd\\fm7$$!+'Qmuk#Fd`q$!+'fsBg$Fd`qF\\`jl7%Fj\\fm7$$!+/ n0:EFd`q$!+\\hC5OFd`qF\\`jl7%F`]fm7$$!+c&fEe#Fd`q$!+iBx#Fd`q$!+!*e*)zOFd`qF\\`jl7%F^ bfm7$$!+dnHf@Fd`q$!+cak#o$Fd`qF\\`jl7%Fdbfm7$$!+$>$GE@Fd`q$!+u2,&o$Fd` qF\\`jl7%Fjbfm7$$!+q8;$4#Fd`q$!+I'*)po$Fd`qF\\`jl7%F`cfm7$$!+?5#*f?Fd` q$!+)zz&)o$Fd`qF\\`jl7%Ffcfm7$$!+y8bE?Fd`q$!+K!z(*o$Fd`qF\\`jl7%F\\dfm 7$$!+;6/$*>Fd`q$!+\\]e!p$Fd`qF\\`jl7%Fbdfm7$$!+x#y$f>Fd`q$!+Fb*4p$Fd`q F\\`jl7%Fhdfm7$$!+--bD>Fd`q$!+*335p$Fd`qF\\`jl7%F^efm7$$!+XMa\"*=Fd`q$ !+(H?1p$Fd`qF\\`jl7%Fdefm7$$!+xOMd=Fd`q$!+S'H)*o$Fd`qF\\`jl7%Fjefm7$$! +\"eNH#=Fd`q$!+CNj)o$Fd`qF\\`jl7%F`ffm7$$!+JFI)y\"Fd`q$!+o#Hqo$Fd`qF\\ `jl7%Ffffm7$$!+auU`OFd`qF\\`jl7%Fj[gm7$$!+Iz%[A\"Fd`q$!+!=)z6OFd`qF\\`jl7%F`\\gm7$$ !+T*RD=\"Fd`q$!+M$*>.OFd`qF\\`jl7%Ff\\gm7$$!+*pd#R6Fd`q$!+D\"HTf$Fd`qF \\`jl7%F\\]gm7$$!+;A)[4\"Fd`q$!+2Ue%e$Fd`qF\\`jl7%Fb]gm7$$!+z:F\\5Fd`q $!+[AcuNFd`qF\\`jl7%Fh]gm7$$!+]UD-5Fd`q$!+EE1kNFd`qF\\`jl7%F^^gm7$$!+m g>O&*Fi]q$!+wv3`NFd`qF\\`jl7%Fd^gm7$$!+`@/J!*Fi]q$!+NTkTNFd`qF\\`jl7%F j^gm7$$!+$*=r.&)Fi]q$!+nvuHNFd`qF\\`jl7%F`_gm7$$!+1)y(\\zFi]q$!+'pFu^$ Fd`qF\\`jl7%Ff_gm7$$!+>qAjtFi]q$!+:8u/NFd`qF\\`jl7%F\\`gm7$$!+*p>bt'Fi ]q$!+@&)z\"\\$Fd`qF\\`jl7%Fb`gm7$$!+<;$Q0'Fi]q$!+lY#)yMFd`qF\\`jl7%Fh` gm7$$!+ybK(H&Fi]q$!+J*GjY$Fd`qF\\`jl7%F^agm7$$!+#4T\"GWFi]q$!+uNnbMFd` qF\\`jl7%Fdagm7$Fci[m$!+7!3=X$Fd`qF\\`jl7%Fjagm7$F]i[m$!+\"GtKX$Fd`qF \\`jl7%F^bgm7$Fgh[m$!+S!GjX$Fd`qF\\`jl7%Fbbgm7$Fah[m$!+6*[$Fd`qF \\`jl7%Fbdgm7$Fcf[m$!+&[+E;$Fd`qF\\`jl7%Ffdgm7$F]f[m$!+=M0MJFd`qF\\`jl 7%Fjdgm7$Fge[m$!+.0;1JFd`qF\\`jl7%F^egm7$Fae[m$!+%Hl%zIFd`qF\\`jl7%Fbe gm7$F[e[m$!+M*fS0$Fd`qF\\`jl7%Ffegm7$Fed[m$!+Pu')HIFd`qF\\`jl7%Fjegm7$ F_d[m$!+!ojn+$Fd`qF\\`jl7%F^fgm7$Fic[m$!+gBi%)HFd`qF\\`jl7%Fbfgm7$Fcc[ m$!+(RHL'HFd`qF\\`jl7%Fffgm7$$!+K`Y5AFgdq$!+aDgmGFd`qF\\`jl7%Fjfgm7$$! +d]QiCFgdq$!+gh3\"y#Fd`qF\\`jl7%F`ggm7$$!+6_E>DFgdq$!+x,#Gq#Fd`qF\\`jl 7%Ffggm7$$!+zRjtCFgdq$!+\\\"['HEFd`qF\\`jl7%F\\hgm7$$!+nqvqBFgdq$!+&*y DgDFd`qF\\`jl7%Fbhgm7$$!+jfgNAFgdq$!+.Zy$\\#Fd`qF\\`jl7%Fhhgm7$$!+`D5$ 3#Fgdq$!+AXiHCFd`qF\\`jl7%F^igm7$$!+HImA>Fgdq$!+ljLnBFd`qF\\`jl7%Fdigm 7$$!+Fd`qF\\`jl7% F^\\hm7$$!+.pDczF^cq$!+=5(z!>Fd`qF\\`jl7%Fd\\hm7$$!+sDwOpF^cq$!+@!yL&= Fd`qF\\`jl7%Fj\\hm7$$!+EO$3,'F^cq$!+.09*z\"Fd`qF\\`jl7%F`]hm7$$!+X1[v^ F^cq$!+:w?X\"F^cq$!+9b[?8Fd`qF\\`j l7%Ff`hm7$$!+Wlf`&*Fabq$!+bq(yE\"Fd`qF\\`jl7%F\\ahm7$$!+![>Db(Fabq$!+e CK:7Fd`qF\\`jl7%Fbahm7$$!++'>*)*eFabq$!+n;\"G;\"Fd`qF\\`jl7%Fhahm7$$!+ s5XZXFabq$!+$=O.6\"Fd`qF\\`jl7%F^bhm7$$!+2R*fX$Fabq$!+q)))y0\"Fd`qF\\` jl7%Fdbhm7$$!+_E&ee#Fabq$!+)yja+\"Fd`qF\\`jl7%Fjbhm7$$!+Sm(=!>Fabq$!+] .cI&*Fi]qF\\`jl7%F`chm7$$!+^p]s8Fabq$!+!3;m+*Fi]qF\\`jl7%Ffchm7$$!+-%e vp*Fa`q$!+OFx#[)Fi]qF\\`jl7%F\\dhm7$$!+&Q]7p'Fa`q$!+SX+fzFi]qF\\`jl7%F bdhm7$$!+\\&*p%\\%Fa`q$!+R7HNuFi]qF\\`jl7%Fhdhm7$$!+vO#Fi]qF\\`jl7%Fdhhm7$$!+07Q&G'F0$!+)3;bn\"Fi]qF\\`jl7%Fjhh m7$$!+`:x[mFeq$!+2t\">:\"Fi]qF\\`jl7%F`ihm7$$!+FU8`Z5FgdqF\\`jl7%F\\jhm7$F($\"+0-z)=%FgdqF\\`jl7%Fh f[m7$F]dgm$\"+Aa7:KFd`q7$$\"#OFh^v$\"$u$Fh^v7%Fdjhm7$$\"+G[;-5Fi]q$\"+ 8?=NKFd`qFgjhm7%F][im7$$\"+n(31D\"Fi]q$\"+qak]KFd`qFgjhm7%Fc[im7$$\"+$ >Hp[\"Fi]q$\"+egiiKFd`qFgjhm7%Fi[im7$$\"+t\">rq\"Fi]q$\"+Ez@sKFd`qFgjh m7%F_\\im7$$\"+\")4O6>Fi]q$\"+-x>!G$Fd`qFgjhm7%Fe\\im7$$\"+ni=,@Fi]q$ \"+'HqqG$Fd`qFgjhm7%F[]im7$$\"+e6NyAFi]q$\"+/9;$H$Fd`qFgjhm7%Fa]im7$$ \"+_4ZWCFi]q$\"+E_o)H$Fd`qFgjhm7%Fg]im7$$\"+S&=3:$Fi]q$\"+I\"*f@LFd`qF gjhm7%F]^im7$$\"+DEy;PFi]q$\"+Ew\"4M$Fd`qFgjhm7%Fc^im7$$\"+*emC>%Fi]q$ \"+Eg\")eLFd`qFgjhm7%Fi^im7$$\"++t!Rg%Fi]q$\"+q2.wLFd`qFgjhm7%F__im7$$ \"+sM]m\\Fi]q$\"+i-(GR$Fd`qFgjhm7%Fe_im7$$\"+8*>-H&Fi]q$\"+K1[4MFd`qFg jhm7%F[`im7$$\"+4[$>e&Fi]q$\"+>j$fU$Fd`qFgjhm7%Fa`im7$$\"+!GIm%eFi]q$ \"+1mFUMFd`qFgjhm7%Fg`im7$$\"+?A0)3'Fi]q$\"+5A_eMFd`qFgjhm7%F]aim7$$\" +b>54jFi]q$\"+&\\$ouMFd`qFgjhm7%Fcaim7$$\"+@<37lFi]q$\"+?[w!\\$Fd`qFgj hm7%Fiaim7$$\"+ra&))p'Fi]q$\"+kpw1NFd`qFgjhm7%F_bim7$$\"+V\"e4(oFi]q$ \"+E&)oANFd`qFgjhm7%Febim7$$\"+CAnHqFi]q$\"+yn_QNFd`qFgjhm7%F[cim7$$\" +DG3wrFi]q$\"+!=yUb$Fd`qFgjhm7%Facim7$$\"+6#=6J(Fi]q$\"+0(Q*pNFd`qFgjh m7%Fgcim7$$\"+5-eNuFi]q$\"+^S]&e$Fd`qFgjhm7%F]dim7$$\"+Uu;]vFi]q$\"+n( p4g$Fd`qFgjhm7%Fcdim7$$\"+BL\\bwFi]q$\"+Z8L;OFd`qFgjhm7%Fidim7$$\"+E-5 _xFi]q$\"+yUeJOFd`qFgjhm7%F_eim7$$\"+_2ZSyFi]q$\"+ySsYOFd`qFgjhm7%Feei m7$$\"+\"zO5#zFi]q$\"+;juhOFd`qFgjhm7%F[fim7$$\"+0p=%*zFi]q$\"+DmkwOFd `qFgjhm7%Fafim7$$\"+mBFg!)Fi]q$\"+02U\"p$Fd`qFgjhm7%Fgfim7$$\"+s@h>\") Fi]q$\"+HV11PFd`qFgjhm7%F]gim7$$\"+8s\\s\")Fi]q$\"+QLd?PFd`qFgjhm7%Fcg im7$$\"+xP>>#)Fi]q$\"+VO%\\t$Fd`qFgjhm7%Figim7$$\"+RFd` qFgjhm7%F]]jm7$$\"+QXF3$)Fi]q$\"+P]AJRFd`qFgjhm7%Fc]jm7$$\"+M['*z#)Fi] q$\"+;m&G%RFd`qFgjhm7%Fi]jm7$$\"+(\\&*zC)Fi]q$\"+ZcGaRFd`qFgjhm7%F_^jm 7$$\"+9DX7#)Fi]q$\"+I%3b'RFd`qFgjhm7%Fe^jm7$$\"+V\"=M<)Fi]q$\"+R7_wRFd `qFgjhm7%F[_jm7$$\"+#>r48)Fi]q$\"+A.K()RFd`qFgjhm7%Fa_jm7$$\"+ms=&3)Fi ]q$\"+#*=!z*RFd`qFgjhm7%Fg_jm7$$\"++!Rh.)Fi]q$\"+D@E3SFd`qFgjhm7%F]`jm 7$$\"+hi*Q)zFi]q$\"+brR=SFd`qFgjhm7%Fc`jm7$$\"+Xj_GzFi]q$\"+kIIGSFd`qF gjhm7%Fi`jm7$$\"+VT4qyFi]q$\"+&)e(z.%Fd`qFgjhm7%F_ajm7$$\"+2Bm3yFi]q$ \"+*e6u/%Fd`qFgjhm7%Feajm7$$\"+$R\"HWxFi]q$\"+$31m0%Fd`qFgjhm7%F[bjm7$ $\"+#**Rqn(Fi]q$\"+._blSFd`qFgjhm7%Fabjm7$$\"+g['pg(Fi]q$\"+2ZDuSFd`qF gjhm7%Fgbjm7$$\"+E57MvFi]q$\"+s-q#3%Fd`qFgjhm7%F]cjm7$$\"+)*=ceuFi]q$ \"+zu)34%Fd`qFgjhm7%Fccjm7$$\"+e$R.Q(Fi]q$\"+8=\"))4%Fd`qFgjhm7%Ficjm7 $$\"+cQ]*H(Fi]q$\"+a'ok5%Fd`qFgjhm7%F_djm7$$\"+#[/h@(Fi]q$\"+jK&Q6%Fd` qFgjhm7%Fedjm7$$\"+^!*=IrFi]q$\"+z2'47%Fd`qFgjhm7%F[ejm7$$\"+nT!=/(Fi] q$\"+2iyFTFd`qFgjhm7%Faejm7$$\"+)G&*4&pFi]q$\"+2WKMTFd`qFgjhm7%Fgejm7$ $\"+'y1y&oFi]q$\"+$3q09%Fd`qFgjhm7%F]fjm7$$\"+'*>GinFi]q$\"+rx^YTFd`qF gjhm7%Fcfjm7$$\"+tKYkmFi]q$\"+E=;_TFd`qFgjhm7%Fifjm7$$\"+I?RklFi]q$\"+ 2k\\dTFd`qFgjhm7%F_gjm7$$\"+!y3@Y'Fi]q$\"+ia^iTFd`qFgjhm7%Fegjm7$$\"+u JldjFi]q$\"+7F@nTFd`qFgjhm7%F[hjm7$$\"+LS1^iFi]q$\"+J;erTFd`qFgjhm7%Fa hjm7$$\"+r$zB9'Fi]q$\"+DahvTFd`qFgjhm7%Fghjm7$$\"+GkjJgFi]q$\"+;qIzTFd `qFgjhm7%F]ijm7$$\"+#or)=fFi]q$\"+2!\\E=%Fd`qFgjhm7%Fcijm7$$\"+n37/eFi ]q$\"+nOj&=%Fd`qFgjhm7%Fiijm7$$\"+&)*=uo&Fi]q$\"+$*GD)=%Fd`qFgjhm7%F_j jm7$$\"+5.!)obFi]q$\"+#=)\\!>%Fd`qFgjhm7%Fejjm7$$\"+\"R)H[aFi]q$\"+\"f gB>%Fd`qFgjhm7%F[[[n7$$\"+Xg%fK&Fi]q$\"+.2$Q>%Fd`qFgjhm7%Fa[[n7$$\"+[` x,_Fi]q$\"+w&)*[>%Fd`qFgjhm7%Fg[[n7$$\"+8w\"e2&Fi]q$\"+'p`b>%Fd`qFgjhm 7%F]\\[n7$$\"+nL5[\\Fi]q$\"+?\\y&>%Fd`qFgjhm7%Fc\\[n7$$\"+;Bm=[Fi]q$\" +:/e&>%Fd`qFgjhm7%Fi\\[n7$$\"+$HBvo%Fi]q$\"+$eF\\>%Fd`qFgjhm7%F_][n7$$ \"+0UraXFi]q$\"+$)H\"Q>%Fd`qFgjhm7%Fe][n7$$\"+c>E?WFi]q$\"+RAA#>%Fd`qF gjhm7%F[^[n7$$\"+bB>%G%Fi]q$\"+P*R,>%Fd`qFgjhm7%Fa^[n7$$\"+1+`YTFi]q$ \"+-&\\v=%Fd`qFgjhm7%Fg^[n7$$\"+m\")H2SFi]q$\"+jIV%=%Fd`qFgjhm7%F]_[n7 $$\"+y&=l'QFi]q$\"+#Hr2=%Fd`qFgjhm7%Fc_[n7$$\"+j7@CPFi]q$\"+BKawTFd`qF gjhm7%Fi_[n7$$\"+gUR!e$Fi]q$\"+OgsrTFd`qFgjhm7%F_`[n7$$\"+>L3NMFi]q$\" +*z%HmTFd`qFgjhm7%Fe`[n7$$\"+.:H)G$Fi]q$\"+\"=A-;%Fd`qFgjhm7%F[a[n7$$ \"+5(G+9$Fi]q$\"+)3yM:%Fd`qFgjhm7%Faa[n7$$\"+j5I!*HFi]q$\"+]#Hg9%Fd`qF gjhm7%Fga[n7$$\"+[,6RGFi]q$\"+/(Qy8%Fd`qFgjhm7%F]b[n7$$\"+X?X'o#Fi]q$ \"+&=l)GTFd`qFgjhm7%Fcb[n7$$\"+&3;B`#Fi]q$\"+lB1>TFd`qFgjhm7%Fib[n7$$ \"+QKowBFi]q$\"+AzP3TFd`qFgjhm7%F_c[n7$$\"+6S_>AFi]q$\"+tBv'4%Fd`qFgjh m7%Fec[n7$$\"+Sczg?Fi]q$\"+0w6%3%Fd`qFgjhm7%F[d[n7$$\"+I$Q/!>Fi]q$\"+) )\\RqSFd`qFgjhm7%Fad[n7$$\"+!4q$Q(Fgdq$\"+v7VLRFd`qFgjhm7%F[g[n7$$\"+S_%))Q&Fgdq$\"+;S^0RFd`qFgjh m7%Fag[n7$$\"+4(fK_$Fgdq$\"+*ewV(QFd`qFgjhm7%Fgg[n7$$\"+_j&>e\"Fgdq$\" +oWFRQFd`qFgjhm7%F]h[n7$$!+RTP1ZF^cq$\"+]j5*z$Fd`qFgjhm7%Fch[n7$$!+!)e 2.FFgdq$\"+AZ7_PFd`qFgjhm7%Fih[n7$$!+#z\"Qq_Fgdq$\"+YGF&p$Fd`qFgjhm7%F _i[n7$$!+@:(G&eFgdq$\"+&yrAo$Fd`qFgjhm7%Fei[n7$$!+&e9,Z'Fgdq$\"+BEeoOF d`qFgjhm7%F[j[n7$$!+*HE08(Fgdq$\"+m67aOFd`qFgjhm7%Faj[n7$$!+L`hXyFgdq$ \"+r')yQOFd`qFgjhm7%Fgj[n7$$!+A2_J')Fgdq$\"+i5ZAOFd`qFgjhm7%F][\\n7$$! +z;[6&*Fgdq$\"+V?/0OFd`qFgjhm7%Fc[\\n7$$!+w3)>0\"Fi]q$\"+'\\xje$Fd`qFg jhm7%Fi[\\n7$$!+(R'zq6Fi]q$\"+T/SmNFd`qFgjhm7%F_\\\\n7$$!+pB?:8Fi]q$\" +FI?XNFd`qFgjhm7%Fe\\\\n7$Fgcgm$\"+;)GL_$Fd`qFgjhm7%F[]\\nF^g[mFgjhm7% Fhf[mF^g[mFgjhm7%Ffcgm7$Ff\\\\n$!+FI?XNFd`q7$Fhjhm$!$u$Fh^v7%Fa]\\n7$F `\\\\n$!+T/SmNFd`qFd]\\n7%Fh]\\n7$Fj[\\n$!+'\\xje$Fd`qFd]\\n7%F\\^\\n7 $Fd[\\n$!+V?/0OFd`qFd]\\n7%F`^\\n7$F^[\\n$!+i5ZAOFd`qFd]\\n7%Fd^\\n7$F hj[n$!+r')yQOFd`qFd]\\n7%Fh^\\n7$Fbj[n$!+m67aOFd`qFd]\\n7%F\\_\\n7$F\\ j[n$!+BEeoOFd`qFd]\\n7%F`_\\n7$Ffi[n$!+&yrAo$Fd`qFd]\\n7%Fd_\\n7$F`i[n $!+YGF&p$Fd`qFd]\\n7%Fh_\\n7$$!+p)\\jr%Fgdq$!+Ztl2PFd`qFd]\\n7%F\\`\\n 7$$!+2)of=%Fgdq$!+.i[>PFd`qFd]\\n7%Fb`\\n7$$!+?4]vOFgdq$!+;:\"3t$Fd`qF d]\\n7%Fh`\\n7$$!+vD*>=$Fgdq$!+'4y;u$Fd`qFd]\\n7%F^a\\n7$$!+tK(e**)F^c q$!+oxK!z$Fd`qFd]\\n7%Fda\\n7$$\"+/W)==\"Fgdq$!+E,pJQFd`qFd]\\n7%Fja\\ n7$$\"+U.\">9$Fgdq$!+6AqnQFd`qFd]\\n7%F`b\\n7$$\"+/'f1-&Fgdq$!+RUc**QF d`qFd]\\n7%Ffb\\n7$$\"+3_'3%oFgdq$!+Ww2GRFd`qFd]\\n7%F\\c\\n7$$\"+Z'Qk h)Fgdq$!+Rz!Q&RFd`qFd]\\n7%Fbc\\n7$$\"+M(Qc.\"Fi]q$!+X8u0%Fd`qFd]\\n7%F\\\\]n7$$\"+mV&z_%Fi]q$!+.M`$>%Fd`qFd] \\n7%Fb\\]n7$$\"+0U4hYFi]q$!+]?u%>%Fd`qFd]\\n7%Fh\\]n7$$\"+U%pDz%Fi]q$ !+Kh[&>%Fd`qFd]\\n7%F^]]n7$$\"+mBNA\\Fi]q$!+p#zd>%Fd`qFd]\\n7%Fd]]n7$$ \"+AVT]]Fi]q$!+@Uj&>%Fd`qFd]\\n7%Fj]]n7$$\"+vdsw^Fi]q$!+oH1&>%Fd`qFd] \\n7%F`^]n7$$\"+hjD,`Fi]q$!+&ywS>%Fd`qFd]\\n7%Ff^]n7$$\"+>\\(RU&Fi]q$! +3jo#>%Fd`qFd]\\n7%F\\_]n7$$\"+D&\\[a&Fi]q$!+)e,4>%Fd`qFd]\\n7%Fb_]n7$ $\"+4v%Qm&Fi]q$!+X@t)=%Fd`qFd]\\n7%Fh_]n7$$\"+pa$4y&Fi]q$!+8q='=%Fd`qF d]\\n7%F^`]n7$$\"+u#zg*eFi]q$!+!yuK=%Fd`qFd]\\n7%Fd`]n7$$\"+pSC4gFi]q$ !+CO+!=%Fd`qFd]\\n7%Fj`]n7$$\"+nUR?hFi]q$!+_8QwTFd`qFd]\\n7%F`a]n7$$\" +RN\\HiFi]q$!+AaTsTFd`qFd]\\n7%Ffa]n7$$\"+,[]OjFi]q$!+wH6oTFd`qFd]\\n7 %F\\b]n7$$\"+$>!RTkFi]q$!+i3[jTFd`qFd]\\n7%Fbb]n7$$\"+g56WlFi]q$!+dc_e TFd`qFd]\\n7%Fhb]n7$$\"+( Fi]q$!+\"4(H:TFd`qFd]\\n7%Fbe]n7$$\"+Uv-$G(Fi]q$!+3v'z5%Fd`qFd]\\n7%Fh e]n7$$\"+0(zVO(Fi]q$!+JZO+TFd`qFd]\\n7%F^f]n7$$\"+G)GJW(Fi]q$!+AN\\#4% Fd`qFd]\\n7%Fdf]n7$$\"+UZA>vFi]q$!+D&eV3%Fd`qFd]\\n7%Fjf]n7$$\"+eeh#f( Fi]q$!+wU'f2%Fd`qFd]\\n7%F`g]n7$$\"+z!\\Km(Fi]q$!+5_JnSFd`qFd]\\n7%Ffg ]n7$$\"+5(p5t(Fi]q$!+rcTeSFd`qFd]\\n7%F\\h]n7$$\"+[8-'z(Fi]q$!+;*p#\\S Fd`qFd]\\n7%Fbh]n7$$\"+!yX!eyFi]q$!+D@))RSFd`qFd]\\n7%Fhh]n7$$\"+^G3l\" )Fi]q$!+v#)pyRFd`qFd]\\n7%F\\[^n7$$\"+\"QA\\?)Fi]q$!+4zsnRFd`qFd]\\n7% Fb[^n7$$\"+3,(\\$)Fi]q$!+=&*y4RFd`qFd]\\ n7%F`]^n7$$\"+e#*Gn$)Fi]q$!+\\.h(*QFd`qFd]\\n7%Ff]^n7$$\"+UF$3Q)Fi]q$! +iDC&)QFd`qFd]\\n7%F\\^^n7$$\"+?8C!R)Fi]q$!+3)*osQFd`qFd]\\n7%Fb^^n7$$ \"+F-S&R)Fi]q$!+Td&*fQFd`qFd]\\n7%Fh^^n7$$\"+4\")='R)Fi]q$!+AS/ZQFd`qF d]\\n7%F^_^n7$$\"+*QwCR)Fi]q$!+G$eR$QFd`qFd]\\n7%Fd_^n7$$\"+h>Q)Fi]q $!+\\Bq?QFd`qFd]\\n7%Fj_^n7$$\"+$Q**4P)Fi]q$!++)zs!QFd`qFd]\\n7%F``^n7 $$\"+nR$HN)Fi]q$!+BWp$z$Fd`qFd]\\n7%Ff`^n7$$\"+XnwH$)Fi]q$!+\"**\\*zPF d`qFd]\\n7%F\\a^n7$$\"+\"H?8I)Fi]q$!+;.0mPFd`qFd]\\n7%Fba^n7$$\"+kRSn# )Fi]q$!+_#**>v$Fd`qFd]\\n7%Fha^n7$$\"+`B\"yA)Fi]q$!+/2!yt$Fd`qFd]\\n7% F^b^n7$$\"+![BB=)Fi]q$!+D'eMs$Fd`qFd]\\n7%Fdb^n7$$\"+1npI\")Fi]q$!+Kq( *3PFd`qFd]\\n7%Fjb^n7$$\"+\"HqE2)Fi]q$!+,+O%p$Fd`qFd]\\n7%F`c^n7$$\"+D &ez+)Fi]q$!+w;hzOFd`qFd]\\n7%Ffc^n7$$\"+X$[i$zFi]q$!+ritkOFd`qFd]\\n7% F\\d^n7$$\"+-`>dyFi]q$!+r!Q(\\OFd`qFd]\\n7%Fbd^n7$$\"+U(=/x(Fi]q$!+J9i MOFd`qFd]\\n7%Fhd^n7$$\"+\")f\\vwFi]q$!+p2R>OFd`qFd]\\n7%F^e^n7$$\"+E_ &>d(Fi]q$!+e00/OFd`qFd]\\n7%Fde^n7$$\"+\")oEfuFi]q$!+3`g)e$Fd`qFd]\\n7 %Fje^n7$$\"+nF$oL(Fi]q$!+I&fId$Fd`qFd]\\n7%F`f^n7$$\"+0E(R?(Fi]q$!+!p< ub$Fd`qFd]\\n7%Fff^n7$$\"+xo!*fqFi]q$!+IToTNFd`qFd]\\n7%F\\g^n7$$\"+tZ t.pFi]q$!+XH'e_$Fd`qFd]\\n7%Fbg^n7$$\"+=aSMnFi]q$!+&zd*4NFd`qFd]\\n7%F hg^n7$$\"+g'z1b'Fi]q$!+/;(R\\$Fd`qFd]\\n7%F^h^n7$$\"+h$y5N'Fi]q$!+\"41 zZ$Fd`qFd]\\n7%Fdh^n7$$\"+d3\"Q8'Fi]q$!+05whMFd`qFd]\\n7%Fjh^n7$$\"+DM n'*eFi]q$!+cG`XMFd`qFd]\\n7%F`i^n7$$\"+T3!pj&Fi]q$!+YF@HMFd`qFd]\\n7%F fi^n7$$\"+Y:%4N&Fi]q$!+ADy7MFd`qFd]\\n7%F\\j^n7$$\"+ZB5M]Fi]q$!+sv?'R$ Fd`qFd]\\n7%Fbj^n7$$\"+$)o&*zYFi]q$!+0DUzLFd`qFd]\\n7%Fhj^n7$$\"+7VHzU Fi]q$!+1-IiLFd`qFd]\\n7%F^[_n7$$\"+?q3=QFi]q$!+brdWLFd`qFd]\\n7%Fd[_n7 $$\"+o95tKFi]q$!+^FkDLFd`qFd]\\n7%Fj[_n7$$\"+zJ$4g#Fi]q$!+?ly.LFd`qFd] \\n7%F`\\_n7$Fh]im$!+E_o)H$Fd`qFd]\\n7%Ff\\_n7$Fb]im$!+/9;$H$Fd`qFd]\\ n7%Fj\\_n7$F\\]im$!+'HqqG$Fd`qFd]\\n7%F^]_n7$Ff\\im$!+-x>!G$Fd`qFd]\\n 7%Fb]_n7$F`\\im$!+Ez@sKFd`qFd]\\n7%Ff]_n7$Fj[im$!+egiiKFd`qFd]\\n7%Fj] _n7$Fd[im$!+qak]KFd`qFd]\\n7%F^^_n7$F^[im$!+8?=NKFd`qFd]\\n7%Fb^_nF\\d gmFd]\\n7%FfcgmF\\dgmFd]\\n-Fd\\q6&Ff\\qF`gx$\"#bFh^vFj\\qFegx-%(SCALI NGG6#%,CONSTRAINEDG-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%+ AXESLABELSG6%%&Re(z)G%&Im(z)G-F`__n6#%(DEFAULTG-%%VIEWG6$;$!$R(Fh^v$\" $R\"Fh^v;$!$z%Fh^v$\"$z%Fh^v" 1 2 0 1 10 0 2 9 1 2 1 1.000000 16.000000 38.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------ ---------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#------------ -----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 " characteristics of the embedded schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6072 "ee := \{c[2]=5/33,\nc[3]=29/115,\nc[4]=814716465/3253796668-23/3253796668*1 05151417455945^(1/2),\nc[5]=317/476,\nc[6]=14/19,\nc[7]=1,\nc[8]=1,\n \na[2,1]=5/33,\na[3,1]=5597/132250,\na[3,2]=27753/132250,\na[4,1]=3793 68932539978890648812440551/7242812371317610711938898070128-\n \+ 2455501968740121696101/7242812371317610711938898070128*10515141745594 5^(1/2),\na[4,2]=9899096170368469289460332229/624380376837725061374042 93708+\n 45207024234619537323/62438037683772506137404293708*1 05151417455945^(1/2),\na[4,3]=285859955732027554698452564025/724281237 1317610711938898070128-\n 53985530215629267813075/72428123713 17610711938898070128*105151417455945^(1/2),\na[5,1]=123752083413065843 138789879/209127510371847411054672640+\n 16088515006383971210 47/90970467011753623808782598400*105151417455945^(1/2),\na[5,2]=-20000 41633670013681/5324522345395547392+\n 15875554484889/38602787 00411771859200*105151417455945^(1/2),\na[5,3]=189029124529327947719767 69428125/17444617282574859822209614750208+\n 1006628842683460 78346663939/505893901194670934844078827756032*105151417455945^(1/2),\n a[5,4]=-516318245298821111959896996980826931/8146967117152301473968518 20173032960-\n 107920362352247138873225158999/488818027029138 088438111092103819776*105151417455945^(1/2),\na[6,1]=-4922558902230123 3744183735779/63282599719433884959882336960-\n 35106877881205 84753272949/82583792633861219872646449732800*105151417455945^(1/2),\na [6,2]=-245632221609852/423282790552285-16879401/3230316033162175*10515 1417455945^(1/2),\na[6,3]=86276540353665039847710695810845/83349355222 1848817093515269462144-\n 216788956050328897851037816063/4592 54947274238698218526913473641344*105151417455945^(1/2),\na[6,4]=359873 158906430802491237985928506042059/246529621131827805043897351525007341 440+\n 230654465754417653852094653348797/44375331803729004907 9015232745013214592*105151417455945^(1/2),\na[6,5]=67/126,\na[7,1]=103 8263320304911199846238164482685063317425776107411696073055841920084894 548437/\n 347736854176255731022079256869405427680335065673735 93703788548575160867886583040+\n 6756803198934452034968261504 7488926280454856996955945138286279519534217777/\n 30253106313 3342485989208953476382722081891507136150265222960372603899550613272448 0*105151417455945^(1/2),\na[7,2]=59734302131229/44926355375720-1441341 /1302864305895880*105151417455945^(1/2),\na[7,3]=152497967976522319889 69792537921142476851854434785659517885170619034922780974618277305/\n \+ 20726650173902839432495268544515964009257243716562331895034391 2141403953196522628760064+\n 31895489414207660020619126425491 808266074209880766698111670849008793214933062155/\n 286225169 0681820683535060894242680744135524132287179166457130215286054591761502 96859136*\n 105151417455945^(1/2),\na[7,4]=-16936601238178391 4299345101325740304220565539747522073844711653458806174655894526676245 216204127995/\n 155162665043814783234249541624593701687526417 086957004644589292251741395415822637890959585918820864-\n 291 8940970688744346217763899407616530392240341519924427396583752120589420 04402007722166107207/\n 2327439975657221748513743124368905525 312896256304355069668839383776120931237339568364393788782312960*\n \+ 105151417455945^(1/2),\na[7,5]=461565570890663210197437065017946 5855337591303167331051987992414499309155/\n 13018916981429931 091924277859236858824633898424165122087931906513654014256-\n \+ 201035901905594888203478188336475409356265197258304205097572636855/\n \+ 1301891698142993109192427785923685882463389842416512208793190 6513654014256*105151417455945^(1/2),\na[7,6]=9714101019226554178088904 960/31957908807350197910133249679+\n 261835787666411012880/31 957908807350197910133249679*105151417455945^(1/2),\na[8,1]=90538070892 31/136378733497104-501797/2045681002456560*105151417455945^(1/2),\na[8 ,2]=0,\na[8,3]=119501137217800311358375/307441845354011196253392-\n \+ 2018586005231875/307441845354011196253392*105151417455945^(1/2) ,\na[8,4]=-37638654822534247872916940674662423328915/98446743335790684 98057764513689180633044624+\n 3121650305280233144636524202663 962531/443010345011058082412599403116013128487008080*105151417455945^( 1/2), a[8,5]=1190204022066367942642960/4085963763691414586732367-\n \+ 2178285514342448/4085963763691414586732367*105151417455945^(1/2 ),\na[8,6]=830592632727775794785/5052174577198328191248+1504077797251/ 5052174577198328191248*105151417455945^(1/2), a[8,7]=91/978,\n\nb[1]=9 053807089231/136378733497104-501797/2045681002456560*105151417455945^( 1/2),\nb[2]=0,\nb[3]=119501137217800311358375/307441845354011196253392 -\n 2018586005231875/307441845354011196253392*105151417455945 ^(1/2),\nb[4]=-37638654822534247872916940674662423328915/9844674333579 068498057764513689180633044624+\n 312165030528023314463652420 2663962531/443010345011058082412599403116013128487008080*\n 1 05151417455945^(1/2),\nb[5]=1190204022066367942642960/4085963763691414 586732367-\n 2178285514342448/4085963763691414586732367*10515 1417455945^(1/2), b[6]=830592632727775794785/5052174577198328191248+15 04077797251/5052174577198328191248*105151417455945^(1/2),\nb[7]=91/978 ,\n\n`b*`[1]=14475354021635991174701/197802622822953213408000-\n \+ 478269920776171/329671038038255355680000*105151417455945^(1/2),\n` b*`[2]=0,\n`b*`[3]=15/53,\n`b*`[4]=64406830726760624192719424755704650 8737/6369286362605991319749019126566156224000+\n 111760904420 113865733188135964353/222925022691209696191215669429815467840000*10515 1417455945^(1/2),\n`b*`[5]=39/125,\n`b*`[6]=65572753075427021134812949 889/496177443284238540373654560000+\n 9841360159811270667/719 0977438902007831502240000*105151417455945^(1/2), `b*`[7]=1685942870904 0130745055637936223921/188074229500551422433322061497920000+\n \+ 244403905585142824289185521/2194199344173099928388757384142400000*10 5151417455945^(1/2), `b*`[8]=113093754206156431292803151127249/1273419 2622433169227256181247255000-\n 52967678767390756540012887153 /99836070159876046741688460978479200000*105151417455945^(1/2)\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "T[5,7];" "6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 127 " denote the vector whose components are the principal error terms of \+ the 7 stage, order 5 scheme (the error terms of order 6)." }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,8];" "6#&%#T*G6$\"\"% \"\")" }{TEXT -1 145 " denote the vector whose components are the pri ncipal error terms of the embedded 8 stage, order 4 scheme (the error \+ terms of order 5) and let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\" \"&\"\")" }{TEXT -1 99 " denote the vector whose components are the e rror terms of order 6 of the embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of t hese vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7]));" "6#-%$absG6#-F$ 6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4 ,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\" \")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG 6#-F)6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/abs(abs(`T*`[4, 8]));" "6#/&%\"BG6#\"\"'*&-%$abs G6#-F*6#&%#T*G6$\"\"&\"\")\"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,8]-T[5,7]))/abs(ab s(`T*`[4,8]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\") \"\"\"&%\"TG6$F2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to ensure that \+ " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 73 " is a minimu m, if the embedded scheme is to be used for error control, " } {XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " should be chos en so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magn itude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "From this point of view the Bo gacki-Shampine scheme is near ideal with respect to the 8 stage, order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_7 := PrincipalErrorTerms(5,7,' expanded'):\n`errterms5_8*` :=subs(b=`b*`,PrincipalErrorTerms(5,8,'exp anded')):\n`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expa nded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "snmB := sqrt(add(simplify(subs(ee,`errterms5_8*`[i]) )^2,i=1..nops(`errterms5_8*`))):\nsdnB := sqrt(add(simplify(subs(ee,`e rrterms4_8*`[i]))^2,i=1..nops(`errterms4_8*`))):\nsnmC := sqrt(add((si mplify(subs(ee,`errterms5_8*`[i])-subs(ee,errterms5_7[i])))^2,i=1..nop s(errterms5_7))):\n`B*`[6]= evalf[8](snmB/sdnB);\n`C*`[6]= evalf[8](sn mC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B*G6#\"\"'$\")*\\CH\" !\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\"\"'$\")ML-5!\"(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#---------------------------- -------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 50 "#======================================== =========" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "The linear stability imaginary axis inclusion " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"# " }{TEXT -1 51 " for a general 7 stage, order 5 Runge-Kutta scheme" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "It is a desirable characteristic of an explicit Runge-Kutta method for the st ability region to intersect the imaginary axis in an interval centred \+ at the origin which has a large radius." }}{PARA 0 "" 0 "" {TEXT -1 46 "The radius of this interval may be called the " }{TEXT 260 41 "lin ear stability imaginary axis inclusion" }{TEXT -1 17 " and denoted by \+ " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^ 3/6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F) *&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F, F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 68 " is the stability function of a 7 stage o rder 5 Runge-Kutta scheme. " }}{PARA 0 "" 0 "" {TEXT -1 61 "The follow ing picture shows how the linear stability radius " }{XPPEDIT 18 0 "r ho" "6#%$rhoG" }{TEXT -1 53 " and the linear stability imaginary axis inclusion " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 13 " depend on " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 140 "This picture has been taken from the the sis defence presentation of Colin B. MacDonald, Simon Fraser Universit y, Burnaby, B.C. Canada, 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "The solid contours labelled in " }{TEXT 256 4 "blue" }{TEXT -1 14 " refer to the " }{TEXT 256 23 "linear stabi lity radius" }{TEXT -1 2 " " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 38 " and the dashed contours labelled in " }{TEXT 265 5 "black" } {TEXT -1 14 " refer to the " }{TEXT 275 41 "linear stability imaginary axis inclusion" }{TEXT -1 2 " " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6# \"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{BITMAP 600 492 492 1 "?TMGt\\U:;JUJNjfD:SY:F:k:;::::yyyyyy::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::tyyyYyuyyiyy yYyyIwiywA:rxqyxmYxmyyyyyyQyAYs=Yqqxnyyyiyywyy=::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::eys=yyuyxsysiyyIyvYyYy^YpIyyy wqykyyInYtIyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::fyyUy:vyy=nYtIJygyx yyweywiyuQyqI:wiywAuqymyy_YpixmyuyxuyyiIyyy:::tyywyy=:::::::<:xI:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::bywQysyyqxwysYyuayxyyyuyVyxiysywlqx[Yv]Q^Qv[YitEiytqxriyvaxyiyoyysay xQxyqymyyxiyuqxryyyyyiIxkAyviYsYy[ivayxyqtmyy]yx_yyiyuayrIyvQyxyywiyxQ ywqytuyyIywqxyiwmyyI:::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::byuyywYyxiyqyyaYlI oiydIximqxtixqAyyyYyQxYYtyxyuyvqyxYyu]YtqYymivqyuyxwIyryymQxvyyuyywQyv yyxyyyayyuwyiyaYyyixyIyoixyiymIysiwyyywYyfYjyyYyyix=lIwax=fYpI:wiyryyA ::Z:Zy=::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::jxqyxuYy;yqxCxuIykIyuyyyyyyQvYyuAYvqyyuUYuuMXk?`W^iuyaYt aYymYsyyxqxvIyvmypyYy=yyeywuYwyYtAysyyyyivuwoiysiylXrAyr]ywQYwiYxyyxqY yeyyyYyyivqyxAyxyyuiyyyytmxSyvyy=::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Zxeyny wuypIsqyeYy?ywqyriyxYq:YrRgeZovdYwwiywqsEqrNGdCOxiyvyyyqayxAytYysUyyYy kyyAir_y>[rAwGhuwyuiyqYxyyk?xlAC@?c@Sr?MJ?LsAAsyuyvyYyUyyyYxqyygYyAyyy wqyuIqyyyyxwyyqiyvYwYytYxuYyiyyyYxuYyqiyqyx]yyY:::<:xI:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::wax=jySyf Y\\IWYveYyyyyyyykIwciu[ywoyn_Ho;S=EbpCHhyYymYwiywSiwgiwyixqqqoyj=oAyay IyruxuiyqIyyIxsqwyyykyoyxyyumvyyyaIpYyeYxuYymYygYpIyyy:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::byyiynywuypIoasMvuyxYYxqYuiYyiRAESwlqxLYtMytLQrpUWEuxe YsCxyyymiypyw]YwuYvyYmIxv=lGsZLVqr`pyqyn qwkixZWfb_kBxjyqywYqsxqan^dnuVatwiykqyyYyeiwgiYpIwqx[xQyyyyuyY:wqyuYx; oauyYtiyuyyA::Z:Zy=::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::jxcyZy=hyxqYuuYyYuVauQQR=?J=dsFicxqxxyypqyyiyv qYbsKItouXme;Cr:gtNYtuSItyyyqyyJyyIsANYZxqYyyyyA::Z:Zy=::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Zxyyxi Ytiyuybxyiy_YyoIqAIr;w_cvpYXmwIxmyyryyYyNY`yy?yviyumxsyyAbyiivYyywyxty xkyoyyEyuiyoyypisEykiyoyy\\yymYwyYyqiyuy:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::jxSy:vyyUyfYjyyI:tIwiyyYyyqwsxoyxiyyyiyyixyyxgyyeywyiu]iwyiyuiuUYt[y w[YxyixqiwkQvyytIyvmYyyypmXnqyyaywqiraywyvcIwEisGQj=Hjkywhqoexndxn]xyy uyJywYsQywYoiyZxqYyyyyA::Z:Zy=:::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::ZxyyxuyoYxuyyxyywyueyUYxaiwyixy iyoqweyyiyxuiyyyyyywmwwysyYYIw`yxuyxsyyAZuQYuIyyyyyyy:tYxqyvyYyqisqYvx vExhYxuYysyxI::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::fYyYy:vyy=uawmyyIJyyix]xy ywmwyiuaYkiyhiuyyoIxqAyuyytqYyAyyxgiycIsQIsuYxwiygyyqyxyiwkQyyywuiuYYy yuaIwuyyyYyNyuIYwqS=UPIR:mqyyywyyyiIwyyyiysiyZxuxyaxqYxyy;::B:ry:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::byyiyxYuMYwuyxyyyyyyiivywwyxlyyqYyyQyyUyoYxuiybyxiyiYlYyuyueYyQyve yxyyyyyy;bydyviyywyy=wqyiyxgxyqytqyriywIyxAyuIyrqysayyiYyay=:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::uatMxwyvmxSyvyy=nYtI::rxsyygiwiyvmYwoYymyyoiy\\y xmYpexnyiuqXaiwwIwwixyqsqysIyveYyyYrAyqQSEMCPkE\\;TUV]uq=YqMMS@milOftx oYxyyywyyyiIwaxaYpIoau]x]yyY:::yyyYyyuxnyyMxSyxYwuiu]x] yyY:::<:xI:::::::::::::::nyyey:vyy=:wax=:jyyI:uawmyyIfYpI:yyyjxSyvyy=: ::::::::::::::::::::::::::::::::uawmyyIfYpIyyyjxSyvyy=uawmyyI::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: uasqywQyvayxAymaylawvIxhyypqxyUxwYyiixyyvgyxayvayyyiyry:::NyviyqIytiyw iYtIqyxsyvsixYqyYYwMyviyuqyxuYxairayyiYyqyyI:::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::Jycyvyy=waxmyyInYtIyyyJycy vyy=:::::::::::::::::::::::pIsiyyixuyyoiylIuayyYZxMy;::rxwQuqyxoysYyyY YoaqaytyyyYyyyysaxvIxryyrqytuvWywmqyUyvqyweYxmYxmyxyinqxqiyyencHo_dN_l :^cDGctAyqyyy]uqYqav]wMwuiuMyuIxyy;::B:ry:::::::::::::::Jyyix]weyuqyqI ryyAwyylIqiy:pyyoYxixEywqvyisMxcywQxcxyiyqqyqIyyywYytYwwytyyyiYvY::::: ::::::::::::::::::::::::::::oQyeixqyyiyxUyoyywixvYxIuawmyyIfYpIyyywyyt I:yyy:::::::::::::::::::::::::::::::::::::nyxqyyyYuAyQycyZymxSyvyy=uaw myyI::::::::::::::::::::::::::::::::By`yuYyxGyqyyoIxxqxjaypixxYysyYiix XYw?ytQyyyyxyYxInYtIyyyZwMxwywyYyyyby^ywqyoYytAxyyyqIypAyvAywqyyEyuYxs yyZy=::::::::::::::::::::::::::::::Ryry:::waxmyyI::::::::::::::::::::: :::::::::::::Ryry:byryJycyvyy=waxmyyIuaw=xyy;:::::::::::::::::::::^yxe xwQxSyyay=nYtI:::rxtYvuyxayyqyxuyx_yxqymqxnyyyuxqYwmYyiyxyQx]yxqiyaxnQ yuaYkywmyywyyciwrymyxHytUYxuYyyiyyYTyuQyyyiyuxZvyyyqyxyywayxyYyQxuIyqy ;::B:vyy=::::::::::::::ZxiyxmwuYxiiyIyxyysqYfywyyxuxiiywIxyyymyyyYx:ty xUyuQYyuYyuxsEyrqyxAywiYveYwqyxGYwIvkIxnysUyNyvyyuaxyiynyywiyyuywqyyYy yyyuIyyy:bynYxyy;::::::::::::::::::::::::::ZxiywmtqyxYyyuytUYseYyaYsqy wMYyuyviywuyy?yuIysqyvYywiYsiycIwcyyyiyyqweyvQyweYuyyyiyxiYwqywwyy]x=j yyI:::::::::::::::::::::::::::::::::bywiykIxkIywiyiIyqYyyqymiyaYeYweYy yQwuitmxyiwMwcx^YwiywAxIyA::::::::::::::::::::::::::::::nyyiyviYoiymIy hywyYyiIkYywywsYysiyyYxqyxcYxiyxyiyyyyxIuawmyyIfYpIryyAtYyoyyiyxuxyiyo yyyyxyYyuYyuqyyYywYyqyyI:::::::::::::::::::::::::::::::sqyeIrwmixwyyui ypYsEYsyyyuiwIytIyoqxqixvayveysyyvYxI::::::::::::::::::::::::::::::::u atMxCxVxmeywiywYyyIysIyyyysYysIxyyywywiywtyxYYvUYyuYysyxiseyqanewkIxky ywiyyqyiYwmYy;uqy;jyyYyyixmyyI:::::::::::::::Ryry::::::bywax]xeyfYpIyy ywyytYxyyxmyyqyyqyyyyxoywyYwyyxyy;piyayxUwmixoiyyaxIY\\ywaYyyywwyumvui syYuiyyqt?ywyyyiYjyyI::Z:Zy=::::::::::::::byVyxyyuayxextyuyiqyyyIyymyu YyOYTYyoXfyxixyyys]wiywayvUxyiy[IywYvuYysyyhYxeyvyYyUYyuYtyyxaiweyhyye YtyyxMIsayqytuywaxmyyI:tIwayyY:::::::::::::::::::::::::::byuIxqqriywYy y_ywIywqytYxxayiYyuixqyxymwyYwYyx]ywqyxiyxuYyoyyyyxyYqyydiyYYhI:wiuYYu xYyyixmyyI:::::::::::::::::::::::::::::::::bywiywav]xyytmtEYtyYxuixqyu uymiyyYxwqxyyyqImasMxKy[ymy;:::::::::::::::::::::::::::::ZxaywEywIxqiy sYyyqxqyxiImiyyYxnwjAxvyywyxxYysUyxyyyyygyyUyvyy]xyYxixiywuyVxkIxyyyyy YwyytYyyyxoywqYtyy[wqyvuyyyy::::::::::::::::::::::::::::::ZwuyeYx`yxwx tEywyyPytyyyuyy;eIw[yyyyxnyymx_YvYTYuUiyyyy=:::::::::::::::::::::::::: ::::pIuagIxlAxnixvqyyiyyuywyyoixoyyxIuiyhyySxoIxwyyuawiyyuvkywmaiyxpyx vYYnyxqyxayvIypAyxiyyqyxUyxywsxfYyYyvyyqyyyYvyy=:::::::::::::::::::::: ::jxky;::::nxqeyyYyRxuQuSxwqvqiwMygyuQypiytixyayqyyyYYtyxyikYxnayxqywm yyI::Z:jyyI::::::::::::::ZxyyxQywqYuyyyyxFyxMyOyyqyweyxywqyxYyxqyuyyya yxqyuoytyYnyxYyuuxyyyWywoiueIxsyyqYywix[YwcIveiwiyywaxeyoYywQvyYx]Yyqi yqyyUyyAyiyvyy=ZxMysyyA:::::::::::::::::::::::::::kiwgYv]YveYyuIxuiyeI w\\yuIYuuyxIitaYyIw_Yvciw[qyayvoynyxuYYqiysyxwYymiywIxyqwsxbweYviYwcyy Jy;yyyfYpIyyy::::::::::::::::::::::::::::::::tYyuyv_yrQyuIxnAyviyyYyoQ yyYyyqywEu]IvmiyyYxVyyyYyqusw^yxyyyiYwayyyyyY::::::::::::::::::::::::: :::::byuqvqyyaIwYyxyxmYyiixyxyuwtiwYyeywdixxYyniytQyyqytayvuxuiyuQymyw yiueylywYyyuyxyiwyxoaXvYwAwQy;xiy]xmysy::::::::::::::::::::::::::::::Z wyyyQyxExqqxyiwmviYxEYtMyyuYw]yuIYyyyxIYtEYryxoaYvMYxuisqywyyyyyy=:::: :::::::::::::::::::::::::::pIqYydYygyyewkqqEigIxlYxnQxwqyvayxqymixoayu yXdyxyYyEYtMyxyityYgqxYyysyPyugxoixxayvyywyyxiyyqyxUyxywyyykxwywmyyYyy ixmyyI:::::::::::::::::::::::::::::::wqxEysAysIyseYyyytminYyxyxraywayx qyyiyfiwcQyqQxyiyryyxQyiAxhayxqywMyuiyyYyA::Z:Zy=::::::::::::::bypyvyY yeyyyiwMwuisMwSyvyyayuYyoiyxQYhYxqywayvaYwmyxeYxoyyiysaytaxqYytuyxyvQy uAyt]YueYyyyvyiuyyxmvSxqYyhiwyYwyywYyxIyiyvyy=ZxMysyyA:::::::::::::::: :::::::::::tYyUytYys]ywaYywyxiysqxlyxvQyrqywYywqyyEyxYu]ywEYueyw]ywyyy yyyaYvMyyOyyyyteyoIw]YuyyyqqyiIwyyuqyyYywYvyymxSyvyy=::::::::::::::::: ::::::::::::::bywiysiyyixuqvuisayyQymQyvixtqywexyqvuyyoYqaumtyyyYiyayx uyuQysyxyy;:::::::::::::::::::::::::::::ZxaYkYyyYxsqxqIsyxyyxsyyiyyXix iyyqysiyraxwQykQyqyxvYyvqyyiYhYw_ywYyyyyu=:wixyQy]YuQyyyYyyixqYwA::yqy yI::::::::::::::::::::::::::::::RyxYwqipIYiYxlYyWywUwiqyOYwixwqxuIyiyo yyNywYyqayuQyvuxqyxyqyyI:::::::::::::::::::::::::::::::WIv]Iywyyuqy_yw AywyytqYtYxqYyGyuYyvqyxyyxQyuQypYxpaxvqYsQyEyuUYvyyyUYyqywyiwmyuiyiywi ytayyiyymyqIqqyiIuawmyyYyyixmyyI:::::::::::::::::::::::::::::::iIxgyyy iysqyeYy?xqQyqqyyIyvUxqIysyypYyeyyGYoYwwYymiyyYyyalaxoaYwyyyqy;::B:ry: ::::::::::::::tyyuiyexxiveyoatMxCytyxuYxmywYIuiyyQwmYxQyveYvqyywYsyyqY ydYyuYyaYwwxryWyYwwyyDYs_yxqYyqyyyvyyygywiywayxuyyyyuIyxiyeyvyy=:::::: :::::::::::::::::::::::tYyuYrqxqqYxuYvCveIxqYxmyyyyxqyyoixyYyoqxqYymYy qysmyxIyvuyxaXqAyxuYxSxoywpYwQyyuIhyyWyxiYkYxiyyyiyyQy;yyy:::::::::::: :::::::::::::::::::::ZxqYiYxoYwmixcYxsIwiyxyyxyQwyYwYyvUisyXwywtYxuYyi yyyytswyyysiyyqxyYyuiyuyyA::::::::::::::::::::::::::::::tIkYy`ixqyvuyq ixwyxsyybwfyyqyoAypqxyaypyyvixqyxxiyyYyvEywqysyhYtIyqyyyiyyWytIYyYysyy JyKyoY::vYyyy::::::::::::::::::::::::::::::ZwuyiixwixuqsYYvqiyiYxyyuYt Qysyyyiyysxoiw[yuQixmayqyvyywAyrAYyyyyA::::::::::::::::::::::::::::::: nyxEytyusxyyygixqisKYtgQvyIeywiIwcIwoqvuIyQymyxiywmYwmyywxyuwVyyUw>ypq XwiyyiywAxiwmwyiumxSynyxqyxmysy:::::::::::::::::::::::::::::::Jyeyxiiu ayqaYmYywYxuiykYyXYxqYnixnqityyxQyy=uYywqsIysmYyyyxSyvyy=::B:ry::::::: ::::::::tyyuYwcxxYywYsav=tyxyyywYuIy:gariyyixuyypqyxyxuixqAyiyyyYysqxk uyayyHixQYwiyuYyuyyyiyvxyUxyYyxYxuiyyyyqyw]yeyvyy=:::::::::::::::::::: :::::::::tYyuivuxsqueyyYyw?ypMx]yrEYqiyyeyuGyxyYwiy_iuYYsoyywywsyysYyq yyhyyuyvOYuIykauMwwyuuywyytI:yyy:::::::::::::::::::::::::::::::::ZxqYs qyeImiydyxSu_yyoyyyYyyIwgywmyydIsQyiiyyXXIsqyyyxoyxuyyyyyA:::::::::::: ::::::::::::::::::tyyOywUxbxsYyqixwAkYy@YuYYyuYyYytMYvyYxwyvqywyytIyuI yvyyiYXYIvWyyhIqYxDyyyywyIqYxkyxmIxyyywaxmyoYwAZymy;:::::::::::::::::: ::::::::::::RyxYvayxeyxyYwqiVAv[qyxqyrIyquyyyytYxaYtyYyayyyYyyixqyyYyx yYayvpyxqiyyyy=::::::::::::::::::::::::::::::juYyuqIiyxsixwqxyiwMwuiiI xlyyyyyxyywQytQYyyyuIykIxiyvaivWIyyQuuyxUYy?ywaywyyyyYuQySywyywiyxyyyI uyyhyygYpIwiywiyvYxI:::::::::::::::::::::::::::::::nywuyoanIYmIyyywtYu iYyaYy_yyuywixtYw]ytAYyqyvyyxmYx_ywyYuqykyyI::Z:Zy=::::::::::::::byyiy oyyjx[weyuAjyuYyyYyuiyexsQuSwqyyWywyyo;IsMdMET@mR=Mb;=rLMTJuYyQiywiugI wSyyyAuyylyyyyvwysqyxqywayxqyxqYyAwyyyiyvyy=:::::::::::::::::::::::::: :::pyxqyxmYyQiuYwkaybQwnacBwbEvZ<>\\Kv[HGdIoyyiylEuev^KVPkdMoxqwXq?yrU eUSE;AJ?XJ;hQvHqxeYuyiuIxjawxUxxYyQysEyyyYyyyyuIyyy::::::::::::::::::: ::::::::::::::ZxqyxqwjIxsqyyIxrYyYQv`QbrnakXnchyrQywyyyIyyqYxYxuYyiIiy yXIsaxqYryyqy;:::::::::::::::::::::::::::::ZxyYqiyNyyqyqYyviyvexsqumyy aXqyxJlkHXV[AVuayuQXo]XGasGUn]hNt]yx]xp=Yr@qqHJ;]NQtLaysYyyyyyaYeyywYxyYy`Yw]yyyYyu iwaxn=yA:::::::::::::::::::::::::::::::NyuUyyyysiyyixwQveYwYyveYvqiVHn ^h^_f^?Yr=qywyYwQwYysyYyaYuIyqqyyuYqUytYq:ARx@R;aywiyyqYy=tGesHQsDyYyq yysyyiywiywYyfYyIytix=yyyyyy:::::::::::::::::::::::::::::::JyiYxeYtUyu eyxyYjUxompsTQtXqwlYyyyxuYwMqrlpZu?imgwdqVnywIyxyyuYY`ipyyyqyyyyxmyyI:::::::::::::: :::::::::::::::::::RyFyxutayv=itUIsIytWCBGShOgeyyxwYxgIxyyyyIywQxuitmt SwVyyeywAxI::::::::::::::::::::::::::::::ZxyxmQxxyywIyxUwfxsqypQyyYtUQ v^qvhQ`]v^GYqv`ynixyqyxUyyYxwQpeyy=SH=D;R=Ob;GR:;bTEeRyIymyyyyxbysyxya yeyvyy=jxSyvyy=::::::::::::::::::::::::::::::Ry^Yyiu_yvmYywIywyy[iv_yy yyyyQyQR<=r>UR;yyyeyyyywCylyweywayywyy=::::::::::::::::::::::::::::::: byyQymywYyyyYyyiyaYwYu]iv[YwkiwsyygyxXYxqYwuyyEQsDq@eR=yYxYixIyuyytixy u_Cgc;R:CR=YB>KB;ARYguVOYruxyuIxyqvyisiYhYymyxyiteyuIytyxoYwyyvY:::::: ::::::::::::::::::::::::JrAIxYwuiqaYmIy`IkqyeYy_qshpqF`f^v[auyUxyyyqiy yQomXpqIuqykyyI::Z:Zy=::::::::::::::bywyyuiyvYyYy^YryyyywyytYyqyvyIyiy uA]ixuiuoyyCivwYS[EXif__vywIwuyymI`qf^yeXQuymityyyiIxqYxsqx;hisaxxqynm xyIyiyyqyyI::::::::::::::::::::::::::::::RyFwhevuyyIYwyIPCETyvaUvyuytE ItSibEWdweYSeyoysyQwyYxuywayvAvYyuUHMVpyvuiyqyxqyryyyYymQxhiyvAyyMxeyv yixyY:yyy:::::::::::::::::::::::::::::::::ZwmtqyxyVfiWChKIlrAISKLKFhVX =vyeyxyyxyYycyZteyeanqyxYyyeywqyyiy=::::::::::::::::::::::::::::::byyi yPYuMyuyikMvqyyMV_yVMtlTqYwEiyqyxyywYysAYwYywiy`ywmyyyyyyI::::::::::::::::::::::::::::::ZxY ysAyxayuiyyyYtyxwyuM\\evb]OjXpotPsXywmyumyyQn^dn>UR;oypqweqvxyyuiwqyxZ iwiQ`s^aLqtQyyxqyoQxhyxmqwyqyseyvysmxmiw]tyYuIyJyuYyyyywY::::::::::::: :::::::::::::::::jxoywyYqYywixwQsCWuEZOuTYExiQhqwg=mRpL[yUWyYyaysMYrmy yqI\\YwiYyoYvY:::<:yyy:::::::::::::::fYyIyjyeIoawMyyyxuiuiywyxyyysyyxQ ywqyvIyxyykaYQiwwYpCItIETGu;CjAhJ?`PmDPpexisvyyyoiypixeyyYyxiuyytIyw]x QyyEywiyyyyxI::::::::::::::::::::::::::::::RyfvgawlqyyawsyyMyu[igiWfEN [:FZ<>Zm_hlgxmAwpixaMwyYx_xoIyXqu^YarFb>>Z?N[@fZGN[?FlHPkxYyqyYqIwaawa YyiyyyywAvyymxyiw=jyyI::::::::::::::::::::::::::::::ZwmucvsIyUiwmYvcIx GEQ?uoMXmcytWQxuyyuYyoYbxtytyIZxqYyay=::::::::::::::::::::::::::::::Zx iyoqX_ywkYwyQuqieQxoIxtqybYwjYeJWdKQr@yyqAyyqyxiyvewuyyhyyuwqEYQFnVHmsqxluywYxiiwxYyyixmyyIuawmyyYycyvyy=:::::::::::::::::::: ::::::::::pI=yrCixwyyGitQyuaIw=uRA]:F\\In[uOipwv`QvyyyvyyxUyqixxYykysx qyxxyyyI::::::::::::::::::::::::::::::ZxYyvQyxuxwyyuyxqyxOiuYYxwqvwyRa u[AghWhwlIypiyyQyvqxxiymyyxiysyxpeygyywIwyiyoyxo IykYxuyxqyyuyxyyyqQxeyvsxxivYyqMxwyuyYwYyqixryyiyvyymxsxVYpIuayyY::::: :::::::::::::::::::::::::RyFyxEucIwiyyyYu]ivhyyuyt]yymyyOyvyyxqYwAlYyG yxyysqYuyyxyy;:::::::::::::::::::::::::::::::FyqqXyyyxidQxlIywqysayyAY `YwuitiyyIyvuxsyywqxmisqYYiyyiuqYyuyyXytMyyuyxyyyiYt=yqyyyuYqmxowyxyyy qYjyyIuawmyyYycyvyy=::::::::::::::::::::::::::::::JxuimixpiyyyyvMxcwkI xtyxsxwapqxpeyuYxZpAYwiywyyyqy;:::::::::::::::::::::::::::::::nylYxkyy xiyyuyeiwoywoaxyylixsAyvyyjiXyyyyiysyxyiyuyxmqyOyvUxkyyyYwyyyYIwiYyyau QyxIypyYqyyuYxyiyoyyyyxwqxyixiyyEytYyqyyqixqYwyyvY:::::::::::::::::::: :::::::::::wyyyqxcxwquiiraywQYlIyiyuqyiyweiyqyt]dYwciyyyyw=jyyI::Z:Zy= ::::::::::::::::pIxI:JycyZyiYpIqyyyYyvyymxmyyqYyyixuyqauMysyxisqywAyuI ypYytQyyqYyiymauyYsaxqY::yyyxI:RyfY:jyyI:::fywiyxqYwaymy;::::::::::::: ::::uyyhImIy`yxgyxayyqytmyyixmwsxnywiywYYsYysyywYyyYytYwyYyoyvQYuixuyx \\YuyixyYsyywixwqxqyyuyyoyweyoyyvYoqyayygYyYyvyy=:::::::::::::::::wiyw iyvYyyy:::::::::::fYpI:yiyyay=::fYpIyyy:::::::::::::::::::Jyuixmysy::: fxo]yyY:`ywiyxqyxyyyOyvaywuyxIVyw=xisIysqyvayoixoiYdyywxqyxxMxiyxqYxuy ymiyiyxeywQy_YdYyoxxqyqyYuayyYZxyYxI:sIysYyuYywaymy;:uYyuYynYtIyay=Jyu ix=::jy;rynYtIryvY::::::::JycyZymy;:::::::::::ZxyYxI:::wiytI:::giwgqyy YweYwyyyyIJxsyyyyyA::nYtIyay=::JycyvYxI::::::::::::::::tyyqyygyy=:yyy: :wqyuyxwyuyYuayyYZxyY::yyy:::ZxqYwaymy;:::waxmysy:::::::::::::::nYtIya y=:::::::pIuAZyyYZxyYwAZyyY:::::<:xI:::::::::::::::::::::::oIxkyxmYxyy xsqxqyyuyxoYuqxyyyoXpyyyyxsxNyyExuIycYxkYwoYxuavqyxqyx=ryvY::pIuA:vyy= :::wiytIyay=:::::::::::::::::ZxyYwqxSyvyy]xuyyyyfyxaYyiybywaxyYwyyjyyy qMyseyyOxqexyiyuyyt JAtJ=::j`qv`qF:BmSHmS@fSpF:::W`H=:::j`qv`qFS`Fm_iv_iFS`F=::j_sl:::^SlF :::::::GhkGxkIxkMXlMxlQxlQ@NsDQsCIsCAsBAsBm^av^aFOhlOhlVsEYsEm^av^aF:: :::::O`D=::::::j_clNS`FM`BM^;::::::::yqyyI:::::::B:ry::::::::::::::::: ::::::VxmYxyqyxyyvYy\\iuVqqrHqrXpnPuOqtxIytiyqAyvuwuywkyyoyxyYywQySyty xyywyisyYiqyYyyGYhIuqyyyy[xeyyiyxyxqyyiyxkyqyygYxyy;Jyoyxmxqiwmyyiy=:J ycyvyyuy:::::::::::::::::::nY:pIry:mix@ysiS?qrpywjyYwmYvIYsuiwiwnIxnyy wIxxaylyywAyxiytmxoyxyywyyxMyoyxuyyqy;waxmxSyrypiyyyy=byZvaYyyyyyy:::: :::^YlIwiyxIyA:::::::::::::::::::::::nYtIuawmysy:Zx=lIyay=::::::JxCynY xIyA::::ZwmvuiuyYyyynYyYywYrmhp[wcOwyrxZviywyxrQyoqxrAyrIyvQypYysisMQt payuixtIyiaxmYxoixdawfiwgiwnQX]yuYIvWiu[iuWyuUYuQytQitOitKitOysMYtKIt \\H:IysIisGisEYsEIsCIsOitOit:FwcIwbMmUXmKmUXmUH:Ea_=jkIxkIH:::;Ir;yqyx q;Ir;Yr=Yr=A>VyywiytqxoYyxaypiytIyyiysyYoavqyxyYwyynY wAryxyy;::uyypIyyy:JycyvYxI:::::::waxmysy::::::::::::::::::::::::::::: nYtIyay=::::::JycyvYxI:::::RyNYwIwmIxuYyuqxuyt=YsyYwqYveXq=rrPquUxnyYw myyeyxyYx?ymUxiywoyxmyvkiwQWf[wueIwiIyqyucyv]iweytUYuaYweywiyw_iv_Yv]Y v[IvxxqyXquXqqxpqXpmXpkpqxpqH:SadQwdIwcUtvvayvbAwbAWjmYh`mlCsFv[Iv[=Jk EXkEH:yxqyxq:::Fv[Iv[Mj=hYyuYyUJj=hZMiuWiuGRquXqu@:::::nuX=sHqshpohpmX p`G:::::ixoXG:::kHpkHpvtQeogpdQGcPdsnaPdCo:::VTHG:::uWiugZ?fZKHlKhqwhq [Hn[Xn]pfawfqv`qVdMWdMGrnVT:jbAwbAG:::axnahn_hn_@FtKItKMb=Wb=G]`K=::Jb Cn::::vsIemWPaSmUP`mfFm`qv`qF:::S`F=::j`Sm:::::::vsIySZGm_iv_iF:UXmUXm :^SlF:U`G=::::::j_sl:::::::^SlFQ`EM_efCIS\\fBm]Yv]YF:::::::yqyyI:::::: :B:ry:::::::::::::::::::::::Zxmyuiy=ZwyYsyysYvcIoIXl]wbIGi]GgUirEyyqYy hiwmyqYyyYyoyx=:yyyxI:tyyyYygYpI:yyy:::jxqyxuYyoYxiyyY:::::::::::::::: :::::JyuIvYpIqawiYyiywyyuYy\\yvyYwiiuUnAXkogeWGx[IvyYysYyqixyQyqixyyyu yJyyixiYyYyyqyyI:::fYpIyyy:::::::::::::JyuYyuIyyyyyy:::::::::::::::::: ::::::::::::::::::::nYtIyay=::pIgyxoqyuYyoYmixyYyyQwqyumxqyYwuYWYTOYXd YwyaYxCyoIweIyoixwyyuIxqQLaTNYYxiYwyYyMI[YwcYsOYtWIv[iu[Iv_iv_Iv[IvYyu YiuWqpqxpmHQytdXoOW:jnssnVTHOitOitVweYweMoSt>wbEtNv\\Qv\\=HxkIXlMh^mlQ xlQH?ir?ir:::^v^av\\Qv\\AvZuqyxqwhqxwi?vZErvUxGw`XmhqgVaUs@fuWiuW=::Ji uG^UhwgOuTUpmpf;::::i`Q=::jfawfaGkHpXweotPeoftOEocpdQwdQG:::c`N=::j:Ar :AtJATgeWgAvZAveYweQwdQW[EV[]Wf]WdMWdMGrnaPcEgLmbSnVTHG:::_`LmbSnZJAT[ `IyS]Xn]Xn:::vsIysI=:::JaufGiSSHmVSdfDM`mV`mF:::Q`E=::J`Cm:::::::FtKIt J=lV`mV_eV_eFQ`EMauVauFU`GM`CmVShFU`G=::::::j_ifE=::::::J`CmVSdFMXl>SK `AyRGhkTF:::::::yay=:::::::tJAtOitOiT@wbAWauVa=Wb=WbsmYPauFNtLQtLMbsmvSTJaSmUP`mfEYSSP`mF:::::::::::::::[Hn[xlQxlKHlKhlOhlQ`DQSO`Fm`;Q@NS:hF MXlpv`wsIulQxlrlVS`FM`Em`SmVsEelNShFbl:::VShF:BmZEM^SlMP_;O`CISIxkIhkG p]sk:::::::vYxI:::::::B:ry::::::::::::::::::::::::ZxqyxyyxuyRyfY:uiwmq wuyyKuUyh=WcqUREuZMUUaYpuxyiYy[tuyuyyyQyAYpqxyuivayviyyyYyQwsueIwSiv]y wsyxuyyyyyyA::::::::::::::::::::::::::wyyyiyvyy=::Jwiyviyv]ixiyiyvbyyu mtYYJyrCqXh=xyeYxeYsIyyqiyUymywlyyiyteYvUyxeyxyyyqiyyyy=:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::RyVyy=uaxqyx=sqywYqiyny tawhIrmhpEqq:ywaAwvqXyYyuyxmqOitOyxqyxqsyxYr=YrmXpmxlQxlMYuUYq=YruXr=Y r=iiywiawhqWUqpixomXocthxoWW:dHMabAWMYtJn;MacmmSs^v^urAqk;BsCqlQh`qV=a [=::JkEhZ]XeqvuYyuY=tG:uXquHqsPh;Rqu@tWiuWiSqVuUYuTQUo`VaUu`WMhspq@dGm Xp`G:BpkPfsoipe;::::ghogxoipe;boftOEocPd;:::eXo\\Wf]W_efTQuTatNatKItKY TlR@mR_efFmayvaAG_`LmbAgJATe`MYT]`KMccnZKMdsnaPcEGFT:DG[Hn[@e`KMauVa=G [`IySWhmJccn>TxFY`J=xF::::[@nSpv`_sFulQP`mfF=::::::::::::::JbslQP^]fDm _clOP_CmfSj_;O@ZEm^afGmaslQ@hv_WS`FM`Em`SmVsEelNShFbl:::VShF:::M`CM_cl ZCISIxkTV][A=::::::jysy:::::::Z:vYxI:::::::::::::::::::::::uayyY:bywqx qiyiywewiYykYxyyxuqxyiwmwmyyeIqiymytYYiWgfOw`yFqUXnyywmyyciwwYwkqxuiwi yxqyxayvEwgqteyvayyuIdIuyyyyyyA:::::::::::::::::::::::::::::jxSyNyyUxl yq=YqyYyqiwqysqwjyysiyT[X ofT@GWhmTJaSmUP`mfEYSSHmlF:::::::::::::::[` EM^]fDm_clOP_CmfSj_;O@ZEm^afGmaslQ@hFQ`Cm^slfSpFQxldFO`E=dF:::Q`E=dFO@ ZCISM@:>sBElvrAuk::::::::vYxI:::::::B:ry::::::::::::::::::::::ZwyyyYYh IRyfYdIuyypIryyqxqyxuYyoYjxiioAxkiypIyuQxoyyuqyy=twd]WWyTSmtJMTjIwdqYs MyymiuaxqqxkYxyiyyEywyyuayyY::::::::::::::::::::::::::::::fyyuxVxtyxry yvQyyqydawfyyyyyyqywavViuyUy`ywowjimIHlmqvbyytuyuamYxyyYuawmyyI::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::nyxey:vyyyyy]xu ywQymiyqyxyywuuKytyIyyYrgXp?G_gVwIYtQXi=xyqqw=ysyyvmYxMytyxm]XuAYsYytQ Yr=Yr=IWIv]YuYIvWiuWYuUqpixoOWdheYWQAZdmnahbAWMYtJn;MacmmSs^VHxkWV:LXl gv_esGqjGv[=::JkEhZMiugXmiygY=tWi;fuWEq^U:pwhOUlWh_UhwgGuSEpkpfagTQUqx pdWgOU:`wfGU::\\Gk`QyTi@::::ntPuoipe;boftOEocPd;:::]`QytQ]QMcEWcIwcIWh mWhAwbWTHG:_hnDGRn]pdQwdIwcGT@G_`L=@Gc`MYT_@FT:DG[Hn[XoepdSnnsHEn>Txva osH=_`LMbsmvSTJaSmUP`mFQP`mfF=::::::::::::::Jbsl>sBelVSdV_OSl FU@VSJ_;rlFsCUmvShv_[EYSQ`Cm^slfSpFQxldFO`E=dF:::Q`E=dFO`CISM`BASM`CM^ ClvrAuk:::::::::vYxI:::::::B:ry::::::::::::::::::::::ZwuyqatuyoQyaYyqY wsxxyw;gQviyxgyyyyveymYxmixoixwiyxYyuiwmxcxsyyoIylyyyYxyiuqyyAyuitBQsq PofXs:YryYyxuxnxsIygYxy]xmywY:::::::::::::::::::::::::::::jxSyFyyiypiy tYxryxtyyxayvayjqwgyywew[YvyYyxYyyiuiynxnchs:QryixqyyviYsayyYry::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::JyuYyuIryxyy;:U YuUyyvywmxSxsyymQxixqetEeSvYxoyYwuyy]YxiYrIyumXqyYtUirIyliwhQxjAX[Qq=y r[hqwgiWQytQitOQostQqo;btFwcEtKqnaH>WjnStvVPHCa]YVAAZ^aVEYsTXmGV@H:::? irUX wewT::::JeUgQyTi@ZPmdQgNaTc@:::NtLQtSIuS]NmayvayFe`UYuUItKItLmcCo^THwc GTSXv]or@ekvRXF::::: :::yay=:::::::<:xI::::::::::::::::::::::RyNyyexyqvwysqyyuywixNYpyygY:h yygYpImyxTxFY` JEnrm::::>TJaSmUP`mFQP`mfF=::::::::::::::JbslQP^]fDm_clOP_CmfsG=Q@NS:h FMXlpFYxmYxlZEm_SlFShFU`Gm_clNShFbl:::VShFO`Dm^afBASIxkXFOhlO@IxkIhkGP ];I`A=:::::::jysy:::::::Z:ryxI:::::::::::::::::::::::uatMwuitqYxIyqxSy :ZsqywIyxiyuyYuawmyyI:wiypyxmYweyymYyyiyUy\\ysQiuuyoiy_yq?yhKwdCxfiGxM YuyyxxyxSywQyuiyuyfYyAryyA:byyay=:::::::::::::::jxSyvyy=::::::oyxmIypI XYxeyuqYxeyy_yyiyuaytyYuaxyxluytiuayfUqgGdOWrchoyIyuqxcytYywyxuyyqycyw iyuyypIryyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::Jy cyZymxsxNyyexFYuqxqiwuywiyyIyuyxaIw`Yxyyuiyw]xoIVRIulYwfuyrIyyEYtyysQY sYxnmYtQYtyXrEYrCvpxqyXpuXqaXqqXmqXoUxoaXoeHQqpiHZfQWQYtMItKqoaxnKn;MY t`HIa_MlsrAqk;BsCqlQh`qV=a[=::JkEhZ]XeqvuYyuY=tG:u`VEqs@ZWiUq`VaUBqNUo XpmPgegUYUm`SMf]gSIUm@::ZRMfsoipe;::::ghoXwewT:TGeXoLWd_T:::jdsna`LUn] PgeWgeG[Hn[xpqpbCn[Hn[`IyS[`KMcOtMmdQwdkbSnNTDGRn^TcxnaPc;]@ZLMb=gOmdS nnsHqsJMbsmnsH=_`LMbsmvSTJaSmUP`mfEYSS`F=::::::::::::::Jbsl>s BelVsEelOP_CmfsG=Q@NS:hFMXlpFYxmhFrlVS`FM`Em`SmVSdFO`E=dF:::Q`EM_SlMP^ ]fBm]YfAM_;IxkTV]gr?ir@qr@yR\\V^]FXF:::::::yay=:::::::<:xI:::::::::::: :::::::::::uyypI::ryyqxSy:::vyy=:::uyxqQyuiwMwiyykxGx`AwKYr?ArepoyYxpy ytMxqYvUytmyyqYscxyiyoqxmyyqyyoYsyxiIx\\yvQywuivUyvyy=:::jxyiw=:jyyI:: ::::::::::::::ZwmwsxfYxyyoyvaYyyyyiywyyqqyqYyyIQyrCYkkgh]xjCPwmIxYy_yq AyyyyxqxqixxUysatMwoyvqyxuyuyytyyqYxmYwaYyuixMxuivMyuiyyYtYyuI:yay=jxS yvyy=:::::::::::::::::::::::::::::::::::::::::::::::::RyfYJxuYyiIyyyyy yFyqyxxexyyxsixcqyqipIweiqiHpRUleysKivGisyixDymaxwIysyxoeXruxpMyuQYqIy rIXpaXiaXmAXoYXneXoqxoowfawjYwfywdYwgAwcawaAwbqv`qwcQwdAwbQwbAWbtOYtMq lYxmIxlQXlsrAqk;BsCqlQh`qV=a[=::JlMXlywiyWhmWhuWiwuYyuXMiCr;Iryxqy@u`V Mh;RquPiuWiKhspqPgegSIUohpdGm`S=::::\\Gk`QyTi@:>uRUpmpf;k`Rmeso:::VtMY tMMdSoepd;cHoLGsHqshlOPeUWeauVaUV]UVixoiX^]fREp]Xn]Xn:::vTihnvsIysKMcS n]pbsnapd;axnHG]`K=:HGa`Kmayvaqv`qv`smxVaufJAT_`LMbsmYxmY`JATrm:ZJ=xF[ `JmbAGvSJacm^sFasF=::::::::::j`qv`qF:::_hn_xlQxlIxkIXl`V_OsCISbmWp`qfF aSQxldV_OS:hFMXlpFY`E=hFQ`C=lFWhmW@Qxl`FM`DM^]V^eFQ`Em^SlNSlFS@VShFO`C m^;IxkXF:::::::::::yay=:::::::<:xI:::::::::::::::::::::::uawmyyIfYpIyy yjxSyvyy=uaw=jyyI:::Vyu=yawYYrybyyiy]yvpXrMtAESa=V[yywqixiyjaweqyxEy>y uIyoAypYyvQyyyyxuywIyyyxsyyqix`yxyIyQyuiyyyyyY::jxyiwmwyiumxSyvyy=:::: :::::::::::::uawmyyI:ZtQYyyyyyybyyyvQIuIXhoGeqfayixsyymixyiyuivWYuyyyw ivaQwqyw_ytIyuYysYyuaywYywqyvuywyyyiywQyKxCynyx=:ryxyy;:uawmyyI::::::: ::::::::::::::::::::::::::::::::::::::::::::ZvayviywiywMvUYuqywiixyyvy xrUyxYwiyxAyruUQytbIV\\?ysqwZQvyyxsUweIwYiu_IvYIvaYvKYuQIrMis;iuOys_yu GytOytQQqmHQqqKn[div`iwbIWFwayvdAwbIwcEtKQlUXmEXlMXlMh]mk;BsCqlQHGA@H: ::CaYMhcqwpiygXMiCr;aY=uXqlWh_U:pwhqWiuG^UhwgiWgSpmPgegTmfSp::::ZRAUk` QyTi@:>uRUpmXpm`RMfsontPeo:::VtMYtNMdSofTLWd_THwcOTtVauVPmTPir?iRmUXma uVwuYyuRAuRIT@G:::i`LQTYxm@G_`KIT]`MYTe@ZKmbCn[PbSnFTDG_`KmayvauVaoSxv ayFw `qVBtKysIqlsrAQlMh]mk;Bsfv_esGqjSr:::^VxGsHqtWiwuYyuXquXqUtJEn:vsI=_@vsIysHqSYxmYXnnsHEnNTDG[`ImaCn[@xF:bmnS::pv`gSJ` mfF=:::::::::::j_ifE=::JbclOhlKHlhv_iV`clOp^afEulU@VSdV_OS:hFMXlMXmvSh FrlVS`FKHldV_eV`mV`mFQ`Dm_Cm>sBASZDm^;MXlhFI`BAsBAS:j^SlvrAuk::::::::: ::vYxI:::::::B:ry:::::::::::::::::::::::::::::::::::fywiyxqYyyyyyyZt]u YyyyyyqYwyw_avqyyIxmQnIPlQarAyysiyyIyuUwgIyoYxsiydyyuYxgyvyywyyyayyyyx iyyyywyYryyAjxSy::::vyy=:::::::::::::::::::::uarQytQytuyuYyyYyxirYygqw ]hlPXxcIwyqyxYymaXoyxyYysqywYwA:tiuayt]y=::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::pyxqIhywGyqyxyiywqyvayiiwgawbyvpQxkyywUy eQwuXqMW]YVN]sFyxjExyqyxAxfawqexnyyyuYoixnIXmMxqMYs=xlIXneXoWWSIuSIufw eUt`Xo]h`qv`Aw`usEYsIYsEAGa`MlMXlEXk_v^urAqk;BsCqlQHFV@H:::CaYMhcqtwiy gXqUw`ZAVyxqy@u`VaUs@ZWiUwhqJhspqPgegSIUohpdGm`S=::::\\Wf?UXwewT:\\Wf? U:XGghoTGcHoLG::::e`O=jcIgLQTUXmUxbAwbQwdQWpmp\\Qv\\qwhqwcIwcyvayvbSn: ::vTDGYxmY@_`KIT]`MuncHoHwcOTJb=gJ=JaufJmbSnvsIysIMb=gLenWhm:DG[`IysIy SS`FI`A=::::::::::jysy:::::::Z:Zy=::::::::::::::::: ::::::::::::::::pyywyy=uYyuixoqxqiyyYyAZwyYcyxkixyYykyvcyysiyHytMwZMvW ETMyypAyyiyxyyxwyyyyqIysiyxayxyywyYwIyuyxuyyyyxVYpIuAVYhIuawmwsxfYpIyy y::::::::::::::::::::jxqit=xIyA^yxqyxqYyYysYxcyvyywkyyuqUqtPuUTMuyqiwy XyyxsQyuyyuYyqywiYymiyyyyyYwiyyYyyawyYuA:::vyy=::::::::::::::::::::::: :::::::::::::::::::::::::::::::RyuAlYx?yrAyvayvixoEr;Ir;iqwhwcIwiyv_yy wakIxiYv]urCIf;WbUyq;yygiwyywiivOitYitMiuOYtOItIYuWIuOitOYtQytOIuSitSI uQytXxnaXn]h^MmUh_iVCIsPxl?w`qvbqVGIsNVCIsHxkWV:LXlgVTXmGV@H:::CaYyUs` XmiygXqUw`ZAVyxqjhCq^U:pwhouX=s`UYUohpoXpmPgegTmfSp::::ZRAUk`QyTi@ZRAU k@vTTWegtOUo^tNEo:::ftOUo:ZNmcIGvnaxn:@j:`uVujAxjghoghlOhlg`HqsHItKItK =::jecnvsIUnNT@wbGTHwc_tN=_`JMbsmYpaCn[PaufJEn]`KmayfIMbcnnsH=bn>TxFY` J=xFbmWPaSmUp`CmSHmpfGM`CmNsDulVSJ`mfF=::::::::::J_cl:::vSdFMXllFU`DQS blfSlFQxlQhlOP_;rlFsCUmvShFrlVS`FIxkdFS`Fm_sl^SpFMp^iFQ`DM^]fAyRI`Cm\\ SknRTFI`BASK@vRXF:::::::::::yay=:::::::<:xI::::::::::::::::::::::::::: ::::::uyypIyyy:tYyuywSyuayuynYvYkyysYwsYxwiypYvayymyx?ytYxVAvhxlUhx^yv yyxsyywqyyiyxYysUxoIxuiysyxwIyqyyyixyqwsx::fYpI::yyy:::::::::::::::::: :::tIsIysyyyqy;uAyAwyypyyqiwAYpyxawayVOysJyyuKyyIByoixsixoyywiyyyywiyp yxyiw=:::jyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::nYw YyuAlIkawiywIxkIxOitOAsDQsyyYqIyoysEIsyYxmyymYxmQpuWbAwMaSGyxjEXy=yruX kEXyMYtYyqyxmywiIyquXoeXoaheEuSitFweYwaUtGItKIs^v`qv]iv_]_iVKis\\Xmov^ aV?a^urVV:LXlgVTXmGV@H:::CaYMhcqwpiywiuWioUTxFY`JATrmnStFUXmlV`mV`;U`G=hv__Shv_iv_;S`F=::::::: :::J_efD=::Jb=gDQSKHlhF:MXlhFUXmUHmVSdV_OS:hFMXlpFYxmhFrlVS`FKHldfDaSl FQ`Dm_CmFsCelZDm]Yv]UV]wR\\V^or@qr@m]sk:ZBm]YfA=::::::::::jysy:::::::Z :Zy=::::::::::::::::::::::::::::::::::::lIuyxqyyuYysIyXyvaYpAyqYYxeYxu YyYyuYyyyYyyYxqivqyrQW=he]Wo;hjkYsKyyoIyyYywyywiyqIyoIyyQwyisyYmANYyYx yqwsxyqxSyZyyY:::::::::::::::::::::uawmyyIZuYyuYYpyxgYlIcqswyxuyyiysIy yYywiykYxaisKI`OF]ayrCyygiwlIuqxiywoyxyyyqyymxwywmwyiu=dIuavqyxYyyuxfY xyy;ZxqyxyYyyy:::::::::::::::::::::::::::::::::::::::::::::::::::Zwiyw qyxqytQytAYkyyyyy_iv^tNytVQudywgaYuqyqYpYXm=Yn]XyEYs?ysAvLYtSXkFXswpQs qwqXqMXf]wxAyrQyoiXpehcYweQwfawbiwfyvfYw_QwcavcAw\\yv`Yvbyv\\yv`UsTXmG wayvfyvbEsEYsAQlMh]mk;Bsfv_iVFV@H:::CIsxwi_uVeqtwiywiuWioUyrAyxqY]Yv]In_hn ]Hpeht]iuoyy\\YwGxiqwlqwgixhYwoawcaw\\QvyiY[irAij_wfeXj;XxIisoitQIu;Ir uywkItGQpmXpeHAYsEYrGYs;YsCyrMIt=ysGYrGYsAYsEisGQo]hdAwbUsTXlov^av]mk; BsCqlQh`es=Yr@H:::?irS\\F:::I`A=::J]Uf@=::jyyiy=:::::::<:xI:::::: :::::::::::::::::::::::::::::::::tIwAhyywxyqypMuSxfwlIxyyyyIymaxyYYEIo ihesVaeIsIiy]yvpimYxmyYyyyZXqUyyyxI:::::::::::::::::::::::::::RyVywiyt yYwyyuiyyaxUyuqyqyy_ywuxsyymyxEyokXdqFayyxuyw]ivyixqQxcywIyoQw]iyyYrxy awUuaivmyywYxfYpIyyyJyoyxmyyiy=::::::::::::::::::::::::::::::::::::::: :::::::::::JycyvYhyw[xMtMYtQweYwcewe]xruXwCx>xkev[yueIvUiwYIuyyxqYrixo wIv]yycYwGhf]wiQgdgisIyyhYhMwdiYuYyp]xnixnaxrAyrMXmUxlexnAxmUhaqvZqv_Y VEQmOwbMnSs\\xkMXmMHAqk;BsCqlQh`mjSr:::Nv\\ErnuXeqvuYyuY=tG:uXqJh;RquP iuG^UhwgOuTUpmPgegTmfSp::::ZRAUk`Qme;:kHpJeUgOiTcHoHwc_TLG:::u`HqsHiuW ukIxkGisTf@Yv]YvNaT]`LMc;]Xn@Ga`OmbSn:::fTHwcGtKUnNTDGRnVTDWc[JATYxmtV awSYxmnSHGa`I=TxFY`J=jbsmYxmUXmhv_[GM`mfF=::::::::::j_ifE=: :J_efD=::Jb?tEM^]fDQSQp_eFO`Fm`;Q@NS:hFMXlpv`wShFrlVS`FKHlhFU`GM`clOP_ slNSJ`;KHlTFI`BElIxkXF:::K`B=::j]sk:::::::vYxI:::::::B:xiy=::::::::::: ::::::::::::::::::::::::::::wqyqyyoyweyqQyeiwQYnywaxyEsYYuoiyTYtMiviYc iwkwbGgs]HomIuWyyhiwewcqvYiyqyxAypqxuIysuyxIuawmyyI::::::::::::::::::: :::::ZwMxuYyeyyuYyyItyyoyxMyuiyqYwYxoawUywyyqQDeR?uWeYwyqYy_yyiwaAwy]w EvWiuoIxkAkYy`yyuyv;uqyqIyyyJycyvyyyyy=::::::::::::::::::::::::::::::: ::::::::::::::::::::tIbxoquYyuyyy?yrAyYIv[Yu\\yufixlQyyiyoQyrqwkywieug iv]yw_YvIir=yyuYysIv[IuwxqyYv_YpIHlkgcIwsiHpyywmytAIs]YtOYwYIvYitQitOi tCQmMxn]xj;JmOwbEuIItQYsGItAAVVAAZ^aVEYsTH=a[=::JkEhZMiugXmiywiyGbq:fu WEq^U:pwhouX=s`UYUohp`wfOuTepFU`G::::BpkPfsoXwe;:kHpJeUgOiTcHoHwc_TLG: ::a`ZAVCHkCxoixocHochmWhm[`NatNITDG_@FtKItKmcSoFT@G:::e`MYT]`KMccnZKmc cn_@TxFY`J=jbsmYp`qfEYSRm^sFEm:::::::: :::VsEul:::NsDel:::>ThFKHldV_WSQhlNSlV`gSj_;O@ZEm^afGmawsE=hv_WS`FKHlh FU`GM`clOP_slNSJ`;KHlTV][BAsByrAuk:::>S\\F:::I`A=::::::jyyiy=:::::::<: xI:::::::::::::::::::::::::::::::::::::::qiysYyyiyyQyyIsyyqYyyiysqxayy OxyiymyyxYwmYxtyYyiywixYYvyYxsYo;xj_GcGwyqYyuYwgyyuiyiaxIytMyyYyoyx=si ywYyyay=:::::::::::::::::::::::::siylIyyyxyy;nYwAyiyuixgauayoywjyywiyy hpoxqnppvIxmyyuiylQweIymaXuiymywiYyuYyuyyyiwqqwyyyiyygYpIyyyJycyvyyyyy =:::::::::::::::::::::::::::::::::::::::::::::::::::tIwauYYiywiiywqxqi bQwdyvhYwmQyqQyyiynIyqqwkyWtxsAYrUYsAyxaYvaXlIXyMYtyYv]YsMxlyysMYquwia s>eRwaXpIxdYWxyxrayrEYoMxlixnaxiAxjQXoaXkYXmAXm;GQo]HIAVVjkWV:LXlgv_es FV@H:::?irtJAtPqtPatNEofTPuTQUeUgDmbcnNTjbAgKmcSo eXo]`K=::jdsnapbSnNTDGRnVTDWc[JATYxmtVawSxFW`Muna`I=TxFY`J= jbsmYxmUXmhv_[GM`mfF=::::::::::j_ifE=::J_efD=::Jbsl>SdV_WSdFO`FaSU@VSJ _;rlFsCUmvShFrlVS`FKHlhFU`GM`clOP_slNSJ`;KHlTV][BElIxkXF:::K`B=::j]sk: ::>S\\F:::yyyxI:::::::B:ry::::::::::::::::::::::::::::::::::::::ZwMwyi tyYuyywyytiyiYyqyoytUyeYysIwyYycyyDYvYyxyyyyYyqyteXpyYwWx_Iw_xlUXwWIvy IYyYyyIymyxyiyyayqyxqyx=xyyyyy;sawyyyuy::::::::::::::::::::::::wqyuIjx yYyqIvYyqwmYwyixyywiyyqyxixjyWuYyKIqsXdqVa_IsGyyoiyyyxwiymyxyQyuYyWYHy tWyyuxyqxyyyqIyyyJyuixmywyy=:::::::::::::::::::::::::::::::::::::::::: :::::::::tIBysIygiwgYyuYyyyysBulQp`Sm^SdV_OShFO@^SJ^]V^UV];\\v]Yf A=::J^Cl:::vRXF:::K`B=::jysy:::::::Z:Zy=:::::::::::::::::::::::::::::: ::::::::pIoateysqxqyyuyxoYvyyqywMwCtWYuaYxkYxyiymIyoyyvYyyyoyx\\ywyixY qgHprhojxyuuyuyyuIxoyyxyyuYyoYryyqy;uawmyyI:::::::::::::::::::::::Jyyi x=fYlYyuYx;oYyqYxsixvyy]waYiiwoIv[iwKitGw_qfxQyuyyxwyx_qyqiyqyxYxmuyry hyy[wmyyInyxqyxmywyy=::::::::::::::::::::::::::::::::::::::::::::::::: ::pyxqiy=gavayvqyxqyqMwqyweImiysIwmyw[yv]ytWIuUiuSYuVwfQwqixnIyoaxnywh IxfYwgiv_AxbAWvwTiuVhjCpwSauuQwgquMiTsYyGIsCir?irCisGyrEqkQh`qVKItKitK a_esCIsCIsHxkWV:LXlgv_esTxjSr:::Nv\\ErnuXquXmiygY=tG:uXqBqs@ZWiUwhqw@s `UYUohp`wfOuTepFU`G::::BpkPfsovT:Jf]GntPUoePdMgMYTcHoLG:::BpkHpOhlOXlM pBAsByuYyUaufNatNITDG_@FtKUnVTPG]`K=::jdsnapbSnNTDGRnVTDWc[JATYxmtVaws IumnSHwcWTxFBn>TxF:_`LMbsmvSTxF:_`LMbsmvSS\\F:::I`A=::::::jysy::::: ::Z:Zy=::::::::::::::::::::::::::::::::::::::pIsiyjvyismwwytyYuayqYyiy uIxcqxiywqyxqywyYy]iwYxxaypqyxayxyysqxryytuyqiyhiwPpoxrvxQeIxyyxwyysyy tixqYryyyyyA::Zw]yUyry::::::::::::::::::::wqyqyyoyyeysQyoyyQyyex:^yyiy xUykIxkYyuqyuIoyxhIeYxUhi?hjawgqYu]yytYyYYwqywiyyqixeyxyyyIRyxyxsxVYjx Syvyy=:::uaumw;uayyY:::::::::::::::::::::::::::::::::::::::::::wiylYxs yyyynxpqxsIysqyxexoQwiyveywaYuayuQYvUYsQysuxryxp=YqyXs=ysUYtmxqqxneXnA Ys=YxeyvIyquXpYXmUYquXveXpIYkQxvMxmMvRMuVitPeypyxyMYuuWeYwoQXmaXlQxnMx l]Xm?wbesKItCa\\]^av[Yv]Yv]=brCIsAYsEIsGis=a[mkAxjIH::Cir?IrnuXquXmiyg Y=tG:uXqlGs@ZWiUwhqJhspqPgegSIUohpdGm`S=::::\\Wf?UXGi@:>uR=ghoPGc`MYTc `N=::jdcogPhmWhaqv`Qgeg`qv`isGisOUo]XnDG_@FtKUnVTPGRn:::ftOunapbSnNTDG RnVTDGBn[payfHqSYxmxFW`MmcsmZJMbsm:NTDG[`ImaCn:FTxvayv`qfEYSRm^sFEm::: ::::::::VsEul:::NsDel:::>Thv_?sBelVSdFO`FEmU@VSJ_;rlFsCUmvSYxlZEm_Sl>s BulfsGUm^SdV_OShFO@^SJ^]V^UV]wR\\FIxkXF:::K`B=::j]sk:::nr@qr@=::jyyiy= :::::::<:xI:::::::::::::::uqyqI:yyyJycy:vyyuyfYpI:yyy:::::::::::::::pI uaumw;uawmyyIbyxYyuYyuIfYhImyy\\IkiyuyxyyyyyypywiiwMwqyjIweQUL]tyyXsyY u]yygyvIyuqyvayyyyyyYwixkywyYyuyyyyyyaxqYqYyhyy_Yhyy[w]y=wax=jyyiy=::: ::::::::::::::pyygY:jyyI::nysYyyayyexPw]iV]IUUUxdawyayxWYkYvgyyoIylYyk xyityYiyyDYweyuyyyaYyyYwyiwyyyyyy=wiytI:RyVY`yywXnYqyypIuAvyy=:::::::: :::::::::::::::::::::::::::::::::rxqyxyyyyIwcIwuYyuIxnyxuyFysuxqyxqarM sEYsyxqyxqSv]Yv[IvTja;BnfS:SdFQ`Dm]Yf@qREXkEhkGhkI`A=::::::: :::::::::::jywyy=:::::::<:xI::::::::::::::RyfYhyygYpIyyyJycy:vyyuyfYpI qaumyyI:::::::::::::::fYpI:yyy:::jxSyZumuSxyQxuYyeIyYyyQwqyyayxIYl]XWa TPqwaAGlyt]yyqixaY_Iv[yywqyuivQYkiwgyyZuMvqityyxYYyyxfYxyyoYwiy:vyyuy: ::::::::::::::::::::::byyYw_qyqyyWyxAyaQwxXohPsuxqyAyvyyrqyyIyyyyvevYY vyyydyy_wnaxoyyyqxxuwyQwSyvyyeywQy;uYyuiywiywawyYqyy`yyuyuswyyyqawmx;y yy::::::::::::::::::::::::::::::::::::::::::lYxwyyyyqyXyyyyYxmqvctOitm YxmyxqqwCxNXxHYakIXUXGi@nTTGroipeCpkPf;ghoghovTXGbo^tNUoePeUgJm cIwcUWeOtLQtJATe`NMfcj?hjmWhmWl`Vf]gQMbsmYpbcn_Pc;]Xn@Ga`OmbSn:::ftOit JEnUXmxva?TxFY`KMc;[Hnja;BnnsHqsJmb;[`HmaSnnsHEnFTvZUr=Qj=X jywiyGJiuGuph;whqxwiguWiuVaUqxpdWgGuSEpFuSUp:ZRMfsovTJeUgPmeYgQ=Jf]gRm eso:nTLWd_THwc_tNUoPWdcn_PcSoepdSnfTDWcWtMUpmXpqxpqhogho[`HqS]pbEG_@Ft KUnVTaXoFT@G:::e`Jm`qfIMbsmYpaSnNTJb=GvS:vZUr=Qj=hYyUBrvUtWiguWUqnuXeqfUl WhWuUepopfagRmfagS=:\\Gk`QyTi@::::>uREpZQMeSoepcIgMMcEgRAUc`JATe`SMdMW dAwbAwcWtRAUm`OmcCocHoe`NmbcnNTjbAgKmcSoFT]Xn:::fT tJ=Y@ZJmbcnVTDG[HnxFbnDWauVa=G]`JMaufFaSS@::::::::ZGm`;BmSp_ifDQSO@::: ::::VShF:::YxmYHmSP_efCISO`Em_clfSj_;O@ZE=xfIATlV`mV_CmfShv_;ZFM_ClKP^ ;IxkXF:::::::::::::::K`B=::jysy:::::::Z:Zy=::::::::::::::piyiYhIRyvyy= wiytI:yyyxIRyfYjyyI:::::::::::::::::::::::VywexsIyyyywqyqIwiysqxiywoyx myaYtUyymyyyYwqyk]WgUtGqSxAysyyxwyuuykQvIytyyxyyyuixqYuYylYyuyxyyyqIqy yhI:uawmyyI:::::::::::::::::::::::^yvUxmYxwiyxYywYXYswyuayyaywAx_yv^Hm UPYYYxSiuliwAytYypAyyiYvyxeywAvyyuyfYdyygYpI:nyweyvYxyyyI::::::::::::: :::::::::::::::::::::::::::::::ZpMyuYykvaicHocHysIyuYYqar=jqSv]QrCvZiy WWiupxpWWQitdxsAyrmXoexp]xnAyomXr]xn?v\\Qx_yvyYxoqwWAvFXixwuHItyixriuI ItmqvbAxbIWGysEIsGirCYsKyrEIt=IsOYrCItAYs?IsEIrEisoHsCIqCIsyxrCYrLxiIh VaVeXsXwlMXhMxkyGxwiyG;IrtJumZKMauf J=UXGi@>uREpvTXG:g`Pm eYgQ=JeCocHoc`MmayfTQuTYtMYtKITohpoHn[P]UV]QwdWTPGghoPGroNtLQtFasFIT]h nNTjbSnVtM=]Xn@G:::epd=GUXmxF[`IySY`KMc;[Hnja;BnFTDGa`LMauVauFYxmY@Whm UXwewTJf]gR=XGi`PMe; :ixoXGbo^tNatOmbspqxpaxna@m`=Yr=ypqxpLQTaxn:Xwe?ULG[`IyS]pbEWcOTjbSnVT aXoZK=::jdCnTja;BnFTDG_`KMaufGisGiStVaoSyuQvNAuBIqwhtAuyqqyiYy KuAiyqyxyYvyxyyyyywmvyItIyywyqwSyvyy=:::::::::::::::::::::::uapAyrQyva yuyyyYYdYxwyyyyuIYlyyoxsYygIxa@mVPvKityIywyYxYxuiuyYuaxqYry`YyGYXyyGyy uxfYbywAvYxI:::::::::::::::::::::::::::::::::::::::::::waxmxqiyyyyQytQ yhqwhyyyUdjviywWY\\I_AvW@yr?XUXwewTJf]gR=XGi`PqTg@fTPGghoXwewT JecoNtLupqxp_hn_xnvT`uVaUPmTPuVauvcWtHqsHisGen_hn_`IysIQtLQtSmbAwbEG_@ FT@Ga`OmbSn:::ftOEnfsGumxVb=gIySY`KUn_@>tJ=Y@ZJMbSnNTUXwewT:\\Wf?UjecogPeSoePdMgN =:::JeUWeYweOT_XofTlUXmuTQuT=WbGT@G:::bnNTjbAgKmckdGT@G:::ghogHnnS AslHmWPu=yrnawfAtXauQYw`ivdupOXmdwjMyZaVDeprDx\\]VqAxk=XhiwmIXkQxj=xky WiuWhmWk=XjuWi_UsHqfuWUqZVaUqXquPhspqhpopfagRmeagQIUohpoxpqPhmGFuS=mXp dGo`SMfsoipe;BpkPf;i`PqTg`OMdCo::::FTHGohpoXquXqAxjAxayvaQwdQgpopbAgK= :::DG_@FtKUnVTPG]`K=::JeUgJMaufJmbcmWp`qfIum_@>tJumYpa;BnfsGen_PccmWhm Qxlpv`gSlV`OsDUmvSxFWhmlV`_Spv`_SlF:::::::W`IySRmfSlFS@NSdF:::::::S`Fm _ifE=J_efDMbsl>sBAsDm_clNSlFYxmY@rlQp_clNShFMXlMXmnSdV_wrAUlFSXv]gr?El ZDm^aFvR\\FIxkXF:::::::::::K`B=j^aFnr@m\\gR:::jysy:::::::Z:vYxI::::::: ::::::::::uyyyYyZuMvqYxExmUXq=YrmYyuyyyiyMyuYyIIyyyxI::::::::::::::::: ::::::wYytyxcyxIuAFyyUx:VyyUynyxeyvywyYuA^yxqyqQyyYYiyu[ifIWdaGdOwysqy qYy;yYxgQv]YvyIyax]yuy::::::::::::::::::::::::fYyYyVYhIuawmyyivaYuAwyy yiyyQxuivMuOx^qvkXn_pw[YvyyxsyyweymixwIytYxwyyyyy]xMygywUyqyxwiywiyvYy yy:::::::::::::::::::::::::::::::::::::::::::^yvEynyxqYvxqeywAPytQyvay viywWY\\I_AvW@yrwWS\\FMXl`FO`BASI`@qRPf?=::jysy:::::: :Z:vYxI:::::::::::::::wiytIyyyxIuawmwcxyqwsxfYfywaYTYuoyxqYvY::::::::: ::::::::::::::JyqIuax=uawmwyitMw;qawmxcywQyqiyiYJxuIwIyiYxwIxmyyuQoywj QTGmStaxo]YrAyymIuawqytAyrQyrAyyqyxyYxI::::::::::::::::::::::::uyy::jy yixeyyIyxyxyiw=JxyiuiypIweqnRPmjUsyywfyxqyrIYvYwYyqqyqyyoyyyYyA::::::: ::::::::::::::::::::::::::::::::::::::wiytIyyypyxgxoixpqxpayvuyyyymYxh ywWY\\I_AvW@yrAIS\\FM`C=J^]fAm\\Sk:::vyyuy:::::::Z:vYxI::::::::::: ::::wiytIyyyxIuyypIqyyhI_ywgyxfYxxrAyysybypywsyyyy:::::::::::::::::::: :::fYpIyyy:jxSy::^yxqywyYuayyYqyxLYtMyyuYyyyxwY`ys_ymqxtywlypd`oaQuVyY qyykYxwywoqxyiwUxcayuy:::::::::::::::::::::::byyyywayyY:nyxqyx=xyy[xmx qiwMyuYyyYywyrYyyQywUxsyhUWf[xjExymyxlIuqyCymQxYYvSyyyyyIyNyvaysYyuyYx I:uqyqIyyy:::::::::::::::::::::::::::::::::::::::::jxqyxayvaYv]YvQytQy tsxqyxmYxmyxqyxqarMs;YakUvBv[Iv@I^WdHMYt`HGQnGwcIwc=btKaaetEis[ytSItAI sSitOIuOit?ir?ytQyt_IuUysehoyYsIyyiHpsyr=YxqhpyinehikeY_ECQcl?F\\yYrAy xoxoqYqsxyIIsainYXuihn_yvOInqxnshruHqyXqsXqqHquxpuXquXqZVaUuXqZUYuVQuT IuSEpvtSuomPgeWgiwg_uVupqpfaGFuSepNU`Gk`QyTi@ZRAUk@vTTGg`OiTc`N=:::Jes oiPb=gqxpq@pj@pOitOqv`qvAyR]Xn]Xn:::ZLMc;]Xn@Ga@NTDG:::g`JMaufJAT]`HqS UXmUxmNT@G[HnxvawS:ThFKHlKhlZDM_CmnSJ`cl NS`FM`DM^]fFasFiS\\V^]V]UV]]FK`@m\\QfBM_clFSXv]wR\\FIxkXF::::::::::::K `Cm^ClKP^skfr?Uk:::vYxI:::::::B:yqyyI:::::::::::::::wiywiyZyyYuyypIqyy hIRvgYwuyyyYxqyxyqyQIyiywyywayuy:::::::::::::::::::::::fYpIyyy::jxSy:Z umw;uayyYiadAWyQyyIdyxWYyYwiIlqWiwWfaGxQYuyIyuixaQwAYtyYxyix=yA::::::: :::::::::::::::byyQysyyiywiywA::vYxIuauiywYyyyyvmyqyyOyvyyuqyqawiquKYt upojxyuiylmyiiyiyxUylYwGyuYyuyYoyydIyay=jxSyvyy=:::::::::::::::::::::: ::::::::::::::::::wAyqwiirMyCysIygasIYmauYYqarMs;YakIXBv[Iv@I^wfetdxna hcMm?WMYt`HVwcetKqoUxmgVOYsGysEYsGisDXlMXjUxliXryxpiHexoYh_mFk=F[yyq?y yuPZUQwdqP@MYYQwxYj=xmmV^=XbevyYxkaYeIwQErypT?URlqWdiWjYWjAxhuwiqWhWuV iuUiUpGBqsphqGrpqhpopfagRAUmXp`Gohphwg_uVupqxpmXp\\GbpNU`Gk`QyTi@ZRAUk @vTTWeoTPwd_TLG::::eXoDGcHoc@[HnLgN]HmbSn:::ZLMc;]Xn]XnVTjbSn:::ntPEnn SWHnFTtVagsGisIum_@>tJumvS:Sj]skfr?Uk:::vYxI:::::::B:yay=::::::::::::::JyuixmyyiymxSyNYwixNiyoi: fZ>>Z;FZ:>Z:NZ:RAqr@;hiwwywqysy::::::::::::::::::::::ZwmwyiumxsyyA:::p IqasyYmiy`IoyyHYvYyxwyyyYSitMIyiIxlyyiYtsxwyyqiyMIqwhkkghkhlSxyqYyyyxd yrMiwqyyqYry:::::::::::::::::::::::byyiyuyyuYytYySyyiyuQyuYymYyuiyUyxY wSxyiyiAwQxwyyiYQYsGyyqqw=YsUynixOqsIQym]xw=YsyYxqyyqYygYmyxlYyuYwuitq YyyyyAuqyqI::yyy:::::::::::::::::::::::::::::::::::::::jwiixqYuanaxnqy xqyymucxoix`yviywiywCxNXjqSv]A sBEm^S\\FEXkXv]YV^?s@qREXk\\FO`Dm^skIp]ClvrAuk:::::::::::^r>Ekfr?ir@qr @yRXFGhkTFE`?=::jysy:::::::Z:vYxI:::::::::::::::uaw=:jwyyyiIehpmXGisGG f[GvywqyyiiiwgyyxEvWiuxiy=::::::::::::::::::::::pIoaumxsyyA:JyuYyyIyaw mwSxyqvuis]yuwuIy_ywgYyyyyyyywQxmYwQYugyyQyquysqyiyxayoqXbEV\\]VWmUpqw jyywuyymyxgYtYyyIyA::::::::::::::::::::::byyiyuqxqixqYpixiYwiywIywqyyi wuyoasyYxIwqwqYymyvwywqynqwiyyviYqYyyIwcIgGwcohmWhxYYvyIyfYuyxqayyyxuw gIxyyyxIuqyqI::ryyA::::::::::::::::::::::::::::::::::::::pIkIxkyyyqySv ]qwiywyIeqt]yvYyvayviywWY\\I_aimqSv]qqCv[Iv@I^wfetFwcUt>wbUtMqnstMqoCu IItQisfVAYrarkiqwxyVIupYuQYyXQuoauTawUAu_auRavWQu\\i uTavWQu]auSQvWQuZauUYuVQuRYuSitTIuPAuRYuSIuUIuSauTQUqPhmgUYuUIuSIuRAuR IuSepoPgSpmPfsoipe;BpkPf;i`PqTg`OMdCo::::FTpwhqwcIgDQSgho\\Wf]Wb=WbMWd [K=:::DG_@FtKItKmcSoepayfI=::JeotJMaufJmbcmWp`qv`yF_@>tJAtIma;BnfsGenN TtVauv_ifGUmU`FasFQsDUmvSxFWPamV`_SpFS`F=::::::j^av^aFO`EYSQ`FaSU`GM_e V_eF:::::::U`FM`slQp_clOP_SlMPb?tEulKHldFQ`DM_CmSp`SmVsE=M`C=j]YfFasFa S\\FEXkXFI`@qRPfBM_clFSj]ClKp]YfA=::::::::::j[Iv[IFCHkCp\\oRTFRkfR:::j ysy:::::::Z:vYxI::::::::::::::RyxywyiuMwuihQxmYyyyyqayuiyyyyvqyxAytupq pnXPmdR=QrsyXqyyyuyyiYwyyyuiyyyy=::::::::::::::::::::::jxSy:vyy=:::uau mw;myy`ix]tmwSyvyy=wiyhyvaYwMytgyxyyuiyXItMYrlppZysLyyvqyvaxoyYNyuyYwI yuyxyqymixYyxayvyywqyxuyvyxiYpIwYyqyxwiywiy:Zy]xqyx=jysyfYpI:yyy:::::: :::::::jxSy:nYtIyYyyqxcyxixyywmwmy[]UWUvViuyIyuyytayyeirIxnyyuuyuyyxyy KxkyGyxiymAyqyxvYyuyYqawmx;yyy:::::jxSyvyy=::::::::::::::::::::::::::n yyey:uqyqIrxVywqywyYuaiIxkyYwIynyvYyLItGYysAywYYqIyoIxmyw_yv]yu[iuUyuY yu]qqAyrqxqyXrAyqAYs=Yqyxp]xnYxnYXmiXpexmYxm]Xn]Xn;RtIysKisXxlGwayv`qv `qVLXmUxk]h_yvaQw^uyyHryyhwW^?cCGF=crbiSIwihqwy;iqeyqoxuGirEYtAyniXncY uOYtcxnsipqxy;yryIogXvqwhaYgegvmwgyypgxxCIquHnKxsuhoSIqgXv;YqQIqmxrsXp qHqoXosXp_hpmxnkXpqHpohqihpshpoXqqxpsHqhwgGuSEpkpfagTQUm`S=\\Gk`QyTi@Z RAUm`QyTeXoexnaPdMgOiT]`NEoohpohj?hj?ir?Ipkp_iv_IwcGUtWiuwgiwgyvayv`qf NMeUgKmb;:::bn_Pc;]`KmcSoepasm:::ftO=U`JAT]`IySWhmar@m^av^eV_GS\\FK`Cm]Yv]YF:::MXl`FBl>SXFI@FS`FK`Bm]ckGp\\Qf?m ];G`?m\\CkCHkC`=Yr=ir?ir?=::jysy:::::::Z:ryxI::::::::::::::RyyyySIv`Yy yys]ywyXtQywyyywY]yvaYxqyxyqyyIYYuSYZuEY[F]QfxryqixmYwyyyyyy:::::::::: :::::::::::::fYpI:yyy:::jxSy:VyyUxyqv;:oawyYmyxryyYywYwYqy]ywWYsytWIpI hl[hjCxyqYykYuYyypYx]yuyyyuyveYvaywiYwqywmyyqYxuYymYxmywmYwuiweytYy;:: :ZyyyyyYjxyyyqI:yyy::::::::::::::::tYyoywyyxiYtIuqyuiyiyxyywUvGqGmRBAY kUxwqxryIeIuxyvmYxeYwqyxqywqyxeyyuywsxZpYytuyhIuawmyyI:::::fYpIyyy:::: :::::::::::::::::::::::wyywYyxYywyxayxUyyQwcxwQueyweYyyyxyioyxqyyyeysy xmiw_IimWgqixpYx]ywmYw]iqmrkpyhiIxkyW]Yu>xkUv_IvVwbIw_IwaqvgYwdyvaEtKQ n;rsIa`mlwvayv`mlQxlUXmssIitKQqIxlyyiwyvhqZOZ@>nufAIgdyyr=Yuihn[Ys=isA iqMXmKxrCIroXochwKYtSYncxyiXpYFPqTSCcCqt>aBP]R=IwTEuymXnyIsohtmHnYYqgH schmQIqmph]wemWgAWhaWcOuNAuSiuRQuXMiaWgqwgCqhwgGU\\WfGuSepopfSpZRMfsoi pe;BpkpfsoipdQgMMdMgOiT[Pbawf[Qm_iv_MWdMwhqg[Iv[iqwhQpvfawfKdMWdYweYwc IgFasFItKItK=:::DG_@FtKItKmcIwcQGY`I=::jd;U`JmbAgIySW`J=@GY`JMbSnFTDGY xmYhnNTpv`WsEemvsIysHM_efGmasmnSlFS`Gm`;:::W`Hm`qv`mV`WsEulNsDel:::::: ::::::VShF:::_`FasFIsCulQxlS`Dm^afFaSS`EM_;KHlXv]YV^?sBM]UfDM`ClZBM^ck fr?ukIP_eV_eFM@>Sj]sk:::FS`FK`B=XFI@FS`FK@nr@ekfRPFGhkTFE`?=J\\Mf>m\\Q f?=::jyyiy=:::::::Z?^:@tKItyyixeyYYuSytIiscDNYdg]Gf_yuQaumyyI::::::::::::::::::::::::::::: :RyfYhIoyydImyy`IqawmuayvyyywyxuyqQxUyu_ywMyayoyxdYv^=UKMTwexoqyq=yyuy xgyvyyyqyy=lyyqixEywqxoytiYuiysyyyqywyy=JyuYyyyy;vY::::::::::::::::::: ::tIuYytYyuyyqIuatiyxyxtUxsIe=GcAiokxyqYyyyx^xpyywqyyyYyarQytQxnaxmixo IyuaywyyyiY:::jyyI:::::::::::::::::::::::::::::::JywyuyYwyytYxuIyyyoiy dIkasiyxiyyewkQxqywyYyqyw]YtiTNMtweYu;TiqQXZiywiIxkywlAxgAxhAxkywk=UIt FwbavgYwd]bAWKAZayVGa_iVMysIItGqlcsGqmYxmehb]vyu\\yyCQsGhcGwsVHmyAw\\Q yaAv]IuLixliwTYuQAV=yvcivwxpiXtuxpayoixyoHquG\\Qvd_UVkiimwn]F]?WMQDdmd OoIl?xyMirQIrkXqiHnCiqmhqwHp_XqoxmuXpeXpohpiXpwxoshqihpqXphwgiWhmgUYUm XpmHpkPfcpohpm`S=\\Gk`QyTi@ZRAUmXpXwegtOun^tNUoePf]gJ=pwhqwbAwbQwdwStT QuTR=URUV]?uRupqPeotOitOQTpv`qvbAgK=:::DG_@FT@Ga`OiTY`I=::jd;U`J=xvayV aCn[@@G[HnxvayvaCn>T@GrnVTtfHYSQXmUPaCmNSpFY`IMa;S`Gm`;::::BmSHmQxlhFO hldF::::::::::::QxlhF:::_`FEmMXlMxlQp_SlMXlM`EM`mfEM_SlMP^]fAyRK`BM]Uf DelS`B=\\V^?STFG`BM_slFSXv]or@ukvR:::j^afBM^;I`A=J^ClZAM]ckfr?Uk^r>Ekf r?Uk:nRTFE`?=::jysy:::::::Z:ryxI::::::::::::::Ryyyy]iwiywyQuiYxuxsMywy iuMtUYuaywiitQyyyyyyYwyyFrvXqAOb;wyyYymQXyyyyyy::::::::::::::::::::::: ::::::::fYpIrxyqw;RyvyyewgqyuIwiydywiytUyymiyYyxyyuiywywkIp[PnQIsDyysY ywyxriysAyyqywyYwIyuqxqyxmYy_yyiYyYygyxmQxeyxyyyyyy;:::::::::::::::::: ::::::nyxqyxmyqyygYJwmiuqYkqYEUT=uKQtyeyxyyw_ypyxtixpUyuyywiyyiypywiYw aywyYyyIyYyfY::jyyI:::::::::::::::::::::::::::::::tyyoyxyyuyYwyy^yxUyy QwuitqYpipuxuyyyyveYwiYwcyyyiyyqwmtNQtweiuayrYY:jpyxqGxk=Zk=UA\\HZbAwb AW:Xxmov`UsEqnYhaiv_iv_=IysIItGqoIXlwu[yyRIufir=Ay_quYqtGIvXqt]YvWiveq vVYvXqsMItQitNawZquyqv`ix=iRcHpyIpkxyYXmsimIxygHnWI^]FZ?sBWYfUgywHpyho UxpwXouxqmHpwXpYhqoxmshpmHpoXqgXpwxopWfewg]WgegTYuUYuTQUkHpkxoiPf]gSIu SIU:\\Wf?UXGi@ZRmfsoipdQgMMdMgOiTqxpqxkIPf]gGisGauVen_hnUWeUW_eV_EXkEH n[peYweUWeWtM=c`LQT]pbAG:::bnNTjbSnVTPwdwSxF:::e@fsGisJmbsmYxmW`J=S\\FMXlXv]wRJ]Uf@M^skvRTFG`?iRE@Vr=Yr>M\\SkEp \\cknRXFEXkPF:::yay=:::::::X@Ys=Yqgwb]bqVdH \\Xn?WJo]HIQmUXmQxlov`=CIsCqloVTHAyrAyr=Ir=Iqmhs;ypIIruXrwhpXxkewk=XbM xiMwlAxaUwcYYkwyUIv\\QjFhkMn]y@dRwyKQtyqtKyy[auyIsBubKWSMb=eHaonjiwlIx gqvduvgyWg]WiaWcqwgAWhewf]gVytSIuRauSAUkHpdWgOU`wfwtQeogpeYweawfagSuoF uSEpkPfsoipe;BpkpfagQyTeXoHGcHofTY`XeqcHodGYxmYxpqxpCF[CvpqPf]WfMWdGtK Un^TTWeWTxvayvbGtK=:::DWcOTjbGtKmcSoe@xF:::epdk`CnSTFG` BM_efEm^afAM]UfAm];:::::G`@=Pv\\or@ekZ?m\\;:CHkLFEXkPF:G`@m\\Sk:::vYyy y:::::::Z:ryxI::::::::::::::RyxytSwmixqyyhIiYxoYyyyywiywYxmqtQyteR?eRD DkBDJ:Txva[GiSSHmlF:QxlSj]sk::::Z@qr@qRj\\Sk::::nRTFE`?=J\\Mf>m\\ Qf?=::jysy:::::::Z:Zy=::::::::::::::JxuitMsYYueyyuYxwywyYkIxxyyyyvaiym u_XyqySwaivxywuyxaiyqyyuyyA::::::::::::::::::::::::::::::bytyxsyyA::wi ywyyyqwiipqxpyyyyywMveYwqyyWxpaxwyyxyyyexkawMwiyyiyxmyqAYRETLyWcMwyiYx cyuyxqyyvYtYygYpIrxyqwcxyQwyIjyeiyiYpIyyyZxuywiyxiy=::::::::::::::pIqA tywsxfYwYytyysyfxmYXwiytI:By^yy]RQsFIsrXqoyvcyytuyqiymIwxYxuyyuIwQyuiy yYuqyeIoiydYygYpIyyy:::::::::::::::::::::::::::::::wqyiIwyytYxoYxYwuit mvqitqYuanqXyQyAyquxtEYqQwcEwvayvaywiIoiw^xmywkmrAyoIysqxtEypIYriXsAyp ;Uis:QaaEtKQn;OItvVTh_iVEa^aVCAfv_UsNv\\Qv[IVCyq;iuKYt?ipqxuCisiIuYiru hq]XpkxtUYukirCyyKxmsxQYEhQcHyid_WfgTPgfJCtlueYEI`qFcadOSE=_Rj_UWsyqox r?IpwhqkhrwXpYypo`UytVQuIaU_hpo@mHpoxpkPfSpmXpohp`Gi`PqTixo\\Wf[S=\\Wf ?UXwewT:\\Gm`QmdQgMMdMgOiTcHotVauVf]WfqFkHpkXV]UV[Gf[gogPcEG]PcEgLmayv auF[HnPG]pbAG:::bnNTjbSnVTPwdwsIum:::fTj`CnFT]xmYPa;[`IySRm^Shv_WSj^af GUm]`HqSSP`qFU`EM_efGUmY`IMa;S`Gm_sl:::NSdFMXldFO@:::::::::::::VsEul:: :nsHUlMP]UfAM^ckGp\\QfBElS`EM_Sl>sBukI@\\FG`DM`Cl\\F\\FK`@M\\Mf@qRMXld FM`BM^SlvrAyrA=::::TV]oRPFE@:^r>UkEp\\;:I@nRPFE`>M\\sjAp\\Sk:::vYxI::: ::::B:ry:::::::::::::::^YlYygYpI:yyy::wyy:jwsxnYxyy;:::::::::::::::::: :::::::::::::::::uaw=dyyuywSxyquoYNyyiyvyywyYwaxUywyysYyXYvUvViuZ=uSeY puxyqIRytyveIyyxoIyyyyyiywiy[YwciwqixoyywYxyqv;uawuyyyytIwAvyyyyy=:::: ::::::::::pIoAFYjwsxfYpIyyy:tIwAZyqYyyxwyyqqkQweMvSUukQweyYwqIwyyoIywY xqayeyvYmyxhyygxtexyiyuawmyyI::::::::::::::::::::::::::::::fYwIyyYyxYy yyyuYxuINyxexwQucxwQomxp=TKETFasEeUVYuqiXoYytgxqyxuyyyQxnaxrYysYxlqwwE xOys?iv[ItgyvQywaiugyvYIv]YuSYuQysOItCYtKYsMitEysIisKItMYtMYt>wbAw`qv` iv_iv]mk;::rs?yrMYr?ityhrGyq?ItyhrSiqCYt;yryxrCItAIssyoqHwAGecF>=cxguZ yypyp\\MVNmoh`PE\\kmDKEpPHxkXiQxxYiqWsmweyWd=wtaxlQXfUWoegfQv\\auYiuIQ uQAu]IvWQv]IuPAuOAtLIuQytVauVaU\\Wf]weYgSIUm`RAUk`QMeUWeUGroixokHp\\Gi `QMecoZQmdQgOMgeWgEWcEwayfRAuRiSaRC`==j]cknRPFE`>m\\Sk: ::vYxI:::::::B:ry:::::::::::::::^YlYygyyUy:vyy=:nyy=:qaumyyI:::::::::: ::::::::::::::::::::::::::uawmwSxyqvswNxyiylyywqxxiYpywyYyqixqyxqyyqYy yxVywauHAtnXn`xw\\avyaywAyjIXuiykyveyyuyyoIxdYweyxyyykwct[yumasmxwywuy yyyJyuYyuIyyyyyy::::::::::::::ZwmwSxFYdyyuywsxfYpIyyy:tIsiylIwiyjw]YwG yvyyuiykYv_Yp_xnVysKyYwiyiixhitYxpyyxyYwqywyoyXuyyyiwkIxuyypIyyy:::::: ::::::::::::::::::::::::jxoyvyywuywyytYxuiwyYoat=Xv_mvJHLLLJ=TJ:sBAsB=Xv]w Rj^Sl>SJ]Uf@m\\Qf?M]ckfRPF:CHkLFIxkIxknRTF:E`?=:::jyyyyyI:::::::B:vyy= ::::::::::::::Jxoyuiywqyxqyx=jyyyyyI:nyyey:VYhIyyy:::::::::::::::::::: :::::::::::fYpI:yyy::jxsxNyyyyqyyxyXyYwyyy]yywYvyiy_aoiyvIYwYyvyyqYwYy piyIytyyviyyYywArl`pyHplxyuiyyyxsyYxYwmyyuyyyyxWyuexoixmiyuyxyyymasmxw ywuyyyyJxCyvYyyy::::::::::::::ZwmwsxZwmx;:yyy:::tYyuixqyyiyxAyiQXqiyya >yps@oUhmyQyvEyuQxqitmxgytayyqYsIyfywiyyyywuyyyy:::::::::::::::::::::: :::::::::wauyYwyyyiysawyYoiydIGYtUiCMSEITLOTjAxjAxj@SC=SBtj@xmYxm<`J?< LLPlQ@mRxKKWTHEa]urAA::Z[Yv]qu]Qv[IVOyqlh\\qvYavZIv_yt_avYaV_ypyxysxrYhYIv YCSEUgQ=uyEHkyyqkXymXndXnUY`qV]esGIxWmuu]wfQx_ivtuWh=YhiwoqWiQxj=xciWg iVfUwbmWgEweUwgQWeYxi=xgUweuWhmWhmgRmeSpmpfCpkPfsontPeo:vtQuoNuTQuSIUb ogpdQgMMcEgLMbsqyxqixotVauVj=XjAwbgr?irUYU]`K=::::::Jb=gKMcSn]Xn[`JmcS ovSxF:::i`KMaufJMbcmWPaCnFTj`gSSp_ifEM`SmVsEumFTpv`[Gm`slZHmbsmYp`Cm^S pFU@ZFm_slNsDel::::::::::::::::^SlFQ`E=J`CmnsHqsCM]UfAM^ckGP]ClZH=HFK` CIsCiR\\FG`D=\\FBl>STF:rlNSPFG`Dm^afBAsBASj]YfA=J^ClZAM]Uf@m\\Qf?M\\Ck fr?Uk:nRTFE`?M]ck:vRXFEXkPF:::yqyyI:::::::B:ry:::::::::::::::nyxqyx=:j yyyyyI:nyyey:tyxoywuyyyy:::::::::::::::::::::::::::::::fYhIqawmxsyyA:: pIoyy:yywxYsuYyEYyuYtyyxUInytmxsyyiymywiyyuYypYyuYvaitayfquYqpbPonqvby yuqyrywlUyyQxGyqmyyyxsxVYjxSyvyy=sQymIyqyyI:::::::::::::::uaw=::jyyI:: ZxqYwavQYpIsyycIuUyyqYymIuWYbaV_aItOix_yvyIyyyylyyqioQXwIysqyyyySyxyyy I::::::::::::::::::::::::::::::tyyyYyuyyiIwyy^YpyyOyxex^naXohF`oFtMatu iywayvUv_IvSf\\MvsBekGP]Cl\\FtFrjZCm\\ClnRdFBlZBM^ck:ZEM_SknRdFKHl\\FM`AyrAyR J]Uf@M^skIp]cknRPv\\gRj[Iv[MV\\_RPv\\gRTFG`AM\\MFfRTFG`Am]Clfr?Uk:::vY xI:::::::B:ry:::::::::::::::nywmxcywQyuYyuIvyyyyy=:wyytYxCyyYywyyuAyqy yI::::::::::::::::::::::::::::::RyNyyqysyyxIyxUxwqxSyvyy=:RyVY`IxI>Y`I oatmvqYywYsawyYyyywyy>xhiWuAwqu]yyUYvQVVeU\\QuVyYwuYtiXquyuwywqyrqXyqy ]IqyyyyxfYpIyyyJyuixmywyy=:::::::::::::::::::::::JvUYuWYyIywyysyygytWI wGitOWayfwMytuiveyyoQu=yryIyiylYxmyyyiy=:::::::jxSyvyy=::::::::::::::: :::::ZxyYwiyyyxnyyMx;oiyyixFmV@n_>_fnwjIx`AwbyyymyqyEQSDmMXlmlaXoaYyyY wC[Gf[=DJsSuOYskXoUxywIweYvGIw_ItaYvMIvYytMytQAAItMIs:\\xnahcMncsfv_ur VV:::@h]iu]QvVQv[Yv[IV=QhAXkYwkIXfMXmyWlQxpQXmaynexvIxlUyeUWimTP]q_\\n FUryLXiYwiwtrawdiy]YvmYtNYqTpmvxm\\pybQw\\QtMqvTQuZYuTYtMYtKqtOauXquUY UkhoghqsHqmxoiHq^U\\Gixo`wfGU\\Wf?UXGghoPwdgT:JdCoftOUoZPqTRoFuSIuppPk HpDG_`MYTs`HqS]`K=::::::jayvaAG]@vSSTV]oR\\FbmZ=M^?sCISE`BM]cl^sFElK@\\FK`@=:hf E]?M]UfDM^]fBm]Yv]YFRkE@TFI`@M];RkEp\\;AxjAHkCP\\SkfRTFG`AM\\;E`@M]skv R\\FEXkPF:::yyyxI:::::::B:ry:::::::::::::::RyfYtYyuYy;vyyyyy=:wYytYxCy wQygyxUyyqyyI::::::::::::::::::::::::::::::RyNyyUxyquwysqYqawmyyI::fYh IoauyYuQyqIvyvqyxQyyex:jxeyxiyyoYqixciv_yyuiydywgxqyvRau`xlUxyqYyyYyxI w]avyaYsixoqyyIyyxfYpIyyyJyuixmyyiy=::::::::::::::::::::::pIoIysYyyyyu YyuYyvywyYxiuqyyeyfyepmZpqj`pyQyxUwiYyeiwyYywQywYxI:::::::RyfYjyyI:::: :::::::::::::::::tyyyYyuiuyYwyy^YpyyOyxexwQIAtLukJPlsQYuYyvayrAyryixQo bxnxpqxPyxusSYuqyyxq:ArdLObDYr]xyuYuIxjYWt=ylIyrYXtAyn=IMqokksr`xlKn]h cUt>WTh`iv_urVV:::@h]qUpGyXr=Yq?Yrkhr?yoAQiov\\av_quSYuKIsDauNatqAw`iy UYu^Yr:yymiwSqtKAuVytYauRiwRyteysIaxXAv;yojHl:XjdIuVywWqujQv]YvUauRYtN YuSIuWYuUAuRepsHqexoiHq^U\\Gixo`wfGU\\Wf?UXGghoPwd_TLG:_`MYTa`NaTeXoTW eoTjfawfIsCIs`qv`ywiywfawfYv]otPUoFT]Xn:::::::vsIEnFTm\\Sk:nRTFE`?M ]ck:vRXFEXkPF:::yay=:::::::STF:rlZ?M]clvRXFbk nRj\\Qf?=:::J]ckfRPF:CHkLFI`AM]ck:fRPF::::yay=:::::::<:xI::::::::::::: ::wiywiyRyyiyuyysiylIyyyyyy:^YlI:wyyuaxyyxqyyqYwiylIuqyqyywyy=:::::::: :::::::::::::::::::RyNY`IoyydIuawmyyI::fYpI:wiytIsAfYwIyyYyoyywYxFY@yu UYvuyxeyy?yuIytQytYyuYyyiYyixsyyqiydYy=wZQvVmTQiYrEyyqiyQyueywQywyyuy: Zwmx;ryyA:::::::::::::::::::::pIiYydyyyYwyisYyuyyyyyyqYyYyfYwYxxywoykY xwywkIoUhmfyu[yytiYuiytyyyIyQy;ryyawuy:ryyA:jxsxFY`yy[tmxSyVYhIuayyY:u aw=xyy;::::::::::::::tIwiyhIwyy^yxuxyQwuyykvmiw@r>Y`on`P>]Rnr@uyyiyeyv _ivaYvkixqiY;VZKbTjasm FTHGrm:::VtMYtHemQxllFU`EYSO`GiSW`H=j`CmSp_;QxlxF]`GiSQ`Gm`slVSpFW`HM` cldV_slNSdFMXldV_OS:::::::::::::::::J]UfAm^aFNS`FIxkTFW`CM]sk>STV]oR\\ FbmZ==`v^gR\\FG`DM`ClZBM^ck:ZE=Pv\\oRdFI`A=TFG`?iRE@:^r>UkEp\\;:IxkI@G `?m\\Ck^RHv[IFXFI`@M]SkfRLFE`?=::jywyy=:::::::<:xI:::::::::::::::::::: ::JyuitiyjUuwiywYxwYyOyuUyxyiyyay=yQyuYysyyayiYtYxCyvYxI:::::::::::::: :::::::::uaw=jyyI:::::nyxey:vYxIuaw=:jyyI:bypYvwyvuysyyKypuHiWGfcxjIxy oiyhYyWywQyoyxyiyyQyruyyyyuqyeyyOYyIxuiyiIyqixsIyyyyyiytYy;:yaymxSy:vy y=::::::::::::::RyfYhIuawmyyI:RywixeiwyayaywaxaIw>qprPt;aryixqyyxYYqIx yiyoaumwYYvoyyiytiYoyywaxYYyYyyQycyvywyyxiYmiyPyyOXwivtIPyyyyuyyy]Ioyy dIuqyKy;oatmxSyxyyyI::::::::::::::tyyyYyuyyiyyoYpyxwyuyYmaqeykYywYxwyy ohqyhfcGgwiyPYuMyyyywiyv]YyiyxQWcMWhawgYytQidEtCyraIvSyv[Iu_yuQIvYyt[i uUYuWIuSIuSYtQIuIYtOisKYtIqm]Xn]xnahcMncsGisEAVV:::jiAxjywjGv[=y`Yml=h _AVwhqwhqNvWqu\\iUAiquxrwXqEYquxrwpjAxjAxiyWnigbYuWAvXquYyuYYvWqucYuWu jIxd[F_?Xe[WusxqGYokXqYXnKypuHt?yr]hoxVg]wfMXkuVeQWaIWcawgiwdIwcmwe]Wh YWfiWeYWgQWeYwd[MYT_HochncHoqxp@wbotPqtMYTY`[Iv[AtJe^_f^KHlKXmUpcIgOiT axnaHn[payfJmbSn:::ZLMc;Rn]Pb=gImbsmYpaCnZGiS[`OUoY`I=::jccmtv_WsFm`sl VSpFU`JmaslQxlO`HMaslVSWMYt`HKa`qVEAAqk;:::y`YIv[I v[=:AYr=yrnuXquXMjuWi=H::ShFO`@qREXk`FM`@M]cl^S\\FG`BM^ckvR`FO`DM ^;G`AM_efCm^skIP]Uf?M\\Mf>m\\Qf?=::::::J]ck:::Fr;IrDQsDQSLV\\MV`mV`yva yVYuUYauVwyyyyy=:::::::TlFQ`Gm`CmlFJauVauFSHmdFQ` EM_cl:::FsCUlNSdF:Q`EM_cl:::>sBUlMXlrlVSJ_clFS`F:::IxkXF:::W`CM]UfAM^] f@qRG`BM^slZ@qREXk`FM`@M]cl^S\\FG`BM^ckvRj_SlMXlI`@qRG`AM^SlZAM]ckZ?m \\;:::::::GhkTF:::=`=YRapcuUYuu[Iv[MV\\GkCHkmWhmwyyyyy=:::::::<:yyy::: :::::::::::::::::::fYyix>yxesCIsyIxsId]V_KYokxyuYyyIyqyyyiyyA::^YlIyyy yyy:::::::::::::::::::::::::::::::::::::::NytuyyyyyiywqxwYyqxsxwaxUxqi i]wfMGbAwysyypYycyWIuSyyyyyoawiywqyxUyuAtIyiywiywQxCyvyyuy:fyyqywyyxyY xI:::::::::::::::nyyeyvyymxwywmyyInyxUyvywyYsQyiyxyywqyyoYqYygIsEyeuFa [irAyysIyuIxiyyuIyyiyuawqyyUytiyyytyYiaomvsxyqywYqyywAqYywiylyyqYwkygy vqYsiyyavqyyQyxyysqYsQyqyywyy=::::::::::::::ZxMyuyyiyyyyymIrxFY`I[XogX Y?V[Yxncxt[yuoixxixIyraxyyYhiyiyxqysQyuqxqAymaxfyvbqv^avmIxkAxhqwjqwhq whUuUQpstQqnCt\\xmCtKItMYt`HKa`qVEAAqk;:::=a[=::jhqwhqG:::rpqxpmXpmxpm XpwXpoHroxp=YpohriXpwXnahu;ir;ym_XscxoAHfaGdeF`?ym_Hv?IsqXogXsAyrrmUXm mxoixquXq[xlSxqoxpoHpkpe?uPuoePdsnaxncHoLGroipcIgLQTeXopv`qv\\Qv\\av^W tMYtMMccn>T[HnFtKUn::::ZLMc;Rn]PbsmYxm]`IysIySTlV`WsEUmfSlF:WhmtFSHmdFQxlhFO`D=::J^]f Cm^clNShFQ`F=dF:::MXl`FO`Em_;O`CM^]fB=::j]YfA=::JaSlnRXFK`@qRG`BM^slZ@ qREXkj^cknRdFS`BM]Cl>STFG`BM_SlvrAeknRXFbkfRTFG@:fRPF:::::::G`@=::J`mV `Qv\\QVSEUSAr:ArsHqsdR?erLQtLyvawYyA:::::::B:ry::::::::::::::::::::::f yy]ymx_ytewiyyiYxGw\\Yvx?isyIywQyuIjyyyyyInyyyyx=jyyI::::::::::::::::: ::::::::::::::::::::::bxkIxwYxoyyuiyyQy_ywYypyxyIytqyrIYuiySIsEybqFaOi rAyyuiyyYyyyxkixpyyyyxCylyviyxqixqyx]uYyuiyyyywmyyI::::::::::::::::::: :fYpI:yyy\\YvGyqyxyqyxeytYxmYyUyuyyuwyuYyyayvirp`plhn]xyuYyxIyryyxiyxI ytqyxqyyyYwiypImQyayy[wyYuIytyxoywUyVyyuxtYxoyxmymiyayxqYwYyNyyexfYxyy ;:::::::::::::::tyyoyxyyuuyyIyfYhImas]xUxVYtxsEypqxpYYvUYxiyu]yuIyqqxp EYr=yt=yraYtQyxYYvEypuxoaxnMYsEYruXq=YquXquhgiWSaeYWMqn]XnwV\\XnGwcUt> WTXmgVjk;:::=qjAH:::uXqpG:::oHq\\WgeWe]Wf?UqXpFUuHpmXqiHp?ipqhsqXqWYrA YpQXmGYokhmuGjuF]YfpMHm[yrEirexoyhncxqchoQyrCirsXqQhkIHqiHpoHpXweYWeUg OiTc`MYTgHochocHoeXoTWeGtKItLmdQgJAtJQtLer?ir]XnSTV]oR\\FK`E=TV]gr?UlFSTFG`DQSS`BM]Cl>STFG` Bm^SlvrAUkEp\\ckGp\\SkZ@M];rkZ?=::::::J]ck:::^SDUSEu=Yr=isGiS\\MV\\iv_ ?sBAsqxpqxyyyyy=:::::::<:xI::::::::::::::::::::::uyypIeixkixyqtQiyayxy yaYwq`mZxykaxyIytyyyyyveysiyyiytYyyyxcyyAjyyI::::::::::::::::::::::::: ::::::::::::::wiynY`irYxnAypqxwiYyyyyyywqyqyxEypuxyqYyyYvayTqTQEVWiuyi YyOywyyuqYsqyuyyqYwmixqyxqYyyy`yvayxyyyqIyyy::::::::::::::::::::jxSy:v yy=tiyiywqyxMyuiuayuixpiyuQYuIyyyweYskhocf\\OvyoixuYwgyyRyJy;qyyhIByty xqixiyyQYTYyqytqisyyxaYwQxuIyiyuqxSy::vyy=::::::::::::::Zxyyyqyxmywyv= yyxFyyUxyqteywiXrAyrQYuqyygywiygqwfyyyYypixkyyxqYwawqyqQXYitQiw]iviqtA yqyXruXq=YquXquhgUuSaeYWMAKqmCtKqnahcMncsGqlQh]mk;:::=a[=::JhmWhmG::bp sPgcoqhpgHpTWf]wfawfmWgewgagVytRItFisqIxlyuNqtUAtLQwWyu\\atPYt?yrZxkLH txHraatQyyfywbytTytDas\\QUwHpmxpgxokHpXweotPUoePdkcoTTWdgtOUn]pcIwcYG_ `GisGAtJeogpcIgJATYxmYHnNtLEocHoc@FtKUn:::ZLMc;Rn]Pb=gImbsmYpaCnZGiSRo vSxF:::a`Hm_ifFm`qfEYSQ`Gm`CnvShFO`HMaslVSaREXkTFG@f RPF:::::::G`@=::j]CmlVTMUTeV_eVb=gohpo`r>UlMXlyyyxI:::::::B:ry:::::::: :::::::fyy=yyy:::::::wYxoqyuinAxiEywixMIu=HdSgyMIuyYylYy_YZyiywyyxyYxI vyy=::::::::::::::::::::::::::::::::::::::JuMYtyYysyxIuaw=jyyInwcQwyey yIyyyoSxmGG`oniEXwmXqyyviyyiIuyypyxcyyQymyxiyxyiuyYuyypIyyyJyuYyuIyqyy I::::::::::::::::::uawMxmYxyyyyyyCy>yruyvYJuayuUxnYxyyywuyvUYyaYvMS@qr qYxmyyxqiwqYwiyyiytYy[vqYuQyyixuyuIywqtyYyAYymyruYxUYyqywyiuyYoatmxyyy yyyoyweyuqwyIjyqIyyy::::::::::::::byyaxuxvyvqyxiYwyxtyvyis]xUxwqsYYvyv bAwOitOYTNUtWeUUITKus`]v^UwbEWsqXquyvGxfiwiiwgYxkIx`av^AxhqwjqwhqwhUu^ wfutQqnKnwV\\XnGwcUt>WTXmgVHxkWV:::jjAh[=::JhmgV=::dGqXquxosxpehpmxomp eagTQUmXpdWfOw_UjqGiAYqwxvQIuwxnexrkhpqhm[HmqgikWbEwnogiyItSyuoXqShi;x qchoMYr?IqiHpkPfQweYWdUWeIwdQWdsnaPecocPf]Wfqv`qVdcpnsHemNtLUlMXlcHoch nDwbAgLMdMgOiTc`KIT]@:::ZLMc;Rn>TxvaGT]xmYxmY`J=pv`?TPGY`I=::jcIgHm_if Fm`slQp_SmfSTlFQ`Gm`CmS@Jacm^sFelOp_slNSdF:::S`Em_clNS`v^ GS\\V^OSdF:::Q`E=:dFKHl\\FM`C=::j]YfA=::JaSlnRXFK`@qRG`BM^slZ@qREXk`FM `@M]cl^S\\FG`BM^ckfRXFBlnRLV\\_RPFGhkTFRkfRLV\\Wr=EkCp\\Sk:::::::nRTF: ::Axj@vZAVauf:Ar:qr@en_pWiuWYv]wyyuy:::::::Z:Zy=:::::::::::::::::::::: JyuYywyyyYwyytyxqiyyyyyyxuyoixafY=nsUyyqivmwuYx]ywayyyYyuyyqyysy:nYwqy yyy;::::::::::::::::::::::::::::::::::::wax=piyYYjuyiqmvsxyyyyqygYpIsi yyyxNyyIstpqCYqwhyjQxyixqUywyytyvYivayvyyyUxmqywyy=wiyyYyyYyuYyqyxxyyy I:::::::::::::::::uqyqIyyyjxSyZumwcxtYwuIryRyqaviyuyYwyytYvIYsaxjAxSat NeYsEyyqixIYyiyuQyeyxmYvyyyiyv]YtyYyiiiqwyYytqysAytyYaiysB=M`DM_SlFS\\FEXkJ]skvR\\FK @nr@ek:::>tJUlMXlEXkXFrkI@lV`?S\\FI`@qRG@vRXFG`Dm_sk^r>ekGP]CkCP^]V^aF O`Cm\\sjAxj?hj?hjfr?ir?=::jXquXUV]OSTFC`@M]sjAp_iv_uUYuU_efImWiuWuUYoS WxkIxkwgi\\WfGpkHpEXkExoixOxV^]V^QFA`mM`loAxkYv[IvrqxpywgiWYqqyhhqwhYWRtMA>wbAwbmnahcMnssRsLXlov`qv^ av^aVJmSsEqkSr@XkEXkkj;:AyrLXlouXquX=::lWiuwg=xiQwgewcaWfUwgeWfqWhQWga We]Wf]WgeWiuWi_uVquVauWYuUquSQuXIuTAVTxfiWqaWhuylaxhMV^MuHAT\\yTSExbMW m]WguwcQWi]wfiV]YviewgYXkIx_av^IwbAWnIXlIwbAweYweMV\\OtLuQytQOhlOxpqxp EXkEhp^tNEo:::ZOiTe@ZNaTaxnDWcOTHwc_tNatN=jccnFtKItK=JcEgMMdSoFT@G:::g hoTJaCn[Hn]@nSlfFaSj`qfG =::::::J`mFfStVawsIemWp`gsGm_ifFM`SmfSlFS`EYSMXlMhlOp^aFNShv_iv_;OhldF :::[`DQsDyrAElNS`FM`FaSU@VSJ_;rlZHqSW`DQSKHl`FO@NSlFU`Em^skIP]UfAM^Cl: ::NSj^af?iRCHkPFRj=XjA`GisGQrvZAvZmjAh\\QV?a]Yv]yuYyuY=:: pwiyWhAXjUWhiwdSpqxplWh]WgwtSIUqPiuWiqwhqWimWhWUuxpqxqopieg]YuVIwRYufy tTyx\\ivZYsHApW@nqpncpuGAt^qtRYuJQteYv^YuMatKYsFivZIvSatOIsAyr`av^AvZA v^av^qghog[Iv[YtMEocpjAxjmV`mvdSo::::ntPeoZOiTcHoHwcWTLGc`OiTe`NMdsnVT JcEgKIT_hnHwcWTLG_hnDG:::g`JMaufKUnBn[payvaAwbAwcIgMMcSn>tJEnFtKUnnsHu naxna`Im`qfIum[`Im`;]`KmayvauVauVasmYxmbm:::::::fsGisHMasmvStFU`GM`Cmf sGisG=J`CmZEm_;:OhldFQxlhF:::YxmYhl>sBulQxlSHmdV_GsCulQPauVauFU@VShFS` Fm_smvShv_?sBelNS`FM`EM`CmNS\\FK`Cm^;:::Q`AyRPfBASS`Em^afCm`qf@mcIgUYu UysIUkEXkGgcGWWeUWYv]YFasFakGhkWoeXobiwgisGi@wbgsGisFaSO`D=\\FGhkHFyyy xI:::::::B:ry:::::::::::::::::::::::JyyyyuIyyytyxoYyYysiywIywiywiyxYvi yyqXsAt=qryEytyYxwyxyYvYxI::::::::::::::::::::::::::::::::::::::::Ryjy wYoiy`yyWYhIuayyYwqxuYycywqxyiw=sYwkyysyyoyu]I`Uv]AIokxyqYyyyxuywYIvmy vcyyuqyoYpyyuyyyYysyyA:::::::::::::::::::wiywqygYyA:byBy^yyyywMwmYyaIq YytYyyYy;qixyywiyyoywxyw_ysuweyykqhmSIeOBLkj]tOyYtMyyIYsyYuUyy]yvyYqAY yAysyYwqyyqyyiYwmYwuyyoYiyxsIywyyyqy;::::::::::::::byyiyuqyiiyyyveyuyy jvuisMiHPm`Na@otaYwqyyyynahoqxn]bqv_Qw_qvwAysisBIsjV_fFo^hnaQv]axlQxlA xjywgiWYaieuWqo;MA>wbUtMQoehdmnaxnYxm_v^av^mlQh^Ml;Ea_mkSr>vZErvuYyuYM j=XjAxjOv\\ernuXquX=::lWiuwg=hPYuTqTkXpqxpoHqZSmgiwgigTQuTYuSepkPg]WfK hYWfuWfawemVaUxfiwkQweUXfeWlIWeUuL]tNUsFYW\\YvquwjAxcQwkUWf=XfawdyVbaW dQWimWhMWcEw^afIySYv]Yvnap`qfVauVquXquGiS_hn_hn::::VtMunNtLenFtKEn[Pb; ]`LMcSn>TShFS`CISRlfSJ_ SlFS:dFI`EYSS`@qr@ir?ukvRPv\\_r>ekGP^ClnRTFrknRTF:::CHk@vZWr=ulnSJ]ClN rSXFI` ?M[EV[=VZ=vyyiy=:::::::<:xI:::::::::::::::::::::::nyyyyx=ZvyYqYyuIywQy uiyQyuyysYY[qKarAyYqEyyYyvoyyuyyA::::::::::::::::::::::::::::::::::::: ::uqyqI:rxyqx;RytyysyfYyAwiywiyFysIyyiywUywaxYxfywyaxqit[CGeSbAwbWhmluXiUeMwcQv[Evaev_m SEYsYdmWTmRDMc=gnQXThv_iV_SmUPaufEYSrlQPacmZGiSSHmlFU` Gm^av^uFW`CM^]fD=dFIxk`v^WShFO`CISOhlhFMXl`F:::O`D=::j^;O@ZCm]Yv]YFG`@ m]ClKP^skIP]ck>Sj]ckvRPFE`>aRI`A=::J`skIxkG`CM_skZAm^SlZB=XFK`Bm\\QfDQ SO`@qRAxjPv\\gRLF?hj?HkCP]ckfr?=E`@qRCHkLF:::I`?M\\SkVr=uiywi;Hj;HkCHk OhlXuUYuYyuYuVaoqxpqmXpmPrtJAtIySY` JAT]Xn@G:[`I=jacmWPasmYxm[`Jm`qfG=::JcSmUXmQxlQXmfSlFS`HqsHAtJumnSpv`g S:tVaWSxva?TlFO`Fm`qfEM_eFZGm_;O`EYSbl:::VShF:::BmSP`slQp_cl>sBAsBm]Yf AM^]fCISM`BASI`A=\\FK`Am];:K`B=::J`skfr?uk>STV]Skj_;MXlJ^;RlvrA=Q`AyRE `@M^Sk^RTFBlnRTFrkI@PF:::Ohlj\\Qf:Ar:Ir;Ir@qr@AS\\FuWiuGpkp`Sj=XjixotV agpohpwhQmWhmghoOb=gSIuSYr=eiwgiEXkEXj=XjEXkEHk^RPv\\wRXFGhkLFyyyyyy:: :::::Z:Zy=::::::::::::::jxyiwmyyI::::::^YlYyuIvYyQyyyyuIJyyyxmyxwyxiYv ypAyyywlqmyojyawlyyviYyyyyA::::::::::::::::::::::::::::::::::::::fYhyy gYlIyyyyyy::ZwmwSxyiykYyvyxyxvYyvyywmyiYuYiviypiwjyysyyyyxwYx\\qv:yolp s:QryIyuyyweysIyuYyuYyyyyxI::::::::::::::ZwmxsxyqyyIZxqyxyyxuynyx=:siy wYyfYhIiyyTlFQxllFU`FaSS` HmawS^SdFO`Em_;:::::::S`Gm`CmVShFMXl\\V^wrAEl>S`v^GS\\FK`Am];K`B=j^Sl> S\\F:::Whm\\V^]v\\ckTV^ckGhkI`CISOhldFMXl\\FK@:vRlV`_S\\V^gRXFI`@M]skF S`FIxkTFrknRTF:::blvrAUj=p]YfIysIYr=Yrhpoh`sFUj=XjE`yxqyXpmXpEYSohp@vZ Av^av^UV]_r>UiuWiqwhq_EYT`v^[?m\\skFS\\FG`>mysy:::::::Z:vYxI:::::::::: :::::uyyyYyZyyY:Zxyyyuy::ZwmxCyxYxoyxqYyyywA:::tyyqyqAyyIxnmS:Uryaywky yyyyI::::::::::::::::::::::::::::::wiywA:::::::uaw=jyyInYpyxqYyqyyuYyw YpIqasyyxIyxyytyyyAYqqxiYymiyiyxYyoqxyaywAykuyoyyhYyOyxatAIspXogXvUauy iYyYywQyuyysy::fYpIryyA::::::::::pIuAvyy=:tIwQyuI:^yxqywyyyiYdyyuYtyyx =YyqYoeyvuXyqYsuYyUYyuywgyxqyxyYuQvailaxcxffWpPcuGV]SVDQcssf`yYxcyysYw yYyoyysIxyyyDYryxwuYx]yxiYxyywyYyiijYxoqtSixZQvyyXpYygywuyyYyry::::::: :::::::ZxiYyiyyyyxIuauyyyiYdYyyYwsxyqsYivqyyyYyIv]yuIWcCwxmIxiYweYxiyw XYvwyyYxhyw]Iv[yyyyy\\AvZywgiwk]ieuWiuQacIWMA>W`xnOwdQwdmnahbMmCsLXlsr VVJlMh^MkOvZErwhqwXqfU:tWi?vZAvZmhqwhqG:::?Ir;IqohpoHpkXqqpiqwhmWgWuSU q^uT=oHp`weGuQ=bokHpaXpkxnqhp_xoLWdMWi]wf=xcQwhmvaIXaAwsuwjMvQEuL=sCIV RIuoqwiAXdUWaUv]mWeYw`qv`qwiyw_ifPeoyxqpv_ivh]wfuv^evd=wbIwbAwcAgKITrm ]XnUhn]Xm_paAgKIT_`LmbSn>tJumYpaCn[PbSn]pbCn>TxFY`J=jbcn^tNEo^sFasF=:: JdMgGisGIsCel^sFulQP`CnnsHqsGisGaShFQ@:NsDumYpaslQ@lFS`EM`SmUPa;O`BASK `CISbl:::FsCUl:::VSJ`slNSj]sknr@ekvR\\FK`AM]ckfRTFG`Am]Cl>SJ]Uf@=::JbS lfRXFrkZFaSMXl\\FI`@M];:blVSXv]_r>eknRLV\\gRJ^ClnRJ\\SkfR:::J[EV[eV_ev ^sjAP^;?gb?wWiuWAxjAxiywiOhlxFCHkCXj=P\\Cj;HjIxkIxjAp]Yv]yva_sFUjvqyxq ?iRI`@M\\CknRTFC`YPlxYwYyrmxnyYyqyyyiy=:::::jxSy:vyy=:::: ::::RyrypIuanqxpayxqydQWwiywIxkQUUUUEYsEyuYyuayvaywayvUYuUYs=Yrqxpqxps uYyuWiuaRIxkPFBkfRPF?hjxuYyuyyyy;: ::::::<:xI::::::::::::::RyyyyoiydIuyypIyyywyytIqqyiI:waxaYwYyyyyoavuyy yyyA:::bywiyuYypyyyyyyIbxqqtwynyx_PlOxytqYkQvUYuuYyuYxqywwYyQyuyyqiwMy KyKxuYymYywyyyyy=::::::::::::::::::::::Jyoyx=::::::jysy::::fYwYytywyyy iIbxyiyoqxeiwayyiyvmxqyxuyyqYyyivmteyvcySiuWyywqymiFyq;anOHmviwkyytayy iyxAymQxyqywixpaxvyywIyxaynIyqQyyqyvMyuiyyYyQySywiyvYyA:nYwiy::Zymxwyw mwsxfYpIyyy::::::lixQyyExvxyqysuyxI^yuiywiywayyuywatqxgaplhgBnZspdLcpW xysyuxYvQYvUyyqyyyyvwysMvyYy]ivyxrUyoQwQyjQSEYqM\\lgQuRyYxUyyuyvmyxayy yyyyI::::::uaw=jyyI::::::::::ZtQytyyyyyiywiqYpI_hnpyxqyuYyuayvGYTI[Iv[ YuUqpsuYyuWiuWyuYyu[Iv[iuWiuQabAwbAW::`xnGWXxmgv_iv^aVCa]MkEh]YVEYsEYs Vv]Ur@Xj=XjuWiuwiywiyGBr;qjAh\\QV;aZ=::jiywiegTAUhGsHqdWgGuSupqphqwhmG qxp`wfGU\\Wf]WfSpmphigWYuUYuUUpvtQytPqTixohwfaWnIXlqv]]VpAxkaxgqwMEsDm vTaukeVaMxbMwaYv]]WdIwgYWeEXiqwcyVaMWbyVoQXlEw`mvgUwdeWeQgPitMqtNItPat JytNysPYtIqtMYtOitMitOitO=::cHoLG:::eXoPGc`NmdSontP=_hn_hn:::vtQyTvsIy sLQTaxn@wbwsIySZLQT]XnsBulQp`cmW@dFQ`EmXquXIwc_tNelOp`qf GM]SkEp`qfHmEYsEePodP aRyyyyyy:::::::Z:vYxI::::::::::::::RyyQwuitqYuyypIyyywyyyiyVYhI:waxaYw YyyyymYywIywyyyyyyA::::::jxuyxYYvyywOoPXmyAyvyywqytQytqyxUxmqyyYyqYymi vYyxayuyyyayyyyvMyyixmyyI:::::::::::::::::::::::uyypI::yyyjxSy:::ZyyY: :nYtIyyyfYpIrxVY\\iyIxyiyjEy_qxcytIyQwoysyYiyyuiugYwmyywyveiwciuYyypyy ]Yx=tAISjyVcayomxymYyyyxuYxeYwiywiyyyyyiiwayyiyvyYtivqyx]wmyyIfYpIyyyJ yuix=:jyyiymxSy:vyy=::::::by^yyiytUyoQyeIuyylYyuiv]yyxwaysQyryyrixpItG ijXpxpgWxxqyXquXqyxqyXr=YruXquxoCtvvaEtKAXHKabmmcsGQlcrDXk;=Yr= ir?QlMXlMH=YrElFSXv]oR\\FO@FsCElvR\\FO@^r>ar:ARKHlKhm>S@vZAvVauVmUXmuSIuSUU UUu_iv_AwbAw[IfxpqxxrAuk::::>S\\FI`AM]Sk>SPFCHkPv\\or@=Axj@vZAvyyyyyI: ::::::B:yay=::::::::::::::pyy_YyixwyyuyypIyyywyytIqyyhI:waxaYrymawyYyy y:::::JyuiyyyyyYJxmiyYyvyyqYybPlQxypAYsYyyYywyywqyoyyMygywqyvyyyuylyy[ xmyyI:::::::::::::::::::::::uaw=:jyyIfYpI::::yyy:pIxYyqixyYpIuA:VYPISq xexyqYquilqYiyyyYxwqvswyiyaiysYrSitiyywyxxyyuYyIypuxyiYyOxpAtw@rIyq:iy jQxlQweyywmxuIuyyyyyyQyqIsQymYyGyxiyuyyyayxEyfyyyyw=jyyixqyxqY:jyyiymx Sy:vyy=::::::by^yyiyvyYuaw=tIuIylYyyyrUityYyiyqQvUyqav\\yysqxymYryyyIy w]ysyyxswuyygiywywwyyqyymixwyywyyyixiiwExjce?orugV`yIw]yyuIxuiyoyxyixs yyuIywiyyqyyI:uaw=:hiyiYpI::yyy::::::::::::\\YvWxmYxpqxpqYwiwgquYyuayv GYTYu?XpxpqxpsuYyuUYuUAYahqWOitOItKItIysXHGisTHIysXHGa_iVHh[Iv[Qv\\Er; Ir;a[IVrrAAxwiywi;whqwhqvuYyuYMj;qxpqxp:::vuYyuTQUkHpkhpohpqxpqXpmXpkH pkxp^uVupqPgeWgawf[RMg]gT]OqtUqtQIuQUoeXo_hntJulQp^av ^iv_gSdV_OShFSHmlFOhl`v^?sBEl:vrAyrFaSU@>sBulZEYSGhkGHlNSdFK`AM^SlMp]Y fA=::j^ClvRXFG`?iRE@::::^RLF:::E`?=::J]ckfRPFC`=Yr=YRJ^sjAxj=XjHv[_r>e j?P[CkCHkG`@m\\;A@^RJZ=VZMV\\oRDV[Gr;Ir>m\\sjAP[EV[MFG`Am\\Qv\\MFA`>Mc Egr@qrxqyxqGiSywiyGj;P`mf@qRgPeyuYyuXquXuVaOShuWiUZ=VZAwbWr=uj:::NrwjywqyYmAkywyyyyYy kYxoivayyqYyyYxHxdYwGqrAeypyxyiyx]YrAyyiYxWYwiywyyrydyyuyvCxwYygiynywQ yyyyuyYqAfYpIyyyJyuIvY:jxSyvyy=::::::bynyyiyxyyweytI:pivqyyIYdyyoyvYYs YxdyuyywQYxeYrqipiywyxyUydyxyYwyyycxpyuiYxyyyuyyUytiu?aRrdLOxYhewyUYro ypYyuixxyyuqYuyytiyuyfYpI:qqyiIbx:tyxwyw=jyyI::::::::::VyuYYyyywiywqxG xkIxmYxmyyyeuWquYyuayvGYTYu?XpxpgWxxqyxpqhhqwhywiyWNwdQwayVIAfVEisGisI ysXHGisLXlOv\\Ur=qjCr;QjSr=QkEh\\mjsqyxqy`XquXqU:xwi?vZ=rq:::nU`wfaweY weeWgewgigSIuSAuRepoPhmWhiGohp\\gRAUXweGuQepkHpqXplWg_uTQuREpghoHwdQWc YWeIWfYw_=waqwiywdIwcUXjAxPETLmVY=VfQfMaq:AmuoijItNYu:Yr^ysKQwZAvDAsAA uYau>ysFyrOItAitKYsNItHYtKItLAtOItJAuJAtQysIYtHysKysIysI=::[HnS\\F:::MXl`FK`Bm]ckGP];S`@qRAxjPFG`?m\\skIxkMXl:XFG`@m]skVr =ElFSPv\\_RTFK`Am\\QfAM_eV_eFKHlJ]sknQdFW`Dm\\grwhQywiygmWPXme=YRWhm\\ FsGis?OhlpuXqU]or@=:::::PFC`=YRA`S\\F:::M`C=::::J^skZBm`skIxkCHkPFG@ZC=XFGhkPFE@Z@=dFQ`Am\\Q fAyRK@fR\\FOhldFM`AyRI@FRxwiOplPpOhlOxrAEn[PUUUUeV_gStVaWSlUXmuVauVuVa uvdQg>aRIxkXF:::G`Am];bkfr?UkVr=ujNrm[If;Ir;yyyuy:::::::Z:Zy =::::::::::::::jxyyyqIyyy:byyyywA:vyy=:::::JyuiyyYyAnYpyxwywiYyqveYvYY vUyyyyyyYyyiwqyyaxpilxOjxIwfUweYyqixsyyuIyyYyyQyuyyqYyyyxyIyqyyI:::::: :::::::::::::::::uyypI:nYtIyay=:::VYbyoAFY`I:qaumxSyvyy=ByNyxyyraywIyw qytaywYywqytmxmYyuyxyYvYyvuyxyyywysYyuIxmYxtQytyyyqYtixiYwQwMytGytixcI w[XlNpxbQwyYywyyxyYwiywawuysyyuixpyvyYyUIrwtyvcxxywsxfYpIyyy:::::::::: jxCywqwuYxmiwqyxuysaxqY^YyIyxyxmiuQYsiyyYxyqucvyYy_yysIuoiw_YyoIxuyxyI yyyykIy_yo;yycYxyIyyqrMyyEXlEPMTlYyr>yyuuxyyYsiyxAyyyyuuyxwyyqYyyyyA:f YpIqaumvyyxaiwyX`IoiydYygyyUyvyy=::::::::::uYyuYyvYhywGysIyuYyu]tQytyx qyxsIYmapMrSuUqpsuYyuSIuSiuWiuWagiWMYtMysIysGisTHrsIqmCtKqmYh_mkIh\\QV ?a[mjcr?qkIh]MkCr;QiuWiuGuXqpGwhqtGyxqu jVR:j\\SlMp^SkEp[;E`>M[Ef?ir?yrAuknRLV\\_RPFQxlQhggggYv]Yf^_f^=XjxvaGr ;Ir=M^Cj;Hj=`FaSY`>aRqwhqGisO\\MV\\MF:::E`?M]Uf@m\\CkVRHF=XjXfWUaiywiawfMqoWlXpmxnaxnKmUh`mmYhaMn]hbmmYh`mlQh\\QV?a[IV=a\\Ml Mh^MkCrnuXquWiUu@nuXeqvuYEr;qhqgW=:::\\WfotPUpmPgegRmeYgSIUqxphGmXp\\W f[QmfYG^uQAuWAUoHpXGcHocXnaPbQGntVYuDAtIytVYu=@kApsJAtTQUexnahqaXoKyne xrKxlwxjGHfkf`YWawvq]hnwHocHnQhlaXn[Xn_Hn]hnShv__ SXv]YV\\cknRLV\\WRTV]UV^ClvRPFE`@mVauVEQsDqFasFiwg?r:UgeWg[Hn[hk^R\\FR lMp^Cj;Hj;`IySEXkPF:::G`Am];:E`>M[Ef;Ir;IRDfuREpVuSupgxom`SyTeXoeHn_hnYXn] xmcpaMwc=v\\MvO=uQEv]UVcYWeevauVcMWdmFuXpkXmEXkeXlMHtkXpwXlQxhEwcmWbAg noGiQiokxuuhqaXkGxqixoEhkChnPwfExiIV`aV_yVamVauV`uFvsIum>tJEnvsIysIMbC nvSxFWhmpv`gStFW`IM`mfGisGysIysKItKYsEYsE=::JbslQxlMXlhv__sFUlMP^]fDQs DiSlV`WSQhlOP_;rlQp_CmfSJ_efCIsCQsDulNSlFWhmWhm^sFulVSlFblNSJ^]fAyRI@n r@ek:::::::vRXF::::::::bknRPv\\gRJaClKP]UfBElMXlTV]gRTFK`A=PFE@Z@MWeUW MWdMW_ef;IRWhmWhm^r>arv`qvxqyxq>aRWhmTWeUwbAwb]UVOlLPlpohpOodPoCISYxmH v[gRPF=XjXv]ckPFE@VRDF?@Fr;Uj:::::::nr@ujApZAvZAF?`ThFMXlhv_iV`mfDQSMXldV_gSlV`WsEelOP_;rlVS lFUp`K_SlZEM_CmnStFSHmhFS`Gm_ifEM_ClKp^ClKp]YfBASGhkTF:::::::I`A=::::: :j\\Qf?M\\Mf>=j[If=m_if@ekCHkTFI`>m[Sk>SXFGhkPv\\gR:TFYxmYXhmWhaxnhv_g pohpAyRSHmlFO`@qRA`>aRIxkXFE`;IRI`GUmEp\\quX_r>aRfRdFbkfRHv[[=YR?hj?xj HvZAf;=::::::J]CkNrEk:VsEekVRTFI`>M\\SkEp^av^aFK`AyRGhkj];EXkd FW`kHpk@sBUoeXo[Gf[wlQxlEXkEHk^RJ]clOhlRn@w\\Qf;IRs_;m]sk^RJ^skvRTFEXk LFC`=m[;:::::::E@Z=m\\SkVr=Uj=pysy:::::::Z:vYxI::::::::::::::ZxyYwauMy cysAyqwcxwqqIYteYygYyIywiywqx[yYymaxluyxI:::::::nyy=::jveYvyyyyyyUYvyY vqY[YtOQWX=vyQytyItYyYIgIxkYyyyyyyyhywwYyyy::::::::::::::::::::wiyyQyy YyOxfiwyiyxUyuIwSIu?GauViiGh=HiqwgYweywgiwrqxptywyYysxqYxWIvyyxxywyirY yyQYwIysQycywqyyYwmYxwyyuyjxqyxuYyuiwqYkIxyIw_yymIwlIyywoiykywyIx_iyki vwQuyYxayyqiyyYyQyoyxmywyxiYuyyyixwQwyitmvuYxeyxiYxqywuyyqYygyyqYsywgq xqyvyxqMW^]VZEuSEYlUxyiyx_ywyyvUykYxlyxyYyuyyuixayvuyyYyyyyuauyYuayyY: ::::::uawmyyixMyCywyykyyPyyyYwyiveyuqyqIyaymxqisqyyqwwyyiqyyIywqyyiywi yxaywqytaxlUweYwoiyyiyuyykYtUyyeyxyyqCytmwhyixeyyiIvsiw]yyyyxsiywIyvyw qyxXpoxpchoyYyxAykaxyiyyYynyxyiyyyyxeywApiyyyyUyfY:jyyI::::::::::fywUy vYXyuGysEasFayyyjuSxFYTIRuUYuUaiywiiwgiwhqwhywiyWNW\\HIysXHEa_MmUh`mlQ h^aV?aZmjAhZAV;a[MkEh\\mjsqyxqy@nuXeqvuYyuYMj=H^uVauV=::jhqwheWgeWf]Wf ]GmXpdWgGU\\Wf]wgiwgeWg[SMfCpFU`GghogHpkHpixoPwdotPUpmPfCocpeQwdYweYwf YgHqSWHmHGa`JQtLQtSqtCAtGIsHaSgHosxogxpYxmwhmYxseho_Xiyg]muX=ImWxraHoo hmWHnKHlwhpmhmSxl_Hn[Hn[Pc=WbEGYxmYxm:nsHem:ZGiSSHmlFQxllV`gSj_slfsGum xv`qv`qF:::[Hn[xlQxlMXlMHmSp`slNsDEmZGM`slZEm_;RmfSlFblNSlFKHldV_[FM_e fCISO`EM`slVSdFM`BASK`AyRKHl\\F:::::::::::::::I`A=J^Cl:^sFasAyrAir?ElK P_skIp]SlM@\\FI`@qREXkTFG@>r:EmSPZ=VZEGEXkDV[EV`mV`eQwdQ\\MV\\av^gsGUm vrAej?pXquXmUXmu^av^EV[Ev[IfDQSM`=mZAf:Ar:yrAeknRPFCHkHv[WRDV[[==::::: :jZ;=Xj=xjfRPFC`iyYyuyexmyYu=i wyXyywuyymIxyIxmQweyyqyyyyycyuYytYyuyyuyyoyyYytuxuiyyyyyIywiysyyuqyuYx yiuqyvaYuyytiyqymqwgyywiYnvRIukHn\\@x]ivyixqEyuQxgYpYvayyuiyeyxyyyIuyy pIyyy::::pIuAvyy=:uawmyyixqyxEywyxuiyeyyyivwiymyywyxxYxoYwqyyYywywiyyy ypqYSiyuiupYwuyyqiwQywYysaytqyxqyxmnuxqaYyoywEvOytyyxxywaYxaYtyyyayyqY wIyquxrQYuEYuYYyiyxyYxgw\\ivFHj>`ymqxyeyvyrYXtYyuYy;vyyyyymxSy:vyy=::: ::::::::waqmtQytItKITyuYyyxqyxuYisIYmap=pxpgWxxqyxpqxpuXquxqyHNW\\Xnwv ausnv`esvvausnVPxlWV@xjGVvZeqwhqyxqyxq:ZZmjSr^uVEq: ::ZTQUkHp\\GmXpdWgewfagRAUq`UMgeGFU:JfsoiPf]gSMecn_PdEWcaweYwdIgLAtJAu PqtKysIYtKItSytQysGisH]LAtHitLQtQitKatLAtHisJasEqtHqsVQtLAwVauGyq:IksO iHaqwhvOqtSYsFAtar >Qrar>M[Cj;pyyiy=:::::::<:xI::::::::::::::ZxuyuiyyyxyQyyixqyxq ywyyyayxyytqYgixqYxwyyoyyyIVeEWcUWxivyyyYyuQypyyyuy:::::::^yxqyveyyyyx I::BxmywhyvyIwiwiyymIyMim_HsQxmwIv_yywyyxyykxqiyyyyyYdYweyyyyyyI:::::: ::::::::::::JyuyyqYxgywYxayvKhj=xywQyrYxjyywaysaxmyywayyqywEyxYyqiwayx Iyryw_ivWAtLaoXPntXoi`uEysxiwjyyumwuIuqxyiwuyxyykxoyxuytiqyxyIv\\IhGN \\PX^`FsfWgyQsAyynIXaiuqIvSyywYxyyyqYywixyyywQyuIxYyyYyuIyAuIysIywIyxY yuyykyGyxiYyQqMysAyuUYyuiwYyuyyuQyyqywMymyqAyZ=uSMw[UvuUxnOyvyyviYuqxi Iwiyry::fYpIyyy:jxsxyqw;uawmyyI:ZxeywyxuyxsyyeyynyxIYwixyiysqxmyyqI:Py xmYomieiyvIyyEy>ywYysqywyyxqyyayxyxuaypYyvQyxyyuyYsyyuyt]IucyyiYvmivgI waqwmyymYyuywcwkqwuisayyAyuamKXlYatPyyviYvyweytIlIyyyxI:uawmyyI::::::: ::::VyuYyrAyrIysIyppppHysIylQXiyw`yvGYTYu?XpxpqxpsuYyuUYutXquxqyxqKoCt vvaus:ZbAWKa`qv`aVHxkIXkEh[mjcr?qkIxkIH?aZMjcqwhqyxqyxqZZAV=YrJhmWhmG: ::whqwHqsPgegSIUohphwgOuTUp^uVauUYU:m@NuRAuTQuTauVauNqtPAuRUquXqiXoeHl GhkSxkKXpcph]wfAW`qvh]wfYWcIweEwcmwe]WaaV_MWcEW_qv_UwgaWeiwfefGyrFQsNy tOYtPElMxkixn_xpgXoixlQH`KV\\ygdSgtchoQynexo?Hkoxm]Xo[XnaXnDWb=Wb=gIys IySJb=gKIT]@ZJATY`HqSW`IysIATJacm>TDWcEWauVauF:::Y`DQsDIsCEmnSlfFYsEUm UPaufGiSSHmj_;BmSP`SmnSj_ifDQSQxllFQ`GmayvayFUXmUHmfSpFS`Fm_clNS`FKHl \\FM`C=::::::J^Cl:::::::^r>ar>=PFE`@qRG@>SHv[IfT Mb =gCIsCyrAuknRPFE`>m[If=M\\Mf>=::::::j[sjZ?M];?hj?hiwgiyyyxI:::::::B:ya y=::::::::::::::pyyWyxaYpyyuyxwyxqywyywayxqywEywyymiyyYvmYxeyy`prwbEtFwcIwcMn]XnYHEYsHHAa\\QV?a]YVEYsEYsVV@xj?vZAvYyuYyUJj=hZ mjAh\\QVs`V=::jiywiqwhqwgiwgiGsHqpwhqWhmWhiwg;NuTepFuSUpVuSIuTQUyxqyHo gho\\WfywiywdWt:qqxHryxqNQtLqv[QvOAtKyuTYuQQtMitJItTitPysDYsXauVquYiuE qsFAsEIsOQuRQuYiuIqtMasMAtPYuSIsDAsVAuQqvVauZUnkgbCWhwVb=IkIHx;irAineX qehooHoeXpntNaTcpcIwc=Wc?tLQtKYtMItMYTVtMen_pbAwb=Wb?T@wbOtL=[`JMcOtNa tNQtLQtL=::JdsmYp`qfJEnBn[PbcnNTjbSnvsIysIMb=gJ=DG_`K=tVawSYHnfsGisIys IItKItKmacmZJMb;rmYxmW`GiSU`FaSUXmpF:::::::S`F=::::::j_ifEM`CmfSpFWhmt F[`EulMXllFW`FaSS`Hm`Sm^SlFOhlOhlVsEulfREXl`v`qfJ@lJHlKHlCIsCyrAunaxnU XmpFBm^SpF[`KMbclOpbAwbkZAfDQS_PcAvZAv_ifE=dV_GsCElKP^sk>sBEl:::::::Vs E=blOP_ClKp[Wrxpqxxyyuy:::::::Z:Zy=:::::::::::::jxyiumuyisyYiIw[Qyqiuy yuIyyAYoiyuyycIysYvyiySiyyiSceVgeXYUyqIvsix_yywixyqxuyyqyyuYyyyyoyxqyx =xIyA::::::::::wyytIqYyohlQHgCWcuIv_yyuqxaYvqyweyyyYyuyyqYwyYyeIByVyxa YuyxyyywyytYyuYy;:xyyyyxqixqYwaymy;:uyyyIytyxyyyqI::yyy:lYxwyyuy^YjyKw iiqAy]XlNxyvqywyxryyxiyrqxoIyqqxyyyyyyxuyJyuI:yiyxIpYyyItyyoyyAylYxMYs FipaHo`XlPpvIYtyIyvUweqw]yyyYyoywyyqyXSIvqWJQDs=EUyyosHhwuYKEDKCR?cB_G Z;vy[Iv`YvgytmxoywEyqQwiiwiypQyuQxxqygIysAyviyyuypiwaYmiyyIwwqryYyEYyu YwwyvqyxqyxmywyxmyiyyuIuYywIv]IjQWe_XgkwygqyuYyqYymYxmYxyYy_yqIywAytyy uqyrYxoqyxqyvQysyywQyrQxgeyoIyyyx]iylXyqyjayumXyqyvuyyyYxwywiyyqyxyYyq xsyyYynyxqyxmwyityYvyyyYbynYlYyOYwYxrywAsavMwcxyQxcywApyygYJwSxpyvyyxK xoYwAjxSyvyyMywYvxtIvMitvXohxyqIYiYxsYwkyysaxqywiywqxpuxyyyiYysywyiyny uyYuayuynyvaYuYyuiywQysyyyyuQyiixIYryvywmyyI:::fywiytQytyyyyyjAxjqyxqy kIxkQytQypqxpQytExkIxSIuSitOitWiuWitOqoixoiHbtOitMYtXxmov`UsEQlcsGisGA fVHH?irsBe lfsGUmZFaSS@VsEulFsCemWhmY`EYSOhlhFQ`Dm]YfCIsCYShFM`BM^SlZB=::j\\Qf@ek G`Am]ClKP^SlvrAuk:::nr@=I`@=J\\Mf>M^]V^]FIxkJ]SkEp\\;KP^]FI@nRPFE@>StV auv`qf>aR?hj\\V^wRxuYyu\\QfCm\\Qf?M^sjVRUf]Wf=Wb=wcIwcqwhq_EISxxayjiyuQY`YuwyvUysiyiaqyyyQxxevwqwKyuYyuIyYywqx qYyOxmywyyyuUweQwiYpAWdUUI=Tt?wpQxkyyxiysyxqyyvuyuyy[YtSyymixyIywyyeqm MwbyywQyyiYuiyyeytgyhqywIWtxxyyvmivIyuiyvqyxqyxyyvqYuyyqyyyixfyyyYyqxy yyqIyyy::jxsxfYpIyyyjxSy:vyy=:uaw=:jyyiwmwcxZvaYtyyWyxyynqxmQv_QoVpmZY tOyyuqYqiylyx]YuUyyyieiwgqxvYYyqvyiqaYkqwqYxcywA:ryyyyuQyiyyuI`Iqqyqiy yyy=::uYyuqyoyxqysIYqyxTYuUyxqyxaisQytQypMwCx\\Yp_wdetWQqeHQytQyt:FwcI wbAWGa_iVEAnv`esfVLXlOv\\QvZMkEH>VJkSr=YryxqyXquphCqsPhspqpgCqsPhSquXq kHpkHp:::NuTEpkpdQgPqTixo\\WfwtQeogPeSoePdMWdIwcIWcEgKIT_hnDGaxnHGeXoe XiuoJADIo[Iv[]WfWtMYtL]Kun]Pc=WbEWbWtKItM=JcCn[payvaqvayv_=waqvaosHemM hl_xkMXqYXnQiomHsQxmSheaGfeV`;ilSHoyWj[qjiwayVhYWeyVbuv^mV_evaqV]qV_Uv `eFXV`ev^mV_aFQHmOxl^sFasGiSU`EYSOhldFU`IumWHmShmSHmSHm::Jb=WbiFKHlKxl NS\\V^wrAUl^sFEmVsEulNSdF:K`GiSWhmdV_eFdfDQS`v^?sBUlVSJ^ckGhkG`Am];::: EXkTFG@ZAm];GhkTF:::EXkJ];Rk^r>ujAp\\Qf?=J\\Ck:fr?Uk:^RLF:QxlQxjAxj=Xj TFBj;HjGhkHgMir?emWpayvamV`osHEkVquTiuWighogHmSpZCj;p]Yv]ifEASLV\\_RHv [gR`FGhkHv[Or=HF?`=YR=`;=J[Ef<=::::jZAvZAF:::AxjDFbjNRJZ=fwh qwxYxI:::::::B:yay=:::::::::::::pyyyyxSxfwsUwtyuUYsOysqytyyvyxyiypiyxi yyYxuYyWawhQvriy^ohqghsogyqyiaypywrYxjeyvyyqYxI:yA::::::::::::wqymiyQy vaxaIwXPlLxyuaYuiy^yyyyxmxSysiynYxiyyyxyYwAvyy=JyuYywYbyvYxI::RyVyyuxf YpI:yyy:pyxqyyyiymwiiyyyyyyxuyqIyYWb?wyoyxyYxoqxuYxqywqyyuYwmYxiyxqyys ybytYy;^YyiyuawyYsiyuixsQxuyyqyxyYwiiwayyqywEyuqveYqMXlMW]YvOaSGawZIvy Yxm?xnyyNqupW]^^uS`myywhyymyWsYxyivYyygYwyyooHW;tKSFSMEx?IsyiyyYxyQtyy xqXv?wmAxsyywexoqtUyyqixqYiYxoixwyyiyyyYxwQxuYyoYxyxyixayyeyyaxiynAXyI yuyx[ivhywyyuYYWYTOQxdUWuuXqyYygxjIxyiyxUywQwiimyxppkJHemF`squTAygIwyy ytayyYyueyqavqyyayuiyvYywIY\\yxuyxmivYywQyxyysqYuyyyYyvyy=:::Ryry::::: ::::pIoyyuixryyqviyxiywyyxuiuqYqanyXswicxkMxymyxmyvcyyuiyeyvayyyAvyxYY pIwiyhywoyvuywiytiyyYyAfYhyyOYjxSyvyy=::uYyuiyZqyxqQxlQxvayvAyrexoixoa pMweirMpctOQquhdMpmXpmHrtQytOit\\XnoVPHEAnv`esZ^aV?a[MkEH>V;Yr=QkcrvuY yuWiuWiUJhmgV=jhqgWmgiwgiG:::ohp\\WfgTTWewtQEpkpecogpfagRAURoeXoe`NatN aTjcCocphqwhEUSEULMTLiv__tNatOitOqTTGLWdWTexna@:ihnTwbEWdEG_pbAgIITDG] P`uVaaweYWfEwcMw]avimvamxeeWkav`UvPAuBmQy`xe]WcEv[]W_iv]yuYAw`mv`iV_]v ^YV`yv`evaqV_CmYXmS@ZHysGqsHemFtIysJqsHisGisGMcEWcIwcgSpF:::_`GisGYsEU mnsHulQP_efFMauVaqFU@^sFasF=hv_WSxvayVb=gFaSOhlO@UXmhFS`HMaCmSP_ClKHlK `CM^]fB=::J_SlMXlM@:>S\\FM`C=::J^ClFS`FBlvrAekGP];EXkTFI`A=J]ckfRj]sk: VSTf@Ir;Ir=m_smvrAyruXQ?hj?hlOhl?hj?XhmWhAxjAXmUPaSn]P]UfAM^ckGp[cj?p[ If?m\\CkCp]YfDm^Ck^RXFI`@M]SkZ>m[If=M\\_r>=j[cjZ==:::J\\Ck:::fr?=A@fRP FC`EkZ?iRGhkTFI`AM\\Mf>=PFbk vRXFW`AyrAYr=ekFSM@CHk@vZOrSLFA`AyRI@nR:j[cj?hjCHkCHkVRHF?`;IR=@NRDF:::A`==:::D F?`>m\\Sk^RDV[wYxI:::::::B:yyyxI::::::::::::::uasyYeqywYiYwcYw[yuyixyq uuyuIYvuyxyyuyidyxsYjPHm>^\\Mvhu?hyuvQIt;yywixyyywYytyxwyw=jyyI::::::: ::::ZxqYsix^ywqyrIyvqxqUywi[;VZoiv_yyoixpYyyYycyxIyQyyyyuIJxoywyYuA::Z yyY::RyfYhIRyVYjyyIZxqYvyymxqiyyYyixqIuCYsyXoexysqyqIvYwYytYwoytmxwywy Ywiywiy:^yxyytmvuisiYwyxuyyoIyliyiyxuyyiyvYwyyqIyoIweyykix@YuyyrMyyYxn yYsIymQuTQvyVtWiuWi;uXqlWh_UpwhouX quXmeYweYG:::qxpqXpmpeYgQMf]gSIUm`QmecoghoeXoLWdOtLQtMYTa@FuSulQpgWupp PWgeWwoipbAwbawfOtLenFtKEn[payvayF_Hn[hn[`GqsLisHAtHqsGiSU`EisGAsGaSUH m[XocxkMXlQXlMXpaHou`RYsGekMHqQhmCIoeHoCXk[FZ=vnAHkwHn]hpSXm]XkEhoOpba v^yV_aVaev^mfCQsEIsBYsCqrEIs@m]av^]V_ev_ifDm^afEYSWhm@GQxlhF:::W`BASIx k`v^aF\\V^]V^clOhlRlZBASK@vRXFGhkhv_iV`mV`]V^or@El>SXFM`DQSQ`Dm]Yf>M\\ ;::::I`AM]skfr?EkCP\\;EXkEXk:::::nr@qr>ar>Yr=ej?pZAf;=DV[Wr=Yr=M\\Mf>m ZAf;=DFrj^RLFSHm@F=`CIsCQR?XgeWgAxjm[;;Hj;HjFr;Uj>r:E jvqyxqxpqxpQjYyuYyE:::;Hj;Hj:::ZylYCjNrar:Ar:AsBAS^r>ej?pZAf:AR?hjDFIxkTFMXlMHmNSTV]gr?UlFSJ^ClvRXFG`@M \\Mf>=:Hv[WRJ\\MV\\MF:::::::EXkEHkCp[;GhkTFC`Ypq@ydawyaY lyyyYyOyrEyuyxyyyyixsIyoyywYxtYwsxxyxyiwmyyI:::::fywUx`yyyyyQv\\oyxEys IyoixoiwNytEx\\Yp_wdMquhdQWUqpqHBuSqoSt>wbusnv`EtKItKAvVPHAyrDXkWv]==a \\QVAa^aV=Yrxwiywicqwhqw`WiUu`XqUw`YyuY]X=::JhmWheWg?uRAuQyTmXp`GroiPh mWhiwgiWgeWgawfaWf]G>uREpntPunaPj=h^av^YuUEpkHpeXoeHn[pfawfkesontPeoFu QUpixoqhodwd?uPMegtOuneXo]hoeXngPaMwcQwgeWeiWgyvbAW`mV`QWbAwhYWfAxfeWm mWhEW]UvgqVaYym]XNUsEYUKMttywj]xcQwjyVbywa_uJAtRATaXn[xm_HnUhn[Xm>T_Xn a`MUn]Xn[Hn[HnVtMUoeXo[`J=::jcsmYxmUXmQpmXrqxpyEyyYynywYxpaxyQytUwgYmiv_;EBsBvowfyIy]Yyyyv]Qt= Yy]YtiYumyyqyxqyxyYxwywuywqwcxyQxuIyqxmYyQYt]Yx]yxeYxqYuuYyYyy[yQyrquj xofW`lvyIasyquUAoy^aUX`h>xfhnyavVyytixuaxkIynIxyayvyyhIxvIv^yywiyyIysy yuMveyyYyvyyuUyyuyvwysiyyIyuyytayyiYxyyyI::::::::::::::^ytQyyiyw]wyyyY yyUxyqvuismxsyyQyqyygYlYyWyxyyqYy^YtQAriPpfyu[yyuqYuqxiywyyyuyyyyymixQ yyqyuMw;uqyqIwiywyyyqy;::::wanaXsIysybAwbeYweywiywoixTIoarAYSadMqctfwg UuZfaWQytdXo?wbusIQmUhayVKIt\\HIa`MlMh\\QVHh\\Mk;Aa]mjsqyxqy`XquXqUpwh gUtWioUxwiywfawfaG:::sHqhwg?uREpFuSUpZRAU:ixoPwdgTLWd_TPwdOuTQuPqtPysI ysLQTQxlQxl>uREn[HncHoHwcOtLQtKIT]@VtKunntKQtOATaXn@wb[HAtJasKAT]p`IWc Yv`mv^uv`uvcEWbEWcqv`qvaqv`]wcMwdqv`eWb=WhuvamW`qvjyVbEuHySM=SC=X^evia v_mW_iVfev_Qw_iVbmv_mv`SmdFShmWHmWhmWhmhv_OsDelfSvsIQsDel:::nS`v^av]Yf EYSS`EYSO`FasFisGEmSP`;rlVSj]YF^S\\V^gr?ElKp^aFvR`FQ`Dm^afAM^;KHl\\F:: :E`?M]ckvrA=K@vrAuk:::nr@ekvRXFG`?iRE`>aRIxkXFbknRPFE`>=XFbknRPFE`>Md_ t:AR?hjPwdQW^]f]Xn]`mV`mlPPMXlMHmSHmrkFSDV[Gr;IrHqsHiRTFK@fr?iraR?hjDFrjAp[;:=Xj=hj?hj?`=m[Ck^Rj[If==::J[Ef<=: :j\\Qf=Yr=]vZAvXquXqUpwhgUlWh_UpwhgUtWiOuTep:::nuXquSIuSytQy tTQUqxpqxoixoghoXwewuYyuLQT;IrlV`mVj=XjEV[_uVUoepdspqPf]gHqSUXmHG]Xn`v ^OtLen::ZNQtLitJUo@Wc=Wb=Wb[GysIYsJul[P`EwbmVcAgKEm]@pvagsIysIySaxmWhn UHmchlOhpSXm=Yn_XqKhl;PPMT@urnIwdIWY=ViuvauWcEWdmV`?sAAtHisHysGasFUmUp aqfHasFYsEulnsHUn]XnOhlOhlFsCUl>sBElvrAukvsIysDQSKHlKxlQxlS`DQSM`Emayf GisGasFulVS:lFKHllV`gS\\V^]V]UF>STFG`BM_eF>SXFI`B=XF:::::::BlZAM]Uf?iR J[EV\\Mf>=::::::j[If@m^Sn]XnKHlPFywiyGkCHkUXm`v^or@ek>S\\FE`;Ir;AsBukZ >aRC`=m[;?`AyRI`=Yr=Ir;EkCp\\Qv\\IF;Hj;HjNRTV]UFr:Ejvyyuy:::::::Z:vYxI::::::::::::::uyypI:sq ymixiYwAsiywIyxYwyiueyfyymy;tyxwY:xyyyIbxoQwsyyqy;::::::::::Ry^yxyYwQx oywmyyixeyyiyxIdywuYvayyiyxEtEasgIwcyYyqivYYwawiyyiywAyrAyyyYuauyYBsOY tkiyuYxfYyYyvyy=wQymix]umxSy:vyy=:pyxuYxmyyyyyyIhYxUYrAy\\iuWqysIyyuIt YwgysUyayxoywyiyoQyqixqYwA:pIuauiYsqwyyyQyywwyqyiMtyiseysaoYxliyuQYsIy yywoikufbsEKMDK?r;at=[bVSSEsvOotygHpyyxoYywIyiixkiywiyyixuyykyxSil[HlE Wd\\YvyytUyhqQtZDIsyAw[yyqqwlQyiQWcYsuyv_yysYxtYyoyyiyyUyyQyuyyyyy qIyyyJyqiwaYbxwqwSynyyiYsixpXq=W?ir@ElL@RDTk^mqvxyraxyiyqyYyAyPYTKIuOi tyMYuOyvyytQynAw`yywYyWXiugfXF^X?[HVeHN\\``ggwynIxyYxgyyxIYyYxsyyqYyyi yyyyxI::::::::::::::wYytyxwyyyywuymqxYIiaqmvoyuyYbywiytiyaYyIy:uQxqyTe tPMvVmuyqYygyxywgmyyyy;:uYyuixoawiyyyYyA:::nYwqy[tQytaxnaxghogpxpExkIx oixoiwNY\\Yv_wfetdXqctOqnstQytSIuSytFwcusvVjlQh_MmUh`mlQh^aV?A;qjAhZAV ;ANv\\Ur=qiywiqwhqwhCqsPhspqpgCqsPhSqupfawfaG:::sHqsHpkHpgho`wfOuTQuSI uSIUhwgiw^af]Yv]Ir;IrTQuTQv\\QVVsEUpmXp?hjaR?hjDF C`A=@vZAvZSkNruRepoPiugRAuRyvaeogPZ=fKItKitOitHqSiwgigl> UPwd[HMdMWdQwdGtKItK=::JeMWdUWdMwdIgNQT@wbuVbgsIysGysIqsJAtHATRmfStFSh mWHmWhmYHnWHo_XneXn[XoUXmgHmSHqWxmyxm[HrWxmUxhsg_CV[_Hj=xoIHl;inaXpUXm uxnaHoSPbiV`uV`_sFulQP_efEYSWhmtJATFsCulQp`qv`eV_OShFB mSHmblNS:hFbmnSdV_?sBUlMP_eFnr@ElNSdFK`AyRKHl`FIxkXF:::K`B=:::`FM@>sBe kfr?Uknr@ek:::::::>qr@qFaSEXkTWewqyxqMYtMIr;Ir;=Hv[IV\\_r?m\\;Ixk`v^GS \\FI`@qRE`?=Tf@]CM_ckGp[;C`vZmjOVDH;QjuWiuwhqwh_Us`UYUq`VauVaUpwhqwf Sp:::fuWiuQytQatNatPqTixoPwdgT\\Wf]wayvaUWeGr;uoixoGisHwcGSlPplPZ=fJEn qxpqxoXwayfHqsHItKItLQtLQT:::jdIgOYtMatLun]payvayFW`EisGQTtVb=wbAgIItK AtJem[Pb=gIysIAsCyrGqsFYtKAtNAtIQtGasLQsDqtDQsPmniWhavWmUxlpp`vU]uiqVa eW^aVb=vZiwaGtEasJqsHqsHEmSp_ifFmayvaAwbAw^afC=dV_OShFQ`FMauVaav^wrAel ZCISKHldFW`GisGasFulVS:j^SmUXmU`CISI`BASM@fr?ukFS`FI`@qRG`AyRbk:::vRXF :::::bkfr?EkCp\\Qf?=::::::JcEWcIv[oRXtMYT]Ufj@PEWcEGXkEXywiygj?hjEXkLF AxjHFG`CM^skI@PFBk:fR`FM`>ar>Ir;ejNRm[cj?P[SjN RS`FK`Bm]ckGp]sk:::::::vpqxpMYT;HjPuTQU:=R:eSGeSKEdwhQEXkEH lNSJ]Uf?iRIxkdV_OS`FK`@qRRk:vRhv_WSTV]Uv[CkCP\\cj?P\\Mf=YREXkEhlvrAyr= YR?hjTV]_r>UkEp\\sjA@DF=Xj=Xj^RLF:::::::::::E`?=LV\\WRJZ?ryxQyyyyA:::: :::B:yay=::::::::::::::byxyxqyyuYvoyrqYxyvoyriyyyytqYoIysIypyxuYwktqYx UYyuyxcyyYyyYydYtQYbQV^yYviIyyyuqyqYyuYxmyyuIyqy;uyy:jyyI:::::::fYwYyt yyyYymyyqYwoyuyyyQyyexyiy]yvaIyyYyyyywyweIwyiydxlUxwQYusywHn_XgIWdyywm yyqIyoIweyytyqyxqqiyiYhyygYpIyyyJyuYyqiyYYhIuaw=jyyIfywiywmyyiy=lYxmyx iywWyvyytYyLQr=EkGxyuiyyiyuIxmywyyyyyy:nYlYyuYyuivMwKyGYyYxZu]yEywiyqy ydIaYwlYyuyyuiyuxVYnxnYXwQy_yoQysywxyycAytIyxyykYxnQyxyytmwqyxyYvstYyW er@UF=ab^or@yiigwymYugIqgHdmv_WYq;yyoqyUYwYYpyxymisAYsiyyayqyxeyryuYyy iymiysYxlYwyyweiwyXPyyuYuuitmyyyyyYygyruysYyyiyyAyQyayuUYyyyxUYugxnqxy uy\\yyOtVYr]ogt?ZD^qwnbIYdSokZN^j@dJwyXiuyqpn`haf>GhjyqrUYtaxyyywyixyY yyy:bynYxyy;::::::::::jxSy::vyy=uaw=:hIqatMv_yweyrweYwyqyOywexDvRQUbyU [AykQxyiYy_yyMyuivyyyyywyYuAvyy=::wiywqyWyuexoarUlMpxqipMwcwZrMpctOQqu hdmoihfaWUYuhHKItTHEa_mmYxmUXmUxmYhaMmSsVv]Er;QkEXkAH=AVv]er>V;iqwhq;I r;yqyxqwhqtG:yxqyHr;QhmWhmG:::BpkPfCqsHqwhqwHqsHqs`WiUMYtMIkCPkcpoPmUX mQwdohpohYyuYQuTQuRAUghoLgNaTPwd_tNunaPdMgN=::jdSontNeoc@PwcgtMYT:^tLQ tNQTaXnHgKatLQtLAtJitPatKQtJasIisIQtJqsMItIQtJItLAtOYtLqsCAs_iuWYuGqsf QuUIoMhlxXn`pwKatWyrCIuFqsMqsIItIysHqSSHmpv`?TDWcEWauVauF:YxmYxm:VtMem ^SxfIAtJemWp`qfIysIqSJ`mfEYSQ@^SlFBn[pbAgGisGQsDQsFM`clOP_CmnsHem^ShFQ `F=hF:::::::MXlMhlOP_slVSdV_GS`F::::::::eXoehggggKGdKw]Yv]uWi`v^Gv[IvK UnSHmSXmUp`CmNSJ_sl^ShFO`BASK`AyRI@>SlV`_S\\V^gr?ekGp]Yf?iRbk>sBulQ@Pv \\Qv\\ClvRTV]oR:j\\Qf?M]ck:::::::::::FS`FBlKP]Uf>aR?hj@vZwyyuy:::::::Z :vYxI::::::::::::::ZxiYpixAyviyriyvmvKwyYyeYy[xyywqyxAyuyyt]nayvYyyyyw qyvyywqyoiwhAsshq\\YtQQwVAvyYyxyY^YtIwqyyiymxyiwmwyiumxSy:vyy=:::::nyy eyvyveyoyywYxyqvgyrqYmyywyx>yuIyrIyqaxlIxyqywyywuyoyxyYxqYwKitOW`sVqYX nyasYxnyyxqyoyxqiyyyywmyyI:nyxqywyyyYYhIuaw=jyyInYlYxmYymYxyyyyyySyuYy wIysqxCxmqtYyUatOQYnoypqxyiyvyywMywYxIwQyuIsiywYyNyyexyqwky_Yyqx_Ywyxn xxqynqyxQXTYy?ywYyxyyuqywIYyQyyyxSytYyYYxiyquyy=yxyYwqixQYuYxwixgyxcYv ayyqIyLxkEUn\\PsXj>yyUwwyxeiyqiwyIyoyyuixpYvyYx]iviYyyyyqxssYiuUYysYuqQxuYymxm]x yuYyyyxyiwMumyy]XsaxOYuvxKPtxBesyuxqaYUIutmSFiT\\@nyUSEqSX`mHaOcxyuuxy uYsyinyyyYYyiyyyyxI:uaw=xyy;::::::::::fYpIyyy:tYyuYy;ryyqxoyweyoQyaixA ywqysyYxyyqxsEyxyyyoYyYyxYwmyxUywyYwqyc=v[aW]aVoaWh?ytUywyyyqwiitayviy yUyvyy=::bytYxmYxmIeYweitOituYyXIoixTIoix\\Yv_wfetOQquhdQWQytpxpqxpstQ qnaxnUXmUxlQxlQHAa^aVEYsPHCIsDXkEXjsqyxq;IrJiugYyuYAVaRAxjXv]_r>UkEP\\MFfRTFG@fRPF:::::::::::I`A=J];A`yyaypyyviX\\yymYxyywuiuIyyiyqqyuayxqyvMteilyYKYyqytuIyoyywyysq yeYxaipyYNxsyyfyxD`cROjnvayawkyyriXeyuhWd=vh`oNAmR@RFPKyEVXyyw=YrYXgyy yQyysytyyyiy=pyyyyxsxVYpIyyy::::::::::jxSyvyy=bywiywAvyyuyfYwaxYYhImqy aIiyywiwsyxaYweYyuYyqIxoiymyxyIywyyqiyfy[AtMYrlhpZQtNyyrQYnyvayyyYvywm yyInYtIyiyxyxqyyyiwiywqyxqyxmsIysyyyoYwIxkIxoixoiwbx\\ISadMquhdMpmXpuX quxpqxpmXpmXn]XnMXlgv_UsFv[er?qkIh]MkEXk=HtWiqwhcqw@sHqsXquXqw`XMhmgTQ uTquXquWmhCq^UpGu`XMgegT=::Ji;i`SIUghogXmUXmGhkGhjCHkcxneXquXqu`IySiwg iGisGiaxnpfGYtMYtM=LWdMwcIgJAtJATDG[Pb=G:::OHmShlSp`qv`qF[hmWXnYxm]xmY hn[HnahmZGqsJemahmxv`qvbmv`yv`qv^eV^ev_av`=Wa=wdIw_EwaIV_YV\\av]qVbuva =Waqv^]Vb]V^UXfawhiV`YwZIv[qtQ=T?mRgYV_Yw^iVdav_Aw^eVa]v^qV^aVamV`yva? sDQs@YsDirDIs?QsCarCAs@AsBirAyr?yrAysIelO`BulnsFasDQSQXlMxlfsEUlMXlK`A yRI@ZBM]UfD]EekGP\\Mf@M]CkZBElOP_av^[Am]ClZA=::j\\Qf?=::J[EV[Iv[_r>Ekn RLFC`=YRBk:::::::fRj^av^K`cj?hjGP]]FI`>M[Ef=M];G`=YRRkEP\\Mf=mZAvZAF:y wiyGkCp\\Sj=@DFBj;PZ;?hj?XkEP\\Cj;pYyuYIv[IF@vZAvZcj?hj?`=m[;=`;=::::: :::::J[cjVRHFbj?hj=Xjuk> SJ\\cjFr;ejVRPF=`@qr@ASLFBkCHkC`M]SkNrTxvayVb=Wb=G:]`KMbsmvStVa gsGEmSHmQxlhFWhmxva[Gm`slUP`CmvsIyS>tHUnYxm]xmYhnUhm]XlOxnExk[HhoW_kuX hUY]XcMwaqUYOrSXFE`Ek?hjXfAASPv\\Wr==C`m[sj:::::::Fr;ujHV\\Sk^r>ujNRLFyyyyyy:::::::Z:Zy=::::::::: :::::::::::::Jy_yyyyyyyxqyyiyxEyxyw?x\\qvMarByrpHqtawhyyueyeIxuywkyywa yyywuyoasmvcxVyyyywmx;:yyy::tixyYxI:::::::uawmwayvyyxuIbycisKymwWjsge[ wytyyMyuyImIwxyyyIdImiyuaxYYjxsyyQycyvYxIuaw=JwSyxyyyilAxjeywyyuQwEysy XlUxWqtQGxmyYwiymqxaYyyyxyYyuYyeYxmYveYwqyyyiy=::wayyYyA:::::::Zx=xIty yyYxqYymYxqyyoxu=bxkIxyYyxyxuiuExOiuoIuWibcf_EIpoPviivyyyiYyYywIyoixoq uwYnY^ywYyyyyxeylYyOyyexfYwYyvYxIvyyiyyEyyQvuyyYYxYxWEvu`LR@X;aryUtPyy XIVWUOePqJlkyev\\yyv=yymYtMUKis]uTOyYtEyyiyvyyyuiwexkIxuiysyyyiysavMwg ytUysiywIynYuyytyyyiy=:::::::::::::::JyuixmyyiyMy;uawMyKywywEuEItyixsi xSiuGG_qVv?isyIxmyyuQxeixyywayyiymAypqxuQysqywaywYYbywYyqyxsIxpyv]Yxey vaYw]YweYwUIoix\\Yv]IdhdqwhetfwgiweYweawfUuUYuMYtMyrAQkEh_iv_YVAIsCQlS sEQlMh\\QV;Qjywiywicqwhqw`YyuYIv[Iv\\Qv\\]XquXqUjhqgW=JiugXmfawfaG:::y XqshnYxluxpoxlShlAhjCXrAIsmHq;igmWi_UWmU=[B?MrM[Ef:AR;@Fr;Uj> r:Ar>ar>qRDV[[pjQ YoexY`avyeysgynixjyvyqyqQy[yuvPn\\HwVQuyQytyyxiYuYynyyyyyixmYxyyYuQyqi xayxeywqyoYnyxqYvYyyy:::::::::::::::nYtIyay=piyiYJveYwqyxWxjQxyiyyEyui xYYvyixwigEWd[xi?xyoIyyYywyyyqueiryyvixwayseysyyxixyyxuyqIyoYwcYykywyY ysixqYxoixTYuOytEx\\IdhdqWdHBuSIuSagiWOitOIsCIsAyrTXmUXlMXlMHEYsPHCIsH H=aYyuYAvZuqyxqy`ZMkEh\\Mj?vXquXqUjhqgW=JiugXmgiwgiG::JheWqexnAWaeveQW d]xnYXcAwc]V_iViAxluwjQXoqxrMuUiugpOi`j?hjFXkEPr;EreMWbMwcyVo]XnmWhosH qsKITmXpmXn]P`mV`]Wf]wccn_pbAwb=Wb?T:@wbAGSlFM`CM^]fB=XFI@>S\\FrkvRTFG@::::VsEehoghQxlXwe?r:AruXquhpot^av^]V^] fcGgcawfaGisGiMXldV_Oruk>STFEXkXFI`@=Hv[cj?hjVr=ujF r;Ir@ekK`>aR?hjLV\\_RDFA`r:ej?p\\ckG@ar>QtLQttPqthlOhlQxl QhqwhQpvbAG^SdV_ev]YfIM`Cm:::vS`v^or@elVsE=KHldV_eFhFO`BAsBASXv]or@ukX Fhv_gS`FIxkIXlFSXFK@FSlV`mV_ckTv]sl>sBAsB=::j^afBM]UFvRTFK`C=\\F:::::: :S`CISI`AM_SlMP]cjvYyyy:::::::Z:vYxI:::::::::::::::::::::::tiuIYmywiyy wiyyyyhyxqYweYwiywiixqykawgeyPvY=vOQTMyywuyyYYwyywuivqYwyywYyyiywiyyaw uyoyywYxvxyqyqmvsxyay=waxmysy:::::::fY\\Yv]yyyyyyItyxqyweyvMytyIyiywyy uiykIv]Iq]xnWWcKwykyxXYvyIyyynyxqY:xyygYpIqyywyxfYpI:yyyjxCuWiuyAqyxqi wiyyuawyYgYxgXlQHraXocYtSyysqyoywYyuiyrIYwiyyAyqx;:::yyy::::::jwiiyyyy yYpyxqiyuyZu]xqywYYyYyuiywiyoIxxIyarqxpyyxuywawqyyQYwyveIsiXpCW`svyeIx _ysMyysiypIsYyiyv_yyyiyyyywqxmyyyyymYyuYxoY:nyxuyxyxYYxqyvuYwuipqywYvh YwgyxkaywIywQyoaysaxyyysuymIyuYxmyyoIveyvUytOqyuyxQytMVXmuXQuTAynexyuy ygxpiyoqxyiyxeytYysyxyyuyyyivyywQYwixyqxyyyqyywyy=::::::::::::::wiytIr yyyyuaw=dIqauyyyiYpyxqYyKygyrQyyQywyyiYxryu]YlZQn\\LvhoghtPqt@vZAvKITMXlMxnaxnDgDQsDatNEo NtLUn]pb;bn_hn[Hn[HnvsIysIm`qfG=lV`mv`qfG=:JaoSUp`yvaqvayVaufH=pFU@>tJ =pfGMacmBmSxmOPkeWgMVWeuiIgKYRIyokhdov`uv\\Svr]hnGioixraHoiXlOxmEXk]Xl MHo[xmaXn[HmUxlShmXv^av\\Yv]]V_ev^ifAIsCqrBASO@\\V_ev_clOP_SlMp_afEM^] fGisGiS`FI`CUlOhlj^ClNSOXmVsEukIp]slQP_clFS`FBl>S:j]sk::::::>sBEhkGhEX kxvayVZ=f@m`qfr@qrXTaxn>sBUj=XjI`AM]CmSHmK`B=::J_SkEp[Iv[Qf?qr@EkCp[IF vrAyrAM]SkEP\\MV\\MFA`==XFM`?iRA`?m\\sj^Rj\\Cl\\V]cj?p[IfBASCHkLF:::EX kLFC`=m[Ck^RPFBk:::::::>sBAsAm\\Qf@M\\Ck>r:eiwgiyyyxI:::::::B:ry:::::: :::::::::::::::::nYwiyZyyYkYysYwoYxuyyyqyyYuUYuyyygyueykYxIIqwx`UV^sYu ]yyqyyhiwaYyIyfyyqYpiyQyxyysqYiyyyywyqvyYyiIyyytIwyyuiypyywyy=:::::::j qqXpyyyuyyyimYxmyyyuyZfQwdyyxyYiIxuivcybgF`WYrAyykixyIywyysiywqyuyxcyv yyiyyUyNyyexZwuyuAnyx=Bv_ivuyyHIpyyuyviyyqyyyYwqYxAiRYsGytDasyyxsyyviy saxoixiAXuay=qqyiIbxyqwsx:fYpIyyyJycyvYxI::::warMvSyuqyuyyoyyMyiYuYiwM ywYyiwkIxqYyuYyxyx;_iv_yyuiywIxmyxaIwyixuiiaWgWHj=XyYIvyIywyyiIx_IuUiy mixxiyqywMywyxayxqyuiYoyydIsayqyyuxDYqIYsAyteYyyYtuijqYYiyoIvsiwmyyoit QysyyyqYvYiuqYxkYyyyygiyDYwaywmYwyYyqYnEhf`n_ptCYsyYywyyuuwYIvyYynyx]y yyyiYwIyyQwcxyqxyiwuyyyy::::::::::::::JyuYyuIryfYpIqyyyixnxuQyuyyyAYpy yyYweiyyyyyyxevSYuyIywQtUyyeYxqSBEsnYWfyywmYyUyuoyxyYwiwmywsiywyu]YvSX mQxoeHomYwcYymIxdYwKweirMpctdXqKo_wfaWVweYwfaWKItDXkwU=irCIsDh]Yv]YVLX lWv]UrvuYyuXquXAvZAvYyuY]YyuYAvZErfuWiuUYuUYUdWgGuSUp:NuTupqxpixoXG::J fUWkuwhaWeEwjyWiIXlExe?u=mYEV]=wcYw[UV_]wgywbIWdAsCIsXhmW`j=TJNPlJS`FS`DQSKHlKHlFSdV_GS`v]OsByrYYuTyryxqNQsDQsvhqPqrA Ysp@q^xkKpsrPqVyrCIvIAtQyrBys?irNasFysEQs>Qr;isGekKHl?xkGXkOXlGxlOxjKx kAHlIXkOXl?xkGhjIhkChkGHkGP]Uf?iRGHkChkCHkCHk\\V^[?m[If?M]Uf>aRIxkTV]? SKxlFsCISZDM]SkfRJ\\Mf>m[If>Ekrj:::::::vR=::J]Uf;Ir;yqydj?hjBj;PZcjNR:jZAvZ=VZwqyxqxpqxp QHv[_R@vZAvYCj>RtUY?r:uiFr;EkCP[EV[uUYuUYcj?hj;Hj;Hj::::vQxE:;`:=::::: :::DFA`aryQyvyywqyyQy uqxlIXyiyyQy;yIywiyoyydIRynywUysqxmilYxqAyyetgiwYYyqYvyyxiiyqIyyYywIyq yyqqyeywiiuiwqyxnyywYyyqyrqxoyYwYyeiwWhj?xhYweQYqyxyqYyfYuiyyiywiywqyo yvyYsiywAvywmyyI::::::::::::::nyxUyuqysyfYpIqyydImasmwswfyyyyxmwiixqyx QyrAyyiYuyw>ytatyPrPyq=qyeqWqaxiYuIysiywiuaivGXlIxm_HnkiweavQytQypqxpe x\\ir]dMqctdXpmHNWhxowV@xjouZAv\\av^Iv[er?qkIh]MkEhZAvZquXquWiuWAvZAVn UjiywiyGuXqdWgeWgSpmPf]gR=jfagTQUkHpkHp:::mxoGir=ilIxjgHoaxnchnQhlShq; YroXq=ilUXnWXneXpsHrsgiyWTQuT[dMWtb=GbKYs=yn_hmqXpmhoghoWhmWhm^tNUmUP_ eV_UWeUwcIwcEWcGtKUn>tJAtJmbAwbAG_hn_@]`JmayvauVagsGisG=::tVa[GMacmZGm `;S`F=dV_?sBUlVsEYsGiSrlQP]av]IV\\EvcyVaOsDYuIysVasGAu=irIIquXlugi=Yog pvJQt\\AtKQt>irMIsCatIqSSPaufAm\\Yf@YsDyrFYs>QsCaRVsDiRTv_?sDQsBQSMXlF SSxkKXlFsAyrFEmUXm`fCyrAyrCIsCIS\\V^wrAekGp^afFaSMXlPv\\oRdFM`C=\\FK@v RXFG`@=Xv][@=::::j]SkEP^ClFr;Uj^R@F?P[av^aFTFM`=Yr=AR;hlOp]sk:::VSnRAx jTFI`?M\\Mf@m\\cknRPFC`=Yr=QruPQw`iv=ArvhuUQuZIvYqsGysPUp]HokHmYxn[xnixm_hoYXn_hm[Hn]xmYXnYh mcXnUhnpVdEWcIwc_tNunaxn]Xn]Hn[pbAgLQTa`LQT]Xn@G[`IysIemtF[Hn[HnvSJaCm SHmS@fSpF:::bmnS:pv`_sFulQp_;::RlQxlIHmOhlUxlYXmSXnSxlchlOxoMhluxlShrS hmOhfcGd[f^Shgkgo=xjwhmYxmAxjchlOHoYhmKhkGhmWPaSkEXk;XkCHlUHmGHmQhiCxj ChlMHlUHmMxlQXkIxkIxklV_osCQsCyrAYsCIsDQSQxlQhlOP^]fAyRKHl`FIxkIXl`V_c lvrAekvRXFSHmlFOP_av^wRTFE`>aRuWixvayV\\MV\\YFqwhdV_ORlV`mvZAvZeV_evay vaMUTMu`qfHMWeUWyqyxqYyuYAFC@>S\\FI`@m]sk:::::::VsEUkEP[Ef>M\\Sj=P[OR^ RXFG`M[Ef>M\\sjAP\\MV\\k\\ClnRGXj=PZ=f>ar>]==:::: :::J\\SkEp\\;C`>m[sj:::Fr;=;`;IRAxjHF=`yxqyxyyyY:::::::Z:Zy=:::::::::: ::::::::::::JyuixmywywmwSxnxxqyqyY_ixmYxyiyyayiywyywyYtitqYWivwhlYhj]W gqIu[ixYivyYytYxmiymyuiyuwnXlYyuYyyyxmyxiyyuiyqyxuyxI::::::::RyxiyUxFY dIjwiYweYwyyxqixqYuiyyYxsiyaiwsiu[IdqVa_XlSPxwywqyrYxneytYy?yrEwcIwciw gYyvyy=syxuywmixqyydixiyyUyyqyyInYtIyiyxiyexuiwUIvGYpqHraptexpyYxyisQY oyxyiyvyyQxnaxpYytYYyIxwYyRxfYpIyyyJyoyxmysy:fYpIyyy::jxSyvyy=wYytyxcy jxcxyqvuyyYyywxyUxyqwSynyxeyuqyqIsiyyYyLYqyhwiyqiwgamLPl=YquxywqyxAysy ywayraxmyYuQyqiyqyaQWuat=dIuIyyiy^yuyyrIywYyvyysayyQyxexyQxuyyuyxoYZxe yVysuytyyqYyyYwiyxYYv?YkitQYsTW^iuXyyxuyxQiyeyiyv_QyqYyuYyUYugYlixaYsY yuYysIysAyyyyyy::::::::::::nyxuyvYjxyiw=hIqatmwsx:ZtmumYweyysxoyyoqyai yAyuIxPQurhm[PwOAuyixqyytYylqwgapoXpvyywawmAxdIxhIyuYyqmwiI^W`HUqp]Xn] xoixomXpowhqweYweyVLHAa]MkEh]YVAa^aVRrvuYyuXquXyuYyUfuWiuXqUy`YmhqgTQU mXp`G:bpohpqxphGkHp\\G::jfYWiewfqWgUWbuv`]wfYw]UV^EXlQXaAwdIWeeWcQwfAW dGtMatJQtLYtKun]HngxnYHoaxmaXn]pcIWdMgMYT]Xn]@[`LQtLUnJbsmYxmWhmpv`gSl V`?Tjacm:^SJ_cl::::ZEYSOhl`v^?sBEl:::Z@AsBQREHkMxkMhlKHmKxkWhkGhnGhkkH lMHqOxlyXlQXoyWjkuREUmqGi=Yn_hq[XnAxgigp[HnqXogxjwwiswgkGlGhkQXmUXjGXk =hkExl]Hn=xkGxjMHlChkGXjAxjExjAHlGhkTv]]v[MV^UV]YF>sAm\\K\\gRTFG`Am^;G P]QFE`@m]sknr@ek^r>ej?P[;ywiyGlKHl=`DmYyuYeV_Gr;UkEXkI@fpohpdQwdapn`pc HocHj;Hj\\Pn\\@r:=AxjAXkEP]SkPv\\CkVr=uj:::::::vR@vZAVYuUY=f:]yxqypqxp q;IR=`=mZAfxLYSjNr<=y_=Yr=ar>Ej;PYue:mZsi>r:EjNrS\\FrkVSPFAxjtVaOR?XlMXlIxkTFUXmdUWeugigdPod@qr@ quWiu_pn`pIysIAsBEl:::vRXF:::::::Q`@qRAxjLV\\gr?=A`?m\\SlM@Hv[WRTFG`=:::::::j\\Sk:^RLFE`?=::J] SkEP\\CkfRPF?hjEkZ=YR=Xj=xiywiyyyyA:::::::B:ry:::::::::::::: :::::::::fYyA:rxfY:pIwiyxyyyyy;tiwYyuQyoqXxivyyyAyuyrvpqq`ohxyuqYqatay wQyvaysYyxYyweywyyfYpIyyy:::::::jxsxZw=xyy[xeyry:fywiytaYqyxqiywiySIuS yyyyy\\yvgYsIwXW`uvXmtQmyouxyiIqavqyuAytyyxqyymycxlyxyyywY:RyyqxCywiyw iyjyuYyuYxwyxqyyiyxqxguoSxmSXi=xyoAlysUiwqyxqyxaYsYyVYhIuawmyyInYwiyvy yuy:::::fYpIyyyjxoyyeyyqyyI:uqyqI::::nyviYyyywaxiysAyyYytEyuIycYw=yoiX cmV`_IsEyyiywpiyqYyYypywgyyyYxYyuYyuIpyymYyyyuyilqyxIxyiyjqyvaxxqysqYs axiyyyywMywyyyysyxpIyqqxyyyxyYwYxoIxkYxiyytyyuIsyxpYyUWayvQETLaypqxyiy xgyxqysQyyiywUydYwgywuyyyyyA::::::::::::::Zw]y=::::wqxairayyiyvUyoyylI yaoywjuxuQtYYqaWhyS>mRriwhyysMyywyoixoiyuYypqXdiqyxqQwdQwbAwbaW\\HMYtM itOitOacIwcqv`EsVv]ur^v^Esfv_Us^v^urFVxwiywjAHvUjjCr;QiuWi_UqxphGohpdG qxphGsHqpwhGU`G::Jg]Wkuwhuv_YvkAXjqWimw]UV^UR>aR\\Yv_=wcYwcUWgEgSQtNit JQtLYtKEo_XngxnYXocHnexnapdQwdCocPcEgKIT_hnLWdMWauVayvayva;WhmtF:axnah nNT@G[Hnxvayva;UXmpF::::W`Hm`qfG=hv_WS:::J^ifDiSOXmQ@Kxl`V`ev`uV`Awauv bqV`=w_ev^AvZEW]UvjIWduSAyrNYSFAWXuUuAXlmw]ev_]uWAw[QvemfHAsCQsBASYxmE hlMHlFsEUkIP^UV]iv^aVaafEqrAYsCelOp_clOP^]V^K^]fCM]UfAm^Clnr@qr?iRE`@m ]cknRj]sk:>SHv[Iv^av^aFE`;IRIxkIXlZCISghotUYuuYyuYMUT?tJEqsHqKHlXFEXkE HkC`=ujZ>aRC@:::::::NsDQs>ar>Ir;ujA@DV[ORLV\\MV\\sknr@ejFr;Ir>ar>ir?ej ?P[ckTv]sjFr;uj^RDF?@^R\\FG`]:Ar:=VR:LV\\gR@Fywi@F=`:AR;`yL[SkEP\\Cj;PZCkFR@F::::::::?`uRAuR=::jeQWdywhmwcAgxxqw@mQHmCYsFItNAuOEq _Xom@]`IMc=WbEWbWtKAtNQtKysIEn[Hn]XnaRDV[ORLFE`AukG`=HFA`Ayr AASLV\\ORj\\;?`;IRC`Bm]sjVRXFrj:::::::^r>Ek:Vr=ej>r:AryxQ?`<=::jYyuYuU YwqyxqyLZ=VZ=FwgiwGisGiyqyyI:::::::B:yay=::::::::::::::::::::::jxwyuyY uawmyyI:nyxeyfyumyyywSxNY\\yxmYtaYvqiyayvMyuyyqyygYgYu[yyoqTMtMaW^iVxu XsyYxSywiysIyxyyyI:Ryyyy::qaumxsyyqxSyvyyUyVYdyygYyYynYwiy:VywEytyxqiu iywQyyExwquyyykuoY^xpqxxqyxaysYyyayxUxwaxyYuIrYhnCxfgWrchoyixqqxuyyuYy yyyyYyyyx[yuyfYlYyuiyUyVywiywyyyyyyEyoixuyxqQySyXYrIyyiI]YwWhhyGk[Wgki tWyyuiydywoYwiyxyywYjxwyw=jyyI:::::::fYpIyyy:tYyoYjxSyZuyYqatMw;eixkYw oIxuaxyywUymYxoiwmyywYxwqvaiwyxryyuYyyYyuExoYiYWf=XgeGvCIshywWYtyyiIki yuIxwqxyiyeyyiy^yxuxxYyyyxyitmwyyyiIuaweysQyeImQxaYyqIlIsIysaxyyyqyoIx jyywuyrxuIyYYv_gdQwo]hnuywkqxuiwqYwaxeywqyyyyyIRyxIhIqawmyyI:::::::fYp Iyyy:Jyuix=:ZweyuqywYJyyix]rIYyqyeIaANxxEwwQwsyuYywyveqsUYtawhiTBMsyMy uyYsMyyGyuixpQwdUsnvfYw\\AwaQvbyv^IwbyvcIwbAwbyv_iv_Yv]iv]Yv^][Iv\\QVE yrAyrAQlEXk=Xj=XjuWiqG;iqw@sHqsXquXqwhqtGsHqdWgewfagRAuRAU:`wfaWgegUYU ixoixo:::nuVYuRqtOItIqshpohpqxpqQIuTIsEasMytRitPQuMmb_tLQT[Hn[xn]Hnchn LWcOtLunaPdMgMYT]Xn]Hn[pbAgLQTUXmUXmZHqSW@:NtLenFtKUn>TxvayVaufHm`qfG= ::Ja;ShmWHmWp`Sm^sFelOhlQp_iF:::ShlMHmUxlSXmQhlUxlGxlMhkUhlbl^rEAs:ir= asDIs>AryhuMYtQYsEIssPq^qsJqsjhp=YnbHr]@oLIqwpuEqsQQsFYs=ar?Ar;AsBAsDa sFIsEukIP]sl`V`OsCyRMXldFblOP^]fAyRK`Cm\\ck>SKxkfr?EkCP\\SkEp]cknR:PFE @ZIySMWdMGYsEYggggGj;Hj;`CISKHlxuYyu[If@M_ciwgi=Xj=HlKP[Or>EkGP]YFI`@M \\Mfm[IfA=DF=XjLFE`Ekfr?ir@qRG`?iRE`>aR?hj?HkCP]orAyRI`>a r>QRRtUYuUY;y_:MZsinqxdivQJYsiFR@FywiyWivYxI:::::::B:xiy=:::::::: :::::::::::jxyiw=jyyIfYpI:yyy:::jxSy:VYhIuQyyixqyxqYvywuyuAVyxywZavXhk KxypQywaxqexyQxuyymIyiyQyt@ysIYwyyxeiwavcAwtyyxQYwAsiynyxuyyyytyyoYjyy I:::::uaw=JyuixmysyZwmx;_iv_yyyyysiwiqwGvLqtdXmYPx]yvuiwkyywqyxQysyyyu vcivkyxmIu]yuxYyayyGYJxyyxyixyYwiyyYywQyqyxqyyyyyyYysxhYrEyy]yv]S?irw= Ys_ywYyouyuqwaYviYxeyyyYymI]ywgYxyqxSyvyy=:::wiywiyfYtYyoYxyyyYwqyweyx iYysytyyoyxyyw]weyoarqY_IxHYvYYxyyxyIyasQyrIyxaYlYxqywyyxmyyqyv]ivYvxY y_iyuiyyyyyexlyvUyqmXvIYswyvyYqyymiwWiokxh;wbSiokhwKYtyYysYxwAmiyyyxui ysawUyuYysqviYxUYwiyvuyyaYygYyYyvyyqyymxwYyA:tYyuYxoyvMyuIvYnYyQyyyx?x \\YvRPkEHwVquyiYyiyyqwmYxYYvyYyuiyyyiAxhaYvywqwnYxnIyr]yyY:::::::::::: ::::::RyfYhyxqyywyxeyBxgiwyyxsqqAyyUXpaT?AsgMV^yyvmytixomWfYWoyxpAYxGx vIyVyv^iu^IVSitMitKyr=IrEYr;is?YrIirAys?yrKyrCItAIsIirAis=QlyWjExhuWly WjKjuWi=XiCr;yqyxqqXquXpqxpqxpJgeWgeGqxphGsHqpwhgUhwgiweCocHoeXo\\Wf?U Pwd_tNEpkHp_hnXv]OsDEpkpgiwgIwcIWf]WfUWeoTPwdQGLGe`OMd;ghoLWdOtLUnFTDW c_tNunNtLenFT@G_`LMdCoVTjbsmYxmY`HqS[HnS`Frk nr@UkEP\\Mf>=PFE@::Z@M];:S`@qr@YRLV\\MV]CkCp\\skZCISBkCp[If@m]SkEXkE`C ISO`@qRC`@m];G`?m]Sl`V^sjAP\\skfRPF:G@fRJ\\Ck:::VRHFEXkEhknRPFAxjHF?hj ?hjVRLFC`=M[cjVRjZAfM[Ef:ARyyyxI:::::::B:ry::::::::::::::::::::: fYyYy:vyy=::::::RyVyyyyumxSyvyyeynywyyyiY:hYxyywqYKMSEaW`uvyiYyyywGymu yyiyyyywAkYx\\ywayxyivqyxQyyex^yxyyvmyyyyyixiYwYyvYxI:::::::wiytIyayMx Syxyx;yyy@yqyxyqYyYyrEyyaYxyYt]yTMTN=xaIwyMYuyYwGyniXwiyyQyyIkyxmIyyYy syyuix^yxqyvqyx]wmxyixqyxYyvQYvywuyoqyUYvWxoal@PktywlyyvuyoyxqYxmIysIy sYyqiypYwoysqywiyyyyyyY:::::::savmycyyYywiypyxqyyoyweyoat=jyyiyMymYxeY xoxxiyniytAyxIyxyywYysIygIxlqvsqXsIxyYyqyyMYtyywsQvmyyqI`YuAYqyYyqiwuy gIxUHhowkeggqiwaYv_iuyYysyweIwUiuOIyyYyuqyqxqyxymiwayxIytqytqyyaYwqyyy y;:::::wQyyYyuyyqYy]ysEiqAYsiy=XgeGgKGd`ywyyxuyyiyxyYyuYymiyyyxQyuIy_A w`ayxqyvqYyyyxI:::::::::::::::::::uYyuiyvYxI;Ir;yxmYxyyyyYwYanyxyIywat ;arjXmXhybQwyqwiUv]IqkxowxquHtUytMIvUyrOysEitKItMysEyr?ysAirHHCqmQxlYx lQXmMXlMXkEXk=XjAxiyWk=H>vXqu[yuYErtIiStu[Av]mV_k]mfBisFas HqsFqsHIsEYsDQSO`EYsEytQejfr?iryLdUWeMv^]vcQwdIwdWU\\G_hnpv`qV]UV]YuUY UHmSHMRSdV_[?iRE`BASK`@qr@ir?e lOP_ck^r>ekvRTFG`?m]clFSj\\SlnRTF:E`?=::::J\\Mf>m\\Qf?=J\\Ck:fRPF:C`>= Hv[WRJ\\CkFr;uiywiyyyyA:::::::B:ry:::::::::::::::::::::fyyyyw=jyyI:::: :::fYhyygYpIyyyJyqyyuyxwyw=QisIax]DiqyxnVxmuYxnUyuYxcyvwixmqyuixaytmyw YwQxoyv=yIytIwYyyiyxIvYyyy:::::::nYtIyaymxwYVYhIpyxyyyyytIIuawyytIxnyy uyyJyq=Qp_xnyaxpEyyqxuYuAYsyYyqyxiYw]yvYyyoyvyywayxqyt]yqys=dyyoxsuy^x luyuqucygywyqyMtEasrUXniysMIwyywQvUixyywaY\\YweyyyyyyI::::::Ryyyyqavmy oyymwiiuMxqiwmxyyymIsauiywQyryxwIYtixqYiaoqyuyxwAywqytyytayuQyxYxxayyU xwhgiGrGwdsYjYxyYIxyywgyyqYwyiwtYwGyxiyxqyxyytUuqHqOwb;gruhpyYxmyyoixy qwYXnMxteyvmYyuYxaywQYrEYyqyyeYxqYyuIyyyyA::::::wiygiv]yywYypYygyxUweI wKItYGcGguGysruMityYywyywQxeyvyixQytuyxyy;::::::::::::::::::::oQweIggg gWweYwyyyyiwgQwaywIYqyxyUYvyyouxEyq:EyeewuywjyYsIymqwhyxpmXmaxmqWk=Xnq h`IwaAvXiuaiv^av\\urAyrGIs^VEyrAYr=Yr=yqyhqwhq=qi=XiuWjuWikiywiyGuXqux pqpgcpNU:hwg_uVUqupdQwdMWdwtQUquXqq`NaTaxn\\gRauVauQytQatNEoVT@wbGTLWd MwdQGVTDG_@ZMYT:a`LMbcmWhmYxmYHn[pbAgJmayfHqSW@ZI=tF:Y@nS:j`qfG=lV`mV` ;U@VsEYsDQSO`EM`SmZEM_;S`FM_eV_]V^]v_WsE=dFBl^sDQsGYsEAS\\F::blKpdAWbq V^Yvdiv_MW\\QvgQv]qUKQtIEr<]wWuup=wcixcQwdIV\\yv[IV]EvZyU[=v]uv_uu]MV[ ]v]QV_aV]iFOP]]V^MV]UV\\UV]Qv]YfAUj=XjKHlKhkKHl]xnaHkGpZIv[eUXmuWiuWiE sGisGj;P_eV_yvayv]YfuldQgEYsEQrM];E`>M\\sjNr=HFM`>=TV]wRLV\\[>M\\ClKHlIxkHF?`?iRGhk Hv[[@qr@yRjZAvZIv[IV\\cj?P[Sj=@XFG`sBAsCm^ ClnRTFI`CM]Uf@=::LF]HpehfiWgchn]XpYxmUHhoWpwWjIhcOWfSF^ahe_wsGhl=ikMXo AHkMXj=xjChjOxmUhhAHj;hkExjKxkCXlKHkMHl?xkGHjLVZQV\\AV]QVWmUXav_OsFasE isGaqxpqA=@wbosHqsHM_ef@qRG`Bm]YfujAP[Ef;IR=@>r:ej?P\\;A`m[Sj=PZSkEp\\Cj;HjywiyWjNR@F=`yxQ?P[Qv\\_Rr:ARnQtE:y_:=JZSjFRym]weyuyv]IvyyyxI::uqyqIyyy:::pIqYyui xsIywyyuyytyyqYyyItyxyyyKxeYvYYpexnEXiqWlyWiIyoixyiYx_ywExfywyyxQYpYxU yuqYuayyayvyyq=ywQtQ=r>\\KkmOrlrADLhXjADtndPnEV[yYtIYuayuUYyqyyuyxyyva ixYYiyv\\yuAuFaSYytPyYvQixiymUvyiyyyyuqwUYvyYxwyxqYyyyyA::::::Jyyxpmhy qyyqywyyuQYwQvEysyywmyliwgIUNeTvixpyYx?xbIWsYypYygyxeywAyA::fYyYyZuyYq AfYpIyyy:::::::::::Jxmiyuywiywywiyw;Gb;WxmYxiYxmIwmIxcyyuYwuIysYyqywYy uQypsxySiuQhdWWhgf`AikMHyGItyivaivKItIyr?IsEyr;yr=iq=yqwHr;Qj=xiAxjywj AXi=XjqwiywgqGq`WiUrpjiqwhuWhmgTQuTAuREpjeso>uRAuRmfagUYUcHoTWeUWdMgKI tKQtLEpkHpsPhmGSHmShmNtLeogPecn_hn]XnHwcgtOitMYTa`LQTRnNTHwcGT:jayvauV aoSxvayVb=WbyvayVaSmfS:J`CmZGisGiS:::lFS@VsEul^sFEmVShF:S`F=hFOhlhv_WS j^afAyrAAS`v^OS`v^wRXFM`EulI`A=:j\\Yf=ASuGlExlgxnKxlMHjqghyxm[xsYXnUHe [GnQweUW[OVnWgfyhjEhrShmexkKXlCxjUhmS`;QrAqr=AsAiRPV_aV[]v]=V]QVZUv\\E v]UV\\aV^UFqWiuglSpZAvZAgKYr=EkNqtPq;IRGhkfRAxj`FOhlhF?`@m^av^]V^wrAUl NSdFG`Am]ckGhkG`?=HFBkVr=uj:Nr=:HFA@>sBujFr;Ir>m\\QFN RJ\\skZ;IR=`>m\\cjNRTFI`=YR=`=YRC`ar;ypqx yrAyyqYgIvxyxoYwIywqywyy=:jxSy::vyyUy^yxqyvmyoyyyYpiyQYeytQysyxqsv_gFh cwfkxnchruhqeHnYHkoghmGeWGiMGeyYsMyyKIuyiwoyytiwayuqxmyYoQueiJQupO^hNv CXlyitYYvT`noG`r^fT>^^XghwyiIxaQVyqxyyuYixYYwyxeisCyykywWirwhe?GbuHo]x ysIyyYxmYwYIvhYyiYveixqyyyyyyY:jxSy::::vwjqWyiyeyvxyyuywaYvyYyqiwMwwyw eyaYwptQivXuewiYuYyoivxyxuyyuYyuYysyuiypixqYyyyyAfYpIqau=`yyOYhyyyyxSy :vyy=::::uawmyyI:::VyuYyyyyyyYXyuYYN]TNqyxWxmYxqIysaveAwkAyoYxrqxymoUx lUYnaxyeXpqWYAvXeSIUx\\affAxyiwjYwYyueqv_AweIwUAvXau[AvWQv[qu]QvXYv\\i u\\IvUAvYIuXiuRiuVIuXiUrpqHqsHqyXquxquXquxpqpfawfagRAUk@ZSIUohphwgGtKI tOitOYtMEn[Pf]Wf=Xj=xcIwcmUXwrAem^tNUoePdMgN=DWcgtO=axnDWcOT:Hwc;^tNen >TtfHysIysJAT[`IySWhmtFUXmpFWhmtF:U`G=lV`gSUXmnStFRmfSlFS@VsEEmSP`SmfS lFS`E=J_efEM`CmNS\\V^wR\\FMXl`FKHl\\FMxlQp`?sBEl::nrBAs>ISWxocHiEhjugj ;Ho_pbYv]AwY=VmiVaAX[QVXisIEvKUTZQTNMWY=vgaV_yv[IVb]V^Yv[EV]Uf@AsByrCU kO@ar=M[Ef;IR?`< =j[sj:::^RLF:O`>MZ=f<=DF?`?iRbkTv\\Qf;mZCk^RDF?`Am]sjHFHFC`==j\\QfAyrA ir?UjFRPFAxjHF?`yq?arxpr?qrFYsqhqv XsJys@YsDYpmXpv_iv_mTPmIisGYr=YrAukMp^skI`DM_skIP]Uf@m_ifJAtJIsCIsyxqy HR=xlnr@ukvRTFEXkLV\\Orr:ej^r>UkPFDF;`xpQywiywi>r:EjFR@F?`=LFAxjAHj;Hjywiygj?p[;=`@M]cjFr;ujAxjA`oBmsfbyyxoQxayyqyvoxlew_ qxaIcivAYrGw\\WfwwHryIxxyveixyyqixoeywIxgAwqy_yvaywiywyxsIywmyyIfYpIbx sIysiywIywiyyQyyyy;:wYytYyuyvwywuyvvUYut`og`t@AsyayxyyvmxuyyuYyyYyWxkI XtyykyyyysxuQveiriyvYyyyyuyYoyyhIuawmyyI::::::::::ZwiYyanaX?gb?GysQw]Y vqywiYw]yuUYtIYvayuMyuQYnixn]xnYxrmxoIymOy^qvgQtOYshHpCqokHxUiupav_Yw[ Iv[auUQw_qv[yuYQv\\Qv\\av^qveQwZiv^ysOatPauUAuXiuUAvYauXquTauVyuWer;Ir ?YqwxpmpgiwgiGohpohpVuUYuUMhmgWiUu`KItKysIysXquXAuRAuOitOij?hjUYuUitOU n]Xnbn_PcSn]Pb=gLQtLiTLWdWTDWcGTjbAgKmcIgNaTeXoHG]XnSj^afAm\\Qf?M]]V^]V _?S\\F:::WxmUhlQXlIHlGhlKxkQXlKxlIhkWhkGhnKHl_hkGXqYHnYXiwGaIV\\aw`uvs SxmyHkIHp?Hk]HkGp[MVc=WbaFAXkEHkMHlqWj;xjMHlq_CysHirDIsRhmVpixoi^`n^xs GElGhkI`@M];:EXkPF::::::::AxjHF:::::::?hjHv[WRLFC`?m\\;GhkDF;HjHv[WRDF =XjHFEXkEHkCp[Ifar>QwdQWhmWhYweYGXwe?uReogPdMGvTj dQwdMGaxnHG:_hnLWd_TDWc?tJAtHqsHAtJEnNtLen:FtKItK=JcEGFTJb=gJ=:::::::: :jayva=WbGtKUnvsIysGisGqsHemZFasFaShv_WsGem[HnUXmUXm::ZIisGqtNYtLqsGis EQsGqsFqsIisCasDIsFQsGqSYXmSHkygiMYquhpKXlQw^eVhsvasxhygu]HoqHlQhpUxme XmWXlCXkchn_hm[HnowiwWmahnMxmWp_GmXhmCpkG@rQrS\\FE`CISM`?iRAxjTV]wRTV]gr?EkCHkI`D=LV\\MV\\sknRTF:::::::E`?=:::: ::J^ck^r>EkfRPFC`;IRyyyyyy:::::::Z:ryxI::::::::::::::RyVyyUxyqv;qau=li xIxkIxtQytyyxqyyqYuqwyYywY`yuYyyyiy=::::::jx;qA::fYpI:qaumxsyyaxqyyiyw yywuypInYMyqyxhYwecitOyywYycyu[yywiyyyyxyyGyq=yA:::::::::Zwmx;:::rx>yx iysyyxQYhyxqixayyqYyYyfyx]XYtOInM`lrQweyyuiyyQyuAymaXwiyqYxkqyuyycxqqw mYwqYymyxuYxyyyuywmYwyyyuYpexnyIyYy`yrgykIxAYqupruhqyiYyIyyyxgawyYsqyu I`YyuYyuIyqyyI::wiywiyfYxyxuYh=Wb=XcIwymyx_yx=wqysyJywyweylYyOyxyytyYu YyxYxqirayyqxuyyhiwyUxXYryYvMYmATLEpFhKumv\\yYxAYx]ypqYyMyyuywyyxqirYy yIyuExkQiAxaAR>QuhHpl]SHUYXquyUiyQxiyypQXdiwIyo@pihqohptQXpiyMuAYsyYyu yuEIwqyuIyqy;uYyuyxqyxxyyyIbyyqyoYyiyuyywYyvYxIgayyyxyywqYoyxsywkaxUyX yq=YOysJMxfiwymyyyYxyYsmxpyyxwYRywIysyyyyyyAfyyUyNYhiyiYxyy;:::::::::: nyxqYsIyyajEv_hn\\Yv]yqyxqayuOyryxraXuIxiyxgYwUYtKytIisQIuOItMysYIvWIt MysGir=YuAyrQYpohbOF]kxhwWy?YsoysOIwIYtAypsXqkXp[iuWYsEYsUhn]XsOYtiXqs hpyhqqXquxpqxpwHqsXryxq=Iqupe_U`GkHp\\GmXp`GohphwgWSQHtKItSHmSXmUXmSHm @r:AR]UfHmfawfMWdMWcOtNMeUgMYtMYTLWd_TDG_`JATYxmxF:]XnDG_`KmayvaqfGysI ysIm`SmZHqSW@::fsGisG=lFS@::::::::fSJ`;U`Hm_ifCIsCYShFRl>sBElFsEm`GsCU l::VsCIsEmd=wayV_av_YV]aV_]V]iv^Ev^UVZaV]Ev_]v]uv_YV^UVeAWb]W_iVa]UWeu H=tTMSFUwTaun=Wd]vT]u`yuZMw^iVa]v^efBiqyxqy`r=yq?arsxqxhqEk:::::::fRPF:Bk^RJ_ef?m[SkEp] Yv]MV\\_RPFrknRJ\\CkZ?M[Ef?M]Uf;IRywiyWj@v[cj?pZAvZ=VZWr=ekfRypqxyqYxiy=::::::pIqau=::jx Sy::vyy=wQyuIyiywYynyvayyqywuwlWcEwPItKeYsEyyqYyyixMxqyweyyyyyeYv]iyyy y=:::::::::jxSyvyyeytyxyyyqI::mas=dyyOywYyqAypYysIyyqyxyywqYBxkYdqFaCx gmwyqYyhYyuYvaYxYYvoywUysywrywyyyiyoyxlyyqYxyyxyixYxnexkqyuIRyuyx[iv=i okXcoVaYIsIyysyyyIypYveywUyuoYtixqY:yyyyyy:nyxexoaxqyyiyxQrnhpyYyvuxwi xiYxyiytYywyyuyJyyixmyoYuIywiywiyyYysQwiyyIYx]sxTrWAtJQyjmwyuytyyvQiJi tsg_kVqA@jyixaUxSiyyYwsyxiyxmIxcyu[yyuYyKImShQQsEsUD]sfgTQaFKITPiB@MX_ evykYwuIw[yxuywuIykyywyx[isGYdwFbEipuhyeIxyYyxYyyiwqysaXtiwYYyiytYy[yy yy=vYwyytIJxgyvUyRyyiyrynyvUyqyxpItyyuIuqyiyxqUSMuRUtNyYxyyyqitaXOIsEy yuQxeiyiyw=uYywYyvYuyylYyuyyyyyyI::::::::::::bv_ivYyuYYxmYxYyuYAmywgYw aivsywgIycIw[ItKIuIisUytOYsKiswXs?IqAIruhr;Iw_YvWirDYhuwP=sCaWXyUvAxlU w\\]vt]XoaYq=Yi]wfEXjWuWiuNIuRAuWauTmgspqPi[YIvV=oHpkHpkpeYweYGBpkpfag TQUaxnPwdQwgig=Yr=qqxpqV`mVHtKItWiuWitOitJAtJqsHEnar< IrIrwpr=YrDyrFysGIrw`q:qppHvIAtTQr>QpO@mcPkGxtsdqEXlSy ooHoAhkCXhqWnQHmWHmSXlQp]mv_QuWeU^qV`QWgawYAvZeUXmu_if@ar>ar>ujAP[EV[E F=XjR>Z:V]Pv\\NysJyyxMyuiyuy::::::ZwmwsxZw=h IuawmyyI:::Jy]yvYyyoyyqywiyuQyoIxjyywaYuiyLytusCQStYxnaYq=yyqYyyivqYtY xcywAyA:::::::::::::ZwmwuYx]yxOyyqyvqYsyyuyxwIyqIyoaxEyqYxcyvyAuiyeXlM HewvayYvayygIxpiyYYciuYQwiiyqYyiy_IwcixyiyoyyuYu_IvsiywQxmYw]yuSygIv]q xuyxMyuqYsQYQ=TLaW`ufnqXvyvqyyaYpyygYwQyuI:yqyyIwyyuyxqyxyiyxyxuybIV\\ IykMxyeyxqYwmyyuYyuIyyywqyuyxyYyqIyqyyIqIxoQxuYyaYwoxtUukIU?TMEVQ?uygY wyIwWyy_iuyIu_Y_QTNuvRutySIryyxYyyyiwyyyqyyuiyqywgyysIyQYn]XYkdP;igiwy iYwyywaiuuwhGEAsBv[Gfyyxgiy\\yyuYwoyuqxfawyUy=hgmweCwcyyxsyykYxwIxmyyu yypivIyuQyrAYuYyyiyuaxEysatyyyqyyey:savqYbxyQxyipIyqYxoixyuyyAwYymYwcy xeiwlyxyywOvNAu[XlQ@x]ivyYywyyvyyuqxrmyyIfyvQytiywiyyyyyyYhywsyyyy:::: ::::::::nyxuvayvuYyuIxkQr=Yr]Yv]yu]YuQys;Zhqv`yweQw_mlmxnExo]XliXnmwiq wtqxpYxe]wyAytaUJMtSmSJyYoqXmIWeyxjIxumXqAW`qvkIxk]WgeWfqWh;VUw`ZquXAv UauUqtQIuQytQyT:BpkpfaGoPb=WbufHAvZAvUYuUAtJAtGiSghogXoeXokHpkxnaxn]Xn ]HocHoeXoDWcEWcSoePdMgNMcEgKUn[HnM[Ef?M]sjAxjA`@ekG@fRLFA`>aRC@VRXFGhk@Fywi@vZOrm[Or=::::::j[sj:::::::fr?Uk:^RJ\\CkFsCIs>M[Ef?M]CkCp[SkvRIxkfRLFC@ :NRj\\CjfqwhqyxQ;`xpQy_ylZSkEp[Ifyxqypqxdj?hj=Xj@F:::::::;Hj_cvtLYtyqyxyYyyy::::::ZwMwcxZumw;uawmyyIZ xeywAZwyyyQYDyuUYwyYymixyYwixgYwwYxoqxuyx]yvyywmiwyygAv\\Yk;Hjyiywyywq yyiyxeywyyxyy;:uqyqI:yyy::jxSyvyy=:wiywiy:ZwiywiYtYyoY:ZyiyuaytQYqIyyQ xuitayvYyxaykaxlyyyuypiuaynawgyytayAIqu`uDYstQweyyvqYsYycIvYyysyxwyyuI yyYyoqx]yygxvIypYysYyxayyeyxYygyxYypAYmIyyyxvxtyylaxjXn_HuFqSvywyYuAJy cyrytIuQyyyySyfYliaRbiwgiqwhqWiuWiu_ v`QuWiuWivQywj>r:EisOXCj;HjywixE:::;`:=::JYci:::vqyxqy<::JZ;uWiugiwgiy ?vQtUYwYyyy:::::::Z:Zy=::::::::::::::pyyGyyqyqyYiAVyyuxyqxSyVmSHm=Gb;w yxuyuiy;ipqHFKSDgHn]xyuiyyyyxI:::::::sAyiysQyiyy[w=jyyIZxqyxqY:pyyOYwY xyQwCxoix`yuwYyiywYyqawqywAYyYxoYyeiwyIwgIcmnciV`yYyyyyqYyyiyyY:pyygYh IRyvyy=::Ryry:tYyuYy;:::yyyjwiywuYyuyyuyxuyyqyyoyvyywayxUxpyvmYwmyyqYu SywIynYsQu=Isiy_irAij[WfUHj?xyuiypyxuyv]YwUytyyyuyvmYweyyqYwCxuyxmIyoI yvYpywcyFyuQyoyxyYyxqypyxvIxmysIsByrJisyPsAqqAar=As@yrBqrCir>Qs?arHqr@ItCIsMYsEqr;ArEyr@ar?Yr=As@YR LSByr=Uqv`XbOtqXqGPhpwqb`o[asIQrk`p[YtOyrvpqGyrBQs?qrCIsCysKItwPr;iptH qCqsGqrBAsAyRRkE`=YraRAxjDF?@Z=M\\skIp]Sj=XjywiyGjFr;==pZ=VZ=v[ckfR=N[xYiqwywXwixiyxgywwyxmyyuYxyiymqxmIsixmYxeIwyywkyyqYykhl SXkcwgEinexymIyyIyuiymixwIyqIyuyxkiyuYwuIykiyuYxyQwyyxiYymyxyYyeyxgyyq yw=:xixyYJyqyygYyiyxYySyyqx;syyaxlShcqVa[YsIyyqqyeyxiYuCywIyqYycYxkYwu IybxNxoYxwqyvmxuYxmYyuYyuIyyypixQyyqysyyxqX\\Yyqyxyyyyyy;::VyuuxvYyysI ysqyxqAUYuUywiywkIxkIwLYv]YvoxpAwdQw`AwbawfetKItQYtMyuOytYitQiuMitKisG IsCIsyHsAYrAyraiuYiswxqaisKIwAisMhgmWfsvaeHjAxyUIvSyooxv;yrIiomxsoHqMy rCYqqxpgxpohqCyrgHqqHrGYsmhquHomHpyXr=IqsHqeXoeHs?iriXoehocHowXpoHrwhq qxpqhogPdUWeawgiWf]WfUWeCocHoftOeogpdQgNaTRo^THwcOtLenVtMunNTLWd_TDG]` JAT]XnHwcIGDG]`JmayvayF[HnmIAtJUTMUTb]V^aw`qVjGtHaqwHpVpmoxm[ptx@rfatPytsBAsB=j]skZ@qRMXl`FK`Bm]sknRTFI`A=:::Hv[WR\\V^]V_OsBAsBqr@=KHl\\FI` @M];:blOp_ifAyrAir?ukIxkK`@M]SkE@dV_eV^CkCHkC`BAsByRXF:::::::::::K`Bm] sknRTF:UXmHFggggGf[Gf]Wf]WgeWge_mXpmxyyyyy=:::::::<:xI::::::::::::::Ry FyyuwyquSxVyyuxyyyuawmAqr@yn`pnwayvqYvgwaAw^n]YnvZIvuqxqyyyuyZvaYoixoi x^yvaywiywyyyyyy=::::::::::pIqasqYoqyiIuawMyuYywyyqYtIyYyfYJxuYseh[IuT yoa@oyiyveyryEisCIyyYyoyywYvgYwiIyoYxwIyqiysixqQwiyvqYymYyuIyqy;:wavqY tI::vYxIuawmyyI::::::nYnYyyxsix_yvhyxuXjEXWaTPEXdUwyiYyqYuayymyxUIpYyw YyawMtUYuqyywyyqYpIxIwQyoYyixqiykYxyiywIlwgiYGcGgwQYuyIywywYqymiyyYeYy syvqixoyyuYyyQuUYuyYwwytiyyQyvqqg@pwywkyyvqyyqyyeyuyyuIyxYyIYuMYwyipmu yyyYyyuyxoyxyywEyuQwwyoyxyIxmyynixyiyweyuIyoYxmixkyyyyyyyxwyygYyGXbMWu IXmuItOyyiywUY_qvYYb?UgI?dsMf\\yipmxyiIwuYvUyywixcyweIxyIycYyoYvoyweIy mIyyYyyyywyywYyyYywiykYxdYxk[EuTIUKQtvuXryyxuYxYYuyyyuYwyirqywywoaxkqx oQyyiyvUycyweYyyiyuQykyyiy=:jxwywmyyIbyyApIuAwyyuiyaf[KvoSxmyyx>yuaxgA xyiyyayxqyrqYGIuQIu]IvuyybyaYvxYyoyxEyvyyUytYwSxnX\\IqyyyqyyyyyI::ZvaY yyyyYpmXp?ir?IxkqqyhtQYiywiIwcIwkIxTYuUymaxnQxmYxoixo]XnWwcEuOitUytQyt MYtGisHxkIXjEXkAxkIXf]Wf=YquXoExkexhyWvQXnIWYAVSasHUvOytqMWfyXeeWwQXnA xcQwgIWdMXkEXaIWcotOAtPitIitNYsKAtJitNIuUUoixo_hn_Xpixo_Hn[HnWhm]HmUhn [HneXoeHpkPeUgNatNitOunaxn]Xn@G_hnHwcOtLUnZLmbCnar@=:j\\If?Yr==:Z=YRRjAxjHF ::fr?ir?M\\Mf>=Hv[Iv[;E`?=LFC`=m[;CHkLF:::I`RLFEXk@vZwqyxq;Ir;QRJ[SjVRPFC`yxqy@r:Ek>R;Hj:::: :::::::Fr;UjZ:AR;@vqytivrA=sGisghoghsGisgiwoYyuYuUYwYyyy:::::::Z:Zy=:: ::::::::::::piyIyyuwFY`IqqyiyygYpIiWgewLMTLnyyiY[hlQxImcHyYywYy`iyyyy= ::::::::::::::pIqasmwsxfYpIyyyJyuYyuIsixoyysYypyygYJwiYsYxnqTKMTp=XjyY uUyyqiqyXoixlYyuYxyyxmiweyjw_ysiyrayyiyweyuqyqIyyyJycy::ZyyYuaw=jyyI:: :::::wqyqyyWyxyX]yyqqyiyxAVTUUNisHyYwCyuaxyYXYtUyyuirAYyyylIqiysIyyYyv ywuyyyywyytiwiYqIyqiwiqxgxdawfxm[PsvhqyYYyYxqyy^xlUuUYuwiywiynxrAyyeyu yxqYyeiwuyu]Yj[wfEqeyyuayyuyxyixQyryXyiysiydyyOyy=uQyyyywyw=pIuYy^yyyy xuyhyv]yyqyxcyyyxwyq?xl?EDassWwgyIvcyyxxyMxnmumhPYupqxyn]xyuwfyynIxyey uyYxaixuucivkyysIxlYxuYyiywOyoqxvyXyixlYweixUw]IvyiyyYuEisQWb=GnGhkyix sIy]yvwIysIysqyyiyyY:sanIyrQyyqytMxSyyqxoywyyxqyyuyyiyuqyiyy[wmyyInyxq yyqyxyywUx_YwyQyYVXyUKYSGyyueyyiiyayyIynmyuyywyyMtQysaYw]IpixYyyqyuMxC ywAdIoAfYyqy;::Zxqyxayvunap`qv`UYuUYsEYsayvaYuUYuIysIyuYyuQytQYn]XnUXm UXoeXoYxmYXn]XnaxnaXoeXo]Xn]xlQxlEXkExjAxjEXkEXn]XnUWdMwnIXlUyoqxn]wgy XgqwlmVbAUFmsWETMuxeiwkuvb]XgmwoIXlEw`uvfawf]WgewfuwhkemwgmWiuWfeWgWuU AVpwhqwFasF]UV]UeMgRAuRItKItJAT_hnLWdMWdcn_Pb=Wb=G]XnDWcGtKEnZKMbsmYxm WhmWhmvsIysIm`qv`uVa[HM`mV`iFSHmSXmUp`CmSHmQ`DQSO@ZEYSbl:::FsCUlZDQSO` Em_;MXlMXlNSdFRl>sBuk>sBElZAM]Uf@=:Xv]?S\\FI`Am^afFaSAxjHF:::cHoxvayvS IuSmV`oqxToeXooghoWgeWgCHkDWcor@qr;IRIxkTFMXlLV\\gr@arAEk]HmQHoOPa=vZy VWmU[eTP]TBESaUUViVU]ujAWcuV[EfxhqOqtNYqx`qu@rxHr;uj=XjA@::>r:Ar:=J[=V ZEF;HjjZSj:::Vr=ujZsBAsr:Ar=YR=Xj=XiuWiu_;=p E:::::::::::wOYuE:uWipEsGisGivqyxqy<T:@wbosHqsIysIAtJumYxmUXmlV`gsGemWPaSmUP`slVS:lV`[E=::::: :J_clVsEulNSdFMXl\\V^GS`FK`B=Xv]wRJ]UfAM^?sB=`FQ`GisGqr@qr@=::J^]V^uVa uVTMUTYv]WtMEeSGe]Xn]@awfaWn]P\\MV\\gRMXlMhj?hjQxlQHisGiyGjwGmOXlaxmWh l?hjUXj=xqYHnqhkKhg=wbgVZAFqIXlQXhoWqYHnUxjAPYqU_Uv\\Yv\\MV\\EV[MV[EF: :DV[Wr=uj:::^r>ar>=:::j[If==DV[ORj[sj:::::::FsCujApYyuYAF=`:AR;`=M[Sj= @xuY?R;@y?Z=M\\CjnqxDj]b:;B:SdKITpaHoyYywAwiyxyyyyy;:Ryuqyyiy=::::::::::pyyW YhI:uawmyuyyoYxyyuYyoYjy;pyy_yxyysQYsauiyIar@Yqh@pyiywyywqYoyxbx>yuIyt yywIYTYyqYuuINYyyxVYpIuAvyy=wQyuIyyyxI:uawmyyI:::::::VyuuywasQyuqxqMyg yxIyjYxnIwfAsuhqQQSyIywIycquyXrgyxyxjYxyuyuQxiYy_yxqytyYoavMxsy^ywYyyq ywUypytUyymYx?xkYrf@pNYqyxygAxyQYqiweyvcivsyyuIyyYytirExtiwIYyQyEw\\Yv `qUYyYwSyxYwcytyyqyyoyyyyx=jyoyxqyyiyuqYuyywYyyiyyQykxqyxuIsyxuyykYyIh ZYFZKcGMY_=wywWluI^mFW_B@uwZCvykIwyquyyxaymeWfusAqroiWhyYxmyymyxyYxqYy aYwmYweIwqyGYTYxiyvyYyeIByoQy]yvWywIyeiw;qokpssXqyyxrMxyytYyyqyxWytawg YwtyywyXwywyqv;RyxyyyYycyqauyyyiYpIyyyZxyyxiYqYxxyxuYxUyuQyq=ivyyyayyI qehoKAr=Ay`yvyQyuEyuyw_yvyyywIyyYygIyqIwyYycyyjvuyywYuqywyxeylYywyyyyy =::JxmYxyyyyyysmYxm]Xn]XyuYyeYweYv]Yvgiwgiv_QuoxnaXKa`Mo;Iys\\XnGwc=CI sRr=AvuWiucyvaqv[Qv_iuXiw\\avbYuWIvMitgauYAuAQsKujOW`sVkiWh;ynehsmxpei tSymChk=iokHrexo?ipqXpaHochn_xq=YrUhnlsJ=T\\ev^yvdMwc]weqvayvbspqxpuxp qHo_hnghoLWdMG_hnVtMunZLQT:]Xnr:uoixoOhlLUTMucIgOitOIqsHqwgiwwnaxnCIS;Hj;HkFSMHkCPauVa=VZWsCAs; yqxHs?arLisFysAyrJar>itAAsQyrBarnhp?XhnGt;QrGiqxPuFisyHpl@t>aRQxlIXj?h k=hjChjLF:?hjj[SjAxjHFZ;mZ;CHkLF::::AxjHFbj?P[;A`==::::::j^sjvqyxq;mZC j;PZsjNR@vZ[yxQ;@vQ:HFC`:MYue:mZAfyxqyxQtUY[=mZAfwhQwgiDF;Hj;Hj::::::: ::::FR@F;@ZylY;RjNrRtEywiywivyyyY:::::::Z:ryxI::::::::::::::RyVYyY xFYjwsxfYlYy[sIYyiywyyuYyiiv@YpqxxiywoyxuyxI::::::::::::::RyfYhIqAfYpI yiyyyywqwiixqYwA:rxpIpioywjmxyIEIpoXeyFbqiu[ixYYvyQyyixuy`Yx?yxiypMtyi nyYiAyQwwyumwSyry:::jxSyvyy=:::::::oIysyyyqysxmAwQyuivuysyymiu[isqXq;W _ivyciwyyxuIvEyslyyyItYyuyyqYxCyNYwixFyxEywQxyiviyuIxgawyqywyywayxQxly yqQyYAsFYm@Hkkqu[yywiyyyyxyywUtOItwiywiy^ywYyyqywMy_yxat>irHqqyxYuyy`I syyuqyuIyay=sAwQwgytUyrxyQxyixUyyqyqIWivayyqyyXXt]w@iSwTLSLP>HK:qkLXvs \\qy=YqyyuAyyqYvaxnIxyaYuyywaykIWdeS?iRveXpqYrEyyiYxoyyYyvQyjixmYxtyxp qYmQyUyweyyyyyyYyuyyiiuixpExoyyiYx=ImWXiKgdSypwxylywAyrqYvYyyoyiQxjIyy iysyYmyydixayxEyyqyyyyqIqyyhIuawmyyI:bywIxgiwyAtywUYvWyxevcIuqhqEHfawu sxqyYxqyysqyqYxwYmaqyYcyyuIwFyxmygYwyyyQxKywyyyyy=:::JxmiuXquHuSIurAyr QxlQxqyXgiwgyvaaqyxoiXKItTXmOWXHKIt`xnaXoehbmlQH?qjAh\\QVUytQinUhmAiqy hw[YvyXpohqkXpMYr?ioQXm?yomHtqXqahkKha[f^qWeYwsgXpihkMHwAisQipuXscxo=y nehriXpMIsEYtMYtyGkAh[Sv\\WhoehnoXpaXpkxmcxnYXn]xoixosPfQwcIWeMWdEWc?T sBukI`BElZAM]ck:ZAyRK`Bm]skFsCEmSp[Wr==::jc Wtr@qrhr?irGisGqn`pnUXmU`eV_ekHpkPqtPqFasFir?ir;Ir;qr@qr>ar>ISXv]Yv\\I V[UV\\IVbmv_=W`gr:ujwWi[hkGHnGhkexkKhjcWgCFWeuoEhkwWdQGrQXmihjCHmsWiKH j=Xk;XjE`=Ir;Yr;==XjJY=VZuUZ=F::jZAf;=::j[Iv[IFbj?pZAf;=::DfynYxoIxjyyxqy_IuSAq_@oyYxpyYqqxqyyuiyqYxiyMx[xAyyaypyyvyxw ayqayuQywYyyyyxyYjxwymqwjyymiyyIxuyySAvuqsKqyLYtyqxmyY_ixyiwcyyiywwYxy yyqiwYIuyiywyx]YvGHgeWo;hj]Yr?yyuYyJymyumYweiwIxnQxyyyxyyxiyrqxoyyxiyy qywuywAdYxgwiqp_@ojxm\\xylYxyiywyX]yywYyuyyvyxmymYyuYxyyxmiweywQygyyUy :::vyyeysIyyiydYv]yyuYyiYuYivuysYymytPULYT[AUTyyvSypyweywmioQyruyuYxVY lYywYxyyoyxqYuAyA::::mYxlTPmToOwoixoAxjAxuYyuyxqyxnMv]ipqxpIwcIwayvaYw eYwbAWMYtdXoWweYwcUtGisHHCir?YsAqjmwhAYpqxh]wfaWdQwryh_Yv]atIAtdiv`ivZ IvOysJivTauaIuUAoQPmm@n]@y]qv[asIyvJat\\YsIiuDis\\itQyuSQu>ar>qnePoYhn ^xqAqrCItJqtVYuRiuVYtOitKItKqtKQtSyTiXoexnaxn]Xn]XnNtLunaPcEgKITbnFTtJEnvSjbAwbKauVaqv`qVaufIysIyStVagSlV`_S:pv`[F=::j`SmZFM`slQp_; :SHmlFQ`EM_efCISOhldFM`C=\\FK@FsCelVsE=KHlKHlNsDEmSHmGhkTF:::WPaMV\\?q r@qYweYwkIxkr@qrXr=YrDQsDqsHemfSDV[Ev\\QfAyRywiyglOhlAXj;hmOXlaXn\\V]Q vYqUXeV^YVcqV`IVYqUfuVaivYyucIV\\=swpqaauWyvUaufAv[AWXuuaEV\\av[MV^IV \\]FGHkTV\\_r>==xjAXjA@xU\\IvYMF:Fr=UjVr==::EXkPFCHkCHkZ=YRA@:Z>aRC`?i r?iRJ\\Ck:::vRDV[EVZ=fEkE`;IRywiy WjNRJ[SjVRPFC`yxQ;PZMF;HjqprPt>iroIwcyYwyysYysayyyxQYTYxGyyUxyiybxxIhIuawmyyI:::::: ::::ZwmtSxyiynysqywiyyyyyyyxqyxqyyqYyixsyyuyylyyuyt]ykUWfAuKQTqWvyYyxE yxyxSywawqYlIoyyuYx>yyExnyyiyyeyyawYyuUylyu]yyyYyyyxqyuQXm]uKQt`yuZyyw wyvqYmyxsiwiIykay]vyyuQyyiywQxYQvRXj@xysYyyQyuMyoYxIyQxuirqYeiyqywyYyk iyuixyiysyyjxuYxeywaYwyYyaywmyyeyyiyoEyyYiyIxjyywIyxQyoAyyAygaynqwxyx` QxfaywYYwYyyiyvYyyxsIs_xnywe[wnIxkiyu[yywyyyyywax]sQyriyuuywYyBynYuIxm ywmIweqxuiuiydauXYqgpoQYsEyysMxqInYtIyqySytYxuivmxyyyqI::nyweyvyuYYyyy yyyfxmYxyeyqIxoyyqIyyIyNYuHmYXhKwdqXn]xyuYyyiypywsywqyuIyyyyAwyyfYpiyy yy=::::qadPodpyxqyrAYWiuWyxqQv]YvMYtCxTYuOwdQwbEtSIu\\Xn]xnaheYweYWdXo eXmUh^av^iv]Yv_YVAiqyhruhqWYtOYuQytsHqsXquXqCIsCYpmXpsHqshtEis?ipqHtqX qiisMYjWWf;W\\UfqEXl[YomHvixpWYpsXv;yrmXn_HdGgcIWdMgp;yqohr=ymihoQXn[h mahngHpkPfOuOqtRqtPQuRAuPqtPatNatNmdQgPqtPitOitNaTRo^THwcOTDGaxnHG_`Na Tc`Len]XntJEnvSxF:::]Xn]Xn>TS\\FI`AM]ck:^sFasCIsCqr@qrBm^sk>r:ArtPqtxyyyyy=:::::: :R lV`mv^CjnsHElKHlEXkXFMXlXv]Yv]Slfr?irBAsBqr@ej?hj=XjLV\\MV\\Sj=XjExj?x kEHkKhkEhkIP\\Qv[_rQr?YrQs?qrx`qvPrIREXjGXk;P[=F?h j::J\\MV\\MFAxjLV\\_RJ[EV[=VZWRAxjNrRjYsi:::>RiolxytayrAxkUywQxqYwQIuauaysIxlUuWIlkW h=IogxyuQuAYrmYvYIwIyyQwcxZvqYmixkIyuyxxYyyyyuYwiiyiyvExkYuyHrwxmUhq]x mGiruXv_iumiysIxuyxkyxmyypyyiiyQyw=uiyyiyuAaYuWyymyxyYxsylkwhof[QFyWIv yyxuyyuAwQyGyxuxyiyoyyuysSitgiyuiwfxtEwcIwyYywqwqytuWkIVP=UbmuZyyviytY xnqysIyygYnyy]weywIysIyuYyuYyyyyuqyKwyiteyqyyyyxyqxSyvyy=wQyuIxiyey]yt SyykixyYyuYpQHmUGayfvCisyQxyywoyqQYwiyWyuUIypYsYitqyyayxqywmyyI::Nytuy yyyyaZAvZayvuwiywoixxxqyxuYyuIys[nMnSsFwcUsEqlssIQncsFVJiuWiywhqwiqwh= Xiuwjywiywi_UwHpuHqgxpoxomXpiXpmHpkHpuHpmXqehoAiokXv;yrWYpsxowgjKw]aFj Iwd=ilWXtgXpAiokhqgxoyhpqHocpbIwcAWeQWcMWduVb=WcEgOitOQuRAuUIuSItKItIy sIisGUmnsHqsH=pv`wsIysHqsH]Fm_ifE=J`cmWhmYxmxFU`EYsE]Fm_slZDM_;M`CM^]f B=`FM`DQSO@>sBElZCM^ClvR\\FGhkGhk:::vrAUj=XjKHlKXiuWi_hn_HlKPWeUW=Wb=W [EfxpQnsGM`Cj;P^]V^UFwgiwwjAxjSHmLV\\MvfagtPqtXquXqDQSQxlTV]Uv[If?iRGh kGHkCP\\skIp\\QFnR@vZAVZ=f>m[Cjbj=XkAp\\UV\\Mv\\IVZIvZ=v[AvZEVZEv[AVZu uXYV[AvbYv]aw_mVg]v^qvUaUUuSJAt?qr`muY]vWquk]g>Iqu`s>irKIsDisAAs>yq:Ir ;UjZyHr;Arr:eiwOZCj:vqyxqy<::: :j]wr;MYuUYkZsiywiy_;IR=`=M[EfyxqypqxpqsBElFS`F:::UXmUxlQP`Cj;p`qf=YRM`FaS kGhkgb?gbwfawVdMWdKHlTUUUuYyeTQuTYsEelOpXquX=VZ=VGeSG]V^?tJAt:EjU`AyrA ar>ukIP^]f@qRG`Bm]SlMXlK`>m[If@M]sjAp[MV[or>yrBqr@yr?Qr?Yr;araR=xj=:::HF?hjHv[IV\\;=XjxuYGr;Uj>R@v ZORDF:=Xj@F:::MXlMxj>r:ej?hjA`;MZ;Rk^r>Ej;PZsjVR@Fy_>ar>qr@qrRJ[SkZyxQy_=ujBj>R:jYyuYyE;HjZ:>ZMV\\M^lN`lKITMXlxuYyUZsjA PPmdsGiswpqxp;Ir;asFas>M`mfAyrAar>ekvrAekGP]sk>SJ^CkVr=Uknr@ujAP[IvZMv \\IV]Yv\\QV]Mv[MV[Ev[AV[IvZQv[or>ek?p\\=f?ARMXjlV\\Mvcav^]W`mvYutR]rud QcAV\\yVYAvcEv\\ivXyUdav_iv[Mv^IFLv]Ev[Qv[IV[EfyHRyWjj[Wr==:::DF=XjDFA xjAxjFr;IryxQy?Z:mZAf;M[cj>r:Ej:::vRDFywi@f;=vQ@FBkCHkriywiy_WPHMA fVXxm?wbAw`=AAwhqpwh_uXquWyuYAvZAvZmiywiywhqWiaWguwfeWgQWeAW`qvfMwdMxi =XimWhiwfawh]wf]WcIWdav_MxeaWgqvaIxdYwGer@YUMatruwjMxcUWhuVb]w_qVeiv`U WbAWdMWduvcEw`IWcgtNAtLQtHATQXmUHmWhmYxmYHn[payvauVaoSpv`_sFulNsDel:ZF aSUXmUXmVsEYsCISOhldF::M`C=\\V^GS`FK`B=:j_ifD=`FK`Am\\Qv\\QF::::?`whqw @sBAsGisGIsC=sEYsU@mR@auVaubAwbWf]WFGcCGsBAsrZAvZYxmYxkIpSIuSqr@qr=Yr= Qv\\OplPpFasFyrAUkNsDUkEXk?`>m\\sjAxjA`@M\\Mf@ekEXkyq:yqyxqxpqxpQJZCj:::FR>Ry wiRhuWiuWcinQpuX;^QjW;w_ xlXSi::::::nQtE:::Ri^QpEywiyGjnqxdivyyyyy=:::::::<:yyy:::::::::::::::: :::::fYpI:yyy:::::::::::::::jxSyZuyYuaw=::jwsx:fYwYytIyIywiysiydyy_YlI qiysyyyiyyYy`YtSywIycYwyiwgIgGwc_YsIyyoIylyxyyveYvEysuyveyyuYyyyxqyyyy yyIuqyqIyyy:::::::::JyuYyuIryyqxSy:NYwixfYxyycyxIyiyuQyWxkAXwYyyixqqxu ytaxnAuKMT^eUXmyqAyymyx_ywUytYxmYwOYyQxCyyYyxyx]yvYYsMYtQYwaYwyitmyyYy yIUYuUyywiysIwVywitytQywy yyiYyysMYtyyyyiv_ivkiwgIxgiwiywHImywiiwcIwKa_mnSsfVXxm?wbesGAxGwhqhwh_ uXquTyuXyuYyuYMjuWiAXhqWkewgAxfewmAXkAxgmWkqWiuWhmWdUWeUwgewfyWiEweUw` yvaAwayvEASCEylYxquwj_tQispPqUPj@ptyXrcatQiwYIv]AuTqtKQtRAuRytVEomHpex oixneXo]xnaXnaxn_HochncHocHoHwcgtOitNatN]LQT]Xn@G:[HnDWc_tN=]Xn]@[`KIT ]@ZJMb;YxmYxm>tJEnZImacmWPa;YxmYxmZHMa;:SHmSHm:::NtLQtr@qrpixoi]Xn]hsG iseXoe@[B>[CHkCxsIystPqtHtKItKMHlKHdOgdOV]UVeV_eV]UfHMbsoixokGhkwmYxm[ `@qr@AT[hlOhlIxkdV_WsEYSMp^slQxlO`GiSQp_YfAqr@UlMXlM`@M]av]YV_]v_mV_mV _aV`]v]iv]Uv_Uv\\mV^YV`]v]iv^]V[Qv[iU[yuWUv[MvamV]yV`QV^UvdeV`yVXyufMv ]UxbMwEErej?hjG`@=j\\Sk:nRTF:::U`A=TV]Uv]SkEXkCHkJ]SlMP^]f >M\\Cl\\V^SkEp[IfAyrAIsCUkE@Pv\\gRHFCHkLFG@>SHv[Iv[skfRPF:G`?=TFE`?=LF C`=YRA@^RLF:::::::EXk::J[siywiyyyyA:::::::B:vyy=:::::::::::::::::::jxS y::vyy=::::::::::::::::uaw=::hIuAZw=JycypixqYyAwIyyYysiywixtyvuYxeyyqY xyyx]IeiwcyyyiyryoIyyixuioAhkyGeYGuwHryIyuqwmyvMytoyxyYxI:uqyqI:yyy::: ::::JyuYyuI:yawmxcxVYyAuAvyyUyuYyuavqYmiyuixyiysayeyyiyxYwiipAY]yvmgcM wqWXnwyveyyqiypIuaxmyuYwyyxeyyqywmYw]YuQyxiYwwYsatmtmyv]yy?ywayvqyxqyw yyyqxsyyrYyoAv]inOplYYtMyYuyyuIylyx_YRyiIx^ypAwcIrm`pS:j]skZ?yrBirAASZAyRqwhqG[?VZ?B:hqv\\Q `ev^YtMYtIxkIHodTo^iV`IwdUwQAUSItKITLLLLHr:=rVYUUaweUwaqfCyr@yr?EkLVau VaAv[MvYAV[MV]Yv]af=uj=XjHv[IV\\cjNRPv\\gRXv]oRDF;HjLf>aR@vZGr>Qr>M]Cl GXkKHkAhk?P]EvZYv[EV[=vYAV[=V]av]aVbqV[iV^UuZuuVEvYMv^Yv^IV\\AW\\os:Qr Bypt@tw@rcplSPnG@lRYr@ItxHrJyqAs=ar@MZEV[uu[?rsBAs=YRywiyWj @V[Ef:MZcjNRPFCHkioX`mwAxhyyvayyiyviysAYhY vIYryYyoylAxuqwfaqihothoetu[yyoyxyixsQsQYgQV^qUNiTxmxpyiwUywQywywqywYY wiylYywxvQytiyuqyxeyvyyqyyyYuYxkqweyyqYyyYxqywuXrmXi=XOmSImWayvy]iyYY` YxyyvmiviyyiytqyrIxoiw`qmGPkiiUyIxmQxcyuyykYwwivWyysiwpYvqyxYyxqyuyyye YwQYsyyyiypEYrqYymYZyTSYVRMuyyxsyYwuYueXryyvuYxiyxeyyuyvSxvxyiytyyyIyr QyuQyviyyyytyYoauyYqyyjxyiwmyyixqyxiyuuywiyyYyfYsYvayvEysmGcGgxEysyywx YxcxqqyyyvayvmYyOYS\\FrkvR::::jbyVaIuSIUaGtBYsIAqw@r@YsHQr?Is?yr BYsCAspXpiXk@Xj=qqupr=yq?YriryPr@Ar=yr:YrAarBIs:QrRJ[Cjar> QRExjAXkAxj?hj?hj^r>arAukChkGxi=XjyWj=xjEXkdv^Ev]UV[Ef?Ar;ys>qrNYr@qsq XqRJ[SkEXkC`yxQy_ioPXmyIxpyyueywQvqYxiXvUYoyYxahuaxyexVyyyy wuyryyA:jxsxuYymyxqQxmyxyyyqIyiyyqxuyymyxOyvIuEisbxlTXyiywyUyoIweIyiIx iYwcyywaxiyVPmStxmywkiw\\YuIYtEyuQytUyrEywUyuaYtWw`=QYtMYtIysIysPh_iva yv_yvaAv\\QvSYuUitTIuTauVAuTMgWuUYuUMgeWgawfawewtOitOatNUquXqyxqyxnaxn _hnS KHl:::>rxhq@qRkghqwh;xjQTNgt@=cCOgeaWlQHmGHkAhnWXlYXlCXpahmGXjwGk?hjOH mUHjCP]evaMV_yv`=WdAv[MvVYUUIVZiUfDo_@MPHlZiUTMvZaU_QV[eV^Yv]aV_IV]evZ QV_Av\\oRQXjGXk?hjHv[IV\\cj?hj=`>M\\ckGhkEXkEhj?@J\\Sj=P[CknRj[Wrqq:ARwgiMXj?xnExkiHkIHoygjugcOg]mEY[xhwWm swiYhjCHksWiSHkExjwwiAxj^r>Qr>aRFr=YryHR;`:MYoqxlZyuYyuY;nqxdi>RuixuYcjFR@F:;`:=::xEw_whQsGihuWiUYci:::^qv`qvlXquXqE:s_ vlWCisGis_wLYci>Rjyyyy;:::::::<:xI:::::::::::::::::::::::::::::::::::: ::::::::::::::::::JyuyyqirQxmyyuqyyayyyywqyyAysQwWyus`ohLiyyxSyyYxciwy IyuyweYw_Iv[yywiyxyyyI:tyyoYxyy;:jxyyyqIyyy::::::::::waxmyoyxyyxqyyMtk x;syyLYuUYneXQysKeXh=xyeYx]ypyxyiYwwyxIxpaxryyvaxwExyIyuyywyytyxqyxyYy uyxkyuYxCywQygyoYXuyxoiymIxyIysyyoyxcgb=WmqGiyixxywiYxEYs?yryyyqyyqywa yqaxqAxgyywiyehmX`nQHmyaxpyysaYmqq]WhUtAISlIWeyyviYyUyuiytUixiyyUymixk IxhyuuywayyqIlIoyxlyxiixiyyeyyQyqYvaivmwqyuUhDypthsvxqmYv^yylYxwIxiyyv aYuiyyyx>xoyylAyiHe[WmCf\\WiaCglQf]BX_f^uYXmyAylyynywyYyqyyxiYuiylyxCy iivxYxuyxexp]uL]Tdav_YYsIiwqyvqY]yywItmiwSIwaixyYyuyyyQyuyyyyyyI:uqyqI rxyyyqavmwqyxiyxqyyuyyyiy]wAYyQyyysMyTYTNEW]Uvy_yyYywyywiYpyxQytQTJysy yYyYYvUyqMYsqXs=ytQytMYsEyvYyuYytQyoUh_Yv]QwbAwbMn]XnYxmYxlcsKIt=yrAip sHqgxpdWhmwgigTepsPhmGq@>uRAuPqTeXoeXquXqyxqyHocHo_hn_XquXqOhlOHawfaox qTwgewaAwbav_iVauVa]WeUwhYWfiwcMwbMV]uWbEWqiWiaVTYUpQwfAT>mrPMSFqW^ivr QxlEW_evd=waiweUW_Uv\\uV`iVhewfyv`mvbEWbAWc=w`=WaSmMhmShlYXmQ`EqsFQsDu lQP`mfEYSMXl>sBISdFrl:^SlF:Q`EM_efCm]Yv]YFGhkGXkEp]Yv]YF:::oHpixhqwhOH mUxmcxo=v[WV=ab@_eWqUjAHksghmGlCxiYXl?xnUHlKxj;XjywiQXmWHhqwiqwiCHjCXl eGhswjEhkIXkChqqhoUeT?UCyB@AXigGk;xhKXk=P]_r@yrQrarylXqE>RpuXqUXmexpqx@R;hkFRhuWiuZAvZEFwgiJYcj^r>ar;mYyuYk ZciwoYwqxlZCkNRtEw_;MZ;BjvQjY;ri>RIpn`q]xnyqvdyysuyYivYHhugiIGeyYxvXYitMyywQwiIyYyqiysixyAuY xwawiyvyYwyyuYxoqvYYuyYywywqYtiviyyQY]yrEilkWhUxgmwyQYuyIuYyy]YwyYxyyy mYyyywuyy;xkCGAKcuctQyIcKwlMUTAE=ORXgsF=yhiwyqIxyYyuyyuQu=yruyveyyuyxW ytUyuiwWiumXocXh]WfIincxyoyxwIxmiywAyyvayvwiytIpyyuIuyypIqyy`yyuyvoYZu mwqyxaIsIytYyuYtMiyyyxyYsyypytixoeW`uvcuUZuynixyeiyQyvYyjQxn`oghwcAwsQ yrAxoYxmAypYxoixlQxlyxqyxnaxnAw`qv\\Qv\\yvayv_iVMYtMysXXm]XnYxnaxkQxlm WiuWgmWhouXauVauWiuWquXquXmhqwhiwgiwfagRAUixoihoghoqxpqHqsPgeWg=Xj=xgi wgQv]YfFYsJqtOIuXEpoPdMWdUWdMwh]wfIxhuWf=wbEw]avdYv^=yiEXfQV^aW]av[utR yTHysayUZywcIwfqv`Qw_iVkeWgUWauv^Mv[=xgeweIWcMWcAwbEWbqVbuv_=WaevaqV_y v`iVbuv^iv_iv`qVauVaqv`WsEYsDQSO`FasFisGUm:::^sFEmVsE=MXlMHlKHlK`AyrAI sCIsC=::J[=vYUV]Wr?qrq`qyPrBysGWdTG[GN\\X@n[`sDIsJasAYs?qqJQs?ItFIsCyr @isIAtvxq=iq;qrn@qwxdToeOxlRHsAqrE@k>p`lf_iGg\\OsAyqEyrm\\sjVRPf?iRXv]gr?UjZ>=DV[Wr==GhkTFCHkDV[ORHFC`=YR E`DQSIxkIxjAxj?hjTV]UvYGr?yrAIsEYsAyRExjAxk?xjIXj?hkwwisXnaXkkghQXiyW \\kuXmGdQWoCxkcxjTw\\]v`OrFIROhj`v[Mv\\Iv[Av[IVYIfwPr;Yq;ArvPr;qq=Qr;Y r=Yr=ejVr;Ir=HF:::G`XJ=XJ<Shv_iV_ef?ir?iR\\FG@OhlKhlOhjCHkyWj=hjCHkIxkdV^gr:IrSI tLarsXqLar@yl>xjnxm\\hvHQtNyqr@ ir;qRAXkEp\\?s=ar@EkGHkLV\\;fRPF:G`@=::J_eV_Iv[IFLV\\MFDF=XjHFA`EYsEir ?iryxqyhr?ir@M[SkNRXv]YV^CkCP[Ef=YRA`;Ir;]qRAxjPFG`>aRHf@qREp\\]V^GSTV][AyRI `=m\\;MXlMHlKP]UV]k\\ck^RLFGhkGhl>sBAs>aRA`AyRA`@m\\ci;HjowiwWi?Xj=hkE p\\WquXq@IrYrwX rYruXrYrwPr;yq;ujnr;Qr?Yr=ir=Yr=YR:bjNRj[sj:::FsCIs>ar>IRLFC`< mZGrr:UjFRxuYGr;Ir:Ar:Qrr:EjvQxE;Hj;HjFR:RxEwgiwgivyyu y:::::::Z:Zy=::::::::::::::::::::::::::::::::::::::::::::::::::::::pyy WywiysMyqyyuyxyYxmYxuYywyweymyxoYwiyyqiyvxxIxmioYHnfyu[yyqQyyYyxUywyyw iyyiy:vYyAZwuyuauyYqasMw[yiYpIyyy:::::::::::xxrAyxyyyiIBv_ivmyxqyyyyyk iwgyysYyhiTISIg_mFv?IsyIyuQyuYxuYyeyyoxxewNYpyyuixqyyqYyAFxiqwyyyxYYoI xuyxcixmIxwYyeixhYymYymYxgyxavTQuf@nZhxbQwyQyvyyvayyqywuywywaqyqYyuysQ yyIYuyyvmyYmtQeuNmTyiXqyYviisayxiwjipbPoxXpoxyqIyyaYpYyyywmyxiywuYy_Yw YxyiymyyJwgyvyywYyvQyxyyweywYx>xxiykiywAyqYyyIyyyyoqyyywpyyuqYmIxyiocX UQcCIH_OvygitDYmyylqWZItI?r;aunHqwN]XVq`hNoqy]YtyisyyvixyQymyyyyxwIynq yrYXqYxyiweImgGhugdUGv?is`iyeylIwyyyQyyyyyYxuisqYiyywywyiykqxeixayxqyx qYRyNYuixnxsyxyyyxayuyxyiyvayqAYtyyqyx]xgawQySJmXhqwqAXkgywyxgiwyQytIy nixlQXiywHysgv_iv[Iv[iv_iv]Yv]Iv[IvZAvZQv\\Qv^av^Qv\\QvVauVaUxwiWuUYuV aUuXquXq^uVepFuSuoixoqxpqhoghomXp`GeXoexpqxpuXquXnaxnIHmQXlYhmYHoaHnep `EwbqvayVcMWdeWgewcAwbYWcIWeAWcQwa=W`Uv]uWeYWiIwdEw\\YvpiWhEwXyuNIsDit DYseEV\\YxdUWg]v^Iw\\UVguVaQw`qvbqV`qv_ev^mV_Uv_aV]iv^YV`ev]av^WSS`EYS `fBm^clNS:`v^GS\\FIxkIxk^r>ar>m\\SknRXv]wR\\FEXkPF:::?HjyWj=pbIWdIUViu WAV^auYUvZQV_ymZDNQllO@uP]T]=vVAw_Mv[uuWyv`opoxp?qrDAr=yrthq;AsDYsAyRw GiswiuGiQHlCxlKxjAhiqWk?hiExj?xjCpZ=V[uUZAVXyu[mUZAV]]V^=V[EvYyuYAF?`: ARxuYEF=Xj=XknRHv[WRLFAp[yuYyELf>iRPFAxj@vZGRDF;`:M[EV[Qv\\_r>EjvQHFxu YUV\\Mv\\SjAxjuwj?P\\_qXjwmimavfyyiiyyIxvyyvuyoIxTYrYtslp= un\\xymqwyYypyYxIyy?yiytQYlj_hlgcK_xOAuyAyrev]yyoiwyyxeyyuiwyyygyweYuw yygIywIx_yvYYywix_yv]ibufa_xkMXy_YwyyxwqxwYwyxqyxyyyyyxuYyoQyqXsAYu]Yv qyyyYxmYxyyxqyyoYJyqyyuyyyiy=:yyyRysyywamYx>apoHri@pyywjyytYytawhYyoqx paxnixmYxmIxkiweYw_Yv]QvZAv^Qv\\av\\Qv_Yv]qv^aVvuYyuWiUs`VmhqgWMiugXmh guVMgeWg]Wf?UTWeUwdQwdYweYWiuWi]Wf]WdMWduVb=w`EwbuvcEwaMwcuvcEw`EwbqVc AwbIwcQwdQWb=WbEWb=WcCoYHncxmjeyVbaWa=Whmv`mXgiWhYv^EUHuSOQsEqVVeUlyvb iw_mveaV_uV\\Mvcuv`Aw`GsDAsBisEirDAs>IsAAsDQsDasFisGUmVSQxl^sFasHqsHAs DukMP^]fAyRI@ZBM^;:::UXmU@GhkGxkIxkQxlQxjAPXmUXYv]YV^Uv\\eV_wq;Qr>IsFi q=Qs=IsJQsHYtIHlL`\\LF\\YWeP_s@yqyHqlHtFIsKysHapmhpDYsIyq=ar:ar?Qr>ir= YRChj?Hk?XjIhk?p^Qv]MVZQf:ir=QRVrIr==GHlKXjAxj=XjLV\\gr?ir=M[Ck Cp[sknr@ej?pZ;E`Yr==Ep\\]FKhkLV[Efar ?YR?`x@r:Qq:yqpHry@q:yqqxqxPq:yqwxQ=xiygkwwiMxhuWmkwhUHgkgmawgyVZGfoqw i_xhywloWiIHiwGkyGj;PZ]UWeUXAVZauYuuVuuXmuZ=vYAfyxqyHrvhqyhqwXr;IR=@:: ::::J]UV]AvZwQFr;QrRLF?hj?hiwOXmeyxqyHR@F w_whQyoYIv[GRpuXqUXme:ARuWipE:::wgitE:::ywiywi:::fqwhqw<:::IRj]av^EV\\MV[k\\gRHv[ORLV\\Mv \\Qv\\YFGhkDF?`>=HF=XjTFI`=M[;ChkDV\\MvZCjVRAhkG`?irISwIGhyYvkyysixyixcyyAYqOu>]ru;WayYuGyy wywyYysQxqiuiyyywmyyXqvmhgnOhUv]w_^afpd?gVg\\LvnVOeyqtNyysaxyayqQytaxv yyuAyyayrEyqiwiqveyYMUUevYAvuyXraysMyyqyy]ytYYxYYvqiuyyyyyyMy_yyqyxMyu YxmYyuYyuIyyywYytYwSxyiygiywIwyYygyysIsMysyyyxYxmyyMYu]yl]X]uTRmVTUupm wiyYtUyyWyjQweaw^ivqawhAxbIwWytRAw\\Yvcav_yv_iv_Yv]MjuWiuwhqwhqGqxplWh mWhSquPhmgTQUkHpXweGv[Iv>aRQxldR?erhqwhQwdQWiuWiiv_iVb=WbyvbOtLQtIItKq sJAtHAtJQtLQtNatNysIysI=:@wayvbCoWxmMYr=InCHkIXhoWpMhl?inaXmqGigWbAGei V`mxkKxkmghQyqyHoSHmsGhiwoeHo]xmWXkGHkOhlOxlQP`mfFM_efCm_iv_K`clZBASIx kTV]oR::::j\\Qv\\EWc_r>aRNsDEj;p_iv_quXqUK=tIyqyxQOmtPQV_qVUmu[QsGEtd` OitMZ@NcUjvQ@F?hj@F:C`AyREXkEXj=P[=f@a r>armsy=YvyYtQyyEx iiYYeurXKEXYO]tyMYqyyyeYxYYvyyx_Y>yuuuiIpeFaEHWiupKv[uIfWwyIhkMwJCTh=U RyYtEyysYwqYw[YyuywkyyoQxaYyuItyyqiTysKapcXo^auWyyvyyyQywUywYyiawiyyqy xqxnuywqyuimYXwiytIyyywYyyqwwysMukyWxvIysyYgyweIw[yuyIwgQtUieiw]qsJap[ `n`AuTyyuiyyAytaxautwHrQIr?yuIItQyrCitCYsMIsEisCIs=YrtWiuwhqgXquX=>vZA vYyuYiuWEqsPgSpmXpaxnaXj=PxlQxtVaWtMYtSIuSiuWUlMpcIgNatNitOitNatNunaxn apdQgQytQItKItK=::HwbAwcyVamweUwgAwbAw\\QveaV_QxdUweQV]iUM]TK=sBiw^eV] ]uVUW_evlqwhYV[gsCAsCyRUXmtVauvayvauVagsGisFasFisGisH=lFQxl`v^aV^ClvrA uk>sBEl:::NrHrxTQZiUVyv`mV_eV_MV]Yv[If>M\\sjApZAfsBASFRLFEXkHf=QrRywij[IFZ>ar>]yxqyhqw <;@nqxR=Pv ]YVZQV\\=f>Qr>EkCp\\k\\QV^UV]iv\\UV`Av[aVXuUdYv^]W_mVEQR>MuKUt\\EuTUxd ]WbEV\\AW\\UVaIv\\]V[Iv\\Qv\\UV^]vYMv[YUXiU\\UV]UV];::fr?Uk:::NSPF?`?M ]UV]Iv[Or= ::j\\Sk:::^RLF:::::::K`>mZcj^RLF=`xpQyyyxI:::::::B:yay=::::::::::::::: :::::::::::::::::::::::::::::::::::::::::jxsxyQwyYycy_ywgixyiyuarQytIx kuwiqyqYyyYxyyxUyvyywuisiygqu\\InNplyIyuYymaxyeypyyuIwiytIjyqiyyyy=::: ::::jxSy::::vyy=::cyweIyyYyoyywyvmIxSyv[IwmywsyyuyytiuYykYwfyyuqYciwkg dSgkmwhyIyyyyuayyywmxsxqyxuYyuiywiywYyyqxyyxeYxuyxyYyuix=lYyuyxSyvyxqy yqYxiyqyyyyqyxyqxpYrm`pPqrAyyxYymqwgexqQy_xiAxpaxnuywqyuiwqYsiy[irCYaY F^=iokxymyxyIwgyyoIyyYxsymwWj[wa?WraXoyyxbyuYwgAdYxuiyixpQyyQyyyYqimWf `]D>mROUr>ydQrDAsAytLI tWYtMAuDYsNQr=Ity@r[asHaoQHmihmXPt>irOIsCquNatLQsDysDYsFaSS`Em^afAM_]V ^av^OSdFMXl\\V^?S:`FM@:::ftOitIryhqv@sBAs?yrAar>UkGHkEX kLV\\Wr=uj^r@yrBQS?p\\EFCHkLF:GHkAhkCxjMXlLV\\ORPv\\oRGHkCp[SkEp\\SlMX lI`=m[ckGhkG`ar ;Yr=Ir=Yr:QR;hj?XjAxj?HkCHkCpZmeBQr=QsyHrHArsBsEF]yiv YyyqyvyIy]yy\\XrER;mRHlk_As?Ux^EvymXnyYtEysexmyiwmy=Yr=yysxmYxyYyyqwuY rQXnOyxauEqsIqq;awYIvyIyuUwgyywYyvykYxmqxxiynyywIyyqyteyuAyQyuYyoYmYwc QyqYimwgMwdIWlYxlyWlEXvyYxeyxiyyaywQymaxyUyvexfmwL=sCIVQAUxaXpMxcUWpyw jiXjEXnExkQXjAXjuWiiwgiwg;ixoiHpkHpk`SIuSqtRAuNqtPatNatMYtMYpohpbPodht Oit>aRAyrAykIxkYYuUij;HjahmYHoYHneHn]`JYTHWayvd=wbQwcIwaqv`yv`qv`qFVsG isE=`Vaqv\\mv^qVdAw`EWbYV^UV^Mv[Aw^avdav^yvZEVk=wb]w[QVgQv]=uGqsJASCyv ZEVgmv`avYyUa]v^qv_iv`iv_mV_ev]Yf@qRG`BAsBir?ujE`?EkCp\\Qv\\UFTV]av]Yv \\Qv\\QF:::KP^UTMUt\\QfEYsEiQuwkIxk=`BAsByr?arn`pnhqy@rFItNQq:irnPqyPr @YssXqvXqs@qvxpnXs>IrOqsEisBiruHqr`R\\UWeUZ=f;Qr:Ir;Ejnqxtiywi;hjAxjEh ksGj=hi;Xj;HjDVZ?r:=?pYEVZyu\\Qf:Ar:qQ@f;QRjYye;mZsjFr;IrxLYSjNRxuYyUY uem[IFnQDFuWilUX_QhuWgquXqx`qvxqwtiuWi=x iyGj;PXuUYiuXqUVqUX]UYquVyUYiUZyUY=f@aRIxi;xhWGfOXhsGsDwHUR?AUFmS]ytSA WXyUcAV\\UvWqUY]uVuUYuuYAvZaUYqexpqxpQ:::sOXmE:::=XjlUXOQlUXgqwhqtPqtH qsHquthpuXyuYyUYchoOWSipuXshqwhoghowiywi=pZquX[wLXchNQJXcinQ`uVauVSiuO Wch:::VquXqu<::JXmUXmE:::o_t<::JXSifQlEmWhmGhkGhiwgiWhmWhyyyyyy::::::: Z:vYxI::::::::::::::::uyypIyyywyyjyyI:nYtyxyiwmyyI:::::::::::::::::::: ::::::::::::nyxqyxmyyyyyIuauyYoQyeIgIyqqyuI\\yxGyuYyxqyxeyuQyqIyQxqyvE ytyYwqirIyuAxlQswxq@YqwxyvyYuaw=tiyey:xyyyIRyfY:jyyI:::fYpI::yyy::ZxyY wQyoywyyyixtAysqyvayxqytayvAyuIYuyxoyydImiyuIwmIxsiywIxgiwyYyxYwwykqxF iptxt>yrpAwdyyuuytYyaYwyyyyyxswsyxkYysQyqYxuiwmyyixqYwAVyxaytYysyYwyxg ywpIqIy\\ywyyvwynqxFYqxPswxqnawgqytIyvQysqyxEywiysABysyyuiy`yvyYsIY`YU WQVRIuyEytyYrMyy]YxyyuCsKYjOgeEHfagwEItyiwoyysiyeYyuiwwqxiiyyyxyywQYsY xyYw]yyiyvxYvayxYyryywAYymysMYvyyuuyyAYwIvuDryhJFLsJhly]V^yYjewyYyqyY] uU>ALIxUihoyAynyYoIXQAs@mW[EvyeywgYyIyuyyyqyiYxoyxyywiywAysMyyyu]yyiyx EYlUx]eUXuvZIVx=YryyxqyyeywwyyqxxiynyYbxtyyyyyyItYycytyxYysWuSAuIAtHAx qaxpyyvquaavhYxkQylYxuqwjqykYxyQxnqvMqtnxm\\pp]pncAuUawXAvfyu[YWXxi=Xk mwhuwfeWiOuVQuTQuRAuTQuTQU\\WgeweawfQweYwcQgLatNYlOhl@ASUxmYXr=YrohpoX n]XngHocxpmXpeHn]ho]peEwcUwbgtJun]XnihnXwd?tHqsJqsHqsH=ShmlFblYhmWxoch jQHl;xkCxkShlUhmSHmXv`]v]]Wauveiv_ywa=whiV`]W]YVWetOmS@qRV]tNQxe]Wdav_ Aw_OtFisIQsEYsCIsAukKHlXv^av]av^UV^]v]SlMP`eV_qv_iVbiV`yv`qV_]V^]V^OSK P^OS>SO@KpLItKqqxpqVauVmV`_RCXn]PXmUXefDYtKAtuXqu@SQXi;Hkmgi=XlWXnuGjC hi;XjOHlKHmKxkWXlKXhcWfSHlIxj=HjCHkCXjAP\\MV\\Mv\\IV\\Mf=M[sjVr?ekKXl= xjCXjAHkA@AxjAxjj\\Or?YraR=Xj=Hkfr?ujAP[CkCP\\skIP]cj?P[SkEXkE` m[siywiy_RpEwgipuX?r:Ar=mZA fwhQu_:mXSi:::nQtE:::ywiywi:::fQpE:::s_yxqyHRqrAIr=ar;IRAxjAXkHv\\[Rlev`QxE; `ylYcinQxeyxQir=itMQtMysHar:yq>AryXrxxqDAsBysGisuppqHmNhlThlPxplXpDir@U ggGlAHkUxlQXlIxkA`@]@qr:QrEkAxjAxiyoYsjVRtLQtFqsHyrFYs@asEyrGasBqsGI sHelSHmYxmYxn]Xn[hlQXnMhlsxm]hpShmgxjGXpCxkGHf_GaGF\\QhgkWqQHmUxi;XpYx m[HlKHmKxkMhlKhlUxlSxmUXkKhkwWjywjGHkKxlMxkKp^afDQSSHmShisGiOGdKwf]WfU xlQHqohpWHmlV_ev^Yv]eFMXl\\V^oRGxjHV]orAyRKP^]Fbk^r>ujAP\\Ck:::Z?iRE`> aRA`=M]clOhlCHkCHk:::VsEUkNrM]or>M\\cknR\\fByrAujAp[cknRHFC`?M]or> M[EV[EFrjApZGr:m[If@=aR?`xpqxpQ@F=`whQywiyGkC HkC`:MYuUY=F=`:mYyuYyE=XjLV\\Mv[ciwgiyoYkZCj>RxEy_:AR;`;mYyuYyE;`:=jZS j>R]P@XKGiNdXwEmsyyWiyyvMyyYypyikyyyIXwIvyIy_ysqXoKGakFqWxmyqx ?xqyysiYHys=xkIS:IR]]UWyYxuipuywiyyqxmImIweiymawEwWyuyyxRy_ypyhhAwcKyo oxysaxqyvMsCxwQwyivqY]weWgvWYusixmIv]yuYiv[iveyvkyxmIweyvmywgywaivMYr= IsohpQyqyXuyHr[irAIwCYssIuUycmf`Uwa;GtsHq[IsCYtyxqwHochswpkiwguwf]wjqW hIwbwuZquMqtNAtOUng`OYtJYTLWcYGgxoeXn_HnaxnaXoepd;axnaHocPeUWe]Wf]WcEW cEG::[hn_HnftOitLQTUxmYhlYhmSXn[Xm_XnUhn]hm[Hn[HnPwcIWbiV`yv^eVhAwcqWb EWhiVaExaEWf=V[QWWqUJ]R?eULUTk=wbqw`uVlUweivZAVcuv`UWdIW_iv^AV\\Ev\\eV ^Ev^UV[aV]uvdEW`AWbYv^afCEmSP_OsJAtJysGisv`Q_Hn[XlIxkExjAHmOhlS`AyREXk \\fBIsCelNShv_iV_eV_av^?sBuk>sBEl:::ZCm^ClnRXFOhlOXmUXmK`B=::JaosBAsBi r?ukIxkM`AyrAyR`FK`EYSbkGhkG`CISM`@qRI`Bm^ClnRLV\\oRXFCHkCHknRdFKHlKHk Vr=ukXv]ckGP\\SkEp\\CkCP]skZ@=Pv\\gRLFC@Nr>ej:^r>ekCHkMxjCxlAPfiv`uVZE VaiUYAX`yvA]QxpUMeTpQwfqVWqUbMV\\iv[IV`MV\\eV]QV^Mv[IvZ=V[IvZMv\\Iv[UV \\IV]MFPV];:NsDQs?ir?QRLV\\gRHv[Iv[ckGP\\_RfRDF=XjLV\\;VRAxk>SJ[Ef>m\\ sj^r>ujHv\\Qv\\afCyRHF?hj?hkGhkE`>M[sjVRDFC`?m[;A`=M\\;E`@qr@aRLF:::A` ==::j\\SkZ>=DF;@yoyyyy;:::::::tJunap`qv`qF::QXmU@[Hn[XmUP_efAIShv_eV`mV_mV`av_ifDem WhmOHlKhlEpbiV`Aw^eVbIv\\]w]evfIv\\AX]avk]v_yTDYsTqsIyw^iVamuXmWayv`Iv [QWbyVZuuXyU[=vgExiYVcyVQaUUIVaiVYMv[uu[EV]]V^UV^]vYAvZMV]UV`mV`Iv\\QV _ef=YRA`EYSShlOHk?hjAXj=hlKHlC`>=:Hv[Iv[cjVr=Yr==:::Lf>aRHF=Xj=hjDv\\C lVRHF:::KP^EV[wqyxqr:EjFRDF;PZyuY yuZGr?iRA`yxqypqxdjDV[Sj>r:Ej:FRDFRj>RxEy?ZwxQowiwghygiBiwgisgitUYOrwp q=iqx@rrHqQQsFqs:QrxhojhvLitSpjCHnHHlIaqyPt:IrJAr;ATlVZ=v\\quXuUWauXiU WyUZuUYAvYmuZyUXcinQRtEywitE=`>aR?P[uUYuUYSj>RxEbi:vQr:EjZytJen_hnQxlhF::MxlQXlnsHqsEYSKHl\\FMp^ ]V_eV^eV_YF`fHqsHUlMHlGhkYhlQxlEhkKHj=HnAXkop]uW]]viYv^]x`=WSYSGmSIR@FBj;PZ;ywiyGj;Hj=Xj@F:::bj?P[Sj @VZCjVRAhkFR@F:::Ep\\yEuWiuwiywi;`xpqxpQRheuXQtEw_wtJAtHqsHisEYsFQSUxlQP `mV`;QxlQXn]XnSHmSp^osFasFyrByrQr=Hv[ORJ]Cl\\v[sj :::vRIXjnqxTjNrar>]yxqydiDF?`:MZSjVRAhjFr;Ej;Hjrj>RxuYOr R@F?`RM\\sjHV[Efqr RyolxyhIxyQy vyyvyyxAysexqQu]YgYV_MtAEss=xkyyvwYnYwamIxjyyxiyyYyteyuIvaYvYYwcIxuiym iyyixfyvuyqavUxaaviYuiy]IsGIhIgdKxfgwyUIvyYwkQxuyyYiYItMqnO@mWasIyyvay yyyxyyyqyxyYyqyuixeyuyywYypiyqxriuDisg@mTxyfAxyYwgyytQyoAxiiysAymIxjyx rqxlYxkqxqixrIyqQv]IvUauTqvaivUIuRiv[Av]yuXavZyu`IvZQvWauWitOiuMYt]itP IvIAttav`qgROeUpjBhwRQu\\AtKiuGqsZQtMiuMYtWitOysDIsNATehocXnchnihpkhm] xmWhmxvawSJaufHMb=gLQTSHmSHm:::>tJAtEYSOHlKxlMXl`V_eV^]V_]V^iFYxmY@Exk IXkIxkShmWxkMXlwwj?xjKxkugj;P_]Vbuv`=w_evaQv\\Iw[MveEV\\YwZIVHUR>YrrHQ ?Qqu\\UMYTi=gQisGyryxqDir>As=ar?Yr=qRGxkMXlAHlI`;iq=QrIsBqrB EkGhkChj?PZ_ry@r?Ar;arUkEp\\ckGp]Yf@qr@ir?ir?m[Wr>aRC@Z?M^?sEYSCHkC@E@: nRTFOP_IF;`=ujC`m\\skXv\\Qf;Ir;IRLV\\MFHF;`?iRI`=mZAvZIv[WR@FEXk HFEXkExkfR@F=`@qRCP\\Iv[OraRGhkGhj?hj;`aR=`==HFA`?M\\ClKp]wr>M\\SkEP]sjN rM[EV[IFC`;mZCj;p[ck^RmZSjZtJAtLQtLasFasF=::Ja=WbYV`iV\\aV^QV_aV^ClKHlO`FQsDisEUlIxk=XjPv]Yv_ =waIv^]VV=vYqU^UVW]V\\iu^QVXYV\\euZuuYEVZeV^Yvcuv`]WauvhyVb=WZAVgQv]UV REUJ]R?=uD]ShAV\\=X^eVeAV\\uVZEVbIv\\Iw_mv`YV^EVZ=vYAf?QsCyq>Yr:ir>YRE xjEXkEp[Qv[IV^IV\\af?ir=ujA@:::J\\CkVRHF?hj@F?hjHFRj>r:Ar:m[SkEP\\;A`< M[SjFRr:Ar= =J[;BkNRtUYuUYcjZxQryPr:ir=QrAQr;Qsqr`pkJ@mw?jGQpoxu@QS;xjMhgmgjcggIxi;HmIHlIxktUZ=VXAfwPr;Ir=M[Wr;uj=P ]AV[YV[Iv[Af;IR:::bj?hj?@Z;UjyoYyE;Hj@F;HjxeyhQtUYGRHv[IF@FBjZyLYuexL \\MV\\KXcivqydinQRtEy?>RDF;PZmUX_QEc_v^Yyq;yysYyuXn]ht;Ir[ItKipgXo?IryHqyXqQsCIrAqr=IsBAsD Qs?yrAAs@qrEIsCYsCIsSytQysGisxpQOHmShnihomwiww`;waSuUUe\\Uv\\UGfWWjChj IhlXV_]fBqr?qr>QsByr?Ary@t@um=hjQXhowqMxl]Xhqg_qeYowaAgouWj;ylWxmwWjGX hqwmIHlUHlMhhoghyWj=xiCxj?HkCxjAxj?`>QrRHFriyw iwgiHFE@vQ@F=`yLZciFRHv[GRpew`qvTjnQtEuWitE:ywixEuWipEs_wqr@qtPqt`QsGiNQoWhZulyyyyyI:::::::B:yay=: ::::::::::::::::wyyjyyIuauyY[YxsySYeU_eVCeywIuyyycyyyIxyyysax]y=:::::: :::::::::::::::::::::::::::::::jxSy:::::vyy=:wiywiwgiwxiyqYciuYqxuyyiY yiYtYyyQYvmXiEXA=R;yywwxfqwyeylinywkyyyyyy=:jxyI:vyy=:uawmyyI:::::::^y xEywiy::wiyJxyivyywAytQyuiywqyxqyyyYwyyuyywYypiqQxmyywyyyYyxQw\\YvcpmZ HwXAvyaYyYyxYyuyyqIuQyuYywYuqykvuyxaiwYywiYJyuyyqYyuiwiywqyyiyyqyw]mqW yyywyyyawiyyiYyixsYj[wfmgcMWySyuyYyuqyuYoixn]ywayteyveyy_YxyywYfYpYyWy uaswXq[hkFxx\\IvyAytyyuYyyiyxeyyyxmawqy:AqsPp]xn\\qsMaybiwyIywuxwIxgAP YuMYsMYtmYyuYwoytmvuyxaIoqxKxeyvWytyywqyyiYQiq;yi]wfKXgkwykyx\\YxGY`im Qv_quCasDipqxykQxyIyviqfxorYwgywayVyiyqHqohyyYyoIv[iwKYt]wa;GfwvaMYpoH uwxqMYquHwOitCIryXryhquHrwhq=yqkXqqXngHo]`LytOqsMItQauTIr>QrUauTYsCAsj YwdQs:upYxmwhn_Htoxp?hfawbaF_ggcIwraHo]Yr?IkkGh_HmUHnWpbyva=Wauv`qfGma yvaAwbWShF::UxlQxlQ`KAt=YsDQrByr>UlIXlMXkIxkMxkIxlMXlGHkChlKHlAXj=xiyo [Qv\\IUVYuC]sEyq:AR\\GrLqsGqtLItHasEir@ar=]Cyr@YsCIqvPqHysGasByr=Yqups RjYSj=p\\Qv\\]fB Ar:Ar;IR=@:NrQjXqENqtThm?hEyyyyyy:::::::Z:Zy=: :::::::::::::::tYyuyyyyyyI:sauyyyaxuqyjohowhsggyyyiyYciyJxyyyuyysy:::: ::::::::::::::::::::::::::::::::::fyyUy:::::vyy=:uavqYgayqYyYywIy\\ivq yyYywyyrYYqqvmXq=TCMsqEXlyYtUYxAYtyYx;xI:pIuauyyyYYpIuayyY:uawmyyI:::: :::^YwIywiy::wiyjt]Yumyyqixqyv=XyvayveYvwyxexqyyuQymYxyYyuInxqQsyHr:qp pxyvMxqIfYyiywqygYdixuy`YyyywoywiYyiywQyqIsyyqixtyxmYw_yryXwiyuyywywsi xeYysywsyxyYxsyykIypXsutH=tZMeyQytyyyqywqyvaYuawiywiyvayvqyxqyxmyoyyQY dyw?xiYtuXqx@odHycQwyAysyyxqypAyrYyuYyyqyyyytqy]isLanGPlLaq;yysuyqYyyY ywawyYwixoixkixoYxwYykiyuAeixkIyyYyyQyYytwyyyyyqyxyywmvGy`Iw]pkKxvPQuy IYTYwyYwwyhIxtivaynLhljYuXypZ`nyQxoaylQx^Qv[qxyayTYvZywlAxqYxlyyoixRqr AApUhmNqq:yx[YvditQYwWqu`auVQvXiuYAvXIUoHnexnWxn]XngHoWXnYxooHpgXpixps hpqHqoHkExj=yqwxmQhlOXkCxoYxmqxmYHpOxlkgcIgaMv\\GXfagv;Yr`v_MW`qVcmv`A W`qvaev_uV`osHqsJAtJQSdF:BmOPaaV_mV__sHqs=IS=hkExj:GhkKhkGhlKHlQXlvRYX mHwcIWSYUUIR>YRAISCUV^_r:yqEqr?isByrCir>IraRGHkPv[_rr:UjFRj\\CjfQxEbiwOYSj=P[;=`v`Qqwhqr@qppppxpqxpsHqsxyyyyy=:::::::xnixgOhj?hkwfyyylyyyaxyqynMwyyy[y=::::::::: ::::::::::::::::::::::::::::::::::::::::tYyOyrAYmixpiyqxrIsr`qKAr>yypQ YtyyYYvyiwqyvExkIxyyylyyqYvyYx]ImQyayyOYlYy_yyUyyiyxyxyixyYxIuawmyyI:: :::::::::::RysaviyvAyxyYwyyuQyqYwEYsEYmMXKEsCaxfewyeYxWyvyYwiyyQteyvYy yuyuyyyuxsMYwuyy]YweyyyyyyixQylQxyqyuyyxQyvYyqyyxiywYytYyyyymYytQxxqyp qyxayuyyyAywyysyyyqxtquGAt:iojpuGAtyayvyyxiyyqYuqwyyxyyyqyyqYwmYyqyyyy yyixiyuaxkyyuYyxAxkarlXpQr=AXjDVZ=V\\Ef>QrR;HjZyxqyhQugiFRHv[IVZSj=pZcj?P[sjVRLV\\MV]Ufar>Yr=uiyOYuEVRJZcjVRHF=pZyEBjNR@FA`?m[Wrwhqw`QxuYyUXuUYaUYOqvD huOWmEVqwhqtxqx`qxth;xi;hj?Hj?@jXyuYuEuWitEAXj=pZWr;Ir=MZuUYyuXqewdinQ xuYyuYCj>RtEmgiuWhyOWSiyoYye>yq:yry@RyWjGXgkWpGXlkwbGg]iuXQg^gfp=Xkwxk QxloWiQXlKXgoGhoWiqwi;P]EvZIVYquWIuSMVWeU]uUYYV[Av\\AVZmuXiuUmUWaUZquU =vXQUYiUUiUWyuZAv[IfylWiev`QwgiwWivqydi>R;hjDVZ=ftlVGqvlYsinQlev`Q:jXS i:::vpqxpn`pnHqsHq>ar>aR`uV_pn`pqxpqxPdUWeuXSiNqtDhkGhk_sHQqwhqGh>Qdet `Qs?s_vlyyyyyI:::::::B:wayyyyyY::::::wYyyiyuqyqyywyyqyyqYVYyyx:::waxyy wyYyYyry:ZxuyxIbyRy>f\\WfO?ePOiw_hlaxbqyvIxyyyoyyyQYlyyyiy=::::::::::: ::::::::::::::::::::::::::::::::::JyuYyuIyay=pIuAvyyMy?xeawwixpyytqyQq rDIpWPntIwgyyuiyyeynYyQsIyrmyyqYwyipiytIyyqyseyoyydIsyypixyYwyyjyyIuaw myyI:::::::::::::byuYydYwiyuqYyeYyyyxyYyqyyiYwoxmyyqAy:qoixsxHryyxsuwi avIyxiyjqxoIyxyyqayyqxxqyqiyvYxhyWeYwyyxmYwKisyiymYysIxuyxoyxsixqyyyYw uIyciyXyyWyvUysyyyyxyQwqyyQyvqXgqwbGr?ippxYoYx_yuyYywQyyyvUYvyyxyyymYy QYsMyyyYyoYXIuiydi^atOqrpxpNqrAiw[IvyaYyywmYxWIvyyxwqvqyy=YtAX^mVNISEM xcUwyaywyYxqyxQYuYysEiyyypIyqiyyqyyyyxYyvQYsqyoywyyx]yMy]yv_xpexuIrYhn mwdSgvGisyIywQxuiumrMYaUUVAvNitFaR?yyuYymIXk]ywayl=ypuxuMynaxnEYoixxex pyYp=Ys=WeeSulqeMUVMyd]wpYwfmYpqxeyvamWfYWeYwdUweQWiewfeWeQweQWdOuRqtU IuJytNQsMAtCQtIasMATaXnchn]Hp[HnwHn]hqSXmchi;xkYWf;G\\OvpExkGYnaxsehoy xnaxlAxjeXn[pfYWcIwbyVbuVayv`Iwb=wcAWbwsHQuLQt[IuSauQAuZquXIsAyrvPqtPv ZAvFQsDIryxqQitOQsBAsMYtMysKItZQv\\qomPpHYtLqsJAtDasFQsDQsHqsHAtHqsFQs DAsBAsCIsCasDQsDASOhlOP_efD=QXlMxlMp`]v^uV^av_AvZUW_evaAV[eVWiU@eqwDvP ytieV`Axa=w_iUXaw`qV_EfBar=qtQitu@rxPokpoHitLQrAqrYr=EkCP]UfCISGhkTF:IxkIhkZAM`mV`UV]Uv[Wr?M]U FVRPFBlKHlIxkIxjVRTFG`>ar>aRTFK@fr?Ek^RPv\\oRPv\\?sBelvrAej?P[SkEXk=xj AhiAhjwwj?Xi?Xjugj=hiAhjw_y`r=Ar?ar:Ek=hkEXjGp[Qv\\Iv\\Or>ar=jZIv[Av[IFJ[_r;M]MfBYr>qs@yr>ypspt=qrHyPKW]]FgcV`whiAXrKHm ?hhoGn]xmKhkEXkAhj;HiqWjswhmXn]xnQxlQxjAhlGXkIXkCxi?HjsWjywhCXjsWk?Xh? xiqGjHv\\?SLV\\?R;XjNR@FAp[EV[Ev\\skIP\\MfyxQy_vZ?vhtGhyyypAxhaVyywyyyoyyly ywyy=:::::::::::::::::::::::::::::::::::::::::::::JyuYyuIyaymxSy:vyyuw iywyiywyyuiyiiu]yytYxqxrQUL]trIxlQyoiXxUyuyYysywYywIyiixkIyyYyfYNYwixy iysayeyyqyyI::::uawmyyI::::::::::JyqIqIyoIyuyxwyyuyytyyuyxyyyOyiIxyQyv quGAtipm[`wZAvyiywaytQyyyyyixnqxlqxqAxxiymqyvyyviyyAytaxOqtyAyrexkyxoi wsyy`YymiyiysyyxeydYxqyqEYuYYvmyxaYyyYxwyuqYtYvmXpywbEwPiSGQykExymYxyy wwyxqywQyvyysayyayxAytQytyyxqyxqyvyxriyoixyiywyyuiyLar@qqaXouQv`yyoAyy QywyyvqywixpyyuuyaYwmyuYyp]Xn;g^aVscxoyixxYwmYvAYtgywiywUxmQyyyxmIoaia xmqxsyxyyyxyYnywyxmQXtYrUXmiuOmTlivayYsUyyQyuyyxuy\\auVIxeYwy]YwAUJIt \\ytRAygqwyqXr]YomxyqyxyxoeXdEgUIuRitGisfquYYsmhpe`kIHwBasxyu\\ItuhqNp jA`uQqtKysHatOYtIQtJItLAtPytOytRqtMatLQtJysLAtIItGasKisFEmQXnSxlixmYxp axnohmWHoCXk]xhuw`AV[=XdQwqOHmqP`iwayv^AfPItJaSMHlMxkKXlI@CXkApZ=V^Mv[ uV_avZAvZauVaUdMWdAwbWr=ukIPXmUXAwbAW]UV]yuYyu[QfDasFIsEUjAxjChkGxkMP] clOhlO`@qr@qr>ukEXkKhkGhkfr?Uk:nRGhk>s@elEpeEwb=VUUucYv]=W[IvfYV^YsuivQDFwGj;hifQyGiwOXuewxqypq:Arxl YAfyHRwGj;hi:vQxE;`:mYyeyRUrxxqvXYvuxyyytyxpQxyyYwcyvyy=::::::::::::::::::::::::::::: ::::::::::::::::JyuiwiywqyxqyymxwYVyyuxNyvaywiyw=wiyyIyxixexwyvCItGW`q vsshqyIyuivUiuyYyfyvuywyywYyyyywqyuYxgYpyysynyxqYvyyyyy=::jxSyvyy=:::: ::::::nyyeybyyYyyYysQyqYyuIwiypYyiysMyyiyxYynixYIuSqv[IvyYivqxqEyuiywq qmYwYyvayyEytUTyqr@iqytgmWhKhl`uWiUXuUYiv`qv[Iv[YfAElHv[Iv[Af>QrmWYUUYWcOs:ArDqqxht?qrLAr;anLXlWXkFXqcPoOir@It?ir:qppXrxhqCir>u iwgiyWiyoYmEqWiuOXCipEwOYyuYwqwtiuWi:::uWitEw_whqw`QJZ=f=mYyuYuexxQjXC i^Qj[IfwhqwPqtDisoX;o_vaRC`?mZyUYIvZ=v_Uv\\uV]UvVqtPYw^evgaV_]UMYtFYR>yu PutbAV[Aw[MvcYv]gqwpqppp?Ary`rAsumyyYYxyYrUywUYxyYuiYyayxuyxy iyyY:::::uarYysqxoIxyUxxyuqYyQyyyywcyyay=::::::::::::::::::::::::::::: :::::::::::::::::tYyuYysyyqxSyVY`YywxvYytyyxyYxISYtQyyuyynymaWyYy^ywIw ]avIxjAxysQYtiwqysYxneywQywyxiyyeyuQymyyOyyexyiyuqyqIyyy:::::::::::::: pIiIyoYxwIyyyywyytixqyyiywqyvaywiYcYxiixyYyuyyxyycxqQtSvayymixlYvyyu=y ywxyav\\ajqVaLpbDwy_ivyYyrUw_yysixyIv_YukhpyiyyyyyyylYymyviywqYyEyuUyr YYuayyuywwyvyYyYysqyiYsyxoyYyqinqXmIymhjIxbSV^wYu]yyiyxLytoYoyy`xr=Yvu yxeIyaqixouxsyyqqyayxyYviym=WduTHqsfyvamyuUivuyiYxoyu]yyuyyyYyuQsIyyAy s]VWmUXuTRYYomxyiyxqytUYu=yryyyyYxuYyAyuUYvuyxqxsEyxyYyuyysxoyyyiysiwi QxuyvixpaTFismaWgqyrAyyqyxyysQihQxvqveuuayx;IsIG^cfhIgd_IsGip_xnixn_ht KysshpmXpcxnEYpmX`IF\\[YokhbAf[]g]_fqDVk=wbuWcEWiMWdyweYWgQWdEWaqvaiV_ yv`mV`GsGqsFisHasEqsFQsIisGQtJEnj_yuYUXdQwc?r`AtLIu=irUhjAhoQHmTirAqTI hnGhkShj?Xk;xiFr?As@IrBirxhrarBAsBIsCEkLV[EV[MV\\MV\\cjVr=uj^r>Ek:Z?McQw cAUS=U[=vYIVYqUhMWdquRAUeUv]uW`qv\\IUTus>iRQmsHAwY=v]YUVavWmUeefBqqxxr ;Qr=IRAXj=HjNr:Ar;mZcj?hj=pZAF:::ywiyGjFR@F;`R;XiuOYuUYkYciFR=HkNRpe whQ@FwGj;XiyoXyevdis_wxQu_x@RwGjpEu_vLXSifQtEy?wOYSiuoXcinQxE;`:=jYciZ ylZyuY=VYuexDiNqydhyoVuespQ=hkEhhsGiwGiPvY=V^qeFqq:Atw@Riwhwwa?G`iuXYh feGkWGfGHhmwnWXmwvauFkQHliGjuwg?xioWjwwgsghhuVAvXmu[uuXyUWauX]uVAvYyU[ ciNqtthqoXSi^qvthqOYuUYAFyoYeetPQtex`qv`qvlWshZvlXquXqEs_v<::JV]eoLWeU WyuYyuYchfP\\UVWpmXpsHqs@r:Ar;MXmeppppPplPpll?ir?UR=UREUSEEUSwqyxqsThk GhDV[wYyyy:::::::Z:^yxqywyyyyyy=::::ZxyYqyy`yyWywyysiyxQywyysqyxAyyqyt ypn@pn>`i>xhiwyyyxiYhyyOyxyYwqxyiyyY:::::uyylyyWytqxyyysmu]YvEyyyywcyy qyyI::::::::::::::::::::::::::::::::::::::::::::::wiywiyZyyYuawmwcxyiy myywixyQvqYxqyyyyyyIRy^yxiyuYyyqyxyywqyyiY;g_gfoMxlyYwiQwqyyqYyyYyuYyu iyyyxiYwIyyYyoatqyyuypIyyy::::::::::::::pIkiy`YyqyxqyxyYyuyyqIuQyoyvqy xyXwiumywiYysYynyuiyyQywewqyyiixlytyYraxymXoAwA=sjDkDtyXyuyeYvyYyaiwQY cyvcXdSw\\AUSuXn]Xv_IvyIywyyqYylIyyysyyRycyxkYucYvkYxiyywYyxioywjyyviy qawiyYuQwuYXUtOMVQIuy]ixYxpyyuUyyQtiYvuxuMYtyyxUYy?ytQyyqyxIymmuayyiyx ?ytyw\\av]PlOpvKatyAyuyytaypqwiyyxiykiwfyyyqyyqywyxiyw:apnXqgpoxAxkyyu iYmix[ytSyyyyyeIxgqyqyu]YuQYuMYyyyxuyyqyvaixayvyytYYmyxaga=gueXpuIsKIw =Issyu]YyeiwuYweItwhqhYxaXdUWPurCyvQMutewhaXdUWp=XjAxfaWNITKMW_evqqWiA Yeawg]UXIT:MRcQUVEWXuUg]v^=XdMwgUwd=wbyv^uV`aVbqvZev]Mv`ev]uv_Uv_aV_qv _yVbuVcuv`yV_aVfuVaivYyUjEwcYw]]vaeUXiw\\YvZitPUTBES`YuVIw[MvfiV`Qw_iV aaV^uV`iv^mV_Uv_WprxpFGdJOrAqrvHRShn]hjKxkwGkAxjMHlIHmQxmcxnAXkaRLMTLeTOetRAuR]UV]uXqewLVcf_gfyyyyyy:::::::Z:ZwmyyI:::::tiyY Y`iyIYZxMyuYvMYtuxpexl=lH\\K^Yv[yYyoyyiyyyyxiyyUyvyy=:::::iYxmYxmQyyIh YxaYtyyyeYyyywyixyYyyy::::::::::::::::::::::::::::::::::::::::::::::ny xqyxmyyiymxSyVyyexyiymQyaIiqwYYxiYxuYymYxmyxmYyyyywYwQxiyyyyyyYxqyyqyx aVXuUbEV\\yysQiviYtixqyxqYyQymIoQyeIuawmyyI:::::::::::::ZwMvkyuyxqIuYy yyxyqxoywyysqywQxwYylyywqXHYvYYyqyyyYwqyuiwjYWUeuyEypyytuxymuV=QJxK]QR RxEu_vlW CinQxEy_xLXshnQtErivQqr@ipohpyPQWhkIhkmwhoHkGHm]oW=tKyru`QJeR@]W]YvbAV[=VTMu\\quXQVZyu[uu XquV]uVeUVmuXiUXquWaUW]UV]UVyuYyuZSiNqtth^QdUWeUYCinqxpq;MYuer@Qe_sXqu pQpew`qvthqOWchFqsdhoghq_u<::jUYuU=VZ?ob@owgiwgI\\TN]TZ=fqlQytQuUYOr]ohpoxyyyyy=:::::::iRSqsI=XayvcUV^QW_mVb]fx@qsP tFisEqRcxm[hj;P^UV]UV]_r@qr>M[YV]Av]UV[Yfar@ekCHkj\\gr;IR=`>aRC`m\\QFNRar;mZ;GHkCxj=P[Ef?yrAAsDQs>IsBqq>YruHr:irEQs Ir=ujAP\\EV[QvY=v^quXaw^eViav_IvPytPQsEuUM]teyuZeW[MV_QuU UW\\Qv`yUZeV[Ev]QV\\QV^UVXIVZMv_]V\\UFKHlI`;IRwgi@FC`>M_eV_AvZAv\\Qv\\ iv_iV]UV]QFGhk@FIxkIHjFR\\V^]V]CisGimWhmwiywiIxkIxjApZGr?m\\cggoGisGUR =WkEXkwgiwWj=XjaxnaHr;UjnQLV\\MV\\;AxjHF?`<=J]Uf?m[Iv[AFC`@mYyuYUUUUuy yyyyI:::::::B:sqymixuyyyy:::::tyyOyyewyiygqyQyyyyuwytIxoywtyyuqxtixaAw ^Y`ln_GisAyyyuyyYxyyypyyayyoYyqy;:::::nYtI:pYymYxmYxqyxwyyuyZwiyw]y=:: ::::::::::::::::::::::::::::::::::::::::JyuYysyyAfyyUyZuyYoQyeIqau=pIu Anyyiyy]xqywYYyiywyyuiy`yvAykMXNisHaWa=wymixQyseyuYyqixkIxkIxFysIyvayv ]xqyxuyxIrwiywuyyyYyvyy=:::::::::RyFywqysiyxiYyiywiypyyuiyiyyyysyyxAxw YylmraYviyyyyy[wyyPauEPcFWtuWiKYhhGj`N^yIrsxylQwtYykqyyYyxUymIweyyqqym yxeYsEyyyYwOyqYupTnCwcgheYwy]IvyIxiyuAYrpYwuyyiyn=yp=YvYyvyYyiIsiyyyys QyiyxyyxsyyApYyOyviwRYu\\hkJxvPIuyQyvyytYyyyyyywoaxkqyvywwYygQyrqxyayv yywQytQyqaxpyytaYrx\\XgiWN]sFIw\\UvyiyxQyq=yyiYxwyxIyqqxnywjyywqYmyxEi m_xk[Gg[hjCxyqqymiyIytUynywqxnYxyyyxyyyqYwqsiXpYYomXu]xneVYwodHoiiwfYw cAwXauUqxkAxrqwhIuCEtaXo]iomhyyhrmhgowdKF]Uim]Xu_Xo=ym[xoMhlGiqwHlwgiI YoghpChksxkMhiSWeaV[GVfMWdoXocXkAhjKXlIHlMxkMP^]wfYw]QV\\yuWeugYWe=wby VUqUWaU\\AVZaV]YV`eV`ev^avZAVa=vZuW`qvcquY]XbEWgMV]=SwlQBeQx\\w[Qvjslq GiKXiywo]hnWXlOXiqwh?hj?XjAxjqGjyGkOP[]v]Ev]Uv[CkBk\\v[Mv^IV\\aV\\QV]M V\\UV]wRXFEXkE@A`>aRA`<];Uj>RR:HFbiwgiuWiqrdiwgiC`tPqtXpmXpqxpqXQPv\\QVSEUS]UV]U\\MflPpl`qv`qv<::J TMUTQuTwpqxpsHqsXquXqwhqwxQxOJDLttlQ@sdPoyMxjyYxaixIyyyy taxoYxwyyxYYyYxNyyUynyweyuQxuYymixyyyEwYiuyIyxYxwsuDeef`yIu[qouxyyyyUY y_xxqylyywawoQxhixlIyxiytyyxIxpixveymywiyyuiyyyxuInAXkSw`yfvCisyyxxYwi iwqYuQu=yr_ywexsyyciwAyoihhSgeoXlSxygIxhyxuYvaiwqytixpqwcQwyiywyxrexuy xIitKYtEIsMxhsggGwcyixsIy]yvcYtQYyeiwEIryxtAyruxm[xtgHpEYmYxu_XoAG[Ivh wVbmyrEyqYHnGHisgqghoaXlMxqaxnIYpohtoxp_hjAXcef_IW_gFoCXkwHn]xoQxloHn[ hh_gfOXkCXtIis\\V]muZyu[uv_EVbmv\\=g@isDir@ar?Ir:At@qrWysJYssPqJYqwxqc XoKYqwpq`@oUXj?XrgxoYQsFIvJYtEqq:qruhqDar?ary@rx`qvpq:ArwPr;Ir@ir:=Df< ararrpPExjApZGqsHqs<::J[EV[=f:iQdEmWh\\UV]UUUUUMUTMuXqu XYuU_pn`psHQ;hj?p]UVX?rnHqrHr@eiAhjmgiuWgoWhaGhHUVYUTaUVUuWeu`EWbqUYmu ZmuW=W\\MVaauWEvL]TUIsEet Q\\EeWg\\UVGqsHqqxpqxPheuXQ\\UV]uyyyyyI:::::::B:yyyyyy:::::nyxqyyyYyAt yyyYvuXpEXDAs;uqwlPEMS;iqvXpt@qilraRfqwhqtlXquX=VZ gpohpuXqupPaqtPqNZq ^awffkWhkgiw_jC@kLYtKYtLqsEQs@qv_IVyYyoYyqyxwIyyyyxI:::::tiyqYwiytIuyy wYyvyyyyy=:tYywYxI::::::::::::::::::::::::::::::::::::::::::::::::uAyA ::jxiYwmYwaiyiyyeyxyxqYymYxqywyyyuywaYvyYyqirAypQwdIp]`n^AuRyytmxuirIy rqxpYyuYyyyyyyYlYxmyvayvyyyyyy;:::::::::::tyyoYwQyiyyGYwYxyQwsxyQwmyxu yxyyyqyyWyyayqIyyYyxyytiyoitPifg>_yQtEyyqiWkyuyyyoauayxyyxiY\\yyyiyEx> xoyyciwyawieuWyy]yuyyuWyt_WfCuDOSxWxlyIxeYywixaIymytgyvmyyjxCywyyuyyJw yYyaIKyv]iwwYyqyxqQxqyvuXryysQIuysaHpyf\\UFyMIuyyxuiw]ivsiywIxwYy[Ywai xyYyoyydIeixdyyuYxgyuIyrQynixyUyaYsIigAgcUHgkwyMYuhyxyYxwyxuv]YvyYywyy uiytyymYtMYjMWdMTEYS`aUXyyvmirAyyqYSweUgrAYryYxoywOIuoYsKyyQyuaw`uvw?I ssIuUYuAyrqhoeHs?YrsXqqhrEyrqHqoXruHqohn_HthWmMwdyrvhqWItLEX`uVbMfiyva aw\\QvKYR_XiyGqChk;yjGxrCxkdtPQSwpQceUXaXdGT?hq_hnaxlQP`ivZEVZ?s@ytTAu QQuRAquHqW@nXXkC@kpOhkwlLHlAQr;iruXqLar?qt@yrKaqxXtx@ROF]OW^efqEXleHjA hj]GgmHmWHnKXloWgggmSHmChkGxj^RHFIhjXFKhj`v[Mv\\SkPf@ek^r>arZ:Faqv`qgHqWhmghko TIuSuUXmURuTQgqw`qvDi:::ZtlVaesLWeet<\\UV]uRIuSuUZ=v\\Yv]iuXquRAuREUSE uXqe>qr@Apj@poxpqhqyxq:Qr[ByQXlyyxkiyyQsqYwAyyiyvyYymYyuyxmYyqYxwyxy ywyYqqx]YyqYtyyxIXqmxxyyy]YuUyyqIuiy\\ywGywyyrYynxm\\pqepoyQxnaypyxyqy wMy]yvYYq=YrmYyuYwcx^yyiyv]tQYuiyhyxyyvgxXQvy@ofHvJEwIYtyYy^ywUywivUiu tiyqyyuwiyyeYwYiq=Ycef_UxjPWbqViUWeuXlMXryWkOywYy^AwyYwiItuhqYQsFyypyx lqv`IxcIwdAwaiq:qqNYuRItRitIaTEisCInQhl?IoextgHpugbEwdaV_yxm[xmAxjohmh xhuwlIWdEX`uVoyVcUx_uvaMuUexaEwGIrm[WrsBAsdPodXpmTn]Xn;Hj\\UV_qvD]SF]wEYse j;HjaWf]whmWhQTLMTr:Ar:MYCiluXci^ppppoxpq`pppPvqyTkEXkwgixtQyTWeetqrDhfpoDhF QXEi_ulWChvyyyyy=:::::::apaHoTQtMYs@yr@qr@ir?ir>M]MV\\]v[?s=UlPV\\Mf>q r@yrAyrAm\\Qf=m\\If@ar>YRDFAXjHvZAVZCjFr;Ir;m[MV[IV\\EV[IF?HjwWjtuZ[wP r:aqyhqu@rxdj;hi>quHqnxpo@rxhqKIsCisthqJIqwxu@QsKAqvPrexoIPiw_rk`pMQr? YswxqBiqyprxxQFRywiy?puYyUY=VZuUZ[wxqyxqyDj;p[AvZMV[wqyxqylZye;mYyE:Fr ;EjvqytiZ:MZ;rjA@tUYuuXCjFRtUY?R;HjFR@F;`ylYciwOUUUU]Of\\Ojpj@pqxPwgixuYWquThVQtE@vZauWiUREUSEUTMuZAvZE VZ=vKAtJqqwhQ;=R:=RytQylQxhqwhq:Ar:AsBAsMYtMMUU_rAirrXqs\\pxpo@qsxp=Ir :yrurumvbyisOyyQytyypoxcOTLgi_gf`I TK]VPctyeIvyIx_YywixiyywivsIymiyyayyY^YJyuixuytyxoysaytaxuAyryyvAxqaxu qyvexoqxwYyYycywyYxuyy_ywmhjGxlgGhUYr=qxqyyiYwoywMyyyxqixUyqixoyxmiyyY ysyylIayvayywyyyYyuQsIYdyuZqUNetyuxryIyixuyyqiyUytQYyyiymiysixuyxyQyQx d_wKatiAtMatqPqqxm[@pUhmf`mWHuGisyiywyvWiulQwdIvVYutohogv]QvMatLIv^QvN ytOQuWYuMItJyv]QvRAtJYuIys`qtQYu@AsVpjBHrfpoqiv`qTDW]UVpEXkYwayveqv`iw ayvd?s`atOiuAIsfHlNXoHHlViq:ywHQtVyqs@irBqr?=HV^OrAir?QsBqrEIs?QsB]CasDAsCyr@Qr;It>q rTIsFisthqYasIAvEqs@IpnxlqrIar@Ys?qrBqRGhkLV]Uv\\ev ^IV^Gr@ir=ir?yrAyrAm^[@As@=OXkGHl>STV];Vr=Yr>ar>aRHF?hjDFCHkChkTV\\MV \\k\\Qf;mZCkCHkC`[p`pnHryxqAQRCxidtLUt=Dj;TqrDqYiUW]V\\IV] EvZuV_av\\Iv[Iv[;::>sBAsEYSKHl`uVauJAtJqr@qRarxdi;Hj;hkGhkCP\\MFA`=M[cj:nR Pv\\WR@vZAV\\ckvqydgjyyI:::::::B:yyyyiywiyyyyyyy::::nYjyqiyyyymxQodPOA tJAPyuYixqYyqysy::JyyixmyyI::::::::uyyyYyvyy=uawmyyI:::::::::::::::::: :::::::::::::::::::::::::::::::::::Zwmx;:yyy:tYycywyyRywAsyu]IewFbYhi? xyoiy`YwmYweiyEytywuYxeyyqYwcx>YqixyYyuyydYxmyysynYwqywY::jxSy::vyy=Ry VYhIuayyYtiyqYbywAxyywY:wqyqYxmivYywayxyywyYoiyxyxSu_iXMTN?fKWtyQIvyIx iyykyvyyv[Isef^kvESs\\OcCaIiiOsmxyqYuayyeYsyyxAyyuyrayvoyyyyuMwyyymiyq YxyyaYw]yyyyyqIuaumwSyvyyMySyuQxqiuuyqiy\\YxEYlUxbIv\\]UOmtyiIqYynYuaw iysIYyQyiYy[smvuyxeYxmIwyytitQyyIXiGbCGrIxlyyu_yyqYy^yvqYoasYyuyyyqyQQ tKquLQtqyvbyysyxciuUaw\\QvkIwcaxiAxLisHIymaxnqwiqlJHlaHn\\xv[QvMqsIytO itPqtPytQytMiTUhn]hlYhmQXmUhnSXmIYogXp=xjMW^aVhCgckHlMHoKHlixmYXmKxkeX n[HlAhjaxlQXoQxlYxjAHmsWi_Xiyw_oUYKHbEwrowiSYlUHrEHl]Xj?hkwgi=Hk?xk[Xm igk?hgIxjkgk?hhAHjTV\\qv^]v^Mv[AvVaudqv`yv[IVeMv\\aw[QVZITMEt:QrR=SDew X=vdIV\\EW^]v^QV\\meuPr?YrvHry@poXpvXr:yp:iqtXr:qq?QrwPr:qQLV[Av[yUY=v Weu\\=vYQVZyEJ[=vYEF;Xjr:ei;pWquXiuXqElUY_qxTiywiuwixuY gqwtiuWiyghqWjmghOXiygmuGjEHfaggyvbovZIfnqwigXjChmuGjugfcwjugiuwhqGiwg ikwisGh=himgjyghyWilewXqt`r:UlCxjAhiuwjwWi=HiqgimGhmwggghkwgqGiogiyWiO hlOxiywikGhkghoghsGisWhFQlUXmUTcjVr=ehoghbhdUUShmWhrfawfYXn\\UWeuUaUWQ uVeUU]uVMUVaUSMuTQoe\\oGhkGptPqTYmuWYV\\=vWYUUyuXmUVeuWiUYEVViuYQTNmtO qTQMUUYURytRAvYyUW]UVatMUT<yqx`rypqHIsBIrvXqppohXsyTo;hjeg ^gFhYF_WXdUgqygj]HiwgmCHk_HnYxkShlwwkChjUhl?xlKhh=higWhiWiqghEXj;HlCxj MxkGHjqghMXiuwqUpnAwcQWViUUYSG=t:MraYuV_qw`rr@qJYsDqrAirrhqu@sGYsrHrxp py`qupr=QrCqr:ir=iq:qq;yQ=XisGjdu[uuXIFEXiHVYOrxhq:qqwpqyhqxxQuWiu?bif qwhqwLXmewlViuWauW[t`qvPqv`quhqwXQ`UYquVyesxqxthqGioghCHilVY=V_Oqir;ir@Iqv@rrXqyPppxp[gf`w\\Qv\\fho ePrxhq?QrxhqrhpyhqvXqv@rhPpqHogPp>AsGqPuGi;XjMHlOxhqOYmUXIr:=rCxJAHQrl pWYUSyv_UvYaUTQsBmRFUsDySJusOitOIuSIETEmGhoGiqWikGhkGgcGgIwcIGasFaIwcI whqwhuoXEUSOqtPqt<::::::J\\Efyhqw`qtPquXQsGiXuVaUU]UVOQwgidv\\UV[YuVqV XyUbauXiUJMtTYSH=UCYSbQUVAwViUmMgvpoiHrvXqEasDyq@Yr`HphPpwPqjXqr`pwPqt Hrxhq:mWeuXaUVyuWeUWOqrlVShZtLWshFqsHqsQ\\Em_s<::::::JXshqOWeesL V]erqq;yQ=xi?x j=@Yhn[XlOHl=HitUUGs;IrNQsDisy@r:YogplsskKwd?ykOhiSWe=XhTv]QvM]TMIV^Gq x`q:yr>AsEIs>yr?IrQR>Mry\\Qtxpu@QpXsCuRRu tPIUTEUWeev`qvPqtPqr@qr@qqHqs@qtDilEo_tlYyuYEV[gqwhqn`pn`QsGi::::::nqv `qxl^Mv\\IVYyUYmUXqewpQwWhqOUiUWiuXquWiuWeV^]vXQUUIVU]ueMv]]wZQv_EUU=u DasDqqy\\W[MVXYtNivXuUYUeuHqrPpoXpphqt@qy`qnhqt@QpU[wq[B>;R:wb>_R>Or;qCHcswgivwyyuyyvY:::::::::::j xyiwmyyIfYpI::yyy::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::JymyweYuQiwqyyaywIyfqwZPlN`vPAuyQyvyyuiyoiwhyywqYyiy]Iw_ixyYyiyyw IucIwqyyyYyfYjyyiwmx;::::yyy::::::jxsxyyymiy`YyOyyqytyYiQyYIoayYYyYvgi qOWfuuGwcnOF^;YdSGwwgiCgQytUSsDYyimwyeyt`YrYvQaT]QUSyysIyymyxYYsMyvQYw yYxyivyYwyyuyylYyuyvuyxIytQyxyyyyyy;fYpI:wiyryyiyxYyuYyiyxmyviYyaywuit YyoqXqYy_isGydyvaQxjAxywYyyixqqxuYsuxpmYwayyyYyyyyuYxqiviYyQyyIsAvyuay yQyveuYymoGiEhfeGyWIvyYyuixgYwwyyuiwiYwIItGipkxo_iuUIuOYtKitOYrGYsUhn` SIqShuWiiv`qvdQwdUgPAuPqTFuQuoepcAwb=Wauvaqv`AW`qV_Uv\\MwauVmmWhYvWiuj AgNUjUVWiemgGh_hhsGowGjsgdQwWMeTuWgewjygiAYr;ihgWgGHiswoMhl=ym[HhAWc]v Y;viUgeOIpmhn;XjUXiwGmwwiQXj=hlEHkKXlIhkMxk?XkAxh;hiowiPV_]fEIsxHryHqx `q>yr?qsDIsNqrAQtqPqqplTxmxOjMAquPuBEkoghihn]xhiggwGiqwj=HjExj?hj;xi?h iuWk=HjGhjPf:ar:yqRjZSj>RxEwOYqEu_vlXSinqxpqxlX;sOYmEmgiughygiq Wiu?;hhqwksgiOXiyWmyWjmHlQXjWWf]hi=h`If\\oV\\SfkkghQhi;hjiWh=XitUXiUZm uW=F?`>AryprZ>>] TV]:>Z:F\\Iv[nN`knkExj=ArvxquppHQs?EjqWikWg=xiwWj?HjDf;mXquXeUWeuV]UWe uViuYyuYmUXWQtUYuuYyuYKWchVQhE::::::uwhpEIhjAXjsWisghlEs_rPqtppuDhuGia WgeWlMXlyghqGicWgOHiwwmwWjcxirfqWcIgZSeU]HiwGn;XjMxiywfWWe?hiPV]MVWquW ]uWauVmUW?rxxQhUXeUXeuVuUXiUZuUYqEs_uLWch:^QlEmwhqWhhUWShmoV;o_t<::jWs h:::NQdEq_u<:jyyyyyI:::::::B:xiy=tYywyyyyy=:::wyyyiywyyuYvYyuUhkMhMIdL GB:=ReMGdqYxkyyyiyyqyyI::::::::::::uaw=::jyyIfYpI::yyy:::::::::::::::: ::::::::::::::::::::::::::::::::::::::jxSy^YtYyoxnaxyUyyIyuqtYyRucfytT yyiQxyIyuyyviyvyxrIytAyqaytqxxiypqYmyywyxyyyqAfYhIqAfYpI:yyy::::::::jx sxvxyqynyywixyyyoyY]YyuYwyiyuqyyYxqyyaywOxqQvfHpN`dTGeXN^FW\\PvkSGemyr @yysYxyyxgyywiXsiwhWfqs@uryIytyywuyyeiyIYgIveyyqyysiy`ivqxwQyryyveywYy yay=uawmwwyuMveYwmYyuyyyixqyxqYjyuiyMyaYwyYyqivMwGsyxql@obHygiwuIxkyyw qyyYYqywgyyyqy_YsIxsYytyyqYwgylyxoyYwQyqYxUyuyyviYwaXpEvQEUZqTRyyviyyi yxywe]wt=yquXpiXkIxjQXnUXlaXnMXoaxZUv\\Uwgew^uv`aWiqW\\UV]awgiwcIwcUWd MWeMWdGtLatHysKYsFQtDYsIQsEIsDAs:Yr;YtKAt[IuSyrq@qXAsDqwKatMPhpOoChkI@ hmgmxj^irBau>qr QQr>atQRyGk?XiGxjygkAHiEhjmGlCXhOhkuwlxU]IvYqUXAvUYuiyVbI wXyuZEtLeswtQ]YTOEX_mvWiTPmWb=w^yuYavYyU]mUX=VV]UZiUWIvYuU\\=vYyuXmUYy uXuuYqE::J[=VZAvYyUYuexlXCilUWOqtLXmev<::jU;s_v\\tDiiGhyWhowjqGiugfcgm =xjMxhuGkaggIg_mF\\ieXIhgmWlqWiMXiuwkuWiGHiswjqwh=HiqWjlU\\=vYMV[AVZwq xxqw`qwDiuwhwwiuwi;hiwwiuGiNrUquxuQ=U]]uVavXquZYuUuUUUUYaUViuUUUW]ex`QsWiTuV YuU]UUaUW]uWiuWeEm_rlUYeq<\\UV]uWiENQ\\erHQJVsgvP:::jVSh:::vpqxprLWeUW eE:m_rlyyyyyI:::::::B:ry:nyxqyyyyyyY:::wYyyiysiywixwyyEItSyVgeX;R:?ryy yyyA:::::::::::::fYpI:::::qaumxSyvyy=::::::::::::::::::::::::::::::::: :::::::::::::::::::::uawMwyYyeIwQyUIryoqyeyxaVTYu^UuVyyrIyyayxyyu]yyqY yuyyqYoyXqayyuyvyIVYhI::uaw=jyyI::::::::fYhIiamyyxIxyqyiey]aoyyxmyyYyq YxYiyixqyy_IwyivbyyJYtyAv[yyhqwyAypyyxyxyewlYtyywUyyUysyytQYPysJMXeeWr =xmyyvyyyeiyayyyxqyxsyytyxxIyqqysIyxaywyyyuyjxSyVYhIsiypIyyyJyuix=jyyY yyyySywYyiIxyixoyuEYssgeLW^wyomxwywiyuqxoiysAyyeyxinixmixtAysqyvQYuiwm ywyyyyyyuiyoYuYyykyxqItQIemFacXjCxygYxqyuYIrqhpkiweYrCirKitOqkIxpyxqyw jAXfeWgAwcIWgyWi=WcEWdUWeYweYwfYgSatOytKQtPqsIytFqsMisHysNQtJytNqq;UmK pniWhuw^ivcUuVMwSYuAmPrDwRQUkQV^IwXyuiyVbuV\\Mv_YV]Mv]Qv`UwcqU]IV[aV]] V]Qv^quXuW^evdUuVusx@r[MtMmvQ?u:QrXIsEatBAsAQr;Ir>Qr=QsAisPYtuHs?yo:aq lXryxp?IruPry`r?YrJYsDiqmXp]itPyuCYsCqokXn>xj`xjDxtyHr@Ipmpt>UoEhk]xjC Hm;HjKHj;HlAP]EvZuuYquZMfy`rRhjFpkIxndpok`prHr>AsW@n^Xqy@rgHplx^av^OodNoplHpwPqp@ru@quxPuw i;xgogiYgfiWgoWiQge_wAACC;dJATiuWi?hjxuX_pjxokpodHuNItlpocppkpoFIs?qqv XQ^qvDifqwTinqxpqxlWiuWiEu_:Ar:yqyxqvLXciwgiu_w<:::::jYquXyEIhjAhjugiu whtEw_t`qv@qw`qu@ryHqxhq;qr?ypsHq:ARMHkxuU]u\\eeDiqyprpxpEIqtPrjHpqwfb op`xnKIRwGf]wkmghOhiwWkqwhCHisWk=Hj=hjhUYmewXqw@rxtiygifQpEs_wlXcivqyt iZ:Ar:qqxpqx=?`=YRbjVr;Ir=Ir;As= ar>yq:ary@rAiRGhkGHj?hjyWj=hjCHkWhmTUTMU^Ev[mvZIVgmVaUvSUUgQV^ISxar>Ir;=m_jlRAuRAf;isGisyxqypq:Irwxq:yq bxVqqppAar:qsDirJisCqsHqsuxq:yptDjAxku gi?HSGuSIcCKSYuMxlQxtoipo`hN_ir=Yr=Yryr?aryPr:iq:qqwPr:ArQqwthmGhAxiwWlEHkAXj;HhmwgiGhgWlOHl= hj;xhiO\\]UVqVVeU^YtOiT>mRVaSHaVQAUfEV\\UW\\Qv^iUXyuUYuZmuW=vZyUXyuXaU ZqeyhqtxQBiyWiwwiugi^qvDiZwlX;::BiVquthfquXqxHqtPrvhqvppqHqnhpvPqthqwh qq`quxpvXqu@ry`ok@px@R;xhsGksWiAhgkwmswiAHe[Wp;HkwvZGfegF`?XeYglowhKhh qWj`U[qUXAV[=fxdgugheghkwgsghkWiqggvojpoa@o`xoixoV`mV@kB@kCHkChn_hnxpQ ywiyWf]WfqwhDF;Hj;HisGiiwgigeWge]WfPv\\QVV]UVMenHr=arIog`xPYup YuYIy\\qvgQuWqtCYsLQsEIuMatNQtLytQytCIsCasFasGisGasFasEYsEIsAyrGYsEYqs DjwWiChj=hkCxjChj=Hk@v_YfJYsDatFas;yPwhmYHgyVbSFWgelYgfUIntVVeU]mUXUV \\Iv\\]V]Ev\\Iv[AVZ]v[?t@qrPQsDIswDpSHmOpWITA=SIUR>AvNiTfEV\\uVZAv_Yf= Ir:AruPqHar>ashrgxoJYqw`uFis>Qqtpqppp?Yr\\J>`J? \\JUEuRMvZqUUEuPMvZqUVUuTauWiUY=vZuUZAvUaUW]TOitGicJ@lJhoghokHpkxpo`pt @qqpqtHqyXqtxqwTjygiqwhhE:::wgiwGisGiqwh`uVGQjWiuWiEu_wLX;oghoghFQ\\UV _qv`qvrvXq>yqxprv`QkWhOhhqwlgGhkGb?gaGF\\sHkGxjUweaXj?xlugiAxhqwjygiq GiTuVYUWeUWeEm_tLWshqOXSiuOWOqtlV;o_uIR;AtFasYeuVIwb=WeY wdevaqVYIvZQUW]uYEVZqwgegohpYySWhgkw_oUYahfgwtSxmgHkEHm?hj]XmSXlIhkgGg agj;xi_HnYhkCxjGHkAHhcwfQXj=hsgxoKHf_W^cuWuFZAfowGjGIoehlsWiQXiwWkmObU V]mWdMw[uuXuUXiUZIvZuu[AVXAvYqUZ?rxhq?iQOHiuGnqWiuxkMHh?GcgFZ?fhKWdCHg egoOp`IfQTEcGhmwfiGhMUTMeBASBEVZ;VhggfEXjsWjwghsghmw hq_u`qt`qwHqu`qrPQmWh\\Eiwg`uVaUYmuWuUXiuXiUWmUW_qt]qxlQvpQxhQ:yqtdqvlq:=RCESCatMYTWMuTMvViU^iu XyVYAV\\MuT=WYyU_]uVAwZEVIurAyr:ER_uUYMW^]V]mUXiuSIUVIUSoqwPr:yqsPqr@q sxpy@rx@sBAs;UjsGisoWWQ`uVGQhEuWiuGj;pYwqtLWSiuOYshqoVSiuoXyUZauWmuT]U XYUWuuUeUYQUVmURMuTyvcMWYoqVhmU@qnXpAar:qqthpvHqoHrypqvhquHQXuUYuTYuU] UWeuYye;=tUYuuZyuY=VYuuXiesLV]UVauVOQdE:::qoWiE:bhFQmGhVquxbIWoIydXqqTEisr=XlyYt]yyUyv_yyyywqyvyXyiywyxqyxxyyqyywyy=uawmyyI^ Ywix^yxUyfYwyytyywyxqyxqYxyywwdYwyiY]YuSYxcyvlYx]W`mVWetOEYlQxymYxyYxi iyqyvyywYyyqysqyvqxuEfaWfoiweysyxqWiqyHwyXrkfZCfwwHriIrCyyKIuOYneHgqva EG][vnsGj]ipwhqQhm_XjCxnGHleXmWhqmhp]v^aVVgUWaGhiggggg]XlMHoOhl]XlMHmK xkCxkvr>iqEyr:qsDqp>ArHytNapoXpGir?it?unuwiMImYHauEZKxdWgqAhk]Xhqglowh qhoexjGHkQGg]wk_xmeGimGjAXjAxiwWjiwgUXiwwqMxlyHi;xlAGdkw^iff]V__Xiywji wgywgiWmKPUAURaV\\IV]AVZQVZ?sxpq?iqwHrw`qupQgwisggyOW]uUyuU?t>arSYsFiq gpoxogi_oVhmDQquXt=arFqqyprvtgYweCHjtUXiu]QV\\]v^Yv^af?ir?Ir;IrQdetXquDi^QdEiwgiggnPXuUG QmghNQJV?qqLVShFQdEo?Zrlyyiy=:::::::<:xIwiyjy;::::byZy=:tIry:::::::::: :ZxMyiyxyIGYpqHb_F_yIxqyymIyqiwkyyuyytix=xIyA::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::nyyiyxyYyyyyYyVyyuwtYucwwA`yyqYxyyxqXq mxyuyxoywYymaxyYywAvLqtsXoipyjaxyayyExqqxyYuAysuyviYyuYyuIryvYJxuIxyxw Yvyy=:::uawmyyI:JyqyyuYwcx^yxqyxqYyIyxYxoYwyyyAxYxgysqywIYyIytyxOyxexw qxcywQySyuAwqyyYvyimqyvywtyxmIyrqyxqyyYyxYygIxsAwfAsnHqOQr?yyuiytywkiy oqxyiYkiwgiywiyxiy=::JxyyxmiwyYwYytYxCyfYyYy^yxuytyyyivQyseywyyuiyeYuW yyqYyyIxoYf=gb;hfawyuYyxYyMyt_yxqyxqYsIxiybwVayYywytAIsWIsCIhWgeluXUXj AXr]xn=xcQwpcyQiugXjChcUV^_W`uFqAxkMingxqOXmiXlQHduva[wdSWoeXoAxjAHo_h n=HiqWl=XjWXlKHnWXmUHnWhkUhl=hlIXiGxjAHmMxgwwhaHo_xmQhlOHj;xpUhm_Hj=Hm cgg=G[IfneGhAYmDW[Iv]quX]V\\Iv\\or=QsAYrDyrw`RCXkAHnKHlShiyGlaodquYqXc QWlYV_QUFis=yprPvSMuiQwd]uSEU]AVZiv[IvaAV[=WZAvayUZUVYuUYyuXYUZquTSgNr AirmHpjXvXiusxoiXroppQir@QpWxmYHkEPsnppPIr=AsrHqDqqxprxpq>yQ_GfYWhiggg OUaEywiyGkCHkywiygeWgeav^aVZ=VZauVaEYsEY=VZ=V[EV[_f^_VdMWdiwgJZ?R=Xj=` :QrArxpqsxpqpqv`q:qqx`qv`QywioWiqWi;PYAvYyuXmeuPq=Yquhr uXq@Qqt`sxxqFaqw`p\\XnJXiwOrfpoyog`oNysHAt>arNQRQXgiWh= wbKFV_ekWGfgXjAhmwwiuV^]V]EV[[F]QVkAhjIxkIxk>SKHlfr?iruXquhogho]Xn]xrA yr@qr@QR?PX_QDV[Ev]YfBMZEV[quYyUXoQ;Hj?HkCxkIxkG`?ir?yrAyr?ir?iRXFriyw isGipuXwonPpxHry`tLIts`pm@t>arRAsCQrfpoXqrBYoMhllPmVPrgxoMYr>atBAs?iqw @rvDk?XjEXk@vZwqyxqYr=Un]Xn;HiqWmOhkCHjqGimGgHvYEvZqUZuu XyuXmuXqeyxqyXr=YryQR:xuYyUYSiuWix ey<::JYsiywi;@vQJZCjfr?iryLXciNr:uiuWiyWiuWjyoZyuYav]Yv[AvZmUWeU[=VZIv ZAVZuUYAFoWiqWhwGiowiugi=xi=hj;xj=HjAhiugjogh]xkIXjLuhiV`yVWmUDmQypvTY u`QuUeW^av]yUZMvYGryxqxLY;y_yLZCjvqydinQjYsinQJZSj>r:EjnqxpqwLYueylY;: w_xLZ?r:=xEuoXmEoOWeEyay=:::::::Qr>yr?isHas;aquXvKQtZPkFpmBPkHIquht?qrFAr:qs>ar=qqwHrypqrHqqxqw` qFQsCYsAqrxPqspqs@qFar>qs@qrKir?qrpxpMqq:AqZPnKHiuopX@nXAsDYt;QrGIRIxi yOZGrypqxxqwpqyhqu@rxXolxoqpquHtPatiR]MuTyvZEvZQuTIw_iV]AVZYv\\Mv[AVZYUVUUUYuTuuY?rxhqtppoPs @ir=qqwPsxpqEYqvhsqHqAaoi`oK`lUhjAhswpqAiqw`rv`q;aQ=Xh@vVauYeevThgwhmw _;Wa[wgcWi;hiUXlKXhYweEXggwmoGiewa=GfoFayV\\Qv^wEZIXggWkegg?hggGksGioG hkghduYquXAVXqUZiUXOQBisOWeUWMUTMuTQuTquXqUYuerLW]UVAvYyUTUUUauWiUY=VZ qEmWh\\UUUUXYUV=vWmU[qUYmUWeuWUuUiuVwPmwfXUUiUWeuYuUTqUWEuW]uZMV[=UQqt \\qe=ypq`stXqIQqvxssXqHAqthpYHnRXj?XnKPlBQquPt@qruXpm@s=arQJUUUUgPcGgHuSIUV]UV]ErgZplUgPeoZAvZeUXmuUShNqr@qqhpoxpoThi wgkwfcwiuWiuGhmggcGgsghlUWeUUMUTUephpt@qpThmgisoYquTeUVEuTIeoXpsxP=Xis gfUGeKhiwWnAHkChfaGlaWg[g`sFcWF^GHgegjcWgwGgcwhgWgkggeWgkgggWhiwgmOToP goUChnppTgiwgcWhkWgoWhewgiwgiOVUerppp\\qxpqHqqxpr@Qu_tdhkGhkgggggcGgcW gfPTUUUUU;kGhkGhFqsTh>Q\\Eyyyyyy:::::::Z:Zy=::::::::::::::::::pyysy::J yuYyqyxqywqyxqyyyyxiywiYveyyqyyWyxiwYYv:yomHUQxygIxsYwkyysqyuYyyYysywi xuyxoyy\\YyqyumiriyviYyYyyqyyixqyxqY:xiyyYjxSyvyy=:::::::::::::::::::: ::::::::::::::::::::::::::RyfYhyyyyxSyvyy=:::uau=pIeywiyyyyygYv_YyiIxp YyAW]]VOisIyyuiyyeyyuyt]yyiiyiyyeyXyuYYweYwuYyuivqysAytAywqyqyyxIY`Yym YxyyxwyweyuyywApyweIuyxyyySixkytuIy]yywYwyyymyydyysyyYyxywSxvxyyyoyYeA FYhiyaYwIyVyyux:Zwmx;yyy:jxsxVYJxsyyqyuiyiyxExmYvCisqHlQHleGhyixwQuayy mYyyYxoyvaYwayuxqyxyQySyqYypIoasyYkapayuIyyqyumxcytixuyRyyqxcyRywIysQw iIuQxqinAyZItOajoGixAxlyyuiyBXiw_tCYshYv_yvVqubYv^ytSQuZiv`YvaAwh@pkhi uWiaPn]HuOUlEHkKHkAhnUhlexlQXkagfMw]YVcAF[Eha;GsCXkoHlMhl;HjMxiyGl;Hj= XhmWj`u^yuYavZAv[uuXIvZ=vY=VYYUW]uTeUViU[yu[ev]]v_aVY]uVaVXuUeQV^iRvhq PesHew^mv]auWevYAVWQUUquWiU[Ef>ar>YquXQeWgyghqWlCXkSxkKxjugiwggiWkoGi] xjEhn=HkCXfcGfyfbseU[EgGGdWGb?G[gEXEFWiU]suYKvWou\\;fZSUUQUdWGeUHo]xh; hiMgeSghuwhGHkAXikwgkGf[WluWiShiyghKWd=w\\UvaMv\\yWdQwpQHmiwf_wfcgfywh ogheGgMXj=xg_WfGhjtuYquVmUWaUXeEjWmUWqet\\ylXiuWiuWgQpEriowhyghJZiUX=v WmuXSiVquDhoghkghJU]UVYuVauWiuWuevThduTUUZeuWeV[wpfpoKYr?M\\EuIEtH]r?q TEeS]AUT=wX=v\\iuWIvVaU^mUXafvXropp:AqrXqvPqk@qo@PoggEXj_GjsGfmggoWiqw iuGiuWhkgiNqtPqxpqxpQdUW?qrDhoghsGisGhoghqwh`esPqtPqxLYCioWhuWhkwiqghV qtPQm_rLV;:m_s<::jWchZqxpqxPjVauVaE::::::::k_rlVchNQ`Ek_rlZAfv`qvDhkgh duWShFQhEqWh@vZAVYmUXUUUWquXqvqtxqwHq:egugheGituZyUXquWAvXmu\\iuWMVTQuYatOYUHySRISEYtAASewr >Qqt`qn`pEqr?apjxoBar=Ao`hnxxQoWiqGhuwhoWidEsWitUXiUZme:MY_qx`Q;xhasAAsmhohPqgxoNYr@Qso @q^XkIhnDpKUWf?YmYHoCXkoGeUGiWwe_hkGHmCHkeggcoWaUTqUWeu\\EvU=vXIUWYuVu UXuuYqUXeuVyuXchtEywiyghoOVshZtlWiesHQo_xLYchduWaewXQs_uLWSh`uV;:o_t<: :JXmEZslV;:o_t<::::::jVShNQhuWWQdEm_sL[EV[mUXGQhuWWQ`esHQhEo_;Ir;aQPuT _qvTiuoVaEnpr@qvpqx@R;HiFQ\\UWeuWchvqwhqt@qrxqwhqxsvxqsavaiuBYsmqu[imDhkdHmV@vMitdav_ypvXqohqvPs OQtrHrxHtLAtIQsCAt>QrLir=Yt>QrKiqv`sn`pFyolXoFxka@kExrbXoVyqirBuhEXiuwiqoU]elxpoppy`qyPsAIrFIsQsvhqoHpjpomHpk@qoXsJq saPob@quHqrXqsPquHqtLXeuVmEwghm?sOWmet<::fQqWi^qtPqvLWeeslV;=HiuWikoWc ilevThqwhpuXyuZAvRGpAQr=QsQkGhvPXEk_r<:::jV;RhZrlUcgTEkGh:::::::v P\\Em_sQ \\Eqwh\\EwOYuEeWgewgvQnppPsphpyqxXsAqrHIsBItCIsLQsDAtDYs FQsDAr?aruhqwps>arCypr@uirBQQ?HgeGmwwioHm UxmAxjEW`mvXYuUyf^cFiWGfugdUWmkwhoXkKHhAgcMhgmgluwi=xhsGj;HjGHlKxl[xmS hmWhkGhkChj?hkCHk?hj?Hj?hj;`>M^UV]qV^aVaaV_=W_?s:IrDyq;arqHqGyQ[xi?Hdi f`uF]YvkgOgev_uVZAV_uUYeVZ=v]EvZ=VYqUYyuXIV\\Ev[MV[[;AryXqtxqqxp=Iqsxs ;QrWAsDAtmppEQohhmr_iLIpo`sp@q>AqrXr;ArvxqwxqPnJHlEuiGhjxuXmU_QV\\]VYuU`CoGxkW hj?HksGiGhj=XjChjgWjwwgEXjkgiuGj?hjLV\\uUZ=VYuUY=f:ARpuX?r:ArxlYyuYIv[ Iv[siywiwgiwGj>RjYyuYyEBj;PZSj=@iry@rvppqPr:Ar:ARbi;Hj;HjtERxEsGi \\Eyyyyyy:::::::Z:Zy=:::hywwyyyyy=tixyyyuy::::::::fYwYyyiyvywuy::yyy:w yytYyoyyuybyvYxIZxiYhyxwyyyyyMyuivQytyyvqyyayyIyfqw^plR`vPQuyIyvuxwqwu ivUyoIytiwmwyyyMYwayuyYyYixIYhiyiYpIwQyuIyay=::::::::::::::::::::::::: ::::::::::::::::::::::::::::::TYtMyxiywoyxAyrAykIxkyyxyYwyykiwgqxwyvuy syyAG`qFm=HkyyxdysUyyqyyuYyuYwyYymYyoyvYyyayvyyqIYUIucItCyywYxxyweywaY wiyu]yvYYyuYyqyxqYyyyx[xuywyywYytyxwyvqYtiwyyyiYwiytIuyydImiyyYxwyyqyy ryyyy:::::::::tYySylYxmYv]YvyYyoxiQxyaYwyveijewg]hjGhxSIvyyxwyyqiylyyy YwmywayvyYyuYyuiyyYyyyjxSyZuyY:qYydiviyyyywMv]YveityyyyyyqyxUxoyyuiyci uYiigwgyhnchoKhluHmYxk]wfQFUUenaGgSin_xpIhkkhkEhnAXjKhhqgm?hjYxkIHlCxj ;xhsWgeoTYesxqxTkAxhAhj]weWGgSGeiv\\Qf[kUXWFZ=fY[UVSEQuTXMeTQW`ofk]wfo XkIXpAXkQXgiwk]wfeXiygqMxlGxfggpMHmwWdSGoygjIHfcGdaf_QG`qFoIXlQXkGXhmW hUweYGe]GfewgiGhoghqWhmwhmoYyuYmUYuuWquXauWiuVGqsxPgGgcgg]gfoGgeGoSXmU hjCXi_GgKxhuGkgGh?G_gf]uEZoGdMGlgwg]Xj?XmyGjExhqgjogh;xhogiooXiUWeuW_Q mWiFrwdjuGi?hhogjkGhQxhsGnoGikxgqwcCF\\wVY=VngWhKhgXvXGRqGj>RtUXiUXqet hQo?BisghsGi:::qWiuwhuWis_vLYmUXquWiuXsioOZiUXuuVWqsHquXqu`qv`qsXQmgiu Wi?XjTuTEv]Uv]Yv]]UUUuT=uRQvXue:Qr=qprXt?qrDQquxnL`lVPkE@rjdlwOZMUTeVY uu]aUWyvZEV_muXAvU_rv`qxxqwHPmgf;HicghmWiywiwGj;xhuWiqwhq?uWi`uV_qv`qt PqtXqutiywiy_uXQoghoGiluWieuyuQykaxyQxpYrjXp_QtOayeqwyIy uuxfYpyxyixYYXyxmyuyyyYyyGYdYygYpyyoyxeyvYxI:::::::::::::::::::::::::: :::::::::::::::::::::::::::::wIysyywiyyatAyriyyyyuMtaYvAYuQyxyyyMYtMyw UyuyyveiuiyQQsFIp]hnwAxlyYsiwkIxbyyYydywyYvei[yvxXkIhmsGiqQvZyyrixxyyu IyxaysqyvqYTYtMYyyiweyqYyuiywqycywiypyyyyxCywApixiYuyyyyxFY\\yxWyyUyvy y=::::::waxmyyiymxsxVYjxcv_ivyiywyysYykYvcyymIyKipsxkgWh[yq?yyaYwyyxwQ xuyxeYxyyxqIwQyuiyuyfYpIqasyyxQYyYxwAsYxwYyrx>ytQywyyyyyyewfyrIykawgyy wqyGyrBqs@yrKqrBitBQsQ`uUYuYeuWquU]UZeuWMVXqUZQ uUIVVeu[UUVaUNiTDyq:UUJETaauWeVVau[IUTIvUYU[]UVeUSEuUIUSMUSAuTIUSiuV]U YyuXuuYqUW]enPpkpqs@q:qpp`rpxP=wbDTMysulqaUtOUwRQu_IuTMVTQuXQuTeUUQUVU uTUuUQUSYuTEEnplxpp<::VprxpoxpqDhkgggggkgg\\EmGgeWhcWgoWggWhcOVUUU]UVK UAuSIuSwptxqxXps@ql@qqXprxpqHqsxPiGg_gfeGf]GhaGgGhj=Xl=XjMXhoGfoFa=GZC VhoVaUxgmWiYGfyWgeGksGiAxgigh[GfsGhiggmwg]ggcwfkgg_GgcWh`evThmoUChvpqx pn`pnxpqxpoTgmoUquWieutgZqtggggPuTQuT;g_plUYuUaespPTE:::k?nPTEe_oqtJIt:Aqs@vRIuKIsCys@Q rHQrvPpUHmL`hm?lk?hOxhpwlwgi[hkFhoY@npHpjhqy@rtPrR;Hj= xiwwiugj;xiIxj?hjuGiyWgeWk`U^euWEW[IVeAv[YVSMUUuSJ]TBESJ]R?eSAT?urA]Qx@VOuTeIV]qvYAVZeuWaesHqudhygikWisggkGhm_vlYAvZeuYopuPq oPqsHquthuWioGihuWGqqxpFyrBQrwpqnpoi@RqwkyGj?xgmwkmO[qtQ]TCMSNESD]VSMu dIV\\UVUYUZEuSivZoQpuV]UW]uUYUVUUTYuTEeo`psxppXqsppsxpq@QmWgcgm?hjQxfe Wo[GgiW\\YVeyU[qxfkgneGhMxgkgjkOYaUVierxpt@qolUauVYuVaEPUWauTeenPqshPP EgwhoggqghiWhmWhmoWaet@qr`qq@qvxPuGhmWikWhwGisoXwpsHqndgiGiqGjEHkkgiug fkwg[WgcwfkwgoGisWiNqttgiwgqgi`uUUU]eUWEWXuu\\ETMITyDrOurBYwZIV\\UuUav YyU[eUWeuT_qtHqu`qtHqqppvPqstgiWiuWi;PZquXWQlEq_qlWiuWauVaUXeUWuUYuUYc hooVGquXQkGh\\E:RhZulVSh:::^qvQtUYuUZ=VZmUXGQdEqwhdEk_rlZGr;m TQuTQEqWiuwhuOSYuTyu[uU]IvYIVWYUZ=uRavSQUeuUZYtAAsJeRAeuL]t_iUX]VXmu[c j\\U\\iUVyUVQuWaUVauVwpsTgfPLERg:vpsThmoWaeulX]euHQmwgighoghu_v`QkGhPE k_tlyyyyyI:::::::B:ry:::::::::::::::byuiyyyxyQywyyiyyuxNyxex:sYyqixsyy yyyciv]yyyiyyyyqixkywyyywiywiyyqyqImyywywyquwyqyYoyyyyxyqyyIwauiyyaYyQ yOywAuAQsq`naxymyxyYyxiyqAYuQyyyxqyyGY`IbxfYlYyuYyuiyqYyyyxI:::::::::: :::::::::::::::::::::::::::::::::::::::::::::wQyuIRyyqwSxvXXiyIYhIyyyf ywUy^YyIyryQYsGyysYyyYysyyoixyYvcYyIhmCVEicdoTRyYv]qwYywMyruYuUyyiYxyY xqYv]YveyyuyvswtysuYxIyyuysyyy=YweyumywqiyqywmxyYyiyxgyyUypiyYYdixiYyY yvyy=::::::wiytIRyVY`ItyyYItiuyYyyytIwYy^YnYqiyyqyewaEWQITMyYuaYxQYvyy xyIyyyoYyyiywYyuqxkyyysQyteyyuytiYwayyuiyevoYxkiyuYwqixqyyyyxvxvYyiHpk HpkD_cF_;iqwxgeWg;xhsgo[XnMxjCHnQhlSXlCHmGxiiGeKgf=gasFZymB=S[ITL=X]UV _muXMvYuuXmUYuUZAv\\EV\\AVXuUZiuXeV\\or:IrAQr;qr;Ar=qqwPrw`q>aqv@rr@qu XpmXRqWjmO\\mUXMVXoqppp>aqvXs;Ir>Qqtxrt`q@Aqt@sqPqFIqvpss`qFyptXsthqHI r=qqsPqsXqu@qw`qrlVuEVQNQ;XkdUZyuTiUWQuWeUWyUYmuZ=vVuuXYEuWi@vYyUYmUXm uU]uWUuU=VWiU[gq>Iqu@txxq?yokHpUhmKXiwoqcXoCyprPstXq;ipoXqmXpw@QwghmGh mwgYGg_WecWfYGheGgu_vHqpXqsPQtuV]uYQuTIvREu_utRAwNutGqpslVIMtjmUZ=wT]u XAuRiuUgquPQuwhoghoWhjUMUUUUUiUWaUYqUUmuW=UUQuTiUWUEkghoGhoOWeevPQ`eyH qtxqtLVaEhuWiuViuWQUWaUVqUXuu[EVUiUWUE:Bh>QJ[IvZiuV]UYMUTavVeucquYqVSQ UVQsFqSxaruLUopsHQu_vlVaeslWcifqwThmoVCi^Q`Em_vlYciwgio?:VQ`Eo_ wlXChkGhioU]UVWqvPqu\\s@qv@QuGhiWjiwgGhgiglewg[W_kfcQf]gw`wFmgGhgp[yuS AUWIURyUX]uXiEqWhmoV]UWeUUieq`quhpuPqnPqsPprxpnHQkghhuWgqudimOXYUVaeqX qsDhkGhk_slWshFQmGhvyyuy:::::::Z:jyyI::::::::::::::ZxyyxiYyyxwqxmYyuYw yiuyYoiynYSyuaixuyysawiywmyUYsAYuIyqyyyiyyoyxyywyYpImyywAiqyYyyGYhiyiY tixyY^yxiyvyyyaYsiu_yyeIxWwaAWwIytyIywYxaayUweYwsiywyxyqwSyfYJySyjyyI: ::::::::::::::::::::::::::::::::::::::::::::::::::::::wax=pyyWY`IiaqMw yyyeixiYtIyAsqxkvmyyYYwyyuainixyYytyyfAXMtM_vIIty_xoyqvoyryyyYYmixaYoo xyeYxyYy`YyqyrYYuYyyuytyYyqxwayqyYyAyyuytsvkywwYyyqxCyxYxoyweywqxwyw=j yyI::::::nyxeyZwmwwYyAXImat=tIwYypYxwyuyYqAvxwyysayC@j=PvSauyQywQykaXs qsuxqyIyqx]YvwyyyywMwcucyvoyyyYwqixmAVYhIcYxkIdOgdgf_gvTQuTUXmUhkCpdIw cqV^aV[qUYmV^YvhqweiV^Iv[yUWaXjmWhiV^GoaPmqojx id_gIiqxHri@pBAqs@svlWiUZiUWquWeUVquWYUYiuSmuVEuW]UTieoPqrHqt@quHQqwgg whawf;Hf]OQ=ud=UTEtvtqLmq:IWSUUgMV]MVUYUXEUSiUUQU]=fyPquPqr@qqxP]wfaGe ]Gf]ggeGhwWi]wggWfiggc_nLUUerpppPQgwheggqggiWhcWggGgcGhkGhewgiWfgWgcWh kghygi_wgggf:ZnLTcggO\\EvZyuWeUWQeyhpohrsDlg_dho:yMAfSMUpChk;HhPtJ=tZq UXEvXmuYQuT=VV]UVIUTmUWeuXSh`uTQen`pnpppdgvpqTgPUUoptPqt@qtDggggeoTYeq HQXeqpppdgfPPEbgnP:::jUsg:::nPTE:::e?nPXEi_pLUSgZp<::jYyerLTMETUV]UVYE e?Ztdh\\UVQuTgP\\erHqsdggOUShVQdetxPJUsg>QTUUUEdUWOQTUU_pn`prpppXqs@qs dggwgeGgiWgaggcwgLUYMUTMVTQuaMuUiUHySLIr=EvKUt[utPmvYquU=uP]uTAuVQUTau TMUVUeqxPiWhLuV]USYUU=UUQURUuTAuUUuTYuUwPmwgiwhgwgoWggWhFqqxppdge?vPXE g_nlysy:::::::Z:vYxI::::::::::::::ZxyYvyuqYlixYY`ImiyyYxwq>urB=J:LJ:qpqpuFisqhn``stXq>PhmOqbHo?YPWhiygkoghuWgZt@qq`qtXptxpjxpnHqx`qupqvxq :qq:qqwHrwDjswhwghmgiVqkHpIyq:ypVxmW`i:@rUxmVYqy`tvpqFaqw`rqHqwPpnhqop p=qqyPr:Arr@QaGhiGhuGiewhoWgqOVeUW]UWOqtDinQowimoYeeuHQqWhpuWiuWieqHQo GisGj?hjkghoGh::NqtDivryxqar>aRRxEBj:::^R@vZ[;MYshqwhu_rxpqp_n^n]gfy iweyypAxriynQyyqxpYylqyyQYuixMim]xyIytyixyyyeyxyywwyytitusGItiyywivyYy OIxiiwyyybywqyqIuQyuixmxyIxIjyyI::::::nYtIyyyxIRyfY:hixiyy]yiYpIryyQyk xqYtyXruYvayyqyy_YRi_OUitPyYxmyxQYuuYvaI@YpqhwqyxayuuyyAxiqyxqayvayyyy yiywiyf`of`hlOhL?dJon^`ngWgeWisGinOhl_oa@ompoiHqm`pk`oelP=uTIuW=vYYUXe UVUuT]UR=U[EuSAx^iVWiSI]Rn`PNYr?IWUaUbiuXEVV]uUeUVaTSytL=uPqTVQuSmuV=u VUuRUUTUuVYUVaerppohqs@qvLVIUSuuV]u[uuXeUUQUU]UVMUUUuUYeuHqs`qq@quhpph qrHqyxqyhptHqkHqr@prxpixppPPgWgNqr@quHqsDilEqwh`uVaUUcg>QkghoOW]UVqeu` qtPqsxpq@qpppuThq_rppp@qphpuxPKHj;xkuWiKhhqWiUge_Xiyg^uuYYg`sfhKWdGhgi glsWiAXhmGiiggwgh@vYuUWUuT]UUQUWUEvprppq@qpppsxpqPQ[WgaWhwGi]ggcGioWhM Xiugd_F_AwX;fl=gcaxfkWosGjWxhwgi[gfeWeWgheggsGiswgmWhaGhiwffpqTgioVYes lWUeupPoGhkggTUUShmOW]UV]UUWQxuYGqqxpqxPm?ioVSh:vqwtiqoX]uUquV]uWmepXq sPpqhpiPpjxqtHq=YpnPv?AsipkLHmr_iBiojprpxp=YquhquXqsThqwhqWiuoWWqqxP`u VuEsGidUWeuWiuWyuYyuYshNQlevXQlUXOQdErhZvlWsh:::NQdE:::q_u<::JW;q_vLXs hVQdErh:::vQlUXmElEo_pdgBisoWciwgiu_sLV]euXQs_slVCivQpEo_sHQm_uqpppt@qrPqr@qp`pnpP\\UU?Qgwfawfeggc GdMwcShlMhfUGekgeWWpQHmwGeUGiIWdafY;FcYV^mGcGGk_wf?HgeWiawfmWf]GhagfoW g`ELuSUUTIuTUenxpohprlVYURIuRetQqtUuuWYUXeURmtOMVTQu]]tOaSudQQurCAVMet a]UXUvT]UWqtQYuQ=UWQUUeUWeuSQuTIuTQEPuTwpotgFqnhpqhPeWgHuSIUTMUTUUUUuU QuTKVUUUuUXmuUQuTYE:::;XjyGhmwgcGfYGg[we_Wgawfowg]GhegeaWfsghmwjmWh?Xe Ygq=HkwWa=w][uVUw_mFiWweogfaWiqwh?Hj;xhqwhgggggg>qrThmoVsgiwgkGhlUXmUX ChkoUYuUaes@QkGhvPXEk_rlVchoOV?qr<::jUsg:::>Q\\E:::i?FQ`E:k_q<\\E:::w_ tPqtxP`Em_pLUchooVshqOWsgnP`uVOQjUchoOX_quXQkGhTUUUuUYesLV;k_vLXShNqrD ioghowgmghiWhmwgvPkwgmGhkoVeevPqsXqtppxXqqPpe@oIYr;qqi`oBipl@oMPl`pkHX rbXo@qojHslppAapqPrnppv`poXqp@qs@qtxpsTgoOU]UVQuUYuUYet@qrhqrHqulWaeql U;kGhXEeoTsfyqyyI:::::::B:xiy=::::::::::::::byyiyoyyyYxfYyiyuatyYmiyfy xAxqQysqyyQywyy\\Qv\\IdG_cmpoexyuqxyqYyyysYyyiyyyyyyyJyuyyqixqyx=xyygy yyywMw[xIyxmyyyxstQytyiywQyuywMIsyyhYyuuQIumMweYYmaxyaYxyyxyYyuYyeYxmy vuisyyxiYpiy=:vyy=:::::::::::::::::::::::::::::::::::::::::::::::::::: ::bynYjyyiwmwyyyaIiaqyyyQyxuykYyyio=I]cSJyd;YbxighyyxiYyiIvyyyiYwaytyy yiyyyywwyyDyuQysuYxyxuYyy]yyyysuySeq;QqVHNngsyyxxqysYyfqwiyxymyWYpIwax =pIoatqYuyypIyyy:::::::ZxyYwayyyxqYwAvYpIqatyYiaqmtyiqmo=yqyyycy`YtQyv MytyyxgyrQyyYyxayoAYqYygGdOWm@vyiyx=YnahwqyyqYyiy\\iryyyyyy=lYweyyclKH luuXqEiooySfYweMwcIwdMWdyWiuGhgggsWhqGeMwdwWhqgiiOQ]tNYqqqpdggoX=VYUuVYuQEURyTRutRMUSQUVUuU]EiWgwGiqWjuGiOhj?XiYGfGHf_ GkIgdGW[MFfaf_]GbAwiiGhKxjChisGieWgdUWeUXmeslXiuWiuVaUXee:qqxpqxdhVqsH qwlVGqs<:dUWshlUXOqtPqt<::jVauVaE:::o_t<::jVShNQhEq_tlVShNQdE:::AxjAHi sGikGhdUWeUXchFQJXciwgiu_slVshqOXchFQlUXwqyxqxdio_r@qrHqsHQFQ\\EbinQhu WOQlEiWho_tXqrTikghuoVuuVeuXiUWmuXiuTiUUiuZmUZMvYetMIt^uUWMw[=VcauUYT> aRUYSGuuJMt^uTSmvTaU^YUWAVUcgmghioViepXqtxpsdgk_rth`UYaEhuWaerlUChkoVS hvPPeoxYyyy:::::::Z:jyyI::::::::::::::ZxyYqaueyuaxiYdixQyxexNxqQytiyyy xsQyvayvYsAqr=?bv^ysixuQypyyyqYtyywyy=wiyyYywiytIryyqxSyVYyixFYyYxyqs] yuqyyyYyoYwqx]IvyuqYaywApTqW`yvumxqyYxyixEysIyuyyyyxfYpIqau=jxSyZyyY:: ::::::::::::::::::::::::::::::::::::::::::::::::::bytYy;:yyyxIuauMvoys qYoayQyxurUYVYcHAYcQwyEisyYymIywIxmiw]yyuIxyIyxYxYYyyyw]yymYruyxUyxyyy uxuyyeEXmHLO@urTqyEYuuYrEyymyxuyxyysYYvaYyyywyiuMxuiuiywyYyAoyydyyuyxw ywmyyI:::::::byyiyuQyuYyuI:yqyyIuaumvuyyYyywXjwcywQweYwyiyyyxEyoixpyyq YveyvMYuyywuiqIyaIuUYnP@myIyuyyrYYoyxnyrmyyiy]wiYqyxyyyyIvWiunyymysgeW gwGisGhgggewfaWgawfmggkweSgeqGgiggUGfMhi`UQ=UXMUTEuPmtGUsDUr;yQxXQuhR; yQOqsG]VTIU^uTQMuDUsDUqvLuEeS_EUT]vSQu[Ae=ippHrohpohogPqptfcgfeOUEuUQU PIUR=uVUUUyUXiTUIuPeUVeD]_nHqr\\qhPoGhkgiVpohpkxP_GhiwfmGhaWhkOV_ppppm hpopqxpqrlTIuSQuSGqqxpuHqsdgg_m@qpthVqsDh>qpdhkoVYemXpmhpt@qePpixrFQsp xPWgdMgjoO]eexpoiHty@rs`n`PnFpk^pkJPq^pnEqpr@spxpt`ogpqpppuppoHqo`pp`p m`poXpkppnHprhplHqpppuHqlXPigfpuTwqohpxHqrPquHqqhpnXsx@r\\xkLHlvwIoVaM hgmWjaWgoWeYGjewguWggwheggo?>QTUU[nppphpqxpl`pnhpoTh>qpDhaGgsggigjsWiq OUKW;k_ppPBgFqvPqoPqr`psxppHQkWh\\Eoggeggcwfg_wpqvXqs@qn@piXqixoKaqxht pHqfhkJPnChkvHofPswxqxppqppkPpx`qvxqyxqqxpq`qtPqt@Qq_yhQuWi`es@qrdhJV] E:RhmOWshjVaE:::::::::::k_rlVchNQ`Ek_rlVSh:::^RCxhqOUUesHQuWiuGisoVGqs LWciwgiuWi`Ek_uXqu`QjVCinQJWeerlVshqwhk_pLV]ewhqwhQdEm_sXQcGhmGgkWhowg luUauXYUVuUVaUXeuV_qtxpxXqkXqoHprhpxPrwTh@VTEu]EuR]WXmUMYR?MuD]s\\YTPu vTaU`aUXMvUaeopps`poPqttg`EkwgvPdEqWhlUWeUUcgfpohpqxpq@qrThvPiWgPuysy: ::::::Z:Zy=:::::::::::::::wyyyiyvyy=pyyyyx;myxsIywyywiywIwcIwshpoH?_R> gHoaxyyiyry:::::fYpI:qqyqixyYxIpiyiYJxyivqyMAsDIrptyuqyqyyiYveyyqixiYe quYYxiyvuyyaYyyiyUxyQwcxVyyyYxyyyIZxqYwAvYxI:::::::::::::::::::::::::: ::::::::::::::::::::::::::RyfY:xYxyigQxyiwmyYcyygywoYyqIwwIxqyyoyydiuA yfyv^qyyUxyYyKiv_yyqyytXwiU==ShHLLxYbmvyQYryyx]yyuyxqyxqyvqYyaItINyyux fYwyytYxyyxmyyqyyyiy=::::::::::::pIuA:ZyyY:::tYymyuwywMxqyouWjmUQ=ucEv \\WywyyuaYwYywqxuIyQyyI:qIyoyyyiyyat@qqhoePot@qqhpk@pv@QqggighegggGf]w fUgeogfagjowhwggiwfUgeuwgkwhiOWQeshPmGgaWiaOZEuSYvRIU^aTPqSwtQKArAryXqs@qs`qtHqvPqgPpjHquThqOWChkGhoGhdE :sOWiuVauVaerDhgGhkWgioTYenpPkOV]E:::m_s<::jUCh>Q`Em_rLVsgZr<::jYyuYeU WeuUYEFQXEi_rLVCjqr@q;MXmUXQEmWhmghoOV]UVmUXsgVquX qwtgmWhewg\\UW[sxpqxqsPqBYqw@so@qIAqvppRhm\\@j@psd@pVujwGf]wlAxjGXj;Hg [we]weWGgiWg_WhttVUuRmuVQUXaeuTimOYaUVuEsGhdeqPrxhqqPpkhqmXpBQqu@q`xnW xjCXoJXlIQpo@uw@rQqtXpipohHpi`pu@qhhplhptHQioVYevPqtDhpuUOrt XqBAr;iqq@qvxP\\uViEiWhmggqoTSgfqyxqt`qvxPmOVGqsHqtLXeevLVSjfqwhqohpoX qsHqw@qspqpxpBQqv`ts`qCQogPn?@KanaqUYUVWeUW=URIvWiu\\=fxHqrHrw`qyhQ]gf [GeYWeaGhgwgsghqWiuwhuWisGihEq_x`qvpqv`QvqwhqyhQuWilEoGi^QoOXeEsGi:::: :::JWeUWieuXQlUX_QhEq_tLXmev<::j[Iv[quXqUW;s_slVCisOWsjnqxpqqxPqwhqGiZ xdioghJZKX[whQs?vqyxqvlXquX=VZ=VZshqoVauViEsgiwwhu?lEuwhqgi^qvaqwXswxq?Apmpu=yr;In`@k`_gqr@qqqyPsxpqLQr=YrkPpIQqvPi_wf>qojHs mhpLiqy@txxq>qpqHrqxp=qqwpqyhqoPqrppuDiAHjiGiJW;mwhdEm_r@Qyyypiyyyy=ua w=jyyI:::Z:Zy=::::::::::::::::::JpmXpiywiywsyyyy[Iv[yyvY::::::::pIqyyy yxfYpIyyywyyyQyuYykxeYxyyxyisaYsyysYvay`[f^qIvaiyeIxlyxqywkyyIbxoixoay yyyyY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::jxcxyiy mQyaIsiylYyGyyUxVywiyyayx]yQyvyxviymqyvQxryxyAytQy^poeG\\QvySityiykayp YwyyypyyvyxyyyuUyuixoixwYyyIyNyyUyxyyyIwYynyyqYvyy=::::::::::::::::jxS yvyy=tIRy:ZuqYsyyyyu_inuGjqgdUwyoIyyqxqxn]XxuYy]yxmYugypqywAyyay_PobXn ^Pnahn^`obxnmxohPob@o_hn_Pn\\Pn_hn_hogholPpl@pj@peXoePpj@pkxoiPpj@pllR AuREUSEUUMUTaUTQuP]tM]UR=uPySJ=vOqTf=V[iTAgkf?huPmXHsdhoIapqXtwxq@ugcG eSwgkggmgisWd_wemVb;G_sv`iVasvaEGcOweWWfawfggggghdUUMUTKWQeslUAUSUURAU VEuSaUTQU[=VZqEgggggf_gfTUUmEiwgiWeUWeRiuWio_n`pn@qrDhVpmXpqlSIer@Qkgh oGggORQelxP_wggOTChkgikoQMTMyVYAVcqUZAVMetZATMERl`P[eTPeUOitZauV=VYqUR IuRaUX?s=QruXpmXqqppsdgcoUIUWYUTiUV]uWwqtHq;IqsHqo`pqrDh :::vpqQXEmoVaE:::yoYeEi_rlVsgioUSh^pn`P^qvdgZtlTQuTKUUUUiEu_s lU;m_qLVsgZvLXShm?dUW?qr@qr<:`Eq_uLUChkGhRhNqrdhkGhsghtUXoplHpepn_XuKI tnpnaXsp@qTPj>XoHHlMqpsXuxHrIypsPrjHpt@pjHqo`pqPqr`puDfmggeghmWgoWhiWh TUVgPLUUOpndfcGguyyyYyvyy=uaw=jyyI:::Z:vyyuy::::::::::::::::::::vwiywy yyy]wiywiYHysIyyyyyyI::::::RyyqwyyyiIuawmyyInyxeyvYFytuxwixaQwIiyQywin T`mYatPyywqYuiydYxuywqyyqyxuYxmYymYxkyuYyyIyAfYpInywyyxyyyuy:::::::::: ::::::::::::::::::::::::::::::::::::::::::::ZvyywYyxayuexoYyqYxwYy_yyD YyuYvoyyYyyqyfYxpAVyIywQyCymyyYyuEXRIuc=dKyirqpxIixixyuwmIw]yywawqyuqx ryywyYuQyKyuiyeyyYyxyxkyyyyyI:::::::::::::::::uaw=jyyirAyrYypyxyiYtYyy yxuYwMytiysMyyeyxyYvmysMXnAuKQTe]v^yyxqywUixqyxAyuIytyywuyuY`sf`MF]Of[ IV[oF`iVcAgbGgcGga;GbqVauva=WbIWdMwdUWeMwdQwcMWdOgdOgdZeXoepof`oihogHp ixOFqt@qh`oexpixoyXpmhqe`o>aohXr`Hoi@lMHmvwigPlOXreho;aohPp`pnrdfwGiqg fewf[ggeW_qf`eUYsESSuTMuUWEY?VZgv`ofdYgeUweYGf[Gfcgf_gg]gfkGf]WhWGfowe ]Gh[WfqggXuTQuRAuR=UR=uSIuSCheWgegeWge[GfhuW?Qkgf_gfgggPuT?pj@pn`pnXpk HppppppprdfcGgWwf_GfeGg[_hXPWGf[WgawfChiyGduvaoGeWwpGXloWbEgjIwdefXuUf sfaSHj;XikwgEWb;Wg]o]EvZauRAuXIes`pmPpqhPiGg]GhegfiWgqwghuUUuSAUR]UUQu WaUVmuUUuVAuRmUR=u[QUUivWmuSITLeRwhQLMsD]VVaUSMtLMUN]T^uUY]UR=uWauVMUU UuTYepLTUellUYev`Qi_sxPoWghuToqmhpy`pphql`puXpn@ruthmWhLenAPmTvJQtaEuTiVU ]U^mUXiUUQuQqTPwpohquPQ[geaWgeWfgWg[WgcGfFPDuUUuT;]wfagfcOTMeohPe_ppPg ?fpoTg:::nPTE:::e_oLUsgiwgi_ppPe_o<:::jXqeqLTMeppppxPPeo`PXE_gfpEo_n`P k_q`hoOsrHqQir@Iqa@oCYqvxqq@qthpp @qpDh\\uUSgnpr@qvpQcgggOUOqoppumRRASDEvMeTeAv[EW\\QvXQu TauTMUZmuWYUV_ppdfioRSfg?DuUUuS]uUMuV]URQUT=enXpohpohP:bggoTSg:::nPTE: :::i_r@Qk_qLUcgfPPE:::wgi\\UVgPXEm_p<\\EcGgtEoghLEk_rlT;bhoghsGis?gOU] er@QTEi_olUshhuWcgfPjTSg^PLERgnPTEa_ohPbg^PLEaWgHUVYuYev]EtO]tT]UUMvYu uVaTOyvUaUbytSMuAISR]r@UvKYTeaUXyvViuYAUSYUSAUUYuTIepXprtf:NpqTf>potec wfqyyyyxfYyYyvyy=:::::B:xyyyI:::::::::::::::::::btOitmYxmyyyqysxqqtQiy yyyyY:::::::pIuauyYRyvyy=bywiyqyxqyywqymyxyYwiyyqYygYlyyaYqAY_quY=XcMw x=YsyywuyyoywQYdyw[xeywAxyygYhIoyydyygYyYyvyy=:::::::::::::::::::::::: :::::::::::::::::::::::::::::JyqImYyyYx:WiytxxyYvmyyyYvyyuqWoiT:qRgtKN lSiLpyQxlyymYxIMLM@Y;=ryUyqIYpIX^GqXAuMqyqaxyiyxYyneykYyyyxyqwSytIuyyy QyqyxyYymyyqyywyy=:::::::::::::::::jxSyZyyY::wiyPysIyxiywkyqiyAysyytEw Ais_W`uvecF`gtHwCdaV_yywmIfOwdKgcEG^aV^UF_]fWyEYAF\\Iv\\[v]ovawfcQgdIW eSwbMGdGwdOgeagfKWeSGdUGeOGeSGeSOQmTPgOageYwfWwe=XisgjwWi]WeSGdCwb[GeQ gdGWcUGdPvXqUZ]UVQuOitJASCmru`QFER<]VREUgEV\\AVPutYEUTUURAUS=URAuRAUQ= URiTRyTQMuSMUWauOAURetQutOqtP=UR=uUEuSUURAuQYTNUTIySTyTRYtLQtLQdlPPiwg iWf]Wf[_r@qr`PpuX?qrTf]WfeWge?BhfpmXpsxpq\\mlQWpk\\px j^hjBPqW@nHqprxropprpoh`pipokxoh@Qvpq`ppdfi?@uUUuRYUUKS]EZnpPgggjUQuTY uTQUUcg:::::::ZqxPi_r@Qk_qlUcgTUU;:::ywiyWhnP\\Ek_pdgg_rQTEm_qtgi_uLWOqohPc_ohPiwgiwgnPPuTgPLEi?ZqLVsgfPeGgVPHeqPom@ pu`r;yonPpgXodPtBAsuHodHr`HoYARLSFMuD]sX=tKeVSMU_]uV=VSIelPpxPqspppDgf pqtfkoScfkORUuT=EmyywYxtywyyy[y=:::::B:ry:::::::::::::::::::vXvyyyYJyo yvayvmyyI:::::::RyfYhIRyvyy=:waxmyuitAYiYweyywiyuIxoyysyyyyxuYnChkAXgi GwGityyxNyvyxjYxyYyveyfywiYxyygYhyyGyyqysmxwywmyyI:::::::::::::::::::: ::::::::::::::::::::::::::::::::::wYyNyvqyriyxQyriyuYxyyykiyyyxtyyyqxy yykmqiyoEuWAqQLm;In[`YMMtHAlJlyAqryYXkyYyYyyeysIvX]UXQUSqytUyyiIqyydyy yywCywyyxyxwyuiywQyyiYjyyI::::::::::::::::::uyypIbxsIykixoYyyyybwkIxqY yuYy^YwYxqyxyyyyyywiyiiu]yykIylXrysxHRfPmVDuCUsy]ywIUM]TQUTMAtIuSMGoMP mQpgmOhmghoOj>`jSpmWPojxoeHpjhnePo]@oaHohhobhof\\d`OW?UOSqTQIUQyTP]tMe TMQTSutPYuTMURIuRopnXpp`pc`odxOvoh@rn`puHod`nFpk\\hjBhqV@nJYppPrg@puho iHqh@pqdfcwe[geWgeDUSYUXiUTaUVyTTIuPEuRIuTQuVaEeWgSwcKGbqFamv_iFYoeXcE Wce_gf_TUUiuWiUTkSIUTMUTmUXwpqxpixoiXpmXPVP\\UV]uSAuR]UUUUVcfcOREUS=uT SgVplxoo`puHr:aok@pcXoepppdg]gfiWf_gjowhqweYWnyGjqga=g_ieXIG]YvlgGhMxh `uQytWYUUiuU?rpppyhpoPqo`PgoPUuSiTUIuS]UUyTS=UTYuTqUZuUUaebXochojpoirG=r;evQEu]=e>efAHfagjcggyGggWhaOW]UVuUYuUSMU TgPaWgPuT?qpppuppqXqpqr dgTUUSh`UV]emXP]oRIuS?qrDhnPPEe?:nPXEmWhXuUgPLE_OTIuTwopXprxqvhol@pxxq wppk@pyXpmHty@rqHn]xsqHq^`kHhlwwiqxm[HtuhqEqprHri@p;iPmwfcggc_oLUUUSCf PUR;myy`ixQYhIxI:::::Z:jyyI::::::::::::::::::::::::::::::RyfYhyyWYjxSy vyy=bytYy;:yay]vuyqiyshm_Hf=WceysQyysyynyyaYlInyuiyryxsiyyyyqayuIxqqxs yyxiyyyywmyyI:::::::::::::::::::::::::::::::::::::::::::::::::::::Zxyy waywIyvqyriYtyyqYwyYyaYxyYxeyyyyuyinyyyYwnyynQyyauXIlNOdlW]QVkPV\\kqkD xyrQxyyytyYoQxayiQwdEvT?vLqtyYY\\yxyywsxuywciwyyyyIxmywqYyuIywiyyyyxI: ::::::::::::::::::uaw=hyy[sqyyIyx=giyuiwwqs]YuyyyuyxiywOyvyyriy>qneHfx NbMpglgxRYut`n`Xp]`nkxna`nZxm`xn_Xn[Hn\\@nZ@mR@mRPmThndHocxohpnfXod@pi `ndHokppopoj@p]hn_hoeXooXpmXphxohHodHpg`oihofHpipojHpihol@pfXpj`omTe_G f[wf]weUGeewdQwiYGfEXeluIEtOeRAiSwtq^uTS=vNqTYaTP=vQEuW=USytNetUEuS]V^ ]VREUSYEiGiqWe_WfKWeSWe_WfgGhkOWeuWiuWQuTQUKETKeR?OjQlUXmUXChvP`Eo_sHQm_rLVsgvPTEe_qlWchZqlU;gwgXUVUUWiuVmUWaEtUWoq pppyhP?hggwmyGjYWawf^seYYw_mFomGicXhTVREU`yuZquUopkPPeWgBgmGhaWhkwguGi myywYxyiyqyyyyxyqyyI:::::Z:jyyI::::::::::::::::::::::::::::::RyfYhIqAf YpIyyy::JyuYyuI:hiyAyuyyuqY_iwEw_mvxWIvyyxwYygYxkyveyyuyypI>wbAwwyyyQY wixtyxyiwuyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::by oiymIywQyyywwywqyxqYsyysawIyyewuqvuisaYmIxyYs=YlEfYeUAirt=w`yIxciyuYxy yyqaxyysIy`YuWitAIsIIr?yyuqYkyxlYyyyxyiuaYyiyayvaIywiyry:::::::::::::: :::::ZwmxsxyqwSxbyXyyuYuuyxUYyOxqixwyyxQytQyyaywexsYjSweCuKOt^cuWGYlSX `=vZMw^eFjWoYAUSEUN]TUutQMUQytOUtM]TN]TTUem@qqPoipocxoh`olHp[xn`PPpuYy US=URytPqtX]uVqUVauTUempPcwgeGgiWgaOUIUWYuTeeqPQowhmwhiggyWgTVVaet`qFa pphrbXovxm\\plrgiAaojPtshq@yom@tw@rEiqyHqh@psXP?Hj;xe]WfMgeUwfmGhuwj?H h;xiWggewd]GfgGhkwiyOZ=VZeUWeUUChkGhqwhhEcGg\\Em_?ir?YoeXoqRb=uTewYIV bauXyUQAUTYtNuV^av\\Iv[QeoXojxomHqrXq;ArqxqxPptHqnThyWj=HisoVaeuXQsGiJ WCisGiogh@vZAVYuUY]EuWiuWi>qpppx@qs`rr@qCiqw`rypqvhquppuHqnhqt`pwPQqGh CXkAxdGWcQXiwwnwGjsF[GVcKv\\eHhqgnigh;Xe[gjewg_xkMHjowhKgcGwjEXk[Xochb KwciWiswguGimgiuWhwoWqewhQwGisgi^qvQ\\E:moVaE::k_rQ\\E:::::;Hj;xhhuUYesTh oOW]UV?QdUWeuUCj;Hjo_ohpoHqsQ \\E:m_sQXuUOqr@qyPqupqs@qxLW]uUUuVkUIUWOpu@qlXqrThggiyWi?Xiswi] gfcga;GeYf^ig_oFnYwfeHi;hligh;hfcwicgg=xhsWjywigGhkGeFyyqysyYqauyYuawm yyI::::Z:Zy=::::::::::::::::::::::::::::::pIuaumw;:uawmyyI:::^yqAYuiyu IxoqxyywMyuyyvmyRItMIylaxy]ywWYsiwiYykYxyiywqyyysIysyyyyyy;::::::::::: ::::::::::::::::::::::::::::::::::::::::::::waxyyw]uyyyIyxUwyyy_iywivy YyiyywyugyvsyyuiyhYrEYpyyyYyyyYvyyvEYyyywqyyuYxmipqxpqyxqYwawqYeyw[YsI Y`WF^qiveyyqqyqihAwcyyxMysyxI:::::::::::::::::::tYyuYywYpiyiY:jwmYwyyy uyxyyymYyuywuYxeyyqYsQys]tOaT[]ryQyteUNaTQISEMrl\\PKyq;YVMeTMyrC ]tEeSKQsEuSEYSLusIETJ=TJ=d^Hob`kK@lU@nYpmZ@nWxmYxn_hnYhmWHnVhmmXpmTdG? yfcAGbEwbAGc?wdIgc[GdKGgOgdRhUWegGeSGfKGdYgdMgfUGeeGfYgh[Gf=he]glawg;w [QF]]UW[w_qfiEWdowbIgnygjewcMgiaWgsggiWeQwdwvb?gaGWcCweUWd]GfkWisGgTUQ uTQ]TN]tNatNqtPqTRseYweY_tPQc_qxPeoTudh`pn`peXoeHlKHlR@mRHn[HncHocteYw e]Wf[gf_geagfyVb=gcQgdYGgaWfgWgUgf]WenprditUUEuRiv]UvIqSHierpplHpyXpm` ts`qHxglwq^xnEqprPoXpmFAsAynd@olTg]weWwiqghHUSEuSAuREURutRyTQAuQyTS=US IuRIUTEuWmUWYuTMuTAURYUSAuVAerpohXqgpo:qoihrhPpGaprPtqXqHXhq?loGiNYqy@ tshqCqpsxpcXoxpPyghqGf_gfGGePTRuTMMUSYTTEuTiUW_pnHpkHplPplPPxtQytQSf@u RShmOSMefpohXpmXpmlRytQIuRAuT=uQgpnhqohptPplppk@pnPpkpomHpaPpi@pyXqO`n Z@psppBIsAaqjTl]wfMhcOG^QuUeW_ovmgghIhfeggEGd;XgiWiiGh[geWwgmORUuTyDYG gageagf[gf_Wfawfc_qhpoTgLUTMuSIuSEUSEUSsfaOTSgfPLE:a_lLS;eWgjT;a?NPjSs f:::^qv`qplSIuSQuTQUUCgcGg:a_tPqthPxtQytTQuTQE_gfLUTMUSEUSYuUYUV]eolSC g^PDEc?ZrDhk_nLS;bfZmqr@qsLTMUTY uUgP`EqwhqoUYEXEi_op jDfZ=MSEe\\dbMWdpuXGr;IrWhmWxokHpi`pmHpmXpiHpkhoghoeXoepohpokHpkPodPoe XoeXn_hnKXlM@jQjUshhuWsgXUUChZplUsg:Zr<:XEeoTMEi_qHQgWhiwfkggaGhggfiWg cwgPUVwprdhkwgBioWhywho?gWf[GjkGhAXhmGiSWeIG][VdIV]gg_qfmawg]hi=Xjcggi we[WhcWgyyyyyxyqw;uawmyyI::::Z:Zy=::::::::::::::::::::::::::::::pIuaum w;uawmwsx::::fxtIytyyxyxqyxxqYyiywIyrxwyyqiyTyvIW^iV[YeyAytevaqxuitYyy qYyyyxI::::::::::::::::::::::::::::::::::::::::::::::::::::::BywQwuyye YyyiyqyuyYqQyeyyOYuIyyQwuyvuyyqyyuyx_ywyyvqywayyYyvyy`IwgayxyyxiYqixsi yuyywiywAyAuYyyyywyvYiuyiyyyyuiyKxhughUweyyxwQwqIxyyyyy;:::::::::::::R yyyyuauyyyyyy=bywiywA^yxEynyx=ryvY:wIyuyxpYxuiwMyqIsyyuYytYsIyrmyxiYfU wdisEUSKysGEtHesDir?=TBASEyq;qS;UrF=RiRPiSI=uKQTOEtKmtM]tNYtMMTLMt XquXuSIusghoghqv`qUQeh@ochpcHoqHodppdTgQwd]WeSwdWGeMGfUwc[WeAgeQWdcWfQ Gg]WcWwdMGfUwfkOTseirZutRYvViuYEUTmUTQEighkGh\\eolSQuT_P_GgPuTwpqxptLW]UViU UYuVMuT]uSMEXEiWgTEewgHUVYuSCgmGhaWgeWge?TEi_q<\\UVoppppp<::JXmUXYeq`p n`poQXeqxPTEg_olT;awfagggggk_sHqsxPiWgRgDuUUuPUuSmtUMURaUUMuV?qsx puxPoWghuSIUWEemXoe@ruPq?YraqwXrwpqaRytQyhqwhq@ir?ySGiS IMSD=TCISK=sBIt@yRLaR@AT;QrGeQyLst\\QZITM=T>erNYSFmTKEtNYdW`n[`kQ`la@p g`m[@nXpmXPnWpmaPn]`OltM]tL=tJmSGesKItKOn\\HmXhmXpn_Hm``nRpn^pnmHpUpn^ @og`ocHocHOptPqTQuTQytPqTRuTQYUSIUN=tJAfphsthqq@n^`m;hjV`i;PpJpl=Aofxs y@rqhogxn\\Hnexn`XpipojpogHod@obHoaPogXoo@qpdgcwe[geewf_wf]GfQGeOgdYWe MggcWdgGg_Gg`uSMUZyTSiUQ=UOyTR]tQEUVAUUeUQMuT=USUuRIuVIUTaE[WgptSEuTIU SUUSAURIuRiTTgnpPp^@qmHor`pi@qohoePooHochriHpFIpnxpYHnZPkFPoRHm?ypr`pc XojXofxpqThwWiSGgagcWWe[gf_GhgggSGcEgi]wfCxgmGi]wfeGf]wfeWgQOSusNYTO=u QIUVYuXyevpqxhpqxpiHpkpPcwhgwgiwe[GjaWgAHfaGd]V_iFY;fhovamxguglYwfCHhk wgagfLe:YqtHrsTg[weigf]giswhkWhiWhsghi_lxpo@poXpkpPaGhgwfgGggGhkggkOV] EnppppmhpoXpmtf>qpppoXPewfaghkGhmWhTEbgkGh_OSMUS_phxovxokxrpHqCQpp@pOH mL`hs_r`XowPn`HsnxpGyq;QQDUR=uTYuUGPewgiggvpohpp`pn@qm`pt`PPUUuUUYUXUu UmEoWggggggga_jpP[ggjSQenpppdg:PuT_pntfnppdg:::fqwDhk?gwgiwgg_ldfawfa? g_tlWieqlTgpqtgk?ZtispPqq`mZxkn?iy`nbprgHps`naPrnpp=AqthpgpoqlSWpkXpmPplTf]_kHpnPpl xplDh_wfk?vpkPppHpl`plTfFPjSIuSIE:bfZj@pjLSEE:::uWiugggoRAuRIEcGgcWf]o QytQkSIeqxpq@qrDgLUSEUSMUTMuTQemtfeWgeGhkOWeUW]er`P@E_gf_?gggLen\\tPQo _pdgc?ZnMtEasyiYy_yxiyvqyxqyxiYyixyiykYywYxwyyqawyYyyy:::<:xI :::::::::::::::::::::::::byvYxI::::::::::::wax=xIuawmwsxfY`yvaYuUYuyYy uyyqyymyrwytqycXmWXsx@ryqyxeydYuQyyuyxyixYxkAxxyywayxEytYyuiyyYyQyyix= jyyI:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::oixPy tQYv]YvyYyuiwqyyYywyyoyxWxkKhw]YvyiywyytQyrYxlqytIYtIwiyvyyuyZviyuyyyy yy=:pIuAvyy=:::::::::::bxdywiywyIyAZkIxkyyyyyv]yE^QjU;bgnPjUsg::FQ`EYgdQWjqo_IV\\qUSMUWqtQ]vWqUZEUTqSBEsyqyyyyxoyx qyxmxcxyqvuisqyyYyyUyxyyyI::Z:vYxI:::::::::::::::::::::::::bywqyyyyyI: :::::::::::wawiYsiypyygYjwsxZhAxj]yiywqysQyyiyxExqIxUyueHlOhoOhlyIywyw YIvyiyuivWYuyyysqwYyyyYyyix=fyyqywmysyjyyI:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::wiywqxwYyyyZvEysaxqyxQyruysyxyYyfyxyo_h nBqqxxyweykQxYYvOyuexqIyoixuyxqayyYwqyqyySyyqx;:::yyy:::::::::::Jyoyyy YoixxyyyyvCisGiQsDQc_gf_eYwei]Wf][F^[f\\Of\\YV^]fcKGdcv`kf^mf_[V_aV^_F ^gf^]VaeV_KGasFf;WbGG_eFbQV]seSKEaCf[_g`dTCMsYadcHmTXnU`m[xmYHocHoghOk f`oFYweYaUWeEYweYgF`kvcIwcSgdOWfSWeAgayfbufaageYwc?wb[wdSgeSoQgof`ofpn b@o^pn`Poiponppp`mXpmihogXqpxPYGfeGdOGitt[mTRMvNqTaqtRySxqrtgvPTUUoPjUsgnPJVChnPPEi?>QjU;bgnPjUsg::ZsLVUUUI UQytVMuTAvWmuYYey`ppProxpv`poxywqYwiyRyyyyqyyyYxyiyiyyyYxwApIyyy:::<:x I:::::::::::::::::::::::::::::::::::::::waxMxSyfYjwcxyqwyiaaweiyyyyyyY _Iv[iymixlyyyyxuyymiBiqy@p]hnyayxixgiwyUyqywgAyQy;jxyYymyyqyyoyy=yyy:: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZvmwiywuIyyyt Yy[y=wyywatYysaxoexoIoOhlQGb?wyqYypYygyx=oyxsayqyxyYwqygYyYyyAhIqAfYpI :yyy::::::::::JweYwwYwiywayYyuuyxYxsbAwbKCDKc`of`yyyDSCESEUSEuSIusDYsE Mr>]RCiSFyRFQsB]sDYSHeSJItJUTLITLySIETF]sOqSIitFesT=dJ@itOlo?ixxm\\HqV xmwxncHp]hn`@n[@o]`nh`ofXoeTaunAAsBmRAuRolPpLR=URJuSIIUQwn\\Dbov`ogfag hcOPUtMetLUtNYtMeTOetNatNUTN]tNutPAUUQuLetNmTQuTPeTOUtJETQUtMyUSMUZEUT ]uL]T\\mTQ]wWuuZMtM=st\\QB]QwpUNeTamuXQuM]tVqtPmUTMUUAURqtN]TPqtOiDKWe QGeaGfEGfWOR=UY=e=qojXqe`ofPoc@mhHoQPpcpni`odhoeHofPof`of@phpoppOswdWW hQWecgfxTS=UT=uQYuPqtViTPeUOwpeXok`oe`pqhpfhplpniDdQWdeWeUgj[WfIhdUgj= WcqVXyUhqFbKXfeWjaWgmWf_gf[OSEUSauVauRAuRQuSIuRmtPmtM]tSudw@qsPpj@pZpm XtfaGeSGesNH]R?eryLr;iRByrKaTN]UYqUWyUYMTMUTW]eIIsDYqjPpmhna@so@qHypuX tpHqgpkK@mw?jh@mTXroppvxppHpnPpnXqrxopXpnHqqHqqpppPpk`pipon@piHplde]oR IuSMUUUer@qnpP_gfxtQ?pj@pllSIuSEErgvPDUSoojdeagf]WhkweoGhWWhiweiOTaUV] UV]uUEuSEUOiTYAuSMvSUU`YUWutDaSBEquxtAMSdMuVAVNmTXytRYuREuTIeoXPawfJRE eiHpkHpmXpmXPgWf_wg]gfqGf_ghYWfswe]ghYWfigfaWga_mQkwgioTSgNqtPqqLVCi^ Q\\UV]uUchooVaesLV;g_pQ:XEi?nPTEi_q<::jSIearyyyyeTOeTJ=TJysIysNCb?gb;wb=WbGGcTTLmsJysEmSG]sHisJMTKiTPetOY TMytMYtUitOWn^Pq\\`nw@NIG\\QF^aeWuGcKWjSwe?xe_WhQWe_wcMGfOwd[geWgeWOU] UVatO?n\\PnkHPQwdQgb;GbeGf^poHQLuQ=uRytQEel\\h@pjPoipoe@pi`ptHqahofxnf Xop@qrpqxpqrppp@pj@PVpnPpsxpphqohp=YpnPtt`qNiprhpJ`lGhfdgrZ`nIqol`sk`p =IplHrsHQoGhWWg_geiGgXUUuuSQud=UTEXRMujgpQiPAxgawhkGfAHhcwjiwfygg_GiFq tpptXqspqrHqvxprxpt@qoPqr@qsxpv@qqxqqDjkGhwWhkGioWh_wgeWh?xikwj;Xfowg[ wf]ghgWgChhowlmgh_HgmGfWF_aW_ofnmWiWXi;hiaWguGhmWiuo^av^QuUYUPooqxPyWi uGiiGhkgfaghegg;xhswhmWhgggFqsXqwhqtxQiWiswfmGhsgj=HhwWiiGiqWhqwhywiLV [EvUAeT`lO`iqohwXpm@vEisxHn^HnxOjjxkMHshHpFQqu@rppp:aquxqwdhqWhmghpUYm uY=VYyUZuUXmUXiEq_whqwpqxpqyxqyhqwhqv`Q;HjlUXmUXciwgiw_tPqtPQpuXqUZEV[ Av[IvXyuYYUXiUVqUXmuZ=fydgk_v`q?yq:Ysx@rByptpsqXqPYqy`qXXnH@howspPqHyP SXhsWlqWiGxhuWjqGiduWeUW_qxpqtxqxXqwhQuoYquX=vWmUZiUXKYmUXuUXgqwTi^Qsw hNqtdhfQpE:::rhqwhq_wLXchdEDV[EVXme:mZAvZquXquW;w_v`qvpqxpqwhqwhQxuYyU Z=VZkYyuYuUYuuYCisOXsiywiy_vLXsi>RxuYyUYSifQ::tEy?fQlEu_ylY;w_xlXSi:nQ tE:::GHlKxdUWe;Hj;XmUXmgGgtUXOs?qrahn_pyxqyymxyiumwSxyiyiyy`YysypIyyy: ::<:yyy::::::::::::::::::::::::nYtIyay=tIry:::::::::::nYtIyaymxSyVYhIm yy`yyuYw;qawmxcynyvaYsYymYv_yyqYyUYr=yeAwb?IpkxywyytivEyyyywYwaivyyywy yvxrmyuyywyy=::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::JxuIfyyqyyeywiywyywiyyIyxYxqyyaywmVY=VY=uRyYuUyyqIuQymywoYvywayxQ ytuxuYyqqyiIoqyaImaumxyiwmyyInYwiyvyyyyy=::::::Jv]YvwYqyxvyyAYjycdV`mV `N`lNhywiyN`lNhkGhkEXkE`lN`lT@mR`kDHkP`lM`kBxjBpj?PmWXmO@mPXkH`kUpmVHl L@lMHlJhlJxkWHlKpnPxljXmW@pRPme@lMxhV?f^HkGHqZ`nfhlSxmHHl^xlSHo[Pnepna @nZ@N=WbuvayvcSwd?wcGwa=WbWGf[wcIoNYtMisFasPitOQtLQTCESCeSHmSMiTOAUUQU MmtOItN]tJYTM=tLMtJUtLEtMUTKYTM=TMQdpDdYGeKWdIGfKGdggcdUKMt[=TLavJUTGE qult=ibS`mv`n`@qdXon`ofpodHogHobhpbPoLqonHvVpncPdVWu;xjRYkIpt\\PnKqodh r^HnPIqo@qWHm:EeuGhpfeojbggq`mZPrg@pjxm\\HoVpmb@n[Xo`xn``Nkf`onIuSEmsGasIusOAURMtNG ocHoWXmUHnYxmcxNawf\\TOetEasFqTSAUNutPUTPiTLetNMTOatLiTOUTPitLiTOETOat P]uTaTSydfPoh`oehpeXorTdwgcKWkKwdSXe[WkGWduFZAV^gUX_WayNRAuUYTN=VQyTYI tMyvIQTyyTVMRHXltpj@pmXP_OSQEmoV]Ec_lPplXPLERfV PaGhkGhk_mlRcfNP:HEa?:nPTE:i_qaryiYyYydytUyyuivQytiyuQyyyyxyyyuy:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::^YwQysyxixMyuyyqixMyuYwQIuqyKwiYAMRhlG hkJXkDxgeOgh_ge_kGXkO@MUf^[V^cf^_v^]F^]v]cf^]Fakv_Aw`qvcmf`Ow^iViyFcqV Y?FYwtRmfZGvg?Wc]ga=WdqVaWwbEGeGOK=TJAtKITHEtJuSMQTOEuReSJysHysI]SFoof `od@ob@mR@mUhmWhnaxnnHqr@n_`nX`n]hm]PnX`n]pm`hnYxn`xm_`nWXn\\Poiho^Hoa Pn_Xn_pn^@o`hnb`n]Xo]Xnlpn`@qdTiQGeGXfaggwFbefXuEayeZuga=GmMWeixecwlkv aqygwgdODOiF?=SredRSwEkS[Ub>wFCOsdGDLSX__Fn_We=hdKGi[geKWeQWbSgdMgeSgf aoWmUWedcppqhp_Xn\\@pfXotppohoePogXodXofPo`Xoc`ngPo[`ocHnfHo`XocPrqHqt hnbPrchoW`jBhnHPlCapqhrnxpq`OeGf]Wf]WfWwf_gea?p j@pjYvYyAZ]Yv]aoftpqxpyyyyy\\Qv\\Q^PplP`mTPmOX lM@nUPmSHlK@lEPk\\hmVPnXhml?hi?gcwfY@nXXmX`mPHmQ@lMHlK`lLPmWXm\\hn]@n[ xmUplO@qaPolXmYXq\\pne@lNPjcggI@ivwpZ`nqPn_poW@nmpnbXo]`nahn_pmZ@nYhn^ `olHpTxmXXn_hnhoghWisGibpn`pPgWdMWdoFasVeYweEwcIwbKwcCWdKWcQgdIWeSgcSw dE_^HobPpmHpfhoeHod@obXocPogXoa`oddeSgeYWeuv`ovkywi;hfagfuva;XcKwe[v^w VXuEhYF_SPMmWOWv_XoAygoglLf]yiiCxyNPmfyhtgrF?dKx_iFlTgd:YmQxrUPmq@lLPp \\XnupPaWf[weUGeswgggfOgdChhoWeAwbiwdQoQytUEuROP]gf[WeFOXTSyTMAUQiTRuT WMuTmUQ=u[utRyTFmsB]QxTvQIUZiTQuUREuUydkHPW_h`pmponXPaoRAuR=Eawf@E]_nl Rse]_n`ppXqtpoj@pgPpkxonXpkppoHPe?[WgcWfaO[Qv[ISEMsSMep`pmpplHp=Iqs@pX xmLQqv@saXoyPmXplhWh?anbxrkXp\\@mRhf`_fSpmV`lPplT`mVHn[Hn`pn`pohpoohpo @QkGgVn]Xnj@PwgiwwiywigggxtQytUQuTEvXuUUutQiUREUYAuSeuQ?qjHprlUQuTQuTs giGgVpohpl`P_GgPEkggtUUOsqPqByomhslpptxm]`kmwhAApnPq_@oJiq:aqh@pxpP;`r @qrHoipo[ddCWfYgemwg_wggWgioUYerLVsgvPTUUOplDgLUUUUUYEk?nPTEsGisggVpmT gPuTsfVPPEe_whqwlUytQytSIuSME_gfXuUYuTsg>QTep\\lLTsgXuTsf^P`uVauVCgVpm XpolU;e_nLTsfVPDEi_pLUCgcGge?^PHEiwgPEe_nLTsfVPDEe_oLT;a_nlTSg^ppTgfpq PHEa_lPP:]?>PjTSg ZnlSsfNPDEa?:FPJR;__llRSf:NplPplLTMUTMEe_orm?hSqnVXsc`mQdetXquXqtPQiwgPEyay=:Jycyv YxI:::B:vyy=:::::::::::::::::::::::JycyvYxIbyZy=::::::::::::jxyiwmyyix My;RypYyuix=hyy_xyiykqyvywtAynayuQyyiyvyywyYwyywYyyyxsQxqiuiylivaYnOpl yqxpyyqAyyiyxeywQywYxI:::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::nYyiy:satyyxIyyExwQvqyweyygyyqyiAxhyyxqyyiYpIoIy]IrCYnqG jghjIxykIyyyxuYxaIwkyvcyyyqxwywmyyI:^yxqywyyyiYjyyI::::fYpI:sIytYyuiyy yxqyx]ymvayvuYyuYySweYwyyyXsEYSLMTLqsHqsEasFiSJySKiTNqREIsEusGITNUTGqs GEtKATO]tM]SDISBmr?uSGasNQTLUTLItH=TIqSK=TKaTMQTQitLide@pipn^`ncHn\\Xq gxoupncHqXPnDQoixnA`k_hjDxnF@ly@of@rg@plhna`pfho_xmZXocHoj@pj@mR@m[HNQ XlMx\\MV\\cdN_tdMWdewfaWfvoixoiLQ=URytRooj@pgxoi`qtHqp`pmPpjxonPpkXpnT f_Gf_wf]WggGgYGfWwdSOS=uQYuSEek@puppo`qn`pmxna@rkxo=yohprm@pYaqrpwg`oU @c@OnBwb^AnSPtrPoFyrrPrthoCypjhsdxn?Yn^Hsu`qwXqt@pkTicOjAV[aWOqTvpm\\ \\ShLpamsKMwRQu\\QUUauUUespphLTaet@qrXrp@q=qPeGf]Gg_gfwwfegjYgfcXhuW`m eYEXcOWmcGhEHgigigOUUekxppxoqppiLS]uUIUVYuSSfgoRUuTIeqhpqxpoLVsiuWiwGi sWiqwhsghogho?:jaqV`MUSAUYmuWeuWauRUUTYUXeUSaUUAV^QvUmeu`qtTgcwiYGfMIm Yxhgf`[gZKVl=gcGXfPVWmU\\uuY=vWmuWauVeUWwpsHqoxpqxpsHqAr:Ar[Hn[hiw gikGhkgiwc\\SF]kfauVcUGe]ggeoTSgpuWGru`qyxprXru`q;YqvPrvhqaR=]R>]r@mrxTr;iRDAS;aR=AR=MrD]SEISDES?ir >uSHiSFMsCUSCAsCyRAESByRFasEmsIqsEmSGurDEsCaSF=SEQsEasFqsGisIMSEItDYSR iSIMuGycPPmSpiuReQuVIVQ=U[eUWEuTIuJetM]tS=ULqTO]tQmtPudx`quXP@ vXmuVQUTuTPiTPuTQeDUweYWf]Wf_gf_WfYweSgcIgc;WbieVaeamf`kwfcgf[GfWOQ]UW euWquXEUTMuRse]OPmTPmTOeTTyTRquU]UZQuUmuQAUgmVa=uH=tCmQ:utDas_=UTEvMit bIUVEvP=UUAUREuTIuP=UQmtQqtTUUTQuSEUTyTQaUTIuVMuSeuTMUWUerppoXpp`pjhpm honHpgPpj`qxpqoxpqXPLEc_lPplPPLUTMElUXmuVsfaOU;]Wf]GgLURsgiwgqwhqGh\\U TCgZlLTseYweggggWiuoTgpmXpm\\n`pnPPHuSWPLEc_ohpohPTUUUEHE_gfDEa_nLUUeq xPc_n?cPAnZ@skxotppk`oexnaPobPodPoe@odHp`Hokpncho`HofhnbXrrPqo @odhp^xnrhnbXpZXnwTeSvWoEjUGfsgeduPyTUutQAuQyTPuTQeDMgeUWdWoRAuRytQyTR CfnOlTPmTSEUSeUWeUQuTQytQ[jabET\\MV\\M^ELAuRA=SB=SCESCAsBAsDQsDQC]V^]v]Yv]cF_cV`mV`m^PplPxmWhm YhmW`mV`mR@M_F_cNEWmUXmYxmYhmRHm`pmYhoYHnrhnbXoLplWxi>@oBhk[Hj@PoIPlgx lShqe`on@obHmNXlTXmS@ogXOOwcAgeQw`GwbwvcCWcSWduFbwv][f]kF_avdIgcewfaWa qv`mv_ifcCGcXtMedb@pe`ogXoexn_hnp`pnHoatcEWcMWdMwc?wbcwcMGlAWc;xU_USQr qrdfvoiDfFPiru@plPrk`pJIqw`q[hn=Xfb_sbpoFao k@spxpwHpk@qj@pn@piPpjxosPqr@qsxp`HoaHpndfewfaggcwfgOUYeoppnxpo`PcwfsG iPuTQUSsfHuUQuTQuS_pndggggcOTquXqUV]ej@pj`pnDg>plPpnpppPplDh\\uWiuWQeo PPHEeWgPEggggWf>qrTiuWie_kLT_pldfeWgPEcOTME:a_mLT;e_ovVhmOQoaPr[pmGApf@s^`n@an_xQIghCgcmgbGWfCgc;gc lsMQTO?orhoiXroxpypokxqhT`Kv\\Cw^gvhUwetdspojxPrevoglO=uQqtQgoghoglQyt Q=URwOlTPwoixoplPqdhLQseYORCf:vP@uRAUQuTQ=URGpkTfZnlQytQuUYuuQSeFpkHpf `OawfLUSEUVIUSyuSIuXiTPuVU]UHqQ:esvpQ\\OoaWhmW@yrAy?hj?@kB@kHpkHxkIxkF`kFh kGhkF`kF`KpB[F^[V^]V^arAyrAqsHqSB=cogho_hn_hDPkDxlOhlUHMs^R @mR@mTPmVpmXHmXhmP@mRHmUXm[xmYHmNhlXplQhnRPmo`NCXe]Ge]V_sW`uV]KEU]fU_M HysVItLeTHqSPMTLitN]tEmSG=tMMtIUtKISGYSFySHITOYTCYsDEqtHQyqrTipuTSfZkLUSgeoTCg^PHuSIuScf^p nTgeOUcgfPLE__klSsf:::>pjdf_gfc_oLUcgfPLEa_m<::jysy:::::::Z:vyyuy::::: ::::::::::fYpI:siylIbyuQyuix=xyyuyycyyyxyqwyInyy=yyy:::::::::::::::::: ::::::fYpIryyAJyuYywYbyvywyYuavmxuyyqifQv\\Ik;PjyiYyqx_yueywQywyyqyyMx uityyxYyyextyxwywmyyI::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::ZwmxoyvyywiYyiv_ivqYxmyywyypiwqxsyyvyYmyx=oDaw_qvymyxmYuYy ymyyOyvyyxqYwAjywYxywkyOYhIqawmyyInyx=yAZuYYyyyyyyZdPod`]V^]IxkIHysE^[ F^_NCyrAyr>ar>ER;Er?ir?iBR]UV]R^]^Jd]Wf]UV]U>_^MXlMXLhBiuWiESCESgV^_Fa iF`oF`kv^av^cF_cf`snG=tI]sHmsEmsGUSF]cQxldhn`t`sv`WF^Uwa=Ggyfbag_mfhmV agVWmE\\MEUIG]YVeiV`IW_gvi]gfUwbAweQgdMwcGWa;ga=GdEwdcWfCWeOw]kV_kUYsU TQuTgEXku_sv`CWdKWbGWcEwcIgcKGd?GcCGf_OKMTLItKccHtJAtOUdh`n^@pWPmVqpnP xg@peImXpt@XkFQjBHuSxmPanbpq[`ntHofxo`HohHoepocXoxkbofShghomp?ICV[Uw^aFf;WbWGeQWf_Gf?GauV bgF`sv]]f];fZEw_mvdwFbAhgkGgIWdgwdUGgSWe_GeUGfWOQudfpohhoiTeY_gDeSoPqt PytQyTPsdQwdWgeWgggggYwexDWgetDreYOR=eklUYuUAEUWeU?beZlPPUWeUGisGiWOQq tPGPhtOwokHpfHPGWfWGeegfChj=PMUt\\qtQMvL]ThYuWyRotpRuRCIVLYT^EUTuVYyuR atNeUUQUTQuSyTS=UQ=URutSEuQMuSutSEuOAURatQutNcdYgeKWeSgdYge[gf_wfeOUCg coQytQeTOopndeSGeSgdDUSEuUSgiwgoGisWjAxj]Wf]wf]WfQUSGuacf_cwcMwgGo\\uT SyW\\]VPirAAUBMSSQSFmuMat_yuYquWeUPAuQmtUME`TTAuPMuRyTS=UUEuRauRAUYEUS quSIuZiUXYuTQUSsfaoTgpmXpm`PTepLTquX?Q]r@qr@qBOf\\Of\\^j>`j>L=UR=ar>aR @mR@qBS^DPkDhkGhkB@kBxJpr@qrBAsBACcF_cv`qv`adMYdW]eVcf^_V`iv__f^_V^av^ aF`iF_mF`cf`mV_qf`Uf^]F^_f^a^WhmW@PXjbxNaXWd= wyyYwyyymyyoyyyyxmyyI:::::::::::::::::::::::::::uaw=hiyiYpIyyywaxUywyx eiwyIyjsQxmaUNiTgmv`yytQyyiYxoYyyydywaYuayumyyqyxSyuYyqyxqIysIyyyyyyy: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyuxvxyuwF YdIuyyxyyoyxqyyiYpIqavqyyYyxUxsIwMIushm[hkmwhyixxYxuisqywAytUywIyoqxyy yUyweyvyYy]YyoxvYysqyvyyyuywaweyuqyyyyayvwyyyyyMxmIfxoixj@PWcEWsSIuSyy yyI_cF_WeUWU]UV]U^BL>;:Of\\O>Wf]WV^]V^Yv]Yf]Wf]SF]hr?_kF`kHpK[F^[f^DsJ msHQQr@QvHaNPlL@lJ@lH@lJplUPmHXlLhlUPmRpmWPlRxlPXmT@lO`lLxlP`ncd`yfamf asf`uv`=Wasfd;GbgWbAWgkv`KwZIF`[UW_g\\YvhkFaeG`qVekf`lSHOn[`n_XnO@mP`o f`oYxmY`n^`nfpohxm^XnPXmThm]Pn``oePnbxnh`pm@n\\PnVpmXXocHofPod@nU`mepn a`n\\PnaPn]`phxoaxnaXm``nVpogPlahnX`oeXo_PokXncxo]Hoe`nchna@oc@odxpapo ManfXv<`kCP^cnpqGi@Ao]`nUPlixpf@nhXn^`pcPneHn_Po[`oc`niHo^@pdhnpPo`pra @oAynb`riPpupojxpfpomXofPpdXojPoe@PVofPpgDfWge[geptP_oftdQwdMoN?OUWeUw dQOPmTPqDRdnOWWgPuQytQyD:::iwgiGffOQpmPsxpqmXoePqlXpnpO]GeUWfYwe[_gxOUgf]WfiggQWf[wd]OPAefLR?PNpj@plLVIU TYUSIuSAuREUR=uQSeYweSWf[We@uPEejPplHpmXPawf[gfFpkXphxoohohppbHo?ipp@s pxprXn_@oL`l[XkGPm?xjH`iw_mFhkqHodxqmXpyxpqHpg`O[we]Gg_GeYWewGj;PTMuPq tPEUSEuTSf@uQytQIempPgWj=XjFpmXpm`qvDhvpqxphpohhOXuUOqtDgcGg[_mXPg_plR ciwgiSGe@vZAVTMemLTMer@qr`of`osHQiwgiWgeWgIwcIWj=PPmTPyuYyUTMUTMErfVPD elPPjUYuUUUU_P:DEawfaGgnPPEe_n`Pc_mlScfNPHEa_nLTsfVPDEyyyyyy:::::::Z:j yyI::::::::::::::::siywYyNyxUxxyvyYyYixaokPptnayFyuYyyyyxIpg`oBgb;gwdi vyayryyyqyyyyxmyyI:::::::::::::::::::::::::::uyypIqatyyxQyyqyuuykyxmyy yiynYqIypYyyyxuiuiYYYvWHj=XnEXkyYywiymIxgiv]yyyiyxiqAYwiywyyyyyyAZxqYr y:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fYjvuiqyYm atmxyiwMycyRypYxuyyqIuavyYgYxaisKismxpQGbAwyqatyyyMvIYtyyxuYweiwyYqau= wYytyysywAfYyyyyYxmqyyyyyIoixoyyyyywaEYsEqcHocof`oFysIyj?hjolPplSHmSXi uWiEXkE@kB@kBl>ar>aBQv\\QV]UV]W^It]nkGhKU>ZIxkIHlKHlLPlL@mPplFxjBhlJHl WXmU@lJ@lH@Lmv`qf_qf`aF`iv_uFaaV`kv_uFa]v_gf^kv_?GdIw_unE=tHaSJqsHuSHE tHmsNqsHyTGmSSYSGMuD]S?iPqhRpxpSUsF=VNeTVIddPmUpmLPlt@pjHqm`p_@n[XmSHm XpmXPn^`ndxohpnePoNPmSxkO`lWPn[xpsHQGONUTM=tGmSRetOUTJAtMATKQtKITP=uRm SN]TCYtLYsQqTAUTLqsOeTPUtOEULetPItNeTLatMUTN]tMatRYtPUvLmTmEtNQUsXQ[=T JauQ]TLYTImtTitEETFAtQMTIitKySPMtKiTLQTPQTOmTMMuOYT\\]TNUvM]tYmtQeuPyt TqdkXofHpe`oiPOYgdtDYWeUoP;^oftdhTOeTOatNgoghoeXOS_ghORdnOPeoxOxD:::io U=ESGeSweYOREUSyD[_jLQuTQmEWgeldf@pjddFPDE_geUgfVqste[geO_k`plhoePooHp jHqixowxojXrh@psXn_PnFpkRPj>xoT`mIiqxxp`HoxHpm@roxpp@pkpocPodPodHohhod @pi`OlTSgOltR=UQ=UR=UROpjP]r>abEl@ qbF`kFhkGT]:nKxrAyrAC^[F^]V^]V^vkGhkRplPpmV`mSxlQxkIxkIHLcf_gf^if_[V_c f_qf``SFasIuSB]sE=SFYsJYdNPmSxlVPmVxmWxmZpmYhmVpmTHm\\hmV@p`pnv@pjPoTX mc`lPhjiGhV`j@PqW@nvxm\\prc`oYxjCpqg@pe@n\\hnWpm^xmZHnYxm^DcAGdIGa=Gbu fb=gYAfZceWgedSGe=wayFdAGcyV`oFh[WfUgbEWd?WcQGf]GaKoEQTLmSPitEItKItM]t RUtNMuL_o_PochndHOhtLetSQtOmuLet_UTOiVPqTVIUR=TNATLytM]tQ]tHqsEuTQ]TLe tLysOQtHqTMysPYTMmtM=uOOqeHoxPocxpd`omXofDeUWfSWebelTPmdcXoePof`ocLPmd firJMr=euGuS_gpAapqpqjPplxN[gdPuSIeotd[weSgf]Ge_WfU_glQAuRGPaWf]wf NPDURWpkHpnPpldfJR=URytRAuPytQuTR=uQCf_OREEvOxtRWoh`ojHpihpk@pvPplPrjH p>Iof@qS`mc@kFPpKhlKapqPpVpmIyq:YpeXol@pixol@pY`n\\HpmT_iv_quXqUZ=VZAw bAGisOUUUUItKITQuTQIv[IVUUekHPgggggg>qr@qhpohHpkHpqxpq@qr@qmtfUWelTP_O htOWpmXpv_I;B:Q^Fl@s]Yv]]V^]V^:VlKHlL@lJ`kF`kJ@lJXmUXmTPmTplPplUhmW`lPplQHmSP lN`lNHmRpkMPlKplOXnbxnUhmW`mTHmUHmRXmVPmVTakVaqv^ov_ifaqfaEgbaf_cvdIgc uv^aff;Wb=g[MvZ;ESXR@qUFms]aTQMuHAtQmsImTI=TMusIYTK_n^`nQHmS@og`o\\xn` pm]PnYHn[xdQwdIPkEXo_pnXplQXo]TdAGcaGf_gcEwcGWbAwdKgdsFb=WcCgcaGcKgfCG dAwcKGbKwcIgcKGeGwc]wbIWfCgc[GdMwcSwd[vcAg`]GeAwdOWc=GbmGdQgf?wbSwcKGc Mwc?wdKwbSoMOoc`od`ogXoh`oeXo`xnfxnbpocPoh`ofhOdDltOUTPitLiTOUDEgdMwcM WdOgdhtOwogDfWgeU_eLPSeZcHocpOPEUoPqD:::gOUytQWOptPOpj@PWOPmTPKPmduXqu xoixohpO_OSiD[Gf[_h`oihoh<:\\uSEuPiTOyTSwnjhodhplXonHpdhoePpg`oyHpl`rf pow@n]`mvGjbPkIxq`PoIQqwhrqHqkpnbHo^hnghOWGf[Ge]?pDBfbfJQudixO]Wf]Ggfp mTgaOTEemHpkPpjLROpjLRGpkdf_OREUSyTTIUTiUVUtPetL]TMMUSOqlPpqHodxqd`oKI quxpVxmvHn^pjh?huhnaxp_pn=YpnHrohpixnahr:IrtPqt`OSwbAoOadqhpoXPaGdKGd_ gfhuWiuRAeklJAtJar>aRKETKQv\\QVTMeolT;__ghogXOdTOgn_hn]V^]vayvaqxpqPpl tgiwgMWdMGWcEWcB?crUYuUKGd@uRWOlQxlQPmTPMUTMuSIuSIE:::]oREUS_pnTgeOUcg fpoDgNP_wfaoSCgcOTSgfPTE_?VPLEc_mlScfvYxI:::::::B:ry:::::::::::::::nyy iyxUyyiytYyOYjuswZeAxkQyyqy`qv`icC?cUplMxyyYyyiysyyyqYyiyvyy=::::::::: ::::::::::::piyQyyUxtyv;uyypIyyy::jxsxxYwsxyQuayuqyyuyy_ywqytuxmyyyAwA uyxqayIymaxyYywyykYxZxkLhx^AwyQxpyywmyyyyCysAxIyasIYyyyyyy:::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::nyxey:ZymxSyNyvayuiywuy yYynYtIyayMwYyvyYxCyuywSiuGYpqHgIWdqIvayymIyyyxwyyuaweyqYywiypyyOyyewq ixyyyxIuYyryyawMweiwmyssIysyyyyiKGdKoeXoU\\MV\\WF]Sf]SF]CF[CV[EV[Iv[`B Q>Sn@S]NkDPK:vkID^[F^]V^DSCEs@qr@mR@mrBAsBUSEUsCIsCyrAyRCESC]SF]sDQsDU SF]SBESCMSEUSA=SBglNXnbxnVpmXHn\\@nSPmRhlP`lQPmRhlT@mLHmPHmZhm\\Ho``oi hoQhlNHo[Hn^t_Uw_mVgkv`YfU_U]SEVWG^eVh;GcugcOwdkv`wV^aFeCWcWWdOW_eV_yV b=wbEWc[gf_wa>ob@oagf_?im_hl`ogxn[PnbPn]Xph@pc`n`xnZXng@od`m[dbGwcWwbG weCgc?GdIgaMwcAGdIWdIwcUWcGweCoSutQasIuS@QtJqSUEuLetNatJEddhogxm[Hoaxn ]@OKGdKGdMwcIgdKGdQWdMGeOwdSgdOGdKOMYtOatNeDMwdQwcSwdbcZ_lLCcMGdIWdMwd QwdWGeSGfWgeU?^ofTeZcTdW_ohPS_f<::JUUUUydiXoeTeUoQytQqtPqtOCe^OlUXmuQy diLSEUSiD[_hdeWweUgeVqsTfYge[GeVplDeOweFoh`oc@pg`okxok`Pggf]ggWgesgeYW k[gfew_svdQF^Kg\\YVgsFbyGeYgi[gfggeYgfUgeWOQmTQgoixofHPDeiHpk@PDUS?pjD fFpkpj @pqLNC[CF[MV\\MVauVaYweYweFpkHpppppxpqDgNP_GgcOGeSG=J:p jT_eV_uUYuecGgcsGidUWeUB=cQv\\Q^f^_fUXM_gfDEawfHE:cOTMEawfHEc_olT;a?VP HEc_nNkDT]UnBAsBIsCOmV`mGxkIPlLPlHpkHHkAxjT@m R`kDPkL?dJwkIxkOhlOXlMXlSHmS`lN`lOhlOXn_hnPXmTplUHmLxlOxlT@mW@nXXmVPmN hlMXmS@mchn^@o^XnYXmT@oZ@neXn]HoYxm[xlQhmKHlf`mW@oKhlGPhpwlv?jw`nbHRYG dif`=G_gfdyVbawdSwdEgcQgcIgb;GbEwbAGc?gbIWcEgawfa=ga;wb;gb]GeWWdCOLAtK UTOiTDQTKmSO]TLETK]TKItJUTL=tNUtIed[Poa`nc@ocxNSwbGgeQWe;GfYw]yfaKwbEW gOWe?GauFeSOIUTMAtLUtPMtMAuKWo`xnb@OMWdhtNgoeXOdTO]TNYtMYtLYtMMtNonePo `XOKWejLetNMD\\TNaDU_hlPsdhTPmdg<`dcpOPEQ_e<::JU;MWdMGeSOPuTQeTPmTOsdV OhuWiuQseVpmDeSOR?pgLRutP=Eighkwd[geIgeQgeegfIGeOWfnodHos`pmpqpppk`OIG dGGeYOQIuREURydYHnDiprxrhPp]PkHhlxOjiPmWXrjXpxhojppcXomXofHPxdg`OhTRyt Osd]Gf_GgcWfawf]oR=E]weYWfYwecwe[Wg[WfkwfNpmXpkdf[Gf]_ixO[Gf[oRopqhpgp ofXofPom`plHpnPpcddOWeLuTIuVYUUetLMtYQuTmVYyu[uTReShhogl@pj`plPpnPpl\\mtfa_nLT;iwgigg^pn`pmLScfVPLEggggWgeoTCg ^PHEa_lLSsfVPLEc_mlScfvYyyy:::::::Z:ryxI:::::::::::::::::uyyyYy:Z>pj@` [F^[CF[Cnhn_h^Qv[yyyyyxayvyyyyyy=::::::::::::::::::::::jxSy:vyy=:::::: ::::wiytIyiywyyRyxywqYZutQMwZMvyIytyywuIpywiiyqY^yxmyuyumywYYxiywyYyei wayyUyyiyyay=tixqY:::xyygYpI:yyy:::::::::::::::::::::::::::::::::::::: :::::::::::::jxSy:vyy=:wax]wyyxiYyiyxYySyyQwqYaAv[ivXyUuYxoyyvmyymYyiY veyymYyyyxqYymYxcytYyktoxrYysayyiybIwaQr=IrIisFyyyuYuEYSf\\MF[?VZCFZyE ]OV\\AvZAvZvj?hjD@kBhkEXkGXkE@kB@k@pj@xkIxkF`kFHkCHkCL?eBnkHd]FL@C::b^ _nghogTQuTqLQdJ@lJXlMXlUXmUhn_d^cF_iV`mV`qv`\\SFisGisIysIasFaSHeSGmTOe THmSHqsHqSKqSIITIysKusIEtHusJiSHMtFisUQTMitDYSEiQylrrLqIYr>mUM]TY]TOYT CMsSIdwxojPoZ@n`pmX@o^Xn\\@nYpnahn[`n\\Hn[Tb^plPpVPMOGdPdTOGPxtQytQqtPytP?qrDeWgeKgdOgeag fMwdysyx`xn_@^Xf]kwghgyuQyAhj>pkFXkIhkFPj:xi>Pj;xjA xjAL?]R>ir>aR?]R>mR?gKUv]Yv]Wf]WF]SN?eR?eBU>vKJB=SB=C:]V^@C_f^HsCIS>]R >IqsHqwhqwDSCEsKItKmSHmsJAtJisGisEaCqv`kF`kv`vmYO\\DOgdOGeSGeS_eLPmdghOlddddYweXeqXoeXO^OpDU_fLPsdfpoD evnaxndPOWgeWgefOpDOgdOWhmWhUoPmD[GfhtO?PxtQwogteUOV?qhpohHoeXojPpl@od Poj@pj@PlTOeTTEUSeUV[ghOWGf[GeWoQUuSetQqTSutPiuRAuVYTNqtDYcFxki@lMPuw@ rwhnapqfhoAiqwPpjxoPhmTHpyXqUHo_homPpjPP]_kp jTf]oRcf_OREeiHP]WfDURgpmXpp`pn`plPplDfc_rXpnXpgpogPn^@m=hjl`mXHrehoih mXxrsPqtPpl@qnXpkPpjHplpowpqtpokxolXpk`poXpglUEuSeuTUuUEUTYtK]tY=V\\iT P=UQGpAGcD?]Ov\\`?f[_\\Ff[h_f^OqhpoGYr=yof@ohxoe@ppHpbHphHoktbSweYGgiG e[Wfif_kv?_r>cuUWuiiOTqTOAVWYU]AVZmUVYUPqtOetPids@qwPr;IqsHqppppXpmXpk Hpk@P:xDY?Fpkdf_oSIenlTQenLSOPVPLEc_mlScf_oysy:::::::Z:Zy=:::::::::::: ::::::jvaivayvyyyuy:fywiyuuxyyyyyy:::::::::::::::::::::::::::::::::::: :::wqyiYxOyvYwRYufHmVxymiXuYyyyxoyxsixmixkyxwIysiywixtywuYxiYymYxyyxyi xiyyyywyYyyyJyuYyuI::yyyxIuaw=jyyI:::::::::::::::::::::::::::::::::::: ::::::::::::::::::::ZxmxcywqyqyygYjwuYyIYuYtFqSe]V_uyrIyyaYxmyu]yyqYyw yxiyuuyyiysYyqyyyiyyyyTUTIuPhLOtpqppyyYymqxhQyxqxtrAqR:Arxlr@ir ]R?c\\>kB@kCHkCXkEXkJ@lJ`kF`kEXkEPkDPkF`kFxkIxkJ@lJ@LxrAwK:JBc^VLaF_cF _yvayfXoeXsDQsd]WNIuSIQsDQSJ=TJisGiSE]cRPmTxlQxlTPmTpmXpmR@mRXmSHmfPod PMo>r`qv`sFaxsIysJysIQtI=TSUTM=UJAtQqSIqTFesJir@QsxxQFqQ:IT>irUysJIvPy TS=tJeTHqSO=TJATHisNQTLon_\\WXmUPpj@peHoc@nXpmswhqwhqC_cF__Gf[WeQwdEWc LTLMtLQtLUTMUTMScEoMYtMYDBcGgcTDfNPDI_a<\\TN]tNSdNoddd>ob@ob<`DM?NO`tN wnaxNZcLN]dbLOeTOmTPWOdDQwdQWeUWeQoOatNatQwoppPUWelDS_elOCeSOPSepTT_pg hogPOjPsd^OtTQuTOeTOauVgogDe>PdD[_hpohhOpDmWhtTQOO@EOgdOweYweY_dPO]_qx P[GfptP?pjDefonHphXpk@pkDgYge=xfcGkYWfag_mf`qeYag_mViPtcuUZauOmTQaTNuu YquK]TMAUVQuPEuRGpkxokDf_gf[gfJSEE>pj@pj`Tf\\P>[:FZKp jBHroDm?hjUgdGWf_geQGfWGcSwdigi=Xd_ggUgfegdWwesv`uF=IB]R>ir?ir@qr@eR?wjAxj?hj?\\Bl>k?wkIxK[nAybHpKW>:>l J@lLd^cF_cV_enCIsCESCESodPotRAur>ar>yBIwcIg`oNDUSEUSF]sDQcR@MsFasV_eV_ mv_iVdIOHmSHisGiSGecSHmSPmTDasFayVaHTIysNAtJItFasKQSEytI=TUMtLqtEasFAr ;eRsLqNAsCauKQTRisHqtHuSREtKUTHmSOETKUtJ=tP]tNITIySTyTRMTKETQmTPmSHmS: =R:]PodPE]SFysJAtNitOmTPWn]dasFa?Gb;WdMWdGgcTDRcEoL;:CGcLDGgcXtMYtNatN aD:IoMYD:KGd\\DOgddDIwcjNadcLNCdVoeXoghOMWdM?M_eXoeHOXDUWeUGgcOQuTQSeQ _dPOO_eXOU_hpohXpmXphpohXoeXofLPcddDxtQyTOchoghUWeUGevoixoclQytQqdg\\g hOkOVuDO_kLOsenohpocHoc@pj@pqxP[_ghO[GfvOEGeMwecgfcWgawfYgeogeWWjYGfuW cxTDYs;]po\\VREuQeSHEVSIuYIuSMuQutTIUSQUUMuPytQuTRoOjQCf_gf[gf@uSIuPyt QytQ[j@P[gf_we]WfBfNplTf]OR=eixoihoixohlRcf_oNatNyDawfagfvoixohpOageYg hYWfQhgkWm_WggW`sVZGUTswcMWiUgeChgiWhUWe_wdKgiqwgUgeSGeYWe[wf]GdUwdWwf _We_WfIwdSge_wf_WgmWb`TMiTPEv]]VFecFN[Dn\\Hv[gPn^ppf`o?Yqu@pgHonhpkHoh `oaxohPnc`o[PohxonxppPquppoxpmn_iNc?_bmXoePs;qqHxj>hhjggPXlI`ogXo]pn^H oeXo>qr@iogteYwe__ohpo@qr@qqxPeWgeGgnpplTWPa?]?VpmDgcOS;a_nLTsfVPDEyay =:::::::<:xI:::::::::::::::::::sIyxyyyyy;uYyuIxkIxgiwgywiywyyyyyy::::: :::::::::::::::::::::::::::::::::ZxqyxyywqyqQXlyxgyxAx`qvWHlK`yjAxyIys yywqyyqYyiylivqyvYyxyywyYwYywqxyyyiyyyywSyyqx;yyyJyuYywyyyY:jxSyvyy=:: :::::::::::::::::::::::::::::::::::::::::::::::::::::uaw=jyyixMy;oIxki ysIyyYy^yyyyuiy]qtRAnMhltywkyywqYuiymIweiymixuYxkyywqyiiexn`p^^^^OgdOg yyqyDxkGHj@`j=@k@@ggWgtwiwgjAxj?hJCF[Cv[Iv[Of\\Ov\\Qv\\Iv[IvZAvZGf[Gf[ ZCHkC`kF`kHpKUV]lR@;fKjAyrAAsBASDMSDQsD_kF`k]XNCF[CF`kF`Iv[If^_f^UWeUg `of`cF_c>_f^_V_eV_mV`mv^av^cF_LTLMTIuSIqsHqsGisGasFasF;of`psGisMQTLitM UTOETKOmThnQ@mfPmUpoSddcf_yUTQuaOF]Kw^eVfyFbcGcEwcmf`QwayFdwfa;G`mFd;W bqWggWakV`=Wb=GcGgcqFbyv\\[v]MuUWeX;vYef`mFdUOP_o^Pn\\HnYxMAwbBcZ_hNE? :Fn[db?gbCGcC?GgcjM;:GgcTDI_b@OOgdhtOwnaxNFocTd>obDdNOlTPwnaxn`<`tNwna TcEWcOgdOgf_gfWgepDQ_dPOO_elPqdhpOawftTQ_of`of<`tNatOitOytQWoedhoghUoP mDYwe`DWgeJP;iwgiWeUOOSfFocHoiLQCdKGd[GfTUUUuQydghO]WflTQutLeTNetQqdmH poHpj`pfdgOgdwoPQVREuXYtNeQlTpX]TOOn]hswpqiHn\\`pdPoo@pihogDeWgeOweWwd [weSWfltR[gPPOweWGeWgeUweYGeWge[Gf[Wf]oR;BfvOtDW_ixO[_klR;awfaWgeOTMek HP]GeUWgOGeMHggWngWh=XdQWiEwcCVTSUguvaChfaGiWweqWf]gkwgicweWweUGeYggaG d]geSGgTTQqtNqTQitQ=URQUUAuVauLytQ]SJ=TXAfZ?fZ?dDWcxppq`o\\Pnyxpq@qmPp sxpp@ogXoeXplxopHqbHpoXojXp`LNYtN]ocLoGTkDPTJySTqTO]vZuuT=uP]ryrAysIysyyyyyy=:::: ::::::::::::::::::::::::::::::::::::ZxUyqayyyyYytyywiyyAyrAmF`kjav^yyt ayyiyxyywMywywaYyiyyQyeyxiywyyyeyyuiyyyuyYuawmyyI:::fYpIyyy::::::::::: ::::::::::::::::::::::::::::::::::::::::::::jxSy:vyy=bywiyyiyxIpyyiYu] yyqieYv^yxdaw_@mTHvSQuyqyyUywyyuiycIvYyypixiytppo`aq>ajoghgyyuYCfZeEYm E\\YF]]EXeEQAERMv\\Qv[Iv[AvZDR;Erar?WkF`kFxjAxj@pj@pJXr=YR?ebEl ?c\\Of\\Yv]Yf]Wf]pbFl@qr@=SB=sCIcOhlOXlMXlSHmSHkCHkUXmUXlMt^^n^d_gnCIs CMSDMsBAsB_LkF`DSCEsCIc]tbuVauVaVMqf`onFacR@mR@MJMatMetOatMITKuSEUsHIs CatGmSSEtKEe[hmBPk]pkJPlwwiThjAhoU`mixmZxo[Pn_xlR@oUhm_PmVHqkPp^xmZxna xnZhn^HnadbOWdwe[EfX?VZ?F\\If_qf`GGdKgdKGdMGc`TLgn]Xn]XNBcCOL;EWcEWcNn \\Pn\\lKId]LL;E?>ob@obLMUTMUDrcFocddOOO;RdMoMYdalNsdQoMScvnaxncLMScEOO eTOAEU_f`OS_eXOQ_f`OS_ghOeWgeweYweOgdlTP_OhDOgdptPWOQghfOJQuTQ]DUWeU?S _flU;MWdyokxoT`mGAqthqe`oqXoePpgpofpOSgedDOwehDQGfY?:UweptQ?PPPtTQ_ofdeWge_gfTUUo PPuT_PDUSOP@uRGPJSEemXpmXPLUT_PHuSWPDUSwYxI:::::::B:ry:::::::::::::::: :::FYmyyvYjxqyxYyuYYv]Yvqyxqyyyiy=:::::::::::::::::::::::::::::::::::: ::jsMYsyyymyvYYuiyuUyyuyxUYquxyiyxYUOmTlEwceYoixymyxoyxeytYywyyqyt=yYx yiymyywAqaumxyyyqIyyy::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::jxsxZw=jyyIbywyxmYxmywiyyuiyiayiyxyyvyYiIxUgbCGq[HnsIv[qxqyy qIuaqyxeXn]P__^^_GfZ_xoYx?@k@HjBhjGplMHipjdenOTuREuZae;ipo@rhxoEqprHpY@n=qojHpSXmLPivwoSXmAippxq mXptPplPqotfYgeaWgeGe]OOyTQ?PYWgcw`AgbOWgaGgyWi?GdIGf_oSAuRauSMUWEuS=V WiUTyTREUSEuRIefHpl`i=pjFOdQ_hqWiNplQpn_xn``n^X`in_N?dIGqnHpqPpihplpoo `pjXpk@pgpof`ohDf_gfDeixORfNplTf]oOitOMTLMTR=UR=E]Wf@EBf>PxdiPpltfaoSC gcOTsfHuScf_oywyy=:::::::]R>ir?_KlB_f^piPojTd^olHpeDfS_ephpoihogTetDWGf[we]_kDfnOWWe:FP]wgiwgg ggxdi\\kLUYeqXpl@qhpopPodprrHq=YpnpqbPoppmZ@mw?jt`n`hqgpox@pkhqkHpp`of PqopprpPiwgiweYweUgf]wjMHlofb=gcWWeYWgcwecwfkgggWdCWc=xhsgikWhOWcGweYO U]UVitQ=u?=sCAkDdK:@j<`j>XkEXkEL@S]Un@c]Wf]UV]UF]lR@;U^HlAyrAyB [F^[F^nkHpkMXlM@lJ@lQxlQ`lN`lOT_AwbAW_enxpaTPMUV]dSGwmYxmR@MGv[Eve[gek F`kF`FmSHmUlHqsHqCqv`XsF_ne@oOpmUplWDa=waufasfbwNMATJUTHmsNqsHIuLUTTEt KqtGmSPesGAtA=sBmQyhRu\\QTySKetEeSWaTPUuNiTM=tJAtIySFeSGusKEtIQTLqsKET HATJysJAtLIdc`n_Po^DdAGc@DCGcIwclD?gb?wcIwcGgcGwbAOLMd^ pgPPiwgiGgYGfmwfpuUWphTevokDe[Wf[WgkgdWwfmFa;gXsUYadNctCIsCOB`j>@KJ?C]SF ]S>:::fkGt]Yv]]V^]V^vkIxkLd^Yv]Yf_gf__f^_F_cnKIdOllPplhpohpEYsEir?iREU SEir?God@oSPmRPmTPmSHMkF`dSGWmUXmV`Mof`ov_mV`:cVaof`=gaoVaqVawFaCGbyVb sv`Ew`qFfCGcWGb=gdov`UWawvd>nHxkZXkGPjg?hg@mUxqfxo^plR@oWxmnpoiXnXTb=W b;gb?gbCGc?GcCWcEoLItKUtJETMATKKtDQsLSDWoeXoR@mRPodt`qNJ=TJMTLMTL;RcfN TTMoN:\\TN?OjNadb@OIwcXDG_`lMYdb@OM_c]rFUSC =uPatWeUUMuSyTQmtOEURytRAuRiDU_ilRAemXpmXP@uRAuQydgLQudmXPUWeUWapsGicd POcGgcWfVPTUUUuUYepLTMUTIuSwpqxpuXQyyyyyy:::::::Z:Zy=::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::pyy_YwaxaYpyyoyyiYpixeyBy syysYydywIYnexGIsCyYxmyyqYyyYwiyxiywuYyoyweyuyyyIyxYyyyyyyyuYyuIpI:yyy :::::::::::::::::::::::::::::::::::::::::::::::::::::::::pIqyyhIRyfY:j yyI:::FyqyxyiYqYwgywYIvyYywImCXkWGcCwyuYyyYyxIya:pixOj:xiH`kEHkD@km_hl GgcGgqwhq_j>`jCHkC@kB@kBl=Yr=Qr]R?er?ibFL@S]Un@C]Sn?ir?iBr ]YNCs^av^Yv]Y>UV]Uv^av^:?gb?W_enyxqydPodPuTQu`sFWLjDQcYxmY`mV`mTPMkF`k F`NMdCmV`TSF_ne@oQpmUpL`SJqSFisFiSHesIqSHqsF]sJmsGitKItNyc`PmT`oXxmfpm YHoSPmh`mWXoPHm?@hmOmAPkk@n]PoVpmfxm[PoYHn^`mWPnWDc?gbKgcGGdGgcIgbAwbw vaCwa;Gc?OF]SFEQsDqr@qr@uRAUHmSHQtLwmYTdMWdCGcCGc:fn_Tc:::nNvNKGdKGdfn _hn_LMccvN`tNaTOedeXOGOMeTOetMYd`pNbcGgcI_ohPQwdhDOgddD:Rd`TTMUTmTPmtM YdclOiDZdLN]TNYuUWOQwdvoiTdnohpof`of`OjUCeFOMweYON]TNqDWge\\D[GfTUUooh tdQ?lDUgeSWeW?ltQqTPgoe@phPoihoflPuTPEUQqtRatNYuOit[IUTUvRIUSeSHYsy@rO UsFUvSUUY=US?odXo_pnEIsCYlOhllxppPojxod\\ipoddeSgeWweYORsf]WfW_hlQsefo ixogLRCf:voiteFpilQgO`tQutQuUWQTUEekhocPpihonHpjHpipphpowxojPrjHpCYpoH rehof`LAg[MVe[v^;XcKglcggwGeQWjcgfuWgegfWWeQGehTRuTQIUSuTR=USEUSytPqTV IuTuuViUTAuSqtQ=UQAuTidmHpnHqjXprPpmxpoxpr@qtXqqHqsHpl@pgXod@piXonXpih pn@plHpgHpipomHP_Gf[weUWeSOPgO[_jLScf:VpmXpohPcGgxtQydgholPplhpoTgNplD gcOQudWhMGf[GVWeUW_eV_UWeUWyyyyyy:::::::Z:Zy=::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::pIsiywixVywiyvqyxiYwyyyiyyqyuYywy xqyy]sQyyiyxqypqxMHkChxgqwyYyvUyxYxqiwyYuAdyuayvYyuwYyawuy:yyy:::::::: :::::::::::::::::::::::::::::::::::::::::::::::::jxSy:::vyy=:::giwgyyy yyyYywqxgyrIyyaywaw\\Qvj`n^xyvqyjAwcyywqy[`mUHlFXkpGhjgis?iGXkD`iv_ixo ix_j>d\\Of\\Q^AxJPbB@KIv[XBEv[If[KF\\MV\\Mv\\QN@mR@mBU^EXkEPkDPkCHKb]W NCESCMCYv]YV^]V^SF]SV^]V^UV]UF^>nZ@nOhlOpmXD^[F^;ER;EZ;FZgf_gf`dS:=R:i tOiTIuSIisGiSF]cQxlQ@mR@mSHMm^QHmShnd@oSpmVplUHmSpmVxlT@mSPmR`mWXmSxlP HnW`mepn_@oZ@n[HmSxnWhmchmWpnTPmd`mWXoRXmlxm\\pmC`kpjteYoQce>phpogTepdahO[giowcD uL=uP]tRuTOIURiTRuTSutP]UQutViTPuUQyT]MuTMVQ=uRUsFus:MrMmrA=vL]T^=URav TMUZEuSIuOwog`ooppnXpohpehogPpj@pgXoe@qmhpwPqvXpl`ph@PYgfgwd[wfQgeaWdS WfQGe[WeYGfYWf_we]WfWweUwdSgdiggDuRqTRytOIEcgf]wf]GfaWfTUTMUQudhMYmYxymyxU QodPBMR<]SAur=yqyxr?eR?]R>ER;ER;s[Iv[Of\\OF\\\\r>GkGhkGxjAxjAL=UR=Yr=? kBd\\Of\\Q^El@qr@ir?GkCHkCL?eR?qr@GlKHlN`L[F^[v^av^SF]S>S^ILJ=dMXLa^SH M;FZ;VN]TNKF\\Kf^_f^sFasfawfauVahsGicRt_iNF]cSHMm^R@MIgdKW`sf`kv`mNI?m SxlSPmRXmVPmQ@mP@nXhmd@oa@o^XnZ`mUXnXhm]XmUpmPpl]HmS@oT`m]@lMXq`Ho_pkL HjdogTXjATa=gfuFb]ga=WesvaMw`uFduvaMwbCGdGgcEWcDTKon`pn_LDMSDAsBAShlOh HsCISN]d]Xn]`n^Dc::NNCWbAwbEWcEGcCGcDd[Hn[Pn\\TcEWcKGdKgdOOOSdvnaTcPtL ;GgcXtM_n^`nblLScFOPDG_mLOeTOeDRd>O\\DI_aLTMef`OIwc\\TNOO\\TNwN:XuUYUP Ce>P`DW_elO;g_fDeM_jPY_ilJ@lLPlLhkGd^_f^Wf]Wv]Yv]wfawf^_>lR@mRB=ctOitwgiwgpj@pghOWgeWW f]oPqtPmDQwdtTQWpmTgeWgawf@uRAuywyy=:::::::<:xI::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::nywmxwyvqyxQyyex:ZpAyrQxmIxyyy xyyxiysYxnuxsqueYdIV]Uxd]WyyxsyYwyyyiyyiYveYyaYwyyyyyy;uaw=:jyyI:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::byxYysyyA:::RyNYuyv kiweiwcyyyiytYyoyyyynqxwAxlyyuiYqYy;ym_HPGtKkf[IV`IF\\yuU[U`UV]GFZ=F[? fZGVZ?vZ=VZGF[CV\\Iv[ON?eBYV]UV]Qv\\`r>eR>]r>Yr=?kBPKOv\\fkGhkF`KS^EXK Q^F`KUV]pBYv]HsCokHpkN`lN@lJD^nmXpmKHlKhlOhlIxkIPkEHk_pn^PitOiAxjA`kEh k`hna@mR@mQxLi^RLGecTlFecQ@mPxnbpnVPmSXmS@mZpmWPmUHmU`mT`mYhmRXMkv`mFc IWc?WcAw`wFayFbwnIqsIqSHUTKc`sFewFbAHfagfqvaKW]]ViyFcYW]af\\KeUKG\\UFe ]V__w`@UJItNATKMtI=tJAtJAtKITGmSHItLQtMYdghogHlIxkbpn`xn_hnUHmS`n^DcNn ^`NAwb=wbphDfnoh=R:mr=Ur@aR>]R WkCHkEl@C]vkGhkH`kFpkF`kGXkEhkEXkDPkDHlIxkMXlMPlJ@lPpL]V^]V^vmYxmKHLMV \\hsGisDUSDuqypQKETKMsCQSDIsEMtLaTEUcP<`sFWmT`mUXmUXmS@mS@mNHob`nXPmSh mS@m\\pmW`mTHmVhmUhmZpmS`mTlFITMMTIItJUsGacVPmSXmUHmUXm]Xn]PnWpm\\PmUx o\\`nipm[hpZXnnXmYPoFHlr@mWpkhWhYHj@xoPPmlxm\\PoVpm]d`CgayfdEgcGgcGgbC Gc=Wb=w`mV`OWcGGdAGcMGcEgcCGcCGcDD;WcCwaCwbyFcAGb?gb?gbjMscnn`dcfn_Tcn n`dc^n^TcE?\\DK_axNRcZdlMscNoddcvNLEKGd`tNGOdTOOOhtOitOCeSGe_gf_wdhTMS dMWdSGeSwdVOlDOgdOGhkGhOgdOWdMoPqDvOYgeNOOgdnPlDM_kHpkPodPoixOBd>P[ggT UQooflQydf`of`oihoepofhOSwdSgdSWeQwdSgdUwdOGeOWdQGemr=EqtTqh@PyPpnDsxxqDar=mrYriR?Yr>]r=ar?ar>mR?erAer?=s?mrBmbLhkHPlGpkIhkGXlHxkMH lK@lE`kNPlLhkGhkIxkIpmWxmK@lLxkFpkNPlLxkIHkMhlIpmXpmI@lNhkFhl`Hoh`lPpl MplNHmUXmUPmVHmR`mU@mTPfYWeY`mQXnZXmlxoexlN@lRhlKxmW`mSxlPHn\\@nTXmS@m TPmaHocPmV`mSXmUPmYpmN@mSPmXxmT`mWxm[HnV`M_WepsE]SNqsI]TGmsPesH=uGusTu SKms;YR@IquPT@=sV=TLmuLaTWQTNYtFiSJmsGQTNUtL_n[daOgb?WfIwcSGcCWdCwb=gb ;gbIWc=gdIwaKWcsVc?gaAWb?wb=WcAgbKGdKgcTtKId]LLMTLkLCcC?TTMwN\\TN]TN;r c^NHtKItLgnatcEoLMDOgdOgfDUN;GgcTDI_bLOedeXOawfaGeSOMUDZaqr@qf`of@ob@oixOKONuTQutOcdFO`uVauMwnhlNatNit OiTQudgTeU_fDeS_elOcd^odPoelOqtPoOQGfYWeYOPudiLWEuSUUNetZeTQaVQEUTQSFY SvhQKeR@UVUYuPQdqHqsHomXpUHobxno`pghpoHob`oiPofpocPonPpk`oiho`Xoc@po`p oxpqPpj@pl`ogPpfhomhohXpfxonhojXph@plxok@piHpk@plHqqxpo@P]GeUWfYwexdhL PuTQGpk`ohpooxpqHpmXpflQydkLRiTPQei`pdPoCYqvprpt_Ef[SfXqug?GcIXfagk[gf ;heYwgMo[]UW=VXiUQIUS]tSAULEURMTS=UNIuRitR_ojHpj@P_GfLuQ=uTyTRUUQyTUut QQuQ=ekPpoXpmtferH]r?QSxQRodPl`r?QREs=arB]r?yR>iRAar?mr>ir?ar>ir>aRAa R?ur>eRCmr@ECaf]Yv^Wv]]F]UF_YF^Yf\\Qf^[F^SF]Sv]YNJysKQSD]sDESDeSFYSBAS >eR@URCESCir?AS>arCasGItFusHmRCqRDYsDIsDUSGUSHmQvpqO\\LLtmWHmPaTLMR:iq G]SDMtJqsEQSD]SEQSBur@iSHesFac`@obPmV`MbayF_kV`mFa;w_qFaqva;w_sv`SgeWg `of`Cwa;gbsVaGw`ufdsfaMg_mFgyfb_W`uv]cEXmfY?VgofaWG^eVh?gcUwbAw`if_EWb ;WeMGd;w`oFbqf`SGdIGdGWcufasvdnngPOSV\\aEYme[Uf\\=Gc?wdSgdof`oVauVaCGc CGd\\tNad`lKCb;GbEWcEgcnN:XDI_blLQd^lLscXtLgn^lNadkHPMWdMGdKoMccG?\\DM _dPO__flMYDZatcrdQOMUdqxpqPodPodlPqtPUDQwdQGdfOUGeSOUUeelMwnhpOGgcGGeS OOeTOatNatNShmWhI?NodDeSGeUWeUWe^OlD:QwdhDUwdQOP_OhtPqdhDeWOQudglOatNu TNaTXuTRQVRIUYATLWkIpnAPkcpkKHsrXq_xmZ@odXojhpppm]pn`HogpqrhqjHof`pfho j`oePoihocPPAwdOwaCwbQWdMghcWgpTQgoh`pg@pnhoj\\mLRytS=uQAuVUUVIuRAuQqt PAuQwoideWOQSf^ohTgVpmXpf`OY_iLQmTPEE]WeSwdGWcGhhowkdUQasG_ixgmYrqHpjPPjpOCyr?Ur=SAur?ir?yrAyRKAtKasDeSCqrAMSBy RDQS@MsEuRDMcIxkOHlKhmVpm``kK@ll?imWkH`kFpkI`lI@mwFayN_`f^Bgb:_[DNZy_i qwmV@mRhlKhlMPlKxkHhmWt^anFasF]Dkf`qF`ov`mFaHSEesFmSJYCcV`of^kv_KWeSG` of`?gb?gbufa?GauFcsVaMGawvbev__wa?Gdaf_iGbCwdcF`wuRIedSV^gw_oVdWnT=TKu TLMTEYsD=tNUTOQUSqREISGqsGQtKEtPaTNmTNYtLITKISEMSx@rylprxpGqsH=TJ=TLMT LETKETIuSIysIysLQtLitOiTMCcCGcRb=Wb?gb?GcCGcGgcGGd>OXtMwN\\TNoNHDKGdxt QGocTd>OXDI_`pNG?FpkDeSoNsdQoOscI?XDbgTuNadcLP_o_hNRdMWdEWcXDcGgcOOwNp tPgN`tNaTN;K_sThreYOP_ohpohPOhdeXOlTP_OptPgOtTQuTQSeUoOWoduQ:QS>erF]r?Msy@rHQrmQ?URwXr=S?irCAsByrAyR@mbBPkDHkCd\\Of\\Qv \\lR@?L@sBAs?ir?=SBgkEXKWnJuSJUsCQsGUSFESAqRCIS?YSGASB=SBUR=mr@qrFqsIY TBQSDmQ;qQ@yr@AsCMSEASGqlNxl]LNZpr?IRYhMUXJ<obtcI?`tNaTMCc>obDf>O\\DrcvNTTMgNPD]Wf]wdQoNsdQwdQ_bL MCdvNTUUUUN]DVoeTcEWcOOOatNoNJTMeeXOK_hLMUddLN]TNYDK_sHQI_iLPceNOOwdQo OCelTPSeUoPcenOptPWOddd`odPogXoepof`oglQAuRitPqtOatNmtM]tRaTOUuNiTXiTQ YVP=UVmrC=uyXR]YsIAwPIUVAtLMuMatVMuT]TMGqixothoeppd@ofhn\\`n_XnbxofHon Hp]pofholHpjxokhphHpphojppg@po`oiXpfxok`oh\\klPqDYweY_hpOW_kHPW_qlSCeS oQse^O]r=AR;eR;IRAIr=r?]Rx`r=eq=QRyTrmR? eRB=cGxkIXkGhkCXkE@kDPkBPkD\\CXkEhkIxkJPlL`kF`kKHlKxkGhkJpkH@nWxmM@lLx mT`mJ`kE@lK@kO@mIxKYV]Uv^iv_AGawfcOv]Wf]cv]_V_aV_iF`HSHELILl[mTPmP@mRPof `oSHmS@nZ@nYhmW@nU`mZXmVHnU`m`pmYpo]hnRhkI`OXUOmtF]r?]uKUtOmrAqRq@qEAr ;AuJETLmSHysGeSNUtLITLEtMeTNqSJuSHqsGQDIwb?WbuFa;WbyvdSgdsEYsE\\KF\\kF `kFcCGc>o`dcA_[HNCGcCGcVn]db?gbAwbPtLgNTDKGdXtMwN`tNatMYd_hNO_kHPGgcGw cIoM;bcGoLSc>pj@peXOI_cLOeTO]TNGocHOvNIWgeWgKGdKgcNOOGcCGcOgd`tNGOdDeW gewdQwdK_hLMsdFocHob@OK_sHQI_ilOidgTeQwdQwd^ofDe:fOpDBeSOPsdQoPidgPd[?FZ;VYquXCvY;f]Sv\\=FZsu_[V]]VYsEb;VZwFVgEZUDOcE KOthaF`LR>yr=]r;]R=uq>URxdR==r?]R=ir>er@iR?mR@eR@mB\\r@mR>qB:vjFXkE@lI XkGt^av^]v]Yf^[F^?WayV_[f^kv^eV^UF]aF_Qv^gV]_f^_v`qVb_f^wvVeUY=dKEdZKv ZMF]OF^_v^gV^oVDEsDQDLCTbyv_CGcwVYsEX=sAwRMUTMmV`mV]UV]iv_iv_mV`kF`kvc IOGeSGasF_mV`mPplP`mUhmRPmUhmWhmSHMQwdXsEYSIuSIisGisHisGuSHmsP]tNqSFas EASCytNetJiSHES?irQYTNMuMatMYSFQTBESDaQxPSwpqQasGQuKMTMeSGUTIuSIaSFMUR ytRMUSaSImSDasEQTNUtPaTNySGaSLETKItJATK=TJETJ=TJ=d\\Pn\\hn_dcGOL_n^qQx\\QFyR @yrwdQNaR@YTyHrr@n]HslhPM=R=iS;URCUR=eR>YRMr@eR==S?aRBiR=]R>Yr>ar> ir?]r@mr>ur@arAuR?yRAeBCV\\KV\\WV]hbL`lNHlKHlNPlL`nYHnPHlMPlGxkPPlKPmU PlK`lEpkHD`kfaiuWAVFasHgr@oR\\Yf\\kv`mf^cV_MF[UFAkbAwsHiS__V]MOLasEUSo dPoTR=URN]dOhlOhlQxlX@nZxlQxl`pNof`onF_mVd_VmT`mTHmUxmYxmTPmT`ofd_gnHq sHeSGgmUXmXLHeSGMTKETNs`mV`;ga\\TMUTJiSHqSFasGUsDeUQutXYtNYSwpq>eppldL @rbPoiPmV@qbHo\\PmTXmVPma`odHoh`oW@nX@nXhmiXod@o`pnahNGGcCwcEWcTd_hNAw bAWb=WbAoKMTLgn_dc:nNPD:Iwc`tNwNTDS_n`pnXn]pjTdfNTDK _cHOSGexDIwcPEI__TcKGdKwbAwbOgdOWdMoPqdeXoe`pn`pb@ob`n^tdQOLMddPOQwddT OGO`esxNtTQGocHoeXOWgeWWeUoPCe^OhDQ_dlOsd^ofTeU?hDQWddTN]tRqTP=UPgoelO Sf[wdgGgGWfYwaOGd=WdP@uRGPDE__mLRCf:::vYxI:::::::B:yay=: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::j xyyyiiyiYpIyyyJyuiymymiyIyvuwoyvEYtaV[Mfy[IwyixyyyoyyliryxuyyvqYsayyyj ixmqxxiypqywaywYyyqywMyIytQyrayveyyyyweYyuywqYykyyI::::::::::::::::::: :::::::::::::::::::::::::::::::::::JyuYyuIyyyXyuYYwmYxqyy_yxqyuyYoasyY BXuEYCV\\KVZAvZiUXmEYsEY?FZyUZyeYAVZ;F\\GV[]V\\Mf_Gv[mnyhss\\Q:MOgtT;] rL=RMR:eRYr=Yr=k>ar=mr?aRAqr=qR@Yr@mr>yRAmrB=s= ar>ir@qR>]R>mR?eRIasGIS@yRFIsDQsB=sCMs?]sGES@mR@YsEqSxlQ>MJ=tJaHOclRCq rD]SEMSEYs?UrAEKALkdDOapSGQsPiTOQrn[xmUHmRHoYxmqhn`hoHHlNxgm?nrgiy`mZPq\\`nd`mWxmTHmaXNWgdMW cAgbuFbwvcOGdsfawvcIoMSb=WbA_ghogHocHoUXmUXlMXl]tbU_]Xn]`mV`mb@OYWeUwb =WbEWcHd]hn_hnaxNCGcCwbAON]DYOMUTMUDRcEoLCcCOL;YweYWdMOLccnNLTLMTLccZn `pnxnaxn`lNatNItKItNatNYtMwNjSIeb@OC_dLKEd`pn`@obtbAoOiD]WfQwd\\TN_ofd dOgdI_b@OOgdlTP_OjOcdOON]TNatNWOptP_of`oflOcdNoeHog`obPpdTfQGeOOPMTQut HyTQuSRyTJqtOytQyTUMtLmvMat`ESEUuu`QnEuRQWHqsqYtPAuj\\pe`JB\\sddodUtN] VPmtMisFAUQetRqTPitMUTTmTP]uSoofTdMOOmTPeTPgpqxph@pjDeSwdUWepdelOSeNOL UTMuRAuReDWgeWgeVoeXohpOlTOEek@qphpm`plhojpog@phxojpoihofxpmPpj@obxpeX o?ipp`rh@pi`lQxmPAlKhLfpOhHsFMs>mR?eR@qRoPPqdJ=lJvPqr`r>IrIomSHmSxnaTXmUXuRAurCIsCsE YsUb=WbkF`kfdOONc_gf_aV^]f_cF_mv_iFbwfaoF`knDOOXCwfadSGesFacSLE]SFYsFa SHmcXpmXhmYxmVpmXXmWhmT`mVPmYhmQPmRXmQplb`mVPq\\`nxXmYXohiw gi?Xj=Xjwoi<`iwoiv_ix?j:pi@hj:XkCHjDxjA@k@Xjm_hbHkFPjQOeHxe\\OpEhk\\pj @plCxjJHk@`lDPkL@kBpk@xjDhj@HkAxjCT\\KV]Sv\\[v]GF]QV\\Yf]IV\\MV]Yv]Qv \\QF^Wf]Aga;G`]F_gV^aV`eF_YF^IV][v[Qv\\Qf]WF_EV[]VPqtRICEQCU_UUoVaqF[G v[oSGwCO:`DdTOYTPeTMyTPatQetSitOYuOitTetPmtOqtGqTQaSQmTLaTOeVTUUiasK=sSlma mPtxYH_tRoe@ibF_jD^\\txd[?uRhmXIpn@s\\PnFan_`t\\pnvHlNHq^Xnvxpn`olhoZx n\\`oihogpof`n`pncXoepPgweYoOidf`OUWeUwdQoOSe^OTUUOP_GeSGe[GfUR;]SFES HuSIQsFesBISDAsCMSFesGisHuSGecN`lNpmXpmYxMaV^[f_cv^OGdIwauFa_f]UnF]sGa sF]SEWocHoRpLuVadSGGmSd`of`ev_if_kF`kf`of`sFasfawVayNH;qfasv`wFayVasVa qf`of_efagf_awbCGm]Gg;x`yV_SUVofViefaf_OG`mfd?gbUwdOW_kf_KgdOGcCoJAtJI tKITLMTLEDC_b@obpmXpmaxN?fZ?vTQuT]V^]VdMWdyvayfb?gbAoKETK[^LKSb=OMceWo LCcCOL;::]Wf]gdvnaxnb@obHoctcvNdTOeTLCgLuMccGoNatNIDM_aO`DMwdQGdUwdEW fQGdYgdcGeSGhOGeiWdUWeOGe;GeWwaQWeQgcKWqiWi[i`Gg[gbBMhNqtyefaExIEd]OR> SXOwtyygksxcOWiqv`ugauVpawfQX_ofb[EWsfZ?nO]tOAUPEtNMtTUUTMUSAUKEd`@obh pohphpOQwdlTP_OhtOWOtTQgOUgggggawfaWeUoRAuR=EUWejNmTO=uSAuSUUTyTS=UPyt PqTRuTRAuQAuQuTPetNAuP_pipor@PoGf[wiawfyWf_WiUgeegbEGhEGdouSOEfyfbmwdx USMUV=uRauSMuSutQEuPuTRuTQytPqtPqdhpO_Gf[gf[OR=EvOxDWgepDU?>pj@pjlQSeU OQcevoiteFpkdf_OS;Rf]OR;yqy;:::::::<:xI::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::uaw=JyuYyyyyyyyuipIxlyywqyyi yyqyqQytYxrYpcxoyYYaiuYyyuIymivYyymiwyYw]YZmDOgdHaCrWxkyYygYxcYvyyxsyx oIyeIyyiwwiymyywYyxyyyI::::::::::::::::::::::::::::::::::::::::::::::: ::::::::vxquyyyyyawiYrvayvmYxmiywiyB[CF[?FZ;VZyuYyUYuuZ=VZIfZCV[;fZyuY yuX;vYweZAVZGV[GV\\dr?arEUSCQR;uq==rxtsByr;uPpXQfXOKeQxXty@rM]r?ET?mRH aR@Qs=ar@QR=iR=YR@eR?uRAuR?ybGPlK\\JPlLxkGhkJpkHHnThmM`kIHlFpkPPlKhlPh kIPlCXlMXlGhkM`kF@m\\`nfPkIpkEGdFwbD?cfn_ivf\\Wg`Pnb`mShlTPmNplR@mN@mS @lMxlQPmXXmWpm:Pj=PhlOh>`j>PKOfb?gbEwb?G`gV_sV`efaqv_ov_anF]sHisG]SEUs OadQhlOhmWhmSHMm^TPMo^UXmU`mV`mVlHqsHqCrauVayNHc`kv`mNImsG=TIqSImsIesF ItDUsWQtMYVOqtIuQ;MrkXPLYR?YuJITPusI=UN]tOUtMItKItJAd\\PN=WbtSIusIwm_T cIwcIgcGON?o@pj@pf`ofF`kFpojt_mV`CGcHtK_NTTMUTL_n\\PNIoMyDAwbA?C?::>pj ddOOMCdFocHoalMcd^NCGgLuMYtMUDM_]XNM_aO`DMwdQWdQwd`tNaDQWdMwdQwdOGeSWdWGeIWfQWdYgdc_qPofH pdhoePofpnepodHogpqcHpMInexxY`ouxbNwpfv`nP_kFokf`LaggGr]wfc@f\\gvs@qCa ob@sgHoLYpj@oxwi=W]Vvjh_gchmShodpn_pn\\xpmTg]GfEgcCGepTTMehpohPodPoeXO SGehtOWOtdhhOTEa_gTe]Wf_bX`mVhweYwt?ir?ipohuGisWj;HjGXkEpj=hj=Pj>xi;Hj:Pj Pk@hjExj@XkDpjoohlolQXl>HjwokaRAqbDPkIHlKxkGhkKxkIPnWxmOd^[v\\Uv]Qf\\ef_Uv^cv\\_f^xrA Qs?arDItLqt?=sAUSHYs@=SAmRAyRJqSMESAMS>Qr:ar>IrOqtPisI=T@yrCASD]S=]r>E N\\PniSFMsLETIusGYSHeSG]SEUCXsDSa`sFGM hsGisHqsHqCBasFas^UXmU`mTPmYPmUHmO`l]HnZxm\\@nMPmQxlXXm_PobXn^PnLpkGhn WhmfpmY`rh@prXmXXkdogHhgkGpNxlo@n\\PoYHnaPn]pn^`n``n^PNDDAwbLDAwbdTOom XTauVaCGcpr@?kDte]WfEWcLTLgn_hn`pn``NCwb>O[WinZxypywyiynuxgyymywyIv_YyKitwiywIuiYwmyywAuyyyyyyA::jxS y:vyy=::::::::::::::::::::::::::::::::::::::::::::::::myxqYyyyymYxxyyw YZwiynPplH^Vf]YPmUxyxqyP`lNPk?pj=piyWisGin_hngi;PjxOj=Pj>hj?`j@Pk>hjD` j?`k>`jEpj?`j=xi]OfXgkDpjWplM@jn?hVHk@xnH`kn?eRwkjWhO@iuOmyWjVhjCxl@Hk KpjBxkCPkIhKEv[IF\\UF]IV\\Mf][F^YV]UF^trJmSIMsAAsEEsCar=URFaSBQsDyrBAs BAcQ`kEhlXhm`pjEPkKxlL`jAhjFpkIHmPxmMHlQXmS@m[@nV`lPplNplQxkJ`lM`lRPkF hk]gf`OdKWd`of`o^^^^l>`i^gbwfUxlPXmPXlS`lKXnXXmVHmOhmUXmQxLMWdLSDMSImc T@mR`mTPmWXmU@nXpmYhmWpmV`mX`mVXmUt`^MxCkF_aVcAgbIWcCg`qV`\\SEqSK=tKet MATNQtEusGmsHiSJ]SFIuJEt[MtMmThjCpjB`k@xjGhj@XKEv\\GV[CfZ=vPodO?fYuE^Mf[cv\\xsCAsYR:QPntr qDqFiQ:ms;YrFUr>Ms=aRCer?AS@qR?ebFpkH@kDPkH@lJ`kF`kH`kFXmSHmGXkEHmQxl? XjOKwcnNXtM GOdDQ_dlNadbLNitNiTPeTRetOAuOmTQSdUgeCWe[gdWweiGdSGl\\tn]tRmTldP\\er?y wVMuRQtFMUT]tTIuOMuNQTTQSDIsndp_ATKAWT=uLETGQuPUTbatMyvCMScqRBUWGis_Qt KEvQetTAUPQtNITRIuReTPmTM]DOgdOGeSgdJPuTQuTOCgFPddd@phpojpOUwdQgeW_iho i`plHppTe[geSWeQ?c_oHpjXoapnl@pi`poXphteMGfUwd_weQgfYGdUwdigfaGhUwecwc Mgj]GgugdUgjUGfYg`uF[YeVgWbCwgCwcsWdSGiWGfqGf_Gh[WfcgeYweUWeSGehtPqtOq tPqtQ_OptQwoide[GfUWepDbeWgeYoQ=UROplPpllQydiLRCfvOxDWgeptPqtywyy=:::: :::<:xI:::::::::::::::wQymYyGyxUxwyyuyyyiynYpyyyyvyYyaixywkIxxqyxQyrAy yiywyyxyyuiyxyyyuy::::::::::::::::::::::::::::::::::::::::::::::::::Zw uyyyywyyyiyvyy=ZxmuiY[Yqy\\ta\\oyqWjyYVeutXkGdvu`qyIyqyYwUyvyYwUyxeyyi YyQyjUx:iojDqaHouixomyxaYvyYv]xq]XxqyvyYyeYwYysyyw]YtmXpyYxiyyuyxqyyyy voyoyyyyxyyysqYwYyyQxuyyyyyyI::::::::::::::::::::::::::::::::::::::::: ::::::giwxyyyYyuiymuYivayvyY\\TN]TGdKGXsEYsyyyyar>aRytQyxqyxQytQydQwdq xxqytQ:=ryxqy@r:Ar<=r:Ur:ER=AR;Qr;IR;=R:Eqs@qbxnahqv\\Q:mqwhSEQs;iqw\\ S?irMyrBerotpxtojpRsPqFMR=]S?iREmR@Es@mr@iR?yRBur=eR>aR@eR;Qr;]r>YRIys H]SF]sAyrAurAqR>ar=]r>YRAmR@er=arAiRBETIITKysJqR@urebIPkExkEpkLpkL XlB`kGHmSHmfPodXlHxkThlP`mTPmJpkHXNyf_]f^Sv[KfZueYQR;GB_Wv]qV_gv__V^iV ^_v`gv_oF`kV_eV_OgdOw_iNIucTPmT@mR@mRlEYsE]SFWmUTauVawfawVauVaqv`dSGeS H_mWhMyVauvafmWt`uVaovawf`yfakVasv_sv`mv`qVaqv`Eg`sFeqfaaGa;ghsVbCWZEV bweZIG\\QveiV`[Gb=WfIwcQgcEGbsv`Ggb=gcCwbIWcCWb?GbsVaqf`uv`yFc?WcOGdo^ ^pn`@nZ@nKHlKPlLPl]XNUwdQGdGgcSgd\\TMUtLQtLKJETKesIusHEtJITNYTQ=eaHoc` n`pn\\`n^Xn_hnWxmY@nZ@n\\Pn\\XNHuSIUNscNodPo\\lLQtLETKGocTd^pn`pdPOEWc EwdVNXtMYTMSb=WbMWdYna@r`pnuPoa`phHom@pe @o]@nwHPywdOWg[f^MGVaunyvaqWcyflSwc[gSGU[=cCgU=[RZACCCXZwemIwawweGWgcg dkWheWe_GfOweWGdUGeQWePtMYTNcdvoghorpppXpkHpeHoc@pe`oiPoe`oa@oh`ofxnaD dOOU]UV=URgO@ekHoaxnfPodhoedcGOQuTQIuTQuPytQaTQqtPEuRmTQutRytQIUQyTTut QYuQ=uVuTRmuQAuYutRyUQoieogdhlRHrfxoxHofpqc`oBipr`rmppupOQwcltQqTQAuQ] tPOoihoidf[wf]wcSgdUGfWoQqtOuTPiTQmTQytPyTRutSAURMUS?phDfWgeWge::jyyiy =:::::::<:xI:::::::::::::::watyYiyyyIwwqwyixMySyVveQwpqywYyyyYvyxqYsIy wYyyywmixyyyxI:::::::::::::::::::::::::::::::::::::::::::::::::::nyyey vyy=ZxMuiyyutUiQB`kyqsKmQJPlUuojxYhiwyeyt]YyaYlEYqMYyuYwUywyYuqyyaXqyY kUxy]YvyYwYijiWsYxyYymYymIwyiyoYwcIvwYymIvYyttityywixyevwiyciyXyy_Ytyw qyywyy=::::::::::::::::::::::::::::::::::::::::::::::JyuiyyyyIysIYyyy: yaghog@^Z>>PSv]YvyqyxqwhqwPr]b=Xj=Xiwgivoixwiywi;l;Ar:Qr;IrirBmRAITI=tC =SCyRA=sDMSFMsDas>mRB]r@=S?ASDUr?urodPp\\QuTqL=tJAtGmSlxOjxQwhQHUsEYsB ESSqTQ]ocLOA\\j>HrxtQKmsHAtGmsJic`Hn\\PmR@mPplPPodD`kF`wfawf`FmSHmSLEU SE]SF]SGeSGmSHomXpmXobXoeXOvnadb?gbG_axN_gf_Wdnn` DeSOLMTLatNatMYtMEDM_j@PK_atcOgdOGd\\TN;RdMoNcdOOO;I_aLNCd:FO`DOwdMgdQ ON]dalMSdMWdMwdSGeQWe[wdSGfOwdaGdHUNetRatNytOetQ_ojpog`qrxpaxmYhqi@p>a n_pqu?jHIj?XvaxnfaobpyWtkmtPoHUSenQUTAFEKShquVEXe=WbyF_iWhaGe]weIOP]tP mTOmTPYtNSdhtOiTQmTPYuTQUS=URitNsd@uOmtP]tNytPqTO]TNeTOeTV]UVytQyTPmTP AuRAEptOiTPmTPYtMwoixopppphoixoehogxokHpgxoi\\hpohpOyyna xyiysqwpywcIylaxyQykQyqQyyiyryyvQxxqykyyxIxwiyqyyyaYpyxqyyyiy=:::::::: :::::::::::::::::::::::::::::::::::::::juwxyyyyqyxuyyyyrdQOLLLLlSHmsyy i=Xj=HisGixoix?j:@juWiu_ixoiugiwgiwcYweYrZAvZGf[GF\\KF\\K^?hj?@j:@jTOe TOj:@jAhj?plN`lAHjpkewgePkHxrdhol`mXXoWpmeXn]Pobxn]pn^XmX`mRXmSxmZpm_pn^ hn_Tb=WbEWcE?KF\\KFV[EV=VZ=VcEOMUdcXoe@odPo]hn_Hn`TcOWd?wcGGawfaQWeUwc XtKId]LLMTLETKONLd^\\ldfIwcXDQwdLDKGdTTMUTLMTLQtLGP]GdKGdRenn`tdhTN]TN MDQwdQWf]WfOgdXDRdMWdKGd\\D:MWd`D::K_bobTdVoeXof`OYWeUweUW e[wdyYvYYxmYumxxMyneYu=yyiyvyyxwywQxyiykyyxq xyyysqyyayxMywyyyyy=::::::::::::::::::::::::::::::::::::::::::::::jxqy xwyyayvuyyqy;cIwcIf[Gfgb?gr^av^wiywy\\Qv\\?fZ?f[Gf[UV]Uv]Yv]uuYyUYyMyt QytA;^[vOitokCPkTd_=EQ uD;;B:wR@oB;;B:OR;Gr@gr?oUWgeamf`yf` `sFOmTt`^mVdawfa;Gb;?sFasFa:ZWhMu>nmXpmXlHqsHqC:yVaxCIgb@uMooSXmqPmWhq V@nUphtokewgsPmWhqaHon`n_Po\\PnaXNCWcAwbCgb;WbyFauv`AWb=Waqv`GGcCGgcGg SGeSw\\UV]qUYuUY?VZUf^]faAgbEgdMwaCwbyFcAGdOgdEwcIgawnOitOUD?gbLTLWn]X n]uBaeRAKGFcCrqUXciakVy?Wa;yUUu^eCF; Xa[fhiWdPuQaTRqTMiTO]tPmtNitOatOitOidhpOOgdOwfawf]weYwdMWd]_lhohxodXok xoipof`of`OggglTPmTN]TNytQGocdeWgeW_b@ob@PXuUYUR=URuTQutRGpglNudcTeQWe UwdnOirAYPotPBxjB`quXQGQSEATHqSxXQvtj>dj_Tn]tmVdMfHOd`RO:::XtMOO`DM_bLNscvNTT Monb@o`@OKGdjNSdNodPoeXOUWeUWe:^oftdFocHoeHO_GeUwhEwcIiegGniEZmvAOcdCC FmHQEUiySKiwD_CeIcDgy\\Kvy?WaAHKEdbqCHgxe=Wc?w^aWgYWdWGeGOOkOgocXoe`of teYweOgdOwfaoRsdvogDfWgeYgdJQ_OSOUsdQwdK_ixOMWdxtQyTQudcHoc@pjdggoQydf LR=URqtQytO=uQeTRytO=egxoipO`j>pj@pJ\\r;Ir;=R:=R;cZZ;l:;;FZxA=VZ=f[Gf[sEYsu[Iv[EV[ Ef\\Of\\wEZ;V\\WV]gUZwE]ef^EF\\HRypQJ=sBUTA=sHmQ:YT;UR?QpnTssXqGuq:It@ yrFiR@AS@irGmSGerAqbGXkFxkGPkF`kGxKHcF`kFxkFpK^kEhKSf][V^ef]Yv^GV\\UV] [F_Yf^gF[Gv[ECCGCHgcGCvY;vXguWMDKEDDACCcuU[UZsUYxQ:=uOmtH]sFQTJAtFMsDQ SBAsCurA]SEUSGeSGUSEGocHoOhlO`mVT`mnFs`qv`q^V`Mqv`hCb`oNG;u^WLHmcVlG;o >^mV`mVlHqcXLI;uValCO \\D:I_aLM]TNUTN?ob]RoieV]xdyVgYgbggg[gdYWeKgdOgdSoOqDVo eteYoOitOME_GfFOpdihogpocPojpOSGeSGenPlDK_ixOM_hlPqtP]TN[pppppohtdQoQc dSGeqRAmrBESBmr@iR@mR@qsHOlKXlGlAASC=SCIs >eRAiR@=sCMSF=SCIS:yq:qlPpL>TJ=DJ:nXpmWhMcv_eva?GbQgeSGb=oCyRAmTKET[Eu S]UJEtWeSIQT:MrBmPrlswxqUysJIvSMURUTMYsAyROysJyTLUtQYTN]tKMTHuSIasIusF MTKiSNUTFQtKesMQtIeTNETNYtKQtLId]Xn[db;GbUWeTd`Xn]Dc^NDd\\Xn]Xn_hn_XND elhNJMUTM=TJ=TMUTMMd^hn_dcGoRscXTLcdOOKscNodPo^lOitOAuRGocHo_hNKGdKgcG ON]dbqQylQ?QREQt`S]rFacN`lNpn`pnLPlLHmSHmQxlQ@mRT`mNDc_gf_kF`hsG_mV`mUXmUPmTT `mnEs_>MdSGeSHmSHqsHsayvaFMdCbawfaynIyC=wayfawfasvauFayVaif`kf_mv_;Wby Vc=GbGGasFesVagGb?WgoVatrBmsxQpmhqe`o:ApkPqhpoohogHpg`oiDe SweUwdfoldeaOPuTQmTQuTQuD::vyyyyy=:::::::[lQuTyyoiXwYyyayeyvyy=::::::: ::::::::::::::::::::::::::::::::::::::::::::::lixIYyYxwqxsypyyuyxuIkQx ]YyqyyuyygxuyyFqtl_]bnpKGe@@_fvyBaryaxhyynYxyYvbYuewhnW[PvyUiuyaytqywI yvyyuaywYysqyvywyqyciyxaWuYvyAhyy_YxiyyY:::::::::::::::::::::::::::::: :::::::::::::::::pyxqyyyyyuYy[yMxmiyyyyiYjycWgeW=VZ=f[Tr;Ir;IB;FZar=QR=MrUr?aR>mR>YRAUR==S=URCUR=]S?iRDEr;yrvhq:apoXRpxPNqRB]TCMSIAsAUsAmRCqR? Es@eRAer>ari^RD`mV`hsG?nXpm[xmY` mV`mVlGicTPMo^VlIysIyCmfasF`uv`sVaqfasv`;GaqVbuFaAgauFcyfaAoH_mVho[Hnh `mWHm;XjCPggomuwi;An^HqZ`nlxm\\Hp]hni`n`XnXxmP@mRpm]Pnihpn`m^Xn\\PocXn c@oY`n]hnaxncHOi^wWiuGqqxpPplPhn_hnkHPyvayvbAwbI_UXmU@nZ@nj@pjhn_hn^`N KGd@tJAtLcb?gbHd[Hn[xpqxp^`NS_b@OIwcXDC_^LMUdqxpqpn`TeUoMCdKoLQd`pn`pN XtMYtMCd>O`tN?obtcIOMUd_hNrcFOdTOWoeddOgdKGdKgcTTMscvN\\DKgdOGdOgdOgd` tNGO\\DK?nodHoHapn@tDHl[HbBwoyvbrqkNxyrxpeqlNHuZ_fXp\\VVkE>\\_Ij;PtUpl vHnQxrj@oxpobpqq`pmHpjHocdcGOPmTP=UR=UQuTQ]TN?O\\UV]uPqdfDeRdfoghodPOb dvpqxpixoiXoeXokHpkxNhtOiTN?obLOshqwhOgd@uRWoeXod`ofpnePoa`oexN\\tPmTN SdWWeQoPuTQ_of`ogXoehoeteOwdYgdEsy=YrYYuEYvuyvQYxYYxqywSy]afUySiyuYvFyx yyvqyyyYyA:::::::::::::::::::::::::::::::::::::::::::::::wiywyxqyxxyyq IqyxqAyA:nYdQwfj?hj=Xj=pixoi>`J=VZ=N:?Jj:Ab=XJEV[Ef[Gf[EV[EfYweYmUXmU[ EV[KN>qr@qr;Ar:]R=UrBqR@yr?er?aR>QR=MR;Qr;Er>mRT`mnGs`^mVDaFmST` >mR@MNmTt`qNJucZpMuValSHmSH;uVau>bawnGuSHmsHiSIWmX`MrauFasvauv`;ga?WdG W_oF`mv_gfeCGcsgcKwf_f_EGY=v`aEXcg_qfemFaUGayfe=wbWgcIGeIGdIwcLTMUtFus HqSKAtHEtJmSIusIAtJQtL?nXpmNPLmTPmTf]WfI_J@lJxnaddOoMsbFn[Hnj@pj`n^`n] XNI_Z@nZhn_tbHTLMTLEd\\ppppp]XNQwdQwcIwcI_b@OGOMQd_pNGgggggEWclTPmtM;b cvnaxna^rImbxyYyvyqwjapqgh]g]WFdE^[Wg\\UntC@lyiw jyyuAyryyrqwrQxoyysayyiyvMuuYyQyyOyy]xYyyEywiyyyyyyy:::::::::::::::::: :::::::::::::::::::::::::::::::^yvuyvYpIwiyJyoyv_I=VZ=FYsEYweYwe]Wf]=V Z@B:bZ?NqsA=sHYr>iSy@RJyq;ernpPEmQxXS;ArNQ SDITDESAAryHS?YrJasFaSDIsByRAyRBuRB=SBAsBAsAAsByBWF^]f][NI=TJAsCISB=SB IsCwkIxkKHlKplPplLPlLXkEXkkHP]V^]FV[EVGf[GF_cF_uVauF_cNCESCAsBASMS_enH qsHasFaSF]cRlFS`>MjFc`onGicV`Mqv`dSGGmS^YYjNtukyyGisyIuOyymhoyYlKxy?iriikKxy OityYymYxuywQivUIxyixwyytIuyyyyxyqxqyxqIwAvY:::::::::::::::::::::::::: :::::::::::::::::::::JyuYyky;uayuyya_iv_aofDpkHpyyyyYZ=VZweYweXomxpqx` r>aR;Eb<<:j;Ib=L=UbB@kBhj?hjAxjAhhogh>`j>hdMWdK`kGhj:xiC`j=HkApj?@k@Xj Dxjbmhpp`baN _mItLyywIyuAymyyyaYtyyyyyyI::::::::::::::::::::::::::::::::::::::::::: :::::::::RyyyyqqyayyuYtuIuYysqyoyxyywayyqywyYyyyuaqyYUYxAIaPpBHLS]PtxY jIxy]YtyysixymytyyvIYlQV[iVRmtwyxnyYyayxuYw=YtyXyyyxyYyqYyqyyuiyiywUyB ypYxuYyuYyyyyyyy;:::::::::::::::::::::::::::::::::::::::::::::::qyxqyy xYweIwiyoixoIj;HjWcEWc`oNweYwIr;IRytQypqxpqwhqw`qv`q:Ar:AB?^<<:Jxla=Xj =hj?d\\ONxlQxlR@mRg\\Ofxr@qR=Mr;]R=QrIQ uXUKQt^MuUYUKQtRusJ]TGmsLeSHUTJATMMTLEd\\@n\\Pn^pn`HpmXpRPmThnaxnqGiso ghogghO]Wf]?A_k<@tJ_NCoIySMUd_lMwn]lTCbvNIGcCGcEWcE?OgdOwcIoLQdmXpmHn[ DdKOMUTMYtMGocDd>OXDI_`pNG__hNGgcTDI_aLN]dbog`o]@oaPoipoeh OlTPmDg_eLNseFOxtQwOlTPOOOWhnNptPqTOCeSORudihogxoghoh`ofpofTeQwdUwdlTO eTPsdQoNitOaDbdSGeQWetTQutTUUTMUSAURmtO=uOetWAe=ipp@qbHobplQpjkWhupnbh qe`ol`n_@sv`qp`O;G`kvhigg]weYweY?tTQuTQSefOjywyy=:::::::<:xI:::::::::: ::::ZxyyxQYyYxpiuyyyiWwyuxywcyoQviyyMYwyY;AsCmMZ@NNhJA@ya=WwuXoyyyeyyy YxCyvyy=::::::::::::::::::::::::::::::::::::::::::::::::::::piyYyyUxtY tuiquysiyyiywiysyyuixpYxuYyyYyyyyYYyyymIyyaxvyXQuuoTlRHSgpooAw_yYyQyyo xyiynyywqxuIw\\ak]R=UR==SB= ShlOhdPodp=eR>YR@eR;YrYRYR?UrkF`kf_gf_OgdLSDMsEs^anDQsDasFolPplQxlQ@mR@mSHMm^TlFacSLF]cRlFc`VmUt `ZTPmTPMhCq^V`MuVapCbasFa?WasVbhSImcRlDc^flZhmRHnXHmX`mWpmV@nVXm\\pmWx n]PNGWc?wb=GaoV`Iwa;WdqVaaGbAGgyfbovZIvZOeUkVY=VjGgd=HdSghEORySKUtGqsJ msH=TJ?n\\PnZhn^Pof`oaHocxlSHm`@Ogf_dR?erSIecHocPn\\PnkHPCOLEDEWcpelOmTQmdh`OUwdptOooftdVO`DM_d`OQWeUGeWgeGwdMwe_GfxtPatMUtQetN yuTQU\\gpvtdewbCgYMuTuv\\SvlmghYWb?GgKGdShj?heGgcageYweYwebeWOQSefOjyw yy=:::::::`j>pixsYxq:Ab=Xj=`j>`j>< Hr;Ir;;w]>`JEV[EF\\fggggSoeW?hsohtgj`jH HkBPk?`JOV\\IV[Cv[KNG]sE]S@mrHAr:QUBEsUESDQssLqwlohlRsDQLuRBAdLXmId_Wf ]]V]Sf]SF]SF]hr@uRAuRAkGqsHyrBASB=cMXLWf]WF^[F^c^HpkH`lN`lV`mVxlQxlGhk G@lJ@lKHLiv_mWd`uv` sVaqvauFa?gauVbyfaqFaoV`sf`m^Q`mTXlTxlNXmRhm^HndhOEwa;GciV`Kw_oFgyfbgW a=gdSV^wfYAF`kUYew_svh?gcoWcKWeyVb?Gaufb;Gb;gb?w`;waOweWgcQgdUweYw`uVa ?GcCWeYwehc[Hn[HNobDdfn_dcGgc:MWd\\TNwNIwc>o`pn bpn`pn`arI=r;uqjLp;ippDuHySIurAeSAuREmR@As?mr@ir?iR?mR@mR@ir@qRHu cGPlKpkJ@lL`lNpkJ@lIxkIHlKHlF`KcF_cfb?o>abcHochkGhkN`LSF]Sv^av^?fZ\\R> wnaT``sHqsHQsDQsBAsBAC[F^HsC?MhsGgmWhmV`mVHmSHmR@M]v^af^cF_cf_gv_mV`of `hsGOmTd`vlQd_gnEYcQlFasFicU`mXpmWPn[@mY`mQpmU@mWXmVLJmsGMTHgnWhm^xmXh n]PnUpmVxlVPmRHnX@m[pmLXmRHlTxlQ`mT@o`pncHn\\xnT`mbxlTpoS`mnXmY`pShml` lSHke?h\\xjE`pWHnjhmZHoVpmeXn^Po^hn[xmY`ok@pT@NmfauVcOWdmfauv_mV`IwcIW b=OJ=TJEUR=uMQdbXn^`oa@o]Hn[Ho^hn^Pn\\pn^Tb=OVUUUItKWocDc^n\\Pn\\PN:CG cCWg^n^tdQwdE?GoLQDnNXtMwN\\TN?O`tN_NPtLQtMYDZb@OIwcTTM?o`tcEWcC_^P@E]GeUGfWgelTPqtOgoetdQ?dDpdflOEe^Xob`ncxnfxogxog` odhn^@p`pnv@pjxpcPosHod`qdXo]HlL`ktWiahlP`rlXpnpna@qgxonpoiXphxojpO[ge WWefOhtPqtxyiw=jwyyyiIRyfYxyy;::Z:Zy=:::::::::::::::wyytIyYyFYwYxyiysQ yiyxqYxuyyeixIyxyyveyyYywyyuYyyiyuiyyqyoYvY::::::::::::::::::::::::::: :::::::::::::::::::::::::::ZweyxIyyxFY`yyOYpiyqywmx;ByxyxuixMygyxayyyY yQyiYxeyyyYyyyxqYuUxmAWXuUoYWgyytYyymyygYmyxqixsIyyqysykYysYxyiybxVywU yuqyyyyyYyuix=ryyyyyAfYpIyyy:::::::::::::::::::::::::::::::::::::::jxq yxKv]iuYyuyxqyXyqySdMOGdKGDSCEsyyyyUR=URwda@pj@XkEXkv_iv?j:@jywiywIj:A r:MRwjAxj>`J=VZ@B?fZPrpj@xfcGgvGj:xjFXk:PjUr?erHMtJ=sEMS?yr@urCAsC]SEAS DEs@uR@er?arA=SAySJuSEMsCUSDISFasEASC=sD]SEEsDISgdOgLROXDMWdNOCGeSgbUwdAwdOWcOWdQgcIwdEgcUwcKgdKGdCWdKGcOWd IWdMgeMgdgGdOWiGgdMHdSg]UUVmGc\\UQuTMMtK]tOatTUUTEtKATMYdnxpohnfHo]`oc HojhOMoRqdjXod`odDeOoTQuTitO?OevSYUXiTQMvViUXMuT=uOmTPedkHpkhpqxpa`oeHyxyyqyYmiyy YxwA:dYygyyUyyqyyI::Z:Zy=::::::::::::::::jxsxfYpI::yyy:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::pIuAZyyY:::RyFYuixtyxyyyiyw wyxqyyaywuxuIyQiugwbGGpIhlyixwIxYivyQyqyxqYy_yiywiQyyexxyvyItYxsxRyJyu YyuIryyqy;uawmyyI:::::::::::::::::::::::::::::::::::::::>Ykyyyqwkyyyxq iymwCdKoOhlO`sFgywiyywiygggggqwhqOkDPkIxkIPjNJ?fZZ>`JGN=MRYr=Yrar>eR?eRF ]SFqbE`kA@kN`kGplCXkUPkG`nIPlYpjD`m;hj_xjFHj]?gKpht?oKhlQ`jAhmJPlJPkEP mR@mIxkIxlVPmFHlIhjD@kA`KSV^xRC=SBIsBUsEQSBAsAySJusCMSCISDESEYsDKDYsDE sDIStLQtdPodP;ER;EBkF`kV_eV_cF_cF`k>PCb_gf_iv_\\SFOmTPmTlEYsEYC:k^SHMo f`dCsFasv`q^TPmSHmSHMdCqv`lSHgmYxmVpmXpmXTauNImSHuSHwmTTaqv`uVanmXpmZ@ nWxmYXmWd`sFaqVauFawfasVb;w`;wawFb;gb?gbxSJItHusPETLYuNiTSySKaULYtJ]R@ iRrLQHArO\\DrcMWdGwdOgcQgdrcMWdK_cxnaPob@ob@OGGdKwcFocDeO_cPotxndPscho oPlPhklgh?ipp`o^`nd@oaxog`obHoa`nahnlxpohN:\\tNOpg`ojlQedgXOggglDKGdKG fFocHoixoipofTdIOQSgeoPqtPitOiTQuTQmTPmTPSeUWeUwdptO[h`OU_gLPmdf<::jQq tPytPqTQmdelO;UWeU?SgdOGeGwccWdQgjcggAhe]gfmVaGw\\YfbMV]Ahe]Wf;wbj:@j:Ll:sZAnar=Yr=uRAuR?eR?eBiv_`r>ar>c]SF]UF\\Mf^Ov\\wF^_FbUF^CW]]ffm nGER=IRotpsdOi`TCUsL=sCMtCUS@yq:ysE]sJqSHQSEMsA=SAmrAqr@=SAurByrBMSCQS FUSAAsAqSJuSBIsBurByRCQsCqRBurCUSDEsDISHmcoghoohpohpKCEcKHlK@lJd_gNE;r _inEC`kNFS`mnEYsEYCk>FMJF]cRlGisGiCBasnGicUO\\D::MWd`D:I wcIWdM_ctc>odTdFO\\TN?OjMadalNadeHOW_o@odPsl`pw@n]hltgi]hkIpqfhogHn[`p jxo`xn_Pnahnk@qohnfdcQWdSweUWdFpg`ohPocpof`of`OiwgiGeSGeOgdOweYweKGdKW eUWeU_b@OYweYWgeWgU_dddUWeUwdhTPmTPitOwoghoihOUWelDrd^OlD::WGetDYgdQ?W OPqtPqtPsdUWeMwddTPmTQuTQAUPqdbHoo@od`rk`pCapq@qYXnYpjDxkr_iyPogPqc`ou hoihpd`on`ogpr;Qr_lxyyyiyyyyxwywmwsxfYpIyyy::::<:xI:::::::::::::::::ua wmyyI:fYpIyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::pIqat=pIwiywYyqyxsYxqyyqYylYyWYsXnaXhSgeyyxsiykYxyYywyxgywyiyw IysIysauiywiyyUyJycyvYxIxyy;fYpIyyy::::::::::::::::::::::::::::::::::: :::::JvsyyQyoyxmyyibAwbYyuuyyyyyaywiyghoghr?irGJPrMkV\\Qv\\Ef\\MF\\UF]GF\\dR>]RCmr@QS?mRHiRAqTGqsJeRAQSyDrEuQ;QQgto HMR>QT@=SSqSJurx:NmTPMZWhMwfafMu^V`mVHMlcVpmXpmYxmYdapCw> yvasVb;g`;OI=TJysJwmYDbwfaKgbAgfOwd=G_gffCgc[Wa;Wg;wb;XdUWaGf\\KvUaecQ v]MXfaWdaV_OW_iViUgeaWdMw`gV_sf`mFdQWdqfbyVdpTJQtK=uSAUGqsGYTNUtfPOdhr =UrHisGMTLMtSIeahn_xmWdcCGc=wayFcCONUdahn_pn^`np`pnLKatN_n^dcGoLQd`pNI wcXDKGdPtLQtKcb?oLQd`LN]TNatNGO\\TNwnadc^NCWcEWcBd>O:jMscZbLOeTOaDOgdd DG_alMCdFO`D::I_alNaDVocdeKWdAXgggfuvawwbGW][uVSW^cviOGeiWeUGcEwb?WdGw ekWgGwdMGdQoQqTPatMUTQmtOitN]TPmdghOkGhkWeUWeMWdxdixnaDeSOQuTQatN[olPg odLPmTPeDRe>phDfWoPCeNO`DSGeSGe::nofdefobHohHOWGeSoPoohpofpohHohhobLOm TP=UR=uQetOitKMtSQtMEvQEu\\=uSqUMetOMSFmrt`qQesHqUOqT\\IUUAVTUuKiSHEvX ueyUyry:pIxItyyyiy=:::<:xI:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::siywIxoixoIysYyyyyxixqYwayq yxYypmxuyymyyYW`uvL]SGyytYYuuxq_YyYyyqx[wayxqyxqywyYqasyYmQyKwSyfYyAyy y:::::::::::::::::::::::::::::::::::::::::tixAyrAyxqyxqYxyywY:`yvgywuX quX?fZLB;^v_ivoixoi:@j:Pjl;Ir;ER;OJ@r:ABLBGN=YBGf[GF[CN:C[fj? hj=XJE^EXkE`kF`kCHkCPkDt\\QnYrAur]r?ir?e rATDUSByQ;mtEeSFArHkdxkMxnL`lUhkH`lB@kJ@kB`lJxk QhlNXlNPlE@lHXkLxkTHnXhkNHlFHlIxkNPlCpkFHlNPlJXlKhkGt_iv_EoLiqwhQ:=R:I sC_mVD_cF_eV_TSEUsEYcR@Mk>vlQd_g>\\CmV`dSGOM\\SFwLTCmV`mV`VmULH_M:hsGi sFacTlGicVlHqcWJDBCN?KHr ;IBlR@mr@qr@UR=Ur?ir?mR@_j>`jB@kB\\B@kB@mTPmFHlJ`jDHKY^G`k?XkD@kG`KKn? ar>qsE]SMisHirxxqIer@yS=erOiRBeQftOCmPsdT=mRVqSJqtEaSJurAQs=]rBar>QsB= SCIsBmrBWkLxkRHnX`K_V]cV^]F`ev\\[f]]^HHlIHlKHlIxkIlJeR?eRDMcPpLiv_LSD_ LPsD?mR@mRlFS`>mR@mQxLi>>MkV`FMJFC`:FMdSGOMhsGCalCqv`hCof`dCq^UlHqcXpM uVapCsFajFasFKIuSIyCkVasv_sv`oFasFawfasFadC=wayvdGwcAWawffOwd=g_kf_QV] YgbCgfEwcEG_iFfmVauG`sFbiEYwVXuuggf`mWbAGeov`IWaufdGWcUGfWW_oF`Gwd`SIm SJQtKetPitFeSFetOatBIsCaSIqSQ=URMTMUtIAtJUTN]tHysIysJATJETKysJAtJAdkHp kHnYxmbpn``n\\PNBcPtLgN:LDC_]lKCcCGcE_`pNIwcXD:G_`lMsc:>obDdvNIwc:>O\\ D:rcvNTDrcZcHocPodddFocTd:>ob@o`@obxopXph`oeXpWxmHYoiho>HkXHiw_q`@omHo c`nahnkLNitNadb\\chn^PoeHo_@o`@odPocXoe@qr@qhdeO_jLN]deXOUWeUGdKoQydo \\gddfoghoeXoe`oftdQwdU_gL=UbAl;Ir; Eb<@j:@j;Hj;`j>`J>kB@kBl?ir?Ir;IrxpqxaR>]R>Yr=YR?ebF`KEnMkV` mNGc`>mRt_vLJFS`VmUXmV`MqVadSHOMJFc`dsHqsHysI_mV`mWhMwfapsHWmUXmTPMsFa s>o^SXmU`MtSIuSHmcTLK=TJkOUtMQSBAsIaSGytNeTM=tJItGmSOycfXmX`to@qxtaGwZ IF`mEYwvZGVgyVbqWdOgckV`uv^aFmIhkAWb;gayVaMGeOGcIWcif`kvbGGc=gcEWcOWd_ wggWcOWdy^`XodXmZxmVHNwvb?Wa?WbyVb=gf_?dTN]d]`nbpNLd^lKsb^n^Dcfn_TcVn] tbZ^LLccGgcXdb@OMWd\\TN]tMccfn_hn`pNI_bLN;I_a<:TDMWdM?:G?vN\\DOgdOgdZc HOM_bLM?o^Hob@opPp`Xoc@qbHo>inb@tbhoN`gk_pQPmrpna`oapnnxpo@oetcOGdSWeQ wcEGcQWdKwcKgcGGdKGdOoUceVoeTf>ob@of`OS_axnapohTgeoPsdQOQuTQmTPmTPSeU? hDBeS?pDSGelD::WGetD]Ge@egpoftdQoOSdQwdMgeU_fPprxpcpogXnbxn_xnaHocDeIG d[gdQWhWGfcwbIwjUgfmGcpUNiTLASDaRvhqNisHuuSQuyqyygY:wYytItixmysy:::Z:Z y=:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::Jv?yxqYnywMwyiseyoqyaYygYyIyVysIyyqyxyyvmueyyUYwquNu TtExnqyqIyyiYyuywqyyuYyiywiYwuyyaYywxyyYxIdIuawMyuYyuIyyyyyy:::::::::: :::::::::::::::::::::::::::::nYjycxoixyyyyyyNYjyCyvYxqyxQ`jK`r>arBAsB]Qv\\qpppphqw>kB@ k?hJCF[Cv\\Qv\\Iv[Iv\\Qv\\G^El=S\\MV\\mnFmR@mrAyrAIsCokHd[KF\\Ef\\MfZI f[IF]QV`wN=]R>=s?_mKXlW`kIXnF@l[XjBPn:pjWHiywj`OgThi:HoKXlaPLq^RpkH`lJ xkLXlKhkJpkUPNYF`eF^mf_eFamV]_F^YF__v\\WF]Of\\Ov]YNN]dLPLiv_if_gNES_eV _]V^]f^_f^gf_`sFOmTd`FmSD`kF`iv_JE;iv_dSGWmUXmV`MkF`k>eV_eV_nlPD`kF`qv `psHqsHysIisHqsFasFYsEYsES`mV`uVau^YPmTPmWhMrafmWt`qnFacSlHCasnHqcYodPobPodxnf<;gcEGdWWeUgf]gdYgeMwdQwdQOQmdjPoehqiXpo`nbxqhdkeGhk gdSwa]v^WVZAv_Sf]yYyfyy]xiY:tixmysy:::Z:Zy=::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JyuixmySyxy vyiqmuwyqyyyQyyyyuyYyyynYhyx_YyyxyqvwnPPmZasKqyjaxyIyvyywuyuYysariyvAy wqyriyx]uyYsiywIywqxqiyqYyay=::::::::::::::::::::::::::::::::::::::JuU YuyyyyYyoyoixoyyyyynaxnyyyyyymvaIyqyCYsEYweYwUYuUYwMy`r>aryxqytQytQy:R Z=VZB[CF[]V^]f[Gf[;FZ;fZ?n]rxpqx@r:ARB=cwgIyuYt RAur>abDPKIv[IF[CNDMSDyr@qR@Ur=Is?mRCqRAIcK@lLPl>HkBpjEPkP`mU@kG`kEhkG HlFhkPpkIXmGxkXXkHHnB`k\\xjE`oIPl;pfc?lrOifplR@nJHlQt\\eV]UV^UF]]f^[fa =waSV^Yv\\[f]ef`PSG]sC]SEUr>YRB=cHpkHhn_hnMXlMXmUXmQxLcF_LCmV`mf]Wf]Aw bAw`qnCIsCMCkF`psHqSF]SF]C:::wfawFasnFasFYsEolPt_iNF]cTPMsFasv`qv`q^TP Mo^UXMq^V`Ms^XpM;Gb;GbnmXpmUXmUXMjIqsHuSHoMO`tNGO\\DGgcGWc>obTdFOhtOitOcddTM]TNITNYTPQUSMtO]T OUtLMUMYtYMtMQVL]tUESEQSt\\Q_MUU=UMUtSMUSiSLAtOMuRITOYTNUtLqtNGocddOOU UUUqtPOO@uROodPoixoixOtTQutPqtPmUXmUN]dixOOgdhtO_ofDe:::voixohLP_oeLOc dVOtTP?pe`okLRuTQuTQWoghod`ofpneTcOoNyTQ]TQqtNutP]DKWe`tOitNidfPodXo_p npXogPqiHpu@plXqjtfQGe=W`oV]IF\\yiywQy[ymySyfY:jyyI::Z:ryxI::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::waxmySyFyyqYtIyywxIwyxfYtIpyy_YyAqarayEQqwptAYsyIyvqxhyWuiywIysy yvxuayraYmiyuyxuYyqYypyxwyyyyy=::::::::::::::::::::::::::::::::::::::: JykywY`yvsylYxwYlXpmxyyiy=LQtLq:Ar:]R>?j:@jtOitGj;Hj:@J;>:FJDb<@kBdZ?f Z;FZLRlCyVaufavMxCwfapsHOn\\PndPodXmWhmUhMwfawva y^Z@nX@nZ\\VHnZhnfHoZhn]Xn[@nfPn\\PpZHnm@mUXpJ`lmpkLHlf?h=An^PsepomXmW xo[PncHn[pmVXmbHoa@nXpMsFaUwdQwdQOOmTPqSKOhqohQgeV?ghggWXn\\pnb@orPqtp mV`mYlOUtMetLUtMQd_hNBc^NHDA?vnaxn`pn`hNHd]XNLTLgn_dcGoMYdaLN]dblMscnN TDKGdKGdvNXD:G_`O`tNOodddZbLQIuRQTO]tM]TMutN]tUYTN=VN etXeSIYT;UrP=sCEWZAVHesF?pgXnhXoWpn]pofTeMGdQoNeDg_fDeMWdMORKNqtPqtQyt QqDQwdheu@obdeWgeMWdhtOiTPmdf<::jPqdgLPmdeXOQ?foghOfoeXohlQqtPgObdSoNi d`XodHohhocLNCd:FOhtOwogholhohHpe`oiHod@pe`ompoipocPo_@n[pyxqYvYpyyWYh I:uawmyyI::Z:Zy=:::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::ZxuywiyxiwmwSxyqvuiqyyyuywixfYtixqYxyx CywyysauqyyYYaXkIxgOGeyIxqQwqyyqYyMYsEyyyyykxsyyAtIxiy=::::::::::::::: :::::::::::::::::::::::::pyxqyxSxmYxyyytIqyxDV[EVLLLLb[Gf[Iv[Pr<_j>d[VlMXlB@kB@lJ@l]Wf]_iv S\\MNytQy\\R>ghogh\\Of\\gj?t[In?ir?eR?erGasFmr=]RCYr>QS>erBMR=As>erD=s Bqr@_kHpkU@nYhjEPk?XkDxjFXkCXkE`kFd^UNDurAaSA=sTQTMesqtHglA@kbpmXXmPhlNPlKPlMHlEpkFHlP`lJhlMplPplLLLMdIxkIplPt^aNDM cOhLYv]YvQytQeV_efeWgeAvZAv]Yv]Ogd@sBOmTPmTlFacR@mRxlQd_gnDQcOLF]cSHM: m>NmTd`:FM`Co^UXMlcWhmWHmSDasNIucXLHmSHmCRanmVTaqv`yVa obtcnn`dcvNXD:K_bpj@phpOQWd\\TMUTPeTOIuRAUTEUSuTQutyyiw yYuau=XixIY`YyyywyiwmyyI::Z:Zy=::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::pIqateyoiy`Imyydyy[yy Ywax]wyyyiYkYwiyyoYy`ywAXdUwTYtOyYwoypyxyiyxqYtYyuYysyyqySyxYxoypAyriy yyyyyY:::::::::::::::::::::::::::::::::::::::rwiywyyyyixoqxqyxyIBdKGdw bAwB`kF`yyyaR>eR?QSGaS>qR@Yr@mr=mr?Ur?er=c\\SNBuRA MSAyrHIsCmTJ=tCIR>mRd_gNEs_iv_i^SHmSPmTt`qnGc`of`mV``C:o^ULHSauVak^TPMsFalCqv`qv `^MtSIuSImCqv`wFasFbwfasNHiC;Gb;GdKoHCawnGqsHisH_mXTaynIAtJqsKEtIuTOUs LAtD=tHiSJusMEtJqtI=TTuSJmuIEtWmsIMvL]tE]qwpQkPpNMSEiuOmTNmSHqtJATSMTM IuLYTXytRitKMTKmsHYtLQtJItKaTQqtJUtLASEQsfhogpsIysKAtJQtKItMEtKUtJETMM d^`N:::?gbDDCGcPtLgN:::JMUd`LN]dblLgNnNXtMwN\\TN?O`tNGO\\TNoNGWcEWcE_` LNSdZblMccGoMCd^odteOwdkwf_geOWdKwdMgcQOMYtLAuNaTZeTP=vJMTKArabwgiwgItQytq:Ab =Xj=hj?T[^j>T[EV[AvZHB:E^B@KOf\\Of\\^j>`jIxK=VZHqsHqkHpkLSDWj=tMYtM=R: =b:?b:cDOcT[E>@Bgf_gv]UV][v\\Sf]Mf\\GVZ?F\\AF[Sf\\Of[Gn=C_gf_Uv]Yf\\Yf ]OF]Sf[Qf\\Gv\\OF\\Of\\Qv\\Qf]SF]GGcAw\\If[;g_gV`QF]yf\\Sv`AV[oUQ;u_oU YIg\\Uvfkv`;w\\UF`Ov\\gf]Wv\\MF\\]f^[F^@sBIsCuRB=sLQtLESCEsFacMXL]V^]F ^[NF]cxoixgeWgeGhkGPkB@kl@pjpj@T_eNDMcNlDQcPpLiv_\\SF?MdCeV_e>kF``sFOm Tt`qnG;mV`mV`:ZTj:@j>`JeUWeE>[B>_cF_CB;CBasFasWiuWKF\\Kf\\drE Yc?hJUv]hr@qr=Yr=YBSF]Sf[Gf[KF\\KF`\\sAyrAuRAur@qr@ar?iR>ebAHkCxjCHKfl T@mXHnYHk>XjUxkIhnM`lYxjCPoEpkApfdokgGhcPkGPoIPlepL?g_gV]Kv[[f]UF^aV^W F^[V]Yv]CGcCw^av^of`oF_cnBAcIxkI@lJt]Yv]Iv[IFIsCIuv`qv]UV]]V^]V_eV__f^ DCcF_PsD?mRT`VmUXmULEUSEYsE?M\\Cm^TPmTXmUXmV`Mof`dCmV``C:o^TlF;o>>mR@m RLGSauNJucYhMyVauVbyvasFahsGwmYxmalKId\\LJ=TJysIySIucWhMy^WhMOwdMWbCgb svauva?Gbwvb=w`;gaqfasVawFa?GbyfeMGdSGb\\UNatZmTQAswpQ;YPoXUGqSTasHMuG ySZatPMuJMTPysJ=UNatNMtLudhPlN`lnppphlQxlj@Pof`oFdGgcYgdQgasFaAwbHDCGc LD:EWcPDAwbAwb:^NLD:GgcTDCGcLDRcvnaxnb@OGOMUDIwcXD:K_bT[EN=C[CF[;FZxqyfJhr?gj?hjtOitGasFaQv\\Qvgg ggGXkEpj@TXmUX_DOcdUr?ebGtZEV[KF\\psGiSAmbH`kF\\F`KQv\\QV\\Qn=ar>QR?arEusGUr@erHiSGQs@q RCEr;eTDUSOiRAIT:MrAmPr@T:MrIEr<]TCMsOiSHeSB=sCyRAiR@eRA=SBmRAurKWnL@l JPmR@mOXlMPlLd^FlMXl@@kBXKhcUXcEWcN`lNxmYxmQxlQPlLPlMXLcF_PsDwlQxlR@Mm V`dSGGmSHmR@Mk^QxLR`VmUDasNHs`qNGeC>M\\C:of`hCR`vlQxlQlFasFiCy^XpMyva< TJ=dWhMqv`?Gb;gdKOJ=TJ=D?GbDD=wayFbwfa=GaufbsVa_gdOgd?gb?gauFb=wauFc=w _?waaVbuv^=WaqfcAwann^Xnb@nZ`p^hn>qojhpRXmFhgjomxWjgxkN@sgHpxxnexoVxma HmUXmM`l[`mWHocd]WNOeTOeDoghoWcEWcsf`ofdKoKItKAtKItJcbCGc?GcCwbEWcPDZ^ `NEwbAGc?gbCgb??EwbAgcCGcKGdXtMgn_hn]tbA_^`NPtLYtM?obDdvNXD:G_`ob@ob`j>L:cYweYw=Z:@j:PjOj[XweWwhqwhA@k@@mR@m CXKwe[EfZOV\\=V\\KF[OV\\IF]QVZAvZKF\\pCQV\\MF]Of\\Uv\\Qv]Un@qbF`kFHkEX k@XKuFc@R>QRMS`j>XlMdb?gb cF_cF_flOd_gNEs_iNF]cRlGS`mnEYcPpLB`ZV`MuVauFasNGeSGasF?mR@mQxLi>VmUXm U<\\SF]SEUSEUCkNFeSG?nZ@nYxmYxMobTdMoNCdnn`TcfN TDIwc`tNatNCdFoaxndhNmWf]WeEWcSwcGgeYWe?wdKwaOwc=GeMWcKOVmTPQvPyTWQsFa srTQ`yTSquPuTFisFaTSytK]tMUTN]TSMecLN]TNyDIwcptPqTN]dalMsgXuLgnf`OKONe DKGdKGdFocTdNodtdQOPmdflQytQuTQOO`DOgddDQ?foixofpOO_eLOCeOgdUwd^ofPpgp oihogHpf<[_ilQedfPod`of`odddW_mHpkXphxoepNOGcEWeKWdyiywQy;vYxIuyypIyyy ::::<:xI:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::uaw=jyyiwiYwYyyiyuaxMywyvuyqiy`ywYW_mV` uU[yywuiuMx[vyxsyyyuucyviYysYxyquqYxQywOyyqyriyvYyyyywyYxIZxeywAvyyuy: ::::::::::::::::::::::::::::::Zu=yyy>yruyyqsIysEUSEUDLKDpr@_yvE[CF[mUX mE[CF[KF\\KFXkEX=n:AB:?^=XJA^>`J;fZ?FYweYKF\\LR\\oeHO=Mr;YQuDqXtMXpKIdk]R;YR=]R?er@yrAar>OkDPkTd`MV\\`BB]SF]Wf]Wf]^kFD\\Of\\mfauV\\WV ]Kf\\OF]SF]Sv[KV^QF]sV^anAAsF]r?]E=g\\WF[QuUKVU]eaAv[_W`ufb]F_Yv\\OF_e v^Kwb?w`]V^mV]Uvbiv_kF^[v^YnCUSD=CYF__F^aV^[v^]f^ev^af_cF_iV_iv_psHqsG isG]R>Wn]XnOhLiv_TSE?mRD`vlSHmP@mRHmSD`kF`ev_iV_iNF]cTPMsFasfawnHqcTPM iv_PsDQsDs_XsGicV`mVXmUt`ZVLF]cXpmX\\WLKEdXpmXpMdTOgmWdawNICb;Gbyva`JC^_b=IES;uZ;FYsv_cV[=FZGf[TR> ]r;Qr]R@mR@UR=UBKF`\\s?ibDPKO^EXKUV]UV]^kFd\\Of\\qVauF]Wn>ibEl=ar> mbF`lLPlMpkIhlGpkbXmWPnIPlY@kF@mtwiB@ggWmu?je@lNt`qf^UF]]WdMwZquXEw`@s =YRJ]SFAs@mREMsCEsDIsBMSCyRDEsBMSCEsDIsCc_iV__f^MTSF]cSHmR@Mb_>mRd`onGisGmSHgmWd`oNF]SFYsEwL\\Co^ULGecTlGcaw nFasFuSIusGisGiC;_V`mV`M`tNwmYxmYLIuSIuCrayva;GbD[B`kNAuRAmR@OkDd\\VKQF]Sn?ir?eR ?_mV`mHd]OF]prAybEXkD`kFpkJ@lH`kFHmNhlZhlQ@mAPkaPlOPoIXlIHhpGlighSxhwg oHHlbPlNPnPplUHmRXjBpjKplN@k@hjShlNxlLHlQXlL`lOXlMD_]f_cV^flP`lQhlN`mV `mFD]CGcPsDQSDMcOhL]V^]f_gf_kf_gf_gnEYcQLF]cSlF;:b`of`o>:>mRerDyq :qs=aRY]TO]tCQsQqSIIvTUUK]SFAuNatRmtOqSFYSHMSDEUOeTGESCeodL_G`k`PNYWeS gbAWb;wb=OLETJWnZxn^pma`nXlHUtKsb?GcLTLgn]TcAoMEDvn_hnbpn`xn_hN>o`tcfn _tbHtKQtLwna@obtcvNTDG_axnaxN\\DK?vNXD:G_`LTMUTYDC_`XodHNGgbKwcKwdMOMQ dbxn;ipohoX`mt@oahl;@jfPM_wb=ggUgdWWdKwd^P`tNon`pnflLQddPOO_b@ob@OXuUY uLsdQwdIwc\\DMWd`D:OgddDrdQoNGOZeLPmdelOcdFocTd:NOhDS_fyrUxmYxxiy=:tQytq:Ar:ABweYwUYuMytay[;tZ VJJ;ER;Ar:?j_kDPkxGitO`g^_LN[DfdGgcs^`mnZqR@]r@mR?ybAHkCpkF`k\\`mWHm FpkS@kEpnK`la`kJPnZTPMqv`lSHGmSHmTLGS`>M\\CsFasfawfauVanMxsIoohDbnM@tJATHmSHqsH[V`Ms^fDe;Gb@jy_mTHmfhoe@n]HnY`n\\`m_PnUpn]Pmb`nT@o^\\Vx n^Hn`hn\\`n^Xn]Xn_XNE_``NIWcfNIWcEwcZ_XN?gb?gb^n^`naxNMWd\\TNoNGWcEoLc cGON]dcHOM_bLNscIoMccnNPDcGgcwc^n`tcMWd=gcEoMUtMeTNQtKETRiTOMUOetWuTPE uKATQmsG=SxhQXetNIUMMtP]tMitN]dnTdnNptPgNEgddtNadaxnaxNXuUYuLCeSON?odP OK_blNSdZdPOO?vnaDdK?`DO_dlOitOeTOetNCdvnadcvnaddOgdSGetTQutPqdf`OQwdJ OsdfogdeWgeO?VOlDS_elOcdvYxI:::::::B:ry::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fYpI:y yytYyuYy;yiyHYqyhwMxyyymyxmyrIYGyrBiXjExyuyyyyxuiqExaYwyyybysixoYyuYyt Yyyyyyyy;uyypInyweyyay=:::::::::::::::::::::::::::::::hIuYyuyyyyyrwiyw sIysyyyyysIylIyavCYyuYyE[Cn:AbywiywItQynIJ:=R:EbYr=iR=YR;qQxDmPllCtj@Pj:OkDt\\ Qv\\SF]dR?gmUXmHL@uRA=SCEs>ur@aRAs]WF[fiywibpn``nXxmPXkGplBPk[`kIPoKhl Yxi>Hmn?iHXgiWoJXLIF\\Ggb=g]]v]Uf^[F]]v][V^Yv^]nEISCesDMSGUsDeSEQsEQSD QSEMS[rHgSGOXtM[blNcdOOOSdvNTDI_cHocXoeXoghOWgeptP_OhDQ?^O lDOgdhtOWOlDS_elOcdvYxI:::::::B:ry:::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Zwuyrxyqw;ua wmyyixMymiyiywYysIYsYyyYytYvaiuuycywIXhqGr_HoyYwkyyqiylixIyuyywmxoyxey ZymxwyuMwyYyeyyuyxwywmyyI::::::::::::::::::::::::::::::nYjwiivayvuyxIw AvY:taxoIj:S[Fj;Hj:@j:xIyuYnixCZNj^KtRB=C`RA_kLHlAhk F`lTHmSpmWxjCHkJpkHhmRHmSHlLXmJPlRPkFXnM`lMhiy?pMpl?`fbOoEpkTxi>mRD`vL\\SFGmSHM>MdCs^VLGeSGasFasFc`o>tSIusHqCsF a;Gb;?uVadTOOMoFanmXpmYxmYxMtSIusHCa^OSgawNHmSH=Dba>n\\PnVpmXhMlTPwMps HGNlSH?n\\PnUHnZxkTxlHhmSxmgHoY@o_Xnbpnh`oeHnVXm_HmSHp[PnihmX@q\\`nuXn _xjiGhKHiuWoOxlrpnaXoYxmeTcKwb?G`cv^kGgaWbuFayv`oveQoOUtLetLMtNMtKYtKE tMITKKNMtKYtLMTMon]LJQTLySLItIMDZZhn^\\[pn_`n`LMwn_hnaPNbcCGcGgcvnaxN \\TN?OXDI_`LM;I_aplPp]hn_XNEWdMOM;QGeOwcKgcGGcAGhgWgSwcCWgSWdaGc ?gfuFacFYsuhIgcWgcGoM;MWdReUoMYdf`ofHocDdvNXeqhNEOPwnaxncHOIwcIwc>O\\D RdMoN;QwdNOM_blMYDFO`DQ?ZblMSdVOhDUWepD:SGelDQ_dLOsdQoOCeSOPsdVOdDyay= :::::::TYuUYoeXouZA vZ?fZDBRZ=VZ;FZxqynixoiwgICF[Cf\\Of\\AvZlQxVj=T[Pr;Ar:Qr;Irypqx\\R=URB =SBAr:ARsDQsDR`j Bl@qbAl=s_Xs>ar>aBB]S^HpkHL@mR@eR?_MtR@mr?ibGxkIt\\`RAqr@ISCarAuR:UrOjNSd^oftdQoNatN]DI_aLNSdVoetdNOdD MWdMWdNoddd^ofTeUWeWgetDUWelDO_cD[EV[En;Ir:Ar:uQyviys[Vj=XJv kKHl?xjAhhqwhHXlLXkGhkpohpWkCHkfwfb_jxwi=XivOlDXkBPj=`jM>SF]IV\\MF\\UF]Sf^]f\\[v]QV^ [V`yfaQV^[V\\WV]WF^[V\\Qv\\aV^]f^[F^iV_cF_Yf][gbAgc[f^SW][VakEYoVVguai EYmg`uvbWF^cF]Sf_iV_ef`kV^cf^_F^Yf_cv^_f^\\SF]cGtWiuWi]GhKkF`kf^_f^cF_ cF_flOLEoLXsEwL:\\SFGmSd`oNGC`knEYcPpLk^TPMqv`hCof`jHqcZ@Nof`tSI_mVDaF n[HnWTayvayFeSoGicV`mV`MpsHgMtSIoMxsIGocdb^mVDafMpC;_^`n^pmXDenmXDavmY xmTPmTHn[HnX@nZPnahnYpn]Hoh`oWPnZhmX`m``N?_bXn\\Xo]XneHNSGb;whYweKGasv eAwbQWawFfwva_fXse_mEYAG[IFgif`wwbGWiEwcoGcTULQTPusIitJAtPQddhN?gauv`m F`wvauFdMwc;GbXtMwNpsIysG=tIATMQtIQTL]SIqSIMtKuC;gb?gb?oLItKYtLQTOoneT dXdaxNBdKOOedcHOOgddDEoLUTMoNXtMwN\\DK_aHocpoj@paHocXn]Xn``n^Ho^hndLOw ngpofpoihojHpi@ochniPoaXp``nqXn\\HqYxmNxix_pdPofPocppn`pcHochNlTPgNdTO _ofddNO\\UV]UL_ncHObcVOQ?O_clNCdKONscIOMUdalNatNeDQwdQwdNO`DK_b<`tNGOd DO?FOjPqtPuTQoOpDSGelDO_elOCeSOPsdhtOcdvyyuy:::::::Z:jyyI::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::RyVyyqysyyw]tmwsxxyxSyvyy=bytYy;rywqymiuaxqyyuiybYuWYmJXlyY yxewmqwcxsyyoqx[wyyyEywiycYxkixyiyuqyqIyyy:::::::::::::::::::::::::::: :::tixuyxIbxvyyEhkGh=SB=C`kF`eYweiXomv`qvhqwNitOiykyBZ;n:SZZ:lyxaxoix_ iv_ixoI;^=Xj=`j>tZAN;Ebir?iR=UR=QrybQhmVPkJxkBpkG`kK@lA`kE@lJ @lHpkH`lOXlJpkGpn[@nUhkHpnJPl\\Hj?`nroi`phvwmmGilHlOXp]`nVxlPpkIhkNxlO PmUHmNhlM`lNT`mV`_f^_^ipghUHmRPpgpoYPmUxlOhl^`n^poj@PSw dyFcAOL;>n\\PnZPNE_`DcOWchTMYTM_n`pn`pN:jMYdaLMccvNXD:KGdTTN]TPAURYtN_ n^`nc`NQWcGgdCWcQwcGwdSgdMGeOwd[geEWdCgeQwc_WcAgg?WbogawV_uUYYWcEwdIOT EUSYtMYTLMTLeTOetKItK]TN]tOitOaDKGdXuUgNEwdQOMUTMKOeddlNSd:>O\\DIwc\\D MWd`DOgdOgd:>ob@ob<`DM_dLO;:QwdlTP_O:JOOOVOlDS_eXOQ_dlysy:::::::Z:Zy=: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::pIqau=pIuA:vyy=::JywywyyrIxnyytiYqiyeitSY^Cf[yy xuyx[yvyIywixiYxyYyyyyuqsAyryyyyyxoykaxliyyuyyA::::::::::::::::::::::: :::::::Jxmiymy;RyuYyyyyhtOitCHkCLSDoyxqyCHkCxjAxj>`j>xiywi;Hj;pixsYyuY ;FZ@r:GJJ:=b:lxpqxpAy];L;Oj=Xj=`JPrhJsUYyUZ?vZIF\\yuZCVXueYmTQwDCK SDCb;Ib:=r:?sA=SPcdOAEQydIeCHCs@wBMAdKSf\\Ov\\QnxrAyr>ibA`KQF^Yf[Qf\\IF]XSHOkIpkCpkG`kKD [MF\\Wf]hr?irBESBqRAmrJySI=s?eRIESCMTA=sLMR>Wi:XmgWh[@iw?oAXkd`lPHnOhl L@lIpkKxKgF__F_cNEUSDMSDyrAwKDSC_lNd^_f^_^MXLcF_TcPt_XC:gf_TCr_>mRT`mn GisGmSHolPpLZRlEYCNmTDasNF]cV`mVHmSHmR@MB`vLTTMomXpmXlHqcWO\\DI_aJxqyvIJ:cZ?fZCF[PB?F[tQ;IrrPQu`r@uR=er?YmWlm?TCiv_\\SFOMov `qnEYsEkFC`kF`i>NMlCkF`psHqsFaCVMjECcCoIysIuCw^WhmWhMlSH_MhsGwnaxn[Hn[ \\XpMw^TPMmV`hsGisGsdhTJ=TJysIWn]XnV`mVPnZtauVa;GaqfbuFaYgdMWc=Gb?gauf cCwbuv`ofb;wayFbwFayVauNIyTS=UGysHEtMQTJQtKYtLMtMqsHytI=TTEtKQUJETRisH aTDUSMAsCMtAAs=EQtDSxtqLyrBUUJETV=TKIuGqsVAtKIvPyTp`OhxRy@RPysJqtLUTNE tKeTN]tMadY`n]Pnapn[pn_XNEgbLtLIDCoMEtKYTKITMMd_hNCGcLDA_\\Pn\\PNjNadb @OK_axNI_`LMsbfolHp`XodHoaxngxnbHp`@of`n_@o^Xn^xNyFdEWc[WeCwdGgdQwc_gc CGiCOYusIYSv`QREtKatLgpmXpeXoe@obdeWgeEWcEGdKGdI?nNXuUgNptPqtNatNitOwN \\TN]TNSdMWdM_dPodPOhDOgd`DK_axNI_bLOedelNadc<::JOSd>OXDRdVOlTPmTOOoel OCe^OhDQ_dlywyy=:::::::j: TZ=n:cZ^JDR;ER:=R:yqyvIar>Ir;GkCHkV`MSN @;j?iBNK`Bmv`qF]^kHt]]V^UV]UF]nKWv[MV\\Qv\\`sGiSA=SBqrAyRBESCar?iR@ubC XKEF\\GV]WF]K_JXkD`lJxkR`lMPmO`lT@lJHnI@l`HkFpmpWiMPgiwmtoikplSPoVhmNx kH`lLPlHpkHxkIxkOT_iv_PsDQSEUcV`MkF`PsDQSDMSDESCESCS_eV_mV`mv`hsGc`oNF ]cQxLgf_TCi^RLFS``SGecSHMkF`\\Co^V`MmV`tSIusFacSLHmcQxlQ\\_hn_xmYxmXpM w^WhMu^V`Ms^UXMIwc@DbawNIc`FMhCq^etd;_YxmYXn]Da>nZ@NuNImsGAtHmsPaTNQtJ =tIqSHITLEtGmSGqSJusG=TIesIqSGysHiTRutG=d]@o`pm]TbGGcCwcEgbAWbwFaqvaoV `EWaufeCGcewcKWhMgdYg`sfaGF\\YfVeu^cuWKw[QfiqvaCXbEwiwfb=hcOw]kMx`PplS A=SUutQmtMwmYDcGgc?wcGGbEGcAWcEwbEWcGGcCWcA_[Pn`Hn\\@o`pnbpNEWcPDC_]lK ;MWdMGdKONscvNTDCOMATM_olHp`<:ttKQTNMd^xNuFdEWcawe?geKGdSwc_gctUMUt[AT KeswlqQATKatLMTTEUSeTOeTN?ohpOE_cHOIwcI?G_qlLSeFoctdQoMYdb@OK_cHOM_dPO O_elOcd>OXtMwN\\DM?FO`D:::O_axNI?ZcHOrd:VOlTPmTPsdVOdDyay=:::::::<:xI: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::uyyTYv]yuYyuyyxqyyqiwqxsyyrYyyyxtyrvpq TysKqymqxyayyyywqyyqyxyyxUvYyuyqy;::::::::::::::::::::::::::::::jwiiyy yyyY:hIsIysYcEOF\\KF\\SF]sxqyxMRyuYxABZFJDBEV[Hr;Gj ;DZvIxA;>=VZEv[KvZIF\\KV]WVVguW=cCIsD[SFIF]Uf[Ov\\gcHqsDUcE_v]]f_]v^Uf [MVYgUX[uSOu>OR=qeWiu[EV[Qv\\lR@_KPrOGGeGwc[Wc IgeAWcOgcEgcMwcuFdEOT=UJuTNYtOUtRQd:Aobxr\\`nUXivooZHnaXN_Gf[wcIOLMddP odPn\\Pn`pn`xnaxn`pNG_qxPEoLqDM_eXOKGd\\DMWd`D:OgddDQwddDMWd\\TN?OjNSd NOdDM_cir?ipoHkEXkUZNJ@r:?j:lyvIJ:=R:AB?^Z>Fg_Og?xi>XinGiLWcJw]OV]sN`o ^jy?jCxjAHkAxjEHkCpkHd\\Of\\SF]dSGes>ar>eR?er?ibFL@;Qv\\dR?OmV`MtR?mR@ qrAWkEXkGhkG`lLPlKxkIT]UVaqv`[f]WF]SF]Yv]`r>oKxr@qrDASB=S@irJySIqRAmRC QsCiRBuR@AsAAcN@lMHlIhkFHmIxk`HlMhmxGjIxffwmighnXkJHoJ`lZXlOHmM`lMHlKH lKD_cF_c^MXlMxjCHkCXK[F^LSDMSEUcQxLkf_gv_eV_mV``C:kF`XsEolPplPLF]SFasF GM\\Ci^QLF]cTPMkF`lCmV``CunHaCqv`\\TN_mVDafmWhmWmrZ]TPqUNiTTITMMuNitPYTN=tGmsLItKM TLMtLQtLYtLwN\\tKMTNEtKUtJ?o]`n`\\^tcEWcEWc:JMUTMUDIwcIwcfNIWcnnb@o^HO QWf[WcOWdGGcCgdCWcYGcGGeCWcQ_c`odPmb`n_`pjxmgxn^@o]po\\@nyxna@s]hnUXiv Gp]`nchN_Gf[gcGgc?gb?GdKGd=_`pn`@ob@oaxNI_qtgE_f`OKGdKgdOOO;M_cJ@r:Oj@kCpi yWjd^_gFZ:^Z=>Z<>[Z=n_a^_?pi;XkCHkEHkChj=XjAhj?xkIxkHpkHhkGd_gN>] R>ar>ar?ir?mR@gkGT]^kF`kEXKoFasNAGkED]Wn?ibHpKev^av^]NAmR@mSGeSAs\\MV \\SF]Xr=YrAqr@uR@mRH]sEuR?aRHesFqRAmrC]SEiRBuR?yr@urCAsBUSDASEWlK@lN`K EG^_FfYv^GGXwE]wDSSW\\UVd]v^Aw^cF`[F^gf^]F`mv_mf`kf^cF_[eV_EXoeXAV[Ef] [F^iv_iF`knFYsEglOplPt_iNFc`oNGS`mV`kF`XsEwL\\Ck^QlDglOlEYcSHMgf_hsGiS F]SFKHmSHaCq^blGicV`mV`MpsHgMtSIuSIsayvaMoNEDsFasFavmWhmYhM;GbLTLMTIuc f`ofpmXDalcYd`dsIAtJuSJScAGbAWbKwdMWa;gamFaofaFmVPmWPnZxMtsJysI=TI=uRy tI=d`xn_Xn[@n_Xn\\pmVDdGWchdcdcEwb?GdHtMId^xmXxn\\Tc;waOGb;weEWc[gclTK ETP=TJytJETVUtM=t?qr;mPrtqnh`?@knpnaxpd`o``mVXoZPndPn\\`o^hnf`NUWcGGeC GcOgbdTKItMATKYTKItMEd``NGGcCWcEoLccGoLQDnNPtMYtLYd^Hob`ncLOWok@PVn]dc CoNMdbpn`@ocxnbxofHmaXn_XpiPnfpn]`nZhoZpmxpn`hr]`nUXIewcKGeKwcDUSYtMWn ]XndPodXN\\TNGoctcvNXuUYuLCelTN;QwddTOOO`dcHO\\DK_axNMWdMgdOgdQwdhDO_c lMccGOOeTOeDRdFO\\DK?vN`DO_f`OS_eXORdZelOCeSGeS_elOcdvYxI:::::::B:xiy= ::::::::::::::tIwAvywyYuayyYJyyixaywYywqyvyyyqyuaytayyiyvyyweywIywiyyy yyyy::JyyixmyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::uawmyyI::>ypqXuiyyIyuIqYXnGXgmgxQyukIu_yyqiyVYyqyo ywqyvYyvQYZv=dyy_yxEyvYwyyZxyyw]y=::::::::::::::::::::::jvkywYjxqiyyYy avaYyyyyyxqALRL;ER ;kwpqx`qwhQ=]bBPKkdPqTAyrAwBB=SLQtLaeVcUZ=nitOjdMV\\m\\DN\\pPn`p;uQ:yr @qR>UR=ir?ir>ar>]R>]R@mR@YBI^OL>]R>Yr=YR=UR=UBIv[IF\\KF\\Of\\OF]SF]gf_ `r>GKtBOf\\hr?gKUv[Iv[Wf]WVaunAyrAqbG@L\\R>gkGt\\QNC=SByr@_n\\PnGhKYV^ hr@qr?qr@qbI`kHd][F^av^aF`aF_HSD=tAESMirAiT>mR=aOhDtudQTMsDqtF]sGMsBQS D=sAokBhkEt]UFREuRYF_aF]]F^=wbCw^[f^Cwa=W^Sf]av^aV^av^av^LSD?mPplT@mR` mTPmWXMmv_iF`gNEUCflQxlQHmSHmUXmT`Mgf_hCmV``Cu^QxlQplPdb?oHqsHmSHOmTPm SHMkF`\\CmV`hcU`n^`NwNGs`qNHmcUXMq^XpMqv`dTOgmWhmWlJAtJeSGesHqcSHmShMx sIWoedb?gbw^[Xn]HMhsHqSJ=dXpMw^hdesFaHtKItIysI=TJ=TK=TJEDAWb=Gc?gbCgbD TJ=TKqSIATHqsJysIETJ=tMATK]TKITNIdbXn^dbXd[t`qFfCgbcgcIGeuv`mV\\Ov]uEY [fYyE`CvZiWbAgg=Gbcwa=gf;wacgbAWgGGcaWcGWfEWcYGc`tJETMusIQtI=tNMtLeTN] daxn`lJAtJ=TKEtKQtLITNYtKaTNysMUtHQTLmSLITOAUQITNoMO`DQwdlTP?OK?M_dLOSdFO\\DyyyxI:::::::B:x iy=::::::::::::::tixqYjykxyYykyyyyoyyey:qayyyxQxoYxqIysayxqyvmywyy=::w yytIyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::nyx=yAZwuy:yyyfywqyrIYciwoytYYccF`yywxywSywAyiyuyyyiyyQymy yqIpyyOYwixyAJxsytyy;vyy=:::::::::::::::::::::::JyuiuMxmIxyy;ByZwgiwOj s_iN?eR?]R>]R>;r[In=C\\FmSD]SF]Qv\\Qf]NKdBS^Al@qbV`MYv]tRA?lJ@lC HkCpKhr?_kFt\\^n\\PnKxKenDqrAyr@gkHpkHhkId][F^_f^_v^av^ev^av_]f^qf][Fc Sf]Yw[MFa[UV_vTUEcEf[Ww`kFbiv^cV^UV^eF^iFaoVVgEWedPmD^_f^Sv\\UfdIWdKvZ EFbuvae^L`lN`lNT_eNEUSEasEYsGasFqsGiSG]SFasEYSEUcPlDYcP@mR@mTPmSXmUHmS HmWhmWHMXsEYsGisGYsE?mR@m_LHmSHisGOmTd`FmST`:nn`Db;OHmSHqsHgMdSGeSGCas Faq^dlHqcV`M=Wb=g`fM`Cu^XpMQwdDDw^]Xn]PmTPmXpmXxmYxmXpMw^hdes^]lISb=Wb yVauNIusJyC;oJAd[lKAtJID;Gb@DCgbLTKWn]Xn]lLId_\\[@nc`n[ho`TexsNeSF]TDI SOQSDuTEUSxDPktRtLQNuRAeUKETWQTLEUJysRETJQuM]tOuSJqTJETQItLutKQtOITLUT JATLusIATHqSUMUTqtPqtKItKasGisDesFasIusIQTLQtOeTNitNAd\\hmZpMAWbwvcCw` Vm`TanlY`mfHpiPqphpwHpk@rehorXmX@oAXkVHj>Hp^pnbpNaWf]?C?QOMkNwNIgb?gbM _j@pj@OTTMUtNadaLMscIwcKGd\\DMWd`DBd\\TNSdFOdTOOO`tNGO\\DK?FO`D:O_d<`D K_b<`DO_eXOKGdjNcdNO`DM_blysy:::::::Z:Zy=::::::::::::::tIwA:vYxInyyeyQ IuOyyyqycuYYuwiywiy^YhYymyyyiy=::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::JyuixmywyymxyivqyxuyhyywyyeysIy yYywyyuQpAXlyYwuyLUSGiXh=xyqYyeytMyyeywwYiIxgyyyiyuiysywoIxoyywix:^yxU yyqyyYyyixmyyI:::::::::::::::::::::::wiytIoixoIysIyxInYxIyAZxoIAvZAf[G f[AvZAvXqMytQytArYBZ@r:Ar;Ib>`j>xiycYwmxpaywIAvZA>C>Fj;T[En]Bfj?hj>`JsFatRA_kF`kIxKOf\\Of\\^KXr=OkDd`of`Yv]Yf]WNCESCeR ?erAybF`KMV\\Mf\\ONLEdMHlKxlQD^[F^Wf]WV]UnAyrAyB[F^[V^]V^_f^_F__f^iV^_ Fa[F^MW_cFbCvZcvU]E_UuUAgY;veav^IG`gf`aF^SV]QF[IV[AvZAFLGdKkv^ev[?F[YF ]WFaqVacf_gv^eV_aV_e>b_gnEYcR@MmV`\\SFwlQt_flQxLknFS`^mVdawnFacQLGecR@ mRPmTPmcHOof`dCqv`hCb`onFacRlNatNEDu^XpMuVadCo^V`Mqv`dDs^VlJc`oNIucTPM s^WhMOgdOWb=WbwfaLd^`mVDb;GbwNIqCu^hpOs^^`n^@nZ@n\\Pn\\`mVDafMtcX@nZ@N ^n^`n`pn`PNDD=Wb=WbZ\\Pn\\XNLTL_n\\Hn\\@nYPnXhMwfaWGb=Gg?wbgGb?wgyVb=W \\Qf\\_EWoeRCUYIUTMFWgu^=fZ?g^affCWc[OLAuKQtRItLuTJETOqsIetIATSYtNaUQ= uKiSHYtIOo_pnfPodhoeXocTdG__Xn]hpmPph`oe@o`dcIWcEGdGgbEwb=Gc?GbAWbcWga wdIgcGGasvfAGcCHeWWhsvaMG\\SvbKv\\]GcGWdIwccgfXDEWcEGeVn]TdvNIWbFocHoi xoiHocdcNodDd>OXtMwNJN]dblNaddPOO_cHOM_b@ob@O`DM_dLO;M_cOj NSd>OjNcdNO`DM_blyyiy=:::::::<:xI::::::::::::::bynY::jyoyy=:tyyqyvaixq yxIyvQyuqyv]y=:::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::Jyoyx=xyyyyxSyVYhIuax]xqyyaYlihYv`yytqy`@mTPvOyt yiYyIyuYw[YvyiyuyyVxkAxyyyxaYdyyOYhyy_YlYysyuyyvYxI::::::::::::::::::: ::::rxqqxqIyyyyawiywIysexoixmYxmyxqyxyyyyIZ>J@B?fZTR=UR:=bywIweYtAyuY< B?fZ?vZVJ@r:wiycY>j:`j>@Khr?iBPBMV\\Mv[Iv[O^R@MMV\\Mf\\ON?;MV\\XBEV[EvZANHmc HL@mbIL?ebEXKYv]Y>M^SHmSpkHt]Yv]_f^hr?iRB=SBmR@mR?eR?eBAWb=W^Yv]B^[nAy rA=SB?LxBY>>LDSCESEMSDasD?lCpj]xlMPoJd`euW[FTOeaiUXMW]UvceF_;g_gV^Sv\\ MF[Ef`ef_uCHosaoFaAGb?w[GF\\ev_PsE_lPpLVLLSEUSDUSEQsEYsFacSLF]cRLE]SFU CkF`jGicYxmYPmTT`mNHS`m>`DFmTPMqv`lSH_MhsGGmSD`>OKWbfmWLIoMlSH_MtSI[eX OsFahsGGn[d`nMdSGWMpsHqsNadYxmYhmWhm^`NsFa@Dw^WlHceWges^^LJ=D?oHSa:vmY db?gbEWcEwcIwcAwbAgb?oJCb;OJSb=oKItKMTLOn_XnZHnYxmW`m\\hmV`oZdf?wbeGb? GgwFbAXeYGbOF]GFV_E[YUVEvVcUYYEV?FWee];VZoV\\SFcaf__WbCWhKwdkOO=uIETM] SG=TCMSVeTPMuMaTPAtK]tHysKiSHMTHqsLqSIYtI=TUuTQuTMUtMysIUtJ=TMAd^LKqSH QtHqSLeSGYuNeT\\MuTQVREUV?nZXj@`n?@ksHn]@p_xncxnaXpkHpaLLcdOoKCdKOMSbF OxtQGocdcGOOCdFocDdvNTTMUTMCdKOOedeXoeXOdDK_alMCdNOhDQ_dLOSdMWdM_bLNsc VoeabPpLGN=YBMV\\Mf\\NK`r>wJTBmV`hBOf\\Of]Wf]Of\\lR@mrBAcG hKOf\\dcTpkHT]UNCESCiBYv]pr@okHD]SnIysIir?OlLPlIxkIHlKHlND_[F^[F^vkId] WnAybLLDMcNxlOpkLhkKHlEHnNPldPlLpmvgiHxfc?mjghjhkKHqXHn`HlM@nMhlGxjBpk CPkQhlOHlKt^aNE]SFISFYSCYSEEsEUsBUsDASEQSCYSEECgF`kf_k>JF]CvlQt_>mRTau NHmcVLJ=TJisGisGCd>MkV`mnGisGmSH_MhsGOmTT`mnLQdYxMsFatSIwmYdawNISb=oHc d^MhCy^TPMw^ULHSaunNCb;oHqc^`NuVaufb?gbw^WhMu^hLHCc>NDTK?nZ@nYxMwfatCy ^[Hn[Xn]DcCOLsbAoJAtJ=TJ?N@tJOn\\Pn[Xn]DdGgbIWc?WcAGc?WbMWb=We=Wb[gbAg f?wbcoLudYHp[PnkXn^pmLPl>PitgjuWiShkGxmHHlQ`jAPkqOi@Xhp?lw?j]@lMHq^xn> yolXpZXnnHn^Xp\\hnnXn`xp`HotHof`qepoxhojPqfhol`n_poZHnkXn^hpa@oqxnbHqa @ow@odXq_xn@Apl@rbPodPlNxm>xj[`jA`pQPmCYpo@p_xNIOTCdKOLMTLeddPN?wcIwcI _[LN]dixOKGdKgcNOOGdKoNadblMccGOMCdKOOeTOitOWOdTO?OXDI_b@OO_elOcdNO`tN GO\\DK_aLOedd ]bTPmTxjAdYweYAn;QplPpsHqsxqyxQGeCVj=XjCHKIv[\\R>_Kj>c_fj?hj@pJI>FKdR? OKJF?mDL>]bF] V^]F_FlKHlJ@lJxkIt]>lJ@lKHL_f^_f^ev^aV`cf]_F]_v]Qfa_F^Iw]YnmR?QQvlqmdpBmqyds<]RMASDEuH=tUEtLUuJMtRmsIqTFiSXgon`n_xoY@nl @n[hpZPnnpmZ`pVpmq`mYpq[`no@mU@mw?jE@hmgnDhkyxnc`rfpot@ocxoa@od@obXpmX pb@obXn]XndPO?OKYda@ob@o\\LOCf[oNad_hNMWdTDKGdXDI?>O\\DMWddTOOO`tNGO\\ DK_clNcdNOjNSd:>O\\DM_cLOcdZcLNscZclNcdNO`DM_blyyiy=:::::::<:xI::::::: :::::::::wiytIyawyYtyyqyygYZGn[FNdJwc\\IvWAyqYxyyyuMyyixmyyI::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::tYyyYyoyxuyyqxSyVYdixIxqqxuyyyYYHytQyyyIqyxtYyYYswyuqYoYyWHj?XiYW fyywqyysYyUysKqxuyyyYyyIByyQwyYyayyuYwcxvYyyy::::::::::::::::::::::::n yxqyrAYuYyxiyIysUsEYsQwdXweYwyyyyQrJ @B?fZHr;Gj;tYyM;ER;yqyxqyrZFK\\R>]R:=R:YquXq>ab=Xj=`j>TUUUUGeSGEXkMCES CYr=Yr]bBL?c\\>MdBKF\\hBb\\vKYV ]UnAyrAqsHqSAubELAubB@KYv]Yf]WNB=SBqB=Wb=G]SF]]^GhKY^KHlKHL:@SEGLvlPPlRxlL@mQ@lPhlLLE]SFGMof`dCqV`mf`knEYcV`Mo^T LIucTd`qv``DmnF;JGecTlGCasnKSauNGSaunHCasnHqcZ@nZpmXtdVMq>w^SlIysImCyv axCOgdOwaynGisGETKWmUTb=OKEdZ@NyvatTQuDDDuVatSIoMxsI?nZDbFn[TbZZ@NAwbA wb:NNDD=_X@nZPmZxmVPn[pm^XnY`N?gbHtJATMATKYd]Db=WbqFaAWawfeKWdgWf_weUW eAWb=WakV`Eg`sfdsfaYga=wf?WciGcIwgCwccWbCGgyfbXc^xkLPm=xjJHiw_koGiGpht GltoiNxiOjNSd:NOdDIwcI?:O?FO\\DK_clNcdNO`DM_blyyiy=:::: :::<:xI::::::::::::::::waveyuqyqyy_YlYygpq@qwva:_yvayym`iV_qOh\\oyyywe yvQyyyYyyyyuIyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::bywyywayuyfYpIqyydImakMuiYwiyyyYu]iyyyyayvMyuY xaywyywuiuiy^qtRarqHqnIweyywqYmixyiyuaxyYwAoqxeIoQymYywYyyy::::::::::: ::::::::::::^YvyyyY`yvoyxuyvvayvyxqyH^[F^KHlKXyuYyAxJmUXmeYnj@pj;Hj;pi xsYyuY;FZDR;OJ@r:?j:tYymxpawkyxqy@r:AR<_j?hj?PjghoghAxjAxlQxlF`Kb\\drar>Yr=WkEt`hs>abB@K Iv[TR=UR=C\\KF\\Of\\hr?OmTt\\QN>]bEt\\Gf[G>U^DPK]^XpMWNAeR?_KjAyrAuBW^ GhK?gbtRAWlMd]Wf]U^HpK]V^@C::[F^DSB=sEQcMPlHHlO@lQXmP`lOXlKHkCpmCXkm`l QHpBxkwWdTwnhdeXMqV`>mR@mRlFacT\\V`mWhMEW cEgawNG;s^TPmTXmUTauVawfahDVmTPMw^R@mRxmYDaFn[Db;OOedYlGSbNMoFb;GbA_[H n[HNxtQytGicZ@nZ`mVTa^MjIyc[HN?gbDD=_[LKsbAOLMd^lKItKETKETJSayvaPTLmSL om^DbGWc?wcGWcIwcAwbAwb=OMMTLUTLgn]Xn\\@nZpMxsJAtLQtLatMYtOQTMyTLUtQET LmtIAtNisHeTHuSRETL]uMeTUITMKUATLUEcOKMuIEtT=tKIUJITLUsEmSB=sFiR@=TAyR NESDqTCQSRISEIE;xbGgkQgeywcMWg;gbLtL=ULMTQMTLAUPitP]tNeTN]D\\DA_clJAtJ UTMUtOitOQtLWoeDf>OLDGgcHtKItLQtLUTM?ob@odPOO_cHoc@obtcvN\\TNOOhtOitOc dOON?oalMCd>O`tNGOdTOeTOsdnn`pnaxNMWdM?rdNO`DK?FOdDO_clNCdvYxI:::::::B :yyyxI:::::::::::::::nyyyyxiYpIbxNyxexwyybu[YvyyyyyytYp]xmEmQhlYiUUyyy iIwyyjyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::wiywiyvyyuyfYpIqaumvyYyaIoyyhIqawmxcybywQvayxQYuWYgXl QHdsFa_ItKyyuYyqYxkiyyYyuQwyixIytAyvyywAyuIyxyywyYyyy::::::::::::::::: ::::::Znaxnyyyuy^YsAyaf`ofP]TN]PplPpyxqy>`j>`iv_i`jywIy]:l:Ar:E R;ER;CZ;nyxaxoixwIJyBZ;N;C[Vj=dZ?>j:Ab:@JR[>kB@kEXK?fZ?F\\KnabCl;Ib>`JIv[IFZ;FZAvZLSDWkEd\\ON>]b@pJG^AxJKF\\`Bgf_hr?WKp r@oj@nZ@n\\Pn\\HNFn[PmYpmT@nYobtcnNTDI_bLOsdQON]TN]D:::yay=:::::::<: xI:::::::::::::::nyyeyuqyqIoQyeixeydYy;nxrIyyiywIw`Qv[f]RNvXIuyqysyyya yyyyxyYyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::nyxuyyqy;uaw=hyyWYdIqau=jxsyyqwayv_ywyywqymYwfyYuyye Ywmv]YfswhqyiyuIyoYxyyywyxmIxyyywivayuyyywyu[iuyay=::::::::::::::::::: ::::PytQyyyyyyIoixxyyyyyseYweQsDQCasnyyyymR@gj?dZ?fZyuYyEYsEYweYweYviy CZFj;TZ:Z:@JweYw==VZHr;WJDR;_JCvZAvZ=VZ=FZNj<`j>\\kBD\\vjAxj@pj@pJJF]cF`kF `KZXXn\\pm]Pn[Xn]xm[HnYLJETKysJ_m[@nWPn[@n_`n\\xn`HocteYwe=wayVaqv`yUYu E]Of\\uv_knTUuNATKmTLoo]ddyVbUGb?wfGGdgWdQgfIWd_WeSwdIgcOgb?WeAOREtK=U JATTysJYUKMtSuSJyuQ=uQItKeTKETNUtLusJySKYtL=DHtKQtLQTSEUSeTOwnaTeUWeCG c\\TN]tMYd[lNaD>obTcEoNad`pNG?ZaxNI_b@OK?ZaLMccZaLNSdM?TD:I?ZdPOK?vnat cZb<`DKGdKGd:::vYyyy:::::::Z:Zy=:::::::::::::::wyytyxyyyqyyyYwyYyaixey yYxwQ^qVbyJCTkgqWiyyyyyaev]eN\\xMtuxnuYwQyyyYxwyxyYxI::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::RyfY::: jyyInYuqxmyuAYsyyxuyyqYyAynexyeywQuPuTgEWcyYyqiwqyyqywqytIywYytMyoYyyy xI:::::::::::::::::::::::wiy\\Yvwyy=oixxyyyyySeUWeqr@qr[InpqxpER;?j:DZ Njj:TZVJB[AvZAVZ=VZyuYyuY>JDB?>fJLRpj@pjB@KO^ AxJOf\\Hr;?j:@JVmUtZAvZI^B@kB`j>`j?hJMV\\LR?kBt[IN=Ub? hJof`of]WN@mbLPLUV]^KQv\\Hr;_kF`kUXMY^HpK]V^dR?okHD]vkIt\\QnIysIiBa^Ix kIxKJ?ebF`KS^HpKWF^[F^_f^]v^af^cF__F^[F_YF^ev][v_YF^gF^]f_]f^aF__f^ev^ SV^YF_ev^uf_af`QV\\]VXmE^]eVwfYyue[V^Iw]xphtOLUSFiQrDqKqSIqsGiS?mR@aSI qsAQSD=SEOlQd^i^N`lN`LinDasEYSEUcUXMiv_XCof`XsEwLLTLGmSHmS<::jLS`mNGec YxMqnGeSGesGicOhlOPmTdddtIycYLKEdUlIycVlHqcXpMM_YlGcb?OGeSGADyvayVauVa u^ghog`mVtbvmYTb=WbwfapsHgMtCyva_fvZB^[=FZ;>bu>aOysDyxkQwyqYyyysAyyyxI::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::uqyq I:::yyyjxoyyeyoIxkqwmYyaYwyywmyyqYyMYpqXV=uR]XjAXwEysyyxuywYYuoywixkAx yeyyqyyI:::::::::::::::::::::::rxqqyyyymYx_YxyyyI\\TNo^`n^SHMwiyXr=oix SYuM;Eb?L:?jywIweYtAyuY`j>Xj =TZ=VZ?^=l]b=XJSF]SV\\MN:=R:mR@gmWt[Iv[Gf[G>C^@l?ib@pJK^TPMGf[Gv [FkCd\\Of\\O^CHKIv[TBsFapr@qR?ebIxkIhkGhkHpKW^@pJSF]Sv`vkI<[F^[f\\ZFL@ C\\KNHmSHar>GlKD^[nAyrAAsBAS@mR@mBUV]pBxrAyB[f^LSDMsCAsBIsBAcHxkMpkIhl I@lO@lKhlMd^ev^eFamf\\]f]QV^SFamV_uF^Wv^quXgVVaeamuXUg]]vgyNi`N`\\tEes DeR@asCMSK=TJeR@mRAMsC=SEQsCs^k^OhlL`lND_XCeV_dCiv_XCq^R@MmV`XDiv_iF`k >dSGOM:JMUTMasF[YlGc`oNHmcQxlQHmSDdKOHmcVLJ=TJasFasHqsHisGisFasFeCG_Wh Mo^]XNu^\\PNw^V`mV`MldfPmTTbfmWtayva;GbxsIoMpsHomXtaynJAtJETKGNxsIoMtC ;GbDTK[[LLUtMItLUtJITL=TKwm[PnXPN=wbCwaAOH=tJuSKITJCbCWcw^WHn\\hM?ga>m Xhm[xn`HmXhmq`qu@mW`mRhmV`hpohpoHwfawfcGgcYWeUweUWeOGdKwcEOMMTLYtLQtQq TPatMwn]Pnb`n]@o^XnaPNIgb=GeCGcGWbPuR=UMMtKATLETKYtLiC:;WcCwbEWc]Wf]Gd KGdCGchtOONXtMYTN]d\\PNO_kHPOgdOwcIoOSdMoMYda<::jNadbLNscIOMcc:>ob@obl MScPTMUTMUDIwcXDK_cHOOgddDMWd\\DG__LN]TN]D:::yyyxI:::::::B:yyyyyy::::: ::::::::::tyyuyxuyyiyxsxyQw[xyxyyypiyyiwkqxhF`o^qwHrxayvyYqQyaiyyxyqys MxyixuyxI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::uqyqIyyypIiIyoixpYxuYyqxpqxyyyyyYyuYlyWjgyxyytY yi`n`HwVquqiwgyywayyqywyywqysaxoyyyyyy=:::::::::::::::::::::::tixyyyux qayyyyMuUYu]TN?]R>]PpLyyyyy[Iv[quXquYyuYAvZDR;Er;IB>j:@jywIweYtAyuYJ@r:Ar:cZfJLRwkIxkGhKYv]hBu^CHkC@lJT]Un@c]WnAyrAyBWf]tBr]>LDSBS^aV^[f^WF^ [V]Yv]tRBIsBAsEEcQXlLXlN@lOXmPxkSXlC@lExlNxkbd_Ko>MspxP@apppTAASmhnatt FGl=hjOPkFPnWxmIpKYF_@SEQsC]cM@MPC^lNT_eV_e^SHMiv_iv_^mV`mRLGecbDdkNFa sFOmTt`qNHmSHmC:M_UXMmV`lSH?MdCwfa`cSHM\\Ds^UXMwfa\\SF]sHqcUlGCavnaxnW hMo^[LGecYyrAyv`qv`^^ ^^MXlMXxmYxv_IoeXoeZ?fZC^ywiy_j>tZAvZ=VZ=FZ;nyxaxoIw=NjNjR[nJTBEV[EF[CF[Iv[JLMTE]SEq RBqcT`lN`lMhlO`lPplNplPPlNd^cF_eV_XsEYsDQcSHMgf_TCqv`XsEGmSdcNmTd`ZUXM q^V`MuVapCMoNeSGwlQd`nlPD`kF`w^UXMq^dddu^V`Myva`sFasIycWhMAwbAWb=WbQwd QGb;GbsFasFb;OF]SFmSH[YxMw^hpohXmUdb?OIsayNKEd[HN;GbJIucWhMu^XpM=WbxcY \\XLJ=d\\PNRbBb?Gc=wbGwbEwcCgcIGcGwb?Gc=GbxSI=tJCb?oJSbAGcyVb?Wb;gb?Wb AGcAWcsVb;GdUGeiFaHTNYtDesFIUVYuJUtLqsIysrHqshqxpQ@ubUhmWHn]XnX@nZxm[H n]Xn]xog`OvN?gcRcAgbAWb?wb=WcGGcyfb;we_Gf?_[`n\\`nada=OLYtLEd]xnaxnaLS EUSYDCGcdTOWN\\TNOodTcEWcQwdQWf>OPDIwcIWc>obDdvNIwc:nNTDIwcXDKGd\\Drcn NPtLwNXDG_`j:@jywItax;J:=bL;sYyM;ER;EBrZ^JPBG^AxJGf[Gf[>j:@jAxJI^@LlLPlPr

]R>Yr=wJ\\R>]R>;b\\ONF]cEXkEHkCHkEXKIv[`bC@LtRAuRACasNAokF`KY^ClA c]WnDQsDyrAyRJ=dCHKr\\Qn?C]Sn@c]WnAyrA=SB=sBAcJ@LYv]tBWF^]v]]f^YV^_f]_ v^Wv^_V]_V^Yf^[f^[nHeSEEs?]RDiR>ATFQSIesDESAaREEs@utJySQISEUtAqt\\tD]sNisI]sAESFISDUSDMsDc^FlMXlL`lNhlQxlQHMcF_\\SF]SEUcOhlOPmT nZDbZYxMw^X lJAdYtaba>nZ@n\\Pn\\Hn[TbNnYTcyFcEwaCGc;WcCGbE?bb=Wc=GbCWb;Gc?WbE?;waA wbyFcGgbIgcEwcwfalSIusNitOmSIuCtSJ=tNitOUTMomXTaunJAtJ]TNOoddcGOMsdMWd ]WeS_`@o]Pnb`NCoK=tKATJItJITMMtHETJqTS=UKWn]`n\\pnahnZHnY@o`dcCwbIwcTT MGpkHp`pn`Pn\\PncHO=oJYtM?obdbZixOKGdPDKGd:\\DrcvNTDG?fNTTMwnaDd>OjMcc nN:jMCdFocddOON]DvnatcZb<`DK_b<::jysy:::::::Z:Zy=::::::::::::::Zxeypix qywuyoatyyxeydYyGywqytqY]yvcyyyyyiyv_yyyyx]iuOyyyyxyyysAyqyyI::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::u yypIyyy:JyuixmxqyxuYyoywyyxiyyeytIRyxytqYxMyyqYtuYyUYyOyyEywqxuYxeYvYY uMYtyyxuyyaywyYu]ysQxmeTHuSlYwfyyxuiwMyuix=yyyxIRyfY:xyy;::::::::::::: ::::fywEysqyyyyyIBwLyyyiy=:pqxpq;IbxoixGisGi;Hj;pixsYyM:=boj@GmST[EV[UV]Uv\\Qn@qbIxKOf \\lR@mrAybKHlKPlLPlKHlK@lJt]YNA=sBqrBESBEsCAsCMsAMsCqRCAS@wkI@lH@lFXK_ v]wF^]F`QV\\gv]Sf`mV__f_]V\\KF[sF]UVbMv\\KW]]V[OeUWVWmebQv]MG_iFbYf^;g ^cVa_F_mv^cv__v^_f^_f^cF_cV`kv_sv`[V_cv_mV`cf_HsDQsFasFQsDolPtbAOHmcTP MkF`TSEoL\\SFOmTDasNKETKUSEUSESaunFacSlGicPplPxlQdcGOFs_iNGecN`Lqv``sF oLjKIdTPmT@M@tJOMnZ@n\\PN=Wb@DAGbDtIAtKqSK=tHysJqSKITJgn[pna`n`HnZ`n[db=GbCGb@ tJqSKkMITNetMaTN]d[LHccGgcwFb;w`uVasFaXtM?pjdcGgc;GbxC=oJuSIusKAtJ]TMU tRqTPmtMUTN;?GcCgaAgb;gb?WbGWcsfb=GeaGf?WcAWcAONUd\\@NQgc`TLITMMtKMTLM TQuTQItKITIuc_hn_pmXDcCGcEWcobDdZatcI_`pNG?^n^` NvnaDdFocTd>obtcfNPDbcGoMYdbLOeddLN]daLMccvN\\DO_eXOKGd\\D:::yay=::::: ::<:xI:::::::::::::::::uyypI::yyy:byyQy;ryyA:::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZwmvyYyYyyu yuwysqyyIyyuxyqwyiwmyyipaxnqypyxyiyxyyviyZitQAuAQsbytTyyviyyiyxYysIyxa yvuywaxixqyxtyYoyywAByVyxaywyyxyYtyyqyysy::::::fYpIyyy:::::::Jyoyyyyyy YdYwwyyyyyMw;uYypIyyytQyFkCD[CF[quXquYyM:sYyMytaxkyxqydZ?fZ?>Z;Hj;HJLRar>qr@qr=Ybv_ivWkET\\MN>]R>yqyvgiwgE XK]V^]V\\MV\\ymyDBKN>=R:=RwjAt[:NmTT]UN@mR@=CSF]Sv \\Qn@C\\KF\\Qv\\hsGisAC^[NCESCeR?oKhr?okH<;Gbpr@WlMXlIxKSF]lBQv\\hBW^I xK]^KWkED[CNB=SBQB;N:Ir;wg iwgggggWj=t[In=C\\KN?eb=XJKF\\lCIv[j>ar>eR?OK\\R>wjAd[Gf[of`pBQv\\xrAW kEt\\nKdR?_KBaY^GhK]V^]f\\ONAC]SNAuRAiBwfahr?WlMt]YNAubGhKQ^El@qbHpK[F ^[f^_nCIsCICUv]YV]Yv]:WF^[>cF_cV_eV_cf^_v^]V^eF^]f^[F^[F^JA=SBuBcv_eV^ Yf]eV]UFb@sJir@iSy@rG]Qx@TxarSidRHmQPhsohThn\\@lUHmO hmVXlS@mLxlP@mTPmNplP@mR@MGoGasFiCoF`dSF]SF]CvlQT_iNMs`qnGsayvamV`mF`k NHC`kNFscIoGc`of`uVaTSEOM\\Cqv`\\SFgOdSGOn\\TauNHmcUXMq^TLGsdQoFGm[HNu VaxsI_mVDa:fmW<=oJETKwmYdawNI;:RaunJAdYj:DZFkCd[Gf[;FZpqxpQytaywiyGj;dZ?fZEV[ TR=oJLRc`dSZKHL _f^HsColPD_cF_aV^]v]Yv]Y^J@LB^_f^eF`gf^aV^af]UnBAsNoldxkL@nyWjM@hnwlpO im`lQ`q[`nVPkEHmIxk\\pM[V^Yf]aV^aF`iv_sv`cV`PsEYSGmSHUSF]sEasFYDiv_i>k F`\\Co>knFacSLMc`onFacWhMiv_jHqcUXMmV`\\TNOmTD`knGicOhLqv``CsFa`sF_of` oSHM;GbpsHqSHmcWhMu^UlGsdhtF;s^XpmXpMpsHgM:tSIwmYDb;oIycYO\\DI_a ]b?hj?Hj;HjywI;FZ@r:OJJL=Ub?hj?Xj=XjwiyS_enDcUW eUMUTMuYyM>C[CNOkDd\\FkCD\\KNGc\\Of\\KF\\lR@?kB @kEXK]^Hl@qbUl@qbDl@qbB@KYv]pr@GLlR@omXD\\KF\\B]Sn@c]WNA;U^GLAC^[NCESC ECR^]V^]>::flOhlN`lNXlMXlKHlK@lJT^]nBkC?lMHlO@mPHmQplSHlKxmI@laPlNHoHH l_@kExklghw?f]_nFpkqPn^XmGT_UnGUsD=THisEQsDmSGeSG]SF?Mu^QHmS@mTTcIwccf _PsEYsDYcR@Mk^UHmSXmSHmVPmTpN`sF?mRd`of`g^RD`w^V`Mof`oFdKOGs_iNGecOhLq ^TPMu^SLPC`knIycV`MuVatSIuSIs`qNHmceXoeHMmvaynGiCvmYdanM::JJ=dZlI;w^Wh MsFalC;_ZlI;w>vmYDb;oJAd\\PNAwbLTL_NHD?_Z@nZxMJKEd]LLCcVNDD;_[lKId^LN] deXoehNlSH?ir?ix_iv_nYHnU@mTXolhpJxlTpnaXOCOAeSGmLUTm;@j:`KDHkPhLOl;Ib;HJ=^=L>L:nMtCy^Z@N=WbDTK?N:LTL_NHD=Wb@Dy>Nn\\ tbAOLMd^lKcbnmXtbAoLQd^lLQtL]TNOodtc^of`o@`j>XfX?fQ`lP@lQPm[@oexlR`mnh pspmdPogv`qF_av^pGhj?]Pf\\VV]QFfW?eJPk?Ho_`nfPoc`ofdcGoKId]LLCcnn`TcEo LSgPuKcdOgdGgc\\TNGocDdKONscvNTd`pNPtLwnaxnb@OM_cLNscfNEGcvnatc::::>O \\DRdFOJMscIoNaddPOQwdhDOgd`Dyay=:::::::j:DZFj;TZZ:@JyuYtQyfiwSYviy wi=l]bEL?ebaR=UBfJXBOf\\lbFHmSt\\ZF`KKN>KAs\\QN@mbTPMS^E XKWf]\\R>gkGhkF`kFHLpr@wmYT\\vkIxkDPKO>>L>lJT]YN?yr@qRC=sAESByRBuRFMsCySGaSJUSEas?GmCPkvOgeojnghAPh nojkWhdhlQpp[Xn_@lL`nK`laplSHnMhlXXlOpmRHmRplPhlQtaEGc_F`iF^kNDaSFMsF] SF?mT@MuF`mVaknMQtLasF?mRt`nlPplQxlQ`mVT`mnFCdNM\\SF]CPCq^SHmS`mV>nZTb=OK Ed\\lIycZ@NRbNNDD=Wb@D:?_]XNA?FNjJScEOMUd[Hn[xmYxm\\PNGgcTDkGhkW`iv_Sv [MVUOEUKv]_F]af_KWdUGauVbYV`mfQKET;s@sR`cv^?GbyfJ=TIIr:;rOadM?gaufcCoM Yd^lJAtJItKgn_Dcfn_obDdZaO\\DRdMoMYdaLNCdFO`DKGd\\Dyay=:::::::<:xI:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::saoixoQytQyyqyxyyxiyvAyqyywqYtynqwharwl qyAysyYwmyymiwiYwaxmyyYymIbxwqxyiwmyyI::::::::::::::>yrAyyyYqayyYuYyxy yWY\\TNO^\\N^TPMyyy`r>arwhqwlQxlQ;Eb?hJweYxqyxQ:=b;l:;;FZ;vYyuYuUYpAwe YwEZ;n;Ib>`JAvZDR;Gj;dZ?N;;B[>KKF]SN>;yuYXbAhj?aR>]b@pJyuYyUZvj Axj=Xj=XJ@sBOkDT\\M>J=s[IN>]bClEwlEXKOf\\pr@?kBd\\nK\\BR`mNAubF`KWf]\\ BU^DPKU>fmWd\\FLlBS^EXKQ^GhK[F^[f^_NCS^]nAybGhkGxkIxkJ@L]V^DSCOLjBC^[N AubIxKR^FLDC[f^_V^av^QF^hRBuBtrByRCQsCYSFUSFMsCQSByRPMTLIqmXp;IQttS@qR @UQvTRqrdhTFCb;oGisGqCRauNIucYxMy>nmXTauNHCafMtC w^YLJ=dZlH;w^Y< OG_`OjLccGOM;I_a<\\DK_alMccnN:XDyyyyyy:::::::Z:Zy=::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::JyuYyuIyyyyyy::^xnuyyqySyVyuYywyY uyydYxmYw]iyaYuIxmyyuiylYx]YoiXYqTRmXi=xymyx_yweywQywYwyy^yxqyvqYuyypI yyy::::::::::::::jxqiyyYkauYyuyxqyxyyYyacHOkB@kb_Tsywy?lyxa?hJCF[CvWiu WuUYueYwM:=b;L;EbD[KN>mB= VZ=V^@sxpqx@r:GkCd[GnBAsBar>WkEt[In;IbAl?cZVJTSEUs>abC]bAxJGf[TBk^E XKOf\\tBO^CHKQ^@pJM^ST`U^F`K[^B@KS^CHkC@kBd[GNH_mDPK_f^_v]Yv]UV]tRAwjA d\\ONAc^_f^cF_HsCGlKt]YNAubIxK]V^DSCWlMd^_f^]V^@CY^Hl@qr@qBW^KHL]v^av^ eV_Sv]UV]_^NPlHXlKHkH`kCpkFhLHTLETBur@msEUs;YquTQmXpMAsCus;UrvloiXT>ir QASE=uDaSMyrCATAAsHESDQcK`LcfaAoDiSGMSGasDesFUSF]cRT`iv_qf_\\SEUSMMTLK GecXpMkF`kf_gf_mV`LSDMsEYsEYdaXmUt`nmXd_gNGecQxLof`\\CU_UXMA_XpMqv`dCu ^UXMq^etdm^ZLHmcWhMBasFawfaxsIySJ=dYxMw^XlGs`^MpCyvaNXj=DphloJETJQTMEtHaSDAuP]tLATJitN] TPmTPatNgNLD?_ZLLMTLKKcf_gf;GbXD?gb?GcC?DDAwbHDCGcPDE?vnadcGoLScnn`pna xNMWddTOon`dcZalMCdKON;E__d[Gf[KF\\Xr=_j>`j=XJ?fZLRvMj:tYyM:=b;L;sZAN=S[EV[CF[H r;Ir;C[CN=s[nj@d\\OnvmY xmCLB=cJlAybF`KQ^F`KU^HpK[F^@sBOlLt^aNB=cJlAs]ZJ@L]^KLCS^]nBC^[NBS^FLD C_V_aV^cf^W^FXlJ`kM@lE`lKhkPXlJxlN@lOXlPxlOpn[@n>@irOoTXmYHK]g_mFbAf[w EREucMf]MG^cfaMF]Ig_mFciV`HSDiTMYtEYCFmSHmSLG]SF]SEUSFUSEasEYsLQdR@Mk^ V`MkF`\\Co^OhLgf_gfcTTGc`nMXsEWM\\Ck^SHMGgcpsHOmTPm[lFacXpMw^WLHmcfDeo >^mVdawfabawNHmcVtCw^YxmYxM@tJOn\\O`tN?OXDG__lyyyyyI:::::::B:ry::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::fyyUy:vyy=::uawmwyisyYkiyuYxyiysyyXYvUyyyYyyYyqYyaYwOYyYywyysiysyu]I jaWgOXiywysiyyYyHYqyxyqYyyYyuyyyiuiywQyvayuiywqYjy;::::::::::::wiywAkI xxyywYJysdQoN`lNTSEWyuYyGhKGf[GfYweYquXqUYuUYCF[CfZ?N:=bxoIuUYtQy>j:TZ NJDB=VZ@BBZ;N:;A^=LglOd[Gn=SYuMrabQxlQxJIf[nJXBMV\\dR?eB\\R>gmWt\\QN?ebIL@mbDPKb]Wn@qbUXMMV\\ `BUn@]R>]RAokEXKUV]lR@wMyF\\KnAyBnKlR@_KpBU^HpkHxkID^[nBAcKlAyB>L:jAyr AuB[F^@C]^LPL_^KHL:Yf^[NC=SAIsBKAQSCqREIsAUsCyBUf^[V^cf^wf`mf_Wf]MWauV `dRMAsCEUGmSNyrCuql`P>ypsxTGqsPesHUcD@nMhlixnbHmM`lVxlRxmTXmWXM`sFisFa sHicWhMEWc\\SFwlQxlTLEUcR@Ms^R@Mi^atcof``sF_mV`mPplP`mVT`mNGeSGisGon`p nV`Mk^YxMkF`psHwmYTa^mVDeSGem^ZlGCasnHqCsNIucXnZtayvay>n mXda:ZYxM=WbJnJTB:;FZ Pr]bBlDQcAt[Gf[pqxpQlLPltQytQ?C]SF]O^PpLOf\\Xr=YR]RAs\\vkId]WnIycAxJR]nkHD]S NAubH<:xrAyrA;W>>lJT^]nBs]YNAS]UnAybJob@oWhMmV`NMlSHgmW dawfayvaxCw^WlGisGeSGoMtC::::y^YnZDbvmYta>nZdb?oIyCFn[TbNn \\dbVn]tb>NobD d:vnatcnNTDyqyyI:::::::B:ry::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Zwy Yoyy`IoatmxyiwmyyI:fYhIqasyYmiyyyxyQxyyxiYymyvmYwmyxiYxqywyyyuIwQyqYx] IqIylyxIXfewRQtLyywWxeaWqiyyiywyyyyyyA:::::::::::::::fyw]yYyuuyyyyRoeX o_eV\\V^]vyyiwgiw?ir?iL:c[Gn;Ir;QBeV_eF\\KnL>]bEXKOf\\`r>OmTD\\KN=UbEXKI^@pJSF]lBUV]dSGOKOV\\fkGD \\\\RAs\\QnAc]WnIycAl@S]nkHD]SF]W^H<:jAyrAk@qB>lJD^ZIxKR]ZIlBAcLPLav^a v^FlKT^^kIhK[f]Yf^[>YF__F^ev^[V_DsDIsD]SEYSG]s@uR@EsCASHesFETIqSEASBis BAsJIsDATBISOYSGes<]R:iPqPSyDrTuSKEuGySNQSFQTDYSIAsC]s@yrEASCisFasFacL `lNhn_t_nlPT`nL\\SFgmWd`>MXDm^R@Mq^PLHs`>nZdatSNSa>MxCk^XpMw^V`mV`MhtO iTFsaNMlSHgmWdawnIycYLISaunGicTLIucX<::jHSanMxCy^XLISa^mVdawnIsanMtC=W bDTKwmYta>nZDbFn[Hn[LKsbAoJSbZ\\LKsbVN@tKITQIUSQSGaSGasGEuRIuobDdvNXDG_`ydawH@jL ]r>abB@KmV`\\R>gj?t\\QN=Ub?hj?HkC>lJT^]NCEcLfMjIycYLISavmYDb;OJ;yvaxCBbFn[TbNn\\ db::ZZPn\\PmYpmcpO?oKUsDYSKYTOYTOuT]xM\\hKD`K:@J;TJ>`jVHMTpSFYSDQsCETM At?ir`j=Xj=Pj< dZZ=L]b?T[I^?hj?PkDD`^kFD]FlKd\\ON@mbAxjAXkEt\\vmYT \\>LjCIcN`L[v^]>[V^YF^ ]v]R^_F^_v^]v^cf^QF]Ov^cf^af_cV\\WF][V_aF\\YF]Wv^]V`sf`;fYwE_Wv]Kw`uFc vmDhkbxkMPn>Hk?xffom=xjdHlOxoThm`hlRpmMhlWxlRPmRd^_nLQdP@M\\cVLF]cRLHe SGYsD?mPplbpn`PMdCwfaXCmv`qf_kF`\\cSHME_UXMkF`tcXhlOd`NM\\SF]sFcdOgdkF `tCs^V<::jHSa^mVt`^mVTanMLD_lBl?c\\ONcD OcXr=Wj=Xj>`j>xjAxjN`lNHKhr?gkGd\\nj@D[fJXBqv`hbEHkCt]YN?eb?hJGf[J>]bR @mR`Kpr@OlLPlDL@c[Gn>ar>aBy^DLCEcKHLYv]YF]Sn?c\\^KpBU^HpKYv]Yv]^KhBQ>n kHD^NlLPlN`lN@lJD^vkIt]>lJd^ZOhlOPlOXlJXL[V^YF^]NAyr@=cIXlK@lOXlLXmVPm KPlJpkKxkDhkEPmYhmC@lGhkL@lOPmRPf^_fIhkGxmSPmUpkJPnK`leplS`oMxlm@mVHj` OgRxi=`oMxlbXlPpmG@lW@lLhmL`lRPlMhn_t_mV``cWhmWPmR@mT@MuF`mNCIsFMsDetL UtHisGisGgmYxmOxlQDaqf^if_iV`mv`hSNsaynGSb=Wbiv_iv`qnFC`kF`k^cHoc@mRTb fMxsIysHCaVMhCsFapsHwMD[fJLRD[>KLR]R>Yr=Yr=S\\Mn@qbEXK Iv[HBkF`kf[Gn;S\\MnYv]tRAgkGtB[NBKDMCFlMXlIHlKxkKHL^i;@jH?dJOkB@k[`mWXnU `mOPkFpnO@mq`naPnExkDXhqWkj_h`XkIhpV@nePlPHnDpkYHlM`odHoOPmR@lMHlTXmSH nW`mUplOXmMXl^PmT@nR@mfxn`hmS@mSPmRPmYhmNXmRplY`mM`mS@mW`mOxlQ@obt^anG icSHmSPMlSHmSF]cRLGecdddgf_psHqsGcawnHqcWLHmcUXMq^WhMyvayFb;_[Db;oIycV D[Vj=tZZab@pJoNGabC\\EX KIv[XBSF]Sv\\QN?ebTd`Q^Al@qbEXK]V^]v\\Qv\\Q>vMXBS^FLAubHL@mbGhKWf]pr@q r?C]S>xrAyRAubHl@S]ZHlAybJ@LcF_cv^aNCEcJ@L[^KHL_f^_v^anC;:_^KHL[F^xrAw K@sBASDMcNlCS^]nBC^_f^YF_afZIf[[eV_u_iv_sf`oFagv_gv\\UFf?Gcuf\\UFcDSFy q;QRotPFeqypuIMtYATMms=erLqsHUsEQsDUSDisF]sHaSFiSEQsIYsEYtIysJacf@oa`m THmQPmRPmYT^ov_efaqf^qF`kVas>MWdLSDWmUt_iNGecUXmUPmTd`VMhdexlQdawnGWmX pMw>fmWDasNH;uVapCyvaxCw>^MlC::u^WLIuCfMJICb;oJAdZ@NwfawVaZZlHSanMxsI? nZ@nZ<:jJcb?oKId]nZtbAoMsbAOKEdixoi\\YxMCGcCWayvaC_\\Pn \\`n^L;ER;EBG^=X JOf\\OVZvlQd[Gf[;^Dd\\AvZHBYv]Yv\\QN=UbRL=Ub?hJE^@L?ebEXKOf\\dBoNGaBGf [lR@?kBd[Gn?ibEL?eBds?s[IN@c\\OnAybB@KU^HLJ=dB@kB`kFt\\Qn?;R]vkIt]fkGh kF`KS^GhK:W^IxKY>:FL@C:_f^DC:R^]>DC:]^KLAubJ@Lav^HC_f^DCav^^lNxkNt`;wa qTQudXoeX;gawvamf`eV]WfeAWcuv]]fcaf_AG][v^oeYQFVce_quYQW]_Vfofaxc\\PmT hmOhlRXlLPmO`lV@mQ@nVXmZpMqFaofcMwckv`mNGGmX`mM@MiFaoV_kf_qVauF`knNadN \\UD`kF`mV`mf`oNHmcVlHqcf`of@MtCqv`tCw>:fmWDasnGicTPMuVapCwfatCRa^MlCq ^V`Ms^WlHcanMxsIgMuvayva;Gbar>Ar:ARF]cAxjAhj?t[IN:=bwgIIn=QBG^OhlO xjAd[GN=;Iv[IF\\KF\\MV\\MV\\>mR`JKF\\Xr=OkDPkTd`Q^AL@S\\Mn@qb AxJU^GlJAdB@KU^EXKQ^DPKQv\\pr@?lJD^nkHpkF`kFXkED]S>JAubIxkI@lJD^vkId]W >xB]V^DSCOL:JB=cJlS H_MhCq^VlHcavmYDb;OHmcXpMra:>NDTKwmYdafMpCw^YlJAd\\PN=_[lIycYLJSb=OLCc NodPoZ@N:uVau>AoKID=_hpoh@nZDbfNEwayvaC_[LKScEWcYweY_Y`NdSGeSN?oblNade XOIwcIWeUoGiccHoc`n^`nXpmX`n^Dc:::FNJLMd_hn_pn`dcfNLDA_]Z=lc[Gn`JI^@pJM^TPMSF] \\R>_kFT\\fkGd[GN?eB>nZD\\nkHD]Sn@qbF`KOf\\tRAwkID^[NAubFL@;O^El@s]Y>< CYv]xBWf]xB]V^DSCGlK<_^KLB=SB=CY^ILBS^]nCIsCMSDMsAyrAkCIcLPLb^^lN`lNpl P`m[@nGPlK`lPplugiwolPt_eV_mF_eVcyFb;G`mVbgF`;g^cfa[nJ=sC=t@=SCqQ:Mrp@ QNqRBAeQHpSXmcxlRxmOhlV@mQ`lQhlE`lKPlWPmYPoaXlVHmNHmQ`mWXmTplOpmS@mU\\ TXmSxlQPodd_gNHmcSHMkF``sFOmTd`ZcHOeV_ev`vLdCkF`k>qv`psHqcXdafmWTaVMlS HgmWtaynIcaw>lSHWmUt`:^MpC;Gb@tJ_mVdawnIycYFocDbFNPtLomXTaVn]Tb>nZdeWge=WbjLCb;OLMdZ@nZxMT[EV[bZ^j>DZ;n=sYyuYiv_if[FkCt[Iv[KN>iBKF \\Kf[GN?s]>kBT\\Mn>C\\Kns[ZTPMS^CLAubDPKW^A xjAxJ`r>wmYT\\vkId]WNAS]UN?C]SNAubHl@S]ZIL?s\\Qn@qbHpKYv]`JGN=MR?kBt]NkDd[>kB@k@pJR`mn@qbDlAybGhK[F^`r>[CT\\y^ClAc]W NAC]NkDt\\QN@mbGhKSF]jAc^_n?ir?iBS>nkHD^[F^]V^DSCOL@sB?lJt]YnA;[F^@C_> vkIxkHLAC^ZMXL_f^HsCWL@sBASBC^VlMT_eV_]V^DC]^YxMQv\\XsEglMXl``n^HmNhlR XlNhmQ@m[XmVXmOplOhkHPmIHlc@mUXm:XjVhi:tZGFd[f^Kg_@tDQsHQsDysGeSEasEEt MQTEasEQSFUSHqsG]SEQcTHmUxlPHmQxlN`LMWdMg_nmXt`qnG;kF`\\CmV``dchLlSHGM lSH[SN: jIyCfMpCyvaxC:w>^MlCYweYWb=?PD?_]XNCGctSIwMtTQutJSbnN@hj?hKHLqv`qfeWgeCGcxsIysKId]VJLBEV[LR<[xoIE^ wgiwglOhl>`JAvZPrabDPkDpJPr]R>MR]R>eR??nZd\\Of\\Y^GhKSF]hr?GkCH kEXKQ^F`KR]FlKD_cN@mR@mBQ^Fl@qbIxK_f^HsC_lNd^_f^[F^tRAgkGd]WNB=cKHL_f^ @sBwkId]WNAs]Yv]]V^]f^VLHCR^FLNtSI[VLGeSGmSHomXtay nISb=oJCbZWlHsaynHcawnISbFNvM@D=_ZOKwcvMyvdQwdof`Hr;_gf_gPpLOgdOWdM oKId]c[GN@mbLOdLWkET`mn= Yr=MBAvZLRlJ@lHpkH`kFd`on?ibB@KSF]` r>GlKHlLPLYv]lR@omXD\\\\r@qr@ar>OkDT]Un?ibEjCC_cnDQcOLDMSDICev^aNEwl QT_eV_gf_`sFWmUd`sFalcAxjA@mRD^[>LTLwlOhlYHmT`oZ@nj@n[HOxrvhQ^mVDafmWTavmYDb;oIycYFn[Tb>NxCs>ZZlJSb>NxCW_Z@nZpMDTKwMD TKwNLDC_kHpklJuC=WblCC_]XNCGcjPqtPAtJgMLTLomXDcfn_Tb=WbG_`L?ebiwgi?j:d cGgcEWcTTM_n^tayNIuc\\LLMd^tZAn:Ab:@j:xiycYweYw=>j:DZFj;Hj;L;Eb;H J=>>JHBGf[TBAvZDR;[;L;c[Nj;iv_iF\\KF\\EV[\\BC^AxJUV]Uf\\On?ibSl?ibCHkChK\\R>wKxB[F^tRAG n[t\\QNB=SBmR@_KtRAur@qbGL@mbFlJ T^]NCEcLLB=cJlBs^FlKd^NLHsC_lND_ZMXLqV`mF__f^UV]Uf^_nGicTPMMv\\\\Qwdqw pqxXtNaTBESC_lNhnaxnMhLPcRplP`obxnPpkHxmMXlnPn]@nD`kBXghwlrWirxlUppQPm JiprPnGxkUxkIpn\\HnJhLgvasv]kV_oFasF`kNOedPLIs`qnGCasNHc`onFasFadchLlC iv_\\SFGmST`NmTd`VmUfoghoYxMw^\\lISbfn_TbVn]Df[Gf?gbJKEdWhMA_\\lK Id]lPcb?OIuSIQtLomXtbAOKCb^B@KYaCHkCPkDPkD[Vj=dZ?fZR[ EN;Eb?hJ;FZ;f[Gn:ABVLav[In>kB@kV_eVwjAd^_f^CF[Pr]bEL EUSE]R>oj@T\\fJTB:Of\\`sF_kFd\\On@s[Iv[U^DPKWf]lR@Gn[D]SnBAcF`KS^GhKWf ]xrAyr@qbF`KS^EXKMV\\dR?_kF`kGLAc]::ZGhKr]>lJD^ZKHL[F^vkIvlQLFGmST`NMdCq>^MjGic ULHCaZWhMu>^mVnZTb=OJ=dZlIcan MJJ;QwdxsIoMHDwfaxC?gbpC=Wb`J?fZ DBA^]b?hJG^`JMV\\\\SF]S@s\\QNAubAxjAPkDt[Iv[R\\ MNIucEXK[^DPkD\\DL@C]:::NKhBU^J@LYv]xBWf]pr@WkEt\\^KpBWf]fmWdawfauValSHWmUt `^MpCyva@tJATIcaZVlGs`fmWhmZ@N;_YxMw^WlHca>nZTb=OKEd[HNyvajHSanMxsIOod daw>HtKwmYda>NdSGeSIuchpohXn]db?oLQdXpMA_Z@N;_[lPCc>NTTMomXdb?OKSbNNLT LMTMUd`lLQd^LKEd[LJ=d\\PNAwbAwb:NNjKCcCoLSc:NNjKCc^NPDE_`pNCGcjLsbAoJc bfNXtMwNjLQd^lKsb:vYxI:::::::B:xiy=::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::JxyivUytixqYuyywYyvYxInyyqYcyu[yyuiyyanIweYnOxlqI weyyuqYqiyqiwkyyoIyyIywyyxyuQytyyywYjxwywmyyI:ZxqyxAyreywQysyvYjvaitQy tyyyuYxq@qr@ar>WiuWitOitgiwgiywI;FZDR;Gj;TZ>j:dYwmyrY:>J@BCF[CV[EN;Eb= tZA^;HJ;^`J;>VlMt[VkET\\Mn?ir?MR eBO^AL;ER;ABAvZAv[>mRt\\QN?ebHL>]bAxJMV\\Hr;IR>C`\\S@C]vkIxkAxJO^Al>C \\KNIuSIeR?wKdBMV\\dR?;VkEt\\fKpBWf]xrAyr>s\\:>lJd]Wf]UV]pBOf\\dBQ^FD] UV]lJ@lJt_iN>?kPLEC^_f^kvasf]ef^iF`gFbsv`sF^[v`Mf\\K G^_Vfcv_wvY?v[OUUKW][fdcf_sF]UFdwfauF`if`gV_iv_iV^nN@sBGmSHmPLEs_inEc_ gNECdK?xsIWMpsHGM`Cb`oNG;:mV``C:o^V`mVpmXtaynHSa^MlCRaunIycZ@NsFatCuVa lCof`hsG[Z@N=Wbj:TZFJJ:=b:l:;yuYxA;^;l;Ib>l;Ib=d[TR;Eb?L:=b@l:Ab= lCWlAL@mR@ar>_kFt[Inv`qvhpohpyxaTL>]bCHkCHK\\R>gj?tZAn;;k^EL?ebHpKMV\\ MN>OKLR]bRD`S^FlAC\\KN@S\\VkES\\NkDt\\fkGd]WnAybJ@ LQv\\QF]SN@S]Un@;B]SF]S>VkE:FmST`NMlCrayNISaunHCasNHSaunHcawNHcanMpCq^UlHqcYxM=WbvMxC; Gb`JAvZLR<_JHB?^;L;Eb>L;Eb@pJ?fZ\\R>_j>T[EnDQsDUBO^>l>abEl;IbxoixW j=t_iN=UR=UBB\\>KXr=gj?dZ?nEYcDPKKF\\hr?wjAt[FKPrZDPKO^EXKb]WNAs]Yv]UV]pBSF]lBQv\\hBSF]lB[NB=C UV]j@c]>lJd^_f^[F^@sBOlLd^>LlJt]Yn=c]WnGicXpMcf^hr >arEQsDqtOitHiceHoc@jxoiD@kBpn^`nL@lJHlIxkRplPhnbpnM@mPpKDsCASFasEItJ= tJmsGyr=YRIESC=UKItQqSIUrrHqVoeXoXpMu^ [lHqcYxMAwbxsIoMptPqtJSbfn_tayNLcb?oJSbVoeTb=OIuc`pN;_`pNw^[LN]TNuTQuT N?oYxMwfaLDOgd@tJGmSt^aN=cZ?>_fddTMUTMQtL_NHDA?^NPtLgNJLCc:VNHDRcEOMUd `LLMTLETKONHtKOn\\oj@T\\fJTBb[G>\\SF_Khr??LdR?eRB=cIxKO^AxJq^@pj@Xk ED\\Kn?c]Wn?s\\^kFD]:fKpBYv]Yf]WN@mbEXKOf\\dBQ^FLCEcKHlKpkHNlLt^aNCEcKlAc]^lNT]UV]U^N`lNhlMXlJpkHplN`l \\@nZpLhr>aRJuSIESB=SBuRAQtKITFISDMSCEsLUTLQSFUSDWLgF_WV^YV^cnFeSFasEU sGYSEuTMUtJ_laPlN`m>pj=pfcwmxOjyPn_T_xSCMSRQtMeSCIsIaSGmsE]sFYsEasFaSG mSH]SGesEaccTde^VlFYsEeSF]sFacS<::JIucVlGicTPMo^V`mVpmXtayvawfajI;w^U< \\SF]sGisGuC;Gb@tJwmYTaunHcawnICb;?@D;GbxsIgMlSH_MpsHwmYDbFN@D;_YLPmdX pMs^Z@NuVa `J;FZ@r:Ojd\\vjAt[fkGhkHpKW^V`MUV]dR?gkGt[INAubGhKav^pr@gMuf[ZB@KK^CLAubHl@qbF `KS^EXKOf\\dBU^HlAybILAubGl?iBVKlR@wKlJT^^lNT_enDC_cF_av^av^fKDSCGlKT^nlPplMXLcF_cVb=oDQsDESCWlMd_gf_eV_Ps CIsH]sFYTKITLETKYC_F^[F_cnDYsEMSEUSDUcR@Mmv_htL_nQHma`lQpoOTbAF\\oFXwE ]OEVAXcOgbMV]mgbGGdeF`Ag_kFacV_oF``sGisDesFYSHisFusH]TOetCIsCiCuF`DTJ= THs`qnGc`oNGS`mnFC`kNEUcSHMqv`qVauNIucWhMsFaJGs`qnHcavmYtanmXTafMlCs^U LHmcWhMyvaar>Ar:ArEc[vjAtZAN;Eb@pJ G^?hJU^QxLE^?Ll]BfkGT]^kFD]:VkEt\\^kFT]UV]Wf]tBR]UN@s\\VK lBWf]xrA?lJt]YnAc]WNA;Y^JlBAcK<:JCEcMXLa>:vkID_cF_]V^jDQsDESC_lN`l[HNq v`HsCISB=cN`Liv_HsCIsCurAISAyrIqcRplPXlKHlLLCs]]V^[f^PsEYSEUSE]tKMTJQs EQTDYsQ]SH]t@ASN]r@=TyLR@ePrDVKYtZITNmsj:D[CN>?kAxJweYwuYyM:SZ=n:;;FZc[GN nJXBiv_if[GnL>C\\ZR@mR`kFd\\On@s[VK`r>aRAc\\Of\\s^B@kBpkHfkGT]^kFD]ZHLAs]YNB=cKHLWf]tBr]:VlMD_cF_]^ KLB=cJlBS^NlLt^aNCEcOhL]V^xrA_lNvMxCQ_YlIsbfmWdaNn\\tanMlTPmTIucWLKEdWlJca>nZTb=oPqd^`N;OJUTMgM DD=Wb@D?gb@uRgMHD;GbHtKOmTtbAoMYd\\PN=WbTTMwnatbAo=YbIxKuVa\\D`d^lJAd \\lKId[HNCGcHDBbfn_DeSGeC_^<:::jLQd_Lyta`J?>>KdR??kBD\\vjAd[Gf[EV[PBCF[ LBof``r>?KtBMn>Yr=GkC^lNd_gf_]V^xrAOlLD^[F^av^@tJOlLt_inGicSHmSXLnZ@nb@obhNDDCGc \\TN?O@tJGNTTM_n^Dc:::fn_Tc:^n^`n]LKEd\\lyyiy=:::::::<:xI::::::::::::: ::wax=JxoYjymIkixqYxqIyyQyyYyqivYyyuxyQxyyyiIwyytIyyy::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::Zxayvey^ywmyoxsyyoAybPmXXvRQuyiyxyywqyrYxnyyxqY yQxmiyyyyyYhywkymIsay=\\TN]TOdLOXsE?yrAy=XJweYxA;^xoIrY:>J@r:GJDR;OJtYyM=CZ>M\\bB\\Fl? ib@d[E^AxJ=VZ\\Qv>kB@KvjAt[nj@pJ^j>D[NM`BK^Hl>s[In>;K^RD`UV]pB]^DPKWf] Wv\\VK\\Bs^BLB=cF`Kr\\FkCd\\dR?s\\Qn@qbHpKW^IxKSF]lB:UV]JAs]vKpr@WkEt \\^KjBs^av^a^LPL[F^xrAokHd]vKkB@kBl>abST`U^F`K[^DPkDxKpr@[CHKsFa\\BY^EXKQ> FKdR?WKhBUV]pBWf]xrAgKpBSF]lBR]UNA;[F^tRAgkGt\\QN@;[F^@sB_lNt^anBAcJ@L Yv]tRAoKJB=CvK:@sBGLDSCOLPsDGlKt]YNDS^]nCIc[lEs^anBS_enGicN`lNhiuWi^Of \\OlGpkQPlMxLXsDQSEUcOxlQxkNXlDxKWv^_vaCwbkF`hsFOnWpmXplQ`nSXm]plRPnNp Laf_kWcKwdkv`MFWiEZMUUega?WbSv]QG`qVciV`=g_lTMYTFUSEmSIuSFqSHiSJysEasF YsFaSGecTj:TZ=n;IB?>D[:Vj=<=^;L ;;EV[DR;GkCTZ@rEc[^j>D\\fj?TZ=n>ar>yrAyrxpaZ?fZOj]bBl=YbAlE wl@l;Ir;aBE^?L?ebDl>abR@MQ^DPKW^ClAc]Wf]W^DLHs[Xr@c\\On?C]SN?eBVKlBS^G hKU^HpKYv]j@qbF`KS>fKtB[F^xrA[GL@;W^IlBAcK:vkID_F lKd^_nDQcLPLcF_@D_f^@sB?LLSDgmWhmYxMaV^]VZyuYsuWkE]IF\\gF_\\SEUsCIcL`L ]f_ef^if_WF_aFbEoDesFGMsFa`SDQSJesG=TFasGISDEtEaSXytRySAASQiSIMsx@Rw@p lNobAwywiyyyyxQw_aven^]^tGQsyqyuyYxixmyyI:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::RyyQvqYxiyygY`IwiyhyvwyuaYoyxpiyAysatFis=Yqvpwb Iwyayw]uIYxinaxnyyyuy^yvuyyyyyaf`OoJGDcLxxquYymxpa?hj?hJhqwVJAfZ?fZ;FZ xqynixoixkyvi:@J=VZ@B;FZ;of`\\R>_j>d\\Of\\Iv[XBU ^HpKM^QxlQHkCD\\KN@_kALAubHLB=cEXKsFaTR=UR@mbB@KO^Gl?ibElJt_iNF]cKHLWf]PsDQSJuSIMTKEtp\\pnTr:ErFYcUHmSPlLD ^_f^ef`dsIuSC]sEAtMUTDacKplOhlQxlJlFYcRpLKwblSEYSOEtKySCMSPaSH=uFqSMqr BmqkXpLmRBYTCQsMasGIuOmTFUSEUsGesCaSFYsHmSFqcSXmUXmUd`NM`CqV`dSF]SHmcV LGc`:VmUvmY TcEoKId]<@uR[\\<@tJGN]rMYtMUTM?nZdcGON]d[HNCGcLD:::?OKEDAwbLTLgn_Tc:vYxI:::::::B:yay=::::: :::::::::Jyuix=JxoYjymYy_kHxk[Wf^gxmYXyiyCyqwXG[SEkXnWxywyxuyxoyyyiyya ymtQIvwiywyuyuYyuixoayyyyMx;yawmywyymt;yavmxqyxky;:::::::sAnyxqyymy;:: wiypyx[ymy;wax=::jysy^YhIwiywAwavayvqyxqyymy;:waw=xIyA^YdYweYxmYxyIuaw =::Jxmix=xI:wavMxcyvYxI:wavmysy:::^YpIwaymy;:sax=tIqyxlYxmyysynYtI:::y ay=tIuAvYxI::byfYlIry:::FYxyywY:lIuaymxSy:::^YtIyay=JyCyvYxI:::uaxmysy :ZxMxcyvYjw;yayMycy::ZvqyxYyyuxyAlYyuYyuIvyxqyyiyxiyrIYsyyyywkYp]hnAxh sWwYIvyQyyYykxuIxirAyryyyuyVweYwxpqx@_b>_IxKkIxlR@?irCV[EVav^tQytQxlaw gIyuYyUZ=n:sYyuYuUYlQxlQytax;::J;Eb<<::J:=b:l:AbTZ=VZCF[XBGf[LB[F^ [f]WN?C\\Kn=YbAL>]R>]BI^LLEUSEqr@_j>`j>l>abCGK dBS^Gl@C]NK`BQv\\`BM^GhK_f^HsCwkId\\Of\\W^H<::j?ibF`KU^HpK[F^DSC[MLCEc L<>nZ@n^`NGgcGgcFNI\\A::::m]sKwdQwhqwhQxlQxlAkEXXAgeWgUWeUWgeWgEXkEXoeXoEYs EYgeWTA:::=VZlQxlquXAFisGiuWiuOitOiqwHoeXoeXFisGirkurWnHPAkEXkEYsmyxqy lQxvHLQtLquXqudARX:::>I`Aq]vKxrXquXmUXmEX>I`Aq]vKxrXFI\\Ak=::::FI`Ao]u KxBYvH:`AgeWguWiuWsEYsUZ=NuTap;::JxlQxdQwdQv\\QvTQuTqtbWvH\\As]vkwbXFI `Ak]r;:::jtPatOitoixcYNitOiq;hqwhqvRX:::>I`Aq]vKxrXFI\\Ak=:::fHTArW>I` Ao]tkubXVIJu:s]; ir:iQxPqqdWgeW_eV_eXomtPqtPplPp[Hn[DPkDp >abOgdOGb;GbsFasFATMUTmXpmXb:?b :;:::jYxmYMTLMDLKDLGdKGDKCDKAtJAtJZ>; h>WdMWtKItK=TJ=dJ?dJETKETK^b>_b?gb?wbAwbB?cB?CXn]Xn\\Pn\\L>C\\?k]BL>CX >E\\>K]:C\\AwbA_cF_cGgcG?cB?c?gb?wBX>:::E\\@ob@oB\\N^\\n^`n^dN_dn_hn_h >CDKCTKfBHn[VBPn\\Pn^`n^tNatN]TN]T>::KdKGDLfeUWel>`jvb?gbMWdMk[Xn^`N] \\N^S\\FKcDOc:j_rL^Cl>OdLOdMWdM[DNxnaxn_hn_dN_dn`pn`S\\HkaRN]TN]\\IwcIocHocFK`:U\\G;t>W:::UTMUTM:::vYxI:::::::B:ry:::::::::::::::nyxey:^Yw AvyvqYev_kVHqCIutPqDYlYxmyysynYwqy OytQyyyYmYxJusyyyysawMy;yaxmxCylIpI:::qau=::JxmYxiIqavMx;uaxMxmivmwsxZ w=tIyA>yruwiqvayvmYxmyxSyvyyyyvmvayvaIqyxqAqasIysyYtIsauMweYwiywiYxSyZ yuyyawiywQYlIByVYlIuaxMySy^YhIqavmxcynYpIsau=ditQYpyxqYysyvYhyw[v=::jw ;sAfYpIwaxMwCy^YJwsxfYxIsav=::jysxvYxIkawyyyiyyewkIxuavqyxYywiysYyuIyu YYNyvayuiywyxvYyuyYxIwaqQyyAyuyysmvmIVYlyygYxyyiyxqI\\yxmyviywKyoyymyc y^YhywiYxcxFysIywiYryZuYYoixdIqavMxSynYtI::::uavmwcxZvMysyZu=::jysynYp IRynYxIyAnYpIsAnYxImavmywywiYoasmwWyxqYkav=tIqyxqixVYpiwaYpIyay=jxCyVY hIRyZvmxSyvYxI:eaxqyxyYhIqA:::fYpI:::yay=:::::::<:xI:::::::::::::::wax =JxuYymIvyvqyyytSQuh>`kVysIyyqywuywiwayuyqwyiuMyyyyuYyoytMuUYukycyvYyy y::FY`IyyyxI]Yv]iwgiwbx>yrAyyMy;yA:Zx=xI:::sIylIuawMycyvYxIbynY:xIyA:: :::^YhI:uavmysy:ZxMy;uayuyyA^YhIbynY:::Jxmiw=xIyaxmxCy^YpIwaymycyfYlIy ay=::JxCyfYpIwaxmysy^Y:pIqavmysy:::::ZxMy;ryvYJycyfYpIsavmysy:::VyuUyv YxIuavMxcytYy;:::sawMysyvYtIuavmysy:::nYtIryvY:JxSynYpIqavmysy::::fyyu xyqwcxyQw;siywyyRy`yvaYv]YvuYywyxuyuYy\\yvuYxgYhIqayuyyAfYlIsaymsIysyy yOxlQxyyYmav=xI:::::::::::::::::::Ry:vY:::::tixuyxI::byZy=:::::::::::t I:waymy;::::::::::::::::maymysxqQy;yawyYyyypIxiw]yUyfYJwcxfYxyySyfYJyS yyiyuqymYygyyuyyYyfY:jyyI::::ZxmysynY:jy;::::::bynYjysy::::::::::::::: Zx=xI:::::::::::uYyxIyaumwsyvY:::::::::::::::::<:xI:::::::::::::::wax= Jxuiv=yIyVuWquT_eWWsCIsyqYMWcCgNUdLwYxgyxkIwkiweYwgIwyyyxIkatQytyYhIiy wdYweYx_ywmvcv_ivsIysyyNytuy^yv=yANYjxSynYxIyaxmxqiwMycy:::vYxI:::wawm x;byvYxI::::::::bynY:pIuAvYxI:byvYxI::::::::::::wax=::jysy:::::::::::: nYpIuAZxmysy:::nYtI:::ryvY::::::::::Jycy:::fYpIbynY:xIyA:::::::fyyUy^y xqyuiYmiyuYxwqwqyxmYygyyUyvyyexZy]xuyvYlYxOYjy;sANYxIyANYjysxfYuAwA:Zw iYby:::vYxI:::::wQyky;waxmysyZxMxmYxmIuAvYxIRyuQwSxfYpIwAvY:tIwA:vYxI: ByVYjysyZxMy;ryvYtIwA:ZymycynY:pIuaxmysyZwmysy:::::nYtI:sawmxcynYpIqat MxCy:::FYxIuatmycx>YxIwIysQxoyvaYsQySynYlYyuYvqirqywQyyqyvqYwiypIyYyui yyYywiyjxiixayvyYxI:::::bywQysyvY:JycyZymy;::::::bynY:::::jxSy:nyxeyvY xIuqxqI:waxmysyfY:xIwaw=xIyAZxMxmYxqIyyyxIsavmxSy:^yvEyvYtIwA::NYxIqAZ yMw;yA:::nYtI:::yay=:::::::<:xI:::::::::::::::wax=Jxuiv=yIyxIdxniXXlMX pj?djOXLMduUIuyuywiYvMyyoyyeyvylmysynxpUyuYyyay=::tXqKyyxqyHyASIuSav]y mxgywMycy:fYpI::::wax=::jxSy:Zx=pIwax=::::::pIyaxqyx]w=:Jycy:::vYxI::: wax=::::::jxSy:::nYtI:::uAnY::::::jxSynYjx;:wax=::jxSy:::nYtI:::yay=:: Jycy:::fYpI::::::waymx;wA::::::VYhyx_yxuxNyvaysqyw]uMwSyyyyxis=xIbxZy= JycxoqxsyvYpIiywiyyvYpIyay=:::::::::::::tIwAvYxIwaxmysyZxmxSyvYxIbyVyu YyumysynYtIyay=JycyvYxI::bynYlYx_YxIyAnYxIyAnYtI:yay=Jycy:vYxI:::::::: ::::wawMxmiwiYyAZuYyuiYuYy:::^YxImAvY\\IpiymyqIuAnyxyyx=wiylYyuiyExwYy mAyyyuyylYyuyxuYxyyyuyyoYjyyI:RyuYyyyyyyy:::nyxuyyqy;::::::wax=::jysy: ::nYtI:::yay=::::::JySyuQycy::vYxIwawmxcyvYxIwax=::jysyZxMySy:vYpImYxj y;kIxvYyA:::fYpI:::yay=:::::::<:xI:::::::::::::::wax=JxCyjymiyixnqxqqx rqyxeuYYuyyywyyhYuIyqsxsixmYxoIx_f^pqxpqywykMIucAxJmYxxxqyXxmIxyycywqR AuRUMUTmsIyswY_aJ@LER;ER;^]V^]GisGyYxI:::::::wax=::jyyyyyI:::::::sIytY yuIxIwawMy;yyyxI:::wAfYJycyZw=tI::::RyfYtIwA:ZwmxcynYjysy:::nYtI:::uA: vYxIwavaYyay=::::::Jycy:::vYxI:::byfYpIwaxmx;wA::::ZwmxcynYjysy::::::: fYjx;yayMyCyZy=::::pyygYjwyiumumywYYxiyxyyyYyvgyyyyrQytyY`Iyay=JxcxvYx I:aAvYLYtWyuExkIxry:::::::::::nYxIyaxMysyvYJycyZymy;:wax=pIuayuyyA::nY tIryvY:::tIRyuQy;yavMxsyvY:tIwaymy;:bynYxIyA:::::::::::nYjy;wAvYtIry:: ::>YpIiywjykxwyxuywQyuIfY:jysyyayexyQw;qyyhIuaw=jyyI::::::::nYwiyvYxI: :::wax=:::jysy:::::::::::::::::::::nYjy;:::::::byfYxyywYJysyZx=::jysy: ::::::::::Z:Zy=::::::::::::::::jxSy:nywyyqqxriywiYxyyoyxyyuqxnYxyqywuy wYyFxmYxJOdLGr=YrViuUib@_b_qv^yyyEyuyxyqyoyxyxqyxxqYoGhkgAsBAQYtKyyhym UXk]LMPL`ev^mYwey?ar>MPkDpyuYyyYyqIyyytyysynyxqyxmyyiy=:pIxiw]y=:::wAv Y:JySyZy=::JysyvY::JycyZymy;::::::::::::::::::::::::::::byZy=::jxqyxuI ry:Zx=xI::::::::waxmysy:::::::::::::::ZxMy;yay=::tIry:::fY:xIuyypIbxfY ^yvuy\\IrveYwxiy=jxqiyuyyAZtQYeYweyyvXxIsAvYtI:wawmxcynYxIby:::vYxI::: :waxmysy::nYtIyay=JycyvYxI:qyx:yyyZxmxCyfywuyvY::Jy;RynYxIyatmwWyymy;w ax=jysy:nYtI:yay=:::::::JySyZxmysy:ZwmwCyvYxI:wax=::jwqIyqwcywQxeiwYyu yYPysIYyuYyuI:RynyvaYsaxUxwYyoIyhyygyyUxwYymyxpyyWywEywqysynYxIyA::::: :nYtIyay=::JycyvYxI:::uYynYwaymy;:::::waxmysynYpIsIylIwQyuIyayMySyZvmy ;RynYxIRyvYyYyfYtIyayMyCysQycyfYpI:waxmysyZxmx;ByvYjysyNYxIyA::::::::: :::::::Z:Zy=:::::::::::::::::::::byyaymwyitqyyAxnixIYtNAdOWdgGhfwq:aqw qytyyyIyqqxlyywayyyYyiywyyuyymGggWDACC@ynyyyIyYquS=o`PnyuYwoYQXkGxDEcC Cyqwxyyyxtyxyiu=wYyvyvYywayv=qYyuYyyqxyyyeIkixqixwyysiylIuawmyyYyyyyui x]wuyxI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::Ryry::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::pIxiw]yUyry:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::<:xI::::::: :::::::::::::::RyvwmaxsqyyYxrYyqaywye^OfFOdGOxpAxwyysyyyyxyyysyYqyyyyy EirCyysiyEhiAXAkr?iYwQYywiuqV`WfKCdHiIw[yyuQv=yruMUXMiaWfyYyiYvQyryXqe XyqyvQysEYvYyvuyxyiyyYZwmviYxUYwiYviyyKwuitqYuawmyyYywYVYwAxI::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::B:ry::::::::::::::::::::::ZwmvqYy=xl]X wuyyQWgaWVpmW`nadnaIw`iYyQyveyrmXpUXyyywayuYivqyyIyxapgPpiN_cfweIvyyyj AlHXj?PjtwyuQyyIyvyuPIu_f]X^tIqSwAqqymyyuyxcymIxoiyuyyxYyyYyKywyxQytQy vqyxayxuxuiyoiyyixfYpIyyywyybynYxiy=:::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::<:xI::::::::::::::::::::::RyNywyysqyyIYtpsDqIMTKuTTyTh_dVypyxyayuyyxeyyYygIxkAqYyLYtMyyuYwyyx ]yw]ysyYyayyyywyyyiixUypyyuyyyYyoyxqYyyyyAJyuyyqyywyy=:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::<:xI::::::::::::::::::::::Ry^YdyxkyaIoyy`Yxgy xQywyYyqvmYxUyyyywqyv]YvIYt]wcQwKpKK`LLPLiuWfqyyYyeYwbMN[@nRuTRIyqMYQ] tPqmUlmyeYxuXpmxxuYxwywqxuQysiyxIywqyxeyaIvQyyLyyyYuyiqyYiatayuyxyeyyy yxYyky;:waxmyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:ry:::::::::: ::::::::::::Zw=hxq=yvmyyeXqyXvqivyYmixgyyyixUyuMIxLyyyywyyyuixqyxYywyv ]Awkn_jvvbavtqyriwhaw^`Nib?ob:;b;USDYspiHqyYyyqyyyviywIYvYYwuyyeiyauZi vjywkIju_hbQvQyyyaxyUwyyyiasyywAyyyyvyyyuy::JywyxmyyI::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::B:ry:::::::::::::::::::::::^yxexwqwyiuMwyYy]yw oyyiyxiysqyyQYdyv]Ytiyv]YyuixMvQyuuYyWyxuviivsiywywmixoYxuYysyyhivAywY xlaxwyyyQYLyv]YuiYxISFis;TJAHJ:lTMasy_xsyxfyyyiXnyuqxxiysyyyyyy= ::waxmyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::B:ry:::::::::::::::: :::::::fYpI:qqymixyyyuytIwavyYwaxqYwYynyxuyxYxgyvyywYyyuxuiyuIywYysqyi IsiyyyypIWyv]ixyiyaIxmYnkxp=F[SVqwhrQET;uysiuuixSyyyIwfwuIykyywAYwixxy yyYyyIyA:nYtIyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::Z:Zy=:::::::: ::::::::::::::jxSy:ZveyryJycy:vYwiywA^yyiyvqyxiyy]yiyvUyvyv=cIwHMULMDX kD " 0 "" {MPLTEXT 1 0 918 "pts := [[1/720,.2067659927e -3],[0.14e-2,.2091511543e-3],[0.15e-2,.2313318473e-3],\n [0.16e-2,. 2544768368e-3],[0.17e-2,.2782784343e-3],[0.18e-2,.3025504122e-3],\n \+ [0.19e-2,.3271757008e-3],[0.20e-2,.3520777628e-3],[0.21e-2,.377204671 1e-3],\n [0.22e-2,.4025199830e-3],[0.23e-2,.4279973289e-3],[0.24e-2 ,.4536170963e-3],\n [0.25e-2,.4793643319e-3],[0.26e-2,.5052273770e- 3],[0.27e-2,.5311969557e-3],\n [0.28e-2,.5572655505e-3],[0.29e-2,.5 834269667e-3],[0.30e-2,.6096760213e-3]]:\np1 := plot(pts,labels=[`t[6] `,`t[7]`],color=COLOR(RGB,.5,0,1)):\np2 := plot([[[1/720,.2067659927e- 3]]$4],style=point,symbol=[diamond,cross,circle$2],symbolsize=[10$3,15 ],\n color=[coral$3,black]):\np3 := plot([[[17291/12418560,269 /1379840]]$4],style=point,symbol=[diamond,cross,circle$2],symbolsize=[ 10$3,15],\n color=[green$3,black]):\nplots[display]([p1,p2,p3] ,axes=boxed,view=[0..3e-3,-2e-4..6.1e-4],font=[HELVETICA,9]);\n" }} {PARA 13 "" 1 "" {GLPLOT2D 595 482 482 {PLOTDATA 2 "6/-%'CURVESG6$747$ $\"3'*)))))))))))))Q\"!#?$\"31+++F*fw1#!#@7$$\"3++++++++9F*$\"3-+++V:^ \"4#F-7$$\"3/+++++++:F*$\"3/+++t%=LJ#F-7$$\"33+++++++;F*$\"3:+++o$oZa# F-7$$\"3#**************p\"F*$\"37+++VVy#y#F-7$$\"3'**************z\"F* $\"3*******>7/b-$F-7$$\"3++++++++>F*$\"3!)*****z+dDSF-7$$\"3'**************H#F*$\"37+++*Gt*zUF-7$ $\"3!)*************R#F*$\"3-+++j4Lk$z%F -7$$\"3))*************f#F*$\"3K+++qPF_]F-7$$\"39+++++++FF*$\"3))*****p bp>J&F-7$$\"3)**************z#F*$\"33+++0blsbF-7$$\"3!)**************G F*$\"3^*****pmpU$eF-7$$\"31+++++++IF*$\"3O+++8-w'4'F--%&COLORG6&%$RGBG $\"\"&!\"\"$\"\"!Feq$\"\"\"Feq-F$6&7#F'-%'COLOURG6&F`q$\"*++++\"!\")$ \")AR!)\\F`rFdq-%'SYMBOLG6$%(DIAMONDG\"#5-%&STYLEG6#%&POINTG-F$6&FjqF[ r-Fdr6$%&CROSSGFgrFhr-F$6&FjqF[r-Fdr6$%'CIRCLEGFgrFhr-F$6&Fjq-F\\r6&F` qFeqFeqFeq-Fdr6$Fes\"#:Fhr-F$6&7#7$$\"3i>]:[9N#R\"F*$\"3=xcY\"R,&\\>F- -F\\r6&F`qFdqF^rFdqFcrFhr-F$6&F_tFetF^sFhr-F$6&F_tFetFcsFhr-F$6&F_tFhs FjsFhr-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6% %%t[6]G%%t[7]G-Fbu6#%(DEFAULTG-%%VIEWG6$;Fdq$\"\"$!\"$;$!\"#!\"%$\"#h! \"&" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The curve has i ts left end point where " }{XPPEDIT 18 0 "t[6]=1/720" "6#/&%\"tG6#\" \"'*&\"\"\"F)\"$?(!\"\"" }{TEXT -1 2 " " }{TEXT 272 1 "~" }{TEXT -1 23 " 0.001388888889 and " }{XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" } {TEXT -1 1 " " }{TEXT 266 1 "~" }{TEXT -1 29 " 0.000206765992657234467 67. " }{XPPEDIT 18 0 "rho[2];" "6#&%$rhoG6#\"\"#" }{TEXT -1 36 " is \+ zero to the left of this point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "At the left end point we have " } {XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 2 " " } {TEXT 267 1 "~" }{TEXT -1 49 " 1.9472383679836230476 and the common v alue of " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 40 " increases on ly slowly along the curve." }}{PARA 0 "" 0 "" {TEXT -1 42 "For example , on reaching the point where " }{XPPEDIT 18 0 "t[6] = 0" "6#/&%\"tG6 #\"\"'\"\"!" }{TEXT -1 10 ".003 and " }{XPPEDIT 18 0 "t[7]" "6#&%\"tG 6#\"\"(" }{TEXT -1 2 " " }{TEXT 269 1 "~" }{TEXT -1 42 " 0.0060967602 13436046224146525 we have " }{XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG& F$6#\"\"#" }{TEXT -1 2 " " }{TEXT 268 1 "~" }{TEXT -1 29 " 1.9690451 83368835194402570." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "Bogacki-Shampine scheme" }{TEXT -1 6 " has " }{XPPEDIT 18 0 "t[6]=17291/12418560" "6#/&%\"tG6#\"\"'*&\"&\" H<\"\"\"\")g&=C\"!\"\"" }{TEXT -1 2 " " }{TEXT 270 1 "~" }{TEXT -1 23 " 0.001392351448 and " }{XPPEDIT 18 0 "t[7]=269/1379840" "6#/&% \"tG6#\"\"(*&\"$p#\"\"\"\"(S)z8!\"\"" }{TEXT -1 1 " " }{TEXT 271 1 "~ " }{TEXT -1 58 " 0.0001949501391, which corresponds to a point (shown in " }{TEXT 257 5 "green" }{TEXT -1 30 ") close to the left end point ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The \+ Bogacki-Shampine scheme has " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 " " }{TEXT 273 1 "~" }{TEXT -1 29 " 1.9939635993630663812 and \+ " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 2 " " }{TEXT 274 1 "~" }{TEXT -1 25 " 1.6643168882207173871. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "determination of values for " } {XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 47 " where stabilit y radius = imag axis inclusion" }}{PARA 0 "" 0 "" {TEXT -1 53 "We can \+ use the following procedure which calculates " }{XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 25 " for a given value for " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 374 "rad_inc := proc(t_7)\n local R,t6,lft,rgt,real_ part,u0;\n t6 := t_6: lft := ll_; rgt := rr_;\n R := z -> 1+z+1/2* z^2+1/6*z^3+1/24*z^4+1/120*z^5+t6*z^6+t_7*z^7:\n real_part := proc(u )\n Re(newton(R(z)=exp(u*Pi*I),z=2*I))\n end proc:\n u0 := bi sect('real_part'(u),u=lft..rgt);\n Im(evalf[Digits+4](newton(R(z)=ex p(u0*Pi*I),z=2*I)))+newton(R(z)=-1,z=-4)/2;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "t_6 := \+ 1/720: ll_ := .56: rr_ := .66:\nbisect('rad_inc'(t_),t_=0.0002..0.0002 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+F*fw1#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "t_6 := 0. 14e-2: ll_ := .61: rr_ := .65:\nbisect('rad_inc'(t_),t_=0.000208..0.00 021);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V:^\"4#!#8" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "t_6 : = 0.15e-2: ll_ := .62: rr_ := .65:\nbisect('rad_inc'(t_),t_=0.00023..0 .000232);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+t%=LJ#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "t_ 6 := 0.16e-2: ll_ := .62: rr_ := .65:\nbisect('rad_inc'(t_),t_=0.00025 4..0.000255);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+o$oZa#!#8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "t_6 := 0.17e-2: ll_ := .62: rr_ := .65:\nbisect('rad_inc'(t_),t_=0 .000278..0.000279);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VVy#y#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "t_6 := 0.18e-2: ll_ := .62: rr_ := .65:\nbisect('rad_inc'(t_), t_=0.000302..0.000303);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+AT]DI!# 8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.19e-2: ll_ := .63: rr_ := .66:\nbisect('rad_ inc'(t_),t_=0.00032..0.00034);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ 3qvrK!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "t_6 := 0.20e-2: ll_ := .63: rr_ := .66:\nbisect('rad_ inc'(t_),t_=0.000352..0.0003522);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+Gwx?N!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "t_6 := 0.21e-2: ll_ := .63: rr_ := .67:\nbisect('r ad_inc'(t_),t_=0.0003771..0.0003773);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6n/sP!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "t_6 := 0.12e-2: ll_ := .63: rr_ := .67:\nbisec t('rad_inc'(t_),t_=0.0004025..0.0004026);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I)*>DS!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "t_6 := 0.23e-2: ll_ := .63: \+ rr_ := .67:\nbisect('rad_inc'(t_),t_=0.000427..0.000428);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*Gt*zU!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.24e-2: ll_ := . 63: rr_ := .67:\nbisect('rad_inc'(t_),t_=0.00045..0.00046);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+j4 " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.25e-2: ll_ := \+ .63: rr_ := .67:\nbisect('rad_inc'(t_),t_=0.00047..0.00048);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>Lk$z%!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.26e-2: ll_ : = .63: rr_ := .67:\nbisect('rad_inc'(t_),t_=0.00049..0.00051);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qPF_]!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.27e -2: ll_ := .63: rr_ := .67:\nbisect('rad_inc'(t_),t_=0.00053..0.00054) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+d&p>J&!#8" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.2 8e-2: ll_ := .63: rr_ := .67:\nbisect('rad_inc'(t_),t_=0.00055..0.0005 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0blsb!#8" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "t_6 := 0.2 9e-2: ll_ := .65: rr_ := .68:\nbisect('rad_inc'(t_),t_=0.00058..0.0005 9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+n'pU$e!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "t_6 := 0. 30e-2: ll_ := .65: rr_ := .68:\nbisect('rad_inc'(t_),t_=0.0006..0.0006 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8-w'4'!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#============================== ===================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "A scheme w ith stability radius = imaginary axis inclusion" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------- ------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1885 "ee := \{c[ 2]=11/63,\nc[3]=4/17,\nc[4]=174/349,\nc[5]=36/49,\nc[6]=97/129,\nc[7]= 1,\nc[8]=1,\n\na[2,1]=11/63,\na[3,1]=244/3179,\na[3,2]=504/3179,\na[4, 1]=184816101/935188078,\na[4,2]=-411995808/467594039,\na[4,3]=10049361 3/85017098,\na[5,1]=2424510108/16025323237,\na[5,2]=12241368/78942479, \na[5,3]=-4360301712/39234412063,\na[5,4]=613806882624/1137797949827, \na[6,1]=166741171875227/3452798773601100,\na[6,2]=1006084/2623731,\na [6,3]=-188382845675879/1408900648998150,\na[6,4]=89559834743712383/245 148712925678100,\na[6,5]=31/350,\na[7,1]=60220124468460112045376943394 19654322731/30099864007383356992178712977537163264000,\na[7,2]=-6561/8 296,\na[7,3]=341104813140161009375674013573715202837301/31331940973160 3971335732895410416320512000,\na[7,4]=95341943376640595142991543684079 745176400281/1318406651709399034894169155355186749556736000,\na[7,5]=- 1154594845209196567962511465161858938950699/87744676451677464497983027 0188996722688000,\na[7,6]=1085508425299375/621834120640512,\na[8,1]=48 27113239/65475838080,\na[8,2]=0,\na[8,3]=797791005101/2404074449920,\n a[8,4]=5703219796216031/30660055749940320,\na[8,5]=-33901013357491/177 576803988480,\na[8,6]=29204563953141/55776754401170,\na[8,7]=34/449,\n \nb[1]=4827113239/65475838080,\nb[2]=0,\nb[3]=797791005101/24040744499 20,\nb[4]=5703219796216031/30660055749940320,\nb[5]=-33901013357491/17 7576803988480,\nb[6]=29204563953141/55776754401170,\nb[7]=34/449,\n\n` b^`[1]=7345858582709/98987482023552,\n`b^`[2]=0,\n`b^`[3]=304858187261 52979/92415623446907904,\n`b^`[4]=2471755716006328519/1300727684721130 9344,\n`b^`[5]=-5627835948231581/26545332727174144,\n`b^`[6]=45/83,\n` b^`[7]=34/449,\n\n`b*`[1]=66873629126427899/902163023828283336,\n`b*`[ 2]=0,\n`b*`[3]=121/366,\n`b*`[4]=3468482659797555651329/18566189507645 514457800,\n`b*`[5]=-65/379,\n`b*`[6]=952822237714091128053/1892592848 332731588736,\n`b*`[7]=129752050156606597/1891022063549226300,\n`b*`[8 ]=56866223980208550227/7201012017995453750400\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 577 "s ubs(ee,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n \+ [c[4],a[4,1],a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[` `$3,a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]], \n [c[7],a[7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n \+ [c[8],a[8,1],a[8,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n [`b`,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],[``$3,b[7]],\n [`b ^`,`b^`[1],`b^`[2],`b^`[3]],[``,`b^`[4],`b^`[5],`b^`[6]],[``$3,`b^`[7] ],\n [`b*`,`b*`[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]], [``$2,`b*`[7],`b*`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#777&#\"#6\"#jF(%!GF+7&#\"\"%\"#<#\"$W#\"%zJ#\"$/&F2F+7&#\"$u\"\"$ \\$#\"*,h\"[=\"*y!)=N*#!*3e*>T\"*RSfn%#\"*8O\\+\"\"))4<])7&#\"#O\"#\\# \"+3,^CC\",PKKDg\"#\")o8C7\")zC%*y#!+7%RVpPX?6g%oW7?-'\"J+SEjr`xHry@*pN$Q2S')*4I#!%hl\"%'H)#\" K,t$G?:Pd8Snv$45;SJ\"[5T$\"K+?^?jT5a*GtN8(RgJ(4%>LJ7&F+#\"M\"G+kM&*\"O+gtc&\\n=b`:pT*[.*R44_%[fa 6\"K+!)oAn**)=q-$)z\\kun^knWx)#\"1v$*HD%3b3\"\"070k?T$=i7&Fbo#\"+RK6F[ \",!3QeZl\"\"!#\"-,^+\"z(z\".?*\\W2/C7&F+#\"1Jg@'z>Kq&\"2?.%*\\d0g1$#! /\"\\dL,,R$\"0![))R!odx\"#\"/TJ&Rc/#H\"/q6Sanxb7&F+F+F+#\"#M\"$\\%7&% \"bGFgpFjpF[qF^qFhq7&%#b^G#\".4FeeeM(\"/_N-#[()*)*Fjp#\"2zH:E(=e[I\"2/ z!pWBcT#*7&F+#\"4>&Gj+;dvrC\"5W$48@ZoF2I\"#!1\"eJ#[f$yi&\"2WT " 0 "" {MPLTEXT 1 0 388 "RK5_7eqs := [ op(RowSumConditions(7,'expanded')),op(OrderConditions(5,7,'expanded')) ]:\nexpand(subs(ee,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`R K4_7eqs^` := subs(b=`b^`,OrderConditions(4,7,'expanded')):\nsubs(ee,`R K4_7eqs^`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\n`RK4_8eqs*` := subs(b =`b*`,OrderConditions(4,8,'expanded')):\nexpand(subs(ee,`RK4_8eqs*`)): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that is, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_7 := PrincipalErrorTerms(5,7,'expande d'):\nsqrt(add(subs(ee,errterms5_7[i])^2,i=1.. nops(errterms5_7))):\ne valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*)z%o(>!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also cal culate the 2 norm of the principal error of the order 4 embedded schem es." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7^` := su bs(b=`b^`,PrincipalErrorTerms(4,7,'expanded')):\nsqrt(add(subs(ee,`err terms4_7^`[i])^2,i=1.. nops(`errterms4_7^`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+w.:R%*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_8*` := subs( b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nsqrt(add(subs(ee,`errter ms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+du[l%*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := StageOrd erConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following r espective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]) ;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" \"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 " c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 7 " ) are " }{TEXT 260 13 " not satisfied" }{TEXT -1 15 ", except when " }{XPPEDIT 18 0 "j=2" "6# /%\"jG\"\"#" }{TEXT -1 33 ", where the condition reduces to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\" \"(\"\"!" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "[Sum(b[i]*a[i,1],i=1+1..7)=b[1],se q(Sum(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6)];\neval(subs(Sum=add ,%)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),expand(subs(ee,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F -F-/F,;\"\"#\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-, &F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F -&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF -/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$ *&F)F-&F/6$F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sum(b[i]*a[i,2],i= 3..7);\neval(subs(Sum=add,%));\nexpand(subs(ee,%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&% \"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F )*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i -1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6 $F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Su m(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=2..7);\neval(subs(Sum=add,%));\ne xpand(subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"b G6#%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+! \"\"F+/F*;F5\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"% \"\"\"&%\"aG6$F(\"\"$F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F. F)F)*&&F+6$F4F(F)&F+6$F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)* &&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 72 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nexpand(subs(ee,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"# F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$ \"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)* (&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(& F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "O ne of the principal error conditions is satisfied by the order 5 schem e. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK5_7err_eqs := Prin cipalErrorConditions(5,7,'expanded'):\nsubs(ee,RK5_7err_eqs):\nmap(u-> `if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"! \"\"\"F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "PrincipalErrorConditions(5,7)[1];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jG F,-F%6$*&&F16$F3%\"kGF,-F%6$*&&F16$F9%\"lGF,-F%6$*&&F16$F?%\"mGF,&%\"c G6#FEF,/FE;F,,&F?F,F,!\"\"F,/F?;F,,&F9F,F,FLF,/F9;F,,&F3F,F,FLF,/F3;F, ,&F+F,F,FLF,/F+;F,\"\"(#F,\"$?(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "The left hand side of this equation coinc ides with the coefficient of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 30 " in the stability polynomial." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "coeff(StabilityFunction(5,7),z^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\" jGF+-F$6$*&&F06$F2%\"kGF+-F$6$*&&F06$F8%\"lGF+-F$6$*&&F06$F>%\"mGF+&% \"cG6#FDF+/FD;F+,&F>F+F+!\"\"F+/F>;F+,&F8F+F+FKF+/F8;F+,&F2F+F+FKF+/F2 ;F+,&F*F+F+FKF+/F*;F+\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "None of the 9 principal error conditions are sa tisfied by the 8 stage embedded order 4 embedded." }}{PARA 0 "" 0 "" {TEXT -1 95 "The largest magnitude of an error term is about 44 times \+ the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4, 8,'expanded')):\nsubs(ee,`errterms4_8*`):\nmap(abs,evalf[8](%));\neval f[4](max(op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\") zd&R)!#7$\")1%3+$F&$\")3Je9F&$\")q!H1#F&$\")KXWd!#8$\")yZ<5F&$\")1*)[6 F&$\")-G+5F&$\")x\"[\">F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%%Q%! \"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 88 "simul taneous construction of the 7 stage, order 5 scheme and the 8 stage, o rder 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 290 "From a theoretical point of view, it is interesting to o bserve that it is possible to construct a 7 stage, order 5 Runge-Kutta scheme which has the stability function that leads to the scheme havi ng maximal stability radius (along with a compatible embedded 8 stage, order 4 [FSAL] scheme.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 123 "The following \"simple\" order conditions (given \+ in abreviated form) are relevant for the construction of the order 5 s cheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "SO5 := SimpleOrderConditions(5):\n[seq([i,SO5[i]],i= [1,2,4,5,8,12,13,15,16])]:\nlinalg[augment](linalg[delcols](%,2..2),ma trix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%%~~~~G/*&%\"bGF (%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7 %\"\"&F)/*&F,F(-%!G6#*&%\"aGF(-F@6#*&FCF(F2F(F(F(#F(\"#C7%\"\")F)/*&F, F()F2F:F(#F(F57%\"#7F)/*&F,F(-F@6#*&FCF(-F@6#*&F8F(FCF(F(F(#F(\"#g7%\" #8F)/*(F,F(F2F(FVF(#F(\"#:7%FjnF)/*&F,F(-F@6#*&FMF(FCF(F(#F(\"#?7%\"#; F)/*&F,F()F2F5F(#F(F " 0 "" {MPLTEXT 1 0 189 "`SO4*` := subs(b=`b*`,SimpleOrderConditions(4)):\n[s eq([i,`SO4*`[i]],i=[1,2,4,7,8])]:\nlinalg[augment](linalg[delcols](%,2 ..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"\"\"%%~~~~G/*&%# b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\" \"$7%\"\"(F)/*&F,F(-%!G6#*&F8F(%\"aGF(F(#F(\"#77%\"\")F)/*&F,F()F2F:F( #F(F5Q)pprint406\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We incorp orate the simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"b G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6 $*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"! " }{TEXT -1 9 " and " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$ *&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"(\" \"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "for the order 5 scheme and use the simple order cond itions in the first table above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 135 "We also need to specify that the linking coefficients for the 8th (FSAL) stage are given by the weights of the order 5 scheme, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[i] = a[8,i];" "6#/&%\"bG6#%\"iG&%\"aG6$\"\")F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG 6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "for the order 4 embedded schem e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "SO5_7 := SimpleOrderConditions(5,7,'expanded'):\n`SO 4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expanded')): \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[1,2,4,7,8 ])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(StageOrderCon ditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3..7)=0,ad d(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n add(b[i]*c[i]* a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [seq(b[i]= a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op(simp_eqns),op(SO_e qs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 11/63;" "6#/&%\"cG6#\"\"#*&\"#6\"\"\"\"#j!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 174/349;" "6#/&%\"cG6#\"\"%*&\"$u\" \"\"\"\"$\\$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 36/49;" "6 #/&%\"cG6#\"\"&*&\"#O\"\"\"\"#\\!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 97/129;" "6#/&%\"cG6#\"\"'*&\"#(*\"\"\"\"$H\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 1;" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "and the weights " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3];" "6#&%\"c G6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\" \"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 27 " are initi ally parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 192 "e1 := \{c[2]=11/63,c[4]=174/349,c[5]=36/49,c[ 6]=97/129,c[7]=1,c[8]=1,b[2]=0,`b*`[2]=0\}:\neqns := subs(e1,cdns):\nn ops(eqns);\nindets(eqns) minus \{c[3],a[6,5],a[7,5],b[7],`b*`[3],`b*`[ 5]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#&F,6$F;F'&F,6$F;FE&F,6$F.F' &F,6$F1F*&F,6$F.FE&F,6$F.F>&F,6$F1F>&F,6$F1F'&F,6$F1FE&F,6$F*F>&F,6$F* F'&F,6$F'F>&F,6$F*FE&F,6$F>FE&F,6$F'FE&F%6#FE&%#b*GF)&FgoF3&Fgo6#F;&Fg o6#F6&FgoFeo" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel [solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "e2 := sol ve(\{op(eqns)\},indets(eqns) minus \{c[3],a[6,5],a[7,5],b[7],`b*`[3],` b*`[5]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6901 "e3 := \{b[5] = 5764801/950706720*(336000*b[7]*c[3]- 55675*c[3]-336000*b[7]+33327)/(49*c[3]-36), `b*`[2] = 0, c[4] = 174/34 9, c[2] = 11/63, `b*`[7] = -1/988251600*(82945940350140*`b*`[3]*c[3]^5 -203323028186376*`b*`[3]*c[3]^4+177763353282228*`b*`[3]*c[3]^3-6499854 7281864*`b*`[3]*c[3]^2-248134453920*c[3]^2*`b*`[5]+9979576425*c[3]^2+2 97951486048*c[3]*`b*`[5]+8292198936672*`b*`[3]*c[3]+332257835105*c[3]- 111975965220-66169187712*`b*`[5])*b[7]/(-1740*c[3]^2+1741*c[3]-348+278 4*b[7]+10440*b[7]*c[3]^2-12536*b[7]*c[3]), c[8] = 1, c[7] = 1, b[2] = \+ 0, a[7,4] = -121801/3379557575962494012240*(-1096171153410248596642500 *c[3]^3*b[7]+181663025831942184302850*c[3]^3+347333040645699883320000* c[3]^3*a[6,5]*b[7]-57596215201578145486800*a[6,5]*c[3]^3+6264135452741 82573600*c[3]^3*b[7]*a[7,5]-12898106137937051177328000*a[6,5]*c[3]^2*b [7]+2117206834564111533258480*a[6,5]*c[3]^2-23106285837928574360640*b[ 7]*a[7,5]*c[3]^2-319190820038463397711995*c[3]^2+235889234253244194130 2000*b[7]*c[3]^2+188422683621447282668507*c[3]+33312178817754519941280 *b[7]*a[7,5]*c[3]-1592422280225430907060500*b[7]*c[3]+2158426752583992 1320600000*a[6,5]*c[3]*b[7]-2798287029539129469243744*a[6,5]*c[3]+9359 09730465449650667136*a[6,5]-38442984537782467558146-122504727010678013 54880*b[7]*a[7,5]+329701091103237562401000*b[7]-9033494428548570026592 000*b[7]*a[6,5])/(45021*c[3]^2-56299*c[3]+16878)/b[7]/(5*c[3]-177), a[ 8,4] = -14835483601/480881705040*(-11635*c[3]+24960*b[7]*c[3]-24960*b[ 7]+4737)/(349*c[3]-174), a[3,1] = -63/22*c[3]^2+c[3], a[2,1] = 11/63, \+ `b*`[6] = 749194461/35407328*`b*`[3]*c[3]^3+186761943/17703664*`b*`[3] *c[3]-507097315647/520704591484*`b*`[5]-1122718347/35407328*`b*`[3]*c[ 3]^2+715563/141629312, a[8,2] = 0, a[6,1] = 1/19730278706292*(49368279 707076435*c[3]^3+46722299680980*a[6,5]*c[3]^3-52541603790919439*c[3]^2 -1688295996227412*a[6,5]*c[3]^2+1215161198233488*a[6,5]*c[3]+157846223 43426356*c[3]-1474629377571504)/(5*c[3]-177)/c[3]^2, c[5] = 36/49, a[8 ,7] = b[7], a[7,3] = -1/212838001769085*(179234670870367977058425*c[3] ^5-1075408025222207862350550*c[3]^5*b[7]-7199526900197268185850*a[6,5] *c[3]^4+3712319484409708736491620*b[7]*c[3]^4+43416630080712485415000* a[6,5]*c[3]^4*b[7]-582923746227231157926870*c[3]^4-1612263267242131397 166000*c[3]^3*a[6,5]*b[7]+742482817030973809401765*c[3]^3-184891080625 19364600*c[3]^3*b[7]*a[7,5]-4992822842471257424898960*c[3]^3*b[7]+2646 50854320513941657310*a[6,5]*c[3]^3+668417088064847354640*b[7]*a[7,5]*c [3]^2+2698033440729990165075000*a[6,5]*c[3]^2*b[7]-3497858786923911836 55468*a[6,5]*c[3]^2-457673775844854604521101*c[3]^2+325495104083989081 8735540*b[7]*c[3]^2+135025465013646540859218*c[3]+11698871630818120633 3392*a[6,5]*c[3]-1020300654277625730939930*b[7]*c[3]-11291868035685712 53324000*a[6,5]*c[3]*b[7]-492154257868829412120*b[7]*a[7,5]*c[3]+12126 0996721491462962280*b[7]-15157624590186432870285)/(45021*c[3]^2-56299* c[3]+16878)/b[7]/(129*c[3]-97)/(5*c[3]-177)/c[3]^2, a[4,2] = 2861082/4 67594039*(349*c[3]-116)/c[3], a[4,1] = -348/467594039*(2199398*c[3]^2- 1287687*c[3]+166518)/c[3]^2, b[3] = -1/20*(1456000*b[7]+123759)/c[3]/( 2853786*c[3]-4379407*c[3]^2+2206029*c[3]^3-607608), a[3,2] = 63/22*c[3 ]^2, a[7,6] = 715563/314001039145940535620*(-44648228838432670920*a[6, 5]*c[3]^2-6649293908722560480*b[7]*a[7,5]*c[3]^2+9152034076070655*c[3] ^2+269250419105193708000*a[6,5]*c[3]^2*b[7]-467067053549825820000*a[6, 5]*c[3]*b[7]+60698307308717196144*a[6,5]*c[3]-330863768350101602*c[3]+ 9885052184822955360*b[7]*a[7,5]*c[3]+243614376824894691-20494672852131 775296*a[6,5]+197816634444632112000*b[7]*a[6,5]-3673364076769368960*b[ 7]*a[7,5])/(16641*c[3]^2-25026*c[3]+9409)/b[7], a[7,2] = -63/88*(-1740 *c[3]^2+1741*c[3]-348+2784*b[7]+10440*b[7]*c[3]^2-12536*b[7]*c[3])/c[3 ]/b[7]/(5*c[3]-177), `b*`[8] = 1/7906012800*(2280747631715*c[3]+275129 2454400*b[7]-455887522020+8292198936672*`b*`[3]*c[3]-66169187712*`b*`[ 5]-69144646750200*`b*`[3]*c[3]^2+201958513907556*`b*`[3]*c[3]^3-227943 7610100*c[3]^2+110594587133520*`b*`[3]*c[3]^5-248957266785948*`b*`[3]* c[3]^4-13764368284800*b[7]*c[3]+13756462272000*b[7]*c[3]^2+33103607990 4*c[3]*`b*`[5]-330845938560*c[3]^2*`b*`[5])/(-1740*c[3]^2+1741*c[3]-34 8+2784*b[7]+10440*b[7]*c[3]^2-12536*b[7]*c[3]), `b*`[1] = 15007/5626*` b*`[3]*c[3]^3-50660/8439*`b*`[3]*c[3]^2+73177/16878*`b*`[3]*c[3]+95364 1/330946637*`b*`[5]+33691/202536-`b*`[3], b[4] = -14835483601/48088170 5040*(-11635*c[3]+24960*b[7]*c[3]-24960*b[7]+4737)/(349*c[3]-174), a[5 ,4] = 1475497314/3411821*(245*c[3]^2-278*c[3]+72)/(349*c[3]-174)/(5*c[ 3]-177), b[6] = -276922881/2412124220*(45500*b[7]*c[3]-7545*c[3]-45500 *b[7]+4714)/(129*c[3]-97), a[5,2] = -2916/184877*(163415*c[3]^2-134336 *c[3]+22968)/c[3]/(5*c[3]-177), a[5,3] = 216/117649*(57031835*c[3]^3-8 9536829*c[3]^2+43157178*c[3]-5994648)/(349*c[3]-174)/(5*c[3]-177)/c[3] ^2, c[6] = 97/129, a[5,1] = 18/37530031*(4748731365*c[3]^3-4911797666* c[3]^2+1440261108*c[3]-131882256)/(5*c[3]-177)/c[3]^2, a[8,1] = 1/3645 6480*(5340605*c[3]+4368000*b[7]*c[3]-371277-4368000*b[7])/c[3], b[1] = 1/36456480*(5340605*c[3]+4368000*b[7]*c[3]-371277-4368000*b[7])/c[3], `b*`[4] = -1827867607/115781050*`b*`[3]*c[3]^3+4803466037/173671575*` b*`[3]*c[3]^2-4123329253/347343150*`b*`[3]*c[3]+552611137/833623560-36 1407683598/6810762375725*`b*`[5], a[7,1] = 1/28738238350868933040*(-40 78927194244299325054350*c[3]^4-57596215201578145486800*a[6,5]*c[3]^4+2 4473580737371222005983700*b[7]*c[3]^4+347333040645699883320000*a[6,5]* c[3]^4*b[7]+201325843347433081200*b[7]*a[7,5]*c[3]^4-52622657807629436 401229880*c[3]^3*b[7]+7953870760484319095915385*c[3]^3+211720683456411 1533258480*a[6,5]*c[3]^3-7426232267872937222880*c[3]^3*b[7]*a[7,5]-128 98106137937051177328000*c[3]^3*a[6,5]*b[7]+10706349734646032435760*b[7 ]*a[7,5]*c[3]^2+37956265737715870306243860*b[7]*c[3]^2+215842675258399 21320600000*a[6,5]*c[3]^2*b[7]-2798287029539129469243744*a[6,5]*c[3]^2 -5282327700151995462043543*c[3]^2-9033494428548570026592000*a[6,5]*c[3 ]*b[7]+935909730465449650667136*a[6,5]*c[3]-10482495467629110937171440 *b[7]*c[3]-3937234062950635296960*b[7]*a[7,5]*c[3]+1370214999285185463 080244*c[3]+970087973771931703698240*b[7]-121260996721491462962280)/b[ 7]/(129*c[3]-97)/(5*c[3]-177)/c[3]^2, a[8,5] = 5764801/950706720*(3360 00*b[7]*c[3]-55675*c[3]-336000*b[7]+33327)/(49*c[3]-36), a[8,6] = -276 922881/2412124220*(45500*b[7]*c[3]-7545*c[3]-45500*b[7]+4714)/(129*c[3 ]-97), a[6,2] = -679/2623731*(10942425*c[3]^2-9343855*c[3]+1654044)/c[ 3]/(5*c[3]-177), a[6,3] = -2/5154200289*(-296470608791175*c[3]^3+47487 1835976300*c[3]^2+780149716380*a[6,5]*c[3]^2-27617299959852*a[6,5]*c[3 ]-234017076432937*c[3]+33514304035716)/(1745*c[3]^2-62643*c[3]+30798)/ c[3]^2, a[8,3] = -1/20*(1456000*b[7]+123759)/c[3]/(2853786*c[3]-437940 7*c[3]^2+2206029*c[3]^3-607608), a[4,3] = -30276/42508549*(349*c[3]-17 4)/c[3]^2, a[6,4] = -121801/1793661700572*(37867593960*a[6,5]*c[3]^2-1 713543170955*c[3]^2+1973895466697*c[3]-515391279326-1368333915624*a[6, 5]*c[3]+984866566176*a[6,5])/(1745*c[3]^2-62643*c[3]+30798)\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "the coefficients in terms of " }{XPPEDIT 18 0 "c[3];" "6#&%\"c G6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\" \"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 " for ii from 2 to 8 do\n print(`===================================== =================`);\n temp_eq := c[ii]=subs(e3,c[ii]);\n if lhs(t emp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a parameter`) ;\n else\n print(temp_eq);\n end if;\n print(``); \n for \+ jj to ii-1 do\n temp_eq := a[ii,jj]=subs(e3,a[ii,jj]);\n i f lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(temp_eq)*` is a \+ parameter`);\n else\n print(temp_eq);\n end if;\n print(``);\n end do:\nend do:\n`############################ ##########################`;\n``;\nfor ii to 7 do\n temp_eq := b[ii] =subs(e3,b[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lh s(temp_eq)*` is a parameter`);\n else\n print(temp_eq);\n end if;\n print(``);\nend do:\n`####################################### ###############`;\n``;\nfor ii to 8 do\n temp_eq := `b*`[ii]=subs(e3 ,`b*`[ii]);\n if lhs(temp_eq)=rhs(temp_eq) then\n print(lhs(tem p_eq)*` is a parameter`);\n else\n print(temp_eq);\n end if; \n print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======== ==============================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"#6\"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#\"#6 \"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"$\"\"\"%0~is~a~par ameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",&*&#\"#j\"#AF(*$)&%\"cG6#F'\"\"#F( F(!\"\"F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,$*&#\"#j\"#A\"\"\"*$)&%\"cG6#F'F( F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"$u\"\"$\\$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"%\"\"\",$*&#\"$[$\"*RSfn%F(*&,(*&\"()R*>#F()&%\"cG6#\"\" $\"\"#F(F(*&\"((o(G\"F(F3F(!\"\"\"'=l;F(F(F3!\"#F(F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" %\"\"#,$*&#\"(#3hG\"*RSfn%\"\"\"*&,&*&\"$\\$F.&%\"cG6#\"\"$F.F.\"$;\"! \"\"F.F3F8F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,$*&#\"&w-$\")\\&3D%\"\"\"*&,&* &\"$\\$F.&%\"cG6#F(F.F.\"$u\"!\"\"F.F3!\"#F.F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W============== ========================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#O\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*&# \"#=\")J+`PF(*(,**&\"+l8t[ZF()&%\"cG6#\"\"$F6F(F(*&\"+mwz6\\F()F3\"\"# F(!\"\"*&\"+36ES9F(F3F(F(\"*cA)=8F;F(,&*&F'F(F3F(F(\"$x\"F;F;F3!\"#F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,$*&#\"%;H\"'x[=\"\"\"*(,(*&\"':M;F.)&%\"cG6# \"\"$F(F.F.*&\"'OV8F.F4F.!\"\"\"&oH#F.F.F4F:,&*&F'F.F4F.F.\"$x\"F:F:F. F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,$*&#\"$;#\"'\\w6\"\"\"**,**&\")N=.d F.)&%\"cG6#F(F(F.F.*&\")Ho`*)F.)F4\"\"#F.!\"\"*&\")yr:VF.F4F.F.\"([Y*f F;F.,&*&\"$\\$F.F4F.F.\"$u\"F;F;,&*&F'F.F4F.F.\"$x\"F;F;F4!\"#F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"%,$*&#\"+9t\\v9\"(@=T$\"\"\"*(,(*&\"$X#F.)&%\"cG6# \"\"$\"\"#F.F.*&\"$y#F.F4F.!\"\"\"#sF.F.,&*&\"$\\$F.F4F.F.\"$u\"F;F;,& *&F'F.F4F.F.\"$x\"F;F;F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%W==================================== ==================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'# \"#(*\"$H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",$*&#F(\"/#H1(y-t>F(*(,0*&\"2Nk22 (z#o$\\F()&%\"cG6#\"\"$F5F(F(*(\"/!)4o*HAn%F(&F%6$F'\"\"&F(F1F(F(*&\"2 R%>4z.;a_F()F2\"\"#F(!\"\"*(\"17uA'*fH)o\"F(F8F(F=F(F?*(\"1)[L#)>h^@\" F(F8F(F2F(F(*&\"2cjUVBi%y:F(F2F(F(\"1/:dx$HYZ\"F?F(,&*&F:F(F2F(F(\"$x \"F?F?F2!\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#,$*&#\"$z'\"(JPi#\"\"\"*(,(*& \")DC%4\"F.)&%\"cG6#\"\"$F(F.F.*&\"(bQM*F.F4F.!\"\"\"(WSl\"F.F.F4F:,&* &\"\"&F.F4F.F.\"$x\"F:F:F.F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\"\"#\"+*G+U: &\"\"\"*(,.*&\"0v6z31Z'HF.)&%\"cG6#F(F(F.!\"\"*&\"0+j(f$=([ZF.)F4F,F.F .*(\"-!Q;(\\,yF.&F%6$F'\"\"&F.F:F.F.*(\"/_)f**HF.F7F.F.\"-E$z7R:&F>*(\".Cc\"RLo8F.F3F.F7F.F>*&\"-whcm[)* F.F3F.F.F.,(*&\"%X\"&)zIF.F>F.F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG 6$\"\"'\"\"&\"\"\"%0~is~a~parameterGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======================= ===============================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"cG6#\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",$*&#F(\"5SI$*o3NQ#Q(GF(*,, N*&\":]V0D$*HWU>F*ySF()&%\"cG6#\"\"$\"\"%F(!\"\"*(\"8+o[X\"y:?:ifdF(&F %6$\"\"'\"\"&F(F1F(F7*(\";+P)f+A7PP2etW#F(&%\"bG6#F'F(F1F(F(**\"9++K$) )*pX1/LtMF(F:F(F1F(F@F(F(**\"6+73LuMVeK,#F(F@F(&F%6$F'F=F(F1F(F(*(\";! ))H7SO%Hw!ylAE&F()F2F5F(F@F(F7*&\":&Q:f4>V[g2(Q&zF(FKF(F(*(\":![eK`6Tc Mo?<@F(F:F(FKF(F(**\"7!)GAPH(yEKiU(F(FKF(F@F(FGF(F7**\";+!Gt<^q$z81\") *G\"F(FKF(F:F(F@F(F7**\"8gdVKgkM(\\jq5F(F@F(FGF()F2\"\"#F(F(*(\";gQC1. (erPdEcz$F(F@F(FVF(F(**\";++g?8#*Re_nUe@F(F:F(FVF(F@F(F(*(\":WPCp%H\"R &HqG)z#F(F:F(FVF(F7*&\":VN/ia*>:+xK#G&F(FVF(F7**\":+?fE+d[&GW\\L!*F(F: F(F2F(F@F(F7*(\"9Orm]'\\aYI(4f$*F(F:F(F2F(F(*(\";S9xtz3q*F(F@F(F(\"9!GiHY\"\\@n*4E@\"F7F(F@F7,&*&\"$H\"F(F2F(F (\"#(*F7F7,&*&F=F(F2F(F(\"$x\"F7F7F2!\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"#,$*&#\"#j\"#))\"\"\"**,.*&\"%SF.F3F.F.*(\"&OD\"F.F>F .F4F.F8F.F4F8F>F8,&*&\"\"&F.F4F.F.\"$x\"F8F8F.F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"$,$*&#\"\"\"\"0&3p<+QG@F,*.,P*&\"9D%eq(zOq3nM#z\"F,)&%\"cG6#F(\"\"&F ,F,*(\":]0Niy?A_-3a2\"F,F2F,&%\"bG6#F'F,!\"\"*(\"7]e=os>+p_*>(F,&F%6$ \"\"'F6F,)F3\"\"%F,F<*(\":?;\\O(3(4W[>Br$F,F9F,FBF,F,**\"8+]T&[723ImTV F,F?F,FBF,F9F,F,*&\"9qo#z:JsAYP#HeF,FBF,F<**\":+g;(RJ@CnKE7;F,)F3F(F,F ?F,F9F,F<*&\"9lD13\"*[=F,FLF,F9F,&F%6$F'F 6F,F<*(\":g*)*[Ud7ZUG#G*\\F,FLF,F9F,F<*(\"95tlTR^?V&3lk#F,F?F,FLF,F,** \"6SYNZ[1)3<%o'F,F9F,FQF,)F3\"\"#F,F,**\":+]2l,**H2WL!)p#F,F?F,FYF,F9F ,F,*(\"9oal$=\"R#pyey\\$F,F?F,FYF,F<*&\"9,6_/Y&[%extwXF,FYF,F<*(\":Sbt =3*)R3/^\\D$F,F9F,FYF,F,*&\"9=#f3aYO,la-N\"F,F3F,F,*(\"9#RLj?\"=3jr))p 6F,F?F,F3F,F,*(\":I*R4tDwFa1I?5F,F9F,F3F,F<**\":+SK`7doN!o=H6F,F?F,F3F ,F9F,F<**\"6?@TH)oyDa@\\F,F9F,FQF,F3F,F<*&\"9!GiHY\"\\@n*4E@\"F,F9F,F, \"8&GqGV'=!fCw::F$e-j;=F.F3F.F.**\"9 ++K$))*pX1/LtMF.F3F.&F%6$\"\"'\"\"&F.F8F.F.*(\"8+o[X\"y:?:ifdF.F@F.F3F .F;**\"6+Od#=u_a8kiF.F3F.F8F.&F%6$F'FCF.F.**\";+!Gt<^q$z81\")*G\"F.F@F .)F4\"\"#F.F8F.F;*(\":![eK`6TcMo?<@F.F@F.FLF.F.**\"8S1Ou&Gz$eG1J#F.F8F .FHF.FLF.F;*&\"9&*>r(Rj%Q+#3>>$F.FLF.F;*(\":+?IT>WKDM#*)eBF.F8F.FLF.F. *&\"92&oEGZ9i$oA%)=F.F4F.F.**\"8!GT*>Xv<)y@JLF.F8F.FHF.F4F.F.*(\":+012 4VD-GACf\"F.F8F.F4F.F;**\";++g?8#*Re_nUe@F.F@F.F4F.F8F.F.*(\":WPCp%H\" R&HqG)z#F.F@F.F4F.F;*&\"9Orm]'\\aYI(4f$*F.F@F.F.\"8Y\"evY#yPX)HWQF;*( \"8!)[N,y1,FZ]A\"F.F8F.FHF.F;*&\"9+5SivB.64,(H$F.F8F.F.*(\":+?fE+d[&GW \\L!*F.F8F.F@F.F;F.,(*&\"&@]%F.FLF.F.*&\"&*HcF.F4F.F;\"&yo\"F.F;F8F;,& *&FCF.F4F.F.\"$x\"F;F;F.F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\"\"%0~is~a~para meterGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*&#\"'jbr\"6?c`Sf9R5+9$\"\"\"*(,:* (\"5?4nK%Q)G#[Y%F.&F%6$F(\"\"&F.)&%\"cG6#\"\"$\"\"#F.!\"\"**\"4![gDs3R H\\mF.&%\"bG6#F'F.&F%6$F'F5F.F6F.F<*&\"1b12wS._\"*F.F6F.F.**\"6+!3P>0 \">/Dp#F.F3F.F6F.F?F.F.**\"6++#e#)\\N0nqYF.F3F.F7F.F?F.F<*(\"5Wh><(3tI )pgF.F3F.F7F.F.*&\"3-;5]$oP'3LF.F7F.F<**\"4g`&H#[=_]))*F.F?F.FBF.F7F.F .\"3\"p%*[#oP9OCF.*&\"5'Hv<8_Gn%\\?F.F3F.F<*(\"6+?6KYWMm\"y>F.F?F.F3F. F.*(\"4g*o$pn2kLn$F.F?F.FBF.F\\(\")GtSN\"\"\"*&&F%6#\"\"$F-)&%\"cGF0F1F-F-F-*&#\"*V>w'=\") kOq#H)F+F1F+F5F+F+*&\"-0^$yDK$F+F5F +F+\"-?_'f(>6F<*&\",7x=ph'F+FFF+F*&\".:#H)F+&F%F3F+F1F+F+*&\",7x=ph'F+&F%6#\"\"&F+F<*(\"/+-vYY9pF+F?F+)F1\" \"#F+F<*(\"0cv!R^e>?F+F?F+)F1F4F+F+*&\".+,hP%zAF+FGF+F<*(\"0?N8(e%f5\" F+F?F+)F1FDF+F+*(\"0[fyms&*[#F+F?F+)F1\"\"%F+F<*(\"/+[GoVw8F+F7F+F1F+F <*(\"/+?Fikv8F+F7F+FGF+F+*(\"-/*zg.J$F+F1F+FBF+F+*(\"-g&Qf%3LF+FGF+FBF +F " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK 5_7eqs := [op(RowSumConditions(7,'expanded')),op(OrderConditions(5,7,' expanded'))]:\nsimplify(subs(e3,RK5_7eqs)):\nmap(u->lhs(u)-rhs(u),%); \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#79\"\"!F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The stability function of a general 7 stage, order 5 Runge-Kutta scheme has the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/ 6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7;" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*& F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F) *&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 7 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[ 5,4]*a[4,3]*a[3,2]*c[2]+b[7]" "6#/&%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&% \"aG6$F'\"\"&F-&F/6$F1\"\"%F-&F/6$F4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F -&F+6#\"\"(F-" }{TEXT -1 2 " (" }{XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]* a[3,2]*c[2]+a[7,6]*a[6,4]*a[4,3]*a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"( \"\"&\"\"\"&F&6$F)\"\"%F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,& F&6$F(\"\"'F*&F&6$F:F-F*&F&6$F-F0F*&F&6$F0F3F*&F56#F3F*F*%!GF*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[7,6]*a[6,5]*a[5,3]*a[3,2]*c[2]+a[7,6] *a[6,5]*a[5,4]*a[4,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,3]*c[3]" "6#,(*,&% \"aG6$\"\"(\"\"'\"\"\"&F&6$F)\"\"&F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG 6#F3F*F**,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-\"\"%F*&F&6$F>F3F*&F56#F3F*F**, &F&6$F(F)F*&F&6$F)F-F*&F&6$F-F>F*&F&6$F>F0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[7]=b[7]*a[7,6]* a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\" \"\"&%\"aG6$F'\"\"'F,&F.6$F0\"\"&F,&F.6$F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9 \"\"#F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 41 "0.000206765992657:\nconvert(%,rational,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#8\"&tG'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Thus the following two equations hold." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "eq7 := b[7]*a[7,6]*a[6,5]*a [5,4]*a[4,3]*a[3,2]*c[2]=13/62873;\neq6 := b[6]*a[6,5]*a[5,4]*a[4,3]*a [3,2]*c[2]+b[7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+\n a[7,6]*(a[6,4]*a [4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3 ]*c[3]))))=1/720;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq7G/*0&%\"bG6 #\"\"(\"\"\"&%\"aG6$F*\"\"'F+&F-6$F/\"\"&F+&F-6$F2\"\"%F+&F-6$F5\"\"$F +&F-6$F8\"\"#F+&%\"cG6#F;F+#\"#8\"&tG'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/,&*.&%\"bG6#\"\"'\"\"\"&%\"aG6$F+\"\"&F,&F.6$F0\"\"%F,&F .6$F3\"\"$F,&F.6$F6\"\"#F,&%\"cG6#F9F,F,*&&F)6#\"\"(F,,&*,&F.6$F@F0F,F 1F,F4F,F7F,F:F,F,*&&F.6$F@F+F,,&**&F.6$F+F3F,F4F,F7F,F:F,F,*&F-F,,&*(& F.6$F0F6F,F7F,F:F,F,*&F1F,,&*&&F.6$F3F9F,F:F,F,*&F4F,&F;6#F6F,F,F,F,F, F,F,F,F,F,#F,\"$?(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 61 "We shall require the following two equations to be sati sfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqn6 := op(solve( simplify(subs(e3,\{eq6\})),a[7,5]));\neqn7 := op(solve(simplify(subs(e 3,\{eq7\})),a[7,5]));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/&% \"aG6$\"\"(\"\"&,$*&#\"\"\"\"=g0XCg^ypH%3/'p>F.*,,J*&\"C_b`\"HRX;w0/ki rU_b)F.&F'6$\"\"'F*F.F.*&\"DbKc.BXO,*G'3XFn`vP\"F.&%\"cG6#\"\"$F.F.*& \"E7\")\\lG9T(e`p#QI\\LJ(>(F.)F4\"\"#F.!\"\"*&\"D:3Oi553>e\"4S!y8OBb#F .)F9F@F.FA*(\"F+Sm\")o@xqC(po#*[4*=p%pF.F?F.&%\"bG6#F)F.F.*(\"D#*4\\\" =x!p-Q*z&**H?6O=&F.F4F.F9F.FA*(\"DOV$3/$pAD0Z3C`ioJQ*F.F4F.FDF.F.*(\"C +O$Gx<1!\\s([Sz^15Y\"F.F4F.)F9\"\"%F.F.\"C:Vil\\&QPTH\\C19d/vyS#=Dv7(F.F?F.FPF.F A*(\"GS5Nt0Y`'*>\"*f:&\\bwUgq\"F.F?F.)F9Ft(R\"=\"F.F?F.FDF.FGF.F.**\" H+S#)*Qk$GX)\\c%QtNj)H5!>\"F.F?F.FYF.FGF.FA*(\"Fcqj9c[1$R2B[QV$*\\%\\^ &F.F?F.F9F.F.*(\"D!3c2@h>Q\"=Pfyr=2HV&F.F4F.FYF.FA**\"G+gd?0$H7NwTKI7k $=)e!\\F.F?F.F9F.FGF.FA*&\"D0'4.$y\\q>h%y/o:k)4\\\"F.FYF.F.*&\"B]%y#4G ZrvK')*ewcC6SF.FPF.FAF.F4FAFGFA,**&\"(Sx<%F.FYF.F.*&\"(T'*Q(F.FDF.FA*& \"(GJ4%F.F9F.F.\"';_nFAFA,&*&\"$H\"F.F9F.F.\"#(*FAFAF.F." }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%%eqn7G/&%\"aG6$\"\"(\"\"&,$*&#\"\"\"\"ASAqc[z= XP-&y$Hp\"[(F.*.,H*&\"Hw$p%Q^@Cav#\\BGK\"='G]\\5F.)&F'6$\"\"'F*\"\"#F. !\"\"*(\"FmVpn[q?P],RI*fLv46lF.F5F.&%\"cG6#\"\"$F.F9*(\"G`#p&etzD\"Q%p U*[@*4%yL3\"F.F5F.)F,') *ey\"F.F4F.FBF.F9*(\"H+UF&>>5aV^Z7F.F5F .F.*(\"FSL#o$))*eF2]k&z&QGEIYfF.F5F.)FF.F4F.FVF.F.*&\"CD$RRi0'[7dN #4?h-k[$F.FVF.F.*&\"Eb::9N7Cq-L%f:dp " 0 "" {MPLTEXT 1 0 38 "factor(simplify(rhs(eqn6 )-rhs(eqn7)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"2z_(3i$4:`\" \":kSq.DZ4mwp_&f\"\"\"*2,&*&\"$H\"F(&%\"cG6#\"\"$F(F(\"#(*!\"\"F(,&*& \"\"&F(F-F(F(\"$x\"F2F(,(*&\"+qt&)eZF()F-\"\"#F(F(*&\"+B!==z#F(F-F(F2 \"*I3U$RF(F(,&*&F5F(F-F(F(F;F2F2,&*&\"#\\F(F-F(F(\"#OF2!\"#,&*&\"%S " 0 "" {MPLTEXT 1 0 146 "denom_factors : = select(has,map(op,select(has,map(denom@rhs,e3),c[3])),c[3]):\nmember (5*c[3]-177,denom_factors);\nmember(129*c[3]-97,denom_factors);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "However, the quadratic factor has two real zeros." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "4758857370*c[3]^2-2791818023 *c[3]+393420830;\nsolve(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"+qt&)eZ\"\"\")&%\"cG6#\"\"$\"\"#F&F&*&\"+B!==z#F&F(F&!\"\" \"*I3U$RF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\")(epi*\"*gq>G$\"\" \"*&F&!\"\"\"0p\")4dh.j$#F'\"\"#F',&F$F'*&F&F)F*F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+Bp$Q^$!#5$\"+^_t_BF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[3] = 4/17;" "6#/&%\"cG6#\"\"$*&\"\"%\"\"\"\"# " 0 "" {MPLTEXT 1 0 94 "(96269587-sq rt(363036157098169))/328197060:\nevalf[10](%);\nconvert(%,rational,5); \nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^_t_B!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+w6%HN#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "Substituting for " }{XPPEDIT 18 0 "c[3]; " "6#&%\"cG6#\"\"$" }{TEXT -1 32 " in each of the equations for " } {XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 29 " now give s an equation for " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "expand(subs(c[3]=4/17,eqn6));\nexpand(subs(c[3]= 4/17,eqn7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,** &#\"*hGl^$\"+!G&QGO\"\"\"*&F.F.&%\"bG6#F'!\"\"F.F.*&#\":v%R#G)RE$fsev. #\"<#R\"\\Y4Oy()>#exJ:F.*&F.F.*&F0F.&F%6$\"\"'F(F.F3F.F3*&#\".*\\WmD

" 0 "" {MPLTEXT 1 0 126 "-2037558725932639828239475/153177582198778360946491392,-43959 403348939462185253837/3303189630546054082054520832:\nevalf[10](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$!+;Q>I8!#6$!+Do\"3L\"F%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We choose the firs t of the two equations to procede further." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 102 "e4 := \{c[3]=4/17,a[7,5]=simplify(subs(c[3]=4/17,r hs(eqn6)))\}:\ne5 := `union`(e4,simplify(subs(e4,e3))):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3008 "e5 := \{`b*`[2] = 0, a [7,2] = 9/2684*(-179+2128*b[7])/b[7], c[4] = 174/349, c[2] = 11/63, a[ 5,4] = 613806882624/1137797949827, b[6] = -8189994205575/136646837063* b[7]+6917256644499/1366468370630, a[8,4] = 504243252114389/75113722327 2480-10028786914276/1564869215151*b[7], c[8] = 1, c[7] = 1, b[2] = 0, \+ a[5,2] = 12241368/78942479, a[5,3] = -4360301712/39234412063, a[2,1] = 11/63, a[8,2] = 0, `b*`[7] = -1/346805722200*(189886468604448*`b*`[3] -17749124038368*`b*`[5]-60208363421605)*b[7]/(-179+2128*b[7]), a[7,6] \+ = 13814933825/1179950893056*1/(b[7]*a[6,5]), c[5] = 36/49, a[5,1] = 24 24510108/16025323237, a[8,7] = b[7], a[8,1] = 15050711/145825920-29575 /75951*b[7], b[1] = 15050711/145825920-29575/75951*b[7], `b*`[8] = 179 /38842240886400*(-6436136194495+38842240886400*b[7]+9994024663392*`b*` [3]-934164423072*`b*`[5])/(-179+2128*b[7]), `b*`[4] = -113837894222/77 568040725*`b*`[3]+552611137/833623560-361407683598/6810762375725*`b*`[ 5], a[4,1] = 184816101/935188078, a[4,2] = -411995808/467594039, a[8,5 ] = 1008840175/15845112*b[7]-1982278707059/395493995520, a[8,6] = -818 9994205575/136646837063*b[7]+6917256644499/1366468370630, a[7,3] = 17/ 635302028948874411924988359357526179840*(31746749822078423292121642838 355534336*a[6,5]-307536269003986937465923921485763706880*b[7]*a[6,5]-5 29784324126395679728869055956553826304*a[6,5]^2+6272617553159915219977 301865533079552000*a[6,5]^2*b[7]-46465343501899179752989249014561625)/ b[7]/a[6,5], a[4,3] = 100493613/85017098, a[3,2] = 504/3179, c[6] = 97 /129, `b*`[6] = 1981393947/1976775028*`b*`[3]-507097315647/52070459148 4*`b*`[5]+715563/141629312, a[6,2] = 1006084/2623731, a[7,4] = -121801 /4560278903716977957508701117466029526794240*(-89395293131557083678668 056803703296000*b[7]*a[6,5]+12487559563262765253695372612399632896*a[6 ,5]+6272617553159915219977301865533079552000*a[6,5]^2*b[7]-52978432412 6395679728869055956553826304*a[6,5]^2-94746707292686471982568752301687 375)/b[7]/a[6,5], a[7,5] = 1/104656160582103779829527058520780308480*( 10142979714098117930840928338720088576*a[6,5]-139212977614405653661451 9232348309625-529784324126395679728869055956553826304*a[6,5]^2+6272617 553159915219977301865533079552000*a[6,5]^2*b[7])/b[7]/a[6,5], b[3] = 1 4616175/7078984*b[7]+10336475439/58897146880, a[7,1] = -1/104113381603 201852865329459906038005760*(1319213784397153583677930647380660736*a[6 ,5]-529784324126395679728869055956553826304*a[6,5]^2+30603826849025181 059421299477921710080*b[7]*a[6,5]+627261755315991521997730186553307955 2000*a[6,5]^2*b[7]-31925551356974997368184850899635125)/b[7]/a[6,5], a [6,1] = 7899289187/57522678444-69992/69629*a[6,5], c[3] = 4/17, a[6,3] = -407293106/1066904433+5251419/1875181*a[6,5], a[8,3] = 14616175/707 8984*b[7]+10336475439/58897146880, a[6,4] = -152007648/54380249*a[6,5] +227562878917/371282742684, a[3,1] = 244/3179, `b*`[1] = -11503349/414 60807*`b*`[3]+953641/330946637*`b*`[5]+33691/202536, b[4] = 5042432521 14389/751137223272480-10028786914276/1564869215151*b[7], b[5] = 100884 0175/15845112*b[7]-1982278707059/395493995520\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We choose values for " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 54 " that minimize the norm of the principal error terms." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "errterms5_7 := PrincipalEr rorTerms(5,7,'expanded'):\nsm := 0:\nfor ct to nops(errterms5_7) do\n \+ sm := sm+subs(e5,errterms5_7[ct])^2;\nend do:\nsm := simplify(sm):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1121 "sm := 1/ 5623370898390421508042439548100312657403983724624408477102903904744940 04602398827420110028800*1/a[6,5]^2*(-115131889799695494681667538101406 3217794337991229344358967630090077428789181973504000*a[6,5]+6170441364 1033431315566774156255955706387987491182731312218586952618665698219477 9586560000*a[6,5]^3*b[7]-673747090776591681931687213396595752825487990 65226705481416143020879726387090781095591936000*a[6,5]^4*b[7]+39885644 3606983846293479509512076532083226056358156814646879865055149907501943 602806784000000*a[6,5]^4*b[7]^2+34915797170330020376480745763721388193 474497166049585870992099136794406527550179246080000*b[7]^2*a[6,5]^2+44 5604055029679378270085107045741055417106101895018369514523003271823822 800683399643136*a[6,5]^2+509847374991318988630907565341127515729631712 28335460587636384249032939548504984375-5288273727319585567134876813638 413572757373331792759563031168617253666616306851171532800*a[6,5]^2*b[7 ]+28452288386320344248472721730234260878829863936606895625026100363796 35985499815081972596736*a[6,5]^4-5211545387453504922507328661535477658 7992072342933337124189622434132262061735396109189120*a[6,5]^3):" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indets(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"aG6$\"\"'\"\"&&%\"bG6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We can determine values o f " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 39 " to minim ize the principal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "plot3d(sm,b[7]=0.07..0.08,a[6,5]=0.05..0.13,labels=[`b[7]`,`a[6 ,5]`,``],style=patch,linestyle=2,\n color=COLOR(RGB,.65,.3,.8),axes=b oxed,grid=[30,30],\n view=[0.07..0.08,0.05..0.13,0..3e-7],orientation =[32,57]);" }}{PARA 13 "" 1 "" {GLPLOT3D 542 453 453 {PLOTDATA 3 "6)-% %GRIDG6&;$\"\"(!\"#$\"\")F);$\"\"&F)$\"#8F)X,%)anythingG6\"6\"[gl'!%\" !!#_cn\"?\"?3E47DF2FF8682D363E40D27F4413E4183E3EA19D9082F1383E43343383 8B415B3E4C5FCB39607DE63E555EAA7ED96FA83E5F1DBF2A177B6B3E65B32ACDEB1F55 3E6D1984CC8B52E23E72DFE34780CA223E77D21A8689BA5C3E7D62B4E31DBDBC3E81C8 903B4461CB3E852E72EE09A2FA3E88E2D172ADAB8A3E8CE5833C6B06BA3E909B3354B5 AB963E92EAAFDD06CD263E95612B8592DA383E97FE9C8227A59B3E9AC2FA97827C9F3E 9DAE3ED6EAE3B83EA06031B34402473EA1FCB1A9EDD89D3EA3AC9D372FA01D3EA56FF2 9456BBC23EA746B03F7CE1F23EA930D4F0EBA9EC3EAB2E5F9242CBCA3EAD3F4F370FF2 453E4A77A3B79730FA3E4167A212E6C7263E3B504CBEAA87B63E3E268B36F9D6FC3E45 8C1D8ACCFFE33E507E36A09BA06A3E58A971213413E43E61A078224DE3173E681FA7E7 E9C9163E6FD020DF82FE483E745814006F889B3E795F2B65BC43753E7EFCC4B8F3A578 3E8298344AAF0F5B3E85FCDA664CE4433E89AC2C228DD8713E8DA607DD9B950A3E90F5 28CBEA8A023E933C78F7071C933E95A8EAA3F156DD3E983A75976681BD3E9AF112E2AE 26C73E9DCCBCABEF6DDF3EA066B7005613D03EA1F991582ED7163EA39EEB96CF5FD63E A556C43A5168583EA72119FAFE87163EA8FDEBC276722C3EAAED38A446D8AE3E4E0009 3F9778F13E430700076321D13E3A4AD78F8FACC13E386AC2FEBFF36D3E4019FEF19AAD 8B3E48BD033269CC4C3E53068DE2BCE3143E5BFE5652D1355D3E63A03B573966CC3E6A 645078E5FF523E7124D7C31A08E43E75A778FF4F05E53E7AB97A0A5DF0AF3E802D31BF C79D7E3E8344E998407D143E86A3BC01D422D43E8A49875AAC30403E8E362FA327A69A 3E9134CEBC0D31373E9371DEA09007DB3E95D23F43D9A3D93E9855E9B7318B033E9AFC D820BEE4013E9DC7058E0381543EA05A36E73C5F173EA1E286AA5CAC903EA37C708E7C 61963EA527F34BE5163E3EA6E50DCC3680A63EA8B3BF22FE4B9F3E513C304834828D3E 45B0992188E9B03E3B913E0332732B3E35350E5E68C6F83E3812DEDB932CD93E41FF16 D14A971A3E4C6A2ADD63DF073E559E87C676A4583E5F367E34F441B63E657C555B2AA6 EC3E6C70CB9D12719E3E723B9DAFD6074C3E76C7406AC7A96F3E7BDAD69AA69CB73E80 BAFF08887B863E83CC32DA3DDD373E8720E5209D272B3E8AB8F9DC054BE33E8E9459A9 4A2ED13E9159787803BC073E938A579CDBE2F43E95DCC354750DEC3E9850B5C4F66826 3E9AE629FBE1C12A3E9D9D1BC8B0748E3EA03AC3CEFEA16C3EA1B7B53BFDCDAE3EA345 60E39F56EE3EA4E3C5AF82F3313EA692E2B3364F613E53F054D505EDA13E49646D6158 09F03E3F23801992C8243E34856D55F462AE3E32B4DDFEB346643E3985503BB325FE3E 446A0F8A2665533E5021849F8BFAFD3E582166635804DC3E61182F8650E58C3E67257C 457A37F43E6E3732EEA27D5E3E732617DA30CFB83E77B1C1E6A443EB3E7CBE356E49B0 713E812590ABCB13BD3E842C212F6E79CC3E8772B0426E05683E8AF926819D9CD93E8E BF705498E8273E9162BEA26D3C6E3E93859FBA78B2023E95C8559895F7EE3E982ADB4A 46E5FF3E9AAD2C9F04D1493E9D4F46096A7BD03EA0089242D5AB273EA17962C22D4925 3EA2FA136C5BCB963EA48AA354EEE0F23E571C72463FF7E93E4E227CC6D0AC363E4280 CEE95855D83E365BDFE562B57E3E3019FB4C95D6623E320F6E302D71B63E3C19939782 97EC3E470E99BC228ABE3E52012F399E0BDC3E5A6FBDF4B185EE3E6267C17F6B625E3E 688ED8AB816E903E6FAC00B132C7103E73DF25638834863E786279482B31693E7D5FAA ECF771103E816B3B872024DE3E846352D661D6403E8798040114ADFD3E8B093B6ED454 F43E8EB6E8A91B632E3E91507EE93C72EA3E9363B79B9D96503E9595197932ED953E97 E4A05175D4603E9A5248970107193E9CDE0F4607F9DF3E9F87F1CF1DD6FA3EA127F702 C109D63EA29B01082800553E5AC0889BE2A44A3E51F563A8F958A33E4695CB974621B2 3E3AB8660CB3D0793E304236C53A8F093E2B390F00083F353E32A4B344699BA23E3F3F 82081A26103E49ABB16F8CCEED3E53B6C76E846ED43E5C6F3695CBE68D3E637E2C9648 D4B53E69ADF3CC02C9C03E706301115264B43E7462C99EB576413E78D602749FAF753E 7DBC684F645DD63E818AE197E0BCE13E8470F227AF623D3E8790523EB9C81F3E8AE8F1 667A8B6C3E8E7AC1C180AC743E9122DBCE0D41853E9324E488A5DBD93E954376E0037B AF3E977E8F46C0E1A33E99D62AB9117F393E9C4A46A78882763E9EDAE0E5655AFB3EA0 C3FBCCE1AE643E5EDC97D5EDF2C43E555EA6815F09F63E4BD0B61692AB6D3E40CD7FE5 F3D9D03E332D9068A1DD413E285938566EB4CD3E28EAFC3602575A3E33ECFE2DAED550 3E413EC5BBA202B13E4C0AF6F43534F93E552A134FD3BD483E5E0A5D5DF15F9D3E6452 0904D1A7813E6A7AA9E005B4B53E70BF2671E87B4B3E74AE27EE8ED4CA3E790A162249 24723E7DD2B90DD56C663E8183F0F56DB8243E8454B4C4493C223E875B977CF7E1DC3E 8A988B4208ADF83E8E0B845FC9F44B3E90DA3C789FAD6D3E92C9B04AADC95A3E94D41A 18AA0D773E96F976DEC7EA463E9939C40D9A91F13E9B94FF78616C363E9E0B274637D7 1A3E61B84FFA30F0393E594D06EC9977173E5118C7339EF4D23E4581D6917F26EF3E38 DC0836CB73413E2B7F58638DD2423E2317E83693474B3E284B4FD206D73E3E357736AE F77C5B3E42B7B42EE811CF3E4E003259DCCB9C3E5647BDEB2208AF3E5F3080B73EC0A1 3E64DC41FF332AF23E6AEF1F8388937D3E70E81B5AC4DD953E74BF8086EEA27D3E78FD 8746FF8DB03E7DA200D49F624A3E815662FF82B5213E840EDAEC9374E73E86FA5A5410 F0933E8A18D5824980603E8D6A429240C8053E90774C9174BB3B3E9252E90CBC888A3E 9447F3B72B390F3E96566A015400BC3E987E49BE764A0B3E9ABF9115AD6C323E643E50 7B9F3C763E5DC084EAA89AD23E54DC2A44A40A5E3E4B793708FBD8253E40A6CF17DBB0 4B3E3255B793B2E8463E23D02A8A8689F53E1FA5FCE85F43713E2888C90C68B8173E36 E78D1A4295883E43E2905A37B6A23E4F68F5A84749453E570133A0D7D79A3E5FD59500 5A583F3E65180B1C91B4783E6B07B97283AF723E70DCA77D54D64E3E74962DDB3FD775 3E78B0410CAA97993E7D2AB9E0CC6BC33E8102BBB54D409F3E83A02EF79974483E866D AB039927493E896B25F44FFCEE3E8C989768B0AB3B3E8FF5F845F0A9D23E91C1A1423B 6F803E93A03882B4D5653E9596BFB7A3F2B23E97A53508241F7E3E67004D6F41D6D93E 615C903DC63BE13E5932843E587EDB3E5159D0A634FAFC3E46412929B2E9F73E39EEBE 50FB49993E2B13C331DAFF733E1CE21083B0141D3E1B9453F3E8D36C3E28FCB83B8424 F93E37F68141703D753E44A12831EC54FB3E50162AC66EB57C3E574C86C5E7707B3E5F F21F5DD8C92D3E6502D8140B6AA73E6AC3160AF7865F3E709CACCA964CC93E7432A292 FD131F3E7823452DE766A83E7C6E73AE4A822D3E808A092CA2355A3E830A04E3B8E907 3E85B723176CFCF73E88915B66B126493E8B98A6B6BAE31B3E8ECCFEFFF1154F3E9117 2F91BD0D1C3E92DE6163EA67F23E94BC131D9BF0FD3E69FE46D518C2493E641B6CCFA2 84583E5E1BD520BC5BAF3E55988AADE4AAE03E4D3D1250EB37F73E42456034D023083E 3471591648D6BF3E24256D3AFFDFFF3E1565204D241B293E18CF55A138CBE33E29290C B5780FA33E387026E666B0CF3E44DECC500671693E501D594F0D96BB3E57245A75311B 6B3E5F83253442111C3E649C563EC6D5823E6A22082A05DB5D3E7029256796D4DD3E73 9667E6566E4A3E7758AAA436F4773E7B6FD1E6566C8D3E7FDBC64551867B3E824E39FB 97C5E03E84D8E51CEAEE103E878DDD6BD7BAE23E8A6D1CE0C513193E8D769E5CD94C07 3E90552EC349A46D3E92042B5614DC673E6D383CAD23FEC63E671CD82AE926CF3E61CC 0E75E7CE143E5A78C99B8D1DDF3E52CD4546C268603E4914DEEED0BCFD3E3E9E7BBD54 E1593E307816AFD391C43E1E83F7248505C23E101FE3E79E1FA13E167CC0014C18703E 28B6DDB0702EDD3E383397165585133E4490193799C52D3E4F8D8EFE5926A73E5687D2 0988DAC53E5E8A1F2C2F135B3E63E667750B6E1B3E69279314EFB3CD3E6F08441432FA 453E72C41C4C5FD81B3E76539C9668EE1B3E7A328B80D177A93E7E60D541A0B1913E81 6F348B5E028E3E83D59C654739593E86639C26F2DA523E89192EB1874D4F3E8BF64FAB 835E9C3E8EFAFB631DCDC03E7057177BB1C4B53E6A60D24F9A24923E64D3ADCFC9227C 3E5FFA8D6F2E3A603E57ACC8EFBFC2D33E50B2EDAB3FC7943E4608A4C708F7083E3A68 A457E71A983E2B786C3D05170F3E17EB1B4A36544C3E08B3F191AA5CB83E1410A0E535 B5F73E27754CF92B0C373E373282B094D2663E43B2CCF79635843E4E276D4FD6B7DE3E 557A8421D359613E5D0CEEEC7AAFF53E62E51DF7404A4E3E67D8E732611B563E6D6191 4D8A4FB93E71BF72697BEF5D3E75185979F19C243E78BB6A0E2D69213E7CA8936414BE F73E806FE3A30954423E82B07CD27A6CEF3E85161021840D393E87A09936A50BC83E8A 501460141BAB3E72300ED9EBB4B93E6DE75B3DB57C563E6824C89E0228B63E630EEB14 6404763E5D3D14236DB31A3E559C2AB5EE45A53E4E64E1443FE4433E43F22FCADD453D 3E37AAF1136CCEBE3E28187DE4815A933E13E9C3C18A79F43E0352984D3597E13E114B 1B4B3072043E2558AC00B4E3EC3E3570DDAC328BCF3E424DA61ED30D6E3E4C13B6BCF4 C5D93E5404705B3731F33E5B15D6EC3EDA7D3E619EB9273755E03E663D5F66CDCCB43E 6B66A6BF1EDBB13E708D3030B1F3EC3E73AC325CD5ADF13E77104921E0168B3E7AB966 4A3C23833E7EA77DC6B78E0B3E816D42ACCF96FD3E83A93A27F84CCA3E8607A1A30CA0 DA3E742704713FCBD13E70D8397A9D96663E6BBF5EE092E0C23E667151E42D407D3E61 BF1370E622713E5B46269773E3603E5431F9AB27A6003E4C75A434A6CC853E42B69753 F027053E363C615900926F3E26961075B5B0963E1237BD250DF0AD3DFFBF818F94843E 3E0C6D62F42348583E2279D2678A4B733E33049D000D34F33E407049C448BB8B3E4966 A66C987A4F3E5231F7231A2D8D3E58B373E56B53CA3E601BA2E48A31A13E645E7EF146 D8113E69221F4A24F9453E6E665C5B3316803E72158A501E03463E753815D70ADB7F3E 789AC4B32DD8FF3E7C3D8CA6D3C6463E8010327F7D236D3E8221A32C0762703E763BF8 41AE0D9C3E72DE0CBB159EE23E6FA370977B4CFA3E6A247B26F2D58A3E653800956DB2 5D3E60D870A7E84B0B3E5A02ED7E9B86C53E535F57B4A823AA3E4B81776DEE84B43E42 45970366A2433E360844119E46073E26DDF9F04821BE3E12D8E93C774B393DFD685CB1 2983203E064DE42F20DA073E1E2B339B8CB4333E301582B344F35F3E3C665DF6E86E9C 3E463D3926254A4C3E5013D329B7E16A3E55F8B78D7EDD923E5CCCDAD2DF9DC23E6247 EFAE267AC13E66A0BB00F1F01F3E6B70ADDD9D01B83E705BD5EC7ED8AA3E733ACE6A57 BABC3E7655362AA5F2473E79AB047A672A1C3E7D3C31F608C0DC3E786EEA4B3675073E 7505276042D26A3E71E87EE15DB5813E6E2866DCB4C7E03E6909517F4D93083E646E2D 6F82343C3E6052A60E3DD74E3E5966A8B46CCEC43E531B0C6BD8A9E53E4B7C527DD38B 2D3E4299123A5673DA3E370D7C164871193E29031AEA715F243E161B8918147B773E00 BABD14798ADD3E016EDF913AE49A3E1718EBE80EC8453E29B9E98C04E0733E376F1CF0 A0790C3E42BD2036A943FC3E4B7E2835B625A23E52FCE44F3380423E5917A31AD0CF21 3E600780AAE7F1BD3E63F15DFBE32FC63E68494D15302DD53E6D0F35D86A66803E7121 81E515B1FF3E73F252C237350C3E76FA06200724813E7ABFDA8DD907253E774D896A25 34E63E74240331299C433E713E8A82B9866B3E6D33062E85BABB3E686449A287AB7A3E 640C8AC264034E3E6028628CD0B6CD3E596A3DC89C660A3E5361318DE362ED3E4C6454 B20355053E43B4B651D83C763E395DF18B4A64D73E2D3D839EB7FA1B3E1C96EA2719A0 283E076E510ED350E43DFF2BC5CCD990383E10C44519FB20543E234BB8F2D5DF473E32 4EAAE90629E43E3E296DC4A40BFF3E469A34AF12D7113E4FB7139FAAFED43E55355AB2 2A24B33E5B5A49F61D216B3E61250F62AD3CE23E650254718C65CC3E6944DFAC45D67F 3E6DEC9FACD4CF503E717CC2D60A00F63E7D2EC90995C0573E79B732D8BCC5AE3E7684 453B215E463E739142D096D78B3E70DA8F518B17A53E6CBAC540F8B0C53E682F24DBC0 4C4C3E640ED6722300163E605727E6A135CF3E5A0BE46EA04D623E5432F4B7EAE3443E 4E4167502E38183E45A5B348A99AE03E3D1E24671732C33E31F4E53B89FA563E23952E EBF8AB523E126C1B6E8D14B33E03159BC2F1DDC43E08F5873549ECDC3E1B323067E639 853E2AD46C8345C4C83E36F5F3AF85568A3E41B726F6B5F9CA3E4969042DE57DD83E51 4805B63E37E23E5695E582E95AAF3E5C9DF1402B543C3E61AFFFFB9B86DD3E656DF76E 0192323E6988D03022B34F3E7FBBB5BE6CA3813E7C4223AC09822A3E790944FF44F933 3E760C5C57F2530A3E7347CD6E7F79543E70B8D0256A9F643E6CBA745A52B3EB3E6866 B00A2D40383E6473A13CE554F43E60DF20F090305C3E5B4EE839E6D0EF3E5596688312 FAC93E50927B21D4A7013E488285069B7F713E41426BDBECF41F3E36C443EE8CB02A3E 2B7E34ED089BD03E1DC76D6E8016673E0ECCD1C2E263323E079F2001721A6B3E12FAA7 89A292183E22706061D9554C3E305F00758569F83E3A31DE10A6808F3E4357DE705368 373E4AEBDCC61B02E83E51EA44C80C090F3E5708C97058638F3E5CD1598FE973D23E61 A1EBCC3A34053E813350562ED6E03E7EEE5BE40B6BA43E7BB3027D946BDC3E78AFD718 CBFC1E3E75E13D6E1FFE863E73446D600EAE4E3E70D73C9F0D99CF3E6D2FEF54EF7A39 3E690A8AE71A9B9A3E653C24F2B1D63B3E61C3026F7AC8323E5D3B79B9DFCC673E5796 60DB5236B33E5294DCB06EB10B3E4C6AC8FF5ADDD83E44ED67C5C5D8933E3D5EF72D70 E7893E335E88F044887F3E27AD30DD077D473E1B12CA6E8C4CD43E109B229BD5F1473E 0FD8F712E24D563E1900FB5E88EBEF3E25E9D7CE768AD53E321819EF7DCF0C3E3BA8E5 971A2F363E43D346F57631FD3E4B083787E02E663E51B97F755C61F03E5689B100B55A 6D3E8297C4E9B4709F3E80DDEDC061420A3E7E817DB60FBAF33E7B7BB31323D0A33E78 A6DF506CAC193E76003A506882783E738599C38CE8FA3E71354A2934D88E3E6E1BE4E5 410F583E6A1CFE3DB525403E666C25538B90D53E6307F3A63ECCF13E5FDE8AD0CD8D4F 3E5A4257A128C01B3E5539C5040798353E50C38E933E56273E49BD4E803454B43E4314 5E1B25FAE43E3B15AB891F94523E323EF7EC57FE183E2745251E927FB83E1CFAFF0F4E 518B3E1448244F0903203E146CC77E2408DF3E1D64B815F992D63E27963458A65D0A3E 32703560E24EC63E3B47DBC7F7EF183E4328C118B7DD803E49C65932100DCC3E840B38 99C71E443E8255514117646D3E80BA5B545B70E43E7E6FF046F9CF893E7B98B315657D 2F3E78EC36F6781F713E7668519AA747743E740B4F814DEDE93E71D3D79BAC4E523E6F 81ACD19A03883E6BA2DCC925B6533E6809D89D7624233E64B57C81233C3C3E61A4D9AA BDEE983E5DAE577B04B37E3E5897A10CB55BC23E540482FBDAF1F73E4FE83A96DADCA8 3E48CB69C417A0B13E42B159F926DA663E3B32079B63EF243E330312D55302373E29A8 BE53CAF6B93E21495FF31F25DE3E19CBE48AA245B23E18F95D2F7BDE343E200C130421 57603E2792D7D4108A3C3E3187F353A114DB3E39412985B1B5D23E858DAB6666E0E63E 83DD5874281DA53E824656AAC4F0C33E80C6475A26FE8A3E7EB6B8BD0A75AC3E7C0863 523D81A83E797F64245CB1FA3E771A07B2C301B63E74D6F4EEB0A8BA3E72B518573044 2F3E70B39468249EC03E6DA36BC295F04A3E6A1DD5B7DE95453E66D577E6B408133E63 C98DF2527A3A3E60F975A7A4057E3E5CC950FE9EA4573E58156CF5A08D773E53D6841A ECEBD13E500BF9AB60E19E3E496A93E5A286353E43A40C0B6CE6113E3D87411A0C1092 3E359166354FF3BF3E2ECAF68DF797B13E260539EB80B6243E20D0149CF1674E3E1E54 7DD7C48C133E2112A2780B2AA83E268852CF1695813E871F1D4F93B6B53E8576035993 6E033E83E4B0DE445DAD3E8268C72D902A7F3E81007823ADC9D03E7F54BF63B8AA023E 7CCAD160AD2BCF3E7A6172BD9410F03E78174A6BAD96E43E75EB44EA0450D33E73DC84 B47B16C93E71EA568ACF21AC3E701428864C607D3E6CB3068476A8913E697409207417 643E666AB6AD7B31EB3E639688A4329A333E60F70F06162D6D3E5D17D88CEB08CF3E58 A9A4077E67D63E54A2FCC7BB6F463E510373E21A4C243E4B9557721ADA5C3E45F0AA49 6482393E41185A393C4EA83E3A17EB29F50B5D3E339637C619AAE93E2D55ECD76600FB 3E26AB37D53921273E232B64135646D23E88BF8E554DA1DE3E871F51F15955343E8595 69EED9B7A13E841F779DB86D3F3E82BBACDA2C6A8A3E8168A59574CD243E80254CA7CC 59AA3E7DE190A1C11E9B3E7B94D812A3133A3E79635C21492F743E774C3F49963FA73E 754ECE4B4781303E736A773FA8E8123E719EC2C202EBF63E6FD69D47E743B43E6C9F93 97E03A173E6997E1EC974DCD3E66BF1847886B3F3E6414D91C03252A3E6198D6087621 153E5E959A659756623E5A550E4A7FFC463E566FAA2218C9413E52E51EC47EE74C3E4F 6A52528F82EB3E49BF20F43EE6803E44C848F47A07503E40857865A70CB13E39ECD35F 65CD533E3435C36C12F6C53E8A6EFE7794A0333E88D9443B79D2913E875881DC84FE0A 3E85EA58AA9FC6403E848CFA82011FF13E833F0353E828583E81FF5DF88FA4423E80CD 30AFA513D93E7F4F9DE39124B63E7D1D5DFCFEDAE23E7B02C427761A923E78FF1D22B4 1B2E3E7711D70804D8FB3E753A7A72B0C91D3E7378A53455FBE33E71CC06336984993E 70345A2C3EB3373E6D62D23F26FA053E6A86088E1F525B3E67D208E3D506903E654691 663276F93E62E36A9F73E9853E60A865C0D279443E5D2AB66CB9B9753E595452F7BB9F 053E55CD684C56B8613E5295C62DC1F7B33E4F5A874A60BF283E4A277BCD318AE63E45 922EC760F4AC3E8C2D6DB668B3E23E8AA3DA37F4E7693E892DF8A74630513E87C96A54 4633E83E8674611B2BE71B3E852D78ED36662B3E83F39CA2A077533E82C5F27B1796DC 3E81A3CDEF3BE4473E808CA53E92AADC3E7F00134E1AA51F3E7CFB431114F6603E7B0A 47DF603BA03E792CAA5444EDB33E77620841611D7F3E75AA108D29DA063E74047FF3F2 7A393E72711E7678EFF53E70EFBD4E64FD3C3E6F006A95DBF4F33E6C44CAFE122AB43E 69AC645FA8B6763E673707EBD8F93B3E64E48E0EB176D03E62B4D543FCF12B3E60A7C1 2091891C3E5D7A730C1527923E59EA53F20369BD3E569F0226E48B4D3E5398601C6DFE C83E8DFADC11C9DB1A3E8C7F13E6CA91C63E8B15CE4F1D4FA43E89BCAC9AABB96C3E88 71E0A5ACC2F13E873406615F87F43E860208A5FED06B3E84DB0DB33819183E83BE6901 AB7DFD3E82AB90D0DE4FF83E81A2165EC1F2103E80A1A00B3504E73E7F53C9C5BB0BCE 3E7D755266BF53D33E7BA777CB14FF4A3E79E9E8D93116633E783C624D66F8A63E769E AC28748E793E751097A4019D583E7391FD8F45812B3E7222BCFD355DF33E70C2BA32EF 4DD83E6EE3BB921FDB103E6C6027EB3D34653E69FA97ED0145B63E67B2EF2C4230023E 6589158A9B51753E637CF68CA6526B3E618E80CC5E863E3E5F7B4B053D58C0-%&COLOR G6&%$RGBG$\"#lF)$\"\"$!\"\"$F+F=-%*LINESTYLEG6#\"\"#-%+AXESLABELSG6%%% b[7]G%'a[6,5]G%!G-%*AXESSTYLEG6#%$BOXG-%&STYLEG6#%&PATCHG-%%VIEWG6%F&F ,;$\"\"!FV$F " 0 "" {MPLTEXT 1 0 75 "evalf[15]( minimize(evalf[20](sm),b[7]=0..0.1,a[6,5]=0.01..0.1,'location'));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"06]E<*[2R!#C<#7$<$/&%\"bG6#\"\"($\" 0FYC\\-Fd(!#;/&%\"aG6$\"\"'\"\"&$\"0\"RC%ybv&))F0F#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use rational app roximations for the values of " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\" (" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"& " }{TEXT -1 54 " that mimimize the norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "[.757270249244627e-1,.88 5755578424391e-1];\nmap(convert,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"0FYC\\-Fd(!#;$\"0\"RC%ybv&))F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$#\"#M\"$\\%#\"#J\"$]$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We set " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7] = 34/449;" "6#/&%\"bG6#\"\"(*&\"#M\"\"\" \"$\\%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,5] = 31/350;" "6#/& %\"aG6$\"\"'\"\"&*&\"#J\"\"\"\"$]$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 121/366;" "6#/&%#b*G6#\"\"$*&\"$@\"\"\"\"\"$m$!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5] = -65/379;" "6#/&%#b*G6# \"\"&,$*&\"#l\"\"\"\"$z$!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The values for " } {XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" }{TEXT -1 179 " are chosen \+ so as to give a compatible stability region for the embedded 8 stage, \+ order 4 scheme and to give a suitable value for the principal error no rm of the embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e6 := \{b[7]=34/449,a[6,5]=31/350, `b*`[3]=121/366,`b*`[5]=-65/379\};\ne7 := `union`(e6,simplify(subs(e6, e5))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e6G<&/&%\"aG6$\"\"'\"\"&# \"#J\"$]$/&%\"bG6#\"\"(#\"#M\"$\\%/&%#b*G6#\"\"$#\"$@\"\"$m$/&F96#F+#! #l\"$z$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1761 " e7 := \{b[5] = -33901013357491/177576803988480, a[8,1] = 4827113239/65 475838080, `b*`[2] = 0, b[4] = 5703219796216031/30660055749940320, a[6 ,1] = 166741171875227/3452798773601100, a[7,5] = -11545948452091965679 62511465161858938950699/877446764516774644979830270188996722688000, c[ 6] = 97/129, c[4] = 174/349, c[5] = 36/49, c[2] = 11/63, b[3] = 797791 005101/2404074449920, a[7,3] = 341104813140161009375674013573715202837 301/313319409731603971335732895410416320512000, a[7,4] = 9534194337664 0595142991543684079745176400281/13184066517093990348941691553551867495 56736000, a[7,1] = 6022012446846011204537694339419654322731/3009986400 7383356992178712977537163264000, c[8] = 1, c[7] = 1, b[2] = 0, a[8,2] \+ = 0, `b*`[6] = 952822237714091128053/1892592848332731588736, `b*`[4] = 3468482659797555651329/18566189507645514457800, `b*`[1] = 66873629126 427899/902163023828283336, a[2,1] = 11/63, a[6,5] = 31/350, b[7] = 34/ 449, a[6,4] = 89559834743712383/245148712925678100, a[6,3] = -18838284 5675879/1408900648998150, a[7,2] = -6561/8296, a[8,5] = -3390101335749 1/177576803988480, b[6] = 29204563953141/55776754401170, a[8,6] = 2920 4563953141/55776754401170, a[7,6] = 1085508425299375/621834120640512, \+ a[8,3] = 797791005101/2404074449920, b[1] = 4827113239/65475838080, a[ 8,4] = 5703219796216031/30660055749940320, a[8,7] = 34/449, a[5,2] = 1 2241368/78942479, `b*`[7] = 129752050156606597/1891022063549226300, `b *`[8] = 56866223980208550227/7201012017995453750400, a[4,2] = -4119958 08/467594039, a[5,3] = -4360301712/39234412063, a[4,3] = 100493613/850 17098, a[5,4] = 613806882624/1137797949827, c[3] = 4/17, `b*`[3] = 121 /366, a[5,1] = 2424510108/16025323237, a[4,1] = 184816101/935188078, a [3,2] = 504/3179, a[6,2] = 1006084/2623731, a[3,1] = 244/3179, `b*`[5] = -65/379\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "subs(e7,matrix([[ c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1] ,a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a[5,4]],\n \+ [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n [c[7],a[7 ,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8],a[8,1],a[8, 2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n [`b`,b[1],b [2],b[3]],[``,b[4],b[5],b[6]],[``$3,b[7]],\n [`b*`,`b*`[1],`b*`[2 ],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``$2,`b*`[7],`b*`[8]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747&#\"#6\"#jF(%!GF+7&#\" \"%\"#<#\"$W#\"%zJ#\"$/&F2F+7&#\"$u\"\"$\\$#\"*,h\"[=\"*y!)=N*#!*3e*>T \"*RSfn%#\"*8O\\+\"\"))4<])7&#\"#O\"#\\#\"+3,^CC\",PKKDg\"#\")o8C7\")z C%*y#!+7%RVpPX?6g%oW7?-' \"J+SEjr`xHry@*pN$Q2S')*4I#!%hl\"%'H)#\"K,t$G?:Pd8Snv$45;SJ\"[5T$\"K+? ^?jT5a*GtN8(RgJ(4%>LJ7&F+#\"M\"G+kM&*\"O+gtc&\\n=b`: pT*[.*R44_%[fa6\"K+!)oAn**)=q-$)z\\kun^knWx) #\"1v$*HD%3b3\"\"070k?T$=i7&Fbo#\"+RK6F[\",!3QeZl\"\"!#\"-,^+\"z(z\".? *\\W2/C7&F+#\"1Jg@'z>Kq&\"2?.%*\\d0g1$#!/\"\\dL,,R$\"0![))R!odx\"#\"/T J&Rc/#H\"/q6Sanxb7&F+F+F+#\"#M\"$\\%7&%\"bGFgpFjpF[qF^qFhq7&%#b*G#\"2* *yUE\"HO(o'\"3OLGGQ-j@!*Fjp#\"$@\"\"$m$7&F+#\"7H8lbvzfE[oM\"8+yX9bk2&* =m&=#!#l\"$z$#\"6`!G649xBAG&*\"7O()eJFL[Gf#*=7&F+F+#\"3(f1m:]?vH\"\"4+ jA\\N1A5*=#\"5F-b3-)RAmo&\"7+/v`a*z,75?(Q)pprint886\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i- 1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1 ..8)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7+7+$\")<.YF*$!)A(4\"))F*$\")+/#=\"!\"(F+F+F+ F+F+7+$\"))QpM(F*$\")V#H^\"F*$\")$p1b\"F*$!)jM66F*$\")9p%R&F*F+F+F+F+7 +$\"))z$>vF*$\"))f\"H[F1$\")ZbMQF*$!)64P8F*$\")fG`OF*$\")H9d))F1F+F+F+ 7+$\"\"\"\"\"!$\")wn+?F*$!)2j3zF*$\")3o)3\"F=$\")OgJsF1$!)u&eJ\"F=$\") flXF*$\")T(fB& F*$\")JQsvF1F+7+%\"bGFaoFcoFdoFfoFhoFjoF\\pF+7+%#b*G$\")\")e7uF1Fco$\" )4,1LF*$\"): " 0 "" {MPLTEXT 1 0 83 "expand(subs(e7 ,StabilityFunction(5,7,'expanded'))): R := unapply(%,z):\n'R(z)'=R(z); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F) \"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F )F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"%vz\") K#)eQF)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 124 "We can find the point where the boundary of the s tability region intersects the negative real axis by solving the equat ion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#z0G$!+s,8&*Q!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "We can determine the weights " } {XPPEDIT 18 0 "`b^`[i], i=1" "6$&%#b^G6#%\"iG/F&\"\"\"" }{TEXT -1 26 " . . 7, ( initially with " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"' " }{TEXT -1 136 " as a parameter) for an auxiliary 7 stage, order 4 e mbedded scheme by using the quadrature conditions along with the requi rements that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b^` [7]=b[7]" "6#/&%#b^G6#\"\"(&%\"bG6#F'" }{TEXT -1 7 " qnd " } {XPPEDIT 18 0 "`b^`[2]=0" "6#/&%#b^G6#\"\"#\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "quad_eqns := subs(b=`b^`,QuadratureConditions(4,7,'expanded')):\n e8 := \{`b^`[2]=0\}:\ne9 := `union`(e7,e8):\ncdns2 := [`b^`[7]=b[7],op (quad_eqns)]:\neqns2 := subs(e9,cdns2):\nnops(eqns2);\nindets(eqns2); \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<(&%#b^G6#\"\"\"&F%6#\"\"%&F%6#\"\"$&F%6#\"\"'&F%6#\" \"(&F%6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 := \+ solve(\{op(eqns2)\},\{`b^`[1],`b^`[3],`b^`[4],`b^`[5],`b^`[7]\}):\ne11 := `union`(e9,e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The coefficients are expressed in terms of " } {XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e11));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<#&%#b^G6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2073 "e11 := \{b[5] = -33901013357491/1775768 03988480, a[8,1] = 4827113239/65475838080, `b*`[2] = 0, b[4] = 5703219 796216031/30660055749940320, a[6,1] = 166741171875227/3452798773601100 , a[7,5] = -1154594845209196567962511465161858938950699/87744676451677 4644979830270188996722688000, c[6] = 97/129, c[4] = 174/349, c[5] = 36 /49, c[2] = 11/63, b[3] = 797791005101/2404074449920, `b^`[2] = 0, a[7 ,3] = 341104813140161009375674013573715202837301/313319409731603971335 732895410416320512000, a[7,4] = 95341943376640595142991543684079745176 400281/1318406651709399034894169155355186749556736000, a[7,1] = 602201 2446846011204537694339419654322731/30099864007383356992178712977537163 264000, c[8] = 1, c[7] = 1, b[2] = 0, a[8,2] = 0, `b*`[6] = 9528222377 14091128053/1892592848332731588736, `b*`[4] = 3468482659797555651329/1 8566189507645514457800, `b*`[1] = 66873629126427899/902163023828283336 , a[2,1] = 11/63, a[6,5] = 31/350, b[7] = 34/449, a[6,4] = 89559834743 712383/245148712925678100, a[6,3] = -188382845675879/1408900648998150, a[7,2] = -6561/8296, a[8,5] = -33901013357491/177576803988480, b[6] = 29204563953141/55776754401170, a[8,6] = 29204563953141/55776754401170 , a[7,6] = 1085508425299375/621834120640512, a[8,3] = 797791005101/240 4074449920, b[1] = 4827113239/65475838080, a[8,4] = 5703219796216031/3 0660055749940320, a[8,7] = 34/449, a[5,2] = 12241368/78942479, `b^`[7] = 34/449, `b*`[7] = 129752050156606597/1891022063549226300, `b*`[8] = 56866223980208550227/7201012017995453750400, a[4,2] = -411995808/4675 94039, a[5,3] = -4360301712/39234412063, a[4,3] = 100493613/85017098, \+ a[5,4] = 613806882624/1137797949827, `b^`[4] = 430739127017/5913221938 272+46292617523431/214177102136856*`b^`[6], c[3] = 4/17, `b^`[5] = 131 624642359/325828998144-147489575537843/129816912805632*`b^`[6], `b*`[3 ] = 121/366, `b^`[1] = 1408730279/53787439584*`b^`[6]+8101481/13500172 8, `b^`[3] = 1808676733/4668105728-53867119513/507236850432*`b^`[6], a [5,1] = 2424510108/16025323237, a[4,1] = 184816101/935188078, a[3,2] = 504/3179, a[6,2] = 1006084/2623731, a[3,1] = 244/3179, `b*`[5] = -65/ 379\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "We choose " }{XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 88 " so that the principal error norms of the two embedded schemes are approximately equal." }}{PARA 0 "" 0 "" {TEXT -1 66 "First we obtain the principal error of the 8 stage order \+ 4 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expanded')):\nprin_err4_8 := \+ sqrt(add(subs(e7,`errterms4_8*`[i])^2,i=1.. nops(`errterms4_8*`))):\ne valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+du[l%*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The principal err or norm of the 7 stage order 4 scheme is expressed in terms of " } {XPPEDIT 18 0 "`b^`[6]" "6#&%#b^G6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "`errterms4_7^` := subs(b=`b^`,Prin cipalErrorTerms(4,7,'expanded')):\nprin_err4_7 := sqrt(add(subs(e11,`e rrterms4_7^`[i])^2,i=1.. nops(`errterms4_7^`)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,prin_err4_7G,$*&#\"\"\"\"M+g#\\p5&e2v#>p6^5)*Q!\\7si jF(*&\"Y)>d-.GF#)RLIDuOa8?%ocbi`&[iB " 0 "" {MPLTEXT 1 0 44 "prin_err4_7=prin_err4 _8:\nsolve(%):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+A\\?A a!#5$\"+5Ku\\]F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "We choose a rational approximation for the larger of the two values since this value gives a larger real stability interval th an the other value." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 ".5422 204922;\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +A\\?Aa!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#X\"#$)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " } {XPPEDIT 18 0 "`b^`[6] = 45/83;" "6#/&%#b^G6#\"\"'*&\"#X\"\"\"\"#$)!\" \"" }{TEXT -1 89 " which gives the following principal error norm for the 7 stage order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eval(prin_err4_7,`b^`[6] =45/83):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+x.:R%*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "e12 := `union`(\{`b^`[6]=45/83\},subs(`b^`[6]=45/83,e11)):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1996 "e12 := \{b [5] = -33901013357491/177576803988480, a[8,1] = 4827113239/65475838080 , `b*`[2] = 0, b[4] = 5703219796216031/30660055749940320, a[6,1] = 166 741171875227/3452798773601100, a[7,5] = -11545948452091965679625114651 61858938950699/877446764516774644979830270188996722688000, c[6] = 97/1 29, c[4] = 174/349, c[5] = 36/49, c[2] = 11/63, b[3] = 797791005101/24 04074449920, `b^`[2] = 0, a[7,3] = 34110481314016100937567401357371520 2837301/313319409731603971335732895410416320512000, a[7,4] = 953419433 76640595142991543684079745176400281/1318406651709399034894169155355186 749556736000, a[7,1] = 6022012446846011204537694339419654322731/300998 64007383356992178712977537163264000, c[8] = 1, c[7] = 1, b[2] = 0, a[8 ,2] = 0, `b*`[6] = 952822237714091128053/1892592848332731588736, `b*`[ 4] = 3468482659797555651329/18566189507645514457800, `b*`[1] = 6687362 9126427899/902163023828283336, a[2,1] = 11/63, a[6,5] = 31/350, b[7] = 34/449, a[6,4] = 89559834743712383/245148712925678100, a[6,3] = -1883 82845675879/1408900648998150, a[7,2] = -6561/8296, a[8,5] = -339010133 57491/177576803988480, b[6] = 29204563953141/55776754401170, a[8,6] = \+ 29204563953141/55776754401170, a[7,6] = 1085508425299375/6218341206405 12, a[8,3] = 797791005101/2404074449920, b[1] = 4827113239/65475838080 , a[8,4] = 5703219796216031/30660055749940320, a[8,7] = 34/449, `b^`[6 ] = 45/83, `b^`[3] = 30485818726152979/92415623446907904, `b^`[1] = 73 45858582709/98987482023552, `b^`[5] = -5627835948231581/26545332727174 144, `b^`[4] = 2471755716006328519/13007276847211309344, a[5,2] = 1224 1368/78942479, `b^`[7] = 34/449, `b*`[7] = 129752050156606597/18910220 63549226300, `b*`[8] = 56866223980208550227/7201012017995453750400, a[ 4,2] = -411995808/467594039, a[5,3] = -4360301712/39234412063, a[4,3] \+ = 100493613/85017098, a[5,4] = 613806882624/1137797949827, c[3] = 4/17 , `b*`[3] = 121/366, a[5,1] = 2424510108/16025323237, a[4,1] = 1848161 01/935188078, a[3,2] = 504/3179, a[6,2] = 1006084/2623731, a[3,1] = 24 4/3179, `b*`[5] = -65/379\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The coefficients fo r the whole scheme including the 7 stage and 8 stage embedded order 4 \+ schemes are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 578 " subs(e12,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1],a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]], [``$3,a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]] ,\n [c[7],a[7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n \+ [c[8],a[8,1],a[8,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]], \n [`b`,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],[``$3,b[7]],\n [ `b^`,`b^`[1],`b^`[2],`b^`[3]],[``,`b^`[4],`b^`[5],`b^`[6]],[``$3,`b^`[ 7]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6] ],[``$2,`b*`[7],`b*`[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#777&#\"#6\"#jF(%!GF+7&#\"\"%\"#<#\"$W#\"%zJ#\"$/&F2F+7&#\"$u\"\" $\\$#\"*,h\"[=\"*y!)=N*#!*3e*>T\"*RSfn%#\"*8O\\+\"\"))4<])7&#\"#O\"#\\ #\"+3,^CC\",PKKDg\"#\")o8C7\")zC%*y#!+7%RVpPX?6g%oW7?-'\"J+SEjr`xHry@*pN$Q2S')*4I#!%hl\"%'H)#\" K,t$G?:Pd8Snv$45;SJ\"[5T$\"K+?^?jT5a*GtN8(RgJ(4%>LJ7&F+#\"M\"G+kM&*\"O+gtc&\\n=b`:pT*[.*R44_%[fa 6\"K+!)oAn**)=q-$)z\\kun^knWx)#\"1v$*HD%3b3\"\"070k?T$=i7&Fbo#\"+RK6F[ \",!3QeZl\"\"!#\"-,^+\"z(z\".?*\\W2/C7&F+#\"1Jg@'z>Kq&\"2?.%*\\d0g1$#! /\"\\dL,,R$\"0![))R!odx\"#\"/TJ&Rc/#H\"/q6Sanxb7&F+F+F+#\"#M\"$\\%7&% \"bGFgpFjpF[qF^qFhq7&%#b^G#\".4FeeeM(\"/_N-#[()*)*Fjp#\"2zH:E(=e[I\"2/ z!pWBcT#*7&F+#\"4>&Gj+;dvrC\"5W$48@ZoF2I\"#!1\"eJ#[f$yi&\"2WT " 0 "" {MPLTEXT 1 0 166 "subs(e12,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b^`,seq(`b^`[i],i=1..7),`` ],[`b*`,seq(`b*`[i],i=1..8)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7+$\")<.YF*$!)A(4 \"))F*$\")+/#=\"!\"(F+F+F+F+F+7+$\"))QpM(F*$\")V#H^\"F*$\")$p1b\"F*$!) jM66F*$\")9p%R&F*F+F+F+F+7+$\"))z$>vF*$\"))f\"H[F1$\")ZbMQF*$!)64P8F*$ \")fG`OF*$\")H9d))F1F+F+F+7+$\"\"\"\"\"!$\")wn+?F*$!)2j3zF*$\")3o)3\"F =$\")OgJsF1$!)u&eJ\"F=$\")flXF*$\")T(fB&F*$\")JQsvF1F+7+%\"bGFaoFcoFdoFfoFhoFjoF\\p F+7+%#b^G$\")v*4U(F1Fco$\")Lx)H$F*$\")pG+>F*$!)\\3?@F*$\")no@aF*F\\pF+ 7+%#b*G$\")\")e7uF1Fco$\")4,1LF*$\"): " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 26 "absolute stability regions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------- --------------------------------------------" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1885 "ee := \{c[ 2]=11/63,\nc[3]=4/17,\nc[4]=174/349,\nc[5]=36/49,\nc[6]=97/129,\nc[7]= 1,\nc[8]=1,\n\na[2,1]=11/63,\na[3,1]=244/3179,\na[3,2]=504/3179,\na[4, 1]=184816101/935188078,\na[4,2]=-411995808/467594039,\na[4,3]=10049361 3/85017098,\na[5,1]=2424510108/16025323237,\na[5,2]=12241368/78942479, \na[5,3]=-4360301712/39234412063,\na[5,4]=613806882624/1137797949827, \na[6,1]=166741171875227/3452798773601100,\na[6,2]=1006084/2623731,\na [6,3]=-188382845675879/1408900648998150,\na[6,4]=89559834743712383/245 148712925678100,\na[6,5]=31/350,\na[7,1]=60220124468460112045376943394 19654322731/30099864007383356992178712977537163264000,\na[7,2]=-6561/8 296,\na[7,3]=341104813140161009375674013573715202837301/31331940973160 3971335732895410416320512000,\na[7,4]=95341943376640595142991543684079 745176400281/1318406651709399034894169155355186749556736000,\na[7,5]=- 1154594845209196567962511465161858938950699/87744676451677464497983027 0188996722688000,\na[7,6]=1085508425299375/621834120640512,\na[8,1]=48 27113239/65475838080,\na[8,2]=0,\na[8,3]=797791005101/2404074449920,\n a[8,4]=5703219796216031/30660055749940320,\na[8,5]=-33901013357491/177 576803988480,\na[8,6]=29204563953141/55776754401170,\na[8,7]=34/449,\n \nb[1]=4827113239/65475838080,\nb[2]=0,\nb[3]=797791005101/24040744499 20,\nb[4]=5703219796216031/30660055749940320,\nb[5]=-33901013357491/17 7576803988480,\nb[6]=29204563953141/55776754401170,\nb[7]=34/449,\n\n` b^`[1]=7345858582709/98987482023552,\n`b^`[2]=0,\n`b^`[3]=304858187261 52979/92415623446907904,\n`b^`[4]=2471755716006328519/1300727684721130 9344,\n`b^`[5]=-5627835948231581/26545332727174144,\n`b^`[6]=45/83,\n` b^`[7]=34/449,\n\n`b*`[1]=66873629126427899/902163023828283336,\n`b*`[ 2]=0,\n`b*`[3]=121/366,\n`b*`[4]=3468482659797555651329/18566189507645 514457800,\n`b*`[5]=-65/379,\n`b*`[6]=952822237714091128053/1892592848 332731588736,\n`b*`[7]=129752050156606597/1891022063549226300,\n`b*`[8 ]=56866223980208550227/7201012017995453750400\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stability regi on for the 7 stage, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "The stability function R for the 7 stage, order 5 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction( 5,7,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F) *&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$ )F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"%vz\")K#)eQF)*$)F'\"\"(F)F )F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 " We can find the point where the boundary of the stability region inter sects the negative real axis by solving the equation: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$ \"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# z0G$!+s,8&*Q!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z ),-1],z=-5.19..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],style =point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0] ,[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2 ,p3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 " " 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7W7$$!3Q++++++ !>&!#<$!3Ma8sH9J*)*)F*7$$!3Ynm;#3'4G^F*$!3W'fe!*ow!4#)F*7$$!3kLLLk@>m] F*$!3aZk]NJ'z[(F*7$$!3E+]U'*)HB,&F*$!3m&**H&3Rk0pF*7$$!3!pm;&GwYe\\F*$ !3SaZS92zijF*7$$!3s+](\\(Q*y*[F*$!3?)G3$)=4mz&F*7$$!3nLLV@,KP[F*$!3_lC brdPu_F*7$$!3'RLLd%[MwZF*$!3/GRYCT6!z%F*7$$!3NLL.q&p`r%F*$!3w!>[\"=ghW VF*7$$!3;nmJG')*Rf%F*$!3#Gtht7u9T#F*7$$!3$QL$)f7eWC%F*$!3-jR5/lrX>F*7$$! 3A++lN]MCTF*$!35`(GFV\\)f:F*7$$!3ummYeRz+SF*$!3]K\"z%GOTK7F*7$$!3_LLV- ,(>*QF*$!3D8ld5baO**!#=7$$!35++S:-YpPF*$!3];)*zm!=!>xFbp7$$!3K+++\"HZk k$F*$!3%H&z`L*HQ\"fFbp7$$!3;++gW:!z_$F*$!3hw@b'4Wi]%Fbp7$$!3hLL)*\\1D? MF*$!39.y58jriMFbp7$$!3'ommSKVAH$F*$!3))[JxKz5hCFbp7$$!3/nmEGV!Q=$F*$! 3)>CGGk$Rz7$$!3-++0xX]BGF*$!3a:3B]!>-#GF[s7$$!3*)***\\\"R>&oq#F*$\"3< 3.u?RO=L!#?7$$!3gmm;\\r8&e#F*$\"3!fLDdpG*yIF[s7$$!3ymmrw\\OtCF*$\"3%*) yn%)4:8D&F[s7$$!3SLL$))e.GN#F*$\"3MyhxI#oGM(F[s7$$!3nLL)**=uvA#F*$\"3% R=U)y7-a$*F[s7$$!3K++:I;c=@F*$\"3EUGa8G206Fbp7$$!31LL.z]#3+#F*$\"3Y!=( 4h^q*G\"Fbp7$$!3M++?,<>z=F*$\"3k;$)\\5b\\)[\"Fbp7$$!3;++!4<(>gFbp7$$!3H++q9zA<:F*$\"3)\\p1 gM:d=#Fbp7$$!3EnmEY;O-9F*$\"3'[\"o'4XogX#Fbp7$$!3#)*****pQ<(z7F*$\"3]g **=Ga8zFFbp7$$!3)RL$efMeo6F*$\"3I3N(o'412JFbp7$$!3I****fAZ3Z5F*$\"3:%p 4>Vf\"4NFbp7$$!3xqm;(zQwK*Fbp$\"3cKYm\"znW$RFbp7$$!3&z***\\)ecE8)Fbp$ \"3x^0l&)Q,MWFbp7$$!3'3nmm0VV'pFbp$\"3_&*>Kk?d$)\\Fbp7$$!3P)***\\iqATd Fbp$\"3[[G6!)H)>j&Fbp7$$!3aFLL*)4AjXFbp$\"3w_[F3k4OjFbp7$$!33LLLO'R&eL Fbp$\"3ow%)[lYFZrFbp7$$!3Uim;`O$Q;#Fbp$\"39c-1)pkU0)Fbp7$$!3?*****>$H- m5Fbp$\"3kgOGQ4$)))*)Fbp7$$\"3v*QLLU?>#>F[s$\"3[-s.#30%>5F*7$$\"3%ymmY ^avJ\"Fbp$\"3S?M-(HH39\"F*7$$\"3E0+]HcU " 0 "" {MPLTEXT 1 0 1295 "R := z -> 1+z+1/2*z ^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+7975/38588232*z^7:\npts := []: \nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton(R(z)=exp(ct*Pi/20 *I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np1 := plot(pts,color=COLOR(RGB,.3,.15,.45)):\np2 := plots[polygonpl ot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=p atchnogrid,color=COLOR(RGB,.6,.3,.9)):\npts := []:\nz0 := 1.54+4.3*I: \nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := p lot(pts,color=COLOR(RGB,.3,.15,.45)):\np4 := plots[polygonplot]([seq([ pts[i-1],pts[i],[1.43,4.21]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.6,.3,.9)):\npts := []:\nz0 := 1.54-4.3*I:\nfor \+ ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts ,color=COLOR(RGB,.3,.15,.45)):\np6 := plots[polygonplot]([seq([pts[i-1 ],pts[i],[1.43,-4.21]],i=2..nops(pts))],\n style=patchnogrid, color=COLOR(RGB,.6,.3,.9)):\np7 := plot([[[-4.79,0],[2.19,0]],[[0,-4.7 9],[0,4.79]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],vi ew=[-4.79..2.19,-4.79..4.79],font=[HELVETICA,9],\n labels =[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 606 615 615 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$F( $\"3++++Fjzq:!#=7$$\"3:+++%fK;.$!#E$\"3s*****4n#fTJF-7$$\"32+++;%z!Qu! #D$\"3C+++:\"*Q7ZF-7$$\"3S+++@b2vp!#C$\"3')*****Hq%=$G'F-7$$\"3l***** \\,%G=Q!#B$\"3)******4QpR&yF-7$$\"3++++k&3%p9!#A$\"3E+++YdpC%*F-7$$\"3 A+++uwkrVFI$\"3#*******G5_*4\"!#<7$$\"3%*******4d9d5!#@$\"3-+++b@^c7FQ 7$$\"37+++Y]o?@FU$\"30+++y!fLT\"FQ7$$\"3y*****fqd\"*[$FU$\"3))*****f_0 *p:FQ7$$\"34+++YzAVVFU$\"3.+++dR*es\"FQ7$$\"3B+++gW*4^#FU$\"3++++nk%4) =FQ7$$!3')*****z0xqP'FU$\"3)******H!)oX.#FQ7$$!3')*****f$*='oH!#?$\"3' )******[F='=#FQ7$$!3%3++g)Rj?yF^p$\"3.+++X-$\"3))*****\\K()3[#FQ7$$!3#******\\w!)44$Fip$\"3')*****fh,Ei#FQ7$$!3? +++Xc7U_Fip$\"3%)*******HV$fFFQ7$$!3-+++>/D)Fip$\"3!******phQ(*)GFQ7 $$!3,+++&>T%=7F-$\"36+++a5*>,$FQ7$$!3))******oUq+o#=CJF Q7$$!3)******zgcuD#F-$\"36+++f)Q[A$FQ7$$!3/+++*y$=oGF-$\"3\")*****RB!Q 8LFQ7$$!3)******pipC^$F-$\"31+++7J9!R$FQ7$$!3#******HwjS<%F-$\"3%)**** **>y0cMFQ7$$!3!*******4M$=%[F-$\"3)******>`zA^$FQ7$$!3]+++We!*3bF-$\"3 5+++\"zU*fNFQ7$$!3i******e$p8<'F-$\"3:+++v)[+g$FQ7$$!3O+++ycvR![()F-$\"33+++-a[*p$FQ7$$!3Y*****4Y4FP*F-$ \"3:+++j#G;r$FQ7$$!3S+++e9&**)**F-$\"3*******>#)p$>PFQ7$$!31+++Y(3+1\" FQ$\"3z*****\\-PHs$FQ7$$!3,+++jhM?6FQ$\"3!*******3,^APFQ7$$!3)******f) Q/!=\"FQ$\"3,+++#yI#=PFQ7$$!3.+++899R7FQ$\"3$*******3'3-r$FQ7$$!3%**** **H+$o(H\"FQ$\"33+++!4E&)p$FQ7$$!35+++F,sb8FQ$\"31+++mGC$o$FQ7$$!32+++ *e9LT\"FQ$\"37+++U%*RkOFQ7$$!3%******ROU0Z\"FQ$\"3;+++r3-UOFQ7$$!3$*** ***R=)\\F:FQ$\"3z*****H)37;OFQ7$$!3#*******f3I%e\"FQ$\"3)******f52ne$F Q7$$!33+++x\"*4T;FQ$\"39+++,#)y`NFQ7$$!35+++&zx!)p\"FQ$\"3%*******oRQ< NFQ7$$!3%******H*>YbcXVV$FQ7$$!3/+++^FQ$\"3&) *****Hm['QLFQ7$$!3)******zm2[*>FQ$\"3%******4D)y'G$FQ7$$!3'*******zybe ?FQ$\"3,+++wy*GB$FQ7$$!36+++RR@C@FQ$\"3y*****zd1w<$FQ7$$!3=+++Zsl\">#F Q$\"32+++sFf@JFQ7$$!3?+++z(\\0E#FQ$\"3/+++3'*[lIFQ7$$!32+++g=PIBFQ$\"3 5+++_(*y4IFQ7$$!30+++-]]+CFQ$\"39+++tjyaHFQ7$$!3#******\\;E.Z#FQ$\"36+ ++kyc+HFQ7$$!3=+++y2GRDFQ$\"3))*****p,cq%GFQ7$$!3$)******f0#pg#FQ$\"3' )*****R&G1%z#FQ7$$!35+++[/\"Hn#FQ$\"3!******HYW8u#FQ7$$!39+++dn,PFFQ$ \"35+++57k)o#FQ7$$!3/+++:y3*z#FQ$\"38+++!*GqNEFQ7$$!3))*****f:M!fGFQ$ \"33+++j=I#e#FQ7$$!3$******Hd6o\"HFQ$\"3)******p=#FQ7$$!3!)*****>v!phKF Q$\"32+++w$R(>@FQ7$$!30+++Q`)HI$FQ$\"31+++'3Ng0#FQ7$$!33+++(fzCM$FQ$\" 3#******R:W3*>FQ7$$!3%)*****z*\\D!Q$FQ$\"3$******4#*\\T#>FQ7$$!3y***** pg*R;MFQ$\"3-+++c,&f&=FQ7$$!3!******RR25X$FQ$\"3!******H\")eiy\"FQ7$$! 3%******RvwT[$FQ$\"3)******f<3^r\"FQ7$$!3A+++*)z+;NFQ$\"34+++=6bU;FQ7$ $!39+++-'*fYNFQ$\"3!******>.j'o:FQ7$$!3!)*****\\PVgd$FQ$\"35+++$*Ga$\\ \"FQ7$$!3\"*******4&=Wg$FQ$\"32+++UGJ<9FQ7$$!34+++z`yJOFQ$\"3*)*****\\ x:,M\"FQ7$$!3-+++S(z\"eOFQ$\"3!*******\\16i7FQ7$$!3!******HU3Oo$FQ$\"3 '******fnlM=\"FQ7$$!3()*****HPZ!3PFQ$\"3(******f/]V5\"FQ7$$!3-+++qDWJP FQ$\"3!******zDD\\-\"FQ7$$!33+++\"y8Pv$FQ$\"3a+++_EP`%*F-7$$!3/+++D0wu PFQ$\"3;+++ZY5d')F-7$$!3++++a#pWz$FQ$\"3w*****feT9'yF-7$$!3))*****\\p? F\"QFQ$\"3/+++I`6nqF-7$$!3:+++ZkRHQFQ$\"3M+++lyguiF-7$$!3.+++)H%QWQFQ$ \"3-+++dw<%[&F-7$$!3/+++c?edQFQ$\"3A+++&=%*ep%F-7$$!3/+++&****)oQFQ$\" 3B+++XZn4RF-7$$!35+++`@EyQFQ$\"3,+++K$>`7$F-7$$!37+++;og&)QFQ$\"3\"*** ***4?TDM#F-7$$!3A+++&[')3*QFQ$\"3)******>A#*4c\"F-7$$!3-+++%fnS*QFQ$\" 3V+++d>$G!yFip7$$!3++++s,8&*QFQF(7$Fgjl$!3V+++d>$G!yFip7$Fbjl$!3)***** *>A#*4c\"F-7$F]jl$!3\"******4?TDM#F-7$Fhil$!3,+++K$>`7$F-7$Fcil$!3B+++ XZn4RF-7$F^il$!3A+++&=%*ep%F-7$Fihl$!3-+++dw<%[&F-7$Fdhl$!3M+++lyguiF- 7$F_hl$!3/+++I`6nqF-7$Fjgl$!3w*****feT9'yF-7$Fegl$!3;+++ZY5d')F-7$F`gl $!3a+++_EP`%*F-7$F[gl$!3!******zDD\\-\"FQ7$Fffl$!3(******f/]V5\"FQ7$Fa fl$!3'******fnlM=\"FQ7$F\\fl$!3!*******\\16i7FQ7$Fgel$!3*)*****\\x:,M \"FQ7$Fbel$!32+++UGJ<9FQ7$F]el$!35+++$*Ga$\\\"FQ7$Fhdl$!3!******>.j'o: FQ7$Fcdl$!34+++=6bU;FQ7$F^dl$!3)******f<3^r\"FQ7$Ficl$!3!******H\")eiy \"FQ7$Fdcl$!3-+++c,&f&=FQ7$F_cl$!3$******4#*\\T#>FQ7$Fjbl$!3#******R:W 3*>FQ7$Febl$!31+++'3Ng0#FQ7$F`bl$!32+++w$R(>@FQ7$F[bl$!3'******4$p)>=# FQ7$Ffal$!3)******\\P?GC#FQ7$Faal$!3++++J_H-BFQ7$F\\al$!3=+++(y![gBFQ7 $Fg`l$!3,+++i;YcXVV$FQ7$Fcz$!3#)*****flRvZ$FQ7$F^z$!3%*******oRQ#)p$>PFQ7$F]v$!3:+++j#G;r$FQ 7$Fhu$!33+++-a[*p$FQ7$Fcu$!3%)*****H/dEo$FQ7$F^u$!3/+++3#)ygOFQ7$Fit$! 3@+++GgVLOFQ7$Fdt$!3:+++v)[+g$FQ7$F_t$!35+++\"zU*fNFQ7$Fjs$!3)******>` zA^$FQ7$Fes$!3%)******>y0cMFQ7$F`s$!31+++7J9!R$FQ7$F[s$!3\")*****RB!Q8 LFQ7$Ffr$!36+++f)Q[A$FQ7$Far$!3#)*****>o#=CJFQ7$F\\r$!36+++a5*>,$FQ7$F gq$!3!******phQ(*)GFQ7$Fbq$!3%)*******HV$fFFQ7$F]q$!3')*****fh,Ei#FQ7$ Fgp$!3))*****\\K()3[#FQ7$Fbp$!3.+++X-T%=7F `_n$\"+a5*>,$FjanFa_n7%Fgfn7$$!+pUq+PFja nFa_n7%F[\\o7$$!+Y(3+1\"Fjan$\"+Dq$Hs$FjanFa_n7%Fa\\o7$$!+jhM?6Fjan$\" +4,^APFjanFa_n7%Fg\\o7$$!+')Q/!=\"Fjan$\"+#yI#=PFjanFa_n7%F]]o7$$!+899 R7Fjan$\"+4'3-r$FjanFa_n7%Fc]o7$$!+.Io(H\"Fjan$\"+!4E&)p$FjanFa_n7%Fi] o7$$!+F,sb8Fjan$\"+mGC$o$FjanFa_n7%F_^o7$$!+*e9LT\"Fjan$\"+U%*RkOFjanF a_n7%Fe^o7$$!+kBaq9Fjan$\"+r3-UOFjanFa_n7%F[_o7$$!+%=)\\F:Fjan$\"+$)37 ;OFjanFa_n7%Fa_o7$$!+g3I%e\"Fjan$\"+1rq'e$FjanFa_n7%Fg_o7$$!+x\"*4T;Fj an$\"+,#)y`NFjanFa_n7%F]`o7$$!+&zx!)p\"Fjan$\"+pRQYbFjan$\"+j'['QLFj anFa_n7%F[bo7$$!+ow![*>Fjan$\"+^#)y'G$FjanFa_n7%Fabo7$$!+!)ybe?Fjan$\" +wy*GB$FjanFa_n7%Fgbo7$$!+RR@C@Fjan$\"+ylgxJFjanFa_n7%F]co7$$!+Zsl\"># Fjan$\"+sFf@JFjanFa_n7%Fcco7$$!+z(\\0E#Fjan$\"+3'*[lIFjanFa_n7%Fico7$$ !+g=PIBFjan$\"+_(*y4IFjanFa_n7%F_do7$$!+-]]+CFjan$\"+tjyaHFjanFa_n7%Fe do7$$!+lhKqCFjan$\"+kyc+HFjanFa_n7%F[eo7$$!+y2GRDFjan$\"+=#FjanFa_n7%Fiio7$$!+_2phKFjan$\"+ w$R(>@FjanFa_n7%F_jo7$$!+Q`)HI$Fjan$\"+'3Ng0#FjanFa_n7%Fejo7$$!+(fzCM$ Fjan$\"+aT%3*>FjanFa_n7%F[[p7$$!+)*\\D!Q$Fjan$\"+@*\\T#>FjanFa_n7%Fa[p 7$$!+2'*R;MFjan$\"+c,&f&=FjanFa_n7%Fg[p7$$!+%R25X$Fjan$\"+8)eiy\"FjanF a_n7%F]\\p7$$!+an<%[$Fjan$\"+w\"3^r\"FjanFa_n7%Fc\\p7$$!+*)z+;NFjan$\" +=6bU;FjanFa_n7%Fi\\p7$$!+-'*fYNFjan$\"+KImo:FjanFa_n7%F_]p7$$!+vL/wNF jan$\"+$*Ga$\\\"FjanFa_n7%Fe]p7$$!+5&=Wg$Fjan$\"+UGJ<9FjanFa_n7%F[^p7$ $!+z`yJOFjan$\"+vd6S8FjanFa_n7%Fa^p7$$!+S(z\"eOFjan$\"+]16i7FjanFa_n7% Fg^p7$$!+B%3Oo$Fjan$\"+wcY$=\"FjanFa_n7%F]_p7$$!+tt/3PFjan$\"+Y+N/6Fja nFa_n7%Fc_p7$$!+qDWJPFjan$\"+e_#\\-\"FjanFa_n7%Fi_p7$$!+\"y8Pv$Fjan$\" +_EP`%*F`_nFa_n7%F_`p7$$!+D0wuPFjan$\"+ZY5d')F`_nFa_n7%Fe`p7$$!+a#pWz$ Fjan$\"+'eT9'yF`_nFa_n7%F[ap7$$!+&p?F\"QFjan$\"+I`6nqF`_nFa_n7%Faap7$$ !+ZkRHQFjan$\"+lyguiF`_nFa_n7%Fgap7$$!+)H%QWQFjan$\"+dw<%[&F`_nFa_n7%F ]bp7$$!+c?edQFjan$\"+&=%*ep%F`_nFa_n7%Fcbp7$$!+&****)oQFjan$\"+XZn4RF` _nFa_n7%Fibp7$$!+`@EyQFjan$\"+K$>`7$F`_nFa_n7%F_cp7$$!+;og&)QFjan$\"+, 7aUBF`_nFa_n7%Fecp7$$!+&[')3*QFjan$\"+AA*4c\"F`_nFa_n7%F[dp7$$!+%fnS*Q Fjan$\"+d>$G!yFaenFa_n7%Fadp7$$!+s,8&*QFjanF(Fa_n7%Fgdp7$Fbdp$!+d>$G!y FaenFa_n7%F[ep7$F\\dp$!+AA*4c\"F`_nFa_n7%F_ep7$Ffcp$!+,7aUBF`_nFa_n7%F cep7$F`cp$!+K$>`7$F`_nFa_n7%Fgep7$Fjbp$!+XZn4RF`_nFa_n7%F[fp7$Fdbp$!+& =%*ep%F`_nFa_n7%F_fp7$F^bp$!+dw<%[&F`_nFa_n7%Fcfp7$Fhap$!+lyguiF`_nFa_ n7%Fgfp7$Fbap$!+I`6nqF`_nFa_n7%F[gp7$F\\ap$!+'eT9'yF`_nFa_n7%F_gp7$Ff` p$!+ZY5d')F`_nFa_n7%Fcgp7$F``p$!+_EP`%*F`_nFa_n7%Fggp7$Fj_p$!+e_#\\-\" FjanFa_n7%F[hp7$Fd_p$!+Y+N/6FjanFa_n7%F_hp7$F^_p$!+wcY$=\"FjanFa_n7%Fc hp7$Fh^p$!+]16i7FjanFa_n7%Fghp7$Fb^p$!+vd6S8FjanFa_n7%F[ip7$F\\^p$!+UG J<9FjanFa_n7%F_ip7$Ff]p$!+$*Ga$\\\"FjanFa_n7%Fcip7$F`]p$!+KImo:FjanFa_ n7%Fgip7$Fj\\p$!+=6bU;FjanFa_n7%F[jp7$Fd\\p$!+w\"3^r\"FjanFa_n7%F_jp7$ F^\\p$!+8)eiy\"FjanFa_n7%Fcjp7$Fh[p$!+c,&f&=FjanFa_n7%Fgjp7$Fb[p$!+@* \\T#>FjanFa_n7%F[[q7$F\\[p$!+aT%3*>FjanFa_n7%F_[q7$Ffjo$!+'3Ng0#FjanFa _n7%Fc[q7$F`jo$!+w$R(>@FjanFa_n7%Fg[q7$Fjio$!+Jp)>=#FjanFa_n7%F[\\q7$F dio$!+v.#GC#FjanFa_n7%F_\\q7$F^io$!+J_H-BFjanFa_n7%Fc\\q7$Fhho$!+(y![g BFjanFa_n7%Fg\\q7$Fbho$!+i;YPFjanFa_n7%F_eq7$Ff[o$!+j#G;r$FjanFa_n7%Fceq7$F`[o$!+-a[*p$Fja nFa_n7%Fgeq7$Fjjn$!+Vql#o$FjanFa_n7%F[fq7$Fdjn$!+3#)ygOFjanFa_n7%F_fq7 $F^jn$!+GgVLOFjanFa_n7%Fcfq7$Fhin$!+v)[+g$FjanFa_n7%Fgfq7$Fbin$!+\"zU* fNFjanFa_n7%F[gq7$F\\in$!+K&zA^$FjanFa_n7%F_gq7$Ffhn$!+?y0cMFjanFa_n7% Fcgq7$F`hn$!+7J9!R$FjanFa_n7%Fggq7$Fjgn$!+M-Q8LFjanFa_n7%F[hq7$Fdgn$!+ f)Q[A$FjanFa_n7%F_hq7$F^gn$!+#o#=CJFjanFa_n7%Fchq7$Fhfn$!+a5*>,$FjanFa _n7%Fghq7$Fbfn$!+<'Q(*)GFjanFa_n7%F[iq7$F\\fn$!++LMfFFjanFa_n7%F_iq7$F fen$!+;;gAEFjanFa_n7%Fciq7$F_en$!+Dt)3[#FjanFa_n7%Fgiq7$Fidn$!+X-.?=VFQ7$ $\"3%******>IGq^\"FQ$\"32+++5I?DVFQ7$$\"3*******4\"4W2:FQ$\"3:+++ccXJV FQ7$$\"31+++I(>t\\\"FQ$\"3++++Fz\"pL%FQ7$$\"3++++E4r'[\"FQ$\"3@+++z%\\ :M%FQ7$$\"3!******RbicZ\"FQ$\"33+++Q(4`M%FQ7$$\"3/+++E]Ak9FQ$\"3E+++&p d\"[VFQ7$$\"3!******pl^CX\"FQ$\"31+++C<0]VFQ7$$\"3-+++,$*RS9FQ$\"3q*** **pV\\4N%FQ7$$\"31+++>\"H\"G9FQ$\"31+++uv!3N%FQ7$$\"3'******RZ2dT\"FQ$ \"39+++]>e\\VFQ7$$\"3.+++=q?.9FQ$\"35+++$\\FsM%FQ7$$\"3-+++\")zq!R\"FQ $\"3S+++/%)pVVFQ7$$\"3))*****>t*Hy8FQ$\"3l*****z][*QVFQ7$$\"3!*******R E3m8FQ$\"3r*****H(=$HL%FQ7$$\"3'*******)HqTN\"FQ$\"3J+++SQgDVFQ7$$\"36 +++T@pU8FQ$\"3B+++PE#pJ%FQ7$$\"3#*******okzJ8FQ$\"3:+++]=&oI%FQ7$$\"34 +++GQl@8FQ$\"3o*****>6k`H%FQ7$$\"31+++Y0Y78FQ$\"3/+++ClW#G%FQ7$$\"3\"* *****p)>W/8FQ$\"3-+++K%3\"oUFQ7$$\"35+++mY&yH\"FQ$\"3o*****>H#R_UFQ7$$ \"3/+++(o&)HH\"FQ$\"3t*****>G)QNUFQ7$$\"3-+++?q9!H\"FQ$\"3!)*******z_s @%FQ7$$\"3!*******Q6m*G\"FQ$\"3K+++v'H#)>%FQ7$$\"3'*******HS$=H\"FQ$\" 3K+++A.nyTFQ7$$\"3#******H08pH\"FQ$\"3A+++^_/fTFQ7$$\"35+++w@.08FQ$\"3 p*****4rN*RTFQ7$$\"3(******pIchJ\"FQ$\"36+++KZ*>7%FQ7$$\"31+++wb/I8FQ$ \"3I+++'zve5%FQ7$$\"3%*******p1EY8FQ$\"3)******RCW@4%FQ7$$\"3)******fP :UO\"FQ$\"3S+++R1?\"3%FQ7$$\"3#******pmfKQ\"FQ$\"39+++[ICtSFQ7$$\"3-++ +.Mw-9FQ$\"3g******>fFoSFQ7$$\"31+++Vj>uSFQ7$$\"3!*******)**f;\\\"FQ$\"3u*****4!* y13%FQ7$$\"3/+++QW)f]\"FQ$\"3G+++Qmf)3%FQ7$$\"3&******z!o\"*=:FQ$\"3++ ++/\")p(4%FQ7$$\"3.+++z1WI:FQ$\"3H+++fQw2TFQ7$$\"31+++8=cS:FQ$\"3f**** *R`+'=TFQ7$$\"33+++(4,$\\:FQ$\"3=+++s#Q+8%FQ7$$\"3)******H[*oc:FQ$\"3T +++ay#>9%FQ7$$\"3/+++;Wwi:FQ$\"3#)*****H(y8aTFQ7$$\"3#******fMnvc\"FQ$ \"3m*****R:_l;%FQ7$$\"33+++I?9r:FQ$\"3)******HYn!zTFQ7$$\"3-+++nL`t:FQ $\"3%)*****f_\"f\">%FQ7$$\"3%******Hk'yu:FQ$\"3g*****pLTS?%FQ7$$\"35++ +Vq%\\d\"FQ$\"3V+++:%zUb:FQ$\"3#)*****RX4NG%FQ7$$\"33++++:;\\:FQ$\"3?++ +Z.5$H%FQFj^rF]^n-Fj^n6jn7%7$$\"+qi9U:Fjan$\"+[v5-VFjan7$$\"+=TUM:Fjan $\"+\\\")[5VFjan7$$\"$V\"Ff^n$\"$@%Ff^n7%F^bs7$$\"+?s.E:Fjan$\"+?.?=VF janFcbs7%Fibs7$$\"+-$Gq^\"Fjan$\"+5I?DVFjanFcbs7%F_cs7$$\"+64W2:Fjan$ \"+ccXJVFjanFcbs7%Fecs7$$\"+I(>t\\\"Fjan$\"+Fz\"pL%FjanFcbs7%F[ds7$$\" +E4r'[\"Fjan$\"+z%\\:M%FjanFcbs7%Fads7$$\"+aDmv9Fjan$\"+Q(4`M%FjanFcbs 7%Fgds7$$\"+E]Ak9Fjan$\"+&pd\"[VFjanFcbs7%F]es7$$\"+d;X_9Fjan$\"+C<0]V FjanFcbs7%Fces7$$\"+,$*RS9Fjan$\"+P%\\4N%FjanFcbs7%Fies7$$\"+>\"H\"G9F jan$\"+uv!3N%FjanFcbs7%F_fs7$$\"+uuq:9Fjan$\"+]>e\\VFjanFcbs7%Fefs7$$ \"+=q?.9Fjan$\"+$\\FsM%FjanFcbs7%F[gs7$$\"+\")zq!R\"Fjan$\"+/%)pVVFjan Fcbs7%Fags7$$\"+K(*Hy8Fjan$\"+3&[*QVFjanFcbs7%Fggs7$$\"+SE3m8Fjan$\"+t =$HL%FjanFcbs7%F]hs7$$\"+*HqTN\"Fjan$\"+SQgDVFjanFcbs7%Fchs7$$\"+T@pU8 Fjan$\"+PE#pJ%FjanFcbs7%Fihs7$$\"+pkzJ8Fjan$\"+]=&oI%FjanFcbs7%F_is7$$ \"+GQl@8Fjan$\"+7TO&H%FjanFcbs7%Feis7$$\"+Y0Y78Fjan$\"+ClW#G%FjanFcbs7 %F[js7$$\"+()>W/8Fjan$\"+K%3\"oUFjanFcbs7%Fajs7$$\"+mY&yH\"Fjan$\"+#H# R_UFjanFcbs7%Fgjs7$$\"+(o&)HH\"Fjan$\"+#G)QNUFjanFcbs7%F][t7$$\"+?q9!H \"Fjan$\"++GD%FjanFcbs7%Fi[ t7$$\"+IS$=H\"Fjan$\"+A.nyTFjanFcbs7%F_\\t7$$\"+`I\"pH\"Fjan$\"+^_/fTF janFcbs7%Fe\\t7$$\"+w@.08Fjan$\"+6d$*RTFjanFcbs7%F[]t7$$\"+2j:;8Fjan$ \"+KZ*>7%FjanFcbs7%Fa]t7$$\"+wb/I8Fjan$\"+'zve5%FjanFcbs7%Fg]t7$$\"+q1 EY8Fjan$\"+WU9#4%FjanFcbs7%F]^t7$$\"+w`@k8Fjan$\"+R1?\"3%FjanFcbs7%Fc^ t7$$\"+n'fKQ\"Fjan$\"+[ICtSFjanFcbs7%Fi^t7$$\"+.Mw-9Fjan$\"+?fFoSFjanF cbs7%F__t7$$\"+VjF2f9Fjan$\"+e'4%pSFjanFcbs7%Fa`t7$$\"+\"y\"*fZ \"Fjan$\"+O>>uSFjanFcbs7%Fg`t7$$\"+***f;\\\"Fjan$\"+,*y13%FjanFcbs7%F] at7$$\"+QW)f]\"Fjan$\"+Qmf)3%FjanFcbs7%Fcat7$$\"+3o\"*=:Fjan$\"+/\")p( 4%FjanFcbs7%Fiat7$$\"+z1WI:Fjan$\"+fQw2TFjanFcbs7%F_bt7$$\"+8=cS:Fjan$ \"+M0g=TFjanFcbs7%Febt7$$\"+(4,$\\:Fjan$\"+s#Q+8%FjanFcbs7%F[ct7$$\"+$ [*oc:Fjan$\"+ay#>9%FjanFcbs7%Fact7$$\"+;Wwi:Fjan$\"+ty8aTFjanFcbs7%Fgc t7$$\"+Ytcn:Fjan$\"+a@bmTFjanFcbs7%F]dt7$$\"+I?9r:Fjan$\"+ju1zTFjanFcb s7%Fcdt7$$\"+nL`t:Fjan$\"+E:f\">%FjanFcbs7%Fidt7$$\"+Vmyu:Fjan$\"+P8// UFjanFcbs7%F_et7$$\"+Vq%\\d\"Fjan$\"+:Q!4c\"Fjan$\"+1#zLF%FjanFcbs7%Fcgt7$$\"+UzUb:Fjan$\"+a%4NG%Fja nFcbs7%Figt7$$\"++:;\\:Fjan$\"+Z.5$H%FjanFcbs7%F_htFiasFcbsF]^rFc^r-F$ 6$7in7$F[_r$!3R+++[v5-VFQ7$Fbas$!3?+++Z.5$H%FQ7$F]as$!3#)*****RX4NG%FQ 7$Fh`s$!3W+++1#zLF%FQ7$Fc`s$!3%******pubFE%FQ7$F^`s$!3)******fX(o^UFQ7 $Fi_s$!3g******QaASUFQ7$Fd_s$!3L+++&3C%GUFQ7$F__s$!3V+++:%FQ7$F`^s$!3)******HYn!zTFQ7$F [^s$!3m*****R:_l;%FQ7$Ff]s$!3#)*****H(y8aTFQ7$Fa]s$!3T+++ay#>9%FQ7$F\\ ]s$!3=+++s#Q+8%FQ7$Fg\\s$!3f*****R`+'=TFQ7$Fb\\s$!3H+++fQw2TFQ7$F]\\s$ !3++++/\")p(4%FQ7$Fh[s$!3G+++Qmf)3%FQ7$Fc[s$!3u*****4!*y13%FQ7$F^[s$!3 1+++O>>uSFQ7$Fijr$!3m*****zl4%pSFQ7$Fdjr$!3G+++*)yimSFQ7$F_jr$!3C+++fFoSFQ7$Feir$!39+++[ICtSFQ7$F`ir$!3S+++R1?\"3%FQ 7$F[ir$!3)******RCW@4%FQ7$Ffhr$!3I+++'zve5%FQ7$Fahr$!36+++KZ*>7%FQ7$F \\hr$!3p*****4rN*RTFQ7$Fggr$!3A+++^_/fTFQ7$Fbgr$!3K+++A.nyTFQ7$F]gr$!3 K+++v'H#)>%FQ7$Fhfr$!3!)*******z_s@%FQ7$Fcfr$!3t*****>G)QNUFQ7$F^fr$!3 o*****>H#R_UFQ7$Fier$!3-+++K%3\"oUFQ7$Fder$!3/+++ClW#G%FQ7$F_er$!3o*** **>6k`H%FQ7$Fjdr$!3:+++]=&oI%FQ7$Fedr$!3B+++PE#pJ%FQ7$F`dr$!3J+++SQgDV FQ7$F[dr$!3r*****H(=$HL%FQ7$Ffcr$!3l*****z][*QVFQ7$Facr$!3S+++/%)pVVFQ 7$F\\cr$!35+++$\\FsM%FQ7$Fgbr$!39+++]>e\\VFQ7$Fbbr$!31+++uv!3N%FQ7$F]b r$!3q*****pV\\4N%FQ7$Fhar$!31+++C<0]VFQ7$Fcar$!3E+++&pd\"[VFQ7$F^ar$!3 3+++Q(4`M%FQ7$Fi`r$!3@+++z%\\:M%FQ7$Fd`r$!3++++Fz\"pL%FQ7$F_`r$!3:+++c cXJVFQ7$Fj_r$!32+++5I?DVFQ7$Fe_r$!3()******>.?=VFQ7$F`_r$!3@+++\\\")[5 VFQFhhtF]^n-Fj^n6jn7%7$Fjas$!+[v5-VFjan7$F`ht$!+Z.5$H%Fjan7$Fdbs$!$@%F f^n7%Fbdu7$Fjgt$!+a%4NG%FjanFedu7%Fidu7$Fdgt$!+1#zLF%FjanFedu7%F]eu7$F ^gt$!+ZdviUFjanFedu7%Faeu7$Fhft$!+cuo^UFjanFedu7%Feeu7$Fbft$!+RaASUFja nFedu7%Fieu7$F\\ft$!+&3C%GUFjanFedu7%F]fu7$Ffet$!+:%FjanFedu7%Fifu7$Fddt$!+ju1z TFjanFedu7%F]gu7$F^dt$!+a@bmTFjanFedu7%Fagu7$Fhct$!+ty8aTFjanFedu7%Feg u7$Fbct$!+ay#>9%FjanFedu7%Figu7$F\\ct$!+s#Q+8%FjanFedu7%F]hu7$Ffbt$!+M 0g=TFjanFedu7%Fahu7$F`bt$!+fQw2TFjanFedu7%Fehu7$Fjat$!+/\")p(4%FjanFed u7%Fihu7$Fdat$!+Qmf)3%FjanFedu7%F]iu7$F^at$!+,*y13%FjanFedu7%Faiu7$Fh` t$!+O>>uSFjanFedu7%Feiu7$Fb`t$!+e'4%pSFjanFedu7%Fiiu7$F\\`t$!+*)yimSFj anFedu7%F]ju7$Ff_t$!+7%FjanFedu7%F i[v7$F\\]t$!+6d$*RTFjanFedu7%F]\\v7$Ff\\t$!+^_/fTFjanFedu7%Fa\\v7$F`\\ t$!+A.nyTFjanFedu7%Fe\\v7$Fj[t$!+v'H#)>%FjanFedu7%Fi\\v7$Fd[t$!++GDe\\VFjanFedu7%Fa`v7$F`fs$!+uv!3N%FjanFedu7%Fe `v7$Fjes$!+P%\\4N%FjanFedu7%Fi`v7$Fdes$!+C<0]VFjanFedu7%F]av7$F^es$!+& pd\"[VFjanFedu7%Faav7$Fhds$!+Q(4`M%FjanFedu7%Feav7$Fbds$!+z%\\:M%FjanF edu7%Fiav7$F\\ds$!+Fz\"pL%FjanFedu7%F]bv7$Ffcs$!+ccXJVFjanFedu7%Fabv7$ F`cs$!+5I?DVFjanFedu7%Febv7$Fjbs$!+?.?=VFjanFedu7%Fibv7$F_bs$!+\\\")[5 VFjanFedu7%F]cvF_duFeduF]^rFc^r-F$6%7$7$$!3/++++++!z%FQF(7$$\"3%****** ********=#FQF(-%'COLOURG6&F`^nF)F)F)-%*LINESTYLEG6#Fb^n-F$6%7$7$F(Fecv 7$F($\"3/++++++!z%FQFjcvF]dv-%%FONTG6$%*HELVETICAGFb^r-%(SCALINGG6#%,C ONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fhdv6#%(DEFAULTG-%*AXESSTY LEG6#%$BOXG-%%VIEWG6$;$!$z%Ff^n$\"$>#Ff^n;F_fv$\"$z%Ff^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 " interval of absolute stability" }{TEXT -1 89 " (or stability interval ) is the intersection of the stability region with the real line." }} {PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-3.8951, 0];" "6#7$,$-%&FloatG6$ \"&^*Q!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 71 "The stability radius is half the leng th of the real stability interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "newton(R(z)=-1,z=-4)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+'3lv%>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1 /120*z^5+1/720*z^6+7975/38588232*z^7:\npts := []:\nz0 := 0:\nfor ct fr om 0 to 360 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := \+ zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,col or=red):\nrd := 1.947565086: \np2 := plot([-rd+rd*cos(t),rd*sin(t),t=0 ..2*Pi],color=blue):\nplots[display]([p1,p2],scaling=constrained,font= [HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 384 384 {PLOTDATA 2 "6(-%'CURVESG6$7eal7$$\"\"!F)F(7$F($\"31+++^v>Z5!#=7$$\"31+++]5h07!#F$ \"3++++/^R%4#F-7$$\"36+++IkjJI!#E$\"3s*****4n#fTJF-7$$\"3%******zdIv%H !#D$\"39+++!H!z)=%F-7$$\"37+++^2w&p\"!#C$\"3s******)*y)fB&F-7$$\"3')** ***f^v](pFC$\"3')*****Hq%=$G'F-7$$\"3/+++9I$zE#!#B$\"3`*****zKx.L(F-7$ $\"3o*****\\VXd='FN$\"3]+++fdbx$)F-7$$\"3++++k&3%p9!#A$\"3E+++YdpC%*F- 7$$\"3t******3*ef6$FY$\"3#*******GY.^*fFY$\"3/+++' 3h=:\"F[o7$$\"3%*******4d9d5!#@$\"3-+++b@^c7F[o7$$\"3))*****p,(e;%Fdo$\"33+++HVqxF[o7$$!34+++_gJ37!#?$\"3'*******3+O& 3#F[o7$$!3')*****f$*='oHF]r$\"3')******[F='=#F[o7$$!3.+++J[hUeF]r$\"37 +++6&QeG#F[o7$$!3-+++W@CB5!#>$\"3')*****4ARTQ#F[o7$$!3')*****\\]\"Qf;F ]s$\"3))*****\\K()3[#F[o7$$!3'******R@vT%=7F-$\"36+++a5*>,$F[o7$$!34+++AoxI:F-$\"3%******f(*4! )3$F[o7$$!3))*****fyg*y=F-$\"3#******\\ul!fJF[o7$$!3)******zgcuD#F-$\" 36+++f)Q[A$F[o7$$!37+++*RK*fEF-$\"3/+++iy?&G$F[o7$$!3A+++**e7!3$F-$\"3 ********y*Q-M$F[o7$$!3)******pipC^$F-$\"31+++7J9!R$F[o7$$!3A+++UjR_RF- $\"3!)******y:ANMF[o7$$!33+++q>Q'R%F-$\"33+++!Q`zA^$F[o7$$!3a+++FZ\"pG&F-$\"30+++l3%\\a$F[o7$$!3M+++ 3*p.t&F-$\"3=+++5k5uNF[o7$$!3i******e$p8<'F-$\"3:+++v)[+g$F[o7$$!3-+++ [!)Q4mF-$\"35+++]$3Ii$F[o7$$!3Q+++!o8T/(F-$\"3')*****>C%>VOF[o7$$!39++ +'zz`Z(F-$\"3/+++3#)ygOF[o7$$!3\\*****p7=J!zF-$\"3'******4OZfn$F[o7$$! 3++++q_KF$)F-$\"35+++\"R2))o$F[o7$$!3s*****>vR![()F-$\"33+++-a[*p$F[o7 $$!3w******HlKl\"*F-$\"3'******pT#33PF[o7$$!3++++$ym#z&*F-$\"3()*****H a&o9PF[o7$$!3S+++e9&**)**F-$\"3*******>#)p$>PF[o7$$!3++++\\xuR5F[o$\"3 ,+++'y*>APF[o7$$!3/+++&f%>!3\"F[o$\"3'******zyIKs$F[o7$$!3,+++jhM?6F[o $\"3!*******3,^APF[o7$$!3#******>O8-;\"F[o$\"3@+++1z2?PF[o7$$!3%****** Ho2)*>\"F[o$\"3-+++4!ofr$F[o7$$!3.+++899R7F[o$\"3$*******3'3-r$F[o7$$! 31+++8zAy7F[o$\"3A+++eG#Gq$F[o7$$!33+++d>3<8F[o$\"3))*****4SHQp$F[o7$$ !35+++F,sb8F[o$\"31+++mGC$o$F[o7$$!3%******zEhTR\"F[o$\"3-+++RV2rOF[o7 $$!33+++8qUK9F[o$\"3%******f%=LdOF[o7$$!3%******ROU0Z\"F[o$\"3;+++r3-U OF[o7$$!3\"******>NO&3:F[o$\"3&******H&\\9DOF[o7$$!3++++`FWY:F[o$\"3,+ ++(Q1ng$F[o7$$!3#*******f3I%e\"F[o$\"3)******f52ne$F[o7$$!3&********Gc @i\"F[o$\"3=+++r(\\^c$F[o7$$!31+++g<1g;F[o$\"32+++o\"R?a$F[o7$$!35+++& zx!)p\"F[o$\"3%*******oRQ\"\\$F[ o7$$!3-+++>_su cXVV$F[o7$$!3)******4)fu_=F[o$\"3*)*****>,kPS$F[o7$$!3/+++$p,D*=F[o$\" 3y*****4sL=P$F[o7$$!35+++$*3)G$>F[o$\"3&)*****Hm['QLF[o7$$!3%******4nr R(>F[o$\"3&******f\")GVI$F[o7$$!3<+++fu%e,#F[o$\"3?+++k6-pKF[o7$$!3'** *****zybe?F[o$\"3,+++wy*GB$F[o7$$!3=+++\"e>@5#F[o$\"31+++^3:'>$F[o7$$! 3#******>v3l9#F[o$\"3')*****fR#)*eJF[o7$$!3=+++Zsl\">#F[o$\"32+++sFf@J F[o7$$!3-+++*p`uB#F[o$\"3%******4UmT3$F[o7$$!3))*****p'*\\PG#F[o$\"37+ ++8$fo/$F[o7$$!32+++g=PIBF[o$\"35+++_(*y4IF[o7$$!3!)*****\\.KrP#F[o$\" 3)******>1MI(HF[o7$$!3z*******pUQU#F[o$\"3*)*****>GFm$HF[o7$$!3#****** \\;E.Z#F[o$\"36+++kyc+HF[o7$$!3/+++KAU;DF[o$\"3=+++kZ#['GF[o7$$!3?+++ \"G#*>c#F[o$\"37+++X`MHGF[o7$$!3$)******f0#pg#F[o$\"3')*****R&G1%z#F[o 7$$!31+++^M6^EF[o$\"3++++oF!*eFF[o7$$!37+++$[(\\%p#F[o$\"3)*******[xyB FF[o7$$!39+++dn,PFF[o$\"35+++57k)o#F[o7$$!3++++--jyFF[o$\"3!)*****Rr*Q `EF[o7$$!3$)*****4)*3$>GF[o$\"3,+++AW'zh#F[o7$$!3))*****f:M!fGF[o$\"33 +++j=I#e#F[o7$$!37+++8Zz(*GF[o$\"3-+++0UMYDF[o7$$!3))*****\\\"feNHF[o$ \"30+++6#R+^#F[o7$$!3++++#)zSsHF[o$\"3++++'**RLZ#F[o7$$!3*)******\\]E3 IF[o$\"3#)******yY?OCF[o7$$!3.+++rV;VIF[o$\"3++++uff)R#F[o7$$!3/+++.d6 xIF[o$\"3=+++(y![gBF[o7$$!3!)******o385JF[o$\"3=+++F)H=K#F[o7$$!3()*** **RWBA9$F[o$\"3#)*****RDC?#F[o7$$!33+++$oBJB$F[o$\"3#****** \\B(Rh@F[o7$$!3!)*****>v!phKF[o$\"32+++w$R(>@F[o7$$!31+++SVU*G$F[o$\"3 !*******=QVx?F[o7$$!3++++WiM;LF[o$\"33+++h5ZM?F[o7$$!33+++(fzCM$F[o$\" 3#******R:W3*>F[o7$$!37+++x)[yO$F[o$\"3/+++<)[l%>F[o7$$!3#******4()zCR $F[o$\"3&******4h$e,>F[o7$$!3y*****pg*R;MF[o$\"3-+++c,&f&=F[o7$$!3(*** ***z@O'RMF[o$\"30+++,Ml4=F[o7$$!3%*******G(=AY$F[o$\"3'******4\">qi.j'o:F[o7$$!3')*****\\5^jc$F[o$\"30+++aFr=:F[o7$$!3%***** *H198F[o7$$!3-+++S(z\"eOF[o$\"3!*******\\16i7F[o7$$!3A+++S.CvOF [o$\"3\"******4CT(47F[o7$$!3%)*****fym=p$F[o$\"3)*******yy8d6F[o7$$!3( )*****HPZ!3PF[o$\"3(******f/]V5\"F[o7$$!3\")*****RhkPs$F[o$\"3'******H 8D90\"F[o7$$!3:+++1e**QPF[o$\"3_*****fHuS)**F-7$$!33+++\"y8Pv$F[o$\"3a +++_EP`%*F-7$$!3;+++=y)yw$F[o$\"3&******zv.D#*)F-7$$!39+++5Y[\"y$F[o$ \"3Y*****R#\\x\"R)F-7$$!3++++a#pWz$F[o$\"3w*****feT9'yF-7$$!3*)*****H> 1o!QF[o$\"3^+++ivqJtF-7$$!3=+++n+Y=QF[o$\"37+++Mqs-oF-7$$!3:+++ZkRHQF[ o$\"3M+++lyguiF-7$$!3'*******QCeRQF[o$\"3))*****\\\"eTZdF-7$$!3-+++ar) *[QF[o$\"3w*****fFz6A&F-7$$!3/+++c?edQF[o$\"3A+++&=%*ep%F-7$$!3')***** z>T`'QF[o$\"3!)*****R%)G:<%F-7$$!3#)*****\\OTA(QF[o$\"3E+++o%G![OF-7$$ !35+++`@EyQF[o$\"3,+++K$>`7$F-7$$!39+++$)fQ$)QF[o$\"3C+++hCJ.EF-7$$!3+ +++!4)f()QF[o$\"3'******z%o!>3#F-7$$!3A+++&[')3*QF[o$\"3)******>A#*4c \"F-7$$!3;+++2>C$*QF[o$\"3$******\\T^//\"F-7$$!33+++zxl%*QF[o$\"38+++h Qi,_F]s7$$!3++++s,8&*QF[oF(7$Fbjm$!38+++hQi,_F]s7$F]jm$!3$******\\T^// \"F-7$Fhim$!3)******>A#*4c\"F-7$Fcim$!3'******z%o!>3#F-7$F^im$!3C+++hC J.EF-7$Fihm$!3,+++K$>`7$F-7$Fdhm$!3E+++o%G![OF-7$F_hm$!3!)*****R%)G:<% F-7$Fjgm$!3A+++&=%*ep%F-7$Fegm$!3w*****fFz6A&F-7$F`gm$!3))*****\\\"eTZ dF-7$F[gm$!3M+++lyguiF-7$Fffm$!37+++Mqs-oF-7$Fafm$!3^+++ivqJtF-7$F\\fm $!3w*****feT9'yF-7$Fgem$!3Y*****R#\\x\"R)F-7$Fbem$!3&******zv.D#*)F-7$ F]em$!3a+++_EP`%*F-7$Fhdm$!3_*****fHuS)**F-7$Fcdm$!3'******H8D90\"F[o7 $F^dm$!3(******f/]V5\"F[o7$Ficm$!3)*******yy8d6F[o7$Fdcm$!3\"******4CT (47F[o7$F_cm$!3!*******\\16i7F[o7$Fjbm$!3\"*******ec>98F[o7$Febm$!3)** ****z-YfO\"F[o7$F`bm$!32+++UGJ<9F[o7$F[bm$!3#******R`\\#o9F[o7$Ffam$!3 0+++aFr=:F[o7$Faam$!3!******>.j'o:F[o7$F\\am$!3/+++\\_1=;F[o7$Fg`m$!3# *******)*)))om\"F[o7$Fb`m$!3)******f<3^r\"F[o7$F]`m$!3'******4\">qiF[o7 $Fi^m$!3/+++<)[l%>F[o7$Fd^m$!3#******R:W3*>F[o7$F_^m$!33+++h5ZM?F[o7$F j]m$!3!*******=QVx?F[o7$Fe]m$!32+++w$R(>@F[o7$F`]m$!3#******\\B(Rh@F[o 7$F[]m$!3(******zR>C?#F[o7$Ff\\m$!3)******\\P?GC#F[o7$Fa\\m$!3#)*****R DGFm$HF[ o7$Fbfl$!3)******>1MI(HF[o7$F]fl$!35+++_(*y4IF[o7$Fhel$!37+++8$fo/$F[o 7$Fcel$!3%******4UmT3$F[o7$F^el$!32+++sFf@JF[o7$Fidl$!3')*****fR#)*eJF [o7$Fddl$!31+++^3:'>$F[o7$F_dl$!3,+++wy*GB$F[o7$Fjcl$!3?+++k6-pKF[o7$F ecl$!3&******f\")GVI$F[o7$F`cl$!3&)*****Hm['QLF[o7$F[cl$!3y*****4sL=P$ F[o7$Ffbl$!3*)*****>,kPS$F[o7$Fabl$!3&)*****>cXVV$F[o7$F\\bl$!35+++4w] jMF[o7$Fgal$!33+++K*)>\"\\$F[o7$Fbal$!3%*******oRQAPF[o7$Fh[l$!3*******>#)p$>PF[o7$Fc[l$! 3()*****Ha&o9PF[o7$F^[l$!3'******pT#33PF[o7$Fiz$!33+++-a[*p$F[o7$Fdz$! 35+++\"R2))o$F[o7$F_z$!3'******4OZfn$F[o7$Fjy$!3/+++3#)ygOF[o7$Fey$!3' )*****>C%>VOF[o7$F`y$!35+++]$3Ii$F[o7$F[y$!3:+++v)[+g$F[o7$Ffx$!3=+++5 k5uNF[o7$Fax$!30+++l3%\\a$F[o7$F\\x$!3)******>`zA^$F[o7$Fgw$!33+++!Q,$F[o7$Fjt$!38+++X7\\JHF[o 7$Fet$!3++++\"e#3ZGF[o7$F`t$!3%)*******HV$fFF[o7$F[t$!3;+++\")HzoEF[o7 $Ffs$!3/+++87'ed#F[o7$Fas$!3))*****\\K()3[#F[o7$F[s$!3')*****4ARTQ#F[o 7$Ffr$!37+++6&QeG#F[o7$Far$!3')******[F='=#F[o7$F[r$!3'*******3+O&3#F[ o7$Ffq$!3!*******QZb$)>F[o7$Faq$!3++++nk%4)=F[o7$F\\q$!33+++HVqxZ5F-F'F*F.F4F:F@F FFKFQFVFfnF\\oFaoFgoF\\pFapFfpF[qF`qFeqFjqF`rFerFjrF`sFesFjsF_tFdtFitF ^uFcuFhuF]vFbvFgvF\\wFawFfwF[xF`xFexFjxF_yFdyFiyF^zFczFhzF][lFb[lFg[lF \\\\lFa\\lFf\\lF[]lF`]lFe]lFj]lF_^lFd^l-%'COLOURG6&%$RGBG$\"*++++\"!\" )F(F(-F$6$7SF'7$$!3&=.$47PlB=F]s$\"3ATo-@\"o*eEF-7$$!3C?eeMk\"HN'F]s$ \"36SjXI^uL\\F-7$$!3yr&=*QnUj9F-$\"3A8j*o)[!oS(F-7$$!34$Hp=0KKi#F-$\"3 3&)\\,=g,i(*F-7$$!3O\"F[o7$$!3]FD:!pw=o&F-$ \"3s5%Q())=*[P\"F[o7$$!3s+WS)>bMc(F-$\"3GAa]w3yS:F[o7$$!3&)pP,Br%or*F- $\"37F)>h\"zU&o\"F[o7$$!3ql%Rk-oN?\"F[o$\"3qEA$H.b)*z\"F[o7$$!3%=vw>nH dX\"F[o$\"31!fXbHQW)=F[o7$$!3+\"=Q[$4f&o\"F[o$\"3PR.2,^')H>F[o7$$!3e*z ,2*zs[>F[o$\"3M82\")QZcZ>F[o7$$!31l\\J$RL^.%H>F[o7$$!3# **>BFGEHY#F[o$\"31H\"\\)H29y=F[o7$$!3Ow:)zg-Co#F[o$\"3I@u7LSh.=F[o7$$! 3Fr6&R!))eHHF[o$\"3GK>?ue&=o\"F[o7$$!31\\:9q\\zBJF[o$\"3/qs,vTD_:F[o7$ $!386n0'eb#GLF[o$\"3;9)GCEtNP\"F[o7$$!3Ed[L+.'p[$F[o$\"3!4F$[(p4I>\"F[ o7$$!3**)*=aL9-MOF[o$\"3q#)eDM--T(*F-7$$!3k^^m0+NXPF[o$\"3'odui6a)*[(F -7$$!3F%Q;6m='HQF[o$\"3::%fLCL)3]F-7$$!3sSqC;&\\EF-7 $$!3g6*>g!37&*QF[o$\"3<\"f%Rz>+SgF]r7$$!3=.g?!)yHxQF[o$!3`D(HYmp%HEF-7 $$!3rH,P+BlJQF[o$!3V$yRP,'yJ\\F-7$$!3=Ym&GpX;v$F[o$!35#Q\"\\II\"pL(F-7 $$!3b)*R.#H>pj$F[o$!3oyQp0mn!p*F-7$$!3'4YLA]8^\\$F[o$!33#*RpOVT#=\"F[o 7$$!3.%*z2#3%RKLF[o$!3eb1j]2Sp8F[o7$$!3'*)3&4hmcDJF[o$!3!G)ffs+\"4b\"F [o7$$!3m['\\y8M&>HF[o$!3qc#)[glo(o\"F[o7$$!3'*Q14N)QBo#F[o$!35hb[9+k.= F[o7$$!3i7vr'o>cX#F[o$!3c)HH4RI,)=F[o7$$!3=R(p<71\"*>#F[o$!3l0F+pEDJ>F [o7$$!3'=b@#R&>N&>F[o$!3-BaX$)fbZ>F[o7$$!3]a&z5gsnp\"F[o$!3LFiV=+NJ>F[ o7$$!3Uc?=aT&*\\9F[o$!3Uj?]L9#H)=F[o7$$!3lH_t%Q=0?\"F[o$!3Yd')f!R\"f)z \"F[o7$$!3%QgGKM(RJ(*F-$!3)[/x3bpio\"F[o7$$!33Ml\"QK$HxvF-$!3jB&>uu\\= a\"F[o7$$!3;)Q*G#H,$\\cF-$!3!yUg&ffhr8F[o7$$!3YXd@Z#R&*3%F-$!3A!3D6r?S >\"F[o7$$!3]r\"\\#pR\"=e#F-$!37OxC;E;!p*F-7$$!3\"\\7+_3h$4:F-$!3*p/_I. Sv^(F-7$$!3%pD(3@!f^s'F]s$!3[oDw)3lP2&F-7$$!3w.-*f\\,@\"=F]s$!3a>$R'3C d]EF-7$F($\"35B9e0\"4yf\"F7-Fago6&FcgoF(F(Fdgo-%(SCALINGG6#%,CONSTRAIN EDG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q!6\"Ffgp-F_gp6#%(DEFAUL TG-%%VIEWG6$FjgpFjgp" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 289 "We can distort the boundary curve horizo ntally by taking the 11th root of the real part of points along the cu rve. In this way we see that the largest interval on the nonnegative i maginary axis that contains the origin and lies inside the stability r egion is [ 0, 1.95 ] approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "R := z -> 1+z+1/2*z^2+ 1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+7975/38588232*z^7:\nDigits := 20: \npts := []: z0 := 0:\nfor ct from 0 to 69 do\n zz := newton(R(z)=ex p(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),1 1),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.6,0,.9),thickness=2,f ont=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 249 293 293 {PLOTDATA 2 "6(-%'CURVESG6#7bo7$$\"\"!F)F(7$$\"5P))H')*>i\\)fJ !#@$\"5w4=)*e`EfTJF-7$$\"5yU/R*4Qq1B&F-$\"5B]QF=2`=$G'F-7$$\"5^w&H!*=5 cN-(F-$\"5iC$QA3'zxC%*F-7$$\"5=1l\\nKr@c')F-$\"5Y5LZZhqjc7!#?7$$\"5-!G YDvUfy,\"F?$\"5#Hs_upK'zq:F?7$$\"5=-n3!\\o(yh6F?$\"5]>%otFfb\\)=F?7$$ \"5'*ypZU6W4*H\"F?$\"5bjIODf[6*>#F?7$$\"5ZG6=*o\\<4V\"F?$\"5%z=xiq7uK^ #F?7$$\"53e2WB<)o!e:F?$\"5MJOi;(RLu#GF?7$$\"5dtOba7@:\"o\"F?$\"5)p$=G# 3n#fTJF?7$$\"5BXr\"3\"\\1j+=F?$\"5*z:nE%\\>vbMF?7$$\"5_XoP*[ooo\">F?$ \"5ZfQ?8M7\"*pPF?7$$\"5CV\\ue9#e,.#F?$\"5<_'ee^_qS3%F?7$$\"5S)*[6:VptS @F?$\"5)Gc1`1#)H#)R%F?7$$\"5rn'G]yx+)[AF?$\"5)RwPg\\6*Q7ZF?7$$\"5i;N(= lv7XN#F?$\"5eb-T8'R[l-&F?7$$\"5fbYA(*z&4!eCF?$\"5*Q)*[$\\UwqS`F?7$$\"5 BDBdc'[1%fDF?$\"5BB#H6!=o'[l&F?7$$\"5m)Hbf,&4!)eEF?$\"5Sh\\]Hme-pfF?7$ $\"5m-5l'R_viv#F?$\"54xvR!Hq%=$G'F?7$$\"5Fd7'o90+>&GF?$\"5!e,%\\s0KM(f 'F?7$$\"5%\\V#)4#*RLd%HF?$\"54LkY6.7]6pF?7$$\"5O(G;!o@[#y.$F?$\"5(p:o, &f%ecA(F?7$$\"5:4o?F#[)F?7$$\"5>3]9+h=KtMF?$\"5[U?b^`%=kz)F?7$$\"59+A\"Q%Q wcbNF?$\"58U*zx\\8g06*F?7$$\"5-y/S3wI@OOF?$\"5JA&G#*eu&pC%*F?7$$\"5,MU ?a3TD:PF?$\"5>+_!H4hB)Q(*F?7$$\"5)[#>;*)44o#z$F?$\"5w\"p*Q\\pTH05!#>7$ $\"5c$)=6d[xZoQF?$\"5`-5akaZqO5Ffu7$$\"5z*[jc7uAE%RF?$\"5OWQldBQ6o5Ffu 7$$\"5nE*)p`(\\(3:SF?$\"5s/WYvG5_*4\"Ffu7$$\"5=**QctUm$e3%F?$\"5%)H/t! 3'f#48\"Ffu7$$\"5KxON*3?F[:%F?$\"5`*y2=29GB;\"Ffu7$$\"5;F=*oK\"y+AUF?$ \"5tO14_6qs$>\"Ffu7$$\"526zD9fxJ(G%F?$\"5s4K)H&H>7D7Ffu7$$\"5e@)4=Wt&o ]VF?$\"5ZHjQ:b@^c7Ffu7$$\"5#3Yo%HG$G?T%F?$\"5Pk.7OVo*yG\"Ffu7$$\"5SGHD V<\"[7Z%F?$\"5ik$esQ.v#>8Ffu7$$\"5^5)3JzEJ#GXF?$\"5c\")3*z6kX1N\"Ffu7$ $\"5)pXmMT\\WGe%F?$\"5d'4i.XW2?Q\"Ffu7$$\"5y2 fLT\"Ffu7$$\"5hsk7\\\\iI%o%F?$\"5]&3l/,-*pW9Ffu7$$\"5h[A%G)*p\\2t%F?$ \"5ugRN'feDgZ\"Ffu7$$\"5*e6-.&)e(*Rx%F?$\"5g)\\W@o\"pL2:Ffu7$$\"5t:RE^ #4IP\"[F?$\"58JQ<(G(4jQ:Ffu7$$\"54liz?;[b\\[F?$\"52Re4jaR2\\F?$\"5bwf.5NiQK;Ff u7$$\"5>O_%p6Q*)z#\\F?$\"5Wa\")3,Zpej;Ffu7$$\"5f))R)H%\\vpT\\F?$\"5PG; Zjjsv%p\"Ffu7$$\"5e+U5JVM0Z\\F?$\"5rqD1(o&R*es\"Ffu7$$\"5\"fJxZS=d>%\\ F?$\"5p5/P>*f$*pv\"Ffu7$$\"5cOd_LfhBB\\F?$\"5DR@XgoD0)y\"Ffu7$$\"5`*)R t;3\\t&)[F?$\"5;N-Uwbq1>=Ffu7$$\"5$ehFX@**\\,#[F?$\"5IyJcGtI.]=Ffu7$$ \"5$\\N'GP!fkmq%F?$\"5!)R\"\\xqYY4)=Ffu7$$\"5I!Q_=803v[%F?$\"5c=OkbHH! =\">Ffu7$$\"5:c,rHXv&>i$F?$\"5'=L_bW,)fU>Ffu7$$!5d@Vc&QvX@^%F?$\"5,)ov mG:FL(>Ffu7$$!5Rhyj#3rL!z[F?$\"5@1*fl+n&)R+#Ffu7$$!5Df9*4v;VG7&F?$\"5K -V#zD!)oX.#Ffu7$$!5uq$*H=`1V;`F?$\"5cv@B)[rr]1#Ffu7$$!5\">dAK9T@A[&F?$ \"5rGqv.:&*[&4#Ffu7$$!5qe\")RI2o;IcF?$\"5],A'[&os\"e7#Ffu7$$!5=U\\;$f= obw&F?$\"5+Tn5m4+0c@Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\"\"# -%+AXESLABELSG6$Q!6\"F\\bl-%&COLORG6&%$RGBG$\"\"'!\"\"F($FdalFdbl-%%VI EWG6$%(DEFAULTGFibl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 123 "The relevant intersection point of the boundary c urve with the imaginary axis can be determined more accurately as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the b oundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits : = 15:\nz0 := 1.95*I:\nfor ct from 61 to 64 do\n newton(R(z)=exp(ct*P i/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Wz'**G8'[\"!#=$\"0Wc&HH!=\">!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0j\"z:$osS\"!#>$\"0_bW,)fU>!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0&)y#)***Qy:!#=$\"0wmG:FL(>!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0)>'pt$oHP!#=$\"0gl+n&)R+#!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bise ction method to calculate the parameter value associated with the inte rsection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_par t := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.95*I))\nend proc:\nDig its := 15:\nu0 := bisect('real_part'(u),u=0.61..0.64);\nnewton(R(z)=ex p(u0*Pi*I),z=1.95*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#u0G$\"0([fNv84i!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0wR%Gz A5I!#H$\"03Ozq3a%>!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnega tive imaginary axis that contains the origin and lies inside the stabi lity region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.9454];" "6#7$\" \"!-%&FloatG6$\"&a%>!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Thus the imaginar y axis inclusion is " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" } {TEXT -1 1 " " }{TEXT 279 1 "~" }{TEXT -1 62 " 1.954 which is approxi mately equal to the stability radius " }{XPPEDIT 18 0 "rho" "6#%$rhoG " }{TEXT -1 1 " " }{TEXT 280 1 "~" }{TEXT -1 6 " 1.948" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------- -----------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "the stabilit y region for the 8 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for th e 8 stage, order 4 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(4,8,'expan ded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/-%#R*G6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F) *&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#\"KPVa>`&Gf@ vO,pg$=#f+\")Ga'\"M++g\"Ga 'eLEVc/yojIe*pF)*$)F'F1F)F)F)*&#\" =h$=[f$p2aU&HJDl\"\"@+Sye6(zK-fbr'HiL)F)*$)F'\"\"(F)F)F)*&#\"7$ecz/Ea \\?\"\\;\"=#ztMpLar+93_/,\"F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point wh ere the boundary of the stability region intersects the negative real \+ axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R *`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+LFXER!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z ),-1],z=-4.49..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],sty le=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_ 0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([ p_1,p_2,p_3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 393 267 267 {PLOTDATA 2 "6+-%'CURVESG6$7U7$ $!3A++++++!\\%!#<$!3PZj7q$zJs#F*7$$!3;+]7'3DdV%F*$!3)f'))[um1)[#F*7$$! 35++Ds,X\"Q%F*$!3mb=q(y(eqAF*7$$!3(*\\P*)feAMVF*$!3KC^)fA<[4#F*7$$!3%) *\\Pva,qG%F*$!3a4yobr\"3$>F*7$$!3I+](4>%y!=%F*$!3c)pW#oM[,;F*7$$!35+]# oviQ2%F*$!3!fn,&)pD%>8F*7$$!3A+ve,&\\u'RF*$!39kC%eH4:3\"F*7$$!3?+v8#z! zoQF*$!3V+\")Gg0YS*)!#=7$$!3;+v.8ajmPF*$!3ernv>BP!H(FU7$$!3C+v$Q<')4m$ F*$!3yw$)\\Q\">F&eFU7$$!3G+v3`dnbNF*$!3=p/zgN$F*$!3[$*)f5%y&f'GFU7$$!3E++:Qn_WKF*$!3%yvGCc e2:#FU7$$!3#****\\s&QnOJF*$!3)zP&o$oFkb\"FU7$$!3&****\\=iPF.$F*$!3+]nQ >-c!3\"FU7$$!3Q+v$Gc`$QHF*$!3=(f5UuJE;(!#>7$$!3+++l(y@h#GF*$!3$GjG]J') G^$F]q7$$!3=++g_n/JFF*$!3!R(\\W5&pW())!#?7$$!3C+vt%)=X?EF*$\"3EnP$)f'p uu\"F]q7$$!3D++5(ocD_#F*$\"3)H#z@\"H)p\"z$F]q7$$!3#**\\P(R,::CF*$\"3-) e[GH!f3eF]q7$$!3&)*\\7sqtGJ#F*$\"3y/6`hW.zvF]q7$$!30+]Pz*eh?#F*$\"3p#) >?V-sM$*F]q7$$!3l*\\([Y:;3@F*$\"3i$e.)[hO\"4\"FU7$$!3$***\\7w!eC+#F*$ \"3+sFanOyi7FU7$$!3%**\\Py(=m#*=F*$\"3Q)3+;()zmW\"FU7$$!3%)*\\iW'R3(z \"F*$\"3r^nTRD1:;FU7$$!3-+]d*>dQp\"F*$\"3()Q1zq@\"*3=FU7$$!3)*****p]Q@ (e\"F*$\"3KgT(3(QkD?FU7$$!3I+]FRT)G[\"F*$\"3O6C5&y(>dAFU7$$!3')*\\P*y( R>Q\"F*$\"3U\"p_>q=F]#FU7$$!3++]Kx#e)p7F*$\"3-Rmu&=)y.GFU7$$!3=++5j![ \"p6F*$\"3MuBUUlG.JFU7$$!3s***\\KT=;1\"F*$\"3A!>'zE'>sX$FU7$$!3'***\\P %o0=k*FU$\"3)zAC$yu%>\"QFU7$$!3](***\\tEbw&)FU$\"3`cX0#\\F5C%FU7$$!39- ]P2EBuvFU$\"3o7@%4,i%)o%FU7$$!3>)*\\(GL>l_'FU$\"3REY!4a5l?&FU7$$!3z++] -\")=-bFU$\"3+\\;:Wj=odFU7$$!3$*)*\\PM#3)HWFU$\"3%f9#e6\"z6U'FU7$$!3;( ****f'*ypR$FU$\"3\"R*=4.(\\)>rFU7$$!3L)**\\U'=wSBFU$\"3:2%o![Y,8zFU7$$ !3?$*\\Pt2H$H\"FU$\"33 \"F*7$$\"3l+]i_F06GFU$\"3)=g#R!=$fC8F*7$$\"3\\.]Pt1&z\"QFU$\"3g+!RkX7 \\Y\"F*7$$\"3!***************[FU$\"3RU#zBC=Bj\"F*-%'COLOURG6&%$RGBG$\" *++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7 $FBFa\\l7$FGFa\\l7$FLFa\\l7$FQFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oF a\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FipFa\\l7$F_qFa\\l7$FdqF a\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsF a\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[v Fa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^x Fa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$Fa zFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7# 7$$!33+++LFXERF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\ l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6# %(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\ lF[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q \"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`c l$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following p icture shows the stability region for the 8 stage, order 4 scheme. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1745 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+\n 6542881005921 83606901367521592855319544337/7773145152086713296563778707579175428160 0000*z^5+\n 958188731532349599236581756432633586190773/69958306368 7804196690740083682125788534400000*z^6+\n 165253129542540769359481 8361/8336229671555902327971158784000*z^7+\n 1649120495426047956583 /1010452081400715433693473792*z^8:\npts := []: z0 := 0: tt := 0: \nwhi le tt<=201/20 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 : = zz:\n if (13/20<=tt and tt<=6/5) or (87/10<=tt and tt<=93/10) then \n hh := 1/40\n else \n hh := 1/20\n end if;\n tt := t t+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts, color=COLOR(RGB,.25,.08,.4)):\np_2 := plots[polygonplot]([seq([pts[i-1 ],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,color= COLOR(RGB,.5,.15,.8)):\npts := []: z0 := 1.5+4.3*I:\nfor ct from 0 to \+ 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=CO LOR(RGB,.25,.08,.4)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i] ,[1.32,4.19]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.5,.15,.8)):\npts := []: z0 := 1.5-4.3*I:\nfor ct from 0 to 60 \+ do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n p ts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR (RGB,.25,.08,.4)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1 .32,-4.19]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.5,.15,.8)):\np_7 := plot([[[-4.79,0],[1.99,0]],[[0,-4.79],[0,4.7 9]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.7 9..1.99,-4.79..4.79],font=[HELVETICA,9],\n labels=[`Re(z) `,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F(7$$!3. +++.2)G`\"!#E$\"37+++[izq:!#=7$$!3i+++`Z'z[*!#D$\"3;+++h-fTJF07$$!3!** ****\\#QEA5!#B$\"3*)*****zJsBr%F07$$!36+++\"*p[:`F:$\"37+++TF7$G'F07$$ !30+++4guP=!#A$\"3y******fX\"Q&yF07$$!3G+++k_2#)[FE$\"3Y+++cKTC%*F07$$ !3)*******zkQ!3\"!#@$\"3/+++&p&[*4\"!#<7$$!31+++9%yJ5#FP$\"3)******z_* \\c7FS7$$!3))*****fRbkw$FP$\"31+++bnW89FS7$$!3S+++&)H7tkFP$\"3%******> X^-d\"FS7$$!3*******z,a'46!#?$\"3*******zfuns\"FS7$$!3/+++$Hkj%>F`o$\" 3++++*\\#z#)=FS7$$!3:+++-i([]$F`o$\"36+++Qg)z.#FS7$$!3k*****pI'\\GZF`o $\"3$******pG\\^6#FS7$$!3-+++/2pyjF`o$\"3#)*****>\"z%>>#FS7$$!3_*****> ug,e)F`o$\"3)******pj?$oAFS7$$!33+++tJ/[6!#>$\"3++++$z-UM#FS7$$!32+++7 R)\\_\"F_q$\"3\"******p]D&>CFS7$$!31+++J&Q!3?F_q$\"35+++>N@%\\#FS7$$!3 &******zm@$=EF_q$\"3)*******)G\"=oDFS7$$!3q*****pML%yLF_q$\"3/+++`sKTE FS7$$!3'******\\&\\z6VF_q$\"34+++[k_8FFS7$$!3*)*****Rwu;W&F_q$\"3')*** **4GAYy#FS7$$!3a*****4dI)*y'F_q$\"3;+++!)*=W&GFS7$$!3S******\\juu$)F_q $\"3=+++=rnAHFS7$$!3-+++Eg&4-\"F0$\"3))*****\\B:\"*)HFS7$$!3/+++*3(**H 7F0$\"3=+++*R?M0$FS7$$!3%*******)[gUY\"F0$\"34+++\"zo_6$FS7$$!33+++A.W AaV<$FS7$$!39+++=4M-?F0$\"3\")*****p`=/B$FS7$$!3/+++PA 4,BF0$\"3)******p)oF$G$FS7$$!3?+++;;X:EF0$\"3))*****zcIGL$FS7$$!3>+++^ m6UHF0$\"3++++6l1zLFS7$$!33+++bG)zF$F0$\"3?+++6`/AMFS7$$!3))*****>W6.i $F0$\"3;+++!4$)=Y$FS7$$!3/+++.j!o'RF0$\"33+++%=K()\\$FS7$$!3y*****\\d< `m%F0$\"3y******[l:kNFS7$$!3o*****>99LO&F0$\"3%)******p?r>OFS7$$!3!)** ****ybAbgF0$\"3)******R_smm$FS7$$!3!******4;g$QnF0$\"3\")******>\"4hq$ FS7$$!3E+++hEm6uF0$\"3()*****fQ*))QPFS7$$!3Y******\\([\\2)F0$\"35+++H^ qlPFS7$$!3C+++kcYG()F0$\"3;+++Mb5(y$FS7$$!3#)*****fc^EP*F0$\"30+++Bn_. QFS7$$!3*)*****>)4!3+\"FS$\"3%)*****4q:`\"QFS7$$!33+++A[]j5FS$\"3)**** **>&*[F#QFS7$$!3,+++zaUD6FS$\"3-+++dm/EQFS7$$!3++++@%4m=\"FS$\"35+++1N QDQFS7$$!33+++U;5Z7FS$\"3<+++qp*3#QFS7$$!3/+++#HZpI\"FS$\"3&******pn$p 7QFS7$$!3/+++aQ>m8FS$\"3:+++sW&3!QFS7$$!3!******\\\"Q*[U\"FS$\"3?+++l& Qay$FS7$$!3)******f\"y5$[\"FS$\"3%)*****4@([mPFS7$$!30+++M$34a\"FS$\"3 :+++(HFSu$FS7$$!3#******4#RQ)f\"FS$\"30+++4`2=PFS7$$!3%******p)Rkb;FS$ \"3-+++MAk)o$FS7$$!3%******z#R#Gr\"FS$\"3%)*****>4Sdl$FS7$$!3'******p8 !4qOFS7$$!3,+++_QkF=FS$\"3/+++'[R'zNFS7$$!3++++h>s& )=FS$\"3)********fpl`$FS7$$!3&******fu\"fW>FS$\"3++++ScK!\\$FS7$$!39++ +Xa`/?FS$\"3>+++df8TMFS7$$!3%)*****z]?e1#FS$\"3/+++aRL*Q$FS7$$!3()**** *\\#FS$\"3')*****4=#yzKFS7$$!3'** *****))*o\"fAFS$\"3\")*****z!zk_$*f#FS$\"3*******\\KX0%H FS7$$!33+++z>%em#FS$\"3?+++>.<&)GFS7$$!3<+++/p#3t#FS$\"3&)*****RtU*HGF S7$$!3')*****\\&e0%z#FS$\"3!******R\"4kuFFS7$$!3%******p,b`&GFS$\"3-++ +S5/>FFS7$$!31+++!G7Y\"HFS$\"3))*****4TKHm#FS7$$!3))*****4Li<(HFS$\"3' ******z!\\71EFS7$$!3<+++4WxEIFS$\"35+++T9X[DFS7$$!3/+++BIkzIFS$\"3;+++ qvw*[#FS7$$!3!******\\G\"QIJFS$\"30+++X(\\*HCFS7$$!3')*****>R;!zJFS$\" 31+++xI*)oBFS7$$!3++++[neDKFS$\"35+++#=4lI#FS7$$!3#)*****pGS,F$FS$\"3& )******zVsUAFS7$$!3\")*****flLFJ$FS$\"3?+++X%yu<#FS7$$!35+++S>V`LFS$\" 3%*******zRs5@FS7$$!33+++o'3BR$FS$\"3)********[EC/#FS7$$!3.+++MeWHMFS$ \"3!*******o^cs>FS7$$!3y******[O$\\Y$FS$\"35+++\"QM6!>FS7$$!3')******Q (p))\\$FS$\"31+++kd9G=FS7$$!3++++euNJNFS$\"3-+++a2j`p>\"FS7$$!3%)*****4xa+v$FS$\"31+++.O696FS7$$!3#******Ho hGx$FS$\"3\"******p2-5.\"FS7$$!37+++$HUXz$FS$\"3e*****>!zQx%*F07$$!3() ******fo)\\\"QFS$\"3I+++#zIXk)F07$$!3\")******)esS$QFS$\"38+++WmY7yF07 $$!3))*****zlr;&QFS$\"3]+++#**G>)pF07$$!3%*******4$ew'QFS$\"3Y+++RJQ`h F07$$!3%*******[W\">)QFS$\"3W+++#4dqK&F07$$!33+++\"fLV*QFS$\"37+++8.)H ]%F07$$!3$******>KB[!RFS$\"3-+++%*R-\"o$F07$$!3=+++hlI8RFS$\"35+++x&Q4 'GF07$$!3z*****4+A(>RFS$\"3.+++tiQU?F07$$!3'******40CS#RFS$\"3-+++;z' \\A\"F07$$!3@+++oD=ERFS$\"33+++>iW#3%F_q7$Fd^m$!33+++>iW#3%F_q7$F_^m$! 3-+++;z'\\A\"F07$Fj]m$!3.+++tiQU?F07$Fe]m$!35+++x&Q4'GF07$F`]m$!3-+++% *R-\"o$F07$F[]m$!37+++8.)H]%F07$Ff\\m$!3W+++#4dqK&F07$Fa\\m$!3Y+++RJQ` hF07$F\\\\m$!3]+++#**G>)pF07$Fg[m$!38+++WmY7yF07$Fb[m$!3I+++#zIXk)F07$ F][m$!3e*****>!zQx%*F07$Fhjl$!3\"******p2-5.\"FS7$Fcjl$!31+++.O696FS7$ F^jl$!3/+++m*>p>\"FS7$Fiil$!3'******p.]#z7FS7$Fdil$!3-+++4N#4O\"FS7$F_ il$!3+++++0wT9FS7$Fjhl$!3.+++N3f@:FS7$Fehl$!34+++[FS7$F\\gl$!3!*******o^cs>FS7$Fgfl$!3)********[EC/#FS7$Fbfl$!3%******* zRs5@FS7$F]fl$!3?+++X%yu<#FS7$Fhel$!3&)******zVsUAFS7$Fcel$!35+++#=4lI #FS7$F^el$!31+++xI*)oBFS7$Fidl$!30+++X(\\*HCFS7$Fddl$!3;+++qvw*[#FS7$F _dl$!35+++T9X[DFS7$Fjcl$!3'******z!\\71EFS7$Fecl$!3))*****4TKHm#FS7$F` cl$!3-+++S5/>FFS7$F[cl$!3!******R\"4kuFFS7$Ffbl$!3&)*****RtU*HGFS7$Fab l$!3?+++>.<&)GFS7$F\\bl$!3*******\\KX0%HFS7$Fgal$!3=+++-8E'*HFS7$Fbal$ !3=+++yfW_IFS7$F]al$!30+++a`64JFS7$Fh`l$!3%)*****\\)38mJFS7$Fc`l$!3\") *****z!z+++df8TMFS7$Fj^l$!3++++ScK!\\$FS7$Fe^l$!3)********fp l`$FS7$F`^l$!3/+++'[R'zNFS7$F[^l$!3$******R'4R>OFS7$Ff]l$!3%)*****>4Sd l$FS7$Fa]l$!3-+++MAk)o$FS7$F\\]l$!30+++4`2=PFS7$Fg\\l$!3:+++(HFSu$FS7$ Fb\\l$!3%)*****4@([mPFS7$F]\\l$!3?+++l&Qay$FS7$Fh[l$!3:+++sW&3!QFS7$Fc [l$!3&******pn$p7QFS7$F^[l$!3<+++qp*3#QFS7$Fiz$!35+++1NQDQFS7$Fdz$!3-+ ++dm/EQFS7$F_z$!3)******>&*[F#QFS7$Fjy$!3%)*****4q:`\"QFS7$Fey$!30+++B n_.QFS7$F`y$!3;+++Mb5(y$FS7$F[y$!35+++H^qlPFS7$Ffx$!3()*****fQ*))QPFS7 $Fax$!3\")******>\"4hq$FS7$F\\x$!3)******R_smm$FS7$Fgw$!3%)******p?r>O FS7$Fbw$!3y******[l:kNFS7$$!3\")*****HL?cJ%F0$!3%******f8kF`$FS7$F]w$! 33+++%=K()\\$FS7$Fhv$!3;+++!4$)=Y$FS7$Fcv$!3?+++6`/AMFS7$F^v$!3++++6l1 zLFS7$Fiu$!3))*****zcIGL$FS7$Fdu$!3)******p)oF$G$FS7$F_u$!3\")*****p`= /B$FS7$Fjt$!3%******4>aV<$FS7$Fet$!34+++\"zo_6$FS7$F`t$!3=+++*R?M0$FS7 $F[t$!3))*****\\B:\"*)HFS7$Ffs$!3=+++=rnAHFS7$Fas$!3;+++!)*=W&GFS7$F\\ s$!3')*****4GAYy#FS7$Fgr$!34+++[k_8FFS7$Fbr$!3/+++`sKTEFS7$F]r$!3)**** ***)G\"=oDFS7$Fhq$!35+++>N@%\\#FS7$Fcq$!3\"******p]D&>CFS7$F]q$!3++++$ z-UM#FS7$Fhp$!3)******pj?$oAFS7$Fcp$!3#)*****>\"z%>>#FS7$F^p$!3$****** pG\\^6#FS7$$!3(*******pr-0EF`o$!33+++;f^g>FS7$$!3#******\\!HHk9F`o$!3) ******>Sk[!=FS7$$!3i*****z$Qgg%)FP$!3#******>,g&[;FS7$$!3E+++nBP]\\FP$ !3++++[Q(=\\\"FS7$$!3')*****RM^k$GFP$!3-+++=])\\L\"FS7$$!3&******4#zIF :FP$!35++++!)*z<\"FS7$$!3L+++0cQ-uFE$!3#******42m4-\"FS7$$!37+++ArExIF E$!3y*****\\JG\"R')F07$$!3'******>`g%G5FE$!3]******pyZoqF07$$!3%****** >)4\\'[#F:$!3!)*****>/`x\\&F07$$!38+++0g$)HN!#C$!3\")*****fz$)p#RF07$$ !3.+++#\\'*=s\"F4$!3)******\\!R>cBF07$$!3.+++\\jA:C!#G$!3g+++4;)R&yF_q 7$$!3*)*****HME_T#Fcen$\"3g+++4;)R&yF_q-%&COLORG6&%$RGBG$\"#D!\"#$\"\" )Fafn$\"\"%!\"\"-%)POLYGONSG6^y7%F'7$$!+.2)G`\"F0$\"+[izq:!#57$$FafnF) F(7%F[gn7$$!+`Z'z[*FS$\"+h-fTJF`gnFagn7%Fdgn7$$!+DQEA5!#:$\"+=BP7ZF`gn Fagn7%Fjgn7$$!+\"*p[:`F]hn$\"+TF7$G'F`gnFagn7%Fahn7$$!+4guP=!#9$\"+gX \"Q&yF`gnFagn7%Fghn7$$!+k_2#)[Fjhn$\"+cKTC%*F`gnFagn7%F^in7$$!+!['Q!3 \"!#8$\"+&p&[*4\"!\"*Fagn7%Fdin7$$!+9%yJ5#Fgin$\"+G&*\\c7FjinFagn7%F\\ jn7$$!+'Rbkw$Fgin$\"+bnW89FjinFagn7%Fbjn7$$!+&)H7tkFgin$\"+_9Dq:FjinFa gn7%Fhjn7$$!+=Sl46!#7$\"+)funs\"FjinFagn7%F^[o7$$!+$Hkj%>Fa[o$\"+*\\#z #)=FjinFagn7%Fe[o7$$!+-i([]$Fa[o$\"+Qg)z.#FjinFagn7%F[\\o7$$!+2j\\GZFa [o$\"+(G\\^6#FjinFagn7%Fa\\o7$$!+/2pyjFa[o$\"+7z%>>#FjinFagn7%Fg\\o7$$ !+U2;!e)Fa[o$\"+P1KoAFjinFagn7%F]]o7$$!+tJ/[6!#6$\"+$z-UM#FjinFagn7%Fc ]o7$$!+7R)\\_\"Ff]o$\"+2b_>CFjinFagn7%Fj]o7$$!+J&Q!3?Ff]o$\"+>N@%\\#Fj inFagn7%F`^o7$$!+o;K=EFf]o$\"+*G\"=oDFjinFagn7%Ff^o7$$!+ZLVyLFf]o$\"+` sKTEFjinFagn7%F\\_o7$$!+b\\z6VFf]o$\"+[k_8FFjinFagn7%Fb_o7$$!+kZnTaFf] o$\"+\"GAYy#FjinFagn7%Fh_o7$$!+r0$)*y'Ff]o$\"+!)*=W&GFjinFagn7%F^`o7$$ !+]juu$)Ff]o$\"+=rnAHFjinFagn7%Fd`o7$$!+Eg&4-\"F`gn$\"+N_6*)HFjinFagn7 %Fj`o7$$!+*3(**H7F`gn$\"+*R?M0$FjinFagn7%F`ao7$$!+*[gUY\"F`gn$\"+\"zo_ 6$FjinFagn7%Ffao7$$!+A.WAaV<$FjinFagn7%F\\bo7$$!+=4M-?F`gn $\"+P&=/B$FjinFagn7%Fbbo7$$!+PA4,BF`gn$\"+()oF$G$FjinFagn7%Fhbo7$$!+;; X:EF`gn$\"+o0$GL$FjinFagn7%F^co7$$!+^m6UHF`gn$\"+6l1zLFjinFagn7%Fdco7$ $!+bG)zF$F`gn$\"+6`/AMFjinFagn7%Fjco7$$!+U9J?OF`gn$\"+!4$)=Y$FjinFagn7 %F`do7$$!+.j!o'RF`gn$\"+%=K()\\$FjinFagn7%Ffdo7$$!+vvJlYF`gn$\"+\\l:kN FjinFagn7%F\\eo7$$!+UTJj`F`gn$\"+q?r>OFjinFagn7%Fbeo7$$!+zbAbgF`gn$\"+ CDnmOFjinFagn7%Fheo7$$!+h,OQnF`gn$\"+?\"4hq$FjinFagn7%F^fo7$$!+hEm6uF` gn$\"+'Q*))QPFjinFagn7%Fdfo7$$!+]([\\2)F`gn$\"+H^qlPFjinFagn7%Fjfo7$$! +kcYG()F`gn$\"+Mb5(y$FjinFagn7%F`go7$$!+m:ls$*F`gn$\"+Bn_.QFjinFagn7%F fgo7$$!+#)4!3+\"Fjin$\"+,dJ:QFjinFagn7%F\\ho7$$!+A[]j5Fjin$\"+_*[F#QFj inFagn7%Fbho7$$!+zaUD6Fjin$\"+dm/EQFjinFagn7%Fhho7$$!+@%4m=\"Fjin$\"+1 NQDQFjinFagn7%F^io7$$!+U;5Z7Fjin$\"+qp*3#QFjinFagn7%Fdio7$$!+#HZpI\"Fj in$\"+xOp7QFjinFagn7%Fjio7$$!+aQ>m8Fjin$\"+sW&3!QFjinFagn7%F`jo7$$!+:Q *[U\"Fjin$\"+l&Qay$FjinFagn7%Ffjo7$$!+;y5$[\"Fjin$\"+6s[mPFjinFagn7%F \\[p7$$!+M$34a\"Fjin$\"+(HFSu$FjinFagn7%Fb[p7$$!+@RQ)f\"Fjin$\"+4`2=PF jinFagn7%Fh[p7$$!+()Rkb;Fjin$\"+MAk)o$FjinFagn7%F^\\p7$$!+GR#Gr\"Fjin$ \"+#4Sdl$FjinFagn7%Fd\\p7$$!+P,4qOFjinFagn7%Fj\\p7$$!+_Qk F=Fjin$\"+'[R'zNFjinFagn7%F`]p7$$!+h>s&)=Fjin$\"++'pl`$FjinFagn7%Ff]p7 $$!+YFjin$\"+ScK!\\$FjinFagn7%F\\^p7$$!+Xa`/?Fjin$\"+df8TMFjinFagn 7%Fb^p7$$!+30#e1#Fjin$\"+aRL*Q$FjinFagn7%Fh^p7$$!+vTlG@Fjin$\"+LrONLFj inFagn7%F^_p7$$!+C\"GJ>#Fjin$\"+\"=#yzKFjinFagn7%Fd_p7$$!+*)*o\"fAFjin $\"+3z%em#Fjin$\"+>.<&)GFjinFagn7%F^bp7$$!+/p#3t#Fjin$\"+MF%*HGFj inFagn7%Fdbp7$$!+be0%z#Fjin$\"+94kuFFjinFagn7%Fjbp7$$!+<]NbGFjin$\"+S5 />FFjinFagn7%F`cp7$$!+!G7Y\"HFjin$\"+6C$Hm#FjinFagn7%Ffcp7$$!+JBwrHFji n$\"+3\\71EFjinFagn7%F\\dp7$$!+4WxEIFjin$\"+T9X[DFjinFagn7%Fbdp7$$!+BI kzIFjin$\"+qvw*[#FjinFagn7%Fhdp7$$!+&G\"QIJFjin$\"+X(\\*HCFjinFagn7%F^ ep7$$!+#R;!zJFjin$\"+xI*)oBFjinFagn7%Fdep7$$!+[neDKFjin$\"+#=4lI#FjinF agn7%Fjep7$$!+(GS,F$Fjin$\"+!QCFC#FjinFagn7%F`fp7$$!+cOt7LFjin$\"+X%yu <#FjinFagn7%Fffp7$$!+S>V`LFjin$\"+!)Rs5@FjinFagn7%F\\gp7$$!+o'3BR$Fjin $\"+!\\EC/#FjinFagn7%Fbgp7$$!+MeWHMFjin$\"+p^cs>FjinFagn7%Fhgp7$$!+\\O $\\Y$Fjin$\"+\"QM6!>FjinFagn7%F^hp7$$!+R(p))\\$Fjin$\"+kd9G=FjinFagn7% Fdhp7$$!+euNJNFjin$\"+a2j`p>\"FjinFagn7%Fd[q7$$!+rZ0]PFjin$\"+.O696FjinFagn7%Fj[q7$$! +$ohGx$Fjin$\"+x?+J5FjinFagn7%F`\\q7$$!+$HUXz$Fjin$\"+-zQx%*F`gnFagn7% Ff\\q7$$!+go)\\\"QFjin$\"+#zIXk)F`gnFagn7%F\\]q7$$!+*esS$QFjin$\"+WmY7 yF`gnFagn7%Fb]q7$$!+e;n^QFjin$\"+#**G>)pF`gnFagn7%Fh]q7$$!+5$ew'QFjin$ \"+RJQ`hF`gnFagn7%F^^q7$$!+\\W\">)QFjin$\"+#4dqK&F`gnFagn7%Fd^q7$$!+\" fLV*QFjin$\"+8.)H]%F`gnFagn7%Fj^q7$$!+AL#[!RFjin$\"+%*R-\"o$F`gnFagn7% F`_q7$$!+hlI8RFjin$\"+x&Q4'GF`gnFagn7%Ff_q7$$!+,?s>RFjin$\"+tiQU?F`gnF agn7%F\\`q7$$!+^S-CRFjin$\"+;z'\\A\"F`gnFagn7%Fb`q7$$!+oD=ERFjin$\"+>i W#3%Ff]oFagn7%Fh`q7$Fi`q$!+>iW#3%Ff]oFagn7%F^aq7$Fc`q$!+;z'\\A\"F`gnFa gn7%Fbaq7$F]`q$!+tiQU?F`gnFagn7%Ffaq7$Fg_q$!+x&Q4'GF`gnFagn7%Fjaq7$Fa_ q$!+%*R-\"o$F`gnFagn7%F^bq7$F[_q$!+8.)H]%F`gnFagn7%Fbbq7$Fe^q$!+#4dqK& F`gnFagn7%Ffbq7$F_^q$!+RJQ`hF`gnFagn7%Fjbq7$Fi]q$!+#**G>)pF`gnFagn7%F^ cq7$Fc]q$!+WmY7yF`gnFagn7%Fbcq7$F]]q$!+#zIXk)F`gnFagn7%Ffcq7$Fg\\q$!+- zQx%*F`gnFagn7%Fjcq7$Fa\\q$!+x?+J5FjinFagn7%F^dq7$F[\\q$!+.O696FjinFag n7%Fbdq7$Fe[q$!+m*>p>\"FjinFagn7%Ffdq7$F_[q$!+P+Dz7FjinFagn7%Fjdq7$Fij p$!+4N#4O\"FjinFagn7%F^eq7$Fcjp$!++0wT9FjinFagn7%Fbeq7$F]jp$!+N3f@:Fji nFagn7%Ffeq7$Fgip$!+[F jinFagn7%Fjfq7$Figp$!+p^cs>FjinFagn7%F^gq7$Fcgp$!+!\\EC/#FjinFagn7%Fbg q7$F]gp$!+!)Rs5@FjinFagn7%Ffgq7$Fgfp$!+X%yu<#FjinFagn7%Fjgq7$Fafp$!+!Q CFC#FjinFagn7%F^hq7$F[fp$!+#=4lI#FjinFagn7%Fbhq7$Feep$!+xI*)oBFjinFagn 7%Ffhq7$F_ep$!+X(\\*HCFjinFagn7%Fjhq7$Fidp$!+qvw*[#FjinFagn7%F^iq7$Fcd p$!+T9X[DFjinFagn7%Fbiq7$F]dp$!+3\\71EFjinFagn7%Ffiq7$Fgcp$!+6C$Hm#Fji nFagn7%Fjiq7$Facp$!+S5/>FFjinFagn7%F^jq7$F[cp$!+94kuFFjinFagn7%Fbjq7$F ebp$!+MF%*HGFjinFagn7%Ffjq7$F_bp$!+>.<&)GFjinFagn7%Fjjq7$Fiap$!+D`aSHF jinFagn7%F^[r7$Fcap$!+-8E'*HFjinFagn7%Fb[r7$F]ap$!+yfW_IFjinFagn7%Ff[r 7$Fg`p$!+a`64JFjinFagn7%Fj[r7$Fa`p$!+&)38mJFjinFagn7%F^\\r7$F[`p$!+3z< BKFjinFagn7%Fb\\r7$Fe_p$!+\"=#yzKFjinFagn7%Ff\\r7$F__p$!+LrONLFjinFagn 7%Fj\\r7$Fi^p$!+aRL*Q$FjinFagn7%F^]r7$Fc^p$!+df8TMFjinFagn7%Fb]r7$F]^p $!+ScK!\\$FjinFagn7%Ff]r7$Fg]p$!++'pl`$FjinFagn7%Fj]r7$Fa]p$!+'[R'zNFj inFagn7%F^^r7$F[]p$!+k4R>OFjinFagn7%Fb^r7$Fe\\p$!+#4Sdl$FjinFagn7%Ff^r 7$F_\\p$!+MAk)o$FjinFagn7%Fj^r7$Fi[p$!+4`2=PFjinFagn7%F^_r7$Fc[p$!+(HF Su$FjinFagn7%Fb_r7$F][p$!+6s[mPFjinFagn7%Ff_r7$Fgjo$!+l&Qay$FjinFagn7% Fj_r7$Fajo$!+sW&3!QFjinFagn7%F^`r7$F[jo$!+xOp7QFjinFagn7%Fb`r7$Feio$!+ qp*3#QFjinFagn7%Ff`r7$F_io$!+1NQDQFjinFagn7%Fj`r7$Fiho$!+dm/EQFjinFagn 7%F^ar7$Fcho$!+_*[F#QFjinFagn7%Fbar7$F]ho$!+,dJ:QFjinFagn7%Ffar7$Fggo$ !+Bn_.QFjinFagn7%Fjar7$Fago$!+Mb5(y$FjinFagn7%F^br7$F[go$!+H^qlPFjinFa gn7%Fbbr7$Fefo$!+'Q*))QPFjinFagn7%Ffbr7$F_fo$!+?\"4hq$FjinFagn7%Fjbr7$ Fieo$!+CDnmOFjinFagn7%F^cr7$Fceo$!+q?r>OFjinFagn7%Fbcr7$F]eo$!+\\l:kNF jinFagn7%Ffcr7$$!+L.i:VF`gn$!+OTwKNFjinFagn7%Fjcr7$Fgdo$!+%=K()\\$Fjin Fagn7%F`dr7$Fado$!+!4$)=Y$FjinFagn7%Fddr7$F[do$!+6`/AMFjinFagn7%Fhdr7$ Feco$!+6l1zLFjinFagn7%F\\er7$F_co$!+o0$GL$FjinFagn7%F`er7$Fibo$!+()oF$ G$FjinFagn7%Fder7$Fcbo$!+P&=/B$FjinFagn7%Fher7$F]bo$!+\">aV<$FjinFagn7 %F\\fr7$Fgao$!+\"zo_6$FjinFagn7%F`fr7$Faao$!+*R?M0$FjinFagn7%Fdfr7$F[a o$!+N_6*)HFjinFagn7%Fhfr7$Fe`o$!+=rnAHFjinFagn7%F\\gr7$F_`o$!+!)*=W&GF jinFagn7%F`gr7$Fi_o$!+\"GAYy#FjinFagn7%Fdgr7$Fc_o$!+[k_8FFjinFagn7%Fhg r7$F]_o$!+`sKTEFjinFagn7%F\\hr7$Fg^o$!+*G\"=oDFjinFagn7%F`hr7$Fa^o$!+> N@%\\#FjinFagn7%Fdhr7$F[^o$!+2b_>CFjinFagn7%Fhhr7$Fd]o$!+$z-UM#FjinFag n7%F\\ir7$F^]o$!+P1KoAFjinFagn7%F`ir7$Fh\\o$!+7z%>>#FjinFagn7%Fdir7$Fb \\o$!+(G\\^6#FjinFagn7%Fhir7$$!+qr-0EFa[o$!+;f^g>FjinFagn7%F\\jr7$$!+0 HHk9Fa[o$!+-W'[!=FjinFagn7%Fbjr7$$!+QQgg%)Fgin$!+7+c[;FjinFagn7%Fhjr7$ $!+nBP]\\Fgin$!+[Q(=\\\"FjinFagn7%F^[s7$$!+W8XOGFgin$!+=])\\L\"FjinFag n7%Fd[s7$$!+@zIF:Fgin$!++!)*z<\"FjinFagn7%Fj[s7$$!+0cQ-uFjhn$!+rg'4-\" FjinFagn7%F`\\s7$$!+ArExIFjhn$!+:$G\"R')F`gnFagn7%Ff\\s7$$!+K0YG5Fjhn$ !+qyZoqF`gnFagn7%F\\]s7$$!+#)4\\'[#F]hn$!+UIv(\\&F`gnFagn7%Fb]s7$$!+0g $)HN!#;$!+'z$)p#RF`gnFagn7%Fh]s7$$!+#\\'*=s\"FS$!+0R>cBF`gnFagn7%F_^s7 $$!+\\jA:CF`o$!+4;)R&yFf]oFagn7%Fe^s7$$!+VjA:CF`o$\"+4;)R&yFf]oFagn-F \\fn6&F^fn$\"\"&Fffn$\"#:Fafn$FcfnFffn-%&STYLEG6#%,PATCHNOGRIDG-F$6$7i n7$$\"35+++$z,t\\\"FS$\"3A+++J%3![VFS7$$\"3#******>Nk&*[\"FS$\"33+++bm >dVFS7$$\"3!*******R:6\"[\"FS$\"3!******R*[rlVFS7$$\"3%******>v&)>Z\"F S$\"3#******RD@NP%FS7$$\"3#******>))HAY\"FS$\"3A+++eUd!Q%FS7$$\"31+++F w)=X\"FS$\"3=+++\\D$oQ%FS7$$\"3,+++HQ+T9FS$\"3;+++5YD#R%FS7$$\"36+++g[ iH9FS$\"3!)*****p\\)z'R%FS7$$\"3-+++n!*z<9FS$\"3k******y:U+WFS7$$\"3&* *****4HxbS\"FS$\"3t*****fH!3.WFS7$$\"3))*****HZ8IR\"FS$\"30+++<*HZS%FS 7$$\"3#*******y`;!Q\"FS$\"3,+++IVK0WFS7$$\"31+++ja4n8FS$\"39+++[e\"[S% FS7$$\"3!******z!>(QN\"FS$\"3#******\\2bJS%FS7$$\"3&******p*)p0M\"FS$ \"3$******f$3H+WFS7$$\"3*******4:tsK\"FS$\"3!)*****pAqhR%FS7$$\"3,+++? e298FS$\"31+++i)Q2R%FS7$$\"3,+++,[3,8FS$\"3;+++)RTRQ%FS7$$\"3)******Hb A%)G\"FS$\"39+++@DsvVFS7$$\"3/+++K0Bw7FS$\"3G+++:'GgO%FS7$$\"3!******z EtYE\"FS$\"3-+++x/\"[N%FS7$$\"3/+++5J%RD\"FS$\"3:+++%fF?M%FS7$$\"36+++ UaEW7FS$\"39+++LZlFVFS7$$\"30+++lQ!fB\"FS$\"3M+++M=p6VFS7$$\"3.+++dT;H 7FS$\"3-+++c%yTH%FS7$$\"33+++mZRC7FS$\"3i*****f,9_F%FS7$$\"3\"******>( )z>A\"FS$\"3O+++WV)\\D%FS7$$\"3'******R;>BA\"FS$\"3!)*****z\"HzLUFS7$$ \"3!******fU(yD7FS$\"37++++/47UFS7$$\"3(******z#*pEB\"FS$\"3'******4$* )[!>%FS7$$\"3%******f%*zIC\"FS$\"3I+++U9upTFS7$$\"33+++RE)oD\"FS$\"3o* ****4vp1:%FS7$$\"3%******RKjOF\"FS$\"3B+++pn/MTFS7$$\"3%******HtoFH\"F S$\"3=+++I,Z?TFS7$$\"3%******z*GT88FS$\"3x*****44z-6%FS7$$\"30+++cK![L \"FS$\"3K+++sTa.TFS7$$\"3-+++=#RiN\"FS$\"3\"******4?=,5%FS7$$\"3/+++;< ;x8FS$\"3)******pg8(*4%FS7$$\"3%*******[*frR\"FS$\"3P+++NY(>5%FS7$$\"3 3+++'=`fT\"FS$\"3!*******[(Hl5%FS7$$\"3'******zejLV\"FS$\"3%)*****\\6B I6%FS7$$\"31+++E&*G\\9FS$\"3l*****\\7I67%FS7$$\"3-+++fLoj9FS$\"3&***** *zvi08%FS7$$\"3#******\\XMlZ\"FS$\"3$******4Ip59%FS7$$\"3!*******po&y[ \"FS$\"3i******Q8V_TFS7$$\"3'******4m!o(\\\"FS$\"3;+++L(fW;%FS7$$\"3/+ ++7f/1:FS$\"3=+++k/*p<%FS7$$\"3\"*******o')*H^\"FS$\"3-+++y2))*=%FS7$$ \"3,+++D#)e=:FS$\"3G+++vd+.UFS7$$\"3'******zKlG_\"FS$\"3&)******)\\bi@ %FS7$$\"3(******f-\")e_\"FS$\"3u*****4ZK&HUFS7$$\"3/+++^foF:FS$\"3-+++ N'\\FC%FS7$$\"3%******p))H$G:FS$\"3S+++O'GeD%FS7$$\"3/+++\\:'y_\"FS$\" 3G+++%R)poUFS7$$\"3.+++)\\Gj_\"FS$\"3Q+++)*QH\"G%FS7$$\"3#******>9xP_ \"FS$\"3!******z?bNH%FS7$$\"31+++7GD?:FS$\"3#)*****>gEaI%FS7$$\"35+++s )*z::FS$\"3x*****>&e&oJ%FS7$$\"32+++n=Y5:FS$\"3=+++1OzFVFS7$$\"3(***** *\\p\"G/:FS$\"3E+++[G>QVFSF^`sF[fn-Fhfn6jn7%7$$\"+$z,t\\\"Fjin$\"+J%3! [VFjin7$$\"+_Vc*[\"Fjin$\"+bm>dVFjin7$$\"$K\"Fafn$\"$>%Fafn7%Fbct7$$\" +S:6\"[\"Fjin$\"+%*[rlVFjinFgct7%F]dt7$$\"+_d)>Z\"Fjin$\"+a7_tVFjinFgc t7%Fcdt7$$\"+#))HAY\"Fjin$\"+eUd!Q%FjinFgct7%Fidt7$$\"+Fw)=X\"Fjin$\"+ \\D$oQ%FjinFgct7%F_et7$$\"+HQ+T9Fjin$\"+5YD#R%FjinFgct7%Feet7$$\"+g[iH 9Fjin$\"+(\\)z'R%FjinFgct7%F[ft7$$\"+n!*z<9Fjin$\"+z:U+WFjinFgct7%Faft 7$$\"+\"HxbS\"Fjin$\"+'H!3.WFjinFgct7%Fgft7$$\"+tM,$R\"Fjin$\"+<*HZS%F jinFgct7%F]gt7$$\"+z`;!Q\"Fjin$\"+IVK0WFjinFgct7%Fcgt7$$\"+ja4n8Fjin$ \"+[e\"[S%FjinFgct7%Figt7$$\"+3>(QN\"Fjin$\"+v]:.WFjinFgct7%F_ht7$$\"+ (*)p0M\"Fjin$\"+O3H+WFjinFgct7%Feht7$$\"+^JFF8Fjin$\"+F-<'R%FjinFgct7% F[it7$$\"+?e298Fjin$\"+i)Q2R%FjinFgct7%Fait7$$\"+,[3,8Fjin$\"+)RTRQ%Fj inFgct7%Fgit7$$\"+`DU)G\"Fjin$\"+@DsvVFjinFgct7%F]jt7$$\"+K0Bw7Fjin$\" +:'GgO%FjinFgct7%Fcjt7$$\"+oKnk7Fjin$\"+x/\"[N%FjinFgct7%Fijt7$$\"+5J% RD\"Fjin$\"+%fF?M%FjinFgct7%F_[u7$$\"+UaEW7Fjin$\"+LZlFVFjinFgct7%Fe[u 7$$\"+lQ!fB\"Fjin$\"+M=p6VFjinFgct7%F[\\u7$$\"+dT;H7Fjin$\"+c%yTH%Fjin Fgct7%Fa\\u7$$\"+mZRC7Fjin$\"+;S@vUFjinFgct7%Fg\\u7$$\"+s)z>A\"Fjin$\" +WV)\\D%FjinFgct7%F]]u7$$\"+k\">BA\"Fjin$\"+=HzLUFjinFgct7%Fc]u7$$\"+E uyD7Fjin$\"++/47UFjinFgct7%Fi]u7$$\"+G*pEB\"Fjin$\"+J*)[!>%FjinFgct7%F _^u7$$\"+Y*zIC\"Fjin$\"+U9upTFjinFgct7%Fe^u7$$\"+RE)oD\"Fjin$\"+^(p1:% FjinFgct7%F[_u7$$\"+CLmt7Fjin$\"+pn/MTFjinFgct7%Fa_u7$$\"+L(oFH\"Fjin$ \"+I,Z?TFjinFgct7%Fg_u7$$\"+)*GT88Fjin$\"+\"4z-6%FjinFgct7%F]`u7$$\"+c K![L\"Fjin$\"+sTa.TFjinFgct7%Fc`u7$$\"+=#RiN\"Fjin$\"+,#=,5%FjinFgct7% Fi`u7$$\"+;<;x8Fjin$\"+2Or*4%FjinFgct7%F_au7$$\"+\\*frR\"Fjin$\"+NY(>5 %FjinFgct7%Feau7$$\"+'=`fT\"Fjin$\"+\\(Hl5%FjinFgct7%F[bu7$$\"+)ejLV\" Fjin$\"+:J-8TFjinFgct7%Fabu7$$\"+E&*G\\9Fjin$\"+D,8@TFjinFgct7%Fgbu7$$ \"+fLoj9Fjin$\"+eFcITFjinFgct7%F]cu7$$\"+bW`w9Fjin$\"+,$p59%FjinFgct7% Fccu7$$\"+qo&y[\"Fjin$\"+R8V_TFjinFgct7%Ficu7$$\"+h1o(\\\"Fjin$\"+L(fW ;%FjinFgct7%F_du7$$\"+7f/1:Fjin$\"+k/*p<%FjinFgct7%Fedu7$$\"+p')*H^\"F jin$\"+y2))*=%FjinFgct7%F[eu7$$\"+D#)e=:Fjin$\"+vd+.UFjinFgct7%Faeu7$$ \"+G`'G_\"Fjin$\"+*\\bi@%FjinFgct7%Fgeu7$$\"+E5)e_\"Fjin$\"+rC`HUFjinF gct7%F]fu7$$\"+^foF:Fjin$\"+N'\\FC%FjinFgct7%Fcfu7$$\"+())H$G:Fjin$\"+ O'GeD%FjinFgct7%Fifu7$$\"+\\:'y_\"Fjin$\"+%R)poUFjinFgct7%F_gu7$$\"+) \\Gj_\"Fjin$\"+)*QH\"G%FjinFgct7%Fegu7$$\"+UrxB:Fjin$\"+3_b$H%FjinFgct 7%F[hu7$$\"+7GD?:Fjin$\"+-mU0VFjinFgct7%Fahu7$$\"+s)*z::Fjin$\"+_e&oJ% FjinFgct7%Fghu7$$\"+n=Y5:Fjin$\"+1OzFVFjinFgct7%F]iu7$$\"+&p\"G/:Fjin$ \"+[G>QVFjinFgct7%FciuF]ctFgctF`_sFg_s-F$6$7in7$F_`s$!3A+++J%3![VFS7$F fbt$!3E+++[G>QVFS7$Fabt$!3=+++1OzFVFS7$F\\bt$!3x*****>&e&oJ%FS7$Fgat$! 3#)*****>gEaI%FS7$Fbat$!3!******z?bNH%FS7$F]at$!3Q+++)*QH\"G%FS7$Fh`t$ !3G+++%R)poUFS7$Fc`t$!3S+++O'GeD%FS7$F^`t$!3-+++N'\\FC%FS7$Fi_t$!3u*** **4ZK&HUFS7$Fd_t$!3&)******)\\bi@%FS7$F__t$!3G+++vd+.UFS7$Fj^t$!3-+++y 2))*=%FS7$Fe^t$!3=+++k/*p<%FS7$F`^t$!3;+++L(fW;%FS7$F[^t$!3i******Q8V_ TFS7$Ff]t$!3$******4Ip59%FS7$Fa]t$!3&******zvi08%FS7$F\\]t$!3l*****\\7 I67%FS7$Fg\\t$!3%)*****\\6BI6%FS7$Fb\\t$!3!*******[(Hl5%FS7$F]\\t$!3P+ ++NY(>5%FS7$Fh[t$!3)******pg8(*4%FS7$Fc[t$!3\"******4?=,5%FS7$F^[t$!3K +++sTa.TFS7$Fijs$!3x*****44z-6%FS7$Fdjs$!3=+++I,Z?TFS7$F_js$!3B+++pn/M TFS7$Fjis$!3o*****4vp1:%FS7$Feis$!3I+++U9upTFS7$F`is$!3'******4$*)[!>% FS7$F[is$!37++++/47UFS7$Ffhs$!3!)*****z\"HzLUFS7$Fahs$!3O+++WV)\\D%FS7 $F\\hs$!3i*****f,9_F%FS7$Fggs$!3-+++c%yTH%FS7$Fbgs$!3M+++M=p6VFS7$F]gs $!39+++LZlFVFS7$Fhfs$!3:+++%fF?M%FS7$Fcfs$!3-+++x/\"[N%FS7$F^fs$!3G+++ :'GgO%FS7$Fies$!39+++@DsvVFS7$Fdes$!3;+++)RTRQ%FS7$F_es$!31+++i)Q2R%FS 7$Fjds$!3!)*****pAqhR%FS7$Feds$!3$******f$3H+WFS7$F`ds$!3#******\\2bJS %FS7$F[ds$!39+++[e\"[S%FS7$Ffcs$!3,+++IVK0WFS7$Facs$!30+++<*HZS%FS7$F \\cs$!3t*****fH!3.WFS7$Fgbs$!3k******y:U+WFS7$Fbbs$!3!)*****p\\)z'R%FS 7$F]bs$!3;+++5YD#R%FS7$Fhas$!3=+++\\D$oQ%FS7$Fcas$!3A+++eUd!Q%FS7$F^as $!3#******RD@NP%FS7$Fi`s$!3!******R*[rlVFS7$Fd`s$!33+++bm>dVFSF\\juF[f n-Fhfn6jn7%7$F^ct$!+J%3![VFjin7$Fdiu$!+[G>QVFjin7$Fhct$!$>%Fafn7%Ffev7 $F^iu$!+1OzFVFjinFiev7%F]fv7$Fhhu$!+_e&oJ%FjinFiev7%Fafv7$Fbhu$!+-mU0V FjinFiev7%Fefv7$F\\hu$!+3_b$H%FjinFiev7%Fifv7$Ffgu$!+)*QH\"G%FjinFiev7 %F]gv7$F`gu$!+%R)poUFjinFiev7%Fagv7$Fjfu$!+O'GeD%FjinFiev7%Fegv7$Fdfu$ !+N'\\FC%FjinFiev7%Figv7$F^fu$!+rC`HUFjinFiev7%F]hv7$Fheu$!+*\\bi@%Fji nFiev7%Fahv7$Fbeu$!+vd+.UFjinFiev7%Fehv7$F\\eu$!+y2))*=%FjinFiev7%Fihv 7$Ffdu$!+k/*p<%FjinFiev7%F]iv7$F`du$!+L(fW;%FjinFiev7%Faiv7$Fjcu$!+R8V _TFjinFiev7%Feiv7$Fdcu$!+,$p59%FjinFiev7%Fiiv7$F^cu$!+eFcITFjinFiev7%F ]jv7$Fhbu$!+D,8@TFjinFiev7%Fajv7$Fbbu$!+:J-8TFjinFiev7%Fejv7$F\\bu$!+ \\(Hl5%FjinFiev7%Fijv7$Ffau$!+NY(>5%FjinFiev7%F][w7$F`au$!+2Or*4%FjinF iev7%Fa[w7$Fj`u$!+,#=,5%FjinFiev7%Fe[w7$Fd`u$!+sTa.TFjinFiev7%Fi[w7$F^ `u$!+\"4z-6%FjinFiev7%F]\\w7$Fh_u$!+I,Z?TFjinFiev7%Fa\\w7$Fb_u$!+pn/MT FjinFiev7%Fe\\w7$F\\_u$!+^(p1:%FjinFiev7%Fi\\w7$Ff^u$!+U9upTFjinFiev7% F]]w7$F`^u$!+J*)[!>%FjinFiev7%Fa]w7$Fj]u$!++/47UFjinFiev7%Fe]w7$Fd]u$! +=HzLUFjinFiev7%Fi]w7$F^]u$!+WV)\\D%FjinFiev7%F]^w7$Fh\\u$!+;S@vUFjinF iev7%Fa^w7$Fb\\u$!+c%yTH%FjinFiev7%Fe^w7$F\\\\u$!+M=p6VFjinFiev7%Fi^w7 $Ff[u$!+LZlFVFjinFiev7%F]_w7$F`[u$!+%fF?M%FjinFiev7%Fa_w7$Fjjt$!+x/\"[ N%FjinFiev7%Fe_w7$Fdjt$!+:'GgO%FjinFiev7%Fi_w7$F^jt$!+@DsvVFjinFiev7%F ]`w7$Fhit$!+)RTRQ%FjinFiev7%Fa`w7$Fbit$!+i)Q2R%FjinFiev7%Fe`w7$F\\it$! +F-<'R%FjinFiev7%Fi`w7$Ffht$!+O3H+WFjinFiev7%F]aw7$F`ht$!+v]:.WFjinFie v7%Faaw7$Fjgt$!+[e\"[S%FjinFiev7%Feaw7$Fdgt$!+IVK0WFjinFiev7%Fiaw7$F^g t$!+<*HZS%FjinFiev7%F]bw7$Fhft$!+'H!3.WFjinFiev7%Fabw7$Fbft$!+z:U+WFji nFiev7%Febw7$F\\ft$!+(\\)z'R%FjinFiev7%Fibw7$Ffet$!+5YD#R%FjinFiev7%F] cw7$F`et$!+\\D$oQ%FjinFiev7%Facw7$Fjdt$!+eUd!Q%FjinFiev7%Fecw7$Fddt$!+ a7_tVFjinFiev7%Ficw7$F^dt$!+%*[rlVFjinFiev7%F]dw7$Fcct$!+bm>dVFjinFiev 7%FadwFcevFievF`_sFg_s-F$6%7$7$$!3/++++++!z%FSF(7$$\"3+++++++!*>FSF(-% 'COLOURG6&F^fnF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(Fidw7$F($\"3/++++++ !z%FSF^ewFaew-%%FONTG6$%*HELVETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%+ AXESLABELSG6%%&Re(z)G%&Im(z)G-F]fw6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%% VIEWG6$;$!$z%Fafn$\"$*>Fafn;Fegw$\"$z%Fafn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stabil ity interval" }{TEXT -1 37 " for this scheme is (approximately) " } {XPPEDIT 18 0 "[-3.9265, 0];" "6#7$,$-%&FloatG6$\"&l#R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The following picture compares the stability region for \+ the 8 stage, order 4 scheme with that of the order 5 scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),p _||(1..6)],view=[-4.79..2.19,-4.79..4.79],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "65-%'CURVESG6$7ew7 $$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3:+++%fK;.$!#E$\"3s*****4n#fTJF-7$ $\"32+++;%z!Qu!#D$\"3C+++:\"*Q7ZF-7$$\"3S+++@b2vp!#C$\"3')*****Hq%=$G' F-7$$\"3l*****\\,%G=Q!#B$\"3)******4QpR&yF-7$$\"3++++k&3%p9!#A$\"3E+++ YdpC%*F-7$$\"3A+++uwkrVFI$\"3#*******G5_*4\"!#<7$$\"3%*******4d9d5!#@$ \"3-+++b@^c7FQ7$$\"37+++Y]o?@FU$\"30+++y!fLT\"FQ7$$\"3y*****fqd\"*[$FU $\"3))*****f_0*p:FQ7$$\"34+++YzAVVFU$\"3.+++dR*es\"FQ7$$\"3B+++gW*4^#F U$\"3++++nk%4)=FQ7$$!3')*****z0xqP'FU$\"3)******H!)oX.#FQ7$$!3')*****f $*='oH!#?$\"3')******[F='=#FQ7$$!3%3++g)Rj?yF^p$\"3.+++X-$\"3))*****\\K()3[#FQ7$$!3#******\\w!)44$Fip$\"3')***** fh,Ei#FQ7$$!3?+++Xc7U_Fip$\"3%)*******HV$fFFQ7$$!3-+++>/D)Fip$\"3!** ****phQ(*)GFQ7$$!3,+++&>T%=7F-$\"36+++a5*>,$FQ7$$!3))******oUq+o#=CJFQ7$$!3)******zgcuD#F-$\"36+++f)Q[A$FQ7$$!3/+++*y$=oGF-$ \"3\")*****RB!Q8LFQ7$$!3)******pipC^$F-$\"31+++7J9!R$FQ7$$!3#******Hwj S<%F-$\"3%)******>y0cMFQ7$$!3!*******4M$=%[F-$\"3)******>`zA^$FQ7$$!3] +++We!*3bF-$\"35+++\"zU*fNFQ7$$!3i******e$p8<'F-$\"3:+++v)[+g$FQ7$$!3O +++ycvR![()F-$\"33+++-a[*p$FQ7$$!3 Y*****4Y4FP*F-$\"3:+++j#G;r$FQ7$$!3S+++e9&**)**F-$\"3*******>#)p$>PFQ7 $$!31+++Y(3+1\"FQ$\"3z*****\\-PHs$FQ7$$!3,+++jhM?6FQ$\"3!*******3,^APF Q7$$!3)******f)Q/!=\"FQ$\"3,+++#yI#=PFQ7$$!3.+++899R7FQ$\"3$*******3'3 -r$FQ7$$!3%******H+$o(H\"FQ$\"33+++!4E&)p$FQ7$$!35+++F,sb8FQ$\"31+++mG C$o$FQ7$$!32+++*e9LT\"FQ$\"37+++U%*RkOFQ7$$!3%******ROU0Z\"FQ$\"3;+++r 3-UOFQ7$$!3$******R=)\\F:FQ$\"3z*****H)37;OFQ7$$!3#*******f3I%e\"FQ$\" 3)******f52ne$FQ7$$!33+++x\"*4T;FQ$\"39+++,#)y`NFQ7$$!35+++&zx!)p\"FQ$ \"3%*******oRQYbcXVV$FQ7$$!3/+++^FQ$\"3&)*****Hm['QLFQ7$$!3)******zm2[*>FQ$\"3%******4D)y'G$FQ7 $$!3'*******zybe?FQ$\"3,+++wy*GB$FQ7$$!36+++RR@C@FQ$\"3y*****zd1w<$FQ7 $$!3=+++Zsl\">#FQ$\"32+++sFf@JFQ7$$!3?+++z(\\0E#FQ$\"3/+++3'*[lIFQ7$$! 32+++g=PIBFQ$\"35+++_(*y4IFQ7$$!30+++-]]+CFQ$\"39+++tjyaHFQ7$$!3#***** *\\;E.Z#FQ$\"36+++kyc+HFQ7$$!3=+++y2GRDFQ$\"3))*****p,cq%GFQ7$$!3$)*** ***f0#pg#FQ$\"3')*****R&G1%z#FQ7$$!35+++[/\"Hn#FQ$\"3!******HYW8u#FQ7$ $!39+++dn,PFFQ$\"35+++57k)o#FQ7$$!3/+++:y3*z#FQ$\"38+++!*GqNEFQ7$$!3)) *****f:M!fGFQ$\"33+++j=I#e#FQ7$$!3$******Hd6o\"HFQ$\"3)******p=#FQ7$$!3 !)*****>v!phKFQ$\"32+++w$R(>@FQ7$$!30+++Q`)HI$FQ$\"31+++'3Ng0#FQ7$$!33 +++(fzCM$FQ$\"3#******R:W3*>FQ7$$!3%)*****z*\\D!Q$FQ$\"3$******4#*\\T# >FQ7$$!3y*****pg*R;MFQ$\"3-+++c,&f&=FQ7$$!3!******RR25X$FQ$\"3!******H \")eiy\"FQ7$$!3%******RvwT[$FQ$\"3)******f<3^r\"FQ7$$!3A+++*)z+;NFQ$\" 34+++=6bU;FQ7$$!39+++-'*fYNFQ$\"3!******>.j'o:FQ7$$!3!)*****\\PVgd$FQ$ \"35+++$*Ga$\\\"FQ7$$!3\"*******4&=Wg$FQ$\"32+++UGJ<9FQ7$$!34+++z`yJOF Q$\"3*)*****\\x:,M\"FQ7$$!3-+++S(z\"eOFQ$\"3!*******\\16i7FQ7$$!3!**** **HU3Oo$FQ$\"3'******fnlM=\"FQ7$$!3()*****HPZ!3PFQ$\"3(******f/]V5\"FQ 7$$!3-+++qDWJPFQ$\"3!******zDD\\-\"FQ7$$!33+++\"y8Pv$FQ$\"3a+++_EP`%*F -7$$!3/+++D0wuPFQ$\"3;+++ZY5d')F-7$$!3++++a#pWz$FQ$\"3w*****feT9'yF-7$ $!3))*****\\p?F\"QFQ$\"3/+++I`6nqF-7$$!3:+++ZkRHQFQ$\"3M+++lyguiF-7$$! 3.+++)H%QWQFQ$\"3-+++dw<%[&F-7$$!3/+++c?edQFQ$\"3A+++&=%*ep%F-7$$!3/++ +&****)oQFQ$\"3B+++XZn4RF-7$$!35+++`@EyQFQ$\"3,+++K$>`7$F-7$$!37+++;og &)QFQ$\"3\"******4?TDM#F-7$$!3A+++&[')3*QFQ$\"3)******>A#*4c\"F-7$$!3- +++%fnS*QFQ$\"3V+++d>$G!yFip7$$!3++++s,8&*QFQF(7$Fgjl$!3V+++d>$G!yFip7 $Fbjl$!3)******>A#*4c\"F-7$F]jl$!3\"******4?TDM#F-7$Fhil$!3,+++K$>`7$F -7$Fcil$!3B+++XZn4RF-7$F^il$!3A+++&=%*ep%F-7$Fihl$!3-+++dw<%[&F-7$Fdhl $!3M+++lyguiF-7$F_hl$!3/+++I`6nqF-7$Fjgl$!3w*****feT9'yF-7$Fegl$!3;+++ ZY5d')F-7$F`gl$!3a+++_EP`%*F-7$F[gl$!3!******zDD\\-\"FQ7$Fffl$!3(***** *f/]V5\"FQ7$Fafl$!3'******fnlM=\"FQ7$F\\fl$!3!*******\\16i7FQ7$Fgel$!3 *)*****\\x:,M\"FQ7$Fbel$!32+++UGJ<9FQ7$F]el$!35+++$*Ga$\\\"FQ7$Fhdl$!3 !******>.j'o:FQ7$Fcdl$!34+++=6bU;FQ7$F^dl$!3)******f<3^r\"FQ7$Ficl$!3! ******H\")eiy\"FQ7$Fdcl$!3-+++c,&f&=FQ7$F_cl$!3$******4#*\\T#>FQ7$Fjbl $!3#******R:W3*>FQ7$Febl$!31+++'3Ng0#FQ7$F`bl$!32+++w$R(>@FQ7$F[bl$!3' ******4$p)>=#FQ7$Ffal$!3)******\\P?GC#FQ7$Faal$!3++++J_H-BFQ7$F\\al$!3 =+++(y![gBFQ7$Fg`l$!3,+++i;YcXVV$FQ7$Fcz$!3#)*****flRvZ$FQ7$F^z$!3%* ******oRQ#)p$>PFQ7$F]v$! 3:+++j#G;r$FQ7$Fhu$!33+++-a[*p$FQ7$Fcu$!3%)*****H/dEo$FQ7$F^u$!3/+++3# )ygOFQ7$Fit$!3@+++GgVLOFQ7$Fdt$!3:+++v)[+g$FQ7$F_t$!35+++\"zU*fNFQ7$Fj s$!3)******>`zA^$FQ7$Fes$!3%)******>y0cMFQ7$F`s$!31+++7J9!R$FQ7$F[s$!3 \")*****RB!Q8LFQ7$Ffr$!36+++f)Q[A$FQ7$Far$!3#)*****>o#=CJFQ7$F\\r$!36+ ++a5*>,$FQ7$Fgq$!3!******phQ(*)GFQ7$Fbq$!3%)*******HV$fFFQ7$F]q$!3')** ***fh,Ei#FQ7$Fgp$!3))*****\\K()3[#FQ7$Fbp$!3.+++X-T%=7F`_n$\"+a5*>,$FjanFa_n7%Fgfn7$$!+pUq+PFjanFa_n7%F[\\o7$$!+Y(3+1\"Fjan$\"+Dq$Hs$FjanFa_n7%Fa\\o7$$!+ jhM?6Fjan$\"+4,^APFjanFa_n7%Fg\\o7$$!+')Q/!=\"Fjan$\"+#yI#=PFjanFa_n7% F]]o7$$!+899R7Fjan$\"+4'3-r$FjanFa_n7%Fc]o7$$!+.Io(H\"Fjan$\"+!4E&)p$F janFa_n7%Fi]o7$$!+F,sb8Fjan$\"+mGC$o$FjanFa_n7%F_^o7$$!+*e9LT\"Fjan$\" +U%*RkOFjanFa_n7%Fe^o7$$!+kBaq9Fjan$\"+r3-UOFjanFa_n7%F[_o7$$!+%=)\\F: Fjan$\"+$)37;OFjanFa_n7%Fa_o7$$!+g3I%e\"Fjan$\"+1rq'e$FjanFa_n7%Fg_o7$ $!+x\"*4T;Fjan$\"+,#)y`NFjanFa_n7%F]`o7$$!+&zx!)p\"Fjan$\"+pRQYbFjan $\"+j'['QLFjanFa_n7%F[bo7$$!+ow![*>Fjan$\"+^#)y'G$FjanFa_n7%Fabo7$$!+! )ybe?Fjan$\"+wy*GB$FjanFa_n7%Fgbo7$$!+RR@C@Fjan$\"+ylgxJFjanFa_n7%F]co 7$$!+Zsl\">#Fjan$\"+sFf@JFjanFa_n7%Fcco7$$!+z(\\0E#Fjan$\"+3'*[lIFjanF a_n7%Fico7$$!+g=PIBFjan$\"+_(*y4IFjanFa_n7%F_do7$$!+-]]+CFjan$\"+tjyaH FjanFa_n7%Fedo7$$!+lhKqCFjan$\"+kyc+HFjanFa_n7%F[eo7$$!+y2GRDFjan$\"+< g0ZGFjanFa_n7%Faeo7$$!+g0#pg#Fjan$\"+aG1%z#FjanFa_n7%Fgeo7$$!+[/\"Hn#F jan$\"+jWMTFFjanFa_n7%F]fo7$$!+dn,PFFjan$\"+57k)o#FjanFa_n7%Fcfo7$$!+: y3*z#Fjan$\"+!*GqNEFjanFa_n7%Fifo7$$!+cT.fGFjan$\"+j=I#e#FjanFa_n7%F_g o7$$!+t:\"o\"HFjan$\"+x\"Q#GDFjanFa_n7%Fego7$$!+#)zSsHFjan$\"+'**RLZ#F janFa_n7%F[ho7$$!+sQ$e-$Fjan$\"+i;Y=#FjanFa_n7%Fiio7$$!+_2 phKFjan$\"+w$R(>@FjanFa_n7%F_jo7$$!+Q`)HI$Fjan$\"+'3Ng0#FjanFa_n7%Fejo 7$$!+(fzCM$Fjan$\"+aT%3*>FjanFa_n7%F[[p7$$!+)*\\D!Q$Fjan$\"+@*\\T#>Fja nFa_n7%Fa[p7$$!+2'*R;MFjan$\"+c,&f&=FjanFa_n7%Fg[p7$$!+%R25X$Fjan$\"+8 )eiy\"FjanFa_n7%F]\\p7$$!+an<%[$Fjan$\"+w\"3^r\"FjanFa_n7%Fc\\p7$$!+*) z+;NFjan$\"+=6bU;FjanFa_n7%Fi\\p7$$!+-'*fYNFjan$\"+KImo:FjanFa_n7%F_]p 7$$!+vL/wNFjan$\"+$*Ga$\\\"FjanFa_n7%Fe]p7$$!+5&=Wg$Fjan$\"+UGJ<9FjanF a_n7%F[^p7$$!+z`yJOFjan$\"+vd6S8FjanFa_n7%Fa^p7$$!+S(z\"eOFjan$\"+]16i 7FjanFa_n7%Fg^p7$$!+B%3Oo$Fjan$\"+wcY$=\"FjanFa_n7%F]_p7$$!+tt/3PFjan$ \"+Y+N/6FjanFa_n7%Fc_p7$$!+qDWJPFjan$\"+e_#\\-\"FjanFa_n7%Fi_p7$$!+\"y 8Pv$Fjan$\"+_EP`%*F`_nFa_n7%F_`p7$$!+D0wuPFjan$\"+ZY5d')F`_nFa_n7%Fe`p 7$$!+a#pWz$Fjan$\"+'eT9'yF`_nFa_n7%F[ap7$$!+&p?F\"QFjan$\"+I`6nqF`_nFa _n7%Faap7$$!+ZkRHQFjan$\"+lyguiF`_nFa_n7%Fgap7$$!+)H%QWQFjan$\"+dw<%[& F`_nFa_n7%F]bp7$$!+c?edQFjan$\"+&=%*ep%F`_nFa_n7%Fcbp7$$!+&****)oQFjan $\"+XZn4RF`_nFa_n7%Fibp7$$!+`@EyQFjan$\"+K$>`7$F`_nFa_n7%F_cp7$$!+;og& )QFjan$\"+,7aUBF`_nFa_n7%Fecp7$$!+&[')3*QFjan$\"+AA*4c\"F`_nFa_n7%F[dp 7$$!+%fnS*QFjan$\"+d>$G!yFaenFa_n7%Fadp7$$!+s,8&*QFjanF(Fa_n7%Fgdp7$Fb dp$!+d>$G!yFaenFa_n7%F[ep7$F\\dp$!+AA*4c\"F`_nFa_n7%F_ep7$Ffcp$!+,7aUB F`_nFa_n7%Fcep7$F`cp$!+K$>`7$F`_nFa_n7%Fgep7$Fjbp$!+XZn4RF`_nFa_n7%F[f p7$Fdbp$!+&=%*ep%F`_nFa_n7%F_fp7$F^bp$!+dw<%[&F`_nFa_n7%Fcfp7$Fhap$!+l yguiF`_nFa_n7%Fgfp7$Fbap$!+I`6nqF`_nFa_n7%F[gp7$F\\ap$!+'eT9'yF`_nFa_n 7%F_gp7$Ff`p$!+ZY5d')F`_nFa_n7%Fcgp7$F``p$!+_EP`%*F`_nFa_n7%Fggp7$Fj_p $!+e_#\\-\"FjanFa_n7%F[hp7$Fd_p$!+Y+N/6FjanFa_n7%F_hp7$F^_p$!+wcY$=\"F janFa_n7%Fchp7$Fh^p$!+]16i7FjanFa_n7%Fghp7$Fb^p$!+vd6S8FjanFa_n7%F[ip7 $F\\^p$!+UGJ<9FjanFa_n7%F_ip7$Ff]p$!+$*Ga$\\\"FjanFa_n7%Fcip7$F`]p$!+K Imo:FjanFa_n7%Fgip7$Fj\\p$!+=6bU;FjanFa_n7%F[jp7$Fd\\p$!+w\"3^r\"FjanF a_n7%F_jp7$F^\\p$!+8)eiy\"FjanFa_n7%Fcjp7$Fh[p$!+c,&f&=FjanFa_n7%Fgjp7 $Fb[p$!+@*\\T#>FjanFa_n7%F[[q7$F\\[p$!+aT%3*>FjanFa_n7%F_[q7$Ffjo$!+'3 Ng0#FjanFa_n7%Fc[q7$F`jo$!+w$R(>@FjanFa_n7%Fg[q7$Fjio$!+Jp)>=#FjanFa_n 7%F[\\q7$Fdio$!+v.#GC#FjanFa_n7%F_\\q7$F^io$!+J_H-BFjanFa_n7%Fc\\q7$Fh ho$!+(y![gBFjanFa_n7%Fg\\q7$Fbho$!+i;YPFjanFa_n7%F_eq7$Ff[o$!+j#G;r$FjanFa_n7%Fceq7$F`[o$! +-a[*p$FjanFa_n7%Fgeq7$Fjjn$!+Vql#o$FjanFa_n7%F[fq7$Fdjn$!+3#)ygOFjanF a_n7%F_fq7$F^jn$!+GgVLOFjanFa_n7%Fcfq7$Fhin$!+v)[+g$FjanFa_n7%Fgfq7$Fb in$!+\"zU*fNFjanFa_n7%F[gq7$F\\in$!+K&zA^$FjanFa_n7%F_gq7$Ffhn$!+?y0cM FjanFa_n7%Fcgq7$F`hn$!+7J9!R$FjanFa_n7%Fggq7$Fjgn$!+M-Q8LFjanFa_n7%F[h q7$Fdgn$!+f)Q[A$FjanFa_n7%F_hq7$F^gn$!+#o#=CJFjanFa_n7%Fchq7$Fhfn$!+a5 *>,$FjanFa_n7%Fghq7$Fbfn$!+<'Q(*)GFjanFa_n7%F[iq7$F\\fn$!++LMfFFjanFa_ n7%F_iq7$Ffen$!+;;gAEFjanFa_n7%Fciq7$F_en$!+Dt)3[#FjanFa_n7%Fgiq7$Fidn $!+X-.?=VFQ7$$\"3%******>IGq^\"FQ$\"32+++5I?DVFQ7$$\"3*******4\"4W2:FQ$ \"3:+++ccXJVFQ7$$\"31+++I(>t\\\"FQ$\"3++++Fz\"pL%FQ7$$\"3++++E4r'[\"FQ $\"3@+++z%\\:M%FQ7$$\"3!******RbicZ\"FQ$\"33+++Q(4`M%FQ7$$\"3/+++E]Ak9 FQ$\"3E+++&pd\"[VFQ7$$\"3!******pl^CX\"FQ$\"31+++C<0]VFQ7$$\"3-+++,$*R S9FQ$\"3q*****pV\\4N%FQ7$$\"31+++>\"H\"G9FQ$\"31+++uv!3N%FQ7$$\"3'**** **RZ2dT\"FQ$\"39+++]>e\\VFQ7$$\"3.+++=q?.9FQ$\"35+++$\\FsM%FQ7$$\"3-++ +\")zq!R\"FQ$\"3S+++/%)pVVFQ7$$\"3))*****>t*Hy8FQ$\"3l*****z][*QVFQ7$$ \"3!*******RE3m8FQ$\"3r*****H(=$HL%FQ7$$\"3'*******)HqTN\"FQ$\"3J+++SQ gDVFQ7$$\"36+++T@pU8FQ$\"3B+++PE#pJ%FQ7$$\"3#*******okzJ8FQ$\"3:+++]=& oI%FQ7$$\"34+++GQl@8FQ$\"3o*****>6k`H%FQ7$$\"31+++Y0Y78FQ$\"3/+++ClW#G %FQ7$$\"3\"******p)>W/8FQ$\"3-+++K%3\"oUFQ7$$\"35+++mY&yH\"FQ$\"3o**** *>H#R_UFQ7$$\"3/+++(o&)HH\"FQ$\"3t*****>G)QNUFQ7$$\"3-+++?q9!H\"FQ$\"3 !)*******z_s@%FQ7$$\"3!*******Q6m*G\"FQ$\"3K+++v'H#)>%FQ7$$\"3'******* HS$=H\"FQ$\"3K+++A.nyTFQ7$$\"3#******H08pH\"FQ$\"3A+++^_/fTFQ7$$\"35++ +w@.08FQ$\"3p*****4rN*RTFQ7$$\"3(******pIchJ\"FQ$\"36+++KZ*>7%FQ7$$\"3 1+++wb/I8FQ$\"3I+++'zve5%FQ7$$\"3%*******p1EY8FQ$\"3)******RCW@4%FQ7$$ \"3)******fP:UO\"FQ$\"3S+++R1?\"3%FQ7$$\"3#******pmfKQ\"FQ$\"39+++[ICt SFQ7$$\"3-+++.Mw-9FQ$\"3g******>fFoSFQ7$$\"31+++Vj>uSFQ7$$\"3!*******)**f;\\\"FQ$ \"3u*****4!*y13%FQ7$$\"3/+++QW)f]\"FQ$\"3G+++Qmf)3%FQ7$$\"3&******z!o \"*=:FQ$\"3++++/\")p(4%FQ7$$\"3.+++z1WI:FQ$\"3H+++fQw2TFQ7$$\"31+++8=c S:FQ$\"3f*****R`+'=TFQ7$$\"33+++(4,$\\:FQ$\"3=+++s#Q+8%FQ7$$\"3)****** H[*oc:FQ$\"3T+++ay#>9%FQ7$$\"3/+++;Wwi:FQ$\"3#)*****H(y8aTFQ7$$\"3#*** ***fMnvc\"FQ$\"3m*****R:_l;%FQ7$$\"33+++I?9r:FQ$\"3)******HYn!zTFQ7$$ \"3-+++nL`t:FQ$\"3%)*****f_\"f\">%FQ7$$\"3%******Hk'yu:FQ$\"3g*****pLT S?%FQ7$$\"35+++Vq%\\d\"FQ$\"3V+++:%zUb:FQ$\"3#)*****RX4NG%FQ7$$\"33+++ +:;\\:FQ$\"3?+++Z.5$H%FQFj^rF]^n-Fj^n6jn7%7$$\"+qi9U:Fjan$\"+[v5-VFjan 7$$\"+=TUM:Fjan$\"+\\\")[5VFjan7$$\"$V\"Ff^n$\"$@%Ff^n7%F^bs7$$\"+?s.E :Fjan$\"+?.?=VFjanFcbs7%Fibs7$$\"+-$Gq^\"Fjan$\"+5I?DVFjanFcbs7%F_cs7$ $\"+64W2:Fjan$\"+ccXJVFjanFcbs7%Fecs7$$\"+I(>t\\\"Fjan$\"+Fz\"pL%FjanF cbs7%F[ds7$$\"+E4r'[\"Fjan$\"+z%\\:M%FjanFcbs7%Fads7$$\"+aDmv9Fjan$\"+ Q(4`M%FjanFcbs7%Fgds7$$\"+E]Ak9Fjan$\"+&pd\"[VFjanFcbs7%F]es7$$\"+d;X_ 9Fjan$\"+C<0]VFjanFcbs7%Fces7$$\"+,$*RS9Fjan$\"+P%\\4N%FjanFcbs7%Fies7 $$\"+>\"H\"G9Fjan$\"+uv!3N%FjanFcbs7%F_fs7$$\"+uuq:9Fjan$\"+]>e\\VFjan Fcbs7%Fefs7$$\"+=q?.9Fjan$\"+$\\FsM%FjanFcbs7%F[gs7$$\"+\")zq!R\"Fjan$ \"+/%)pVVFjanFcbs7%Fags7$$\"+K(*Hy8Fjan$\"+3&[*QVFjanFcbs7%Fggs7$$\"+S E3m8Fjan$\"+t=$HL%FjanFcbs7%F]hs7$$\"+*HqTN\"Fjan$\"+SQgDVFjanFcbs7%Fc hs7$$\"+T@pU8Fjan$\"+PE#pJ%FjanFcbs7%Fihs7$$\"+pkzJ8Fjan$\"+]=&oI%Fjan Fcbs7%F_is7$$\"+GQl@8Fjan$\"+7TO&H%FjanFcbs7%Feis7$$\"+Y0Y78Fjan$\"+Cl W#G%FjanFcbs7%F[js7$$\"+()>W/8Fjan$\"+K%3\"oUFjanFcbs7%Fajs7$$\"+mY&yH \"Fjan$\"+#H#R_UFjanFcbs7%Fgjs7$$\"+(o&)HH\"Fjan$\"+#G)QNUFjanFcbs7%F] [t7$$\"+?q9!H\"Fjan$\"++GD% FjanFcbs7%Fi[t7$$\"+IS$=H\"Fjan$\"+A.nyTFjanFcbs7%F_\\t7$$\"+`I\"pH\"F jan$\"+^_/fTFjanFcbs7%Fe\\t7$$\"+w@.08Fjan$\"+6d$*RTFjanFcbs7%F[]t7$$ \"+2j:;8Fjan$\"+KZ*>7%FjanFcbs7%Fa]t7$$\"+wb/I8Fjan$\"+'zve5%FjanFcbs7 %Fg]t7$$\"+q1EY8Fjan$\"+WU9#4%FjanFcbs7%F]^t7$$\"+w`@k8Fjan$\"+R1?\"3% FjanFcbs7%Fc^t7$$\"+n'fKQ\"Fjan$\"+[ICtSFjanFcbs7%Fi^t7$$\"+.Mw-9Fjan$ \"+?fFoSFjanFcbs7%F__t7$$\"+VjF2f9Fjan$\"+e'4%pSFjanFcbs7%Fa`t7$ $\"+\"y\"*fZ\"Fjan$\"+O>>uSFjanFcbs7%Fg`t7$$\"+***f;\\\"Fjan$\"+,*y13% FjanFcbs7%F]at7$$\"+QW)f]\"Fjan$\"+Qmf)3%FjanFcbs7%Fcat7$$\"+3o\"*=:Fj an$\"+/\")p(4%FjanFcbs7%Fiat7$$\"+z1WI:Fjan$\"+fQw2TFjanFcbs7%F_bt7$$ \"+8=cS:Fjan$\"+M0g=TFjanFcbs7%Febt7$$\"+(4,$\\:Fjan$\"+s#Q+8%FjanFcbs 7%F[ct7$$\"+$[*oc:Fjan$\"+ay#>9%FjanFcbs7%Fact7$$\"+;Wwi:Fjan$\"+ty8aT FjanFcbs7%Fgct7$$\"+Ytcn:Fjan$\"+a@bmTFjanFcbs7%F]dt7$$\"+I?9r:Fjan$\" +ju1zTFjanFcbs7%Fcdt7$$\"+nL`t:Fjan$\"+E:f\">%FjanFcbs7%Fidt7$$\"+Vmyu :Fjan$\"+P8//UFjanFcbs7%F_et7$$\"+Vq%\\d\"Fjan$\"+:Q!4c\"Fjan$\"+1#zLF%FjanFcbs7%Fcgt7$$\"+UzUb:Fjan $\"+a%4NG%FjanFcbs7%Figt7$$\"++:;\\:Fjan$\"+Z.5$H%FjanFcbs7%F_htFiasFc bsF]^rFc^r-F$6$7in7$F[_r$!3R+++[v5-VFQ7$Fbas$!3?+++Z.5$H%FQ7$F]as$!3#) *****RX4NG%FQ7$Fh`s$!3W+++1#zLF%FQ7$Fc`s$!3%******pubFE%FQ7$F^`s$!3)** ****fX(o^UFQ7$Fi_s$!3g******QaASUFQ7$Fd_s$!3L+++&3C%GUFQ7$F__s$!3V+++: %FQ7$F`^s$!3)**** **HYn!zTFQ7$F[^s$!3m*****R:_l;%FQ7$Ff]s$!3#)*****H(y8aTFQ7$Fa]s$!3T+++ ay#>9%FQ7$F\\]s$!3=+++s#Q+8%FQ7$Fg\\s$!3f*****R`+'=TFQ7$Fb\\s$!3H+++fQ w2TFQ7$F]\\s$!3++++/\")p(4%FQ7$Fh[s$!3G+++Qmf)3%FQ7$Fc[s$!3u*****4!*y1 3%FQ7$F^[s$!31+++O>>uSFQ7$Fijr$!3m*****zl4%pSFQ7$Fdjr$!3G+++*)yimSFQ7$ F_jr$!3C+++fFoSFQ7$Feir$!39+++[ICtSFQ7$F`ir$!3 S+++R1?\"3%FQ7$F[ir$!3)******RCW@4%FQ7$Ffhr$!3I+++'zve5%FQ7$Fahr$!36++ +KZ*>7%FQ7$F\\hr$!3p*****4rN*RTFQ7$Fggr$!3A+++^_/fTFQ7$Fbgr$!3K+++A.ny TFQ7$F]gr$!3K+++v'H#)>%FQ7$Fhfr$!3!)*******z_s@%FQ7$Fcfr$!3t*****>G)QN UFQ7$F^fr$!3o*****>H#R_UFQ7$Fier$!3-+++K%3\"oUFQ7$Fder$!3/+++ClW#G%FQ7 $F_er$!3o*****>6k`H%FQ7$Fjdr$!3:+++]=&oI%FQ7$Fedr$!3B+++PE#pJ%FQ7$F`dr $!3J+++SQgDVFQ7$F[dr$!3r*****H(=$HL%FQ7$Ffcr$!3l*****z][*QVFQ7$Facr$!3 S+++/%)pVVFQ7$F\\cr$!35+++$\\FsM%FQ7$Fgbr$!39+++]>e\\VFQ7$Fbbr$!31+++u v!3N%FQ7$F]br$!3q*****pV\\4N%FQ7$Fhar$!31+++C<0]VFQ7$Fcar$!3E+++&pd\"[ VFQ7$F^ar$!33+++Q(4`M%FQ7$Fi`r$!3@+++z%\\:M%FQ7$Fd`r$!3++++Fz\"pL%FQ7$ F_`r$!3:+++ccXJVFQ7$Fj_r$!32+++5I?DVFQ7$Fe_r$!3()******>.?=VFQ7$F`_r$! 3@+++\\\")[5VFQFhhtF]^n-Fj^n6jn7%7$Fjas$!+[v5-VFjan7$F`ht$!+Z.5$H%Fjan 7$Fdbs$!$@%Ff^n7%Fbdu7$Fjgt$!+a%4NG%FjanFedu7%Fidu7$Fdgt$!+1#zLF%FjanF edu7%F]eu7$F^gt$!+ZdviUFjanFedu7%Faeu7$Fhft$!+cuo^UFjanFedu7%Feeu7$Fbf t$!+RaASUFjanFedu7%Fieu7$F\\ft$!+&3C%GUFjanFedu7%F]fu7$Ffet$!+:%FjanFedu7%Fifu7 $Fddt$!+ju1zTFjanFedu7%F]gu7$F^dt$!+a@bmTFjanFedu7%Fagu7$Fhct$!+ty8aTF janFedu7%Fegu7$Fbct$!+ay#>9%FjanFedu7%Figu7$F\\ct$!+s#Q+8%FjanFedu7%F] hu7$Ffbt$!+M0g=TFjanFedu7%Fahu7$F`bt$!+fQw2TFjanFedu7%Fehu7$Fjat$!+/\" )p(4%FjanFedu7%Fihu7$Fdat$!+Qmf)3%FjanFedu7%F]iu7$F^at$!+,*y13%FjanFed u7%Faiu7$Fh`t$!+O>>uSFjanFedu7%Feiu7$Fb`t$!+e'4%pSFjanFedu7%Fiiu7$F\\` t$!+*)yimSFjanFedu7%F]ju7$Ff_t$!+7 %FjanFedu7%Fi[v7$F\\]t$!+6d$*RTFjanFedu7%F]\\v7$Ff\\t$!+^_/fTFjanFedu7 %Fa\\v7$F`\\t$!+A.nyTFjanFedu7%Fe\\v7$Fj[t$!+v'H#)>%FjanFedu7%Fi\\v7$F d[t$!++GDe\\VFjanFedu7%Fa`v7$F`fs$!+uv!3N% FjanFedu7%Fe`v7$Fjes$!+P%\\4N%FjanFedu7%Fi`v7$Fdes$!+C<0]VFjanFedu7%F] av7$F^es$!+&pd\"[VFjanFedu7%Faav7$Fhds$!+Q(4`M%FjanFedu7%Feav7$Fbds$!+ z%\\:M%FjanFedu7%Fiav7$F\\ds$!+Fz\"pL%FjanFedu7%F]bv7$Ffcs$!+ccXJVFjan Fedu7%Fabv7$F`cs$!+5I?DVFjanFedu7%Febv7$Fjbs$!+?.?=VFjanFedu7%Fibv7$F_ bs$!+\\\")[5VFjanFedu7%F]cvF_duFeduF]^rFc^r-F$6%7$7$$!3/++++++!z%FQF(7 $$\"3%**************=#FQF(-%'COLOURG6&F`^nF)F)F)-%*LINESTYLEG6#Fb^n-F$ 6%7$7$F(Fecv7$F($\"3/++++++!z%FQFjcvF]dv-F$6$7]yF'7$$!3.+++.2)G`\"F1$ \"37+++[izq:F-7$$!3i+++`Z'z[*F7$\"3;+++h-fTJF-7$$!3!******\\#QEA5FC$\" 3*)*****zJsBr%F-7$$!36+++\"*p[:`FC$\"37+++TF7$G'F-7$$!30+++4guP=FI$\"3 y******fX\"Q&yF-7$$!3G+++k_2#)[FI$\"3Y+++cKTC%*F-7$$!3)*******zkQ!3\"F U$\"3/+++&p&[*4\"FQ7$$!31+++9%yJ5#FU$\"3)******z_*\\c7FQ7$$!3))*****fR bkw$FU$\"31+++bnW89FQ7$$!3S+++&)H7tkFU$\"3%******>X^-d\"FQ7$$!3******* z,a'46F^p$\"3*******zfuns\"FQ7$$!3/+++$Hkj%>F^p$\"3++++*\\#z#)=FQ7$$!3 :+++-i([]$F^p$\"36+++Qg)z.#FQ7$$!3k*****pI'\\GZF^p$\"3$******pG\\^6#FQ 7$$!3-+++/2pyjF^p$\"3#)*****>\"z%>>#FQ7$$!3_*****>ug,e)F^p$\"3)******p j?$oAFQ7$$!33+++tJ/[6Fip$\"3++++$z-UM#FQ7$$!32+++7R)\\_\"Fip$\"3\"**** **p]D&>CFQ7$$!31+++J&Q!3?Fip$\"35+++>N@%\\#FQ7$$!3&******zm@$=EFip$\"3 )*******)G\"=oDFQ7$$!3q*****pML%yLFip$\"3/+++`sKTEFQ7$$!3'******\\&\\z 6VFip$\"34+++[k_8FFQ7$$!3*)*****Rwu;W&Fip$\"3')*****4GAYy#FQ7$$!3a**** *4dI)*y'Fip$\"3;+++!)*=W&GFQ7$$!3S******\\juu$)Fip$\"3=+++=rnAHFQ7$$!3 -+++Eg&4-\"F-$\"3))*****\\B:\"*)HFQ7$$!3/+++*3(**H7F-$\"3=+++*R?M0$FQ7 $$!3%*******)[gUY\"F-$\"34+++\"zo_6$FQ7$$!33+++A.WAaV< $FQ7$$!39+++=4M-?F-$\"3\")*****p`=/B$FQ7$$!3/+++PA4,BF-$\"3)******p)oF $G$FQ7$$!3?+++;;X:EF-$\"3))*****zcIGL$FQ7$$!3>+++^m6UHF-$\"3++++6l1zLF Q7$$!33+++bG)zF$F-$\"3?+++6`/AMFQ7$$!3))*****>W6.i$F-$\"3;+++!4$)=Y$FQ 7$$!3/+++.j!o'RF-$\"33+++%=K()\\$FQ7$$!3y*****\\d<`m%F-$\"3y******[l:k NFQ7$$!3o*****>99LO&F-$\"3%)******p?r>OFQ7$$!3!)******ybAbgF-$\"3)**** **R_smm$FQ7$$!3!******4;g$QnF-$\"3\")******>\"4hq$FQ7$$!3E+++hEm6uF-$ \"3()*****fQ*))QPFQ7$$!3Y******\\([\\2)F-$\"35+++H^qlPFQ7$$!3C+++kcYG( )F-$\"3;+++Mb5(y$FQ7$$!3#)*****fc^EP*F-$\"30+++Bn_.QFQ7$$!3*)*****>)4! 3+\"FQ$\"3%)*****4q:`\"QFQ7$$!33+++A[]j5FQ$\"3)******>&*[F#QFQ7$$!3,++ +zaUD6FQ$\"3-+++dm/EQFQ7$$!3++++@%4m=\"FQ$\"35+++1NQDQFQ7$$!33+++U;5Z7 FQ$\"3<+++qp*3#QFQ7$$!3/+++#HZpI\"FQ$\"3&******pn$p7QFQ7$$!3/+++aQ>m8F Q$\"3:+++sW&3!QFQ7$$!3!******\\\"Q*[U\"FQ$\"3?+++l&Qay$FQ7$$!3)******f \"y5$[\"FQ$\"3%)*****4@([mPFQ7$$!30+++M$34a\"FQ$\"3:+++(HFSu$FQ7$$!3#* *****4#RQ)f\"FQ$\"30+++4`2=PFQ7$$!3%******p)Rkb;FQ$\"3-+++MAk)o$FQ7$$! 3%******z#R#Gr\"FQ$\"3%)*****>4Sdl$FQ7$$!3'******p8!4qOFQ7$$!3,+++_QkF=FQ$\"3/+++'[R'zNFQ7$$!3++++h>s&)=FQ$\"3)********fp l`$FQ7$$!3&******fu\"fW>FQ$\"3++++ScK!\\$FQ7$$!39+++Xa`/?FQ$\"3>+++df8 TMFQ7$$!3%)*****z]?e1#FQ$\"3/+++aRL*Q$FQ7$$!3()*****\\#FQ$\"3')*****4=#yzKFQ7$$!3'*******))*o\"fAFQ$\"3 \")*****z!zk_$*f#FQ$\"3*******\\KX0%HFQ7$$!33+++z>%em#FQ $\"3?+++>.<&)GFQ7$$!3<+++/p#3t#FQ$\"3&)*****RtU*HGFQ7$$!3')*****\\&e0% z#FQ$\"3!******R\"4kuFFQ7$$!3%******p,b`&GFQ$\"3-+++S5/>FFQ7$$!31+++!G 7Y\"HFQ$\"3))*****4TKHm#FQ7$$!3))*****4Li<(HFQ$\"3'******z!\\71EFQ7$$! 3<+++4WxEIFQ$\"35+++T9X[DFQ7$$!3/+++BIkzIFQ$\"3;+++qvw*[#FQ7$$!3!***** *\\G\"QIJFQ$\"30+++X(\\*HCFQ7$$!3')*****>R;!zJFQ$\"31+++xI*)oBFQ7$$!3+ +++[neDKFQ$\"35+++#=4lI#FQ7$$!3#)*****pGS,F$FQ$\"3&)******zVsUAFQ7$$!3 \")*****flLFJ$FQ$\"3?+++X%yu<#FQ7$$!35+++S>V`LFQ$\"3%*******zRs5@FQ7$$ !33+++o'3BR$FQ$\"3)********[EC/#FQ7$$!3.+++MeWHMFQ$\"3!*******o^cs>FQ7 $$!3y******[O$\\Y$FQ$\"35+++\"QM6!>FQ7$$!3')******Q(p))\\$FQ$\"31+++kd 9G=FQ7$$!3++++euNJNFQ$\"3-+++a2j`p>\" FQ7$$!3%)*****4xa+v$FQ$\"31+++.O696FQ7$$!3#******HohGx$FQ$\"3\"******p 2-5.\"FQ7$$!37+++$HUXz$FQ$\"3e*****>!zQx%*F-7$$!3()******fo)\\\"QFQ$\" 3I+++#zIXk)F-7$$!3\")******)esS$QFQ$\"38+++WmY7yF-7$$!3))*****zlr;&QFQ $\"3]+++#**G>)pF-7$$!3%*******4$ew'QFQ$\"3Y+++RJQ`hF-7$$!3%*******[W\" >)QFQ$\"3W+++#4dqK&F-7$$!33+++\"fLV*QFQ$\"37+++8.)H]%F-7$$!3$******>KB [!RFQ$\"3-+++%*R-\"o$F-7$$!3=+++hlI8RFQ$\"35+++x&Q4'GF-7$$!3z*****4+A( >RFQ$\"3.+++tiQU?F-7$$!3'******40CS#RFQ$\"3-+++;z'\\A\"F-7$$!3@+++oD=E RFQ$\"33+++>iW#3%Fip7$Fagx$!33+++>iW#3%Fip7$F\\gx$!3-+++;z'\\A\"F-7$Fg fx$!3.+++tiQU?F-7$Fbfx$!35+++x&Q4'GF-7$F]fx$!3-+++%*R-\"o$F-7$Fhex$!37 +++8.)H]%F-7$Fcex$!3W+++#4dqK&F-7$F^ex$!3Y+++RJQ`hF-7$Fidx$!3]+++#**G> )pF-7$Fddx$!38+++WmY7yF-7$F_dx$!3I+++#zIXk)F-7$Fjcx$!3e*****>!zQx%*F-7 $Fecx$!3\"******p2-5.\"FQ7$F`cx$!31+++.O696FQ7$F[cx$!3/+++m*>p>\"FQ7$F fbx$!3'******p.]#z7FQ7$Fabx$!3-+++4N#4O\"FQ7$F\\bx$!3+++++0wT9FQ7$Fgax $!3.+++N3f@:FQ7$Fbax$!34+++[FQ7$Fi_x$!3!*******o^ cs>FQ7$Fd_x$!3)********[EC/#FQ7$F__x$!3%*******zRs5@FQ7$Fj^x$!3?+++X%y u<#FQ7$Fe^x$!3&)******zVsUAFQ7$F`^x$!35+++#=4lI#FQ7$F[^x$!31+++xI*)oBF Q7$Ff]x$!30+++X(\\*HCFQ7$Fa]x$!3;+++qvw*[#FQ7$F\\]x$!35+++T9X[DFQ7$Fg \\x$!3'******z!\\71EFQ7$Fb\\x$!3))*****4TKHm#FQ7$F]\\x$!3-+++S5/>FFQ7$ Fh[x$!3!******R\"4kuFFQ7$Fc[x$!3&)*****RtU*HGFQ7$F^[x$!3?+++>.<&)GFQ7$ Fijw$!3*******\\KX0%HFQ7$Fdjw$!3=+++-8E'*HFQ7$F_jw$!3=+++yfW_IFQ7$Fjiw $!30+++a`64JFQ7$Feiw$!3%)*****\\)38mJFQ7$F`iw$!3\")*****z!z +++df8TMFQ7$Fggw$!3++++ScK!\\$FQ7$Fbgw$!3)********fpl`$FQ7$F]gw$!3/+++ '[R'zNFQ7$Fhfw$!3$******R'4R>OFQ7$Fcfw$!3%)*****>4Sdl$FQ7$F^fw$!3-+++M Ak)o$FQ7$Fiew$!30+++4`2=PFQ7$Fdew$!3:+++(HFSu$FQ7$F_ew$!3%)*****4@([mP FQ7$Fjdw$!3?+++l&Qay$FQ7$Fedw$!3:+++sW&3!QFQ7$F`dw$!3&******pn$p7QFQ7$ F[dw$!3<+++qp*3#QFQ7$Ffcw$!35+++1NQDQFQ7$Facw$!3-+++dm/EQFQ7$F\\cw$!3) ******>&*[F#QFQ7$Fgbw$!3%)*****4q:`\"QFQ7$Fbbw$!30+++Bn_.QFQ7$F]bw$!3; +++Mb5(y$FQ7$Fhaw$!35+++H^qlPFQ7$Fcaw$!3()*****fQ*))QPFQ7$F^aw$!3\")** ****>\"4hq$FQ7$Fi`w$!3)******R_smm$FQ7$Fd`w$!3%)******p?r>OFQ7$F_`w$!3 y******[l:kNFQ7$$!3\")*****HL?cJ%F-$!3%******f8kF`$FQ7$Fj_w$!33+++%=K( )\\$FQ7$Fe_w$!3;+++!4$)=Y$FQ7$F`_w$!3?+++6`/AMFQ7$F[_w$!3++++6l1zLFQ7$ Ff^w$!3))*****zcIGL$FQ7$Fa^w$!3)******p)oF$G$FQ7$F\\^w$!3\")*****p`=/B $FQ7$Fg]w$!3%******4>aV<$FQ7$Fb]w$!34+++\"zo_6$FQ7$F]]w$!3=+++*R?M0$FQ 7$Fh\\w$!3))*****\\B:\"*)HFQ7$Fc\\w$!3=+++=rnAHFQ7$F^\\w$!3;+++!)*=W&G FQ7$Fi[w$!3')*****4GAYy#FQ7$Fd[w$!34+++[k_8FFQ7$F_[w$!3/+++`sKTEFQ7$Fj jv$!3)*******)G\"=oDFQ7$Fejv$!35+++>N@%\\#FQ7$F`jv$!3\"******p]D&>CFQ7 $F[jv$!3++++$z-UM#FQ7$Ffiv$!3)******pj?$oAFQ7$Faiv$!3#)*****>\"z%>>#FQ 7$F\\iv$!3$******pG\\^6#FQ7$$!3(*******pr-0EF^p$!33+++;f^g>FQ7$$!3#*** ***\\!HHk9F^p$!3)******>Sk[!=FQ7$$!3i*****z$Qgg%)FU$!3#******>,g&[;FQ7 $$!3E+++nBP]\\FU$!3++++[Q(=\\\"FQ7$$!3')*****RM^k$GFU$!3-+++=])\\L\"FQ 7$$!3&******4#zIF:FU$!35++++!)*z<\"FQ7$$!3L+++0cQ-uFI$!3#******42m4-\" FQ7$$!37+++ArExIFI$!3y*****\\JG\"R')F-7$$!3'******>`g%G5FI$!3]******py ZoqF-7$$!3%******>)4\\'[#FC$!3!)*****>/`x\\&F-7$$!38+++0g$)HNF=$!3\")* ****fz$)p#RF-7$$!3.+++#\\'*=s\"F7$!3)******\\!R>cBF-7$$!3.+++\\jA:C!#G $!3g+++4;)R&yFip7$$!3*)*****HME_T#F_^z$\"3g+++4;)R&yFip-F^^n6&F`^n$\"# DFf^n$\"\")Ff^n$\"\"%Fc^n-Fj^n6^y7%F'7$$!+.2)G`\"F-$\"+[izq:F`_nFa_n7% Fb_z7$$!+`Z'z[*FQ$\"+h-fTJF`_nFa_n7%Fh_z7$$!+DQEA5Fj`n$\"+=BP7ZF`_nFa_ n7%F^`z7$$!+\"*p[:`Fj`n$\"+TF7$G'F`_nFa_n7%Fd`z7$$!+4guP=Faan$\"+gX\"Q &yF`_nFa_n7%Fj`z7$$!+k_2#)[Faan$\"+cKTC%*F`_nFa_n7%F`az7$$!+!['Q!3\"F_ bn$\"+&p&[*4\"FjanFa_n7%Ffaz7$$!+9%yJ5#F_bn$\"+G&*\\c7FjanFa_n7%F\\bz7 $$!+'Rbkw$F_bn$\"+bnW89FjanFa_n7%Fbbz7$$!+&)H7tkF_bn$\"+_9Dq:FjanFa_n7 %Fhbz7$$!+=Sl46Fddn$\"+)funs\"FjanFa_n7%F^cz7$$!+$Hkj%>Fddn$\"+*\\#z#) =FjanFa_n7%Fdcz7$$!+-i([]$Fddn$\"+Qg)z.#FjanFa_n7%Fjcz7$$!+2j\\GZFddn$ \"+(G\\^6#FjanFa_n7%F`dz7$$!+/2pyjFddn$\"+7z%>>#FjanFa_n7%Ffdz7$$!+U2; !e)Fddn$\"+P1KoAFjanFa_n7%F\\ez7$$!+tJ/[6Faen$\"+$z-UM#FjanFa_n7%Fbez7 $$!+7R)\\_\"Faen$\"+2b_>CFjanFa_n7%Fhez7$$!+J&Q!3?Faen$\"+>N@%\\#FjanF a_n7%F^fz7$$!+o;K=EFaen$\"+*G\"=oDFjanFa_n7%Fdfz7$$!+ZLVyLFaen$\"+`sKT EFjanFa_n7%Fjfz7$$!+b\\z6VFaen$\"+[k_8FFjanFa_n7%F`gz7$$!+kZnTaFaen$\" +\"GAYy#FjanFa_n7%Ffgz7$$!+r0$)*y'Faen$\"+!)*=W&GFjanFa_n7%F\\hz7$$!+] juu$)Faen$\"+=rnAHFjanFa_n7%Fbhz7$$!+Eg&4-\"F`_n$\"+N_6*)HFjanFa_n7%Fh hz7$$!+*3(**H7F`_n$\"+*R?M0$FjanFa_n7%F^iz7$$!+*[gUY\"F`_n$\"+\"zo_6$F janFa_n7%Fdiz7$$!+A.WAaV<$FjanFa_n7%Fjiz7$$!+=4M-?F`_n$\"+ P&=/B$FjanFa_n7%F`jz7$$!+PA4,BF`_n$\"+()oF$G$FjanFa_n7%Ffjz7$$!+;;X:EF `_n$\"+o0$GL$FjanFa_n7%F\\[[l7$$!+^m6UHF`_n$\"+6l1zLFjanFa_n7%Fb[[l7$$ !+bG)zF$F`_n$\"+6`/AMFjanFa_n7%Fh[[l7$$!+U9J?OF`_n$\"+!4$)=Y$FjanFa_n7 %F^\\[l7$$!+.j!o'RF`_n$\"+%=K()\\$FjanFa_n7%Fd\\[l7$$!+vvJlYF`_n$\"+\\ l:kNFjanFa_n7%Fj\\[l7$$!+UTJj`F`_n$\"+q?r>OFjanFa_n7%F`][l7$$!+zbAbgF` _n$\"+CDnmOFjanFa_n7%Ff][l7$$!+h,OQnF`_n$\"+?\"4hq$FjanFa_n7%F\\^[l7$$ !+hEm6uF`_n$\"+'Q*))QPFjanFa_n7%Fb^[l7$$!+]([\\2)F`_n$\"+H^qlPFjanFa_n 7%Fh^[l7$$!+kcYG()F`_n$\"+Mb5(y$FjanFa_n7%F^_[l7$$!+m:ls$*F`_n$\"+Bn_. QFjanFa_n7%Fd_[l7$$!+#)4!3+\"Fjan$\"+,dJ:QFjanFa_n7%Fj_[l7$$!+A[]j5Fja n$\"+_*[F#QFjanFa_n7%F``[l7$$!+zaUD6Fjan$\"+dm/EQFjanFa_n7%Ff`[l7$$!+@ %4m=\"Fjan$\"+1NQDQFjanFa_n7%F\\a[l7$$!+U;5Z7Fjan$\"+qp*3#QFjanFa_n7%F ba[l7$$!+#HZpI\"Fjan$\"+xOp7QFjanFa_n7%Fha[l7$$!+aQ>m8Fjan$\"+sW&3!QFj anFa_n7%F^b[l7$$!+:Q*[U\"Fjan$\"+l&Qay$FjanFa_n7%Fdb[l7$$!+;y5$[\"Fjan $\"+6s[mPFjanFa_n7%Fjb[l7$$!+M$34a\"Fjan$\"+(HFSu$FjanFa_n7%F`c[l7$$!+ @RQ)f\"Fjan$\"+4`2=PFjanFa_n7%Ffc[l7$$!+()Rkb;Fjan$\"+MAk)o$FjanFa_n7% F\\d[l7$$!+GR#Gr\"Fjan$\"+#4Sdl$FjanFa_n7%Fbd[l7$$!+P,4qO FjanFa_n7%Fhd[l7$$!+_QkF=Fjan$\"+'[R'zNFjanFa_n7%F^e[l7$$!+h>s&)=Fjan$ \"++'pl`$FjanFa_n7%Fde[l7$$!+YFjan$\"+ScK!\\$FjanFa_n7%Fje[l7$$!+X a`/?Fjan$\"+df8TMFjanFa_n7%F`f[l7$$!+30#e1#Fjan$\"+aRL*Q$FjanFa_n7%Fff [l7$$!+vTlG@Fjan$\"+LrONLFjanFa_n7%F\\g[l7$$!+C\"GJ>#Fjan$\"+\"=#yzKFj anFa_n7%Fbg[l7$$!+*)*o\"fAFjan$\"+3z%em#Fjan$\"+>.<&)GFjanF a_n7%F\\j[l7$$!+/p#3t#Fjan$\"+MF%*HGFjanFa_n7%Fbj[l7$$!+be0%z#Fjan$\"+ 94kuFFjanFa_n7%Fhj[l7$$!+<]NbGFjan$\"+S5/>FFjanFa_n7%F^[\\l7$$!+!G7Y\" HFjan$\"+6C$Hm#FjanFa_n7%Fd[\\l7$$!+JBwrHFjan$\"+3\\71EFjanFa_n7%Fj[\\ l7$$!+4WxEIFjan$\"+T9X[DFjanFa_n7%F`\\\\l7$$!+BIkzIFjan$\"+qvw*[#FjanF a_n7%Ff\\\\l7$$!+&G\"QIJFjan$\"+X(\\*HCFjanFa_n7%F\\]\\l7$$!+#R;!zJFja n$\"+xI*)oBFjanFa_n7%Fb]\\l7$$!+[neDKFjan$\"+#=4lI#FjanFa_n7%Fh]\\l7$$ !+(GS,F$Fjan$\"+!QCFC#FjanFa_n7%F^^\\l7$$!+cOt7LFjan$\"+X%yu<#FjanFa_n 7%Fd^\\l7$$!+S>V`LFjan$\"+!)Rs5@FjanFa_n7%Fj^\\l7$$!+o'3BR$Fjan$\"+!\\ EC/#FjanFa_n7%F`_\\l7$$!+MeWHMFjan$\"+p^cs>FjanFa_n7%Ff_\\l7$$!+\\O$\\ Y$Fjan$\"+\"QM6!>FjanFa_n7%F\\`\\l7$$!+R(p))\\$Fjan$\"+kd9G=FjanFa_n7% Fb`\\l7$$!+euNJNFjan$\"+a2j`p>\"FjanFa_n7%Fbc\\l7$$!+rZ0]PFjan$\"+.O69 6FjanFa_n7%Fhc\\l7$$!+$ohGx$Fjan$\"+x?+J5FjanFa_n7%F^d\\l7$$!+$HUXz$Fj an$\"+-zQx%*F`_nFa_n7%Fdd\\l7$$!+go)\\\"QFjan$\"+#zIXk)F`_nFa_n7%Fjd\\ l7$$!+*esS$QFjan$\"+WmY7yF`_nFa_n7%F`e\\l7$$!+e;n^QFjan$\"+#**G>)pF`_n Fa_n7%Ffe\\l7$$!+5$ew'QFjan$\"+RJQ`hF`_nFa_n7%F\\f\\l7$$!+\\W\">)QFjan $\"+#4dqK&F`_nFa_n7%Fbf\\l7$$!+\"fLV*QFjan$\"+8.)H]%F`_nFa_n7%Fhf\\l7$ $!+AL#[!RFjan$\"+%*R-\"o$F`_nFa_n7%F^g\\l7$$!+hlI8RFjan$\"+x&Q4'GF`_nF a_n7%Fdg\\l7$$!+,?s>RFjan$\"+tiQU?F`_nFa_n7%Fjg\\l7$$!+^S-CRFjan$\"+;z '\\A\"F`_nFa_n7%F`h\\l7$$!+oD=ERFjan$\"+>iW#3%FaenFa_n7%Ffh\\l7$Fgh\\l $!+>iW#3%FaenFa_n7%F\\i\\l7$Fah\\l$!+;z'\\A\"F`_nFa_n7%F`i\\l7$F[h\\l$ !+tiQU?F`_nFa_n7%Fdi\\l7$Feg\\l$!+x&Q4'GF`_nFa_n7%Fhi\\l7$F_g\\l$!+%*R -\"o$F`_nFa_n7%F\\j\\l7$Fif\\l$!+8.)H]%F`_nFa_n7%F`j\\l7$Fcf\\l$!+#4dq K&F`_nFa_n7%Fdj\\l7$F]f\\l$!+RJQ`hF`_nFa_n7%Fhj\\l7$Fge\\l$!+#**G>)pF` _nFa_n7%F\\[]l7$Fae\\l$!+WmY7yF`_nFa_n7%F`[]l7$F[e\\l$!+#zIXk)F`_nFa_n 7%Fd[]l7$Fed\\l$!+-zQx%*F`_nFa_n7%Fh[]l7$F_d\\l$!+x?+J5FjanFa_n7%F\\\\ ]l7$Fic\\l$!+.O696FjanFa_n7%F`\\]l7$Fcc\\l$!+m*>p>\"FjanFa_n7%Fd\\]l7$ F]c\\l$!+P+Dz7FjanFa_n7%Fh\\]l7$Fgb\\l$!+4N#4O\"FjanFa_n7%F\\]]l7$Fab \\l$!++0wT9FjanFa_n7%F`]]l7$F[b\\l$!+N3f@:FjanFa_n7%Fd]]l7$Fea\\l$!+[< E+;FjanFa_n7%Fh]]l7$F_a\\l$!+FKkx;FjanFa_n7%F\\^]l7$Fi`\\l$!+a2j`FjanFa_n7% Fh^]l7$Fg_\\l$!+p^cs>FjanFa_n7%F\\_]l7$Fa_\\l$!+!\\EC/#FjanFa_n7%F`_]l 7$F[_\\l$!+!)Rs5@FjanFa_n7%Fd_]l7$Fe^\\l$!+X%yu<#FjanFa_n7%Fh_]l7$F_^ \\l$!+!QCFC#FjanFa_n7%F\\`]l7$Fi]\\l$!+#=4lI#FjanFa_n7%F``]l7$Fc]\\l$! +xI*)oBFjanFa_n7%Fd`]l7$F]]\\l$!+X(\\*HCFjanFa_n7%Fh`]l7$Fg\\\\l$!+qvw *[#FjanFa_n7%F\\a]l7$Fa\\\\l$!+T9X[DFjanFa_n7%F`a]l7$F[\\\\l$!+3\\71EF janFa_n7%Fda]l7$Fe[\\l$!+6C$Hm#FjanFa_n7%Fha]l7$F_[\\l$!+S5/>FFjanFa_n 7%F\\b]l7$Fij[l$!+94kuFFjanFa_n7%F`b]l7$Fcj[l$!+MF%*HGFjanFa_n7%Fdb]l7 $F]j[l$!+>.<&)GFjanFa_n7%Fhb]l7$Fgi[l$!+D`aSHFjanFa_n7%F\\c]l7$Fai[l$! +-8E'*HFjanFa_n7%F`c]l7$F[i[l$!+yfW_IFjanFa_n7%Fdc]l7$Feh[l$!+a`64JFja nFa_n7%Fhc]l7$F_h[l$!+&)38mJFjanFa_n7%F\\d]l7$Fig[l$!+3zOFjanFa_n7%F`f]l7$Fcd[l$!+#4Sdl$FjanFa_n7%F df]l7$F]d[l$!+MAk)o$FjanFa_n7%Fhf]l7$Fgc[l$!+4`2=PFjanFa_n7%F\\g]l7$Fa c[l$!+(HFSu$FjanFa_n7%F`g]l7$F[c[l$!+6s[mPFjanFa_n7%Fdg]l7$Feb[l$!+l&Q ay$FjanFa_n7%Fhg]l7$F_b[l$!+sW&3!QFjanFa_n7%F\\h]l7$Fia[l$!+xOp7QFjanF a_n7%F`h]l7$Fca[l$!+qp*3#QFjanFa_n7%Fdh]l7$F]a[l$!+1NQDQFjanFa_n7%Fhh] l7$Fg`[l$!+dm/EQFjanFa_n7%F\\i]l7$Fa`[l$!+_*[F#QFjanFa_n7%F`i]l7$F[`[l $!+,dJ:QFjanFa_n7%Fdi]l7$Fe_[l$!+Bn_.QFjanFa_n7%Fhi]l7$F__[l$!+Mb5(y$F janFa_n7%F\\j]l7$Fi^[l$!+H^qlPFjanFa_n7%F`j]l7$Fc^[l$!+'Q*))QPFjanFa_n 7%Fdj]l7$F]^[l$!+?\"4hq$FjanFa_n7%Fhj]l7$Fg][l$!+CDnmOFjanFa_n7%F\\[^l 7$Fa][l$!+q?r>OFjanFa_n7%F`[^l7$F[][l$!+\\l:kNFjanFa_n7%Fd[^l7$$!+L.i: VF`_n$!+OTwKNFjanFa_n7%Fh[^l7$Fe\\[l$!+%=K()\\$FjanFa_n7%F^\\^l7$F_\\[ l$!+!4$)=Y$FjanFa_n7%Fb\\^l7$Fi[[l$!+6`/AMFjanFa_n7%Ff\\^l7$Fc[[l$!+6l 1zLFjanFa_n7%Fj\\^l7$F][[l$!+o0$GL$FjanFa_n7%F^]^l7$Fgjz$!+()oF$G$Fjan Fa_n7%Fb]^l7$Fajz$!+P&=/B$FjanFa_n7%Ff]^l7$F[jz$!+\">aV<$FjanFa_n7%Fj] ^l7$Feiz$!+\"zo_6$FjanFa_n7%F^^^l7$F_iz$!+*R?M0$FjanFa_n7%Fb^^l7$Fihz$ !+N_6*)HFjanFa_n7%Ff^^l7$Fchz$!+=rnAHFjanFa_n7%Fj^^l7$F]hz$!+!)*=W&GFj anFa_n7%F^_^l7$Fggz$!+\"GAYy#FjanFa_n7%Fb_^l7$Fagz$!+[k_8FFjanFa_n7%Ff _^l7$F[gz$!+`sKTEFjanFa_n7%Fj_^l7$Fefz$!+*G\"=oDFjanFa_n7%F^`^l7$F_fz$ !+>N@%\\#FjanFa_n7%Fb`^l7$Fiez$!+2b_>CFjanFa_n7%Ff`^l7$Fcez$!+$z-UM#Fj anFa_n7%Fj`^l7$F]ez$!+P1KoAFjanFa_n7%F^a^l7$Fgdz$!+7z%>>#FjanFa_n7%Fba ^l7$Fadz$!+(G\\^6#FjanFa_n7%Ffa^l7$$!+qr-0EFddn$!+;f^g>FjanFa_n7%Fja^l 7$$!+0HHk9Fddn$!+-W'[!=FjanFa_n7%F`b^l7$$!+QQgg%)F_bn$!+7+c[;FjanFa_n7 %Ffb^l7$$!+nBP]\\F_bn$!+[Q(=\\\"FjanFa_n7%F\\c^l7$$!+W8XOGF_bn$!+=])\\ L\"FjanFa_n7%Fbc^l7$$!+@zIF:F_bn$!++!)*z<\"FjanFa_n7%Fhc^l7$$!+0cQ-uFa an$!+rg'4-\"FjanFa_n7%F^d^l7$$!+ArExIFaan$!+:$G\"R')F`_nFa_n7%Fdd^l7$$ !+K0YG5Faan$!+qyZoqF`_nFa_n7%Fjd^l7$$!+#)4\\'[#Fj`n$!+UIv(\\&F`_nFa_n7 %F`e^l7$$!+0g$)HNFc`n$!+'z$)p#RF`_nFa_n7%Ffe^l7$$!+#\\'*=s\"FQ$!+0R>cB F`_nFa_n7%F\\f^l7$$!+\\jA:CF^p$!+4;)R&yFaenFa_n7%Fbf^l7$$!+VjA:CF^p$\" +4;)R&yFaenFa_n-F^^n6&F`^n$\"\"&Fc^nFd^n$F\\_zFc^nFc^r-F$6$7in7$$\"35+ ++$z,t\\\"FQ$\"3A+++J%3![VFQ7$$\"3#******>Nk&*[\"FQ$\"33+++bm>dVFQ7$$ \"3!*******R:6\"[\"FQ$\"3!******R*[rlVFQ7$$\"3%******>v&)>Z\"FQ$\"3#** ****RD@NP%FQ7$$\"3#******>))HAY\"FQ$\"3A+++eUd!Q%FQ7$$\"31+++Fw)=X\"FQ $\"3=+++\\D$oQ%FQ7$$\"3,+++HQ+T9FQ$\"3;+++5YD#R%FQ7$$\"36+++g[iH9FQ$\" 3!)*****p\\)z'R%FQ7$$\"3-+++n!*z<9FQ$\"3k******y:U+WFQ7$$\"3&******4Hx bS\"FQ$\"3t*****fH!3.WFQ7$$\"3))*****HZ8IR\"FQ$\"30+++<*HZS%FQ7$$\"3#* ******y`;!Q\"FQ$\"3,+++IVK0WFQ7$$\"31+++ja4n8FQ$\"39+++[e\"[S%FQ7$$\"3 !******z!>(QN\"FQ$\"3#******\\2bJS%FQ7$$\"3&******p*)p0M\"FQ$\"3$***** *f$3H+WFQ7$$\"3*******4:tsK\"FQ$\"3!)*****pAqhR%FQ7$$\"3,+++?e298FQ$\" 31+++i)Q2R%FQ7$$\"3,+++,[3,8FQ$\"3;+++)RTRQ%FQ7$$\"3)******HbA%)G\"FQ$ \"39+++@DsvVFQ7$$\"3/+++K0Bw7FQ$\"3G+++:'GgO%FQ7$$\"3!******zEtYE\"FQ$ \"3-+++x/\"[N%FQ7$$\"3/+++5J%RD\"FQ$\"3:+++%fF?M%FQ7$$\"36+++UaEW7FQ$ \"39+++LZlFVFQ7$$\"30+++lQ!fB\"FQ$\"3M+++M=p6VFQ7$$\"3.+++dT;H7FQ$\"3- +++c%yTH%FQ7$$\"33+++mZRC7FQ$\"3i*****f,9_F%FQ7$$\"3\"******>()z>A\"FQ $\"3O+++WV)\\D%FQ7$$\"3'******R;>BA\"FQ$\"3!)*****z\"HzLUFQ7$$\"3!**** **fU(yD7FQ$\"37++++/47UFQ7$$\"3(******z#*pEB\"FQ$\"3'******4$*)[!>%FQ7 $$\"3%******f%*zIC\"FQ$\"3I+++U9upTFQ7$$\"33+++RE)oD\"FQ$\"3o*****4vp1 :%FQ7$$\"3%******RKjOF\"FQ$\"3B+++pn/MTFQ7$$\"3%******HtoFH\"FQ$\"3=++ +I,Z?TFQ7$$\"3%******z*GT88FQ$\"3x*****44z-6%FQ7$$\"30+++cK![L\"FQ$\"3 K+++sTa.TFQ7$$\"3-+++=#RiN\"FQ$\"3\"******4?=,5%FQ7$$\"3/+++;<;x8FQ$\" 3)******pg8(*4%FQ7$$\"3%*******[*frR\"FQ$\"3P+++NY(>5%FQ7$$\"33+++'=`f T\"FQ$\"3!*******[(Hl5%FQ7$$\"3'******zejLV\"FQ$\"3%)*****\\6BI6%FQ7$$ \"31+++E&*G\\9FQ$\"3l*****\\7I67%FQ7$$\"3-+++fLoj9FQ$\"3&******zvi08%F Q7$$\"3#******\\XMlZ\"FQ$\"3$******4Ip59%FQ7$$\"3!*******po&y[\"FQ$\"3 i******Q8V_TFQ7$$\"3'******4m!o(\\\"FQ$\"3;+++L(fW;%FQ7$$\"3/+++7f/1:F Q$\"3=+++k/*p<%FQ7$$\"3\"*******o')*H^\"FQ$\"3-+++y2))*=%FQ7$$\"3,+++D #)e=:FQ$\"3G+++vd+.UFQ7$$\"3'******zKlG_\"FQ$\"3&)******)\\bi@%FQ7$$\" 3(******f-\")e_\"FQ$\"3u*****4ZK&HUFQ7$$\"3/+++^foF:FQ$\"3-+++N'\\FC%F Q7$$\"3%******p))H$G:FQ$\"3S+++O'GeD%FQ7$$\"3/+++\\:'y_\"FQ$\"3G+++%R) poUFQ7$$\"3.+++)\\Gj_\"FQ$\"3Q+++)*QH\"G%FQ7$$\"3#******>9xP_\"FQ$\"3! ******z?bNH%FQ7$$\"31+++7GD?:FQ$\"3#)*****>gEaI%FQ7$$\"35+++s)*z::FQ$ \"3x*****>&e&oJ%FQ7$$\"32+++n=Y5:FQ$\"3=+++1OzFVFQ7$$\"3(******\\p\"G/ :FQ$\"3E+++[G>QVFQFeg^lFg^z-Fj^n6jn7%7$$\"+$z,t\\\"Fjan$\"+J%3![VFjan7 $$\"+_Vc*[\"Fjan$\"+bm>dVFjan7$$\"$K\"Ff^n$\"$>%Ff^n7%Fij_l7$$\"+S:6\" [\"Fjan$\"+%*[rlVFjanF^[`l7%Fd[`l7$$\"+_d)>Z\"Fjan$\"+a7_tVFjanF^[`l7% Fj[`l7$$\"+#))HAY\"Fjan$\"+eUd!Q%FjanF^[`l7%F`\\`l7$$\"+Fw)=X\"Fjan$\" +\\D$oQ%FjanF^[`l7%Ff\\`l7$$\"+HQ+T9Fjan$\"+5YD#R%FjanF^[`l7%F\\]`l7$$ \"+g[iH9Fjan$\"+(\\)z'R%FjanF^[`l7%Fb]`l7$$\"+n!*z<9Fjan$\"+z:U+WFjanF ^[`l7%Fh]`l7$$\"+\"HxbS\"Fjan$\"+'H!3.WFjanF^[`l7%F^^`l7$$\"+tM,$R\"Fj an$\"+<*HZS%FjanF^[`l7%Fd^`l7$$\"+z`;!Q\"Fjan$\"+IVK0WFjanF^[`l7%Fj^`l 7$$\"+ja4n8Fjan$\"+[e\"[S%FjanF^[`l7%F`_`l7$$\"+3>(QN\"Fjan$\"+v]:.WFj anF^[`l7%Ff_`l7$$\"+(*)p0M\"Fjan$\"+O3H+WFjanF^[`l7%F\\``l7$$\"+^JFF8F jan$\"+F-<'R%FjanF^[`l7%Fb``l7$$\"+?e298Fjan$\"+i)Q2R%FjanF^[`l7%Fh``l 7$$\"+,[3,8Fjan$\"+)RTRQ%FjanF^[`l7%F^a`l7$$\"+`DU)G\"Fjan$\"+@DsvVFja nF^[`l7%Fda`l7$$\"+K0Bw7Fjan$\"+:'GgO%FjanF^[`l7%Fja`l7$$\"+oKnk7Fjan$ \"+x/\"[N%FjanF^[`l7%F`b`l7$$\"+5J%RD\"Fjan$\"+%fF?M%FjanF^[`l7%Ffb`l7 $$\"+UaEW7Fjan$\"+LZlFVFjanF^[`l7%F\\c`l7$$\"+lQ!fB\"Fjan$\"+M=p6VFjan F^[`l7%Fbc`l7$$\"+dT;H7Fjan$\"+c%yTH%FjanF^[`l7%Fhc`l7$$\"+mZRC7Fjan$ \"+;S@vUFjanF^[`l7%F^d`l7$$\"+s)z>A\"Fjan$\"+WV)\\D%FjanF^[`l7%Fdd`l7$ $\"+k\">BA\"Fjan$\"+=HzLUFjanF^[`l7%Fjd`l7$$\"+EuyD7Fjan$\"++/47UFjanF ^[`l7%F`e`l7$$\"+G*pEB\"Fjan$\"+J*)[!>%FjanF^[`l7%Ffe`l7$$\"+Y*zIC\"Fj an$\"+U9upTFjanF^[`l7%F\\f`l7$$\"+RE)oD\"Fjan$\"+^(p1:%FjanF^[`l7%Fbf` l7$$\"+CLmt7Fjan$\"+pn/MTFjanF^[`l7%Fhf`l7$$\"+L(oFH\"Fjan$\"+I,Z?TFja nF^[`l7%F^g`l7$$\"+)*GT88Fjan$\"+\"4z-6%FjanF^[`l7%Fdg`l7$$\"+cK![L\"F jan$\"+sTa.TFjanF^[`l7%Fjg`l7$$\"+=#RiN\"Fjan$\"+,#=,5%FjanF^[`l7%F`h` l7$$\"+;<;x8Fjan$\"+2Or*4%FjanF^[`l7%Ffh`l7$$\"+\\*frR\"Fjan$\"+NY(>5% FjanF^[`l7%F\\i`l7$$\"+'=`fT\"Fjan$\"+\\(Hl5%FjanF^[`l7%Fbi`l7$$\"+)ej LV\"Fjan$\"+:J-8TFjanF^[`l7%Fhi`l7$$\"+E&*G\\9Fjan$\"+D,8@TFjanF^[`l7% F^j`l7$$\"+fLoj9Fjan$\"+eFcITFjanF^[`l7%Fdj`l7$$\"+bW`w9Fjan$\"+,$p59% FjanF^[`l7%Fjj`l7$$\"+qo&y[\"Fjan$\"+R8V_TFjanF^[`l7%F`[al7$$\"+h1o(\\ \"Fjan$\"+L(fW;%FjanF^[`l7%Ff[al7$$\"+7f/1:Fjan$\"+k/*p<%FjanF^[`l7%F \\\\al7$$\"+p')*H^\"Fjan$\"+y2))*=%FjanF^[`l7%Fb\\al7$$\"+D#)e=:Fjan$ \"+vd+.UFjanF^[`l7%Fh\\al7$$\"+G`'G_\"Fjan$\"+*\\bi@%FjanF^[`l7%F^]al7 $$\"+E5)e_\"Fjan$\"+rC`HUFjanF^[`l7%Fd]al7$$\"+^foF:Fjan$\"+N'\\FC%Fja nF^[`l7%Fj]al7$$\"+())H$G:Fjan$\"+O'GeD%FjanF^[`l7%F`^al7$$\"+\\:'y_\" Fjan$\"+%R)poUFjanF^[`l7%Ff^al7$$\"+)\\Gj_\"Fjan$\"+)*QH\"G%FjanF^[`l7 %F\\_al7$$\"+UrxB:Fjan$\"+3_b$H%FjanF^[`l7%Fb_al7$$\"+7GD?:Fjan$\"+-mU 0VFjanF^[`l7%Fh_al7$$\"+s)*z::Fjan$\"+_e&oJ%FjanF^[`l7%F^`al7$$\"+n=Y5 :Fjan$\"+1OzFVFjanF^[`l7%Fd`al7$$\"+&p\"G/:Fjan$\"+[G>QVFjanF^[`l7%Fj` alFdj_lF^[`lF]g^lFc^r-F$6$7in7$Ffg^l$!3A+++J%3![VFQ7$F]j_l$!3E+++[G>QV FQ7$Fhi_l$!3=+++1OzFVFQ7$Fci_l$!3x*****>&e&oJ%FQ7$F^i_l$!3#)*****>gEaI %FQ7$Fih_l$!3!******z?bNH%FQ7$Fdh_l$!3Q+++)*QH\"G%FQ7$F_h_l$!3G+++%R)p oUFQ7$Fjg_l$!3S+++O'GeD%FQ7$Feg_l$!3-+++N'\\FC%FQ7$F`g_l$!3u*****4ZK&H UFQ7$F[g_l$!3&)******)\\bi@%FQ7$Fff_l$!3G+++vd+.UFQ7$Faf_l$!3-+++y2))* =%FQ7$F\\f_l$!3=+++k/*p<%FQ7$Fge_l$!3;+++L(fW;%FQ7$Fbe_l$!3i******Q8V_ TFQ7$F]e_l$!3$******4Ip59%FQ7$Fhd_l$!3&******zvi08%FQ7$Fcd_l$!3l***** \\7I67%FQ7$F^d_l$!3%)*****\\6BI6%FQ7$Fic_l$!3!*******[(Hl5%FQ7$Fdc_l$! 3P+++NY(>5%FQ7$F_c_l$!3)******pg8(*4%FQ7$Fjb_l$!3\"******4?=,5%FQ7$Feb _l$!3K+++sTa.TFQ7$F`b_l$!3x*****44z-6%FQ7$F[b_l$!3=+++I,Z?TFQ7$Ffa_l$! 3B+++pn/MTFQ7$Faa_l$!3o*****4vp1:%FQ7$F\\a_l$!3I+++U9upTFQ7$Fg`_l$!3'* *****4$*)[!>%FQ7$Fb`_l$!37++++/47UFQ7$F]`_l$!3!)*****z\"HzLUFQ7$Fh__l$ !3O+++WV)\\D%FQ7$Fc__l$!3i*****f,9_F%FQ7$F^__l$!3-+++c%yTH%FQ7$Fi^_l$! 3M+++M=p6VFQ7$Fd^_l$!39+++LZlFVFQ7$F_^_l$!3:+++%fF?M%FQ7$Fj]_l$!3-+++x /\"[N%FQ7$Fe]_l$!3G+++:'GgO%FQ7$F`]_l$!39+++@DsvVFQ7$F[]_l$!3;+++)RTRQ %FQ7$Ff\\_l$!31+++i)Q2R%FQ7$Fa\\_l$!3!)*****pAqhR%FQ7$F\\\\_l$!3$***** *f$3H+WFQ7$Fg[_l$!3#******\\2bJS%FQ7$Fb[_l$!39+++[e\"[S%FQ7$F][_l$!3,+ ++IVK0WFQ7$Fhj^l$!30+++<*HZS%FQ7$Fcj^l$!3t*****fH!3.WFQ7$F^j^l$!3k**** **y:U+WFQ7$Fii^l$!3!)*****p\\)z'R%FQ7$Fdi^l$!3;+++5YD#R%FQ7$F_i^l$!3=+ ++\\D$oQ%FQ7$Fjh^l$!3A+++eUd!Q%FQ7$Feh^l$!3#******RD@NP%FQ7$F`h^l$!3!* *****R*[rlVFQ7$F[h^l$!33+++bm>dVFQFcaalFg^z-Fj^n6jn7%7$Fej_l$!+J%3![VF jan7$F[aal$!+[G>QVFjan7$F_[`l$!$>%Ff^n7%F]]bl7$Fe`al$!+1OzFVFjanF`]bl7 %Fd]bl7$F_`al$!+_e&oJ%FjanF`]bl7%Fh]bl7$Fi_al$!+-mU0VFjanF`]bl7%F\\^bl 7$Fc_al$!+3_b$H%FjanF`]bl7%F`^bl7$F]_al$!+)*QH\"G%FjanF`]bl7%Fd^bl7$Fg ^al$!+%R)poUFjanF`]bl7%Fh^bl7$Fa^al$!+O'GeD%FjanF`]bl7%F\\_bl7$F[^al$! +N'\\FC%FjanF`]bl7%F`_bl7$Fe]al$!+rC`HUFjanF`]bl7%Fd_bl7$F_]al$!+*\\bi @%FjanF`]bl7%Fh_bl7$Fi\\al$!+vd+.UFjanF`]bl7%F\\`bl7$Fc\\al$!+y2))*=%F janF`]bl7%F``bl7$F]\\al$!+k/*p<%FjanF`]bl7%Fd`bl7$Fg[al$!+L(fW;%FjanF` ]bl7%Fh`bl7$Fa[al$!+R8V_TFjanF`]bl7%F\\abl7$F[[al$!+,$p59%FjanF`]bl7%F `abl7$Fej`l$!+eFcITFjanF`]bl7%Fdabl7$F_j`l$!+D,8@TFjanF`]bl7%Fhabl7$Fi i`l$!+:J-8TFjanF`]bl7%F\\bbl7$Fci`l$!+\\(Hl5%FjanF`]bl7%F`bbl7$F]i`l$! +NY(>5%FjanF`]bl7%Fdbbl7$Fgh`l$!+2Or*4%FjanF`]bl7%Fhbbl7$Fah`l$!+,#=,5 %FjanF`]bl7%F\\cbl7$F[h`l$!+sTa.TFjanF`]bl7%F`cbl7$Feg`l$!+\"4z-6%Fjan F`]bl7%Fdcbl7$F_g`l$!+I,Z?TFjanF`]bl7%Fhcbl7$Fif`l$!+pn/MTFjanF`]bl7%F \\dbl7$Fcf`l$!+^(p1:%FjanF`]bl7%F`dbl7$F]f`l$!+U9upTFjanF`]bl7%Fddbl7$ Fge`l$!+J*)[!>%FjanF`]bl7%Fhdbl7$Fae`l$!++/47UFjanF`]bl7%F\\ebl7$F[e`l $!+=HzLUFjanF`]bl7%F`ebl7$Fed`l$!+WV)\\D%FjanF`]bl7%Fdebl7$F_d`l$!+;S@ vUFjanF`]bl7%Fhebl7$Fic`l$!+c%yTH%FjanF`]bl7%F\\fbl7$Fcc`l$!+M=p6VFjan F`]bl7%F`fbl7$F]c`l$!+LZlFVFjanF`]bl7%Fdfbl7$Fgb`l$!+%fF?M%FjanF`]bl7% Fhfbl7$Fab`l$!+x/\"[N%FjanF`]bl7%F\\gbl7$F[b`l$!+:'GgO%FjanF`]bl7%F`gb l7$Fea`l$!+@DsvVFjanF`]bl7%Fdgbl7$F_a`l$!+)RTRQ%FjanF`]bl7%Fhgbl7$Fi`` l$!+i)Q2R%FjanF`]bl7%F\\hbl7$Fc``l$!+F-<'R%FjanF`]bl7%F`hbl7$F]``l$!+O 3H+WFjanF`]bl7%Fdhbl7$Fg_`l$!+v]:.WFjanF`]bl7%Fhhbl7$Fa_`l$!+[e\"[S%Fj anF`]bl7%F\\ibl7$F[_`l$!+IVK0WFjanF`]bl7%F`ibl7$Fe^`l$!+<*HZS%FjanF`]b l7%Fdibl7$F_^`l$!+'H!3.WFjanF`]bl7%Fhibl7$Fi]`l$!+z:U+WFjanF`]bl7%F\\j bl7$Fc]`l$!+(\\)z'R%FjanF`]bl7%F`jbl7$F]]`l$!+5YD#R%FjanF`]bl7%Fdjbl7$ Fg\\`l$!+\\D$oQ%FjanF`]bl7%Fhjbl7$Fa\\`l$!+eUd!Q%FjanF`]bl7%F\\[cl7$F[ \\`l$!+a7_tVFjanF`]bl7%F`[cl7$Fe[`l$!+%*[rlVFjanF`]bl7%Fd[cl7$Fjj_l$!+ bm>dVFjanF`]bl7%Fh[clFj\\blF`]blF]g^lFc^r-%%FONTG6$%*HELVETICAGFb^r-%( SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F]\\cl6#%(DEF AULTG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$z%Ff^n$\"$>#Ff^n;Fd]cl$\"$z%Ff ^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#------------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "th e stability region for the 7 stage, order 4 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R ^ for the 7 stage, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b^`,StabilityFunction(4,7,'expanded'))):\n`R^` := unapply(%,z ):\n'`R^`(z)'=`R^`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R^G6#%\" zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)* &#F)\"#CF)*$)F'\"\"%F)F)F)*&#\">nVDcND*43%4%zd,C\"@+!H1z;fs9!=ys([N)[#)f#y()pR8;lC\"A+SxV2]b$)33pO1dQ " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R^`(z)=-1,z =-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+4r#*oR!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "zo0 := newton(`R^`(z)=-1,z=-4):\npo1 := plot([`R^`(z),-1],z=-4.49 ..0.49,color=[red,blue]):\npo2 := plot([[[zo0,-1]]$3],style=point,symb ol=[circle,cross,diamond],color=black):\npo3 := plot([[zo0,0],[zo0,-1] ],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([po1,po2,po3], view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%! #<$!3SE@F(4]=n#F*7$$!3;+]7'3DdV%F*$!3&3y[>#)e\"GCF*7$$!35++Ds,X\"Q%F*$ !3&)=h\\<%HO?#F*7$$!3(*\\P*)feAMVF*$!3AhmM<2&G-#F*7$$!3%)*\\Pva,qG%F*$ !3Ka!*QIQza=F*7$$!3I+](4>%y!=%F*$!3;d./HNP>:F*7$$!35+]#oviQ2%F*$!3k%4# 3[CsM7F*7$$!3A+ve,&\\u'RF*$!33-\"*fE,!)p**!#=7$$!3?+v8#z!zoQF*$!3gfH,` '\\c6)FP7$$!3;+v.8ajmPF*$!3Y*f%\\0t3,lFP7$$!3C+v$Q<')4m$F*$!3&>uq#RX<6 ^FP7$$!3G+v3`dnbNF*$!3!4**)eu.N$F*$!3QkECoP$eH#FP7$$!3E++:Qn_WKF*$!3pFI^GU*Gk\"FP7$$!3#*** *\\s&QnOJF*$!3H6Os$3.%46FP7$$!3&****\\=iPF.$F*$!3'e)pG%)4@%*o!#>7$$!3Q +v$Gc`$QHF*$!3K=g*pE+'GPFhp7$$!3+++l(y@h#GF*$!3K#*4\"f,VV.'!#?7$$!3=++ g_n/JFF*$\"3'RPnRRVqh\"Fhp7$$!3C+vt%)=X?EF*$\"3@3C!*z1$z#QFhp7$$!3D++5 (ocD_#F*$\"3Fmp!)p'3!RbFhp7$$!3#**\\P(R,::CF*$\"3NS)[QsoZB(Fhp7$$!3&)* \\7sqtGJ#F*$\"3ggQF1pfS()Fhp7$$!30+]Pz*eh?#F*$\"3AQr<@3-E5FP7$$!3l*\\( [Y:;3@F*$\"3'fxD8#=bl6FP7$$!3$***\\7w!eC+#F*$\"39m!Qe7'Q?8FP7$$!3%**\\ Py(=m#*=F*$\"3-J*HQ59-\\\"FP7$$!3%)*\\iW'R3(z\"F*$\"3eDV*R^^'[;FP7$$!3 -+]d*>dQp\"F*$\"3(R6?(HW%Q$=FP7$$!3)*****p]Q@(e\"F*$\"3'Qt`&efdV?FP7$$ !3I+]FRT)G[\"F*$\"3oZ[HX3))pAFP7$$!3')*\\P*y(R>Q\"F*$\"3&G\"p%yWi:^#FP 7$$!3++]Kx#e)p7F*$\"3DlvT[!=&4GFP7$$!3=++5j![\"p6F*$\"3G>P<1B.2JFP7$$! 3s***\\KT=;1\"F*$\"3#)*Ht7,)\\fMFP7$$!3'***\\P%o0=k*FP$\"3r/#))*eTL8QF P7$$!3](***\\tEbw&)FP$\"33_birayTUFP7$$!39-]P2EBuvFP$\"3]G0oL8'))o%FP7 $$!3>)*\\(GL>l_'FP$\"33bxC=gp1_FP7$$!3z++]-\")=-bFP$\"3EHfUaMEodFP7$$! 3$*)*\\PM#3)HWFP$\"3k*f\">([/7U'FP7$$!3;(****f'*ypR$FP$\"3HnN/Ti&)>rFP 7$$!3L)**\\U'=wSBFP$\"3l`H$=j:I\"zFP7$$!3?$*\\Pt2H$H\"FP$\"3ki/'[kZoy) FP7$$!33R****pit2LFhp$\"3#>:346PYn*FP7$$\"3Z^+]FkzBxFhp$\"3)QV'G=\"*H! 3\"F*7$$\"35*****z2`!f\"F*7$$\"3l+]i_F06GFP$\"3wtHdyHf C8F*7$$\"3\\.]Pt1&z\"QFP$\"3i\\)Hsc6\\Y\"F*7$$\"3!***************[FP$ \"3$yD*H8`JK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7 S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\ l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l 7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l 7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l 7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\ l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\ l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lF a\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3-+++4r#*oRF*Fa\\l-%'SYMBOLG6# %'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\ `l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[ \\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%V IEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture shows the stability regio n for the 8 stage, order 4 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1582 "`R^` := z -> 1+z+1/2*z^2+ 1/6*z^3+1/24*z^4+\n 24015779409408099253556254367/28976177278180 14725916790629000*z^5+\n 24651613396987782598248835487/173857063 66908088355500743774000*z^6+7975/38588232*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=201/20 do\n zz := newton(`R^`(z)=exp(tt*Pi*I),z=z 0):\n z0 := zz:\n if (13/20<=tt and tt<=6/5) or (87/10<=tt and tt< =93/10) then\n hh := 1/40\n else \n hh := 1/20\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo1 \+ := plot(pts,color=COLOR(RGB,.2,.05,.38)):\npo2 := plots[polygonplot]([ seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.4,.1,.75)):\npts := []: z0 := 1.6+4.3*I:\nfor c t from 0 to 60 do\n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo3 := plot( pts,color=COLOR(RGB,.2,.05,.38)):\npo4 := plots[polygonplot]([seq([pts [i-1],pts[i],[1.45,4.18]],i=2..nops(pts))],\n style=patchnogr id,color=COLOR(RGB,.4,.1,.75)):\npts := []: z0 := 1.6-4.3*I:\nfor ct f rom 0 to 60 do\n zz := newton(`R^`(z)=exp(ct*Pi/30*I),z=z0):\n z0 \+ := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\npo5 := plot(pts ,color=COLOR(RGB,.2,.05,.38)):\npo6 := plots[polygonplot]([seq([pts[i- 1],pts[i],[1.45,-4.18]],i=2..nops(pts))],\n style=patchnogrid ,color=COLOR(RGB,.4,.1,.75)):\npo7 := plot([[[-4.79,0],[2.19,0]],[[0,- 4.79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([po||(1..7) ],view=[-4.79..2.19,-4.79..4.79],font=[HELVETICA,9],\n la bels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 555 586 586 {PLOTDATA 2 "6/-%'CURVESG6$7]y7$$\"\"!F)F (7$$\"3++++lcX>6!#E$\"3')******ojzq:!#=7$$\"3\\*****HZ*QMs!#D$\"3\"*** ***4!RfTJF07$$\"3[+++eX)yM)!#C$\"3'******\\.(R7ZF07$$\"3++++bo%fu%!#B$ \"3=+++F+@$G'F07$$\"38+++4@R>=!#A$\"3e******Gn,ayF07$$\"3q*****z4+cQ&F F$\"3U+++4'GZU*F07$$\"31+++*\\`qJ\"!#@$\"3!******fB3&*4\"!#<7$$\"3%*** ****yO!Rv#FQ$\"31+++cqWc7FT7$$\"3y*****pytu)\\FQ$\"3,+++g-<89FT7$$\"3' )******>()GexFQ$\"3%*******GCZp:FT7$$\"3)4++]-1#*))*FQ$\"3!*******=t.D b)FQ$\"3/+++B#H%z=FT7$$!35+++[5d^:FQ$\"3<+++l76K?FT 7$$!33+++ptzQ7!#?$\"3>+++2hi2@FT7$$!3C+++.5(H)GF`p$\"3%******\\b)[#=#F T7$$!3$*******Qf5W_F`p$\"3/+++>AicAFT7$$!3#3++!zX_'\\)F`p$\"3$)*****>; ]*HBFT7$$!3%*******R;#HG\"!#>$\"3@+++lHR-CFT7$$!3%)*****>!z9W=Feq$\"3! )*****HHlQZ#FT7$$!3))*****>lJPb#Feq$\"3#*******)4rUa#FT7$$!3o*****z#p$ =V$Feq$\"3A+++))z\\8EFT7$$!3))*****4a4v\\%Feq$\"3=+++u:T\"o#FT7$$!3'** ****>E/ww&Feq$\"3++++/6&yu#FT7$$!3W*****pVXbD(Feq$\"3!)*****R*yi7GFT7$ $!3[+++q%=+(*)Feq$\"3%******H@Gb(GFT7$$!3-+++u\"y84\"F0$\"38+++KW'RF0$\"3!******H %>y%R$FT7$$!3()*****p5C7j%F0$\"3#)*****pVpQX$FT7$$!3@+++nj@+`F0$\"37++ +\"yRT]$FT7$$!3?+++[LcmfF0$\"3')******eSgYNFT7$$!3u*****fq\"eFmF0$\"3% ******R#o7#e$FT7$$!3-+++&z4>G(F0$\"3@+++DCU6OFT7$$!3*)*****H2i*GzF0$\" 3%******\\Or]j$FT7$$!3))*****Ri#fo&)F0$\"3#)******f=a`OFT7$$!3')*****H N'*3?*F0$\"3%)*****RN5sm$FT7$$!3@+++nh4E)*F0$\"39+++]*yjn$FT7$$!3$**** **\\F[W/\"FT$\"3))*****fZ*G\"o$FT7$$!3++++['Qc5\"FT$\"35+++'eM@o$FT7$$ !3%******\\%p@m6FT$\"35+++]k1zOFT7$$!36+++lIAE7FT$\"35+++pN?sOFT7$$!31 +++h?q&G\"FT$\"33+++DijhOFT7$$!3.+++#Q1ZM\"FT$\"3')*****4)4VZOFT7$$!34 +++:))H.9FT$\"3:+++fYjHOFT7$$!3-+++Zhbh9FT$\"3:+++=#y#3OFT7$$!3/+++uNd >:FT$\"3!)*****z-\"Q$e$FT7$$!3)******>!)pud\"FT$\"3?+++mh&\\b$FT7$$!35 +++'o#RN;FT$\"3'******>_)3_FT$\"3)*******4ax7LF T7$$!3-+++x$)H'*>FT$\"3%)*****Hm*)=E$FT7$$!3%)*****H>z71#FT$\"3-+++%\\ b*3KFT7$$!3-+++)3w\"G@FT$\"3%******pH!faJFT7$$!3')*****H=%)o>#FT$\"3#) *****>`k%*4$FT7$$!39+++#H$3nAFT$\"3A+++\"Q6U/$FT7$$!32+++h,FQBFT$\"3)* *****>&GL*)HFT7$$!3++++qi$)4CFT$\"3)******z3Q^$HFT7$$!3++++D-;\"[#FT$ \"3++++LJt\")GFT7$$!31+++EAo^DFT$\"3!******4za!HGFT7$$!3#)*****>'\\%4i #FT$\"3%******\\NDpx#FT7$$!3-+++=Mg)o#FT$\"33+++Qk5DFFT7$$!3A+++3\\TaF FT$\"3#*******)HSLn#FT7$$!3>+++P0A=GFT$\"3)*******yjP@EFT7$$!3'******R ^D*zGFT$\"3++++6c)*oDFT7$$!3')*****p'>[RHFT$\"3\"*******)Qmf^#FT7$$!3' ******fTvo*HFT$\"3/+++Pb9iCFT7$$!3!)*****z![6_IFT$\"3A+++mmP2CFT7$$!3, +++QbA0JFT$\"3;+++Eu`^BFT7$$!3%******pxWi:$FT$\"33+++Hl_%H#FT7$$!3(*** *****\\=_?$FT$\"39+++45EOAFT7$$!3-+++C(*>_KFT$\"3)******\\'Rnw@FT7$$!3 )*******yvC(H$FT$\"3$******p)Gr:@FT7$$!3/+++#)oUSLFT$\"3#)******='QL0# FT7$$!3%)*****H)z!=Q$FT$\"3%*******e\\_*)>FT7$$!34+++LmY@MFT$\"34+++t) eU#>FT7$$!3)*******eO[fMFT$\"3#******\\GTv&=FT7$$!3))*****>AWf\\$FT$\" 31+++J$)Q*y\"FT7$$!3!******zVO4`$FT$\"3!******\\$H$)>Yr$FT$\"3'******f(*4tF\"FT7$$!3=+ ++'ow8u$FT$\"3'******z:`/?\"FT7$$!3;+++y01nPFT$\"34+++&e'4B6FT7$$!31++ +yei\"z$FT$\"31+++8yQX5FT7$$!3!******z*R+:QFT$\"3'*******o7iu'*F07$$!3 :+++.z5PQFT$\"3s*****H:sV*))F07$$!3'******R$z$y&QFT$\"3!)******Qu49\") F07$$!39+++jz3xQFT$\"3-+++m(RXL(F07$$!3%*******H6v%*QFT$\"3K+++MBAclF0 7$$!32+++gYs5RFT$\"3C+++htYzdF07$$!35+++5N\"\\#RFT$\"3i*****z+BW+&F07$ $!3\")*****H&GBPRFT$\"3%)*****R\"Q4JUF07$$!3++++V&4w%RFT$\"3')*****R!3 PfMF07$$!3#)*****p!G)f&RFT$\"3!)*****f!)e!*o#F07$$!3')*****4T/B'RFT$\" 3))*****zz**)>>F07$$!3=+++C&Ql'RFT$\"3'******40#f^6F07$$!3#******fYh'o RFT$\"3)******HSb!QQFeq7$Fe^m$!3)******HSb!QQFeq7$F`^m$!3'******40#f^6 F07$F[^m$!3))*****zz**)>>F07$Ff]m$!3!)*****f!)e!*o#F07$Fa]m$!3')*****R !3PfMF07$F\\]m$!3%)*****R\"Q4JUF07$Fg\\m$!3i*****z+BW+&F07$Fb\\m$!3C++ +htYzdF07$F]\\m$!3K+++MBAclF07$Fh[m$!3-+++m(RXL(F07$Fc[m$!3!)******Qu4 9\")F07$F^[m$!3s*****H:sV*))F07$Fijl$!3'*******o7iu'*F07$Fdjl$!31+++8y QX5FT7$F_jl$!34+++&e'4B6FT7$Fjil$!3'******z:`/?\"FT7$Feil$!3'******f(* 4tF\"FT7$F`il$!35+++!*Q^`8FT7$F[il$!3.+++SL\"*G9FT7$Ffhl$!3-+++J^O.:FT 7$Fahl$!3#******z)*Rnd\"FT7$F\\hl$!3#******HWE*[;FT7$Fggl$!3!******\\$ H$)>FT7$Fcfl$!3%*******e\\_*)>FT7$F^fl$!3#)******='QL0#FT7$Fiel$!3$*** ***p)Gr:@FT7$Fdel$!3)******\\'Rnw@FT7$F_el$!39+++45EOAFT7$Fjdl$!33+++H l_%H#FT7$Fedl$!3;+++Eu`^BFT7$F`dl$!3A+++mmP2CFT7$F[dl$!3/+++Pb9iCFT7$F fcl$!3\"*******)Qmf^#FT7$Facl$!3++++6c)*oDFT7$F\\cl$!3)*******yjP@EFT7 $Fgbl$!3#*******)HSLn#FT7$Fbbl$!33+++Qk5DFFT7$F]bl$!3%******\\NDpx#FT7 $Fhal$!3!******4za!HGFT7$Fcal$!3++++LJt\")GFT7$F^al$!3)******z3Q^$HFT7 $Fi`l$!3)******>&GL*)HFT7$Fd`l$!3A+++\"Q6U/$FT7$F_`l$!3#)*****>`k%*4$F T7$Fj_l$!3%******pH!faJFT7$Fe_l$!3-+++%\\b*3KFT7$F`_l$!3%)*****Hm*)=E$ FT7$F[_l$!3)*******4ax7LFT7$Ff^l$!3%*******Qm5hLFT7$Fa^l$!3))******3v] 1MFT7$F\\^l$!3#)******)4A([MFT7$Fg]l$!3!)*****\\t*e([$FT7$Fb]l$!3'**** **>_y%R$FT7$F iv$!3=+++gqehLFT7$Fdv$!3%)******RExDLFT7$F_v$!3!******z!*4sG$FT7$Fju$! 3++++T'*yXKFT7$Feu$!3;+++#pL9?$FT7$F`u$!3?+++Ip5aJFT7$F[u$!3;+++&RFQ5$ FT7$Fft$!3?+++=?n]IFT7$Fat$!37+++Obx%*HFT7$F\\t$!38+++K;]*HBFT7$Fip$!3/+++>AicAFT7$F dp$!3%******\\b)[#=#FT7$F^p$!3>+++2hi2@FT7$$\"3!******4YyO/&FQ$!31+++% R?g&>FT7$$\"3%*)****4ip\"=**FQ$!3.+++(\\6C!=FT7$$\"3S+++4y5K!*FQ$!3)** ****>`rtk\"FT7$$\"3S+++943]jFQ$!3!*******H4R\"\\\"FT7$$\"3#)*****RTfIx $FQ$!3)******R4Y[L\"FT7$$\"3!******4['pR>FQ$!31+++pa*z<\"FT7$$\"3(**** **4U\"4-')FF$!3.+++mx*4-\"FT7$$\"3,++++H<;KFF$!3k*****RD&RR')F07$$\"3o *****>gXsl*F@$!3Q+++%4<'oqF07$$\"3%******Hih%=@F@$!3k******=C!y\\&F07$ $\"39+++6))=xFF:$!35+++tV*p#RF07$$\"33+++<$H0G\"F4$!3-+++,[>cBF07$$\"3 #******>\")*\\W6F0$\"+pjzq:!#57$$FbfnF)F(7%Fifn7$$\"+t%*QMsFT$\"+,RfTJF^gnF_gn7%Fbg n7$$\"+eX)yM)!#;$\"+NqR7ZF^gnF_gn7%Fhgn7$$\"+bo%fu%!#:$\"+F+@$G'F^gnF_ gn7%F_hn7$$\"+4@R>=!#9$\"+Hn,ayF^gnF_gn7%Ffhn7$$\"+)4+cQ&Fihn$\"+4'GZU *F^gnF_gn7%F]in7$$\"+*\\`qJ\"!#8$\"+O#3&*4\"!\"*F_gn7%Fcin7$$\"+zO!Rv# Ffin$\"+cqWc7FiinF_gn7%F[jn7$$\"+(ytu)\\Ffin$\"+g-<89FiinF_gn7%Fajn7$$ \"+?()GexFfin$\"+HCZp:FiinF_gn7%Fgjn7$$\"+Dg?*))*Ffin$\"+>t.Db)Ffin$\"+B#H%z=FiinF_gn7%Fc[o7$$!+[5d^:Ffin$\"+l76K?F iinF_gn7%Fi[o7$$!+ptzQ7!#7$\"+2hi2@FiinF_gn7%F_\\o7$$!+.5(H)GFb\\o$\"+ b&)[#=#FiinF_gn7%Ff\\o7$$!+Rf5W_Fb\\o$\"+>AicAFiinF_gn7%F\\]o7$$!+zX_' \\)Fb\\o$\"+i,&*HBFiinF_gn7%Fb]o7$$!+S;#HG\"!#6$\"+lHR-CFiinF_gn7%Fh]o 7$$!+-z9W=F[^o$\"+$HlQZ#FiinF_gn7%F_^o7$$!+_;t`DF[^o$\"+*4rUa#FiinF_gn 7%Fe^o7$$!+Gp$=V$F[^o$\"+))z\\8EFiinF_gn7%F[_o7$$!+T&4v\\%F[^o$\"+u:T \"o#FiinF_gn7%Fa_o7$$!+iUgndF[^o$\"+/6&yu#FiinF_gn7%Fg_o7$$!+PaabsF[^o $\"+%*yi7GFiinF_gn7%F]`o7$$!+q%=+(*)F[^o$\"+8#Gb(GFiinF_gn7%Fc`o7$$!+u \"y84\"F^gn$\"+K5P#F^gn$\"+#p L9?$FiinF_gn7%Fgbo7$$!+(oSTn#F^gn$\"+T'*yXKFiinF_gn7%F]co7$$!+.:?()HF^ gn$\"+3*4sG$FiinF_gn7%Fcco7$$!+Iu*yI$F^gn$\"+SExDLFiinF_gn7%Fico7$$!+P n?MOF^gn$\"+gqehLFiinF_gn7%F_do7$$!+z(>W'RF^gn$\"+V>y%R$FiinF_gn7%Fedo 7$$!+2TAJYF^gn$\"+P%pQX$FiinF_gn7%F[eo7$$!+nj@+`F^gn$\"+\"yRT]$FiinF_g n7%Faeo7$$!+[LcmfF^gn$\"+fSgYNFiinF_gn7%Fgeo7$$!+1G(F^gn$\"+DCU6OFiinF_gn7%Fcfo7$$!+t?'*GzF^gn$\"+l 82NOFiinF_gn7%Fifo7$$!+CEfo&)F^gn$\"+g=a`OFiinF_gn7%F_go7$$!+`j*3?*F^g n$\"+a.@nOFiinF_gn7%Fego7$$!+nh4E)*F^gn$\"+]*yjn$FiinF_gn7%F[ho7$$!+v# [W/\"Fiin$\"+w%*G\"o$FiinF_gn7%Faho7$$!+['Qc5\"Fiin$\"+'eM@o$FiinF_gn7 %Fgho7$$!+Xp@m6Fiin$\"+]k1zOFiinF_gn7%F]io7$$!+lIAE7Fiin$\"+pN?sOFiinF _gn7%Fcio7$$!+h?q&G\"Fiin$\"+DijhOFiinF_gn7%Fiio7$$!+#Q1ZM\"Fiin$\"+\" )4VZOFiinF_gn7%F_jo7$$!+:))H.9Fiin$\"+fYjHOFiinF_gn7%Fejo7$$!+Zhbh9Fii n$\"+=#y#3OFiinF_gn7%F[[p7$$!+uNd>:Fiin$\"+G5Q$e$FiinF_gn7%Fa[p7$$!+-) pud\"Fiin$\"+mh&\\b$FiinF_gn7%Fg[p7$$!+'o#RN;Fiin$\"+Av,BNFiinF_gn7%F] \\p7$$!+a[_$p\"Fiin$\"+N(*e([$FiinF_gn7%Fc\\p7$$!+'>)3_Fiin$\"+5ax7LFiinF_gn7%F[^p7$$!+x$)H'*>Fi in$\"+j'*)=E$FiinF_gn7%Fa^p7$$!+$>z71#Fiin$\"+%\\b*3KFiinF_gn7%Fg^p7$$ !+)3w\"G@Fiin$\"+(H!faJFiinF_gn7%F]_p7$$!+$=%)o>#Fiin$\"+KXY*4$FiinF_g n7%Fc_p7$$!+#H$3nAFiin$\"+\"Q6U/$FiinF_gn7%Fi_p7$$!+h,FQBFiin$\"+_GL*) HFiinF_gn7%F_`p7$$!+qi$)4CFiin$\"+)3Q^$HFiinF_gn7%Fe`p7$$!+D-;\"[#Fiin $\"+LJt\")GFiinF_gn7%F[ap7$$!+EAo^DFiin$\"+\"za!HGFiinF_gn7%Faap7$$!+i \\%4i#Fiin$\"+b`#px#FiinF_gn7%Fgap7$$!+=Mg)o#Fiin$\"+Qk5DFFiinF_gn7%F] bp7$$!+3\\TaFFiin$\"+*HSLn#FiinF_gn7%Fcbp7$$!+P0A=GFiin$\"+zjP@EFiinF_ gn7%Fibp7$$!+9b#*zGFiin$\"+6c)*oDFiinF_gn7%F_cp7$$!+n>[RHFiin$\"+*Qmf^ #FiinF_gn7%Fecp7$$!+;a(o*HFiin$\"+Pb9iCFiinF_gn7%F[dp7$$!+3[6_IFiin$\" +mmP2CFiinF_gn7%Fadp7$$!+QbA0JFiin$\"+Eu`^BFiinF_gn7%Fgdp7$$!+xZCcJFii n$\"+Hl_%H#FiinF_gn7%F]ep7$$!++&=_?$Fiin$\"+45EOAFiinF_gn7%Fcep7$$!+C( *>_KFiin$\"+lRnw@FiinF_gn7%Fiep7$$!+zvC(H$Fiin$\"+()Gr:@FiinF_gn7%F_fp 7$$!+#)oUSLFiin$\"+>'QL0#FiinF_gn7%Fefp7$$!+$)z!=Q$Fiin$\"+f\\_*)>Fiin F_gn7%F[gp7$$!+LmY@MFiin$\"+t)eU#>FiinF_gn7%Fagp7$$!+fO[fMFiin$\"+>v &=FiinF_gn7%Fggp7$$!+AU%f\\$Fiin$\"+J$)Q*y\"FiinF_gn7%F]hp7$$!+Qk$4`$F iin$\"+NH$)>Yr$Fiin$\"+w*4tF\"FiinF_gn7%Fgjp7$$!+'ow8u$Fi in$\"+eJX+7FiinF_gn7%F][q7$$!+y01nPFiin$\"+&e'4B6FiinF_gn7%Fc[q7$$!+ye i\"z$Fiin$\"+8yQX5FiinF_gn7%Fi[q7$$!+)*R+:QFiin$\"+p7iu'*F^gnF_gn7%F_ \\q7$$!+.z5PQFiin$\"+`@P%*))F^gnF_gn7%Fe\\q7$$!+Mz$y&QFiin$\"+Ru49\")F ^gnF_gn7%F[]q7$$!+jz3xQFiin$\"+m(RXL(F^gnF_gn7%Fa]q7$$!+I6v%*QFiin$\"+ MBAclF^gnF_gn7%Fg]q7$$!+gYs5RFiin$\"+htYzdF^gnF_gn7%F]^q7$$!+5N\"\\#RF iin$\"+3IU/]F^gnF_gn7%Fc^q7$$!+`GBPRFiin$\"+9Q4JUF^gnF_gn7%Fi^q7$$!+V& 4w%RFiin$\"+/3PfMF^gnF_gn7%F__q7$$!+2G)f&RFiin$\"+1)e!*o#F^gnF_gn7%Fe_ q7$$!+6WIiRFiin$\"+)z**)>>F^gnF_gn7%F[`q7$$!+C&Ql'RFiin$\"+^?f^6F^gnF_ gn7%Fa`q7$$!+m9moRFiin$\"+.a0QQF[^oF_gn7%Fg`q7$Fh`q$!+.a0QQF[^oF_gn7%F ]aq7$Fb`q$!+^?f^6F^gnF_gn7%Faaq7$F\\`q$!+)z**)>>F^gnF_gn7%Feaq7$Ff_q$! +1)e!*o#F^gnF_gn7%Fiaq7$F`_q$!+/3PfMF^gnF_gn7%F]bq7$Fj^q$!+9Q4JUF^gnF_ gn7%Fabq7$Fd^q$!+3IU/]F^gnF_gn7%Febq7$F^^q$!+htYzdF^gnF_gn7%Fibq7$Fh]q $!+MBAclF^gnF_gn7%F]cq7$Fb]q$!+m(RXL(F^gnF_gn7%Facq7$F\\]q$!+Ru49\")F^ gnF_gn7%Fecq7$Ff\\q$!+`@P%*))F^gnF_gn7%Ficq7$F`\\q$!+p7iu'*F^gnF_gn7%F ]dq7$Fj[q$!+8yQX5FiinF_gn7%Fadq7$Fd[q$!+&e'4B6FiinF_gn7%Fedq7$F^[q$!+e JX+7FiinF_gn7%Fidq7$Fhjp$!+w*4tF\"FiinF_gn7%F]eq7$Fbjp$!+!*Q^`8FiinF_g n7%Faeq7$F\\jp$!+SL\"*G9FiinF_gn7%Feeq7$Ffip$!+J^O.:FiinF_gn7%Fieq7$F` ip$!+))*Rnd\"FiinF_gn7%F]fq7$Fjhp$!+Vk#*[;FiinF_gn7%Fafq7$Fdhp$!+NH$)> FiinF_gn7%Fagq7$F\\gp$!+f\\_*)>FiinF_gn7%Fegq7$Fff p$!+>'QL0#FiinF_gn7%Figq7$F`fp$!+()Gr:@FiinF_gn7%F]hq7$Fjep$!+lRnw@Fii nF_gn7%Fahq7$Fdep$!+45EOAFiinF_gn7%Fehq7$F^ep$!+Hl_%H#FiinF_gn7%Fihq7$ Fhdp$!+Eu`^BFiinF_gn7%F]iq7$Fbdp$!+mmP2CFiinF_gn7%Faiq7$F\\dp$!+Pb9iCF iinF_gn7%Feiq7$Ffcp$!+*Qmf^#FiinF_gn7%Fiiq7$F`cp$!+6c)*oDFiinF_gn7%F]j q7$Fjbp$!+zjP@EFiinF_gn7%Fajq7$Fdbp$!+*HSLn#FiinF_gn7%Fejq7$F^bp$!+Qk5 DFFiinF_gn7%Fijq7$Fhap$!+b`#px#FiinF_gn7%F][r7$Fbap$!+\"za!HGFiinF_gn7 %Fa[r7$F\\ap$!+LJt\")GFiinF_gn7%Fe[r7$Ff`p$!+)3Q^$HFiinF_gn7%Fi[r7$F`` p$!+_GL*)HFiinF_gn7%F]\\r7$Fj_p$!+\"Q6U/$FiinF_gn7%Fa\\r7$Fd_p$!+KXY*4 $FiinF_gn7%Fe\\r7$F^_p$!+(H!faJFiinF_gn7%Fi\\r7$Fh^p$!+%\\b*3KFiinF_gn 7%F]]r7$Fb^p$!+j'*)=E$FiinF_gn7%Fa]r7$F\\^p$!+5ax7LFiinF_gn7%Fe]r7$Ff] p$!+Rm5hLFiinF_gn7%Fi]r7$F`]p$!+4v]1MFiinF_gn7%F]^r7$Fj\\p$!+*4A([MFii nF_gn7%Fa^r7$Fd\\p$!+N(*e([$FiinF_gn7%Fe^r7$F^\\p$!+Av,BNFiinF_gn7%Fi^ r7$Fh[p$!+mh&\\b$FiinF_gn7%F]_r7$Fb[p$!+G5Q$e$FiinF_gn7%Fa_r7$F\\[p$!+ =#y#3OFiinF_gn7%Fe_r7$Ffjo$!+fYjHOFiinF_gn7%Fi_r7$F`jo$!+\")4VZOFiinF_ gn7%F]`r7$Fjio$!+DijhOFiinF_gn7%Fa`r7$Fdio$!+pN?sOFiinF_gn7%Fe`r7$F^io $!+]k1zOFiinF_gn7%Fi`r7$Fhho$!+'eM@o$FiinF_gn7%F]ar7$Fbho$!+w%*G\"o$Fi inF_gn7%Faar7$F\\ho$!+]*yjn$FiinF_gn7%Fear7$Ffgo$!+a.@nOFiinF_gn7%Fiar 7$F`go$!+g=a`OFiinF_gn7%F]br7$Fjfo$!+l82NOFiinF_gn7%Fabr7$Fdfo$!+DCU6O FiinF_gn7%Febr7$F^fo$!+Co7#e$FiinF_gn7%Fibr7$Fheo$!+fSgYNFiinF_gn7%F]c r7$Fbeo$!+\"yRT]$FiinF_gn7%Facr7$F\\eo$!+P%pQX$FiinF_gn7%Fecr7$$!+%RLr H%F^gn$!+@e\\DMFiinF_gn7%Ficr7$Ffdo$!+V>y%R$FiinF_gn7%F_dr7$F`do$!+gqe hLFiinF_gn7%Fcdr7$Fjco$!+SExDLFiinF_gn7%Fgdr7$Fdco$!+3*4sG$FiinF_gn7%F [er7$F^co$!+T'*yXKFiinF_gn7%F_er7$Fhbo$!+#pL9?$FiinF_gn7%Fcer7$Fbbo$!+ Ip5aJFiinF_gn7%Fger7$F\\bo$!+&RFQ5$FiinF_gn7%F[fr7$Ffao$!+=?n]IFiinF_g n7%F_fr7$F`ao$!+Obx%*HFiinF_gn7%Fcfr7$Fj`o$!+KAicAFiinF_gn7 %F_ir7$Fg\\o$!+b&)[#=#FiinF_gn7%Fcir7$F`\\o$!+2hi2@FiinF_gn7%Fgir7$$\" +h%yO/&Ffin$!+%R?g&>FiinF_gn7%F[jr7$$\"+@'p\"=**Ffin$!+(\\6C!=FiinF_gn 7%Fajr7$$\"+4y5K!*Ffin$!+K:PZ;FiinF_gn7%Fgjr7$$\"+943]jFfin$!+I4R\"\\ \"FiinF_gn7%F][s7$$\"+9%fIx$Ffin$!+%4Y[L\"FiinF_gn7%Fc[s7$$\"+\"['pR>F fin$!+pa*z<\"FiinF_gn7%Fi[s7$$\"+@94-')Fihn$!+mx*4-\"FiinF_gn7%F_\\s7$ $\"++H<;KFihn$!+a_RR')F^gnF_gn7%Fe\\s7$$\"+-cCd'*Fbhn$!+%4<'oqF^gnF_gn 7%F[]s7$$\"+B;Y=@Fbhn$!+>C!y\\&F^gnF_gn7%Fa]s7$$\"+6))=xFF[hn$!+tV*p#R F^gnF_gn7%Fg]s7$$\"+<$H0G\"FT$!+,[>cBF^gnF_gn7%F]^s7$$\"+7)*\\Wc:F T$\"3g******3&f%zUFT7$$\"31+++L)o$[:FT$\"3u*****fdgvG%FT7$$\"3\"****** fC!*)R:FT$\"35+++Bk)\\H%FT7$$\"3!******R$p!3`\"FT$\"36+++,hp,VFT7$$\"3 '******z$G;@:FT$\"3O+++F\"\\wI%FT7$$\"3.+++(=.5^\"FT$\"3d*****4B0GJ%FT 7$$\"3)*******f[P+:FT$\"3L+++OT7a9FT$\"33+++h5@DVFT7$$\"30+++!e=?W\"FT$\"3=+++ QlsCVFT7$$\"3$******>!zrH9FT$\"3'******RV`JK%FT7$$\"3#*******o)otT\"FT $\"3?+++KvW?VFT7$$\"33+++&oa]S\"FT$\"3O+++XUc;VFT7$$\"3-+++[\"oGR\"FT$ \"3L+++)3f9J%FT7$$\"3!*******eN\"4Q\"FT$\"30+++y%)30VFT7$$\"3-+++c*3$p 8FT$\"3))*****R)4T(H%FT7$$\"3++++G()=e8FT$\"3m*****zC*Q)G%FT7$$\"31+++ ]lqZ8FT$\"3++++XG*zF%FT7$$\"35+++c&Q!Q8FT$\"3++++&[-iE%FT7$$\"3-+++dkQ H8FT$\"3E+++Rg,`UFT7$$\"3'******R0!)>K\"FT$\"3m*****H,d%QUFT7$$\"36+++ 8\"ygJ\"FT$\"30+++zgeAUFT7$$\"31+++ne'>J\"FT$\"3-+++$3;b?%FT7$$\"3\"** ****feY*48FT$\"3/+++%>Iu=%FT7$$\"3\"*******RQK58FT$\"31+++/8goTFT7$$\" 3*******pHrLJ\"FT$\"3S+++8!3%\\TFT7$$\"3-+++8#*G>8FT$\"3++++P\"R.8%FT7 $$\"3/+++eE:G8FT$\"3>+++rk(>6%FT7$$\"35+++6]')R8FT$\"3!)******)Q[\\4%F T7$$\"3-+++)HOTN\"FT$\"3++++Mu&)zSFT7$$\"3++++')=]q8FT$\"3=+++8,?nSFT7 $$\"35+++c2Q)Q\"FT$\"3))*****zj0t0%FT7$$\"3*******z$R:29FT$\"3?+++cLJ] SFT7$$\"30+++inBE9FT$\"3W+++Y+>YSFT7$$\"3!******\\\"p7X9FT$\"3l*****pX sZ/%FT7$$\"3#******zoCMY\"FT$\"3!*******fj\"e/%FT7$$\"3/+++X>$3[\"FT$ \"3!)*****H,R!\\SFT7$$\"3%******R@Rr\\\"FT$\"3q******G(\\T0%FT7$$\"3/+ ++c&3A^\"FT$\"3g*****H')o31%FT7$$\"3,+++!ycf_\"FT$\"3S+++Jy$*oSFT7$$\" 3(******f3T$Q:FT$\"3!)*****>;B\"ySFT7$$\"3)******\\mZ$\\:FT$\"3q*****4 &p@)3%FT7$$\"3/+++iI)*e:FT$\"3o*****pSN!*4%FT7$$\"31+++lyve\"F T$\"3y*****R6X*eTFT7$$\"3'******>>_'*e\"FT$\"3s*****z`%HrTFT7$$\"3-+++ [2h!f\"FT$\"3++++&4bN=%FT7$$\"3!******4G)\\!f\"FT$\"3')*****4M`c>%FT7$ $\"31+++%\\e$*e\"FT$\"3N+++,@_2UFT7$$\"3%******foMse\"FT$\"3M+++)[*4>U FT7$$\"3)******HppTe\"FT$\"3T+++z!G.B%FT7$$\"3#******\\!f?!e\"FT$\"3j* ****4?a6C%FT7$$\"3)******>L&Qv:FT$\"3i*****RJF:D%FT7$$\"3-+++3(\\(p:FT $\"3%******H[*RhUFT7$$\"3!*******H1Mj:FT$\"3))*****f!\\sqUFTF[`sFien-F ffn6jn7%7$$\"+B(*>c:Fiin$\"+4&f%zUFiin7$$\"+L)o$[:Fiin$\"+w0c(G%Fiin7$ $\"$X\"Fbfn$\"$=%Fbfn7%F_ct7$$\"+Y-*)R:Fiin$\"+Bk)\\H%FiinFdct7%Fjct7$ $\"+Mp!3`\"Fiin$\"+,hp,VFiinFdct7%F`dt7$$\"+QG;@:Fiin$\"+F\"\\wI%FiinF dct7%Ffdt7$$\"+(=.5^\"Fiin$\"+J_!GJ%FiinFdct7%F\\et7$$\"+g[P+:Fiin$\"+ OT7a9Fiin$\"+h5@DVFiinFdct7%Fjft7$$\"+!e=?W\"Fiin$\"+QlsCVFiin Fdct7%F`gt7$$\"+-zrH9Fiin$\"+MM:BVFiinFdct7%Ffgt7$$\"+p)otT\"Fiin$\"+K vW?VFiinFdct7%F\\ht7$$\"+&oa]S\"Fiin$\"+XUc;VFiinFdct7%Fbht7$$\"+[\"oG R\"Fiin$\"+)3f9J%FiinFdct7%Fhht7$$\"+fN\"4Q\"Fiin$\"+y%)30VFiinFdct7%F ^it7$$\"+c*3$p8Fiin$\"+%)4T(H%FiinFdct7%Fdit7$$\"+G()=e8Fiin$\"+[#*Q)G %FiinFdct7%Fjit7$$\"+]lqZ8Fiin$\"+XG*zF%FiinFdct7%F`jt7$$\"+c&Q!Q8Fiin $\"+&[-iE%FiinFdct7%Ffjt7$$\"+dkQH8Fiin$\"+Rg,`UFiinFdct7%F\\[u7$$\"+a +)>K\"Fiin$\"+8qXQUFiinFdct7%Fb[u7$$\"+8\"ygJ\"Fiin$\"+zgeAUFiinFdct7% Fh[u7$$\"+ne'>J\"Fiin$\"+$3;b?%FiinFdct7%F^\\u7$$\"+'eY*48Fiin$\"+%>Iu =%FiinFdct7%Fd\\u7$$\"+SQK58Fiin$\"+/8goTFiinFdct7%Fj\\u7$$\"+(HrLJ\"F iin$\"+8!3%\\TFiinFdct7%F`]u7$$\"+8#*G>8Fiin$\"+P\"R.8%FiinFdct7%Ff]u7 $$\"+eE:G8Fiin$\"+rk(>6%FiinFdct7%F\\^u7$$\"+6]')R8Fiin$\"+*Q[\\4%Fiin Fdct7%Fb^u7$$\"+)HOTN\"Fiin$\"+Mu&)zSFiinFdct7%Fh^u7$$\"+')=]q8Fiin$\" +8,?nSFiinFdct7%F^_u7$$\"+c2Q)Q\"Fiin$\"+QcIdSFiinFdct7%Fd_u7$$\"+QR:2 9Fiin$\"+cLJ]SFiinFdct7%Fj_u7$$\"+inBE9Fiin$\"+Y+>YSFiinFdct7%F``u7$$ \"+:p7X9Fiin$\"+dCxWSFiinFdct7%Ff`u7$$\"+)oCMY\"Fiin$\"+gj\"e/%FiinFdc t7%F\\au7$$\"+X>$3[\"Fiin$\"+8!R!\\SFiinFdct7%Fbau7$$\"+9#Rr\\\"Fiin$ \"+H(\\T0%FiinFdct7%Fhau7$$\"+c&3A^\"Fiin$\"+j)o31%FiinFdct7%F^bu7$$\" +!ycf_\"Fiin$\"+Jy$*oSFiinFdct7%Fdbu7$$\"+'3T$Q:Fiin$\"+iJ7ySFiinFdct7 %Fjbu7$$\"+lwM\\:Fiin$\"+^p@)3%FiinFdct7%F`cu7$$\"+iI)*e:Fiin$\"+2a.*4 %FiinFdct7%Ffcu7$$\"+_'*e\"Fiin$\"+QXHrTFiinFdct7%Fjeu7$$\"+[2h!f\"Fiin$\"+&4b N=%FiinFdct7%F`fu7$$\"+\"G)\\!f\"Fiin$\"+TLl&>%FiinFdct7%Fffu7$$\"+%\\ e$*e\"Fiin$\"+,@_2UFiinFdct7%F\\gu7$$\"+'oMse\"Fiin$\"+)[*4>UFiinFdct7 %Fbgu7$$\"+$ppTe\"Fiin$\"+z!G.B%FiinFdct7%Fhgu7$$\"+0f?!e\"Fiin$\"+,U: TUFiinFdct7%F^hu7$$\"+K`Qv:Fiin$\"+9t_^UFiinFdct7%Fdhu7$$\"+3(\\(p:Fii n$\"+$[*RhUFiinFdct7%Fjhu7$$\"+I1Mj:Fiin$\"+1\\sqUFiinFdct7%F`iuFjbtFd ctF\\_sFd_s-F$6$7in7$F\\`s$!3g******3&f%zUFT7$Fcbt$!3))*****f!\\sqUFT7 $F^bt$!3%******H[*RhUFT7$Fiat$!3i*****RJF:D%FT7$Fdat$!3j*****4?a6C%FT7 $F_at$!3T+++z!G.B%FT7$Fj`t$!3M+++)[*4>UFT7$Fe`t$!3N+++,@_2UFT7$F``t$!3 ')*****4M`c>%FT7$F[`t$!3++++&4bN=%FT7$Ff_t$!3s*****z`%HrTFT7$Fa_t$!3y* ****R6X*eTFT7$F\\_t$!3k******oweYTFT7$Fg^t$!3N+++&z6V8%FT7$Fb^t$!3#*** ***fR;B\"ySFT7$Fi\\t$!3S+++Jy$*oSFT7$Fd\\t$!3g*** **H')o31%FT7$F_\\t$!3q******G(\\T0%FT7$Fj[t$!3!)*****H,R!\\SFT7$Fe[t$! 3!*******fj\"e/%FT7$F`[t$!3l*****pXsZ/%FT7$F[[t$!3W+++Y+>YSFT7$Ffjs$!3 ?+++cLJ]SFT7$Fajs$!3))*****zj0t0%FT7$F\\js$!3=+++8,?nSFT7$Fgis$!3++++M u&)zSFT7$Fbis$!3!)******)Q[\\4%FT7$F]is$!3>+++rk(>6%FT7$Fhhs$!3++++P\" R.8%FT7$Fchs$!3S+++8!3%\\TFT7$F^hs$!31+++/8goTFT7$Figs$!3/+++%>Iu=%FT7 $Fdgs$!3-+++$3;b?%FT7$F_gs$!30+++zgeAUFT7$Fjfs$!3m*****H,d%QUFT7$Fefs$ !3E+++Rg,`UFT7$F`fs$!3++++&[-iE%FT7$F[fs$!3++++XG*zF%FT7$Ffes$!3m***** zC*Q)G%FT7$Faes$!3))*****R)4T(H%FT7$F\\es$!30+++y%)30VFT7$Fgds$!3L+++) 3f9J%FT7$Fbds$!3O+++XUc;VFT7$F]ds$!3?+++KvW?VFT7$Fhcs$!3'******RV`JK%F T7$Fccs$!3=+++QlsCVFT7$F^cs$!33+++h5@DVFT7$Fibs$!3C+++#f]YK%FT7$Fdbs$! 3$)*****R)z3BVFT7$F_bs$!3%******4Nl0K%FT7$Fjas$!3L+++OT7UFiinFfev7%Fjfv7$F]gu$!+,@_2UFiinFfev7%F^gv7 $Fgfu$!+TLl&>%FiinFfev7%Fbgv7$Fafu$!+&4bN=%FiinFfev7%Ffgv7$F[fu$!+QXHr TFiinFfev7%Fjgv7$Feeu$!+9^%*eTFiinFfev7%F^hv7$F_eu$!+pweYTFiinFfev7%Fb hv7$Fidu$!+&z6V8%FiinFfev7%Ffhv7$Fcdu$!+'RYSFiinFfev7%Ff[w7$F[`u$!+cLJ]SFiinFfev7%Fj[w7$Fe_u$!+QcIdS FiinFfev7%F^\\w7$F__u$!+8,?nSFiinFfev7%Fb\\w7$Fi^u$!+Mu&)zSFiinFfev7%F f\\w7$Fc^u$!+*Q[\\4%FiinFfev7%Fj\\w7$F]^u$!+rk(>6%FiinFfev7%F^]w7$Fg]u $!+P\"R.8%FiinFfev7%Fb]w7$Fa]u$!+8!3%\\TFiinFfev7%Ff]w7$F[]u$!+/8goTFi inFfev7%Fj]w7$Fe\\u$!+%>Iu=%FiinFfev7%F^^w7$F_\\u$!+$3;b?%FiinFfev7%Fb ^w7$Fi[u$!+zgeAUFiinFfev7%Ff^w7$Fc[u$!+8qXQUFiinFfev7%Fj^w7$F][u$!+Rg, `UFiinFfev7%F^_w7$Fgjt$!+&[-iE%FiinFfev7%Fb_w7$Fajt$!+XG*zF%FiinFfev7% Ff_w7$F[jt$!+[#*Q)G%FiinFfev7%Fj_w7$Feit$!+%)4T(H%FiinFfev7%F^`w7$F_it $!+y%)30VFiinFfev7%Fb`w7$Fiht$!+)3f9J%FiinFfev7%Ff`w7$Fcht$!+XUc;VFiin Ffev7%Fj`w7$F]ht$!+KvW?VFiinFfev7%F^aw7$Fggt$!+MM:BVFiinFfev7%Fbaw7$Fa gt$!+QlsCVFiinFfev7%Ffaw7$F[gt$!+h5@DVFiinFfev7%Fjaw7$Feft$!+#f]YK%Fii nFfev7%F^bw7$F_ft$!+%)z3BVFiinFfev7%Fbbw7$Fiet$!+^`c?VFiinFfev7%Ffbw7$ Fcet$!+OT7#Fbfn;Fbgw$ \"$z%Fbfn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 23 "real stability interval" }{TEXT -1 37 " for this scheme is (approximately) " }{XPPEDIT 18 0 "[-3.9689, 0];" "6#7$,$-% &FloatG6$\"&*oR!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "The following picture co mpares the stability region for the 7 stage, order 4 scheme with that \+ of the order 5 scheme.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p||(1..7),po||(1..6)],view=[-4.79..2.19,-4.79..4.79], font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,sc aling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 623 575 575 {PLOTDATA 2 "65-%'CURVESG6$7ew7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3:+ ++%fK;.$!#E$\"3s*****4n#fTJF-7$$\"32+++;%z!Qu!#D$\"3C+++:\"*Q7ZF-7$$\" 3S+++@b2vp!#C$\"3')*****Hq%=$G'F-7$$\"3l*****\\,%G=Q!#B$\"3)******4QpR &yF-7$$\"3++++k&3%p9!#A$\"3E+++YdpC%*F-7$$\"3A+++uwkrVFI$\"3#*******G5 _*4\"!#<7$$\"3%*******4d9d5!#@$\"3-+++b@^c7FQ7$$\"37+++Y]o?@FU$\"30+++ y!fLT\"FQ7$$\"3y*****fqd\"*[$FU$\"3))*****f_0*p:FQ7$$\"34+++YzAVVFU$\" 3.+++dR*es\"FQ7$$\"3B+++gW*4^#FU$\"3++++nk%4)=FQ7$$!3')*****z0xqP'FU$ \"3)******H!)oX.#FQ7$$!3')*****f$*='oH!#?$\"3')******[F='=#FQ7$$!3%3++ g)Rj?yF^p$\"3.+++X-$\"3))*****\\K()3[#FQ7$ $!3#******\\w!)44$Fip$\"3')*****fh,Ei#FQ7$$!3?+++Xc7U_Fip$\"3%)******* HV$fFFQ7$$!3-+++>/D)Fip$\"3!******phQ(*)GFQ7$$!3,+++&>T%=7F-$\"36+++ a5*>,$FQ7$$!3))******oUq+o#=CJFQ7$$!3)******zgcuD#F-$\" 36+++f)Q[A$FQ7$$!3/+++*y$=oGF-$\"3\")*****RB!Q8LFQ7$$!3)******pipC^$F- $\"31+++7J9!R$FQ7$$!3#******HwjS<%F-$\"3%)******>y0cMFQ7$$!3!*******4M $=%[F-$\"3)******>`zA^$FQ7$$!3]+++We!*3bF-$\"35+++\"zU*fNFQ7$$!3i***** *e$p8<'F-$\"3:+++v)[+g$FQ7$$!3O+++ycvR![()F-$\"33+++-a[*p$FQ7$$!3Y*****4Y4FP*F-$\"3:+++j#G;r$FQ7$$!3S+++ e9&**)**F-$\"3*******>#)p$>PFQ7$$!31+++Y(3+1\"FQ$\"3z*****\\-PHs$FQ7$$ !3,+++jhM?6FQ$\"3!*******3,^APFQ7$$!3)******f)Q/!=\"FQ$\"3,+++#yI#=PFQ 7$$!3.+++899R7FQ$\"3$*******3'3-r$FQ7$$!3%******H+$o(H\"FQ$\"33+++!4E& )p$FQ7$$!35+++F,sb8FQ$\"31+++mGC$o$FQ7$$!32+++*e9LT\"FQ$\"37+++U%*RkOF Q7$$!3%******ROU0Z\"FQ$\"3;+++r3-UOFQ7$$!3$******R=)\\F:FQ$\"3z*****H) 37;OFQ7$$!3#*******f3I%e\"FQ$\"3)******f52ne$FQ7$$!33+++x\"*4T;FQ$\"39 +++,#)y`NFQ7$$!35+++&zx!)p\"FQ$\"3%*******oRQYbcXVV$FQ7$$!3/+++^ FQ$\"3&)*****Hm['QLFQ7$$!3)***** *zm2[*>FQ$\"3%******4D)y'G$FQ7$$!3'*******zybe?FQ$\"3,+++wy*GB$FQ7$$!3 6+++RR@C@FQ$\"3y*****zd1w<$FQ7$$!3=+++Zsl\">#FQ$\"32+++sFf@JFQ7$$!3?++ +z(\\0E#FQ$\"3/+++3'*[lIFQ7$$!32+++g=PIBFQ$\"35+++_(*y4IFQ7$$!30+++-]] +CFQ$\"39+++tjyaHFQ7$$!3#******\\;E.Z#FQ$\"36+++kyc+HFQ7$$!3=+++y2GRDF Q$\"3))*****p,cq%GFQ7$$!3$)******f0#pg#FQ$\"3')*****R&G1%z#FQ7$$!35+++ [/\"Hn#FQ$\"3!******HYW8u#FQ7$$!39+++dn,PFFQ$\"35+++57k)o#FQ7$$!3/+++: y3*z#FQ$\"38+++!*GqNEFQ7$$!3))*****f:M!fGFQ$\"33+++j=I#e#FQ7$$!3$***** *Hd6o\"HFQ$\"3)******p=#FQ7$$!3!)*****>v!phKFQ$\"32+++w$R(>@FQ7$$!30+++ Q`)HI$FQ$\"31+++'3Ng0#FQ7$$!33+++(fzCM$FQ$\"3#******R:W3*>FQ7$$!3%)*** **z*\\D!Q$FQ$\"3$******4#*\\T#>FQ7$$!3y*****pg*R;MFQ$\"3-+++c,&f&=FQ7$ $!3!******RR25X$FQ$\"3!******H\")eiy\"FQ7$$!3%******RvwT[$FQ$\"3)***** *f<3^r\"FQ7$$!3A+++*)z+;NFQ$\"34+++=6bU;FQ7$$!39+++-'*fYNFQ$\"3!****** >.j'o:FQ7$$!3!)*****\\PVgd$FQ$\"35+++$*Ga$\\\"FQ7$$!3\"*******4&=Wg$FQ $\"32+++UGJ<9FQ7$$!34+++z`yJOFQ$\"3*)*****\\x:,M\"FQ7$$!3-+++S(z\"eOFQ $\"3!*******\\16i7FQ7$$!3!******HU3Oo$FQ$\"3'******fnlM=\"FQ7$$!3()*** **HPZ!3PFQ$\"3(******f/]V5\"FQ7$$!3-+++qDWJPFQ$\"3!******zDD\\-\"FQ7$$ !33+++\"y8Pv$FQ$\"3a+++_EP`%*F-7$$!3/+++D0wuPFQ$\"3;+++ZY5d')F-7$$!3++ ++a#pWz$FQ$\"3w*****feT9'yF-7$$!3))*****\\p?F\"QFQ$\"3/+++I`6nqF-7$$!3 :+++ZkRHQFQ$\"3M+++lyguiF-7$$!3.+++)H%QWQFQ$\"3-+++dw<%[&F-7$$!3/+++c? edQFQ$\"3A+++&=%*ep%F-7$$!3/+++&****)oQFQ$\"3B+++XZn4RF-7$$!35+++`@EyQ FQ$\"3,+++K$>`7$F-7$$!37+++;og&)QFQ$\"3\"******4?TDM#F-7$$!3A+++&[')3* QFQ$\"3)******>A#*4c\"F-7$$!3-+++%fnS*QFQ$\"3V+++d>$G!yFip7$$!3++++s,8 &*QFQF(7$Fgjl$!3V+++d>$G!yFip7$Fbjl$!3)******>A#*4c\"F-7$F]jl$!3\"**** **4?TDM#F-7$Fhil$!3,+++K$>`7$F-7$Fcil$!3B+++XZn4RF-7$F^il$!3A+++&=%*ep %F-7$Fihl$!3-+++dw<%[&F-7$Fdhl$!3M+++lyguiF-7$F_hl$!3/+++I`6nqF-7$Fjgl $!3w*****feT9'yF-7$Fegl$!3;+++ZY5d')F-7$F`gl$!3a+++_EP`%*F-7$F[gl$!3!* *****zDD\\-\"FQ7$Fffl$!3(******f/]V5\"FQ7$Fafl$!3'******fnlM=\"FQ7$F\\ fl$!3!*******\\16i7FQ7$Fgel$!3*)*****\\x:,M\"FQ7$Fbel$!32+++UGJ<9FQ7$F ]el$!35+++$*Ga$\\\"FQ7$Fhdl$!3!******>.j'o:FQ7$Fcdl$!34+++=6bU;FQ7$F^d l$!3)******f<3^r\"FQ7$Ficl$!3!******H\")eiy\"FQ7$Fdcl$!3-+++c,&f&=FQ7$ F_cl$!3$******4#*\\T#>FQ7$Fjbl$!3#******R:W3*>FQ7$Febl$!31+++'3Ng0#FQ7 $F`bl$!32+++w$R(>@FQ7$F[bl$!3'******4$p)>=#FQ7$Ffal$!3)******\\P?GC#FQ 7$Faal$!3++++J_H-BFQ7$F\\al$!3=+++(y![gBFQ7$Fg`l$!3,+++i;YcXVV$FQ7$F cz$!3#)*****flRvZ$FQ7$F^z$!3%*******oRQ#)p$>PFQ7$F]v$!3:+++j#G;r$FQ7$Fhu$!33+++-a[*p$FQ7$Fcu$! 3%)*****H/dEo$FQ7$F^u$!3/+++3#)ygOFQ7$Fit$!3@+++GgVLOFQ7$Fdt$!3:+++v)[ +g$FQ7$F_t$!35+++\"zU*fNFQ7$Fjs$!3)******>`zA^$FQ7$Fes$!3%)******>y0cM FQ7$F`s$!31+++7J9!R$FQ7$F[s$!3\")*****RB!Q8LFQ7$Ffr$!36+++f)Q[A$FQ7$Fa r$!3#)*****>o#=CJFQ7$F\\r$!36+++a5*>,$FQ7$Fgq$!3!******phQ(*)GFQ7$Fbq$ !3%)*******HV$fFFQ7$F]q$!3')*****fh,Ei#FQ7$Fgp$!3))*****\\K()3[#FQ7$Fb p$!3.+++X-T%=7F`_n$\"+a5*>,$FjanFa_n7%Fgfn7 $$!+pUq+PFjanFa_n7%F[\\o7$$!+Y(3+1\"Fjan $\"+Dq$Hs$FjanFa_n7%Fa\\o7$$!+jhM?6Fjan$\"+4,^APFjanFa_n7%Fg\\o7$$!+') Q/!=\"Fjan$\"+#yI#=PFjanFa_n7%F]]o7$$!+899R7Fjan$\"+4'3-r$FjanFa_n7%Fc ]o7$$!+.Io(H\"Fjan$\"+!4E&)p$FjanFa_n7%Fi]o7$$!+F,sb8Fjan$\"+mGC$o$Fja nFa_n7%F_^o7$$!+*e9LT\"Fjan$\"+U%*RkOFjanFa_n7%Fe^o7$$!+kBaq9Fjan$\"+r 3-UOFjanFa_n7%F[_o7$$!+%=)\\F:Fjan$\"+$)37;OFjanFa_n7%Fa_o7$$!+g3I%e\" Fjan$\"+1rq'e$FjanFa_n7%Fg_o7$$!+x\"*4T;Fjan$\"+,#)y`NFjanFa_n7%F]`o7$ $!+&zx!)p\"Fjan$\"+pRQYbFjan$\"+j'['QLFjanFa_n7%F[bo7$$!+ow![*>Fjan$ \"+^#)y'G$FjanFa_n7%Fabo7$$!+!)ybe?Fjan$\"+wy*GB$FjanFa_n7%Fgbo7$$!+RR @C@Fjan$\"+ylgxJFjanFa_n7%F]co7$$!+Zsl\">#Fjan$\"+sFf@JFjanFa_n7%Fcco7 $$!+z(\\0E#Fjan$\"+3'*[lIFjanFa_n7%Fico7$$!+g=PIBFjan$\"+_(*y4IFjanFa_ n7%F_do7$$!+-]]+CFjan$\"+tjyaHFjanFa_n7%Fedo7$$!+lhKqCFjan$\"+kyc+HFja nFa_n7%F[eo7$$!+y2GRDFjan$\"+=#FjanFa_n7%Fiio7$$!+_2phKFjan$\"+w$R(>@FjanFa_n7%F_jo7$$!+Q`)H I$Fjan$\"+'3Ng0#FjanFa_n7%Fejo7$$!+(fzCM$Fjan$\"+aT%3*>FjanFa_n7%F[[p7 $$!+)*\\D!Q$Fjan$\"+@*\\T#>FjanFa_n7%Fa[p7$$!+2'*R;MFjan$\"+c,&f&=Fjan Fa_n7%Fg[p7$$!+%R25X$Fjan$\"+8)eiy\"FjanFa_n7%F]\\p7$$!+an<%[$Fjan$\"+ w\"3^r\"FjanFa_n7%Fc\\p7$$!+*)z+;NFjan$\"+=6bU;FjanFa_n7%Fi\\p7$$!+-'* fYNFjan$\"+KImo:FjanFa_n7%F_]p7$$!+vL/wNFjan$\"+$*Ga$\\\"FjanFa_n7%Fe] p7$$!+5&=Wg$Fjan$\"+UGJ<9FjanFa_n7%F[^p7$$!+z`yJOFjan$\"+vd6S8FjanFa_n 7%Fa^p7$$!+S(z\"eOFjan$\"+]16i7FjanFa_n7%Fg^p7$$!+B%3Oo$Fjan$\"+wcY$= \"FjanFa_n7%F]_p7$$!+tt/3PFjan$\"+Y+N/6FjanFa_n7%Fc_p7$$!+qDWJPFjan$\" +e_#\\-\"FjanFa_n7%Fi_p7$$!+\"y8Pv$Fjan$\"+_EP`%*F`_nFa_n7%F_`p7$$!+D0 wuPFjan$\"+ZY5d')F`_nFa_n7%Fe`p7$$!+a#pWz$Fjan$\"+'eT9'yF`_nFa_n7%F[ap 7$$!+&p?F\"QFjan$\"+I`6nqF`_nFa_n7%Faap7$$!+ZkRHQFjan$\"+lyguiF`_nFa_n 7%Fgap7$$!+)H%QWQFjan$\"+dw<%[&F`_nFa_n7%F]bp7$$!+c?edQFjan$\"+&=%*ep% F`_nFa_n7%Fcbp7$$!+&****)oQFjan$\"+XZn4RF`_nFa_n7%Fibp7$$!+`@EyQFjan$ \"+K$>`7$F`_nFa_n7%F_cp7$$!+;og&)QFjan$\"+,7aUBF`_nFa_n7%Fecp7$$!+&[') 3*QFjan$\"+AA*4c\"F`_nFa_n7%F[dp7$$!+%fnS*QFjan$\"+d>$G!yFaenFa_n7%Fad p7$$!+s,8&*QFjanF(Fa_n7%Fgdp7$Fbdp$!+d>$G!yFaenFa_n7%F[ep7$F\\dp$!+AA* 4c\"F`_nFa_n7%F_ep7$Ffcp$!+,7aUBF`_nFa_n7%Fcep7$F`cp$!+K$>`7$F`_nFa_n7 %Fgep7$Fjbp$!+XZn4RF`_nFa_n7%F[fp7$Fdbp$!+&=%*ep%F`_nFa_n7%F_fp7$F^bp$ !+dw<%[&F`_nFa_n7%Fcfp7$Fhap$!+lyguiF`_nFa_n7%Fgfp7$Fbap$!+I`6nqF`_nFa _n7%F[gp7$F\\ap$!+'eT9'yF`_nFa_n7%F_gp7$Ff`p$!+ZY5d')F`_nFa_n7%Fcgp7$F ``p$!+_EP`%*F`_nFa_n7%Fggp7$Fj_p$!+e_#\\-\"FjanFa_n7%F[hp7$Fd_p$!+Y+N/ 6FjanFa_n7%F_hp7$F^_p$!+wcY$=\"FjanFa_n7%Fchp7$Fh^p$!+]16i7FjanFa_n7%F ghp7$Fb^p$!+vd6S8FjanFa_n7%F[ip7$F\\^p$!+UGJ<9FjanFa_n7%F_ip7$Ff]p$!+$ *Ga$\\\"FjanFa_n7%Fcip7$F`]p$!+KImo:FjanFa_n7%Fgip7$Fj\\p$!+=6bU;FjanF a_n7%F[jp7$Fd\\p$!+w\"3^r\"FjanFa_n7%F_jp7$F^\\p$!+8)eiy\"FjanFa_n7%Fc jp7$Fh[p$!+c,&f&=FjanFa_n7%Fgjp7$Fb[p$!+@*\\T#>FjanFa_n7%F[[q7$F\\[p$! +aT%3*>FjanFa_n7%F_[q7$Ffjo$!+'3Ng0#FjanFa_n7%Fc[q7$F`jo$!+w$R(>@FjanF a_n7%Fg[q7$Fjio$!+Jp)>=#FjanFa_n7%F[\\q7$Fdio$!+v.#GC#FjanFa_n7%F_\\q7 $F^io$!+J_H-BFjanFa_n7%Fc\\q7$Fhho$!+(y![gBFjanFa_n7%Fg\\q7$Fbho$!+i;Y PFjanFa_n7%F_eq7$Ff[ o$!+j#G;r$FjanFa_n7%Fceq7$F`[o$!+-a[*p$FjanFa_n7%Fgeq7$Fjjn$!+Vql#o$Fj anFa_n7%F[fq7$Fdjn$!+3#)ygOFjanFa_n7%F_fq7$F^jn$!+GgVLOFjanFa_n7%Fcfq7 $Fhin$!+v)[+g$FjanFa_n7%Fgfq7$Fbin$!+\"zU*fNFjanFa_n7%F[gq7$F\\in$!+K& zA^$FjanFa_n7%F_gq7$Ffhn$!+?y0cMFjanFa_n7%Fcgq7$F`hn$!+7J9!R$FjanFa_n7 %Fggq7$Fjgn$!+M-Q8LFjanFa_n7%F[hq7$Fdgn$!+f)Q[A$FjanFa_n7%F_hq7$F^gn$! +#o#=CJFjanFa_n7%Fchq7$Fhfn$!+a5*>,$FjanFa_n7%Fghq7$Fbfn$!+<'Q(*)GFjan Fa_n7%F[iq7$F\\fn$!++LMfFFjanFa_n7%F_iq7$Ffen$!+;;gAEFjanFa_n7%Fciq7$F _en$!+Dt)3[#FjanFa_n7%Fgiq7$Fidn$!+X-.?=VFQ7$$\"3%******>IGq^\"FQ$\"32++ +5I?DVFQ7$$\"3*******4\"4W2:FQ$\"3:+++ccXJVFQ7$$\"31+++I(>t\\\"FQ$\"3+ +++Fz\"pL%FQ7$$\"3++++E4r'[\"FQ$\"3@+++z%\\:M%FQ7$$\"3!******RbicZ\"FQ $\"33+++Q(4`M%FQ7$$\"3/+++E]Ak9FQ$\"3E+++&pd\"[VFQ7$$\"3!******pl^CX\" FQ$\"31+++C<0]VFQ7$$\"3-+++,$*RS9FQ$\"3q*****pV\\4N%FQ7$$\"31+++>\"H\" G9FQ$\"31+++uv!3N%FQ7$$\"3'******RZ2dT\"FQ$\"39+++]>e\\VFQ7$$\"3.+++=q ?.9FQ$\"35+++$\\FsM%FQ7$$\"3-+++\")zq!R\"FQ$\"3S+++/%)pVVFQ7$$\"3))*** **>t*Hy8FQ$\"3l*****z][*QVFQ7$$\"3!*******RE3m8FQ$\"3r*****H(=$HL%FQ7$ $\"3'*******)HqTN\"FQ$\"3J+++SQgDVFQ7$$\"36+++T@pU8FQ$\"3B+++PE#pJ%FQ7 $$\"3#*******okzJ8FQ$\"3:+++]=&oI%FQ7$$\"34+++GQl@8FQ$\"3o*****>6k`H%F Q7$$\"31+++Y0Y78FQ$\"3/+++ClW#G%FQ7$$\"3\"******p)>W/8FQ$\"3-+++K%3\"o UFQ7$$\"35+++mY&yH\"FQ$\"3o*****>H#R_UFQ7$$\"3/+++(o&)HH\"FQ$\"3t***** >G)QNUFQ7$$\"3-+++?q9!H\"FQ$\"3!)*******z_s@%FQ7$$\"3!*******Q6m*G\"FQ $\"3K+++v'H#)>%FQ7$$\"3'*******HS$=H\"FQ$\"3K+++A.nyTFQ7$$\"3#******H0 8pH\"FQ$\"3A+++^_/fTFQ7$$\"35+++w@.08FQ$\"3p*****4rN*RTFQ7$$\"3(****** pIchJ\"FQ$\"36+++KZ*>7%FQ7$$\"31+++wb/I8FQ$\"3I+++'zve5%FQ7$$\"3%***** **p1EY8FQ$\"3)******RCW@4%FQ7$$\"3)******fP:UO\"FQ$\"3S+++R1?\"3%FQ7$$ \"3#******pmfKQ\"FQ$\"39+++[ICtSFQ7$$\"3-+++.Mw-9FQ$\"3g******>fFoSFQ7 $$\"31+++Vj >uSFQ7$$\"3!*******)**f;\\\"FQ$\"3u*****4!*y13%FQ7$$\"3/+++QW)f]\"FQ$ \"3G+++Qmf)3%FQ7$$\"3&******z!o\"*=:FQ$\"3++++/\")p(4%FQ7$$\"3.+++z1WI :FQ$\"3H+++fQw2TFQ7$$\"31+++8=cS:FQ$\"3f*****R`+'=TFQ7$$\"33+++(4,$\\: FQ$\"3=+++s#Q+8%FQ7$$\"3)******H[*oc:FQ$\"3T+++ay#>9%FQ7$$\"3/+++;Wwi: FQ$\"3#)*****H(y8aTFQ7$$\"3#******fMnvc\"FQ$\"3m*****R:_l;%FQ7$$\"33++ +I?9r:FQ$\"3)******HYn!zTFQ7$$\"3-+++nL`t:FQ$\"3%)*****f_\"f\">%FQ7$$ \"3%******Hk'yu:FQ$\"3g*****pLTS?%FQ7$$\"35+++Vq%\\d\"FQ$\"3V+++:%zUb:F Q$\"3#)*****RX4NG%FQ7$$\"33++++:;\\:FQ$\"3?+++Z.5$H%FQFj^rF]^n-Fj^n6jn 7%7$$\"+qi9U:Fjan$\"+[v5-VFjan7$$\"+=TUM:Fjan$\"+\\\")[5VFjan7$$\"$V\" Ff^n$\"$@%Ff^n7%F^bs7$$\"+?s.E:Fjan$\"+?.?=VFjanFcbs7%Fibs7$$\"+-$Gq^ \"Fjan$\"+5I?DVFjanFcbs7%F_cs7$$\"+64W2:Fjan$\"+ccXJVFjanFcbs7%Fecs7$$ \"+I(>t\\\"Fjan$\"+Fz\"pL%FjanFcbs7%F[ds7$$\"+E4r'[\"Fjan$\"+z%\\:M%Fj anFcbs7%Fads7$$\"+aDmv9Fjan$\"+Q(4`M%FjanFcbs7%Fgds7$$\"+E]Ak9Fjan$\"+ &pd\"[VFjanFcbs7%F]es7$$\"+d;X_9Fjan$\"+C<0]VFjanFcbs7%Fces7$$\"+,$*RS 9Fjan$\"+P%\\4N%FjanFcbs7%Fies7$$\"+>\"H\"G9Fjan$\"+uv!3N%FjanFcbs7%F_ fs7$$\"+uuq:9Fjan$\"+]>e\\VFjanFcbs7%Fefs7$$\"+=q?.9Fjan$\"+$\\FsM%Fja nFcbs7%F[gs7$$\"+\")zq!R\"Fjan$\"+/%)pVVFjanFcbs7%Fags7$$\"+K(*Hy8Fjan $\"+3&[*QVFjanFcbs7%Fggs7$$\"+SE3m8Fjan$\"+t=$HL%FjanFcbs7%F]hs7$$\"+* HqTN\"Fjan$\"+SQgDVFjanFcbs7%Fchs7$$\"+T@pU8Fjan$\"+PE#pJ%FjanFcbs7%Fi hs7$$\"+pkzJ8Fjan$\"+]=&oI%FjanFcbs7%F_is7$$\"+GQl@8Fjan$\"+7TO&H%Fjan Fcbs7%Feis7$$\"+Y0Y78Fjan$\"+ClW#G%FjanFcbs7%F[js7$$\"+()>W/8Fjan$\"+K %3\"oUFjanFcbs7%Fajs7$$\"+mY&yH\"Fjan$\"+#H#R_UFjanFcbs7%Fgjs7$$\"+(o& )HH\"Fjan$\"+#G)QNUFjanFcbs7%F][t7$$\"+?q9!H\"Fjan$\"++GD%FjanFcbs7%Fi[t7$$\"+IS$=H\"Fjan$\"+A.ny TFjanFcbs7%F_\\t7$$\"+`I\"pH\"Fjan$\"+^_/fTFjanFcbs7%Fe\\t7$$\"+w@.08F jan$\"+6d$*RTFjanFcbs7%F[]t7$$\"+2j:;8Fjan$\"+KZ*>7%FjanFcbs7%Fa]t7$$ \"+wb/I8Fjan$\"+'zve5%FjanFcbs7%Fg]t7$$\"+q1EY8Fjan$\"+WU9#4%FjanFcbs7 %F]^t7$$\"+w`@k8Fjan$\"+R1?\"3%FjanFcbs7%Fc^t7$$\"+n'fKQ\"Fjan$\"+[ICt SFjanFcbs7%Fi^t7$$\"+.Mw-9Fjan$\"+?fFoSFjanFcbs7%F__t7$$\"+VjF2 f9Fjan$\"+e'4%pSFjanFcbs7%Fa`t7$$\"+\"y\"*fZ\"Fjan$\"+O>>uSFjanFcbs7%F g`t7$$\"+***f;\\\"Fjan$\"+,*y13%FjanFcbs7%F]at7$$\"+QW)f]\"Fjan$\"+Qmf )3%FjanFcbs7%Fcat7$$\"+3o\"*=:Fjan$\"+/\")p(4%FjanFcbs7%Fiat7$$\"+z1WI :Fjan$\"+fQw2TFjanFcbs7%F_bt7$$\"+8=cS:Fjan$\"+M0g=TFjanFcbs7%Febt7$$ \"+(4,$\\:Fjan$\"+s#Q+8%FjanFcbs7%F[ct7$$\"+$[*oc:Fjan$\"+ay#>9%FjanFc bs7%Fact7$$\"+;Wwi:Fjan$\"+ty8aTFjanFcbs7%Fgct7$$\"+Ytcn:Fjan$\"+a@bmT FjanFcbs7%F]dt7$$\"+I?9r:Fjan$\"+ju1zTFjanFcbs7%Fcdt7$$\"+nL`t:Fjan$\" +E:f\">%FjanFcbs7%Fidt7$$\"+Vmyu:Fjan$\"+P8//UFjanFcbs7%F_et7$$\"+Vq% \\d\"Fjan$\"+:Q!4c\"Fjan$\"+1# zLF%FjanFcbs7%Fcgt7$$\"+UzUb:Fjan$\"+a%4NG%FjanFcbs7%Figt7$$\"++:;\\:F jan$\"+Z.5$H%FjanFcbs7%F_htFiasFcbsF]^rFc^r-F$6$7in7$F[_r$!3R+++[v5-VF Q7$Fbas$!3?+++Z.5$H%FQ7$F]as$!3#)*****RX4NG%FQ7$Fh`s$!3W+++1#zLF%FQ7$F c`s$!3%******pubFE%FQ7$F^`s$!3)******fX(o^UFQ7$Fi_s$!3g******QaASUFQ7$ Fd_s$!3L+++&3C%GUFQ7$F__s$!3V+++:%FQ7$F`^s$!3)******HYn!zTFQ7$F[^s$!3m*****R:_l;%FQ7$Ff ]s$!3#)*****H(y8aTFQ7$Fa]s$!3T+++ay#>9%FQ7$F\\]s$!3=+++s#Q+8%FQ7$Fg\\s $!3f*****R`+'=TFQ7$Fb\\s$!3H+++fQw2TFQ7$F]\\s$!3++++/\")p(4%FQ7$Fh[s$! 3G+++Qmf)3%FQ7$Fc[s$!3u*****4!*y13%FQ7$F^[s$!31+++O>>uSFQ7$Fijr$!3m*** **zl4%pSFQ7$Fdjr$!3G+++*)yimSFQ7$F_jr$!3C+++fF oSFQ7$Feir$!39+++[ICtSFQ7$F`ir$!3S+++R1?\"3%FQ7$F[ir$!3)******RCW@4%FQ 7$Ffhr$!3I+++'zve5%FQ7$Fahr$!36+++KZ*>7%FQ7$F\\hr$!3p*****4rN*RTFQ7$Fg gr$!3A+++^_/fTFQ7$Fbgr$!3K+++A.nyTFQ7$F]gr$!3K+++v'H#)>%FQ7$Fhfr$!3!)* ******z_s@%FQ7$Fcfr$!3t*****>G)QNUFQ7$F^fr$!3o*****>H#R_UFQ7$Fier$!3-+ ++K%3\"oUFQ7$Fder$!3/+++ClW#G%FQ7$F_er$!3o*****>6k`H%FQ7$Fjdr$!3:+++]= &oI%FQ7$Fedr$!3B+++PE#pJ%FQ7$F`dr$!3J+++SQgDVFQ7$F[dr$!3r*****H(=$HL%F Q7$Ffcr$!3l*****z][*QVFQ7$Facr$!3S+++/%)pVVFQ7$F\\cr$!35+++$\\FsM%FQ7$ Fgbr$!39+++]>e\\VFQ7$Fbbr$!31+++uv!3N%FQ7$F]br$!3q*****pV\\4N%FQ7$Fhar $!31+++C<0]VFQ7$Fcar$!3E+++&pd\"[VFQ7$F^ar$!33+++Q(4`M%FQ7$Fi`r$!3@+++ z%\\:M%FQ7$Fd`r$!3++++Fz\"pL%FQ7$F_`r$!3:+++ccXJVFQ7$Fj_r$!32+++5I?DVF Q7$Fe_r$!3()******>.?=VFQ7$F`_r$!3@+++\\\")[5VFQFhhtF]^n-Fj^n6jn7%7$Fj as$!+[v5-VFjan7$F`ht$!+Z.5$H%Fjan7$Fdbs$!$@%Ff^n7%Fbdu7$Fjgt$!+a%4NG%F janFedu7%Fidu7$Fdgt$!+1#zLF%FjanFedu7%F]eu7$F^gt$!+ZdviUFjanFedu7%Faeu 7$Fhft$!+cuo^UFjanFedu7%Feeu7$Fbft$!+RaASUFjanFedu7%Fieu7$F\\ft$!+&3C% GUFjanFedu7%F]fu7$Ffet$!+:%FjanFedu7%Fifu7$Fddt$!+ju1zTFjanFedu7%F]gu7$F^dt$!+a @bmTFjanFedu7%Fagu7$Fhct$!+ty8aTFjanFedu7%Fegu7$Fbct$!+ay#>9%FjanFedu7 %Figu7$F\\ct$!+s#Q+8%FjanFedu7%F]hu7$Ffbt$!+M0g=TFjanFedu7%Fahu7$F`bt$ !+fQw2TFjanFedu7%Fehu7$Fjat$!+/\")p(4%FjanFedu7%Fihu7$Fdat$!+Qmf)3%Fja nFedu7%F]iu7$F^at$!+,*y13%FjanFedu7%Faiu7$Fh`t$!+O>>uSFjanFedu7%Feiu7$ Fb`t$!+e'4%pSFjanFedu7%Fiiu7$F\\`t$!+*)yimSFjanFedu7%F]ju7$Ff_t$!+7%FjanFedu7%Fi[v7$F\\]t$!+6d$*RTFjanFe du7%F]\\v7$Ff\\t$!+^_/fTFjanFedu7%Fa\\v7$F`\\t$!+A.nyTFjanFedu7%Fe\\v7 $Fj[t$!+v'H#)>%FjanFedu7%Fi\\v7$Fd[t$!++GD e\\VFjanFedu7%Fa`v7$F`fs$!+uv!3N%FjanFedu7%Fe`v7$Fjes$!+P%\\4N%FjanFed u7%Fi`v7$Fdes$!+C<0]VFjanFedu7%F]av7$F^es$!+&pd\"[VFjanFedu7%Faav7$Fhd s$!+Q(4`M%FjanFedu7%Feav7$Fbds$!+z%\\:M%FjanFedu7%Fiav7$F\\ds$!+Fz\"pL %FjanFedu7%F]bv7$Ffcs$!+ccXJVFjanFedu7%Fabv7$F`cs$!+5I?DVFjanFedu7%Feb v7$Fjbs$!+?.?=VFjanFedu7%Fibv7$F_bs$!+\\\")[5VFjanFedu7%F]cvF_duFeduF] ^rFc^r-F$6%7$7$$!3/++++++!z%FQF(7$$\"3%**************=#FQF(-%'COLOURG6 &F`^nF)F)F)-%*LINESTYLEG6#Fb^n-F$6%7$7$F(Fecv7$F($\"3/++++++!z%FQFjcvF ]dv-F$6$7]yF'7$$\"3++++lcX>6F1$\"3')******ojzq:F-7$$\"3\\*****HZ*QMsF7 $\"3\"******4!RfTJF-7$$\"3[+++eX)yM)F=$\"3'******\\.(R7ZF-7$$\"3++++bo %fu%FC$\"3=+++F+@$G'F-7$$\"38+++4@R>=FI$\"3e******Gn,ayF-7$$\"3q*****z 4+cQ&FI$\"3U+++4'GZU*F-7$$\"31+++*\\`qJ\"FU$\"3!******fB3&*4\"FQ7$$\"3 %*******yO!Rv#FU$\"31+++cqWc7FQ7$$\"3y*****pytu)\\FU$\"3,+++g-<89FQ7$$ \"3')******>()GexFU$\"3%*******GCZp:FQ7$$\"3)4++]-1#*))*FU$\"3!******* =t.Db)FU$\"3/+++B#H%z=FQ7$$!35+++[5d^:FU$\"3<+++l76 K?FQ7$$!33+++ptzQ7F^p$\"3>+++2hi2@FQ7$$!3C+++.5(H)GF^p$\"3%******\\b)[ #=#FQ7$$!3$*******Qf5W_F^p$\"3/+++>AicAFQ7$$!3#3++!zX_'\\)F^p$\"3$)*** **>;]*HBFQ7$$!3%*******R;#HG\"Fip$\"3@+++lHR-CFQ7$$!3%)*****>!z9W=Fip$ \"3!)*****HHlQZ#FQ7$$!3))*****>lJPb#Fip$\"3#*******)4rUa#FQ7$$!3o***** z#p$=V$Fip$\"3A+++))z\\8EFQ7$$!3))*****4a4v\\%Fip$\"3=+++u:T\"o#FQ7$$! 3'******>E/ww&Fip$\"3++++/6&yu#FQ7$$!3W*****pVXbD(Fip$\"3!)*****R*yi7G FQ7$$!3[+++q%=+(*)Fip$\"3%******H@Gb(GFQ7$$!3-+++u\"y84\"F-$\"38+++KW'RF-$\"3!*** ***H%>y%R$FQ7$$!3()*****p5C7j%F-$\"3#)*****pVpQX$FQ7$$!3@+++nj@+`F-$\" 37+++\"yRT]$FQ7$$!3?+++[LcmfF-$\"3')******eSgYNFQ7$$!3u*****fq\"eFmF-$ \"3%******R#o7#e$FQ7$$!3-+++&z4>G(F-$\"3@+++DCU6OFQ7$$!3*)*****H2i*GzF -$\"3%******\\Or]j$FQ7$$!3))*****Ri#fo&)F-$\"3#)******f=a`OFQ7$$!3')** ***HN'*3?*F-$\"3%)*****RN5sm$FQ7$$!3@+++nh4E)*F-$\"39+++]*yjn$FQ7$$!3$ ******\\F[W/\"FQ$\"3))*****fZ*G\"o$FQ7$$!3++++['Qc5\"FQ$\"35+++'eM@o$F Q7$$!3%******\\%p@m6FQ$\"35+++]k1zOFQ7$$!36+++lIAE7FQ$\"35+++pN?sOFQ7$ $!31+++h?q&G\"FQ$\"33+++DijhOFQ7$$!3.+++#Q1ZM\"FQ$\"3')*****4)4VZOFQ7$ $!34+++:))H.9FQ$\"3:+++fYjHOFQ7$$!3-+++Zhbh9FQ$\"3:+++=#y#3OFQ7$$!3/++ +uNd>:FQ$\"3!)*****z-\"Q$e$FQ7$$!3)******>!)pud\"FQ$\"3?+++mh&\\b$FQ7$ $!35+++'o#RN;FQ$\"3'******>_)3_FQ$\"3)*******4a x7LFQ7$$!3-+++x$)H'*>FQ$\"3%)*****Hm*)=E$FQ7$$!3%)*****H>z71#FQ$\"3-++ +%\\b*3KFQ7$$!3-+++)3w\"G@FQ$\"3%******pH!faJFQ7$$!3')*****H=%)o>#FQ$ \"3#)*****>`k%*4$FQ7$$!39+++#H$3nAFQ$\"3A+++\"Q6U/$FQ7$$!32+++h,FQBFQ$ \"3)******>&GL*)HFQ7$$!3++++qi$)4CFQ$\"3)******z3Q^$HFQ7$$!3++++D-;\"[ #FQ$\"3++++LJt\")GFQ7$$!31+++EAo^DFQ$\"3!******4za!HGFQ7$$!3#)*****>' \\%4i#FQ$\"3%******\\NDpx#FQ7$$!3-+++=Mg)o#FQ$\"33+++Qk5DFFQ7$$!3A+++3 \\TaFFQ$\"3#*******)HSLn#FQ7$$!3>+++P0A=GFQ$\"3)*******yjP@EFQ7$$!3'** ****R^D*zGFQ$\"3++++6c)*oDFQ7$$!3')*****p'>[RHFQ$\"3\"*******)Qmf^#FQ7 $$!3'******fTvo*HFQ$\"3/+++Pb9iCFQ7$$!3!)*****z![6_IFQ$\"3A+++mmP2CFQ7 $$!3,+++QbA0JFQ$\"3;+++Eu`^BFQ7$$!3%******pxWi:$FQ$\"33+++Hl_%H#FQ7$$! 3(********\\=_?$FQ$\"39+++45EOAFQ7$$!3-+++C(*>_KFQ$\"3)******\\'Rnw@FQ 7$$!3)*******yvC(H$FQ$\"3$******p)Gr:@FQ7$$!3/+++#)oUSLFQ$\"3#)******= 'QL0#FQ7$$!3%)*****H)z!=Q$FQ$\"3%*******e\\_*)>FQ7$$!34+++LmY@MFQ$\"34 +++t)eU#>FQ7$$!3)*******eO[fMFQ$\"3#******\\GTv&=FQ7$$!3))*****>AWf\\$ FQ$\"31+++J$)Q*y\"FQ7$$!3!******zVO4`$FQ$\"3!******\\$H$)>Yr$FQ$\"3'******f(*4tF\"FQ7$ $!3=+++'ow8u$FQ$\"3'******z:`/?\"FQ7$$!3;+++y01nPFQ$\"34+++&e'4B6FQ7$$ !31+++yei\"z$FQ$\"31+++8yQX5FQ7$$!3!******z*R+:QFQ$\"3'*******o7iu'*F- 7$$!3:+++.z5PQFQ$\"3s*****H:sV*))F-7$$!3'******R$z$y&QFQ$\"3!)******Qu 49\")F-7$$!39+++jz3xQFQ$\"3-+++m(RXL(F-7$$!3%*******H6v%*QFQ$\"3K+++MB AclF-7$$!32+++gYs5RFQ$\"3C+++htYzdF-7$$!35+++5N\"\\#RFQ$\"3i*****z+BW+ &F-7$$!3\")*****H&GBPRFQ$\"3%)*****R\"Q4JUF-7$$!3++++V&4w%RFQ$\"3')*** **R!3PfMF-7$$!3#)*****p!G)f&RFQ$\"3!)*****f!)e!*o#F-7$$!3')*****4T/B'R FQ$\"3))*****zz**)>>F-7$$!3=+++C&Ql'RFQ$\"3'******40#f^6F-7$$!3#****** fYh'oRFQ$\"3)******HSb!QQFip7$Fagx$!3)******HSb!QQFip7$F\\gx$!3'****** 40#f^6F-7$Fgfx$!3))*****zz**)>>F-7$Fbfx$!3!)*****f!)e!*o#F-7$F]fx$!3') *****R!3PfMF-7$Fhex$!3%)*****R\"Q4JUF-7$Fcex$!3i*****z+BW+&F-7$F^ex$!3 C+++htYzdF-7$Fidx$!3K+++MBAclF-7$Fddx$!3-+++m(RXL(F-7$F_dx$!3!)******Q u49\")F-7$Fjcx$!3s*****H:sV*))F-7$Fecx$!3'*******o7iu'*F-7$F`cx$!31+++ 8yQX5FQ7$F[cx$!34+++&e'4B6FQ7$Ffbx$!3'******z:`/?\"FQ7$Fabx$!3'******f (*4tF\"FQ7$F\\bx$!35+++!*Q^`8FQ7$Fgax$!3.+++SL\"*G9FQ7$Fbax$!3-+++J^O. :FQ7$F]ax$!3#******z)*Rnd\"FQ7$Fh`x$!3#******HWE*[;FQ7$Fc`x$!3!****** \\$H$)>FQ7$F__x$!3%*******e\\_*)>FQ7$Fj^x$!3#)******='QL0#FQ7$Fe^x$!3$ ******p)Gr:@FQ7$F`^x$!3)******\\'Rnw@FQ7$F[^x$!39+++45EOAFQ7$Ff]x$!33+ ++Hl_%H#FQ7$Fa]x$!3;+++Eu`^BFQ7$F\\]x$!3A+++mmP2CFQ7$Fg\\x$!3/+++Pb9iC FQ7$Fb\\x$!3\"*******)Qmf^#FQ7$F]\\x$!3++++6c)*oDFQ7$Fh[x$!3)*******yj P@EFQ7$Fc[x$!3#*******)HSLn#FQ7$F^[x$!33+++Qk5DFFQ7$Fijw$!3%******\\ND px#FQ7$Fdjw$!3!******4za!HGFQ7$F_jw$!3++++LJt\")GFQ7$Fjiw$!3)******z3Q ^$HFQ7$Feiw$!3)******>&GL*)HFQ7$F`iw$!3A+++\"Q6U/$FQ7$F[iw$!3#)*****>` k%*4$FQ7$Ffhw$!3%******pH!faJFQ7$Fahw$!3-+++%\\b*3KFQ7$F\\hw$!3%)***** Hm*)=E$FQ7$Fggw$!3)*******4ax7LFQ7$Fbgw$!3%*******Qm5hLFQ7$F]gw$!3))** ****3v]1MFQ7$Fhfw$!3#)******)4A([MFQ7$Fcfw$!3!)*****\\t*e([$FQ7$F^fw$! 3'******>_y%R$FQ7$Fe_w$!3=+++gqehLFQ7$F`_w$!3%)******RExDLFQ7$F[_w$!3!*** ***z!*4sG$FQ7$Ff^w$!3++++T'*yXKFQ7$Fa^w$!3;+++#pL9?$FQ7$F\\^w$!3?+++Ip 5aJFQ7$Fg]w$!3;+++&RFQ5$FQ7$Fb]w$!3?+++=?n]IFQ7$F]]w$!37+++Obx%*HFQ7$F h\\w$!38+++K;]*HBFQ7$Ffiv$!3/+++>AicAFQ7$Faiv$!3%******\\b)[#=#FQ7$F\\iv$ !3>+++2hi2@FQ7$$\"3!******4YyO/&FU$!31+++%R?g&>FQ7$$\"3%*)****4ip\"=** FU$!3.+++(\\6C!=FQ7$$\"3S+++4y5K!*FU$!3)******>`rtk\"FQ7$$\"3S+++943]j FU$!3!*******H4R\"\\\"FQ7$$\"3#)*****RTfIx$FU$!3)******R4Y[L\"FQ7$$\"3 !******4['pR>FU$!31+++pa*z<\"FQ7$$\"3(******4U\"4-')FI$!3.+++mx*4-\"FQ 7$$\"3,++++H<;KFI$!3k*****RD&RR')F-7$$\"3o*****>gXsl*FC$!3Q+++%4<'oqF- 7$$\"3%******Hih%=@FC$!3k******=C!y\\&F-7$$\"39+++6))=xFF=$!35+++tV*p# RF-7$$\"33+++<$H0G\"F7$!3-+++,[>cBF-7$$\"3#******>\")*\\W6F-$\"+pjzq:F`_nFa_n7%F`_z7$$\"+t%*QMsFQ $\"+,RfTJF`_nFa_n7%Ff_z7$$\"+eX)yM)Fc`n$\"+NqR7ZF`_nFa_n7%F\\`z7$$\"+b o%fu%Fj`n$\"+F+@$G'F`_nFa_n7%Fb`z7$$\"+4@R>=Faan$\"+Hn,ayF`_nFa_n7%Fh` z7$$\"+)4+cQ&Faan$\"+4'GZU*F`_nFa_n7%F^az7$$\"+*\\`qJ\"F_bn$\"+O#3&*4 \"FjanFa_n7%Fdaz7$$\"+zO!Rv#F_bn$\"+cqWc7FjanFa_n7%Fjaz7$$\"+(ytu)\\F_ bn$\"+g-<89FjanFa_n7%F`bz7$$\"+?()GexF_bn$\"+HCZp:FjanFa_n7%Ffbz7$$\"+ Dg?*))*F_bn$\"+>t.Db)F_bn$\"+B#H%z=FjanFa_n7 %Fbcz7$$!+[5d^:F_bn$\"+l76K?FjanFa_n7%Fhcz7$$!+ptzQ7Fddn$\"+2hi2@FjanF a_n7%F^dz7$$!+.5(H)GFddn$\"+b&)[#=#FjanFa_n7%Fddz7$$!+Rf5W_Fddn$\"+>Ai cAFjanFa_n7%Fjdz7$$!+zX_'\\)Fddn$\"+i,&*HBFjanFa_n7%F`ez7$$!+S;#HG\"Fa en$\"+lHR-CFjanFa_n7%Ffez7$$!+-z9W=Faen$\"+$HlQZ#FjanFa_n7%F\\fz7$$!+_ ;t`DFaen$\"+*4rUa#FjanFa_n7%Fbfz7$$!+Gp$=V$Faen$\"+))z\\8EFjanFa_n7%Fh fz7$$!+T&4v\\%Faen$\"+u:T\"o#FjanFa_n7%F^gz7$$!+iUgndFaen$\"+/6&yu#Fja nFa_n7%Fdgz7$$!+PaabsFaen$\"+%*yi7GFjanFa_n7%Fjgz7$$!+q%=+(*)Faen$\"+8 #Gb(GFjanFa_n7%F`hz7$$!+u\"y84\"F`_n$\"+K5P#F`_n$\"+#pL9?$FjanFa_n7%Fdjz7$$!+(oSTn#F`_n$\"+T'*yXKF janFa_n7%Fjjz7$$!+.:?()HF`_n$\"+3*4sG$FjanFa_n7%F`[[l7$$!+Iu*yI$F`_n$ \"+SExDLFjanFa_n7%Ff[[l7$$!+Pn?MOF`_n$\"+gqehLFjanFa_n7%F\\\\[l7$$!+z( >W'RF`_n$\"+V>y%R$FjanFa_n7%Fb\\[l7$$!+2TAJYF`_n$\"+P%pQX$FjanFa_n7%Fh \\[l7$$!+nj@+`F`_n$\"+\"yRT]$FjanFa_n7%F^][l7$$!+[LcmfF`_n$\"+fSgYNFja nFa_n7%Fd][l7$$!+1G(F`_n$\"+ DCU6OFjanFa_n7%F`^[l7$$!+t?'*GzF`_n$\"+l82NOFjanFa_n7%Ff^[l7$$!+CEfo&) F`_n$\"+g=a`OFjanFa_n7%F\\_[l7$$!+`j*3?*F`_n$\"+a.@nOFjanFa_n7%Fb_[l7$ $!+nh4E)*F`_n$\"+]*yjn$FjanFa_n7%Fh_[l7$$!+v#[W/\"Fjan$\"+w%*G\"o$Fjan Fa_n7%F^`[l7$$!+['Qc5\"Fjan$\"+'eM@o$FjanFa_n7%Fd`[l7$$!+Xp@m6Fjan$\"+ ]k1zOFjanFa_n7%Fj`[l7$$!+lIAE7Fjan$\"+pN?sOFjanFa_n7%F`a[l7$$!+h?q&G\" Fjan$\"+DijhOFjanFa_n7%Ffa[l7$$!+#Q1ZM\"Fjan$\"+\")4VZOFjanFa_n7%F\\b[ l7$$!+:))H.9Fjan$\"+fYjHOFjanFa_n7%Fbb[l7$$!+Zhbh9Fjan$\"+=#y#3OFjanFa _n7%Fhb[l7$$!+uNd>:Fjan$\"+G5Q$e$FjanFa_n7%F^c[l7$$!+-)pud\"Fjan$\"+mh &\\b$FjanFa_n7%Fdc[l7$$!+'o#RN;Fjan$\"+Av,BNFjanFa_n7%Fjc[l7$$!+a[_$p \"Fjan$\"+N(*e([$FjanFa_n7%F`d[l7$$!+'>)3_Fjan$\"+5ax7LFjanFa_n7%Fhe[l7$$!+x$)H'*>Fjan$\"+j'* )=E$FjanFa_n7%F^f[l7$$!+$>z71#Fjan$\"+%\\b*3KFjanFa_n7%Fdf[l7$$!+)3w\" G@Fjan$\"+(H!faJFjanFa_n7%Fjf[l7$$!+$=%)o>#Fjan$\"+KXY*4$FjanFa_n7%F`g [l7$$!+#H$3nAFjan$\"+\"Q6U/$FjanFa_n7%Ffg[l7$$!+h,FQBFjan$\"+_GL*)HFja nFa_n7%F\\h[l7$$!+qi$)4CFjan$\"+)3Q^$HFjanFa_n7%Fbh[l7$$!+D-;\"[#Fjan$ \"+LJt\")GFjanFa_n7%Fhh[l7$$!+EAo^DFjan$\"+\"za!HGFjanFa_n7%F^i[l7$$!+ i\\%4i#Fjan$\"+b`#px#FjanFa_n7%Fdi[l7$$!+=Mg)o#Fjan$\"+Qk5DFFjanFa_n7% Fji[l7$$!+3\\TaFFjan$\"+*HSLn#FjanFa_n7%F`j[l7$$!+P0A=GFjan$\"+zjP@EFj anFa_n7%Ffj[l7$$!+9b#*zGFjan$\"+6c)*oDFjanFa_n7%F\\[\\l7$$!+n>[RHFjan$ \"+*Qmf^#FjanFa_n7%Fb[\\l7$$!+;a(o*HFjan$\"+Pb9iCFjanFa_n7%Fh[\\l7$$!+ 3[6_IFjan$\"+mmP2CFjanFa_n7%F^\\\\l7$$!+QbA0JFjan$\"+Eu`^BFjanFa_n7%Fd \\\\l7$$!+xZCcJFjan$\"+Hl_%H#FjanFa_n7%Fj\\\\l7$$!++&=_?$Fjan$\"+45EOA FjanFa_n7%F`]\\l7$$!+C(*>_KFjan$\"+lRnw@FjanFa_n7%Ff]\\l7$$!+zvC(H$Fja n$\"+()Gr:@FjanFa_n7%F\\^\\l7$$!+#)oUSLFjan$\"+>'QL0#FjanFa_n7%Fb^\\l7 $$!+$)z!=Q$Fjan$\"+f\\_*)>FjanFa_n7%Fh^\\l7$$!+LmY@MFjan$\"+t)eU#>Fjan Fa_n7%F^_\\l7$$!+fO[fMFjan$\"+>v&=FjanFa_n7%Fd_\\l7$$!+AU%f\\$Fjan$ \"+J$)Q*y\"FjanFa_n7%Fj_\\l7$$!+Qk$4`$Fjan$\"+NH$)>Yr$Fjan$\"+w*4tF\"FjanFa_n7%Fdb\\l7$$!+'ow8u$Fjan$\"+eJX+7FjanF a_n7%Fjb\\l7$$!+y01nPFjan$\"+&e'4B6FjanFa_n7%F`c\\l7$$!+yei\"z$Fjan$\" +8yQX5FjanFa_n7%Ffc\\l7$$!+)*R+:QFjan$\"+p7iu'*F`_nFa_n7%F\\d\\l7$$!+. z5PQFjan$\"+`@P%*))F`_nFa_n7%Fbd\\l7$$!+Mz$y&QFjan$\"+Ru49\")F`_nFa_n7 %Fhd\\l7$$!+jz3xQFjan$\"+m(RXL(F`_nFa_n7%F^e\\l7$$!+I6v%*QFjan$\"+MBAc lF`_nFa_n7%Fde\\l7$$!+gYs5RFjan$\"+htYzdF`_nFa_n7%Fje\\l7$$!+5N\"\\#RF jan$\"+3IU/]F`_nFa_n7%F`f\\l7$$!+`GBPRFjan$\"+9Q4JUF`_nFa_n7%Fff\\l7$$ !+V&4w%RFjan$\"+/3PfMF`_nFa_n7%F\\g\\l7$$!+2G)f&RFjan$\"+1)e!*o#F`_nFa _n7%Fbg\\l7$$!+6WIiRFjan$\"+)z**)>>F`_nFa_n7%Fhg\\l7$$!+C&Ql'RFjan$\"+ ^?f^6F`_nFa_n7%F^h\\l7$$!+m9moRFjan$\"+.a0QQFaenFa_n7%Fdh\\l7$Feh\\l$! +.a0QQFaenFa_n7%Fjh\\l7$F_h\\l$!+^?f^6F`_nFa_n7%F^i\\l7$Fig\\l$!+)z**) >>F`_nFa_n7%Fbi\\l7$Fcg\\l$!+1)e!*o#F`_nFa_n7%Ffi\\l7$F]g\\l$!+/3PfMF` _nFa_n7%Fji\\l7$Fgf\\l$!+9Q4JUF`_nFa_n7%F^j\\l7$Faf\\l$!+3IU/]F`_nFa_n 7%Fbj\\l7$F[f\\l$!+htYzdF`_nFa_n7%Ffj\\l7$Fee\\l$!+MBAclF`_nFa_n7%Fjj \\l7$F_e\\l$!+m(RXL(F`_nFa_n7%F^[]l7$Fid\\l$!+Ru49\")F`_nFa_n7%Fb[]l7$ Fcd\\l$!+`@P%*))F`_nFa_n7%Ff[]l7$F]d\\l$!+p7iu'*F`_nFa_n7%Fj[]l7$Fgc\\ l$!+8yQX5FjanFa_n7%F^\\]l7$Fac\\l$!+&e'4B6FjanFa_n7%Fb\\]l7$F[c\\l$!+e JX+7FjanFa_n7%Ff\\]l7$Feb\\l$!+w*4tF\"FjanFa_n7%Fj\\]l7$F_b\\l$!+!*Q^` 8FjanFa_n7%F^]]l7$Fia\\l$!+SL\"*G9FjanFa_n7%Fb]]l7$Fca\\l$!+J^O.:FjanF a_n7%Ff]]l7$F]a\\l$!+))*Rnd\"FjanFa_n7%Fj]]l7$Fg`\\l$!+Vk#*[;FjanFa_n7 %F^^]l7$Fa`\\l$!+NH$)>FjanFa_n7%F^_]l7$Fi ^\\l$!+f\\_*)>FjanFa_n7%Fb_]l7$Fc^\\l$!+>'QL0#FjanFa_n7%Ff_]l7$F]^\\l$ !+()Gr:@FjanFa_n7%Fj_]l7$Fg]\\l$!+lRnw@FjanFa_n7%F^`]l7$Fa]\\l$!+45EOA FjanFa_n7%Fb`]l7$F[]\\l$!+Hl_%H#FjanFa_n7%Ff`]l7$Fe\\\\l$!+Eu`^BFjanFa _n7%Fj`]l7$F_\\\\l$!+mmP2CFjanFa_n7%F^a]l7$Fi[\\l$!+Pb9iCFjanFa_n7%Fba ]l7$Fc[\\l$!+*Qmf^#FjanFa_n7%Ffa]l7$F][\\l$!+6c)*oDFjanFa_n7%Fja]l7$Fg j[l$!+zjP@EFjanFa_n7%F^b]l7$Faj[l$!+*HSLn#FjanFa_n7%Fbb]l7$F[j[l$!+Qk5 DFFjanFa_n7%Ffb]l7$Fei[l$!+b`#px#FjanFa_n7%Fjb]l7$F_i[l$!+\"za!HGFjanF a_n7%F^c]l7$Fih[l$!+LJt\")GFjanFa_n7%Fbc]l7$Fch[l$!+)3Q^$HFjanFa_n7%Ff c]l7$F]h[l$!+_GL*)HFjanFa_n7%Fjc]l7$Fgg[l$!+\"Q6U/$FjanFa_n7%F^d]l7$Fa g[l$!+KXY*4$FjanFa_n7%Fbd]l7$F[g[l$!+(H!faJFjanFa_n7%Ffd]l7$Fef[l$!+% \\b*3KFjanFa_n7%Fjd]l7$F_f[l$!+j'*)=E$FjanFa_n7%F^e]l7$Fie[l$!+5ax7LFj anFa_n7%Fbe]l7$Fce[l$!+Rm5hLFjanFa_n7%Ffe]l7$F]e[l$!+4v]1MFjanFa_n7%Fj e]l7$Fgd[l$!+*4A([MFjanFa_n7%F^f]l7$Fad[l$!+N(*e([$FjanFa_n7%Fbf]l7$F[ d[l$!+Av,BNFjanFa_n7%Fff]l7$Fec[l$!+mh&\\b$FjanFa_n7%Fjf]l7$F_c[l$!+G5 Q$e$FjanFa_n7%F^g]l7$Fib[l$!+=#y#3OFjanFa_n7%Fbg]l7$Fcb[l$!+fYjHOFjanF a_n7%Ffg]l7$F]b[l$!+\")4VZOFjanFa_n7%Fjg]l7$Fga[l$!+DijhOFjanFa_n7%F^h ]l7$Faa[l$!+pN?sOFjanFa_n7%Fbh]l7$F[a[l$!+]k1zOFjanFa_n7%Ffh]l7$Fe`[l$ !+'eM@o$FjanFa_n7%Fjh]l7$F_`[l$!+w%*G\"o$FjanFa_n7%F^i]l7$Fi_[l$!+]*yj n$FjanFa_n7%Fbi]l7$Fc_[l$!+a.@nOFjanFa_n7%Ffi]l7$F]_[l$!+g=a`OFjanFa_n 7%Fji]l7$Fg^[l$!+l82NOFjanFa_n7%F^j]l7$Fa^[l$!+DCU6OFjanFa_n7%Fbj]l7$F [^[l$!+Co7#e$FjanFa_n7%Ffj]l7$Fe][l$!+fSgYNFjanFa_n7%Fjj]l7$F_][l$!+\" yRT]$FjanFa_n7%F^[^l7$Fi\\[l$!+P%pQX$FjanFa_n7%Fb[^l7$$!+%RLrH%F`_n$!+ @e\\DMFjanFa_n7%Ff[^l7$Fc\\[l$!+V>y%R$FjanFa_n7%F\\\\^l7$F]\\[l$!+gqeh LFjanFa_n7%F`\\^l7$Fg[[l$!+SExDLFjanFa_n7%Fd\\^l7$Fa[[l$!+3*4sG$FjanFa _n7%Fh\\^l7$F[[[l$!+T'*yXKFjanFa_n7%F\\]^l7$Fejz$!+#pL9?$FjanFa_n7%F`] ^l7$F_jz$!+Ip5aJFjanFa_n7%Fd]^l7$Fiiz$!+&RFQ5$FjanFa_n7%Fh]^l7$Fciz$!+ =?n]IFjanFa_n7%F\\^^l7$F]iz$!+Obx%*HFjanFa_n7%F`^^l7$Fghz$!+KAicAFjanFa_n7%F\\a^l7$Fedz$!+b&)[#=#FjanFa_n7%F`a^l7$F_dz $!+2hi2@FjanFa_n7%Fda^l7$$\"+h%yO/&F_bn$!+%R?g&>FjanFa_n7%Fha^l7$$\"+@ 'p\"=**F_bn$!+(\\6C!=FjanFa_n7%F^b^l7$$\"+4y5K!*F_bn$!+K:PZ;FjanFa_n7% Fdb^l7$$\"+943]jF_bn$!+I4R\"\\\"FjanFa_n7%Fjb^l7$$\"+9%fIx$F_bn$!+%4Y[ L\"FjanFa_n7%F`c^l7$$\"+\"['pR>F_bn$!+pa*z<\"FjanFa_n7%Ffc^l7$$\"+@94- ')Faan$!+mx*4-\"FjanFa_n7%F\\d^l7$$\"++H<;KFaan$!+a_RR')F`_nFa_n7%Fbd^ l7$$\"+-cCd'*Fj`n$!+%4<'oqF`_nFa_n7%Fhd^l7$$\"+B;Y=@Fj`n$!+>C!y\\&F`_n Fa_n7%F^e^l7$$\"+6))=xFFc`n$!+tV*p#RF`_nFa_n7%Fde^l7$$\"+<$H0G\"FQ$!+, [>cBF`_nFa_n7%Fje^l7$$\"+7)*\\Wc:FQ$\"3g******3&f%zUFQ7$$\"31+++L)o$[:FQ$\"3u*****fdgvG %FQ7$$\"3\"******fC!*)R:FQ$\"35+++Bk)\\H%FQ7$$\"3!******R$p!3`\"FQ$\"3 6+++,hp,VFQ7$$\"3'******z$G;@:FQ$\"3O+++F\"\\wI%FQ7$$\"3.+++(=.5^\"FQ$ \"3d*****4B0GJ%FQ7$$\"3)*******f[P+:FQ$\"3L+++OT7a9FQ$\"33+++h5@DVFQ7$$\"30+++ !e=?W\"FQ$\"3=+++QlsCVFQ7$$\"3$******>!zrH9FQ$\"3'******RV`JK%FQ7$$\"3 #*******o)otT\"FQ$\"3?+++KvW?VFQ7$$\"33+++&oa]S\"FQ$\"3O+++XUc;VFQ7$$ \"3-+++[\"oGR\"FQ$\"3L+++)3f9J%FQ7$$\"3!*******eN\"4Q\"FQ$\"30+++y%)30 VFQ7$$\"3-+++c*3$p8FQ$\"3))*****R)4T(H%FQ7$$\"3++++G()=e8FQ$\"3m*****z C*Q)G%FQ7$$\"31+++]lqZ8FQ$\"3++++XG*zF%FQ7$$\"35+++c&Q!Q8FQ$\"3++++&[- iE%FQ7$$\"3-+++dkQH8FQ$\"3E+++Rg,`UFQ7$$\"3'******R0!)>K\"FQ$\"3m***** H,d%QUFQ7$$\"36+++8\"ygJ\"FQ$\"30+++zgeAUFQ7$$\"31+++ne'>J\"FQ$\"3-+++ $3;b?%FQ7$$\"3\"******feY*48FQ$\"3/+++%>Iu=%FQ7$$\"3\"*******RQK58FQ$ \"31+++/8goTFQ7$$\"3*******pHrLJ\"FQ$\"3S+++8!3%\\TFQ7$$\"3-+++8#*G>8F Q$\"3++++P\"R.8%FQ7$$\"3/+++eE:G8FQ$\"3>+++rk(>6%FQ7$$\"35+++6]')R8FQ$ \"3!)******)Q[\\4%FQ7$$\"3-+++)HOTN\"FQ$\"3++++Mu&)zSFQ7$$\"3++++')=]q 8FQ$\"3=+++8,?nSFQ7$$\"35+++c2Q)Q\"FQ$\"3))*****zj0t0%FQ7$$\"3*******z $R:29FQ$\"3?+++cLJ]SFQ7$$\"30+++inBE9FQ$\"3W+++Y+>YSFQ7$$\"3!******\\ \"p7X9FQ$\"3l*****pXsZ/%FQ7$$\"3#******zoCMY\"FQ$\"3!*******fj\"e/%FQ7 $$\"3/+++X>$3[\"FQ$\"3!)*****H,R!\\SFQ7$$\"3%******R@Rr\\\"FQ$\"3q**** **G(\\T0%FQ7$$\"3/+++c&3A^\"FQ$\"3g*****H')o31%FQ7$$\"3,+++!ycf_\"FQ$ \"3S+++Jy$*oSFQ7$$\"3(******f3T$Q:FQ$\"3!)*****>;B\"ySFQ7$$\"3)****** \\mZ$\\:FQ$\"3q*****4&p@)3%FQ7$$\"3/+++iI)*e:FQ$\"3o*****pSN!*4%FQ7$$ \"31+++lyve\"FQ$\"3y*****R6X*eTFQ7$$\"3'******>>_'*e\"FQ$\"3s* ****z`%HrTFQ7$$\"3-+++[2h!f\"FQ$\"3++++&4bN=%FQ7$$\"3!******4G)\\!f\"F Q$\"3')*****4M`c>%FQ7$$\"31+++%\\e$*e\"FQ$\"3N+++,@_2UFQ7$$\"3%******f oMse\"FQ$\"3M+++)[*4>UFQ7$$\"3)******HppTe\"FQ$\"3T+++z!G.B%FQ7$$\"3#* *****\\!f?!e\"FQ$\"3j*****4?a6C%FQ7$$\"3)******>L&Qv:FQ$\"3i*****RJF:D %FQ7$$\"3-+++3(\\(p:FQ$\"3%******H[*RhUFQ7$$\"3!*******H1Mj:FQ$\"3))** ***f!\\sqUFQFdg^lFe^z-Fj^n6jn7%7$$\"+B(*>c:Fjan$\"+4&f%zUFjan7$$\"+L)o $[:Fjan$\"+w0c(G%Fjan7$$\"$X\"Ff^n$\"$=%Ff^n7%Fhj_l7$$\"+Y-*)R:Fjan$\" +Bk)\\H%FjanF][`l7%Fc[`l7$$\"+Mp!3`\"Fjan$\"+,hp,VFjanF][`l7%Fi[`l7$$ \"+QG;@:Fjan$\"+F\"\\wI%FjanF][`l7%F_\\`l7$$\"+(=.5^\"Fjan$\"+J_!GJ%Fj anF][`l7%Fe\\`l7$$\"+g[P+:Fjan$\"+OT7a9Fjan$\"+h5@DVFj anF][`l7%Fc^`l7$$\"+!e=?W\"Fjan$\"+QlsCVFjanF][`l7%Fi^`l7$$\"+-zrH9Fja n$\"+MM:BVFjanF][`l7%F__`l7$$\"+p)otT\"Fjan$\"+KvW?VFjanF][`l7%Fe_`l7$ $\"+&oa]S\"Fjan$\"+XUc;VFjanF][`l7%F[``l7$$\"+[\"oGR\"Fjan$\"+)3f9J%Fj anF][`l7%Fa``l7$$\"+fN\"4Q\"Fjan$\"+y%)30VFjanF][`l7%Fg``l7$$\"+c*3$p8 Fjan$\"+%)4T(H%FjanF][`l7%F]a`l7$$\"+G()=e8Fjan$\"+[#*Q)G%FjanF][`l7%F ca`l7$$\"+]lqZ8Fjan$\"+XG*zF%FjanF][`l7%Fia`l7$$\"+c&Q!Q8Fjan$\"+&[-iE %FjanF][`l7%F_b`l7$$\"+dkQH8Fjan$\"+Rg,`UFjanF][`l7%Feb`l7$$\"+a+)>K\" Fjan$\"+8qXQUFjanF][`l7%F[c`l7$$\"+8\"ygJ\"Fjan$\"+zgeAUFjanF][`l7%Fac `l7$$\"+ne'>J\"Fjan$\"+$3;b?%FjanF][`l7%Fgc`l7$$\"+'eY*48Fjan$\"+%>Iu= %FjanF][`l7%F]d`l7$$\"+SQK58Fjan$\"+/8goTFjanF][`l7%Fcd`l7$$\"+(HrLJ\" Fjan$\"+8!3%\\TFjanF][`l7%Fid`l7$$\"+8#*G>8Fjan$\"+P\"R.8%FjanF][`l7%F _e`l7$$\"+eE:G8Fjan$\"+rk(>6%FjanF][`l7%Fee`l7$$\"+6]')R8Fjan$\"+*Q[\\ 4%FjanF][`l7%F[f`l7$$\"+)HOTN\"Fjan$\"+Mu&)zSFjanF][`l7%Faf`l7$$\"+')= ]q8Fjan$\"+8,?nSFjanF][`l7%Fgf`l7$$\"+c2Q)Q\"Fjan$\"+QcIdSFjanF][`l7%F ]g`l7$$\"+QR:29Fjan$\"+cLJ]SFjanF][`l7%Fcg`l7$$\"+inBE9Fjan$\"+Y+>YSFj anF][`l7%Fig`l7$$\"+:p7X9Fjan$\"+dCxWSFjanF][`l7%F_h`l7$$\"+)oCMY\"Fja n$\"+gj\"e/%FjanF][`l7%Feh`l7$$\"+X>$3[\"Fjan$\"+8!R!\\SFjanF][`l7%F[i `l7$$\"+9#Rr\\\"Fjan$\"+H(\\T0%FjanF][`l7%Fai`l7$$\"+c&3A^\"Fjan$\"+j) o31%FjanF][`l7%Fgi`l7$$\"+!ycf_\"Fjan$\"+Jy$*oSFjanF][`l7%F]j`l7$$\"+' 3T$Q:Fjan$\"+iJ7ySFjanF][`l7%Fcj`l7$$\"+lwM\\:Fjan$\"+^p@)3%FjanF][`l7 %Fij`l7$$\"+iI)*e:Fjan$\"+2a.*4%FjanF][`l7%F_[al7$$\"+_'*e\"Fjan$ \"+QXHrTFjanF][`l7%Fc]al7$$\"+[2h!f\"Fjan$\"+&4bN=%FjanF][`l7%Fi]al7$$ \"+\"G)\\!f\"Fjan$\"+TLl&>%FjanF][`l7%F_^al7$$\"+%\\e$*e\"Fjan$\"+,@_2 UFjanF][`l7%Fe^al7$$\"+'oMse\"Fjan$\"+)[*4>UFjanF][`l7%F[_al7$$\"+$ppT e\"Fjan$\"+z!G.B%FjanF][`l7%Fa_al7$$\"+0f?!e\"Fjan$\"+,U:TUFjanF][`l7% Fg_al7$$\"+K`Qv:Fjan$\"+9t_^UFjanF][`l7%F]`al7$$\"+3(\\(p:Fjan$\"+$[*R hUFjanF][`l7%Fc`al7$$\"+I1Mj:Fjan$\"+1\\sqUFjanF][`l7%Fi`alFcj_lF][`lF if^lFc^r-F$6$7in7$Feg^l$!3g******3&f%zUFQ7$F\\j_l$!3))*****f!\\sqUFQ7$ Fgi_l$!3%******H[*RhUFQ7$Fbi_l$!3i*****RJF:D%FQ7$F]i_l$!3j*****4?a6C%F Q7$Fhh_l$!3T+++z!G.B%FQ7$Fch_l$!3M+++)[*4>UFQ7$F^h_l$!3N+++,@_2UFQ7$Fi g_l$!3')*****4M`c>%FQ7$Fdg_l$!3++++&4bN=%FQ7$F_g_l$!3s*****z`%HrTFQ7$F jf_l$!3y*****R6X*eTFQ7$Fef_l$!3k******oweYTFQ7$F`f_l$!3N+++&z6V8%FQ7$F [f_l$!3#******fR;B\"ySFQ7$Fbd_l$!3S+++Jy$*oS FQ7$F]d_l$!3g*****H')o31%FQ7$Fhc_l$!3q******G(\\T0%FQ7$Fcc_l$!3!)***** H,R!\\SFQ7$F^c_l$!3!*******fj\"e/%FQ7$Fib_l$!3l*****pXsZ/%FQ7$Fdb_l$!3 W+++Y+>YSFQ7$F_b_l$!3?+++cLJ]SFQ7$Fja_l$!3))*****zj0t0%FQ7$Fea_l$!3=++ +8,?nSFQ7$F`a_l$!3++++Mu&)zSFQ7$F[a_l$!3!)******)Q[\\4%FQ7$Ff`_l$!3>++ +rk(>6%FQ7$Fa`_l$!3++++P\"R.8%FQ7$F\\`_l$!3S+++8!3%\\TFQ7$Fg__l$!31+++ /8goTFQ7$Fb__l$!3/+++%>Iu=%FQ7$F]__l$!3-+++$3;b?%FQ7$Fh^_l$!30+++zgeAU FQ7$Fc^_l$!3m*****H,d%QUFQ7$F^^_l$!3E+++Rg,`UFQ7$Fi]_l$!3++++&[-iE%FQ7 $Fd]_l$!3++++XG*zF%FQ7$F_]_l$!3m*****zC*Q)G%FQ7$Fj\\_l$!3))*****R)4T(H %FQ7$Fe\\_l$!30+++y%)30VFQ7$F`\\_l$!3L+++)3f9J%FQ7$F[\\_l$!3O+++XUc;VF Q7$Ff[_l$!3?+++KvW?VFQ7$Fa[_l$!3'******RV`JK%FQ7$F\\[_l$!3=+++QlsCVFQ7 $Fgj^l$!33+++h5@DVFQ7$Fbj^l$!3C+++#f]YK%FQ7$F]j^l$!3$)*****R)z3BVFQ7$F hi^l$!3%******4Nl0K%FQ7$Fci^l$!3L+++OT7UFjanF_]bl7%Fc^bl7$Ff^al$!+,@_2UFja nF_]bl7%Fg^bl7$F`^al$!+TLl&>%FjanF_]bl7%F[_bl7$Fj]al$!+&4bN=%FjanF_]bl 7%F__bl7$Fd]al$!+QXHrTFjanF_]bl7%Fc_bl7$F^]al$!+9^%*eTFjanF_]bl7%Fg_bl 7$Fh\\al$!+pweYTFjanF_]bl7%F[`bl7$Fb\\al$!+&z6V8%FjanF_]bl7%F_`bl7$F\\ \\al$!+'RYSFjanF_]bl7%F_cbl7$Fdg`l$!+cLJ]SFjanF_]bl7%Fccbl7$F^g`l$ !+QcIdSFjanF_]bl7%Fgcbl7$Fhf`l$!+8,?nSFjanF_]bl7%F[dbl7$Fbf`l$!+Mu&)zS FjanF_]bl7%F_dbl7$F\\f`l$!+*Q[\\4%FjanF_]bl7%Fcdbl7$Ffe`l$!+rk(>6%Fjan F_]bl7%Fgdbl7$F`e`l$!+P\"R.8%FjanF_]bl7%F[ebl7$Fjd`l$!+8!3%\\TFjanF_]b l7%F_ebl7$Fdd`l$!+/8goTFjanF_]bl7%Fcebl7$F^d`l$!+%>Iu=%FjanF_]bl7%Fgeb l7$Fhc`l$!+$3;b?%FjanF_]bl7%F[fbl7$Fbc`l$!+zgeAUFjanF_]bl7%F_fbl7$F\\c `l$!+8qXQUFjanF_]bl7%Fcfbl7$Ffb`l$!+Rg,`UFjanF_]bl7%Fgfbl7$F`b`l$!+&[- iE%FjanF_]bl7%F[gbl7$Fja`l$!+XG*zF%FjanF_]bl7%F_gbl7$Fda`l$!+[#*Q)G%Fj anF_]bl7%Fcgbl7$F^a`l$!+%)4T(H%FjanF_]bl7%Fggbl7$Fh``l$!+y%)30VFjanF_] bl7%F[hbl7$Fb``l$!+)3f9J%FjanF_]bl7%F_hbl7$F\\``l$!+XUc;VFjanF_]bl7%Fc hbl7$Ff_`l$!+KvW?VFjanF_]bl7%Fghbl7$F`_`l$!+MM:BVFjanF_]bl7%F[ibl7$Fj^ `l$!+QlsCVFjanF_]bl7%F_ibl7$Fd^`l$!+h5@DVFjanF_]bl7%Fcibl7$F^^`l$!+#f] YK%FjanF_]bl7%Fgibl7$Fh]`l$!+%)z3BVFjanF_]bl7%F[jbl7$Fb]`l$!+^`c?VFjan F_]bl7%F_jbl7$F\\]`l$!+OT7#Ff^n;Fc]cl$\"$z%Ff^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#--------------------------------------------------- ------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#-----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded schemes " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the \+ combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1885 "ee := \{c[2]=11/63,\nc[3]=4/17,\nc[4]=174/34 9,\nc[5]=36/49,\nc[6]=97/129,\nc[7]=1,\nc[8]=1,\n\na[2,1]=11/63,\na[3, 1]=244/3179,\na[3,2]=504/3179,\na[4,1]=184816101/935188078,\na[4,2]=-4 11995808/467594039,\na[4,3]=100493613/85017098,\na[5,1]=2424510108/160 25323237,\na[5,2]=12241368/78942479,\na[5,3]=-4360301712/39234412063, \na[5,4]=613806882624/1137797949827,\na[6,1]=166741171875227/345279877 3601100,\na[6,2]=1006084/2623731,\na[6,3]=-188382845675879/14089006489 98150,\na[6,4]=89559834743712383/245148712925678100,\na[6,5]=31/350,\n a[7,1]=6022012446846011204537694339419654322731/3009986400738335699217 8712977537163264000,\na[7,2]=-6561/8296,\na[7,3]=341104813140161009375 674013573715202837301/313319409731603971335732895410416320512000,\na[7 ,4]=95341943376640595142991543684079745176400281/131840665170939903489 4169155355186749556736000,\na[7,5]=-1154594845209196567962511465161858 938950699/877446764516774644979830270188996722688000,\na[7,6]=10855084 25299375/621834120640512,\na[8,1]=4827113239/65475838080,\na[8,2]=0,\n a[8,3]=797791005101/2404074449920,\na[8,4]=5703219796216031/3066005574 9940320,\na[8,5]=-33901013357491/177576803988480,\na[8,6]=292045639531 41/55776754401170,\na[8,7]=34/449,\n\nb[1]=4827113239/65475838080,\nb[ 2]=0,\nb[3]=797791005101/2404074449920,\nb[4]=5703219796216031/3066005 5749940320,\nb[5]=-33901013357491/177576803988480,\nb[6]=2920456395314 1/55776754401170,\nb[7]=34/449,\n\n`b^`[1]=7345858582709/9898748202355 2,\n`b^`[2]=0,\n`b^`[3]=30485818726152979/92415623446907904,\n`b^`[4]= 2471755716006328519/13007276847211309344,\n`b^`[5]=-5627835948231581/2 6545332727174144,\n`b^`[6]=45/83,\n`b^`[7]=34/449,\n\n`b*`[1]=66873629 126427899/902163023828283336,\n`b*`[2]=0,\n`b*`[3]=121/366,\n`b*`[4]=3 468482659797555651329/18566189507645514457800,\n`b*`[5]=-65/379,\n`b*` [6]=952822237714091128053/1892592848332731588736,\n`b*`[7]=12975205015 6606597/1891022063549226300,\n`b*`[8]=56866223980208550227/72010120179 95453750400\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5,7];" "6#&%\"TG6$\"\"&\" \"(" }{TEXT -1 127 " denote the vector whose components are the princ ipal error terms of the 7 stage, order 5 scheme (the error terms of or der 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,8 ];" "6#&%#T*G6$\"\"%\"\")" }{TEXT -1 145 " denote the vector whose co mponents are the principal error terms of the embedded 8 stage, order \+ 4 scheme (the error terms of order 5) and let " }{XPPEDIT 18 0 "`T*`[ 5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 99 " denote the vector whose \+ components are the error terms of order 6 of the embedded order 4 sche me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "De note the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[5, 7] ));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\")" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6 #-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5, 7]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&\"\"(" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5, 8]))/abs(abs(`T*`[4, 8]));" " 6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\")\"\"\"-F*6#-F*6#&F/6 $\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T* `[5,8]-T[5,7]))/abs(abs(`T*`[4,8]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6 #,&&%#T*G6$\"\"&\"\")\"\"\"&%\"TG6$F2\"\"(!\"\"F4-F*6#-F*6#&F06$\"\"%F 3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "Dormand and Prince have suggested that as well as attem pting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" } {TEXT -1 73 " is a minimum, if the embedded scheme is to be used for \+ error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "From this point o f view the Bogacki-Shampine scheme is near ideal with respect to the 8 stage, order 4 embedded scheme." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_7 := PrincipalErr orTerms(5,7,'expanded'):\n`errterms5_8*` :=subs(b=`b*`,PrincipalErrorT erms(5,8,'expanded')):\n`errterms4_8*` := subs(b=`b*`,PrincipalErrorTe rms(4,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "snmB := sqrt(add(simplify(subs(ee, `errterms5_8*`[i]))^2,i=1..nops(`errterms5_8*`))):\nsdnB := sqrt(add(s implify(subs(ee,`errterms4_8*`[i]))^2,i=1..nops(`errterms4_8*`))):\nsn mC := sqrt(add((simplify(subs(ee,`errterms5_8*`[i])-subs(ee,errterms5_ 7[i])))^2,i=1..nops(errterms5_7))):\n`B*`[6]= evalf[8](snmB/sdnB);\n`C *`[6]= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B*G 6#\"\"'$\")U(G/\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C*G6#\"\" '$\")c-E6!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The corresponding characterstics for the auxiliary 7 stag e, order 4 embedded scheme are as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 134 "`errterms5_7^` :=subs(b=`b^`,PrincipalErrorTerms(5 ,7,'expanded')):\n`errterms4_7^` := subs(b=`b^`,PrincipalErrorTerms(4, 7,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 337 "snmB := sqrt(add(simplify(subs(ee,`errterms5_ 7^`[i]))^2,i=1..nops(`errterms5_7^`))):\nsdnB := sqrt(add(simplify(sub s(ee,`errterms4_7^`[i]))^2,i=1..nops(`errterms4_7^`))):\nsnmC := sqrt( add((simplify(subs(ee,`errterms5_7^`[i])-subs(ee,errterms5_7[i])))^2,i =1..nops(errterms5_7))):\n`B^`[6]= evalf[8](snmB/sdnB);\n`C^`[6]= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#B^G6#\"\"'$\") 6!\\8\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C^G6#\"\"'$\")m*>9 \"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#----------------- ------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#================================== ===============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "#==============================" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Abreviated calculations .. 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 727 "SO5_ 7 := SimpleOrderConditions(5,7,'expanded'):\n`SO4_8*` := subs(b=`b*`,S impleOrderConditions(4,8,'expanded')): \ncdns := [seq(SO5_7[i],i=[1,2, 4,5,8,12,13,15,16]),seq(`SO4_8*`[i],i=[1,2,4,7,8])]:\n\nSO_eqs := [op( RowSumConditions(7,'expanded')),op(StageOrderConditions(2,7,'expanded' ))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3..7)=0,add(b[i]*add(a[i,j]*a[j, 2],j=3..i-1),i=2..7)=0,\n add(b[i]*c[i]*a[i,2],i=3..7)=0,add(` b*`[i]*a[i,2],i=3..7)=0]:\nwt_eqns := [seq(b[i]=a[8,i],i=1..7)]: \ncdn s := [op(wt_eqns),op(cdns),op(simp_eqns),op(SO_eqs)]:\n\nerrterms5_7 : = PrincipalErrorTerms(5,7,'expanded'):\n`errterms5_8*` :=subs(b=`b*`,P rincipalErrorTerms(5,8,'expanded')):\n`errterms4_8*` := subs(b=`b*`,Pr incipalErrorTerms(4,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 14 "calc_RKcoeffsA" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2002 "calc_RKcoeffsA : = proc()\n local eqns,sol,sm,ct,Rz,stb5,stb4,nmB,snmB,dnB,sdnB,nmC,s nmC,B_6,C_6;\n global e1,e2,e3,e4;\n\n e1 := \{c[2]=c_2,c[3]=c_3,c [4]=c_4,c[5]=c_5,c[6]=c_6,c[7]=1,c[8]=1,\n b[2]=0,b[7]=b_7,`b*`[2]=0 \};\n eqns := subs(e1,cdns);\n sol := solve(\{op(eqns)\},indets(eq ns) minus \{`b*`[3],`b*`[5],a[6,5],a[7,5]\});\n e3 := \{a[6,5]=a6_5, a[7,5]=a7_5,`b*`[3]=bs_3,`b*`[5]=bs_5\}:\n e4 := `union`(e1,e3,subs( e3,sol)); ## combined scheme\n sm := 0;\n for ct to nops(errterms5 _7) do\n sm := sm+(evalf(subs(e4,errterms5_7[ct])))^2;\n end do ;\n Rz := subs(e4,StabilityFunction(5,7,'expanded'));\n stb5 := ev alf[8](max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7)));\n R z := subs(e4,subs(b=`b*`,StabilityFunction(4,8,'expanded')));\n stb4 := evalf[8](max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7))); \n nmB := 0;\n for ct to nops(`errterms5_8*`) do\n nmB := nmB +evalf(subs(e4,`errterms5_8*`[ct]))^2;\n end do:\n snmB := sqrt(nm B);\n dnB := 0;\n for ct to nops(`errterms4_8*`) do\n dnB := \+ dnB+evalf(subs(e4,`errterms4_8*`[ct]))^2;\n end do;\n sdnB := sqrt (dnB);\n nmC := 0;\n for ct to nops(errterms5_7) do\n nmC := \+ nmC+(evalf(subs(e4,`errterms5_8*`[ct]))-evalf(subs(e4,errterms5_7[ct]) ))^2;\n end do;\n snmC := sqrt(nmC);\n B_6 := evalf[8](snmB/sdnB );\n C_6 := evalf[8](snmC/sdnB);\n print(`nodes:`,c[2]=c_2,c[3]=c_ 3,c[4]=c_4,c[5]=c_5,c[6]=c_6);\n print(`linking coefficients:`,a[6,5 ]=a6_5,a[7,5]=a7_5);\n print(`order 5 weights:`,seq(b[i]=evalf[6](su bs(e4,b[i])),i=[1,$3..7]));\n print(`order 4 weights:`,seq(`b*`[i]=e valf[6](subs(e4,`b*`[i])),i=[1,$3..8]));\n print(infinity*`-norm of \+ linking coeffs`=evalf(max(seq(seq(subs(e4,abs(a[i,j])),j=1..i-1),i=2.. 7))));\n print(`2-norm of principal error of order 5 scheme` = sqrt( sm));\n print(`2-norm of principal error of order 4 scheme` = sdnB); \n print(`order 5 stability interval` = [stb5,0]);\n print(`order \+ 4 stability interval` = [stb4,0]);\n print('B[6]'=B_6,'C[6]'=C_6);\n end proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Bogacki-Shampine scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "c_2 := 1/6: c_3 := 2/9: c_4 := 3/7: c_5 := 2/3: c_6 \+ := 3/4: \na6_5 := 4617/20480: a7_5 := -29421/29068: b_7 := 7267/94080: bs_3 := 123/416: bs_5 := 43/1440:\ncalc_RKcoeffsA();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$#F( \"\"*/&F&6#\"\"%#F/\"\"(/&F&6#\"\"&#F(F//&F&6#F+#F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%6linking~coefficients:G/&%\"aG6$\"\"'\"\"&#\"%F2/&F&6#\"\"&$\"'RQ')!\")/&F&6#\"\"'$\"'m&f$F2/&F&6#\"\"($ \"'GCxF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G 6#\"\"\"$\"'[%3(!\"(/&F&6#\"\"$$\"'tcH!\"'/&F&6#\"\"%$\"'d'z\"F2/&F&6# \"\"&$\"'6')HF+/&F&6#\"\"'$\"'*GY$F2/&F&6#\"\"($\"'CwrF+/&F&6#\"\")$\" ']7f!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-nor m~of~linking~coeffsGF&$\"+U:vj6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%L2-norm~of~principal~error~of~order~5~schemeG$\"+zF$p@#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 4~schemeG$\"+y(\\:1\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5 ~stability~intervalG7$$!)s#z)R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)yG)*R!\"(\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")%fb.\"!\"(/&%\"CGF&$\")/%[/\"F *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(e4,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]])) ;\n``;\nevalf[6](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 +7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F+ F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#F.F6#\"#o\"$ (H#!\"%\"#6#\"#U\"$V\"#\"%g>\"%hQF+F+F+F+7+#F6F3#\"$(f\"&GD##\"#\")\"$ _$#\"&*4j\"'Gde#\"&`'e\"'!3m$#\"%u\"\"'W#f*#!& U4$\"&P*z#\"(P@:)\")RWu>#\"'1hm\"(\"=R5#!&@%H\"&o!H#\"'[?[\"'>UTF+F+7+ F)#\"$(e\"%k!)\"\"!#\"(R.W%\")S=\\:#\"&`V#\"'+[7#\"$(Q\"&+[%#\"%_@\"%& )f#\"%ns\"&!3%*F+7+%\"bGF]pF`pFapFdpFgpFjpF]qF+7+%#b*G#\"%zC\"&#*\\$F` p#\"$B\"\"$;%#\"'THh\"(?-;F*$\"'RaAF*F+F+ F+7+$\"\"\"\"\"!$\"')f\"=F*$!'!3(QF*$\"'$)GTF*$\"'\"*4kF*$!'975!\"&$\" 'vj6F]oF+F+7+FV$\"'FzsF1$FXFX$\"'CmGF*$\"'O^>F*$\"'RQ')!\")$\"'m&f$F*$ \"'GCxF1F+7+%\"bGFaoFcoFdoFfoFhoF[pF]pF+7+%#b*G$\"'[%3(F1Fco$\"'tcHF*$ \"'d'z\"F*$\"'6')HF1$\"'*GY$F*$\"'CwrF1$\"']7fFjoQ)pprint876\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 31 "#============================== " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Abreviated calculations .. 2 " }}{PARA 0 "" 0 "" {TEXT -1 97 "The method of construction in this se ction involves specifying the coefficients of the terms in " } {XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 50 " in the stability function. \nEither of the nodes " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 6 " or " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 113 " can be chosen as a parameter, by including the first argument \+ 3 or 4 respectively. The default is to choose " }{XPPEDIT 18 0 "c[3] " "6#&%\"cG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 119 "Th e value obtained for the unspecified node is the root of a quadratic p olynomial and, as such, may involve a radical. " }}{PARA 0 "" 0 "" {TEXT -1 176 "There are two possible choices for the node parameter gi ven by the roots of a quadratic polynomial. The minimum value will be \+ chosen unless the argument \"max_node\" is included." }}{PARA 0 "" 0 " " {TEXT -1 145 "Including the argument \"rational_node\" causes a rati onal approximation to be taken for the unspecified node, whereby the s tability function will " }{TEXT 260 27 "not be exactly as requested" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 727 "SO5_7 := SimpleOrderConditions(5,7 ,'expanded'):\n`SO4_8*` := subs(b=`b*`,SimpleOrderConditions(4,8,'expa nded')): \ncdns := [seq(SO5_7[i],i=[1,2,4,5,8,12,13,15,16]),seq(`SO4_8 *`[i],i=[1,2,4,7,8])]:\n\nSO_eqs := [op(RowSumConditions(7,'expanded') ),op(StageOrderConditions(2,7,'expanded'))]:\nsimp_eqns := [add(b[i]*a [i,2],i=3..7)=0,add(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..7)=0,\n \+ add(b[i]*c[i]*a[i,2],i=3..7)=0,add(`b*`[i]*a[i,2],i=3..7)=0]:\nwt_ eqns := [seq(b[i]=a[8,i],i=1..7)]: \ncdns := [op(wt_eqns),op(cdns),op( simp_eqns),op(SO_eqs)]:\n\nerrterms5_7 := PrincipalErrorTerms(5,7,'exp anded'):\n`errterms5_8*` :=subs(b=`b*`,PrincipalErrorTerms(5,8,'expand ed')):\n`errterms4_8*` := subs(b=`b*`,PrincipalErrorTerms(4,8,'expande d')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 14 "ca lc_RKcoeffsB" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3900 "calc_RKcoeffsB := proc()\n local omit,vars,eqn s,eq6,eq7,eqn6,eqn7,fctrs,sol,cc,dd,sm,mn,e_temp,ct,\n Rz,stb5,stb4, nmB,snmB,dnB,sdnB,nmC,snmC,B_6,C_6;\n global e1,e2,e3,e4,e5,e6;\n\n \+ omit := 3:\n if nargs>0 then omit := args[1] end if;\n if not me mber(omit,\{3,4\}) then error \"invalid choice of parameter node\" end if;\n vars := \{c[2],c[3],c[4],c[5],a[6,5],a[7,5],b[7],`b*`[3],`b*` [5]\}:\n vars := remove(s_ -> op(0,s_)='c' and op(1,s_)<>omit,vars); \n e1 := \{c[2]=c_2,c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_6,c[7]=1,c[8]= 1,b[2]=0,`b*`[2]=0\}:\n e1 := remove(s_ -> op(0,lhs(s_))='c' and op( 1,lhs(s_))=omit,e1);\n eqns := subs(e1,cdns):\n e2 := solve(\{op(e qns)\},indets(eqns) minus vars):\n e3 := `union`(e1,e2):\n eq6 := \+ b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2 ]*c[2]+\n a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2 ]*c[2]+\n a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))))=t_6;\n eq7 := b[ 7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=t_7;\n eqn6 := op(solve(s implify(subs(e3,\{eq6\})),a[7,5]));\n eqn7 := op(solve(simplify(subs (e3,\{eq7\})),a[7,5]));\n fctrs := [op(`minus`(select(has,\{op(numer (factor(simplify(rhs(eqn6)-rhs(eqn7)))))\},c[omit]),\n select (has,map(op,select(has,map(denom@rhs,e3),c[omit])),c[omit])))];\n so l := solve(mul(fctrs[i],i=1..nops(fctrs)));\n if member('max_node', \{args\}) then\n cc := max(sol);\n else\n cc := min(sol); \n end if;\n if member('rational_node',\{args\}) then cc := conver t(evalf[10](cc),rational,5);\n dd := subs(c[omit]=cc,rhs(eqn6)); \n e4 := \{c[omit]=cc,a[7,5]=dd\}:\n e5 := `union`(e4,simpli fy(subs(e4,e3))):\n else\n dd := rationalize(subs(c[omit]=cc,rh s(eqn6)));\n e4 := \{c[omit]=cc,a[7,5]=dd\}:\n e5 := `union` (e4,expand(rationalize(subs(e4,e3)))):\n end if;\n sm := 0:\n fo r ct to nops(errterms5_7) do\n sm := sm+expand(rationalize(subs(e 5,errterms5_7[ct])))^2;\n end do:\n sm := expand(rationalize(sm)): \n mn := evalf[14](minimize(evalf[14](sm),b[7]=0..0.1,a[6,5]=1e-10.. 2,'location'));\n e_temp := op(1,op(1,op(2,[mn])));\n e_temp := \{ b[7]=convert(subs(e_temp,b[7]),rational,5),a[6,5]=convert(subs(e_temp, a[6,5]),rational,5),\n `b*`[3]=bs_3,`b*`[5]=bs_5\};\n e6 := `union`(e_temp,simplify(subs(e_temp,e5))): ##combined scheme\n s m := 0;\n for ct to nops(errterms5_7) do\n sm := sm+(evalf(subs (e6,errterms5_7[ct])))^2;\n end do;\n Rz := subs(e6,StabilityFunct ion(5,7,'expanded'));\n stb5 := evalf[8](max(fsolve(Rz=1,z=-8..-1e-7 ),fsolve(Rz=-1,z=-8..-1e-7)));\n Rz := subs(e6,subs(b=`b*`,Stability Function(4,8,'expanded')));\n stb4 := evalf[8](max(fsolve(Rz=1,z=-8. .-1e-7),fsolve(Rz=-1,z=-8..-1e-7)));\n nmB := 0;\n for ct to nops( `errterms5_8*`) do\n nmB := nmB+evalf(subs(e6,`errterms5_8*`[ct]) )^2;\n end do:\n snmB := sqrt(nmB);\n dnB := 0;\n for ct to no ps(`errterms4_8*`) do\n dnB := dnB+evalf(subs(e6,`errterms4_8*`[c t]))^2;\n end do;\n sdnB := sqrt(dnB);\n nmC := 0;\n for ct to nops(errterms5_7) do\n nmC := nmC+(evalf(subs(e6,`errterms5_8*`[ ct]))-evalf(subs(e6,errterms5_7[ct])))^2;\n end do;\n snmC := sqrt (nmC);\n B_6 := evalf[8](snmB/sdnB);\n C_6 := evalf[8](snmC/sdnB); \n if omit=3 then\n print(`nodes:`,c[2]=c_2,c[3]=subs(e6,c[3]), c[4]=c_4,c[5]=c_5,c[6]=c_6);\n else\n print(`nodes:`,c[2]=c_2,c [3]=c_3,c[4]=subs(e6,c[4]),c[5]=c_5,c[6]=c_6);\n end if;\n print(` order 5 weights:`,seq(b[i]=evalf[6](subs(e6,b[i])),i=[1,$3..7]));\n \+ print(`order 4 weights:`,seq(`b*`[i]=evalf[6](subs(e6,`b*`[i])),i=[1,$ 3..8]));\n print(infinity*`-norm of linking coeffs`=evalf(max(seq(se q(subs(e6,abs(a[i,j])),j=1..i-1),i=2..7))));\n print(`2-norm of prin cipal error of order 5 scheme` = sqrt(sm));\n print(`2-norm of princ ipal error of order 4 scheme` = sdnB);\n print(`order 5 stability in terval` = [stb5,0]);\n print(`order 4 stability interval` = [stb4,0] );\n print('B[6]'=B_6,'C[6]'=C_6);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------------- -----------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 48 "minor variat ions of the Bogacki-Shampine scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The schemes in this subsection are just to test the construction process. " }}{PARA 0 "" 0 "" {TEXT -1 89 "Th e parameters are extracted from the Bogacki-Shampine scheme (with mino r modifications)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "The scheme obtained has (exactly) the same stability fun ction as the Bogacki-Shampine scheme, but, because of the minimization step, it has a lower principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 48 "However, one of the order 5 weights is negative." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "c_2 := 1/6: c_4 := 3/7: c_5 := 2/3 : c_6 := 3/4: bs_3 := 123/416: bs_5 := 43/1440:\nt_6 := 17291/12418560 : t_7 := 269/1379840:\ncalc_RKcoeffsB(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$#F(\"\"*/ &F&6#\"\"%#F/\"\"(/&F&6#\"\"&#F(F//&F&6#F+#F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\"\"$\"'KRt!\"(/&F&6#\"\" $$\"'1KG!\"'/&F&6#\"\"%$\"'JD?F2/&F&6#\"\"&$!')e&f!\")/&F&6#\"\"'$\"'b .PF2/&F&6#\"\"($\"'1ZwF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~ weights:G/&%#b*G6#\"\"\"$\"'[%3(!\"(/&F&6#\"\"$$\"'tcH!\"'/&F&6#\"\"%$ \"'d'z\"F2/&F&6#\"\"&$\"'6')HF+/&F&6#\"\"'$\"'*GY$F2/&F&6#\"\"($\"'60j F+/&F&6#\"\")$\"'Qi9F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinit yG\"\"\"%8-norm~of~linking~coeffsGF&$\"+Q/!3>\"!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+xd k7:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~4~schemeG$\"+YTMu;!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%; order~5~stability~intervalG7$$!)s#z)R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)7m\"3%!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")@Qb5!\"(/&%\"CGF&$\" )y,;5F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 33 "In this scheme the coefficients " } {XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 6 " of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^7 " "6#*$%\"zG\"\"(" }{TEXT -1 105 " in the stability function are simp ler rational approximations of those in the Bogacki-Shampine scheme. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "c_2 := 1/6: c_4 := 3/7: c_5 := 2/3: c_6 := 3/4: bs_3 := 123/416: bs_5 := 43/1440:\nt_6 := 1/7 18: t_7 := 1/5129:\ncalc_RKcoeffsB(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$,&#\"(#[iD\"(bl?*F **(\"\"%F*\")l'>w#!\"\"\"-lem50:#F*F(F7/&F&6#F5#F/\"\"(/&F&6#\"\"&#F(F //&F&6#F+#F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/ &%\"bG6#\"\"\"$\"'FOt!\"(/&F&6#\"\"$$\"'fJG!\"'/&F&6#\"\"%$\"')e-#F2/& F&6#\"\"&$!&:$eF+/&F&6#\"\"'$\"'R-PF2/&F&6#\"\"($\"'=[wF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"'vzq!\"(/& F&6#\"\"$$\"'tcH!\"'/&F&6#\"\"%$\"'#pz\"F2/&F&6#\"\"&$\"'6')HF+/&F&6# \"\"'$\"'5jMF2/&F&6#\"\"($\"'MPjF+/&F&6#\"\")$\"'IH9F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+_>\")*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~princ ipal~error~of~order~5~schemeG$\"+r&3Ia\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~4~schemeG$\"+^\\v p;!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interval G7$$!)&)e))R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~st ability~intervalG7$$!)j+xS!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/&%\"BG6#\"\"'$\")`A`5!\"(/&%\"CGF&$\")If95F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#------------------------ --------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 79 "This \+ scheme is similar to the previous scheme, except that the irrational n ode " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 43 " is repla ced by a rational approximation. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "c_2 := 1/6: c_4 := 3/7: c_5 := 2/3: c_6 := 3/4: bs_3 := 123/416: bs_5 := 43/1440:\nt_6 := 1/718: t_7 := 1/5129:\ncalc_RKco effsB(3,'rational_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G /&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$#\"$D$\"%j9/&F&6#\"\"%#F/\"\"(/&F &6#\"\"&#F(F//&F&6#F+#F/F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~ 5~weights:G/&%\"bG6#\"\"\"$\"'EOt!\"(/&F&6#\"\"$$\"'fJG!\"'/&F&6#\"\"% $\"'*e-#F2/&F&6#\"\"&$!'&=$e!\")/&F&6#\"\"'$\"'R-PF2/&F&6#\"\"($\"'=[w F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\" \"$\"'tzq!\"(/&F&6#\"\"$$\"'tcH!\"'/&F&6#\"\"%$\"'#pz\"F2/&F&6#\"\"&$ \"'6')HF+/&F&6#\"\"'$\"'5jMF2/&F&6#\"\"($\"'ZPjF+/&F&6#\"\")$\"' " 0 "" {MPLTEXT 1 0 143 "c_2 := 1/6: c_4 := 3/7: c_5 := 2/3: c_6 := 3/4: bs_3 := 13/45 : bs_5 := 33/3335:\nt_6 := 1/718: t_7 := 1/5129:\ncalc_RKcoeffsB(3,'ra tional_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\" \"##\"\"\"\"\"'/&F&6#\"\"$#\"$D$\"%j9/&F&6#\"\"%#F/\"\"(/&F&6#\"\"&#F( F//&F&6#F+#F/F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G /&%\"bG6#\"\"\"$\"'EOt!\"(/&F&6#\"\"$$\"'fJG!\"'/&F&6#\"\"%$\"'*e-#F2/ &F&6#\"\"&$!'&=$e!\")/&F&6#\"\"'$\"'R-PF2/&F&6#\"\"($\"'=[wF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"' `F2/&F&6#\"\"&$\"'0&*) *!\")/&F&6#\"\"'$\"'C)f$F2/&F&6#\"\"($\"':6rF+/&F&6#\"\")$\"'rbeF?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+([B)*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-nor m~of~principal~error~of~order~5~schemeG$\"+\"3r@a\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~4~schemeG$\" +?3(pQ(!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~int ervalG7$$!)(o&))R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order ~4~stability~intervalG7$$!)5&z,%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")**H46!\"(/&%\"CGF&$\")As95F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1549 "e6 := \{`b *`[2] = 0, `b*`[3] = 13/45, `b*`[5] = 33/3335, b[1] = 13467397/1835730 00, a[6,3] = 1846824078266629/19188332223840000, a[6,1] = 280830778264 7/95306285880000, a[8,3] = 1751973914877571/6187242902711000, b[3] = 1 751973914877571/6187242902711000, a[8,4] = 259184863/1279362600, b[5] \+ = -1190133/204074600, b[6] = 242246272/654296535, a[8,1] = 13467397/18 3573000, b[4] = 259184863/1279362600, a[7,2] = -165316551/371956000, a [8,6] = 242246272/654296535, a[7,3] = 21602867795640134192625864309976 78007/5127830892523219551826042203264000000, `b*`[8] = 231071586717784 5173/394609099162552884456, `b*`[7] = 3507653968593450820/493261373953 19110557, a[7,6] = 423381778393131/355835788360000, c[8] = 1, c[7] = 1 , b[2] = 0, a[8,2] = 0, a[7,4] = 84613978331807886478139956214497/1213 06570023811525887932236800000, a[4,3] = 568062/739375, a[7,1] = 105504 58392762450202355661024979/52218059944024829024606592000000, a[7,5] = \+ -8652906571663520059689760027/8115450927552358158412800000, a[8,5] = - 1190133/204074600, c[5] = 2/3, c[4] = 3/7, c[6] = 3/4, c[3] = 325/1463 , a[8,7] = 40/523, a[3,1] = 158600/2140369, a[3,2] = 316875/2140369, a [4,2] = -7533/15925, a[5,4] = 88838785/174976686, a[6,2] = 34472709/14 8782400, a[4,1] = 689916/5175625, a[5,3] = 1807262507897/6160637486250 , a[5,1] = 14003870242/61198385625, a[5,2] = -7599794/20922525, `b*`[1 ] = 8721770037031/120841413181335, a[6,4] = 92640839137/544994051328, \+ c[2] = 1/6, `b*`[6] = 26088955318928/72504847908801, `b*`[4] = 1137881 8791667/59187630945960, a[2,1] = 1/6, a[6,5] = 65/292, b[7] = 40/523\} :" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(e6,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(` b*`[i],i=1..8)]]));\n``;\nevalf[6](%%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"$D$\"%j9#\"'+' e\"\"(p.9##\"'voJF2F+F+F+F+F+F+7+#\"\"$\"\"(#\"';**o\"(Dc<&#!%Lv\"&Df \"#\"'i!o&\"'v$R(F+F+F+F+F+7+#\"\"#F7#\",U-(Q+9\",DcQ)>h#!(%z*f(\")DD# 4##\".(*y]is!=\".]i[P1;'#\")&yQ)))\"*'ow\\#\",P\"R3k#*\" -G80%*\\a#\"#l\"$#HF+F+F+7+F)#\"Az\\-hcN--XiFRe/b5\"A+++#f1Y-H[-W*f!=A &#!*^lJl\"\"*+g&>P#\"F2!yw*4V'ei#>M,k&znGg@\"F+++kK?Ug#=b>K_#*3$y7&#\" A(\\9i&*R\"yk)y!=LyRh%)\"B++!oBKz)e_6Q-qlI@\"#!=F+w*of+_j;d1Hl)\"=++!G Te\"eBbF4X:\")#\"0JJRyv\"\"1+5r-HC(='#\"*j[=f#\"++EOz7#!(L,>\"\"*+Y2/##\"*siCU# \"*NlHa'#\"#S\"$B&F+7+%\"bGFapFdpFepFhpF[qF^qFaqF+7+%#b*G#\".Jq.q`RPhK\\#\"4t^%yF *$\"'0&*)*Fjo$\"'C)f$F*$\"':6rF1$\"'rbeFjoQ)pprint146\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 55 "#--------------------------------------- ---------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 24 "an efficient s cheme .. A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The following scheme again has approximately the same stabilit y function as the Bogacki-Shampine scheme." }}{PARA 0 "" 0 "" {TEXT -1 17 "The two weights " }{XPPEDIT 18 0 "`b*`[3]" "6#&%#b*G6#\"\"$" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]" "6#&%#b*G6#\"\"&" } {TEXT -1 181 " have been adjusted so that the ratio of the principal \+ error norm of the order 5 scheme to that of the order 4 scheme is app roximately the same as for the Bogacki-Shampine scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "c_2 := 1/6: c_3 := 16/63: c_5 := 5 0/69: c_6 := 63/83: bs_3 := 71/190: bs_5 := -62/241:\nt_6 := 1/718: t_ 7 := 1/5129:\ncalc_RKcoeffsB(4,'max_node','rational_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\" \"$#\"#;\"#j/&F&6#\"\"%#F2\"$1\"/&F&6#\"\"&#\"#]\"#p/&F&6#F+#F2\"#$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\"\"$ \"'Arx!\"(/&F&6#\"\"$$\"')*RP!\"'/&F&6#\"\"%$\"'A$H#F2/&F&6#\"\"&$!'55 DF2/&F&6#\"\"'$\"'j`\\F2/&F&6#\"\"($\"'LhuF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"'Wyx!\"(/&F&6#\"\" $$\"'%ot$!\"'/&F&6#\"\"%$\"'g6BF2/&F&6#\"\"&$!'hsDF2/&F&6#\"\"'$\"''3+ &F2/&F&6#\"\"($\"'Z=vF+/&F&6#\"\")$!'h(Q'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+0c2 k=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~5~schemeG$\"+)pNfb%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% L2-norm~of~principal~error~of~order~4~schemeG$\"+pBuX@!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)rS))R!\"(\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7 $$!)$4P+%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$ \")uBk5!\"(/&%\"CGF&$\")[!G+\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "a[6,5]=subs(e6,a[6,5]),a[7,5 ]=convert(evalf(subs(e6,a[7,5])),rational,5),b[7]=subs(e6,b[7]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"'\"\"&#\"#:\"$k\"/&F%6$\" \"(F(#!$&=\"$A\"/&%\"bG6#F/#\"##)\"%*4\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "#--------------------------------- ---------" }}{PARA 0 "" 0 "" {TEXT -1 115 "The coefficients can be sim plified slightly by using the earlier (simpler) calculation method whe re the values of " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 40 " are extracted from the previous scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "c_2 := 1/6: c_3 := 16/63: c_4 := 63/106: c_5 := 50/69: c_6 := 63/83: bs_3 := 71/1 90: bs_5 := -62/241: \na6_5 := 15/164: a7_5 :=-185/122: b_7 := 82/1099 : \ncalc_RKcoeffsA();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&% \"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$#\"#;\"#j/&F&6#\"\"%#F2\"$1\"/&F&6# \"\"&#\"#]\"#p/&F&6#F+#F2\"#$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%6li nking~coefficients:G/&%\"aG6$\"\"'\"\"&#\"#:\"$k\"/&F&6$\"\"(F)#!$&=\" $A\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\" \"\"$\"'Arx!\"(/&F&6#\"\"$$\"')*RP!\"'/&F&6#\"\"%$\"'A$H#F2/&F&6#\"\"& $!'55DF2/&F&6#\"\"'$\"'j`\\F2/&F&6#\"\"($\"'LhuF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"'Wyx!\"(/&F&6#\"\" $$\"'%ot$!\"'/&F&6#\"\"%$\"'g6BF2/&F&6#\"\"&$!'hsDF2/&F&6#\"\"'$\"''3+ &F2/&F&6#\"\"($\"'Z=vF+/&F&6#\"\")$!'h(Q'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+0c2 k=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~5~schemeG$\"+]m+XX!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L 2-norm~of~principal~error~of~order~4~schemeG$\"+pBuX@!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)7N))R!\"(\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$ !)k5!\"(/&%\"CGF&$\")_!G+\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------------------------------" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 24 "an efficient scheme .. B" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "c_2 := 1/6: c_4 := 12/19: c_5 := 66/85: c_6 := 27/34: bs_3 := 71/ 183: bs_5 := 1/57:\nt_6 := 1/717: t_7 := 1/5338:\ncalc_RKcoeffsB(3,'ra tional_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\" \"##\"\"\"\"\"'/&F&6#\"\"$#\"$b\"\"$%f/&F&6#\"\"%#\"#7\"#>/&F&6#\"\"&# \"#m\"#&)/&F&6#F+#\"#F\"#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~ 5~weights:G/&%\"bG6#\"\"\"$\"'7Qz!\"(/&F&6#\"\"$$\"'h\")Q!\"'/&F&6#\" \"%$\"'d!R#F2/&F&6#\"\"&$\"'Nh@F+/&F&6#\"\"'$\"'q$)>F2/&F&6#\"\"($\"'x TtF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\" \"\"$\"'(H%z!\"(/&F&6#\"\"$$\"'yzQ!\"'/&F&6#\"\"%$\"'$))R#F2/&F&6#\"\" &$\"'Ra " 0 "" {MPLTEXT 1 0 92 "a[6,5]=subs(e6,a[6,5]),a[7,5]=convert(evalf(subs(e6,a[7,5])),ratio nal,6),b[7]=subs(e6,b[7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6 $\"\"'\"\"&#\"#b\"$1%/&F%6$\"\"(F(#!$a\"\"$H&/&%\"bG6#F/#\"#H\"$&R" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "#------- -----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 115 "T he coefficients can be simplified slightly by using the earlier (simpl er) calculation method where the values of " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&% \"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[7]" "6#&%\"b G6#\"\"(" }{TEXT -1 40 " are extracted from the previous scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "c_2 := 1/6: c_3 := 155/594: c_4 := 12/19: c_5 := 66/85: c_6 := 27/34: \na6_5 := 55/406: a7_5 := - 154/529: b_7 := 29/395: bs_3 := 71/183: bs_5 := 1/57:\ncalc_RKcoeffsA ();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\" \"\"'/&F&6#\"\"$#\"$b\"\"$%f/&F&6#\"\"%#\"#7\"#>/&F&6#\"\"&#\"#m\"#&)/ &F&6#F+#\"#F\"#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%6linking~coeffici ents:G/&%\"aG6$\"\"'\"\"&#\"#b\"$1%/&F&6$\"\"(F)#!$a\"\"$H&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\"\"$\"'7Qz!\" (/&F&6#\"\"$$\"'h\")Q!\"'/&F&6#\"\"%$\"'d!R#F2/&F&6#\"\"&$\"'Nh@F+/&F& 6#\"\"'$\"'q$)>F2/&F&6#\"\"($\"'xTtF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"'(H%z!\"(/&F&6#\"\"$$\"'yzQ!\" '/&F&6#\"\"%$\"'$))R#F2/&F&6#\"\"&$\"'Ra " 0 "" {MPLTEXT 1 0 140 "subs(e4,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..7),``],[`b*`,seq(` b*`[i],i=1..8)]]));\n``;\nevalf[6](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"$b\"\"$%f#\"%l m\"'7w6#\"&DS#F2F+F+F+F+F+F+7+#\"#7\"#>#\"*s$p#4\"\"*vuyk\"#!(s=M#\"(X J1\"#\"*)3(zd$F;F+F+F+F+F+7+#\"#m\"#&)#!,**oF&)y'\"-]([+uN\"#\")OR'y$ \")X[^<#!0k;sX'zJQ\"0D1'>J+RG#\",Hax%)[&\"-]Uqo\"=\"F+F+F+F+7+#\"#F\"# M#!/,'QpGxG#\"/+@4RulG#\"+6%=ny#\"*S>x5*#!2)3fE_(o1w&\"2vNcIp^o*H#\".z (\\)**[)R\"/I1U))QZ7#\"#b\"$1%F+F+F+7+F)#\"9\"Rng-%o&z3T!yN\"9++\\$)4% Go2!\\hQ#!(Hib(\"(5!)o##\"=n\\+*H28#!$a\"\"$H&#\"6z/m'e#35%3#Q\"\"6+].sne li#>DF+F+7+F)#\")FV'>&\"*+xha'\"\"!#\"16$*oX\"Qc'p\"2+RLdpBXz\"#\",%)p PdX\"\",vO'\\*3'#\"+D6;f?\",7Y2s_*#\"+*o'*Rz\"\"++nqV!*#\"#H\"$&RF+7+% \"bGFbpFepFfpFipF\\qF_qFbqF+7+%#b*G#\"1JrQR>9dW\"2+bT8#oU6cFep#\"#r\"$ $=#\"3zH)QRU^Vm$\"4+v(H9z]bF:#F)\"#d#\"1>j&fh8\"*R#\"2v$>\\&)HF*=\"#\" 7Pzr3%*e_)ppI$\"8v$[Xy'e$))=_7X#\"4Jecnw%eT#=)\"8]!e99/.mi-:aQ)pprint1 56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+$\"'nm;!\"'F(%!GF+F+F+F+F+F+7+$\"'V4EF* $\"'%pm&!\"($\"'tU?F*F+F+F+F+F+F+7+$\"'z:jF*$\"'$4j'F*$!'y-A!\"&$\"'Er @F;F+F+F+F+F+7+$\"'rkxF*$!'7,]F*$\"'#=;#F;$!'q\\8F;$\"'hWYF*F+F+F+F+7+ $\"'=TzF*$!'-$)zF*$\"'tfIF;$!'CA>F;$\"'f%>$F*$\"'oa8F*F+F+F+7+$\"\"\" \"\"!$\"''fE*F*$!'46GF;$\"''*GCF;$\"'W!)>F*$!':6HF*$\"'2'[&F*F+F+7+FW$ \"'7QzF1$FYFY$\"'h\")QF*$\"'d!R#F*$\"'Nh@F1$\"'q$)>F*$\"'xTtF1F+7+%\"b GFaoFcoFdoFfoFhoFjoF\\pF+7+%#b*G$\"'(H%zF1Fco$\"'yzQF*$\"'$))R#F*$\"'R a " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#---------------------------------------------- --------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 25 "an efficient scheme . . C " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "c_2 := 9/53: c_4 := 167/250: c_5 := 34/43: c_6 := 22 /27: bs_3 := 635/1588: bs_5 := 31/307:\nt_6 := 1/718: t_7 := 1/5450:\n calc_RKcoeffsB(3,'rational_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6( %'nodes:G/&%\"cG6#\"\"##\"\"*\"#`/&F&6#\"\"$#\"#$)\"$6$/&F&6#\"\"%#\"$ n\"\"$]#/&F&6#\"\"&#\"#M\"#V/&F&6#\"\"'#\"#A\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\"\"$\"'&z2)!\"(/&F&6#\" \"$$\"'J**R!\"'/&F&6#\"\"%$\"'w@GF2/&F&6#\"\"&$\"'==5F2/&F&6#\"\"'$\"' NcfF+/&F&6#\"\"($\"':tvF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4 ~weights:G/&%#b*G6#\"\"\"$\"'az!)!\"(/&F&6#\"\"$$\"'u)*R!\"'/&F&6#\"\" %$\"'mCGF2/&F&6#\"\"&$\"'x45F2/&F&6#\"\"'$\"'P8gF+/&F&6#\"\"($\"'?^vF+ /&F&6#\"\")$\"'/:C!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinit yG\"\"\"%8-norm~of~linking~coeffsGF&$\"+8t'QD(!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+U] hZR!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~4~schemeG$\"+Hc1YD!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%; order~5~stability~intervalG7$$!)DF!4%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)5^LT!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")8S]6!\"(/&%\"CGF&$\" )[?*4\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "a[6,5]=subs(e6,a[6,5]),a[7,5]=convert(evalf(subs(e6 ,a[7,5])),rational,6),b[7]=subs(e6,b[7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"'\"\"&#\"#G\"#()/&F%6$\"\"(F(#\"#I\"%.5/&% \"bG6#F/#\"#W\"$\"e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 43 "#------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 115 "The coefficients can be simplified slightly by u sing the earlier (simpler) calculation method where the values of " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 40 " are extracted fr om the previous scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 " c_2 := 9/53: c_3 := 83/311: c_4 := 167/250: c_5 := 34/43: c_6 := 22/27 : \na6_5 := 28/87: a7_5 := 30/1003: b_7 := 44/581: bs_3 := 635/1588: \+ bs_5 := 31/307:\ncalc_RKcoeffsA();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6( %'nodes:G/&%\"cG6#\"\"##\"\"*\"#`/&F&6#\"\"$#\"#$)\"$6$/&F&6#\"\"%#\"$ n\"\"$]#/&F&6#\"\"&#\"#M\"#V/&F&6#\"\"'#\"#A\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%6linking~coefficients:G/&%\"aG6$\"\"'\"\"&#\"#G\"#()/& F&6$\"\"(F)#\"#I\"%.5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~wei ghts:G/&%\"bG6#\"\"\"$\"'&z2)!\"(/&F&6#\"\"$$\"'J**R!\"'/&F&6#\"\"%$\" 'w@GF2/&F&6#\"\"&$\"'==5F2/&F&6#\"\"'$\"'NcfF+/&F&6#\"\"($\"':tvF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"' az!)!\"(/&F&6#\"\"$$\"'u)*R!\"'/&F&6#\"\"%$\"'mCGF2/&F&6#\"\"&$\"'x45F 2/&F&6#\"\"'$\"'P8gF+/&F&6#\"\"($\"'?^vF+/&F&6#\"\")$\"'/:C!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+8t'QD(!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~ of~principal~error~of~order~5~schemeG$\"+-%>#[R!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~4~schemeG$\"+Hc 1YD!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interva lG7$$!)iF!4%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~st ability~intervalG7$$!)\\^LT!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$/&%\"BG6#\"\"'$\")4T]6!\"(/&%\"CGF&$\")[?*4\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The principal error norm \+ of the order 4 scheme is roughly 6.45 times that of the order 5 scheme ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 ".2546065629e-4/.3948219 402e-5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ " 0 "" {MPLTEXT 1 0 140 "subs (e4,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,se q(b[i],i=1..7),``],[`b*`,seq(`b*`[i],i=1..8)]]));\n``;\nevalf[6](%%); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"*\"#`F(%!GF+F +F+F+F+F+7+#\"#$)\"$6$#\"&<&**\"(y4u\"#\"'<^OF2F+F+F+F+F+F+7+#\"$n\"\" $]##\",()*GsjI\",+]i](Q#!+^Fk!p(\"+](oz\"H#\"-t&*y*\\q#\"-+]iSw5F+F+F+ F+F+7+#\"#M\"#V#!0yc5q`#=O\"0`/m!4GuF#\"-44#H+&\\\"-lL1v+5#!2\"yyQKCv$ ))*\"2&o.unLHyG#\"/+D,P`G\")\"06dY`&[&R\"F+F+F+F+7+#\"#A\"#F#!4=WI23n1 _c(\"4(*)*RG3xTFo$#\"-jt&*o&)*)\"-D%oW(Q7#!6*f&pMR=HZ'*G(\"6D]4Zv.$Qap 9#\"1+vALMw=S\"2n2/(G%*G$e\"#\"#G\"#()F+F+F+7+\"\"\"#\"?,Fm-+$#\"?vV'*4*oS;8'3O5nwR\"@_5^m\"=@w*e,p1Slc\"#\"#I \"%.5#\":&HC&f0$=N]pv&p)\";_>\\&4cn%#\"1+vVtxJ[L\"2j\\zZF/m=\"#\"+*o$zr;\",)G w$>k\"#\"+@\")H%o%\",ob#Qky#\"#W\"$\"eF+7+%\"bGFdpFgpFhpF[qF^qFaqFdqF+ 7+%#b*G#\"6v&*GV\\E3Ydf#\"7Yv(*GjD#zTF@$Fgp#\"$N'\"%)e\"#\":]P%[r;N8C; !HR$\";zxO\")[EdER6<,7#\"#J\"$2$#\"6kde2lYHgOb*\"8:F^&G7*e3O()e\"#\"=# )[zON:b8>tY4'3#\">NXkZV1cy#Gx&)fiF#\"9Je#f0$[3d>J:K\"=!e5fI/L0H$3GPJ8Q (pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7+7+$\"'6)p\"!\"'F(%!GF+F+F+F+F+F+7+$\"' \")oEF*$\"':;d!\"($\"'>(4#F*F+F+F+F+F+F+7+$\"'+!o'F*$\"'D1zF*$!'iNE!\" &$\"'*H^#F;F+F+F+F+F+7+$\"')p!zF*$!'@/8F;$\"'KY\\F;$!'*QV$F;$\"')[#eF* F+F+F+F+7+$\"':[\")F*$!'Ba?F;$\"'(QD(F;$!'[g\\F;$\"'CQDF*$\"'R=KF*F+F+ F+7+$\"\"\"\"\"!$\"'/)=\"F;$!'bUOF;$\"'QEIF;$\"'^QDF*$\"'.\"*HF1$\"'yV 9F*F+F+7+FW$\"'&z2)F1$FYFY$\"'J**RF*$\"'w@GF*$\"'==5F*$\"'NcfF1$\"':tv F1F+7+%\"bGFaoFcoFdoFfoFhoFjoF\\pF+7+%#b*G$\"'az!)F1Fco$\"'u)*RF*$\"'m CGF*$\"'x45F*$\"'P8gF1$\"'?^vF1$\"'/:C!\"*Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 55 "#---------------------------------------- --------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 38 "a scheme with m aximal stability radius" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 305 "This scheme demonstrates that it is possible to con struct a scheme which has the maximal possible stability radius for a \+ 7 stage, order 5 scheme along with an embedded FSAL 8 stage, order 4 s cheme with a compatible stability region. Three of the nodes are the s ame as those in the Bogacki-Shampine scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The values " }{XPPEDIT 18 0 "t [6]=0" "6#/&%\"tG6#\"\"'\"\"!" }{TEXT -1 17 ".001103840 and " } {XPPEDIT 18 0 "t[7]=0" "6#/&%\"tG6#\"\"(\"\"!" }{TEXT -1 42 ".00006537 663796 for the coefficients of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\" '" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 29 " in the stability function: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6+t[7] *z^7" "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F ,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86# \"\"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 62 "giv e a (near) maximal value for the stability radius, namely " } {XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 281 1 "~" } {TEXT -1 12 " 3.17386983." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 33 "In this scheme the coefficients " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[7]" " 6#&%\"tG6#\"\"(" }{TEXT -1 6 " of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG \"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" } {TEXT -1 111 " in the stability function are rational approximations \+ to the values which give the maximal stability radius. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "[0.1103 840e-2,0.6537663796e-4]:\nmap(convert,%,rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"#8\"&x<\"#\"\"\"\"&'H:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "c_2 := 5/33 : c_3 := 29/115: c_5 := 317/476: c_6 := 14/19: bs_3 := 15/53: bs_5 := \+ 39/125:\nt_6 := 13/11777: t_7 := 1/15296:\ncalc_RKcoeffsB(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"&\"#L/&F&6 #\"\"$#\"#H\"$:\"/&F&6#\"\"%,&#\"*lkr9)\"+omz`K\"\"\"*(\"#BF;F:!\"\"\" 0XfX<9:0\"#F;F(F>/&F&6#F*#\"$<$\"$w%/&F&6#\"\"'#\"#9\"#>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\"\"$\"'>(Q'!\"(/& F&6#\"\"$$\"'o8K!\"'/&F&6#\"\"%$\"'KVoF+/&F&6#\"\"&$\"'CeGF2/&F&6#\"\" '$\"'cu;F2/&F&6#\"\"($\"'q/$*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1o rder~4~weights:G/&%#b*G6#\"\"\"$\"'WIe!\"(/&F&6#\"\"$$\"'>IG!\"'/&F&6# \"\"%$\"'ii5F2/&F&6#\"\"&$\"'+?JF2/&F&6#\"\"'$\"'!>Y\"F2/&F&6#\"\"($\" 'Yy!*F+/&F&6#\"\")$\"'sSM!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%) infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+hPw*y'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG $\"+Gq=-c!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal ~error~of~order~4~schemeG$\"+xmcly!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)C/[j!\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)$f@!o!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\"))\\CH\"!\"(/&%\"CGF& $\")ML-5F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "#----------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#---------------------------------------------- --------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 43 "a scheme with near ma ximal stability radius" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "t[6]=0" "6#/&%\"tG6#\"\"'\"\" !" }{TEXT -1 16 ".00117605 and " }{XPPEDIT 18 0 "t[7]=0" "6#/&%\"tG6 #\"\"(\"\"!" }{TEXT -1 42 ".00007827904795 are the coefficients of \+ " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 29 " in the stability \+ function: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = \+ 1+z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7" "6#/-%\"RG6#%\"zG,2 \"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F' \"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "the stability radius \+ " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 282 1 "~" } {TEXT -1 61 " 3.04298260 and the associated real stability interval i s: " }{XPPEDIT 18 0 "[-2*rho,0]" "6#7$,$*&\"\"#\"\"\"%$rhoGF'!\"\"\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 156 "We construct a 7 stage, order 5 scheme with this stabi lity function along with an embedded FSAL 8 stage, order 4 scheme wit h a compatible stability region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "[0.117605e-2,0.7827904795e-4 ]:\nmap(convert,%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$# \"\"$\"%^D#\"\"\"\"&uF\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "c_2 := 1/6: c_4 := 8/21: c_5 := 2/ 3: c_6 := 3/4: bs_3 := 20/561: bs_5 := 2/67:\nt_6 := 3/2551: t_7 := 1/ 12774:\ncalc_RKcoeffsB(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes: G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$,&#\"(VXJ)\")&Rbr#F**&\")!z5V&! \"\"\".:35F(\\q#F*F(F6/&F&6#\"\"%#\"\")\"#@/&F&6#\"\"&#F(F//&F&6#F+#F/ F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\" \"$\"';%=)!\"(/&F&6#\"\"$$\"'+JL!\"'/&F&6#\"\"%$\"&x!)*F2/&F&6#\"\"&$ \"'xZ&*F+/&F&6#\"\"'$\"'%)GJF2/&F&6#\"\"($\"'khyF+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6*%1order~4~weights:G/&%#b*G6#\"\"\"$\"'Fr7!\"'/&F&6#\" \"$$\"'1lN!\"(/&F&6#\"\"%$\"'MQVF+/&F&6#\"\"&$\"'2&)HF2/&F&6#\"\"'$\"' A:FF+/&F&6#\"\"($\"'86#)F2/&F&6#\"\")$\"'L!*>F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+\"R T.^\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~er ror~of~order~5~schemeG$\"+I[prp!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%L2-norm~of~principal~error~of~order~4~schemeG$\"+P?>w6!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)[7%3'!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~interval G7$$!)^f!3'!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"' $\")0Zq;!\"(/&%\"CGF&$\")-C%o\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "c_2 := 1/6: c_4 := 8/21: c_ 5 := 2/3: c_6 := 3/4: bs_3 := 20/561: bs_5 := 2/67:\nt_6 := 3/2551: t_ 7 := 1/12774:\ncalc_RKcoeffsB(3,'rational_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$#\"#W\"$r \"/&F&6#\"\"%#\"\")\"#@/&F&6#\"\"&#F(F//&F&6#F+#F/F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~5~weights:G/&%\"bG6#\"\"\"$\"'8$=)!\"(/&F&6# \"\"$$\"'sKL!\"'/&F&6#\"\"%$\"'$Hy*F+/&F&6#\"\"&$\"'Qx&*F+/&F&6#\"\"'$ \"'hEJF2/&F&6#\"\"($\"'DjyF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1orde r~4~weights:G/&%#b*G6#\"\"\"$\"'Gr7!\"'/&F&6#\"\"$$\"'1lN!\"(/&F&6#\" \"%$\"'LQVF+/&F&6#\"\"&$\"'2&)HF2/&F&6#\"\"'$\"'A:FF+/&F&6#\"\"($\"'26 #)F2/&F&6#\"\")$\"'U!*>F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infi nityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+eu[5:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+! pue'p!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~err or~of~order~4~schemeG$\"+`j-x6!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %;order~5~stability~intervalG7$$!)a$G1'!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)x\\rg!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")>8q;!\"(/&%\"CGF&$\" )o#Ro\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#--------------- ---------------------------------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 58 "a scheme with stability radius = imaginary axis inclusion " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "c_2 := 11/63: c_4 := 174/349: c_5 := 36/49: c_6 := 9 7/129: bs_3 := 121/366: bs_5 := -65/379:\nt_6 := 1/720: t_7 := 13/6287 3:\ncalc_RKcoeffsB(3,'rational_node');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"#6\"#j/&F&6#\"\"$#\"\"%\"#F2/&F&6#\"\"'$\" '(fB&F2/&F&6#\"\"($\"'QsvF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order ~4~weights:G/&%#b*G6#\"\"\"$\"'f7u!\"(/&F&6#\"\"$$\"',1L!\"'/&F&6#\"\" %$\"'!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~4~schemeG$\"+bu[l%*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% ;order~5~stability~intervalG7$$!)-8&*Q!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)FXER!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")U(G/\"!\"(/&%\"CGF&$ \")c-E6F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 ".9465487455e-4/.1976847989e-4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H;<)y%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 31 "#==============================" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "#==============================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Test-bed procedures for examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK5step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2662 "rk5step := proc(x_rk5step::realco ns)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,a51,a52,a53,a5 4,\n a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n f1,f2,f3,f4,f5 ,f6,f7,b1,b2,b3,b4,b5,b6,b7,\n xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,sa veDigits;\n options `Copyright 2004 by Peter Stone`;\n \n data : = SOLN_;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Di gits)),Digits+5);\n\n # procedure to evaluate the slope field\n fn := proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_) );\n if val=lasterror or not type(val,numeric) then\n err or \"evaluation of slope field failed at %1\",evalf([X_,Y_],saveDigits );\n end if;\n val;\n end proc;\n\n xx := evalf(x_rk5 step);\n n := nops(data);\n\n if (data[1,1]data [n,1] or xxdata[1,1])) then\n error \"independent variable is outsi de the interpolation interval: %1\",evalf(data[1,1])..evalf(data[n,1]) ;\n end if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; c5 := c5_; c6 := \+ c6_; c7 := c7_; \n a21 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a43_;\n a51 := a51_; a52 := a52_; a53 := a53_; \+ a54 := a54_;\n a61 := a61_; a62 := a62_; a63 := a63_; a64 := a64_; a 65 := a65_;\n a71 := a71_; a72 := a72_; a73 := a73_; a74 := a74_; a7 5 := a75_; a76 := a76_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_ ; b5 := b5_; b6 := b6_; b7 := b7_;\n # Perform a binary search for t he interval containing x.\n n := nops(data);\n jF := 0;\n jS := \+ n+1;\n\n if data[1,1]1 do\n \+ jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] then jF := jM els e jS := jM end if;\n end do;\n if jM = n then jF := n-1; jS \+ := n end if;\n else\n while jS-jF> 1 do\n jM := trunc((j F+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := jM end i f;\n end do;\n if jM = n then jF := n-1; jS := n end if;\n \+ end if;\n \n # Get the data needed from the list.\n xk := data[j F,1];\n yk := data[jF,2];\n\n # Do one step with step-size ..\n \+ h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2 *h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t* h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h );\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h, yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n ys \+ := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n\n evalf[saveDigits](ys);\nend proc: # of rk7_5step" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "RK5_1 Bogacki-Shampine scheme " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2863 "RK5_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5 ,c6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a6 5,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6, f7,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits,SQRT5;\n \n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n f n := unapply(fxy,x,y);\n \n A := matrix([[1/6,1/6,0,0,0,0,0,0],\n [2/9,2/27,4/27,0,0,0,0,0],\n [3/7,183/1372,-162/3 43,1053/1372,0,0,0,0],\n [2/3,68/297,-4/11,42/143,1960/3861, 0,0,0],\n [3/4,597/22528,81/352,63099/585728,58653/366080,46 17/20480,0,0],\n [1,174197/959244,-30942/79937,8152137/19744 439,666106/1039181,-29421/29068,482048/414219,0],\n [0,587/8 064,0,4440339/15491840,24353/124800,387/44800,2152/5985,7267/94080]]); \n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf( A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := e valf(A[6,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := eval f(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]); \n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf( A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 : = evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[ 6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := e valf(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n \+ soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk) ;\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41 *f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t : = a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t* h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 \+ := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74* f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n \+ yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 )*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n \+ if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_= c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41 _=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54 ,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a7 1_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1, b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,e val(rk5step)); \n else \n return evalf[saveDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "RK5_2 a minor variation of the Bogacki-Shampine scheme \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2978 "RK5_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a 63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3, f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits,SQRT5;\n \n s aveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5) ;\n\n fn := unapply(fxy,x,y);\n \n A := matrix([[1/6,1/6,0,0,0, 0,0,0],\n [2/9,2/27,4/27,0,0,0,0,0],\n [3/7,183/1372 ,-162/343,1053/1372,0,0,0,0],\n [2/3,68/297,-4/11,42/143,1960 /3861,0,0,0],\n [3/4,32575/1096128,81/352,38673/395824,242133 5/14249664,77/346,0,0],\n [1,2218605592257973/110037136578969 60,-63/143,\n 1473430904732051/3532056235868160,2549 732718751057/3649190753894400,\n -13054508705469277/1 2226348508774400,21360483/17937920,0],\n [0,539/7344,0,190269 /671840,362551/1790100,-81/13600,48448/130815,13/170]]);\n \n c2 := \+ evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c 5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]); \n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n \+ a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3 ,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := e valf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n \+ a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5 ]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := eva lf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a7 5 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7,2]); \n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7 ,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := eval f(A[7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk ,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32 *f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 \+ + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a5 2*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t \+ := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6 *h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n x k := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true t hen\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n \+ a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a 61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a 72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_=b3 ,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk5step)) ; \n else \n return evalf[saveDigits]([soln]);\n end if;\ne nd proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "RK5_3 a n efficient scheme .. A " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3452 "RK5_3 := proc(fxy,x,y,xx,yy,h,stp s,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52 ,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4, b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits ,SQRT5;\n \n saveDigits := Digits;\n Digits := max(trunc(evalhf( Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n \n A := matri x([[1/6,1/6,0,0,0,0,0,0],\n [16/63,80/1323,256/1323,0,0,0,0,0 ],\n [63/106,155816577/304900096,-16967475/9528128,568356831/ 304900096,0,0,0,0],\n [50/69,64921911575/789872295744,1250725 /43533526,\n 995353041375/3166454547136,365139662800/121 9687315371,0,0,0],\n [63/83,-30354668306338585/17891002053168 6528,3896053749/4091707772,\n -2492114394031368057/645495 9947119420288,861399479018673064/3177050598972839673,15/164,0,0],\n \+ [1,22116285273625596605656418138617/387755997939521400532489825 99680,-197127/114496,\n 24118537135561993104813811090103 /14776247724722745081046875555840,\n 8864759094911693189 04581727562843/1979638263308617118988089415360030,-185/122,\n \+ 4082588597964313509378858016009/2571428541481836530593320891180, 0],\n [0,16270811903/209372688000,0,76925739070593/2056846311 94240,747543949006532/3259794500756955, -35320882628403/ 140715258838000,1551373170541891/3131789077344060,82/1099]]);\n \n c 2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]); \n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6 ,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3] );\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := eval f(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]); \n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf( A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 : = evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7 ,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := eval f(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n soln : = [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n \+ t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 \+ + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a4 2*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n \+ t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75 *f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n yk \+ := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n \+ xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=t rue then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c 3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n \+ a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a 72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b 3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk5s tep)); \n else \n return evalf[saveDigits]([soln]);\n end i f;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "RK5 _4 an efficient scheme .. B" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3379 "RK5_4 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n \+ a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2 ,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,sav eDigits,SQRT5;\n \n saveDigits := Digits;\n Digits := max(trunc( evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n \n A : = matrix([[1/6,1/6,0,0,0,0,0,0],\n [155/594,6665/117612,24025 /117612,0,0,0,0,0],\n [12/19,109269372/164787475,-2341872/106 3145,357797088/164787475,0,0,0,0],\n [66/85,-67885276899/1357 40048750,37863936/17514845,\n -383179645721664/28390 0311960625,54884775429/118168704250,0,0,0],\n [27/34,-2287728 6938601/28657439092100,2786718411/910771940,\n -5760 6687522659088/29968516930563575,3984899849779/12473888420630,55/406,0, 0],\n [1,357804108795684026067391/386149007682840983490000,-7 556229/2688010,\n 1811005935036153778848695517/7455 90435061329862990895000,\n 258902691791384553326881 /1307299004967194685810000,-154/529,\n 138208410082 586660479/251926265586772035000,0],\n [0,51964327/654617700,0 ,6965638145689311/17945236957333900,14557376984/60894963675,\n \+ 2059161125/95272074612,1793996689/9043706700,29/395]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3 ,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := eval f(A[6,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A [2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n \+ a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5 ,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := e valf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n \+ a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5 ]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := eval f(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n \+ b7 := evalf(A[7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n s oln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a3 1*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a 51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h); \n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := \+ fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 \+ + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n \+ yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h ;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2, c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a 41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_= a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_ =b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval (rk5step)); \n else \n return evalf[saveDigits]([soln]);\n \+ end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "RK5_5 an efficient scheme .. C" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3493 "RK5_5 := proc(fxy,x,y,xx ,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n \+ a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1, b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,s aveDigits,SQRT5;\n \n saveDigits := Digits;\n Digits := max(trun c(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n \n A := matrix([[9/53,9/53,0,0,0,0,0,0],\n [83/311,99517/1740978, 365117/1740978,0,0,0,0,0],\n [167/250,30637228987/38750625000 ,-7690642751/2917968750,270499789573/107640625000,0,0,0,0],\n \+ [34/43,-361825370105678/277428090660453,495002920909/100075063365,\n \+ -98837524323878781/28782933677403685,81285337012500/1395 48553465711,0,0,0],\n [22/27,-7565206670807304418/36827417708 28399897,898568957363/123874468425,\n -72896472918393469 5599/146954383037547095025,4018763433227500/15832894287040767,28/87,0, 0],\n [1,930898979175170148742163494117/783561862284207348259 733379696,-4034678053/1107651600,\n 9079937523748774469 8230149845605767/30002662701192241345485814174532400,\n \+ 397667103608613164068909964375/1566540066901589762118166511052,30/100 3,\n 8695756950351830559524295/602289358456417560954919 52,0],\n [0,70352057/870914352,0,186846453120293/467197050726 588,3348317773437500/11866042747794963,\n 1671793689/16 419376288,4684298121/78643825568,44/581]]);\n\n c2 := evalf(A[1,1]); \n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4 ,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n a21 := c2; \n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf( A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 : = evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[ 5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := \+ evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n \+ a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6, 6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7,2]);\n b2 := eval f(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]);\n\n \+ xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for \+ k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 \+ := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62 *f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n \+ f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2* f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n xk := xk + h:\n \+ soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4, c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a4 3_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a6 2,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74 _=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5, b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk5step)); \n else \+ \n return evalf[saveDigits]([soln]);\n end if;\nend proc: " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "RK5_6 a scheme with st ability radius = imag axis inclusion " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3582 "RK5_6 := proc(fxy,x,y, xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43, \n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n \+ b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns, A,saveDigits,SQRT5;\n \n saveDigits := Digits;\n Digits := max(t runc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n \n \+ A := matrix([[11/63,11/63,0,0,0,0,0,0],\n [4/17,244/3179,50 4/3179,0,0,0,0,0],\n [174/349,184816101/935188078,-411995808/ 467594039,100493613/85017098,0,0,0,0],\n [64/87,664243232/441 1311597,134272/804837,-450364000/3600035901,\n 170170 613120/313203123387,0,0,0],\n [208/277,1173093471427355749/20 510309752258061736,5262525216/14261389043,\n -29099809 3809856214/2092287345417129861,\n 6983637996204390922 4/182028999051290297907,25/312,0,0],\n [1,1055737917969239666 776120789763037627565/5375218998249906870255258530319640363008,\n \+ -80409/101992,\n 111225467315817129272785 63605967453549877/10144181256036282577701554712032827035648,\n \+ 20938581734779343733162630745803921131771/351824880994825935 617176218492327710635008,\n -2396370256144923749263546 015554136471/1619941963842719664253235178425745408,\n \+ 22045290267/11510033408,0],\n [0,97909739/1328005120,0,16629 0311/501139200,8560074037777/45981232702440,\n -349257 479643/1457934745600,3345792666338623/5848339366072320,114/1505]]);\n \+ \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3 ,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := eval f(A[6,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A [2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n \+ a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5 ,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := e valf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n \+ a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5 ]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := eval f(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n \+ b7 := evalf(A[7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n s oln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a3 1*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a 51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h); \n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := \+ fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 \+ + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n \+ yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h ;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2, c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a 41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_= a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_ =b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval (rk5step)); \n else \n return evalf[saveDigits]([soln]);\n \+ end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 54 "#----- ------------------------------------------------" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 46 "RK5_7 a scheme with maximal stability radius " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6390 "RK5_7 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c 6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65, a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7 ,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits,SQRT5;\n \n saveDigits : = Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn \+ := unapply(fxy,x,y);\n \n A := matrix([[5/33,5/33,0,0,0,0,0,0],\n [29/115,5597/132250,27753/132250,0,0,0,0,0],\n[814716465/3253796668-23 /3253796668*105151417455945^(1/2),379368932539978890648812440551/72428 12371317610711938898070128-2455501968740121696101/72428123713176107119 38898070128*105151417455945^(1/2),9899096170368469289460332229/6243803 7683772506137404293708+45207024234619537323/62438037683772506137404293 708*105151417455945^(1/2),285859955732027554698452564025/7242812371317 610711938898070128-53985530215629267813075/724281237131761071193889807 0128*105151417455945^(1/2),0,0,0,0],\n[317/476,12375208341306584313878 9879/209127510371847411054672640+1608851500638397121047/90970467011753 623808782598400*105151417455945^(1/2),-2000041633670013681/53245223453 95547392+15875554484889/3860278700411771859200*105151417455945^(1/2),1 8902912452932794771976769428125/17444617282574859822209614750208+10066 2884268346078346663939/505893901194670934844078827756032*1051514174559 45^(1/2),-516318245298821111959896996980826931/81469671171523014739685 1820173032960-107920362352247138873225158999/4888180270291380884381110 92103819776*105151417455945^(1/2),0,0,0],\n[14/19,-4922558902230123374 4183735779/63282599719433884959882336960-3510687788120584753272949/825 83792633861219872646449732800*105151417455945^(1/2),-245632221609852/4 23282790552285-16879401/3230316033162175*105151417455945^(1/2),8627654 0353665039847710695810845/833493552221848817093515269462144-2167889560 50328897851037816063/459254947274238698218526913473641344*105151417455 945^(1/2),359873158906430802491237985928506042059/24652962113182780504 3897351525007341440+230654465754417653852094653348797/4437533180372900 49079015232745013214592*105151417455945^(1/2),67/126,0,0],\n[1,1038263 3203049111998462381644826850633174257761074116960730558419200848945484 37/3477368541762557310220792568694054276803350656737359370378854857516 0867886583040+67568031989344520349682615047488926280454856996955945138 286279519534217777/302531063133342485989208953476382722081891507136150 2652229603726038995506132724480*105151417455945^(1/2),59734302131229/4 4926355375720-1441341/1302864305895880*105151417455945^(1/2),152497967 9765223198896979253792114247685185443478565951788517061903492278097461 8277305/20726650173902839432495268544515964009257243716562331895034391 2141403953196522628760064+31895489414207660020619126425491808266074209 880766698111670849008793214933062155/286225169068182068353506089424268 074413552413228717916645713021528605459176150296859136*105151417455945 ^(1/2),-16936601238178391429934510132574030422056553974752207384471165 3458806174655894526676245216204127995/15516266504381478323424954162459 3701687526417086957004644589292251741395415822637890959585918820864-29 1894097068874434621776389940761653039224034151992442739658375212058942 004402007722166107207/232743997565722174851374312436890552531289625630 4355069668839383776120931237339568364393788782312960*105151417455945^( 1/2),46156557089066321019743706501794658553375913031673310519879924144 99309155/1301891698142993109192427785923685882463389842416512208793190 6513654014256-20103590190559488820347818833647540935626519725830420509 7572636855/13018916981429931091924277859236858824633898424165122087931 906513654014256*105151417455945^(1/2),9714101019226554178088904960/319 57908807350197910133249679+261835787666411012880/319579088073501979101 33249679*105151417455945^(1/2),0],\n[0,9053807089231/136378733497104-5 01797/2045681002456560*105151417455945^(1/2),0,11950113721780031135837 5/307441845354011196253392-2018586005231875/307441845354011196253392*1 05151417455945^(1/2),-37638654822534247872916940674662423328915/984467 4333579068498057764513689180633044624+31216503052802331446365242026639 62531/443010345011058082412599403116013128487008080*105151417455945^(1 /2),1190204022066367942642960/4085963763691414586732367-21782855143424 48/4085963763691414586732367*105151417455945^(1/2),8305926327277757947 85/5052174577198328191248+1504077797251/5052174577198328191248*1051514 17455945^(1/2),91/978]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf( A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := e valf(A[5,1]);\n c7 := evalf(A[6,1]);\n a21 := c2;\n a31 := evalf (A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 \+ := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]); \n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf( A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 : = evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[ 6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := \+ evalf(A[6,7]);\n b1 := evalf(A[7,2]);\n b2 := evalf(A[7,3]);\n b 3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]); \n b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]);\n\n xk := evalf(x x);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stp s do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + \+ c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h ,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n \+ f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + \+ a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71* f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + \+ c7*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4 *f4 + b5*f5 + b6*f6 + b7*f7)*h;\n xk := xk + h:\n soln := so ln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[so ln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c 7_=c7,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51 ,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_ =a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75, a76_=a76,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\} ;\n return subs(eqns,eval(rk5step)); \n else \n return e valf[saveDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Testing the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 80 "These tests do not \+ make use of the embedded order 4 method for error correction." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 1 of 7 stage, order 5 Runge- Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%# dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F +F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6 #\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x* cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/ 289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ \"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX$\\'FJ$ \"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$3_DG1qF J$\"31le1yn9(R(FJ7$$\"3]***\\il'pisFJ$\"3E!)4GzFfsmFJ7$$\"3+MLe*[!)y_( FJ$\"3CJpN=**=vfFJ7$$\"3Qnm\"HKkIz(FJ$\"3'oU:>LtrL&FJ7$$\"3!3+]i:[#e!) FJ$\"31b0R&QB=w%FJ7$$\"3>MLe*)>VB$)FJ$\"3qsV#=-7'\\UFJ7$$\"3wmmTg()4_) )FJ$\"3,.mLb#*p3MFJ7$$\"3Y++DJbw!Q*FJ$\"3)=h%pn^Z#y#FJ7$$\"3+N$ekGkX#* *FJ$\"3i0nI\\:'RK#FJ7$$\"3%ommTIOo/\"F2$\"3\"GyFJ7$$\"3E+]7GTt%4\"F2$\"3YFEp[WPZ=FJ7$$\"3(p;/,/$o=6F2 $\"3;\"e:UqpMz\"FJ7$$\"3YLL3_>jU6F2$\"3EC/vOKMe(4+7ES\"F2$\"3U)GYYI=FJ7$$\"35+++v\"=YI\"F2$\"3/ >xDBH;*)>FJ7$$\"33++](=h(e8F2$\"3'>M4q'>VoAFJ7$$\"3&*****\\7!Q4T\"F2$ \"3Ig`=6c[gEFJ7$$\"3/++]P[6j9F2$\"31r>cjB_'>$FJ7$$\"3%o;HKR'\\5:F2$\"3 XwZW,h_FQFJ7$$\"3UL$e*[z(yb\"F2$\"3CY!yD$)***3YFJ7$$\"3w;/Ev&[ge\"F2$ \"3#G<.XQ`j9&FJ7$$\"34+Dc,#>Uh\"F2$\"3t/<(f[0bt&FJ7$$\"3V$eky#)*QU;F2$ \"3y9nBQhKrjFJ7$$\"3wmm;a/cq;F2$\"3A`yB*3;b/(FJ7$$\"3\"pm;a)))G=F2$\"3 )3HOInxF>\"F27$$\"3KLe9;0?E>F2$\"3`!yI!pI]77F27$$\"3pTg-gl[Q>F2$\"3>kb 3F2$\"3OF\\_#G6DA\"F27$$\"3WekyZ'eI'>F2$\"3Te$z >cCQA\"F27$$\"3gmmm\"pW`(>F2$\"3C(*f_UYpA7F27$$\"3dLe9TOEH?F2$\"3mC!>8 `I->\"F27$$\"3K+]i!f#=$3#F2$\"3w9E]:+C>6F27$$\"3/++D\"=EX8#F2$\"3f+))G PMfE5F27$$\"3?+](=xpe=#F2$\"3ES3-I\\16#*FJ7$$\"3mLeRA9WRAF2$\"3IMhv&[? ^3)FJ7$$\"37nm\"H28IH#F2$\"3H\\m$Q)R4@qFJ7$$\"3$p;a8d3AM#F2$\"39j2HRJ+ ZhFJ7$$\"3um;zpSS\"R#F2$\"3#>07(=j$QR&FJ7$$\"3-+v$41oWW#F2$\"3QVRl9U0B ZFJ7$$\"3GLL3_?`(\\#F2$\"3/\\cKWs=$>%FJ7$$\"3AL3_D1l_DF2$\"3o6E$fFc$yP FJ7$$\"3fL$e*)>pxg#F2$\"3ym)*p(*f`&[$FJ7$$\"3%omm\"z+vbEF2$\"3jG&[,$f< =LFJ7$$\"33+]Pf4t.FF2$\"3%R>3YHT'HFF2$\"3/C%)f*f*e+K FJ7$$\"3om\"zWi^bv#F2$\"3-Gu$[oUh>$FJ7$$\"3)*\\7.d>Y\"y#F2$\"3#p*R$)o? n4KFJ7$$\"3uLLe*Gst!GF2$\"3>.X!=mk1C$FJ7$$\"3)om\"H2\"34'GF2$\"3'[>IF2$\"3a-&\\&*p%H, TFJ7$$\"3F+]i!RU07$F2$\"3'fkDHe#=P[FJ7$$\"3+++v=S2LKF2$\"3K%\\5FaXpw&F J7$$\"3Jmmm\"p)=MLF2$\"3))zmB`6`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3#o,C;(=8fo FJ7$$\"3B++](=]@W$F2$\"3#G%=QV$\\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3u,muc\"4C(FJ7$$\"3hTgx.2vFNF2$\"3/^M\"Q[;l C(FJ7$$\"35L$e*[$z*RNF2$\"3=wJ%fi2nC(FJ7$$\"3)*\\PMFwrmNF2$\"3R[i&\\xl (GsFJ7$$\"3%o;Hd!fX$f$F2$\"3IEKi0hy'=(FJ7$$\"3r$e9T=%>?OF2$\"3(>gS`&3d ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shamp ine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`eff icient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme wit h stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 25 :\nfor ct to 6 do\n Fn_RK5_||ct := RK5_||ct(F(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Fn_RK5_||ct):\n for ii to n umpts do\n sm := sm+(Fn_RK5_||ct[ii,2]-f(Fn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF, -%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~p oint:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ +&Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+-xdj H!#@7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+HEV_?F +7$%3efficient~scheme~AG$\"+$>gt\"Q!#A7$%3efficient~scheme~BG$\"+TD#e& **F47$%3efficient~scheme~CG$\"+l9tBNF47$%Sscheme~with~stability~radius ~=~imag~axis~inclusionG$\"+nQ&p*GF+Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\! \"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: n umsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,F(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modi fication of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficie nt scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 25:\nfor ct to 6 do\n fn_ RK5_||ct := RK5_||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx \+ := 4.999: fxx := evalf(f(xx)):\nfor ct to 6 do\n errs := [op(errs),a bs(fn_RK5_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7& 7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,- %$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~ width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint126\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$%8Bogacki-Shampine~schemeG$\"+\"R3??$!#@7$%Ta~minor~modificatio n~of~the~Bogacki-Shampine~schemeG$\"+q6@zAF+7$%3efficient~scheme~AG$\" +`c#R=%!#A7$%3efficient~scheme~BG$\"+u#yt3\"F+7$%3efficient~scheme~CG$ \"+e.#[%QF47$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+ W+:KKF+Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sh ampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sc heme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs : = []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((f(x)-'fn_RK5_||c t'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [o p(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8B ogacki-Shampine~schemeG$\"+Ly1lH!#@7$%Ta~minor~modification~of~the~Bog acki-Shampine~schemeG$\"+.0d_?F+7$%3efficient~scheme~AG$\"+%o%)o\"Q!#A 7$%3efficient~scheme~BG$\"+oB5c**F47$%3efficient~scheme~CG$\"+#R!*G_$F 47$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+IS-(*GF+Q) pprint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical \+ procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "evalf[20](plot(['fn_RK5_1'(x)-f(x),'fn_RK5_2'(x)-f(x),'fn_RK5_3 '(x)-f(x),'fn_RK5_4'(x)-f(x),\n'fn_RK5_5'(x)-f(x),'fn_RK5_6'(x)-f(x)], x=0..5,\nfont=[HELVETICA,9],color=[blue,magenta,COLOR(RGB,.1,.8,0),red ,brown,COLOR(RGB,.6,0,.8)],\nlegend=[`Bogacki-Shampine scheme`,`a mino r modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`e fficient scheme B`,`efficient scheme C`,`scheme with stability radius \+ = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 745 499 499 {PLOTDATA 2 "6,-%'CURVESG6%7`r7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$!'g?P! #>7$$\"5MLLLL3x&)*3\"!#?$!'0I_F07$$\"5nmm\"z%\\v#pK\"F4$!'_DXF07$$\"5+ ++]i!R(*Rc\"F4$!'AbIF07$$\"5MLL3xJs1,=F4$\"&\"*=&F07$$\"5nmmm\"H2P\"Q? F4$\"'r1QF07$$\"5MLLek.pu/BF4$\"'Tw#*F07$$\"5+++]PMnNrDF4$\"(*G(H\"F07 $$\"5ML$eR(\\;m/FF4$\"(iK_\"F07$$\"5nmmT5ll'z$GF4$\"()z/;F07$$\"5++](o /[r7(HF4$\"($pP;F07$$\"5MLLL$eRwX5$F4$\"($>E:F07$$\"5MLLLL$eI8k$F4$\"& Q[%F07$$\"5NLLL$3x%3yTF4$!(/u_$F07$$\"5-++]PfyG7ZF4$!(v4'*)F07$$\"5omm m\"z%4\\Y_F4$!)!=HI\"F07$$\"5+++]P4'4.P&F4$!)IVi8F07$$\"5NLLL$3FGT\\&F 4$!);mP9F07$$\"5qmm;HKp%zh&F4$!)Enr9F07$$\"5++++v$flF47$$\"5OLL Le*)>VB$)F4$\")%=.a%F47$$\"5qmm;H#o)fb%)F4$\")>WHZF47$$\"50++++v`w(e)F 4$\")A<#)\\F47$$\"5SLL$3x1K*>()F4$\");NZ^F47$$\"5qmmmTg()4_))F4$\")%zj 5&F47$$\"5+++]7`aE%)*)F4$\")g$>3&F47$$\"5NLLL$e9Kk6*F4$\")ubL]F47$$\"5 qmm;aQ))f[#*F4$\")]E\")[F47$$\"5.+++DJbw!Q*F4$\")m:PZF47$$\"5NLL$ekGkX #**F4$\")%GA5%F47$$\"5nmmm;/j$o/\"F0$\")ESiOF47$$\"5+++]7GTt%4\"F0$\") O5_NF47$$\"5MLLL3_>jU6F0$\")wCsOF47$$\"5nmm;HdNb'>\"F0$\")\"=q(RF47$$ \"5++++]i^Z]7F0$\")#33^%F47$$\"5+++++v\"=YI\"F0$\")c)o=&F47$$\"5++++]( =h(e8F0$\")c9$='F47$$\"5++++]7!Q4T\"F0$\")@K@xF47$$\"5++++]P[6j9F0$\"* 2h8-\"F47$$\"5nmm\"HKR'\\5:F0$\"*D$z,9F47$$\"5MLL$e*[z(yb\"F0$\"*f*[C> F47$$\"5nm;/Ev&[ge\"F0$\"*HD*RBF47$$\"5+++Dc,#>Uh\"F0$\"*kE$HGF47$$\"5 ML$eky#)*QU;F0$\"*Vs3P$F47$$\"5nmmm;a/cq;F0$\"*nq#QRF47$$\"5nmm;zpYU%p \"F0$\"*+w**Q%F47$$\"5nmmmT&)))G=F0$\")*f7i'F07$$\"5MLLe9; 0?E>F0$\")3`/mF07$$\"5++]i!Rgs2&>F0$\")&\\E['F07$$\"5nmmmm\"pW`(>F0$\" )WkqjF07$$\"5MLLe9TOEH?F0$\")Oe_fF07$$\"5,++]i!f#=$3#F0$\")U`paF07$$\" 5,+++D\"=EX8#F0$\")yO)*\\F07$$\"5,++](=xpe=#F0$\"*,.Fa%F47$$\"5MLLeRA9 WRAF0$\"*+=%*4%F47$$\"5nmmm\"H28IH#F0$\"*o_cm$F47$$\"5nmmTNr&3AM#F0$\" *kaeG$F47$$\"5nmm;zpSS\"R#F0$\"*q(**HHF47$$\"5MLLL3_?`(\\#F0$\"*\"yN/B F47$$\"5MLL3_D1l_DF0$\"*OMu2#F47$$\"5MLL$e*)>pxg#F0$\"*xot\">F47$$\"5o mmm;z+vbEF0$\"*t\"yF=F47$$\"5,++]Pf4t.FF0$\"*G)\\zHT'HF F0$\"*5d!pY\"y#F0$\"*@ nyx\"F47$$\"5MLLLe*Gst!GF0$\"*T'G'z\"F47$$\"5omm;H2\"34'GF0$\"*0&*R'=F 47$$\"5,++++DRW9HF0$\"*Y\"pp>F47$$\"5,+++DJE>>IF0$\"*#[z&G#F47$$\"5,++ ]i!RU07$F0$\"*JeBr#F47$$\"5,+++v=S2LKF0$\"*@0rC$F47$$\"5ommmm\"p)=MLF0 $\"**\\)*pOF47$$\"5MLLLeR%p\")Q$F0$\"*y;n$QF47$$\"5,+++](=]@W$F0$\"*a= y%RF47$$\"5ML$ekyZ2mY$F0$\"*^T2)RF47$$\"5omm\"H#oZ1\"\\$F0$\"*!H'*)*RF 47$$\"5MLe9T8MH.NF0$\"*T'40SF47$$\"5,+]Pfe?_:NF0$\"*9-#3SF47$$\"5omTgx .2vFNF0$\"*enz+%F47$$\"5MLL$e*[$z*RNF0$\"*znM+%F47$$\"5,+]PMFwrmNF0$\" *LX())RF47$$\"5omm\"Hd!fX$f$F0$\"*s'>gRF47$$\"5ML$e9T=%>?OF0$\"*ra3#RF 47$$\"5,+++]iC$pk$F0$\"*(yisQF47$$\"5MLLe*[t\\sp$F0$\"*LD(ePF47$$\"5om m;H2qcZPF0$\"*WzMi$F47$$\"5,++]7.\"fF&QF0$\"*ZA(3LF47$$\"5ommm;/OgbRF0 $\"*hP,,$F47$$\"5,++]ilAFjSF0$\"*+ffv#F47$$\"5omm\"zW7@^6%F0$\"*'\\FjE F47$$\"5NLLLL$)*pp;%F0$\"*k,=f#F47$$\"5NLL$3-$H**>UF0$\"*ST2a#F47$$\"5 NLLL3xe,tUF0$\"*2R<^#F47$$\"5-++v=n(*fDVF0$\"*dZN]#F47$$\"5omm;HdO=yVF 0$\"*))=W^#F47$$\"5MLLe9\"z-lU%F0$\"*1`(RDF47$$\"5,++++D>#[Z%F0$\"*ZN \"yDF47$$\"5ommmT&G!e&e%F0$\"*kO\\q#F47$$\"5NLLLL$)Qk%o%F0$\"*F_c%GF47 $$\"5-++]iSjE!z%F0$\"*sQr*HF47$$\"5-++]P40O\"*[F0$\"*Xt'=JF47$$\"\"&F) $\"*&GW-KF4-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-S hampine~schemeG-F$6%7erF'7$F+$\"'E,FF07$F2$\"'\"Rh$F07$F8$\"'yDLF07$F= $\"'!\\s#F07$FB$\"')HA\"F07$FG$!&%3FF07$FQ$!'$\\/'F07$F_o$!('=g9F07$$ \"5MLLLe*[`HP$F4$!(g/u\"F07$Fdo$!(Jv*>F07$$\"5MLL$3-8>bx$F4$!(#*p0#F07 $$\"5MLLL3xwq4RF4$!(s*)4#F07$$\"5MLL$eRA'*Q/%F4$!(;j4#F07$Fio$!(%[s?F0 7$$\"5ommT5:j=XWF4$!(w+)>F07$F^p$!('e1>F07$$\"5om;/^J'Qe%[F4$!(Da*=F07 $$\"5NLLek.%*Qz\\F4$!(r<\">F07$$\"5++]7yv,%H6&F4$!($f')>F07$Fcp$!(`F.# F07$F]q$!(mZB#F07$Fgq$!($fpBF07$F\\r$!(AgR#F07$Far$!(ssS#F07$Ffr$!(qWN #F07$F[s$!)<8PAF47$F`s$!)hfp5F47$Fes$\"(G07*F47$Fjs$\")9x$4$F47$F_t$\" )dCdUF47$Fdt$\")7l-VF47$Fit$\")z#zR%F47$F^u$\");SQWF47$Fcu$\")40`VF47$ F]v$\")o7!>%F47$Fgv$\")%3,*QF47$F\\w$\")[uTKF47$Faw$\")nY\\FF47$$\"5ML Le9;_yq5F0$\")LU.EF47$Ffw$\")AT0DF47$$\"5nmmT5SIo=6F0$\")5PZCF47$F[x$ \")yNECF47$F`x$\")cRvCF47$Fex$\")$HIl#F47$Fjx$\")*4f%HF47$F_y$\")+\\ZM F47$Fdy$\")#fQN%F47$Fiy$\")s%H'fF47$F^z$\")2_J&)F47$Fcz$\"*YpH?\"F47$F hz$\"*-x$y9F47$F][l$\"*Vliz\"F47$Fb[l$\"*!))[Q@F47$Fg[l$\"*Z.l[#F47$F \\\\l$\"*:\"GfFF47$Fa\\l$\"*\"G&*HIF47$Ff\\l$\"*O'y&G$F47$F[]l$\"*\"o \\7NF47$F`]l$\")F0$\")^yRWF07$Fd^l$\")l+bWF07$$\"5MLekyZ'eI'>F0$\")& \\5Y%F07$Fi^l$\")!*>eWF07$F^_l$\")!f!RVF07$Fc_l$\")vYjSF07$Fh_l$\")%)Q 5PF07$F]`l$\"*$e6HLF47$Fb`l$\"*z%3\\HF47$Fg`l$\"*0%\\,EF47$F\\al$\"*'e &fJ#F47$Faal$\"*H(Gh?F47$$\"5+++v$41oWW#F0$\"*3&\\A=F47$Ffal$\"*sBpi\" F47$F[bl$\"*\\O,Z\"F47$F`bl$\"*O)Qe8F47$Febl$\"*BD_H\"F47$Fjbl$\"*F47$Fgel$\"*&>*)oAF47$F\\fl$\"*:\"RmDF47$Fafl$\"*po8p#F47$Fffl$\" *LdDy#F47$F[gl$\"*'346GF47$F`gl$\"*Ca8$GF47$Fegl$\"*fR\"QGF47$Fjgl$\"* PkF%GF47$F_hl$\"*,q_%GF47$Fdhl$\"*'=xXGF47$Fihl$\"*HE\"RGF47$F^il$\"*& ['H#GF47$Fcil$\"*&>v(z#F47$Fhil$\"*&GYkFF47$F]jl$\"*C[Lo#F47$Fbjl$\"*S >Xe#F47$Fgjl$\"*yWIN#F47$F\\[m$\"*TZp8#F47$Fa[m$\"*U6j&>F47$Ff[m$\"*%= @\"*=F47$F[\\m$\"*_Q8%=F47$F`\\m$\"*T'31=F47$Fe\\m$\"*)zA'y\"F47$Fj\\m $\"*Y#*4y\"F47$F_]m$\"*.@\"*y\"F47$Fd]m$\"*2&H2=F47$Fi]m$\"*8_X$=F47$F ^^m$\"*STS#>F47$Fc^m$\"*;\\K-#F47$Fh^m$\"*w-28#F47$F]_m$\"*\"y0=AF47$F b_m$\"*E4'zAF4-Fg_m6&Fi_mFj_mF(Fj_m-F^`m6#%Ta~minor~modification~of~th e~Bogacki-Shampine~schemeG-F$6%7esF'7$F+$\"'V=;F07$F2$\"'(oI$F07$$\"5+ +]i!*GER37F4$\"'T$f$F07$F8$\"'!Ht$F07$$\"5ML$3_+ZiaW\"F4$\"'&[\"QF07$F =$\"'7JQF07$FB$\"'n2MF07$FG$\"'\\BGF07$FQ$\"&:u&F07$F_o$!'s<5F07$$\"5M LL$3F%\\wQKF4$!&*e%*F07$F]bm$!&R7(F07$$\"5MLL$ek.Ur]$F4$\"%9FF07$Fdo$ \"&[&eF07$Fjbm$\"'JFFF07$Fio$\"'/@WF07$$\"5-+](oHaN;J%F4$\"'\"[l&F07$F gcm$\"'tmhF07$$\"5NL$eRs3P(yXF4$\"',rlF07$F^p$\"')fu'F07$$\"5NL3FWXK1z ZF4$\"'+qmF07$F_dm$\"'D&\\'F07$$\"5-+D\"yv,9E\"\\F4$\"'OugF07$Fddm$\"' >%*eF07$Fidm$\"'n'[%F07$Fcp$\"'dRLF07$Fgq$!'C%Q%F07$F[s$!)#=P>\"F47$$ \"5qmmmm\"HZ_O'F4$!)$4;H\"F47$$\"5++++vVVX$\\'F4$!)cJ;9F47$$\"5qmm;zpy bdlF4$!)\\]C9F47$$\"5NLLL$eRh;i'F4$!);)GZ\"F47$$\"5+++](=#\\w&o'F4$!)Y 1f9F47$F`s$!)eAl9F47$$\"5NLLL3_DG1qF4$!)F)>Q\"F47$Fes$!)-*f>\"F47$Fjs$ !(U`%pF47$F_t$!(T\\R%F47$Fdt$!(NAA%F47$Fit$!(8fB%F47$F^u$!(lQj%F47$Fcu $!(6\"Q[F47$F]v$!(Le*fF47$Fgv$!(jFu'F47$$\"5qmmT&)3\\m_'*F4$!(xJu(F47$ F\\w$!(,$H&)F47$$\"5nmTg_(R^g+\"F0$!(8Vg)F47$$\"5++]iSmjk>5F0$!(4F'))F 47$$\"5MLekGN8CL5F0$!(=U)))F47$Faw$!(A5$))F47$Ffw$!(t2e)F47$F[x$!(B&># )F47$$\"5+++voaFfp6F0$!(UO8)F47$F`x$!(P2<)F47$$\"5MLLe*)fV^B7F0$!(0&[$ )F47$Fex$!(C_j)F47$$\"5++++vomax7F0$!(wI(*)F47$Fjx$!(zhE*F47$$\"5+++]7 GR:=8F0$!(KsP*F47$$\"5++++D\"o*oJ8F0$!(JZK*F47$$\"5+++]PMaAX8F0$!(U2O* F47$F_y$!(j#)=*F47$Fdy$!(w3%oF47$Fiy$!(,h!=F47$F^z$\"(nT_'F47$Fcz$\")v C$f\"F47$Fhz$\")hvvAF47$F][l$\")'>!))HF47$Fb[l$\")fJ!o$F47$Fg[l$\")YXK VF47$Fa\\l$\")l%pJ&F47$F[]l$\")N#4F'F47$$\"5++]ilsqx$z\"F0$\"(&)Q\"oF0 7$F`]l$\"(!eQtF07$$\"5om;aj%e'H\\=F0$\"(M&3yF07$Fe]l$\"(.L@)F07$Fj]l$ \"(!)R_)F07$F_^l$\"(l*>()F07$Fc]n$\"(o))y)F07$Fd^l$\"(lP$))F07$F[^n$\" ((4Y))F07$Fi^l$\"(8\\$))F07$F^_l$\"(Fa[)F07$Fc_l$\"(bax(F07$Fh_l$\"(-H 'pF07$F]`l$\")$[2<'F47$Fb`l$\")FE`aF47$Fg`l$\")u&H$[F47$F\\al$\")3G=VF 47$Faal$\")86QQF47$F[`n$\")'H2P$F47$Ffal$\")1L!)HF47$F[bl$\")65nEF47$F `bl$\")l(>X#F47$$\"5,++D1R'f0uzEF0$\")b$pH#F47$Fjbl$\"),DsAF47$F_cl$\")%e'fAF47$Fdcl$\")jtfAF47 $Ficl$\")TrtAF47$F^dl$\")m&))H#F47$Fcdl$\")r&>R#F47$Fhdl$\")k\"\\`#F47 $F]el$\")tk`HF47$Fbel$\")A(z[$F47$Fgel$\")y3PTF47$F\\fl$\")aVtYF47$Faf l$\")WH.\\F47$Fffl$\")?0s]F47$F[gl$\")(*QC^F47$F`gl$\")+3h^F47$Fjgl$\" )8C\"=&F47$Fdhl$\")-F&=&F47$Fihl$\")@'><&F47$F^il$\")S&39&F47$Fcil$\") Vn$4&F47$Fhil$\")HlK]F47$F]jl$\")2'f)[F47$Fbjl$\")T#*4ZF47$Fgjl$\")[!> I%F47$F\\[m$\")6pZ`$F47$Fc^m$\")Vr ;PF47$Fh^m$\")338RF47$F]_m$\")2^sSF47$Fb_m$\")`i%=%F4-%&COLORG6&Fi_m$ \"\"\"!\"\"$\"\")FcfpF(-F^`m6#%3efficient~scheme~AG-F$6%7jrF'7$$\"5NLL L$3FWYs#F-$\"'uZ7F07$F+$\"'uoCF07$$\"5SLLL3x1h6oF-$\"'1JFF07$$\"50+++] 7G$R<)F-$\"'3OGF07$$\"5qmmm\"z%\\DO&*F-$\"'JCFF07$F2$\"'!oH#F07$F=$!&5 $eF07$FG$!'\">v&F07$FQ$!(*Q76F07$F_o$!(i!)R\"F07$Fdo$!(;/K\"F07$Fio$!( 99<\"F07$Fg\\o$!(2^;\"F07$Fgcm$!(9#z6F07$F_]o$!(W$>7F07$F^p$!(%[:8F07$ Fddm$!(MG[\"F07$Fcp$!(Lbv\"F07$F]q$!(u.-#F07$Fgq$!(fL=#F07$F\\r$!((o8A F07$Far$!(euB#F07$Ffr$!(HiA#F07$F[s$!)\"p%[@F47$F`s$!)B>>9F47$Fes$!(hy e#F47$$\"5NLLLe*[!)y_(F4$\"(hZ\"RF47$Fjs$\"(SZb)F47$$\"50+++Dc\"[#e!)F 4$\")F47$Fgco$\"(ora\"F47$F\\do$\"($e?5F47$Fado$\"'piwF47$Faw$\"'Z =kF47$Ffw$\"'pL')F47$F[x$\"(t2$=F47$Fex$\"(#**fUF47$Fjx$\"(SDs&F47$F_y $\"(5Ka)F47$Fdy$\")j,@9F47$Fiy$\")\"[RN#F47$F^z$\")GtCPF47$Fcz$\")_T1a F47$Fhz$\")BJqmF47$F][l$\")4L(4)F47$Fb[l$\")oHU'*F47$Fg[l$\"*p]u7\"F47 $F\\\\l$\"*V!)3E\"F47$Fa\\l$\"**\\]-9F47$Ff\\l$\"*N)y]:F47$F[]l$\"*4rb o\"F47$Fgio$\")'p?%=F07$F`]l$\")0$=)>F07$F_jo$\")Fr&4#F07$Fe]l$\")lk$= #F07$Fj]l$\")mFOAF07$F_^l$\")AdlAF07$Fc]n$\")m/qAF07$Fd^l$\")_TmAF07$F [^n$\")\"Q9E#F07$Fi^l$\")DN^AF07$F^_l$\")%eq9#F07$Fc_l$\")%*=))>F07$Fh _l$\")2c9=F07$F]`l$\"*,2qj\"F47$Fb`l$\"*'*R\"f9F47$Fg`l$\"*z-\")G\"F47 $F\\al$\"*(*y/9\"F47$Faal$\"*;!)\\+\"F47$F[`n$\")g]$y)F47$Ffal$\")iEix F47$F[bl$\")g3qpF47$F`bl$\")sWOkF47$F]^p$\")^5xiF47$Febl$\")h<_hF47$Fe ^p$\")\"3w1'F47$Fjbl$\")d46gF47$F_cl$\")yH()fF47$Fdcl$\")44$*fF47$Ficl $\")TeMgF47$F^dl$\")a8-hF47$Fcdl$\")$H)QjF47$Fhdl$\")A#Qp'F47$F]el$\") dL?xF47$Fbel$\")#ol2*F47$Fgel$\"*SE:3\"F47$F\\fl$\"*O2wA\"F47$Fafl$\"* !R!*)G\"F47$Fffl$\"*q$fK8F47$F[gl$\"*T]eM\"F47$F`gl$\"*#)>ZN\"F47$Fegl $\"*lLwN\"F47$Fjgl$\"*O\"\\f8F47$F_hl$\"*]t-O\"F47$Fdhl$\"*8%))f8F47$F ihl$\"*g))[L\"F47$Fhil$\"*'>z=8F47 $F]jl$\"*>M,G\"F47$Fbjl$\"*WlNB\"F47$Fgjl$\"**>VC6F47$F\\[m$\"*Qi7-\"F 47$Fa[m$\")R&>M*F47$Ff[m$\")M!y-*F47$F[\\m$\")MU(y)F47$F`\\m$\")n$yh)F 47$Fe\\m$\")]sA&)F47$Fj\\m$\")=r(\\)F47$F_]m$\"):VO&)F47$Fd]m$\")(eGi) F47$Fi]m$\")2K_()F47$F^^m$\"))ou<*F47$Fc^m$\")&***\\'*F47$Fh^m$\"*hkj, \"F47$F]_m$\"*Hx\"e5F47$Fb_m$\"*:mv3\"F4-Fg_m6&Fi_mFj_mF(F(-F^`m6#%3ef ficient~scheme~BG-F$6%7fsF'7$F+$\"'$))4#F07$F2$\"'d$[$F07$Fein$\"'^vMF 07$F8$\"'mhLF07$F]jn$\"'yYJF07$F=$\"'08GF07$FB$\"'@k7F07$FG$!&\\j\"F07 $FQ$!'pyTF07$F_o$!'AVgF07$Fdo$!'**>KF07$Fio$\"'fIAF07$Fgcm$\"'_JYF07$F ^p$\"'F?cF07$Fg]o$\"'\\mbF07$F_dm$\"'PaaF07$F_^o$\"'T%3&F07$Fddm$\"'wB \\F07$Fidm$\"'<\\NF07$Fcp$\"'p7CF07$Fgq$!'t/aF07$F[s$!))4yI\"F47$Fc_o$ !)d$eS\"F47$Fh_o$!)qSL:F47$$\"5NLL3x1h]DlF4$!)BRd:F47$F]`o$!)#\\:a\"F4 7$$\"5+++D\"Gj4'*e'F4$!)xyi:F47$Fb`o$!)*3Af\"F47$Fg`o$!)W%pd\"F47$F`s$ !)^v$e\"F47$F_ao$!)i/.:F47$Fes$!)VO;8F47$Fjs$!()eb$)F47$F_t$!(F%ziF47$ Fdt$!(MU?'F47$Fit$!(W8S'F47$$\"5qmmTN@([Ql)F4$!(R`e'F47$F^u$!(#[2qF47$ $\"50++D19a,'y)F4$!(z#**pF47$Fcu$!(HCI(F47$F]v$!((=3))F47$Fgv$!(T*z'*F 47$F_co$!)A[!3\"F47$F\\w$!)^3i6F47$Fgco$!)(HS;\"F47$F\\do$!)E0'=\"F47$ Fado$!)q(4=\"F47$Faw$!)<9n6F47$Ffw$!)!o(36F47$F[x$!)*>(R5F47$F_eo$!),- ;5F47$F`x$!)CF45F47$Fgeo$!)16:5F47$Fex$!):=J5F47$F_fo$!)qLc5F47$Fjx$!) #o\"p5F47$F\\go$!)T-_5F47$F_y$!)Di<5F47$Fdy$!(f>H(F47$Fiy$!(+c2#F47$F^ z$\"(Dyn&F47$Fcz$\")3@s8F47$Fhz$\")&)fE>F47$F][l$\")9M!\\#F47$Fb[l$\") P*4/$F47$Fg[l$\")SC&e$F47$F\\\\l$\")1g@SF47$Fa\\l$\")!zP\\%F47$Ff\\l$ \")M>B]F47$F[]l$\")#))H_&F47$Fgio$\"(L&ohF07$F`]l$\"(*y-oF07$F_jo$\"(* RZtF07$Fe]l$\"(%='z(F07$Fj]l$\"(647)F07$F_^l$\"(&4*H)F07$$\"5+]Pf3QNMK >F0$\"(?(H$)F07$Fc]n$\"(s([$)F07$$\"5M$e9m>eHY%>F0$\"(`CO)F07$Fd^l$\"( vIO)F07$F[^n$\"()p`$)F07$Fi^l$\"(4)>$)F07$F^_l$\"(M'fyF07$Fc_l$\"(0h6( F07$Fh_l$\"(dxM'F07$F]`l$\")T$)RcF47$Fb`l$\")&*R;]F47$Fg`l$\")i1oWF47$ F\\al$\")vM'*RF47$Faal$\")7gUNF47$Ffal$\")'GVs#F47$F[bl$\")g)4V#F47$F` bl$\")jVOAF47$F]^p$\")N&)y@F47$Febl$\")LsM@F47$Fe^p$\")nK0@F47$Fjbl$\" )lM'3#F47$F_cl$\")1Yz?F47$Fdcl$\")l6$3#F47$Ficl$\")+O+@F47$F^dl$\")0JE @F47$Fcdl$\")<&y@#F47$Fhdl$\")Hx^BF47$F]el$\")6mIFF47$Fbel$\")6y/KF47$ Fgel$\")cq$z$F47$F\\fl$\")`$yH%F47$Fafl$\")Ut:XF47$Fffl$\")h$Qn%F47$F[ gl$\")Wr@ZF47$F`gl$\")C.aZF47$Fegl$\")I$F47$Ff[m$\")%4>>$F47$F[\\m$\")DG2 JF47$F`\\m$\")IF 47$Fd]m$\")Mu]IF47$Fi]m$\")#oq4$F47$F^^m$\")*QzC$F47$Fc^m$\")pU9MF47$F h^m$\")*o[f$F47$F]_m$\")#R>u$F47$Fb_m$\")gZXQF4-Fg_m6&Fi_m$\")#)eqkF\\ `m$\"))eqk\"F\\`mFbjs-F^`m6#%3efficient~scheme~CG-F$6%7arF'7$F]gp$!'lN 6F07$F+$!'a*Q\"F07$Fjgp$!&K;#F07$F2$\"'aDR\"*F47$Fcu$!) 'G#G!*F47$Fhu$!)g'=$*)F47$F]v$!)%H!)*))F47$Fbv$!)'yao)F47$Fgv$!)T`=&)F 47$F\\w$!)&)e%*yF47$Faw$!)]K&G(F47$Ffw$!)M`kpF47$F[x$!)Wz\\oF47$F_eo$! )lv$)oF47$F`x$!)phzpF47$Fgeo$!)^2zrF47$Fex$!)T3nuF47$Fjx$!)T?.$)F47$F_ y$!)#\\Re*F47$Fdy$!*yA98\"F47$Fiy$!*^byO\"F47$F^z$!*;h*e;F47$Fcz$!*ngm .#F47$Fhz$!*=,=J#F47$F][l$!*)o(yi#F47$Fb[l$!*'*47)HF47$Fg[l$!*onWO$F47 $F\\\\l$!*e(G$p$F47$Fa\\l$!*!4;OSF47$Ff\\l$!*@lPQ%F47$F[]l$!*<<0r%F47$ Fgio$!)k(\\2&F07$F`]l$!)s6+aF07$F_jo$!)[cucF07$Fe]l$!)1u$*eF07$Fj]l$!) &))e.'F07$F_^l$!)avHhF07$Fc]n$!)9%p:'F07$Fd^l$!)MkqhF07$F[^n$!)y6uhF07 $Fi^l$!)/WlhF07$F^_l$!)R4%*fF07$Fc_l$!)Xd[cF07$Fh_l$!)kI5_F07$F]`l$!*> %4%F47$Fg`l$!*#Hf(o$F47$F\\al$!*&43kKF47$Faal$!*a%4 \"*GF47$F[`n$!*1Q6b#F47$Ffal$!*R=)zAF47$F[bl$!*pur1#F47$F`bl$!*NIi\">F 47$Febl$!*q[7$=F47$Fjbl$!*M`Zy\"F47$F_cl$!*.hWx\"F47$Fdcl$!*ZnNx\"F47$ Ficl$!*ZCEy\"F47$F^dl$!*+(e+=F47$Fcdl$!*1rk'=F47$Fhdl$!*Gl'p>F47$F]el$ !*wXWF#F47$Fbel$!*!4s\"o#F47$Fgel$!*Cv??$F47$F\\fl$!*h&\\NOF47$Fafl$!* QUv\"QF47$Fffl$!*GV\"\\RF47$F[gl$!*hW(*)RF47$F`gl$!*/>#RF47$F]jl$!*\"QW2QF47$Fbjl$!*&>$ym$F 47$Fgjl$!*QW!RLF47$F\\[m$!*RY1.$F47$Fa[m$!*R\")Hx#F47$Ff[m$!*;%R!o#F47 $F[\\m$!*))p&4EF47$F`\\m$!*lb&fDF47$Fe\\m$!*hw8`#F47$Fj\\m$!*(4*Q_#F47 $F_]m$!*]-``#F47$Fd]m$!*\\R4c#F47$Fi]m$!*]c%*f#F47$F^^m$!*Bjis#F47$Fc^ m$!*qFB$F4-F_fp6&Fi _m$\"\"'FcfpF(Fdfp-F^`m6#%Sscheme~with~stability~radius~=~imag~axis~in clusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~7~sta ge~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F^bu-%%VIEWG6$ ;F(Fb_m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modification of the B ogacki-Shampine scheme" "efficient scheme A" "efficient scheme B" "eff icient scheme C" "scheme with stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 2 of 7 stage, \+ order 5 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF( " }{TEXT -1 10 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"! \"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/% \"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code c onstructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based o n each of the methods and gives the " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 760 "G := (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bog acki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`effic ient scheme C`,`scheme with stability radius = imag axis inclusion`]: \nerrs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 do\n Gn_RK5_||ct := RK5_||ct(G(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn_RK5_||ct):\n for ii to numpts do\n sm \+ := sm+(Gn_RK5_||ct[ii,2]-g(Gn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G- %!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)p print156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+#RtNV$!#A 7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+6lUZ5F+7$% 3efficient~scheme~AG$\"+c$>@n\"!#B7$%3efficient~scheme~BG$\"+kB_t6F+7$ %3efficient~scheme~CG$\"+m(4#)e#F+7$%Sscheme~with~stability~radius~=~i mag~axis~inclusionG$\"+.=j5_F4Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 690 "G : = (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix ([[`slope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-S hampine scheme`,`a minor modification of the Bogacki-Shampine scheme`, `efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]: errs := []:\nDigits := 30:\nfor ct to 6 do\n gn_RK5_||ct := RK5_||ct(G(x,y),x,y,x0,y0,hh,n umsteps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := ev alf(g(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(gn_RK5_||ct(xx)- gxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~ G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~widt h:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint196\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$%8Bogacki-Shampine~schemeG$\"+&y%3[z!#B7$%Ta~minor~modification ~of~the~Bogacki-Shampine~schemeG$\"+#4h/%RF+7$%3efficient~scheme~AG$\" +sgmr\\!#D7$%3efficient~scheme~BG$\"+cCV9WF+7$%3efficient~scheme~CG$\" +JIIi'*F+7$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+'* 4,A6F+Q)pprint206\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sh ampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sc heme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs : = []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 do\n sm := NCint((g(x)-'gn_RK5_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,fa ctor=100);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+I02DM!#A7 $%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+Qe(\\/\"F+7 $%3efficient~scheme~AG$\"+0Otn;!#B7$%3efficient~scheme~BG$\"+VvTq6F+7$ %3efficient~scheme~CG$\"+CeE\"e#F+7$%Sscheme~with~stability~radius~=~i mag~axis~inclusionG$\"+$)o['>&F4Q)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "evalf[20](plot(['gn_RK5_1' (x)-g(x),'gn_RK5_2'(x)-g(x),'gn_RK5_3'(x)-g(x),'gn_RK5_4'(x)-g(x),\n'g n_RK5_5'(x)-g(x),'gn_RK5_6'(x)-g(x)],x=0..10,font=[HELVETICA,9],\ncolo r=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nleg end=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sh ampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sc heme C`,`scheme with stability radius = imag axis inclusion`],\ntitle= `error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 822 552 552 {PLOTDATA 2 "6,-%'CURVESG6%7_p7$$\"\"!F) F(7$$\"5ILLL3x1h6o!#A$\"#7$$\"5mmmmTN@Ki8!#@$\"%z5F07$$\"5******\\ 7.K[V?F4$\"&gA\"F07$$\"5KLLL$3FWYs#F4$\"&$foF07$$\"5lmm;aQ`!eS$F4$\"'U -EF07$$\"5)******\\iSmp3%F4$\"'->xF07$$\"5KLL$eRZF\"oZF4$\"(n5$>F07$$ \"5lmmmmT&)G\\aF4$\"(a$fDF07$$\"5+++]P4'\\/8'F4$\"('oeDF07$$F,F4$\"($) Gc#F07$$\"5lmm;zW!z'F07$$\"5LLLLe*)4D2>F_p$\"(#pasF07$$\"5mmmmm;arz@F_p$\"(NxJ)F07$$ \"5+++D\"yD&y;CF_p$\"((y6()F07$$\"5LLL$e*)4bQl#F_p$\"(BoV*F07$$\"5mmmT 5S\\#4*GF_p$\"(Z>g*F07$$\"5++++D\"y%*z7$F_p$\")X8<5F07$$\"5mmm;ajW8-OF _p$\")\"zo0\"F07$$\"5LLLL$e9ui2%F_p$\")V(*o5F07$$\"5++++voMrU^F_p$\")a XI5F07$$\"5mmmmm\"z_\"4iF_p$\"(KQ]*F07$$\"5lmmmmm6m#G(F_p$\"(bRe)F07$$ \"5lmmmmT&phN)F_p$\"(`'*p(F07$$\"5++++v=ddC%*F_p$\"(`S\"pF07$$\"5LLLLe *=)H\\5F0$\"(WiB'F07$$\"5++++v=JN[6F0$\"(Cpw&F07$$\"5mmmm\"z/3uC\"F0$ \"(N#p`F07$$\"5++++DJ$RDX\"F0$\"($)zq%F07$$\"5mmmm\"zR'ok;F0$\"(kH?%F0 7$$\"5++++D1J:w=F0$\"(]F!QF07$$\"5LLLLL3En$4#F0$\"(4YY$F07$$\"5mmmm;/R E&G#F0$\"(\\R@$F07$$\"5+++++D.&4]#F0$\"('[qHF07$$\"5+++++vB_F07$$\"5*******\\7=lj;%F0$\"(D E'=F07$$\"5*******\\PaR&z+Y2aF0$\"(K6X\"F07$$\"5mmmm;zXu9 cF0$\"(I#*R\"F07$$\"5**********\\y))GeF0$\"(<$\\8F07$$\"5********\\i_Q QgF0$\"(rPI\"F07$$\"5*******\\7y%3TiF0$\"(-DE\"F07$$\"5********\\P![hY 'F0$\"(,'>7F07$$\"5KLLLL$Qx$omF0$\"(RM=\"F07$$\"5*********\\P+V)oF0$\" ()4Z6F07$$\"5mmmm\"zpe*zqF0$\"(Eg6\"F07$$\"5*********\\#\\'QH(F0$\"(4R 3\"F07$$\"5KLLLe9S8&\\(F0$\"(<`0\"F07$$\"5*******\\i?=bq(F0$\"(()p-\"F 07$$\"5KLLLL3s?6zF0$\"(62+\"F07$$\"5*******\\7`Wl7)F0$\"'&fu*F07$$\"5l mmmmm*RRL)F0$\"'!p]*F07$$\"5lmmm;a<.Y&)F0$\"'=u#*F07$$\"5KLLLe9tOc()F0 $\"'Ja!*F07$$\"5**********\\Qk\\*)F0$\"'@h))F07$$\"5KLLL$3dg6<*F0$\"'t \\')F07$$\"5lmmmmmxGp$*F0$\"'*)o%)F07$$\"5)******\\7oK0e*F0$\"'>%G)F07 $$\"5)******\\(=5s#y*F0$\"'x9\")F07$$\"#5F)$\"&-%z!#=-%'COLOURG6&%$RGB GF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7crF'7$F 2$!$W\"F07$F=$!%P&*F07$FB$!&9u$F07$FG$!'$[:\"F07$FL$!'(Q-$F07$FQ$!'TvS F07$FV$!'FuSF07$Fen$!'W!3%F07$Fin$!'zITF07$F^o$!'(HM%F07$Fho$!'k!*F07$Fhp$!'A;!*F07$F]q$!')p.*F07$Fbq$!'7l\"*F07$Fgq$!'\"zi*F 07$F[r$!(`N4\"F07$F`r$!(f\"49F07$Fer$!(vvT\"F07$Fjr$!(hhT\"F07$$\"5LLL 3FpE!Hq\"F_p$!(wdT\"F07$$\"5mmm;/w(=5x\"F_p$!(S7U\"F07$$\"5+++D\"G)[8R =F_p$!(HxW\"F07$F_s$!(@D`\"F07$$\"5mmmTN'4n`(>F_p$!(HOv\"F07$$\"5+++]7 .K[V?F_p$!(?5*=F07$$\"5LLLe*)4$*f6@F_p$!(s%))=F07$Fds$!(!>')=F07$$\"5+ +DJ&p(G)*QAF_p$!(:g)=F07$$\"5LL$eRsL]#)H#F_p$!(-E*=F07$$\"5mmTg_(zF07$Fis$!(B1*>F07$$\"5LLe*)4=F0wCF_p$!(O'f@F07$$\"5mm;aQy,KN DF_p$!('*=F#F07$$\"5++v=nQwe%f#F_p$!(d'oAF07$F^t$!((RlAF07$Fct$!(F,I#F 07$Fht$!(/P_#F07$$\"5mm;H#=qHlC$F_p$!(o^^#F07$$\"5LLLeRAY1lLF_p$!(U'=D F07$$\"5++](oHa*f$[$F_p$!(>Xi#F07$F]u$!(nGl#F07$$\"5LL$e9TQp1s$F_p$!(D ?k#F07$$\"5+++vo/V?RQF_p$!(.$HEF07$$\"5mm;/ED#Rx&RF_p$!(&*)REF07$Fbu$! ()4jEF07$$\"5mmm;H2Q\\4YF_p$!(]:d#F07$Fgu$!(jIT#F07$$\"5NLL$3-8Lfn&F_p $!(lx@#F07$F\\v$!(3>,#F07$$\"5lmmm;zp!fu'F_p$!(S_\"=F07$Fav$!(tyF_p$!(E$*\\\"F07$Ffv$!(<7R\"F07$$\"5ILL$3-js.*))F_p$!(e zJ\"F07$F[w$!(/tF\"F07$$\"5lm;a)3\\E\"e&*F_p$!(,EF\"F07$$\"5ILL3-jsn\" p*F_p$!(6VE\"F07$$\"5++]i:N!G_#)*F_p$!(!**e7F07$$\"5lmm;H2)y(e**F_p$!( OkE\"F07$$\"5LL3F%z&HB45F0$!(HvE\"F07$$\"5++]i:N!)eA5F0$!(R'f7F07$$\"5 L$3-jPdl#H5F0$!(osD\"F07$$\"5mm\"zpB6Vf.\"F0$!(C\"e7F07$$\"5+]il(4l?E/ \"F0$!(Y^E\"F07$F`w$!(NAG\"F07$Few$!(OZK\"F07$Fjw$!(%)4Q\"F07$$\"5LLLL e*ot*\\8F0$!(q\\X\"F07$F_x$!(j2]\"F07$$\"5mmmm\"z4wb]\"F0$!(Xo^\"F07$$ \"5LLLLekGhe:F0$!(f\"H:F07$$\"5++++DJ'\\;h\"F0$!(9y`\"F07$Fdx$!(8Ia\"F 07$$\"5+++++vIb</x\"F0$!(`Va\"F07$$\"5mmmm; HkGB=F0$!($*4a\"F07$Fix$!(v``\"F07$$\"5mmm;HdG\"\\)>F0$!(4y^\"F07$F^y$ !(Wt\\\"F07$Fcy$!(GBW\"F07$Fhy$!(JzP\"F07$F]z$!(jOI\"F07$Fbz$!(7bB\"F0 7$Fgz$!(%ex6F07$F\\[l$!(XA6\"F07$Fa[l$!(,71\"F07$Ff[l$!(yn+\"F07$F[\\l $!'x?'*F07$F`\\l$!'Ln\"*F07$Fe\\l$!'(4x)F07$Fj\\l$!'m*Q)F07$F_]l$!'0m! )F07$Fd]l$!'dUxF07$Fi]l$!'\\JuF07$F^^l$!'fzrF07$Fc^l$!'?DpF07$Fh^l$!'0 !o'F07$F]_l$!'6ckF07$Fb_l$!'&HD'F07$Fg_l$!'cTgF07$F\\`l$!'%F07$Fbdl$!'+2TF07$Fgdl$!'0BS F07$F\\el$!&l$RF`el-Fbel6&FdelFeelF(Feel-Fiel6#%Ta~minor~modification~ of~the~Bogacki-Shampine~schemeG-F$6%7jqF'7$FQ$!&00#F07$Fcp$!&hc*F07$Fh p$!&Ec*F07$F]q$!&tf*F07$Fbq$!&<$)*F07$Fgq$!'*p2\"F07$F[r$!')\\O\"F07$F `r$!')o5#F07$Fer$!'WG@F07$Fjr$!'ME@F07$F\\jl$!'aE@F07$Fajl$!'^T@F07$Ff jl$!'S9AF07$F_s$!'fdCF07$F^[m$!'[:JF07$Fc[m$!'*R`$F07$Fh[m$!'DHNF07$Fd s$!'_DNF07$Fe\\m$!'bcNF07$Fis$!'VBRF07$Fb]m$!'U[XF07$Fg]m$!'^l\\F07$F \\^m$!'We\\F07$F^t$!'j^\\F07$$\"5mm\"zW#fD78FF_p$!'oZ\\F07$$\"5++]7`>+ RsFF_p$!'(p&\\F07$$\"5LL3x\")zulJGF_p$!'X3]F07$Fct$!'rn^F07$$\"5++D1R+ C>]HF_p$!'[jbF07$$\"5LL$3x1')f%4IF_p$!'^SiF07$$\"5mmTN'4KF(oIF_p$!'BIi F07$Fht$!'!*>iF07$$\"5LLek`TAE(=$F_p$!'d5iF07$Fj^m$!'@2iF07$$\"5++v$4@ ;(z0LF_p$!'UDiF07$F__m$!'\"GI'F07$Fd_m$!'K0qF07$F]u$!'b%>(F07$F\\`m$!' 9qrF07$Fa`m$!'&Q>(F07$Ff`m$!'\\8vF07$Fbu$!'F-yF07$$\"5mm;z>h!z&4UF_p$! '.nxF07$$\"5+++DcwR)GM%F_p$!'CixF07$$\"5LL$3F>*))=wWF_p$!'JezF07$F^am$ !'L7!)F07$$\"5LLL3-QO5w[F_p$!'gYzF07$Fgu$!'#f(yF07$Ffam$!';huF07$F\\v$ !'k[oF07$$\"5lmmmT&))HvZ'F_p$!'UijF07$F^bm$!'t:hF07$$\"5lmmm\"H2%G9qF_ p$!'12aF07$Fav$!']I`F07$$\"5lmmmTg#Q5b(F_p$!'FCYF07$Ffbm$!'sVXF07$$\"5 lmmm\"zW#z(3)F_p$!'x%)QF07$Ffv$!'X*y$F07$F^cm$!'U\"4$F07$F[w$!'6dCF07$ Fedm$!'2))=F07$F`w$!',\"Q\"F07$Few$!&Su)F07$Fjw$!&3V&F07$F\\gm$!&n,$F0 7$F_x$!&M!>F07$Figm$!&\">#F07$Fis$!'\"[I#F07$F^t$!'i9EF07$Fct$!'(ee#F07$F ht$!'OTDF07$F]u$!'J`=F07$Fbu$!&!)H&F07$F^am$\"'zf8F07$Fgu$\"'vyOF07$$ \"5qmm\"z%*HB$4aF_p$\"'=BVF07$Ffam$\"''RE'F07$$\"5+++v$4'HaUfF_p$\"'IF uF07$F\\v$\"'e^*)F07$F^bm$\"(x)f6F07$Fav$\"(b)49F07$Ffbm$\"(Q\"Q;F07$F fv$\"(-8%=F07$F^cm$\"(hu,#F07$F[w$\"(f!o@F07$Fedm$\"(_`H#F07$F`w$\"(/A S#F07$$\"5mmmm;ac#))4\"F0$\"())zV#F07$Few$\"(]&eCF07$$\"5mm;H#=0WX:\"F 0$\"(RzY#F07$$\"5LLLe*[)\\tg6F0$\"(%RgCF07$$\"5++](oz\"f#p;\"F0$\"(\"F0$\"(0EW#F07$$\"5LLLLL$e!)y >\"F0$\"(OiY#F07$$\"5++]iS;:2/7F0$\"($HwCF07$$\"5mmm\"z%\\CE57F0$\"(v( oCF07$$\"5LL$3_DQ`k@\"F0$\"(78Y#F07$$\"5+++]i:VkA7F0$\"(tSX#F07$$\"5mm ;zp[_$)G7F0$\"($oZCF07$$\"5LLL3x\"=E]B\"F0$\"(mQW#F07$$\"5++]P%[6<7C\" F0$\"(*\\YCF07$Fjw$\"(.KY#F07$$\"5LLL3_+%=QD\"F0$\"(`FZ#F07$$\"5+++]7` (G-E\"F0$\"(V]Y#F07$$\"5mmm\"Hd5RmE\"F0$\"(!RdCF07$$\"5LLLLLe%\\IF\"F0 $\"(\\*\\CF07$$\"5+++v$4\")f%z7F0$\"(JLW#F07$$\"5mmm;aj,(eG\"F0$\"(P#R CF07$$\"5LLLe9;0G#H\"F0$\"(S:W#F07$$\"5++++vo3p)H\"F0$\"(?zX#F07$$\"5L LL$eRd6:J\"F0$\"(Y/X#F07$$\"5mmmm;zALC8F0$\"(McV#F07$$\"5LLL3xJEuI8F0$ \"(7#HCF07$$\"5+++]P%)H:P8F0$\"(?`U#F07$$\"5mmm\"zpLjNM\"F0$\"(VwU#F07 $F\\gm$\"(>LW#F07$$\"5mmmmT5lD,9F0$\"(dbT#F07$F_x$\"(YLQ#F07$Fdx$\"(DU A#F07$Fix$\"(*Re?F07$F^y$\"(Q))*=F07$Fcy$\"(w6x\"F07$Fhy$\"(0Lk\"F07$F ]z$\"(6'H:F07$Fbz$\"(DGV\"F07$Fgz$\"(%Ha8F07$F\\[l$\"(@3F\"F07$Fa[l$\" (8u?\"F07$Ff[l$\"(\")39\"F07$F[\\l$\"(=w3\"F07$F`\\l$\"(fW.\"F07$Fe\\l $\"'I$))*F07$Fj\\l$\"'(GW*F07$F_]l$\"'\"42*F07$Fd]l$\"'f+()F07$Fi]l$\" '0Y$)F07$F^^l$\"'xf!)F07$Fc^l$\"']rxF07$Fh^l$\"'G%\\(F07$F]_l$\"'JTsF0 7$Fb_l$\"'57qF07$Fg_l$\"'\"Qx'F07$F\\`l$\"''Hd'F07$Fa`l$\"'6rjF07$Ff`l $\"'`)>'F07$F[al$\"'9?gF07$F`al$\"'MheF07$Feal$\"'*Rq&F07$Fjal$\"'0ebF 07$F_bl$\"'*HT&F07$Fdbl$\"'A!G&F07$Fibl$\"'(4:&F07$F^cl$\"'&)G]F07$Fcc l$\"'g@\\F07$Fhcl$\"'9/[F07$F]dl$\"'q.ZF07$Fbdl$\"'6,YF07$Fgdl$\"',2XF 07$F\\el$\"&,T%F`el-Fbel6&FdelFeelF(F(-Fiel6#%3efficient~scheme~BG-F$6 %7hqF'7$F=$!$y\"F07$FQ$FghoF07$Fen$!#YF07$F^o$!%L@F07$Fho$!&RA#F07$Fcp $!&K>%F07$Fjr$!'RB5F07$Fds$!'l^9F07$Fis$!'QI9F07$F^t$!'T=8F07$Fgfn$!'q .8F07$Fct$!'/]6F07$Fdgn$!&U\\)F07$Fign$!&;.$F07$F^hn$!&m-$F07$Fht$!&/- $F07$F]u$\"'I$y\"F07$Fbu$\"'8g\\F07$F^am$\"'l!3*F07$Fgu$\"(^.R\"F07$Ff am$\"($=8>F07$F\\v$\"(\")oW#F07$F^bm$\"(ib'HF07$Fav$\"(Z.X$F07$Ffbm$\" (a#*)QF07$Ffv$\"(BpF%F07$F^cm$\"(X3h%F07$F[w$\"(^X*[F07$Fedm$\"('4L^F0 7$F`w$\"(eDL&F07$$\"5++]ilA\"*[b5F0$\"(`\"Q`F07$$\"5mmm\"Hd0!oh5F0$\"( !p@`F07$$\"5LL$3-)))4(y1\"F0$\"(>aI&F07$$\"5+++](=#>1u5F0$\"(9,H&F07$$ \"5mm;z%\\&GD!3\"F0$\"(S&y_F07$$\"5LLL3-)yVk3\"F0$\"(q!y_F07$$\"5++]P4 @Zj#4\"F0$\"(**\\I&F07$Fe]p$\"(/5R&F07$$\"5LL$eRse;]5\"F0$\"(oXS&F07$$ \"5+++DJ?v?66F0$\"(dzQ&F07$$\"5mm;aQ`%)R<6F0$\"(/:P&F07$$\"5LLL$ekQ*eB 6F0$\"($yb`F07$$\"5++]7`>.yH6F0$\"(BHM&F07$$\"5mmmTg_7(f8\"F0$\"(\"oQ` F07$$\"5LL$3xc=i@9\"F0$\"(LfN&F07$Few$\"(\"*)>aF07$F]^p$\"(QwV&F07$Fb^ p$\"(95U&F07$Fg^p$\"(@XS&F07$F\\_p$\"('e)Q&F07$Fa_p$\"(L[P&F07$Ff_p$\" (NxO&F07$F[`p$\"(1uP&F07$F``p$\"(8QU&F07$Fe`p$\"(QIW&F07$Fj`p$\"(6lU&F 07$F_ap$\"((45aF07$Fdap$\"(5TR&F07$Fiap$\"(#yz`F07$F^bp$\"(t0P&F07$Fcb p$\"(%Ru`F07$Fjw$\"(')pS&F07$F^ep$\"(UiQ&F07$F\\gm$\"(auM&F07$F_gp$\"( PCG&F07$F_x$\"(6*3_F07$Fdx$\"(6h&[F07$Fix$\"(NT\\%F07$F^y$\"($RZTF07$F cy$\"(\"**pQF07$Fhy$\"(c@f$F07$F]z$\"(\"eWLF07$Fbz$\"(fO8$F07$Fgz$\"(= C'HF07$F\\[l$\"(D-y#F07$Fa[l$\"(cF0 7$Fd]l$\"(BU!>F07$Fi]l$\"(em#=F07$F^^l$\"(@Sw\"F07$Fc^l$\"(X4q\"F07$Fh ^l$\"(#GS;F07$F]_l$\"(C\\e\"F07$Fb_l$\"(jZ`\"F07$Fg_l$\"(;E[\"F07$F\\` l$\"(g'Q9F07$Fa`l$\"('[%R\"F07$Ff`l$\"(:nN\"F07$F[al$\"(twJ\"F07$F`al$ \"(ShOF_p$\"'O^oF07$F\\`m$\"'ABoF07$$\"5mmT5 SWo$*zPF_p$\"'R[nF07$Fa`m$\"'IWlF07$Ff`m$\"'(*H\\F07$Fbu$\"'2\"[$F07$$ \"5++Dc^.m#H9%F_p$\"'\\rMF07$Fhjn$\"'M\\MF07$$\"5LL3-))=:BwUF_p$\"'5rL F07$F][o$\"'s>JF07$Fb[o$\"&\\g)F07$F^am$!%7lF07$$\"5LLeR(\\EYhn%F_p$!% SqF07$$\"5++]ilA()zUZF_p$!&N0\"F07$$\"5mmT&Q.=^%4[F_p$!&fW#F07$Fj[o$!& Eg'F07$$\"5***\\7.d4cF%\\F_p$!'7(o\"F07$$\"5lm;aQ`&3%4]F_p$!'g)[$F07$$ \"5IL3x1651w]F_p$!'IzMF07$Fgu$!'_rMF07$F][p$!'QxVF07$Ffam$!'iNlF07$Fe[ p$!'Z*z(F07$F\\v$!'\"z8*F07$F^bm$!(_H7\"F07$Fav$!(N9G\"F07$Ffbm$!(`GR \"F07$Ffv$!(UCY\"F07$$\"5++vVtF\\%HU)F_p$!(BcZ\"F07$$\"5IL$3-QJ?(*[)F_ p$!(W_]\"F07$$\"5lm\"zp)*p&\\c&)F_p$!(=v]\"F07$$\"5+++v$f3rKi)F_p$!(qD ]\"F07$$\"5lm;H2e=#ov)F_p$!(tR\\\"F07$F^cm$!($f&\\\"F07$$\"5lmm\"zWm@\"F07$F_x$!'o,**F07$Fdx$!'W]\")F07$Fix$!'-^oF07$F^ y$!'Q\\eF07$Fcy$!'![?&F07$Fhy$!'G>YF07$F]z$!'%=>%F07$Fbz$!'YaQF07$Fgz$ !'_%f$F07$F\\[l$!'NPLF07$Fa[l$!'+\\JF07$Ff[l$!'vbHF07$F[\\l$!'h1GF07$F `\\l$!'JhEF07$Fe\\l$!'jODF07$Fj\\l$!',>CF07$F_]l$!'T?BF07$Fd]l$!'&GA#F 07$Fi]l$!'F07$Fh^l$!'T4>F07$F]_l$!'AW= F07$Fb_l$!'D&y\"F07$Fg_l$!'2CJ:F07$F`al$!'j!\\\"F07$Feal$!'Z]9F07$Fjal$!'D89F07$F_ bl$!'Fw8F07$Fdbl$!'VU8F07$Fibl$!']48F07$F^cl$!'Ry7F07$Fccl$!'3^7F07$Fh cl$!'=@7F07$F]dl$!'i&>\"F07$Fbdl$!'_p6F07$Fgdl$!'cX6F07$F\\el$!&47\"F` el-Faho6&Fdel$\"\"'FehoF(Ffho-Fiel6#%Sscheme~with~stability~radius~=~i mag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curve s~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fg \\t-%%VIEWG6$;F(F\\el%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modificatio n of the Bogacki-Shampine scheme" "efficient scheme A" "efficient sche me B" "efficient scheme C" "scheme with stability radius = imag axis i nclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 3 o f 7 stage, order 5 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&% \"xGF&%\"yGF&F(" }{TEXT -1 11 ", " }{XPPEDIT 18 0 "y(0)=1" "6 #/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp (-x^2/2);" "6#/%\"yG-%$expG6#,$*&%\"xG\"\"#F+!\"\"F," }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The f ollowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 760 "H := (x,y) -> -x*y: hh := 0.1: num steps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[` initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modifi cation of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag a xis inclusion`]:\nerrs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\n for ct to 6 do\n Hn_RK5_||ct := RK5_||ct(H(x,y),x,y,x0,y0,hh,numstep s,false);\n sm := 0: numpts := nops(Hn_RK5_||ct):\n for ii to nump ts do\n sm := sm+(Hn_RK5_||ct[ii,2]-h(Hn_RK5_||ct[ii,1]))^2;\n \+ end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10 :\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"yG\"\"\"%\"xGF ,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no. ~of~steps:~~~G\"$+\"Q)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampi ne~schemeG$\"+`?y8r!#?7$%Ta~minor~modification~of~the~Bogacki-Shampine ~schemeG$\"+Fp.ZbF+7$%3efficient~scheme~AG$\"+DX?m5!#>7$%3efficient~sc heme~BG$\"+c))o, " 0 "" {MPLTEXT 1 0 690 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bog acki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`effic ient scheme C`,`scheme with stability radius = imag axis inclusion`]: \nerrs := []:\nDigits := 20:\nfor ct to 6 do\n hn_RK5_||ct := RK5_|| ct(H(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nh := x -> exp(-x^2/2) :\nxx := 9.99: hxx := evalf(h(xx)):\nfor ct to 6 do\n errs := [op(er rs),abs(hn_RK5_||ct(xx)-hxx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point: ~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pp rint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+@UR)p(!#N 7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+Z/4FxF+7$% 3efficient~scheme~AG$\"+JPY[kF+7$%3efficient~scheme~BG$\"+!es#GX!#O7$% 3efficient~scheme~CG$\"+JK]KqF87$%Sscheme~with~stability~radius~=~imag ~axis~inclusionG$\"+LL`9 " 0 "" {MPLTEXT 1 0 456 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`effici ent scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with s tability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\n h := x -> exp(-x^2/2):\nfor ct to 6 do\n sm := NCint((h(x)-'hn_RK5_| |ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n errs := \+ [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %8Bogacki-Shampine~schemeG$\"+Hy`Jn!#?7$%Ta~minor~modification~of~the~ Bogacki-Shampine~schemeG$\"+)*o\\J_F+7$%3efficient~scheme~AG$\"+ZCA)** *F+7$%3efficient~scheme~BG$\"+3+C%f\"!#>7$%3efficient~scheme~CG$\"+#*z %[X\"F87$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+<9Fm &*F+Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the nume rical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 512 "evalf[20](plot([h(x)-'hn_RK5_1'(x),h(x)-'hn_RK5_2'(x ),h(x)-'hn_RK5_3'(x),h(x)-'hn_RK5_4'(x),\nh(x)-'hn_RK5_5'(x),h(x)-'hn_ RK5_6'(x)],x=0..5,numpoints=100,font=[HELVETICA,9],\ncolor=[blue,magen ta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlegend=[`Bogacki -Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme `,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`sche me with stability radius = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 835 584 584 {PLOTDATA 2 "6,-%'CURVESG6%7^_l7$$\"\"!F)F(7$$\" 5#HHHHHHz?k#!#@$\"'K5V!#?7$$\"5&eeeeeeeTG&F-$\")HmUFF07$$\"5IFFFF_L()e eF-$\")jf'3&F07$$\"5qoooo=\")eLkF-$\")EI+*)F07$$\"555555&)GI3qF-$\"*=e R[\"F07$$\"5]^^^^^w,$e(F-$\"*(=fvBF07$$\"5NMMMM%=ZCt)F-$\"*Bt4^&F07$$ \"5;<<<<>>>>WIX<6F0$\"+#o()GB\"F07$$\"5mmmm m;%=nC\"F0$\"+JbGJ7F07$$\"599999*y$)fP\"F0$\"+!pfCB\"F07$$\"5hhhhhh\" \\_]\"F0$\"+OkTZ7F07$$\"5CDDDDv-Jq:F0$\"+#)GIo7F07$$\"5)))))))))))Qr`j \"F0$\"+z7&fI\"F07$$\"5_____-DV+Qm#F0$\"+YXX.DF07$$\"533333eI&Qs#F0$ \"+%>l)pDF07$$\"5[[[[[[j)Qy#F0$\"+%fn=n#F07$$\"5HHHHHHH&R!HF0$\"+0#[#Q IF07$$\"5555555&>S-$F0$\"+eJ]nNF07$$\"5!**)*)*)*)*[jEF$F0$\"+c!o,a$F07 $$\"5ppppppuI@NF0$\"+bUQLNF07$$\"5efffffWf&e$F0$\"+x1)*[NF07$$\"5[\\\\ \\\\\\9))\\OF0$\"+t$\\:e$F07$$\"5QRRRRR%oTr$F0$\"+_)y(ROF07$$\"5GHHHHH aXyPF0$\"+I\"z_t$F07$$\"5344444%Hq!RF0$\"+&>$f-TF07$$\"5)))))))))))Q.c .%F0$\"+)eJ)*e%F07$$\"5rrrrrrm#>H%F0$\"++2yTXF07$$\"5aaaaaa*\\#[XF0$\" +!fM&>XF07$$\"5nnnn\"[F0$\"+-_CYZF07$$ \"5KLLLLe*RP%\\F0$\"+8g\"H<&F07$$\"5ffffff*pb2&F0$\"+ZB8daF07$$\"5NLLL LL=!yI&F0$\"+K1b#R&F07$$\"5222222P.SbF0$\"+44N[`F07$$\"5b````GKR0cF0$ \"+:*fGN&F07$$\"5+++++]FvqcF0$\"+B^Nu`F07$$\"5XYYYYrA6OdF0$\"+*)yc@aF0 7$$\"5!HHHHHzr9!eF0$\"+8SB1bF07$$\"5&eeeee$3>KfF0$\"+;)pP&eF07$$\"5yyy yyy)4H1'F0$\"+s(*f^hF07$$\"5baaaaa9UDjF0$\"+_6J`gF07$$\"5IIIIIII$ze'F0 $\"+[s8%)fF07$$\"5lmmmmmw<^mF0$\"+QX8))fF07$$\"5+.....BU9nF0$\"+udZ5gF 07$$\"5SRRRRRpmxnF0$\"+$)>wfgF07$$\"5vvvvvv:\"4%oF0$\"+FCRZhF07$$\"557 7777i:/pF0$\"+O?;)G'F07$$\"5][[[[[3SnpF0$\"+p`$4]'F07$$\"5gdddd2?@$)pF 0$\"+0y.olF07$$\"5qmmmmmJ-**pF0$\"+ld^TmF07$$\"5!eddddKM[,(F0$\"+9\"p$ RmF07$$\"5&[[[[[[X1.(F0$\"+25+KmF07$$\"50.....yEiqF0$\"+7(Rsh'F07$$\"5 @@@@@@,*Q4(F0$\"+LbW-mF07$$\"5DEEEEw)=OK(F0$\"+XGi%\\'F07$$\"5JJJJJJwM `vF0$\"+B#*o+kF07$$\"556666'[K**o(F0$\"+\"fwPQ'F07$$\"5!44444M-J'zF0$\"+U)4H s'F07$$\"5gggg5BYRJ!)F0$\"+Q_4LoF07$$\"5]]]]]]qo*4)F0$\"+>6c&z'F07$$\" 5DEEEEEw4J$)F0$\"+;$zym'F07$$\"5,-----#3Dc)F0$\"+jy!ya'F07$$\"5NLLLLe4 5(p)F0$\"+_U'>]'F07$$\"5lkkkk9PpJ))F0$\"+G/%f^'F07$$\"5IIII!G4!**)*))F 0$\"+e=DolF07$$\"5&ffff4Z'Gm*)F0$\"+D>yqmF07$$\"5ghhh6\\GeL!*F0$\"+0$ \"+w&)ySiF07$$\"5&[[[[[y@\\.\"Fbfl$\"+#>rF3'F07$$\"5(pppppk:)f5Fbfl$\" +#f)40fF07$$\"5999999Rw66Fbfl$\"+t)GKU&F07$$\"5-----xihN6Fbfl$\"+8\\Ox _F07$$\"5!**)*)*)*)R'o%f6Fbfl$\"+gEJ$4&F07$$\"5gggggNm>&=\"Fbfl$\"+w]- *p%F07$$\"5JJJJJJY#4@\"Fbfl$\"+@e&*\\UF07$$\"588888)y[wB\"Fbfl$\"+oN/3 TF07$$\"5&\\\\\\\\\\%HPk7Fbfl$\"+-$43(QF07$$\"5------kj(G\"Fbfl$\"+GV* *3LF07$$\"544444f)**3J\"Fbfl$\"+Ua=1FF07$$\"52222dWCYB8Fbfl$\"+W!R8m#F 07$$\"500000I]-O8Fbfl$\"+j*[Ah#F07$$\"5....`:we[8Fbfl$\"+(=-La#F07$$\" 5,,,,,,-:h8Fbfl$\"+q[q)F07$$\"544444M0wD9Fbfl$\"*G#)3/)F07$$\"5(pppppMd%Q9F bfl$\"*\\]M!yF07$$\"5\"4444Mv0[W\"Fbfl$\"*$=\"ee(F07$$\"5&[[[[)fT:^9Fb fl$\"*`-3B(F07$$\"5zyyyGmD]d9Fbfl$\"*a^8 F07$$\"5&[[[[tWIm_\"Fbfl$!+FN!\\K\"F07$$\"5qppppW1FS:Fbfl$!+AEU78F07$$ \"57777PV24Z:Fbfl$!+[%GIK\"F07$$\"5aaaa/U3\"Rb\"Fbfl$!+&QbsN\"F07$$\"5 'ppp>2%4tg:Fbfl$!+(yx&H9F07$$\"5RRRRRR5bn:Fbfl$!+;_Df:F07$$\"5PPPPP(oj ?f\"Fbfl$!+c&*4:HF07$$\"5NNNNNNjd;;Fbfl$!+ROkDPF07$$\"5FFFFFxDmH;Fbfl$ !+W+Y]OF07$$\"5>>>>>>)[Fk\"Fbfl$!+/%4tf$F07$$\"5:::::S>H\\;Fbfl$!+=T6% f$F07$$\"566666h]$el\"Fbfl$!+c?Y@OF07$$\"522222#=yBm\"Fbfl$!+p#\\hp$F0 7$$\"5......8#*o;Fbfl$!+^!G+%QF07$$\"5QQQQQjuj#p\"Fbfl$!+ai*RT&F07$$\" 5uttttBON;nF?/'z)F07$$ \"5UUUUUUFvH=Fbfl$!+u4!zf)F07$$\"5\\\\\\\\\\*H.D%=Fbfl$!+:(*[G%)F07$$ \"5-....y&y)[=Fbfl$!+g$fBP)F07$$\"5ccccccQDb=Fbfl$!+$>vFN)F07$$\"55555 5N\"H;'=Fbfl$!+(Qr$*Q)F07$$\"5jjjjj8W+o=Fbfl$!+1Wv2&)F07$$\"5))))))))) QTOH*=Fbfl$!,c%*zP/\"F07$$\"5999999%oy\">Fbfl$!,8X3m7\"F07$$\"5[[[[)4; >4$>Fbfl$!,iN3#*4\"F07$$\"5#GGGGy!*pR%>Fbfl$!,%*G`c2\"F07$$\"5******* \\7G&\\]>Fbfl$!,sy+w1\"F07$$\"5;<<Fbfl$!,'yIDk5F07$$\"5vvvDQTLG g>Fbfl$!,eA)3l5F07$$\"5MMMM4Ggaj>Fbfl$!,7s#3o5F07$$\"5#HHH/[r3o'>Fbfl$ !,J)*fO2\"F07$$\"5^^^^^,92q>Fbfl$!,6g!H#3\"F07$$\"5<<<<Fbfl$!,A &3]98F07$$\"5$GGGGG3\\.-#Fbfl$!,#))Q(3N\"F07$$\"5cbbbbIJ?L?Fbfl$!,u2%* pJ\"F07$$\"5GGGGGyr0Y?Fbfl$!,g#[>)G\"F07$$\"5kkkk9-U[_?Fbfl$!,a5(Ry7F0 7$$\"5,,,,,E7\"*e?Fbfl$!,:#4=u7F07$$\"5?>>>%ztC@1#Fbfl$!,K5))\\F\"F07$ $\"5QPPP()\\#Q`1#Fbfl$!,`So#y7F07$$\"5cbbb!=w^&o?Fbfl$!,D\\xWG\"F07$$ \"5uttttt_wr?Fbfl$!,\"pu6%H\"F07$$\"5'ffffM)G^%3#Fbfl$!,N*pYx8F07$$\"5 =====$\\gs4#Fbfl$!,fe'pj:F07$$\"5'ff4(RWR&))4#Fbfl$!,+oojf\"F07$$\"5ut tBh&RZ/5#Fbfl$!,?,,)>;F07$$\"5_^^w#o%3/-@Fbfl$!,_h'Q9;F07$$\"5HHHH/)HM O5#Fbfl$!,HB')*3;F07$$\"5%[[[t/?@o5#Fbfl$!,/]F#)f\"F07$$\"5SSSS!H53+6# Fbfl$!,,>Dve\"F07$$\"5^^^^w2>Q;@Fbfl$!,_d*Hm:F07$$\"5iiiii7dvA@Fbfl$!, *f$o`a\"F07$$\"5aaaa/U%pW8#Fbfl$!,ty(Q3:F07$$\"5YYYYYrJ=Y@Fbfl$!-,))oR w9F-7$$\"5UUUU>>>>9hk>#Fbfl$!-E$Gv\\s\"F-7$$ \"5---_*))HR\")>#Fbfl$!-5?phe#Fbfl$!-\">!z*[z\"F-7 $$\"5AAAs%Gh&\\,AFbfl$!-#os&4$z\"F-7$$\"5KKKK#)pP<.AFbfl$!-7A4['y\"F-7 $$\"5____x$3Il?#Fbfl$!-y/&4Lx\"F-7$$\"5sssss(R'))4AFbfl$!-ekf@g`B#Fbfl$!-%e-_[m\"F-7$$\"5@@@@@rqKZAFbfl$!-%3$e#pi\"F-7$$\"58888jD 4L`AFbfl$!-jt;z7;F-7$$\"500000!yM$fAFbfl$!-iUU;/;F-7$$\"5(ppppWjQ`E#Fb fl$!-HnWV.;F-7$$\"5*))))))))))[U8F#Fbfl$!-]x/d8;F-7$$\"5[[[[)4@XTG#Fbf l$!-u%3b$)o\"F-7$$\"533333Lz%pH#Fbfl$!-/W'y'p=F-7$$\"5...GMt#[&)H#Fbfl $!-fd6)>!>F-7$$\"5)zzz/Oh[,I#Fbfl$!-&HB+F$>F-7$$\"5$HHzmQ&*[F-7$$\"5)yyyGTH\\LI#Fbfl$!-_OS^=>F-7$$\"5yxxFlu*\\lI#Fbfl$!-x;DT />F-7$$\"5onnn>kz%y\"F-7$$\"5 aaaaaa51ZBFbfl$!-)ySj/u\"F-7$$\"5OOOO'Q(z=`BFbfl$!-&Q1yKs\"F-7$$\"5=== ==$*[JfBFbfl$!-4!R/=r\"F-7$$\"5++++]7=WlBFbfl$!-?VZb3F-7$$\"5MLLL$3n(Q7CFbfl$!-H8F-7$$\"5%QQQQQ)R*fU #Fbfl$!-k;CR(*=F-7$$\"5kjjj8^gxPCFbfl$!-b[LeX=F-7$$\"5WVVVV=\"e&\\CFbf l$!-tk*)*4!=F-7$$\"5MLLL3_\"\\aX#Fbfl$!-UY4=%y\"F-7$$\"5CBBBt&=S8Y#Fbf l$!-x?zLt7BnCFbfl$!-7o*>3x\"F-7$$\"5.....`A7tCFbfl$!-A* Rl%zF-7$$\"555555gKhEDFbfl$!-a%*HNC> F-7$$\"54444fr5!)QDFbfl$!-%H'[(y'=F-7$$\"533333$))))4b#Fbfl$!-:2Zp>=F- 7$$\"5eddd#))y#3dDFbfl$!-TAF'=!=F-7$$\"52222d%pwJc#Fbfl$!-:,)G3z\"F-7$ $\"5ccccJ+1FpDFbfl$!-(z8:#*y\"F-7$$\"5111111XOvDFbfl$!-eN09+=F-7$$\"5% RRRRk0'p)e#Fbfl$!-)fjE_)=F-7$$\"5#====ogF?g#Fbfl$!-$y\"GPN?F-7$$\"5qpp p>d\"f`h#Fbfl$!-o,B\"e'>F-7$$\"5ddddd22pGEFbfl$!-3[]j)*=F-7$$\"5HHHHHa kp^EFbfl$!-4#[Cxz\"F-7$$\"5,,,,,,AquEFbfl$!-yh&y3x\"F-7$$\"511111JE!yo #Fbfl$!-yolkQ=F-7$$\"566666hI!4q#Fbfl$!->Y%4b)>F-7$$\"5;;;;;\"\\.Sr#Fb fl$!-j*zCj\">F-7$$\"5@@@@@@R5FFFbfl$!-D7TY\\=F-7$$\"5!**)*)*)*)*Ge(RFF bfl$!-_pVk)y\"F-7$$\"5eeeeeeET_FFbfl$!-+S<yDS\"=F-7 $$\"5`````.eIGGFbfl$!-=6<;F-7$$\"5%QQQQ8P33*GFbfl$!-AzKjx;F-7$$\"5ZZZZZAE\"H!HF bfl$!-:e21a*HFbfl$!-!)z@]]:F-7$$\"5-----FJS/IFbfl$!-] !Qktf\"F-7$$\"5eeee3'4uo,$Fbfl$!-/iG^Q:F-7$$\"5:::::l]MHIFbfl$!-b!=h=[ \"F-7$$\"5ONNN&GFN]8F]`p7$$\"5'fffff\\X=3$Fbfl$!.@&*)3 Kf8F]`p7$$\"5/////zos$4$Fbfl$!.pj$*4'39F]`p7$$\"577777i#3c5$Fbfl$!.h?Y ryU\"F]`p7$$\"5?????X'*[iJFbfl$!.Nc=]\\@\"F]`p7$$\"5'eeeee[k(oJFbfl$! .%Q=a\\/7F]`p7$$\"577777i+LvJFbfl$!.1[Ma9?\"F]`p7$$\"5QQQQQQc*==$Fbfl$ !.`XLPv?\"F]`p7$$\"577777i]?KKFbfl$!.u$HqOa6F]`p7$$\"5JJJJJJ?R#G$Fbfl$ !.D8!>\\`5F]`p7$$\"5::::::D\"[L$Fbfl$!-%Fbfl$!.DtbMlL\"F`hp7$$\"5'oooooyUAC%Fbfl$!.9[$))4'4\"F` hp7$$\"5666666dt\"H%Fbfl$!-,@EQS'*F`hp7$$\"588888jlQXVFbfl$!.(>&4UFr(! #C7$$\"5mmmmmm%4=R%Fbfl$!.c\\N&H3oF]\\q7$$\"544444fy7XWFbfl$!._YDw$z`F ]\\q7$$\"5,,,,,^*\\_\\%Fbfl$!.7l6)=GYF]\\q7$$\"5======z)[a%Fbfl$!.G5R) \\iOF]\\q7$$\"5ssssss`c%f%Fbfl$!.26O\"4?JF]\\q7$$\"5\"=====BL_k%Fbfl$! .KT8sAU#F]\\q7$$\"5jjjjj8.%*)p%Fbfl$!.H=we)[>F]\\q7$$\"5[[[[[[W=[ZFbfl $!.Q*\\PQP:F]\\q7$$\"5#GGGGGt%z'z%Fbfl$!.,k$oEd7F]\\q7$$\"5wwwwwE$e([[ Fbfl$!.B=/%\\6&*!#D7$$\"5]]]]]+)31!\\Fbfl$!.69[Pe5(F`_q7$$\"5zzzzzzw9Z \\Fbfl$!.e\\1tIo&F`_q7$$\"\"&F)$!.)Q,Iu,SF`_q-%'COLOURG6&%$RGBGF(F($\" *++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7`_lF'7$F2$\"(.9 L\"F07$FP$\")el.NF07$F^o$\")h!*)y$F07$F\\q$\")CiU#*F07$Faq$\")))[(>*F0 7$Ffq$\")R*RV*F07$F`r$\"*2:x+\"F07$Fjr$\"*rOI;\"F07$F_s$\"*^a7[\"F07$F ds$\"*^F07$Fis$\"*^u9!>F07$F^t$\"*4Y#>>F07$Fht$\"*a$\\0?F07$Fbu$ \"*KhvC#F07$Fgu$\"*530!GF07$F\\v$\"*f-8_$F07$Fav$\"*S%H&[$F07$Ffv$\"*P `w]$F07$F`w$\"*@9vl$F07$Fjw$\"*m&)**3%F07$F_x$\"*D_%*4&F07$Fdx$\"*p$G% z&F07$Fix$\"*fiqs&F07$F^y$\"**3zAdF07$Fhy$\"*$))[xeF07$Fbz$\"*?O@P'F07 $Fgz$\"*`Ege(F07$F\\[l$\"*8'>n')F07$Fa[l$\"*:,1`)F07$Ff[l$\"*QS;^)F07$ F`\\l$\"*2^7v)F07$Fj\\l$\"*Uc*z%*F07$F_]l$\"+@#Gx,\"F07$Fd]l$\"+-()o@6 F07$Fh^l$\"+e<,!>\"F07$Fb_l$\"+N&3Z=\"F07$Fg_l$\"+E&3b;\"F07$F\\`l$\"+ p_``6F07$Fa`l$\"+1\")fn6F07$Ff`l$\"+3j)QB\"F07$F[al$\"+-fD18F07$F`al$ \"+!=%y@9F07$Feal$\"+e6m/:F07$Fjal$\"+wgR'\\\"F07$F_bl$\"+&yn$o9F07$Fd bl$\"+*G_gW\"F07$Fibl$\"+Uy!=X\"F07$F^cl$\"+5wN2:F07$Fccl$\"+W&RUd\"F0 7$Fhcl$\"+4'3bo\"F07$F]dl$\"+T&\\(e:!z=F07$Fffl$\"+B.;J=F07$F[gl$\"+N6Wwg$*RF07$F_]m$\"(4%[ _F07$Fd]m$!)'38/&F07$Fi]m$!*\\:&fgF07$F^^m$!+FEp-6F07$Fc^m$!+G?:\"3\"F 07$Fh^m$!+#=w)p5F07$F]_m$!+-.Yw5F07$Fb_m$!+P@w*4\"F07$Fg_m$!+!))e$\\6F 07$F\\`m$!+hAOP7F07$Fa`m$!+dw>&4#F07$Ff`m$!++j)Qb#F07$F[am$!+n\\k-DF07 $F`am$!+Nk\"\\#F07$F_bm$!+>(R![DF07 $Fdbm$!+gRa`EF07$Fibm$!+h)HM1%F07$Fbdm$!+4SysSF07$Fgdm$!+B6,@TF07$F\\em$!+% o/^A%F07$Faem$!+r]q!\\&F07$Ffem$!+r\"R53'F07$F[fm$!+5xdWfF07$F`fm$!+(4 **4$eF07$Fefm$!+/v\"pz&F07$Fjfm$!+KNc\"z&F07$F_gm$!+,j\")HeF07$Fdgm$!+ .`cIfF07$Figm$!+DY**)Q(F07$F^hm$!+66c(F07$F\\jm$!+6340wF07$Ffjm$!+ZVLIx F07$$\"5MMMM%=#3k#)>Fbfl$!+RcO5$)F07$F[[n$!+5LoO&*F07$$\"5)yyy`sf_$)*> Fbfl$!+mX=r**F07$$\"5eeeeL_\\\\,?Fbfl$!,)[G5>5F07$$\"5HHHzT2tj/?Fbfl$! ,dT3F,\"F07$$\"5++++]i'zx+#Fbfl$!,^HWj+\"F07$$\"5UTTTmsV19?Fbfl$!+Ut5P **F07$F`[n$!+QrI7)*F07$Fe[n$!+*>*Gn&*F07$Fj[n$!+$4DUO*F07$F_\\n$!+;up+ $*F07$Fd\\n$!+Wg(GG*F07$F^]n$!+n5MK$*F07$Fh]n$!+0I%eZ*F07$F]^n$!,/?Nl, \"F07$Fb^n$!,!>1Fi6F07$Fg^n$!,&f,<(=\"F07$F\\_n$!,J\\V\\?\"F07$Fa_n$!, =\"e\"4?\"F07$Ff_n$!,gb)*o>\"F07$F[`n$!,0K&*))=\"F07$F``n$!,b8M4=\"F07 $Fe`n$!,F[Y^;\"F07$Fj`n$!,[%pe\\6F07$F_an$!,ST1A7\"F07$Fdan$!-vo%4!*4 \"F-7$Fian$!-E$)=&34\"F-7$F^bn$!-F![#om@Fbfl$!-.a(394\"F-7$Fhbn$ !-\")HF.(4\"F-7$F]cn$!--&y`-;\"F-7$Fbcn$!-c:q,58F-7$Fgcn$!--')yUO8F-7$ F\\dn$!-1MUrk8F-7$Fadn$!-nO2Tj8F-7$Ffdn$!-'y0\"Qe8F-7$F[en$!-![&fO[8F- 7$F`en$!--T,TQ8F-7$Feen$!-iSgo=8F-7$Fjen$!-s5:F*H\"F-7$F_fn$!-OMt0m7F- 7$Fdfn$!-)z$Q*yB\"F-7$Fifn$!-B(HBzA\"F-7$F^gn$!-uGWhA7F-7$$\"5,,,,E2nL iAFbfl$!-G0tIA7F-7$Fcgn$!-*z0RRA\"F-7$$\"5$HHHz;cS$oAFbfl$!-d%)H#yA\"F -7$Fhgn$!-%Gf'HM7F-7$F]hn$!-m/Mp*H\"F-7$Fbhn$!-PW!f'[9F-7$Fghn$!-CxR_u 9F-7$F\\in$!-.;<)*)\\\"F-7$Fain$!-Uk@Z$\\\"F-7$Ffin$!-J`!zz[\"F-7$F[jn $!-#>2UqZ\"F-7$F`jn$!->Q4==ob-dGkCFbfl$!- n2(=?S\"F-7$Fe_o$!-#R(\\*HS\"F-7$$\"5333e?On$[\"F-7$Fd`o$!-ejL)yk\"F-7$Fi`o$!-)>A+if\"F- 7$F^ao$!-7g[_V:F-7$Fcao$!-(yqC%)\\\"F-7$Fhao$!-M\">-1Y\"F-7$F]bo$!-cZ \"*=Z9F-7$Fbbo$!-7^jrR9F-7$$\"5#===VukBic#Fbfl$!-A&e)))Q9F-7$Fgbo$!-xi SEFbfl$!-i(>B/]\"F-7$Fedo$!-#41nLY\"F-7$$\"5AAAA(4*yWdEFb fl$!-\\oS!)\\9F-7$$\"5::::lF$*>jEFbfl$!-(yr/9W\"F-7$$\"53333Lk2&*oEFbf l$!-kw(Q*R9F-7$Fjdo$!-AH!HuW\"F-7$F_eo$!-7D@]4:F-7$Fdeo$!-(e1/`j\"F-7$ Fieo$!-o=FKy:F-7$F^fo$!-2*fxK_\"F-7$Fcfo$!-p.QPt9F-7$Fhfo$!-&>A29V\"F- 7$F]go$!-lC&>jT\"F-7$Fbgo$!-iDI_29F-7$$\"5WWWWW>1BoFFbfl$!-V!>vhS\"F-7 $Fggo$!-He\"psS\"F-7$$\"5zyyyy.ybuFFbfl$!-4Bw769F-7$F\\ho$!-kI%y!=9F-7 $Faho$!-)4KCH[\"F-7$Ffho$!-EZ#y^c\"F-7$F[io$!-t&*eb5:F-7$F`io$!-=mX&yX \"F-7$Feio$!-K_tt39F-7$Fjio$!-iV=ln8F-7$F_jo$!-Zw*4HN\"F-7$Fdjo$!-M+6B W8F-7$$\"5???q#eVa#pGFbfl$!-W/LzU8F-7$Fijo$!-2zLlV8F-7$$\"5```.TDUbvGF bfl$!-^sC4Z8F-7$F^[p$!-vU(*Q`8F-7$Fc[p$!-fw6n39F-7$Fh[p$!-M_(*yu9F-7$F ]\\p$!-Et-xB9F-7$Fb\\p$!-K*=*[u8F-7$Fg\\p$!-\")3KUC8F-7$F\\]p$!-Bl50#G \"F-7$Fa]p$!-&)Ri[m7F-7$Ff]p$!-.vR%oD\"F-7$F[^p$!-Mg#G^D\"F-7$F`^p$!-_ EKYj7F-7$Fe^p$!-q]C!QJ\"F-7$Fj^p$!-abaWa8F-7$F__p$!-p=Za/8F-7$Fd_p$!-g $*>`c7F-7$Fi_p$!.f!yp\\47F]`p7$F_`p$!.1%\\Mjq6F]`p7$Fd`p$!.;&R>(p:\"F] `p7$Fi`p$!.\"3$\\+#\\6F]`p7$$\"5\"333$=l;+sIFbfl$!.u#zn2[6F]`p7$F^ap$! .^(fb4\\6F]`p7$$\"5\"444M!>UcyIFbfl$!.u*z@Z_6F]`p7$Fcap$!.M2U+%e6F]`p7 $Fhap$!.gX]+J?\"F]`p7$F]bp$!.#psun>7F]`p7$Fbbp$!.l:n?a<\"F]`p7$Fgbp$!. OTkAG8\"F]`p7$F\\cp$!.HV9J*)3\"F]`p7$Facp$!.!p\">sB0\"F]`p7$Ffcp$!.rmI *=R5F]`p7$F[dp$!.*)fV[6.\"F]`p7$F`dp$!.&4JjsH5F]`p7$Fedp$!.y&enIO5F]`p 7$Fjdp$!-![t%yB**F]`p7$F_ep$!-2`***o4*F]`p7$Fdep$!-s(3\">G&)F]`p7$Fiep $!-g_#4P$yF]`p7$F^fp$!-'\\\\OXI(F]`p7$Fcfp$!-gUHaXmF]`p7$Fhfp$!-*=#y_4 gF]`p7$F]gp$!-;4xn(\\&F]`p7$Fbgp$!-*fYuL)[F]`p7$Fggp$!-NRW]fWF]`p7$F\\ hp$!.A\\M\"Q`RF`hp7$Fbhp$!.n-Y2bb$F`hp7$Fghp$!.OkJ$GuIF`hp7$F\\ip$!.nu H\\3y#F`hp7$Faip$!.X(=[mfBF`hp7$Ffip$!.*3:9$*H@F`hp7$F[jp$!.m#=hx,=F`h p7$F`jp$!.vWX8xf\"F`hp7$Fejp$!.Zh=%*>J\"F`hp7$Fjjp$!.BTofU<\"F`hp7$F_[ q$!-*z$z$=e*F`hp7$Fd[q$!--%)G&fW)F`hp7$Fi[q$!.=B`Vqr'F]\\q7$F_\\q$!.>W %yvRfF]\\q7$Fd\\q$!.*)>(='*eYF]\\q7$Fi\\q$!.nM*o)H+%F]\\q7$F^]q$!.lRvb u9$F]\\q7$Fc]q$!.^u_*owEF]\\q7$Fh]q$!.N%)HS$f?F]\\q7$F]^q$!.LMT5jj\"F] \\q7$Fb^q$!..ED\"o(G\"F]\\q7$Fg^q$!.\"3a*)QT5F]\\q7$F\\_q$!.Q8&>D&z(F` _q7$Fb_q$!.')37Wai&F`_q7$Fg_q$!.%zK0*[]%F`_q7$F\\`q$!.zMDUq*HF`_q-Fa`q 6&Fc`qFd`qF(Fd`q-Fh`q6#%Ta~minor~modification~of~the~Bogacki-Shampine~ schemeG-F$6%7j]lF'7$F2$!(yFE\"F07$FP$!)a2'G(F07$F^o$!)VCXyF07$F\\q$!)p EvYF07$Faq$!)X(yk%F07$Ffq$!)f\\=WF07$F`r$!)zVKPF07$Fjr$!))y%o;F07$F_s$ \")c\\%f$F07$Fds$\"*,m+B\"F07$Fis$\"*5/3A\"F07$F^t$\"*^JPC\"F07$Fht$\" *R_[O\"F07$Fbu$\"*SL]x\"F07$Fgu$\"*.%o&*GF07$F\\v$\"*I/Le%F07$Fav$\"*) zjNXF07$Ffv$\"*5bLc%F07$F`w$\"*(*Qt\"[F07$Fjw$\"*5c:o&F07$F_x$\"*AW[,) F07$Fdx$\"*V!o)y*F07$Fix$\"*[ANn*F07$F^y$\"*D4Ul*F07$Fhy$\"*NH*\\**F07 $Fbz$\"+0_z.6F07$Fgz$\"+4ZW.9F07$F\\[l$\"+&yaHp\"F07$Fa[l$\"+*4rgm\"F0 7$Ff[l$\"+5$=Am\"F07$F[\\l$\"+.6q!o\"F07$F`\\l$\"+)3i*=F07$Fd]l$\"+%*p4'Q#F07$Fb_l$\"+S7AsDF07$Fg_l $\"+D,ZIDF07$F\\`l$\"+#oSm]#F07$$\"5?@@@re+k@wF0$\"+Qb\"z^#F07$Fa`l$\" +ANG^DF07$$\"5+,,,^8\\AexF0$\"+%\\9\"Fbfl$\"+gK]biF07$$ \"5'ffffMjg!)>\"Fbfl$\"+s%4sb'F07$$\"5kjjjQKE\\/7Fbfl$\"+S12VmF07$Fdhl $\"+Krm\"f'F07$Fihl$\"+6OwzjF07$F^il$\"+,Q_\">'F07$$\"5[[[[[tY+w7Fbfl$ \"+`VkchF07$Fcil$\"+)zo`@'F07$$\"5zyyyGmAX$H\"Fbfl$\"+zS-5jF07$$\"5cbb bbI\"o#*H\"Fbfl$\"+'=`IZ'F07$$\"5KKKK#[*R308Fbfl$\"+yDuckF07$Fhil$\"+U 7\"yS'F07$Fbjl$\"+D*pw>'F07$F\\[m$\"+N)RH)fF07$Ff[m$\"+74;qbF07$F`\\m$ \"+MR!)p`F07$Fd]m$\"+ly-F^F07$Fi]m$\"+.d\"Gi%F07$F^^m$\"+\\ycVSF07$Fh^ m$\"+*e(>tQF07$F\\`m$\"+7?5xNF07$Fa`m$\"+@MTrDF07$Ff`m$\"+xMN!y\"F07$F [am$\"+$=w%p\"F07$Feam$\"+=Nde;F07$Fjam$\"+lr70;F07 $F_bm$\"+(3,3_\"F07$Fdbm$\"+W)RgQ\"F07$Fibm$!)Wy***)F07$F^cm$!+%[+w8\" F07$Fccm$!+5V#R6\"F07$Fhcm$!+y:o-6F07$F]dm$!+okf76F07$Fbdm$!+1)Hd9\"F0 7$Fgdm$!+)**o!>7F07$F\\em$!+1y9e8F07$Faem$!+iqY%H$F07$Ffem$!+:nDAYF07$ F[fm$!+vC.=XF07$F`fm$!+S*G6V%F07$Fefm$!+e.w2WF07$Fjfm$!+z`i8WF07$F_gm$ !+=#o%pWF07$Fdgm$!+;iG1YF07$Figm$!+V,kAqF07$F^hm$!+J[#f])F07$Fchm$!+Fi ,)H)F07$Fhhm$!+1Y7<\")F07$F]im$!+$3vm0)F07$Fbim$!+#QM#R!)F07$F\\jm$!+n 6z%4)F07$Ffjm$!+\\K+o#)F07$F[[n$!,@G3]:\"F07$F`[n$!,$yNia7F07$Fe[n$!,U K0HA\"F07$Fj[n$!,3fi_>\"F07$F_\\n$!,7'pi&=\"F07$Fd\\n$!,IHD<=\"F07$Fi \\n$!,3%z-$=\"F07$F^]n$!,1z%G(=\"F07$Fc]n$!,wCL_>\"F07$Fh]n$!,H()[x?\" F07$F]^n$!,V?Y_K\"F07$Fb^n$!,1RI/j\"F07$Fg^n$!,-)f$*)o\"F07$F\\_n$!,*o ,\"Ht\"F07$Fa_n$!,)*oF-7$Fgcn$!-y 5=xg?F-7$F\\dn$!-Y;@[J@F-7$Fadn$!-1RLdK@F-7$Ffdn$!-w'=1Z7#F-7$F[en$!-q ^5/4@F-7$F`en$!-5W\"oM4#F-7$Feen$!-\")\\Zgi?F-7$Fjen$!-B%>i@.#F-7$F_fn $!-g=8Vz>F-7$Fdfn$!-?sfSK>F-7$Fifn$!-5\"o?S\">F-7$F^gn$!-gqR9->F-7$Fcg n$!-NMk^+>F-7$Fhgn$!-FUSV9>F-7$F]hn$!-7C(3E.#F-7$Fbhn$!--(pr%pBF-7$Fgh n$!-#RyM_V#F-7$F\\in$!-t-7y*\\#F-7$Fain$!-&H=$f!\\#F-7$Ffin$!-DzDV\"[# F-7$F[jn$!-0LM>jCF-7$F`jn$!-I*fj]W#F-7$Fejn$!-f'HO\"4CF-7$Fjjn$!-HH$)o tBF-7$F_[o$!-M>!GyI#F-7$Fd[o$!-=fbB[AF-7$Fi[o$!-t+T+CAF-7$F^\\o$!-:Ba& o?#F-7$Fc\\o$!-_kb!4?#F-7$Fh\\o$!-[!>4?@#F-7$$\"52222K))=PyBFbfl$!-'*Q y=aAF-7$F]]o$!-'H^pBM#F-7$$\"5edddK,#y>R#Fbfl$!-J)RCm\\#F-7$Fb]o$!-\\` P:VFF-7$Fg]o$!-pstG\\#F-7$Fj_o$! -\"e$[@bCF-7$$\"5UTTT\"*G'3)zCFbfl$!-_W,7,DF-7$F_`o$!-ui!*f&f#F-7$$\"5 ====o!Q\"=$\\#Fbfl$!-c3\"p(eFF-7$Fd`o$!-+gwa;IF-7$Fi`o$!-[_\\8CHF-7$F^ ao$!-9lGUFGF-7$Fcao$!-N.))HVFF-7$Fhao$!-wxY')oEF-7$F]bo$!-GakmREF-7$Fb bo$!-IkL9?EF-7$Fgbo$!-R)ei`h#F-7$F\\co$!-m,+KKEF-7$Faco$!-5Mj(\\z#F-7$ Ffco$!-5H**3YJF-7$F[do$!-0JAcQIF-7$F`do$!-)*34ZMHF-7$Fedo$!-L&e5Lx#F-7 $Fjdo$!-*eVQ=s#F-7$F_eo$!-!e#['\\&GF-7$Fdeo$!-T#RA$4KF-7$Fieo$!-BU**[( 4$F-7$F^fo$!-e:QB*)HF-7$Fhfo$!-**)H#R,GF-7$F\\ho$!-V)4KFv#F-7$Faho$!-$ )QM+0HF-7$Ffho$!-TLYhiJF-7$F[io$!-oLYB_IF-7$F`io$!-x#Qqa%HF-7$Fjio$!-J o?AbFF-7$F^[p$!-$)))[T/FF-7$Fc[p$!-zK)>0%GF-7$Fh[p$!-69!)[hIF-7$F]\\p$ !-9x1dbHF-7$Fb\\p$!-%Q7mI&GF-7$F\\]p$!-Bn2r`EF-7$F`^p$!-V4*)R$f#F-7$Fe ^p$!-3e7NDFF-7$Fj^p$!-XQyL%)GF-7$F__p$!-'z))f!yFF-7$Fd_p$!-y;k_vEF-7$F _`p$!.e!H$*>%[#F]`p7$Fcap$!.+.r^%RCF]`p7$Fhap$!.d@Q())oDF]`p7$F]bp$!.8 vT9@m#F]`p7$Fbbp$!.?[y.bc#F]`p7$Fgbp$!.dG!pEsCF]`p7$Facp$!.TZhc*)G#F]` p7$Fedp$!.:wTmjB#F]`p7$$\"5#====V*HZ%>$Fbfl$!.)>8W+aBF]`p7$$\"5DDDDD]. 02KFbfl$!.(*zj6bS#F]`p7$$\"5oooo=1xi>KFbfl$!.bqXZ-J#F]`p7$Fjdp$!.exRR( =AF]`p7$$\"5srrrrY&)HdKFbfl$!.'Gi@He?F]`p7$F_ep$!.2dR\\6,#F]`p7$$\"5HH HHz\"fW*)G$Fbfl$!.S9q%=Z?F]`p7$$\"5FFFFF_r\\&H$Fbfl$!.I-:Bz6#F]`p7$$\" 5EEEE^KMx)H$Fbfl$!.(4B1[p@F]`p7$$\"5DDDDv7(\\?I$Fbfl$!.3of'=x@F]`p7$$ \"5CCCC*H*fK0LFbfl$!.o)G#[P:#F]`p7$$\"5BBBBBtAg3LFbfl$!.\\#*RR08#F]`p7 $$\"5>>>>>%R2F]`p7$$\"5!)zzzz/J& *eLFbfl$!.Tm'f.:=F]`p7$Fiep$!.HAM#[uF]`p7$$\"5)*)*)*[6_ls/MFbf l$!.y1i;g)=F]`p7$$\"5NNNNNgp\"yS$Fbfl$!.e78ni'=F]`p7$$\"5!3333Lfy,U$Fb fl$!.L1cf\"*y\"F]`p7$F^fp$!.7MwN_r\"F]`p7$$\"5)yyyyy$*)=YMFbfl$!.Gr*=& )Q;F]`p7$$\"5\\\\\\\\\\\\w$)fMFbfl$!.W+U#ot:F]`p7$$\"5IIIII0?mmMFbfl$! .*zwsE\\:F]`p7$$\"556666hj[tMFbfl$!.`*oX(Q`\"F]`p7$$\"5\">>>>pr5.[$Fbf l$!.yU!)\\8`\"F]`p7$Fcfp$!.3Kf)fY:F]`p7$Fhfp$!.X[*p_Z9F]`p7$F]gp$!.kt$ pZ78F]`p7$Fbgp$!.$RkJ@37F]`p7$Fggp$!.lF+XA4\"F]`p7$F\\hp$!/-iDbr15F`hp 7$Fbhp$!.'H:w]i*)F`hp7$Fghp$!.(>&GE?2)F`hp7$F\\ip$!.-a-)3LsF`hp7$Faip$ !.'Q-Mn1kF`hp7$Ffip$!.gX?sGw&F`hp7$F[jp$!.Ci2\"*z2&F`hp7$F`jp$!.,0(\\= )\\%F`hp7$Fejp$!.9HT(Q^QF`hp7$Fjjp$!.,X9$e]MF`hp7$F_[q$!.g_!y^ZHF`hp7$ Fd[q$!.;N@+Zg#F`hp7$Fi[q$!/\"z(egmx@F]\\q7$F_\\q$!/m**e0#\\$>F]\\q7$Fd \\q$!/1BN=(fg\"F]\\q7$Fi\\q$!/rC6pt<9F]\\q7$F^]q$!/I6Igzl6F]\\q7$Fc]q$ !/zFiV!)=5F]\\q7$Fh]q$!.=y]F]\\q7$F\\_q$!/^5>M[rRF`_q7$Fb_q$!/^Ek9,=MF`_ q7$Fg_q$!/)GGw\"RAFF`_q7$F\\`q$!/BcnF\"eI#F`_q-%&COLORG6&Fc`q$\"\"\"! \"\"$\"\")Fi`yF(-Fh`q6#%3efficient~scheme~AG-F$6%7^^lF'7$F2$\"(X<9%F07 $FP$\"*_3ka\"F07$FZ$\"*e&\\[;F07$F^o$\"*AP=o\"F07$Fho$\"*Ag9!=F07$Fbp$ \"*^&R[@F07$Fgp$\"*o!)Q)HF07$F\\q$\"*-KsH%F07$Faq$\"*`7YF%F07$Ffq$\"*L yKK%F07$F`r$\"*%)*R2XF07$Fjr$\"*o7u,&F07$F_s$\"*1G)4iF07$Fds$\"*\"[b\\ !)F07$Fis$\"*>qy)zF07$F^t$\"*]=/*zF07$Fht$\"*k9W=)F07$Fbu$\"*S)pT))F07 $Fgu$\"+r.&p0\"F07$F\\v$\"+!3uwI\"F07$Fav$\"+;f(RH\"F07$Ffv$\"+1)R6H\" F07$F`w$\"+$)eNA8F07$Fjw$\"+ch*HV\"F07$F_x$\"+\"Hs#HF07$Fix$\"+6/WE>F07$F^y$\"+$[O[\">F07$Fhy$\"+,!pY%>F07$Fbz$\"+k]/n?F0 7$Fgz$\"+M1/5CF07$F\\[l$\"+2f%*QFF07$Fa[l$\"+Ul@&p#F07$Ff[l$\"+&opfn#F 07$F`\\l$\"+\\ACIFF07$Fj\\l$\"+8!3p#HF07$F_]l$\"+Cn;HJF07$Fd]l$\"++;`V MF07$Fh^l$\"+/n&fl$F07$Fb_l$\"+VQmROF07$Fg_l$\"+u@K!e$F07$F\\`l$\"++r? QNF07$Fa`l$\"+D!\\Fd$F07$Ff`l$\"+/anoPF07$F`al$\"+XJWiVF07$Fjal$\"+l:q 1YF07$F_bl$\"+$H@/_%F07$Fdbl$\"+#36SX%F07$Fibl$\"+u[h\"[%F07$F^cl$\"+V Y7&o%F07$Fccl$\"+@J>C\\F07$Fhcl$\"+mc)zJ&F07$$\"5SPP()\\l06$)*)F0$\"+> oH[aF07$$\"5!)yyy.gY$****)F0$\"+W`\"Qf&F07$$\"5???qda(en,*F0$\"+dC(fe& F07$F]dl$\"+Mr\\xbF07$$\"5XWWW>Q5Bn!*F0$\"+?y`gbF07$Fbdl$\"+kscVbF07$F gdl$\"+6E`BaF07$F\\el$\"+@loD`F07$Fael$\"+**p\"3M&F07$Ffel$\"+]L\"3`&F 07$Fg[r$\"+=2(*fdF07$F[fl$\"+c#p&QhF07$F_\\r$\"+Eb?hjF07$F`fl$\"+br[>j F07$Fffl$\"+j/)3;'F07$F[gl$\"+#oVU-'F07$F]]r$\"+Z'*o?gF07$Fb]r$\"+zBl( ='F07$Fg]r$\"+k/r,kF07$F\\^r$\"+fu&4w'F07$Fa^r$\"+ui/5oF07$F`gl$\"+m5? hnF07$Fegl$\"+@.3#e'F07$Fjgl$\"+[!f;T'F07$F^av$\"+UWhcjF07$F_hl$\"+&z* o2kF07$Ffav$\"+)=E)=lF07$F[bv$\"+#=)3GnF07$F`bv$\"+Z\")>!y'F07$Fdhl$\" +,CtFnF07$Fihl$\"+3=l4lF07$F^il$\"+k'fxF'F07$Fhil$\"+QA(o-'F07$Fbjl$\" +luQFeF07$F\\[m$\"+QIP&e&F07$Fa[m$\"+bTqz]F07$Ff[m$\"+!yS%4XF07$F`\\m$ \"+m-)32%F07$Fi]m$\"+=P/zJF07$F^^m$\"+ejH;@F07$Fc^m$ \"+`^,s?F07$Fh^m$\"+B[.>?F07$F]_m$\"+j/3!)>F07$Fb_m$\"+Shk@>F07$Fg_m$ \"+C]&)G=F07$F\\`m$\"+<)4$z;F07$Fa`m$\")n\\rRF07$Ff`m$!+Frx#>\"F07$F[a m$!+6\\))p6F07$F`am$!+%zcV;\"F07$Feam$!+:[p#=\"F07$Fjam$!+a@#=B\"F07$F _bm$!+]hKL8F07$Fdbm$!+m&H\">:F07$Fibm$!+,E\\/QF07$F^cm$!+yMah`F07$Fccm $!+\"z!oX_F07$Fhcm$!+#p@?:&F07$F]dm$!+6oLJ^F07$Fbdm$!+(p@([^F07$Fgdm$! +'eP5B&F07$F\\em$!+uSC=aF07$Faem$!+%)*yJL)F07$Ffem$!,S=wk-\"F07$F[fm$! ,UGDK+\"F07$F`fm$!+\"eu!H)*F07$Fefm$!+N`1g(*F07$Fjfm$!+1z^P(*F07$F_gm$ !+3hW$z*F07$Fdgm$!+[oIv**F07$Figm$!,LHAnN\"F07$F^hm$!,wxb+d\"F07$Fchm$ !,NB<;`\"F07$Fhhm$!,e?$e(\\\"F07$F]im$!,id3`[\"F07$Fbim$!,_mF(z9F07$F \\jm$!,Xj-a[\"F07$Ffjm$!,fX)34:F07$F[[n$!,&*y0j*>F07$F`[n$!,2n-[8#F07$ Fe[n$!,;=!y!3#F07$Fj[n$!,qSFK.#F07$F_\\n$!,h\\ef,#F07$Fd\\n$!,*pW]2?F0 7$Fi\\n$!,*[=D3?F07$F^]n$!,H()*\\8?F07$Fc]n$!,IxeV-#F07$Fh]n$!,8GX@/#F 07$F]^n$!,Uny_@#F07$Fb^n$!,Z,a8n#F07$Fg^n$!,w#3-fFF07$F\\_n$!,6/DZ#GF0 7$Fa_n$!,z-$G:GF07$Ff_n$!,FXle!GF07$F[`n$!,E`.ry#F07$F``n$!,([%R%oFF07 $Fe`n$!,y389t#F07$Fj`n$!,3DW[p#F07$F_an$!,/un'HEF07$Fdan$!-QVUQrDF-7$F ian$!-i#o[%[DF-7$F^bn$!-N!>xL`#F-7$Fcbn$!-\")zEnIDF-7$Fhbn$!-O%)=rYDF- 7$F]cn$!-!Rl$p1FF-7$Fbcn$!-?$4&R*=$F-7$Fgcn$!-xDmj&G$F-7$F\\dn$!-4k%Q@ R$F-7$Fadn$!-Q'y)=$R$F-7$Ffdn$!-'4f'QW2$F-7$Fifn$!-QH+zWIF-7$F^gn$!-wg94DIF-7$Fcgn$!-,1/$F-7$F]hn$!-`$=bp@$F-7$Fbhn$!-6TaTEPF-7$Fghn$!-VKf@EQF-7$F\\in$ !-Odh>CRF-7$Fain$!-AeEx4RF-7$Ffin$!-R(>#R&*QF-7$F[jn$!-j)4gn'QF-7$F`jn $!-t:'*HQQF-7$Fejn$!-Q=1!>y$F-7$Fjjn$!-4aYDEPF-7$Fd[o$!-_W&p\"HNF-7$Fh \\o$!-wWG*)oMF-7$Fagw$!-TN+3KNF-7$F]]o$!-tb1zlOF-7$Figw$!-m'Ho5!RF-7$F b]o$!-!*[?qyUF-7$Fg]o$!-OL-,PUF-7$F\\^o$!-(o!Q1+TF-7$Ff^o$!-a?@\\&)QF- 7$Fj_o$!-(=Y]g#QF-7$F]iw$!-'z5Pg*QF-7$F_`o$!-yQ-GTSF-7$Feiw$!-C%fCMH%F -7$Fd`o$!-\"z.(f$p%F-7$Fi`o$!-.,m!)\\XF-7$F^ao$!-C$4H$*R%F-7$Fcao$!-a] #G%oUF-7$Fhao$!-j!f9D:%F-7$F]bo$!-NCn%p5%F-7$Fbbo$!-@(=Xj2%F-7$Fh\\t$! -^*H6!pSF-7$Fgbo$!-h:3goSF-7$F`]t$!-^]VewSF-7$F\\co$!-xW0n%4%F-7$$\"5+ +++D\"GI?e#Fbfl$!-?$)\\[zTF-7$Faco$!-w\"f8$[VF-7$$\"5)yyyG;$=O&f#Fbfl$ !-+&*\\2NYF-7$Ffco$!-13*o4!\\F-7$F[do$!-Pf;2=]F-7$Fieo $!-aQ8@V[F-7$F^fo$!-')3A%Rn%F-7$Fhfo$!-U[]:!Q%F-7$F\\ho$!-,8ys0VF-7$Fa ho$!-%HT2?b%F-7$Ffho$!-(fT-M(\\F-7$F[io$!-GsJ#)*z%F-7$F`io$!-Ajo#>j%F- 7$Fjio$!-Sd]lKVF-7$F^[p$!-q.TcbUF-7$Fc[p$!-fD(f.[%F-7$Fh[p$!-qT&4+&[F- 7$F]\\p$!-*R?:Ao%F-7$Fb\\p$!-)3cB)>XF-7$F\\]p$!-W0M)Q?%F-7$F`^p$!-,1v. 7TF-7$Fe^p$!-!\\V=^L%F-7$Fj^p$!-C\"G4(4YF-7$F__p$!-_&Q,5*RF]`p7$Facp$! .'ze$y[p$F]`p7$Fedp$!.[rL>ch$F]`p7$Fabx$!.%)zXka#QF]`p7$Ffbx$!.:Ql(*o# RF]`p7$F[cx$!.ddn\"QrPF]`p7$Fjdp$!.Sj=y>i$F]`p7$Fccx$!.F'>D\")fLF]`p7$ F_ep$!.4W@2()G$F]`p7$F`dx$!.)H6-D&[$F]`p7$Fdex$!.4[p`3_$F]`p7$Fiex$!.+ 5a17P$F]`p7$Fdep$!.G!H%*4GKF]`p7$Fafx$!.0m'R:**HF]`p7$Fiep$!.kFZ#>QHF] `p7$Fifx$!.p\"z>j\"*HF]`p7$F^gx$!.8ZmZT4$F]`p7$Fcgx$!.]kCX%oJF]`p7$Fhg x$!.jPwf<>$F]`p7$F]hx$!.YXBn$eJF]`p7$Fbhx$!.3fF%HDJF]`p7$Fghx$!.k)[l;' *HF]`p7$F^fp$!.VWsZB(GF]`p7$F_ix$!.wO[*HWFF]`p7$Fdix$!.z+1v\\j#F]`p7$F iix$!.+YQvUf#F]`p7$F^jx$!.BZ#*=%pDF]`p7$$\"5]^^^,R&)*oZ$Fbfl$!.F,vQ^c# F]`p7$Fcjx$!.i*y.wnDF]`p7$$\"5KKKK#[*Gs$[$Fbfl$!.Qg(GfyDF]`p7$Fcfp$!.8 -'p7*f#F]`p7$Fhfp$!.jqR>)eCF]`p7$F]gp$!.kFXUwB#F]`p7$Fbgp$!.WNXHQ3#F]` p7$Fggp$!.yj?1)*)=F]`p7$F\\hp$!/XjS*F`hp7$F`jp$!./d`WmR)F`hp7$Fejp$!.>@weIG(F`hp7 $Fjjp$!.U)=!R(zlF`hp7$F_[q$!.d%\\w!yp&F`hp7$Fd[q$!.])\\vG!3&F`hp7$Fi[q $!/G?Wk\"*3VF]\\q7$F_\\q$!/ecxC7lQF]\\q7$Fd\\q$!/5w)*3tdKF]\\q7$Fi\\q$ !/(z,E/4#HF]\\q7$F^]q$!/!RLTf$GCF]\\q7$Fc]q$!/%o(*=VW:#F]\\q7$Fh]q$!/1 (pY8/y\"F]\\q7$F]^q$!/ePvey)e\"F]\\q7$Fb^q$!/=)p'y3s7F]\\q7$Fg^q$!/**4 9M&Q8\"F]\\q7$F\\_q$!/%z.X[01*F`_q7$Fb_q$!/rv_d$*y!)F`_q7$Fg_q$!/\\s\" \\\"QKkF`_q7$F\\`q$!/3&y&p6gcF`_q-Fa`q6&Fc`qFd`qF(F(-Fh`q6#%3efficient ~scheme~BG-F$6%7d^lF'7$F2$!'(y#\\F07$FP$!)s%fi%F07$F^o$!)+&3'\\F07$F\\ q$\")\\'**[#F07$Faq$\")5OzCF07$Ffq$\")t6yFF07$F`r$\")G$fu$F07$Fjr$\")U F/mF07$F_s$\"*()R]P\"F07$Fds$\"*SU%RDF07$Fis$\"*h!4?DF07$F^t$\"*I#\\TD F07$Fht$\"*Ya-p#F07$Fbu$\"*WF3?$F07$Fgu$\"*q3uf%F07$F\\v$\"*3Exp'F07$F av$\"*ouxi'F07$Ffv$\"*20vk'F07$F`w$\"*$QdZpF07$Fjw$\"*hx3*zF07$F_x$\"+ O5;$3\"F07$Fdx$\"+!\\$\\*H\"F07$Fix$\"+([eTG\"F07$F^y$\"+Fl**z7F07$Fhy $\"+*y7VJ\"F07$Fbz$\"+#*))eW9F07$Fgz$\"+!)*yy!=F07$F\\[l$\"+ad`g@F07$F a[l$\"+eI:E@F07$Ff[l$\"+t8#)=@F07$F[\\l$\"+qoMS@F07$F`\\l$\"+p$pe=#F07 $Fe\\l$\"+pNppAF07$Fj\\l$\"+=oL7CF07$Fd]l$\"+2H)))*HF07$Fb_l$\"+3k)pA$ F07$Fg_l$\"+&zLX<$F07$F\\`l$\"+uRAVJF07$Fhiu$\"+3<3cJF07$Fa`l$\"+l(Rf> $F07$F`ju$\"+!4$Q\"G$F07$Ff`l$\"+@e#\\'eF07$Fg[r$\"+Cp?\"='F07$F [fl$\"+;'HXp'F07$$\"5&\\\\\\u!)p0f)**F0$\"+W4HioF07$$\"5CCCuOxWA+5Fbfl $\"+LKO?qF07$$\"5**)*)RF\\Qe=+\"Fbfl$\"+\\+*)3qF07$F_\\r$\"+MpT(*pF07$ $\"5CBBtg2,w15Fbfl$\"+m6ZupF07$F`fl$\"+_l_^pF07$Fffl$\"+cu$yx'F07$F[gl $\"+>&)3ZmF07$F\\_v$\"+dZI\\mF07$F]]r$\"+G,#**o'F07$Fd_v$\"+tj(>z'F07$ Fb]r$\"+YNc))pF07$Fg]r$\"+'H_eK(F07$F\\^r$\"+&*QqmyF07$Fa^r$\"+*Rm9&zF 07$F`gl$\"+vfV%*yF07$Fegl$\"+%\\Wlo(F07$Fjgl$\"+!G=G^(F07$F^av$\"+37g1 vF07$F_hl$\"+M:t.xF07$Ffav$\"+,k+_zF07$F[bv$\"+J6hi$)F07$F`bv$\"+m!4>[ )F07$Fdhl$\"+'*oF;%)F07$Fihl$\"+pa(f9)F07$F^il$\"+Q`38zF07$F^cv$\"+F.z #)yF07$Fcil$\"+JBB*)zF07$Ffcv$\"+&F07$Fjfm$!+oXn,_F07$F_gm$!+!)*f:F&F07$Fdgm$!+Bh!QW&F07$Figm$!+wJ:'e )F07$F^hm$!,kiO%e5F07$Fchm$!,]2AD.\"F07$Fhhm$!,Z()*z45F07$F]im$!,%*4&3 -5F07$Fbim$!+yoK(***F07$F\\jm$!,i'4m15F07$Ffjm$!,%R2#)G5F07$F[[n$!,vg) fl9F07$F`[n$!,(**=90;F07$Fe[n$!,&)\\![k:F07$Fj[n$!,u*yvG:F07$F_\\n$!,j \"f5;:F07$Fd\\n$!,.\\m2^\"F07$Fi\\n$!,i&*4B^\"F07$F^]n$!,LMCx^\"F07$Fc ]n$!,-y=!G:F07$Fh]n$!,ic-Wa\"F07$F]^n$!,L6z9q\"F07$Fb^n$!,#)Gt&>@F07$F g^n$!,qwq1?#F07$F\\_n$!,n-;>E#F07$Fa_n$!,jFbVD#F07$Ff_n$!,X49oC#F07$F[ `n$!,!e.zJAF07$F``n$!,h'[%o@#F07$Fe`n$!,A$Q>(=#F07$Fj`n$!,=m+z:#F07$F_ an$!,j%Qi0@F07$Fdan$!-$Rw#ye?F-7$Fian$!-?kYWS?F-7$F^bn$!-[(oX(G?F-7$Fc bn$!-QNMsF?F-7$Fhbn$!-]qS@V?F-7$F]cn$!-Se**H!>#F-7$Fbcn$!-(4ke&QEF-7$F gcn$!-Y]pwGFF-7$F\\dn$!-_*4\\)GGF-7$Fadn$!-2;9.JGF-7$Ffdn$!-bAwe?GF-7$ F[en$!-W\\=z*z#F-7$F`en$!-x5%=\"zFF-7$Feen$!-yGV9QFF-7$Fjen$!-y-mr(p#F -7$F_fn$!-^decFEF-7$Fdfn$!-F<4fkDF-7$Fifn$!-.$3/(RDF-7$F^gn$!-0\\;RBDF -7$Fcgn$!-B$oL3_#F-7$Fhgn$!-J31^RDF-7$F]hn$!-ng$[Kq#F-7$Fbhn$!-%e?KA=$ F-7$Fghn$!-s4q\"pF$F-7$F\\in$!-LCO@qLF-7$Fain$!-S/j#yN$F-7$Ffin$!-vYfZ XLF-7$F[jn$!-pbe)3K$F-7$F`jn$!-kTIW'H$F-7$Fejn$!-'o`.![KF-7$Fjjn$!-4I \"*>+KF-7$F_[o$!-)\\7^76$F-7$Fd[o$!-s#H0..$F-7$Fi[o$!-#f,#4(*HF-7$F^\\ o$!-Fy.MtHF-7$Fc\\o$!-\\eK%['HF-7$Fh\\o$!-dEy,!)HF-7$Fagw$!-%H?$4RIF-7 $F]]o$!-\"o0([kJF-7$Figw$!-d9g%pQ$F-7$Fb]o$!-yt()[ZPF-7$$\"5999R&p9#[+ CFbfl$!-&\\q4h#QF-7$$\"5YXX&zg$H=-CFbfl$!-!o7:0\"QF-7$$\"5xww^?DP)QS#F bfl$!-5LJ(\\z$F-7$$\"53333L9Xe0CFbfl$!-W2O[zPF-7$$\"5rqq?e#4')*3CFbfl$ !-dF;m[PF-7$Fg]o$!-XD#\\!=PF-7$$\"5feeeLF3>>CFbfl$!-!4Hnul$F-7$F\\^o$! ->wz$yf$F-7$Ff^o$!-[a'['3MF-7$Fj_o$!-v&f%HdLF-7$F]iw$!-L([iJU$F-7$F_`o $!-*[aJ1c$F-7$Feiw$!-;mJQ,QF-7$Fd`o$!-OB:?(=%F-7$Fi`o$!-!zpX'fSF-7$F^a o$!-\")pTLDRF-7$Fcao$!-tNqC3QF-7$Fhao$!-H.#*)Rq$F-7$F]bo$!-$\\bYFm$F-7 $Fbbo$!-11d(\\j$F-7$Fgbo$!-#eXj\"GOF-7$F\\co$!-ApH*Gl$F-7$Fhh[l$!-,zkK LPF-7$Faco$!-oHZn%*QF-7$F`i[l$!-MwmHrTF-7$Ffco$!-x-'[;V%F-7$F[do$!-wI3 =!G%F-7$F`do$!-vfz[LTF-7$Fedo$!-Cv[;0RF-7$Fjdo$!-*HCG@$QF-7$Fdj[l$!-y] f_'*QF-7$F_eo$!-!4jdV.%F-7$F\\[\\l$!-#HL\"GwUF-7$Fdeo$!-$Hb_ue%F-7$Fie o$!-FPkfFWF-7$F^fo$!-4aM!GF%F-7$Fhfo$!-`H9(G+%F-7$F\\ho$!--(f,V$RF-7$F aho$!-(4$pXsTF-7$Ffho$!-,WQd)e%F-7$F[io$!-&>^B%GWF-7$F`io$!-e***eMF%F- 7$Fjio$!-5EG*e*RF-7$F^[p$!-c))p&R#RF-7$Fc[p$!-x`CtUTF-7$Fh[p$!-4q<25XF -7$F]\\p$!-5^l.aVF-7$Fb\\p$!-xYE)H?%F-7$F\\]p$!-#pH/y!RF-7$F`^p$!-D)*[ C@QF-7$Fe^p$!-#)3)\\)RSF-7$Fj^p$!-!HJ;hJ%F-7$F__p$!-?o!zq:%F-7$Fd_p$!- u)pxN+%F-7$F_`p$!.*G!)**e:PF]`p7$Fcap$!.Zm!fH`OF]`p7$Fhap$!.$RN\"H](QF ]`p7$F]bp$!.'GfZI[SF]`p7$Fbbp$!.p#3VQ,RF]`p7$Fgbp$!.P F]`p7$F[cx$!.d;Am@d$F]`p7$Fjdp$!.Heds0V$F]`p7$Fccx$!.bhTT3=$F]`p7$F_ep $!.Q@8aF6$F]`p7$F`dx$!.xkul%4LF]`p7$Fdex$!.`vNG:N$F]`p7$Fiex$!.p$Q&p!4 KF]`p7$Fdep$!.C$=qtsIF]`p7$Fafx$!.K*y+V`GF]`p7$Fiep$!.ArJx[z#F]`p7$F^g x$!.W-C6>&HF]`p7$Fbhx$!.-9?+())HF]`p7$Fghx$!.>&*G6_'GF]`p7$F^fp$!.l;AR nu#F]`p7$F_ix$!.5fmFRi#F]`p7$Fdix$!.-qnR%=DF]`p7$Fiix$!.Z32%*)yCF]`p7$ F^jx$!.nqNaYX#F]`p7$F`e\\l$!.\\vP>0X#F]`p7$Fcjx$!.e%fYA`CF]`p7$Fhe\\l$ !.YcbPTY#F]`p7$Fcfp$!.11HI[[#F]`p7$Fhfp$!.U-Qe8O#F]`p7$F]gp$!.>)[')H[@ F]`p7$Fbgp$!.0$GnK4?F]`p7$Fggp$!.zI:28#=F]`p7$F\\hp$!/*\\U;*G3JKF]\\q7$Fi\\q$!/O\"*HiH,HF]\\q7$F^]q$!/V\"RFfnT#F]\\q7$Fc]q$!/ di+HxY@F]\\q7$Fh]q$!/JNN@&zx\"F]\\q7$F]^q$!/zj#\\y8f\"F]\\q7$Fb^q$!/EO cDlu7F]\\q7$Fg^q$!/CK+bbQ6F]\\q7$F\\_q$!/U#3#HX5\"*F`_q7$Fb_q$!/a7G8q` \")F`_q7$Fg_q$!/&eWwc4\\'F`_q7$F\\`q$!/ru-G;LdF`_q-Fa`q6&Fc`q$\")#)eqk Ff`q$\"))eqk\"Ff`qF^h`l-Fh`q6#%3efficient~scheme~CG-F$6%7a]lF'7$F+$!'Y \\5F07$F2$!(ar&oF07$F<$!)k5jAF07$FF$!))\\D;'F07$FK$!*sTHY\"F07$FP$!*bt Q8$F07$FZ$!*nZSO$F07$F^o$!*Rd#)R$F07$Fho$!*%QJHNF07$Fbp$!*)zx-RF07$Fgp $!*nFky%F07$F\\q$!*Ki%fhF07$Faq$!*z%REhF07$Ffq$!*O5=9'F07$F`r$!*\\;NC' F07$Fjr$!*_9#4lF07$F_s$!*_p%zqF07$Fds$!*Y;\"*)yF07$Fis$!*#H=GyF07$F^t$ !*@?Ly(F07$F\\v$!*@;6/)F07$Fav$!*ZVe&zF07$Ffv$!*1OO&yF07$F`w$!*\"[TTxF 07$Fjw$!*'*\\TX(F07$F_x$!*(=%)4nF07$Fdx$!*9MP5'F07$Fix$!*]1$HgF07$F^y$ !*_&)o*eF07$Fhy$!*=WKl&F07$Fbz$!*(fV#)\\F07$Fgz$!*x'=%H$F07$F\\[l$!*l \\ym\"F07$Fa[l$!*cSnj\"F07$Ff[l$!*Yc3T\"F07$Fj\\l$\")g(zO'F07$Fb_l$\"* -8,f&F07$Fg_l$\"*wu\\]&F07$F\\`l$\"*7Vth&F07$Ff`l$\"*k/JG)F07$Fjal$\"+ /\"[,e\"F07$F_bl$\"+(Q(\\^:F07$Fdbl$\"+o#)Q`:F07$F^cl$\"+]E`7>F07$Fbdl $\"+#fk<*GF07$$\"5XZZZZAm,?#*F0$\"+S\"*[gGF07$Fgdl$\"+a0jIGF07$$\"5&yy yyGT\"He%*F0$\"+bE&*3GF07$F\\el$\"+[%fn\"GF07$Ffel$\"+!)QxxKF07$F`fl$ \"+*)QPoWF07$$\"5zyyyy`[ZA5Fbfl$\"+?,P7WF07$Fffl$\"+)GL'eVF07$$\"5\"44 44frot/\"Fbfl$\"+ukmI&\\F07$F`gl$\" +aA,`iF07$$\"53333e&4!pB6Fbfl$\"+])z/<'F07$Fegl$\"+936\"4'F07$$\"5'fff f%eCaZ6Fbfl$\"+\">0p-'F07$Fjgl$\"+q$GO,'F07$F_hl$\"+E\\BzmF07$Fdhl$\"+ 3mS9#)F07$Fb_r$\"+s%3A3)F07$Fihl$\"+,()pbzF07$Fj_r$\"+q$>z&yF07$F^il$ \"+mg#='yF07$Fcil$\"+2i!)\\()F07$Fhil$\",%*=#QE5F07$Fbjl$\"+0!3I$**F07 $F\\[m$\"+&*\\)pt*F07$Fa[m$\",NomW1\"F07$Ff[m$\",V-r)H7F07$F`\\m$\",T4 Yp=\"F07$Fd]m$\",/\\ZM;\"F07$Fi]m$\",!>#z*p7F07$F^^m$\",iB')>V\"F07$Fh ^m$\",[WLZP\"F07$F\\`m$\",sowyM\"F07$Fa`m$\",-\"oM-:F07$Ff`m$\",;yuUh \"F07$F`am$\",T@a'[:F07$Fdbm$\",$*G#3>:F07$Fibm$\",_Y+_o\"F07$F^cm$\", &R&Q)*y\"F07$Fhcm$\",b$448F07$F`fm$\",Y@^q&=F07$Fdgm$\",TpFq!=F07$Figm$\",*pe)H)>F 07$F^hm$\",y](Hv?F07$Fhhm$\",E:)fv>F07$Ffjm$\",:T$z@>F07$F[[n$\",'))=+ V@F07$F`[n$\",a^Lk<#F07$Fj[n$\",r>(oo?F07$Fh]n$\",7$4q9?F07$F]^n$\",/B ;K2#F07$Fb^n$\",yCF-7$F[en$\"-GV#*=,CF-7$F `en$\"-W[*eMQ#F-7$Feen$\"-\"z9:$[BF-7$Fjen$\"-V8ih8BF-7$Fdfn$\"-/c!ei> #F-7$Fhgn$\"-DV$z18#F-7$F]hn$\"-MJ#*[\"=#F-7$Fbhn$\"-W+$*f$R#F-7$Fghn$ \"->Z\">'QCF-7$F\\in$\"-!>*3F$[#F-7$Fain$\"-6cN9uCF-7$Ffin$\"-GbM/lCF- 7$F[jn$\"-\"yvCpW#F-7$F`jn$\"-,-X\"*GCF-7$Fejn$\"-b*f?KR#F-7$Fjjn$\"-R G$yzN#F-7$Fd[o$\"-6x!*=IAF-7$Fh\\o$\"-?t'zh:#F-7$Fagw$\"-$y:i*o@F-7$F] ]o$\"-H%3jL@#F-7$Figw$\"-)e<bbBF-7$Ff^o$\"-'4J]&GAF-7$Fj_o$\"-(=O.1;#F-7$F]iw$\"-,e &)4x@F-7$F_`o$\"-!zUVsA#F-7$Feiw$\"-*z8&*zK#F-7$Fd`o$\"-Td%GI]#F-7$Fi` o$\"-ze+fDCF-7$F^ao$\"-R(Q!HXBF-7$Fhao$\"-'*Gtu4AF-7$F\\co$\"-i_\"o_9# F-7$Faco$\"-1?V%>B#F-7$Ffco$\"-%psv&pCF-7$F[do$\"-)[en^Q#F-7$F`do$\"-w PnM.BF-7$Fedo$\"-S\"ppH<#F-7$Fjdo$\"-T+L?1@F-7$Fdj[l$\"-mZF-7$Fc[p$\"-otn\")o?F-7$Fh[p$\"-`U*z $[AF-7$F]\\p$\"-2N:fq@F-7$Fb\\p$\"-;vWC&4#F-7$F\\]p$\"-T3%4a%>F-7$F`^p $\"-]LEa$)=F-7$Fe^p$\"-jTgr\")>F-7$Fj^p$\"-=8)pZ7#F-7$F__p$\"-hTcZY?F- 7$Fd_p$\"-UgN%3(>F-7$F_`p$\".@u?dg#=F]`p7$Fcap$\".[z%f()zF]`p7$Fbbp$\".F]`p7$Fgbp$\"..[ 04W%=F]`p7$Facp$\".2C,$=/cl\"F]`p7$Fabx$\".uSTPp;F]`p7$Fiex$\".**z'fS)f\"F]`p7$Fdep$\".Gx-h.`\"F]`p7$Faf x$\".v?yG%=9F]`p7$Fiep$\".L84eUQ\"F]`p7$F^gx$\"..!yE=y9F]`p7$Fbhx$\".( 4E^-8:F]`p7$Fghx$\".mcF/0X\"F]`p7$F^fp$\".RY#[W!R\"F]`p7$F_ix$\".%f-+o F8F]`p7$Fdix$\".8!o&3BF\"F]`p7$Fiix$\".$=b;p]7F]`p7$F^jx$\".v#z;&oB\"F ]`p7$F`e\\l$\".;Qf\"HM7F]`p7$Fcjx$\".yI![pN7F]`p7$Fhe\\l$\".a8,!3U7F]` p7$Fcfp$\".=T&[oa7F]`p7$$\"5#HHHHz'RJ$\\$Fbfl$\".=)o$e$)H\"F]`p7$$\"57 8888jG\\*\\$Fbfl$\".#4z))**y8F]`p7$$\"5KLLLLe4c06 F]`p7$$\"58888j+2[uNFbfl$\".#3vTt%4\"F]`p7$$\"5OOO')[R-kxNFbfl$\".G>#G %G4\"F]`p7$$\"5gfffMy(*z!e$Fbfl$\".'[n[2%4\"F]`p7$$\"5$GGG.sJfRe$Fbfl$ \"..QJj\"*4\"F]`p7$F]gp$\".n(=c)*36F]`p7$Fbgp$\".@4%>=k5F]`p7$$\"5#GGG GGoV.l$Fbfl$\".8@:k0-\"F]`p7$$\"5XXXXX&zd@m$Fbfl$\"-1eB_D)*F]`p7$$\"5w wwww^[1oOFbfl$\"-%*=s2x'*F]`p7$$\"5333333>(Rn$Fbfl$\"-NG_GN'*F]`p7$$\"5????X^F3#p$ Fbfl$\"-_Y^F/**F]`p7$$\"5pppp>#[z$)p$Fbfl$\".DzH.V/\"F]`p7$$\"5122K)[m `**p$Fbfl$\".$\\X@Hj5F]`p7$$\"5WWW%pv%y_,PFbfl$\".7NG\"*y0\"F]`p7$$\"5 \"==ob-.-Jq$Fbfl$\".PzpV<0\"F]`p7$$\"5=>>>%H@wYq$Fbfl$\".mgCHc/\"F]`p7 $$\"5$RRR9$yX#yq$Fbfl$\".V%)4*\\L5F]`p7$$\"5oooooVH(4r$Fbfl$\".u!y&*\\ @5F]`p7$$\"5nnnn<0kcBPFbfl$\".GPM\")yu*F`hp7$F\\hp$\".^P\\yF`hp7$F\\ip$\".C&[)zo1(F`hp7$Faip$\".')R\"pAxlF`hp7$Ffip$\".Re t*\\'*fF`hp7$F[jp$\".rwZ?-_&F`hp7$F`jp$\".S)y-9t\\F`hp7$Fejp$\".)[$[.J W%F`hp7$Fjjp$\".LKg4i0%F`hp7$F_[q$\".H6x7'=OF`hp7$Fd[q$\".%=4?%HE$F`hp 7$Fi[q$\"/]-[dq]GF]\\q7$F_\\q$\"/2AHRz(e#F]\\q7$Fd\\q$\"/.FfROZAF]\\q7 $Fi\\q$\"/.CgrUg?F]\\q7$F^]q$\"/8C!HEzu\"F]\\q7$Fc]q$\"/2@#fgCe\"F]\\q 7$$\"5********\\PBB2YFbfl$\"/:L:XC%f\"F]\\q7$$\"5EFFFF-$**)>YFbfl$\"/9 Q^*[P]\"F]\\q7$$\"5aaaa/nicKYFbfl$\"/J/zd@=9F]\\q7$Fh]q$\"/'y1IEyL\"F] \\q7$$\"5EFFFF-+meYFbfl$\"/*)pMdEg7F]\\q7$$\"5ssssssn3sYFbfl$\"/Z7#ol% )>\"F]\\q7$$\"5=====VN^&o%Fbfl$\"/KiFvju6F]\\q7$F]^q$\"/mN8)z1C\"F]\\q 7$$\"511111\"QiNs%Fbfl$\"/C)z[o27\"F]\\q7$Fb^q$\".9!y-,\")**F]\\q7$$\" 51222d>qLgZFbfl$\".;aC!Q]%*F]\\q7$$\"5lllll!f*[sZFbfl$\".B&f%zh-*F]\\q 7$$\"5CCCCuh@k%y%Fbfl$\".)Q@q%p$))F]\\q7$Fg^q$\".oHYV+;*F]\\q7$$\"5zzz zzHlxA[Fbfl$\"/?uy!\\bT)F`_q7$F\\_q$\"/-bC@QFuF`_q7$$\"5?????X4sh[Fbfl $\"/db.oH+qF`_q7$$\"5jjjjjjNou[Fbfl$\"/e0WG'=n'F`_q7$$\"512222#=Yw)[Fb fl$\"/TYmIZrlF`_q7$Fb_q$\"/Dn%zMw#pF`_q7$$\"59::::S#yQ#\\Fbfl$\"/C#egu $zhF`_q7$Fg_q$\"/)p[am>^&F`_q7$$\"5%[[[[)f2Og\\Fbfl$\"/YI!Hc(y^F`_q7$$ \"5!**)*)*)*)RQdt\\Fbfl$\"/HY!4fZ\"\\F`_q7$$\"5&\\\\\\\\*>py')\\Fbfl$ \"/enZ\"z3\"[F`_q7$F\\`q$\"/y&)f4Ru]F`_q-Fe`y6&Fc`q$\"\"'Fi`yF(Fj`y-Fh `q6#%Sscheme~with~stability~radius~=~imag~axis~inclusionG-%&TITLEG6#%U error~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%%FONTG6$%*HELVE TICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fdgdl-%%VIEWG6$;F(F\\`q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shamp ine scheme" "a minor modification of the Bogacki-Shampine scheme" "eff icient scheme A" "efficient scheme B" "efficient scheme C" "scheme wit h stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "evalf[20](plot([1 -'hn_RK5_1'(x)/h(x),1-'hn_RK5_2'(x)/h(x),1-'hn_RK5_3'(x)/h(x),1-'hn_RK 5_4'(x)/h(x),\n1-'hn_RK5_5'(x)/h(x),1-'hn_RK5_6'(x)/h(x)],x=0..10,\nfo nt=[HELVETICA,9],color=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLO R(RGB,.6,0,.8)],\nlegend=[`Bogacki-Shampine scheme`,`a minor modificat ion of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient sc heme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`],\ntitle=`relative error curves for 7 stage order 5 Runge- Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 822 619 619 {PLOTDATA 2 "6,-%'CURVESG6%7[o7$$\"\"!F)F(7$$\"5mmmmm;arz@!#?$\"+'>$\",;ooz.\"F-7$$\"5mmmm\"z/3u C\"FB$\"+C%>^!))F-7$$\"5++++DJ$RDX\"FB$\"+bZ&o/#F-7$$\"5mmmm\"zR'ok;FB $!+7s.%\\\"FB7$$\"5++++D1J:w=FB$!+Pr'>8&FB7$$\"5LLLLL3En$4#FB$!,M&=9T8 FB7$$\"5mmmm;/RE&G#FB$!,acHPJ#FB7$$\"5+++++D.&4]#FB$!,Ln'=#o%FB7$$\"5+ ++++vB_wFB7$$\"5+++++v'Hi#HFB$!-`tpn&=\"FB7$$\"5mmmm\"z*ev :JFB$!-UHcKu\"FB7$$\"5LLLLeR\"3Gy%FB$!.NZ``/G\"FB7$$\" 5mmmm;/T1&*\\FB$!.z$Qw8\"=\"FB7$$\"5mmmm\"zRQb@&FB$!-aahN(o'FB7$$\"5** *****\\(=>Y2aFB$\"./;Vz!\\;F-7$$\"5mmmm;zXu9cFB$\"/^8+WTU<#F-7$$\"5KLLLL3s?6zFB$\"1[smD[s%\\.*)G(QF-7$$\"5lmmmmm*RRL)FB$\"1T'=a0_/4&F-7$$\"5lmmm;a<.Y&) FB$\"1+(H(*G&=JmF-7$$\"5KLLLe9tOc()FB$\"1pmdk=qq&)F-7$$\"5**********\\ Qk\\*)FB$\"2f8Gzkg!*4\"F-7$$\"5KLLL$3dg6<*FB$\"201ZC%=o09F-7$$\"5++++v oTAq#*FB$\"2vM5#>op\"e\"F-7$$\"5lmmmmmxGp$*FB$\"2txo8O6rx\"F-7$$\"5ILL $eRA5\\Z*FB$\"27rEd>1C+#F-7$$\"5)******\\7oK0e*FB$\"2K!pR\\&G3E#F-7$$ \"5+++++]oi\"o*FB$\"2kkK,$36NDF-7$$\"5)******\\(=5s#y*FB$\"2Ye!4LpMRGF -7$$\"5+++D1k2/P)*FB$\"2*zhxrWw6JF-7$$\"5+++]P40O\"*)*FB$\"2h=@2fi:E$F -7$$\"5+]i!RvZ]\")*)*FB$\"2Vd&*>8;/T$F-7$$\"5++DJqX/%\\!**FB$\"2J634Wd *oMF-7$$\"5+](=nQTI<\"**FB$\"2q'oWRt&*oMF-7$$\"5++]7.#Q?&=**FB$\"2(RE* p(e&*oMF-7$$\"5++v$f$=.5K**FB$\"26yNucD*oMF-7$$\"5+++voa-oX**FB$\"2,.5 xrA*oMF-7$$\"5++Dc,\">g#f**FB$\"2n>\"[+Y\"3Z$F-7$$\"5++]PMF,%G(**FB$\" 2'\\r$*[>h'[$F-7$$\"5++v=nj+U')**FB$\"20D]C6!GlNF-7$$\"#5F)$\"2Ex>MVrA 'QF--%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine ~schemeG-F$6%7[oF'7$F+$\")/jM%*F-7$F1$\"*cU+#QF-7$F6$\"+:@iT5F-7$F;$\" +1\"*ox?F-7$F@$\"+(\\_I7$F-7$FF$\"+^z/oJF-7$FK$\")R*)3(*F-7$FP$!+iE#3. \"FB7$FU$!+h\"[ff$FB7$FZ$!+*=5F&**FB7$Fin$!,#pN@#y\"FB7$F^o$!,D9Bbv$FB 7$Fco$!,$G&\\dF'FB7$Fho$!,*4I6p**FB7$F]p$!-\"f!oh::FB7$Fbp$!-XM*o*=AFB 7$Fgp$!-QIr:BJFB7$F\\q$!-Cec@vUFB7$Faq$!-yo\\^!f&FB7$Ffq$!-8gd@srFB7$F [r$!-xK=pU()FB7$F`r$!.qP:4+.\"FB7$Fer$!.0qh6z2\"FB7$Fjr$!-vq@R#>*FB7$F _s$!-$zPz,Q$FB7$Fds$\".Q(***fKo&F-7$Fis$\"/P;&[*GH@F-7$F^t$\"/qy-TsLYF -7$Fct$\"/0$[)>pj%)F-7$Fht$\"0PBGOYHT\"F-7$F]u$\"0Lq\\ZxKA#F-7$Fbu$\"0 jyBGgbP$F-7$Fgu$\"05>1jFL6&F-7$F\\v$\"0y0)y2lPsF-7$Fav$\"1i5z*[um4\"F- 7$Ffv$\"1?gd'4/6`\"F-7$F[w$\"1CgO'4'>.AF-7$F`w$\"1%)=?xIN^HF-7$Few$\"1 >()pPe$=\"RF-7$Fjw$\"1Mkqo!Qa8&F-7$F_x$\"1KI*oGDIo'F-7$Fdx$\"1!Gn,#F-7$Fbz$\"2n]>0%[6rAF-7$Fgz$ \"2TP()QK#4YDF-7$F\\[l$\"2Npf(pd1^GF-7$Fa[l$\"2a(=@%))[S7$F-7$Ff[l$\"2 MOOiR`UF$F-7$F[\\l$\"2')f\"3&H\"RBMF-7$F`\\l$\"2!Rjm'pN?[$F-7$Fe\\l$\" 2*f$QdfN?[$F-7$Fj\\l$\"2Hc(4UT.#[$F-7$F_]l$\"2PG'\\UT+#[$F-7$Fd]l$\"2) \\+5(G/?[$F-7$Fi]l$\"2:Wb=Y7R[$F-7$F^^l$\"2yaASlt(*\\$F-7$Fc^l$\"2.s!e LUkyNF-7$Fh^l$\"2\\$f\"*y*)=wQF--F]_l6&F__lF`_lF(F`_l-Fd_l6#%Ta~minor~ modification~of~the~Bogacki-Shampine~schemeG-F$6%7\\oF'7$F+$!(r@x%FB7$ F1$\"*#=9s\\F-7$F6$\"+OdcM?F-7$F;$\"+583f\\F-7$F@$\"+w+8_#*F-7$FF$\",I C*es8F-7$FK$\",d#HC2:F-7$FP$\"+>T)o\"fF-7$FU$!+)f,(*)GFB7$FZ$!,;Z8%e8F B7$Fin$!,+>hGz#FB7$F^o$!,gG$))zoFB7$Fco$!-#=vA;B\"FB7$Fho$!-w]NMp?FB7$ F]p$!-ucI.3LFB7$Fbp$!-$RX-/3&FB7$Fgp$!-G8YKCvFB7$F\\q$!.NvmDb3\"FB7$Fa q$!.(4\"y4`^\"FB7$Ffq$!.kVLQt3#FB7$F[r$!.Cm9[tz#FB7$F`r$!.S=F)**oPFB7$ Fer$!.e/'fuDZFB7$Fjr$!.JI)fF\"3'FB7$F_s$!.zG![z%Q(FB7$Fds$!.t<:7\"e&)F B7$Fis$!.K<\"F-7$ F`w$\"10$e@f&oB xspQl%F-7$Fdx$\"1**y0(3srC'F-7$Fix$\"1mkc#QDTF)F-7$$\"5lmmmT5ASg!*FB$ \"1&yqtS\\H[*F-7$F^y$\"2#>Fc7&3r3\"F-7$Fcy$\"2=H_B(eJQ7F-7$Fhy$\"2/C)H ;dD29F-7$F]z$\"2_t0'**3h-;F-7$Fbz$\"2dakaq>!G=F-7$Fgz$\"2/x!4131p?F-7$ F\\[l$\"2QO&*fF/yL#F-7$Fa[l$\"2!\\+;!zWDe#F-7$Ff[l$\"2y1U3FIGr#F-7$F[ \\l$\"2YMZ2D\\w%GF-7$F`\\l$\"2H;tS:)3,HF-7$Fe\\l$\"2>Q$*=/)3,HF-7$Fj\\ l$\"2DD]nN'3,HF-7$F_]l$\"2qn)p(zX5!HF-7$Fd]l$\"2y8KHQD4!HF-7$Fi]l$\"2Q +Szz/@!HF-7$F^^l$\"2iaj/j;\\\"HF-7$Fc^l$\"2i]&f@4u$)HF-7$Fh^l$\"2\"*>> ]]/LD$F--%&COLORG6&F__l$\"\"\"!\"\"$\"\")F`hmF(-Fd_l6#%3efficient~sche me~AG-F$6%7bqF'7$F+$\"*c!R'Q%F-7$F1$\"+(H9'=9F-7$F6$\"+mhg\"H$F-7$F;$ \"+5cE'R'F-7$F@$\",&GfF`5F-7$FF$\",aX,%*R\"F-7$FK$\",%3u587F-7$FP$!*%y ,XbFB7$FU$!+T!>Q5'FB7$FZ$!,*zz#*RAFB7$Fin$!,&*4E#=WFB7$F^o$!-sV^Zq5FB7 $Fco$!-*H$evD>FB7$Fho$!-Xf=DyKFB7$F]p$!-2u-?S`FB7$Fbp$!-0t4Y&R)FB7$Fgp $!.)pz87y7FB7$F\\q$!.B%f%GE!>FB7$Faq$!.\\x^bBv#FB7$Ffq$!.+h.Ec%RFB7$F[ r$!.P$=o!H`&FB7$F`r$!.S0(y?$)yFB7$Fer$!/gnb&ye/\"FB7$Fjr$!/e#**e#3n9FB 7$F_s$!/%3!*\\BZ(>FB7$Fds$!/2liCFNDFB7$Fis$!/V$o`Yg@$FB7$F^t$!/V?Q!*)H .%FB7$Fct$!/Aq4FB7$F`w$!0/fbv)e*>#FB7$Few$!0-D)yZC#R#FB7 $Fjw$!0^')FB$!0&)z(>%\\tm#FB7$$\"5lmT 5!p]XVm)FB$!0a)Htgt)o#FB7$$\"5IL$3F%z9\\x')FB$!0Z=(45GIFFB7$$\"5l;/,pl W1%o)FB$!0s;z#*z+v#FB7$$\"5++DJ&>XP1p)FB$!0^KW@j2v#FB7$$\"5I$e9;#Q/@(p )FB$!0z$[or%)*p#FB7$$\"5lmm\"zWU$y.()FB$!0^\\w_&[\\EFB7$$\"5++]7`p`2I( )FB$!0dB1*G!)\\EFB7$Fdx$!0)Hlx/QkEFB7$$\"5lmm;H#e0I&))FB$!0%)*HX;U>EFB 7$Fix$!0BM#*p]&QDFB7$F^y$!0M<#oje,BFB7$Fcy$!0f,C[#3u?FB7$Fhy$!0*GZc+_ \"y\"FB7$F]z$!0L`k07lV\"FB7$Fbz$!/<-COW5**FB7$$\"5++]PM_%pJf*FB$!/V\") QaT\"*yFB7$$\"5+++vVBi!eg*FB$!/@Zp&G\\h$FB7$$\"5++]7`%*HW='*FB$!/D=Cv> :OFB7$$\"5+++]il(z5j*FB$!/4[t]0AOFB7$$\"5+++D\"yI`jl*FB$!/\"H#zeUlQFB7 $Fgz$!/(eY_WjQ%FB7$$\"5++]P4@OE%p*FB$!/+_y^*HZ\"FB7$$\"5+++v=#R+pq*FB$ \"0S\">FO*z!HF-7$$\"5++]7Gjr`>(*FB$\"0Tp.,#e2HF-7$$\"5+++]PMRF-7$Fe\\l$\"1&fVG1L=(>F-7$Fj\\l$\"1Fl$*R:!=(>F -7$F_]l$\"151!\\k$yq>F-7$Fd]l$\"1+/\")G>$G'>F-7$Fi]l$\"1lDAby1M>F-7$F^ ^l$\"1%o%[\"=u/*=F-7$$\"5+]7y]&4I'z**FB$\"1E3Kl+Z+>F-7$Fc^l$\"1?uG&*eu **>F-7$$\"5+]Pf$=.5K***FB$\"16/.\"\\DaI#F-7$Fh^l$\"1F1Ord9JIF--F]_l6&F __lF`_lF(F(-Fd_l6#%3efficient~scheme~BG-F$6%7`rF'7$F+$\")^qTDF-7$F1$\" *ZYfE(F-7$F6$\"+51\\'f#F-7$F;$\"+uLN%='F-7$F@$\",W&*4(f6F-7$FF$\",.grF v\"F-7$FK$\",lI&4&*>F-7$FP$\"+36R?%*F-7$FU$!+6)yYV$FB7$FZ$!,[04zv\"FB7 $Fin$!,:lzhr$FB7$F^o$!,)ol]^&*FB7$Fco$!-1Az\\gT=FB7$Faq $!.Yn\")yKo#FB7$Ffq$!.<3=W:(QFB7$F[r$!.Yd;-^Y&FB7$F`r$!.drG-<%yFB7$Fer $!/%RI)*ys/\"FB7$Fjr$!/ZJ7\")o#[\"FB7$F_s$!/.*)\\E>:?FB7$Fds$!/Nu9ij2E FB7$Fis$!/=>5D`NLFB7$F^t$!/V5_[B?UFB7$Fct$!/0Cx7FB7$F\\v$!0+T* )G^'R9FB7$Fav$!05oF2z*f1#FB7$F[w$!0E>eIa2T#FB7$F`w$! 0`'*)[V!)\\FFB7$Few$!0HnN@:65$FB7$Fjw$!0r*FB$!06?Ny:oi%FB7$$ \"5lm\"H#=Xz6-#*FB$!0(Q6N8FVXFB7$$\"5IL$3_+U4$3#*FB$!004GWrKa%FB7$$\"5 ++v=#\\*3]9#*FB$!0\"eF**\\FVXFB7$$\"5lmm;zpBp?#*FB$!0mAM:2La%FB7$$\"5+ +]7`>`2L#*FB$!0<8<81Sa%FB7$$\"5ILL3Fp#eaC*FB$!0F%ena#)[XFB7$$\"5lm;/,> 7%yD*FB$!0O3Z\"GUnXFB7$Fcy$!0ul'**>y8YFB7$$\"5lm\"z>Ok:kF*FB$!0ZJJWexk %FB7$$\"5IL$e*[=rg#G*FB$!09c$oH7\"o%FB7$$\"5++v$fLf)z)G*FB$!0P0t\\*z&p %FB7$$\"5lmm\"H#o+*\\H*FB$!08z^N:^l%FB7$$\"5ILe*)4V:=,$*FB$!0Q$)G+%=PX FB7$$\"5++](oz,ttI*FB$!038t0%=PXFB7$$\"5lmT&QG\\kNJ*FB$!0Y2\\c'=PXFB7$ $\"5ILL$3x'fv>$*FB$!0cpBZ7s`%FB7$$\"5lm;zW<*Q@L*FB$!0M,FpPy`%FB7$$\"5+ ++v=n=_W$*FB$!0'oGzrQUXFB7$$\"5IL$3Fp\"[!pN*FB$!0^u8>$[gXFB7$Fhy$!07Mu lFkg%FB7$$\"5I$eRZ-<*)eP*FB$!04*Q*FB$!0vqQ1]ao%FB7$$\"5IL$e*)4Q$p&R*FB$!0=)Q7P\"f h%FB7$$\"5+]7.d%y%H-%*FB$!0(pa^KE&[%FB7$$\"5lmT5:)='*)3%*FB$!0y62Wj_[% FB7$$\"5I$3xJ6.8\")[*FB$!0$[aeHeJYFB7$$\"5I$3_+ZV9Z\\*FB$ !0$)H7r.)fXFB7$$\"5++]7GQeJ,&*FB$!0ICB+=*zVFB7$$\"5IL3FWX'=X^*FB$!07AD _A*zVFB7$$\"5lmmTg_9sF&*FB$!0XFs`1-Q%FB7$$\"5++DcwfU#4a*FB$!0L.aB-MQ%F B7$$\"5IL$3Fp1FTb*FB$!00U)R@W*R%FB7$$\"5lmT&)3u)Htc*FB$!0b\"*Q^(*fW%FB 7$Fbz$!0v(z[=u;XFB7$Fgz$!0%y%G6*3bVFB7$F\\[l$!0m6)[7j=TFB7$Fgjn$!0n4Dq Oa)QFB7$F\\[o$!0L<\"3U]aOFB7$Fa[o$!0xkRbBYl$FB7$Fa[l$!0\"3TDhucOFB7$F^ \\o$!0U!\\f27qOFB7$Fh\\o$!0z$FB7$Fh^l$!0J) pbWuBFFB-F]_l6&F__l$\")#)eqkFb_l$\"))eqk\"Fb_lF_\\q-Fd_l6#%3efficient~ scheme~CG-F$6%7YF'7$F+$!)(HsG'FB7$F1$!)pIB()FB7$F6$!)(oS+#FB7$F;$\"+\\ 24'>#F-7$F@$\"+oaG\"[(F-7$FF$\",CP'H:[Dor>F-7$Fin$\"-sDPJ%)HF-7$F^o$\"-:nH\"pq &F-7$Fco$\"-l#y`tJ*F-7$Fho$\".Tc\"3q>:F-7$F]p$\".:a!30oCF-7$Fbp$\".sU- )>zRF-7$Fgp$\".r8f3UM'F-7$F\\q$\"/SF-7$Fds$\"0Iv6BpHVffrF-7$F `w$\"1)3!y#H!\\/\"*F-7$Few$\"2?qc\"[[V^6F-7$Fjw$\"2!3PN5#G'[9F-7$F_x$ \"2!4VN]+w8=F-7$Fdx$\"2jOhL&H!GF-7$F^y$\"2&zkk$H Y9[$F-7$Fcy$\"2>HxbkV)fQF-7$Fhy$\"2?9H9 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 " Test 4 of 7 stage, order 5 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathematics of Computation, Vol. 20, no. 95 , (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&% #dyG\"\"\"%#dxG!\"\",$**\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"* &F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2 );" "6#/%\"yG)\"\"#,&\"#8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=-32*x*y(x)*ln(2);\nic := y(-1)=1/ 8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^simplify(log[2](rhs(%)));\nk := un apply(rhs(%),x):\nplot(k(x),x=-1..1,font=[HELVETICA,9],labels=[`x`,`y( x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\" xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" } }{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo 7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3w KL$3$3(F-7$$!3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3 :mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$ $!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$ FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7 $$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn 7$$!3Q+++]$*4)*\\F-$\"3e;N4:OF ap7$$!3]++]PYx\"\\#F-$\"37]-4,Tp9TFap7$$!3QnmTNz>&H#F-$\"3y(*QMk^JnXFa p7$$!3EMLLL7i)4#F-$\"3yCPsPtXE]Fap7$$!3#pm;aVXH)=F-$\"3_cYryDpGbFap7$$ !3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3s*****\\F&*=Y\"F-$\"3K`b3X@Jjk Fap7$$!3')****\\74_c7F-$\"3co2Qfx9woFap7$$!3ZmmT5VBU5F-$\"3E!>K?nVAE(F ap7$$!3)3LLL3x%z#)!#>$\"3C'Q/NU&H#f(Fap7$$!3gKL$e9d;J'Fft$\"3@erx_1&z$ yFap7$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o %4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")Fap7$$!3=o;/^wj!p\"Fft$\"3]3j^O K2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4%yz\")Fap7$$!3S<+D\"GyNH'!#?$\" 3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#)*eHY6>>)Fap7$$\"3o'H$3x\"yY_%F jv$\"3Q>*>/AS,>)Fap7$$\"3&yK$3_Nl.5Fft$\"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b \"Fft$\"3-l$f@nl+<)Fap7$$\"3@CL$3-Dg5#Fft$\"3kJX?)*G!=:)Fap7$$\"3e?Le* ['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ezw5VFft$\"3U_-I6(**[-)Fap7$$\"3L `mmmJ+IiFft$\"3%pB(\\hv&o%yFap7$$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wF ap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\ \\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$ \"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"31))p(30[[c&Fap7$$\"3#z****\\_q n2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%zAF-$\"3ZUhzOe!Rg%Fap7$$\"3U)** *\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\(=GB2FF-$\"3WV]5@%**Rj$Fap7$$ \"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7 $$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\" Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-? C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v^Fgn7$$\"3y)***\\7k.6aF-$\"3# pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9+3>Fgn7$$\"33****\\i!*3`iF-$ \"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4*yfd(\\\"fFM7$$\"3wKLL3N1#4( F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\"3%o[)olFVm:FM7$$\"3Y*******p (G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$\"35omE\\#[Ck$F=7$$\"3fKLLLb dQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 767 "K := (x,y) -> -32*x*y(x)*ln (2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slop e field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine \+ scheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficie nt scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with st ability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\nf or ct to 6 do\n Kn_RK5_||ct := RK5_||ct(evalf(K(x,y)),x,y,x0,evalf(y 0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK5_||ct):\n f or ii to numpts do\n sm := sm+(Kn_RK5_||ct[ii,2]-k(Kn_RK5_||ct[ii ,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\" %\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F, \"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint276\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+!HGR&>!#97$%Ta~min or~modification~of~the~Bogacki-Shampine~schemeG$\"+ " 0 "" {MPLTEXT 1 0 698 "K := (x,y) -> -32*x *y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix ([[`slope field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-S hampine scheme`,`a minor modification of the Bogacki-Shampine scheme`, `efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits : = 20:\nfor ct to 6 do\n kn_RK5_||ct := RK5_||ct(evalf(K(x,y)),x,y,x0 ,evalf(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx )):\nfor ct to 6 do\n errs := [op(errs),abs(kn_RK5_||ct(xx)-kxx)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$** \"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G- %!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)p print296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+;omK=!#=7 $%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+xP'\\v\"F+7 $%3efficient~scheme~AG$\"+nfyf9F+7$%3efficient~scheme~BG$\"+;rL([#F+7$ %3efficient~scheme~CG$\"+xi2c:F+7$%Sscheme~with~stability~radius~=~ima g~axis~inclusionG$\"+!Qu$4v!#>Q)pprint306\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[-1,1 ]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint " }{TEXT -1 98 " to perform numerical integration by the 7 point Newt on-Cotes method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "mthds := [`Bogacki-Shampine scheme`,`a minor modi fication of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficie nt scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n sm \+ := NCint((k(x)-'kn_RK5_||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7, factor=100);\n errs := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10 :\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+)p'ze>!#9 7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+dnz!)=F+7$ %3efficient~scheme~AG$\"+RT " 0 "" {MPLTEXT 1 0 499 "evalf[20](plot(['kn_RK5_1'( x)-k(x),'kn_RK5_2'(x)-k(x),'kn_RK5_3'(x)-k(x),'kn_RK5_4'(x)-k(x),\n'kn _RK5_5'(x)-k(x),'kn_RK5_6'(x)-k(x)],x=-1..1,\nfont=[HELVETICA,9],color =[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlege nd=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sha mpine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sch eme C`,`scheme with stability radius = imag axis inclusion`],\ntitle=` error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 716 585 585 {PLOTDATA 2 "6,-%'CURVESG6%7eo7$$!\"\"\" \"!$F*F*7$$!5nmmmm;p0k&*!#?$\",!)=uEY$F/7$$!5LLLL$37$$!5nmmmm\"4m(G$)F/$\"-H5bVl7F<7$$ !5LLLL$3i.9!zF/$\"-[Qi)e?$F<7$$!5mmmm;/R=0vF/$\",4r\"R&*o!#=7$$!5++++] P8#\\4(F/$\"-7;s*)R9FL7$$!5mmmm;/siqmF/$\"-A*od'*)GFL7$$!5++++](y$pZiF /$\",k;\\%*[&!#<7$$!5LLLLL$yaE\"eF/$\"-Ejy;55Ffn7$$!5mmmmm\">s%HaF/$\" -=H..c;Ffn7$$!5+++++]$*4)*\\F/$\"-nWxroFFfn7$$!5+++++]_&\\c%F/$\"-Qr** )*=WFfn7$$!5+++++]1aZTF/$\",%o'y/l'!#;7$$!5mmmm;/#)[oPF/$\",db-(3$*F`p 7$$!5LLLLL$=exJ$F/$\"-MD_fH8F`p7$$!5LLLLLeW%o7$F/$\"-<%>p\\_\"F`p7$$!5 LLLLLL2$f$HF/$\"-%QVLZt\"F`p7$$!5mmmmT&o_Qr#F/$\"-/R+h%*>F`p7$$!5***** ***\\PYx\"\\#F/$\"-qwq>oAF`p7$$!5mmmmTNz>&H#F/$\"-Q%z$4=DF`p7$$!5LLLLL L7i)4#F/$\"-f2u\\rFF`p7$$!5mmmmTNa%H)=F/$\"--F\\d[IF`p7$$!5********\\P 'psm\"F/$\"-!HNA)=LF`p7$$!5*********\\F&*=Y\"F/$\"-8HF(Rc$F`p7$$!5**** ****\\74_c7F/$\"-*z,I:z$F`p7$$!5mmmmT5VBU5F/$\"-\\wiG/SF`p7$$!4LLLLL3x %z#)F/$\"-LQ**3'=%F`p7$$!5ILLL$e9d;J'!#@$\"-Q47M@VF`p7$$!4LLLL$3s$QM%F /$\"-UY**zAWF`p7$$!5immmT&QdDG$Fjt$\"-meMciWF`p7$$!5&*******\\ivF@AFjt $\"-\"pIX9\\%F`p7$$!5imm;/^wj!p\"Fjt$\"-Z_/r,XF`p7$$!5GLLLeRx**f6Fjt$ \"-0<%H#4XF`p7$$!5S*****\\7GyNH'!#A$\"-M_r!R^%F`p7$$!2mmmmm\"zr)*F/$\" -gY9y:XF`p7$$\"5]LLL3x\"yY_%Fiv$\"-PT=v9XF`p7$$\"5ILLL3_Nl.5Fjt$\"-k'> B1^%F`p7$$\"5DLL$ekGR[b\"Fjt$\"-jX4d.XF`p7$$\"5?LLL$3-Dg5#Fjt$\"-q1[R$ \\%F`p7$$\"55LLLe*['R3KFjt$\"-e\")H=kWF`p7$$\"3LLLLezw5VF<$\"-Z5CABWF` p7$$\"5]mmmmmJ+IiFjt$\"-CKC([K%F`p7$$\"3++++v$Q#\\\")F<$\"-qLMI%>%F`p7 $$\"5lmmm\"z\\1A-\"F/$\"-X\\Om?SF`p7$$\"4LLLL$e\"*[H7F<$\"-H^lm:F`p7$$\"4nmmmmwanL$F<$\"-P\\Y$[J\"F`p7$$\"4+++++v+'oPF<$\",z &R5q$*F`p7$$\"4LLLLeR<*fTF<$\",&Ri(*emF`p7$$\"4+++++&)Hxe%F<$\"-!zMVzT %Ffn7$$\"4nmmm\"H!o-*\\F<$\"-g;&*=,HFfn7$$\"4++++DTO5T&F<$\"-8wN(o!=Ff n7$$\"4nmmmmT9C#eF<$\"-Au;g*4\"Ffn7$$\"4++++D1*3`iF<$\",$z+lFL7$$\"4nmmm \"HYt7vF<$\"-\"e!eFj5FL7$$\"4+++++q(G**yF<$\"-&[/uQ*eF<7$$\"4nmmm;9@BM )F<$\"-D#Hhw*GF<7$$\"4LLLLL`v&Q()F<$\"-?0*>j^\"F<7$$\"4++++DOl5;*F<$\" -7D))[#R(F/7$$\"4++++v.Uac*F<$\"-bH'Q/l$F/7$$\"\"\"F*$\"-.\"**[Lq\"F/- %'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~sche meG-F$6%7eoF'7$F-$\",o+J$eLF/7$F3$\"-FA.'>O\"F/7$F8$\",XM#o)R%F<7$F>$ \"-fvjgC7F<7$FC$\"-(H)3Z*4$F<7$FH$\",5F-9m'FL7$FN$\"-k&eV)*Q\"FL7$FS$ \"-c(4Iqy#FL7$FX$\",wb10H&Ffn7$Fhn$\",CQL!G(*Ffn7$F]o$\"-**)[KQf\"Ffn7 $Fbo$\"-?hq'Gm#Ffn7$Fgo$\"-K@&[zC%Ffn7$F\\p$\",,=$Q!R'F`p7$Fbp$\",?Z9; %*)F`p7$Fgp$\"-.t6!oF\"F`p7$F\\q$\"-\"R4xUY\"F`p7$Faq$\"-h&o_bm\"F`p7$ Ffq$\"-8uy!\\\">F`p7$F[r$\"-DDfTx@F`p7$F`r$\"-,a[ACx&=$F`p7$Fds$\"-cg&G6U$F`p7$Fis$\"-! HgK'ROF`p7$F^t$\"-gTn'R%QF`p7$Fct$\"-%)[*G'=SF`p7$Fht$\"-2cMi[TF`p7$F^ u$\"-t]=?YUF`p7$Fcu$\"-)yaxWG%F`p7$Fhu$\"--9KJ7VF`p7$F]v$\"-D$fYAK%F`p 7$Fbv$\"-JY8]HVF`p7$Fgv$\"-Y\"zeSL%F`p7$F]w$\"-1lI\"fL%F`p7$Fbw$\"-w2H (\\L%F`p7$Fgw$\"-Hf/6JVF`p7$F\\x$\"-%>&=MCVF`p7$Fax$\"-N/(pYJ%F`p7$Ffx $\"-%3A?nG%F`p7$F[y$\"-Q_g[ZUF`p7$F`y$\"--O&GK:%F`p7$Fey$\"-w1&3!GSF`p 7$Fjy$\"-o2oRhQF`p7$F_z$\"-zGsbmOF`p7$Fdz$\"-,t9(4W$F`p7$Fiz$\"-Y\\be' >$F`p7$F^[l$\"-c(3%=XHF`p7$Fc[l$\"-$fH\"H(o#F`p7$Fh[l$\"-/L)RmV#F`p7$F ]\\l$\"-5CGQ*=#F`p7$Fb\\l$\"-u#y9O#>F`p7$Fg\\l$\"-wQSHr;F`p7$F\\]l$\"- !4+I!f9F`p7$Fa]l$\"-'>!QOi7F`p7$Ff]l$\",.x.N**)F`p7$F[^l$\",xl\")))Q'F `p7$F`^l$\"-V6NROUFfn7$Fe^l$\"-[XF?!y#Ffn7$Fj^l$\"-VCi5IcF<7$F]al$\"-*>?5Kw#F<7$Fbal$\"-qldqY9F<7 $Fgal$\"-\\*\\O'fqF/7$F\\bl$\"-\"=N\"\\!\\$F/7$Fabl$\"-![?!yJ;F/-Ffbl6 &FhblFiblF+Fibl-F]cl6#%Ta~minor~modification~of~the~Bogacki-Shampine~s chemeG-F$6%7eoF'7$F-$\"+.;V>`F/7$F3$\",#zMTAAF/7$F8$\"+0B1[tF<7$F>$\", U)p^\"4#F<7$FC$\",X1&R4`F<7$FH$\",F$elY6FL7$FN$\",:mz\\Q#FL7$FS$\",S;X Px%FL7$FX$\"+CihN!*Ffn7$Fhn$\",(>0xb;Ffn7$F]o$\",jnu@q#Ffn7$Fbo$\",13& y%\\%Ffn7$Fgo$\",)[$[9:(Ffn7$F\\p$\",%*fTP2\"F`p7$Fbp$\",9S&[+:F`p7$Fg p$\",QHx59#F`p7$F\\q$\",pgj]X#F`p7$Faq$\",Kx5Bz#F`p7$Ffq$\",=Nd.@$F`p7 $F[r$\",PJa2l$F`p7$F`r$\",sMWK0%F`p7$Fer$\",]Ms:Y%F`p7$Fjr$\",SL(=3\\F `p7$F_s$\",5%y&RM&F`p7$Fds$\",7*=VRdF`p7$Fis$\",bq]m5'F`p7$F^t$\",-K$3 ]kF`p7$Fct$\",\"e**pVnF`p7$Fht$\",-(F`p7$Fhu$\",ZIPwB(F`p7$F]v$\",E7,WD(F`p7$Fbv$\",*4'3mE(F`p7$Fgv $\",A9BVF(F`p7$F]w$\",j!eZxsF`p7$Fbw$\",0?OfF(F`p7$Fgw$\",jq7&psF`p7$F \\x$\",eK`\"esF`p7$Fax$\",`yd>C(F`p7$Ffx$\",1Ek]>(F`p7$F[y$\",L)f?HrF` p7$F`y$\",w*f!4(pF`p7$Fey$\",*[#40w'F`p7$Fjy$\",<2$\\!['F`p7$F_z$\",T# H%H:'F`p7$Fdz$\",_)=ktdF`p7$Fiz$\",'pGmi`F`p7$F^[l$\",$)pH)R\\F`p7$Fc[ l$\",Jf)31XF`p7$Fh[l$\",%H'\\Y3%F`p7$F]\\l$\",EDc\"pOF`p7$Fb\\l$\",H'[ )GA$F`p7$Fg\\l$\",\\V^+!GF`p7$F\\]l$\",VHz\\W#F`p7$Fa]l$\",QJOn6#F`p7$ Ff]l$\",d$y58:F`p7$F[^l$\",@Ffn7$Fd_l$\",Gz#eT8Ffn7$Fi_ l$\",X\"Q^_')FL7$F^`l$\",)G;n&f&FL7$Fc`l$\",_pk\\f$FL7$Fh`l$\"-oJ#Q)[B F<7$F]al$\"-6l_Bi8F<7$Fbal$\",[)e'RJ)F<7$Fgal$\"-)pD*ypYF/7$F\\bl$\"- \"H'>58EF/7$Fabl$\"-Z;=C!R\"F/-%&COLORG6&Fhbl$FbblF)$\"\")F)F+-F]cl6#% 3efficient~scheme~AG-F$6%7eoF'7$F-$\"+#=Kw;)F/7$F3$\",O!zB\"[$F/7$F8$ \",-?Q7<\"F<7$F>$\",eK)[%Q$F<7$FC$\",L:5Cu)F<7$FH$\",Pu)z4>FL7$FN$\",l Kn:-%FL7$FS$\",kb0u7)FL7$FX$\",<@+7b\"Ffn7$Fhn$\",*yL=kGFfn7$F]o$\",X# oh*p%Ffn7$Fbo$\",0R#HhyFfn7$Fgo$\"-iz8,b7Ffn7$F\\p$\",crA!*)=F`p7$Fbp$ \",oW\"RWEF`p7$Fgp$\",U1Zzx$F`p7$F\\q$\",JulNL%F`p7$Faq$\",f@9.$\\F`p7 $Ffq$\",yIH*pcF`p7$F[r$\",_zx*[kF`p7$F`r$\",$ofjgrF`p7$Fer$\",M<_C)yF` p7$Fjr$\",Al]K&R6F`p7$Fct$\"-X\\)e8>\"F`p7$Fht$\"-^)RA*H7F`p7$F^ u$\"-_a%e)e7F`p7$Fcu$\"-z!G.-F\"F`p7$Fhu$\"-do*[%y7F`p7$F]v$\"-.LvQ\"G \"F`p7$Fbv$\"-E?T`$G\"F`p7$Fgv$\"--!=y[G\"F`p7$F]w$\"-<[0U&G\"F`p7$Fbw $\"-@tb8&G\"F`p7$Fgw$\"-'G9vRG\"F`p7$F\\x$\"-9t!o>G\"F`p7$Fax$\"-waR3z 7F`p7$Ffx$\"-s9&z2F\"F`p7$F[y$\"-Fc%G\"f7F`p7$F`y$\"-Xf@9J7F`p7$Fey$\" -%)3D(R>\"F`p7$Fjy$\"-Rz!GX9\"F`p7$F_z$\"-eV5r'3\"F`p7$Fdz$\"-BDix>5F` p7$Fiz$\",h:oEZ*F`p7$F^[l$\",-;-os)F`p7$Fc[l$\",oI!zhzF`p7$Fh[l$\",f`x %=sF`p7$F]\\l$\",Gpxb['F`p7$Fb\\l$\",u&\\F`p7$ F\\]l$\",&[A$[K%F`p7$Fa]l$\",()zUZu$F`p7$Ff]l$\",j(ouwEF`p7$F[^l$\",Z+ )z9>F`p7$F`^l$\"-Dg.B*G\"Ffn7$Fe^l$\",x+x$y')Ffn7$Fj^l$\",/*Q\"\\l&Ffn 7$F__l$\",,jD/m$Ffn7$Fd_l$\",;;j?H#Ffn7$Fi_l$\"-P&oQ'\\ FF/7$F3$!-Wy^!30\"F/7$F8$!,PJnE@$F<7$F>$!,(yK5#[)F<7$FC$!-zpmB^?F<7$FH $!,-'4IiUFL7$FN$!,th#oQ')FL7$FS$!-\"z!>*>p\"FL7$FX$!,dwl5:$Ffn7$Fhn$!, t:%G*p&Ffn7$F]o$!,JxYYC*Ffn7$Fbo$!-g[J[H:Ffn7$Fgo$!-%zI1gU#Ffn7$F\\p$! ,!*pK[j$F`p7$Fbp$!,?l@@2&F`p7$Fgp$!,\"pxkGsF`p7$F\\q$!,eXB_G)F`p7$Faq$ !,h[)\\>%*F`p7$Ffq$!-*z9xC3\"F`p7$F[r$!-&)[wSI7F`p7$F`r$!->p7jl8F`p7$F er$!-d/zy-:F`p7$Fjr$!-a;r\"Gl\"F`p7$F_s$!-$[[#=*z\"F`p7$Fds$!-+;Y)>$>F `p7$Fis$!-9F>Hb?F`p7$F^t$!-WXfhq@F`p7$Fct$!-&z7*>pAF`p7$Fht$!-1`IdUBF` p7$F^u$!-R>7l(R#F`p7$Fcu$!-X%4c#>CF`p7$Fhu$!-^Jv'\\V#F`p7$F]v$!-HOEdSC F`p7$Fbv$!-,*\\oYW#F`p7$Fgv$!-W!yRsW#F`p7$F]w$!-w6mG[CF`p7$Fbw$!-Wl^vZ CF`p7$Fgw$!-1kbdXCF`p7$F\\x$!-B`PvTCF`p7$Fax$!-49hHOCF`p7$Ffx$!-tZ6_?C F`p7$F[y$!-1>wP)R#F`p7$F`y$!-!p<(=XBF`p7$Fey$!-V+D`uAF`p7$Fjy$!-PDj_!= #F`p7$F_z$!-bI6gq?F`p7$Fdz$!-P\"HIL%>F`p7$Fiz$!-m/@X0=F`p7$F^[l$!-%pk4 Om\"F`p7$Fc[l$!-B9I3=:F`p7$Fh[l$!-e(*4gw8F`p7$F]\\l$!-%=3spB\"F`p7$Fb \\l$!-N=bx'3\"F`p7$Fg\\l$!,\"\\PeS%*F`p7$F\\]l$!,z'p`Q#)F`p7$Fa]l$!,sf ,M7(F`p7$Ff]l$!,fAp51&F`p7$F[^l$!,'))RKvNF`p7$F`^l$!-au31UBFfn7$Fe^l$! -8B,(e]\"Ffn7$Fj^l$!,!G#QY,*Ffn7$F__l$!,R:J`<&Ffn7$Fd_l$!,U;h3r#Ffn7$F i_l$!-#3@k4L\"FL7$F^`l$!,4bMr_&FL7$Fc`l$!,a*)*QRs$!+OJ9MGF<7$FC$!+*)R\\0#)F<7$FH$!+AX(e$>FL7$FN$!+s];QWFL7$ FS$!+bA<*e*FL7$FX$!+!Gm*R>Ffn7$Fhn$!+TOHzPFfn7$F]o$!+TX&*QkFfn7$Fbo$!, CXL.7\"Ffn7$Fgo$!,JAqF$=Ffn7$F\\p$!+H7-5GF`p7$Fbp$!+^(Gf)RF`p7$Fgp$!+) H,.v&F`p7$F\\q$!+.0p:mF`p7$Faq$!+/;SXvF`p7$Ffq$!+:fg(p)F`p7$F[r$!+j>[5 **F`p7$F`r$!,@E*[,6F`p7$Fer$!,8H\\L@\"F`p7$Fjr$!,K=SaL\"F`p7$F_s$!,l?$ Ra9F`p7$Fds$!,mo)=i:F`p7$Fis$!,/k5@m\"F`p7$F^t$!,r*yQbF`p7$Fcu$!,rk(Gb>F`p7$Fhu$!,X&Hq n>F`p7$F]v$!,7A@?(>F`p7$Fbv$!,M`M_(>F`p7$Fgv$!,_)*Hr(>F`p7$F]w$!,!Qn#y (>F`p7$Fbw$!,S)REx>F`p7$Fgw$!,&frAv>F`p7$F\\x$!,=`K@(>F`p7$Fax$!,@'>Yn >F`p7$Ffx$!,Y[mW&>F`p7$F[y$!,F`p7$F`y$!,yCFI*=F`p7$Fey$!,@5!yN=F `p7$Fjy$!,_f;*fS0f9F`p 7$F^[l$!,wW2fM\"F`p7$Fc[l$!,]G(>I7F`p7$Fh[l$!,I4B!=6F`p7$F]\\l$!,InFw+ \"F`p7$Fb\\l$!+G:W+*)F`p7$Fg\\l$!+p9ksxF`p7$F\\]l$!+n*H$GoF`p7$Fa]l$!+ 7dR`fF`p7$Ff]l$!+V4%4L%F`p7$F[^l$!+1;!f;$F`p7$F`^l$!,k&))o&>#Ffn7$Fe^l $!,pOSo_\"Ffn7$Fj^l$!,!4R'R8.$FL7$F^`l$!,c&*GU.#FL7$Fc`l$!,2:#Gk8FL7$Fh`l$!,znaSG*F<7$F]a l$!,&\\x8TcF<7$Fbal$!,7DI(=OF<7$Fgal$!-l5#[@9#F/7$F\\bl$!-'\\I$Rn7F/7$ Fabl$!,p:S4E(F/-F\\_n6&Fhbl$\"\"'F)F+F__n-F]cl6#%Sscheme~with~stabilit y~radius~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#% Uerror~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$ Q\"x6\"Q!Fgjp-%%VIEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor mo dification of the Bogacki-Shampine scheme" "efficient scheme A" "effic ient scheme B" "efficient scheme C" "scheme with stability radius = im ag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 " Test 5 of 7 stage, order 5 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&\"#;F&*&,&*&F*F&%\"xGF&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 " , " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-% %sqrtG6#,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F+%\"xGF+F+F+F+F+F+F+F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y (0)=1;\ndsolve(\{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x= 0..0.5,0..2.6,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*& F/F0F,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-% \"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$, &*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 " " 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$ \"\"\"F)7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\" 3?25A!pa&\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F 6$\"3]_vibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3CLL$3x%3yTF6$ \"31#\\\\E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$$\"3)HL$eR-/ PiF6$\"3.fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$$\"3`KLe*)>V B$)F6$\"3K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3s mm;/j$o/\"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za =F27$$\"3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA \\>F27$$\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW`(>Fco$\"3So# znsrC'>F27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=xpe=#Fco$\"3? *eB@.[<+#F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;zpSS\"R#Fco$ \"3)4US+%ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fc o$\"3ab\\mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GF co$\"3Q>IFco$\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3 b***\\(=S2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\" 3!*****\\(=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F 27$$\"3#*****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY% \\6AF27$$\"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG( yA8CB#F27$$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IIm U*yHDD#F27$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1 xfrAF27$$\"3))*****\\#>#[Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G!e&e%Fco$\"3 )zsS%e\"3%*G#F27$$\"3;LLL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco $\"3e%4h.zwhI#F27$$\"35+]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$ \"3m'>())[`fABF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG \"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F($\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 750 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005 : numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y )],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of step s: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor m odification of the Bogacki-Shampine scheme`,`efficient scheme A`,`effi cient scheme B`,`efficient scheme C`,`scheme with stability radius = i mag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n \+ Sn_RK5_||ct := RK5_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Sn_RK5_||ct):\n for ii to numpts do\n sm := \+ sm+(Sn_RK5_||ct[ii,2]-s(Sn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose ]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0 F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no .~of~steps:~~~G\"$+\"Q)pprint326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shamp ine~schemeG$\"+!oind\"!#?7$%Ta~minor~modification~of~the~Bogacki-Shamp ine~schemeG$\"+Ga\"o6#!#>7$%3efficient~scheme~AG$\"+XWaB')F+7$%3effici ent~scheme~BG$\"+RB#=O\"F07$%3efficient~scheme~CG$\"+t:clAF07$%Sscheme ~with~stability~radius~=~imag~axis~inclusionG$\"+MT'[*p!#@Q)pprint336 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Th e following code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/% \"xG\"\"!" }{TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: num steps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[` initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modifi cation of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag a xis inclusion`]: errs := []:\nDigits := 25:\nfor ct to 6 do\n sn_RK5 _||ct := RK5_||ct(S(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := \+ 0.4995: sxx := evalf(s(xx)):\nfor ct to 6 do\n errs := [op(errs),abs (sn_RK5_||ct(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$ %0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$% 0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~ steps:~~~G\"$+\"Q)pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~s chemeG$\"+b*\\gA\"!#?7$%Ta~minor~modification~of~the~Bogacki-Shampine~ schemeG$\"+ZLv`7$%3efficient~scheme~AG$\"+pNwArF+7$%3efficient~sch eme~BG$\"+/'))R7\"F07$%3efficient~scheme~CG$\"+V[;k=F07$%Sscheme~with~ stability~radius~=~imag~axis~inclusionG$\"+F*yi7'!#@Q)pprint376\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interva l [0, 0.5] of each Runge-Kutta method is estimated as follows using \+ the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equa l subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shamp ine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient schem e C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := [ ]:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((s(x)-'sn_RK5_||ct'( x))^2,x=0..0.5,adaptive=false,numpoints=7,factor=50);\n errs := [op( errs),sqrt(sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8B ogacki-Shampine~schemeG$\"+H&R\\d\"!#?7$%Ta~minor~modification~of~the~ Bogacki-Shampine~schemeG$\"+5l<;@!#>7$%3efficient~scheme~AG$\"+t " 0 "" {MPLTEXT 1 0 500 "evalf[20](plot(['sn_RK5_1'(x)-s(x),'sn_RK5_2'(x)-s(x ),'sn_RK5_3'(x)-s(x),'sn_RK5_4'(x)-s(x),\n'sn_RK5_5'(x)-s(x),'sn_RK5_6 '(x)-s(x)],x=0..0.5,\nfont=[HELVETICA,9],color=[blue,magenta,COLOR(RGB ,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlegend=[`Bogacki-Shampine sc heme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stab ility radius = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 832 551 551 {PLOTDATA 2 "6,-%'CURVESG6%7eo7$$\"\"!F)F(7$$\"5SLLL3x1h6o!#B$ !%+y!#>7$$\"5ommmTN@Ki8!#A$!'#pc$F07$$\"5-++]7.K[V?F4$!(W&pEF07$$\"5NL LL$3FWYs#F4$!(&y\"G)F07$$\"5-++vo/[AlIF4$!)Eos5F07$$\"5omm;aQ`!eS$F4$! ([Zi*F07$$\"5NLLeRseQYPF4$\"''3f#F07$$\"5-+++D1k'p3%F4$\")%)GUFF07$$\" 5pmmT5SpaFWF4$\")]*eY)F07$$\"5OLL$eRZF\"oZF4$\"*1k>!>F07$$\"50++D\"y+3 (3^F4$\"*\\no,$F07$$\"5qmmmmT&)G\\aF4$\"*p[Q+$F07$$F,F4$\"*pWr%HF07$$ \"50+++]7G$R<)F4$\"*#*4@#GF07$$\"5qmmTNYL^9&)F4$\"*+fNy#F07$$\"5SLL$3- )Q4b))F4$\"*UN=w#F07$$\"50++D19Wn&>*F4$\"*jgHy#F07$$\"5qmmm\"z%\\DO&*F 4$\"*WAz)GF07$$\"5NLL3x\"[No()*F4$\"*3Qq8$F07$$\"5+++Dc,;u@5!#@$\"*:$f qKF07$$\"5nm;z%\\l*zb5Fhq$\"*D@*eKF07$$\"5MLLLL3x&)*3\"Fhq$\"*\"fOZKF0 7$$\"5+++]i!R(*Rc\"Fhq$\"*D2q.$F07$$\"5nmmm\"H2P\"Q?Fhq$\"*\"p=&y#F07$ $\"5+++]PMnNrDFhq$\"*rrKc#F07$$\"5MLLL$eRwX5$Fhq$\"*%pY!R#F07$$\"5MLLL L$eI8k$Fhq$\"*e%3cAF07$$\"5NLLL$3x%3yTFhq$\"*)p[]@F07$$\"5ommm\"z%4\\Y _Fhq$\"*vEq*>F07$$\"5NLLLeR-/PiFhq$\"*M:g*=F07$$\"5-+++DcmpisFhq$\"*;R '==F07$$\"5OLLLe*)>VB$)Fhq$\"*'fDcjU6F_v$\"*es$H;F07$$\"5++ ++]i^Z]7F_v$\"*Ltnf\"F07$$\"5++++](=h(e8F_v$\"*\"GPo:F07$$\"5++++]P[6j 9F_v$\"*X`Qa\"F07$$\"5MLL$e*[z(yb\"F_v$\"*u\"fB:F07$$\"5nmmm;a/cq;F_v$ \"*cJ<]\"F07$$\"5nmmmm;t,mF_v$\"*6\"e^9F07$$\"5,++]i!f#=$3#F_v$\"*#yNO9F07 $$\"5,++](=xpe=#F_v$\"*wTGU\"F07$$\"5nmmm\"H28IH#F_v$\"*pxg#F_v$\"*Bq`P\"F07$$\"5,++]Pf4t.FF_v$\"*cXgO\"F07$$\"5MLLLe*Gst !GF_v$\"*4zkN\"F07$$\"5,++++DRW9HF_v$\"*N(4Z8F07$$\"5,+++DJE>>IF_v$\"* fp$Q8F07$$\"5,++]i!RU07$F_v$\"*+:.L\"F07$$\"5,+++v=S2LKF_v$\"*8(y@8F07 $$\"5ommmm\"p)=MLF_v$\"*AnWJ\"F07$$\"5,+++](=]@W$F_v$\"*c$)pI\"F07$$\" 5MLL$e*[$z*RNF_v$\"*sy/I\"F07$$\"5,+++]iC$pk$F_v$\"*VXOH\"F07$$\"5omm; H2qcZPF_v$\"*tluG\"F07$$\"5,++]7.\"fF&QF_v$\"*BZ7G\"F07$$\"5ommm;/OgbR F_v$\"*^\"Rv7F07$$\"5,++]ilAFjSF_v$\"*B$[p7F07$$\"5NLLLL$)*pp;%F_v$\"* L%*RE\"F07$$\"5NLLL3xe,tUF_v$\"*ht&e7F07$$\"5omm;HdO=yVF_v$\"*\")yLD\" F07$$\"5,++++D>#[Z%F_v$\"*Rb([7F07$$\"5ommmT&G!e&e%F_v$\"*%QiV7F07$$\" 5NLLLL$)Qk%o%F_v$\"*Vx\"R7F07$$\"5-++]iSjE!z%F_v$\"*%ydM7F07$$\"5-++]P 40O\"*[F_v$\"*<0.B\"F07$$\"\"&!\"\"$\"*cZeA\"F0-%'COLOURG6&%$RGBGF(F($ \"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7`pF'7$F+$\"&@ 0\"F07$F2$\"'ZEtF07$$\"5NLL3FpE!Hq\"F4$\"(N)))GF07$F8$\"(#*G())F07$$\" 5omm\"zptjSQ#F4$\")W%=H#F07$F=$\")y'>@&F07$FB$\"*H&*\\2\"F07$FG$\"*%zD _?F07$FL$\"*w&ezOF07$FQ$\"*,8JE'F07$FV$\"+frS?5F07$Fen$\"+9H^,;F07$Fjn $\"+odOL@F07$F_o$\"+X=;C@F07$$\"50++]P4'\\/8'F4$\"+?_rF4$\"+za([3#F07$$\"5qmm;zW#F07$F\\q$ \"+n$frX#F07$Faq$\"+Zm!3p#F07$Ffq$\"++]k#z#F07$F\\r$\"+HIo#y#F07$Far$ \"+A-)Gx#F07$$\"5nm\"z>'o^7\\6Fhq$\"+=8McFF07$$\"5++]i!*GER37Fhq$\"+&e l8u#F07$$\"5ML3F>*3gwE\"Fhq$\"+.I0JFF07$$\"5nmm\"z%\\v#pK\"Fhq$\"+Ce%G t#F07$$\"5ML$3_+ZiaW\"Fhq$\"+!4hI%GF07$Ffr$\"+j:3$)HF07$$\"5nm;z>6B`#o \"Fhq$\"+V$GG&HF07$$\"5MLL3xJs1,=Fhq$\"+$fR-$HF07$$\"5++]PM_@g>>Fhq$\" +>aZ\\HF07$F[s$\"+x2/GIF07$F`s$\"+X1*p*HF07$Fes$\"+=B3WHF07$F_t$\"+1;f CGF07$Fdt$\"+qG.:FF07$Fit$\"+1p=FEF07$F^u$\"+uyJ[DF07$Fcu$\"+hR(zZ#F07 $Fhu$\"+[4@BF07$Fhv$\"+^,#oF#F07$ F]w$\"+@./QAF07$Fbw$\"+J\\F/AF07$Fgw$\"+&3Mi<#F07$F\\x$\"+kemX@F07$Fax $\"+$3U=7#F07$Ffx$\"+Zy>'4#F07$F[y$\"+fJ9v?F07$F`y$\"+Nwg`?F07$Fey$\"+ %[eW.#F07$Fjy$\"+m%Qd,#F07$F_z$\"+$Gm&**>F07$Fdz$\"+O56$)>F07$Fiz$\"+U /+n>F07$F^[l$\"+Zur`>F07$Fc[l$\"+h*y+%>F07$Fh[l$\"+IppE>F07$F]\\l$\"+U SC9>F07$Fb\\l$\"+z)[F!>F07$Fg\\l$\"+/Md!*=F07$F\\]l$\"+d:7!)=F07$Fa]l$ \"+4VVp=F07$Ff]l$\"+;A9g=F07$F[^l$\"+\">\"Q]=F07$F`^l$\"+L8bT=F07$Fe^l $\"+U`mK=F07$Fj^l$\"+JhHC=F07$F__l$\"+P3&e\"=F07$Fd_l$\"+pY+3=F07$Fi_l $\"+'Qb-!=F07$F^`l$\"+/(GGz\"F07$Fc`l$\"+v(=iy\"F07$Fh`l$\"++=))y(F07$FL$\"*(>=h8F07$FQ$\"*VD>V#F07$FV$\"*#)y=9%F07$Fen$\"*3HBx'F07$Fj n$\"*yJ4E*F07$F_o$\"*bw4A*F07$F^fl$\"*!zIV\"*F07$Fdo$\"*Ap82*F07$F[gl$ \"*&GEC!*F07$Fho$\"*:fA2*F07$Fbp$\"*Bq-S*F07$F\\q$\"+3\\VS5F07$Faq$\"+ H*3t8\"F07$Ffq$\"+HOl!=\"F07$F\\r$\"+F=Ww6F07$Far$\"+(4&Hs6F07$F_il$\" +F,ne6F07$Fiil$\"+[Y7_6F07$F^jl$\"+5r.*=\"F07$Ffr$\"+hNPU7F07$Ffjl$\"+ FIqH7F07$F[[m$\"+)G&G>7F07$$\"5nm\"Hd?pM.'=Fhq$\"+s$)p<7F07$F`[m$\"+cV !HA\"F07$$\"5ML3-j7'p)y>Fhq$\"+\"QPS%o5F0 7$F^u$\"+GB'e.\"F07$Fcu$\"+Qh(p+\"F07$Fhu$\"**R0@)*F07$F]v$\"*pg!)f*F0 7$Fcv$\"*K!oB%*F07$Fhv$\"*au$[#*F07$F]w$\"*gq04*F07$Fbw$\"*[>K&*)F07$F gw$\"*bx\"R))F07$F\\x$\"*!*4\\r)F07$Fax$\"*lk!=')F07$Ffx$\"*#[%Q^)F07$ F[y$\"*\">GG%)F07$F`y$\"*ev2M)F07$Fey$\"*tqHE)F07$Fjy$\"*78p=)F07$F_z$ \"*N777)F07$Fdz$\"*-iV0)F07$Fiz$\"*@;*))zF07$F^[l$\"*Ke\\$zF07$Fc[l$\" *Ie&zyF07$Fh[l$\"*?,_#yF07$F]\\l$\"*()=Yx(F07$Fb\\l$\"*?Fzs(F07$Fg\\l$ \"*1t%ywF07$F\\]l$\"*4?gj(F07$Fa]l$\"*E6Ef(F07$Ff]l$\"*np[b(F07$F[^l$ \"*PB_^(F07$F`^l$\"*jf$zuF07$Fe^l$\"*.oKW(F07$Fj^l$\"*pv#4uF07$F__l$\" *^u\\P(F07$Fd_l$\"*y1JM(F07$Fi_l$\"*nK;J(F07$F^`l$\"*))o9G(F07$Fc`l$\" *RAYD(F07$Fh`l$\"*3B[A(F07$F]al$\"*'=+*>(F07$Fbal$\"*Q!HsrF07$Fgal$\"* Vwu9(F07$F\\bl$\"*I*e@rF0-%&COLORG6&Fdbl$\"\"\"F^bl$\"\")F^blF(-Fibl6# %3efficient~scheme~AG-F$6%7^pF'7$F+$\"%9DF07$F2$\"'3WCF07$Ffcl$\"(zZ3 \"F07$F8$\"(M4o$F07$F^dl$\")xFO5F07$F=$\")QcUDF07$FB$\")!QMh&F07$FG$\" *'*Q*R6F07$FL$\"*NLI;#F07$FQ$\"*Gw+)QF07$FV$\"*\"RcQmF07$Fen$\"+dr(34 \"F07$Fjn$\"+#f^q\\\"F07$F_o$\"+^Ff!\\\"F07$F^fl$\"+L*Q!y9F07$Fdo$\"+& f0kY\"F07$F[gl$\"+J[je9F07$Fho$\"+?QZl9F07$Fbp$\"+A.N::F07$F\\q$\"+SU$ )p;F07$Faq$\"+Ex#)>=F07$Ffq$\"+ge,()=F07$F\\r$\"+YUG!)=F07$Far$\"+Wplt =F07$F_il$\"+Q<)=&=F07$Fiil$\"+[k*3%=F07$F^jl$\"+f3(\\*=F07$Ffr$\"+3=o t>F07$Ffjl$\"+A*eN&>F07$F[[m$\"+Om\"p$>F07$Fgjm$\"+N,*R$>F07$F`[m$\"+/ -.T>F07$F_[n$\"+Fj'z'>F07$F[s$\"+=9Tz>F07$F`s$\"+@bAX>F07$Fes$\"+m\"RE !>F07$F_t$\"+#QBt\"=F07$Fdt$\"+**e[Vv8F07$Fax$\"+>g\"*f8F07$Ffx$\"+siZV8F07$F[y$\"+#>z*H8F07$F` y$\"+HY<;8F07$Fey$\"+<,!RI\"F07$Fjy$\"+T4!>H\"F07$F_z$\"+$>N:G\"F07$Fd z$\"+n!))4F\"F07$Fiz$\"+&3i1E\"F07$F^[l$\"+;&[@D\"F07$Fc[l$\"+(>2MC\"F 07$Fh[l$\"+2.$[B\"F07$F]\\l$\"+j*[oA\"F07$Fb\\l$\"+s9[>7F07$Fg\\l$\"+# *zn67F07$F\\]l$\"+F#z\\?\"F07$Fa]l$\"+.'H\")>\"F07$Ff]l$\"+vT<#>\"F07$ F[^l$\"+-#=f=\"F07$F`^l$\"+W!f-=\"F07$Fe^l$\"+;Rcu6F07$Fj^l$\"+2+?p6F0 7$F__l$\"+Wtyj6F07$Fd_l$\"+n'e(e6F07$Fi_l$\"+)3#z`6F07$F^`l$\"+rA.\\6F 07$Fc`l$\"+g8\"F07$Fbal$\"+WR!=8 \"F07$Fgal$\"+*H))y7\"F07$F\\bl$\"+7L!Q7\"F0-Fbbl6&FdblFeblF(F(-Fibl6# %3efficient~scheme~BG-F$6%7^pF'7$F+$\"%*e#F07$F2$\"'zeKF07$Ffcl$\"(\\* R:F07$F8$\"(dmY&F07$F^dl$\")^'Gf\"F07$F=$\")P[;SF07$FB$\")\\eo!*F07$FG $\"*_Em(=F07$FL$\"*/A*=OF07$FQ$\"*VmPe'F07$FV$\"+b`aS6F07$Fen$\"+'eU^* =F07$Fjn$\"+,c')=EF07$F_o$\"+rnc2EF07$F^fl$\"+b-f&e#F07$Fdo$\"+:w+lDF0 7$F[gl$\"+>f6]DF07$Fho$\"+[XpdDF07$Fbp$\"+73:MEF07$F\\q$\"+1zL$)GF07$F aq$\"+[)*\\IJF07$Ffq$\"+\">()=C$F07$F\\r$\"+\">A.B$F07$Far$\"+GZ$*=KF0 7$F_il$\"+1(\\7=$F07$Fiil$\"+*)4*)eJF07$F^jl$\"+MWWMKF07$Ffr$\"+W;&=N$ F07$Ffjl$\"+2(3wJ$F07$F[[m$\"+d&z!)G$F07$Fgjm$\"+6#H1G$F07$F`[m$\"+TM? (G$F07$F_[n$\"+/GhALF07$F[s$\"+XhtOLF07$F`s$\"+\"4dKE$F07$Fes$\"+;'>:= $F07$F_t$\"+-d`FIF07$Fdt$\"+$e*3**GF07$Fit$\"+Yv)**z#F07$F^u$\"+B$\\Jr #F07$Fcu$\"+-QpOEF07$Fhu$\"+k\\6rDF07$F]v$\"+GyV7DF07$Fcv$\"+P'omY#F07 $Fhv$\"+DGm?CF07$F]w$\"+\"31$zBF07$Fbw$\"+E6KVBF07$Fgw$\"+E([MJ#F07$F \\x$\"+Rx!4G#F07$Fax$\"+\"3\\bD#F07$Ffx$\"+,QEGAF07$F[y$\"+YR'e?#F07$F `y$\"+kq&H=#F07$Fey$\"+@0fi@F07$Fjy$\"+X>oU@F07$F_z$\"+k\\[D@F07$Fdz$ \"+Bs)z5#F07$Fiz$\"+&ed34#F07$F^[l$\"+u\\tw?F07$Fc[l$\"++^Bi?F07$Fh[l$ \"+6$3![?F07$F]\\l$\"+a&pZ.#F07$Fb\\l$\"+O\"\\D-#F07$Fg\\l$\"+#y0'4?F0 7$F\\]l$\"+F[\\)*>F07$Fa]l$\"+CP8()>F07$Ff]l$\"+-fDx>F07$F[^l$\"+x&zo' >F07$F`^l$\"+jK\\d>F07$Fe^l$\"+F07$Fj^l$\"+83:R>F07$F__l$\"+*[t,$ >F07$Fd_l$\"+aI$=#>F07$Fi_l$\"+Tcf8>F07$F^`l$\"+r6q0>F07$Fc`l$\"+1\\n) *=F07$Fh`l$\"+Ne(3*=F07$F]al$\"+1z6%)=F07$Fbal$\"+&)p7x=F07$Fgal$\"+%o K1(=F07$F\\bl$\"+7v&Q'=F0-Fbbl6&Fdbl$\")#)eqkFgbl$\"))eqk\"FgblFbdp-Fi bl6#%3efficient~scheme~CG-F$6%7\\sF'7$F+$!%WkF07$F2$!'`\"=$F07$F8$!((* ))o#F07$F=$!)s>`5F07$FB$!)'Grr\"F07$FG$!)z%p\\#F07$FL$!)5k4KF07$FQ$!) \"Q/]$F07$FV$!)V0yFF07$Fen$!(N;T\"F07$Fjn$\")3C(Q$F07$F_o$\")ZgsLF07$F ^fl$\")MfRLF07$Fdo$\")v\"\\D$F07$F[gl$\")q5mHF07$Fho$\")z!fC#F07$F]p$ \")1k!o\"F07$Fbp$\")<[15F07$Fgp$\"(XD2$F07$F\\q$!(7[b#F07$Faq$!(\"Q_TF 07$Ffq$!(o/+$F07$F\\r$!(;,*HF07$Far$!(t[)HF07$F_il$!(-Sd$F07$Fiil$!(r* o\"*F07$$\"5M$eRA'z7cc8Fhq$!)&z\\A\"F07$$\"5++Dcw4]>'Q\"Fhq$!)N,:;F07$ $\"5n;a)3*R(GeT\"Fhq$!)#o+3#F07$F^jl$!)-8)f#F07$$\"5+]7`>+i4v9Fhq$!)ij EJF07$$\"5nmT&Q.$*HZ]\"Fhq$!)G`FNF07$$\"5M$3x\"[gOOM:Fhq$!)?<=NF07$Ffr $!)\"e*3NF07$$\"5MLe9\"4&[EB;Fhq$!)x\\#\\$F07$Ffjl$!)#e!*[$F07$$\"5++v V[r(*zTX`F07$F[s$!)O\"Gl&F07$$\"5MLe*)fI&*y/@Fhq$!)ZZCcF07$$\"5++]7G ))>Wr@Fhq$!)JR-cF07$$\"5M$eRAr@oZ?#Fhq$!)#y$*f&F07$$\"5nmTN'fW%4QAFhq$ !)Jy2cF07$$\"5+](o/[n?9F#Fhq$!)WQMcF07$$\"5MLLek.pu/BFhq$!)kg(o&F07$$ \"5nm;/,>=0QCFhq$!)$eyN'F07$F`s$!)>ZmpF07$$\"5nm\"Hd?>4!QEFhq$!)wqOpF0 7$$\"5ML$eR(\\;m/FFhq$!)ji:pF07$$\"5n;H2eyy)zt#Fhq$!)-59pF07$$\"5++v=U 2TJrFFhq$!)1$R#pF07$$\"5M$3-jiLSY!GFhq$!),l]pF07$$\"5nmmT5ll'z$GFhq$!) ^%3+(F07$$\"5++](o/[r7(HFhq$!)t-!e(F07$Fes$!)g#*exF07$Fjs$!)\">]A)F07$ F_t$!))o\"*[)F07$$\"5ommT5:j=XWFhq$!)Zjv&)F07$$\"5-++]PfyG7ZFhq$!)>rF' )F07$$\"5NL3FWXK1zZFhq$!)C#=h)F07$$\"5om;/^J'Qe%[Fhq$!)dU7')F07$$\"5-+ D\"yv,9E\"\\Fhq$!)'*pW')F07$$\"5NLLek.%*Qz\\Fhq$!)fnG()F07$$\"5qmTNr*y kh/&Fhq$!)t._()F07$$\"5++]7yv,%H6&Fhq$!)VbG()F07$$\"5NLe*[=c:(z^Fhq$!) f&fq)F07$Fdt$!)0M'o)F07$$\"5NLLeky-S3`Fhq$!)6=v')F07$$\"5+++]P4'4.P&Fh q$!)m/y')F07$$\"5qmmT5S*=AV&Fhq$!)lI0()F07$$\"5NLLL$3FGT\\&Fhq$!)g/q() F07$$\"5qmm;HKp%zh&Fhq$!)]>S()F07$$\"5++++v$fl@p)F07$$\"5+++v$fe$\\FfFhq$!)$HAq)F07$$\"5NL$3-8D[%efFhq$!))3 #>()F07$$\"5qmmmm;HS*)fFhq$!)`[W()F07$$\"5+++]7y:A8hFhq$!)')>@()F07$Fi t$!)&>jo)F07$$\"5qmmm\"zWo)\\nFhq$!)mcV')F07$F^u$!)Zf(e)F07$Fcu$!)TO\\ %)F07$Fhu$!);!\\I)F07$F]v$!)B;i\")F07$Fcv$!)&\\Z.)F07$Fhv$!)`o1zF07$F] w$!)Xw#y(F07$Fbw$!);5twF07$Fgw$!)s=\"e(F07$F\\x$!)$\\*yuF07$Fax$!)l5*R (F07$Ffx$!).;7tF07$F[y$!)&)fSsF07$F`y$!)D)p;(F07$Fey$!)#Q85(F07$Fjy$!) ?-PqF07$F_z$!)_D\")pF07$Fdz$!)p\\CpF07$Fiz$!)lvooF07$F^[l$!)9uAoF07$Fc [l$!))=ax'F07$Fh[l$!)'Q*GnF07$F]\\l$!)/m&o'F07$Fb\\l$!)=pXmF07$Fg\\l$! )PK.mF07$F\\]l$!)o%pc'F07$Fa]l$!)&R(HlF07$Ff]l$!)0P(\\'F07$F[^l$!)-Pjk F07$F`^l$!)dfKkF07$Fe^l$!)2i,kF07$Fj^l$!)bVsjF07$F__l$!)!yHM'F07$Fd_l$ !)zg:jF07$Fi_l$!);d)G'F07$F^`l$!)\"eEE'F07$Fc`l$!)QfRiF07$Fh`l$!)#))R@ 'F07$F]al$!)(*z\">'F07$Fbal$!)[%)ohF07$Fgal$!)(=v9'F07$F\\bl$!)?FDhF0- Fddn6&Fdbl$\"\"'F^blF(Fhdn-Fibl6#%Sscheme~with~stability~radius~=~imag ~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~f or~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fecr-% %VIEWG6$;F(F\\bl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modification of the B ogacki-Shampine scheme" "efficient scheme A" "efficient scheme B" "eff icient scheme C" "scheme with stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 6 of 7 stage, \+ order 5 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*sin(3*x))*exp(-y);" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xGF&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F& F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0; " "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x +2/9*sin(3*x)-2/3*x*cos(3*x)-2/3*cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\" xG\"\"\"*(\"\"#F*\"\"*!\"\"-%$sinG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$ cosG6#*&F3F*F)F*F*F.*(F,F*F3F.-F66#*&F3F*F)F*F*F.*&\"\"&F*F3F.F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff(y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y( x));\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nu := unapply(rhs(%),x): \nplot(u(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/ *(\"\"#F/,&F,F/F/F/F/-%$sinG6#,$*&\"\"$F/F,F/F/F/F/F/-%$expG6#,$F)!\" \"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,,F'\"\"\"*&#\" \"#\"\"*F,-%$sinG6#,$*&\"\"$F,F'F,F,F,F,*&#F/F6F,*&F'F,-%$cosGF3F,F,! \"\"*&#F/F6F,F:F,F<#\"\"&F6F," }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\" 3QWK+t!=.P\"F-7$$\"3umm\"H2P\"Q?F-$\"3pUCE&GmM$HF-7$$\"3MLL$eRwX5$F-$ \"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF-$\"3dz%)zauhMpF-7$$\"3emm\"z%4\\Y_F -$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF-$\"36YrjIBvP5!#<7$$\"3]***\\il'pisF- $\"3wGtPF*HL<\"FI7$$\"3>MLe*)>VB$)F-$\"3kc:o\"[JMG\"FI7$$\"3Y++DJbw!Q* F-$\"3/$y^LV#)3O\"FI7$$\"3+N$ekGkX#**F-$\"3'R4-T\"[w(Q\"FI7$$\"3%ommTI Oo/\"FI$\"3i$3slO!o09FI7$$\"3E+]7GTt%4\"FI$\"3qC'f&H6%RT\"FI7$$\"3YLL3 _>jU6FI$\"3#[%e$e^!3:9FI7$$\"3ym;HdNb'>\"FI$\"3CNMY.Kv29FI7$$\"37++]i^ Z]7FI$\"3=<^j^\"=7R\"FI7$$\"35+++v\"=YI\"FI$\"3![4*)z8y_O\"FI7$$\"33++ ](=h(e8FI$\"3yj8C4s%*H8FI7$$\"3/++]P[6j9FI$\"3![\"=po,dN7FI7$$\"3UL$e* [z(yb\"FI$\"3%eEi/R377\"FI7$$\"3wmm;a/cq;FI$\"3!frK?-E\\b*F-7$$\"3%omm mJFI$\"3yKBC%\\'\\`dF-7$$\"3gmmm\"pW`(>FI$\"3$)>nU%[*3T`F-7$$\"3_ek.HW #)))>FI$\"3kKE$Q%*GSE&F-7$$\"3?]iSmTI-?FI$\"3NxPe(47T?&F-7$$\"3*=/wP!R y:?FI$\"3MId=-(\\?;&F-7$$\"3dLe9TOEH?FI$\"3+V!)\\%**R%Q^F-7$$\"3EDc^yL uU?FI$\"3=k(\\#*\\cP8&F-7$$\"3'pT&)e6Bi0#FI$\"3[P(R22Q$[^F-7$$\"3k3_D` Gqp?FI$\"3m1F&*)f\"Q#=&F-7$$\"3K+]i!f#=$3#FI$\"3z7KO0x$fB&F-7$$\"3/++D \"=EX8#FI$\"31#)Gask8:cF-7$$\"3?+](=xpe=#FI$\"3qCq#fO$Q]iF-7$$\"37nm\" H28IH#FI$\"3sUF;\"\\Mi=)F-7$$\"3$p;a8d3AM#FI$\"3NM9xIK\")Q#*F-7$$\"3um ;zpSS\"R#FI$\"3r*)ek&)o0L5FI7$$\"3-+v$41oWW#FI$\"3if$>8HM6:\"FI7$$\"3G LL3_?`(\\#FI$\"31,>EjcGm7FI7$$\"3AL3_D1l_DFI$\"3+C:%e=o-Q\"FI7$$\"3fL$ e*)>pxg#FI$\"3%R*>K#Rqn[\"FI7$$\"33+]Pf4t.FFI$\"3A89rRoB^;FI7$$\"3uLLe *Gst!GFI$\"3eR;lLLE'z\"FI7$$\"30+++DRW9HFI$\"36)ejqPI#4>FI7$$\"3K+]7y# =o'HFI$\"3?c()=o%e2&>FI7$$\"3:++DJE>>IFI$\"3s.\"e&4:F$)>FI7$$\"3A+v$4^ n)pIFI$\"39;D$z^_h+#FI7$$\"3F+]i!RU07$FI$\"3vuj8s:f??FI7$$\"3?]il(Hv'[ JFI$\"3&\\3XiGc\\-#FI7$$\"39+vo/#3o<$FI$\"3(>zS&>rqE?FI7$$\"32](=<6T\\ ?$FI$\"3X%zI>hOe-#FI7$$\"3+++v=S2LKFI$\"3'[xpLqNB-#FI7$$\"3;L$3_NJOG$F I$\"3:>!e))R4?FI7$$\"3Jmmm\"p)=MLFI$\"3OrPOD8'y)>FI7$$\"3GLLeR%p\")Q $FI$\"3S(z'\\,LGb>FI7$$\"3B++](=]@W$FI$\"3K:aBe2s7>FI7$$\"35L$e*[$z*RN FI$\"3!>**fc?u*4=FI7$$\"3e++]iC$pk$FI$\"3'e\\^`F#Qg;FI7$$\"3[m;H2qcZPF I$\"351&p,mkr[\"FI7$$\"3O+]7.\"fF&QFI$\"3o;2PnR[#G\"FI7$$\"3Ymm;/OgbRF I$\"3/lp%G(QG#3\"FI7$$\"3*G$e*[$zV4SFI$\"3P9Az61+7**F-7$$\"3w**\\ilAFj SFI$\"3!p>ERLl'*>*F-7$$\"3#G3_]p'>*3%FI$\"3O9*z2=!e_*)F-7$$\"3ym\"zW7@ ^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59% FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp; %FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>% FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**> UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,t UFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z- lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5, -`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+] P%37^j%FI$\"3_eaMDR_K " 0 " " {MPLTEXT 1 0 762 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := \+ 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U (x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of \+ steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a min or modification of the Bogacki-Shampine scheme`,`efficient scheme A`,` efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 25:\nfor ct to 6 do \n Un_RK5_||ct := RK5_||ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n \+ sm := 0: numpts := nops(Un_RK5_||ct):\n for ii to numpts do\n s m := sm+(Un_RK5_||ct[ii,2]-u(Un_RK5_||ct[ii,1]))^2;\n end do:\n er rs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sin G6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~G- %!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)ppri nt396\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+j*e#)o\"! #A7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+=(e%4s!# B7$%3efficient~scheme~AG$\"+OOU3AF07$%3efficient~scheme~BG$\"+2`1h%)!# C7$%3efficient~scheme~CG$\"+$oJVC\"F07$%Sscheme~with~stability~radius~ =~imag~axis~inclusionG$\"+5!p.)pF0Q)pprint406\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\! \"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`ef ficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 25:\nfor ct to 6 do\n \+ un_RK5_||ct := RK5_||ct(U(x,y),x,y,x0,y0,hh,numsteps,true);\nend do: \nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 6 do\n errs := [op(er rs),abs(un_RK5_||ct(xx)-uxx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sinG6# ,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G 6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint4 16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+\"Hzp4\"! #A7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+:$QHG)!# B7$%3efficient~scheme~AG$\"+U&z^D#F07$%3efficient~scheme~BG$\"+7$pV\"H !#C7$%3efficient~scheme~CG$\"+kkSs:F07$%Sscheme~with~stability~radius~ =~imag~axis~inclusionG$\"+TBY%R)F0Q)pprint426\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor mo dification of the Bogacki-Shampine scheme`,`efficient scheme A`,`effic ient scheme B`,`efficient scheme C`,`scheme with stability radius = im ag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((u(x)-'un_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7 ,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 1 0:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+yiwz;!#A7 $%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+gHd0s!#B7$% 3efficient~scheme~AG$\"+u0W0AF07$%3efficient~scheme~BG$\"+o%=0T)!#C7$% 3efficient~scheme~CG$\"+&RvzB\"F07$%Sscheme~with~stability~radius~=~im ag~axis~inclusionG$\"+(><\\'pF0Q)pprint436\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "evalf[20](plot(['un_RK5_1' (x)-u(x),'un_RK5_2'(x)-u(x),'un_RK5_3'(x)-u(x),'un_RK5_4'(x)-u(x),\n'u n_RK5_5'(x)-u(x),'un_RK5_6'(x)-u(x)],x=0..5,\nfont=[HELVETICA,9],color =[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlege nd=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sha mpine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sch eme C`,`scheme with stability radius = imag axis inclusion`],\ntitle=` error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 858 583 583 {PLOTDATA 2 "6,-%'CURVESG6%7es7$$\"\"!F)F (7$$\"5qmmmmT&)G\\a!#@$\")`!e@#F-7$$\"5MLLLL3x&)*3\"!#?$\"((ySRF37$$\" 5nm\"z>'o^7\\6F3$\"(u*RSF37$$\"5++]i!*GER37F3$\"(@9D%F37$$\"5ML3F>*3gw E\"F3$\"(Q_B%F37$$\"5nmm\"z%\\v#pK\"F3$\"(P[R%F37$$\"5ML$3_+ZiaW\"F3$ \"(2[]%F37$$\"5+++]i!R(*Rc\"F3$\"(/#)e%F37$$\"5nm;z>6B`#o\"F3$\"(3ol%F 37$$\"5MLL3xJs1,=F3$\"(-]r%F37$$\"5++]PM_@g>>F3$\"(Wan%F37$$\"5nmmm\"H 2P\"Q?F3$\"(gBh%F37$$\"5+++]PMnNrDF3$\"(#*)oSF37$$\"5MLLL$eRwX5$F3$\"( ]8?$F37$$\"5NLLL$3x%3yTF3$\"(ovp\"F37$$\"5ommm\"z%4\\Y_F3$\"'w]yF37$$ \"5NLLLeR-/PiF3$\"&NF%!#>7$$\"5-+++DcmpisF3$\"&78$F[q7$$\"5OLLLe*)>VB$ )F3$\"&^(HF[q7$$\"5.+++DJbw!Q*F3$\"&e0$F[q7$$\"5nmmm;/j$o/\"F[q$\"&m>$ F[q7$$\"5MLLL3_>jU6F[q$\"&(*[$F[q7$$\"5++++]i^Z]7F[q$\"&AM%F[q7$$\"5++ ++](=h(e8F[q$\"&,5'F[q7$$\"5++++]P[6j9F[q$\"&6!))F[q7$$\"5nmm\"HKR'\\5 :F[q$\"&A&**F[q7$$\"5MLL$e*[z(yb\"F[q$\"'[X5F[q7$$\"5nm;/Ev&[ge\"F[q$ \"'6;5F[q7$$\"5+++Dc,#>Uh\"F[q$\"&i+*F[q7$$\"5ML$eky#)*QU;F[q$\"'E2mF3 7$$\"5nmmm;a/cq;F[q$\"'%>`#F37$$\"5nmmmT&)))G=$)=F[q$!(%[B\")F37$$\"5ML3_]%QU$*)=F[q$!(-mF)F37$$ \"5,]7`W1a[&*=F[q$!(2:Q)F37$$\"5om;aQG%G;!>F[q$!(;eY)F37$$\"5,+DcEsW\" R\">F[q$!(**y[)F37$$\"5MLLe9;0?E>F[q$!(1fM)F37$$\"5++]i!Rgs2&>F[q$!(CE (oF37$$\"5nmmmm\"pW`(>F[q$!(bO#\\F37$$\"5++]iSmTI-?F[q$!'[(*[F37$$\"5M LLe9TOEH?F[q$\"(o**y%F37$$\"5,+Dc^yLuU?F[q$\"(d')*yF37$$\"5om;a)e6Bi0# F[q$\"):cT5F37$$\"5ML3_D`Gqp?F[q$\")l;w9F37$$\"5,++]i!f#=$3#F[q$\")F(y v\"F37$$\"5,++v$fQa)3@F[q$\")pCABF37$$\"5,+++D\"=EX8#F[q$\")bq1GF37$$ \"5,++Dcwz>g@F[q$\")cxGKF37$$\"5,++](=xpe=#F[q$\")p$[O$F37$$\"5M3_D.+! =#*=#F[q$\")V&HS$F37$$\"5o;/,>Gic#>#F[q$\")F;@MF37$$\"5,DcwMcW\"f>#F[q $\")$RtS$F37$$\"5ML3_]%oi#*>#F[q$\")@\"\\V$F37$$\"5,]7.#39ff?#F[q$\")P @GMF37$$\"5om;a8(fbE@#F[q$\")UsXMF37$$\"5,+Dcw4&[gA#F[q$\")em5MF37$$\" 5MLLeRA9WRAF[q$\")NkjLF37$$\"5++]ilZsAmAF[q$\")/])=$F37$$\"5nmmm\"H28I H#F[q$\"):!>%HF37$$\"5nmmTNr&3AM#F[q$\")L#\\V#F37$$\"5nmm;zpSS\"R#F[q$ \"(B0'>F[q7$$\"5+++v$41oWW#F[q$\"(#>j:F[q7$$\"5MLLL3_?`(\\#F[q$\"(AZE \"F[q7$$\"5MLL3_D1l_DF[q$\"(36/\"F[q7$$\"5MLL$e*)>pxg#F[q$\"'cK*)F[q7$ $\"5,++]Pf4t.FF[q$\"'H%H(F[q7$$\"5MLLLe*Gst!GF[q$\"'r%G'F[q7$$\"5,++++ DRW9HF[q$\"'PXcF[q7$$\"5,+++DJE>>IF[q$\"'wb_F[q7$$\"5,++]i!RU07$F[q$\" 'a`]F[q7$$\"5,+++v=S2LKF[q$\"'qM]F[q7$$\"5ommmm\"p)=MLF[q$\"'4O_F[q7$$ \"5,+++](=]@W$F[q$\"'%eu&F[q7$$\"5MLL$e*[$z*RNF[q$\"'o6lF[q7$$\"5,+++] iC$pk$F[q$\"'Y]vF[q7$$\"5om;zp)4\"4sOF[q$\"''pu(F[q7$$\"5MLLe*[t\\sp$F [q$\"'OMzF[q7$$\"5,+]P4r$3Cs$F[q$\"'x\")zF[q7$$\"5omm;H2qcZPF[q$\"'6.! )F[q7$$\"5MLL3F>grgPF[q$\"'!)>yF[q7$$\"5,+++DJ]'Qx$F[q$\"'#Rw(F[q7$$\" 5ML$eRs`R/y$F[q$\"'dzvF[q7$$\"5omm\"HK/9qy$F[q$\"'\\SwF[q7$$\"5,+](=# \\&)e$z$F[q$\"'O`uF[q7$$\"5MLL$3_0j,!QF[q$\"'0%>(F[q7$$\"5,++v=n?J8QF[ q$\"'O$)pF[q7$$\"5ommm;z5YEQF[q$\"'&>p'F[q7$$\"5MLLe9\"45'RQF[q$\"'?-g F[q7$$\"5,++]7.\"fF&QF[q$\"'4saF[q7$$\"5MLLek`8=/RF[q$\"'LFCF[q7$$\"5o mmm;/OgbRF[q$!'$zY\"F[q7$$\"5,+]7`p2_#)RF[q$!'+HMF[q7$$\"5MLLe*[$zV4SF [q$!(p%4VF37$$\"5o;zpBEs;;SF[q$!($GiVF37$$\"5,+D\"yv^'*G-%F[q$!(z$fUF3 7$$\"5M$3F>*3eiHSF[q$!(#)R*RF37$$\"5om;/E+^NOSF[q$!(sZ!RF37$$\"5ML3F%H o8)\\SF[q$!(*3xCF37$$\"5,++]ilAFjSF[q$!(=[6\"F37$$\"5ML$3_]p'>*3%F[q$ \"(*ogQF37$$\"5omm\"zW7@^6%F[q$\")#Ht0\"F37$$\"5NL3F>RL3GTF[q$\")H!=[ \"F37$$\"5-+]i!RbX59%F[q$\")7q<@F37$$\"5om\"z>'ox+aTF[q$\")al8DF37$$\" 5NLLLL$)*pp;%F[q$\")\"G\"fHF37$$\"5NL3Fp^yftTF[q$\")xa%F[q$ \")yP*4%F37$$\"5NL$e*[$>Pn?%F[q$\")y-!\\%F37$$\"5NLL$3-$H**>UF[q$\")aO D^F37$$\"5NLLek.W]YUF[q$\")+;AcF37$$\"5NLLL3xe,tUF[q$\")]!33'F37$$\"5- +v$4'\\=;'G%F[q$\")'f'GhF37$$\"5om;a8AyI*H%F[q$\")*e4='F37$$\"5NLe9m%z `CJ%F[q$\"(y.9'F[q7$$\"5-++v=n(*fDVF[q$\"(5g.'F[q7$$\"5NL$eRAr\"*=N%F[ q$\"(,Fs&F[q7$$\"5omm;HdO=yVF[q$\"(BRJ&F[q7$$\"5,+](=UAVBS%F[q$\"(hE'[ F[q7$$\"5MLLe9\"z-lU%F[q$\"(h#[WF[q7$$\"5om;H2eBm]WF[q$\"(C#))RF[q7$$ \"5,++++D>#[Z%F[q$\"('\\GOF[q7$$\"5ommT5::^-XF[q$\"(>!>KF[q7$$\"5MLL$3 _5,-`%F[q$\"(i*oGF[q7$$\"5,++DJ&p!*yb%F[q$\"(=Cg#F[q7$$\"5ommmT&G!e&e% F[q$\"(YzN#F[q7$$\"5-++]P%37^j%F[q$\"(Pj,#F[q7$$\"5NLLLL$)Qk%o%F[q$\"( 1qw\"F[q7$$\"5-++]iSjE!z%F[q$\"(VQU\"F[q7$$\"5-++]P40O\"*[F[q$\"(j*G7F [q7$$\"\"&F)$\"((4'4\"F[q-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGEND G6#%8Bogacki-Shampine~schemeG-F$6%7buF'7$F1$\"'SxMF37$FP$\"'%ol&F37$F^ o$\"'assF37$$\"5++]7G))>Wr@F3$\"'R?uF37$$\"5MLLek.pu/BF3$\"'q!f(F37$$ \"5nm;/,>=0QCF3$\"'-0vF37$Fco$\"'ratF37$$\"5nmmT5ll'z$GF3$\"'uDnF37$Fh o$\"'VFcF37$$\"5MLLLL$eI8k$F3$\"'VNIF37$F]p$\"%#*pF37$Fbp$!'b&)\\F37$$ \"5++++v$fl])F[q7$$\"5+++++v\"=YI\"F[q$!&H!*)F[q7$F[s$!&wb*F[q7$$\"5 ++++]7!Q4T\"F[q$!'Pg5F[q7$F`s$!'4=7F[q7$Fes$!'zc9F[q7$Fjs$!'Mt%F37$Fjx$!( F:q#F37$$\"5MLek.HW#)))>F[q$!(YpQ#F37$F_y$!(5b8#F37$$\"5M$e9\"4NS/4?F[ q$!(&H#4#F37$$\"5nmTgx.Ry:?F[q$!(4p/#F37$$\"5+]P4YsP_A?F[q$!(gy+#F37$F dy$!(f+-#F37$Fiy$!(L;3#F37$F^z$!(4()=#F37$Fcz$!(S)RCF37$Fhz$!(%3BEF37$ F][l$!(-h(HF37$Fb[l$!(\\r<$F37$$\"5,+]i!*y?OZ@F[q$!(@1<$F37$Fg[l$!(]I4 $F37$$\"5,+](=U(Q.t@F[q$!(U\"pHF37$F\\\\l$!(*o/GF37$Fd^l$!(YXt\"F37$F^ _l$!(JI7\"F37$$\"5nmTg_Z>J0BF[q$!(BW4\"F37$$\"5nm;a8A3hF[q7$Fb`l$!'?\"\\$F[q7$$\"5ML$3-)Q84DDF[q$!'f$z$F[q 7$Fg`l$!'yVSF[q7$$\"5M$3_+sWS&fDF[q$!'J&4%F[q7$$\"5ML3-))o-VmDF[q$!'A$ 3%F[q7$$\"5M$e*)f04?Ld#F[q$!'N]TF[q7$$\"5ML$eRA\"*4-e#F[q$!'>>UF[q7$$ \"5MLe*)fb&*)Rf#F[q$!'LDUF[q7$F\\al$!'IZUF[q7$$\"5om;/,>Ww>EF[q$!'(RJ% F[q7$$\"5,++D1R'f0uzEF[ q$!'k5VF[q7$$\"5om;HKRdt\"p#F[q$!''3H%F[q7$Faal$!'egUF[q7$$\"5omm\"zWi ^bv#F[q$!'^OTF[q7$Ffal$!'B*)RF[q7$F[bl$!'_#p$F[q7$F`bl$!'d![$F[q7$Febl $!'8!Q$F[q7$Fjbl$!'W&R$F[q7$$\"5MLL$3_NJOG$F[q$!'S\\MF[q7$F_cl$!'yONF[ q7$$\"5MLLLeR%p\")Q$F[q$!'FuOF[q7$Fdcl$!'buQF[q7$$\"5omm\"H#oZ1\"\\$F[ q$!'pHTF[q7$Ficl$!'-)[%F[q7$$\"5omm\"Hd!fX$f$F[q$!'sK]F[q7$F^dl$!'yLeF [q7$Fhdl$!'$o&pF[q7$Fbel$!'Df&)F[q7$F\\fl$!'7s'*F[q7$F`gl$!(w95\"F[q7$ Fjgl$!(BPA\"F[q7$Fdhl$!(*4$Q\"F[q7$$\"5om;aQG-ZyQF[q$!([t_\"F[q7$Fihl$ !(OTn\"F[q7$$\"5,+]i!*yC*)HRF[q$!(p7!=F[q7$F^il$!(/%*)=F[q7$$\"5MLe*[o =i!pRF[q$!(%=:>F[q7$Fcil$!(me#>F[q7$$\"5M$eRs31]#*)RF[q$!(et#>F[q7$$\" 5omTN@_$zf*RF[q$!(5T$>F[q7$$\"5,](oaNk3F+%F[q$!())f\">F[q7$Fhil$!)mW.> F37$F\\[m$!)lY[=F37$Ff[m$!)w5Hpd\"F37$$ \"5N3_v:y;O=TF[q$!)aOu:F37$$\"5-]Pf$=B-;7%F[q$!)\\%Rc\"F37$$\"5o\"HK9b yU[7%F[q$!)<1m:F37$Fe\\m$!)$plc\"F37$$\"5o;z%\\lWkX8%F[q$!)hij:F37$Fj \\m$!)'y#o:F37$F_]m$!)[3#e\"F37$Fd]m$!)P&Gg\"F37$Fh^m$!)IMq;F37$Fb_m$! )P0?VUF[q$!)AJ4Z6F[q7$$\"5NLekGg6but%F[q$!(ov1\"F [q7$Faem$!(?x,\"F[q7$Ffem$!'4Y\"*F[q7$F[fm$!'gz#)F[q-F`fm6&FbfmFcfmF(F cfm-Fgfm6#%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG-F$6%7 [sF'7$F1$\"';8;F37$FF$\"'^Q?F37$FP$\"'6>BF37$FU$\"'-gCF37$FZ$\"'m7EF37 $Fin$\"'RaEF37$F^o$\"'KtEF37$Fggm$\"'MqEF37$F\\hm$\"'mPEF37$Fahm$\"')H c#F37$Fco$\"'%zY#F37$Fihm$\"'Ab@F37$Fho$\"'aG=F[q7$Fc\\n$!&v,#F[q7$F`s$!&qQ#F[q7$Fe s$!&C+$F[q7$Fjs$!&J#QF[q7$Fdt$!&FP&F[q7$F^u$!'FeuF37$Fcu$!'Wt\"*F37$Fh u$!(^_2\"F37$Fj]n$!(i*R6F37$F]v$!('Gu6F37$Fg^n$!(<&)=\"F37$Fbv$!(wI>\" F37$F\\w$!(Q**=\"F37$Ffw$!(cS>\"F37$F[x$!(So?\"F37$F`x$!(V\\A\"F37$Fex $!(LaI\"F37$Fjx$!(*)zS\"F37$Fdy$!(#H!z\"F37$Fhz$!(V50#F37$$\"5,+]7G)[= g4#F[q$!(yd0#F37$F][l$!(t\"R?F37$$\"5,+]Pf$G!p@@F[q$!(x\"*)>F37$Fb[l$! (W)H>F37$Fg[l$!(z+r\"F37$F\\\\l$!(v5^\"F37$Fd^l$!(\")Q0\"F37$F^_l$!'o. ))F37$F\\dn$!'w]()F37$Fadn$!'(>#))F37$Ffdn$!'&3C*F37$Fc_l$!'[q%*F37$F^ en$!(GD+\"F37$Fh_l$!'i86F[q7$F]`l$!'lj7F[q7$Fb`l$!'$HO\"F[q7$Fien$!'F# R\"F[q7$Fg`l$!'u29F[q7$Fffn$!'a)R\"F[q7$F`gn$!'+29F[q7$Fegn$!'d\"R\"F[ q7$F\\al$!'wz8F[q7$F\\in$!'1N8F[q7$Faal$!'#*z7F[q7$Ffal$!'Wa6F[q7$F[bl $!'a`5F[q7$F`bl$!&&)*)*F[q7$Febl$!&$y&*F[q7$Fjbl$!&]^*F[q7$Fg[o$!&'*e* F[q7$F_cl$!&Tu*F[q7$F_\\o$!'\"G+\"F[q7$Fdcl$!'G\\5F[q7$Fg\\o$!'o:6F[q7 $Ficl$!'P;7F[q7$F_]o$!'vv8F[q7$F^dl$!'#=h\"F[q7$Fhdl$!'RL>F[q7$Fbel$!' 1iBF[q7$F`gl$!'KWHF[q7$Fdhl$!'WENF[q7$Fihl$!'*G/%F[q7$F^il$!'I3WF[q7$F hil$!(#H_YF37$Ff[m$!(!3V]F37$F`\\m$!(JXp&F37$Fd]m$!((f5jF37$Fi]m$!(fEP 'F37$F^^m$!(M!>kF37$Fc^m$!(1&3kF37$Fh^m$!(11U'F37$F]_m$!(@PQ'F37$Fb_m$ !(d(piF37$Fg_m$!()emfF37$F\\`m$!(n4Y&F37$Ff`m$!(Gz%\\F37$F`am$!'D*\\%F [q7$Feam$!'B8TF[q7$Fjam$!'3qQF[q7$F_bm$!'Y[PF[q7$Fdbm$!'omOF[q7$$\"5o; /w(GoUDV%F[q$!'o$o$F[q7$F[ho$!'XhOF[q7$$\"5M$e9TjYAYW%F[q$!'GgOF[q7$Fi bm$!'=#p$F[q7$Fcho$!'^vOF[q7$F^cm$!'DhOF[q7$$\"5ML$3_+sm')[%F[q$!'-nOF [q7$Fccm$!'_(o$F[q7$$\"5ML3-j7RV4XF[q$!'o(o$F[q7$$\"5,+]i:5jN;XF[q$!'w eOF[q7$$\"5om\"H#o2(yK_%F[q$!'ywOF[q7$Fhcm$!')pp$F[q7$F]dm$!'3FOF[q7$F bdm$!'OwNF[q7$Fgdm$!'([V$F[q7$F\\em$!'B_KF[q7$Faem$!'OTGF[q7$Ffem$!'z1 DF[q7$F[fm$!'(RD#F[q-%&COLORG6&Fbfm$\"\"\"!\"\"$\"\")FihqF(-Fgfm6#%3ef ficient~scheme~AG-F$6%7]vF'7$F+$\"(UwE\"F-7$F1$\"')yb#F37$F<$\"'sVGF37 $FF$\"'t&)HF37$FK$\"'q+JF37$FP$\"'(3>$F37$FU$\"':kKF37$FZ$\"'DBLF37$Fi n$\"'F'G$F37$F^o$\"'_>KF37$Fco$\"'EADF37$Fho$\"'5_7F37$Faim$\"&\"Q7F37 $F]p$!&T(yF37$$\"5ommT5:j=XWF3$!'VJ6F37$Fcbp$!'@*Q\"F37$$\"5NLLek.%*Qz \\F3$!'r%\\\"F37$Fbp$!':i:F37$$\"5NLLL$3FGT\\&F3$!'vn:F37$F\\jm$!'F?:F 37$$\"5qmmmm;HS*)fF3$!&pW\"F[q7$Fgp$!&[M\"F[q7$F]q$!%F()F[q7$Fbq$!%*) \\F[q7$Fgq$!%PPF[q7$F\\r$!%)>%F[q7$Far$!%Z_F[q7$Ffr$!%0iF[q7$F[s$!%woF [q7$F`s$!%;%)F[q7$Fes$!%.)*F[q7$Fjs$!&K7\"F[q7$F_t$!&/?\"F[q7$Fdt$!&'H 7F[q7$$\"5nmTNr9XIG;F[q$!&w@\"F[q7$Fit$!'-i6F37$$\"5n;/,W%[K%\\;F[q$!' @F6F37$$\"5++Dc,T^Zc;F[q$!'v66F37$$\"5M$e9\"f(z()F37$F][l$\"(8w:\"F37$Fb[l$\"(QI\\\"F37$Ffbn$\"( @ck\"F37$Fg[l$\"(64\">F37$F^cn$\"(=I.#F37$F\\\\l$\"(**)[@F37$Fj]l$\"(a lW#F37$Fd^l$\"(gZe#F37$$\"5nT&Q`0l*yUAF[q$\"(uZf#F37$$\"5+]P4ryy8YAF[q $\"(F1e#F37$$\"5Me*[oo5'[\\AF[q$\"(]_g#F37$$\"5nmTg-NV$GD#F[q$\"(g!3EF 37$$\"5+v$f$=jD=cAF[q$\"(*z#f#F37$$\"5M$e9T8zI&fAF[q$\"(qog#F37$$\"5n \"zp)\\>!zGE#F[q$\"(@Kg#F37$Fi^l$\"(jpe#F37$$\"5MLekGg,izAF[q$\"(\"zbD F37$F^_l$\"(I\")[#F37$Fadn$\"(V\"F[q7$F\\fl$!&hz\"F [q7$F`gl$!&$QAF[q7$Fegl$!&:M#F[q7$Fjgl$!&2T#F[q7$F_hl$!&QP#F[q7$Fdhl$! &&zAF[q7$Fihl$!&'*G\"F[q7$F^il$!%#>#F[q7$$\"5,]7y]&*GLiRF[q$!%[y9zvRF[q$!%.CF[q7$Fcil$!%@TF[q7$Ff`o$!%UtF[q7$F hil$!'LU:F37$F\\[m$!'_/LF37$Ff[m$!'<`gF37$F[\\m$!'%44)F37$F`\\m$!'qL\" *F37$Fj\\m$!'\"36)F37$Fd]m$!'tzcF37$F^^m$!';cCF37$Fh^m$!&%\\OF37$$\"5N L3-8D$4,?%F[q$\"'\"e#>F37$F]_m$\"'eC@F37$$\"5NLe*[=1lL@%F[q$\"'C\"Q%F3 7$Fb_m$\"'0lpF37$Fjdo$\"'&oh*F37$Fg_m$\"(?EC\"F37$$\"5NL3_+sA8`UF[q$\" (ms\\\"F37$Fafo$\"(;Is\"F37$$\"5NLeRs3!)QmUF[q$\"(FNw\"F37$F\\`m$\"(*) f+#F37$Fa`m$\"(*zOAF37$Ff`m$\"((RkDF37$F[am$\"'T)z#F[q7$F`am$\"'B8HF[q 7$$\"5omTNrRduQVF[q$\"'LOIF[q7$Feam$\"'-DJF[q7$$\"5-+Dcw%oP]O%F[q$\"'E 0JF[q7$Fjam$\"'xbIF[q7$F_bm$\"'_bGF[q7$Fdbm$\"'ptDF[q7$F[ho$\"'^!R#F[q 7$Fibm$\"'&p;#F[q7$Fcho$\"'G-?F[q7$F^cm$\"'tR=F[q7$Ffeq$\"'YN;F[q7$Fcc m$\"'-A9F[q7$Fcfq$\"'Aw7F[q7$Fhcm$\"'Xn5F[q7$F]dm$\"&rS)F[q7$Fbdm$\"&3 N'F[q7$Fgdm$\"&N6%F[q7$F\\em$\"&W9$F[q7$$\"5om\"z%\\!pYyp%F[q$\"&E,$F[ q7$$\"5-+]il(\\\\5r%F[q$\"&,%HF[q7$$\"5NL3x\"[I_Us%F[q$\"&W)GF[q7$Fc^p $\"&-%GF[q7$$\"5NL$3-jsgQw%F[q$\"&-&GF[q7$Faem$\"&X!HF[q7$$\"5-++++DM \"3%[F[q$\"&r)HF[q7$Ffem$\"&=.$F[q7$$\"5,++voa-oX\\F[q$\"&***HF[q7$F[f m$\"&\\\"HF[q-F`fm6&FbfmFcfmF(F(-Fgfm6#%3efficient~scheme~BG-F$6%7jvF' 7$F1$\"'a>TF37$FP$\"'O:jF37$F^o$\"'3_yF37$Fggm$\"'3!)zF37$F\\hm$\"'5\" >)F37$Fahm$\"',N\")F37$Fco$\"'WQ!)F37$Fihm$\"',#o(F37$Fho$\"')42(F37$F aim$\"'_>dF37$F]p$\"'d8XF37$F]\\r$\"'d`SF37$Fcbp$\"'cAPF37$Fe\\r$\"'jB NF37$Fbp$\"'!RU$F37$F\\jm$\"'#pV$F37$Fgp$\"&=g$F[q7$F]q$\"&9(RF[q7$Fbq $\"&t8%F[q7$Fgq$\"&L-%F[q7$F\\r$\"&'fQF[q7$Far$\"&#>QF[q7$Ffr$\"&N%RF[ q7$F[\\n$\"&2.%F[q7$F[s$\"&I2%F[q7$Fc\\n$\"&C)RF[q7$F`s$\"&+v$F[q7$Fes $\"&eI$F[q7$Fjs$\"&%RGF[q7$F_t$\"&M\\#F[q7$Fdt$\"&O<#F[q7$Fa`r$\"&O2#F [q7$Fit$\"'By>F37$F^ar$\"'9o>F37$F^u$\"'Z.?F37$$\"5nmm\"z>c#\\#o\"F[q$ \"'U%3#F37$$\"5nmm;zpYU%p\"F[q$\"'g-AF37$$\"5nmmTgxnN1))F37$Fdy$!(C!y6F37$F^z$!(+kQ\"F37$Fhz$!(VF[q$\"'Fo6F[q7$F`am$\"'Z/9F[q7$F[is$ \"'I\")>F[q7$Feam$\"'IrDF[q7$Fcis$\"'0YGF[q7$Fjam$\"'_\\$F[q7$$\"5o\"z%*z_%G[BWF[q$\"'vnMF[q7$Fdbm$\"'/XMF[q7$F ibm$\"'P]KF[q7$F^cm$\"'&\\'HF[q7$Fccm$\"'giDF[q7$Fhcm$\"''[>#F[q7$F]dm $\"'hV>F[q7$Fbdm$\"'()Qg\"F[q7$Faem$\"'mM;F[q7$$\"5-+](=<6.H![F[q $\"'tL;F[q7$$\"5-++D\"G))Rb\"[F[q$\"'YK;F[q7$$\"5-+]i!Rlw\"G[F[q$\"'/O ;F[q7$F_^t$\"'f_;F[q7$$\"5-+]P4'>]M&[F[q$\"'AZ;F[q7$$\"5-++v=np3m[F[q$ \"'tT;F[q7$$\"5-+]7GQPsy[F[q$\"'\"=k\"F[q7$Ffem$\"'[X;F[q7$Fg^t$\"'%>h \"F[q7$F[fm$\"'/u:F[q-F`fm6&Fbfm$\")#)eqkFefm$\"))eqk\"FefmF]gv-Fgfm6# %3efficient~scheme~CG-F$6%7dqF'7$F1$\"'5aIF37$F^o$\"'^(e%F37$Fho$\"']f rF37$F]p$\"(o4,\"F37$Fbp$\"(F7D\"F37$Fgp$\"'T_8F[q7$F]q$\"'Wh8F[q7$Fbq $\"'@L8F[q7$Fgq$\"'0&H\"F[q7$F\\r$\"'h\"G\"F[q7$Far$\"'Q28F[q7$Ffr$\"' \\&R\"F[q7$F[s$\"'Bp:F[q7$F`s$\"',k=F[q7$Fjs$\"'WoAF[q7$F^u$\"(;H#HF37 $Fcu$\"(%zzJF37$Fhu$\"(&R]LF37$$\"5MLe9m%>(*)zN$F37$$\" 5+]P%['*)of9=F[q$\"(n_K$F37$F]v$\"(aVF$F37$Fg^n$\"(%\\FJF37$Fbv$\"(Li# HF37$F`x$\"(ciQ#F37$Fjx$\"(C\\)>F37$Fj_n$\"(DY&>F37$F_y$\"(NF(>F37$Fb` n$\"(Z))*>F37$Fg`n$\"(:N-#F37$F\\an$\"(X.4#F37$Fdy$\"(N(\\@F37$F^z$\"( `TV#F37$Fhz$\"(Li'GF37$F][l$\"(MK@$F37$Fb[l$\"(^&yMF37$Ffbn$\"(8cb$F37 $Fg[l$\"(<^l$F37$F^cn$\"(Mtl$F37$F\\\\l$\"(\"pVOF37$Fj]l$\"(*[)e$F37$F d^l$\"(M')\\$F37$F^hr$\"(*QdMF37$Fi^l$\"(FyT$F37$Feir$\"(mBV$F37$F^_l$ \"(;5V$F37$Fc_l$\"(O)GOF37$Fh_l$\"'`!*RF[q7$F]`l$\"'/LVF[q7$Fb`l$\"'i3 YF[q7$Fien$\"'f>ZF[q7$Fg`l$\"'31[F[q7$Fffn$\"'N*y%F[q7$F`gn$\"'f_[F[q7 $Fegn$\"'w<[F[q7$F\\al$\"''yz%F[q7$F\\in$\"'SRZF[q7$Faal$\"'lQYF[q7$Ff al$\"']HVF[q7$F[bl$\"'SSSF[q7$F`bl$\"'ATQF[q7$$\"5,++v$4^n)pIF[q$\"':& y$F[q7$Febl$\"'GcPF[q7$Fj`u$\"'$yv$F[q7$Fjbl$\"'q/QF[q7$Fg[o$\"'k%)QF[ q7$F_cl$\"'!o+%F[q7$F_\\o$\"'*3>%F[q7$Fdcl$\"'_ZWF[q7$Ficl$\"'mP^F[q7$ F_]o$\"'KmcF[q7$F^dl$\"'#oM'F[q7$Fhdl$\"'ulrF[q7$Fbel$\"'$f<)F[q7$F`gl $\"''zZ*F[q7$Fdhl$\"(/v3\"F[q7$Fihl$\"(&3D7F[q7$F^il$\"(B%R8F[q7$Fhil$ \")?\"pS\"F37$Ff[m$\")wcW9F37$F`\\m$\")sm-:F37$Fd]m$\")EI/;F37$Fh^m$\" )7Ag;F37$Fb_m$\");#[p\"F37$Fedo$\")f#po\"F37$Fjdo$\")BN\"p\"F37$F_eo$ \")$[qo\"F37$Fdeo$\")0k#p\"F37$Fieo$\")'G\"F[q7$Fbdm$\"(JZA\"F[q7$F\\em$\"(``8\"F[ q7$Faem$\"(.C-\"F[q7$Ffem$\"'56#*F[q7$F[fm$\"'J$R)F[q-Fehq6&Fbfm$\"\"' FihqF(Fjhq-Fgfm6#%Sscheme~with~stability~radius~=~imag~axis~inclusionG -%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~7~stage~order ~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fh\\x-%%VIEWG6$;F(F[fm %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "B ogacki-Shampine scheme" "a minor modification of the Bogacki-Shampine \+ scheme" "efficient scheme A" "efficient scheme B" "efficient scheme C " "scheme with stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 7 of 7 stage, order 5 Runge- Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy /dx=-(1+4*cos(3*x))*(y-1/3)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&*&\" \"%F&-%$cosG6#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF&*&F&F&F2F(F(F&F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin(3/2*x)*c os(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F+F* %\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+F*F7F,F1F *F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x) -x)" "6#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0 F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*cos(3*x))*(y(x)-1 /3);\nic := y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x)));\nv := unapply(r hs(%),x):\nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9],labels=[`x`,`y(x )`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,$*&,&\"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0F0F0,&F)F0#F0F8! \"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+-% $expG6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*&#\"\")F,F+*&-F56# ,$*(F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F.6#,&*&#F3F,F+F4F +F9F'F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&- %'CURVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3W+7cSy5h&*!#=7$ $\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F 27$$\"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$z x(F27$$\"3GLLL3x&)*3\"F2$\"3IM[S(o-#HsF27$$\"3em\"z%\\v#pK\"F2$\"3i=)H '*Q$=:oF27$$\"3))**\\i!R(*Rc\"F2$\"3w,'pRB0LX'F27$$\"3&edVF27$$\"3%QL$3_DG1qF2$\" 3'fN^hMLe*)>VB$)F2$\"3DB(Rfp)*\\j%F27$$\"3Y++DJbw!Q*F2$ \"3%GsCu$*)zK]F27$$\"3+N$ekGkX#**F2$\"3u>+\\,YW?`F27$$\"3%ommTIOo/\"!# <$\"3q]2x8ZEqcF27$$\"3E+]7GTt%4\"Fgt$\"39b$=$pWlHgF27$$\"3YLL3_>jU6Fgt $\"3nwYdkc=KkF27$$\"3ym;HdNb'>\"Fgt$\"3l[hQOW]BpF27$$\"37++]i^Z]7Fgt$ \"3IVnF)*yXIuF27$$\"35+++v\"=YI\"Fgt$\"3ahS!3L%e=zF27$$\"33++](=h(e8Fg t$\"3l&QV-<82M)F27$$\"3&*****\\7!Q4T\"Fgt$\"3^]H\"3wS2k)F27$$\"3/++]P[ 6j9Fgt$\"3ur)[IAj$)z)F27$$\"3'=HKkAg\\Z\"Fgt$\"33z^;ogY6))F27$$\"3W$ek `h0o[\"Fgt$\"3h=q?g>u:))F27$$\"3/voH/5l)\\\"Fgt$\"3p\\\\U!)G36))F27$$ \"3%o;HKR'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr=M:Fgt$\"3Erd.MaCV()F27$ $\"3UL$e*[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y 3J)F27$$\"3wmm;a/cq;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF27$$\"3%ommmJ>IFgt$ \"3N!o4Joz]`$F27$$\"3F+]i!RU07$Fgt$\"3=,?;D0\"Qg$F27$$\"3+++v=S2LKFgt$ \"3wRH=fZn5PF27$$\"3Jmmm\"p)=MLFgt$\"3RsXuk([b#QF27$$\"3B++](=]@W$Fgt$ \"3%4[=*QOMSRF27$$\"3mm\"H#oZ1\"\\$Fgt$\"3QK??D+QyRF27$$\"35L$e*[$z*RN Fgt$\"3UAxt;S)>+%F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$p k$Fgt$\"3eIRs#H!Q\"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m; H2qcZPFgt$\"3w)))[$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ym m;/OgbRFgt$\"3W-Tml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3S nmT&G!e&e%Fgt$\"35gRzc#\\LF2 7$$\"\"&F)$\"3Ii#4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fiel;F($\"#6F cfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "V := (x,y) -> -(1+4*cos(3*x ))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shamp ine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`eff icient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme wit h stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 30 :\nfor ct to 6 do\n Vn_RK5_||ct := RK5_||ct(V(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Vn_RK5_||ct):\n for ii to n umpts do\n sm := sm+(Vn_RK5_||ct[ii,2]-v(Vn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"% F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initia l~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~ ~~G\"$]#Q)pprint446\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$ \"+ClU%=\"!#?7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$ \"+U\"=hP%!#@7$%3efficient~scheme~AG$\"+\"*R'*=lF07$%3efficient~scheme ~BG$\"++r\"RX\"F07$%3efficient~scheme~CG$\"++3hkoF07$%Sscheme~with~sta bility~radius~=~imag~axis~inclusionG$\"+uI`eEF+Q)pprint456\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%& FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh \+ := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. \+ of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a \+ minor modification of the Bogacki-Shampine scheme`,`efficient scheme A `,`efficient scheme B`,`efficient scheme C`,`scheme with stability rad ius = imag axis inclusion`]:\nerrs := []:\nDigits := 30:\nfor ct to 6 \+ do\n vn_RK5_||ct := RK5_||ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\nen d do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 6 do\n errs := [ op(errs),abs(vn_RK5_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\" $F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\" \"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint46 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+t:C*f&!#A 7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+%)*Gcc#F+7 $%3efficient~scheme~AG$\"+R`#p2#F+7$%3efficient~scheme~BG$\"+e&*Qx5F+7 $%3efficient~scheme~CG$\"+!HbX5#F+7$%Sscheme~with~stability~radius~=~i mag~axis~inclusionG$\"+qs\"Q@\"!#@Q)pprint476\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor mo dification of the Bogacki-Shampine scheme`,`efficient scheme A`,`effic ient scheme B`,`efficient scheme C`,`scheme with stability radius = im ag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((v(x)-'vn_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7 ,factor=100);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 1 0:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+7?g'=\"!# ?7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+jV.iV!#@7 $%3efficient~scheme~AG$\"+`x(e^'F07$%3efficient~scheme~BG$\"+&f*4U9F07 $%3efficient~scheme~CG$\"+F1AhoF07$%Sscheme~with~stability~radius~=~im ag~axis~inclusionG$\"+)Hk\\l#F+Q)pprint486\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "evalf[20](plot([v(x)-'vn_R K5_1'(x),v(x)-'vn_RK5_2'(x),v(x)-'vn_RK5_3'(x),v(x)-'vn_RK5_4'(x),\nv( x)-'vn_RK5_5'(x),v(x)-'vn_RK5_6'(x)],x=0..5,\nfont=[HELVETICA,9],color =[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlege nd=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sha mpine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sch eme C`,`scheme with stability radius = imag axis inclusion`],\ntitle=` error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 898 547 547 {PLOTDATA 2 "6,-%'CURVESG6%7ds7$$\"\"!F)F (7$$\"5NLLL$3FWYs#!#@$!*7&\\*=$!#?7$$\"5qmmmmT&)G\\aF-$!*/A8h&F07$$\"5 SLLL3x1h6oF-$!*wcD\"oF07$$\"50+++]7G$R<)F-$!**\\^HyF07$$\"5qmmTNYL^9&) F-$!*^F8q(F07$$\"5SLL$3-)Q4b))F-$!*7!e(e(F07$$\"50++D19Wn&>*F-$!*$3,Nv F07$$\"5qmmm\"z%\\DO&*F-$!*)\\4RwF07$$\"5NLL3x\"[No()*F-$!*5ZS+)F07$$ \"5+++Dc,;u@5F0$!*S1\">\")F07$$\"5nm;z%\\l*zb5F0$!*+j\"))zF07$$\"5MLLL L3x&)*3\"F0$!*1:8(yF07$$\"5+]ilZQ9\\>6F0$!*XL)4yF07$$\"5nm\"z>'o^7\\6F 0$!*[a0$yF07$$\"5M$3-j()*)e(y6F0$!*se/(zF07$$\"5++]i!*GER37F0$!*af75)F 07$$\"5ML3F>*3gwE\"F0$!*>h%zyF07$$\"5nmm\"z%\\v#pK\"F0$!*k`5s(F07$$\"5 ++Dcw4]>'Q\"F0$!*hs;x(F07$$\"5ML$3_+ZiaW\"F0$!*HjPj(F07$$\"5nmT&Q.$*HZ ]\"F0$!*-$)HW(F07$$\"5+++]i!R(*Rc\"F0$!*\"R*oM(F07$$\"5MLL3xJs1,=F0$!* 3r*ymF07$$\"5nmmm\"H2P\"Q?F0$!*!G(=%fF07$$\"5+++]PMnNrDF0$!*IIEW%F07$$ \"5MLLL$eRwX5$F0$!*zE5F$F07$$\"5MLLLe*[`HP$F0$!*F@[(GF07$$\"5MLLLL$eI8 k$F0$!*jBZd#F07$$\"5MLLL3xwq4RF0$!*s*z$R#F07$$\"5NLLL$3x%3yTF0$!*;MbH# F07$$\"5ommT5:j=XWF0$!*7qWE#F07$$\"5-++]PfyG7ZF0$!*l#\\YAF07$$\"5NLLek .%*Qz\\F0$!*h\"*[I#F07$$\"5ommm\"z%4\\Y_F0$!*DL'*Q#F07$$\"5++++v$flVB$)F0$!*rz!oVF07$$\"5qmmmTg()4_))F0$!*klu?&F07$$\"5.+++DJ bw!Q*F0$!*M!39kF07$$\"5NLL$ekGkX#**F0$!*-DfA)F07$$\"5nmmm;/j$o/\"!#>$! +m'eU5\"F07$$\"5MLLe9;_yq5Ffx$!+T#R[C\"F07$$\"5+++]7GTt%4\"Ffx$!+x#=)3 9F07$$\"5nmmT5SIo=6Ffx$!+=6A:;F07$$\"5MLLL3_>jU6Ffx$!+)>%oE=F07$$\"5++ +voaFfp6Ffx$!+'3rB/#F07$$\"5nmm;HdNb'>\"Ffx$!+\"HbaG#F07$$\"5MLLe*)fV^ B7Ffx$!+4t5^DF07$$\"5++++]i^Z]7Ffx$!+h*4,x#F07$$\"5++++vomax7Ffx$!+OU4 !*HF07$$\"5+++++v\"=YI\"Ffx$!+bqa$=$F07$$\"5++++D\"o*oJ8Ffx$!+z0(yL$F0 7$$\"5++++](=h(e8Ffx$!+([NaW$F07$$\"5++++++'\\[Q\"Ffx$!+K,e5NF07$$\"5+ +++]7!Q4T\"Ffx$!+m:%ya$F07$$\"5+++D1k!*>99Ffx$!+CY>^NF07$$\"5+++]i:,Y< 9Ffx$!+RxtSNF07$$\"5+++v=n6s?9Ffx$!+?**G9NF07$$\"5++++v=A)RU\"Ffx$!+;: eANF07$$\"5+++](=K/0V\"Ffx$!+a^_ONF07$$\"5+++++Dk-P9Ffx$!+DaTCNF07$$\" 5++++DJ12]9Ffx$!+)[n.]$F07$$\"5++++]P[6j9Ffx$!+Z5BOMF07$$\"5nmm\"HKR' \\5:Ffx$!+:PJ_KF07$$\"5MLL$e*[z(yb\"Ffx$!+yzTCHF07$$\"5nm;/Ev&[ge\"Ffx $!+)y$z(p#F07$$\"5+++Dc,#>Uh\"Ffx$!+s#QY\\#F07$$\"5ML$eky#)*QU;Ffx$!+c gaIAF07$$\"5nmmm;a/cq;Ffx$!+w:#H/#F07$$\"5nmmmT&)))G=I$)=Ffx$ !+B9yn8F07$$\"5,Dc^`t3F')=Ffx$!+suC]8F07$$\"5ML3_]%QU$*)=Ffx$!+Mn/M8F0 7$$\"5oTg_Z&*QT#*=Ffx$!+9'*zA8F07$$\"5,]7`W1a[&*=Ffx$!+2+pC8F07$$\"5Me k`TFfx$!+nCbt8F07$$\"5M$3_D.Xrx!>F fx$!+zRpP8F07$$\"5,+DcEsW\"R\">Ffx$!+)*[U;8F07$$\"5M3xcB$)f)p\">Ffx$!+ 8\\rH8F07$$\"5o;Hd?%\\d+#>Ffx$!+/'*yy8F07$$\"5,D\"yv^+HJ#>Ffx$!+'fs*f8 F07$$\"5MLLe9;0?E>Ffx$!+6BZT8F07$$\"5++]i!Rgs2&>Ffx$!+6bU38F07$$\"5nmm mm\"pW`(>Ffx$!+l_d(G\"F07$$\"5MLLe9TOEH?Ffx$!+I'f$>7F07$$\"5,++]i!f#=$ 3#Ffx$!+#=5*36F07$$\"5,++](=xpe=#Ffx$!*SkC&yF07$$\"5nmmm\"H28IH#Ffx$!* 1mF'\\F07$$\"5nmm;zpSS\"R#Ffx$!*T*4*G$F07$$\"5+++v$41oWW#Ffx$!*6B^t#F0 7$$\"5MLLL3_?`(\\#Ffx$!*F(HpBF07$$\"5MLL3_D1l_DFfx$!*&=sG@F07$$\"5MLL$ e*)>pxg#Ffx$!*c^\"**>F07$$\"5ommm;z+vbEFfx$!*d#Q\\>F07$$\"5,++]Pf4t.FF fx$!*c_%e>F07$$\"5MLLLe*Gst!GFfx$!*c8=8#F07$$\"5,++++DRW9HFfx$!*([J!e# F07$$\"5,+++DJE>>IFfx$!*8jOT$F07$$\"5,++]i!RU07$Ffx$!*\"3\\1[F07$$\"5, +++v=S2LKFfx$!*$*e-:(F07$$\"5ommmm\"p)=MLFfx$!*_*Q!z*F07$$\"5MLLLeR%p \")Q$Ffx$!+`m686F07$$\"5,+++](=]@W$Ffx$!+'Gg/A\"F07$$\"5ML$ekyZ2mY$Ffx $!+VH2e7F07$$\"5omm\"H#oZ1\"\\$Ffx$!+A'*[(G\"F07$$\"5MLe9T8MH.NFfx$!+D Ek&H\"F07$$\"5,+]Pfe?_:NFfx$!+E5n18F07$$\"5omTgx.2vFNFfx$!+3Z,68F07$$ \"5MLL$e*[$z*RNFfx$!+EDY58F07$$\"5omT5:)[[Lb$Ffx$!+'[-kJ\"F07$$\"5,+]P MFwrmNFfx$!+'4s7J\"F07$$\"5MLek`mn3!e$Ffx$!+?'f?OFfx$!+5?Pm7F07$$\"5,+++]iC$pk$Ffx$!+ .:FL7F07$$\"5MLLe*[t\\sp$Ffx$!+%f<*R6F07$$\"5omm;H2qcZPFfx$!+)*QaA5F07 $$\"5,++]7.\"fF&QFfx$!*AA`m(F07$$\"5ommm;/OgbRFfx$!*j7Ri&F07$$\"5,++]i lAFjSFfx$!*!Hr6TF07$$\"5NLLLL$)*pp;%Ffx$!*(H&o(GF07$$\"5NLLL3xe,tUFfx$ !*(>so=F07$$\"5omm;HdO=yVFfx$!*Ika<\"F07$$\"5,++++D>#[Z%Ffx$!)qK$)yF07 $$\"5ommmT&G!e&e%Ffx$!)KIkbF07$$\"5NLLLL$)Qk%o%Ffx$!)4LcYF07$$\"5-++]i SjE!z%Ffx$!)>N,WF07$$\"5-++]P40O\"*[Ffx$!)P>\"p%F07$$\"\"&F)$!)g)Gh&F0 -%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~sch emeG-F$6%7jwF'7$$\"5qmm;aQ`!eS$!#A$!&hC\"F07$$F8F[gm$!'2eqF07$$\"5,++D c,;u@5F-$!(//*pF07$$\"5ommmTN@Ki8F-$!)M'fL$F07$$\"5NLL3FpE!Hq\"F-$!*#) >h/\"F07$$\"5-++]7.K[V?F-$!*I04A#F07$$\"5omm\"zptjSQ#F-$!**Ro$=#F07$F+ $!*y(4b@F07$$\"5-++vo/[AlIF-$!*A+M=#F07$$\"5omm;aQ`!eS$F-$!*aOkR#F07$$ \"5NLLeRseQYPF-$!*Q/w)HF07$$\"5-+++D1k'p3%F-$!*hcEu$F07$$\"5pmmT5SpaFW F-$!*h)>!o$F07$$\"5OLL$eRZF\"oZF-$!*zZxi$F07$$\"50++D\"y+3(3^F-$!*8^0j $F07$F2$!*%f3&z$F07$F7$!*Z[T`%F07$F<$!*&3VQ^F07$FA$!*y8W0&F07$FF$!*u88 )\\F07$FK$!*<&[_\\F07$FP$!*,.O-&F07$FU$!*:R4@&F07$FZ$!*JtzA&F07$Fin$!* e*yV^F07$F^o$!*2@*p]F07$F\\q$!*jLn%[F07$F`r$!*n%[@WF07$Fer$!*CY^l$F07$ Fjr$!*brhF07$F^t$\"(\"H?$*F07$Fht$\")@b\\OF07$$\"5 -+](oHaN;J%F0$\")-D#)RF07$F]u$\")QfPWF07$$\"5NL$eRs3P(yXF0$\")\\h5Ffx$!*],qW\"F07$Fdx$!*A7t)>F07$Fjx$!*'GdhDF07$F_y$!*u(\\cKF 07$Fdy$!*2\\z=%F07$Fiy$!*VW]4&F07$F^z$!*J/?#fF07$Fcz$!*G'>_oF07$Fhz$!* N)*z$yF07$F][l$!*jisb)F07$Fb[l$!*4!*GB*F07$Fg[l$!*_$)Hx*F07$F\\\\l$!+j -G>5F07$Fa\\l$!+&z$4Y5F07$Ff\\l$!+D-2m5F07$F[]l$!+l)fJ3\"F07$Fe]l$!+,Z Q&3\"F07$F_^l$!+jq!f3\"F07$Fd^l$!+qD`!4\"F07$Fi^l$!+;me#4\"F07$F^_l$!+ %Rqd4\"F07$Fc_l$!+lE;(4\"F07$$\"5nm\"HKkAg\\Z\"Ffx$!+#y$[*4\"F07$$\"5M L$ek`h0o[\"Ffx$!+n&)>*4\"F07$$\"5++voH/5l)\\\"Ffx$!+z$\\\")4\"F07$Fh_l $!+,;Y&4\"F07$$\"5++]P4rr=M:Ffx$!+7Py%3\"F07$F]`l$!+NFBl5F07$Fb`l$!+.v EH5F07$Fg`l$!*ZaC\")*F07$F\\al$!*Y)GA\"*F07$Faal$!*`_7Y)F07$$\"5nmm;zp YU%p\"Ffx$!*!*Gi$yF07$Ffal$!*$QR'=(F07$$\"5nmm;/,J:U]hF07$$\"5MLe9m%>(*)z3:dF07$$\"5M$ek`;,<2!=Ffx$!*#oTPdF07$ $\"5+vVB:\")p=/=Ffx$!*%\\*pm&F07$$\"5nmT5l]pl2=Ffx$!*G_#)f&F07$$\"5MeR (\\,#p76=Ffx$!*S!\\RbF07$$\"5+]P%['*)of9=Ffx$!*F^K^&F07$$\"5nTNr9fo1== Ffx$!*#f*pb&F07$F`bl$!*%Q)Rf&F07$$\"5o;HKknnZG=Ffx$!*kgKX&F07$$\"5,+D1 k1nTN=Ffx$!*BDYS&F07$$\"5o\"HKRhn'))Q=Ffx$!*m))[^&F07$$\"5M$3-QckcB%=F fx$!*qq;`&F07$$\"5,v=n8:m#e%=Ffx$!*>ItX&F07$$\"5om;aj%e'H\\=Ffx$!*4C'* Q&F07$$\"5Me9T8alw_=Ffx$!*YJ5N&F07$$\"5,]7GjBlBc=Ffx$!*%4P\"R&F07$$\"5 oT5:8$\\1(f=Ffx$!*&ea%e&F07$$\"5ML3-jikFfx$!*(oA9dF 07$$\"5+]Pf3QNMK>Ffx$!*v!*\\n&F07$$\"5Me*)f0\\]TN>Ffx$!*()GWr&F07$$\"5 nmTg-gl[Q>Ffx$!*-$f/fF07$$\"5+v$4'*42e:%>Ffx$!*G.f.'F07$$\"5M$e9m>eHY% >Ffx$!*)**[^fF07$$\"5n\"z>OH4,x%>Ffx$!*\\S5(eF07$F`gl$!*vh!3eF07$$\"5n ;aj%ei:p&>Ffx$!*r#4(*eF07$$\"5MLekyZ'eI'>Ffx$!*]>C5'F07$$\"5nT5lve,8m> Ffx$!*wJe,'F07$$\"5+]ilsp;?p>Ffx$!*aDy$fF07$$\"5Me9mp!=tA(>Ffx$!*YP/*e F07$Fegl$!*\")3$>fF07$$\"5MLek.HW#)))>Ffx$!*#*oV,'F07$$\"5++]iSmTI-?Ff x$!*j5bA'F07$$\"5n\"zp[25uc+#Ffx$!*]H]7'F07$$\"5M$e9\"4NS/4?Ffx$!*cFE. 'F07$$\"5+v$fL%pRT7?Ffx$!*(oLsfF07$$\"5nmTgx.Ry:?Ffx$!*y\\s*fF07$$\"5M e*[=\"QQ:>?Ffx$!*Gmq='F07$$\"5+]P4YsP_A?Ffx$!*wUM>'F07$$\"5nT&Q.oq$*e- #Ffx$!*)GP#4'F07$Fjgl$!*lQ(**fF07$$\"5om;a)e6Bi0#Ffx$!*h9s'eF07$F_hl$! *#>1mdF07$$\"5,+++D\"=EX8#Ffx$!*n\\R,&F07$Fdhl$!*mkkC%F07$$\"5MLLeRA9W RAFfx$!*@drP$F07$Fihl$!*6yNh#F07$$\"5nmmTNr&3AM#Ffx$!*$eZ??F07$F^il$!* S,]g\"F07$Fcil$!*a%zl7F07$Fhil$!*Y'Gb5F07$F]jl$!)T.#=*F07$Fbjl$!)nO;%) F07$Fgjl$!)$[s5)F07$F\\[m$!)t))e!)F07$$\"5omm\"zWi^bv#Ffx$!)]XL#)F07$F a[m$!)fRi')F07$$\"5omm;H2\"34'GFfx$!)Z&yQ*F07$Ff[m$!*='[W5F07$$\"5,++] 7y#=o'HFfx$!**zj&>\"F07$F[\\m$!*@k)49F07$$\"5,++v$4^n)pIFfx$!*lhOo\"F0 7$F`\\m$!*CqK2#F07$$\"5,++vo/#3o<$Ffx$!*N![xDF07$Fe\\m$!*9g'F07$F]dm$!)x\"\\+%F07$Fbdm$!),g) o#F07$Fgdm$!)Ml*=#F07$F\\em$!)/dS?F07$Faem$!)))Gd@F07$Ffem$!)\\-sDF0-F [fm6&F]fmF^fmF(F^fm-Fbfm6#%Ta~minor~modification~of~the~Bogacki-Shampi ne~schemeG-F$6%7esF'7$Fifm$!%*)fF07$F_gm$!'sVIF07$Fcgm$!(N>a#F07$Fhgm$ !(q)R*)F07$F]hm$!)yb=8F07$Fbhm$\")/GP9F07$Fghm$\")U*>T\"F07$F+$\")M(HN \"F07$F_im$\")!H$>6F07$Fdim$\"([M'oF07$Fiim$\")&fIL\"F07$F^jm$\")S%pX& F07$Fhjm$\")'H]B&F07$F2$\")Hc#y%F07$F<$\"*yrB&=F07$F^o$\"*tF&QDF07$F\\ q$\"*47.E$F07$F`r$\"*p`A5%F07$Fer$\"*3:$paF07$Fjr$\"*i\"z_eF07$F_^n$\" *aX/0'F07$F_s$\"*ih1M'F07$$\"5ML$eR(\\;m/FF0$\"*qeMX'F07$Fg^n$\"*d%>sm F07$$\"5+]7`%Rz#HrGF0$\"*%\\M#f'F07$$\"5MLekyA!>Y!HF0$\"*3-v^'F07$$\"5 n;/wi^_%z$HF0$\"*W@nY'F07$$\"5++](o/[r7(HF0$\"*d%e1lF07$$\"5M$e*)4$4xf /IF0$\"*\\'GInF07$$\"5nmT5:QR#z.$F0$\"*e%)Hl'F07$$\"5+](=#*p;]72$F0$\" *!=VxlF07$Fds$\"*y.j]'F07$Fis$\"*5)G#R'F07$F^t$\"*BTyE'F07$Fct$\"*/=a( fF07$Fht$\"*yios&F07$Fbu$\"*2G#R_F07$F\\v$\"*ofw&[F07$$\"5NLLL$3FGT\\& F0$\"*,F'RZF07$Fav$\"*())>dYF07$$\"5NLL$3_D%eleF0$\"*39Li%F07$$\"5qmmm m;HS*)fF0$\"*1!)Qg%F07$$\"5+++]7y:A8hF0$\"*_*=(f%F07$Ffv$\"*dGAf%F07$$ \"5qmmmm\"HZ_O'F0$\"*9E`g%F07$$\"5++++vVVX$\\'F0$\"*Ly?i%F07$$\"5NLLL$ eRh;i'F0$\"*\"RDZYF07$$\"5qmmm\"zWo)\\nF0$\"*Z%o(o%F07$$\"5NLLL3_DG1qF 0$\"*DBKy%F07$F[w$\"*@@\"=\\F07$Fdan$\"*#)48^5F07$Fcz$\"+%H&H)>\"F07$F][l$\"+1'y)\\8F07$Fg[l$\"+*Q^()[\"F07$Fa\\ l$\"+n)>pf\"F07$Ff\\l$\"+?sLJ;F07$F[]l$\"+M?qe;F07$Fe]l$\"+@**)=m\"F07 $F_^l$\"+3[dg;F07$Fd^l$\"+QXnn;F07$Fi^l$\"+]\"\\%p;F07$F^_l$\"+#)=xp;F 07$Fc_l$\"+f*))Hm\"F07$Fh_l$\"+<>JR;F07$F]`l$\"+BxV(e\"F07$Faal$\"+70f Q9F07$Ffal$\"+9Kzz8F07$F[bl$\"+oJX)H\"F07$Fc[o$\"+>CLA7F07$F`bl$\"+@\" y'f6F07$Fg_o$\"+X$>&f5F07$Febl$\"*?*[`&*F07$F]el$\"*l/fe)F07$F[gl$\"*# \\8/wF07$F`gl$\"*3Mjl'F07$Fegl$\"*Z!4NdF07$Fcio$\"*o#>eZF07$Fjgl$\"*.7 Z-%F07$F^\\p$\"*X5zO$F07$F_hl$\"*Rq'fGF07$$\"5,++v$fQa)3@Ffx$\"*e$G0DF 07$Ff\\p$\"*KPTA#F07$$\"5,++Dcwz>g@Ffx$\"*o&41@F07$Fdhl$\"*pL>'>F07$F^ ]p$\"*'p1W=F07$Fihl$\"*j3@q\"F07$F^il$\"*B9.^\"F07$Fhil$\"*Rl#R7F07$F] jl$\"*l&G>6F07$Fbjl$\"*n#*>.\"F07$Fgjl$\")RMo)*F07$F\\[m$\")E()*o*F07$ F`_p$\")UjB)*F07$Fa[m$\"*1@e-\"F07$Ff[m$\"*!QZ37F07$F[\\m$\"*VYC\\\"F0 7$F`\\m$\"*!yZ&)=F07$Fe\\m$\"*oTf_#F07$Fj\\m$\"*5Q5C$F07$F_]m$\"*e*G\\?>%F 07$F`am$\"*RW67%F07$Feam$\"*A!RERF07$Fjam$\"*raJo$F07$F_bm$\"*TGj.$F07 $Fdbm$\"*iwqB#F07$Fibm$\"*U'>U8F07$$\"5omm\"zW7@^6%Ffx$\"*FQM+\"F07$F^ cm$\")Zx]uF07$$\"5NLL$3-$H**>UFfx$\");$ys&F07$Fccm$\")'foe%F07$$\"5-++ v=n(*fDVFfx$\").4rRF07$Fhcm$\")\"o)zMF07$F]dm$\")>(\\z#F07$Fbdm$\")BX& >#F07$Fgdm$\")C\"\\$=F07$F\\em$\")=q\"p\"F07$Faem$\")p2sF07$Fiim$!*NNAL#F07$F^jm$!*OK(>GF07$Fcjm$ !*e7Fx#F07$Fhjm$!*T:St#F07$F][n$!*I,%RFF07$F2$!*d\\.'GF07$F7$!*S5a>$F0 7$F<$!*)G(RJ$F07$FP$!*\"R#*GKF07$F^o$!*=<\"3HF07$Fbp$!*q'\\oCF07$F\\q$ !*WP!fBF07$Faq$!*l&Rr?F07$Ffq$!*hOKv\"F07$F[r$!*y^Sr\"F07$F`r$!*i5Sf\" F07$$\"5n;H#o27JOf\"F0$!*()e]?\"F07$$\"5MLe9\"4&[EB;F0$!*O/0.\"F07$$\" 5+](oa5e)*Gl\"F0$!*#=,<5F07$$\"5nm;z>6B`#o\"F0$!*b!Q05F07$$\"5M$e9T8/m @r\"F0$!)s,Z**F07$$\"5++vV[r(*zT>F0$!)h uiFF07$$\"5ML3-j7'p)y>F0$!'8)f*F07$Fjr$\")uCKSF07$$\"5++]7G))>Wr@F0$\" )w%*fcF07$F_^n$\")e6L**F07$$\"5++D\"G8O*RrBF0$\"*@6S9\"F07$$\"5nm;/,>= 0QCF0$\"*([(Ga\"F07$$\"5+]i:&y/y8Z#F0$\"*z(\\A:F07$$\"5ML3FpwUq/DF0$\" *MiZ]\"F07$$\"5n;aQ`00.QDF0$\"*rj5^\"F07$F_s$\"*h%*zi\"F07$Ff\\q$\"*S! z=>F07$Fg^n$\"*%of(G#F07$F^]q$\"*3g,E#F07$Fc]q$\"*B5aB#F07$Fh]q$\"*8)> HAF07$F]^q$\"*Os=I#F07$Fb^q$\"*c\"GWDF07$Fg^q$\"*5b]^#F07$F\\_q$\"*X^k [#F07$Fds$\"*2g-Y#F07$Fis$\"*!H#fi#F07$F^t$\"*e+ct#F07$Fct$\"*KzGl#F07 $Fht$\"*MWpb#F07$Fbu$\"*SBqG#F07$F\\v$\"*33L*>F07$Fav$\"*@'*f\"=F07$Ff v$\"*iQ&z;F07$F`bq$\"*#*))\\n\"F07$Febq$\"*E&\\e;F07$Fjbq$\"*!*H#R;F07 $F_cq$\"*w#R\\;F07$$\"5++++++b2yoF0$\"*9yEk\"F07$Fdcq$\"*]Nhj\"F07$$\" 5qmmm;/'*[MrF0$\"*Sxsl\"F07$F[w$\"*2i!e;F07$Fdan$\"*#*HOq\"F07$F`w$\"* fyXx\"F07$F\\bn$\"*+ofw\"F07$Few$\"*%*p(fM\"F07$F_y$\"*)=9e6F07$Fiy$\")S8f$)F07$Fcz$\")79!z&F0 7$F][l$\")8'*)y\"F07$Fb[l$!(,\"*)pF07$Fg[l$!)#[)oPF07$F\\\\l$!)%[l&fF0 7$Fa\\l$!)&[-G*F07$F[]l$!*gYaQ\"F07$Fc_l$!*t&=F`P!p9Ffx $!*Rt#HO\"e9F07$F[bl$\"*(G(Qi\"F07$$\" 5Mek`;'G([pF07$Fgel$\")< \"3>\"F07$F[gl$!)#R4,$F07$F[fo$!)daviF07$F`gl$!)CqXzF07$Fggo$!*U4;H\"F 07$Fegl$!*K`[M\"F07$F^io$!*sLPp\"F07$Fcio$!*eD#H@F07$Fhio$!*F0 7$F^]p$!*BtDl\"F07$$\"5++]ilZsAmAFfx$!*2qWP\"F07$Fihl$!*7RD9\"F07$Ff]p $!)NAxrF07$F^il$!);kh[F07$$\"5ML$ek`1OzT#Ffx$!)mblQF07$Fcil$!)&zB6$F07 $$\"5nm;/^c++rCFfx$!)yr.FF07$Fhil$!)U6!R#F07$$\"5ML$3-)Q84DDFfx$!)b_*= #F07$F]jl$!)Y%>4#F07$$\"5ML$eRA\"*4-e#Ffx$!)`#**3#F07$Fbjl$!)\\nz?F07$ Fgjl$!)\"Q<9#F07$F\\[m$!).\\vAF07$Fa[m$!)j)=i#F07$Fh_p$!)*HU%HF07$Ff[m $!)XB\"Q$F07$F``p$!)^p#3%F07$F[\\m$!)Xvs]F07$F`\\m$!)TGGzF07$Fe\\m$!*S 2&G7F07$Fhap$!*D_IZ\"F07$Fj\\m$!*egqr\"F07$F_]m$!*$*>s*>F07$Fd]m$!*L!) [E#F07$F^^m$!*(z!3X#F07$Fb_m$!*5#H&e#F07$Fg_m$!*\"o3*f#F07$F\\`m$!*')f Wh#F07$Fa`m$!*\\.*>EF07$Ff`m$!*aaph#F07$F[am$!*L%R&e#F07$F`am$!*,@)GDF 07$Feam$!*G!>QBF07$Fjam$!*yrF0#F07$F^ep$!*(z](o\"F07$F_bm$!*`sKP\"F07$ Ffep$!*#fZ)4\"F07$Fdbm$!)[g!H*F07$$\"5,+]7`p2_#)RFfx$!)RYM*)F07$$\"5ML Le*[$zV4SFfx$!)8:p%)F07$$\"5o;zpBEs;;SFfx$!)N#fR)F07$$\"5,+D\"yv^'*G-% Ffx$!)-!fd)F07$$\"5M$3F>*3eiHSFfx$!)W*zK)F07$$\"5om;/E+^NOSFfx$!)_6!G) F07$$\"5ML3F%Ho8)\\SFfx$!)^'QB)F07$Fibm$!)8U3%)F07$F^br$!)ME#*yF07$F^c m$!)0*=W(F07$Ffbr$!))Ru_'F07$Fccm$!):+L_F07$Fhcm$!)AWZHF07$$\"5MLLe9\" z-lU%Ffx$!)3\"F0-F[fm6&F]fmF^fmF(F(-Fbfm6#%3effici ent~scheme~BG-F$6%7[sF'7$Fifm$!%NcF07$F_gm$!'$*eHF07$Fcgm$!(&*=g#F07$F hgm$!)P*G,\"F07$Fbhm$!)9,s7F07$F+$!)U'\\E\"F07$F_im$!)s9i9F07$Fdim$!)m Ld>F07$Fiim$!)IkP=F07$F^jm$\")gxW5F07$Fhjm$\"(&*Rq*F07$F2$\"(\"G\"p&F0 7$F<$\"*#[U>8F07$F^o$\"*$pVo?F07$F\\q$\"*\"*zI)GF07$F`r$\"*dxo&QF07$Fe r$\"*\"4MMaF07$Fjr$\"*u@%QfF07$F_s$\"*B8g*[F0 7$Fcaq$\"*tw\"o[F07$Fhaq$\"*@1x&[F07$Ffv$\"*r#zU[F07$F`bq$\"*]=R&[F07$ Febq$\"*uIY'[F07$Fjbq$\"*_*z\")[F07$F_cq$\"*<')G#\\F07$Fdcq$\"*X`_+&F0 7$F[w$\"*'\\1O^F07$Fdan$\"*\\_D[&F07$F`w$\"*Z\"yhfF07$Few$\"*\\3AZ'F07 $Fjw$\"*7yQ6(F07$F_x$\"*0Ft\"zF07$Fdx$\"*u$=l))F07$F_y$\"***o#y**F07$F iy$\"+qjQC6F07$Fcz$\"+BP;!H\"F07$F][l$\"+3qo`9F07$Fg[l$\"+2yY!f\"F07$F a\\l$\"+`1D)o\"F07$Ff\\l$\"+_Ea:')>,A=#F07$Fdhl$\"*F07$Fe\\m$\"*q[.h#F07$Fj\\m$\"*\"[RlLF07$F_]m$\"*PPJu$F07$ Fd]m$\"*N@V/%F07$Fi]m$\"*t=j:%F07$F^^m$\"*$)H]C%F07$Fc^m$\"*ievE%F07$F h^m$\"*0Y_I%F07$F]_m$\"*f4!>VF07$Fb_m$\"*z=KK%F07$Fg_m$\"*=([WVF07$F\\ `m$\"*:@zL%F07$Fa`m$\"*'prCVF07$Ff`m$\"*%*G*>VF07$F[am$\"*AnVE%F07$F`a m$\"*1pH>%F07$Feam$\"*mk0+%F07$Fjam$\"*P`aw$F07$F_bm$\"*8%yTJF07$Fdbm$ \"*6QmM#F07$Fibm$\"*wJ;S\"F07$F^br$\"*'GXO5F07$F^cm$\")nAFvF07$Ffbr$\" ),zjcF07$Fccm$\")!4[]%F07$F^cr$\")xd;RF07$Fhcm$\")'\\GZ$F07$F]dm$\")s? ^GF07$Fbdm$\")mIkAF07$Fgdm$\")Y?()=F07$F\\em$\"),\"3t\"F07$Faem$\")0C0 =F07$Ffem$\")=?1@F0-F[fm6&F]fm$\")#)eqkF`fm$\"))eqk\"F`fmF\\]w-Fbfm6#% 3efficient~scheme~CG-F$6%7]qF'7$F+$\"*yfp&eF07$F2$\"+(3_(p5F07$F<$\"+` '[Ss\"F07$F^o$\"+\\Q2d=F07$F\\q$\"+%o!Gi>F07$F`r$\"+ARl]?F07$Fc[s$\"+R kI-@F07$Fer$\"+`7/\"=#F07$Fj\\s$\"+(y9[7#F07$F_]s$\"+***=c2#F07$Fd]s$ \"+%4_D4#F07$Fjr$\"+qAd@@F07$F_s$\"+Ke@_>F07$Fds$\"+]1az)e%F07$F][l$\"+N;' pF&F07$Fg[l$\"+!4*pPfF07$Fa\\l$\"+#[A.\\'F07$F[]l$\"+h.\"Q'oF07$Fc_l$ \"+k/;EqF07$Fhgn$\"+:UPSqF07$F]hn$\"+\"*)*eJqF07$Fbhn$\"+R*4Z,(F07$Fh_ l$\"+l\"[3*pF07$Fjhn$\"+>L9/pF07$F]`l$\"+s$yEx'F07$Fg`l$\"+NntFjF07$Fa al$\"+!=5Tv&F07$Ffal$\"+s4NU_F07$F[bl$\"+.;MOZF07$F`bl$\"+nn\\;UF07$Fe bl$\"+!z#QoOF07$F[gl$\"+or\\$H$F07$Fegl$\"+]'*f&)GF07$Fjgl$\"+V#4/_#F0 7$F_hl$\"+TSn)=#F07$Ff\\p$\"+TA(p$=F07$Fdhl$\"+^(Q&p:F07$Fihl$\"+$fK/3 \"F07$F^il$\"*\"y@`yF07$Fcil$\"*H)HinF07$Fhil$\"*0k-$fF07$F]jl$\"*U&[G `F07$Fbjl$\"*YSv%\\F07$Fgjl$\"*VSkw%F07$F\\[m$\"*K1tr%F07$F`_p$\"*Pz5! [F07$Fa[m$\"*w%yJ]F07$Fh_p$\"*b-8V&F07$Ff[m$\"*.w2,'F07$F[\\m$\"*\\m$f xF07$F`\\m$\"+(pe,/\"F07$Fe\\m$\"+$f.jX\"F07$Fj\\m$\"+(3y&4>F07$Fd]m$ \"+g>>hBF07$F^^m$\"+tb*z]#F07$Fb_m$\"+:(*G&f#F07$Fg_m$\"+4Iy3EF07$F\\` m$\"+R([`h#F07$Fa`m$\"+@iJ;EF07$Ff`m$\"+e%yPh#F07$F[am$\"+2b%))e#F07$F `am$\"+;*)fWDF07$Feam$\"+W " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 8 of 7 stage, order 5 Rung e-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx=x*(9-x^2)/(1+y^2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(%\"xGF&,&\"\"*F&* $F*\"\"#F(F&,&F&F&*$%\"yGF.F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0 )=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = rho(x)/2-2/rho(x);" "6#/%\"yG,&*&-%$rhoG6#%\"xG\"\"\"\"\"#!\"\"F+*&F, F+-F(6#F*F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where \+ " }{XPPEDIT 18 0 "rho(x) = (54*x^2-3*x^4+sqrt(64+9*x^8-324*x^6+2916*x^ 4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*&\"#a\"\"\"*$F'\"\"#F,F,*&\"\"$F,*$F '\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\"\"*F,*$F'\"\")F,F,*&\"$C$F,*$F'\"\" 'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,F0F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := di ff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x) );\nw := unapply(rhs(%),x):\nplot(w(x),x=0..4,0..3.7,numpoints=75,font =[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\" F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#! \"\",(*&\"\"$\"\"\")F'\"\"%F/F+*&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F' \"\")F/F/*&\"$C$F/)F'\"\"'F/F+*&\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+ F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVE SG6$7io7$$\"\"!F)F(7$$\"3()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$y ZbcF-$\"3ZF^'eEW*Q9F-7$$\"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$ \"3c?q**e5wz?F>7$$\"3$4\"3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=l B$F>$\"3L\\!fpl0?S%F>7$$\"3G(H(H(p](oPF>$\"3XzNO%Rmzr&F>7$$\"3/I(H(Hj= >VF>$\"3n)f#*4g%))4rF>7$$\"373\"3\"3n&y'[F>$\"3[oK(4I9f[)F>7$$\"3oKCVK ;BKaF>$\"3y!44&)e3J')*F>7$$\"3etH(HPL$HfF>$\"3xCnB_*)['F>$\"3g[#>`Yr+B\"Fho7$$\"3Ul['[')o30(F>$\"3?yA^q$e=N\"Fho7$$\"3 )yH(H(pzBf(F>$\"3)GF.C0:VY\"Fho7$$\"3m53\"3TBT3)F>$\"3ipKusL^i:Fho7$$ \"3u\\'['[U&)o')F>$\"3_Iow_)\\Zn\"Fho7$$\"3ynvcnz>k\"*F>$\"31*=-!RxBm< Fho7$$\"3nq-Fq/SS(*F>$\"3+Ep.yT!)o=Fho7$$\"3\">*=*=tV]-\"Fho$\"3eT(\\) >LNc>Fho7$$\"3!z$y$y&G+\"3\"Fho$\"3!4zI&*RE\"\\?Fho7$$\"37>*=*y\"*GM6F ho$\"3tVizMbXM@Fho7$$\"3'\\'['['y))*=\"Fho$\"3fa-R]1_?AFho7$$\"3s%f%f9 [%4C\"Fho$\"3#y2[?x6qH#Fho7$$\"3[KCVKm,'H\"Fho$\"3)*e,ywZ!pP#Fho7$$\"3 c8N^t2A`8Fho$\"3WUCG,38dCFho7$$\"3Y'['[Yr,.9Fho$\"3@#)=I[fvCDFho7$$\"3 q8N^$f)zc9Fho$\"3%3DX;Fho$\"3iG:8:1'ez#F ho7$$\"3c%f%fuKqx;Fho$\"3[Z4\"[C.I'GFho7$$\"3=^8N\"ft,t\"Fho$\"3!\\dJr u-7#HFho7$$\"3EaS0ao>'y\"Fho$\"3!*)Rs,)f8\")HFho7$$\"3tcnvcA'p$=Fho$\" 3`s'\\F\\'[LIFho7$$\"3363\"3DiC*=Fho$\"3wC-.o]f)3$Fho7$$\"3$)******>Mo W>Fho$\"3Mca,!\\@%QJFho7$$\"3!*['[')ep#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31 Yf%faPE0#Fho$\"3!)Qt2wy;NKFho7$$\"3C^8N^)3&3@Fho$\"3Ik#>&=s*=G$Fho7$$ \"3E>*=*e&>B;#Fho$\"3HYm&4R?ZK$Fho7$$\"3_f%f%z([t@#Fho$\"3-!>y6]piO$Fh o7$$\"3oq-FIB#>F#Fho$\"3%>b4%)\\*>0MFho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%* QMFho7$$\"3eCVKkVazBFho$\"3))))QO[-#QDFho$\"3] n/[Ud8hNFho7$$\"3AdnvO#pkf#Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\" 3e\"*)HR3Wug$Fho7$$\"3$ovcn0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fh o$\"3)*3G:eW%Hk$Fho7$$\"3[aS09&4M\"GFho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUli GFho$\"3?xe_?\\\"fm$Fho7$$\"3163\"3DQ(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=* =HE\"H(HFho$\"3IP\"\\xLemn$Fho7$$\"3%pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\" 3-dnvO9*43$Fho$\"3y,I^j&GHn$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7 $$\"3'>*=*=tG))=$Fho$\"3#R\\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOF ho7$$\"3!>*=*=h2%)H$Fho$\"3%>QOHhJ`h$Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f 0f$Fho7$$\"3)\\'['[I)[0MFho$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59 (yGT$>NFho7$$\"3['['[1m/8NFho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\" 3s<,a='3,U$Fho7$$\"37.Fq-Y#3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$ \"3?^'pWBqHH$Fho7$$\"32dnv'*p'o+i*\\/FFFho7$$\"\" %F)$\"3CxC=rRoRDFho-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETI CAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 751 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.0 1: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x, y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of ste ps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor \+ modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`eff icient scheme B`,`efficient scheme C`,`scheme with stability radius = \+ imag axis inclusion`]:\nerrs := []:\nDigits := 30:\nfor ct to 6 do\n \+ Wn_RK5_||ct := RK5_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm : = 0: numpts := nops(Wn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Wn_RK5_||ct[ii,2]-w(Wn_RK5_||ct[ii,1]))^2;\n end do:\n errs : = [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpos e]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F +*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$ F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint496\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %8Bogacki-Shampine~schemeG$\"+U))>wV!#B7$%Ta~minor~modification~of~the ~Bogacki-Shampine~schemeG$\"+I'f9j#F+7$%3efficient~scheme~AG$\"+qRX&*o !#C7$%3efficient~scheme~BG$\"+t>q=;F+7$%3efficient~scheme~CG$\"+;I'*4= F+7$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+a*[4;$F+Q )pprint506\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical \+ procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Ku tta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value ob tained by each of the methods at the point where " }{XPPEDIT 18 0 "x \+ = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is also giv en." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "W := (x,y) -> x*(9-x ^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[ `slope field: `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Sham pine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`ef ficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme wi th stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 3 0:\nfor ct to 6 do\n wn_RK5_||ct := RK5_||ct(W(x,y),x,y,x0,y0,hh,num steps,true);\nend do:\nxx := 3.499: wxx := evalf(w(xx)):\nfor ct to 6 \+ do\n errs := [op(errs),abs(wn_RK5_||ct(xx)-wxx)];\nend do:\nDigits : = 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F +*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G6$\" \"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint516\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+]r#=j*!#C7$%Ta~min or~modification~of~the~Bogacki-Shampine~schemeG$\"+nJZ25!#B7$%3efficie nt~scheme~AG$\"+g$*exLF+7$%3efficient~scheme~BG$\"+#p3,Z'F+7$%3efficie nt~scheme~CG$\"+D-&*HXF+7$%Sscheme~with~stability~radius~=~imag~axis~i nclusionG$\"+!=R+O\"F0Q)pprint526\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square er ror" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$ \"\"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes met hod over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of \+ the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B` ,`efficient scheme C`,`scheme with stability radius = imag axis inclus ion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((w( x)-'wn_RK5_||ct'(x))^2,x=0..4,adaptive=false,numpoints=7,factor=200); \n errs := [op(errs),sqrt(sm/4)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+2aeKU!#B7$%Ta~minor~modifica tion~of~the~Bogacki-Shampine~schemeG$\"+.\"ymX#F+7$%3efficient~scheme~ AG$\"+_\"\\\\(o!#C7$%3efficient~scheme~BG$\"+)[+Gf\"F+7$%3efficient~sc heme~CG$\"+gp+'p\"F+7$%Sscheme~with~stability~radius~=~imag~axis~inclu sionG$\"+JGc4JF+Q)pprint536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 517 "evalf[20](plot(['wn_RK5_1'(x)-w(x),'wn_RK5 _2'(x)-w(x),'wn_RK5_3'(x)-w(x),'wn_RK5_4'(x)-w(x),\n'wn_RK5_5'(x)-w(x) ,'wn_RK5_6'(x)-w(x)],x=0..4,-1.2e-13..1.95e-13,\nfont=[HELVETICA,9],co lor=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nl egend=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki- Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient \+ scheme C`,`scheme with stability radius = imag axis inclusion`],\ntitl e=`error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 968 632 632 {PLOTDATA 2 "6,-%'CURVESG6%7cp7$$\"\" !F)F(7$$\"5LLLLLL3VfV!#@$!(EL@\"!#?7$$\"5mmmmmm;')=()F-$!(\\Y=#F07$$\" 50+++DJ59n'*F-$!(LtG#F07$$\"5MLLLeR?ah5F0$!(A1J#F07$$\"5+++D\"y+c*36F0 $!(PVB#F07$$\"5nmm;/w*pj:\"F0$!()\\HAF07$$\"5++]i:gp2!=\"F0$!(4$*=#F07 $$\"5MLL3FWRy.7F0$!($QB?F07$$\"5nm;aQG4\\F7F0$!(AE-#F07$$\"5++++]7z>^7 F0$!(#p(>%F0$\"( I=!RF07$$\"5+++++vCT$f%F0$\"',n@F07$$\"5mmmmm\">K'*)\\F0$!(U`%>F07$$\" 5+++++vMw%>&F0$!(QSn#F07$$\"5NLLLLeZ*)*R&F0$!(+o:$F07$$\"5++++++/Y-bF0 $!(1SH$F07$$\"5qmmmmTg-0cF0$!(eiR$F07$$\"5NLLLL$o\"f2dF0$!(M,Z$F07$$\" 5+++++Dt:5eF0$!(c+_$F07$$\"5qmmmm\"R/B-'F0$!(eub$F07$$\"5NLLLLe9XMiF0$ !(%[RNF07$$\"5+++++D&)fYkF0$!(!z%[$F07$$\"5mmmmm\"fX(emF0$!(1oS$F07$$ \"5+++++DCh/vF0$!(/-,$F07$$\"5LLLLLL/pu$)F0$!()p*f#F07$$\"5mmmmm;c0T\" *F0$!(i\"*H#F07$$\"5+++++I,Q+5!#>$!(5N-#F07$$\"5+++++]*3q3\"F\\x$!(I5! =F07$$\"5+++++q=\\q6F\\x$!(9ii\"F07$$\"5nmmm;fBIY7F\\x$!(eP\\\"F07$$\" 5LLLLLj$[kL\"F\\x$!()ph8F07$$\"5LLLLL`Q\"GT\"F\\x$!(ArE\"F07$$\"5++++] s]k,:F\\x$!(QJ<\"F07$$\"5LLLLL`dF!e\"F\\x$!(I<5\"F07$$\"5++++]sgam;F\\ x$!(UQ.\"F07$$\"5++++]F\\x$!'me))F07$$\"5nmmmmTc-)*>F\\x$!'oq%)F07$$\"5nmmm ;f`@'3#F\\x$!'+?\")F07$$\"5++++]nZ)H;#F\\x$!'#Q&yF07$$\"5nmmmmJy*eC#F \\x$!'=-wF07$$\"5+++++S^bJBF\\x$!'sztF07$$\"5+++++0TN:CF\\x$!'_%>(F07$ $\"5++++]7RV'\\#F\\x$!'qXqF07$$\"5+++++:#fke#F\\x$!'w9pF07$$\"5LLLLL`4 NnEF\\x$!'kFoF07$$\"5+++++],s`FF\\x$!'?pnF07$$\"5nmmm;zM)>$GF\\x$!'+\\ nF07$$\"5+++++qfa2_#F0-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki -Shampine~schemeG-F$6%7apF'7$F2$\"(-H!=F07$FZ$\"(O/u#F07$F^o$\"(w0^$F0 7$$\"5+++]i!f`rt\"F0$\"(#p3OF07$Fco$\"(Htk$F07$$\"5LLL$3_XT/&>F0$\"(bl h$F07$Fho$\"(,$3NF07$F]p$\"(n?/$F07$Fbp$\"($=^AF07$F\\q$\"'ndPF07$Ffq$ !'QSRF07$$\"5+++]PM%3$\\MF0$!'L8WF07$F[r$!'wYXF07$$\"5nmm;zp'*)Hm$F0$! '=8XF07$F`r$!''3\\%F07$$\"5LLL$3_!4nwQF0$!'q_YF07$Fer$!'\"*e^F07$$\"5+ ++]iS@N!4%F0$!'UbhF07$Fjr$!'YrxF07$F_s$!(0'\\:F07$Fds$!(2e]#F07$F^t$!( $o&[$F07$Fbu$!(O0.%F07$Fgu$!([k=%F07$F\\v$!(u2G%F07$Fav$!(5WK%F07$Ffv$ !(O$GVF07$$\"5NLLLL3!z;3(F0$!(olD%F07$F[w$!(%y?F07$Fdy$!(s8)=F07$Fiy$!(3Iu\"F07$F^z$!(![P;F07$Fcz$!(Ao`\"F07$F hz$!(mMX\"F07$F][l$!(9uP\"F07$Fb[l$!(cgJ\"F07$Fg[l$!(3xD\"F07$F\\\\l$! (gY?\"F07$Fa\\l$!(KS;\"F07$Ff\\l$!(G`7\"F07$F[]l$!(_/4\"F07$F`]l$!(U41 \"F07$Fe]l$!(!QO5F07$Fj]l$!(mM,\"F07$F_^l$!'ul**F07$Fd^l$!'5B)*F07$Fi^ l$!'+F(*F07$F^_l$!'Ue'*F07$Fc_l$!'SG'*F07$Fh_l$!'IL'*F07$F]`l$!'Uw'*F0 7$Fb`l$!'/c(*F07$Fg`l$!'in)*F07$F\\al$!']%***F07$Faal$!(5s+\"F07$Ffal$ !'Ev**F07$F[bl$!'!\\N*F07$F`bl$!'U'f)F07$Febl$!'5BsF07$Fjbl$!'=,SF07$F _cl$\"'GE7F07$Fdcl$\"()[<6F07$Ficl$\"(#R4JF07$$\"5+++DccB&R#RF\\x$\"(S 3*QF07$F^dl$\"(+\"[[F07$$\"5+++voa-oXRF\\x$\"(yk.'F07$Fcdl$\"('GUvF07$ $\"5++]7Gyh(>'RF\\x$\"(;c2*F07$Fhdl$\"()H/&*F07$$\"5++]PMF,%G(RF\\x$\" )ePC6F07$F]el$\")ms:7F07$Fbel$\")O*pR\"F07$Fgel$\")%*)4f\"F07$$\"5++DJ ?Q?&=*RF\\x$\")_gG,\"f#F0$\"(#F07$Fiil$!'/6EF07$F[r$!'>$y%F07$ Fajl$!'N3nF07$F`r$!'w#Q)F07$Fer$!(sY5\"F07$Fjr$!(d*)H\"F07$F_s$!(f&F07$Fcz$!'-z[F07$Fhz$!'c;YF07$F][l$!'WwVF07$Fb[l$!'c#=%F07$ Fg[l$!')z*RF07$F\\\\l$!'+IQF07$Fa\\l$!'U,PF07$Ff\\l$!'))yNF07$F[]l$!'s oMF07$F`]l$!'UvLF07$Fe]l$!'!zH$F07$Fj]l$!'1EKF07$F_^l$!'MtJF07$Fd^l$!' ]HJF07$Fi^l$!'!35$F07$F^_l$!'U\"3$F07$Fc_l$!'IvIF07$Fh_l$!'S#3$F07$F]` l$!'#R5$F07$Fb`l$!'9VJF07$Fg`l$!'#**>$F07$F\\al$!']zKF07$Faal$!'S#Q$F0 7$Ffal$!'Y(\\$F07$F[bl$!'I]OF07$Febl$!'!)oPF07$Fjbl$!')[w$F07$F_cl$!'_ ,PF07$$\"5++++D1>V_QF\\x$!'%*4OF07$Fdcl$!'_dMF07$$\"5++++vt&pG*QF\\x$! 'Y8KF07$Ficl$!'eJGF07$F^dl$!'!)QAF07$Fcdl$!'9k7F07$Fhdl$!&-E&F07$F]el$ \"&;(\\F07$Fgel$\"'W&)>F07$Fafl$\"'-mUF0-%&COLORG6&Fhfl$\"\"\"!\"\"$\" \")FajnF(-F]gl6#%3efficient~scheme~AG-F$6%7cpF'7$F+$\"'*RB'F07$F2$\"(m aH\"F07$FZ$\"(g&f>F07$F^o$\"($4WDF07$F]hl$\"(&oKEF07$Fco$\"(I0o#F07$$ \"5+++D1*[>r*=F0$\"(wAo#F07$Fehl$\"(7tn#F07$$\"5++]7GQC5x>F0$\"(;Bm#F0 7$$\"5mmmTN@Mw.?F0$\"(:_i#F07$$\"5LL$3FWSC/.#F0$\"(L3i#F07$Fho$\"(BBh# F07$F]p$\"(!oaAF07$Fbp$\"((HC:F07$Ff[n$\"'s6)*F07$Fgp$\"'D#)HF07$F^\\n $!'b-IF07$F\\q$!'m+&)F07$Ff\\n$!(hXQ\"F07$Faq$!(Ki)=F07$F^]n$!(!*3M#F0 7$Ffq$!(P'QFF07$F[r$!(doM$F07$F`r$!(>hr$F07$Fijl$!(*yDQF07$Fer$!(Mh*QF 07$$\"5LLL3xJ=$p.%F0$!(dN$RF07$Fa[m$!(Fc$RF07$$\"5mmm\"z%\\CxVTF0$!('z MRF07$Fjr$!(WB&RF07$F_s$!(];\"QF07$Fds$!(L`j$F07$Fbu$!(OPI$F07$Ffv$!(' fGHF07$F[w$!(%y$e#F07$F`w$!(y5E#F07$Few$!(sP-#F07$Fjw$!(S,!=F07$F`x$!( gKh\"F07$Fex$!(WVY\"F07$Fjx$!()4\\8F07$F_y$!(=JB\"F07$Fdy$!(A$\\6F07$F iy$!([`1\"F07$F^z$!(g6+\"F07$Fcz$!'#zR*F07$Fhz$!'E*)))F07$F][l$!'WC%)F 07$Fb[l$!'1\\!)F07$Fg[l$!')>p(F07$F\\\\l$!'+ntF07$Fa\\l$!'-=rF07$Ff\\l $!')3)oF07$F[]l$!'-nmF07$F`]l$!'7'['F07$Fe]l$!'gNjF07$Fj]l$!';&>'F07$F _^l$!'%=4'F07$Fd^l$!'+0gF07$Fi^l$!'+ZfF07$F^_l$!'s1fF07$Fc_l$!'S\"*eF0 7$Fh_l$!'S+fF07$F]`l$!'-PfF07$Fb`l$!'M2gF07$Fg`l$!'#H6'F07$F\\al$!'+ni F07$Faal$!'!3['F07$Ffal$!''\\v'F07$F[bl$!'!QA(F07$F`bl$!'iJvF07$Febl$! 'q^zF07$Fjbl$!'G#o)F07$F_cl$!'#)=)*F07$Fdcl$!(-\\=\"F07$Ficl$!(=rf\"F0 7$F^dl$!(gH(>F07$Fcdl$!(aVd#F07$Fhdl$!(Ka-$F07$F]el$!(%eaOF07$Fbel$!(/ @7%F07$Fgel$!('Q&e%F07$Fjgm$!(y7'\\F07$F\\fl$!(K++&F07$Fbhm$!()[Y^F07$ Fafl$!()GfgF0-Fffl6&FhflFiflF(F(-F]gl6#%3efficient~scheme~BG-F$6%7epF' 7$$\"5mmmmm;arz@F-$!%_qF07$F+$!%7$*F07$$\"5+++++]i9RlF-$\"$'=F07$F2$\" &>]$F07$FZ$\"'F$>#F07$F^o$\"'n]eF07$F]hl$\"'^?oF07$Fco$\"'@fwF07$Fehl$ \"'/e#)F07$Fho$\"'q$\\)F07$$\"5MLLek`tS5@F0$\"'s#H)F07$$\"5nmm;z>$HP;# F0$\"'OL#)F07$$\"5ML$ekGI!R!>#F0$\"'K()yF07$$\"5+++v$fG^q@#F0$\"':\"[( F07$$\"5nm;/,pArVAF0$\"'mhuF07$F]p$\"'?OtF07$$\"5+++]P%=N&F07$F\\\\l$!'?B^F07$Fa\\l$!'_[\\F07$Ff\\l$!'e#y%F 07$F[]l$!'KLYF07$F`]l$!'U2XF07$Fe]l$!'!HS%F07$Fj]l$!'w0VF07$F_^l$!'aMU F07$Fd^l$!'+vTF07$Fi^l$!'qNTF07$F^_l$!'i3TF07$Fc_l$!'+*4%F07$Fh_l$!'S1 TF07$F]`l$!'-LTF07$Fb`l$!'M$=%F07$Fg`l$!'AfUF07$F\\al$!'IsVF07$Faal$!' 5RXF07$Ffal$!'wyZF07$F[bl$!'Sp_F07$F`bl$!'UjcF07$Febl$!'?(G'F07$Fjbl$! 'o>5F07$ Fjgm$!)[+A6F07$F\\fl$!)AyI6F07$Fbhm$!)[`n6F07$Fafl$!)GZ<9F0-Fffl6&Fhfl $\")#)eqkF[gl$\"))eqk\"F[glF\\^q-F]gl6#%3efficient~scheme~CG-F$6%7_qF' 7$Fa\\p$!'CA(*F07$F+$!(9P$>F07$Fi\\p$!(kk'GF07$F2$!(L/u$F07$F<$!(JBO%F 07$FZ$!(72s%F07$$\"5mmm\"H2)=h)H\"F0$!('4eZF07$$\"5LLL$e*[e-Y8F0$!(Tcv %F07$$\"5+++v=<)RMR\"F0$!(6.r%F07$Fin$!(49n%F07$$\"5+++](=s\"oN:F0$!(* o`WF07$F^o$!(*=#4%F07$Fco$!(*[=HF07$Fho$!(55>\"F07$Fh^p$!'Q0;F07$F]p$ \"'ek%*F07$F_`p$\"(\\f4#F07$Fbp$\"(*zYKF07$Ff[n$\"(&esVF07$Fgp$\"(*p-a F07$F^\\n$\"(]].'F07$F\\q$\"(&\\LkF07$Ff\\n$\"(lHn'F07$Faq$\"(V$onF07$ $\"5nmmmTg:W\"=$F0$\"(T2v'F07$F^]n$\"(`Bu'F07$$\"5nmmm\"zR#z)G$F0$\"(t ep'F07$Ffq$\"(wCi'F07$F`r$\"(]iw&F07$Fjr$\"(RQL&F07$$\"5++++](oZiH%F0$ \"(F(R`F07$$\"5mmmmm;EI&R%F0$\"(dZP&F07$$\"5LLLL$eadV\\%F0$\"(Q'F07$F]u$\"(;4S'F07$Fbu $\"(W!3kF07$Fgu$\"(K'zjF07$F\\v$\"(OYJ'F07$Fav$\"(+4A'F07$Ffv$\"(ai5'F 07$F[w$\"(Y'ebF07$F`w$\"(-s\"\\F07$Few$\"(30U%F07$Fjw$\"(5*RRF07$F`x$ \"(IM`$F07$Fex$\"(w&3KF07$Fjx$\"(_m&HF07$F_y$\"(#=.FF07$Fdy$\"(3,_#F07 $Fiy$\"(_mL#F07$F^z$\"(Ik>#F07$Fcz$\"()Qi?F07$Fhz$\"(%G^>F07$F][l$\"(E )\\=F07$Fb[l$\"(ayw\"F07$Fg[l$\"(i)*o\"F07$F\\\\l$\"(5*=;F07$Fa\\l$\"( 3Xc\"F07$Ff\\l$\"(UE^\"F07$F[]l$\"([eY\"F07$F`]l$\"(3iU\"F07$Fe]l$\"(q JR\"F07$Fj]l$\"(MAO\"F07$F_^l$\"(m$R8F07$Fd^l$\"(!))>8F07$Fi^l$\"(glI \"F07$F^_l$\"(QoH\"F07$Fc_l$\"(5AH\"F07$Fh_l$\"(?@H\"F07$F]`l$\"(GrH\" F07$Fb`l$\"(msI\"F07$Fg`l$\"(=CK\"F07$F\\al$\"(q\"F0-F]jn6&Fhfl$\"\"'FajnF(Fbjn-F]gl6#%Sscheme~with~stability~radi us~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror ~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6 \"Q!Fbcr-%%VIEWG6$;F(Fafl;$!#7!#9$\"$&>!#:" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor mo dification of the Bogacki-Shampine scheme" "efficient scheme A" "effic ient scheme B" "efficient scheme C" "scheme with stability radius = im ag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 " Test 9 of 7 stage, order 5 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x))*y^3" "6#/*&%#dyG \"\"\"%#dxG!\"\",$*&,&F&F&-%$cosG6#*&\"\"#F&%\"xGF&F&F&*$%\"yG\"\"$F&F (" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2);" "6#/-%\"yG6#\"\" !-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solut ion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(s in(2*x)+2*x+1/2)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\" xGF&F&*&F/F&F0F&F&*&F&F&F/!\"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := dif f(y(x),x)=-(1+cos(2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\} ,y(x));\nm := unapply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVE TICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#de G/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F 0)F)\"\"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!*$\"\"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG* &\"\"\"F)*$,(*(\"\"#F)-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F- !\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURV ESG6$7Y7$$\"\"!F)$\"3:&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H 8F,7$$\"3')*****\\7t&pKF0$\"3!G<)\\ef9f7F,7$$\"3z****\\(ofV!\\F0$\"3oe \"F,7$$\"3s******\\i9RlF0$\"3kESFh\"zh9\"F,7$$\"33++vVV)RQ*F0$ \"3'f)*)e-w\\p5F,7$$\"3/++vVA)GA\"!#=$\"3V)o6<$fq15F,7$$\"3;+]iSS\"Ga \"FJ$\"3IyW%eHk>[*FJ7$$\"3+++]Peui=FJ$\"3#4`!o2+#G**)FJ7$$\"37+++]$)z% =#FJ$\"3OGH4wwYu&)FJ7$$\"3A++]i3&o]#FJ$\"3=1g%=M2W@)FJ7$$\"3%)***\\(oX *y9$FJ$\"31u2v$Q9&GwFJ7$$\"3z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\ P*zhdVFJ$\"3P$G(zQ8#4%oFJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)** *\\(=$f%GcFJ$\"3mYY%G?7\"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"3 3++]7[X$ocbFJ7 $$\"3))***\\PpnsM*FJ$\"3!\\;$Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aF J7$$\"3-+++!R5'f5F,$\"3q`:6QhHm`FJ7$$\"3)***\\P/QBE6F,$\"3@Igj*yDKK&FJ 7$$\"3!******\\\"o?&=\"F,$\"3i/K.-M\\%H&FJ7$$\"31+]Pa&4*\\7F,$\"3OjcS# )ygr_FJ7$$\"33+]7j=_68F,$\"3'e4m\")R`oD&FJ7$$\"33++vVy!eP\"F,$\"3a@U-1 /NZ_FJ7$$\"34+](=WU[V\"F,$\"3Nrr*HO\"oU_FJ7$$\"3)****\\7B>&)\\\"F,$\"3 'HX%)zwR1C&FJ7$$\"3)***\\P>:mk:F,$\"3<^\"Q\"4\"y-C&FJ7$$\"3'***\\iv&QA i\"F,$\"3:*4?^OZ,C&FJ7$$\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"3!****** \\Nm'[F,$\"3[h+0^h(R>&FJ 7$$\"3z*****\\@80+#F,$\"3!zBIi>A%o^FJ7$$\"31++]7,Hl?F,$\"3<)30`]&>L^FJ 7$$\"3()**\\P4w)R7#F,$\"3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#) G/&FJ7$$\"3!)**\\P/-a[AF,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>; ?IA\\FJ7$$\"3')****\\i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3% Gjf])o8tZFJ7$$\"3>+++!*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!** R*=7x[1YFJ7$$\"3=+]P%>5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVv c$ovV%FJ7$$\"33++Dr\"[8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q% zK7E%FJ7$$\"31+]P/)fT(GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]& )H\\$4%FJ7$$\"\"$F)$\"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%F ONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l; F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solut ion" }{TEXT -1 44 " based on each of the methods and gives the " } {TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 764 "M := (x,y) -> -(1+cos(2 *x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix ([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-S hampine scheme`,`a minor modification of the Bogacki-Shampine scheme`, `efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits : = 20:\nfor ct to 6 do\n Mn_RK5_||ct := RK5_||ct(M(x,y),x,y,x0,evalf( y0),hh,numsteps,false);\n sm := 0: numpts := nops(Mn_RK5_||ct):\n \+ for ii to numpts do\n sm := sm+(Mn_RK5_||ct[ii,2]-m(Mn_RK5_||ct[i i,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F, -%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~ G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\" $+$Q)pprint546\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+@( *zM!)!#A7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+*) pSNJ!#@7$%3efficient~scheme~AG$\"+2)z&pdF07$%3efficient~scheme~BG$\"+k n`P;!#?7$%3efficient~scheme~CG$\"+)fFKh%F97$%Sscheme~with~stability~ra dius~=~imag~axis~inclusionG$\"+%e2sg\"F+Q)pprint556\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code c onstructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for so lutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999;" "6#/%\"xG-%&FloatG6$\" %**H!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numstep s := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modi fication of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficie nt scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 25:\nfor ct to 6 do\n mn_ RK5_||ct := RK5_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend d o:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 6 do\n errs := [op( errs),abs(mn_RK5_||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F, F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step ~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint586\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7(7$%8Bogacki-Shampine~schemeG$\"+'> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the \+ Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`ef ficient scheme C`,`scheme with stability radius = imag axis inclusion` ]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((m(x)-' mn_RK5_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7,factor=150);\n \+ errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$%8Bogacki-Shampine~schemeG$\"++>0\\z!#A7$%Ta~minor~modification ~of~the~Bogacki-Shampine~schemeG$\"+d4+*4$!#@7$%3efficient~scheme~AG$ \"+(R=8p&F07$%3efficient~scheme~BG$\"+7?P:;!#?7$%3efficient~scheme~CG$ \"+jJ]^XF97$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+/ Sly:F+Q)pprint606\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "The following error graphs are constructed using the nu merical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "evalf[20](plot([m(x)-'mn_RK5_1'(x),m(x)-'mn_RK5_2'(x ),m(x)-'mn_RK5_3'(x),m(x)-'mn_RK5_4'(x),\nm(x)-'mn_RK5_5'(x),m(x)-'mn_ RK5_6'(x)],x=0..0.5,\nfont=[HELVETICA,9],color=[blue,magenta,COLOR(RGB ,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlegend=[`Bogacki-Shampine sc heme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stab ility radius = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 968 645 645 {PLOTDATA 2 "6,-%'CURVESG6%7[s7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#A$ \"$r$!#>7$$\"5NLLL$3FWYs#F-$\"&%*)>F07$$\"5omm;aQ`!eS$F-$\"&$4pF07$$\" 5-+++D1k'p3%F-$\"'\"3(=F07$$\"5OLL$eRZF\"oZF-$\"'KeUF07$$\"5qmmmmT&)G \\aF-$\"'=?&)F07$$\"50++]P4'\\/8'F-$\"(1A\"F07$$\"5++Dcw4]>'Q\"Fjo$\")rP>7F0 7$$\"5ML$3_+ZiaW\"Fjo$\")D(=A\"F07$$\"5nmT&Q.$*HZ]\"Fjo$\")5;K7F07$$\" 5+++]i!R(*Rc\"Fjo$\")sd`7F07$$\"5nm;z>6B`#o\"Fjo$\")w,X8F07$$\"5MLL3xJ s1,=Fjo$\")`\"f_\"F07$$\"5++]PM_@g>>Fjo$\")y79=F07$$\"5nmmm\"H2P\"Q?Fj o$\");^d?F07$$\"5++]7G))>Wr@Fjo$\")/gH?F07$$\"5MLLek.pu/BFjo$\")aq.?F0 7$$\"5++D\"G8O*RrBFjo$\")_I$*>F07$$\"5nm;/,>=0QCFjo$\")(Rn)>F07$$\"5ML 3FpwUq/DFjo$\"),b')>F07$$\"5+++]PMnNrDFjo$\")2.'*>F07$$\"5ML$eR(\\;m/F Fjo$\")a0g?F07$$\"5nmmT5ll'z$GFjo$\")C?7AF07$$\"5++](o/[r7(HFjo$\")7Mw CF07$$\"5MLLL$eRwX5$Fjo$\")?aADF07$$\"5MLL$3F%\\wQKFjo$\")Ni!\\#F07$$ \"5MLLLe*[`HP$Fjo$\")>)>Y#F07$$\"5MLL3-jx/SMFjo$\")X7^CF07$$\"5MLL$ek. Ur]$Fjo$\")f![W#F07$$\"5MLLe*)4jBuNFjo$\")zUXCF07$$\"5MLLLL$eI8k$Fjo$ \")V#fX#F07$$\"5MLLL3xwq4RFjo$\")%RRm#F07$$\"5NLLL$3x%3yTFjo$\")QWqFF0 7$$\"5-+](oHaN;J%Fjo$\")4GQFF07$$\"5ommT5:j=XWFjo$\")U!4r#F07$$\"5-+v= <,<'>^%Fjo$\")K],FF07$$\"5NL$eRs3P(yXFjo$\")D7(p#F07$$\"5om\"H2LZ7bk%F jo$\")\"4**p#F07$$\"5-++]PfyG7ZFjo$\")%eBr#F07$$\"5om;/^J'Qe%[Fjo$\")X (zGF07$$\"5NLLL3_DG1qF jo$\")g;YHF07$$\"5qmmm;/'*[MrFjo$\")Z[)Fjo$\")sBXG F07$$\"5OLLLe*)>VB$)Fjo$\")6\\=GF07$$\"5.+++DJbw!Q*Fjo$\")'4#GFF07$$\" 5nmmm;/j$o/\"!#?$\")*pCi#F07$$\"5MLLL3_>jU6F]gl$\")f+KDF07$$\"5++++]i^ Z]7F]gl$\")&3@U#F07$$\"5++++](=h(e8F]gl$\"*'GU9BF]gl7$$\"5++++]P[6j9F] gl$\"*%***\\@#F]gl7$$\"5MLL$e*[z(yb\"F]gl$\"*R_*G@F]gl7$$\"5nmmm;a/cq; F]gl$\"*P\"))H?F]gl7$$\"5nmmmm;t,mF]gl7$$\"5,++]iSj0x=F] gl$\"*Z!*Q'=F]gl7$$\"5nmmmm\"pW`(>F]gl$\"*0J5z\"F]gl7$$\"5,++]i!f#=$3# F]gl$\"*\">I;pxg#F]gl$\"*7#H79F]gl7$$\"5,++]Pf4 t.FF]gl$\"*abfO\"F]gl7$$\"5MLLLe*Gst!GF]gl$\"*51'=8F]gl7$$\"5,++++DRW9 HF]gl$\"*vjDF\"F]gl7$$\"5,+++DJE>>IF]gl$\"*.>-B\"F]gl7$$\"5,++]i!RU07$ F]gl$\"*10;>\"F]gl7$$\"5,+++v=S2LKF]gl$\"*lG6:\"F]gl7$$\"5ommmm\"p)=ML F]gl$\"*q6p6\"F]gl7$$\"5,+++](=]@W$F]gl$\"*%*\\B3\"F]gl7$$\"5MLL$e*[$z *RNF]gl$\"*.XF0\"F]gl7$$\"5,+++]iC$pk$F]gl$\"*%o.A5F]gl7$$\"5omm;H2qcZ PF]gl$\")!3n%**F]gl7$$\"5,++]7.\"fF&QF]gl$\")W;v'*F]gl7$$\"5ommm;/OgbR F]gl$\")oBB%*F]gl7$$\"5,++]ilAFjSF]gl$\")>%G<*F]gl7$$\"5NLLLL$)*pp;%F] gl$\")^$Q%*)F]gl7$$\"5NLLL3xe,tUF]gl$\")!e6s)F]gl7$$\"5omm;HdO=yVF]gl$ \")966&)F]gl7$$\"5,++++D>#[Z%F]gl$\")8(pK)F]gl7$$\"5ommmT&G!e&e%F]gl$ \")Y\"e7)F]gl7$$\"5NLLLL$)Qk%o%F]gl$\")ZEazF]gl7$$\"5-++]iSjE!z%F]gl$ \")eezxF]gl7$$\"5-++]P40O\"*[F]gl$\")_*)>wF]gl7$$\"\"&!\"\"$\")+&fX(F] gl-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~s chemeG-F$6%7\\rF'7$F+$\"$&yF07$F2$\"&]^%F07$F7$\"'sJ;F07$F<$\"')Qh%F07 $FA$\"(V75\"F07$FF$\"('\\@BF07$FK$\"(:0X%F07$FP$\"((R:zF07$FU$\")%[ZK \"F07$FZ$\")SS3@F07$$\"5qmmTNYL^9&)F-$\")P\"ph#F07$Fin$\")^g;KF07$$\"5 0++D19Wn&>*F-$\")UE=RF07$F^o$\")CBLZF07$Fco$\")(pLn&F07$Fho$\")7(>.'F0 7$F^p$\")dD4gF07$Fcp$\")io')fF07$Fhp$\")az4fF07$F]q$\")KRTeF07$Fbq$\") r*o\"eF07$Fgq$\")%zg!eF07$F\\r$\")=gFjo$\")f\"[&*)F07$Fes$\")*p,K*F07$F_t$\")O1t!*F07$Fcu$\")m[` *)F07$Fhu$\")[m4\"*F07$F]v$\")X:='*F07$Fbv$\"*(\\7t5F07$Fgv$\"*Q\"*p4 \"F07$Faw$\"*bN+2\"F07$Fex$\"*O$\\e5F07$Fjx$\"*t8Y7\"F07$F_y$\"*X'Go6F 07$Fdy$\"*d\"ea6F07$Fiy$\"*Vl@9\"F07$F^z$\"*f@r8\"F07$Fcz$\"*#3^L6F07$ Fhz$\"*nk?8\"F07$F][l$\"*yRP8\"F07$Fb[l$\"*OW::\"F07$Fg[l$\"*2q)*>\"F0 7$F\\\\l$\"*m%y47F07$Fa\\l$\"*7Tj?\"F07$Ff\\l$\"*/9H?\"F07$F[]l$\"*^.& *>\"F07$F`]l$\"*\"[t#>\"F07$Fe]l$\"*:^g=\"F07$Fj]l$\"*'42u6F07$F_^l$\" *'=Rj6F07$$\"5+++Dc,w.cbFjo$\"*Tz!f6F07$Fd^l$\"*r?f:\"F07$$\"5NLL3-ji& )zcFjo$\"*63W:\"F07$Fi^l$\"*@j^:\"F07$F^_l$\"*(*zm;\"F07$Fc_l$\"*6R%)> \"F07$Fh_l$\"*:,:>\"F07$F]`l$\"*_m(z6F07$Fg`l$\"*`Cs:\"F07$F[bl$\"*:)) f9\"F07$F`bl$\"*aZH:\"F07$Febl$\"*))yU<\"F07$Fjbl$\"*yYG;\"F07$F_cl$\" *BA;:\"F07$Fafl$\"*bx06\"F07$Fffl$\"*'=?l5F07$F[gl$\"*VZj,\"F07$Fagl$ \")_%ev*F07$Ffgl$\")9@(G*F07$F[hl$\"*gQ*Q))F]gl7$F`hl$\"*hMAV)F]gl7$Fe hl$\"*6E]3)F]gl7$Fjhl$\"*F!G\"p(F]gl7$F_il$\"*:F,Q(F]gl7$Fdil$\"*C\\1/ (F]gl7$Fiil$\"*Ph&enF]gl7$F^jl$\"*?P.Z'F]gl7$Fcjl$\"*-&y7iF]gl7$Fhjl$ \"*BRH'fF]gl7$F][m$\"*;uZu&F]gl7$Fb[m$\"*%*=o_&F]gl7$Fg[m$\"*h@HJ&F]gl 7$F\\\\m$\"*,W!Q^F]gl7$Fa\\m$\"*%Rkf\\F]gl7$Ff\\m$\"*m!Q'y%F]gl7$F[]m$ \"*q!>FYF]gl7$F`]m$\"*Gc@[%F]gl7$Fe]m$\"*!\\?IVF]gl7$Fj]m$\"*#='=?%F]g l7$F_^m$\"*UjA2%F]gl7$Fd^m$\"***Q8'RF]gl7$Fi^m$\"*-yi%QF]gl7$F^_m$\"*1 &zVPF]gl7$Fc_m$\"*52@k$F]gl7$Fh_m$\"**4zZNF]gl7$F]`m$\"*5[SX$F]gl7$Fb` m$\"*5N$oLF]gl7$Fg`m$\"*Z+]G$F]gl7$F\\am$\"*01k?$F]gl7$Faam$\"*g&[PJF] gl7$Ffam$\"*._A1$F]gl7$F[bm$\"*RP!)*HF]gl7$F`bm$\"*2/F$HF]gl7$Febm$\"* (e$H(GF]gl7$Fjbm$\"*(*)o6GF]gl-F`cm6&FbcmFccmF(Fccm-Fgcm6#%Ta~minor~mo dification~of~the~Bogacki-Shampine~schemeG-F$6%7\\qF'7$F+$!#MF07$F2$\" %*H(F07$F7$\"&ai%F07$F<$\"'`N>F07$FA$\"'\")*G'F07$FF$\"(5yr\"F07$FK$\" (*GGTF07$FP$\"(S8**)F07$FU$\")`#4\"=F07$FZ$\")*=?U$F07$F\\fm$\")G()3YF 07$Fin$\")>`KhF07$Fdfm$\")4Dp!)F07$F^o$\"*#\\&30\"F07$Fco$\"*!y[b8F07$ Fho$\"*@;$z9F07$F^p$\"*FXPZ\"F07$Fcp$\"*@4#o9F07$F]q$\"*:*yI9F07$Far$ \"*-\"o19F07$Ffr$\"**y9F9F07$F[s$\"*FmD_\"F07$F`im$\"*H>ci\"F07$F`s$\" *sm()y\"F07$$\"5n;zp[#)eB\\>Fjo$\"*$)f.!>F07$Fhim$\"*?bo.#F07$$\"5+]PM xUL]3?Fjo$\"*Uh)\\@F07$Fes$\"*5%HV@F07$F_t$\"*ONd3#F07$Fcu$\"*%e:Q?F07 $$\"5nm\"Hd?>4!QEFjo$\"*x![M?F07$Fhu$\"*P;,/#F07$$\"5++v=U2TJrFFjo$\"* G`81#F07$F]v$\"*([b2@F07$Fbv$\"*2nRL#F07$Fgv$\"*SfXR#F07$Faw$\"*e*HMBF 07$Fex$\"*f1oG#F07$Fjx$\"*e0:O#F07$F_y$\"*=TUW#F07$Fiy$\"*NcrQ#F07$F][ l$\"**=lXBF07$$\"5NL3FWXK1zZFjo$\"*hqbM#F07$Fb[l$\"*Mx_N#F07$$\"5-+D\" yv,9E\"\\Fjo$\"*J()*zBF07$Fg[l$\"*KmpU#F07$F\\\\l$\"*fXZW#F07$Fa\\l$\" *&oyPCF07$Ff\\l$\"*Ih3V#F07$F[]l$\"*poRU#F07$F`]l$\"*M\"G5CF07$Fe]l$\" *=CnR#F07$F]`l$\"*+2qJ#F07$F_cl$\"*7h!4AF07$Fafl$\"*`C!*3#F07$Fffl$\"* n'=r>F07$F[gl$\"*r8]&=F07$Fagl$\"*z(\\g\"[y7F]gl7$Fdil$\"+Qr\\97F]gl7$Fiil$\"+e*)>i6F]gl7$F^j l$\"+,'z!46F]gl7$Fcjl$\"+8Q@i5F]gl7$Fhjl$\"+(f-p,\"F]gl7$F][m$\"*9)f!y *F]gl7$Fb[m$\"*J15R*F]gl7$Fg[m$\"*;^Q,*F]gl7$F\\\\m$\"*%Q*oq)F]gl7$Fa \\m$\"*!Hz&R)F]gl7$Ff\\m$\"*p_\\4)F]gl7$F[]m$\"*KL#>yF]gl7$F`]m$\"*QP& ovF]gl7$Fe]m$\"*M+rI(F]gl7$Fj]m$\"*-7j3(F]gl7$F_^m$\"*5qS'oF]gl7$Fd^m$ \"*t/Rn'F]gl7$Fi^m$\"*\"zBxkF]gl7$F^_m$\"*1*>-jF]gl7$Fc_m$\"**\\%)GhF] gl7$Fh_m$\"*.B#ofF]gl7$F]`m$\"*)4!)3eF]gl7$Fb`m$\"*#*[Jm&F]gl7$Fg`m$\" *Wn;_&F]gl7$F\\am$\"*FN$)Q&F]gl7$Faam$\"*ud:F&F]gl7$Ffam$\"*^nS9&F]gl7 $F[bm$\"*`Ha.&F]gl7$F`bm$\"*Cz[#\\F]gl7$Febm$\"*cnQ#[F]gl7$Fjbm$\"*\\g -s%F]gl-%&COLORG6&Fbcm$\"\"\"F\\cm$\"\")F\\cmF(-Fgcm6#%3efficient~sche me~AG-F$6%7\\qF'7$F+$!#SF07$F2$\"&7V#F07$F7$\"'rV9F07$F<$\"'EfeF07$FA$ \"(`&p=F07$FF$\"('pU]F07$FK$\")*G2?\"F07$FP$\")MD'f#F07$FU$\")J0)>&F07 $FZ$\")![Kx*F07$F\\fm$\"*2ILJ\"F07$Fin$\"*9JQu\"F07$Fdfm$\"*S9+H#F07$F ^o$\"*(*4n(HF07$Fco$\"*=VG$QF07$Fho$\"*#RS!=%F07$F^p$\"*)4mkTF07$Fcp$ \"*i;!\\TF07$F]q$\"*pNL/%F07$Far$\"*/Zl(RF07$Ffr$\"*#fiOSF07$F[s$\"*b$ o4VF07$F`im$\"*TADg%F07$F`s$\"*snV1&F07$Fg^o$\"*_0&z`F07$Fhim$\"*&>Pkd F07$F__o$\"*3&e#3'F07$Fes$\"*>/S1'F07$F_t$\"*#*z6!fF07$Fcu$\"*Vpuw&F07 $F]`o$\"*\")*yddF07$Fhu$\"*-cZx&F07$Fe`o$\"*5Uh$eF07$F]v$\"*p!HofF07$F bv$\"*nr.h'F07$Fgv$\"*3>8y'F07$Faw$\"*))=2h'F07$Fex$\"*ccsZ'F07$Fjx$\" *iICp'F07$F_y$\"*ftr#pF07$Fiy$\"*%o[lnF07$F][l$\"*Wz*[mF07$Febo$\"*9!Q \\mF07$Fb[l$\"*KXwn'F07$F]co$\"*/`%[nF07$Fg[l$\"*A$=#)oF07$F\\\\l$\"*8 dE$pF07$Fa\\l$\"*+CH\"pF07$Ff\\l$\"*'\\G$*oF07$F[]l$\"*MRP(oF07$F`]l$ \"**e#\\$oF07$Fe]l$\"*.$['z'F07$F]`l$\"*bRMd'F07$F_cl$\"*IV%piF07$Fafl $\"*hO/$fF07$Fffl$\"*gOrf&F07$F[gl$\"*\\m\"o_F07$Fagl$\"*C#R+]F07$Ffgl $\"*QCTr%F07$F[hl$\"+l()>\\WF]gl7$F`hl$\"+3/i9UF]gl7$Fehl$\"+.E7=SF]gl 7$Fjhl$\"+i!*H,QF]gl7$F_il$\"+N*)\\KOF]gl7$Fdil$\"+LDv]MF]gl7$Fiil$\"+ F\"z@I$F]gl7$F^jl$\"+kxD^JF]gl7$Fcjl$\"+V?4=IF]gl7$Fhjl$\"+4KL*)GF]gl7 $F][m$\"+!p_*yFF]gl7$Fb[m$\"+FHBoEF]gl7$Fg[m$\"+?#[5c#F]gl7$F\\\\m$\"+ i7\"QZ#F]gl7$Fa\\m$\"+cxR&Q#F]gl7$Ff\\m$\"+@2!**H#F]gl7$F[]m$\"+XBa@AF ]gl7$F`]m$\"+(*\\H]@F]gl7$Fe]m$\"+Ww*f2#F]gl7$Fj]m$\"+)f]K,#F]gl7$F_^m $\"+G@4]>F]gl7$Fd^m$\"+\"R[g*=F]gl7$Fi^m$\"+[*e,%=F]gl7$F^_m$\"+\\bT!z \"F]gl7$Fc_m$\"+E!*R\"F ]gl7$Febm$\"+0TKq8F]gl7$Fjbm$\"+'>$)3M\"F]gl-F`cm6&FbcmFccmF(F(-Fgcm6# %3efficient~scheme~BG-F$6%7aqF'7$F+$\"$:(F07$F2$\"'pB6F07$F7$\"'ojbF07 $F<$\"(=P/#F07$FA$\"(a-7'F07$FF$\")zhz:F07$FK$\")g'4k$F07$FP$\")9-ywF0 7$FU$\"*PUp]\"F07$FZ$\"*s[wy#F07$F\\fm$\"*Z\"o>PF07$Fin$\"*Y\"32\\F07$ Fdfm$\"*T,dS'F07$F^o$\"*!3x!G)F07$Fco$\"+nH!31\"F07$Fho$\"+CC'\\:\"F07 $F^p$\"+lHh]6F07$Fcp$\"+$y!HY6F07$F]q$\"+x><<6F07$Far$\"+,i4+6F07$Ffr$ \"+5ez=6F07$F[s$\"+w%es>\"F07$F`im$\"+L4cz7F07$F`s$\"+PM)yS\"F07$Fhim$ \"+`)R0g\"F07$Fes$\"+\\kY#o\"F07$F_t$\"+ZYKP;F07$Fcu$\"+,I@,;F07$F]`o$ \"+!4c#*f\"F07$Fhu$\"+:y'\\g\"F07$Fe`o$\"+CzCB;F07$F]v$\"+(=q7m\"F07$F bv$\"+/?.T=F07$Fgv$\"+G:;))=F07$Faw$\"+/-tS=F07$Fex$\"+t^s/=F07$Fjx$\" +^DUo=F07$F_y$\"+sg[M>F07$Fiy$\"+p\"f%*)=F07$F][l$\"+FY@e=F07$Febo$\"+ *HE!f=F07$Fb[l$\"+[zvn=F07$F]co$\"+\"\\A%))=F07$Fg[l$\"+g\"Ql#>F07$F\\ \\l$\"+a'*zS>F07$Fa\\l$\"+f`FN>F07$Ff\\l$\"+2uxH>F07$F[]l$\"+KcIC>F07$ F`]l$\"+f-W8>F07$Fe]l$\"+$y!o->F07$F_^l$\"+8xCk=F07$Fi^l$\"+y4&p$=F07$ $\"5qmm\"zW#\\n.eFjo$\"+qp;O=F07$F^_l$\"+()**[S=F07$$\"5+++v$fe$\\FfFj o$\"+vG?_=F07$Fc_l$\"+XJNu=F07$Fh_l$\"+isRi=F07$F]`l$\"+-7.W=F07$F_cl$ \"+%**z*o;F07$Fffl$\"+42.x:F07$F[gl$\"+(yje[\"F07$F agl$\"+jV`69F07$Ffgl$\"+**3qJ8F07$F[hl$\",r[`wD\"F]gl7$F`hl$\",^Yx>>\" F]gl7$Fehl$\",ci*)o8\"F]gl7$Fjhl$\",;%o)f2\"F]gl7$F_il$\",caA&G5F]gl7$ Fdil$\"+v\")ot(*F]gl7$Fiil$\"+!4J]N*F]gl7$F^jl$\"+z#p&H*)F]gl7$Fcjl$\" +g8#Qb)F]gl7$Fhjl$\"+FST!>)F]gl7$F][m$\"+#Rs%yyF]gl7$Fb[m$\"+z@llvF]gl 7$Fg[m$\"+,!GDE(F]gl7$F\\\\m$\"+bht:qF]gl7$Fa\\m$\"+V+]lnF]gl7$Ff\\m$ \"+wQWBlF]gl7$F[]m$\"+N(e:I'F]gl7$F`]m$\"+g.y*4'F]gl7$Fe]m$\"+#e&H*)eF ]gl7$Fj]m$\"+V0`6dF]gl7$F_^m$\"+E,cKbF]gl7$Fd^m$\"+VXTz`F]gl7$Fi^m$\"+ ac+@_F]gl7$F^_m$\"+=$4+3&F]gl7$Fc_m$\"+h[NS\\F]gl7$Fh_m$\"+)**[4\"[F]g l7$F]`m$\"+?'*\\#o%F]gl7$Fb`m$\"+q#R^c%F]gl7$Fg`m$\"+XL8^WF]gl7$F\\am$ \"+S%*oVVF]gl7$Faam$\"+51e\\UF]gl7$Ffam$\"+*[Po9%F]gl7$F[bm$\"+_8GfSF] gl7$F`bm$\"+?P=qRF]gl7$Febm$\"+I(p())QF]gl7$Fjbm$\"+ngE0QF]gl-F`cm6&Fb cm$\")#)eqkFecm$\"))eqk\"FecmF]cr-Fgcm6#%3efficient~scheme~CG-F$6%7_qF '7$F+$!#$*F07$F2$!%d\\F07$F7$!&5o\"F07$F<$!&NN%F07$FA$!&U?*F07$FF$!''e j\"F07$FK$!',PCF07$FP$!'JjGF07$FU$!'(y!>F07$FZ$\"'+*H#F07$F\\fm$\"'=Ul F07$Fin$\"(e1H\"F07$Fdfm$\"(/t?#F07$F^o$\"(!f([$F07$Fco$\"(\"pK_F07$Fh o$\"(=m)fF07$F^p$\"(tS'fF07$Fcp$\"(k;%fF07$F]q$\"(X7y&F07$Far$\"(B&=bF 07$$\"5MLe9\"4&[EB;Fjo$\"(JqU&F07$Ffr$\"(bTM&F07$$\"5M$e9T8/m@r\"Fjo$ \"(RaJ&F07$$\"5++vV[r(*zTcF07$F`s$\"(x!=iF07$Fg^o$\"(L&4nF07$Fhim$\" (3HP(F07$F__o$\"(&)f'zF07$Fes$\"(];%zF07$F_t$\"(QYs(F07$Fcu$\"(n[V(F07 $F]`o$\"(\"[OtF07$Fhu$\"(83C(F07$$\"5n;H2eyy)zt#Fjo$\"(.@?(F07$Fe`o$\" (5m<(F07$$\"5M$3-jiLSY!GFjo$\"(F?<(F07$F]v$\"((R)>(F07$Fbv$\"(*fBzF07$ Fgv$\"(T#=#)F07$Faw$\"(6R+)F07$Fex$\"($=8xF07$$\"5MLL3xc[U3PFjo$\"(3[i (F07$$\"5MLL$3-8>bx$Fjo$\"(v$\\vF07$$\"5ML$3FpEm!4QFjo$\"(!QDvF07$$\"5 MLLek.MhUQFjo$\"(\"y!o(RFjo$\"(\")G$yF07$$\"5MLL$eRA'*Q/%Fjo$\"(R?&zF07$$\"5ML LeR(\\!*46%Fjo$\"(xR!zF07$F_y$\"(Rj&yF07$Fe]l$\"(D')G(F07$F]`l$\"(g%Gn F07$F_cl$\"('3vhF07$Fafl$\"(-+m&F07$Fffl$\"(Yo?&F07$F[gl$\"(d)*z%F07$F agl$\"(YD[%F07$Ffgl$\"(2-<%F07$F[hl$\")#fU*QF]gl7$F`hl$\")f\"*eOF]gl7$ Fehl$\")3%zY$F]gl7$Fjhl$\")OnjKF]gl7$F_il$\")Oo3JF]gl7$Fdil$\")5^WHF]g l7$Fiil$\")z38GF]gl7$F^jl$\")g)4o#F]gl7$Fcjl$\")+,mDF]gl7$Fhjl$\")&QfX #F]gl7$F][m$\")#RAO#F]gl7$Fb[m$\")j0pAF]gl7$Fg[m$\")f5z@F]gl7$F\\\\m$ \")$zg5#F]gl7$Fa\\m$\")28K?F]gl7$Ff\\m$\")^og>F]gl7$F[]m$\")SI&*=F]gl7 $F`]m$\")=%f$=F]gl7$Fe]m$\")g!Rx\"F]gl7$Fj]m$\")fl@'=8F]gl7$Faam$\")4w!H\"F]gl7$Ffam$\") [Sg7F]gl7$F[bm$\")GVM7F]gl7$F`bm$\")w037F]gl7$Febm$\")=)Q=\"F]gl7$Fjbm $\")n@f6F]gl-Fh\\p6&Fbcm$\"\"'F\\cmF(F\\]p-Fgcm6#%Sscheme~with~stabili ty~radius~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6# %Uerror~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6 $Q\"x6\"Q!F[is-%%VIEWG6$;F(Fjbm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor mo dification of the Bogacki-Shampine scheme" "efficient scheme A" "effic ient scheme B" "efficient scheme C" "scheme with stability radius = im ag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 " Test 10 of 7 stage, order 5 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -(2*sin(5*x)+3*cos(7*x))*sinh(y );" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&*&\"\"#F&-%$sinG6#*&\"\"&F&%\"xGF &F&F&*&\"\"$F&-%$cosG6#*&\"\"(F&F3F&F&F&F&-%%sinhG6#%\"yGF&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y(0)=sqrt(5)/2" "6#/-%\"yG6#\"\"!*&-%%sqr tG6#\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de := diff(y(x),x)=-(2*sin(5*x)+3*cos(7*x))*sin h(y(x));\nic := y(0)=sqrt(5)/2;\ndsolve(\{de,ic\},y(x));\nsimplify(con vert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x),x=0..5,font=[HELVETI CA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ -%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F2F,F2F2 F2F2*&\"\"$F2-%$cosG6#,$*&\"\"(F2F,F2F2F2F2F2-%%sinhG6#F)F2!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!,$*&\"\"#!\"\"\"\" &#\"\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6# -%%tanhG6#,**&#\"\"\"\"\"&F1-%$cosG6#,$*&F2F1F'F1F1F1!\"\"*&#\"\"$\"#9 F1-%$sinG6#,$*&\"\"(F1F'F1F1F1F1#F1F2F1*&#F1\"\"#F1-F)6#,$*&,&-%$expG6 #,$*&FFF8F2FEF1F1F1F1F1,&FLF1F1F8F8F8F1F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,$*&,*-%$expG6#,4#\"\"#\"\"&\"\" \"*&#\"\"$\"\"(F5-%$sinGF&F5!\"\"*&#\"$#>F9F5*&)-%$cosGF&\"\"'F5F:F5F5 F5*&#\"$S#F9F5*&)FB\"\"%F5F:F5F5F<*&#\"#sF9F5*&)FBF3F5F:F5F5F5*&#\"#KF 4F5*$)FBF4F5F5F<*&\"\")F5)FBF8F5F5*&F3F5FBF5F<*&F3FF5F@F5F5*&#FGF9F5FHF5F<*&FLF5FNF5F5*&#FRF4F5FSF 5F<*&FVF5FWF5F5*&F3F5FBF5F$\"3`uw, WSt45F,7$$\"3WmmmT&)G\\aF0$\"3cSB%H69F5*!#=7$$\"3m****\\7G$R<)F0$\"3OF 6@7@G;#)F87$$\"3GLLL3x&)*3\"F8$\"3_]\"Hlv:eW(F87$$\"3))**\\i!R(*Rc\"F8 $\"3WS]N\"zi(yjF87$$\"3umm\"H2P\"Q?F8$\"35U:-NK+KcF87$$\"3YLek.pu/BF8$ \"3:Vt%)yf.P`F87$$\"3!***\\PMnNrDF8$\"3!okU]7kD7&F87$$\"37$eR(\\;m/FF8 $\"3q$>&4'\\tL/&F87$$\"3MmT5ll'z$GF8$\"3gBk>%*z$=)\\F87$$\"37](o/[r7(H F8$\"3dP2*>TRs$\\F87$$\"3MLL$eRwX5$F8$\"3*H\">\\xg!*3\\F87$$\"3:L$3F% \\wQKF8$\"3wA2?_M<'*[F87$$\"3_LLe*[`HP$F8$\"3-d)o(RAn)*[F87$$\"3*QLek. Ur]$F8$\"3K()))eR[\"e\"\\F87$$\"3rLLL$eI8k$F8$\"3kdF/H[-Z\\F87$$\"3*QL $3xwq4RF8$\"3!**RBYN;$\\]F87$$\"33ML$3x%3yTF8$\"3r5fX,$G1?&F87$$\"3h+] PfyG7ZF8$\"3C'yxl/-Si&F87$$\"3emm\"z%4\\Y_F8$\"3#GR!R(Q;g:'F87$$\"32++ v$flMLe*)>VB$)F8$\"3R-PW?nJGpF87$$\"3wmm Tg()4_))F8$\"3XS!**oz*pDlF87$$\"3Y++DJbw!Q*F8$\"3mrIZl^+ohF87$$\"3=nT& )3\\m_'*F8$\"3P'=[TB`#HgF87$$\"3+N$ekGkX#**F8$\"30<=#[laH$fF87$$\"3nTg _(R^g+\"F,$\"3/eYLpC*H!fF87$$\"31]iSmjk>5F,$\"3K[/nW4A')eF87$$\"3XekGN 8CL5F,$\"3S5jkI7P$)eF87$$\"3%ommTIOo/\"F,$\"3]\"*RMEK8&*eF87$$\"3cLe9; _yq5F,$\"3_[.xr\")Q`fF87$$\"3E+]7GTt%4\"F,$\"3_%yw2JYC1'F87$$\"3(p;/,/ $o=6F,$\"3[(3')fymbA'F87$$\"3YLL3_>jU6F,$\"3aRGM!H;eW'F87$$\"3ym;HdNb' >\"F,$\"3IR8#3QfS;(F87$$\"37++]i^Z]7F,$\"3/nQ!Q)F87$$\"35+++v\"=Y I\"F,$\"3&441r_RP_*F87$$\"33++](=h(e8F,$\"3G(\\BrLl\"*4\"F,7$$\"3&**** *\\7!Q4T\"F,$\"39o3wy`oD7F,7$$\"3/++]P[6j9F,$\"3;)>\\NC[mH\"F,7$$\"3'= HKkAg\\Z\"F,$\"3Scb1EJ%4I\"F,7$$\"3W$ek`h0o[\"F,$\"3fbt7ruA+8F,7$$\"3/ voH/5l)\\\"F,$\"3Otx0vuR%H\"F,7$$\"3%o;HKR'\\5:F,$\"3Mz4l5FZ$G\"F,7$$ \"3-]P4rr=M:F,$\"31-%*Q\"yBqC\"F,7$$\"3UL$e*[z(yb\"F,$\"3W\")o!f;2L>\" F,7$$\"3w;/Ev&[ge\"F,$\"3Q?R!QDA@6\"F,7$$\"34+Dc,#>Uh\"F,$\"39\"HI%*[n *=5F,7$$\"3V$eky#)*QU;F,$\"3mms=Lk!y?*F87$$\"3wmm;a/cq;F,$\"3Pe)[!HQiL #)F87$$\"3\"pm;a)))G=F,$\"3U@p0?YWZMF87$$\"3KLe9;0?E>F,$\"3w;XWhm;,LF87$$\" 3pTg-gl[Q>F,$\"3qsZpoe$*\\KF87$$\"31]i!Rgs2&>F,$\"3VKI#4&[)H@$F87$$\"3 WekyZ'eI'>F,$\"3=g(Q?ez,>$F87$$\"3gmmm\"pW`(>F,$\"3q(HR/(>[\"=$F87$$\" 3_ek.HW#)))>F,$\"3SdFyRyC)=$F87$$\"3?]iSmTI-?F,$\"3]%GeCR0B@$F87$$\"3* =/wP!Ry:?F,$\"3/I\"o!ej/aKF87$$\"3dLe9TOEH?F,$\"3u\"yA?()GSJ$F87$$\"3' pT&)e6Bi0#F,$\"3#fXkrX5=\\$F87$$\"3K+]i!f#=$3#F,$\"3?\\niL?f`PF87$$\"3 /++D\"=EX8#F,$\"3)=7zi$ePDXF87$$\"3?+](=xpe=#F,$\"3#GkoxRDBu&F87$$\"3$ pTNrfbE@#F,$\"3_(**Rb6vOf'F87$$\"3mLeRA9WRAF,$\"3o9cI=h\\:wF87$$\"3S]i lZsAmAF,$\"3.'zb%4IS@))F87$$\"37nm\"H28IH#F,$\"3v'yDR:j=-\"F,7$$\"3!oT N@#3hF,7$$\"3W eRseStdCF,$\"3B.$en#[n??F,7$$\"3)oT5l0+5Z#F,$\"31^_;#*)R!\\?F,7$$\"35Y OSbIjxCF,$\"3'f>URy8d0#F,7$$\"3IvoHagE%[#F,$\"3K3\\y.l?d?F,7$$\"3_/,>` !**3\\#F,$\"3yIUb4&*\\`?F,7$$\"3GLL3_?`(\\#F,$\"3u\")GRRomW?F,7$$\"3.$ 3-)Q84DDF,$\"3;`^lOB%z&>F,7$$\"3AL3_D1l_DF,$\"3VH*H8v%))4=F,7$$\"3I3-) )o-VmDF,$\"3o]!f;TM@s\"F,7$$\"3S$eRA\"*4-e#F,$\"3ag!)pZ(f(H;F,7$$\"3[e *)fb&*)Rf#F,$\"38o>0Ar[N:F,7$$\"3fL$e*)>pxg#F,$\"3I\\_^!fj9W\"F,7$$\"3 V+D1R'f:&H/%HG\"F,7$$\"3%omm\"z+vbEF,$\"3Q#=@:,\\a8\"F,7$$ \"3AL3F>0uzEF,$\"3C!3Cbi?=+\"F,7$$\"33+]Pf4t.FF,$\"3(3T(3()HVJ))F87$$ \"3Q$3F>HT'HFF,$\"3W)y5tq@xr(F87$$\"3om\"zWi^bv#F,$\"3?:fooRnqnF87$$\" 3)*\\7.d>Y\"y#F,$\"3Eo_f3vnxfF87$$\"3uLLe*Gst!GF,$\"3eVA%fwpRK&F87$$\" 3)om\"H2\"34'GF,$\"3cu]3R$*[ONF87$$\"3C+voaa+$*HF,$\"3_,\"*[foPONF87$$\"3> ](oH/*41IF,$\"33E#)e$)41_NF87$$\"3:++DJE>>IF,$\"37lJ^K5P$e$F87$$\"3A+v $4^n)pIF,$\"3q;[s.$y8&QF87$$\"3F+]i!RU07$F,$\"3%Q(yiwjedVF87$$\"39+vo/ #3o<$F,$\"3.u`zNU*o?&F87$$\"3+++v=S2LKF,$\"3rC&z7e?,M'F87$$\"3;L$3_NJO G$F,$\"3'oVq=V<'GvF87$$\"3Jmmm\"p)=MLF,$\"3**GlY8#))ft)F87$$\"3GLLeR%p \")Q$F,$\"3\"\\w,G=fT\")*F87$$\"3B++](=]@W$F,$\"3p.a#\\O(RU5F,7$$\"3u \"H#oK)yVX$F,$\"37R>JG%fz/\"F,7$$\"3C$ekyZ2mY$F,$\"31>$4cdg.0\"F,7$$\" 3=vo/Bh$)yMF,$\"3i,C<-Qm\\5F,7$$\"3mm\"H#oZ1\"\\$F,$\"3;\"3uAU\"*f/\"F ,7$$\"36]Pfe?_:NF,$\"3t?:\"y&yYI5F,7$$\"35L$e*[$z*RNF,$\"3AZcwFIo05F,7 $$\"3%o;Hd!fX$f$F,$\"3\\R'zK#G91$*F87$$\"3e++]iC$pk$F,$\"3!e-txpVQY)F8 7$$\"3ILe*[t\\sp$F,$\"3!)*z!\\!\\E.v(F87$$\"3[m;H2qcZPF,$\"3Yq*ep\"3v- sF87$$\"3s***\\7.lQx$F,$\"3_n\"\\:&*zo*pF87$$\"3UL$3_0j,!QF,$\"3aavs*p f#\\oF87$$\"3F+v=n?J8QF,$\"35S^fg!Quz'F87$$\"36nm;z5YEQF,$\"3-)RYLJ)=g nF87$$\"3^Le9\"45'RQF,$\"39oisVpNPnF87$$\"3O+]7.\"fF&QF,$\"3VEh1S\\sGn F87$$\"3i3_vlYhlQF,$\"33cF9)GTPt'F87$$\"3)oT&QG-ZyQF,$\"3kS=Q3Gt^nF87$ $\"39Dc,\"zD8*QF,$\"3e/VIG#3By'F87$$\"3TLek`8=/RF,$\"3IQP7dc+DoF87$$\" 3$*\\i!*yC*)HRF,$\"3reG3V'=X%pF87$$\"3Ymm;/OgbRF,$\"3'GUsy#>%[5(F87$$ \"3*G$e*[$zV4SF,$\"3O_/NuwfQvF87$$\"3w**\\ilAFjSF,$\"3D8xFXpB@!)F87$$ \"3#G3_]p'>*3%F,$\"3D\")e-u54K#)F87$$\"3ym\"zW7@^6%F,$\"31QL&Rp%o0%)F8 7$$\"3w3F>RL3GTF,$\"3-T$yF^SJZ)F87$$\"3t]i!RbX59%F,$\"3t*pQA'3/D&)F87$ $\"3#=z>'ox+aTF,$\"3M&4T=0@'f&)F87$$\"3yLLL$)*pp;%F,$\"3eA(p'zRPv&)F87 $$\"3!Q3_+sD-=%F,$\"3se;6C(p2d)F87$$\"3#Q$3xc9[$>%F,$\"3MZZFn2MW&)F87$ $\"3'Qe*[$>Pn?%F,$\"31`G!HhJc\\)F87$$\"3)QL3-$H**>UF,$\"3[:#3'R6kC%)F8 7$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$\"3)RL$3xe,tUF,$\"3]-#zi!3VLzF87$ $\"3Cn;HdO=yVF,$\"3/\">X)HduejF87$$\"3a+++D>#[Z%F,$\"3gxAO;Hx\")\\F87$ $\"3TM$3_5,-`%F,$\"3%GSzCT^zW%F87$$\"3SnmT&G!e&e%F,$\"3=g#R;G9D:%F87$$ \"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe*[=Y.h%F,$\"33)>WD]()H5%F87$$ \"386but%F,$\"3_@+X\\xqwZF87$$\"37+]iSjE!z%F,$\"3<\"z**[W(=JcF87$ $\"3y*\\7G))Rb\"[F,$\"3OXJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$\"3!**fZR-9n(o F87$$\"3)3](=np3m[F,$\"3#eM]7.#Q?&=\\F,$\"3'4(*[g6ply*F87$$\"3s+voa-oX\\F,$\"3?(Q=b([U56 F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3.ug')yo>49F,-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\" xG%%y(x)G-%%VIEWG6$;F(Ficn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 788 "P := (x,y) -> -(2*sin(5*x)+3*cos(7 *x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\n matrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bog acki-Shampine scheme`,`a minor modification of the Bogacki-Shampine sc heme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,` scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDi gits := 30:\nfor ct to 6 do\n Pn_RK5_||ct := RK5_||ct(P(x,y),x,y,x0, evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Pn_RK5_||ct) :\n for ii to numpts do\n sm := sm+(Pn_RK5_||ct[ii,2]-evalf(p(P n_RK5_||ct[ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpt s)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6# ,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6 $\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G \"$+&Q)pprint776\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+e q#Rc#!#?7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+D1 nBVF+7$%3efficient~scheme~AG$\"+'*Rt6;F+7$%3efficient~scheme~BG$\"+$pM 4<#F+7$%3efficient~scheme~CG$\"+H%=$eDF+7$%Sscheme~with~stability~radi us~=~imag~axis~inclusionG$\"+])QPH$F+Q)pprint786\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh : = 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope fi eld: `,P(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine sch eme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient \+ scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stabi lity radius = imag axis inclusion`]:\nerrs := []:\nDigits := 30:\nfor \+ ct to 6 do\n pn_RK5_||ct := RK5_||ct(P(x,y),x,y,x0,evalf(y0),hh,nums teps,true);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 6 d o\n errs := [op(errs),abs(pn_RK5_||ct(xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-% $sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F. F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F- F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint636\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+?$QK+(!#?7$%Ta~minor~m odification~of~the~Bogacki-Shampine~schemeG$\"+7\"pKK\"!#>7$%3efficien t~scheme~AG$\"+vR.fSF+7$%3efficient~scheme~BG$\"+7e'4)\\F+7$%3efficien t~scheme~CG$\"+**RWvLF+7$%Sscheme~with~stability~radius~=~imag~axis~in clusionG$\"+'*e6m5F0Q)pprint646\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$ \"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes met hod over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of \+ the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B` ,`efficient scheme C`,`scheme with stability radius = imag axis inclus ion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((p( x)-'pn_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200); \n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+:Lz#e#!#?7$%Ta~minor~modific ation~of~the~Bogacki-Shampine~schemeG$\"+f3p)H%F+7$%3efficient~scheme~ AG$\"+u)R]g\"F+7$%3efficient~scheme~BG$\"+2=mm@F+7$%3efficient~scheme~ CG$\"+1kYeDF+7$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$ \"+1X8sKF+Q)pprint656\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 92 "The following error graphs are constructed using t he numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 500 "evalf[20](plot(['pn_RK5_1'(x)-p(x),'pn_RK5_2'(x)-p (x),'pn_RK5_3'(x)-p(x),'pn_RK5_4'(x)-p(x),\n'pn_RK5_5'(x)-p(x),'pn_RK5 _6'(x)-p(x)],x=0..2.2,\ncolor=[blue,magenta,COLOR(RGB,.1,.8,0),red,bro wn,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9],\nlegend=[`Bogacki-Shampine \+ scheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficie nt scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with st ability radius = imag axis inclusion`],\ntitle=`error curves for 7 sta ge order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 859 509 509 {PLOTDATA 2 "6,-%'CURVESG6%7jr7$$\"\"!F)F(7$$\"5LLLLLepo(R#!#@ $\"(\"zG%)!#>7$$\"5mmmmm;RP&z%F-$\"*!G=(R\"!#?7$$\"5NLL$3x@\"[QeF-$\"* LcDc\"F67$$\"5++++v=&)e\")oF-$\"*DR!y;F67$$\"5lmm;z>epCzF-$\"*UT=v\"F6 7$$\"5LLLL$37.y'*)F-$\"*!y0\"z\"F67$$\"5&****\\P%[ZMa&*F-$\"*L.Mw\"F67 $$\"5mmmTgP')395F6$\"*4#p(y\"F67$$\"5LL$ek.!Gus5F6$\"*f)p`$F6$\"*[(o-7F67$$\"5mmm mTv+JiOF6$\"*:mDB\"F67$$\"5++++vLo`FTF6$\"*,33I\"F67$$\"5LLLLLQ(zgg%F6 $\"*g!pM9F67$$\"5mmmm;*e!eF]F6$\"*pa:h\"F67$$\"5+++++:24-bF6$\"*/B8%=F 67$$\"5+++++q*>.u&F6$\"*%GGE>F67$$\"5+++++D#\\&yfF6$\"*%e2t>F67$$\"5++ +++bs73iF6$\"*>#4R>F67$$\"5+++++&G0xV'F6$\"*#R^t=F67$$\"5NLL$3-8%=YmF6 $\"*HD%ew/x(F6$\"**=Y45F67$$\"5 lmm;/Y?w9!)F6$\")By?&*F67$$\"5*******\\()*y/f#)F6$\")**z\"Q*F67$$\"5lm m;/rAGv%)F6$\"))GNX*F67$$\"5LLLLLVm^\"p)F6$\")2GU'*F67$$\"5*******\\() R.g;*F6$\")!>)G(*F67$$\"5*******\\i*p#yh*F6$\")/J&Q*F67$$\"5LLLL3_d#*3 5F0$\");ip()F67$$\"5LLL$32z)F67 $$\"5+++]7A;k*=\"F0$\")]c]&)F67$$\"5MLL3x9FW77F0$\")aO*H*F67$$\"5nmmmT 2QCN7F0$\"*K@X0\"F67$$\"5MLL$3sc*ze7F0$\"*!p-\"H\"F67$$\"5+++++F`N#G\" F0$\"*Lx8p\"F67$$\"5+++]PU+S08F0$\"*nc$)>#F67$$\"5++++vdZWG8F0$\")Q*o' HF07$$\"5+++D\"[qT2N\"F0$\")F`=SF07$$\"5+++](=lQIP\"F0$\")8lgF07$$\"5 +]ilZ[zX>9F0$\")3V&)fF07$$\"5+D\"Gj$e_+@9F0$\")Z$)RfF07$$\"5++++DoDbA9 F0$\")g?ffF07$$\"5LLL3F#=vOV\"F0$\")[7'o&F07$$\"5mmm;H'z(zW9F0$\")#R7< &F07$$\"5+++DJ5/#fX\"F0$\")3W\"R%F07$$\"5LLLLLCI/n9F0$\")_o>LF07$$\"5m mmTg14)HZ\"F0$\")**z!H#F07$$\"5+++]())y=*y9F0$\")dQ[;F07$$\"5LL3FWfKS! [\"F0$\")ZB+5F07$$\"5mm;/,Ix)=[\"F0$\")d0+5F07$$\"5++D\"y0?sL[\"F0$\"( 2'z**F07$$\"5LLLe9rm&[[\"F0$\"(#H@)*F07$$\"5++]7G7c#y[\"F0$\"((z*)oF07 $$\"5mmmmT`Xz!\\\"F0$!(_!*f$F07$$\"5++vV)R-zA\\\"F0$!(%=)f$F07$$\"5LL$ 3_X\\jP\\\"F0$!(z?j$F07$$\"5mm\"z>^'zC&\\\"F0$!(re(QF07$$\"5+++voNCt' \\\"F0$!([+'[F07$$\"5mm;H#oP,(*\\\"F0$!)k:#\\\"F07$$\"5LLL$ezJqE]\"F0$ !)B\\8gkF07$$\"5n;a)3aa9kc\"F0$! ))>mW'F07$$\"5M3_+s]^)yc\"F0$!)>VlkF07$$\"5++]7.cdNp:F0$!)lhZlF07$$\"5 nmTgFx\"Q_d\"F0$!)q3'\\'F07$$\"5MLL3_)f?6e\"F0$!)[/kkF07$$\"5nm;/,Ta)G f\"F0$!)@m`hF07$$\"5++++]$G]Yg\"F0$!)1UNdF07$$\"5mmmTN$))*yE;F0$!*@pts %F67$$\"5LLL$3K[H*[;F0$!*4%=SNF67$$\"5mmm;HW<2s;F0$!*gV3S#F67$$\"5+++] P0S@&p\"F0$!*i;Qd\"F67$$\"5mmmTg&zRyr\"F0$!)gy#z)F67$$\"5LLLL$eel/u\"F 0$!)Yp)y#F67$$\"5mmmTN\"p_Tw\"F0$\"(Rqd\"F67$$\"5+++](ozRyy\"F0$\")yaE @F67$$\"5nm;H#eaY#*z\"F0$\")E\"*fGF67$$\"5MLL3x%H`1\"=F0$\")6b(H$F67$$ \"5++](=P/g?#=F0$\")$[*RMF67$$\"5nmmmm#zmM$=F0$\")a%)*\\$F67$$\"5nmm;H *)ozc=F0$\")M:([$F67$$\"5nmmm\"f)p7!)=F0$\")cv9NF67$$\"5+++Dc(*QE.>F0$ \")F0$\")uGF0$\")Fo)[%F67$$ \"5+++++Z;#*o>F0$\")0Th`F67$$\"5LLLLeD`l_@F0$\"*1:rK#F67$$\"5++]P4o*[T;#F0$\"*rn!REF67$$ \"5+++D17$*4w@F0$\"*o=&GIF67$$\"5++]7.c'\\!)=#F0$\"*3pjb$F67$$\"#A!\" \"$\"*T=7T%F6-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki -Shampine~schemeG-F$6%7btF'7$F+$\"(oa`&F07$F2$\"*d(**\\5F67$F=$\"*0N$) Q\"F67$FG$\"**\\.+;F67$$\"5lm;ajMR2h#*F-$\"*&z/,;F67$FL$\"*C&[%e\"F67$ $\"5IL$eRAc:w%)*F-$\"*[-Lg\"F67$FQ$\"*S')yk\"F67$$\"5++vV)*=dTV5F6$\"* M[$G;F67$FV$\"*piBi\"F67$$\"5+](oa5M1u3\"F6$\"*yd#R;F67$$\"5mm\"zW<))p ?5\"F6$\"*-'yv;F67$$\"5L$e*[VAMt;6F6$\"*TEdm\"F67$Fen$\"*(f'el\"F67$Fj n$\"*dO-l\"F67$F_o$\"*oiij\"F67$$\"5nmmmTN&*4%[\"F6$\"*WH'>;F67$$\"5nm mmmca=-;F6$\"*U`Dg\"F67$$\"5nmmm\"zPr-s\"F6$\"*[[Mc\"F67$Fdo$\"*=4__\" F67$Fio$\"*YY*39F67$$\"5++++DhoPEDF6$\"*lIPP\"F67$F^p$\"*%GR`8F67$$\"5 LLLL3<@%*pHF6$\"*9LmM\"F67$Fcp$\"**\\Gc8F67$Fhp$\"*j+nR\"F67$F]q$\"*([ so9F67$Fbq$\"*]mac\"F67$Fgq$\"*hxGm\"F67$F\\r$\"*Ue=v\"F67$Ffr$\"*Z^\" ==F67$F`s$\"*E`L&=F67$Fjs$\"*^FR!>F67$$\"5+++](o[kD5(F6$\"*FmD%>F67$F_ t$\"*'eJg>F67$Fdt$\"*Q.M'>F67$Fit$\"*))G'[>F67$F^u$\"*v+)3>F67$Fcu$\"* %R*)\\=F67$F]v$\"*0+)>*o9F67$$\"5nmm \"H_FmJ7\"F0$\"*;y1`\"F67$F[x$\"*CDXh\"F67$Fex$\"*2BR&=F67$Fjx$\"*/\"* =/#F67$F_y$\"*DM@G#F67$Fdy$\"*.]Bj#F67$Fiy$\"*%F07$Fhz$\")y^%y%F07$F][l$\")>F(=&F07$$\"5++v=U\"*yAz8F0 $\")lSd_F07$Fb[l$\")J$HL&F07$$\"5+](=U2v6&)Q\"F0$\")9VY`F07$$\"5++Dc^q jg\"R\"F0$\")v9E`F07$$\"5+vVBS!o`JR\"F0$\")7_\\`F07$$\"5+]i!*G!*4q%R\" F0$\")fir`F07$$\"5+D\"yv,I[iR\"F0$\")a?)Q&F07$Fg[l$\")O1(Q&F07$F[]l$\" )k3K_F07$Fi^l$\");2Z^F07$Fc_l$\")qOk[F07$F]`l$\")B(*=YF07$$\"5mmT5!\\ \\F&o9F0$\")`g.YF07$$\"5++](oa'>,q9F0$\")*)obXF07$$\"5LLek.Ok\\r9F0$\" )s1eXF07$Fb`l$\")=0gXF07$$\"5LL$eRx%)\\fZ\"F0$\")9;jXF07$Fg`l$\")YshXF 07$Faal$\")E#[b%F07$F[bl$\");@`XF07$$\"5mmTNrT6M'[\"F0$\")'fTb%F07$F`b l$\")KHgXF07$$\"5LLe*[G35$*[\"F0$\")RcyXF07$Febl$\")Lt#f%F07$Fcdl$\")7 NUYF07$F]el$\")>%4o%F07$$\"5M3_+ZL7*e^\"F0$\")WetYF07$$\"5n;/,W%QOs^\" F0$\")5\"=n%F07$$\"5+Dc,TN:e=:F0$\")Sf&o%F07$$\"5ML3-Q'oE*>:F0$\")'odt %F07$$\"5+]7.K))phA:F0$\")/M=ZF07$Fbel$\")%>^p%F07$$\"5+vo/BTClE:F0$\" )?\\&o%F07$$\"5M$3_+Af(*z_\"F0$\")[F$o%F07$$\"5n\"HdqJuU$H:F0$\")@w,ZF 07$$\"5++D19%*yoI:F0$\")f)*=ZF07$$\"5n;H23'>yL`\"F0$\");?\"p%F07$Fgel$ \")#R=m%F07$F\\fl$\")-^rXF07$Fafl$\")zCCWF07$Fcil$\")LzaRF07$F]jl$\")n :8NF07$Fbjl$\"*>4W?$F67$Fgjl$\"*(4utIF67$F\\[m$\"*g8v1$F67$Fa[m$\"*7$z ZIF67$$\"5LL$e*[+p_1< F0$\"*a9D5$F67$$\"5+]7GQ>!o1s\"F0$\"*u(4MJF67$$\"5mTN@FJ@3At \"F0$\"*&y)H5$F67$$\"5+v=nQE]RLF67$$\"5++]il=5.P>F0$\"*DNL\">F67$Fb_m$\"*ja*)*=F67$$\"5LL$ 3_vV\"He>F0$\"*F\\K*=F67$Fg_m$\"*qai*=F67$$\"5LLLekm]5\")>F0$\"*(p*4\" >F67$$\"5mmm;H'[)G$*>F0$\"*MO+$>F67$$\"5+++v$f!>Z0?F0$\"**=fe>F67$F\\` m$\"*M:\")*>F67$$\"5+++]7<$\\%R?F0$\"*-nr4#F67$Fa`m$\"*C*4GAF67$$\"5ML L3F\\-[%3#F0$\"*It6T#F67$Ff`m$\"*en*oEF67$F[am$\"*c6a1$F67$F`am$\"*MwK g$F67$Feam$\"*e)HvRF67$Fjam$\"*)fzNWF67$$\"5++vo/%[u?=#F0$\"*k1A\"[F67 $F_bm$\"*k[z/&F67$$\"5++Dc,G[-%>#F0$\"*'oh]aF67$Fdbm$\"*%zi3gF6-Fjbm6& F\\cmF]cmF(F]cm-Facm6#%Ta~minor~modification~of~the~Bogacki-Shampine~s chemeG-F$6%7gtF'7$F+$\"(=?@\"F07$F2$\");WQ>F67$$\"5++voHHBwD\\F-$\")(H &>?F67$$\"5NL$3F>u]h0&F-$\")>>y?F67$$\"5qm\"HdX:Rl=&F-$\")%z^1#F67$$\" 5+++v=nv#pJ&F-$\")ex_?F67$$\"5qm;zW#R/xd&F-$\")]CT?F67$F8$\")Qd(3#F67$ $\"5qmm\"H#o[.gjF-$\")'[>6#F67$F=$\")*p(G@F67$$\"5ILL3Fp@9.uF-$\")FB#4 #F67$FB$\")c*p2#F67$$\"5+++DJq%\\iW)F-$\"),\\6?F67$FG$\")zZZ>F67$Fen$ \")x\">d\"F67$F_o$\")XE)>\"F67$Fdgm$\"(F')***F67$Figm$\"(lwh(F67$F^hm$ \"(Y$)R'F67$Fdo$\"((o4aF67$$\"5MLLeky>)e&>F6$\"(zPk%F67$$\"5+++]7emSt? F6$\"(G^2%F67$$\"5mmmTgP8$4>#F6$\"(Aqq$F67$Fio$\"(v7l$F67$Fihm$\"(Tr,% F67$F^p$\"(#oqZF67$Fcp$\"(V&etF67$Fhp$\")!))>+\"F67$F]q$\")D:!G\"F67$$ \"5mmm;/'G3oO%F6$\")g?y8F67$Fbq$\")nqX9F67$$\"5++++vj,$o\"[F6$\")YQf9F 67$Fgq$\")G\"QV\"F67$$\"5+++]i?\"3i9&F6$\")\"=*49F67$$\"5NLLL3_c$[E&F6 $\")&*Qu8F67$$\"5qmm;a$=jMQ&F6$\")(=iJ\"F67$F\\r$\")u>87F67$Ffr$\")t$f +\"F67$F`s$\"(]0\"))F67$Fjs$\"(b%y()F67$F_t$\"(E$)e)F67$Fit$\"(ECN)F67 $Fcu$\"(ApS*F67$F]v$\"):>R*F6$\")+ sR9F67$Fgv$\")&)4Q9F67$F^]n$\")IR%R\"F67$F\\w$\")GY78F67$Faw$\")?#=6\" F67$F^^n$\")I[G5F67$Ffw$\"(5H]*F67$$\"5nm;H2*=S56\"F0$\"(rp>*F67$Ff^n$ \"()Q\">*F67$$\"5nm;aQhBHN6F0$\"(9pK*F67$F[x$\"(iml*F67$$\"5+++v=TU(z: \"F0$\")*er,\"F67$F`x$\")&F07$Fhz$\"(n#z^F07$$\"5++]PMy,*=O\"F0 $\"(jEA&F07$F][l$\"(QcB&F07$Fb[l$\"(z0E&F07$Fg[l$\"(:FI&F07$F[]l$\"(X! 4bF07$Fi^l$\"(+]*eF07$Fc_l$\"()[CrF07$F]`l$\"(Pu)))F07$Fg`l$\").J+5F07 $Febl$\")rP%4\"F07$F_cl$\")2m#4\"F07$Ficl$\")'\\K4\"F07$$\"5LL3_D1p@) \\\"F0$\")N.*4\"F07$F^dl$\")Ca:6F07$$\"5++D1RZe=,:F0$\"))*y>6F07$Fcdl$ \")k>=6F07$Fhdl$\")Fu86F07$F]el$\")Zh16F07$Fbel$\")\\%41\"F07$Fgel$\"( bN#)*F07$F\\fl$\"(w-w)F07$Fafl$\"(u+\\(F07$Fihl$\"(3rs&F07$Fcil$\"($f! >%F07$Fhil$\"(*zwJF07$F]jl$\"(Y:N#F07$$\"5LL$3FM3?dh\"F0$\"(w:t\"F07$F bjl$\")\\+-8F67$$\"5+]PfLLtbH;F0$\")`Kb6F67$$\"5LL3xJ$yCBj\"F0$\")%p)* 3\"F67$$\"5m;z%*HLA4N;F0$\")W`v5F67$$\"5++]7G$ofyj\"F0$\")nw^5F67$$\"5 mm\"zWKe%RV;F0$\"(X9r*F67$Fgjl$\"(y=r*F67$F\\[m$\")Z\\a6F67$Fa[m$\")i7 #R\"F67$F_\\o$\")Kt$\\\"F67$Ff[m$\")Dpl:F67$Ff]o$\")k\"3d\"F67$Fj^o$\" )-qu:F67$F^`o$\")AIF:F67$F[\\m$\")[Eo9F67$F`\\m$\")GcH6F67$Fe\\m$\"(23 N'F67$Fi]m$!(d;S'F67$F^^m$!)C_r5F67$Fc^m$!)F-J:F67$Fh^m$!)u/$p\"F67$F] _m$!)WX'z\"F67$Fg_m$!)Z'z*>F67$F\\`m$!)\\%>T#F67$$\"5mmmTN@BbG?F0$!)y* Qa#F67$Fido$!)O\\uEF67$$\"5nm;/,:y*[/#F0$!)L6=FF67$$\"5MLLe*GJY.0#F0$! )#>#oFF67$$\"5++]7y5[zb?F0$!)]'Q!GF67$Fa`m$!)Gu6GF67$$\"5ML3x\"Qa_q1#F 0$!)U\\\\GF67$$\"5++](o*y<'G2#F0$!).*\\!GF67$$\"5nm\"z>T,r'y?F0$!)]>!z #F67$Faeo$!):L=FF67$$\"5nm;Hd>()4'4#F0$!)O&=^#F67$Ff`m$!):0*3#F67$$\"5 +++vV[v$)=@F0$!)+/@8F67$F[am$\"(GT-\"F67$$\"5+++Dcl#y59#F0$\")u&oW\"F6 7$F`am$\")RoXJF67$$\"5++v$4hzt\"e@F0$\")T&=t$F67$Feam$\")#=uK&F67$$\"5 +]Pf3al8n@F0$\"))*=ScF67$$\"5++D\"y+9C,<#F0$\").i.yF67$$\"5+]7.2E<6t@F 0$\")*y)GzF67$Fjam$\")Qk`\")F67$$\"5+](oa!)*o3z@F0$\"):@6'*F67$Fefo$\" *KFv5\"F67$$\"5+]i!R+2i]=#F0$\"*_#HH6F67$F_bm$\"*vs%=7F67$$\"5+vVt-\\M a*=#F0$\"*3[dQ\"F67$$\"5+]PM-Us.\">#F0$\"*)*=J[\"F67$$\"5+DJ&>].JD>#F0 $\"*vod\\\"F67$F]go$\"*\"RQ4:F67$$\"5+v=<,@'=b>#F0$\"*!\\UG:F67$$\"5+] 7y+9C,(>#F0$\"*:,'o:F67$$\"5+D1R+2i])>#F0$\"*8]7n\"F67$Fdbm$\"*99p#>F6 -%&COLORG6&F\\cm$\"\"\"Ffbm$\"\")FfbmF(-Facm6#%3efficient~scheme~AG-F$ 6%7esF'7$F+$\"([M@#F07$F2$\"))Qk@$F67$Fbio$\")2;'H$F67$F8$\")[noKF67$F _jo$\")lm(=$F67$F=$\")b\"z4$F67$FB$\")41[FF67$FG$\");ohAF67$F_o$\"(([: LF67$Fdgm$!(&)H;\"F67$Figm$!(!eJiF67$F^hm$!(F67$F^z$!)-)R8#F67$$\"5+++D1+C#pJ\"F0$!)i-iBF67$Fc z$!(ldo#F07$Fhz$!(M%\\NF07$F][l$!(LR=%F07$Fi`n$!((o%G%F07$Fb[l$!(hFQ%F 07$Ffan$!(0)eVF07$Fg[l$!(>))Q%F07$F[]l$!(Pn%QF07$Fi^l$!(;>M$F07$Fc_l$! ((eY>F07$F]`l$!'vHhF07$Ficn$!'Rs_F07$F^dn$!'2!G#F07$Fcdn$!'K\"G#F07$Fb `l$!'>&G#F07$F[en$!'qhBF07$Fg`l$!'%e/#F07$Faal$!'S\\5F07$F[bl$!'P16F07 $F`bl$!''fr\"F07$Febl$!'PCIF07$Fcdl$!'#3R)F07$F]el$!(l'4I+(e\"F0$!)D:+6F07$$\"5n\"H#o2DOZ)e\"F0$!)\"f v6\"F07$$\"5M$3-)QIU%**e\"F0$!)1qy6F07$$\"5+v=#*pN[T\"f\"F0$!)5zv6F07$ Fhil$!)z#)o6F07$$\"5ML3_Diyw)f\"F0$!)Uam6F07$F]jl$!)#o)y6F07$Fbjl$!*l^ \"*3\"F67$Fgjl$!)fXh')F67$F\\[m$!)6l%*eF67$Fa[m$!)YNwQF67$F_\\o$!)\\3( 3$F67$Ff[m$!)G8_CF67$Ff]o$!)Vt'=#F67$Fj^o$!)d\"=,#F67$F^`o$!)hV>>F67$F [\\m$!)@Mf=F67$$\"5mmTN@itQYF67$$\"5LL$e9TC'*fx\"F 0$!)hN%>#F67$Fe\\m$!)%RF_#F67$F_]m$!)%[*)e$F67$Fi]m$!))=!=UF67$F^^m$!) \"[(>ZF67$Fc^m$!)TWF_F67$$\"5ML$eRF0$!)Ri'=&F67$F]_m$!)!R36&F67$Fb_m$!)?P4\\F67$Fg_m$! )aF^YF67$F\\`m$!)p'**>%F67$Fido$!)r6pRF67$Fa`m$!)\\rbOF67$Ffdq$!)\"F-$\"'$=Z)F07$F+ $\"((>o7F07$$\"5mmmm\"zp3r*HF-$\")iS/8F67$$\"5++++]P/`'f$F-$\")z-m7F67 $$\"5mmm;H28C'*QF-$\")'*)*H6F67$$\"5LLLL3x@&f>%F-$\"(NAL*F67$$\"5+++]( o/jc\\%F-$\"(<@=*F67$F2$\"(+>.)F67$F=$!(F$z%*F67$FG$!)'Q9M$F67$Fen$!)i T!H&F67$F_o$!)#Rce'F67$Fdgm$!)tN$=(F67$Figm$!)i$>(yF67$F^hm$!)ed8!)F67 $Fdo$!)c%)y!)F67$Fi\\p$!)C4&3)F67$F^]p$!)OW]!)F67$Fc]p$!)[b\"*zF67$Fio $!)Z\"G(yF67$Fihm$!)[@avF67$F^p$!)vxHsF67$Fcp$!)eM?mF67$F\\ar$!)E*)H5F67$F]v$!)Hb-%*F67$Fbv$!)Ou 1&)F67$Fdbp$!)(y*)=)F67$Fgv$!)_:!*zF67$F^]n$!)]v!*yF67$F\\w$!)sP/zF67$ Ff]n$!)nS=!)F67$Faw$!)5Q%*3\"F67$Fiy$!*S?b5\"F67$F^z$!*EF@@\"F67$Fcz$!)%fBQ\"F07$F hz$!)h@:;F07$F][l$!)I:-=F07$Fi`n$!)>BQ=F07$Fb[l$!)Z%H(=F07$Faan$!)kr\" )=F07$Ffan$!)<:z=F07$F`bn$!)RQ&*=F07$Fg[l$!)O`,>F07$F[]l$!)Z^9=F07$Fi^ l$!)%*QQ;F0$!*y;HZ#F67$$\"5++D1R$)\\D@;F0$!*\\S`Y#F67$$\"5L$eRsLVASi\"F0$ !*<2VV#F67$Fbjl$!*QwkS#F67$Fg^q$!*vAsK#F67$Fgjl$!*W4q?#F67$F\\[m$!*(3V 6>F67$Fa[m$!*#*oxn\"F67$F_\\o$!*g]]f\"F67$Ff[m$!*Ay$Q:F67$Ff]o$!*04;` \"F67$Fj^o$!*p%==:F67$F__o$!*Fxb`\"F67$Fd_o$!*9s[_\"F67$Fi_o$!*=gU^\"F 67$F^`o$!*fCR]\"F67$Fc`o$!*NW]\\\"F67$Fh`o$!*b1=\\\"F67$F]ao$!*'e:0:F6 7$F[\\m$!*lm4`\"F67$Fj`s$!*X=>_\"F67$F`\\m$!*@6,_\"F67$$\"5+v=#\\a8Lcw \"F0$!*JV1^\"F67$$\"5L$3FW&zN6ny\"F0$!* Zowc\"F67$$\"5+vV)*ek%)R$y\"F0$!*w'Hd:F67$$\"5L$e*[o3*y[y\"F0$!*:*QZ:F 67$$\"5m\"z%*zFNfjy\"F0$!*bH)R:F67$Fe\\m$!*uY5a\"F67$Fj\\m$!*ON,e\"F67 $F_]m$!*'e7;;F67$Fd]m$!*!fc8;F67$Fi]m$!*\"4o2;F67$$\"5nmm\"z4%=8X=F0$! *v&\\(f\"F67$F^^m$!*%fe'e\"F67$$\"5nm;zW8%HE'=F0$!*2Imf\"F67$$\"5nmmTg P>Yo=F0$!*]\"[z:F67$$\"5nm;/whWHu=F0$!*Tnrd\"F67$Fc^m$!*p\"*3e\"F67$Fh ^m$!*'4@O:F67$F]_m$!*71*)\\\"F67$Fgbo$!*w^m[\"F67$Fb_m$!*,#zx9F67$F_co $!*t0JZ\"F67$Fg_m$!*NQLZ\"F67$Fgco$!*Fi3[\"F67$F\\do$!**oZ#\\\"F67$Fad o$!*m'36:F67$F\\`m$!*(>!p`\"F67$Fido$!*<7Xf\"F67$Fa`m$!*a*G[;F67$Ffdq$ !*cONn\"F67$Faeo$!*o76p\"F67$$\"5+v$fe!eD$f3#F0$!*)>v(p\"F67$$\"5n;aj% o'[Q(3#F0$!*@j-q\"F67$$\"5Me9Tjvr$))3#F0$!*u&R!p\"F67$$\"5++v=U%[*G!4# F0$!*pKMm\"F67$$\"5M$eR(*>5%>$4#F0$!*k`*z;F67$Fceq$!*shWp\"F67$$\"5+]P %[rL.!*4#F0$!*JE[n\"F67$$\"5MLeRsaz!>5#F0$!*#=y[;F67$$\"5n;z%*HsD\"[5# F0$!*%o(fm\"F67$Ff`m$!*SVom\"F67$F[am$!*k?!*Q\"F67$F`am$!)o;N)*F67$F[g q$!)B,I#*F67$Feam$!)\"e&=nF67$Fcgq$!)on-kF67$Fhgq$!)009DF67$F]hq$!)m:_ DF67$Fjam$!)\"GvT#F67$Fehq$\"(9f8\"F67$Fefo$\")j'[r#F67$$\"5+voH/x#oN= #F0$\")'=Fu#F67$F]iq$\")t]>GF67$$\"5+Dc^.jeb'=#F0$\")vIJJF67$F_bm$\")V _KUF67$Feiq$\")r&\\W(F67$Fjiq$\")*)3n#*F67$F_jq$\")O_Y$*F67$F]go$\")xo S%*F67$Fgjq$\")i>M'*F67$F\\[r$\"*mgW-\"F67$Fa[r$\"*pWX@\"F67$Fdbm$\"*b o8t\"F6-Fjbm6&F\\cm$\")#)eqkF_cm$\"))eqk\"F_cmF]bv-Facm6#%3efficient~s cheme~CG-F$6%7isF'7$F+$!(o3M(F07$F2$!*^;(=8F67$F=$!*wr[q\"F67$FG$!*y4B &>F67$Fddm$!*Qsc&>F67$FL$!*x_V$>F67$$\"5gmT&Q`:!)4q*F-$!*m>M$>F67$F\\e m$!*3lU&>F67$$\"5&**\\iS\"p4D%***F-$!*EAG-#F67$FQ$!*]>\\,#F67$Fdem$!*< z2*>F67$FV$!*F67$F\\fm$!*kBK+#F67$Fafm$!*[4e0#F67$Fffm$!*pnM/#F67 $Fen$!*=Q8.#F67$Fjn$!*=#3N?F67$F_o$!*'zYK?F67$Fdgm$!**ejL?F67$Figm$!*< zy/#F67$F^hm$!*%)\\'>?F67$Fdo$!*(H)>*>F67$Fio$!*(ziG>F67$F^p$!*&)ep)=F 67$Fcp$!**\\i))=F67$Fhp$!*!oo:>F67$F]q$!*mEq'>F67$Fbq$!*A8^0#F67$Fgq$! */Av<#F67$F\\r$!*]?aN#F67$Ffr$!*Vk,c#F67$F`s$!*yhav#F67$Fes$!*+OA$GF67 $Fjs$!*nBn*GF67$$\"5NLLe9JPhypF6$!*!4UFHF67$Fa[n$!*CI'fHF67$$\"5lmmTgU _^EsF6$!*1E=(HF67$F_t$!*gOh(HF67$Fdt$!*!=nmHF67$Fit$!*NE2$HF67$F^u$!*= 9Y&GF67$Fcu$!*Z\\ov#F67$F]v$!**p(y`#F67$Fbv$!*jd(4BF67$Fdbp$!*Zy\\@#F6 7$Fgv$!*wpH9#F67$F^]n$!*S*[&4#F67$F\\w$!*bZE2#F67$Ff]n$!*O&ey?F67$Faw$ !*5nW5#F67$F^^n$!*42S:#F67$Ffw$!*x4AB#F67$F[x$!*^\")*fCF67$Fex$!*(H!Ru #F67$F_y$!*9Qv:$F67$Fiy$!*+\"4qPF67$Fcz$!)T>VXF07$F][l$!)FT/`F07$Fg[l$ !)X#Gj&F07$Fi^l$!)MtBeF07$Fc_l$!):1XfF07$F]`l$!)x!f,'F07$Fg`l$!)!*4TgF 07$Febl$!)E#e0'F07$Ficl$!)RmPgF07$Fcdl$!)*[V.'F07$Fhdl$!)!H&4gF07$F]el $!)2;yfF07$Fgel$!)!)\\*y&F07$Fafl$!)>$o^&F07$Fcil$!)E8o_F07$F]jl$!)(*p h]F07$F`[u$!)t4Q]F07$Fe[u$!)QDH]F07$Fj[u$!)Fhb]F07$F_\\u$!)ATJ^F07$Fd \\u$!)p9p]F07$F`]q$!)ia8]F07$F\\]u$!)/w,]F07$Fa]u$!)D)R-&F07$Ff]u$!*d* f=^F67$F[^u$!**p%Q5&F67$F`^u$!*@z-/&F67$Fbjl$!*-(G(*\\F67$$\"5L3_]M3O< G;F0$!*z$H8]F67$Fh]q$!*=Vy4&F67$$\"5m\"H#oKe5%4j\"F0$!*].j7&F67$F]^q$! *2aI4&F67$$\"5+v$f3$3&3Pj\"F0$!*`8/1&F67$Fb^q$!*/(yI]F67$$\"5Lek.HefZO ;F0$!*#Rd6]F67$Fg^q$!*7p,-&F67$$\"5mTN@F3MCR;F0$!*[I74&F67$$\"5L$3-jK8 F1k\"F0$!*pe;;&F67$$\"5+D1RDe3,U;F0$!*-Au7&F67$F\\_q$!*hUO4&F67$$\"5L3 xcB3$yZk\"F0$!*ru@1&F67$$\"5+]ilAL?;Y;F0$!*b9#R]F67$$\"5m\"zW<#edaZ;F0 $!*.=+/&F67$Fgjl$!*Tne4&F67$$\"5mmTg_ueP];F0$!*U&3-_F67$$\"5++]P%eEA=l \"F0$!*s'Ql^F67$$\"5LLe9;d'oKl\"F0$!*1Q\"H^F67$$\"5mmm\"z%[]ra;F0$!**[ @&4&F67$$\"5++vozR9;c;F0$!*mF/2&F67$$\"5LL$e96$ygd;F0$!*uMB2&F67$$\"5m m\"HKCAa!f;F0$!*gxz8&F67$$\"5++++v81]g;F0$!*m7MB&F67$$\"5mm;aQ'R$Rj;F0 $!*eF\"f^F67$$\"5LLL3-zhGm;F0$!*#Q%45&F67$$\"5++]ilh*y\"p;F0$!*zp!z^F6 7$F\\[m$!*.b>A&F67$$\"5LLLL$[(Gk$o\"F0$!*&4o,_F67$Fa[m$!*5fL<&F67$F_\\ o$!*Ziu9&F67$Ff[m$!*0Y!G^F67$Fj^o$!*!3'H8&F67$F[\\m$!*]!pD^F67$F`\\m$! *NaQ)[F67$Fe\\m$!*'*Q![YF67$Fi]m$!*N1AA%F67$Fc^m$!*Yx<'QF67$Fh^m$!*%)p qs$F67$F]_m$!*(y_ZOF67$Fgbo$!**>5LOF67$Fb_m$!*O]uTF67$Fa` m$!*\\[6Y%F67$Faeo$!*h--$[F67$Ff`m$!*$Rl%H&F67$F[am$!*cS)QeF67$F`am$!* z&z!\\'F67$Feam$!*Md:!pF67$Fjam$!*(G*fO(F67$F_bm$!*w@*)*yF67$Fdbm$!*&)F6-Fi[r6&F\\cm$\"\"'FfbmF(F]\\r-Facm6#%Sscheme~with~stability~radi us~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror ~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6 \"Q!Faax-%%VIEWG6$;F(Fdbm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modificatio n of the Bogacki-Shampine scheme" "efficient scheme A" "efficient sche me B" "efficient scheme C" "scheme with stability radius = imag axis i nclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 502 "evalf[20](plot(['pn_RK5_1'(x)-p(x),'pn_RK5_2'(x)-p (x),'pn_RK5_3'(x)-p(x),'pn_RK5_4'(x)-p(x),\n'pn_RK5_5'(x)-p(x),'pn_RK5 _6'(x)-p(x)],x=2.2..2.8,\ncolor=[blue,magenta,COLOR(RGB,.1,.8,0),red,b rown,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9],\nlegend=[`Bogacki-Shampin e scheme`,`a minor modification of the Bogacki-Shampine scheme`,`effic ient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with \+ stability radius = imag axis inclusion`],\ntitle=`error curves for 7 s tage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 835 542 542 {PLOTDATA 2 "6,-%'CURVESG6%7br7$$\"#A!\"\"$\"*T=7T%!#?7$$ \"5++++]#HyI@#!#>$\"*x?L@&F-7$$\"5+++]([kdWA#F1$\"*'[mJhF-7$$\"5++++v; \\DPAF1$\"*\"[B$R(F-7$$\"5++++Dq0]BF1$\"*BM3^&F17$$\"5++++]U80jBF1$\"*@\"epnF17$$\"5++++]hDJpBF1$ \"*7\"G)e(F17$$\"5++++]!ytbP#F1$\"*Kf4F)F17$$\"5+++v=n&f7Q#F1$\"*pgPZ* F17$$\"5+++](QNXpQ#F1$\"*\\]$y**F17$$\"5+++v=/jq$R#F1$\"+=U;K6F17$$\"5 ++++]asY+CF1$\"+c%zlF\"F17$$\"5++++DmY>1CF1$\"+TwdK8F17$$\"5+++++y?#>T #F1$\"+A6wb9F17$$\"5+++vV>We=CF1$\"+*opw`\"F17$$\"5+++](3wY_U#F1$\"+$p @wi\"F17$$\"5+++v$>1W6V#F1$\"+XEsx;F17$$\"5+++++j8/PCF1$\"+v)>Ls\"F17$ $\"5+](=Us9]yV#F1$\"+m**4B)*\\CF1$\"+@iE7;F17$$\"5++]Pf=A_^CF1$\"+\\Ww>;F17$$\"5+++DJED1`C F1$\"+&G#[G;F17$$\"5++]7.MGgaCF1$\"+>\")QN;F17$$\"5++++vTJ9cCF1$\"+q^% [j\"F17$$\"5+++v=dPAfCF1$\"+3u%y_\"F17$$\"5+++]isVIiCF1$\"+fd2_9F17$$ \"5+++voqHtoCF1$\"+K(3AN\"F17$$\"5++++vo:;vCF1$\"+2*\\:?\"F17$$\"5++DJ qLujwCF1$\"+$H4N=\"F17$$\"5++]il)H8\"yCF1$\"+u$3a7\"F17$$\"5++v$4O;*ez CF1$\"*Tp+z*F17$$\"5+++DcG]1\"[#F1$\"*B,p7*F17$$\"5++Dc^$*3a#[#F1$\"*> Ry7*F17$$\"5++](o%en,%[#F1$\"*Aob6*F17$$\"5+]7`%4paZ[#F1$\"*yLF4*F17$$ \"5++v=UBE\\&[#F1$\"*Y!QW!*F17$$\"5+]P%)*ebIi[#F1$\"*M%f^*)F17$$\"5+++ ]P)[op[#F1$\"*Iuky)F17$$\"5+](=U#[Ww([#F1$\"*D&*=[)F17$$\"5++v$4\"3/c) [#F1$\"*P;,)zF17$$\"5+]il(zOc$*[#F1$\"*E,;>(F17$$\"5++]P%yK_,\\#F1$\"* k$GliF17$$\"5++D\"yvCW<\\#F1$\"*E!QbiF17$$\"5+++DJnhL$\\#F1$\"*53+C'F1 7$$\"5+](ozr7KT\\#F1$\"*QoQA'F17$$\"5++vo/(3G\\\\#F1$\"*`nB>'F17$$\"5+ ]iS\"p/Cd\\#F1$\"*SO78'F17$$\"5++]7y1+_'\\#F1$\"*kd\"=gF17$$\"5++Dc^E> 6)\\#F1$\"*x]L\\&F17$$\"5++++DYQq*\\#F1$\"*%Gh%>%F17$$\"5+++D1v!=j]#F1 $\"*UFQj$F17$$\"5+++](QIKH^#F1$\"*!Q(z+#F17$$\"5++++]4+p=DF1$\"*f\"=$ \\\"F17$$\"5+++]7:xWCDF1$\")._l$*F17$$\"5+]i:5D]ADDF1$\")KUY#*F17$$\"5 ++D\"y]L-g_#F1$\")A1\\!*F17$$\"5+](oa]kzn_#F1$\")29L()F17$$\"5++]7.bpb FDF1$\")'3?D)F17$$\"5++vV)\\d6\"HDF1$\")s82mF17$$\"5+++v$\\>m1`#F1$\") :be^F17$$\"5++]P%[VvP`#F1$\")n%)o]F17$$\"5++++vuY)o`#F1$\")*HP)[F17$$ \"5++]7G\\2\\QDF1$\")f=D[F17$$\"5+++D\"Q#o4SDF1$\")'zr=&F17$$\"5++]PM) *GqTDF1$\")S2R^F17$$\"5+++](G(*3La#F1$\")9_#4&F17$$\"5++]iSZ]\"\\a#F1$ \")]To]F17$$\"5+++v$>7@la#F1$\")X-l^F17$$\"5++DJ?fTKZDF1$\")W%)Q`F17$$ \"5++](ok>F\"[DF1$\")pNzcF17$$\"5++vVtL-$*[DF1$\")*[mG'F17$$\"5+++++rK t\\DF1$\")x_,tF17$$\"5++++v:JIiDF1$\"*$p==6F17$$\"5+++](o3lWd#F1$\"*@? WX\"F17$$\"5++++D#))oze#F1$\"*DY!4=F17$$\"5+++]iid.%f#F1$\"*NxB)>F17$$ \"5+++++VE5+EF1$\"*Au**>#F17$$\"5+++](y[TLg#F1$\"*7.t9#F17$$\"5++++vK. e1EF1$\"*A(y@@F17$$\"5++](oR/!R2EF1$\"*^LJ8#F17$$\"5+++v=b(*>3EF1$\"*! \\3f@F17$$\"5++]iSm%4!4EF1$\"*n_[?#F17$$\"5+++]ix\">)4EF1$\"*:DiF#F17$ $\"5+++D1+'Q9h#F1$\"*!4RrAF17$$\"5++++]A!eIh#F1$\"*4$*RC#F17$$\"5+++]( =_(zCEF1$\"*E[wD#F17$$\"5++++]&*=jPEF1$\"*;6AC#F17$$\"5+++](3/3(\\EF1$ \"*FtfC#F17$$\"5+++]P#4JBm#F1$\"*;Eb:#F17$$\"5++++]KCnuEF1$\"*s&>]?F17 $$\"5+++](=n#f(o#F1$\"+lkyY>F-7$$\"5+++++)RO+q#F1$\"+g$*zr=F-7$$\"5+++ +D0>w7FF1$\"+HZPe$[x#F1$\"+Wqs58F-7$$\"5+++]7hK'py#F1$\"+tu\"yB\"F-7$$ \"#GF*$\"+SOhn6F--%'COLOURG6&%$RGBG$\"\"!Fg`mFf`m$\"*++++\"!\")-%'LEGE NDG6#%8Bogacki-Shampine~schemeG-F$6%7gp7$F($\"*%zi3gF-7$F/$\"*?3N&pF-7 $F5$\"*jk]-)F-7$F:$\"*zS&)\\*F-7$F?$\"+pS0.7F-7$FD$\"+z&3#G9F-7$FI$\"+ ^%>Kp\"F-7$FN$\"+Nb#)R?F-7$FS$\"*X\")Gj#F17$FX$\"*V7*)=$F17$Fgn$\"*Wo^ 'QF17$$\"5+++]P&4o8L#F1$\"*1V/T%F17$F\\o$\"*p+&fYF17$$\"5+++](oU'eVBF1 $\"*xIkH&F17$Fao$\"*Gcn1'F17$$\"5+++++\"=alN#F1$\"*=:vT'F17$Ffo$\"*69b F(F17$F[p$\"*Lq%3!)F17$F`p$\"*D#*)R')F17$Fep$\"*#>F-'*F17$Fjp$\"+-5W25 F17$F_q$\"+)eM(36F17$Fdq$\"+J;S07F17$Fiq$\"+L9lb7F17$F^r$\"+8Q(\\K\"F1 7$Fhr$\"+!HPDV\"F17$$\"5++D19'3@nU#F1$\"+(eZFW\"F17$$\"5++]iS6a>GCF1$ \"+#>myW\"F17$$\"5+]i!RSdK*GCF1$\"+&4,kW\"F17$$\"5++v=nO(p'HCF1$\"+wpM S9F17$$\"5+](o/$**oSICF1$\"+nU#)Q9F17$F]s$\"+)RDYW\"F17$$\"5++](oCr#4M CF1$\"+U4Fn9F17$Fbs$\"+P2Z$[\"F17$Fgs$\"+cE2$[\"F17$F\\t$\"+s')px9F17$ Fat$\"+P5_k9F17$Fft$\"+a%*f^9F17$F[u$\"+14[i9F17$F`u$\"+m],t9F17$Fju$ \"+8P[#[\"F17$Fdv$\"+tQ\"z[\"F17$$\"5+]P%[\"ebbZCF1$\"+#Q&z'[\"F17$Fiv $\"+5kv![\"F17$$\"5+]7GjEJ<\\CF1$\"+=W%pY\"F17$F^w$\"+*f98W\"F17$Fhw$ \"+Cnec9F17$Fbx$\"+o,\")o9F17$Fgx$\"++0_Z9F17$F\\y$\"+vRBL9F17$Ffy$\"+ lV\\F9F17$Fjz$\"+:\\3H9F17$Fh\\l$\"+)z>,V\"F17$Ff^l$\"+!*[yW9F17$Fd`l$ \"+4G\"4Y\"F17$$\"5++DJX.uN,DF1$\"+hQ1j9F17$$\"5++]ilg4,.DF1$\"+oK]d9F 17$$\"5++v$fy^kY]#F1$\"+y&\\;X\"F17$Fi`l$\"+41SY9F17$$\"5+]iSm`[92DF1$ \"+IP\"\\W\"F17$$\"5++DcEK;(z]#F1$\"+9'*GX9F17$$\"5+](=n3T)z3DF1$\"+GC .\\9F17$$\"5++](o%*=D'4DF1$\"+[#p%e9F17$$\"5+]7.2o>X5DF1$\"+N\\Rj9F17$ $\"5++v=nY(y7^#F1$\"+ZPYf9F17$$\"5+]PMFDb57DF1$\"+6(QaX\"F17$F^al$\"+l (>8X\"F17$Fcal$\"+4$e!G9F17$Fhal$\"+'R%))39F17$F`dl$\"+xF-7$F/$\"*)pM3DF -7$F5$\"*bR,=$F-7$F:$\"*B'\\ySF-7$F?$\"*.d[n&F-7$FD$\"*,3.&pF-7$FI$\"* ])\\6%)F-7$FN$\"+(pdV-\"F-7$FS$\"*ei5K\"F17$FX$\"*\\2:e\"F17$Fgn$\"*lZ 2)=F17$F\\o$\"*%z!f?#F17$Fao$\"*Fsaq#F17$Ffo$\"*N1R7$F17$F`p$\"*Nd?c$F 17$Fjp$\"*+Zg)RF17$Fdq$\"*s'R?XF17$F^r$\"*MTr$\\F17$Fhr$\"*x_fV&F17$F] s$\"*9u(RdF17$Fbs$\"*K#>LfF17$F`u$\"*[sEI'F17$F^w$\"*%>xXnF17$Fbx$\"*` \"o/pF17$F\\y$\"*+poL(F17$Ffy$\"*RF!)z(F17$F`z$\"*s_9'yF17$Fjz$\"*4`m, )F17$Fd[l$\"*$3zGt F17$Fhal$\"*H$3FnF17$Ffcl$\"*&fieiF17$F`dl$\"*Q)pOgF17$Fdel$\"*S+>h&F1 7$Fbgl$\"*8\"Re_F17$$\"5+++]P$>=gb#F1$\"*wx#>]F17$Fggl$\"*ntpr%F17$F\\ hl$\"*<\"G%G%F17$Fahl$\"*3<<(QF17$F[il$\"*sV4b$F17$Fc[m$\"*WE1B$F17$Fh [m$\"*_kL'HF17$F]\\m$\"*NeVp#F17$Fb\\m$\"*)H2`CF17$Fg\\m$\"*OuZA#F17$F \\]m$\"*)e\\D?F17$Fa]m$\"+)Ghv$=F-7$Ff]m$\"+)Qn+m\"F-7$F[^m$\"+hGW7:F- 7$F`^m$\"+IWn\"Q\"F-7$Fe^m$\"+%**R@F\"F-7$Fj^m$\"+-4O\\6F-7$F__m$\"+9H ih5F-7$Fd_m$\"*jH.z*F-7$Fi_m$\"*Wgw2*F-7$F^`m$\"*$4\\E$)F--%&COLORG6&F e`m$\"\"\"F*$\"\")F*Ff`m-F\\am6#%3efficient~scheme~AG-F$6%7[o7$F($\"*/ F17$FX$ \"*>[zJ#F17$Fgn$\"*$>6ZFF17$F\\o$\"*H!R,KF17$Fao$\"*5***fQF17$Ffo$\"*% 3m4WF17$F`p$\"*&*fs(\\F17$Fjp$\"**R>HbF17$Fdq$\"*%4K'H'F17$F^r$\"*v)R' )pF17$Fhr$\"*THZ)yF17$F]s$\"**p0=\"F17$F^\\l$\"+p1pz6F17$Fh\\l$\"+Q'os<\"F17$Ff^l$\" +[`4]6F17$Fd`l$\"+_@[06F17$F^al$\"+Mk!z+\"F17$Fcal$\"***>Ck*F17$Fhal$ \"*)4nb!*F17$Ffcl$\"*GmY@)F17$F`dl$\"*^)R4zF17$Fdel$\"*#>$f:(F17$Fbgl$ \"*X\"H`lF17$F]^o$\"*0M%>iF17$Fggl$\"*\\8,s&F17$F\\hl$\"**\\DN^F17$Fah l$\"*lm]k%F17$F[il$\"*\")QbL%F17$Fc[m$\"*\\Bu*RF17$Fh[m$\"*#)>Cr$F17$F ]\\m$\"*#z*HT$F17$Fb\\m$\"*(obOJF17$Fg\\m$\"*Q%*4&GF17$F\\]m$\"*y$e$f# F17$Fa]m$\"+x[gXBF-7$Ff]m$\"+2A#35#F-7$F[^m$\"+NM*H!>F-7$F`^m$\"+*=Qws \"F-7$Fe^m$\"+tN(*z:F-7$Fj^m$\"+_vT59F-7$F__m$\"+&R*3&H\"F-7$Fd_m$\"+4 >q(=\"F-7$Fi_m$\"+DjD&4\"F-7$F^`m$\"*W7E'**F--Fc`m6&Fe`mFh`mFf`mFf`m-F \\am6#%3efficient~scheme~BG-F$6%7]o7$F($\"*bo8t\"F-7$F/$\"*aqlx#F-7$F5 $\"*2Ws/%F-7$F:$\"*M%)**y&F-7$F?$\"*ITP;*F-7$FD$\"+$*=dx6F-7$FI$\"+l*) Q![\"F-7$FN$\"+.f$)e=F-7$FS$\"*!\\&fZ#F17$FX$\"*--g)HF17$Fgn$\"*SkZa$F 17$F\\o$\"*-nT6%F17$Fao$\"*I\"fo[F17$Ffo$\"*f$praF17$F`p$\"*%[YdgF17$F jp$\"*0&e'f'F17$Fdq$\"*H-[P(F17$F^r$\"*R'*>D)F17$Fhr$\"*f<5`*F17$F]s$ \"+h%\\y1\"F17$Fbs$\"+H@756F17$F`u$\"+6(G9C\"F17$F^w$\"+Q,i\"Q\"F17$Fb x$\"+Pu%eT\"F17$F\\y$\"+Ip*y_\"F17$Fay$\"++uAr:F17$Ffy$\"+>U?>;F17$F`z $\"+\"3R$G;F17$Fjz$\"+l5LY;F17$Fd[l$\"+^A_Y;F17$Fh\\l$\"+%3B[k\"F17$F \\^l$\"+6n/N;F17$Ff^l$\"+]kTH;F17$F`_l$\"+NarD;F17$Fj_l$\"+NUb?;F17$F_ `l$\"+L&z3h\"F17$Fd`l$\"+\")oD)e\"F17$F^al$\"+D(f%p9F17$Fcal$\"+pY519F 17$Fhal$\"+\\HC=8F17$Ffcl$\"+GaTr6F17$F`dl$\"+]I/D6F17$Fdel$\"*Tw*)z*F 17$Fbgl$\"*/&y$e)F17$F]^o$\"*ol/0)F17$Fggl$\"*D?z*pF17$F\\hl$\"*\\]u(f F17$Fahl$\"*bB(*>&F17$F[il$\"*P+Us%F17$Fc[m$\"*'Hz;VF17$Fh[m$\"*xnI'RF 17$F]\\m$\"*!eRpNF17$Fb\\m$\"*ZF!RJF17$Fg\\m$\"*dKRt#F17$F\\]m$\"*3f)z BF17$Fa]m$\"+I(>>.#F-7$Ff]m$\"+[wgN;F-7$F[^m$\"+Y(4-R\"F-7$F`^m$\"+S*> &z6F-7$Fe^m$\"+W\"Q?+\"F-7$Fj^m$\"*3Y:!yF-7$F__m$\"*nS1n'F-7$Fd_m$\"*1 +/q&F-7$Fi_m$\"*Lgz)[F-7$F^`m$\"*0SO$RF--Fc`m6&Fe`m$\")#)eqkFj`m$\"))e qk\"Fj`mFf[q-F\\am6#%3efficient~scheme~CG-F$6%7]r7$F($!*&)F-7$F/$! *9l\\F*F-7$F5$!+mcB/5F-7$F:$!+F17$FX$!**RxAAF17$Fgn$!*g%eEDF 17$F\\o$!*A<&[GF17$Fao$!*jcoI$F17$Ffo$!*'QN\\PF17$F`p$!*wEuA%F17$Fjp$! *h&32ZF17$Fdq$!*8GBK&F17$F^r$!*E`rz&F17$Fhr$!**>h9jF17$Fbs$!*[Tjr'F17$ F^w$!*u*yXqF17$F\\y$!*.4+J(F17$Fay$!*JU0U(F17$Ffy$!*9rY\\(F17$F`z$!*77 s_(F17$Fjz$!*A$**ovF17$Fd[l$!*l'>qvF17$Fh\\l$!*3$3rvF17$Fb]l$!*YD4e(F1 7$F\\^l$!*<`2g(F17$Fa^l$!*iN))e(F17$Ff^l$!*u@Zd(F17$Fj_l$!*=)zVvF17$Fd `l$!*nYR_(F17$Fi`l$!*+y+T(F17$F^al$!*b+WD(F17$Fhal$!*VVc!pF17$F`dl$!*& GF*['F17$Fbgl$!*V)G'='F17$Fggl$!*>mM%fF17$F\\hl$!*\\g$\\dF17$$\"5+]Pf3 u!4`d#F1$!*w_&=dF17$$\"5++voHhI:wDF1$!*\\TLp&F17$$\"5+]7y][q*pd#F1$!*3 w&ycF17$$\"5++](=d.Tyd#F1$!*`c>o&F17$$\"5+](oHH-&oyDF1$!*z&3:dF17$$\"5 ++D195!H&zDF1$!*VQZz&F17$$\"5+v$4YP+^*zDF1$!*_H!feF17$$\"5+]i:N(*HP!e# F1$!*_F<&eF17$$\"5+DJq&4*\\z!e#F1$!*q+O$eF17$$\"5+++Dc%)p@\"e#F1$!*<7b \"eF17$$\"5++vV)*e\\!He#F1$!*3-Pu&F17$$\"5++]iSLHf%e#F1$!*Qh\\n&F17$$ \"5+](=<1#pV&e#F1$!*.xYk&F17$$\"5++D\"Gy!4G'e#F1$!*k95i&F17$$\"5+]i!R] *[7(e#F1$!*)zZ4cF17$Fahl$!*D'o=cF17$$\"5++]PM-c[*e#F1$!*i(oJdF17$$\"5+ ++vVAB+\"f#F1$!*Hw;x&F17$$\"5++]7`U!>Df#F1$!*[Skq&F17$Ffhl$!*iUHk&F17$ $\"5++](=F[_bf#F1$!*N8ne&F17$$\"5+++D\"G?pqf#F1$!*Tq_b&F17$$\"5++v$fGc Fyf#F1$!*Cm0c&F17$$\"5++]i!H#fe)f#F1$!*3n1f&F17$$\"5++DJ&HGW$*f#F1$!*[ (odcF17$F[il$!*(4O`dF17$F`il$!*\"[\\9cF17$Feil$!*HoJ]&F17$Fjil$!**3P' \\&F17$F_jl$!*A41^&F17$Fdjl$!*:$fdbF17$Fijl$!*$**f`cF17$F^[m$!*pSAi&F1 7$Fc[m$!*0ARb&F17$Fh[m$!*.-@R&F17$$\"5+]ilAq'*fDEF1$!*R@WO&F17$$\"5++D \"y&==SEEF1$!*(*o@M&F17$$\"5+](oHp'R?FEF1$!*K#HH`F17$$\"5++]7G:h+GEF1$ !*qOT5'HEF1$!*8V-U&F17$$ \"5+Dc,;'[6+j#F1$!*zJvY&F17$$\"5+]PfLgDTIEF1$!*yW6X&F17$$\"5+v=<^MO\"3 j#F1$!*f8[V&F17$$\"5+++vo3Z@JEF1$!*lQ&=aF17$$\"5++]P4-LUMEF1$!*26>H&F1 7$F]\\m$!*>>@@&F17$$\"5+++v=o*pOk#F1$!**H[&=&F17$Fb\\m$!*7tG9&F17$$\"5 ++++v.Q'Gl#F1$!*`ZH1&F17$$\"5+++]im&>gl#F1$!*x!ec\\F17$$\"5++]PM2&3ol# F1$!*[Ex$\\F17$$\"5+++D1[ufdEF1$!*Un@KF-7$Fd_m$!+J!o(R IF-7$Fi_m$!+'3sL)GF-7$F^`m$!+gX`WFF--F^bo6&Fe`m$\"\"'F*Ff`mFbbo-F\\am6 #%Sscheme~with~stability~radius~=~imag~axis~inclusionG-%%FONTG6$%*HELV ETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~7~stage~order~5~Runge-Kutta~m ethodsG-%+AXESLABELSG6$Q\"x6\"Q!Fahr-%%VIEWG6$;F(F^`m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine \+ scheme" "a minor modification of the Bogacki-Shampine scheme" "efficie nt scheme A" "efficient scheme B" "efficient scheme C" "scheme with st ability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "evalf[20](plot(['pn_RK5 _1'(x)-p(x),'pn_RK5_2'(x)-p(x),'pn_RK5_3'(x)-p(x),'pn_RK5_4'(x)-p(x), \n'pn_RK5_5'(x)-p(x),'pn_RK5_6'(x)-p(x)],x=2.8..5,\ncolor=[blue,magent a,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9], \nlegend=[`Bogacki-Shampine scheme`,`a minor modification of the Bogac ki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficie nt scheme C`,`scheme with stability radius = imag axis inclusion`],\nt itle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 813 529 529 {PLOTDATA 2 "6,-%'CURVESG6%7ho7 $$\"#G!\"\"$\"+SOhn6!#?7$$\"5nmmm;RP&z%G!#>$\"*9\"eb&*F-7$$\"5LLL$37.y '*)GF1$\"*c\">8%)F-7$$\"5+++DJ'pRJ\"HF1$\"*cw`,)F-7$$\"5nmmmTh8gOHF1$ \"*/wvw(F-7$$\"5nmmmmX&=-'HF1$\"*)p3'o(F-7$$\"5nmmm\"*Hd$Q)HF1$\"*_%[! p(F-7$$\"5LLL$3JF1$\"**>R\\**F-7$$\"5nmm;a25BmJF1$\"+bAI$=\"F-7$$\"5+++] P$o`F@$F1$\"+=x+k9F-7$$\"5LLLL$Q(zggKF1$\"+(ptA&=F-7$$\"5nmmm\"*e!eFI$ F1$\"+K=.SAF-7$$\"5++++]r!4-N$F1$\"+'*oRUEF-7$$\"5+++++(*>.uLF1$\"+#)H ;\"z#F-7$$\"5++++]A\\&yR$F1$\"+krD#)GF-7$$\"5++++DtVf.MF1$\"*ftr)GF17$ $\"5+++++CQL4MF1$\"*:R7*GF17$$\"5++++vuK2:MF1$\"*\"Rn.HF17$$\"5++++]DF \"3U$F1$\"*gIw)GF17$$\"5+++++F;HKMF1$\"*0l\\)GF17$$\"5++++]G0xVMF1$\"* ^m9(GF17$$\"5MLL3-8%=YY$F1$\"*!GOw/xNF1$\"+o.^xBF-7$$\"5+++]()* y/fi$F1$\"+'*yEt@F-7$$\"5LLLLLk;:pOF1$\"+#R*p$*>F-7$$\"5+++]()R.g;PF1$ \"+>,!z\"=F-7$$\"5+++]i*p#yhPF1$\"+r(e$*o\"F-7$$\"5LLLL3_d#*3QF1$\"+Qq K1;F-7$$\"5LLLeRr8LUF-7$$\"5+++]PU+S0TF1$\"+aY `U?F-7$$\"5++++vdZWGTF1$\"+uPdt?F-7$$\"5+++D\"[qT2:%F1$\"+gwqx?F-7$$\" 5+++](=lQI<%F1$\"+D?Ji?F-7$$\"5+++D15cz(>%F1$\"+&\\d$=?F-7$$\"5++++DoD bAUF1$\"+9xdT>F-7$$\"5LLLLLCI/nUF1$\"+G3Ry[z)F-7$$\"5MLL3x%H`1h%F1$\"*g$\\:()F-7$$\"5nmmmm#zmMj%F1$\" *sWTu)F-7$$\"5nmm;H*)ozcYF1$\"*O/R*))F-7$$\"5nmmm\"f)p7!o%F1$\"*H(['=* F-7$$\"5LLL$3#43SEZF1$\"+;>f:5F-7$$\"5+++++Z;#*oZF1$\"+0=wn6F-7$$\"5LL LLeD`l<[F1$\"+*y'e_9F-7$$\"5+++]7<$\\%R[F1$\"+ToEY;F-7$$\"5nmmmm3LCh[F 1$\"+=.\\&*=F-7$$\"5MLL3F\\-[%)[F1$\"+K6WHAF-7$$\"5+++]()*=_\\F1$\"*%4+eTF17$$\"5+ +]P4o*[T'\\F1$\"*L,Ap%F17$$\"5+++D17$*4w\\F1$\"*8i]K&F17$$\"5++vo/%[u? )\\F1$\"*v4I!eF17$$\"5++]7.c'\\!))\\F1$\"*/UX6'F17$$\"5++Dc,G[-%*\\F1$ \"*bR@g'F17$$\"\"&\"\"!$\"*]4QA(F1-%'COLOURG6&%$RGBG$F^clF^clFecl$\"*+ +++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7^p7$F($\"+M:F-7$FD$\"+ !)GD:>F-7$FI$\"+&40)**=F-7$$\"5+++D\"emSt+$F1$\"+5:Y8>F-7$FN$\"+`5&e&> F-7$$\"5+++]7'oPE0$F1$\"+8x^>?F-7$FS$\"+h*G\"3@F-7$FX$\"+G,DyBF-7$Fgn$ \"+1!pUz#F-7$F\\o$\"+Y#p)pLF-7$Fao$\"+z\"=+7%F-7$Ffo$\"+^FYi[F-7$F[p$ \"+zc4$o&F-7$Fep$\"+;KmdjF-7$Fiq$\"*R6ie'F17$Fcr$\"*7+>u'F17$$\"5nm;/w qW>aMF1$\"*&e6%y'F17$Fhr$\"*UEw!oF17$$\"5++]7GbB/vMF1$\"*7Q?\"oF17$F]s $\"*ZKtz'F17$$\"5+++vo[kD5NF1$\"*.z(*o'F17$Fbs$\"*Hq/\\'F17$Fgs$\"+FPH @gF-7$F\\t$\"+\"397T&F-7$Fat$\"+lu?;\\F-7$Fft$\"+R/;wWF-7$F[u$\"+@0IxT F-7$$\"5mmmT&eAa`y$F1$\"+()*p%pSF-7$F`u$\"+o=e%*RF-7$Feu$\"+YV:aRF-7$F ju$\"+RluRRF-7$F_v$\"+B%eC&RF-7$Fdv$\"+mrG$*RF-7$$\"5nmm\"H_FmJ#RF1$\" +#f4M1%F-7$Fiv$\"+&f$GfTF-7$F^w$\"+&*G.xVF-7$Fcw$\"+1qwdYF-7$Fhw$\"+qS ZU\\F-7$F]x$\"+#Gb*f]F-7$Fbx$\"+v:)*\\^F-7$Fgx$\"+)*po._F-7$F\\y$\"+!y '4?_F-7$$\"5++](o48kG)*F17$Fhal$\"+`f)[3\"F1 7$F]bl$\"+$Hx-:\"F17$Fbbl$\"+M6A-7F17$Fgbl$\"+#RD)o7F17$F\\cl$\"+D-9W8 F1-Fbcl6&FdclFfclFeclFfcl-Fjcl6#%Ta~minor~modification~of~the~Bogacki- Shampine~schemeG-F$6%7_p7$F($\"*$4\\E$)F-7$F/$\"*tj0`'F-7$$\"5+++v=&)e \")oGF1$\"*(=P**fF-7$F5$\"*x\"Q+cF-7$F:$\"*5\\oG&F-7$F?$\"*T=73&F-7$FD $\"*`36(\\F-7$FI$\"*3#RO\\F-7$Ffel$\"*5tn(\\F-7$FN$\"*nq\\4&F-7$F^fl$ \"*AX#p_F-7$FS$\"*m;Q^&F-7$FX$\"*>k%ziF-7$Fgn$\"*]bQX(F-7$F\\o$\"*pRb/ *F-7$Fao$\"+kSZ-6F-7$Ffo$\"+uL([H\"F-7$F[p$\"+3Vz8:F-7$Fep$\"+2!=Uq\"F -7$Fiq$\"*V*er1%o@\"F-7$F[u$ \"+/56O6F-7$Fhjl$\"+-*\\m5\"F-7$F`u$\"+5/5'3\"F-7$Feu$\"+g$Q]2\"F-7$Fj u$\"++3@r5F-7$F_v$\"+6&4\\2\"F-7$Fdv$\"+u&*\\'3\"F-7$F\\\\m$\"+:OA16F- 7$Fiv$\"+9q)G8\"F-7$F^w$\"+s^%H>\"F-7$Fcw$\"+\"pb#p7F-7$Fhw$\"+#*)3nM \"F-7$F]x$\"+'Rl#z8F-7$Fbx$\"+)3s\\S\"F-7$Fgx$\"+ZyN@9F-7$F\\y$\"+,S+F 9F-7$Fi]m$\"+8-&[U\"F-7$Fay$\"+7F-7$F_`l$\"+R#*>5@F-7$Fd`l$\"*W vCL#F17$Fi`l$\"*a5Qb#F17$F^al$\"*$4t&z#F17$Fcal$\"**fmwIF17$Fhal$\"*Ps ZQ$F17$F]bl$\"*>p2d$F17$Fbbl$\"*:4Zs$F17$Fgbl$\"*$*R)4RF17$F\\cl$\"*'o k0TF1-%&COLORG6&Fdcl$\"\"\"F*$\"\")F*Fecl-Fjcl6#%3efficient~scheme~AG- F$6%7]p7$F($\"*W7E'**F-7$F/$\"*4E'HxF-7$F[fm$\"*'p>\"4(F-7$F5$\"*Gc'Hm F-7$F:$\"*Z\"y\"G'F-7$F?$\"*h-!ogF-7$FD$\"*4,^)fF-7$FI$\"*6$[wfF-7$Ffe l$\"*_it0'F-7$FN$\"*LsKB'F-7$FS$\"*CT=w'F-7$FX$\"*Qhim(F-7$Fgn$\"*DL7. *F-7$F\\o$\"+!f.%)3\"F-7$Fao$\"+[6=D8F-7$Ffo$\"+=*eDc\"F-7$F[p$\"+q`#G %=F-7$Fep$\"+a&Qd3#F-7$Fiq$\"*@Oz;#F17$Fcr$\"*uXh@#F17$Fahl$\"*&>BEAF1 7$Fhr$\"*6r!HAF17$Fihl$\"*c#)\\A#F17$F]s$\"*)oV9AF17$Fail$\"*g8];#F17$ Fbs$\"*eK_4#F17$Fgs$\"+d_8W>F-7$F\\t$\"+=67b$ei\"F-7$Fbx$\"+aa&fl\"F-7$Fgx$\"+vm2v;F-7$F\\y$\"+c%48o \"F-7$Fi]m$\"+KnWy;F-7$Fay$\"+%yS3n\"F-7$Fa^m$\"+yCMe;F-7$Ffy$\"+ZK1T; F-7$Fi^m$\"+=Mn)f\"F-7$F[z$\"+-dfV:F-7$F`z$\"+?u)zR\"F-7$Fez$\"+1]Ya7F -7$Fjz$\"+t^[/6F-7$F_[l$\"*S\")3y*F-7$Fd[l$\"*1nLo)F-7$Fi[l$\"*7]\"*)y F-7$F^\\l$\"*2T&3wF-7$Fc\\l$\"*bx*>uF-7$Fh\\l$\"**zqEtF-7$F]]l$\"*&RyG tF-7$Fb]l$\"*VB!HuF-7$Fg]l$\"*-r$QwF-7$Feam$\"*kY0'zF-7$F\\^l$\"*W9nT) F-7$Fa^l$\"*>\\/t*F-7$Ff^l$\"+bc7L7F-7$F[_l$\"+4Um;9F-7$F`_l$\"+&)4%4l \"F-7$Fe_l$\"+D1db>F-7$Fj_l$\"+GK;`BF-7$F_`l$\"+(HrCf#F-7$Fd`l$\"*t(=w GF17$Fi`l$\"*+tT:$F17$F^al$\"**=NbMF17$Fcal$\"*:f:!QF17$Fhal$\"*1Sp<%F 17$F]bl$\"*CgwR%F17$Fbbl$\"*Sr\\e%F17$Fgbl$\"*%)3J![F17$F\\cl$\"*hZ9.& F1-Fbcl6&FdclFfclFeclFecl-Fjcl6#%3efficient~scheme~BG-F$6%7go7$F($\"*0 SO$RF-7$$\"5MLLLepo(R#GF1$\"*8b\"yIF-7$F/$\"*$3\\\"\\#F-7$F[fm$\"*X3D9 #F-7$F5$\"*(fCD>F-7$F:$\"*Q\\kz\"F-7$F?$\"*yJ3t\"F-7$FI$\"*)fZBxIXF-7$F dv$\"*\"RG.YF-7$Fiv$\"*@*)=![F-7$F^w$\"*3\\//&F-7$Fcw$\"*u6:L&F-7$Fhw$ \"*IGSi&F-7$Fbx$\"*,A6(eF-7$Fgx$\"*_2p&fF-7$F\\y$\"*qcg*fF-7$Fay$\"*9; d(fF-7$Ffy$\"*$))>\")eF-7$Fi^m$\"*npws&F-7$F[z$\"*N/,_&F-7$F`z$\"*d,%o \\F-7$Fez$\"*4wVY%F-7$Fjz$\"*:`0)RF-7$F_[l$\"*,)HtNF-7$Fd[l$\"**yX(>$F -7$Fi[l$\"*_BL!HF-7$Fc\\l$\"*9J2r#F-7$F]]l$\"*kZmj#F-7$Fg]l$\"*T&>)p#F -7$Feam$\"*3t:!GF-7$F\\^l$\"*,i!oHF-7$$\"5mmmT5G7mZZF1$\"*cIa?$F-7$Fa^ l$\"*.qBd$F-7$$\"5mmm;H'[)G$z%F1$\"*FQo?%F-7$Ff^l$\"*E:z4&F-7$F[_l$\"* !\\VNkF-7$F`_l$\"*!fYe#)F-7$$\"5++](o*y<'G([F1$\"*0J4O*F-7$Fe_l$\"+gj. k5F-7$$\"5nm;Hd>()4'*[F1$\"+%f%=87F-7$Fj_l$\"+rx@#R\"F-7$F_`l$\"+0(f/g \"F-7$Fd`l$\"*&34c=F17$Fi`l$\"*jCK3#F17$F^al$\"*9\\;K#F17$Fcal$\"*ARRe #F17$Fhal$\"*&GGbGF17$Fbbl$\"*(p*38$F17$F\\cl$\"*\"\\1'R$F1-Fbcl6&Fdcl $\")#)eqkFhcl$\"))eqk\"FhclFfdp-Fjcl6#%3efficient~scheme~CG-F$6%7`p7$F ($!+gX`WFF-7$F/$!+#\\WxG#F-7$F5$!+U98L?F-7$F:$!+eUXM>F-7$F?$!+JaVm=F-7 $FD$!+)e3'H=F-7$FI$!+eWO<=F-7$Ffel$!+KC/K=F-7$FN$!+O4Ht=F-7$F^fl$!+v%o Q$>F-7$FS$!+5mVw@D' F17$Fcr$!*)R&yS'F17$Fahl$!*u4#\\kF17$Fhr$!*$*H8Z'F17$$\"5nmT5:%QI)pMF1 $!*-AhZ'F17$Fihl$!*E&>ukF17$$\"5MLe9TEVD![$F1$!*6m$pkF17$F]s$!*>#=ekF1 7$Fail$!*Eo!\\jF17$Fbs$!*Q*QchF17$Fgs$!+x^*pq&F-7$F\\t$!+gb;A^F-7$Fat$ !+u&[F-7$$\"5++]7GJKfRTF1$!+Msq$)[F-7$Fgx $!+IV22\\F-7$$\"5++]PMy,*=;%F1$!+yd4@\\F-7$F\\y$!+CK9D\\F-7$Fi]m$!+[/C <\\F-7$Fay$!+5&Hi*[F-7$Fa^m$!+&*Qli[F-7$Ffy$!+TZW9[F-7$Fi^m$!+F'pfp%F- 7$F[z$!+(\\?0a%F-7$F`z$!+^)f&=TF-7$Fez$!+CMF'o$F-7$Fjz$!+_QdDKF-7$F_[l $!+U$[\\%GF-7$Fd[l$!++nxFDF-7$Fi[l$!+8aB5BF-7$F^\\l$!+h9rMAF-7$Fc\\l$! +Y`N'=#F-7$Fh\\l$!+R]An@F-7$F]]l$!+R!=K<#F-7$Fb]l$!+)oXz?#F-7$Fg]l$!+< DlvAF-7$Feam$!+K2/tBF-7$F\\^l$!+VY`1DF-7$Fa^l$!+\\xaiGF-7$Ff^l$!+:Zp# \\$F-7$F[_l$!+O7+zQF-7$F`_l$!+\\/A[VF-7$Fe_l$!+gTXd\\F-7$Fj_l$!+<\"Q8r &F-7$F_`l$!+3(zn8'F-7$Fd`l$!*\"p48mF17$Fi`l$!*s\\a7(F17$F^al$!*bhto(F1 7$Fcal$!*d7uM)F17$Fhal$!**p6t!*F17$Fbbl$!*$Qrr)*F17$F\\cl$!+o<$[2\"F1- F`en6&Fdcl$\"\"'F*FeclFden-Fjcl6#%Sscheme~with~stability~radius~=~imag ~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~f or~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fffq-% %VIEWG6$;F(F\\cl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modification of the B ogacki-Shampine scheme" "efficient scheme A" "efficient scheme B" "eff icient scheme C" "scheme with stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 11 of 7 stage, order 5 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that appe ars in an article by F. G. Lether: Mathematics of Computation, Vol. 2 0, no. 95, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2 " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F( " }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\" \"\"F(,&%\"xGF(F(!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6#/-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG 6#,$%\"xG!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "de := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y( 0)=sin(1);\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x ),x=0..1-1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6 #,$F,!\"\"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-% $sinG6#*&F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#> =\"*)>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7 $$\"3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^ !)F,7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j? zF,7$$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6 wxF,7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3Dw F,7$$\"3?0sMKLNtDF,$\"3,;%fG`C(F, 7$$\"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$ $\"3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$ \"3\"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$ $\"3`G+(=Gs!HXF,$\"3S.Rv)o&))[hF,7$$\"3&\\%Rcec>>`F,7$$\" 39pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"3QD(p)Qdl>bF,$\"34_#)R=svWXF,7$$\"3 #)Qm@o*Q!=dF,$\"3Iba)Q0\")Q2%F,7$$\"3#oP&GV\\)*4fF,$\"3;HPYk\"[Jb$F,7$ $\"3qUjqA#3J7'F,$\"3iI9Us9n*)GF,7$$\"3-koIo*3YJ'F,$\"3G0-$>\\f\"3AF,7$ $\"3fQ6D*)o2>lF,$\"3CG]Vmkt$Q\"F,7$$\"35Y]`:_N/nF,$\"33Dv;1s&pZ&F07$$ \"3.\"Q#ekL\"p!pF,$!3OST&zF\\gd%F07$$\"3%yB5rz/v4(F,$!3cn!**p^J2Z\"F,7 $$\"33-p_gxs'H(F,$!3A9e\"Q#e(4b#F,7$$\"3%324>i/:\\(F,$!3S!=W0GJd`$F,7$ $\"39FEN$GhMf(F,$!3U'\\`Wd^A(RF,7$$\"3c%='zWzT&p(F,$!3f\\'**z(4Q8VF,7$ $\"3,%e*>Oh^WxF,$!3G(>*pycjJWF,7$$\"3K#)HgFVh$z(F,$!3)Q%ReLd!G^%F,7$$ \"3h#o/LUj\"=yF,$!3p>V&oBmw`%F,7$$\"3y\"Q1!>DrUyF,$!3MH+5_!>5b%F,7$$\" 3%433Zhhs'yF,$!3KtI_'\\k?b%F,7$$\"3?\"y4/r5=*yF,$!3)*fMuc7)*RXF,7$$\"3 @o'GDq??%zF,$!3)R70$\\E%3Z%F,7$$\"3=bvk%pIA*zF,$!3(yFHocWfL%F,7$$\"3IV kw'oSC/)F,$!3&[P'\\m/)z7%F,7$$\"3II`))y1l#4)F,$!3E`XP\"[()*RQF,7$$\"3? gE&\\8RA>)F,$!3'*pxl2UD4IF,7$$\"37!**>5fF=H)F,$!3Fu3OFE=:=F,7$$\"3$)fW ::LeP$)F,$!374'4ii(p\\6F,7$$\"3kI*)GR!RLQ)F,$!3q\\t;%4rO@%F07$$\"3Z,MU jZ4H%)F,$\"3c]$R4W&G]NF07$$\"3=ryb([][Z)F,$\"37Ccp$Rpv:\"F,7$$\"3%QzY4 r\"HF&)F,$\"3CcG)))y\"yp?F,7$$\"3g()F,$\"3y_cgRCTxTF,7$$\"3Wx*okV>:t)F,$\"37VUul#=X<%F,7$$\"3 q%oYTYr\\v)F,$\"3Ri.A'QV25%F,7$$\"3'>RC=\\B%y()F,$\"3x'=W7Fx&HRF,7$$\" 3]1)zraF`#))F,$\"3QFG%plB4F$F,7$$\"39A_`-;Bs))F,$\"3700_RQTz@F,7$$\"31 ^1[bjB(*))F,$\"3OuNH?R:M9F,7$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF07$$\"37 6:PheCZ*)F,$!3ocHHl.]EIF07$$\"39TpJ91Ds*)F,$!38U'y=D^^A\"F,7$$\"3=rBEn `D(**)F,$!3j&4Nt%[@=@F,7$$\"3@,y??,EA!*F,$!3!e@9jLE#=HF,7$$\"3EJK:t[EZ !*F,$!3I(>(y]J<_NF,7$$\"3Gh')4E'pA2*F,$!3U`$=&457URF,7$$\"3'3]u3\"GDy! *F,$!3U'ykJ>F2*RF,7$$\"3cT.l&*fB%3*F,$!3AH^N$RP+-%F,7$$\"3E#=E/=>-4*F, $!3&Rn/8H]\"HSF,7$$\"3'=--_O-i4*F,$!39=u$p*4B>*F,$!39yV6L@mxvF07$$\"3oYr;\"p**Q?*F,$!3 _9%**Gult/#!#?7$$\"3(p#)=21me@*F,$\"3=-0hZ^rztF07$$\"3E20FIC$yA*F,$\"3 MLbSaq^*[\"F,7$$\"3b(=A)*z)zR#*F,$\"3Qccz;X$>?#F,7$$\"3%*oQPp^w^#*F,$ \"3\\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$ \"3Ma)HpV]]I!H!G$*F,$\"3a %4t07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$ \"33n1kl[]%*F,$ !3m(=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3 U$3!Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"# 5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VI EWG6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 804 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sha mpine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sch eme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 30:\nfor ct to 6 do\n Qn_RK5_||ct := RK5_||ct(Q(x,y) ,x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,false);\n sm := 0: nump ts := nops(Qn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Qn_R K5_||ct[ii,2]-q(Qn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(err s),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#* &F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step ~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~G FFQ)pprint666\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+E1([ Z\"!#=7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+X:RT 6F+7$%3efficient~scheme~AG$\"+C_IS8F+7$%3efficient~scheme~BG$\"+k<(RI \"F+7$%3efficient~scheme~CG$\"+\"\\b+`\"F+7$%Sscheme~with~stability~ra dius~=~imag~axis~inclusionG$\"+?dzo:F+Q)pprint676\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 728 "Q := (x,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000 *Pi): numsteps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: \+ `,Q(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no . of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme`,` a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability r adius = imag axis inclusion`]:\nerrs := []:\nDigits := 30:\nfor ct to \+ 6 do\n qn_RK5_||ct := RK5_||ct(Q(x,y),x,y,x0,evalf(y0),evalf(hh),num steps,true);\nend do:\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(qn_RK5_||ct(xx)-qxx)];\nend do:\nDigits \+ := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG! \"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~poin t:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+I F1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint686\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %8Bogacki-Shampine~schemeG$\"+6qtF+7$%3efficient~scheme~AG$\"+uU&[N# F+7$%3efficient~scheme~BG$\"+)y^%GAF+7$%3efficient~scheme~CG$\"+iNezEF +7$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+X=KmFF+Q)p print696\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"F &*&F&F&*&\"\"'F&%#PiGF&!\"\"F+" }{TEXT -1 82 " of each Runge-Kutta me thod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 po int Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 452 "mthds := [`Bogacki-Shampine scheme`,`a m inor modification of the Bogacki-Shampine scheme`,`efficient scheme A` ,`efficient scheme B`,`efficient scheme C`,`scheme with stability radi us = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((q(x)-'qn_RK5_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=f alse,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi )))];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine ~schemeG$\"+:G4o!)!#>7$%Ta~minor~modification~of~the~Bogacki-Shampine~ schemeG$\"+'>())QrF+7$%3efficient~scheme~AG$\"+E(42e(F+7$%3efficient~s cheme~BG$\"+GfgZyF+7$%3efficient~scheme~CG$\"+NqsC()F+7$%Sscheme~with~ stability~radius~=~imag~axis~inclusionG$\"+- " 0 "" {MPLTEXT 1 0 514 "evalf[20]( plot(['qn_RK5_1'(x)-q(x),'qn_RK5_2'(x)-q(x),'qn_RK5_3'(x)-q(x),'qn_RK5 _4'(x)-q(x),\n'qn_RK5_5'(x)-q(x),'qn_RK5_6'(x)-q(x)],x=0..0.7,-9e-16.. 9e-16,\ncolor=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6, 0,.8)],font=[HELVETICA,9],\nlegend=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`ef ficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Runge -Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 786 534 534 {PLOTDATA 2 "6,-%'CURVESG6%7fn7$$\"\"!F)F(7$$\"5mmmmm\"z+e_\"!#@$!\")! #?7$$\"5LLLL3->R`GF-$!#6F07$$\"5mmmm;apSYVF-$!#>F07$$\"5lmmm;z'=$\\eF- $!#DF07$$\"5KLLL3Ft3XtF-$!#LF07$$\"5lmmmTNj&=t)F-$!#VF07$$\"5+++](=`xn ,\"F0$!#`F07$$\"5mmm;ay/Gl6F0$!#mF07$$\"5+++]PurI88F0$!#zF07$$\"5LLLL$ e#3dl9F0$!#&*F07$$\"5mmmm\"Ht%o*f\"F0$!$3\"F07$$\"5++++]F_m]F0$!$]\"F07$$\"5++++]s2O[?F0$!$y\"F07$$\"5mmm;aG\"H5= #F0$!$-#F07$$\"5LLLL$ej%yQBF0$!$N#F07$$\"5LLLLLVUUsCF0$!$n#F07$$\"5*** ***\\(o()yyi#F0$!$3$F07$$\"5LLLLLoD[lFF0$!$b$F07$$\"5******\\(oibk\"HF 0$!$2%F07$$\"5******\\i!o<-1$F0$!$f%F07$$\"5LLLL3-$=-@$F0$!$D&F07$$\"5 LLL$3xplzM$F0$!$#fF07$$\"5mmmm\"H([a'\\$F0$!$s'F07$$\"5mmm;ayo(3l$F0$! $h(F07$$\"5******\\7VLA&y$F0$!$Y)F07$$\"5mmmmT07KIRF0$!$T*F07$$\"5**** *****\\\\@-3%F0$!%U5F07$$\"5*******\\PopoA%F0$!%R6F07$$\"5******\\(oMf (oVF0$!%77F07$$\"5*******\\ii.j_%F0$!%t7F07$$\"5KLLLLoT'ym%F0$!%x7F07$ $\"5********\\i-,>[F0$!%*>\"F07$$\"5mmm;a)3rf&\\F0$!%<5F07$$\"5******* *\\Zaq0^F0$!$9'F07$$\"5KLL$3-\"QfY_F0$\"$3\"F07$$\"5******\\PWF'QR&F0$ \"%$>\"F07$$\"5KLLL$e/Xy`&F0$\"%=IF07$$\"5******\\(=<\"e)o&F0$\"%wfF07 $$\"5lmmmmwzvLeF0$\"&\"e5F07$$\"5lmmm\"zAAA)fF0$\"&5q\"F07$$\"5KLL$3-7 d%HhF0$\"&kF0$\"&bT'F07$$\"5+++]7=p:*['F0$\"&qe(F07$$\"5lmm mmO9]elF0$\"&*)H*F07$$\"5ILL3xcrVKmF0$\"(2\"Q6F-7$$\"5)*****\\(o(GP1nF 0$\"(ImQ\"F-7$$\"5+++v$f$evTnF0$\"(vh^\"F-7$$\"5+++++&zQrx'F0$\"(A\"G; F-7$$\"5+++D1a<_7oF0$\")(4,x\"!#A7$$\"5)*****\\78Z!z%oF0$\"(h[$>F-7$$ \"5++]P%[`Gf)oF0$\"()=?@F-7$$\"5+++DccB&R#pF0$\"(9)=BF-7$$\"5++]7Gyh(> 'pF0$\"(.2`#F-7$$\"\"(!\"\"$\"(a`v#F--%'COLOURG6&%$RGBGF(F($\"*++++\"F /-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7fnF'7$F+$\"\"#F07$F2$Fa]l F07$F7$\"#6F07$F<$\"#F07$Faq$\"$B# F07$Ffq$\"$c#F07$F[r$\"$(GF07$F`r$\"$G$F07$Fer$\"$o$F07$Fjr$\"$;%F07$F _s$\"$u%F07$Fds$\"$H&F07$Fis$\"$#fF07$F^t$\"$i'F07$Fct$\"$P(F07$Fht$\" $.)F07$F]u$\"$$))F07$Fbu$\"$M*F07$Fgu$\"$s*F07$F\\v$\"$r*F07$Fav$\"$7* F07$Ffv$\"$[(F07$F[w$\"$c%F07$F`w$!#tF07$Few$!$$**F07$Fjw$!%\"[#F07$F_ x$!%IYF07$Fdx$!%_zF07$Fix$!&#f7F07$F^y$!&hi\"F07$Fcy$!&%*4#F07$Fhy$!&o ^#F07$F]z$!&28$F07$Fbz$!':*)QF-7$Fgz$!'R2[F-7$F\\[l$!'6'H&F-7$Fa[l$!'B CdF-7$Ff[l$!(DIE'Fj[l7$F\\\\l$!'&[*oF-7$Fa\\l$!'j>wF-7$Ff\\l$!'\")4%)F -7$F[]l$!'^o#*F-7$F`]l$!(]*>5F--Ff]l6&Fh]lFi]lF(Fi]l-F\\^l6#%Ta~minor~ modification~of~the~Bogacki-Shampine~schemeG-F$6%7fnF'7$F+$!\"$F07$F2F eil7$F7$!\"'F07$FF07$Fis$!$@#F07$F^ t$!$V#F07$Fct$!$j#F07$Fht$!$x#F07$F]u$!$%GF07$Fbu$!$z#F07$Fgu$!$[#F07$ F\\v$!$\">F07$Fav$!#rF07$Ffv$\"$N\"F07$F[w$\"$Q%F07$F`w$\"$Y*F07$Few$ \"%aF 07$Faq$\"$=#F07$Ffq$\"$\\#F07$F[r$\"$\"GF07$F`r$\"$?$F07$Fer$\"$g$F07$ Fjr$\"$3%F07$F_s$\"$m%F07$Fds$\"$A&F07$Fis$\"$&eF07$F^t$\"$f'F0Fecl7$F ht$\"$4)F07$F]u$\"$-*F07$Fbu$\"$o*F07$Fgu$\"%K5F07$F\\v$\"%l5F07$Fav$ \"%h5F07$Ffv$\"$\")*F07$F[w$\"$,)F07$F`w$\"$Y%F07$Few$!$4#F07$Fjw$!%,8 F07$F_x$!%:HF07$Fdx$!%TaF07$Fix$!%)F--Ff]l6&Fh]lFi]lF(F(-F\\^l6#%3efficient~scheme~BG-F $6%7fnF'Fdil7$F2$Fb]lF07$F7Feil7$F%F--Fgcm6&Fh]l$\"\"'Fb]lF(F[dm-F\\^l6#%Sscheme~with~stability~radi us~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror ~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6 \"Q!F`co-%%VIEWG6$;F(F`]l;$!\"*!#;$FgboFico" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor mo dification of the Bogacki-Shampine scheme" "efficient scheme A" "effic ient scheme B" "efficient scheme C" "scheme with stability radius = im ag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 519 "evalf[20](plot(['qn_RK5_1'(x)-q(x),'qn_RK5 _2'(x)-q(x),'qn_RK5_3'(x)-q(x),'qn_RK5_4'(x)-q(x),\n'qn_RK5_5'(x)-q(x) ,'qn_RK5_6'(x)-q(x)],x=0.7..0.87,-5e-13..3.9e-12,\ncolor=[blue,magenta ,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9], \nlegend=[`Bogacki-Shampine scheme`,`a minor modification of the Bogac ki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficie nt scheme C`,`scheme with stability radius = imag axis inclusion`],\nt itle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 823 527 527 {PLOTDATA 2 "6,-%'CURVESG6%7ao7 $$\"\"(!\"\"$\"(a`v#!#@7$$\"5LLLL3i^0Pq!#?$\"'Q*)HF17$$\"5nmm\"z/m'Hpq F1$\"'ALKF17$$\"5LLL$eufbb5(F1$\"'q([$F17$$\"5LLL$3A)[0UrF1$\"'kYPF17$ $\"5nmm\"HAp!QyrF1$\"'Z0SF17$$\"5LLLe9o$f?@(F1$\"'giTF17$$\"5+++DJ'oJp C(F1$\"'*=Q%F17$$\"5LLLek(o'*HG(F1$\"'R,YF17$$\"5+++D1Gg%*=tF1$\"'b\"z %F17$$\"5nmmmTVV#fN(F1$\"'#R$\\F17$$\"5LLL$3P'[\\)Q(F1$\"'g3]F17$$\"5+ +++Db:;DuF1$\"'.&)\\F17$$\"5++++v.)y>Y(F1$\"'fJ[F17$$\"5++++vW!fu\\(F1 $\"'j8YF17$$\"5LLLek-&y'HvF1$\"'/^UF17$$\"5nmmmTa0*zc(F1$\"'XgNF17$$\" 5nmmmm()eW+wF1$\"'PsDF17$$\"5+++D\"e:*>QwF1$\"'1c6F17$$\"5nmmm;&><;n(F 1$!&Mv(F17$$\"5+++D\"33#G3xF1$!'r]LF17$$\"5+++vVAd>VxF1$!'OfbF17$$\"5n mm;zWWizxF1$!'O-#*F17$$\"5nmm\"HPQxI\"yF1$!(A:P\"F17$$\"5LLL$3x*3;\\yF 1$!(!o))>F17$$\"5LLLekF:k')yF1$!()\\lFF17$$\"5+++v=E&o#>zF1$!(\"QFPF17 $$\"5LLL$e%yl]azF1$!(K\"3\\F17$$\"5++++]M4\"4*zF1$!(H#fgF17$$\"5+++]i% \\Dl-)F1$!(xuQ(F17$$\"5+++D\"GT%)41)F1$!(qp1*F17$$\"5+++]Pm^C*4)F1$!)< \"o7\"F17$$\"5nmmm;bTiL\")F1$!)d/q8F17$$\"5++++vj5Lq\")F1$!)+(*\\;F17$ $\"5LLLekyHf.#)F1$!)Bba>F17$$\"5++++DPq&*R#)F1$!)V!)pAF17$$\"5nmm\"zCy sTF)F1$!)NP%\\#F17$$\"5+++D1&4Q*4$)F1$!)5EIFF17$$\"5nmmmTD_!\\M)F1$!)# 46\"HF17$$\"5NL$eky*)3KO)F1$!*'*R&eHF-7$$\"5+++DJqD^\"Q)F1$!*N#3dHF-7$ $\"5lm;HK#=T\"*R)F1$!*?3D*GF-7$$\"5LLLLL%zpnT)F1$!*#)enu#F-7$$\"5NLL3F 'f(zM%)F1$!*nZ$)\\#F-7$$\"5LLL$3#)RDGX)F1$!**3=B@F-7$$\"5++]P%3#Rqq%)F 1$!)U6$f\"F17$$\"5nmm\"zMW#e)[)F1$!(+sv)F17$$\"5++++]a%R9_)F1$\"(6!)z) F17$$\"5nmm;/(H(4f&)F1$\")fPfRF17$$\"5+++vo3\"Qfd)F1$\"),\")*G&F17$$\" 5LLLLL?*yFf)F1$\").OVsF17$$\"5lm;H2QZt5')F1$\")$G3\"**F17$$\"5+++D\"eb !pG')F1$\"*^8/J\"F17$$\"5++vV)z0()Hj)F1$\"*cEAT\"F17$$\"5++]i:gNGP')F1 $\"*4N!y;F17$$\"5++D\"GB1!eT')F1$\"*'[[x;F17$$\"5++++]kl(ek)F1$\"*:=Io \"F17$$\"5++v=nmI<]')F1$\"*!*R0u\"F17$$\"5++]P%)o&pWl)F1$\"*e)[J?F17$$ \"5+v=nQ%pVbl)F1$\"*n(**=@F17$$\"5+](oH*>yhc')F1$\"*pd')F1$\"*oW&=@F17$$\"5++Dc,rgwe')F1$\"*,L$=@F17$$\"5+]i:5AV\"4m)F1$ \"*mm\"=@F17$$\"5+++v=tD1j')F1$\"*%Gb>@F17$$\"5++D1*)HpHs')F1$\"*(f](Q #F17$$\"5++]Pf'GJ:o)F1$\"*P.*>EF17$$\"5++voHVcw!p)F1$\"*kB;(GF17$$\"#( )!\"#$\"*p\"yyJF1-%'COLOURG6&%$RGBG$\"\"!FcalFbal$\"*++++\"!\")-%'LEGE NDG6#%8Bogacki-Shampine~schemeG-F$6%7]q7$F($!(]*>5F-7$F/$!'Q>6F17$F5$! '1E7F17$F:$!'*>M\"F17$F?$!'ql9F17$FD$!'$pf\"F17$FI$!'`#o\"F17$FN$!'X4= F17$FS$!'j_>F17$FX$!'\\,@F17$Fgn$!'g_AF17$F\\o$!'z-CF17$Fao$!'%fa#F17$ Ffo$!'!on#F17$F[p$!'>]FF17$F`p$!'^GGF17$Fep$!'y()GF17$Fjp$!'(H!HF17$F_ q$!'yXGF17$Fdq$!'j)p#F17$Fiq$!'&)HCF17$F^r$!'%y6#F17$Fcr$!'06;F17$Fhr$ !&,Z*F17$F]s$\"%4\")F17$Fbs$\"'H*[\"F17$Fgs$\"''RN$F17$F\\t$\"'D3eF17$ Fat$\"'Ch%)F17$Fft$\"(4+;\"F17$F[u$\"(R(o:F17$F`u$\"(?&e@F17$Feu$\"('R gGF17$Fju$\"(!pkPF17$F_v$\"(?\\)[F17$Fdv$\"(L0E'F17$Fiv$\"(TOZ(F17$F^w $\"(5Q4*F17$Fcw$\")P876F17$F]x$\"*k-aO\"F-7$Fbx$\"*z6k]\"F-7$Fgx$\"*&4 dd;F-7$F\\y$\"*bTv\"=F-7$Fay$\"*bYY)>F-7$Ffy$\")fuc@F17$F[z$\")&*pIBF1 7$$\"5NL$e*)*[4,0&)F1$\")ja-DF17$F`z$\")P2#f#F17$Fez$\")jq^GF17$Fjz$\" )ng,HF17$F_[l$\")>8qHF17$$\"5:Hd?%*e6!Rf)F1$\")f@gHF17$$\"5+D\"y]vRB]f )F1$\")xGaHF17$$\"5!3_]fhjXhf)F1$\")'3#eHF17$$\"5l;H#oZ(yE(f)F1$\")O3 \")HF17$$\"5]7`pP8,R)f)F1$\")dmHIF17$$\"5I3xc)>N7&*f)F1$\")cKHIF17$$\" 5:/,Wf!fM1g)F1$\")=)*GIF17$$\"5++DJ?Hov,')F1$\")!3'GIF17$$\"5l\"Hd?kI, Sg)F1$\")\"3t-$F17$$\"5I$3-QOyXig)F1$\")<@BIF17$$\"5+voa&3E!\\3')F1$\" )(>=,$F17$Fd[l$\")sD))HF17$$\"5IL3F%pk7(>')F1$\")%Gf'HF17$Fi[l$\")B8HH F17$F^\\l$\")6jIGF17$Fc\\l$\")e)3z#F17$$\"5+](=U7\"=VR')F1$\")sC!z#F17 $Fh\\l$\")a&*)y#F17$$\"5+]iST8$GPk)F1$\")'*G%y#F17$F]]l$\")3()pFF17$Fb ]l$\")^gsEF17$Fg]l$\")na%[#F17$F\\^l$\")X6lCF17$Fa^l$\")$\\[Y#F17$Ff^l $\")KdkCF17$F[_l$\")#HUY#F17$F`_l$\")ZZiCF17$Fe_l$\")'*\\bCF17$$\"5+]i !R:vzwm)F1$\")T+tBF17$Fj_l$\")zx*3#F17$$\"5]i:&yWA^Mn)F1$\")x+\"*>F17$ $\"5+D1k1>bgu')F1$\")))pS>F17$$\"5](oHaO\")fdn)F1$\")NZS>F17$$\"5+](=U #3T\"pn)F1$\")^AS>F17$$\"5+vozT(pA#z')F1$\")h**Q>F17$F_`l$\")@rK>F17$$ \"5+DJ&pd()RQo)F1$\")!o#4>F17$$\"5+]7`%\\Y[ho)F1$\")Z7Y=F17$$\"5+v$4@T 0d%)o)F1$\")Ta7>p)F1$\")CDB8F17$$\" 5+DcEZKU2$p)F1$\")f/e6F17$$\"5](oagq_GUp)F1$\")?\"z:\"F17$$\"5+]P%[;#G Q&p)F1$\")$ox:\"F17$$\"5+v=U#3T\"p(p)F1$\")'pp:\"F17$Fi`l$\")D!=:\"F1- F_al6&FaalFdalFbalFdal-Fhal6#%Ta~minor~modification~of~the~Bogacki-Sha mpine~schemeG-F$6%7jn7$F($\"'bqpF-7$F/$\"&>\\(F17$F5$\"&f,)F17$F:$\"&W `)F17$F?$\"&i-*F17$FD$\"&1Z*F17$FI$\"&Lq*F17$FN$\"&;)**F17$FS$\"'#f,\" F17$FX$\"'y:5F17$Fgn$\"&'3**F17$F\\o$\"&PL*F17$Fao$\"&?K)F17$Ffo$\"&8v 'F17$F[p$\"&-A&F17$F`p$\"&p&HF17$Fep$!%uzF17$Fjp$!&1t&F17$F_q$!'8J7F17 $Fdq$!'O&3#F17$Fiq$!'=zJF17$F^r$!'SoSF17$Fcr$!',PbF17$Fhr$!'MKtF17$F]s $!'([r*F17$Fbs$!(rVE\"F17$Fgs$!(x&>;F17$F\\t$!(*\\X?F17$Fat$!(%>XCF17$ Fft$!(y8!HF17$F[u$!(91Z$F17$F`u$!(tT=%F17$Feu$!([?%\\F17$Fju$!(wfv&F17 $F_v$!(]\"flF17$Fdv$!(=(fsF17$Fiv$!(([EX\\\"F17$Fi`l$\"*8b]i\"F1-%&COLORG6&Faal $\"\"\"F*$\"\")F*Fbal-Fhal6#%3efficient~scheme~AG-F$6%7eo7$F($!'T*>)F- 7$F/$!&(f!*F17$F5$!&e***F17$F:$!'6.6F17$F?$!'z:7F17$FD$!'6Q8F17$FI$!') *>9F17$FN$!'\"Qa\"F17$FS$!'T)o\"F17$FX$!'WX=F17$Fgn$!'(Q,#F17$F\\o$!'; $>#F17$Fao$!'#4Q#F17$Ffo$!'7wDF17$F[p$!'Y6FF17$F`p$!'FvGF17$Fep$!'SvIF 17$Fjp$!'%4F$F17$F_q$!'>_MF17$Fdq$!'S6OF17$Fiq$!'#>t$F17$F^r$!'5VPF17$ Fcr$!'WbPF17$Fhr$!''zr$F17$F]s$!'zFNF17$Fbs$!'H_JF17$Fgs$!'yUDF17$F\\t $!'F-;F17$Fat$!&7-%F17$Fft$\"'3$3\"F17$F[u$\"'R4JF17$F`u$\"'%HN'F17$Fe u$\"(@)\\5F17$Fju$\"(R=j\"F17$F_v$\"((R8CF17$Fdv$\"(#ycMF17$Fiv$\"(tGX %F17$F^w$\"(zW(eF17$Fcw$\"(uH\"yF17$F]x$\"*$oz[5F-7$Fgx$\"*w:WR\"F-7$F ay$\"*!>RO=F-7$F[z$\")#oxQ#F17$Fijl$\"),q2FF17$F`z$\")#zN#HF17$F^an$\" )mGuKF17$Fez$\")!fvk$F17$Fjz$\")k'\\*QF17$F_[l$\")E.]UF17$Fd[l$\")5KcY F17$Fi[l$\")c1e]F17$F]]l$\");]KaF17$Fe_l$\")\"y%RdF17$$\"5+D\"GjB;r`m) F1$\")Y1FdF17$Fjbm$\")>(oq&F17$$\"5]7`p7YS$)o')F1$\")!zyp&F17$$\"5+vV[ rS$))*p')F1$\")N&fp&F17$$\"5]PMFINE9r')F1$\")!3.r&F17$Fj_l$\")\"Hav&F1 7$Fbcm$\")*fG&eF17$Fgcm$\")9#H$fF17$F\\dm$\")dBKfF17$Fadm$\")L`JfF17$F fdm$\")_jHfF17$F_`l$\"):lCfF17$Fcem$\")9*>)eF17$Fd`l$\")yeGeF17$F`fm$ \")WWgeF17$Fefm$\")KaPfF17$Fjfm$\")x&o$fF17$F_gm$\")^;OfF17$Fdgm$\")AU MfF17$Fi`l$\")\"Q)HfF1-F_al6&FaalFdalFbalFbal-Fhal6#%3efficient~scheme ~BG-F$6%7in7$F($\"'aP[F-7$F/$\"&_9&F17$F5$\"&(QaF17$F:$\"&^q&F17$F?$\" &g#fF17$FD$\"&/3'F17$FI$\"&A7'F17$FN$\"&\\6'F17$FS$\"&e'fF17$FX$\"&*=c F17$Fgn$\"&0,&F17$F\\o$\"&v1%F17$Fao$\"&$*o#F17$Ffo$\"%;wF17$F[p$!%()* *F17$F`p$!&p^$F17$Fep$!&\"RvF17$Fjp$!'Yn7F17$F_q$!'?N>F17$Fdq$!'I&y#F1 7$Fiq$!'hbQF17$F^r$!'p2ZF17$Fcr$!'A7hF17$Fhr$!'8F17$Fat$!(N[M#F17$Fft$!(q'[FF 17$F[u$!(=)[KF17$F`u$!(k7'QF17$Feu$!(Pr\\%F17$Fju$!(!z^^F17$F_v$!($*\\ v&F17$Fdv$!(0l?'F17$Fiv$!(/:M'F17$F^w$!(pj?'F17$Fcw$!(UfZ&F17$F]x$!)!p \"yOF-7$Fgx$!(!y'3$F-7$Fay$\")PIw`F-7$F[z$\")V/H9F17$Fijl$\")TpI?F17$F `z$\")y&e]#F17$F^an$\")r.*H$F17$Fez$\")#*QNUF17$Fjz$\")Z8M\\F17$F_[l$ \")>LafF17$Fd[l$\")w5*G(F17$Fi[l$\")TEC))F17$F^\\l$\")$z[D*F17$Fc\\l$ \"*=`O0\"F17$Fh\\l$\"*QJK0\"F17$F]]l$\"*WdX0\"F17$Fg]l$\"*$=D07F17$Fe_ l$\"*(oVY7F17$Fj_l$\"*?Y8N\"F17$F_`l$\"*QtRX\"F17$Fd`l$\"*&eDW:F17$Fi` l$\"*]'yq;F1-F_al6&Faal$\")#)eqkFfal$\"))eqk\"FfalFb_p-Fhal6#%3efficie nt~scheme~CG-F$6%7ao7$F($\"(fQ>%F-7$F/$\"'hiXF17$F5$\"':\\\\F17$F:$\"' *eN&F17$F?$\"'(Rx&F17$FD$\"'R(>'F17$FI$\"'vekF17$FN$\"'.HoF17$FS$\"'I6 sF17$FX$\"'UgvF17$Fgn$\"'d]yF17$F\\o$\"'9a!)F17$Fao$\"'+G\")F17$Ffo$\" 'AG!)F17$F[p$\"'o2yF17$F`p$\"'(HS(F17$Fep$\"'\"4d'F17$Fjp$\"'4G`F17$F_ q$\"'2#\\$F17$Fdq$\"&/R*F17$Fiq$!';;DF17$F^r$!'gBbF17$Fcr$!(S(\\5F17$F hr$!(9#p;F17$F]s$!(+K_#F17$Fbs$!(4]g$F17$Fgs$!(,=&\\F17$F\\t$!(wUh'F17 $Fat$!(HtC)F17$Fft$!)()o85F17$F[u$!)TM`7F17$F`u$!)#4*p:F17$Feu$!)C7A>F 17$Fju$!)$y#F17$Fg]l$\"*-Jbe#F17$ F\\^l$\"*uBCq#F17$Fa^l$\"*iL@q#F17$Ff^l$\"*eY=q#F17$F[_l$\"*R!e,FF17$F `_l$\"*iG9q#F17$Fe_l$\"*KJNq#F17$Fj_l$\"*h#)H2$F17$F_`l$\"*SHvQ$F17$Fd `l$\"*hu)QPF17$Fi`l$\"*mS$fTF1-F`dn6&Faal$\"\"'F*FbalFddn-Fhal6#%Ssche me~with~stability~radius~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG \"\"*-%&TITLEG6#%Uerror~curves~for~7~stage~order~5~Runge-Kutta~methods G-%+AXESLABELSG6$Q\"x6\"Q!F]^q-%%VIEWG6$;F(Fi`l;$!\"&!#8$\"#RFf^q" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampin e scheme" "a minor modification of the Bogacki-Shampine scheme" "effic ient scheme A" "efficient scheme B" "efficient scheme C" "scheme with \+ stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 505 "evalf[20](plot(['qn_ RK5_1'(x)-q(x),'qn_RK5_2'(x)-q(x),'qn_RK5_3'(x)-q(x),'qn_RK5_4'(x)-q(x ),\n'qn_RK5_5'(x)-q(x),'qn_RK5_6'(x)-q(x)],x=0.87..0.925,\ncolor=[blue ,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],font=[HELVET ICA,9],\nlegend=[`Bogacki-Shampine scheme`,`a minor modification of th e Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,` efficient scheme C`,`scheme with stability radius = imag axis inclusio n`],\ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`)); " }}{PARA 13 "" 1 "" {GLPLOT2D 747 615 615 {PLOTDATA 2 "6,-%'CURVESG6% 7ds7$$\"#()!\"#$\"*p\"yyJ!#?7$$\"5nmm;zM%))>r)F-$\"*Cm:z$F-7$$\"5LL$3- y]>Cs)F-$\"*.*G/QF-7$$\"5nmmTNS.:M()F-$\"*uJ=W%F-7$$\"5nmm\"zC$*efu)F- $\"*rPIg%F-7$$\"5LL$3F/S6xv)F-$\"*&G+8^F-7$$\"5nm;ajUugo()F-$\"*AXCu&F -7$$\"5++](=Km*))z()F-$\"*b&p*y&F-7$$\"5nm;a)=vd:z)F-$\"*q\"QkjF-7$$\" 5++]P%3U)=.))F-$\"*/)46lF-7$$\"5LLL$eM*>::))F-$\"*7zO$oF-7$$\"5nmm\"HZ ^*oD))F-$\"*[Rx3(F-7$$\"5+++](yE_v$))F-$\"*TE)*3(F-7$$\"5+++]iIPY\\))F -$\"*G?;+(F-7$$\"5+++]7KE%4'))F-$\"*9>V\"oF-7$$\"5nm;aQulOr))F-$\"*U/# HkF-7$$\"5LLL$e*\\;w$)))F-$\"*B>T>&F-7$$\"5LLLL$[!>E%*))F-$\"*P_\\5&F- 7$$\"5++](ou>wk!*)F-$\"*6!)H0$F-7$$\"5LLLL3;zG<*)F-$\"+YD0C?!#@7$$\"5+ +](o\\3]\"H*)F-$!*@\\rs#Fgq7$$\"5++]i!\\nX/%*)F-$!,AW!*z(\\!#A7$$\"5lm TNYc&Qj%*)F-$!+YP*f)\\Fgq7$$\"5LLL3-Q9B_*)F-$!+'p9S^&Fgq7$$\"5NekyA9Ve `*)F-$!+Ir(3,'Fgq7$$\"5N$e*[V!>P\\&*)F-$!+nZLcoFgq7$$\"5N3F>km+Hc*)F-$ !+hICM#)Fgq7$$\"5NLe*[G%Hkd*)F-$!,.\\e*R5Fgq7$$\"5N$3-j_p[.'*)F-$!,_9 \"fV6Fgq7$$\"5LL$3xwWaI'*)F-$!,9FCM9\"Fgq7$$\"5nmm\"Hd_GZ(*)F-$!+gm\\% \\\"F-7$$\"5l\"zp)\\'GWi(*)F-$!+`;F- 7$$\"5lTgx.3eFz*)F-$!+xf,')>F-7$$\"5lm\"H2)o:z!)*)F-$!+hsu&)>F-7$$\"5l ;ajM!4BQ)*)F-$!+L\"*\\')>F-7$$\"5nm;a)=hao)*)F-$!+AZ5**>F-7$$\"5++voHg N\\*)*)F-$!+_+m^?F-7$$\"5NLL$3(3D8#**)F-$!+!=L%>AF-7$$\"5qm\"z>rXrZ**) F-$!+zYtgEF-7$$\"5++]7`0/T(**)F-$!+R;)*QIF-7$$\"5NL3Fpy16.!*F-$!+*yZ'R IF-7$$\"5nmmT&=&4\")3!*F-$!+I=/OJF-7$$\"5N$e9To=$G5!*F-$!+OGNPKF-7$$\" 5++D\"G=Ub<,*F-$!+9)QcT$F-7$$\"5q;/^\"olFK,*F-$!+G(4Qr$F-7$$\"5NL$3-=* )*p9!*F-$!+zh)=>%F-7$$\"5qmTgxhVk(F -7$$\"5+++D1U\"Qc0*F-$!+$ycH>(F-7$$\"5lm;H2\\%*>h!*F-$!+GqO)>(F-7$$\"5 LLLL3c2wm!*F-$!+tkD(R(F-7$$\"5++](=spH(p!*F-$!+?R\"R(yF-7$$\"5lmmTNQ') ps!*F-$!+\\hAi&)F-7$$\"5IL$e*[zvmv!*F-$!+\">u(f&)F-7$$\"5+++]i?ljy!*F- $!+9izf&)F-7$$\"5nm;aQGxR*3*F-$!+g!G%3#*F-7$$\"5+++](3di65*F-$!+g,7R&* F-7$$\"5LL$3-3PKA6*F-$!+nd\"Gm*F-7$$\"5lm;H2Oz,=\"*F-$!+(>5xl*F-7$$\"5 ++]PM,N!Q7*F-$!+;iOW&*F-7$$\"5I$eRA^sJm7*F-$!+\"4YC8*F-7$$\"5lmT5!*[*f %H\"*F-$!+s:_w$)F-7$$\"5+](ozE<)GK\"*F-$!+*QMTP)F-7$$\"5LLL$ekR;^8*F-$ !+ONRq$)F-7$$\"5+]P%[YGx!Q\"*F-$!+NnKZ$)F-7$$\"5lmT&QGLH![\"*F -$!+E*G&3\\F-7$$\"5NeRs$Gz&Q[\"*F-$!+DRN3\\F-7$$\"5]P%[YYr)4\\\"*F-$!+ TO+3\\F-7$$\"5q;HdXO;\")\\\"*F-$!+-+l2\\F-7$$\"5+v=U2![P7:*F-$!+Fq&o! \\F-7$$\"5NL3FpBLm_\"*F-$!+lST0\\F-7$$\"5+](oH4,:b:*F-$!+.^N))[F-7$$\" 5mmmm;)pm$e\"*F-$!+MA)*pZF-7$$\"5lm;HKA#*>k\"*F-$!+#G9*QAF-7$$\"5mmm\" zkuJ+<*F-$\"+CEi2;F-7$$\"5N$e*[VgQ#H<*F-$\"+zKe9;F-7$$\"5++D1Ruf\"e<*F -$\"+`,#Rp\"F-7$$\"5Ie*[o8.is<*F-$\"+W8#[$=F-7$$\"5l;ajM)33(y\"*F-$\"+ G5TQ@F-7$$\"5+v=UKXT:!=*F-$\"+aw')QFF-7$$\"5LL$3-B?+;=*F-$\"+%z9%[QF-7 $$\"5+]iSmaxD%=*F-$\"+8TO)Q)F-7$$\"5lmTg-2`\"p=*F-$\",p-%[+7F-7$$\"5I$ 3-)QfGd*=*F-$\"-[K,C+7Fgq7$$\"5++++v6/B#>*F-$\"-mg,g,7Fgq7$$\"5l;Hd?RL v$>*F-$\"-MAW817Fgq7$$\"5ILe9mmiF&>*F-$\"-Xxs#)=7Fgq7$$\"5+](=*z'>* F-$\"-b3t/\\7Fgq7$$\"5lm;Hd@@K)>*F-$\"-VjlJ88Fgq7$$\"5I$ekG!\\]%)*>*F- $\"-UZg$)Q9Fgq7$$\"5++vV[wzO,#*F-$\"-zTh/o;Fgq7$$\"5l;/,%R!4*G?*F-$\". I]nnV1#Fbr7$$\"5LLLeRJQT/#*F-$\".$*))\\;(yEFbr7$$\"5lmT&Q.3Qr?*F-$\"-) >7'*zn#Fgq7$$\"5++]7GHB')4#*F-$\"-_Ph!zn#Fgq7$$\"5q;/Ev`WA6#*F-$\"-.7) >&zEFgq7$$\"5NLeRAyle7#*F-$\"-yc(e[o#Fgq7$$\"5+]7`p-([R@*F-$\"-)=\"HW) p#Fgq7$$\"5mmmm;F3J:#*F-$\"-r\"3W$GFFgq7$$\"5+](=UZW:#=#*F-$\"-.QY.=HF gq7$$\"5NL3xJi+7@#*F-$\",06k?_$F-7$$\"5+voa5rBdA#*F-$\",@^J59%F-7$$\"5 q;HK*)zY-C#*F-$\",C=j\"fXF-7$$\"5Ne*)4o))pZD#*F-$\",x%R]eXF-7$$\"5++]( ouHHpA*F-$\",QJ.zb%F-7$$\"5+]i!*GU$>$G#*F-$\",7qzwb%F-7$$\"5++v$4rQ4(H #*F-$\",e>T(eXF-7$$\"5+](oH>V*4J#*F-$\",dncLc%F-7$$\"5++++vw%*[K#*F-$ \",;-bhd%F-7$$\"5++D1Rm&p_B*F-$\",.W[bm%F-7$$\"5++]7.c'\\!Q#*F-$\",vX' 4w\\F-7$$\"5+vVt-\\MaR#*F-$\",&\\c!yL&F-7$$\"5+]PM-Us.T#*F-$\",'eE!Q$f F-7$$\"5]P%[@&QTyT#*F-$\",.8cJN'F-7$$\"5+DJ&>].JDC*F-$\",D@47k'F-7$$\" 5]7yv^JzFV#*F-$\",VC82k'F-7$$\"5++Dc,G[-W#*F-$\",aF=-k'F-7$$\"5+]7y+9C ,Z#*F-$\",S()H'QmF-7$$\"$D*!\"$$\",1\")))Rk'F--%'COLOURG6&%$RGBG$\"\"! FcfmFbfm$\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7[r7$ F($\")D!=:\"F-7$F/$\"'F\"G%F-7$F4$!'$y/#F-7$F9$!)%y`\\\"F-7$F>$!)#*)R3 #F-7$FC$!)%z;e$F-7$FH$!)%oK>'F-7$FM$!)GzVkF-7$FR$!)5>b)*F-7$FW$!*))zV6 \"F-7$Ffn$!*@CIW\"F-7$F[o$!*&Q>9?F-7$F`o$!*)=-Y?F-7$Feo$!*A@cs#F-7$Fjo $!*O450$F-7$F_p$!*V*o(f$F-7$Fdp$!*GjWk%F-7$Fip$!*%)=%yYF-7$F^q$!*mhC(e F-7$Fcq$!+S3(3D'Fgq7$Fiq$!+N)**zF(Fgq7$F^r$!,fRJd$))Fbr7$Fir$!+HaE%))) Fgq7$Fgt$!,AU5f/\"Fgq7$F\\u$!+no)H4\"F-7$Fjv$!+_$o/?\"F-7$F^x$!+[ASK8F -7$Fhx$!+nn)>K\"F-7$Ffz$!+-;M)R\"F-7$F[[l$!+p<;[8F-7$$\"5++DJ?EaoP!*F- $!+eC?_8F-7$F`[l$!+NKIX8F-7$Fd\\l$!+rZ]g7F-7$Fh]l$!+>1$)>6F-7$F]^l$!+& zWZ6\"F-7$Fb^l$!+ws`K5F-7$Fg^l$!*)HKQ&)F-7$F\\_l$!*')oW.'F-7$Fa_l$!*'o 4KgF-7$Ff_l$!*f#Q8gF-7$F[`l$!),6!y\"F-7$$\"5]i:5/JI8!4*F-$\"*\\#4t:F-7 $$\"5Ne9mpL$o34*F-$\"*&*Q)HKF-7$$\"5?a8ANOOg\"4*F-$\"*c,'HKF-7$$\"5+]7 y+R*QB4*F-$\"*7n$HKF-7$$\"5qT5!>Va4Q4*F-$\"*%G/HKF-7$$\"5NL3-j\\,G&4*F -$\"*Xs+B$F-7$$\"5q;/EDg8A)4*F-$\"*amQE$F-7$F``l$\"*!>?BNF-7$$\"5lmT&Q 3Z(p1\"*F-$\"*f.K*zF-7$Fe`l$\"+))[64=F-7$$\"5+++vV`^7:\"*F-$\"+%y;\"4= F-7$Fj`l$\"+a?a;=F-7$$\"5++D1RFVY>\"*F-$\"+SwIK=F-7$$\"5ILL$3(=2\"47*F -$\"+c^Ip=F-7$$\"5lmTg-5rNA\"*F-$\"+P:^Z>F-7$F_al$\"+*p=-5#F-7$Fdal$\" +W>8SGF-7$Fial$\"+Znt;SF-7$F^bl$\"+;Qj:SF-7$Fcbl$\"+G-6;SF-7$$\"5lT&Q` 0%ofO\"*F-$\"+6Yr?SF-7$Fhbl$\"+0jxNSF-7$$\"5Ie*[V(GxbR\"*F-$\"+)HJ_2%F -7$F]cl$\"+]hWlTF-7$Fgcl$\"+BuF7ZF-7$Fadl$\"+PI5&['F-7$Ffdl$\"+i&>C&oF -7$F[el$\"+E'pQ3(F-7$F`el$\"+'4YO$3(F-7$Fjel$\"+l&fG3 (F-7$F_fl$\"+aVN#3(F-7$Fdfl$\"+%fb83(F-7$Fifl$\"+(yK03(F-7$$\"5q\"z>6t ;*3a\"*F-$\"+'zL23(F-7$F^gl$\"+b&RZ3(F-7$$\"5N3x\"[X&3%p:*F-$\"+REg*4( F-7$Fcgl$\"+0M[SrF-7$Fhgl$\"+rmjM')F-7$F]hl$\",f%\\#e5\"F-7$Fbhl$\",>D hb5\"F-7$Fghl$\",aHhq5\"F-7$F\\il$\",oBg66\"F-7$Fail$\",&3be@6F-7$Ffil $\",qzGZ9\"F-7$F[jl$\",:](Q\">\"F-7$F`jl$\",iv>@S\"F-7$Fejl$\",Up&R!e \"F-7$Fjjl$\"-O,$f*z:Fgq7$F_[m$\"-eQ&y%z:Fgq7$Fd[m$\"-AOz_z:Fgq7$Fi[m$ \"-x!>l4e\"Fgq7$$\"5l\"HK*QIx.'>*F-$\"-]L9%He\"Fgq7$F^\\m$\"-lO/X'e\"F gq7$$\"5I3_]%ylgv>*F-$\"- *[#F-7$Fgbm$\",KP>U[#F-7$Facm$\",N#=`xCF-7$F[dm$\",](ySPDF-7$F_em$\",k :%H/EF-7$Fdem$\",2AELg#F-7$Fiem$\",)>Pj*f#F--F_fm6&FafmFdfmFbfmFdfm-Fh fm6#%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG-F$6%7fs7$F( $\"*8b]i\"F-7$F/$\"*&>Gf=F-7$F4$\"*G?='=F-7$F9$\"*#*)z'3#F-7$F>$\"*r+7 8#F-7$FC$\"*$)[EH#F-7$FH$\"*PuwW#F-7$FM$\"*+:hX#F-7$FR$\"*P\"4QDF-7$FW $\"*\\9X`#F-7$Ffn$\"*\\l6]#F-7$F[o$\"*soNH#F-7$F`o$\"*$z$eF#F-7$Feo$\" *d-*Q=F-7$Fjo$\"*Z:F-7$Fip$!)c+'=# F-7$F^q$!*dL/\">F-7$Fcq$!+?&*)Rj#Fgq7$Fiq$!+g%frM%Fgq7$F^r$!,Mjn]a(Fbr 7$Fdr$!+wsGXvFgq7$Fir$!+'RWp$yFgq7$F^s$!+LoZM\")Fgq7$Fcs$!+dj1e')Fgq7$ Fhs$!+[VbO&*Fgq7$F]t$!,k$[E&4\"Fgq7$Fbt$!,t$R(R;\"Fgq7$Fgt$!,%>cqj6Fgq 7$F\\u$!+:r)GO\"F-7$Fjv$!+6FGr;F-7$F_w$!+n)[lp\"F-7$Fdw$!+g8+&y\"F-7$F iw$!+hC\"33\"*F-$!+:JV)4%F-7$$\"5+]7.#3#\\Y4\"*F-$!+ %Ga\"zPF-7$$\"5l\"z>6ek[36*F-$!+%pTXs$F-7$Fe`l$!++d-CPF-7$Fjan$!+!eWFs $F-7$Fj`l$!+$>:sr$F-7$Fbbn$!+7^M1PF-7$Fgbn$!+BFuzOF-7$F\\cn$!+#Qu1i$F- 7$F_al$!+ST***\\$F-7$Fdal$!+\\#)=zGF-7$Fial$!+Z2/X=F-7$F^bl$!+i+\\W=F- 7$Fcbl$!+?.NU=F-7$F`dn$!+7'[m$=F-7$Fhbl$!+NYn>=F-7$Fhdn$!+NPZF-7$Fdem$\",I$R-OZF-7$F iem$\",?%odOZF--%&COLORG6&Fafm$\"\"\"!\"\"$\"\")F`jpFbfm-Fhfm6#%3effic ient~scheme~AG-F$6%7ir7$F($\")\"Q)HfF-7$F/$\")))*yn&F-7$F4$\")%HCj&F-7 $F9$\")BHV]F-7$F>$\")'3\"yYF-7$FC$\")8*H)QF-7$FH$\")_XK@F-7$FM$\")YNX> F-7$FR$!(!\\.pF-7$FW$!)lq*y\"F-7$Ffn$!)6n]YF-7$F[o$!*]Mf+\"F-7$F`o$!*0 o$Q5F-7$Feo$!*?rit\"F-7$Fjo$!*l%H$4#F-7$F_p$!*-Psp#F-7$Fdp$!*Y8v#RF-7$ Fip$!*k=g(RF-7$F^q$!*gL)oaF-7$Fcq$!+!oc<+'Fgq7$Fiq$!+dy*yN(Fgq7$F^r$!, r'\\'ff*Fbr7$Fir$!+nRME(*Fgq7$Fgt$!,^Fm_@\"Fgq7$F\\u$!+lG.98F-7$Fjv$!+ \"4.Q\\\"F-7$F_w$!+c8v+:F-7$Fdw$!+:*zi`\"F-7$Fiw$!+>n*zl\"F-7$F^x$!+%Q @lx\"F-7$Fcx$!+Hn@vj)F-7$ Fban$!**)eYd'F-7$Fjjo$!*r%f^JF-7$F_[p$\"*n5jM#F-7$Fd[p$\"*%)yBD$F-7$Fe `l$\"*ro>D$F-7$Fjan$\"*RAgD$F-7$Fj`l$\"*FSSL$F-7$Fbbn$\"*7qg\\$F-7$Fgb n$\"*!>FyQF-7$$\"5+](=nV\"Rj@\"*F-$\"*akJ@%F-7$F\\cn$\"*0@Bp%F-7$$\"5I $e*[o0.3B\"*F-$\"*Vq]O&F-7$F_al$\"*c3QH'F-7$Fdal$\"+>1W89F-7$Fial$\"+9 amqEF-7$F^bl$\"+Bx%*pEF-7$Fcbl$\"+(eO5n#F-7$F`dn$\"+H'F-7$Fcg l$\"+B6&>D'F-7$Fhgl$\"+&RFgq7$Fb]m $\".HUN]s5#Fbr7$Fg]m$\".NJ'G/CCFbr7$Fa^m$\"-98XoACFgq7$Fe_m$\"-*yQr/V# Fgq7$Fj_m$\"-Mq5%H[#Fgq7$F_`m$\",y]pMp#F-7$Fd`m$\",J!\\TMHF-7$Fi`m$\", EU:\\5$F-7$F^am$\",/cjW5$F-7$Fcam$\",yY(*R5$F-7$F]bm$\",&QF*G5$F-7$Fgb m$\",,-h>5$F-7$F[hp$\",2X^K5$F-7$F\\cm$\",V8u(3JF-7$Fhhp$\",IkbN7$F-7$ Facm$\",hZxh:$F-7$F[dm$\",0Ni&fLF-7$F_em$\",x(\\uJNF-7$Fdem$\",H[l0`$F -7$Fiem$\",(HLzFNF--F_fm6&FafmFdfmFbfmFbfm-Fhfm6#%3efficient~scheme~BG -F$6%7fs7$F($\"*]'yq;F-7$F/$\"*JN$*)=F-7$F4$\"*Xm3*=F-7$F9$\"*,**Q4#F- 7$F>$\"*\")z!H@F-7$FC$\"*h&QnAF-7$FH$\"*$zq!Q#F-7$FM$\"*ue]Q#F-7$FR$\" *C(=6CF-7$FW$\"*Y8gQ#F-7$Ffn$\"*q\"p(H#F-7$F[o$\"*TrF*>F-7$F`o$\"*+'Qp >F-7$Feo$\"*#)*p99F-7$Fjo$\"*h3e2\"F-7$F_p$\")S?hZF-7$Fdp$!)\"y@?*F-7$ Fip$!)MK4**F-7$F^q$!*9.)**GF-7$Fcq$!+y42'p$Fgq7$Fiq$!+M\")*zf&Fgq7$F^r $!,T#pW\"4*Fbr7$Fdr$!+<\\m!4*Fgq7$Fir$!+e^M'R*Fgq7$F^s$!+S=_8(*Fgq7$Fc s$!,cuNv-\"Fgq7$Fhs$!,#*)yJA6Fgq7$F]t$!,[%H\"eF\"Fgq7$Fbt$!,_-p/N\"Fgq 7$Fgt$!,y'\\9]8Fgq7$F\\u$!+&)z))f:F-7$Fjv$!+57k*)=F-7$F_w$!+VG=:>F-7$F dw$!+-yq1?F-7$Fiw$!+!R+'oAF-7$F^x$!+'3)p.DF-7$Fcx$!+f3)H]#F-7$Fhx$!+wc ]WDF-7$F]y$!+p(RSf#F-7$Fby$!+:\"*='o#F-7$Fgy$!+7g!y%GF-7$F\\z$!+w`(y6$ F-7$Faz$!+M#>p<$F-7$Ffz$!+O78wJF-7$F[[l$!+z_h8NF-7$F`[l$!+,AM\\QF-7$Fj [l$!+f+=wQF-7$Fd\\l$!+Az,#*RF-7$Fi\\l$!+,XFGTF-7$F^]l$!+S'G*eVF-7$Fc]l $!+CVA9WF-7$Fh]l$!+0Aa8WF-7$F]^l$!+rl[7WF-7$Fb^l$!+XF-7$F``l$!+_EF/XF-7$Fhio$!+$y=`[%F- 7$F]jo$!+R//YWF-7$Fbjo$!+5:upVF-7$Fban$!+*>h'HUF-7$Fjjo$!+`3i%)RF-7$F_ [p$!+]$F-7$Fj`l$!+ \"3'4&\\$F-7$Fbbn$!+%*Gb\"[$F-7$Fgbn$!+h?q[MF-7$F\\cn$!+SjXwLF-7$F_al$ !+N14IKF-7$Fdal$!+Y77&[#F-7$Fial$!+c%HRD\"F-7$F^bl$!+f'RND\"F-7$Fcbl$! +uZF^7F-7$F`dn$!+,UtW7F-7$Fhbl$!+sp3D7F-7$Fhdn$!+W(GX<\"F-7$F]cl$!+L)[ \"f5F-7$Fgcl$!*b2'pNF-7$Fadl$\"+So^J>F-7$Ffdl$\"+w6l1CF-7$F[el$\"+a=C1 FF-7$F`el$\"+n`91FF-7$Feel$\"+L*[gq#F-7$Fjel$\"+gh&eq#F-7$F_fl$\"+t]m0 FF-7$Fdfl$\"+0kL0FF-7$Fifl$\"+gEZ0FF-7$F^gl$\"+F%Rsr#F-7$Fcgl$\"+.,n=F gq7$Fh\\m$\"-RC!*43>Fgq7$F]]m$\"-y8`Ov?Fgq7$Fb]m$\".eQP/QP#Fbr7$Fg]m$ \".EW=_'[GFbr7$F\\^m$\"-1(>yy%GFgq7$Fa^m$\"-`\"QBt%GFgq7$Ff^m$\"-#o1`x %GFgq7$F[_m$\"-'H>.+&GFgq7$F`_m$\"-C?=mcGFgq7$Fe_m$\"-%p>*osGFgq7$Fj_m $\"-)HfMa)HFgq7$F_`m$\",NrzpP$F-7$Fd`m$\",A5#4(z$F-7$Fi`m$\",0R(o'3%F- 7$F^am$\",7I%4'3%F-7$Fcam$\",@E8b3%F-7$F]bm$\",2,7]3%F-7$Fgbm$\",wCj54 %F-7$Ffgp$\",Js=f4%F-7$F[hp$\",z*zT.TF-7$F`hp$\",F^,Y6%F-7$F\\cm$\",'3 +\"38%F-7$Fhhp$\",_CDb=%F-7$Facm$\",&*G)o'G%F-7$F[dm$\",r'pP5[F-7$F_em $\",L.UL@&F-7$Fdem$\",cjW=@&F-7$Fiem$\",74[=@&F--F_fm6&Fafm$\")#)eqkFf fm$\"))eqk\"FffmF^]t-Fhfm6#%3efficient~scheme~CG-F$6%7`t7$F($\"*mS$fTF -7$F/$\"*$[`:]F-7$F4$\"*cU[.&F-7$F9$\"*wG#QfF-7$F>$\"*1j[<'F-7$FC$\"*1 y*4pF-7$FH$\"*>#oYyF-7$FM$\"*<:\">zF-7$FR$\"*\"*zS\"))F-7$FW$\"*[7(e!* F-7$Ffn$\"*\"o,9'*F-7$F[o$\"+FW=<5F-7$F`o$\"+(=/(=5F-7$Feo$\"+&o%=L5F- 7$Fjo$\"+!R-4-\"F-7$F_p$\"*Vxw!**F-7$Fdp$\"*G<-n)F-7$Fip$\"*\\T`c)F-7$ F^q$\"*9j()H'F-7$Fcq$\"+d*H#z]Fgq7$Fiq$\"+ea/ACFgq7$F^r$!,c1*HHKFbr7$F dr$!+9@NUKFgq7$Fir$!+4L(4#RFgq7$F^s$!+!pwVa%Fgq7$Fcs$!+\"RcPf&Fgq7$Fhs $!+6cU(G(Fgq7$F]t$!+0A[D**Fgq7$Fbt$!,X_'==6Fgq7$Fgt$!,KA)3=6Fgq7$F\\u$ !+gC$fc\"F-7$Fau$!+'[V&4>F-7$Ffu$!+A?2x@F-7$F[v$!+'3Wn<#F-7$F`v$!+U#F-7$F_w$!+\"[X^E#F-7$Fdw$!+Poj$[#F-7$ Fiw$!+@,`ZIF-7$F^x$!+SiTDNF-7$Fcx$!+O_$o_$F-7$Fhx$!+v5-eOF-7$F]y$!+V&* *Gz$F-7$Fby$!+UJsFSF-7$Fgy$!+x@j;WF-7$F\\z$!+#QKY.&F-7$Faz$!+OP*o;&F-7 $Ffz$!+ETPm^F-7$F[[l$!+Wer$='F-7$$\"5]7.K.pz!G.*F-$!+[:8wkF-7$$\"5+D1* G'[Z]L!*F-$!+16[OoF-7$$\"5]P4YAG:?M!*F-$!+]MRnqF-7$$\"5+]7.#yI)*[.*F-$ !+_5!p1(F-7$$\"5+v=<,n=HO!*F-$!+GH#f1(F-7$Fi\\n$!+bz.lqF-7$$\"5+]PfeWD ZS!*F-$!+aM+mqF-7$F`[l$!+&4,v3(F-7$Fe[l$!+\\-(y7(F-7$Fj[l$!+Y$\\H@(F-7 $F_\\l$!+'\\SeP(F-7$Fd\\l$!+d]&fm(F-7$Fi\\l$!+T0&R:)F-7$F^]l$!+nYRP*)F -7$Fc]l$!+E++@\"*F-7$Fh]l$!+ayg>\"*F-7$F]^l$!+%p!pG\"*F-7$Fb^l$!+*Q0NU *F-7$Fg^l$!,7f^B,\"F-7$F\\_l$!,!4C=86F-7$Fa_l$!,J/mG6\"F-7$Ff_l$!,8BRH 6\"F-7$F[`l$!,-(y)4A\"F-7$F``l$!,noF)y7F-7$Fban$!,*G_v48F-7$Fe`l$!,_,) **\\8F-7$Fj`l$!,)>9g\\8F-7$F_al$!,i$*>_M\"F-7$$\"5l\"H2LKh<_7*F-$!,#Q* )oO8F-7$Fdal$!,R(QJ=8F-7$$\"5+v=<,Pe/G\"*F-$!,&H'eDG\"F-7$Fial$!,W?6@E \"F-7$F^bl$!,%RJvh7F-7$Fcbl$!,Jsv7E\"F-7$Fhbl$!,mOe*e7F-7$F]cl$!,M7-^C \"F-7$$\"5!3x1')[Ry<9*F-$!,52AnB\"F-7$Fbcl$!,'4LyC7F-7$$\"5:z>6)*Q)eK9 *F-$!,,xx!37F-7$Fgcl$!,]I#3&=\"F-7$F\\dl$!,'yG=76F-7$Fadl$!+SE'4&)*F-7 $Ffdl$!+'4m@V*F-7$F[el$!+_`_n\"*F-7$F`el$!+$f)>n\"*F-7$Feel$!+'yro;*F- 7$Fjel$!+ox@m\"*F-7$F_fl$!+!Gfb;*F-7$Fdfl$!+eO8k\"*F-7$Fifl$!+x\\\">;* F-7$F^gl$!+M4!49*F-7$Fcgl$!+-.&[+*F-7$Fhgl$!+]$[]?'F-7$F]hl$!+#Gq`+#F- 7$Fbhl$!+X7<&*>F-7$Fghl$!+/dj&*=F-7$F\\il$!+)R@Ts\"F-7$Fail$!+`BRg8F-7 $Ffil$!*V>?]'F-7$F[jl$\"*mg*ykF-7$F`jl$\"+aq*o)eF-7$Fejl$\",bC:A+\"F-7 $Fjjl$\"--KW0-5Fgq7$F_[m$\"-lVZ./5Fgq7$Fd[m$\"-GTb-55Fgq7$Fi[m$\"-8:#= j-\"Fgq7$F^\\m$\"-&4k_V1\"Fgq7$Fc\\m$\"-U+!))R9\"Fgq7$Fh\\m$\"-Bfl[(H \"Fgq7$F]]m$\"-b[ehu:Fgq7$Fb]m$\".g7t>!\\?Fbr7$Fg]m$\".&Q\"es!yFFbr7$F \\^m$\"-S[&Htx#Fgq7$Fa^m$\"-GvAZxFFgq7$Ff^m$\"-?N%G%QF-7$Fd`m$\",:x=yg%F-7$Fi`m$\",^LP;7&F-7$F^am$\",7)y*37&F -7$Fcam$\",$*)\\C?^F-7$Fham$\",aWF,7&F-7$F]bm$\",)e#G=7&F-7$Fbbm$\",_^ n%G^F-7$Fgbm$\",%)*yHY^F-7$F\\cm$\",7'*)\\n_F-7$Facm$\",ery&zcF-7$Ffcm $\",nPkP:'F-7$F[dm$\",t)>**GpF-7$F`dm$\",]*eoruF-7$Fedm$\",A'p`VyF-7$F jdm$\",$>7&H%yF-7$F_em$\",j+nB%yF-7$Fdem$\",6v=1%yF-7$Fiem$\",3p'4\\yF --F\\jp6&Fafm$\"\"'F`jpFbfmFajp-Fhfm6#%Sscheme~with~stability~radius~= ~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~cur ves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F j[v-%%VIEWG6$;F(Fiem%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modificatio n of the Bogacki-Shampine scheme" "efficient scheme A" "efficient sche me B" "efficient scheme C" "scheme with stability radius = imag axis i nclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 513 "evalf[20](plot(['qn_RK5_1'(x)-q(x),'qn_RK5_2'(x)-q (x),'qn_RK5_3'(x)-q(x),'qn_RK5_4'(x)-q(x),\n'qn_RK5_5'(x)-q(x),'qn_RK5 _6'(x)-q(x)],x=0.925..0.9469483523,\ncolor=[blue,magenta,COLOR(RGB,.1, .8,0),red,brown,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9],\nlegend=[`Boga cki-Shampine scheme`,`a minor modification of the Bogacki-Shampine sch eme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`s cheme with stability radius = imag axis inclusion`],\ntitle=`error cur ves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 777 601 601 {PLOTDATA 2 "6,-%'CURVESG6%7_s7$$\"$D*!\"$$\",1 \")))Rk'!#?7$$\"5#HT(y=;Tya#*F-$\",Lub6y'F-7$$\"5x:8C5]n%*e#*F-$\",AZ' *o[(F-7$$\"5/&Gg&Hn!GOE*F-$\",0s!pY%)F-7$$\"5H&H!zH:/Mo#*F-$\",')>^mW) F-7$$\"5->FFLm..t#*F-$\",\"4EK3&)F-7$$\"55GIvH^&ytF*F-$\",I!pX'y)F-7$$ \"5%*R\")=(\\%3)=G*F-$\",JEeF6*F-7$$\"5g)H\\IN7PlG*F-$\",YuY#4\"*F-7$$ \"5>L9ZMp%y6H*F-$\",rB6]4*F-7$$\"5e\"*f&3Sm_fH*F-$\",>/4Sr)F-7$$\"5zVB rExx:+$*F-$\",9+JxI(F-7$$\"5v\\**y\")Q<*[I*F-$\",+3J*F-$\",WO'RWpF-7$$\"5D]9w\")>fA9$*F -$\",N[)))>hF-7$$\"5!ph%H(R\"eI;$*F-$\",Q\\]3F%F-7$$\"5g$yFG\"3dQ=$*F- $\",kG$>M8F-7$$\"55#pj]M!*e3K*F-$\",j')HQL\"F-7$$\"5e+'*Hx)4KLK*F-$\", wVP:L\"F-7$$\"5LK,#o/MAvK*F-$\",\">KwK7F-7$$\"5S'p3&3k%f*H$*F-$\"+\")G GC&)F-7$$\"5Wgs>q(e'RK$*F-$!+>#)egSF-7$$\"50*ozz8AvML*F-$!,)[ZzW:F-7$$ \"5l<@w0bQbM$*F-$!,r>\\jF$F-7$$\"5DYXat)[KcL*F-$!,)[*\\;%eF-7$$\"5$[(p KTA6rO$*F-$!,_eaw8F -7$$\"5g(=KoxB:GN*F-$!.4Qd%o#e\"!#@7$$\"5bA-'**y*\\*QN*F-$!.-ZWB&))=F] v7$$\"5_d#)3.eZ(\\N*F-$!.Z\"\\zSJBF]v7$$\"50**Qph\"4dbN*F-$!.-`ZhMk#F] v7$$\"5gS&*H?D%RhN*F-$!.qVVxJ#GF]v7$$\"55#=0*yenUL'f$*F-$!.&*y_eA#GF]v7$$\"5?C$\\b/&H0i$*F-$!.[FoPI#GF]v7$$\"5g e_;>uCZk$*F-$!.H6wEM$GF]v7$$\"55y0mrw'ylO*F-$!.&=A&Q>(GF]v7$$\"5c(*e:C z[oo$*F-$!.WM2*G&)HF]v7$$\"5?a&obuDA)p$*F-$!.gCnLt5$F]v7$$\"5!3@\")pcj f4P*F-$!./q)Q())H$F]v7$$\"5SnQR)Q,(4s$*F-$!.gbeX%*e$F]v7$$\"5/Cl!)4#RM KP*F-$!.CF#H\\f.>#QP*F-$!-yX%)33VF-7$$\"5!>jz&3:%4WP*F -$!-RDEDcYF-7$$\"5Sqi#fHHcXP*F-$!-P*H;Lv%F-7$$\"5&)3HF$3<.ZP*F-$!-l1([+&[P*F-$!-!p88d\"\\F-7$$\"5!e=mzl#p*\\P*F-$!-`O4k:\\F -7$$\"5!GYfEBo!Hv$*F-$!-/Rl\\:\\F-7$$\"5vRFN2QWev$*F-$!-f^@N:\\F-7$$\" 5gZe71h%fnP*F-$!-+ahx9\\F-7$$\"5]b*)*[S[MzP*F-$!-SXPA9\\F-7$$\"5qwLnbX NB!Q*F-$!-czZ$Q\"\\F-7$$\"5)yzZkqgKDQ*F-$!-KY'R'=\\F-7$$\"5lqg()Gnqv%Q *F-$!-4s**[U\\F-7$$\"5WVVI^F:)pQ*F-$!-m29<=]F-7$$\"5&yM`A%pk@)Q*F-$!-2 yU\"Q5&F-7$$\"5I_B?L69X*Q*F-$!-H&ov\"R_F-7$$\"5qc8:C`jo!R*F-$!-lhuWVaF -7$$\"57h.5:&H@>R*F-$!-oHE^RdF-7$$\"5buSHQX4.$R*F-$!-=#o4_5'F-7$$\"5+) y([h&fSTR*F-$!-bWP<#R'F-7$$\"5S,:o%eC]_R*F-$!-?!yk9R'F-7$$\"5$[@vyg*)f jR*F-$!-t+Iw!R'F-7$$\"508PT-t%H()R*F-$!-@[Qb*Q'F-7$$\"5D6A&p*\\!*4,%*F -$!-LHbE!R'F-7$$\"5gFC8xFMR0%*F-$!-plCX*R'F-7$$\"5lZ@\"z+:nlS*F-$!-2n] '[R'F-7$$\"5qn=pQs3u2%*F-$!-6V#eQP'F-7$$\"5q(er%p%f9*3%*F-$!-c<@%*=jF- 7$$\"5v28D+<$)35%*F-$!-e\"*HN,iF-7$$\"5D*=$QA)p#>6%*F-$!--,Ed$*fF-7$$ \"5vq]^WzqH7%*F-$!-#4rSaj&F-7$$\"55T![]Z)R6Ol`78%*F-$!-$f$\\#*oaF-7$$\"5D_pkmg9S8 %*F-$!-'\\%RxoaF-7$$\"5+$*GrF^O&RT*F-$!-1,?ZoaF-7$$\"5xL)y()=%e]9%*F-$ !-[g,1R[CGM7>%*F-$!-%H'f) RY&F-7$$\"5!z#>ee&p!Q@%*F-$!-I&fpxW&F-7$$\"5e\\*zEH'zjB%*F-$!-=9B3o`F- 7$$\"5:'zZ$yP&>[U*F-$!-asw;d_F-7$$\"5vUc,k76+E%*F-$!-\"e@![\\]F-7$$\"5 N*[$o\\(o#=F%*F-$!-PR)Q6o%F-7$$\"5%fL^`BEk$G%*F-$!-\"*)\\!4dSF-7$$\"5! 3=[%Rc-kI%*F-$!-b>WZ\"e\"F-7$$\"5nD]aV]i\"HV*F-$\",sM/S&GF-7$$\"5]iV92 $yV_V*F-$\",S0<[&GF-7$$\"5H*pV2dJrvV*F-$\",.&3h$*GF-7$$\"5lsZFXXasQ%*F -$\",\"[^!)))HF-7$$\"50Ye!)>v&z)R%*F-$\",1+rRB$F-7$$\"5S>pL%\\qL5W*F-$ \",2%*p0z$F-7$$\"5x#*z')oMy=U%*F-$\",SS,(Q\\F-7$$\"5X.U81g$[KW*F-$\", \"QAk-pF-7$$\"5:9/SV&))3VW*F-$\"-ra%y'G5F-7$$\"5![im13Tp`W*F-$\"-HW()) oe\"F-7$$\"5]NG$zh$*HkW*F-$\"-74b0uCF-7$$\"5D/Em$zTXwW*F-$\"-x`ET!4%F- 7$$\"5+tBRp**3')[%*F-$\"-\\ze7XmF-7$$\"5SdsDdS'o%\\%*F-$\"-%4xX0S)F-7$ $\"5vT@7X\"Qw+X*F-$\".fx:Ne0\"F-7$$\"5!Rea!*=D!Q]%*F-$\".\\GW26=\"F-7$ $\"55Eq)HB7%o]%*F-$\".A<4ZTD\"F-7$$\"5Io%>pF*z)4X*F-$\".cD)*3TD\"F-7$$ \"5Y5>&3K'=H^%*F-$\".Kh(32a7F-7$$\"50l*pbf9mMX*F-$\".*f%Q1QD\"F-7$$\"5 n>!)GqG/kb%*F-$\".!)e!4\"QD\"F-7$$\"5&)z%fa\"[&*zc%*F-$\".)RqtXa7F-7$$ \"50S4jgn'ezX*F-$\".EI*=]c7F-7$$\"5D+C!eqy<\"f%*F-$\".^eUk:E\"F-7$$\"5 WgQ(4l!pFg%*F-$\".](z9\\s7F-7$$\"5&H*Q`sJjQh%*F-$\".'=UMr#H\"F-7$$\"5] DR4%pv&\\i%*F-$\"/;\\GMyG8F]v7$$\"50eRl:#=0OY*F-$\"/fXu3v*Q\"F]v7$$\"5 c!*R@P2Yrk%*F-$\"/x-01I)[\"F]v7$$\"5&H\\5zi$o!fY*F-$\"/ocuX#fl\"F]v7$$ \"5I&*pg=l!*4n%*F-$\"/3W>Oj:>F]v7$$\"5]Y_&R'z^pn%*F-$\"/Dl&)HU\"4#F]v7 $$\"5l(\\.$4%H\"Ho%*F-$\"/tZ/T?0BF]v7$$\"5!)[;#GB(p2CtG!#F-%'COLOURG6&%$RGBG$\"\"!F[emFjdm$\" *++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7as7$F($\",)>Pj* f#F-7$F/$\",0LNVb#F-7$F4$\",abyxN#F-7$F9$\",uS%[%4#F-7$F>$\",3N:/4#F-7 $FC$\",b:4.-#F-7$FH$\",t)>#[`\"F-7$FM$\"+e@A9]F-7$FR$\"+Q@u!*\\F-7$FW$ \"+$f6MB%F-7$$\"5S7Pmnmbc$H*F-$\"+!ze+1#F-7$Ffn$!+-EBhTF-7$$\"5qnTyj?_ 0)H*F-$!,.pkVq\"F-7$F[o$!,zxtGr#F-7$$\"5vY6D/eZ_-$*F-$!,_nnAr#F-7$F`o$ !,$G(fGr#F-7$Feo$!,`rBFz#F-7$Fjo$!,@[X9/$F-7$F_p$!,7iU1y$F-7$Fdp$!,.'H _8aF-7$Fip$!,D4<5)zF-7$F^q$!,8JW!zzF-7$Fcq$!,*e@fxzF-7$Fhq$!,$Ql.>!)F- 7$F]r$!,$))\\'*=#)F-7$Fbr$!,_S,N%*)F-7$Fgr$!,W.[ti*F-7$F\\s$!-_0M;p5F- 7$Fas$!-zm'\\+B\"F-7$Ffs$!-!z._rY\"F-7$F[t$!-h*)e?@:F-7$F`t$!-h-s$3_\" F-7$Fet$!-[%\\bD_\"F-7$Fjt$!-Go_!e`\"F-7$F_u$!-ymp%*)e\"F-7$Fdu$!-XULW V;F-7$Fiu$!.OG-r5t\"F]v7$F_v$!./o@_q'=F]v7$Fdv$!.zE\\O<2#F]v7$Fiv$!.&G Q$>(>AF]v7$F^w$!.&=)31gI#F]v7$Fcw$!.@5!=(eI#F]v7$Fhw$!.4'Rut0BF]v7$F]x $!.FLOmaI#F]v7$Fbx$!.qKy!=0BF]v7$Fgx$!.rs\"4V/BF]v7$F\\y$!./rYkPI#F]v7 $Fay$!.#og??1BF]v7$Ffy$!.d'y7[@BF]v7$F[z$!.Y]DkFM#F]v7$F`z$!.hdlq4Q#F] v7$Fez$!.K]N1`W#F]v7$Fjz$!.IS/A$[DF]v7$F_[l$!-G.f&>i#F-7$Fd[l$!-/Ea-8F F-7$Fh\\l$!-M&fpCy#F-7$Fb]l$!-kAhI#y#F-7$Fg]l$!-&RNy>y#F-7$F\\^l$!-(\\ !*Q;y#F-7$Fa^l$!-[<1p!y#F-7$Ff^l$!-Q!*)ezx#F-7$F[_l$!-d`q`pFF-7$F`_l$! -v8/CYFF-7$Fe_l$!-[fJP?FF-7$Fj_l$!-B[v^yEF-7$F_`l$!-,P#R?h#F-7$Fd`l$!- qU+-3DF-7$Fi`l$!-F8s%pO#F-7$F^al$!-2QEQYAF-7$Fcal$!-&Q:LhC#F-7$Fhal$!- _Rx(eC#F-7$F]bl$!-q^$4]C#F-7$Fbbl$!-'[qh2C#F-7$$\"5S>B/()QiC.%*F-$!-3% QpEA#F-7$Fgbl$!-%Quc)e@F-7$F\\cl$!-\"R!yxz?F-7$Facl$!-E,$*3V>F-7$Ffcl$ !-6j4n:DZBV*F-$\"-C*\\njz&F-7$Feil$\"-Djw .'z&F-7$Fjil$\"-6F*=Zz&F-7$F_jl$\"-q%**z[z&F-7$Fijl$\"-q@J45eF-7$Fc[m$ \"-D)G*p-fF-7$Fh[m$\"-$[1ek,'F-7$F]\\m$\"-eF,4>iF-7$Fb\\m$\"-pj'*\\ilF -7$Fg\\m$\"-q5[x?rF-7$F\\]m$\"-*>y%>f\")F-7$Fa]m$\"-H/yDJ)*F-7$Ff]m$\" .,G^#Q*4\"F-7$F[^m$\".;L(*\\LC\"F-7$F`^m$\".;(>9OF8F-7$Fe^m$\".XW#>Yw8 F-7$Fj^m$\".N%)4?kP\"F-7$F__m$\".iGFyjP\"F-7$Fd_m$\".tKHxgP\"F-7$Fi_m$ \".P'HPxv8F-7$Fc`m$\".8;:OeP\"F-7$F]am$\".Ti,W#z8F-7$$\"5qwQv6>;$3Y*F- $\".`!=M]\"Q\"F-7$Fbam$\".vpN(y%Q\"F-7$$\"5?4RJLW5%>Y*F-$\".8vHO%*Q\"F -7$Fgam$\"/eQ8H(eR\"F]v7$F\\bm$\"/6mT2H;9F]v7$Fabm$\"/$[cy!f^9F]v7$Ffb m$\"/Na`I4::F]v7$F[cm$\"/<\"R-t\"=;F]v7$F`cm$\"/j')\\>&**o\"F]v7$Fecm$ \"/E-m7vyUk(>5#**>??Fedm-Fgdm6&Fi dmF\\emFjdmF\\em-F`em6#%Ta~minor~modification~of~the~Bogacki-Shampine~ schemeG-F$6%7fr7$F($\",?%odOZF-7$F/$\",J1b'yZF-7$F4$\",V,,r,&F-7$F9$\" ,qyLbM&F-7$F>$\",`B2KM&F-7$FC$\",eS4TL&F-7$FH$\",m)[L+_F-7$FM$\",s`l)) y%F-7$FR$\",&\\>z&y%F-7$FW$\",7k%3OZF-7$Ffn$\",UXB&zSF-7$F]hm$\",u+C2* HF-7$F[o$\",lPs96#F-7$Fehm$\",&e:%46#F-7$F`o$\",#3dA4@F-7$Feo$\",Gj98- #F-7$Fjo$\",v4jyu\"F-7$F_p$\"+ryp&G*F-7$Fdp$!+j]4A*)F-7$Fip$!,yFakw$F- 7$F^q$!,Q:Sbw$F-7$Fcq$!,'3#zew$F-7$Fhq$!,%>e.QQF-7$F]r$!,eEOy8%F-7$Fbr $!,RzOP;&F-7$Fgr$!,mfNr5'F-7$F\\s$!,TW)oavF-7$Fas$!,c4Ejr*F-7$Ffs$!-un y\"oG\"F-7$F[t$!-)zDA%e8F-7$F`t$!-;gB;e8F-7$Fet$!-];Z\"\\O\"F-7$Fjt$!- )*of^%R\"F-7$F_u$!-/%=7f\\\"F-7$Fdu$!-,2'4Jf\"F-7$Fiu$!.iT[fQu\"F]v7$F _v$!.1W4L3(>F]v7$Fdv$!.TV:GQI#F]v7$Fiv$!.@8$paSDF]v7$F^w$!.>jPnun#F]v7 $Fcw$!.3VY6tn#F]v7$Fhw$!.:sfbrn#F]v7$F]x$!.^slXon#F]v7$Fbx$!.xGEaln#F] v7$Fgx$!.F%RF-7$Fh\\ l$!-f&[E9)RF-7$F]]l$!-XI&48)RF-7$Fb]l$!-vzD>\")RF-7$Fg]l$!-c!3D2)RF-7$ F\\^l$!-k\\EE!)RF-7$Fa^l$!-D6'R&zRF-7$Ff^l$!-:\\9Q!)RF-7$F`_l$!-F@*))> ,%F-7$Fd`l$!-h]SJSUF-7$Fhal$!-tQ6kBWF-7$Fbbl$!-.:y^?WF-7$Fhcn$!-HyVf7W F-7$Fgbl$!-5Ry[zVF-7$F\\cl$!-Fh0XLVF-7$Facl$!-:v)GwC%F-7$Ffcl$!-?jjJ&4 %F-7$F[dl$!-QD=@OQF-7$F`dl$!-h35QUMF-7$Fedl$!-%G`+E$GF-7$Fjdl$!-=\\u$f j#F-7$F_el$!->H=lkDF-7$Fdel$!-P@5ekDF-7$Fiel$!-B:-^kDF-7$F^fl$!-+!foVc #F-7$Fcfl$!-pVnAkDF-7$Fhfl$!-Uk+ajDF-7$F]gl$!-fqXugDF-7$Fbgl$!-14LRUDF -7$Fggl$!-PKs&\\X#F-7$F\\hl$!-%GJ.iL#F-7$Fahl$!-XT(fs6#F-7$Ffhl$!-9Lv# Rt\"F-7$F[il$!-p*H294\"F-7$F`il$\"-N`e!RU\"F-7$Feil$\"-e-\"RRI$F-7$Fji l$\"-KI1E.LF-7$F_jl$\"-LdPH0LF-7$Fdjl$\"-oS&o?J$F-7$Fijl$\"-AFvlILF-7$ F^[m$\"-;#GbVP$F-7$Fc[m$\"-DRyhmMF-7$Fh[m$\"-2hH3FOF-7$F]\\m$\"-D-)fr! RF-7$Fb\\m$\"-N&f%4uVF-7$Fg\\m$\"-*H1PH7&F-7$F\\]m$\"-P(>e()\\'F-7$Fa] m$\"-\"*yd#**o)F-7$Ff]m$\".<4#>H?5F-7$F[^m$\".Hu%z%o?\"F-7$F`^m$\".2sL (Q:8F-7$Fe^m$\".&G2\\ty8F-7$Fj^m$\".iA,$py8F-7$F__m$\".I)=6ly8F-7$Fd_m $\".yPXa$y8F-7$Fi_m$\".vXa*=y8F-7$Fc`m$\".YhAf&z8F-7$F]am$\".'RbJQ*Q\" F-7$Fbam$\".Mo9F]v7$Fabm $\"/#G%3'*zO:F]v7$Ffbm$\"/-Rp#zal\"F]v7$F[cm$\"/9a%)))3U=F]v7$F`cm$\"/ D1AKep>F]v7$Fecm$\"/I_D?`D@F]v7$Fjcm$\"0oeEd\"*[J#F^dm7$F`dm$\"5>WF--! fRKa#Fedm-%&COLORG6&Fidm$\"\"\"!\"\"$\"\")FdgpFjdm-F`em6#%3efficient~s cheme~AG-F$6%7dr7$F($\",(HLzFNF-7$F/$\",/j1Z]$F-7$F4$\",k:IGU$F-7$F9$ \",\"f$[wJ$F-7$F>$\",RdvOJ$F-7$FC$\",AXY]D$F-7$FH$\",,1;,$GF-7$FM$\",S 2'G\"*=F-7$FR$\",Y'GW))=F-7$FW$\",QjaF\"=F-7$Ffn$\"+>'3%p%*F-7$F]hm$!+ FVbBSF-7$F[o$!,izx`Y\"F-7$Fehm$!,iMq]Y\"F-7$F`o$!,Z')*4m9F-7$Feo$!,*=I $pb\"F-7$Fjo$!,3(QYQ=F-7$F_p$!,ov0]n#F-7$Fdp$!,=3HE_%F-7$Fip$!,gp6rU(F -7$F^q$!,lj\"GDuF-7$Fcq$!,*G+FCuF-7$Fhq$!,E>j(zuF-7$F]r$!,<]i5t(F-7$Fb r$!,VND1i)F-7$Fgr$!,Vl_9X*F-7$F\\s$!-ag[rt5F-7$Fas$!-+GN7n7F-7$Ffs$!-` Ik%4b\"F-7$F[t$!-#omabh\"F-7$F`t$!-E)R&=:;F-7$Fet$!-jz6n=;F-7$Fjt$!-Wj rOQ;F-7$F_u$!-b(\\0=r\"F-7$Fdu$!-'3wR\\y\"F-7$Fiu$!.#[^2w+>F]v7$F_v$!. &))>QEy?F]v7$Fdv$!.[V[#oUBF]v7$Fiv$!.4AomD`#F]v7$F^w$!.XN!H$Hk#F]v7$Fc w$!.5!)**yFk#F]v7$Fhw$!.r&[]iUEF]v7$F]x$!.gT(eJUEF]v7$Fbx$!.#[%=,?k#F] v7$Fgx$!.^!)e/9k#F]v7$F\\y$!._aKxBk#F]v7$Fay$!.WPJ(o]EF]v7$Ffy$!.pk)4% Ho#F]v7$F[z$!.]aFMBs#F]v7$F`z$!.tuG=))y#F]v7$Fez$!.z\"4j)e*GF]v7$Fjz$! .<>l\"ohIF]v7$F_[l$!-+@%zy<$F-7$Fd[l$!-[\"[;*>LF-7$Fh\\l$!-k6UKFMF-7$F b]l$!-AbG7FMF-7$Fg]l$!-V!f>nU$F-7$F\\^l$!-t!p3jU$F-7$Fa^l$!-PSsLDMF-7$ Ff^l$!-#G:\\MU$F-7$F`_l$!-6q%G&3MF-7$Fd`l$!-')HAF(H$F-7$Fhal$!-vVk5cJF -7$Fbbl$!-,&=)3^JF-7$Fhcn$!-;`LQMJF-7$Fgbl$!-pka2uIF-7$F\\cl$!-4aG'z*H F-7$Facl$!-yV\")okGF-7$Ffcl$!-6_$[.k#F-7$F[dl$!-B&Q;aF#F-7$F`dl$!-%*oS zSJ\"fF-7$F]gl$!,))z!3&)eF-7$Fbgl$!,YGmAo&F-7$Fggl$!,7+vCt%F-7$F\\hl$ !,$)y44Y$F-7$Fahl$!,HMQ%Q6F-7$Ffhl$\",2e=c*GF-7$F[il$\",K#)=?h*F-7$F`i l$\"-$p/O%oNF-7$Feil$\"-/f;/1bF-7$Fjil$\"-S'R8[]&F-7$F_jl$\"-\"*G,h0bF -7$Fijl$\"-=!)GcDbF-7$Fc[m$\"-TMA\"3k&F-7$Fh[m$\"-'eFb,y&F-7$F]\\m$\"- A#))*QEgF-7$Fb\\m$\"-q*>V5W'F-7$Fg\\m$\"-H*)ph6rF-7$F\\]m$\"-a.^0`$)F- 7$Fa]m$\".rQusV.\"F-7$Ff]m$\".%3.KUs6F-7$F[^m$\".8ve/JM\"F-7$F`^m$\".X ()y%fU9F-7$Fe^m$\".l.\">r+:F-7$Fj^m$\".O)3jm+:F-7$F__m$\".Z\"32i+:F-7$ Fd_m$\".jB3%H+:F-7$Fi_m$\".9yK2+]\"F-7$Fc`m$\".Goa#\\+:F-7$F]am$\".A/U ki]\"F-7$Fb]o$\".?tws(4:F-7$Fbam$\".d%)eVZ^\"F-7$Fj]o$\".@YrO;_\"F-7$F gam$\"/r2#f=5`\"F]v7$F\\bm$\"/f\"[][,c\"F]v7$Fabm$\"/O9Q:Y4;F]v7$Ffbm$ \"/!f0(pr'p\"F]v7$F[cm$\"/.GP-`O=F]v7$F`cm$\"/'HF(G;L>F]v7$Fecm$\"/e]% R%>_?F]v7$Fjcm$\"0`c8L\\w>#F^dm7$F`dm$\"5>2j3^\"\\mSP#Fedm-Fgdm6&FidmF \\emFjdmFjdm-F`em6#%3efficient~scheme~BG-F$6%7fr7$F($\",74[=@&F-7$F/$ \",>Q$zZ_F-7$F4$\",c=%zgaF-7$F9$\",*zRobdF-7$F>$\",jv'p_dF-7$FC$\",eG: Gt&F-7$FH$\",T()yQ_&F-7$FM$\",UYq9&\\F-7$FR$\",ec!)z%\\F-7$FW$\",%GY\" f)[F-7$Ffn$\",P=O@4%F-7$F]hm$\",w=\"e#z#F-7$F[o$\",Z$RI[$e'F-7$Fgr$!,!*o<_k (F-7$F\\s$!,o*>uw#*F-7$Fas$!-;nacr6F-7$Ffs$!-nBv]F:F-7$F[t$!-KQOR3;F-7 $F`t$!-/\"ou!3;F-7$Fet$!-)=N;_h\"F-7$Fjt$!-Co]MZ;F-7$F_u$!-Uai]eRmWF-7$F\\^l$!-)3pqeY%F-7$Fa^l$!-*>n$* \\Y%F-7$Ff^l$!-lJcWlWF-7$F`_l$!-`zy'H\\%F-7$Fd`l$!-zHCH\"p%F-7$Fhal$!- ^qQ(4%[F-7$Fbbl$!-qm#Gq$[F-7$Fhcn$!-8J7FE[F-7$Fgbl$!-Y&*38$y%F-7$F\\cl $!-];UhCZF-7$Facl$!-\"[bmrh%F-7$Ffcl$!-`hDzGWF-7$F[dl$!-lR?_6TF-7$F`dl $!-6#3.Jj$F-7$Fedl$!-`#\\Xr*GF-7$Fjdl$!-o**\\fgEF-7$F_el$!-;Yj$\\d#F-7 $Fdel$!-Qa_'[d#F-7$Fiel$!-.kTzuDF-7$F^fl$!-]h>luDF-7$Fcfl$!-\"*f%4Xd#F -7$Fhfl$!-8R#*ztDF-7$F]gl$!-vC_eqDF-7$Fbgl$!-iVH9\\DF-7$Fggl$!-Ef7KZCF -7$F\\hl$!-?(*eR4BF-7$Fahl$!-Sj.ab?F-7$Ffhl$!-mP8s6;F-7$F[il$!,X:Oro)F -7$F`il$\"-&)4xgN?F-7$Feil$\"-[\">'>/UF-7$Fjil$\"-?gZJ.UF-7$F_jl$\"-0$ p>a?%F-7$Fdjl$\"->TV(G@%F-7$Fijl$\"-nTN`LUF-7$F^[m$\"--AyM#G%F-7$Fc[m$ \"-JX]v&Q%F-7$Fh[m$\"-=f&=gc%F-7$F]\\m$\"-\"))3/7)[F-7$Fb\\m$\"-6y%GuS &F-7$Fg\\m$\"-[**)zBD'F-7$F\\]m$\"-s&e9l!yF-7$Fa]m$\".'\\(p<%G5F-7$Ff] m$\".&4g/h*>\"F-7$F[^m$\".@A@#y59F-7$F`^m$\".@9PyO`\"F-7$Fe^m$\".=Ys9a g\"F-7$Fj^m$\".k:%fO0;F-7$F__m$\".(>grJ0;F-7$Fd_m$\".@\"G5(\\g\"F-7$Fi _m$\".,y**eZg\"F-7$Fc`m$\".QI-mhg\"F-7$F]am$\".Ke2&o;;F-7$Fbam$\".\"4) )>(3j\"F-7$Fgam$\"/&\\?oSql\"F]v7$F\\bm$\"/OM)p!\\-F]v7$F[cm$\"/\"oE'QZ8@F]v7$F`cm$\"/y+VI2aAF]v7 $Fecm$\"/;fUI@ECF]v7$Fjcm$\"08S['HTNEF^dm7$F`dm$\"5>b'G(3Rk)y)GFedm-Fg dm6&Fidm$\")#)eqkF^em$\"))eqk\"F^emF_fs-F`em6#%3efficient~scheme~CG-F$ 6%7]s7$F($\",3p'4\\yF-7$F/$\",,dtz/)F-7$F4$\",.B-%e!*F-7$F9$\"-z?2>V5F -7$F>$\"-SYkLV5F-7$FC$\"-ls)HT0\"F-7$FH$\"-'o8u\"46F-7$FM$\"-a5[<\">\" F-7$FR$\"-k&*fz!>\"F-7$FW$\"-P9js\">\"F-7$Ffn$\"-lZ)\\C<\"F-7$F[o$\"-O [1^r5F-7$F`o$\"-9H5\"42\"F-7$Feo$\"-8pcMi5F-7$Fjo$\"-2JT_N5F-7$F_p$\", L#>5O&*F-7$Fdp$\",yx%))*o(F-7$Fip$\",e9C#[ZF-7$F^q$\",P)f+ZZF-7$Fcq$\" ,!=dUVZF-7$Fhq$\",;=7yi%F-7$F]r$\",q`^u>%F-7$Fbr$\",\"zL`'z#F-7$Fgr$\" ,Yk(HR:F-7$F\\s$!+nIcMOF-7$Fas$!,^oa0<$F-7$Ffs$!,Z*e_>sF-7$F[t$!,)Hk@O \")F-7$F`t$!,0)y4O\")F-7$Fet$!,q-')[G)F-7$Fjt$!,tPK9$))F-7$F_u$!-Hm].c 5F-7$Fdu$!-)*o;?:7F-7$Fiu$!.j3kCjX\"F]v7$F_v$!.'>zL)=\"=F]v7$Fdv$!.Rt6 ERK#F]v7$Fiv$!.x4!GL$o#F]v7$F^w$!.mRpK**)GF]v7$Fcw$!.B1Uk(*)GF]v7$Fhw$ !.%f1kf*)GF]v7$F]x$!.^9fp#*)GF]v7$Fbx$!.W@m>!*)GF]v7$Fgx$!.vi8I.*GF]v7 $F\\y$!.]+AdV!HF]v7$Fay$!.&Q0]baHF]v7$Ffy$!.zH\"\\M*4$F]v7$F[z$!.jS)\\ [`KF]v7$F`z$!.Hkz!\\&F-7$Ff^l $!-\"e%3=)\\&F-7$F[_l$!-97=&H`&F-7$F`_l$!-NKi2UcF-7$Fe_l$!-X^nVldF-7$F j_l$!-'Qu^3'fF-7$F_`l$!-l0_6diF-7$Fd`l$!-#\\p#e*o'F-7$Fi`l$!-%QeU(GsF- 7$F^al$!-J9npbwF-7$Fcal$!-XMx%[l(F-7$Fhal$!-?x/,awF-7$F]bl$!-\"e+EFl(F -7$Fbbl$!-)>fT_l(F-7$Fgbl$!-(ekcyp(F-7$Facl$!-Y))*[Uu(F-7$F[dl$!-[8n\" ot(F-7$$\"5][sJh20k5%*F-$!-#**>Cyq(F-7$F`dl$!-j3UqfwF-7$$\"5+I\"\\M))) [u6%*F-$!-#[72ke(F-7$Fedl$!-2@;e![(F-7$F_el$!-\\%oCrQ(F-7$Fiel$!-J)y;n Q(F-7$F^fl$!-6r*3jQ(F-7$Fcfl$!-`n9!fQ(F-7$Fhfl$!-8\"\\$=%Q(F-7$F]gl$!- 9zGE\"Q(F-7$Fbgl$!-*G3(emtF-7$Fggl$!-s7i%GH(F-7$F\\hl$!-94^Q)=(F-7$Fah l$!-`O$o1*pF-7$Ffhl$!-#\\Wfpj'F-7$F[il$!-6`l[LgF-7$F`il$!-TH\"4$>OF-7$ Feil$!-'>Iv()y\"F-7$Fjil$!-wqW<)y\"F-7$F_jl$!-0AZ3$y\"F-7$Fdjl$!-\"ym8 =x\"F-7$Fijl$!-uH0cV!)Qc_6F-7$Fj^m$\".Qy&)GD:\"F-7$F__ m$\".Z*QQ\\_6F-7$Fd_m$\".=lsaA:\"F-7$Fi_m$\".R*eKP_6F-7$F^`m$\".fdOmK: \"F-7$Fc`m$\".t#)G?f:\"F-7$Fh`m$\".#*eA9B;\"F-7$F]am$\".kw#*ye<\"F-7$F bam$\".*yU(p1?\"F-7$Fgam$\"/s]UKXW7F]v7$F\\bm$\"/7&*f*[yJ\"F]v7$Fabm$ \"/![HkecV\"F]v7$Ffbm$\"/!*>)G)zM;F]v7$F[cm$\"/$oZ-h;%>F]v7$F`cm$\"/*[ 6(fl[@F]v7$Fecm$\"/Jj&[c)*R#F]v7$Fjcm$\"0`.vKoEq#F^dm7$F`dm$\"5>?sX48H SlIFedm-F`gp6&Fidm$\"\"'FdgpFjdmFegp-F`em6#%Sscheme~with~stability~rad ius~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerro r~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6 \"Q!Fb`u-%%VIEWG6$;F(F`dm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modificatio n of the Bogacki-Shampine scheme" "efficient scheme A" "efficient sche me B" "efficient scheme C" "scheme with stability radius = imag axis i nclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 12 \+ of 7 stage, order 5 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*& -%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "5*y*sin^7*7*x;" "6#*,\"\"&\"\"\"%\"yGF%%$sinG\"\"(F(F%%\"xGF%" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(16/49+5/3136*cos*49 *x-cos*35*x/64+5/64*cos*21*x-25/64*cos*7*x);" "6#/%\"yG-%$expG6#,,*&\" #;\"\"\"\"#\\!\"\"F+*,\"\"&F+\"%OJF-%$cosGF+F,F+%\"xGF+F+**F1F+\"#NF+F 2F+\"#kF-F-*,F/F+F5F-F1F+\"#@F+F2F+F+*,\"#DF+F5F-F1F+\"\"(F+F2F+F-" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "de := diff(y(x),x)=5*y(x)*sin(7*x)^7;\nic := y( 0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=combine((numer(rhs(%))/convert(de nom(rhs(%)),exp)));\nr := unapply(rhs(%),x):\nplot(r(x),x=0..5,font=[H ELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"&\"\"\"F)F0)-%$sinG6#,$*&\"\"(F 0F,F0F0F7F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&# \"#D\"#k\"\"\"-%$cosG6#,$*&\"\"(F0F'F0F0F0!\"\"*&#F0F/F0-F26#,$*&\"#NF 0F'F0F0F0F7*&#\"\"&\"%OJF0-F26#,$*&\"#\\F0F'F0F0F0F0*&#FAF/F0-F26#,$*& \"#@F0F'F0F0F0F0#\"#;FGF0" }}{PARA 13 "" 1 "" {GLPLOT2D 806 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7_^l7$$\"\"!F)$\"\"\"F)7$$\"3ALL$3FWYs#!#>$ \"3RX36^,++5!#<7$$\"3WmmmT&)G\\aF/$\"3/h:lL\\.+5F27$$\"3MKL3x1h6oF/$\" 3N>!>$zG>+5F27$$\"3m****\\7G$R<)F/$\"3^$*H^L`v+5F27$$\"3±z%\\DO&*F/ $\"3J?*f5+AB+\"F27$$\"3GLLL3x&)*3\"!#=$\"3x]lWM_&f+\"F27$$\"3em\"z%\\v #pK\"FJ$\"3$\\ClDT`A-\"F27$$\"3))**\\i!R(*Rc\"FJ$\"3%z>L#y^ah5F27$$\"3 _;z>6B`#o\"FJ$\"3c]f\"H$G$Q4\"F27$$\"3!**HO6F27$$ \"34]PM_@g>>FJ$\"33RW!o::'*=\"F27$$\"3umm\"H2P\"Q?FJ$\"3?q+*)GVj`7F27$ $\"3C]7G))>Wr@FJ$\"3L$)HfnG\"pL\"F27$$\"3YLek.pu/BFJ$\"3Gv`o46\")G9F27 $$\"33D\"G8O*RrBFJ$\"3+PI&HSekZ\"F27$$\"3o;/,>=0QCFJ$\"3mjV0*4cU_\"F27 $$\"3c3FpwUq/DFJ$\"3#pc5I%eTr:F27$$\"3!***\\PMnNrDFJ$\"3U+AysC:<;F27$$ \"37$eR(\\;m/FFJ$\"3)f:eA`w9q\"F27$$\"3MmT5ll'z$GFJ$\"35TSzD!3Dx\"F27$ $\"37](o/[r7(HFJ$\"3e:[6>4vF=F27$$\"3MLL$eRwX5$FJ$\"3ICf`%)[>n=F27$$\" 3_LLe*[`HP$FJ$\"3!eZ8'p.63>F27$$\"3rLLL$eI8k$FJ$\"3%[F&)3\"\\`>>F27$$ \"3_L$3-8>bx$FJ$\"3IprJEM*3#>F27$$\"3*QL$3xwq4RFJ$\"3rrv?(f28#>F27$$\" 3EM$eRA'*Q/%FJ$\"3X^'fP/+9#>F27$$\"33ML$3x%3yTFJ$\"3cOdy3HT@>F27$$\"3h +]PfyG7ZFJ$\"3VH&=@l89#>F27$$\"3emm\"z%4\\Y_FJ$\"3ml4uC:e?>F27$$\"3C+] P4'4.P&FJ$\"3`*yZBqt)=>F27$$\"3'QLL3FGT\\&FJ$\"3t^RtM))*[\">F27$$\"3Um ;HKp%zh&FJ$\"3!*H'3:A\\o!>F27$$\"32++v$flV;F27$$\"3I++vVVX$\\'FJ$\"3s/S7/D%eb\"F27$$\"3p LL$eRh;i'FJ$\"3)z%[YI??k9F27$$\"31nm\"zWo)\\nFJ$\"31\")Q9\"48QP\"F27$$ \"3W++++b2yoFJ$\"3k#Re\"fbk*G\"F27$$\"3%QL$3_DG1qFJ$\"3jY.gupS:7F27$$ \"3Anm;/'*[MrFJ$\"3y@x$3A]K:\"F27$$\"3]***\\il'pisFJ$\"3_zleRG$Q5\"F27 $$\"3+MLe*[!)y_(FJ$\"3:2l<5ljQ5F27$$\"3Qnm\"HKkIz(FJ$\"3z[=lFuU55F27$$ \"35MLeRilDzFJ$\"3tR>*>I/Y+\"F27$$\"3!3+]i:[#e!)FJ$\"3ZybuD9w,5F27$$\" 3[nm\"H2S3>)FJ$\"36f+OZ/c+5F27$$\"3>MLe*)>VB$)FJ$\"3q[OmV#R,+\"F27$$\" 3wmmTg()4_))FJ$\"3!z[$G++++5F27$$\"3Y++DJbw!Q*FJ$\"3(f!Q*>W.++\"F27$$ \"3E$3_+A:n^*FJ$\"3$o[9!4H.+5F27$$\"3=nT&)3\\m_'*FJ$\"3qlL!>[$=+5F27$$ \"33^il(f9')y*FJ$\"39t!)=KMs+5F27$$\"3+N$ekGkX#**FJ$\"3)[%4Vj`B-5F27$$ \"31]iSmjk>5F2$\"3?UoC!*)fG,\"F27$$\"3%ommTIOo/\"F2$\"3%pJ)[*fok/\"F27 $$\"3?]i:g2\")e5F2$\"3'ob!*>J(\\t5F27$$\"3cLe9;_yq5F2$\"3!f)3IJ$>-6\"F 27$$\"3!pTN@nfF3\"F2$\"3O\\W6vrpd6F27$$\"3E+]7GTt%4\"F2$\"3&p\"fsjqH;7 F27$$\"3i$e9Te3n5\"F2$\"3e&f$*[!\\V&G\"F27$$\"3(p;/,/$o=6F2$\"3K1G\">I $Qj8F27$$\"3L]P4'\\d18\"F2$\"3mAQZ9DAZ9F27$$\"3YLL3_>jU6F2$\"3A//J![AI `\"F27$$\"3)\\PMF:s$\\6F2$\"3%**=P`q)\\!e\"F27$$\"3u;aQ`B6c6F2$\"3vn0o ImOE;F27$$\"3[ek.aD&G;\"F2$\"3q#GT!4W()p;F27$$\"3C+voaFfp6F2$\"3Hrm)p6 m.r\"F27$$\"3]$e*)f:tI=\"F2$\"3C=5u0pN!y\"F27$$\"3ym;HdNb'>\"F2$\"3oD; 8U'oS$=F27$$\"3NLe*)fV^B7F2$\"3Q\\+g\"\\,f*=F27$$\"37++]i^Z]7F2$\"3!y& =TMDx;>F27$$\"37+]i:4,k7F2$\"34bF!Hpw)>>F27$$\"37++vomax7F2$\"3s-.5q=, @>F27$$\"35+](=U#3\"H\"F2$\"3EfS@r$Q8#>F27$$\"35+++v\"=YI\"F2$\"3=\"f$ >R_S@>F27$$\"34++D\"o*oJ8F2$\"36A3*=r89#>F27$$\"33++](=h(e8F2$\"3#z)y \\8PT@>F27$$\"3!********f\\[Q\"F2$\"34cw3'H49#>F27$$\"3&*****\\7!Q4T\" F2$\"3n&=!o'on6#>F27$$\"3(****\\(=A)RU\"F2$\"3bRtq7'G/#>F27$$\"3****** *\\UEqV\"F2$\"3yBj&H)zK=>F27$$\"3-++DJ12]9F2$\"32T7QjDP8>F27$$\"3/++]P [6j9F2$\"3+dVO')[G.>F27$$\"3W$ek`h0o[\"F2$\"3I$zM;9\\?'=F27$$\"3%o;HKR '\\5:F2$\"3-heuX3wwd\"F2$\"3yrf8;r3y8F27$$\"3#f*[o=f+z:F2$\"3lUh \"F2$\"3=x^]S+%)R6F27$$\"3V$eky#)*QU;F2$\"3k!o/)p1f`5F27$$\"3wmm;a/cq; F2$\"3nOMw%)=y95F27$$\"3im\"z>c#\\#o\"F2$\"3VZh#ea.v+\"F27$$\"3sm;zpYU %p\"F2$\"3He\"pN-SM+\"F27$$\"3#o;/wxcjq\"F2$\"3/ne\\$[\"R,5F27$$\"3\"p m;a)))G=F2$\"3:l7r\"p]]+\"F27$$\"3%\\ilAZ9R\">F2$\"3^hCaoQo55F27$$\"3K Le9;0?E>F2$\"3:4Zh8k[?5F27$$\"31]i!Rgs2&>F2$\"3wO#Q8N_'f5F27$$\"3gmmm \"pW`(>F2$\"3,$4k)\\,ZO6F27$$\"3_ek.HW#)))>F2$\"3Q;r;:R*z>\"F27$$\"3?] iSmTI-?F2$\"3_fyny')>t7F27$$\"3EY64NS/4?F2$\"3+2!G,Td`J\"F27$$\"3*=/wP !Ry:?F2$\"3/M$G\"4m+g8F27$$\"3^P4YsP_A?F2$\"3$R*>L\"f'e19F27$$\"3dLe9T OEH?F2$\"3%\\u#Gw;Va9F27$$\"3kH2$)4N+O?F2$\"3\\;?B!3#z-:F27$$\"3EDc^yL uU?F2$\"37f,s,Z'3b\"F27$$\"3)3_+sC$[\\?F2$\"3C:ye37$yf\"F27$$\"3'pT&)e 6Bi0#F2$\"3VLP%*y0!Hk\"F27$$\"3k3_D`Gqp?F2$\"3;-/UsbdC+v$fQa)3@F2$\"3_4a5;!3X(=F27$$\"3/++D\" =EX8#F2$\"3XR)Rsy(o4>F27$$\"3?]i!*y?OZ@F2$\"3#=X-%G8_;>F27$$\"3M+Dcwz> g@F2$\"3\"HQ**f-s'>>F27$$\"3/](=U(Q.t@F2$\"3jDRpm*34#>F27$$\"3?+](=xpe =#F2$\"3.\"p)[NLI@>F27$$\"3mLeRA9WRAF2$\"3HST.9PT@>F27$$\"37nm\"H28IH# F2$\"3`7Ny#=%Q@>F27$$\"3=Ug_Z>J0BF2$\"3F27$$\"3!oTN@#3hF27$$\"3(=zWnp4*HBF2$\"3aFU&R3A$>>F27$$\"3$p;a8d3AM#F2$\" 3/F6dAB#f\">F27$$\"31gc(z*f*=F27$$\"3um;zpSS\"R#F2$ \"3->e1#==?%=F27$$\"3uC\"yv1qYS#F2$\"3[KdPD&>Az\"F27$$\"3;$ek`1OzT#F2$ \"3m9DqP!=ls\"F27$$\"3fT5:j??JCF2$\"3!\\()pJuklk\"F27$$\"3-+v$41oWW#F2 $\"3y%HsO_ejb\"F27$$\"3CH2$)f55^CF2$\"3(>FHM)z84:F27$$\"3WeRseStdCF2$ \"3/?hWfY`h9F27$$\"3m(=9\"F27$$\"3.$3-)Q84DDF2 $\"3St1.wX6#4\"F27$$\"37e9;#)4()QDF2$\"3A!)Gqgm(f0\"F27$$\"3AL3_D1l_DF 2$\"3/w0[YW]J5F27$$\"3I3-))o-VmDF2$\"3hGW>'Q(=;5F27$$\"3S$eRA\"*4-e#F2 $\"38vW*f>\\u+\"F27$$\"3%4F>Rt*4(e#F2$\"34I&o$G5\"[+\"F27$$\"3[e*)fb&* )Rf#F2$\"3*Rd232\"*H+\"F27$$\"3/Y'ysPz3g#F2$\"3EN#z,7\"y,5F27$$\"3fL$e *)>pxg#F2$\"3cLdJH'45+\"F27$$\"3%omm\"z+vbEF2$\"3VA\\(Rr,++\"F27$$\"33 +]Pf4t.FF2$\"3EB[5++++5F27$$\"3Q$3F>HT'HFF2$\"3PG;$Qk,++\"F27$$\"3om\" zWi^bv#F2$\"3tr+Z;M5+5F27$$\"3M3_v!z1&oFF2$\"33DW)H[F/+\"F27$$\"3)*\\7 .d>Y\"y#F2$\"3?pw0F@P,5F27$$\"3#3Fp,aRzy#F2$\"3V/:%[T%G-5F27$$\"3k\"H2 L7ST$GF2$\"3>]6Z^*QK.\"F27$$\"3)om\"H2 \"34'GF2$\"3c89;GxR$4\"F27$$\"3=](=<1#HuGF2$\"3H$eWMp\"*>9\"F27$$\"3YL e9;gn()GF2$\"3LgFbL%HW?\"F27$$\"3w;Hdq*f5!HF2$\"337;\"\\lV-G\"F27$$\"3 0+++DRW9HF2$\"3/%p0(oZBn8F27$$\"3-D19>2*4#HF2$\"3.aKDn)*p79F27$$\"3X]7 G8v`FHF2$\"39c[>!pr#f9F27$$\"3Uv=U2V3MHF2$\"3FI^JzcD1:F27$$\"3S+Dc,6jS HF2$\"3)fI*>\"*G\"Hb\"F27$$\"3O]P%)*oCP&HF2$\"3kq@\"z(y\"F27$$\"3C+v oaa+$*HF2$\"3](oH/*41IF2$\"3/1YYU*QP(=F27$$\"3:++DJ E>>IF2$\"3w5uUgAb'*=F27$$\"3Fv=<^8'=.$F2$\"3=[IQQI\\4>F27$$\"3S]P4r+`W IF2$\"3bz\"Rx1kj\">F27$$\"33Dc,\"z)>dIF2$\"3\")Hman(y&>>F27$$\"3A+v$4^ n)pIF2$\"3P!f*z`l'3#>F27$$\"3-]7y]\\?&4$F2$\"3W+7R*\\%R@>F27$$\"3F+]i! RU07$F2$\"3Lg5nwOT@>F27$$\"39+vo/#3o<$F2$\"3oIo*Q]69#>F27$$\"3+++v=S2L KF2$\"39dX:D?6=>F27$$\"3Om\"zpo_$eKF2$\"3#zYTO,TK!>F27$$\"3;L$3_NJOG$F 2$\"3=J`$Q^]x&=F27$$\"3y;HK*oqiH$F2$\"3h#pBbd4n\"=F27$$\"3'**\\PM-5*3L F2$\"3uG&fHPn7w\"F27$$\"39$3_vN\\:K$F2$\"3Y(o1g2R=p\"F27$$\"3Jmmm\"p)= MLF2$\"3896'o\"f'3h\"F27$$\"3&)\\i:&GO4M$F2$\"3=*)4prhIk:F27$$\"3%H$ek yQoZLF2$\"3IceZ)o3k^\"F27$$\"3/;a8s9VaLF2$\"3G6b_$H\"*zY\"F27$$\"3e** \\il!z6O$F2$\"3q?/^g#o)>9F27$$\"35$e9\"fm#zO$F2$\"3#o/$41/\"GP\"F27$$ \"3kmTg_UnuLF2$\"3\\t!fI&=^F8F27$$\"3=]P4Y=U\"Q$F2$\"3]X:oIac%G\"F27$$ \"3GLLeR%p\")Q$F2$\"3F&[%Q!RZWC\"F27$$\"3!**\\ilik;S$F2$\"3_Z%zlm()R< \"F27$$\"3)pmTN\")f^T$F2$\"331)eDsVv6\"F27$$\"3hL3_+]lGMF2$\"3CD\\786* [2\"F27$$\"3B++](=]@W$F2$\"3.\">ZcV5Y/\"F27$$\"3C$ekyZ2mY$F2$\"3bMcTbf 795F27$$\"3mm\"H#oZ1\"\\$F2$\"38K^#3O0J+\"F27$$\"35+5F2 7$$\"35L$e*[$z*RNF2$\"3:J>88\">++\"F27$$\"3%o;Hd!fX$f$F2$\"3+/++++++5F 27$$\"3e++]iC$pk$F2$\"3>3i#3WY++\"F27$$\"3)*e*)f!y6&fOF2$\"3/sYmoa@+5F 27$$\"3%p\"zp)4\"4sOF2$\"3/b\">zh_2+\"F27$$\"3!\\(oz;/n%o$F2$\"3FA+oZ! R@+\"F27$$\"3ILe*[t\\sp$F2$\"3]TEK8J>05F27$$\"36]P4r$3Cs$F2$\"3cyY&>9K :-\"F27$$\"3[m;H2qcZPF2$\"3yA^kd)oL1\"F27$$\"3KL3F>grgPF2$\"3kx1i!p\" \\+6F27$$\"3s***\\7.lQx$F2$\"3WMY`\\)G.:\"F27$$\"3em\"HK/9qy$F2$\"3j!G cKQeN@\"F27$$\"3UL$3_0j,!QF2$\"3%*G.x'3G&*G\"F27$$\"3i;z>hvt1QF2$\"3uM !o>\"3kJ8F27$$\"3F+v=n?J8QF2$\"3yKp#[4RfP\"F27$$\"3\"R3xJd'))>QF2$\"3/ CVy4j'=U\"F27$$\"36nm;z5YEQF2$\"3%GUA'*pv(o9F27$$\"3J]i:&eNI$QF2$\"3eJ %\\YU_f^\"F27$$\"3^Le9\"45'RQF2$\"3][!)RoHki:F27$$\"3s;a8(f%=YQF2$\"3Z k-ND-43;F27$$\"3O+]7.\"fF&QF2$\"3]5UW.;d^;F27$$\"3i3_vlYhlQF2$\"3S)o\\ Z'>`GF27$$\"3Ymm;/OgbRF2$\"3e5rkQQq>>F27$$\"3*G$e*[$zV4SF2$\"3^Ar&38 89#>F27$$\"3w**\\ilAFjSF2$\"3Q')**40OT@>F27$$\"3#G3_]p'>*3%F2$\"3!o$)[ rAz8#>F27$$\"3ym\"zW7@^6%F2$\"3371C6\\d?>F27$$\"3@Qf$=B-;7%F2$\"39!e*f [J()>>F27$$\"3w3F>RL3GTF2$\"3A(\\vy/K(=>F27$$\"3Iz%\\lWkX8%F2$\"38-([e tep\">F27$$\"3t]i!RbX59%F2$\"38M:lS7J9>F27$$\"3#=z>'ox+aTF2$\"3&>*GfbB =0>F27$$\"3yLLL$)*pp;%F2$\"3C=Lu(zB&))=F27$$\"3!Q3_+sD-=%F2$\"3ug(ptnA /'=F27$$\"3#Q$3xc9[$>%F2$\"3:,22Ku<==F27$$\"3'Qe*[$>Pn?%F2$\"3i6,,)=G, w\"F27$$\"3)QL3-$H**>UF2$\"33wUWwLu'o\"F27$$\"3X3xc)z?mA%F2$\"3/e\"zYY >^k\"F27$$\"3\"R3Fpm[KB%F2$\"3x%eAK2J4g\"F27$$\"3OfkGNl()RUF2$\"3/aZW5 %e[b\"F27$$\"3#R$ek.W]YUF2$\"3_pHqN-l2:F27$$\"3]3_+sA8`UF2$\"3?puckU3g 9F27$$\"3'Rek.9g(fUF2$\"30z2N6y#HT\"F27$$\"3SfRs3!)QmUF2$\"3E%zqA:-pO \"F27$$\"3)RL$3xe,tUF2$\"3Hzu(eD`EK\"F27$$\"3kv$4'\\=;'G%F2$\"32(*p'*e u>U7F27$$\"3G#[Z%F2$\"3S)\\h/++++\"F27$$\"3TM$3_5,-`%F2$\"3 y/++\"F27$$\"3SnmT&G!e&e%F2$\"3$eA?a3RF+\"F27$$\"3fLe*[=Y.h%F2$\"3M:+/ vy285F27$$\"3m+]P%37^j%F2$\"356M^f6but%F2$\"3%\\.QNQyVc\"F27$$\"3ID19>zl]ZF2$\" 3S[]>-K'z\"F27$$\"37+]iSjE!z%F2$\"3&)4L&Q*oyW=F27$$\"3y*\\7G))Rb\"[F2$\" 3)fCVDB`!)*=F27$$\"3L+++DM\"3%[F2$\"3$Gxb\\1an\">F27$$\"3i]P4'>]M&[F2$ \"3p-qe54u>>F27$$\"3)3](=np3m[F2$\"3zS9J5F#4#>F27$$\"3G]7GQPsy[F2$\"3j :7^7UI@>F27$$\"3a+]P40O\"*[F2$\"3L:$**HR(R@>F27$$\"3s+voa-oX\\F2$\"3%e \"e*Rr89#>F27$$\"\"&F)$\"3n\\kX'z.7#>F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F (-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F _[q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solut ion" }{TEXT -1 44 " based on each of the methods and gives the " } {TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 748 "R := (x,y) -> 5*y*sin(7 *x)^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh ],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine s cheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficien t scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with sta bility radius = imag axis inclusion`]:\nerrs := []:\nDigits := 30:\nfo r ct to 6 do\n Rn_RK5_||ct := RK5_||ct(R(x,y),x,y,x0,y0,hh,numsteps, false);\n sm := 0: numpts := nops(Rn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Rn_RK5_||ct[ii,2]-r(Rn_RK5_||ct[ii,1]))^2;\n en d do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF ,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint716\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+N8W\"*f!#>7$%Ta~minor~m odification~of~the~Bogacki-Shampine~schemeG$\"+FnE:5F+7$%3efficient~sc heme~AG$\"+#[REl%!#?7$%3efficient~scheme~BG$\"+p6!>v\"F+7$%3efficient~ scheme~CG$\"+O6..5F+7$%Sscheme~with~stability~radius~=~imag~axis~inclu sionG$\"+ukavgF4Q)pprint726\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 " R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sha mpine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sch eme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 25:\nfor ct to 6 do\n rn_RK5_||ct := RK5_||ct(R(x,y) ,x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx )):\nfor ct to 6 do\n errs := [op(errs),abs(rn_RK5_||ct(xx)-rxx)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*( \"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~poin t:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q )pprint736\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+5HY,8!# =7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+m-25?!#>7 $%3efficient~scheme~AG$\"+m>`_A!#?7$%3efficient~scheme~BG$\"+R-S6NF07$ %3efficient~scheme~CG$\"+uBcy:F07$%Sscheme~with~stability~radius~=~ima g~axis~inclusionG$\"+5\"4'o%*F5Q)pprint746\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor mo dification of the Bogacki-Shampine scheme`,`efficient scheme A`,`effic ient scheme B`,`efficient scheme C`,`scheme with stability radius = im ag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((r(x)-'rn_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7 ,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 1 0:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+]?([(f!#> 7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+zg&G,\"F+7 $%3efficient~scheme~AG$\"+R@._Y!#?7$%3efficient~scheme~BG$\"+1`9[ " 0 "" {MPLTEXT 1 0 498 "evalf[20](plot([r(x)-'rn_RK 5_1'(x),r(x)-'rn_RK5_2'(x),r(x)-'rn_RK5_3'(x),r(x)-'rn_RK5_4'(x),\nr(x )-'rn_RK5_5'(x),r(x)-'rn_RK5_6'(x)],x=0..5,\ncolor=[blue,magenta,COLOR (RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9],\nlegen d=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sham pine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sche me C`,`scheme with stability radius = imag axis inclusion`],\ntitle=`e rror curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 977 637 637 {PLOTDATA 2 "6,-%'CURVESG6%7^^l7$$\"\"!F) F(7$$\"5ommmTN@Ki8!#@$!(\\t8&!#>7$$\"5NLLL$3FWYs#F-$!)#yuG#F07$$\"5-++ +D1k'p3%F-$!)*G+M'F07$$\"5qmmmmT&)G\\aF-$!)PgHvF07$$\"50+++]7G$R<)F-$! '/qPF07$$\"5MLLLL3x&)*3\"!#?$\"*oux9\"F07$$\"5++]i!*GER37FH$\"*M&=F8F0 7$$\"5nmm\"z%\\v#pK\"FH$\"*ol*\\6F07$$\"5ML$3_+ZiaW\"FH$\")VDn&)F07$$ \"5+++]i!R(*Rc\"FH$\")XIKdF07$$\"5nm;z>6B`#o\"FH$\")t;kOF07$$\"5MLL3xJ s1,=FH$\"(5![VF07$$\"5++]PM_@g>>FH$!)#\\&))RF07$$\"5nmmm\"H2P\"Q?FH$!* Uuc<\"F07$$\"5+]7yv,LYr?FH$!*+cBJ\"F07$$\"5MLe*)fI&*y/@FH$!*!fl/@F07$$ \"5n;/,Wfd6Q@FH$!*'4gU@F07$$\"5++]7G))>Wr@FH$!*D]WH#F07$$\"5nmTN'fW%4Q AFH$!*P6C*GF07$$\"5MLLek.pu/BFH$!*BD9w#F07$$\"5nm;/,>=0QCFH$!*0^)38F07 $$\"5+++]PMnNrDFH$\"*Q?kJ#F07$$\"5nmmT5ll'z$GFH$\"+EB=4:F07$$\"5MLLL$e RwX5$FH$\"+pp*[w\"F07$$\"5MLLLe*[`HP$FH$\"+P()fP9F07$$\"5MLLLL$eI8k$FH $\"+NLhs7F07$$\"5MLL$3-8>bx$FH$\"+8Q(fK\"F07$$\"5MLLL3xwq4RFH$\"+&QP([ 9F07$$\"5MLL$eRA'*Q/%FH$\"+jl%p\\\"F07$$\"5NLLL$3x%3yTFH$\"+\\:Y@:F07$ $\"5-++]PfyG7ZFH$\"+go:D:F07$$\"5ommm\"z%4\\Y_FH$\"+@(3Da\"F07$$\"5NLL L$3FGT\\&FH$\"+ZtF^6F07$$\"5++++v$flF07$$\"5-+++DcmpisFH$\"+PJk\\5F07$$\"5NLLLe*[!)y_(FH$\"+C'yD=\"F0 7$$\"5qmmm\"HKkIz(FH$\"+t5\"=F\"F07$$\"5SLLLeRilDzFH$\"+f?,.8F07$$\"50 +++Dc\"[#e!)FH$\"+Jvo18F07$$\"5qmmm\"H2S3>)FH$\"+q*o3G\"F07$$\"5OLLLe* )>VB$)FH$\"+\"o5BB\"F07$$\"5qmmmTg()4_))FH$\"+2ytv6F07$$\"5.+++DJbw!Q* FH$\"+[Z%Q8\"F07$$\"5NLL$ekGkX#**FH$\"+J)*p`7F07$$\"5nmmm;/j$o/\"F0$\" +_K0>8F07$$\"5++]i:g2\")e5F0$\"+G)p,K\"F07$$\"5MLLe9;_yq5F0$\"+&H%fQ8F 07$$\"5nm;a8s'fF3\"F0$\"+XiIq8F07$$\"5+++]7GTt%4\"F0$\"+%z*p'Q\"F07$$ \"5ML$e9Te3n5\"F0$\"+m'=]P\"F07$$\"5nmmT5SIo=6F0$\"+AlN[8F07$$\"5++]P4 '\\d18\"F0$\"+-v-i9F07$$\"5MLLL3_>jU6F0$\"+!zL*[\"F0$\"+G'zT*RF07$$\"5MLLe*)fV^B7F0$\"+I:&y#QF07$ $\"5++++]i^Z]7F0$\"+KCoSNF07$$\"5+++D1RICd7F0$\"+$G#RTNF07$$\"5+++]i:4 ,k7F0$\"+Q%H'eNF07$$\"5+++v=#zy2F\"F0$\"+J.m6OF07$$\"5++++vomax7F0$\"+ ([X5i$F07$$\"5+++DJXXJ%G\"F0$\"+2xL!o$F07$$\"5+++](=U#3\"H\"F0$\"+]KKU PF07$$\"5+++vV)H]yH\"F0$\"+?$e=pPF07$$\"5++++ v$R0=P\"F0$\"+,^'\\\"QF07$$\"5++++++'\\[Q\"F0$\"+$3'oeQF07$$\"5++++D1Q *yR\"F0$\"+cIP)*QF07$$\"5++++]7!Q4T\"F0$\"+5<1xQF07$$\"5+++++Dk-P9F0$ \"+%fFVl$F07$$\"5++++]P[6j9F0$\"+7+A_JF07$$\"5M$e9m>`P!p9F0$\"+Y^N$4$F 07$$\"5nm\"HKkAg\\Z\"F0$\"+mL(o/$F07$$\"5+]P%)*3#H)3[\"F0$\"+^!G!yIF07 $$\"5ML$ek`h0o[\"F0$\"+cTcoIF07$$\"5++voH/5l)\\\"F0$\"+8'pIN$F07$$\"5n mm\"HKR'\\5:F0$\"+BKo(*RF07$$\"5MLe9;#yTB_\"F0$\"+\">z6P%F07$$\"5++]P4 rr=M:F0$\"+l]A)e%F07$$\"5n\"H#oK=&[r`\"F0$\"+26KnXF07$$\"5M$e*)fb')4,a \"F0$\"+O\\cwYF07$$\"5+voHz772V:F0$\"+@jLb\"F0$\"+f_?*R%F07$$\"5MLL$e*[z(yb\"F0$\"+*y*yQUF 07$$\"5nm;/Ev&[ge\"F0$\"+?@VeHF07$$\"5+++Dc,#>Uh\"F0$\"+>.d\"R#F07$$\" 5nmTNr9XIG;F0$\"+W6NjBF07$$\"5ML$eky#)*QU;F0$\"+YgK.CF07$$\"5n;/,W%[K% \\;F0$\"+,0L8CF07$$\"5++Dc,T^Zc;F0$\"+K(R-T#F07$$\"5M$e9\"f(z/RCF07$$\"5nmmm;a/cq;F0$\"+It#zY#F07$$\"5nmm\"z>c#\\#o\"F0$\"+VQ- &[#F07$$\"5nmm;zpYU%p\"F0$\"+,'Re\\#F07$$\"5nmmTgxnN1N#F07$$\"5MLLekGo`@=F0$\"+.=+VBF07$$\"5,++]iSj0x=F0$\"+mawaBF 07$$\"5MLLe9;0?E>F0$\"+'z(Q[DF07$$\"5nmmmm\"pW`(>F0$\"+*=uVr#F07$$\"5+ +]iSmTI-?F0$\"+A0#)fGF07$$\"5MLLe9TOEH?F0$\"+xZY&=$F07$$\"5,+Dc^yLuU?F 0$\"+QCB!o$F07$$\"5om;a)e6Bi0#F0$\"+ORu1VF07$$\"5ML3_D`Gqp?F0$\"+'z'z< `F07$$\"5,++]i!f#=$3#F0$\"+kS#Q!fF07$$\"5,+++D\"=EX8#F0$\"+=gp?fF07$$ \"5,++](=xpe=#F0$\"+_Q+yfF07$$\"5MLLeRA9WRAF0$\"+w(Qu/'F07$$\"5nmmm\"H 28IH#F0$\"+\"e(4!='F07$$\"5nmmTNr&3AM#F0$\"+X:XMdF07$$\"5nmm;zpSS\"R#F 0$\"+pxg#F0$\"+'=&*en$F07$$\"5,++D1R'f0uzEF0$\"+;p!>b$F07$$\"5,++]Pf4t. FF0$\",?Wtub$FH7$$\"5omT5lDho;FF0$\"+@'eNa$F07$$\"5ML$3F>HT'HFF0$\"+n! pn]$F07$$\"5,+DJ?ekfUFF0$\"+-sz'[$F07$$\"5omm\"zWi^bv#F0$\"+Y86([$F07$ $\"5,+]7.d>Y\"y#F0$\"+4!HAg$F07$$\"5MLLLe*Gst!GF0$\"+<;X:PF07$$\"5om;/ ,Wiv?GF0$\"+Rh?WPF07$$\"5,++vV)>ST$GF0$\"+dXssPF07$$\"5ML$ekG:Cv%GF0$ \"+r,MGQF07$$\"5omm;H2\"34'GF0$\"+>$**4#RF07$$\"5MLLe9;gn()GF0$\"+Fr,_ UF07$$\"5,++++DRW9HF0$\"+VQO#f%F07$$\"5,+]7G8v`FHF0$\"+OM5/\\F07$$\"5, ++Dc,6jSHF0$\"+q$efg&F07$$\"5,+]P%)*oCP&HF0$\"+!\\#HPjF07$$\"5,++]7y#= o'HF0$\"+^\\*)\\rF07$$\"5,+]iSm=\"*zHF0$\"+\"*RT[!)F07$$\"5,++voaa+$*H F0$\"+Qk>!Q)F07$$\"5,+D\"G)[Ab**HF0$\"+)Gx4V)F07$$\"5,+](oH/*41IF0$\"+ \"GC`Z)F07$$\"5,+v$4r$ek7IF0$\"+m@[!Q)F07$$\"5,+++DJE>>IF0$\"+;:!>K)F0 7$$\"5,]P4@\"*p_DIF0$\"+V%oXF)F07$$\"5,+v=<^8'=.$F0$\"+YSVd\")F07$$\"5 ,]7G86d>QIF0$\"+%[^f9)F07$$\"5,+]P4r+`WIF0$\"+\"H,Q5)F07$$\"5,](oa5Vk3 0$F0$\"+EQ'f4)F07$$\"5,+Dc,\"z)>dIF0$\"+`_e,\")F07$$\"5,]il(4:LN1$F0$ \"+\\a#)Q\")F07$$\"5,++v$4^n)pIF0$\"+&*=)p>)F07$$\"5,+v$f3BOD3$F0$\"+# 3j$F0$\"+( e]TY)F07$$\"5,]P4'\\)y!z>$F0$\"+VFjk%)F07$$\"5,+](=<6T\\?$F0$\"+s'oqX) F07$$\"5,+vVBlv+>KF0$\"+%=W]N)F07$$\"5,+++v=S2LKF0$\"+r&pu9)F07$$\"5om m\"zpo_$eKF0$\"+oG**)p(F07$$\"5MLL$3_NJOG$F0$\"+v00RvF07$$\"5,++vVB+\" *3LF0$\"+6]/F\")F07$$\"5ommmm\"p)=MLF0$\"+Q#>rW)F07$$\"5MLLekyQoZLF0$ \"+Ef'3-)F07$$\"5,++]il!z6O$F0$\"+m(G)=rF07$$\"5ommTg_UnuLF0$\"+ODi`jF 07$$\"5MLLLeR%p\")Q$F0$\"+\"zu7o&F07$$\"5,++DcEYm,MF0$\"+p:AO_F07$$\"5 omm;a8)f^T$F0$\"+_TT@]F07$$\"5,+]7.2u!>U$F0$\"+!eX\\)\\F07$$\"5MLL3_+] lGMF0$\"+TY\\B\\F07$$\"5om;/,%f-aV$F0$\"+QeL#*[F07$$\"5,+++](=]@W$F0$ \"+xb)=*[F07$$\"5om\"H#oK)yVX$F0$\"+q6Gg[F07$$\"5ML$ekyZ2mY$F0$\"+)>:Q &[F07$$\"5,+vo/Bh$)yMF0$\"+[)>/([F07$$\"5omm\"H#oZ1\"\\$F0$\"+niMz[F07 $$\"5MLe9T8MH.NF0$\"+R8tj[F07$$\"5,+]Pfe?_:NF0$\"+E^RO[F07$$\"5omTgx.2 vFNF0$\"+sWg'z%F07$$\"5MLL$e*[$z*RNF0$\"+W')HUZF07$$\"5omT5:)[[Lb$F0$ \"+IvtFZF07$$\"5,+]PMFwrmNF0$\"+Q*ois%F07$$\"5MLek`mn3!e$F0$\",QJpAu%F H7$$\"5omm\"Hd!fX$f$F0$\",Y!\\!ou%FH7$$\"5ML$e9T=%>?OF0$\"+wI`0ZF07$$ \"5,+++]iC$pk$F0$\"+B0qrYF07$$\"5MLLe*[t\\sp$F0$\"+#GrN([F07$$\"5omm;H 2qcZPF0$\"+NQQ/^F07$$\"5,+++DJ]'Qx$F0$\"+IO%fY&F07$$\"5MLL$3_0j,!QF0$ \"+ymm9fF07$$\"5,++v=n?J8QF0$\"+GYvWiF07$$\"5ommm;z5YEQF0$\"+,.())o'F0 7$$\"5MLLe9\"45'RQF0$\"+?j)o](F07$$\"5,++]7.\"fF&QF0$\"+S$Q)R%)F07$$\" 5ML3_vlYhlQF0$\"+@%G`I*F07$$\"5om;aQG-ZyQF0$\",;aD:,\"F07$$\"5,+Dc,\"z D8*QF0$\",\"y\">s0\"F07$$\"5MLLek`8=/RF0$\",&*yow1\"F07$$\"5oT&Q.VC&R2 RF0$\",0Wq%p5F07$$\"5,]P4'\\841\"RF0$\",rTbz0\"F07$$\"5Me*[=c-BQ\"RF0$ \",o9931\"F07$$\"5omTgF;p.h5F07$$\"5,++]ilAFjSF0$\",M BEW1\"F07$$\"5omT&Q.[Mi2%F0$\",3$)f)o5F07$$\"5ML$3_]p'>*3%F0$\",0SFR2 \"F07$$\"5o;a)3C!yn&4%F0$\",7E;$Q(*F07$$\"5N$e9\"fAv;!>%F0$\"+=W-%** *F07$$\"5NLL3xc9[$>%F0$\"+W'))*G**F07$$\"5NL$e*[$>Pn?%F0$\",F&3Y75F07$ $\"5NLL$3-$H**>UF0$\",'G8Qh5F07$$\"5NL$3Fpm[KB%F0$\",*eqnL5F07$$\"5NLL ek.W]YUF0$\"+e$)*pv*F07$$\"5NL$ek.9g(fUF0$\"+VhYH()F07$$\"5NLLL3xe,tUF 0$\"+:pfMyF07$$\"5om;a8AyI*H%F0$\"+G@$Rj'F07$$\"5-++v=n(*fDVF0$\"+])*G +iF07$$\"5N$3_]Mvs@L%F0$\"+?[MuhF07$$\"5omTNrRduQVF0$\"+;xC?hF07$$\"5- ]il(fs=`M%F0$\"+@\\J\"4'F07$$\"5NL$eRAr\"*=N%F0$\"+L%H04'F07$$\"5-+Dcw %oP]O%F0$\"+SN'=1'F07$$\"5omm;HdO=yVF0$\"+NBOjgF07$$\"5MLLe9\"z-lU%F0$ \"+\"z\"3zfF07$$\"5,++++D>#[Z%F0$\",H!p9AfFH7$$\"5ML$3_+sm')[%F0$\",.% y'=$fFH7$$\"5ommT5::^-XF0$\"+=n3EfF07$$\"5,+]i:5jN;XF0$\"+#o/!3fF07$$ \"5MLL$3_5,-`%F0$\"++.tmeF07$$\"5om;/E+f/WXF0$\"+**f*z&eF07$$\"5,++DJ& p!*yb%F0$\"+s,`teF07$$\"5ML$ek.\\N%\\(F07$$\"5-+]il(\\\\5r%F0$\"+VR!y(yF07$$\" 5NL3x\"[I_Us%F0$\"+\\!f$f%)F07$$\"5omm\"z>6but%F0$\"+*>'\\X#*F07$$\"5- +D19>zl]ZF0$\",ib8?0\"F07$$\"5NL$3-jsgQw%F0$\",aw:p9\"F07$$\"5omTNYLN1 xZF0$\",ayxwA\"F07$$\"5-++]iSjE!z%F0$\",Y)fXx7F07$$\"5-]P%)RLbU$z%F0$ \",^27RG\"F07$$\"5-+v=*F07$Fio$!*S3\"RGF07$F^p$!*lz)=GF07$Fcp$!*6PoF#F07$Fhp$!*<&o6BF07 $F]q$!*nJ$*>#F07$Fbq$!)B9*z)F07$Fgq$\"*#[G6:F07$$\"5n;zp[KJ2QBFH$\"*A0 6a\"F07$$\"5++D\"G8O*RrBFH$\"*$pbn=F07$$\"5M$3Fp,fDZS#FH$\"*N&)=e%F07$ F\\r$\"*l9$fYF07$$\"5+]i:&y/y8Z#FH$\"*GpZ-&F07$$\"5ML3FpwUq/DFH$\"*^>O n(F07$$\"5n;aQ`00.QDFH$\"*=c%*y(F07$Far$\"*Mx95)F07$$\"5nm\"Hd?>4!QEFH $\"+s)zq+\"F07$$\"5ML$eR(\\;m/FFH$\"+[&4C2\"F07$$\"5n;H2eyy)zt#FH$\"+e JF%3\"F07$$\"5++v=U2TJrFFH$\"+O%[T3\"F07$$\"5M$3-jiLSY!GFH$\"*-fX()*F0 7$Ffr$\"*#\\di**F07$$\"5MLekyA!>Y!HFH$\"*D_E\"yF07$$\"5++](o/[r7(HFH$ \"*ABDl(F07$$\"5nmT5:QR#z.$FH$\"*TnYM&F07$F[s$\"*(o#==$F07$$\"5MLL3Fp1 nrJFH$\"*eAa2$F07$$\"5MLL$3F%\\wQKFH$\"*_n2$>F07$$\"5ML$3F%z?JsKFH$\"* XFv\">F07$$\"5MLLe9;#feI$FH$\"*+#H'p\"F07$$\"5ML$ekGN1%RLFH$\"*QU:q\"F 07$F`s$\"*W2\"*y\"F07$$\"5MLL$ek.Ur]$FH$\"*v!o(RFH$\"*,X0k(F07$Fdt$\" *k&H^xF07$$\"5MLLeR(\\!*46%FH$\"*!fDrvF07$Fit$\"*C'G.vF07$F^u$\"*F[u#o F07$Fcu$\"*Wa/k&F07$$\"5+++]P4'4.P&FH$\"*]$Hx[F07$Fhu$\"*#>+\">%F07$$ \"5qmm;HKp%zh&FH$\"*\\]dQ%F07$F]v$\"*ehcH&F07$$\"5NL$e9\"f-ssdFH$\"*Dz nX&F07$$\"5qmm\"zW#\\n.eFH$\"*8RMz'F07$$\"5++]P%)*eHY$eFH$\"*rHBx'F07$ $\"5NLL$3_D%eleFH$\"*7K)\\oF07$$\"5+++v$fe$\\FfFH$\"*+qfd)F07$Fbv$\"*Q ;OL*F07$$\"5NLLeRZAJ^gFH$\"+&[dS-\"F07$$\"5+++]7y:A8hFH$\"+ei+Z6F07$$ \"5NL$e*[Vi#[u\"F07$$\"5NLLL3_DG1qFH$\"*,/JT\"F07$$\"5qmmm; /'*[MrFH$\"*b)eP;F07$Faw$\"*[r6#>F07$Ffw$\"*0UYj#F07$F[x$\"*Qc\\)HF07$ Fex$\"*?!3?CF07$F_y$\"*!Gj[5F07$$\"50++v$fL:&*Q)FH$\")S0;%*F07$$\"5qmm ;H#o)fb%)FH$\")0)*=#)F07$$\"5SLLekG?o@&)FH$\")3LIyF07$$\"50++++v`w(e)F H$\")V.%H)F07$$\"5SLL$3x1K*>()FH$\"*Z%z#3\"F07$Fdy$\"*]6>F\"F07$$\"5++ ]PfL/9&)))FH$\"*kKFJ\"F07$$\"5NLL3x1@==*)FH$\"+>'R5F0$\"*'R$f-#F07$Fcz$\")hN@25\"F0$!)&QYo*F07 $$\"5+](=<,(\\r.6F0$!)j=4)*F07$F\\\\l$!)8t.%*F07$$\"5++v$4@\"ep76F0$!) 574;F07$$\"5n;aj4#)QmF6F0$\"*KYK0#F07$Ff\\l$\"*XP=E%F07$$\"5M$ e9\"456lL6F0$\"*y\"fGVF07$$\"5nmT&)3CZkO6F0$\"*Q!\\!e%F07$$\"5+]Pf3Q$Q 'R6F0$\"*X#[8oF07$F[]l$\"*XE]e(F07$$\"5++vVt_@P\\6F0$\"*U+rd*F07$F`]l$ \"+P:V*3\"F07$$\"5MLek.aD&G;\"F0$\"+ObLu7F07$Fe]l$\"+;^r18F07$Fj]l$\"+ v-h07F07$F_^l$\"*mH2w*F07$$\"5ML3F%zv$H.7F0$\"*#et*R(F07$$\"5++]PfeR.5 7F0$\"*]aNW&F07$$\"5M$3F>*eSS87F0$\"*/bRY&F07$$\"5nm\"zW#fTx;7F0$\"*c` NT&F07$$\"5+]7.dfU9?7F0$\"*,U=Y%F07$Fd^l$\"*uHRZ%F07$$\"5++voagXDI7F0$ \"*#*QO[%F07$$\"5nm;z>hZ*pB\"F0$\"*8h%zXF07$$\"5MLe*[='\\tV7F0$\"*H=`I &F07$Fi^l$\"*`n>c'F07$Fc_l$\"*bc^$zF07$F]`l$\"*V6vD*F07$Fb`l$\"*%yEz** F07$Fg`l$\"+e%*GT5F07$F\\al$\"+J\\7U5F07$Faal$\"+$*)fe/\"F07$$\"5+++]7 GR:=8F0$\"+m]'G,\"F07$Ffal$\"*G>D]*F07$$\"5+++D\"ybd%Q8F0$\"*$G)3V*F07 $$\"5+++]PMaAX8F0$\"*Jq*)G*F07$$\"5+++v$4J$*>N\"F0$\"*l.[E*F07$F[bl$\" *O=ZJ*F07$F`bl$\"*E5Lg*F07$Febl$\"**)=ju*F07$$\"5+++]7.\"R*)) F07$$\"5++++v=A)RU\"F0$\"*ceJ>)F07$Fdcl$\"*HTdW(F07$$\"5+++]7G&[NW\"F0 $\"*rFq&pF07$$\"5++++DJ12]9F0$\"*vw&HoF07$$\"5+++D\"GoJLX\"F0$\"*(RSBo F07$$\"5+++]PMFfc9F0$\"*1]E&oF07$$\"5+++v$fy`)f9F0$\"*)o**=sF07$Ficl$ \"*a\")oC(F07$Fcdl$\"*-$))*G)F07$F]el$\"*5'[K**F07$Fbel$\"+$3[\\@\"F07 $Fgel$\"+Q$omT\"F07$F\\fl$\"+V@s)R\"F07$Fafl$\"+P?&GF\"F07$F[gl$\"+cK4 &4\"F07$Fegl$\"+Vq2i5F07$Fjgl$\"*y^H^)F07$F_hl$\"*JbC&zF07$$\"5n;aQ`01 #\\c\"F0$\"*Mw'*='F07$$\"5++v$4@Ej>d\"F0$\"*WiRd%F07$$\"5M$e*[o=f+z:F0 $\"*&fVWSF07$Fdhl$\"*eFPZ$F07$$\"5M3x\"[N!*p&*e\"F0$\"*RkcC$F07$$\"5+] Pf$=B\"4$f\"F0$\"*Dm/7$F07$$\"5n\"zpB,c7mf\"F0$\"**=y&3$F07$$\"5MLe9T) )Q8+;F0$\"*X_G<$F07$$\"5+v=#*p;_l.;F0$\"*llk7$F07$$\"5n;zp)\\awrg\"F0$ \"*+K$=JF07$$\"5MeRZFtyp5;F0$\"*r^QO$F07$Fihl$\"*^T8K$F07$Fcil$\"*4iV' RF07$Fgjl$\"*h^1X%F07$F[\\m$\"*m\"Q/JF07$F_]m$\"*uX8O#F07$$\"5+]iSmbs& Hx\"F0$\"*.0/b#F07$$\"5MLe9m%>(*)zMF07$$\"5o;/^ci$*>$)=F0$\"*3(o1PF07$$ \"5ML3_]%QU$*)=F0$\"*AS(>QF07$$\"5,]7`W1a[&*=F0$\"*>\")\\*QF07$$\"5om; aQG%G;!>F0$\"*Khr)QF07$$\"5,+DcEsW\"R\">F0$\"*NSSn$F07$F^^m$\"*N@%yKF0 7$$\"5++]i!Rgs2&>F0$\"*A3S)=F07$Fc^m$\"*N0O4\"F07$$\"5Me9\"4gi9(y>F0$ \")(edp*F07$$\"5+]i:NgX3#)>F0$\")FFLvF07$$\"5nT5Sp%\\aa)>F0$\")t(\\h(F 07$$\"5MLek.HW#)))>F0$\")QMhqF07$$\"5+D1*yLO%>#*>F0$\")K*e?'F07$$\"5n; a8s(Hkb*>F0$\")\"y3O'F07$$\"5M3-Q1KU$*)*>F0$\")]3gxF07$Fh^m$\")YN(\\*F 07$$\"5M$e9\"4NS/4?F0$\"*vQW]\"F07$$\"5nmTgx.Ry:?F0$\"*Dm)R@F07$$\"5+] P4YsP_A?F0$\"*lZf:%F07$F]_m$\"*O?S$fF07$$\"5,D\"G)[vNjK?F0$\"*F1J;(F07 $$\"5o;H2$)4N+O?F0$\"*,WxP(F07$$\"5M3xJsUk2#F0$\"+!f`*f:F07$$\"5M3_DGcE\")z?F0$\"+)GZSU\"F07$Fa`m$\"+i.( QT\"F07$$\"5,+DJXR0g*3#F0$\"+`7pQ7F07$$\"5,+]7G)[=g4#F0$\"+&R.F=\"F07$ $\"5,+v$4rVOC5#F0$\"*cLm\\*F07$$\"5,++v$fQa)3@F0$\"*esHy)F07$$\"5,+Dcw MBF:@F0$\"*QM9\"yF07$$\"5,+]Pf$G!p@@F0$\"*OFr2(F07$$\"5,+v=UK#3\"G@F0$ \"*bz%prF07$Ff`m$\"*O!QWtF07$$\"5,+D\"y+8W49#F0$\"*52:H)F07$$\"5,+]i!* y?OZ@F0$\"*rE(z%)F07$$\"5,+vVtF+y`@F0$\"*y5=i*F07$$\"5,++Dcwz>g@F0$\"+ OCi'4\"F07$$\"5,+](=U(Q.t@F0$\"+18Y47F07$F[am$\"+3np&G\"F07$$\"5M3_D.+ !=#*=#F0$\"+(3mUI\"F07$$\"5o;/,>Gic#>#F0$\"+Z8V=8F07$$\"5,DcwMcW\"f>#F 0$\"+x8E=8F07$$\"5ML3_]%oi#*>#F0$\"+L]Y;8F07$$\"5oTgFm74h-AF0$\"+Dp::8 F07$$\"5,]7.#39ff?#F0$\"+t(4UJ\"F07$$\"5Meky(*otI4AF0$\"+WCl(H\"F07$$ \"5om;a8(fbE@#F0$\"+d(=oG\"F07$$\"5,+Dcw4&[gA#F0$\"+r/uZ7F07$F`am$\"+$ zzX?\"F07$$\"5++]ilZsAmAF0$\"+i$*3=7F07$Feam$\"+\"ee&R7F07$$\"5nm;a8A3 hYe8\"F07$Fjam$\"*!)>Hg*F07$$\"5n;a)e'3!e$[BF0$\"*@0Ch*F07$$ \"5nmTN'fW2XN#F0$\"*ac-e*F07$$\"5nT&)ehk@edBF0$\"*a*4r'*F07$$\"5n;H#oK )olgBF0$\"+wkJ:5F07$$\"5n\"Hd?>gJPO#F0$\"+)z3P,\"F07$$\"5nm;Hd?j!oO#F0 $\"+)=X7-\"F07$$\"5n;/w(yvbHP#F0$\"+@mFI6F07$$\"5nm\"H#=&>0\"zBF0$\"+$ Qn[@\"F07$$\"5n;zp[KYD&Q#F0$\"+Ut'**G\"F07$F_bm$\"+Ge&)o9F07$$\"5++D\" yv1qYS#F0$\"+'Rjqf\"F07$Fdbm$\"+Ui]P;F07$$\"5nmT5:j??JCF0$\"+q&3.[\"F0 7$Fibm$\"+:9FS7F07$$\"5n;H2$)f55^CF0$\"+\\PR75F07$F^cm$\"*\"f(f`*F07$$ \"5+](=Ww>EF0$\"*?0e9%F07$Fjfm $\"*e_av$F07$$\"5omT&)3\\svPEF0$\"*R'fJPF07$$\"5ML$e9\"f[vVEF0$\"*440i $F07$$\"5,+D19pCv\\EF0$\"*MOUl$F07$F_gm$\"*ETYm$F07$$\"5,+](=#*HXxm#F0 $\"*>7\\&QF07$Fdgm$\"**p&G:%F07$$\"5om;HKRdt\"p#F0$\"+V.7bUFH7$Figm$\" *<#HMUF07$F^hm$\"*Vc!RTF07$Fchm$\"*_!e0SF07$$\"5o;/^1v)=ht#F0$\"*OXp*R F07$Fhhm$\"**3lbSF07$$\"5M$e9T8/u!\\FF0$\"*&3daTF07$F]im$\"*VyTD%F07$$ \"5ML3_v!z1&oFF0$\"*476m%F07$Fbim$\"*2Yt=&F07$$\"5oTg-g[2q%y#F0$\"*D,& *=&F07$$\"5M$3Fp,aRzy#F0$\"*:#*[@&F07$$\"5,D\"GQ2*4#HF0$\"*^=\\x'F07$Fj[n$\"*\"HFAuF07$$\"5,]i:N54\"3$HF0$ \"+F!y2+\"F07$$\"5,+v=U2V3MHF0$\"+T#)Q<5F07$$\"5,](=#\\/xNPHF0$\"+DkAq 5F07$F_\\n$\"+\"z*\\Z8F07$$\"5,]7Gj)\\/R%HF0$\"+r2'yO\"F07$$\"5,+DJq&* y;F07$Fc]n $\"+j\"RlQ\"F07$F]^n$\"+\\\"[e;\"F07$Fg^n$\"*FqI(**F07$Fa_n$\"+n)z>-\" F07$F[`n$\"+N7_K6F07$Fe`n$\"+vXD\"QJF0$\"+:7%*z9F07$$ \"5,]P%)*32U;9$F0$\"+.N#4Z\"F07$$\"5,vVtF%oe^9$F0$\"+m5%4Z\"F07$F]cn$ \"+`vss9F07$Fbcn$\"+I._KF0$\"+w(osB\"F07$Fjen$\"+++gd7F07$$\"5,+] P4@?*4F$F0$\"+#H%zN9F07$F_fn$\"+pD_'f\"F07$$\"5om;HK*oqiH$F0$\"+rmKc+K:(F07$F_[o$\"*9jx#oF07$Fi[o$\"*Vs=$eF07$Fc\\o$\"*>&QO]F07 $F]]o$\"*u(H*H&F07$Fg]o$\"++T3ycFH7$$\"5M$3_D`ZS,g$F0$\"+QMyOcFH7$$\"5 ,+v=#\\/Dog$F0$\"*iT3j&F07$$\"5o;H#=Xh4Nh$F0$\"*cex`&F07$F\\^o$\"*A/CW &F07$$\"5,]P4r`(yoi$F0$\"*&y&=W&F07$$\"5om\"H2LKjNj$F0$\"*q-qT&F07$$\" 5M$ek.H*yCSOF0$\"*n1'4bF07$Fa^o$\"*Xq!GbF07$$\"5om;zp)4\"4sOF0$\"*S$f% R'F07$Ff^o$\"*j\")*enF07$$\"5,+]P4r$3Cs$F0$\"*bLu'fF07$F[_o$\"*%y/)3&F 07$$\"5MLL3F>grgPF0$\"*j2^S%F07$F`_o$\"*d:Z=%F07$$\"5om\"zWUG_rx$F0$\" *'GX4UF07$$\"5ML$eRs`R/y$F0$\"*PFt*RF07$$\"5,+vVB!zEPy$F0$\"*7SC0%F07$ $\"5omm\"HK/9qy$F0$\"*l=I6%F07$$\"5,+](=#\\&)e$z$F0$\"*MZaD%F07$Fe_o$ \"*'))=8]F07$$\"5om;z>hvt1QF0$\"*F^wF&F07$Fj_o$\"*r+y&oF07$$\"5om\"H#= ?$*f;QF0$\"*&3-7rF07$$\"5ML$3xJd'))>QF0$\"*9jHH*F07$$\"5,+v=J\"F07$$\"5omT5:QGKOQF0$\"+7zUX8F07$Fd`o$\"+BwEt:F07$ $\"5om;a8(f%=YQF0$\"+H$=kp\"F07$Fi`o$\"+**yQM>F07$$\"5o;/,W%)o=fQF0$\" +mf@4?F07$F^ao$\"+la>q?F07$$\"5,]7.2ZC/sQF0$\"+lE^)*>F07$Fcao$\"+@\")H e>F07$$\"5M$3_+(4!)*[)QF0$\"+(\\!fG=F07$Fhao$\"+A;G(e\"F07$$\"5o;H2LsN v(*QF0$\"+vGIj:F07$F]bo$\"+DT\"3R\"F07$Fbbo$\"+()[;z8F07$Fgbo$\"+Ov/n7 F07$F\\co$\"+]&Q0F\"F07$Faco$\"+YL\"=F\"F07$$\"5,v$fLp!3D?RF0$\"+xW+[7 F07$Ffco$\"+sbO]7F07$$\"5o\"zp[#)eym#RF0$\"+YA5d7F07$F[do$\"+*Q>,J\"F0 7$F`do$\"+%p\\\\V\"F07$Fedo$\"+X$Qhd\"F07$Fjdo$\"+)ez*\\v\"F07$$\"5,D\"G8'G$yJ0%F 0$\"+^5K`w!*fSF0$\"+Gtkr< F07$F]go$\"+E$QHx\"F07$F[io$\"+m&*4d;F07$Feio$\"+;w!*3;F07$F_jo$\"+rZ# et\"F07$Fijo$\"+\\H#*4>F07$F^[p$\"+%Hi1!>F07$Fc[p$\"+%z`K!>F07$Fh[p$\" +.owm?F07$F]\\p$\"+(*fC`?F07$$\"5N$3_]4R&z'>%F0$\"+^%3p/#F07$$\"5NL3-8 D$4,?%F0$\"+-F`k@F07$$\"5N$e*)4$fKU.UF0$\"+f?NY@F07$Fb\\p$\"++iGH@F07$ $\"5N$3Fpw7^+@%F0$\"+BP3l@F07$$\"5NLe*[=1lL@%F0$\"+j*\\F9#F07$$\"5N$ek Gg**ym@%F0$\"+&Q!*e6#F07$Fg\\p$\"+Chfa?F07$$\"5NL3xc)z?mA%F0$\"+))Q<'* >F07$F\\]p$\"+8Q4B=F07$$\"5NLekGNl()RUF0$\"+qcH/;F07$Fa]p$\"+yoGK:F07$ $\"5NL3_+sA8`UF0$\"+>6)[I\"F07$Ff]p$\"+AH_J6F07$$\"5NLeRs3!)QmUF0$\"+L Czm5F07$F[^p$\"*O)fv$*F07$$\"5-+v$4'\\=;'G%F0$\"*%fvL%)F07$F`^p$\"*^!) e?)F07$$\"5NLe9m%z`CJ%F0$\"*`LLF)F07$Fe^p$\"*_(Qz#)F07$F__p$\"*EF!>%)F 07$Fi_p$\"*u!*fm)F07$F^`p$\"*CPPt)F07$Fc`p$\"*LK_l)F07$$\"5,+](=UAVBS% F0$\"*vg\"\\wF07$Fh`p$\"*i.us'F07$$\"5,v=<,PF_HWF0$\"*!QrrlF07$$\"5o;/ w(GoUDV%F0$\"*bxe]'F07$$\"5Me*[V(GEcNWF0$\"*w#G0lF07$$\"5,+v$4Yd#eQWF0 $\"*x!y*['F07$$\"5M$e9TjYAYW%F0$\"*v'GgkF07$$\"5om;H2eBm]WF0$\"*Doac'F 07$$\"5MLek`T@uiWF0$\"*>qIv'F07$F]ap$\"*X_H%pF07$$\"5M$3-jP7$GyWF0$\"+ %)oMypFH7$$\"5omTg_AVu\"[%F0$\"+LK%o1(FH7$$\"5,]i!*G@b?&[%F0$\"+oxTnqF H7$Fbap$\"+8[*)zqFH7$$\"5,+D\"yv6*e&\\%F0$\"+n'oZ4(FH7$Fgap$\"*(QwNqF0 7$F\\bp$\"*H*yGpF07$Fabp$\"*$eoYoF07$F[cp$\"*%)eoG(F07$Fecp$\"*S\\28)F 07$F_dp$\"*W&*Q\"pF07$Fidp$\"*DN4!eF07$$\"5-]iS\"pGX7p%F0$\"**p$Q6'F07 $F^ep$\"*;Z]Vr%F0$\"*\\E-Z*F07$$\"5o;zpB,4lK\"F07$$\"5- ]P%)R3P&3t%F0$\"+VE)3g\"F07$$\"5N3-))=5W:MZF0$\"+#4!\\E;F07$F]fp$\"+T0 a)o\"F07$$\"5-DJ&pP\"evSZF0$\"+_b()\\>F07$$\"5N$e*)fb^cSu%F0$\"+hE,y>F 07$$\"5oTg-N#F07$$\"5o;a8sA$fsv%F0$\" +Z1B`AF07$Fgfp$\"+BIw3BF07$$\"5o\"zW#4G9;nZF0$\"+\\@>]M&[F0$ \"+)Hv\"y=F07$F^jp$\"+WHc-?F07$$\"5-+]7GQPsy[F0$\"+'Q,U5#F07$Fcjp$\"+Z -XP@F07$$\"5-+]7.#Q?&=\\F0$\"+&p.c1#F07$$\"5,++voa-oX\\F0$\"+$*zTg#f\\F0$\"+7n%=/#F07$$\"5++]PMF,%G(\\F0$\"+73Ul?F07$$\"5+ ]7y]&4I'z\\F0$\"+\"3\"Hj?F07$$\"5++v=nj+U')\\F0$\"+syPh?F07$$\"5+]Pf$= .5K*\\F0$\"+]9GN?F07$Fhjp$\"+-Uw#)>F0-F][q6&F_[qF`[qF(F`[q-Fd[q6#%Ta~m inor~modification~of~the~Bogacki-Shampine~schemeG-F$6%7e`mF'7$F2$!)Ihp >F07$F<$!)dAGYF07$FA$\")FE&\\%F07$FF$\"*I$e:9F07$Fen$\")(G0!HF07$Fio$! *$\\VB9F07$Fcp$!)fY\"G'F07$F]q$!)0*f3&F07$$\"5M$eRAr@oZ?#FH$\")7Tl(*F0 7$Fbq$\")HP`**F07$$\"5+](o/[n?9F#FH$\"*4x:A\"F07$Fgq$\"*QP@M$F07$Fj_q$ \"*2I3S$F07$F_`q$\"*j'F07$F\\aq $\"*V'*Q_'F07$Faaq$\"*NCNo)F07$Ffaq$\"*]xO\"))F07$Far$\"*6015*F07$F^bq $\"+9[\"42\"F07$Fcbq$\"+'=ej9\"F07$Fhbq$\"+%*3Ff6F07$F]cq$\"+Dn**o6F07 $Fbcq$\"+U'>q8\"F07$Ffr$\"+BN]Z6F07$$\"5+]7`%Rz#HrGFH$\"+^zzZ6F07$Fjcq $\"+NMg^5F07$$\"5n;/wi^_%z$HFH$\"+5\"4#f5F07$F_dq$\"+a(yG0\"F07$Fddq$ \"*,*R&G*F07$F[s$\"*]/8)yF07$Faeq$\"*R+]y'F07$F`s$\"*v`#4hF07$Fhfq$\"* il?V'F07$Fes$\"*JC=?(F07$Fjs$\"***epA)F07$F_t$\"*7.mc*F07$Ffgq$\"*V0Ii *F07$Fdt$\"*&z/\"))*F07$F^hq$\"*%He(*)*F07$Fit$\"*+!*p')*F07$F^u$\"*)o i-&*F07$Fcu$\"*9?7)*)F07$F\\iq$\"*)pL)*zF07$Fhu$\"*,3Xb'F07$$\"5+++Dc, w.cbFH$\"*(fOciF07$Fdiq$\"*/M%ydF07$$\"5NLL3-ji&)zcFH$\"*P&[ZdF07$F]v$ \"*LjFH$\"+AUHL6F07$Fi]r$\"+^o@?6F07$F^^r$\"+(=T13\"F07$Fc^r$ \"*:%>?)*F07$$\"5NLL3x1h]DlFH$\"*PKc@*F07$Fh^r$\"*i-7/*F07$$\"5+++D\"G j4'*e'FH$\"*e6i+)F07$F]_r$\"*m(H^nF07$$\"5qmmT&)eJr`mFH$\"*3RZh'F07$Fb _r$\"*jr:o&F07$$\"5NLLe*[o;yr'FH$\"*k0?\"RF07$F\\w$\"*S\\/$QF07$$\"5++ +v$4@?>y'FH$\"*rBi<$F07$$\"5NLL$eR(>(R\"oFH$\"*E`\"38F07$$\"5qmm\"zptB g%oFH$\"*$=8y7F07$Fj_r$\")K.(=*F07$$\"5qmm;/E!z@%pFH$!)J\\#*fF07$F_`r$ !*d))4o\"F07$$\"5qmmT5:VLQqFH$!*')>!f;F07$$\"5+++]7ygQqqFH$!*%)Q\")o\" F07$$\"5NLLe9TyV-rFH$!*D/80#F07$Fd`r$!*'>OE?F07$$\"5NLL$3-8$f)>(FH$!*c =!p>F07$Faw$!*a!o;>F07$Ffw$!)2+ooF07$F[x$\")iTTRF07$$\"50+++D\"Gg$fyFH $\")^c))\\F07$F`x$\")v+qbF07$$\"5qmmm\"z>_>*zFH$\")51MZF07$Fex$\")YA%* QF07$Fjx$!)AJ::F07$F_y$!)>gevF07$Fdy$!)2l,#*F07$Fiy$!*X*e\"=\"F07$$\"5 qmT5lv.u[%*FH$!*J\"3a7F07$$\"5NL$3_+A:n^*FH$!*qpKA\"F07$$\"5++DJXk+p%e *FH$!*&zum6F07$Fjer$!*!p)Q-\"F07$$\"5++]il(f9')y*FH$!)_v(z%F07$F^z$\") 8b)R#F07$$\"5++v$f38RD***FH$\")\\O&\\%F07$$\"5nmTg_(R^g+\"F0$\")&H/+'F 07$$\"5+v$4Y(R,X45F0$\")j&)GqF07$$\"5M$e9m>))[G,\"F0$\")7^&[(F07$$\"5n \"z>'=CwC;5F0$\")_P\"\\(F07$Fbfr$\")')3roF07$$\"5MLekGN8CL5F0$\")rF*f$ F07$Fcz$!)\"=7S\"F07$F][l$!*x)Gc6e%F07$Fj^s$\"*5`ee%F07$$\"5n;zWWo:F0$\"*k@K,%F07$F[hs$\"*)Qph@F07$F`hs$\")e>D&*F07$Fdhl$!)$)p] IF07$F]is$!*j$\\,>F07$Fgis$!*A\\6x#F07$F\\js$!*-24t#F07$Fajs$!*F*zOFF0 7$Ffjs$!*)z\"4'HF07$Fihl$!*g'>BHF07$Fcil$!*>#*o)=F07$Fgjl$!)>Yi[F07$F[ \\m$!)n^6$*F07$F_]m$!*zbw$>F07$F_\\t$!*9W5o\"F07$Fi\\t$!*%[Rx:F07$F^]t $!*[M\\d\"F07$Fc]t$!*4#z(e\"F07$Fh]t$!*#*p$o;F07$Fd]m$!*cL4#=F07$F`^t$ !*?,&p=F07$Fe^t$!*Ny>(>F07$Fj^t$!*MQ?/#F07$F__t$!*/.]+#F07$Fd_t$!*'*)H PF 0$!)&)3F6F07$F`at$!(\"*G\"RF07$$\"5M3xcB$)f)p\">F0$!(*>%p%F07$$\"5o;Hd ?%\\d+#>F0$!)9s]=F07$$\"5,D\"yv^+HJ#>F0$!)e8c=F07$F^^m$!)MHo>F07$Fhat$ !*ZHhg\"F07$Fc^m$!*CTY!GF07$Febt$!*%=GNKF07$F_ct$!*2RIP$F07$Fdct$!*!\\ +SMF07$Fict$!*B8#*[$F07$F^dt$!*bz&eLF07$Fh^m$!*fxv7$F07$Ffdt$!*;h[l#F0 7$F[et$!*LJ/0#F07$$\"5Me*[=\"QQ:>?F0$!*,q#p5F07$F`et$!)6\\R9F07$$\"5nT &Q.oq$*e-#F0$!(&*)[&)F07$F]_m$\"*'\\w;8F07$Fhet$\"*bUxQ#F07$F]ft$\"*)o m,DF07$Fbft$\"*ra&ySF07$Fb_m$\"*Zi60&F07$Fjft$\"*>%f\"e'F07$Fg_m$\"*kl XT(F07$Fggt$\"*'*)\\'f)F07$F\\`m$\"*v58#))F07$Fdht$\"*4e/!*)F07$Fiht$ \"*yP'f*)F07$F^it$\"*/C'\\$)F07$Fa`m$\"*Eq'4$)F07$Ffit$\"*%*>^T(F07$F[ jt$\"*`)[:rF07$F`jt$\"*!=b\"p&F07$Fejt$\"*4rt8&F07$F_[u$\"*;]rU$F07$Ff `m$\"*joL-$F07$F[]u$\"*Q))zi%F07$F[am$\"*wA)*G'F07$F]^u$\"*iMHv'F07$Fg ^u$\"*P(3doF07$F\\_u$\"*Z;!=pF07$Fa_u$\"*UXq\"pF07$Ff_u$\"*`(HjoF07$F[ `u$\"*%RC=oF07$F``u$\"*Klhb'F07$F`am$\"*54K='F07$$\"5+]P4ryy8YAF0$\"*T u<8'F07$$\"5nmTg-NV$GD#F0$\"*m:2:'F07$$\"5M$e9T8zI&fAF0$\"*JW\"*G'F07$ Fh`u$\"*U4'ejF07$$\"5MLekGg,izAF0$\"*JK='oF07$Feam$\"*\"yoNqF07$$\"5nm Tg_Z>J0BF0$\"*U\"pfoF07$F`au$\"*+9,D'F07$$\"5nm\"zWnp4*HBF0$\"*+?[n%F0 7$Fjam$\"*h?So$F07$F]bu$\"*sS)*)HF07$Facu$\"*cDjt#F07$F[du$\"*YI*4MF07 $F_bm$\"*f-!R]F07$Fhdu$\"*OQcS'F07$Fdbm$\"*l[`%yF07$$\"5+]7yDk!pXU#F0$ \"*H\\2a)F07$F`eu$\"*$36B()F07$$\"5M$3FW?1NyV#F0$\"*rCFT)F07$Fibm$\"*8 '4cwF07$$\"5Me9TQgXyZCF0$\"*?Z/H(F07$Fheu$\"*B(H0eF07$$\"5+vVtFfvTaCF0 $\"*sjrq&F07$F^cm$\"*BooC&F07$$\"5n\"Hdq\"e00hCF0$\"*QOF<$F07$F`fu$\"* ZxK6$F07$$\"5M3-Q1dNonCF0$\"*F07$$\"5n;/,>`!**3\\#F0 $!*V@`;$F07$$\"5+v=nj_b@%\\#F0$!*D`67$F07$F]dm$!**G'))>$F07$Faem$!*\"R Ig?F07$Fefm$!*9bpP\"F07$Fjfm$!*D7nZ#F07$F_gm$!*G$QiFF07$Fdju$!*Yuen#F0 7$Fdgm$!*gJkW#F07$$\"5,+voHH\"Qdo#F0$!*Px=U#F07$F\\[v$!*s`%\\BF07$$\"5 MLe*[$\\Lt(p#F0$!*Vm6N#F07$Figm$!*]%4rBF07$F^hm$!*f9B\\#F07$Fchm$!*LN% \\FF07$Fj[v$!*qB>#GF07$Fb\\v$!*q1>x#F07$F]im$!*$yY5FF0 7$Fbim$!*\\h!e:F07$Fgim$!)%pkF07$F`cv$\"*Ny=O(F07$Fi\\n$\"*#*)oIvF07$F^]n$\"*UPym'F07$F c]n$\"*$e&*[`F07$F]^n$\"*=I)oQF07$Fg^n$\"*h9C:#F07$F\\_n$\"*Y.#)y\"F07 $Fa_n$\"*j?_^\"F07$Ff_n$\"*PQhc\"F07$F[`n$\"*qqby\"F07$Fe`n$\"*gyra#F0 7$F_an$\"*Oko9%F07$Fdan$\"*-Co\"\\F07$Fian$\"*oD/K&F07$$\"5,v=QN&F07$$\"5,](=U2JR:5$F0$\"*S!y7aF07$$\"5,DcEs!\\1Z5$F0$\"*zZC T&F07$F^bn$\"*'>'**Q&F07$$\"5,]iSmI!3U6$F0$\"*\")3OE&F07$Fcbn$\"*iaM* \\F07$Fhbn$\"*=s)QZF07$F]cn$\"*w*=OYF07$Fbcn$\"*`VF\"\\F07$Fgcn$\"*a7' R_F07$$\"5,D1k1=[K!=$F0$\"*#*)>qaF07$F\\dn$\"*_$RqaF07$$\"5,v=U#[/et=$ F0$\"*%=;\"[&F07$Fadn$\"*\\&R@bF07$$\"5,DJ?er7R%>$F0$\"*!R$4_&F07$Ffdn $\"*()Q4[&F07$$\"5,vV)R$)\\C9?$F0$\"*Q@FF&F07$F[en$\"*/J!p_F07$F`en$\" *+[@H%F07$Feen$\"*g)z!)GF07$Faiv$\"*%ogu>F07$Fjen$\"*[1oL\"F07$F^jv$\" *Qw()o\"F07$F_fn$\"*TH(>EF07$$\"5o;zpt)=\"z'G$F0$\"**[<\"p#F07$$\"5,+D cEA5&**G$F0$\"*tb#**QF07$$\"5M$3F%zb36$H$F0$\"*>1N#RF07$Ffjv$\"*O9f&RF 07$$\"5ML3-Qc.f-LF0$\"*bUQO&F07$Fdfn$\"*xe%>fF07$$\"5om\"z%\\!pH_J$F0$ \"*tR0f'F07$F^[w$\"*2'H4uF07$$\"5,+v$4Y-pyK$F0$\"*!>t)H(F07$Fifn$\"*SX !HsF07$Ff[w$\"*pn-<'F07$F^gn$\"*=$*[r&F07$$\"5,+D1Rvw0^LF0$\"*@Jx/%F07 $F^\\w$\"*WOZ(RF07$$\"5ML3-))o_!yN$F0$\"*$>%R\\$F07$Fcgn$\"*A`OR\"F07$ $\"5om\"zpB'GbkLF0$\"*8)3g8F07$$\"5ML$e9\"fm#zO$F0$\")E#eZ)F07$$\"5,+v $feX+8P$F0$!*'f\\$=\"F07$Fhgn$!*<(=v6F07$$\"5MLe*[$\\![!yLF0$!*)>'Gh\" F07$$\"5,+]P4Y=U\"Q$F0$!*7um;$F07$$\"5omT&QGk&z%Q$F0$!*-Qi7$F07$F]hn$! *z'[9MF07$Fa]w$!*L2FF%F07$Fbhn$!*59yu%F07$F^^w$!*!fMtYF07$Fghn$!*>!)[g %F07$Fain$!*iT+4%F07$F[jn$!*9[rF$F07$Fejn$!*6_+C#F07$F_[o$!*x&z!*=F07$ Fi[o$!*.l7f#F07$Fc\\o$!*1*y1NF07$$\"5,](oa&=RmYNF0$!*ukF^$F07$Fh\\o$!* 8A#RNF07$$\"5,v=#\\Hx!pcNF0$!*Y-_`$F07$$\"5M$eRZx0L+c$F0$!*X;.X$F07$$ \"5o\"HdXDMvLc$F0$!*Le-X$F07$F]]o$!*W$F07$$\"5o;/,%p>-Md$F0$!*%=k1L F07$Fb]o$!*26,=$F07$$\"5,]7G8O8x'e$F0$!*,Ld<$F07$Fg]o$!*R(>FJF07$F^aw$ !*VhC<$F07$Fcaw$!*y*o!=$F07$Fhaw$!*(4*3I$F07$F\\^o$!*w)QhMF07$$\"5oTgF \"*ok`BOF0$!*v-:Y$F07$F`bw$!*xO9Z$F07$$\"5Me9\"4&Q5AIOF0$!*TKDe$F07$Fe bw$!*%*)f#e$F07$Fjbw$!*^IHf$F07$Fa^o$!*UHSe$F07$Fbcw$!*_8$)p#F07$Ff^o$ !*1l)==F07$$\"5o\"Hd?Wc%R+PF0$!*8,Dj\"F07$$\"5,]7`%RRRNq$F0$!*.;Yj\"F0 7$$\"5M3_+ZBUo1PF0$!*l9]j\"F07$$\"5om\"z%*H0H)4PF0$!*[*)el\"F07$$\"5,D J&>D)Q(Hr$F0$!*'p1k;F07$$\"5M$3FW?r=hr$F0$!*G*=u;F07$$\"5oT5!p:aj#>PF0 $!*,`]\"=F07$Fjcw$!*b\"*=$>F07$$\"5ML3F>*o()\\t$F0$!*=*\\+CF07$F[_o$!* (4#z1$F07$Fbdw$!*3#*y=%F07$F`_o$!*7I6$[F07$F_ew$!*?2F?&F07$Fiew$!*z\"H \\`F07$$\"5MLeRA'H,.z$F0$!*1VqE&F07$F^fw$!*!)zlM&F07$$\"5omTN@-e(oz$F0 $!*\\s]Q&F07$Fe_o$!*@.]q%F07$$\"5,+DJ?3.X.QF0$!*n.,y%F07$Fffw$!*A1$pZF 07$$\"5ML3F>9[-5QF0$!*lOnL$F07$Fj_o$!*&R1\"R$F07$F^gw$!*Ep'GLF07$Fcgw$ !*\"3f$R\"F07$Fhgw$!*Q?gB\"F07$F_`o$!*v%>L6F07$F`hw$\")/Yn))F07$Fehw$ \"*)ehB8F07$Fjhw$\"*[9cW\"F07$Fd`o$\"*r3.8$F07$Fbiw$\"*)f\"[%QF07$Fi`o $\"*quVW&F07$Fjiw$\"*JrR!fF07$F^ao$\"*h7*QiF07$Fbjw$\"*(*\\7$fF07$Fcao $\"*(y%fp&F07$Fjjw$\"*iI#o\\F07$Fhao$\"*&=qZNF07$Fb[x$\"*9-mK$F07$F]bo $\"*A6@5#F07$Faco$\")bAX$)F07$F[do$\"()*zc%F07$F`do$\")7&G<%F07$Fedo$ \"*yt%y6F07$Fjdo$\"*mDB\\#F07$F_eo$\"*-q)RNF07$Fb^x$\"*3-lt$F07$Fdeo$ \"*J@'oQF07$$\"5Me9T)y*RM**RF0$\"*?y%*)QF07$Fj^x$\"*'4E$*QF07$$\"5oTg_ A*Gtg+%F0$\"*rR*))QF07$Fieo$\"*G'3qPF07$F^fo$\"*vqPU$F07$Fcfo$\"*h,\") =$F07$$\"5Me*)4$fu>(RSF0$\"*$45DJF07$Fh_x$\"*5RF6$F07$$\"5oTN@FP!\\k/% F0$\"*H*H;JF07$Fhfo$\"*v1q?$F07$F``x$\"*r+?A$F07$Fe`x$\"*BVWB$F07$Fj`x $\"*\"4XlMF07$F]go$\"*VUB[$F07$Fbgo$\"*'*y()z$F07$Fggo$\"*3RL3GTF0$\"*9#>)*=F07$F`io$\")t XwJF07$$\"5N$3-j7mEv9%F0$\")YRtBF07$$\"5om\"z>'ox+aTF0$!)()y\\6#F07$Ff]p$\"((HU\\F07$$\"5N$3FWX2uIE%F0$!)#p[\\$F07$Fffx$ !)_HAXF07$$\"5N$ek.H%>qpUF0$!*ZgjK#F07$F[^p$!*iX%[FF07$F^gx$!*E$4kWF07 $F`^p$!*$GbkbF07$Ffgx$!*:,IX&F07$Fe^p$!*%y;#)\\F07$Fi_p$!*!3$RU$F07$Fc `p$!*c0Uj#F07$Fjhx$!*kgT0$F07$Fh`p$!*s)*\\)QF07$Fgix$!*%ovlTF07$Fajx$! *\\N-A%F07$$\"5oTg_Z?DgTWF0$!*B)f6VF07$Ffjx$!*&e!=J%F07$$\"5,DJq?7CkZW F0$!*nYTJ%F07$F[[y$!*p>%3VF07$F`[y$!*z6)*>%F07$F]ap$!*\\FK0%F07$Fh[y$! *4uC-%F07$F]\\y$!*y$*3%RF07$Fb\\y$!**=NSRF07$Fbap$!*U#\\FRF07$Fj\\y$!* u#f6RF07$Fgap$!*.'))yRF07$F\\bp$!*JAw7%F07$Fabp$!*Bn#zVF07$F[cp$!*/=G5 %F07$Fecp$!*qDO'F07$Fd^y$!** Hc*>'F07$$\"5N3FWq))fa%p%F0$!**o&QH'F07$F^ep$!*0NuB'F07$F\\_y$!*Wb[e&F 07$Fcep$!*WI$zRF07$Fd_y$!*\"[))QSF07$Fi_y$!*#3q\\PF07$F^`y$!*:?Ow\"F07 $Fhep$!*`cby\"F07$Ff`y$!*[#3_9F07$F[ay$\")(*\\7tF07$F`ay$\")2Q#[(F07$F ]fp$\"*eQ>0\"F07$Fhay$\"*L$3RHF07$F]by$\"*sfW)HF07$Fbby$\"*1$o+KF07$Fb fp$\"*=3uP%F07$Fjby$\"*0s'RXF07$Fgfp$\"*qth([F07$Fbcy$\"*E<=)[F07$Fgcy $\"*QhgM%F07$F\\dy$\"*v9MQ%F07$F\\gp$\"*4V%>VF07$Fddy$\"*SHaA$F07$Fagp $\"*F%=Wi\"F07$Fbgy$\"*%GaN>F07$Fggy$\"*n-FN#F07$$\"5+D\"yD96Ni(\\F0$ \"*(*G2O#F07$F\\hy$\"*;Xq[#F07$$\"5+vV)*ez]-$)\\F0$\"*p#)H_#F07$Fahy$ \"*60A_#F07$Ffhy$\"*L,*QCF07$Fhjp$\"*S?5.#F0-%&COLORG6&F_[q$\"\"\"!\" \"$\"\")Fj\\blF(-Fd[q6#%3efficient~scheme~AG-F$6%7]jlF'7$F2$!)?A*z\"F0 7$F<$!)4*[s#F07$FA$\");9WtF07$FF$\"*pnwN\"F07$FQ$\")7#z$oF07$Fen$!)Qb- [F07$Fjn$!*/W>-\"F07$F_o$!*gG\"[7F07$Fdo$!),L\\#)F07$Fio$\")PRLEF07$Fc p$\"*t+@8#F07$F]q$\"*UJKX#F07$Fbq$\"*p+F&[F07$Fgq$\"*5*3TyF07$F\\r$\"+ FW1'4\"F07$Far$\"+\\YIX8F07$$\"5M$e9;K'Ho/EFH$\"+do)*=9F07$F^bq$\"+&=S uV\"F07$$\"5+]P%)*3UN8n#FH$\"+hi<]9F07$Fcbq$\"+Vg]G9F07$Fhbq$\"+.lNW9F 07$F]cq$\"+P&yuW\"F07$Fbcq$\"+lyFe8F07$Ffr$\"+$po1P\"F07$F_dq$\"+kI&* \\7F07$F[s$\"+#*z,K5F07$Fes$\"+3?2i7F07$Fit$\"+%o]4a\"F07$F^u$\"+?DLw9 F07$Fcu$\"+@`Nr8F07$F\\iq$\"+%f-\\G\"F07$Fhu$\"+2([S>\"F07$Fhbz$\"+j\\ yy6F07$Fdiq$\"+min%=\"F07$F`cz$\"+G6;&>\"F07$F]v$\"+]D!yB\"F07$Fbv$\"+ qaR,:F07$Fgv$\"+kf\")y;F07$Fi]r$\"+$4s)4;F07$Fc^r$\"+1LVO8F07$Fh^r$\"+ =#*GC7F07$F]_r$\"*#\\r@$*F07$Fb_r$\"*!GWV!)F07$F\\w$\"*JFE\"fF07$Fj_r$ \"*Y_)))GF07$F_`r$\")by!z(F07$Fejz$\")3y[xF07$Fd`r$\")XX'H)F07$$\"5+++ v=n8amrFH$\")zVr&)F07$Fb[[l$\"*xkIG\"F07$$\"5qmm\"HK*[kIsFH$\"*jZgJ\"F 07$Faw$\"*DAPL\"F07$Ffw$\"*Q4@;$F07$F[x$\"*))>+(RF07$Fex$\"*X&R(R$F07$ F_y$\"*-\\!p=F07$Fhar$\"*,![Qf\"F07$Fbbr$\"*oJ^^\"F0 7$Fgbr$\"*/>.b\"F07$F\\cr$\"*9,Py\"F07$Fdy$\"*!G^x>F07$Fcdr$\"+Q'zt6#F H7$F]er$\"*>9X1#F07$Fber$\"*a@t$>F07$Fiy$\"*YUz\"=F07$F\\^[l$\"*i_fz\" F07$Fa^[l$\"**[20>F07$Ff^[l$\"*@Ul)>F07$Fjer$\"*8BD<#F07$F^_[l$\"*eY'H FF07$F^z$\"*\"R;MLF07$Ff_[l$\"*tyPW$F07$F[`[l$\"*N9!HNF07$F``[l$\"**\\ 2*\\$F07$Fe`[l$\"*()p.\\$F07$Fj`[l$\"*Y0L[$F07$Fbfr$\"*W^\"oKF07$Fba[l $\"*k0tu#F07$Fcz$\"*\\nQ<#F07$Fhz$\"*^Qe_\"F07$F][l$\"*vCH5\"F07$$\"5+ +D19WCxw5F0$\"*w\"oP6F07$Fb[l$\"*([Qq7F07$$\"5ML3-8+pu)3\"F0$\"*gs*\\: F07$Fg[l$\"*yY%p>F07$F\\\\l$\"*5T6`$F07$Fa\\l$\"*-Oks'F07$F_ir$\"*%o)) >')F07$Ff\\l$\"+@mco6F07$Fajr$\"+PwQ<7F07$F[]l$\"+1M6![\"F07$F`]l$\"+< /@@F07$Fd`s$\"+Xr:U>F07$Ffal$\"+Gc.z=F07 $F\\as$\"+*Q)Rr=F07$Faas$\"+:q*f&=F07$Ffas$\"+,l/`=F07$F[bl$\"+*=H(e=F 07$F`bl$\"+.oA$*=F07$Febl$\"++G(Q\">F07$Fdbs$\"+6hk;>F07$Fjbl$\"+!e*)= \">F07$F\\cs$\"+b)y<*=F07$F_cl$\"+C)Q$Q=F07$Fdcl$\"+>9sw;F07$Ficl$\"+! oW&*f\"F07$F]el$\"+1huik^\"F0$\"+W*eM-#F07$$\"5nT&QG\\V!Q>:F0$\"+NHnW?F07$F \\fl$\"+,\\=]?F07$$\"5n;/wiwWEG:F0$\"+Ih^/?F07$Fafl$\"+#Qu^&>F07$Fegl$ \"+JM9]F0$\"*k4Ym%F07 $Fhat$\"*#*>%GQF07$$\"5M3-j([6WQ&>F0$\"*-xV&QF07$$\"5n;aj%ei:p&>F0$\"* q=`&QF07$$\"5+D1k\"o8()*f>F0$\"*z`1`$F07$$\"5MLekyZ'eI'>F0$\"*lBsb$F07 $$\"5+]ilsp;?p>F0$\"*#)eha$F07$Fc^m$\"*Gzlb$F07$F_ct$\"*OD))H%F07$Fh^m $\"*.aq]'F07$Ffdt$\"*!zSDyF07$F[et$\"*=lSG*F07$F`et$\"+*pk*H7F07$F]_m$ \"+:k1N9F07$F]ft$\"+1!*G0;F07$Fb_m$\"+=Qit=F07$Fjft$\"+WAK8?F07$Fg_m$ \"+wUS2@F07$Fbgt$\"+#4`/9#F07$Fggt$\"+>*)[p@F07$F\\ht$\"+Oc'4>#F07$F\\ `m$\"+i(o9:#F07$Fdht$\"+L<\"*e@F07$Fiht$\"+#zlI<#F07$F^it$\"+:e)33#F07 $Fa`m$\"+#\\dB3#F07$F[jt$\"+OM$R)>F07$Fejt$\"+s/[_=F07$Fjjt$\"+=F07$F[]u$\"+'ze= 9#F07$F[am$\"+#H*GOBF07$Fh]u$\"+(\\5*eBF07$F]^u$\"+\")Q>wBF07$Fb^u$\"+ GsEwBF07$Fg^u$\"+D'y!yBF07$F\\_u$\"+UO5zBF07$Fa_u$\"+')>MyBF07$Ff_u$\" +1g)QO#F07$F[`u$\"+*RHTN#F07$F``u$\"+\"48bJ#F07$F`am$\"+%HN*pAF07$Fd_] l$\"+'*4:kAF07$Fi_]l$\"+&)e0mAF07$F^`]l$\"+@@:zAF07$Fh`u$\"+>:c&G#F07$ $\"5n;a8(RqBHF#F0$\"+tT))4BF07$Ff`]l$\"+ma=BBF07$$\"5+]i:g;mJ'G#F0$\"+ Uu=FBF07$Feam$\"+0uF07$Fbbu$\"+_OO%*>F07$Fgbu$\"+2cn>?F07$F\\cu $\"+SY>;?F07$Facu$\"+G9u=?F07$F[du$\"+!Qv#R@F07$F_bm$\"+LXa*H#F07$Fhdu $\"+%*)37P#F07$Fdbm$\"+\">!)eR#F07$F`eu$\"+Edx8'F07$$\"5M3xJ#H'p(4]#F0$\"*t6O\"eF07$ $\"5M$3-jP(=U/DF0$\"*&o1NdF07$$\"5MekGg%ymy]#F0$\"*ry2t&F07$Fbdm$\"*r! H>fF07$$\"5M3_DG1mv9DF0$\"*]:H&eF07$$\"5M$eRAr^,#=DF0$\"*]O([fF07$$\"5 MeRA'zUY;_#F0$\"*zSxL'F07$Fgdm$\"*G0>G'F07$F\\em$\"*'p^RqF07$Faem$\"*_ OW\"yF07$$\"5M3x1OOb4cDF0$\"*<4Kz(F07$Fdgu$\"*#)*=j!)F07$$\"5Mek./e`)H c#F0$\"*Zhn:)F07$Ffem$\"*6**39)F07$$\"5M3_+sz^()pDF0$\"*r[%=$)F07$F\\h u$\"*NG&=$)F07$$\"5MeR(*R,]wwDF0$\"*UZvH)F07$F[fm$\"*6**zD)F07$F`fm$\" *7$=MzF07$Fefm$\"*eF07$Fdju$\" *8'F07$F]im$\"*X5[H'F07$Fbim$\"*+mGW(F07$Fgim$\"*w$HNyF07$Fajm$ \"*5Cu!pF07$F[[n$\"*YIp$eF07$$\"5ML3_]%3+w'GF0$\"*@$>_fF07$Fd_v$\"*pyh -'F07$$\"5om\"HK*QS)4)GF0$\"*!f\"\\^'F07$F`[n$\"*3h=&oF07$F\\`v$\"*St* R(*F07$Fe[n$\"+X:w#G\"F07$Fd`v$\"+DQp/;F07$Fj[n$\"+_t!ep\"F07$Faav$\"+ Q\\B&)>F07$F_\\n$\"+Mm[YAF07$Fd\\n$\"+Fz]pCF07$Fi\\n$\"+A`>jDF07$Fc]n$ \"+X2cWBF07$Fg^n$\"+u\\D+AF07$F[`n$\"+-RPPBF07$F_an$\"+?D[yEF07$Fdan$ \"+DgZfFF07$Fian$\"+)p(z!z#F07$Fgb^l$\"+hs)ey#F07$F^bn$\"+uW6!y#F07$Fd c^l$\"+lhHcFF07$Fcbn$\"+JKT=FF07$F\\fv$\"+=zL=FF07$Fafv$\"+t8b8FF07$Ff fv$\"+UCH(o#F07$Fhbn$\"+gB?(o#F07$Fcgv$\"+![iNn#F07$F]cn$\"+&)o]vEF07$ Fbcn$\"+a!o9q#F07$Fgcn$\"+yH*os#F07$F\\dn$\"+x6oRFF07$Fadn$\"+Ds0IFF07 $Fhe^l$\"+.L))HFF07$Ffdn$\"+8_[AFF07$F`f^l$\"+qd0\"p#F07$F[en$\"+t]Z!p #F07$F`en$\"+7q\"pe#F07$Feen$\"+$>kvY#F07$F\\iv$\"+bT9MCF07$Faiv$\"+j$ fGT#F07$Ffiv$\"+ahO-CF07$Fjen$\"+(Q%=5CF07$F^jv$\"+>jE;DF07$F_fn$\"+jK ,3EF07$Ffjv$\"+=O%\\p#F07$Fdfn$\"+$\\'RwFF07$$\"5M$e9mp&)p?J$F0$\"+AKR /GF07$Fdi^l$\"+Md;xFF07$$\"5,]PM-C&*Q=LF0$\"+VT)pv#F07$F^[w$\"+z>x_FF0 7$F\\j^l$\"+)H>un#F07$Fifn$\"+&Q9%oDF07$Ff[w$\"+&Ql(RBF07$F^gn$\"+idlG AF07$F^\\w$\"+qJka>F07$Fcgn$\"+_@20;F07$F_\\_l$\"+v._*\\\"F07$Fhgn$\"+ A4/S7F07$Fa]_l$\"+lJh75F07$F]hn$\"*Pt=f*F07$Fa]w$\"*n[zh)F07$Fbhn$\"*( RT*G)F07$F^^w$\"*%G2>#)F07$Fghn$\"*3KPK)F07$Fain$\"*%4'z*))F07$F[jn$\" *\\qIr*F07$F`jn$\"+M:!)35F07$Fejn$\"+;!)pO5F07$Fb_w$\"+y$G'[5F07$Fjjn$ \"+w&y5/\"F07$$\"5o\"zWUVG$*=[$F0$\"+\\]vM5F07$Fj_w$\"+dELL5F07$$\"5,v $fLpg2!)[$F0$\"+(p1]-\"F07$F_[o$\"*tS'z**F07$Fi[o$\"*fBn())F07$Fc\\o$ \"*)zYLzF07$F]]o$\"*6K_:)F07$Fg]o$\"+\"Qz6b)FH7$F^aw$\"+Uxw0&)FH7$Fcaw $\"*Sq&)\\)F07$Fhaw$\"*cU')Q)F07$F\\^o$\"*c[ME)F07$Febw$\"*1\"R/#)F07$ Fa^o$\"*7O'*G)F07$Fbcw$\"*3bcF*F07$Ff^o$\"*e9a%**F07$Fjcw$\"*d$\\R%*F0 7$F[_o$\"*SIke)F07$$\"5,+]7G8:9aPF0$\"*7K?J)F07$Fbdw$\"*jHB=)F07$$\"5o m;/ED0HnPF0$\"*kcF07$F_`o$\" +:&4O1#F07$Fehw$\"+G3mxBF07$Fd`o$\"+\"39Ze#F07$Fbiw$\"+'o.\\q#F07$Fi`o $\"+TfqTGF07$Fjiw$\"+L/)o)GF07$F^ao$\"+;O3IHF07$$\"5oTgFTc&G)oQF0$\"+K gB7HF07$Fbjw$\"+jr-vGF07$$\"5MekysPjDvQF0$\"+%\\*=)*GF07$Fcao$\"+/.'f( GF07$Fhao$\"+-iM3FF07$F]bo$\"+s:$Hk#F07$Fbbo$\"+,)GZk#F07$Fgbo$\"+(G>N f#F07$F\\co$\"+I!p1g#F07$Faco$\"+ML32EF07$Ff\\x$\"+*QiUg#F07$Ffco$\"+% 3h!4EF07$F^]x$\"+Bf+fjEF07$F`do$\"+d#3Jx#F07$Fedo$\"+P j,,HF07$Fjdo$\"+f$372$F07$F_eo$\"+)e[9=$F07$Fb^x$\"+1Pa%>$F07$Fdeo$\"+ #oDI?$F07$Fj^x$\"+7EB\">$F07$Fieo$\"+zz+pJF07$F^fo$\"+(\\h07$F07$Fcfo$ \"+c#)Q#4$F07$Fh_x$\"+0`6%3$F07$Fhfo$\"+[g[$4$F07$F``x$\"+Sb(\\4$F07$F e`x$\"+$\\Ng4$F07$Fj`x$\"+OA&o6$F07$F]go$\"+?]Q=JF07$F[io$\"+B@T;IF07$ Feio$\"+of:cGF07$Fijo$\"+yqFVIF07$F]\\p$\"+p*[S6$F07$Fjbx$\"+k[()*4$F0 7$F_cx$\"+@/Y\">$F07$Fdcx$\"+1?skJF07$Fb\\p$\"+XJtSJF07$F\\dx$\"+z&R)) =$F07$Fadx$\"+(\\Dg:$F07$Ffdx$\"+v-d@JF07$Fg\\p$\"+F\"465$F07$F\\]p$\" +0/JnGF07$Fa]p$\"+fW#G^#F07$F^fx$\"+u\\M*=#F07$Ff]p$\"+Ac$\\(=F07$Fffx $\"+oh1fVF0$\"+[\\C )4\"F07$Fe^p$\"+ijI86F07$F__p$\"+h\\It6F07$Fi_p$\"+XKJR7F07$$\"5o;/E]) pk%eVF0$\"+n%H8C\"F07$F^`p$\"+wkIe7F07$$\"5N$ekG5n5;P%F0$\"+!HUQE\"F07 $Fc`p$\"+/Tlb7F07$Fjhx$\"+)e\\c9\"F07$Fh`p$\"+HL0U5F07$Fbix$\"+(=uJ-\" F07$Fgix$\"+Xh>:5F07$F\\jx$\"+vv.:5F07$Fajx$\"+VG375F07$Ffjx$\"+Fzl15F 07$F[[y$\"+V_w95F07$F`[y$\"+pkrK5F07$F]ap$\"+Fz8_5F07$Fbap$\",\"=i)o1 \"FH7$Fgap$\"+U$**>1\"F07$F\\bp$\"+Kz+\\5F07$Fabp$\"+G9^M5F07$F[cp$\"+ KU&z2\"F07$Fecp$\"+J(3,>\"F07$F^^al$\"+aG[(>\"F07$Fc^al$\"+4z=67F07$Fh ^al$\"+j$QA@\"F07$F]_al$\"+j\"oB@\"F07$Fb_al$\"+8k067F07$Fg_al$\"+Xj\" G@\"F07$F\\`al$\"+Aie67F07$Fjcp$\"+wVG)=\"F07$Fd`al$\"+/$fF:\"F07$F_dp $\"+8n'Q6\"F07$F\\aal$\"+2&\\+3\"F07$Fddp$\"+z3dd5F07$Fdaal$\"+e;'>6\" F07$Fidp$\"+FEuI7F07$Fd^y$\"+>SX$R\"F07$F^ep$\"+Dhmv9F07$F\\_y$\"+i'*4 \"p\"F07$Fcep$\"+yk.5?F07$Fi_y$\"+Tb(=7#F07$Fhep$\"+R#[VV#F07$F[ay$\"+ 107bFF07$F]fp$\"+.2mfGF07$Fbfp$\"+5dV(>$F07$Fgfp$\"+To['G$F07$Fbcy$\"+ &4qiI$F07$Fgcy$\"+,[uDKF07$F\\dy$\"++!))RD$F07$F\\gp$\"+3lcmKF07$Fddy$ \"+]XSyJF07$Fagp$\"+!p#)z2$F07$Fehp$\"+)=!oEIF07$F_ip$\"+jyC2IF07$Fdip $\"+f\"4(\\IF07$Fiip$\"+!GI3@$F07$F^jp$\"+@l*3Y$F07$Fcjp$\"+(p[Rh$F07$ Fhfy$\"+%[jta$F07$F]gy$\"+?w/'\\$F07$Fbgy$\"+&)*pZ_$F07$Fggy$\"+8>RcNF 07$F\\hy$\"+fNF07$Fahy$\"+ia*)eNF07$Ffhy$\"+-MZONF07$Fhjp$\"+v=#Q[$ F0-F][q6&F_[qF`[qF(F(-Fd[q6#%3efficient~scheme~BG-F$6%7ibmF'7$F2$!)saG AF07$F<$!)5!R&\\F07$FA$\")andeF07$FF$\"*E$z!p\"F07$Fen$\")*f)RKF07$Fio $!)0p[GF07$Fcp$\"*\"e?$H\"F07$F]q$\"*+Yub\"F07$F^[z$\"*EZ8!QF07$Fbq$\" *#)4&oQF07$Ff[z$\"*h')4@%F07$Fgq$\"*>&z!*pF07$Fj_q$\"*1F16(F07$F_`q$\" *s)o6vF07$Fd`q$\"+EPKJ5F07$F\\r$\"+\\p.[5F07$F\\aq$\"+PHR(3\"F07$Faaq$ \"+_C488F07$Ffaq$\"+&z$eK8F07$Far$\"+G40l8F07$F^bq$\"+TYx=:F07$Fcbq$\" +S%*ex:F07$Fhbq$\"+Y=H&f\"F07$F]cq$\"+Ah72;F07$Fbcq$\"+,NMl:F07$Ffr$\" +$3\"yz:F07$Fh^z$\"+#eXJe\"F07$Fjcq$\"+9ED\"\\\"F07$F`_z$\"+3e5-:F07$F _dq$\"+#4e'*\\\"F07$Fddq$\"+Td6#R\"F07$F[s$\"+?t@s7F07$Faeq$\"+Kzo$=\" F07$F`s$\"+!yrT8\"F07$Fhfq$\"+0&pX>\"F07$Fes$\"+=L&)*G\"F07$Fjs$\"+*fR =T\"F07$F_t$\"+%>Tic\"F07$Ffgq$\"+'G.Bd\"F07$Fdt$\"+;4>+;F07$F^hq$\"+N zI*f\"F07$Fit$\"+fw?&f\"F07$F^u$\"+*oT\"[:F07$Fcu$\"+&e\"zz9F07$F\\iq$ \"+pIGr8F07$Fhu$\"+Z$y3A\"F07$Fhbz$\"+0HH!>\"F07$Fdiq$\"+j]JZ6F07$F`cz $\"+?;wW6F07$F]v$\"+,*p1:\"F07$F[[r$\"+5CR17F07$Fbv$\"+/*RrM\"F07$Fh[r $\"+_so09F07$F]\\r$\"+FEOG:F07$Fb\\r$\"+:0U;:F07$Fg\\r$\"+q(y/_\"F07$F \\]r$\"+FbX<;F07$Fgv$\"+>nw,;F07$F`ez$\"+gf&4f\"F07$Fd]r$\"+KV#Rk\"F07 $Fhez$\"+,$>Xi\"F07$Fi]r$\"+AYY/;F07$F^^r$\"+P%zoa\"F07$Fc^r$\"+bt!*>9 F07$Fffz$\"+31\"[M\"F07$Fh^r$\"+*zr)>8F07$F^gz$\"+wle)=\"F07$F]_r$\"+T A\"=.\"F07$Ffgz$\"+1Dl65F07$Fb_r$\"*5#=1*)F07$F^hz$\"*TWFk'F07$F\\w$\" *K@B^'F07$Ffhz$\"*#RMVcF07$F[iz$\"*_L`>$F07$F`iz$\"*FGM8$F07$Fj_r$\"*U %*Hk#F07$$\"5NLL3-js75pFH$\")kq?gF07$Fhiz$\")Y&p)eF07$$\"5+++D1*yIU(pF H$\")X];QF07$F_`r$!)_2$)*)F07$F`jz$!)p2r))F07$Fejz$!)([LP*F07$Fjjz$!*2 =8W\"F07$Fd`r$!*qbPU\"F07$Fb[[l$!*&ew_8F07$Faw$!*&zQ/8F07$Ffw$\")!f88$ F07$F[x$\"*1K-b\"F07$$\"5SLLL3-B@EyFH$\"*uhCl\"F07$F`\\[l$\"*wl;l\"F07 $$\"5qmmmTg#3D*yFH$\"*uS:m\"F07$F`x$\"*7r+m\"F07$Fh\\[l$\"*Uj__\"F07$F ex$\"*\\i.S\"F07$Fjx$\")Ir*3(F07$F_y$!(#[89F07$F]br$!)/Y.NF07$Fgbr$!)G E1\\F07$F\\cr$!)`\"[J$F07$Fdy$!)t()R:F07$Fdcr$!)cA86F07$Ficr$!(&R#\\#F 07$F^dr$!(A!RCF07$Fcdr$!(=4I\"F07$Fhdr$\"']c_F07$F]er$!(11j(F07$Fber$! )])=R#F07$Fiy$!)lA]VF07$F\\^[l$!)Cr(3&F07$Fa^[l$!)i^$e%F07$Ff^[l$!)k(G )QF07$Fjer$!)*pw9#F07$F^_[l$\")8+HUF07$F^z$\"*+ciD\"F07$Ff_[l$\"*&y)\\ \\\"F07$F[`[l$\"*w=&p;F07$F``[l$\"*HnEy\"F07$Fe`[l$\"*_tb$=F07$Fj`[l$ \"**>kO=F07$Fbfr$\"*0l^v\"F07$Fba[l$\"*7xXO\"F07$Fcz$\")A\"F07$Fbhr$\"*'\\;Y=F07$F a\\l$\"*$*[$eGF07$Fjhr$\"*:Q;c%F07$F_ir$\"*A)*\\k%F07$Fdir$\"*j&es^F07 $Ff\\l$\"*#3V@yF07$F\\jr$\"*AG*RzF07$Fajr$\"*CJPA)F07$Ffjr$\"+WgBR5F07 $F[]l$\"+(>S%=6F07$F^[s$\"+K:h)G\"F07$F`]l$\"+)4CfS\"F07$Ff[s$\"+e9*Ga \"F07$Fe]l$\"+nSV!e\"F07$Fj]l$\"+*z:Wc\"F07$F_^l$\"+&ow*z9F07$Fd\\s$\" +41qk8F07$Fi\\s$\"+vMR[7F07$F^]s$\"+QT7`7F07$Fc]s$\"++fs]7F07$Fh]s$\"+ (*pSl6F07$Fd^l$\"+'o(Ro6F07$Fbf[l$\"+kw:n6F07$F`^s$\"+-\\6F07$Fi^l$\"+mnS;7F07$Fc _l$\"+L1;<8F07$F]`l$\"+')G5Q9F07$Fg`l$\"+o%Q&z:F07$Faal$\"+vmd/;F07$Ff al$\"+4)3W`\"F07$F[bl$\"+son9:F07$F`bl$\"+](*=e:F07$Febl$\"+)H?1f\"F07 $F`[\\l$\"+ZSt%f\"F07$Fdbs$\"+w+k1;F07$Fh[\\l$\"+'*3h1;F07$Fjbl$\"+\\` q/;F07$F\\cs$\"+,Vy#f\"F07$F_cl$\"+Y!\\Ga\"F07$Fdcs$\"+d\"4%f9F07$Fdcl $\"+7))pZ8F07$Fads$\"+nt$)z6F07$Ficl$\"+t%)pW6F07$Fcdl$\"+;Amg6F07$F]e l$\"+0.nG7F07$Fh]\\l$\"+iHnE8F07$Fbel$\"+)=#*GO\"F07$F`^\\l$\"+f5^N9F0 7$Fgel$\"+l^cd:F07$Fbdcl$\"+Y2h#R\"F07$Fjgl$\"+-.m#G\" F07$$\"5nTg_s,m\"\\b\"F0$\"+v$4IE\"F07$F_hl$\"+X?X'>\"F07$Fj_\\l$\"*4t 7]*F07$Ffgs$\"*@\"y9$*F07$Fb`\\l$\"*\"H%*y\")F07$F[hs$\"*\\4'pdF07$F`h s$\"*K%>)4%F07$Fdhl$\"*^!ygBF07$Fhhs$\")8ROtF07$F]is$\")0[6:F07$Fbis$ \"(?-\"pF07$Fgis$!*S$e'4\"F07$F\\js$!*EU53\"F07$Fajs$!*JZ#=6F07$Ffjs$! *Vw-X\"F07$Fihl$!*&ypJ9F07$Fcil$!)%y3b\"F07$Fgjl$\"*)y\"3[\"F07$F[\\m$ \")K^.sF07$F_]m$!).8LVF07$F_\\t$!)*>H?\"F07$Fi\\t$\"(D^%F07$Fd_t$!)@YV7F07$Fi]m$\")8ZP PF07$Fa`t$\"*0Yh=\"F07$F[at$\"*\\vat\"F07$Fhe\\l$\"*4_>w\"F07$F`at$\"* ANg%=F07$F`f\\l$\"*2f\"Q=F07$Fef\\l$\"*[]zm\"F07$Fjf\\l$\"*;4@n\"F07$F ^^m$\"*A.Lm\"F07$Fhat$\"(*e[sF07$Fc^m$!)r.`$*F07$F`bt$!*\")f:+\"F07$Fe bt$!*;6G2\"F07$Fjbt$!*C,Y3\"F07$F_ct$!)s34'*F07$Fict$!)i)oB(F07$Fh^m$ \")%R(>QF07$Ffdt$\"*\\d!p8F07$F[et$\"*[$GMDF07$F`i\\l$\"*$oJ&*RF07$F`e t$\"*uUmN&F07$Fhi\\l$\"*#yJBbF07$F]_m$\"*'o/MtF07$Fhet$\"*y!yG()F07$F] ft$\"*dND&*)F07$Fbft$\"+CgSw5F07$Fb_m$\"+JZk\">\"F07$Fjft$\"+w5ud8F07$ Fg_m$\"++q?c9F07$Fggt$\"+ao%pc\"F07$F\\`m$\"+/(pde\"F07$Fdht$\"+8cG(f \"F07$Fiht$\"+!4W#4;F07$F^it$\"+MD$ya\"F07$Fa`m$\"+(e>$\\:F07$F[jt$\"+ 4+&oX\"F07$Fejt$\"+jy'[H\"F07$F_[u$\"+\")>za6F07$Ff`m$\"+z'z@8\"F07$Fa \\u$\"+)f@?<\"F07$F[]u$\"+@fVW8F07$F`]u$\"+I&H3X\"F07$F[am$\"+c<()Q:F0 7$F]^u$\"+mEm!f\"F07$Fg^u$\"+L!45g\"F07$F\\_u$\"+Hp/2;F07$Fa_u$\"++E\" og\"F07$Ff_u$\"+/G&*)f\"F07$F[`u$\"+pT!Gf\"F07$F``u$\"+F)=/c\"F07$F`am $\"+ZS2;:F07$Fh`u$\"+RF%e`\"F07$Feam$\"+<#y\"3;F07$F^a]l$\"+W0M&e\"F07 $F`au$\"+ZlA9:F07$Ffa]l$\"+x*G,M\"F07$Fjam$\"+064O7F07$F]bu$\"+3@1n6F0 7$Facu$\"+'R\")\\9\"F07$F[du$\"+*4MF@\"F07$F_bm$\"+`?Ee8F07$Fhdu$\"+8k rj9F07$Fdbm$\"+m02p:F07$$\"5n\"z>6[c_7U#F0$\"+LgfQ;F07$F`c]l$\"+:`+?;F 07$$\"5M3FWqjb)yU#F0$\"+2Zp3;F07$F`eu$\"+2>?<;F07$Fhc]l$\"+0qwi:F07$Fi bm$\"+>=^c9F07$F`d]l$\"+7[+.9F07$Fheu$\"+XWo:7F07$Fhd]l$\"+z+i&>\"F07$ F^cm$\"+:x&)H6F07$F`e]l$\"*h[!R')F07$F`fu$\"*x?z[)F07$Fhe]l$\"*oE(RyF0 7$Fccm$\"*rs7&\\F07$F`f]l$\"*V2:'[F07$Fef]l$\"*.YJM%F07$Fjf]l$\"*@(fV= F07$Fhcm$\"*i#G3=F07$Fbg]l$\"*6k#y9F07$Fgg]l$!)\"=\\J#F07$F\\h]l$!)xN8 BF07$F]dm$!)wdZQF07$Faem$\")#)GiiF07$Fefm$\"**f)=:\"F07$Fjfm$!)*=jx\"F 07$F_gm$!)#R#[[F07$Fdju$!)=r#p$F07$Fdgm$!(Sw>*F07$$\"5o;aQGC$RFo#F0$!( ]L]'F07$Ffi]l$!(/WG'F07$$\"5M$e*)4V$pt)o#F0$!(Yr(GF07$F\\[v$\")W3,AFH7 $F^j]l$\")WL/?FH7$Figm$!'KrGF07$F^hm$!)!y@S\"F07$Fchm$!)5O?UF07$Fj[v$! )rniXF07$Fhhm$!)u*y)[F07$Fb\\v$!)&e*GUF07$F]im$!)(H/X$F07$Fbim$\"*3qj+ \"F07$Fgim$\"*.@#3=F07$F\\jm$\"*Srrf\"F07$Fajm$\"*#>&)Q6F07$Ffjm$\")c: #p%F07$F[[n$!)kj:gF07$Ffcel$!)h\"es'F07$Fd_v$!)$>I))*F07$F^del$!*h3+. \"F07$F`[n$!))o@*)*F07$$\"5,+v$fL*zO%*GF0$!)7C9`F07$F\\`v$\")a+LzF07$$ \"5,](oz\"ffS/HF0$\")*eW7)F07$$\"5MLekyZ>v2HF0$\"*e_i<\"F07$$\"5o;HKRO z46HF0$\"*f#=gIF07$Fe[n$\"*9F57$F07$Ff]^l$\"*k'oEOF07$Fd`v$\"*$HI7hF07 $F^^^l$\"*[.1A'F07$Fj[n$\"*&>rRnF07$F\\av$\"*%y)[Y*F07$Faav$\"*\"*[-i* F07$Ffav$\"+WFu25F07$F_\\n$\"+a8rZ7F07$F^bv$\"+#)=Wm7F07$Fcbv$\"+\\bm, 8F07$Fhbv$\"+I4Ji9F07$Fd\\n$\"+Mh!>[\"F07$F`cv$\"+m[Gp:F07$Fi\\n$\"+H: \"Gg\"F07$F^]n$\"+w%*\\D:F07$Fc]n$\"+g/dF9F07$F]^n$\"+CTK88F07$Fg^n$\" +ko/v6F07$F\\_n$\"+U\\xZ6F07$Fa_n$\"+EmAN6F07$Ff_n$\"+O!*QY6F07$F[`n$ \"+\\*H-=\"F07$F``n$\"+Ga]l7F07$Fe`n$\"+2Vqw7F07$Fj`n$\"+[)QFP\"F07$F_ an$\"+&*3dm9F07$Fdan$\"+lHCb:F07$Fian$\"+VPx*f\"F07$Fgb^l$\"+,*>xg\"F0 7$F^bn$\"+\\EX/;F07$Fdc^l$\"+#=qye\"F07$Fcbn$\"+rV%[b\"F07$Fhbn$\"+vgb C:F07$F]cn$\"+5L]7:F07$Fbcn$\"+_S#Ra\"F07$Fgcn$\"+.u3!e\"F07$Fhd^l$\"+ ;ml/;F07$F\\dn$\"+7!oYg\"F07$F`e^l$\"+e:[0;F07$Fadn$\"+gq$yg\"F07$Fhe^ l$\"+)Haxg\"F07$Ffdn$\"+-2c-;F07$F`f^l$\"+$\\))od\"F07$F[en$\"+\"z1kd \"F07$F`en$\"+2*[XY\"F07$Feen$\"+0\\458F07$Faiv$\"+?zg:7F07$Fjen$\"+^= Ja6F07$F^jv$\"+NJ\"Q>\"F07$F_fn$\"+69!QF\"F07$Ffjv$\"+yD\")z8F07$Fdfn$ \"+4?pM:F07$Fdi^l$\"+$Hb8e\"F07$F^[w$\"+#p#>U;F07$F\\j^l$\"+[vI3;F07$F ifn$\"+=XBx:F07$Ff[w$\"+Oc6H9F07$F^gn$\"+7iq^8F07$Fjj^l$\"+B*y%Q6F07$F ^\\w$\"+uT->6F07$Fb[_l$\"+#)3+[5F07$Fcgn$\"*Pq`u(F07$Fj[_l$\"*NjFg(F07 $F_\\_l$\"*;tl%oF07$Fd\\_l$\"*.UO8%F07$Fhgn$\"*8>30%F07$F\\]_l$\"*^#)) *Q$F07$Fa]_l$\"*m_9D\"F07$Ff]_l$\"*N:%>7F07$F]hn$\")))o%f(F07$Fa]w$!)B !H^&F07$Fbhn$!*SF+L\"F07$F^^w$!*&**oW8F07$Fghn$!*K-)f8F07$Fain$!)Rzs!) F07$F[jn$\")n6g;F07$Fejn$\"*Z8;J\"F07$F_[o$\"*z,0`\"F07$Fi[o$\")V<.iF0 7$Fc\\o$!)U&>W%F07$F\\`_l$!)Gj(\\%F07$Fh\\o$!)#>%=F07$Fh e_l$\"*?K`%=F07$F]f_l$\"*fvq\"=F07$Fbf_l$\"*#pU9=F07$Fgf_l$\"*F!Q5=F07 $F\\g_l$\"*'3mX;F07$Fjcw$\"*ae[^\"F07$Fdg_l$\"*q)*>,\"F07$F[_o$\")FeoK F07$Fbdw$!)L@4qF07$F`_o$!*flo-\"F07$F_ew$!)Neu&*F07$Fiew$!)b\\h$*F07$F ^fw$!):%y$GF07$Fe_o$\"*>:xD\"F07$Fhi_l$\"*9n\"z7F07$Fffw$\"*CMCW\"F07$ F`j_l$\"*XNws$F07$Fj_o$\"*HP0z$F07$F^gw$\"*%yF7SF07$Fcgw$\"*A^Em'F07$F hgw$\"*56U+(F07$F_`o$\"*Ab8E(F07$F`hw$\"*e&[H(*F07$Fehw$\"+Pl&o.\"F07$ Fjhw$\"+PZ8j5F07$Fd`o$\"+&oh)[7F07$Fbiw$\"+pCdZ8F07$Fi`o$\"+UGM8:F07$F jiw$\"+q))3m:F07$F^ao$\"+h&))*3;F07$Fbjw$\"+$4\\0e\"F07$Fcao$\"+Y)yCd \"F07$Fjjw$\"+mv<::F07$Fhao$\"+*GewR\"F07$Fb[x$\"+?A1*Q\"F07$F]bo$\"+B Qn)G\"F07$Faco$\"+i#>0>\"F07$F[do$\"+_H\"=9\"F07$F`do$\"+)>:))>\"F07$F edo$\"+;_N%H\"F07$Fjdo$\"+:UY]9F07$F_eo$\"+fP.r:F07$Fb^x$\"+0VL#f\"F07 $Fdeo$\"+e+f1;F07$Ff_`l$\"+tkD2;F07$Fj^x$\"+#[`ng\"F07$F^``l$\"+Br71;F 07$Fieo$\"+b[x!f\"F07$F^fo$\"+a&f%[:F07$Fcfo$\"+\"ya0_\"F07$F\\a`l$\"+ :l?8:F07$Fh_x$\"+Qlw6:F07$Fda`l$\"+9W<7:F07$Fhfo$\"+wFdA:F07$F``x$\"+z 9HC:F07$Fe`x$\"+\"*\\oD:F07$Fj`x$\"+(*4x^:F07$F]go$\"+p/o`:F07$Fbgo$\" +ja[)e\"F07$Fggo$\"+=M63;F07$Faho$\"+rVjl:F07$F[io$\"+5gs*[\"F07$Fdc`l $\"+vhdn8F07$F`io$\"+'ReB?\"F07$F\\d`l$\"+\\;9$>\"F07$Fad`l$\"+oqv`6F0 7$Ffd`l$\"+4\\Ff6F07$Feio$\"+b\"=z:\"F07$Fijo$\"+!**[jI\"F07$F]\\p$\"+ 5iS69F07$Fb\\p$\"+%ylp^\"F07$Fg\\p$\"+vm.^;F07$F^ex$\"+gN)3h\"F07$F\\] p$\"+Z;$fb\"F07$Ffex$\"+3uI*Q\"F07$Fa]p$\"+,`;C8F07$F\\g`l$\"+'pzU4\"F 07$F^fx$\"++)\\e/\"F07$Fdg`l$\"+Sr`95F07$Ff]p$\"*zjrT(F07$F\\h`l$\"*PT Iz'F07$Fffx$\"*9#3UlF07$Fdh`l$\"*:;***RF07$F[^p$\"*CFxL$F07$$\"5-vV[@q BIwUF0$\"*6z&zJF07$$\"5o;ajMj))ezUF0$\"*f\\\"f8F07$$\"5NekyZc`(GG%F0$ \")vc?sF07$F^gx$\")U!Q_'F07$$\"5N$eRse$[t#H%F0$!)/T&3)F07$F`^p$!*-?t< \"F07$Fhagl$!*BN*z8F07$F]bgl$!*b)Qm8F07$Fbbgl$!*9a$[8F07$Ffgx$!*;!)QI \"F07$Fjbgl$!*3)H)4\"F07$Fe^p$!)v&>j)F07$F__p$!(sgZ&F07$Fi_p$\")oHL#*F 07$F^`p$\"*_3WS\"F07$Fc`p$\"*0\"R!p\"F07$Fjhx$\"*P:t-\"F07$Fh`p$\"(je( >F07$Fgix$!)'zw,$F07$Fajx$!)Mj,OF07$Fjj`l$!)-X%e%F07$Ffjx$!)?i&e%F07$F b[al$!)EC&e%F07$F[[y$!)O&HS%F07$F`[y$!)ArCIF07$F]ap$!)-#QD\"F07$Fh[y$! (dX)))F07$F]\\y$\"'EStF07$Fb\\y$\"(#\\xzFH7$Fbap$\")AU(H#FH7$Fj\\y$\"( J0:%F07$Fgap$!(>=b$F07$F\\bp$!)v4F?F07$Fabp$!)f4;ZF07$F[cp$!)e&G@\"F07 $Fecp$\"*UC5G\"F07$F^^al$\"*#ohH9F07$Fc^al$\"*?jXo\"F07$Fh^al$\"*b_xo \"F07$F]_al$\"*$*3%Hr=F07$F\\`al$ \"*5Hu'=F07$Fjcp$\"*K%\\r8%>F07$Fddp$!).g4vF07$$\"5NLeRse%pgm%F0$!)vKI\")F07$Fdaal$!*h,% R5F07$$\"5N$ekG5nc`n%F0$!*wy80\"F07$$\"5omTNY3CXyYF0$!*V#=95F07$$\"5-] P%)*e9[:o%F0$!)`T^&)F07$Fidp$!)jNP')F07$Fd^y$\"(Z>[\"F07$F^ep$\")uj%>$ F07$$\"5-Dc^G#RZ6q%F0$\"*H$z*y\"F07$F\\_y$\"**pBF=F07$$\"5oT&)e'ez[xq% F0$\"*J#QqAF07$Fcep$\"*lfhZ%F07$Fd_y$\"*-`%fXF07$Fi_y$\"*-4<4&F07$F^`y $\"*0O=v(F07$Fhep$\"*.&=')yF07$Ff`y$\"*Nj7U)F07$F[ay$\"+\\vl-6F07$F`ay $\"+bwG?6F07$F]fp$\"+5W'e;\"F07$Fhay$\"+Xy%yO\"F07$F]by$\"+B@c(Q\"F07$ Fbby$\"+^c`?9F07$Fbfp$\"+#f#yJ:F07$Fjby$\"+Np:r:F07$Fgfp$\"+'f9ng\"F07 $Fbcy$\"+%fnjh\"F07$Fgcy$\"+9EKi:F07$F\\dy$\"+2%))fd\"F07$F\\gp$\"+9(3 'y:F07$Fddy$\"+f)p%*[\"F07$Fagp$\"+uLKp8F07$Fehp$\"+'35KE\"F07$F_ip$\" +4#o%y6F07$Fdgal$\"+Ruvi6F07$Figal$\"+\"Rrq8\"F07$F^hal$\"+[\">(Q6F07$ Fdip$\"+p.qT6F07$F[ial$\"+)R$\\a6F07$Fiip$\"++c!*>7F07$Fhey$\"+j-j>8F0 7$F^jp$\"+#431V\"F07$F`fy$\"+C1`Y:F07$Fcjp$\"+/>86;F07$Fhfy$\"+Zo4l:F0 7$F]gy$\"+,\")y9:F07$Fbgy$\"+jVG]:F07$Fggy$\"+[eG'f\"F07$F`[bl$\"+(Hkq f\"F07$F\\hy$\"+%fU(4;F07$Fh[bl$\"+X#fLh\"F07$Fahy$\"+`.28;F07$Ffhy$\" +')**)4g\"F07$Fhjp$\"+=KP_:F0-F][q6&F_[q$\")#)eqkFb[q$\"))eqk\"Fb[qFc[ `m-Fd[q6#%3efficient~scheme~CG-F$6%7c^mF'7$F2$!)S$eY#F07$F<$!),Q)Q*F07 $FA$!)F07$Fjn$\"*K7#\\5F07$F_o$!)g6+rF07$Fdo$!*$*>_&>F07$Fio $!*'f[uJF07$Fcp$!*j+h$RF07$F]q$!*/-)3TF07$F^[z$!*Tx5-%F07$Fbq$!*dI!*3% F07$Ff[z$!**3(*)3%F07$Fgq$!*YoV6$F07$Fj_q$!*FAT;$F07$F_`q$!*4`&GIF07$F d`q$!*.5v=\"F07$F\\r$!*7_=?\"F07$F\\aq$!)s_^$*F07$Faaq$\"*3))4]\"F07$F faq$\"*k$[H:F07$Far$\"*@Cc)=F07$Fd`bl$\"*t(4mWF07$F^bq$\"*q[<`%F07$F\\ abl$\"*[q&3\\F07$Fcbq$\"*n#>LrF07$F]cq$\"*dS5a(F07$Ffr$\"*\"zvz!*F07$F jcq$\"*\"yQA(*F07$F_dq$\"*;e5&)*F07$Fddq$\"*M*3E$*F07$F[s$\"*oK\\(yF07 $Faeq$\"*jNm&fF07$F`s$\"*\"3`KPF07$$\"5MLL3-jx/SMFH$\"*G;(eBF07$Fhfq$ \"*^>jN\"F07$$\"5MLLe*)4jBuNFH$\"*#e%GD\"F07$Fes$\"*0!z.5F07$F_t$\"*-U *GCF07$Fit$\"*BQ(QMF07$$\"5omT5!p:g[C%FH$\"*@mH_$F07$$\"5-+](oHaN;J%FH $\"*@wyX$F07$$\"5o;/E+OK-XVFH$\"*B)*yX$F07$$\"5NLek.H4TyVFH$\"*j+$QMF0 7$$\"5-]7.2A')z6WFH$\"*\">TDLF07$$\"5ommT5:j=XWFH$\"*^%FH$\"*>HkE$F07$$\"5NL$eRs3P(yXFH$\"*]BgG$F07$$\"5om\"H2LZ7bk%FH $\"*x;?R$F07$F^u$\"*^!>LPF07$$\"5om;/^J'Qe%[FH$\"*()R)RUF07$$\"5NLLek. %*Qz\\FH$\"*!3%f%[F07$$\"5+](ozm4xF,&FH$\"*$*4')3&F07$$\"5qmTNr*ykh/&F H$\"*&*y04&F07$$\"5N$eRZF[_&z]FH$\"*C>W5&F07$$\"5++]7yv,%H6&FH$\"*i0=/ &F07$$\"5NLe*[=c:(z^FH$\"*PC&R\\F07$Fcu$\"*zA5Y%F07$$\"5NLLeky-S3`FH$ \"*Am$=LF07$F\\iq$\"*&=UcJF07$$\"5qmmT5S*=AV&FH$\"*h71n\"F07$Fhu$\")J& )*f$F07$Fhbz$!)lCDJF07$Fdiq$!*Y17:#F07$F`cz$!*gWia#F07$F]v$!*w35h$F07$ F\\jq$!*,u\"QPF07$Fajq$!*\"*4=E%F07$Ffjq$!*Gf([UF07$F[[r$!*_%oXUF07$F` [r$!*Un*4QF07$Fbv$!* #F07$$\"5qm;/w7pE#3'FH$!*]%G^:F07$F]\\r$\")blCBF07$Fb\\r$\")*[lY#F07$F g\\r$\")W(f#pF07$F\\]r$\"*R;24$F07$Fgv$\"*&)yn1$F07$F`ez$\"*6%e2LF07$F d]r$\"*WA_j&F07$Fhez$\"*v^;d&F07$Fi]r$\"*b]#ecF07$F^^r$\"*_'e)3(F07$Fc ^r$\"*'>bdsF07$Fh^r$\"*;+D5(F07$F]_r$\"*i*pPfF07$Fb_r$\"*+H()4&F07$F\\ w$\"*7Y6h$F07$Ffhz$\"*Y%4WHF07$F[iz$\"*-Hn5\"F07$F`iz$\"*EQP2\"F07$Fj_ r$\")M\"\\2'F07$Fhiz$!*'Gj97F07$F_`r$!*EP4%HF07$Fd`r$!*w4J\"RF07$Faw$! *(oI?VF07$Ffw$!*@?tb$F07$F[x$!*kg-o\"F07$F`x$!)g=p\")F07$Fex$!)7b\\VF0 7$$\"5SLLLe9TaC\")FH$!)GxJMF07$Fjx$!)+z\\RF07$$\"50+++DJg8d#)FH$!)*zb% [F07$F_y$!)v3mxF07$Fdy$!*^[dg\"F07$Fiy$!*6xF7#F07$F^z$!)9^I$)F07$Fcz$ \")#fBA(F07$Fhz$!(#Q?**F07$F][l$!*e4,a\"F07$Fb[l$!*p;4w#F07$Fg[l$!*!>p OTF07$F\\\\l$!*I>c[&F07$Fa\\l$!*a=p@'F07$Ff\\l$!*KCF$\\F07$F[]l$!*G!)> #HF07$F^[s$!*'\\-o6F07$F`]l$!(<8*oF07$$\"5+]P4r`C[f6F0$\"*ol3%=F07$Ff[ s$\"*\"='en#F07$$\"5n;z>OaEAm6F0$\"*&=[YGF07$Fe]l$\"*k,'fXF07$Fj]l$\"* Hzvk'F07$F_^l$\"*wB]/(F07$Fd\\s$\"*%R=WiF07$Fi\\s$\"*4!3/YF07$Fc]s$\"* LQTZ%F07$Fd^l$\"*3krf#F07$Fe^s$\")pEEZF07$Fi^l$!*FWuo\"F07$F^_l$!*H7Ew \"F07$Fc_l$!*H'R(*=F07$Fh_l$!*[,Cf\"F07$F]`l$!*(z!3_\"F07$Fg`l$!)_1RKF 07$Faal$\")\"QnE#F07$$\"5++]7G8@+38F0$\")\")G[JF07$F^i[l$\")5!yQ&F07$$ \"5++]P%)*)*pZJ\"F0$\")Ki:aF07$Fd`s$\"))Q4y&F07$Ffi[l$\")^=FhF07$Ffal$ \")Zr$=&F07$F\\as$\")P%e\"[F07$Faas$\")?J/RF07$Ffas$\")y<#o$F07$F[bl$ \")CsVVF07$Febl$\"*2%)HY\"F07$F_cl$\"*csv/#F07$Fdcs$\"*z+JL\"F07$Fdcl$ !)v)4:\"F07$F\\ds$!*J$4!o\"F07$Fads$!*)f&zi$F07$F[es$!*a^^w$F07$Ficl$! *o-tV&F07$Fcdl$!*FCYr'F07$F]el$!*Fve.(F07$Fh]\\l$!*1KZK'F07$Fbel$!*%3& fi&F07$$\"5n\"z%*H:N7;]\"F0$!*D[/Y%F07$F`^\\l$!*1+iT%F07$$\"5+v$4'*f/N v]\"F0$!*R$*4+%F07$Fgel$!*=u(zF07$Fafl$\"*ZY ma$F07$Fffl$\"*n>Bt$F07$F[gl$\"*Hq-\"\\F07$F`gl$\"*%>5\\[F07$Fegl$\"*- zp#[F07$Fjgl$\"*)GONZF07$F_hl$\"*uF6V%F07$Ffgs$\"*+-V@$F07$F[hs$\")tbo )*F07$F`hs$!)aYBEF07$Fdhl$!*rk9a\"F07$F]is$!*Xuqc$F07$Fgis$!*m)zx]F07$ F\\js$!*+pa+&F07$Fajs$!*-?*y]F07$Ffjs$!*.$R]eF07$Fihl$!*y2!zdF07$$\"5n Tg-&)H0u<;F0$!**)yy#eF07$$\"5M$3-Q\"e=E@;F0$!*0xX3'F07$$\"5+D\"yDk=$yC ;F0$!*&)oI-'F07$F^il$!*$oP3gF07$$\"5+]i!*GrrMN;F0$!**4<'*eF07$Fcil$!*y X[b&F07$F]jl$!*AN%=\\F07$Fgjl$!*6t\\]$F07$Fa[m$!*!zp>AF07$F[\\m$!*/>Q! >F07$Fe\\m$!*VXSr#F07$F_]m$!*\\ut8$F07$Fd]m$!*$QoeLF07$Fi]m$!*TXtY$F07 $F[at$!*7WIU\"F07$F^^m$!)C4KaF07$Fhat$!*)fsZ:F07$Fc^m$!*#=2#e$F07$F_ct $!*-zx^&F07$Fh^m$!*UxxY(F07$Ffdt$!*&Q-czF07$F[et$!*\\87K)F07$F`et$!*o' H9!)F07$F]_m$!*X??Q(F07$F]ft$!*F7\\%oF07$Fb_m$!*.U\"fYF07$Fjft$!*zi'eF F07$Fg_m$!*v')4$=F07$Fbgt$\").^#R\"F07$Fggt$\")VR?xF07$F\\ht$\"),Gr#*F 07$F\\`m$\"*&fUwDF07$Fiht$\"*92X3$F07$Fa`m$\"*G&pNTF07$Ffit$\"*JE??%F0 7$F[jt$\"*kH8;%F07$F`jt$\"*L,e4$F07$Fejt$\"*VLyK#F07$Fjjt$\"*.K/F\"F07 $F_[u$!)4/QwF07$Fd[u$!*(y.*G\"F07$Ff`m$!*87il#F07$Fa\\u$!*,2D:%F07$F[] u$!*w3yw%F07$F`]u$!*K*HuVF07$F[am$!*6[)=PF07$F`am$!*M6$*\\#F07$Feam$!) 4?k')F07$$\"5n;a8A5D;*H#F0$!)%\\]X(F07$F^a]l$!)H3:pF07$$\"5n;H2$[Qh9J# F0$!)R?'y*F07$F`au$!*5by4\"F07$Ffa]l$!*PGp9$F07$Fjam$!*()*3Q]F07$F]bu$ !*1,%))pF07$Facu$!*s;mz)F07$Ffcu$!*sB2v*F07$F[du$!*^'>o)*F07$$\"5nTNY$ Q\"*z@Q#F0$!*[+3%)*F07$F`du$!*jk\")z*F07$$\"5n\"HKR6NH$)Q#F0$!*s_Xa*F0 7$F_bm$!*w-=x)F07$$\"5M$e*[ooq.)R#F0$!*A6TA)F07$Fhdu$!*n6r`'F07$$\"5Me RZ-nl)zS#F0$!*Wme(eF07$$\"5n;a8ZmII6CF0$!*zget$F07$$\"5+voz\"fc>YT#F0$ !*$[uwOF07$Fdbm$!*?Cw)HF07$F`c]l$!)@;]vF07$F`eu$\"*xj0_\"F07$Fhc]l$\"* LHo\"=F07$Fibm$\"*Rn!pDF07$Fheu$\"*Y%pDAF07$F^cm$\"*%o!G&>F07$F`fu$\") CAlfF07$Fccm$!*DTGo\"F07$Fhcm$!*&)yJ'RF07$F]dm$!*>%>$*fF07$F_[el$!*TmU ,(F07$Fbdm$!*J=tm(F07$Fa\\el$!*xfSi(F07$Fgdm$!*UIAn(F07$F\\em$!*@9XE(F 07$Faem$!*gv*yjF07$F[fm$!*R-e:%F07$Fefm$!*&[RuLF07$Fjfm$!*S@2(RF07$F_g m$!*I5Ab%F07$$\"5ML3F>*oZ#F07$Fajm$!*KAn?#F0 7$Ffjm$!*[!4eFF07$F[[n$!*C(*oI%F07$Fd_v$!*E#=$e&F07$F`[n$!*m#fXtF07$F \\`v$!*gm9f*F07$Fe[n$!+\\,:Q5F07$Ff]^l$!+[')=Z5F07$Fd`v$!*wK()*)*F07$F ^^^l$!+?P!e+\"F07$Fj[n$!*C(3#***F07$F\\av$!*tO^b)F07$Faav$!*(R\\%o)F07 $Ffav$!*$pc*\\)F07$F_\\n$!*a-LN'F07$F^bv$!*G!>SkF07$Fcbv$!*,cU>'F07$Fh bv$!*+`pr$F07$Fd\\n$!*]'GgPF07$$\"5,]iS\"p3)*p&HF0$!*xnW^$F07$F`cv$!*j !))*>\"F07$$\"5,](oa5)[ajHF0$!*dE$47F07$Fi\\n$!*wRJ,\"F07$$\"5,+DcEs]O tHF0$\")&edv'F07$F^]n$\"*P)Qz9F07$$\"5,]ilZj_=$)HF0$\"*Bb5_\"F07$$\"5, +voag'ek)HF0$\"*Dmlb\"F07$$\"5,](=K'F07$F[`n$!*<,-9(F07$$\" 5,v=U2^spZIF0$!*f4RH(F07$F``n$!*@>*pwF07$$\"5,Dc^.6;.aIF0$!*.R`n(F07$F e`n$!*j,-r(F07$Fj`n$!*m$3;wF07$F_an$!*fZ0>(F07$Fian$!*GH`%eF07$Fcbn$!* u$yO_F07$Fhbn$!*w&*[O&F07$F]cn$!*Y!)yV&F07$Fbcn$!*'yN3^F07$Fgcn$!*iwIf %F07$F\\dn$!*yHd5%F07$Fadn$!*+5**p$F07$Ffdn$!*s#peOF07$F[en$!*_<@i$F07 $F`en$!*[_$4XF07$Feen$!*B(\\RlF07$F\\iv$!*\\dYy(F07$Faiv$!*ZNdX)F07$Ff iv$!+XyuJ5F07$Fjen$!+*pS^3\"F07$$\"5MLek.aBnkKF0$!+9Lg)=\"F07$F^jv$!+: $)Hu7F07$$\"5M$eRAY&=:uKF0$!+K&*pq7F07$$\"5omT5:)o6tF$F0$!+l:\"4F\"F07 $$\"5,](oz;_r/G$F0$!+1#4OD\"F07$F_fn$!+;F8[7F07$F_h^l$!+f#H\">6F07$Ffj v$!+Km(>4\"F07$F\\i^l$!*+?Ul)F07$Fdfn$!*dJ!zrF07$Fd\\fl$!*#**oTcF07$Fd i^l$!*uFg`&F07$F\\]fl$!*(*\\re%F07$F^[w$!*^xwq#F07$F\\j^l$!*?*yC@F07$F ifn$!)//_aF07$$\"5MLe9T)[ivL$F0$!))=@=$F07$Ff[w$\")yMjFF07$$\"5omT5!>3 5VM$F0$\")r^lFF07$F^gn$\")0?\"e#F07$F^\\w$!)'G_@$F07$Fcgn$!*'Q#*\\?F07 $F_\\_l$!*&e2jCF07$Fhgn$!*5$>[UF07$Fa]_l$!*)GPdkF07$F]hn$!*PIb!oF07$Fb hn$!*-TK0*F07$Fghn$!*T#\\F$*F07$$\"5MLekG5O`=MF0$!*`]aM*F07$F\\in$!*5j KO*F07$$\"5omTgx.7GDMF0$!*@-+G*F07$Fain$!*ZJo=*F07$Ffin$!*w![X*)F07$F[ jn$!*c&=k%)F07$Fejn$!*r7c!pF07$F_[o$!*XZ)y]F07$$\"5MeRZ_H>7%\\$F0$!*FP P2&F07$$\"5,]7.#34zr\\$F0$!*!>@L]F07$$\"5oT&)e6_iB+NF0$!*$*HP*[F07$Fd[ o$!*iJ<*[F07$$\"5o;/E+OxS4NF0$!*S#fA\\F07$Fi[o$!*%fgm\\F07$F^\\o$!*HSV G&F07$Fc\\o$!*`R1)eF07$$\"5oT5lvL;KVNF0$!*cM])eF07$F\\`_l$!*^>V!fF07$$ \"5MekGN.i+]NF0$!*D9+5'F07$Fh\\o$!*tX-5'F07$Fi`_l$!*rTe='F07$F]]o$!*s( H*='F07$Fb]o$!*A#R'4'F07$Fg]o$!*gd]0'F07$F\\^o$!*nDa^'F07$Fa^o$!*\\3$) *pF07$Fbcw$!*^PeG'F07$Ff^o$!*<(>ZZF07$F]f_l$!*MPjw$F07$Fjcw$!*1s&=OF07 $Fdg_l$!*Y]K'QF07$F[_o$!*4%HLXF07$Fbdw$!*G=wB'F07$F`_o$!*];8k(F07$Fiew $!*B;(H%*F07$Fe_o$!+EMxr6F07$Fffw$!+-Sy87F07$Fj_o$!+z$QCC\"F07$F^gw$!+ \"yQ*f7F07$Fcgw$!+8NA'=\"F07$Fhgw$!+2IG'>\"F07$F_`o$!+'pzn?\"F07$Fehw$ !+W3NZ5F07$Fd`o$!*55-r)F07$$\"5,+D19Wt*G%QF0$!*oYv>)F07$Fbiw$!*#\\\"z< )F07$$\"5ML3-8]=Z\\QF0$!*#yIujF07$Fi`o$!*u$o2cF07$Fjiw$!*,C(GUF07$F^ao $!*\"R+RKF07$Fbjw$!*fX,p\"F07$Fcao$!*c[=U\"F07$Fhao$!*x4u\"=F07$F]bo$! *GW)yLF07$Fgbo$!*/g8S&F07$Faco$!*-jNl&F07$Ffco$!*\"y[#[(F07$F[do$!*a+( [!*F07$F`do$!+mr:A5F07$Fedo$!+.kZi5F07$F_eo$!*\\..B*F07$Fieo$!*hj/9)F0 7$$\"5o;zpBEs;;SF0$!*t?)3\")F07$F^fo$!*fw8<)F07$$\"5M$3F>*3eiHSF0$!*EO .F)F07$Fcfo$!*ntrI)F07$Fh_x$!*zz\"o$)F07$Fhfo$!*[?$o#)F07$F``x$!*s7CD) F07$Fe`x$!*!H+N#)F07$Fj`x$!*C()*QzF07$F]go$!*a)fW*zlF07$Ffho$!**3\"3\"F07$Ffgx$!+>Zh66F07$Fjbgl$!+@@Q+6 F07$Fe^p$!+l%GX3\"F07$Fi_p$!*V2.9*F07$Fc`p$!*G\\-N(F07$$\"5,]PM-\\NA%Q %F0$!*N5R\"pF07$$\"5ML3_vSME!R%F0$!*ogw_'F07$$\"5o;zp[KLI'R%F0$!*NSR]' F07$Fjhx$!*W&o.kF07$$\"5om\"H#o2IU9WF0$!*%[5HlF07$Fh`p$!*,6s$oF07$Fajx $!*'yoxsF07$F[[y$!*g6hk(F07$F`[y$!*b/Dq(F07$F]ap$!*`lMm(F07$Fbap$!*faU e(F07$Fgap$!*jQSk(F07$F\\bp$!*iwW%yF07$Fabp$!*EKDO)F07$$\"5o;a8(RIiO`% F0$!*!)fIO)F07$$\"5,+vVt-N7PXF0$!*RXuQ)F07$$\"5M$eR(\\,ZeSXF0$!*+%>E&) F07$Ffbp$!*;So_)F07$$\"5MLeky(Ho4b%F0$!*;v-\\)F07$F[cp$!*u`BW)F07$F`cp $!*I&G\\wF07$Fecp$!*hIi\"pF07$F]_al$!*&R/IgF07$Fjcp$!*%G&4?&F07$$\"5-+ Dcwfw`;YF0$!*Pk'3_F07$Fd`al$!*Y4]>&F07$$\"5NL3_]41#*GYF0$!*A5+L&F07$F_ dp$!*6xxa&F07$F\\aal$!*7Q(HjF07$Fddp$!*8JD6)F07$Fdaal$!*%*f4p*F07$Fidp $!+nD-V6F07$F^ep$!+%pV!Q8F07$Fcep$!+\"\\$fN9F07$Fhep$!+u<)yR\"F07$F]fp $!+tK>:7F07$Fhay$!*iZ&=**F07$F]by$!+P9'\\+\"F07$Fbby$!*OZ5y*F07$Fbfp$! *`ZlY(F07$Fjby$!*n(R*H(F07$Fgfp$!*EULV&F07$Fbcy$!*C+,E&F07$Fgcy$!*d$RK TF07$F\\dy$!*)\\3lTF07$F\\gp$!*R=`5%F07$$\"5-D1RDNUO!y%F0$!*EGr(RF07$F ddy$!*>WQ+%F07$$\"5oTNY$)Qc'py%F0$!*rW%eSF07$Fagp$!*YKn!\\F07$Fehp$!*V 0Ln'F07$F_ip$!*d(ze))F07$Figal$!+U.$33\"F07$Fdip$!+b>\"48\"F07$F[ial$! +(z`)Q7F07$Fiip$!+j\\DF8F07$$\"5-+voa5=8Z[F0$!+#G^cL\"F07$Fhey$!+weJ_8 F07$$\"5-+D1k\"eo(f[F0$!+-1#*H8F07$F^jp$!+<[-B8F07$F`fy$!+`i]Q7F07$Fcj p$!+:c%G9\"F07$$\"5-]i!RvZ]\")*[F0$!+[sTK6F07$$\"5-+DJqX/%\\!\\F0$!+0E $*46F07$$\"5-](=nQTI<\"\\F0$!+[K8-6F07$Fhfy$!+,Pq.6F07$$\"5-]7`>].JD\\ F0$!+<7666F07$$\"5-+v$f$=.5K\\F0$!+(G;R7\"F07$$\"5-]PM_'G!*)Q\\F0$!+Ro cD6F07$F]gy$!+>AdE6F07$Fggy$!+\\#e(>5F07$Fhjp$!*&*[*o&*F0-Ff\\bl6&F_[q $\"\"'Fj\\blF(F[]bl-Fd[q6#%Sscheme~with~stability~radius~=~imag~axis~i nclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~7~st age~order~5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fecgm-%%VIEWG 6$;F(Fhjp%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modification of the B ogacki-Shampine scheme" "efficient scheme A" "efficient scheme B" "eff icient scheme C" "scheme with stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 13 of 7 stage, order 5 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathematica in Action\" by Stan W agon, Springer-Verlag, page 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y; " "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"! ,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y = 1/5;" "6#/%\"yG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the different ial equation " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\" \"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " c ontains an exponential term, but with the initial condition " } {XPPEDIT 18 0 "y(0) = -2/5" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\" F-" }{TEXT -1 23 " this term disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=co s(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF& F-!\"\"*&#F-F,F--%$sinGF&F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any sli ght deviation of a numerical solution from the correct solution tends \+ to become rapidly magnified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2 *y(x);\nic := y(0)=-2/5;\ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%) ,x):\nplot(e(x),x=0..8,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$c osGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!#!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\" xG,&*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }} {PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7 $$\"\"!F)$!3A+++++++S!#=7$$\"3ELLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLLLLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\ o'F,$!31c1[)*fT**=F,7$$\"3)emmm^&Q%R)F,$!39J7$$\"3))*****\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LL L=\"\\g**FK7$$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F ,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g(y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&H F,7$$\"3&)******f-w+?FO$\"375@EVOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n #=i\"RF,7$$\"3)*******RP)4M#FO$\"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3: @(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO$\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+ <'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$\"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35 ^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\"3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3! [MOl&\\jZVF,7$$\"3!)*****\\9!H.IFO$\"37X%)HL)HvB%F,7$$\"3Immm1:bgJFO$ \"37d#H1rl8'RF,7$$\"3<+++X@4LLFO$\"3G,Fnxt@YNF,7$$\"31+++N;R(\\$FO$\"3 +2Ml_]z_IF,7$$\"3wmmm;4#)oOFO$\"3?6K>$)*R0X#F,7$$\"3jmmm6lCEQFO$\"3xp` %>:UP$=F,7$$\"3ELLL$G^g*RFO$\"38$\\Qkcw!=6F,7$$\"3oKLL=2VsTFO$\"39U#4i *[S4MFK7$$\"3f*****\\`pfK%FO$!3E&Q)=hjJ]MFK7$$\"3!HLLLm&z\"\\%FO$!3u$z \"\\\">,j2\"F,7$$\"3s******z-6jYFO$!3_=%f%oq`+=F,7$$\"3<******4#32$[FO $!3Gvm#oI!>eCF,7$$\"3O*****\\#y'G*\\FO$!3Ak5yX#4\"HIF,7$$\"3G******H%= H<&FO$!3EIq1&[C$pNF,7$$\"35mmm1>qM`FO$!3%z'[2h*Gn&RF,7$$\"3%)*******HS u]&FO$!3%*oc=HW4cUF,7$$\"3'fmm\"HOq&e&FO$!3oqc`'[F/N%F,7$$\"3'HLL$ep'R m&FO$!3$e%**GFr7=WF,7$$\"3D***\\P?[nq&FO$!3TlAsE+sVWF,7$$\"3Umm;\\%H& \\dFO$!3y[ey96=hWF,7$$\"3eLLe%p5Bz&FO$!3=)zg%Q%y/Z%F,7$$\"3')******R>4 NeFO$!3waa0%)\\frWF,7$$\"3HLL$ed*f:fFO$!3]_J$4k<:X%F,7$$\"3#emm;@2h*fF O$!3V5vHeMg-WF,7$$\"37LLL))3E!3'FO$!3=l`a'y%*4K%F,7$$\"3]*****\\c9W;'F O$!3>=$e-d.)3UF,7$$\"3Lmmmmd'*GjFO$!3Gy*y<4!G/RF,7$$\"3j*****\\iN7]'FO $!3;B6I^7jsMF,7$$\"3aLLLt>:nmFO$!37+2hu:afHF,7$$\"35LLL.a#o$oFO$!3;\"e /Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4`1I$pa!o\"F,7$$\"3y******z]rfrFO$!3pfL '*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3?j;%3XMsQ#FK7$$\"3/LLL8-V&\\(FO$\"3qi( R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3ZX^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXE SLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(Fg]l%(DE FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "T he following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 754 "E := (x,y) -> cos(x)+2*y: h h := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope fiel d: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine scheme `,`a minor modification of the Bogacki-Shampine scheme`,`efficient sch eme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stabilit y radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct \+ to 6 do\n En_RK5_||ct := RK5_||ct(E(x,y),x,y,x0,evalf(y0),hh,numstep s,false);\n sm := 0: numpts := nops(En_RK5_||ct):\n for ii to nump ts do\n sm := sm+(En_RK5_||ct[ii,2]-e(En_RK5_||ct[ii,1]))^2;\n \+ end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10 :\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\" \"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~ width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint766\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7(7$%8Bogacki-Shampine~schemeG$\"+1()3M6!#;7$%Ta~minor~modification~o f~the~Bogacki-Shampine~schemeG$\"+fEDd6F+7$%3efficient~scheme~AG$\"+A4 jB[!#<7$%3efficient~scheme~BG$\"+uF)[S(F47$%3efficient~scheme~CG$\"+%* ReMRF47$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+@s2 " 0 "" {MPLTEXT 1 0 685 "E := (x,y) -> cos(x)+ 2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slop e field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampine \+ scheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficie nt scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with st ability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\nf or ct to 6 do\n en_RK5_||ct := RK5_||ct(E(x,y),x,y,x0,evalf(y0),hh,n umsteps,true);\nend do:\nxx := 7.999: exx := evalf(e(xx)):\nfor ct to \+ 6 do\n errs := [op(errs),abs(en_RK5_||ct(xx)-exx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\" \"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~ width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7(7$%8Bogacki-Shampine~schemeG$\"+L]X%G'!#;7$%Ta~minor~modification~o f~the~Bogacki-Shampine~schemeG$\"+#4;GT'F+7$%3efficient~scheme~AG$\"+o N(Hn#F+7$%3efficient~scheme~BG$\"+n?N.TF+7$%3efficient~scheme~CG$\"+he J!=#F+7$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+k`oZC F+Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Sh ampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient sc heme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs : = []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((e(x)-'en_RK5_||c t'(x))^2,x=0..8,adaptive=false,numpoints=7,factor=200);\n errs := [o p(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8B ogacki-Shampine~schemeG$\"+C$pJ6\"!#;7$%Ta~minor~modification~of~the~B ogacki-Shampine~schemeG$\"+%*f!f8\"F+7$%3efficient~scheme~AG$\"+1VlMZ! #<7$%3efficient~scheme~BG$\"+HCHosF47$%3efficient~scheme~CG$\"+Bn+iQF4 7$%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+%))*fNVF4Q) pprint806\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical \+ procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "evalf[20](plot(['en_RK5_1'(x)-e(x),'en_RK5_2'(x)-e(x),'en_RK5_3 '(x)-e(x),'en_RK5_4'(x)-e(x),\n'en_RK5_5'(x)-e(x),'en_RK5_6'(x)-e(x)], x=0..2,\ncolor=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6 ,0,.8)],font=[HELVETICA,9],\nlegend=[`Bogacki-Shampine scheme`,`a mino r modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`e fficient scheme B`,`efficient scheme C`,`scheme with stability radius \+ = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 805 579 579 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5MLLLLL3VfV!#@$\"))[QV*! #A7$$\"5nmmmm\"H[D:)F-$\")O2R>F-7$$\"5LLLLLe0$=C\"!#?$\")#f$=IF-7$$\"5 LLLLL3RBr;F9$\")eT!=%F-7$$\"5nmmm;zjf)4#F9$\")]wIaF-7$$\"5MLLL$e4;[\\# F9$\")QrCnF-7$$\"5++++]i'y]!HF9$\")ouE\")F-7$$\"5MLLL$ezs$HLF9$\")w4l' *F-7$$\"5++++]7iI_PF9$\"*UpL8\"F-7$$\"5nmmmm;_M(=%F9$\"*#)3sK\"F-7$$\" 5MLLLL3y_qXF9$\"*7[A\\\"F-7$$\"5+++++]1!>+&F9$\"*M]Nr\"F-7$$\"5+++++]Z /NaF9$\")nZI>F97$$\"5+++++]$fC&eF9$\")HCd@F97$$\"5MLLL$ez6:B'F9$\")!*3 #Q#F97$$\"5nmmmm;=C#o'F9$\")o\")eEF97$$\"5nmmmmm#pS1(F9$\")!px\"HF97$$ \"5,+++]i`A3vF9$\")&eRB$F97$$\"5nmmmmm(y8!zF9$\")\"p!RNF97$$\"5,+++]i. tK$)F9$\")G0'*QF97$$\"5,+++](3zMu)F9$\")bBjUF97$$\"5nmmmm\"H_?<*F9$\") $Gpn%F97$$\"5nmmm;zihl&*F9$\")B+$3&F97$$\"5MLLLL3#G,***F9$\")%pqb&F97$ $\"5LLLLezw5V5!#>$\")E4'3'F97$$\"5++++v$Q#\\\"3\"Fbs$\")B\"=e'F97$$\"5 LLLL$e\"*[H7\"Fbs$\")XxcrF97$$\"5+++++qvxl6Fbs$\")*o')z(F97$$\"5++++]_ qn27Fbs$\")Y+y%)F97$$\"5++++Dcp@[7Fbs$\")z[(=*F97$$\"5++++]2'HKH\"Fbs$ \"*)H;/5F97$$\"5nmmmmwanL8Fbs$\"*xNs3\"F97$$\"5+++++v+'oP\"Fbs$\"*R'4$ =\"F97$$\"5LLLLeR<*fT\"Fbs$\"*uVrF\"F97$$\"5+++++&)Hxe9Fbs$\"*\"H\"yQ \"F97$$\"5nmmm\"H!o-*\\\"Fbs$\"*#y#3]\"F97$$\"5++++DTO5T:Fbs$\"*Kf%G;F 97$$\"5nmmmmT9C#e\"Fbs$\"*0\\Sw\"F97$$\"5++++D1*3`i\"Fbs$\"*Dm#=>F97$$ \"5LLLLL$*zym;Fbs$\"*pM$z?F97$$\"5LLLL$3N1#4)zGF97$$\"5LLLLL`v&Q(=Fbs$\"*x>26$F97$$\"5++++DOl5;>Fbs$\"* \\6uP$F97$$\"5++++v.Uac>Fbs$\"*Z7Wl$F97$$\"\"#F)$\"*n,g(RF9-%'COLOURG6 &%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7S F'7$F+$\"))=/d*F07$F2$\")E^n>F-7$F7$\")A>jIF-7$F=$\")[GVUF-7$FB$\")]W8 bF-7$FG$\"))3$GoF-7$FL$\")eS`#)F-7$FQ$\")1]<)*F-7$FV$\"*A_9:\"F-7$Fen$ \"*#fm[8F-7$Fjn$\"*_*e;:F-7$F_o$\"*Co=u\"F-7$Fdo$\")Spi>F97$Fio$\")(*e $>#F97$F^p$\")yfACF97$Fcp$\")7W/FF97$Fhp$\")XFoHF97$F]q$\")&=/H$F97$Fb q$\")2O,OF97$Fgq$\")7>lRF97$F\\r$\")O[RVF97$Far$\")QFhZF97$Ffr$\")'F97$Ffs$\")L#Rq'F97$F[t$\")\"R.H(F97$F` t$\")v.XzF97$Fet$\")_)zj)F97$Fjt$\");vh$*F97$F_u$\"*j1L-\"F97$Fdu$\"*m g!36F97$Fiu$\"*9qe?\"F97$F^v$\"*YD=I\"F97$Fcv$\"*EhZT\"F97$Fhv$\"*W\"3 I:F97$F]w$\"*LD.m\"F97$Fbw$\"*r!o)z\"F97$Fgw$\"*)3/c>F97$F\\x$\"*)*)R? @F97$Fax$\"*'3*GI#F97$Ffx$\"*1^$*\\#F97$F[y$\"*0NVp#F97$F`y$\"*!*es$HF 97$Fey$\"*g!*G<$F97$Fjy$\"*'40XMF97$F_z$\"*GJxs$F97$Fdz$\"*TXf0%F9-Fiz 6&F[[lF\\[lF(F\\[l-F`[l6#%Ta~minor~modification~of~the~Bogacki-Shampin e~schemeG-F$6%7SF'7$F+$\"))QJ:%F07$F2$\"(Yt_)F-7$F7$\")#[fK\"F-7$F=$\" ))*[M=F-7$FB$\")qm!Q#F-7$FG$\")=%[%HF-7$FL$\")39bNF-7$FQ$\")mqBUF-7$FV $\")iiZ\\F-7$Fen$\")Kt'y&F-7$Fjn$\")K\"4]'F-7$F_o$\")uqbuF-7$Fdo$\"(!o \"R)F97$Fio$\"(t(o$*F97$F^p$\")/fL5F97$Fcp$\")kj_6F97$Fhp$\")*)zj7F97$ F]q$\")xa*R\"F97$Fbq$\")NKI:F97$Fgq$\")cF$o\"F97$F\\r$\")\"G/%=F97$Far $\")!os,#F97$Ffr$\")3'3>#F97$F[s$\"):7$R#F97$F`s$\")5!*=EF97$Ffs$\")IL IGF97$F[t$\")&4c2$F97$F`t$\")dR\\LF97$Fet$\")))**QOF97$Fjt$\")dGTRF97$ F_u$\")EE0VF97$Fdu$\")c)*eYF97$Fiu$\")q$p1&F97$F^v$\")VFnaF97$Fcv$\")' Hy$fF97$Fhv$\")Uf=kF97$F]w$\")iJhpF97$Fbw$\")h6QvF97$Fgw$\")QA%>)F97$F \\x$\")EIz))F97$Fax$\")`'*R'*F97$Ffx$\"*(G(e/\"F97$F[y$\"*I1r7\"F97$F` y$\"*`P$G7F97$Fey$\"*p3lK\"F97$Fjy$\"*\"p))R9F97$F_z$\"**)\\wb\"F97$Fd z$\"*U\"H%p\"F9-%&COLORG6&F[[l$\"\"\"!\"\"$\"\")Fc^mF(-F`[l6#%3efficie nt~scheme~AG-F$6%7SF'7$F+$\"))=cR'F07$F2$\")'RIJ\"F-7$F7$\")7]T?F-7$F= $\")=CCGF-7$FB$\")g)[m$F-7$FG$\")[*G`%F-7$FL$\"))f=Z&F-7$FQ$\")1d+lF-7 $FV$\")7K9wF-7$Fen$\")#=W!*)F-7$Fjn$\"*#[F+5F-7$F_o$\"*MHp9\"F-7$Fdo$ \")F!3H\"F97$Fio$\")$y4W\"F97$F^p$\")$)f*e\"F97$Fcp$\")XasF97$F]q$\")&>>:#F97$Fbq$\")\\\"GN#F97$Fgq$\")Ez(e#F97$F\\r$\")a=HG F97$Far$\")2z+JF97$Ffr$\")[RnLF97$F[s$\")U'zn$F97$F`s$\")ulCSF97$Ffs$ \")eJ\\VF97$F[t$\"))\\fs%F97$F`t$\")cNY^F97$Fet$\")l-\"f&F97$Fjt$\")([ ^0'F97$F_u$\")U)Rh'F97$Fdu$\")K/drF97$Fiu$\")8J$y(F97$F^v$\")C#zR)F97$ Fcv$\")gE?\"*F97$Fhv$\")GJe)*F97$F]w$\"*#z9p5F97$Fbw$\"*M&pd6F97$Fgw$ \"*g=%e7F97$F\\x$\"*!oej8F97$Fax$\"*Zd.[\"F97$Ffx$\"*\"\\/1;F97$F[y$\" *yW2t\"F97$F`y$\"*gQh)=F97$Fey$\"*qQo.#F97$Fjy$\"*Yw3@#F97$F_z$\"*&)[; R#F97$Fdz$\"*M.9g#F9-Fiz6&F[[lF\\[lF(F(-F`[l6#%3efficient~scheme~BG-F$ 6%7SF'7$F+$\"))oIS$F07$F2$\"(1j)pF-7$F7$\")s<'3\"F-7$F=$\")=d-:F-7$FB$ \")!R(\\>F-7$FG$\")3U6CF-7$FL$\")e\"3\"HF-7$FQ$\")O\"zX$F-7$FV$\")i>]S F-7$Fen$\")#zht%F-7$Fjn$\")#*=?`F-7$F_o$\")M()*4'F-7$Fdo$\"(&zkoF97$Fi o$\"(zJm(F97$F^p$\"(OKX)F97$Fcp$\"(udU*F97$Fhp$\")>OL5F97$F]q$\")^BW6F 97$Fbq$\")\\,^7F97$Fgq$\")k!fP\"F97$F\\r$\")B?/:F97$Far$\")va[;F97$Ffr $\")nB!z\"F97$F[s$\")1Gb>F97$F`s$\")l_R@F97$Ffs$\")N07BF97$F[t$\")E?7D F97$F`t$\")6hNFF97$Fet$\")!4>(HF97$Fjt$\")y`=KF97$F_u$\")J\\:NF97$Fdu$ \")&fS!QF97$Fiu$\")#Ho8%F97$F^v$\")TTjWF97$Fcv$\")!Gs%[F97$Fhv$\")HRR_ F97$F]w$\")+5#o&F97$Fbw$\"))*f_hF97$Fgw$\")()z(o'F97$F\\x$\"))4mC(F97$ Fax$\")(pq'yF97$Ffx$\")S!\\`)F97$F[y$\")q[(>*F97$F`y$\"*l:B+\"F97$Fey$ \"*%*)Q#3\"F97$Fjy$\"*[h[<\"F97$F_z$\"*'H\"4F\"F97$Fdz$\"*!HO#Q\"F9-Fi z6&F[[l$\")#)eqkF^[l$\"))eqk\"F^[lFhan-F`[l6#%3efficient~scheme~CG-F$6 %7SF'7$F+$!)7u;OF07$F2$!(CxV(F-7$F7$!)3Me6F-7$F=$!)7>0;F-7$FB$!)?w'3#F -7$FG$!)s&\\e#F-7$FL$!)_eDJF-7$FQ$!)a,?PF-7$FV$!)QVlVF-7$Fen$!)G-9^F-7 $Fjn$!)Ge_dF-7$F_o$!)OR0mF-7$Fdo$!(I[W(F97$Fio$!(,HK)F97$F^p$!(1U>*F97 $Fcp$!)NmE5F97$Fhp$!)@4F6F97$F]q$!)vw\\7F97$Fbq$!)mF97$F[s$!)EN_@F97$F`s$!)q, eBF97$Ffs$!)2&3b#F97$F[t$!)1\\uFF97$F`t$!)E=CIF97$Fet$!)3])G$F97$Fjt$! )qhkNF97$F_u$!)M)p*QF97$Fdu$!)>O?UF97$Fiu$!)^d$f%F97$F^v$!)xsf\\F97$Fc v$!)^$3R&F97$Fhv$!)&y4$eF97$F]w$!)#R#GjF97$Fbw$!)()HcoF97$Fgw$!)_(oX(F 97$F\\x$!)L?%3)F97$Fax$!)#H2y)F97$Ffx$!)/dI&*F97$F[y$!*^\"\\F5F97$F`y$ !*E+-7\"F97$Fey$!*U[,@\"F97$Fjy$!*LHSJ\"F97$F_z$!*eN>U\"F97$Fdz$!*#GGZ :F9-F_^m6&F[[l$\"\"'Fc^mF(Fd^m-F`[l6#%Sscheme~with~stability~radius~=~ imag~axis~inclusionG-%+AXESLABELSG6$Q\"x6\"Q!F[\\o-%%FONTG6$%*HELVETIC AG\"\"*-%&TITLEG6#%Uerror~curves~for~7~stage~order~5~Runge-Kutta~metho dsG-%%VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modificatio n of the Bogacki-Shampine scheme" "efficient scheme A" "efficient sche me B" "efficient scheme C" "scheme with stability radius = imag axis i nclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 498 "evalf[20](plot(['en_RK5_1'(x)-e(x),'en_RK5_2'(x)-e (x),'en_RK5_3'(x)-e(x),'en_RK5_4'(x)-e(x),\n'en_RK5_5'(x)-e(x),'en_RK5 _6'(x)-e(x)],x=2..8,\ncolor=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown ,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9],\nlegend=[`Bogacki-Shampine sc heme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stab ility radius = imag axis inclusion`],\ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 835 594 594 {PLOTDATA 2 "6,-%'CURVESG6%7Z7$$\"\"#\"\"!$\"*n,g(R!#?7$$\"5++ +++DHyI@!#>$\"*H\"pP^F-7$$\"5++++v[kdWAF1$\"*8Y>U'F-7$$\"5++++]n\"\\DP #F1$\"*#4aj#)F-7$$\"5++++]s,P,DF1$\"+(H/a1\"F-7$$\"5++++v8*y&HEF1$\",c tLMP\"!#@7$$\"5++++vG[W[FF1$\",ga8'Q/A#FH 7$$\"5++++vQ=\"))*HF1$\",'oVTgGFH7$$\"5++++vj=pDJF1$\"-'eNqKo$!#A7$$\" 5+++++lN?cKF1$\",%*y*eyZFH7$$\"5++++]U$e6P$F1$\",e+P5,'FH7$$\"5+++++&> q0]$F1$\",UvxOy(FH7$$\"5+++++DM^IOF1$\"-W/<945FH7$$\"5+++++0ytbPF1$\", $G\"QhH\"F-7$$\"5++++vQNXpQF1$\",yjxpi\"F-7$$\"5+++++XDn/SF1$\",V,q?8# F-7$$\"5+++++!y?#>TF1$\",nv@4o#F-7$$\"5++++v3wY_UF1$\",@Hv&*\\$F-7$$\" 5+++++IOTqVF1$\",cv$fIWF-7$$\"5++++v3\">)*\\%F1$\",'3BRRdF-7$$\"5++++D EP/BYF1$\",hj\"\\VtF-7$$\"5++++](o:;v%F1$\",olgp\\*F-7$$\"5++++v$)[op[ F1$\"-:&\\jE?\"F-7$$\"5++++]i%Qq*\\F1$\"-B!)zb^:F-7$$\"5++++vQIKH^F1$ \"-#Hx.:-#F-7$$\"5++++D^rZW_F1$\"-gU]2XDF-7$$\"5++++]Zn%)o`F1$\"-Gn@&Q E$F-7$$\"5+++++5FL(\\&F1$\"-h\\GA?UF-7$$\"5++++]d6.BcF1$\"-*H%R[EaF-7$ $\"5++++vo3lWdF1$\"-e!\\H3#pF-7$$\"5++++]A))ozeF1$\".c#[#on1*FH7$$\"5+ ++++Ik-,gF1$\"/kTtJqb6FH7$$\"5+++++D-eIhF1$\"/ey!fKv\\\"FH7$$\"5++++v= _(zC'F1$\"0i_VdZQ*=Fgn7$$\"5+++++b*=jP'F1$\"/?m_%o![CFH7$$\"5++++v3/3( \\'F1$\"/i)*f4&o6$FH7$$\"5++++vB4JBmF1$\"/S2Z-)>,%FH7$$\"5+++++DVsYnF1 $\"/!Q<>m^8&FH7$$\"5++++v=n#f(oF1$\".%z6\"Q$\\mF-7$$\"5+++++!)RO+qF1$ \".d,><$G&)F-7$$\"5++++]_!>w7(F1$\"/gq`x++6F-7$$\"5++++v)Q?QD(F1$\"/\" 34rNeT\"F-7$$\"5+++++5jyptF1$\"/mr,nT&y\"F-7$$\"5++++DE8COuF1$\"/LBi(3 #R?F-7$$\"5++++]Ujp-vF1$\"/9\\]m2HBF-7$$\"5++++D,X8ivF1$\"/Hf8u2BEF-7$ $\"5+++++gEd@wF1$\"/pIk+>aHF-7$$\"5+++]PMh%\\o(F1$\"/@0`*)Q`LF-7$$\"5+ +++v3'>$[xF1$\"/iOs9`1QF-7$$\"5+++++5h(*3yF1$\"/KU76\\(H%F-7$$\"5++++D 6EjpyF1$\"/$[([Qx^[F-7$$\"5+++vVeWA-zF1$\"/*G\"*Ql&y^F-7$$\"5+++]i0j\" [$zF1$\"/b=9!ot_&F-7$$\"5+++D\"G:3u'zF1$\"/1!pCk'**eF-7$$\"\")F*$\"/H' H^OqH'F--%'COLOURG6&%$RGBG$F*F*Fb]l$\"*++++\"!\")-%'LEGENDG6#%8Bogacki -Shampine~schemeG-F$6%7Z7$F($\"*TXf0%F-7$F/$\"*y;9C&F-7$F5$\"*y7?b'F-7 $F:$\"*EA8V)F-7$F?$\"+e<3(3\"F-7$FD$\",Ol:9S\"FH7$FJ$\",]mpSx\"FH7$FO$ \",ebKdE#FH7$FT$\",w.=)=HFH7$FY$\"-'Qu)\\ePFgn7$Fin$\",k+3i([FH7$F^o$ \",eOWQ8'FH7$Fco$\",Kj4F%zFH7$Fho$\"-uX,wH5FH7$F]p$\",c/?EK\"F-7$Fbp$ \",yF=-m\"F-7$Fgp$\",GbGc<#F-7$F\\q$\",vk\"pNFF-7$Faq$\",ZTn5d$F-7$Ffq $\",et-6_%F-7$F[r$\",<-Mm&eF-7$F`r$\",1n(\\$\\(F-7$Fer$\",9%>&4p*F-7$F jr$\"-!y]HsA\"F-7$F_s$\"-:&=]Ke\"F-7$Fds$\"-9PYzi?F-7$Fis$\"-(>&)fqf#F -7$F^t$\"-%)Q!=0L$F-7$Fct$\"-.REU1VF-7$Fht$\"-Rs;KPbF-7$F]u$\"-N[$*=iq F-7$Fbu$\".Ex4f>D*FH7$Fgu$\"//B['3$z6FH7$F\\v$\"/3?u*>\"G:FH7$Fav$\"0i DCuHD$>Fgn7$Ffv$\"/!)ef12)\\#FH7$F[w$\"/_SZJ^!=$FH7$F`w$\"/SK'eDR4%FH7 $Few$\"/!y3y_+C&FH7$Fjw$\".Vl!>:&y'F-7$F_x$\".$*=W4Dq)F-7$Fdx$\"/u\\!f vC7\"F-7$Fix$\"/()z'QaZW\"F-7$F^y$\"/y)>8%)=#=F-7$Fcy$\"/-\\Q*f33#F-7$ Fhy$\"/Q&zU[mP#F-7$F]z$\"/K)o?amn#F-7$Fbz$\"/B:!*)HX,$F-7$Fgz$\"/))\\% [#)=U$F-7$F\\[l$\"/JQ<0G%)QF-7$Fa[l$\"/9K#4o_Q%F-7$Ff[l$\"/jp[@(3&\\F- 7$F[\\l$\"/pF-7$Fgp$\"+tz3o!*F-7$F\\q$\",n/]-9\"F-7$Faq$\",[.Y%) [\"F-7$Ffq$\",+VNW)=F-7$F[r$\",%zs5TCF-7$F`r$\",ElzL7$F-7$Fer$\",H4;$R SF-7$Fjr$\",:Yl_6&F-7$F_s$\",p!GA*f'F-7$Fds$\",*R&\\!)f)F-7$Fis$\"-@li \\#3\"F-7$F^t$\"-&zm9#)Q\"F-7$Fct$\"-4h!*)\\z\"F-7$Fht$\"-'3n\\!3BF-7$ F]u$\"-j&\\TO%HF-7$Fbu$\".Y8@wj&QFH7$Fgu$\".W&oGc:\\FH7$F\\v$\".3Cup%p jFH7$Fav$\"/iQV17b!)Fgn7$Ffv$\"/g]G%R7/\"FH7$F[w$\"/i'GS$pD8FH7$F`w$\" /q@!o>kq\"FH7$Few$\"/5R>b9%=#FH7$Fjw$\".fQ\")p\"GGF-7$F_x$\".XZ*yNFOF- 7$Fdx$\".mXAu'yYF-7$Fix$\".(\\d%))>-'F-7$F^y$\".Sk*4$Rf(F-7$Fcy$\".+BJ (Qt')F-7$Fhy$\".2/I&G1**F-7$F]z$\"/^xohn:6F-7$Fbz$\"/$)fV'3lD\"F-7$Fgz $\"/#phI+jU\"F-7$F\\[l$\"/H'o'e.>;F-7$Fa[l$\"/c'4Acy#=F-7$Ff[l$\"/np]) 4O1#F-7$F[\\l$\"/[HAWg-AF-7$F`\\l$\"/pm\"*4'4N#F-7$Fe\\l$\"/$>'>,J4DF- 7$Fj\\l$\"/@fh[KyEF--%&COLORG6&Fa]l$\"\"\"!\"\"$F[]lFjcmFb]l-Fg]l6#%3e fficient~scheme~AG-F$6%7Z7$F($\"*M.9g#F-7$F/$\"*t#ofLF-7$F5$\"*qgx>%F- 7$F:$\"*ky(*R&F-7$F?$\"*^J*fpF-7$FD$\"+'[C1(*)FH7$FJ$\",qRPa8\"FH7$FO$ \",=6f*\\9FH7$FT$\",c\"evn=FH7$FY$\"-'=cb\\S#Fgn7$Fin$\",%RP0?JFH7$F^o $\",G0(pCRFH7$Fco$\",#GS/#3&FH7$Fho$\",u6#y)e'FH7$F]p$\"+8F-7$Fis$\"-eo!oEFH7$ Few$\"/qS$GRHN$FH7$Fjw$\".I9$zfTVF-7$F_x$\".O%)e_%obF-7$Fdx$\".\\e&fN# =(F-7$Fix$\".%e()H^W#*F-7$F^y$\"/YlMVwl6F-7$Fcy$\"/TP9XZJ8F-7$Fhy$\"/, *)G*R2_\"F-7$F]z$\"/wu\\Qq7F-7$Fgz$\"/s4P6b*=#F-7 $F\\[l$\"//5oXU&[#F-7$Fa[l$\"/Fo)=!*f!GF-7$Ff[l$\"/SR*y,z;$F-7$F[\\l$ \"/lt8gF\"Q$F-7$F`\\l$\"/oI(3A!4OF-7$Fe\\l$\"/N[+!3@&QF-7$Fj\\l$\"/>Iy pc6TF--F_]l6&Fa]lFc]lFb]lFb]l-Fg]l6#%3efficient~scheme~BG-F$6%7Z7$F($ \"*!HO#Q\"F-7$F/$\"*op_y\"F-7$F5$\"*Pt0B#F-7$F:$\"*Rc#pGF-7$F?$\"*0I#) p$F-7$FD$\"+E0gmZFH7$FJ$\"+SF>LgFH7$FO$\"+Q**Q/xFH7$FT$\"+mgLC**FH7$FY $\"-'=CtyF\"Fgn7$Fin$\",9.Ryl\"FH7$F^o$\",e*[Q&3#FH7$Fco$\",KiR.q#FH7$ Fho$\",MSV4]$FH7$F]p$\"+Yrg'\\%F-7$Fbp$\"+7fQWcF-7$Fgp$\"+aGr'R(F-7$F \\q$\"+)Ql3I*F-7$Faq$\",v(f597F-7$Ffq$\",<55r`\"F-7$F[r$\",eF\"=\"*>F- 7$F`r$\",vV.xa#F-7$Fer$\",U!H#[H$F-7$Fjr$\",BIjC<%F-7$F_s$\",&e:\"HQ&F -7$Fds$\",B=LL,(F-7$Fis$\",[a-)H))F-7$F^t$\"-$))\\^B8\"F-7$Fct$\"-W[I: k9F-7$Fht$\"-+86l#)=F-7$F]u$\"-`\"e'4,CF-7$Fbu$\".Yfa/c9$FH7$Fgu$\".uy xr&4SFH7$F\\v$\".G#f!3b>&FH7$Fav$\"/i8caZqlFgn7$Ffv$\".!pF$yK\\)FH7$F[ w$\"/K$>*QN\"3\"FH7$F`w$\"/!\\\\B3>R\"FH7$Few$\"/g0!3%e\"y\"FH7$Fjw$\" .!4\"z2pI#F-7$F_x$\".=h$pzeHF-7$Fdx$\".\"y)QWj\"QF-7$Fix$\".hsRr?\"\\F -7$F^y$\".)fg!)G%>'F-7$Fcy$\".K`**)yuqF-7$Fhy$\".A^l]/3)F-7$F]z$\".*fE 3X+\"*F-7$Fbz$\"/4@@1#\\-\"F-7$Fgz$\"/VZeyTj6F-7$F\\[l$\"/&=s=I1K\"F-7 $Fa[l$\"/KJ4F'4\\\"F-7$Ff[l$\"/FJ:WE$o\"F-7$F[\\l$\"/f\"ezSmz\"F-7$F` \\l$\"/u^dOl<>F-7$Fe\\l$\"/[+Ut\"o/#F-7$Fj\\l$\"/sK`3o%=#F--F_]l6&Fa]l $\")#)eqkFe]l$\"))eqk\"Fe]lF^jn-Fg]l6#%3efficient~scheme~CG-F$6%7Z7$F( $!*#GGZ:F-7$F/$!*X(y**>F-7$F5$!*w.,]#F-7$F:$!*[bu@$F-7$F?$!*I0([TF-7$F D$!+%p_&[`FH7$FJ$!+!\\C5x'FH7$FO$!+-ksZ')FH7$FT$!,WKcS6\"FH7$FY$!-9iic M9Fgn7$Fin$!,1'y=h=FH7$F^o$!,#HXATBFH7$Fco$!,=kc;.$FH7$Fho$!,;:(\\IRFH 7$F]p$!+L " 0 "" {MPLTEXT 1 0 2 " ;" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 14 of 7 stage, order 5 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 10*x*cos*x-10*y;" "6#/*&%#dyG \"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F&%\"yGF&F(" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5);" "6#/-%\"yG6#\" \"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Sol ution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101 " "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\"$!**F&\"&,-\"! \"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+10/101" "6#,&*&%$cosG\" \"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x* sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F&F&*&\"$+#F&\"&,-\"!\"\" F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990/10201+sqrt(5))*exp(-10* x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F&\"&,-\"!\"\"F&-%%sqrtG6# \"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := \+ diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic \},y(x));\nb := unapply(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9 ],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-%$cosGF+F0F0*&F/F0F)F0!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"&#\"\"\"\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&#\"$+\"\"$,\" \"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,-\"F-F/F-!\"\"*&#\"#5F,F-*&-%$ sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$expG6#,$*&F8F-F'F-F5F-,&#F3F4F- *$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\")*y*\\xz1OA!#<7$$\"3ALL$ 3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$ \"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x&)*3\"!#=$\"3(**[ro!GyV zF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\"3umm\"H2P\"Q?F@$\"3/$)o qvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3!***\\PMnNrDF@$\"3M:4%* 3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3MLL$eRwX5$F@$\"3)GTJ!oG 0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33ML$3x%3yTF@$\"3n$**Q]` \"yRLF@7$$\"3emm\"z%4\\Y_F@$\"3%)*)G8T#p2%RF@7$$\"3`LLeR-/PiF@$\"3PZ.% R2Cm^%F@7$$\"3]***\\il'pisF@$\"31e'*fKlL9]F@7$$\"3>MLe*)>VB$)F@$\"3%3) yy-pAk`F@7$$\"3Y++DJbw!Q*F@$\"3Kg$RQBm7^&F@7$$\"3%ommTIOo/\"F,$\"3xrTB 'zz$GaF@7$$\"3YLL3_>jU6F,$\"3!p\\3Dp!RX^F@7$$\"37++]i^Z]7F,$\"3Kvxv\"* =%>e%F@7$$\"33++](=h(e8F,$\"3C^UQ-(*\\]PF@7$$\"3/++]P[6j9F,$\"3icMTpV \"zp#F@7$$\"3UL$e*[z(yb\"F,$\"3C-jVJM+L:F@7$$\"3wmm;a/cq;F,$!33XIKjA5+ 5F07$$\"3%ommmJF,$!3!\\F1VDv$)p&F@7$$\"3K+]i!f#=$3#F,$!3V$*[LN4F4!)F @7$$\"3?+](=xpe=#F,$!3$[6ral`?.\"F,7$$\"37nm\"H28IH#F,$!3s6ToLL*4G\"F, 7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F,$!3[Tb]!*H%e)HF,7$$\"3+++v =S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$!3N`)=cl>V?$F,7$$\"3B++](=] @W$F,$!3Y_BiN[odKF,7$$\"3mm\"H#oZ1\"\\$F,$!3)o4&)z%=-oKF,7$$\"35L$e*[$ z*RNF,$!3%)Q61)ek$pKF,7$$\"3%o;Hd!fX$f$F,$!3#*y!45Ut-E$F,7$$\"3e++]iC$ pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D1>x`HA5KF,7$$\"3[m;H2qcZPF ,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX&>E0$F,7$$\"3Ymm;/OgbRF,$! 3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!yEF,7$$\"3yLLL$)*pp;%F,$!30 ,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F,7$$\"3Cn;HdO=yVF,$!35Q!)* 4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$$\"3SnmT&G!e&e%F,$!3]X/0\" RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@7$$\"37+]iSjE!z%F,$!3+r[gf MO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$$\"\"&F)$\"3&Q8`\">jC3#*F@ -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 770 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: \+ x0 := 0: y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modification o f the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme \+ B`,`efficient scheme C`,`scheme with stability radius = imag axis incl usion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n Bn_RK5_||ct : = RK5_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: nu mpts := nops(Bn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Bn _RK5_||ct[ii,2]-evalf(b(Bn_RK5_||ct[ii,1])))^2;\n end do:\n errs : = [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpos e]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"y GF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~ ~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint796\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %8Bogacki-Shampine~schemeG$\"+W8C;m!#@7$%Ta~minor~modification~of~the~ Bogacki-Shampine~schemeG$\"+wW`;mF+7$%3efficient~scheme~AG$\"+wozIsF+7 $%3efficient~scheme~BG$\"+B7EC7!#?7$%3efficient~scheme~CG$\"+'fskO*F+7 $%Sscheme~with~stability~radius~=~imag~axis~inclusionG$\"+^[F,GF+Q)ppr int806\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999; " "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 694 "B := (x,y) -> 10*x*cos(x)-1 0*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`s lope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Bogacki-Shampi ne scheme`,`a minor modification of the Bogacki-Shampine scheme`,`effi cient scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with stability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20: \nfor ct to 6 do\n bn_RK5_||ct := RK5_||ct(B(x,y),x,y,x0,evalf(y0),h h,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfor ct \+ to 6 do\n errs := [op(errs),abs(bn_RK5_||ct(xx)-bxx)];\nend do:\nDig its := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xG F,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\" \"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprin t126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\")dy'\\)!#? 7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"),B6&)F+7$% 3efficient~scheme~AG$\")80)e%F+7$%3efficient~scheme~BG$\"*Mx'R5F+7$%3e fficient~scheme~CG$\")^TyiF+7$%Sscheme~with~stability~radius~=~imag~ax is~inclusionG$\"*X)fU7F+Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6 #7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "mthds := [`Bogacki-Shampine scheme`,`a minor modific ation of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient \+ scheme B`,`efficient scheme C`,`scheme with stability radius = imag ax is inclusion`]:\nerrs := []:\nDigits := 20:\nfor ct to 6 do\n sm := \+ NCint((b(x)-'bn_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,fact or=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nl inalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+#o0DR'!#@7$%Ta~min or~modification~of~the~Bogacki-Shampine~schemeG$\"+'3*y#R'F+7$%3effici ent~scheme~AG$\"+[32')pF+7$%3efficient~scheme~BG$\"+5tc#=\"!#?7$%3effi cient~scheme~CG$\"+'o6e/*F+7$%Sscheme~with~stability~radius~=~imag~axi s~inclusionG$\"+c?U-FF+Q)pprint856\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constr ucted using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 515 "evalf[20](plot(['bn_RK5_1'(x)-b(x) ,'bn_RK5_2'(x)-b(x),'bn_RK5_3'(x)-b(x),'bn_RK5_4'(x)-b(x),\n'bn_RK5_5' (x)-b(x),'bn_RK5_6'(x)-b(x)],x=0..0.65,numpoints=100,\ncolor=[blue,mag enta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],font=[HELVETICA, 9],\nlegend=[`Bogacki-Shampine scheme`,`a minor modification of the Bo gacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`effi cient scheme C`,`scheme with stability radius = imag axis inclusion`], \ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 811 476 476 {PLOTDATA 2 "6,-%'CURVESG6%7d\\ l7$$\"\"!F)F(7$$\"5SSSSSS:N<7$$\"5\"333333.ZV$F-$\"'1k8F0 7$$\"5+,,,,^)yLH%F-$\"'gA_F07$$\"5?@@@@@Y0_^F-$\"(!fj:F07$$\"5STTTT\"R I2,'F-$\"(Q#\\RF07$$\"5ihhhhhhSpoF-$\"(@_!))F07$$\"5XXXX&zNNlh(F-$\")` _N;F07$$\"5IHHHHaXmj$)F-$\")Z_kGF07$$\"55888j]Pz5\"*F-$\")7(zx%F07$$\" 5&ppppp%H#z&)*F-$\")!zvk(F07$$\"5HHHagC0Z/5!#@$\")EwC$)F07$$\"5*)))))Q ^a([J-\"F[o$\")!G#4$)F07$$\"5[[[BU%)p#=/\"F[o$\")Bs$H)F07$$\"53333L9_] g5F[o$\")fCy#)F07$$\"5FFFx9u;'y4\"F[o$\")^QZ#)F07$$\"5YYYY'R8=_8\"F[o$ \")@n;#)F07$$\"5%[[[)f`5$*47F[o$\")_5c\")F07$$\"5BBBBBtRk%G\"F[o$\")yu )4)F07$$\"53444Mlkmo8F[o$\")&Gf/)F07$$\"5%\\\\\\\\u&*)o_9F[o$\")I&[-)F 07$$\"5(yyy.N?+Z\\\"F[o$\")xJP!)F07$$\"5!333e&\\9rO:F[o$\")oF[o$\"*?XgL\"F07$$\"5GGGG.s(oz(>F[o$ \"*Q2^T\"F07$$\"5YYYY'Rj8\"**>F[o$\"*ZaK]\"F07$$\"5kkkk*e\\e--#F[o$\"* A4T]\"F07$$\"5$GGGGyN.9/#F[o$\"*;K4]\"F07$$\"5?>>>p\"3$p$3#F[o$\"*R)f% \\\"F07$$\"5cbbbb0G)f7#F[o$\"*d$H)[\"F07$$\"5GGGGG`Ac5AF[o$\"*!\\\"eZ \"F07$$\"5,,,,,,<9&H#F[o$\"*EvPY\"F07$$\"5utttt[6szBF[o$\"*`WIX\"F07$$ \"5YYYYY'f+VY#F[o$\"*]GcW\"F07$$\"5>>>>>W+))[DF[o$\"*>$RX9F07$$\"5#>>> >>\\fMj#F[o$\"*we*e9F07$$\"5aaaaH[pjt:F07$$\"5!)zzz/h=*f)GF[o$\"*[2/r\"F07$$\"5UUUUU<$p,(HF [o$\"*w\"*p$>F07$$\"5333$=:o87*HF[o$\"*j6=,#F07$$\"5uttBhX!eA,$F[o$\"* hUI/#F07$$\"5SRRkq4CILIF[o$\"*pZ(Q?F07$$\"50000!QxYV0$F[o$\"*!=YM?F07$ $\"5OOO'))>]Nk4$F[o$\"*YF07$$\"5YXXXX?C#\\Q$ F[o$\"*:7.(>F07$$\"5)zzzzzplHY$F[o$\"*k$4f>F07$$\"5]]]]]v*35a$F[o$\"*n eL&>F07$$\"5.....`A0>OF[o$\"*!Rnd>F07$$\"5333333)Q^x$F[o$\"*^^v-#F07$$ \"588888j`AJRF[o$\"*bNpE#F07$$\"5%[[[[)4+jrRF[o$\"*7fdP#F07$$\"5cccccc Y.7SF[o$\"*)\\zkCF07$$\"5GGGGG.$RC0%F[o$\"*:c[X#F07$$\"5+++++]R%G4%F[o $\"*vd\\W#F07$$\"5WVVVVVKltTF[o$\"*g%HDCF07$$\"5(ooooo`iWD%F[o$\"*L5fS #F07$$\"5uttttB63;WF[o$\"*`)3qBF07$$\"5hgggg5(*pxXF[o$\"*1\"4]BF07$$\" 533333e?*>\"\\F[o$\"*)QvuDF07$$\"5cbbbb0WGY_F[o$\"*[NKs#F07$$\"5SRRRRR X*GT&F[o$\"*PK1o#F07$$\"5DBBBBtY]zbF[o$\"*Q)4`EF07$$\"5522222[6YdF[o$ \"*tzzn#F07$$\"5\"44444%\\s7fF[o$\"*NZ.%GF07$$\"5XWW%>8ppb&fF[o$\"*./< #HF07$$\"5+)zzHf7/'F[o$\"*#[%y,$F07$ $\"50000bUR5%3'F[o$\"*wU\\+$F07$$\"55777PVMzphF[o$\"*?9$zHF07$$\"5?>>> >WH[biF[o$\"*,4S&HF07$$\"5NLLL$e%>'oU'F[o$\"*]ck!HF07$$\"5[ZZZZZ4C)f'F [o$\"*l+a(GF07$$\"5NLLLL$QU,!pF[o$\"*BZK,$F07$$\"5?>>>>>Q/-sF[o$\"*2YZ 8$F07$$\"5+++++v&y>P(F[o$\"*\\\"*H3$F07$$\"5!33333L8>a(F[o$\"*.x//$F07 $$\"5?@@@r32)oi(F[o$\"*cZ&HIF07$$\"5ghhhh'3[=r(F[o$\"*d$*H.$F07$$\"5+- --_ka\"oz(F[o$\"*IE+1$F07$$\"5VUUUUUGy\")yF[o$\"*^yO7$F07$$\"5baaaa/\" *4nzF[o$\"*:'GUKF07$$\"5lmmmmm`T_!)F[o$\"+;WJ6QJFigl7$$\"5][[[[[]=$*))F[o $\"+O=F:KFigl7$$\"5edddddr0A#*F[o$\"+c(RDH$Figl7$$\"5&eeee$[3Qr$*F[o$ \"+t_eWKFigl7$$\"5:9999RXq?&*F[o$\"+'>%Q-KFigl7$$\"5IGGG`%Qm`f*F[o$\"+ $f\")p=$Figl7$$\"5XUUU#*H#G+n*F[o$\"+=#=!zJFigl7$$\"5]\\\\*>E:ftq*F[o$ \"+vl6zJFigl7$$\"5gcccJv+pW(*F[o$\"+>&=G=$Figl7$$\"5ljj8,)*4-#y*F[o$\" +'oF4>$Figl7$$\"5rqqqq?>N>)*F[o$\"+vRP/KFigl7$$\"5gdddKw?83**F[o$\"+=X NkKFigl7$$\"5XWWW%>B7p***F[o$\"+6m$)yLFigl7$$\"5888jvQ#p&35Figl$\"+uV> bLFigl7$$\"5#====VDZu,\"Figl$\"+HraDLFigl7$$\"5?>>>W&G._.\"Figl$\"+_9f nKFigl7$$\"5dcccc;$fH0\"Figl$\"+X>G;KFigl7$$\"5yxxxFg,[g5Figl$\"+*=,0? $Figl7$$\"5+**)*)*)R+,!o5Figl$\"+>$Figl7$$\"5gfff%eUh<2\"Figl$\"+H fr\">$Figl7$$\"5@???qZ=_v5Figl$\"+v3%\\>$Figl7$$\"5#333e&pAGz5Figl$\"+ 2KW-KFigl7$$\"5UTTTT\"pUI3\"Figl$\"+qy6:KFigl7$$\"5jiii7NNc!4\"Figl$\" +)yK-E$Figl7$$\"5%QQQQ)yV3)4\"Figl$\"+OqFSLFigl7$$\"5999kw*ek**4\"Figl $\"+\"GjrO$Figl7$$\"5WWWWp+[%=5\"Figl$\"+VC\\hLFigl7$$\"5uuuCi6]s.6Fig l$\"+n!y^N$Figl7$$\"50000bA_g06Figl$\"+gb()[LFigl7$$\"5mlllSWcO46Figl$ \"+(>1jL$Figl7$$\"5EEEEEmg786Figl$\"+Y]yBLFigl7$$\"5LLLLeCJiI6Figl$\"+ :(fjE$Figl7$$\"5SSSS!H=?\"[6Figl$\"+u3#H@$Figl7$$\"5%RRRk?roo:\"Figl$ \"+*R:4>$Figl7$$\"5[ZZZATshl6Figl$\"+Yd*f<$Figl7$$\"5DCC\\!e]\"**p6Fig l$\"++dssJFigl7$$\"5-,,^QqdOu6Figl$\"+XCNtJFigl7$$\"5yxx_'\\.S(y6Figl$ \"+'=;*yJFigl7$$\"5baaaa*H9J=\"Figl$\"+39l!>$Figl7$$\"5'eeeL-Be3>\"Fig l$\"+n#o6B$Figl7$$\"5=<<<#4;-')>\"Figl$\"+1#HcI$Figl7$$\"5%GGyli7uC?\" Figl$\"+P]S:LFigl7$$\"5\\[[)4;4Yj?\"Figl$\"+RFf-LFigl7$$\"5999R&p0=-@ \"Figl$\"+--$)*G$Figl7$$\"5!)zzzHA+497Figl$\"+p$=rF$Figl7$$\"5UUUUn$)y dH7Figl$\"+\"==pA$Figl7$$\"500000Xd1X7Figl$\"+RjDzJFigl7$$\"5uttBOk?c` 7Figl$\"+b=?cJFigl7$$\"5UUUUn$Qe?E\"Figl$\"+^O(z8$Figl7$$\"5666h)Hqa0F \"Figl$\"+uw>GJFigl7$$\"5!)zzzHA50z7Figl$\"+zNHKJFigl7$$\"5\\[[)4;MZvG \"Figl$\"+Tt&z:$Figl7$$\"5=<<<#4mVgH\"Figl$\"+Ayn:KFigl7$$\"5NM%o\\2un \")H\"Figl$\"+V`lOKFigl7$$\"5_^^wd?=H+8Figl$\"+68GcKFigl7$$\"5po=cS+fT -8Figl$\"+[?P\\KFigl7$$\"5'eeeL-)*RXI\"Figl$\"+^uZUKFigl7$$\"5???&*))R \")y38Figl$\"+1AtGKFigl7$$\"5baaaa*HOIJ\"Figl$\"+9h/:KFigl7$$\"5IIIII? $)RX8Figl$\"+!*Rk9JFigl7$$\"511111T.wx8Figl$\"+sPujIFigl7$$\"5555gZM>? 'Q\"Figl$\"+(Gp73$Figl7$$\"59999*y_VYR\"Figl$\"+%Q]d7$Figl7$$\"5::l_CE Rv'R\"Figl$\"+2$*QUJFigl7$$\"5;;;\"*fCV'))R\"Figl$\"+z^khJFigl7$$\"5<< nH&Hsu4S\"Figl$\"+K#y+<$Figl7$$\"5===oI@^3.9Figl$\"+g^RjJFigl7$$\"5??? X,=fI29Figl$\"+!Gr+:$Figl7$$\"5AAAAs9n_69Figl$\"+FR!o8$Figl7$$\"5IIIIb ,*4%G9Figl$\"+i_R%3$Figl7$$\"5QQQQQ)3$HX9Figl$\"+(H;W.$Figl7$$\"5]]]]D Crzg9Figl$\"+0CN'*HFigl7$$\"5iiii7g6Iw9Figl$\"+iLb!)HFigl7$$\"5uuuu*f> 0=\\\"Figl$\"+AwQR')GFigl7$$\"5!*)))))Q Ess:f\"Figl$\"+i#[`\"HFigl7$$\"53222dCM%*3;Figl$\"+Nw!3%HFigl7$$\"5EDD D]ETJE;Figl$\"+L&>-*GFigl7$$\"5WVVVVG[oV;Figl$\"+C4bTGFigl7$$\"5/...`v g!)e;Figl$\"+KCo/GFigl7$$\"5jiiiiAt#Rn\"Figl$\"+')z6$y#Figl7$$\"5AAAAs p&[!*o\"Figl$\"+_Fz(z#Figl7$$\"5#====o\")pTq\"Figl$\"+)=\"oRGFigl7$$\" 5cccc1RD$ot\"Figl$\"+\"z,)[FFigl7$$\"5JJJJJh_\\ps#Figl7 $$\"5,,,,^V46?=Figl$\"+N!zjn#Figl7$$\"5eddddPG)p$=Figl$\"+n#R>j#Figl7$ $\"5#====Vp)[`=Figl$\"+*\\*p\"f#Figl7$$\"511111^X**p=Figl$\"+$G'=iDFig l7$$\"5====VzuCy=Figl$\"+?(Qlb#Figl7$$\"5IIII!yS+l)=Figl$\"+quehDFigl7 $$\"5UUUUFigl$\"+^t)*)f#Figl7 $$\"5GFFFx%zY\\$>Figl$\"+Vx]Figl$\"+X[<\\CFigl7 $$\"5:::lF`Mvv>Figl$\"+we!*RCFigl7$$\"5IIIIb\"e>Y)>Figl$\"+Kb\\SCFigl7 $$\"5XXX&H)4d[$*>Figl$\"+ez?cCFigl7$$\"5gggg5Q=N-?Figl$\"+7#4cZ#Figl7$ $\"5!444fY4%3??Figl$\"+;^5KCFigl7$$\"5@@@@@^j\"y.#Figl$\"+BGn*Q#Figl7$ $\"5!**)*)*)Rs&\\P0#Figl$\"+y3DaBFigl7$$\"5eeeee$z#op?Figl$\"+0D7FBFig l7$$\"5#HHHzTS\\w2#Figl$\"+hPR?BFigl7$$\"5FFFFx9gh&3#Figl$\"+Yhb@BFigl 7$$\"5ihhhODEe$4#Figl$\"+UYBMBFigl7$$\"5'fffffB\\:5#Figl$\"+1EN_BFigl7 $$\"5&\\\\\\\\\\Ytb8#Figl$\"+f@'QF#Figl7$$\"5%RRRRRp(fp@Figl$\"+zb(*3A Figl7$$\"5$HHHagW0t<#Figl$\"+H'*4-AFigl7$$\"5#>>>p\")>8]=#Figl$\"+b_x, AFigl7$$\"5\"444%G]4s#>#Figl$\"+&=:4@#Figl7$$\"5!**)*)*)R-(G/?#Figl$\" +sGnIAFigl7$$\"5)yyyGm?We@#Figl$\"+-5b'>#Figl7$$\"5'eeee3rf7B#Figl$\"+ '3JI;#Figl7$$\"5VUUUU#es')H#Figl$\"+`:-0@Figl7$$\"5'ffff4%G5iBFigl$\"+ wW)o)>Figl7$$\"5tssssPdSGCFigl$\"+Mk*Q$>Figl7$$\"5RQQQQQ*GK\\#Figl$\"+ ]_Vg=Figl7$$\"5(pppp>#G4hDFigl$\"+!G$flQ\\?GFigl$\"+4/M;:Figl7$$\"5gffff\\^I!*GFigl$\"+o#=) R9Figl7$$\"5cbbbbN_u_HFigl$\"+:^Tu8Figl7$$\"5YXXXX!R>$>IFigl$\"+XNtE8F igl7$$\"5POOOO^$RI3$Figl$\"+77Bd7Figl7$$\"5**)*)*)*)*y@z`JFigl$\",x&** e'>\"F[o7$$\"5%RRRR*G*e]@$Figl$\",DH0\"e6F[o7$$\"588888Qsf%G$Figl$\",? h#H\"4\"F[o7$$\"5)yyyyy&Q(zM$Figl$\",zwII/\"F[o7$$\"5&[[[[)>zHR1#***F[o7$$\"5JJJJJhG6xMFigl$\"+$z**3Y*F[o7$$\"5edddd(4N_a$Figl$ \"+&Q&R?!*F[o7$$\"5vuuuu9y.6OFigl$\"+Q-B^')F[o7$$\"5gffffWvzwOFigl$\"+ ^,KP\")F[o7$$\"5EEEEEc`JUPFigl$\"+[\"fQw(F[o7$$\"5<<<<CSF[o7$$\"5lkkkk/h`(f%Figl$\"+x#yN!QF[ o7$$\"5)yyyyG^aKm%Figl$\"+`nynNF[o7$$\"5FEEEET%)3IZFigl$\"+)GP3O$F[o7$ $\"5#>>>>p#=_\"z%Figl$\"+dO=oJF[o7$$\"5nmmmmEy+d[Figl$\"+Ue!\\(HF[o7$$ \"5tssssU`*>#\\Figl$\"+/-.&z#F[o7$$\"5NMMMM%y6:*\\Figl$\"+m?30EF[o7$$ \"5jiiiiKq&G0&Figl$\"+&*f#4X#F[o7$$\"5XWWWW9njB^Figl$\"+:\">hF#F[o7$$ \"5TSSSS?*4v=&Figl$\"+@%)[E@F[o7$$\"5/////>\"42D&Figl$\"+Ecy*)>F[o7$$ \"5^]]]]g&o'=`Figl$\"+\"p/\"R=F[o7$$\"5#>>>>>\"H!pQ&Figl$\"+H))*Rq\"F[ o7$$\"5WVVVVog)*[aFigl$\",&Qe@)e\"F-7$$\"5$HHHHHi:\\^&Figl$\",`S2gX\"F -7$$\"5XWWWWCkDzbFigl$\",=SERN\"F-7$$\"522222KD+\\cFigl$\",2v=sB\"F-7$ $\"5nmmmm1BN4dFigl$\",!)fNW7\"F-7$$\"5#====o@m'ydFigl$\",-gRg.\"F-7$$ \"5KJJJJO\\#Q%eFigl$\"+h!*)eQ*F-7$$\"5kjjjj$Ha$3fFigl$\"+$GS3J)F-7$$\" 5baaaa%)\\$H(fFigl$\"+dA%=p(F-7$$\"5POOOO,K!)QgFigl$\"+FcAxnF-7$$\"5ts sss2Ci3hFigl$\"+4=bedF-7$$\"5.....$yRE<'Figl$\"+G4*QH&F-7$$\"5onnnn_J$ eB'Figl$\"+&>C#)[%F-7$$\"5!)zzzzCeQ.jFigl$\"+*Q&*pc$F-7$$\"5mllllS9zqj Figl$\"+aO@LKF-7$$\"5utttt$)>HJkFigl$\"+_h[(\\#F-7$$\"#l!\"#$\"+S!3]k \"F--%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine ~schemeG-F$6%7d\\lF'F*F17$F7$\"'fA_F07$F<$\"()ej:F07$FA$\"(K#\\RF07$FF $\"(2_!))F07$FK$\")G_N;F07$FP$\")._kGF07$FU$\")Q'zx%F07$FZ$\")rcZwF07$ Fin$\")'\\ZK)F07$F_o$\")]@4$)F07$Fdo$\")%4PH)F07$Fio$\")IBy#)F07$F^p$ \")BPZ#)F07$Fcp$\")$fm@)F07$Fhp$\")D4c\")F07$F]q$\")^t)4)F07$Fbq$\")g \"f/)F07$Fgq$\")/%[-)F07$F\\r$\")_IP!)F07$Far$\")T;u!)F07$Ffr$\")6tW\" )F07$F[s$\")1wg#)F07$F`s$\")6s!p)F07$Fes$\")U$=_*F07$Fjs$\"*oIx4\"F07$ F_t$\"*$G/O8F07$Fdt$\"*%[5:9F07$Fit$\"*u^K]\"F07$F^u$\"*\\1T]\"F07$Fcu $\"*WH4]\"F07$Fhu$\"*o&f%\\\"F07$F]v$\"*(3H)[\"F07$Fbv$\"*B7eZ\"F07$Fg v$\"*hsPY\"F07$F\\w$\"*!>/`9F07$Faw$\"*)eiX9F07$Ffw$\"*c!RX9F07$F[x$\" *3c*e9F07$F`x$\"*G(['\\\"F07$Fex$\"*4*=t:F07$Fjx$\"*6//r\"F07$F_y$\"*x ()p$>F07$Fdy$\"*V2=,#F07$Fiy$\"*LQI/#F07$F^z$\"*TV(Q?F07$Fcz$\"*`dW.#F 07$Fhz$\"*A8f-#F07$F][l$\"*93u,#F07$Fb[l$\"*on0+#F07$Fg[l$\"*8PU)>F07$ F\\\\l$\"*,3.(>F07$Fa\\l$\"*`*3f>F07$Ff\\l$\"*caL&>F07$F[]l$\"*wpw&>F0 7$F`]l$\"*5Zv-#F07$Fe]l$\"*IIpE#F07$Fj]l$\"*\\`dP#F07$F_^l$\"*/*ykCF07 $Fd^l$\"*B][X#F07$Fi^l$\"*&=&\\W#F07$F^_l$\"*v)GDCF07$Fc_l$\"*_/fS#F07 $Fh_l$\"*\"G3qBF07$F]`l$\"*O&3]BF07$Fb`l$\"*6ZZd#F07$Fg`l$\"*&zABFF07$ F\\al$\"*&\\i!o#F07$Faal$\"*+\"4`EF07$Ffal$\"*8szn#F07$F[bl$\"*#)Q.%GF 07$F`bl$\"*0&p@HF07$Febl$\"*bgf-$F07$Fjbl$\"*HNy,$F07$F_cl$\"*GL\\+$F0 7$Fdcl$\"*z/$zHF07$Ficl$\"*o**R&HF07$F^dl$\"*JZk!HF07$Fcdl$\"*^\"RvGF0 7$Fhdl$\"*)pB8IF07$F]el$\"*![tMJF07$Fbel$\"*S!)H3$F07$Fgel$\"*1m//$F07 $F\\fl$\"*dO&HIF07$Fafl$\"*Y#)H.$F07$Fffl$\"**[,gIF07$F[gl$\"*`mO7$F07 $F`gl$\"*=tAC$F07$Fegl$\"+w4;!H$Figl7$F[hl$\"+mC@iKFigl7$F`hl$\"+a>aMK Figl7$Fehl$\"+IQ;\"=$Figl7$Fjhl$\"+Y-5QJFigl7$F_il$\"+%yd_@$Figl7$Fdil $\"+?X_#H$Figl7$Fiil$\"+a-dWKFigl7$F^jl$\"+Q$pB?$Figl7$Fcjl$\"+en'p=$F igl7$Fhjl$\"+FL+zJFigl7$F][m$\"+8;5zJFigl7$Fb[m$\"+[M!G=$Figl7$Fg[m$\" +gC\"4>$Figl7$F\\\\m$\"+O&eV?$Figl7$Fa\\m$\"+u#QVE$Figl7$Ff\\m$\"+n*=) yLFigl7$F[]m$\"+=o>$Figl7$Fi^m$\"+D*)p\">$ Figl7$F^_m$\"+XP#\\>$Figl7$Fc_m$\"+&*eU-KFigl7$Fh_m$\"+8.5:KFigl7$F]`m $\"+)[9-E$Figl7$Fb`m$\"+VvDSLFigl7$Fg`m$\"+,M9nLFigl7$F\\am$\"+#fs9O$F igl7$Faam$\"+`#e^N$Figl7$Ffam$\"+%yb)[LFigl7$F[bm$\"+&\\'GOLFigl7$F`bm $\"+=awBLFigl7$Febm$\"+C/MmKFigl7$Fjbm$\"+q=!H@$Figl7$F_cm$\"+tk*3>$Fi gl7$Fdcm$\"+&zwf<$Figl7$Ficm$\"+wmqsJFigl7$F^dm$\"+*GLL<$Figl7$Fcdm$\" +Go*)yJFigl7$Fhdm$\"+a$Figl7$F]em$\"+%zZ6B$Figl7$Fbem$\"+*R2cI$Fig l7$Fgem$\"+vHQ:LFigl7$F\\fm$\"+j2d-LFigl7$Fafm$\"+6$3)*G$Figl7$Fffm$\" +jl4xKFigl7$F[gm$\"+2n*oA$Figl7$F`gm$\"+i^BzJFigl7$Fegm$\"+!z!=cJFigl7 $Fjgm$\"+?P1$Figl7$F[\\n$\"+A]C\"3$Figl7$F `\\n$\"+y[sDJFigl7$Fe\\n$\"+wLOUJFigl7$Fj\\n$\"+o(=;;$Figl7$F_]n$\"+l: 0qJFigl7$Fd]n$\"+]&oL;$Figl7$Fi]n$\"+\"yW+:$Figl7$F^^n$\"+SvxOJFigl7$F c^n$\"+9$pV3$Figl7$Fh^n$\"+V2RMIFigl7$F]_n$\"+TqK'*HFigl7$Fb_n$\"+sw_! )HFigl7$Fg_n$\"+a.Oj#Figl7$F[fn$\"+L]m\"f#F igl7$F`fn$\"+E=:iDFigl7$Fefn$\"+&)Q]cDFigl7$Fjfn$\"+'z^:c#Figl7$F_gn$ \"+K'pBe#Figl7$Fdgn$\"+9)[*)f#Figl7$Fign$\"+1/ZMBFigl7$Fj[o$\"+(44BN#Figl7$F_\\o$\"+$4?QF#Figl7$Fd\\o$\"+$=M*3A Figl7$Fi\\o$\"+')y0-AFigl7$F^]o$\"+NFt,AFigl7$Fc]o$\"+h7(3@#Figl7$Fh]o $\"+NoiIAFigl7$F]^o$\"+pc]'>#Figl7$Fb^o$\"+Sk)H;#Figl7$Fg^o$\"+\"[t\\5 #Figl7$F\\_o$\"+Y%Qo)>Figl7$Fa_o$\"+9o%Q$>Figl7$Ff_o$\"+=RQg=Figl7$F[` o$\"+4Dal\"F[o7$F]co$\",N@N!e6F [o7$Fbco$\",g%QA\"4\"F[o7$Fgco$\",RXgH/\"F[o7$F\\do$\"+zaJ\"***F[o7$Fa do$\"+tV!*F[o7$F[eo$\"+)zF/l)F[o7$F`eo$\"+,zaO\") F[o7$Feeo$\"+Qq0jxF[o7$Fjeo$\"+vF[o7$F]]p$\"+J!4z$=F[o7$Fb]p$\"+*QAG q\"F[o7$Fg]p$\",&oB.(e\"F-7$F\\^p$\",`uiZX\"F-7$Fa^p$\",=+CFN\"F-7$Ff^ p$\",2a#*fB\"F-7$F[_p$\",!3*RJ7\"F-7$F`_p$\",--'zM5F-7$Fe_p$\"+h/:t$*F -7$Fj_p$\"+$QOuH)F-7$F_`p$\"+d30zwF-7$Fd`p$\"+F1,knF-7$Fi`p$\"+4[sWdF- 7$F^ap$\"+Gbp!G&F-7$Fcap$\"+&RgXZ%F-7$Fhap$\"+*ouEb$F-7$F]bp$\"+a$G'>K F-7$Fbbp$\"+_hN$[#F-7$Fgbp$\"+STAI;F--F]cp6&F_cpF`cpF(F`cp-Fdcp6#%Ta~m inor~modification~of~the~Bogacki-Shampine~schemeG-F$6%7d\\lF'7$F+$\"%W BF07$F2$\"'$R^\"F07$F7$\"'&Rz&F07$F<$\"(\"4M8=F07$FP$\")d&e<$F07$FU$\")A!zH&F07$FZ$\")_U\"[)F07$Fi n$\")W'GB*F07$F_o$\")_j:#*F07$Fdo$\")\"Q%)>*F07$Fio$\")MF\"=*F07$F^p$ \")j/Z\"*F07$Fcp$\")J)H6*F07$Fhp$\").\"e/*F07$F]q$\"))*>#)*)F07$Fbq$\" )oiB*)F07$Fgq$\")dD+*)F07$F\\r$\")D29*)F07$Far$\")7$\\&*)F07$Ffr$\")Y: L!*F07$F[s$\")?xh\"*F07$F`s$\")6RQ'*F07$Fes$\"*()zf0\"F07$Fjs$\"*'\\P< 7F07$F_t$\"*vB<[\"F07$Fdt$\"*^X%p:F07$Fit$\"*;fsm\"F07$F^u$\"*k4#o;F07 $Fcu$\"*(fok;F07$Fhu$\"*MF07$Fgel$\"*4vFP$F07$ F\\fl$\"*r`1O$F07$Fafl$\"*()yWO$F07$Fffl$\"*EtWR$F07$F[gl$\"*Q6^Y$F07$ F`gl$\"*Lxnf$F07$Fegl$\"+w-(*\\OFigl7$F[hl$\"+O`'*=OFigl7$F`hl$\"+9*o# )e$Figl7$Fehl$\"+![b!HNFigl7$Fjhl$\"+%p*G\"[$Figl7$F_il$\"+UN'pc$Figl7 $Fdil$\"+J'\\Gl$Figl7$Fiil$\"+p!\\'*f$Figl7$F^jl$\"++k$Gb$Figl7$Fcjl$ \"+cWvNNFigl7$Fhjl$\"+)yGp_$Figl7$F][m$\"+wU/FNFigl7$Fb[m$\"+]'f6`$Fig l7$Fg[m$\"+%[n,a$Figl7$F\\\\m$\"+T45bNFigl7$Fa\\m$\"+Q`r@OFigl7$Ff\\m$ \"+=/*)[PFigl7$F[]m$\"++pmAPFigl7$F`]m$\"+@Ex*o$Figl7$Fe]m$\"+14ZDOFig l7$Fj]m$\"+[4boNFigl7$F_^m$\"+X00^NFigl7$Fd^m$\"+P`iTNFigl7$Fi^m$\"+Fl KTNFigl7$F^_m$\"+@t\"\\a$Figl7$Fc_m$\"+k!eKb$Figl7$Fh_m$\"+VMMnNFigl7$ F]`m$\"+0FZEJ@NFigl7$Fcdm $\"+![4v_$Figl7$Fhdm$\"+bHcSNFigl7$F]em$\"+c6h&e$Figl7$Fbem$\"+$R)RoOF igl7$Fgem$\"+jjGzOFigl7$F\\fm$\"+by1lOFigl7$Fafm$\"+qX!4l$Figl7$Fffm$ \"+jvzOOFigl7$F[gm$\"+\\$)3\"e$Figl7$F`gm$\"+,5?GNFigl7$Fegm$\"+o[i-NF igl7$Fjgm$\"+d@T#[$Figl7$F_hm$\"++:frMFigl7$Fdhm$\"+rh=wMFigl7$Fihm$\" +\"4aZ]$Figl7$F^im$\"+<(o*oNFigl7$Fcim$\"+ZfI#f$Figl7$Fhim$\"+,G99OFig l7$F]jm$\"+=UZ1OFigl7$Fbjm$\"+8>#))f$Figl7$Fgjm$\"+#3mNe$Figl7$F\\[n$ \"+7fPoNFigl7$Fa[n$\"+]p%pX$Figl7$Ff[n$\"+!ei0S$Figl7$F[\\n$\"+Tg5?MFi gl7$F`\\n$\"+E`jpMFigl7$Fe\\n$\"+P)e\")[$Figl7$Fj\\n$\"+``f4NFigl7$F_] n$\"+(H$**=NFigl7$Fd]n$\"+_Yd6NFigl7$Fi]n$\"+yUy'\\$Figl7$F^^n$\"+bm0# [$Figl7$Fc^n$\"+]-)QU$Figl7$Fh^n$\"+*f2%oLFigl7$F]_n$\"+=J=ELFigl7$Fb_ n$\"+!=I(3LFigl7$Fg_n$\"+\\M%)\\LFigl7$F\\`n$\"+#='f*Q$Figl7$Fa`n$\"+ \\H/zKFigl7$Ff`n$\"+JSv/KFigl7$F[an$\"+7y;PKFigl7$F`an$\"+j^plKFigl7$F ean$\"+r(=&4KFigl7$Fjan$\"+V2[bJFigl7$F_bn$\"+a)pX6$Figl7$Fdbn$\"+Sar! 4$Figl7$Fibn$\"+#*[C2JFigl7$F^cn$\"+_&3T:$Figl7$Fccn$\"+\"opJ0$Figl7$F hcn$\"+9DntHFigl7$F]dn$\"+(*fknHFigl7$Fbdn$\"+iI?vHFigl7$Fgdn$\"+(*p)H +$Figl7$F\\en$\"+`Q2CIFigl7$Faen$\"+S**[tHFigl7$Ffen$\"+$G?T#HFigl7$F[ fn$\"+ptVzGFigl7$F`fn$\"+c*Qn%GFigl7$Fefn$\"+cIcSGFigl7$Fjfn$\"++7LYGF igl7$F_gn$\"+j.rpGFigl7$Fdgn$\"+-EN))GFigl7$Fign$\"+CS!yz#Figl7$F^hn$ \"+t9'>s#Figl7$Fchn$\"+e$e&*y.FFigl7$F\\jn$\"+HJ\"e#Fi gl7$Fe[o$\"+\"*3n&f#Figl7$Fj[o$\"+8X6;EFigl7$F_\\o$\"+jm#)GDFigl7$Fd\\ o$\"+Jz!oX#Figl7$Fi\\o$\"+bXF\\CFigl7$F^]o$\"+r,4\\CFigl7$Fc]o$\"+p*=& fCFigl7$Fh]o$\"+k2&=[#Figl7$F]^o$\"+Un)QW#Figl7$Fb^o$\"+_Tf1CFigl7$Fg^ o$\"+fX5VBFigl7$F\\_o$\"+R)o<@#Figl7$Fa_o$\"+D1(Q:#Figl7$Ff_o$\"+1E$H2 #Figl7$F[`o$\"+'zCx'>Figl7$F``o$\"+2O;7>Figl7$Fe`o$\"++/*G$=Figl7$Fj`o $\"+A\\#)RCy,\"F[o7$F[eo$\"+)*=4$y*F[ o7$F`eo$\"+J()[2#*F[o7$Feeo$\"+)3\"p,))F[o7$Fjeo$\"+&G)Ro%)F[o7$F_fo$ \"+TCXRzF[o7$Fdfo$\"+av[\"f(F[o7$Fifo$\"+J1+_sF[o7$F^go$\"+QJoJoF[o7$F cgo$\"+k%=p]'F[o7$Fhgo$\"+8=(=A'F[o7$F]ho$\"+G%*yQeF[o7$Fbho$\"+EeE`bF [o7$Fgho$\"+NH7-`F[o7$F\\io$\"+D%zu)\\F[o7$Faio$\"+P(4Ts%F[o7$Ffio$\"+ (4mL\\%F[o7$F[jo$\"+B#=8A%F[o7$F`jo$\"+otW1SF[o7$Fejo$\"+n$G')z$F[o7$F jjo$\"+#Q'*4e$F[o7$F_[p$\"+C!3&)R$F[o7$Fd[p$\"+O'48>$F[o7$Fi[p$\"+N')G F-7$Fa^p$\",=LzU%=F-7$Ff^p$\",2*p'>s\"F-7$F[_p$\",![)F-7$F]bp$\"+aaEDwF-7$Fbbp$\"+_#[f(pF-7$Fgbp$\"+S,4XiF- -%&COLORG6&F_cp$\"\"\"!\"\"$\"\")F[]wF(-Fdcp6#%3efficient~scheme~AG-F$ 6%7d\\lF'7$F+$\"%%f$F07$F2$\"'1kBF07$F7$\"'@L\"*F07$F<$\"(J*fFF07$FA$ \"(y\"QqF07$FF$\")k)[e\"F07$FK$\")s>qHF07$FP$\")K4]_F07$FU$\")HGS))F07 $FZ$\"*_\"F07$F[s$\"*q2Ea\"F07$F`s$\"*Cz/i\"F07$Fes$\"*+)H t]`k#F07$Fit$\"*x(z8GF0 7$F^u$\"*rjb\"GF07$Fcu$\"*a;'4GF07$Fhu$\"*@gxz#F07$F]v$\"*qdfy#F07$Fbv $\"*n*eiFF07$Fgv$\"*-'**RFF07$F\\w$\"*E0(>FF07$Faw$\"*+'H0FF07$Ffw$\"* 3zPq#F07$F[x$\"*ipts#F07$F`x$\"**f=&z#F07$Fex$\"*tej$HF07$Fjx$\"*NUE>$ F07$F_y$\"*\"[u@OF07$Fdy$\"*%yfkPF07$Fiy$\"*ZyW#QF07$F^z$\"*dQk\"QF07$ Fcz$\"*f:%3QF07$Fhz$\"*#4U#z$F07$F][l$\"*8*\\wPF07$Fb[l$\"*Qj\\u$F07$F g[l$\"*\"3L9PF07$F\\\\l$\"*yn!)o$F07$Fa\\l$\"*g?mm$F07$Ff\\l$\"*3=]l$F 07$F[]l$\"*wv;m$F07$F`]l$\"*-'R)y$F07$Fe]l$\"*azoB%F07$Fj]l$\"*IbPW%F0 7$F_^l$\"*V$49YF07$Fd^l$\"*&z[&f%F07$Fi^l$\"*4epd%F07$F^_l$\"*BY,a%F07 $Fc_l$\"*GQQ]%F07$Fh_l$\"*XSkV%F07$F]`l$\"*#oS(R%F07$Fb`l$\"**\\78[F07 $Fg`l$\"*RH!)4&F07$F\\al$\"*ci&F07$Fdcl$\"*GQwd&F07$Ficl$\"*PY-`&F07$F^dl$\"*D-4 W&F07$Fcdl$\"*(o@\"Q&F07$Fhdl$\"*m,aj&F07$F]el$\"*6s)oeF07$Fbel$\"*'o$ =x&F07$Fgel$\"*]I8p&F07$F\\fl$\"*9?)pcF07$Fafl$\"*wB[n&F07$Fffl$\"*YvR s&F07$F[gl$\"*UZF%eF07$F`gl$\"*@?x1'F07$Fegl$\"+'3x,;'Figl7$F[hl$\"+w& [y5'Figl7$F`hl$\"+MU.cgFigl7$Fehl$\"+!QBf&fFigl7$Fjhl$\"+INJueFigl7$F_ il$\"++N@:gFigl7$Fdil$\"+5'e\\;'Figl7$Fiil$\"+O#e]2'Figl7$F^jl$\"+_dX& *fFigl7$Fcjl$\"+$4qf'fFigl7$Fhjl$\"+Kr8]fFigl7$F][m$\"+^Hy\\fFigl7$Fb[ m$\"+DS;cfFigl7$Fg[m$\"+Wz$3(fFigl7$F\\\\m$\"+#*Qi&*fFigl7$Fa\\m$\"+aF ?3hFigl7$Ff\\m$\"+@YrEjFigl7$F[]m$\"+#30FG'Figl7$F`]m$\"+>$)=FiFigl7$F e]m$\"+7ze=hFigl7$Fj]m$\"+$G+>-'Figl7$F_^m$\"+2Xt\"*fFigl7$Fd^m$\"+_C% \\(fFigl7$Fi^m$\"+$fHR(fFigl7$F^_m$\"+#px%zfFigl7$Fc_m$\"+4<4$*fFigl7$ Fh_m$\"+.v_;gFigl7$F]`m$\"+vXI,hFigl7$Fb`m$\"+$)e#RD'Figl7$Fg`m$\"+PMe 0jFigl7$F\\am$\"+8:*\\H'Figl7$Faam$\"+An;$G'Figl7$Ffam$\"+cTOriFigl7$F [bm$\"+id#yC'Figl7$F`bm$\"+xuPCiFigl7$Febm$\"+Yk!o6'Figl7$Fjbm$\"+RiQ; gFigl7$F_cm$\"+w.oufFigl7$Fdcm$\"+'=Rf%fFigl7$Ficm$\"+1*>$RfFigl7$F^dm $\"+f!z*RfFigl7$Fcdm$\"+U<$*\\fFigl7$Fhdm$\"+WbfrfFigl7$F]em$\"+\"H9y/ 'Figl7$Fbem$\"+c!>+>'Figl7$Fgem$\"+,^C4iFigl7$F\\fm$\"+H'\\_='Figl7$Fa fm$\"+4tMhhFigl7$Fffm$\"+>(Rv8'Figl7$F[gm$\"+q!*\\VgFigl7$F`gm$\"+<&=S &fFigl7$Fegm$\"+yi\\5fFigl7$Fjgm$\"+J]wveFigl7$F_hm$\"+/:jceFigl7$Fdhm $\"+(ymM'eFigl7$Fihm$\"+t;I6fFigl7$F^im$\"+^>2@gFigl7$Fcim$\"+m(=81'Fi gl7$Fhim$\"+-'Figl7$Fa[n$\"+'oIO$eFigl7$Ff[n$\"+#e(3OdFi gl7$F[\\n$\"+*z\"fodFigl7$F`\\n$\"+?S0`eFigl7$Fe\\n$\"+MV$\\)eFigl7$Fj \\n$\"+w:'>#fFigl7$F_]n$\"+$[!RQfFigl7$Fd]n$\"+I8(e#fFigl7$Fi]n$\"+$>7 4!fFigl7$F^^n$\"+y)eg(eFigl7$Fc^n$\"+&eoyx&Figl7$Fh^n$\"+,<1%o&Figl7$F ]_n$\"+u!*47cFigl7$Fb_n$\"+'*)38e&Figl7$Fg_n$\"+([A/l&Figl7$F\\`n$\"+V %>)>dFigl7$Fa`n$\"+$*f:LbFigl7$Ff`n$\"+H/.1aFigl7$F[an$\"+n&p-Y&Figl7$ F`an$\"+kP]5bFigl7$Fean$\"+vOq:aFigl7$Fjan$\"+'*eQC`Figl7$F_bn$\"+\\.& [D&Figl7$Fdbn$\"+iWc8_Figl7$Fibn$\"+L*Q3C&Figl7$F^cn$\"+Gl%>K&Figl7$Fc cn$\"+@/e^^Figl7$Fhcn$\"+o/G;]Figl7$F]dn$\"+L5`0]Figl7$Fbdn$\"+'pTz,&F igl7$Fgdn$\"+\"Rx`1&Figl7$F\\en$\"+37C-^Figl7$Faen$\"+RT*o,&Figl7$Ffen $\"+_7bL\\Figl7$F[fn$\"+F\\(y&[Figl7$F`fn$\"+Yg!>![Figl7$Fefn$\"+Z$o4z %Figl7$Fjfn$\"+!Q*R+[Figl7$F_gn$\"+`7KS[Figl7$Fdgn$\"+lN#H([Figl7$Fign $\"+H\"G,s%Figl7$F^hn$\"+e)[8f%Figl7$Fchn$\"+WiitXFigl7$Fhhn$\"+I`huXF igl7$F]in$\"+1xw/YFigl7$Fbin$\"+WHyUYFigl7$Fgin$\"+3R>hXFigl7$F\\jn$\" +(y!e\"[%Figl7$Fajn$\"+d0'\\T%Figl7$Ffjn$\"+\"e?OO%Figl7$F[[o$\"+(el2N %Figl7$F`[o$\"+$3]HN%Figl7$Fe[o$\"+N*)RxVFigl7$Fj[o$\"+5&)*GT%Figl7$F_ \\o$\"+REEITFigl7$F^]o$\"+SOnH TFigl7$Fc]o$\"+^.QZTFigl7$Fh]o$\"+=\\*f=%Figl7$F]^o$\"+OH'>7%Figl7$Fb^ o$\"+O90fSFigl7$Fg^o$\"+V,H^RFigl7$F\\_o$\"+x*3'HPFigl7$Fa_o$\"+Ij$ Figl7$Ff_o$\"+\"G6V\\$Figl7$F[`o$\"+(4-sJ$Figl7$F``o$\"+zsHBKFigl7$Fe` o$\"+:'f')3$Figl7$Fj`o$\"+dn)>$HFigl7$F_ao$\"++t3aGFigl7$Fdao$\"+LIn5F Figl7$Fiao$\"+jC%))e#Figl7$F^bo$\"+dV7,DFigl7$Fcbo$\"+6WZqBFigl7$Fhbo$ \",Z&Q)yD#F[o7$F]co$\",:KSv=#F[o7$Fbco$\",S([)>1#F[o7$Fgco$\",fw$\\s>F [o7$F\\do$\",f-!3#*=F[o7$Fado$\",t\\S>z\"F[o7$Ffdo$\",bd91r\"F[o7$F[eo $\",o'=HV;F[o7$F`eo$\",hr?ia\"F[o7$Feeo$\",GWzuZ\"F[o7$Fjeo$\",lHA0U\" F[o7$F_fo$\",^m9:L\"F[o7$Fdfo$\",Cl'Qs7F[o7$Fifo$\",rA'Q97F[o7$F^go$\" ,QoEQ9\"F[o7$Fcgo$\",%fy\\)3\"F[o7$Fhgo$\",8l?'R5F[o7$F]ho$\"+=)QYv*F[ o7$Fbho$\"+O\\(pE*F[o7$Fgho$\"+&38j$))F[o7$F\\io$\"+&>N)4$)F[o7$Faio$ \"+di#*eyF[o7$Ffio$\"+2K6juF[o7$F[jo$\"+8.h3qF[o7$F`jo$\"+ya$)QmF[o7$F ejo$\"+2T(eG'F[o7$Fjjo$\"+U>T>fF[o7$F_[p$\"+aF@.cF[o7$Fd[p$\"+Yef__F[o 7$Fi[p$\"+l()Qf\\F[o7$F^\\p$\"+&QQ=l%F[o7$Fc\\p$\"+TeXsVF[o7$Fh\\p$\"+ Ywn;TF[o7$F]]p$\"+@BDcQF[o7$Fb]p$\"+f+A,OF[o7$Fg]p$\",&o7I&Q$F-7$F\\^p $\",`&**4hJF-7$Fa^p$\",='>>fHF-7$Ff^p$\",2C^zu#F-7$F[_p$\",!QGF-7 $Fd`p$\",FScJu\"F-7$Fi`p$\",4FDld\"F-7$F^ap$\",GQ$>q9F-7$Fcap$\",&\\%= ^L\"F-7$Fhap$\",*)4R&)=\"F-7$F]bp$\",a![`-6F-7$Fbbp$\"+_XZN)*F-7$Fgbp$ \"+S!o\"*\\)F--F]cp6&F_cpF`cpF(F(-Fdcp6#%3efficient~scheme~BG-F$6%7j\\ lF'7$F+$\"%pCF07$F2$\"'+q;F07$F7$\"'^MlF07$F<$\"(u&)*>F07$FA$\"(m^:&F0 7$FF$\")g_t6F07$FK$\")**=>AF07$FP$\")cocRF07$FU$\")C(zr'F07$FZ$\"*VAX4 \"F07$Fin$\"*goa>\"F07$F_o$\"*wPK>\"F07$Fdo$\"*465>\"F07$Fio$\"*g)y)= \"F07$F^p$\"*qcV=\"F07$Fcp$\"*8X*z6F07$Fhp$\"*bK7<\"F07$F]q$\"*D?H;\"F 07$Fbq$\"*#=/b6F07$Fgq$\"*CZ7:\"F07$F\\r$\"*BqB:\"F07$Far$\"*q^n:\"F07 $Ffr$\"*C)pl6F07$F[s$\"*:()3=\"F07$F`s$\"*!e'*Q7F07$Fes$\"*@XXN\"F07$F js$\"*02@c\"F07$F_t$\"*\\c,\">F07$Fdt$\"*OOs-#F07$Fit$\"*.U&e@F07$F^u$ \"*A'**f@F07$Fcu$\"*xLa:#F07$Fhu$\"*+Qj9#F07$F]v$\"*?$GP@F07$Fbv$\"**) [$>@F07$Fgv$\"*Us>5#F07$F\\w$\"*3bi3#F07$Faw$\"*cC[2#F07$Ffw$\"*6@H2#F 07$F[x$\"*/G)*3#F07$F`x$\"*K7.9#F07$Fex$\"*P#=ZAF07$Fjx$\"*(feVCF07$F_ y$\"*\"o#fx#F07$Fdy$\"*e8s)GF07$Fiy$\"*&H+MHF07$F^z$\"*/Ny#HF07$Fcz$\" *4!o@HF07$Fhz$\"*[4%4HF07$F][l$\"*J%>(*GF07$Fb[l$\"*O#*H(GF07$Fg[l$\"* GW%\\GF07$F\\\\l$\"*%p:HGF07$Fa\\l$\"*S%Q7GF07$Ff\\l$\"*d&)G!GF07$F[]l $\"*o[q!GF07$F`]l$\"*K:KS$F07$F]`l$\"*%Q>>p1s:@cF[o$\" *)RG0QF07$$\"5?::::S(4Gm&F[o$\"*rR%3QF07$$\"5:666htAY/dF[o$\"*K())=QF0 7$Ffal$\"*qG(QQF07$$\"5+**)*)*)R()>%HeF[o$\"*Ses\"RF07$F[bl$\"*@Y22%F0 7$F`bl$\"*3U/>%F07$Febl$\"*CedM%F07$Fjbl$\"*5gVL%F07$F_cl$\"*cHeJ%F07$ Fdcl$\"*)y,zUF07$Ficl$\"*DYEC%F07$F^dl$\"*'y'Q<%F07$Fcdl$\"*5>q7%F07$F hdl$\"*l#o>VF07$F]el$\"*@xC]%F07$Fbel$\"*vFzU%F07$Fgel$\"*OBbO%F07$F\\ fl$\"*C>$[VF07$Fafl$\"*MF7N%F07$Fffl$\"*EX!)Q%F07$F[gl$\"*%*z*yWF07$F` gl$\"*6+Ml%F07$Fegl$\"+Y-,EZFigl7$F[hl$\"+;T'eo%Figl7$F`hl$\"+ut5YYFig l7$Fehl$\"++_v`[Fig l7$F[]m$\"+,A9?[Figl7$F`]m$\"+^\"[vx%Figl7$Fe]m$\"+7+<%p%Figl7$Fj]m$\" +`Ob>YFigl7$F_^m$\"+,())ff%Figl7$Fd^m$\"+2[`#e%Figl7$Fi^m$\"+>,W\"e%Fi gl7$F^_m$\"+zfQ&e%Figl7$Fc_m$\"+))>c&f%Figl7$Fh_m$\"+#RkLh%Figl7$F]`m$ \"+<(4'yYFigl7$Fb`m$\"+%*oN(z%Figl7$Fg`m$\"+<]wP[Figl7$F\\am$\"+=]lH[F igl7$Faam$\"+!y#e?[Figl7$Ffam$\"+%fF:\"[Figl7$F[bm$\"+'RoMz%Figl7$F`bm $\"+(=yax%Figl7$Febm$\"+$=AHp%Figl7$Fjbm$\"+aFi:YFigl7$F_cm$\"+6oF$e%F igl7$Fdcm$\"+;VogXFigl7$Ficm$\"+s+GbXFigl7$F^dm$\"+fSXbXFigl7$Fcdm$\"+ (f&ziXFigl7$Fhdm$\"+d[AzXFigl7$F]em$\"+J7#zj%Figl7$Fbem$\"+4(y'[ZFigl7 $Fgem$\"+r/(Rw%Figl7$F\\fm$\"+B-cXZFigl7$Fafm$\"+59AFZFigl7$Fffm$\"+$3 b*3ZFigl7$F[gm$\"+k[yOYFigl7$F`gm$\"+(Rpzc%Figl7$Fegm$\"+`1LMXFigl7$Fj gm$\"+6QF2XFigl7$F_hm$\"++H/#\\%Figl7$Fdhm$\"+[nu'\\%Figl7$Fihm$\"+T7I LXFigl7$F^im$\"+k6_=YFigl7$Fcim$\"+sg'*\\YFigl7$Fhim$\"+z!='zYFigl7$F] jm$\"+V()opYFigl7$Fbjm$\"+#[!yfYFigl7$Fgjm$\"+%4F+k%Figl7$F\\[n$\"+u%e .i%Figl7$Fa[n$\"+qzsvWFigl7$Ff[n$\"+W;Q*R%Figl7$F[\\n$\"+`^5CWFigl7$F` \\n$\"+Fge*[%Figl7$Fe\\n$\"+\\oY9XFigl7$Fj\\n$\"+kwVVXFigl7$F_]n$\"+-< TcXFigl7$Fd]n$\"+)*f!oa%Figl7$Fi]n$\"+L`lFXFigl7$F^^n$\"+-ee3XFigl7$Fc ^n$\"+KSBLWFigl7$Fh^n$\"+>$=6O%Figl7$F]_n$\"+4.V0VFigl7$Fb_n$\"+[1*4G% Figl7$Fg_n$\"+]U.MVFigl7$F\\`n$\"+6#f))Q%Figl7$Fa`n$\"+&y^bC%Figl7$Ff` n$\"+Q*))o9%Figl7$F[an$\"+3jT)=%Figl7$F`an$\"+L4UGUFigl7$Fean$\"+`1nbT Figl7$Fjan$\"+FH]&3%Figl7$F_bn$\"+([5=.%Figl7$Fdbn$\"+EG]**RFigl7$$\"5 UUUU4mp\"Fig l$\"+_(*znSFigl7$F^cn$\"+1^#R3%Figl7$Fccn$\"+;H:`RFigl7$Fhcn$\"+aXe[QF igl7$F]dn$\"+ml,SQFigl7$Fbdn$\"+?jU\\QFigl7$Fgdn$\"+@WL')QFigl7$F\\en$ \"+J/a:RFigl7$Faen$\"+$eU+&QFigl7$Ffen$\"+V/0'y$Figl7$F[fn$\"+96yFPFig l7$F`fn$\"+E(RVo$Figl7$Fefn$\"+!Hrcn$Figl7$Fjfn$\"+\"4CGo$Figl7$F_gn$ \"+cZ#Rr$Figl7$Fdgn$\"+S1zRPFigl7$Fign$\"+HF]AOFigl7$F^hn$\"+'[YJ_$Fig l7$Fchn$\"+g\"o#4NFigl7$Fhhn$\"+d*z)4NFigl7$F]in$\"+!3\\L`$Figl7$Fbin$ \"+@'>Mc$Figl7$Fgin$\"+Jwz+NFigl7$F\\jn$\"+6CmRMFigl7$Fajn$\"+/EP)Q$Fi gl7$Ffjn$\"+'>*e[LFigl7$F[[o$\"+$o6&QLFigl7$F`[o$\"+AJ6SLFigl7$Fe[o$\" +B5>fLFigl7$Fj[o$\"+VXD(Q$Figl7$F_\\o$\"+`\\>uKFigl7$Fd\\o$\"+J\\%)zJF igl7$Fi\\o$\"+!z@'pJFigl7$F^]o$\"+>%*4pJFigl7$Fc]o$\"+T>$H=$Figl7$Fh]o $\"+QcQ8KFigl7$F]^o$\"+$>JU;$Figl7$Fb^o$\"+o\"Gf6$Figl7$Fg^o$\"+*GUL.$ Figl7$F\\_o$\"+VF3jGFigl7$Fa_o$\"+:*e')y#Figl7$Ff_o$\"+z&fFo#Figl7$F[` o$\"+nM=ZDFigl7$F``o$\"+r7kvCFigl7$Fe`o$\"+SZ5sBFigl7$Fj`o$\"+k-;_AFig l7$F_ao$\"+AW)G>#Figl7$Fdao$\"+?Yb#3#Figl7$Fiao$\"+m\"p%*)>Figl7$F^bo$ \"+6\")fA>Figl7$Fcbo$\"+0\")*>#=Figl7$Fhbo$\",(>!fgt\"F[o7$F]co$\",N)R b#o\"F[o7$Fbco$\",q17fe\"F[o7$Fgco$\",*4(ow^\"F[o7$F\\do$\",f)4Sc9F[o7 $Fado$\",.td#z8F[o7$Ffdo$\",XO,tJ\"F[o7$F[eo$\",Q)*3hE\"F[o7$F`eo$\",r \"QI\">\"F[o7$Feeo$\",)*)e**Q6F[o7$Fjeo$\",DJ\"z&4\"F[o7$F_fo$\",\"4w8 F5F[o7$Fdfo$\"+%4-A#)*F[o7$Fifo$\"+h'G?Q*F[o7$F^go$\"+=M/P))F[o7$Fcgo$ \"+9&QqT)F[o7$Fhgo$\"+VkDZ!)F[o7$F]ho$\"+)4U4b(F[o7$Fbho$\"+OUN\"=(F[o 7$Fgho$\"+D'z^&oF[o7$F\\io$\"+l(f\"[kF[o7$Faio$\"+P\"Ro5'F[o7$Ffio$\"+ du<2eF[o7$F[jo$\"+.?PbaF[o7$F`jo$\"+eHlw^F[o7$Fejo$\"+2I+2\\F[o7$Fjjo$ \"+s_^DYF[o7$F_[p$\"+W)e%)Q%F[o7$Fd[p$\"+'=\")*>TF[o7$Fi[p$\"+D*RZ*QF[ o7$F^\\p$\"+DW0kOF[o7$Fc\\p$\"+hg1\\MF[o7$Fh\\p$\"+c\\&QD$F[o7$F]]p$\" +\"\\\"zfIF[o7$Fb]p$\"+\\iyiGF[o7$Fg]p$\",&yGM)p#F-7$F\\^p$\",`.uC`#F- 7$Fa^p$\",=-5TP#F-7$Ff^p$\",2_'z9AF-7$F[_p$\",!e?&*z?F-7$F`_p$\",-Ccw$ >F-7$Fe_p$\",hgj!3=F-7$Fj_p$\",$eQ5!o\"F-7$F_`p$\",dl:\\CF07$Fgq$!)a#fT#F07$F[s$!)&3AW# F07$F`s$!)&oA`#F07$Fes$!)6ZTFF07$Fjs$!)+&R;$F07$F_t$!)'Gu%RF07$Fdt$!)N uDUF07$Fit$!)J'\\a%F07$F^u$!)93]XF07$Fcu$!)/ZSXF07$Fhu$!)&48_%F07$F]v$ !)-B-XF07$Fbv$!)@MkWF07$Fgv$!)J0FWF07$F\\w$!)EY\"R%F07$Faw$!)y$4O%F07$ Ffw$!)%3PM%F07$F[x$!)I3dVF07$F`x$!)!)=LWF07$Fex$!)J@GYF07$Fjx$!)6BM]F0 7$F_y$!)r&[z&F07$Fdy$!)itjgF07$Fiy$!)t!)zhF07$F^z$!)e\"o;'F07$Fcz$!)=& Q:'F07$Fhz$!)d+GhF07$F][l$!)\"pA5'F07$Fb[l$!)%e60'F07$Fg[l$!)0!3+'F07$ Fa\\l$!)E(\\\"fF07$F[]l$!)c_veF07$F`]l$!)-%R-'F07$Fe]l$!)va_nF07$Fj]l$ !)8dHrF07$F_^l$!)>^auF07$Fd^l$!)KXCuF07$Fi^l$!)b^%R(F07$F^_l$!)#3]L(F0 7$Fc_l$!)]1wsF07$Fh_l$!)#)\\irF07$F]`l$!)1SxqF07$Fb`l$!):j!o(F07$Fg`l$ !)fQM#)F07$F\\al$!)$z45)F07$Faal$!)_\"[*zF07$Fi_[l$!)1oxzF07$Ffal$!)-3 =!)F07$Ff`[l$!)Q-k\")F07$F[bl$!)4S#\\)F07$F`bl$!)uuo()F07$Febl$!)6'\\9 *F07$Fjbl$!)5Figl7$F[]m$!+l=s75Figl7$F`]m$ !+d5x.5Figl7$Fe]m$!*Ir:')*Figl7$Fj]m$!*+Xqp*Figl7$F_^m$!*Lq&R'*Figl7$F d^m$!*%[&**f*Figl7$F^_m$!*skDf*Figl7$Fh_m$!*1\"GS'*Figl7$F]`m$!*Q$)zx* Figl7$Fb`m$!+U`p05Figl7$Fg`m$!+iqn:5Figl7$F\\am$!+*G1S,\"Figl7$Faam$!+ M:575Figl7$Ffam$!+c.?55Figl7$F[bm$!+:(3k+\"Figl7$F`bm$!+R8j-5Figl7$Fjb m$!*O_eo*Figl7$Fhdm$!*Huhc*Figl7$F]em$!*pG+p*Figl7$Fbem$!*U3:&**Figl7$ Fgem$!*Z\")Q***Figl7$F\\fm$!*Ng_&**Figl7$Fafm$!*^)y;**Figl7$Fffm$!*$eY y)*Figl7$F[gm$!*Epns*Figl7$F`gm$!*P,'z&*Figl7$Fegm$!*O4X]*Figl7$Fjgm$! *@=)R%*Figl7$F_hm$!*R^kR*Figl7$Fdhm$!*BoPR*Figl7$Fihm$!*L?RY*Figl7$F^i m$!*5Dwl*Figl7$Fcim$!*>^Ot*Figl7$Fhim$!*hQu!)*Figl7$F]jm$!*(*Gmy*Figl7 $Fbjm$!*Vjew*Figl7$Fgjm$!*]kWs*Figl7$F\\[n$!*OTKo*Figl7$Fa[n$!*i#>x$*F igl7$Ff[n$!*uXb=*Figl7$F`\\n$!*'y.v$*Figl7$F^^n$!*iu$Q%*Figl7$Fc^n$!*! \\V!G*Figl7$Fh^n$!*hCq7*Figl7$F]_n$!*wL8+*Figl7$Fb_n$!*%)3?$*)Figl7$Fg _n$!**[fN!*Figl7$F\\`n$!*+&yv\"*Figl7$Fa`n$!**\\\"[())Figl7$Ff`n$!*@> \\k)Figl7$F[an$!*yu7s)Figl7$F`an$!*`!4F))Figl7$Fean$!*p9^n)Figl7$Fjan$ !*RPp_)Figl7$F_bn$!*@(G3%)Figl7$Fdbn$!*kQlK)Figl7$Ffd\\l$!*aa,K)Figl7$ Fibn$!*(\\%zN)Figl7$F^e\\l$!*'ezl%)Figl7$F^cn$!*lM2^)Figl7$Fccn$!*@mvB )Figl7$Fhcn$!*Y8R+)Figl7$F]dn$!*H`t(zFigl7$Fbdn$!*1x,*zFigl7$Fgdn$!*nS +2)Figl7$F\\en$!*uyO9)Figl7$Faen$!*iNu+)Figl7$Ffen$!*SfP(yFigl7$F[fn$! *,'z[xFigl7$F`fn$!*Vfsk(Figl7$Fefn$!*$y>@wFigl7$Fjfn$!*H;'HwFigl7$F_gn $!*;3gp(Figl7$Fdgn$!*'\\`gxFigl7$Fign$!*]Zn^(Figl7$F^hn$!*,]\"*H(Figl7 $Fchn$!*kCDE(Figl7$Fhhn$!*k^hD(Figl7$F]in$!*9kLI(Figl7$Fbin$!*rAaP(Fig l7$Fgin$!*/&zXsFigl7$F\\jn$!*h0(=rFigl7$Fajn$!*#\\L4qFigl7$Ffjn$!*3\"o D_'Figl 7$F^]o$!*f@^^'Figl7$Fc]o$!*TH1a'Figl7$Fh]o$!**f34mFigl7$F]^o$!*3&)z]'F igl7$Fb^o$!*7y%3kFigl7$Fg^o$!**>Y8iFigl7$F\\_o$!*)f%*feFigl7$Fa_o$!*AN ))o&Figl7$Ff_o$!*fybW&Figl7$F[`o$!*udd;&Figl7$F``o$!*%=[(*\\Figl7$Fe`o $!*v84w%Figl7$Fj`o$!*GhP^%Figl7$F_ao$!*4psO%Figl7$Fdao$!*,w@7%Figl7$Fi ao$!*x!>GRFigl7$F^bo$!*>$*Hw$Figl7$Fcbo$!*w7#[NFigl7$Fhbo$!+.*>1O$F[o7 $F]co$!+l?G?KF[o7$Fbco$!++!Hf,$F[o7$Fgco$!+,fykGF[o7$F\\do$!+\"*)zzq#F [o7$Fado$!+F$zCb#F[o7$Ffdo$!+&QVhS#F[o7$F[eo$!+-XHnAF[o7$F`eo$!+fZb@@F [o7$Feeo$!+_Z*H*>F[o7$Fjeo$!+&[Xy'=F[o7$F_fo$!+REqTRi \"F[o7$Fifo$!+p=l(\\\"F[o7$F^go$!+-Xh/9F[o7$Fcgo$!+'>#[*G\"F[o7$Fhgo$! +nv4v6F[o7$F]ho$!+UG/(4\"F[o7$Fbho$!*M#41**F[o7$Fgho$!*vsk!*)F[o7$F\\i o$!*N2cF)F[o7$Faio$!*8*R]sF[o7$Ffio$!*$yrDjF[o7$F[jo$!*(zL8eF[o7$F`jo$ !*sO'*)[F[o7$Fejo$!*`ChA%F[o7$Fjjo$!*)*ywn$F[o7$F_[p$!*1JWz#F[o7$Fd[p$ !*M=$*=#F[o7$Fi[p$!*bKzu\"F[o7$F^\\p$!)lOX*)F[o7$Fc\\p$!)f$=$\\F[o7$Fh \\p$!(%[19F[o7$$\"5IFFFxR))o%G&Figl$\");qzEF[o7$F]]p$\")J7mzF[o7$$\"5& eeee$[rsN`Figl$\")6=XyF[o7$$\"5?@@@@Ody_`Figl$\")H3cyF[o7$$\"5bccc1CV% )p`Figl$\")7,t%)F[o7$Fb]p$\"*fF<5\"F[o7$Fg]p$\"+&e\"RP:F-7$$\"5!333eqX oaY&Figl$\"+yMXk:F-7$$\"5?===oX3&>[&Figl$\"+oB6L!\\ &Figl$\"+])QP$>F-7$$\"5bbbbIMKV)\\&Figl$\"+31zoAF-7$$\"5DCCuhGWn1bFigl $\"+fr#pL#F-7$F\\^p$\"+`Tv mQ#F-7$$\"5IFFx*youze&Figl$\"+(y%HnDF-7$$\"55555N^Hp'f&Figl$\"+d(**3kcFigl$\" +J;4GHF-7$$\"5&oooo$>u>>p'*e&\\z&Figl$\"+L')\\.TF-7$$\"5bccccwbC6eFigl$\"+m,\"fJ%F-7$$\"5 &RRRRkDNv#eFigl$\"+09\\YUF-7$Fe_p$\"+hK=$=%F-7$$\"5SRRRkvs&*feFigl$\"+ P+^]TF-7$$\"5]ZZZ(\\h*3weFigl$\"+]ihBUF-7$$\"5bbbbIa>A#*eFigl$\"+3;J'e %F-7$Fj_p$\"+$e\\X#\\F-7$$\"554444RYkSfFigl$\"+;wmrZF-7$F_`p$\"+d)Rlv% F-7$$\"5++++vQ?S*)fFigl$\"+2>%40&F-7$$\"5XXXX&H4pe+'Figl$\"+\\tk.bF-7$ $\"5!444fr9OB-'Figl$\"++?&QT&F-7$Fd`p$\"+F;4G`F-7$$\"5XXXX&H+ei0'Figl$ \"+k+3h_F-7$$\"5baaaa/GrtgFigl$\"+oMs&H&F-7$$\"5ljjj81w;\"4'Figl$\"+6( o[k&F-7$Fi`p$\"+4(e^-'F-7$$\"5!zyyy`4J19'Figl$\"+=b3ReF-7$F^ap$\"+Gt.% z&F-7$$\"5?>>>WD\"Q%)='Figl$\"+5M#31'F-7$$\"5NNNN&yYOU?'Figl$\"+Ay!Qc' F-7$$\"5]^^^E5[.?iFigl$\"+lR(4Y'F-7$Fcap$\"+&>)RhjF-7$$\"5qqqqq?8s_iFi gl$\"+'QjCF'F-7$$\"5vtttt)[4'piFigl$\"+gJZfiF-7$$\"5IDDDvsN0yiFigl$\"+ yQ?EjF-7$$\"5!onnnnl(\\'G'Figl$\"+S6z'['F-7$$\"5IGGGyS<%\\H'Figl$\"+VC 2&z'F-7$Fhap$\"+*GRp0(F-7$$\"5vssssK')3PjFigl$\"+n%f_#oF-7$F]bp$\"+aE' Gs'F-7$$\"5qmm;a3SNyjFigl$\"+\"\\Eqy'F-7$$\"5qnnnUwl\"fQ'Figl$\"+sh:Hp F-7$$\"5qoo=JW\"zMR'Figl$\"+FvX)=(F-7$$\"5qppp>7 " 0 " " {MPLTEXT 1 0 501 "evalf[25](plot(['bn_RK5_1'(x)-b(x),'bn_RK5_2'(x)-b (x),'bn_RK5_3'(x)-b(x),'bn_RK5_4'(x)-b(x),\n'bn_RK5_5'(x)-b(x),'bn_RK5 _6'(x)-b(x)],x=0.65..5,\ncolor=[blue,magenta,COLOR(RGB,.1,.8,0),red,br own,COLOR(RGB,.6,0,.8)],font=[HELVETICA,9],\nlegend=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`effici ent scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with s tability radius = imag axis inclusion`],\ntitle=`error curves for 7 st age order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 788 509 509 {PLOTDATA 2 "6,-%'CURVESG6%7_v7$$\"#l!\"#$\"0Gxi33]k\"!#F7$$\" :++++++DJ5)3up!#D$!0@JN7$$\":+++++D1*e.3P()F1$!1_!*>;C5Y%)F>7$$\":++++++vVY\")4?*F1$!1jL?nrl [(*F>7$$\":+++++Dc,]DxJ*F1$!1w2!GP@_v*F>7$$\":+++++]Pf`pWV*F1$!1WgO78p `(*F>7$$\":+++++v=7$$\":+++++++vgdzm*F1$!1s2.9= 8[)*F>7$$\":+++++D\"GV;q%y*F1$!2Ucs4LbW,\"F>7$$\":+++++]i!zcW,**F1$!2 \"flIs&y!z5F>7$$\":++++]P%[r*==+\"!#C$!3i#>pNvC7Qn5F\\p7$$\":+++++voCrWk5\"Fio$!34'zH(zROG6F\\p7$$\":+ +++]i:g$p@<6Fio$!3-nhhbl_O6F\\p7$$\":+++++]i&f\"*)z7\"Fio$!37kvw3#=.: \"F\\p7$$\":++++]P4JQh(Q6Fio$!3i!QrIym@<\"F\\p7$$\":+++++DcmgL&\\6Fio$ !3T2\"3Ufp\\?\"F\\p7$$\":+++](oHaimA_6Fio$!3&fj\\:\"Fio$!3e1#Hf'[0r6F\\p7$$\":+++]iS;Vx7w:\"Fio$!3(3#eT*Qh^;\"F \\p7$$\":++++]7.-$eIg6Fio$!3v[\\v$))*4I7F\\p7$$\":+++]P%)*3')))*H;\"Fi o$!3-zNxx>a(>\"F\\p7$$\":++++Dcw>%>pl6Fio$!3Ff#>mf')3<\"F\\p7$$\":+++] 7Gjy*\\Qo6Fio$!3COsamks\"=\"F\\p7$$\":++++++]P0y5<\"Fio$!3&[g%*=')HkA \"F\\p7$$\":++++](oHx-&==\"Fio$!3^doqbBYA7F\\p7$$\":+++++vV3]AE>\"Fio$ !3Hm3PnkC=7F\\p7$$\":++++++vV^wP?\"Fio$!3$Qy%Q/:^47F\\p7$$\":+++++D1z_ I\\@\"Fio$!3\"4UHCWj>?\"F\\p7$$\":+++++]P9a%3E7Fio$!3IUU$*>:'y>\"F\\p7 $$\":+++++vo\\bQsB\"Fio$!4tg#=hwi@,7!#I7$$\":+++++++&oDR[7Fio$!4g_'*>h >*>=7Fav7$$\":+++++DJ?eY&f7Fio$!4*\\')*fL^8tD\"Fav7$$\":++++D19a3NBE\" Fio$!4t%HR7`HC]7Fav7$$\":++++](oz)eB^E\"Fio$!4e`@H,7f(=7Fav7$$\":++++v oz@47zE\"Fio$!4F9*HCfC'p@\"Fav7$$\":+++++]ibf+2F\"Fio$!4T,H9yW**QF\"Fa v7$$\":++++DJX*)4*[t7Fio$!4s!HivM4=R7Fav7$$\":++++]7GBgxiF\"Fio$!4KdEb BKxT@\"Fav7$$\":++++v$4r0h1z7Fio$!4Q9\\e\"ze.X7Fav7$$\":+++++v$44Y&=G \"Fio$!4)z%3'3#G%3i7Fav7$$\":+++++vVe5!R$H\"Fio$!4,y62w/8cC\"Fav7$$\": +++++v$f-c#\\I\"Fio$!4_i%=)f;C4B\"Fav7$$\":+++++vV$*4hkJ\"Fio$!4*)*yS? dC'GA\"Fav7$$\":+++++v$4'f'*zK\"Fio$!4je\"pg:wnJ7Fav7$$\":+++++voWMkPL \"Fio$!4ZQ`9T=3([7Fav7$$\":+++++vVG4K&R8Fio$!4*zA]Q$f#[u7Fav7$$\":++++ +DJq'fTU8Fio$!4&fg6SWK\\m7Fav7$$\":+++++v=7%)*HX8Fio$!47/cqL%>+M7Fav7$ $\":+++++D1ar$=[8Fio$!4>j)*)H\"[mqB\"Fav7$$\":+++++v$f*en5N\"Fio$!4!\\ GkP$Q7ZG\"Fav7$$\":+++++vVj3.EO\"Fio$!40v!4Zvc*fE\"Fav7$$\":+++++v$4$e QTP\"Fio$!4&GKY9Lo/[7Fav7$$\":++++]Pf`3PcQ\"Fio$!4MMlc>0!*RB\"Fav7$$\" :++++++DweNrR\"Fio$!4(GC&>)>I$*H7Fav7$$\":++++]i!*)*3M'39Fio$!4\"3D5\\ r8*zC\"Fav7$$\":+++++Dc@fK,U\"Fio$!4gj$)45bktH\"Fav7$$\":+++++](o'fHJW \"Fio$!4_GFn]IRsD\"Fav7$$\":+++++v=7gEhY\"Fio$!5,ynD]:Y/D7!#J7$$\":+++ +](=#H4PM^\"Fio$!5SjYD9P(e\"R7F^_l7$$\":++++++DYeZ2c\"Fio$!5HSz_,[&H\\ D\"F^_l7$$\":+++++vo9p=Cg\"Fio$!5()e%zNxk&*f@\"F^_l7$$\":+++++]7$)z*3W ;Fio$!5&[cJbe0z\\<\"F^_l7$$\":++++++v8R8zt\"Fio$!6/\\,E)*p>O\"*4\"!#K7 $$\":++++++]A\"y,&y\"Fio$!6d'pH02xBqi5Fh`l7$$\":++++++DJBA@$=Fio$!6faA F*4ffZV5Fh`l7$$\":+++++++?:>[&=Fio$!5y7]]AKuMo**Fh`l7$$\":++++++v32;v( =Fio$!5%e-:7D%))\\h'*Fh`l7$$\":+++++v$f+.>$)=Fio$!5KB]!R;Q')Gz*Fh`l7$$ \":+++++]7.`k)))=Fio$!5k=luZ'zk1p*Fh`l7$$\":++++](=Vvk'*e&*Fh`l7$$\":++++]i!*[(eP (*=Fio$!5(QXt\"p5@$yV*Fh`l7$$\":++++++](*)H@+>Fio$!5y^D6*y_xJ&)*Fh`l7$ $\":+++++](=\\9c6>Fio$!5fm^:]D\\Z-'*Fh`l7$$\":++++++D'3*4H#>Fio$!6<.b/ 4jzF`N*!#L7$$\":+++++v$f:QN0?Fio$!6NQ=\\uT0$[[\")Fedl7$$\":++++++D^f(Q .@Fio$!6pXB`%eV'z[%oFedl7$$\":+++++++b1Nk=#Fio$!7J+YPEOyVwFa!#M7$$\":+ ++++vVj,RIG#Fio$!79&45.d^87X$QFeel7$$\":++++++]<))\\&oBFio$!7;$=5/)[<* )fEAFeel7$$\":+++++vVQ&)oBY#Fio$!7$G-%\\dCMW$Qp$!#N7$$\":+++++DJ:q1d8Ha(y%GL$!#O7$$ \":+++++v=2a@0t#Fio$\"92y/(44i$)>)=`aF`gl7$$\":+++++]7`)G&G#GFio$\"9p \\YcL/yhS_rsF`gl7$$\":+++++v$4.#f(=HFio$\":&*>$z&>(*ei4=b@*!#P7$$\":++ +++DcY$fC-IFio$\":(o**=:e_UMW\"e6\"F`gl7$$\":+++++]P>*QT#4$Fio$\":^,zj =#o.uU$yH\"F`gl7$$\":++++++]Z@mb=$Fio$\":#oSK'**z?5DjPV\"F`gl7$$\":+++ ++](=*e(pwKFio$\":L^^miwvty/*)e\"F`gl7$$\":+++++vV)z=([O$Fio$\":m;k@S& p$)p_a]F`gl7$$\":++++v=#* phqPZ$Fio$\":-J6uHzX+*R6J>F`gl7$$\":++++](=nPow%[$Fio$\":`sz F`gl7$$\":++++Dc^$eIw&\\$Fio$\":-F2MpYJ*GIRG>F`gl7$$\":+++++DJ!z#fn]$F io$\":Xi;(*3HjSR&[O>F`gl7$$\":++++]i!R?48;'p$)G1!Q,?F`gl7 $$\":++++++]<;W2b$Fio$\":A\"Q2yO8`UgH'4#F`gl7$$\":++++](=Uu\\[iNFio$\" :Hjgl#zOZmuGs?F`gl7$$\":+++++v$4(yDUd$Fio$\":+^kYgt\\&o9w\\?F`gl7$$\": +++](='4!e$Fio$\":!zx* GR%[%G1rj9#F`gl7$$\":+++]il(f'RJIe$Fio$\":Ox8d:G#HkO_%3#F`gl7$$\":++++ ]il(*fmfe$Fio$\":U<$354WLIk7O?F`gl7$$\":+++]PfLH!=!*)e$Fio$\":N2L()f' \\&\\G1#F`gl7$$\":+++++]PCT2xf$Fio$\":%QY?O&z*f Bus^?F`gl7$$\":++++]P4^A[%4OFio$\":Q9zz^rDMX$*f8#F`gl7$$\":+++++D\"yP! *=@OFio$\":vV9fG>C)phun@F`gl7$$\":++++]7`/&)HHj$Fio$\":5i$eSbU'*H5\"49 #F`gl7$$\":++++++DJmqYk$Fio$\":jU;v&Gm7vWv;@F`gl7$$\":+++]PfLWe4`l$Fio $\":lS'*p?XDz9&G;@F`gl7$$\":++++v=Ud][fm$Fio$\":')4/SCav+x'3=@F`gl7$$ \":+++]7y]qU(ewOFio$\":\\l#=CY1rA\"[M7#F`gl7$$\":++++]Pf$[jA(o$Fio$\": Ni[2(Q\"GrWJS8#F`gl7$$\":++++Dcw4>/&3PFio$\":*H%p'peU%QvX&z@F`gl7$$\": +++++v$fL?yHPFio$\":0QNf3@r'4=8wAF`gl7$$\":++++D1*yNV/`PFio$\":=.'p$H9 Ej1[JC#F`gl7$$\":++++]P%)zj1jx$Fio$\":\">'zDx#eS>+y,AF`gl7$$\":++]i!*y edU9#zPFio$\":>\\+#pm()G&)3&zE#F`gl7$$\":+++DJ?LN@A@y$Fio$\":)eDZa8VYT u6%G#F`gl7$$\":++](=AF`gl7$$\":+++v$f3jkLv$z$Fio$\"::u4q'R:LWV;aAF`g l7$$\":++++voz,%*o&*z$Fio$\":]K$yzB1BF`gl7$$\":++]i:5azswC!QFio$ \":%zN%f.^rG\\o#)G#F`gl7$$\":+++Dc^Gd^%Q0QFio$\":!Rp;&>CG\"[-vHAF`gl7$ $\":++](oHH].BH3QFio$\":R(Gl6c2#4Lg6C#F`gl7$$\":+++]PMx74+7\"QFio$\":Y LzF.\"zT9>MBBF`gl7$$\":++]7yv^!zy59QFio$\":?JpbA%3?IoCeAF`gl7$$\":+++v =QFio$\":)yQ!Q&\\4$ )eVD\\BF`gl7$$\":++++++vBCJG#QFio$\":H()**3nPIm>(H\"H#F`gl7$$\":+++]P4 @\"\\_xLQFio$\":JOH$GG@]*\\jbF#F`gl7$$\":++++v=neD>Z%QFio$\":Wz3&p,yT* [\\8E#F`gl7$$\":+++]7G8EEjc&QFio$\":)zPT+,E[N3x]AF`gl7$$\":++++]Pf$psg mQFio$\":FNJ6DYQ8&*=tC#F`gl7$$\":++++Dc^GG&\\))QFio$\":W!fEq(oa(f+S%G# F`gl7$$\":+++++vVjH$Q5RFio$\":&GWn&4oD>!eP%Q#F`gl7$$\":+++++](=xDELRFi o$\":3)*z`azL#o\"*y@BF`gl7$$\":+++++DJ!e=9cRFio$\":\")[-5%4!)[GM;uAF`g l7$$\":++++++v)Q6-zRFio$\":s%R'*>^bdy`>GBF`gl7$$\":+++++v=(>/!>+%Fio$ \":j+\"*R9`M^6w/O#F`gl7$$\":++++DJ&p)4pU-%Fio$\":Ban]`HV@k3kI#F`gl7$$ \":++++](=nxxjYSFio$\":t=UC%\\'y1l2AF#F`gl7$$\":++++vV[mX1!pSFio$\":d7 ;+#QO2%p89K#F`gl7$$\":++++++Dc8v84%Fio$\":'f&yU9/[so<,O#F`gl7$$\":+++] PfeD5%3.TFio$\":NAT*G#4H8]quJ#F`gl7$$\":++++v=#\\pIz9TFio$\":&GwCb[7Pc LJyAF`gl7$$\":++]PfeD7J?x6%Fio$\":LsV(>a*z6\"[.jAF`gl7$$\":+++vV)*eHbZ 17%Fio$\":5q.$*[mtLe0uO#F`gl7$$\":++]7GQ#p%zuN7%Fio$\":a9o/Z$f%*HTv*H# F`gl7$$\":+++]7yDk.-l7%Fio$\":F)oma4,1V_>`AF`gl7$$\":++](oz\"f\"y#H%HT Fio$\":.lx<+)[X\\;YIBF`gl7$$\":+++D\"yD*)>lNKTFio$\":$[eTUKs!p\\lMK#F` gl7$$\":++]il(fihPGNTFio$\":/k%=/3,C\")GfiAF`gl7$$\":++++]PfL+6#QTFio$ \"::#Q'p1N&)[j3oE#F`gl7$$\":+++v=*\\TFio$\":@Bjj2lS#=3DiBF`gl7$$\":++](=nj-7s%G:%Fio$ \":6T!**>F())=i\\NI#F`gl7$$\":+++DcwfPXud:%Fio$\":f.G-h?`j)Q?ZAF`gl7$$ \":++]iS;$\\&p,(eTFio$\":HjK4o'4&HX.eF#F`gl7$$\":++++DcEs$*G;;%Fio$\": P40*o&42GTonK#F`gl7$$\":+++]i:gT!zLtTFio$\":mcA/uf5NszEG#F`gl7$$\":+++ ++v$4ro/&=%Fio$\":T71G&Ru?*>nHC#F`gl7$$\":+++++++b)QEvUFio$\":.PnN;6:B wiy<#F`gl7$$\":+++++]iI\"Q_nVFio$\":;jBi8vL=)\\(33#F`gl7$$\":+++++vV=y >!fWFio$\":&[Rjc*=m.S,!3?F`gl7$$\":++++++]Z2&4VXFio$\":Xl4.)*Qiej%Q#)= F`gl7$$\":+++++]7$[[XRYFio$\":zV#)3'oJ!3BNbs\"F`gl7$$\":+++++++&y,kDZF io$\":2cDlco!zLR([]\"F`gl7$$\":+++++vVjrJv\"[Fio$\":^?*oi(*yrlDG)H\"F` gl7$$\":+++++DcJk$[0\\Fio$\":*Gh,pk%yd9Rj3\"F`gl7$$\"\"&\"\"!$\":E7f)Q +I([M#zg')F`hl-%'COLOURG6&%$RGBG$F\\dnF\\dnFcdn$\"*++++\"!\")-%'LEGEND G6#%8Bogacki-Shampine~schemeG-F$6%7_v7$F($\"0GF1DC-j\"F-7$F/$!0@!*zWA4 c\"F-7$F5$!06CRgye@%F-7$F:$!1mc,CdKIeF>7$F@$!1TP]Tp*38(F>7$FE$!1_gZ!=t TY)F>7$FJ$!1j$fnJ<\"o(*F>7$FO$!1w22>S\\u(*F>7$FT$!1W5()oyxs(*F>7$FY$!1 L@1)3bDy*F>7$Fhn$!1sn&e@Jq')*F>7$F]o$!2Uc>p>\"Q;5F>7$Fbo$!2\"f%>G@963 \"F>7$Fgo$!3i#*3BLATv5F\\p7$F^p$!3>p9Y9FPp5F\\p7$Fcp$!34'\\r/)pRI6F\\p 7$Fhp$!3-nvGdocQ6F\\p7$F]q$!37k\\;.hP_6F\\p7$Fbq$!3i![W8)yDu6F\\p7$Fgq $!3T2j$4i;r?\"F\\p7$F\\r$!3&ft.MrVG?\"F\\p7$Far$!3e19'e5WJ<\"F\\p7$Ffr $!3(3#fTxSBn6F\\p7$F[s$!3v['))e[\"HK7F\\p7$F`s$!3-zuR.bn*>\"F\\p7$Fes$ !3Ff$pjhqH<\"F\\p7$Fjs$!3CO0%p'H#Q=\"F\\p7$F_t$!3&[+qh96'G7F\\p7$Fdt$! 3^dxOZIjC7F\\p7$Fit$!3Hm?QVjS?7F\\p7$F^u$!3$QelNg_;@\"F\\p7$Fcu$!3\"47 a\"\\v3/7F\\p7$Fhu$!3IU[U5R(**>\"F\\p7$F]v$!4tgi(4g\"HL?\"Fav7$Fcv$!4g _OYE#pL?7Fav7$Fhv$!4*\\'oWG\"*>&f7Fav7$F]w$!4t%HP*42TCD\"Fav7$Fbw$!4e` ,\"o!p+4A\"Fav7$Fgw$!4F9**R;n%4>7Fav7$F\\x$!4T,H'4?r8w7Fav7$Fax$!4s!H \\$QYd8C\"Fav7$Ffx$!4Kd'fmFpI;7Fav7$F[y$!4Q9*Q\"[v9sC\"Fav7$F`y$!4)z%Q $='H*Hk7Fav7$Fey$!4,yJFmF(zZ7Fav7$Fjy$!4_ia7P=!3L7Fav7$F_z$!4*)*ydh;.+ D7Fav7$Fdz$!4je@*)4qDQB\"Fav7$Fiz$!4ZQV())=>*3D\"Fav7$F^[l$!4*zAwR%*fq w7Fav7$Fc[l$!4&fgtx0rqo7Fav7$Fh[l$!47/c.0jdhB\"Fav7$F]\\l$!4>j3sLH@#R7 Fav7$Fb\\l$!4!\\G*4n+dpG\"Fav7$Fg\\l$!40vI!*3e1#o7Fav7$F\\]l$!4&GK*y8; D-D\"Fav7$Fa]l$!4MM05XgThB\"Fav7$Ff]l$!4(GC*[(=O2K7Fav7$F[^l$!4\"3D\\$ eCg,D\"Fav7$F`^l$!4gjji?IE'*H\"Fav7$Fe^l$!4_GP(=#[I%f7Fav7$Fj^l$!5,yn( Q-'p8C\"F^_l7$Fe_l$!5HSzYyI67d7F^_l7$Fj_l$!5( )e%fkG`B\"=7F^_l7$F_`l$!5&[c@U-eSq<\"F^_l7$Fd`l$!6/\\,'[Orz2,6Fh`l7$Fj `l$!6d'pHG>Swfk5Fh`l7$F_al$!6faAn*R\"za`/\"Fh`l7$Fdal$!5y7]!f^tmj)**Fh `l7$Fial$!5%e-:e(zb,z'*Fh`l7$F^bl$!5KB]!\\9b>2\")*Fh`l7$Fcbl$!5k=lk9VI G3(*Fh`l7$Fhbl$!5`?$4mR7[?%)*Fh`l7$F]cl$!5O4@4JS^Tw&*Fh`l7$Fbcl$!5(QXt 3V=K]X*Fh`l7$Fgcl$!5y^D\"foh:7()*Fh`l7$F\\dl$!5fm^X#)*H.,i*Fh`l7$Fadl$ !6<.b/HYK_DP*Fedl7$Fgdl$!6NQ=\\F,37R;)Fedl7$F\\el$!6pXB`,5qe&eoFedl7$F ael$!7J+YPEt$4k$RaFeel7$Fgel$!79&45.(G0s,WQFeel7$F\\fl$!7;$=5/3qg6QB#F eel7$Fafl$!7$G-%\\du*3P7u$Fefl7$Fgfl$\"8ajz6H\"Rz'H:]\"Fefl7$F\\gl$\"9 7r(>d8Hhl5LL$F`gl7$Fbgl$\"92y/(44i#)\\(RcaF`gl7$Fggl$\"9p\\YcL/s(eStF( F`gl7$F\\hl$\":&*>$z&>(*eMeBSA*F`hl7$Fbhl$\":(o**=:es$oKJp6\"F`gl7$Fgh l$\":^,zj=#)H:70#*H\"F`gl7$F\\il$\":#oSK'**z^?,N`V\"F`gl7$Fail$\":L^^m iw_g\"\\p!f\"F`gl7$Ffil$\":m;k@S&>9,Vc_F` gl7$F`jl$\":-J6uHzk\"z-RL>F`gl7$Fejl$\":`szF`gl7$Fjjl$\":-F 2MpY*>\")>nI>F`gl7$F_[m$\":Xi;(*3HzRiv(Q>F`gl7$Fd[m$\":*>48;'pw4G_P+#F `gl7$Fi[m$\":A\"Q2yO8;/'*z)4#F`gl7$F^\\m$\":Hjgl#zEbegwu?F`gl7$Fc\\m$ \":+^kYgtBVH;A0#F`gl7$Fh\\m$\":jJ.D7Av!G/KD?F`gl7$F]]m$\":!zx*GR%GGhe% *[@F`gl7$Fb]m$\":Ox8d:GB$*=Cq3#F`gl7$Fg]m$\":U<$354W'\\Gm&Q?F`gl7$F\\^ m$\":N2L()f'\\Kzds#3#F`gl7$Fa^m$\":gn([w%evQdcL7#F`gl7$Ff^m$\":?\"RM\\ `2]kpHl?F`gl7$F[_m$\":%QY?O&zczM&=a?F`gl7$F`_m$\":Q9zz^rWyTf&Q@F`gl7$F e_m$\":vV9fG>_7Lh.<#F`gl7$Fj_m$\":5i$eSb-*yF(\\V@F`gl7$F_`m$\":jU;v&G' Qj?9$>@F`gl7$Fd`m$\":lS'*p?X%p_p%)=@F`gl7$Fi`m$\":')4/SCa.f)Hl?@F`gl7$ F^am$\":\\l#=CYcLqE-E@F`gl7$Fcam$\":Ni[2(QhjS/iO@F`gl7$Fham$\":*H%p'pe #Qdn%>#=#F`gl7$F]bm$\":0QNf3@08O8*yAF`gl7$Fbbm$\":=.'p$H9t+[**eC#F`gl7 $Fgbm$\":\">'zDx#o>\"\\$[/AF`gl7$F\\cm$\":>\\+#pmF7MKtqAF`gl7$Facm$\": )eDZa88y+#HpG#F`gl7$Ffcm$\":&=,i69V\\]W$\\r24BF`gl7$Fjdm$\": %zN%f.^dB%>4\"H#F`gl7$F_em$\":!Rp;&>Cq@o*\\KAF`gl7$Fdem$\":R(Gl6c@EBF`gl7$F^fm$\":?JpbA%obJg.hAF`gl7$Fcf m$\":M?'R8T[&*\\R7FAF`gl7$Fhfm$\":)yQ!Q&\\>ky$e@N#F`gl7$F]gm$\":H()**3 nPcj*38%H#F`gl7$Fbgm$\":JOH$GG@-N2QyAF`gl7$Fggm$\":Wz3&p,e4f;:kAF`gl7$ F\\hm$\":)zPT+,'**4nhND#F`gl7$Fahm$\":FNJ6DY#e)R1,D#F`gl7$Ffhm$\":W!fE q(o35#fB(G#F`gl7$F[im$\":&GWn&4oqm%[N(Q#F`gl7$F`im$\":3)*z`az*zmjpCBF` gl7$Feim$\":\")[-5%4q5WS,xAF`gl7$Fjim$\":s%R'*>^X)RB;6L#F`gl7$F_jm$\": j+\"*R9`fc3bMO#F`gl7$Fdjm$\":Ban]`HUpND$4BF`gl7$Fijm$\":t=UC%\\')GeM3v AF`gl7$F^[n$\":d7;+#Q'*eGeNCBF`gl7$Fc[n$\":'f&yU9/%4#)o7jBF`gl7$Fh[n$ \":NAT*G#4&Ql'G/K#F`gl7$F]\\n$\":&GwCb[UC-SA\"G#F`gl7$Fb\\n$\":LsV(>aR +@$>fE#F`gl7$Fg\\n$\":5q.$*[m>*)pM/P#F`gl7$F\\]n$\":a9o/Z$4(QY'p-BF`gl 7$Fa]n$\":F)oma4hJGR2cAF`gl7$Ff]n$\":.lx<+)eQ+!RML#F`gl7$F[^n$\":$[eTU KAL**=WEBF`gl7$F`^n$\":/k%=/3^FW'*[lAF`gl7$Fe^n$\"::#Q'p1N(yaSqpAF`gl7 $Fj^n$\":)pKx])HASM+UG#F`gl7$F__n$\":@Bjj2lS1q#GlBF`gl7$Fd_n$\":6T!**> Fn,fv]1BF`gl7$Fi_n$\":f.G-h?tm)p3]AF`gl7$F^`n$\":HjK4o'z3PwryAF`gl7$Fc `n$\":P40*o&4GI&)f(HBF`gl7$Fh`n$\":mcA/uf4f#zh&G#F`gl7$F]an$\":T71G&R% RPnceC#F`gl7$Fban$\":.PnN;6`D/,2=#F`gl7$Fgan$\":;jBi8vOQj7$F@$!1TxZm5G*[#F>7$FE$!1_!>Pi7tc$F>7$FJ$!1jj!=Qs<]%F> 7$FT$!1WIcu$=\\g%F>7$Fhn$!1sn]*\\(*Qt%F>7$F]o$!1U'\\8aP-#\\F>7$Fbo$!1 \"f#3Yc$oG&F>7$Fgo$!2i#>hlPn'G&F\\p7$F^p$!2>p!GW#**=G&F\\p7$Fcp$!24'*z $z$>Qt&F\\p7$Fhp$!2-nASwmcy&F\\p7$F]q$!27kms/9f'eF\\p7$Fbq$!2i!oH9ok() fF\\p7$Fgq$!2T2+M$ypmhF\\p7$F\\r$!2&f.ObJ>[hF\\p7$Far$!2e1T%yd7'*fF\\p 7$Ffr$!2(3s&)p=XlfF\\p7$F[s$!2v[:$)f*31jF\\p7$F`s$!2-zc\"G%e\"RhF\\p7$ Fes$!2Ff+`Q:?+'F\\p7$Fjs$!2COASyqy0'F\\p7$F_t$!2&[!*zdb/%H'F\\p7$Fdt$! 2^dJ/$e#*ziF\\p7$Fit$!2Hm*GjO-kiF\\p7$F^u$!2$QGo)*)\\VA'F\\p7$Fcu$!2\" 4_asq'F\\p7$Fhu$!2IUu=WAJ<'F\\p7$F]v$!3tg_S(\\!4%>'Fav7$Fcv$!3g_'>w r%y&G'Fav7$Fhv$!3*\\'=%o\\iT\\'Fav7$F]w$!3t%H#Qq()3gkFav7$Fbw$!3e`JesK *pH'Fav7$Fgw$!3F9*)p&*eU'G'Fav7$F\\x$!3T,pMX'[ee'Fav7$Fax$!3s!HX;H(H1k Fav7$Ffx$!3Kdm*H<&3wiFav7$F[y$!3Q9R$)o,)fV'Fav7$F`y$!3)z%oe]X0GlFav7$F ey$!3,y,DS\"yeW'Fav7$Fjy$!3_i%3iBLCP'Fav7$F_z$!3*)*yW690EL'Fav7$Fdz$!3 jeJ]1o2!Q'Fav7$Fiz$!3ZQB%eTDFZ'Fav7$F^[l$!3*zA+i+Emg'Fav7$Fc[l$!3&fgF7 peuc'Fav7$Fh[l$!37/;VMpX)R'Fav7$F]\\l$!3>jeebg%GT'Fav7$Fb\\l$!3!\\Gx'* oLVm'Fav7$Fg\\l$!30v5PW`\\plFav7$F\\]l$!3&GK%fk<&Fh`l7$F\\ dl$!5fm^:*)))fBg]Fh`l7$Fadl$!6<.b/6Eu'*=$\\Fedl7$Fgdl$!6NQ=\\GaK(o5VFe dl7$F\\el$!6pXB`X(y/YVOFedl7$Fael$!7J+YPE+%Hi0\"HFeel7$Fgel$!79&45.(y8 Y>\"4#Feel7$F\\fl$!7;$=5/=-d#3e7Feel7$Fafl$!7$G-%\\dk'Q]f(HFefl7$Fgfl$ \"7ajz6H,4vWSnFefl7$F\\gl$\"97r(>d8\\u!)[!G;F`gl7$Fbgl$\"92y/(44s$*ygw s#F`gl7$Fggl$\"9p\\YcL/$*3I[yOF`gl7$F\\hl$\":(*>$z&>(*G&\\\"[Ap%F`hl7$ Fbhl$\":to**=:eAB1e!4dF`hl7$Fghl$\":1:!zj=#[\"4r\"Hm'F`hl7$F\\il$\":Co SK'**z*yZ_%ztF`hl7$Fail$\":E8:lEmniC'p%>)F`hl7$Ffil$\":fmT;-ap)>b/Z!*F `hl7$F[jl$\":\"fLHLyV?OzG-5F`gl7$F`jl$\":-J6uHz%RtY0+5F`gl7$Fejl$\":HD (z&)**F`hl7$Fjjl$\":;qsS$pYD4`a))**F`hl7$F_[m$\":Xi;(*3H.(H?3. 5F`gl7$Fd[m$\":*>48;'p2V+'*o.\"F`gl7$Fi[m$\":A\"Q2yOV*zPFq3\"F`gl7$F^ \\m$\":Hjgl#zYh)eVZ2\"F`gl7$Fc\\m$\":+^kYgtgJ5%>j5F`gl7$Fh\\m$\":jJ.D7 An-N`*[5F`gl7$F]]m$\":!zx*GR%Q%3e5N6\"F`gl7$Fb]m$\":Ox8d:G3R>>93\"F`gl 7$Fg]m$\":U<$354CpcP9c5F`gl7$F\\^m$\":N2L()f'4(fK\"zy5F`gl7$Fa^m$\":gn ([w%e'3ulT+6F`gl7$Ff^m$\":?\"RM\\`Z5fkEq5F`gl7$F[_m$\":%QY?O&zpjBaT1\" F`gl7$F`_m$\":Q9zz^r;@\"HB36F`gl7$Fe_m$\":vV9fG>^@Zy_7\"F`gl7$Fj_m$\": 5i$eSb#)z(R3:6\"F`gl7$F_`m$\":jU;v&Gw1=q2*4\"F`gl7$Fd`m$\":lS'*p?X)\\f b%*)4\"F`gl7$Fi`m$\":')4/SCa&e43(**4\"F`gl7$F^am$\":\\l#=CYYKcZ#G5\"F` gl7$Fcam$\":Ni[2(QTmAEQ36F`gl7$Fham$\":*H%p'pe#e!)zWA8\"F`gl7$F]bm$\": 0QNf3@>pZDK=\"F`gl7$Fbbm$\":=.'p$H9.!oBfm6F`gl7$Fgbm$\":\">'zDx#QaxG\\+#pmPU.bZz6F`gl7$Facm$\":)eDZa8LVL7S)=\"F`gl7$Ffcm$ \":&=,i69.58Wuc6F`gl7$F[dm$\":hNaYg4wTV]U:\"F`gl7$F`dm$\"::u4q'RDod6'H <\"F`gl7$Fedm$\":]1s-(*)*>\"F`gl7$Fjdm$\":%zN%f.^$zr%*)3>\"F`gl7$ F_em$\":!Rp;&>Cr)Q+Kg6F`gl7$Fdem$\":R(Gl6cnT6h)e;\"F`gl7$Fiem$\":YLzF. \"*z$)o6$47F`gl7$F^fm$\":?JpbA%=]=')Rv6F`gl7$Fcfm$\":M?'R8T)p>\"F`gl7$Fb gm$\":JOH$GG@`I\"\\Z=\"F`gl7$Fggm$\":Wz3&p,)))=/ou<\"F`gl7$F\\hm$\":)z PT+,EPM`.s6F`gl7$Fahm$\":FNJ6DY#>_kEq6F`gl7$Ffhm$\":W!fEq(oP&Qpp*=\"F` gl7$F[im$\":&GWn&4o@X#\\yU7F`gl7$F`im$\":3)*z`az[Vy'[57F`gl7$Feim$\": \")[-5%4I8'R$y&=\"F`gl7$Fjim$\":s%R'*>^NU*H%497F`gl7$F_jm$\":j+\"*R9`) [vt&=B\"F`gl7$Fdjm$\":Ban]`H#)))=TR?\"F`gl7$Fijm$\":t=UC%\\wlM*zh=\"F` gl7$F^[n$\":d7;+#QETI>477F`gl7$Fc[n$\":'f&yU9/uBmUKB\"F`gl7$Fh[n$\":NA T*G#4v,3F6@\"F`gl7$F]\\n$\":&GwCb[7z)>y2>\"F`gl7$Fb\\n$\":LsV(>a*)\\vq U#=\"F`gl7$Fg\\n$\":5q.$*[mj@w\"fP7F`gl7$F\\]n$\":a9o/Z$\\cP<@-7F`gl7$ Fa]n$\":F)oma46H&fMw<\"F`gl7$Ff]n$\":.lx<+))3]#4/=7F`gl7$F[^n$\":$[eTU Ki&z4*z97F`gl7$F`^n$\":/k%=/3T)>+rG=\"F`gl7$Fe^n$\"::#Q'p1Ny\"[Qr%=\"F `gl7$Fj^n$\":)pKx])H@Z^!*G>\"F`gl7$F__n$\":@Bjj2lC-Wq`B\"F`gl7$Fd_n$\" :6T!**>FPQO?u/7F`gl7$Fi_n$\":f.G-h?*H6[7v6F`gl7$F^`n$\":HjK4o'*\\%fGz* =\"F`gl7$Fc`n$\":P40*o&4v2>sq@\"F`gl7$Fh`n$\":mcA/ufrYk\"=%>\"F`gl7$F] an$\":T71G&RuUPL^t6F`gl7$Fban$\":.PnN;6!\\6h1T6F`gl7$Fgan$\":;jBi8v0?t -<4\"F`gl7$F\\bn$\":&[Rjc*=n'H')Qb5F`gl7$Fabn$\":[a'4.)*QwFJU;**F`hl7$ Ffbn$\":(yV#)3'oyt'*\\@6*F`hl7$F[cn$\":pgb_m&[())z73(zF`hl7$F`cn$\":;0 #*oi(RO)Q'p.pF`hl7$Fecn$\":#*Gh,pk-!fAa5eF`hl7$Fjcn$\":E7f)Q+S=%G90o%F `hl-%&COLORG6&Fbdn$\"\"\"!\"\"$\"\")FairFcdn-Fhdn6#%3efficient~scheme~ AG-F$6%7ev7$F($\"0G^-2o\"*\\)F-7$Fcgp$\"0PW-$GmS`F-7$F/$\"0z,J*z(Qj#F- 7$F[hp$!/]c)\\*)o%>F-7$F5$!06m9O&p#*=F-7$F:$!1m'3Q9=[i%F>7$F@$!1TF+Mq$ =t'F>7$FE$!1_!yJ%z;v()F>7$FJ$!2j`KlVf41\"F>7$FT$!2WISg3nT2\"F>7$Fhn$!2 sZ^)=!\\M4\"F>7$F]o$!2U10%oiZI6F>7$Fbo$!2\"fg`$>RI@\"F>7$Fgo$!3i#\\*3. dC57F\\p7$F^p$!3>pH[e$Qm?\"F\\p7$Fcp$!34'*[!=8.VH\"F\\p7$Fhp$!3-n0zrHD /8F\\p7$F]q$!37kL9_[q?8F\\p7$Fbq$!3i!)3(>#\\5Z8F\\p7$Fgq$!3T2MhL4)zQ\" F\\p7$F\\r$!3&fjyXy_eQ\"F\\p7$Far$!3e1_'*f%Q6N\"F\\p7$Ffr$!3(3_,wy(pT8 F\\p7$F[s$!3v[K(>b;4U\"F\\p7$F`s$!3-zr^e@F$Q\"F\\p7$Fes$!3Ff'[)\\H\\^8 F\\p7$Fjs$!3COJS&4K8O\"F\\p7$F_t$!3&[g))3sGxT\"F\\p7$Fdt$!3^d1;4f799F \\p7$Fit$!3HmzK[-;59F\\p7$F^u$!3$Q3z1p03S\"F\\p7$Fcu$!3\"4st\\A)[#R\"F \\p7$Fhu$!3IU)p)Q%4wQ\"F\\p7$F]v$!4tgK#o)\\Q3R\"Fav7$Fcv$!4g_c7^#p<59F av7$Fhv$!4*\\'[4[@l\"e9Fav7$F]w$!4t%Hcm$QQEX\"Fav7$Fbw$!4e`@?h_FaT\"Fa v7$Fgw$!4F9pQ!H0N59Fav7$F\\x$!4T,4lTH#z![\"Fav7$Fax$!4s!H`#Gs\\.W\"Fav 7$Ffx$!4Kdm\\+ED(49Fav7$F[y$!4Q9z%\\H2vV9Fav7$F`y$!4)z%G@`O%on9Fav7$Fe y$!4,yh&*G3e!\\9Fav7$Fjy$!4_ikL(Rx3K9Fav7$F_z$!4*)*yy+:%>>U\"Fav7$Fdz$ !4jeJ3\"[\"\\3V\"Fav7$Fiz$!4ZQVRctG[X\"Fav7$F^[l$!4*zA/o'H!z#[\"Fav7$F c[l$!4&fgCR&)z>w9Fav7$Fh[l$!47/O]l!*zvV\"Fav7$F]\\l$!4>je-#)eE!Q9Fav7$ Fb\\l$!4!\\GpNF*oz\\\"Fav7$Fg\\l$!40v5)*=lMmZ\"Fav7$F\\]l$!4&GK3!G^\"* fX\"Fav7$Fa]l$!4MMbf'=kER9Fav7$Ff]l$!4(GC*pM]&HL9Fav7$$\":+++]iS\">8-, +9Fio$!4:$z$)Q%=G$=:Fav7$$\":++++D\"yvQ[)GS\"Fio$!4n&G9)p*>Rv9Fav7$$\" :+++](=UKk%fdS\"Fio$!4w8=tub\"**Q9Fav7$F[^l$!4\"3Df+1Dg`9Fav7$$\":+++] 7.dar3:T\"Fio$!4.+U7xU]d\\\"Fav7$$\":++++vVB5M$Q99Fio$!47i]@Y%4Wa9Fav7 $$\":+++]P%)*emzD<9Fio$!4#=s0z!=\\QV\"Fav7$F`^l$!4gjj&yEb=;:Fav7$Fe^l$ !4_G(QM-l:q9Fav7$Fj^l$!5,yn$4-6u>V\"F^_l7$F`_l$!5SjYvp4=1_9F^_l7$Fe_l$ !5HSz/'3(4&GZ\"F^_l7$Fj_l$!5()e%>VnI:*G9F^_l7$F_`l$!5&[c^U+Q7BQ\"F^_l7 $Fd`l$!6/\\,E_OnTUH\"Fh`l7$Fj`l$!6d'pHx,f7@c7Fh`l7$F_al$!6faA2=\"Fh`l7$Fial$!6%e-:us*R(yW6Fh`l7$F^bl$!6KB] !HUAa9j6Fh`l7$Fcbl$!6k=l%y/!Q+&[6Fh`l7$Fhbl$!6`?$4-Y&4xu;\"Fh`l7$F]cl$ !6O4@4#z]NtN6Fh`l7$Fbcl$!6(QXt68Ln`>6Fh`l7$Fgcl$!6y^D^C_eP:<\"Fh`l7$F \\dl$!6fm^&>]D[LU6Fh`l7$Fadl$!7<.b/H[=8_86Fedl7$Fgdl$!6NQ=\\lT`aLu*Fed l7$F\\el$!6pXB`<0wg'f#)Fedl7$Fael$!7J+YPEi\"z:\\h'Feel7$Fgel$!79&45.nk d8\\Fd@P`$F`gl7$Fbgl$\"92y/(44#)pBl Z*fF`gl7$Fggl$\"9p\\YcL/n[`,H\")F`gl7$F\\hl$\":+Kz&>(*=T1^RP5F`gl7$Fbh l$\":(o**=:e#4\"yORo7F`gl7$Fghl$\":^,zj=#3'*[ad#[\"F`gl7$F\\il$\":#oSK '**z>5*R#Hk\"F`gl7$Fail$\":L^^miw;%z`sC=F`gl7$Ffil$\":m;k@S&4?*>)G=?F` gl7$F[jl$\":\"fLHLyj'f>4$QAF`gl7$F`jl$\":-J6uHz>*[DLLAF`gl7$Fejl$\":`s z48;'pAcVr2J#F`gl7$Fi[m$\":A\"Q2yO.zLf,HCF`gl7$F^\\m$\":Hjg l#zmd#>)p,CF`gl7$Fc\\m$\":+^kYgtL`'QrvBF`gl7$Fg]m$\":U<$3549GpP`eBF`gl 7$F[_m$\":%QY?O&zxsbwLP#F`gl7$F`_m$\":Q9zz^rW;E))HZ#F`gl7$Fe_m$\":vV9f G>.Dt]b^#F`gl7$Fj_m$\":5i$eSb#HP(f'[[#F`gl7$F_`m$\":jU;v&GYeW[ncCF`gl7 $Fd`m$\":lS'*p?X#>tQ)fX#F`gl7$Fi`m$\":')4/SCat]%RgdCF`gl7$F^am$\":\\l# =CYc=o#HIY#F`gl7$Fcam$\":Ni[2(Q\"zxs-VZ#F`gl7$Fham$\":*H%p'peK_WF+EDF` gl7$F]bm$\":0QNf3@<=NVXk#F`gl7$Fbbm$\":=.'p$H9d.KV(4EF`gl7$Fgbm$\":\"> 'zDx#e,aHdfDF`gl7$F\\cm$\":>\\+#pmdD&p.Nj#F`gl7$Facm$\":)eDZa8t5]8,fEF `gl7$Ffcm$\":&=,i69.*z-)H(e#F`gl7$F[dm$\":hNaYg4?+g(zwDF`gl7$F`dm$\":: u4q'RX-y,UCEF`gl7$Fedm$\":]WKG\">\"o#F`gl7$Fjdm$\":%zN%f.^^/?][m# F`gl7$F_em$\":!Rp;&>CK#RH@&f#F`gl7$Fdem$\":R(Gl6c(RuZOEg#F`gl7$Fiem$\" :YLzF.\"\\?b_B1FF`gl7$F^fm$\":?JpbA%Q!GaJ+j#F`gl7$Fcfm$\":M?'R8T3\"fm# >'e#F`gl7$Fhfm$\":)yQ!Q&\\fYM7#ft#F`gl7$F]gm$\":H()**3nP\"Rn#o$pEF`gl7 $Fggm$\":Wz3&p,y$))GTZj#F`gl7$Fahm$\":FNJ6DY$QJ%fjh#F`gl7$$\":++]P%)*e /rKMpQFio$\":B&e2F:!R')yxLq#F`gl7$$\":+++v=UKFFz?(QFio$\":JQ'y&Hy'*)4$ \\&=FF`gl7$$\":++]7`%*=WF:[(QFio$\":Gv/z.!3<5Gn[EF`gl7$$\":+++](oa5w7b xQFio$\":WdBn\"R\\lA)y[i#F`gl7$$\":++](=#*>zxsG!)QFio$\":#)[QqlK:>>'fs FF`gl7$$\":+++Dc^y%zK-$)QFio$\":TZd)p*e-\"[Y)zp#F`gl7$$\":++]i!R];\"Gf d)QFio$\":nH8Ac9jd-!=NEF`gl7$Ffhm$\":W!fEq(o13/jol#F`gl7$$\":+++]il(f* GR%**QFio$\":?-sgs\\IJ)=)es#F`gl7$F[im$\":&GWn&4oK%R_f#y#F`gl7$F`im$\" :3)*z`az$>'pr/r#F`gl7$Feim$\":\")[-5%4!Qgg)=`EF`gl7$Fjim$\":s%R'*>^0T' [)>8FF`gl7$F_jm$\":j+\"*R9`Spo=%fFF`gl7$Fdjm$\":Ban]`Hc\"G,z'p#F`gl7$F ijm$\":t=UC%\\wLefWaEF`gl7$F^[n$\":d7;+#Q;<)pY+r#F`gl7$Fc[n$\":'f&yU9/ u6Z]Qw#F`gl7$Fh[n$\":NAT*G#4LO*3P9FF`gl7$F]\\n$\":&GwCb[Ul(3(GoEF`gl7$ Fb\\n$\":LsV(>a\\LR?!\\k#F`gl7$Fg\\n$\":5q.$*[mKC?`Sx#F`gl7$F\\]n$\":a 9o/Z$zf:\\g%p#F`gl7$Fa]n$\":F)oma4T9opvOEF`gl7$Ff]n$\":.lx<+)e43l\"es# F`gl7$F[^n$\":$[eTUK_8f.5BFF`gl7$F`^n$\":/k%=/3@F]#H/l#F`gl7$Fe^n$\":: #Q'p1N(HBah\\EF`gl7$Fj^n$\":)pKx])H_+#\\'Qn#F`gl7$F__n$\":@Bjj2lz1-s'o FF`gl7$Fd_n$\":6T!**>FZBM9$3q#F`gl7$Fi_n$\":f.G-h?)y\\uxKEF`gl7$F^`n$ \":HjK4o'\\\\\"=j6m#F`gl7$Fc`n$\":P40*o&4GBlm'GFF`gl7$Fh`n$\":mcA/uf&* [_-un#F`gl7$F]an$\":T71G&R/)>!pVIEF`gl7$Fban$\":.PnN;6ASr&*)eDF`gl7$Fg an$\":;jBi8v\">*)G?YCF`gl7$F\\bn$\":&[Rjc*=*3H=&eO#F`gl7$Fabn$\":Xl4.) *Q!zS\"y\"HAF`gl7$Ffbn$\":zV#)3'o9dD^^Z?F`gl7$F[cn$\":2cDlcoz:hGZz\"F` gl7$F`cn$\":^?*oi(f49iN_b\"F`gl7$Fecn$\":*Gh,pk@*=s$Q88F`gl7$Fjcn$\":B \"f)Q+q]e\"yai5F`gl-F`dn6&FbdnFddnFcdnFcdn-Fhdn6#%3efficient~scheme~BG -F$6%7ev7$F($\"0GudH4l)yF-7$Fcgp$\"0P_*4ZVlaF-7$F/$\"0z:/MgkT$F-7$F[hp $\"0]`./4?Q\"F-7$F5$\".*))H)e]Q(F-7$F:$!1m1L*e(3U>F>7$F@$!1T(f*)[w1Z$F >7$FE$!1_5S?6f.\\F>7$FJ$!1j$3%*)eUVhF>7$FT$!1Wg`aOEtiF>7$Fhn$!1sZTiIrG kF>7$F]o$!1UO8'**4\"omF>7$Fbo$!1\"f'Gz-3%=(F>7$Fgo$!2i#>F;?V\"=(F\\p7$ F^p$!2>pB*fMQsrF\\p7$Fcp$!24'H'=e%HlxF\\p7$Fhp$!2-n(>NwDHyF\\p7$F]q$!2 7k=Bz9@$zF\\p7$Fbq$!2i!Q==i8&4)F\\p7$Fgq$!2T2)*p@]kM)F\\p7$F\\r$!2&ft4 TV;O$)F\\p7$Far$!2e1%G!>&*p7)F\\p7$Ffr$!2(3_M%fI(o!)F\\p7$F[s$!2v[%\\5 !y,b)F\\p7$F`s$!2-zeF\")HOK)F\\p7$Fes$!2Ff^*QbrJ\")F\\p7$Fjs$!2CO\"))f !=(*=)F\\p7$F_t$!2&[SB#e\"zL&)F\\p7$Fdt$!2^dD:zIY^)F\\p7$Fit$!2Hm:yDEI \\)F\\p7$F^u$!2$Q)z(z!e'Q%)F\\p7$Fcu$!2\"4-\"zQ<,R)F\\p7$Fhu$!2IU&Hz]w h$)F\\p7$F]v$!3tg7'GwO=Q)Fav7$Fcv$!3g_1_nBE*\\)Fav7$Fhv$!3*\\'==?Ib\"z )Fav7$F]w$!3t%Hx0$>Cg()Fav7$Fbw$!3e`,r56ON&)Fav7$Fgw$!3F9H*)*)fw-&)Fav 7$F\\x$!3T,*)[$3P6$*)Fav7$Fax$!3s!H\\`nYro)Fav7$Ffx$!3KdmM37I,&)Fav7$F [y$!3Q9f1m.g0()Fav7$F`y$!3)z%eFc(pI&))Fav7$Fey$!3,yh*o<\\:u)Fav7$Fjy$! 3_iukCOiR')Fav7$F_z$!3*)*ytyVB!y&)Fav7$Fdz$!3jeT-GnIJ')Fav7$Fiz$!3ZQLM Oq8z()Fav7$F^[l$!3*zAgTM1m%*)Fav7$Fc[l$!3&fgG)[Du3*)Fav7$Fh[l$!37/Y6,8 6v')Fav7$F]\\l$!3>j)o?@]cn)Fav7$Fb\\l$!3!\\G.79]1/*Fav7$Fg\\l$!30vSR1d L7*)Fav7$F\\]l$!3&GK+;,%*zy)Fav7$Fa]l$!3MMXXX6!oo)Fav7$Ff]l$!3(GCH&\\# e)\\')Fav7$Fids$!3:$z,x)Fav7$F[fs$!3.+imq5XH!*Fav7$F`fs$!37iI(e*) [)z()Fav7$Fefs$!3#=s$*>P]Ol)Fav7$F`^l$!3gj.lug1`\"*Fav7$Fe^l$!3_GP7\\= dv))Fav7$Fj^l$!4,yn*R]bIW')F^_l7$F`_l$!4SjYxRCc%Fefl7$Fgfl$\"7ajz6H,A&fFf)Fe fl7$F\\gl$\"97r(>d8z#e9=_@F`gl7$Fbgl$\"92y/(44U8\"3kTOF`gl7$Fggl$\"9p \\YcL/(y8.A$\\F`gl7$F\\hl$\":&*>$z&>(*3&4>A(G'F`hl7$Fbhl$\":to**=:e#>i sL(o(F`hl7$Fghl$\":1:!zj=#y.K9F)*)F`hl7$F\\il$\":AoSK'**zn$4\\5&**F`hl 7$Fail$\":L^^miw2)RF!\\5\"F`gl7$Ffil$\":m;k@S&p?#HX@A\"F`gl7$F[jl$\": \"fLHLy$RfQt_N\"F`gl7$F`jl$\":-J6uHzJ#*eIAN\"F`gl7$Fejl$\":`sz\\a8F`gl7$Fd[m $\":*>48;'pEj%3b)R\"F`gl7$Fi[m$\":A\"Q2yO$34(*41Z\"F`gl7$F^\\m$\":Hjgl #zY=o\\0a9F`gl7$Fc\\m$\":+^kYgt[:jx#Q9F`gl7$Fg]m$\":U<$354ascerF9F`gl7 $F[_m$\":%QY?O&z\"*\\O3kV\"F`gl7$F`_m$\":Q9zz^roW%Gz'\\\"F`gl7$Fe_m$\" :vV9fG>*o\\C!H_\"F`gl7$Fj_m$\":5i$eSbKW&*zI/:F`gl7$F_`m$\":jU;v&G;_7`< ([\"F`gl7$Fd`m$\":lS'*p?X$y@Lp'[\"F`gl7$Fi`m$\":')4/SCac-['e([\"F`gl7$ F^am$\":\\l#=CY'*yy)f2\\\"F`gl7$Fcam$\":Ni[2(Q\"\\!3#fu\\\"F`gl7$Fham$ \":*H%p'pesgjKdG:F`gl7$F]bm$\":0QNf3@\"GR)o1g\"F`gl7$Fbbm$\":=.'p$H9#f E0vz:F`gl7$Fgbm$\":\">'zDx#=O'z?\"\\:F`gl7$F\\cm$\":>\\+#pmn#fxUOf\"F` gl7$Facm$\":)eDZa8$=Q?U&4;F`gl7$Ffcm$\":&=,i69jLhr/m:F`gl7$F[dm$\":hNa Yg4R/re#f:F`gl7$F`dm$\"::u4q'Rv'y*Hd)e\"F`gl7$Fedm$\":]\"e!HHEi\" F`gl7$Fjdm$\":%zN%f.^bz*408;F`gl7$F_em$\":!Rp;&>C')*Q1yq:F`gl7$Fdem$\" :R(Gl6cZF)pQ[d\"F`gl7$Fiem$\":YLzF.\"RbV84Q;F`gl7$F^fm$\":?JpbA%o\"y\" Q$>f\"F`gl7$Fcfm$\":M?'R8T[,F@1l:F`gl7$Fhfm$\":)yQ!Q&\\ftK)yfl\"F`gl7$ F]gm$\":H()**3nPRLWfdh\"F`gl7$Fggm$\":Wz3&p,[\"oq;Zf\"F`gl7$Fahm$\":FN J6DY:=\\NLe\"F`gl7$F[et$\":B&e2F:q\"z?oej\"F`gl7$F`et$\":JQ'y&HyUrdrak \"F`gl7$Feet$\":Gv/z.!GJGs5.;F`gl7$Fjet$\":WdBn\"Rz(*f4J)e\"F`gl7$F_ft $\":#)[QqlK\"Q'Qt\"y;F`gl7$Fdft$\":TZd)p*e$H!42Ij\"F`gl7$Fift$\":nH8Ac 9NQ,M[f\"F`gl7$Ffhm$\":W!fEq(oa7%fb2;F`gl7$Fagt$\":?-sgs\\)pMF\\\\;F`g l7$F[im$\":&GWn&4oE%f*)=%o\"F`gl7$F`im$\":3)*z`az!RfF]S;F`gl7$Feim$\": \")[-5%4!pRH*f0;F`gl7$Fjim$\":s%R'*>^&ym%>fT;F`gl7$F_jm$\":j+\"*R9`HVc _+n\"F`gl7$Fdjm$\":Ban]`HQbQ(3K;F`gl7$Fijm$\":t=UC%\\1V!exhg\"F`gl7$F^ [n$\":d7;+#Qm!pV\"eR;F`gl7$Fc[n$\":'f&yU9/^@=DEn\"F`gl7$Fh[n$\":NAT*G# 4DjRiEk\"F`gl7$F]\\n$\":&GwCb[U'[0,Zh\"F`gl7$Fb\\n$\":LsV(>a*)pMr8+;F` gl7$Fg\\n$\":5q.$*[m!*H)>wy;F`gl7$F\\]n$\":a9o/Z$*zp*zmI;F`gl7$Fa]n$\" :F)oma4TOUfS&f\"F`gl7$Ff]n$\":.lx<+)o&R8%=\\;F`gl7$F[^n$\":$[eTUK7q3J \"zk\"F`gl7$F`^n$\":/k%=/3\"z>nHQg\"F`gl7$Fe^n$\"::#Q'p1N^b@4Hg\"F`gl7 $Fj^n$\":)pKx])H\"\\a\"y!=;F`gl7$F__n$\":@Bjj2l?$HsRv;F`gl7$Fd_n$\":6T !**>Fx!z04Wj\"F`gl7$Fi_n$\":f.G-h?F'\\&pIf\"F`gl7$F^`n$\":HjK4o'z[bl') 4;F`gl7$Fc`n$\":P40*o&4%)y![C^;F`gl7$Fh`n$\":mcA/uf:/2+-i\"F`gl7$F]an$ \":T71G&RCe/@p\"f\"F`gl7$Fban$\":.PnN;6c\"p*p#[:F`gl7$Fgan$\":;jBi8vDQ If'z9F`gl7$F\\bn$\":&[Rjc*=.N@!)3V\"F`gl7$Fabn$\":Xl4.)*Q)p*Qi%[8F`gl7 $Ffbn$\":zV#)3'o*o$R69Q7F`gl7$F[cn$\":2cDlcooAyI_3\"F`gl7$F`cn$\":;0#* oi(p!\\t4/S*F`hl7$Fecn$\":#*Gh,pkR'p;=PzF`hl7$Fjcn$\":E7f)Q+g2\"QLwT'F `hl-F`dn6&Fbdn$\")#)eqkFfdn$\"))eqk\"FfdnFhaw-Fhdn6#%3efficient~scheme ~CG-F$6%7\\x7$F($\"0Gi,a'[!*zF-7$F5$\"1*yE^(ynp5F-7$F@$\"2fA`BzWOD\"F> 7$FE$\"2[p`e\\(oz8F>7$FJ$\"2PEX:#oB?:F>7$FO$\"2Cs,x*)*G6:F>7$FT$\"2c4& HOX(>]\"F>7$FY$\"2n))R#)[&3%\\\"F>7$Fhn$\"2G_(R*)[C&\\\"F>7$F]o$\"2e88 '*QSw_\"F>7$Fbo$\"24M(zyCRE;F>7$Fgo$\"3Q2Bmbmz7;F\\p7$F^p$\"3\"3j>4,Y! *f\"F\\p7$$\":+++]7GQAH7^-\"Fio$\"3'zRhKk$z(e\"F\\p7$$\":++++D19%47tO5 Fio$\"3\\om%HZ<`e\"F\\p7$$\":++]7y+3(QfjR5Fio$\"3ITs]W'f@m\"F\\p7$$\": +++DJ&>+o1aU5Fio$\"3_E7O;F\\p7$$\":++++]ilP/p*f5Fio$\"3nCNn;LX8OR#QGHr\"F\\p7$Fgq$ \"3f#**[=^INw\"F\\p7$F\\r$\"30kMntzpj'GfGD $=Fav7$F]w$\"4F0P#e6BxG=Fav7$Fbw$\"4UY=+k&30\"y\"Fav7$Fgw$\"4t&3*\\SJ0 ,x\"Fav7$F\\x$\"4f)40>x`/j=Fav7$Fax$\"4G4xe.BZ?\"=Fav7$Ffx$\"4oUBBsT08 x\"Fav7$F[y$\"4i&3oD#eHSe-4#= Fav7$Fjy$\"4[PXZ,,.%y)=<*y\"Fav7$$\":++++DcEU$y))z8Fio$\"4e%*y&R8 -t))=Fav7$$\":+++](oHzfCw#Q\"Fio$\"4`uk$[t@\"y%=Fav7$Fa]l$\"4mlWP(e&z+ !=Fav7$$\":+++]7yD4r6&)Q\"Fio$\"4Wz3AzV*o3=Fav7$$\":++++v=#\\OjQ\"R\"F io$\"4!>(e2RO8M(=Fav7$$\":+++]Pfe?'4E%R\"Fio$\"4QT&)>^]L6#=Fav7$Ff]l$ \"48d(o6$4D&*y\"Fav7$Fids$\"4&o?;\\ZCTB.A\"=Fav7$F[fs$\"4(**z#*>IAVb()**Rcy\"F^_l7$F`_l$\"5 gO`IJ%*>+5=F^_l7$Fe_l$\"5rf?Tp%H%\\L=F^_l7$Fj_l$\"58T0y2G)fox\"F^_l7$F _`l$\"5:N%)[O;#3or\"F^_l7$Fd`l$\"6'4&)R(491)>*f\"Fh`l7$Fj`l$\"6V..Z08* HT`:Fh`l7$F_al$\"6TXxs8/-=r_\"Fh`l7$Fdal$\"6A()\\4Nv6u#e9Fh`l7$Fial$\" 6;u\\yYp(Re39Fh`l7$F^bl$\"6ow\\fd*e!HQV\"Fh`l7$Fcbl$\"6O\"[`<$>yY#*R\"Fh`l7$Fbcl$\"68 YlsHq:_mP\"Fh`l7$Fgcl$\"6A[uG;:7!3V9Fh`l7$F\\dl$\"6TL[u-v)3`19Fh`l7$Fa dl$\"7$o\\a*em^3Xq8Fedl7$Fgdl$\"7lh\"3Dxv2\"*R>\"Fedl7$F\\el$\"7JawYoQ WT/15Fedl7$Fael$\"7p*RDOx#RX!f)zFeel7$Fgel$\"7'[!**oH`h*4$*o&Feel7$F\\ fl$\"7%o\")*e>$*z,2[LFeel7$Fafl$\"7aOIDZF`gl7$Fbgl$!9$>_H!4zh7JO5yF`gl7$Fggl$!:I ]`Vm&H0VN#e/\"F`gl7$F\\hl$!:+o?/G5x*p@I>8F`gl7$Fbhl$!:8.5[=uP/,/:h\"F` gl7$Fghl$!:\\)4i8yrM*\\%ew=F`gl7$F\\il$!:=$fn.+7TTuxr?F`gl7$Fail$!:n[[ LPB#HY'RJH#F`gl7$Ffil$!:M$e$yf/\"Hrk\"[`#F`gl7$F[jl$!:4k1n;iHs![g1GF`g l7$F`jl$!:)*o)e-2i!e82'*z#F`gl7$Fejl$!:ZF?#e7))\\)4LPz#F`gl7$Fjjl$!:)H Ff1Ll#)4Ah\"z#F`gl7$F_[m$!:bP$G54x_$Q^&)z#F`gl7$Fd[m$!:,3pQQIZWdCY)GF` gl7$Fi[m$!:y=E>Kmd))oY3/$F`gl7$F^\\m$!:rORM2K$=N351IF`gl7$Fc\\m$!:+\\N `RE#3;![D(HF`gl7$Fh\\m$!:Po'\\xy2x%of?#HF`gl7$F]]m$!:5A-rg:'z'>1R6$F`g l7$Fb]m$!:kA'GW=FKBV+CIF`gl7$Fg]m$!:e#o\"**3f@$)f))z%HF`gl7$F\\^m$!:l# pE,M!['Q\"*z**HF`gl7$Fa^m$!:SK7N_TI/!p%p2$F`gl7$Ff^m$!:!)3c1lC+=W*z!*H F`gl7$F[_m$!:;O&zj/U_AgvgHF`gl7$F`_m$!:i&3-#[G$)RE&)f3$F`gl7$Fe_m$!:Dc &392)*zMTYXJF`gl7$Fj_m$!:!zjTfWZ)*\\6^1JF`gl7$F_`m$!:Pd$[Ur$oA7)))pIF` gl7$Fd`m$!:Nf.Izajc0_w1$F`gl7$Fi`m$!:9!f*fvX&p%>dy1$F`gl7$F^am$!:^MY19*yYx?=vH$F`gl7$Fbbm$!:#oRI1d[Y'o5kD$F`gl7$Fgbm$!:4Q?uA

Y]q\"\\ib*3MF`gl7$F]gm$!:r7+\"HB'e*)[ssK$F`gl7$Fggm$!:c?\"\\I)>E_ $)H@G$F`gl7$Fahm$!:tko)[P0i,%3TD$F`gl7$F[et$!:x9CHZ)z]Vs?fLF`gl7$F`et$ !:ph8Uq@UoefgQ$F`gl7$Feet$!:sC&4i*>5H&H(yH$F`gl7$Fjet$!:cUwK31,v(*z4E$ F`gl7$F_ft$!:=^hHMn*ema'GX$F`gl7$Fdft$!:f_U,.T.[K`)fLF`gl7$Fift$!:Lq'y PaQzC+#*yKF`gl7$Ffhm$!:c4M(H7``DJ3)H$F`gl7$Fagt$!:!yz#RF]Ub;6kQ$F`gl7$ F[im$!::dDV!>BU1JujMF`gl7$F`im$!:#>+ia/#)*3t'zsLF`gl7$Feim$!:>^(**e!*z gGy(oH$F`gl7$Fjim$!:G0O+)[/F%GAUO$F`gl7$F_jm$!:P**3g&o9p.Vrgt\"zVP$F`gl7$F[^n$!:<:%edn*Gj'*y 9'G$F`gl7$Fe^n$!:&yh.L\\')zL&)*pF$F`gl7$Fj^n$!:-tE#\\,xck2,;LF`gl7$F__ n$!:zwOO#\\j_b1:KMF`gl7$Fd_n$!:*)e4+GFe=_.$\\LF`gl7$Fi_n$!:T'>x*Qzsy@Q AE$F`gl7$F^`n$!:rOn!>LSm1t\"**G$F`gl7$Fc`n$!:j!\\4J/pW5TK$Q$F`gl7$Fh`n $!:MVx&f-u-,P0>LF`gl7$F]an$!:f(Q>Zg&RO2N!fKF`gl7$Fban$!:(HEVO)))pP14^; $F`gl7$Fgan$!:%ojxj[ilu;*[,$F`gl7$F\\bn$!::0mL/\"eM[k&*3HF`gl7$Fabn$!: bM!p>5YS![NCu#F`gl7$Ffbn$!:@c<\"RJK$z#Q&p]#F`gl7$F[cn$!:$RWZL9NUv2L$># F`gl7$F`cn$!:\\z5tB+wRXK&*)=F`gl7$Fecn$!:6(Q)4`L^&=vV)e\"F`gl7$Fjcn$!: x39h**p7lcK;F\"F`gl-F]ir6&Fbdn$\"\"'FairFcdnFbir-Fhdn6#%Sscheme~with~s tability~radius~=~imag~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TI TLEG6#%Uerror~curves~for~7~stage~order~5~Runge-Kutta~methodsG-%+AXESLA BELSG6$Q\"x6\"Q!Fj\\z-%%VIEWG6$;F(Fjcn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a mino r modification of the Bogacki-Shampine scheme" "efficient scheme A" "e fficient scheme B" "efficient scheme C" "scheme with stability radius \+ = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 15 of 7 stage, order 5 Runge-Kutta methods " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Com paring Numerical Methods for Ordinary Differential Equations, Hull, En right, Fellen and Sedgwick,\n Siam Journal on Numerical Analysi s, Vol. 9, No. 4 (Dec. 1972), page 617, Example A5." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d y/dx = (y-x)/(y+x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&%\"yGF&%\"xGF(F&,& F+F&F,F&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(1) = 1;" "6#/-%\" yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/( x^2))+4*arctan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6# *&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*&F0F' F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x);\nic := y(1)=1;\ndsolve( \{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$- %\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,&F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\"#\"\"\"-%#lnG6#*&,&*$)F'F-F.F. *$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-%'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\" \"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#Pi GF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The solutio n (for " }{TEXT 285 1 "x" }{TEXT -1 47 " increasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,& *&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\"\"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ln((x^2+y^2))+2*arctan(y/x)= ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-F-F-*&F,F--%'arctanG6#*&F 0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',&F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot( [sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4],coords=polar,thickness=2,color=r ed):\np2 := plot([sqrt(2)*exp(Pi/4-t),t,t=Pi/4..2*Pi],coords=polar,col or=black,linestyle=2):\np3 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi /4],coords=polar,color=black,linestyle=2):\np4 := plot([[[1,1],[uu,-uu ]]$4],style=point,symbol=[circle$2,diamond,cross],\n \+ symbolsize=[12,10$3],color=[black,green$3]):\nplots[display]([p1,p2,p3 ,p4],font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"3F_`'4Qx/\"[!# <$!3\"o*>&>Qx/\"[F*7$$\"3.\"H4?nl\\![F*$!3+OcK;[q'[%F*7$$\"3[WM^6Oe\"z %F*$!34+VrCq:9UF*7$$\"3m#o2Wj>ww%F*$!3B=.>Xj!y\"RF*7$$\"3bf)4;S8Yt%F*$ !3:ymd`!*HIOF*7$$\"3U*G$3T;c$p%F*$!3i\"z4>R][N$F*7$$\"3)>>'Q`Nt[YF*$!3 pIdcz?N/uDVF*$!3Mk<%4i8_'>F*7$$\"3neCR^2fVUF*$!3U`V:L2,dAqSF*$!3]0%Rz+R[P\"F*7 $$\"3CpGt*G2)))RF*$!3Vje1<#>j@\"F*7$$\"31&*>Xy@@*)QF*$!3KrYY*z8q.\"F*7 $$\"3m&*QGM(*y-QF*$!3Z+5yDwaF*)!#=7$$\"3x^E')y`D+PF*$!3%[->)G'=aL(F\\q 7$$\"3v7s?m@*zg$F*$!3c,0?p.L,gF\\q7$$\"3'*yXFYgX0NF*$!3H?*=wI4nh%F\\q7 $$\"3W4K\"RWEoS$F*$!3%3mwkU\\IP$F\\q7$$\"3sJbIp3<.LF*$!3OMj)e)4)4:#F\\ q7$$\"3$p#*\\.zXv?$F*$!3gM#f*yWs%4\"F\\q7$$\"368R(=GrT5$F*$!3oLM#4e`nS #!#?7$$\"3O349%H!z'*HF*$\"3?)\\7Z#>+:5F\\q7$$\"3_&4uxb.N!HF*$\"3?;dqg# **3'=F\\q7$$\"3Q!f))4wMJ!GF*$\"3Rn1jA(f_r#F\\q7$$\"3`xr$)p%\\+q#F*$\"3 7>6@EK]NNF\\q7$$\"3;#Rcs]s**f#F*$\"3]^E27l!)yUF\\q7$$\"3Y 0BF*$\"3I%H$\\j$zL='F\\q7$$\"3**>nP`dw1AF*$\"30SJ7l:PHnF\\q7$$\"30Xcw+ 8\"*=@F*$\"3CV>J\"e5-=(F\\q7$$\"3Xb+T*Q]zR\"\\ti(F\\q7$$ \"3\"GM^D8wq$>F*$\"3\\$HR%\\7H1!)F\\q7$$\"3OsIgb1\\Z=F*$\"3#Q*3z#yPy=*F\\q7$$\"3tj&H(oDW3:F*$\"3 !)e5j3rW%R*F\\q7$$\"3Hm+#y!yfG9F*$\"3U,&*3jpEm&*F\\q7$$\"3YmX)QzoqN\"F *$\"3]k9op*4mp*F\\q7$$\"3s5rrvvGx7F*$\"3P5fR'>0`\")*F\\q7$$\"3XZ9deE%z ?\"F*$\"3Gc$*zYS?&*)*F\\q7$$\"30pa^\"yv$))F\\q$\"3]-mK&*3^l**F\\q7$$\"3eM?/#4dlu(F\\q$\"3uEm +DD_n)*F\\q7$$\"3%*yjGB$o\\&oF\\q$\"3AaVWo?rO(*F\\q7$$\"3[>li88H;gF\\q $\"3w@=8&*eNF\\q$\"3yNhC*4%R#y)F\\q7$$ \"3#*HGP\"4-3'GF\\q$\"3/%=\\C*>ai%)F\\q7$$\"3E\"4(yFv\\CAF\\q$\"3&z[> \"H4QC\")F\\q7$$\"3&3B;*phCW;F\\q$\"3u\"3vLbP3x(F\\q7$$\"3SY8\"*=2Ba6F \\q$\"3unuMyI\"GV(F\\q7$$\"3'Qw\"f#\\*)H3(!#>$\"3)pj-cf?z3(F\\q7$$\"3f (3R5[t$3HFc_l$\"3f7r>seQEnF\\q7$$!37`UU,+,&Q)F`s$\"3yN-Jd*yHO'F\\q7$$! 3$)3\"RF\\q$\"3#4c;J=K@(HF\\q7$$!3iuBr]$3f1#F\\ q$\"3Gm]`0-<5CF\\q7$$!3G4mSr>(e2#F\\q$\"30UK<,&yQ#>F\\q7$$!3#o3.QX91/# F\\q$\"3P$*fjX!f&H:F\\q7$$!3#y(eD!G(*f&>F\\q$\"3UEc#G.d\"=6F\\q7$$!3=q le`Y)o&=F\\q$\"37JSjs:(p;)Fc_l7$$!30utN:SE>B.UUNGWgF`s7$$!3zWz&))>R47\"F\\q$!3Unq>(fHx'=F c_l7$$!3_y,&)zs9)3t![Fc_l$!3_$o;W4rzI%Fc_l7$$!3E\\<<**)*)Rz$Fc_l$!3 e#*))[vi%\\I%Fc_l7$$!3>P#z)H$>z\"HFc_l$!3)y!G_rWw*>%Fc_l7$$!3At>+u$>i< #Fc_l$!3'oQ0i'\\&Q-%Fc_l7$$!3Q^mSZRTp9Fc_l$!3_uFr$)*ycw$Fc_l7$$!3-[9Nr!)oebDFc_l7$$ \"3Wcg/C\"='=WF`s$!37)o9o#3(GD#Fc_l7$$\"33dhy:jnDiF`s$!3?g29(Qlz%>Fc_l 7$$\"3-K8`c=f([(F`s$!35q&z[NHem\"Fc_l7$$\"3ckwjCv\"HN)F`s$!3EJf'zD?4R \"Fc_l7$$\"3*>iL/L$Q;))F`s$!3I#Q\\]N!=[6Fc_l7$$\"3[Kf]CL_\")*)F`s$!36H bzi()GSn;F`s7$$\"3/Xu8A%33 ['F`s$!3;Is^'z2(yxFehl7$$\"3)Q4tG5rBz&F`s$\"3c0'H(*oS@v%!#H-Fjz6&F\\[l Fa[lFa[lFa[l-%*LINESTYLEGFd[l-F$6%7S7$$\"3W'4ORO![>WF*$!3Ar[\"*HXwawF* 7$$\"3VdsTJRpPWF*$!3M'>\"=)=2ge(F*7$$\"3'G4%*HMRJX%F*$!38,ZT))oTEvF*7$ $\"3_L#[)Gh1qWF*$!3SWWx3PmfuF*7$$\"3_-)f6JNm[%F*$!3#o7ACd\\FR(F*7$$\"3 akH:B>m-XF*$!3:r-1\\\\VEtF*7$$\"3C*o#*G!46\" Q)[RgXF*$!35r.'eb*[sqF*7$$\"3I*G!GQ%[pF*7$$\"3u*H#3E-h*f%F*$!3wdnf&y\"pUX'F*7$$\"3CLFg!*\\T'o%F*$!3WZZJr$p:R'F*7$$\"39!\\a\\())R&p% F*$!3JUA:p?9KjF*7$$\"3*yZODQ#Q/ZF*$!3eR\\wA`UqiF*7$$\"3#Gs'H4VG7ZF*$!3 @Fo#z)3,9iF*7$$\"3zqGL?zV?ZF*$!3[?z6=2W`hF*7$$\"3,4vY0\\]GZF*$!3')*GM% 4S$34'F*7$$\"3<`G9ym>NZF*$!3T9'Qi<\"fOgF*7$$\"3J@Tq\"p\"3UZF*$!3!y4%3& *fFyfF*7$$\"3w'3pk%[#)[ZF*$!3<#*G&H-C$=fF*7$$\"31ni$>nh]v%F*$!3CB5%zdh *feF*7$$\"3G2i'*30wgZF*$!3$H7(43_w.eF*7$$\"3)Q\"Q9p_qmZF*$!3a]Y_qGoTdF *7$$\"3[]LaAnqrZF*$!3kKO%\\U\"='o&F*7$$\"3!zPO+/&pwZF*$!3R2\")Hxq@FcF* 7$$\"3oO**41X!4y%F*$!3Ayw?\\-0ubF*7$$\"3G3]Qj><&y%F*$!3t<&zB%>@;bF*7$$ \"3?1\"fYJr))y%F*$!3Crd]bY1iaF*7$$\"3eA/XIST#z%F*$!3Z>]2+%\\dS&F*7$$\" 392#[Idgbz%F*$!3Y'QI)[?(4N&F*7$$\"3gZd(yDA&)z%F*$!3#*=D?KK#RH&F*7$$\"3 ZJ;\"pbc5![F*$!34Ww)4ym#R_F*7$$\"3d)\\>CyGL![F*$!3AMPAGam$=&F*7$$\"3B_ c%y]m_![F*$!3C%yzJ*p\")G^F*7$$\"3'*o#foxsn![F*$!3?qs#R*Qny]F*7$$\"37Ot >[%z\"3[F*$!3@8Kvdp]@]F*7$$\"3$**\\xVK^\"4[F*$!3[*Q1)f,lq\\F*7$$\"3s=V +(*H*)4[F*$!3ES2X[7r;\\F*7$$\"3@WmjN6K5[F*$!3i,7R(Rg`'[F*F'F_]mFa]m-F$ 6&7$7$$\"\"\"Fa[lFj\\n7$$\"3>+++#Qx/\"[F*$!3>+++#Qx/\"[F*F_]m-%'SYMBOL G6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fh\\n-Fjz6&F\\[lF`[lF][lF`[l-F b]n6$Fd]n\"#5Ff]n-F$6&Fh\\nF\\^n-Fb]n6$%(DIAMONDGF`^nFf]n-F$6&Fh\\nF\\ ^n-Fb]n6$%&CROSSGF`^nFf]n-%+AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAU LTG-Fa_n6$%*HELVETICAG\"\"*-%%VIEWG6$Fc_nFc_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "The following procedure uses " }{TEXT 0 6 "fsolve" }{TEXT -1 23 " to solve the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" " 6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F, F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 "for " }{TEXT 283 1 "y" }{TEXT -1 25 " numerically i n terms of " }{TEXT 284 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) lo cal y;\n fsolve(ln(x^2+y^2)+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/ 2-x);\n end proc:\nuu := evalf(exp(Pi/2)):\nplot('phi'(x),x=1..u u,numpoints=100,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 13 " " 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\" \"!$\"+++++5!\"*7$$\"+M.FS5F-$\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F 37$$\"+eVr96F-$\"+Zyon**F37$$\"+#e!Qa6F-$\"+z@\"=%**F37$$\"+!GeQ>\"F-$ \"+^qu3**F37$$\"+f\"f/B\"F-$\"+$)4mr)*F37$$\"+\\sNo7F-$\"+m$=o#)*F37$$ \"+4:b28F-$\"+-Eit(*F37$$\"+s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig 9X'*F37$$\"+jM?A9F-$\"+N0$)y&*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?] \"F-$\"+2v=4%*F37$$\"+d@iS:F-$\"+c3U<$*F37$$\"+1sjv:F-$\"+R4cG#*F37$$ \"+'[tsh\"F-$\"+gf2;\"*F37$$\"+x[a_;F-$\"+=V+:!*F37$$\"+*)Qd$p\"F-$\"+ F-$\"+rA&\\1)F37$$\"+-Hdj>F-$\"+oiz%*yF37$$\"+`3.**>F-$\"+4o( 3u(F37$$\"+6kKP?F-$\"+KCbovF37$$\"+q%*)o2#F-$\"+LT&QQ(F37$$\"+0Uf:@F-$ \"+Okc'>(F37$$\"+EI/`@F-$\"+7w54qF37$$\"+cNi%>#F-$\"+Snr$z'F37$$\"+8b) >B#F-$\"+;Bi$f'F37$$\"+wt(=F#F-$\"+v62tjF37$$\"+L`-3BF-$\"+'*\\+nhF37$ $\"+)fWvM#F-$\"+7i*[$fF37$$\"+Y\"HZQ#F-$\"+Yr#*4dF37$$\"+9xfBCF-$\"+\\ X)yY&F37$$\"+V))fhCF-$\"+8POC_F37$$\"+.DQ,DF-$\"+iJ/i\\F37$$\"+(*))pRD F-$\"+qG<-ZF37$$\"+VH))yDF-$\"+%H()*GWF37$$\"+EDuLBF37$$\"+#GbZ)GF-$\"+E=zC?F37$$\"+O0_DHF-$\"+Z[&em\"F37$$\"+SOni HF-$\"+<&>.L\"F37$$\"+(H8L+$F-$\"+L)p\"R&*!#67$$\"+[%y$QIF-$\"+H`(4@'F ]\\l7$$\"+iNJyIF-$\"+j(QwK#F]\\l7$$\"+6*))o6$F-$!+[(['>:F]\\l7$$\"+f!R a:$F-$!+RiBhaF]\\l7$$\"+)QZQ>$F-$!+d^2([*F]\\l7$$\"+oduIKF-$!+O[*\\M\" F37$$\"+%*QjqKF-$!+%Q!>%y\"F37$$\"+r,l3LF-$!+B!>N@#F37$$\"+70m[LF-$!+? 8/xEF37$$\"+0&z[Q$F-$!+_aG2JF37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F -$!+\")RRuSF37$$\"+>h\\,NF-$!+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+ ,/CyNF-$!+4mR*e&F37$$\"+zL#fh$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$ \"+I6?&p$F-$!+q5-gsF37$$\"+aqsLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F 37$$\"+a3#*3QF-$!+'*3rE!*F37$$\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6 MXI5F-7$$\"+C/;ERF-$!+B0!=5\"F-7$$\"+$oA@'RF-$!+qNzm6F-7$$\"+kch.SF-$! +'>,VC\"F-7$$\"+!))f5/%F-$!+-[r;8F-7$$\"+v*3\"ySF-$!+Z4$3R\"F-7$$\"+b' [z6%F-$!+zwYt9F-7$$\"+$\\\\z:%F-$!+'fX(f:F-7$$\"+MVM%>%F-$!+DqOT;F-7$$ \"+QS*HB%F-$!+A:fJF- 7$$\"+G\"ypM%F-$!+AUgA?F-7$$\"+M=h(Q%F-$!+:96P@F-7$$\"+`'4eU%F-$!+)[X5 D#F-7$$\"+C&QOY%F-$!+d+zqBF-7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$! +'GZ)QEF-7$$\"+d6/\"e%F-$!+$pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhb YF-$!+9%4S9$F-7$$\"+(y;_p%F-$!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M `Y_ZF-$!+El5vPF-7$$\"+o))>qZF-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$ \"+D\"Q.z%F-$!+!p\\]>%F-7$$\"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 " " {MPLTEXT 1 0 815 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modification o f the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme \+ B`,`efficient scheme C`,`scheme with stability radius = imag axis incl usion`]:\nerrs := []: vals := []:\nDigits := 25:\nfor ct to 6 do\n \+ Cn_RK5_||ct := RK5_||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Cn_RK5_||ct):\n for ii to numpts do\n if ct= 1 then vals := [op(vals),phi(Cn_RK5_||ct[ii,1])] end if;\n sm := \+ sm+(Cn_RK5_||ct[ii,2]-vals[ii])^2;\n end do:\n errs := [op(errs),s qrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope ~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~G -%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint 136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+y*z&*G$!# @7$%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+Fg(Hd#F+7 $%3efficient~scheme~AG$\"+#3:M?*!#A7$%3efficient~scheme~BG$\"+&4`oR#F+ 7$%3efficient~scheme~CG$\"+C!oz)eF+7$%Sscheme~with~stability~radius~=~ imag~axis~inclusionG$\"+\"=))yU$F+Q)pprint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6#/%\"xG-%&FloatG6$\" %\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 678 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Bogacki-Shampine scheme`,`a minor modification of \+ the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B` ,`efficient scheme C`,`scheme with stability radius = imag axis inclus ion`]:\nerrs := []:\nDigits := 25:\nfor ct to 6 do\n cn_RK5_||ct := \+ RK5_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749: cx x := evalf(phi(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(cn_RK5_ ||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~ field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~G- %!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint1 56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine~schemeG$\"+MtMAT!#?7 $%Ta~minor~modification~of~the~Bogacki-Shampine~schemeG$\"+R^$pM$F+7$% 3efficient~scheme~AG$\"+La)H9\"F+7$%3efficient~scheme~BG$\"+Y+X6IF+7$% 3efficient~scheme~CG$\"+_Zb1uF+7$%Sscheme~with~stability~radius~=~imag ~axis~inclusionG$\"+Y,,SVF+Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75 ];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" }{TEXT -1 82 " of each Runge-Kut ta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "mthds := [`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`effici ent scheme A`,`efficient scheme B`,`efficient scheme C`,`scheme with s tability radius = imag axis inclusion`]:\nerrs := []:\nDigits := 20:\n for ct to 6 do\n sm := NCint(('phi'(x)-'cn_RK5_||ct'(x))^2,x=1..4.75 ,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm /5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%8Bogacki-Shampine ~schemeG$\"+Ly1lH!#@7$%Ta~minor~modification~of~the~Bogacki-Shampine~s chemeG$\"+.0d_?F+7$%3efficient~scheme~AG$\"+%o%)o\"Q!#A7$%3efficient~s cheme~BG$\"+oB5c**F47$%3efficient~scheme~CG$\"+#R!*G_$F47$%Sscheme~wit h~stability~radius~=~imag~axis~inclusionG$\"+IS-(*GF+Q)pprint146\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The foll owing error graphs are constructed using the numerical procedures for \+ the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "evalf[20] (plot(['phi'(x)-'cn_RK5_1'(x),'phi'(x)-'cn_RK5_2'(x),'phi'(x)-'cn_RK5_ 3'(x),'phi'(x)-'cn_RK5_4'(x),\n'phi'(x)-'cn_RK5_5'(x),'phi'(x)-'cn_RK5 _6'(x)],x=1..3.75,-2.6e-16..4.6e-16,\nfont=[HELVETICA,9],color=[blue,m agenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlegend=[`Bog acki-Shampine scheme`,`a minor modification of the Bogacki-Shampine sc heme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,` scheme with stability radius = imag axis inclusion`],\ntitle=`error cu rves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 745 499 499 {PLOTDATA 2 "6,-%'CURVESG6%7[p7$$\"\"\"\"\"!$F*F *7$$\"5LLL$eR$\"$K&!#?7$$\"5nm;/,Rv476F/$\"$5*F27$$\"5LLL3x,F27$ $\"5LL$3xJ@PIM\"F/$\"%`@F27$$\"5++]P4;$[%*R\"F/$\"%*Q#F27$$\"5LL$3F%f( )yd9F/$\"%4EF27$$\"5++](=U5Uf^\"F/$\"%@GF27$$\"5nmm;Hn*fdd\"F/$\"%KIF2 7$$\"5LLLektvWG;F/$\"%6KF27$$\"5+++]PR8w(o\"F/$\"%2MF27$$\"5+++]7`'=tu \"F/$\"%,OF27$$\"5+++]igJr/=F/$\"%&y$F27$$\"5LL$3F>(G$o&=F/$\"%_RF27$$ \"5nmm;z\\#3)=>F/$\"%`TF27$$\"5nmmm;C&48(>F/$\"%CVF27$$\"5++]PM()4QK?F /$\"%AXF27$$\"5nmmmT!eRk3#F/$\"%!p%F27$$\"5++]P%[U]d9#F/$\"%()[F27$$\" 5++]7`u$GA?#F/$\"%r]F27$$\"5nmmT5!>d6E#F/$\"%r_F27$$\"5nm;aQQAF:BF/$\" %]aF27$$\"5LLLekGEktBF/$\"%[cF27$$\"5LL$3F%fIFMCF/$\"%eeF27$$\"5++]ilF ?0([#F/$\"%QgF27$$\"5LLL3FfZ0WDF/$\"%LiF27$$\"5++++veT%Hg#F/$\"%PkF27$ $\"5+++v=ZfbgEF/$\"%SmF27$$\"5++]P%[J)H;FF/$\"%OoF27$$\"5+++DJ52>yFF/$ \"%YqF27$$\"5nmmmT!y.Q$GF/$\"%SsF27$$\"5+++]7.E=$*GF/$\"%MuF27$$\"5LL$ 3F>k))p%HF/$\"%:wF27$$\"5+++]PaG\"e+$F/$\"&1x(!#@7$$\"5nm;/,a=;hIF/$\" &U#zFgv7$$\"5++](=n]JF/$\"&+1)Fgv7$$\"5nmm;H#)>evJF/$\"&m<)Fgv7$$\" 5ML3F>k3>0KF/$\"%3#)F27$$\"5++]P4Y(*zMKF/$\"%h#)F27$$\"5mmTNY=mJjKF/$ \"%l#)F27$$\"5LLLL$3\\L=H$F/$\"%d#)F27$$\"5L$ekGg9C\"*H$F/$\"%a#)F27$$ \"5LLeRA,[T1LF/$\"%z#)F27$$\"5L$3F>kX0PJ$F/$\"%!G)F27$$\"5LL$e9;6'*4K$ F/$\"%o#)F27$$\"5L$e*)4ow'GGLF/$\"%n#)F27$$\"5LL3_+AudNLF/F]z7$$\"5L$3 _+s2oGM$F/$\"%S#)F27$$\"5LLLeRK(e,N$F/Fjz7$$\"5++DJ&>()z!zLF/$\"%0#)F2 7$$\"5nm;/^65+3MF/$\"%^\")F27$$\"5Mek`T#*[k9MF/$\"%H\")F27$$\"5+]7.Kt( )G@MF/$\"%!4)F27$$\"5nTg_AaE$zU$F/$\"%B\")F27$$\"5ML3-8NldMMF/$\"%(3)F 27$$\"5EJXkN0vBOMF/$\"%-\")F27$$\"5*3 eWf&RMF/$\"%X!)F27$$\"5+Dc^.;/ATMF/$\"%>!)F27$$\"5%3-j([cBaWMF/$\"%T!) F27$$\"5n;/,%pHkyW$F/$\"%`!)F27$$\"5e9Tj;n__\\MF/$\"%')zF27$$\"5]7yDRP i=^MF/$\"%()zF27$$\"5U5:)=w?ZGX$F/$\"%%*zF27$$\"5M3_]%y<3XX$F/$\"%B!)F 27$$\"5E1*Gr![\"phX$F/$\"%@!)F27$$\"5%F2-%'COLOURG6&%$RGBGF+F +$\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6%7YF'7$F-$!$i &F27$F4$!$\\*F27$F9$!%D8F27$F>$!%G;F27$FC$!%o=F27$FH$!%u?F27$FM$!%_AF2 7$FR$!%0CF27$FW$!%YDF27$Ffn$!%uEF27$F[o$!%wFF27$F`o$!%yGF27$Feo$!%vHF2 7$Fjo$!%jIF27$F_p$!%OJF27$Fdp$!%>KF27$Fip$!%)G$F27$F^q$!%mLF27$Fcq$!%U MF27$Fhq$!%F27$F>$\"$u#F27$FC$\"$_$F27$FH$\"$J%F27$FM$\"$<&F27$FR$\"$/'F27$FW$ \"$'pF27$Ffn$\"$&zF27$F[o$\"$&))F27$F`o$\"$!**F27$Feo$\"%+6F27$Fjo$\"% 57F27$F_p$\"%88F27$Fdp$\"%Z9F27$Fip$\"%m:F27$F^q$\"%2F27$F]r$\"%N@F27$Fbr$\"%2BF27$Fgr$\"%hCF27$F\\s$\"%ZEF27$Fa s$\"%aGF27$Ffs$\"%QIF27$F[t$\"%cKF27$F`t$\"%+NF27$Fet$\"%hPF27$Fjt$\"% 8SF27$F_u$\"%FVF27$Fdu$\"%ZYF27$Fiu$\"%1]F27$F^v$\"%b`F27$Fcv$\"&ev&Fg v7$Fiv$\"&x@'Fgv7$F^w$\"&Sp'Fgv7$Fcw$\"&4A(Fgv7$F]x$\"%\\yF27$Fgx$\"%6 &)F27$F][l$\"%3$*F27$Fe[l$\"&*35F27$F_al$\"&.5\"F27$Fdal$\"&f@\"F27$Fi al$\"&6L\"F27$Fcbl$\"&rZ\"F27$Fhbl$\"&&f:F27$F]cl$\"&gk\"F27$Fbcl$\"&g u\"F27$Fgcl$\"&J&=F2-%&COLORG6&F_dl$F)!\"\"$\"\")FdimF+-Fddl6#%3effici ent~scheme~AG-F$6%7WF'7$F-$!$,&F27$F4$!$Q)F27$F9$!%c6F27$F>$!%.9F27$FC $!%!f\"F27$FH$!%TF27$FW$!%O?F27$Ffn$!%'4#F27$F[ o$!%O@F27$F`o$!%l@F27$Feo$!%$=#F27$Fjo$!%)=#F27$F_p$!%%=#F27$Fdp$!%k@F 27$Fip$!%Q@F27$F^q$!%,@F27$Fcq$!%q?F27$Fhq$!%7?F27$F]r$!%Y>F27$Fbr$!%j =F27$Fgr$!%#z\"F27$F\\s$!%*o\"F27$Fas$!%n:F27$Ffs$!%d9F27$F[t$!%48F27$ F`t$!%O6F27$Fet$!$N*F27$Fjt$!$](F27$F_u$!$\"\\F27$Fdu$!$3#F27$Fiu$\"$> \"F27$F^v$\"$W%F27$Fcv$\"%I%)Fgv7$Fiv$\"&wL\"Fgv7$F^w$\"&v%=Fgv7$Fcw$ \"&2W#Fgv7$F]x$\"%*=$F27$Fgx$\"%KSF27$F][l$\"%v]F27$Fe[l$\"%BhF27$F_al $\"%fuF27$Fdal$\"%#=*F27$F_]m$\"&Y+\"F27$Fial$\"&a4\"F27$F^bl$\"&*37F2 7$Fcbl$\"&zL\"F27$Fhbl$\"&/[\"F27$F]cl$\"&nj\"F27$Fbcl$\"&v\"=F27$Fgcl $\"&&=?F2-F]dl6&F_dlF`dlF+F+-Fddl6#%3efficient~scheme~BG-F$6%7VF'7$F-$ \"#^F27$F4$\"$8\"F27$F9$\"$1#F27$F>$\"$7$F27$FC$\"$@%F27$FH$\"$U&F27$F M$\"$s'F27$FR$\"$4)F27$FW$\"$e*F27$Ffn$\"%>6F27$F[o$\"%p7F27$F`o$\"%W9 F27$Feo$\"%H;F27$Fjo$\"%==F27$F_p$\"%&*>F27$Fdp$\"%CAF27$Fip$\"%KCF27$ F^q$\"%yEF27$Fcq$\"%%*GF27$Fhq$\"%lJF27$F]r$\"%TMF27$Fbr$\"%\\PF27$Fgr $\"%GSF27$F\\s$\"%rVF27$Fas$\"%]ZF27$Ffs$\"%$4&F27$F[t$\"%2bF27$F`t$\" %pfF27$Fet$\"%qkF27$Fjt$\"%apF27$F_u$\"%svF27$Fdu$\"%.#)F27$Fiu$\"%E*) F27$F^v$\"%H'*F27$Fcv$\"'=[5Fgv7$Fiv$\"'#[9\"Fgv7$F^w$\"'oZ7Fgv7$Fcw$ \"'7j8Fgv7$F]x$\"&N]\"F27$Fgx$\"&#f;F27$F][l$\"&_%=F27$Fe[l$\"&u.#F27$ F_al$\"&(pAF27$Fdal$\"&yc#F27$Fial$\"&G(GF27$F^bl$\"&o1$F27$Fcbl$\"&WG $F27$Fhbl$\"&G_$F27$F]cl$\"&ay$F27$Fbcl$\"&()3%F27$Fgcl$\"&rU%F2-F]dl6 &F_dl$\")#)eqkFbdl$\"))eqk\"FbdlF^^o-Fddl6#%3efficient~scheme~CG-F$6%7 SF'7$F-$\"%99F27$F4$\"%HCF27$F9$\"%pMF27$F>$\"%_VF27$FC$\"%#4&F27$FH$ \"%cdF27$FM$\"%ujF27$FR$\"%OpF27$FW$\"%xuF27$Ffn$\"%,!)F27$F[o$\"%U%)F 27$F`o$\"%:*)F27$Feo$\"%w$*F27$Fjo$\"%8)*F27$F_p$\"&+-\"F27$Fdp$\"&l1 \"F27$Fip$\"&d5\"F27$F^q$\"&4:\"F27$Fcq$\"&/>\"F27$Fhq$\"&bB\"F27$F]r$ \"&(y7F27$Fbr$\"&^K\"F27$Fgr$\"&!o8F27$F\\s$\"&eT\"F27$Fas$\"&sY\"F27$ Ffs$\"&G^\"F27$F[t$\"&Qc\"F27$F`t$\"&(=;F27$Fet$\"&\\n\"F27$Fjt$\"&7t \"F27$F_u$\"&lz\"F27$Fdu$\"&#f=F27$Fiu$\"&(G>F27$F^v$\"&e*>F27$Fcv$\"' xq?Fgv7$Fiv$\"'F\\@Fgv7$F^w$\"'9MAFgv7$Fcw$\"'6CBFgv7$F]x$\"&lU#F27$Fg x$\"&;`#F27$F][l$\"&Ql#F27$Fe[l$\"&.y#F27$F_al$\"&`\"HF27$Fdal$\"&$*3$ F27$Fial$\"&[E$F27$Fcbl$\"&K[$F27$F]cl$\"&it$F27$Fgcl$\"&*eSF2-Faim6&F _dl$\"\"'FdimF+Feim-Fddl6#%Sscheme~with~stability~radius~=~imag~axis~i nclusionG-%&TITLEG6#%Uerror~curves~for~7~stage~order~5~Runge-Kutta~met hodsG-%+AXESLABELSG6$Q\"x6\"Q!Feho-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6 $;F(Fgcl;$!#E!#<$\"#YFcio" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modification of the B ogacki-Shampine scheme" "efficient scheme A" "efficient scheme B" "eff icient scheme C" "scheme with stability radius = imag axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 545 "evalf[20](plot(['phi'(x)-'cn_RK5_1'(x),'phi'(x)-'cn_RK5_2'(x) ,'phi'(x)-'cn_RK5_3'(x),'phi'(x)-'cn_RK5_4'(x),\n'phi'(x)-'cn_RK5_5'(x ),'phi'(x)-'cn_RK5_6'(x)],x=3.75..4.6,-1.3e-13..2.3e-13,\nfont=[HELVET ICA,9],color=[blue,magenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0 ,.8)],\nlegend=[`Bogacki-Shampine scheme`,`a minor modification of the Bogacki-Shampine scheme`,`efficient scheme A`,`efficient scheme B`,`e fficient scheme C`,`scheme with stability radius = imag axis inclusion `],\ntitle=`error curves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 823 617 617 {PLOTDATA 2 "6,-%'CURVESG6%7e o7$$\"$v$!\"#$\"%*>%!#?7$$\"5nmm;/\"eF&oP!#>$\"%zQF-7$$\"5LL$eR-L[Yy$F 1$\"%#Q$F-7$$\"5nmm\"H()zxF!QF1$\"%FFF-7$$\"5nmmT5Tu-@QF1$\"%&*>F-7$$ \"5LL$e9hM!>RQF1$\"%p8F-7$$\"5nm;H2%oHg&QF1$\"$^(F-7$$\"5++]i:VeYtQF1$ !#8F17$$\"5nm;H#QM)\\\"*QF1$!$8\"F17$$\"5++]7.9IZ4RF1$!$7#F17$$\"5LLL$ 3<%F1$!%7QF17$$\"5++]P 4jUj4UF1$!%AUF17$$\"5nmm\"H#*G`sA%F1$!%'e%F17$$\"5++++DnaXXUF1$!%k]F17 $$\"5+++DJZFEjUF1$!%zbF17$$\"5++]iS1A\\!G%F1$!%*4'F17$$\"5+++v=$eA'*H% F1$!%&e'F17$$\"5LLLLex?\"oJ%F1$!%ZpF17$$\"5+++](=`l^L%F1$!%_tF17$$\"5n m;HK*['z^VF1$!%$e(F17$$\"5+++]i=&y*pVF1$!%cvF17$$\"5LL$eR7R'3(Q%F1$!%R uF17$$\"5++]7`Z!p\\S%F1$!%UmF17$$\"5LLL$3Fh_CU%F1$!%e[F17$$\"5++]i:&Gc 2W%F1$!%B:F17$$\"5mmmm;(*[QeWF1$\"%&4#F17$$\"5mmmT5*p7kZ%F1$\"%\"4*F17 $$\"5LL$eR2^%F1$\"&jL%F17$$\"5LLL3_[ '[&HXF1$\"&:\\(F17$$\"5++]PMa!pz`%F1$\"&7u)F17$$\"5mmmm;g%*QYXF1$\"'K1 6F17$$\"5LLek.ptObXF1$\"'8`9F17$$\"5++]i!zFXVc%F1$\"'C=>F17$$\"5++D\"y +yT'oXF1$\"'$=;#F17$$\"5++++D#GQHd%F1$\"'!e`#F17$$\"5+voHz2C,uXF1$\"'a XDF17$$\"5+]PfLLl3vXF1$\"'WhDF17$$\"5+D1*y)e1;wXF1$\"'w$f#F17$$\"5++v= U%yMsd%F1$\"')Hm#F17$$\"5+]7y]NIQzXF1$\"'Q(3$F17$$\"5++]Pf'GJ:e%F1$\"' $=O$F17$$\"5]iS;=\"e&o#e%F1$\"'HtLF17$$\"5+DJ&pd()RQe%F1$\"'#oQ$F17$$ \"5](=Ud.<%*\\e%F1$\"'\\3MF17$$\"5+]7`%\\Y[he%F1$\"'t`MF17$$\"5]7.K`fF I(e%F1$\"'TcNF17$$\"5+v$4@T0d%)e%F1$\"'=!y$F17$$\"5]P%)*3([8h*e%F1$\"' ZNUF17$$\"5++voHVcw!f%F1$\"'C![%F17$$\"5+DcEZKU2$f%F1$\"'A7XF17$$\"5+] P%[;#GQ&f%F1$\"'cqXF17$$\"5]7GjB;r`'f%F1$\"'p^YF17$$\"5+v=U#3T\"p(f%F1 $\"'`M[F17$$\"5D1k\"=\"e&o#)f%F1$\"'`%*\\F17$$\"5]P4@T0d%))f%F1$\"'WC_ F17$$\"5voagq_GU*f%F1$\"'<\\bF17$$\"#Y!\"\"$\"'W+gF1-%'COLOURG6&%$RGBG $\"\"!FgblFfbl$\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6 %7[o7$F($!&mv#F-7$F/$!&+\"HF-7$F5$!&m4$F-7$F:$!&=M$F-7$F?$!&jh$F-7$FD$ !&1(QF-7$FI$!&B7%F-7$FN$!%sWF17$FS$!%u[F17$FX$!%z_F17$Fgn$!%&o&F17$F\\ o$!%shF17$Fao$!%\"z'F17$Ffo$!%*\\(F17$F[p$!%-\")F17$F`p$!%+))F17$Fep$! %\"z*F17$Fjp$!&$*3\"F17$F_q$!&d>\"F17$Fdq$!&\\I\"F17$Fiq$!&bY\"F17$F^r $!&>l\"F17$Fcr$!&6'=F17$Fhr$!&9-#F17$F]s$!&]H#F17$Fbs$!&ti#F17$Fgs$!&& yHF17$F\\t$!&NI$F17$Fat$!&&)z$F17$Fft$!&PT%F17$F[u$!&5;&F17$F`u$!&T+'F 17$Feu$!&`v'F17$Fju$!&a*zF17$F_v$!&wa*F17$Fdv$!'V_6F17$Fiv$!'%RI\"F17$ F^w$!'1\"e\"F17$Fcw$!'t]>F17$Fhw$!'-SCF17$F]x$!'c!*GF17$Fbx$!',\"e$F17 $Fgx$!'\"ff%F17$F\\y$!'(4,'F17$Fay$!'kByF17$Ffy$!'X<&)F17$F[z$!'jp'*F1 7$F`z$!(:n7\"F17$Fez$!(vbK\"F17$Fjz$!(%)yU\"F17$F_[l$!(K/d\"F17$Fc\\l$ !(vpi\"F17$F]]l$!(o\\(=F17$Fg]l$!(Z'))=F17$Fa^l$!(L\"=>F17$F[_l$!(6W.# F17$Fe_l$!(e*fAF17$Fj_l$!(+hF#F17$F_`l$!(n=I#F17$Fd`l$!(HCL#F17$Fi`l$! (WZR#F17$Fcal$!(#pF-7$F5$\"&L)>F-7$F:$\"&d1#F-7$F?$\"&O:#F-7$FD$\"&iB#F-7$ FI$\"&fJ#F-7$FN$\"%ACF17$FS$\"%QDF17$FX$\"%YEF17$Fgn$\"%eFF17$F\\o$\"% #)GF17$Fao$\"%LIF17$Ffo$\"%$>$F17$F[p$\"%QLF17$F`p$\"%&\\$F17$Fep$\"%, PF17$Fjp$\"%;RF17$F_q$\"%BTF17$Fdq$\"%PVF17$Fiq$\"%@YF17$F^r$\"%P\\F17 $Fcr$\"%x_F17$Fhr$\"%\\bF17$F]s$\"%lfF17$Fbs$\"%^kF17$Fgs$\"%OpF17$F\\ t$\"%%Q(F17$Fat$\"%S!)F17$Fft$\"%2))F17$F[u$\"%$p*F17$F`u$\"&i1\"F17$F eu$\"&K:\"F17$Fju$\"&vG\"F17$F_v$\"&yW\"F17$Fdv$\"&gk\"F17$Fiv$\"&nz\" F17$F^w$\"&[1#F17$Fcw$\"CF17$Fhw$\"&C&GF17$F]x$\"&zC$F17$Fbx$\"&s'Q F17$Fgx$\"&^w%F17$F\\y$\"&r+'F17$Fay$\"&Ke(F17$Ffy$\"&A=)F17$F[z$\"&CC *F17$F`z$\"'%32\"F17$Fez$\"'\\b7F17$Fjz$\"'bR8F17$F_[l$\"'W'[\"F17$Fc \\l$\"'2H:F17$F]]l$\"'!*zF17$Fe_l$\"'lg@F17$Fj_l$\"'lv@F17$F_`l$\"'t&>#F17$Fd`l$\"'P=AF17$Fi `l$\"'LrAF17$Fcal$\"'f$R#F17$F]bl$\"'OdEF1-%&COLORG6&Febl$\"\"\"F_bl$ \"\")F_blFfbl-F\\cl6#%3efficient~scheme~AG-F$6%7[o7$F($\"&&=?F-7$F/$\" &C8#F-7$F5$\"&(fAF-7$F:$\"&\\U#F-7$F?$\"&[g#F-7$FD$\"&$pFF-7$FI$\"&6$H F-7$FN$\"%cJF17$FS$\"%1MF17$FX$\"%WOF17$Fgn$\"%()QF17$F\\o$\"%rTF17$Fa o$\"%CXF17$Ffo$\"%8\\F17$F[p$\"%Z_F17$F`p$\"%JcF17$Fep$\"%_hF17$Fjp$\" %=nF17$F_q$\"%_sF17$Fdq$\"%6yF17$Fiq$\"%'f)F17$F^r$\"%!\\*F17$Fcr$\"&r /\"F17$Fhr$\"&I7\"F17$F]s$\"&$[7F17$Fbs$\"&xR\"F17$Fgs$\"&7b\"F17$F\\t $\"&Cp\"F17$Fat$\"&l!>F17$Fft$\"&h;#F17$F[u$\"&^Z#F17$F`u$\"&u\"GF17$F eu$\"&O7$F17$Fju$\"&6i$F17$F_v$\"&LB%F17$Fdv$\"&T+&F17$Fiv$\"&me&F17$F ^w$\"&6m'F17$Fcw$\"&Z2)F17$Fhw$\"&U$**F17$F]x$\"']g6F17$Fbx$\"'CC9F17$ Fgx$\"'%>\"=F17$F\\y$\"'%HN#F17$Fay$\"'2UIF17$Ffy$\"'>.LF17$F[z$\"'1gP F17$F`z$\"'h#R%F17$Fez$\"'n&=&F17$Fjz$\"'EfbF17$F_[l$\"'rrhF17$Fc\\l$ \"'#[O'F17$F]]l$\"'-8uF17$Fg]l$\"''[Y(F17$Fa^l$\"')Hc(F17$F[_l$\"'#H+) F17$Fe_l$\"'%f+*F17$Fj_l$\"'Ep!*F17$F_`l$\"'Ve\"*F17$Fd`l$\"'qh#*F17$F i`l$\"'O#\\*F17$Fcal$\"(\"4+5F17$F]bl$\"(Dc5\"F1-Fcbl6&FeblFhblFfblFfb l-F\\cl6#%3efficient~scheme~BG-F$6%7]o7$F($\"&rU%F-7$F/$\"&%=YF-7$F5$ \"&U$[F-7$F:$\"&-6&F-7$F?$\"&9T&F-7$FD$\"&$)o&F-7$FI$\"&R'fF-7$FN$\"%R jF17$FS$\"%enF17$FX$\"%hrF17$Fgn$\"%tvF17$F\\o$\"%d!)F17$Fao$\"%d')F17 $Ffo$\"%B$*F17$F[p$\"%#*)*F17$F`p$\"&a0\"F17$Fep$\"&c9\"F17$Fjp$\"&OC \"F17$F_q$\"&pL\"F17$Fdq$\"&^V\"F17$Fiq$\"&Rd\"F17$F^r$\"&Et\"F17$Fcr$ \"&y!>F17$Fhr$\"&S/#F17$F]s$\"&:F#F17$Fbs$\"&Wa#F17$Fgs$\"&l#GF17$F\\t $\"&$)3$F17$Fat$\"&'*[$F17$Fft$\"&6)RF17$F[u$\"&Bd%F17$F`u$\"&9B&F17$F eu$\"&b#eF17$Fju$\"&p!oF17$F_v$\"&'H!)F17$Fdv$\"&de*F17$Fiv$\"':w5F17$ F^w$\"'!zH\"F17$Fcw$\"'X$f\"F17$Fhw$\"'Q()>F17$F]x$\"'9F17$F]]l$\"(Umm\"F17$Fg]l$\"(+$y;F17$Fa^l$\"()e+F17$Fft$\"&MF#F17$F[u$\"&sg#F17$F`u$\"&z)HF17$Feu$\"&;L$F17$Fj u$\"&*)*QF17$F_v$\"&jh%F17$Fdv$\"&Ca&F17$Fiv$\"&&fiF17$F^w$\"&ke(F17$F cw$\"&AQ*F17$Fhw$\"''*z6F17$F]x$\"'209F17$Fbx$\"'gc& fF17$Fez$\"'C(4(F17$Fjz$\"'x%o(F17$F_[l$\"'FK&)F17$Fc\\l$\"'=_))F17$F] ]l$\"(N].\"F17$Fg]l$\"(nD/\"F17$Fa^l$\"(`#f5F17$F[_l$\"(C)G6F17$Fe_l$ \"(\\#p7F17$Fj_l$\"()Gy7F17$F_`l$\"(KHH\"F17$Fd`l$\"(B3J\"F17$Fi`l$\"( y%[8F17$Fcal$\"(@]U\"F17$F]bl$\"(7 " 0 "" {MPLTEXT 1 0 545 "evalf[20](p lot(['phi'(x)-'cn_RK5_1'(x),'phi'(x)-'cn_RK5_2'(x),'phi'(x)-'cn_RK5_3' (x),'phi'(x)-'cn_RK5_4'(x),\n'phi'(x)-'cn_RK5_5'(x),'phi'(x)-'cn_RK5_6 '(x)],x=4.6..4.75,-4.3e-11..6.3e-11,\nfont=[HELVETICA,9],color=[blue,m agenta,COLOR(RGB,.1,.8,0),red,brown,COLOR(RGB,.6,0,.8)],\nlegend=[`Bog acki-Shampine scheme`,`a minor modification of the Bogacki-Shampine sc heme`,`efficient scheme A`,`efficient scheme B`,`efficient scheme C`,` scheme with stability radius = imag axis inclusion`],\ntitle=`error cu rves for 7 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 831 591 591 {PLOTDATA 2 "6,-%'CURVESG6%7cq7$$\"#Y!\"\"$\"'W+ g!#>7$$\"5+++]7t&pKg%F-$\"'#R1'F-7$$\"5++](=7T9hg%F-$\"'?+iF-7$$\"5+++ v=HPJ4YF-$\"'+\"R(F-7$$\"5+++DJaU`7YF-$\"'gx\")F-7$$\"5++]P%GZRdh%F-$ \"'>T$)F-7$$\"5++](=276(=YF-$\"'DU%*F-7$$\"5++](o**3)y@YF-$\"(.@6\"F-7 $$\"5++](ofHq\\i%F-$\"(c(G6F-7$$\"5++]Pf'HU\"GYF-$\"(IqB\"F-7$$\"5+++] 7*309j%F-$\"(M(H:F-7$$\"5+++Dce*yUj%F-$\"(@\"[:F-7$$\"5+++]([D9vj%F-$ \"(m(\\;F-7$$\"5+++]iNGwSYF-$\"(mx7#F-7$$\"5+++]7XM*Qk%F-$\"(1^:#F-7$$ \"5++](o%QjtYYF-$\"((zQAF-7$$\"5+++]i8o6]YF-$\"(F=+$F-7$$\"5++++]>0)Hl %F-$\"(8o.$F-7$$\"5++](=-p6jl%F-$\"(m\"QJF-7$$\"5++++vS.EfYF-$\"(`^$QF -7$$\"5++](=xZ&\\iYF-$\"(!ehVF-7$$\"5++]i:$4wbm%F-$\"(L4Y%F-7$$\"5+++v =#R!zoYF-$\"(l`E&F-7$$\"5++]P4A@urYF-$\"(!*>Q'F-7$$\"5+++Dchf#\\n%F-$ \"(IV]'F-7$$\"5++](of2L#yYF-$\"(+sP(F-7$$\"5++]7yG>6\"o%F-$\"(JHc*F-7$ $\"5+++voo6A%o%F-$\"($\\D(*F-7$$\"5++++v&[\"F-7$$\"5++](=t/ZF-$\")\\z>CF-7$$\"5++](o/Q*>1ZF-$\")Z_*[#F-7$$ \"5+]7.#)G:+2ZF-$\")!*QtDF-7$$\"5++v=ZF-$\")\"H=)fF-7$$\"5++vo/QJG?ZF-$\")yUAoF-7$$\"5+]7y+`14@ZF-$\")1>f oF-7$$\"5++](oz;)*=s%F-$\")6c'*oF-7$$\"5++vVtLOXBZF-$\")CcupF-7$$\"5++ ++]*44]s%F-$\")?3)4(F-7$$\"5++DJqVW!es%F-$\")aJBsF-7$$\"5++]i!zy*fEZF- $\")7R[uF-7$$\"5++v$4@8&RFZF-$\")<,cyF-7$$\"5+++DJw/>GZF-$\")mZy&)F-7$ $\"5+]iSmM#z*GZF-$\")Mx0)*F-7$$\"5++Dc,$*zwHZF-$\"*4-O=\"F-7$$\"5DJq&4 yem)HZF-$\"*Xkl@\"F-7$$\"5]i:Ng#=l*HZF-$\"*R>:D\"F-7$$\"5v$4Y(RxP1IZF- $\"*._\\E\"F-7$$\"5+D19>sB;IZF-$\"*%3(eE\"F-7$$\"5](oHzj&p7F-7$$\"5+voHaI6&4t%F-$\"*w$Gt7F-7$$\" 5++](=(4bMJZF-$\"*5MqF\"F-7$$\"5++vVt$3&zKZF-$\"*k98H\"F-7$$\"5++++vdY CMZF-$\"*2q#48F-7$$\"5++DJXX`2NZF-$\"*d#3E8F-7$$\"5++]i:Lg!ft%F-$\"*@: bN\"F-7$$\"5++v$f3sOnt%F-$\"*0A0T\"F-7$$\"5+++Dc3ucPZF-$\"*0&[8:F-7$$ \"5++voa&Q5$QZF-$\"*$)Gdn\"F-7$$\"5++]7`iL0RZF-$\"*i_`%>F-7$$\"5++Dc^R jzRZF-$\"*w([!Q#F-7$$\"5++++];$R0u%F-$\"*1K.b#F-7$$\"5++v$fLlB@u%F-$\" *,R[e#F-7$$\"5++](=-*zqVZF-$\"*O()Ri#F-7$$\"5++vV['>mWu%F-$\"*9T+l#F-7 $$\"5++++v-WAXZF-$\"*Z)))*o#F-7$$\"5++Dc,4E)fu%F-$\"*P3#eFF-7$$\"5++]7 G:3uYZF-$\"*073)GF-7$$\"5+]P4Y6cbZZF-$\"*7DD7$F-7$$\"5++D1k2/P[ZF-$\"* !=ziNF-7$$\"5+vo/t0yx[ZF-$\"*)GA(*QF-7$$\"5+]7.#Q?&=\\ZF-$\"*a))pL%F-7 $$\"5]PM_'G!*)Q\\ZF-$\"*g#f\\ZF-$\"*)z>5\\F-7$$\" 5]7y]&4I'z\\ZF-$\"*TnsD&F-7$$\"$v%!\"#$\"*ze5l&F--%'COLOURG6&%$RGBG$\" \"!F\\\\mF[\\m$\"*++++\"!\")-%'LEGENDG6#%8Bogacki-Shampine~schemeG-F$6 %7dq7$F($!(Mmu#F-7$F/$!(#evFF-7$F4$!(Z!GGF-7$F9$!(E<=$F-7$F>$!(0]S$F-7 $FC$!(TfY$F-7$FH$!(e;y$F-7$FM$!(&H>UF-7$FR$!(#fzUF-7$FW$!(cF-7$$ \"5+]il(*eB.!o%F-$!),ET?F-7$$\"5++D\"yb@#R!o%F-$!)rvW?F-7$$\"5+](oz@2_ 2o%F-$!)`E[?F-7$Ffs$!)4z^?F-7$$\"5++vVt[lm#o%F-$!)\"Qu1#F-7$F[t$!)Y$o3 #F-7$$\"5++](=KCFeo%F-$!).xE@F-7$F`t$!)&4PB#F-7$Fet$!)&H>]#F-7$Fjt$!)p jSGF-7$F_u$!)Q0jGF-7$Fdu$!)%Q\"))GF-7$Fiu$!)_WNHF-7$F^v$!)Y_iIF-7$Fcv$ !)Y9!=$F-7$Fhv$!)9@jLF-7$F]w$!);;SOF-7$Fbw$!)DgJSF-7$Fgw$!)w.oSF-7$F\\ x$!)&=t5%F-7$Fax$!)0$*eTF-7$Ffx$!)=HrUF-7$F[y$!)LK$R%F-7$F`y$!)o!Hf%F- 7$Fey$!)T25\\F-7$Fjy$!)'fyR&F-7$F_z$!)P60dF-7$Fdz$!)HD7fF-7$Fiz$!)MrDf F-7$F^[l$!)7DRfF-7$Fc[l$!)HimfF-7$Fh[l$!)Qx%*fF-7$F]\\l$!)QykgF-7$Fb\\ l$!)Ey)>'F-7$Fg\\l$!)2\"yK'F-7$F\\]l$!)7\\PlF-7$Fa]l$!)^#>(oF-7$Ff]l$! )C;!R(F-7$F[^l$!)Ju7#)F-7$F`^l$!)A5F-7$Fh`l$!*cY#*3\"F-7$F]al$!*'[&\\>\"F-7$Fbal$!*d89F-7$Fjcl$!*\"))3C9F-7$F _dl$!*;t,W\"F-7$Fddl$!*ji7Y\"F-7$Fidl$!*%)o4[\"F-7$F^el$!*;qK^\"F-7$Fc el$!*Bqwc\"F-7$Fhel$!*S*yd;F-7$Fbfl$!*r%eo>F-7$F\\gl$!**3\\JBF-7$Fagl$ !*a@JO#F-7$Ffgl$!*Gq/S#F-7$F`hl$!*J3_Y#F-7$Fjhl$!*d^ai#F-7$F_il$!*J4Xz #F-7$Fdil$!*S&pdIF-7$Fiil$!*8(4NKF-7$F^jl$!*#\\'*[MF-7$Fhjl$!*b\"p.PF- 7$Fb[m$!*uyL+%F--Fh[m6&Fj[mF]\\mF[\\mF]\\m-Fa\\m6#%Ta~minor~modificati on~of~the~Bogacki-Shampine~schemeG-F$6%7cq7$F($\"'OdEF-7$F/$\"'%[o#F-7 $F4$\"'/DFF-7$F9$\"'A%3$F-7$F>$\"'N^LF-7$FC$\"'R+MF-7$FH$\"'t5PF-7$FM$ \"'b\\UF-7$FR$\"'!HI%F-7$FW$\"'Y*e%F-7$Ffn$\"'))*[&F-7$F[o$\"')*\\bF-7 $F`o$\"'h0eF-7$Feo$\"'*HA(F-7$Fjo$\"'86tF-7$F_p$\"'*R^(F-7$Fdp$\"'[1(* F-7$Fip$\"'D=)*F-7$F^q$\"(Hl+\"F-7$Fcq$\"(1a>\"F-7$Fhq$\"(^EN\"F-7$F]r $\"(BvP\"F-7$Fbr$\"(The\"F-7$Fgr$\"(U9\">F-7$F\\s$\"('fV>F-7$Fas$\"(f2 ;#F-7$Ffs$\"(mWy#F-7$F[t$\"(l)GGF-7$F`t$\"(\"=?IF-7$Fet$\"(lE\\$F-7$Fj t$\"(u&)>%F-7$F_u$\"(*fJUF-7$Fdu$\"(*RmUF-7$Fiu$\"(\"\\CVF-7$F^v$\"(hP ]%F-7$Fcv$\"(f@q%F-7$Fhv$\"(\"3_]F-7$F]w$\"(0tk&F-7$Fbw$\"(PMe'F-7$Fgw $\"(&)Gk'F-7$F\\x$\"(BYq'F-7$Fax$\"(*=wnF-7$Ffx$\"(^q$pF-7$F[y$\"((oSr F-7$F`y$\"(AM_(F-7$Fey$\"(;(>#)F-7$Fjy$\"(scV*F-7$F_z$\")X%p-\"F-7$Fdz $\")H+&3\"F-7$Fiz$\")GZ(3\"F-7$F^[l$\")#e**3\"F-7$Fc[l$\")&o\\4\"F-7$F h[l$\")d0+6F-7$F]\\l$\")-s66F-7$Fb\\l$\")7TK6F-7$Fg\\l$\")\"F-7$Fa]l$\")C#4F\"F-7$Ff]l$\")0N/9F-7$F[^l$\")yrX;F-7$F`^l$ \")$4*))=F-7$Fe^l$\")y3**=F-7$Fj^l$\")yS4>F-7$F__l$\")SGI>F-7$Fd_l$\") _F-7$Fi_l$\")2!**)>F-7$F^`l$\")NCX?F-7$Fc`l$\")=C]@F-7$Fh`l$\")phWB F-7$F]al$\")0(oo#F-7$Fbal$\")v1pKF-7$Fgal$\")LtkLF-7$F\\bl$\")'QkY$F-7 $Fabl$\")>]0NF-7$Ffbl$\")z/3NF-7$F[cl$\")\\:8NF-7$F`cl$\")*z#=NF-7$Fec l$\");fGNF-7$Fjcl$\")&y*QNF-7$F_dl$\")j5yNF-7$Fddl$\")B9COF-7$Fidl$\") <*[m$F-7$F^el$\")ZyOPF-7$Fcel$\")*yj(QF-7$Fhel$\")rL\\TF-7$F]fl$\")$H] f%F-7$Fbfl$\")Kzb`F-7$Fgfl$\")PJ3mF-7$F\\gl$\")VP+rF-7$Fagl$\")+C'>(F- 7$Ffgl$\")f:,tF-7$F[hl$\")c-ntF-7$F`hl$\")40muF-7$Fehl$\")`kQwF-7$Fjhl $\")h()ezF-7$F_il$\")$RIh)F-7$Fdil$\")TLS)*F-7$Fiil$\"*Nw(y5F-7$F^jl$ \"*\"zV/7F-7$Fcjl$\"*/'R\"G\"F-7$Fhjl$\"*)=Lp8F-7$F][m$\"*tc&p9F-7$Fb[ m$\"*`4Ne\"F--%&COLORG6&Fj[m$\"\"\"F*$\"\")F*F[\\m-Fa\\m6#%3efficient~ scheme~AG-F$6%7dq7$F($\"(Dc5\"F-7$F/$\"(Gr6\"F-7$F4$\"(S^8\"F-7$F9$\"( qFG\"F-7$F>$\"(2zQ\"F-7$FC$\"(4&49F-7$FH$\"(yt`\"F-7$FM$\"(2ku\"F-7$FR $\"(F\"pQF-7$Fip$\"( .Q'QF-7$F^q$\"(dW'RF-7$Fcq$\"(KUl%F-7$Fhq$\"(+z?&F-7$F]r$\"(MqI&F-7$Fb r$\"(1&fgF-7$Fgr$\"(\"HrrF-7$F\\s$\"(+\\H(F-7$Fas$\"(t%o!)F-7$Ffs$\")p t85F-7$F[t$\")u6I5F-7$Fjcm$\")n&o/\"F-7$F`t$\")f&y4\"F-7$Fet$\")f!GD\" F-7$Fjt$\")+#oZ\"F-7$F_u$\")F-7$Fbw$\" )A'pA#F-7$Fgw$\")Q2ZAF-7$F\\x$\")()4oAF-7$Fax$\")A'GH#F-7$Ffx$\")>'oM# F-7$F[y$\")TA7CF-7$F`y$\")qeF-7$Fe^l$\")8K^eF-7$Fj^l$\"):9$)eF-7$F__l$\")O\"z%fF-7$F d_l$\")J?TgF-7$Fi_l$\")9^JhF-7$F^`l$\")M>%H'F-7$Fc`l$\")S:$f'F-7$Fh`l$ \")>UJrF-7$F]al$\")W(o0)F-7$Fbal$\")si)f*F-7$Fgal$\")ET\\)*F-7$F\\bl$ \"*:T:,\"F-7$Fabl$\"*y\"z@5F-7$Ffbl$\"*'R`A5F-7$F[cl$\"*_AS-\"F-7$F`cl $\"*_;b-\"F-7$Fecl$\"*u@&G5F-7$Fjcl$\"*!)\\:.\"F-7$F_dl$\"*$R)H/\"F-7$ Fddl$\"*mil0\"F-7$Fidl$\"*g$\\o5F-7$F^el$\"*Jz*)3\"F-7$Fcel$\"*oHu7\"F -7$Fhel$\"*ef-?\"F-7$F]fl$\"*p/hJ\"F-7$Fbfl$\"*^5&4:F-7$Fgfl$\"*e#e@=F -7$F\\gl$\"*ecO%>F-7$Fagl$\"*t6*p>F-7$Ffgl$\"*IL))*>F-7$F[hl$\"*7Yq,#F -7$F`hl$\"*eXS/#F-7$Fehl$\"*xc)*3#F-7$Fjhl$\"*^#Hs@F-7$F_il$\"*K)oNBF- 7$Fdil$\"*A1Qj#F-7$Fiil$\"*$\"($e/KF-7$FC$\"(EZD$F-7$FH$\"(!4hNF -7$FM$\"(pr1%F-7$FR$\"(A,7%F-7$FW$\"(\"Q/WF-7$Ffn$\"(s2C&F-7$F[o$\"(z$ *H&F-7$F`o$\"(&e`bF-7$Feo$\"(S<&oF-7$Fjo$\"(*QOpF-7$F_p$\"(\"QQrF-7$Fd p$\"(dI6*F-7$Fip$\"(a#=#*F-7$F^q$\"(h1Y*F-7$Fcq$\")]M:6F-7$Fhq$\")ma^7 F-7$F]r$\")m[v7F-7$Fbr$\")8\\g9F-7$Fgr$\")!y\\t\"F-7$F\\s$\")$\\\\w\"F -7$Fas$\")9Yb>F-7$Ffs$\")lWnCF-7$F[t$\")k$F-7$Fdu$\")+IsOF-7$ Fiu$\")*GQs$F-7$F^v$\")#Q](QF-7$Fcv$\")*z].%F-7$Fhv$\")j\"*4VF-7$F]w$ \")#=qw%F-7$Fbw$\")1_saF-7$Fgw$\")F%>_&F-7$F\\x$\")'QOd&F-7$Fax$\")$=Z j&F-7$Ffx$\")xYodF-7$F[y$\")1bIfF-7$F`y$\")wcDiF-7$Fey$\")5wZnF-7$Fjy$ \")IvQwF-7$F_z$\")R+T#)F-7$Fdz$\")nOe')F-7$Fiz$\")'y!y')F-7$F^[l$\")\\ !zp)F-7$Fc[l$\")f\"zt)F-7$Fh[l$\")5iy()F-7$F]\\l$\")g9t))F-7$Fb\\l$\") $Q//*F-7$Fg\\l$\")Y)H@*F-7$F\\]l$\")0RA&*F-7$Fa]l$\"*2fs+\"F-7$Ff]l$\" *s8B5\"F-7$F[^l$\"**[Tq7F-7$F`^l$\"*i5tV\"F-7$Fe^l$\"*&[0X9F-7$Fj^l$\" *,9HX\"F-7$F__l$\"*2E*o9F-7$Fd_l$\"*?j?\\\"F-7$Fi_l$\"*9#[9:F-7$F^`l$ \"*aG[b\"F-7$Fc`l$\"*(e')G;F-7$Fh`l$\"*ex>w\"F-7$F]al$\"*))f0*>F-7$Fba l$\"*5!*4P#F-7$Fgal$\"*XWGV#F-7$F\\bl$\"*7X%)\\#F-7$Fabl$\"*dDP_#F-7$F fbl$\"*reb_#F-7$F[cl$\"*FN#HDF-7$F`cl$\"*UDH`#F-7$Fecl$\"*mZ.a#F-7$Fjc l$\"*%o#ya#F-7$F_dl$\"*yzgd#F-7$Fddl$\"*K3(4EF-7$Fidl$\"*H0$REF-7$F^el $\"*kx+p#F-7$Fcel$\"**[;&y#F-7$Fhel$\"*7T['HF-7$F]fl$\"*6l+D$F-7$Fbfl$ \"*>![DPF-7$Fgfl$\"*-e:\\%F-7$F\\gl$\"*J>6z%F-7$Fagl$\"*ZWe&[F-7$Ffgl$ \"*-]s#\\F-7$F[hl$\"*$eIs\\F-7$F`hl$\"*xy!R]F-7$Fehl$\"*gF@:&F-7$Fjhl$ \"*VS\\N&F-7$F_il$\"*#oobdF-7$Fdil$\"*.>[['F-7$Fiil$\"*1ok.(F-7$F^jl$ \"*3\\%exF-7$Fcjl$\"*aFk>)F-7$Fhjl$\"*()3Pp)F-7$F][m$\"*6XpD*F-7$Fb[m$ \"*8oL*)*F--Fh[m6&Fj[m$\")#)eqkF_\\m$\"))eqk\"F_\\mFc]r-Fa\\m6#%3effic ient~scheme~CG-F$6%7dq7$F($\"(7$\"(N&))>F-7$FC$\"(.X-#F-7$FH$\"(ZkA#F-7$FM$\"([%>DF-7$ FR$\"(8bb#F-7$FW$\"(j*\\FF-7$Ffn$\"()RQKF-7$F[o$\"(hmF$F-7$F`o$\"(DhX$ F-7$Feo$\"(\"*)>UF-7$Fjo$\"(OPF%F-7$F_p$\"(e'>WF-7$Fdp$\"(B$*e&F-7$Fip $\"(2Wl&F-7$F^q$\"(^u#eF-7$Fcq$\"(I0(oF-7$Fhq$\"(mzj(F-7$F]r$\"(YV!yF- 7$Fbr$\"(4l(*)F-7$Fgr$\")Re_5F-7$F\\s$\")9Us5F-7$Fas$\")Yd'>\"F-7$Ffs$ \")n(p[\"F-7$F[t$\")`;7:F-7$Fjcm$\")\">7a\"F-7$F`t$\")^yC;F-7$Fet$\")6 X`=F-7$Fjt$\")8jh@F-7$F_u$\")(z'y@F-7$Fdu$\")#ov>#F-7$Fiu$\")o2LAF-7$F ^v$\")&R\\L#F-7$Fcv$\")t_NCF-7$Fhv$\")3!)*f#F-7$F]w$\")$)zgGF-7$Fbw$\" )*>![KF-7$Fgw$\")!otF$F-7$F\\x$\"))e(3LF-7$Fax$\")/L\\LF-7$Ffx$\")$y4W $F-7$F[y$\")HwYNF-7$F`y$\")$H'HPF-7$Fey$\")8lPSF-7$Fjy$\")(z0a%F-7$F_z $\").Kr[F-7$Fdz$\")+)z4&F-7$Fiz$\")he4^F-7$F^[l$\")5E@^F-7$Fc[l$\")e%[ 9&F-7$Fh[l$\")3-p^F-7$F]\\l$\")s0G_F-7$Fb\\l$\")$f8M&F-7$Fg\\l$\")pDca F-7$F\\]l$\"),r`cF-7$Fa]l$\"),()))fF-7$Ff]l$\")\")QVlF-7$F[^l$\")%fp[( F-7$F`^l$\")L&)*R)F-7$Fe^l$\")l6X%)F-7$Fj^l$\")h9\"\\)F-7$F__l$\"):a(e )F-7$Fd_l$\")LRR()F-7$Fi_l$\")=;*)))F-7$F^`l$\")wd]\"*F-7$Fc`l$\")F45' *F-7$Fh`l$\"**4=S5F-7$F]al$\"*;$>r6F-7$Fbal$\"*F.FQ\"F-7$Fgal$\"*LEmT \"F-7$F\\bl$\"*\"z\\_9F-7$Fabl$\"*rOjY\"F-7$Ffbl$\"*z,uY\"F-7$F[cl$\"* &z`p9F-7$F`cl$\"**>or9F-7$Fecl$\"*#\\*fZ\"F-7$Fjcl$\"*&GM![\"F-7$F_dl$ \"*6Ip\\\"F-7$Fddl$\"*.5z^\"F-7$Fidl$\"*!*yt`\"F-7$F^el$\"*ar1d\"F-7$F cel$\"*_W5j\"F-7$Fhel$\"*&QiSF-7$Fbfl$\"*a]5=#F-7$Fg fl$\"*)[u5EF-7$F\\gl$\"*Wuvx#F-7$Fagl$\"*6v^\"GF-7$Ffgl$\"*xB\"eGF-7$F [hl$\"*HFo)GF-7$F`hl$\"*xo-$HF-7$Fehl$\"*)o7.IF-7$Fjhl$\"*gP.8$F-7$F_i l$\"*czQP$F-7$Fdil$\"*U(*[!QF-7$Fiil$\"*=\"*f7%F-7$F^jl$\"*EqFa%F-7$Fc jl$\"*QmVz%F-7$Fhjl$\"*X_#z]F-7$F][m$\"*q\">,aF-7$Fb[m$\"*I+Vw&F--F[eo 6&Fj[m$\"\"'F*F[\\mF_eo-Fa\\m6#%Sscheme~with~stability~radius~=~imag~a xis~inclusionG-%&TITLEG6#%Uerror~curves~for~7~stage~order~5~Runge-Kutt a~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fbbs-%%FONTG6$%*HELVETICAG\"\"*-%%V IEWG6$;F(Fb[m;$!#V!#7$\"#jF`cs" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Bogacki-Shampine scheme" "a minor modificatio n of the Bogacki-Shampine scheme" "efficient scheme A" "efficient sche me B" "efficient scheme C" "scheme with stability radius = imag axis i nclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 31 "#==============================" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }