{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 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 267 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 " System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 274 "" 1 18 0 0 0 0 0 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 "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 "Times" 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 "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 "Normal " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 2 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 103 "Derivation of 11 stage, order 7 \+ Runge-Kutta schemes, each with a 12 stage, order 6 FSAL embedded schem e" }}{PARA 0 "" 0 "" {TEXT -1 46 "by Peter Stone, Gabriola Island, B.C ., Canada " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 3.12.2011" }} {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 6 " 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 that follow, where e ach 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:\\\\Maple/procdrs/intg.m\" ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "#========= =================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Relations be tween the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The Runge-Kutta schemes considered in this worksheet are \+ such that the nodes nodes " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " satisfy the cer tain relations." }}{PARA 0 "" 0 "" {TEXT -1 63 "These arise from the s tage order conditions for stages 4 to 6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "SO_eqs := [op(StageO rderConditions(2,4..6,'expanded')),op(StageOrderConditions(3,4..6,'exp anded')),\n op(StageOrderConditions(4,6..6,'expanded'))]: \n\ne_1 := \{seq(a[i,2]=0,i=4..6),seq(a[i,3]=0,i=6..6)\}:\nnode_eqs := subs(e_1,SO_eqs):\nsol := solve(\{op(node_eqs)\},indets(node_eqs) min us \{c[4],c[6]\}):\nnode_eqns := \{seq(c[i]=subs(sol,c[i]),i=[3,5])\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*node_eqnsG<$/&%\"cG6#\"\"&,$*&# \"\"\"\"\"#F.*(&F(6#\"\"'F.,&*&\"\"%F.&F(6#F6F.F.*&\"\"$F.F1F.!\"\"F., &*&F:F.F7F.F.*&F/F.F1F.F;F;F.F./&F(6#F:,$*&#F/F:F.F7F.F." }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we have " } {XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 16 " \+ and the nodes " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3* c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6#/,**&\"\"$\"\"\"*$& %\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F '&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 274 20 "____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "#=================== ==============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Sharp-Verner sc heme with " }{XPPEDIT 18 0 "c[10]=11/12" "6#/&%\"cG6#\"#5*&\"#6\"\"\" \"#7!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 161 "See: Completely Imbedded Runge-Kutta Pairs, by P. W.Sharp and J.H.Verner,\n Siam Journal on Numerical Analysis, V ol.31, No.4. (August 1994) pages 1169-1190." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/ 12, 1/12, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [4/27, 4/243, 3 2/243, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [2/9, 1/18, 0, 1/6, `` , ``, ``, ``, ``, ``, ``, ``, ``], [5/9, 5/9, 0, -25/12, 25/12, ``, `` , ``, ``, ``, ``, ``, ``], [2/3, 1/15, 0, 0, 1/3, 4/15, ``, ``, ``, `` , ``, ``, ``], [1/6, 319/3840, 0, 0, 161/1536, -41/960, 11/512, ``, `` , ``, ``, ``, ``], [4/9, 245/5184, 0, 0, 1627/10368, 151/1296, -445/10 368, 1/6, ``, ``, ``, ``, ``], [3/4, -556349853/7539261440, 0, 0, -435 6175383/3015704576, -814787343/1884815360, 831004641/3015704576, 35545 2237/235601920, 107943759/117800960, ``, ``, ``, ``], [11/12, -6899869 8967/1063035863040, 0, 0, -767387292485/425214345216, -205995991597/26 5758965760, -22181208863/141738115072, 26226796959/15502606336, 161420 0643/1107329024, 187/329, ``, ``, ``], [1, 24511479161/17979371520, 0, 0, 3889847115/217931776, 22028391/3681280, 614528179/217931776, -1484 01247/10215552, -3122234829/318384704, -4160/1221, 15040/20757, ``, `` ], [1, 5519/110880, 0, 0, 0, 0, 83/560, 932/3675, 282123/1047200, 2624 /24255, 3008/19635, 37/2100, ``], [``, 15509/341880, 0, 0, 0, 0, 6827/ 15540, 22138/81585, 78003/387464, -64144/299145, 623408/2179485, 0, -1 5/518]])" "6#-%'matrixG6#7.7/*&\"\"\"F)\"#7!\"\"*&F)F)F*F+%!GF-F-F-F-F -F-F-F-F-F-7/*&\"\"%F)\"#FF+*&F0F)\"$V#F+*&\"#KF)F3F+F-F-F-F-F-F-F-F-F -F-7/*&\"\"#F)\"\"*F+*&F)F)\"#=F+\"\"!*&F)F)\"\"'F+F-F-F-F-F-F-F-F-F-7 /*&\"\"&F)F9F+*&FAF)F9F+F<,$*&\"#DF)F*F+F+*&FEF)F*F+F-F-F-F-F-F-F-F-7/ *&F8F)\"\"$F+*&F)F)\"#:F+FF+*&\"$>$F)\"%SQF+FF+F-F-F-F-F-7/*&FIF)F0F+, $*&\"*`)\\jbF)\"+S9ERvF+F+FgN#F+*&\"*fP%z5F )\"*g4!y6F+F-F-F-F-7/*&FenF)F*F+,$*&\",n*)p)**oF)\".SI'e.j5F+F+FbF)\"'!)36F+FF+*&\"#PF)\"%+@F+F-7/F- *&\"&4b\"F)\"'!)=MF+F " 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 1760 "# coefficients by Sharp and Verne r 1994\nee := \{c[2]=1/12,\nc[3]=4/27,\nc[4]=2/9,\nc[5]=5/9,\nc[6]=2/3 ,\nc[7]=1/6,\nc[8]=4/9,\nc[9]=3/4,\nc[10]=11/12,\nc[11]=1,\nc[12]=1,\n \na[2,1]=1/12,\na[3,1]=4/243,\na[3,2]=32/243,\na[4,1]=1/18,\na[4,2]=0, \na[4,3]=1/6,\na[5,1]=5/9,\na[5,2]=0,\na[5,3]=-25/12,\na[5,4]=25/12,\n a[6,1]=1/15,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1/3,\na[6,5]=4/15,\na[7,1]= 319/3840,\na[7,2]=0,\na[7,3]=0,\na[7,4]=161/1536,\na[7,5]=-41/960,\na[ 7,6]=11/512,\na[8,1]=245/5184,\na[8,2]=0,\na[8,3]=0,\na[8,4]=1627/1036 8,\na[8,5]=151/1296,\na[8,6]=-445/10368,\na[8,7]=1/6,\na[9,1]=-5563498 53/7539261440,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-4356175383/3015704576,\n a[9,5]=-814787343/1884815360,\na[9,6]=831004641/3015704576,\na[9,7]=35 5452237/235601920,\na[9,8]=107943759/117800960,\na[10,1]=-68998698967/ 1063035863040,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-767387292485/42521434 5216,\na[10,5]=-205995991597/265758965760,\na[10,6]=-22181208863/14173 8115072,\na[10,7]=26226796959/15502606336,\na[10,8]=1614200643/1107329 024,\na[10,9]=187/329,\na[11,1]=24511479161/17979371520,\na[11,2]=0,\n a[11,3]=0,\na[11,4]=3889847115/217931776,\na[11,5]=22028391/3681280,\n a[11,6]=614528179/217931776,\na[11,7]=-148401247/10215552,\na[11,8]=-3 122234829/318384704,\na[11,9]=-4160/1221,\na[11,10]=15040/20757,\na[12 ,1]=5519/110880,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12 ,6]=83/560,\na[12,7]=932/3675,\na[12,8]=282123/1047200,\na[12,9]=2624/ 24255,\na[12,10]=3008/19635,\na[12,11]=37/2100,\n\nb[1]=5519/110880,\n b[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=83/560,\nb[7]=932/3675,\nb[8] =282123/1047200,\nb[9]=2624/24255,\nb[10]=3008/19635,\nb[11]=37/2100, \n\n`b*`[1]=15509/341880,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5] =0,\n`b*`[6]=6827/15540,\n`b*`[7]=22138/81585,\n`b*`[8]=78003/387464, \n`b*`[9]=-64144/299145,\n`b*`[10]=623408/2179485,\n`b*`[11]=0,\n`b*`[ 12]=-15/518\}:" }}}{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 0 126 "subs(ee,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[j],j=1..11),``],[`b*`,se q(`b*`[j],j=1..12)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7/7/#\"\"\"\"#7F(%!GF+F+F+F+F+F+F+F+F+F+7/#\"\"%\"#F#F.\"$V##\"#KF1F+ F+F+F+F+F+F+F+F+F+7/#\"\"#\"\"*#F)\"#=\"\"!#F)\"\"'F+F+F+F+F+F+F+F+F+7 /#\"\"&F7F>F:#!#DF*#\"#DF*F+F+F+F+F+F+F+F+7/#F6\"\"$#F)\"#:F:F:#F)FF#F .FHF+F+F+F+F+F+F+7/F;#\"$>$\"%SQF:F:#\"$h\"\"%O:#!#T\"$g*#\"#6\"$7&F+F +F+F+F+F+7/#F.F7#\"$X#\"%%=&F:F:#\"%F;\"&o.\"#\"$^\"\"%'H\"#!$X%FinF;F +F+F+F+F+7/#FFF.#!*`)\\jb\"+S9ERvF:F:#!+$QvhN%\"+wXq:I#!*Vty9)\"+g`\"[ )=#\"*TY+J)Ffo#\"*PAXb$\"*?>gN##\"*fP%z5\"*g4!y6F+F+F+F+7/#FVF*#!,n*)p )**o\".SI'e.j5F:F:#!-&[#H(Qn(\"-;_M9_U#!-(f\"*f*f?\"-gd'*edE#!,j)37=A \"-s]6Q<9#\",fpzEi#\",Ojg-b\"#\"+V1?9;\"+C!Ht5\"#\"$(=\"$H$F+F+F+7/F)# \",h\"z9^C\",?:Pzz\"F:F:#\"+:r%)*)Q\"*w<$z@#\")\"RG?#\"(!G\"o$#\"*z\"G XhF_r#!*Z7S[\"\")_b@5#!+H[BAJ\"*/ZQ=$#!%gT\"%@7#\"&S]\"\"&d2#F+F+7/F)# \"%>b\"'!)36F:F:F:F:#\"#$)\"$g&#\"$K*\"%vO#\"'B@G\"(+s/\"#\"%CE\"&bU## \"%3I\"&N'>#\"#P\"%+@F+7/%\"bGFbsF:F:F:F:FesFhsF[tF^tFatFdtF+7/%#b*G# \"&4b\"\"'!)=MF:F:F:F:#\"%Fo\"&Sb\"#\"&Q@#\"&&e\")#\"&.!y\"'kuQ#!&WT' \"'X\"*H#\"'3Mi\"(&[z@F:#!#:\"$=&Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs : = [op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'expand ed'))]:\n`RK6_12eqs*` := subs(b=`b*`,OrderConditions(6,12,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Chec k: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(ee,RK7 _11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(ee,`RK6_1 2eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#P" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we s et-up stage-order condtions to check for stage-orders from 2 to 6 incl usive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to \+ 6 do\n so||ct||_11 := StageOrderConditions(ct,11,'expanded');\nend d o:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "S tages 6 to 11 have stage-order 4, while stages 4 and 5 have stage-orde r 3 has and stage 3 has stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(subs(ee,so||i||_11[j])),i=2..6)],j= 1..9)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) t hen break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#\"\"$F%\"\"%F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j] ,i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 10 " }}{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 0 151 "[Sum(b[i]*a[i,1],i=1+1..11)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b [j]*(1-c[j]),j=2..10)];\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=ad d,%))));\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,/-%$SumG6$*&&% \"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"#6&F*6#F-/-F&6$*&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,;\"\"&F 4*&&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,;\"\"(F4*&&F*6#FeoF-,& F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FE FdqFFF-/-F&6$*&F)F-&F/6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF- /-F&6$*&F)F-&F/6$F,F_rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-/-F&6$*& F)F-&F/6$F,F]sF-/F,;F4F4*&&F*6#F]sF-,&F-F-&FEF]tFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 53 "None of the principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "RK7_11err_eqs := Principa lErrorConditions(7,11,'expanded'):\nexpand(subs(ee,RK7_11err_eqs)):\nm ap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 _r\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms7_11 := PrincipalErrorTerms(7,11,'expan ded'):\nsm := 0:\nfor ct to nops(errterms7_11) do\n sm := sm+(evalf( subs(ee,errterms7_11[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+)y$*G;#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm of the order 6 e mbedded scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms6_12*` := subs(b=`b*`,PrincipalErrorTerms(6 ,12,'expanded')):\nsm := 0:\nfor ct to nops(`errterms6_12*`) do\n sm := sm+(evalf(subs(ee,`errterms6_12*`[ct])))^2;\nend do:\nsqrt(sm);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+YNd]R!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "We incorporate the stage-order equations to ensure that stage \+ 3 has stage-order 2 and stages 4 and 5 have stage order 3 and stages 6 to 11 have stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 57 "These condi tions lead to relationships between the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\" \"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c [6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6#/,**&\"\"$\"\"\"*$&% \"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F' &F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 68 "which we include explicitly to make the compu tation more efficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j +1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6# /%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and " }{XPPEDIT 18 0 "j = 6;" "6# /%\"jG\"\"'" }{TEXT -1 9 " . . 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 134 "The simplifying conditions allow us to \+ manage with just 10 \"simple\" order conditions which are given (in ab reviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These incl ude the 7 order 7 quadrature conditions and 3 additional order 7 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]] ,i=[1,2,4,8,16,32,38,48,61,64])]:\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(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#KF)/*&F,F ()F2FGF(#F(\"\"'7%\"#QF)/*&F,F(-%!G6#*&%\"aGF(-FT6#*&FWF(-FT6#*&FWF(-F T6#*&F8F(FWF(F(F(F(F(#F(\"%?D7%\"#[F)/*&F,F(-FT6#*&FWF(-FT6#*&FWF(-FT6 #*&F?F(FWF(F(F(F(#F(\"$S)7%\"#hF)/*(F,F(F2F(-FT6#*&FEF(FWF(F(#F(\"#N7% \"#kF)/*&F,F()F2FNF(#F(\"\"(Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "SO7_11 := SimpleO rderConditions(7,11,'expanded'):\nSO_eqs := [op(RowSumConditions(11,'e xpanded')),op(StageOrderConditions(2,11,'expanded')),\n op(StageOrder Conditions(3,4..11,'expanded')),op(StageOrderConditions(4,6..11,'expan ded'))]:\nord_cdns := [seq(SO7_11[i],i=[1,2,4,8,16,32,38,48,61,64])]: \nsimp_eqs := [add(b[i]*a[i,1],i=2..11)=b[1],seq(add(b[i]*a[i,j],i=j+1 ..11)=b[j]*(1-c[j]),j=[$8..10])]:\nnode_eqs := [3*c[3]=2*c[4],3*c[6]^2 -4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4]=0]:\ncdns := [op(node_eqs),op(sim p_eqs),op(SO_eqs),op(ord_cdns)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The scheme can be obtained by specifying the nodes: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/12;" "6#/&% \"cG6#\"\"#*&\"\"\"F)\"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 2/9;" "6#/&%\"cG6#\"\"%*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 2/3;" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1/6;" "6#/&%\"cG6#\"\"(*&\"\" \"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 4/9;" "6#/&% \"cG6#\"\")*&\"\"%\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[9] = 3/4;" "6#/&%\"cG6#\"\"**&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[10]=11/12" "6#/&%\"cG6#\"#5*&\"#6\"\"\"\"#7!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\"#6\"\"\"" } {TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 45 "together with the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[5,2] = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\" \"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10, 2] = 0;" "6#/&%\"aG6 $\"#5\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/& %\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[6,3] = 0;" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3] = 0;" "6#/&%\"aG6$\"\"(\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,3] = 0;" "6#/&%\"aG6$\"\"* \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,3] = 0;" "6#/&%\"aG 6$\"#5\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/ &%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[4] = 0;" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[5] = 0;" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "e1 := \{c[2]=1/12,c[4]=2/9,c[6]=2/3,c[7]=1/6,c[8]=4/9,c[9]=3/4,c[ 10]=11/12,c[11]=1,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),seq (b[i]=0,i=2..5)\}:\neqns := subs(e1,cdns):\nnops(eqns);\nindets(eqns) \+ minus \{a[8,7],a[9,7],a[9,8]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\\" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[8,7],a[9,7],a[9,8 ]\}):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(ma p(rhs,e3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\")\"\"(&F% 6$\"\"*F(&F%6$F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1936 "e3 := \{a[8,5] = 9/40*a[8,7]+32/405, b[9] = 2624/24 255, a[9,5] = 9/40*a[9,7]-16/15*a[9,8]+1053/5120, a[11,10] = 15040/207 57, b[8] = 282123/1047200, b[7] = 932/3675, a[9,4] = -63/64*a[9,7]-1/3 *a[9,8]+2835/8192, b[6] = 83/560, a[8,1] = -21/160*a[8,7]+28/405, a[7, 4] = 161/1536, a[5,1] = 5/9, a[2,1] = 1/12, a[11,7] = -5272/111+282123 /3256*a[8,7]+104960/8547*a[9,7], a[5,3] = -25/12, a[11,8] = -1165671/5 5352+104960/8547*a[9,8], a[8,4] = -63/64*a[8,7]+26/81, a[6,1] = 1/15, \+ b[4] = 0, a[10,9] = 187/329, b[11] = 37/2100, a[10,4] = -8687921/11550 72+17773749/1540096*a[8,7]+6273/3008*a[9,7]+697/987*a[9,8], b[1] = 551 9/110880, a[10,7] = 180081/26320-282123/24064*a[8,7]-697/329*a[9,7], a [11,5] = 7821/740+2539107/130240*a[8,7]+7872/2849*a[9,7]-335872/25641* a[9,8], a[7,6] = 11/512, b[10] = 3008/19635, a[10,5] = -1217887/721920 -2539107/962560*a[8,7]-6273/13160*a[9,7]+11152/4935*a[9,8], a[5,4] = 2 5/12, a[6,5] = 4/15, b[3] = 0, a[11,6] = 3167/1184-1974861/208384*a[8, 7]-1640/1221*a[9,7]+104960/25641*a[9,8], a[10,1] = -1765687/2887680+59 24583/3850240*a[8,7]+2091/7520*a[9,7]-697/4935*a[9,8], a[6,4] = 1/3, c [2] = 1/12, c[4] = 2/9, c[5] = 5/9, c[6] = 2/3, c[7] = 1/6, c[8] = 4/9 , c[9] = 3/4, a[11,4] = 63999/1184-17773749/208384*a[8,7]-4920/407*a[9 ,7]-104960/25641*a[9,8], a[4,1] = 1/18, a[7,1] = 319/3840, a[8,6] = -7 /64*a[8,7]-2/81, a[9,1] = -21/160*a[9,7]+1/15*a[9,8]+1293/20480, a[9,6 ] = -7/64*a[9,7]+1/3*a[9,8]+1107/8192, b[5] = 0, c[3] = 4/27, a[11,9] \+ = -4160/1221, a[10,8] = 408969/120320-697/329*a[9,8], a[11,1] = 482563 /97680-5924583/520960*a[8,7]-656/407*a[9,7]+20992/25641*a[9,8], a[7,5] = -41/960, a[3,2] = 32/243, a[3,1] = 4/243, a[4,3] = 1/6, b[2] = 0, c [10] = 11/12, c[11] = 1, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = \+ 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[6,3] = 0, a[7,3 ] = 0, a[8,3] = 0, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[10,6] = -27 995/385024+1974861/1540096*a[8,7]+697/3008*a[9,7]-697/987*a[9,8]\}:" } {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 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "subs(e3,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(8-i)],i=2..7)]));\nfor ii from 8 to 11 do\n print(` `);\n print(c[ii]=subs(e3,c[ii])); \n for jj to ii-1 do\n coe ff_val := subs(e3,a[ii,jj]);\n if coeff_val=a[ii,jj] then \n \+ print(a[ii,jj]*` is a parameter.`)\n else\n print(a[i i,jj]=coeff_val);\n end if;\n end do:\nend do:\nprint(``);\nseq (b[i]=subs(e3,b[i]),i=1..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7*#\"\"\"\"#7F(%!GF+F+F+F+F+7*#\"\"%\"#F#F.\"$V##\"#KF1F+F+F+ F+F+7*#\"\"#\"\"*#F)\"#=\"\"!#F)\"\"'F+F+F+F+7*#\"\"&F7F>F:#!#DF*#\"#D F*F+F+F+7*#F6\"\"$#F)\"#:F:F:#F)FF#F.FHF+F+7*F;#\"$>$\"%SQF:F:#\"$h\" \"%O:#!#T\"$g*#\"#6\"$7&F+Q)pprint186\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"\"%\"\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&*&#\"#@\"$g \"F(&F%6$F'\"\"(F(!\"\"#\"#G\"$0%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"% ,&*&#\"#j\"#k\"\"\"&F%6$F'\"\"(F.!\"\"#\"#E\"#\")F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\"\"*\"#S\"\"\"&F%6$F'\"\"(F. F.#\"#K\"$0%F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"', &*&#\"\"(\"#k\"\"\"&F%6$F'F,F.!\"\"#\"\"#\"#\")F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"\",(*&#\"#@\"$g\"F(&F%6$F'\"\"(F(!\"\"*&#F(\"#:F(&F%6$F'\"\" )F(F(#\"%$H\"\"&![?F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,(*&#\"#j\"#k \"\"\"&F%6$F'\"\"(F.!\"\"*&#F.\"\"$F.&F%6$F'\"\")F.F2#\"%NG\"%#>)F." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,(*&#F'\"#S\"\"\"& F%6$F'\"\"(F-F-*&#\"#;\"#:F-&F%6$F'\"\")F-!\"\"#\"%`5\"%?^F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',(*&#\"\"(\"#k\"\"\"&F%6$ F'F,F.!\"\"*&#F.\"\"$F.&F%6$F'\"\")F.F.#\"%26\"%#>)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\"(\"\"\"%1~is~a~parameter.GF)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\")\"\"\"%1~is~a~par ameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#6\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",**&#\"($eCf\"(S-&QF(&F%6$\"\")\"\"(F(F(#\"((ol< \"(!o()G!\"\"*&#\"$(p\"%N\\F(&F%6$\"\"*F0F(F5*&#\"%\"4#\"%?vF(&F%6$FsF3*&#\"&_6\"\"%N\\F.&F%6$\"\"*F1F.F.*&#\"%ti\"&gJ\"F.&F%6$F=F2F.F3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',**&#\"(h[(>\"(' 4S:\"\"\"&F%6$\"\")\"\"(F.F.#\"&&*z#\"'C]Q!\"\"*&#\"$(p\"$()*F.&F%6$\" \"*F1F.F6*&#F9\"%3IF.&F%6$F=F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#5\"\"(,(*&#\"'B@G\"&kS#\"\"\"&F%6$\"\")F(F.!\"\"#\"'\"3!= \"&?j#F.*&#\"$(p\"$H$F.&F%6$\"\"*F(F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"),&#\"'p*3%\"'?.7\"\"\"*&#\"$(p\"$H$F-&F%6$\"\" *F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"$(= \"$H$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#6\"\"\",**&#\"($eCf\"'g4_F(&F%6$\"\")\"\"(F(!\"\"*&#\"$c' \"$2%F(&F%6$\"\"*F1F(F2#\"'jD[\"&!o(*F(*&#\"&#*4#\"&Tc#F(&F%6$F9F0F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,**&#\")\\Px<\"'%Q3#\"\"\"&F%6$\"\") \"\"(F.!\"\"*&#\"%?\\\"$2%F.&F%6$\"\"*F2F.F3#\"&**R'\"%%=\"F.*&#\"'g\\ 5\"&Tc#F.&F%6$F:F1F.F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"&,**&#\"(2\"RD\"'S-8\"\"\"&F%6$\"\")\"\"(F.F.*&#\"%sy\"%\\GF.&F%6 $\"\"*F2F.F.#\"%@y\"$S(F.*&#\"'seL\"&Tc#F.&F%6$F9F1F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',**&#\"(h[(>\"'%Q3#\"\"\"& F%6$\"\")\"\"(F.!\"\"*&#\"%S;\"%@7F.&F%6$\"\"*F2F.F3#\"%nJ\"%%=\"F.*&# \"'g\\5\"&Tc#F.&F%6$F:F1F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#6\"\"(,(*&#\"'B@G\"%cK\"\"\"&F%6$\"\")F(F.F.*&#\"'g\\5\"%Z&)F.&F %6$\"\"*F(F.F.#\"%s_\"$6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"),&*&#\"'g\\5\"%Z&)\"\"\"&F%6$\"\"*F(F.F.#\"(rc;\"\"&_`& !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!%gT\"%@7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#\"&S]\"\"&d2#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6-/&%\"bG6#\"\"\"#\"%>b\"'!)36/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"% F//&F%6#\"\"&F//&F%6#\"\"'#\"#$)\"$g&/&F%6#\"\"(#\"$K*\"%vO/&F%6#\"\") #\"'B@G\"(+s/\"/&F%6#\"\"*#\"%CE\"&bU#/&F%6#\"#5#\"%3I\"&N'>/&F%6#\"#6 #\"#P\"%+@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK7_11eqs := [op(RowSumConditions(11,'exp anded')),op(OrderConditions(7,11,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simplify(subs(e3,RK7_11eqs)):\nmap(u->lhs(u )-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ 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#\"#&*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We now \+ turn our attention to the embedded 12 stage, order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 210 "The first 11 stages of the embedded order 6 sc heme coincide with those of the order 7 scheme and the weights of the \+ order 7 scheme provide the linking coefficients for the 12th stage of \+ the order 6 scheme. Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[12,j] = b[j];" "6#/&%\"aG6$\"#7%\"jG&%\"bG6#F(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 11." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplif ying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*& &%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and \+ " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The simplifying condit ions allow us to manage with just 7 \"simple\" order conditions which \+ are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 6 order 6 quadrature conditions and 1 additio nal order 6 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO6*` := subs(b=`b*`,Simpl eOrderConditions(6)):\n[seq([i,`SO6*`[i]],i=[1,2,4,8,16,31,32])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7)7%\"\"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7 %\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#JF)/*&F,F(-%!G6#*&FEF(%\"aGF(F(#F(\"#I7%\"#KF)/ *&F,F()F2FGF(#F(\"\"'Q)pprint346\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "`SO6_12*` := subs(b=`b*`,S impleOrderConditions(6,12,'expanded')):\n`cdns*` := [seq(`SO6_12*`[i], i=[1,2,4,8,16,31,32])]:\n`simp_eqs*` := [seq(add(`b*`[i]*a[i,j],i=j+1. .12)=`b*`[j]*(1-c[j]),j=[4,5])]:\nwt_eqns := [seq(b[i]=a[12,i],i=1..11 )]:\n`cdns*` := [op(`cdns*`),op(wt_eqns),op(`simp_eqs*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0; " "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = \+ 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] \+ = 0;" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5 ] = 0;" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "`b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e4 := \+ \{c[12]=1,seq(`b*`[i]=0,i=2..5),`b*`[11]=0\}:\ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqns2 := [op(remove(_U->type(rhs(_U),realcons),e5)),op(subs(e 5,`cdns*`))]:\nnops(eqns2);\nindets(eqns2) minus \{a[8,7]\};\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "e6 := solve(\{op(eqns2)\},indets(eqns2) \+ minus \{a[8,7]\}):\ne7 := simplify(subs(select(u_->member(lhs(u_),\{b[ 8],a[8,7],a[9,7],a[9,8]\}),e6),e5)):\ne8 := `union`(e7,e6):\ninfolevel [solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a[1 0,4]=subs(e8,a[10,4]);\na[11,1]=subs(e8,a[11,1]);\n`b*`[12]=subs(e8,`b *`[12]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"+^] e[z\"+sq)>K$!\"\"*&#\"-VSfJ+D\",Ov0p3(\"\"\"&F%6$\"\")\"\"(F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"*.,^D'\"*!oF4G F(*&#\"+@_,69\"*?ZTs#F(&F%6$\"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#!#:\"$=&" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Note that some of the coefficients depend on " }{XPPEDIT 18 0 "a[8,7 ]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2350 "e8 := \{b[10] = 3008/19635, b[11] = 37/2100, `b*`[1] = 15509/341880, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] \+ = 0, `b*`[5] = 0, `b*`[6] = 6827/15540, `b*`[7] = 22138/81585, `b*`[8] = 78003/387464, `b*`[9] = -64144/299145, `b*`[10] = 623408/2179485, ` b*`[11] = 0, `b*`[12] = -15/518, a[10,6] = -245637799/1107329024+27781 288227/70869057536*a[8,7], a[8,5] = 9/40*a[8,7]+32/405, b8[2] = 0, b8[ 3] = 0, b8[4] = 0, b8[5] = 0, a[8,4] = -63/64*a[8,7]+26/81, a[9,7] = 9 88911/460160-90522117/23560192*a[8,7], c[12] = 1, a[9,1] = -18593823/1 17800960+1900964457/3769630720*a[8,7], c[2] = 1/12, a[10,4] = -7948585 051/3321987072+250031594043/70869057536*a[8,7], a[10,1] = -2380352557/ 16609935360+83343864681/177172643840*a[8,7], a[8,1] = -21/160*a[8,7]+2 8/405, a[11,4] = 41412987/1702592-4233045663/108965888*a[8,7], a[6,2] \+ = 0, a[6,3] = 0, a[6,4] = 1/3, a[6,5] = 4/15, a[7,1] = 319/3840, a[7,2 ] = 0, a[7,3] = 0, a[7,4] = 161/1536, a[7,5] = -41/960, a[7,6] = 11/51 2, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[9,3] = 0, a[9,8] = 107943759/ 117800960, a[10,2] = 0, a[10,3] = 0, a[10,8] = 1614200643/1107329024, \+ a[11,5] = 1198197/266030+604720809/68103680*a[8,7], a[10,5] = -6651452 83/1038120960-35718799149/44293160960*a[8,7], a[11,8] = -3122234829/31 8384704, c[3] = 4/27, c[4] = 2/9, a[11,7] = -1684316/79809+67191201/17 02592*a[8,7], a[9,4] = -48883905/23560192+5702893371/1507852288*a[8,7] , a[10,7] = 69310989/30278528-3968755461/1107329024*a[8,7], a[9,5] = - 2121957/7362560-814699053/942407680*a[8,7], a[9,6] = 4842081/23560192+ 633654819/1507852288*a[8,7], c[5] = 5/9, c[6] = 2/3, c[7] = 1/6, c[8] \+ = 4/9, c[9] = 3/4, c[10] = 11/12, c[11] = 1, a[2,1] = 1/12, a[3,1] = 4 /243, a[3,2] = 32/243, a[4,1] = 1/18, a[4,2] = 0, a[4,3] = 1/6, a[5,1] = 5/9, a[5,2] = 0, a[5,3] = -25/12, a[5,4] = 25/12, a[6,1] = 1/15, a[ 11,2] = 0, a[11,3] = 0, a[10,9] = 187/329, a[11,6] = 6025841/1702592-4 70338407/108965888*a[8,7], a[11,1] = 625510103/280927680-1411015221/27 2414720*a[8,7], a[11,9] = -4160/1221, a[11,10] = 15040/20757, a[12,1] \+ = 5519/110880, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[1 2,6] = 83/560, a[12,7] = 932/3675, a[12,8] = 282123/1047200, a[12,9] = 2624/24255, a[12,10] = 3008/19635, a[12,11] = 37/2100, b[1] = 5519/11 0880, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[6] = 83/560, b[7] = 93 2/3675, b[8] = 282123/1047200, b[9] = 2624/24255, a[8,6] = -7/64*a[8,7 ]-2/81\}:" }{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 65 "The coefficients for the embedded ord er 6 scheme are as follows. " }}{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 0 477 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7)] ));\nfor ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e8,c[i i])); \n for jj to ii-1 do\n coeff_val := subs(e8,a[ii,jj]);\n \+ if coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a param eter.`)\n else\n print(a[ii,jj]=coeff_val);\n end if ;\n end do:\nend do:\nprint(``);\nc[12]=subs(e8,c[12]);\nseq(a[12,i] =subs(e8,a[12,i]),i=1..11);\nprint(``);\nseq(`b*`[i]=subs(e8,`b*`[i]), i=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\" #7F(%!GF+F+F+F+F+7*#\"\"%\"#F#F.\"$V##\"#KF1F+F+F+F+F+7*#\"\"#\"\"*#F) \"#=\"\"!#F)\"\"'F+F+F+F+7*#\"\"&F7F>F:#!#DF*#\"#DF*F+F+F+7*#F6\"\"$#F )\"#:F:F:#F)FF#F.FHF+F+7*F;#\"$>$\"%SQF:F:#\"$h\"\"%O:#!#T\"$g*#\"#6\" $7&F+Q)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"\"%\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&*&#\"#@\"$g\"F(&F%6$F'\"\"(F(!\" \"#\"#G\"$0%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\"#j\"#k\"\"\" &F%6$F'\"\"(F.!\"\"#\"#E\"#\")F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\")\"\"&,&*&#\"\"*\"#S\"\"\"&F%6$F'\"\"(F.F.#\"#K\"$0%F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"\"(\"#k\"\" \"&F%6$F'F,F.!\"\"#\"\"#\"#\")F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& &%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\",&# \")BQf=\"*g4!y6!\"\"*&#\"+dW'4!>\"+?2jpPF(&F%6$\"\")\"\"(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,&#\")0R))[\")#>gN#!\"\"*&#\"+rL*Gq& \"+)G_y]\"\"\"\"&F%6$\"\")\"\"(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\"&,&#\"(d>7#\"(gDO(!\"\"*&#\"*`!*p9)\"*!o2C%*\"\"\"& F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"',&#\"(\"3U[\")#>gN#\"\"\"*&#\"*>[lL'\"+)G_y]\"F-&F%6$\"\")\"\"(F-F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"'6*))*\"'g ,Y\"\"\"*&#\")<@_!*\")#>gN#F-&F%6$\"\")F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"*fP%z5\"*g4!y6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5# \"#6\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",&#\"+d DN!Q#\",g`$*4m\"!\"\"*&#\",\"okQM$)\"-SQksrK$!\"\"*&#\"-VSfJ+D \",Ov0p3(\"\"\"&F%6$\"\")\"\"(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#5\"\"&,&#\"*$GX^m\"+g47Q5!\"\"*&#\",\\\"*z=d$\",g4;$HW\"\" \"&F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"',&#\"**zPcC\"+C!Ht5\"!\"\"*&#\",F#)G\"yF\",Ov0p3(\"\"\"&F%6$\"\") \"\"(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\")* )4Jp\")G&y-$\"\"\"*&#\"+havoR\"+C!Ht5\"F-&F%6$\"\")F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"+V1?9;\"+C!Ht5\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"$(=\"$H$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"\"\",&#\"*.,^D'\"*!oF4GF(*&#\"+@_,69\"*?ZTs#F(&F%6$\"\")\"\"(F(!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,&#\")()HTT\"(#f-<\"\"\"*&#\"+jc/LU\" *))e'*3\"F-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"\"&,&#\"((>)>\"\"'IgE\"\"\"*&#\"*43s/'\")!o.\"oF-&F%6$\" \")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',&# \"(Te-'\"(#f-<\"\"\"*&#\"*2%Q.Z\"*))e'*3\"F-&F%6$\"\")\"\"(F-!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"(;Vo\"\"&4)z!\" \"*&#\"),7>n\"(#f-<\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#!+H[BAJ\"*/ZQ=$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!%gT\"%@7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#\"&S]\"\"&d2#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-/&%\"aG6$\"#7\"\"\"#\"%>b\"'!) 36/&F%6$F'\"\"#\"\"!/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6 $F'\"\"'#\"#$)\"$g&/&F%6$F'\"\"(#\"$K*\"%vO/&F%6$F'\"\")#\"'B@G\"(+s/ \"/&F%6$F'\"\"*#\"%CE\"&bU#/&F%6$F'\"#5#\"%3I\"&N'>/&F%6$F'\"#6#\"#P\" %+@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"#\"&4b\"\"'!)=M/&F%6#\"\"#\"\"!/&F%6#\" \"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"%Fo\"&Sb\"/&F%6#\"\"(#\"& Q@#\"&&e\")/&F%6#\"\")#\"&.!y\"'kuQ/&F%6#\"\"*#!&WT'\"'X\"*H/&F%6#\"#5 #\"'3Mi\"(&[z@/&F%6#\"#6F//&F%6#\"#7#!#:\"$=&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "R K7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7, 11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,OrderConditions(6,12,'e xpanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(s ubs(e8,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs (e8,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#P" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "The Sharp-Verner choice of " }{XPPEDIT 18 0 "a[8,7] = 1/6;" "6#/&%\"aG6$\"\")\"\"(*&\"\"\"F*\"\"'! \"\"" }{TEXT -1 80 " makes the principal error norm for the order 7 s cheme approximately a minimum." }}{PARA 0 "" 0 "" {TEXT -1 34 "A sligh tly better value would be " }{XPPEDIT 18 0 "a[8,7]=9/53" "6#/&%\"aG6$ \"\")\"\"(*&\"\"*\"\"\"\"#`!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "errterms := PrincipalErrorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errte rms) do\n sm := sm+(evalf(subs(e8,errterms[ct])))^2;\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(sqrt(sm),a[8,7]=-.01..0.35,color=COLOR(RGB,.5,0,1),thickness= 2,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 277 277 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$!3-+++++++5!#>$\"3JeE#p%)\\As#!#A7$$!3 S,+++X-`@!#?$\"3M$GYBxN!zEF-7$$\"3))*****\\#peuYF1$\"3py')z\"4kCk#F-7$ $\"3/+++0]HN7F*$\"3CEH[o:`-EF-7$$\"3V+++N5A3?F*$\"3OsQ&)34njDF-7$$\"3U ****\\#[tux#F*$\"3mgg=(e(QEDF-7$$\"3a****\\s*o1\\$F*$\"3[s^9\"y;J\\#F- 7$$\"3A****\\#fT\"HUF*$\"3W%3,*3e.gCF-7$$\"3M****\\K5(G*\\F*$\"3k*o%** 3%[tU#F-7$$\"3c****\\#=^Tv&F*$\"37)3O\"z8P'R#F-7$$\"3O+++!R@s`'F*$\"39 zRx'\\YiO#F-7$$\"3V+++b+&pA(F*$\"3=*=5@vE7M#F-7$$\"3.,++q6U.!)F*$\"3ce o1Tt\"[J#F-7$$\"3y******\\03$y)F*$\"3opQ5CTB!H#F-7$$\"3p+++IoUM&*F*$\" 3)[oW_yO%oAF-7$$\"31++DB@n@5!#=$\"3)RNiFg)H]AF-7$$\"3,+++FN!G5\"Fbp$\" 3D3Z!*)pO3B#F-7$$\"39+++oC`r6Fbp$\"3flqf;B=;AF-7$$\"3'****\\_c![^7Fbp$ \"3^^)G;d(H,AF-7$$\"3x*****z<[AK\"Fbp$\"3%eXJi)z4!>#F-7$$\"3'****\\_Y \"*)*R\"Fbp$\"3!>z%[m/)*z@F-7$$\"3#)***\\dBEQZ\"Fbp$\"3!=/eX#\\[s@F-7$ $\"35++]7%p4b\"Fbp$\"3ugx/-@\"p;#F-7$$\"3,++DI4\"=i\"Fbp$\"3s.wgV;$Q;# F-7$$\"3%)****\\xIA)p\"Fbp$\"3'f#)3=I0F;#F-7$$\"3#****\\K#QfxFbp$ \"39)>IXVeC<#F-7$$\"3w*****fi*R)*>Fbp$\"3/&oMRk6.=#F-7$$\"3y****\\%p=Q 2#Fbp$\"3SDrH)=*=!>#F-7$$\"3#****\\7_!zY@Fbp$\"3IMy,EZy,AF-7$$\"3u**** \\$H8yA#Fbp$\"3*o6B$G(pp@#F-7$$\"3o*****z&eh+BFbp$\"3#*Hi*Q8[EB#F-7$$ \"3#******\\8[$yBFbp$\"3/vLA^kY^AF-7$$\"3w***\\78&y[CFbp$\"3=c'\\bjG.F #F-7$$\"3y*****HP\"zDDFbp$\"3!)4u>;t'GH#F-7$$\"3U***\\_C[#)f#Fbp$\"3$G Q!pxB&eJ#F-7$$\"3k***\\Ub')Rn#Fbp$\"3ZsWPl[mTBF-7$$\"3e*****\\fM![FFbp $\"3c]: " 0 "" {MPLTEXT 1 0 162 "evalf[12](minimize(sqrt(sm),a[8,7]=0..1,'location')) ;\nconvert(subs(op(1,op(op(2,[%]))),a[8,7]),rational,4);\nconvert(subs (op(1,op(op(2,[%%]))),a[8,7]),rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-;__qi@!#;<#7$<#/&%\"aG6$\"\")\"\"($\"-B+#>xp\"!#7F# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#t\"$I%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "evalf(eval(sqrt(sm),a[8,7]=1/6)); \nevalf(eval(sqrt(sm),a[8,7]=9/53));\nevalf(eval(sqrt(sm),a[8,7]=73/43 0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!z$*G;#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$G0F;#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+`_qi@!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e9 := `union`(subs(a[8,7]=1/6,e8),\{a[8,7]=1/6\}): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1961 " e9 := \{b[10] = 3008/19635, b[11] = 37/2100, `b*`[1] = 15509/341880, ` b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, `b*`[6] = 6827/1554 0, `b*`[7] = 22138/81585, `b*`[8] = 78003/387464, `b*`[9] = -64144/299 145, `b*`[10] = 623408/2179485, `b*`[11] = 0, `b*`[12] = -15/518, a[11 ,1] = 24511479161/17979371520, b8[2] = 0, b8[3] = 0, b8[4] = 0, b8[5] \+ = 0, c[12] = 1, c[2] = 1/12, a[6,2] = 0, a[6,3] = 0, a[6,4] = 1/3, a[6 ,5] = 4/15, a[7,1] = 319/3840, a[7,2] = 0, a[7,3] = 0, a[7,4] = 161/15 36, a[7,5] = -41/960, a[7,6] = 11/512, a[8,1] = 245/5184, a[8,2] = 0, \+ a[8,3] = 0, a[8,4] = 1627/10368, a[8,5] = 151/1296, a[8,6] = -445/1036 8, a[8,7] = 1/6, a[9,1] = -556349853/7539261440, a[9,2] = 0, a[9,3] = \+ 0, a[9,4] = -4356175383/3015704576, a[9,5] = -814787343/1884815360, a[ 9,6] = 831004641/3015704576, a[9,7] = 355452237/235601920, a[9,8] = 10 7943759/117800960, a[10,2] = 0, a[10,3] = 0, a[10,4] = -767387292485/4 25214345216, a[10,5] = -205995991597/265758965760, a[10,6] = -22181208 863/141738115072, a[10,7] = 26226796959/15502606336, a[10,8] = 1614200 643/1107329024, a[11,8] = -3122234829/318384704, c[3] = 4/27, c[4] = 2 /9, c[5] = 5/9, c[6] = 2/3, c[7] = 1/6, c[8] = 4/9, c[9] = 3/4, a[10,1 ] = -68998698967/1063035863040, c[10] = 11/12, c[11] = 1, a[2,1] = 1/1 2, a[3,1] = 4/243, a[3,2] = 32/243, a[4,1] = 1/18, a[4,2] = 0, a[4,3] \+ = 1/6, a[5,1] = 5/9, a[5,2] = 0, a[5,3] = -25/12, a[5,4] = 25/12, a[6, 1] = 1/15, a[11,2] = 0, a[11,3] = 0, a[11,4] = 3889847115/217931776, a [11,5] = 22028391/3681280, a[11,6] = 614528179/217931776, a[11,7] = -1 48401247/10215552, a[10,9] = 187/329, a[11,9] = -4160/1221, a[11,10] = 15040/20757, a[12,1] = 5519/110880, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[12,6] = 83/560, a[12,7] = 932/3675, a[12,8] = 282 123/1047200, a[12,9] = 2624/24255, a[12,10] = 3008/19635, a[12,11] = 3 7/2100, b[1] = 5519/110880, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[ 6] = 83/560, b[7] = 932/3675, b[8] = 282123/1047200, b[9] = 2624/24255 \}:" }{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 62 "The coefficients for the combined scheme \+ are given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "su bs(e9,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[`b `,seq(b[j],j=1..11),``],[`b*`,seq(`b*`[j],j=1..12)]]));\n``;\nevalf[6] (%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/#\"\"\"\"#7F( %!GF+F+F+F+F+F+F+F+F+F+7/#\"\"%\"#F#F.\"$V##\"#KF1F+F+F+F+F+F+F+F+F+F+ 7/#\"\"#\"\"*#F)\"#=\"\"!#F)\"\"'F+F+F+F+F+F+F+F+F+7/#\"\"&F7F>F:#!#DF *#\"#DF*F+F+F+F+F+F+F+F+7/#F6\"\"$#F)\"#:F:F:#F)FF#F.FHF+F+F+F+F+F+F+7 /F;#\"$>$\"%SQF:F:#\"$h\"\"%O:#!#T\"$g*#\"#6\"$7&F+F+F+F+F+F+7/#F.F7# \"$X#\"%%=&F:F:#\"%F;\"&o.\"#\"$^\"\"%'H\"#!$X%FinF;F+F+F+F+F+7/#FFF.# !*`)\\jb\"+S9ERvF:F:#!+$QvhN%\"+wXq:I#!*Vty9)\"+g`\"[)=#\"*TY+J)Ffo#\" *PAXb$\"*?>gN##\"*fP%z5\"*g4!y6F+F+F+F+7/#FVF*#!,n*)p)**o\".SI'e.j5F:F :#!-&[#H(Qn(\"-;_M9_U#!-(f\"*f*f?\"-gd'*edE#!,j)37=A\"-s]6Q<9#\",fpzEi #\",Ojg-b\"#\"+V1?9;\"+C!Ht5\"#\"$(=\"$H$F+F+F+7/F)#\",h\"z9^C\",?:Pzz \"F:F:#\"+:r%)*)Q\"*w<$z@#\")\"RG?#\"(!G\"o$#\"*z\"GXhF_r#!*Z7S[\"\")_ b@5#!+H[BAJ\"*/ZQ=$#!%gT\"%@7#\"&S]\"\"&d2#F+F+7/F)#\"%>b\"'!)36F:F:F: F:#\"#$)\"$g&#\"$K*\"%vO#\"'B@G\"(+s/\"#\"%CE\"&bU##\"%3I\"&N'>#\"#P\" %+@F+7/%\"bGFbsF:F:F:F:FesFhsF[tF^tFatFdtF+7/%#b*G#\"&4b\"\"'!)=MF:F:F :F:#\"%Fo\"&Sb\"#\"&Q@#\"&&e\")#\"&.!y\"'kuQ#!&WT'\"'X\"*H#\"'3Mi\"(&[ z@F:#!#:\"$=&Q)pprint276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'LL$)!\"(F(%!GF+F+F +F+F+F+F+F+F+F+7/$\"'[\"[\"!\"'$\"'4Y;F*$\"'(oJ\"F/F+F+F+F+F+F+F+F+F+F +7/$\"'AAAF/$\"'cbbF*$\"\"!F:$\"'nm;F/F+F+F+F+F+F+F+F+F+7/$F8F/F>F9$!' L$3#!\"&$\"'L$3#FAF+F+F+F+F+F+F+F+7/$\"'nmmF/$FFF*F9F9$\"'LLLF/$\"'nmE F/F+F+F+F+F+F+F+7/F;$\"'H2$)F*F9F9$\"'=[5F/$!'$3F%F*$\"'W[@F*F+F+F+F+F +F+7/$\"'WWWF/$\"'3EZF*F9F9$\"'Dp:F/$\"'7l6F/$!'0#H%F*F;F+F+F+F+F+7/$ \"'++vF/$!'PztF*F9F9$!']W9FA$!'!HK%F/$\"'fbFF/$\"'q3:FA$\"'Bj\"*F/F+F+ F+F+7/$\"'nm\"*F/$!'s!\\'F*F9F9$!'r/=FA$!'B^xF/$!'%\\c\"F/$\"'x\"p\"FA $\"'ud9FA$\"'*Qo&F/F+F+F+7/$\"\"\"F:$\"'Jj8FAF9F9$\"'*[y\"!\"%$\"'*Q)f FA$\"'#)>GFA$!'q_9Faq$!'[1)*FA$!'/2MFA$\"'vXsF/F+F+7/F[q$\"'Xx\\F*F9F9 F9F9$\"'9#[\"F/$\"'0ODF/$\"'2%p#F/$\"'%=3\"F/$\"''>`\"F/$\"'!>w\"F*F+7 /%\"bGF_rF9F9F9F9FarFcrFerFgrFirF[sF+7/%#b*G$\"'ROXF*F9F9F9F9$\"'=$R%F /$\"'\\8FF/$\"'<8?F/$!'CW@F/$\"'NgGF/F9$!'v&*GF*Q)pprint286\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "a[8,7] = 9/53;" "6#/&%\"aG6$\"\")\"\"(*&\"\"*\"\"\"\"# `!\"\"" }{TEXT -1 36 " gives a slightly different scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "e10 \+ := `union`(subs(a[8,7]=9/53,e8),\{a[8,7]=9/53\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "subs(e10,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[ j],j=1..11),``],[`b*`,seq(`b*`[j],j=1..12)]]));\n``;\nevalf[6](%%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/#\"\"\"\"#7F(%!GF+F+F +F+F+F+F+F+F+F+7/#\"\"%\"#F#F.\"$V##\"#KF1F+F+F+F+F+F+F+F+F+F+7/#\"\"# \"\"*#F)\"#=\"\"!#F)\"\"'F+F+F+F+F+F+F+F+F+7/#\"\"&F7F>F:#!#DF*#\"#DF* F+F+F+F+F+F+F+F+7/#F6\"\"$#F)\"#:F:F:#F)FF#F.FHF+F+F+F+F+F+F+7/F;#\"$> $\"%SQF:F:#\"$h\"\"%O:#!#T\"$g*#\"#6\"$7&F+F+F+F+F+F+7/#F.F7#\"&z@$\"' !)ooF:F:#\"&lA%\"'_ZF#\"&H,#\"'?<<#!&()=\"Fin#F7\"#`F+F+F+F+F+7/#FFF.# !+R()G&)G\",Kc3e*RF:F:#!-@a;)[9\"\",k7<;*z#!+`&HbM%\"+39_*)**#\",B@BF@ #Fho#\"+$=\\SM*\"+!)3XVi#\"*fP%z5\"*g4!y6F+F+F+F+7/#FVF*#!-d!=(etN\".7 T2!4McF:F:#!-tK!f,I%\"-#z8&[(R##!.z`j]`4\"\".G&=D_39#!-l,#=<$e\".3%\\+ 1cP#\"-4Z!*z.p\"-/z1>3T#\"+V1?9;\"+C!Ht5\"#\"$(=\"$H$F+F+F+7/F)#\"-^Sh $zT'\"-!GXLXw%F:F:#\"-P4WvB5\"+k?>vd#\",xTi*p@\"+S]\\4O#\",4qg1i\"Far# !+Xu.**Q\"*G@rq##!+H[BAJ\"*/ZQ=$#!%gT\"%@7#\"&S]\"\"&d2#F+F+7/F)#\"%>b \"'!)36F:F:F:F:#\"#$)\"$g&#\"$K*\"%vO#\"'B@G\"(+s/\"#\"%CE\"&bU##\"%3I \"&N'>#\"#P\"%+@F+7/%\"bGFdsF:F:F:F:FgsFjsF]tF`tFctFftF+7/%#b*G#\"&4b \"\"'!)=MF:F:F:F:#\"%Fo\"&Sb\"#\"&Q@#\"&&e\")#\"&.!y\"'kuQ#!&WT'\"'X\" *H#\"'3Mi\"(&[z@F:#!#:\"$=&Q)pprint296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'LL$)! \"(F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"'[\"[\"!\"'$\"'4Y;F*$\"'(oJ\"F/F+F+F+ F+F+F+F+F+F+F+7/$\"'AAAF/$\"'cbbF*$\"\"!F:$\"'nm;F/F+F+F+F+F+F+F+F+F+7 /$F8F/F>F9$!'L$3#!\"&$\"'L$3#FAF+F+F+F+F+F+F+F+7/$\"'nmmF/$FFF*F9F9$\" 'LLLF/$\"'nmEF/F+F+F+F+F+F+F+7/F;$\"'H2$)F*F9F9$\"'=[5F/$!'$3F%F*$\"'W [@F*F+F+F+F+F+F+7/$\"'WWWF/$\"'\"[o%F*F9F9$\"'IQ:F/$\"'?s6F/$!'XEVF*$ \"'6)p\"F/F+F+F+F+F+7/$\"'++vF/$!'z?sF*F9F9$!'gK9FA$!'4]VF/$\"'\")oFF/ $\"'i'\\\"FA$\"'Bj\"*F/F+F+F+F+7/$\"'nm\"*F/$!'zUjF*F9F9$!'h$z\"FA$!'f wxF/$!'i_:F/$\"']!o\"FA$\"'ud9FA$\"'*Qo&F/F+F+F+7/$\"\"\"F:$\"'-Z8FAF9 F9$\"'os`\"F/$\"'!>w\"F*F+7/%\"bGFarF9F9F9F9FcrFerFgrFirF[sF]sF+7/%#b*G$\"'RO XF*F9F9F9F9$\"'=$R%F/$\"'\\8FF/$\"'<8?F/$!'CW@F/$\"'NgGF/F9$!'v&*GF*Q) pprint306\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------- -------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded 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 sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1760 "# coefficients by Sharp and Verner 1994\nee := \{c[ 2]=1/12,\nc[3]=4/27,\nc[4]=2/9,\nc[5]=5/9,\nc[6]=2/3,\nc[7]=1/6,\nc[8] =4/9,\nc[9]=3/4,\nc[10]=11/12,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/12,\na[ 3,1]=4/243,\na[3,2]=32/243,\na[4,1]=1/18,\na[4,2]=0,\na[4,3]=1/6,\na[5 ,1]=5/9,\na[5,2]=0,\na[5,3]=-25/12,\na[5,4]=25/12,\na[6,1]=1/15,\na[6, 2]=0,\na[6,3]=0,\na[6,4]=1/3,\na[6,5]=4/15,\na[7,1]=319/3840,\na[7,2]= 0,\na[7,3]=0,\na[7,4]=161/1536,\na[7,5]=-41/960,\na[7,6]=11/512,\na[8, 1]=245/5184,\na[8,2]=0,\na[8,3]=0,\na[8,4]=1627/10368,\na[8,5]=151/129 6,\na[8,6]=-445/10368,\na[8,7]=1/6,\na[9,1]=-556349853/7539261440,\na[ 9,2]=0,\na[9,3]=0,\na[9,4]=-4356175383/3015704576,\na[9,5]=-814787343/ 1884815360,\na[9,6]=831004641/3015704576,\na[9,7]=355452237/235601920, \na[9,8]=107943759/117800960,\na[10,1]=-68998698967/1063035863040,\na[ 10,2]=0,\na[10,3]=0,\na[10,4]=-767387292485/425214345216,\na[10,5]=-20 5995991597/265758965760,\na[10,6]=-22181208863/141738115072,\na[10,7]= 26226796959/15502606336,\na[10,8]=1614200643/1107329024,\na[10,9]=187/ 329,\na[11,1]=24511479161/17979371520,\na[11,2]=0,\na[11,3]=0,\na[11,4 ]=3889847115/217931776,\na[11,5]=22028391/3681280,\na[11,6]=614528179/ 217931776,\na[11,7]=-148401247/10215552,\na[11,8]=-3122234829/31838470 4,\na[11,9]=-4160/1221,\na[11,10]=15040/20757,\na[12,1]=5519/110880,\n a[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=83/560,\na[12, 7]=932/3675,\na[12,8]=282123/1047200,\na[12,9]=2624/24255,\na[12,10]=3 008/19635,\na[12,11]=37/2100,\n\nb[1]=5519/110880,\nb[2]=0,\nb[3]=0,\n b[4]=0,\nb[5]=0,\nb[6]=83/560,\nb[7]=932/3675,\nb[8]=282123/1047200,\n b[9]=2624/24255,\nb[10]=3008/19635,\nb[11]=37/2100,\n\n`b*`[1]=15509/3 41880,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=6827/1 5540,\n`b*`[7]=22138/81585,\n`b*`[8]=78003/387464,\n`b*`[9]=-64144/299 145,\n`b*`[10]=623408/2179485,\n`b*`[11]=0,\n`b*`[12]=-15/518\}:" }}} {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[7,11];" "6#&%\"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose components are the principal error terms of \+ the 11 stage, order 7 scheme (the error terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6, 12];" "6#&%#T*G6$\" \"'\"#7" }{TEXT -1 146 " denote the vector whose components are the p rincipal error terms of the embedded 12 stage, order 6 scheme (the err or terms of order 7) and let " }{XPPEDIT 18 0 "`T*`[7,12];" "6#&%#T*G 6$\"\"(\"#7" }{TEXT -1 99 " denote the vector whose components are th e error terms of order 8 of the embedded order 6 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[7, 11]));" "6#-%$absG 6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T *`[6,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"#7" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "abs(abs(`T*`[7,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\" (\"#7" }{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[8] = abs(abs(T[7, 11]));" "6#/&%\"AG6#\"\")-%$ absG6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[8 ] = abs(abs(`T*`[7,12]))/abs(abs(`T*`[6, 12]));" "6#/&%\"BG6#\"\")*&-% $absG6#-F*6#&%#T*G6$\"\"(\"#7\"\"\"-F*6#-F*6#&F/6$\"\"'F2!\"\"" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7,12]-T[7,11]) )/abs(abs(`T*`[6,12]));" "6#/&%\"CG6#\"\")*&-%$absG6#-F*6#,&&%#T*G6$\" \"(\"#7\"\"\"&%\"TG6$F2\"#6!\"\"F4-F*6#-F*6#&F06$\"\"'F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "D ormand and Prince have suggested that as well as attempting to ensure \+ that " }{XPPEDIT 18 0 "A[8];" "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a \+ minimum, if the embedded scheme is to be used for error control, " } {XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\")" }{TEXT -1 27 " should be chos en so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magn itude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := P rincipalErrorTerms(7,11,'expanded'):\n`errterms7_12*` :=subs(b=`b*`,Pr incipalErrorTerms(7,12,'expanded')):\n`errterms6_12*` := subs(b=`b*`,P rincipalErrorTerms(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf( subs(ee,`errterms7_12*`[i]))^2,i=1..nops(`errterms7_12*`))):\nsdnB := \+ sqrt(add(evalf(subs(ee,`errterms6_12*`[i]))^2,i=1..nops(`errterms6_12* `))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms7_12*`[i])-subs(ee,err terms7_11[i])))^2,i=1..nops(errterms7_11))):\n'B[8]'= evalf[8](snmB/sd nB);\n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"BG6#\"\")$\")43H9!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6 #\"\")$\")M$=V\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----- ----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{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 1760 "# coefficients by Sharp and Verner 1994\nee := \{c[2]=1/12,\nc[3 ]=4/27,\nc[4]=2/9,\nc[5]=5/9,\nc[6]=2/3,\nc[7]=1/6,\nc[8]=4/9,\nc[9]=3 /4,\nc[10]=11/12,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/12,\na[3,1]=4/243,\n a[3,2]=32/243,\na[4,1]=1/18,\na[4,2]=0,\na[4,3]=1/6,\na[5,1]=5/9,\na[5 ,2]=0,\na[5,3]=-25/12,\na[5,4]=25/12,\na[6,1]=1/15,\na[6,2]=0,\na[6,3] =0,\na[6,4]=1/3,\na[6,5]=4/15,\na[7,1]=319/3840,\na[7,2]=0,\na[7,3]=0, \na[7,4]=161/1536,\na[7,5]=-41/960,\na[7,6]=11/512,\na[8,1]=245/5184, \na[8,2]=0,\na[8,3]=0,\na[8,4]=1627/10368,\na[8,5]=151/1296,\na[8,6]=- 445/10368,\na[8,7]=1/6,\na[9,1]=-556349853/7539261440,\na[9,2]=0,\na[9 ,3]=0,\na[9,4]=-4356175383/3015704576,\na[9,5]=-814787343/1884815360, \na[9,6]=831004641/3015704576,\na[9,7]=355452237/235601920,\na[9,8]=10 7943759/117800960,\na[10,1]=-68998698967/1063035863040,\na[10,2]=0,\na [10,3]=0,\na[10,4]=-767387292485/425214345216,\na[10,5]=-205995991597/ 265758965760,\na[10,6]=-22181208863/141738115072,\na[10,7]=26226796959 /15502606336,\na[10,8]=1614200643/1107329024,\na[10,9]=187/329,\na[11, 1]=24511479161/17979371520,\na[11,2]=0,\na[11,3]=0,\na[11,4]=388984711 5/217931776,\na[11,5]=22028391/3681280,\na[11,6]=614528179/217931776, \na[11,7]=-148401247/10215552,\na[11,8]=-3122234829/318384704,\na[11,9 ]=-4160/1221,\na[11,10]=15040/20757,\na[12,1]=5519/110880,\na[12,2]=0, \na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=83/560,\na[12,7]=932/367 5,\na[12,8]=282123/1047200,\na[12,9]=2624/24255,\na[12,10]=3008/19635, \na[12,11]=37/2100,\n\nb[1]=5519/110880,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb [5]=0,\nb[6]=83/560,\nb[7]=932/3675,\nb[8]=282123/1047200,\nb[9]=2624/ 24255,\nb[10]=3008/19635,\nb[11]=37/2100,\n\n`b*`[1]=15509/341880,\n`b *`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=6827/15540,\n`b* `[7]=22138/81585,\n`b*`[8]=78003/387464,\n`b*`[9]=-64144/299145,\n`b*` [10]=623408/2179485,\n`b*`[11]=0,\n`b*`[12]=-15/518\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT 259 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stab ility function R for the 11 stage, order 7 scheme is given as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunctio n(7,11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,:\"\"\"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)*&#F)\"%S]F)*$)F'\"\"(F)F)F) *&#\"+nxtiN\"0+7t\">tU9F)*$)F'\"\")F)F)F)*&#\"+FqZPw\"1+;;^u\"pf#F)*$) F'\"\"*F)F)F)*&#\"*^p8=$FVF)*$)F'\"#5F)F)F)*&#\"&hv(\".gt#421KF)*$)F' \"#6F)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,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+&y%3AY!\"*" }}}{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=[-5.19..0.49,-1.47..1.47],font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVE SG6$7V7$$!3Q++++++!>&!#<$!3Gn^!p\"*30B$F*7$$!3Ynm;#3'4G^F*$!3IE(Ghhh8' GF*7$$!3kLLLk@>m]F*$!3\"*yEbaztIDF*7$$!3E+]U'*)HB,&F*$!3[DF=B\"[:F#F*7 $$!3!pm;&GwYe\\F*$!3Tw9)pcFl.#F*7$$!3s+](\\(Q*y*[F*$!39v(pbv\\&)z\"F*7 $$!3nLLV@,KP[F*$!3u2]%f@lee\"F*7$$!3NLL.q&p`r%F*$!3x_'=W6hYA\"F*7$$!3; nmJG')*Rf%F*$!31SD+*[`&*R*!#=7$$!3uLLyGAZ\"[%F*$!3=53vDc*4I(FU7$$!3%3+ ])fw&\\O%F*$!3e!\\\")))HmDd&FU7$$!3$QL$)f7eWC%F*$!3+P(oz$euoTFU7$$!3A+ +lN]MCTF*$!3sfdDM%p#zIFU7$$!3ummYeRz+SF*$!3(\\yw>EUK@#FU7$$!3_LLV-,(>* QF*$!3>j40H3:>;FU7$$!35++S:-YpPF*$!3Cs;-)*Q0)4\"FU7$$!3K+++\"HZkk$F*$! 3#\\4pgF&Qnp!#>7$$!3;++gW:!z_$F*$!3-7Lyv$*zxRFhp7$$!3hLL)*\\1D?MF*$!3n P!oew)zF=Fhp7$$!3'ommSKVAH$F*$\"3$3a!o:L01@!#?7$$!3/nmEGV!Q=$F*$\"34dA \"f..nh\"Fhp7$$!39++0(*RmdIF*$\"3uYG.*Rsy*HFhp7$$!39nmEI%3g%HF*$\"3udd 6))\\iqSFhp7$$!3-++0xX]BGF*$\"3QhP%*f([h:&Fhp7$$!3*)***\\\"R>&oq#F*$\" 3;))oS>.CahFhp7$$!3gmm;\\r8&e#F*$\"3A7Uq%Q5^?(Fhp7$$!3ymmrw\\OtCF*$\"3 D2+TyGQ7#)Fhp7$$!3SLL$))e.GN#F*$\"3q/QT$>FcP*Fhp7$$!3nLL)**=uvA#F*$\"3 o?umR:$*p5FU7$$!3K++:I;c=@F*$\"3u!=.9)\\6(>\"FU7$$!31LL.z]#3+#F*$\"3-( 3O$p(y$\\8FU7$$!3M++?,<>z=F*$\"3F(Q6!*zfb_\"FU7$$!3;++!4<(>gFU7$$!3H++q9zA<:F*$\"3U\"[4QD!)H >#FU7$$!3EnmEY;O-9F*$\"3[n!pe5_+Y#FU7$$!3#)*****pQ<(z7F*$\"3w]C4ae6\"y #FU7$$!3)RL$efMeo6F*$\"3H)o!\\*R]!3JFU7$$!3I****fAZ3Z5F*$\"3C'y;gB(e4N FU7$$!3xqm;(zQwK*FU$\"3jy%**yzWY$RFU7$$!3&z***\\)ecE8)FU$\"3Tw\")oAh2M WFU7$$!3'3nmm0VV'pFU$\"39$Rnf=\"f$)\\FU7$$!3P)***\\iqATdFU$\"3<5-f(R() >j&FU7$$!3aFLL*)4AjXFU$\"3bL7#z=(4OjFU7$$!33LLLO'R&eLFU$\"3+dOoVZFZrFU 7$$!3Uim;`O$Q;#FU$\"3k>10,ZEa!)FU7$$!3?*****>$H-m5FU$\"3%[k+$Q4$)))*)F U7$$\"3v*QLLU?>#>Fhp$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\"FU$\"3d2+-(HH39\" F*7$$\"3E0+]HcUiJK;F*-%'COLOURG6&%$RGBG$\"*++++ \"!\")$\"\"!Fa\\lF`\\l-F$6$7S7$F($!\"\"Fa\\l7$F3Ff\\l7$F=Ff\\l7$FGFf\\ l7$FLFf\\l7$FQFf\\l7$FWFf\\l7$FfnFf\\l7$F[oFf\\l7$F`oFf\\l7$FeoFf\\l7$ FjoFf\\l7$F_pFf\\l7$FdpFf\\l7$FjpFf\\l7$F_qFf\\l7$FdqFf\\l7$FjqFf\\l7$ F_rFf\\l7$FdrFf\\l7$FirFf\\l7$F^sFf\\l7$FcsFf\\l7$FhsFf\\l7$F]tFf\\l7$ FbtFf\\l7$FgtFf\\l7$F\\uFf\\l7$FauFf\\l7$FfuFf\\l7$F[vFf\\l7$F`vFf\\l7 $FevFf\\l7$FjvFf\\l7$F_wFf\\l7$FdwFf\\l7$FiwFf\\l7$F^xFf\\l7$FcxFf\\l7 $FhxFf\\l7$F]yFf\\l7$FbyFf\\l7$FgyFf\\l7$F\\zFf\\l7$FazFf\\l7$FfzFf\\l 7$F[[lFf\\l7$F`[lFf\\l7$Fe[lFf\\l-Fj[l6&F\\\\lF`\\lF`\\lF]\\l-F$6&7#7$ $!3?+++&y%3AYF*Ff\\l-%'SYMBOLG6#%'CIRCLEG-Fj[l6&F\\\\lFa\\lFa\\lFa\\l- %&STYLEG6#%&POINTG-F$6&F\\`l-Fa`l6#%&CROSSGFd`lFf`l-F$6&F\\`l-Fa`l6#%( DIAMONDGFd`lFf`l-F$6%7$7$F^`lF`\\lF]`l-%&COLORG6&F\\\\lF`\\l$\"\"&Fg\\ lF`\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Febl-%%FONTG6#%(DEF AULTG-Fhbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$>&!\"#$\"#\\Fecl;$!$Z\"Fecl $\"$Z\"Fecl" 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 50 "The following p icture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1442 "R := z -> 1+z+1/2*z^2+1/6 *z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 3562737767/14 4273191731200*z^8+7637477027/2596917451161600*z^9+\n 31813695 1/2596917451161600*z^10+77561/3206070927360*z^11:\npts := []:\nz0 := 0 :\nfor ct from 0 to 280 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,.08,.18,.48)):\np2 := plots[polygonplot]([seq ([pts[i-1],pts[i],[-2.3,0]],i=2..nops(pts))],\n style=patchno grid,color=COLOR(RGB,.15,.35,.95)):\npts := []: z0 := 2.8+6.2*I:\nfor \+ ct from 0 to 40 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:\np3 := plot(pts ,color=COLOR(RGB,.08,.18,.48)):\np4 := plots[polygonplot]([seq([pts[i- 1],pts[i],[2.75,6.26]],i=2..nops(pts))],\n style=patchnogrid, color=COLOR(RGB,.15,.35,.95)):\npts := []: z0 := 2.8-6.2*I:\nfor ct fr om 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := z z:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,colo r=COLOR(RGB,.08,.18,.48)):\np6 := plots[polygonplot]([seq([pts[i-1],pt s[i],[2.75,-6.26]],i=2..nops(pts))],\n style=patchnogrid,colo r=COLOR(RGB,.15,.35,.95)):\np7 := plot([[[-5.39,0],[3.19,0]],[[0,-6.59 ],[0,6.59]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],vie w=[-5.39..3.19,-6.59..6.59],font=[HELVETICA,9],\n labels= [`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 442 554 554 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F(7$ F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$\"3t*****pPb#Qi!#G$\"3,+++ y*)Q7ZF-7$$!3%******>T35R(!#F$\"3q*****zF&=$G'F-7$$!3%******\\*oJa8!#D $\"3++++r9)R&yF-7$$!3++++Y='f\")*F@$\"3s*****fXxZU*F-7$$!3:+++6$31i%!# C$\"35+++ttb*4\"!#<7$$!3'******4KGCj\"!#B$\"37+++xtjc7FN7$$!3y*****Hur mi%FR$\"3)******z.>PT\"FN7$$!3#*******p>Q&3\"!#A$\"3!******p)e!3d\"FN7 $$!34+++*H[h6#Fgn$\"3)*******yf!zs\"FN7$$!3!********pS?I$Fgn$\"33+++'= N])=FN7$$!3u*****4?\"4QMFgn$\"3?+++m5AU?FN7$$\"3%******>A6pj(FR$\"3y** ***\\&f]*>#FN7$$\"33+++w@p^;!#@$\"3))*****f]YpN#FN7$$\"3m*****>F*[mdFa p$\"3)*******zZg9DFN7$$\"3#******\\qSA[\"!#?$\"3')*****f)G_sEFN7$$\"31 +++y%HFE$F\\q$\"3(******\\)*o1$GFN7$$\"3q******fbflkF\\q$\"3))*****>0R )))HFN7$$\"3'******\\-cz<\"!#>$\"3-+++vT]YJFN7$$\"35+++Qa+*)>F\\r$\"3* ******pIGEI$FN7$$\"3!******fG&f:JF\\r$\"3')*****pxobX$FN7$$\"3H+++\"3d #=XF\\r$\"3)******\\7eKg$FN7$$\"3M+++\\$R-1'F\\r$\"3*******f0YPu$FN7$$ \"3_******)eue`(F\\r$\"3)******4u%zvQFN7$$\"3p+++I1)Ht)F\\r$\"37+++$Qs !**RFN7$$\"3U+++#3AH[*F\\r$\"3m*****pk&*Q6%FN7$$\"3%******\\a&*on*F\\r $\"3>+++!3R3A%FN7$$\"3++++6MIf#*F\\r$\"3k*****p#)f/K%FN7$$\"3c*****p]F X@)F\\r$\"31+++C)yJT%FN7$$\"3Q+++RFDblF\\r$\"3A+++obE*\\%FN7$$\"3)**** **RgRRJ%F\\r$\"3G+++'or)yXFN7$$\"33+++vLjO:F\\r$\"3w*****>Iz?l%FN7$$!3 1+++L+\">s\"F\\r$\"3$)******Gk%*=ZFN7$$!3!)******QpN-aF\\r$\"3#******z 8T&zZFN7$$!3c+++i\\LW%*F\\r$\"3Q+++B>'R$[FN7$$!3!******zqI*y8F-$\"3W++ +'HXB)[FN7$$!3.+++\\&o#Q=F-$\"3U+++oO'[#\\FN7$$!3,+++ur^LF-$\"3P+++pfI >]FN7$$!3))*****\\wvV$QF-$\"3$******p;&[S]FN7$$!3!******>nCeN%F-$\"3C+ ++s0(o0&FN7$$!3++++aVA\")[F-$\"3!)*****>,b'o]FN7$$!3[+++n#p)3aF-$\"3-+ ++*e5g2&FN7$$!3e******3EVPfF-$\"3G+++:f3z]FN7$$!3a*****z\"z#fY'F-$\"3I +++%H/!y]FN7$$!3))*****\\#Ho$*pF-$\"33+++([iG2&FN7$$!3'******zxF._(F-$ \"3q******R)HP1&FN7$$!3q*****z1+e/)F-$\"3++++;wk]]FN7$$!3q*****R+q.d)F -$\"3\"******H`GO.&FN7$$!3q*****4r!p%4*F-$\"3#******zncE,&FN7$$!3.+++6 3))>'*F-$\"3S+++7$)o()\\FN7$$!30+++8\\w95FN$\"3?+++XYle\\FN7$$!37+++ct /o5FN$\"3U+++b'oa#\\FN7$$!3)********[%=A6FN$\"3#)*****p')R!))[FN7$$!3! ******zTdw<\"FN$\"3K+++$)RIY[FN7$$!3,+++,>6N7FN$\"3f*****Rky-![FN7$$!3 3+++4uP&H\"FN$\"3++++1/;]ZFN7$$!3%********fM%f8FN$\"3M+++`JZ'p%FN7$$!3 /+++^TEG9FN$\"3I+++-MCSYFN7$$!3(*******\\t^-:FN$\"3')*****f*34$e%FN7$$ !33+++XI3#e\"FN$\"3q******Qn.FXFN7$$!3)******\\xhem\"FN$\"3G+++\">^RZ% FN7$$!3!******RVT?v\"FN$\"3z*****p&*[\\U%FN7$$!3$******f?6(Q=FN$\"3/++ +k#)=!Q%FN7$$!3&******4t\\V#>FN$\"3:+++9Y7RVFN7$$!3!)*****>htz+#FN$\"3 >+++#p-4I%FN7$$!3'*******)Rc!*3#FN$\"3o*****R_RYE%FN7$$!35+++MOQn@FN$ \"3n*****RRe&HUFN7$$!32+++B'GHC#FN$\"3g*****4qD]>%FN7$$!3*)******pxw:B FN$\"3q*****p4V0;%FN7$$!3?+++4*GgQ#FN$\"3T+++h\\sDTFN7$$!3%)******)yhQ X#FN$\"3=+++=XF!4%FN7$$!3'******p6A%>DFN$\"31+++kK'R0%FN7$$!3%*******G N'Ge#FN$\"3;+++#o9m,%FN7$$!3$)*****p3MVk#FN$\"3=+++r=4yRFN7$$!3!****** \\FvRq#FN$\"33+++)R)GQRFN7$$!3-+++oB#>w#FN$\"37+++'p@r*QFN7$$!3#)***** RY0$=GFN$\"3%)*****fVGX&QFN7$$!3)******R!3DtGFN$\"3z*****RZh/\"QFN7$$! 38+++$4#)o#HFN$\"35+++g!))[w$FN7$$!3A+++S;KzHFN$\"3>+++!3*y6\"3$FN$\"3++++uu+>OFN7$$!3-+++w WqIJFN$\"3++++T!ftc$FN7$$!3#)*****R6(ezJFN$\"32+++ \"oIK$FN$\"3++++Fl$pM$FN7$$!3))******4]9qLFN$\"3/+++c&H))G$FN7$$!3'*** ****[I&pT$FN$\"34+++f#o(HKFN7$$!33+++*)z^jMFN$\"3y******z$)*)pJFN7$$!3 &)*****>oN)4NFN$\"3!)*****z$*o$4JFN7$$!3!)******Q;(eb$FN$\"3')******>V K[IFN7$$!32+++jKc,OFN$\"3!)*****>R'*o)HFN7$$!3)******ReCok$FN$\"3;+++M x>DHFN7$$!3#******H\"Gb\"p$FN$\"35+++zoJjGFN7$$!34+++gdjNPFN$\"38+++Cg J,GFN7$$!3?+++y!f*yPFN$\"32+++\"=J#RFFN7$$!30+++`DT@QFN$\"3%******RJuq n#FN7$$!3-+++Z\\*G'QFN$\"3y******Ho$[h#FN7$$!3?+++xsJ.RFN$\"3++++BP\\_ DFN7$$!3$)*****Rj/E%RFN$\"3')*****\\!y+!\\#FN7$$!3A+++Xop!)RFN$\"3**** ****QOLFCFN7$$!3e*****zF[v,%FN$\"3;+++Z4UkBFN7$$!3K+++7r7`SFN$\"3))*** ***=u@,BFN7$$!3u******)H9u3%FN$\"3$*******p4nPAFN7$$!3l*****4U-/7%FN$ \"3=+++&\\JP<#FN7$$!3q*****H_%4_TFN$\"3%)*****pD_$4@FN7$$!37+++1I]#=%F N$\"3y*****R*4\\W?FN7$$!3t*****Hd[;@%FN$\"3'******H%36z>FN7$$!3\"***** *fAf&RUFN$\"3-+++p6=8>FN7$$!3G+++o#piE%FN$\"3'******4Kym%=FN7$$!3g**** *>E==H%FN$\"3!******HP'ezO%FN$\"35+++Cmuu:FN7$$!3G++ +R1J$Q%FN$\"30+++XgJ0:FN7$$!36+++GFu.WFN$\"3)******\\F``V\"FN7$$!3G+++ JeGBWFN$\"33+++w&)*[O\"FN7$$!31+++)*\\(>W%FN$\"3)******4z+SH\"FN7$$!3s *****fKP)fWFN$\"3%*******ffrA7FN7$$!3)*******\\/*oZ%FN$\"3-+++ca5^6FN7 $$!3O+++.:9$\\%FN$\"35+++>PBz5FN7$$!3P+++*o'e3XFN$\"3++++Xd;25FN7$$!3g *****>^6K_%FN$\"3c+++y_k\\$*F-7$$!3++++r;*p`%FN$\"3[+++ml)oi)F-7$$!3#* *****H^%*)\\XFN$\"3B+++\\F!R!zF-7$$!3i*****R\"4)=c%FN$\"3y*****pDN6=(F -7$$!3=+++!Q2Hd%FN$\"3g*****\\mN*ekF-7$$!3U+++@#GHe%FN$\"3$******p`hvt &F-7$$!3g*****Hg(*=f%FN$\"3()******3u<<]F-7$$!3M+++19x*f%FN$\"3'****** f)*fyH%F-7$$!3p*****>y3lg%FN$\"3#******HV1'zNF-7$$!3<+++DM27YFN$\"3C++ +%pWB'GF-7$$!3()*****f`Mkh%FN$\"3)*******4\\%f9#F-7$$!34+++Qvc>YFN$\"3 4+++h7BI9F-7$$!3r*****fca9i%FN$\"3e+++))y$*\\rF\\r7$$!3?+++&y%3AYFNF(7 $F^gm$!3e+++))y$*\\rF\\r7$Fifm$!34+++h7BI9F-7$Fdfm$!3)*******4\\%f9#F- 7$F_fm$!3C+++%pWB'GF-7$Fjem$!3#******HV1'zNF-7$Feem$!3'******f)*fyH%F- 7$F`em$!3()******3u<<]F-7$F[em$!3$******p`hvt&F-7$Ffdm$!3g*****\\mN*ek F-7$Fadm$!3y*****pDN6=(F-7$F\\dm$!3B+++\\F!R!zF-7$Fgcm$!3[+++ml)oi)F-7 $Fbcm$!3c+++y_k\\$*F-7$F]cm$!3++++Xd;25FN7$Fhbm$!35+++>PBz5FN7$Fcbm$!3 -+++ca5^6FN7$F^bm$!3%*******ffrA7FN7$Fiam$!3)******4z+SH\"FN7$Fdam$!33 +++w&)*[O\"FN7$F_am$!3)******\\F``V\"FN7$Fj`m$!30+++XgJ0:FN7$Fe`m$!35+ ++Cmuu:FN7$F``m$!36+++,UhV;FN7$F[`m$!3'******z\"y*=r\"FN7$Ff_m$!3!**** **HP'ezFN7$Fg^m$!3'***** *H%36z>FN7$Fb^m$!3y*****R*4\\W?FN7$F]^m$!3%)*****pD_$4@FN7$Fh]m$!3=+++ &\\JP<#FN7$Fc]m$!3$*******p4nPAFN7$F^]m$!3))******=u@,BFN7$Fi\\m$!3;++ +Z4UkBFN7$Fd\\m$!3********QOLFCFN7$F_\\m$!3')*****\\!y+!\\#FN7$Fj[m$!3 ++++BP\\_DFN7$Fe[m$!3y******Ho$[h#FN7$F`[m$!3%******RJuqn#FN7$F[[m$!32 +++\"=J#RFFN7$Ffjl$!38+++CgJ,GFN7$Fajl$!35+++zoJjGFN7$F\\jl$!3;+++Mx>D HFN7$Fgil$!3!)*****>R'*o)HFN7$Fbil$!3')******>VK[IFN7$F]il$!3!)*****z$ *o$4JFN7$Fhhl$!3y******z$)*)pJFN7$Fchl$!34+++f#o(HKFN7$F^hl$!3/+++c&H) )G$FN7$Figl$!3++++Fl$pM$FN7$Fdgl$!3/+++BO&RS$FN7$F_gl$!3%)******42wfMF N7$Fjfl$!32+++OFN7$F[fl$! 31+++h\"f\"pOFN7$Ffel$!3>+++!3*y%FN7$Feal$!3n*****RRe&HUFN7$F`al$!3o*****R_RYE%FN7$F[a l$!3>+++#p-4I%FN7$Ff`l$!3:+++9Y7RVFN7$Fa`l$!3/+++k#)=!Q%FN7$F\\`l$!3z* ****p&*[\\U%FN7$Fg_l$!3G+++\">^RZ%FN7$Fb_l$!3q******Qn.FXFN7$F]_l$!3') *****f*34$e%FN7$Fh^l$!3I+++-MCSYFN7$Fc^l$!3M+++`JZ'p%FN7$F^^l$!3++++1/ ;]ZFN7$Fi]l$!3f*****Rky-![FN7$Fd]l$!3K+++$)RIY[FN7$F_]l$!3#)*****p')R! ))[FN7$Fj\\l$!3U+++b'oa#\\FN7$Fe\\l$!3?+++XYle\\FN7$F`\\l$!3S+++7$)o() \\FN7$F[\\l$!3#******zncE,&FN7$Ff[l$!3\"******H`GO.&FN7$Fa[l$!3++++;wk ]]FN7$F\\[l$!3q******R)HP1&FN7$Fgz$!33+++([iG2&FN7$Fbz$!3I+++%H/!y]FN7 $F]z$!3G+++:f3z]FN7$Fhy$!3-+++*e5g2&FN7$Fcy$!3!)*****>,b'o]FN7$F^y$!3C +++s0(o0&FN7$Fix$!3$******p;&[S]FN7$Fdx$!3P+++pfI>]FN7$F_x$!3I+++zM7$* \\FN7$Fjw$!3Q+++rsrh\\FN7$Few$!3U+++oO'[#\\FN7$F`w$!3W+++'HXB)[FN7$F[w $!3Q+++B>'R$[FN7$Ffv$!3#******z8T&zZFN7$Fav$!3$)******Gk%*=ZFN7$F\\v$! 3w*****>Iz?l%FN7$Fgu$!3G+++'or)yXFN7$Fbu$!3A+++obE*\\%FN7$F]u$!31+++C) yJT%FN7$Fht$!3k*****p#)f/K%FN7$Fct$!3>+++!3R3A%FN7$F^t$!3m*****pk&*Q6% FN7$Fis$!37+++$Qs!**RFN7$Fds$!3)******4u%zvQFN7$F_s$!3*******f0YPu$FN7 $Fjr$!3)******\\7eKg$FN7$Fer$!3')*****pxobX$FN7$F`r$!3*******pIGEI$FN7 $Fjq$!3-+++vT]YJFN7$Feq$!3))*****>0R)))HFN7$F`q$!3(******\\)*o1$GFN7$F jp$!3')*****f)G_sEFN7$Fep$!3)*******zZg9DFN7$F_p$!3))*****f]YpN#FN7$$ \"3i*****4A6pj(FR$!3y*****\\&f]*>#FN7$Feo$!3?+++m5AU?FN7$F`o$!33+++'=N ])=FN7$F[o$!3)*******yf!zs\"FN7$$!33+++r>Q&3\"Fgn$!3!******p)e!3d\"FN7 $FV$!3)******z.>PT\"FN7$FP$!37+++xtjc7FN7$FI$!35+++ttb*4\"FN7$$!3Z+++W 6'f\")*F@$!3s*****fXxZU*F-7$$!3'******R*oJa8F@$!3++++r9)R&yF-7$$!35+++ m$35R(F:$!3q*****zF&=$G'F-7$$\"3V+++tjDQiF4$!3,+++y*)Q7ZF-7$F($!3)**** **Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"\")!\"#$\"#=Fhbo$\"# [Fhbo-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$FR!\"\"F(7%Faco7$F($\"+aE fTJFdcoFeco7%Fico7$$\"+x`DQiF\\q$\"+y*)Q7ZFdcoFeco7%F]do7$$!+7%35R(F\\ r$\"+y_=$G'FdcoFeco7%Fcdo7$$!+&*oJa8FN$\"+r9)R&yFdcoFeco7%Fido7$$!+Y=' f\")*FN$\"+cuxC%*FdcoFeco7%F_eo7$$!+6$31i%!#;$\"+ttb*4\"!\"*Feco7%Feeo 7$$!+@$GCj\"!#:$\"+xtjc7F[foFeco7%F]fo7$$!+VPT\"F[foFec o7%Fdfo7$$!+q>Q&3\"!#9$\"+()e!3d\"F[foFeco7%Fjfo7$$!+*H[h6#F]go$\"+zf! zs\"F[foFeco7%Fago7$$!++2/-LF]go$\"+'=N])=F[foFeco7%Fggo7$$!+,74QMF]go $\"+m5AU?F[foFeco7%F]ho7$$\"+A7\"pj(F`fo$\"+bf]*>#F[foFeco7%Fcho7$$\"+ w@p^;!#8$\"+1l%pN#F[foFeco7%Fiho7$$\"+s#*[mdF\\io$\"+!y/Y^#F[foFeco7%F `io7$$\"+02C#[\"!#7$\"+')G_sEF[foFeco7%Ffio7$$\"+y%HFE$Fiio$\"+&)*o1$G F[foFeco7%F]jo7$$\"+gbflkFiio$\"+_!R)))HF[foFeco7%Fcjo7$$\"+Dg&z<\"!#6 $\"+vT]YJF[foFeco7%Fijo7$$\"+Qa+*)>F\\[p$\"+2$GEI$F[foFeco7%F`[p7$$\"+ 'G&f:JF\\[p$\"+x(obX$F[foFeco7%Ff[p7$$\"+\"3d#=XF\\[p$\"+D\"eKg$F[foFe co7%F\\\\p7$$\"+\\$R-1'F\\[p$\"+cguVPF[foFeco7%Fb\\p7$$\"+*eue`(F\\[p$ \"+TZzvQF[foFeco7%Fh\\p7$$\"+I1)Ht)F\\[p$\"+$Qs!**RF[foFeco7%F^]p7$$\" +#3AH[*F\\[p$\"+Zc*Q6%F[foFeco7%Fd]p7$$\"+Xb*on*F\\[p$\"+!3R3A%F[foFec o7%Fj]p7$$\"+6MIf#*F\\[p$\"+F)f/K%F[foFeco7%F`^p7$$\"+2v_9#)F\\[p$\"+C )yJT%F[foFeco7%Ff^p7$$\"+RFDblF\\[p$\"+obE*\\%F[foFeco7%F\\_p7$$\"+/'R RJ%F\\[p$\"+'or)yXF[foFeco7%Fb_p7$$\"+vLjO:F\\[p$\"+-$z?l%F[foFeco7%Fh _p7$$!+L+\">s\"F\\[p$\"+Hk%*=ZF[foFeco7%F^`p7$$!+RpN-aF\\[p$\"+Q6azZF[ foFeco7%Fd`p7$$!+i\\LW%*F\\[p$\"+B>'R$[F[foFeco7%Fj`p7$$!+32$*y8Fdco$ \"+'HXB)[F[foFeco7%F`ap7$$!+\\&o#Q=Fdco$\"+oO'[#\\F[foFeco7%Ffap7$$!+u r^LFdco$\"+pfI>]F[foFeco7%Fhbp7$$!+ldPMQFdco$\"+n^[S]F[foFeco 7%F^cp7$$!+sY#eN%Fdco$\"+s0(o0&F[foFeco7%Fdcp7$$!+aVA\")[Fdco$\"+7]lo] F[foFeco7%Fjcp7$$!+n#p)3aFdco$\"+*e5g2&F[foFeco7%F`dp7$$!+4EVPfFdco$\" +:f3z]F[foFeco7%Ffdp7$$!+=z#fY'Fdco$\"+%H/!y]F[foFeco7%F\\ep7$$!+DHo$* pFdco$\"+([iG2&F[foFeco7%Fbep7$$!+yxK?vFdco$\"+S)HP1&F[foFeco7%Fhep7$$ !+o+!e/)Fdco$\"+;wk]]F[foFeco7%F^fp7$$!+/+Pq&)Fdco$\"+L&GO.&F[foFeco7% Fdfp7$$!+62p%4*Fdco$\"+yml7]F[foFeco7%Fjfp7$$!+63))>'*Fdco$\"+7$)o()\\ F[foFeco7%F`gp7$$!+8\\w95F[fo$\"+XYle\\F[foFeco7%Ffgp7$$!+ct/o5F[fo$\" +b'oa#\\F[foFeco7%F\\hp7$$!+!\\%=A6F[fo$\"+n)R!))[F[foFeco7%Fbhp7$$!+= ulx6F[fo$\"+$)RIY[F[foFeco7%Fhhp7$$!+,>6N7F[fo$\"+W'y-![F[foFeco7%F^ip 7$$!+4uP&H\"F[fo$\"+1/;]ZF[foFeco7%Fdip7$$!++YVf8F[fo$\"+`JZ'p%F[foFec o7%Fjip7$$!+^TEG9F[fo$\"+-MCSYF[foFeco7%F`jp7$$!+]t^-:F[fo$\"+'*34$e%F [foFeco7%Ffjp7$$!+XI3#e\"F[fo$\"+Rn.FXF[foFeco7%F\\[q7$$!+v<'em\"F[fo$ \"+\">^RZ%F[foFeco7%Fb[q7$$!+M9/_F[fo$\"+9Y7RVF[foFeco7%Fd \\q7$$!+7O(z+#F[fo$\"+#p-4I%F[foFeco7%Fj\\q7$$!+*Rc!*3#F[fo$\"+C&RYE%F [foFeco7%F`]q7$$!+MOQn@F[fo$\"+%Re&HUF[foFeco7%Ff]q7$$!+B'GHC#F[fo$\"+ ,d-&>%F[foFeco7%F\\^q7$$!+qxw:BF[fo$\"+(4V0;%F[foFeco7%Fb^q7$$!+4*GgQ# F[fo$\"+h\\sDTF[foFeco7%Fh^q7$$!+*yhQX#F[fo$\"+=XF!4%F[foFeco7%F^_q7$$ !+<@U>DF[fo$\"+kK'R0%F[foFeco7%Fd_q7$$!+HN'Ge#F[fo$\"+#o9m,%F[foFeco7% Fj_q7$$!+(3MVk#F[fo$\"+r=4yRF[foFeco7%F``q7$$!+v_(Rq#F[fo$\"+)R)GQRF[f oFeco7%Ff`q7$$!+oB#>w#F[fo$\"+'p@r*QF[foFeco7%F\\aq7$$!+kaI=GF[fo$\"+O %GX&QF[foFeco7%Fbaq7$$!+/3DtGF[fo$\"+u9Y5QF[foFeco7%Fhaq7$$!+$4#)o#HF[ fo$\"+g!))[w$F[foFeco7%F^bq7$$!+S;KzHF[fo$\"+!3*y6\"3$F[fo$\"+uu+>OF[foFeco7% F`cq7$$!+wWqIJF[fo$\"+T!ftc$F[foFeco7%Ffcq7$$!+9rezJF[fo$\"+\"oIK$F[fo$\"+Fl$pM$F[foFeco7%F^eq7$$!+5]9qLF[f o$\"+c&H))G$F[foFeco7%Fdeq7$$!+\\I&pT$F[fo$\"+f#o(HKF[foFeco7%Fjeq7$$! +*)z^jMF[fo$\"+!Q)*)pJF[foFeco7%F`fq7$$!+#oN)4NF[fo$\"+Q*o$4JF[foFeco7 %Fffq7$$!+R;(eb$F[fo$\"+?VK[IF[foFeco7%F\\gq7$$!+jKc,OF[fo$\"+#R'*o)HF [foFeco7%Fbgq7$$!+%eCok$F[fo$\"+Mx>DHF[foFeco7%Fhgq7$$!+8Gb\"p$F[fo$\" +zoJjGF[foFeco7%F^hq7$$!+gdjNPF[fo$\"+CgJ,GF[foFeco7%Fdhq7$$!+y!f*yPF[ fo$\"+\"=J#RFF[foFeco7%Fjhq7$$!+`DT@QF[fo$\"+9V2xEF[foFeco7%F`iq7$$!+Z \\*G'QF[fo$\"+Io$[h#F[foFeco7%Ffiq7$$!+xsJ.RF[fo$\"+BP\\_DF[foFeco7%F \\jq7$$!+MYgURF[fo$\"+0y+!\\#F[foFeco7%Fbjq7$$!+Xop!)RF[fo$\"+ROLFCF[f oFeco7%Fhjq7$$!+y#[v,%F[fo$\"+Z4UkBF[foFeco7%F^[r7$$!+7r7`SF[fo$\"+>u@ ,BF[foFeco7%Fd[r7$$!+*H9u3%F[fo$\"+q4nPAF[foFeco7%Fj[r7$$!+@CS?TF[fo$ \"+&\\JP<#F[foFeco7%F`\\r7$$!+BX4_TF[fo$\"+dAN4@F[foFeco7%Ff\\r7$$!+1I ]#=%F[fo$\"+%*4\\W?F[foFeco7%F\\]r7$$!+t&[;@%F[fo$\"+V36z>F[foFeco7%Fb ]r7$$!+E#f&RUF[fo$\"+p6=8>F[foFeco7%Fh]r7$$!+o#piE%F[fo$\"+@$ym%=F[foF eco7%F^^r7$$!+i#==H%F[fo$\"+tjezO%F[f o$\"+Cmuu:F[foFeco7%Ff_r7$$!+R1J$Q%F[fo$\"+XgJ0:F[foFeco7%F\\`r7$$!+GF u.WF[fo$\"+vKNN9F[foFeco7%Fb`r7$$!+JeGBWF[fo$\"+w&)*[O\"F[foFeco7%Fh`r 7$$!+)*\\(>W%F[fo$\"+\"z+SH\"F[foFeco7%F^ar7$$!+Et$)fWF[fo$\"+gfrA7F[f oFeco7%Fdar7$$!+]/*oZ%F[fo$\"+ca5^6F[foFeco7%Fjar7$$!+.:9$\\%F[fo$\"+> PBz5F[foFeco7%F`br7$$!+*o'e3XF[fo$\"+Xd;25F[foFeco7%Ffbr7$$!+7:@BXF[fo $\"+y_k\\$*FdcoFeco7%F\\cr7$$!+r;*p`%F[fo$\"+ml)oi)FdcoFeco7%Fbcr7$$!+ 8X*)\\XF[fo$\"+\\F!R!zFdcoFeco7%Fhcr7$$!+94)=c%F[fo$\"+d_8\"=(FdcoFeco 7%F^dr7$$!+!Q2Hd%F[fo$\"+lc$*ekFdcoFeco7%Fddr7$$!+@#GHe%F[fo$\"+P:cPdF dcoFeco7%Fjdr7$$!+.w*=f%F[fo$\"+4u<<]FdcoFeco7%F`er7$$!+19x*f%F[fo$\"+ ')*fyH%FdcoFeco7%Ffer7$$!+#y3lg%F[fo$\"+LkgzNFdcoFeco7%F\\fr7$$!+DM27Y F[fo$\"+%pWB'GFdcoFeco7%Fbfr7$$!+OXV;YF[fo$\"+5\\%f9#FdcoFeco7%Fhfr7$$ !+Qvc>YF[fo$\"+h7BI9FdcoFeco7%F^gr7$$!+mXX@YF[fo$\"+))y$*\\rF\\[pFeco7 %Fdgr7$$!+&y%3AYF[foF(Feco7%Fjgr7$Fegr$!+))y$*\\rF\\[pFeco7%F^hr7$F_gr $!+h7BI9FdcoFeco7%Fbhr7$Fifr$!+5\\%f9#FdcoFeco7%Ffhr7$Fcfr$!+%pWB'GFdc oFeco7%Fjhr7$F]fr$!+LkgzNFdcoFeco7%F^ir7$Fger$!+')*fyH%FdcoFeco7%Fbir7 $Faer$!+4u<<]FdcoFeco7%Ffir7$F[er$!+P:cPdFdcoFeco7%Fjir7$Fedr$!+lc$*ek FdcoFeco7%F^jr7$F_dr$!+d_8\"=(FdcoFeco7%Fbjr7$Ficr$!+\\F!R!zFdcoFeco7% Ffjr7$Fccr$!+ml)oi)FdcoFeco7%Fjjr7$F]cr$!+y_k\\$*FdcoFeco7%F^[s7$Fgbr$ !+Xd;25F[foFeco7%Fb[s7$Fabr$!+>PBz5F[foFeco7%Ff[s7$F[br$!+ca5^6F[foFec o7%Fj[s7$Fear$!+gfrA7F[foFeco7%F^\\s7$F_ar$!+\"z+SH\"F[foFeco7%Fb\\s7$ Fi`r$!+w&)*[O\"F[foFeco7%Ff\\s7$Fc`r$!+vKNN9F[foFeco7%Fj\\s7$F]`r$!+Xg J0:F[foFeco7%F^]s7$Fg_r$!+Cmuu:F[foFeco7%Fb]s7$Fa_r$!+,UhV;F[foFeco7%F f]s7$F[_r$!+=y*=r\"F[foFeco7%Fj]s7$Fe^r$!+tjezF[foFeco7%Ff^s7$Fc]r$!+V36z>F[foFeco 7%Fj^s7$F]]r$!+%*4\\W?F[foFeco7%F^_s7$Fg\\r$!+dAN4@F[foFeco7%Fb_s7$Fa \\r$!+&\\JP<#F[foFeco7%Ff_s7$F[\\r$!+q4nPAF[foFeco7%Fj_s7$Fe[r$!+>u@,B F[foFeco7%F^`s7$F_[r$!+Z4UkBF[foFeco7%Fb`s7$Fijq$!+ROLFCF[foFeco7%Ff`s 7$Fcjq$!+0y+!\\#F[foFeco7%Fj`s7$F]jq$!+BP\\_DF[foFeco7%F^as7$Fgiq$!+Io $[h#F[foFeco7%Fbas7$Faiq$!+9V2xEF[foFeco7%Ffas7$F[iq$!+\"=J#RFF[foFeco 7%Fjas7$Fehq$!+CgJ,GF[foFeco7%F^bs7$F_hq$!+zoJjGF[foFeco7%Fbbs7$Figq$! +Mx>DHF[foFeco7%Ffbs7$Fcgq$!+#R'*o)HF[foFeco7%Fjbs7$F]gq$!+?VK[IF[foFe co7%F^cs7$Fgfq$!+Q*o$4JF[foFeco7%Fbcs7$Fafq$!+!Q)*)pJF[foFeco7%Ffcs7$F [fq$!+f#o(HKF[foFeco7%Fjcs7$Feeq$!+c&H))G$F[foFeco7%F^ds7$F_eq$!+Fl$pM $F[foFeco7%Fbds7$Fidq$!+BO&RS$F[foFeco7%Ffds7$Fcdq$!+52wfMF[foFeco7%Fj ds7$F]dq$!+OF[foFeco7%Ffes7$F[cq$!+h\"f\"pOF[foFeco7%Fjes7$Febq$!+!3*y%F[foFeco7% F^is7$Fg]q$!+%Re&HUF[foFeco7%Fbis7$Fa]q$!+C&RYE%F[foFeco7%Ffis7$F[]q$! +#p-4I%F[foFeco7%Fjis7$Fe\\q$!+9Y7RVF[foFeco7%F^js7$F_\\q$!+k#)=!Q%F[f oFeco7%Fbjs7$Fi[q$!+d*[\\U%F[foFeco7%Ffjs7$Fc[q$!+\">^RZ%F[foFeco7%Fjj s7$F][q$!+Rn.FXF[foFeco7%F^[t7$Fgjp$!+'*34$e%F[foFeco7%Fb[t7$Fajp$!+-M CSYF[foFeco7%Ff[t7$F[jp$!+`JZ'p%F[foFeco7%Fj[t7$Feip$!+1/;]ZF[foFeco7% F^\\t7$F_ip$!+W'y-![F[foFeco7%Fb\\t7$Fihp$!+$)RIY[F[foFeco7%Ff\\t7$Fch p$!+n)R!))[F[foFeco7%Fj\\t7$F]hp$!+b'oa#\\F[foFeco7%F^]t7$Fggp$!+XYle \\F[foFeco7%Fb]t7$Fagp$!+7$)o()\\F[foFeco7%Ff]t7$F[gp$!+yml7]F[foFeco7 %Fj]t7$Fefp$!+L&GO.&F[foFeco7%F^^t7$F_fp$!+;wk]]F[foFeco7%Fb^t7$Fiep$! +S)HP1&F[foFeco7%Ff^t7$Fcep$!+([iG2&F[foFeco7%Fj^t7$F]ep$!+%H/!y]F[foF eco7%F^_t7$Fgdp$!+:f3z]F[foFeco7%Fb_t7$Fadp$!+*e5g2&F[foFeco7%Ff_t7$F[ dp$!+7]lo]F[foFeco7%Fj_t7$Fecp$!+s0(o0&F[foFeco7%F^`t7$F_cp$!+n^[S]F[f oFeco7%Fb`t7$Fibp$!+pfI>]F[foFeco7%Ff`t7$Fcbp$!+zM7$*\\F[foFeco7%Fj`t7 $F]bp$!+rsrh\\F[foFeco7%F^at7$Fgap$!+oO'[#\\F[foFeco7%Fbat7$Faap$!+'HX B)[F[foFeco7%Ffat7$F[ap$!+B>'R$[F[foFeco7%Fjat7$Fe`p$!+Q6azZF[foFeco7% F^bt7$F_`p$!+Hk%*=ZF[foFeco7%Fbbt7$Fi_p$!+-$z?l%F[foFeco7%Ffbt7$Fc_p$! +'or)yXF[foFeco7%Fjbt7$F]_p$!+obE*\\%F[foFeco7%F^ct7$Fg^p$!+C)yJT%F[fo Feco7%Fbct7$Fa^p$!+F)f/K%F[foFeco7%Ffct7$F[^p$!+!3R3A%F[foFeco7%Fjct7$ Fe]p$!+Zc*Q6%F[foFeco7%F^dt7$F_]p$!+$Qs!**RF[foFeco7%Fbdt7$Fi\\p$!+TZz vQF[foFeco7%Ffdt7$Fc\\p$!+cguVPF[foFeco7%Fjdt7$F]\\p$!+D\"eKg$F[foFeco 7%F^et7$Fg[p$!+x(obX$F[foFeco7%Fbet7$Fa[p$!+2$GEI$F[foFeco7%Ffet7$Fjjo $!+vT]YJF[foFeco7%Fjet7$Fdjo$!+_!R)))HF[foFeco7%F^ft7$F^jo$!+&)*o1$GF[ foFeco7%Fbft7$Fgio$!+')G_sEF[foFeco7%Ffft7$Faio$!+!y/Y^#F[foFeco7%Fjft 7$Fjho$!+1l%pN#F[foFeco7%F^gt7$$\"+@7\"pj(F`fo$!+bf]*>#F[foFeco7%Fbgt7 $F^ho$!+m5AU?F[foFeco7%Fhgt7$Fhgo$!+'=N])=F[foFeco7%F\\ht7$Fbgo$!+zf!z s\"F[foFeco7%F`ht7$$!+r>Q&3\"F]go$!+()e!3d\"F[foFeco7%Fdht7$Fefo$!+Q!> PT\"F[foFeco7%Fjht7$F^fo$!+xtjc7F[foFeco7%F^it7$Ffeo$!+ttb*4\"F[foFeco 7%Fbit7$$!+W6'f\")*FN$!+cuxC%*FdcoFeco7%Ffit7$$!+%*oJa8FN$!+r9)R&yFdco Feco7%F\\jt7$$!+m$35R(F\\r$!+y_=$G'FdcoFeco7%Fbjt7$$\"+tjDQiF\\q$!+y*) Q7ZFdcoFeco7%Fhjt7$F($!+aEfTJFdcoFeco7%F^[u7$F($!+Fjzq:FdcoFeco7%Fb[uF 'Feco-Fcbo6&Febo$\"#:Fhbo$\"#NFhbo$\"#&*Fhbo-%&STYLEG6#%,PATCHNOGRIDG- F$6$7K7$$\"3#)*****>%Q?vFFN$\"3?+++A_;,iFN7$$\"33+++DnX%y#FN$\"3m***** fLfa?'FN7$$\"3%)*****z\"3z#z#FN$\"31+++*z@5@'FN7$$\"3;+++&em+!GFN$\"3: +++n\\mDiFN7$$\"3!******Hm_5\"GFN$ \"3p*****RFHMB'FN7$$\"3.+++$pQY\"GFN$\"3q*****fJt@C'FN7$$\"3>+++h+\"p \"GFN$\"3#******HE[7D'FN7$$\"3>+++6\"fy\"GFN$\"3M+++;'y/E'FN7$$\"3&)** ***f7&\\9h.GFN$\"3')*****R0GKI'FN7$$\"3?+++\\;L(z#FN$\"3G++ +>Z-5jFN7$$\"3))******f\")4!z#FN$\"3B+++YD!fJ'FN7$$\"3%)*****RwB?y#FN$ \"3))*****>?H2K'FN7$$\"3!)*****R\\RKx#FN$\"3h******[)zVK'FN7$$\"3!)*** **pE'*Qw#FN$\"3w******f%RnK'FN7$$\"3!*******ek;aFFN$\"38+++wgqFjFN7$$ \"3#******p3XUu#FN$\"3k*****>=&>FjFN7$$\"3$)*****zA]Vt#FN$\"3m*****REX ^K'FN7$$\"3,+++_@sCFFN$\"3W+++^X_@jFN7$$\"3)******\\d?cr#FN$\"3g*****z wQjJ'FN7$$\"3#******fK>tq#FN$\"3V+++[%R'4jFN7$$\"3z******=y4+FFN$\"3g* ****p+K:I'FN7$$\"3;+++_)GUp#FN$\"3m*****pO#=#H'FN7$$\"3%)*****Hzi**o#F N$\"3o*****H\\?=G'FN7$$\"3!*******H*3vo#FN$\"3A+++$)*Q2F'FN7$$\"3%**** **R#e,(o#FN$\"3Q+++-]GfiFN7$$\"3!******4]`&)o#FN$\"3m*****foWyC'FN7$$ \"3#)*****R&45#p#FN$\"3p*****fa?oB'FN7$$\"3)******R6Uvp#FN$\"3n*****pI /mA'FN7$$\"36+++]?n/FFN$\"3t*******=\\v@'FN7$$\"3*)*****R(G@8FFN$\"3d* ****fAY*4iFN7$$\"3z*****fcOGs#FN$\"3u*****HP3S?'FN7$$\"3;+++<0>LFFN$\" 3m*****H@k)*>'FN7$$\"3)*******f:#Ru#FN$\"3K+++@2c(>'FN7$$\"3%*******Ha paFFN$\"3M+++[G2(>'FN7$$\"3')*****4@5_w#FN$\"3w******RpJ)>'FNFe\\uFbbo -F^co6L7%7$$\"+UQ?vFF[fo$\"+A_;,iF[fo7$$\"+DnX%y#F[fo$\"+O$fa?'F[fo7$$ \"$v#Fhbo$\"$E'Fhbo7%Feiu7$$\"+=3z#z#F[fo$\"+*z@5@'F[foFjiu7%F`ju7$$\" +&em+!GF[fo$\"+n\\mDiF[foFjiu 7%F\\[v7$$\"+jE06GF[fo$\"+u#HMB'F[foFjiu7%Fb[v7$$\"+$pQY\"GF[fo$\"+;L< UiF[foFjiu7%Fh[v7$$\"+h+\"p\"GF[fo$\"+j#[7D'F[foFjiu7%F^\\v7$$\"+6\"fy \"GF[fo$\"+;'y/E'F[foFjiu7%Fd\\v7$$\"+E^\\9h.GF[f o$\"+a!GKI'F[foFjiu7%Fb^v7$$\"+\\;L(z#F[fo$\"+>Z-5jF[foFjiu7%Fh^v7$$\" +g\")4!z#F[fo$\"+YD!fJ'F[foFjiu7%F^_v7$$\"+kP-#y#F[fo$\"+-#H2K'F[foFji u7%Fd_v7$$\"+%\\RKx#F[fo$\"+\\)zVK'F[foFjiu7%Fj_v7$$\"+ni*Qw#F[fo$\"+g %RnK'F[foFjiu7%F``v7$$\"+fk;aFF[fo$\"+wgqFjF[foFjiu7%Ff`v7$$\"+(3XUu#F [fo$\"+#=&>FjF[foFjiu7%F\\av7$$\"+G-NMFF[fo$\"+k_9DjF[foFjiu7%Fbav7$$ \"+_@sCFF[fo$\"+^X_@jF[foFjiu7%Fhav7$$\"+v0i:FF[fo$\"+o(QjJ'F[foFjiu7% F^bv7$$\"+E$>tq#F[fo$\"+[%R'4jF[foFjiu7%Fdbv7$$\"+>y4+FF[fo$\"+2?`,jF[ foFjiu7%Fjbv7$$\"+_)GUp#F[fo$\"+nB=#H'F[foFjiu7%F`cv7$$\"+$zi**o#F[fo$ \"+$\\?=G'F[foFjiu7%Ffcv7$$\"+I*3vo#F[fo$\"+$)*Q2F'F[foFjiu7%F\\dv7$$ \"+Ce,(o#F[fo$\"+-]GfiF[foFjiu7%Fbdv7$$\"+,Nb)o#F[fo$\"+'oWyC'F[foFjiu 7%Fhdv7$$\"+a45#p#F[fo$\"+Y0#oB'F[foFjiu7%F^ev7$$\"+9@a(p#F[fo$\"+2VgE iF[foFjiu7%Fdev7$$\"+]?n/FF[fo$\"+!>\\v@'F[foFjiu7%Fjev7$$\"+uG@8FF[fo $\"+Ei%*4iF[foFjiu7%F`fv7$$\"+ml$Gs#F[fo$\"+t$3S?'F[foFjiu7%Fffv7$$\"+ <0>LFF[fo$\"+8U')*>'F[foFjiu7%F\\gv7$$\"+g:#Ru#F[fo$\"+@2c(>'F[foFjiu7 %Fbgv7$$\"+IapaFF[fo$\"+[G2(>'F[foFjiu7%Fhgv7$$\"+6-@lFF[fo$\"+SpJ)>'F [foFjiu7%F^hvF`iuFjiuFf[uF^\\u-F$6$7K7$Ff\\u$!3?+++A_;,iFN7$Fihu$!3w** ****RpJ)>'FN7$Fdhu$!3M+++[G2(>'FN7$F_hu$!3K+++@2c(>'FN7$Fjgu$!3m*****H @k)*>'FN7$Fegu$!3u*****HP3S?'FN7$F`gu$!3d*****fAY*4iFN7$F[gu$!3t****** *=\\v@'FN7$Fffu$!3n*****pI/mA'FN7$Fafu$!3p*****fa?oB'FN7$F\\fu$!3m**** *foWyC'FN7$Fgeu$!3Q+++-]GfiFN7$Fbeu$!3A+++$)*Q2F'FN7$F]eu$!3o*****H\\? =G'FN7$Fhdu$!3m*****pO#=#H'FN7$Fcdu$!3g*****p+K:I'FN7$F^du$!3V+++[%R'4 jFN7$Ficu$!3g*****zwQjJ'FN7$Fdcu$!3W+++^X_@jFN7$F_cu$!3m*****REX^K'FN7 $Fjbu$!3k*****>=&>FjFN7$Febu$!38+++wgqFjFN7$F`bu$!3w******f%RnK'FN7$F[ bu$!3h******[)zVK'FN7$Ffau$!3))*****>?H2K'FN7$Faau$!3B+++YD!fJ'FN7$F\\ au$!3G+++>Z-5jFN7$Fg`u$!3')*****R0GKI'FN7$Fb`u$!3U+++$)3l&H'FN7$F]`u$! 37+++smV(G'FN7$Fh_u$!3!)*****\\nL(yiFN7$Fc_u$!3s*****z%\\ppiFN7$F^_u$! 3M+++;'y/E'FN7$Fi^u$!3#******HE[7D'FN7$Fd^u$!3q*****fJt@C'FN7$F_^u$!3p *****RFHMB'FN7$Fj]u$!3M+++)e(>DiFN7$Fe]u$!3:+++n\\m'F[fo7$F[ju$!$E'Fhbo7%Fe`w7$Figv$!+[G2(>'F[foFh`w7%F\\aw7$Fcgv $!+@2c(>'F[foFh`w7%F`aw7$F]gv$!+8U')*>'F[foFh`w7%Fdaw7$Fgfv$!+t$3S?'F[ foFh`w7%Fhaw7$Fafv$!+Ei%*4iF[foFh`w7%F\\bw7$F[fv$!+!>\\v@'F[foFh`w7%F` bw7$Feev$!+2VgEiF[foFh`w7%Fdbw7$F_ev$!+Y0#oB'F[foFh`w7%Fhbw7$Fidv$!+'o WyC'F[foFh`w7%F\\cw7$Fcdv$!+-]GfiF[foFh`w7%F`cw7$F]dv$!+$)*Q2F'F[foFh` w7%Fdcw7$Fgcv$!+$\\?=G'F[foFh`w7%Fhcw7$Facv$!+nB=#H'F[foFh`w7%F\\dw7$F [cv$!+2?`,jF[foFh`w7%F`dw7$Febv$!+[%R'4jF[foFh`w7%Fddw7$F_bv$!+o(QjJ'F [foFh`w7%Fhdw7$Fiav$!+^X_@jF[foFh`w7%F\\ew7$Fcav$!+k_9DjF[foFh`w7%F`ew 7$F]av$!+#=&>FjF[foFh`w7%Fdew7$Fg`v$!+wgqFjF[foFh`w7%Fhew7$Fa`v$!+g%Rn K'F[foFh`w7%F\\fw7$F[`v$!+\\)zVK'F[foFh`w7%F`fw7$Fe_v$!+-#H2K'F[foFh`w 7%Fdfw7$F__v$!+YD!fJ'F[foFh`w7%Fhfw7$Fi^v$!+>Z-5jF[foFh`w7%F\\gw7$Fc^v $!+a!GKI'F[foFh`w7%F`gw7$F]^v$!+$)3l&H'F[foFh`w7%Fdgw7$Fg]v$!+smV(G'F[ foFh`w7%Fhgw7$Fa]v$!+vOtyiF[foFh`w7%F\\hw7$F[]v$!+[\\ppiF[foFh`w7%F`hw 7$Fe\\v$!+;'y/E'F[foFh`w7%Fdhw7$F_\\v$!+j#[7D'F[foFh`w7%Fhhw7$Fi[v$!+; LDiF[foFh` w7%Fdiw7$Fgju$!+n\\m$Fhbo;$!$f'Fhbo$\"$f'Fhbo" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute \+ stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " } {XPPEDIT 18 0 "[-4.6221, 0];" "6#7$,$-%&FloatG6$\"&@i%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve." }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way we see that the intersection of the \+ stability region with the non-negative imaginary axis consists of the \+ union of two disjoint intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 465 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 3562737767/1442 73191731200*z^8+7637477027/2596917451161600*z^9+\n 318136951/ 2596917451161600*z^10+77561/3206070927360*z^11:\nDigits := 18:\npts := []: z0 := 0:\nfor ct from 0 to 230 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(z z)]]:\nend do:\nplot(pts,color=COLOR(RGB,0,.35,.95),thickness=2,font=[ HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7cy7$$\"\"!F)F(7$$\"3)e/RDu1h(=!#>$\"3C $z*e`EfTJF-7$$\"35W(>'[(GG5$F-$\"3.'ezrI&=$G'F-7$$\"3fyU-6+/gTF-$\"3pi $p2'zxC%*F-7$$\"3#\\.cGTQf6&F-$\"3x%*eVhqjc7!#=7$$\"3e(\\([T#G%)*fF-$ \"36IZzEjzq:F?7$$\"3\\>M76+u@oF-$\"3JDI:#fb\\)=F?7$$\"3!ogu\"y\")p$f(F -$\"3A*[4v&[6*>#F?7$$\"3>4&z)QLu#GF?7$$\"3%y5IS0D+j*F-$\"3#GC5Nl#fTJF?7$$\"3TRsBV]I@5 F?$\"3e6\\w=>vbMF?7$$\"3b8Kx3n,u5F?$\"3\"zE9R=6*pPF?7$$\"3E26507**>6F? $\"3Fvw()[/2%3%F?7$$\"3GH,Ucy6d6F?$\"3H2%HNrH#)R%F?7$$\"3B3[S(px5=\"F? $\"3Y]>ox*)Q7ZF?7$$\"3**pn=$)*R2=\"F?$\"3y=m1T#[l-&F?7$$\"35Lzmy/R-6F? $\"38#R7L]22M&F?7$$!3I2`$pF%)e>\"F?$\"3/*eCRwm[l&F?7$$!3s@!3MJh8O\"F?$ \"3bkeEAg-pfF?7$$!3zJw6(32(y9F?$\"3x*4VvF&=$G'F?7$$!33A/3@\\%*z:F?$\"3 oJ]!)GXM(f'F?7$$!37*p&*G%*zGn\"F?$\"3%p'p&\\x.:\"pF?7$$!3xy5'3zR1w\"F? $\"3,N1![,jcA(F?7$$!3bM:o?`yW=F?$\"3jg\"4rCA)RvF?7$$!3!et<*o??E>F?$\"3 tn#\\2Z\")R&yF?7$$!3g.@&=eOa+#F?$\"3a]^'[oS\"o\")F?7$$!3!GJ7#ol%G3#F?$ \"3&[%49*))*H#[)F?7$$!3P(fi_:w'e@F?$\"3]v8;%3fkz)F?7$$!3_/rK_p4LAF?$\" 3E6H*=F=16*F?7$$!3a(30e'>B1BF?$\"3r%e5jXxZU*F?7$$!3KwBIh-Wm$*Q(*F?7$$!3(Hi?.Lw*[CF?$\"3tsah%e4`+\"!#<7$$!3y[qkuip=DF?$\"3<[My2 bsO5Ffu7$$!3-Wp\"p*>O(e#F?$\"371n9O99o5Ffu7$$!3)e5B_4%*\\l#F?$\"3L_&oF Pd&*4\"Ffu7$$!3U%zWl)Qg@FF?$\"3=24x@L(48\"Ffu7$$!3Q-\\ED[>(y#F?$\"3%R, #f)G*Qi6Ffu7$$!3j\"=_v[j<&GF?$\"3v,zG!G0Q>\"Ffu7$$!3**pC\"GB+`\"HF?$\" 3YF>)eI@_A\"Ffu7$$!3M%=H\\h*yxHF?$\"3n$omnPPmD\"Ffu7$$!3zRe=!f5#RIF?$ \"3PQC;2N0)G\"Ffu7$$!3vt9h]l`*4$F?$\"3*G0OYrp%>8Ffu7$$!31>^%z>N(eJF?$ \"38)*Ro?g)3N\"Ffu7$$!3CG4[i#on@$F?$\"3xTEQ^CI#Q\"Ffu7$$!3dBPKh5ftKF?$ \"3p+%3\"Q!>PT\"Ffu7$$!3c8.%4)=:HLF?$\"3DJ'H$=e8X9Ffu7$$!3#*)o/$e6R$Q$ F?$\"3p`7ZOGbw9Ffu7$$!3kueGA/COMF?$\"3%f1_[9qz]\"Ffu7$$!3fWC#3%4i([$F? $\"3!\\[$p/yQR:Ffu7$$!3_\\.'z*>WPNF?$\"3d]q>()e!3d\"Ffu7$$!3'>6D_p)f&e $F?$\"31IkouWA-;Ffu7$$!3(*Qc%)H\"p>j$F?$\"3Sby!=mVOj\"Ffu7$$!3sh[XT2Tw OF?$\"3074ycN1l;Ffu7$$!3Ffu7$$!3#3)RS'*\\?XRF?$\"3!>,(H<2!z%>Ffu7$$! 3koyVn^L]RF?$\"3&=QO\\bP$z>Ffu7$$!3ao\\^C:%e%RF?$\"3)\\.8)\\vx5?Ffu7$$ !37D=\"*)\\a$GRF?$\"38'>=h1@A/#Ffu7$$!3(R?yC<(3#*QF?$\"3,:2g$\\oO2#Ffu 7$$!3k0Qof$pc#QF?$\"3zV[%pC?^5#Ffu7$$!3%oq$fPy\"4q$F?$\"3xqTlkndO@Ffu7 $$!3*p9^@YTmR$F?$\"3@]7O2&Q!o@Ffu7$$\"35?l5nT=EMF?$\"3OXTZbf]*>#Ffu7$$ \"3l\"p%eB<;QQF?$\"3(44iggz4B#Ffu7$$\"3(3NCCVk22%F?$\"3aUF()o*fCE#Ffu7 $$\"3mrwy>d/[UF?$\"3N%)HOhv%RH#Ffu7$$\"3@=E^weu(R%F?$\"3z3Sh-HWDBFfu7$ $\"3(>9]y-03`%F?$\"3Qy$*)f]YpN#Ffu7$$\"3E\"zaP>&o_YF?$\"3<.@Qq)e%)Q#Ff u7$$\"3im*y9(*4lw%F?$\"3TON))p/)*>CFfu7$$\"3s*G(G2EBu[F?$\"32;LpT<^^CF fu7$$\"3M)zkuZar(\\F?$\"3Itf2sI0$[#Ffu7$$\"3y&fn&4Y=w]F?$\"3QWm:!y/Y^# Ffu7$$\"3yD1U)\\\")><&F?$\"3d$>O$*4nha#Ffu7$$\"3=U?!R)f.l_F?$\"3ONe4c, uxDFfu7$$\"3IA`*[KAdN&F?$\"3_tT&f%RK4EFfu7$$\"3A-)z[0JVW&F?$\"3Y?pL1$= 4k#Ffu7$$\"3;$zGx(34JbF?$\"36,L7')G_sEFfu7$$\"3BMA>y&F?$\"3$[\"f5i3 RnFFfu7$$\"3H@zsB\"zF'eF?$\"3,m'=wwF!*z#Ffu7$$\"3O+%4GX+C%fF?$\"3roVm% )*o1$GFfu7$$\"3#o6[9NX3-'F?$\"3*)p$)\\\"=7B'GFfu7$$\"3cab\\vcjgJ4mF?$\"3P@XNKE0:JFfu7$$\"3p5#)*G)e -ymF?$\"3)))4ea\"oF?$\"3jK1ULz3%*)=xoF?$\"3id]\\FaXSKFfu7$$\"3 )fF\">K97TpF?$\"3p*eu4=(frKFfu7$$\"3FU_40&[P+(F?$\"3O_&*e1$GEI$Ffu7$$ \"3yo?Wwz,lqF?$\"3<\"p[:jLNL$Ffu7$$\"3#*Q%R3,v[7(F?$\"38:Zh)=(HkLFfu7$ $\"3`&)*H5&HE$=(F?$\"3<)HWWN-\\R$Ffu7$$\"3Ar!)*3AC,C(F?$\"30`y<=?LDMFf u7$$\"3vy,(RG,aH(F?$\"3w4![uxobX$Ffu7$$\"3FK5?Lw.\\tF?$\"3RSt1d^f&[$Ff u7$$\"3i'3Y=yy4S(F?$\"3)HEGQ%QR:NFfu7$$\"3]R^+@K<^uF?$\"3=]gHHz%\\a$Ff u7$$\"3M\"QwvIt&*\\(F?$\"3@&3!R^6CuNFfu7$$\"3@w&os/Oha(F?$\"3^h\\3D\"e Kg$Ffu7$$\"3I'\\k5uB3f(F?$\"3#G`9Gb%)>j$Ffu7$$\"3]0=&)pWgLwF?$\"3'p5Qh S2/m$Ffu7$$\"3'>3a-S_Wn(F?$\"3RR&[=2:&)o$Ffu7$$\"3Y1QA\\yF?$\"3\\ ?`\"y'z-CQFfu7$$\"3^u!**4Zk#yyF?$\"3t#)=6Jz3]QFfu7$$\"3OJSnQDN0zF?$\"3 i,XXTZzvQFfu7$$\"3NM/RR+]IzF?$\"37MLJ!f[6!RFfu7$$\"3MI+iwHs`zF?$\"3L!p \")Ht]h#RFfu7$$\"3!*oQN?(Q](zF?$\"3*GsiJM.3&RFfu7$$\"3LZIxU_Y%*zF?$\"3 uNFFn$4^(RFfu7$$\"3efb3x.-7!)F?$\"3WH1t#Qs!**RFfu7$$\"3QfX/16sF!)F?$\" 3DOrB)GeT!)F?$\"37ONxbg)f/%Ffu7$$\"3lmOWz*=O0)F?$\" 3\\@/,ke%*oSFfu7$$\"3=M7?n)RQ1)F?$\"3r\"[L0v!e\"4%Ffu7$$\"3)4H?_g_A2)F ?$\"3d>=nYc*Q6%Ffu7$$\"3;.1@U-')y!)F?$\"3B#z$owa*e8%Ffu7$$\"3^#\\['37m $3)F?$\"31g+]$4&edTFfu7$$\"3O6Dp'o[m3)F?$\"3]\")*=cAp*yTFfu7$$\"3_U\"> (Q*4y3)F?$\"3qf.FMC0+UFfu7$$\"39.i]sb7(3)F?$\"3CMN))z!R3A%Ffu7$$\"3))4 %oproX3)F?$\"3_V(poHL8C%Ffu7$$\"3NJ!=q2/,3)F?$\"330Buv*Q:E%Ffu7$$\"3SW n1Hoot!)F?$\"3YB%R;vf9G%Ffu7$$\"3pb-tZ))4!)F?$\"3[(*>/h!**oP% Ffu7$$\"3)pn$HH.2!*zF?$\"3$=S()>Cs^R%Ffu7$$\"3<*4'))pjcnzF?$\"3.`(\\S# )yJT%Ffu7$$\"3ub@Zv!o@%zF?$\"3*e$=M)\\?4V%Ffu7$$\"3YEzt)>GO\"zF?$\"3Qy ))Hk()R[WFfu7$$\"3.SEQGjk\")yF?$\"3kV^cX\\hlWFfu7$$\"37hEFgV&e%yF?$\"3 EOpJ1-d#[%Ffu7$$\"3NBf;Ykz0yF?$\"3@a%*ynbE*\\%Ffu7$$\"3(*4ew!p+4w(F?$ \"3a&\\0[#>q:XFfu7$$\"3(=BJ`9Q/r(F?$\"3g%*48i+)=`%Ffu7$$\"3RFj/_:Y`wF? $\"3&\\B10n+ya%Ffu7$$\"3q@>_e/r)e(F?$\"31RGAiVYjXFfu7$$\"3aQCthwX9vF?$ \"35(oNhor)yXFfu7$$\"3dzb[VpCGuF?$\"3XFh,UJ-%f%Ffu7$$\"3rlG/$e8kK(F?$ \"3/3&oT>>*3YFfu7$$\"3:!Qha9EJ?(F?$\"3T@kC%GgNi%Ffu7$$\"3rH>J_IA[qF?$ \"3>/$=K%o%zj%Ffu7$$\"3h8\"\\e$[OToF?$\"3V.VW-$z?l%Ffu7$$\"3R_#p3ee1`' F?$\"3l;x$Q5efm%Ffu7$$\"3S$*RQ!*f#[(eF?$\"3K%*[44PezYFfu7$$!3IgBJW&HD+ 'F?$\"3#)*H()zgcHp%Ffu7$$!3Ytmn`$f'*f'F?$\"3Xq%=dKxgq%Ffu7$$!3UX'ey`MD \"pF?$\"3\"f\"pNHk%*=ZFfu7$$!3a'p\"zj:SJrF?$\"3(>_jL`k:t%Ffu7$$!3^_y9: [--tF?$\"3m'yQUIKRu%Ffu7$$!3k)ei)y'zHW(F?$\"3(\\>@VY]gv%Ffu7$$!3C&finF >Pc(F?$\"3zw%Ffu7$$!3_#*f!G#RtpwF?$\"3#e!fzP6azZFfu7$$!3$[<%fS^ ]kxF?$\"3-zj6)R:4z%Ffu7$$!3U`v1CUQ]yF?$\"3!o%fzVN/-[Ffu7$$!3!z+')*4l.H zF?$\"3PzzS1m#H\"[Ffu7$$!3SaZ([F'o,!)F?$\"3\"RjO1olN#[Ffu7$$!3C!3mA)4E p!)F?$\"3?\")zMB>'R$[Ffu7$$!3Z%obA%3[K\")F?$\"3Yt_=_l6W[Ffu7$$!3soG&p& o\">>)F?$\"3#oiRP%3.a[Ffu7$$!3'QXB\"4)H![#)F?$\"3fDkMJhqj[Ffu7$$!3>F6_ `s>,$)F?$\"3gQjb-Q9t[Ffu7$$!3P%\\,EpKo\"\\Ffu 7$$!3JxX!p\\1Hd)F?$\"3&[P&*zmj[#\\Ffu7$$!3N>c:pa-7')F?$\"3NkZ\\ Ffu7$$!3\\6e_O,L@()F?$\"3*y2/]X*ya\\Ffu7$$!3NYooiiPb()F?$\"3j\"zg5F<<' \\Ffu7$$!3!zh2x)*\\Ffu7$$!36lR7#>&)p'*)F?$\"3Eq)*e73@/]Ffu7$$!3-c2#)))3*R**) F?$\"3E]9QSbW4]Ffu7$$!323kl^'4.-*F?$\"3\"RW&[tmZ9]Ffu7$$!3\"*=eBtP(f/* F?$\"3Qa%*>pfI>]Ffu7$$!3o\")*fNC852*F?$\"3)*=]-p^$R-&Ffu7$$!3p()fnufX& 4*F?$\"3#e[tk*fOG]Ffu7$$!3m4!=z*zK>\"*F?$\"36`***\\:+E.&Ffu7$$!3N35qPO lU\"*F?$\"3bxC1E$Rm.&Ffu7$$!37`_/Id4#*F?$\"3;;erhLgZ]Ffu7$$!3)e(\\;Xs#4 B*F?$\"3yByM'*)y30&Ffu7$$!3/A/wPf$=D*F?$\"3WUgAhu'R0&Ffu7$$!3w\"[`&eeJ s#*F?$\"3&fX-;dqo0&Ffu7$$!3$\\:v&eGQ#H*F?$\"3`5wJ5(*ef]Ffu7$$!3q-c%Q$> 07$*F?$\"3OD)3nKE@1&Ffu7$$!36>Z:*GP8L*F?$\"3K;?gN=[k]Ffu7$$!39Z3(eR_-N *F?$\"3TE>Q;wlm]Ffu7$$!3)=@2&Ffu7$$!30MFj2pXA%*F ?$\"3'QBG&f(*et]Ffu7$$!34B)H()f+(R%*F?$\"3$f7)z$H')[2&Ffu7$$!3*>0ri5Um X*F?$\"3B4``)e5g2&Ffu7$$!3qeLL25Ht%*F?$\"3wZb`aP'p2&Ffu7$$!36]auekl*[* F?$\"3YJ[%>'oux]Ffu7$$!3OAAK:f3z]Ffu-%*THICKNESSG6#\" \"#-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBGF($\"#N!\"#$\"#&*Fddo-% +AXESLABELSG6$Q!6\"Fjdo-%%VIEWG6$%(DEFAULTGF_eo" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The relevant intersectio n points of the boundary curve with the imaginary axis can be determin ed as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for point s on the boundary curve either side of each intersection point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Digits := 15:\nz0 := 0.55*I:\nfor ct from 16 to 19 do\n newton( R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 2.2*I:\nfor ct from 68 to 71 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 4 .7*I:\nfor ct from 166 to 169 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0 );\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0+o %RAm=i!#D$\"0im5C[l-&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0$eX( yl?#H!#D$\"0R7L]22M&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0dJ'>N_ ar!#D$\"0fCRwm[l&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0#QsN2^wH! #C$\"0'eEAg-pf!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$!0g5L4PSy\"!#>$\"0#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0$>'e#o!Hm#!#>$\"04iggz4B#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0gPS1n c@*!#<$\"0sPQ5efm%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0UuneMq( G!#<$\"0*[44PezY!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!04okxI[k$!# <$\"0I()zgcHp%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0FC+#[^M5!#;$ \"0Z=dKxgq%!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to calculate the para meter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "Digits := 15:\nreal_part := proc(u)\n \+ Re(newton(R(z)=exp(u*Pi*I),z=0.55*I))\nend proc:\nu0 := bisect('real_p art'(u),u=0.14..0.19);\nnewton(R(z)=exp(u0*Pi*I),z=0.55*I);``;\nreal_p art := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.2*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.68..0.71);\nnewton(R(z)=exp(u0*Pi*I),z=2 .2*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=4.7*I ))\nend proc:\nu0 := bisect('real_part'(u),u=1.66..1.69);\nnewton(R(z) =exp(u0*Pi*I),z=4.7*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0XWq[N'R%#u0G$\"09(ex]2^p!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0dPxeyA=$!#H$\"0_Ery4T=#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0)Gd?cWu;!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0 " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,StabilityFuncti on(6,12,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\".JrJ_\"37\" 1+W0%43\"Q`F)*$)F'\"\"(F)F)F)*&#\"/4]`&3Ka$\"3+W&3&)>;gs'F)*$)F'\"\")F )F)F)*&#\"-`'*eD'=\"\"3+Ori\\S]\"o\"F)*$)F'\"\"*F)F)!\"\"*&#\",PJ'=(4% \"2?R6!)f@!o*)F)*$)F'\"#5F)F)F)*&#\"*^p8=$FinF)*$)F'\"#6F)F)Fen*&#\"&h v(\"0K[-;jr5\"F)*$)F'\"#7F)F)Fen" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R* `(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z_0 := newton(`R*`(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Cc^$e$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R* `(z)=1,z=-4):\np_1 := plot([`R*`(z),1],z=-4.19..0.49,color=[red,blue]) :\np_2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,cross,diamond] ,color=black):\np_3 := plot([[z_0,0],[z_0,1]],linestyle=3,color=COLOR( RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.19..0.49,-0.07..1 .47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 366 261 261 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!>%!#<$\"3yBEF\")>'R%QF*7$$!3 3+]iHt\\kTF*$\"3gO4zo.oWOF*7$$!3l++DfY**QTF*$\"3I\"*QzLkqaMF*7$$!3M+]( )))>\\8TF*$\"3M'37wIfOF$F*7$$!3-++]=$*)z3%F*$\"3!QrJ1kr65$F*7$$!3O+vBW )4O/%F*$\"3d`#3Y;s.#GF*7$$!3E+](*p.B**RF*$\"3;HSI>3liDF*7$$!3W+Dm45K\\ RF*$\"3E[@\"z*)=$)H#F*7$$!3;++N\\;T**QF*$\"3?_pZ80')e?F*7$$!3A++Sd9<\\ QF*$\"3pke;G\"e3%=F*7$$!3E++Xl7$*)z$F*$\"3,'\\w%Qn.W;F*7$$!3U+DOc)H*[P F*$\"35!)z_/jRn9F*7$$!37+]FZ%G*)p$F*$\"3MAT%*G(4$38F*7$$!3X+]dLI@1OF*$ \"3/N;ZQ5pa5F*7$$!3=+](Hf6-^$F*$\"3jf!zef2.T)!#=7$$!3?+]xln#4T$F*$\"3t J$F*$\"3=acSEG&pB&F]p7$$!3?++I>7;5KF*$\"3_FA] 7L&)3TF]p7$$!33++&G\\'\\?JF*$\"3+:$fNs06L$F]p7$$!32++!zCb&>IF*$\"3Mird w'4Kl#F]p7$$!3>++]G&*>=HF*$\"3V1R4XF,W@F]p7$$!3******47X_?GF*$\"3+r;#) [TK%y\"F]p7$$!3<+]xRi#=t#F*$\"3S^]Hk@&za\"F]p7$$!3\"******[Tbji#F*$\"3 KBNu4()4d8F]p7$$!3y****f\"z2q`#F*$\"3!)p?#[4]wD\"F]p7$$!3$)**\\\"F]p7$$!3:++got2TBF*$\"3A/'f#*e-4>\"F]p7$$!3#)**\\<& 4T,C#F*$\"31\\%e!)*yy=7F]p7$$!3%)**\\_$*e-W@F*$\"3wt$f&zz/w7F]p7$$!3m* **\\PwRP/#F*$\"3Sx\"[1xfJO\"F]p7$$!3w**\\n!zX;&>F*$\"31#**f&*)f,l9F]p7 $$!31++D***4B&=F*$\"3Ij/TH6X'f\"F]p7$$!3/+]xH!G\"\\F]p7$$!3!****\\p`*Hi:F*$\"3!ReA#zk>-@F] p7$$!33++?'[!3i9F*$\"3CFt,7Z_?BF]p7$$!3%)***\\rpNSO\"F*$\"3))\\^Y=HxdD F]p7$$!3y**\\PABp5F*$\"3l+Y7#)*HHV$F]p7$$!3)o****\\Cu9o*F]p$\"3SKY G%ozyz$F]p7$$!3g,+v$H$zl()F]p$\"3w)zww1R?;%F]p7$$!3+&****4:7Zw(F]p$\"3 3dZ7(*yD+YF]p7$$!3%>+]<\"GxAoF]p$\"3EP%R!*)4ka]F]p7$$!37&**\\Zzu\"QeF] p$\"3_7()R(*zkxbF]p7$$!3M*****\\E]b([F]p$\"3:_AKQ*f79'F]p7$$!3!z**\\Pf rx'QF]p$\"3YRrv$>CCz'F]p7$$!3M$****f&4;(*GF]p$\"3YwT3(HgZ[(F]p7$$!3N)* **\\!RrX!>F]p$\"3]hM?[=\"eE)F]p7$$!3#Q$**\\x'4??*!#>$\"3\"3**eN>o37*F] p7$$!3_r'****>ymc\"!#?$\"3;bk*R[XV)**F]p7$$\"3S-+]r9.@5F]p$\"3,o;2,x\\ 26F*7$$\"352++[nE[>F]p$\"3BGx2h.5:7F*7$$\"3-0+D[H*o$HF]p$\"3st?(f2n8M \"F*7$$\"3)H+]xOMJ)QF]p$\"3]#)[tA>\\u9F*7$$\"3!***************[F]p$\"3 l(Q6k\\;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fe]lFd]l-F$6$7S7$F ($\"\"\"Fe]l7$F=Fj]l7$FGFj]l7$FQFj]l7$FenFj]l7$F_oFj]l7$FdoFj]l7$FioFj ]l7$F_pFj]l7$FdpFj]l7$FipFj]l7$F^qFj]l7$FcqFj]l7$FhqFj]l7$F]rFj]l7$Fbr Fj]l7$FgrFj]l7$F\\sFj]l7$FasFj]l7$FfsFj]l7$F[tFj]l7$F`tFj]l7$FetFj]l7$ FjtFj]l7$F_uFj]l7$FduFj]l7$FiuFj]l7$F^vFj]l7$FcvFj]l7$FhvFj]l7$F]wFj]l 7$FbwFj]l7$FgwFj]l7$F\\xFj]l7$FaxFj]l7$FfxFj]l7$F[yFj]l7$F`yFj]l7$FeyF j]l7$FjyFj]l7$F_zFj]l7$FdzFj]l7$FizFj]l7$F_[lFj]l7$Fe[lFj]l7$Fj[lFj]l7 $F_\\lFj]l7$Fd\\lFj]l7$Fi\\lFj]l-F^]l6&F`]lFd]lFd]lFa]l-F$6&7#7$$!3!** ****Ri:Ne$F*Fj]l-%'SYMBOLG6#%'CIRCLEG-F^]l6&F`]lFe]lFe]lFe]l-%&STYLEG6 #%&POINTG-F$6&F`al-Feal6#%&CROSSGFhalFjal-F$6&F`al-Feal6#%(DIAMONDGFha lFjal-F$6%7$7$FbalFd]lFaal-%&COLORG6&F`]lFd]l$\"\"&!\"\"Fd]l-%*LINESTY LEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Fjcl-%%FONTG6#%(DEFAULTG-F]dl6$%*HE LVETICAG\"\"*-%%VIEWG6$;$!$>%!\"#$\"#\\Fjdl;$!\"(Fjdl$\"$Z\"Fjdl" 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 6 scheme. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1523 "`R*` := \+ z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 1208152 317131/5338108094054400*z^7+35432085535009/672601619850854400*z^8-\n \+ 118625589653/168150404962713600*z^9+40971863137/89680215980113920 *z^10-\n 318136951/89680215980113920*z^11-77561/110716316024832* z^12:\npts := []:\nz0 := 0:\nfor ct from 0 to 280 do\n zz := newton( `R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(z z),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.12,.38)):\np _2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.4,0]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,0,.23,.75)):\npts := []: z0 := 2.8+4.5*I:\nfor ct from 0 to 40 do\n zz := newton(`R*`(z) =exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,0,.12,.38)):\np_4 := p lots[polygonplot]([seq([pts[i-1],pts[i],[2.76,4.46]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,0,.23,.75)):\npts := []: z0 := 2.8-4.5*I:\nfor ct from 0 to 40 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_5 := plot(pts,color=COLOR(RGB,0,.12,.38)):\np_6 := plots [polygonplot]([seq([pts[i-1],pts[i],[2.76,-4.46]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,0,.23,.75)):\np_7 := plot([ [[-4.19,0],[3.19,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\n plots[display]([p_||(1..7)],view=[-4.19..3.19,-5.19..5.19],font=[HELVE TICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=cons trained);" }}{PARA 13 "" 1 "" {GLPLOT2D 457 554 554 {PLOTDATA 2 "6/-%' CURVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3'******>*4qH7!#F$ \"38+++VFfTJF-7$$\"3!******RuXh(o!#E$\"3#)*****fg!R7ZF-7$$\"3'******\\ L1#)=\"!#C$\"3A+++M%)>$G'F-7$$\"3)*******y'pI2\"!#B$\"3!)*****p@\\S&yF -7$$\"3z*****zbf)3kFC$\"3^+++=y.D%*F-7$$\"3))******3g3rG!#A$\"35+++(RQ '*4\"!#<7$$\"3.+++K9JS5!#@$\"3/+++M8&oD\"FQ7$$\"36+++:#)=,KFU$\"30+++0 @@99FQ7$$\"3')*****R#)f&[')FU$\"31+++IK\"=d\"FQ7$$\"3;+++/[')*4#!#?$\" 3.+++P*Q(H$\"37+++b)oM?#FQ7$$\"3&)***** Ry,L7$F_p$\"3))*****f(\\pdBFQ7$$\"3!)*****4\\;l%\\F_p$\"3!******\\)Qp0 DFQ7$$\"3]******=nE'4(F_p$\"3-+++TK8WEFQ7$$\"3s*****>%p:a#*F_p$\"3')** ***f@s5x#FQ7$$\"3(*******)y-/6\"F-$\"3;+++(=_k)GFQ7$$\"3++++\"pQNC\"F- $\"3/+++PtL\"*HFQ7$$\"3/+++w9)\\J\"F-$\"3)******z\\Jq3$FQ7$$\"38+++\"o *eA8F-$\"3-+++4tkuJFQ7$$\"31+++&Q/'o7F-$\"35+++\"f<]D$FQ7$$\"3/+++)z5v :\"F-$\"3(*******e=uGLFQ7$$\"3e*****H5)3[**F_p$\"3#)*****f%)eiR$FQ7$$ \"3Q+++P<;jyF_p$\"3A+++*R2zX$FQ7$$\"3w******eLmx`F_p$\"3')*****4HoR^$F Q7$$\"3))******4>wXDF_p$\"31+++9%*okNFQ7$$!3y*****H$H,GeF_o$\"3@+++T!* H5OFQ7$$!3I+++mKFjRF_p$\"31+++[:,^OFQ7$$!3c******oJ)fb(F_p$\"3%)*****f ()Gqo$FQ7$$!3/+++%)RhK6F-$\"3%******H(*R&=PFQ7$$!34+++^gMC:F-$\"3/+++- &>du$FQ7$$!3*)*****zIs\"G>F-$\"3/+++5lsoPFQ7$$!3-+++I0$=M#F-$\"3/+++AL q(y$FQ7$$!3G+++c^PjFF-$\"3*******fsuF!QFQ7$$!3E+++I\"R6>$F-$\"3;+++it/ 9QFQ7$$!3*)*****pJ-Pi$F-$\"3.+++?#4;#QFQ7$$!39+++ms')fSF-$\"3&)*****f5 Hb#QFQ7$$!3!)*****\\+V')\\%F-$\"3!)*****p(f&e#QFQ7$$!3$)*****HnP#R\\F- $\"3-+++-!=E#QFQ7$$!3]*****fOh5Q&F-$\"3:+++u7#e\"QFQ7$$!3P+++g9uBeF-$ \"30+++j![a!QFQ7$$!3A+++tT:niF-$\"3#)*****Hca9z$FQ7$$!3d*****>Sy9r'F-$ \"3#)*****47oPx$FQ7$$!3g*****RX\"GdrF-$\"3?+++0TG_PFQ7$$!3-+++R&\\4<$FQ7$$!3/+++[x*zf\"FQ$\"3<+++B. 'e8$FQ7$$!35+++P(y!y;FQ$\"3/+++G5;/JFQ7$$!3!******R&=gaFQ$\"3y*****\\!pH*)HFQ7$$!3$******\\tyo-#FQ$ \"3/+++CdDgHFQ7$$!3;+++1ub(3#FQ$\"37+++<N**GFQ7$$!3')*****R[>:?#FQ$\"3'*******oi8nGFQ7$$!37+++(R^^D#FQ$ \"33+++]AeLGFQ7$$!3!)*****R!Rv1BFQ$\"3#******f6/')z#FQ7$$!3y*****f?ikN #FQ$\"35++++++9.xQD FQ$\"3#)*****>zd/g#FQ7$$!3;+++z&*y!e#FQ$\"33+++P3%fb#FQ7$$!3!)*****4]< ;i#FQ$\"34+++GVv4DFQ7$$!3!******f3x8m#FQ$\"3/+++)))**=Y#FQ7$$!3))***** RZ(>+FFQ$\"3)******4p#R7CFQ7$$!3*******zx8#QFFQ$\"3)******>Cl7O#FQ7$$! 3;+++e`cvFFQ$\"3$******4yr&3BFQ7$$!3))******y?R7GFQ$\"39+++5GRaAFQ7$$! 35+++#=G)[GFQ$\"3++++O*Q))>#FQ7$$!3#)******oP*\\)GFQ$\"3')*****p'G0U@F Q7$$!3.+++=X)4#HFQ$\"31+++b#3U3#FQ7$$!3%)******>5'o&HFQ$\"30+++sU]D?FQ 7$$!3-+++P)RE*HFQ$\"3%******4fbh'>FQ7$$!32+++m&)GGIFQ$\"3!******4gyj!> FQ7$$!35+++.msjIFQ$\"3!******>,wj%=FQ7$$!3')*****R!>$))4$FQ$\"33+++=FK 'y\"FQ7$$!3%*******pCXLJFQ$\"33+++EbNE+++!H@FB$FQ$\"3++++'>,xa\" FQ7$$!3A+++scujKFQ$\"3))*****ze<')[\"FQ7$$!3#******f'4^$H$FQ$\"3%***** *\\>@(H9FQ7$$!3#)*****\\38>K$FQ$\"33+++wf&4P\"FQ7$$!3!******zep)[LFQ$ \"3'******pIcAJ\"FQ7$$!3-+++a\"=VP$FQ$\"35+++\\\"FQ7$$!3-+++^.`?MFQ$\"30+++C<%e8\"FQ7$$!3)******RD^7 W$FQ$\"35+++>rpw5FQ7$$!3')*****4=v.Y$FQ$\"3&******fzrs,\"FQ7$$!39+++b6 \"zZ$FQ$\"3=+++O*[]d*F-7$$!3-+++(RyQ\\$FQ$\"3!)*****4)**Rt*)F-7$$!3$** ****>x0$3NFQ$\"3w*****pqbsO)F-7$$!3$)******)[J7_$FQ$\"3))*****R1bhv(F- 7$$!3()******eHqKNFQ$\"3[+++#G*pRrF-7$$!3z******zmxUNFQ$\"3T+++(*HcpRF-7$$!3!)*****HNGZd$FQ$\"3/+++e%evJ$F-7$$!3))*****4$Q-yNFQ$\"3#*** ***4&e%3m#F-7$$!3A+++_p[!e$FQ$\"3++++QOw**>F-7$$!3')*****pU*=#e$FQ$\"3 ))*****\\F-7$F\\`m$!3#******4&e%3m#F-7$Fg_m$!3/+++e%evJ$F-7$Fb_m$!3)***** *HE(>pRF-7$F]_m$!35+++MMD:YF-7$Fh^m$!3)******zq1aD&F-7$Fc^m$!3W+++7!4& *)eF-7$F^^m$!3T+++(*Hcrpw5FQ7$Ff[m$!30+++C<%e8\" FQ7$Fa[m$!3!******R\\qZ>\"FQ7$F\\[m$!35+++\\@(H9FQ7$Fhil$!3))*****ze <')[\"FQ7$Fcil$!3++++'>,xa\"FQ7$F^il$!35+++IB,2;FQ7$Fihl$!3'******zFol m\"FQ7$Fdhl$!33+++EbNE,wj%=F Q7$Fegl$!3!******4gyj!>FQ7$F`gl$!3%******4fbh'>FQ7$F[gl$!30+++sU]D?FQ7 $Fffl$!31+++b#3U3#FQ7$Fafl$!3')*****p'G0U@FQ7$F\\fl$!3++++O*Q))>#FQ7$F gel$!39+++5GRaAFQ7$Fbel$!3$******4yr&3BFQ7$F]el$!3)******>Cl7O#FQ7$Fhd l$!3)******4p#R7CFQ7$Fcdl$!3/+++)))**=Y#FQ7$F^dl$!34+++GVv4DFQ7$Ficl$! 33+++P3%fb#FQ7$Fdcl$!3#)*****>zd/g#FQ7$F_cl$!3\")*****\\?9Lk#FQ7$Fjbl$ !31+++nx_%o#FQ7$Febl$!38+++QQ7CFFQ7$F`bl$!35+++N **GFQ7$Fg`l$!37+++<&\\4<$FQ7$Fj]l$ !3/+++%pS4@$FQ7$Fe]l$!3!)*****\\*y4dKFQ7$F`]l$!3.+++&GU&4LFQ7$F[]l$!3; +++)R'[mLFQ7$Ff\\l$!3@+++GWsCMFQ7$Fa\\l$!3%)*****RFo6[$FQ7$F\\\\l$!30+ ++3$[Q`$FQ7$Fg[l$!37+++HQ)=e$FQ7$Fb[l$!3%)*****f$*e]i$FQ7$F][l$!3!)*** ****GqMm$FQ7$Fhz$!3%******zHGtp$FQ7$Fcz$!3!*******)*G'os$FQ7$F^z$!3?++ +0TG_PFQ7$Fiy$!3#)*****47oPx$FQ7$Fdy$!3#)*****Hca9z$FQ7$F_y$!30+++j![a !QFQ7$Fjx$!3:+++u7#e\"QFQ7$Fex$!3-+++-!=E#QFQ7$F`x$!3!)*****p(f&e#QFQ7 $F[x$!3&)*****f5Hb#QFQ7$Ffw$!3.+++?#4;#QFQ7$Faw$!3;+++it/9QFQ7$F\\w$!3 *******fsuF!QFQ7$Fgv$!3/+++ALq(y$FQ7$Fbv$!3/+++5lsoPFQ7$F]v$!3/+++-&>d u$FQ7$Fhu$!3%******H(*R&=PFQ7$Fcu$!3%)*****f()Gqo$FQ7$F^u$!31+++[:,^OF Q7$Fit$!3@+++T!*H5OFQ7$Fdt$!31+++9%*okNFQ7$F_t$!3')*****4HoR^$FQ7$Fjs$ !3A+++*R2zX$FQ7$Fes$!3#)*****f%)eiR$FQ7$F`s$!3(*******e=uGLFQ7$F[s$!35 +++\"f<]D$FQ7$Ffr$!3-+++4tkuJFQ7$Far$!3)******z\\Jq3$FQ7$F\\r$!3/+++Pt L\"*HFQ7$Fgq$!3;+++(=_k)GFQ7$Fbq$!3')*****f@s5x#FQ7$F]q$!3-+++TK8WEFQ7 $Fhp$!3!******\\)Qp0DFQ7$Fcp$!3))*****f(\\pdBFQ7$F]p$!37+++b)oM?#FQ7$F ho$!3)*******e$3i/#FQ7$Fco$!31+++h\\(z)=FQ7$F]o$!3.+++P*Q(H$G'F-7$$ \"3)*******Qd9woF7$!3#)*****fg!R7ZF-7$F/$!38+++VFfTJF-7$F($!3++++Fjzq: F-F'F*F.F4F:F@FFFKFRFXFgnF\\oFboFgoF\\pFbpFgpF\\qFaqFfqF[rF`rFerFjrF_s FdsFisF^tFctFhtF]uFbuFguF\\vFavFfvF[wF`wFewFjwF_x-%&COLORG6&%$RGBGF($ \"#7!\"#$\"#QF_hn-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$F=!\"\"F(7%Ff hn7$$\"+#*4qH7F_p$\"+VFfTJFihnFjhn7%F^in7$$\"+Wd9woF-$\"+11R7ZFihnFjhn 7%Fdin7$$\"+Nj?)=\"!#;$\"+M%)>$G'FihnFjhn7%Fjin7$$\"+z'pI2\"!#:$\"+<# \\S&yFihnFjhn7%Fajn7$$\"+e&f)3kFdjn$\"+=y.D%*FihnFjhn7%Fhjn7$$\"+4g3rG !#9$\"+(RQ'*4\"!\"*Fjhn7%F^[o7$$\"+K9JS5!#8$\"+M8&oD\"Fd[oFjhn7%Ff[o7$ $\"+:#)=,KFi[o$\"+0@@99Fd[oFjhn7%F]\\o7$$\"+C)f&[')Fi[o$\"+IK\"=d\"Fd[ oFjhn7%Fc\\o7$$\"+/[')*4#!#7$\"+P*Q(HnE'4(F_^o$\" +TK8WEFd[oFjhn7%F__o7$$\"+Up:a#*F_^o$\"+;A2rFFd[oFjhn7%Fe_o7$$\"+*y-/6 \"Fihn$\"+(=_k)GFd[oFjhn7%F[`o7$$\"+\"pQNC\"Fihn$\"+PtL\"*HFd[oFjhn7%F a`o7$$\"+w9)\\J\"Fihn$\"+)\\Jq3$Fd[oFjhn7%Fg`o7$$\"+\"o*eA8Fihn$\"+4tk uJFd[oFjhn7%F]ao7$$\"+&Q/'o7Fihn$\"+\"f<]D$Fd[oFjhn7%Fcao7$$\"+)z5v:\" Fihn$\"+f=uGLFd[oFjhn7%Fiao7$$\"+.\")3[**F_^o$\"+Y)eiR$Fd[oFjhn7%F_bo7 $$\"+P<;jyF_^o$\"+*R2zX$Fd[oFjhn7%Febo7$$\"+fLmx`F_^o$\"+\"HoR^$Fd[oFj hn7%F[co7$$\"+5>wXDF_^o$\"+9%*okNFd[oFjhn7%Faco7$$!+LH,GeF\\]o$\"+T!*H 5OFd[oFjhn7%Fgco7$$!+mKFjRF_^o$\"+[:,^OFd[oFjhn7%F]do7$$!+pJ)fb(F_^o$ \"+w)Gqo$Fd[oFjhn7%Fcdo7$$!+%)RhK6Fihn$\"+t*R&=PFd[oFjhn7%Fido7$$!+^gM C:Fihn$\"+-&>du$Fd[oFjhn7%F_eo7$$!+3BFihn$\"+5lsoPFd[oFjhn7%Feeo7$$ !+I0$=M#Fihn$\"+ALq(y$Fd[oFjhn7%F[fo7$$!+c^PjFFihn$\"+EZx-QFd[oFjhn7%F afo7$$!+I\"R6>$Fihn$\"+it/9QFd[oFjhn7%Fgfo7$$!+&\\4<$Fd[oFjhn7%F[_ p7$$!+[x*zf\"Fd[o$\"+B.'e8$Fd[oFjhn7%Fa_p7$$!+P(y!y;Fd[o$\"+G5;/JFd[oF jhn7%Fg_p7$$!+a=gaFd[o$ \"+0pH*)HFd[oFjhn7%F_ap7$$!+N(yo-#Fd[o$\"+CdDgHFd[oFjhn7%Feap7$$!+1ub( 3#Fd[o$\"+<N**GFd[oFjhn7%Fabp7$ $!+%[>:?#Fd[o$\"+pi8nGFd[oFjhn7%Fgbp7$$!+(R^^D#Fd[o$\"+]AeLGFd[oFjhn7% F]cp7$$!+/Rv1BFd[o$\"+;Tg)z#Fd[oFjhn7%Fccp7$$!+1AYcBFd[o$\"++FFd[o$\"+\"p#R7CFd[oFjhn7%Fifp7$$!+yP@QFFd[o$\"+U_EhBFd[oFj hn7%F_gp7$$!+e`cvFFd[o$\"+\"yr&3BFd[oFjhn7%Fegp7$$!+z?R7GFd[o$\"+5GRaA Fd[oFjhn7%F[hp7$$!+#=G)[GFd[o$\"+O*Q))>#Fd[oFjhn7%Fahp7$$!+pP*\\)GFd[o $\"+nG0U@Fd[oFjhn7%Fghp7$$!+=X)4#HFd[o$\"+b#3U3#Fd[oFjhn7%F]ip7$$!+?5' o&HFd[o$\"+sU]D?Fd[oFjhn7%Fcip7$$!+P)RE*HFd[o$\"+\"fbh'>Fd[oFjhn7%Fiip 7$$!+m&)GGIFd[o$\"+,'yj!>Fd[oFjhn7%F_jp7$$!+.msjIFd[o$\"+7gPY=Fd[oFjhn 7%Fejp7$$!+/>$))4$Fd[o$\"+=FK'y\"Fd[oFjhn7%F[[q7$$!+qCXLJFd[o$\"+EbNE< Fd[oFjhn7%Fa[q7$$!+?,UnJFd[o$\"+y#olm\"Fd[oFjhn7%Fg[q7$$!+FRc+KFd[o$\" +IB,2;Fd[oFjhn7%F]\\q7$$!+!H@FB$Fd[o$\"+'>,xa\"Fd[oFjhn7%Fc\\q7$$!+scu jKFd[o$\"+)e<')[\"Fd[oFjhn7%Fi\\q7$$!+m4^$H$Fd[o$\"+&>@(H9Fd[oFjhn7%F_ ]q7$$!+&38>K$Fd[o$\"+wf&4P\"Fd[oFjhn7%Fe]q7$$!+)ep)[LFd[o$\"+2jD78Fd[o Fjhn7%F[^q7$$!+a\"=VP$Fd[o$\"+\\\"Fd[oFjhn7%Fg^q7$$!+^.`?MFd[o$\"+C<%e8\"Fd[oFjhn7%F]_q7$$!+a7DTM Fd[o$\"+>rpw5Fd[oFjhn7%Fc_q7$$!+\"=v.Y$Fd[o$\"+'zrs,\"Fd[oFjhn7%Fi_q7$ $!+b6\"zZ$Fd[o$\"+O*[]d*FihnFjhn7%F_`q7$$!+(RyQ\\$Fd[o$\"+\")**Rt*)Fih nFjhn7%Fe`q7$$!+sdI3NFd[o$\"+2dDn$)FihnFjhn7%F[aq7$$!+*[J7_$Fd[o$\"+k] :cxFihnFjhn7%Faaq7$$!+fHqKNFd[o$\"+#G*pRrFihnFjhn7%Fgaq7$$!+!owFa$Fd[o $\"+(*HcpRFihnFjhn7%Fecq7$$!+`$GZd$Fd[o$\"+e%evJ$FihnFjhn 7%F[dq7$$!+JQ-yNFd[o$\"+^e%3m#FihnFjhn7%Fadq7$$!+_p[!e$Fd[o$\"+QOw**>F ihnFjhn7%Fgdq7$$!+F%*=#e$Fd[o$\"+vT?N8FihnFjhn7%F]eq7$$!+Hr=$e$Fd[o$\" +QjB#o'F_^oFjhn7%Fceq7$$!+Cc^$e$Fd[oF(Fjhn7%Fieq7$Fdeq$!+QjB#o'F_^oFjh n7%F]fq7$F^eq$!+vT?N8FihnFjhn7%Fafq7$Fhdq$!+QOw**>FihnFjhn7%Fefq7$Fbdq $!+^e%3m#FihnFjhn7%Fifq7$F\\dq$!+e%evJ$FihnFjhn7%F]gq7$Ffcq$!+js>pRFih nFjhn7%Fagq7$F`cq$!+MMD:YFihnFjhn7%Fegq7$Fjbq$!+3nSb_FihnFjhn7%Figq7$F dbq$!+7!4&*)eFihnFjhn7%F]hq7$F^bq$!+(*Hcrpw5Fd[oFjhn7%F]jq7$F^_q$!+C<%e8\"Fd [oFjhn7%Fajq7$Fh^q$!+%\\qZ>\"Fd[oFjhn7%Fejq7$Fb^q$!+\\@(H9Fd[oFjhn7%Fe[r7$Fj\\q$!+)e<')[\"Fd[oFjhn7%Fi[r7$Fd\\q$!+'>,xa\"F d[oFjhn7%F]\\r7$F^\\q$!+IB,2;Fd[oFjhn7%Fa\\r7$Fh[q$!+y#olm\"Fd[oFjhn7% Fe\\r7$Fb[q$!+EbNEFd[oFjhn7%Fe]r7$Fjip$!+\"fbh'>Fd [oFjhn7%Fi]r7$Fdip$!+sU]D?Fd[oFjhn7%F]^r7$F^ip$!+b#3U3#Fd[oFjhn7%Fa^r7 $Fhhp$!+nG0U@Fd[oFjhn7%Fe^r7$Fbhp$!+O*Q))>#Fd[oFjhn7%Fi^r7$F\\hp$!+5GR aAFd[oFjhn7%F]_r7$Ffgp$!+\"yr&3BFd[oFjhn7%Fa_r7$F`gp$!+U_EhBFd[oFjhn7% Fe_r7$Fjfp$!+\"p#R7CFd[oFjhn7%Fi_r7$Fdfp$!+)))**=Y#Fd[oFjhn7%F]`r7$F^f p$!+GVv4DFd[oFjhn7%Fa`r7$Fhep$!+P3%fb#Fd[oFjhn7%Fe`r7$Fbep$!+#zd/g#Fd[ oFjhn7%Fi`r7$F\\ep$!+0UJVEFd[oFjhn7%F]ar7$Ffdp$!+nx_%o#Fd[oFjhn7%Faar7 $F`dp$!+QQ7CFFd[oFjhn7%Fear7$Fjcp$!+N**GFd[oFjhn7%Fibr7$F\\bp$!+<&\\4<$Fd[oF jhn7%F]er7$Ff^p$!+%pS4@$Fd[oFjhn7%Faer7$F`^p$!+&*y4dKFd[oFjhn7%Feer7$F j]p$!+&GU&4LFd[oFjhn7%Fier7$Fd]p$!+)R'[mLFd[oFjhn7%F]fr7$F^]p$!+GWsCMF d[oFjhn7%Fafr7$Fh\\p$!+u#o6[$Fd[oFjhn7%Fefr7$Fb\\p$!+3$[Q`$Fd[oFjhn7%F ifr7$F\\\\p$!+HQ)=e$Fd[oFjhn7%F]gr7$Ff[p$!+O*e]i$Fd[oFjhn7%Fagr7$F`[p$ !+!HqMm$Fd[oFjhn7%Fegr7$Fjjo$!+)HGtp$Fd[oFjhn7%Figr7$Fdjo$!+**G'os$Fd[ oFjhn7%F]hr7$F^jo$!+0TG_PFd[oFjhn7%Fahr7$Fhio$!+@\"oPx$Fd[oFjhn7%Fehr7 $Fbio$!+jXX\"z$Fd[oFjhn7%Fihr7$F\\io$!+j![a!QFd[oFjhn7%F]ir7$Ffho$!+u7 #e\"QFd[oFjhn7%Fair7$F`ho$!+-!=E#QFd[oFjhn7%Feir7$Fjgo$!+xf&e#QFd[oFjh n7%Fiir7$Fdgo$!+1\"Hb#QFd[oFjhn7%F]jr7$F^go$!+?#4;#QFd[oFjhn7%Fajr7$Fh fo$!+it/9QFd[oFjhn7%Fejr7$Fbfo$!+EZx-QFd[oFjhn7%Fijr7$F\\fo$!+ALq(y$Fd [oFjhn7%F][s7$Ffeo$!+5lsoPFd[oFjhn7%Fa[s7$F`eo$!+-&>du$Fd[oFjhn7%Fe[s7 $Fjdo$!+t*R&=PFd[oFjhn7%Fi[s7$Fddo$!+w)Gqo$Fd[oFjhn7%F]\\s7$F^do$!+[:, ^OFd[oFjhn7%Fa\\s7$Fhco$!+T!*H5OFd[oFjhn7%Fe\\s7$Fbco$!+9%*okNFd[oFjhn 7%Fi\\s7$F\\co$!+\"HoR^$Fd[oFjhn7%F]]s7$Ffbo$!+*R2zX$Fd[oFjhn7%Fa]s7$F `bo$!+Y)eiR$Fd[oFjhn7%Fe]s7$Fjao$!+f=uGLFd[oFjhn7%Fi]s7$Fdao$!+\"f<]D$ Fd[oFjhn7%F]^s7$F^ao$!+4tkuJFd[oFjhn7%Fa^s7$Fh`o$!+)\\Jq3$Fd[oFjhn7%Fe ^s7$Fb`o$!+PtL\"*HFd[oFjhn7%Fi^s7$F\\`o$!+(=_k)GFd[oFjhn7%F]_s7$Ff_o$! +;A2rFFd[oFjhn7%Fa_s7$F`_o$!+TK8WEFd[oFjhn7%Fe_s7$Fj^o$!+&)Qp0DFd[oFjh n7%Fi_s7$Fd^o$!+w\\pdBFd[oFjhn7%F]`s7$F]^o$!+b)oM?#Fd[oFjhn7%Fa`s7$Fg] o$!+f$3i/#Fd[oFjhn7%Fe`s7$Fa]o$!+h\\(z)=Fd[oFjhn7%Fi`s7$Fj\\o$!+P*Q(H< Fd[oFjhn7%F]as7$Fd\\o$!+IK\"=d\"Fd[oFjhn7%Faas7$F^\\o$!+0@@99Fd[oFjhn7 %Feas7$Fg[o$!+M8&oD\"Fd[oFjhn7%Fias7$F_[o$!+(RQ'*4\"Fd[oFjhn7%F]bs7$Fi jn$!+=y.D%*FihnFjhn7%Fabs7$Fbjn$!+<#\\S&yFihnFjhn7%Febs7$F[jn$!+M%)>$G 'FihnFjhn7%Fibs7$$\"+Rd9woF-$!+11R7ZFihnFjhn7%F]cs7$F_in$!+VFfTJFihnFj hn7%Fccs7$F($!+Fjzq:FihnFjhn7%FgcsF'FjhnFehnF]inFcinFiinF`jnFgjnF][oFe [oF\\\\oFb\\oFh\\oF_]oFe]oF[^oFb^oFh^oF^_oFd_oFj_oF``oFf`oF\\aoFbaoFha oF^boFdboFjboF`coFfcoF\\doFbdoFhdoF^eoFdeoFjeoF`foFffoF\\goFbgoFhgo-Fj gn6&F\\hnF($\"#BF_hn$\"#vF_hn-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3#)* *****Hv76GFQ$\"31+++[@:&[%FQ7$$\"3*******zw)y2GFQ$\"3S+++@@W\"\\%FQ7$$ \"3*)*****4)ye.GFQ$\"3++++td=(\\%FQ7$$\"3%)*****f)Gg)z#FQ$\"3W+++q`F-X FQ7$$\"3;+++iG#Hz#FQ$\"3>+++x7h1XFQ7$$\"31+++jzk'y#FQ$\"3P+++6D55XFQ7$ $\"39+++y&*))zFFQ$\"31+++'enE^%FQ7$$\"3A+++c0xsFFQ$\"3i*****>rNU^%FQ7$ $\"37+++4aUlFFQ$\"3&*******G%[Z^%FQ7$$\"3-+++\\.+eFFQ$\"3l******H<;9XF Q7$$\"33+++)*Hl]FFQ$\"3&)*****zs[C^%FQ7$$\"3,+++_:bVFFQ$\"3()*****z)Gg 4XFQ7$$\"3y*****z4tot#FQ$\"3/+++w+++=%>bs#FQ$\"3E+++*H*e%\\%FQ7$$\"3\"******\\I47s#FQ$ \"3')*****f]!p([%FQ7$$\"33+++kz.=FFQ$\"3o******=z1![%FQ7$$\"36+++9+:;F FQ$\"3/+++SQ\">Z%FQ7$$\"3?+++Wdl:FFQ$\"3q*****p+dMY%FQ7$$\"3!******4c> mr#FQ$\"3S+++nd&\\X%FQ7$$\"3#)*****f2]!>FFQ$\"3E+++7ooYWFQ7$$\"36+++V%FQ7$$\"3 0+++5wHMFFQ$\"3:+++KH)fU%FQ7$$\"3*******R?j9u#FQ$\"3j*****\\a87U%FQ7$$ \"3=+++i'z#\\FFQ$\"3R+++WOx[FHW%FQ7$$\"3\")*****fj!)H\"GFQ$\"31+++v7\")\\WFQ7$$\"3&)*****\\T1Z \"GFQ$\"39+++i9&pX%FQ7$$\"35+++dEO:GFQ$\"3g*****>(o>kWFQ7$$\"3\"****** *3L'\\\"GFQ$\"3m*****Rv-9Z%FQ7$$\"3!)*****zRPN\"GFQ$\"3I+++T8VyWFQFhds Fign-Fchn6L7%7$$\"+Iv76GFd[o$\"+[@:&[%Fd[o7$$\"+o()y2GFd[o$\"+@@W\"\\% Fd[o7$$\"$w#F_hn$\"$Y%F_hn7%Fhat7$$\"+\")ye.GFd[o$\"+td=(\\%Fd[oF]bt7% Fcbt7$$\"+')Gg)z#Fd[o$\"+q`F-XFd[oF]bt7%Fibt7$$\"+iG#Hz#Fd[o$\"+x7h1XF d[oF]bt7%F_ct7$$\"+jzk'y#Fd[o$\"+6D55XFd[oF]bt7%Fect7$$\"+y&*))zFFd[o$ \"+'enE^%Fd[oF]bt7%F[dt7$$\"+c0xsFFd[o$\"+7dB9XFd[oF]bt7%Fadt7$$\"+4aU lFFd[o$\"+H%[Z^%Fd[oF]bt7%Fgdt7$$\"+\\.+eFFd[o$\"+I<;9XFd[oF]bt7%F]et7 $$\"+)*Hl]FFd[o$\"+G([C^%Fd[oF]bt7%Fcet7$$\"+_:bVFFd[o$\"+))Gg4XFd[oF] bt7%Fiet7$$\"+)4tot#Fd[o$\"+wbs#Fd[o$\"+*H*e%\\%Fd[oF]bt7%F[gt7$$\"+0$47s# Fd[o$\"+10p([%Fd[oF]bt7%Fagt7$$\"+kz.=FFd[o$\"+>z1![%Fd[oF]bt7%Fggt7$$ \"+9+:;FFd[o$\"+SQ\">Z%Fd[oF]bt7%F]ht7$$\"+Wdl:FFd[o$\"+2qXjWFd[oF]bt7 %Fcht7$$\"+h&>mr#Fd[o$\"+nd&\\X%Fd[oF]bt7%Fiht7$$\"+w+0>FFd[o$\"+7ooYW Fd[oF]bt7%F_it7$$\"+V%Fd[oF]bt7%F[jt7$$\"+5wHMFFd[o$\"+KH)fU%Fd[oF]bt7%Fajt7$$\"+/ KYTFFd[o$\"+XN@@WFd[oF]bt7%Fgjt7$$\"+i'z#\\FFd[o$\"+WOxkWFd[oF]bt7%Fe_u7$$\"+4L' \\\"GFd[o$\"+aFSrWFd[oF]bt7%F[`u7$$\"+)RPN\"GFd[o$\"+T8VyWFd[oF]bt7%Fa `uFcatF]btF[dsFads-F$6$7K7$Fids$!31+++[@:&[%FQ7$F\\at$!3I+++T8VyWFQ7$F g`t$!3m*****Rv-9Z%FQ7$Fb`t$!3g*****>(o>kWFQ7$F]`t$!39+++i9&pX%FQ7$Fh_t $!31+++v7\")\\WFQ7$Fc_t$!3)******>[FHW%FQ7$F^_t$!3$)*****flckV%FQ7$Fi^ t$!3k*****f4f0V%FQ7$Fd^t$!3M+++RiRDWFQ7$F_^t$!30+++,p7@WFQ7$Fj]t$!3E++ +\"3-zT%FQ7$Fe]t$!3++++r\"eeT%FQ7$F`]t$!3N+++0'3^T%FQ7$F[]t$!3q*****4H MdT%FQ7$Ff\\t$!3R+++WOxV%FQ7$Fb[t$!3'*******)**H*QWFQ7$F][t$!3E+++7ooYW FQ7$Fhjs$!3S+++nd&\\X%FQ7$Fcjs$!3q*****p+dMY%FQ7$F^js$!3/+++SQ\">Z%FQ7 $Fiis$!3o******=z1![%FQ7$Fdis$!3')*****f]!p([%FQ7$F_is$!3E+++*H*e%\\%F Q7$Fjhs$!3&)******y6h+XFQ7$Fehs$!3/+++wrNU^%FQ7$Fgfs$!31+++'enE^%FQ7$Fbfs$!3P+++6D55XFQ7$F] fs$!3>+++x7h1XFQ7$Fhes$!3W+++q`F-XFQ7$Fces$!3++++td=(\\%FQ7$F^es$!3S++ +@@W\"\\%FQFj`uFign-Fchn6L7%7$Fdat$!+[@:&[%Fd[o7$Fb`u$!+T8VyWFd[o7$F^b t$!$Y%F_hn7%Fhhu7$F\\`u$!+aFSrWFd[oF[iu7%F_iu7$Ff_u$!+so>kWFd[oF[iu7%F ciu7$F`_u$!+i9&pX%Fd[oF[iu7%Fgiu7$Fj^u$!+v7\")\\WFd[oF[iu7%F[ju7$Fd^u$ !+#[FHW%Fd[oF[iu7%F_ju7$F^^u$!+cmXOWFd[oF[iu7%Fcju7$Fh]u$!+'4f0V%Fd[oF [iu7%Fgju7$Fb]u$!+RiRDWFd[oF[iu7%F[[v7$F\\]u$!+,p7@WFd[oF[iu7%F_[v7$Ff \\u$!+\"3-zT%Fd[oF[iu7%Fc[v7$F`\\u$!+r\"eeT%Fd[oF[iu7%Fg[v7$Fj[u$!+0'3 ^T%Fd[oF[iu7%F[\\v7$Fd[u$!+\"HMdT%Fd[oF[iu7%F_\\v7$F^[u$!+WOxV%Fd[oF[iu7%F_]v7$Ffit$!+***H*QWFd[oF[iu7%Fc]v7$F`it$!+7ooY WFd[oF[iu7%Fg]v7$Fjht$!+nd&\\X%Fd[oF[iu7%F[^v7$Fdht$!+2qXjWFd[oF[iu7%F _^v7$F^ht$!+SQ\">Z%Fd[oF[iu7%Fc^v7$Fhgt$!+>z1![%Fd[oF[iu7%Fg^v7$Fbgt$! +10p([%Fd[oF[iu7%F[_v7$F\\gt$!+*H*e%\\%Fd[oF[iu7%F__v7$Ffft$!+z6h+XFd[ oF[iu7%Fc_v7$F`ft$!+w%FQF(7$$\"3%************** =$FQF(-%'COLOURG6&F\\hnF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++ !>&FQ7$F($\"3Q++++++!>&FQF`cvFccv-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELV ETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fedv6#%(DEFAULTG-%(SCALING G6#%,CONSTRAINEDG-%%VIEWG6$;$!$>%F_hn$\"$>$F_hn;$!$>&F_hn$\"$>&F_hn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stabi lity interval for the embedded order 6 scheme is (approximately) " } {XPPEDIT 18 0 "[-3.5835, 0];" "6#7$,$-%&FloatG6$\"&Ne$!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two schemes can be displaye d in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "p lots[display]([p_||(1..6),p||(1..7)],view=[-5.39..3.19,-6.59..6.59],fo nt=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scal ing=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 501 606 606 {PLOTDATA 2 "65-%'CURVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3'******>*4 qH7!#F$\"38+++VFfTJF-7$$\"3!******RuXh(o!#E$\"3#)*****fg!R7ZF-7$$\"3'* *****\\L1#)=\"!#C$\"3A+++M%)>$G'F-7$$\"3)*******y'pI2\"!#B$\"3!)*****p @\\S&yF-7$$\"3z*****zbf)3kFC$\"3^+++=y.D%*F-7$$\"3))******3g3rG!#A$\"3 5+++(RQ'*4\"!#<7$$\"3.+++K9JS5!#@$\"3/+++M8&oD\"FQ7$$\"36+++:#)=,KFU$ \"30+++0@@99FQ7$$\"3')*****R#)f&[')FU$\"31+++IK\"=d\"FQ7$$\"3;+++/[')* 4#!#?$\"3.+++P*Q(H$\"37+++b)oM?#FQ7$$\" 3&)*****Ry,L7$F_p$\"3))*****f(\\pdBFQ7$$\"3!)*****4\\;l%\\F_p$\"3!**** **\\)Qp0DFQ7$$\"3]******=nE'4(F_p$\"3-+++TK8WEFQ7$$\"3s*****>%p:a#*F_p $\"3')*****f@s5x#FQ7$$\"3(*******)y-/6\"F-$\"3;+++(=_k)GFQ7$$\"3++++\" pQNC\"F-$\"3/+++PtL\"*HFQ7$$\"3/+++w9)\\J\"F-$\"3)******z\\Jq3$FQ7$$\" 38+++\"o*eA8F-$\"3-+++4tkuJFQ7$$\"31+++&Q/'o7F-$\"35+++\"f<]D$FQ7$$\"3 /+++)z5v:\"F-$\"3(*******e=uGLFQ7$$\"3e*****H5)3[**F_p$\"3#)*****f%)ei R$FQ7$$\"3Q+++P<;jyF_p$\"3A+++*R2zX$FQ7$$\"3w******eLmx`F_p$\"3')***** 4HoR^$FQ7$$\"3))******4>wXDF_p$\"31+++9%*okNFQ7$$!3y*****H$H,GeF_o$\"3 @+++T!*H5OFQ7$$!3I+++mKFjRF_p$\"31+++[:,^OFQ7$$!3c******oJ)fb(F_p$\"3% )*****f()Gqo$FQ7$$!3/+++%)RhK6F-$\"3%******H(*R&=PFQ7$$!34+++^gMC:F-$ \"3/+++-&>du$FQ7$$!3*)*****zIs\"G>F-$\"3/+++5lsoPFQ7$$!3-+++I0$=M#F-$ \"3/+++ALq(y$FQ7$$!3G+++c^PjFF-$\"3*******fsuF!QFQ7$$!3E+++I\"R6>$F-$ \"3;+++it/9QFQ7$$!3*)*****pJ-Pi$F-$\"3.+++?#4;#QFQ7$$!39+++ms')fSF-$\" 3&)*****f5Hb#QFQ7$$!3!)*****\\+V')\\%F-$\"3!)*****p(f&e#QFQ7$$!3$)**** *HnP#R\\F-$\"3-+++-!=E#QFQ7$$!3]*****fOh5Q&F-$\"3:+++u7#e\"QFQ7$$!3P++ +g9uBeF-$\"30+++j![a!QFQ7$$!3A+++tT:niF-$\"3#)*****Hca9z$FQ7$$!3d***** >Sy9r'F-$\"3#)*****47oPx$FQ7$$!3g*****RX\"GdrF-$\"3?+++0TG_PFQ7$$!3-++ +R&\\4<$FQ7$$!3/+++[x*zf\"F Q$\"3<+++B.'e8$FQ7$$!35+++P(y!y;FQ$\"3/+++G5;/JFQ7$$!3!******R&=gaFQ$\"3y*****\\!pH*)HFQ7$$!3$**** **\\tyo-#FQ$\"3/+++CdDgHFQ7$$!3;+++1ub(3#FQ$\"37+++<N**GFQ7$$!3')*****R[>:?#FQ$\"3'*******oi8nGFQ7$$!37 +++(R^^D#FQ$\"33+++]AeLGFQ7$$!3!)*****R!Rv1BFQ$\"3#******f6/')z#FQ7$$! 3y*****f?ikN#FQ$\"35++++++9.xQDFQ$\"3#)*****>zd/g#FQ7$$!3;+++z&*y!e#FQ$\"33+++P3%fb#FQ7$$ !3!)*****4]<;i#FQ$\"34+++GVv4DFQ7$$!3!******f3x8m#FQ$\"3/+++)))**=Y#FQ 7$$!3))*****RZ(>+FFQ$\"3)******4p#R7CFQ7$$!3*******zx8#QFFQ$\"3)****** >Cl7O#FQ7$$!3;+++e`cvFFQ$\"3$******4yr&3BFQ7$$!3))******y?R7GFQ$\"39++ +5GRaAFQ7$$!35+++#=G)[GFQ$\"3++++O*Q))>#FQ7$$!3#)******oP*\\)GFQ$\"3') *****p'G0U@FQ7$$!3.+++=X)4#HFQ$\"31+++b#3U3#FQ7$$!3%)******>5'o&HFQ$\" 30+++sU]D?FQ7$$!3-+++P)RE*HFQ$\"3%******4fbh'>FQ7$$!32+++m&)GGIFQ$\"3! ******4gyj!>FQ7$$!35+++.msjIFQ$\"3!******>,wj%=FQ7$$!3')*****R!>$))4$F Q$\"33+++=FK'y\"FQ7$$!3%*******pCXLJFQ$\"33+++EbNE+++!H@FB$FQ$\" 3++++'>,xa\"FQ7$$!3A+++scujKFQ$\"3))*****ze<')[\"FQ7$$!3#******f'4^$H$ FQ$\"3%******\\>@(H9FQ7$$!3#)*****\\38>K$FQ$\"33+++wf&4P\"FQ7$$!3!**** **zep)[LFQ$\"3'******pIcAJ\"FQ7$$!3-+++a\"=VP$FQ$\"35+++\\\"FQ7$$!3-+++^.`?MFQ$\"30+++C<%e8\"FQ7$$!3 )******RD^7W$FQ$\"35+++>rpw5FQ7$$!3')*****4=v.Y$FQ$\"3&******fzrs,\"FQ 7$$!39+++b6\"zZ$FQ$\"3=+++O*[]d*F-7$$!3-+++(RyQ\\$FQ$\"3!)*****4)**Rt* )F-7$$!3$******>x0$3NFQ$\"3w*****pqbsO)F-7$$!3$)******)[J7_$FQ$\"3))** ***R1bhv(F-7$$!3()******eHqKNFQ$\"3[+++#G*pRrF-7$$!3z******zmxUNFQ$\"3 T+++(*HcpRF-7$$!3!)*****HNGZd$FQ$\"3/+++e%evJ$F-7$$!3))*****4$Q- yNFQ$\"3#******4&e%3m#F-7$$!3A+++_p[!e$FQ$\"3++++QOw**>F-7$$!3')*****p U*=#e$FQ$\"3))*****\\F-7$F\\`m$!3#******4&e%3m#F-7$Fg_m$!3/+++e%evJ$F-7$F b_m$!3)******HE(>pRF-7$F]_m$!35+++MMD:YF-7$Fh^m$!3)******zq1aD&F-7$Fc^ m$!3W+++7!4&*)eF-7$F^^m$!3T+++(*Hcrpw5FQ7$Ff[m$! 30+++C<%e8\"FQ7$Fa[m$!3!******R\\qZ>\"FQ7$F\\[m$!35+++\\@(H9FQ7$Fhil $!3))*****ze<')[\"FQ7$Fcil$!3++++'>,xa\"FQ7$F^il$!35+++IB,2;FQ7$Fihl$! 3'******zFolm\"FQ7$Fdhl$!33+++EbNE,wj%=FQ7$Fegl$!3!******4gyj!>FQ7$F`gl$!3%******4fbh'>FQ7$F[gl$!3 0+++sU]D?FQ7$Fffl$!31+++b#3U3#FQ7$Fafl$!3')*****p'G0U@FQ7$F\\fl$!3++++ O*Q))>#FQ7$Fgel$!39+++5GRaAFQ7$Fbel$!3$******4yr&3BFQ7$F]el$!3)******> Cl7O#FQ7$Fhdl$!3)******4p#R7CFQ7$Fcdl$!3/+++)))**=Y#FQ7$F^dl$!34+++GVv 4DFQ7$Ficl$!33+++P3%fb#FQ7$Fdcl$!3#)*****>zd/g#FQ7$F_cl$!3\")*****\\?9 Lk#FQ7$Fjbl$!31+++nx_%o#FQ7$Febl$!38+++QQ7CFFQ7$F`bl$!35+++N**GFQ7$Fg`l$!37+++<&\\ 4<$FQ7$Fj]l$!3/+++%pS4@$FQ7$Fe]l$!3!)*****\\*y4dKFQ7$F`]l$!3.+++&GU&4L FQ7$F[]l$!3;+++)R'[mLFQ7$Ff\\l$!3@+++GWsCMFQ7$Fa\\l$!3%)*****RFo6[$FQ7 $F\\\\l$!30+++3$[Q`$FQ7$Fg[l$!37+++HQ)=e$FQ7$Fb[l$!3%)*****f$*e]i$FQ7$ F][l$!3!)*******GqMm$FQ7$Fhz$!3%******zHGtp$FQ7$Fcz$!3!*******)*G'os$F Q7$F^z$!3?+++0TG_PFQ7$Fiy$!3#)*****47oPx$FQ7$Fdy$!3#)*****Hca9z$FQ7$F_ y$!30+++j![a!QFQ7$Fjx$!3:+++u7#e\"QFQ7$Fex$!3-+++-!=E#QFQ7$F`x$!3!)*** **p(f&e#QFQ7$F[x$!3&)*****f5Hb#QFQ7$Ffw$!3.+++?#4;#QFQ7$Faw$!3;+++it/9 QFQ7$F\\w$!3*******fsuF!QFQ7$Fgv$!3/+++ALq(y$FQ7$Fbv$!3/+++5lsoPFQ7$F] v$!3/+++-&>du$FQ7$Fhu$!3%******H(*R&=PFQ7$Fcu$!3%)*****f()Gqo$FQ7$F^u$ !31+++[:,^OFQ7$Fit$!3@+++T!*H5OFQ7$Fdt$!31+++9%*okNFQ7$F_t$!3')*****4H oR^$FQ7$Fjs$!3A+++*R2zX$FQ7$Fes$!3#)*****f%)eiR$FQ7$F`s$!3(*******e=uG LFQ7$F[s$!35+++\"f<]D$FQ7$Ffr$!3-+++4tkuJFQ7$Far$!3)******z\\Jq3$FQ7$F \\r$!3/+++PtL\"*HFQ7$Fgq$!3;+++(=_k)GFQ7$Fbq$!3')*****f@s5x#FQ7$F]q$!3 -+++TK8WEFQ7$Fhp$!3!******\\)Qp0DFQ7$Fcp$!3))*****f(\\pdBFQ7$F]p$!37++ +b)oM?#FQ7$Fho$!3)*******e$3i/#FQ7$Fco$!31+++h\\(z)=FQ7$F]o$!3.+++P*Q( H$G'F-7$$\"3)*******Qd9woF7$!3#)*****fg!R7ZF-7$F/$!38+++VFfTJF-7$F($! 3++++Fjzq:F-F'F*F.F4F:F@FFFKFRFXFgnF\\oFboFgoF\\pFbpFgpF\\qFaqFfqF[rF` rFerFjrF_sFdsFisF^tFctFhtF]uFbuFguF\\vFavFfvF[wF`wFewFjwF_x-%&COLORG6& %$RGBGF($\"#7!\"#$\"#QF_hn-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$F=! \"\"F(7%Ffhn7$$\"+#*4qH7F_p$\"+VFfTJFihnFjhn7%F^in7$$\"+Wd9woF-$\"+11R 7ZFihnFjhn7%Fdin7$$\"+Nj?)=\"!#;$\"+M%)>$G'FihnFjhn7%Fjin7$$\"+z'pI2\" !#:$\"+<#\\S&yFihnFjhn7%Fajn7$$\"+e&f)3kFdjn$\"+=y.D%*FihnFjhn7%Fhjn7$ $\"+4g3rG!#9$\"+(RQ'*4\"!\"*Fjhn7%F^[o7$$\"+K9JS5!#8$\"+M8&oD\"Fd[oFjh n7%Ff[o7$$\"+:#)=,KFi[o$\"+0@@99Fd[oFjhn7%F]\\o7$$\"+C)f&[')Fi[o$\"+IK \"=d\"Fd[oFjhn7%Fc\\o7$$\"+/[')*4#!#7$\"+P*Q(HnE'4 (F_^o$\"+TK8WEFd[oFjhn7%F__o7$$\"+Up:a#*F_^o$\"+;A2rFFd[oFjhn7%Fe_o7$$ \"+*y-/6\"Fihn$\"+(=_k)GFd[oFjhn7%F[`o7$$\"+\"pQNC\"Fihn$\"+PtL\"*HFd[ oFjhn7%Fa`o7$$\"+w9)\\J\"Fihn$\"+)\\Jq3$Fd[oFjhn7%Fg`o7$$\"+\"o*eA8Fih n$\"+4tkuJFd[oFjhn7%F]ao7$$\"+&Q/'o7Fihn$\"+\"f<]D$Fd[oFjhn7%Fcao7$$\" +)z5v:\"Fihn$\"+f=uGLFd[oFjhn7%Fiao7$$\"+.\")3[**F_^o$\"+Y)eiR$Fd[oFjh n7%F_bo7$$\"+P<;jyF_^o$\"+*R2zX$Fd[oFjhn7%Febo7$$\"+fLmx`F_^o$\"+\"HoR ^$Fd[oFjhn7%F[co7$$\"+5>wXDF_^o$\"+9%*okNFd[oFjhn7%Faco7$$!+LH,GeF\\]o $\"+T!*H5OFd[oFjhn7%Fgco7$$!+mKFjRF_^o$\"+[:,^OFd[oFjhn7%F]do7$$!+pJ)f b(F_^o$\"+w)Gqo$Fd[oFjhn7%Fcdo7$$!+%)RhK6Fihn$\"+t*R&=PFd[oFjhn7%Fido7 $$!+^gMC:Fihn$\"+-&>du$Fd[oFjhn7%F_eo7$$!+3BFihn$\"+5lsoPFd[oFjhn7% Feeo7$$!+I0$=M#Fihn$\"+ALq(y$Fd[oFjhn7%F[fo7$$!+c^PjFFihn$\"+EZx-QFd[o Fjhn7%Fafo7$$!+I\"R6>$Fihn$\"+it/9QFd[oFjhn7%Fgfo7$$!+&\\4<$Fd[oFjh n7%F[_p7$$!+[x*zf\"Fd[o$\"+B.'e8$Fd[oFjhn7%Fa_p7$$!+P(y!y;Fd[o$\"+G5;/ JFd[oFjhn7%Fg_p7$$!+a=gaFd[o$\"+0pH*)HFd[oFjhn7%F_ap7$$!+N(yo-#Fd[o$\"+CdDgHFd[oFjhn7%Feap7$ $!+1ub(3#Fd[o$\"+<N**GFd[oFjhn7 %Fabp7$$!+%[>:?#Fd[o$\"+pi8nGFd[oFjhn7%Fgbp7$$!+(R^^D#Fd[o$\"+]AeLGFd[ oFjhn7%F]cp7$$!+/Rv1BFd[o$\"+;Tg)z#Fd[oFjhn7%Fccp7$$!+1AYcBFd[o$\"++FFd[o$\"+\"p#R7CFd[oFjhn7%Fifp7$$!+yP@QFFd[o$\"+U_E hBFd[oFjhn7%F_gp7$$!+e`cvFFd[o$\"+\"yr&3BFd[oFjhn7%Fegp7$$!+z?R7GFd[o$ \"+5GRaAFd[oFjhn7%F[hp7$$!+#=G)[GFd[o$\"+O*Q))>#Fd[oFjhn7%Fahp7$$!+pP* \\)GFd[o$\"+nG0U@Fd[oFjhn7%Fghp7$$!+=X)4#HFd[o$\"+b#3U3#Fd[oFjhn7%F]ip 7$$!+?5'o&HFd[o$\"+sU]D?Fd[oFjhn7%Fcip7$$!+P)RE*HFd[o$\"+\"fbh'>Fd[oFj hn7%Fiip7$$!+m&)GGIFd[o$\"+,'yj!>Fd[oFjhn7%F_jp7$$!+.msjIFd[o$\"+7gPY= Fd[oFjhn7%Fejp7$$!+/>$))4$Fd[o$\"+=FK'y\"Fd[oFjhn7%F[[q7$$!+qCXLJFd[o$ \"+EbNE,xa\"Fd[oFjhn7%Fc\\q 7$$!+scujKFd[o$\"+)e<')[\"Fd[oFjhn7%Fi\\q7$$!+m4^$H$Fd[o$\"+&>@(H9Fd[o Fjhn7%F_]q7$$!+&38>K$Fd[o$\"+wf&4P\"Fd[oFjhn7%Fe]q7$$!+)ep)[LFd[o$\"+2 jD78Fd[oFjhn7%F[^q7$$!+a\"=VP$Fd[o$\"+\\\"Fd[oFjhn7%Fg^q7$$!+^.`?MFd[o$\"+C<%e8\"Fd[oFjhn7%F]_q7$ $!+a7DTMFd[o$\"+>rpw5Fd[oFjhn7%Fc_q7$$!+\"=v.Y$Fd[o$\"+'zrs,\"Fd[oFjhn 7%Fi_q7$$!+b6\"zZ$Fd[o$\"+O*[]d*FihnFjhn7%F_`q7$$!+(RyQ\\$Fd[o$\"+\")* *Rt*)FihnFjhn7%Fe`q7$$!+sdI3NFd[o$\"+2dDn$)FihnFjhn7%F[aq7$$!+*[J7_$Fd [o$\"+k]:cxFihnFjhn7%Faaq7$$!+fHqKNFd[o$\"+#G*pRrFihnFjhn7%Fgaq7$$!+!o wFa$Fd[o$\"+(*HcpRFihnFjhn7%Fecq7$$!+`$GZd$Fd[o$\"+e%evJ$ FihnFjhn7%F[dq7$$!+JQ-yNFd[o$\"+^e%3m#FihnFjhn7%Fadq7$$!+_p[!e$Fd[o$\" +QOw**>FihnFjhn7%Fgdq7$$!+F%*=#e$Fd[o$\"+vT?N8FihnFjhn7%F]eq7$$!+Hr=$e $Fd[o$\"+QjB#o'F_^oFjhn7%Fceq7$$!+Cc^$e$Fd[oF(Fjhn7%Fieq7$Fdeq$!+QjB#o 'F_^oFjhn7%F]fq7$F^eq$!+vT?N8FihnFjhn7%Fafq7$Fhdq$!+QOw**>FihnFjhn7%Fe fq7$Fbdq$!+^e%3m#FihnFjhn7%Fifq7$F\\dq$!+e%evJ$FihnFjhn7%F]gq7$Ffcq$!+ js>pRFihnFjhn7%Fagq7$F`cq$!+MMD:YFihnFjhn7%Fegq7$Fjbq$!+3nSb_FihnFjhn7 %Figq7$Fdbq$!+7!4&*)eFihnFjhn7%F]hq7$F^bq$!+(*Hcrpw5Fd[oFjhn7%F]jq7$F^_q$!+C <%e8\"Fd[oFjhn7%Fajq7$Fh^q$!+%\\qZ>\"Fd[oFjhn7%Fejq7$Fb^q$!+\\@(H9Fd[oFjhn7%Fe[r7$Fj\\q$!+)e<')[\"Fd[oFjhn7%Fi[r7$Fd\\q$!+ '>,xa\"Fd[oFjhn7%F]\\r7$F^\\q$!+IB,2;Fd[oFjhn7%Fa\\r7$Fh[q$!+y#olm\"Fd [oFjhn7%Fe\\r7$Fb[q$!+EbNEFd[oFjhn7%Fe]r7$Fjip$!+ \"fbh'>Fd[oFjhn7%Fi]r7$Fdip$!+sU]D?Fd[oFjhn7%F]^r7$F^ip$!+b#3U3#Fd[oFj hn7%Fa^r7$Fhhp$!+nG0U@Fd[oFjhn7%Fe^r7$Fbhp$!+O*Q))>#Fd[oFjhn7%Fi^r7$F \\hp$!+5GRaAFd[oFjhn7%F]_r7$Ffgp$!+\"yr&3BFd[oFjhn7%Fa_r7$F`gp$!+U_EhB Fd[oFjhn7%Fe_r7$Fjfp$!+\"p#R7CFd[oFjhn7%Fi_r7$Fdfp$!+)))**=Y#Fd[oFjhn7 %F]`r7$F^fp$!+GVv4DFd[oFjhn7%Fa`r7$Fhep$!+P3%fb#Fd[oFjhn7%Fe`r7$Fbep$! +#zd/g#Fd[oFjhn7%Fi`r7$F\\ep$!+0UJVEFd[oFjhn7%F]ar7$Ffdp$!+nx_%o#Fd[oF jhn7%Faar7$F`dp$!+QQ7CFFd[oFjhn7%Fear7$Fjcp$!+N**GFd[oFjhn7%Fibr7$F\\bp$!+<& \\4<$Fd[oFjhn7%F]er7$Ff^p$!+%pS4@$Fd[oFjhn7%Faer7$F`^p$!+&*y4dKFd[oFjh n7%Feer7$Fj]p$!+&GU&4LFd[oFjhn7%Fier7$Fd]p$!+)R'[mLFd[oFjhn7%F]fr7$F^] p$!+GWsCMFd[oFjhn7%Fafr7$Fh\\p$!+u#o6[$Fd[oFjhn7%Fefr7$Fb\\p$!+3$[Q`$F d[oFjhn7%Fifr7$F\\\\p$!+HQ)=e$Fd[oFjhn7%F]gr7$Ff[p$!+O*e]i$Fd[oFjhn7%F agr7$F`[p$!+!HqMm$Fd[oFjhn7%Fegr7$Fjjo$!+)HGtp$Fd[oFjhn7%Figr7$Fdjo$!+ **G'os$Fd[oFjhn7%F]hr7$F^jo$!+0TG_PFd[oFjhn7%Fahr7$Fhio$!+@\"oPx$Fd[oF jhn7%Fehr7$Fbio$!+jXX\"z$Fd[oFjhn7%Fihr7$F\\io$!+j![a!QFd[oFjhn7%F]ir7 $Ffho$!+u7#e\"QFd[oFjhn7%Fair7$F`ho$!+-!=E#QFd[oFjhn7%Feir7$Fjgo$!+xf& e#QFd[oFjhn7%Fiir7$Fdgo$!+1\"Hb#QFd[oFjhn7%F]jr7$F^go$!+?#4;#QFd[oFjhn 7%Fajr7$Fhfo$!+it/9QFd[oFjhn7%Fejr7$Fbfo$!+EZx-QFd[oFjhn7%Fijr7$F\\fo$ !+ALq(y$Fd[oFjhn7%F][s7$Ffeo$!+5lsoPFd[oFjhn7%Fa[s7$F`eo$!+-&>du$Fd[oF jhn7%Fe[s7$Fjdo$!+t*R&=PFd[oFjhn7%Fi[s7$Fddo$!+w)Gqo$Fd[oFjhn7%F]\\s7$ F^do$!+[:,^OFd[oFjhn7%Fa\\s7$Fhco$!+T!*H5OFd[oFjhn7%Fe\\s7$Fbco$!+9%*o kNFd[oFjhn7%Fi\\s7$F\\co$!+\"HoR^$Fd[oFjhn7%F]]s7$Ffbo$!+*R2zX$Fd[oFjh n7%Fa]s7$F`bo$!+Y)eiR$Fd[oFjhn7%Fe]s7$Fjao$!+f=uGLFd[oFjhn7%Fi]s7$Fdao $!+\"f<]D$Fd[oFjhn7%F]^s7$F^ao$!+4tkuJFd[oFjhn7%Fa^s7$Fh`o$!+)\\Jq3$Fd [oFjhn7%Fe^s7$Fb`o$!+PtL\"*HFd[oFjhn7%Fi^s7$F\\`o$!+(=_k)GFd[oFjhn7%F] _s7$Ff_o$!+;A2rFFd[oFjhn7%Fa_s7$F`_o$!+TK8WEFd[oFjhn7%Fe_s7$Fj^o$!+&)Q p0DFd[oFjhn7%Fi_s7$Fd^o$!+w\\pdBFd[oFjhn7%F]`s7$F]^o$!+b)oM?#Fd[oFjhn7 %Fa`s7$Fg]o$!+f$3i/#Fd[oFjhn7%Fe`s7$Fa]o$!+h\\(z)=Fd[oFjhn7%Fi`s7$Fj\\ o$!+P*Q(H$G'FihnFjhn7%Fibs7$$\"+Rd9woF-$!+11R7ZFihnFjhn7%F]cs7$F_in$!+V FfTJFihnFjhn7%Fccs7$F($!+Fjzq:FihnFjhn7%FgcsF'FjhnFehnF]inFcinFiinF`jn FgjnF][oFe[oF\\\\oFb\\oFh\\oF_]oFe]oF[^oFb^oFh^oF^_oFd_oFj_oF``oFf`oF \\aoFbaoFhaoF^boFdboFjboF`coFfcoF\\doFbdoFhdoF^eoFdeoFjeoF`foFffoF\\go FbgoFhgo-Fjgn6&F\\hnF($\"#BF_hn$\"#vF_hn-%&STYLEG6#%,PATCHNOGRIDG-F$6$ 7K7$$\"3#)******Hv76GFQ$\"31+++[@:&[%FQ7$$\"3*******zw)y2GFQ$\"3S+++@@ W\"\\%FQ7$$\"3*)*****4)ye.GFQ$\"3++++td=(\\%FQ7$$\"3%)*****f)Gg)z#FQ$ \"3W+++q`F-XFQ7$$\"3;+++iG#Hz#FQ$\"3>+++x7h1XFQ7$$\"31+++jzk'y#FQ$\"3P +++6D55XFQ7$$\"39+++y&*))zFFQ$\"31+++'enE^%FQ7$$\"3A+++c0xsFFQ$\"3i*** **>rNU^%FQ7$$\"37+++4aUlFFQ$\"3&*******G%[Z^%FQ7$$\"3-+++\\.+eFFQ$\"3l ******H<;9XFQ7$$\"33+++)*Hl]FFQ$\"3&)*****zs[C^%FQ7$$\"3,+++_:bVFFQ$\" 3()*****z)Gg4XFQ7$$\"3y*****z4tot#FQ$\"3/+++w+++=%>bs#FQ$\"3E+++*H*e%\\%FQ7$$\"3\"**** **\\I47s#FQ$\"3')*****f]!p([%FQ7$$\"33+++kz.=FFQ$\"3o******=z1![%FQ7$$ \"36+++9+:;FFQ$\"3/+++SQ\">Z%FQ7$$\"3?+++Wdl:FFQ$\"3q*****p+dMY%FQ7$$ \"3!******4c>mr#FQ$\"3S+++nd&\\X%FQ7$$\"3#)*****f2]!>FFQ$\"3E+++7ooYWF Q7$$\"36+++V%FQ7$$\"30+++5wHMFFQ$\"3:+++KH)fU%FQ7$$\"3*******R?j9u#FQ$\"3j**** *\\a87U%FQ7$$\"3=+++i'z#\\FFQ$\"3R+++WOx[FHW%FQ7$$\"3\")*****fj!)H\"GFQ$\"31+++v7\")\\WFQ7$$\" 3&)*****\\T1Z\"GFQ$\"39+++i9&pX%FQ7$$\"35+++dEO:GFQ$\"3g*****>(o>kWFQ7 $$\"3\"*******3L'\\\"GFQ$\"3m*****Rv-9Z%FQ7$$\"3!)*****zRPN\"GFQ$\"3I+ ++T8VyWFQFhdsFign-Fchn6L7%7$$\"+Iv76GFd[o$\"+[@:&[%Fd[o7$$\"+o()y2GFd[ o$\"+@@W\"\\%Fd[o7$$\"$w#F_hn$\"$Y%F_hn7%Fhat7$$\"+\")ye.GFd[o$\"+td=( \\%Fd[oF]bt7%Fcbt7$$\"+')Gg)z#Fd[o$\"+q`F-XFd[oF]bt7%Fibt7$$\"+iG#Hz#F d[o$\"+x7h1XFd[oF]bt7%F_ct7$$\"+jzk'y#Fd[o$\"+6D55XFd[oF]bt7%Fect7$$\" +y&*))zFFd[o$\"+'enE^%Fd[oF]bt7%F[dt7$$\"+c0xsFFd[o$\"+7dB9XFd[oF]bt7% Fadt7$$\"+4aUlFFd[o$\"+H%[Z^%Fd[oF]bt7%Fgdt7$$\"+\\.+eFFd[o$\"+I<;9XFd [oF]bt7%F]et7$$\"+)*Hl]FFd[o$\"+G([C^%Fd[oF]bt7%Fcet7$$\"+_:bVFFd[o$\" +))Gg4XFd[oF]bt7%Fiet7$$\"+)4tot#Fd[o$\"+wbs#Fd[o$\"+*H*e%\\%Fd[oF]bt7%F[g t7$$\"+0$47s#Fd[o$\"+10p([%Fd[oF]bt7%Fagt7$$\"+kz.=FFd[o$\"+>z1![%Fd[o F]bt7%Fggt7$$\"+9+:;FFd[o$\"+SQ\">Z%Fd[oF]bt7%F]ht7$$\"+Wdl:FFd[o$\"+2 qXjWFd[oF]bt7%Fcht7$$\"+h&>mr#Fd[o$\"+nd&\\X%Fd[oF]bt7%Fiht7$$\"+w+0>F Fd[o$\"+7ooYWFd[oF]bt7%F_it7$$\"+V%Fd[oF]bt7%F[jt7$$\"+5wHMFFd[o$\"+KH)fU%Fd[oF]bt 7%Fajt7$$\"+/KYTFFd[o$\"+XN@@WFd[oF]bt7%Fgjt7$$\"+i'z#\\FFd[o$\"+WOxkWFd[oF]bt7% Fe_u7$$\"+4L'\\\"GFd[o$\"+aFSrWFd[oF]bt7%F[`u7$$\"+)RPN\"GFd[o$\"+T8Vy WFd[oF]bt7%Fa`uFcatF]btF[dsFads-F$6$7K7$Fids$!31+++[@:&[%FQ7$F\\at$!3I +++T8VyWFQ7$Fg`t$!3m*****Rv-9Z%FQ7$Fb`t$!3g*****>(o>kWFQ7$F]`t$!39+++i 9&pX%FQ7$Fh_t$!31+++v7\")\\WFQ7$Fc_t$!3)******>[FHW%FQ7$F^_t$!3$)***** flckV%FQ7$Fi^t$!3k*****f4f0V%FQ7$Fd^t$!3M+++RiRDWFQ7$F_^t$!30+++,p7@WF Q7$Fj]t$!3E+++\"3-zT%FQ7$Fe]t$!3++++r\"eeT%FQ7$F`]t$!3N+++0'3^T%FQ7$F[ ]t$!3q*****4HMdT%FQ7$Ff\\t$!3R+++WOxV%FQ7$Fb[t$!3'*******)**H*QWFQ7$F][ t$!3E+++7ooYWFQ7$Fhjs$!3S+++nd&\\X%FQ7$Fcjs$!3q*****p+dMY%FQ7$F^js$!3/ +++SQ\">Z%FQ7$Fiis$!3o******=z1![%FQ7$Fdis$!3')*****f]!p([%FQ7$F_is$!3 E+++*H*e%\\%FQ7$Fjhs$!3&)******y6h+XFQ7$Fehs$!3/+++wrNU^%FQ7$Fgfs$!31+++'enE^%FQ7$Fbfs$!3P+ ++6D55XFQ7$F]fs$!3>+++x7h1XFQ7$Fhes$!3W+++q`F-XFQ7$Fces$!3++++td=(\\%F Q7$F^es$!3S+++@@W\"\\%FQFj`uFign-Fchn6L7%7$Fdat$!+[@:&[%Fd[o7$Fb`u$!+T 8VyWFd[o7$F^bt$!$Y%F_hn7%Fhhu7$F\\`u$!+aFSrWFd[oF[iu7%F_iu7$Ff_u$!+so> kWFd[oF[iu7%Fciu7$F`_u$!+i9&pX%Fd[oF[iu7%Fgiu7$Fj^u$!+v7\")\\WFd[oF[iu 7%F[ju7$Fd^u$!+#[FHW%Fd[oF[iu7%F_ju7$F^^u$!+cmXOWFd[oF[iu7%Fcju7$Fh]u$ !+'4f0V%Fd[oF[iu7%Fgju7$Fb]u$!+RiRDWFd[oF[iu7%F[[v7$F\\]u$!+,p7@WFd[oF [iu7%F_[v7$Ff\\u$!+\"3-zT%Fd[oF[iu7%Fc[v7$F`\\u$!+r\"eeT%Fd[oF[iu7%Fg[ v7$Fj[u$!+0'3^T%Fd[oF[iu7%F[\\v7$Fd[u$!+\"HMdT%Fd[oF[iu7%F_\\v7$F^[u$! +WOxV%Fd[oF[iu7%F_]v7$Ffit$!+***H*QWFd[oF[iu7%Fc]v7 $F`it$!+7ooYWFd[oF[iu7%Fg]v7$Fjht$!+nd&\\X%Fd[oF[iu7%F[^v7$Fdht$!+2qXj WFd[oF[iu7%F_^v7$F^ht$!+SQ\">Z%Fd[oF[iu7%Fc^v7$Fhgt$!+>z1![%Fd[oF[iu7% Fg^v7$Fbgt$!+10p([%Fd[oF[iu7%F[_v7$F\\gt$!+*H*e%\\%Fd[oF[iu7%F__v7$Fff t$!+z6h+XFd[oF[iu7%Fc_v7$F`ft$!+wT35R(F1$\"3q*****z F&=$G'F-7$$!3%******\\*oJa8!#D$\"3++++r9)R&yF-7$$!3++++Y='f\")*F[dv$\" 3s*****fXxZU*F-7$$!3:+++6$31i%F=$\"35+++ttb*4\"FQ7$$!3'******4KGCj\"FC $\"37+++xtjc7FQ7$$!3y*****Hurmi%FC$\"3)******z.>PT\"FQ7$$!3#*******p>Q &3\"FN$\"3!******p)e!3d\"FQ7$$!34+++*H[h6#FN$\"3)*******yf!zs\"FQ7$$!3 !********pS?I$FN$\"33+++'=N])=FQ7$$!3u*****4?\"4QMFN$\"3?+++m5AU?FQ7$$ \"3%******>A6pj(FC$\"3y*****\\&f]*>#FQ7$$\"33+++w@p^;FU$\"3))*****f]Yp N#FQ7$$\"3m*****>F*[mdFU$\"3)*******zZg9DFQ7$$\"3#******\\qSA[\"F_o$\" 3')*****f)G_sEFQ7$$\"31+++y%HFE$F_o$\"3(******\\)*o1$GFQ7$$\"3q******f bflkF_o$\"3))*****>0R)))HFQ7$$\"3'******\\-cz<\"F_p$\"3-+++vT]YJFQ7$$ \"35+++Qa+*)>F_p$\"3*******pIGEI$FQ7$$\"3!******fG&f:JF_p$\"3')*****px obX$FQ7$$\"3H+++\"3d#=XF_p$\"3)******\\7eKg$FQ7$$\"3M+++\\$R-1'F_p$\"3 *******f0YPu$FQ7$$\"3_******)eue`(F_p$\"3)******4u%zvQFQ7$$\"3p+++I1)H t)F_p$\"37+++$Qs!**RFQ7$$\"3U+++#3AH[*F_p$\"3m*****pk&*Q6%FQ7$$\"3%*** ***\\a&*on*F_p$\"3>+++!3R3A%FQ7$$\"3++++6MIf#*F_p$\"3k*****p#)f/K%FQ7$ $\"3c*****p]FX@)F_p$\"31+++C)yJT%FQ7$$\"3Q+++RFDblF_p$\"3A+++obE*\\%FQ 7$$\"3)******RgRRJ%F_p$\"3G+++'or)yXFQ7$$\"33+++vLjO:F_p$\"3w*****>Iz? l%FQ7$$!31+++L+\">s\"F_p$\"3$)******Gk%*=ZFQ7$$!3!)******QpN-aF_p$\"3# ******z8T&zZFQ7$$!3c+++i\\LW%*F_p$\"3Q+++B>'R$[FQ7$$!3!******zqI*y8F-$ \"3W+++'HXB)[FQ7$$!3.+++\\&o#Q=F-$\"3U+++oO'[#\\FQ7$$!3,+++ur^LF-$\"3P +++pfI>]FQ7$$!3))*****\\wvV$QF-$\"3$******p;&[S]FQ7$$!3!******>nCeN%F- $\"3C+++s0(o0&FQ7$$!3++++aVA\")[F-$\"3!)*****>,b'o]FQ7$$!3[+++n#p)3aF- $\"3-+++*e5g2&FQ7$$!3e******3EVPfF-$\"3G+++:f3z]FQ7$$!3a*****z\"z#fY'F -$\"3I+++%H/!y]FQ7$$!3))*****\\#Ho$*pF-$\"33+++([iG2&FQ7$$!3'******zxF ._(F-$\"3q******R)HP1&FQ7$$!3q*****z1+e/)F-$\"3++++;wk]]FQ7$$!3q*****R +q.d)F-$\"3\"******H`GO.&FQ7$$!3q*****4r!p%4*F-$\"3#******zncE,&FQ7$$! 3.+++63))>'*F-$\"3S+++7$)o()\\FQ7$$!30+++8\\w95FQ$\"3?+++XYle\\FQ7$$!3 7+++ct/o5FQ$\"3U+++b'oa#\\FQ7$$!3)********[%=A6FQ$\"3#)*****p')R!))[FQ 7$$!3!******zTdw<\"FQ$\"3K+++$)RIY[FQ7$$!3,+++,>6N7FQ$\"3f*****Rky-![F Q7$$!33+++4uP&H\"FQ$\"3++++1/;]ZFQ7$$!3%********fM%f8FQ$\"3M+++`JZ'p%F Q7$$!3/+++^TEG9FQ$\"3I+++-MCSYFQ7$$!3(*******\\t^-:FQ$\"3')*****f*34$e %FQ7$$!33+++XI3#e\"FQ$\"3q******Qn.FXFQ7$$!3)******\\xhem\"FQ$\"3G+++ \">^RZ%FQ7$$!3!******RVT?v\"FQ$\"3z*****p&*[\\U%FQ7$$!3$******f?6(Q=FQ $\"3/+++k#)=!Q%FQ7$$!3&******4t\\V#>FQ$\"3:+++9Y7RVFQ7$$!3!)*****>htz+ #FQ$\"3>+++#p-4I%FQ7$$!3'*******)Rc!*3#FQ$\"3o*****R_RYE%FQ7$$!35+++MO Qn@FQ$\"3n*****RRe&HUFQ7$$!32+++B'GHC#FQ$\"3g*****4qD]>%FQ7$$!3*)***** *pxw:BFQ$\"3q*****p4V0;%FQ7$$!3?+++4*GgQ#FQ$\"3T+++h\\sDTFQ7$$!3%)**** **)yhQX#FQ$\"3=+++=XF!4%FQ7$$!3'******p6A%>DFQ$\"31+++kK'R0%FQ7$$!3%** *****GN'Ge#FQ$\"3;+++#o9m,%FQ7$$!3$)*****p3MVk#FQ$\"3=+++r=4yRFQ7$$!3! ******\\FvRq#FQ$\"33+++)R)GQRFQ7$$!3-+++oB#>w#FQ$\"37+++'p@r*QFQ7$$!3# )*****RY0$=GFQ$\"3%)*****fVGX&QFQ7$$!3)******R!3DtGFQ$\"3z*****RZh/\"Q FQ7$$!38+++$4#)o#HFQ$\"35+++g!))[w$FQ7$$!3A+++S;KzHFQ$\"3>+++!3*y6\"3$FQ$\"3++++uu+>OFQ7$$! 3-+++wWqIJFQ$\"3++++T!ftc$FQ7$$!3#)*****R6(ezJFQ$\"32+++\"oIK$FQ$\"3++++Fl$pM$FQ7$$!3))******4]9qLFQ$\"3/+++c&H))G$FQ7$$ !3'*******[I&pT$FQ$\"34+++f#o(HKFQ7$$!33+++*)z^jMFQ$\"3y******z$)*)pJF Q7$$!3&)*****>oN)4NFQ$\"3!)*****z$*o$4JFQ7$$!3!)******Q;(eb$FQ$\"3')** ****>VK[IFQ7$$!32+++jKc,OFQ$\"3!)*****>R'*o)HFQ7$$!3)******ReCok$FQ$\" 3;+++Mx>DHFQ7$$!3#******H\"Gb\"p$FQ$\"35+++zoJjGFQ7$$!34+++gdjNPFQ$\"3 8+++CgJ,GFQ7$$!3?+++y!f*yPFQ$\"32+++\"=J#RFFQ7$$!30+++`DT@QFQ$\"3%**** **RJuqn#FQ7$$!3-+++Z\\*G'QFQ$\"3y******Ho$[h#FQ7$$!3?+++xsJ.RFQ$\"3+++ +BP\\_DFQ7$$!3$)*****Rj/E%RFQ$\"3')*****\\!y+!\\#FQ7$$!3A+++Xop!)RFQ$ \"3********QOLFCFQ7$$!3e*****zF[v,%FQ$\"3;+++Z4UkBFQ7$$!3K+++7r7`SFQ$ \"3))******=u@,BFQ7$$!3u******)H9u3%FQ$\"3$*******p4nPAFQ7$$!3l*****4U -/7%FQ$\"3=+++&\\JP<#FQ7$$!3q*****H_%4_TFQ$\"3%)*****pD_$4@FQ7$$!37+++ 1I]#=%FQ$\"3y*****R*4\\W?FQ7$$!3t*****Hd[;@%FQ$\"3'******H%36z>FQ7$$!3 \"******fAf&RUFQ$\"3-+++p6=8>FQ7$$!3G+++o#piE%FQ$\"3'******4Kym%=FQ7$$ !3g*****>E==H%FQ$\"3!******HP'ezO%FQ$\"35+++Cmuu:FQ7 $$!3G+++R1J$Q%FQ$\"30+++XgJ0:FQ7$$!36+++GFu.WFQ$\"3)******\\F``V\"FQ7$ $!3G+++JeGBWFQ$\"33+++w&)*[O\"FQ7$$!31+++)*\\(>W%FQ$\"3)******4z+SH\"F Q7$$!3s*****fKP)fWFQ$\"3%*******ffrA7FQ7$$!3)*******\\/*oZ%FQ$\"3-+++c a5^6FQ7$$!3O+++.:9$\\%FQ$\"35+++>PBz5FQ7$$!3P+++*o'e3XFQ$\"3++++Xd;25F Q7$$!3g*****>^6K_%FQ$\"3c+++y_k\\$*F-7$$!3++++r;*p`%FQ$\"3[+++ml)oi)F- 7$$!3#******H^%*)\\XFQ$\"3B+++\\F!R!zF-7$$!3i*****R\"4)=c%FQ$\"3y***** pDN6=(F-7$$!3=+++!Q2Hd%FQ$\"3g*****\\mN*ekF-7$$!3U+++@#GHe%FQ$\"3$**** **p`hvt&F-7$$!3g*****Hg(*=f%FQ$\"3()******3u<<]F-7$$!3M+++19x*f%FQ$\"3 '******f)*fyH%F-7$$!3p*****>y3lg%FQ$\"3#******HV1'zNF-7$$!3<+++DM27YFQ $\"3C+++%pWB'GF-7$$!3()*****f`Mkh%FQ$\"3)*******4\\%f9#F-7$$!34+++Qvc> YFQ$\"34+++h7BI9F-7$$!3r*****fca9i%FQ$\"3e+++))y$*\\rF_p7$$!3?+++&y%3A YFQF(7$Fh]y$!3e+++))y$*\\rF_p7$Fc]y$!34+++h7BI9F-7$F^]y$!3)*******4\\% f9#F-7$Fi\\y$!3C+++%pWB'GF-7$Fd\\y$!3#******HV1'zNF-7$F_\\y$!3'******f )*fyH%F-7$Fj[y$!3()******3u<<]F-7$Fe[y$!3$******p`hvt&F-7$F`[y$!3g**** *\\mN*ekF-7$F[[y$!3y*****pDN6=(F-7$Ffjx$!3B+++\\F!R!zF-7$Fajx$!3[+++ml )oi)F-7$F\\jx$!3c+++y_k\\$*F-7$Fgix$!3++++Xd;25FQ7$Fbix$!35+++>PBz5FQ7 $F]ix$!3-+++ca5^6FQ7$Fhhx$!3%*******ffrA7FQ7$Fchx$!3)******4z+SH\"FQ7$ F^hx$!33+++w&)*[O\"FQ7$Figx$!3)******\\F``V\"FQ7$Fdgx$!30+++XgJ0:FQ7$F _gx$!35+++Cmuu:FQ7$Fjfx$!36+++,UhV;FQ7$Fefx$!3'******z\"y*=r\"FQ7$F`fx $!3!******HP'ezFQ7$Faex$! 3'******H%36z>FQ7$F\\ex$!3y*****R*4\\W?FQ7$Fgdx$!3%)*****pD_$4@FQ7$Fbd x$!3=+++&\\JP<#FQ7$F]dx$!3$*******p4nPAFQ7$Fhcx$!3))******=u@,BFQ7$Fcc x$!3;+++Z4UkBFQ7$F^cx$!3********QOLFCFQ7$Fibx$!3')*****\\!y+!\\#FQ7$Fd bx$!3++++BP\\_DFQ7$F_bx$!3y******Ho$[h#FQ7$Fjax$!3%******RJuqn#FQ7$Fea x$!32+++\"=J#RFFQ7$F`ax$!38+++CgJ,GFQ7$F[ax$!35+++zoJjGFQ7$Ff`x$!3;+++ Mx>DHFQ7$Fa`x$!3!)*****>R'*o)HFQ7$F\\`x$!3')******>VK[IFQ7$Fg_x$!3!)** ***z$*o$4JFQ7$Fb_x$!3y******z$)*)pJFQ7$F]_x$!34+++f#o(HKFQ7$Fh^x$!3/++ +c&H))G$FQ7$Fc^x$!3++++Fl$pM$FQ7$F^^x$!3/+++BO&RS$FQ7$Fi]x$!3%)******4 2wfMFQ7$Fd]x$!32+++OFQ7$ Fe\\x$!31+++h\"f\"pOFQ7$F`\\x$!3>+++!3*y%FQ7$F_hw$!3n*****RRe&HUFQ7$Fjgw$!3o*****R_RY E%FQ7$Fegw$!3>+++#p-4I%FQ7$F`gw$!3:+++9Y7RVFQ7$F[gw$!3/+++k#)=!Q%FQ7$F ffw$!3z*****p&*[\\U%FQ7$Fafw$!3G+++\">^RZ%FQ7$F\\fw$!3q******Qn.FXFQ7$ Fgew$!3')*****f*34$e%FQ7$Fbew$!3I+++-MCSYFQ7$F]ew$!3M+++`JZ'p%FQ7$Fhdw $!3++++1/;]ZFQ7$Fcdw$!3f*****Rky-![FQ7$F^dw$!3K+++$)RIY[FQ7$Ficw$!3#)* ****p')R!))[FQ7$Fdcw$!3U+++b'oa#\\FQ7$F_cw$!3?+++XYle\\FQ7$Fjbw$!3S+++ 7$)o()\\FQ7$Febw$!3#******zncE,&FQ7$F`bw$!3\"******H`GO.&FQ7$F[bw$!3++ ++;wk]]FQ7$Ffaw$!3q******R)HP1&FQ7$Faaw$!33+++([iG2&FQ7$F\\aw$!3I+++%H /!y]FQ7$Fg`w$!3G+++:f3z]FQ7$Fb`w$!3-+++*e5g2&FQ7$F]`w$!3!)*****>,b'o]F Q7$Fh_w$!3C+++s0(o0&FQ7$Fc_w$!3$******p;&[S]FQ7$F^_w$!3P+++pfI>]FQ7$Fi ^w$!3I+++zM7$*\\FQ7$Fd^w$!3Q+++rsrh\\FQ7$F_^w$!3U+++oO'[#\\FQ7$Fj]w$!3 W+++'HXB)[FQ7$Fe]w$!3Q+++B>'R$[FQ7$F`]w$!3#******z8T&zZFQ7$F[]w$!3$)** ****Gk%*=ZFQ7$Ff\\w$!3w*****>Iz?l%FQ7$Fa\\w$!3G+++'or)yXFQ7$F\\\\w$!3A +++obE*\\%FQ7$Fg[w$!31+++C)yJT%FQ7$Fb[w$!3k*****p#)f/K%FQ7$F][w$!3>+++ !3R3A%FQ7$Fhjv$!3m*****pk&*Q6%FQ7$Fcjv$!37+++$Qs!**RFQ7$F^jv$!3)****** 4u%zvQFQ7$Fiiv$!3*******f0YPu$FQ7$Fdiv$!3)******\\7eKg$FQ7$F_iv$!3')** ***pxobX$FQ7$Fjhv$!3*******pIGEI$FQ7$Fehv$!3-+++vT]YJFQ7$F`hv$!3))**** *>0R)))HFQ7$F[hv$!3(******\\)*o1$GFQ7$Ffgv$!3')*****f)G_sEFQ7$Fagv$!3) *******zZg9DFQ7$F\\gv$!3))*****f]YpN#FQ7$$\"3i*****4A6pj(FC$!3y*****\\ &f]*>#FQ7$Fbfv$!3?+++m5AU?FQ7$F]fv$!33+++'=N])=FQ7$Fhev$!3)*******yf!z s\"FQ7$$!33+++r>Q&3\"FN$!3!******p)e!3d\"FQ7$F^ev$!3)******z.>PT\"FQ7$ Fidv$!37+++xtjc7FQ7$Fddv$!35+++ttb*4\"FQ7$$!3Z+++W6'f\")*F[dv$!3s***** fXxZU*F-7$$!3'******R*oJa8F[dv$!3++++r9)R&yF-7$$!35+++m$35R(F1$!3q**** *zF&=$G'F-7$$\"3V+++tjDQiF`cv$!3,+++y*)Q7ZF-7$F($!3)******Rl#fTJF-Ffgn F'-Fjgn6&F\\hn$\"\")F_hn$\"#=F_hn$\"#[F_hn-Fchn6f\\l7%F'Ffhn7$$FCF\\in F(7%Ffhn7$F($\"+aEfTJFihnFdiz7%Fgiz7$$\"+x`DQiF_o$\"+y*)Q7ZFihnFdiz7%F [jz7$$!+7%35R(F_p$\"+y_=$G'FihnFdiz7%Fajz7$$!+&*oJa8FQ$\"+r9)R&yFihnFd iz7%Fgjz7$$!+Y='f\")*FQ$\"+cuxC%*FihnFdiz7%F][[l7$$!+6$31i%F]jn$\"+ttb *4\"Fd[oFdiz7%Fc[[l7$$!+@$GCj\"Fdjn$\"+xtjc7Fd[oFdiz7%Fi[[l7$$!+VPT\"Fd[oFdiz7%F_\\[l7$$!+q>Q&3\"Fa[o$\"+()e!3d\"Fd[oFdiz7%Fe \\[l7$$!+*H[h6#Fa[o$\"+zf!zs\"Fd[oFdiz7%F[][l7$$!++2/-LFa[o$\"+'=N])=F d[oFdiz7%Fa][l7$$!+,74QMFa[o$\"+m5AU?Fd[oFdiz7%Fg][l7$$\"+A7\"pj(Fdjn$ \"+bf]*>#Fd[oFdiz7%F]^[l7$$\"+w@p^;Fi[o$\"+1l%pN#Fd[oFdiz7%Fc^[l7$$\"+ s#*[mdFi[o$\"+!y/Y^#Fd[oFdiz7%Fi^[l7$$\"+02C#[\"F\\]o$\"+')G_sEFd[oFdi z7%F__[l7$$\"+y%HFE$F\\]o$\"+&)*o1$GFd[oFdiz7%Fe_[l7$$\"+gbflkF\\]o$\" +_!R)))HFd[oFdiz7%F[`[l7$$\"+Dg&z<\"F_^o$\"+vT]YJFd[oFdiz7%Fa`[l7$$\"+ Qa+*)>F_^o$\"+2$GEI$Fd[oFdiz7%Fg`[l7$$\"+'G&f:JF_^o$\"+x(obX$Fd[oFdiz7 %F]a[l7$$\"+\"3d#=XF_^o$\"+D\"eKg$Fd[oFdiz7%Fca[l7$$\"+\\$R-1'F_^o$\"+ cguVPFd[oFdiz7%Fia[l7$$\"+*eue`(F_^o$\"+TZzvQFd[oFdiz7%F_b[l7$$\"+I1)H t)F_^o$\"+$Qs!**RFd[oFdiz7%Feb[l7$$\"+#3AH[*F_^o$\"+Zc*Q6%Fd[oFdiz7%F[ c[l7$$\"+Xb*on*F_^o$\"+!3R3A%Fd[oFdiz7%Fac[l7$$\"+6MIf#*F_^o$\"+F)f/K% Fd[oFdiz7%Fgc[l7$$\"+2v_9#)F_^o$\"+C)yJT%Fd[oFdiz7%F]d[l7$$\"+RFDblF_^ o$\"+obE*\\%Fd[oFdiz7%Fcd[l7$$\"+/'RRJ%F_^o$\"+'or)yXFd[oFdiz7%Fid[l7$ $\"+vLjO:F_^o$\"+-$z?l%Fd[oFdiz7%F_e[l7$$!+L+\">s\"F_^o$\"+Hk%*=ZFd[oF diz7%Fee[l7$$!+RpN-aF_^o$\"+Q6azZFd[oFdiz7%F[f[l7$$!+i\\LW%*F_^o$\"+B> 'R$[Fd[oFdiz7%Faf[l7$$!+32$*y8Fihn$\"+'HXB)[Fd[oFdiz7%Fgf[l7$$!+\\&o#Q =Fihn$\"+oO'[#\\Fd[oFdiz7%F]g[l7$$!+ur^LFihn$\"+pfI>]Fd[oFdi z7%F_h[l7$$!+ldPMQFihn$\"+n^[S]Fd[oFdiz7%Feh[l7$$!+sY#eN%Fihn$\"+s0(o0 &Fd[oFdiz7%F[i[l7$$!+aVA\")[Fihn$\"+7]lo]Fd[oFdiz7%Fai[l7$$!+n#p)3aFih n$\"+*e5g2&Fd[oFdiz7%Fgi[l7$$!+4EVPfFihn$\"+:f3z]Fd[oFdiz7%F]j[l7$$!+= z#fY'Fihn$\"+%H/!y]Fd[oFdiz7%Fcj[l7$$!+DHo$*pFihn$\"+([iG2&Fd[oFdiz7%F ij[l7$$!+yxK?vFihn$\"+S)HP1&Fd[oFdiz7%F_[\\l7$$!+o+!e/)Fihn$\"+;wk]]Fd [oFdiz7%Fe[\\l7$$!+/+Pq&)Fihn$\"+L&GO.&Fd[oFdiz7%F[\\\\l7$$!+62p%4*Fih n$\"+yml7]Fd[oFdiz7%Fa\\\\l7$$!+63))>'*Fihn$\"+7$)o()\\Fd[oFdiz7%Fg\\ \\l7$$!+8\\w95Fd[o$\"+XYle\\Fd[oFdiz7%F]]\\l7$$!+ct/o5Fd[o$\"+b'oa#\\F d[oFdiz7%Fc]\\l7$$!+!\\%=A6Fd[o$\"+n)R!))[Fd[oFdiz7%Fi]\\l7$$!+=ulx6Fd [o$\"+$)RIY[Fd[oFdiz7%F_^\\l7$$!+,>6N7Fd[o$\"+W'y-![Fd[oFdiz7%Fe^\\l7$ $!+4uP&H\"Fd[o$\"+1/;]ZFd[oFdiz7%F[_\\l7$$!++YVf8Fd[o$\"+`JZ'p%Fd[oFdi z7%Fa_\\l7$$!+^TEG9Fd[o$\"+-MCSYFd[oFdiz7%Fg_\\l7$$!+]t^-:Fd[o$\"+'*34 $e%Fd[oFdiz7%F]`\\l7$$!+XI3#e\"Fd[o$\"+Rn.FXFd[oFdiz7%Fc`\\l7$$!+v<'em \"Fd[o$\"+\">^RZ%Fd[oFdiz7%Fi`\\l7$$!+M9/_Fd[o$\"+9Y7RV Fd[oFdiz7%F[b\\l7$$!+7O(z+#Fd[o$\"+#p-4I%Fd[oFdiz7%Fab\\l7$$!+*Rc!*3#F d[o$\"+C&RYE%Fd[oFdiz7%Fgb\\l7$$!+MOQn@Fd[o$\"+%Re&HUFd[oFdiz7%F]c\\l7 $$!+B'GHC#Fd[o$\"+,d-&>%Fd[oFdiz7%Fcc\\l7$$!+qxw:BFd[o$\"+(4V0;%Fd[oFd iz7%Fic\\l7$$!+4*GgQ#Fd[o$\"+h\\sDTFd[oFdiz7%F_d\\l7$$!+*yhQX#Fd[o$\"+ =XF!4%Fd[oFdiz7%Fed\\l7$$!+<@U>DFd[o$\"+kK'R0%Fd[oFdiz7%F[e\\l7$$!+HN' Ge#Fd[o$\"+#o9m,%Fd[oFdiz7%Fae\\l7$$!+(3MVk#Fd[o$\"+r=4yRFd[oFdiz7%Fge \\l7$$!+v_(Rq#Fd[o$\"+)R)GQRFd[oFdiz7%F]f\\l7$$!+oB#>w#Fd[o$\"+'p@r*QF d[oFdiz7%Fcf\\l7$$!+kaI=GFd[o$\"+O%GX&QFd[oFdiz7%Fif\\l7$$!+/3DtGFd[o$ \"+u9Y5QFd[oFdiz7%F_g\\l7$$!+$4#)o#HFd[o$\"+g!))[w$Fd[oFdiz7%Feg\\l7$$ !+S;KzHFd[o$\"+!3*y6\"3$Fd[o$\"+uu+>OFd[oFdiz7%Fgh\\l7$$!+wWqIJFd[o$\"+T! ftc$Fd[oFdiz7%F]i\\l7$$!+9rezJFd[o$\"+\"oIK$Fd[o$\"+Fl$pM$Fd[oFdiz7%Fej\\l7$$!+5]9qLFd[o$\"+c&H))G$Fd[ oFdiz7%F[[]l7$$!+\\I&pT$Fd[o$\"+f#o(HKFd[oFdiz7%Fa[]l7$$!+*)z^jMFd[o$ \"+!Q)*)pJFd[oFdiz7%Fg[]l7$$!+#oN)4NFd[o$\"+Q*o$4JFd[oFdiz7%F]\\]l7$$! +R;(eb$Fd[o$\"+?VK[IFd[oFdiz7%Fc\\]l7$$!+jKc,OFd[o$\"+#R'*o)HFd[oFdiz7 %Fi\\]l7$$!+%eCok$Fd[o$\"+Mx>DHFd[oFdiz7%F_]]l7$$!+8Gb\"p$Fd[o$\"+zoJj GFd[oFdiz7%Fe]]l7$$!+gdjNPFd[o$\"+CgJ,GFd[oFdiz7%F[^]l7$$!+y!f*yPFd[o$ \"+\"=J#RFFd[oFdiz7%Fa^]l7$$!+`DT@QFd[o$\"+9V2xEFd[oFdiz7%Fg^]l7$$!+Z \\*G'QFd[o$\"+Io$[h#Fd[oFdiz7%F]_]l7$$!+xsJ.RFd[o$\"+BP\\_DFd[oFdiz7%F c_]l7$$!+MYgURFd[o$\"+0y+!\\#Fd[oFdiz7%Fi_]l7$$!+Xop!)RFd[o$\"+ROLFCFd [oFdiz7%F_`]l7$$!+y#[v,%Fd[o$\"+Z4UkBFd[oFdiz7%Fe`]l7$$!+7r7`SFd[o$\"+ >u@,BFd[oFdiz7%F[a]l7$$!+*H9u3%Fd[o$\"+q4nPAFd[oFdiz7%Faa]l7$$!+@CS?TF d[o$\"+&\\JP<#Fd[oFdiz7%Fga]l7$$!+BX4_TFd[o$\"+dAN4@Fd[oFdiz7%F]b]l7$$ !+1I]#=%Fd[o$\"+%*4\\W?Fd[oFdiz7%Fcb]l7$$!+t&[;@%Fd[o$\"+V36z>Fd[oFdiz 7%Fib]l7$$!+E#f&RUFd[o$\"+p6=8>Fd[oFdiz7%F_c]l7$$!+o#piE%Fd[o$\"+@$ym% =Fd[oFdiz7%Fec]l7$$!+i#==H%Fd[o$\"+tjezO%Fd[o$\"+Cmuu:Fd[oFdiz7%F]e]l7$$!+R1J$Q%Fd[o$\"+XgJ0:Fd[oFdiz7%F ce]l7$$!+GFu.WFd[o$\"+vKNN9Fd[oFdiz7%Fie]l7$$!+JeGBWFd[o$\"+w&)*[O\"Fd [oFdiz7%F_f]l7$$!+)*\\(>W%Fd[o$\"+\"z+SH\"Fd[oFdiz7%Fef]l7$$!+Et$)fWFd [o$\"+gfrA7Fd[oFdiz7%F[g]l7$$!+]/*oZ%Fd[o$\"+ca5^6Fd[oFdiz7%Fag]l7$$!+ .:9$\\%Fd[o$\"+>PBz5Fd[oFdiz7%Fgg]l7$$!+*o'e3XFd[o$\"+Xd;25Fd[oFdiz7%F ]h]l7$$!+7:@BXFd[o$\"+y_k\\$*FihnFdiz7%Fch]l7$$!+r;*p`%Fd[o$\"+ml)oi)F ihnFdiz7%Fih]l7$$!+8X*)\\XFd[o$\"+\\F!R!zFihnFdiz7%F_i]l7$$!+94)=c%Fd[ o$\"+d_8\"=(FihnFdiz7%Fei]l7$$!+!Q2Hd%Fd[o$\"+lc$*ekFihnFdiz7%F[j]l7$$ !+@#GHe%Fd[o$\"+P:cPdFihnFdiz7%Faj]l7$$!+.w*=f%Fd[o$\"+4u<<]FihnFdiz7% Fgj]l7$$!+19x*f%Fd[o$\"+')*fyH%FihnFdiz7%F][^l7$$!+#y3lg%Fd[o$\"+LkgzN FihnFdiz7%Fc[^l7$$!+DM27YFd[o$\"+%pWB'GFihnFdiz7%Fi[^l7$$!+OXV;YFd[o$ \"+5\\%f9#FihnFdiz7%F_\\^l7$$!+Qvc>YFd[o$\"+h7BI9FihnFdiz7%Fe\\^l7$$!+ mXX@YFd[o$\"+))y$*\\rF_^oFdiz7%F[]^l7$$!+&y%3AYFd[oF(Fdiz7%Fa]^l7$F\\] ^l$!+))y$*\\rF_^oFdiz7%Fe]^l7$Ff\\^l$!+h7BI9FihnFdiz7%Fi]^l7$F`\\^l$!+ 5\\%f9#FihnFdiz7%F]^^l7$Fj[^l$!+%pWB'GFihnFdiz7%Fa^^l7$Fd[^l$!+LkgzNFi hnFdiz7%Fe^^l7$F^[^l$!+')*fyH%FihnFdiz7%Fi^^l7$Fhj]l$!+4u<<]FihnFdiz7% F]_^l7$Fbj]l$!+P:cPdFihnFdiz7%Fa_^l7$F\\j]l$!+lc$*ekFihnFdiz7%Fe_^l7$F fi]l$!+d_8\"=(FihnFdiz7%Fi_^l7$F`i]l$!+\\F!R!zFihnFdiz7%F]`^l7$Fjh]l$! +ml)oi)FihnFdiz7%Fa`^l7$Fdh]l$!+y_k\\$*FihnFdiz7%Fe`^l7$F^h]l$!+Xd;25F d[oFdiz7%Fi`^l7$Fhg]l$!+>PBz5Fd[oFdiz7%F]a^l7$Fbg]l$!+ca5^6Fd[oFdiz7%F aa^l7$F\\g]l$!+gfrA7Fd[oFdiz7%Fea^l7$Fff]l$!+\"z+SH\"Fd[oFdiz7%Fia^l7$ F`f]l$!+w&)*[O\"Fd[oFdiz7%F]b^l7$Fje]l$!+vKNN9Fd[oFdiz7%Fab^l7$Fde]l$! +XgJ0:Fd[oFdiz7%Feb^l7$F^e]l$!+Cmuu:Fd[oFdiz7%Fib^l7$Fhd]l$!+,UhV;Fd[o Fdiz7%F]c^l7$Fbd]l$!+=y*=r\"Fd[oFdiz7%Fac^l7$F\\d]l$!+tjezFd[oFdiz7%F]d^l7$Fjb ]l$!+V36z>Fd[oFdiz7%Fad^l7$Fdb]l$!+%*4\\W?Fd[oFdiz7%Fed^l7$F^b]l$!+dAN 4@Fd[oFdiz7%Fid^l7$Fha]l$!+&\\JP<#Fd[oFdiz7%F]e^l7$Fba]l$!+q4nPAFd[oFd iz7%Fae^l7$F\\a]l$!+>u@,BFd[oFdiz7%Fee^l7$Ff`]l$!+Z4UkBFd[oFdiz7%Fie^l 7$F``]l$!+ROLFCFd[oFdiz7%F]f^l7$Fj_]l$!+0y+!\\#Fd[oFdiz7%Faf^l7$Fd_]l$ !+BP\\_DFd[oFdiz7%Fef^l7$F^_]l$!+Io$[h#Fd[oFdiz7%Fif^l7$Fh^]l$!+9V2xEF d[oFdiz7%F]g^l7$Fb^]l$!+\"=J#RFFd[oFdiz7%Fag^l7$F\\^]l$!+CgJ,GFd[oFdiz 7%Feg^l7$Ff]]l$!+zoJjGFd[oFdiz7%Fig^l7$F`]]l$!+Mx>DHFd[oFdiz7%F]h^l7$F j\\]l$!+#R'*o)HFd[oFdiz7%Fah^l7$Fd\\]l$!+?VK[IFd[oFdiz7%Feh^l7$F^\\]l$ !+Q*o$4JFd[oFdiz7%Fih^l7$Fh[]l$!+!Q)*)pJFd[oFdiz7%F]i^l7$Fb[]l$!+f#o(H KFd[oFdiz7%Fai^l7$F\\[]l$!+c&H))G$Fd[oFdiz7%Fei^l7$Ffj\\l$!+Fl$pM$Fd[o Fdiz7%Fii^l7$F`j\\l$!+BO&RS$Fd[oFdiz7%F]j^l7$Fji\\l$!+52wfMFd[oFdiz7%F aj^l7$Fdi\\l$!+OFd[oFdiz7%F][_l7$Fbh\\l$!+h\"f\"pOFd[oFdiz7%Fa[_l7$F\\h\\ l$!+!3*y%Fd[oFdiz7%Fe^_l7$F^c\\ l$!+%Re&HUFd[oFdiz7%Fi^_l7$Fhb\\l$!+C&RYE%Fd[oFdiz7%F]__l7$Fbb\\l$!+#p -4I%Fd[oFdiz7%Fa__l7$F\\b\\l$!+9Y7RVFd[oFdiz7%Fe__l7$Ffa\\l$!+k#)=!Q%F d[oFdiz7%Fi__l7$F`a\\l$!+d*[\\U%Fd[oFdiz7%F]`_l7$Fj`\\l$!+\">^RZ%Fd[oF diz7%Fa`_l7$Fd`\\l$!+Rn.FXFd[oFdiz7%Fe`_l7$F^`\\l$!+'*34$e%Fd[oFdiz7%F i`_l7$Fh_\\l$!+-MCSYFd[oFdiz7%F]a_l7$Fb_\\l$!+`JZ'p%Fd[oFdiz7%Faa_l7$F \\_\\l$!+1/;]ZFd[oFdiz7%Fea_l7$Ff^\\l$!+W'y-![Fd[oFdiz7%Fia_l7$F`^\\l$ !+$)RIY[Fd[oFdiz7%F]b_l7$Fj]\\l$!+n)R!))[Fd[oFdiz7%Fab_l7$Fd]\\l$!+b'o a#\\Fd[oFdiz7%Feb_l7$F^]\\l$!+XYle\\Fd[oFdiz7%Fib_l7$Fh\\\\l$!+7$)o() \\Fd[oFdiz7%F]c_l7$Fb\\\\l$!+yml7]Fd[oFdiz7%Fac_l7$F\\\\\\l$!+L&GO.&Fd [oFdiz7%Fec_l7$Ff[\\l$!+;wk]]Fd[oFdiz7%Fic_l7$F`[\\l$!+S)HP1&Fd[oFdiz7 %F]d_l7$Fjj[l$!+([iG2&Fd[oFdiz7%Fad_l7$Fdj[l$!+%H/!y]Fd[oFdiz7%Fed_l7$ F^j[l$!+:f3z]Fd[oFdiz7%Fid_l7$Fhi[l$!+*e5g2&Fd[oFdiz7%F]e_l7$Fbi[l$!+7 ]lo]Fd[oFdiz7%Fae_l7$F\\i[l$!+s0(o0&Fd[oFdiz7%Fee_l7$Ffh[l$!+n^[S]Fd[o Fdiz7%Fie_l7$F`h[l$!+pfI>]Fd[oFdiz7%F]f_l7$Fjg[l$!+zM7$*\\Fd[oFdiz7%Fa f_l7$Fdg[l$!+rsrh\\Fd[oFdiz7%Fef_l7$F^g[l$!+oO'[#\\Fd[oFdiz7%Fif_l7$Fh f[l$!+'HXB)[Fd[oFdiz7%F]g_l7$Fbf[l$!+B>'R$[Fd[oFdiz7%Fag_l7$F\\f[l$!+Q 6azZFd[oFdiz7%Feg_l7$Ffe[l$!+Hk%*=ZFd[oFdiz7%Fig_l7$F`e[l$!+-$z?l%Fd[o Fdiz7%F]h_l7$Fjd[l$!+'or)yXFd[oFdiz7%Fah_l7$Fdd[l$!+obE*\\%Fd[oFdiz7%F eh_l7$F^d[l$!+C)yJT%Fd[oFdiz7%Fih_l7$Fhc[l$!+F)f/K%Fd[oFdiz7%F]i_l7$Fb c[l$!+!3R3A%Fd[oFdiz7%Fai_l7$F\\c[l$!+Zc*Q6%Fd[oFdiz7%Fei_l7$Ffb[l$!+$ Qs!**RFd[oFdiz7%Fii_l7$F`b[l$!+TZzvQFd[oFdiz7%F]j_l7$Fja[l$!+cguVPFd[o Fdiz7%Faj_l7$Fda[l$!+D\"eKg$Fd[oFdiz7%Fej_l7$F^a[l$!+x(obX$Fd[oFdiz7%F ij_l7$Fh`[l$!+2$GEI$Fd[oFdiz7%F][`l7$Fb`[l$!+vT]YJFd[oFdiz7%Fa[`l7$F\\ `[l$!+_!R)))HFd[oFdiz7%Fe[`l7$Ff_[l$!+&)*o1$GFd[oFdiz7%Fi[`l7$F`_[l$!+ ')G_sEFd[oFdiz7%F]\\`l7$Fj^[l$!+!y/Y^#Fd[oFdiz7%Fa\\`l7$Fd^[l$!+1l%pN# Fd[oFdiz7%Fe\\`l7$$\"+@7\"pj(Fdjn$!+bf]*>#Fd[oFdiz7%Fi\\`l7$Fh][l$!+m5 AU?Fd[oFdiz7%F_]`l7$Fb][l$!+'=N])=Fd[oFdiz7%Fc]`l7$F\\][l$!+zf!zs\"Fd[ oFdiz7%Fg]`l7$$!+r>Q&3\"Fa[o$!+()e!3d\"Fd[oFdiz7%F[^`l7$F`\\[l$!+Q!>PT \"Fd[oFdiz7%Fa^`l7$Fj[[l$!+xtjc7Fd[oFdiz7%Fe^`l7$Fd[[l$!+ttb*4\"Fd[oFd iz7%Fi^`l7$$!+W6'f\")*FQ$!+cuxC%*FihnFdiz7%F]_`l7$$!+%*oJa8FQ$!+r9)R&y FihnFdiz7%Fc_`l7$$!+m$35R(F_p$!+y_=$G'FihnFdiz7%Fi_`l7$$\"+tjDQiF_o$!+ y*)Q7ZFihnFdiz7%F_``l7$F($!+aEfTJFihnFdiz7%Fe``lFgcsFdiz7%FgcsF'Fdiz-F jgn6&F\\hn$\"#:F_hn$\"#NF_hn$\"#&*F_hnFads-F$6$7K7$$\"3#)*****>%Q?vFFQ $\"3?+++A_;,iFQ7$$\"33+++DnX%y#FQ$\"3m*****fLfa?'FQ7$$\"3%)*****z\"3z# z#FQ$\"31+++*z@5@'FQ7$$\"3;+++&em+!GFQ$\"3:+++n\\mDiFQ7$$\"3!******Hm_5\"GFQ$\"3p*****RFHMB'FQ7$$\"3.+++$ pQY\"GFQ$\"3q*****fJt@C'FQ7$$\"3>+++h+\"p\"GFQ$\"3#******HE[7D'FQ7$$\" 3>+++6\"fy\"GFQ$\"3M+++;'y/E'FQ7$$\"3&)*****f7&\\9h.GFQ$\"3 ')*****R0GKI'FQ7$$\"3?+++\\;L(z#FQ$\"3G+++>Z-5jFQ7$$\"3))******f\")4!z #FQ$\"3B+++YD!fJ'FQ7$$\"3%)*****RwB?y#FQ$\"3))*****>?H2K'FQ7$$\"3!)*** **R\\RKx#FQ$\"3h******[)zVK'FQ7$$\"3!)*****pE'*Qw#FQ$\"3w******f%RnK'F Q7$$\"3!*******ek;aFFQ$\"38+++wgqFjFQ7$$\"3#******p3XUu#FQ$\"3k*****>= &>FjFQ7$$\"3$)*****zA]Vt#FQ$\"3m*****REX^K'FQ7$$\"3,+++_@sCFFQ$\"3W+++ ^X_@jFQ7$$\"3)******\\d?cr#FQ$\"3g*****zwQjJ'FQ7$$\"3#******fK>tq#FQ$ \"3V+++[%R'4jFQ7$$\"3z******=y4+FFQ$\"3g*****p+K:I'FQ7$$\"3;+++_)GUp#F Q$\"3m*****pO#=#H'FQ7$$\"3%)*****Hzi**o#FQ$\"3o*****H\\?=G'FQ7$$\"3!** *****H*3vo#FQ$\"3A+++$)*Q2F'FQ7$$\"3%******R#e,(o#FQ$\"3Q+++-]GfiFQ7$$ \"3!******4]`&)o#FQ$\"3m*****foWyC'FQ7$$\"3#)*****R&45#p#FQ$\"3p*****f a?oB'FQ7$$\"3)******R6Uvp#FQ$\"3n*****pI/mA'FQ7$$\"36+++]?n/FFQ$\"3t** *****=\\v@'FQ7$$\"3*)*****R(G@8FFQ$\"3d*****fAY*4iFQ7$$\"3z*****fcOGs# FQ$\"3u*****HP3S?'FQ7$$\"3;+++<0>LFFQ$\"3m*****H@k)*>'FQ7$$\"3)******* f:#Ru#FQ$\"3K+++@2c(>'FQ7$$\"3%*******HapaFFQ$\"3M+++[G2(>'FQ7$$\"3')* ****4@5_w#FQ$\"3w******RpJ)>'FQFea`lFihz-Fchn6L7%7$$\"+UQ?vFFd[o$\"+A_ ;,iFd[o7$$\"+DnX%y#Fd[o$\"+O$fa?'Fd[o7$$\"$v#F_hn$\"$E'F_hn7%Fe^al7$$ \"+=3z#z#Fd[o$\"+*z@5@'Fd[oFj^al7%F`_al7$$\"+&em+!GFd[o$\"+n\\mDiFd[oFj^al7%F\\`al7$$\"+jE06GFd[o $\"+u#HMB'Fd[oFj^al7%Fb`al7$$\"+$pQY\"GFd[o$\"+;L9h.GFd[o$\"+a!GK I'Fd[oFj^al7%Fbcal7$$\"+\\;L(z#Fd[o$\"+>Z-5jFd[oFj^al7%Fhcal7$$\"+g\") 4!z#Fd[o$\"+YD!fJ'Fd[oFj^al7%F^dal7$$\"+kP-#y#Fd[o$\"+-#H2K'Fd[oFj^al7 %Fddal7$$\"+%\\RKx#Fd[o$\"+\\)zVK'Fd[oFj^al7%Fjdal7$$\"+ni*Qw#Fd[o$\"+ g%RnK'Fd[oFj^al7%F`eal7$$\"+fk;aFFd[o$\"+wgqFjFd[oFj^al7%Ffeal7$$\"+(3 XUu#Fd[o$\"+#=&>FjFd[oFj^al7%F\\fal7$$\"+G-NMFFd[o$\"+k_9DjFd[oFj^al7% Fbfal7$$\"+_@sCFFd[o$\"+^X_@jFd[oFj^al7%Fhfal7$$\"+v0i:FFd[o$\"+o(QjJ' Fd[oFj^al7%F^gal7$$\"+E$>tq#Fd[o$\"+[%R'4jFd[oFj^al7%Fdgal7$$\"+>y4+FF d[o$\"+2?`,jFd[oFj^al7%Fjgal7$$\"+_)GUp#Fd[o$\"+nB=#H'Fd[oFj^al7%F`hal 7$$\"+$zi**o#Fd[o$\"+$\\?=G'Fd[oFj^al7%Ffhal7$$\"+I*3vo#Fd[o$\"+$)*Q2F 'Fd[oFj^al7%F\\ial7$$\"+Ce,(o#Fd[o$\"+-]GfiFd[oFj^al7%Fbial7$$\"+,Nb)o #Fd[o$\"+'oWyC'Fd[oFj^al7%Fhial7$$\"+a45#p#Fd[o$\"+Y0#oB'Fd[oFj^al7%F^ jal7$$\"+9@a(p#Fd[o$\"+2VgEiFd[oFj^al7%Fdjal7$$\"+]?n/FFd[o$\"+!>\\v@' Fd[oFj^al7%Fjjal7$$\"+uG@8FFd[o$\"+Ei%*4iFd[oFj^al7%F`[bl7$$\"+ml$Gs#F d[o$\"+t$3S?'Fd[oFj^al7%Ff[bl7$$\"+<0>LFFd[o$\"+8U')*>'Fd[oFj^al7%F\\ \\bl7$$\"+g:#Ru#Fd[o$\"+@2c(>'Fd[oFj^al7%Fb\\bl7$$\"+IapaFFd[o$\"+[G2( >'Fd[oFj^al7%Fh\\bl7$$\"+6-@lFFd[o$\"+SpJ)>'Fd[oFj^al7%F^]blF`^alFj^al Fj``lFads-F$6$7K7$Ffa`l$!3?+++A_;,iFQ7$Fi]al$!3w******RpJ)>'FQ7$Fd]al$ !3M+++[G2(>'FQ7$F_]al$!3K+++@2c(>'FQ7$Fj\\al$!3m*****H@k)*>'FQ7$Fe\\al $!3u*****HP3S?'FQ7$F`\\al$!3d*****fAY*4iFQ7$F[\\al$!3t*******=\\v@'FQ7 $Ff[al$!3n*****pI/mA'FQ7$Fa[al$!3p*****fa?oB'FQ7$F\\[al$!3m*****foWyC' FQ7$Fgj`l$!3Q+++-]GfiFQ7$Fbj`l$!3A+++$)*Q2F'FQ7$F]j`l$!3o*****H\\?=G'F Q7$Fhi`l$!3m*****pO#=#H'FQ7$Fci`l$!3g*****p+K:I'FQ7$F^i`l$!3V+++[%R'4j FQ7$Fih`l$!3g*****zwQjJ'FQ7$Fdh`l$!3W+++^X_@jFQ7$F_h`l$!3m*****REX^K'F Q7$Fjg`l$!3k*****>=&>FjFQ7$Feg`l$!38+++wgqFjFQ7$F`g`l$!3w******f%RnK'F Q7$F[g`l$!3h******[)zVK'FQ7$Fff`l$!3))*****>?H2K'FQ7$Faf`l$!3B+++YD!fJ 'FQ7$F\\f`l$!3G+++>Z-5jFQ7$Fge`l$!3')*****R0GKI'FQ7$Fbe`l$!3U+++$)3l&H 'FQ7$F]e`l$!37+++smV(G'FQ7$Fhd`l$!3!)*****\\nL(yiFQ7$Fcd`l$!3s*****z% \\ppiFQ7$F^d`l$!3M+++;'y/E'FQ7$Fic`l$!3#******HE[7D'FQ7$Fdc`l$!3q***** fJt@C'FQ7$F_c`l$!3p*****RFHMB'FQ7$Fjb`l$!3M+++)e(>DiFQ7$Feb`l$!3:+++n \\m'Fd[o7$F[_al$!$E'F_hn7%Feebl7$Fi\\b l$!+[G2(>'Fd[oFhebl7%F\\fbl7$Fc\\bl$!+@2c(>'Fd[oFhebl7%F`fbl7$F]\\bl$! +8U')*>'Fd[oFhebl7%Fdfbl7$Fg[bl$!+t$3S?'Fd[oFhebl7%Fhfbl7$Fa[bl$!+Ei%* 4iFd[oFhebl7%F\\gbl7$F[[bl$!+!>\\v@'Fd[oFhebl7%F`gbl7$Fejal$!+2VgEiFd[ oFhebl7%Fdgbl7$F_jal$!+Y0#oB'Fd[oFhebl7%Fhgbl7$Fiial$!+'oWyC'Fd[oFhebl 7%F\\hbl7$Fcial$!+-]GfiFd[oFhebl7%F`hbl7$F]ial$!+$)*Q2F'Fd[oFhebl7%Fdh bl7$Fghal$!+$\\?=G'Fd[oFhebl7%Fhhbl7$Fahal$!+nB=#H'Fd[oFhebl7%F\\ibl7$ F[hal$!+2?`,jFd[oFhebl7%F`ibl7$Fegal$!+[%R'4jFd[oFhebl7%Fdibl7$F_gal$! +o(QjJ'Fd[oFhebl7%Fhibl7$Fifal$!+^X_@jFd[oFhebl7%F\\jbl7$Fcfal$!+k_9Dj Fd[oFhebl7%F`jbl7$F]fal$!+#=&>FjFd[oFhebl7%Fdjbl7$Fgeal$!+wgqFjFd[oFhe bl7%Fhjbl7$Faeal$!+g%RnK'Fd[oFhebl7%F\\[cl7$F[eal$!+\\)zVK'Fd[oFhebl7% F`[cl7$Fedal$!+-#H2K'Fd[oFhebl7%Fd[cl7$F_dal$!+YD!fJ'Fd[oFhebl7%Fh[cl7 $Fical$!+>Z-5jFd[oFhebl7%F\\\\cl7$Fccal$!+a!GKI'Fd[oFhebl7%F`\\cl7$F]c al$!+$)3l&H'Fd[oFhebl7%Fd\\cl7$Fgbal$!+smV(G'Fd[oFhebl7%Fh\\cl7$Fabal$ !+vOtyiFd[oFhebl7%F\\]cl7$F[bal$!+[\\ppiFd[oFhebl7%F`]cl7$Feaal$!+;'y/ E'Fd[oFhebl7%Fd]cl7$F_aal$!+j#[7D'Fd[oFhebl7%Fh]cl7$Fi`al$!+;LDiFd[oFhebl7 %Fd^cl7$Fg_al$!+n\\m$F_hn;$!$f'F_hn$\"$f'F_hn" 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" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 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 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for 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 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1760 "# coefficients by Sharp and Verne r 1994\nee := \{c[2]=1/12,\nc[3]=4/27,\nc[4]=2/9,\nc[5]=5/9,\nc[6]=2/3 ,\nc[7]=1/6,\nc[8]=4/9,\nc[9]=3/4,\nc[10]=11/12,\nc[11]=1,\nc[12]=1,\n \na[2,1]=1/12,\na[3,1]=4/243,\na[3,2]=32/243,\na[4,1]=1/18,\na[4,2]=0, \na[4,3]=1/6,\na[5,1]=5/9,\na[5,2]=0,\na[5,3]=-25/12,\na[5,4]=25/12,\n a[6,1]=1/15,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1/3,\na[6,5]=4/15,\na[7,1]= 319/3840,\na[7,2]=0,\na[7,3]=0,\na[7,4]=161/1536,\na[7,5]=-41/960,\na[ 7,6]=11/512,\na[8,1]=245/5184,\na[8,2]=0,\na[8,3]=0,\na[8,4]=1627/1036 8,\na[8,5]=151/1296,\na[8,6]=-445/10368,\na[8,7]=1/6,\na[9,1]=-5563498 53/7539261440,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-4356175383/3015704576,\n a[9,5]=-814787343/1884815360,\na[9,6]=831004641/3015704576,\na[9,7]=35 5452237/235601920,\na[9,8]=107943759/117800960,\na[10,1]=-68998698967/ 1063035863040,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-767387292485/42521434 5216,\na[10,5]=-205995991597/265758965760,\na[10,6]=-22181208863/14173 8115072,\na[10,7]=26226796959/15502606336,\na[10,8]=1614200643/1107329 024,\na[10,9]=187/329,\na[11,1]=24511479161/17979371520,\na[11,2]=0,\n a[11,3]=0,\na[11,4]=3889847115/217931776,\na[11,5]=22028391/3681280,\n a[11,6]=614528179/217931776,\na[11,7]=-148401247/10215552,\na[11,8]=-3 122234829/318384704,\na[11,9]=-4160/1221,\na[11,10]=15040/20757,\na[12 ,1]=5519/110880,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12 ,6]=83/560,\na[12,7]=932/3675,\na[12,8]=282123/1047200,\na[12,9]=2624/ 24255,\na[12,10]=3008/19635,\na[12,11]=37/2100,\n\nb[1]=5519/110880,\n b[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=83/560,\nb[7]=932/3675,\nb[8] =282123/1047200,\nb[9]=2624/24255,\nb[10]=3008/19635,\nb[11]=37/2100, \n\n`b*`[1]=15509/341880,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5] =0,\n`b*`[6]=6827/15540,\n`b*`[7]=22138/81585,\n`b*`[8]=78003/387464, \n`b*`[9]=-64144/299145,\n`b*`[10]=623408/2179485,\n`b*`[11]=0,\n`b*`[ 12]=-15/518\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..12);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6-/&%\"cG6#\"\"##\"\"\"\"#7/&F%6#\"\"$#\"\"%\"#F/&F%6#F 0#F'\"\"*/&F%6#\"\"&#F:F6/&F%6#\"\"'#F'F./&F%6#\"\"(#F)F?/&F%6#\"\")#F 0F6/&F%6#F6#F.F0/&F%6#\"#5#\"#6F*/&F%6#FTF)/&F%6#F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficient s" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq (a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6^o/&%\"aG6$\"\"#\"\"\"#F(\"#7/&F%6$\"\"$F(#\"\"%\"$V#/&F %6$F.F'#\"#KF1/&F%6$F0F(#F(\"#=/&F%6$F0F'\"\"!/&F%6$F0F.#F(\"\"'/&F%6$ \"\"&F(#FH\"\"*/&F%6$FHF'F?/&F%6$FHF.#!#DF*/&F%6$FHF0#\"#DF*/&F%6$FDF( #F(\"#:/&F%6$FDF'F?/&F%6$FDF.F?/&F%6$FDF0#F(F./&F%6$FDFH#F0Ffn/&F%6$\" \"(F(#\"$>$\"%SQ/&F%6$FhoF'F?/&F%6$FhoF.F?/&F%6$FhoF0#\"$h\"\"%O:/&F%6 $FhoFH#!#T\"$g*/&F%6$FhoFD#\"#6\"$7&/&F%6$\"\")F(#\"$X#\"%%=&/&F%6$Fgq F'F?/&F%6$FgqF.F?/&F%6$FgqF0#\"%F;\"&o.\"/&F%6$FgqFH#\"$^\"\"%'H\"/&F% 6$FgqFD#!$X%Ffr/&F%6$FgqFhoFC/&F%6$FJF(#!*`)\\jb\"+S9ERv/&F%6$FJF'F?/& F%6$FJF.F?/&F%6$FJF0#!+$QvhN%\"+wXq:I/&F%6$FJFH#!*Vty9)\"+g`\"[)=/&F%6 $FJFD#\"*TY+J)Fft/&F%6$FJFho#\"*PAXb$\"*?>gN#/&F%6$FJFgq#\"*fP%z5\"*g4 !y6/&F%6$\"#5F(#!,n*)p)**o\".SI'e.j5/&F%6$FavF'F?/&F%6$FavF.F?/&F%6$Fa vF0#!-&[#H(Qn(\"-;_M9_U/&F%6$FavFH#!-(f\"*f*f?\"-gd'*edE/&F%6$FavFD#!, j)37=A\"-s]6Q<9/&F%6$FavFho#\",fpzEi#\",Ojg-b\"/&F%6$FavFgq#\"+V1?9;\" +C!Ht5\"/&F%6$FavFJ#\"$(=\"$H$/&F%6$FbqF(#\",h\"z9^C\",?:Pzz\"/&F%6$Fb qF'F?/&F%6$FbqF.F?/&F%6$FbqF0#\"+:r%)*)Q\"*w<$z@/&F%6$FbqFH#\")\"RG?# \"(!G\"o$/&F%6$FbqFD#\"*z\"GXhF`z/&F%6$FbqFho#!*Z7S[\"\")_b@5/&F%6$Fbq Fgq#!+H[BAJ\"*/ZQ=$/&F%6$FbqFJ#!%gT\"%@7/&F%6$FbqFav#\"&S]\"\"&d2#/&F% 6$F*F(#\"%>b\"'!)36/&F%6$F*F'F?/&F%6$F*F.F?/&F%6$F*F0F?/&F%6$F*FHF?/&F %6$F*FD#\"#$)\"$g&/&F%6$F*Fho#\"$K*\"%vO/&F%6$F*Fgq#\"'B@G\"(+s/\"/&F% 6$F*FJ#\"%CE\"&bU#/&F%6$F*Fav#\"%3I\"&N'>/&F%6$F*Fbq#\"#P\"%+@" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights \+ for the 11 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..11);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"%>b\"'!)36/&F%6#\"\"#\"\"!/&F% 6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"#$)\"$g&/&F%6#\"\"(# \"$K*\"%vO/&F%6#\"\")#\"'B@G\"(+s/\"/&F%6#\"\"*#\"%CE\"&bU#/&F%6#\"#5# \"%3I\"&N'>/&F%6#\"#6#\"#P\"%+@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 39 "weights for the 12 stage order 6 scheme " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*` [i]=subs(ee,`b*`[i]),i=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%# b*G6#\"\"\"#\"&4b\"\"'!)=M/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//& F%6#\"\"&F//&F%6#\"\"'#\"%Fo\"&Sb\"/&F%6#\"\"(#\"&Q@#\"&&e\")/&F%6#\" \")#\"&.!y\"'kuQ/&F%6#\"\"*#!&WT'\"'X\"*H/&F%6#\"#5#\"'3Mi\"(&[z@/&F%6 #\"#6F//&F%6#\"#7#!#:\"$=&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 " #---------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#===== ============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "s cheme with " }{XPPEDIT 18 0 "c[10] = 15/16;" "6#/&%\"cG6#\"#5*&\"#:\" \"\"\"#;!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 140 "The weights of this scheme are all still positive. The largest magnitude of a linking coefficient is lower tha n for the Sharp-Verner scheme." }}{PARA 0 "" 0 "" {TEXT -1 36 "The nod es of the order 7 scheme are:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[2] = 1/16;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#;!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=4/25" "6#/&%\"cG6#\"\"$*&\"\"%\" \"\"\"#D!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[4] = 6/25;" "6#/&%\" cG6#\"\"%*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5 ]=915/1696" "6#/&%\"cG6#\"\"&*&\"$:*\"\"\"\"%'p\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 5/8;" "6#/&%\"cG6#\"\"'*&\"\"&\"\"\"\"\")! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 35/209;" "6#/&%\"cG6#\" \"(*&\"#N\"\"\"\"$4#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 29 /72;" "6#/&%\"cG6#\"\")*&\"#H\"\"\"\"#s!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9] = 11/13;" "6#/&%\"cG6#\"\"**&\"#6\"\"\"\"#8!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 15/16;" "6#/&%\"cG6#\"#5*&\" #:\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\" cG6#\"#6\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "The 2-norm of the principal error is appr oximately: 0.169788 " }{TEXT 268 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-4)" "6#)\"#5,$\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "ch ecking 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 3315 "ee := \{c[2]=1/1 6,\nc[3]=4/25,\nc[4]=6/25,\nc[5]=915/1696,\nc[6]=5/8,\nc[7]=35/209,\nc [8]=29/72,\nc[9]=11/13,\nc[10]=15/16,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/ 16,\na[3,1]=-28/625,\na[3,2]=128/625,\na[4,1]=3/50,\na[4,2]=0,\na[4,3] =9/50,\na[5,1]=15090136995/39027212288,\na[5,2]=0,\na[5,3]=-5310099562 5/39027212288,\na[5,4]=29533111875/19513606144,\na[6,1]=32755/421632, \na[6,2]=0,\na[6,3]=0,\na[6,4]=9765625/29258496,\na[6,5]=1488770/69717 51,\na[7,1]=1072887507055/12570100065468,\na[7,2]=0,\na[7,3]=0,\na[7,4 ]=1075426748046875/9595107694235844,\na[7,5]=-26411117095731200/385766 943434163819,\na[7,6]=23430395520/608661493759,\na[8,1]=23241845542381 9/11076629098233600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-232407582514046875 /2367427911414660864,\na[8,5]=881809586743097626/5311459993015086075, \na[8,6]=-14411638756714/175988393502075,\na[8,7]=19/48,\na[9,1]=-8046 942294592811005033672093485229/31693413191384942205414737020389900,\na [9,2]=0,\na[9,3]=0,\na[9,4]=-536231038652840856428143497005609375/1382 42660274515884207093807951326924,\na[9,5]=-312405090743882815734153055 386047660032/138949582969101476796789174014714391225,\na[9,6]=46374757 254514641197506707162751264/33542814156299871502725237873062925,\na[9, 7]=2051784331177468089852258884/674022310351068617436737421,\na[9,8]=1 762562147235459382311840/629339225351137831406851,\na[10,1]=2157203756 95479264063749229806841635/440720278308021223336256911460990976,\na[10 ,2]=0,\na[10,3]=0,\na[10,4]=12320199720948281040487332306693759765625/ 2354898495606471238613818261942735405056,\na[10,5]=1866488369998347663 2333388794677364275/6604186914582246254039171103542098296,\na[10,6]=-7 8458209082510636098652961448851403235/65709391590090207335818975096879 372288,\na[10,7]=-284269998655149797778753043937075/785609142574129477 34286452523008,\na[10,8]=-40415483223195085710169801875/13000342692048 285899311310848,\na[10,9]=1620538162595625/5204697537445888,\na[11,1]= -975767605592872353548579494188313/1054764737809245464809448652511500, \na[11,2]=0,\na[11,3]=0,\na[11,4]=-5137868354683248477586346693453125/ 495465427355479410622167542061972,\na[11,5]=-4056777797262682827528926 49892020224/71142722483104145892095323206788025,\na[11,6]=245921960510 918251629683797176094768/59908811922681865929060892977735375,\na[11,7] =367953427734636456312437471451631/44380048256994921706877045210625,\n a[11,8]=808591598711928461147452272/133856729103497890946741285,\na[11 ,9]=-3049240467/3093372235,\na[11,10]=3687441170432/6749831213125,\na[ 12,1]=18009539/351697500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5] =0,\na[12,6]=1351744/5131875,\na[12,7]=898538649982788221/365552354571 3900000,\na[12,8]=986955378624/4938547627475,\na[12,9]=397569776903/30 90028048800,\na[12,10]=2957049856/29666896875,\na[12,11]=71669/6284880 ,\n\nb[1]=18009539/351697500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6 ]=1351744/5131875,\nb[7]=898538649982788221/3655523545713900000,\nb[8] =986955378624/4938547627475,\nb[9]=397569776903/3090028048800,\nb[10]= 2957049856/29666896875,\nb[11]=71669/6284880,\n\n`b*`[1]=9132998184216 727/177194863388812500,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0 ,\n`b*`[6]=5971284900143728/23270216609390625,\n`b*`[7]=46560109562892 125961981671/190526192880008411317500000,\n`b*`[8]=1178122040476134609 6/57864557705069066875,\n`b*`[9]=222323369516196533611/155684102950608 0780000,\n`b*`[10]=1246959894769246208/14946997743616640625,\n`b*`[11] =-5978523233/2539290474000,\n`b*`[12]=1/50\}:" }}}{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 0 941 "s ubs(ee,matrix([[c[2],a[2,1],``$2],\n[c[3],a[3,1],a[3,2],``],[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[c[9],a[9,1],a[9,2],a[9,3]],[``,a[9,4], a[9,5],a[9,6]],[``$2,a[9,7],a[9,8]],\n[c[10],a[10,1],a[10,2],a[10,3]], [``,a[10,4],a[10,5],a[10,6]],\n[``,a[10,7],a[10,8],a[10,9]],\n[c[11],a [11,1],a[11,2],a[11,3]],[``,a[11,4],a[11,5],a[11,6]],\n[``,a[11,7],a[1 1,8],a[11,9]],[``$3,a[11,10]],\n[c[12],a[12,1],a[12,2],a[12,3]],[``,a[ 12,4],a[12,5],a[12,6]],\n[``,a[12,7],a[12,8],a[12,9]],[``$2,a[12,11],a [12,10]],\n[``,`_________________________`$3],[`b`,b[1],b[2],b[3]],[`` ,b[4],b[5],b[6]],\n[``,b[7],b[8],b[9]],[``$2,b[10],b[11]],[`b*`,`b*`[1 ],`b*`[2],`b*`[3]],\n[``,`b*`[4],`b*`[5],`b*`[6]],[``,`b*`[7],`b*`[8], `b*`[9]],[``,`b*`[10],`b*`[11],`b*`[12]]]));;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7E7&#\"\"\"\"#;F(%!GF+7&#\"\"%\"#D#!#G\"$D' #\"$G\"F2F+7&#\"\"'F/#\"\"$\"#]\"\"!#\"\"*F:7&#\"$:*\"%'p\"#\",&*p8!4: \",)G7s-RF;#!,Dc*45`FD7&F+F+F+#\",v=6L&H\",Whg8&>7&#\"\"&\"\")#\"&bF$ \"'K;UF;F;7&F+F+#\"(Dcw*\")'\\e#H#\"(q()[\"\"(^<(p7&#\"#N\"$4##\".bq]( )G2\"\"/oa1+,d7F;F;7&F+#\"1vo/[nUv5\"1WeB%p2^f*#!2+7t&4<6TE\"3>Q;MM%pw &Q#\",?bRIM#\"-fP\\h'3'7&#\"#H\"#s#\"0>QUb%=CB\"2+OB)4Hm26F;F;7&F+#!3v o/9De2CB\"4k3m99\"zUnB#\"3Ew4Vne4=))\"4vg3:I**f9J&#!/9nvQ;T9\"0v?]$R)) f<7&F+F+F+#\"#>\"#[7&#\"#6\"#8#!CH_[$4sO.05\"Gf%HUp/)\"D+**Q?qt9a?U\\Q \">8MpJF;F;7&F+#!Ev$4c+(\\V\"Gk&3%GlQ5BO&\"ECpK^z!Q42U)e^u-mU#Q\"#!HK+ mZgQbI:Md\"G)Qu!40CJ\"HD7R9Z,u\"*y'zw95pHe\\*Q\"#\"Dk7virq1v>TY^asvuj% \"DDH1tyBDF]r)*HcT\"GaL7&F+F+#\"=%))eA&)*3ou?B\"\"Ic]SNF%>E=QhQ7Z 1c\\)*[N##\"GvUOxYz)QLBjwM)**p$)[m=\"F'H)4UN5r\"RSDYAe9p=/m#!GNKS^)[9' Hl)4O1^#34#e%y\"G)Gs$zo4v*=eL2-4!f\"R4d'7&F+#!Bvq$R/`(yxz\\^l)**pUG\"A 3I__kGMx%HTdU\"4cy#!>v=!)p,r&3&>BK[:/%\">[3J6$**eG[?pU.+8#\"1Dcfi\"Q0i \"\"1))eWPvp/_7&F)#!B8$)=%\\z&[NNsGf0wwv*\"C+:^_'[%4[YX#4ytkZ0\"F;F;7& F+#!CDJX$pYjex%[Koa$oy8&\"Bs>1Uv;A1TzaNFaY&\\#!EC--#*)\\E*Gv#Goiszxn0% \"DD!)y1KK&4#*e9/J[AF9r#\"EoZ4wrz$oH;D=4^g>#fC\"Dv`txH*31Hf'=oA>\")3*f 7&F+#\"BJ;XruV7jXOYtFM&zO\"AD1@Xq(oq@\\*pD[+QW#\"r)f\"f3)\"<& GTn%4*y\\.\"Hn&Q8#!+n/C\\I\"+NAP$4$7&F+F+F+#\".K/x\"F;F;7&F+ F;F;#\"1GP9+\\Grf\"2D1R4m@qK#7&F+#\";r;)>'f7#*Gc4,cY\"<++]<8T3+)G>E0># \"5'4Y8w//A\"y6\"5vo1p]qdX'y&#\"66O`'>;&pLKA#\"7++y!31&H5%ob\"7&F+#\"4 3iCpZ*)fpC\"\"5D1k;Ou(*p%\\\"#!+LK_yf\".+SZ!HRD#F)F:Q)pprint276\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12 ),\n[`b`,seq(b[j],j=1..11),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[ 6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'+]i!\"(F (%!GF+F+F+F+F+F+F+F+F+F+7/$\"'++;!\"'$!'+![%F*$\"'+[?F/F+F+F+F+F+F+F+F +F+F+7/$\"'++CF/$\"'++gF*$\"\"!F:$\"'++=F/F+F+F+F+F+F+F+F+F+7/$\"'0&R& F/$\"'dmQF/F9$!'hg8!\"&$\"'Y8:FDF+F+F+F+F+F+F+F+7/$F)F/$\"'ioxF*F9F9$ \"'rPLF/$\"'VN@F/F+F+F+F+F+F+F+7/$\"'ku;F/$\"'BN&)F*F9F9$\"'\"37\"F/$! 'RYoF*$\"']\\QF*F+F+F+F+F+F+7/$\"'yFSF/$\"'G)4#F*F9F9$!')o\")*F*$\"'?g ;F/$!'(*)=)F*$\"'LeRF/F+F+F+F+F+7/$\"'ah%)F/$!'+RDF/F9F9$!'\"*yQFD$!'L [AFD$\"'b#Q\"FD$\"'4WIFD$\"'m+GFDF+F+F+F+7/$\"'+v$*F/$\"'s%*[F/F9F9$\" 'tJ_FD$\"'AEGFD$!'-%>\"FD$!'Z=OFD$!'!)3JFD$\"'h8JF/F+F+F+7/$\"\"\"F:$! '/^#*F/F9F9$!')p.\"!\"%$!'J-dFD$\"'%\\5%FD$\"''4H)FD$\"'sSgFD$!'Ld)*F/ $\"',jaF/F+F+7/Fbq$\"'v?^F*F9F9F9F9$\"'-MEF/$\"'.eCF/$\"'Z)*>F/$\"'i'G \"F/$\"'^n**F*$\"'MS6F*F+7/%\"bGFfrF9F9F9F9FhrFjrF\\sF^sF`sFbsF+7/%#b* G$\"'@a^F*F9F9F9F9$\"'1mDF/$\"'wVCF/$\"'+O?F/$\"'/G9F/$\"'aU$)F*$!'TaB !\")$\"'++?F*Q)pprint616\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions( 11,'expanded')),op(OrderConditions(7,11,'expanded'))]:\n`RK6_12eqs*` : = subs(b=`b*`,OrderConditions(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(ee,RK7_11eqs)):\nmap(u->lhs( u)-rhs(u),%);\nnops(%);\nsimplify(subs(ee,`RK6_12eqs*`)):\nmap(u->lhs( u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $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#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F $F$F$F$F$F$F$F$F$F$F$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#\"#P" }}}{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 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_11 := Stag eOrderConditions(ct,11,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Stages 6 to 11 have stag e-order 4, while stages 4 and 5 have stage-order 3 has and stage 3 has stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([s eq(expand(subs(ee,so||i||_11[j])),i=2..6)],j=1..9)]:\nmap(proc(L) loca l 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&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 .. 11) = b[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 10 " }}{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 0 151 "[Sum(b[i]*a[i,1] ,i=1+1..11)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=2..10) ];\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#7,/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\" aG6$F,F-F-/F,;\"\"#\"#6&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,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$* &F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F /6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_ rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-/-F&6$*&F)F-&F/6$F,F]sF-/F,;F 4F4*&&F*6#F]sF-,&F-F-&FEF]tFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7, \"\"!F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of t he principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "RK7_11err_eqs := PrincipalErrorConditions(7,11,'e xpanded'):\nexpand(subs(ee,RK7_11err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u) ,0,1),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"\"F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 93 "We can calculate the principal error norm, that is, the 2-norm of the pr incipal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "er rterms7_11 := PrincipalErrorTerms(7,11,'expanded'):\nsm := 0:\nfor ct \+ to nops(errterms7_11) do\n sm := sm+(evalf(subs(ee,errterms7_11[ct]) ))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S_(yp \"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm of the order 6 embedded scheme is given as f ollows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms6_12*` \+ := subs(b=`b*`,PrincipalErrorTerms(6,12,'expanded')):\nsm := 0:\nfor c t to nops(`errterms6_12*`) do\n sm := sm+(evalf(subs(ee,`errterms6_1 2*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+rbxF8!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------- -------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "We incorporate the stage -order equations to ensure that stage 3 has stage-order 2, stages 4 an d 5 have stage order 3 and stages 6 to 11 have stage-order 4." }} {PARA 0 "" 0 "" {TEXT -1 57 "These conditions lead to relationships be tween the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c [3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ", " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6] +6*c[5]*c[4] = 0" "6#/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F' &F*6#F,F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F* 6#F/F'F'\"\"!" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 68 "which \+ we include explicitly to make the computation more efficient. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We also i ncorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j]); " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"# 6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" } {TEXT -1 9 " ), and " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 " . . 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "The simplifying conditions allow us to manage with just \+ 10 \"simple\" order conditions which are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 7 order 7 quadrature conditions and 3 additional order 7 order conditions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,3 2,38,48,61,64])]:\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()F 2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#KF)/*&F,F()F2FGF(#F(\"\"' 7%\"#QF)/*&F,F(-%!G6#*&%\"aGF(-FT6#*&FWF(-FT6#*&FWF(-FT6#*&F8F(FWF(F(F (F(F(#F(\"%?D7%\"#[F)/*&F,F(-FT6#*&FWF(-FT6#*&FWF(-FT6#*&F?F(FWF(F(F(F (#F(\"$S)7%\"#hF)/*(F,F(F2F(-FT6#*&FEF(FWF(F(#F(\"#N7%\"#kF)/*&F,F()F2 FNF(#F(\"\"(Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "SO7_11 := SimpleOrderConditions(7, 11,'expanded'):\nSO_eqs := [op(RowSumConditions(11,'expanded')),op(Sta geOrderConditions(2,11,'expanded')),\n op(StageOrderConditions(3,4..1 1,'expanded')),op(StageOrderConditions(4,6..11,'expanded'))]:\nord_cdn s := [seq(SO7_11[i],i=[1,2,4,8,16,32,38,48,61,64])]:\nsimp_eqs := [add (b[i]*a[i,1],i=2..11)=b[1],seq(add(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j] ),j=[$8..10])]:\nnode_eqs := [3*c[3]=2*c[4],3*c[6]^2-4*c[6]*c[4]-4*c[5 ]*c[6]+6*c[5]*c[4]=0]:\ncdns := [op(node_eqs),op(simp_eqs),op(SO_eqs), op(ord_cdns)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The schem e can be obtained by specifying the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/16;" "6#/&%\"cG6#\"\"#*&\"\"\" F)\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 6/25;" "6#/&%\"c G6#\"\"%*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 5/8;" "6#/&%\"cG6#\"\"'*&\"\"&\"\"\"\"\")!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 35/209;" "6#/&%\"cG6#\"\"(*&\"#N\"\"\"\"$4#!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 29/72;" "6#/&%\"cG6#\"\")*& \"#H\"\"\"\"#s!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 11/13;" "6#/&%\"cG6#\"\"**&\"#6\"\"\"\"#8!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 15/16;" "6#/&%\"cG6#\"#5*&\"#:\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 45 "together with the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[ 4,2] = 0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10, 2] = 0;" "6#/&%\"aG6$\"#5 \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG 6$\"#6\"\"#\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[6,3] = 0;" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3] = 0;" "6#/&%\"aG6$\"\"(\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,3] = 0;" "6#/&%\"aG6$\"\"* \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,3] = 0;" "6#/&%\"aG 6$\"#5\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/ &%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[4] = 0;" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[5] = 0;" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "e1 := \{c[2]=1/16,c[4]=6/25,c[6]=5/8,c[7]=35/209,c[8]=29/72,c[9]= 11/13,c[10]=15/16,c[11]=1,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6. .11),seq(b[i]=0,i=2..5)\}:\neqns := subs(e1,cdns):\nnops(eqns);\nindet s(eqns) minus \{a[8,7],a[9,7],a[9,8]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\\" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[8,7],a[9,7],a[9,8 ]\}):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(ma p(rhs,e3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\")\"\"(&F% 6$\"\"*F(&F%6$F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3769 "e3 := \{a[11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = \+ 0, b[5] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[8,2] = 0, a[9,3] = \+ 0, a[9,2] = 0, b[2] = 0, a[10,3] = 0, a[11,2] = 0, a[11,1] = 535698859 7689/413135950500-268347837888821030400/11730163276510776383*a[8,7]-29 9105887061812807275/118882031015078867932*a[9,7]+1793837837508259699/1 779943259178587250*a[9,8], a[11,4] = 2080842453125/16931983068-2711557 0949175000000000/175567853630398997339*a[8,7]-30223559711707958984375/ 1779332693717491908556*a[9,7]-554834655372648584375/760860965035971025 38*a[9,8], a[10,1] = -25100954023655/15693063585792+337400235626813551 40625/11542391006634044699648*a[8,7]+1405132056651148477640625/2898439 166054833206198272*a[9,7]-67416367485825654205/347171542189021790208*a [9,8], a[8,1] = -169934125/1113778138*a[8,7]+1875958409/23052729600, a [9,5] = 1617799909376/4021295967249*a[9,7]-44667864064/42353387325*a[9 ,8]-74966396403712/144362130725475, a[5,3] = -53100995625/39027212288, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, c[11] = 1, a[8,6] = - 639752000/2912255951*a[8,7]+63974/12629925, a[4,3] = 9/50, b[9] = 3975 69776903/3090028048800, a[7,1] = 1072887507055/12570100065468, a[7,4] \+ = 1075426748046875/9595107694235844, a[6,1] = 32755/421632, a[5,1] = 1 5090136995/39027212288, c[2] = 1/16, c[4] = 6/25, c[6] = 5/8, c[7] = 3 5/209, c[10] = 15/16, c[8] = 29/72, c[9] = 11/13, a[11,5] = -386604784 525312/362252356253775+8515713717845669160419328/141172515032807193769 405*a[8,7]+81825736712737654827008/12334010717814432547945*a[9,7]-6551 7648402575560974848/3767249908051479914625*a[9,8], a[5,4] = 2953311187 5/19513606144, b[8] = 986955378624/4938547627475, a[2,1] = 1/16, a[6,4 ] = 9765625/29258496, a[3,2] = 128/625, a[11,9] = -3049240467/30933722 35, c[5] = 915/1696, b[10] = 2957049856/29666896875, a[3,1] = -28/625, b[6] = 1351744/5131875, a[8,4] = -51513671875/50010464262*a[8,7]+1525 245484375/4927101467904, b[7] = 898538649982788221/3655523545713900000 , a[7,5] = -26411117095731200/385766943434163819, a[4,1] = 3/50, c[3] \+ = 4/25, a[10,6] = -66994138012435/60511150791168+198470726839302089062 5/471570534210586611064*a[8,7]+82654826861832263390625/118417276379545 567722496*a[9,7]-9630909640832236315/6956078258796429312*a[9,8], b[1] \+ = 18009539/351697500, b[11] = 71669/6284880, a[10,4] = -47645821689453 125/2891471361343488+3409306405963165283203125/17275742605011283296358 4*a[8,7]+141983472923936798095703125/43381556698492831430475776*a[9,7] +372355454863661013671875/265005713425367567499264*a[9,8], a[9,4] = -5 1513671875/50010464262*a[9,7]-945671875/2138498901*a[9,8]+45257609375/ 91399427028, a[10,5] = 3108615032725/8108976826008-2091213495583782013 71000/27131381985320454679739*a[8,7]-1088629619095899060191375/8516287 93053464786049712*a[9,7]+4809174193669030149055/1435133923912235583936 *a[9,8], a[8,5] = 1617799909376/4021295967249*a[8,7]+2581824934/381180 485925, a[11,6] = 188799182421376/23465407090125-202050136763347599360 0/61342985003392420757*a[8,7]-19414580904620709600/5359435824450276833 *a[9,7]+1793837837508259699/249645958026869250*a[9,8], a[7,6] = 234303 95520/608661493759, a[9,6] = -639752000/2912255951*a[9,7]+611491/14033 25*a[9,8]+120436096/144947075, a[6,5] = 1488770/6971751, a[11,8] = 880 0642221267/533688115070*a[9,8]-325539030347136/8109537631195, a[10,8] \+ = 1140060703753125/196902256961024-8268686802319125/2602348768722944*a [9,8], a[11,7] = -272251330110856964813/2688708097267749375+1579128605 7984/105318670535*a[8,7]+8800642221267/533688115070*a[9,7], a[10,7] = \+ 6714597760818558325/492363296482394112-3772405617123375/19690225696102 4*a[8,7]-8268686802319125/2602348768722944*a[9,7], a[11,10] = 36874411 70432/6749831213125, a[9,1] = -169934125/1113778138*a[9,7]+611491/1000 5525*a[9,8]+185280623/4703996700, a[10,9] = 1620538162595625/520469753 7445888\}:" }{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 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "subs(e3,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7)]));\nfor ii from 8 to 11 do \n print(``);\n print(c[ii]=subs(e3,c[ii])); \n for jj to ii-1 d o\n coeff_val := subs(e3,a[ii,jj]);\n if coeff_val=a[ii,jj] \+ then \n print(a[ii,jj]*` is a parameter.`)\n else\n \+ print(a[ii,jj]=coeff_val);\n end if;\n end do:\nend do:\npri nt(``);\nseq(b[i]=subs(e3,b[i]),i=1..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\"#;F(%!GF+F+F+F+F+7*#\"\"%\"#D# !#G\"$D'#\"$G\"F2F+F+F+F+F+7*#\"\"'F/#\"\"$\"#]\"\"!#\"\"*F:F+F+F+F+7* #\"$:*\"%'p\"#\",&*p8!4:\",)G7s-RF;#!,Dc*45`FD#\",v=6L&H\",Whg8&>F+F+F +7*#\"\"&\"\")#\"&bF$\"'K;UF;F;#\"(Dcw*\")'\\e#H#\"(q()[\"\"(^<(pF+F+7 *#\"#N\"$4##\".bq]()G2\"\"/oa1+,d7F;F;#\"1vo/[nUv5\"1WeB%p2^f*#!2+7t&4 <6TE\"3>Q;MM%pw&Q#\",?bRIM#\"-fP\\h'3'F+Q(pprint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\" )#\"#H\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&*& #\"*DT$*p\"\"+Q\"yP6\"F(&F%6$F'\"\"(F(!\"\"#\"+4%ef(=\",+'HF0BF(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\",v=n8:&\",iUY5+&\"\"\"&F%6$F' \"\"(F.!\"\"#\".vV[X__\"\"./zY,r#\\F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\".w$4**z<;\".\\s'fH@S\"\"\"&F%6$F'\"\"(F.F .#\"+M\\#=e#\"-Df[!=\"QF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"',&*&#\"*+?vR'\"+^fD7H\"\"\"&F%6$F'\"\"(F.!\"\"#\"&uR'\")D*HE \"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"(\"\"\"%1~i s~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#6\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\",(*&#\"*DT$*p\"\"+Q\"yP6\"F(&F%6$F' \"\"(F(!\"\"*&#\"'\"\\6'\")Db+5F(&F%6$F'\"\")F(F(#\"*B1G&=\"++n*Rq%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,(*&#\",v=n8:&\",iUY5+&\"\"\"&F%6$ F'\"\"(F.!\"\"*&#\"*v=nX*\"+,*)\\Q@F.&F%6$F'\"\")F.F2#\",v$4wDX\",GqU* R\"*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,(*&#\".w $4**z<;\".\\s'fH@S\"\"\"&F%6$F'\"\"(F.F.*&#\",kS'ymW\",DtQ`B%F.&F%6$F' \"\")F.!\"\"#\"/7PS'Rm\\(\"0vasI@OW\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',(*&#\"*+?vR'\"+^fD7H\"\"\"&F%6$F'\"\"(F.!\" \"*&#\"'\"\\6'\"(DLS\"F.&F%6$F'\"\")F.F.#\"*'4O/7\"*vq%\\9F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\"(\"\"\"%1~is~a~parameter. GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\")\"\"\"%1~is ~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",*#\"/bO-a45D\"/#z&ejIp:!\"\"*&#\"8D 19b8oiN-SP$\"8['*pWSj15RU:\"F(&F%6$\"\")\"\"(F(F(*&#\":D1kx%[6lc?809\" :s#)>1K$[0m\"R%)*GF(&F%6$\"\"*F5F(F(*&#\"50UlDe[njTn\"63-z@!*=U:9\";wdZI9$G\\)pc:QVF2&F%6$\"\"*F6F2F2*&#\"9v=n85mj[X bBP\"9k#*\\nvODMr0]EF2&F%6$F=F5F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#5\"\"&,*#\".DF.:'3J\".3g#o(*3\")\"\"\"*&#\"9+5P,#y$e&\\87 4#\"8R(zYX?`)>QJr#F-&F%6$\"\")\"\"(F-!\"\"*&#\":v8>g!**e4>'H')3\"\"97( \\gykM0$zG;&)F-&F%6$\"\"*F5F-F6*&#\"7b!\\,.pO>u\"4[\"7OReNA\"R#R8N9F-& F%6$F=F4F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',*# \"/NC,QT*p'\"/o6z]6^g!\"\"*&#\"7D1*3-$Rosq%)>\"6k5h'e5U`q:Z\"\"\"&F%6$ \"\")\"\"(F2F2*&#\"8D1RjA$='o#[l#)\"9'\\AxcX&zjF<%=\"F2&F%6$\"\"*F6F2F 2*&#\"4:jBK3k44j*\"47$Hkze#yg&pF2&F%6$F=F5F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,(#\"4D$e&=3w(f9n\"37TR#['HjB\\\"\"\" *&#\"1vL7F-&F%6$\"\")F(F-!\"\"*&#\"1D\">B!ooo#)\"1WHso ([Bg#F-&F%6$\"\"*F(F-F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"),&#\"1DJv.21S6\"0C5'pD-p>\"\"\"*&#\"1D\">B!ooo#)\"1WHso([Bg#F-& F%6$\"\"*F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"*#\"1Dcfi\"Q0i\"\"1))eWPvp/_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",*#\".*o(f))pN&\"-+0&f88%F(*& #\"6+/.@)))y$yMo#\"5$Qw2^wK;I<\"F(&F%6$\"\")\"\"(F(!\"\"*&#\"6vs!G\"=1 ()e5*H\"6Kz')y],J?))=\"F(&F%6$\"\"*F4F(F5*&#\"4*pf#3v$y$Qz\"\"4]sey\"f K%*z$p\"\"\"\"*&#\"8++++]<\\4d:r#\"6Rt**)RIO&ycv\"F-&F%6$\"\")\"\"(F-! \"\"*&#\"8vV)*ezq6(fNAI\"7c&3>\\/;pc%yr8d^)\"9 0%pP>2G.:D\"4LoF]W#eVf`F-&F%6$\"\"*F5F-F6*&#\"4*pf#3v$y$Qz \"\"3]#po-efk\\#F-&F%6$F=F4F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"(,(#\"68['p&36I8Ds#\"4v$\\xE(43()o#!\"\"*&#\"/%)z0'G\"z: \"-N0n=`5\"\"\"&F%6$\"\")F(F2F2*&#\".n7AU1!))\"-q]6)oL&F2&F%6$\"\"*F(F 2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&*&#\".n7AU1 !))\"-q]6)oL&\"\"\"&F%6$\"\"*F(F.F.#\"0OrMI!RbK\".&>Jw`4\")!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!+n/C\\I\"+NAP$4$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#\".K/ \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK7_11eqs := [op(RowSumConditions(1 1,'expanded')),op(OrderConditions(7,11,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simplify(subs(e3,RK7_11eqs)):\nmap( u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q \"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$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#\"#&*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We now turn our attention to the embedded 12 stage, order 6 scheme." }} {PARA 0 "" 0 "" {TEXT -1 210 "The first 11 stages of the embedded orde r 6 scheme coincide with those of the order 7 scheme and the weights o f the order 7 scheme provide the linking coefficients for the 12th sta ge of the order 6 scheme. Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[12,j] = b[j];" "6#/&%\"aG6$\"#7%\"jG&%\"bG6#F(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 11." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplif ying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*& &%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and \+ " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The simplifying condit ions allow us to manage with just 7 \"simple\" order conditions which \+ are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 6 order 6 quadrature conditions and 1 additio nal order 6 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO6*` := subs(b=`b*`,Simpl eOrderConditions(6)):\n[seq([i,`SO6*`[i]],i=[1,2,4,8,16,31,32])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#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()F2F:F(#F(F57%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#JF)/*&F,F(-%!G6#*&FEF(%\"aGF(F(#F(\"#I7%\"#KF)/ *&F,F()F2FGF(#F(\"\"'Q)pprint346\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "`SO6_12*` := subs(b=`b*`,S impleOrderConditions(6,12,'expanded')):\n`cdns*` := [seq(`SO6_12*`[i], i=[1,2,4,8,16,31,32])]:\n`simp_eqs*` := [seq(add(`b*`[i]*a[i,j],i=j+1. .12)=`b*`[j]*(1-c[j]),j=[4,5])]:\nwt_eqns := [seq(b[i]=a[12,i],i=1..11 )]:\n`cdns*` := [op(`cdns*`),op(wt_eqns),op(`simp_eqs*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] = 0;" "6#/&%#b*G6#\"\"%\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5] = 0;" "6#/&%#b*G6#\"\"& \"\"!" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "`b*`[12] = 1/50;" "6#/&% #b*G6#\"#7*&\"\"\"F)\"#]!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e4 := \{c[12 ]=1,seq(`b*`[i]=0,i=2..5),`b*`[12]=1/50\}:\ne5 := `union`(e3,e4):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqns2 := [op(remove(_U->type(rhs(_U),realcons),e5)),op(subs(e5,`c dns*`))]:\nnops(eqns2);\nindets(eqns2) minus \{a[8,7]\};\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "e6 := solve(\{op(eqns2)\},indets(e qns2) minus \{a[8,7]\}):\ne7 := simplify(subs(select(u_->member(lhs(u_ ),\{b[8],a[8,7],a[9,7],a[9,8]\}),e6),e5)):\ne8 := `union`(e7,e6):\ninf olevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a [10,4]=subs(e8,a[10,4]);\na[11,1]=subs(e8,a[11,1]);\n`b*`[11]=subs(e8, `b*`[11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"? vV[Y^=#zZD1.?oZ&\">W\")>o&)zT&\\+AwosZ\"\"\"*&#\"GD\"G)[DhYK$f8fEt%)oD 'H5\"EcajRe11[Bs'*\\;0BkFlF-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"9p&RB++!)HBc\"H\\\"9+&**3$)z) 3CE/V=!\"\"*&#\">+Sii!QS,q@N$Q%4s\">TrD\"GC$Qb'zu2IJ;F(&F%6$\"\")\"\"( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!+LK_yf\".+SZ!HR D" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Note that some of the coefficients \+ depend on " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4401 " e8 := \{a[11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[8,2] = 0, a[9,3] = 0, a[9,2] = 0, b[2] = 0, a[10,3] = 0, a[11,2] = 0, `b*`[11] = -5978523233/2539290474000, \+ `b*`[12] = 1/50, a[9,8] = 1762562147235459382311840/629339225351137831 406851, a[9,7] = 170562943443328508109024724/2324214863279546956678404 9-6827754158759389403649024/629339225351137831406851*a[8,7], a[11,8] = 808591598711928461147452272/133856729103497890946741285, `b*`[1] = 91 32998184216727/177194863388812500, `b*`[9] = 222323369516196533611/155 6841029506080780000, `b*`[10] = 1246959894769246208/149469977436166406 25, a[11,4] = -6458988830053302378921875/291207242196632211806412+7284 874669393505027062500000000/244160263506435024427357176553*a[8,7], a[8 ,1] = -169934125/1113778138*a[8,7]+1875958409/23052729600, a[5,3] = -5 3100995625/39027212288, a[10,7] = -26222759385858556907304576558175/27 08997043359067163251256983552+599622770614193912143828125/391576587109 88812949732864*a[8,7], a[9,1] = -14329409333116286465358973/1576184895 4170417001476300+52739474031085814720153122752000/31861103208270576820 258201237429*a[8,7], a[9,6] = 298555581540982504349536/679814328429286 135309575+397097216234057899069388218368000/16661790038423465583053059 3356391*a[8,7], a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[ 9,4] = -23261499144285579905328125/2801772708064423410075876+532913162 6931785497195875000000000/476871921789361256342553077537257*a[8,7], a[ 9,5] = -16128244056341868829994958848/30977099503537281327651404025-33 4725456341895372539772294270025728/76689675422307278406361490378491603 *a[8,7], a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, `b*`[6] = 597 1284900143728/23270216609390625, a[10,5] = 51887968719257740266665725/ 133847472404607027828332904+631555770806919887321338571941875000/10251 5973347273430414006740986356101*a[8,7], `b*`[2] = 0, `b*`[3] = 0, `b*` [4] = 0, `b*`[5] = 0, a[11,7] = 876798545717856496590326725071631/4438 0048256994921706877045210625-9333478097749585663488/322224111078073470 979*a[8,7], c[11] = 1, c[12] = 1, a[8,6] = -639752000/2912255951*a[8,7 ]+63974/12629925, a[4,3] = 9/50, a[10,6] = 183149557746430472892673735 /1331736380470759888706701312-5993904167905809120028723916015625/17818 30072004752493863834328816776*a[8,7], a[11,6] = 5585246007267121095021 0128/35211094336889194396661625+542828298181044811580706816000/8530900 7731164044679438049639*a[8,7], b[9] = 397569776903/3090028048800, a[7, 1] = 1072887507055/12570100065468, a[7,4] = 1075426748046875/959510769 4235844, a[11,5] = -501160340702676161290502144/4599514387909306988263 77525-457566667294004865331285916123136/392654096493530507465449859383 87*a[8,7], a[6,1] = 32755/421632, a[5,1] = 15090136995/39027212288, a[ 11,1] = -492915623298000002339569/184304262408879830899500+72094383352 170014038062624000/16313007747965538324281257141*a[8,7], c[2] = 1/16, \+ c[4] = 6/25, c[6] = 5/8, c[7] = 35/209, c[10] = 15/16, c[8] = 29/72, c [9] = 11/13, a[12,7] = 898538649982788221/3655523545713900000, a[5,4] \+ = 29533111875/19513606144, b[8] = 986955378624/4938547627475, a[2,1] = 1/16, a[12,9] = 397569776903/3090028048800, a[6,4] = 9765625/29258496 , a[3,2] = 128/625, a[11,9] = -3049240467/3093372235, c[5] = 915/1696, `b*`[8] = 11781220404761346096/57864557705069066875, a[10,4] = 547682 003062547792185146484375/47726876220049541798568198144-102962568847326 59135933246612548828125/652764230516499672234806065839635456*a[8,7], a [12,10] = 2957049856/29666896875, a[10,1] = 36634479472387668019235558 5/259031064006269967391850496-101896370854398755040488306572265625/436 12944207564600225983756863327232*a[8,7], `b*`[7] = 4656010956289212596 1981671/190526192880008411317500000, b[10] = 2957049856/29666896875, a [10,8] = -40415483223195085710169801875/13000342692048285899311310848, a[3,1] = -28/625, b[6] = 1351744/5131875, a[12,6] = 1351744/5131875, \+ a[8,4] = -51513671875/50010464262*a[8,7]+1525245484375/4927101467904, \+ a[12,8] = 986955378624/4938547627475, b[7] = 898538649982788221/365552 3545713900000, a[7,5] = -26411117095731200/385766943434163819, a[4,1] \+ = 3/50, c[3] = 4/25, b[1] = 18009539/351697500, b[11] = 71669/6284880, a[8,5] = 1617799909376/4021295967249*a[8,7]+2581824934/381180485925, \+ a[12,1] = 18009539/351697500, a[7,6] = 23430395520/608661493759, a[6,5 ] = 1488770/6971751, a[11,10] = 3687441170432/6749831213125, a[12,11] \+ = 71669/6284880, a[10,9] = 1620538162595625/5204697537445888\}:" } {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 65 "The coefficients for the embedded order 6 scheme a re as follows. " }}{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 0 477 "subs (e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7)]));\nfor \+ ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e8,c[ii])); \n \+ for jj to ii-1 do\n coeff_val := subs(e8,a[ii,jj]);\n if c oeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`)\n else\n print(a[ii,jj]=coeff_val);\n end if;\n end do:\nend do:\nprint(``);\nc[12]=subs(e8,c[12]);\nseq(a[12,i]=subs(e8, a[12,i]),i=1..11);\nprint(``);\nseq(`b*`[i]=subs(e8,`b*`[i]),i=1..12); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\"#;F(%!GF+ F+F+F+F+7*#\"\"%\"#D#!#G\"$D'#\"$G\"F2F+F+F+F+F+7*#\"\"'F/#\"\"$\"#]\" \"!#\"\"*F:F+F+F+F+7*#\"$:*\"%'p\"#\",&*p8!4:\",)G7s-RF;#!,Dc*45`FD#\" ,v=6L&H\",Whg8&>F+F+F+7*#\"\"&\"\")#\"&bF$\"'K;UF;F;#\"(Dcw*\")'\\e#H# \"(q()[\"\"(^<(pF+F+7*#\"#N\"$4##\".bq]()G2\"\"/oa1+,d7F;F;#\"1vo/[nUv 5\"1WeB%p2^f*#!2+7t&4<6TE\"3>Q;MM%pw&Q#\",?bRIM#\"-fP\\h'3'F+Q(pprint8 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#H\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&*&#\"*DT$*p\"\"+Q\"yP6\"F(&F%6$F'\"\"(F(! \"\"#\"+4%ef(=\",+'HF0BF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\",v= n8:&\",iUY5+&\"\"\"&F%6$F'\"\"(F.!\"\"#\".vV[X__\"\"./zY,r#\\F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\".w$4**z<;\". \\s'fH@S\"\"\"&F%6$F'\"\"(F.F.#\"+M\\#=e#\"-Df[!=\"QF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"*+?vR'\"+^fD7H\"\"\"&F%6 $F'\"\"(F.!\"\"#\"&uR'\")D*HE\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &&%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \"#6\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\",&#\"; t*e`Y'G;JL4%HV\"\";+jZ,qTqT&*[=w:!\"\"*&#\"A+?vAJ:?Z\"e3JSZRF&\"AHuB,# e-#odq#3K5h=$F(&F%6$\"\")\"\"(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"% ,&#\";D\"G`!*zbGW\"*\\hK#\":we25MUk!3Fx,G!\"\"*&#\"C++++](e>(\\&yJpiJ \"H`\"Bds`xIbUjDh$*y@>(oZ\"\"\"&F%6$\"\")\"\"(F2F2" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,&#\">[)e\\**H)o=McSCGh\"\">DSS^wK \"GPN]*4x4$!\"\"*&#\"EGd-qUHs(RDP&*=McasM$\"D.;\\y.\\hjSysIAan*ow\"\" \"&F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"',&#\"9O&\\V]#)4a\"eb&)H\"9v&4`8'GH%GV\")z'\"\"\"*&#\"B+!o$=#)Qp! **y0Mi@(4(R\"B\"RcLfI0$elMUQ+zhm\"F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"!owQs%zWjO\"<'\\]=Rn *pi+k5.f#F(*&#\"EDcEslI)[S]v)Ra3P'*=5\"DKsKjov$)fA+Yc2U%HhVF(&F%6$\"\" )\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"?vV[Y^=#zZD1.?o Z&\">W\")>o&)zT&\\+AwosZ\"\"\"*&#\"GD\"G)[DhYK$f8fEt%)oD'H5\"EcajRe11[ Bs'*\\;0BkFlF-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#5\"\"&,&#\";Ddmm-ud#>(oz)=&\"dQ8K())>p!3xb:j\"E,hN')4u1STIMFZL(f^-\"F-&F%6$\"\")\"\"(F-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"91xA'*f\";kGt\\H\")))4rew:R\"\"\"&F%6$\"\")F(F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!>v=!)p,r&3&>BK[:/% \">[3J6$**eG[?pU.+8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"*#\"1Dcfi\"Q0i\"\"1))eWPvp/_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"9p&RB++!)HBc\"H\\\"9+&** 3$)z)3CE/V=!\"\"*&#\">+Sii!QS,q@N$Q%4s\">TrD\"GC$Qb'zu2IJ;F(&F%6$\"\") \"\"(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,&#\":v=#*yBI`+$)))*ek\"97k!= @Km>Us?\"H!\"\"*&#\"@++++D1F]]$RpY([G(\"?`l*)oL%46 _$\"\"\"*&#\"?+g\"oq!e6[/\"=)HGGa\">R'\\!Q%zY/k6t2!4`)F-&F%6$\"\")\"\" (F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"BJ;2Dn K!f'\\cyrX&)zw)\"AD1@Xq(oq@\\*pD[+QW\"\"\"*&#\"7)[jce\\x4yML*\"6z4Zt!y 56CAKF-&F%6$\"\")F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\")#\"r)f\"f3)\"<>n%4*y\\.\"Hn&Q8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!+n/C\\I\"+NAP$4$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#\".K/x\"/&F%6#\"\" #\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"1GP9+\\Grf \"2D1R4m@qK#/&F%6#\"\"(#\";r;)>'f7#*Gc4,cY\"<++]<8T3+)G>E0>/&F%6#\"\") #\"5'4Y8w//A\"y6\"5vo1p]qdX'y&/&F%6#\"\"*#\"66O`'>;&pLKA#\"7++y!31&H5% ob\"/&F%6#\"#5#\"43iCpZ*)fpC\"\"5D1k;Ou(*p%\\\"/&F%6#\"#6#!+LK_yf\".+S Z!HRD/&F%6#\"#7#F'\"#]" }}}{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 0 151 "RK7_11eqs := [op(RowSum Conditions(11,'expanded')),op(OrderConditions(7,11,'expanded'))]:\n`RK 6_12eqs*` := subs(b=`b*`,OrderConditions(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(e8,RK7_11eqs) ):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(e8,`RK6_12eqs*`) ):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$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#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "We choose a value for " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 85 " tha t makes the principal error norm for the order 7 scheme approximately \+ a minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 140 "errterms := PrincipalErrorTerms(7,11,'expanded'): \nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e8,er rterms[ct])))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot(sqrt(sm),a[8,7]=.25..0.55,colo r=COLOR(RGB,.5,0,1),thickness=2,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 277 277 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$\"3++++++++D! #=$\"3(RH@D1n(4A!#A7$$\"3;++]i9RlDF*$\"3uzw6W(y(p@F-7$$\"34+]PC#)GAEF* $\"3&4L_D*R!f8#F-7$$\"3;++v$euio#F*$\"3s@pnnP)))4#F-7$$\"3')***\\i3&o] FF*$\"3)Rm*)=vFG1#F-7$$\"3#)**\\(oX*y9GF*$\"3&=u]W37#G?F-7$$\"3/+]P9CA uGF*$\"3%RHvwP2t*>F-7$$\"3:+]P*zhd$HF*$\"3%=z4wNrl'>F-7$$\"3++]P>fS**H F*$\"3_!><'\\7?O>F-7$$\"35+](=$f%G1$F*$\"3!p_1e@Iu!>F-7$$\"3:++]#y,\"G JF*$\"3'[c$)o\\s%z=F-7$$\"3M++Dr\"zb=$F*$\"3g#o&f.9Gc=F-7$$\"3G++](4&G ]KF*$\"33B82\"HZ=$=F-7$$\"3()****\\7nD:LF*$\"3fMiA%)e;4=F-7$$\"3G++]-* oyP$F*$\"3!3Y^&>58*y\"F-7$$\"3****\\PpnsMMF*$\"3a8Mnv#RDx\"F-7$$\"38++ ]siL-NF*$\"37Xo$Q5c[v\"F-7$$\"3M+++!R5'fNF*$\"3U\\;iKEmT\\3@^(f+r\"F-7$$\"3I+]7j=_6QF*$\"3z&)*RF*$\"3S(Q+O7x#)p\"F-7$$\"3?+]P>:mkSF*$\"3UH-j#oy3q\"F-7$$ \"3=+]iv&QA7%F*$\"3))zv2_0503W-w\"F-7$$\"3Y+++: K^+XF*$\"3/d:^z1lwQymDhz\"F-7$$\"3')**\\P4 w)Ri%F*$\"32_pimbY:=F-7$$\"3u++]x%f\")o%F*$\"3WFY)3t)QQ=F-7$$\"3C+]P/- a[ZF*$\"3A#y2)z_fh=F-7$$\"3r+](=Yb;\"[F*$\"3y6I+`_[()=F-7$$\"35++]i@Ot [F*$\"3J0drVlM9>F-7$$\"3`+]PfL'z$\\F*$\"3tX(\\nFRS%>F-7$$\"33,++!*>=+] F*$\"3U#y^f#)*3u>F-7$$\"3[++DE&4Q1&F*$\"3&zNV!ywA1?F-7$$\"3i+]P%>5p7&F *$\"3oII-8'Q%R?F-7$$\"3-,++bJ*[=&F*$\"3)**>:tkw52#F-7$$\"3)4+]7<[8D&F* $\"3REo`rKf3@F-7$$\"3L,++Ijy5`F*$\"3AG#*[M*HK9#F-7$$\"3_+]P/)fTP&F*$\" 3gKMk8xA\"=#F-7$$\"3&4+DcI;[V&F*$\"3NI\"p\"yud=AF-7$$\"3U+++++++bF*$\" 3/t'Q2RH(fAF--%+AXESLABELSG6$Q'a[8,7]6\"Q!F\\[l-%*THICKNESSG6#\"\"#-%% FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"!F_\\l$\"\"\" F_\\l-%%VIEWG6$;$\"#D!\"#$\"#bFh\\l%(DEFAULTG" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "evalf[12](minimiz e(sqrt(sm),a[8,7]=0..1,'location'));\nconvert(subs(op(1,op(op(2,[%]))) ,a[8,7]),rational,4);\nconvert(subs(op(1,op(op(2,[%%]))),a[8,7]),ratio nal,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-8YT(yp\"!#;<#7$<#/&%\"a G6$\"\")\"\"($\"-kz;KgR!#7F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#> \"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#S\"$,\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "evalf(eval (sqrt(sm),a[8,7]=19/48));\nevalf(eval(sqrt(sm),a[8,7]=40/101));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q_(yp\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[T(yp\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e9 := `union`(subs(a[8,7]=19 /48,e8),\{a[8,7]=19/48\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3512 "e9 := \{a[11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = \+ 0, b[5] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[8,2] = 0, a[9,3] = \+ 0, a[9,2] = 0, b[2] = 0, a[10,3] = 0, a[11,2] = 0, a[11,7] = 367953427 734636456312437471451631/44380048256994921706877045210625, `b*`[11] = \+ -5978523233/2539290474000, a[10,5] = 186648836999834766323333887946773 64275/6604186914582246254039171103542098296, a[9,5] = -312405090743882 815734153055386047660032/138949582969101476796789174014714391225, a[9, 4] = -536231038652840856428143497005609375/138242660274515884207093807 951326924, `b*`[12] = 1/50, a[9,8] = 1762562147235459382311840/6293392 25351137831406851, a[11,8] = 808591598711928461147452272/1338567291034 97890946741285, `b*`[1] = 9132998184216727/177194863388812500, `b*`[9] = 222323369516196533611/1556841029506080780000, `b*`[10] = 1246959894 769246208/14946997743616640625, a[5,3] = -53100995625/39027212288, a[1 2,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[4,2] = 0, a[5,2] = \+ 0, a[6,2] = 0, a[7,2] = 0, `b*`[6] = 5971284900143728/2327021660939062 5, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, c[11] = 1, c[12 ] = 1, a[8,7] = 19/48, a[4,3] = 9/50, a[11,4] = -513786835468324847758 6346693453125/495465427355479410622167542061972, a[9,1] = -80469422945 92811005033672093485229/31693413191384942205414737020389900, a[8,6] = \+ -14411638756714/175988393502075, a[11,5] = -40567777972626828275289264 9892020224/71142722483104145892095323206788025, a[10,4] = 123201997209 48281040487332306693759765625/2354898495606471238613818261942735405056 , a[9,6] = 46374757254514641197506707162751264/33542814156299871502725 237873062925, a[10,7] = -284269998655149797778753043937075/78560914257 412947734286452523008, a[8,1] = 232418455423819/11076629098233600, a[9 ,7] = 2051784331177468089852258884/674022310351068617436737421, a[8,5] = 881809586743097626/5311459993015086075, a[8,4] = -23240758251404687 5/2367427911414660864, a[10,1] = 215720375695479264063749229806841635/ 440720278308021223336256911460990976, a[11,1] = -975767605592872353548 579494188313/1054764737809245464809448652511500, a[10,6] = -7845820908 2510636098652961448851403235/65709391590090207335818975096879372288, b [9] = 397569776903/3090028048800, a[7,1] = 1072887507055/1257010006546 8, a[7,4] = 1075426748046875/9595107694235844, a[6,1] = 32755/421632, \+ a[5,1] = 15090136995/39027212288, c[2] = 1/16, c[4] = 6/25, c[6] = 5/8 , c[7] = 35/209, c[10] = 15/16, c[8] = 29/72, c[9] = 11/13, a[12,7] = \+ 898538649982788221/3655523545713900000, a[5,4] = 29533111875/195136061 44, b[8] = 986955378624/4938547627475, a[2,1] = 1/16, a[12,9] = 397569 776903/3090028048800, a[6,4] = 9765625/29258496, a[3,2] = 128/625, a[1 1,9] = -3049240467/3093372235, c[5] = 915/1696, `b*`[8] = 117812204047 61346096/57864557705069066875, a[12,10] = 2957049856/29666896875, `b*` [7] = 46560109562892125961981671/190526192880008411317500000, b[10] = \+ 2957049856/29666896875, a[10,8] = -40415483223195085710169801875/13000 342692048285899311310848, a[3,1] = -28/625, a[11,6] = 2459219605109182 51629683797176094768/59908811922681865929060892977735375, b[6] = 13517 44/5131875, a[12,6] = 1351744/5131875, a[12,8] = 986955378624/49385476 27475, b[7] = 898538649982788221/3655523545713900000, a[7,5] = -264111 17095731200/385766943434163819, a[4,1] = 3/50, c[3] = 4/25, b[1] = 180 09539/351697500, b[11] = 71669/6284880, a[12,1] = 18009539/351697500, \+ a[7,6] = 23430395520/608661493759, a[6,5] = 1488770/6971751, a[11,10] \+ = 3687441170432/6749831213125, a[12,11] = 71669/6284880, a[10,9] = 162 0538162595625/5204697537445888\}:" }{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 62 "The coefficient s for the combined scheme are given as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 940 "subs(e9,matrix([[c[2],a[2,1],``$2],\n[c[3],a[ 3,1],a[3,2],``],[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[c[9],a[9,1] ,a[9,2],a[9,3]],[``,a[9,4],a[9,5],a[9,6]],[``$2,a[9,7],a[9,8]],\n[c[10 ],a[10,1],a[10,2],a[10,3]],[``,a[10,4],a[10,5],a[10,6]],\n[``,a[10,7], a[10,8],a[10,9]],\n[c[11],a[11,1],a[11,2],a[11,3]],[``,a[11,4],a[11,5] ,a[11,6]],\n[``,a[11,7],a[11,8],a[11,9]],[``$3,a[11,10]],\n[c[12],a[12 ,1],a[12,2],a[12,3]],[``,a[12,4],a[12,5],a[12,6]],\n[``,a[12,7],a[12,8 ],a[12,9]],[``$2,a[12,11],a[12,10]],\n[``,`_________________________`$ 3],[`b`,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],\n[``,b[7],b[8],b[9]],[``$ 2,b[10],b[11]],[`b*`,`b*`[1],`b*`[2],`b*`[3]],\n[``,`b*`[4],`b*`[5],`b *`[6]],[``,`b*`[7],`b*`[8],`b*`[9]],[``,`b*`[10],`b*`[11],`b*`[12]]])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7E7&#\"\"\"\"#;F(%!GF +7&#\"\"%\"#D#!#G\"$D'#\"$G\"F2F+7&#\"\"'F/#\"\"$\"#]\"\"!#\"\"*F:7&# \"$:*\"%'p\"#\",&*p8!4:\",)G7s-RF;#!,Dc*45`FD7&F+F+F+#\",v=6L&H\",Whg8 &>7&#\"\"&\"\")#\"&bF$\"'K;UF;F;7&F+F+#\"(Dcw*\")'\\e#H#\"(q()[\"\"(^< (p7&#\"#N\"$4##\".bq]()G2\"\"/oa1+,d7F;F;7&F+#\"1vo/[nUv5\"1WeB%p2^f*# !2+7t&4<6TE\"3>Q;MM%pw&Q#\",?bRIM#\"-fP\\h'3'7&#\"#H\"#s#\"0>QUb%=CB\" 2+OB)4Hm26F;F;7&F+#!3vo/9De2CB\"4k3m99\"zUnB#\"3Ew4Vne4=))\"4vg3:I**f9 J&#!/9nvQ;T9\"0v?]$R))f<7&F+F+F+#\"#>\"#[7&#\"#6\"#8#!CH_[$4sO.05\"Gf% HUp/)\"D+**Q?qt9a?U\\Q\">8MpJF;F;7&F+#!Ev$4c+(\\V\"Gk&3%GlQ5BO&\"ECpK^ z!Q42U)e^u-mU#Q\"#!HK+mZgQbI:Md\"G)Qu!40CJ\"HD7R9Z,u\"*y'zw95pHe\\*Q\" #\"Dk7virq1v>TY^asvuj%\"DDH1tyBDF]r)*HcT\"GaL7&F+F+#\"=%))eA&)*3ou?B\"\"Ic]SNF%>E=QhQ7Z1c\\)*[N##\"GvUOxYz)QLBjwM)**p$)[m=\"F'H)4UN5r\" RSDYAe9p=/m#!GNKS^)[9'Hl)4O1^#34#e%y\"G)Gs$zo4v*=eL2-4!f\"R4d'7&F+#!Bv q$R/`(yxz\\^l)**pUG\"A3I__kGMx%HTdU\"4cy#!>v=!)p,r&3&>BK[:/%\">[3J6$** eG[?pU.+8#\"1Dcfi\"Q0i\"\"1))eWPvp/_7&F)#!B8$)=%\\z&[NNsGf0wwv*\"C+:^_ '[%4[YX#4ytkZ0\"F;F;7&F+#!CDJX$pYjex%[Koa$oy8&\"Bs>1Uv;A1TzaNFaY&\\#!E C--#*)\\E*Gv#Goiszxn0%\"DD!)y1KK&4#*e9/J[AF9r#\"EoZ4wrz$oH;D=4^g>#fC\" Dv`txH*31Hf'=oA>\")3*f7&F+#\"BJ;XruV7jXOYtFM&zO\"AD1@Xq(oq@\\*pD[+QW# \"r)f\"f3)\"<>n%4*y\\.\"Hn&Q8#!+n/C\\I\"+NAP$4$7&F+F+F+#\". K/x\"F;F;7&F+F;F;#\"1GP9+\\Grf\"2D1R4m@qK#7&F+#\";r;)>'f7#* Gc4,cY\"<++]<8T3+)G>E0>#\"5'4Y8w//A\"y6\"5vo1p]qdX'y&#\"66O`'>;&pLKA# \"7++y!31&H5%ob\"7&F+#\"43iCpZ*)fpC\"\"5D1k;Ou(*p%\\\"#!+LK_yf\".+SZ!H RD#F)F:Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(e9,matrix([seq([c[i],seq(a[i,j],j=1 ..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[j],j=1..11),``],[`b*`,seq(`b*` [j],j=1..12)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7/7/$\"'+]i!\"(F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"'++;!\"'$!'+![%F *$\"'+[?F/F+F+F+F+F+F+F+F+F+F+7/$\"'++CF/$\"'++gF*$\"\"!F:$\"'++=F/F+F +F+F+F+F+F+F+F+7/$\"'0&R&F/$\"'dmQF/F9$!'hg8!\"&$\"'Y8:FDF+F+F+F+F+F+F +F+7/$F)F/$\"'ioxF*F9F9$\"'rPLF/$\"'VN@F/F+F+F+F+F+F+F+7/$\"'ku;F/$\"' BN&)F*F9F9$\"'\"37\"F/$!'RYoF*$\"']\\QF*F+F+F+F+F+F+7/$\"'yFSF/$\"'G)4 #F*F9F9$!')o\")*F*$\"'?g;F/$!'(*)=)F*$\"'LeRF/F+F+F+F+F+7/$\"'ah%)F/$! '+RDF/F9F9$!'\"*yQFD$!'L[AFD$\"'b#Q\"FD$\"'4WIFD$\"'m+GFDF+F+F+F+7/$\" '+v$*F/$\"'s%*[F/F9F9$\"'tJ_FD$\"'AEGFD$!'-%>\"FD$!'Z=OFD$!'!)3JFD$\"' h8JF/F+F+F+7/$\"\"\"F:$!'/^#*F/F9F9$!')p.\"!\"%$!'J-dFD$\"'%\\5%FD$\"' '4H)FD$\"'sSgFD$!'Ld)*F/$\"',jaF/F+F+7/Fbq$\"'v?^F*F9F9F9F9$\"'-MEF/$ \"'.eCF/$\"'Z)*>F/$\"'i'G\"F/$\"'^n**F*$\"'MS6F*F+7/%\"bGFfrF9F9F9F9Fh rFjrF\\sF^sF`sFbsF+7/%#b*G$\"'@a^F*F9F9F9F9$\"'1mDF/$\"'wVCF/$\"'+O?F/ $\"'/G9F/$\"'aU$)F*$!'TaB!\")$\"'++?F*Q)pprint116\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded 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 3315 "ee := \{c[2]=1/16,\nc[3] =4/25,\nc[4]=6/25,\nc[5]=915/1696,\nc[6]=5/8,\nc[7]=35/209,\nc[8]=29/7 2,\nc[9]=11/13,\nc[10]=15/16,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/16,\na[3 ,1]=-28/625,\na[3,2]=128/625,\na[4,1]=3/50,\na[4,2]=0,\na[4,3]=9/50,\n a[5,1]=15090136995/39027212288,\na[5,2]=0,\na[5,3]=-53100995625/390272 12288,\na[5,4]=29533111875/19513606144,\na[6,1]=32755/421632,\na[6,2]= 0,\na[6,3]=0,\na[6,4]=9765625/29258496,\na[6,5]=1488770/6971751,\na[7, 1]=1072887507055/12570100065468,\na[7,2]=0,\na[7,3]=0,\na[7,4]=1075426 748046875/9595107694235844,\na[7,5]=-26411117095731200/385766943434163 819,\na[7,6]=23430395520/608661493759,\na[8,1]=232418455423819/1107662 9098233600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-232407582514046875/23674279 11414660864,\na[8,5]=881809586743097626/5311459993015086075,\na[8,6]=- 14411638756714/175988393502075,\na[8,7]=19/48,\na[9,1]=-80469422945928 11005033672093485229/31693413191384942205414737020389900,\na[9,2]=0,\n a[9,3]=0,\na[9,4]=-536231038652840856428143497005609375/13824266027451 5884207093807951326924,\na[9,5]=-3124050907438828157341530553860476600 32/138949582969101476796789174014714391225,\na[9,6]=463747572545146411 97506707162751264/33542814156299871502725237873062925,\na[9,7]=2051784 331177468089852258884/674022310351068617436737421,\na[9,8]=17625621472 35459382311840/629339225351137831406851,\na[10,1]=21572037569547926406 3749229806841635/440720278308021223336256911460990976,\na[10,2]=0,\na[ 10,3]=0,\na[10,4]=12320199720948281040487332306693759765625/2354898495 606471238613818261942735405056,\na[10,5]=18664883699983476632333388794 677364275/6604186914582246254039171103542098296,\na[10,6]=-78458209082 510636098652961448851403235/65709391590090207335818975096879372288,\na [10,7]=-284269998655149797778753043937075/7856091425741294773428645252 3008,\na[10,8]=-40415483223195085710169801875/130003426920482858993113 10848,\na[10,9]=1620538162595625/5204697537445888,\na[11,1]=-975767605 592872353548579494188313/1054764737809245464809448652511500,\na[11,2]= 0,\na[11,3]=0,\na[11,4]=-5137868354683248477586346693453125/4954654273 55479410622167542061972,\na[11,5]=-40567777972626828275289264989202022 4/71142722483104145892095323206788025,\na[11,6]=2459219605109182516296 83797176094768/59908811922681865929060892977735375,\na[11,7]=367953427 734636456312437471451631/44380048256994921706877045210625,\na[11,8]=80 8591598711928461147452272/133856729103497890946741285,\na[11,9]=-30492 40467/3093372235,\na[11,10]=3687441170432/6749831213125,\na[12,1]=1800 9539/351697500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12, 6]=1351744/5131875,\na[12,7]=898538649982788221/3655523545713900000,\n a[12,8]=986955378624/4938547627475,\na[12,9]=397569776903/309002804880 0,\na[12,10]=2957049856/29666896875,\na[12,11]=71669/6284880,\n\nb[1]= 18009539/351697500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=1351744/ 5131875,\nb[7]=898538649982788221/3655523545713900000,\nb[8]=986955378 624/4938547627475,\nb[9]=397569776903/3090028048800,\nb[10]=2957049856 /29666896875,\nb[11]=71669/6284880,\n\n`b*`[1]=9132998184216727/177194 863388812500,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6] =5971284900143728/23270216609390625,\n`b*`[7]=465601095628921259619816 71/190526192880008411317500000,\n`b*`[8]=11781220404761346096/57864557 705069066875,\n`b*`[9]=222323369516196533611/1556841029506080780000,\n `b*`[10]=1246959894769246208/14946997743616640625,\n`b*`[11]=-59785232 33/2539290474000,\n`b*`[12]=1/50\}:" }}}{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[7 ,11];" "6#&%\"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose \+ components are the principal error terms of the 11 stage, order 7 sche me (the error terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "`T*`[6, 12];" "6#&%#T*G6$\"\"'\"#7" }{TEXT -1 146 " \+ denote the vector whose components are the principal error terms of t he embedded 12 stage, order 6 scheme (the error terms of order 7) and \+ let " }{XPPEDIT 18 0 "`T*`[7,12];" "6#&%#T*G6$\"\"(\"#7" }{TEXT -1 99 " denote the vector whose components are the error terms of order \+ 8 of the embedded order 6 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[7, 11]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"# 6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,12]));" "6#-%$absG 6#-F$6#&%#T*G6$\"\"'\"#7" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(ab s(`T*`[7,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\"#7" }{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[8 ] = abs(abs(T[7, 11]));" "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"( \"#6" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,12])) /abs(abs(`T*`[6, 12]));" "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\" (\"#7\"\"\"-F*6#-F*6#&F/6$\"\"'F2!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7,12]-T[7,11]))/abs(abs(`T*`[6,12]) );" "6#/&%\"CG6#\"\")*&-%$absG6#-F*6#,&&%#T*G6$\"\"(\"#7\"\"\"&%\"TG6$ F2\"#6!\"\"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[8];" "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embed ded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "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 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := PrincipalErrorTerms(7,11,'exp anded'):\n`errterms7_12*` :=subs(b=`b*`,PrincipalErrorTerms(7,12,'expa nded')):\n`errterms6_12*` := subs(b=`b*`,PrincipalErrorTerms(6,12,'exp anded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms7_12*`[i]) )^2,i=1..nops(`errterms7_12*`))):\nsdnB := sqrt(add(evalf(subs(ee,`err terms6_12*`[i]))^2,i=1..nops(`errterms6_12*`))):\nsnmC := sqrt(add((ev alf(subs(ee,`errterms7_12*`[i])-subs(ee,errterms7_11[i])))^2,i=1..nops (errterms7_11))):\n'B[8]'= evalf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC /sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")c*4f\"!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")Wie:!\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "co efficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3315 "ee := \{c[2]=1/16,\nc[3]=4 /25,\nc[4]=6/25,\nc[5]=915/1696,\nc[6]=5/8,\nc[7]=35/209,\nc[8]=29/72, \nc[9]=11/13,\nc[10]=15/16,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/16,\na[3,1 ]=-28/625,\na[3,2]=128/625,\na[4,1]=3/50,\na[4,2]=0,\na[4,3]=9/50,\na[ 5,1]=15090136995/39027212288,\na[5,2]=0,\na[5,3]=-53100995625/39027212 288,\na[5,4]=29533111875/19513606144,\na[6,1]=32755/421632,\na[6,2]=0, \na[6,3]=0,\na[6,4]=9765625/29258496,\na[6,5]=1488770/6971751,\na[7,1] =1072887507055/12570100065468,\na[7,2]=0,\na[7,3]=0,\na[7,4]=107542674 8046875/9595107694235844,\na[7,5]=-26411117095731200/38576694343416381 9,\na[7,6]=23430395520/608661493759,\na[8,1]=232418455423819/110766290 98233600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-232407582514046875/2367427911 414660864,\na[8,5]=881809586743097626/5311459993015086075,\na[8,6]=-14 411638756714/175988393502075,\na[8,7]=19/48,\na[9,1]=-8046942294592811 005033672093485229/31693413191384942205414737020389900,\na[9,2]=0,\na[ 9,3]=0,\na[9,4]=-536231038652840856428143497005609375/1382426602745158 84207093807951326924,\na[9,5]=-312405090743882815734153055386047660032 /138949582969101476796789174014714391225,\na[9,6]=46374757254514641197 506707162751264/33542814156299871502725237873062925,\na[9,7]=205178433 1177468089852258884/674022310351068617436737421,\na[9,8]=1762562147235 459382311840/629339225351137831406851,\na[10,1]=2157203756954792640637 49229806841635/440720278308021223336256911460990976,\na[10,2]=0,\na[10 ,3]=0,\na[10,4]=12320199720948281040487332306693759765625/235489849560 6471238613818261942735405056,\na[10,5]=1866488369998347663233338879467 7364275/6604186914582246254039171103542098296,\na[10,6]=-7845820908251 0636098652961448851403235/65709391590090207335818975096879372288,\na[1 0,7]=-284269998655149797778753043937075/785609142574129477342864525230 08,\na[10,8]=-40415483223195085710169801875/13000342692048285899311310 848,\na[10,9]=1620538162595625/5204697537445888,\na[11,1]=-97576760559 2872353548579494188313/1054764737809245464809448652511500,\na[11,2]=0, \na[11,3]=0,\na[11,4]=-5137868354683248477586346693453125/495465427355 479410622167542061972,\na[11,5]=-405677779726268282752892649892020224/ 71142722483104145892095323206788025,\na[11,6]=245921960510918251629683 797176094768/59908811922681865929060892977735375,\na[11,7]=36795342773 4636456312437471451631/44380048256994921706877045210625,\na[11,8]=8085 91598711928461147452272/133856729103497890946741285,\na[11,9]=-3049240 467/3093372235,\na[11,10]=3687441170432/6749831213125,\na[12,1]=180095 39/351697500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6] =1351744/5131875,\na[12,7]=898538649982788221/3655523545713900000,\na[ 12,8]=986955378624/4938547627475,\na[12,9]=397569776903/3090028048800, \na[12,10]=2957049856/29666896875,\na[12,11]=71669/6284880,\n\nb[1]=18 009539/351697500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=1351744/51 31875,\nb[7]=898538649982788221/3655523545713900000,\nb[8]=98695537862 4/4938547627475,\nb[9]=397569776903/3090028048800,\nb[10]=2957049856/2 9666896875,\nb[11]=71669/6284880,\n\n`b*`[1]=9132998184216727/17719486 3388812500,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=5 971284900143728/23270216609390625,\n`b*`[7]=46560109562892125961981671 /190526192880008411317500000,\n`b*`[8]=11781220404761346096/5786455770 5069066875,\n`b*`[9]=222323369516196533611/1556841029506080780000,\n`b *`[10]=1246959894769246208/14946997743616640625,\n`b*`[11]=-5978523233 /2539290474000,\n`b*`[12]=1/50\}:" }}}{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 78 "The stability function R for th e 11 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(7,11,'expanded')):\n R := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/- %\"RG6#%\"zG,:\"\"\"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)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\"@`Z$3IKU@\\sOh^i` !)\"E++S=x9Hd)p*Qo'p,iqG$F)*$)F'\"\")F)F)F)*&#\"Cf*GvANtw\"H_*HA:-#[7 \"H++7l08EC(p[3J#[A!=!Qb%F)*$)F'\"\"*F)F)F)*&#\"A.(>3Bba*)o\\7xmlXw#\" H+!G&)zjFBgpke?z&Qv`D1\"F)*$)F'\"#5F)F)F)*&#\";\\!\\yUm\\1vn`1\"*)\"C[ +(f&f(>HL#*3$zl8lX9F)*$)F'\"#6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boun dary of the stability region intersects the negative real axis by solv ing 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 29 "z0 := newton(R(z)=-1,z=-4.6) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+#Hq)>Y!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 : = newton(R(z)=-1,z=-4.6):\np1 := plot([R(z),-1],z=-5.19..0.49,color=[r ed,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=[-5.19..0.49,-1.47 ..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7W7$$!3Q++++++!>&!#<$!3esAN1?_*e$F*7$$! 3Ynm;#3'4G^F*$!3?3C?25(p9$F*7$$!3kLLLk@>m]F*$!3)pwpu%>daFF*7$$!3E+]U'* )HB,&F*$!3)plddJ)))\\CF*7$$!3!pm;&GwYe\\F*$!39\\Ho-75w@F*7$$!3s+](\\(Q *y*[F*$!3)>o`)R>b,>F*7$$!3nLLV@,KP[F*$!3oyF!\\DT(e;F*7$$!3'RLLd%[MwZF* $!3/.&*o&*)oHW\"F*7$$!3NLL.q&p`r%F*$!33d-$elTGD\"F*7$$!3;nmJG')*Rf%F*$ !3wA1`3n'*)R*!#=7$$!3uLLyGAZ\"[%F*$!3[Mb^zQPUrFZ7$$!3%3+])fw&\\O%F*$!3 &=5>+d7VK&FZ7$$!3$QL$)f7eWC%F*$!3GK^B[Q*=)QFZ7$$!3A++lN]MCTF*$!3I]g?=* R%*y#FZ7$$!3ummYeRz+SF*$!3So89Y:dU>FZ7$$!3_LLV-,(>*QF*$!3'*HX*)p&RbP\" FZ7$$!35++S:-YpPF*$!312*>a:Ax*))!#>7$$!3K+++\"HZkk$F*$!3-_s!Gb_cC&Fhp7 $$!3;++gW:!z_$F*$!3+J*yP!Qq#e#Fhp7$$!3hLL)*\\1D?MF*$!37F0D$Qu$))p!#?7$ $!3'ommSKVAH$F*$\"3yw\"pjp[w1\"Fhp7$$!3/nmEGV!Q=$F*$\"39QQM?<>#G#Fhp7$ $!39++0(*RmdIF*$\"3yIF*)[1%G[$Fhp7$$!39nmEI%3g%HF*$\"3Z\"e3:(e')HWFhp7 $$!3-++0xX]BGF*$\"34DSM`\"o*3aFhp7$$!3*)***\\\"R>&oq#F*$\"3[N#QH+%>JjF hp7$$!3gmm;\\r8&e#F*$\"3Lnq'**RITK(Fhp7$$!3ymmrw\\OtCF*$\"39)Rk\"HJ<$H )Fhp7$$!3SLL$))e.GN#F*$\"3;)ekuj:uU*Fhp7$$!3nLL)**=uvA#F*$\"3m3i@G>4t5 FZ7$$!3K++:I;c=@F*$\"3;)3AT.7\"*>\"FZ7$$!31LL.z]#3+#F*$\"3C%p[;\"\\b]8 FZ7$$!3M++?,<>z=F*$\"3&e9JV08i_\"FZ7$$!3;++!4<(>gFZ7$$!3H++q9zA<:F*$\"3'z#FZ7$$!3En mEY;O-9F*$\"3+2lnv,4gCFZ7$$!3#)*****pQ<(z7F*$\"3GyFzA48\"y#FZ7$$!3)RL$ efMeo6F*$\"3E!>.-Ec!3JFZ7$$!3I****fAZ3Z5F*$\"3'\\Cmo-*e4NFZ7$$!3xqm;(z QwK*FZ$\"3GPMtq_kMRFZ7$$!3&z***\\)ecE8)FZ$\"3wSV%FZ7$$!3'3nmm0VV 'pFZ$\"3'3@hf=\"f$)\\FZ7$$!3P)***\\iqATdFZ$\"3C@G$GR()>j&FZ7$$!3aFLL*) 4AjXFZ$\"3u(ysk=(4OjFZ7$$!33LLLO'R&eLFZ$\"3WM@]VZFZrFZ7$$!3Uim;`O$Q;#F Z$\"3I'pV5qkU0)FZ7$$!3?*****>$H-m5FZ$\"3'[h+$Q4$)))*)FZ7$$\"3v*QLLU?># >Fhp$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\"FZ$\"3.))*>qHH39\"F*7$$\"3E0+]HcU S$*=iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Ff\\lF e\\l-F$6$7S7$F($!\"\"Ff\\l7$F3F[]l7$F=F[]l7$FGF[]l7$FQF[]l7$FVF[]l7$Ff nF[]l7$F[oF[]l7$F`oF[]l7$FeoF[]l7$FjoF[]l7$F_pF[]l7$FdpF[]l7$FjpF[]l7$ F_qF[]l7$FdqF[]l7$FjqF[]l7$F_rF[]l7$FdrF[]l7$FirF[]l7$F^sF[]l7$FcsF[]l 7$FhsF[]l7$F]tF[]l7$FbtF[]l7$FgtF[]l7$F\\uF[]l7$FauF[]l7$FfuF[]l7$F[vF []l7$F`vF[]l7$FevF[]l7$FjvF[]l7$F_wF[]l7$FdwF[]l7$FiwF[]l7$F^xF[]l7$Fc xF[]l7$FhxF[]l7$F]yF[]l7$FbyF[]l7$FgyF[]l7$F\\zF[]l7$FazF[]l7$FfzF[]l7 $F[[lF[]l7$F`[lF[]l7$Fe[lF[]l7$Fj[lF[]l-F_\\l6&Fa\\lFe\\lFe\\lFb\\l-F$ 6&7#7$$!3))*****>Hq)>YF*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-F f`l6#%(DIAMONDGFi`lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLORG6&Fa\\lFe\\l$\" \"&F\\]lFe\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Fjbl-%%FONTG 6#%(DEFAULTG-F]cl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$>&!\"#$\"#\\Fjcl;$!$ Z\"Fjcl$\"$Z\"Fjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1617 "R := z -> \+ 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 8 053625161367249214232300834753/328706201696683896985729147718400000*z^ 8+\n 1248202152229952291767335227528959/455380180224823108486972 426130565120000*z^9+\n 27645656677124968895455230819703/10625537 5385792058646960232763798528000*z^10+\n 891065367750649664278490 49/1445651365793089233291975955970048*z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 280 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(p ts,color=COLOR(RGB,.55,.15,0)):\np2 := plots[polygonplot]([seq([pts[i- 1],pts[i],[-2.3,0]],i=2..nops(pts))],\n style=patchnogrid,col or=COLOR(RGB,.9,.45,.1)):\npts := []: z0 := 1.3+5.1*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=COL OR(RGB,.55,.15,0)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1 .18,5.07]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.9,.45,.1)):\npts := []: z0 := 1.3-5.1*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,. 55,.15,0)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.18,-5.0 7]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.9, .45,.1)):\np7 := plot([[[-5.39,0],[1.69,0]],[[0,-5.69],[0,5.69]]],colo r=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.39..1.69,-5 .69..5.69],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`] ,axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 336 514 514 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:! #=7$F($\"3)******Rl#fTJF-7$$\"3I+++TgRgl!#F$\"3#******p(*)Q7ZF-7$$\"3% )*****>N_h0'!#E$\"31+++p_=$G'F-7$$\"31+++M@s)G$!#D$\"35+++l8)R&yF-7$$ \"3'*******RQn!G\"!#C$\"3F+++rmxC%*F-7$$\"3x*****p*e$Q+%FF$\"35+++\\pb *4\"!#<7$$\"35+++:sa#3\"!#B$\"31+++tbjc7FN7$$\"3%******4'*[]m#FR$\"3(* *****Ri7PT\"FN7$$\"3K+++n1FohFR$\"3%******z7'yq:FN7$$\"33+++h#o,O\"!#A $\"36+++\\=&ys\"FN7$$\"39+++&4/9&GF\\o$\"3/+++#z+\\)=FN7$$\"3s******)[ qQh&F\\o$\"3++++tV\">/#FN7$$\"32+++D$4$=5!#@$\"39+++#)e&))>#FN7$$\"3++ ++X(3;l\"F\\p$\"3?+++;olbBFN7$$\"35+++>KJ_AF\\p$\"3++++Su>7DFN7$$\"3%* *****fT$*36#F\\p$\"3)*******=IGoEFN7$$!36+++UE=,\\F\\o$\"39+++I4iBGFN7 $$!3x*****pcju,*F\\p$\"3)******p\"y!y(HFN7$$!35+++%>:$eH!#?$\"3@+++jaL IJFN7$$!3))*****px^s<(F[r$\"31+++K?h!G$FN7$$!3%******R2@2\\\"!#>$\"3&* *****pLpzU$FN7$$!3#******f0P_y#Ffr$\"37+++h4grNFN7$$!3K+++F62\"z%Ffr$ \"3()*****f;%Q5PFN7$$!3y********QG!o(Ffr$\"3%********GGE%QFN7$$!3-+++% R'Ra6F-$\"3!)******y&))f'RFN7$$!30+++X6nK;F-$\"3S+++-^\"z2%FN7$$!3#*** ***HtX@=#F-$\"3-+++R2hwTFN7$$!35+++ix*>MVFN7$$!32+++#)y*=+%F-$\"3;+++*=&e&R%FN7$$!3-+ ++Z]m4YF-$\"3'*******4*4vW%FN7$$!3J+++'oNi?&F-$\"3?+++j7S\"\\%FN7$$!3Q +++oD4!z&F-$\"3l*****f.C%GXFN7$$!3/+++Au>hjF-$\"3'******pz*\\fXFN7$$!3 p*****p_4-#pF-$\"3k*****ft]`e%FN7$$!3=+++))32ouF-$\"3@+++CHa1YFN7$$!3! *******y\")z0!)F-$\"3x******\\>_BYFN7$$!3w*****\\/'QM&)F-$\"3F+++f.kOY FN7$$!3s*****f#*oZ0*F-$\"3q*****\\Dyhk%FN7$$!3a******GK\"yc*F-$\"3=+++ T'eBl%FN7$$!3#******H^Ku+\"FN$\"3=+++l)e`l%FN7$$!3/+++_m]d5FN$\"3%)*** ***G#>`l%FN7$$!32+++ev226FN$\"3w*****H`\\Bl%FN7$$!3#******f/>i:\"FN$\" 3G+++lT`YYFN7$$!3%******pd2]?\"FN$\"3Q+++*=Ozj%FN7$$!3/+++M__`7FN$\"3n *****R\"3gEYFN7$$!34+++(3j=I\"FN$\"3u*****4jeDh%FN7$$!3%******f-D,N\"F N$\"3!)*****\\)*Gef%FN7$$!3++++9@V)R\"FN$\"3;+++XRUwXFN7$$!3!*******Hu #pW\"FN$\"3y*****>[`Vb%FN7$$!31+++k6y&\\\"FN$\"3N+++!eK'HXFN7$$!3-+++k `>X:FN$\"3'******>E\"H-XFN7$$!3/+++iqS&f\"FN$\"3G+++X&)QsWFN7$$!31+++ \"[(oY;FN$\"3g*****>9J+W%FN7$$!3#*******HOL*p\"FN$\"3/+++\"o'R0WFN7$$! 3-+++Auk`FN$\"3E+++M=G^UFN7$$!3'***** **e]G\"*>FN$\"3V+++$GE:@%FN7$$!3?+++Y?Sb?FN$\"33+++.IVsTFN7$$!3')***** \\d617#FN$\"33+++t%)RMTFN7$$!3!)******=2K'=#FN$\"3s*****zV>w4%FN7$$!3- +++De&>D#FN$\"3g*****zX*4iSFN7$$!33+++xi-`6Q#FN$\"3))*****\\!\\;%*RFN7$$!3!******pupSW#FN$\"3A+++cwBhRFN7$ $!3/+++1cg0DFN$\"3$******z_J'GRFN7$$!3%)*****H.mcc#FN$\"3;+++^T3'*QFN7 $$!3\"******z$=@CEFN$\"3%)*****f:hL'QFN7$$!3!******R#GC\"o#FN$\"3*)*** **zDf-$QFN7$$!3*******HB&yOFFN$\"3%******zM0mz$FN7$$!37+++gF)3z#FN$\"3 ;+++'p`Aw$FN7$$!3@+++\"R!fVGFN$\"3.+++HD3FPFN7$$!35+++j.(\\*GFN$\"3;++ +l1*4p$FN7$$!3=+++;'*3XHFN$\"3#******Ra%*Ql$FN7$$!3-++++$=S*HFN$\"3A++ +)yDdh$FN7$$!3>+++m*G=/$FN$\"3#******pMHkd$FN7$$!3')*****>='f)3$FN$\"3 %******z?if`$FN7$$!3))*****>P'RMJFN$\"3'******fj#H%\\$FN7$$!31+++#p2$z JFN$\"3!******>!**R^MFN7$$!35+++/)4MA$FN$\"3$******HWusS$FN7$$!39+++2M ymKFN$\"3)******zO=>O$FN7$$!3A+++'Q4&4LFN$\"3$)*****H?Y`J$FN7$$!3')*** **>[n;N$FN$\"33+++EdenKFN7$$!3')*****>RMLR$FN$\"31+++-(y'=KFN7$$!3)*** ****)H\"eMMFN$\"3'*******e7ooJFN7$$!3-+++74ZvMFN$\"3.+++dOm?ag^$FN$\"3++++x\"4d1$FN7$$!3#)*****p\"H,$F N7$$!3-+++>\"Gkf$FN$\"3/+++?OPfHFN7$$!3,+++=dBOOFN$\"3*******>>*>0HFN7 $$!31+++D&odn$FN$\"3?+++X?\\]GFN7$$!3<+++/f)\\r$FN$\"3')*********[`z#F N7$$!31+++\"[IQv$FN$\"34+++2e&)RFFN7$$!3)******zzJAz$FN$\"3?+++q5,$QFN$\"3?+++&z'3GEFN7$$!3!)******\\RQnQFN$\"3%)***** >:->d#FN7$$!3>+++>&oR!RFN$\"36+++%[\\b^#FN7$$!36+++u_yRRFN$\"3++++UJ.f CFN7$$!3/+++&zhZ(RFN$\"3)******\\EWBS#FN7$$!3J+++BV$)3SFN$\"3#******pM jaM#FN7$$!3E+++4*[>/%FN$\"37+++OGO)G#FN7$$!3++++3>1uSFN$\"3z*****\\e45 B#FN7$$!3&)*****>)*R^5%FN$\"31+++NqOt@FN7$$!3Q+++p(f^8%FN$\"3()*****>3 (R:@FN7$$!3w*****fM2T;%FN$\"3/+++b;1d?FN7$$!3m*****4fx>>%FN$\"3'****** H,C$)*>FN7$$!3K+++\"\\t(=UFN$\"3.+++^)\\\"R>FN7$$!3-+++Qa]WUFN$\"35+++ Q#3&z=FN7$$!3G+++=0>pUFN$\"3.+++aCP>=FN7$$!3++++w<&GH%FN$\"33+++l2se:Pvp\"FN7$$!3u*****p/?sL%FN$\"30++++? \"ej\"FN7$$!3C+++)[&*zN%FN$\"3!******\\uUNd\"FN7$$!3'******R&=)yP%FN$ \"35+++DWt5:FN7$$!3g*****H;=pR%FN$\"35+++.,SZ9FN7$$!3()*****f*G9:WFN$ \"3'******p2hNQ\"FN7$$!3n******=CfKWFN$\"3'******4\"oC>8FN7$$!3Q+++G$* H\\WFN$\"3++++EX\\a7FN7$$!3o******Q3HlWFN$\"3)*******z$[$*=\"FN7$$!3W+ ++=te![%FN$\"3)*******[#eQ7\"FN7$$!3e*****\\5,_\\%FN$\"3#*******=\"y!e 5FN7$$!3q*****4uN\"4XFN$\"3#)*****f&=k?**F-7$$!3)*******)p&QAXFN$\"3O+ ++qxse#*F-7$$!3\"******>kO\\`%FN$\"3%)*******o#e&f)F-7$$!3G+++4iwYXFN$ \"3O+++X9rJzF-7$$!3W+++P_%yb%FN$\"3))******yMcnsF-7$$!3k*****HJR\"oXFN $\"3*******HQ?Ng'F-7$$!3C+++30hxXFN$\"3k*****R%z))RfF-7$$!3/+++`\">ie% FN$\"3_*****ft#*oF&F-7$$!3=+++Wb#Rf%FN$\"3()*****4$Go9YF-7$$!3\")***** pZ\"p+YFN$\"3!)*****4$>L`RF-7$$!3<+++l:[1YFN$\"3?+++lg%GH$F-7$$!3T+++C VE6YFN$\"3#)*****z/sJj#F-7$$!36+++4I,:YFN$\"3))*****4'p?u>F-7$$!3E+++ \\iqYFN$\"3]+++:9.ylFfr7$$!3))*** **>Hq)>YFNF(7$F^gm$!3]+++:9.ylFfr7$Fifm$!30+++:\"3eJ\"F-7$Fdfm$!3))*** **4'p?u>F-7$F_fm$!3#)*****z/sJj#F-7$Fjem$!3?+++lg%GH$F-7$Feem$!3!)**** *4$>L`RF-7$F`em$!3()*****4$Go9YF-7$F[em$!3_*****ft#*oF&F-7$Ffdm$!3k*** **R%z))RfF-7$Fadm$!3*******HQ?Ng'F-7$F\\dm$!3))******yMcnsF-7$Fgcm$!3O +++X9rJzF-7$Fbcm$!3%)*******o#e&f)F-7$F]cm$!3O+++qxse#*F-7$Fhbm$!3#)** ***f&=k?**F-7$Fcbm$!3#*******=\"y!e5FN7$F^bm$!3)*******[#eQ7\"FN7$Fiam $!3)*******z$[$*=\"FN7$Fdam$!3++++EX\\a7FN7$F_am$!3'******4\"oC>8FN7$F j`m$!3'******p2hNQ\"FN7$Fe`m$!35+++.,SZ9FN7$F``m$!35+++DWt5:FN7$F[`m$! 3!******\\uUNd\"FN7$Ff_m$!30++++?\"ej\"FN7$Fa_m$!3)******>:Pvp\"FN7$F \\_m$!33+++l2se=FN7$Fb^m$!35+++Q#3&z=FN7$F]^m$!3.+ ++^)\\\"R>FN7$Fh]m$!3'******H,C$)*>FN7$Fc]m$!3/+++b;1d?FN7$F^]m$!3()** ***>3(R:@FN7$Fi\\m$!31+++NqOt@FN7$Fd\\m$!3z*****\\e45B#FN7$F_\\m$!37++ +OGO)G#FN7$Fj[m$!3#******pMjaM#FN7$Fe[m$!3)******\\EWBS#FN7$F`[m$!3+++ +UJ.fCFN7$F[[m$!36+++%[\\b^#FN7$Ffjl$!3%)*****>:->d#FN7$Fajl$!3?+++&z' 3GEFN7$F\\jl$!3?+++>*>0HFN7$Fchl$!3/+++?OPfHFN7 $F^hl$!3()******Q>\"H,$FN7$Figl$!3++++x\"4d1$FN7$Fdgl$!3.+++dOmO$FN7$Ffel$!3$******HWusS$FN7$Fael $!3!******>!**R^MFN7$F\\el$!3'******fj#H%\\$FN7$Fgdl$!3%******z?if`$FN 7$Fbdl$!3#******pMHkd$FN7$F]dl$!3A+++)yDdh$FN7$Fhcl$!3#******Ra%*Ql$FN 7$Fccl$!3;+++l1*4p$FN7$F^cl$!3.+++HD3FPFN7$Fibl$!3;+++'p`Aw$FN7$Fdbl$! 3%******zM0mz$FN7$F_bl$!3*)*****zDf-$QFN7$Fjal$!3%)*****f:hL'QFN7$Feal $!3;+++^T3'*QFN7$F`al$!3$******z_J'GRFN7$F[al$!3A+++cwBhRFN7$Ff`l$!3)) *****\\!\\;%*RFN7$Fa`l$!32+++&Q$pFSFN7$F\\`l$!3g*****zX*4iSFN7$Fg_l$!3 s*****zV>w4%FN7$Fb_l$!33+++t%)RMTFN7$F]_l$!33+++.IVsTFN7$Fh^l$!3V+++$G E:@%FN7$Fc^l$!3E+++M=G^UFN7$F^^l$!3m******G([6H%FN7$Fi]l$!3q*****f#o]I VFN7$Fd]l$!3K+++f(e(oVFN7$F_]l$!3/+++\"o'R0WFN7$Fj\\l$!3g*****>9J+W%FN 7$Fe\\l$!3G+++X&)QsWFN7$F`\\l$!3'******>E\"H-XFN7$F[\\l$!3N+++!eK'HXFN 7$Ff[l$!3y*****>[`Vb%FN7$Fa[l$!3;+++XRUwXFN7$F\\[l$!3!)*****\\)*Gef%FN 7$Fgz$!3u*****4jeDh%FN7$Fbz$!3n*****R\"3gEYFN7$F]z$!3Q+++*=Ozj%FN7$Fhy $!3G+++lT`YYFN7$Fcy$!3w*****H`\\Bl%FN7$F^y$!3%)******G#>`l%FN7$Fix$!3= +++l)e`l%FN7$Fdx$!3=+++T'eBl%FN7$F_x$!3q*****\\Dyhk%FN7$Fjw$!3F+++f.kO YFN7$Few$!3x******\\>_BYFN7$F`w$!3@+++CHa1YFN7$F[w$!3k*****ft]`e%FN7$F fv$!3'******pz*\\fXFN7$Fav$!3l*****f.C%GXFN7$F\\v$!3?+++j7S\"\\%FN7$Fg u$!3'*******4*4vW%FN7$Fbu$!3;+++*=&e&R%FN7$F]u$!3e*******>*>MVFN7$Fht$ !3))******QpshUFN7$Fct$!3-+++R2hwTFN7$F^t$!3S+++-^\"z2%FN7$Fis$!3!)*** ***y&))f'RFN7$Fds$!3%********GGE%QFN7$F_s$!3()*****f;%Q5PFN7$Fjr$!37++ +h4grNFN7$Fdr$!3&******pLpzU$FN7$F_r$!31+++K?h!G$FN7$Fiq$!3@+++jaLIJFN 7$Fdq$!3)******p\"y!y(HFN7$F_q$!39+++I4iBGFN7$Fjp$!3)*******=IGoEFN7$F ep$!3++++Su>7DFN7$F`p$!3?+++;olbBFN7$Fjo$!39+++#)e&))>#FN7$Feo$!3++++t V\">/#FN7$F`o$!3/+++#z+\\)=FN7$Fjn$!36+++\\=&ys\"FN7$$\"3i******[1FohF R$!3%******z7'yq:FN7$FV$!3(******Ri7PT\"FN7$FP$!31+++tbjc7FN7$FJ$!35++ +\\pb*4\"FN7$FD$!3F+++rmxC%*F-7$$\"3C+++L@s)G$F@$!35+++l8)R&yF-7$$\"3w *****fM_h0'F:$!31+++p_=$G'F-7$$\"3%******R0;/c'F4$!3#******p(*)Q7ZF-7$ F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#b!\"#$\"# :FdboF(-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$FdboF)F(7%F[co7$F($\"+a EfTJF^coF_co7%Fbco7$$\"+TgRglFfr$\"+x*)Q7ZF^coF_co7%Ffco7$$\"+_B:cgF-$ \"+p_=$G'F^coF_co7%F\\do7$$\"+M@s)G$FN$\"+l8)R&yF^coF_co7%Fbdo7$$\"+SQ n!G\"!#;$\"+rmxC%*F^coF_co7%Fhdo7$$\"+(*e$Q+%F[eo$\"+\\pb*4\"!\"*F_co7 %F_eo7$$\"+:sa#3\"!#:$\"+tbjc7FdeoF_co7%Ffeo7$$\"+h*[]m#Fieo$\"+CEr89F deoF_co7%F]fo7$$\"+n1FohFieo$\"+Ghyq:FdeoF_co7%Fcfo7$$\"+h#o,O\"!#9$\" +\\=&ys\"FdeoF_co7%Fifo7$$\"+&4/9&GF\\go$\"+#z+\\)=FdeoF_co7%F`go7$$\" +*[qQh&F\\go$\"+tV\">/#FdeoF_co7%Ffgo7$$\"+D$4$=5!#8$\"+#)e&))>#FdeoF_ co7%F\\ho7$$\"+X(3;l\"F_ho$\"+;olbBFdeoF_co7%Fcho7$$\"+>KJ_AF_ho$\"+Su >7DFdeoF_co7%Fiho7$$\"+;M*36#F_ho$\"+>IGoEFdeoF_co7%F_io7$$!+UE=,\\F\\ go$\"+I4iBGFdeoF_co7%Feio7$$!+nNY:$eH!#7$\"+jaLIJFdeoF_co7%Fajo7$$!+xMVFdeoF_co7%F_ ^p7$$!+#)y*=+%F^co$\"+*=&e&R%FdeoF_co7%Fe^p7$$!+Z]m4YF^co$\"+5*4vW%Fde oF_co7%F[_p7$$!+'oNi?&F^co$\"+j7S\"\\%FdeoF_co7%Fa_p7$$!+oD4!z&F^co$\" +OSUGXFdeoF_co7%Fg_p7$$!+Au>hjF^co$\"+(z*\\fXFdeoF_co7%F]`p7$$!+F&4-#p F^co$\"+O2N&e%FdeoF_co7%Fc`p7$$!+))32ouF^co$\"+CHa1YFdeoF_co7%Fi`p7$$! +z\")z0!)F^co$\"+]>_BYFdeoF_co7%F_ap7$$!+XgQM&)F^co$\"+f.kOYFdeoF_co7% Feap7$$!+E*oZ0*F^co$\"+b#yhk%FdeoF_co7%F[bp7$$!+HK\"yc*F^co$\"+T'eBl%F deoF_co7%Fabp7$$!+8DV25Fdeo$\"+l)e`l%FdeoF_co7%Fgbp7$$!+_m]d5Fdeo$\"+H #>`l%FdeoF_co7%F]cp7$$!+ev226Fdeo$\"+L&\\Bl%FdeoF_co7%Fccp7$$!+Y!>i:\" Fdeo$\"+lT`YYFdeoF_co7%Ficp7$$!+xv+07Fdeo$\"+*=Ozj%FdeoF_co7%F_dp7$$!+ M__`7Fdeo$\"+93gEYFdeoF_co7%Fedp7$$!+(3j=I\"Fdeo$\"+J'eDh%FdeoF_co7%F[ ep7$$!+E]7]8Fdeo$\"+&)*Gef%FdeoF_co7%Faep7$$!+9@V)R\"Fdeo$\"+XRUwXFdeo F_co7%Fgep7$$!+Iu#pW\"Fdeo$\"+#[`Vb%FdeoF_co7%F]fp7$$!+k6y&\\\"Fdeo$\" +!eK'HXFdeoF_co7%Fcfp7$$!+k`>X:Fdeo$\"+i7H-XFdeoF_co7%Fifp7$$!+iqS&f\" Fdeo$\"+X&)QsWFdeoF_co7%F_gp7$$!+\"[(oY;Fdeo$\"+U6.SWFdeoF_co7%Fegp7$$ !+IOL*p\"Fdeo$\"+\"o'R0WFdeoF_co7%F[hp7$$!+Auk`Fdeo$\"+M=G^UFdeoF_co7%Fcip7$$!+f]G\"*>Fdeo$\"+$ GE:@%FdeoF_co7%Fiip7$$!+Y?Sb?Fdeo$\"+.IVsTFdeoF_co7%F_jp7$$!+v:h?@Fdeo $\"+t%)RMTFdeoF_co7%Fejp7$$!+>2K'=#Fdeo$\"+Q%>w4%FdeoF_co7%F[[q7$$!+De &>D#Fdeo$\"+e%*4iSFdeoF_co7%Fa[q7$$!+xi-`6Q#Fdeo$\"+0\\;%*RFdeoF_co7%F]\\q7$$!+Z(pSW#Fdeo$\"+cwBhRFdeoF _co7%Fc\\q7$$!+1cg0DFdeo$\"+G:jGRFdeoF_co7%Fi\\q7$$!+LgmlDFdeo$\"+^T3' *QFdeoF_co7%F_]q7$$!+Q=@CEFdeo$\"+c6OjQFdeoF_co7%Fe]q7$$!+CGC\"o#Fdeo$ \"+e#f-$QFdeoF_co7%F[^q7$$!+L_yOFFdeo$\"+[`g'z$FdeoF_co7%Fa^q7$$!+gF)3 z#Fdeo$\"+'p`Aw$FdeoF_co7%Fg^q7$$!+\"R!fVGFdeo$\"+HD3FPFdeoF_co7%F]_q7 $$!+j.(\\*GFdeo$\"+l1*4p$FdeoF_co7%Fc_q7$$!+;'*3XHFdeo$\"+WX*Ql$FdeoF_ co7%Fi_q7$$!++$=S*HFdeo$\"+)yDdh$FdeoF_co7%F_`q7$$!+m*G=/$Fdeo$\"+Z$Hk d$FdeoF_co7%Fe`q7$$!+#='f)3$Fdeo$\"+3A'f`$FdeoF_co7%F[aq7$$!+sjRMJFdeo $\"+OEH%\\$FdeoF_co7%Faaq7$$!+#p2$zJFdeo$\"+-**R^MFdeoF_co7%Fgaq7$$!+/ )4MA$Fdeo$\"+VWF2MFdeoF_co7%F]bq7$$!+2MymKFdeo$\"+o$=>O$FdeoF_co7%Fcbq 7$$!+'Q4&4LFdeo$\"+.iM:LFdeoF_co7%Fibq7$$!+#[n;N$Fdeo$\"+EdenKFdeoF_co 7%F_cq7$$!+#RMLR$Fdeo$\"+-(y'=KFdeoF_co7%Fecq7$$!+*H\"eMMFdeo$\"+f7ooJ FdeoF_co7%F[dq7$$!+74ZvMFdeo$\"+dOm\"H,$FdeoF_co7%F]eq7$$!+>\"Gkf $Fdeo$\"+?OPfHFdeoF_co7%Fceq7$$!+=dBOOFdeo$\"+#>*>0HFdeoF_co7%Fieq7$$! +D&odn$Fdeo$\"+X?\\]GFdeoF_co7%F_fq7$$!+/f)\\r$Fdeo$\"++!\\`z#FdeoF_co 7%Fefq7$$!+\"[IQv$Fdeo$\"+2e&)RFFdeoF_co7%F[gq7$$!+)zJAz$Fdeo$\"+d#FdeoF_co7%F]hq7$$!+>&oR!RFdeo$\"+%[\\b^#FdeoF_co7%Fchq7$$!+u_yR RFdeo$\"+UJ.fCFdeoF_co7%Fihq7$$!+&zhZ(RFdeo$\"+lUM-CFdeoF_co7%F_iq7$$! +BV$)3SFdeo$\"+ZLYXBFdeoF_co7%Feiq7$$!+4*[>/%Fdeo$\"+OGO)G#FdeoF_co7%F [jq7$$!+3>1uSFdeo$\"+&e45B#FdeoF_co7%Fajq7$$!+#)*R^5%Fdeo$\"+NqOt@Fdeo F_co7%Fgjq7$$!+p(f^8%Fdeo$\"+#3(R:@FdeoF_co7%F][r7$$!+Yt5kTFdeo$\"+b;1 d?FdeoF_co7%Fc[r7$$!+\"fx>>%Fdeo$\"+8SK)*>FdeoF_co7%Fi[r7$$!+\"\\t(=UF deo$\"+^)\\\"R>FdeoF_co7%F_\\r7$$!+Qa]WUFdeo$\"+Q#3&z=FdeoF_co7%Fe\\r7 $$!+=0>pUFdeo$\"+aCP>=FdeoF_co7%F[]r7$$!+w<&GH%Fdeo$\"+l2seCfKWFdeo$\"+6oC>8FdeoF_co7%F[`r7$$! +G$*H\\WFdeo$\"+EX\\a7FdeoF_co7%Fa`r7$$!+R3HlWFdeo$\"+!Q[$*=\"FdeoF_co 7%Fg`r7$$!+=te![%Fdeo$\"+\\#eQ7\"FdeoF_co7%F]ar7$$!+06?&\\%Fdeo$\"+>\" y!e5FdeoF_co7%Fcar7$$!+Td84XFdeo$\"+c=k?**F^coF_co7%Fiar7$$!+*p&QAXFde o$\"+qxse#*F^coF_co7%F_br7$$!+Um$\\`%Fdeo$\"+!p#e&f)F^coF_co7%Febr7$$! +4iwYXFdeo$\"+X9rJzF^coF_co7%F[cr7$$!+P_%yb%Fdeo$\"+zMcnsF^coF_co7%Fac r7$$!+8$R\"oXFdeo$\"+$Q?Ng'F^coF_co7%Fgcr7$$!+30hxXFdeo$\"+Wz))RfF^coF _co7%F]dr7$$!+`\">ie%Fdeo$\"+OF*oF&F^coF_co7%Fcdr7$$!+Wb#Rf%Fdeo$\"+JG o9YF^coF_co7%Fidr7$$!+x9p+YFdeo$\"+J>L`RF^coF_co7%F_er7$$!+l:[1YFdeo$ \"+lg%GH$F^coF_co7%Feer7$$!+CVE6YFdeo$\"+[?F^coF_co7%Fafr7$$!+\\iqYFdeo$\"+:9.ylFa[pF_co7%F]gr7$$!+#Hq)>YFdeoF(F_co7%Fcgr7$F^gr $!+:9.ylFa[pF_co7%Fggr7$Fhfr$!+:\"3eJ\"F^coF_co7%F[hr7$Fbfr$!+hp?u>F^c oF_co7%F_hr7$F\\fr$!+[?L`RF^coF_co7%F[ir7$Fjdr$!+JGo9YF^coF_co7%F_ir7$Fddr$!+OF*oF& F^coF_co7%Fcir7$F^dr$!+Wz))RfF^coF_co7%Fgir7$Fhcr$!+$Q?Ng'F^coF_co7%F[ jr7$Fbcr$!+zMcnsF^coF_co7%F_jr7$F\\cr$!+X9rJzF^coF_co7%Fcjr7$Ffbr$!+!p #e&f)F^coF_co7%Fgjr7$F`br$!+qxse#*F^coF_co7%F[[s7$Fjar$!+c=k?**F^coF_c o7%F_[s7$Fdar$!+>\"y!e5FdeoF_co7%Fc[s7$F^ar$!+\\#eQ7\"FdeoF_co7%Fg[s7$ Fh`r$!+!Q[$*=\"FdeoF_co7%F[\\s7$Fb`r$!+EX\\a7FdeoF_co7%F_\\s7$F\\`r$!+ 6oC>8FdeoF_co7%Fc\\s7$Ff_r$!+x5c$Q\"FdeoF_co7%Fg\\s7$F`_r$!+.,SZ9FdeoF _co7%F[]s7$Fj^r$!+DWt5:FdeoF_co7%F_]s7$Fd^r$!+XFat:FdeoF_co7%Fc]s7$F^^ r$!++?\"ej\"FdeoF_co7%Fg]s7$Fh]r$!+_r`(p\"FdeoF_co7%F[^s7$Fb]r$!+l2se< FdeoF_co7%F_^s7$F\\]r$!+aCP>=FdeoF_co7%Fc^s7$Ff\\r$!+Q#3&z=FdeoF_co7%F g^s7$F`\\r$!+^)\\\"R>FdeoF_co7%F[_s7$Fj[r$!+8SK)*>FdeoF_co7%F__s7$Fd[r $!+b;1d?FdeoF_co7%Fc_s7$F^[r$!+#3(R:@FdeoF_co7%Fg_s7$Fhjq$!+NqOt@FdeoF _co7%F[`s7$Fbjq$!+&e45B#FdeoF_co7%F_`s7$F\\jq$!+OGO)G#FdeoF_co7%Fc`s7$ Ffiq$!+ZLYXBFdeoF_co7%Fg`s7$F`iq$!+lUM-CFdeoF_co7%F[as7$Fjhq$!+UJ.fCFd eoF_co7%F_as7$Fdhq$!+%[\\b^#FdeoF_co7%Fcas7$F^hq$!+_@!>d#FdeoF_co7%Fga s7$Fhgq$!+&z'3GEFdeoF_co7%F[bs7$Fbgq$!+*>0HFdeoF_co7%F_cs7$Fdeq$!+?OPfHFdeoF_co7%Fccs7$F^e q$!+R>\"H,$FdeoF_co7%Fgcs7$Fhdq$!+x\"4d1$FdeoF_co7%F[ds7$Fbdq$!+dOm O$FdeoF_co7%Fces7$F^bq$!+VWF2MFdeoF_co7%Fges7$Fhaq$!+-**R^MFdeoF_co7%F [fs7$Fbaq$!+OEH%\\$FdeoF_co7%F_fs7$F\\aq$!+3A'f`$FdeoF_co7%Fcfs7$Ff`q$ !+Z$Hkd$FdeoF_co7%Fgfs7$F``q$!+)yDdh$FdeoF_co7%F[gs7$Fj_q$!+WX*Ql$Fdeo F_co7%F_gs7$Fd_q$!+l1*4p$FdeoF_co7%Fcgs7$F^_q$!+HD3FPFdeoF_co7%Fggs7$F h^q$!+'p`Aw$FdeoF_co7%F[hs7$Fb^q$!+[`g'z$FdeoF_co7%F_hs7$F\\^q$!+e#f-$ QFdeoF_co7%Fchs7$Ff]q$!+c6OjQFdeoF_co7%Fghs7$F`]q$!+^T3'*QFdeoF_co7%F[ is7$Fj\\q$!+G:jGRFdeoF_co7%F_is7$Fd\\q$!+cwBhRFdeoF_co7%Fcis7$F^\\q$!+ 0\\;%*RFdeoF_co7%Fgis7$Fh[q$!+&Q$pFSFdeoF_co7%F[js7$Fb[q$!+e%*4iSFdeoF _co7%F_js7$F\\[q$!+Q%>w4%FdeoF_co7%Fcjs7$Ffjp$!+t%)RMTFdeoF_co7%Fgjs7$ F`jp$!+.IVsTFdeoF_co7%F[[t7$Fjip$!+$GE:@%FdeoF_co7%F_[t7$Fdip$!+M=G^UF deoF_co7%Fc[t7$F^ip$!+H([6H%FdeoF_co7%Fg[t7$Fhhp$!+Eo]IVFdeoF_co7%F[\\ t7$Fbhp$!+f(e(oVFdeoF_co7%F_\\t7$F\\hp$!+\"o'R0WFdeoF_co7%Fc\\t7$Ffgp$ !+U6.SWFdeoF_co7%Fg\\t7$F`gp$!+X&)QsWFdeoF_co7%F[]t7$Fjfp$!+i7H-XFdeoF _co7%F_]t7$Fdfp$!+!eK'HXFdeoF_co7%Fc]t7$F^fp$!+#[`Vb%FdeoF_co7%Fg]t7$F hep$!+XRUwXFdeoF_co7%F[^t7$Fbep$!+&)*Gef%FdeoF_co7%F_^t7$F\\ep$!+J'eDh %FdeoF_co7%Fc^t7$Ffdp$!+93gEYFdeoF_co7%Fg^t7$F`dp$!+*=Ozj%FdeoF_co7%F[ _t7$Fjcp$!+lT`YYFdeoF_co7%F__t7$Fdcp$!+L&\\Bl%FdeoF_co7%Fc_t7$F^cp$!+H #>`l%FdeoF_co7%Fg_t7$Fhbp$!+l)e`l%FdeoF_co7%F[`t7$Fbbp$!+T'eBl%FdeoF_c o7%F_`t7$F\\bp$!+b#yhk%FdeoF_co7%Fc`t7$Ffap$!+f.kOYFdeoF_co7%Fg`t7$F`a p$!+]>_BYFdeoF_co7%F[at7$Fj`p$!+CHa1YFdeoF_co7%F_at7$Fd`p$!+O2N&e%Fdeo F_co7%Fcat7$F^`p$!+(z*\\fXFdeoF_co7%Fgat7$Fh_p$!+OSUGXFdeoF_co7%F[bt7$ Fb_p$!+j7S\"\\%FdeoF_co7%F_bt7$F\\_p$!+5*4vW%FdeoF_co7%Fcbt7$Ff^p$!+*= &e&R%FdeoF_co7%Fgbt7$F`^p$!++#*>MVFdeoF_co7%F[ct7$Fj]p$!+RpshUFdeoF_co 7%F_ct7$Fd]p$!+R2hwTFdeoF_co7%Fcct7$F^]p$!+-^\"z2%FdeoF_co7%Fgct7$Fh\\ p$!+z&))f'RFdeoF_co7%F[dt7$Fb\\p$!+!HGE%QFdeoF_co7%F_dt7$F\\\\p$!+mTQ5 PFdeoF_co7%Fcdt7$Ff[p$!+h4grNFdeoF_co7%Fgdt7$F_[p$!+P$pzU$FdeoF_co7%F[ et7$Fijo$!+K?h!G$FdeoF_co7%F_et7$Fbjo$!+jaLIJFdeoF_co7%Fcet7$F\\jo$!+< y!y(HFdeoF_co7%Fget7$Ffio$!+I4iBGFdeoF_co7%F[ft7$F`io$!+>IGoEFdeoF_co7 %F_ft7$Fjho$!+Su>7DFdeoF_co7%Fcft7$Fdho$!+;olbBFdeoF_co7%Fgft7$F]ho$!+ #)e&))>#FdeoF_co7%F[gt7$Fggo$!+tV\">/#FdeoF_co7%F_gt7$Fago$!+#z+\\)=Fd eoF_co7%Fcgt7$Fjfo$!+\\=&ys\"FdeoF_co7%Fggt7$$\"+\\1FohFieo$!+Ghyq:Fde oF_co7%F[ht7$F^fo$!+CEr89FdeoF_co7%Faht7$Fgeo$!+tbjc7FdeoF_co7%Feht7$F `eo$!+\\pb*4\"FdeoF_co7%Fiht7$Fido$!+rmxC%*F^coF_co7%F]it7$$\"+L@s)G$F N$!+l8)R&yF^coF_co7%Fait7$$\"+YB:cgF-$!+p_=$G'F^coF_co7%Fgit7$$\"+agTg lFfr$!+x*)Q7ZF^coF_co7%F]jt7$F($!+aEfTJF^coF_co7%Fcjt7$F($!+Fjzq:F^coF _co7%FgjtF'F_co-F_bo6&Fabo$\"\"*!\"\"$\"#XFdbo$\"\"\"F_[u-%&STYLEG6#%, PATCHNOGRIDG-F$6$7in7$$\"32+++@xF?8FN$\"3%******R2*>&4&FN7$$\"34+++!*e N=8FN$\"3@+++;rU2^FN7$$\"3,+++\\=X:8FN$\"3C+++L*R#>^FN7$$\"35+++xPi68F N$\"3V+++!*zeI^FN7$$\"3++++Uw#pI\"FN$\"3u*****H`E99&FN7$$\"33+++gtT,8F N$\"3!)*****4![r^^FN7$$\"3))******z\\9&H\"FN$\"3K+++\\`Th^FN7$$\"31+++ %)3;)G\"FN$\"3;+++pM\\q^FN7$$\"35+++-S^!G\"FN$\"3Q+++!o;*y^FN7$$\"3/++ +O?Ds7FN$\"3;+++sUl'=&FN7$$\"3#******zp@ME\"FN$\"3u******4pn$>&FN7$$\" 3!*******f*oSD\"FN$\"3i*****4Ec**>&FN7$$\"3/+++C$RUC\"FN$\"3'******Rmk a?&FN7$$\"35+++>\"yRB\"FN$\"32+++8[<5_FN7$$\"3'******\\\"3LB7FN$\"3J++ +o%fS@&FN7$$\"30+++&QVB@\"FN$\"3y******p64<_FN7$$\"3)*******4F1,7FN$\" 3M+++y>C>_FN7$$\"3!******H.P&*=\"FN$\"3=+++4K[?_FN7$$\"31+++.l\"y<\"FN $\"3'********=&y?_FN7$$\"3)******Rz`f;\"FN$\"3g*****f-<,A&FN7$$\"33+++ a[+a6FN$\"3%******RQY%=_FN7$$\"32+++$zH?9\"FN$\"3&)*****zIRd@&FN7$$\"3 &*******fR4I6FN$\"3m*****z3g>@&FN7$$\"3-+++`#p#=6FN$\"3K+++(>rq?&FN7$$ \"3,+++Fdj16FN$\"3H+++rL.,_FN7$$\"33+++qNG&4\"FN$\"3O+++#)e!Q>&FN7$$\" 3#******R`:V3\"FN$\"3;+++HrM&=&FN7$$\"3++++O*\\Q2\"FN$\"3m*****4u:c<&F N7$$\"31+++6V-k5FN$\"3M+++PCdk^FN7$$\"3'******p!)**\\0\"FN$\"3#)***** \\4$=_^FN7$$\"3-++++j'p/\"FN$\"30+++HOUQ^FN7$$\"3&******>,[,/\"FN$\"3E +++*e(GB^FN7$$\"3$*******Q*3[.\"FN$\"3M+++Jxz1^FN7$$\"35+++glDJ5FN$\"3 I+++yI-*3&FN7$$\"3&*******Q2%)H5FN$\"3!)******[H5q]FN7$$\"35+++%[U4.\" FN$\"3I+++4$y-0&FN7$$\"3!******RnW\\.\"FN$\"3q*****pEF*H]FN7$$\"35+++p j(\\FN7$$\"31+++,e\"R3\"FN$\"3y*****z=Hj&\\FN 7$$\"3\"******z3iL5\"FN$\"3*******4V'zV\\FN7$$\"3$******zxhV7\"FN$\"3R +++j-yM\\FN7$$\"3/+++y!Rg9\"FN$\"3k*****p(eQH\\FN7$$\"3/+++\\.hn6FN$\" 3h*****f$4XF\\FN7$$\"35+++FqX)=\"FN$\"3()*******zQ'G\\FN7$$\"3-+++N*Q \"37FN$\"3')*****H@JD$\\FN7$$\"33+++w;PE7FN$\"3?+++@dpQ\\FN7$$\"35+++s 8*HC\"FN$\"3q*****RyCn%\\FN7$$\"3!******H_?zD\"FN$\"3w*****4``i&\\FN7$ $\"31+++)QS6F\"FN$\"3A+++3X'p'\\FN7$$\"3%******fgqEG\"FN$\"3G+++rcey\\ FN7$$\"31+++V\\b#H\"FN$\"3M+++.b)3*\\FN7$$\"3*)*****Hk^3I\"FN$\"3L+++D tm.]FN7$$\"30+++Pqi28FN$\"3&******\\#Qw;]FN7$$\"3(******4Q^HJ\"FN$\"33 +++!=K+.&FN7$$\"36+++\">'*oJ\"FN$\"3))******4*\\L/&FN7$$\"3$******pdK& >8FN$\"3k*****>T6m0&FN7$$\"3/+++C-$4K\"FN$\"3C+++*4D(p]FN7$$\"3$****** \\\"o:@8FN$\"3o******\\5h#3&FNF[\\uF^bo-Fhbo6jn7%7$$\"+@xF?8Fdeo$\"+u! *>&4&Fdeo7$$\"+!*eN=8Fdeo$\"+;rU2^Fdeo7$$\"$=\"Fdbo$\"$2&Fdbo7%F__v7$$ \"+\\=X:8Fdeo$\"+L*R#>^FdeoFd_v7%Fj_v7$$\"+xPi68Fdeo$\"+!*zeI^FdeoFd_v 7%F``v7$$\"+Uw#pI\"Fdeo$\"+LlUT^FdeoFd_v7%Ff`v7$$\"+gtT,8Fdeo$\"+,[r^^ FdeoFd_v7%F\\av7$$\"+!)\\9&H\"Fdeo$\"+\\`Th^FdeoFd_v7%Fbav7$$\"+%)3;)G \"Fdeo$\"+pM\\q^FdeoFd_v7%Fhav7$$\"+-S^!G\"Fdeo$\"+!o;*y^FdeoFd_v7%F^b v7$$\"+O?Ds7Fdeo$\"+sUl'=&FdeoFd_v7%Fdbv7$$\"+)p@ME\"Fdeo$\"+5pn$>&Fde oFd_v7%Fjbv7$$\"+g*oSD\"Fdeo$\"+hi&**>&FdeoFd_v7%F`cv7$$\"+C$RUC\"Fdeo $\"+kYY0_FdeoFd_v7%Ffcv7$$\"+>\"yRB\"Fdeo$\"+8[<5_FdeoFd_v7%F\\dv7$$\" +:3LB7Fdeo$\"+o%fS@&FdeoFd_v7%Fbdv7$$\"+&QVB@\"Fdeo$\"+q64<_FdeoFd_v7% Fhdv7$$\"+5F1,7Fdeo$\"+y>C>_FdeoFd_v7%F^ev7$$\"+Lq`*=\"Fdeo$\"+4K[?_Fd eoFd_v7%Fdev7$$\"+.l\"y<\"Fdeo$\"+!>&y?_FdeoFd_v7%Fjev7$$\"+%z`f;\"Fde o$\"+Eq6?_FdeoFd_v7%F`fv7$$\"+a[+a6Fdeo$\"+%QY%=_FdeoFd_v7%Fffv7$$\"+$ zH?9\"Fdeo$\"+3$Rd@&FdeoFd_v7%F\\gv7$$\"+gR4I6Fdeo$\"+)3g>@&FdeoFd_v7% Fbgv7$$\"+`#p#=6Fdeo$\"+(>rq?&FdeoFd_v7%Fhgv7$$\"+Fdj16Fdeo$\"+rL.,_Fd eoFd_v7%F^hv7$$\"+qNG&4\"Fdeo$\"+#)e!Q>&FdeoFd_v7%Fdhv7$$\"+MbJ%3\"Fde o$\"+HrM&=&FdeoFd_v7%Fjhv7$$\"+O*\\Q2\"Fdeo$\"+Tdhv^FdeoFd_v7%F`iv7$$ \"+6V-k5Fdeo$\"+PCdk^FdeoFd_v7%Ffiv7$$\"+2)**\\0\"Fdeo$\"+&4$=_^FdeoFd _v7%F\\jv7$$\"++j'p/\"Fdeo$\"+HOUQ^FdeoFd_v7%Fbjv7$$\"+7![,/\"Fdeo$\"+ *e(GB^FdeoFd_v7%Fhjv7$$\"+R*3[.\"Fdeo$\"+Jxz1^FdeoFd_v7%F^[w7$$\"+glDJ 5Fdeo$\"+yI-*3&FdeoFd_v7%Fd[w7$$\"+R2%)H5Fdeo$\"+\\H5q]FdeoFd_v7%Fj[w7 $$\"+%[U4.\"Fdeo$\"+4$y-0&FdeoFd_v7%F`\\w7$$\"+uY%\\.\"Fdeo$\"+ns#*H]F deoFd_v7%Ff\\w7$$\"+pj(\\FdeoFd_v7%Fh]w7$ $\"+,e\"R3\"Fdeo$\"+)=Hj&\\FdeoFd_v7%F^^w7$$\"+)3iL5\"Fdeo$\"+JkzV\\Fd eoFd_v7%Fd^w7$$\"+y'*oJ\"Fdeo$\"+5*\\L/&FdeoFd_v7%F^dw 7$$\"+xD`>8Fdeo$\"+79hc]FdeoFd_v7%Fddw7$$\"+C-$4K\"Fdeo$\"+*4D(p]FdeoF d_v7%Fjdw7$$\"+:o:@8Fdeo$\"+]5h#3&FdeoFd_v7%F`ewFj^vFd_vF[[uFd[u-F$6$7 in7$F\\\\u$!3%******R2*>&4&FN7$Fc^v$!3o******\\5h#3&FN7$F^^v$!3C+++*4D (p]FN7$Fi]v$!3k*****>T6m0&FN7$Fd]v$!3))******4*\\L/&FN7$F_]v$!33+++!=K +.&FN7$Fj\\v$!3&******\\#Qw;]FN7$Fe\\v$!3L+++Dtm.]FN7$F`\\v$!3M+++.b)3 *\\FN7$F[\\v$!3G+++rcey\\FN7$Ff[v$!3A+++3X'p'\\FN7$Fa[v$!3w*****4``i& \\FN7$F\\[v$!3q*****RyCn%\\FN7$Fgju$!3?+++@dpQ\\FN7$Fbju$!3')*****H@JD $\\FN7$F]ju$!3()*******zQ'G\\FN7$Fhiu$!3h*****f$4XF\\FN7$Fciu$!3k***** p(eQH\\FN7$F^iu$!3R+++j-yM\\FN7$Fihu$!3*******4V'zV\\FN7$Fdhu$!3y***** z=Hj&\\FN7$F_hu$!39+++B!f>(\\FN7$Fjgu$!3q*******z$***)\\FN7$Fegu$!37++ +VWf4]FN7$F`gu$!3q*****pEF*H]FN7$F[gu$!3I+++4$y-0&FN7$Fffu$!3!)******[ H5q]FN7$Fafu$!3I+++yI-*3&FN7$F\\fu$!3M+++Jxz1^FN7$Fgeu$!3E+++*e(GB^FN7 $Fbeu$!30+++HOUQ^FN7$F]eu$!3#)*****\\4$=_^FN7$Fhdu$!3M+++PCdk^FN7$Fcdu $!3m*****4u:c<&FN7$F^du$!3;+++HrM&=&FN7$Ficu$!3O+++#)e!Q>&FN7$Fdcu$!3H +++rL.,_FN7$F_cu$!3K+++(>rq?&FN7$Fjbu$!3m*****z3g>@&FN7$Febu$!3&)***** zIRd@&FN7$F`bu$!3%******RQY%=_FN7$F[bu$!3g*****f-<,A&FN7$Ffau$!3'***** ***=&y?_FN7$Faau$!3=+++4K[?_FN7$F\\au$!3M+++y>C>_FN7$Fg`u$!3y******p64 <_FN7$Fb`u$!3J+++o%fS@&FN7$F]`u$!32+++8[<5_FN7$Fh_u$!3'******Rmka?&FN7 $Fc_u$!3i*****4Ec**>&FN7$F^_u$!3u******4pn$>&FN7$Fi^u$!3;+++sUl'=&FN7$ Fd^u$!3Q+++!o;*y^FN7$F_^u$!3;+++pM\\q^FN7$Fj]u$!3K+++\\`Th^FN7$Fe]u$!3 !)*****4![r^^FN7$F`]u$!3u*****H`E99&FN7$F[]u$!3V+++!*zeI^FN7$Ff\\u$!3C +++L*R#>^FN7$Fa\\u$!3@+++;rU2^FNFiewF^bo-Fhbo6jn7%7$F[_v$!+u!*>&4&Fdeo 7$Faew$!+]5h#3&Fdeo7$Fe_v$!$2&Fdbo7%Fcax7$F[ew$!+*4D(p]FdeoFfax7%Fjax7 $Fedw$!+79hc]FdeoFfax7%F^bx7$F_dw$!+5*\\L/&FdeoFfax7%Fbbx7$Ficw$!+!=K+ .&FdeoFfax7%Ffbx7$Fccw$!+DQw;]FdeoFfax7%Fjbx7$F]cw$!+Dtm.]FdeoFfax7%F^ cx7$Fgbw$!+.b)3*\\FdeoFfax7%Fbcx7$Fabw$!+rcey\\FdeoFfax7%Ffcx7$F[bw$!+ 3X'p'\\FdeoFfax7%Fjcx7$Feaw$!+JNDc\\FdeoFfax7%F^dx7$F_aw$!+%yCn%\\Fdeo Ffax7%Fbdx7$Fi`w$!+@dpQ\\FdeoFfax7%Ffdx7$Fc`w$!+87`K\\FdeoFfax7%Fjdx7$ F]`w$!++)Q'G\\FdeoFfax7%F^ex7$Fg_w$!+O4XF\\FdeoFfax7%Fbex7$Fa_w$!+xeQH \\FdeoFfax7%Ffex7$F[_w$!+j-yM\\FdeoFfax7%Fjex7$Fe^w$!+JkzV\\FdeoFfax7% F^fx7$F_^w$!+)=Hj&\\FdeoFfax7%Fbfx7$Fi]w$!+B!f>(\\FdeoFfax7%Fffx7$Fc]w $!++Q***)\\FdeoFfax7%Fjfx7$F]]w$!+VWf4]FdeoFfax7%F^gx7$Fg\\w$!+ns#*H]F deoFfax7%Fbgx7$Fa\\w$!+4$y-0&FdeoFfax7%Ffgx7$F[\\w$!+\\H5q]FdeoFfax7%F jgx7$Fe[w$!+yI-*3&FdeoFfax7%F^hx7$F_[w$!+Jxz1^FdeoFfax7%Fbhx7$Fijv$!+* e(GB^FdeoFfax7%Ffhx7$Fcjv$!+HOUQ^FdeoFfax7%Fjhx7$F]jv$!+&4$=_^FdeoFfax 7%F^ix7$Fgiv$!+PCdk^FdeoFfax7%Fbix7$Faiv$!+Tdhv^FdeoFfax7%Ffix7$F[iv$! +HrM&=&FdeoFfax7%Fjix7$Fehv$!+#)e!Q>&FdeoFfax7%F^jx7$F_hv$!+rL.,_FdeoF fax7%Fbjx7$Figv$!+(>rq?&FdeoFfax7%Ffjx7$Fcgv$!+)3g>@&FdeoFfax7%Fjjx7$F ]gv$!+3$Rd@&FdeoFfax7%F^[y7$Fgfv$!+%QY%=_FdeoFfax7%Fb[y7$Fafv$!+Eq6?_F deoFfax7%Ff[y7$F[fv$!+!>&y?_FdeoFfax7%Fj[y7$Feev$!+4K[?_FdeoFfax7%F^\\ y7$F_ev$!+y>C>_FdeoFfax7%Fb\\y7$Fidv$!+q64<_FdeoFfax7%Ff\\y7$Fcdv$!+o% fS@&FdeoFfax7%Fj\\y7$F]dv$!+8[<5_FdeoFfax7%F^]y7$Fgcv$!+kYY0_FdeoFfax7 %Fb]y7$Facv$!+hi&**>&FdeoFfax7%Ff]y7$F[cv$!+5pn$>&FdeoFfax7%Fj]y7$Febv $!+sUl'=&FdeoFfax7%F^^y7$F_bv$!+!o;*y^FdeoFfax7%Fb^y7$Fiav$!+pM\\q^Fde oFfax7%Ff^y7$Fcav$!+\\`Th^FdeoFfax7%Fj^y7$F]av$!+,[r^^FdeoFfax7%F^_y7$ Fg`v$!+LlUT^FdeoFfax7%Fb_y7$Fa`v$!+!*zeI^FdeoFfax7%Ff_y7$F[`v$!+L*R#>^ FdeoFfax7%Fj_y7$F`_v$!+;rU2^FdeoFfax7%F^`yF`axFfaxF[[uFd[u-F$6%7$7$$!3 o*************Q&FNF(7$$\"3%**************o\"FNF(-%'COLOURG6&FaboF)F)F) -%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!p&FN7$F($\"3Q++++++!p&FNF[ay F^ay-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONT G6#%(DEFAULTG-Feby6$%*HELVETICAGF^[u-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$! $R&Fdbo$\"$p\"Fdbo;$!$p&Fdbo$\"$p&Fdbo" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the in tersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approxima tely) " }{XPPEDIT 18 0 "[-4.6199, 0];" "6#7$,$-%&FloatG6$\"&*>Y!\"%! \"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally b y 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 that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[ 0, 2.8 ]" "6#7$\"\"!-%&FloatG6$\"#G!\"\"" } {TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 651 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 805362516136724921 4232300834753/328706201696683896985729147718400000*z^8+\n 124820 2152229952291767335227528959/455380180224823108486972426130565120000*z ^9+\n 27645656677124968895455230819703/1062553753857920586469602 32763798528000*z^10+\n 89106536775064966427849049/14456513657930 89233291975955970048*z^11:\nDigits := 15:\npts := []: z0 := 0:\nfor ct from 0 to 100 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,.9,.45,0),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CU RVESG6#7aq7$$\"\"!F)F(7$F($\"0z*e`EfTJ!#;7$$\"0qLz#F=7$$\"0j%R$*=@D$*F-$\"0jgG7uK^#F=7$$\"02'\\P(e^, \"F=$\"0l)>)QLu#GF=7$$\"0!H7Bx3&4\"F=$\"0H2Nl#fTJF=7$$\"0'=if'4E<\"F=$ \"0ab(=>vbMF=7$$\"0Du.lizC\"F=$\"0*z)Q=6*pPF=7$$\"0fpG$*Q8K\"F=$\"0#3 \")[/2%3%F=7$$\"01-Jr'*GR\"F=$\"0CsLrH#)R%F=7$$\"0AhhVoFY\"F=$\"0\"fLx *)Q7ZF=7$$\"00sSum5`\"F=$\"00Y.C[l-&F=7$$\"0CcG))))yf\"F=$\"0N#)=]22M& F=7$$\"0W9./?Lm\"F=$\"04.7wm[l&F=7$$\"0)oWGfVF&=F=$\"0lUN^Wtf'F=7$$\"01><.ZD \">F=$\"0VZ$\\P]6pF=7$$\"0!>a\"*H.s>F=$\"0M^G(HmDsF=7$$\"0]:bE*\\I?F=$ \"0p&))z@#)RvF=7$$\"0@w:R'*z3#F=$\"0Gl^O\")R&yF=7$$\"0\\lt!RdW@F=$\"0/ i?_S\"o\")F=7$$\"0,@%34G+AF=$\"0oXBk*H#[)F=7$$\"0]eb^m^D#F=$\"0_()erek z)F=7$$\"020!e-G4BF=$\"0\"HKIxh5\"*F=7$$\"0t>_PsEO#F=$\"0*pmqmxC%*F=7$ $\"0tgv.%R:CF=$\"0@x)=b$*Q(*F=7$$\"0!)QR[(\\nCF=$\"0ZN`U4`+\"!#97$$\"0 S)Htg.>DF=$\"0XP[GDn.\"Fdu7$$\"0u3iCk+d#F=$\"0RVt7T\"o5Fdu7$$\"0;bIRP1 i#F=$\"0D]#\\pb*4\"Fdu7$$\"0*GLE;\"3n#F=$\"0%fJYF(48\"Fdu7$$\"0GrjXV1s #F=$\"0SfN^)Qi6Fdu7$$\"0gECP*=qFF=$\"0\"[9XU!Q>\"Fdu7$$\"0:#QM`]>GF=$ \"0uYU$*>_A\"Fdu7$$\"0s8Fdu7$$\"0_txgNb,$F=$\"0A&4& 3#)3N\"Fdu7$$\"0:kH@yW1$F=$\"04U@T(H#Q\"Fdu7$$\"0!f#y*)yM6$F=$\"0YMSi7 PT\"Fdu7$$\"0l*e_5diJF=$\"0fm2qF^W\"Fdu7$$\"0\"fD2@y6KF=$\"0-G!>Eaw9Fd u7$$\"08ME,L6E$F=$\"0Pe;Ndz]\"Fdu7$$\"0kwf#yj5LF=$\"0^9q'=PR:Fdu7$$\"0 k#46MIgLF=$\"0vP#Ghyq:Fdu7$$\"07&Hc#H,T$F=$\"0&zS#4+Ag\"Fdu7$$\"0OLoS2 ,Y$F=$\"01q&4PhL;Fdu7$$\"0[4@]A-^$F=$\"0a<;#p-l;Fdu7$$\"017C!>XgNF=$\" 03;5mRkp\"Fdu7$$\"0KVa)ew5OF=$\"0-]$\\=&ys\"Fdu7$$\"0EN(*)y7hOF=$\"0ps cRj#f:!y%\\6PF=$\"0TTY>u1z\"Fdu7$$\"0!*>*4r\"=w$F=$\"0+\"RF=$\"0t*H\"=2j\">Fdu7$$\"0aRCMi9'RF=$\"0,Wv*>rZ>Fdu 7$$\"0ay+\">h5SF=$\"0a_`*\\6z>Fdu7$$\"0xW\\m&HfSF=$\"0u=S*e^5?Fdu7$$\" 0_o5%=U2TF=$\"0,JLP9>/#Fdu7$$\"0P-$y8*[:%F=$\"0xv)o+Jt?Fdu7$$\"0h`$>qf ,UF=$\"0WXpc-Z5#Fdu7$$\"0,w&3@UZUF=$\"0YX!*R\"4O@Fdu7$$\"0\\j_-RAH%F=$ \"0!*)*f.wu;#Fdu7$$\"0_,1;2fL%F=$\"0*Ql\")e&))>#Fdu7$$\"0:JNJq#yVF=$\" 0A0iEI-B#Fdu7$$\"0>k;L`\">WF=$\"0L]\"R%)fhAFdu7$$\"0vj7vd$eWF=$\"0I3>c fHH#Fdu7$$\"0I;D3cc\\%F=$\"0MT,q7VK#Fdu7$$\"0*\\\"=A%yIXF=$\"0)*Ge\"ol bBFdu7$$\"0*oa:7VjXF=$\"0xd\"f2*pQ#Fdu7$$\"0cYF=$\"0\\'=@Hi\\CFdu7$$\"0&pj]WLUYF=$\"0&)Gt?=4[#Fdu7$$\"0 `K1Iu.m%F=$\"0j())Ru>7DFdu7$$\"0M-Q>-Hn%F=$\"0AO!)yeMa#Fdu7$$\"0&H<1Um yYF=$\"0I4=E+Zd#Fdu7$$\"06kS].fn%F=$\"0Mqbq>fg#Fdu7$$\"08NloK?m%F=$\"0 $oD#z9rj#Fdu7$$\"0AInd\")Hj%F=$\"0xh&=IGoEFdu7$$\"0'H#Q8J9P%\\%F=$\"0<_W(z_IFFdu7$$\"0)G\"GQ(zNVF=$\"0'R(*)y) fhFFdu7$$\"0qD9i^e%RF=$\"0M3A\"4j#z#Fdu7$$!0&)))o*)Qq0%F=$\"0rL-$4iBGF du7$$!0KH;'z\"*4XF=$\"0$3@^_caGFdu7$$!0$e0@(*>rZF=$\"0SZ.6ga)GFdu7$$!0 yl<*\\Sq\\F=$\"0X'Gw:I;HFdu7$$!0-?1\")f\"Q^F=$\"0^S(eb3ZHFdu7$$!0q+Nse nG&F=$\"0$)3r\"y!y(HFdu7$$!08x.n$RAaF=$\"0P@(pRY3IFdu7$$!0y`gg@'[bF=$ \"0tcK]\\!RIFdu7$$!0;&fl4nncF=$\"0:#y\"yf&pIFdu7$$!0/XU`E5y&F=$\"0!pwa +***4$Fdu7$$!0mS&ySs*)eF=$\"0ZILYN.8$Fdu-%*THICKNESSG6#\"\"#-%%FONTG6$ %*HELVETICAG\"\"*-%&COLORG6&%$RGBG$Fa[m!\"\"$\"#X!\"#F(-%+AXESLABELSG6 $Q!6\"F^\\m-%%VIEWG6$%(DEFAULTGFc\\m" 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 b oundary curve with the imaginary axis can be determined more accuratel y 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 := 2.8*I:\nfor ct from 88 to 91 do\n newton(R (z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"0Y41wD!=5!#=$\"0'R(*)y)fhF!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"0zP^BT.h$!#>$\"0M3A\"4j#z#!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0\"[RUE=,\\!#>$\"0rL-$4iBG!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0e$G0)Q)p:!#=$\"0$3@^_caG!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisect ion method to calculate the parameter value associated with the inters ection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.8*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.88..0.91);\nnewton(R(z)=exp(u 0*Pi*I),z=2.8*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #u0G$\"0PM%yqZX*)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0[`*)pHn! G!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 260 112 "largest interval on the nonnegative imaginary axi s that contains the origin and lies inside the stability region" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.8067];" "6#7$\"\"!-%&FloatG6$ \"&n!G!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 12 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs (b=`b*`,StabilityFunction(6,12,'expanded'))):\n`R*` := unapply(%,z):\n '`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,< \"\"\"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'F1 F)F)F)*&#\"K*y\"Hme>;2Bt%)QwlyQ4(H&3A\"O+++!)G?&3wU^AAYkVyiRt#\\i5F)*$ )F'\"\"(F)F)F)*&#\"N6g\\Z6uCPd.EE7m0e#)>n6iw\"\"R+++!3ig^'*pv[dG)or\\z x)\\D4XZF)*$)F'\"\")F)F)F)*&#\"L4KB%3\"\\Nn7[q8#oZm^o)*fxY\"R+++S]L8\\ d(H$RT&G9l$=K$*o3j9Q*=R\"F)*$)F'\"#6F)F)Fen*&#\"; \\!\\yUm\\1vn`1\"*)\"D+C])z(z)fk;Ya'*GoDGsF)*$)F'\"#7F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find th e point where the boundary of the stability region intersects the nega tive 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 30 "z_0 := new ton(`R*`(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+/GS0 O!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-3.6):\np_1 := plot([`R*`( z),1],z=-4.19..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],styl e=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=[-4.19..0.49,-0.07..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 396 281 281 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$ $!3Q++++++!>%!#<$\"3EvAHs)4F*QF*7$$!33+]iHt\\kTF*$\"3W*o0eOy.o$F*7$$!3 l++DfY**QTF*$\"3y7^PE\\8TF*$\"3!4e3P78pG$F*7$$!3-++ ]=$*)z3%F*$\"3$G9mY(R'[5$F*7$$!3O+vBW)4O/%F*$\"3s:')Q#*\\x4GF*7$$!3E+] (*p.B**RF*$\"3Zu(3B%3VSDF*7$$!3W+Dm45K\\RF*$\"3_rX()*Q&zlAF*7$$!3;++N \\;T**QF*$\"3)pZjaL^&=?F*7$$!3A++Sd9<\\QF*$\"3iP54I'p[z\"F*7$$!3E++Xl7 $*)z$F*$\"3QY-W3@>%f\"F*7$$!3U+DOc)H*[PF*$\"3c0U-6^A:9F*7$$!37+]FZ%G*) p$F*$\"3CMLr&[e]D\"F*7$$!3X+]dLI@1OF*$\"3F]l#z)H)>+\"F*7$$!3=+](Hf6-^$ F*$\"3YC5l**yF8z!#=7$$!3?+]xln#4T$F*$\"3=Q-z!evu='F]p7$$!3D+]FY.'>J$F* $\"3u!HJtgA6%[F]p7$$!3?++I>7;5KF*$\"3)[6m)o\\IrPF]p7$$!33++&G\\'\\?JF* $\"31mWn;@\\VIF]p7$$!32++!zCb&>IF*$\"3++]G&*>=HF*$ \"3;:U-g?`b>F]p7$$!3******47X_?GF*$\"3Ba.^S\\,N;F]p7$$!3<+]xRi#=t#F*$ \"32r!=9?Z!H9F]p7$$!3\"******[Tbji#F*$\"39=)*)o4C#o7F]p7$$!3y****f\"z2 q`#F*$\"3k&p\"))y9V*=\"F]p7$$!3$)**\\o]6F]p7$$!3:++g ot2TBF*$\"3#H\"=0\"f3]:\"F]p7$$!3#)**\\<&4T,C#F*$\"31*)*enTQR>\"F]p7$$ !3%)**\\_$*e-W@F*$\"3E\\N$pG+!f7F]p7$$!3m***\\PwRP/#F*$\"3Q.qU'G5?N\"F ]p7$$!3w**\\n!zX;&>F*$\"3QygHL(4xX\"F]p7$$!31++D***4B&=F*$\"3+Ag(*z+,# f\"F]p7$$!3/+]xH!G\"\\F]p7$$!3!****\\p`*Hi:F*$\"3lR&G0u\"\\,@F]p7$$!33++?'[!3i9F*$\"3!=!yAY (Q-K#F]p7$$!3%)***\\rpNSO\"F*$\"3_nY!>H'pdDF]p7$$!3y**\\PAB5;%3P6GF]p7$$!3w***\\%=F%Q;\"F*$\"3sBqH42;BJF]p7$$!33++g%Q*>p5F*$\"3u Tn@!*[(HV$F]p7$$!3)o****\\Cu9o*F]p$\"3%Qia*\\G\"zz$F]p7$$!3g,+v$H$zl() F]p$\"3El^)>gg?;%F]p7$$!3+&****4:7Zw(F]p$\"3=![%)4]p-g%F]p7$$!3%>+]<\" GxAoF]p$\"3[.NJ5mka]F]p7$$!37&**\\Zzu\"QeF]p$\"3;3qP<-lxbF]p7$$!3M**** *\\E]b([F]p$\"3%HtI0mg79'F]p7$$!3!z**\\Pfrx'QF]p$\"35cn7cVU#z'F]p7$$!3 M$****f&4;(*GF]p$\"3MUw3@.w%[(F]p7$$!3N)***\\!RrX!>F]p$\"3OK]h\\=\"eE) F]p7$$!3#Q$**\\x'4??*!#>$\"3I9&oN>o37*F]p7$$!3_r'****>ymc\"!#?$\"3;bk* R[XV)**F]p7$$\"3S-+]r9.@5F]p$\"3k\\$p5q(\\26F*7$$\"352++[nE[>F]p$\"3P7 *[3O+^@\"F*7$$\"3-0+D[H*o$HF]p$\"3K*RP;2n8M\"F*7$$\"3)H+]xOMJ)QF]p$\"3 YI'z,*=\\u9F*7$$\"3!***************[F]p$\"3lO8/?jJK;F*-%'COLOURG6&%$RG BG$\"*++++\"!\")$\"\"!Fe]lFd]l-F$6$7S7$F($\"\"\"Fe]l7$F=Fj]l7$FGFj]l7$ FQFj]l7$FenFj]l7$F_oFj]l7$FdoFj]l7$FioFj]l7$F_pFj]l7$FdpFj]l7$FipFj]l7 $F^qFj]l7$FcqFj]l7$FhqFj]l7$F]rFj]l7$FbrFj]l7$FgrFj]l7$F\\sFj]l7$FasFj ]l7$FfsFj]l7$F[tFj]l7$F`tFj]l7$FetFj]l7$FjtFj]l7$F_uFj]l7$FduFj]l7$Fiu Fj]l7$F^vFj]l7$FcvFj]l7$FhvFj]l7$F]wFj]l7$FbwFj]l7$FgwFj]l7$F\\xFj]l7$ FaxFj]l7$FfxFj]l7$F[yFj]l7$F`yFj]l7$FeyFj]l7$FjyFj]l7$F_zFj]l7$FdzFj]l 7$FizFj]l7$F_[lFj]l7$Fe[lFj]l7$Fj[lFj]l7$F_\\lFj]l7$Fd\\lFj]l7$Fi\\lFj ]l-F^]l6&F`]lFd]lFd]lFa]l-F$6&7#7$$!38+++/GS0OF*Fj]l-%'SYMBOLG6#%'CIRC LEG-F^]l6&F`]lFe]lFe]lFe]l-%&STYLEG6#%&POINTG-F$6&F`al-Feal6#%&CROSSGF halFjal-F$6&F`al-Feal6#%(DIAMONDGFhalFjal-F$6%7$7$FbalFd]lFaal-%&COLOR G6&F`]lFd]l$\"\"&!\"\"Fd]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q !Fjcl-%%FONTG6#%(DEFAULTG-F]dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$>%!\"#$ \"#\\Fjdl;$!\"(Fjdl$\"$Z\"Fjdl" 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 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1874 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+1/720*z^6+\n 22085297093878657638847323071619586 6291789/1062492733962784364462222514276085202880000000*z^7+\n 17662 1167198258056612262603573724741147496011/47450925498777949716882857487 56996516062080000000*z^8-\n 467759986851664768213704812673549108423 3209/5494317689332183651428541393297574913335040000000*z^9+\n 39105 03137998182153864907011530132904686161/3796074039902235977350628599005 597212849664000000*z^10-\n 1047039846242779711464722609454311495623 9/1391893814630819858361897152968718978044876800000*z^11+\n 8910653 6775064966427849049/72282568289654461664598797798502400*z^12:\npts := \+ []:\nz0 := 0:\nfor ct from 0 to 180 do\n zz := newton(`R*`(z)=exp(ct *Pi/15*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\n end do:\np_1 := plot(pts,color=COLOR(RGB,.35,.1,0)):\np_2 := plots[pol ygonplot]([seq([pts[i-1],pts[i],[-1.8,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.72,.36,0)):\npts := []: z0 := 1.5+ 5*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,.35,.1,0)):\np_4 := plots[polygonplot]( [seq([pts[i-1],pts[i],[1.36,4.87]],i=2..nops(pts))],\n style= patchnogrid,color=COLOR(RGB,.72,.36,0)):\npts := []: z0 := 1.5-5*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,.1,0)):\np_6 := plots[polygonplot]([seq([ pts[i-1],pts[i],[1.36,-4.87]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,.72,.36,0)):\np_7 := plot([[[-4.19,0],[1.79,0]] ,[[0,-5.39],[0,5.39]]],color=black,linestyle=3):\nplots[display]([p_|| (1..7)],view=[-4.19..1.79,-5.39..5.39],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 421 452 452 {PLOTDATA 2 "6/-%'CURVESG6$7av7$$\"\" !F)F(7$F($\"3++++/^R%4#!#=7$$!3/+++/Tdg8!#E$\"3')*****4Y!z)=%F-7$$\"3% )*****Hdg\"p()F1$\"3K+++Q/>$G'F-7$$\"3)********\\O2X'!#C$\"3k******R]i x$)F-7$$\"3T+++n1t2w!#B$\"3%******HI@s/\"!#<7$$\"3!)*****fW!yQ\\!#A$\" 33+++Clsc7FE7$$\"3%******>2tWB#!#@$\"35+++m;Lm9FE7$$\"3'*******=ToJyFO $\"3!******RDvgn\"FE7$$\"3'******p!4$3D#!#?$\"3-+++^'Qe)=FE7$$\"3#)*** **>]%QkaFZ$\"3')******yk,&4#FE7$$\"3%******\\j\")>8\"!#>$\"3\"******z] (*>I#FE7$$\"31+++GUs\"*>F_o$\"3@+++ue!Q]#FE7$$\"3$)******\\09SHF_o$\"3 #)******GIn'p#FE7$$\"3G+++nf/\"e$F_o$\"3'*******GxdxGFE7$$\"3;+++Ld(4X $F_o$\"3!)*****fLc^/$FE7$$\"35+++)3%)*z@F_o$\"33+++*z`!*>$FE7$$!3?+++X kWUVFZ$\"3A+++4L!)QLFE7$$!3&*******e5&pS%F_o$\"3#******>mzLY$FE7$$!3#) *****p>hQd*F_o$\"33+++auhrNFE7$$!33+++QPWk:F-$\"3/+++#****Hm$FE7$$!37+ ++xo^HAF-$\"33+++!*y3QPFE7$$!3$)*****f)y0DHF-$\"3!)*****>86$)z$FE7$$!3 %)*****z%e`JOF-$\"3?+++XLYXQFE7$$!3B+++:DtOVF-$\"3/+++;HH\")QFE7$$!3F+ ++XnlL]F-$\"3:+++!)\\I2RFE7$$!3J+++4Jh=dF-$\"3,+++rMrCRFE7$$!3g*****RR _)*Q'F-$\"3y*****4\"HYMRFE7$$!31+++hOuYqF-$\"3#*******z5FPRFE7$$!3a*** **zS1$*o(F-$\"3(*******[5nLRFE7$$!3!******4_qzJ)F-$\"3))*****RWXS#RFE7 $$!3v*****fV![L*)F-$\"3=++++>l3RFE7$$!3G+++M#**o`*F-$\"3=+++y2k()QFE7$ $!32+++p'pH,\"FE$\"3=+++\\o1hQFE7$$!3)*******QFRr5FE$\"3#)*****\\<(*)G QFE7$$!35+++bDDH6FE$\"3++++v%=5z$FE7$$!3-+++tj(p=\"FE$\"3?+++?-DZPFE7$ $!35+++8O>X7FE$\"3#******fUwtp$FE7$$!3)******\\WH[I\"FE$\"3()*****z+B7 k$FE7$$!3-+++bg?n8FE$\"3;+++N6$)yNFE7$$!3&*******4j1M9FE$\"3)******4y% z5NFE7$$!36+++gUH2:FE$\"3-+++QhvQMFE7$$!3-+++z.0)e\"FE$\"3%)*****zx1dO $FE7$$!33+++)Q6cn\"FE$\"3'******p@B`H$FE7$$!37+++86@n&oCrJFE7$$!3!******Rys&\\>FE$\"3#**** **\\d'>#FE$\"3))*****\\tD,(HFE7$$!3%******H/t1F#FE$\"3')* ****pN1A#HFE7$$!3$)*****R5p1M#FE$\"3!)*****zygN(GFE7$$!37+++(>tpS#FE$ \"3++++62sBGFE7$$!3%******H<[)pCFE$\"3;+++j3MsFFE7$$!3:+++h*[&HDFE$\"3 6+++]A;>FFE7$$!3%)*****HT@je#FE$\"3?+++5p)Rm#FE7$$!3%******>(fSSEFE$\" 3;+++c:m1EFE7$$!37+++rN/#p#FE$\"35+++p\"oqa#FE7$$!31+++0A[TFFE$\"35+++ ,/7&[#FE7$$!3'******>'H)*)y#FE$\"32+++R_w?CFE7$$!3@+++*)R#[$GFE$\"3/++ +i'*)RN#FE7$$!3?+++#R+$zGFE$\"3#******\\#=$[G#FE7$$!3#*******QurAHFE$ \"3*)*****pP&R8AFE7$$!3=+++SWPlHFE$\"3=+++UP')R@FE7$$!3')*****z.Qv+$FE $\"3A+++s)3X1#FE7$$!3;+++wsS\\IFE$\"3-+++FE7$$!3;+++s!y54$FE$\"3 %******pn<)4>FE7$$!3>+++e!>D8$FE$\"3/+++?#R8$=FE7$$!3!******RupN<$FE$ \"35+++p,m_+9!e 5FE7$$!3%)*****>We)*\\$FE$\"3X*****\\7C3\")*F-7$$!3#)*****\\P\"*)>NFE$ \"3D+++C)=o.*F-7$$!3z******ziXPNFE$\"3#)*****p2&zc#)F-7$$!3/+++eli_NFE $\"3i******eqcpuF-7$$!3?+++HW]lNFE$\"35+++!=DTn'F-7$$!3=+++'3?id$FE$\" 3%)*****R'GopeF-7$$!31+++R+$\\e$FE$\"3))*****>ZSd0&F-7$$!35+++'Q8=f$FE $\"3z*****>4l@B%F-7$$!3'******4hkqf$FE$\"3')******pxF*R$F-7$$!3-+++^=) 3g$FE$\"33+++Y1$zb#F-7$$!31+++%\\]Mg$FE$\"3()*****>&ya4&ya44l@B%F-7$F[fl$!3))*****>ZSd0&F-7$Ffel$!3%)*****R'GopeF-7$F ael$!35+++!=DTn'F-7$F\\el$!3i******eqcpuF-7$Fgdl$!3#)*****p2&zc#)F-7$F bdl$!3D+++C)=o.*F-7$F]dl$!3X*****\\7C3\")*F-7$Fhcl$!3'******>+9!e5FE7$ Fccl$!3*)*****zH@Y8\"FE7$F^cl$!3'******p&z067FE7$Fibl$!33+++EE[(G\"FE7 $Fdbl$!3/+++)H_SO\"FE7$F_bl$!3&******pb64W\"FE7$Fjal$!32+++_k<=:FE7$Fe al$!3#******zu7ff\"FE7$F`al$!31+++%[5Tn\"FE7$F[al$!35+++p,m_FE7$F\\`l$!3-+++FE7$Fg_l$!3A +++s)3X1#FE7$Fb_l$!3=+++UP')R@FE7$F]_l$!3*)*****pP&R8AFE7$Fh^l$!3#**** **\\#=$[G#FE7$Fc^l$!3/+++i'*)RN#FE7$F^^l$!32+++R_w?CFE7$Fi]l$!35+++,/7 &[#FE7$Fd]l$!35+++p\"oqa#FE7$F_]l$!3;+++c:m1EFE7$Fj\\l$!3?+++5p)Rm#FE7 $Fe\\l$!36+++]A;>FFE7$F`\\l$!3;+++j3MsFFE7$F[\\l$!3++++62sBGFE7$Ff[l$! 3!)*****zygN(GFE7$Fa[l$!3')*****pN1A#HFE7$F\\[l$!3))*****\\tD,(HFE7$Fg z$!3;+++C6'z,$FE7$Fbz$!3-+++UAfmIFE7$F]z$!3#******\\d'>&oCrJFE7$Fcy$!3?+++GdHIKFE7$F^y$!3'******p@B`H$FE7$Fix$!3%)*****z x1dO$FE7$Fdx$!3-+++QhvQMFE7$F_x$!3)******4y%z5NFE7$Fjw$!3;+++N6$)yNFE7 $Few$!3()*****z+B7k$FE7$F`w$!3#******fUwtp$FE7$F[w$!3?+++?-DZPFE7$Ffv$ !3++++v%=5z$FE7$Fav$!3#)*****\\<(*)GQFE7$F\\v$!3=+++\\o1hQFE7$Fgu$!3=+ ++y2k()QFE7$Fbu$!3=++++>l3RFE7$F]u$!3))*****RWXS#RFE7$Fht$!3(*******[5 nLRFE7$Fct$!3#*******z5FPRFE7$F^t$!3y*****4\"HYMRFE7$Fis$!3,+++rMrCRFE 7$Fds$!3:+++!)\\I2RFE7$F_s$!3/+++;HH\")QFE7$Fjr$!3?+++XLYXQFE7$Fer$!3! )*****>86$)z$FE7$F`r$!33+++!*y3QPFE7$F[r$!3/+++#****Hm$FE7$Ffq$!33+++a uhrNFE7$Faq$!3#******>mzLY$FE7$F\\q$!3A+++4L!)QLFE7$Fgp$!33+++*z`!*>$F E7$Fbp$!3!)*****fLc^/$FE7$F]p$!3'*******GxdxGFE7$Fho$!3#)******GIn'p#F E7$Fco$!3@+++ue!Q]#FE7$F]o$!3\"******z](*>I#FE7$Fhn$!3')******yk,&4#FE 7$FX$!3-+++^'Qe)=FE7$FS$!3!******RDvgn\"FE7$FM$!35+++m;Lm9FE7$FG$!33++ +Clsc7FE7$F@$!3%******HI@s/\"FE7$F:$!3k******R]ix$)F-7$$\"3Y+++7^;p()F 1$!3K+++Q/>$G'F-7$$!3'******44u0O\"F1$!3')*****4Y!z)=%F-7$F($!3++++/^R %4#F-F'-%&COLORG6&%$RGBG$\"#N!\"#$\"\"\"!\"\"F(-%)POLYGONSG6bv7%F'7$F( $\"+/^R%4#!#57$$F-FcimF(7%Fhim7$$!+/Tdg8F-$\"+h/z)=%F[jmF\\jm7%F_jm7$$ \"+t0;p()F-$\"+Q/>$G'F[jmF\\jm7%Fejm7$$\"++lt]k!#;$\"+S]ix$)F[jmF\\jm7 %F[[n7$$\"+n1t2w!#:$\"+.8AZ5!\"*F\\jm7%Fb[n7$$\"+Y/yQ\\!#9$\"+Clsc7Fh[ nF\\jm7%Fj[n7$$\"+sIZMA!#8$\"+m;Lm9Fh[nF\\jm7%Fa\\n7$$\"+>ToJyFd\\n$\" +a_2w;Fh[nF\\jm7%Fh\\n7$$\"+24$3D#!#7$\"+^'Qe)=Fh[nF\\jm7%F^]n7$$\"+-X QkaFa]n$\"+zk,&4#Fh[nF\\jm7%Fe]n7$$\"+N;)>8\"!#6$\"+3v*>I#Fh[nF\\jm7%F [^n7$$\"+GUs\"*>F^^n$\"+ue!Q]#Fh[nF\\jm7%Fb^n7$$\"+]09SHF^^n$\"+HIn'p# Fh[nF\\jm7%Fh^n7$$\"+nf/\"e$F^^n$\"+HxdxGFh[nF\\jm7%F^_n7$$\"+Ld(4X$F^ ^n$\"+Oj:XIFh[nF\\jm7%Fd_n7$$\"+)3%)*z@F^^n$\"+*z`!*>$Fh[nF\\jm7%Fj_n7 $$!+XkWUVFa]n$\"+4L!)QLFh[nF\\jm7%F``n7$$!+f5&pS%F^^n$\"+i'zLY$Fh[nF\\ jm7%Ff`n7$$!+(>hQd*F^^n$\"+auhrNFh[nF\\jm7%F\\an7$$!+QPWk:F[jm$\"+#*** *Hm$Fh[nF\\jm7%Fban7$$!+xo^HAF[jm$\"+!*y3QPFh[nF\\jm7%Fhan7$$!+')y0DHF [jm$\"+K6J)z$Fh[nF\\jm7%F^bn7$$!+[e`JOF[jm$\"+XLYXQFh[nF\\jm7%Fdbn7$$! +:DtOVF[jm$\"+;HH\")QFh[nF\\jm7%Fjbn7$$!+XnlL]F[jm$\"+!)\\I2RFh[nF\\jm 7%F`cn7$$!+4Jh=dF[jm$\"+rMrCRFh[nF\\jm7%Ffcn7$$!+%R_)*Q'F[jm$\"+6HYMRF h[nF\\jm7%F\\dn7$$!+hOuYqF[jm$\"+!3rs$RFh[nF\\jm7%Fbdn7$$!+3kI*o(F[jm$ \"+\\5nLRFh[nF\\jm7%Fhdn7$$!+@0(zJ)F[jm$\"+Wa/CRFh[nF\\jm7%F^en7$$!+O/ [L*)F[jm$\"++>l3RFh[nF\\jm7%Fden7$$!+M#**o`*F[jm$\"+y2k()QFh[nF\\jm7%F jen7$$!+p'pH,\"Fh[n$\"+\\o1hQFh[nF\\jm7%F`fn7$$!+RFRr5Fh[n$\"+vr*)GQFh [nF\\jm7%Fffn7$$!+bDDH6Fh[n$\"+v%=5z$Fh[nF\\jm7%F\\gn7$$!+tj(p=\"Fh[n$ \"+?-DZPFh[nF\\jm7%Fbgn7$$!+8O>X7Fh[n$\"+EkP(p$Fh[nF\\jm7%Fhgn7$$!+X%H [I\"Fh[n$\"+3IATOFh[nF\\jm7%F^hn7$$!+bg?n8Fh[n$\"+N6$)yNFh[nF\\jm7%Fdh n7$$!+5j1M9Fh[n$\"+\"y%z5NFh[nF\\jm7%Fjhn7$$!+gUH2:Fh[n$\"+QhvQMFh[nF \\jm7%F`in7$$!+z.0)e\"Fh[n$\"+ynqlLFh[nF\\jm7%Ffin7$$!+)Q6cn\"Fh[n$\"+ Fh[n$\"+vl>#Fh[n$\"+Nd7qHFh[nF\\jm7%F`\\o7$$!+VInqAFh[n$\"+dj ?AHFh[nF\\jm7%Ff\\o7$$!+/\"p1M#Fh[n$\"+)ygN(GFh[nF\\jm7%F\\]o7$$!+(>tp S#Fh[n$\"+62sBGFh[nF\\jm7%Fb]o7$$!+t\"[)pCFh[n$\"+j3MsFFh[nF\\jm7%Fh]o 7$$!+h*[&HDFh[n$\"+]A;>FFh[nF\\jm7%F^^o7$$!+89K'e#Fh[n$\"+5p)Rm#Fh[nF \\jm7%Fd^o7$$!+sfSSEFh[n$\"+c:m1EFh[nF\\jm7%Fj^o7$$!+rN/#p#Fh[n$\"+p\" oqa#Fh[nF\\jm7%F`_o7$$!+0A[TFFh[n$\"+,/7&[#Fh[nF\\jm7%Ff_o7$$!+iH)*)y# Fh[n$\"+R_w?CFh[nF\\jm7%F\\`o7$$!+*)R#[$GFh[n$\"+i'*)RN#Fh[nF\\jm7%Fb` o7$$!+#R+$zGFh[n$\"+D=$[G#Fh[nF\\jm7%Fh`o7$$!+RurAHFh[n$\"+x`R8AFh[nF \\jm7%F^ao7$$!+SWPlHFh[n$\"+UP')R@Fh[nF\\jm7%Fdao7$$!+Q!Qv+$Fh[n$\"+s) 3X1#Fh[nF\\jm7%Fjao7$$!+wsS\\IFh[n$\"+Fh[nF\\jm7%F`bo7$$!+s!y54$ Fh[n$\"+xw\")4>Fh[nF\\jm7%Ffbo7$$!+e!>D8$Fh[n$\"+?#R8$=Fh[nF\\jm7%F\\c o7$$!+W(pN<$Fh[n$\"+p,m_NFh[n$\"+C)=o.*F[jmF\\jm7%Fdgo7 $$!+!Gcu`$Fh[n$\"+x]zc#)F[jmF\\jm7%Fjgo7$$!+eli_NFh[n$\"+fqcpuF[jmF\\j m7%F`ho7$$!+HW]lNFh[n$\"+!=DTn'F[jmF\\jm7%Ffho7$$!+'3?id$Fh[n$\"+kGope F[jmF\\jm7%F\\io7$$!+R+$\\e$Fh[n$\"+s/ub]F[jmF\\jm7%Fbio7$$!+'Q8=f$Fh[ n$\"+#4l@B%F[jmF\\jm7%Fhio7$$!+6Y1(f$Fh[n$\"+qxF*R$F[jmF\\jm7%F^jo7$$! +^=)3g$Fh[n$\"+Y1$zb#F[jmF\\jm7%Fdjo7$$!+%\\]Mg$Fh[n$\"+_ya4Fh[nF\\jm7%Ffap7$Fabo$!+Fh[nF\\jm7%Fjap7$F[bo$!+ s)3X1#Fh[nF\\jm7%F^bp7$Feao$!+UP')R@Fh[nF\\jm7%Fbbp7$F_ao$!+x`R8AFh[nF \\jm7%Ffbp7$Fi`o$!+D=$[G#Fh[nF\\jm7%Fjbp7$Fc`o$!+i'*)RN#Fh[nF\\jm7%F^c p7$F]`o$!+R_w?CFh[nF\\jm7%Fbcp7$Fg_o$!+,/7&[#Fh[nF\\jm7%Ffcp7$Fa_o$!+p \"oqa#Fh[nF\\jm7%Fjcp7$F[_o$!+c:m1EFh[nF\\jm7%F^dp7$Fe^o$!+5p)Rm#Fh[nF \\jm7%Fbdp7$F_^o$!+]A;>FFh[nF\\jm7%Ffdp7$Fi]o$!+j3MsFFh[nF\\jm7%Fjdp7$ Fc]o$!+62sBGFh[nF\\jm7%F^ep7$F]]o$!+)ygN(GFh[nF\\jm7%Fbep7$Fg\\o$!+dj? AHFh[nF\\jm7%Ffep7$Fa\\o$!+Nd7qHFh[nF\\jm7%Fjep7$F[\\o$!+C6'z,$Fh[nF\\ jm7%F^fp7$Fe[o$!+UAfmIFh[nF\\jm7%Fbfp7$F_[o$!+vl>l3RFh[nF\\jm7%Fbjp7$F_en$!+Wa/CRFh[nF\\jm7%Ffjp7$Fidn$!+\\5nLRFh[nF \\jm7%Fjjp7$Fcdn$!+!3rs$RFh[nF\\jm7%F^[q7$F]dn$!+6HYMRFh[nF\\jm7%Fb[q7 $Fgcn$!+rMrCRFh[nF\\jm7%Ff[q7$Facn$!+!)\\I2RFh[nF\\jm7%Fj[q7$F[cn$!+;H H\")QFh[nF\\jm7%F^\\q7$Febn$!+XLYXQFh[nF\\jm7%Fb\\q7$F_bn$!+K6J)z$Fh[n F\\jm7%Ff\\q7$Fian$!+!*y3QPFh[nF\\jm7%Fj\\q7$Fcan$!+#****Hm$Fh[nF\\jm7 %F^]q7$F]an$!+auhrNFh[nF\\jm7%Fb]q7$Fg`n$!+i'zLY$Fh[nF\\jm7%Ff]q7$Fa`n $!+4L!)QLFh[nF\\jm7%Fj]q7$F[`n$!+*z`!*>$Fh[nF\\jm7%F^^q7$Fe_n$!+Oj:XIF h[nF\\jm7%Fb^q7$F__n$!+HxdxGFh[nF\\jm7%Ff^q7$Fi^n$!+HIn'p#Fh[nF\\jm7%F j^q7$Fc^n$!+ue!Q]#Fh[nF\\jm7%F^_q7$F\\^n$!+3v*>I#Fh[nF\\jm7%Fb_q7$Ff]n $!+zk,&4#Fh[nF\\jm7%Ff_q7$F_]n$!+^'Qe)=Fh[nF\\jm7%Fj_q7$Fi\\n$!+a_2w;F h[nF\\jm7%F^`q7$Fb\\n$!+m;Lm9Fh[nF\\jm7%Fb`q7$F[\\n$!+Clsc7Fh[nF\\jm7% Ff`q7$Fc[n$!+.8AZ5Fh[nF\\jm7%Fj`q7$F\\[n$!+S]ix$)F[jmF\\jm7%F^aq7$$\"+ 7^;p()F-$!+Q/>$G'F[jmF\\jm7%Fbaq7$$!+\"4u0O\"F-$!+h/z)=%F[jmF\\jm7%Fha q7$F($!+/^R%4#F[jmF\\jm7%F^bqF'F\\jm-F[im6&F]im$\"#sF`im$\"#OF`imF(-%& STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"31+++:8:c9FE$\"3*******4mSD$\\FE7$$ \"3#******4P'=^9FE$\"3.+++]5#4%\\FE7$$\"33+++ZibX9FE$\"3!)*****pm^([\\ FE7$$\"3\"*******>PJR9FE$\"3k*****4!f*f&\\FE7$$\"3/+++P5^K9FE$\"3c**** *\\i?E'\\FE7$$\"3/+++o**>D9FE$\"3g******Rdfo\\FE7$$\"3-+++1>V<9FE$\"3/ +++!y$*Q(\\FE7$$\"34+++0!e#49FE$\"3#)******p&*[y\\FE7$$\"3)******zNH2S \"FE$\"3%)*****R#*fB)\\FE7$$\"3%******R?(*=R\"FE$\"33+++%R$[&)\\FE7$$ \"3/+++uJ\"GQ\"FE$\"3e*****f0Sy)\\FE7$$\"3++++p&HNP\"FE$\"3k*****RM6%* )\\FE7$$\"31+++$o*4k8FE$\"3'*******G*z,*\\FE7$$\"3/+++e\"yXN\"FE$\"3w* ****fkH,*\\FE7$$\"3.+++#R@]M\"FE$\"3'******zZX#*)\\FE7$$\"3/+++&*z[N8F E$\"3=+++0O^()\\FE7$$\"3)******zHRgK\"FE$\"3k*****H`@\\)\\FE7$$\"31+++ G*RnJ\"FE$\"3')*****4Ke9)\\FE7$$\"3)*******R%ewI\"FE$\"3E+++B\\6x\\FE7 $$\"3!******f,o))H\"FE$\"3A+++)p%)=(\\FE7$$\"3%******fAZ/H\"FE$\"3!)** ***R5kd'\\FE7$$\"3)*******Q3[#G\"FE$\"3K+++^Ove\\FE7$$\"33+++W11v7FE$ \"31+++z\"f3&\\FE7$$\"35+++XiGo7FE$\"3M+++KN4U\\FE7$$\"3)******ztlAE\" FE$\"3u*****>vyC$\\FE7$$\"3/+++D\"FE$\"3U+++V(3o%[FE7$$ \"3%******p-wlD\"FE$\"3t*****pmON$[FE7$$\"3++++,t.j7FE$\"3H+++)Qz1#[FE 7$$\"3#******\\)yIr7FE$\"3'******RL`&3[FE7$$\"31+++'H/8G\"FE$\"3))**** *\\=*[(z%FE7$$\"3)******\\pfGH\"FE$\"3'******RM2yy%FE7$$\"3/+++\"HEdI \"FE$\"3u******y%*yzZFE7$$\"3,+++k+f>8FE$\"3O+++[>ltZFE7$$\"3*)*****4t &4M8FE$\"3))*****>NF&pZFE7$$\"35+++\\l()[8FE$\"39+++A#euw%FE7$$\"35+++ .Cej8FE$\"3I+++FISnZFE7$$\"3-+++M6!zP\"FE$\"39+++j7DpZFE7$$\"3-+++[4d \"R\"FE$\"31+++3G%Gx%FE7$$\"36+++@XQ/9FE$\"3W+++/s)zx%FE7$$\"3)******* Gp=;9FE$\"3w*****z0![%y%FE7$$\"3$******H#*poU\"FE$\"39+++Va6#z%FE7$$\" 3.+++EYOO9FE$\"3V+++FVp+[FE7$$\"32+++zSjW9FE$\"3))*****\\pH+\"[FE7$$\" 3/+++Bkm^9FE$\"3L+++%>\\*>[FE7$$\"3#******>1puX\"FE$\"3V+++%3'HI[FE7$$ \"3%*******))Q1i9FE$\"3:+++)4H4%[FE7$$\"3'******fZ$[l9FE$\"37+++t;s^[F E7$$\"35+++4\"oxY\"FE$\"3k*****4%3ci[FE7$$\"37+++6N'*o9FE$\"3!)******* )fMt[FE7$$\"3/+++2\">\"p9FE$\"3G+++,w)R)[FE7$$\"35+++MnGo9FE$\"3E+++Ng S%*[FE7$$\"37+++o'>lY\"FE$\"37+++1/`/\\FE7$$\"35+++G>(QY\"FE$\"3w***** 4b(H9\\FE7$$\"3'******\\w(Rg9FE$\"3<+++/6lB\\FEF_cqFjhm-Feim6jn7%7$$\" +:8:c9Fh[n$\"+h1aK\\Fh[n7$$\"+rj=^9Fh[n$\"+]5#4%\\Fh[n7$$\"$O\"F`im$\" $([F`im7%Fcfr7$$\"+ZibX9Fh[n$\"+n;v[\\Fh[nFhfr7%F^gr7$$\"+?PJR9Fh[n$\" +,f*f&\\Fh[nFhfr7%Fdgr7$$\"+P5^K9Fh[n$\"+D1ii\\Fh[nFhfr7%Fjgr7$$\"+o** >D9Fh[n$\"+Sdfo\\Fh[nFhfr7%F`hr7$$\"+1>V<9Fh[n$\"+!y$*Q(\\Fh[nFhfr7%Ff hr7$$\"+0!e#49Fh[n$\"+q&*[y\\Fh[nFhfr7%F\\ir7$$\"+e$H2S\"Fh[n$\"+C*fB) \\Fh[nFhfr7%Fbir7$$\"+/s*=R\"Fh[n$\"+%R$[&)\\Fh[nFhfr7%Fhir7$$\"+uJ\"G Q\"Fh[n$\"+c+%y)\\Fh[nFhfr7%F^jr7$$\"+p&HNP\"Fh[n$\"+W8T*)\\Fh[nFhfr7% Fdjr7$$\"+$o*4k8Fh[n$\"+H*z,*\\Fh[nFhfr7%Fjjr7$$\"+e\"yXN\"Fh[n$\"+Y'H ,*\\Fh[nFhfr7%F`[s7$$\"+#R@]M\"Fh[n$\"+yaC*)\\Fh[nFhfr7%Ff[s7$$\"+&*z[ N8Fh[n$\"+0O^()\\Fh[nFhfr7%F\\\\s7$$\"+)HRgK\"Fh[n$\"+L:#\\)\\Fh[nFhfr 7%Fb\\s7$$\"+G*RnJ\"Fh[n$\"+@$e9)\\Fh[nFhfr7%Fh\\s7$$\"+S%ewI\"Fh[n$\" +B\\6x\\Fh[nFhfr7%F^]s7$$\"+;!o))H\"Fh[n$\"+)p%)=(\\Fh[nFhfr7%Fd]s7$$ \"+EsW!H\"Fh[n$\"+/Twl\\Fh[nFhfr7%Fj]s7$$\"+R3[#G\"Fh[n$\"+^Ove\\Fh[nF hfr7%F`^s7$$\"+W11v7Fh[n$\"+z\"f3&\\Fh[nFhfr7%Ff^s7$$\"+XiGo7Fh[n$\"+K N4U\\Fh[nFhfr7%F\\_s7$$\"+QdEi7Fh[n$\"+_(yC$\\Fh[nFhfr7%Fb_s7$$\"+D\"Fh[n$\"+V(3o%[Fh[nFhfr7%F\\bs7$ $\"+Fgdc7Fh[n$\"+nm`L[Fh[nFhfr7%Fbbs7$$\"+,t.j7Fh[n$\"+)Qz1#[Fh[nFhfr7 %Fhbs7$$\"+&)yIr7Fh[n$\"+MLb3[Fh[nFhfr7%F^cs7$$\"+'H/8G\"Fh[n$\"+&=*[( z%Fh[nFhfr7%Fdcs7$$\"+&pfGH\"Fh[n$\"+Wt!yy%Fh[nFhfr7%Fjcs7$$\"+\"HEdI \"Fh[n$\"+z%*yzZFh[nFhfr7%F`ds7$$\"+k+f>8Fh[n$\"+[>ltZFh[nFhfr7%Ffds7$ $\"+Jd4M8Fh[n$\"+_t_pZFh[nFhfr7%F\\es7$$\"+\\l()[8Fh[n$\"+A#euw%Fh[nFh fr7%Fbes7$$\"+.Cej8Fh[n$\"+FISnZFh[nFhfr7%Fhes7$$\"+M6!zP\"Fh[n$\"+j7D pZFh[nFhfr7%F^fs7$$\"+[4d\"R\"Fh[n$\"+3G%Gx%Fh[nFhfr7%Fdfs7$$\"+@XQ/9F h[n$\"+/s)zx%Fh[nFhfr7%Fjfs7$$\"+Hp=;9Fh[n$\"+e+[%y%Fh[nFhfr7%F`gs7$$ \"+B*poU\"Fh[n$\"+Va6#z%Fh[nFhfr7%Ffgs7$$\"+EYOO9Fh[n$\"+FVp+[Fh[nFhfr 7%F\\hs7$$\"+zSjW9Fh[n$\"+&pH+\"[Fh[nFhfr7%Fbhs7$$\"+Bkm^9Fh[n$\"+%>\\ *>[Fh[nFhfr7%Fhhs7$$\"+i!puX\"Fh[n$\"+%3'HI[Fh[nFhfr7%F^is7$$\"+*)Q1i9 Fh[n$\"+)4H4%[Fh[nFhfr7%Fdis7$$\"+wM[l9Fh[n$\"+t;s^[Fh[nFhfr7%Fjis7$$ \"+4\"oxY\"Fh[n$\"+T3ci[Fh[nFhfr7%F`js7$$\"+6N'*o9Fh[n$\"+!*fMt[Fh[nFh fr7%Ffjs7$$\"+2\">\"p9Fh[n$\"+,w)R)[Fh[nFhfr7%F\\[t7$$\"+MnGo9Fh[n$\"+ NgS%*[Fh[nFhfr7%Fb[t7$$\"+o'>lY\"Fh[n$\"+1/`/\\Fh[nFhfr7%Fh[t7$$\"+G>( QY\"Fh[n$\"+^vH9\\Fh[nFhfr7%F^\\t7$$\"+lxRg9Fh[n$\"+/6lB\\Fh[nFhfr7%Fd \\tF^frFhfrFbbqFhbq-F$6$7in7$F`cq$!3*******4mSD$\\FE7$Fger$!3<+++/6lB \\FE7$Fber$!3w*****4b(H9\\FE7$F]er$!37+++1/`/\\FE7$Fhdr$!3E+++NgS%*[FE 7$Fcdr$!3G+++,w)R)[FE7$F^dr$!3!)*******)fMt[FE7$Ficr$!3k*****4%3ci[FE7 $Fdcr$!37+++t;s^[FE7$F_cr$!3:+++)4H4%[FE7$Fjbr$!3V+++%3'HI[FE7$Febr$!3 L+++%>\\*>[FE7$F`br$!3))*****\\pH+\"[FE7$F[br$!3V+++FVp+[FE7$Ffar$!39+ ++Va6#z%FE7$Faar$!3w*****z0![%y%FE7$F\\ar$!3W+++/s)zx%FE7$Fg`r$!31+++3 G%Gx%FE7$Fb`r$!39+++j7DpZFE7$F]`r$!3I+++FISnZFE7$Fh_r$!39+++A#euw%FE7$ Fc_r$!3))*****>NF&pZFE7$F^_r$!3O+++[>ltZFE7$Fi^r$!3u******y%*yzZFE7$Fd ^r$!3'******RM2yy%FE7$F_^r$!3))*****\\=*[(z%FE7$Fj]r$!3'******RL`&3[FE 7$Fe]r$!3H+++)Qz1#[FE7$F`]r$!3t*****pmON$[FE7$F[]r$!3U+++V(3o%[FE7$Ff \\r$!3Q+++!e5-'[FE7$Fa\\r$!3E+++`w\\t[FE7$F\\\\r$!3O+++h$ok)[FE7$Fg[r$ !35+++s='*)*[FE7$Fb[r$!3B+++gM&3\"\\FE7$F][r$!3$******p()[?#\\FE7$Fhjq $!3u*****>vyC$\\FE7$Fcjq$!3M+++KN4U\\FE7$F^jq$!31+++z\"f3&\\FE7$Fiiq$! 3K+++^Ove\\FE7$Fdiq$!3!)*****R5kd'\\FE7$F_iq$!3A+++)p%)=(\\FE7$Fjhq$!3 E+++B\\6x\\FE7$Fehq$!3')*****4Ke9)\\FE7$F`hq$!3k*****H`@\\)\\FE7$F[hq$ !3=+++0O^()\\FE7$Ffgq$!3'******zZX#*)\\FE7$Fagq$!3w*****fkH,*\\FE7$F\\ gq$!3'*******G*z,*\\FE7$Fgfq$!3k*****RM6%*)\\FE7$Fbfq$!3e*****f0Sy)\\F E7$F]fq$!33+++%R$[&)\\FE7$Fheq$!3%)*****R#*fB)\\FE7$Fceq$!3#)******p&* [y\\FE7$F^eq$!3/+++!y$*Q(\\FE7$Fidq$!3g******Rdfo\\FE7$Fddq$!3c*****\\ i?E'\\FE7$F_dq$!3k*****4!f*f&\\FE7$Fjcq$!3!)*****pm^([\\FE7$Fecq$!3.++ +]5#4%\\FEF]]tFjhm-Feim6jn7%7$F_fr$!+h1aK\\Fh[n7$Fe\\t$!+/6lB\\Fh[n7$F ifr$!$([F`im7%Fght7$F_\\t$!+^vH9\\Fh[nFjht7%F^it7$Fi[t$!+1/`/\\Fh[nFjh t7%Fbit7$Fc[t$!+NgS%*[Fh[nFjht7%Ffit7$F][t$!+,w)R)[Fh[nFjht7%Fjit7$Fgj s$!+!*fMt[Fh[nFjht7%F^jt7$Fajs$!+T3ci[Fh[nFjht7%Fbjt7$F[js$!+t;s^[Fh[n Fjht7%Ffjt7$Feis$!+)4H4%[Fh[nFjht7%Fjjt7$F_is$!+%3'HI[Fh[nFjht7%F^[u7$ Fihs$!+%>\\*>[Fh[nFjht7%Fb[u7$Fchs$!+&pH+\"[Fh[nFjht7%Ff[u7$F]hs$!+FVp +[Fh[nFjht7%Fj[u7$Fggs$!+Va6#z%Fh[nFjht7%F^\\u7$Fags$!+e+[%y%Fh[nFjht7 %Fb\\u7$F[gs$!+/s)zx%Fh[nFjht7%Ff\\u7$Fefs$!+3G%Gx%Fh[nFjht7%Fj\\u7$F_ fs$!+j7DpZFh[nFjht7%F^]u7$Fies$!+FISnZFh[nFjht7%Fb]u7$Fces$!+A#euw%Fh[ nFjht7%Ff]u7$F]es$!+_t_pZFh[nFjht7%Fj]u7$Fgds$!+[>ltZFh[nFjht7%F^^u7$F ads$!+z%*yzZFh[nFjht7%Fb^u7$F[ds$!+Wt!yy%Fh[nFjht7%Ff^u7$Fecs$!+&=*[(z %Fh[nFjht7%Fj^u7$F_cs$!+MLb3[Fh[nFjht7%F^_u7$Fibs$!+)Qz1#[Fh[nFjht7%Fb _u7$Fcbs$!+nm`L[Fh[nFjht7%Ff_u7$F]bs$!+V(3o%[Fh[nFjht7%Fj_u7$Fgas$!+!e 5-'[Fh[nFjht7%F^`u7$Faas$!+`w\\t[Fh[nFjht7%Fb`u7$F[as$!+h$ok)[Fh[nFjht 7%Ff`u7$Fe`s$!+s='*)*[Fh[nFjht7%Fj`u7$F_`s$!+gM&3\"\\Fh[nFjht7%F^au7$F i_s$!+x)[?#\\Fh[nFjht7%Fbau7$Fc_s$!+_(yC$\\Fh[nFjht7%Ffau7$F]_s$!+KN4U \\Fh[nFjht7%Fjau7$Fg^s$!+z\"f3&\\Fh[nFjht7%F^bu7$Fa^s$!+^Ove\\Fh[nFjht 7%Fbbu7$F[^s$!+/Twl\\Fh[nFjht7%Ffbu7$Fe]s$!+)p%)=(\\Fh[nFjht7%Fjbu7$F_ ]s$!+B\\6x\\Fh[nFjht7%F^cu7$Fi\\s$!+@$e9)\\Fh[nFjht7%Fbcu7$Fc\\s$!+L:# \\)\\Fh[nFjht7%Ffcu7$F]\\s$!+0O^()\\Fh[nFjht7%Fjcu7$Fg[s$!+yaC*)\\Fh[n Fjht7%F^du7$Fa[s$!+Y'H,*\\Fh[nFjht7%Fbdu7$F[[s$!+H*z,*\\Fh[nFjht7%Ffdu 7$Fejr$!+W8T*)\\Fh[nFjht7%Fjdu7$F_jr$!+c+%y)\\Fh[nFjht7%F^eu7$Fiir$!+% R$[&)\\Fh[nFjht7%Fbeu7$Fcir$!+C*fB)\\Fh[nFjht7%Ffeu7$F]ir$!+q&*[y\\Fh[ nFjht7%Fjeu7$Fghr$!+!y$*Q(\\Fh[nFjht7%F^fu7$Fahr$!+Sdfo\\Fh[nFjht7%Fbf u7$F[hr$!+D1ii\\Fh[nFjht7%Fffu7$Fegr$!+,f*f&\\Fh[nFjht7%Fjfu7$F_gr$!+n ;v[\\Fh[nFjht7%F^gu7$Fdfr$!+]5#4%\\Fh[nFjht7%FbguFdhtFjhtFbbqFhbq-F$6% 7$7$$!3Q++++++!>%FEF(7$$\"3/++++++!z\"FEF(-%'COLOURG6&F]imF)F)F)-%*LIN ESTYLEG6#\"\"$-F$6%7$7$F($!3o*************Q&FE7$F($\"3o*************Q& FEF_huFbhu-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G- %%FONTG6#%(DEFAULTG-Fiiu6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%%VIE WG6$;$!$>%F`im$\"$z\"F`im;$!$R&F`im$\"$R&F`im" 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 75 "The stability interval for the \+ embedded order 6 scheme is (approximately) " }{XPPEDIT 18 0 "[-3.6054 , 0];" "6#7$,$-%&FloatG6$\"&ag$!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stabi lity regions for the two schemes can be displayed in the same picture. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "pp := plot([[[-5.39,0], [1.79,0]],[[0,-5.69],[0,5.69]]],color=black,linestyle=3):\nplots[displ ay]([pp,p_||(1..6),p||(1..6)],view=[-5.39..1.79,-5.69..5.69],font=[HEL VETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=con strained);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 580 580 {PLOTDATA 2 "65-% 'CURVESG6%7$7$$!3o*************Q&!#<$\"\"!F,7$$\"3/++++++!z\"F*F+-%'CO LOURG6&%$RGBGF,F,F,-%*LINESTYLEG6#\"\"$-F$6%7$7$F+$!3Q++++++!p&F*7$F+$ \"3Q++++++!p&F*F0F4-F$6$7av7$F+F+7$F+$\"3++++/^R%4#!#=7$$!3/+++/Tdg8!# E$\"3')*****4Y!z)=%FH7$$\"3%)*****Hdg\"p()FL$\"3K+++Q/>$G'FH7$$\"3)*** *****\\O2X'!#C$\"3k******R]ix$)FH7$$\"3T+++n1t2w!#B$\"3%******HI@s/\"F *7$$\"3!)*****fW!yQ\\!#A$\"33+++Clsc7F*7$$\"3%******>2tWB#!#@$\"35+++m ;Lm9F*7$$\"3'*******=ToJyFco$\"3!******RDvgn\"F*7$$\"3'******p!4$3D#!# ?$\"3-+++^'Qe)=F*7$$\"3#)*****>]%QkaF^p$\"3')******yk,&4#F*7$$\"3%**** **\\j\")>8\"!#>$\"3\"******z](*>I#F*7$$\"31+++GUs\"*>Fip$\"3@+++ue!Q]# F*7$$\"3$)******\\09SHFip$\"3#)******GIn'p#F*7$$\"3G+++nf/\"e$Fip$\"3' *******GxdxGF*7$$\"3;+++Ld(4X$Fip$\"3!)*****fLc^/$F*7$$\"35+++)3%)*z@F ip$\"33+++*z`!*>$F*7$$!3?+++XkWUVF^p$\"3A+++4L!)QLF*7$$!3&*******e5&pS %Fip$\"3#******>mzLY$F*7$$!3#)*****p>hQd*Fip$\"33+++auhrNF*7$$!33+++QP Wk:FH$\"3/+++#****Hm$F*7$$!37+++xo^HAFH$\"33+++!*y3QPF*7$$!3$)*****f)y 0DHFH$\"3!)*****>86$)z$F*7$$!3%)*****z%e`JOFH$\"3?+++XLYXQF*7$$!3B+++: DtOVFH$\"3/+++;HH\")QF*7$$!3F+++XnlL]FH$\"3:+++!)\\I2RF*7$$!3J+++4Jh=d FH$\"3,+++rMrCRF*7$$!3g*****RR_)*Q'FH$\"3y*****4\"HYMRF*7$$!31+++hOuYq FH$\"3#*******z5FPRF*7$$!3a*****zS1$*o(FH$\"3(*******[5nLRF*7$$!3!**** **4_qzJ)FH$\"3))*****RWXS#RF*7$$!3v*****fV![L*)FH$\"3=++++>l3RF*7$$!3G +++M#**o`*FH$\"3=+++y2k()QF*7$$!32+++p'pH,\"F*$\"3=+++\\o1hQF*7$$!3)** *****QFRr5F*$\"3#)*****\\<(*)GQF*7$$!35+++bDDH6F*$\"3++++v%=5z$F*7$$!3 -+++tj(p=\"F*$\"3?+++?-DZPF*7$$!35+++8O>X7F*$\"3#******fUwtp$F*7$$!3)* *****\\WH[I\"F*$\"3()*****z+B7k$F*7$$!3-+++bg?n8F*$\"3;+++N6$)yNF*7$$! 3&*******4j1M9F*$\"3)******4y%z5NF*7$$!36+++gUH2:F*$\"3-+++QhvQMF*7$$! 3-+++z.0)e\"F*$\"3%)*****zx1dO$F*7$$!33+++)Q6cn\"F*$\"3'******p@B`H$F* 7$$!37+++86@n&oCrJF* 7$$!3!******Rys&\\>F*$\"3#******\\d'>#F*$\"3))*****\\tD,( HF*7$$!3%******H/t1F#F*$\"3')*****pN1A#HF*7$$!3$)*****R5p1M#F*$\"3!)** ***zygN(GF*7$$!37+++(>tpS#F*$\"3++++62sBGF*7$$!3%******H<[)pCF*$\"3;++ +j3MsFF*7$$!3:+++h*[&HDF*$\"36+++]A;>FF*7$$!3%)*****HT@je#F*$\"3?+++5p )Rm#F*7$$!3%******>(fSSEF*$\"3;+++c:m1EF*7$$!37+++rN/#p#F*$\"35+++p\"o qa#F*7$$!31+++0A[TFF*$\"35+++,/7&[#F*7$$!3'******>'H)*)y#F*$\"32+++R_w ?CF*7$$!3@+++*)R#[$GF*$\"3/+++i'*)RN#F*7$$!3?+++#R+$zGF*$\"3#******\\# =$[G#F*7$$!3#*******QurAHF*$\"3*)*****pP&R8AF*7$$!3=+++SWPlHF*$\"3=+++ UP')R@F*7$$!3')*****z.Qv+$F*$\"3A+++s)3X1#F*7$$!3;+++wsS\\IF*$\"3-+++< to()>F*7$$!3;+++s!y54$F*$\"3%******pn<)4>F*7$$!3>+++e!>D8$F*$\"3/+++?# R8$=F*7$$!3!******RupN<$F*$\"35+++p,m_+9!e5F*7$$!3%)*****>We)*\\$F*$\"3X*****\\7C3\") *FH7$$!3#)*****\\P\"*)>NF*$\"3D+++C)=o.*FH7$$!3z******ziXPNF*$\"3#)*** **p2&zc#)FH7$$!3/+++eli_NF*$\"3i******eqcpuFH7$$!3?+++HW]lNF*$\"35+++! =DTn'FH7$$!3=+++'3?id$F*$\"3%)*****R'GopeFH7$$!31+++R+$\\e$F*$\"3))*** **>ZSd0&FH7$$!35+++'Q8=f$F*$\"3z*****>4l@B%FH7$$!3'******4hkqf$F*$\"3' )******pxF*R$FH7$$!3-+++^=)3g$F*$\"33+++Y1$zb#FH7$$!31+++%\\]Mg$F*$\"3 ()*****>&ya4&ya44l@B%FH7$Fegl$!3))*****>ZSd0&FH 7$F`gl$!3%)*****R'GopeFH7$F[gl$!35+++!=DTn'FH7$Fffl$!3i******eqcpuFH7$ Fafl$!3#)*****p2&zc#)FH7$F\\fl$!3D+++C)=o.*FH7$Fgel$!3X*****\\7C3\")*F H7$Fbel$!3'******>+9!e5F*7$F]el$!3*)*****zH@Y8\"F*7$Fhdl$!3'******p&z0 67F*7$Fcdl$!33+++EE[(G\"F*7$F^dl$!3/+++)H_SO\"F*7$Ficl$!3&******pb64W \"F*7$Fdcl$!32+++_k<=:F*7$F_cl$!3#******zu7ff\"F*7$Fjbl$!31+++%[5Tn\"F *7$Febl$!35+++p,m_F*7$Ff al$!3-+++F*7$Faal$!3A+++s)3X1#F*7$F\\al$!3=+++UP')R@F*7$Fg`l$!3* )*****pP&R8AF*7$Fb`l$!3#******\\#=$[G#F*7$F]`l$!3/+++i'*)RN#F*7$Fh_l$! 32+++R_w?CF*7$Fc_l$!35+++,/7&[#F*7$F^_l$!35+++p\"oqa#F*7$Fi^l$!3;+++c: m1EF*7$Fd^l$!3?+++5p)Rm#F*7$F_^l$!36+++]A;>FF*7$Fj]l$!3;+++j3MsFF*7$Fe ]l$!3++++62sBGF*7$F`]l$!3!)*****zygN(GF*7$F[]l$!3')*****pN1A#HF*7$Ff\\ l$!3))*****\\tD,(HF*7$Fa\\l$!3;+++C6'z,$F*7$F\\\\l$!3-+++UAfmIF*7$Fg[l $!3#******\\d'>&oCrJF*7$F][l$!3?+++GdHIKF*7$Fhz$! 3'******p@B`H$F*7$Fcz$!3%)*****zx1dO$F*7$F^z$!3-+++QhvQMF*7$Fiy$!3)*** ***4y%z5NF*7$Fdy$!3;+++N6$)yNF*7$F_y$!3()*****z+B7k$F*7$Fjx$!3#******f Uwtp$F*7$Fex$!3?+++?-DZPF*7$F`x$!3++++v%=5z$F*7$F[x$!3#)*****\\<(*)GQF *7$Ffw$!3=+++\\o1hQF*7$Faw$!3=+++y2k()QF*7$F\\w$!3=++++>l3RF*7$Fgv$!3) )*****RWXS#RF*7$Fbv$!3(*******[5nLRF*7$F]v$!3#*******z5FPRF*7$Fhu$!3y* ****4\"HYMRF*7$Fcu$!3,+++rMrCRF*7$F^u$!3:+++!)\\I2RF*7$Fit$!3/+++;HH\" )QF*7$Fdt$!3?+++XLYXQF*7$F_t$!3!)*****>86$)z$F*7$Fjs$!33+++!*y3QPF*7$F es$!3/+++#****Hm$F*7$F`s$!33+++auhrNF*7$F[s$!3#******>mzLY$F*7$Ffr$!3A +++4L!)QLF*7$Far$!33+++*z`!*>$F*7$F\\r$!3!)*****fLc^/$F*7$Fgq$!3'***** **GxdxGF*7$Fbq$!3#)******GIn'p#F*7$F]q$!3@+++ue!Q]#F*7$Fgp$!3\"******z ](*>I#F*7$Fbp$!3')******yk,&4#F*7$F\\p$!3-+++^'Qe)=F*7$Fgo$!3!******RD vgn\"F*7$Fao$!35+++m;Lm9F*7$F[o$!33+++Clsc7F*7$Fen$!3%******HI@s/\"F*7 $FU$!3k******R]ix$)FH7$$\"3Y+++7^;p()FL$!3K+++Q/>$G'FH7$$!3'******44u0 O\"FL$!3')*****4Y!z)=%FH7$F+$!3++++/^R%4#FHFD-%&COLORG6&F3$\"#N!\"#$\" \"\"!\"\"F+-%)POLYGONSG6bv7%FD7$F+$\"+/^R%4#!#57$$FHF\\[nF+7%Fa[n7$$!+ /Tdg8FH$\"+h/z)=%Fd[nFe[n7%Fh[n7$$\"+t0;p()FH$\"+Q/>$G'Fd[nFe[n7%F^\\n 7$$\"++lt]k!#;$\"+S]ix$)Fd[nFe[n7%Fd\\n7$$\"+n1t2w!#:$\"+.8AZ5!\"*Fe[n 7%F[]n7$$\"+Y/yQ\\!#9$\"+Clsc7Fa]nFe[n7%Fc]n7$$\"+sIZMA!#8$\"+m;Lm9Fa] nFe[n7%Fj]n7$$\"+>ToJyF]^n$\"+a_2w;Fa]nFe[n7%Fa^n7$$\"+24$3D#!#7$\"+^' Qe)=Fa]nFe[n7%Fg^n7$$\"+-XQkaFj^n$\"+zk,&4#Fa]nFe[n7%F^_n7$$\"+N;)>8\" !#6$\"+3v*>I#Fa]nFe[n7%Fd_n7$$\"+GUs\"*>Fg_n$\"+ue!Q]#Fa]nFe[n7%F[`n7$ $\"+]09SHFg_n$\"+HIn'p#Fa]nFe[n7%Fa`n7$$\"+nf/\"e$Fg_n$\"+HxdxGFa]nFe[ n7%Fg`n7$$\"+Ld(4X$Fg_n$\"+Oj:XIFa]nFe[n7%F]an7$$\"+)3%)*z@Fg_n$\"+*z` !*>$Fa]nFe[n7%Fcan7$$!+XkWUVFj^n$\"+4L!)QLFa]nFe[n7%Fian7$$!+f5&pS%Fg_ n$\"+i'zLY$Fa]nFe[n7%F_bn7$$!+(>hQd*Fg_n$\"+auhrNFa]nFe[n7%Febn7$$!+QP Wk:Fd[n$\"+#****Hm$Fa]nFe[n7%F[cn7$$!+xo^HAFd[n$\"+!*y3QPFa]nFe[n7%Fac n7$$!+')y0DHFd[n$\"+K6J)z$Fa]nFe[n7%Fgcn7$$!+[e`JOFd[n$\"+XLYXQFa]nFe[ n7%F]dn7$$!+:DtOVFd[n$\"+;HH\")QFa]nFe[n7%Fcdn7$$!+XnlL]Fd[n$\"+!)\\I2 RFa]nFe[n7%Fidn7$$!+4Jh=dFd[n$\"+rMrCRFa]nFe[n7%F_en7$$!+%R_)*Q'Fd[n$ \"+6HYMRFa]nFe[n7%Feen7$$!+hOuYqFd[n$\"+!3rs$RFa]nFe[n7%F[fn7$$!+3kI*o (Fd[n$\"+\\5nLRFa]nFe[n7%Fafn7$$!+@0(zJ)Fd[n$\"+Wa/CRFa]nFe[n7%Fgfn7$$ !+O/[L*)Fd[n$\"++>l3RFa]nFe[n7%F]gn7$$!+M#**o`*Fd[n$\"+y2k()QFa]nFe[n7 %Fcgn7$$!+p'pH,\"Fa]n$\"+\\o1hQFa]nFe[n7%Fign7$$!+RFRr5Fa]n$\"+vr*)GQF a]nFe[n7%F_hn7$$!+bDDH6Fa]n$\"+v%=5z$Fa]nFe[n7%Fehn7$$!+tj(p=\"Fa]n$\" +?-DZPFa]nFe[n7%F[in7$$!+8O>X7Fa]n$\"+EkP(p$Fa]nFe[n7%Fain7$$!+X%H[I\" Fa]n$\"+3IATOFa]nFe[n7%Fgin7$$!+bg?n8Fa]n$\"+N6$)yNFa]nFe[n7%F]jn7$$!+ 5j1M9Fa]n$\"+\"y%z5NFa]nFe[n7%Fcjn7$$!+gUH2:Fa]n$\"+QhvQMFa]nFe[n7%Fij n7$$!+z.0)e\"Fa]n$\"+ynqlLFa]nFe[n7%F_[o7$$!+)Q6cn\"Fa]n$\"+Fa]n$\"+vl>#Fa]n$\"+Nd7qHFa]nFe[n7%Fi]o7$$!+VInqAFa]n$\"+dj?AHFa]nFe[n7%F_^o 7$$!+/\"p1M#Fa]n$\"+)ygN(GFa]nFe[n7%Fe^o7$$!+(>tpS#Fa]n$\"+62sBGFa]nFe [n7%F[_o7$$!+t\"[)pCFa]n$\"+j3MsFFa]nFe[n7%Fa_o7$$!+h*[&HDFa]n$\"+]A;> FFa]nFe[n7%Fg_o7$$!+89K'e#Fa]n$\"+5p)Rm#Fa]nFe[n7%F]`o7$$!+sfSSEFa]n$ \"+c:m1EFa]nFe[n7%Fc`o7$$!+rN/#p#Fa]n$\"+p\"oqa#Fa]nFe[n7%Fi`o7$$!+0A[ TFFa]n$\"+,/7&[#Fa]nFe[n7%F_ao7$$!+iH)*)y#Fa]n$\"+R_w?CFa]nFe[n7%Feao7 $$!+*)R#[$GFa]n$\"+i'*)RN#Fa]nFe[n7%F[bo7$$!+#R+$zGFa]n$\"+D=$[G#Fa]nF e[n7%Fabo7$$!+RurAHFa]n$\"+x`R8AFa]nFe[n7%Fgbo7$$!+SWPlHFa]n$\"+UP')R@ Fa]nFe[n7%F]co7$$!+Q!Qv+$Fa]n$\"+s)3X1#Fa]nFe[n7%Fcco7$$!+wsS\\IFa]n$ \"+Fa]nFe[n7%Fico7$$!+s!y54$Fa]n$\"+xw\")4>Fa]nFe[n7%F_do7$$!+e! >D8$Fa]n$\"+?#R8$=Fa]nFe[n7%Fedo7$$!+W(pN<$Fa]n$\"+p,m_NFa]n$\"+C)=o. *Fd[nFe[n7%F]io7$$!+!Gcu`$Fa]n$\"+x]zc#)Fd[nFe[n7%Fcio7$$!+eli_NFa]n$ \"+fqcpuFd[nFe[n7%Fiio7$$!+HW]lNFa]n$\"+!=DTn'Fd[nFe[n7%F_jo7$$!+'3?id $Fa]n$\"+kGopeFd[nFe[n7%Fejo7$$!+R+$\\e$Fa]n$\"+s/ub]Fd[nFe[n7%F[[p7$$ !+'Q8=f$Fa]n$\"+#4l@B%Fd[nFe[n7%Fa[p7$$!+6Y1(f$Fa]n$\"+qxF*R$Fd[nFe[n7 %Fg[p7$$!+^=)3g$Fa]n$\"+Y1$zb#Fd[nFe[n7%F]\\p7$$!+%\\]Mg$Fa]n$\"+_ya4< Fd[nFe[n7%Fc\\p7$$!+LO#\\g$Fa]n$\"+R7,h&)Fg_nFe[n7%Fi\\p7$$!+/GS0OFa]n F+Fe[n7%F_]p7$Fj\\p$!+R7,h&)Fg_nFe[n7%Fc]p7$Fd\\p$!+_ya4Fa]nFe[n7%F_cp7$Fjco$!+Fa]nFe[n7%Fccp7$Fdco$!+s)3X1#Fa]nFe[n7 %Fgcp7$F^co$!+UP')R@Fa]nFe[n7%F[dp7$Fhbo$!+x`R8AFa]nFe[n7%F_dp7$Fbbo$! +D=$[G#Fa]nFe[n7%Fcdp7$F\\bo$!+i'*)RN#Fa]nFe[n7%Fgdp7$Ffao$!+R_w?CFa]n Fe[n7%F[ep7$F`ao$!+,/7&[#Fa]nFe[n7%F_ep7$Fj`o$!+p\"oqa#Fa]nFe[n7%Fcep7 $Fd`o$!+c:m1EFa]nFe[n7%Fgep7$F^`o$!+5p)Rm#Fa]nFe[n7%F[fp7$Fh_o$!+]A;>F Fa]nFe[n7%F_fp7$Fb_o$!+j3MsFFa]nFe[n7%Fcfp7$F\\_o$!+62sBGFa]nFe[n7%Fgf p7$Ff^o$!+)ygN(GFa]nFe[n7%F[gp7$F`^o$!+dj?AHFa]nFe[n7%F_gp7$Fj]o$!+Nd7 qHFa]nFe[n7%Fcgp7$Fd]o$!+C6'z,$Fa]nFe[n7%Fggp7$F^]o$!+UAfmIFa]nFe[n7%F [hp7$Fh\\o$!+vl>l3RFa]nFe[n7%F[\\q7$Fhfn$!+Wa/CRFa]nFe[n7% F_\\q7$Fbfn$!+\\5nLRFa]nFe[n7%Fc\\q7$F\\fn$!+!3rs$RFa]nFe[n7%Fg\\q7$Ff en$!+6HYMRFa]nFe[n7%F[]q7$F`en$!+rMrCRFa]nFe[n7%F_]q7$Fjdn$!+!)\\I2RFa ]nFe[n7%Fc]q7$Fddn$!+;HH\")QFa]nFe[n7%Fg]q7$F^dn$!+XLYXQFa]nFe[n7%F[^q 7$Fhcn$!+K6J)z$Fa]nFe[n7%F_^q7$Fbcn$!+!*y3QPFa]nFe[n7%Fc^q7$F\\cn$!+#* ***Hm$Fa]nFe[n7%Fg^q7$Ffbn$!+auhrNFa]nFe[n7%F[_q7$F`bn$!+i'zLY$Fa]nFe[ n7%F__q7$Fjan$!+4L!)QLFa]nFe[n7%Fc_q7$Fdan$!+*z`!*>$Fa]nFe[n7%Fg_q7$F^ an$!+Oj:XIFa]nFe[n7%F[`q7$Fh`n$!+HxdxGFa]nFe[n7%F_`q7$Fb`n$!+HIn'p#Fa] nFe[n7%Fc`q7$F\\`n$!+ue!Q]#Fa]nFe[n7%Fg`q7$Fe_n$!+3v*>I#Fa]nFe[n7%F[aq 7$F__n$!+zk,&4#Fa]nFe[n7%F_aq7$Fh^n$!+^'Qe)=Fa]nFe[n7%Fcaq7$Fb^n$!+a_2 w;Fa]nFe[n7%Fgaq7$F[^n$!+m;Lm9Fa]nFe[n7%F[bq7$Fd]n$!+Clsc7Fa]nFe[n7%F_ bq7$F\\]n$!+.8AZ5Fa]nFe[n7%Fcbq7$Fe\\n$!+S]ix$)Fd[nFe[n7%Fgbq7$$\"+7^; p()FH$!+Q/>$G'Fd[nFe[n7%F[cq7$$!+\"4u0O\"FH$!+h/z)=%Fd[nFe[n7%Facq7$F+ $!+/^R%4#Fd[nFe[n7%FgcqFDFe[n-Fejm6&F3$\"#sFijm$\"#OFijmF+-%&STYLEG6#% ,PATCHNOGRIDG-F$6$7in7$$\"31+++:8:c9F*$\"3*******4mSD$\\F*7$$\"3#***** *4P'=^9F*$\"3.+++]5#4%\\F*7$$\"33+++ZibX9F*$\"3!)*****pm^([\\F*7$$\"3 \"*******>PJR9F*$\"3k*****4!f*f&\\F*7$$\"3/+++P5^K9F*$\"3c*****\\i?E' \\F*7$$\"3/+++o**>D9F*$\"3g******Rdfo\\F*7$$\"3-+++1>V<9F*$\"3/+++!y$* Q(\\F*7$$\"34+++0!e#49F*$\"3#)******p&*[y\\F*7$$\"3)******zNH2S\"F*$\" 3%)*****R#*fB)\\F*7$$\"3%******R?(*=R\"F*$\"33+++%R$[&)\\F*7$$\"3/+++u J\"GQ\"F*$\"3e*****f0Sy)\\F*7$$\"3++++p&HNP\"F*$\"3k*****RM6%*)\\F*7$$ \"31+++$o*4k8F*$\"3'*******G*z,*\\F*7$$\"3/+++e\"yXN\"F*$\"3w*****fkH, *\\F*7$$\"3.+++#R@]M\"F*$\"3'******zZX#*)\\F*7$$\"3/+++&*z[N8F*$\"3=++ +0O^()\\F*7$$\"3)******zHRgK\"F*$\"3k*****H`@\\)\\F*7$$\"31+++G*RnJ\"F *$\"3')*****4Ke9)\\F*7$$\"3)*******R%ewI\"F*$\"3E+++B\\6x\\F*7$$\"3!** ****f,o))H\"F*$\"3A+++)p%)=(\\F*7$$\"3%******fAZ/H\"F*$\"3!)*****R5kd' \\F*7$$\"3)*******Q3[#G\"F*$\"3K+++^Ove\\F*7$$\"33+++W11v7F*$\"31+++z \"f3&\\F*7$$\"35+++XiGo7F*$\"3M+++KN4U\\F*7$$\"3)******ztlAE\"F*$\"3u* ****>vyC$\\F*7$$\"3/+++D\"F*$\"3U+++V(3o%[F*7$$\"3%***** *p-wlD\"F*$\"3t*****pmON$[F*7$$\"3++++,t.j7F*$\"3H+++)Qz1#[F*7$$\"3#** ****\\)yIr7F*$\"3'******RL`&3[F*7$$\"31+++'H/8G\"F*$\"3))*****\\=*[(z% F*7$$\"3)******\\pfGH\"F*$\"3'******RM2yy%F*7$$\"3/+++\"HEdI\"F*$\"3u* *****y%*yzZF*7$$\"3,+++k+f>8F*$\"3O+++[>ltZF*7$$\"3*)*****4t&4M8F*$\"3 ))*****>NF&pZF*7$$\"35+++\\l()[8F*$\"39+++A#euw%F*7$$\"35+++.Cej8F*$\" 3I+++FISnZF*7$$\"3-+++M6!zP\"F*$\"39+++j7DpZF*7$$\"3-+++[4d\"R\"F*$\"3 1+++3G%Gx%F*7$$\"36+++@XQ/9F*$\"3W+++/s)zx%F*7$$\"3)*******Gp=;9F*$\"3 w*****z0![%y%F*7$$\"3$******H#*poU\"F*$\"39+++Va6#z%F*7$$\"3.+++EYOO9F *$\"3V+++FVp+[F*7$$\"32+++zSjW9F*$\"3))*****\\pH+\"[F*7$$\"3/+++Bkm^9F *$\"3L+++%>\\*>[F*7$$\"3#******>1puX\"F*$\"3V+++%3'HI[F*7$$\"3%******* ))Q1i9F*$\"3:+++)4H4%[F*7$$\"3'******fZ$[l9F*$\"37+++t;s^[F*7$$\"35+++ 4\"oxY\"F*$\"3k*****4%3ci[F*7$$\"37+++6N'*o9F*$\"3!)*******)fMt[F*7$$ \"3/+++2\">\"p9F*$\"3G+++,w)R)[F*7$$\"35+++MnGo9F*$\"3E+++NgS%*[F*7$$ \"37+++o'>lY\"F*$\"37+++1/`/\\F*7$$\"35+++G>(QY\"F*$\"3w*****4b(H9\\F* 7$$\"3'******\\w(Rg9F*$\"3<+++/6lB\\F*FhdqFdjm-F^[n6jn7%7$$\"+:8:c9Fa] n$\"+h1aK\\Fa]n7$$\"+rj=^9Fa]n$\"+]5#4%\\Fa]n7$$\"$O\"Fijm$\"$([Fijm7% F\\hr7$$\"+ZibX9Fa]n$\"+n;v[\\Fa]nFahr7%Fghr7$$\"+?PJR9Fa]n$\"+,f*f&\\ Fa]nFahr7%F]ir7$$\"+P5^K9Fa]n$\"+D1ii\\Fa]nFahr7%Fcir7$$\"+o**>D9Fa]n$ \"+Sdfo\\Fa]nFahr7%Fiir7$$\"+1>V<9Fa]n$\"+!y$*Q(\\Fa]nFahr7%F_jr7$$\"+ 0!e#49Fa]n$\"+q&*[y\\Fa]nFahr7%Fejr7$$\"+e$H2S\"Fa]n$\"+C*fB)\\Fa]nFah r7%F[[s7$$\"+/s*=R\"Fa]n$\"+%R$[&)\\Fa]nFahr7%Fa[s7$$\"+uJ\"GQ\"Fa]n$ \"+c+%y)\\Fa]nFahr7%Fg[s7$$\"+p&HNP\"Fa]n$\"+W8T*)\\Fa]nFahr7%F]\\s7$$ \"+$o*4k8Fa]n$\"+H*z,*\\Fa]nFahr7%Fc\\s7$$\"+e\"yXN\"Fa]n$\"+Y'H,*\\Fa ]nFahr7%Fi\\s7$$\"+#R@]M\"Fa]n$\"+yaC*)\\Fa]nFahr7%F_]s7$$\"+&*z[N8Fa] n$\"+0O^()\\Fa]nFahr7%Fe]s7$$\"+)HRgK\"Fa]n$\"+L:#\\)\\Fa]nFahr7%F[^s7 $$\"+G*RnJ\"Fa]n$\"+@$e9)\\Fa]nFahr7%Fa^s7$$\"+S%ewI\"Fa]n$\"+B\\6x\\F a]nFahr7%Fg^s7$$\"+;!o))H\"Fa]n$\"+)p%)=(\\Fa]nFahr7%F]_s7$$\"+EsW!H\" Fa]n$\"+/Twl\\Fa]nFahr7%Fc_s7$$\"+R3[#G\"Fa]n$\"+^Ove\\Fa]nFahr7%Fi_s7 $$\"+W11v7Fa]n$\"+z\"f3&\\Fa]nFahr7%F_`s7$$\"+XiGo7Fa]n$\"+KN4U\\Fa]nF ahr7%Fe`s7$$\"+QdEi7Fa]n$\"+_(yC$\\Fa]nFahr7%F[as7$$\"+D\"Fa]n$\"+V(3o%[Fa]nFahr7%Fecs7$$\"+Fgdc7Fa] n$\"+nm`L[Fa]nFahr7%F[ds7$$\"+,t.j7Fa]n$\"+)Qz1#[Fa]nFahr7%Fads7$$\"+& )yIr7Fa]n$\"+MLb3[Fa]nFahr7%Fgds7$$\"+'H/8G\"Fa]n$\"+&=*[(z%Fa]nFahr7% F]es7$$\"+&pfGH\"Fa]n$\"+Wt!yy%Fa]nFahr7%Fces7$$\"+\"HEdI\"Fa]n$\"+z%* yzZFa]nFahr7%Fies7$$\"+k+f>8Fa]n$\"+[>ltZFa]nFahr7%F_fs7$$\"+Jd4M8Fa]n $\"+_t_pZFa]nFahr7%Fefs7$$\"+\\l()[8Fa]n$\"+A#euw%Fa]nFahr7%F[gs7$$\"+ .Cej8Fa]n$\"+FISnZFa]nFahr7%Fags7$$\"+M6!zP\"Fa]n$\"+j7DpZFa]nFahr7%Fg gs7$$\"+[4d\"R\"Fa]n$\"+3G%Gx%Fa]nFahr7%F]hs7$$\"+@XQ/9Fa]n$\"+/s)zx%F a]nFahr7%Fchs7$$\"+Hp=;9Fa]n$\"+e+[%y%Fa]nFahr7%Fihs7$$\"+B*poU\"Fa]n$ \"+Va6#z%Fa]nFahr7%F_is7$$\"+EYOO9Fa]n$\"+FVp+[Fa]nFahr7%Feis7$$\"+zSj W9Fa]n$\"+&pH+\"[Fa]nFahr7%F[js7$$\"+Bkm^9Fa]n$\"+%>\\*>[Fa]nFahr7%Faj s7$$\"+i!puX\"Fa]n$\"+%3'HI[Fa]nFahr7%Fgjs7$$\"+*)Q1i9Fa]n$\"+)4H4%[Fa ]nFahr7%F][t7$$\"+wM[l9Fa]n$\"+t;s^[Fa]nFahr7%Fc[t7$$\"+4\"oxY\"Fa]n$ \"+T3ci[Fa]nFahr7%Fi[t7$$\"+6N'*o9Fa]n$\"+!*fMt[Fa]nFahr7%F_\\t7$$\"+2 \">\"p9Fa]n$\"+,w)R)[Fa]nFahr7%Fe\\t7$$\"+MnGo9Fa]n$\"+NgS%*[Fa]nFahr7 %F[]t7$$\"+o'>lY\"Fa]n$\"+1/`/\\Fa]nFahr7%Fa]t7$$\"+G>(QY\"Fa]n$\"+^vH 9\\Fa]nFahr7%Fg]t7$$\"+lxRg9Fa]n$\"+/6lB\\Fa]nFahr7%F]^tFggrFahrF[dqFa dq-F$6$7in7$Fidq$!3*******4mSD$\\F*7$F`gr$!3<+++/6lB\\F*7$F[gr$!3w**** *4b(H9\\F*7$Fffr$!37+++1/`/\\F*7$Fafr$!3E+++NgS%*[F*7$F\\fr$!3G+++,w)R )[F*7$Fger$!3!)*******)fMt[F*7$Fber$!3k*****4%3ci[F*7$F]er$!37+++t;s^[ F*7$Fhdr$!3:+++)4H4%[F*7$Fcdr$!3V+++%3'HI[F*7$F^dr$!3L+++%>\\*>[F*7$Fi cr$!3))*****\\pH+\"[F*7$Fdcr$!3V+++FVp+[F*7$F_cr$!39+++Va6#z%F*7$Fjbr$ !3w*****z0![%y%F*7$Febr$!3W+++/s)zx%F*7$F`br$!31+++3G%Gx%F*7$F[br$!39+ ++j7DpZF*7$Ffar$!3I+++FISnZF*7$Faar$!39+++A#euw%F*7$F\\ar$!3))*****>NF &pZF*7$Fg`r$!3O+++[>ltZF*7$Fb`r$!3u******y%*yzZF*7$F]`r$!3'******RM2yy %F*7$Fh_r$!3))*****\\=*[(z%F*7$Fc_r$!3'******RL`&3[F*7$F^_r$!3H+++)Qz1 #[F*7$Fi^r$!3t*****pmON$[F*7$Fd^r$!3U+++V(3o%[F*7$F_^r$!3Q+++!e5-'[F*7 $Fj]r$!3E+++`w\\t[F*7$Fe]r$!3O+++h$ok)[F*7$F`]r$!35+++s='*)*[F*7$F[]r$ !3B+++gM&3\"\\F*7$Ff\\r$!3$******p()[?#\\F*7$Fa\\r$!3u*****>vyC$\\F*7$ F\\\\r$!3M+++KN4U\\F*7$Fg[r$!31+++z\"f3&\\F*7$Fb[r$!3K+++^Ove\\F*7$F][ r$!3!)*****R5kd'\\F*7$Fhjq$!3A+++)p%)=(\\F*7$Fcjq$!3E+++B\\6x\\F*7$F^j q$!3')*****4Ke9)\\F*7$Fiiq$!3k*****H`@\\)\\F*7$Fdiq$!3=+++0O^()\\F*7$F _iq$!3'******zZX#*)\\F*7$Fjhq$!3w*****fkH,*\\F*7$Fehq$!3'*******G*z,* \\F*7$F`hq$!3k*****RM6%*)\\F*7$F[hq$!3e*****f0Sy)\\F*7$Ffgq$!33+++%R$[ &)\\F*7$Fagq$!3%)*****R#*fB)\\F*7$F\\gq$!3#)******p&*[y\\F*7$Fgfq$!3/+ ++!y$*Q(\\F*7$Fbfq$!3g******Rdfo\\F*7$F]fq$!3c*****\\i?E'\\F*7$Fheq$!3 k*****4!f*f&\\F*7$Fceq$!3!)*****pm^([\\F*7$F^eq$!3.+++]5#4%\\F*Ff^tFdj m-F^[n6jn7%7$Fhgr$!+h1aK\\Fa]n7$F^^t$!+/6lB\\Fa]n7$Fbhr$!$([Fijm7%F`jt 7$Fh]t$!+^vH9\\Fa]nFcjt7%Fgjt7$Fb]t$!+1/`/\\Fa]nFcjt7%F[[u7$F\\]t$!+Ng S%*[Fa]nFcjt7%F_[u7$Ff\\t$!+,w)R)[Fa]nFcjt7%Fc[u7$F`\\t$!+!*fMt[Fa]nFc jt7%Fg[u7$Fj[t$!+T3ci[Fa]nFcjt7%F[\\u7$Fd[t$!+t;s^[Fa]nFcjt7%F_\\u7$F^ [t$!+)4H4%[Fa]nFcjt7%Fc\\u7$Fhjs$!+%3'HI[Fa]nFcjt7%Fg\\u7$Fbjs$!+%>\\* >[Fa]nFcjt7%F[]u7$F\\js$!+&pH+\"[Fa]nFcjt7%F_]u7$Ffis$!+FVp+[Fa]nFcjt7 %Fc]u7$F`is$!+Va6#z%Fa]nFcjt7%Fg]u7$Fjhs$!+e+[%y%Fa]nFcjt7%F[^u7$Fdhs$ !+/s)zx%Fa]nFcjt7%F_^u7$F^hs$!+3G%Gx%Fa]nFcjt7%Fc^u7$Fhgs$!+j7DpZFa]nF cjt7%Fg^u7$Fbgs$!+FISnZFa]nFcjt7%F[_u7$F\\gs$!+A#euw%Fa]nFcjt7%F__u7$F ffs$!+_t_pZFa]nFcjt7%Fc_u7$F`fs$!+[>ltZFa]nFcjt7%Fg_u7$Fjes$!+z%*yzZFa ]nFcjt7%F[`u7$Fdes$!+Wt!yy%Fa]nFcjt7%F_`u7$F^es$!+&=*[(z%Fa]nFcjt7%Fc` u7$Fhds$!+MLb3[Fa]nFcjt7%Fg`u7$Fbds$!+)Qz1#[Fa]nFcjt7%F[au7$F\\ds$!+nm `L[Fa]nFcjt7%F_au7$Ffcs$!+V(3o%[Fa]nFcjt7%Fcau7$F`cs$!+!e5-'[Fa]nFcjt7 %Fgau7$Fjbs$!+`w\\t[Fa]nFcjt7%F[bu7$Fdbs$!+h$ok)[Fa]nFcjt7%F_bu7$F^bs$ !+s='*)*[Fa]nFcjt7%Fcbu7$Fhas$!+gM&3\"\\Fa]nFcjt7%Fgbu7$Fbas$!+x)[?#\\ Fa]nFcjt7%F[cu7$F\\as$!+_(yC$\\Fa]nFcjt7%F_cu7$Ff`s$!+KN4U\\Fa]nFcjt7% Fccu7$F``s$!+z\"f3&\\Fa]nFcjt7%Fgcu7$Fj_s$!+^Ove\\Fa]nFcjt7%F[du7$Fd_s $!+/Twl\\Fa]nFcjt7%F_du7$F^_s$!+)p%)=(\\Fa]nFcjt7%Fcdu7$Fh^s$!+B\\6x\\ Fa]nFcjt7%Fgdu7$Fb^s$!+@$e9)\\Fa]nFcjt7%F[eu7$F\\^s$!+L:#\\)\\Fa]nFcjt 7%F_eu7$Ff]s$!+0O^()\\Fa]nFcjt7%Fceu7$F`]s$!+yaC*)\\Fa]nFcjt7%Fgeu7$Fj \\s$!+Y'H,*\\Fa]nFcjt7%F[fu7$Fd\\s$!+H*z,*\\Fa]nFcjt7%F_fu7$F^\\s$!+W8 T*)\\Fa]nFcjt7%Fcfu7$Fh[s$!+c+%y)\\Fa]nFcjt7%Fgfu7$Fb[s$!+%R$[&)\\Fa]n Fcjt7%F[gu7$F\\[s$!+C*fB)\\Fa]nFcjt7%F_gu7$Ffjr$!+q&*[y\\Fa]nFcjt7%Fcg u7$F`jr$!+!y$*Q(\\Fa]nFcjt7%Fggu7$Fjir$!+Sdfo\\Fa]nFcjt7%F[hu7$Fdir$!+ D1ii\\Fa]nFcjt7%F_hu7$F^ir$!+,f*f&\\Fa]nFcjt7%Fchu7$Fhhr$!+n;v[\\Fa]nF cjt7%Fghu7$F]hr$!+]5#4%\\Fa]nFcjt7%F[iuF]jtFcjtF[dqFadq-F$6$7e\\lFD7$F +$\"3++++Fjzq:FH7$F+$\"3)******Rl#fTJFH7$$\"3I+++TgRgl!#F$\"3#******p( *)Q7ZFH7$$\"3%)*****>N_h0'FL$\"31+++p_=$G'FH7$$\"31+++M@s)G$!#D$\"35++ +l8)R&yFH7$$\"3'*******RQn!G\"FW$\"3F+++rmxC%*FH7$$\"3x*****p*e$Q+%FW$ \"35+++\\pb*4\"F*7$$\"35+++:sa#3\"Fgn$\"31+++tbjc7F*7$$\"3%******4'*[] m#Fgn$\"3(******Ri7PT\"F*7$$\"3K+++n1FohFgn$\"3%******z7'yq:F*7$$\"33+ ++h#o,O\"F]o$\"36+++\\=&ys\"F*7$$\"39+++&4/9&GF]o$\"3/+++#z+\\)=F*7$$ \"3s******)[qQh&F]o$\"3++++tV\">/#F*7$$\"32+++D$4$=5Fco$\"39+++#)e&))> #F*7$$\"3++++X(3;l\"Fco$\"3?+++;olbBF*7$$\"35+++>KJ_AFco$\"3++++Su>7DF *7$$\"3%******fT$*36#Fco$\"3)*******=IGoEF*7$$!36+++UE=,\\F]o$\"39+++I 4iBGF*7$$!3x*****pcju,*Fco$\"3)******p\"y!y(HF*7$$!35+++%>:$eHF^p$\"3@ +++jaLIJF*7$$!3))*****px^s<(F^p$\"31+++K?h!G$F*7$$!3%******R2@2\\\"Fip $\"3&******pLpzU$F*7$$!3#******f0P_y#Fip$\"37+++h4grNF*7$$!3K+++F62\"z %Fip$\"3()*****f;%Q5PF*7$$!3y********QG!o(Fip$\"3%********GGE%QF*7$$!3 -+++%R'Ra6FH$\"3!)******y&))f'RF*7$$!30+++X6nK;FH$\"3S+++-^\"z2%F*7$$! 3#******HtX@=#FH$\"3-+++R2hwTF*7$$!35+++ix*>MVF*7$$!32+++#)y*=+%FH$\"3;+++*=&e&R%F*7$ $!3-+++Z]m4YFH$\"3'*******4*4vW%F*7$$!3J+++'oNi?&FH$\"3?+++j7S\"\\%F*7 $$!3Q+++oD4!z&FH$\"3l*****f.C%GXF*7$$!3/+++Au>hjFH$\"3'******pz*\\fXF* 7$$!3p*****p_4-#pFH$\"3k*****ft]`e%F*7$$!3=+++))32ouFH$\"3@+++CHa1YF*7 $$!3!*******y\")z0!)FH$\"3x******\\>_BYF*7$$!3w*****\\/'QM&)FH$\"3F+++ f.kOYF*7$$!3s*****f#*oZ0*FH$\"3q*****\\Dyhk%F*7$$!3a******GK\"yc*FH$\" 3=+++T'eBl%F*7$$!3#******H^Ku+\"F*$\"3=+++l)e`l%F*7$$!3/+++_m]d5F*$\"3 %)******G#>`l%F*7$$!32+++ev226F*$\"3w*****H`\\Bl%F*7$$!3#******f/>i:\" F*$\"3G+++lT`YYF*7$$!3%******pd2]?\"F*$\"3Q+++*=Ozj%F*7$$!3/+++M__`7F* $\"3n*****R\"3gEYF*7$$!34+++(3j=I\"F*$\"3u*****4jeDh%F*7$$!3%******f-D ,N\"F*$\"3!)*****\\)*Gef%F*7$$!3++++9@V)R\"F*$\"3;+++XRUwXF*7$$!3!**** ***Hu#pW\"F*$\"3y*****>[`Vb%F*7$$!31+++k6y&\\\"F*$\"3N+++!eK'HXF*7$$!3 -+++k`>X:F*$\"3'******>E\"H-XF*7$$!3/+++iqS&f\"F*$\"3G+++X&)QsWF*7$$!3 1+++\"[(oY;F*$\"3g*****>9J+W%F*7$$!3#*******HOL*p\"F*$\"3/+++\"o'R0WF* 7$$!3-+++Auk`F*$\"3E+++M=G^UF*7$$!3'* ******e]G\"*>F*$\"3V+++$GE:@%F*7$$!3?+++Y?Sb?F*$\"33+++.IVsTF*7$$!3')* ****\\d617#F*$\"33+++t%)RMTF*7$$!3!)******=2K'=#F*$\"3s*****zV>w4%F*7$ $!3-+++De&>D#F*$\"3g*****zX*4iSF*7$$!33+++xi-`6Q#F*$\"3))*****\\!\\;%*RF*7$$!3!******pupSW#F*$\"3A+++cwBhR F*7$$!3/+++1cg0DF*$\"3$******z_J'GRF*7$$!3%)*****H.mcc#F*$\"3;+++^T3'* QF*7$$!3\"******z$=@CEF*$\"3%)*****f:hL'QF*7$$!3!******R#GC\"o#F*$\"3* )*****zDf-$QF*7$$!3*******HB&yOFF*$\"3%******zM0mz$F*7$$!37+++gF)3z#F* $\"3;+++'p`Aw$F*7$$!3@+++\"R!fVGF*$\"3.+++HD3FPF*7$$!35+++j.(\\*GF*$\" 3;+++l1*4p$F*7$$!3=+++;'*3XHF*$\"3#******Ra%*Ql$F*7$$!3-++++$=S*HF*$\" 3A+++)yDdh$F*7$$!3>+++m*G=/$F*$\"3#******pMHkd$F*7$$!3')*****>='f)3$F* $\"3%******z?if`$F*7$$!3))*****>P'RMJF*$\"3'******fj#H%\\$F*7$$!31+++# p2$zJF*$\"3!******>!**R^MF*7$$!35+++/)4MA$F*$\"3$******HWusS$F*7$$!39+ ++2MymKF*$\"3)******zO=>O$F*7$$!3A+++'Q4&4LF*$\"3$)*****H?Y`J$F*7$$!3' )*****>[n;N$F*$\"33+++EdenKF*7$$!3')*****>RMLR$F*$\"31+++-(y'=KF*7$$!3 )*******)H\"eMMF*$\"3'*******e7ooJF*7$$!3-+++74ZvMF*$\"3.+++dOm?ag^$F*$\"3++++x\"4d1$F*7$$!3#)*****p\" H,$F*7$$!3-+++>\"Gkf$F*$\"3/+++?OPfHF*7$$!3,+++=dBOOF*$\"3*******>>*>0 HF*7$$!31+++D&odn$F*$\"3?+++X?\\]GF*7$$!3<+++/f)\\r$F*$\"3')*********[ `z#F*7$$!31+++\"[IQv$F*$\"34+++2e&)RFF*7$$!3)******zzJAz$F*$\"3?+++q5,$QF*$\"3?+++&z'3GEF*7$$!3!)******\\RQnQF*$\"3%)* ****>:->d#F*7$$!3>+++>&oR!RF*$\"36+++%[\\b^#F*7$$!36+++u_yRRF*$\"3++++ UJ.fCF*7$$!3/+++&zhZ(RF*$\"3)******\\EWBS#F*7$$!3J+++BV$)3SF*$\"3#**** **pMjaM#F*7$$!3E+++4*[>/%F*$\"37+++OGO)G#F*7$$!3++++3>1uSF*$\"3z***** \\e45B#F*7$$!3&)*****>)*R^5%F*$\"31+++NqOt@F*7$$!3Q+++p(f^8%F*$\"3()** ***>3(R:@F*7$$!3w*****fM2T;%F*$\"3/+++b;1d?F*7$$!3m*****4fx>>%F*$\"3'* *****H,C$)*>F*7$$!3K+++\"\\t(=UF*$\"3.+++^)\\\"R>F*7$$!3-+++Qa]WUF*$\" 35+++Q#3&z=F*7$$!3G+++=0>pUF*$\"3.+++aCP>=F*7$$!3++++w<&GH%F*$\"33+++l 2se:Pvp\"F*7$$!3u*****p/?sL%F*$\"30+ +++?\"ej\"F*7$$!3C+++)[&*zN%F*$\"3!******\\uUNd\"F*7$$!3'******R&=)yP% F*$\"35+++DWt5:F*7$$!3g*****H;=pR%F*$\"35+++.,SZ9F*7$$!3()*****f*G9:WF *$\"3'******p2hNQ\"F*7$$!3n******=CfKWF*$\"3'******4\"oC>8F*7$$!3Q+++G $*H\\WF*$\"3++++EX\\a7F*7$$!3o******Q3HlWF*$\"3)*******z$[$*=\"F*7$$!3 W+++=te![%F*$\"3)*******[#eQ7\"F*7$$!3e*****\\5,_\\%F*$\"3#*******=\"y !e5F*7$$!3q*****4uN\"4XF*$\"3#)*****f&=k?**FH7$$!3)*******)p&QAXF*$\"3 O+++qxse#*FH7$$!3\"******>kO\\`%F*$\"3%)*******o#e&f)FH7$$!3G+++4iwYXF *$\"3O+++X9rJzFH7$$!3W+++P_%yb%F*$\"3))******yMcnsFH7$$!3k*****HJR\"oX F*$\"3*******HQ?Ng'FH7$$!3C+++30hxXF*$\"3k*****R%z))RfFH7$$!3/+++`\">i e%F*$\"3_*****ft#*oF&FH7$$!3=+++Wb#Rf%F*$\"3()*****4$Go9YFH7$$!3\")*** **pZ\"p+YF*$\"3!)*****4$>L`RFH7$$!3<+++l:[1YF*$\"3?+++lg%GH$FH7$$!3T++ +CVE6YF*$\"3#)*****z/sJj#FH7$$!36+++4I,:YF*$\"3))*****4'p?u>FH7$$!3E++ +\\iqYF*$\"3]+++:9.ylFip7$$!3))** ***>Hq)>YF*F+7$Fcdx$!3]+++:9.ylFip7$F^dx$!30+++:\"3eJ\"FH7$Ficx$!3))** ***4'p?u>FH7$Fdcx$!3#)*****z/sJj#FH7$F_cx$!3?+++lg%GH$FH7$Fjbx$!3!)*** **4$>L`RFH7$Febx$!3()*****4$Go9YFH7$F`bx$!3_*****ft#*oF&FH7$F[bx$!3k** ***R%z))RfFH7$Ffax$!3*******HQ?Ng'FH7$Faax$!3))******yMcnsFH7$F\\ax$!3 O+++X9rJzFH7$Fg`x$!3%)*******o#e&f)FH7$Fb`x$!3O+++qxse#*FH7$F]`x$!3#)* ****f&=k?**FH7$Fh_x$!3#*******=\"y!e5F*7$Fc_x$!3)*******[#eQ7\"F*7$F^_ x$!3)*******z$[$*=\"F*7$Fi^x$!3++++EX\\a7F*7$Fd^x$!3'******4\"oC>8F*7$ F_^x$!3'******p2hNQ\"F*7$Fj]x$!35+++.,SZ9F*7$Fe]x$!35+++DWt5:F*7$F`]x$ !3!******\\uUNd\"F*7$F[]x$!30++++?\"ej\"F*7$Ff\\x$!3)******>:Pvp\"F*7$ Fa\\x$!33+++l2se=F*7$Fg[x$!35+++Q#3&z=F*7$Fb[x$! 3.+++^)\\\"R>F*7$F][x$!3'******H,C$)*>F*7$Fhjw$!3/+++b;1d?F*7$Fcjw$!3( )*****>3(R:@F*7$F^jw$!31+++NqOt@F*7$Fiiw$!3z*****\\e45B#F*7$Fdiw$!37++ +OGO)G#F*7$F_iw$!3#******pMjaM#F*7$Fjhw$!3)******\\EWBS#F*7$Fehw$!3+++ +UJ.fCF*7$F`hw$!36+++%[\\b^#F*7$F[hw$!3%)*****>:->d#F*7$Ffgw$!3?+++&z' 3GEF*7$Fagw$!3?+++>*>0HF*7$Fhew$!3/+++?OPfHF*7 $Fcew$!3()******Q>\"H,$F*7$F^ew$!3++++x\"4d1$F*7$Fidw$!3.+++dOmO$F*7$F[cw$!3$******HWusS$F*7$Ffbw $!3!******>!**R^MF*7$Fabw$!3'******fj#H%\\$F*7$F\\bw$!3%******z?if`$F* 7$Fgaw$!3#******pMHkd$F*7$Fbaw$!3A+++)yDdh$F*7$F]aw$!3#******Ra%*Ql$F* 7$Fh`w$!3;+++l1*4p$F*7$Fc`w$!3.+++HD3FPF*7$F^`w$!3;+++'p`Aw$F*7$Fi_w$! 3%******zM0mz$F*7$Fd_w$!3*)*****zDf-$QF*7$F__w$!3%)*****f:hL'QF*7$Fj^w $!3;+++^T3'*QF*7$Fe^w$!3$******z_J'GRF*7$F`^w$!3A+++cwBhRF*7$F[^w$!3)) *****\\!\\;%*RF*7$Ff]w$!32+++&Q$pFSF*7$Fa]w$!3g*****zX*4iSF*7$F\\]w$!3 s*****zV>w4%F*7$Fg\\w$!33+++t%)RMTF*7$Fb\\w$!33+++.IVsTF*7$F]\\w$!3V++ +$GE:@%F*7$Fh[w$!3E+++M=G^UF*7$Fc[w$!3m******G([6H%F*7$F^[w$!3q*****f# o]IVF*7$Fijv$!3K+++f(e(oVF*7$Fdjv$!3/+++\"o'R0WF*7$F_jv$!3g*****>9J+W% F*7$Fjiv$!3G+++X&)QsWF*7$Feiv$!3'******>E\"H-XF*7$F`iv$!3N+++!eK'HXF*7 $F[iv$!3y*****>[`Vb%F*7$Ffhv$!3;+++XRUwXF*7$Fahv$!3!)*****\\)*Gef%F*7$ F\\hv$!3u*****4jeDh%F*7$Fggv$!3n*****R\"3gEYF*7$Fbgv$!3Q+++*=Ozj%F*7$F ]gv$!3G+++lT`YYF*7$Fhfv$!3w*****H`\\Bl%F*7$Fcfv$!3%)******G#>`l%F*7$F^ fv$!3=+++l)e`l%F*7$Fiev$!3=+++T'eBl%F*7$Fdev$!3q*****\\Dyhk%F*7$F_ev$! 3F+++f.kOYF*7$Fjdv$!3x******\\>_BYF*7$Fedv$!3@+++CHa1YF*7$F`dv$!3k**** *ft]`e%F*7$F[dv$!3'******pz*\\fXF*7$Ffcv$!3l*****f.C%GXF*7$Facv$!3?+++ j7S\"\\%F*7$F\\cv$!3'*******4*4vW%F*7$Fgbv$!3;+++*=&e&R%F*7$Fbbv$!3e** *****>*>MVF*7$F]bv$!3))******QpshUF*7$Fhav$!3-+++R2hwTF*7$Fcav$!3S+++- ^\"z2%F*7$F^av$!3!)******y&))f'RF*7$Fi`v$!3%********GGE%QF*7$Fd`v$!3() *****f;%Q5PF*7$F_`v$!37+++h4grNF*7$Fj_v$!3&******pLpzU$F*7$Fe_v$!31+++ K?h!G$F*7$F`_v$!3@+++jaLIJF*7$F[_v$!3)******p\"y!y(HF*7$Ff^v$!39+++I4i BGF*7$Fa^v$!3)*******=IGoEF*7$F\\^v$!3++++Su>7DF*7$Fg]v$!3?+++;olbBF*7 $Fb]v$!39+++#)e&))>#F*7$F]]v$!3++++tV\">/#F*7$Fh\\v$!3/+++#z+\\)=F*7$F c\\v$!36+++\\=&ys\"F*7$$\"3i******[1FohFgn$!3%******z7'yq:F*7$Fi[v$!3( ******Ri7PT\"F*7$Fd[v$!31+++tbjc7F*7$F_[v$!35+++\\pb*4\"F*7$Fjju$!3F++ +rmxC%*FH7$$\"3C+++L@s)G$Ffju$!35+++l8)R&yFH7$$\"3w*****fM_h0'FL$!31++ +p_=$G'FH7$$\"3%******R0;/c'F[ju$!3#******p(*)Q7ZFH7$F+$!3)******Rl#fT JFH7$F+$!3++++Fjzq:FHFD-Fejm6&F3$\"#bFijm$\"#:FijmF+-F^[n6f\\l7%FD7$F+ $\"+Fjzq:Fd[n7$$FijmF,F+7%F\\`z7$F+$\"+aEfTJFd[nF_`z7%Fb`z7$$\"+TgRglF ip$\"+x*)Q7ZFd[nF_`z7%Ff`z7$$\"+_B:cgFH$\"+p_=$G'Fd[nF_`z7%F\\az7$$\"+ M@s)G$F*$\"+l8)R&yFd[nF_`z7%Fbaz7$$\"+SQn!G\"Fg\\n$\"+rmxC%*Fd[nF_`z7% Fhaz7$$\"+(*e$Q+%Fg\\n$\"+\\pb*4\"Fa]nF_`z7%F^bz7$$\"+:sa#3\"F^]n$\"+t bjc7Fa]nF_`z7%Fdbz7$$\"+h*[]m#F^]n$\"+CEr89Fa]nF_`z7%Fjbz7$$\"+n1FohF^ ]n$\"+Ghyq:Fa]nF_`z7%F`cz7$$\"+h#o,O\"Ff]n$\"+\\=&ys\"Fa]nF_`z7%Ffcz7$ $\"+&4/9&GFf]n$\"+#z+\\)=Fa]nF_`z7%F\\dz7$$\"+*[qQh&Ff]n$\"+tV\">/#Fa] nF_`z7%Fbdz7$$\"+D$4$=5F]^n$\"+#)e&))>#Fa]nF_`z7%Fhdz7$$\"+X(3;l\"F]^n $\"+;olbBFa]nF_`z7%F^ez7$$\"+>KJ_AF]^n$\"+Su>7DFa]nF_`z7%Fdez7$$\"+;M* 36#F]^n$\"+>IGoEFa]nF_`z7%Fjez7$$!+UE=,\\Ff]n$\"+I4iBGFa]nF_`z7%F`fz7$ $!+nNY:$eHFj^n$\"+jaLIJFa]nF_`z7 %F\\gz7$$!+xMVFa]nF_`z7%Fhjz7$$!+#)y*=+%Fd[n$\"+*=&e &R%Fa]nF_`z7%F^[[l7$$!+Z]m4YFd[n$\"+5*4vW%Fa]nF_`z7%Fd[[l7$$!+'oNi?&Fd [n$\"+j7S\"\\%Fa]nF_`z7%Fj[[l7$$!+oD4!z&Fd[n$\"+OSUGXFa]nF_`z7%F`\\[l7 $$!+Au>hjFd[n$\"+(z*\\fXFa]nF_`z7%Ff\\[l7$$!+F&4-#pFd[n$\"+O2N&e%Fa]nF _`z7%F\\][l7$$!+))32ouFd[n$\"+CHa1YFa]nF_`z7%Fb][l7$$!+z\")z0!)Fd[n$\" +]>_BYFa]nF_`z7%Fh][l7$$!+XgQM&)Fd[n$\"+f.kOYFa]nF_`z7%F^^[l7$$!+E*oZ0 *Fd[n$\"+b#yhk%Fa]nF_`z7%Fd^[l7$$!+HK\"yc*Fd[n$\"+T'eBl%Fa]nF_`z7%Fj^[ l7$$!+8DV25Fa]n$\"+l)e`l%Fa]nF_`z7%F`_[l7$$!+_m]d5Fa]n$\"+H#>`l%Fa]nF_ `z7%Ff_[l7$$!+ev226Fa]n$\"+L&\\Bl%Fa]nF_`z7%F\\`[l7$$!+Y!>i:\"Fa]n$\"+ lT`YYFa]nF_`z7%Fb`[l7$$!+xv+07Fa]n$\"+*=Ozj%Fa]nF_`z7%Fh`[l7$$!+M__`7F a]n$\"+93gEYFa]nF_`z7%F^a[l7$$!+(3j=I\"Fa]n$\"+J'eDh%Fa]nF_`z7%Fda[l7$ $!+E]7]8Fa]n$\"+&)*Gef%Fa]nF_`z7%Fja[l7$$!+9@V)R\"Fa]n$\"+XRUwXFa]nF_` z7%F`b[l7$$!+Iu#pW\"Fa]n$\"+#[`Vb%Fa]nF_`z7%Ffb[l7$$!+k6y&\\\"Fa]n$\"+ !eK'HXFa]nF_`z7%F\\c[l7$$!+k`>X:Fa]n$\"+i7H-XFa]nF_`z7%Fbc[l7$$!+iqS&f \"Fa]n$\"+X&)QsWFa]nF_`z7%Fhc[l7$$!+\"[(oY;Fa]n$\"+U6.SWFa]nF_`z7%F^d[ l7$$!+IOL*p\"Fa]n$\"+\"o'R0WFa]nF_`z7%Fdd[l7$$!+Auk`Fa]n$\"+M=G^UFa]nF_`z7%F\\f[l7$$!+f]G\"* >Fa]n$\"+$GE:@%Fa]nF_`z7%Fbf[l7$$!+Y?Sb?Fa]n$\"+.IVsTFa]nF_`z7%Fhf[l7$ $!+v:h?@Fa]n$\"+t%)RMTFa]nF_`z7%F^g[l7$$!+>2K'=#Fa]n$\"+Q%>w4%Fa]nF_`z 7%Fdg[l7$$!+De&>D#Fa]n$\"+e%*4iSFa]nF_`z7%Fjg[l7$$!+xi-`6Q#Fa]n$\"+0\\;%*RFa]nF_`z7%Ffh[l7$$!+Z(pSW#Fa ]n$\"+cwBhRFa]nF_`z7%F\\i[l7$$!+1cg0DFa]n$\"+G:jGRFa]nF_`z7%Fbi[l7$$!+ LgmlDFa]n$\"+^T3'*QFa]nF_`z7%Fhi[l7$$!+Q=@CEFa]n$\"+c6OjQFa]nF_`z7%F^j [l7$$!+CGC\"o#Fa]n$\"+e#f-$QFa]nF_`z7%Fdj[l7$$!+L_yOFFa]n$\"+[`g'z$Fa] nF_`z7%Fjj[l7$$!+gF)3z#Fa]n$\"+'p`Aw$Fa]nF_`z7%F`[\\l7$$!+\"R!fVGFa]n$ \"+HD3FPFa]nF_`z7%Ff[\\l7$$!+j.(\\*GFa]n$\"+l1*4p$Fa]nF_`z7%F\\\\\\l7$ $!+;'*3XHFa]n$\"+WX*Ql$Fa]nF_`z7%Fb\\\\l7$$!++$=S*HFa]n$\"+)yDdh$Fa]nF _`z7%Fh\\\\l7$$!+m*G=/$Fa]n$\"+Z$Hkd$Fa]nF_`z7%F^]\\l7$$!+#='f)3$Fa]n$ \"+3A'f`$Fa]nF_`z7%Fd]\\l7$$!+sjRMJFa]n$\"+OEH%\\$Fa]nF_`z7%Fj]\\l7$$! +#p2$zJFa]n$\"+-**R^MFa]nF_`z7%F`^\\l7$$!+/)4MA$Fa]n$\"+VWF2MFa]nF_`z7 %Ff^\\l7$$!+2MymKFa]n$\"+o$=>O$Fa]nF_`z7%F\\_\\l7$$!+'Q4&4LFa]n$\"+.iM :LFa]nF_`z7%Fb_\\l7$$!+#[n;N$Fa]n$\"+EdenKFa]nF_`z7%Fh_\\l7$$!+#RMLR$F a]n$\"+-(y'=KFa]nF_`z7%F^`\\l7$$!+*H\"eMMFa]n$\"+f7ooJFa]nF_`z7%Fd`\\l 7$$!+74ZvMFa]n$\"+dOm\"H,$Fa]nF_`z7%Ffa\\l7$$!+>\"Gkf$Fa]n$\" +?OPfHFa]nF_`z7%F\\b\\l7$$!+=dBOOFa]n$\"+#>*>0HFa]nF_`z7%Fbb\\l7$$!+D& odn$Fa]n$\"+X?\\]GFa]nF_`z7%Fhb\\l7$$!+/f)\\r$Fa]n$\"++!\\`z#Fa]nF_`z7 %F^c\\l7$$!+\"[IQv$Fa]n$\"+2e&)RFFa]nF_`z7%Fdc\\l7$$!+)zJAz$Fa]n$\"+d#Fa]nF_`z7%Ffd\\l7$$!+>&oR!RFa]n$\"+%[\\b^#Fa]nF_`z7%F\\e \\l7$$!+u_yRRFa]n$\"+UJ.fCFa]nF_`z7%Fbe\\l7$$!+&zhZ(RFa]n$\"+lUM-CFa]n F_`z7%Fhe\\l7$$!+BV$)3SFa]n$\"+ZLYXBFa]nF_`z7%F^f\\l7$$!+4*[>/%Fa]n$\" +OGO)G#Fa]nF_`z7%Fdf\\l7$$!+3>1uSFa]n$\"+&e45B#Fa]nF_`z7%Fjf\\l7$$!+#) *R^5%Fa]n$\"+NqOt@Fa]nF_`z7%F`g\\l7$$!+p(f^8%Fa]n$\"+#3(R:@Fa]nF_`z7%F fg\\l7$$!+Yt5kTFa]n$\"+b;1d?Fa]nF_`z7%F\\h\\l7$$!+\"fx>>%Fa]n$\"+8SK)* >Fa]nF_`z7%Fbh\\l7$$!+\"\\t(=UFa]n$\"+^)\\\"R>Fa]nF_`z7%Fhh\\l7$$!+Qa] WUFa]n$\"+Q#3&z=Fa]nF_`z7%F^i\\l7$$!+=0>pUFa]n$\"+aCP>=Fa]nF_`z7%Fdi\\ l7$$!+w<&GH%Fa]n$\"+l2seCfKWFa]n$\"+6oC>8Fa]nF_`z7%Fd\\]l7$$!+G$*H\\WFa]n$\"+EX\\a 7Fa]nF_`z7%Fj\\]l7$$!+R3HlWFa]n$\"+!Q[$*=\"Fa]nF_`z7%F`]]l7$$!+=te![%F a]n$\"+\\#eQ7\"Fa]nF_`z7%Ff]]l7$$!+06?&\\%Fa]n$\"+>\"y!e5Fa]nF_`z7%F\\ ^]l7$$!+Td84XFa]n$\"+c=k?**Fd[nF_`z7%Fb^]l7$$!+*p&QAXFa]n$\"+qxse#*Fd[ nF_`z7%Fh^]l7$$!+Um$\\`%Fa]n$\"+!p#e&f)Fd[nF_`z7%F^_]l7$$!+4iwYXFa]n$ \"+X9rJzFd[nF_`z7%Fd_]l7$$!+P_%yb%Fa]n$\"+zMcnsFd[nF_`z7%Fj_]l7$$!+8$R \"oXFa]n$\"+$Q?Ng'Fd[nF_`z7%F``]l7$$!+30hxXFa]n$\"+Wz))RfFd[nF_`z7%Ff` ]l7$$!+`\">ie%Fa]n$\"+OF*oF&Fd[nF_`z7%F\\a]l7$$!+Wb#Rf%Fa]n$\"+JGo9YFd [nF_`z7%Fba]l7$$!+x9p+YFa]n$\"+J>L`RFd[nF_`z7%Fha]l7$$!+l:[1YFa]n$\"+l g%GH$Fd[nF_`z7%F^b]l7$$!+CVE6YFa]n$\"+[?Fd[nF_`z7%Fjb]l7$$!+\\iqYFa]n$\"+:9.ylFg_nF_`z7%Ffc]l7$$!+#Hq)>YFa]nF+F_`z7%F\\d]l7$F gc]l$!+:9.ylFg_nF_`z7%F`d]l7$Fac]l$!+:\"3eJ\"Fd[nF_`z7%Fdd]l7$F[c]l$!+ hp?u>Fd[nF_`z7%Fhd]l7$Feb]l$!+[?L`RFd[nF_`z7%Fde]l7$Fca]l$!+JGo9YFd[nF_`z7%Fhe ]l7$F]a]l$!+OF*oF&Fd[nF_`z7%F\\f]l7$Fg`]l$!+Wz))RfFd[nF_`z7%F`f]l7$Fa` ]l$!+$Q?Ng'Fd[nF_`z7%Fdf]l7$F[`]l$!+zMcnsFd[nF_`z7%Fhf]l7$Fe_]l$!+X9rJ zFd[nF_`z7%F\\g]l7$F__]l$!+!p#e&f)Fd[nF_`z7%F`g]l7$Fi^]l$!+qxse#*Fd[nF _`z7%Fdg]l7$Fc^]l$!+c=k?**Fd[nF_`z7%Fhg]l7$F]^]l$!+>\"y!e5Fa]nF_`z7%F \\h]l7$Fg]]l$!+\\#eQ7\"Fa]nF_`z7%F`h]l7$Fa]]l$!+!Q[$*=\"Fa]nF_`z7%Fdh] l7$F[]]l$!+EX\\a7Fa]nF_`z7%Fhh]l7$Fe\\]l$!+6oC>8Fa]nF_`z7%F\\i]l7$F_\\ ]l$!+x5c$Q\"Fa]nF_`z7%F`i]l7$Fi[]l$!+.,SZ9Fa]nF_`z7%Fdi]l7$Fc[]l$!+DWt 5:Fa]nF_`z7%Fhi]l7$F][]l$!+XFat:Fa]nF_`z7%F\\j]l7$Fgj\\l$!++?\"ej\"Fa] nF_`z7%F`j]l7$Faj\\l$!+_r`(p\"Fa]nF_`z7%Fdj]l7$F[j\\l$!+l2se=Fa]nF_`z7%F\\[^l7$F_i\\l$!+Q#3&z=Fa]nF_`z7%F`[^l 7$Fih\\l$!+^)\\\"R>Fa]nF_`z7%Fd[^l7$Fch\\l$!+8SK)*>Fa]nF_`z7%Fh[^l7$F] h\\l$!+b;1d?Fa]nF_`z7%F\\\\^l7$Fgg\\l$!+#3(R:@Fa]nF_`z7%F`\\^l7$Fag\\l $!+NqOt@Fa]nF_`z7%Fd\\^l7$F[g\\l$!+&e45B#Fa]nF_`z7%Fh\\^l7$Fef\\l$!+OG O)G#Fa]nF_`z7%F\\]^l7$F_f\\l$!+ZLYXBFa]nF_`z7%F`]^l7$Fie\\l$!+lUM-CFa] nF_`z7%Fd]^l7$Fce\\l$!+UJ.fCFa]nF_`z7%Fh]^l7$F]e\\l$!+%[\\b^#Fa]nF_`z7 %F\\^^l7$Fgd\\l$!+_@!>d#Fa]nF_`z7%F`^^l7$Fad\\l$!+&z'3GEFa]nF_`z7%Fd^^ l7$F[d\\l$!+*>0HFa]nF_`z7%Fh_^l7$F]b\\l$!+?OPfHFa]nF_`z7%F\\`^l7$Fga\\l$!+R>\" H,$Fa]nF_`z7%F``^l7$Faa\\l$!+x\"4d1$Fa]nF_`z7%Fd`^l7$F[a\\l$!+dOmO$Fa]nF_`z7%F\\b^l7$Fg^\\l$!+VWF2MFa]nF_`z7%F`b^l7$Fa^\\l $!+-**R^MFa]nF_`z7%Fdb^l7$F[^\\l$!+OEH%\\$Fa]nF_`z7%Fhb^l7$Fe]\\l$!+3A 'f`$Fa]nF_`z7%F\\c^l7$F_]\\l$!+Z$Hkd$Fa]nF_`z7%F`c^l7$Fi\\\\l$!+)yDdh$ Fa]nF_`z7%Fdc^l7$Fc\\\\l$!+WX*Ql$Fa]nF_`z7%Fhc^l7$F]\\\\l$!+l1*4p$Fa]n F_`z7%F\\d^l7$Fg[\\l$!+HD3FPFa]nF_`z7%F`d^l7$Fa[\\l$!+'p`Aw$Fa]nF_`z7% Fdd^l7$F[[\\l$!+[`g'z$Fa]nF_`z7%Fhd^l7$Fej[l$!+e#f-$QFa]nF_`z7%F\\e^l7 $F_j[l$!+c6OjQFa]nF_`z7%F`e^l7$Fii[l$!+^T3'*QFa]nF_`z7%Fde^l7$Fci[l$!+ G:jGRFa]nF_`z7%Fhe^l7$F]i[l$!+cwBhRFa]nF_`z7%F\\f^l7$Fgh[l$!+0\\;%*RFa ]nF_`z7%F`f^l7$Fah[l$!+&Q$pFSFa]nF_`z7%Fdf^l7$F[h[l$!+e%*4iSFa]nF_`z7% Fhf^l7$Feg[l$!+Q%>w4%Fa]nF_`z7%F\\g^l7$F_g[l$!+t%)RMTFa]nF_`z7%F`g^l7$ Fif[l$!+.IVsTFa]nF_`z7%Fdg^l7$Fcf[l$!+$GE:@%Fa]nF_`z7%Fhg^l7$F]f[l$!+M =G^UFa]nF_`z7%F\\h^l7$Fge[l$!+H([6H%Fa]nF_`z7%F`h^l7$Fae[l$!+Eo]IVFa]n F_`z7%Fdh^l7$F[e[l$!+f(e(oVFa]nF_`z7%Fhh^l7$Fed[l$!+\"o'R0WFa]nF_`z7%F \\i^l7$F_d[l$!+U6.SWFa]nF_`z7%F`i^l7$Fic[l$!+X&)QsWFa]nF_`z7%Fdi^l7$Fc c[l$!+i7H-XFa]nF_`z7%Fhi^l7$F]c[l$!+!eK'HXFa]nF_`z7%F\\j^l7$Fgb[l$!+#[ `Vb%Fa]nF_`z7%F`j^l7$Fab[l$!+XRUwXFa]nF_`z7%Fdj^l7$F[b[l$!+&)*Gef%Fa]n F_`z7%Fhj^l7$Fea[l$!+J'eDh%Fa]nF_`z7%F\\[_l7$F_a[l$!+93gEYFa]nF_`z7%F` [_l7$Fi`[l$!+*=Ozj%Fa]nF_`z7%Fd[_l7$Fc`[l$!+lT`YYFa]nF_`z7%Fh[_l7$F]`[ l$!+L&\\Bl%Fa]nF_`z7%F\\\\_l7$Fg_[l$!+H#>`l%Fa]nF_`z7%F`\\_l7$Fa_[l$!+ l)e`l%Fa]nF_`z7%Fd\\_l7$F[_[l$!+T'eBl%Fa]nF_`z7%Fh\\_l7$Fe^[l$!+b#yhk% Fa]nF_`z7%F\\]_l7$F_^[l$!+f.kOYFa]nF_`z7%F`]_l7$Fi][l$!+]>_BYFa]nF_`z7 %Fd]_l7$Fc][l$!+CHa1YFa]nF_`z7%Fh]_l7$F]][l$!+O2N&e%Fa]nF_`z7%F\\^_l7$ Fg\\[l$!+(z*\\fXFa]nF_`z7%F`^_l7$Fa\\[l$!+OSUGXFa]nF_`z7%Fd^_l7$F[\\[l $!+j7S\"\\%Fa]nF_`z7%Fh^_l7$Fe[[l$!+5*4vW%Fa]nF_`z7%F\\__l7$F_[[l$!+*= &e&R%Fa]nF_`z7%F`__l7$Fijz$!++#*>MVFa]nF_`z7%Fd__l7$Fcjz$!+RpshUFa]nF_ `z7%Fh__l7$F]jz$!+R2hwTFa]nF_`z7%F\\`_l7$Fgiz$!+-^\"z2%Fa]nF_`z7%F``_l 7$Faiz$!+z&))f'RFa]nF_`z7%Fd`_l7$F[iz$!+!HGE%QFa]nF_`z7%Fh`_l7$Fehz$!+ mTQ5PFa]nF_`z7%F\\a_l7$F_hz$!+h4grNFa]nF_`z7%F`a_l7$Figz$!+P$pzU$Fa]nF _`z7%Fda_l7$Fcgz$!+K?h!G$Fa]nF_`z7%Fha_l7$F]gz$!+jaLIJFa]nF_`z7%F\\b_l 7$Fgfz$!+I GoEFa]nF_`z7%Fhb_l7$Feez$!+Su>7DFa]nF_`z7%F\\c_l7$F_ez$!+;olbBFa]nF_`z 7%F`c_l7$Fidz$!+#)e&))>#Fa]nF_`z7%Fdc_l7$Fcdz$!+tV\">/#Fa]nF_`z7%Fhc_l 7$F]dz$!+#z+\\)=Fa]nF_`z7%F\\d_l7$Fgcz$!+\\=&ys\"Fa]nF_`z7%F`d_l7$$\"+ \\1FohF^]n$!+Ghyq:Fa]nF_`z7%Fdd_l7$F[cz$!+CEr89Fa]nF_`z7%Fjd_l7$Febz$! +tbjc7Fa]nF_`z7%F^e_l7$F_bz$!+\\pb*4\"Fa]nF_`z7%Fbe_l7$Fiaz$!+rmxC%*Fd [nF_`z7%Ffe_l7$$\"+L@s)G$F*$!+l8)R&yFd[nF_`z7%Fje_l7$$\"+YB:cgFH$!+p_= $G'Fd[nF_`z7%F`f_l7$$\"+agTglFip$!+x*)Q7ZFd[nF_`z7%Fff_l7$F+$!+aEfTJFd [nF_`z7%F\\g_l7$F+$!+Fjzq:Fd[nF_`z7%F`g_lFDF_`z-Fejm6&F3$\"\"*F\\[n$\" #XFijmFjjmFadq-F$6$7in7$$\"32+++@xF?8F*$\"3%******R2*>&4&F*7$$\"34+++! *eN=8F*$\"3@+++;rU2^F*7$$\"3,+++\\=X:8F*$\"3C+++L*R#>^F*7$$\"35+++xPi6 8F*$\"3V+++!*zeI^F*7$$\"3++++Uw#pI\"F*$\"3u*****H`E99&F*7$$\"33+++gtT, 8F*$\"3!)*****4![r^^F*7$$\"3))******z\\9&H\"F*$\"3K+++\\`Th^F*7$$\"31+ ++%)3;)G\"F*$\"3;+++pM\\q^F*7$$\"35+++-S^!G\"F*$\"3Q+++!o;*y^F*7$$\"3/ +++O?Ds7F*$\"3;+++sUl'=&F*7$$\"3#******zp@ME\"F*$\"3u******4pn$>&F*7$$ \"3!*******f*oSD\"F*$\"3i*****4Ec**>&F*7$$\"3/+++C$RUC\"F*$\"3'******R mka?&F*7$$\"35+++>\"yRB\"F*$\"32+++8[<5_F*7$$\"3'******\\\"3LB7F*$\"3J +++o%fS@&F*7$$\"30+++&QVB@\"F*$\"3y******p64<_F*7$$\"3)*******4F1,7F*$ \"3M+++y>C>_F*7$$\"3!******H.P&*=\"F*$\"3=+++4K[?_F*7$$\"31+++.l\"y<\" F*$\"3'********=&y?_F*7$$\"3)******Rz`f;\"F*$\"3g*****f-<,A&F*7$$\"33+ ++a[+a6F*$\"3%******RQY%=_F*7$$\"32+++$zH?9\"F*$\"3&)*****zIRd@&F*7$$ \"3&*******fR4I6F*$\"3m*****z3g>@&F*7$$\"3-+++`#p#=6F*$\"3K+++(>rq?&F* 7$$\"3,+++Fdj16F*$\"3H+++rL.,_F*7$$\"33+++qNG&4\"F*$\"3O+++#)e!Q>&F*7$ $\"3#******R`:V3\"F*$\"3;+++HrM&=&F*7$$\"3++++O*\\Q2\"F*$\"3m*****4u:c <&F*7$$\"31+++6V-k5F*$\"3M+++PCdk^F*7$$\"3'******p!)**\\0\"F*$\"3#)*** **\\4$=_^F*7$$\"3-++++j'p/\"F*$\"30+++HOUQ^F*7$$\"3&******>,[,/\"F*$\" 3E+++*e(GB^F*7$$\"3$*******Q*3[.\"F*$\"3M+++Jxz1^F*7$$\"35+++glDJ5F*$ \"3I+++yI-*3&F*7$$\"3&*******Q2%)H5F*$\"3!)******[H5q]F*7$$\"35+++%[U4 .\"F*$\"3I+++4$y-0&F*7$$\"3!******RnW\\.\"F*$\"3q*****pEF*H]F*7$$\"35+ ++pj(\\F*7$$\"31+++,e\"R3\"F*$\"3y*****z=Hj& \\F*7$$\"3\"******z3iL5\"F*$\"3*******4V'zV\\F*7$$\"3$******zxhV7\"F*$ \"3R+++j-yM\\F*7$$\"3/+++y!Rg9\"F*$\"3k*****p(eQH\\F*7$$\"3/+++\\.hn6F *$\"3h*****f$4XF\\F*7$$\"35+++FqX)=\"F*$\"3()*******zQ'G\\F*7$$\"3-+++ N*Q\"37F*$\"3')*****H@JD$\\F*7$$\"33+++w;PE7F*$\"3?+++@dpQ\\F*7$$\"35+ ++s8*HC\"F*$\"3q*****RyCn%\\F*7$$\"3!******H_?zD\"F*$\"3w*****4``i&\\F *7$$\"31+++)QS6F\"F*$\"3A+++3X'p'\\F*7$$\"3%******fgqEG\"F*$\"3G+++rce y\\F*7$$\"31+++V\\b#H\"F*$\"3M+++.b)3*\\F*7$$\"3*)*****Hk^3I\"F*$\"3L+ ++Dtm.]F*7$$\"30+++Pqi28F*$\"3&******\\#Qw;]F*7$$\"3(******4Q^HJ\"F*$ \"33+++!=K+.&F*7$$\"36+++\">'*oJ\"F*$\"3))******4*\\L/&F*7$$\"3$****** pdK&>8F*$\"3k*****>T6m0&F*7$$\"3/+++C-$4K\"F*$\"3C+++*4D(p]F*7$$\"3$** ****\\\"o:@8F*$\"3o******\\5h#3&F*F]h_lFc_z-F^[n6jn7%7$$\"+@xF?8Fa]n$ \"+u!*>&4&Fa]n7$$\"+!*eN=8Fa]n$\"+;rU2^Fa]n7$$\"$=\"Fijm$\"$2&Fijm7%Fa [al7$$\"+\\=X:8Fa]n$\"+L*R#>^Fa]nFf[al7%F\\\\al7$$\"+xPi68Fa]n$\"+!*ze I^Fa]nFf[al7%Fb\\al7$$\"+Uw#pI\"Fa]n$\"+LlUT^Fa]nFf[al7%Fh\\al7$$\"+gt T,8Fa]n$\"+,[r^^Fa]nFf[al7%F^]al7$$\"+!)\\9&H\"Fa]n$\"+\\`Th^Fa]nFf[al 7%Fd]al7$$\"+%)3;)G\"Fa]n$\"+pM\\q^Fa]nFf[al7%Fj]al7$$\"+-S^!G\"Fa]n$ \"+!o;*y^Fa]nFf[al7%F`^al7$$\"+O?Ds7Fa]n$\"+sUl'=&Fa]nFf[al7%Ff^al7$$ \"+)p@ME\"Fa]n$\"+5pn$>&Fa]nFf[al7%F\\_al7$$\"+g*oSD\"Fa]n$\"+hi&**>&F a]nFf[al7%Fb_al7$$\"+C$RUC\"Fa]n$\"+kYY0_Fa]nFf[al7%Fh_al7$$\"+>\"yRB \"Fa]n$\"+8[<5_Fa]nFf[al7%F^`al7$$\"+:3LB7Fa]n$\"+o%fS@&Fa]nFf[al7%Fd` al7$$\"+&QVB@\"Fa]n$\"+q64<_Fa]nFf[al7%Fj`al7$$\"+5F1,7Fa]n$\"+y>C>_Fa ]nFf[al7%F`aal7$$\"+Lq`*=\"Fa]n$\"+4K[?_Fa]nFf[al7%Ffaal7$$\"+.l\"y<\" Fa]n$\"+!>&y?_Fa]nFf[al7%F\\bal7$$\"+%z`f;\"Fa]n$\"+Eq6?_Fa]nFf[al7%Fb bal7$$\"+a[+a6Fa]n$\"+%QY%=_Fa]nFf[al7%Fhbal7$$\"+$zH?9\"Fa]n$\"+3$Rd@ &Fa]nFf[al7%F^cal7$$\"+gR4I6Fa]n$\"+)3g>@&Fa]nFf[al7%Fdcal7$$\"+`#p#=6 Fa]n$\"+(>rq?&Fa]nFf[al7%Fjcal7$$\"+Fdj16Fa]n$\"+rL.,_Fa]nFf[al7%F`dal 7$$\"+qNG&4\"Fa]n$\"+#)e!Q>&Fa]nFf[al7%Ffdal7$$\"+MbJ%3\"Fa]n$\"+HrM&= &Fa]nFf[al7%F\\eal7$$\"+O*\\Q2\"Fa]n$\"+Tdhv^Fa]nFf[al7%Fbeal7$$\"+6V- k5Fa]n$\"+PCdk^Fa]nFf[al7%Fheal7$$\"+2)**\\0\"Fa]n$\"+&4$=_^Fa]nFf[al7 %F^fal7$$\"++j'p/\"Fa]n$\"+HOUQ^Fa]nFf[al7%Fdfal7$$\"+7![,/\"Fa]n$\"+* e(GB^Fa]nFf[al7%Fjfal7$$\"+R*3[.\"Fa]n$\"+Jxz1^Fa]nFf[al7%F`gal7$$\"+g lDJ5Fa]n$\"+yI-*3&Fa]nFf[al7%Ffgal7$$\"+R2%)H5Fa]n$\"+\\H5q]Fa]nFf[al7 %F\\hal7$$\"+%[U4.\"Fa]n$\"+4$y-0&Fa]nFf[al7%Fbhal7$$\"+uY%\\.\"Fa]n$ \"+ns#*H]Fa]nFf[al7%Fhhal7$$\"+pj(\\Fa ]nFf[al7%Fjial7$$\"+,e\"R3\"Fa]n$\"+)=Hj&\\Fa]nFf[al7%F`jal7$$\"+)3iL5 \"Fa]n$\"+JkzV\\Fa]nFf[al7%Ffjal7$$\"+y'*oJ\"Fa]n$\"+5*\\L/&Fa]nFf[al7%F``bl7$$\"+xD`>8Fa]n$ \"+79hc]Fa]nFf[al7%Ff`bl7$$\"+C-$4K\"Fa]n$\"+*4D(p]Fa]nFf[al7%F\\abl7$ $\"+:o:@8Fa]n$\"+]5h#3&Fa]nFf[al7%FbablF\\[alFf[alFdg_lFadq-F$6$7in7$F ^h_l$!3%******R2*>&4&F*7$Fej`l$!3o******\\5h#3&F*7$F`j`l$!3C+++*4D(p]F *7$F[j`l$!3k*****>T6m0&F*7$Ffi`l$!3))******4*\\L/&F*7$Fai`l$!33+++!=K+ .&F*7$F\\i`l$!3&******\\#Qw;]F*7$Fgh`l$!3L+++Dtm.]F*7$Fbh`l$!3M+++.b)3 *\\F*7$F]h`l$!3G+++rcey\\F*7$Fhg`l$!3A+++3X'p'\\F*7$Fcg`l$!3w*****4``i &\\F*7$F^g`l$!3q*****RyCn%\\F*7$Fif`l$!3?+++@dpQ\\F*7$Fdf`l$!3')*****H @JD$\\F*7$F_f`l$!3()*******zQ'G\\F*7$Fje`l$!3h*****f$4XF\\F*7$Fee`l$!3 k*****p(eQH\\F*7$F`e`l$!3R+++j-yM\\F*7$F[e`l$!3*******4V'zV\\F*7$Ffd`l $!3y*****z=Hj&\\F*7$Fad`l$!39+++B!f>(\\F*7$F\\d`l$!3q*******z$***)\\F* 7$Fgc`l$!37+++VWf4]F*7$Fbc`l$!3q*****pEF*H]F*7$F]c`l$!3I+++4$y-0&F*7$F hb`l$!3!)******[H5q]F*7$Fcb`l$!3I+++yI-*3&F*7$F^b`l$!3M+++Jxz1^F*7$Fia `l$!3E+++*e(GB^F*7$Fda`l$!30+++HOUQ^F*7$F_a`l$!3#)*****\\4$=_^F*7$Fj`` l$!3M+++PCdk^F*7$Fe``l$!3m*****4u:c<&F*7$F```l$!3;+++HrM&=&F*7$F[``l$! 3O+++#)e!Q>&F*7$Ff_`l$!3H+++rL.,_F*7$Fa_`l$!3K+++(>rq?&F*7$F\\_`l$!3m* ****z3g>@&F*7$Fg^`l$!3&)*****zIRd@&F*7$Fb^`l$!3%******RQY%=_F*7$F]^`l$ !3g*****f-<,A&F*7$Fh]`l$!3'********=&y?_F*7$Fc]`l$!3=+++4K[?_F*7$F^]`l $!3M+++y>C>_F*7$Fi\\`l$!3y******p64<_F*7$Fd\\`l$!3J+++o%fS@&F*7$F_\\`l $!32+++8[<5_F*7$Fj[`l$!3'******Rmka?&F*7$Fe[`l$!3i*****4Ec**>&F*7$F`[` l$!3u******4pn$>&F*7$F[[`l$!3;+++sUl'=&F*7$Ffj_l$!3Q+++!o;*y^F*7$Faj_l $!3;+++pM\\q^F*7$F\\j_l$!3K+++\\`Th^F*7$Fgi_l$!3!)*****4![r^^F*7$Fbi_l $!3u*****H`E99&F*7$F]i_l$!3V+++!*zeI^F*7$Fhh_l$!3C+++L*R#>^F*7$Fch_l$! 3@+++;rU2^F*F[bblFc_z-F^[n6jn7%7$F][al$!+u!*>&4&Fa]n7$Fcabl$!+]5h#3&Fa ]n7$Fg[al$!$2&Fijm7%Fe]cl7$F]abl$!+*4D(p]Fa]nFh]cl7%F\\^cl7$Fg`bl$!+79 hc]Fa]nFh]cl7%F`^cl7$Fa`bl$!+5*\\L/&Fa]nFh]cl7%Fd^cl7$F[`bl$!+!=K+.&Fa ]nFh]cl7%Fh^cl7$Fe_bl$!+DQw;]Fa]nFh]cl7%F\\_cl7$F__bl$!+Dtm.]Fa]nFh]cl 7%F`_cl7$Fi^bl$!+.b)3*\\Fa]nFh]cl7%Fd_cl7$Fc^bl$!+rcey\\Fa]nFh]cl7%Fh_ cl7$F]^bl$!+3X'p'\\Fa]nFh]cl7%F\\`cl7$Fg]bl$!+JNDc\\Fa]nFh]cl7%F``cl7$ Fa]bl$!+%yCn%\\Fa]nFh]cl7%Fd`cl7$F[]bl$!+@dpQ\\Fa]nFh]cl7%Fh`cl7$Fe\\b l$!+87`K\\Fa]nFh]cl7%F\\acl7$F_\\bl$!++)Q'G\\Fa]nFh]cl7%F`acl7$Fi[bl$! +O4XF\\Fa]nFh]cl7%Fdacl7$Fc[bl$!+xeQH\\Fa]nFh]cl7%Fhacl7$F][bl$!+j-yM \\Fa]nFh]cl7%F\\bcl7$Fgjal$!+JkzV\\Fa]nFh]cl7%F`bcl7$Fajal$!+)=Hj&\\Fa ]nFh]cl7%Fdbcl7$F[jal$!+B!f>(\\Fa]nFh]cl7%Fhbcl7$Feial$!++Q***)\\Fa]nF h]cl7%F\\ccl7$F_ial$!+VWf4]Fa]nFh]cl7%F`ccl7$Fihal$!+ns#*H]Fa]nFh]cl7% Fdccl7$Fchal$!+4$y-0&Fa]nFh]cl7%Fhccl7$F]hal$!+\\H5q]Fa]nFh]cl7%F\\dcl 7$Fggal$!+yI-*3&Fa]nFh]cl7%F`dcl7$Fagal$!+Jxz1^Fa]nFh]cl7%Fddcl7$F[gal $!+*e(GB^Fa]nFh]cl7%Fhdcl7$Fefal$!+HOUQ^Fa]nFh]cl7%F\\ecl7$F_fal$!+&4$ =_^Fa]nFh]cl7%F`ecl7$Fieal$!+PCdk^Fa]nFh]cl7%Fdecl7$Fceal$!+Tdhv^Fa]nF h]cl7%Fhecl7$F]eal$!+HrM&=&Fa]nFh]cl7%F\\fcl7$Fgdal$!+#)e!Q>&Fa]nFh]cl 7%F`fcl7$Fadal$!+rL.,_Fa]nFh]cl7%Fdfcl7$F[dal$!+(>rq?&Fa]nFh]cl7%Fhfcl 7$Fecal$!+)3g>@&Fa]nFh]cl7%F\\gcl7$F_cal$!+3$Rd@&Fa]nFh]cl7%F`gcl7$Fib al$!+%QY%=_Fa]nFh]cl7%Fdgcl7$Fcbal$!+Eq6?_Fa]nFh]cl7%Fhgcl7$F]bal$!+!> &y?_Fa]nFh]cl7%F\\hcl7$Fgaal$!+4K[?_Fa]nFh]cl7%F`hcl7$Faaal$!+y>C>_Fa] nFh]cl7%Fdhcl7$F[aal$!+q64<_Fa]nFh]cl7%Fhhcl7$Fe`al$!+o%fS@&Fa]nFh]cl7 %F\\icl7$F_`al$!+8[<5_Fa]nFh]cl7%F`icl7$Fi_al$!+kYY0_Fa]nFh]cl7%Fdicl7 $Fc_al$!+hi&**>&Fa]nFh]cl7%Fhicl7$F]_al$!+5pn$>&Fa]nFh]cl7%F\\jcl7$Fg^ al$!+sUl'=&Fa]nFh]cl7%F`jcl7$Fa^al$!+!o;*y^Fa]nFh]cl7%Fdjcl7$F[^al$!+p M\\q^Fa]nFh]cl7%Fhjcl7$Fe]al$!+\\`Th^Fa]nFh]cl7%F\\[dl7$F_]al$!+,[r^^F a]nFh]cl7%F`[dl7$Fi\\al$!+LlUT^Fa]nFh]cl7%Fd[dl7$Fc\\al$!+!*zeI^Fa]nFh ]cl7%Fh[dl7$F]\\al$!+L*R#>^Fa]nFh]cl7%F\\\\dl7$Fb[al$!+;rU2^Fa]nFh]cl7 %F`\\dlFb]clFh]clFdg_lFadq-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%% &Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-F^]dl6$%*HELVETICAGFgg_l-%*AXESSTY LEG6#%$BOXG-%%VIEWG6$;$!$R&Fijm$\"$z\"Fijm;$!$p&Fijm$\"$p&Fijm" 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 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for t he combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "co efficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3315 "ee := \{c[2]=1/16,\nc[3]=4 /25,\nc[4]=6/25,\nc[5]=915/1696,\nc[6]=5/8,\nc[7]=35/209,\nc[8]=29/72, \nc[9]=11/13,\nc[10]=15/16,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/16,\na[3,1 ]=-28/625,\na[3,2]=128/625,\na[4,1]=3/50,\na[4,2]=0,\na[4,3]=9/50,\na[ 5,1]=15090136995/39027212288,\na[5,2]=0,\na[5,3]=-53100995625/39027212 288,\na[5,4]=29533111875/19513606144,\na[6,1]=32755/421632,\na[6,2]=0, \na[6,3]=0,\na[6,4]=9765625/29258496,\na[6,5]=1488770/6971751,\na[7,1] =1072887507055/12570100065468,\na[7,2]=0,\na[7,3]=0,\na[7,4]=107542674 8046875/9595107694235844,\na[7,5]=-26411117095731200/38576694343416381 9,\na[7,6]=23430395520/608661493759,\na[8,1]=232418455423819/110766290 98233600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-232407582514046875/2367427911 414660864,\na[8,5]=881809586743097626/5311459993015086075,\na[8,6]=-14 411638756714/175988393502075,\na[8,7]=19/48,\na[9,1]=-8046942294592811 005033672093485229/31693413191384942205414737020389900,\na[9,2]=0,\na[ 9,3]=0,\na[9,4]=-536231038652840856428143497005609375/1382426602745158 84207093807951326924,\na[9,5]=-312405090743882815734153055386047660032 /138949582969101476796789174014714391225,\na[9,6]=46374757254514641197 506707162751264/33542814156299871502725237873062925,\na[9,7]=205178433 1177468089852258884/674022310351068617436737421,\na[9,8]=1762562147235 459382311840/629339225351137831406851,\na[10,1]=2157203756954792640637 49229806841635/440720278308021223336256911460990976,\na[10,2]=0,\na[10 ,3]=0,\na[10,4]=12320199720948281040487332306693759765625/235489849560 6471238613818261942735405056,\na[10,5]=1866488369998347663233338879467 7364275/6604186914582246254039171103542098296,\na[10,6]=-7845820908251 0636098652961448851403235/65709391590090207335818975096879372288,\na[1 0,7]=-284269998655149797778753043937075/785609142574129477342864525230 08,\na[10,8]=-40415483223195085710169801875/13000342692048285899311310 848,\na[10,9]=1620538162595625/5204697537445888,\na[11,1]=-97576760559 2872353548579494188313/1054764737809245464809448652511500,\na[11,2]=0, \na[11,3]=0,\na[11,4]=-5137868354683248477586346693453125/495465427355 479410622167542061972,\na[11,5]=-405677779726268282752892649892020224/ 71142722483104145892095323206788025,\na[11,6]=245921960510918251629683 797176094768/59908811922681865929060892977735375,\na[11,7]=36795342773 4636456312437471451631/44380048256994921706877045210625,\na[11,8]=8085 91598711928461147452272/133856729103497890946741285,\na[11,9]=-3049240 467/3093372235,\na[11,10]=3687441170432/6749831213125,\na[12,1]=180095 39/351697500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6] =1351744/5131875,\na[12,7]=898538649982788221/3655523545713900000,\na[ 12,8]=986955378624/4938547627475,\na[12,9]=397569776903/3090028048800, \na[12,10]=2957049856/29666896875,\na[12,11]=71669/6284880,\n\nb[1]=18 009539/351697500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=1351744/51 31875,\nb[7]=898538649982788221/3655523545713900000,\nb[8]=98695537862 4/4938547627475,\nb[9]=397569776903/3090028048800,\nb[10]=2957049856/2 9666896875,\nb[11]=71669/6284880,\n\n`b*`[1]=9132998184216727/17719486 3388812500,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=5 971284900143728/23270216609390625,\n`b*`[7]=46560109562892125961981671 /190526192880008411317500000,\n`b*`[8]=11781220404761346096/5786455770 5069066875,\n`b*`[9]=222323369516196533611/1556841029506080780000,\n`b *`[10]=1246959894769246208/14946997743616640625,\n`b*`[11]=-5978523233 /2539290474000,\n`b*`[12]=1/50\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2.. 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-/&%\"cG6#\"\"##\"\"\"\"#;/&F%6 #\"\"$#\"\"%\"#D/&F%6#F0#\"\"'F1/&F%6#\"\"&#\"$:*\"%'p\"/&F%6#F6#F:\" \")/&F%6#\"\"(#\"#N\"$4#/&F%6#FB#\"#H\"#s/&F%6#\"\"*#\"#6\"#8/&F%6#\"# 5#\"#:F*/&F%6#FUF)/&F%6#\"#7F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j ]),j=1..i-1),i=1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6^o/&%\"aG6$\" \"#\"\"\"#F(\"#;/&F%6$\"\"$F(#!#G\"$D'/&F%6$F.F'#\"$G\"F1/&F%6$\"\"%F( #F.\"#]/&F%6$F:F'\"\"!/&F%6$F:F.#\"\"*F/&F%6 $\"\"'F(#\"&bF$\"'K;U/&F%6$FhnF'F@/&F%6$FhnF.F@/&F%6$FhnF:#\"(Dcw*\")' \\e#H/&F%6$FhnFI#\"(q()[\"\"(^<(p/&F%6$\"\"(F(#\".bq]()G2\"\"/oa1+,d7/ &F%6$FapF'F@/&F%6$FapF.F@/&F%6$FapF:#\"1vo/[nUv5\"1WeB%p2^f*/&F%6$FapF I#!2+7t&4<6TE\"3>Q;MM%pw&Q/&F%6$FapFhn#\",?bRIM#\"-fP\\h'3'/&F%6$\"\") F(#\"0>QUb%=CB\"2+OB)4Hm26/&F%6$F`rF'F@/&F%6$F`rF.F@/&F%6$F`rF:#!3vo/9 De2CB\"4k3m99\"zUnB/&F%6$F`rFI#\"3Ew4Vne4=))\"4vg3:I**f9J&/&F%6$F`rFhn #!/9nvQ;T9\"0v?]$R))f\"#[/&F%6$FEF(#!CH_[$4sO.05\"Gf %HUp/)\"D+**Q?qt9a?U\\Q\">8MpJ/&F%6$FEF'F@/&F%6$FEF.F@/&F%6$FEF:#!Ev$4 c+(\\V\"Gk&3%GlQ5BO&\"ECpK^z!Q42U)e^u-mU#Q\"/&F%6$FEFI#!HK+mZgQbI:Md\" G)Qu!40CJ\"HD7R9Z,u\"*y'zw95pHe\\*Q\"/&F%6$FEFhn#\"Dk7virq1v>TY^asvuj% \"DDH1tyBDF]r)*HcT\"GaL/&F%6$FEFap#\"=%))eA&)*3ou?B\"\"Ic]SNF%>E=QhQ7Z1c\\)*[N#/&F%6$F_wFI# \"GvUOxYz)QLBjwM)**p$)[m=\"F'H)4UN5r\"RSDYAe9p=/m/&F%6$F_wFhn#!GNKS^)[ 9'Hl)4O1^#34#e%y\"G)Gs$zo4v*=eL2-4!f\"R4d'/&F%6$F_wFap#!Bvq$R/`(yxz\\^ l)**pUG\"A3I__kGMx%HTdU\"4cy/&F%6$F_wF`r#!>v=!)p,r&3&>BK[:/%\">[3J6$** eG[?pU.+8/&F%6$F_wFE#\"1Dcfi\"Q0i\"\"1))eWPvp/_/&F%6$\"#6F(#!B8$)=%\\z &[NNsGf0wwv*\"C+:^_'[%4[YX#4ytkZ0\"/&F%6$F`zF'F@/&F%6$F`zF.F@/&F%6$F`z F:#!CDJX$pYjex%[Koa$oy8&\"Bs>1Uv;A1TzaNFaY&\\/&F%6$F`zFI#!EC--#*)\\E*G v#Goiszxn0%\"DD!)y1KK&4#*e9/J[AF9r/&F%6$F`zFhn#\"EoZ4wrz$oH;D=4^g>#fC \"Dv`txH*31Hf'=oA>\")3*f/&F%6$F`zFap#\"BJ;XruV7jXOYtFM&zO\"AD1@Xq(oq@ \\*pD[+QW/&F%6$F`zF`r#\"r)f\"f3)\"<>n%4*y\\.\"Hn&Q8/&F%6$F` zFE#!+n/C\\I\"+NAP$4$/&F%6$F`zF_w#\".K/ \+ " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..11);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\")R&4!=\"*+vp^$/&F%6#\"\"#\"\"! /&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"(W " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b *`[i]),i=1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"#\" 1Fn@%=)*H8*\"3+D\"))Qj[>x\"/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F// &F%6#\"\"&F//&F%6#\"\"'#\"1GP9+\\Grf\"2D1R4m@qK#/&F%6#\"\"(#\";r;)>'f7 #*Gc4,cY\"<++]<8T3+)G>E0>/&F%6#\"\")#\"5'4Y8w//A\"y6\"5vo1p]qdX'y&/&F% 6#\"\"*#\"66O`'>;&pLKA#\"7++y!31&H5%ob\"/&F%6#\"#5#\"43iCpZ*)fpC\"\"5D 1k;Ou(*p%\\\"/&F%6#\"#6#!+LK_yf\".+SZ!HRD/&F%6#\"#7#F'\"#]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------ ---------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 34 "#=================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[ 10] = 19/20;" "6#/&%\"cG6#\"#5*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "As for \+ the previous scheme, the weights of this scheme are all still positive . The largest magnitude of a linking coefficient is lower than for the Sharp-Verner scheme." }}{PARA 0 "" 0 "" {TEXT -1 36 "The nodes of the order 7 scheme are:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/20;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#?!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[3] = 8/57" "6#/&%\"cG6#\"\"$*&\"\")\"\"\"\"#d!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 4/19;" "6#/&%\"cG6#\"\"%* &F'\"\"\"\"#>!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 44881/822 08" "6#/&%\"cG6#\"\"&*&\"&\")[%\"\"\"\"&3A)!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 37/56;" "6#/&%\"cG6#\"\"'*&\"#P\"\"\"\"#c!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 61/367;" "6#/&%\"cG6#\"\"(*& \"#h\"\"\"\"$n$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 25/57; " "6#/&%\"cG6#\"\")*&\"#D\"\"\"\"#d!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 39/50;" "6#/&%\"cG6#\"\"**&\"#R\"\"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 19/20;" "6#/&%\"cG6#\"#5*&\"#>\"\" \"\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\" #6\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 62 "The 2-norm of the principal error is approximately : 0.161522 " }{TEXT 265 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-4) " "6#)\"#5,$\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the c ombined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffi cients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3835 "ee := \{c[2]=1/20,\nc[3]=8 /57,\nc[4]=4/19,\nc[5]=44881/82208,\nc[6]=37/56,\nc[7]=61/367,\nc[8]=2 5/57,\nc[9]=39/50,\nc[10]=19/20,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/20,\n a[3,1]=-184/3249,\na[3,2]=640/3249,\na[4,1]=1/19,\na[4,2]=0,\na[4,3]=3 /19,\na[5,1]=10879382999084569/17778381694173184,\na[5,2]=0,\na[5,3]=- 41275080377096907/17778381694173184,\na[5,4]=20050852951463513/8889190 847086592,\na[6,1]=5533979/91295232,\na[6,2]=0,\na[6,3]=0,\na[6,4]=128 54870299/39431336448,\na[6,5]=3657883862/13345483011,\na[7,1]=15847549 9801357/1970033827325064,\na[7,2]=0,\na[7,3]=0,\na[7,4]=31153019468948 131/297713925429454536,\na[7,5]=-3016545448677376/81418799560147215,\n a[7,6]=150256844634176/8265656149993815,\na[8,1]=2924917590401981135/6 6605023970341440552,\na[8,2]=0,\na[8,3]=0,\na[8,4]=445876617351108665/ 3098009485599691752,\na[8,5]=196990339933642293248/1651616643670523695 197,\na[8,6]=-7149897627279951488/167672519638445079477,\na[8,7]=35/20 1,\na[9,1]=-42211185912602462564070166033950198162012423/3422058241988 61190074733210927141821875000000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-23188 59757393123763522345863545224823245606023/8477797275669363171509163830 49577334375000000,\na[9,5]=-311125308777988754259235660223610292832986 88/28981348290292110337421437961130838134765625,\na[9,6]=1529210724327 0980188039355727714753150306881614/29901512151935859805913259351031993 989013671875,\na[9,7]=40769840728265525109900718286661374039/155493752 65121654840821192853125000000,\na[9,8]=945612121233663016296797907/598 971783761117752865000000,\na[10,1]=53421576995301575930555089451778526 350776610941/205195002329815636484038978294464618799958400000,\na[10,2 ]=0,\na[10,3]=0,\na[10,4]=50289680792358926711421997672063760549618034 3553/195519042445520666638376728307225007182256000000,\na[10,5]=363012 698911770750340098104804203047936446648/417738629670365256911027966408 581558444453125,\na[10,6]=-1158134489450994429486838471458315247610516 70902703/191939521001616690365566010466177813315410464375000,\na[10,7] =-28699281021898330563484694969577298696532859/13463308844970886630335 729997542445904000000,\na[10,8]=-16493845602934685528675361413563797/2 8001674400224052367433014064000000,\na[10,9]=1285791152889375/22454128 45835068,\na[11,1]=89462997577799701289200380185298019046438429/841085 84887856406378631238816963629525265400,\na[11,2]=0,\na[11,3]=0,\na[11, 4]=6492689767645528199626511179748230475508923/64113959080891227714176 9373352511460044088,\na[11,5]=6940719860472767235945668133931589468815 36/175338845029991694153905751089388426975345,\na[11,6]=-1878242334637 16665067125996081330354271085997696/3021128362720770665523180268214868 4012056060045,\na[11,7]=-197008013321007326029676937491820383240505/20 203428941091626061502271171655020032789,\na[11,8]=-1729758243576400360 0121633126256/9677847661048496307295108424975,\na[11,9]=16563897900000 0000/39827589068734103,\na[11,10]=-845624320000/1531259401089,\na[12,1 ]=125174203/2508655500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0 ,\na[12,6]=56446289600512/342606012403635,\na[12,7]=621432520470476499 77/247198834682622733680,\na[12,8]=9636489994384107/36666550775008000, \na[12,9]=5373710937500/40041548453907,\na[12,10]=13212880000/87750675 837,\na[12,11]=-98471/7223040,\n\nb[1]=125174203/2508655500,\nb[2]=0, \nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=56446289600512/342606012403635,\nb[7 ]=62143252047047649977/247198834682622733680,\nb[8]=9636489994384107/3 6666550775008000,\nb[9]=5373710937500/40041548453907,\nb[10]=132128800 00/87750675837,\nb[11]=-98471/7223040,\n\n`b*`[1]=185777316884981989/3 681484912241059950,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n` b*`[6]=22385886671596155610112/158772260638005081292791,\n`b*`[7]=5379 78856569572663706487529/2159330585189973467574291759,\n`b*`[8]=3249856 2725533982078667/120108582798663659157650,\n`b*`[9]=123545029357825703 1250/7631363446505532525519,\n`b*`[10]=498401087642132308000/386325809 7409608192999,\n`b*`[11]=0,\n`b*`[12]=-274194775/132076182681\}:" }}} {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 1011 "subs(ee,matrix([[c[2],a[2,1],``],\n[c[3],a[3,1],a[3 ,2]],[c[4],a[4,1],a[4,2]],[``$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]],[``$2,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]],[``$2,a[8,7]],\n[c[9 ],a[9,1],a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6]],[``,a[9,7],a[9, 8]],\n[c[10],a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[``,a[10,5],a[10,6] ],\n[``,a[10,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[11,1],a[11,2]],[``, a[11,3],a[11,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7],a[11,8]],[``,a[11, 9],a[11,10]],\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a[12,4]],[``,a[12,5 ],a[12,6]],\n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,10]],[``$2,a[12,11] ],\n[``,`________________________________`$2],[`b`,b[1],b[2]],[``,b[3] ,b[4]],[``,b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10]],[``$2,b[11]],\n [`b*`,`b*`[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[6]],\n[`` ,`b*`[7],`b*`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11],`b*`[12]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\"\"\"\"#?F(%!G7%#\" \")\"#d#!$%=\"%\\K#\"$S'F27%#\"\"%\"#>#F)F8\"\"!7%F+F+#\"\"$F87%#\"&\" )[%\"&3A)#\"2pX3**HQz3\"\"2%=tTp\"Qyx\"F:7%F+#!22p4x.3v7%FD#\"28NY^H&3 0?\"1#f'3Z3>*)))7%#\"#P\"#c#\"(zR`&\")K_H\"*F:7%F+F:#\",*Hq[&G\"\",[kL J%R7%F+F+#\"+iQ)yl$\",6I[XL\"7%#\"#h\"$n$#\"0d8!)*\\v%e\"\"1k]KFQ.q>F: 7%F+F:#\"2J\"[*o%>I:J\"3OXXHa#Rr(H7%F+#!1wtn[aa;I\"2:s9g&*z=9)#\"0wTjW oD]\"\"1:Q**\\hll#)7%#\"#DF/#\"4N6)>S!f<\\#H\"5_0WT.(R-0m'F:7%F+F:#\"3 l'36N\"7(>&pB0nVmh^;#!4)[^*zsi (*)\\r\"6x%z]WQ'>Dnn\"7%F+F+#\"#N\"$,#7%#\"#R\"#]#!MBC,i\")>]R.m,2kDY- E\"f=6A%\"N+++v=#=9F4@LZ2!>h))>Ce?U$F:7%F+F:#!OBggXK#[AXN'eMANwBJRd(f) =B\"N+++vVLx&\\IQ;4:.^$fK \"f!)fe$>:7:!*H7%F+#\"GRSPhmG=2!*4^_l#G2%)p2%\"G+++DJ&G>@3%[l@^Ev$\\b \"#\"<2zz'H;ImL777c%*\"<+++lGv<6w$yr*)f7%#F8F*#\"PT4hw2NE&y0#F:7%F+F:#\"Q`NM!='\\0wj?n(*>U6n# *eBz!o*G]\"Q+++cA=2]A2$GnPQmm?bWU!>b>7%F+#\"N[mWOz/.U![5)4S.vq<\"*)p7I O\"NDJXW%e:e3k'z-6pDl.nH'QxT#!T.F!4n^5wC:$e9ZQo[HW*4X*[M\"e6\"T+]Pk/T: L\"y>7%F+#!MfG`'p)Hx&p\\p%[j0L)*=-\"G*pG\"M+++/fWUv**H dLIm)3(\\%)3LY8#!D(zjNTh`nGboMHgXQ\\;\"D+++kS,LuO_SA+Wn,!G7%F+F+#\"1v$ *)G:\"z&G\"\"1o]$e%GTXA7%F)#\"MH%Qk/>!)H&=!Q+#*G,(*zxv*HY*)\"M+aED&HO' p\")Q7jyjScy)[e3T)F:7%F+F:#\"LB*3bZI#[(z6^E'*>Gbkn(*o#\\'\"K)3W+Y6DNt$ p<9xA\"*33fR6k7%F+#\"KO:)o%*eJR8oc%fBnFZg)>2%p\"KX`(pU)Q*3^d!R:%p\"**H ]%)Q`<#!Q'p(*f3rUNI83'*f7n]m;PYLU#y=\"PX+1c?,%o[@o-=Bbmq2si$G6-$7%F+#! K00C$Q?=\\PpnHgK25K8!3q>\"J*yK+-b;Z##\"12TQ%***[O'*\"2+!3]x]lmO7%F+#\".+v$4 rt`\"/2RX[:/S#\",++)G@8\",Pen]x)7%F+F+#!&r%)*\"(SIA(7%F+%A____________ ____________________GFhw7%%\"bGF]vF:F`vFavFevF\\wFcw7%%#b*G#\"3*)>)\\) oJxd=\"4]*f5C7\\[\"o$F:F`v7%F+F:#\"87,hbhfrm)eQA\"9\"z#H\"30!Q1Es(e\"7 %F+#\"b_Kb]YMOJw#\"6+!3B8Uw3,%)\\\"7**H>3'4u4e K'Q7%F+F:#!*vZ>u#\"-\"o#=w?8Q)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(ee,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[j],j=1. .11),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'++]!\"(F(%!GF+F+F+F+F+F+F+F+F+ F+7/$\"'^.9!\"'$!'GjcF*$\"'%)p>F/F+F+F+F+F+F+F+F+F+F+7/$\"'E0@F/$\"';j _F*$\"\"!F:$\"'&*y:F/F+F+F+F+F+F+F+F+F+7/$\"'WfaF/$\"'W>hF/F9$!'k@B!\" &$\"'kbAFDF+F+F+F+F+F+F+F+7/$\"'92mF/$\"'jhgF*F9F9$\"'1gKF/$\"'#4u#F/F +F+F+F+F+F+F+7/$\"'8i;F/$\"'IW!)F*F9F9$\"'TY5F/$!'(\\q$F*$\"'&y\"=F*F+ F+F+F+F+F+7/$\"''fQ%F/$\"'W\"R%F*F9F9$\"'CR9F/$\"'r#>\"F/$!'?kUF*$\"'H Ti# FD$\"'ty:FDF+F+F+F+7/$\"'++&*F/$\"'X.EF/F9F9$\"'6sDFD$\"'&**o)F/$!'&Q. 'F/$!'nJ@FD$!'J!*eF/$\"'IEdF/F+F+F+7/$\"\"\"F:$\"'mj5FDF9F9$\"'o75!\"% $\"'YeRFD$!'- " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderCon ditions(7,11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,OrderConditio ns(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(ee,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsi mplify(subs(ee,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 #\"#P" }}}{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 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct \+ from 2 to 6 do\n so||ct||_11 := StageOrderConditions(ct,11,'expanded ');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Stages 6 to 11 have stage-order 4, while stages 4 and 5 \+ have stage-order 3 has and stage 3 has stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(subs(ee,so||i||_1 1[j])),i=2..6)],j=1..9)]:\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+\"\"#\"\"$F%\"\"%F&F&F&F&F&" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sim plifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&% \"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,& %\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\" \"" }{TEXT -1 8 " . . 10 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1],i=1+1..11)=b[1],seq(Sum(b[i]*a[i ,j],i=j+1..11)=b[j]*(1-c[j]),j=2..10)];\nmap(u->lhs(u)-rhs(u),subs(ee, eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 7,/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"#6&F*6#F-/-F &6$*&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,;\"\"( F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F* 6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF- ,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&F EF`sFFF-/-F&6$*&F)F-&F/6$F,F]sF-/F,;F4F4*&&F*6#F]sF-,&F-F-&FEF]tFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "None of the principal error conditions ar e satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "RK7_11err_e qs := PrincipalErrorConditions(7,11,'expanded'):\nexpand(subs(ee,RK7_1 1err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal \+ error norm, that is, the 2-norm of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms7_11 := PrincipalEr rorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errterms7_11) do \n sm := sm+(evalf(subs(ee,errterms7_11[ct])))^2;\nend do:\nsqrt(sm) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g+A:;!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm \+ of the order 6 embedded scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms6_12*` := subs(b=`b*`,PrincipalEr rorTerms(6,12,'expanded')):\nsm := 0:\nfor ct to nops(`errterms6_12*`) do\n sm := sm+(evalf(subs(ee,`errterms6_12*`[ct])))^2;\nend do:\nsq rt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q&=Cr#!#8" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------- ------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of t he combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "We incorporate the stage-order equations to ensure that \+ stage 3 has stage-order 2, stages 4 and 5 have stage order 3 and stage s 6 to 11 have stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 57 "These co nditions lead to relationships between the nodes:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\" \"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c [6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6#/,**&\"\"$\"\"\"*$&% \"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F' &F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 68 "which we include explicitly to make the compu tation more efficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j +1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6# /%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and " }{XPPEDIT 18 0 "j = 6;" "6# /%\"jG\"\"'" }{TEXT -1 9 " . . 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 134 "The simplifying conditions allow us to \+ manage with just 10 \"simple\" order conditions which are given (in ab reviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These incl ude the 7 order 7 quadrature conditions and 3 additional order 7 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]] ,i=[1,2,4,8,16,32,38,48,61,64])]:\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(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#KF)/*&F,F ()F2FGF(#F(\"\"'7%\"#QF)/*&F,F(-%!G6#*&%\"aGF(-FT6#*&FWF(-FT6#*&FWF(-F T6#*&F8F(FWF(F(F(F(F(#F(\"%?D7%\"#[F)/*&F,F(-FT6#*&FWF(-FT6#*&FWF(-FT6 #*&F?F(FWF(F(F(F(#F(\"$S)7%\"#hF)/*(F,F(F2F(-FT6#*&FEF(FWF(F(#F(\"#N7% \"#kF)/*&F,F()F2FNF(#F(\"\"(Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "SO7_11 := SimpleO rderConditions(7,11,'expanded'):\nSO_eqs := [op(RowSumConditions(11,'e xpanded')),op(StageOrderConditions(2,11,'expanded')),\n op(StageOrder Conditions(3,4..11,'expanded')),op(StageOrderConditions(4,6..11,'expan ded'))]:\nord_cdns := [seq(SO7_11[i],i=[1,2,4,8,16,32,38,48,61,64])]: \nsimp_eqs := [add(b[i]*a[i,1],i=2..11)=b[1],seq(add(b[i]*a[i,j],i=j+1 ..11)=b[j]*(1-c[j]),j=[$8..10])]:\nnode_eqs := [3*c[3]=2*c[4],3*c[6]^2 -4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4]=0]:\ncdns := [op(node_eqs),op(sim p_eqs),op(SO_eqs),op(ord_cdns)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The scheme can be obtained by specifying the nodes: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/20;" "6#/&% \"cG6#\"\"#*&\"\"\"F)\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 4/19;" "6#/&%\"cG6#\"\"%*&F'\"\"\"\"#>!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 37/56;" "6#/&%\"cG6#\"\"'*&\"#P\"\"\"\"#c!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 61/367;" "6#/&%\"cG6#\"\"(*& \"#h\"\"\"\"$n$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 25/57; " "6#/&%\"cG6#\"\")*&\"#D\"\"\"\"#d!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 39/50;" "6#/&%\"cG6#\"\"**&\"#R\"\"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 19/20;" "6#/&%\"cG6#\"#5*&\"#>\"\" \"\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\" #6\"\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 45 "together w ith the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\" \"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\" \")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"a G6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10, 2] = 0;" " 6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[6,3] = 0;" "6#/&%\"aG6$\"\"'\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3] = 0;" "6#/&%\"aG6$\"\"( \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6 $\"\")\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,3] = 0;" "6#/& %\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,3] = 0; " "6#/&%\"aG6$\"#5\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3 ] = 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[4] = 0;" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[5] = 0;" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "e1 := \{c[2]=1/20,c[4]=4/19,c[6]=37/56,c[7]=61/367,c[8]=25/57,c[9 ]=39/50,c[10]=19/20,c[11]=1,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i= 6..11),seq(b[i]=0,i=2..5)\}:\neqns := subs(e1,cdns):\nnops(eqns);\nind ets(eqns) minus \{a[8,7],a[9,7],a[9,8]\};\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#\\" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[8,7],a[9,7],a[9,8 ]\}):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(ma p(rhs,e3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\")\"\"(&F% 6$\"\"*F(&F%6$F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4504 "e3 := \{a[11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = \+ 0, b[5] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[8,2] = 0, a[9,3] = \+ 0, a[9,2] = 0, b[2] = 0, a[10,3] = 0, a[11,2] = 0, b[8] = 963648999438 4107/36666550775008000, a[4,3] = 3/19, b[9] = 5373710937500/4004154845 3907, a[6,4] = 12854870299/39431336448, a[11,10] = -845624320000/15312 59401089, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[5,3] = -41 275080377096907/17778381694173184, c[11] = 1, a[10,9] = 12857911528893 75/2245412845835068, a[3,2] = 640/3249, a[11,4] = -75394187713185085/5 93074572680712+11512379887383625696340659035927/5946878025730703904366 9904100*a[8,7]+2602153181372959975627500000000/79187555978816472785409 997003*a[9,7]+58260845467310562500000000/5291360124504215303912643*a[9 ,8], b[1] = 125174203/2508655500, a[7,4] = 31153019468948131/297713925 429454536, a[5,1] = 10879382999084569/17778381694173184, a[9,4] = -132 740585168029/135201601012468*a[9,7]-8915976325/27102756924*a[9,8]+1124 171812678239/3136893162500000, a[9,5] = 12815779106816/73949863360715* a[9,7]-457207991281233920/432473590906133463*a[9,8]+407550302467959705 6/28886665375279296875, a[9,1] = -98032836639/894656597332*a[9,7]+4636 307689/64743355836*a[9,8]+1056555417801/20757462500000, b[6] = 5644628 9600512/342606012403635, a[11,8] = -78413190000000000/2342799356984359 *a[9,8]+13091576709544123104/256434085324276475, a[10,4] = 30706970873 6183448198494398465280871/15959091657161337450403168496000000*a[8,7]+3 2728571880643973944854972646875/8500335527344169121360709579072*a[9,7] -1027387825736045988223/79578994423551360000+2442585217956426991430468 75/189332511059050905152478144*a[9,8], b[7] = 62143252047047649977/247 198834682622733680, a[7,5] = -3016545448677376/81418799560147215, a[11 ,7] = 115090942908121789754016/1024491433082177744453-8672840994945696 3/439852633489325*a[8,7]-78413190000000000/2342799356984359*a[9,7], a[ 10,8] = -246560400793884375/62871559683381904*a[9,8]+18893635968227236 3359/33725491705192000000, a[10,7] = -2313306878581865574099/118039220 968172000000*a[8,7]-246560400793884375/62871559683381904*a[9,7]+157887 152551067184260037/13654161986887876160000, b[10] = 13212880000/877506 75837, a[11,1] = -663712793820800179/77803123931574600+850223204453533 6700540067357/393517060405078814645480900*a[8,7]+192176686140321710250 0000000/524000225237806047921282547*a[9,7]-122654411510127500000000/51 174322679348969845941*a[9,8], a[8,5] = 12815779106816/73949863360715*a [8,7]+2196255160461824000/24650994681649607391, c[2] = 1/20, c[4] = 4/ 19, c[6] = 37/56, c[7] = 61/367, c[8] = 25/57, c[9] = 39/50, c[10] = 1 9/20, a[8,1] = -98032836639/894656597332*a[8,7]+464948078675/738074256 5304, a[8,6] = -615083784896/7507407947315*a[8,7]-527229416000/1858039 5726279, a[10,5] = -139025125497932970756872098608/4093348712204365036 0004140625*a[8,7]-806087661495521038967520000/118605695098498209915057 5383*a[9,7]-75638830658444077462592/22900562592223218965625+6468172811 6346170033644000000/15601210662956302996519106961*a[9,8], a[5,4] = 200 50852951463513/8889190847086592, a[4,1] = 1/19, a[8,4] = -132740585168 029/135201601012468*a[8,7]+108099948125/343301587704, a[2,1] = 1/20, a [11,5] = -671637983617704685568/21845750984978900295-22683513146890274 387354529792/663817186640535477583201375*a[8,7]-4101739273880788992000 000000/707141601349242086724204313*a[9,7]+9873898231803062272000000000 00/27904895499397014653039754813*a[9,8], a[11,9] = 165638979000000000/ 39827589068734103, b[11] = -98471/7223040, a[10,1] = 22678003532389103 5813505216613261/105604567783104828192117104000000*a[8,7]+241710154926 73232844629615625/56248455655290209498903480128*a[9,7]-831974799305329 8729601/10439655737557488000000-514228466938195156090625/1831091210321 865095166528*a[9,8], a[10,6] = 26689629923921169611072495217/166223099 03798879454304062500*a[8,7]+38687577685147225649182500/120408787455895 574292310303*a[9,7]+3074609994843206240265663/138471384950564266471437 50-14505739546916647562500/11759228042126758797609*a[9,8], a[9,6] = -6 15083784896/7507407947315*a[9,7]+102534985280/325971854847*a[9,8]+5000 011290456/21772982421875, a[11,6] = 7942958607377342618048/19813998337 331956891095+1088678339790283527793906752/67390880740922253555406375*a [8,7]+196859925269261952000000000/71789185761587136770412433*a[9,7]-22 1435326844800000000000/21032984988034658282997*a[9,8], a[6,5] = 365788 3862/13345483011, a[7,1] = 158475499801357/1970033827325064, c[3] = 8/ 57, a[7,6] = 150256844634176/8265656149993815, a[3,1] = -184/3249, c[5 ] = 44881/82208, a[6,1] = 5533979/91295232\}:" }{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 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7)]));\nfor ii from 8 to 11 do\n print(``);\n print(c[ii ]=subs(e3,c[ii])); \n for jj to ii-1 do\n coeff_val := subs(e3, a[ii,jj]);\n if coeff_val=a[ii,jj] then \n print(a[ii,jj] *` is a parameter.`)\n else\n print(a[ii,jj]=coeff_val); \n end if;\n end do:\nend do:\nprint(``);\nseq(b[i]=subs(e3,b[i ]),i=1..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\" \"\"#?F(%!GF+F+F+F+F+7*#\"\")\"#d#!$%=\"%\\K#\"$S'F2F+F+F+F+F+7*#\"\"% \"#>#F)F8\"\"!#\"\"$F8F+F+F+F+7*#\"&\")[%\"&3A)#\"2pX3**HQz3\"\"2%=tTp \"Qyx\"F:#!22p4x.3v7%FC#\"28NY^H&30?\"1#f'3Z3>*)))F+F+F+7*#\"#P\"#c#\" (zR`&\")K_H\"*F:F:#\",*Hq[&G\"\",[kLJ%R#\"+iQ)yl$\",6I[XL\"F+F+7*#\"#h \"$n$#\"0d8!)*\\v%e\"\"1k]KFQ.q>F:F:#\"2J\"[*o%>I:J\"3OXXHa#Rr(H#!1wtn [aa;I\"2:s9g&*z=9)#\"0wTjWoD]\"\"1:Q**\\hll#)F+Q)pprint136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\")#\"#D\"#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"\",&*&#\",Rm$G.)*\"-KtfcY*)F(&F%6$F'\"\"(F(!\"\"#\"-v'y![\\Y\"./`cU2 Q(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\"0H!o^eSF8\"0oC,,;?N\" \"\"\"&F%6$F'\"\"(F.!\"\"#\"-D\"[**43\"\"-/xe,LMF." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\"/;o5zd\"G\"\"/:2Oj)\\R(\"\" \"&F%6$F'\"\"(F.F.#\"4+S#=Yg^D'>#\"5\"R2'\\;o%*4lCF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"-'*[y$3:'\".:t%zS2v\"\"\"& F%6$F'\"\"(F.!\"\"#\"-+gTHs_\"/zis&R!e=F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\"#R\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"\",(*&#\",Rm$G.)*\"-KtfcY*)F(&F%6$F'\"\"(F(!\"\"*&#\"+*o2jj% \",OeNVZ'F(&F%6$F'\"\")F(F(#\".,yTbl0\"\"/++]iuv?F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,(*&#\"0H!o^eSF8\"0oC,,;?N\"\"\"\"&F%6$F'\"\" (F.!\"\"*&#\"+Dj(f\"*)\",Cpv-r#F.&F%6$F'\"\")F.F2#\"1R#yE\"=\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",**&#\"Bh Kh;_]8e.\"*QKN+yE#\"B+++/r6#>G[5$ynXg0\"F(&F%6$\"\")\"\"(F(F(*&#\">Dch HY%GBtE\\:5G,[.*)\\4-HbcX[i&F(&F%6$\"\"*F1F(F(#\"7,'H()H`I*zu>$) \"8+++)[dvtb'R/\"!\"\"*&#\"9D14c^>QpYGU^\":Gl;&4l=K574J=F(&F%6$F8F0F(F <" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,**&#\"Er3Gl%)R%\\)>[M=O(3(pqI\"D+++' \\oJS]uLhrl\"4ff\"\"\"\"&F%6$\"\")\"\"(F.F.*&#\"Avoks\\&[%R(Rk!)=dGF$ \"@s!z&42O@\"pTMFbL+&)F.&F%6$\"\"*F2F.F.#\"7B#))f/Od#yQF5\"5++O^NU%**y &z!\"\"*&#\"D19/+O]O/Ar[L4%\"\"\"&F%6$\"\")\"\"(F.!\"\"*&#\"<++_n*Q5_& \\hw31)\"=$Qv0:*4#)\\)4&p0'=\"F.&F%6$\"\"*F2F.F3#\"8#fiu2W%e1$)Qc(\"8D c'*=KA#fi0!H#F3*&#\">+++WO.qhM;\"Ghp5>l*HIcHm57g:F.&F%6$F:F1F.F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',**&#\"><_\\s5hp 6#R#*H'*oE\">+D1/VXz))z.*4Bi;\"\"\"&F%6$\"\")\"\"(F.F.*&#\";+D=\\cAZ^o xvoQ\"<..J#Hub*eX(y3/7F.&F%6$\"\"*F2F.F.#\":jcESi?V[**4Y2$\";]P9ZmUc] \\Qr%Q\"F.*&#\"8+DcZm\"paRd]9\"84wzen7U!G#f<\"F.&F%6$F9F1F.!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,(*&#\"7*4ub'=eyoI8B \"6+++s\"o4AR!=\"\"\"\"&F%6$\"\")F(F.!\"\"*&#\"3vV)Qz+/cY#\"2/>Q$of:(G 'F.&F%6$\"\"*F(F.F2#\"9P+E%=n5b_r)y:\"8++;wy)o)>;aO\"F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"),&*&#\"3vV)Qz+/cY#\"2/>Q$of:(G '\"\"\"&F%6$\"\"*F(F.!\"\"#\"6fLOsAofj$*)=\"5+++#>0<\\DP$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"1v$*)G:\"z&G\"\"1o]$e%G TXA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#6\"\"\",*#\"3z,!3#Qz7Pm\"2+YdJR7.y(!\"\"*&#\"=dt1S0qO``W?B -&)\"<+4[XY\")y]Sgq^$RF(&F%6$\"\")\"\"(F(F(*&#\"=++++D5\"+T!*pO/RqId-yo%f\"\"\"&F%6$\"\")\"\"(F2F2 *&#\"@++++viv*fHP\"=`@g#\">.q**4aysk\")yfb(=zF2&F%6$\"\"*F6F2F2*&#\";+ +++Dc5tYX3Ee\":VE\"RI:U]C,O\"H&F2&F%6$F=F5F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&,*#\"6obo/xh$)zjr'\"5&H+*y\\)4vX=#!\" \"*&#\">#zHXN(Qu-*o98NoA\"8[vRIl9qR*\\&*[!z#F2&F%6$F=F5F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',*#\"7[!=EMxtgeH%z\"8&4\"*o&>LP$)*R\" )>\"\"\"*&#\"=_n!RzFNG!zR$y')3\"\";vjSbNDA4u!)3RnF-&F%6$\"\")\"\"(F-F- *&#\"<++++?&>Ep_#*fo>\";LCTqn8(ehd=*yrF-&F%6$\"\"*F5F-F-*&#\"9+++++![% oKN9A\"8(*HGeY.))\\)H.@F-&F%6$F8 %y\"1fV)pN*zUBF-&F%6$\"\"*F(F-F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#6\"\"),&*&#\"2+++++>8%y\"1fV)pN*zUB\"\"\"&F%6$\"\"*F(F.!\"\" #\"5/J7W&4nd\"48\"3vkFC`3MkDF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"*#\"3++++!z*Qc;\"2.Tto!*eF)R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!-++KCc%)\".*3,%f7`\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\" \"\"#\"*.UZ#/&F%6#\"\")#\"12TQ%***[O'*\"2+!3]x]lmO/&F%6#\"\"*#\".+v$ 4rt`\"/2RX[:/S/&F%6#\"#5#\",++)G@8\",Pen]x)/&F%6#\"#6#!&r%)*\"(SIA(" } }}{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 88 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op( OrderConditions(7,11,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simplify(subs(e3,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#&*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We now turn our attent ion to the embedded 12 stage, order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 210 "The first 11 stages of the embedded order 6 scheme coinc ide with those of the order 7 scheme and the weights of the order 7 sc heme provide the linking coefficients for the 12th stage of the order \+ 6 scheme. Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1 2,j] = b[j];" "6#/&%\"aG6$\"#7%\"jG&%\"bG6#F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 11." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j ],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F ," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1 ]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and " }{XPPEDIT 18 0 " j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=5" "6# /%\"jG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The simplifying conditions allow us to manage \+ with just 7 \"simple\" order conditions which are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the \+ 6 order 6 quadrature conditions and 1 additional order 6 order conditi ons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO6*` := subs(b=`b*`,SimpleOrderConditions(6)):\n[s eq([i,`SO6*`[i]],i=[1,2,4,8,16,31,32])]:\nlinalg[augment](linalg[delco ls](%,2..2),matrix([[` `]$(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( \"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#JF )/*&F,F(-%!G6#*&FEF(%\"aGF(F(#F(\"#I7%\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)pp rint446\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 291 "`SO6_12*` := subs(b=`b*`,SimpleOrderConditions(6,1 2,'expanded')):\n`cdns*` := [seq(`SO6_12*`[i],i=[1,2,4,8,16,31,32])]: \n`simp_eqs*` := [seq(add(`b*`[i]*a[i,j],i=j+1..12)=`b*`[j]*(1-c[j]),j =[4,5])]:\nwt_eqns := [seq(b[i]=a[12,i],i=1..11)]:\n`cdns*` := [op(`cd ns*`),op(wt_eqns),op(`simp_eqs*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[4] = 0;" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[5] = 0;" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "`b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "e4 := \{c[12]=1,seq(`b*`[i]=0,i=2..5),`b*`[11]=0\}:\n e5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqns2 := [op(remove(_U->type(rhs(_ U),realcons),e5)),op(subs(e5,`cdns*`))]:\nnops(eqns2);\nindets(eqns2) \+ minus \{a[8,7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" } }{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "e6 := solve(\{op(eqns2)\},indets(eqns2) minus \{a[8,7]\}):\ne7 := simplify(subs(select(u_->member(lhs(u_),\{b[8],a[8,7],a[9,7],a[9,8]\} ),e6),e5)):\ne8 := `union`(e7,e6):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a[10,4]=subs(e8,a[10,4]);\na [11,1]=subs(e8,a[11,1]);\n`b*`[12]=subs(e8,`b*`[12]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&*&#\"QR\"\\f,wn\\\"z/LjAXq[;.s tS4.T\"P+++?R(z!*o>6f\"R&y*oG(=\")374o\"\"\"&F%6$\"\")\"\"(F.!\"\"#\"F J\"=?l;-w^OfOs!QB#Q'>\"\"E++g&=W8h\"=,=bW/5l.LF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"A\"*=frfizjIihZ\")3?[\"A+mrE/m' o6FdUKF*HKF(*&#\"HF5>:M7j!)G2%yXd\\lIh==\"G#4hjaaU66$)[v**f0hFvQ(F(&F% 6$\"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#! *vZ>u#\"-\"o#=w?8" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Note that some of t he coefficients depend on " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\") \"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5166 "e8 := \{a[12,6] = 56446289600512/342606012403635, a [11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[6,3] = 0, a[7 ,3] = 0, a[8,3] = 0, a[8,2] = 0, a[9,3] = 0, a[9,2] = 0, b[2] = 0, a[1 0,3] = 0, a[11,2] = 0, a[10,6] = -424388894487179026647624789861336421 5704307/8439587331801172146385267466516964528125000*a[8,7]-14826092371 952216560740591870317445411813/287425804568540924838464708083053172812 50, b[8] = 9636489994384107/36666550775008000, `b*`[8] = 3249856272553 3982078667/120108582798663659157650, a[4,3] = 3/19, b[9] = 53737109375 00/40041548453907, `b*`[12] = -274194775/132076182681, a[10,4] = -4103 09407372031648704522633304791496776015949139/6809120881187286897853915 9111968907973920000000*a[8,7]+1196382233807236593651760216652018131/33 0365100444551801181611344185600000, `b*`[1] = 185777316884981989/36814 84912241059950, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[ 6,4] = 12854870299/39431336448, a[12,7] = 62143252047047649977/2471988 34682622733680, a[11,10] = -845624320000/1531259401089, a[4,2] = 0, a[ 5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[9,5] = -26247632715714732966326831 7632986112/299801039751138545315942419580078125-2459704493997354551957 270578321439936522112/2162787185842694801300107310532152099609375*a[8, 7], a[9,6] = 680274018217754802615237904530015016/16282526994557884856 10506294873046875+118051687477115850770641843641549601646922/219566676 251252787077452886245482177734375*a[8,7], a[9,4] = -502294453404303308 853524229913961/130225322928489017117203750000000+81525153928407755463 5364051662025117688415571/12653428769655765927625617657456378125000000 0*a[8,7], a[9,1] = -13065812676765907847765267699811/52565380505383583 723758750000000+602087303360619801630327730141720609670961/83730321555 8750159223717178681531250000000*a[8,7], a[10,1] = -3030255972838314131 95771246325112087992057649/4505734305486155149185575426967471380800000 00*a[8,7]+3271711353957440009060853580969749473/8667860008354218618485 479358496000000, a[11,7] = -5811078648183956835870888149494861200/4253 08484539747511978238662224596763+1855106031658227496118552493/82573890 726237436739317181*a[8,7], `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*` [5] = 0, a[5,3] = -41275080377096907/17778381694173184, c[11] = 1, c[1 2] = 1, a[10,9] = 1285791152889375/2245412845835068, a[3,2] = 640/3249 , a[12,11] = -98471/7223040, b[1] = 125174203/2508655500, a[10,7] = 30 91062216221538842660921549436591/503627237414101661284766440000000*a[8 ,7]-181411728433522712513358955747884512127/56684036144962998675181483 264393600000, `b*`[11] = 0, a[7,4] = 31153019468948131/297713925429454 536, a[9,7] = 246460913039148240547744185473827/6546691899510201393941 9375000000-6141690111220283584987447668399/935893412126746488851562500 000*a[8,7], a[5,1] = 10879382999084569/17778381694173184, a[11,1] = 48 200881476162306379625971591891/32299273242572711686660426716600-181861 306549574578407288063123415191027/738752761055999754883111142545463610 92*a[8,7], a[11,4] = 3438952521757345724100377600797/24620962126255184 6487324872952-246247860191053249291362195219332661846297/1116412222801 5885422866630879272587612708*a[8,7], a[11,5] = 29751067600489707004494 739502821376/9069068754534716527351654194729945+1997867993399483445343 18947694002941952/51313680137545125593768144890075629785*a[8,7], a[10, 5] = 11053116788091617894915746521089821705917384/10391508200755354649 528058865885113394140625*a[8,7]+32503310618234572391238775719607387072 /47534821945284281551609311321854671875, b[6] = 56446289600512/3426060 12403635, b[7] = 62143252047047649977/247198834682622733680, a[7,5] = \+ -3016545448677376/81418799560147215, b[10] = 13212880000/87750675837, \+ a[8,5] = 12815779106816/73949863360715*a[8,7]+2196255160461824000/2465 0994681649607391, a[12,1] = 125174203/2508655500, c[2] = 1/20, c[4] = \+ 4/19, c[6] = 37/56, c[7] = 61/367, c[8] = 25/57, c[9] = 39/50, c[10] = 19/20, a[8,1] = -98032836639/894656597332*a[8,7]+464948078675/7380742 565304, a[8,6] = -615083784896/7507407947315*a[8,7]-527229416000/18580 395726279, a[5,4] = 20050852951463513/8889190847086592, a[4,1] = 1/19, `b*`[10] = 498401087642132308000/3863258097409608192999, a[9,8] = 945 612121233663016296797907/598971783761117752865000000, `b*`[9] = 123545 0293578257031250/7631363446505532525519, `b*`[6] = 2238588667159615561 0112/158772260638005081292791, a[8,4] = -132740585168029/1352016010124 68*a[8,7]+108099948125/343301587704, a[2,1] = 1/20, a[10,8] = -1649384 5602934685528675361413563797/28001674400224052367433014064000000, `b*` [7] = 537978856569572663706487529/2159330585189973467574291759, a[11,9 ] = 165638979000000000/39827589068734103, b[11] = -98471/7223040, a[11 ,6] = -48502378223249528686115831379127226816/822560475705585509695924 5808290236745-9588618817947406448833751399321819712/520937717209138915 6003566088789263185*a[8,7], a[12,9] = 5373710937500/40041548453907, a[ 12,8] = 9636489994384107/36666550775008000, a[6,5] = 3657883862/133454 83011, a[12,10] = 13212880000/87750675837, a[11,8] = -1729758243576400 3600121633126256/9677847661048496307295108424975, a[7,1] = 15847549980 1357/1970033827325064, c[3] = 8/57, a[7,6] = 150256844634176/826565614 9993815, a[3,1] = -184/3249, c[5] = 44881/82208, a[6,1] = 5533979/9129 5232\}:" }{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 65 "The coefficients for the embedded ord er 6 scheme are as follows. " }}{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 0 477 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7)] ));\nfor ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e8,c[i i])); \n for jj to ii-1 do\n coeff_val := subs(e8,a[ii,jj]);\n \+ if coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a param eter.`)\n else\n print(a[ii,jj]=coeff_val);\n end if ;\n end do:\nend do:\nprint(``);\nc[12]=subs(e8,c[12]);\nseq(a[12,i] =subs(e8,a[12,i]),i=1..11);\nprint(``);\nseq(`b*`[i]=subs(e8,`b*`[i]), i=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\" #?F(%!GF+F+F+F+F+7*#\"\")\"#d#!$%=\"%\\K#\"$S'F2F+F+F+F+F+7*#\"\"%\"#> #F)F8\"\"!#\"\"$F8F+F+F+F+7*#\"&\")[%\"&3A)#\"2pX3**HQz3\"\"2%=tTp\"Qy x\"F:#!22p4x.3v7%FC#\"28NY^H&30?\"1#f'3Z3>*)))F+F+F+7*#\"#P\"#c#\"(zR` &\")K_H\"*F:F:#\",*Hq[&G\"\",[kLJ%R#\"+iQ)yl$\",6I[XL\"F+F+7*#\"#h\"$n $#\"0d8!)*\\v%e\"\"1k]KFQ.q>F:F:#\"2J\"[*o%>I:J\"3OXXHa#Rr(H#!1wtn[aa; I\"2:s9g&*z=9)#\"0wTjWoD]\"\"1:Q**\\hll#)F+Q)pprint146\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\")#\"#D\"#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\" ,&*&#\",Rm$G.)*\"-KtfcY*)F(&F%6$F'\"\"(F(!\"\"#\"-v'y![\\Y\"./`cU2Q(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\"0H!o^eSF8\"0oC,,;?N\"\"\"\" &F%6$F'\"\"(F.!\"\"#\"-D\"[**43\"\"-/xe,LMF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\"/;o5zd\"G\"\"/:2Oj)\\R(\"\"\" &F%6$F'\"\"(F.F.#\"4+S#=Yg^D'>#\"5\"R2'\\;o%*4lCF." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"-'*[y$3:'\".:t%zS2v\"\"\"&F %6$F'\"\"(F.!\"\"#\"-+gTHs_\"/zis&R!e=F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"*#\"#R\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\",& #\"A6)*pn_wZy!fwwE\"e18\"A+++](ePs$e$Q00QlD&!\"\"*&#\"Kh4n41sT,tF.j,)> 1O.t3-'\"K+++]7`\"oyrrB#f,veb@.t$)F(&F%6$\"\")\"\"(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,&#\"BhR\"*HU_`)3LI/MX%H-&\"B+++]P?C%fJX&Q6vR5!)*H!\"\"*&#\"L7@_O*R 9Ky0Fd>bat*R\\/(fC\"Lv$4'*4_@`5t5+8![pUe=(yi@\"\"\"&F%6$\"\")\"\"(F2F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&#\"E;],IX!zB: E![v<#=SF!o\"Fvo/t[H10h&[)yb%*p_#G;\"\"\"*&#\"KApk,'\\:kV=kq2&e6xuo^!= \"\"KvVtx@[Xi)GXxqy_7Dwmc>#F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"BFQZ&=WxaS#[\"RI\"4YY#\"A+++v$> %RR,-^**=pYl\"\"\"*&#\"@*RowW()\\e$G?76!pTh\"?++]i:&))[Yn77M*e$*F-&F%6 $\"\")F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\") #\"<2zz'H;ImL777c%*\"<+++lGv<6w$yr*)f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#>\"#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",&*&#\"N\\w0#*z37^K Y7x&>89$QG(fDII\"N+++!3Qru'pUvb=\\^:'[0Vt0XF(&F%6$\"\")\"\"(F(!\"\"#\" Ft%\\(p4e`314+WdRN66f\"R& y*oG(=\")374o\"\"\"&F%6$\"\")\"\"(F.!\"\"#\"FJ\"=?l;-w^OfOs!QB#Q'>\"\" E++g&=W8h\"=,=bW/5l.LF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"&,&*&#\"M%QdxQ7RsXB=1J.D$\"Gv=na=K6$4;b\"G%GX >#[`ZF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&*&#\"L2 Vq:UO8')*yCwkE!zr[%*))QC%\"L+]7GX'p^muE&QY@<,=L(eR%)\"\"\"&F%6$\"\")\" \"(F.!\"\"#\"J8=TXuJq=fS2c;A&>P#4E[\"\"J]7G<`I33ZYQ[#4aoX!eU(GF3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&*&#\"C\"fO%\\:#4mU )Q:A;A1\"4$\"B+++Skw%Gh;59uBFO]\"\"\"&F%6$\"\")F(F.F.#\"HF@^%)yub*eL^7 F_L%G:M7j!)G2%yXd\\lIh==\"G#4hjaaU66$)[v**f0hFvQ(F(&F% 6$\"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,&#\"@(z+wP+TsXtv@D &*QM\"?_H([K([Y=bi7i4iC\"\"\"*&#\"K(HY=mK$>_>i8H\\K0\">gyCY#\"J3Fh(es# z3jmGU&)e,GA7k6\"F-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&,&#\"Dw8#G]RZ\\/qq*[+w1^(H\"CX*HZ>a;N FlrMXvo!p!*\"\"\"*&#\"H_>%H+%pZ*=V`W$[*R$*z'y*>\"G&yHc2!*[9oPfD^aP,o88 &F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"',&#\"G;oAF\"z8$e6'oG&\\KAyB][\"FXnB!H3eCfp4be0dZgD#)!\"\"*&#\"F7 (>=K*R^P$)[kSZz\")=')e*\"F&=j#*y)3mN+c\"*Q\"4srP4_\"\"\"&F%6$\"\")\"\" (F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"F+7'[ \\\\\"))3(e$o&R=['y5\"e\"EjnfCAmQ#y>^Z(RX[3`U!\"\"*&#\"=$\\_&=h\\F#e;. 1^&=\";\"=<$RnVPis!*Qd#)\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#!Aci7L;7+O+kdV#e(H<\"@v\\U3^H2j\\[5m Zyn*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"3++++!z*Q c;\"2.Tto!*eF)R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#! -++KCc%)\".*3,%f7`\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"aG6$\"#7\"\"\"#\"*.UZ#/&F%6$F'\"\")# \"12TQ%***[O'*\"2+!3]x]lmO/&F%6$F'\"\"*#\".+v$4rt`\"/2RX[:/S/&F%6$F'\" #5#\",++)G@8\",Pen]x)/&F%6$F'\"#6#!&r%)*\"(SIA(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"# \"3*)>)\\)oJxd=\"4]*f5C7\\[\"o$/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\" %F//&F%6#\"\"&F//&F%6#\"\"'#\"87,hbhfrm)eQA\"9\"z#H\"30!Q1Es(e\"/&F%6# \"\"(#\"b_Kb]YMOJw/&F%6#\"# 5#\"6+!3B8Uw3,%)\\\"7**H>3'4u4eK'Q/&F%6#\"#6F//&F%6#\"#7#!*vZ>u#\"-\"o #=w?8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expa nded')),op(OrderConditions(7,11,'expanded'))]:\n`RK6_12eqs*` := subs(b =`b*`,OrderConditions(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 130 "simplify(subs(e8,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u ),%);\nnops(%);\nsimplify(subs(e8,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u ),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "We choose a value for " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 85 " that makes the principal error \+ norm for the order 7 scheme approximately a minimum." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "errterms \+ := PrincipalErrorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(err terms) do\n sm := sm+(evalf(subs(e8,errterms[ct])))^2;\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot(sqrt(sm),a[8,7]=0..0.35,color=COLOR(RGB,.5,0,1),thickness=2,f ont=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 277 277 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$\"\"!F)$\"3CJ;r4s(H)=!#A7$$\"3[KLLeR+H w!#?$\"3s7v!GWP:'=F,7$$\"3lm;/^fpE9!#>$\"3'[[MYitM%=F,7$$\"3?LL3xM?t@F 6$\"3M#=9W1PQ#=F,7$$\"3FLLeR$fY#HF6$\"3\"3'y\"R<=[!=F,7$$\"3Sm;ajOasOF 6$\"3'p.[.Memy\"F,7$$\"3#GL3x;GfO%F6$\"3]0T?:v_qy_Qn\"F,7$$\"3*****\\iQ!=C5!#=$\"3wZAZx>mj;F,7$$\"3=L 3FkX^!4\"F]p$\"3'z'Remk@b;F,7$$\"3amm\"zJ#Rp6F]p$\"3/Q#\\7o#>Y;F,7$$\" 3hmmm@@@O7F]p$\"3N*[5n0F%R;F,7$$\"3s*\\P%Q%RRJ\"F]p$\"3zpI9e^eK;F,7$$ \"3Umm;%GTFQ\"F]p$\"3iaZb-@YF;F,7$$\"3')*\\PM\"yAe9F]p$\"35Rrgw(fGi\"F ,7$$\"3))*\\7.%)3,`\"F]p$\"3f#40Y%eZ>;F,7$$\"3_m;/^\"4^g\"F]p$\"3Z'3r* e-*ph\"F,7$$\"3mmT&)[G)Rn\"F]p$\"3T\">t`0_ch\"F,7$$\"3CL$ekVs#[F]p$\"3R@.Zl,,?;F,7$$\"3m****\\Z26S?F] p$\"3hkGV6GuB;F,7$$\"3e**\\(=%[V8@F]p$\"3\\sb\"4(yTG;F,7$$\"3%)*\\PMnz V=#F]p$\"3!Hu,AX'*Qj\"F,7$$\"3u**\\78=:jAF]p$\"3F@_Va&o5k\"F,7$$\"34mm ;%3KRL#F]p$\"3izLj2'y%[;F,7$$\"3))***\\780&4CF]p$\"3!)R&[V-&Qd;F,7$$\" 3!H$3FWb)zZ#F]p$\"3=ufY!>Hjm\"F,7$$\"3%)***\\Ps_Gb#F]p$\"3'\\ZmHvTqn\" F,7$$\"3-mT50pHBEF]p$\"33ILqdj*zo\"F,7$$\"3))*\\(=s8$pp#F]p$\"3P@*)=3d L+1*4KF]p$\"3b;:F#e+)4=F,7$$\"3/LLL=2DzKF]p $\"3SHxs(p5v#=F,7$$\"3')*\\P%Qk=`LF]p$\"3coVM?C4Z=F,7$$\"3j*\\ilN_RU$F ]p$\"3e&pyV\")*[m=F,7$$\"3w*************\\$F]p$\"3w6(*RFk-))=F,-%+AXES LABELSG6$Q'a[8,7]6\"Q!F^[l-%*THICKNESSG6#\"\"#-%%FONTG6$%*HELVETICAG\" \"*-%&COLORG6&%$RGBG$\"\"&!\"\"F($\"\"\"F)-%%VIEWG6$;F($\"#N!\"#%(DEFA ULTG" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "evalf[12](minimize(sqrt(sm),a[8,7]=0..1,'location')) ;\nconvert(subs(op(1,op(op(2,[%]))),a[8,7]),rational,4);\nconvert(subs (op(1,op(op(2,[%%]))),a[8,7]),rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-^j+A:;!#;<#7$<#/&%\"aG6$\"\")\"\"($\"-eSd>T " 0 "" {MPLTEXT 1 0 75 "evalf(eval(sqrt(sm),a[8,7]=35/201)) ;\nevalf(eval(sqrt(sm),a[8,7]=183/1051));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l+A:;!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i+A :;!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "e9 := `union`(subs(a[8,7]=35/201,e8),\{a[8,7]=35/201 \}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4032 " e9 := \{a[12,6] = 56446289600512/342606012403635, a[11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0 , a[9,7] = 40769840728265525109900718286661374039/15549375265121654840 821192853125000000, a[8,2] = 0, a[9,3] = 0, a[9,2] = 0, b[2] = 0, a[10 ,3] = 0, a[11,2] = 0, a[8,1] = 2924917590401981135/6660502397034144055 2, a[10,6] = -115813448945099442948683847145831524761051670902703/1919 39521001616690365566010466177813315410464375000, a[9,5] = -31112530877 798875425923566022361029283298688/289813482902921103374214379611308381 34765625, a[9,4] = -2318859757393123763522345863545224823245606023/847 779727566936317150916383049577334375000000, a[9,6] = 15292107243270980 188039355727714753150306881614/299015121519358598059132593510319939890 13671875, b[8] = 9636489994384107/36666550775008000, a[10,7] = -286992 81021898330563484694969577298696532859/1346330884497088663033572999754 2445904000000, a[8,4] = 445876617351108665/3098009485599691752, a[11,7 ] = -197008013321007326029676937491820383240505/2020342894109162606150 2271171655020032789, a[9,1] = -422111859126024625640701660339501981620 12423/342205824198861190074733210927141821875000000, `b*`[8] = 3249856 2725533982078667/120108582798663659157650, a[10,1] = 53421576995301575 930555089451778526350776610941/205195002329815636484038978294464618799 958400000, a[11,4] = 6492689767645528199626511179748230475508923/64113 9590808912277141769373352511460044088, a[4,3] = 3/19, a[8,6] = -714989 7627279951488/167672519638445079477, a[11,1] = 89462997577799701289200 380185298019046438429/84108584887856406378631238816963629525265400, a[ 10,5] = 363012698911770750340098104804203047936446648/4177386296703652 56911027966408581558444453125, a[11,5] = 69407198604727672359456681339 3158946881536/175338845029991694153905751089388426975345, a[8,5] = 196 990339933642293248/1651616643670523695197, b[9] = 5373710937500/400415 48453907, a[11,6] = -187824233463716665067125996081330354271085997696/ 30211283627207706655231802682148684012056060045, `b*`[12] = -274194775 /132076182681, `b*`[1] = 185777316884981989/3681484912241059950, a[12, 2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[6,4] = 12854870299/39 431336448, a[12,7] = 62143252047047649977/247198834682622733680, a[11, 10] = -845624320000/1531259401089, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[5,3 ] = -41275080377096907/17778381694173184, c[11] = 1, c[12] = 1, a[10,9 ] = 1285791152889375/2245412845835068, a[3,2] = 640/3249, a[12,11] = - 98471/7223040, b[1] = 125174203/2508655500, `b*`[11] = 0, a[8,7] = 35/ 201, a[7,4] = 31153019468948131/297713925429454536, a[5,1] = 108793829 99084569/17778381694173184, b[6] = 56446289600512/342606012403635, a[1 0,4] = 502896807923589267114219976720637605496180343553/19551904244552 0666638376728307225007182256000000, b[7] = 62143252047047649977/247198 834682622733680, a[7,5] = -3016545448677376/81418799560147215, b[10] = 13212880000/87750675837, a[12,1] = 125174203/2508655500, c[2] = 1/20, c[4] = 4/19, c[6] = 37/56, c[7] = 61/367, c[8] = 25/57, c[9] = 39/50, c[10] = 19/20, a[5,4] = 20050852951463513/8889190847086592, a[4,1] = \+ 1/19, `b*`[10] = 498401087642132308000/3863258097409608192999, a[9,8] \+ = 945612121233663016296797907/598971783761117752865000000, `b*`[9] = 1 235450293578257031250/7631363446505532525519, `b*`[6] = 22385886671596 155610112/158772260638005081292791, a[2,1] = 1/20, a[10,8] = -16493845 602934685528675361413563797/28001674400224052367433014064000000, `b*`[ 7] = 537978856569572663706487529/2159330585189973467574291759, a[11,9] = 165638979000000000/39827589068734103, b[11] = -98471/7223040, a[12, 9] = 5373710937500/40041548453907, a[12,8] = 9636489994384107/36666550 775008000, a[6,5] = 3657883862/13345483011, a[12,10] = 13212880000/877 50675837, a[11,8] = -17297582435764003600121633126256/9677847661048496 307295108424975, a[7,1] = 158475499801357/1970033827325064, c[3] = 8/5 7, a[7,6] = 150256844634176/8265656149993815, a[3,1] = -184/3249, c[5] = 44881/82208, a[6,1] = 5533979/91295232\}:" }{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 62 "The c oefficients for the combined scheme are given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1011 "subs(e9,matrix([[c[2],a[2,1],``], \n[c[3],a[3,1],a[3,2]],[c[4],a[4,1],a[4,2]],[``$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] ],[``$2,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]],[` `$2,a[8,7]],\n[c[9],a[9,1],a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6 ]],[``,a[9,7],a[9,8]],\n[c[10],a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[ ``,a[10,5],a[10,6]],\n[``,a[10,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[1 1,1],a[11,2]],[``,a[11,3],a[11,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7], a[11,8]],[``,a[11,9],a[11,10]],\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a [12,4]],[``,a[12,5],a[12,6]],\n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,1 0]],[``$2,a[12,11]],\n[``,`________________________________`$2],[`b`,b [1],b[2]],[``,b[3],b[4]],[``,b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10 ]],[``$2,b[11]],\n[`b*`,`b*`[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*` [5],`b*`[6]],\n[``,`b*`[7],`b*`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11] ,`b*`[12]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\" \"\"\"#?F(%!G7%#\"\")\"#d#!$%=\"%\\K#\"$S'F27%#\"\"%\"#>#F)F8\"\"!7%F+ F+#\"\"$F87%#\"&\")[%\"&3A)#\"2pX3**HQz3\"\"2%=tTp\"Qyx\"F:7%F+#!22p4x .3v7%FD#\"28NY^H&30?\"1#f'3Z3>*)))7%#\"#P\"#c#\"(zR`&\")K_H\"*F:7%F+F: #\",*Hq[&G\"\",[kLJ%R7%F+F+#\"+iQ)yl$\",6I[XL\"7%#\"#h\"$n$#\"0d8!)*\\ v%e\"\"1k]KFQ.q>F:7%F+F:#\"2J\"[*o%>I:J\"3OXXHa#Rr(H7%F+#!1wtn[aa;I\"2 :s9g&*z=9)#\"0wTjWoD]\"\"1:Q**\\hll#)7%#\"#DF/#\"4N6)>S!f<\\#H\"5_0WT. (R-0m'F:7%F+F:#\"3l'36N\"7(>&p B0nVmh^;#!4)[^*zsi(*)\\r\"6x%z]WQ'>Dnn\"7%F+F+#\"#N\"$,#7%#\"#R\"#]#!M BC,i\")>]R.m,2kDY-E\"f=6A%\"N+++v=#=9F4@LZ2!>h))>Ce?U$F:7%F+F:#!OBggXK #[AXN'eMANwBJRd(f)=B\"N+++vVLx&\\IQ;4:.^$fK\"f!)fe$>:7:!*H7%F+#\"GRSPhmG=2!*4^_l#G2%)p2%\"G+++D J&G>@3%[l@^Ev$\\b\"#\"<2zz'H;ImL777c%*\"<+++lGv<6w$yr*)f7%#F8F*#\"PT4h w2NE&y0#F:7%F+F:#\"Q`NM !='\\0wj?n(*>U6n#*eBz!o*G]\"Q+++cA=2]A2$GnPQmm?bWU!>b>7%F+#\"N[mWOz/.U ![5)4S.vq<\"*)p7IO\"NDJXW%e:e3k'z-6pDl.nH'QxT#!T.F!4n^5wC:$e9ZQo[HW*4X *[M\"e6\"T+]Pk/T:L\"y>7%F+#!MfG`'p)Hx&p\\p%[j0L)*=-\"G *pG\"M+++/fWUv**HdLIm)3(\\%)3LY8#!D(zjNTh`nGboMHgXQ\\;\"D+++kS,LuO_SA+ Wn,!G7%F+F+#\"1v$*)G:\"z&G\"\"1o]$e%GTXA7%F)#\"MH%Qk/>!)H&=!Q+#*G,(*zx v*HY*)\"M+aED&HO'p\")Q7jyjScy)[e3T)F:7%F+F:#\"LB*3bZI#[(z6^E'*>Gbkn(*o #\\'\"K)3W+Y6DNt$p<9xA\"*33fR6k7%F+#\"KO:)o%*eJR8oc%fBnFZg)>2%p\"KX`(p U)Q*3^d!R:%p\"**H]%)Q`<#!Q'p(*f3rUNI83'*f7n]m;PYLU#y=\"PX+1c?,%o[@o-=B bmq2si$G6-$7%F+#!K00C$Q?=\\PpnHgK25K8!3q>\"J*yK+-b;Z##\"12TQ%***[O'*\"2+!3] x]lmO7%F+#\".+v$4rt`\"/2RX[:/S#\",++)G@8\",Pen]x)7%F+F+#!&r%)*\"(SIA(7 %F+%A________________________________GFhw7%%\"bGF]vF:F`vFavFevF\\wFcw7 %%#b*G#\"3*)>)\\)oJxd=\"4]*f5C7\\[\"o$F:F`v7%F+F:#\"87,hbhfrm)eQA\"9\" z#H\"30!Q1Es(e\"7%F+#\"b_Kb]YMOJw#\"6+!3B8Uw3, %)\\\"7**H>3'4u4eK'Q7%F+F:#!*vZ>u#\"-\"o#=w?8Q)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "s ubs(e9,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[` b`,seq(b[j],j=1..11),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[6](%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'++]!\"(F(%!GF+ F+F+F+F+F+F+F+F+F+7/$\"'^.9!\"'$!'GjcF*$\"'%)p>F/F+F+F+F+F+F+F+F+F+F+7 /$\"'E0@F/$\"';j_F*$\"\"!F:$\"'&*y:F/F+F+F+F+F+F+F+F+F+7/$\"'WfaF/$\"' W>hF/F9$!'k@B!\"&$\"'kbAFDF+F+F+F+F+F+F+F+7/$\"'92mF/$\"'jhgF*F9F9$\"' 1gKF/$\"'#4u#F/F+F+F+F+F+F+F+7/$\"'8i;F/$\"'IW!)F*F9F9$\"'TY5F/$!'(\\q $F*$\"'&y\"=F*F+F+F+F+F+F+7/$\"''fQ%F/$\"'W\"R%F*F9F9$\"'CR9F/$\"'r#> \"F/$!'?kUF*$\"'HTi#FD$\"'ty:FDF+F+F+F+7/$\"'++&*F/$\"'X.EF/F9F9$\"'6sDFD $\"'&**o)F/$!'&Q.'F/$!'nJ@FD$!'J!*eF/$\"'IEdF/F+F+F+7/$\"\"\"F:$\"'mj5 FDF9F9$\"'o75!\"%$\"'YeRFD$!'- " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------- ------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded 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 sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3835 "ee := \{c[2]=1/20,\nc[3]=8/57,\nc[4]=4/19,\nc[5]=44 881/82208,\nc[6]=37/56,\nc[7]=61/367,\nc[8]=25/57,\nc[9]=39/50,\nc[10] =19/20,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/20,\na[3,1]=-184/3249,\na[3,2] =640/3249,\na[4,1]=1/19,\na[4,2]=0,\na[4,3]=3/19,\na[5,1]=108793829990 84569/17778381694173184,\na[5,2]=0,\na[5,3]=-41275080377096907/1777838 1694173184,\na[5,4]=20050852951463513/8889190847086592,\na[6,1]=553397 9/91295232,\na[6,2]=0,\na[6,3]=0,\na[6,4]=12854870299/39431336448,\na[ 6,5]=3657883862/13345483011,\na[7,1]=158475499801357/1970033827325064, \na[7,2]=0,\na[7,3]=0,\na[7,4]=31153019468948131/297713925429454536,\n a[7,5]=-3016545448677376/81418799560147215,\na[7,6]=150256844634176/82 65656149993815,\na[8,1]=2924917590401981135/66605023970341440552,\na[8 ,2]=0,\na[8,3]=0,\na[8,4]=445876617351108665/3098009485599691752,\na[8 ,5]=196990339933642293248/1651616643670523695197,\na[8,6]=-71498976272 79951488/167672519638445079477,\na[8,7]=35/201,\na[9,1]=-4221118591260 2462564070166033950198162012423/34220582419886119007473321092714182187 5000000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-231885975739312376352234586354 5224823245606023/847779727566936317150916383049577334375000000,\na[9,5 ]=-31112530877798875425923566022361029283298688/2898134829029211033742 1437961130838134765625,\na[9,6]=15292107243270980188039355727714753150 306881614/29901512151935859805913259351031993989013671875,\na[9,7]=407 69840728265525109900718286661374039/1554937526512165484082119285312500 0000,\na[9,8]=945612121233663016296797907/598971783761117752865000000, \na[10,1]=53421576995301575930555089451778526350776610941/205195002329 815636484038978294464618799958400000,\na[10,2]=0,\na[10,3]=0,\na[10,4] =502896807923589267114219976720637605496180343553/19551904244552066663 8376728307225007182256000000,\na[10,5]=3630126989117707503400981048042 03047936446648/417738629670365256911027966408581558444453125,\na[10,6] =-115813448945099442948683847145831524761051670902703/1919395210016166 90365566010466177813315410464375000,\na[10,7]=-28699281021898330563484 694969577298696532859/13463308844970886630335729997542445904000000,\na [10,8]=-16493845602934685528675361413563797/28001674400224052367433014 064000000,\na[10,9]=1285791152889375/2245412845835068,\na[11,1]=894629 97577799701289200380185298019046438429/8410858488785640637863123881696 3629525265400,\na[11,2]=0,\na[11,3]=0,\na[11,4]=6492689767645528199626 511179748230475508923/641139590808912277141769373352511460044088,\na[1 1,5]=694071986047276723594566813393158946881536/1753388450299916941539 05751089388426975345,\na[11,6]=-18782423346371666506712599608133035427 1085997696/30211283627207706655231802682148684012056060045,\na[11,7]=- 197008013321007326029676937491820383240505/202034289410916260615022711 71655020032789,\na[11,8]=-17297582435764003600121633126256/96778476610 48496307295108424975,\na[11,9]=165638979000000000/39827589068734103,\n a[11,10]=-845624320000/1531259401089,\na[12,1]=125174203/2508655500,\n a[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=56446289600512 /342606012403635,\na[12,7]=62143252047047649977/247198834682622733680, \na[12,8]=9636489994384107/36666550775008000,\na[12,9]=5373710937500/4 0041548453907,\na[12,10]=13212880000/87750675837,\na[12,11]=-98471/722 3040,\n\nb[1]=125174203/2508655500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0 ,\nb[6]=56446289600512/342606012403635,\nb[7]=62143252047047649977/247 198834682622733680,\nb[8]=9636489994384107/36666550775008000,\nb[9]=53 73710937500/40041548453907,\nb[10]=13212880000/87750675837,\nb[11]=-98 471/7223040,\n\n`b*`[1]=185777316884981989/3681484912241059950,\n`b*`[ 2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=2238588667159615561 0112/158772260638005081292791,\n`b*`[7]=537978856569572663706487529/21 59330585189973467574291759,\n`b*`[8]=32498562725533982078667/120108582 798663659157650,\n`b*`[9]=1235450293578257031250/763136344650553252551 9,\n`b*`[10]=498401087642132308000/3863258097409608192999,\n`b*`[11]=0 ,\n`b*`[12]=-274194775/132076182681\}:" }}}{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[7 ,11];" "6#&%\"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose \+ components are the principal error terms of the 11 stage, order 7 sche me (the error terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "`T*`[6, 12];" "6#&%#T*G6$\"\"'\"#7" }{TEXT -1 146 " \+ denote the vector whose components are the principal error terms of t he embedded 12 stage, order 6 scheme (the error terms of order 7) and \+ let " }{XPPEDIT 18 0 "`T*`[7,12];" "6#&%#T*G6$\"\"(\"#7" }{TEXT -1 99 " denote the vector whose components are the error terms of order \+ 8 of the embedded order 6 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[7, 11]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"# 6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,12]));" "6#-%$absG 6#-F$6#&%#T*G6$\"\"'\"#7" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(ab s(`T*`[7,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\"#7" }{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[8 ] = abs(abs(T[7, 11]));" "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"( \"#6" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,12])) /abs(abs(`T*`[6, 12]));" "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\" (\"#7\"\"\"-F*6#-F*6#&F/6$\"\"'F2!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7,12]-T[7,11]))/abs(abs(`T*`[6,12]) );" "6#/&%\"CG6#\"\")*&-%$absG6#-F*6#,&&%#T*G6$\"\"(\"#7\"\"\"&%\"TG6$ F2\"#6!\"\"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[8];" "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embed ded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "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 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := PrincipalErrorTerms(7,11,'exp anded'):\n`errterms7_12*` :=subs(b=`b*`,PrincipalErrorTerms(7,12,'expa nded')):\n`errterms6_12*` := subs(b=`b*`,PrincipalErrorTerms(6,12,'exp anded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms7_12*`[i]) )^2,i=1..nops(`errterms7_12*`))):\nsdnB := sqrt(add(evalf(subs(ee,`err terms6_12*`[i]))^2,i=1..nops(`errterms6_12*`))):\nsnmC := sqrt(add((ev alf(subs(ee,`errterms7_12*`[i])-subs(ee,errterms7_11[i])))^2,i=1..nops (errterms7_11))):\n'B[8]'= evalf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC /sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")c$o]\"!\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")8z,:!\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "co efficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3835 "ee := \{c[2]=1/20,\nc[3]=8 /57,\nc[4]=4/19,\nc[5]=44881/82208,\nc[6]=37/56,\nc[7]=61/367,\nc[8]=2 5/57,\nc[9]=39/50,\nc[10]=19/20,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/20,\n a[3,1]=-184/3249,\na[3,2]=640/3249,\na[4,1]=1/19,\na[4,2]=0,\na[4,3]=3 /19,\na[5,1]=10879382999084569/17778381694173184,\na[5,2]=0,\na[5,3]=- 41275080377096907/17778381694173184,\na[5,4]=20050852951463513/8889190 847086592,\na[6,1]=5533979/91295232,\na[6,2]=0,\na[6,3]=0,\na[6,4]=128 54870299/39431336448,\na[6,5]=3657883862/13345483011,\na[7,1]=15847549 9801357/1970033827325064,\na[7,2]=0,\na[7,3]=0,\na[7,4]=31153019468948 131/297713925429454536,\na[7,5]=-3016545448677376/81418799560147215,\n a[7,6]=150256844634176/8265656149993815,\na[8,1]=2924917590401981135/6 6605023970341440552,\na[8,2]=0,\na[8,3]=0,\na[8,4]=445876617351108665/ 3098009485599691752,\na[8,5]=196990339933642293248/1651616643670523695 197,\na[8,6]=-7149897627279951488/167672519638445079477,\na[8,7]=35/20 1,\na[9,1]=-42211185912602462564070166033950198162012423/3422058241988 61190074733210927141821875000000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-23188 59757393123763522345863545224823245606023/8477797275669363171509163830 49577334375000000,\na[9,5]=-311125308777988754259235660223610292832986 88/28981348290292110337421437961130838134765625,\na[9,6]=1529210724327 0980188039355727714753150306881614/29901512151935859805913259351031993 989013671875,\na[9,7]=40769840728265525109900718286661374039/155493752 65121654840821192853125000000,\na[9,8]=945612121233663016296797907/598 971783761117752865000000,\na[10,1]=53421576995301575930555089451778526 350776610941/205195002329815636484038978294464618799958400000,\na[10,2 ]=0,\na[10,3]=0,\na[10,4]=50289680792358926711421997672063760549618034 3553/195519042445520666638376728307225007182256000000,\na[10,5]=363012 698911770750340098104804203047936446648/417738629670365256911027966408 581558444453125,\na[10,6]=-1158134489450994429486838471458315247610516 70902703/191939521001616690365566010466177813315410464375000,\na[10,7] =-28699281021898330563484694969577298696532859/13463308844970886630335 729997542445904000000,\na[10,8]=-16493845602934685528675361413563797/2 8001674400224052367433014064000000,\na[10,9]=1285791152889375/22454128 45835068,\na[11,1]=89462997577799701289200380185298019046438429/841085 84887856406378631238816963629525265400,\na[11,2]=0,\na[11,3]=0,\na[11, 4]=6492689767645528199626511179748230475508923/64113959080891227714176 9373352511460044088,\na[11,5]=6940719860472767235945668133931589468815 36/175338845029991694153905751089388426975345,\na[11,6]=-1878242334637 16665067125996081330354271085997696/3021128362720770665523180268214868 4012056060045,\na[11,7]=-197008013321007326029676937491820383240505/20 203428941091626061502271171655020032789,\na[11,8]=-1729758243576400360 0121633126256/9677847661048496307295108424975,\na[11,9]=16563897900000 0000/39827589068734103,\na[11,10]=-845624320000/1531259401089,\na[12,1 ]=125174203/2508655500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0 ,\na[12,6]=56446289600512/342606012403635,\na[12,7]=621432520470476499 77/247198834682622733680,\na[12,8]=9636489994384107/36666550775008000, \na[12,9]=5373710937500/40041548453907,\na[12,10]=13212880000/87750675 837,\na[12,11]=-98471/7223040,\n\nb[1]=125174203/2508655500,\nb[2]=0, \nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=56446289600512/342606012403635,\nb[7 ]=62143252047047649977/247198834682622733680,\nb[8]=9636489994384107/3 6666550775008000,\nb[9]=5373710937500/40041548453907,\nb[10]=132128800 00/87750675837,\nb[11]=-98471/7223040,\n\n`b*`[1]=185777316884981989/3 681484912241059950,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n` b*`[6]=22385886671596155610112/158772260638005081292791,\n`b*`[7]=5379 78856569572663706487529/2159330585189973467574291759,\n`b*`[8]=3249856 2725533982078667/120108582798663659157650,\n`b*`[9]=123545029357825703 1250/7631363446505532525519,\n`b*`[10]=498401087642132308000/386325809 7409608192999,\n`b*`[11]=0,\n`b*`[12]=-274194775/132076182681\}:" }}} {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 78 "The stability function R for the 11 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,Stabili tyFunction(7,11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,:\"\"\"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)*&#F)\"%S]F)*$)F '\"\"(F)F)F)*&#\"L`sSyVLl>zW2CMB!3KjuZ>U\"Q++%ebz.&R;)pv\"RymCW$p89&[0 $*F)*$)F'\"#5F)F)F)*&#\">p'3=dS(yenr>Yom\"\"E#RzV Axhw6Y'pvVN)4q/)F)*$)F'\"#6F)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#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=-1,z=-4.6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+\\4E>Y!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := new ton(R(z)=-1,z=-4.6):\np1 := plot([R(z),-1],z=-5.19..0.49,color=[red,bl ue]):\np2 := plot([[[z0,-1]]$3],style=point,symbol=[circle,cross,diamo nd],color=black):\np3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR (RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-5.19..0.49,-1.47..1.4 7],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7V7$$!3Q++++++!>&!#<$!3!y$>WLXJ*G$F*7$$!3Yn m;#3'4G^F*$!33HU^M(Q/\"HF*7$$!3kLLLk@>m]F*$!3iNzvOVMrDF*7$$!3E+]U'*)HB ,&F*$!3`3j$f6/dI#F*7$$!3!pm;&GwYe\\F*$!3-W8&4s/]1#F*7$$!3s+](\\(Q*y*[F *$!3D-'4(R,Z@=F*7$$!3nLLV@,KP[F*$!3O'=H'3T,/;F*7$$!3NLL.q&p`r%F*$!3\\x ]Z.bBN7F*7$$!3;nmJG')*Rf%F*$!35djAAP$=X*!#=7$$!3uLLyGAZ\"[%F*$!3lY!p( \\A\")=tFU7$$!3%3+])fw&\\O%F*$!3Yycqd'*RmbFU7$$!3$QL$)f7eWC%F*$!3_Q#pK 3\"=ZTFU7$$!3A++lN]MCTF*$!3G\"42&*z&H\\IFU7$$!3ummYeRz+SF*$!3AqO;)e*pz @FU7$$!3_LLV-,(>*QF*$!3kzYGazP&e\"FU7$$!35++S:-YpPF*$!3YQXDH[7m5FU7$$! 3K+++\"HZkk$F*$!3[I;'p(H$)zm!#>7$$!3;++gW:!z_$F*$!3$*4%ey^zos$Fhp7$$!3 hLL)*\\1D?MF*$!3_f2*edz:h\"Fhp7$$!3'ommSKVAH$F*$\"3K7I(4Hz(oQ!#?7$$!3/ nmEGV!Q=$F*$\"37(Ge6L_=w\"Fhp7$$!39++0(*RmdIF*$\"3Gc5\\kv,6JFhp7$$!39n mEI%3g%HF*$\"3Yd\"*e\\=dfTFhp7$$!3-++0xX]BGF*$\"3P)GB`z'*HA&Fhp7$$!3*) ***\\\"R>&oq#F*$\"39M'Gk^)4/iFhp7$$!3gmm;\\r8&e#F*$\"3^OAHx/+TsFhp7$$! 3ymmrw\\OtCF*$\"3-CXAk3OQ#)Fhp7$$!3SLL$))e.GN#F*$\"3gHm*Rc?NR*Fhp7$$!3 nLL)**=uvA#F*$\"35Wf=>B6r5FU7$$!3K++:I;c=@F*$\"3_&QHvg;z>\"FU7$$!31LL. z]#3+#F*$\"3?I:@38*)\\8FU7$$!3M++?,<>z=F*$\"3z_#\\+mre_\"FU7$$!3;++!4< (>gFU7$$!3H++q9zA<:F*$ \"3#QF'3Ka.$>#FU7$$!3EnmEY;O-9F*$\"3)GFIR'43gCFU7$$!3#)*****pQ<(z7F*$ \"3my(*\\k$H6y#FU7$$!3)RL$efMeo6F*$\"3AWD#)Hn03JFU7$$!3I****fAZ3Z5F*$ \"3@'p?ev*e4NFU7$$!3xqm;(zQwK*FU$\"3nyHf[dkMRFU7$$!3&z***\\)ecE8)FU$\" 3zhKg?k2MWFU7$$!3'3nmm0VV'pFU$\"3#[;AeE\"f$)\\FU7$$!3P)***\\iqATdFU$\" 3P1'RIT()>j&FU7$$!3aFLL*)4AjXFU$\"3TY46!>(4OjFU7$$!33LLLO'R&eLFU$\"3EZ e%Quus9(FU7$$!3Uim;`O$Q;#FU$\"3E%ea5qkU0)FU7$$!3?*****>$H-m5FU$\"3%[l+ $Q4$)))*)FU7$$\"3v*QLLU?>#>Fhp$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\"FU$\"3d 4+-(HH39\"F*7$$\"3E0+]HcUiJK;F*-%'COLOURG6&%$RG BG$\"*++++\"!\")$\"\"!Fa\\lF`\\l-F$6$7S7$F($!\"\"Fa\\l7$F3Ff\\l7$F=Ff \\l7$FGFf\\l7$FLFf\\l7$FQFf\\l7$FWFf\\l7$FfnFf\\l7$F[oFf\\l7$F`oFf\\l7 $FeoFf\\l7$FjoFf\\l7$F_pFf\\l7$FdpFf\\l7$FjpFf\\l7$F_qFf\\l7$FdqFf\\l7 $FjqFf\\l7$F_rFf\\l7$FdrFf\\l7$FirFf\\l7$F^sFf\\l7$FcsFf\\l7$FhsFf\\l7 $F]tFf\\l7$FbtFf\\l7$FgtFf\\l7$F\\uFf\\l7$FauFf\\l7$FfuFf\\l7$F[vFf\\l 7$F`vFf\\l7$FevFf\\l7$FjvFf\\l7$F_wFf\\l7$FdwFf\\l7$FiwFf\\l7$F^xFf\\l 7$FcxFf\\l7$FhxFf\\l7$F]yFf\\l7$FbyFf\\l7$FgyFf\\l7$F\\zFf\\l7$FazFf\\ l7$FfzFf\\l7$F[[lFf\\l7$F`[lFf\\l7$Fe[lFf\\l-Fj[l6&F\\\\lF`\\lF`\\lF] \\l-F$6&7#7$$!31+++\\4E>YF*Ff\\l-%'SYMBOLG6#%'CIRCLEG-Fj[l6&F\\\\lFa\\ lFa\\lFa\\l-%&STYLEG6#%&POINTG-F$6&F\\`l-Fa`l6#%&CROSSGFd`lFf`l-F$6&F \\`l-Fa`l6#%(DIAMONDGFd`lFf`l-F$6%7$7$F^`lF`\\lF]`l-%&COLORG6&F\\\\lF` \\l$\"\"&Fg\\lF`\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Febl-% %FONTG6#%(DEFAULTG-Fhbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$>&!\"#$\"#\\Fe cl;$!$Z\"Fecl$\"$Z\"Fecl" 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 50 "T he following picture shows the stability region. " }}{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+1/120*z^5+1/720*z^6+1/5040*z^7+\n 421947 7463320802334240744791965334378407253/17054851413693442466783917569816 3950379555840000*z^8+\n 120078756371563200341377781859286350072342367 /43489871104918278290298989803031807346786739200000*z^9+\n 5957524976 74215431143806770225583918755471/9319258093911059633635497814935387288 597158400000*z^10+\n 16668461971675878740571808669/804700983543756964 611766177224379392*z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 280 d o\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, .1,.35,0)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.3,0]], i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.1,.75, 0)):\npts := []: z0 := 2.5+5.6*I:\nfor ct from 0 to 40 do\n zz := ne wton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re (zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.1,.35,0)):\np 4 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.48,5.65]],i=2..nops(p ts))],\n style=patchnogrid,color=COLOR(RGB,.1,.75,0)):\npts : = []: z0 := 2.5-5.6*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=e xp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.1,.35,0)):\np6 := plots[ polygonplot]([seq([pts[i-1],pts[i],[2.48,-5.65]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.1,.75,0)):\np7 := plot([[[- 5.39,0],[3.19,0]],[[0,-6.19],[0,6.19]]],color=black,linestyle=3):\nplo ts[display]([p||(1..7)],view=[-5.39..3.19,-6.19..6.19],font=[HELVETICA ,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constr ained);" }}{PARA 13 "" 1 "" {GLPLOT2D 395 487 487 {PLOTDATA 2 "6/-%'CU RVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F ($\"3<+++!)*)Q7ZF-7$$!3I+++2UH2]!#F$\"3/+++4`=$G'F-7$$!3W*****pfnPo(!# E$\"3!******zp\")R&yF-7$$!3))******en;>]!#D$\"36+++h&yZU*F-7$$!3)***** ***GB#y?!#C$\"31+++$yd&*4\"!#<7$$!3<+++?z^GgFI$\"33+++>'QmD\"FL7$$!3/+ ++q$H6=\"!#B$\"33+++AAs89FL7$$!3%)*****p@o-y)FI$\"3/+++gH\"3d\"FL7$$\" 3A+++p!>De%FU$\"3++++)p>zs\"FL7$$\"3/+++$4jLu#!#A$\"31+++1#e])=FL7$$\" 3'******H[wWn*F_o$\"39+++jHDU?FL7$$\"3%******fgDDt#!#@$\"35+++bh`*>#FL 7$$\"3J+++*[))H[IFep$\"3A+++K+3sEFL7$$\"3%)*******Q@Z!eFep$ \"3>+++R#[&HGFL7$$\"3'******z\\z].\"!#>$\"3?+++IDQ')HFL7$$\"36+++CO_H< Feq$\"3A+++(\\N;9$FL7$$\"39+++/@y*p#Feq$\"3/+++<\\z$H$FL7$$\"3@++++SU= RFeq$\"3;+++db*3W$FL7$$\"3w*****4Z9!p_Feq$\"3#)*****4.o4e$FL7$$\"3O+++ I8%Qc'Feq$\"37+++(R;Er$FL7$$\"3++++q*Q'*f(Feq$\"3))******)\\&HNQFL7$$ \"3I+++7>B5#)Feq$\"31+++ac9\\RFL7$$\"3;+++,dv)G)Feq$\"3M+++NvjaSFL7$$ \"3a+++N$4Xy(Feq$\"3++++,KH_TFL7$$\"3I+++)[#***o'Feq$\"3h*****HZcDC%FL 7$$\"3*******\\+Mw-&Feq$\"37+++^+xDVFL7$$\"34+++k00RGFeq$\"31+++s:?-WF L7$$\"35+++o5LowUsUuz%FL7$$ !3++++8_B!=$F-$\"3e*****H,jM#[FL7$$!39+++9G$>l$F-$\"3-+++.`/X[FL7$$!3& ******p*>jITF-$\"3m*****>6pB'[FL7$$!32+++'=)y9YF-$\"31+++y.fv[FL7$$!35 +++yh:.^F-$\"3++++y5%[)[FL7$$!3e*****f)fv%f&F-$\"3Q+++b#H-*[FL7$$!3u** ***pk\\))3'F-$\"3y*******=R=*[FL7$$!3%******pbH\\e'F-$\"35+++n:t*)[FL7 $$!3s*****p@>F3(F-$\"3;+++XN%R)[FL7$$!3]*****4,)=#e(F-$\"3!)*****Hs)[u [FL7$$!3M+++zFe$3)F-$\"3g*****>3d8'[FL7$$!3E+++Q/[(e)F-$\"30+++O^^W[FL 7$$!3O+++Yz'[4*F-$\"3'******\\\"o!R#[FL7$$!3[+++J\"esg*F-$\"3e*****f.c %*z%FL7$$!3********y^o75FL$\"3q******)zs5x%FL7$$!3++++^bnl5FL$\"3N+++i hmQZFL7$$!3,+++sa7?6FL$\"3p*****\\$*p@q%FL7$$!3)*******32hw6FL$\"3S+++ M1fhYFL7$$!3$*******)GteB\"FL$\"39+++I24q^\"FL$\"3w*****>H!)eT%FL7$$!3'*******R9N*f\"F L$\"3'******4!fWnVFL7$$!31+++U=Q%o\"FL$\"3G+++/hwAVFL7$$!35+++7@DqFL $\"3w*****4^05@%FL7$$!3'******RZ3)>?FL$\"3m*****\\7?)yTFL7$$!3!)*****R o6#)4#FL$\"3w*****pi?y9%FL7$$!3?+++&3RR<#FL$\"3\")*****RNht6%FL7$$!3'* *****\\@[qC#FL$\"3O+++O<$p3%FL7$$!3$*******fZlX#FL$\"3!)*****zt-B*RF L7$$!35+++hH!f^#FL$\"39+++pB&)eRFL7$$!31+++'4^zd#FL$\"39+++z;=CRFL7$$! 3&)*****Hr8#QEFL$\"3A+++(*Q=))QFL7$$!3#)*******)*>op#FL$\"3;+++Lmx]QFL 7$$!3++++!R$*Qv#FL$\"3********po*=\"QFL7$$!3))*****z%Hb4GFL$\"37+++`v \\rPFL7$$!3++++>Y\"R'GFL$\"3&******pXX&HPFL7$$!3!)*****f_#4pHFL$\"3@+++BM\"4k$FL7$$!33+++7)\\.-$FL$\"3!* *****RWIUf$FL7$$!37+++/[lqIFL$\"3-+++K,*fa$FL7$$!39+++skA?JFL$\"3)**** **p\"pA'\\$FL7$$!3#)*****Hit\"pJFL$\"3:+++KA*\\W$FL7$$!3z*****Hk*f@tSo$FL$\"32+++!ztY\"GFL7$$!3@+++OCNGPFL$\"3'******pXzWv#FL7$$!31++ +GO%=x$FL$\"35+++uv:%p#FL7$$!3$******>XSW\"QFL$\"3++++AXrLEFL7$$!3!)** ***>WYg&QFL$\"37+++*)z8tDFL7$$!39+++3xd'*QFL$\"3=+++<5S7DFL7$$!3,+++IV 'f$RFL$\"3))*****4-m9X#FL7$$!3?+++=2:uRFL$\"3y*****zV)G!R#FL7$$!3n**** *>C&46SFL$\"39+++U)>)GBFL7$$!3#******\\Upn/%FL$\"37+++,0,nAFL7$$!3%)** ****))o:\"3%FL$\"37+++&R6[?#FL7$$!3c*****fB_U6%FL$\"3&)*****R!eFL7$$!3n*****z#y#RB%FL$\"3#******\\:]8i\"FL7$$!3O+++\\+:dVFL$\"3++++l2b`:FL7$$!3s******e6pyVFL$ \"3#******4iw^[\"FL7$$!3;+++c:I*R%FL$\"3'******fKgiT\"FL7$$!3o******Q, ->WFL$\"33+++3N%oM\"FL7$$!3C+++$ozyV%FL$\"33+++>f(pF\"FL7$$!3W+++=^!fX %FL$\"3#*******fSr17FL7$$!3i******=@6tWFL$\"3!******4M>h8\"FL7$$!3Q+++ rh]*[%FL$\"33+++veDl5FL7$$!3R+++6A30XFL$\"3m*****4f!)=%**F-7$$!3Q+++> \\#)>XFL$\"3c******>7yH#*F-7$$!3p******='4P`%FL$\"3^+++]%Qo^)F-7$$!3#* ******=PqYXFL$\"3#******HJmN!yF-7$$!3I+++L&o(eXFL$\"35+++^0S!4(F-7$$!3 9+++>8')pXFL$\"3-+++F&*oxjF-7$$!3'******>NP*zXFL$\"3M+++p0plcF-7$$!3)) *****H!>&*)e%FL$\"3#)*****4=rX&\\F-7$$!3U+++(*>'of%FL$\"3u*****>A9WC%F -7$$!3\"******p!zi.YFL$\"3$********)fANNF-7$$!3,+++eV@4YFL$\"3z******4 i%p#GF-7$$!3@+++[9f8YFL$\"30+++N!f%>@F-7$$!3q*****fKNnh%FL$\"3-+++UWg7 9F-7$$!3(******HmG'=YFL$\"3s*****H&f*=1(Feq7$$!31+++\\4E>YFLF(7$F\\gm$ !3s*****H&f*=1(Feq7$Fgfm$!3-+++UWg79F-7$Fbfm$!30+++N!f%>@F-7$F]fm$!3z* *****4i%p#GF-7$Fhem$!3$********)fANNF-7$Fcem$!3u*****>A9WC%F-7$F^em$!3 #)*****4=rX&\\F-7$Fidm$!3M+++p0plcF-7$Fddm$!3-+++F&*oxjF-7$F_dm$!35+++ ^0S!4(F-7$Fjcm$!3#******HJmN!yF-7$Fecm$!3^+++]%Qo^)F-7$F`cm$!3c******> 7yH#*F-7$F[cm$!3m*****4f!)=%**F-7$Ffbm$!33+++veDl5FL7$Fabm$!3!******4M >h8\"FL7$F\\bm$!3#*******fSr17FL7$Fgam$!33+++>f(pF\"FL7$Fbam$!33+++3N% oM\"FL7$F]am$!3'******fKgiT\"FL7$Fh`m$!3#******4iw^[\"FL7$Fc`m$!3++++l 2b`:FL7$F^`m$!3*******>:]8i\"FL7$Fi_m$!33+++Q;b)o\"FL7$Fd_m$!31+++&eT^ v\"FL7$F__m$!3\"******4i:6#=FL7$Fj^m$!3#******\\FL7$F`^m$!3++++\"*pU:?FL7$F[^m$!3=+++[01z?FL7$Ff]m$!3&)*****R! e)GBFL 7$Fb\\m$!3y*****zV)G!R#FL7$F]\\m$!3))*****4-m9X#FL7$Fh[m$!3=+++<5S7DFL 7$Fc[m$!37+++*)z8tDFL7$F^[m$!3++++AXrLEFL7$Fijl$!35+++uv:%p#FL7$Fdjl$! 3'******pXzWv#FL7$F_jl$!32+++!ztY\"GFL7$Fjil$!3=+++$*=ruGFL7$Feil$!31+ ++y-aMHFL7$F`il$!3++++d)zS*HFL7$F[il$!3!)*****pPFK0$FL7$Ffhl$!3#****** fBf=6$FL7$Fahl$!37+++&QP)pJFL7$F\\hl$!37+++/m,FKFL7$Fggl$!3++++G@D$G$F L7$Fbgl$!39+++YmSQLFL7$F]gl$!3!)******eeN#R$FL7$Fhfl$!3:+++KA*\\W$FL7$ Fcfl$!3)******p\"pA'\\$FL7$F^fl$!3-+++K,*fa$FL7$Fiel$!3!******RWIUf$FL 7$Fdel$!3@+++BM\"4k$FL7$F_el$!3#)*****H3?go$FL7$Fjdl$!3&******pXX&HPFL 7$Fedl$!37+++`v\\rPFL7$F`dl$!3********po*=\"QFL7$F[dl$!3;+++Lmx]QFL7$F fcl$!3A+++(*Q=))QFL7$Facl$!39+++z;=CRFL7$F\\cl$!39+++pB&)eRFL7$Fgbl$!3 !)*****zt-B*RFL7$Fbbl$!3m*****p&4nCSFL7$F]bl$!3%)******Ri8cSFL7$Fhal$! 3O+++O<$p3%FL7$Fcal$!3\")*****RNht6%FL7$F^al$!3w*****pi?y9%FL7$Fi`l$!3 m*****\\7?)yTFL7$Fd`l$!3w*****4^05@%FL7$F_`l$!3s******H'e^C%FL7$Fj_l$! 3U+++Jl9#G%FL7$Fe_l$!3G+++/hwAVFL7$F`_l$!3'******4!fWnVFL7$F[_l$!3w*** **>H!)eT%FL7$Ff^l$!3I+++S;'oY%FL7$Fa^l$!3M+++p;c=XFL7$F\\^l$!3I+++Su6p XFL7$Fg]l$!39+++I243d8'[FL7 $Fjz$!3!)*****Hs)[u[FL7$Fez$!3;+++XN%R)[FL7$F`z$!35+++n:t*)[FL7$F[z$!3 y*******=R=*[FL7$Ffy$!3Q+++b#H-*[FL7$Fay$!3++++y5%[)[FL7$F\\y$!31+++y. fv[FL7$Fgx$!3m*****>6pB'[FL7$Fbx$!3-+++.`/X[FL7$F]x$!3e*****H,jM#[FL7$ Fhw$!3a*****>sUuz%FL7$Fcw$!3/+++'Q#ymZFL7$F^w$!3D+++w&f7t%FL7$Fiv$!35+ ++^]j!p%FL7$Fdv$!3!)*****H.eYk%FL7$F_v$!3#)*****46tIf%FL7$Fju$!37+++B! Gc`%FL7$Feu$!3?+++g03sWFL7$F`u$!31+++s:?-WFL7$F[u$!37+++^+xDVFL7$Fft$! 3h*****HZcDC%FL7$Fat$!3++++,KH_TFL7$F\\t$!3M+++NvjaSFL7$Fgs$!31+++ac9 \\RFL7$Fbs$!3))******)\\&HNQFL7$F]s$!37+++(R;Er$FL7$Fhr$!3#)*****4.o4e $FL7$Fcr$!3;+++db*3W$FL7$F^r$!3/+++<\\z$H$FL7$Fiq$!3A+++(\\N;9$FL7$Fcq $!3?+++IDQ')HFL7$F^q$!3>+++R#[&HGFL7$Fip$!3A+++K+3sEFL7$Fcp$!3&)*****f stW^#FL7$F^p$!3%******RuRpN#FL7$Fho$!35+++bh`*>#FL7$Fco$!39+++jHDU?FL7 $F]o$!31+++1#e])=FL7$Fhn$!3++++)p>zs\"FL7$$!3i*****\\Fo-y)FI$!3/+++gH \"3d\"FL7$FS$!33+++AAs89FL7$FN$!33+++>'QmD\"FL7$FG$!31+++$yd&*4\"FL7$$ !3q******fn;>]FC$!36+++h&yZU*F-7$$!3O+++\"enPo(F=$!3!******zp\")R&yF-7 $$!3]*****f5%H2]F7$!3/+++4`=$G'F-7$F($!3<+++!)*)Q7ZF-7$F($!3)******Rl# fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"\"\"!\"\"$\"#N!\"#F(-%)PO LYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$FUFbboF(7%Fjbo7$F($\"+aEfTJF]coF^co 7%Faco7$F($\"+!)*)Q7ZF]coF^co7%Feco7$$!+2UH2]Feq$\"+4`=$G'F]coF^co7%Fi co7$$!+(fnPo(F-$\"+)p\")R&yF]coF^co7%F_do7$$!+fn;>]FL$\"+h&yZU*F]coF^c o7%Fedo7$$!+!HB#y?!#;$\"+$yd&*4\"!\"*F^co7%F[eo7$$!+?z^GgF^eo$\"+>'QmD \"FaeoF^co7%Fceo7$$!+q$H6=\"!#:$\"+AAs89FaeoF^co7%Fieo7$$!+<#o-y)F^eo$ \"+gH\"3d\"FaeoF^co7%F`fo7$$\"+p!>De%F\\fo$\"+)p>zs\"FaeoF^co7%Fffo7$$ \"+$4jLu#!#9$\"+1#e])=FaeoF^co7%F\\go7$$\"+$[wWn*F_go$\"+jHDU?FaeoF^co 7%Fcgo7$$\"+1c_KF!#8$\"+bh`*>#FaeoF^co7%Figo7$$\"+*[B5#)F\\jo$\"+ac9\\RFaeoF^co7%Fd\\p7$$\"+,dv)G)F\\jo$\"+Nvj aSFaeoF^co7%Fj\\p7$$\"+N$4Xy(F\\jo$\"+,KH_TFaeoF^co7%F`]p7$$\"+)[#***o 'F\\jo$\"+tkbUUFaeoF^co7%Ff]p7$$\"+0SjF]F\\jo$\"+^+xDVFaeoF^co7%F\\^p7 $$\"+k00RGF\\jo$\"+s:?-WFaeoF^co7%Fb^p7$$\"+o5LowUl$F]co$\"+.`/X[FaeoF^co7 %F^bp7$$!+(*>jITF]co$\"+7\"pB'[FaeoF^co7%Fdbp7$$!+'=)y9YF]co$\"+y.fv[F aeoF^co7%Fjbp7$$!+yh:.^F]co$\"+y5%[)[FaeoF^co7%F`cp7$$!+')fv%f&F]co$\" +b#H-*[FaeoF^co7%Ffcp7$$!+Z'\\))3'F]co$\"+!>R=*[FaeoF^co7%F\\dp7$$!+d& H\\e'F]co$\"+n:t*)[FaeoF^co7%Fbdp7$$!+<#>F3(F]co$\"+XN%R)[FaeoF^co7%Fh dp7$$!+6!)=#e(F]co$\"+B()[u[FaeoF^co7%F^ep7$$!+zFe$3)F]co$\"+#3d8'[Fae oF^co7%Fdep7$$!+Q/[(e)F]co$\"+O^^W[FaeoF^co7%Fjep7$$!+Yz'[4*F]co$\"+:o !R#[FaeoF^co7%F`fp7$$!+J\"esg*F]co$\"+OgX*z%FaeoF^co7%Fffp7$$!+z^o75Fa eo$\"+*zs5x%FaeoF^co7%F\\gp7$$!+^bnl5Faeo$\"+ihmQZFaeoF^co7%Fbgp7$$!+s a7?6Faeo$\"+N*p@q%FaeoF^co7%Fhgp7$$!+42hw6Faeo$\"+M1fhYFaeoF^co7%F^hp7 $$!+*GteB\"Faeo$\"+I24q^\"Faeo$\"+#H!)eT%FaeoF^co7%F\\jp7$$!+S9N*f \"Faeo$\"+,fWnVFaeoF^co7%Fbjp7$$!+U=Q%o\"Faeo$\"+/hwAVFaeoF^co7%Fhjp7$ $!+7@DqFaeo$\"+6b+6UFaeoF^co7%Fj[q7$$!+u%3)>?Faeo$\"+D,#)yTFaeo F^co7%F`\\q7$$!+%o6#)4#Faeo$\"+F1#y9%FaeoF^co7%Ff\\q7$$!+&3RR<#Faeo$\" +a8OX#Faeo$\"+QFI#*RFaeoF^co7%Fd^q7$$!+hH!f^#Faeo$\"+pB&)eRFaeoF^co7% Fj^q7$$!+'4^zd#Faeo$\"+z;=CRFaeoF^co7%F`_q7$$!+8P@QEFaeo$\"+(*Q=))QFae oF^co7%Ff_q7$$!+!**>op#Faeo$\"+Lmx]QFaeoF^co7%F\\`q7$$!+!R$*Qv#Faeo$\" +qo*=\"QFaeoF^co7%Fb`q7$$!+[Hb4GFaeo$\"+`v\\rPFaeoF^co7%Fh`q7$$!+>Y\"R 'GFaeo$\"+daaHPFaeoF^co7%F^aq7$$!+ED4pHFaeo$\"+BM\"4k$FaeoF^co7%Fjaq7$$!+7)\\.-$Faeo$\"+W/B%f$FaeoF^c o7%F`bq7$$!+/[lqIFaeo$\"+K,*fa$FaeoF^co7%Ffbq7$$!+skA?JFaeo$\"+)GBFa eoF^co7%Fdjq7$$!+D%pn/%Faeo$\"+,0,nAFaeoF^co7%Fjjq7$$!+*)o:\"3%Faeo$\" +&R6[?#FaeoF^co7%F`[r7$$!+OAD9TFaeo$\"+/eFaeoF^co7%Fh\\r7$$!+Gy#RB%Faeo$\"+v\"yk)=FaeoF^ co7%F^]r7$$!+b3zgUFaeo$\"+@c6@=FaeoF^co7%Fd]r7$$!+05]'G%Faeo$\"+&eT^v \"FaeoF^co7%Fj]r7$$!+^256VFaeo$\"+Q;b)o\"FaeoF^co7%F`^r7$$!+z]jMVFaeo$ \"+_,N@;FaeoF^co7%Ff^r7$$!+\\+:dVFaeo$\"+l2b`:FaeoF^co7%F\\_r7$$!+f6py VFaeo$\"+@m<&[\"FaeoF^co7%Fb_r7$$!+c:I*R%Faeo$\"+E.E;9FaeoF^co7%Fh_r7$ $!+R,->WFaeo$\"+3N%oM\"FaeoF^co7%F^`r7$$!+$ozyV%Faeo$\"+>f(pF\"FaeoF^c o7%Fd`r7$$!+=^!fX%Faeo$\"+gSr17FaeoF^co7%Fj`r7$$!+>@6tWFaeo$\"+T$>h8\" FaeoF^co7%F`ar7$$!+rh]*[%Faeo$\"+veDl5FaeoF^co7%Ffar7$$!+6A30XFaeo$\"+ \"f!)=%**F]coF^co7%F\\br7$$!+>\\#)>XFaeo$\"+?7yH#*F]coF^co7%Fbbr7$$!+> '4P`%Faeo$\"+]%Qo^)F]coF^co7%Fhbr7$$!+>PqYXFaeo$\"+8jc.yF]coF^co7%F^cr 7$$!+L&o(eXFaeo$\"+^0S!4(F]coF^co7%Fdcr7$$!+>8')pXFaeo$\"+F&*oxjF]coF^ co7%Fjcr7$$!+_t$*zXFaeo$\"+p0plcF]coF^co7%F`dr7$$!+.>&*)e%Faeo$\"+\"=r X&\\F]coF^co7%Ffdr7$$!+(*>'of%Faeo$\"+AUTWUF]coF^co7%F\\er7$$!+2zi.YFa eo$\"+!*fANNF]coF^co7%Fber7$$!+eV@4YFaeo$\"+5i%p#GF]coF^co7%Fher7$$!+[ 9f8YFaeo$\"+N!f%>@F]coF^co7%F^fr7$$!+E`t;YFaeo$\"+UWg79F]coF^co7%Fdfr7 $$!+j'G'=YFaeo$\"+`f*=1(F\\joF^co7%Fjfr7$$!+\\4E>YFaeoF(F^co7%F`gr7$F[ gr$!+`f*=1(F\\joF^co7%Fdgr7$Fefr$!+UWg79F]coF^co7%Fhgr7$F_fr$!+N!f%>@F ]coF^co7%F\\hr7$Fier$!+5i%p#GF]coF^co7%F`hr7$Fcer$!+!*fANNF]coF^co7%Fd hr7$F]er$!+AUTWUF]coF^co7%Fhhr7$Fgdr$!+\"=rX&\\F]coF^co7%F\\ir7$Fadr$! +p0plcF]coF^co7%F`ir7$F[dr$!+F&*oxjF]coF^co7%Fdir7$Fecr$!+^0S!4(F]coF^ co7%Fhir7$F_cr$!+8jc.yF]coF^co7%F\\jr7$Fibr$!+]%Qo^)F]coF^co7%F`jr7$Fc br$!+?7yH#*F]coF^co7%Fdjr7$F]br$!+\"f!)=%**F]coF^co7%Fhjr7$Fgar$!+veDl 5FaeoF^co7%F\\[s7$Faar$!+T$>h8\"FaeoF^co7%F`[s7$F[ar$!+gSr17FaeoF^co7% Fd[s7$Fe`r$!+>f(pF\"FaeoF^co7%Fh[s7$F_`r$!+3N%oM\"FaeoF^co7%F\\\\s7$Fi _r$!+E.E;9FaeoF^co7%F`\\s7$Fc_r$!+@m<&[\"FaeoF^co7%Fd\\s7$F]_r$!+l2b`: FaeoF^co7%Fh\\s7$Fg^r$!+_,N@;FaeoF^co7%F\\]s7$Fa^r$!+Q;b)o\"FaeoF^co7% F`]s7$F[^r$!+&eT^v\"FaeoF^co7%Fd]s7$Fe]r$!+@c6@=FaeoF^co7%Fh]s7$F_]r$! +v\"yk)=FaeoF^co7%F\\^s7$Fi\\r$!+&*=C^>FaeoF^co7%F`^s7$Fc\\r$!+\"*pU:? FaeoF^co7%Fd^s7$F]\\r$!+[01z?FaeoF^co7%Fh^s7$Fg[r$!+/e)GBFaeoF^co7%Fh_s7$F_jq$!+Q%)G!R#FaeoF^co7%F\\`s7$Fiiq$!+@gY^CFaeoF^c o7%F``s7$Fciq$!+<5S7DFaeoF^co7%Fd`s7$F]iq$!+*)z8tDFaeoF^co7%Fh`s7$Fghq $!+AXrLEFaeoF^co7%F\\as7$Fahq$!+uv:%p#FaeoF^co7%F`as7$F[hq$!+d%zWv#Fae oF^co7%Fdas7$Fegq$!+!ztY\"GFaeoF^co7%Fhas7$F_gq$!+$*=ruGFaeoF^co7%F\\b s7$Fifq$!+y-aMHFaeoF^co7%F`bs7$Fcfq$!+d)zS*HFaeoF^co7%Fdbs7$F]fq$!+xtA `IFaeoF^co7%Fhbs7$Fgeq$!+O#f=6$FaeoF^co7%F\\cs7$Faeq$!+&QP)pJFaeoF^co7 %F`cs7$F[eq$!+/m,FKFaeoF^co7%Fdcs7$Fedq$!+G@D$G$FaeoF^co7%Fhcs7$F_dq$! +YmSQLFaeoF^co7%F\\ds7$Ficq$!+feN#R$FaeoF^co7%F`ds7$Fccq$!+KA*\\W$Faeo F^co7%Fdds7$F]cq$!+R= *[FaeoF^co7%Fh^t7$Fgcp$!+b#H-*[FaeoF^co7%F\\_t7$Facp$!+y5%[)[FaeoF^co7 %F`_t7$F[cp$!+y.fv[FaeoF^co7%Fd_t7$Febp$!+7\"pB'[FaeoF^co7%Fh_t7$F_bp$ !+.`/X[FaeoF^co7%F\\`t7$Fiap$!+8IYB[FaeoF^co7%F``t7$Fcap$!+AFW(z%FaeoF ^co7%Fd`t7$F]ap$!+'Q#ymZFaeoF^co7%Fh`t7$Fg`p$!+w&f7t%FaeoF^co7%F\\at7$ Fa`p$!+^]j!p%FaeoF^co7%F`at7$F[`p$!+L!eYk%FaeoF^co7%Fdat7$Fe_p$!+6J2$f %FaeoF^co7%Fhat7$F__p$!+B!Gc`%FaeoF^co7%F\\bt7$Fi^p$!+g03sWFaeoF^co7%F `bt7$Fc^p$!+s:?-WFaeoF^co7%Fdbt7$F]^p$!+^+xDVFaeoF^co7%Fhbt7$Fg]p$!+tk bUUFaeoF^co7%F\\ct7$Fa]p$!+,KH_TFaeoF^co7%F`ct7$F[]p$!+NvjaSFaeoF^co7% Fdct7$Fe\\p$!+ac9\\RFaeoF^co7%Fhct7$F_\\p$!+*\\&HNQFaeoF^co7%F\\dt7$Fi [p$!+(R;Er$FaeoF^co7%F`dt7$Fc[p$!+J!o4e$FaeoF^co7%Fddt7$F][p$!+db*3W$F aeoF^co7%Fhdt7$Fgjo$!+<\\z$H$FaeoF^co7%F\\et7$Fajo$!+(\\N;9$FaeoF^co7% F`et7$Fjio$!+IDQ')HFaeoF^co7%Fdet7$Fdio$!+R#[&HGFaeoF^co7%Fhet7$F^io$! +K+3sEFaeoF^co7%F\\ft7$Fgho$!+EPZ9DFaeoF^co7%F`ft7$Faho$!+W(RpN#FaeoF^ co7%Fdft7$Fjgo$!+bh`*>#FaeoF^co7%Fhft7$Fdgo$!+jHDU?FaeoF^co7%F\\gt7$F] go$!+1#e])=FaeoF^co7%F`gt7$Fgfo$!+)p>zs\"FaeoF^co7%Fdgt7$$!+v#o-y)F^eo $!+gH\"3d\"FaeoF^co7%Fhgt7$Fjeo$!+AAs89FaeoF^co7%F^ht7$Fdeo$!+>'QmD\"F aeoF^co7%Fbht7$F\\eo$!+$yd&*4\"FaeoF^co7%Ffht7$$!+gn;>]FL$!+h&yZU*F]co F^co7%Fjht7$$!+\"enPo(F-$!+)p\")R&yF]coF^co7%F`it7$$!+1TH2]Feq$!+4`=$G 'F]coF^co7%Ffit7$F($!+!)*)Q7ZF]coF^co7%F\\jt7$F($!+aEfTJF]coF^co7%F`jt 7$F($!+Fjzq:F]coF^co7%FdjtF'F^co-F]bo6&F_boF`bo$\"#vFeboF(-%&STYLEG6#% ,PATCHNOGRIDG-F$6$7K7$$\"3*)*****pUK=`#FL$\"3e*****p!e]AcFL7$$\"3/+++< )\\l`#FL$\"3!******\\102j&FL7$$\"3>+++`v%*RDFL$\"3Q+++9(z$RcFL7$$\"3%) *****z#4,UDFL$\"3k*****4UN$[cFL7$$\"3*)*****R\\ZFa#FL$\"3g*****4!yQdcF L7$$\"3-+++(p$=UDFL$\"3!)*****H'GOmcFL7$$\"3'******pDi.a#FL$\"3?+++[p4 vcFL7$$\"3()*****R@St`#FL$\"3i*****4VOMo&FL7$$\"3y*****zc(=LDFL$\"31++ +6tB\"p&FL7$$\"3@+++Bk)z_#FL$\"3w******e[O)p&FL7$$\"3&******>bI=_#FL$ \"3B+++\"Q$p/dFL7$$\"33+++Q`#[^#FL$\"30+++(=1,r&FL7$$\"3,+++&*z32DFL$ \"3$******R&e\\9dFL7$$\"3)******H/[()\\#FL$\"3s*****\\#[wdFL7$$\"31+++JQ%3[#FL$\"3F+++#e11s&FL7$$\"3!) *****\\<1;Z#FL$\"3V+++[d/?dFL7$$\"35+++$H?CY#FL$\"3x*****z:-\"=dFL7$$ \"3++++nj[`CFL$\"3J+++;bv9dFL7$$\"3:+++)p=]W#FL$\"3U+++#z6+r&FL7$$\"3@ +++*oVsV#FL$\"3++++5*3Rq&FL7$$\"31+++!*fRICFL$\"3E+++vJ_'p&FL7$$\"3<++ +:@rCCFL$\"3o*****\\'f(zo&FL7$$\"3'******\\)3U?CFL$\"3k*****>#)R%ycFL7 $$\"36++++.t+++x5\"oT#FL$\"3;+++PtOdcFL7 $$\"3$******z3yxT#FL$\"33+++csWYcFL7$$\"32+++iAn?CFL$\"3#)*****>y^dj&F L7$$\"30+++MmWDCFL$\"37+++ydmDcFL7$$\"3&)*****>^g>V#FL$\"3e*****pKilh& FL7$$\"3%)*****4a%)*RCFL$\"3j******oOx3cFL7$$\"3A+++*=N3cF L7$$\"3>+++vk$e_#FL$\"3V+++&)\\)\\h&FLFc[uF\\bo-Fgbo6L7%7$$\"+FC$=`#Fa eo$\"+2e]AcFaeo7$$\"+<)\\l`#Faeo$\"+l]qIcFaeo7$$\"$[#Febo$\"$l&Febo7%F chu7$$\"+`v%*RDFaeo$\"+9(z$RcFaeoFhhu7%F^iu7$$\"+G4,UDFaeo$\"+@aL[cFae oFhhu7%Fdiu7$$\"+%\\ZFa#Faeo$\"+,yQdcFaeoFhhu7%Fjiu7$$\"+(p$=UDFaeo$\" +jGOmcFaeoFhhu7%F`ju7$$\"+dAOSDFaeo$\"+[p4vcFaeoFhhu7%Ffju7$$\"+9-MPDF aeo$\"+JkV$o&FaeoFhhu7%F\\[v7$$\"+ov=LDFaeo$\"+6tB\"p&FaeoFhhu7%Fb[v7$ $\"+Bk)z_#Faeo$\"+f[O)p&FaeoFhhu7%Fh[v7$$\"+_0$=_#Faeo$\"+\"Q$p/dFaeoF hhu7%F^\\v7$$\"+Q`#[^#Faeo$\"+(=1,r&FaeoFhhu7%Fd\\v7$$\"+&*z32DFaeo$\" +ae\\9dFaeoFhhu7%Fj\\v7$$\"+V![()\\#Faeo$\"+D[wdFaeoFhhu7%Ff]v7$$\"+JQ%3[#Faeo$\"+#e11s&FaeoFhhu7% F\\^v7$$\"+vhgrCFaeo$\"+[d/?dFaeoFhhu7%Fb^v7$$\"+$H?CY#Faeo$\"+e@5=dFa eoFhhu7%Fh^v7$$\"+nj[`CFaeo$\"+;bv9dFaeoFhhu7%F^_v7$$\"+)p=]W#Faeo$\"+ #z6+r&FaeoFhhu7%Fd_v7$$\"+*oVsV#Faeo$\"+5*3Rq&FaeoFhhu7%Fj_v7$$\"+!*fR ICFaeo$\"+vJ_'p&FaeoFhhu7%F``v7$$\"+:@rCCFaeo$\"+lf(zo&FaeoFhhu7%Ff`v7 $$\"+&)3U?CFaeo$\"+A)R%ycFaeoFhhu7%F\\av7$$\"++.tV#Faeo$\"+FBc;cFaeoFhhu7%F`cv7$$\"+T X)*RCFaeo$\"+pOx3cFaeoFhhu7%Ffcv7$$\"+*=N3cFL7$F]gu$!3D+++%)y\"Gg&FL7$Fhfu$!3%******f!**e)f&FL7$Fcfu$!3))* ****4@hef&FL7$F^fu$!3#*******3!*z%f&FL7$Fieu$!3%******4JHbf&FL7$Fdeu$! 3#******H;?\")f&FL7$F_eu$!3\")*****fflDg&FL7$Fjdu$!3j******oOx3cFL7$Fe du$!3e*****pKilh&FL7$F`du$!37+++ydmDcFL7$F[du$!3#)*****>y^dj&FL7$Ffcu$ !33+++csWYcFL7$Facu$!3;+++PtOdcFL7$F\\cu$!3#)******GD9ocFL7$Fgbu$!3k** ***>#)R%ycFL7$Fbbu$!3o*****\\'f(zo&FL7$F]bu$!3E+++vJ_'p&FL7$Fhau$!3+++ +5*3Rq&FL7$Fcau$!3U+++#z6+r&FL7$F^au$!3J+++;bv9dFL7$Fi`u$!3x*****z:-\" =dFL7$Fd`u$!3V+++[d/?dFL7$F_`u$!3F+++#e11s&FL7$Fj_u$!3n*****RXE)>dFL7$ Fe_u$!3s*****\\#[wdFaeoFf_w7%Fjew7$Fa]v$!+D[w'FL7$F($\"3Q++++++!>'FLF[jwF^jw-%(SCALINGG6#%,CONSTRAINEDG-%+AXES LABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fe[x6$%*HELVETICAG\"\"*- %*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$R&Febo$\"$>$Febo;$!$>'Febo$\"$>'Febo " 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 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 "[-4.6193, 0];" "6#7$, $-%&FloatG6$\"&$>Y!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can distort the bound ary curve horizontally by taking the 11th root of the real part of poi nts along the curve." }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way we s ee that the intersection of the stability region with the non-negative imaginary axis consists of the union of two disjoint intervals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040 *z^7+\n 4219477463320802334240744791965334378407253/17054851413693442 4667839175698163950379555840000*z^8+\n 120078756371563200341377781859 286350072342367/43489871104918278290298989803031807346786739200000*z^9 +\n 595752497674215431143806770225583918755471/9319258093911059633635 497814935387288597158400000*z^10+\n 16668461971675878740571808669/804 700983543756964611766177224379392*z^11:\nDigits := 18:\npts := []: z0 \+ := 0:\nfor ct from 0 to 220 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)]]:\ne nd do:\nplot(pts,color=COLOR(RGB,0,.75,0),thickness=2,font=[HELVETICA, 9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7ix7$$\"\"!F)F(7$$\"3%RW#RL*)=#y\"!#>$\"3C$ z*e`EfTJF-7$$\"3lOzb\"exr%HF-$\"3Q'ezrI&=$G'F-7$$\"3RdL@$z=1&RF-$\"3%f Pp2'zxC%*F-7$$\"3DF+w5b4d[F-$\"3R7fVhqjc7!#=7$$\"3`\"o#Rc;!Hp&F-$\"3(F-$\"3z%=7v &[6*>#F?7$$\"3^euBP!*)3)yF-$\"3gK(pG7uK^#F?7$$\"30c9^l<^;&)F-$\"3?j^A) QLu#GF?7$$\"3Q]?auJ)[5*F-$\"3%GSwNl#fTJF?7$$\"3k^UIBx\"=k*F-$\"3/$H?*= >vbMF?7$$\"3Y\"\\%QO8#>,\"F?$\"3S%z_U=6*pPF?7$$\"3FpU]i4A_5F?$\"3H+,d \\/2%3%F?7$$\"3aZ*HH?*=#3\"F?$\"3r'[r[rH#)R%F?7$$\"33811]\\?&4\"F?$\"3 XS^;!)*)Q7ZF?7$$\"3@#[%z3P6p5F?$\"3CT(yaC[l-&F?7$$!3Wd)fokJ-u*F-$\"3&) Rs(3^22M&F?7$$!3KP!o0@?j?\"F?$\"3c$3\"\\wn'[l&F?7$$!33>fI'e/yK\"F?$\"3 h.\"fD/E!pfF?7$$!3;oM#Gj!GF9F?$\"3.-9\\4`=$G'F?7$$!3Ye6$[*zo;:F?$\"31g )ezdWtf'F?7$$!3Mpr)H.A,g\"F?$\"3sPt,\\Q]6pF?7$$!3Mo'QISi%z;F?$\"31NiFC JmDsF?7$$!3\\(pPci9dv\"F?$\"32KG71C#)RvF?7$$!3ZQaqp%o%H=F?$\"3a'p:!)p \")R&yF?7$$!3]\"3)Hl(*3,>F?$\"3GDC'[+T\"o\")F?7$$!3oHlv\\?\"3(>F?$\"3q L!*\\L.I#[)F?7$$!3b4od8YyQ?F?$\"3c%R!H$pfkz)F?7$$!383h4.%*40@F?$\"3R/K (o4>16*F?7$$!37R/qan!)p@F?$\"3\"Q(\\1h&yZU*F?7$$!3!QX'zxZ#HB#F?$\"3*eV kz5Q*Q(*F?7$$!39d#\\'e[W%H#F?$\"3xZqiw(4`+\"!#<7$$!3CbCz*fMVN#F?$\"3$ \\\"[GddsO5Ffu7$$!31U7k!*)QDT#F?$\"3*39duvT\"o5Ffu7$$!3\\h^')R%z*oCF?$ \"3R$HZIyd&*4\"Ffu7$$!38;P+_GbBDF?$\"3rYJUTQ(48\"Ffu7$$!3tds%Q7Fhd#F?$ \"3enEqT**Qi6Ffu7$$!3g#yNl@Oli#F?$\"3aE!z]41Q>\"Ffu7$$!3WK=!G2sXn#F?$ \"3$***z=:BAD7Ffu7$$!3PF@*)=I(*>FF?$\"3X@+_='QmD\"Ffu7$$!3evpH'y1Cw#F? $\"3KE)y[-b!)G\"Ffu7$$!3!yV)pe^W,GF?$\"33$4))ybr%>8Ffu7$$!3$yLT\"y\\_O GF?$\"31;oaX#))3N\"Ffu7$$!3K$QUBz&)o'GF?$\"3P@L$370BQ\"Ffu7$$!3*>1y)z] Y\"*GF?$\"3lfoNAAs89Ffu7$$!3%Q0B+n5(3HF?$\"3&eba]fR^W\"Ffu7$$!3#RHA(Gf @;HF?$\"3Ud,$4HdlZ\"Ffu7$$!3O]&RzyD*4HF?$\"3P&[H)p`(z]\"Ffu7$$!3-%os6E .@)GF?$\"37WdC+RRR:Ffu7$$!3kzT2&QbX\"GF?$\"3tte9gH\"3d\"Ffu7$$!3h2CSG) y(REF?$\"3&)=p!yjKAg\"Ffu7$$\"3R$\\'3\"p@K^#F?$\"3wH*[E.`Oj\"Ffu7$$\"3 =6h[0KP2HF?$\"39)=[gDu]m\"Ffu7$$\"3P\"\\Z\\!R78JF?$\"39]:6Kk\\'p\"Ffu7 $$\"3q#p8alQ2F$F?$\"3$eS*Q)p>zs\"Ffu7$$\"3s]iXw:i0MF?$\"34_6]1UMfFNF?$\"3*e9VE7q2z\"Ffu7$$\"3?W$o$zw+SOF?$\"3gG[%zi(>A=Ffu 7$$\"3\\)[1iy2mu$F?$\"3wlSj=pi`=Ffu7$$\"3[i\"HQ%>c[QF?$\"3w2x'f?e])=Ff u7$$\"3?zFfu7$$\"3U:JPM/MUSF?$\"3!z*)y#)oFz% >Ffu7$$\"3Tq;y97RNTF?$\"3!eFsfPm$z>Ffu7$$\"3c$eab0MkA%F?$\"36nC3V!33,# Ffu7$$\"3m0wN!fddJ%F?$\"30Lf%H'HDU?Ffu7$$\"3'\\clZY&e.WF?$\"3k#=q`T,P2 #Ffu7$$\"3hfR^kQ4!\\%F?$\"3Ch(pJo`^5#Ffu7$$\"3mZbz)\\Bad%F?$\"3$>@xy/5 m8#Ffu7$$\"3nW4m5!)ofYF?$\"3DDr^%yq!o@Ffu7$$\"3&>q5M,!)Hu%F?$\"3\"y\"4 Kbh`*>#Ffu7$$\"3M9qo1bPD[F?$\"3H]*[AS15B#Ffu7$$\"3R(R.,;Pp!\\F?$\"380^ $[aK#Ffu7$$\"3#oT*zNQ4Z^F?$\"3&)3=1W(RpN#Ffu7$$\"3)[YeW^bdA&F?$\" 3i:;'feO%)Q#Ffu7$$\"3Dh+3ycw.`F?$\"3ZF!Q?sQ*>CFfu7$$\"3jy,[yG9\"Q&F?$ \"34I3U4fW^CFfu7$$\"3\"RrvVL,zX&F?$\"3zbLTix&H[#Ffu7$$\"3+.()RF80MbF?$ \"3D\"*RWEPZ9DFfu7$$\"3NJm!y\\*f4cF?$\"3QLk%*[I*fa#Ffu7$$\"3Hb>zd!\\Xo &F?$\"3yR&zpu9vd#Ffu7$$\"3xvIPO***)edF?$\"3rFVUrv.4EFfu7$$\"3y]*Gq')[E $eF?$\"31vmjn*f0k#Ffu7$$\"3Z,)[sU*y0fF?$\"3VkJ\\K+3sEFfu7$$\"3+v*>y.7$ yfF?$\"38K$[yY&f.FFfu7$$\"3(HCT)GR?]gF?$\"3r>(G0`.^t#Ffu7$$\"3ZbV.z!\\ 97'F?$\"39(3F\"z4gmFFfu7$$\"3Q\"4n[8G?>'F?$\"3iKK.=S3)z#Ffu7$$\"3)3FWi J=>E'F?$\"3Jc'\\$R#[&HGFfu7$$\"3JkG-;L4JjF?$\"3\")eHfj&))4'GFfu7$$\"3N a^J,K_*R'F?$\"3*4=v.=*R#*GFfu7$$\"3W?ESDV0/^&HFfu7$$\"3*39\"pen)**f'F?$\"3E7lPIDQ')HFfu7$ $\"3Z%\\#ei<1lmF?$\"3fM\\8f'*f&*R&3Y&=HnF?$\"3b2z&)4_u[IFf u7$$\"3r+7aV#z3tB <$Ffu7$$\"3/)py'GieLqF?$\"39xaI&ptH?$Ffu7$$\"31o()oANz!4(F?$\"3.ke)=t@ MB$Ffu7$$\"3)>IMAICm9(F?$\"3$*4jk85qjKFfu7$$\"3WIcX[<,,sF?$\"3cMv0<\\z $H$Ffu7$$\"3(y#)QHm*)QD(F?$\"30WJ#4c'oBLFfu7$$\"3g2[()QL>0tF?$\"3B6.Z3 !fLN$Ffu7$$\"3ZS;%GWg[N(F?$\"36L;v$[&z#Q$Ffu7$$\"3S5<#Q(=$GS(F?$\"3\"p 8d]fz>T$Ffu7$$\"3mkKRkD0\\uF?$\"3z<)zqb&*3W$Ffu7$$\"3&zP>)*=sM\\(F?$\" 3:4m;.%G&pMFfu7$$\"3NH45ad/OvF?$\"39.4?zT'y\\$Ffu7$$\"3\"f.jx-Mnd(F?$ \"3,lL^6,*e_$Ffu7$$\"3*=86(>Q]:wF?$\"3,E(QEu%f`NFfu7$$\"30Cw>u!GBl(F?$ \"3tS@fI!o4e$Ffu7$$\"3EL<$Hz&=(o(F?$\"3#>![W49+3OFfu7$$\"3vQDz!zh+s(F? $\"3SO(G6\"yoMOFfu7$$\"3mBct+j%4v(F?$\"3eAzR];-hOFfu7$$\"3#G+#z[W$)zxF ?$\"3TE_bw()*po$Ffu7$$\"3.gBFBcs1yF?$\"3L(H8oR;Er$Ffu7$$\"3R'\\!RjFiJy F?$\"36XN%p(H(yt$Ffu7$$\"3x[,q\"fJX&yF?$\"3rE\")=D\"oFw$Ffu7$$\"38o&RB \")fa(yF?$\"374#=`Y-ty$Ffu7$$\"3Y;N^$H;W*yF?$\"3%\\4+V]x9\"QFfu7$$\"35 &HM$[-T6zF?$\"3I-_P*\\&HNQFfu7$$\"3vzpYB/XEzF?$\"3HPhcG$f(eQFfu7$$\"3z !fs-IW&RzF?$\"3FBVxnB(=)QFfu7$$\"3>[7^3tp]zF?$\"3I$H.oPQY!RFfu7$$\"3?o &[#[?\"*fzF?$\"3!zEveRhq#RFfu7$$\"31#))*R6v=nzF?$\"3uW\"QNlX\"\\RFfu7$ $\"3&H))e-H=D(zF?$\"3n33I'[&*3(RFfu7$$\"3)='>AcP*e(zF?$\"3P@'*))p_J#*R Ffu7$$\"3GUF\\(=(HxzF?$\"3zh@oh$4M,%Ffu7$$\"3b.r]>[qwzF?$\"3TMVj`?=MSF fu7$$\"3T^2t&y%3uzF?$\"3#fef[`PY0%Ffu7$$\"3qMmy3fRpzF?$\"3b7\"HH')zZ2% Ffu7$$\"3Ky,\"=H'eizF?$\"3&=7DH%Hh%4%Ffu7$$\"3WdxXY;f`zF?$\"3$\\Q#G809 9TFfu7$$\"3#e9>cFLB%zF?$\"3Vz_[PhOLTFfu7$$\"3@Q>W.crGzF?$\"3Hd!R2?$H_T Ffu7$$\"3EqIa4Ji7zF?$\"3'3&yz5\\#4<%Ffu7$$\"3L_AOtk\"R*yF?$\"3*HM-AIk# *=%Ffu7$$\"35*z**3mFC(yF?$\"3u;HDVUJ2UFfu7$$\"3*G]-vd`z%yF?$\"37>l6Xu2 DUFfu7$$\"3v$e\"GZxC?yF?$\"3Z@a`skbUUFfu7$$\"3zolRH#4!*y(F?$\"3?qAnbPv fUFfu7$$\"32vU_(enQv(F?$\"3Nqvo-;nwUFfu7$$\"3D1PZ\"4iVr(F?$\"3q%p-F@7L H%Ffu7$$\"3E&=4`E7*pwF?$\"3l\"f`))ox'4VFfu7$$\"38Ln!>0v(>wF?$\"3Yxw=^+ xDVFfu7$$\"3,JC*)H0)Hc(F?$\"3!*[\"*>\\7fTVFfu7$$\"3_%)*pXyJ#)\\(F?$\"3 _+k$Qx/X,MsF?$\"3_C#*pr:?-WFfu7$$\"3PL\"*>jft3rF?$ \"3)>F+@_%p;WFfu7$$\"3:D'38h_,&pF?$\"31xk#[9F4V%Ffu7$$\"3a70`s=vNnF?$ \"3wK<6*=,\\W%Ffu7$$\"30s,u(*4;0kF?$\"3c4pG2%='eWFfu7$$\"3Hh!\\h+Y0i&F ?$\"3b!R()*f03sWFfu7$$!3XSYIQlhjgF?$\"3Ps()[?%*G&[%Ffu7$$!3=!4>GZASe'F ?$\"34>#4#ynC)\\%Ffu7$$!34Ht'Rxbe(oF?$\"3rjTUTW&4^%Ffu7$$!3h7Ll()*\\U3 (F?$\"38eXCRUTBXFfu7$$!3`zbW6MP[sF?$\"3^PQ\"H-Gc`%Ffu7$$!3qNL$Q,*y%Q(F ?$\"3G\\8[mwfZXFfu7$$!3@G*f[A;@](F?$\"3pa8\"p1D$fXFfu7$$!3v;T%zZLag(F? $\"3A'*poV@\"3d%Ffu7$$!322aS*z$*zp(F?$\"3y#)z'z$31#e%Ffu7$$!3\"y'o=L3, #y(F?$\"3d>$f86tIf%Ffu7$$!3H#ehMsh!fyF?$\"3KMjMW4&Qg%Ffu7$$!3lT7Cv9JIz F?$\"35_'[WL'R9YFfu7$$!38RM**ohk'*zF?$\"3![wUTH6Zi%Ffu7$$!3CSZ`2ive!)F ?$\"39M&*f[yzMYFfu7$$!3UL(\\%f0><\")F?$\"33c%zK.eYk%Ffu7$$!3;DWIeLRs\" )F?$\"3U%)[U\"*QHaYFfu7$$!3m?&*\\P$HZA)F?$\"3XPZ[ruqjYFfu7$$!3Py3fg>]u #)F?$\"3GdN]C3!Hn%Ffu7$$!3AW[tKm'>K)F?$\"3e6K],g(=o%Ffu7$$!3azOe:0Mn$) F?$\"3=l?)30N1p%Ffu7$$!3+DQ]W*43T)F?$\"3MSJ&e,!=*p%Ffu7$$!3IAE#)>h`_%) F?$\"3,c'R>$H^2ZFfu7$$!314EwL&fE\\)F?$\"3$ySVX#ej:ZFfu7$$!3\\y(4mX.8`) F?$\"3$\\t)f12bBZFfu7$$!3J))Q)Grw&o&)F?$\"3rBBKw&f7t%Ffu7$$!3o/.u9fd/' )F?$\"3-Eo1:WwQZFfu7$$!3]E9bvsQR')F?$\"3Q;UG&=ngu%Ffu7$$!35$zf_4)3t')F ?$\"3IwHgG)pJv%Ffu7$$!30Agc')zu0()F?$\"3+l:-kU2gZFfu7$$!3i[+h%*)Hut)F? $\"3o/#>iQ#ymZFfu7$$!3\"z(3r]4>o()F?$\"3hkP)R1'HtZFfu7$$!3$f,8<>$3)z)F ?$\"3IeavQrhzZFfu7$$!3HYIppT:F))F?$\"3EfTGBuu&y%Ffu7$$!3fJ#4;ZZa&))F?$ \"3N!Q(R+()o\"z%Ffu7$$!3,jh]kJ+$)))F?$\"3UN`(=sUuz%Ffu7$$!3G\"z5)y\"e) 4*)F?$\"3](**Gu?6I![Ffu7$$!3m(oa)Rm/O*)F?$\"38smxVeR3[Ffu7$$!3Om\"oe:+ ;'*)F?$\"3k0V\"QG)f8[Ffu7$$!3]J^>p![l)*)F?$\"3*\\`ue9?'=[Ffu7$$!3'Qu$G ww\"4,*F?$\"3rY$pI,jM#[Ffu7$$!3;#*)\\5UMZ.*F?$\"3LR6rK%G\"G[Ffu7$$!3-' *oVl?-e!*F?$\"30lp!e\"zhK[Ffu7$$!3&**ym.%G!33*F?$\"3Qc2iOH$p$[Ffu7$$!3 `]ewzv4.\"*F?$\"3G7%)HK\\2T[Ffu7$$!3i1k#*Re#\\7*F?$\"3U@2b-`/X[Ffu7$$! 3a:-71gIY\"*F?$\"3_H#*Q4a%)[[Ffu7$$!3i4:4)Qbs;*F?$\"3U4.\"pdwC&[Ffu7$$ !33R*=eI!z(=*F?$\"3OhG7\"4Sf&[Ffu7$$!3G-htmh#z?*F?$\"3r__\")*>P#f[Ffu7 $$!3F=IVNvnF#*F?$\"3\"yHdC6pB'[Ffu7$$!3AD+s'>eqC*F?$\"3qwK8+qLl[Ffu7$$ !31jl\"G@\"3m#*F?$\"3h$>-b*>9o[Ffu7$$!3?yjKv*eZG*F?$\"30!o\"y#>&yq[Ffu 7$$!3)3$[d_K5.$*F?$\"3!=:_xknK([Ffu7$$!3122\"GBD6K*F?$\"3>K!yxP!fv[Ffu 7$$!3t+Arjb$)Q$*F?$\"3FbI%=Oax([Ffu7$$!3mgGe)QWiN*F?$\"3TgofS0wz[Ffu7$ $!3'\\!pdz8Ot$*F?$\"3\"[G5k\")4;)[Ffu7$$!3IYKDod>!R*F?$\"3M^pV`II$)[Ff u7$$!3l')Gstjv1%*F?$\"3_O*yw2T[)[Ffu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HE LVETICAG\"\"*-%&COLORG6&%$RGBGF($\"#v!\"#F(-%+AXESLABELSG6$Q!6\"Ffao-% %VIEWG6$%(DEFAULTGF[bo" 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 0 287 "Digits := 15:\nz0 := 0.5*I:\nfor ct from 15 to 18 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0 );\nend do;``;\nz0 := 1.6*I:\nfor ct from 50 to 53 do\n newton(R(z)= exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 4.5*I:\nfor ct from 159 to \+ 162 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0F!#D$\"09l,)*)Q7Z!#: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0QK]-`d3#!#D$\"0uyaC[l-&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0\"[hWB@'[(!#E$\"0Cx3^22M&!#:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0*3![]k?(y!#D$\"03\"\\wn'[l&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0&H=z$o-y)!#@$\"0(e9gH\"3d\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0a#=LY`PV!#@$\"0#p!yjKAg\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02lxQ8m_#!#@$\"0$*[E.`Oj\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0J@&R'4YD\"!#?$\"0>[gDu]m\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0chLl9WW(!# <$\"0\"pG2%='eW!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0)y6o5Lo " 0 "" {MPLTEXT 1 0 479 "Digits := 15:\nreal_part := proc(u)\n \+ Re(newton(R(z)=exp(u*Pi*I),z=0.5*I))\nend proc:\nu0 := bisect('real_pa rt'(u),u=0.15..0.18);\nnewton(R(z)=exp(u0*Pi*I),z=0.5*I);``;\nreal_par t := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.6*I))\nend proc:\nu0 : = bisect('real_part'(u),u=0.5..0.53);\nnewton(R(z)=exp(u0*Pi*I),z=1.6* I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=4.5*I)) \nend proc:\nu0 := bisect('real_part'(u),u=1.59..1.62);\nnewton(R(z)=e xp(u0*Pi*I),z=4.5*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#u0G$\"0#yO.G4#o\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0>tM R]WG&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"04Gw,gw;&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^#$\"0Lsuf\"\\B;!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"00%Q#\\cIg\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0Bd\"e&)*zv*!#H$\"07kOjWhZ%!#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 35 " is [ 0, 0.52 84 ] (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "intersection of the stability regi on with the nonnegative imaginary axis" }{TEXT -1 18 " is [ 0, 0.5284 ]" }{XPPEDIT 18 0 "`union`(``,[1.6031, 4.4761]);" "6#-%&unionG6$%!G7$ -%&FloatG6$\"&Jg\"!\"%-F)6$\"&hZ%F," }{TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------ ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 12 st age, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,StabilityFunction(6,12,'expanded' ))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,<\"\"\"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)*&#\"WtG]!3&[GP%eCO>V*QW7muHA?s(H '*H)\"en++%y1!eog!)px$H>t\">m_CON5Q&[yKUv$F)*$)F'\"\"(F)F)F)*&#\"YLr% \\,YBVDaH@*Q??;)e]Wc:(4)ydn['\"hn++!)[f(=B@3&pp)HYP$=V\"Qh-Y#e1-/%*fV \"F)*$)F'\"\")F)F)F)*&#\"W^z-pexXdck8;%QmZCZp#[&H6ZFv0#\"gn++g0h?sha#G *RkteDdR2i#=$H%oj]4)yLF)*$)F'\"\"*F)F)!\"\"*&#\"V.(>sYJ$[:Ff\"RgD2^OHH \\Pd`fjB!)\"gn+!3=:+rRJ@^\"z2%*R$4H5#=k$>`IVY!f(H#F)*$)F'\"#5F)F)F)*&# \"RhX.')4(QFF#>-gj,wP4y=A)))3Ml\"fn+g\"ouNAWU(4'p;IrGmxI`P')R6&y83M#\\ F)*$)F'\"#6F)F)Fen*&#\"FvW]R$\\/P)>WRs>*z^SqX\"Q_*4Pk/T:W^'=oH^h>c51T$ =G1\"F)*$)F'\"#7F)F)Fen" }}}{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#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-3.7);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!+2SX'o$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1 ,z=-3.7):\np_1 := plot([`R*`(z),1],z=-4.19..0.19,color=[red,blue]):\np _2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,cross,diamond],col or=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=[-4.19..0.49,-0.07..1.47] ,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 339 241 241 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3Q++++++!>%!#<$\"3Dx%=$=APFIF*7$$!33+ ]PKUEUTF*$\"3o))RV1vCF*7$$!3c]7e GQ*H0%F*$\"3E+a#)R:nkAF*7$$!3\\+DT#>f9,%F*$\"3_*f%)3(QVq?F*7$$!3i](o+: \\Z'RF*$\"3KE$\\no#3q=F*7$$!3I+]s2\"R!=RF*$\"3)GEmammuo\"F*7$$!3]++!4W >5(QF*$\"3\\[MId<;?:F*7$$!3D+]2u(**R#QF*$\"3;*z&HE+8o8F*7$$!3I](=Ne.sx $F*$\"392%z'e(z2B\"F*7$$!3N+D'HR2/t$F*$\"3w%odV\"yB16F*7$$!3^+D,v_jVOF *$\"3m\\2A?xEd!*!#=7$$!3[+D\"Hx(y`NF*$\"3x\"eDB8jYM(F^o7$$!3U+Drdt'3Y$ F*$\"3n3m7M)eB!fF^o7$$!3[+DYR\\CoLF*$\"3q4R=8g&Qu%F^o7$$!35++b(RrHF$F* $\"3ak=RH]K%z$F^o7$$!3A+](*4W0*=$F*$\"3jMtjW3hFJF^o7$$!39++ldPe%4$F*$ \"3%*Hdg]l%[`#F^o7$$!3I++v*>D(**HF*$\"3ABe-m!>)y?F^o7$$!3&)***\\BS6$3H F*$\"3#y5`xPwvu\"F^o7$$!3A+Dr;*)HDGF*$\"3EB4#f$pkA:F^o7$$!34++:U!*eEFF *$\"3!yJ>L$ePK8F^o7$$!35++gI)oHk#F*$\"3)zNn[fZ`A\"F^o7$$!39+DhX')pXDF* $\"3;`*p\"yT)>:\"F^o7$$!3K++5]!)ffCF*$\"3DS/)\\G#4D6F^o7$$!3)**\\710K^ O#F*$\"3Lp&=Pzv'H6F^o7$$!35+v$)z!y^F#F*$\"31DmPxs,i6F^o7$$!3)***\\7[0K \"=#F*$\"3xFcXlPj?7F^o7$$!3;+D'[.I^4#F*$\"3!HlW>f;- ?F*$\"3#*znUw!4CR\"F^o7$$!3&**\\7<=%f0>F*$\"31g&[lYfR^\"F^o7$$!3H+vez; `@=F*$\"3_[BHoEFN;F^o7$$!3/+]Ku=uIF^o7$$!35+]-bs=X:F*$\"3(zGvAwYj8#F^o7$$!3%**\\7e'[S c9F*$\"3p!pJ$[%QFL#F^o7$$!3A+]dfr#yN\"F*$\"3Gp#)Qsh;tDF^o7$$!3C++512Dp 7F*$\"3t&pi0d*)3\"GF^o7$$!3!****\\dVwY<\"F*$\"3]D#H)RmP*3$F^o7$$!39+DJ !4y*)3\"F*$\"3)e=3kiocO$F^o7$$!3=&***\\G;(G&**F^o$\"3ou%fD(H='p$F^o7$$ !3S/]7;IJr!*F^o$\"3]0!>!R#*zOSF^o7$$!3;**\\ic-$)\\\")F^o$\"3`8vf'RfkU% F^o7$$!3a++]FC\"*[sF^o$\"3u=rB\\gwV[F^o7$$!3a**\\7`\\t0jF^o$\"3(Q]lZ9h GK&F^o7$$!3k'****f%HM(R&F^o$\"3Qg&))*)pH!HeF^o7$$!3g'**\\n\"4QoWF^o$\" 3I$GCL=ukR'F^o7$$!3A'*\\7i66ZNF^o$\"3aVY0&>gP,(F^o7$$!3v(****p$*f0q#F^ o$\"3y]U>$\\nLj(F^o7$$!3e)**\\(*p;.t\"F^o$\"3CK\"3Pw46T)F^o7$$!3&*3++? e>D')!#>$\"3!)=#>&\\1jt\"*F^o7$$\"3UG-](Q9LF'!#?$\"3/!)f$G.$H15F*7$$\" 3[U+D@1=$[*F`[l$\"3CJL*[\"RZ*4\"F*7$$\"3-+++++++>F^o$\"3')GL*yf\\#47F* -%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F[]lFj\\l-F$6$7S7$F($\"\"\"F[]l7 $F3F`]l7$F=F`]l7$FGF`]l7$FQF`]l7$FenF`]l7$FjnF`]l7$F`oF`]l7$FeoF`]l7$F joF`]l7$F_pF`]l7$FdpF`]l7$FipF`]l7$F^qF`]l7$FcqF`]l7$FhqF`]l7$F]rF`]l7 $FbrF`]l7$FgrF`]l7$F\\sF`]l7$FasF`]l7$FfsF`]l7$F[tF`]l7$F`tF`]l7$FetF` ]l7$FjtF`]l7$F_uF`]l7$FduF`]l7$FiuF`]l7$F^vF`]l7$FcvF`]l7$FhvF`]l7$F]w F`]l7$FbwF`]l7$FgwF`]l7$F\\xF`]l7$FaxF`]l7$FfxF`]l7$F[yF`]l7$F`yF`]l7$ FeyF`]l7$FjyF`]l7$F_zF`]l7$FdzF`]l7$FizF`]l7$F^[lF`]l7$Fd[lF`]l7$Fj[lF `]l7$F_\\lF`]l-Fd\\l6&Ff\\lFj\\lFj\\lFg\\l-F$6&7#7$$!3-+++2SX'o$F*F`]l -%'SYMBOLG6#%'CIRCLEG-Fd\\l6&Ff\\lF[]lF[]lF[]l-%&STYLEG6#%&POINTG-F$6& Ff`l-F[al6#%&CROSSGF^alF`al-F$6&Ff`l-F[al6#%(DIAMONDGF^alF`al-F$6%7$7$ Fh`lFj\\lFg`l-%&COLORG6&Ff\\lFj\\l$\"\"&!\"\"Fj\\l-%*LINESTYLEG6#\"\"$ -%+AXESLABELSG6%Q\"z6\"Q!F`cl-%%FONTG6#%(DEFAULTG-Fccl6$%*HELVETICAG\" \"*-%%VIEWG6$;$!$>%!\"#$\"#\\F`dl;$!\"(F`dl$\"$Z\"F`dl" 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 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1984 "`R*` := z -> 1+z+1/ 2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+ 82996297722022297466 1244389431936245843728485080502873/37542327848538103536245266191731929 37769806068580067840000*z^7+648675778809715564450588162020389212954254 32346014947133/1435994040206582460261381431833746298696950821231875948 800000*z^8-205752747112954826947244766384161364565745775869027951/3378 80950636842931826207395725587364399282546172206105600000*z^9+802363595 35737492929365107256039159271548331467219703/2297590464330531936418210 29093399407791512131397100151808000*z^10-65340888822187809377601636002 19227273870986034561/4923408137851139863753307766287130166960974244223 5746816000*z^11-4570405179919723944198370449339504475/1062818341061056 19615129681865144154104643709952*z^12:\npts := []:\nz0 := 0:\nfor ct f rom 0 to 280 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(pt s,color=COLOR(RGB,0,.2,0)):\np_2 := plots[polygonplot]([seq([pts[i-1], pts[i],[-1.8,0]],i=2..nops(pts))],\n style=patchnogrid,color= COLOR(RGB,.05,.55,.1)):\npts := []: z0 := 2.6+4.6*I:\nfor ct from 0 to 40 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_3 := plot(pts,color=C OLOR(RGB,0,.2,0)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[2 .57,4.57]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.05,.55,.1)):\npts := []: z0 := 2.6-4.6*I:\nfor ct from 0 to 40 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_5 := plot(pts,color=COLOR(R GB,0,.2,0)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.57,-4 .57]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,. 05,.55,.1)):\np_7 := plot([[[-4.39,0],[3.19,0]],[[0,-5.19],[0,5.19]]], color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.39..3. 19,-5.19..5.19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z )`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 474 474 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fj zq:!#=7$$\"3++++!>%R$3$!#F$\"3$)*****fs#fTJF-7$$\"3/+++&z'\\t5!#D$\"38 +++\"G!R7ZF-7$$\"3/+++oO+`9!#C$\"3Y+++8d>$G'F-7$$\"32+++f)HD9\"!#B$\"3 C+++xY.ayF-7$$\"3w*****R1^)oiFC$\"3g******f(z\\U*F-7$$\"33+++a.(ol#!#A $\"33+++1'>'*4\"!#<7$$\"3K+++H2#RE*FN$\"3++++,'*zc7FQ7$$\"3#)*****pRp8 x#!#@$\"3#******RB(399FQ7$$\"3\")*****4&[`CtFZ$\"3-+++&*Qar:FQ7$$\"3%* *****f')*>YI_BV\"!#>$\"3#)*** ***)\\-;?#FQ7$$\"3%)*****4ln$pCF_p$\"37+++Gh#fN#FQ7$$\"3G+++4Or1RF_p$ \"3#******>F\\_]#FQ7$$\"3)******paF#QcF_p$\"3)******z&)>nk#FQ7$$\"3L** ***zN\"GDuF_p$\"33+++;zAyFFQ7$$\"3W+++l0R!**)F_p$\"3-+++;93**GFQ7$$\"3 )******4!z\\65F-$\"3++++C>t4IFQ7$$\"3-+++i#)Gn5F-$\"31+++S<56JFQ7$$\"3 -+++!Qx:1\"F-$\"35+++Y?6/KFQ7$$\"3U+++>6(p%**F_p$\"3')*****zw1&*G$FQ7$ $\"3T******oMD+()F_p$\"3%******p'Q%yO$FQ7$$\"3*******f()pd#pF_p$\"3z** ****)*QaRMFQ7$$\"3I+++tP2\"o%F_p$\"3$)*****H\"H%\\]$FQ7$$\"3'******R8& RD?F_p$\"3)******R\"=LkNFQ7$$!3Z++++<\"*R)*F_o$\"3&)*****R#)zzh$FQ7$$! 3G+++'oTRH%F_p$\"3;+++f`9mOFQ7$$!3i+++-AdcyF_p$\"34+++(zy!4PFQ7$$!31++ +3X'H;\"F-$\"3++++#)4-ZPFQ7$$!37+++wnkd:F-$\"35+++A,?!y$FQ7$$!3!****** 4H]l'>F-$\"39+++B'G)3QFQ7$$!3%******\\$R)pQ#F-$\"3<+++M05LQFQ7$$!3=+++ bDm;GF-$\"3-+++F)*=`QFQ7$$!3%******f&Hl`KF-$\"3#)*****\\Q\\#pQFQ7$$!3y *****49Ejp$F-$\"3')*****>`59)QFQ7$$!3%******pkSNH*QFQ7$$!3Y*****z@\")\\&fF-$\"3#)*** **R))4o)QFQ7$$!3o*****R3Y2T'F-$\"35+++u56xQFQ7$$!3)******zNur'oF-$\"3' ******H*4\"Q'QFQ7$$!37+++7n[CtF-$\"3%)*****p&[&o%QFQ7$$!3A+++?iE$y(F-$ \"34+++j#fh#QFQ7$$!3b+++CPdW#)F-$\"3(*******=3h,QFQ7$$!33+++A.75()F-$ \"3)******RVmIx$FQ7$$!3c*****R>@D=*F-$\"36+++OzNSPFQ7$$!3k*****p&)fcm* F-$\"36+++\\lI.PFQ7$$!3%******zF=l,\"FQ$\"3))*****4Ie,\\l\"FQ$\"3$)*****RT57`t\"FQ$\"3/+++#*zw#= $FQ7$$!3#******HUqE\"=FQ$\"3y*****f5[1:$FQ7$$!3-+++\\?y')=FQ$\"3;+++9] ')>JFQ7$$!31+++D-od>FQ$\"3%*******RAl*3$FQ7$$!3++++`\\]D?FQ$\"3')***** \\XP%fIFQ7$$!33+++wwV!4#FQ$\"3++++%H)zGIFQ7$$!3%)*****HSoE:#FQ$\"37+++ #HBu*HFQ7$$!3()*****HV!Q7AFQ$\"3#)*****4d\"3lHFQ7$$!3'******fFW(pAFQ$ \"3>+++20gJHFQ7$$!3<+++Li\"\\K#FQ$\"3%******>i\\o*GFQ7$$!3!)*******GR! yBFQ$\"3'*******y!H2'GFQ7$$!31+++s]CHCFQ$\"3z*****\\^hJ#GFQ7$$!3++++ui lyCFQ$\"3%)*****>K'3%y#FQ7$$!3;+++A#*QEDFQ$\"3%******4vbMu#FQ7$$!3)*** ***HUcDd#FQ$\"32+++SCB,FFQ7$$!3&******>,psh#FQ$\"3?+++9\")QdEFQ7$$!3%) *****\\)*Q1m#FQ$\"3++++8L!>h#FQ7$$!3A+++\\6y-FFQ$\"3++++w!oZc#FQ7$$!39 +++(e9Qu#FQ$\"30+++RN)f^#FQ7$$!3-+++QM'Qy#FQ$\"3'******pTkbY#FQ7$$!3#* *****4lcI#GFQ$\"3(******4DUNT#FQ7$$!3')*****pME:'GFQ$\"33+++t\"p*fBFQ7 $$!3/+++-WS**GFQ$\"3&)*****Hw@\\I#FQ7$$!3\"******>(p\"o$HFQ$\"3#****** fD/&[AFQ7$$!3;+++)*p(Q(HFQ$\"3%******>9]3>#FQ7$$!3\"******H?v1,$FQ$\"3 \"******f0@@8#FQ7$$!3=+++o2FZIFQ$\"3#)*****f0-D2#FQ7$$!3#******RR$o$3$ FQ$\"3.+++kH>7?FQ7$$!3(******fn)))>JFQ$\"3)*******[kR^>FQ7$$!37+++K#=e :$FQ$\"33+++AXI!*=FQ7$$!31+++)*\\O\">$FQ$\"3&******p\"e3H=FQ7$$!3***** **4x#REKFQ$\"3#*******))e(yw\"FQ7$$!3#)*****f6[2E$FQ$\"34+++k>x1$G\"FQ7$$!3')*****4q7[\\$FQ$ \"3)******Rp9GA\"FQ7$$!3>+++%fvv^$FQ$\"3/+++2[Gi6FQ7$$!3\"******>2Y(QN FQ$\"30+++v%R:5\"FQ7$$!39+++5(>$eNFQ$\"30+++eq^S5FQ7$$!3!)*****fI.jd$F Q$\"3e*****>O&f\"z*F-7$$!39+++')Rr#f$FQ$\"39+++x48u\"*F-7$$!3<+++&eyvg $FQ$\"37+++SpG_&)F-7$$!3\"******pDL4i$FQ$\"3`*****f!piDzF-7$$!34+++)>B Gj$FQ$\"3m*****45xPH(F-7$$!3(*******)R-Lk$FQ$\"3K+++*RPkl'F-7$$!3?+++Y KV_OFQ$\"34+++aaR8gF-7$$!3!******f$eGgOFQ$\"3;+++,Qak`F-7$$!3/+++)zOpm $FQ$\"3%)*****R?**)4ZF-7$$!3-+++guYsOFQ$\"3C+++GFi\\SF-7$$!3%)******=7 'pn$FQ$\"37+++e!QSQ$F-7$$!35+++n+]!o$FQ$\"39+++RZk8FF-7$$!3=+++0/;$o$F Q$\"3&******pXA\"R?F-7$$!31++++%3]o$FQ$\"3#*******)pA8O\"F-7$$!37++++a 4'o$FQ$\"3N+++@NX7oF_p7$$!3-+++2SX'o$FQF(7$F[am$!3N+++@NX7oF_p7$Ff`m$! 3#*******)pA8O\"F-7$Fa`m$!3&******pXA\"R?F-7$F\\`m$!39+++RZk8FF-7$Fg_m $!37+++e!QSQ$F-7$Fb_m$!3C+++GFi\\SF-7$F]_m$!3%)*****R?**)4ZF-7$Fh^m$!3 ;+++,Qak`F-7$Fc^m$!34+++aaR8gF-7$F^^m$!3K+++*RPkl'F-7$Fi]m$!3m*****45x PH(F-7$Fd]m$!3`*****f!piDzF-7$F_]m$!37+++SpG_&)F-7$Fj\\m$!39+++x48u\"* F-7$Fe\\m$!3e*****>O&f\"z*F-7$F`\\m$!30+++eq^S5FQ7$F[\\m$!30+++v%R:5\" FQ7$Ff[m$!3/+++2[Gi6FQ7$Fa[m$!3)******Rp9GA\"FQ7$F\\[m$!3#******fh$>$G \"FQ7$Fgjl$!31+++fx[V8FQ7$Fbjl$!3-+++(HjPS\"FQ7$F]jl$!3!******f(Q3k9FQ 7$Fhil$!3&********32X_\"FQ7$Fcil$!3#******z!)z]e\"FQ7$F^il$!33+++pB$ek \"FQ7$Fihl$!34+++k>x1FQ7$F`gl$!3.+++kH>7?F Q7$F[gl$!3#)*****f0-D2#FQ7$Fffl$!3\"******f0@@8#FQ7$Fafl$!3%******>9]3 >#FQ7$F\\fl$!3#******fD/&[AFQ7$Fgel$!3&)*****Hw@\\I#FQ7$Fbel$!33+++t\" p*fBFQ7$F]el$!3(******4DUNT#FQ7$Fhdl$!3'******pTkbY#FQ7$Fcdl$!30+++RN) f^#FQ7$F^dl$!3++++w!oZc#FQ7$Ficl$!3++++8L!>h#FQ7$Fdcl$!3?+++9\")QdEFQ7 $F_cl$!32+++SCB,FFQ7$Fjbl$!3%******4vbMu#FQ7$Febl$!3%)*****>K'3%y#FQ7$ F`bl$!3z*****\\^hJ#GFQ7$F[bl$!3'*******y!H2'GFQ7$Ffal$!3%******>i\\o*G FQ7$Faal$!3>+++20gJHFQ7$F\\al$!3#)*****4d\"3lHFQ7$Fg`l$!37+++#HBu*HFQ7 $Fb`l$!3++++%H)zGIFQ7$F]`l$!3')*****\\XP%fIFQ7$Fh_l$!3%*******RAl*3$FQ 7$Fc_l$!3;+++9]')>JFQ7$F^_l$!3y*****f5[1:$FQ7$Fi^l$!3/+++#*zw#=$FQ7$Fd ^l$!3$)*****RTSNH*QFQ7$F`x$!35+++$[!\\&*QFQ7$F[x$!3A+++6UX%*QFQ7$F fw$!3;+++oGy*)QFQ7$Faw$!3')*****>`59)QFQ7$F\\w$!3#)*****\\Q\\#pQFQ7$Fg v$!3-+++F)*=`QFQ7$Fbv$!3<+++M05LQFQ7$F]v$!39+++B'G)3QFQ7$Fhu$!35+++A,? !y$FQ7$Fcu$!3++++#)4-ZPFQ7$F^u$!34+++(zy!4PFQ7$Fit$!3;+++f`9mOFQ7$Fdt$ !3&)*****R#)zzh$FQ7$F_t$!3)******R\"=LkNFQ7$Fjs$!3$)*****H\"H%\\]$FQ7$ Fes$!3z******)*QaRMFQ7$F`s$!3%******p'Q%yO$FQ7$F[s$!3')*****zw1&*G$FQ7 $Ffr$!35+++Y?6/KFQ7$Far$!31+++S<56JFQ7$F\\r$!3++++C>t4IFQ7$Fgq$!3-+++; 93**GFQ7$Fbq$!33+++;zAyFFQ7$F]q$!3)******z&)>nk#FQ7$Fhp$!3#******>F\\_ ]#FQ7$Fcp$!37+++Gh#fN#FQ7$F]p$!3#)******)\\-;?#FQ7$Fho$!3%)*****p!)zZ/ #FQ7$Fco$!3++++u`0()=FQ7$F]o$!3********eZ@H'*4\"FQ7$FG$!3g**** **f(z\\U*F-7$FA$!3C+++xY.ayF-7$$\"3%******>x.IX\"F=$!3Y+++8d>$G'F-7$$ \"3++++y_\\t5F7$!38+++\"G!R7ZF-7$$\"3%*******prW$3$F1$!3$)*****fs#fTJF -7$F($!3++++Fjzq:F-F'F*F.F4F:F@FFFKFRFWFgnF\\oFboFgoF\\pFbpFgpF\\qFaqF fqF[rF`rFerFjrF_sFdsFisF^tFctFhtF]uFbuFguF\\vFavFfvF[wF`wFewFjwF_x-%&C OLORG6&%$RGBGF($\"\"#!\"\"F(-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$F- FchnF(7%Fhhn7$$\"+!>%R$3$F_p$\"+EFfTJF[inF\\in7%F_in7$$\"+&z'\\t5FQ$\" +\"G!R7ZF[inF\\in7%Fein7$$\"+oO+`9!#;$\"+8d>$G'F[inF\\in7%F[jn7$$\"+f) HD9\"!#:$\"+xY.ayF[inF\\in7%Fbjn7$$\"+k5&)oiFejn$\"+g(z\\U*F[inF\\in7% Fijn7$$\"+a.(ol#!#9$\"+1'>'*4\"!\"*F\\in7%F_[o7$$\"+H2#RE*Fb[o$\"+,'*z c7Fe[oF\\in7%Fg[o7$$\"+(Rp8x#!#8$\"+Ms399Fe[oF\\in7%F]\\o7$$\"+^[`CtF` \\o$\"+&*Qar:Fe[oF\\in7%Fd\\o7$$\"+m)*>Ynk#Fe[oF\\in7%F`_o7$$\"+e8GDuF`^o$\"+;zAyFFe[oF\\in7%Ff_ o7$$\"+l0R!**)F`^o$\"+;93**GFe[oF\\in7%F\\`o7$$\"+,z\\65F[in$\"+C>t4IF e[oF\\in7%Fb`o7$$\"+i#)Gn5F[in$\"+S<56JFe[oF\\in7%Fh`o7$$\"+!Qx:1\"F[i n$\"+Y?6/KFe[oF\\in7%F^ao7$$\"+>6(p%**F`^o$\"+on]*G$Fe[oF\\in7%Fdao7$$ \"+pMD+()F`^o$\"+nQ%yO$Fe[oF\\in7%Fjao7$$\"+w)pd#pF`^o$\"+**QaRMFe[oF \\in7%F`bo7$$\"+tP2\"o%F`^o$\"+8H%\\]$Fe[oF\\in7%Ffbo7$$\"+M^RD?F`^o$ \"+9=LkNFe[oF\\in7%F\\co7$$!++<\"*R)*F]]o$\"+C)zzh$Fe[oF\\in7%Fbco7$$! +'oTRH%F`^o$\"+f`9mOFe[oF\\in7%Fhco7$$!+-AdcyF`^o$\"+(zy!4PFe[oF\\in7% F^do7$$!+3X'H;\"F[in$\"+#)4-ZPFe[oF\\in7%Fddo7$$!+wnkd:F[in$\"+A,?!y$F e[oF\\in7%Fjdo7$$!+\"H]l'>F[in$\"+B'G)3QFe[oF\\in7%F`eo7$$!+NR)pQ#F[in $\"+M05LQFe[oF\\in7%Ffeo7$$!+bDm;GF[in$\"+F)*=`QFe[oF\\in7%F\\fo7$$!+c Hl`KF[in$\"+&Q\\#pQFe[oF\\in7%Fbfo7$$!+ThK'p$F[in$\"+K0T\")QFe[oF\\in7 %Fhfo7$$!+ZwJVTF[in$\"+oGy*)QFe[oF\\in7%F^go7$$!+'z'\\$f%F[in$\"+6UX%* QFe[oF\\in7%Fdgo7$$!+F`%f/&F[in$\"+$[!\\&*QFe[oF\\in7%Fjgo7$$!+re%**\\ &F[in$\"+-a$H*QFe[oF\\in7%F`ho7$$!+=7)\\&fF[in$\"+%))4o)QFe[oF\\in7%Ff ho7$$!+%3Y2T'F[in$\"+u56xQFe[oF\\in7%F\\io7$$!+eV3h,QFe[oF\\in7%Fdjo7$$!+A.75 ()F[in$\"+Mk1tPFe[oF\\in7%Fjjo7$$!+%>@D=*F[in$\"+OzNSPFe[oF\\in7%F`[p7 $$!+d)fcm*F[in$\"+\\lI.PFe[oF\\in7%Ff[p7$$!+y#=l,\"Fe[o$\"+,$e,\\l\"F e[o$\"+9u@JFe[oF\\ in7%Fd`p7$$!+D-od>Fe[o$\"+SAl*3$Fe[oF\\in7%Fj`p7$$!+`\\]D?Fe[o$\"+buVf IFe[oF\\in7%F`ap7$$!+wwV!4#Fe[o$\"+%H)zGIFe[oF\\in7%Ffap7$$!+.%oE:#Fe[ o$\"+#HBu*HFe[oF\\in7%F\\bp7$$!+L/Q7AFe[o$\"+r:3lHFe[oF\\in7%Fbbp7$$!+ wUupAFe[o$\"+20gJHFe[oF\\in7%Fhbp7$$!+Li\"\\K#Fe[o$\"+A'\\o*GFe[oF\\in 7%F^cp7$$!+!HR!yBFe[o$\"+z!H2'GFe[oF\\in7%Fdcp7$$!+s]CHCFe[o$\"+::;BGF e[oF\\in7%Fjcp7$$!+uilyCFe[o$\"+Aj3%y#Fe[oF\\in7%F`dp7$$!+A#*QEDFe[o$ \"+^dXVFFe[oF\\in7%Ffdp7$$!+BkbsDFe[o$\"+SCB,FFe[oF\\in7%F\\ep7$$!+7!p sh#Fe[o$\"+9\")QdEFe[oF\\in7%Fbep7$$!+&)*Q1m#Fe[o$\"+8L!>h#Fe[oF\\in7% Fhep7$$!+\\6y-FFe[o$\"+w!oZc#Fe[oF\\in7%F^fp7$$!+(e9Qu#Fe[o$\"+RN)f^#F e[oF\\in7%Fdfp7$$!+QM'Qy#Fe[o$\"+#Fe[oF\\in7%Fhhp7$$!+._n5IFe[o$\"+c57K@Fe[ oF\\in7%F^ip7$$!+o2FZIFe[o$\"+c?]s?Fe[oF\\in7%Fdip7$$!+%R$o$3$Fe[o$\"+ kH>7?Fe[oF\\in7%Fjip7$$!+w')))>JFe[o$\"+\\kR^>Fe[oF\\in7%F`jp7$$!+K#=e :$Fe[o$\"+AXI!*=Fe[oF\\in7%Ffjp7$$!+)*\\O\">$Fe[o$\"+x1< Fe[oF\\in7%Fh[q7$$!+>CF%H$Fe[o$\"+pB$ek\"Fe[oF\\in7%F^\\q7$$!+fA\"oK$F e[o$\"+3)z]e\"Fe[oF\\in7%Fd\\q7$$!+xqAeLFe[o$\"+!42X_\"Fe[oF\\in7%Fj\\ q7$$!+;RR)Q$Fe[o$\"+wQ3k9Fe[oF\\in7%F`]q7$$!+7'4sT$Fe[o$\"+(HjPS\"Fe[o F\\in7%Ff]q7$$!+f5fWMFe[o$\"+fx[V8Fe[oF\\in7%F\\^q7$$!+aUZqMFe[o$\"+;O >$G\"Fe[oF\\in7%Fb^q7$$!+,F\"[\\$Fe[o$\"+%p9GA\"Fe[oF\\in7%Fh^q7$$!+%f vv^$Fe[o$\"+2[Gi6Fe[oF\\in7%F^_q7$$!+sguQNFe[o$\"+v%R:5\"Fe[oF\\in7%Fd _q7$$!+5(>$eNFe[o$\"+eq^S5Fe[oF\\in7%Fj_q7$$!+1LIwNFe[o$\"+i`f\"z*F[in F\\in7%F``q7$$!+')Rr#f$Fe[o$\"+x48u\"*F[inF\\in7%Ff`q7$$!+&eyvg$Fe[o$ \"+SpG_&)F[inF\\in7%F\\aq7$$!+dK$4i$Fe[o$\"+1piDzF[inF\\in7%Fbaq7$$!+) >BGj$Fe[o$\"+,rx$H(F[inF\\in7%Fhaq7$$!+*R-Lk$Fe[o$\"+*RPkl'F[inF\\in7% F^bq7$$!+YKV_OFe[o$\"+aaR8gF[inF\\in7%Fdbq7$$!+OeGgOFe[o$\"+,Qak`F[inF \\in7%Fjbq7$$!+)zOpm$Fe[o$\"+/#**)4ZF[inF\\in7%F`cq7$$!+guYsOFe[o$\"+G Fi\\SF[inF\\in7%Ffcq7$$!+>7'pn$Fe[o$\"+e!QSQ$F[inF\\in7%F\\dq7$$!+n+]! o$Fe[o$\"+RZk8FF[inF\\in7%Fbdq7$$!+0/;$o$Fe[o$\"+dC7R?F[inF\\in7%Fhdq7 $$!++%3]o$Fe[o$\"+*pA8O\"F[inF\\in7%F^eq7$$!++a4'o$Fe[o$\"+@NX7oF`^oF \\in7%Fdeq7$$!+2SX'o$Fe[oF(F\\in7%Fjeq7$Feeq$!+@NX7oF`^oF\\in7%F^fq7$F _eq$!+*pA8O\"F[inF\\in7%Fbfq7$Fidq$!+dC7R?F[inF\\in7%Fffq7$Fcdq$!+RZk8 FF[inF\\in7%Fjfq7$F]dq$!+e!QSQ$F[inF\\in7%F^gq7$Fgcq$!+GFi\\SF[inF\\in 7%Fbgq7$Facq$!+/#**)4ZF[inF\\in7%Ffgq7$F[cq$!+,Qak`F[inF\\in7%Fjgq7$Fe bq$!+aaR8gF[inF\\in7%F^hq7$F_bq$!+*RPkl'F[inF\\in7%Fbhq7$Fiaq$!+,rx$H( F[inF\\in7%Ffhq7$Fcaq$!+1piDzF[inF\\in7%Fjhq7$F]aq$!+SpG_&)F[inF\\in7% F^iq7$Fg`q$!+x48u\"*F[inF\\in7%Fbiq7$Fa`q$!+i`f\"z*F[inF\\in7%Ffiq7$F[ `q$!+eq^S5Fe[oF\\in7%Fjiq7$Fe_q$!+v%R:5\"Fe[oF\\in7%F^jq7$F__q$!+2[Gi6 Fe[oF\\in7%Fbjq7$Fi^q$!+%p9GA\"Fe[oF\\in7%Ffjq7$Fc^q$!+;O>$G\"Fe[oF\\i n7%Fjjq7$F]^q$!+fx[V8Fe[oF\\in7%F^[r7$Fg]q$!+(HjPS\"Fe[oF\\in7%Fb[r7$F a]q$!+wQ3k9Fe[oF\\in7%Ff[r7$F[]q$!+!42X_\"Fe[oF\\in7%Fj[r7$Fe\\q$!+3)z ]e\"Fe[oF\\in7%F^\\r7$F_\\q$!+pB$ek\"Fe[oF\\in7%Fb\\r7$Fi[q$!+k>x1Fe[oF\\in7%Ff]r7$F[j p$!+kH>7?Fe[oF\\in7%Fj]r7$Feip$!+c?]s?Fe[oF\\in7%F^^r7$F_ip$!+c57K@Fe[ oF\\in7%Fb^r7$Fihp$!+U,&3>#Fe[oF\\in7%Ff^r7$Fchp$!+cU][AFe[oF\\in7%Fj^ r7$F]hp$!+j<#\\I#Fe[oF\\in7%F^_r7$Fggp$!+t\"p*fBFe[oF\\in7%Fb_r7$Fagp$ !+^Aa8CFe[oF\\in7%Ff_r7$F[gp$!+h#Fe[oF\\in7%Ff` r7$Fcep$!+9\")QdEFe[oF\\in7%Fj`r7$F]ep$!+SCB,FFe[oF\\in7%F^ar7$Fgdp$!+ ^dXVFFe[oF\\in7%Fbar7$Fadp$!+Aj3%y#Fe[oF\\in7%Ffar7$F[dp$!+::;BGFe[oF \\in7%Fjar7$Fecp$!+z!H2'GFe[oF\\in7%F^br7$F_cp$!+A'\\o*GFe[oF\\in7%Fbb r7$Fibp$!+20gJHFe[oF\\in7%Ffbr7$Fcbp$!+r:3lHFe[oF\\in7%Fjbr7$F]bp$!+#H Bu*HFe[oF\\in7%F^cr7$Fgap$!+%H)zGIFe[oF\\in7%Fbcr7$Faap$!+buVfIFe[oF\\ in7%Ffcr7$F[ap$!+SAl*3$Fe[oF\\in7%Fjcr7$Fe`p$!+9]')>JFe[oF\\in7%F^dr7$ F_`p$!+1\"[1:$Fe[oF\\in7%Fbdr7$Fi_p$!+#*zw#=$Fe[oF\\in7%Ffdr7$Fc_p$!+9 u@3h,QFe[oF\\in7%F^hr7$F_jo$!+j#fh #QFe[oF\\in7%Fbhr7$Fiio$!+d[&o%QFe[oF\\in7%Ffhr7$Fcio$!+$*4\"Q'QFe[oF \\in7%Fjhr7$F]io$!+u56xQFe[oF\\in7%F^ir7$Fgho$!+%))4o)QFe[oF\\in7%Fbir 7$Faho$!+-a$H*QFe[oF\\in7%Ffir7$F[ho$!+$[!\\&*QFe[oF\\in7%Fjir7$Fego$! +6UX%*QFe[oF\\in7%F^jr7$F_go$!+oGy*)QFe[oF\\in7%Fbjr7$Fifo$!+K0T\")QFe [oF\\in7%Ffjr7$Fcfo$!+&Q\\#pQFe[oF\\in7%Fjjr7$F]fo$!+F)*=`QFe[oF\\in7% F^[s7$Fgeo$!+M05LQFe[oF\\in7%Fb[s7$Faeo$!+B'G)3QFe[oF\\in7%Ff[s7$F[eo$ !+A,?!y$Fe[oF\\in7%Fj[s7$Fedo$!+#)4-ZPFe[oF\\in7%F^\\s7$F_do$!+(zy!4PF e[oF\\in7%Fb\\s7$Fico$!+f`9mOFe[oF\\in7%Ff\\s7$Fcco$!+C)zzh$Fe[oF\\in7 %Fj\\s7$F]co$!+9=LkNFe[oF\\in7%F^]s7$Fgbo$!+8H%\\]$Fe[oF\\in7%Fb]s7$Fa bo$!+**QaRMFe[oF\\in7%Ff]s7$F[bo$!+nQ%yO$Fe[oF\\in7%Fj]s7$Feao$!+on]*G $Fe[oF\\in7%F^^s7$F_ao$!+Y?6/KFe[oF\\in7%Fb^s7$Fi`o$!+S<56JFe[oF\\in7% Ff^s7$Fc`o$!+C>t4IFe[oF\\in7%Fj^s7$F]`o$!+;93**GFe[oF\\in7%F^_s7$Fg_o$ !+;zAyFFe[oF\\in7%Fb_s7$Fa_o$!+e)>nk#Fe[oF\\in7%Ff_s7$F[_o$!+s#\\_]#Fe [oF\\in7%Fj_s7$Fe^o$!+Gh#fN#Fe[oF\\in7%F^`s7$F^^o$!+*\\-;?#Fe[oF\\in7% Fb`s7$Fh]o$!+2)zZ/#Fe[oF\\in7%Ff`s7$Fb]o$!+u`0()=Fe[oF\\in7%Fj`s7$F[]o $!+fZ@H'*4\"Fe[oF\\in 7%F^bs7$Fjjn$!+g(z\\U*F[inF\\in7%Fbbs7$Fcjn$!+xY.ayF[inF\\in7%Ffbs7$$ \"+sP+`9F^jn$!+8d>$G'F[inF\\in7%Fjbs7$$\"+y_\\t5FQ$!+\"G!R7ZF[inF\\in7 %F`cs7$$\"+qrW$3$F_p$!+EFfTJF[inF\\in7%Ffcs7$F($!+Fjzq:F[inF\\in7%F\\d sF'F\\inFghnF^inFdinFjinFajnFhjnF^[oFf[oF\\\\oFc\\oFi\\oF`]oFf]oF\\^oF c^oFi^oF__oFe_oF[`oFa`oFg`oF]aoFcaoFiaoF_boFeboF[coFacoFgcoF]doFcdoFid oF_eoFeeoF[foFafoFgfoF]goFcgoFigo-F^hn6&F`hn$\"\"&!\"#$\"#bFdds$\"\"\" Fchn-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3-+++w=MBEFQ$\"3Q+++4/0(f%FQ7 $$\"3?+++5*H*>EFQ$\"3z*****fZ2Vg%FQ7$$\"33+++zV`:EFQ$\"3S+++6B*4h%FQ7$ $\"3?+++2%Q-h#FQ$\"3W+++sg)ph%FQ7$$\"3)******p5OTg#FQ$\"3r*****4jy@i%F Q7$$\"3;+++uDL(f#FQ$\"31+++%yoki%FQ7$$\"3;+++DT%**e#FQ$\"3Q+++[ZwHYFQ7 $$\"3A+++(o)4#e#FQ$\"3?+++@])>j%FQ7$$\"3!******fGORd#FQ$\"3E+++(ofIj%F Q7$$\"31+++8&4cc#FQ$\"3t*****4DKHj%FQ7$$\"37+++BQGdDFQ$\"3V+++0@cJYFQ7 $$\"3;+++pu8\\DFQ$\"3r*****RpF*GYFQ7$$\"3#)*****fyg8a#FQ$\"3g******[/. DYFQ7$$\"3&*******\\W:MDFQ$\"3I+++_%**)>YFQ7$$\"33+++lisFDFQ$\"34+++&o 'f8YFQ7$$\"3\")*****4%fGADFQ$\"3v*****H\\Aig%FQ7$$\"30+++qt.=DFQ$\"3%* *****pe?zf%FQ7$$\"32+++-\"o^^#FQ$\"3c*****\\a\")))e%FQ7$$\"3y*****>8OQ ^#FQ$\"3k*****pSV$zXFQ7$$\"3:+++m[:9DFQ$\"3#********y(epXFQ7$$\"3\"*** ***H1yh^#FQ$\"35+++5-$*fXFQ7$$\"3=+++Hu))>DFQ$\"3%******R:/2b%FQ7$$\"3 *******f*e=DDFQ$\"3T+++8*RAa%FQ7$$\"3))******R!**=`#FQ$\"3S+++'*3%[`%F Q7$$\"3)******HW&yRDFQ$\"3G+++L8wGXFQ7$$\"3%*******HYb[DFQ$\"3z*****>K *=CXFQ7$$\"3%******HK*)yb#FQ$\"3*******4#zB@XFQ7$$\"3)******H-ouc#FQ$ \"3W+++y[%*>XFQ7$$\"3%)*****\\X%)pd#FQ$\"3f*****Hhz-_%FQ7$$\"35+++E=;' e#FQ$\"3t*****H*[:AXFQ7$$\"3=+++$RhZf#FQ$\"3N+++;/WDXFQ7$$\"3?+++2ge-E FQ$\"3f*****p4w*HXFQ7$$\"33+++)))z%4EFQ$\"3;+++ZOeNXFQ7$$\"3')*****4(f K:EFQ$\"3O+++Ye2UXFQ7$$\"3@+++%)>/?EFQ$\"3%******>Xj#\\XFQ7$$\"3>++++h dBEFQ$\"3U+++D+'pb%FQ7$$\"3)******zp-fi#FQ$\"3#)*****>2&)\\c%FQ7$$\"3? +++l&=qi#FQ$\"3y*****f\"f;tXFQ7$$\"35+++\\'Rpi#FQ$\"3++++!oQ8e%FQ7$$\" 33+++K%)pDEFQ$\"3w*****4n[$*e%FQF`esF]hn-Fehn6L7%7$$\"+w=MBEFe[o$\"+4/ 0(f%Fe[o7$$\"+5*H*>EFe[o$\"+wuI/YFe[o7$$\"$d#Fdds$\"$d%Fdds7%F`bt7$$\" +zV`:EFe[o$\"+6B*4h%Fe[oFebt7%F[ct7$$\"+2%Q-h#Fe[o$\"+sg)ph%Fe[oFebt7% Fact7$$\"+2h8/EFe[o$\"+J'y@i%Fe[oFebt7%Fgct7$$\"+uDL(f#Fe[o$\"+%yoki%F e[oFebt7%F]dt7$$\"+DT%**e#Fe[o$\"+[ZwHYFe[oFebt7%Fcdt7$$\"+(o)4#e#Fe[o $\"+@])>j%Fe[oFebt7%Fidt7$$\"+'GORd#Fe[o$\"+(ofIj%Fe[oFebt7%F_et7$$\"+ 8&4cc#Fe[o$\"+^A$Hj%Fe[oFebt7%Feet7$$\"+BQGdDFe[o$\"+0@cJYFe[oFebt7%F[ ft7$$\"+pu8\\DFe[o$\"+%pF*GYFe[oFebt7%Faft7$$\"+'yg8a#Fe[o$\"+\\/.DYFe [oFebt7%Fgft7$$\"+]W:MDFe[o$\"+_%**)>YFe[oFebt7%F]gt7$$\"+lisFDFe[o$\" +&o'f8YFe[oFebt7%Fcgt7$$\"+TfGADFe[o$\"+$\\Aig%Fe[oFebt7%Figt7$$\"+qt. =DFe[o$\"+(e?zf%Fe[oFebt7%F_ht7$$\"+-\"o^^#Fe[o$\"+X:)))e%Fe[oFebt7%Fe ht7$$\"+Kh$Q^#Fe[o$\"+2MMzXFe[oFebt7%F[it7$$\"+m[:9DFe[o$\"+!z(epXFe[o Febt7%Fait7$$\"+j!yh^#Fe[o$\"+5-$*fXFe[oFebt7%Fgit7$$\"+Hu))>DFe[o$\"+ aTq]XFe[oFebt7%F]jt7$$\"+'*e=DDFe[o$\"+8*RAa%Fe[oFebt7%Fcjt7$$\"+S!**= `#Fe[o$\"+'*3%[`%Fe[oFebt7%Fijt7$$\"+VayRDFe[o$\"+L8wGXFe[oFebt7%F_[u7 $$\"+IYb[DFe[o$\"+A$*=CXFe[oFebt7%Fe[u7$$\"+B$*)yb#Fe[o$\"+@zB@XFe[oFe bt7%F[\\u7$$\"+B!ouc#Fe[o$\"+y[%*>XFe[oFebt7%Fa\\u7$$\"+bW)pd#Fe[o$\"+ 8'z-_%Fe[oFebt7%Fg\\u7$$\"+E=;'e#Fe[o$\"+$*[:AXFe[oFebt7%F]]u7$$\"+$Rh Zf#Fe[o$\"+;/WDXFe[oFebt7%Fc]u7$$\"+2ge-EFe[o$\"+(4w*HXFe[oFebt7%Fi]u7 $$\"+)))z%4EFe[o$\"+ZOeNXFe[oFebt7%F_^u7$$\"+rfK:EFe[o$\"+Ye2UXFe[oFeb t7%Fe^u7$$\"+%)>/?EFe[o$\"+_ME\\XFe[oFebt7%F[_u7$$\"++hdBEFe[o$\"+D+'p b%Fe[oFebt7%Fa_u7$$\"+)p-fi#Fe[o$\"+s])\\c%Fe[oFebt7%Fg_u7$$\"+l&=qi#F e[o$\"+;f;tXFe[oFebt7%F]`u7$$\"+\\'Rpi#Fe[o$\"+!oQ8e%Fe[oFebt7%Fc`u7$$ \"+K%)pDEFe[o$\"+r'[$*e%Fe[oFebt7%Fi`uF[btFebtF`dsFids-F$6$7K7$Faes$!3 Q+++4/0(f%FQ7$Fdat$!3w*****4n[$*e%FQ7$F_at$!3++++!oQ8e%FQ7$Fj`t$!3y*** **f\"f;tXFQ7$Fe`t$!3#)*****>2&)\\c%FQ7$F``t$!3U+++D+'pb%FQ7$F[`t$!3%** ****>Xj#\\XFQ7$Ff_t$!3O+++Ye2UXFQ7$Fa_t$!3;+++ZOeNXFQ7$F\\_t$!3f*****p 4w*HXFQ7$Fg^t$!3N+++;/WDXFQ7$Fb^t$!3t*****H*[:AXFQ7$F]^t$!3f*****Hhz-_ %FQ7$Fh]t$!3W+++y[%*>XFQ7$Fc]t$!3*******4#zB@XFQ7$F^]t$!3z*****>K*=CXF Q7$Fi\\t$!3G+++L8wGXFQ7$Fd\\t$!3S+++'*3%[`%FQ7$F_\\t$!3T+++8*RAa%FQ7$F j[t$!3%******R:/2b%FQ7$Fe[t$!35+++5-$*fXFQ7$F`[t$!3#********y(epXFQ7$F [[t$!3k*****pSV$zXFQ7$Ffjs$!3c*****\\a\")))e%FQ7$Fajs$!3%******pe?zf%F Q7$F\\js$!3v*****H\\Aig%FQ7$Fgis$!34+++&o'f8YFQ7$Fbis$!3I+++_%**)>YFQ7 $F]is$!3g******[/.DYFQ7$Fhhs$!3r*****RpF*GYFQ7$Fchs$!3V+++0@cJYFQ7$F^h s$!3t*****4DKHj%FQ7$Figs$!3E+++(ofIj%FQ7$Fdgs$!3?+++@])>j%FQ7$F_gs$!3Q +++[ZwHYFQ7$Fjfs$!31+++%yoki%FQ7$Fefs$!3r*****4jy@i%FQ7$F`fs$!3W+++sg) ph%FQ7$F[fs$!3S+++6B*4h%FQ7$Ffes$!3z*****fZ2Vg%FQFbauF]hn-Fehn6L7%7$F \\bt$!+4/0(f%Fe[o7$Fj`u$!+r'[$*e%Fe[o7$Ffbt$!$d%Fdds7%F`iu7$Fd`u$!+!oQ 8e%Fe[oFciu7%Fgiu7$F^`u$!+;f;tXFe[oFciu7%F[ju7$Fh_u$!+s])\\c%Fe[oFciu7 %F_ju7$Fb_u$!+D+'pb%Fe[oFciu7%Fcju7$F\\_u$!+_ME\\XFe[oFciu7%Fgju7$Ff^u $!+Ye2UXFe[oFciu7%F[[v7$F`^u$!+ZOeNXFe[oFciu7%F_[v7$Fj]u$!+(4w*HXFe[oF ciu7%Fc[v7$Fd]u$!+;/WDXFe[oFciu7%Fg[v7$F^]u$!+$*[:AXFe[oFciu7%F[\\v7$F h\\u$!+8'z-_%Fe[oFciu7%F_\\v7$Fb\\u$!+y[%*>XFe[oFciu7%Fc\\v7$F\\\\u$!+ @zB@XFe[oFciu7%Fg\\v7$Ff[u$!+A$*=CXFe[oFciu7%F[]v7$F`[u$!+L8wGXFe[oFci u7%F_]v7$Fjjt$!+'*3%[`%Fe[oFciu7%Fc]v7$Fdjt$!+8*RAa%Fe[oFciu7%Fg]v7$F^ jt$!+aTq]XFe[oFciu7%F[^v7$Fhit$!+5-$*fXFe[oFciu7%F_^v7$Fbit$!+!z(epXFe [oFciu7%Fc^v7$F\\it$!+2MMzXFe[oFciu7%Fg^v7$Ffht$!+X:)))e%Fe[oFciu7%F[_ v7$F`ht$!+(e?zf%Fe[oFciu7%F__v7$Fjgt$!+$\\Aig%Fe[oFciu7%Fc_v7$Fdgt$!+& o'f8YFe[oFciu7%Fg_v7$F^gt$!+_%**)>YFe[oFciu7%F[`v7$Fhft$!+\\/.DYFe[oFc iu7%F_`v7$Fbft$!+%pF*GYFe[oFciu7%Fc`v7$F\\ft$!+0@cJYFe[oFciu7%Fg`v7$Ff et$!+^A$Hj%Fe[oFciu7%F[av7$F`et$!+(ofIj%Fe[oFciu7%F_av7$Fjdt$!+@])>j%F e[oFciu7%Fcav7$Fddt$!+[ZwHYFe[oFciu7%Fgav7$F^dt$!+%yoki%Fe[oFciu7%F[bv 7$Fhct$!+J'y@i%Fe[oFciu7%F_bv7$Fbct$!+sg)ph%Fe[oFciu7%Fcbv7$F\\ct$!+6B *4h%Fe[oFciu7%Fgbv7$Fabt$!+wuI/YFe[oFciu7%F[cvF]iuFciuF`dsFids-F$6%7$7 $$!3o*************Q%FQF(7$$\"3%**************=$FQF(-%'COLOURG6&F`hnF)F )F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>&FQ7$F($\"3Q++++++!>&FQF hcvF[dv-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%F ONTG6#%(DEFAULTG-Fbev6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6 $;$!$R%Fdds$\"$>$Fdds;$!$>&Fdds$\"$>&Fdds" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stability interval for the \+ embedded order 6 scheme is (approximately) " }{XPPEDIT 18 0 "[-3.6865 , 0];" "6#7$,$-%&FloatG6$\"&lo$!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stabi lity regions for the two schemes can be displayed in the same picture. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p_||(1. .6),p||(1..7)],view=[-5.39..3.19,-6.19..6.19],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 501 556 556 {PLOTDATA 2 "65-%'CURVESG6$7e\\ l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3++++!>%R$3$!#F$\"3$)*****fs#fTJ F-7$$\"3/+++&z'\\t5!#D$\"38+++\"G!R7ZF-7$$\"3/+++oO+`9!#C$\"3Y+++8d>$G 'F-7$$\"32+++f)HD9\"!#B$\"3C+++xY.ayF-7$$\"3w*****R1^)oiFC$\"3g******f (z\\U*F-7$$\"33+++a.(ol#!#A$\"33+++1'>'*4\"!#<7$$\"3K+++H2#RE*FN$\"3++ ++,'*zc7FQ7$$\"3#)*****pRp8x#!#@$\"3#******RB(399FQ7$$\"3\")*****4&[`C tFZ$\"3-+++&*Qar:FQ7$$\"3%******f')*>YI_BV\"!#>$\"3#)******)\\-;?#FQ7$$\"3%)*****4ln$pCF_p$\"37+++G h#fN#FQ7$$\"3G+++4Or1RF_p$\"3#******>F\\_]#FQ7$$\"3)******paF#QcF_p$\" 3)******z&)>nk#FQ7$$\"3L*****zN\"GDuF_p$\"33+++;zAyFFQ7$$\"3W+++l0R!** )F_p$\"3-+++;93**GFQ7$$\"3)******4!z\\65F-$\"3++++C>t4IFQ7$$\"3-+++i#) Gn5F-$\"31+++S<56JFQ7$$\"3-+++!Qx:1\"F-$\"35+++Y?6/KFQ7$$\"3U+++>6(p%* *F_p$\"3')*****zw1&*G$FQ7$$\"3T******oMD+()F_p$\"3%******p'Q%yO$FQ7$$ \"3*******f()pd#pF_p$\"3z******)*QaRMFQ7$$\"3I+++tP2\"o%F_p$\"3$)***** H\"H%\\]$FQ7$$\"3'******R8&RD?F_p$\"3)******R\"=LkNFQ7$$!3Z++++<\"*R)* F_o$\"3&)*****R#)zzh$FQ7$$!3G+++'oTRH%F_p$\"3;+++f`9mOFQ7$$!3i+++-Adcy F_p$\"34+++(zy!4PFQ7$$!31+++3X'H;\"F-$\"3++++#)4-ZPFQ7$$!37+++wnkd:F-$ \"35+++A,?!y$FQ7$$!3!******4H]l'>F-$\"39+++B'G)3QFQ7$$!3%******\\$R)pQ #F-$\"3<+++M05LQFQ7$$!3=+++bDm;GF-$\"3-+++F)*=`QFQ7$$!3%******f&Hl`KF- $\"3#)*****\\Q\\#pQFQ7$$!3y*****49Ejp$F-$\"3')*****>`59)QFQ7$$!3%***** *pkSNH*QFQ7$$! 3Y*****z@\")\\&fF-$\"3#)*****R))4o)QFQ7$$!3o*****R3Y2T'F-$\"35+++u56xQ FQ7$$!3)******zNur'oF-$\"3'******H*4\"Q'QFQ7$$!37+++7n[CtF-$\"3%)***** p&[&o%QFQ7$$!3A+++?iE$y(F-$\"34+++j#fh#QFQ7$$!3b+++CPdW#)F-$\"3(****** *=3h,QFQ7$$!33+++A.75()F-$\"3)******RVmIx$FQ7$$!3c*****R>@D=*F-$\"36++ +OzNSPFQ7$$!3k*****p&)fcm*F-$\"36+++\\lI.PFQ7$$!3%******zF=l,\"FQ$\"3) )*****4Ie,\\l\"FQ$\"3$)*****RT57`t\"FQ$\"3/+++#*zw#=$FQ7$$!3#******HUqE\"=FQ$\"3y*****f5[1:$FQ7$ $!3-+++\\?y')=FQ$\"3;+++9]')>JFQ7$$!31+++D-od>FQ$\"3%*******RAl*3$FQ7$ $!3++++`\\]D?FQ$\"3')*****\\XP%fIFQ7$$!33+++wwV!4#FQ$\"3++++%H)zGIFQ7$ $!3%)*****HSoE:#FQ$\"37+++#HBu*HFQ7$$!3()*****HV!Q7AFQ$\"3#)*****4d\"3 lHFQ7$$!3'******fFW(pAFQ$\"3>+++20gJHFQ7$$!3<+++Li\"\\K#FQ$\"3%******> i\\o*GFQ7$$!3!)*******GR!yBFQ$\"3'*******y!H2'GFQ7$$!31+++s]CHCFQ$\"3z *****\\^hJ#GFQ7$$!3++++uilyCFQ$\"3%)*****>K'3%y#FQ7$$!3;+++A#*QEDFQ$\" 3%******4vbMu#FQ7$$!3)******HUcDd#FQ$\"32+++SCB,FFQ7$$!3&******>,psh#F Q$\"3?+++9\")QdEFQ7$$!3%)*****\\)*Q1m#FQ$\"3++++8L!>h#FQ7$$!3A+++\\6y- FFQ$\"3++++w!oZc#FQ7$$!39+++(e9Qu#FQ$\"30+++RN)f^#FQ7$$!3-+++QM'Qy#FQ$ \"3'******pTkbY#FQ7$$!3#******4lcI#GFQ$\"3(******4DUNT#FQ7$$!3')*****p ME:'GFQ$\"33+++t\"p*fBFQ7$$!3/+++-WS**GFQ$\"3&)*****Hw@\\I#FQ7$$!3\"** ****>(p\"o$HFQ$\"3#******fD/&[AFQ7$$!3;+++)*p(Q(HFQ$\"3%******>9]3>#FQ 7$$!3\"******H?v1,$FQ$\"3\"******f0@@8#FQ7$$!3=+++o2FZIFQ$\"3#)*****f0 -D2#FQ7$$!3#******RR$o$3$FQ$\"3.+++kH>7?FQ7$$!3(******fn)))>JFQ$\"3)** *****[kR^>FQ7$$!37+++K#=e:$FQ$\"33+++AXI!*=FQ7$$!31+++)*\\O\">$FQ$\"3& ******p\"e3H=FQ7$$!3*******4x#REKFQ$\"3#*******))e(yw\"FQ7$$!3#)*****f 6[2E$FQ$\"34+++k>x1$G\" FQ7$$!3')*****4q7[\\$FQ$\"3)******Rp9GA\"FQ7$$!3>+++%fvv^$FQ$\"3/+++2[ Gi6FQ7$$!3\"******>2Y(QNFQ$\"30+++v%R:5\"FQ7$$!39+++5(>$eNFQ$\"30+++eq ^S5FQ7$$!3!)*****fI.jd$FQ$\"3e*****>O&f\"z*F-7$$!39+++')Rr#f$FQ$\"39++ +x48u\"*F-7$$!3<+++&eyvg$FQ$\"37+++SpG_&)F-7$$!3\"******pDL4i$FQ$\"3`* ****f!piDzF-7$$!34+++)>BGj$FQ$\"3m*****45xPH(F-7$$!3(*******)R-Lk$FQ$ \"3K+++*RPkl'F-7$$!3?+++YKV_OFQ$\"34+++aaR8gF-7$$!3!******f$eGgOFQ$\"3 ;+++,Qak`F-7$$!3/+++)zOpm$FQ$\"3%)*****R?**)4ZF-7$$!3-+++guYsOFQ$\"3C+ ++GFi\\SF-7$$!3%)******=7'pn$FQ$\"37+++e!QSQ$F-7$$!35+++n+]!o$FQ$\"39+ ++RZk8FF-7$$!3=+++0/;$o$FQ$\"3&******pXA\"R?F-7$$!31++++%3]o$FQ$\"3#** *****)pA8O\"F-7$$!37++++a4'o$FQ$\"3N+++@NX7oF_p7$$!3-+++2SX'o$FQF(7$F[ am$!3N+++@NX7oF_p7$Ff`m$!3#*******)pA8O\"F-7$Fa`m$!3&******pXA\"R?F-7$ F\\`m$!39+++RZk8FF-7$Fg_m$!37+++e!QSQ$F-7$Fb_m$!3C+++GFi\\SF-7$F]_m$!3 %)*****R?**)4ZF-7$Fh^m$!3;+++,Qak`F-7$Fc^m$!34+++aaR8gF-7$F^^m$!3K+++* RPkl'F-7$Fi]m$!3m*****45xPH(F-7$Fd]m$!3`*****f!piDzF-7$F_]m$!37+++SpG_ &)F-7$Fj\\m$!39+++x48u\"*F-7$Fe\\m$!3e*****>O&f\"z*F-7$F`\\m$!30+++eq^ S5FQ7$F[\\m$!30+++v%R:5\"FQ7$Ff[m$!3/+++2[Gi6FQ7$Fa[m$!3)******Rp9GA\" FQ7$F\\[m$!3#******fh$>$G\"FQ7$Fgjl$!31+++fx[V8FQ7$Fbjl$!3-+++(HjPS\"F Q7$F]jl$!3!******f(Q3k9FQ7$Fhil$!3&********32X_\"FQ7$Fcil$!3#******z!) z]e\"FQ7$F^il$!33+++pB$ek\"FQ7$Fihl$!34+++k>x1FQ7$F`gl$!3.+++kH>7?FQ7$F[gl$!3#)*****f0-D2#FQ7$Fffl$!3\"******f0@ @8#FQ7$Fafl$!3%******>9]3>#FQ7$F\\fl$!3#******fD/&[AFQ7$Fgel$!3&)***** Hw@\\I#FQ7$Fbel$!33+++t\"p*fBFQ7$F]el$!3(******4DUNT#FQ7$Fhdl$!3'***** *pTkbY#FQ7$Fcdl$!30+++RN)f^#FQ7$F^dl$!3++++w!oZc#FQ7$Ficl$!3++++8L!>h# FQ7$Fdcl$!3?+++9\")QdEFQ7$F_cl$!32+++SCB,FFQ7$Fjbl$!3%******4vbMu#FQ7$ Febl$!3%)*****>K'3%y#FQ7$F`bl$!3z*****\\^hJ#GFQ7$F[bl$!3'*******y!H2'G FQ7$Ffal$!3%******>i\\o*GFQ7$Faal$!3>+++20gJHFQ7$F\\al$!3#)*****4d\"3l HFQ7$Fg`l$!37+++#HBu*HFQ7$Fb`l$!3++++%H)zGIFQ7$F]`l$!3')*****\\XP%fIFQ 7$Fh_l$!3%*******RAl*3$FQ7$Fc_l$!3;+++9]')>JFQ7$F^_l$!3y*****f5[1:$FQ7 $Fi^l$!3/+++#*zw#=$FQ7$Fd^l$!3$)*****RTSNH*QFQ7$F`x$!35+++$[!\\&*Q FQ7$F[x$!3A+++6UX%*QFQ7$Ffw$!3;+++oGy*)QFQ7$Faw$!3')*****>`59)QFQ7$F\\ w$!3#)*****\\Q\\#pQFQ7$Fgv$!3-+++F)*=`QFQ7$Fbv$!3<+++M05LQFQ7$F]v$!39+ ++B'G)3QFQ7$Fhu$!35+++A,?!y$FQ7$Fcu$!3++++#)4-ZPFQ7$F^u$!34+++(zy!4PFQ 7$Fit$!3;+++f`9mOFQ7$Fdt$!3&)*****R#)zzh$FQ7$F_t$!3)******R\"=LkNFQ7$F js$!3$)*****H\"H%\\]$FQ7$Fes$!3z******)*QaRMFQ7$F`s$!3%******p'Q%yO$FQ 7$F[s$!3')*****zw1&*G$FQ7$Ffr$!35+++Y?6/KFQ7$Far$!31+++S<56JFQ7$F\\r$! 3++++C>t4IFQ7$Fgq$!3-+++;93**GFQ7$Fbq$!33+++;zAyFFQ7$F]q$!3)******z&)> nk#FQ7$Fhp$!3#******>F\\_]#FQ7$Fcp$!37+++Gh#fN#FQ7$F]p$!3#)******)\\-; ?#FQ7$Fho$!3%)*****p!)zZ/#FQ7$Fco$!3++++u`0()=FQ7$F]o$!3********eZ@H'*4\"FQ7$FG$!3g******f(z\\U*F-7$FA$!3C+++xY.ayF-7$$\"3%******>x. IX\"F=$!3Y+++8d>$G'F-7$$\"3++++y_\\t5F7$!38+++\"G!R7ZF-7$$\"3%*******p rW$3$F1$!3$)*****fs#fTJF-7$F($!3++++Fjzq:F-F'F*F.F4F:F@FFFKFRFWFgnF\\o FboFgoF\\pFbpFgpF\\qFaqFfqF[rF`rFerFjrF_sFdsFisF^tFctFhtF]uFbuFguF\\vF avFfvF[wF`wFewFjwF_x-%&COLORG6&%$RGBGF($\"\"#!\"\"F(-%)POLYGONSG6f\\l7 %F'7$F($\"+Fjzq:!#57$$F-FchnF(7%Fhhn7$$\"+!>%R$3$F_p$\"+EFfTJF[inF\\in 7%F_in7$$\"+&z'\\t5FQ$\"+\"G!R7ZF[inF\\in7%Fein7$$\"+oO+`9!#;$\"+8d>$G 'F[inF\\in7%F[jn7$$\"+f)HD9\"!#:$\"+xY.ayF[inF\\in7%Fbjn7$$\"+k5&)oiFe jn$\"+g(z\\U*F[inF\\in7%Fijn7$$\"+a.(ol#!#9$\"+1'>'*4\"!\"*F\\in7%F_[o 7$$\"+H2#RE*Fb[o$\"+,'*zc7Fe[oF\\in7%Fg[o7$$\"+(Rp8x#!#8$\"+Ms399Fe[oF \\in7%F]\\o7$$\"+^[`CtF`\\o$\"+&*Qar:Fe[oF\\in7%Fd\\o7$$\"+m)*>Ynk#Fe[oF\\in7%F`_o7$$\"+e8GDuF` ^o$\"+;zAyFFe[oF\\in7%Ff_o7$$\"+l0R!**)F`^o$\"+;93**GFe[oF\\in7%F\\`o7 $$\"+,z\\65F[in$\"+C>t4IFe[oF\\in7%Fb`o7$$\"+i#)Gn5F[in$\"+S<56JFe[oF \\in7%Fh`o7$$\"+!Qx:1\"F[in$\"+Y?6/KFe[oF\\in7%F^ao7$$\"+>6(p%**F`^o$ \"+on]*G$Fe[oF\\in7%Fdao7$$\"+pMD+()F`^o$\"+nQ%yO$Fe[oF\\in7%Fjao7$$\" +w)pd#pF`^o$\"+**QaRMFe[oF\\in7%F`bo7$$\"+tP2\"o%F`^o$\"+8H%\\]$Fe[oF \\in7%Ffbo7$$\"+M^RD?F`^o$\"+9=LkNFe[oF\\in7%F\\co7$$!++<\"*R)*F]]o$\" +C)zzh$Fe[oF\\in7%Fbco7$$!+'oTRH%F`^o$\"+f`9mOFe[oF\\in7%Fhco7$$!+-Adc yF`^o$\"+(zy!4PFe[oF\\in7%F^do7$$!+3X'H;\"F[in$\"+#)4-ZPFe[oF\\in7%Fdd o7$$!+wnkd:F[in$\"+A,?!y$Fe[oF\\in7%Fjdo7$$!+\"H]l'>F[in$\"+B'G)3QFe[o F\\in7%F`eo7$$!+NR)pQ#F[in$\"+M05LQFe[oF\\in7%Ffeo7$$!+bDm;GF[in$\"+F) *=`QFe[oF\\in7%F\\fo7$$!+cHl`KF[in$\"+&Q\\#pQFe[oF\\in7%Fbfo7$$!+ThK'p $F[in$\"+K0T\")QFe[oF\\in7%Fhfo7$$!+ZwJVTF[in$\"+oGy*)QFe[oF\\in7%F^go 7$$!+'z'\\$f%F[in$\"+6UX%*QFe[oF\\in7%Fdgo7$$!+F`%f/&F[in$\"+$[!\\&*QF e[oF\\in7%Fjgo7$$!+re%**\\&F[in$\"+-a$H*QFe[oF\\in7%F`ho7$$!+=7)\\&fF[ in$\"+%))4o)QFe[oF\\in7%Ffho7$$!+%3Y2T'F[in$\"+u56xQFe[oF\\in7%F\\io7$ $!+eV3h ,QFe[oF\\in7%Fdjo7$$!+A.75()F[in$\"+Mk1tPFe[oF\\in7%Fjjo7$$!+%>@D=*F[i n$\"+OzNSPFe[oF\\in7%F`[p7$$!+d)fcm*F[in$\"+\\lI.PFe[oF\\in7%Ff[p7$$!+ y#=l,\"Fe[o$\"+,$e,\\l\"Fe[o$\"+9u@JFe[oF\\in7%Fd`p7$$!+D-od>Fe[o$\"+SAl*3$Fe[oF\\in7%F j`p7$$!+`\\]D?Fe[o$\"+buVfIFe[oF\\in7%F`ap7$$!+wwV!4#Fe[o$\"+%H)zGIFe[ oF\\in7%Ffap7$$!+.%oE:#Fe[o$\"+#HBu*HFe[oF\\in7%F\\bp7$$!+L/Q7AFe[o$\" +r:3lHFe[oF\\in7%Fbbp7$$!+wUupAFe[o$\"+20gJHFe[oF\\in7%Fhbp7$$!+Li\"\\ K#Fe[o$\"+A'\\o*GFe[oF\\in7%F^cp7$$!+!HR!yBFe[o$\"+z!H2'GFe[oF\\in7%Fd cp7$$!+s]CHCFe[o$\"+::;BGFe[oF\\in7%Fjcp7$$!+uilyCFe[o$\"+Aj3%y#Fe[oF \\in7%F`dp7$$!+A#*QEDFe[o$\"+^dXVFFe[oF\\in7%Ffdp7$$!+BkbsDFe[o$\"+SCB ,FFe[oF\\in7%F\\ep7$$!+7!psh#Fe[o$\"+9\")QdEFe[oF\\in7%Fbep7$$!+&)*Q1m #Fe[o$\"+8L!>h#Fe[oF\\in7%Fhep7$$!+\\6y-FFe[o$\"+w!oZc#Fe[oF\\in7%F^fp 7$$!+(e9Qu#Fe[o$\"+RN)f^#Fe[oF\\in7%Fdfp7$$!+QM'Qy#Fe[o$\"+#Fe[oF\\in7%Fhh p7$$!+._n5IFe[o$\"+c57K@Fe[oF\\in7%F^ip7$$!+o2FZIFe[o$\"+c?]s?Fe[oF\\i n7%Fdip7$$!+%R$o$3$Fe[o$\"+kH>7?Fe[oF\\in7%Fjip7$$!+w')))>JFe[o$\"+\\k R^>Fe[oF\\in7%F`jp7$$!+K#=e:$Fe[o$\"+AXI!*=Fe[oF\\in7%Ffjp7$$!+)*\\O\" >$Fe[o$\"+x1CF%H$Fe[o$\"+pB$ek\"Fe [oF\\in7%F^\\q7$$!+fA\"oK$Fe[o$\"+3)z]e\"Fe[oF\\in7%Fd\\q7$$!+xqAeLFe[ o$\"+!42X_\"Fe[oF\\in7%Fj\\q7$$!+;RR)Q$Fe[o$\"+wQ3k9Fe[oF\\in7%F`]q7$$ !+7'4sT$Fe[o$\"+(HjPS\"Fe[oF\\in7%Ff]q7$$!+f5fWMFe[o$\"+fx[V8Fe[oF\\in 7%F\\^q7$$!+aUZqMFe[o$\"+;O>$G\"Fe[oF\\in7%Fb^q7$$!+,F\"[\\$Fe[o$\"+%p 9GA\"Fe[oF\\in7%Fh^q7$$!+%fvv^$Fe[o$\"+2[Gi6Fe[oF\\in7%F^_q7$$!+sguQNF e[o$\"+v%R:5\"Fe[oF\\in7%Fd_q7$$!+5(>$eNFe[o$\"+eq^S5Fe[oF\\in7%Fj_q7$ $!+1LIwNFe[o$\"+i`f\"z*F[inF\\in7%F``q7$$!+')Rr#f$Fe[o$\"+x48u\"*F[inF \\in7%Ff`q7$$!+&eyvg$Fe[o$\"+SpG_&)F[inF\\in7%F\\aq7$$!+dK$4i$Fe[o$\"+ 1piDzF[inF\\in7%Fbaq7$$!+)>BGj$Fe[o$\"+,rx$H(F[inF\\in7%Fhaq7$$!+*R-Lk $Fe[o$\"+*RPkl'F[inF\\in7%F^bq7$$!+YKV_OFe[o$\"+aaR8gF[inF\\in7%Fdbq7$ $!+OeGgOFe[o$\"+,Qak`F[inF\\in7%Fjbq7$$!+)zOpm$Fe[o$\"+/#**)4ZF[inF\\i n7%F`cq7$$!+guYsOFe[o$\"+GFi\\SF[inF\\in7%Ffcq7$$!+>7'pn$Fe[o$\"+e!QSQ $F[inF\\in7%F\\dq7$$!+n+]!o$Fe[o$\"+RZk8FF[inF\\in7%Fbdq7$$!+0/;$o$Fe[ o$\"+dC7R?F[inF\\in7%Fhdq7$$!++%3]o$Fe[o$\"+*pA8O\"F[inF\\in7%F^eq7$$! ++a4'o$Fe[o$\"+@NX7oF`^oF\\in7%Fdeq7$$!+2SX'o$Fe[oF(F\\in7%Fjeq7$Feeq$ !+@NX7oF`^oF\\in7%F^fq7$F_eq$!+*pA8O\"F[inF\\in7%Fbfq7$Fidq$!+dC7R?F[i nF\\in7%Fffq7$Fcdq$!+RZk8FF[inF\\in7%Fjfq7$F]dq$!+e!QSQ$F[inF\\in7%F^g q7$Fgcq$!+GFi\\SF[inF\\in7%Fbgq7$Facq$!+/#**)4ZF[inF\\in7%Ffgq7$F[cq$! +,Qak`F[inF\\in7%Fjgq7$Febq$!+aaR8gF[inF\\in7%F^hq7$F_bq$!+*RPkl'F[inF \\in7%Fbhq7$Fiaq$!+,rx$H(F[inF\\in7%Ffhq7$Fcaq$!+1piDzF[inF\\in7%Fjhq7 $F]aq$!+SpG_&)F[inF\\in7%F^iq7$Fg`q$!+x48u\"*F[inF\\in7%Fbiq7$Fa`q$!+i `f\"z*F[inF\\in7%Ffiq7$F[`q$!+eq^S5Fe[oF\\in7%Fjiq7$Fe_q$!+v%R:5\"Fe[o F\\in7%F^jq7$F__q$!+2[Gi6Fe[oF\\in7%Fbjq7$Fi^q$!+%p9GA\"Fe[oF\\in7%Ffj q7$Fc^q$!+;O>$G\"Fe[oF\\in7%Fjjq7$F]^q$!+fx[V8Fe[oF\\in7%F^[r7$Fg]q$!+ (HjPS\"Fe[oF\\in7%Fb[r7$Fa]q$!+wQ3k9Fe[oF\\in7%Ff[r7$F[]q$!+!42X_\"Fe[ oF\\in7%Fj[r7$Fe\\q$!+3)z]e\"Fe[oF\\in7%F^\\r7$F_\\q$!+pB$ek\"Fe[oF\\i n7%Fb\\r7$Fi[q$!+k>x1Fe[oF\\in7%Ff]r7$F[jp$!+kH>7?Fe[oF\\in7%Fj]r7$Feip$!+c?]s?Fe[oF \\in7%F^^r7$F_ip$!+c57K@Fe[oF\\in7%Fb^r7$Fihp$!+U,&3>#Fe[oF\\in7%Ff^r7 $Fchp$!+cU][AFe[oF\\in7%Fj^r7$F]hp$!+j<#\\I#Fe[oF\\in7%F^_r7$Fggp$!+t \"p*fBFe[oF\\in7%Fb_r7$Fagp$!+^Aa8CFe[oF\\in7%Ff_r7$F[gp$!+h#Fe[oF\\in7%Ff`r7$Fcep$!+9\")QdEFe[oF\\in7%Fj`r7$F]ep$!+ SCB,FFe[oF\\in7%F^ar7$Fgdp$!+^dXVFFe[oF\\in7%Fbar7$Fadp$!+Aj3%y#Fe[oF \\in7%Ffar7$F[dp$!+::;BGFe[oF\\in7%Fjar7$Fecp$!+z!H2'GFe[oF\\in7%F^br7 $F_cp$!+A'\\o*GFe[oF\\in7%Fbbr7$Fibp$!+20gJHFe[oF\\in7%Ffbr7$Fcbp$!+r: 3lHFe[oF\\in7%Fjbr7$F]bp$!+#HBu*HFe[oF\\in7%F^cr7$Fgap$!+%H)zGIFe[oF\\ in7%Fbcr7$Faap$!+buVfIFe[oF\\in7%Ffcr7$F[ap$!+SAl*3$Fe[oF\\in7%Fjcr7$F e`p$!+9]')>JFe[oF\\in7%F^dr7$F_`p$!+1\"[1:$Fe[oF\\in7%Fbdr7$Fi_p$!+#*z w#=$Fe[oF\\in7%Ffdr7$Fc_p$!+9u@3h, QFe[oF\\in7%F^hr7$F_jo$!+j#fh#QFe[oF\\in7%Fbhr7$Fiio$!+d[&o%QFe[oF\\in 7%Ffhr7$Fcio$!+$*4\"Q'QFe[oF\\in7%Fjhr7$F]io$!+u56xQFe[oF\\in7%F^ir7$F gho$!+%))4o)QFe[oF\\in7%Fbir7$Faho$!+-a$H*QFe[oF\\in7%Ffir7$F[ho$!+$[! \\&*QFe[oF\\in7%Fjir7$Fego$!+6UX%*QFe[oF\\in7%F^jr7$F_go$!+oGy*)QFe[oF \\in7%Fbjr7$Fifo$!+K0T\")QFe[oF\\in7%Ffjr7$Fcfo$!+&Q\\#pQFe[oF\\in7%Fj jr7$F]fo$!+F)*=`QFe[oF\\in7%F^[s7$Fgeo$!+M05LQFe[oF\\in7%Fb[s7$Faeo$!+ B'G)3QFe[oF\\in7%Ff[s7$F[eo$!+A,?!y$Fe[oF\\in7%Fj[s7$Fedo$!+#)4-ZPFe[o F\\in7%F^\\s7$F_do$!+(zy!4PFe[oF\\in7%Fb\\s7$Fico$!+f`9mOFe[oF\\in7%Ff \\s7$Fcco$!+C)zzh$Fe[oF\\in7%Fj\\s7$F]co$!+9=LkNFe[oF\\in7%F^]s7$Fgbo$ !+8H%\\]$Fe[oF\\in7%Fb]s7$Fabo$!+**QaRMFe[oF\\in7%Ff]s7$F[bo$!+nQ%yO$F e[oF\\in7%Fj]s7$Feao$!+on]*G$Fe[oF\\in7%F^^s7$F_ao$!+Y?6/KFe[oF\\in7%F b^s7$Fi`o$!+S<56JFe[oF\\in7%Ff^s7$Fc`o$!+C>t4IFe[oF\\in7%Fj^s7$F]`o$!+ ;93**GFe[oF\\in7%F^_s7$Fg_o$!+;zAyFFe[oF\\in7%Fb_s7$Fa_o$!+e)>nk#Fe[oF \\in7%Ff_s7$F[_o$!+s#\\_]#Fe[oF\\in7%Fj_s7$Fe^o$!+Gh#fN#Fe[oF\\in7%F^` s7$F^^o$!+*\\-;?#Fe[oF\\in7%Fb`s7$Fh]o$!+2)zZ/#Fe[oF\\in7%Ff`s7$Fb]o$! +u`0()=Fe[oF\\in7%Fj`s7$F[]o$!+fZ@H'*4\"Fe[oF\\in7%F^bs7$Fjjn$!+g(z\\U*F[inF\\in7%Fbbs7$Fcj n$!+xY.ayF[inF\\in7%Ffbs7$$\"+sP+`9F^jn$!+8d>$G'F[inF\\in7%Fjbs7$$\"+y _\\t5FQ$!+\"G!R7ZF[inF\\in7%F`cs7$$\"+qrW$3$F_p$!+EFfTJF[inF\\in7%Ffcs 7$F($!+Fjzq:F[inF\\in7%F\\dsF'F\\inFghnF^inFdinFjinFajnFhjnF^[oFf[oF\\ \\oFc\\oFi\\oF`]oFf]oF\\^oFc^oFi^oF__oFe_oF[`oFa`oFg`oF]aoFcaoFiaoF_bo FeboF[coFacoFgcoF]doFcdoFidoF_eoFeeoF[foFafoFgfoF]goFcgoFigo-F^hn6&F`h n$\"\"&!\"#$\"#bFdds$\"\"\"Fchn-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3- +++w=MBEFQ$\"3Q+++4/0(f%FQ7$$\"3?+++5*H*>EFQ$\"3z*****fZ2Vg%FQ7$$\"33+ ++zV`:EFQ$\"3S+++6B*4h%FQ7$$\"3?+++2%Q-h#FQ$\"3W+++sg)ph%FQ7$$\"3)**** **p5OTg#FQ$\"3r*****4jy@i%FQ7$$\"3;+++uDL(f#FQ$\"31+++%yoki%FQ7$$\"3;+ ++DT%**e#FQ$\"3Q+++[ZwHYFQ7$$\"3A+++(o)4#e#FQ$\"3?+++@])>j%FQ7$$\"3!** ****fGORd#FQ$\"3E+++(ofIj%FQ7$$\"31+++8&4cc#FQ$\"3t*****4DKHj%FQ7$$\"3 7+++BQGdDFQ$\"3V+++0@cJYFQ7$$\"3;+++pu8\\DFQ$\"3r*****RpF*GYFQ7$$\"3#) *****fyg8a#FQ$\"3g******[/.DYFQ7$$\"3&*******\\W:MDFQ$\"3I+++_%**)>YFQ 7$$\"33+++lisFDFQ$\"34+++&o'f8YFQ7$$\"3\")*****4%fGADFQ$\"3v*****H\\Ai g%FQ7$$\"30+++qt.=DFQ$\"3%******pe?zf%FQ7$$\"32+++-\"o^^#FQ$\"3c***** \\a\")))e%FQ7$$\"3y*****>8OQ^#FQ$\"3k*****pSV$zXFQ7$$\"3:+++m[:9DFQ$\" 3#********y(epXFQ7$$\"3\"******H1yh^#FQ$\"35+++5-$*fXFQ7$$\"3=+++Hu))> DFQ$\"3%******R:/2b%FQ7$$\"3*******f*e=DDFQ$\"3T+++8*RAa%FQ7$$\"3))*** ***R!**=`#FQ$\"3S+++'*3%[`%FQ7$$\"3)******HW&yRDFQ$\"3G+++L8wGXFQ7$$\" 3%*******HYb[DFQ$\"3z*****>K*=CXFQ7$$\"3%******HK*)yb#FQ$\"3*******4#z B@XFQ7$$\"3)******H-ouc#FQ$\"3W+++y[%*>XFQ7$$\"3%)*****\\X%)pd#FQ$\"3f *****Hhz-_%FQ7$$\"35+++E=;'e#FQ$\"3t*****H*[:AXFQ7$$\"3=+++$RhZf#FQ$\" 3N+++;/WDXFQ7$$\"3?+++2ge-EFQ$\"3f*****p4w*HXFQ7$$\"33+++)))z%4EFQ$\"3 ;+++ZOeNXFQ7$$\"3')*****4(fK:EFQ$\"3O+++Ye2UXFQ7$$\"3@+++%)>/?EFQ$\"3% ******>Xj#\\XFQ7$$\"3>++++hdBEFQ$\"3U+++D+'pb%FQ7$$\"3)******zp-fi#FQ$ \"3#)*****>2&)\\c%FQ7$$\"3?+++l&=qi#FQ$\"3y*****f\"f;tXFQ7$$\"35+++\\' Rpi#FQ$\"3++++!oQ8e%FQ7$$\"33+++K%)pDEFQ$\"3w*****4n[$*e%FQF`esF]hn-Fe hn6L7%7$$\"+w=MBEFe[o$\"+4/0(f%Fe[o7$$\"+5*H*>EFe[o$\"+wuI/YFe[o7$$\"$ d#Fdds$\"$d%Fdds7%F`bt7$$\"+zV`:EFe[o$\"+6B*4h%Fe[oFebt7%F[ct7$$\"+2%Q -h#Fe[o$\"+sg)ph%Fe[oFebt7%Fact7$$\"+2h8/EFe[o$\"+J'y@i%Fe[oFebt7%Fgct 7$$\"+uDL(f#Fe[o$\"+%yoki%Fe[oFebt7%F]dt7$$\"+DT%**e#Fe[o$\"+[ZwHYFe[o Febt7%Fcdt7$$\"+(o)4#e#Fe[o$\"+@])>j%Fe[oFebt7%Fidt7$$\"+'GORd#Fe[o$\" +(ofIj%Fe[oFebt7%F_et7$$\"+8&4cc#Fe[o$\"+^A$Hj%Fe[oFebt7%Feet7$$\"+BQG dDFe[o$\"+0@cJYFe[oFebt7%F[ft7$$\"+pu8\\DFe[o$\"+%pF*GYFe[oFebt7%Faft7 $$\"+'yg8a#Fe[o$\"+\\/.DYFe[oFebt7%Fgft7$$\"+]W:MDFe[o$\"+_%**)>YFe[oF ebt7%F]gt7$$\"+lisFDFe[o$\"+&o'f8YFe[oFebt7%Fcgt7$$\"+TfGADFe[o$\"+$\\ Aig%Fe[oFebt7%Figt7$$\"+qt.=DFe[o$\"+(e?zf%Fe[oFebt7%F_ht7$$\"+-\"o^^# Fe[o$\"+X:)))e%Fe[oFebt7%Feht7$$\"+Kh$Q^#Fe[o$\"+2MMzXFe[oFebt7%F[it7$ $\"+m[:9DFe[o$\"+!z(epXFe[oFebt7%Fait7$$\"+j!yh^#Fe[o$\"+5-$*fXFe[oFeb t7%Fgit7$$\"+Hu))>DFe[o$\"+aTq]XFe[oFebt7%F]jt7$$\"+'*e=DDFe[o$\"+8*RA a%Fe[oFebt7%Fcjt7$$\"+S!**=`#Fe[o$\"+'*3%[`%Fe[oFebt7%Fijt7$$\"+VayRDF e[o$\"+L8wGXFe[oFebt7%F_[u7$$\"+IYb[DFe[o$\"+A$*=CXFe[oFebt7%Fe[u7$$\" +B$*)yb#Fe[o$\"+@zB@XFe[oFebt7%F[\\u7$$\"+B!ouc#Fe[o$\"+y[%*>XFe[oFebt 7%Fa\\u7$$\"+bW)pd#Fe[o$\"+8'z-_%Fe[oFebt7%Fg\\u7$$\"+E=;'e#Fe[o$\"+$* [:AXFe[oFebt7%F]]u7$$\"+$RhZf#Fe[o$\"+;/WDXFe[oFebt7%Fc]u7$$\"+2ge-EFe [o$\"+(4w*HXFe[oFebt7%Fi]u7$$\"+)))z%4EFe[o$\"+ZOeNXFe[oFebt7%F_^u7$$ \"+rfK:EFe[o$\"+Ye2UXFe[oFebt7%Fe^u7$$\"+%)>/?EFe[o$\"+_ME\\XFe[oFebt7 %F[_u7$$\"++hdBEFe[o$\"+D+'pb%Fe[oFebt7%Fa_u7$$\"+)p-fi#Fe[o$\"+s])\\c %Fe[oFebt7%Fg_u7$$\"+l&=qi#Fe[o$\"+;f;tXFe[oFebt7%F]`u7$$\"+\\'Rpi#Fe[ o$\"+!oQ8e%Fe[oFebt7%Fc`u7$$\"+K%)pDEFe[o$\"+r'[$*e%Fe[oFebt7%Fi`uF[bt FebtF`dsFids-F$6$7K7$Faes$!3Q+++4/0(f%FQ7$Fdat$!3w*****4n[$*e%FQ7$F_at $!3++++!oQ8e%FQ7$Fj`t$!3y*****f\"f;tXFQ7$Fe`t$!3#)*****>2&)\\c%FQ7$F`` t$!3U+++D+'pb%FQ7$F[`t$!3%******>Xj#\\XFQ7$Ff_t$!3O+++Ye2UXFQ7$Fa_t$!3 ;+++ZOeNXFQ7$F\\_t$!3f*****p4w*HXFQ7$Fg^t$!3N+++;/WDXFQ7$Fb^t$!3t***** H*[:AXFQ7$F]^t$!3f*****Hhz-_%FQ7$Fh]t$!3W+++y[%*>XFQ7$Fc]t$!3*******4# zB@XFQ7$F^]t$!3z*****>K*=CXFQ7$Fi\\t$!3G+++L8wGXFQ7$Fd\\t$!3S+++'*3%[` %FQ7$F_\\t$!3T+++8*RAa%FQ7$Fj[t$!3%******R:/2b%FQ7$Fe[t$!35+++5-$*fXFQ 7$F`[t$!3#********y(epXFQ7$F[[t$!3k*****pSV$zXFQ7$Ffjs$!3c*****\\a\")) )e%FQ7$Fajs$!3%******pe?zf%FQ7$F\\js$!3v*****H\\Aig%FQ7$Fgis$!34+++&o' f8YFQ7$Fbis$!3I+++_%**)>YFQ7$F]is$!3g******[/.DYFQ7$Fhhs$!3r*****RpF*G YFQ7$Fchs$!3V+++0@cJYFQ7$F^hs$!3t*****4DKHj%FQ7$Figs$!3E+++(ofIj%FQ7$F dgs$!3?+++@])>j%FQ7$F_gs$!3Q+++[ZwHYFQ7$Fjfs$!31+++%yoki%FQ7$Fefs$!3r* ****4jy@i%FQ7$F`fs$!3W+++sg)ph%FQ7$F[fs$!3S+++6B*4h%FQ7$Ffes$!3z*****f Z2Vg%FQFbauF]hn-Fehn6L7%7$F\\bt$!+4/0(f%Fe[o7$Fj`u$!+r'[$*e%Fe[o7$Ffbt $!$d%Fdds7%F`iu7$Fd`u$!+!oQ8e%Fe[oFciu7%Fgiu7$F^`u$!+;f;tXFe[oFciu7%F[ ju7$Fh_u$!+s])\\c%Fe[oFciu7%F_ju7$Fb_u$!+D+'pb%Fe[oFciu7%Fcju7$F\\_u$! +_ME\\XFe[oFciu7%Fgju7$Ff^u$!+Ye2UXFe[oFciu7%F[[v7$F`^u$!+ZOeNXFe[oFci u7%F_[v7$Fj]u$!+(4w*HXFe[oFciu7%Fc[v7$Fd]u$!+;/WDXFe[oFciu7%Fg[v7$F^]u $!+$*[:AXFe[oFciu7%F[\\v7$Fh\\u$!+8'z-_%Fe[oFciu7%F_\\v7$Fb\\u$!+y[%*> XFe[oFciu7%Fc\\v7$F\\\\u$!+@zB@XFe[oFciu7%Fg\\v7$Ff[u$!+A$*=CXFe[oFciu 7%F[]v7$F`[u$!+L8wGXFe[oFciu7%F_]v7$Fjjt$!+'*3%[`%Fe[oFciu7%Fc]v7$Fdjt $!+8*RAa%Fe[oFciu7%Fg]v7$F^jt$!+aTq]XFe[oFciu7%F[^v7$Fhit$!+5-$*fXFe[o Fciu7%F_^v7$Fbit$!+!z(epXFe[oFciu7%Fc^v7$F\\it$!+2MMzXFe[oFciu7%Fg^v7$ Ffht$!+X:)))e%Fe[oFciu7%F[_v7$F`ht$!+(e?zf%Fe[oFciu7%F__v7$Fjgt$!+$\\A ig%Fe[oFciu7%Fc_v7$Fdgt$!+&o'f8YFe[oFciu7%Fg_v7$F^gt$!+_%**)>YFe[oFciu 7%F[`v7$Fhft$!+\\/.DYFe[oFciu7%F_`v7$Fbft$!+%pF*GYFe[oFciu7%Fc`v7$F\\f t$!+0@cJYFe[oFciu7%Fg`v7$Ffet$!+^A$Hj%Fe[oFciu7%F[av7$F`et$!+(ofIj%Fe[ oFciu7%F_av7$Fjdt$!+@])>j%Fe[oFciu7%Fcav7$Fddt$!+[ZwHYFe[oFciu7%Fgav7$ F^dt$!+%yoki%Fe[oFciu7%F[bv7$Fhct$!+J'y@i%Fe[oFciu7%F_bv7$Fbct$!+sg)ph %Fe[oFciu7%Fcbv7$F\\ct$!+6B*4h%Fe[oFciu7%Fgbv7$Fabt$!+wuI/YFe[oFciu7%F [cvF]iuFciuF`dsFids-F$6$7e\\lF'F*7$F($\"3)******Rl#fTJF-7$F($\"3<+++!) *)Q7ZF-7$$!3I+++2UH2]F1$\"3/+++4`=$G'F-7$$!3W*****pfnPo(!#E$\"3!****** zp\")R&yF-7$$!3))******en;>]F7$\"36+++h&yZU*F-7$$!3)********GB#y?F=$\" 31+++$yd&*4\"FQ7$$!3<+++?z^GgF=$\"33+++>'QmD\"FQ7$$!3/+++q$H6=\"FC$\"3 3+++AAs89FQ7$$!3%)*****p@o-y)F=$\"3/+++gH\"3d\"FQ7$$\"3A+++p!>De%FC$\" 3++++)p>zs\"FQ7$$\"3/+++$4jLu#FN$\"31+++1#e])=FQ7$$\"3'******H[wWn*FN$ \"39+++jHDU?FQ7$$\"3%******fgDDt#FZ$\"35+++bh`*>#FQ7$$\"3J+++*[))H[IF_o$\"3A+++K+3sEFQ7$$\"3%)*******Q@Z!eF_o$\"3>+++R#[&HGFQ7$$\" 3'******z\\z].\"F_p$\"3?+++IDQ')HFQ7$$\"36+++CO_HB5#)F_p$\"31 +++ac9\\RFQ7$$\"3;+++,dv)G)F_p$\"3M+++NvjaSFQ7$$\"3a+++N$4Xy(F_p$\"3++ ++,KH_TFQ7$$\"3I+++)[#***o'F_p$\"3h*****HZcDC%FQ7$$\"3*******\\+Mw-&F_ p$\"37+++^+xDVFQ7$$\"34+++k00RGF_p$\"31+++s:?-WFQ7$$\"35+++o5LowUsUuz%FQ7$$!3++++8_B!=$F-$\"3e*** **H,jM#[FQ7$$!39+++9G$>l$F-$\"3-+++.`/X[FQ7$$!3&******p*>jITF-$\"3m*** **>6pB'[FQ7$$!32+++'=)y9YF-$\"31+++y.fv[FQ7$$!35+++yh:.^F-$\"3++++y5%[ )[FQ7$$!3e*****f)fv%f&F-$\"3Q+++b#H-*[FQ7$$!3u*****pk\\))3'F-$\"3y**** ***=R=*[FQ7$$!3%******pbH\\e'F-$\"35+++n:t*)[FQ7$$!3s*****p@>F3(F-$\"3 ;+++XN%R)[FQ7$$!3]*****4,)=#e(F-$\"3!)*****Hs)[u[FQ7$$!3M+++zFe$3)F-$ \"3g*****>3d8'[FQ7$$!3E+++Q/[(e)F-$\"30+++O^^W[FQ7$$!3O+++Yz'[4*F-$\"3 '******\\\"o!R#[FQ7$$!3[+++J\"esg*F-$\"3e*****f.c%*z%FQ7$$!3********y^ o75FQ$\"3q******)zs5x%FQ7$$!3++++^bnl5FQ$\"3N+++ihmQZFQ7$$!3,+++sa7?6F Q$\"3p*****\\$*p@q%FQ7$$!3)*******32hw6FQ$\"3S+++M1fhYFQ7$$!3$*******) GteB\"FQ$\"39+++I24q^\"FQ$\"3w*****>H!)eT%FQ7$$!3'*******R9N*f\"FQ$\"3'******4!fWnVFQ7 $$!31+++U=Q%o\"FQ$\"3G+++/hwAVFQ7$$!35+++7@DqFQ$\"3w*****4^05@%FQ7$$ !3'******RZ3)>?FQ$\"3m*****\\7?)yTFQ7$$!3!)*****Ro6#)4#FQ$\"3w*****pi? y9%FQ7$$!3?+++&3RR<#FQ$\"3\")*****RNht6%FQ7$$!3'******\\@[qC#FQ$\"3O++ +O<$p3%FQ7$$!3$*******fZlX#FQ$\"3!)*****zt-B*RFQ7$$!35+++hH!f^#FQ$\" 39+++pB&)eRFQ7$$!31+++'4^zd#FQ$\"39+++z;=CRFQ7$$!3&)*****Hr8#QEFQ$\"3A +++(*Q=))QFQ7$$!3#)*******)*>op#FQ$\"3;+++Lmx]QFQ7$$!3++++!R$*Qv#FQ$\" 3********po*=\"QFQ7$$!3))*****z%Hb4GFQ$\"37+++`v\\rPFQ7$$!3++++>Y\"R'G FQ$\"3&******pXX&HPFQ7$$!3!)*****f_#4pHFQ$\"3@+++BM\"4k$FQ7$$!33+++7)\\.-$FQ$\"3!******RWIUf$FQ7$$!37+++ /[lqIFQ$\"3-+++K,*fa$FQ7$$!39+++skA?JFQ$\"3)******p\"pA'\\$FQ7$$!3#)** ***Hit\"pJFQ$\"3:+++KA*\\W$FQ7$$!3z*****Hk*f@tSo$FQ$\"32+++!ztY \"GFQ7$$!3@+++OCNGPFQ$\"3'******pXzWv#FQ7$$!31+++GO%=x$FQ$\"35+++uv:%p #FQ7$$!3$******>XSW\"QFQ$\"3++++AXrLEFQ7$$!3!)*****>WYg&QFQ$\"37+++*)z 8tDFQ7$$!39+++3xd'*QFQ$\"3=+++<5S7DFQ7$$!3,+++IV'f$RFQ$\"3))*****4-m9X #FQ7$$!3?+++=2:uRFQ$\"3y*****zV)G!R#FQ7$$!3n*****>C&46SFQ$\"39+++U)>)G BFQ7$$!3#******\\Upn/%FQ$\"37+++,0,nAFQ7$$!3%)******))o:\"3%FQ$\"37+++ &R6[?#FQ7$$!3c*****fB_U6%FQ$\"3&)*****R!eFQ7$$!3n*****z#y#RB%FQ$\"3#******\\:]8i\"FQ7$$!3 O+++\\+:dVFQ$\"3++++l2b`:FQ7$$!3s******e6pyVFQ$\"3#******4iw^[\"FQ7$$! 3;+++c:I*R%FQ$\"3'******fKgiT\"FQ7$$!3o******Q,->WFQ$\"33+++3N%oM\"FQ7 $$!3C+++$ozyV%FQ$\"33+++>f(pF\"FQ7$$!3W+++=^!fX%FQ$\"3#*******fSr17FQ7 $$!3i******=@6tWFQ$\"3!******4M>h8\"FQ7$$!3Q+++rh]*[%FQ$\"33+++veDl5FQ 7$$!3R+++6A30XFQ$\"3m*****4f!)=%**F-7$$!3Q+++>\\#)>XFQ$\"3c******>7yH# *F-7$$!3p******='4P`%FQ$\"3^+++]%Qo^)F-7$$!3#*******=PqYXFQ$\"3#****** HJmN!yF-7$$!3I+++L&o(eXFQ$\"35+++^0S!4(F-7$$!39+++>8')pXFQ$\"3-+++F&*o xjF-7$$!3'******>NP*zXFQ$\"3M+++p0plcF-7$$!3))*****H!>&*)e%FQ$\"3#)*** **4=rX&\\F-7$$!3U+++(*>'of%FQ$\"3u*****>A9WC%F-7$$!3\"******p!zi.YFQ$ \"3$********)fANNF-7$$!3,+++eV@4YFQ$\"3z******4i%p#GF-7$$!3@+++[9f8YFQ $\"30+++N!f%>@F-7$$!3q*****fKNnh%FQ$\"3-+++UWg79F-7$$!3(******HmG'=YFQ $\"3s*****H&f*=1(F_p7$$!31+++\\4E>YFQF(7$F]^y$!3s*****H&f*=1(F_p7$Fh]y $!3-+++UWg79F-7$Fc]y$!30+++N!f%>@F-7$F^]y$!3z******4i%p#GF-7$Fi\\y$!3$ ********)fANNF-7$Fd\\y$!3u*****>A9WC%F-7$F_\\y$!3#)*****4=rX&\\F-7$Fj[ y$!3M+++p0plcF-7$Fe[y$!3-+++F&*oxjF-7$F`[y$!35+++^0S!4(F-7$F[[y$!3#*** ***HJmN!yF-7$Ffjx$!3^+++]%Qo^)F-7$Fajx$!3c******>7yH#*F-7$F\\jx$!3m*** **4f!)=%**F-7$Fgix$!33+++veDl5FQ7$Fbix$!3!******4M>h8\"FQ7$F]ix$!3#*** ****fSr17FQ7$Fhhx$!33+++>f(pF\"FQ7$Fchx$!33+++3N%oM\"FQ7$F^hx$!3'***** *fKgiT\"FQ7$Figx$!3#******4iw^[\"FQ7$Fdgx$!3++++l2b`:FQ7$F_gx$!3****** *>:]8i\"FQ7$Fjfx$!33+++Q;b)o\"FQ7$Fefx$!31+++&eT^v\"FQ7$F`fx$!3\"***** *4i:6#=FQ7$F[fx$!3#******\\FQ7$Faex$!3++ ++\"*pU:?FQ7$F\\ex$!3=+++[01z?FQ7$Fgdx$!3&)*****R!e)GBFQ7$Fccx$!3y*****zV)G!R# FQ7$F^cx$!3))*****4-m9X#FQ7$Fibx$!3=+++<5S7DFQ7$Fdbx$!37+++*)z8tDFQ7$F _bx$!3++++AXrLEFQ7$Fjax$!35+++uv:%p#FQ7$Feax$!3'******pXzWv#FQ7$F`ax$! 32+++!ztY\"GFQ7$F[ax$!3=+++$*=ruGFQ7$Ff`x$!31+++y-aMHFQ7$Fa`x$!3++++d) zS*HFQ7$F\\`x$!3!)*****pPFK0$FQ7$Fg_x$!3#******fBf=6$FQ7$Fb_x$!37+++&Q P)pJFQ7$F]_x$!37+++/m,FKFQ7$Fh^x$!3++++G@D$G$FQ7$Fc^x$!39+++YmSQLFQ7$F ^^x$!3!)******eeN#R$FQ7$Fi]x$!3:+++KA*\\W$FQ7$Fd]x$!3)******p\"pA'\\$F Q7$F_]x$!3-+++K,*fa$FQ7$Fj\\x$!3!******RWIUf$FQ7$Fe\\x$!3@+++BM\"4k$FQ 7$F`\\x$!3#)*****H3?go$FQ7$F[\\x$!3&******pXX&HPFQ7$Ff[x$!37+++`v\\rPF Q7$Fa[x$!3********po*=\"QFQ7$F\\[x$!3;+++Lmx]QFQ7$Fgjw$!3A+++(*Q=))QFQ 7$Fbjw$!39+++z;=CRFQ7$F]jw$!39+++pB&)eRFQ7$Fhiw$!3!)*****zt-B*RFQ7$Fci w$!3m*****p&4nCSFQ7$F^iw$!3%)******Ri8cSFQ7$Fihw$!3O+++O<$p3%FQ7$Fdhw$ !3\")*****RNht6%FQ7$F_hw$!3w*****pi?y9%FQ7$Fjgw$!3m*****\\7?)yTFQ7$Feg w$!3w*****4^05@%FQ7$F`gw$!3s******H'e^C%FQ7$F[gw$!3U+++Jl9#G%FQ7$Fffw$ !3G+++/hwAVFQ7$Fafw$!3'******4!fWnVFQ7$F\\fw$!3w*****>H!)eT%FQ7$Fgew$! 3I+++S;'oY%FQ7$Fbew$!3M+++p;c=XFQ7$F]ew$!3I+++Su6pXFQ7$Fhdw$!39+++I24< YFQ7$Fcdw$!3S+++M1fhYFQ7$F^dw$!3p*****\\$*p@q%FQ7$Ficw$!3N+++ihmQZFQ7$ Fdcw$!3q******)zs5x%FQ7$F_cw$!3e*****f.c%*z%FQ7$Fjbw$!3'******\\\"o!R# [FQ7$Febw$!30+++O^^W[FQ7$F`bw$!3g*****>3d8'[FQ7$F[bw$!3!)*****Hs)[u[FQ 7$Ffaw$!3;+++XN%R)[FQ7$Faaw$!35+++n:t*)[FQ7$F\\aw$!3y*******=R=*[FQ7$F g`w$!3Q+++b#H-*[FQ7$Fb`w$!3++++y5%[)[FQ7$F]`w$!31+++y.fv[FQ7$Fh_w$!3m* ****>6pB'[FQ7$Fc_w$!3-+++.`/X[FQ7$F^_w$!3e*****H,jM#[FQ7$Fi^w$!3a***** >sUuz%FQ7$Fd^w$!3/+++'Q#ymZFQ7$F_^w$!3D+++w&f7t%FQ7$Fj]w$!35+++^]j!p%F Q7$Fe]w$!3!)*****H.eYk%FQ7$F`]w$!3#)*****46tIf%FQ7$F[]w$!37+++B!Gc`%FQ 7$Ff\\w$!3?+++g03sWFQ7$Fa\\w$!31+++s:?-WFQ7$F\\\\w$!37+++^+xDVFQ7$Fg[w $!3h*****HZcDC%FQ7$Fb[w$!3++++,KH_TFQ7$F][w$!3M+++NvjaSFQ7$Fhjv$!31+++ ac9\\RFQ7$Fcjv$!3))******)\\&HNQFQ7$F^jv$!37+++(R;Er$FQ7$Fiiv$!3#)**** *4.o4e$FQ7$Fdiv$!3;+++db*3W$FQ7$F_iv$!3/+++<\\z$H$FQ7$Fjhv$!3A+++(\\N; 9$FQ7$Fehv$!3?+++IDQ')HFQ7$F`hv$!3>+++R#[&HGFQ7$F[hv$!3A+++K+3sEFQ7$Ff gv$!3&)*****fstW^#FQ7$Fagv$!3%******RuRpN#FQ7$F\\gv$!35+++bh`*>#FQ7$Fg fv$!39+++jHDU?FQ7$Fbfv$!31+++1#e])=FQ7$F]fv$!3++++)p>zs\"FQ7$$!3i***** \\Fo-y)F=$!3/+++gH\"3d\"FQ7$Fcev$!33+++AAs89FQ7$F^ev$!33+++>'QmD\"FQ7$ Fidv$!31+++$yd&*4\"FQ7$$!3q******fn;>]F7$!36+++h&yZU*F-7$$!3O+++\"enPo (F`dv$!3!******zp\")R&yF-7$$!3]*****f5%H2]F1$!3/+++4`=$G'F-7$F($!3<+++ !)*)Q7ZF-7$F($!3)******Rl#fTJF-FjgnF'-F^hn6&F`hnFgds$\"#NFddsF(-Fehn6f \\l7%F'Fhhn7$$FCFchnF(7%Fhhn7$F($\"+aEfTJF[inFaiz7%Fdiz7$F($\"+!)*)Q7Z F[inFaiz7%Fhiz7$$!+2UH2]F_p$\"+4`=$G'F[inFaiz7%F\\jz7$$!+(fnPo(F-$\"+) p\")R&yF[inFaiz7%Fbjz7$$!+fn;>]FQ$\"+h&yZU*F[inFaiz7%Fhjz7$$!+!HB#y?F^ jn$\"+$yd&*4\"Fe[oFaiz7%F^[[l7$$!+?z^GgF^jn$\"+>'QmD\"Fe[oFaiz7%Fd[[l7 $$!+q$H6=\"Fejn$\"+AAs89Fe[oFaiz7%Fj[[l7$$!+<#o-y)F^jn$\"+gH\"3d\"Fe[o Faiz7%F`\\[l7$$\"+p!>De%Fejn$\"+)p>zs\"Fe[oFaiz7%Ff\\[l7$$\"+$4jLu#Fb[ o$\"+1#e])=Fe[oFaiz7%F\\][l7$$\"+$[wWn*Fb[o$\"+jHDU?Fe[oFaiz7%Fb][l7$$ \"+1c_KFF`\\o$\"+bh`*>#Fe[oFaiz7%Fh][l7$$\"+*[B5#)F`^o$\"+ac9\\RFe[oFaiz7%F`b[l7$$\"+,dv)G)F`^o$\"+Nvja SFe[oFaiz7%Ffb[l7$$\"+N$4Xy(F`^o$\"+,KH_TFe[oFaiz7%F\\c[l7$$\"+)[#***o 'F`^o$\"+tkbUUFe[oFaiz7%Fbc[l7$$\"+0SjF]F`^o$\"+^+xDVFe[oFaiz7%Fhc[l7$ $\"+k00RGF`^o$\"+s:?-WFe[oFaiz7%F^d[l7$$\"+o5LowUl$F[in$\"+.`/X[Fe[o Faiz7%Fjg[l7$$!+(*>jITF[in$\"+7\"pB'[Fe[oFaiz7%F`h[l7$$!+'=)y9YF[in$\" +y.fv[Fe[oFaiz7%Ffh[l7$$!+yh:.^F[in$\"+y5%[)[Fe[oFaiz7%F\\i[l7$$!+')fv %f&F[in$\"+b#H-*[Fe[oFaiz7%Fbi[l7$$!+Z'\\))3'F[in$\"+!>R=*[Fe[oFaiz7%F hi[l7$$!+d&H\\e'F[in$\"+n:t*)[Fe[oFaiz7%F^j[l7$$!+<#>F3(F[in$\"+XN%R)[ Fe[oFaiz7%Fdj[l7$$!+6!)=#e(F[in$\"+B()[u[Fe[oFaiz7%Fjj[l7$$!+zFe$3)F[i n$\"+#3d8'[Fe[oFaiz7%F`[\\l7$$!+Q/[(e)F[in$\"+O^^W[Fe[oFaiz7%Ff[\\l7$$ !+Yz'[4*F[in$\"+:o!R#[Fe[oFaiz7%F\\\\\\l7$$!+J\"esg*F[in$\"+OgX*z%Fe[o Faiz7%Fb\\\\l7$$!+z^o75Fe[o$\"+*zs5x%Fe[oFaiz7%Fh\\\\l7$$!+^bnl5Fe[o$ \"+ihmQZFe[oFaiz7%F^]\\l7$$!+sa7?6Fe[o$\"+N*p@q%Fe[oFaiz7%Fd]\\l7$$!+4 2hw6Fe[o$\"+M1fhYFe[oFaiz7%Fj]\\l7$$!+*GteB\"Fe[o$\"+I24q^ \"Fe[o$\"+#H!)eT%Fe[oFaiz7%Fh_\\l7$$!+S9N*f\"Fe[o$\"+,fWnVFe[oFaiz7%F^ `\\l7$$!+U=Q%o\"Fe[o$\"+/hwAVFe[oFaiz7%Fd`\\l7$$!+7@DqFe[o$ \"+6b+6UFe[oFaiz7%Ffa\\l7$$!+u%3)>?Fe[o$\"+D,#)yTFe[oFaiz7%F\\b\\l7$$! +%o6#)4#Fe[o$\"+F1#y9%Fe[oFaiz7%Fbb\\l7$$!+&3RR<#Fe[o$\"+a8OX# Fe[o$\"+QFI#*RFe[oFaiz7%F`d\\l7$$!+hH!f^#Fe[o$\"+pB&)eRFe[oFaiz7%Ffd\\ l7$$!+'4^zd#Fe[o$\"+z;=CRFe[oFaiz7%F\\e\\l7$$!+8P@QEFe[o$\"+(*Q=))QFe[ oFaiz7%Fbe\\l7$$!+!**>op#Fe[o$\"+Lmx]QFe[oFaiz7%Fhe\\l7$$!+!R$*Qv#Fe[o $\"+qo*=\"QFe[oFaiz7%F^f\\l7$$!+[Hb4GFe[o$\"+`v\\rPFe[oFaiz7%Fdf\\l7$$ !+>Y\"R'GFe[o$\"+daaHPFe[oFaiz7%Fjf\\l7$$!+ED4pHFe[o$\"+BM\"4k$Fe[oFaiz7%Ffg\\l7$$!+7)\\.-$Fe[o$\"+ W/B%f$Fe[oFaiz7%F\\h\\l7$$!+/[lqIFe[o$\"+K,*fa$Fe[oFaiz7%Fbh\\l7$$!+sk A?JFe[o$\"+)GBFe[oFaiz7%F``]l7$$!+D%pn/% Fe[o$\"+,0,nAFe[oFaiz7%Ff`]l7$$!+*)o:\"3%Fe[o$\"+&R6[?#Fe[oFaiz7%F\\a] l7$$!+OAD9TFe[o$\"+/eFe[oFaiz7%Fdb]l7$$!+Gy#RB%Fe[o$\"+v\"yk)=Fe[oFaiz7%Fjb]l7$$!+b3zgUF e[o$\"+@c6@=Fe[oFaiz7%F`c]l7$$!+05]'G%Fe[o$\"+&eT^v\"Fe[oFaiz7%Ffc]l7$ $!+^256VFe[o$\"+Q;b)o\"Fe[oFaiz7%F\\d]l7$$!+z]jMVFe[o$\"+_,N@;Fe[oFaiz 7%Fbd]l7$$!+\\+:dVFe[o$\"+l2b`:Fe[oFaiz7%Fhd]l7$$!+f6pyVFe[o$\"+@m<&[ \"Fe[oFaiz7%F^e]l7$$!+c:I*R%Fe[o$\"+E.E;9Fe[oFaiz7%Fde]l7$$!+R,->WFe[o $\"+3N%oM\"Fe[oFaiz7%Fje]l7$$!+$ozyV%Fe[o$\"+>f(pF\"Fe[oFaiz7%F`f]l7$$ !+=^!fX%Fe[o$\"+gSr17Fe[oFaiz7%Fff]l7$$!+>@6tWFe[o$\"+T$>h8\"Fe[oFaiz7 %F\\g]l7$$!+rh]*[%Fe[o$\"+veDl5Fe[oFaiz7%Fbg]l7$$!+6A30XFe[o$\"+\"f!)= %**F[inFaiz7%Fhg]l7$$!+>\\#)>XFe[o$\"+?7yH#*F[inFaiz7%F^h]l7$$!+>'4P`% Fe[o$\"+]%Qo^)F[inFaiz7%Fdh]l7$$!+>PqYXFe[o$\"+8jc.yF[inFaiz7%Fjh]l7$$ !+L&o(eXFe[o$\"+^0S!4(F[inFaiz7%F`i]l7$$!+>8')pXFe[o$\"+F&*oxjF[inFaiz 7%Ffi]l7$$!+_t$*zXFe[o$\"+p0plcF[inFaiz7%F\\j]l7$$!+.>&*)e%Fe[o$\"+\"= rX&\\F[inFaiz7%Fbj]l7$$!+(*>'of%Fe[o$\"+AUTWUF[inFaiz7%Fhj]l7$$!+2zi.Y Fe[o$\"+!*fANNF[inFaiz7%F^[^l7$$!+eV@4YFe[o$\"+5i%p#GF[inFaiz7%Fd[^l7$ $!+[9f8YFe[o$\"+N!f%>@F[inFaiz7%Fj[^l7$$!+E`t;YFe[o$\"+UWg79F[inFaiz7% F`\\^l7$$!+j'G'=YFe[o$\"+`f*=1(F`^oFaiz7%Ff\\^l7$$!+\\4E>YFe[oF(Faiz7% F\\]^l7$Fg\\^l$!+`f*=1(F`^oFaiz7%F`]^l7$Fa\\^l$!+UWg79F[inFaiz7%Fd]^l7 $F[\\^l$!+N!f%>@F[inFaiz7%Fh]^l7$Fe[^l$!+5i%p#GF[inFaiz7%F\\^^l7$F_[^l $!+!*fANNF[inFaiz7%F`^^l7$Fij]l$!+AUTWUF[inFaiz7%Fd^^l7$Fcj]l$!+\"=rX& \\F[inFaiz7%Fh^^l7$F]j]l$!+p0plcF[inFaiz7%F\\_^l7$Fgi]l$!+F&*oxjF[inFa iz7%F`_^l7$Fai]l$!+^0S!4(F[inFaiz7%Fd_^l7$F[i]l$!+8jc.yF[inFaiz7%Fh_^l 7$Feh]l$!+]%Qo^)F[inFaiz7%F\\`^l7$F_h]l$!+?7yH#*F[inFaiz7%F``^l7$Fig]l $!+\"f!)=%**F[inFaiz7%Fd`^l7$Fcg]l$!+veDl5Fe[oFaiz7%Fh`^l7$F]g]l$!+T$> h8\"Fe[oFaiz7%F\\a^l7$Fgf]l$!+gSr17Fe[oFaiz7%F`a^l7$Faf]l$!+>f(pF\"Fe[ oFaiz7%Fda^l7$F[f]l$!+3N%oM\"Fe[oFaiz7%Fha^l7$Fee]l$!+E.E;9Fe[oFaiz7%F \\b^l7$F_e]l$!+@m<&[\"Fe[oFaiz7%F`b^l7$Fid]l$!+l2b`:Fe[oFaiz7%Fdb^l7$F cd]l$!+_,N@;Fe[oFaiz7%Fhb^l7$F]d]l$!+Q;b)o\"Fe[oFaiz7%F\\c^l7$Fgc]l$!+ &eT^v\"Fe[oFaiz7%F`c^l7$Fac]l$!+@c6@=Fe[oFaiz7%Fdc^l7$F[c]l$!+v\"yk)=F e[oFaiz7%Fhc^l7$Feb]l$!+&*=C^>Fe[oFaiz7%F\\d^l7$F_b]l$!+\"*pU:?Fe[oFai z7%F`d^l7$Fia]l$!+[01z?Fe[oFaiz7%Fdd^l7$Fca]l$!+/e)GBFe[oFaiz7%Fde^l7$F[`]l$!+Q%)G!R#Fe[oFaiz7%Fhe^l7$Fe_]l$!+@gY^CFe [oFaiz7%F\\f^l7$F__]l$!+<5S7DFe[oFaiz7%F`f^l7$Fi^]l$!+*)z8tDFe[oFaiz7% Fdf^l7$Fc^]l$!+AXrLEFe[oFaiz7%Fhf^l7$F]^]l$!+uv:%p#Fe[oFaiz7%F\\g^l7$F g]]l$!+d%zWv#Fe[oFaiz7%F`g^l7$Fa]]l$!+!ztY\"GFe[oFaiz7%Fdg^l7$F[]]l$!+ $*=ruGFe[oFaiz7%Fhg^l7$Fe\\]l$!+y-aMHFe[oFaiz7%F\\h^l7$F_\\]l$!+d)zS*H Fe[oFaiz7%F`h^l7$Fi[]l$!+xtA`IFe[oFaiz7%Fdh^l7$Fc[]l$!+O#f=6$Fe[oFaiz7 %Fhh^l7$F][]l$!+&QP)pJFe[oFaiz7%F\\i^l7$Fgj\\l$!+/m,FKFe[oFaiz7%F`i^l7 $Faj\\l$!+G@D$G$Fe[oFaiz7%Fdi^l7$F[j\\l$!+YmSQLFe[oFaiz7%Fhi^l7$Fei\\l $!+feN#R$Fe[oFaiz7%F\\j^l7$F_i\\l$!+KA*\\W$Fe[oFaiz7%F`j^l7$Fih\\l$!+< pA'\\$Fe[oFaiz7%Fdj^l7$Fch\\l$!+K,*fa$Fe[oFaiz7%Fhj^l7$F]h\\l$!+W/B%f$ Fe[oFaiz7%F\\[_l7$Fgg\\l$!+BM\"4k$Fe[oFaiz7%F`[_l7$Fag\\l$!+$3?go$Fe[o Faiz7%Fd[_l7$F[g\\l$!+daaHPFe[oFaiz7%Fh[_l7$Fef\\l$!+`v\\rPFe[oFaiz7%F \\\\_l7$F_f\\l$!+qo*=\"QFe[oFaiz7%F`\\_l7$Fie\\l$!+Lmx]QFe[oFaiz7%Fd\\ _l7$Fce\\l$!+(*Q=))QFe[oFaiz7%Fh\\_l7$F]e\\l$!+z;=CRFe[oFaiz7%F\\]_l7$ Fgd\\l$!+pB&)eRFe[oFaiz7%F`]_l7$Fad\\l$!+QFI#*RFe[oFaiz7%Fd]_l7$F[d\\l $!+d4nCSFe[oFaiz7%Fh]_l7$Fec\\l$!+Si8cSFe[oFaiz7%F\\^_l7$F_c\\l$!+O<$p 3%Fe[oFaiz7%F`^_l7$Fib\\l$!+a8OR=*[Fe[oFaiz7%Fdd_l7$Fci[l$!+b#H-*[Fe[oFaiz7 %Fhd_l7$F]i[l$!+y5%[)[Fe[oFaiz7%F\\e_l7$Fgh[l$!+y.fv[Fe[oFaiz7%F`e_l7$ Fah[l$!+7\"pB'[Fe[oFaiz7%Fde_l7$F[h[l$!+.`/X[Fe[oFaiz7%Fhe_l7$Feg[l$!+ 8IYB[Fe[oFaiz7%F\\f_l7$F_g[l$!+AFW(z%Fe[oFaiz7%F`f_l7$Fif[l$!+'Q#ymZFe [oFaiz7%Fdf_l7$Fcf[l$!+w&f7t%Fe[oFaiz7%Fhf_l7$F]f[l$!+^]j!p%Fe[oFaiz7% F\\g_l7$Fge[l$!+L!eYk%Fe[oFaiz7%F`g_l7$Fae[l$!+6J2$f%Fe[oFaiz7%Fdg_l7$ F[e[l$!+B!Gc`%Fe[oFaiz7%Fhg_l7$Fed[l$!+g03sWFe[oFaiz7%F\\h_l7$F_d[l$!+ s:?-WFe[oFaiz7%F`h_l7$Fic[l$!+^+xDVFe[oFaiz7%Fdh_l7$Fcc[l$!+tkbUUFe[oF aiz7%Fhh_l7$F]c[l$!+,KH_TFe[oFaiz7%F\\i_l7$Fgb[l$!+NvjaSFe[oFaiz7%F`i_ l7$Fab[l$!+ac9\\RFe[oFaiz7%Fdi_l7$F[b[l$!+*\\&HNQFe[oFaiz7%Fhi_l7$Fea[ l$!+(R;Er$Fe[oFaiz7%F\\j_l7$F_a[l$!+J!o4e$Fe[oFaiz7%F`j_l7$Fi`[l$!+db* 3W$Fe[oFaiz7%Fdj_l7$Fc`[l$!+<\\z$H$Fe[oFaiz7%Fhj_l7$F]`[l$!+(\\N;9$Fe[ oFaiz7%F\\[`l7$Fg_[l$!+IDQ')HFe[oFaiz7%F`[`l7$Fa_[l$!+R#[&HGFe[oFaiz7% Fd[`l7$F[_[l$!+K+3sEFe[oFaiz7%Fh[`l7$Fe^[l$!+EPZ9DFe[oFaiz7%F\\\\`l7$F _^[l$!+W(RpN#Fe[oFaiz7%F`\\`l7$Fi][l$!+bh`*>#Fe[oFaiz7%Fd\\`l7$Fc][l$! +jHDU?Fe[oFaiz7%Fh\\`l7$F]][l$!+1#e])=Fe[oFaiz7%F\\]`l7$Fg\\[l$!+)p>zs \"Fe[oFaiz7%F`]`l7$$!+v#o-y)F^jn$!+gH\"3d\"Fe[oFaiz7%Fd]`l7$F[\\[l$!+A As89Fe[oFaiz7%Fj]`l7$Fe[[l$!+>'QmD\"Fe[oFaiz7%F^^`l7$F_[[l$!+$yd&*4\"F e[oFaiz7%Fb^`l7$$!+gn;>]FQ$!+h&yZU*F[inFaiz7%Ff^`l7$$!+\"enPo(F-$!+)p \")R&yF[inFaiz7%F\\_`l7$$!+1TH2]F_p$!+4`=$G'F[inFaiz7%Fb_`l7$F($!+!)*) Q7ZF[inFaiz7%Fh_`l7$F($!+aEfTJF[inFaiz7%F\\``lF\\dsFaiz7%F\\dsF'Faiz-F ^hn6&F`hnFgds$\"#vFddsF(Fids-F$6$7K7$$\"3*)*****pUK=`#FQ$\"3e*****p!e] AcFQ7$$\"3/+++<)\\l`#FQ$\"3!******\\102j&FQ7$$\"3>+++`v%*RDFQ$\"3Q+++9 (z$RcFQ7$$\"3%)*****z#4,UDFQ$\"3k*****4UN$[cFQ7$$\"3*)*****R\\ZFa#FQ$ \"3g*****4!yQdcFQ7$$\"3-+++(p$=UDFQ$\"3!)*****H'GOmcFQ7$$\"3'******pDi .a#FQ$\"3?+++[p4vcFQ7$$\"3()*****R@St`#FQ$\"3i*****4VOMo&FQ7$$\"3y**** *zc(=LDFQ$\"31+++6tB\"p&FQ7$$\"3@+++Bk)z_#FQ$\"3w******e[O)p&FQ7$$\"3& ******>bI=_#FQ$\"3B+++\"Q$p/dFQ7$$\"33+++Q`#[^#FQ$\"30+++(=1,r&FQ7$$\" 3,+++&*z32DFQ$\"3$******R&e\\9dFQ7$$\"3)******H/[()\\#FQ$\"3s*****\\#[ wdFQ7$$\"31+++JQ%3[#FQ$\"3F+++#e 11s&FQ7$$\"3!)*****\\<1;Z#FQ$\"3V+++[d/?dFQ7$$\"35+++$H?CY#FQ$\"3x**** *z:-\"=dFQ7$$\"3++++nj[`CFQ$\"3J+++;bv9dFQ7$$\"3:+++)p=]W#FQ$\"3U+++#z 6+r&FQ7$$\"3@+++*oVsV#FQ$\"3++++5*3Rq&FQ7$$\"31+++!*fRICFQ$\"3E+++vJ_' p&FQ7$$\"3<+++:@rCCFQ$\"3o*****\\'f(zo&FQ7$$\"3'******\\)3U?CFQ$\"3k** ***>#)R%ycFQ7$$\"36++++.t+++x5\"oT#FQ$\" 3;+++PtOdcFQ7$$\"3$******z3yxT#FQ$\"33+++csWYcFQ7$$\"32+++iAn?CFQ$\"3# )*****>y^dj&FQ7$$\"30+++MmWDCFQ$\"37+++ydmDcFQ7$$\"3&)*****>^g>V#FQ$\" 3e*****pKilh&FQ7$$\"3%)*****4a%)*RCFQ$\"3j******oOx3cFQ7$$\"3A+++*=N3cFQ7$$\"3>+++vk$e_#FQ$\"3V+++&)\\)\\h&FQFh``lFjhz-Fehn6L7%7$ $\"+FC$=`#Fe[o$\"+2e]AcFe[o7$$\"+<)\\l`#Fe[o$\"+l]qIcFe[o7$$\"$[#Fdds$ \"$l&Fdds7%Fh]al7$$\"+`v%*RDFe[o$\"+9(z$RcFe[oF]^al7%Fc^al7$$\"+G4,UDF e[o$\"+@aL[cFe[oF]^al7%Fi^al7$$\"+%\\ZFa#Fe[o$\"+,yQdcFe[oF]^al7%F__al 7$$\"+(p$=UDFe[o$\"+jGOmcFe[oF]^al7%Fe_al7$$\"+dAOSDFe[o$\"+[p4vcFe[oF ]^al7%F[`al7$$\"+9-MPDFe[o$\"+JkV$o&Fe[oF]^al7%Fa`al7$$\"+ov=LDFe[o$\" +6tB\"p&Fe[oF]^al7%Fg`al7$$\"+Bk)z_#Fe[o$\"+f[O)p&Fe[oF]^al7%F]aal7$$ \"+_0$=_#Fe[o$\"+\"Q$p/dFe[oF]^al7%Fcaal7$$\"+Q`#[^#Fe[o$\"+(=1,r&Fe[o F]^al7%Fiaal7$$\"+&*z32DFe[o$\"+ae\\9dFe[oF]^al7%F_bal7$$\"+V![()\\#Fe [o$\"+D[wdFe[oF]^al7%F[cal 7$$\"+JQ%3[#Fe[o$\"+#e11s&Fe[oF]^al7%Facal7$$\"+vhgrCFe[o$\"+[d/?dFe[o F]^al7%Fgcal7$$\"+$H?CY#Fe[o$\"+e@5=dFe[oF]^al7%F]dal7$$\"+nj[`CFe[o$ \"+;bv9dFe[oF]^al7%Fcdal7$$\"+)p=]W#Fe[o$\"+#z6+r&Fe[oF]^al7%Fidal7$$ \"+*oVsV#Fe[o$\"+5*3Rq&Fe[oF]^al7%F_eal7$$\"+!*fRICFe[o$\"+vJ_'p&Fe[oF ]^al7%Feeal7$$\"+:@rCCFe[o$\"+lf(zo&Fe[oF]^al7%F[fal7$$\"+&)3U?CFe[o$ \"+A)R%ycFe[oF]^al7%Fafal7$$\"++.tV#Fe[o$\"+FBc;cFe[oF]^al7%Fehal7$$\"+TX)* RCFe[o$\"+pOx3cFe[oF]^al7%F[ial7$$\"+*=N3cFQ7$Fb\\al$!3D+++%)y\"Gg&FQ7$F]\\al$!3%* *****f!**e)f&FQ7$Fh[al$!3))*****4@hef&FQ7$Fc[al$!3#*******3!*z%f&FQ7$F ^[al$!3%******4JHbf&FQ7$Fij`l$!3#******H;?\")f&FQ7$Fdj`l$!3\")*****ffl Dg&FQ7$F_j`l$!3j******oOx3cFQ7$Fji`l$!3e*****pKilh&FQ7$Fei`l$!37+++ydm DcFQ7$F`i`l$!3#)*****>y^dj&FQ7$F[i`l$!33+++csWYcFQ7$Ffh`l$!3;+++PtOdcF Q7$Fah`l$!3#)******GD9ocFQ7$F\\h`l$!3k*****>#)R%ycFQ7$Fgg`l$!3o*****\\ 'f(zo&FQ7$Fbg`l$!3E+++vJ_'p&FQ7$F]g`l$!3++++5*3Rq&FQ7$Fhf`l$!3U+++#z6+ r&FQ7$Fcf`l$!3J+++;bv9dFQ7$F^f`l$!3x*****z:-\"=dFQ7$Fie`l$!3V+++[d/?dF Q7$Fde`l$!3F+++#e11s&FQ7$F_e`l$!3n*****RXE)>dFQ7$Fjd`l$!3s*****\\#[wdFe[oF[ebl7%F_[cl7$Ffbal$!+D[w'FQ7$F($\"3Q++++ ++!>'FQF`_clFc_cl-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%& Im(z)G-%%FONTG6#%(DEFAULTG-Fj`cl6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BO XG-%%VIEWG6$;$!$R&Fdds$\"$>$Fdds;$!$>'Fdds$\"$>'Fdds" 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 1 0" "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 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the com bined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffici ents of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3835 "ee := \{c[2]=1/20,\nc[3]=8/57,\nc [4]=4/19,\nc[5]=44881/82208,\nc[6]=37/56,\nc[7]=61/367,\nc[8]=25/57,\n c[9]=39/50,\nc[10]=19/20,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/20,\na[3,1]= -184/3249,\na[3,2]=640/3249,\na[4,1]=1/19,\na[4,2]=0,\na[4,3]=3/19,\na [5,1]=10879382999084569/17778381694173184,\na[5,2]=0,\na[5,3]=-4127508 0377096907/17778381694173184,\na[5,4]=20050852951463513/88891908470865 92,\na[6,1]=5533979/91295232,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1285487029 9/39431336448,\na[6,5]=3657883862/13345483011,\na[7,1]=158475499801357 /1970033827325064,\na[7,2]=0,\na[7,3]=0,\na[7,4]=31153019468948131/297 713925429454536,\na[7,5]=-3016545448677376/81418799560147215,\na[7,6]= 150256844634176/8265656149993815,\na[8,1]=2924917590401981135/66605023 970341440552,\na[8,2]=0,\na[8,3]=0,\na[8,4]=445876617351108665/3098009 485599691752,\na[8,5]=196990339933642293248/1651616643670523695197,\na [8,6]=-7149897627279951488/167672519638445079477,\na[8,7]=35/201,\na[9 ,1]=-42211185912602462564070166033950198162012423/34220582419886119007 4733210927141821875000000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-231885975739 3123763522345863545224823245606023/84777972756693631715091638304957733 4375000000,\na[9,5]=-31112530877798875425923566022361029283298688/2898 1348290292110337421437961130838134765625,\na[9,6]=15292107243270980188 039355727714753150306881614/299015121519358598059132593510319939890136 71875,\na[9,7]=40769840728265525109900718286661374039/1554937526512165 4840821192853125000000,\na[9,8]=945612121233663016296797907/5989717837 61117752865000000,\na[10,1]=534215769953015759305550894517785263507766 10941/205195002329815636484038978294464618799958400000,\na[10,2]=0,\na [10,3]=0,\na[10,4]=502896807923589267114219976720637605496180343553/19 5519042445520666638376728307225007182256000000,\na[10,5]=3630126989117 70750340098104804203047936446648/4177386296703652569110279664085815584 44453125,\na[10,6]=-11581344894509944294868384714583152476105167090270 3/191939521001616690365566010466177813315410464375000,\na[10,7]=-28699 281021898330563484694969577298696532859/134633088449708866303357299975 42445904000000,\na[10,8]=-16493845602934685528675361413563797/28001674 400224052367433014064000000,\na[10,9]=1285791152889375/224541284583506 8,\na[11,1]=89462997577799701289200380185298019046438429/8410858488785 6406378631238816963629525265400,\na[11,2]=0,\na[11,3]=0,\na[11,4]=6492 689767645528199626511179748230475508923/641139590808912277141769373352 511460044088,\na[11,5]=694071986047276723594566813393158946881536/1753 38845029991694153905751089388426975345,\na[11,6]=-18782423346371666506 7125996081330354271085997696/30211283627207706655231802682148684012056 060045,\na[11,7]=-197008013321007326029676937491820383240505/202034289 41091626061502271171655020032789,\na[11,8]=-17297582435764003600121633 126256/9677847661048496307295108424975,\na[11,9]=165638979000000000/39 827589068734103,\na[11,10]=-845624320000/1531259401089,\na[12,1]=12517 4203/2508655500,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12 ,6]=56446289600512/342606012403635,\na[12,7]=62143252047047649977/2471 98834682622733680,\na[12,8]=9636489994384107/36666550775008000,\na[12, 9]=5373710937500/40041548453907,\na[12,10]=13212880000/87750675837,\na [12,11]=-98471/7223040,\n\nb[1]=125174203/2508655500,\nb[2]=0,\nb[3]=0 ,\nb[4]=0,\nb[5]=0,\nb[6]=56446289600512/342606012403635,\nb[7]=621432 52047047649977/247198834682622733680,\nb[8]=9636489994384107/366665507 75008000,\nb[9]=5373710937500/40041548453907,\nb[10]=13212880000/87750 675837,\nb[11]=-98471/7223040,\n\n`b*`[1]=185777316884981989/368148491 2241059950,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=2 2385886671596155610112/158772260638005081292791,\n`b*`[7]=537978856569 572663706487529/2159330585189973467574291759,\n`b*`[8]=324985627255339 82078667/120108582798663659157650,\n`b*`[9]=1235450293578257031250/763 1363446505532525519,\n`b*`[10]=498401087642132308000/38632580974096081 92999,\n`b*`[11]=0,\n`b*`[12]=-274194775/132076182681\}:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" } {TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=s ubs(ee,c[i]),i=2..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-/&%\"cG6#\" \"##\"\"\"\"#?/&F%6#\"\"$#\"\")\"#d/&F%6#\"\"%#F5\"#>/&F%6#\"\"&#\"&\" )[%\"&3A)/&F%6#\"\"'#\"#P\"#c/&F%6#\"\"(#\"#h\"$n$/&F%6#F0#\"#DF1/&F%6 #\"\"*#\"#R\"#]/&F%6#\"#5#F7F*/&F%6#\"#6F)/&F%6#\"#7F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficient s" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq (a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6^o/&%\"aG6$\"\"#\"\"\"#F(\"#?/&F%6$\"\"$F(#!$%=\"%\\K/&F %6$F.F'#\"$S'F1/&F%6$\"\"%F(#F(\"#>/&F%6$F:F'\"\"!/&F%6$F:F.#F.F*)))/&F%6$\"\"'F(#\"(zR`&\")K_H\" */&F%6$FgnF'F@/&F%6$FgnF.F@/&F%6$FgnF:#\",*Hq[&G\"\",[kLJ%R/&F%6$FgnFH #\"+iQ)yl$\",6I[XL\"/&F%6$\"\"(F(#\"0d8!)*\\v%e\"\"1k]KFQ.q>/&F%6$F`pF 'F@/&F%6$F`pF.F@/&F%6$F`pF:#\"2J\"[*o%>I:J\"3OXXHa#Rr(H/&F%6$F`pFH#!1w tn[aa;I\"2:s9g&*z=9)/&F%6$F`pFgn#\"0wTjWoD]\"\"1:Q**\\hll#)/&F%6$\"\") F(#\"4N6)>S!f<\\#H\"5_0WT.(R-0m'/&F%6$F_rF'F@/&F%6$F_rF.F@/&F%6$F_rF:# \"3l'36N\"7(>&pB0nVmh^; /&F%6$F_rFgn#!4)[^*zsi(*)\\r\"6x%z]WQ'>Dnn\"/&F%6$F_rF`p#\"#N\"$,#/&F% 6$\"\"*F(#!MBC,i\")>]R.m,2kDY-E\"f=6A%\"N+++v=#=9F4@LZ2!>h))>Ce?U$/&F% 6$FdtF'F@/&F%6$FdtF.F@/&F%6$FdtF:#!OBggXK#[AXN'eMANwBJRd(f)=B\"N+++vVL x&\\IQ;4:.^$fK\"f!)fe$>:7:!*H/&F%6$FdtF`p#\"GRSPhmG=2!*4^_l#G2%)p2%\"G+++DJ& G>@3%[l@^Ev$\\b\"/&F%6$FdtF_r#\"<2zz'H;ImL777c%*\"<+++lGv<6w$yr*)f/&F% 6$\"#5F(#\"PT4hw2NE&y0# /&F%6$F_wF'F@/&F%6$F_wF.F@/&F%6$F_wF:#\"Q`NM!='\\0wj?n(*>U6n#*eBz!o*G] \"Q+++cA=2]A2$GnPQmm?bWU!>b>/&F%6$F_wFH#\"N[mWOz/.U![5)4S.vq<\"*)p7IO \"NDJXW%e:e3k'z-6pDl.nH'QxT/&F%6$F_wFgn#!T.F!4n^5wC:$e9ZQo[HW*4X*[M\"e 6\"T+]Pk/T:L\"y>/&F%6$F_wF`p#!MfG`'p)Hx&p\\p%[j0L)*=- \"G*pG\"M+++/fWUv**HdLIm)3(\\%)3LY8/&F%6$F_wF_r#!D(zjNTh`nGboMHgXQ\\; \"D+++kS,LuO_SA+Wn,!G/&F%6$F_wFdt#\"1v$*)G:\"z&G\"\"1o]$e%GTXA/&F%6$\" #6F(#\"MH%Qk/>!)H&=!Q+#*G,(*zxv*HY*)\"M+aED&HO'p\")Q7jyjScy)[e3T)/&F%6 $F`zF'F@/&F%6$F`zF.F@/&F%6$F`zF:#\"LB*3bZI#[(z6^E'*>Gbkn(*o#\\'\"K)3W+ Y6DNt$p<9xA\"*33fR6k/&F%6$F`zFH#\"KO:)o%*eJR8oc%fBnFZg)>2%p\"KX`(pU)Q* 3^d!R:%p\"**H]%)Q`\"J*yK+-b;Z#/&F%6$Fg]lF_r#\"12TQ%***[O'*\"2+!3]x]lmO/&F% 6$Fg]lFdt#\".+v$4rt`\"/2RX[:/S/&F%6$Fg]lF_w#\",++)G@8\",Pen]x)/&F%6$Fg ]lF`z#!&r%)*\"(SIA(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 39 "weights for the 11 stage order 7 scheme" }{TEXT -1 1 " :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]), i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"*.UZ #/&F%6#\"\")#\"12TQ%***[O'*\"2+!3]x]lmO/&F%6#\"\"*#\".+v$4rt`\"/2RX[:/ S/&F%6#\"#5#\",++)G@8\",Pen]x)/&F%6#\"#6#!&r%)*\"(SIA(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 12 \+ stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..12);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"#\")L&pt\"\"*+?V[$/&F%6#\"\"#\"\" !/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"(jVQ*\")+7 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#================= ================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with \+ " }{XPPEDIT 18 0 "c[10] = 32/33;" "6#/&%\"cG6#\"#5*&\"#K\"\"\"\"#L!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "One of the weights for the order 7 scheme is negative and all the other weights are positive." }}{PARA 0 "" 0 "" {TEXT -1 36 "T he nodes of the order 7 scheme are:" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 1/100" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$+\"!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 4/27" "6#/&%\"cG6#\"\"$*&\" \"%\"\"\"\"#F!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 2/9" "6#/ &%\"cG6#\"\"%*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 11417/21528" "6#/&%\"cG6#\"\"&*&\"&<9\"\"\"\"\"&G:#!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 49/78" "6#/&%\"cG6#\"\"'*&\"# \\\"\"\"\"#y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 47/283" "6 #/&%\"cG6#\"\"(*&\"#Z\"\"\"\"$$G!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 30/73" "6#/&%\"cG6#\"\")*&\"#I\"\"\"\"#t!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[9] = 37/46" "6#/&%\"cG6#\"\"**&\"#P\"\"\"\"#Y! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 32/33" "6#/&%\"cG6#\"# 5*&\"#K\"\"\"\"#L!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6# /&%\"cG6#\"#6\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 62 "The 2-norm of the principal error is app roximately: 0.142653 " }{TEXT 266 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-4)" "6#)\"#5,$\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "ch ecking 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 4124 "ee := \{c[2]=1/1 00,\nc[3]=4/27,\nc[4]=2/9,\nc[5]=11417/21528,\nc[6]=49/78,\nc[7]=47/28 3,\nc[8]=30/73,\nc[9]=37/46,\nc[10]=32/33,\nc[11]=1,\nc[12]=1,\n\na[2, 1]=1/100,\na[3,1]=-692/729,\na[3,2]=800/729,\na[4,1]=1/18,\na[4,2]=0, \na[4,3]=1/6,\na[5,1]=451814999377/985407860736,\na[5,2]=0,\na[5,3]=-5 52805397249/328469286912,\na[5,4]=96066394193/54744881152,\na[6,1]=131 173/1890096,\na[6,2]=0,\na[6,3]=0,\na[6,4]=5764801/17935632,\na[6,5]=4 769464/20091357,\na[7,1]=1176539682457387/14353367828755172,\na[7,2]=0 ,\na[7,3]=0,\na[7,4]=17717342174781/163306752068660,\na[7,5]=-52806284 1353883264/10338013178660912633,\na[7,6]=1679528554701504/629114888202 41285,\na[8,1]=110903909289926737793/4320958290769994040132,\na[8,2]=0 ,\na[8,3]=0,\na[8,4]=-11607238985710021183/180261023052662924020,\na[8 ,5]=8332987977461879955626624/57056453830329950469093005,\na[8,6]=-397 914630242458427722/6312988060727977701695,\na[8,7]=11/30,\na[9,1]=-199 116158962708264981792850380273725319682574083/212209570379395954037673 5652096650388444524173920,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-902399027478 634049318468354365831042971264299/376720159407433656819366352813041281 670901920,\na[9,5]=-1932841049216489668013274675445061103749096456/122 5034474225538046212066937516775761976271555,\na[9,6]=61212379964740003 623393766579069964177763490780547/672922034904977555273490117957644252 82435991108860,\na[9,7]=1767352010455801522759919865162797424677/85363 1298904311486745705433404624991760,\na[9,8]=48803138404357581919875739 936453/25803619795436169827228351014320,\na[10,1]=14681288291210349822 3477710702327669160817645499822332/\n 3471363275502643502810190 50691094921672653636486171345,\na[10,2]=0,\na[10,3]=0,\na[10,4]=838090 7279399940341415520095001429689250499016988/16655291405980027768151637 90462148830898906131435,\na[10,5]=645991781238548549725468179173866591 8061301836694665216/\n 2108701812611736155789874398208067694599 974390240225435,\na[10,6]=-3296231019369759137139402385395942992019576 68540750512/\n 21636919327312331153389809685281753145015484225 8383315,\na[10,7]=-239812294208127777232571199557076199094738401624/62 716593235272943126739970926235455172660665115,\na[10,8]=-1920990949078 2204629345787487654324166296/6932257046340021610507379175381008191035, \na[10,9]=210642399729505665392/369807601102457012649,\na[11,1]=221939 233494467949726864332982580511239150575217/255387155540276753091910917 079234490443699161270,\na[11,2]=0,\na[11,3]=0,\na[11,4]=18116326319248 7297546324132938257205801916729/17154547141864477699072019121010975163 450940,\na[11,5]=5035138254118716342864368937874683246617259598976/775 682801056824859328708920179990438628643868105,\na[11,6]=-7722480175483 5239618997430260090636543172193252536784/\n 202652428901707058 12536717982284413527106230272155465,\na[11,7]=-20417460493227662758342 87576167164751357072803/242398642004974788232613894826395428361025315, \na[11,8]=-69880579855847721894868113496992143638/12013145933881997013 005874021379315455,\na[11,9]=667711875409843102380/5475725578726647254 29,\na[11,10]=-1808859196027797903/20206200178110899692,\na[12,1]=2354 923/46428480,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6] =31816481086962/143829479014165,\na[12,7]=7273629826007016071/29687570 322557889840,\na[12,8]=131953911267751793/602866646471570040,\na[12,9] =29769129062566/205676274838545,\na[12,10]=30049446213369/166412182583 360,\na[12,11]=-67996657/1112423760,\n\nb[1]=2354923/46428480,\nb[2]=0 ,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=31816481086962/143829479014165,\nb[ 7]=7273629826007016071/29687570322557889840,\nb[8]=131953911267751793/ 602866646471570040,\nb[9]=29769129062566/205676274838545,\nb[10]=30049 446213369/166412182583360,\nb[11]=-67996657/1112423760,\n\n`b*`[1]=182 127300329983916501/3507184399433109696000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b *`[4]=0,\n`b*`[5]=0,\n`b*`[6]=285550402807929509723935449/148289942336 7801818133032950,\n`b*`[7]=6443295529403196642395281317251/26849338691 229488574870087988000,\n`b*`[8]=143483908435216987345410922193/6154087 57915355446585175892000,\n`b*`[9]=139711798081481043314568703/77683420 0093629660855997950,\n`b*`[10]=2259083734163473774459617/1788149915600 3594882624000,\n`b*`[11]=-2987435988158249671/113556675995246584800,\n `b*`[12]=1/400\}:" }}}{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 1011 "subs(ee,matrix([[c[2],a[2, 1],``],\n[c[3],a[3,1],a[3,2]],[c[4],a[4,1],a[4,2]],[``$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]],[``$2,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]],[``$2,a[8,7]],\n[c[9],a[9,1],a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5 ],a[9,6]],[``,a[9,7],a[9,8]],\n[c[10],a[10,1],a[10,2]],[``,a[10,3],a[1 0,4]],[``,a[10,5],a[10,6]],\n[``,a[10,7],a[10,8]],[``$2,a[10,9]],\n[c[ 11],a[11,1],a[11,2]],[``,a[11,3],a[11,4]],[``,a[11,5],a[11,6]],\n[``,a [11,7],a[11,8]],[``,a[11,9],a[11,10]],\n[c[12],a[12,1],a[12,2]],[``,a[ 12,3],a[12,4]],[``,a[12,5],a[12,6]],\n[``,a[12,7],a[12,8]],[``,a[12,9] ,a[12,10]],[``$2,a[12,11]],\n[``,`________________________________`$2] ,[`b`,b[1],b[2]],[``,b[3],b[4]],[``,b[5],b[6]],\n[``,b[7],b[8]],[``,b[ 9],b[10]],[``$2,b[11]],\n[`b*`,`b*`[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[ ``,`b*`[5],`b*`[6]],\n[``,`b*`[7],`b*`[8]],[``,`b*`[9],`b*`[10]],[``,` b*`[11],`b*`[12]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 S7%#\"\"\"\"$+\"F(%!G7%#\"\"%\"#F#!$#p\"$H(#\"$+)F27%#\"\"#\"\"*#F)\"# =\"\"!7%F+F+#F)\"\"'7%#\"&<9\"\"&G:##\"-x$**\\\"=X\"-O2'yS&)*F;7%F+#!- \\sR0Gb\"-7pGp%G$#\",$>%Rmg*\",_6)[ua7%#\"#\\\"#y#\"'t68\"('4!*=F;7%F+ F;#\"(,[w&\")Kc$z\"7%F+F+#\"(k%pZ\")d84?7%#\"#Z\"$$G#\"1(QdCoRl<\"\"2s ^vGyO`V\"F;7%F+F;#\"/\"yu@M`st-Q]Gz\")\\E3F'*eh6*>\"R?R!4n\"GTI\"GNm$>olLuSf,sw$7%F+#!Ock4\\ P5h]WvYF8!o'*[;#\\5%G$>\"Ob:Fw>wvn^Pp17i/QbAuW.D7#\"SZ0y!\\jx*fF_,e X5?Nn<\"Hg<*\\i/MV0du'[6V!*)HJO&)#\"A`k$*Rd()>>edVSQJ!)[\"A?V,^$Gs#)ph V&z>O!e#7%#\"#K\"#L#\"WKB#)*\\Xw\"3;pwK-2rxMA)\\.@\"H)G\"o9\"WX8<'[OOl s;#\\4\"p]!>5G]VE]vKOrMF;7%F+F;#\"R))p,*\\]#*oH9+&4?bTT.%**Rzs!4Q)\"RN 981*)*3$)[@Y!zj^\"ox-!)fS\"Hbm\"7%F+#\"X;_m%pO=Ih!=fmQ?*H%fR&Q-%Rr8f(p$ >5B'H$\"W:LQeA%[:]9`pj@7%F+#!QC;SQZ4*>wqb*>rDBxx73UH7)R #\"P:^mgE\"IN5>35Qv\"z t]5;-Sj/dA$p7%F+F+#\"6#Rlc]H(*RU1@\"6\\E,dC5,w!)p$7%F)#\"Q<_d]\"R7^!e# )HLkos\\zY%\\L#R>A\"Qq7;*pV/\\M#zq\"4\">4`nFSb:(Qb#F;7%F+F;#\"NHn\">!e ?d#QH8Cja(H([#>jK;\"=\"MS4Xj^(4,@\">?2*pxW'=9ZX:<7%F+#\"Rw*)ffshYKouy$ *oV'GM;(=TDQ^.&\"Q0\"oQkG'Q/**z,#*3(G$f[#o0,Gov(#!V%yODD$>sJaO14g-V(** ='R_$[v,[Ax\"Vla:s-B1r_8WG#)zrOD\"eqq,*GCl-#7%F+#!O.G2d8vkr;wvGMeFmFK \\guT?\"N:`-h$GaRE[*QhK#)yu\\+U')RU##!GQO9#*p\\8\"o[*=sZe&)z0))p\"GbaJ z8-ue+8q*>)Q$f98?\"7%F+#\"6!Q-J%)4a(=rn'\"6HaskE(ybsva#!4.zzFg>f)3=\"5 #p**36y,?1-#7%F)#\"(B\\N#\")![Gk%F;7%F+F;F;7%F+F;#\"/ip3\"[;=$\"0lT,z% HQ97%F+#\"4rg,2g#)HOF(\"5S)*)ybA.d(oH#\"3$z^xE6R&>8\"3S+drkkmGg7%F+#\" /mD1H\"p(H\"0X&Q[Fwc?#\"/pL@Y%\\+$\"0gLe#=7k;7%F+F+#!)dm*z'\"+gPU767%F +%A________________________________GF\\x7%%\"bGFavF;FdvFevFivF`wFgw7%% #b*G#\"6,l\"R)*H.IF@=\"7+gp4JV*R%=2NF;Fdv7%F+F;#\"<\\a$Rs4&Hz!GS]bG\"= ]H.L\"==!yOB%**G[\"7%F+#\"@^sJ\"G&RUm>.%HbHVk\"A+!))z3q[d)[H7pQ$\\o## \"?$>A4TXt)p@N%3R[V\"\"?+?*e<&eYaN:zv3ah7%F+#\"<.(oXJV5[\"3)z6(R\"\"<] z*f&3mHO4+U$ox#\":<'fWxtM;MP3fA\";+Si#)[f.g:*\\\")y\"7%F+#!4r'\\#e\")) fV()H\"6+[eY_*fncN6#F)\"$+%Q(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(ee,matrix([seq( [c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[j],j=1..11 ),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[6](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'++5!\"(F(%!GF+F+F+F+F+F+F+F+F+F+7 /$\"'[\"[\"!\"'$!'Y#\\*F/$\"'R(4\"!\"&F+F+F+F+F+F+F+F+F+F+7/$\"'AAAF/$ \"'cbbF*$\"\"!F;$\"'nm;F/F+F+F+F+F+F+F+F+F+7/$\"'L.`F/$\"'1&e%F/F:$!'( Ho\"F4$\"'![v\"F4F+F+F+F+F+F+F+F+7/$\"'0#G'F/$\"'-SpF*F:F:$\"';9KF/$\" '*QP#F/F+F+F+F+F+F+F+7/$\"'yg;F/$\"''p>)F*F:F:$\"'\"\\3\"F/$!'(z5&F*$ \"'npEF*F+F+F+F+F+F+7/$\"'f4TF/$\"'lmDF*F:F:$!'8RkF*$\"'[g9F/$!'6.jF*$ \"'nmOF/F+F+F+F+F+7/$\"'[V!)F/$!'+$Q*F*F:F:$!'T&R#F4$!'zx:F4$\"']'4*F/ $\"'Rq?F4$\"'L\"*=F4F+F+F+F+7/$\"'(pp*F/$\"'EHUF/F:F:$\"')>.&F4$\"'YjI F4$!'VB:F4$!'vBQF4$!'4rFF4$\"'+'p&F/F+F+F+7/$\"\"\"F;$\"'J!p)F/F:F:$\" '2c5!\"%$\"'B\"\\'F4$!'q5QF4$!'4B%)F4$!',7F4$!'+_*)F*F+F+7/F cq$\"':s]F*F:F:F:F:$\"'57AF/$\"'1]CF/$\"'x)=#F/$\"'QZ9F/$\"'s0=F/$!'[7 hF*F+7/%\"bGFgrF:F:F:F:FirF[sF]sF_sFasFcsF+7/%#b*G$\"')H>&F*F:F:F:F:$ \"'iD>F/$\"'!)*R#F/$\"'_JBF/$\"'[)z\"F/$\"'Oj7F/$!'zIEF*$\"'++D!\")Q(p print96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')), op(OrderConditions(7,11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,Or derConditions(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(ee,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u), %);\nnops(%);\nsimplify(subs(ee,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u), %);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 #\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F $F$F$F$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#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for st age-orders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_11 := StageOrderCo nditions(ct,11,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 110 "Stages 6 to 11 have stage-order 4, w hile stages 4 and 5 have stage-order 3 has and stage 3 has stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(su bs(ee,so||i||_11[j])),i=2..6)],j=1..9)]:\nmap(proc(L) local i; for i t o 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&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 .. 11) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 10 " }}{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 0 151 "[Sum(b[i]*a[i,1],i=1+1..1 1)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=2..10)];\nmap(u ->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F -F-/F,;\"\"#\"#6&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,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F /6$F,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,Fa qF-/F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,; \"#5F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-/-F&6$*&F)F-&F/6$F,F]sF-/F,;F4F4*&&F *6#F]sF-,&F-F-&FEF]tFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F $F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the princi pal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "RK7_11err_eqs := PrincipalErrorConditions(7,11,'expa nded'):\nexpand(subs(ee,RK7_11err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0, 1),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"\"F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 93 "We c an calculate the principal error norm, that is, the 2-norm of the prin cipal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errt erms7_11 := PrincipalErrorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errterms7_11) do\n sm := sm+(evalf(subs(ee,errterms7_11[ct]))) ^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ce_E9!# 9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm of the order 6 embedded scheme is given as follo ws." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms6_12*` := s ubs(b=`b*`,PrincipalErrorTerms(6,12,'expanded')):\nsm := 0:\nfor ct to nops(`errterms6_12*`) do\n sm := sm+(evalf(subs(ee,`errterms6_12*`[ ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+s 'oS3\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------- ------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "We incorporate the stage-order equations to ensure that stage 3 has stage-order 2, stages 4 and 5 ha ve stage order 3 and stages 6 to 11 have stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 57 "These conditions lead to relationships between the n odes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6 #/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c [4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0 " "6#/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/ F'!\"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" } {TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 68 "which we include expli citly to make the computation more efficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the s implifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and \+ " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 " . . 10. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "The simp lifying conditions allow us to manage with just 10 \"simple\" order co nditions which are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 7 order 7 quadrature conditions and 3 additional order 7 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO7 := SimpleOrde rConditions(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,32,38,48,61,64])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7 %\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#KF)/*&F,F()F2FGF(#F(\"\"'7%\"#QF)/*&F,F(-%!G6#* &%\"aGF(-FT6#*&FWF(-FT6#*&FWF(-FT6#*&F8F(FWF(F(F(F(F(#F(\"%?D7%\"#[F)/ *&F,F(-FT6#*&FWF(-FT6#*&FWF(-FT6#*&F?F(FWF(F(F(F(#F(\"$S)7%\"#hF)/*(F, F(F2F(-FT6#*&FEF(FWF(F(#F(\"#N7%\"#kF)/*&F,F()F2FNF(#F(\"\"(Q)pprint16 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "SO7_11 := SimpleOrderConditions(7,11,'expanded'):\nS O_eqs := [op(RowSumConditions(11,'expanded')),op(StageOrderConditions( 2,11,'expanded')),\n op(StageOrderConditions(3,4..11,'expanded')),op( StageOrderConditions(4,6..11,'expanded'))]:\nord_cdns := [seq(SO7_11[i ],i=[1,2,4,8,16,32,38,48,61,64])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2.. 11)=b[1],seq(add(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=[$8..10])]:\nn ode_eqs := [3*c[3]=2*c[4],3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] =0]:\ncdns := [op(node_eqs),op(simp_eqs),op(SO_eqs),op(ord_cdns)]:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The scheme can be obtained by s pecifying the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/100;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$+\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 2/9;" "6#/&%\"cG6#\"\"%*&\"\"#\"\"\" \"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 49/78;" "6#/&%\"c G6#\"\"'*&\"#\\\"\"\"\"#y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 47/283;" "6#/&%\"cG6#\"\"(*&\"#Z\"\"\"\"$$G!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[8] = 30/73;" "6#/&%\"cG6#\"\")*&\"#I\"\"\"\"#t!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 37/46;" "6#/&%\"cG6#\"\"* *&\"#P\"\"\"\"#Y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 32/33 ;" "6#/&%\"cG6#\"#5*&\"#K\"\"\"\"#L!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 2 ", " }}{PARA 257 " " 0 "" {TEXT -1 45 "together with the zero linking coefficients: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"a G6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] = 0" "6#/ &%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2] = 0 " "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2 ] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[10, 2] = 0;" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG6$\"#6\"\"#\"\"!" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[6 ,3] = 0;" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3] = 0;" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[9,3] = 0;" "6#/&%\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,3] = 0;" "6#/&%\"aG6$\"#5\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/&%\"aG6$\"#6\"\"$\" \"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weig hts: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/ &%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\" bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4] = 0;" "6#/&%\"b G6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5] = 0;" "6#/&%\"bG 6#\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "e1 := \{c[2]=1/100,c[4]=2/9 ,c[6]=49/78,c[7]=47/283,c[8]=30/73,c[9]=37/46,c[10]=32/33,c[11]=1,\n \+ seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),seq(b[i]=0,i=2..5)\}:\neq ns := subs(e1,cdns):\nnops(eqns);\nindets(eqns) minus \{a[8,7],a[9,7], a[9,8]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\\" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e2 := solve(\{op(eqns)\} ,indets(eqns) minus \{a[8,7],a[9,7],a[9,8]\}):\ne3 := `union`(e1,e2): \ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,e3));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\")\"\"(&F%6$\"\"*F(&F%6$F+F'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4657 "e3 := \{a[ 11,3] = 0, a[10,2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[6,3] = 0, a[7, 3] = 0, a[8,3] = 0, a[8,2] = 0, a[9,3] = 0, a[9,2] = 0, b[2] = 0, a[10 ,3] = 0, a[11,2] = 0, a[11,4] = 1076949412070027939593173940554/162185 66593827392077508520301*a[8,7]+22106062081267147922145563976/170119438 1068282266185358017*a[9,7]-953293781187117/19043143759420+159108764406 2925531731283024/321185476208091271684232017*a[9,8], a[4,2] = 0, a[5,2 ] = 0, a[6,2] = 0, a[7,2] = 0, a[7,1] = 1176539682457387/1435336782875 5172, c[2] = 1/100, c[11] = 1, a[8,4] = -58035621009/57705566102*a[8,7 ]+121036973895/397660568723, a[5,3] = -552805397249/328469286912, c[6] = 49/78, c[7] = 47/283, c[8] = 30/73, c[10] = 32/33, c[9] = 37/46, a[ 7,4] = 17717342174781/163306752068660, a[6,1] = 131173/1890096, a[5,1] = 451814999377/985407860736, a[3,1] = -692/729, a[2,1] = 1/100, a[9,4 ] = -58035621009/57705566102*a[9,7]-2088561933/5447405051*a[9,8]+56151 35577/13629960080, b[1] = 2354923/46428480, a[11,7] = -371133932349243 81012/562114461026499751*a[8,7]-761810133050493328/58961188529413667*a [9,7]+1861600006297665932531220/43764748762518925696667, b[8] = 131953 911267751793/602866646471570040, a[9,6] = -6049609011732/4446041612737 9*a[9,7]+283058392032/763102360489*a[9,8]+58451505762/119334970195, a[ 8,5] = 9842725887631104/36530081903395451*a[8,7]+23790493530086400/503 472164869673923, a[10,5] = -509118282119038504679935918528643072/36511 0863696512123034276884301515127*a[9,7]-32849636449807790452736/1614382 9320358807939179-476116112943292897677224500694586877952/7499614584825 0747554428817358565691277*a[8,7]+125292801366309804927256755693813760/ 22977642047675533145916366477269709*a[9,8], a[10,6] = 1361105026272894 80461125703424/193289677440050548714737004317*a[9,7]-59301468683108912 560329328/112698776436420869771522985+12728751985086013260821472424438 4/39702956777272831858034421245967*a[8,7]-8857404606793940584340641792 /4614068660225625236508612201*a[9,8], c[3] = 4/27, c[4] = 2/9, a[4,1] \+ = 1/18, a[4,3] = 1/6, a[7,5] = -528062841353883264/1033801317866091263 3, a[9,5] = 9842725887631104/36530081903395451*a[9,7]-7266814468392960 /6896878970817451*a[9,8]-58252998024/407766797669, b[11] = -67996657/1 112423760, a[10,1] = 2075790122491393350551973649120/31420810355952403 15153515488781*a[9,7]-1015164807748568926552/724803063367390521129+194 1232831579209684784682238946420/645403878774737429506187452347231*a[8, 7]-76988010530132391107159062144/225016325127747351650199064779*a[9,8] , b[10] = 30049446213369/166412182583360, a[10,9] = 210642399729505665 392/369807601102457012649, a[10,8] = -155176001424206208704/2998440008 9388406431*a[9,8]+475391497736844987934268/67748898552626475331191, a[ 10,4] = 78997680774976103219386851424/15177778784096643825057126333*a[ 9,7]-1159420967514605044/63754908382697895+738768769912735669513198908 74884/3117614468713141306171702227183*a[8,7]+6535484764866926239996524 8/32937522181516284874585659*a[9,8], a[10,7] = -155176001424206208704/ 29984400089388406431*a[9,7]+855025103327566231767390976/55055134941555 472888120803-145117150994199446465864/6158990777511497757381*a[8,7], a [11,10] = -1808859196027797903/20206200178110899692, c[5] = 11417/2152 8, a[3,2] = 800/729, a[11,6] = 224521518169951979121757146032784/24991 842848455543884450927782629*a[8,7]+107177987119326008661793947072/6096 3697148432025688224546251*a[9,7]-327295609648964093633050932/194795509 556464188137958955-5014808169650153964837775872/1046358654512384183600 869841*a[9,8], a[11,9] = 667711875409843102380/547572557872664725429, \+ b[9] = 29769129062566/205676274838545, b[6] = 31816481086962/143829479 014165, a[11,8] = -761810133050493328/58961188529413667*a[9,8]+7043993 548022046530805/378303032270834332423, a[6,4] = 5764801/17935632, a[9, 1] = -3237235171885/25359307117942*a[9,7]+14387871094/217628947361*a[9 ,8]+4947161257/108905940976, a[11,5] = -365296956371223288398397374198 197248/20534087300381026083166496506032701*a[8,7]-17437879809385549720 5466987843584/50089698746535387000993336396019*a[9,7]-6661495818150376 320/1205512133786408371+128742625512085832565846310748160/945693444797 4796020638156314019*a[9,8], a[7,6] = 1679528554701504/6291148882024128 5, a[8,1] = -3237235171885/25359307117942*a[8,7]+1151376026505/1588691 3157353, a[5,4] = 96066394193/54744881152, a[6,5] = 4769464/20091357, \+ a[11,1] = 60072390964048012375179525123810/712741662630372304170999531 6221*a[8,7]+1233079278504724244026062841640/74760744397814010449243785 6657*a[9,7]-5658889499485414/1407464231172797-109608259924334869963710 60832/12831661394809763935736982787*a[9,8], b[7] = 7273629826007016071 /29687570322557889840, a[8,6] = -6049609011732/44460416127379*a[8,7]-7 31969812080/55706472315697\}:" }{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 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "s ubs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7)]));\nf or ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e3,c[ii])); \+ \n for jj to ii-1 do\n coeff_val := subs(e3,a[ii,jj]);\n i f coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.` )\n else\n print(a[ii,jj]=coeff_val);\n end if;\n \+ end do:\nend do:\nprint(``);\nseq(b[i]=subs(e3,b[i]),i=1..11);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\"$+\"F(%!GF+F+ F+F+F+7*#\"\"%\"#F#!$#p\"$H(#\"$+)F2F+F+F+F+F+7*#\"\"#\"\"*#F)\"#=\"\" !#F)\"\"'F+F+F+F+7*#\"&<9\"\"&G:##\"-x$**\\\"=X\"-O2'yS&)*F;#!-\\sR0Gb \"-7pGp%G$#\",$>%Rmg*\",_6)[uaF+F+F+7*#\"#\\\"#y#\"'t68\"('4!*=F;F;#\" (,[w&\")Kc$z\"#\"(k%pZ\")d84?F+F+7*#\"#Z\"$$G#\"1(QdCoRl<\"\"2s^vGyO`V \"F;F;#\"/\"yu@M3Il$\"\"\"&F%6$F'\"\"(F.F.#\"2 +k3IN\\!zB\"3BRnp[;sM]F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"',&*&#\".K<,4'\\g\"/zt7;/YW\"\"\"&F%6$F'\"\"(F.!\"\"#\"-!37)p >t\"/(p:BZ1d&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\" (\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#P\"#Y" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\",(*&#\".&)=c)3#\"+^]SZaF.&F%6$F'\"\")F.F2#\" +xb8:c\",!3g*HO\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"* \"\"&,(*&#\"1/6j()esU)*\"2^aR.>3Il$\"\"\"&F%6$F'\"\"(F.F.*&#\"1gHRoW\" oE(\"1^u\"3(*yo*oF.&F%6$F'\"\")F.!\"\"#\",C!)*HDe\"-pwzmxSF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',(*&#\".K<,4'\\g\"/zt7;/Y W\"\"\"&F%6$F'\"\"(F.!\"\"*&#\"-K?ReIG\"-*[gB5j(F.&F%6$F'\"\")F.F.#\", id]^%e\"-&>q\\L>\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\" *\"\"(\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &&%\"aG6$\"\"*\"\")\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\" #K\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",**&#\"@? \"\\O(>b]LR\"\\A,zv?\"@\"y)[:N::.C&fN53UJF(&F%6$\"\"*\"\"(F(F(#\"7_l#* o&[x![;:5\"6H6_!RnL1.[s!\"\"*&#\"C?k%*QAo%y%o4#z:$GBT>\"BJsM_u=1&Hutu( yQSX'F(&F%6$\"\")F1F(F(*&#\">W@1fr5\"RK,`5!))p(\"?zZ1*>];NZx7Dj,D#F(&F %6$F0FC9&oQ>K5w\\x!o(**y \">Lj7d]#Qk'4%yyx<:\"\"\"&F%6$\"\"*\"\"(F.F.#\"4W]g9v'4Uf6\"2&*yp#Q3\\ vj!\"\"*&#\"A%)[(3*)>8&pct7*p(o(Q(\"@$=FAqrhITJroWh@G=\"4& \"EF^^,V)oFMI77lpj36l$\"\"\"&F%6$\"\"*\"\"(F.!\"\"#\"8OFX!z2)\\kj\\G$ \"8z\"Rz!)e.KHQ9;F3*&#\"H_z(oe%p+XAxw*GHVH6;hZ\"Gx7pl&et\")GWbZ2D[e9'* \\(F.&F%6$\"\")F2F.F3*&#\"EgP\"QpbnDF\\!)4jO,GHD\"\"D4(psZmj\"f9LbnZ?k xH#F.&F%6$F1F=F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"',**&#\"?CMqD6Y![*GFE]5h8\"?\"\"\"&F%6$\"\"*\"\"(F.F .#\";G$H.c7*3Joo9If\"<&)H_r(p3UOkx)p7\"!\"\"*&#\"B%QWUs9#3E8g3&)>vGF\" \"AnfC@W.e=$GFxn&HqRF.&F%6$\"\")F2F.F.*&#\"=#zT1M%eSRz1YSd))\"=,Ah3lBD cAg'oSh%F.&F%6$F1F=F.F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"(,(*&#\"6/(3i?C9+w^:\"5JkS)Q*3+W)*H\"\"\"&F%6$\"\"*F(F.!\"\"#\"< w4Rn%*4:<^9\"7\"Qdx\\6vx!* *ehF.&F%6$\"\")F(F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"),&*&#\"6/(3i?C9+w^:\"5JkS)Q*3+W)*H\"\"\"&F%6$\"\"*F(F.!\"\"#\"9oU $z)\\%ot(\\\"Rv%\"8\">J`ZEEb)*)[x'F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#5\"\"*#\"6#Rlc]H(*RU1@\"6\\E,dC5,w!)p$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6 \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",**&#\"A5 Q7D&z^P7![S'4Rs+'\"@@iJ&**4K31rj*p[LC*f#3'4\"\">(y#)ptNRw4[Rh;$G\"F(&F%6$F8F0F (F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,**&#\"@a0%R<$fRz-q?T\\p2\"\"> ,._3v2#RFQfm&=i\"\"\"\"&F%6$\"\")\"\"(F.F.*&#\">wRcX@#z9n73ig5A\"=,!\"\"*&#\"=CIGJ< `DH1Ww3\"f\"\"<)>utR)R)GB7PcpHl$\"D,F.1l \\mJ3E5Q+t3M0#\"\"\"&F%6$\"\")\"\"(F.!\"\"*&#\"B%eVy)pY0s\\bQ4)zyV<\"A >gROL*4+(QNlu)p*3]F.&F%6$\"\"*F2F.F3#\"4?jP]\"=e\\hm\"4r$3kyL@^07F3*&# \"Bg\"[2JYecKe37biU(G\"\"@>SJc\"Q1-'zuzWMpX*F.&F%6$F:F1F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',**&#\"B%yKg9d<7z>&*p\"=:_ C#\"AHEyF4X%)Qab%[G%=*\\#\"\"\"&F%6$\"\")\"\"(F.F.*&#\"?sq%Rzh'3gK>r)z <2\"\">^iaC#)oD?V[rpj4'F.&F%6$\"\"*F2F.F.#\"!\"\"*&#\"=sexP['R:]'p\"3[,&\"=T)p3g$=%Q7Xlej/\"F.&F%6$F9F1F.F =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,(*&#\"575QC\\B $R8r$\"3^(*\\E5Y9@c\"\"\"&F%6$\"\")F(F.!\"\"*&#\"3GL\\]I85=w\"2nOTH&)= h*eF.&F%6$\"\"*F(F.F2#\":?7`Kfm(H1+gh=\"8nmpD*=Dw[ZwVF." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&*&#\"3GL\\]I85=w\"2nOTH&)=h* e\"\"\"&F%6$\"\"*F(F.!\"\"#\"703`Y?-[N*R/(\"6BCLM3FKIIy$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"6!Q-J%)4a(=rn'\"6HaskE( ybsva" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!4.zzFg>f)3 =\"5#p**36y,?1-#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"(B\\N#\")![Gk%/&F%6#\"\"#\"\"! /&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"/ip3\"[;=$\"0lT,z %HQ9/&F%6#\"\"(#\"4rg,2g#)HOF(\"5S)*)ybA.d(oH/&F%6#\"\")#\"3$z^xE6R&>8 \"3S+drkkmGg/&F%6#\"\"*#\"/mD1H\"p(H\"0X&Q[Fwc?/&F%6#\"#5#\"/pL@Y%\\+$ \"0gLe#=7k;/&F%6#\"#6#!)dm*z'\"+gPU76" }}}{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 88 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'expande d'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simplify(subs(e3 ,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#&*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 67 "We now turn our attention to the embedded 12 stage, ord er 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 210 "The first 11 stages of t he embedded order 6 scheme coincide with those of the order 7 scheme a nd the weights of the order 7 scheme provide the linking coefficients \+ for the 12th stage of the order 6 scheme. Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[12,j] = b[j];" "6#/&%\"aG6$\"#7%\"jG& %\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 8 " . . 11." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorpora te the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6* &&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The simp lifying conditions allow us to manage with just 7 \"simple\" order con ditions which are given (in abreviated form) as follows. " }}{PARA 0 " " 0 "" {TEXT -1 92 "These include the 6 order 6 quadrature conditions \+ and 1 additional order 6 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO6*` := subs(b= `b*`,SimpleOrderConditions(6)):\n[seq([i,`SO6*`[i]],i=[1,2,4,8,16,31,3 2])]:\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()F2F:F(#F(F5 7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#JF)/*&F,F(-%!G6#*&FEF(%\"aGF(F(#F( \"#I7%\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)pprint446\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "`SO6_12*` : = subs(b=`b*`,SimpleOrderConditions(6,12,'expanded')):\n`cdns*` := [se q(`SO6_12*`[i],i=[1,2,4,8,16,31,32])]:\n`simp_eqs*` := [seq(add(`b*`[i ]*a[i,j],i=j+1..12)=`b*`[j]*(1-c[j]),j=[4,5])]:\nwt_eqns := [seq(b[i]= a[12,i],i=1..11)]:\n`cdns*` := [op(`cdns*`),op(wt_eqns),op(`simp_eqs*` )]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " We also specify that " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\" \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\" #\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\" \"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] = 0;" "6#/&%#b*G6# \"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5] = 0;" "6#/&%#b*G 6#\"\"&\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 1/400; " "6#/&%#b*G6#\"#7*&\"\"\"F)\"$+%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := \+ \{c[12]=1,seq(`b*`[i]=0,i=2..5),`b*`[12]=1/400\}:\ne5 := `union`(e3,e4 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqns2 := [op(remove(_U->type(rhs(_U),realcons),e5)), op(subs(e5,`cdns*`))]:\nnops(eqns2);\nindets(eqns2) minus \{a[8,7]\}; \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#&F%6$FCF'&F%6$FCF1&F%6$FC\"#5&F%6$FC\"#6&F%6$FQF+&F%6$ FQF.&F%6$FQF>&F%6$FQF'&F%6$FCF2&F%6$FC\"\"#&F%6$FC\"\"$&F%6$FCF(&F%6$F 1F(&F%6$F1F+&F%6$F1F.&F%6$F1F>&F%6$F1F'&F%6$FNF2&F%6$FNF(&F%6$FNF+&F%6 $FNF.&F%6$FNF>&F%6$FNF'&F%6$FQF2&F%6$FQF(&F66#FN&F66#FQ" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "e6 := solve(\{op(eqns2)\},indets(e qns2) minus \{a[8,7]\}):\ne7 := simplify(subs(select(u_->member(lhs(u_ ),\{b[8],a[8,7],a[9,7],a[9,8]\}),e6),e5)):\ne8 := `union`(e7,e6):\ninf olevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a [10,4]=subs(e8,a[10,4]);\na[11,1]=subs(e8,a[11,1]);\n`b*`[11]=subs(e8, `b*`[11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"C )G:Z*)>p@Gj]NlG\"pTY\"B0ox@yDXk')4ut&Q0\\V\"\"\"*&#\"N'*=;Y_F7Ccu;t`&* Qhi$f/M=G\"M*3w>&4*GS)3&zZRvQHkq(H'>$=F-&F%6$\"\")\"\"(F-!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&*&#\"Nlo\"Qx'3SPv (p)eLCcT8&yPib8\"M!4`[&>cxH4@E([\"C:v.!p*)Q[2MB2\\`U15#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!4r'\\#e\"))fV()H\"6+[eY_*fncN6" }}} {PARA 0 "" 0 "" {TEXT -1 46 "Note that some of the coefficients depend on " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5371 "e8 := \{a[ 9,4] = -571512529090087463288523889599/110665928940900124157224563980+ 297263750179869977755976442750848783743433/393646979527098909947091277 75657396203856*a[8,7], a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[9,3] = 0, a[8,4] = -58035621009/57705566102*a[8,7]+121036973895/397660568723, c [2] = 1/100, a[7,1] = 1176539682457387/14353367828755172, c[6] = 49/78 , a[7,2] = 0, a[7,3] = 0, a[10,7] = -106274711684783049476906346301176 74305201888/1126679120367788433068175171584217285056331+55360967416461 8852646443487290280016/36191271222637091077853137254331923*a[8,7], a[2 ,1] = 1/100, a[3,1] = -692/729, a[5,1] = 451814999377/985407860736, a[ 6,1] = 131173/1890096, a[7,4] = 17717342174781/163306752068660, c[9] = 37/46, c[10] = 32/33, c[8] = 30/73, c[7] = 47/283, a[5,3] = -55280539 7249/328469286912, a[8,5] = 9842725887631104/36530081903395451*a[8,7]+ 23790493530086400/503472164869673923, `b*`[11] = -2987435988158249671/ 113556675995246584800, b[8] = 131953911267751793/602866646471570040, b [1] = 2354923/46428480, b[10] = 30049446213369/166412182583360, b[11] \+ = -67996657/1112423760, a[7,5] = -528062841353883264/10338013178660912 633, a[4,3] = 1/6, a[4,1] = 1/18, c[4] = 2/9, c[3] = 4/27, a[3,2] = 80 0/729, c[5] = 11417/21528, a[11,10] = -1808859196027797903/20206200178 110899692, a[10,9] = 210642399729505665392/369807601102457012649, a[5, 4] = 96066394193/54744881152, a[8,1] = -3237235171885/25359307117942*a [8,7]+1151376026505/15886913157353, a[7,6] = 1679528554701504/62911488 820241285, a[6,4] = 5764801/17935632, b[6] = 31816481086962/1438294790 14165, a[9,6] = 26166361196504118873975398678680836738/489057042870130 91434070008625140606285+1106666062942934099972208741601767167420603/10 83190168105444369927173368588005478751254*a[8,7], b[9] = 2976912906256 6/205676274838545, a[11,9] = 667711875409843102380/5475725578726647254 29, a[6,5] = 4769464/20091357, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, \+ `b*`[5] = 0, a[8,6] = -6049609011732/44460416127379*a[8,7]-73196981208 0/55706472315697, b[7] = 7273629826007016071/29687570322557889840, a[1 2,10] = 30049446213369/166412182583360, `b*`[12] = 1/400, a[9,8] = 488 03138404357581919875739936453/25803619795436169827228351014320, a[9,1] = -1110338580414865926277697473946663/2493561191216863718125674195427 920+7106320350044612082221138973359956486754105/7413952778513641268828 339629307376544892304*a[8,7], a[9,7] = 9246216557605518509697562923477 305/1916894141076779589386745336846818-1536627393719649295788806828881 1/2046493983775972089745696804584*a[8,7], `b*`[9] = 139711798081481043 314568703/776834200093629660855997950, a[11,8] = -69880579855847721894 868113496992143638/12013145933881997013005874021379315455, a[12,1] = 2 354923/46428480, a[12,6] = 31816481086962/143829479014165, a[11,6] = - 168888848900806749206547597491690227898351512/400517166568863471525394 04570568433034829611*a[8,7]-933871567713417010043913936493438936846764 /412386664295689537639473423323461393071245, `b*`[10] = 22590837341634 73774459617/17881499156003594882624000, c[11] = 1, a[6,2] = 0, c[12] = 1, a[12,5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[10,1] = 132 733153056306860778654221331091277392/116543239091879570070280632795824 410545-7405639292154953534018268469884662977212977480/3792502321998306 829630175224999566236808994473*a[8,7], a[12,2] = 0, a[12,3] = 0, a[12, 4] = 0, a[11,4] = -810099397204456889304344411242947753130247/25991738 093734057119806089577289356308259*a[8,7]+88647229794631536206594074208 1/40314782155561698427819545380, a[9,5] = -481343662216757058043820824 680/575789002799497974196573606897-73993091323366409916890019689479551 245024/36573652014495836578954080833460987071989*a[8,7], a[11,5] = 274 782492882505349882735351252636913822280651264/329077551963501455472785 60250060200426669739859*a[8,7]+8752511250149405021648618545129344/2552 097472637344600980331537932869, `b*`[6] = 285550402807929509723935449/ 1482899423367801818133032950, a[10,6] = -75904483548629929107610481585 80052904592/9984116401664855423056635496639385493495-48559113727827959 8352748545447403014383544896/23330128733322917549942422700283454675649 0361*a[8,7], a[12,11] = -67996657/1112423760, `b*`[1] = 18212730032998 3916501/3507184399433109696000, a[12,9] = 29769129062566/2056762748385 45, a[10,8] = -19209909490782204629345787487654324166296/6932257046340 021610507379175381008191035, a[11,7] = 8375194921189822011069688539149 8/2702519689118850934218611462427*a[8,7]-86160967122337538265775125080 650306256980/4354596999999547080438586092273339232211, a[11,1] = -1355 62377851341562433588697753740086773816865/3426701339432503875107104503 6508365595282617*a[8,7]+4872621092977561954853090194296904/21006425349 07233407483889690037515, a[10,2] = 0, a[10,3] = 0, a[11,2] = 0, a[11,3 ] = 0, a[10,5] = 24419663614742996977897053162624126976/15732207910296 489915861663935989058823+181634275718103812007845846240505855585493969 0405888/440690033983643919705302904536691263239284094094091*a[8,7], a[ 4,2] = 0, a[5,2] = 0, a[6,3] = 0, a[12,7] = 7273629826007016071/296875 70322557889840, a[10,4] = 4641691286535506328216919894715288/434905385 737409866445257821776805-281834045936261389553731674562412275246161896 /18319629770642938753947795088402890951976089*a[8,7], a[12,8] = 131953 911267751793/602866646471570040, `b*`[8] = 143483908435216987345410922 193/615408757915355446585175892000, `b*`[7] = 644329552940319664239528 1317251/26849338691229488574870087988000\}:" }}}{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 65 "The coefficients for the embedded order 6 scheme are as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printe d coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1), ``$(8-i)],i=2..7)]));\nfor ii from 8 to 11 do\n print(``);\n print (c[ii]=subs(e8,c[ii])); \n for jj to ii-1 do\n coeff_val := sub s(e8,a[ii,jj]);\n if coeff_val=a[ii,jj] then \n print(a[i i,jj]*` is a parameter.`)\n else\n print(a[ii,jj]=coeff_v al);\n end if;\n end do:\nend do:\nprint(``);\nc[12]=subs(e8,c[ 12]);\nseq(a[12,i]=subs(e8,a[12,i]),i=1..11);\nprint(``);\nseq(`b*`[i] =subs(e8,`b*`[i]),i=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7(7*#\"\"\"\"$+\"F(%!GF+F+F+F+F+7*#\"\"%\"#F#!$#p\"$H(#\"$+)F2F+F +F+F+F+7*#\"\"#\"\"*#F)\"#=\"\"!#F)\"\"'F+F+F+F+7*#\"&<9\"\"&G:##\"-x$ **\\\"=X\"-O2'yS&)*F;#!-\\sR0Gb\"-7pGp%G$#\",$>%Rmg*\",_6)[uaF+F+F+7*# \"#\\\"#y#\"'t68\"('4!*=F;F;#\"(,[w&\")Kc$z\"#\"(k%pZ\")d84?F+F+7*#\"# Z\"$$G#\"1(QdCoRl<\"\"2s^vGyO`V\"F;F;#\"/\"yu@M3Il$ \"\"\"&F%6$F'\"\"(F.F.#\"2+k3IN\\!zB\"3BRnp[;sM]F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\".K<,4'\\g\"/zt7;/YW\"\"\"&F %6$F'\"\"(F.!\"\"#\"-!37)p>t\"/(p:BZ1d&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\"#P\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"\",&#\"Cjm%RZ(pxi#f'[T!eQ.6\"\"C?zU&>uc7=P'o@\">hN\\#!\"\"*&# \"L0Tv'[c*fL(*Q6A#37Y/].K1r\"L/B*[awtIH'R$G)o7k8&yF&RT(F(&F%6$\"\")\" \"(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,&#\"?*f*)Q_)Gju3!4HD^r&\"?! )RcCs:C,!4%*Gfm5\"!\"\"*&#\"KLMu$y[3vUk(fvx*p)z,vjsH\"JcQ?'RdcxF\"4Z*4 *)4F&zpk$R\"\"\"&F%6$\"\")\"\"(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\"&,&#\"?!oC3#Q/eqv;AmV8[\"?(*ogtl>uz\\*z-!*yv&!\"\"* &#\"JC]C^&z%*o>+*o\"*4kOB84$*R(\"J*)>2()4YL33a*yl$e\\9?ltl$\"\"\"&F%6$ \"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"', &#\"GQn$3oy')RvR()=T]'>hj;E\"G&G119D'3+2M948qG/d!*[\"\"\"*&#\"L.1Unrw, ;u3A(**4MH%H1mm5\"\"La7vya+)eoL$3\"F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"C0tZBHc(p4&= bgdl@Y#*\"C=o%oLXnQ*ezn2TT*o\">\"\"\"*&#\"A6))Go!))y&H\\'>PRFm`\"\"@%e /opX(*3sfx$)R\\Y?F-&F%6$\"\")F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"A`k$*Rd()>>edVSQJ!)[\"A?V,^$Gs#)phV&z>O!e# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#K\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#5\"\"\",&#\"H#Rx74J8Aa'y2'oIcI:LF8\"HX0TCezK1Gq+dz=4RKa;\"F(*&# \"O![xH@xHm%))p%o#=S``\\:#HRcS(\"OtW**3oBm&**\\Av,jHoI)*>K-Dz$F(&F%6$ \"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"C)G:Z*)>p@Gj]Nl G\"pTY\"B0ox@yDXk')4ut&Q0\\V\"\"\"*&#\"N'*=;Y_F7Ccu;t`&*Qhi$f/M=G\"M*3 w>&4*GS)3&zZRvQHkq(H'>$=F-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&,&#\"Gwp7CE;`q*y(p*Hu9Om>W#\"GB)e!*)f $Rmhe\"**['H5z?Kd\"\"\"\"*&#\"U))eS!pR\\&ebe]Si%e%y+7Q5=dFM;=\"T\"4%4% 4%GRKE\"pOX!HI0(>Rk$)R.!pS%F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"I#f/H0!ee\"[5w5H*H'[N[/f(\"I&\\$ \\&QRm\\Nm0Ba&[m,k6%)**!\"\"*&#\"N'*[a$Q9ISZaa[FN)fz#ys8\"f&[\"Nh.\\cn aMG+FUU*\\v\"HKL(G,LB\"\"\"&F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"M))=?0Vn<,jM1pZ\\Iy%o6ZF1\"\"LJj 0&GLas8`y2\"4PE Ar7>O\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\")#!J'HmTKaw[(yX$HY?#y!\\4*4#>\"IN5>35Qv\"zt]5;-Sj/dA$p" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"6#Rlc]H(*RU1@\"6 \\E,dC5,w!)p$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&*&#\"Nlo\"Qx'3SPv(p)eLCcT8&yPib8\"M!4`[&>cxH4@E([\"C:v.!p*) Q[2MB2\\`U15#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,&*&#\"KZ-8`x%HC6WM/ $*)oX/sR*45)\"Jf#3jN*Gx&*31)>r0MP4Q<*f#\"\"\"&F%6$\"\")\"\"(F.!\"\"#\" ?\"3U2%f1i`JYzHsk))\">!QX&>yU)phb:#y9.%F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&,&*&#\"Qk7l!GAQ\"pj_7NNF))\\`]#)G\\#y u#\"Pf)R(pmU+-1]-cysaX,N'>bx!H$\"\"\"&F%6$\"\")\"\"(F.F.#\"CW$H^a='[;- 0%\\,D6Dv)\"CpG$z`J.)4gWtjsu4_DF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"\"',&*&#\"N7:N)*yA!p\"\\(fZl?\\n!3!*[))))o\"\"M6'H[.L%o0 d/%RD:Zj)ol;<0S\"\"\"&F%6$\"\")\"\"(F.!\"\"#\"Kkn%o$*QM\\OR\"R/5qT8xcr Q$*\"KX72$RhMKBMZRw`*o&HkmQ7%F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"(,&*&#\"A)\\\"R&)op5,A)*=@\\>v$)\"@FCY6'=U$4&)=\"*o>Dq# \"\"\"&F%6$\"\")F(F.F.#\"J!)pD1.l!3D^xl#QvLAr'4;')\"I6ABRLF#4'eQ/3Z&** ***pfaV!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#!GQ O9#*p\\8\"o[*=sZe&)z0))p\"GbaJz8-ue+8q*>)Q$f98?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"6!Q-J%)4a(=rn'\"6HaskE(ybsva" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!4.zzFg>f)3=\"5#p**3 6y,?1-#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6- /&%\"aG6$\"#7\"\"\"#\"(B\\N#\")![Gk%/&F%6$F'\"\"#\"\"!/&F%6$F'\"\"$F0/ &F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'#\"/ip3\"[;=$\"0lT,z%HQ9/&F% 6$F'\"\"(#\"4rg,2g#)HOF(\"5S)*)ybA.d(oH/&F%6$F'\"\")#\"3$z^xE6R&>8\"3S +drkkmGg/&F%6$F'\"\"*#\"/mD1H\"p(H\"0X&Q[Fwc?/&F%6$F'\"#5#\"/pL@Y%\\+$ \"0gLe#=7k;/&F%6$F'\"#6#!)dm*z'\"+gPU76" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"#\"6,l\"R) *H.IF@=\"7+gp4JV*R%=2N/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6# \"\"&F//&F%6#\"\"'#\"<\\a$Rs4&Hz!GS]bG\"=]H.L\"==!yOB%**G[\"/&F%6#\"\" (#\"@^sJ\"G&RUm>.%HbHVk\"A+!))z3q[d)[H7pQ$\\o#/&F%6#\"\")#\"?$>A4TXt)p @N%3R[V\"\"?+?*e<&eYaN:zv3ah/&F%6#\"\"*#\"<.(oXJV5[\"3)z6(R\"\"<]z*f&3 mHO4+U$ox/&F%6#\"#5#\":<'fWxtM;MP3fA\";+Si#)[f.g:*\\\")y\"/&F%6#\"#6#! 4r'\\#e\"))fV()H\"6+[eY_*fncN6/&F%6#\"#7#F'\"$+%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "R K7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7, 11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,OrderConditions(6,12,'e xpanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(s ubs(e8,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs (e8,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#P" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "We choos e a value for " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" } {TEXT -1 85 " that makes the principal error norm for the order 7 sch eme approximately a minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "errterms := PrincipalErrorT erms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm : = sm+(evalf(subs(e8,errterms[ct])))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot(sqrt(sm ),a[8,7]=0.1..0.65,color=COLOR(RGB,.5,0,1),thickness=2,font=[HELVETICA ,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 277 277 {PLOTDATA 2 "6(-%'CURV ESG6#7S7$$\"\"\"!\"\"$\"gn]#)[qkQ!)>5-.PjN4'\\^<&R8!pvKL&QB;@!#k7$$\"g nnmmmmmmmmmmmmmmmmmmmmmmm\"zM%))>6!#g$\"gn3[90qDNLWL&Rl%H'zp\"yr*p2IfY eL3\\1#F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLL3-y]>C7F1$\"gnA`,Nn?!))**3z6Jxv %4Cx49$=K%\\#z!\\<@?F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmmm;a.M]T8F1$\"gnG%p^ ruA0p5mPcK/\"f:%)H!=n>rv/f%4t>F-7$$\"gnmmmmmmmmmmmmmmmmmmmmmmm;zC$*ef9 F1$\"gnL+YjPE`e5?U9xE_zEo/&)yJf(p/^ff#>F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLL L3F/S6x:F1$\"gnS-'zp7GPbpe=3sMuc)fXaGyOzrv=S!)=F-7$$\"gnmmmmmmmmmmmmmm mmmmmmmmmTNEW2'o\"F1$\"gn-$[8=\\R`vojKmbTqX&4rqWxB9%*\\zXR=F-7$$\"gn++ +++++++++++++++++++++v=Km*))z\"F1$\"gnuiyt;_xH#*[^Ki&G1^*Qh=3!fA]uzn%) z\"F-7$$\"gnmmmmmmmmmmmmmmmmmmmmmmmT&)=vd:>F1$\"gni7)3^[eWD^^.9%*o*3(p K'f#>$R8l#R%od::#F1$\"g nhM1-D!3ZM&HR.nt2qVM-lxKUF;3Bp!o\"F-7$$\"gnmmmmmmmmmmmmmmmmmmmmmmm;HZ^ *oD#F1$\"gnbRoAH^a`@pseyI_QQta2Fy!=$*H(*)))[;F-7$$\"gn++++++++++++++++ ++++++++vyE_vBF1$\"gnn./%*[4517ZaXj!yV*eG$fJx^h@<=g^h\"F-7$$\"gn++++++ ++++++++++++++++++D1tj%\\#F1$\"gnRq))p^gH%p>EpoKH\\JM.cv$yTgv9$ROe\"F- 7$$\"gn++++++++++++++++++++++++D@jU4EF1$\"gn?;7uF+]kNS-t]\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLLLL[!>E %HF1$\"gnwkIF1$\"gnoZ%zt^'GY:sa5TCg(y,09? YAm#>Ljtp9F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLLL$3;zG<$F1$\"gnEkkr^r*f5hcyr uQ;$yBWm*)[fy2K:zb9F-7$$\"gn****************************************** ***\\(o\\3]\"H$F1$\"gnWAmP<>aJ#H[68u5D]@K^kVQ-+zEONW\"F-7$$\"gn******* **************************************\\i!\\nX/MF1$\"gnJt!yCh6W(Q;bQ#* )*fN[D\\bva/:)GV**[V\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLL$3-Q9B_$F1$\"gnD &oI#\\v(Q')fM-R*[(RDU3%>rz/qpi55H9F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLL3xwW aIOF1$\"gnn_\"))*>!pm4%H!GY\"y/Vz(Hdj&oGCd%p+nU\"F-7$$\"gnmmmmmmmmmmmm mmmmmmmmmmm;Hd_GZPF1$\"gn%3VS*f5c'*)=e-],'[q_()z/JsvISkqEF9F-7$$\"gnmm mmmmmmmmmmmmmmmmmmmmmT&)=haoQF1$\"gn#R]1ZF;incc-^:=8V \"F-7$$\"gn*********************************************\\7`0/T(RF1$\" gnr?r`?%4D5eKZI\"f_I#Q%3Y_Z&R!*Q!\\pP9F-7$$\"gnmmmmmmmmmmmmmmmmmmmmmmm ;a=&4\")3%F1$\"gn`->A)R/+4t5[5x)[)peY:'\\19)4wx>vW\"F-7$$\"gn********* ***************************************\\<$))e?%F1$\"gnRE)z]miuf67Z(z) 3/'pp^)>aD]nr^43Y\"F-7$$\"gn****************************************** ****\\P%*=6@VF1$\"gnG)e;-vqC)QU[U\\#*)=Oit&*H*G\"yK6t;oZ\"F-7$$\"gn*** ******************************************\\(oHmfKWF1$\"gn'*y\"*pMqi'4 \\%o\"=]8+sT7![[^#)H)z)Q/&\\\"F-7$$\"gn******************************* ***************\\i?9QcXF1$\"gnRqxFJ%o)3F&fKM;,=-*H-<'o'*3'4@!>$=:F-7$$ \"gnKLLLLLLLLLLLLLLLLLLLLLLL$3c2wm%F1$\"gnS5!)*Ro*f.\"p`%Q$e'3)ej$\\x7 \"[R^$eM%=a\"F-7$$\"gn*********************************************** \\i?ljy%F1$\"gn>&))3o&*GUe^+L=SD@#f]_'pO>AQQye&p:F-7$$\"gnmmmmmmmmmmmm mmmmmmmmmmmT&QGxR*[F1$\"gn=U#)o3OEw**o)Q%Gbm6+\"p/1*4VBmE(Gh;F-7$$\"gn*********************** **********************\\PM,N!Q_F1$\"gn?Oo\"HAH4HoQv::9K&*euWRLBK!)*y0- (p\"F-7$$\"gnKLLLLLLLLLLLLLLLLLLLLLLLekR;^`F1$\"gnXi+uA8#*e[J_0.]p&*Rr wXTD&)*yF-7$$\"gn********************************************* ***\\>ON%>F-7$$\"gnKLLLLLLL LLLLLLLLLLLLLLL$eRJQT/'F1$\"gnm^L.P;\"Gf')o#ek)pYC+1YihM@#G'R\"p#*>F-7 $$\"gnlmmmmmmmmmmmmmmmmmmmmmmmmr#3J:'F1$\"gnt@(eJ8Z_5%*yYS_o.vH#y\\x'z Na;Opx.#F-7$$\"gn)********************************************\\(ouHHp iF1$\"gn*fj4$*Q=')>$*)\\`i8>S&*Q'RXc$[(=^))4p3#F-7$$\"gn)************* *******************************\\7.c'\\!Q'F1$\"gn,'4&eE*Qz$z$*)*p-FtR \"*o1ZvK7I'>A;\\8#F-7$$\"#l!\"#$\"gn21J+2F(>,?)z%**faz#[L!Qr>7m]ET$\\( =#F--%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\"&F*$\"\"!Ff[l$F)F f[l-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q'a[8,7]6\"Q!F`\\l-%%VIEWG6$;F( Fez%(DEFAULTG" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "evalf[12](minimize(sqrt(sm),a[8,7]=0..1,'location '));\nconvert(subs(op(1,op(op(2,[%]))),a[8,7]),rational,4);\nconvert(s ubs(op(1,op(op(2,[%%]))),a[8,7]),rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-h5`_E9!#;<#7$<#/&%\"aG6$\"\")\"\"($\"-6EetoO!#7F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#6\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#^\"$R\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "evalf[10](eval(sqrt(sm),a[8,7]=11/3 0));\nevalf[10](eval(sqrt(sm),a[8,7]=51/139));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ee_E9!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+C`_ E9!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e9 := `union`(subs(a[8,7]=11/30,e8),\{a[8,7]=11/30\}) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4283 " e9 := \{a[8,7] = 11/30, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[9,3] = 0 , c[2] = 1/100, a[7,1] = 1176539682457387/14353367828755172, c[6] = 49 /78, a[7,2] = 0, a[7,3] = 0, a[11,5] = 5035138254118716342864368937874 683246617259598976/775682801056824859328708920179990438628643868105, a [2,1] = 1/100, a[3,1] = -692/729, a[5,1] = 451814999377/985407860736, \+ a[6,1] = 131173/1890096, a[7,4] = 17717342174781/163306752068660, c[9] = 37/46, c[10] = 32/33, c[8] = 30/73, c[7] = 47/283, a[5,3] = -552805 397249/328469286912, `b*`[11] = -2987435988158249671/11355667599524658 4800, b[8] = 131953911267751793/602866646471570040, b[1] = 2354923/464 28480, b[10] = 30049446213369/166412182583360, b[11] = -67996657/11124 23760, a[7,5] = -528062841353883264/10338013178660912633, a[4,3] = 1/6 , a[4,1] = 1/18, c[4] = 2/9, c[3] = 4/27, a[3,2] = 800/729, c[5] = 114 17/21528, a[11,10] = -1808859196027797903/20206200178110899692, a[10,9 ] = 210642399729505665392/369807601102457012649, a[5,4] = 96066394193/ 54744881152, a[7,6] = 1679528554701504/62911488820241285, a[6,4] = 576 4801/17935632, b[6] = 31816481086962/143829479014165, b[9] = 297691290 62566/205676274838545, a[11,9] = 667711875409843102380/547572557872664 725429, a[6,5] = 4769464/20091357, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, b[7] = 7273629826007016071/29687570322557889840, a[12 ,10] = 30049446213369/166412182583360, `b*`[12] = 1/400, a[9,8] = 4880 3138404357581919875739936453/25803619795436169827228351014320, `b*`[9] = 139711798081481043314568703/776834200093629660855997950, a[11,1] = \+ 221939233494467949726864332982580511239150575217/255387155540276753091 910917079234490443699161270, a[11,8] = -698805798558477218948681134969 92143638/12013145933881997013005874021379315455, a[12,1] = 2354923/464 28480, a[12,6] = 31816481086962/143829479014165, `b*`[10] = 2259083734 163473774459617/17881499156003594882624000, c[11] = 1, a[6,2] = 0, c[1 2] = 1, a[10,4] = 8380907279399940341415520095001429689250499016988/16 65529140598002776815163790462148830898906131435, a[10,5] = 64599178123 85485497254681791738665918061301836694665216/2108701812611736155789874 398208067694599974390240225435, a[12,5] = 0, b[2] = 0, b[3] = 0, b[4] \+ = 0, b[5] = 0, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[8,4] = -116072 38985710021183/180261023052662924020, a[9,4] = -9023990274786340493184 68354365831042971264299/376720159407433656819366352813041281670901920, a[8,1] = 110903909289926737793/4320958290769994040132, a[9,6] = 61212 379964740003623393766579069964177763490780547/672922034904977555273490 11795764425282435991108860, a[8,5] = 8332987977461879955626624/5705645 3830329950469093005, a[9,1] = -199116158962708264981792850380273725319 682574083/2122095703793959540376735652096650388444524173920, a[8,6] = \+ -397914630242458427722/6312988060727977701695, a[9,7] = 17673520104558 01522759919865162797424677/853631298904311486745705433404624991760, `b *`[6] = 285550402807929509723935449/1482899423367801818133032950, a[12 ,11] = -67996657/1112423760, a[10,1] = 1468128829121034982234777107023 27669160817645499822332/3471363275502643502810190506910949216726536364 86171345, `b*`[1] = 182127300329983916501/3507184399433109696000, a[10 ,7] = -239812294208127777232571199557076199094738401624/62716593235272 943126739970926235455172660665115, a[12,9] = 29769129062566/2056762748 38545, a[9,5] = -1932841049216489668013274675445061103749096456/122503 4474225538046212066937516775761976271555, a[10,8] = -19209909490782204 629345787487654324166296/6932257046340021610507379175381008191035, a[1 0,2] = 0, a[10,3] = 0, a[11,2] = 0, a[11,3] = 0, a[11,6] = -7722480175 4835239618997430260090636543172193252536784/20265242890170705812536717 982284413527106230272155465, a[4,2] = 0, a[5,2] = 0, a[6,3] = 0, a[11, 4] = 181163263192487297546324132938257205801916729/1715454714186447769 9072019121010975163450940, a[12,7] = 7273629826007016071/2968757032255 7889840, a[11,7] = -2041746049322766275834287576167164751357072803/242 398642004974788232613894826395428361025315, a[10,6] = -329623101936975 913713940238539594299201957668540750512/216369193273123311533898096852 817531450154842258383315, a[12,8] = 131953911267751793/602866646471570 040, `b*`[8] = 143483908435216987345410922193/615408757915355446585175 892000, `b*`[7] = 6443295529403196642395281317251/26849338691229488574 870087988000\}:" }{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 62 "The coefficients for the combin ed scheme are given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1011 "subs(e9,matrix([[c[2],a[2,1],``],\n[c[3],a[3,1],a[3,2]],[c[4 ],a[4,1],a[4,2]],[``$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]],[``$2,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]],[``$2,a[8,7]],\n[c[9],a[9,1], a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6]],[``,a[9,7],a[9,8]],\n[c[ 10],a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[``,a[10,5],a[10,6]],\n[``,a [10,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[11,1],a[11,2]],[``,a[11,3],a [11,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7],a[11,8]],[``,a[11,9],a[11,1 0]],\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a[12,4]],[``,a[12,5],a[12,6] ],\n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,10]],[``$2,a[12,11]],\n[``,` ________________________________`$2],[`b`,b[1],b[2]],[``,b[3],b[4]],[` `,b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10]],[``$2,b[11]],\n[`b*`,`b* `[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[6]],\n[``,`b*`[7], `b*`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11],`b*`[12]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\"\"\"\"$+\"F(%!G7%#\"\"%\"#F# !$#p\"$H(#\"$+)F27%#\"\"#\"\"*#F)\"#=\"\"!7%F+F+#F)\"\"'7%#\"&<9\"\"&G :##\"-x$**\\\"=X\"-O2'yS&)*F;7%F+#!-\\sR0Gb\"-7pGp%G$#\",$>%Rmg*\",_6) [ua7%#\"#\\\"#y#\"'t68\"('4!*=F;7%F+F;#\"(,[w&\")Kc$z\"7%F+F+#\"(k%pZ \")d84?7%#\"#Z\"$$G#\"1(QdCoRl<\"\"2s^vGyO`V\"F;7%F+F;#\"/\"yu@M`st-Q]Gz\")\\E3F'*eh6*> \"R?R!4n\"GTI\"GNm$>olLuSf,sw$7%F+#!Ock4\\P5h]WvYF8!o'*[;#\\5%G$>\"Ob:Fw>w vn^Pp17i/QbAuW.D7#\"SZ0y!\\jx*fF_,eX5?Nn<\"Hg<*\\i/MV0du'[6V!*)HJO &)#\"A`k$*Rd()>>edVSQJ!)[\"A?V,^$Gs#)phV&z>O!e#7%#\"#K\"#L#\"WKB#)*\\X w\"3;pwK-2rxMA)\\.@\"H)G\"o9\"WX8<'[OOls;#\\4\"p]!>5G]VE]vKOrMF;7%F+F; #\"R))p,*\\]#*oH9+&4?bTT.%**Rzs!4Q)\"RN981*)*3$)[@Y!zj^\"ox-!)fS\"Hbm \"7%F+#\"X;_m%pO=Ih!=fmQ?*H%fR&Q-%Rr8f(p$>5B'H$\"W:LQeA%[:]9`pj@7%F+#!QC;SQZ4*>wqb*>rDBxx73UH7)R#\"P:^mgE\"IN5>35Qv\"zt]5;-Sj/dA$p7%F+F+#\"6#Rlc]H(*R U1@\"6\\E,dC5,w!)p$7%F)#\"Q<_d]\"R7^!e#)HLkos\\zY%\\L#R>A\"Qq7;*pV/\\M #zq\"4\">4`nFSb:(Qb#F;7%F+F;#\"NHn\">!e?d#QH8Cja(H([#>jK;\"=\"MS4Xj^(4 ,@\">?2*pxW'=9ZX:<7%F+#\"Rw*)ffshYKouy$*oV'GM;(=TDQ^.&\"Q0\"oQkG'Q/**z ,#*3(G$f[#o0,Gov(#!V%yODD$>sJaO14g-V(**='R_$[v,[Ax\"Vla:s-B1r_8WG#)zrO D\"eqq,*GCl-#7%F+#!O.G2d8vkr;wvGMeFmFK\\guT?\"N:`-h$GaRE[*QhK#)yu\\+U' )RU##!GQO9#*p\\8\"o[*=sZe&)z0))p\"GbaJz8-ue+8q*>)Q$f98?\"7%F+#\"6!Q-J% )4a(=rn'\"6HaskE(ybsva#!4.zzFg>f)3=\"5#p**36y,?1-#7%F)#\"(B\\N#\")![Gk %F;7%F+F;F;7%F+F;#\"/ip3\"[;=$\"0lT,z%HQ97%F+#\"4rg,2g#)HOF(\"5S)*)ybA .d(oH#\"3$z^xE6R&>8\"3S+drkkmGg7%F+#\"/mD1H\"p(H\"0X&Q[Fwc?#\"/pL@Y%\\ +$\"0gLe#=7k;7%F+F+#!)dm*z'\"+gPU767%F+%A_____________________________ ___GF\\x7%%\"bGFavF;FdvFevFivF`wFgw7%%#b*G#\"6,l\"R)*H.IF@=\"7+gp4JV*R %=2NF;Fdv7%F+F;#\"<\\a$Rs4&Hz!GS]bG\"=]H.L\"==!yOB%**G[\"7%F+#\"@^sJ\" G&RUm>.%HbHVk\"A+!))z3q[d)[H7pQ$\\o##\"?$>A4TXt)p@N%3R[V\"\"?+?*e<&eYa N:zv3ah7%F+#\"<.(oXJV5[\"3)z6(R\"\"<]z*f&3mHO4+U$ox#\":<'fWxtM;MP3fA\" ;+Si#)[f.g:*\\\")y\"7%F+#!4r'\\#e\"))fV()H\"6+[eY_*fncN6#F)\"$+%Q(ppri nt66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(e9,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(1 3-i)],i=2..12),\n[`b`,seq(b[j],j=1..11),``],[`b*`,seq(`b*`[j],j=1..12) ]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7 /$\"'++5!\"(F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"'[\"[\"!\"'$!'Y#\\*F/$\"'R(4 \"!\"&F+F+F+F+F+F+F+F+F+F+7/$\"'AAAF/$\"'cbbF*$\"\"!F;$\"'nm;F/F+F+F+F +F+F+F+F+F+7/$\"'L.`F/$\"'1&e%F/F:$!'(Ho\"F4$\"'![v\"F4F+F+F+F+F+F+F+F +7/$\"'0#G'F/$\"'-SpF*F:F:$\"';9KF/$\"'*QP#F/F+F+F+F+F+F+F+7/$\"'yg;F/ $\"''p>)F*F:F:$\"'\"\\3\"F/$!'(z5&F*$\"'npEF*F+F+F+F+F+F+7/$\"'f4TF/$ \"'lmDF*F:F:$!'8RkF*$\"'[g9F/$!'6.jF*$\"'nmOF/F+F+F+F+F+7/$\"'[V!)F/$! '+$Q*F*F:F:$!'T&R#F4$!'zx:F4$\"']'4*F/$\"'Rq?F4$\"'L\"*=F4F+F+F+F+7/$ \"'(pp*F/$\"'EHUF/F:F:$\"')>.&F4$\"'YjIF4$!'VB:F4$!'vBQF4$!'4rFF4$\"'+ 'p&F/F+F+F+7/$\"\"\"F;$\"'J!p)F/F:F:$\"'2c5!\"%$\"'B\"\\'F4$!'q5QF4$!' 4B%)F4$!',7F4$!'+_*)F*F+F+7/Fcq$\"':s]F*F:F:F:F:$\"'57AF/$\" '1]CF/$\"'x)=#F/$\"'QZ9F/$\"'s0=F/$!'[7hF*F+7/%\"bGFgrF:F:F:F:FirF[sF] sF_sFasFcsF+7/%#b*G$\"')H>&F*F:F:F:F:$\"'iD>F/$\"'!)*R#F/$\"'_JBF/$\"' [)z\"F/$\"'Oj7F/$!'zIEF*$\"'++D!\")Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the e mbedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffi cients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4124 "ee := \{c[2]=1/100,\nc[3]= 4/27,\nc[4]=2/9,\nc[5]=11417/21528,\nc[6]=49/78,\nc[7]=47/283,\nc[8]=3 0/73,\nc[9]=37/46,\nc[10]=32/33,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/100, \na[3,1]=-692/729,\na[3,2]=800/729,\na[4,1]=1/18,\na[4,2]=0,\na[4,3]=1 /6,\na[5,1]=451814999377/985407860736,\na[5,2]=0,\na[5,3]=-55280539724 9/328469286912,\na[5,4]=96066394193/54744881152,\na[6,1]=131173/189009 6,\na[6,2]=0,\na[6,3]=0,\na[6,4]=5764801/17935632,\na[6,5]=4769464/200 91357,\na[7,1]=1176539682457387/14353367828755172,\na[7,2]=0,\na[7,3]= 0,\na[7,4]=17717342174781/163306752068660,\na[7,5]=-528062841353883264 /10338013178660912633,\na[7,6]=1679528554701504/62911488820241285,\na[ 8,1]=110903909289926737793/4320958290769994040132,\na[8,2]=0,\na[8,3]= 0,\na[8,4]=-11607238985710021183/180261023052662924020,\na[8,5]=833298 7977461879955626624/57056453830329950469093005,\na[8,6]=-3979146302424 58427722/6312988060727977701695,\na[8,7]=11/30,\na[9,1]=-1991161589627 08264981792850380273725319682574083/2122095703793959540376735652096650 388444524173920,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-9023990274786340493184 68354365831042971264299/376720159407433656819366352813041281670901920, \na[9,5]=-1932841049216489668013274675445061103749096456/1225034474225 538046212066937516775761976271555,\na[9,6]=612123799647400036233937665 79069964177763490780547/6729220349049775552734901179576442528243599110 8860,\na[9,7]=1767352010455801522759919865162797424677/853631298904311 486745705433404624991760,\na[9,8]=48803138404357581919875739936453/258 03619795436169827228351014320,\na[10,1]=146812882912103498223477710702 327669160817645499822332/\n 34713632755026435028101905069109492 1672653636486171345,\na[10,2]=0,\na[10,3]=0,\na[10,4]=8380907279399940 341415520095001429689250499016988/166552914059800277681516379046214883 0898906131435,\na[10,5]=6459917812385485497254681791738665918061301836 694665216/\n 21087018126117361557898743982080676945999743902402 25435,\na[10,6]=-32962310193697591371394023853959429920195766854075051 2/\n 216369193273123311533898096852817531450154842258383315,\n a[10,7]=-239812294208127777232571199557076199094738401624/627165932352 72943126739970926235455172660665115,\na[10,8]=-19209909490782204629345 787487654324166296/6932257046340021610507379175381008191035,\na[10,9]= 210642399729505665392/369807601102457012649,\na[11,1]=2219392334944679 49726864332982580511239150575217/2553871555402767530919109170792344904 43699161270,\na[11,2]=0,\na[11,3]=0,\na[11,4]=181163263192487297546324 132938257205801916729/17154547141864477699072019121010975163450940,\na [11,5]=5035138254118716342864368937874683246617259598976/7756828010568 24859328708920179990438628643868105,\na[11,6]=-77224801754835239618997 430260090636543172193252536784/\n 2026524289017070581253671798 2284413527106230272155465,\na[11,7]=-204174604932276627583428757616716 4751357072803/242398642004974788232613894826395428361025315,\na[11,8]= -69880579855847721894868113496992143638/120131459338819970130058740213 79315455,\na[11,9]=667711875409843102380/547572557872664725429,\na[11, 10]=-1808859196027797903/20206200178110899692,\na[12,1]=2354923/464284 80,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=318164810 86962/143829479014165,\na[12,7]=7273629826007016071/296875703225578898 40,\na[12,8]=131953911267751793/602866646471570040,\na[12,9]=297691290 62566/205676274838545,\na[12,10]=30049446213369/166412182583360,\na[12 ,11]=-67996657/1112423760,\n\nb[1]=2354923/46428480,\nb[2]=0,\nb[3]=0, \nb[4]=0,\nb[5]=0,\nb[6]=31816481086962/143829479014165,\nb[7]=7273629 826007016071/29687570322557889840,\nb[8]=131953911267751793/6028666464 71570040,\nb[9]=29769129062566/205676274838545,\nb[10]=30049446213369/ 166412182583360,\nb[11]=-67996657/1112423760,\n\n`b*`[1]=1821273003299 83916501/3507184399433109696000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n `b*`[5]=0,\n`b*`[6]=285550402807929509723935449/1482899423367801818133 032950,\n`b*`[7]=6443295529403196642395281317251/268493386912294885748 70087988000,\n`b*`[8]=143483908435216987345410922193/61540875791535544 6585175892000,\n`b*`[9]=139711798081481043314568703/776834200093629660 855997950,\n`b*`[10]=2259083734163473774459617/17881499156003594882624 000,\n`b*`[11]=-2987435988158249671/113556675995246584800,\n`b*`[12]=1 /400\}:" }}}{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[7,11];" "6#&%\"TG6$\"\"(\"#6" } {TEXT -1 128 " denote the vector whose components are the principal e rror terms of the 11 stage, order 7 scheme (the error terms of order 8 )." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6, 12]; " "6#&%#T*G6$\"\"'\"#7" }{TEXT -1 146 " denote the vector whose compo nents are the principal error terms of the embedded 12 stage, order 6 \+ scheme (the error terms of order 7) and let " }{XPPEDIT 18 0 "`T*`[7, 12];" "6#&%#T*G6$\"\"(\"#7" }{TEXT -1 99 " denote the vector whose co mponents are the error terms of order 8 of the embedded order 6 scheme ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Deno te the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[7, 11]) );" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"#7" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[7,12]));" "6#-%$absG 6#-F$6#&%#T*G6$\"\"(\"#7" }{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[8] = abs(abs(T[7, 11])); " "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,12]))/abs(abs(`T*`[6, 12])); " "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\"#7\"\"\"-F*6#-F*6#&F /6$\"\"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(` T*`[7,12]-T[7,11]))/abs(abs(`T*`[6,12]));" "6#/&%\"CG6#\"\")*&-%$absG6 #-F*6#,&&%#T*G6$\"\"(\"#7\"\"\"&%\"TG6$F2\"#6!\"\"F4-F*6#-F*6#&F06$\" \"'F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as at tempting to ensure that " }{XPPEDIT 18 0 "A[8];" "6#&%\"AG6#\"\")" } {TEXT -1 73 " is a minimum, if the embedded scheme is to be used for \+ error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\")" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errt erms7_11 := PrincipalErrorTerms(7,11,'expanded'):\n`errterms7_12*` :=s ubs(b=`b*`,PrincipalErrorTerms(7,12,'expanded')):\n`errterms6_12*` := \+ subs(b=`b*`,PrincipalErrorTerms(6,12,'expanded')):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := s qrt(add(evalf(subs(ee,`errterms7_12*`[i]))^2,i=1..nops(`errterms7_12*` ))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms6_12*`[i]))^2,i=1..nops( `errterms6_12*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms7_12*`[i ])-subs(ee,errterms7_11[i])))^2,i=1..nops(errterms7_11))):\n'B[8]'= ev alf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")Kx&e\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")w>x:!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" } }{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4124 "ee := \{c[2]=1/100,\nc[3]=4/27,\nc[4]=2/9,\nc[ 5]=11417/21528,\nc[6]=49/78,\nc[7]=47/283,\nc[8]=30/73,\nc[9]=37/46,\n c[10]=32/33,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/100,\na[3,1]=-692/729,\na [3,2]=800/729,\na[4,1]=1/18,\na[4,2]=0,\na[4,3]=1/6,\na[5,1]=451814999 377/985407860736,\na[5,2]=0,\na[5,3]=-552805397249/328469286912,\na[5, 4]=96066394193/54744881152,\na[6,1]=131173/1890096,\na[6,2]=0,\na[6,3] =0,\na[6,4]=5764801/17935632,\na[6,5]=4769464/20091357,\na[7,1]=117653 9682457387/14353367828755172,\na[7,2]=0,\na[7,3]=0,\na[7,4]=1771734217 4781/163306752068660,\na[7,5]=-528062841353883264/10338013178660912633 ,\na[7,6]=1679528554701504/62911488820241285,\na[8,1]=1109039092899267 37793/4320958290769994040132,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-116072389 85710021183/180261023052662924020,\na[8,5]=8332987977461879955626624/5 7056453830329950469093005,\na[8,6]=-397914630242458427722/631298806072 7977701695,\na[8,7]=11/30,\na[9,1]=-1991161589627082649817928503802737 25319682574083/2122095703793959540376735652096650388444524173920,\na[9 ,2]=0,\na[9,3]=0,\na[9,4]=-9023990274786340493184683543658310429712642 99/376720159407433656819366352813041281670901920,\na[9,5]=-19328410492 16489668013274675445061103749096456/1225034474225538046212066937516775 761976271555,\na[9,6]=612123799647400036233937665790699641777634907805 47/67292203490497755527349011795764425282435991108860,\na[9,7]=1767352 010455801522759919865162797424677/853631298904311486745705433404624991 760,\na[9,8]=48803138404357581919875739936453/258036197954361698272283 51014320,\na[10,1]=146812882912103498223477710702327669160817645499822 332/\n 347136327550264350281019050691094921672653636486171345, \na[10,2]=0,\na[10,3]=0,\na[10,4]=838090727939994034141552009500142968 9250499016988/1665529140598002776815163790462148830898906131435,\na[10 ,5]=6459917812385485497254681791738665918061301836694665216/\n \+ 2108701812611736155789874398208067694599974390240225435,\na[10,6]=-329 623101936975913713940238539594299201957668540750512/\n 2163691 93273123311533898096852817531450154842258383315,\na[10,7]=-23981229420 8127777232571199557076199094738401624/62716593235272943126739970926235 455172660665115,\na[10,8]=-19209909490782204629345787487654324166296/6 932257046340021610507379175381008191035,\na[10,9]=21064239972950566539 2/369807601102457012649,\na[11,1]=221939233494467949726864332982580511 239150575217/255387155540276753091910917079234490443699161270,\na[11,2 ]=0,\na[11,3]=0,\na[11,4]=18116326319248729754632413293825720580191672 9/17154547141864477699072019121010975163450940,\na[11,5]=5035138254118 716342864368937874683246617259598976/775682801056824859328708920179990 438628643868105,\na[11,6]=-7722480175483523961899743026009063654317219 3252536784/\n 202652428901707058125367179822844135271062302721 55465,\na[11,7]=-2041746049322766275834287576167164751357072803/242398 642004974788232613894826395428361025315,\na[11,8]=-6988057985584772189 4868113496992143638/12013145933881997013005874021379315455,\na[11,9]=6 67711875409843102380/547572557872664725429,\na[11,10]=-180885919602779 7903/20206200178110899692,\na[12,1]=2354923/46428480,\na[12,2]=0,\na[1 2,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=31816481086962/14382947901416 5,\na[12,7]=7273629826007016071/29687570322557889840,\na[12,8]=1319539 11267751793/602866646471570040,\na[12,9]=29769129062566/20567627483854 5,\na[12,10]=30049446213369/166412182583360,\na[12,11]=-67996657/11124 23760,\n\nb[1]=2354923/46428480,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\n b[6]=31816481086962/143829479014165,\nb[7]=7273629826007016071/2968757 0322557889840,\nb[8]=131953911267751793/602866646471570040,\nb[9]=2976 9129062566/205676274838545,\nb[10]=30049446213369/166412182583360,\nb[ 11]=-67996657/1112423760,\n\n`b*`[1]=182127300329983916501/35071843994 33109696000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]= 285550402807929509723935449/1482899423367801818133032950,\n`b*`[7]=644 3295529403196642395281317251/26849338691229488574870087988000,\n`b*`[8 ]=143483908435216987345410922193/615408757915355446585175892000,\n`b*` [9]=139711798081481043314568703/776834200093629660855997950,\n`b*`[10] =2259083734163473774459617/17881499156003594882624000,\n`b*`[11]=-2987 435988158249671/113556675995246584800,\n`b*`[12]=1/400\}:" }}}{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 78 "The sta bility function R for the 11 stage, order 7 scheme is given as follows ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFuncti on(7,11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,:\"\"\"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)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F )F)*&#\"Mf6q'HUp*4jp;cF:<6OWondJ\"R+G@6+Osd.9,qK4*f#e,v(y+!RG\"F)*$)F' \"\")F)F)F)*&#\"P\\h$R!R&y+C,nVB^4&fmXs*y!)>8\"U+g`$RVK.R%=8W\\&4\\eY7 Jf[I$po\\F)*$)F'\"\"*F)F)F)*&#\"O`71-ibkpV[.s*RCWbu[;L2#>\"V++%Q[3Je(4 'H.O(QFik6y#[@ELP\"P++7F -=&GdRdHT'oq'\\7^bq#)f%)F)*$)F'\"#6F)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,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=- 1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+NYj " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-5.19..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=[-5.19..0.4 9,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7V7$$!3Q++++++!>&!#<$!3@%)\\CSa([]$ F*7$$!3Ynm;#3'4G^F*$!3E\"Gk6x:@3$F*7$$!3kLLLk@>m]F*$!3LlsVeo41FF*7$$!3 E+]U'*)HB,&F*$!3i%HkPAoKT#F*7$$!3!pm;&GwYe\\F*$!3Gi(*ecKS\\@F*7$$!3s+] (\\(Q*y*[F*$!3!ewx?7;S)=F*7$$!3nLLV@,KP[F*$!3%Gtk\\ee&[;F*7$$!3NLL.q&p `r%F*$!3/H+S1l1`7F*7$$!3;nmJG')*Rf%F*$!3)>Z&3Sq?h%*!#=7$$!3uLLyGAZ\"[% F*$!3%)Q6))e.VLsFU7$$!3%3+])fw&\\O%F*$!3*H/ZTf')pU&FU7$$!3$QL$)f7eWC%F *$!3Ia$)>f(*3%)RFU7$$!3A++lN]MCTF*$!3E;/P2wr$)GFU7$$!3ummYeRz+SF*$!3cf 4rv\"zZ-#FU7$$!3_LLV-,(>*QF*$!3))4`#R<2fW\"FU7$$!35++S:-YpPF*$!3]Pd^H7 ep%*!#>7$$!3K+++\"HZkk$F*$!3CNuqI17'p&Fcp7$$!3;++gW:!z_$F*$!35g[`ibHJH Fcp7$$!3hLL)*\\1D?MF*$!3'e1td_\\8p*!#?7$$!3'ommSKVAH$F*$\"3z#[9R.O6t)F cq7$$!3/nmEGV!Q=$F*$\"3&oq#F*$\"3(p*QA9zO-jFcp7$$!3gmm;\\r8&e#F*$\"39v6gy0n1tFc p7$$!3ymmrw\\OtCF*$\"3YK>]l[j#G)Fcp7$$!3SLL$))e.GN#F*$\"3q@-R(yh;U*Fcp 7$$!3nLL)**=uvA#F*$\"3uQV'[)>\"G2\"FU7$$!3K++:I;c=@F*$\"3czh'>Jz*)>\"F U7$$!31LL.z]#3+#F*$\"3iTX?J(30N\"FU7$$!3M++?,<>z=F*$\"3Z:)3gFU7$$!3H+ +q9zA<:F*$\"3'*z%44'[2$>#FU7$$!3EnmEY;O-9F*$\"3uJ][r\")4gCFU7$$!3#)*** **pQ<(z7F*$\"3q1xxvc8\"y#FU7$$!3)RL$efMeo6F*$\"3)ec5?#*e!3JFU7$$!3I*** *fAZ3Z5F*$\"3=&Q(G!G!f4NFU7$$!3xqm;(zQwK*FU$\"3C>>5:ekMRFU7$$!3&z***\\ )ecE8)FU$\"3gUW![RwSV%FU7$$!3'3nmm0VV'pFU$\"3?abIX7f$)\\FU7$$!3P)***\\ iqATdFU$\"3-[P*eS()>j&FU7$$!3aFLL*)4AjXFU$\"3ff[d)=(4OjFU7$$!33LLLO'R& eLFU$\"3ev7oVZFZrFU7$$!3Uim;`O$Q;#FU$\"3%)Q)[5qkU0)FU7$$!3?*****>$H-m5 FU$\"3iK1IQ4$)))*)FU7$$\"3v*QLLU?>#>Fcp$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ \"FU$\"39&**>qHH39\"F*7$$\"3E0+]HcUiJK;F*-%'COLO URG6&%$RGBG$\"*++++\"!\")$\"\"!Fa\\lF`\\l-F$6$7S7$F($!\"\"Fa\\l7$F3Ff \\l7$F=Ff\\l7$FGFf\\l7$FLFf\\l7$FQFf\\l7$FWFf\\l7$FfnFf\\l7$F[oFf\\l7$ F`oFf\\l7$FeoFf\\l7$FjoFf\\l7$F_pFf\\l7$FepFf\\l7$FjpFf\\l7$F_qFf\\l7$ FeqFf\\l7$FjqFf\\l7$F_rFf\\l7$FdrFf\\l7$FirFf\\l7$F^sFf\\l7$FcsFf\\l7$ FhsFf\\l7$F]tFf\\l7$FbtFf\\l7$FgtFf\\l7$F\\uFf\\l7$FauFf\\l7$FfuFf\\l7 $F[vFf\\l7$F`vFf\\l7$FevFf\\l7$FjvFf\\l7$F_wFf\\l7$FdwFf\\l7$FiwFf\\l7 $F^xFf\\l7$FcxFf\\l7$FhxFf\\l7$F]yFf\\l7$FbyFf\\l7$FgyFf\\l7$F\\zFf\\l 7$FazFf\\l7$FfzFf\\l7$F[[lFf\\l7$F`[lFf\\l7$Fe[lFf\\l-Fj[l6&F\\\\lF`\\ lF`\\lF]\\l-F$6&7#7$$!3$******\\jMwh%F*Ff\\l-%'SYMBOLG6#%'CIRCLEG-Fj[l 6&F\\\\lFa\\lFa\\lFa\\l-%&STYLEG6#%&POINTG-F$6&F\\`l-Fa`l6#%&CROSSGFd` lFf`l-F$6&F\\`l-Fa`l6#%(DIAMONDGFd`lFf`l-F$6%7$7$F^`lF`\\lF]`l-%&COLOR G6&F\\\\lF`\\l$\"\"&Fg\\lF`\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z 6\"Q!Febl-%%FONTG6#%(DEFAULTG-Fhbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$>&! \"#$\"#\\Fecl;$!$Z\"Fecl$\"$Z\"Fecl" 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 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1710 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/504 0*z^7+\n 31576768443611171527561669630996942296701159/128390007877501 5825990932700114035772360011212800*z^8+\n 131980789724566595095123436 70124007853903936149/4968693304859311246584909549441318439033243393536 000*z^9+\n 1920733164874554424399720348436964556202061253/12421733262 148278116462273873603296097583108483840000*z^10+\n 371932666109016984 8664404410447096508831/84598270555112496706864129573957285180227120000 *z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 280 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,.45,0,.45)):\np2 : = plots[polygonplot]([seq([pts[i-1],pts[i],[-2.3,0]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.85,.13,.85)):\npts := [ ]: z0 := 1.5+5.5*I:\nfor ct from 0 to 40 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:\np3 := plot(pts,color=COLOR(RGB,.45,0,.45)):\np4 := plots[po lygonplot]([seq([pts[i-1],pts[i],[1.43,5.47]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.85,.13,.85)):\npts := []: z0 : = 1.5-5.5*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/2 0*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend d o:\np5 := plot(pts,color=COLOR(RGB,.45,0,.45)):\np6 := plots[polygonpl ot]([seq([pts[i-1],pts[i],[1.43,-5.47]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.85,.13,.85)):\np7 := plot([[[-5.39,0 ],[2.19,0]],[[0,-6.19],[0,6.19]]],color=black,linestyle=3):\nplots[dis play]([p||(1..7)],view=[-5.39..2.19,-6.19..6.19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 391 543 543 {PLOTDATA 2 "6/-%'CURVESG6 $7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$\"3)** ****R15([W!#F$\"34+++z*)Q7ZF-7$$\"3s*****pxN!*4%!#E$\"3C+++)H&=$G'F-7$ $\"3-+++\"=G&fA!#D$\"3g*****pe\")R&yF-7$$\"3Z+++UmYN#*F@$\"3H+++_yxC%* F-7$$\"3;+++Jas'=$!#C$\"3%******zVd&*4\"!#<7$$\"3$*******)[y_+\"!#B$\" 3'*******[sjc7FN7$$\"3.+++'R\")p+$FR$\"30+++&f#FN7$$\"31+++I(Hr+'Fgo$\"3#******> iJhN#FN7$$\"3!******RcS(R6!#?$\"3!)*****RE')H^#FN7$$\"3$)*****H.m(=?Fg p$\"3;+++a9]pEFN7$$\"33+++x%G2L$Fgp$\"3,+++%>^`#GFN7$$\"3++++4)Q)*3&Fg p$\"3#******HF>+)HFN7$$\"3v*****z\\]m7(Fgp$\"3$)*****pquF8$FN7$$\"3O** ****zLUk*)Fgp$\"3!******RT/FG$FN7$$\"3_*****fX30M#GFgp$\"3++++. @&fq$FN7$$!3(******pz1y$yFgp$\"3.+++*oga$QFN7$$!37+++lUn1D!#>$\"31+++u *pz&RFN7$$!3K+++FCLm\\Fjs$\"3#******pkRE2%FN7$$!3=+++#\\Z$y\")Fjs$\"31 +++g5`yTFN7$$!3.+++sQv37F-$\"3')******o@yuUFN7$$!35+++cH-e;F-$\"3/+++e Y&3O%FN7$$!3')*****R%pG^@F-$\"3;+++6fpOWFN7$$!35+++TpNuEF-$\"3U+++`Is- XFN7$$!3,+++GiK:KF-$\"37+++T=nfXFN7$$!3w*****pN'HlPF-$\"3$******RP;%3Y FN7$$!3#******\\#*[\"=VF-$\"3;+++.B$)\\YFN7$$!3E+++g\\$*p[F-$\"3C+++i+ s%o%FN7$$!3u*****H8W#=aF-$\"3P+++JUx8ZFN7$$!3!)*****HE(phfF-$\"37+++/) zvt%FN7$$!3#)*****4G$e*\\'F-$\"3i*****>4@mv%FN7$$!3C+++WhhJqF-$\"35+++ ()\\HrZFN7$$!3s*****H)*yxb(F-$\"3m******)[C>y%FN7$$!39+++%*zAy!)F-$\"3 K+++/3x)y%FN7$$!3o*****>COKf)F-$\"37++++N/#z%FN7$$!3b+++yT;.\"*F-$\"3M +++j(3>z%FN7$$!3]*****4b[%3'*F-$\"3S+++,g\\)y%FN7$$!30+++![g4,\"FN$\"3 C+++-J!>y%FN7$$!31+++LSsg5FN$\"33+++!o+Ax%FN7$$!3)******zw2-6\"FN$\"3# ******RjN%fZFN7$$!3)******py)\\f6FN$\"3S+++zVjVZFN7$$!3-+++&[.(37FN$\" 38+++%>1[s%FN7$$!31+++dK&zD\"FN$\"3H+++ss%Hq%FN7$$!36+++p:T28FN$\"3E++ +-j/yYFN7$$!3!******4M#Gd8FN$\"3))*****\\!G4]YFN7$$!3.+++r!>yS\"FN$\"3 k*****H%**3>YFN7$$!3%*******QJLf9FN$\"3e*****\\3u]e%FN7$$!3*******\\t( >7:FN$\"3%)*****RgV\"[XFN7$$!3%******znSoc\"FN$\"3-+++'G)\\3XFN7$$!32+ ++*f:Pi\"FN$\"3#******f\\'[mWFN7$$!3#******H[RKo\"FN$\"3!*******ewkAWF N7$$!3'******RV#pX)G%FN7$$!3%******H01#\\>FN $\"3;+++>%[_C%FN7$$!36+++B7V??FN$\"3%******f^)>/UFN7$$!3%*******[n&>4# FN$\"3%******4Ie^;%FN7$$!3%******f0DI;#FN$\"3L+++7L'z7%FN7$$!38+++D&eI B#FN$\"3O+++k7G#4%FN7$$!39+++!)[m,BFN$\"3E+++p*4x0%FN7$$!3?+++)z2'oBFN $\"3y******GX%Q-%FN7$$!36+++8wwLCFN$\"3++++*y8.*RFN7$$!3)******pX0r\\# FN$\"33+++a]zcRFN7$$!3+++++ajeDFN$\"3,+++Ti,BRFN7$$!3z*****fi/%=EFN$\" 3/+++R>v))QFN7$$!3++++K,[wEFN$\"3=+++.s\"Q&QFN7$$!33+++(=SHt#FN$\"3/++ +Z51=QFN7$$!3;+++-\"pyy#FN$\"3%)******o0O\"y$FN7$$!30++++UNTGFN$\"3!)* ****4-;Ou$FN7$$!33+++QU[$*GFN$\"3#******\\!pu/PFN7$$!3A+++D([V%HFN$\"3 1+++K!*okOFN7$$!3%)*****4yPS*HFN$\"3))******\\CRBOFN7$$!3#)*****>DUE/$ FN$\"3)******R:?3e$FN7$$!3y*****>*QD!4$FN$\"3/+++su%p`$FN7$$!3:+++:a'p 8$FN$\"3#)*****\\.i<\\$FN7$$!39+++h-(G=$FN$\"33+++WUEXMFN7$$!3&)*****R $>1GKFN$\"37+++$3ouR$FN7$$!3%*******eFjsKFN$\"3%******p7-%[LFN7$$!3$** *****\\>n;LFN$\"3'******4^5\")H$FN7$$!3;+++nHEgLFN$\"3!******4e`mC$FN7 $$!31+++/-[.MFN$\"3)******p(y5%>$FN7$$!3')*****fA&QYMFN$\"3.+++]^cSJFN 7$$!33+++EH-*[$FN$\"35++++,8'3$FN7$$!3!)*****46=9`$FN$\"32+++pn\"4.$FN 7$$!37+++'3tNd$FN$\"37+++zO/vHFN7$$!39+++jnY:OFN$\"3=+++J\"G'=HFN7$$!3 %******Hfbql$FN$\"3#)*****pF!yhGFN7$$!3!)*****pLw#)p$FN$\"3*)******)p( f/GFN7$$!3<+++V.0RPFN$\"3-+++f6;ZFFN7$$!3;+++n()GzPFN$\"3?+++^>`*o#FN7 $$!3-+++R!)*)=QFN$\"3')*****\\,^)=*>G#FN7$$!3/+++yAjrSFN$ \"3(******>HbHA#FN7$$!3M+++1=`.TFN$\"39+++\"e-O;#FN7$$!3I+++JwGMTFN$\" 3@+++27*Q5#FN7$$!3-+++F[*Q;%FN$\"32+++s.yV?FN7$$!3S+++irN#>%FN$\"3/+++ r@B$)>FN7$$!32+++mio>UFN$\"30+++XD@A>FN7$$!3O+++73!fC%FN$\"3#******\\7 #pg=FN7$$!3))*****zpD5F%FN$\"33+++*)pk)z\"FN7$$!3O+++&>\"4&H%FN$\"3-++ +\"Qfgt\"FN7$$!3'******z-K\"=VFN$\"3%******\\R=Hn\"FN7$$!3I+++Yj=SVFN$ \"3-+++61A4;FN7$$!35+++QXHhVFN$\"3&******Rkq\\a\"FN7$$!3E+++*)y\\\"Q%F N$\"3'******zb\"=![\"FN7$$!3K+++nq$3S%FN$\"34+++v]([T\"FN7$$!36+++n/N> WFN$\"3!******H\\\"3\\8FN7$$!3i*****H\\sqV%FN$\"33+++R#RGG\"FN7$$!3K++ +n=.aWFN$\"3/+++9S>;7FN7$$!3U+++,,DqWFN$\"3)******R](>\\6FN7$$!3k***** fESd[%FN$\"3!******po0>3\"FN7$$!38+++)=10]%FN$\"3*)*****R$oP95FN7$$!38 +++y@a9XFN$\"3A+++_Kpm%*F-7$$!3*)*****zJLy_%FN$\"3#******pf%R)y)F-7$$! 3$*******)Hc.a%FN$\"3a+++CUR4\")F-7$$!3O+++A23_XFN$\"3i*****pob,V(F-7$ $!3#*******R2(Hc%FN$\"3I+++<82^nF-7$$!3%******p(o)Hd%FN$\"35+++.TXsgF- 7$$!3K++++z3#e%FN$\"3Y*****fMMXR&F-7$$!3=+++*eK-f%FN$\"31+++X1Y^[&3YFN$\"3))******Q))o\"p#F-7$$!32+++bn]7YFN$\"3)*** ***z!*e!=?F-7$$!3t*****\\;]`h%FN$\"3/+++^b-X8F-7$$!38+++-F13\"FN7$F^bm$!3)******R]( >\\6FN7$Fiam$!3/+++9S>;7FN7$Fdam$!33+++R#RGG\"FN7$F_am$!3!******H\\\"3 \\8FN7$Fj`m$!34+++v]([T\"FN7$Fe`m$!3'******zb\"=![\"FN7$F``m$!3&****** Rkq\\a\"FN7$F[`m$!3-+++61A4;FN7$Ff_m$!3%******\\R=Hn\"FN7$Fa_m$!3-+++ \"Qfgt\"FN7$F\\_m$!33+++*)pk)z\"FN7$Fg^m$!3#******\\7#pg=FN7$Fb^m$!30+ ++XD@A>FN7$F]^m$!3/+++r@B$)>FN7$Fh]m$!32+++s.yV?FN7$Fc]m$!3@+++27*Q5#F N7$F^]m$!39+++\"e-O;#FN7$Fi\\m$!3(******>HbHA#FN7$Fd\\m$!3!)*****>)=*> G#FN7$F_\\m$!3!******RCb2M#FN7$Fj[m$!3%******f!pG*R#FN7$Fe[m$!3!****** p#\\idCFN7$F`[m$!3=+++39!e^#FN7$F[[m$!3-+++R'RQd#FN7$Ffjl$!3')*****\\, ^`*o#FN7$F\\jl$!3-+++f6;ZFFN7$Fgil$!3*)******)p(f/ GFN7$Fbil$!3#)*****pF!yhGFN7$F]il$!3=+++J\"G'=HFN7$Fhhl$!37+++zO/vHFN7 $Fchl$!32+++pn\"4.$FN7$F^hl$!35++++,8'3$FN7$Figl$!3.+++]^cSJFN7$Fdgl$! 3)******p(y5%>$FN7$F_gl$!3!******4e`mC$FN7$Fjfl$!3'******4^5\")H$FN7$F efl$!3%******p7-%[LFN7$F`fl$!37+++$3ouR$FN7$F[fl$!33+++WUEXMFN7$Ffel$! 3#)*****\\.i<\\$FN7$Fael$!3/+++su%p`$FN7$F\\el$!3)******R:?3e$FN7$Fgdl $!3))******\\CRBOFN7$Fbdl$!31+++K!*okOFN7$F]dl$!3#******\\!pu/PFN7$Fhc l$!3!)*****4-;Ou$FN7$Fccl$!3%)******o0O\"y$FN7$F^cl$!3/+++Z51=QFN7$Fib l$!3=+++.s\"Q&QFN7$Fdbl$!3/+++R>v))QFN7$F_bl$!3,+++Ti,BRFN7$Fjal$!33++ +a]zcRFN7$Feal$!3++++*y8.*RFN7$F`al$!3y******GX%Q-%FN7$F[al$!3E+++p*4x 0%FN7$Ff`l$!3O+++k7G#4%FN7$Fa`l$!3L+++7L'z7%FN7$F\\`l$!3%******4Ie^;%F N7$Fg_l$!3%******f^)>/UFN7$Fb_l$!3;+++>%[_C%FN7$F]_l$!3%*******Q*)>)G% FN7$Fh^l$!3s*******[0EL%FN7$Fc^l$!3c******=PsxVFN7$F^^l$!3!*******ewkA WFN7$Fi]l$!3#******f\\'[mWFN7$Fd]l$!3-+++'G)\\3XFN7$F_]l$!3%)*****RgV \"[XFN7$Fj\\l$!3e*****\\3u]e%FN7$Fe\\l$!3k*****H%**3>YFN7$F`\\l$!3))** ***\\!G4]YFN7$F[\\l$!3E+++-j/yYFN7$Ff[l$!3H+++ss%Hq%FN7$Fa[l$!38+++%>1 [s%FN7$F\\[l$!3S+++zVjVZFN7$Fgz$!3#******RjN%fZFN7$Fbz$!33+++!o+Ax%FN7 $F]z$!3C+++-J!>y%FN7$Fhy$!3S+++,g\\)y%FN7$Fcy$!3M+++j(3>z%FN7$F^y$!37+ +++N/#z%FN7$Fix$!3K+++/3x)y%FN7$Fdx$!3m******)[C>y%FN7$F_x$!35+++()\\H rZFN7$Fjw$!3i*****>4@mv%FN7$Few$!37+++/)zvt%FN7$F`w$!3P+++JUx8ZFN7$F[w $!3C+++i+s%o%FN7$Ffv$!3;+++.B$)\\YFN7$Fav$!3$******RP;%3YFN7$F\\v$!37+ ++T=nfXFN7$Fgu$!3U+++`Is-XFN7$Fbu$!3;+++6fpOWFN7$F]u$!3/+++eY&3O%FN7$F ht$!3')******o@yuUFN7$Fct$!31+++g5`yTFN7$F^t$!3#******pkRE2%FN7$Fhs$!3 1+++u*pz&RFN7$Fcs$!3.+++*oga$QFN7$F^s$!3++++.@&fq$FN7$Fir$!3;+++hw;qNF N7$Fdr$!3'******\\a:)GMFN7$F_r$!3!******RT/FG$FN7$Fjq$!3$)*****pquF8$F N7$Feq$!3#******HF>+)HFN7$F`q$!3,+++%>^`#GFN7$F[q$!3;+++a9]pEFN7$Fep$! 3!)*****RE')H^#FN7$F`p$!3#******>iJhN#FN7$F[p$!3!******R&H7*>#FN7$Feo$ !3A+++p[0U?FN7$F`o$!3)******4tp\\)=FN7$Fjn$!3(*******=K)ys\"FN7$$\"3_+ ++'H\\bc)FR$!3/+++W#*zq:FN7$FV$!30+++&f$!3g*****pe\")R&yF-7$$\"3 C+++vd.*4%F:$!3C+++)H&=$G'F-7$$\"3++++7,r[WF4$!34+++z*)Q7ZF-7$F($!3)** ****Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#X!\"#F(F`bo-%)POL YGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$FR!\"\"F(7%Fgbo7$F($\"+aEfTJFjboF[co 7%F_co7$$\"+k+r[WFjs$\"+z*)Q7ZFjboF[co7%Fcco7$$\"+xd.*4%F-$\"+)H&=$G'F jboF[co7%Fico7$$\"+\"=G&fAFN$\"+(e\")R&yFjboF[co7%F_do7$$\"+UmYN#*FN$ \"+_yxC%*FjboF[co7%Fedo7$$\"+Jas'=$!#;$\"+Qub*4\"!\"*F[co7%F[eo7$$\"+* [y_+\"!#:$\"+\\sjc7FaeoF[co7%Fceo7$$\"+'R\")p+$Ffeo$\"+&fK)y s\"FaeoF[co7%Fffo7$$\"+;ev)y&Fifo$\"+J(p\\)=FaeoF[co7%F]go7$$\"+-Yfb8! #8$\"+p[0U?FaeoF[co7%Fcgo7$$\"+mB_bHFfgo$\"+aH7*>#FaeoF[co7%Fjgo7$$\"+ I(Hr+'Ffgo$\"+A;8cBFaeoF[co7%F`ho7$$\"+k0uR6!#7$\"+ki)H^#FaeoF[co7%Ffh o7$$\"+Lgw=?Fiho$\"+a9]pEFaeoF[co7%F]io7$$\"+x%G2L$Fiho$\"+%>^`#GFaeoF [co7%Fcio7$$\"+4)Q)*3&Fiho$\"+t#>+)HFaeoF[co7%Fiio7$$\"+)\\]m7(Fiho$\" +2ZxKJFaeoF[co7%F_jo7$$\"+!QBW'*)Fiho$\"+9Wq#G$FaeoF[co7%Fejo7$$\"+cu@ K(*Fiho$\"+Xb\")GMFaeoF[co7%F[[p7$$\"+..n!=)Fiho$\"+hw;qNFaeoF[co7%Fa[ p7$$\"+#30M#GFiho$\"+.@&fq$FaeoF[co7%Fg[p7$$!+(z1y$yFiho$\"+*oga$QFaeo F[co7%F]\\p7$$!+lUn1D!#6$\"+u*pz&RFaeoF[co7%Fc\\p7$$!+FCLm\\Ff\\p$\"+Z 'RE2%FaeoF[co7%Fj\\p7$$!+#\\Z$y\")Ff\\p$\"+g5`yTFaeoF[co7%F`]p7$$!+sQv 37Fjbo$\"+p@yuUFaeoF[co7%Ff]p7$$!+cH-e;Fjbo$\"+eY&3O%FaeoF[co7%F\\^p7$ $!+WpG^@Fjbo$\"+6fpOWFaeoF[co7%Fb^p7$$!+TpNuEFjbo$\"+`Is-XFaeoF[co7%Fh ^p7$$!+GiK:KFjbo$\"+T=nfXFaeoF[co7%F^_p7$$!+djHlPFjbo$\"+ujT3YFaeoF[co 7%Fd_p7$$!+D*[\"=VFjbo$\"+.B$)\\YFaeoF[co7%Fj_p7$$!+g\\$*p[Fjbo$\"+i+s %o%FaeoF[co7%F``p7$$!+LTC=aFjbo$\"+JUx8ZFaeoF[co7%Ff`p7$$!+jsphfFjbo$ \"+/)zvt%FaeoF[co7%F\\ap7$$!+\"G$e*\\'Fjbo$\"+#4@mv%FaeoF[co7%Fbap7$$! +WhhJqFjbo$\"+()\\HrZFaeoF[co7%Fhap7$$!+$)*yxb(Fjbo$\"+*[C>y%FaeoF[co7 %F^bp7$$!+%*zAy!)Fjbo$\"+/3x)y%FaeoF[co7%Fdbp7$$!+UiB$f)Fjbo$\"++N/#z% FaeoF[co7%Fjbp7$$!+yT;.\"*Fjbo$\"+j(3>z%FaeoF[co7%F`cp7$$!+^&[%3'*Fjbo $\"+,g\\)y%FaeoF[co7%Ffcp7$$!+![g4,\"Faeo$\"+-J!>y%FaeoF[co7%F\\dp7$$! +LSsg5Faeo$\"+!o+Ax%FaeoF[co7%Fbdp7$$!+ox?56Faeo$\"+McVfZFaeoF[co7%Fhd p7$$!+(y)\\f6Faeo$\"+zVjVZFaeoF[co7%F^ep7$$!+&[.(37Faeo$\"+%>1[s%FaeoF [co7%Fdep7$$!+dK&zD\"Faeo$\"+ss%Hq%FaeoF[co7%Fjep7$$!+p:T28Faeo$\"+-j/ yYFaeoF[co7%F`fp7$$!+TBGd8Faeo$\"+0G4]YFaeoF[co7%Fffp7$$!+r!>yS\"Faeo$ \"+V**3>YFaeoF[co7%F\\gp7$$!+RJLf9Faeo$\"+&3u]e%FaeoF[co7%Fbgp7$$!+Nx> 7:Faeo$\"+/O9[XFaeoF[co7%Fhgp7$$!+y1%oc\"Faeo$\"+'G)\\3XFaeoF[co7%F^hp 7$$!+*f:Pi\"Faeo$\"+'\\'[mWFaeoF[co7%Fdhp7$$!+$[RKo\"Faeo$\"+fwkAWFaeo F[co7%Fjhp7$$!+MCpXPsxVFaeoF[co7%F`ip7$$!+x(*46=Faeo$\"+!\\0 EL%FaeoF[co7%Ffip7$$!+%e\\\"z=Faeo$\"+R*)>)G%FaeoF[co7%F\\jp7$$!+`g?\\ >Faeo$\"+>%[_C%FaeoF[co7%Fbjp7$$!+B7V??Faeo$\"+;&)>/UFaeoF[co7%Fhjp7$$ !+\\n&>4#Faeo$\"+,$e^;%FaeoF[co7%F^[q7$$!+c]-j@Faeo$\"+7L'z7%FaeoF[co7 %Fd[q7$$!+D&eIB#Faeo$\"+k7G#4%FaeoF[co7%Fj[q7$$!+!)[m,BFaeo$\"+p*4x0%F aeoF[co7%F`\\q7$$!+)z2'oBFaeo$\"+HX%Q-%FaeoF[co7%Ff\\q7$$!+8wwLCFaeo$ \"+*y8.*RFaeoF[co7%F\\]q7$$!+da5(\\#Faeo$\"+a]zcRFaeoF[co7%Fb]q7$$!++a jeDFaeo$\"+Ti,BRFaeoF[co7%Fh]q7$$!+EYS=EFaeo$\"+R>v))QFaeoF[co7%F^^q7$ $!+K,[wEFaeo$\"+.s\"Q&QFaeoF[co7%Fd^q7$$!+(=SHt#Faeo$\"+Z51=QFaeoF[co7 %Fj^q7$$!+-\"pyy#Faeo$\"+p0O\"y$FaeoF[co7%F`_q7$$!++UNTGFaeo$\"+@ghVPF aeoF[co7%Ff_q7$$!+QU[$*GFaeo$\"+0pu/PFaeoF[co7%F\\`q7$$!+D([V%HFaeo$\" +K!*okOFaeoF[co7%Fb`q7$$!+\"yPS*HFaeo$\"+]CRBOFaeoF[co7%Fh`q7$$!+_AkUI Faeo$\"+a,#3e$FaeoF[co7%F^aq7$$!+#*QD!4$Faeo$\"+su%p`$FaeoF[co7%Fdaq7$ $!+:a'p8$Faeo$\"+N?w\"\\$FaeoF[co7%Fjaq7$$!+h-(G=$Faeo$\"+WUEXMFaeoF[c o7%F`bq7$$!+M>1GKFaeo$\"+$3ouR$FaeoF[co7%Ffbq7$$!+fFjsKFaeo$\"+F@S[LFa eoF[co7%F\\cq7$$!+]>n;LFaeo$\"+606)H$FaeoF[co7%Fbcq7$$!+nHEgLFaeo$\"+ \"e`mC$FaeoF[co7%Fhcq7$$!+/-[.MFaeo$\"+xy5%>$FaeoF[co7%F^dq7$$!+E_QYMF aeo$\"+]^cSJFaeoF[co7%Fddq7$$!+EH-*[$Faeo$\"++,8'3$FaeoF[co7%Fjdq7$$!+ 6\"=9`$Faeo$\"+pn\"4.$FaeoF[co7%F`eq7$$!+'3tNd$Faeo$\"+zO/vHFaeoF[co7% Ffeq7$$!+jnY:OFaeo$\"+J\"G'=HFaeoF[co7%F\\fq7$$!+$fbql$Faeo$\"+x-yhGFa eoF[co7%Fbfq7$$!+PjF)p$Faeo$\"+*p(f/GFaeoF[co7%Fhfq7$$!+V.0RPFaeo$\"+f 6;ZFFaeoF[co7%F^gq7$$!+n()GzPFaeo$\"+^>`*o#FaeoF[co7%Fdgq7$$!+R!)*)=QF aeo$\"+:5vJEFaeoF[co7%Fjgq7$$!+x\\ydQFaeo$\"+R'RQd#FaeoF[co7%F`hq7$$!+ !3he*QFaeo$\"+39!e^#FaeoF[co7%Ffhq7$$!+ud/LRFaeo$\"+F\\idCFaeoF[co7%F \\iq7$$!+\"eo#pRFaeo$\"+1pG*R#FaeoF[co7%Fbiq7$$!+9.Z/SFaeo$\"+W_vSBFae oF[co7%Fhiq7$$!+4NgQSFaeo$\"+#)=*>G#FaeoF[co7%F^jq7$$!+yAjrSFaeo$\"+#H bHA#FaeoF[co7%Fdjq7$$!+1=`.TFaeo$\"+\"e-O;#FaeoF[co7%Fjjq7$$!+JwGMTFae o$\"+27*Q5#FaeoF[co7%F`[r7$$!+F[*Q;%Faeo$\"+s.yV?FaeoF[co7%Ff[r7$$!+ir N#>%Faeo$\"+r@B$)>FaeoF[co7%F\\\\r7$$!+mio>UFaeo$\"+XD@A>FaeoF[co7%Fb \\r7$$!+73!fC%Faeo$\"+D@pg=FaeoF[co7%Fh\\r7$$!+)pD5F%Faeo$\"+*)pk)z\"F aeoF[co7%F^]r7$$!+&>\"4&H%Faeo$\"+\"Qfgt\"FaeoF[co7%Fd]r7$$!+G?8=VFaeo $\"+&R=Hn\"FaeoF[co7%Fj]r7$$!+Yj=SVFaeo$\"+61A4;FaeoF[co7%F`^r7$$!+QXH hVFaeo$\"+W1(\\a\"FaeoF[co7%Ff^r7$$!+*)y\\\"Q%Faeo$\"+e:=![\"FaeoF[co7 %F\\_r7$$!+nq$3S%Faeo$\"+v]([T\"FaeoF[co7%Fb_r7$$!+n/N>WFaeo$\"+$\\\"3 \\8FaeoF[co7%Fh_r7$$!+$\\sqV%Faeo$\"+R#RGG\"FaeoF[co7%F^`r7$$!+n=.aWFa eo$\"+9S>;7FaeoF[co7%Fd`r7$$!+,,DqWFaeo$\"+/v>\\6FaeoF[co7%Fj`r7$$!+m- u&[%Faeo$\"+(o0>3\"FaeoF[co7%F`ar7$$!+)=10]%Faeo$\"+MoP95FaeoF[co7%Ffa r7$$!+y@a9XFaeo$\"+_Kpm%*FjboF[co7%F\\br7$$!+=L$y_%Faeo$\"+(f%R)y)Fjbo F[co7%Fbbr7$$!+*Hc.a%Faeo$\"+CUR4\")FjboF[co7%Fhbr7$$!+A23_XFaeo$\"+(o b,V(FjboF[co7%F^cr7$$!+S2(Hc%Faeo$\"+<82^nFjboF[co7%Fdcr7$$!+xo)Hd%Fae o$\"+.TXsgFjboF[co7%Fjcr7$$!++z3#e%Faeo$\"+YV`%R&FjboF[co7%F`dr7$$!+*e K-f%Faeo$\"+X1Y3\"FaeoF[co7%F`[s7$F[ar$!+/ v>\\6FaeoF[co7%Fd[s7$Fe`r$!+9S>;7FaeoF[co7%Fh[s7$F_`r$!+R#RGG\"FaeoF[c o7%F\\\\s7$Fi_r$!+$\\\"3\\8FaeoF[co7%F`\\s7$Fc_r$!+v]([T\"FaeoF[co7%Fd \\s7$F]_r$!+e:=![\"FaeoF[co7%Fh\\s7$Fg^r$!+W1(\\a\"FaeoF[co7%F\\]s7$Fa ^r$!+61A4;FaeoF[co7%F`]s7$F[^r$!+&R=Hn\"FaeoF[co7%Fd]s7$Fe]r$!+\"Qfgt \"FaeoF[co7%Fh]s7$F_]r$!+*)pk)z\"FaeoF[co7%F\\^s7$Fi\\r$!+D@pg=FaeoF[c o7%F`^s7$Fc\\r$!+XD@A>FaeoF[co7%Fd^s7$F]\\r$!+r@B$)>FaeoF[co7%Fh^s7$Fg [r$!+s.yV?FaeoF[co7%F\\_s7$Fa[r$!+27*Q5#FaeoF[co7%F`_s7$F[[r$!+\"e-O;# FaeoF[co7%Fd_s7$Fejq$!+#HbHA#FaeoF[co7%Fh_s7$F_jq$!+#)=*>G#FaeoF[co7%F \\`s7$Fiiq$!+W_vSBFaeoF[co7%F``s7$Fciq$!+1pG*R#FaeoF[co7%Fd`s7$F]iq$!+ F\\idCFaeoF[co7%Fh`s7$Fghq$!+39!e^#FaeoF[co7%F\\as7$Fahq$!+R'RQd#FaeoF [co7%F`as7$F[hq$!+:5vJEFaeoF[co7%Fdas7$Fegq$!+^>`*o#FaeoF[co7%Fhas7$F_ gq$!+f6;ZFFaeoF[co7%F\\bs7$Fifq$!+*p(f/GFaeoF[co7%F`bs7$Fcfq$!+x-yhGFa eoF[co7%Fdbs7$F]fq$!+J\"G'=HFaeoF[co7%Fhbs7$Fgeq$!+zO/vHFaeoF[co7%F\\c s7$Faeq$!+pn\"4.$FaeoF[co7%F`cs7$F[eq$!++,8'3$FaeoF[co7%Fdcs7$Fedq$!+] ^cSJFaeoF[co7%Fhcs7$F_dq$!+xy5%>$FaeoF[co7%F\\ds7$Ficq$!+\"e`mC$FaeoF[ co7%F`ds7$Fccq$!+606)H$FaeoF[co7%Fdds7$F]cq$!+F@S[LFaeoF[co7%Fhds7$Fgb q$!+$3ouR$FaeoF[co7%F\\es7$Fabq$!+WUEXMFaeoF[co7%F`es7$F[bq$!+N?w\"\\$ FaeoF[co7%Fdes7$Feaq$!+su%p`$FaeoF[co7%Fhes7$F_aq$!+a,#3e$FaeoF[co7%F \\fs7$Fi`q$!+]CRBOFaeoF[co7%F`fs7$Fc`q$!+K!*okOFaeoF[co7%Fdfs7$F]`q$!+ 0pu/PFaeoF[co7%Fhfs7$Fg_q$!+@ghVPFaeoF[co7%F\\gs7$Fa_q$!+p0O\"y$FaeoF[ co7%F`gs7$F[_q$!+Z51=QFaeoF[co7%Fdgs7$Fe^q$!+.s\"Q&QFaeoF[co7%Fhgs7$F_ ^q$!+R>v))QFaeoF[co7%F\\hs7$Fi]q$!+Ti,BRFaeoF[co7%F`hs7$Fc]q$!+a]zcRFa eoF[co7%Fdhs7$F]]q$!+*y8.*RFaeoF[co7%Fhhs7$Fg\\q$!+HX%Q-%FaeoF[co7%F\\ is7$Fa\\q$!+p*4x0%FaeoF[co7%F`is7$F[\\q$!+k7G#4%FaeoF[co7%Fdis7$Fe[q$! +7L'z7%FaeoF[co7%Fhis7$F_[q$!+,$e^;%FaeoF[co7%F\\js7$Fijp$!+;&)>/UFaeo F[co7%F`js7$Fcjp$!+>%[_C%FaeoF[co7%Fdjs7$F]jp$!+R*)>)G%FaeoF[co7%Fhjs7 $Fgip$!+!\\0EL%FaeoF[co7%F\\[t7$Faip$!+>PsxVFaeoF[co7%F`[t7$F[ip$!+fwk AWFaeoF[co7%Fd[t7$Fehp$!+'\\'[mWFaeoF[co7%Fh[t7$F_hp$!+'G)\\3XFaeoF[co 7%F\\\\t7$Figp$!+/O9[XFaeoF[co7%F`\\t7$Fcgp$!+&3u]e%FaeoF[co7%Fd\\t7$F ]gp$!+V**3>YFaeoF[co7%Fh\\t7$Fgfp$!+0G4]YFaeoF[co7%F\\]t7$Fafp$!+-j/yY FaeoF[co7%F`]t7$F[fp$!+ss%Hq%FaeoF[co7%Fd]t7$Feep$!+%>1[s%FaeoF[co7%Fh ]t7$F_ep$!+zVjVZFaeoF[co7%F\\^t7$Fidp$!+McVfZFaeoF[co7%F`^t7$Fcdp$!+!o +Ax%FaeoF[co7%Fd^t7$F]dp$!+-J!>y%FaeoF[co7%Fh^t7$Fgcp$!+,g\\)y%FaeoF[c o7%F\\_t7$Facp$!+j(3>z%FaeoF[co7%F`_t7$F[cp$!++N/#z%FaeoF[co7%Fd_t7$Fe bp$!+/3x)y%FaeoF[co7%Fh_t7$F_bp$!+*[C>y%FaeoF[co7%F\\`t7$Fiap$!+()\\Hr ZFaeoF[co7%F``t7$Fcap$!+#4@mv%FaeoF[co7%Fd`t7$F]ap$!+/)zvt%FaeoF[co7%F h`t7$Fg`p$!+JUx8ZFaeoF[co7%F\\at7$Fa`p$!+i+s%o%FaeoF[co7%F`at7$F[`p$!+ .B$)\\YFaeoF[co7%Fdat7$Fe_p$!+ujT3YFaeoF[co7%Fhat7$F__p$!+T=nfXFaeoF[c o7%F\\bt7$Fi^p$!+`Is-XFaeoF[co7%F`bt7$Fc^p$!+6fpOWFaeoF[co7%Fdbt7$F]^p $!+eY&3O%FaeoF[co7%Fhbt7$Fg]p$!+p@yuUFaeoF[co7%F\\ct7$Fa]p$!+g5`yTFaeo F[co7%F`ct7$F[]p$!+Z'RE2%FaeoF[co7%Fdct7$Fd\\p$!+u*pz&RFaeoF[co7%Fhct7 $F^\\p$!+*oga$QFaeoF[co7%F\\dt7$Fh[p$!+.@&fq$FaeoF[co7%F`dt7$Fb[p$!+hw ;qNFaeoF[co7%Fddt7$F\\[p$!+Xb\")GMFaeoF[co7%Fhdt7$Ffjo$!+9Wq#G$FaeoF[c o7%F\\et7$F`jo$!+2ZxKJFaeoF[co7%F`et7$Fjio$!+t#>+)HFaeoF[co7%Fdet7$Fdi o$!+%>^`#GFaeoF[co7%Fhet7$F^io$!+a9]pEFaeoF[co7%F\\ft7$Fgho$!+ki)H^#Fa eoF[co7%F`ft7$Faho$!+A;8cBFaeoF[co7%Fdft7$F[ho$!+aH7*>#FaeoF[co7%Fhft7 $Fdgo$!+p[0U?FaeoF[co7%F\\gt7$F^go$!+J(p\\)=FaeoF[co7%F`gt7$Fgfo$!+>K) ys\"FaeoF[co7%Fdgt7$$\"+'H\\bc)Ffeo$!+W#*zq:FaeoF[co7%Fhgt7$F[fo$!+&f< PT\"FaeoF[co7%F^ht7$Fdeo$!+\\sjc7FaeoF[co7%Fbht7$F\\eo$!+Qub*4\"FaeoF[ co7%Ffht7$Ffdo$!+_yxC%*FjboF[co7%Fjht7$F`do$!+(e\")R&yFjboF[co7%F^it7$ $\"+vd.*4%F-$!+)H&=$G'FjboF[co7%Fbit7$$\"+7,r[WFjs$!+z*)Q7ZFjboF[co7%F hit7$F($!+aEfTJFjboF[co7%F^jt7$F($!+Fjzq:FjboF[co7%FbjtF'F[co-F]bo6&F_ bo$\"#&)Fbbo$\"#8FbboFhjt-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3#****** pkPfa\"FN$\"3m******fSBeaFN7$$\"3!******z(z^Z:FN$\"31++++,-vaFN7$$\"35 +++[ZyY:FN$\"3K+++_OT\"\\&FN7$$\"33+++h^!Ra\"FN$\"3G+++>n<2bFN7$$\"3%* *****z[[!R:FN$\"35+++h_5AbFN7$$\"3/+++@QQK:FN$\"3<+++OH-ObFN7$$\"3.+++ #)*yS_\"FN$\"3q******Hfx[bFN7$$\"3(******zA+V^\"FN$\"3I+++'oG-c&FN7$$ \"3/+++[D@.:FN$\"3-+++_.EqbFN7$$\"3++++>/)4\\\"FN$\"3-+++:=wybFN7$$\"3 1+++')*oxZ\"FN$\"3%)*****zALce&FN7$$\"3/+++Tcuj9FN$\"3A+++]>y!f&FN7$$ \"3'******z/#3\\9FN$\"37+++!*37%f&FN7$$\"3)******fncRV\"FN$\"3!******p Dnbf&FN7$$\"3/+++Rzb=9FN$\"3U+++(yT]f&FN7$$\"3/+++$4)3.9FN$\"3m*****Hd oCf&FN7$$\"35+++^!oxQ\"FN$\"3U+++gdx(e&FN7$$\"3#******fKVGP\"FN$\"3)** ****4\\(*3e&FN7$$\"3/+++l7fe8FN$\"3;+++vxxrbFN7$$\"33+++s'H`M\"FN$\"3g *****pHx.c&FN7$$\"3/+++clUL8FN$\"33+++`ZoYbFN7$$\"3%*******G+JB8FN$\"3 R+++t\\tIbFN7$$\"3\"*******4eZ:8FN$\"3d*******)oj7bFN7$$\"3'*******zn[ 58FN$\"31+++?Xh#\\&FN7$$\"3.+++sU&*38FN$\"3A+++PB1raFN7$$\"3'******R)[ [68FN$\"3G+++L$4'[aFN7$$\"35+++j\\d=8FN$\"3C+++:.,Y\"FN$\"3'******za$*3O&FN7$$\"3-+++ZR_![\"FN$\"3C+++:=1p` FN7$$\"3!******>U/%)\\\"FN$\"3/+++VU>!Q&FN7$$\"3*******fME`YU&FN7$$\"36+++#f#)=a\"FN$\"3I+++t]LTaFNFc[uF\\bo-Fdbo6L7 %7$$\"+Zw$fa\"Faeo$\"+gSBeaFaeo7$$\"+yz^Z:Faeo$\"++,-vaFaeo7$$\"$V\"Fb bo$\"$Z&Fbbo7%Fchu7$$\"+[ZyY:Faeo$\"+_OT\"\\&FaeoFhhu7%F^iu7$$\"+h^!Ra \"Faeo$\"+>n<2bFaeoFhhu7%Fdiu7$$\"+)[[!R:Faeo$\"+h_5AbFaeoFhhu7%Fjiu7$ $\"+@QQK:Faeo$\"+OH-ObFaeoFhhu7%F`ju7$$\"+#)*yS_\"Faeo$\"+Ifx[bFaeoFhh u7%Ffju7$$\"+G-I9:Faeo$\"+'oG-c&FaeoFhhu7%F\\[v7$$\"+[D@.:Faeo$\"+_.Eq bFaeoFhhu7%Fb[v7$$\"+>/)4\\\"Faeo$\"+:=wybFaeoFhhu7%Fh[v7$$\"+')*oxZ\" Faeo$\"+GKj&e&FaeoFhhu7%F^\\v7$$\"+Tcuj9Faeo$\"+]>y!f&FaeoFhhu7%Fd\\v7 $$\"+[?3\\9Faeo$\"+!*37%f&FaeoFhhu7%Fj\\v7$$\"+wm&RV\"Faeo$\"+dsc&f&Fa eoFhhu7%F`]v7$$\"+Rzb=9Faeo$\"+(yT]f&FaeoFhhu7%Ff]v7$$\"+$4)3.9Faeo$\" +t&oCf&FaeoFhhu7%F\\^v7$$\"+^!oxQ\"Faeo$\"+gdx(e&FaeoFhhu7%Fb^v7$$\"+E L%GP\"Faeo$\"+\"\\(*3e&FaeoFhhu7%Fh^v7$$\"+l7fe8Faeo$\"+vxxrbFaeoFhhu7 %F^_v7$$\"+s'H`M\"Faeo$\"+(Hx.c&FaeoFhhu7%Fd_v7$$\"+clUL8Faeo$\"+`ZoYb FaeoFhhu7%Fj_v7$$\"+H+JB8Faeo$\"+t\\tIbFaeoFhhu7%F``v7$$\"+5eZ:8Faeo$ \"+!*oj7bFaeoFhhu7%Ff`v7$$\"+!y'[58Faeo$\"+?Xh#\\&FaeoFhhu7%F\\av7$$\" +sU&*38Faeo$\"+PB1raFaeoFhhu7%Fbav7$$\"+%)[[68Faeo$\"+L$4'[aFaeoFhhu7% Fhav7$$\"+j\\d=8Faeo$\"+:.)z8O&FaeoFhhu7%Ffc v7$$\"+^H!RT\"Faeo$\"+F.Vc`FaeoFhhu7%F\\dv7$$\"+9#ewV\"Faeo$\"+W[Wc`Fa eoFhhu7%Fbdv7$$\"+R!>,Y\"Faeo$\"+[N*3O&FaeoFhhu7%Fhdv7$$\"+ZR_![\"Faeo $\"+:=1p`FaeoFhhu7%F^ev7$$\"+AWS)\\\"Faeo$\"+VU>!Q&FaeoFhhu7%Fdev7$$\" +Yt^8:Faeo$\"+5\"yNR&FaeoFhhu7%Fjev7$$\"+'exd_\"Faeo$\"+)yz&3aFaeoFhhu 7%F`fv7$$\"+/V?N:Faeo$\"+iKlCaFaeoFhhu7%Fffv7$$\"+#f#)=a\"Faeo$\"+t]LT aFaeoFhhu7%F\\gvF^huFhhuFfjtF\\[u-F$6$7K7$Fd[u$!3m******fSBeaFN7$Fggu$ !3I+++t]LTaFN7$Fbgu$!3k*****>E`YU&FN7$F]gu$!3%******zyz&3aFN7$Fhfu$!3) *******4\"yNR&FN7$Fcfu$!3/+++VU>!Q&FN7$F^fu$!3C+++:=1p`FN7$Fieu$!3'*** ***za$*3O&FN7$Fdeu$!3K+++W[Wc`FN7$F_eu$!3)******pKIkN&FN7$Fjdu$!3s**** **=)z8O&FN7$Fedu$!3?+++77Xr`FN7$F`du$!3=+++W\\F'Q&FN7$F[du$!3A+++S*R\\ S&FN7$Ffcu$!3C+++:.y! f&FN7$Ff^u$!3%)*****zALce&FN7$Fa^u$!3-+++:=wybFN7$F\\^u$!3-+++_.EqbFN7 $Fg]u$!3I+++'oG-c&FN7$Fb]u$!3q******Hfx[bFN7$F]]u$!3<+++OH-ObFN7$Fh\\u $!35+++h_5AbFN7$Fc\\u$!3G+++>n<2bFN7$F^\\u$!3K+++_OT\"\\&FN7$Fi[u$!31+ +++,-vaFNFegvF\\bo-Fdbo6L7%7$F_hu$!+gSBeaFaeo7$F]gv$!+t]LTaFaeo7$Fihu$ !$Z&Fbbo7%Fc_w7$Fgfv$!+iKlCaFaeoFf_w7%Fj_w7$Fafv$!+)yz&3aFaeoFf_w7%F^` w7$F[fv$!+5\"yNR&FaeoFf_w7%Fb`w7$Feev$!+VU>!Q&FaeoFf_w7%Ff`w7$F_ev$!+: =1p`FaeoFf_w7%Fj`w7$Fidv$!+[N*3O&FaeoFf_w7%F^aw7$Fcdv$!+W[Wc`FaeoFf_w7 %Fbaw7$F]dv$!+F.Vc`FaeoFf_w7%Ffaw7$Fgcv$!+>)z8O&FaeoFf_w7%Fjaw7$Facv$! +77Xr`FaeoFf_w7%F^bw7$F[cv$!+W\\F'Q&FaeoFf_w7%Fbbw7$Febv$!+S*R\\S&Faeo Ff_w7%Ffbw7$F_bv$!+:.y!f&FaeoFf_w7%Fffw7$F_\\v$!+GKj &e&FaeoFf_w7%Fjfw7$Fi[v$!+:=wybFaeoFf_w7%F^gw7$Fc[v$!+_.EqbFaeoFf_w7%F bgw7$F][v$!+'oG-c&FaeoFf_w7%Ffgw7$Fgju$!+Ifx[bFaeoFf_w7%Fjgw7$Faju$!+O H-ObFaeoFf_w7%F^hw7$F[ju$!+h_5AbFaeoFf_w7%Fbhw7$Feiu$!+>n<2bFaeoFf_w7% Ffhw7$F_iu$!+_OT\"\\&FaeoFf_w7%Fjhw7$Fdhu$!++,-vaFaeoFf_w7%F^iwF`_wFf_ wFfjtF\\[u-F$6%7$7$$!3o*************Q&FNF(7$$\"3%**************=#FNF(- %'COLOURG6&F_boF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>'FN7$F ($\"3Q++++++!>'FNF[jwF^jw-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$B OXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fd[x6#% (DEFAULTG-%%VIEWG6$;$!$R&Fbbo$\"$>#Fbbo;$!$>'Fbbo$\"$>'Fbbo" 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 "[-4.6176, 0];" "6#7$,$-%&Float G6$\"&wh%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the largest interval on the nonneg ative imaginary axis that contains the origin and lies inside the stab ility region is " }{XPPEDIT 18 0 "[0, 3.75];" "6#7$\"\"!-%&FloatG6$\" $v$!\"#" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 738 "R := z -> 1+z+1/2*z^ 2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 3157676844361117 1527561669630996942296701159/12839000787750158259909327001140357723600 11212800*z^8+\n 13198078972456659509512343670124007853903936149/49686 93304859311246584909549441318439033243393536000*z^9+\n 19207331648745 54424399720348436964556202061253/1242173326214827811646227387360329609 7583108483840000*z^10+\n 3719326661090169848664404410447096508831/845 98270555112496706864129573957285180227120000*z^11:\nDigits := 18:\npts := []: z0 := 0:\nfor ct from 0 to 126 do\n zz := newton(R(z)=exp(ct *Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),I m(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.8,0,.8),thickness=2,font= [HELVETICA,9]);\nDigits := 10:\n" }}{PARA 13 "" 1 "" {GLPLOT2D 244 265 265 {PLOTDATA 2 "6(-%'CURVESG6#7[s7$$\"\"!F)F(7$$\"3B')*G+FcC*>!#> $\"3C$z*e`EfTJF-7$$\"3s%fTfJ*)zH$F-$\"3J'ezrI&=$G'F-7$$\"3A4wjxl!zU%F- $\"3Yt$p2'zxC%*F-7$$\"32B4B#3;jX&F-$\"3.4fVhqjc7!#=7$$\"3@It%eJ6YT'F-$ \"3qN[zEjzq:F?7$$\"3HdI#*Hk&)>tF-$\"3jlN:#fb\\)=F?7$$\"3pLfO79j#=)F-$ \"3bJ;^d[6*>#F?7$$\"3$>Fl'yd15!*F-$\"3QOy'G7uK^#F?7$$\"37A+@kDD2)*F-$ \"3c1&>#)QLu#GF?7$$\"32R0dS$*zd5F?$\"3#\\FhNl#fTJF?7$$\"3)o#4MKF_K6F?$ \"3>_K))=>vbMF?7$$\"3f&owRYH^?\"F?$\"3Wm%oT=6*pPF?7$$\"3#*f`\")Q3\"eF \"F?$\"3')*\\*Q\\/2%3%F?7$$\"3Y,\"p;2FZM\"F?$\"3WRV]9(H#)R%F?7$$\"3)GQ M@]:?T\"F?$\"3v)e^%z*)Q7ZF?7$$\"3<=P:)e'zx9F?$\"3'oOXTC[l-&F?7$$\"34i6 'HTy@a\"F?$\"30_/Z3vqS`F?7$$\"30Kpgl,E0;F?$\"3&H^yAxm[l&F?7$$\"3hPt,\" *[8n;F?$\"3))3mQNg-pfF?7$$\"3!HeeDq\"*ys\"F?$\"3.I&zvH&=$G'F?7$$\"3uf% *)GeF?$\"3W'4ki2jcA(F?7$$\"37!yCKSw9'>F?$\"3&[(4[KB#)RvF?7$$\"3]Mbo 99&z,#F?$\"3(eOloe\")R&yF?7$$\"3@_7@(QSQ2#F?$\"34yYeR39o\")F?7$$\"3wM* e#R!R#H@F?$\"3n\"pD64+B[)F?7$$\"3U&[wmqXU=#F?$\"3cY'QCMfkz)F?7$$\"3x%* zp\\4'*QAF?$\"35G(=^f=16*F?7$$\"3,s6#>(o[$H#F?$\"3)\\sH;&yxC%*F?7$$\"3 B8LDfh#zM#F?$\"3;@Ud:r$*Q(*F?7$$\"3sRkIi3Q-CF?$\"3H&R,#R'4`+\"!#<7$$\" 3cVg%R'4&pX#F?$\"3X&G%yobsO5Ffu7$$\"3`/7o#zK<^#F?$\"3vw8A,:9o5Ffu7$$\" 3B!y*zft\"oc#F?$\"3'Hx,xVd&*4\"Ffu7$$\"3vI,%3r)GAEF?$\"3277uzL(48\"Ffu 7$$\"3'=1NwQA#yEF?$\"3)e1R#H$*Qi6Ffu7$$\"3wAV.%4%oMFF?$\"32A%Q&)G0Q>\" Ffu7$$\"3`*GBgkG8Ffu7$$\"3T3>%3k@k-$F?$\"3$[CAFN&)3N\"Ffu7$$\"3)HhW$y9$F?$\"3&4\\l^fDOmz]\"Ffu7$$\"34]kzQ*p\")R$F?$\"3&eKF,w#QR:Ffu7$$\"3(o4>U:9@ Y$F?$\"37l`sV#*zq:Ffu7$$\"3d+lDLe^ENF?$\"3XvF6Ae@-;Ffu7$$\"3UaYzKxK\"f $F?$\"39%e?L]KOj\"Ffu7$$\"3.yr=o<]cOF?$\"3'[>:SH\\]m\"Ffu7$$\"3fS^$en* )>s$F?$\"3t.)\\()>mkp\"Ffu7$$\"3o046VNu(y$F?$\"3V4&R!>K)ys\"Ffu7$$\"3^ x\"\\i@;P&QF?$\"3MRF?$\"35Ea:!f<2z\"Ffu7$$\" 3Q]T`f[8')RF?$\"3?Q,[<\\8A=Ffu7$$\"3YPxd]D\\_SF?$\"3U$[8)4Bb`=Ffu7$$\" 3sH/-sE*)=TF?$\"3k=#Q9tp\\)=Ffu7$$\"3h/zqsYH&=%F?$\"3d6xoKrQ;>Ffu7$$\" 3l(Gr;Tf;D%F?$\"3pyj!pW/y%>Ffu7$$\"39Ry&\\3\\zJ%F?$\"3Y))Q&pe@#z>Ffu7$ $\"3:!\\li7FTQ%F?$\"3LI$37WQ1,#Ffu7$$\"3I6dq(3e,X%F?$\"3Ls?-p[0U?Ffu7$ $\"3>[^t0v+;XF?$\"3c%)He&pqM2#Ffu7$$\"3>`l#Ffu7$$\"3&yIlCEv8%[F?$\"37jef6:`IAF fu7$$\"3tAvG+xW0\\F?$\"3)\\$*erKP>E#Ffu7$$\"39)3S\\&>6p\\F?$\"3SWH&HyR LH#Ffu7$$\"31\"**pvQ#Ffu7$$\"3F:)\\==L!>_F?$\" 3'R=xb\")**)=CFfu7$$\"3z6+4P)f,G&F?$\"3r`cW\"=s-X#Ffu7$$\"3QvMwEOnS`F? $\"3erP5s[j\"[#Ffu7$$\"3m6Gx[l`+aF?$\"3g)\\uQE')H^#Ffu7$$\"3(GE7n'*3(f aF?$\"3;j))REXKWDFfu7$$\"3Tyd5+%\\\"=bF?$\"3W0P00wkvDFfu7$$\"37;i/2V\" ed&F?$\"3@0h8.K&pg#Ffu7$$\"3v#\\<#eylKcF?$\"3Pg@0k(Q#QEFfu7$$\"3%z!o1, t#Ffu7$$\"3ji>\"zJ33&eF?$\"3af(GpW>Jw#Ffu7$$\"3vZw9 &QgF!fF?$\"3!3$[e&>cUz#Ffu7$$\"3'4'HY_Rb`fF?$\"3Hu'*z$>^`#GFfu7$$\"3k \\!\\&f(>J+'F?$\"3/E%*=Q'*RcGFfu7$$\"3\"4g**o)QQ^gF?$\"3Y*oSjO'R()GFfu 7$$\"30U^4@tE)4'F?$\"3r1)Hn&eL=HFfu7$$\"39rQoeaoVhF?$\"37-&GIB7#\\HFfu 7$$\"3K)o**3`Zv='F?$\"3vlH@t#>+)HFfu7$$\"3W[^*f#fvHiF?$\"3R)*f!QU]2,$F fu7$$\"38s2&)y`?qiF?$\"3lv\\Y>))RTIFfu7$$\"3p53?))>y3jF?$\"3KFD)zId>2$ Ffu7$$\"3Q9b]4?OXjF?$\"3AgWf![=C5$Ffu7$$\"30(*o1b.\")zjF?$\"3=0aH2ZxKJ Ffu7$$\"3JZsv5(y>T'F?$\"3&\\?am<=I;$Ffu7$$\"37S/?[JqTkF?$\"3?PxSR49$>$ Ffu7$$\"3<*p!fo5!)okF?$\"3=P^!\\&\\8BKFfu7$$\"3,,.h`s1$\\'F?$\"3'*4pGR @*HD$Ffu7$$\"3EVoq5(oU^'F?$\"3Ha\"GTT/FG$Ffu7$$\"3xh\")3py8KlF?$\"3f6U ;aPE7LFfu7$$\"3&*GR'R[jja'F?$\"3(zrgEBi;M$Ffu7$$\"3r46(oHylb'F?$\"3!H \\&*H1#*3P$Ffu7$$\"36Pl,/?MilF?$\"3W\\.4Nc%**R$Ffu7$$\"3Rb*[bEY\\dMFfu7$$\"3^\"[cYTVoa 'F?$\"3Zpb!y#f(f[$Ffu7$$\"3et-I3\"Gy_'F?$\"3'[$z5)y_U^$Ffu7$$\"3[hrG'H 'o*\\'F?$\"3s%>Tfw=Ba$Ffu7$$\"3=e'3hj7.Y'F?$\"3&R2c2mn,d$Ffu7$$\"3?o6+ 5Wh1kF?$\"3**pL(f[$z(f$Ffu7$$\"3PA9>GtvLjF?$\"3g6=*4W!>DOFfu7$$\"3J.)) H+$3NB'F?$\"3+u?^wGN_OFfu7$$\"3Aq**G)=N**3'F?$\"3hV$zVDv#zOFfu7$$\"3,A plq^ykeF?$\"3s*\\LL5_fq$Ffu7$$\"3@Dy\"HqX&4aF?$\"3,G.?szPKPFfu7$$!30,$ [_)*zN=&F?$\"3]o!f@QZ&ePFfu7$$!3d$p=?F%RpeF?$\"3D&3#yzZX%y$Ffu7$$!3&yX %R$=d7?'F?$\"3(*3Ic#\\%45QFfu7$$!3$)eO*e=;_V'F?$\"3Z`:j)oga$QFfu7$$!3) ='pKfVI@mF?$\"3?bnHUtagQFfu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HELVETICAG \"\"*-%&COLORG6&%$RGBG$\"\")!\"\"F(Fjcm-%+AXESLABELSG6$Q!6\"F`dm-%%VIE WG6$%(DEFAULTGFedm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 122 "The relevant ntersection point of the boundary cu rve with the imaginary axis can be determined more accurately as follo ws." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the bo undary curve either side of the intersection point. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Digits := 15:\nz0 := 3.75*I:\nfor ct from 120 to 123 do\n newton(R(z)=exp(ct* Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0\\u<30M#G!#<$\"0]LL5_fq$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0 " 0 "" {MPLTEXT 1 0 177 "real_part := proc(u) \n Re(newton(R(z)=exp(u*Pi*I),z=3.75*I))\nend proc:\nDigits := 15:\n u0 := bisect('real_part'(u),u=1.2..1.23);\nnewton(R(z)=exp(u0*Pi*I),z= 3.75*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"00 .'GYH;7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"03G;!4))[P!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 112 "largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.7489];" "6#7$\"\"!-%&FloatG6$\"&*[P!\" %" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stab ility function R* for the 12 stage, order 6 scheme is given as follo ws." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,S tabilityFunction(6,12,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)' =`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,<\"\"\"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)*&# \"Y\"*Q8:s')[wUZw>>5G&G$zNru;i$*QWV#*\"gn++g``V[kj!yDOM;(yVIj!z6MJ1x_Y $>XF)*$)F'\"\"(F)F)F)*&#\"gn\"oV$)\\'HpR!poM`1W#G/6Xu?Z= W\"H#)=.qz$)*>(y`hu[C#)\\FPF)*$)F'\"\"*F)F)F)*&#\"fnjSB(RIjP8*>L^\"*z2 $H%fc(>NTJ>1))30Z\"]o+++c!p%*z:sl@FC!\\qm%3I\\D0T3\"f*3;_7W,\"F)*$)F' \"#5F)F)F)*&#\"Zfu'z9H-L:'y#3>TC[%z#o;1`q_US_$z*\"\\o+++GXt**ygG3O@^CN LU]YFE0UbzW!3E1s]F)*$)F'\"#6F)F)F)*&#\"IJ)3l4Z/T/Wm[)p,4hmK>P\"S+++[34 sS\"HeH=lXFo)*\\/A#3$RQ$F)*$)F'\"#7F)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 32 "z_0 := newton(`R*`(z)=1, z=-3.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+_ZpAR!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-3.9):\np_1 := plot([`R*`(z),1],z=-4.39 ..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbo l=[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],vi ew=[-4.39..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 250 250 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o************* Q%!#<$\"3O4'*[c$\\_n#F*7$$!3Amm\">Z2MO%F*$\"3s8cDoZcNDF*7$$!3lLL$Q%\\ \"oL%F*$\"33>yCw2[-CF*7$$!3?++v:CA5VF*$\"3sS0PlQsvAF*7$$!3ummm())HOG%F *$\"3OErgl..b@F*7$$!3\")**\\naQNPUF*$\"3y\"\\PL32(e>F*7$$!3'GL$o@y2\"> %F*$\"3Yv3?([2(y`gBFco7$$!3W****RTi`pJF*$\"3@(=_s<7`!>Fco7$$!3')*****4 3\\Q1$F*$\"3EjQ4sygn:Fco7$$!3m****f=***>'HF*$\"3m$o+G7XNL\"Fco7$$!3Amm \"=76&pGF*$\"3'[jh/@lT=\"Fco7$$!3'GLLn*H`fFF*$\"3!=r')ymb+2\"Fco7$$!3y KL$*)4njm#F*$\"3gH=h*Q%f<5Fco7$$!3!)***\\:,$*zb#F*$\"3/<(4M7,=)**!#>7$ $!35LL$4ej?Y#F*$\"3K)Rw)f9475Fco7$$!3g***\\:z8oN#F*$\"3a<2C'3@c0\"Fco7 $$!3S***\\E5\"fcAF*$\"3*QS!*>Nx37\"Fco7$$!3'HLL3C>?:#F*$\"3(3pn$*e!)4@ \"Fco7$$!3sKL)yi*)f0#F*$\"32%\\5lu4?J\"Fco7$$!3?mm;<(3C&>F*$\"3>F;**3c vR9Fco7$$!3\\mm\"=E<[%=F*$\"3SL`sP#zGf\"Fco7$$!3m***\\Oee6v\"F*$\"3?aA sEOZVFco7$$!3))****>HF]X:F*$\"3wvR# o`lX8#Fco7$$!3b*****=*zEV9F*$\"3%=\\t\\j+HO#Fco7$$!3$)***\\ni]VM\"F*$ \"3@%eMF^&z2EFco7$$!3y****pd(>XB\"F*$\"3rQl?QY15HFco7$$!3MLL$p$=$e8\"F *$\"3%G=^*zSp6KFco7$$!3r*****pTh/.\"F*$\"3G8tg<\"z%oNFco7$$!3qlm;a:!) \\$*Fco$\"3bkxOwv&f#RFco7$$!3r$****f;RfI)Fco$\"3at68\"e4zN%Fco7$$!3$RL L)3guBtFco$\"3)3Z749hw![Fco7$$!3o&***\\`62(H'Fco$\"3xq&GY/yuK&Fco7$$!3 =HLLB)3LH&Fco$\"3+$[8&G$*)**)eFco7$$!36(***\\(oiCC%Fco$\"3vU`h=tiUlFco 7$$!3jcmmiHPIKFco$\"3b-pQ@&o%RsFco7$$!3_jmmR]O&>#Fco$\"36$HGN[3*G!)Fco 7$$!3kBL$[]F*o6Fco$\"3['o-'eg!o*))Fco7$$!3%)=****>,QdAFes$\"3ONQT13zw( *Fco7$$\"3Ozmmcej_&)Fes$\"3)R<'4t-H*3\"F*7$$\"3-TLL,V7A=Fco$\"3#**G-_0 p)*>\"F*7$$\"330+]%[**H&GFco$\"3$\\Ku#\\4;I8F*7$$\"3C4+]r&y'RQFco$\"3' HiDIE)4o9F*7$$\"3!***************[Fco$\"3\")om%\\G;Bj\"F*-%'COLOURG6&% $RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi] l7$FQFi]l7$FenFi]l7$F_oFi]l7$FeoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi]l7$FipFi ]l7$F^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l7$FbrFi]l7$FgrFi]l7$F\\sFi]l7$Fa sFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7 $FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi] l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$Fez Fi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh \\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!3&******>v%pARF*Fi]l-%'SYMBOL G6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6# %&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`a l-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HEL VETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^dl-Ffcl6#%(DEFAULTG-%%VIEWG6$;$ !$R%!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 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 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2035 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+ 924344389362167471357932852 81019197647427648867215133891/4519346527706313411790633043787163436257 80636448435353600000*z^7+237962808064424754592245367341744065334686903 969296498343681/724580372577827934578946780720333360644590868980112853 3504000000*z^8+4982244874615378719983797003188229144184720744511042821 947/7245803725778279345789467807203333606445908689801128533504000000*z ^9+47050888061931413519756594293077991513319913376330397234063/1014412 52160895910841052549300846670490242721657215799469056000000*z^10+97935 2404252705306166827944824411908278615330229147967459/50720626080447955 420526274650423335245121360828607899734528000000*z^11+3719326661090169 848664404410447096508831/338393082220449986827456518295829140720908480 00000*z^12:\npts := []: z0 := 0:\nfor ct from 0 to 180 do\n zz := ne wton(`R*`(z)=exp(ct*Pi/15*I),z=z0):\n z0 := zz:\n pts := [op(pts), [Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.25,0,.25) ):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.96,0]],i=2..no ps(pts))],\n style=patchnogrid,color=COLOR(RGB,.6,0,.6)):\npt s := []: z0 := 1.8+5.3*I:\nfor ct from 0 to 40 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_3 := plot(pts,color=COLOR(RGB,.25,0,.25)):\np_4 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[1.73,5.24]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,.6,0,.6)):\npts := [ ]: z0 := 1.8-5.3*I:\nfor ct from 0 to 40 do\n zz := newton(`R*`(z)=e xp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.25,0,.25)):\np_6 := plo ts[polygonplot]([seq([pts[i-1],pts[i],[1.73,-5.24]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.6,0,.6)):\np_7 := plot( [[[-5.19,0],[2.19,0]],[[0,-6.19],[0,6.19]]],color=black,linestyle=3): \nplots[display]([p_||(1..7)],view=[-5.19..2.19,-6.19..6.19],font=[HEL VETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=co nstrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 478 539 539 {PLOTDATA 2 "6/- %'CURVESG6$7av7$$\"\"!F)F(7$F($\"3++++/^R%4#!#=7$$!3#)*****H19_o*!#F$ \"35+++q.z)=%F-7$$\"3(******H**fI-\"!#E$\"3I+++8')=$G'F-7$$\"39+++t%\\ )[M!#C$\"3%******4L4wP)F-7$$\"3I+++[o\")fV!#B$\"3!******p98s/\"!#<7$$ \"3:+++'*f=DH!#A$\"3++++nrpc7FF7$$\"3!*******e*H#e8!#@$\"3)*******GXDm 9FF7$$\"3m********)fi)[FP$\"3-+++vD$fn\"FF7$$\"37+++7TrZ9!#?$\"31+++k8 r&)=FF7$$\"3%)*****>bWtl$Fen$\"3)******zx5`4#FF7$$\"3p*****f+EM,)Fen$ \"3$)******yI#QI#FF7$$\"3,+++R/PF:!#>$\"3#******>5[#4DFF7$$\"3&******z ;%G:DFeo$\"3'******Hgp$3FFF7$$\"3m*****z%p6VNFeo$\"3))******yoe(*GFF7$ $\"3o*****\\%))3BUFeo$\"3A+++5tLuIFF7$$\"3#)*****\\M;#[TFeo$\"3$****** Ry`wB$FF7$$\"3%)*****H73+-$Feo$\"39+++I?[(Q$FF7$$\"3$)******G9d1pFen$ \"3<+++Z&)zBNFF7$$!3,+++-L\"R%GFeo$\"3?+++?tEYOFF7$$!3g+++'4#=ouFeo$\" 3#******49@Wv$FF7$$!3++++_Rb)H\"F-$\"3#)*****>$=/[QFF7$$!3/+++Y!zn\">F -$\"3%)*****fQ:u#RFF7$$!3=+++N;**zDF-$\"31+++egK$*RFF7$$!3;+++_#31F$F- $\"3!)*****pVpo/%FF7$$!33+++%o&fvRF-$\"37+++xeB*3%FF7$$!3#)******)>z+h F-$\"3i******GD%)fTFF7$$!3y*****\\\"p>*z'F-$\"3%)*****>g#HnTFF7$$!3!** ****fE-&*[(F-$\"3I+++!Hxw;%FF7$$!3s*****>e_6<)F-$\"3g*****f'fPhTFF7$$! 3@+++*z#>W))F-$\"3-+++TBl[TFF7$$!3u******oAB4&*F-$\"3D+++7pmHTFF7$$!3/ +++S3v;5FF$\"3@+++Ll[/TFF7$$!3#******ft0@3\"FF$\"3m*****zu#4tSFF7$$!3) *******>9FF$\"3#******HP#>>QFF7$$!3)******\\'R$p\\\"F F$\"3'******z0;)\\PFF7$$!31+++*=>3e\"FF$\"3!)*****4O=pn$FF7$$!3-+++%\\ E@n\"FF$\"3'******4SgOg$FF7$$!3%******HA$epFF$\"3A+++#ff)3MFF7$$ !3=+++M_;p?FF$\"3!)*****f[*f`LFF7$$!3-+++k,uj@FF$\"3')*****4nj6I$FF7$$ !3/+++fT3aAFF$\"3-+++XH;]KFF7$$!37+++$p+,M#FF$\"35+++@x[*>$FF7$$!3-+++ zH\">U#FF$\"33+++;MI[JFF7$$!39+++e*R(*\\#FF$\"35+++,d*f4$FF7$$!35+++gT $Qd#FF$\"3<+++QN6UIFF7$$!3)******fm`Wk#FF$\"3#)*****p]Bj)HFF7$$!3&**** ***\\=&=r#FF$\"3++++-nPGHFF7$$!3?+++'3wix#FF$\"3#)*****f<'3oGFF7$$!3>+ ++#yqz$GFF$\"3-+++aFJ0GFF7$$!35+++e@=(*GFF$\"3)******f_d*RFFF7$$!3#*** ***43jT&HFF$\"35+++\\*f>n#FF7$$!3++++]k<4IFF$\"34+++J3I,EFF7$$!3'***** *\\Z&\\iIFF$\"38+++`%4!GDFF7$$!3.+++')))R9JFF$\"3-+++iN<_CFF7$$!3'**** **fFF7$$!3#*** ***pavWX$FF$\"3)******4')=:(=FF7$$!3')*****>[O(*\\$FF$\"3!******>]yby \"FF7$$!33+++)QENa$FF$\"3/+++z6q*p\"FF7$$!31+++Hd`&e$FF$\"3!******f4cR h\"FF7$$!3%)*****4v#[DOFF$\"3)******zgM$G:FF7$$!3#)*****fyHJm$FF$\"3-+ ++s.xU9FF7$$!37+++Z**G)p$FF$\"3/+++_$frN\"FF7$$!3y*****f@F3t$FF$\"35++ +klPr7FF7$$!35+++eHlgPFF$\"3)******fl)G&=\"FF7$$!37+++'>@xy$FF$\"3'*** ***fri()4\"FF7$$!3++++US-7QFF$\"3++++*Hs;,\"FF7$$!3&)*****z&zeLQFF$\"3 x*****>8=!R#*F-7$$!3!******R#4Z_QFF$\"3#)*****4x-NN)F-7$$!3=+++W-woQFF $\"3[*****Hy_$fuF-7$$!33+++t.d#)QFF$\"3O+++(*p(fb'F-7$$!31+++P*RS*QFF$ \"3F+++#fuIk&F-7$$!3'******f%pK.RFF$\"3,+++&e+2s%F-7$$!3y*****pC,1\"RF F$\"3C+++%eJ$*y$F-7$$!30+++1O.;RFF$\"3-+++[u\"*\\GF-7$$!3>+++?;y>RFF$ \"3)******fY**Q!>F-7$$!3'******H%Q(>#RFF$\"3U+++ v%pARFFF(7$Fegl$!3U+++F-7$F[gl$!3-+++[ u\"*\\GF-7$Fffl$!3C+++%eJ$*y$F-7$Fafl$!3,+++&e+2s%F-7$F\\fl$!3F+++#fuI k&F-7$Fgel$!3O+++(*p(fb'F-7$Fbel$!3[*****Hy_$fuF-7$F]el$!3#)*****4x-NN )F-7$Fhdl$!3x*****>8=!R#*F-7$Fcdl$!3++++*Hs;,\"FF7$F^dl$!3'******fri() 4\"FF7$Ficl$!3)******fl)G&=\"FF7$Fdcl$!35+++klPr7FF7$F_cl$!3/+++_$frN \"FF7$Fjbl$!3-+++s.xU9FF7$Febl$!3)******zgM$G:FF7$F`bl$!3!******f4cRh \"FF7$F[bl$!3/+++z6q*p\"FF7$Ffal$!3!******>]yby\"FF7$Faal$!3)******4') =:(=FF7$F\\al$!31+++u.Od>FF7$Fg`l$!3<+++Hd%G/#FF7$Fb`l$!3$)*****z[Ow7# FF7$F]`l$!35+++TTM6AFF7$Fh_l$!3;+++k4d$H#FF7$Fc_l$!3-+++U)\\RP#FF7$F^_ l$!3-+++iN<_CFF7$Fi^l$!38+++`%4!GDFF7$Fd^l$!34+++J3I,EFF7$F_^l$!35+++ \\*f>n#FF7$Fj]l$!3)******f_d*RFFF7$Fe]l$!3-+++aFJ0GFF7$F`]l$!3#)*****f <'3oGFF7$F[]l$!3++++-nPGHFF7$Ff\\l$!3#)*****p]Bj)HFF7$Fa\\l$!3<+++QN6U IFF7$F\\\\l$!35+++,d*f4$FF7$Fg[l$!33+++;MI[JFF7$Fb[l$!35+++@x[*>$FF7$F ][l$!3-+++XH;]KFF7$Fhz$!3')*****4nj6I$FF7$Fcz$!3!)*****f[*f`LFF7$F^z$! 3A+++#ff)3MFF7$Fiy$!3)******fNZ%oMFF7$Fdy$!3(******RxaM`$FF7$F_y$!3'** ****4SgOg$FF7$Fjx$!3!)*****4O=pn$FF7$Fex$!3'******z0;)\\PFF7$F`x$!3#** ****HP#>>QFF7$F[x$!3')******Rx/$)QFF7$Ffw$!38+++XtZSRFF7$Faw$!3!****** \\!*Q7*RFF7$F\\w$!3$******f^#RNSFF7$Fgv$!3m*****zu#4tSFF7$Fbv$!3@+++Ll [/TFF7$F]v$!3D+++7pmHTFF7$Fhu$!3-+++TBl[TFF7$Fcu$!3g*****f'fPhTFF7$F^u $!3I+++!Hxw;%FF7$Fit$!3%)*****>g#HnTFF7$Fdt$!3i******GD%)fTFF7$F_t$!3V +++]k\"[9%FF7$Fjs$!3l*****p8b:7%FF7$Fes$!37+++xeB*3%FF7$F`s$!3!)*****p Vpo/%FF7$F[s$!31+++egK$*RFF7$Ffr$!3%)*****fQ:u#RFF7$Far$!3#)*****>$=/[ QFF7$F\\r$!3#******49@Wv$FF7$Fgq$!3?+++?tEYOFF7$Fbq$!3<+++Z&)zBNFF7$F] q$!39+++I?[(Q$FF7$Fhp$!3$******Ry`wB$FF7$Fcp$!3A+++5tLuIFF7$F^p$!3))** ****yoe(*GFF7$Fio$!3'******Hgp$3FFF7$Fco$!3#******>5[#4DFF7$F^o$!3$)** ****yI#QI#FF7$Fin$!3)******zx5`4#FF7$FY$!31+++k8r&)=FF7$FT$!3-+++vD$fn \"FF7$FN$!3)*******GXDm9FF7$FH$!3++++nrpc7FF7$FA$!3!******p98s/\"FF7$F ;$!3%******4L4wP)F-7$$\"3#******4+gI-\"F7$!3I+++8')=$G'F-7$$!3+,++2T@& o*F1$!35+++q.z)=%F-7$F($!3++++/^R%4#F-F'-%&COLORG6&%$RGBG$\"#D!\"#F(F_ im-%)POLYGONSG6bv7%F'7$F($\"+/^R%4#!#57$$!$'>FaimF(7%Ffim7$$!+jS@&o*Fe o$\"+q.z)=%FiimFjim7%F^jm7$$\"+$**fI-\"F-$\"+8')=$G'FiimFjim7%Fdjm7$$ \"+t%\\)[M!#;$\"+J$4wP)FiimFjim7%Fjjm7$$\"+[o\")fV!#:$\"+ZJ@Z5!\"*Fjim 7%Fa[n7$$\"+'*f=DH!#9$\"+nrpc7Fg[nFjim7%Fi[n7$$\"+f*H#e8!#8$\"+HXDm9Fg [nFjim7%F`\\n7$$\"++*fi)[Fc\\n$\"+vD$fn\"Fg[nFjim7%Fg\\n7$$\"+7TrZ9!#7 $\"+k8r&)=Fg[nFjim7%F]]n7$$\"+_XMdOF`]n$\"+y2J&4#Fg[nFjim7%Fd]n7$$\"+1 gU8!)F`]n$\"+zI#QI#Fg[nFjim7%Fj]n7$$\"+R/PF:!#6$\"+-\"[#4DFg[nFjim7%F` ^n7$$\"+oTG:DFc^n$\"+.'p$3FFg[nFjim7%Fg^n7$$\"+[p6VNFc^n$\"+zoe(*GFg[n Fjim7%F]_n7$$\"+X))3BUFc^n$\"+5tLuIFg[nFjim7%Fc_n7$$\"+Xj@[TFc^n$\"+%y `wB$Fg[nFjim7%Fi_n7$$\"+B\"3+-$Fc^n$\"+I?[(Q$Fg[nFjim7%F_`n7$$\"+H9d1p F`]n$\"+Z&)zBNFg[nFjim7%Fe`n7$$!+-L\"R%GFc^n$\"+?tEYOFg[nFjim7%F[an7$$ !+'4#=ouFc^n$\"+T6UaPFg[nFjim7%Faan7$$!+_Rb)H\"Fiim$\"+K=/[QFg[nFjim7% Fgan7$$!+Y!zn\">Fiim$\"+'Q:u#RFg[nFjim7%F]bn7$$!+N;**zDFiim$\"+egK$*RF g[nFjim7%Fcbn7$$!+_#31F$Fiim$\"+P%po/%Fg[nFjim7%Fibn7$$!+%o&fvRFiim$\" +xeB*3%Fg[nFjim7%F_cn7$$!+*>z+hFiim$\"+HD%)fTFg[nFjim7%Fadn7$ $!+:p>*z'Fiim$\"+-EHnTFg[nFjim7%Fgdn7$$!+mA]*[(Fiim$\"+!Hxw;%Fg[nFjim7 %F]en7$$!+#e_6<)Fiim$\"+mfPhTFg[nFjim7%Fcen7$$!+*z#>W))Fiim$\"+TBl[TFg [nFjim7%Fien7$$!+pAB4&*Fiim$\"+7pmHTFg[nFjim7%F_fn7$$!+S3v;5Fg[n$\"+Ll [/TFg[nFjim7%Fefn7$$!+Od5#3\"Fg[n$\"+[F4tSFg[nFjim7%F[gn7$$!+?9Fg[n$\"+tB>>QFg[nFjim7%Fihn7$$!+lR$p\\\"Fg[n$\"+eg\")\\PFg [nFjim7%F_in7$$!+*=>3e\"Fg[n$\"+h$=pn$Fg[nFjim7%Fein7$$!+%\\E@n\"Fg[n$ \"+,/m.OFg[nFjim7%F[jn7$$!+BKepFg[n$\"+#ff)3MFg[nFjim7%F][o7$$!+ M_;p?Fg[n$\"+'[*f`LFg[nFjim7%Fc[o7$$!+k,uj@Fg[n$\"+rO;,LFg[nFjim7%Fi[o 7$$!+fT3aAFg[n$\"+XH;]KFg[nFjim7%F_\\o7$$!+$p+,M#Fg[n$\"+@x[*>$Fg[nFji m7%Fe\\o7$$!+zH\">U#Fg[n$\"+;MI[JFg[nFjim7%F[]o7$$!+e*R(*\\#Fg[n$\"+,d *f4$Fg[nFjim7%Fa]o7$$!+gT$Qd#Fg[n$\"+QN6UIFg[nFjim7%Fg]o7$$!+mOXWEFg[n $\"+2NK')HFg[nFjim7%F]^o7$$!+]=&=r#Fg[n$\"+-nPGHFg[nFjim7%Fc^o7$$!+'3w ix#Fg[n$\"+wh3oGFg[nFjim7%Fi^o7$$!+#yqz$GFg[n$\"+aFJ0GFg[nFjim7%F__o7$ $!+e@=(*GFg[n$\"+Ev&*RFFg[nFjim7%Fe_o7$$!+\"3jT&HFg[n$\"+\\*f>n#Fg[nFj im7%F[`o7$$!+]k<4IFg[n$\"+J3I,EFg[nFjim7%Fa`o7$$!+va\\iIFg[n$\"+`%4!GD Fg[nFjim7%Fg`o7$$!+')))R9JFg[n$\"+iN<_CFg[nFjim7%F]ao7$$!+w\"f^;$Fg[n$ \"+U)\\RP#Fg[nFjim7%Fcao7$$!+VG-:KFg[n$\"+k4d$H#Fg[nFjim7%Fiao7$$!+oQ= kKFg[n$\"+TTM6AFg[nFjim7%F_bo7$$!+a^v7LFg[n$\"+)[Ow7#Fg[nFjim7%Febo7$$ !+AfugLFg[n$\"+Hd%G/#Fg[nFjim7%F[co7$$!+!p`!3MFg[n$\"+u.Od>Fg[nFjim7%F aco7$$!+ZbZaMFg[n$\"+h)=:(=Fg[nFjim7%Fgco7$$!+#[O(*\\$Fg[n$\"+-&yby\"F g[nFjim7%F]do7$$!+)QENa$Fg[n$\"+z6q*p\"Fg[nFjim7%Fcdo7$$!+Hd`&e$Fg[n$ \"+'4cRh\"Fg[nFjim7%Fido7$$!+^F[DOFg[n$\"+3YLG:Fg[nFjim7%F_eo7$$!+'yHJ m$Fg[n$\"+s.xU9Fg[nFjim7%Feeo7$$!+Z**G)p$Fg[n$\"+_$frN\"Fg[nFjim7%F[fo 7$$!+;s#3t$Fg[n$\"+klPr7Fg[nFjim7%Fafo7$$!+eHlgPFg[n$\"+c')G&=\"Fg[nFj im7%Fgfo7$$!+'>@xy$Fg[n$\"+;Fw)4\"Fg[nFjim7%F]go7$$!+US-7QFg[n$\"+*Hs; ,\"Fg[nFjim7%Fcgo7$$!+ezeLQFg[n$\"+K\"=!R#*FiimFjim7%Figo7$$!+C4Z_QFg[ n$\"+rF]`$)FiimFjim7%F_ho7$$!+W-woQFg[n$\"+$y_$fuFiimFjim7%Feho7$$!+t. d#)QFg[n$\"+(*p(fb'FiimFjim7%F[io7$$!+P*RS*QFg[n$\"+#fuIk&FiimFjim7%Fa io7$$!+YpK.RFg[n$\"+&e+2s%FiimFjim7%Fgio7$$!+Z7g5RFg[n$\"+%eJ$*y$FiimF jim7%F]jo7$$!+1O.;RFg[n$\"+[u\"*\\GFiimFjim7%Fcjo7$$!+?;y>RFg[n$\"+m%* *Q!>FiimFjim7%Fijo7$$!+VQ(>#RFg[n$\"+FiimFjim7%F] \\p7$Fdjo$!+[u\"*\\GFiimFjim7%Fa\\p7$F^jo$!+%eJ$*y$FiimFjim7%Fe\\p7$Fh io$!+&e+2s%FiimFjim7%Fi\\p7$Fbio$!+#fuIk&FiimFjim7%F]]p7$F\\io$!+(*p(f b'FiimFjim7%Fa]p7$Ffho$!+$y_$fuFiimFjim7%Fe]p7$F`ho$!+rF]`$)FiimFjim7% Fi]p7$Fjgo$!+K\"=!R#*FiimFjim7%F]^p7$Fdgo$!+*Hs;,\"Fg[nFjim7%Fa^p7$F^g o$!+;Fw)4\"Fg[nFjim7%Fe^p7$Fhfo$!+c')G&=\"Fg[nFjim7%Fi^p7$Fbfo$!+klPr7 Fg[nFjim7%F]_p7$F\\fo$!+_$frN\"Fg[nFjim7%Fa_p7$Ffeo$!+s.xU9Fg[nFjim7%F e_p7$F`eo$!+3YLG:Fg[nFjim7%Fi_p7$Fjdo$!+'4cRh\"Fg[nFjim7%F]`p7$Fddo$!+ z6q*p\"Fg[nFjim7%Fa`p7$F^do$!+-&yby\"Fg[nFjim7%Fe`p7$Fhco$!+h)=:(=Fg[n Fjim7%Fi`p7$Fbco$!+u.Od>Fg[nFjim7%F]ap7$F\\co$!+Hd%G/#Fg[nFjim7%Faap7$ Ffbo$!+)[Ow7#Fg[nFjim7%Feap7$F`bo$!+TTM6AFg[nFjim7%Fiap7$Fjao$!+k4d$H# Fg[nFjim7%F]bp7$Fdao$!+U)\\RP#Fg[nFjim7%Fabp7$F^ao$!+iN<_CFg[nFjim7%Fe bp7$Fh`o$!+`%4!GDFg[nFjim7%Fibp7$Fb`o$!+J3I,EFg[nFjim7%F]cp7$F\\`o$!+ \\*f>n#Fg[nFjim7%Facp7$Ff_o$!+Ev&*RFFg[nFjim7%Fecp7$F`_o$!+aFJ0GFg[nFj im7%Ficp7$Fj^o$!+wh3oGFg[nFjim7%F]dp7$Fd^o$!+-nPGHFg[nFjim7%Fadp7$F^^o $!+2NK')HFg[nFjim7%Fedp7$Fh]o$!+QN6UIFg[nFjim7%Fidp7$Fb]o$!+,d*f4$Fg[n Fjim7%F]ep7$F\\]o$!+;MI[JFg[nFjim7%Faep7$Ff\\o$!+@x[*>$Fg[nFjim7%Feep7 $F`\\o$!+XH;]KFg[nFjim7%Fiep7$Fj[o$!+rO;,LFg[nFjim7%F]fp7$Fd[o$!+'[*f` LFg[nFjim7%Fafp7$F^[o$!+#ff)3MFg[nFjim7%Fefp7$Fhjn$!+ctWoMFg[nFjim7%Fi fp7$Fbjn$!+uZXLNFg[nFjim7%F]gp7$F\\jn$!+,/m.OFg[nFjim7%Fagp7$Ffin$!+h$ =pn$Fg[nFjim7%Fegp7$F`in$!+eg\")\\PFg[nFjim7%Figp7$Fjhn$!+tB>>QFg[nFji m7%F]hp7$Fdhn$!+Sx/$)QFg[nFjim7%Fahp7$F^hn$!+XtZSRFg[nFjim7%Fehp7$Fhgn $!+0*Q7*RFg[nFjim7%Fihp7$Fbgn$!+;DRNSFg[nFjim7%F]ip7$F\\gn$!+[F4tSFg[n Fjim7%Faip7$Fffn$!+Ll[/TFg[nFjim7%Feip7$F`fn$!+7pmHTFg[nFjim7%Fiip7$Fj en$!+TBl[TFg[nFjim7%F]jp7$Fden$!+mfPhTFg[nFjim7%Fajp7$F^en$!+!Hxw;%Fg[ nFjim7%Fejp7$Fhdn$!+-EHnTFg[nFjim7%Fijp7$Fbdn$!+HD%)fTFg[nFjim7%F][q7$ F\\dn$!+]k\"[9%Fg[nFjim7%Fa[q7$Ffcn$!+P^b@TFg[nFjim7%Fe[q7$F`cn$!+xeB* 3%Fg[nFjim7%Fi[q7$Fjbn$!+P%po/%Fg[nFjim7%F]\\q7$Fdbn$!+egK$*RFg[nFjim7 %Fa\\q7$F^bn$!+'Q:u#RFg[nFjim7%Fe\\q7$Fhan$!+K=/[QFg[nFjim7%Fi\\q7$Fba n$!+T6UaPFg[nFjim7%F]]q7$F\\an$!+?tEYOFg[nFjim7%Fa]q7$Ff`n$!+Z&)zBNFg[ nFjim7%Fe]q7$F``n$!+I?[(Q$Fg[nFjim7%Fi]q7$Fj_n$!+%y`wB$Fg[nFjim7%F]^q7 $Fd_n$!+5tLuIFg[nFjim7%Fa^q7$F^_n$!+zoe(*GFg[nFjim7%Fe^q7$Fh^n$!+.'p$3 FFg[nFjim7%Fi^q7$Fa^n$!+-\"[#4DFg[nFjim7%F]_q7$F[^n$!+zI#QI#Fg[nFjim7% Fa_q7$Fe]n$!+y2J&4#Fg[nFjim7%Fe_q7$F^]n$!+k8r&)=Fg[nFjim7%Fi_q7$Fh\\n$ !+vD$fn\"Fg[nFjim7%F]`q7$Fa\\n$!+HXDm9Fg[nFjim7%Fa`q7$Fj[n$!+nrpc7Fg[n Fjim7%Fe`q7$Fb[n$!+ZJ@Z5Fg[nFjim7%Fi`q7$F[[n$!+J$4wP)FiimFjim7%F]aq7$$ \"+,+1B5F-$!+8')=$G'FiimFjim7%Faaq7$$!+2T@&o*Feo$!+q.z)=%FiimFjim7%Fga q7$F($!+/^R%4#FiimFjim7%F]bqF'Fjim-F\\im6&F^im$\"\"'!\"\"F(Fcbq-%&STYL EG6#%,PATCHNOGRIDG-F$6$7K7$$\"3\"******43G]#=FF$\"3M+++Ca'3E&FF7$$\"3! *******pz?@=FF$\"3G+++61Ru_FF7$$\"3)******HlCc\"=FF$\"3;+++#4oqG&FF7$$ \"36+++Q^V3=FF$\"3')*****z!er)H&FF7$$\"3#******4e.)*z\"FF$\"3H+++LR<4` FF7$$\"3-+++_)**)*y\"FF$\"3V+++%H-$=`FF7$$\"3%*******y()*)y4)p^;FF$\"3/+++bo&eI&FF7$$\"3(******43$QU;FF$\"3#)******) \\TNH&FF7$$\"30+++5t%[j\"FF$\"3!)******ppbz_FF7$$\"3)******H6Y%H;FF$\" 3i*****\\W2TE&FF7$$\"3++++l'Qli\"FF$\"3q*****\\<'[Z_FF7$$\"31+++AQYE;F F$\"3g*****4&[4I_FF7$$\"3.+++6?]H;FF$\"3#)*****>:dC@&FF7$$\"3-+++rK#ej \"FF$\"3E+++qr@&>&FF7$$\"3/+++0$Gak\"FF$\"3-+++/;6z^FF7$$\"3-+++t')4e; FF$\"3I+++q%3\\;&FF7$$\"3%******fNyLn\"FF$\"3O+++atJ`^FF7$$\"31+++6Pg! p\"FF$\"3*)*****\\/$*[9&FF7$$\"35+++p+)*3Jw\"FF$\"3%)*****zMde9&FF7$$\"35+++>Vry=FF$\"3 w*****4)e(G?&FF7$$\"3-+++J<]C=FF$\"3Q+++(G?u@&FF7$$\"3*******p3,o#=FF$ \"3A+++a*G@B&FF7$$\"3\"******4SPp#=FF$\"3%******Hr*pY_FFF]cqF[im-Fcim6 L7%7$$\"+\"3G]#=Fg[n$\"+Ca'3E&Fg[n7$$\"+qz?@=Fg[n$\"+61Ru_Fg[n7$$\"$t \"Faim$\"$C&Faim7%F]`r7$$\"+`Yi:=Fg[n$\"+#4oqG&Fg[nFb`r7%Fh`r7$$\"+Q^V 3=Fg[n$\"+3er)H&Fg[nFb`r7%F^ar7$$\"+\"e.)*z\"Fg[n$\"+LR<4`Fg[nFb`r7%Fd ar7$$\"+_)**)*y\"Fg[n$\"+%H-$=`Fg[nFb`r7%Fjar7$$\"+z()*)y&Fg[nFb`r7%Ffhr7 $$\"+0$Gak\"Fg[n$\"+/;6z^Fg[nFb`r7%F\\ir7$$\"+t')4e;Fg[n$\"+q%3\\;&Fg[ nFb`r7%Fbir7$$\"+c$yLn\"Fg[n$\"+atJ`^Fg[nFb`r7%Fhir7$$\"+6Pg!p\"Fg[n$ \"+XI*[9&Fg[nFb`r7%F^jr7$$\"+p+)*3Jw\"Fg[n$\"+[t&e9&Fg[nFb`r7%Ff[s7$$\"+>Vry=Fg[n$\"+\")e(G?&Fg[nFb`r7%Fd]s7$$\"+J<]C=Fg[n$\"+(G?u@&Fg [nFb`r7%Fj]s7$$\"+(3,o#=Fg[n$\"+a*G@B&Fg[nFb`r7%F`^s7$$\"+,u$p#=Fg[n$ \"+8(*pY_Fg[nFb`r7%Ff^sFh_rFb`rFabqFfbq-F$6$7K7$F^cq$!3M+++Ca'3E&FF7$F a_r$!3%******Hr*pY_FF7$F\\_r$!3A+++a*G@B&FF7$Fg^r$!3Q+++(G?u@&FF7$Fb^r $!3w*****4)e(G?&FF7$F]^r$!37+++qe$))=&FF7$Fh]r$!3M+++#4!ov^FF7$Fc]r$!3 q*****zAFQ;&FF7$F^]r$!3K+++i\"HP:&FF7$Fi\\r$!3%)*****zMde9&FF7$Fd\\r$! 3g*****HGx19&FF7$F_\\r$!3&*******QwgQ^FF7$Fj[r$!31+++SQ'*R^FF7$Fe[r$!3 *)*****\\/$*[9&FF7$F`[r$!3O+++atJ`^FF7$F[[r$!3I+++q%3\\;&FF7$Ffjq$!3-+ ++/;6z^FF7$Fajq$!3E+++qr@&>&FF7$F\\jq$!3#)*****>:dC@&FF7$Fgiq$!3g***** 4&[4I_FF7$Fbiq$!3q*****\\<'[Z_FF7$F]iq$!3i*****\\W2TE&FF7$Fhhq$!3!)*** ***ppbz_FF7$Fchq$!3#)******)\\TNH&FF7$F^hq$!3/+++bo&eI&FF7$Figq$!3g*** **R_ojJ&FF7$Fdgq$!3M+++.\\*\\K&FF7$F_gq$!3))*****\\/%pJ`FF7$Fjfq$!3<++ +OPXO`FF7$Fefq$!3h*****HU%GR`FF7$F`fq$!3,+++\"Q9-M&FF7$F[fq$!31+++HmGR `FF7$Ffeq$!3\"*******frbO`FF7$Faeq$!3V+++\"=%4K`FF7$F\\eq$!37+++H\"yfK &FF7$Fgdq$!3V+++%H-$=`FF7$Fbdq$!3H+++LR<4`FF7$F]dq$!3')*****z!er)H&FF7 $Fhcq$!3;+++#4oqG&FF7$Fccq$!3G+++61Ru_FFF__sF[im-Fcim6L7%7$Fi_r$!+Ca'3 E&Fg[n7$Fg^s$!+8(*pY_Fg[n7$Fc`r$!$C&Faim7%F]gs7$Fa^s$!+a*G@B&Fg[nF`gs7 %Fdgs7$F[^s$!+(G?u@&Fg[nF`gs7%Fhgs7$Fe]s$!+\")e(G?&Fg[nF`gs7%F\\hs7$F_ ]s$!+qe$))=&Fg[nF`gs7%F`hs7$Fi\\s$!+#4!ov^Fg[nF`gs7%Fdhs7$Fc\\s$!+Gs#Q ;&Fg[nF`gs7%Fhhs7$F]\\s$!+i\"HP:&Fg[nF`gs7%F\\is7$Fg[s$!+[t&e9&Fg[nF`g s7%F`is7$Fa[s$!+$Gx19&Fg[nF`gs7%Fdis7$F[[s$!+RwgQ^Fg[nF`gs7%Fhis7$Fejr $!+SQ'*R^Fg[nF`gs7%F\\js7$F_jr$!+XI*[9&Fg[nF`gs7%F`js7$Fiir$!+atJ`^Fg[ nF`gs7%Fdjs7$Fcir$!+q%3\\;&Fg[nF`gs7%Fhjs7$F]ir$!+/;6z^Fg[nF`gs7%F\\[t 7$Fghr$!+qr@&>&Fg[nF`gs7%F`[t7$Fahr$!+_rX7_Fg[nF`gs7%Fd[t7$F[hr$!+^[4I _Fg[nF`gs7%Fh[t7$Fegr$!+vh[Z_Fg[nF`gs7%F\\\\t7$F_gr$!+Xu5k_Fg[nF`gs7%F `\\t7$Fifr$!+qpbz_Fg[nF`gs7%Fd\\t7$Fcfr$!+*\\TNH&Fg[nF`gs7%Fh\\t7$F]fr $!+bo&eI&Fg[nF`gs7%F\\]t7$Fger$!+C&ojJ&Fg[nF`gs7%F`]t7$Faer$!+.\\*\\K& Fg[nF`gs7%Fd]t7$F[er$!+XSpJ`Fg[nF`gs7%Fh]t7$Fedr$!+OPXO`Fg[nF`gs7%F\\^ t7$F_dr$!+BWGR`Fg[nF`gs7%F`^t7$Ficr$!+\"Q9-M&Fg[nF`gs7%Fd^t7$Fccr$!+Hm GR`Fg[nF`gs7%Fh^t7$F]cr$!+grbO`Fg[nF`gs7%F\\_t7$Fgbr$!+\"=%4K`Fg[nF`gs 7%F`_t7$Fabr$!+H\"yfK&Fg[nF`gs7%Fd_t7$F[br$!+%H-$=`Fg[nF`gs7%Fh_t7$Fea r$!+LR<4`Fg[nF`gs7%F\\`t7$F_ar$!+3er)H&Fg[nF`gs7%F``t7$Fi`r$!+#4oqG&Fg [nF`gs7%Fd`t7$F^`r$!+61Ru_Fg[nF`gs7%Fh`tFjfsF`gsFabqFfbq-F$6%7$7$$!3Q+ +++++!>&FFF(7$$\"3%**************=#FFF(-%'COLOURG6&F^imF)F)F)-%*LINEST YLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>'FF7$F($\"3Q++++++!>'FFFeatFhat-%(SC ALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"* -%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F^ct6#%(DEFAULTG-%%VIEWG6$;$!$>&Faim$ \"$>#Faim;$!$>'Faim$\"$>'Faim" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "The stability interval for the embedded o rder 6 scheme is (approximately) " }{XPPEDIT 18 0 "[-3.9227, 0];" "6# 7$,$-%&FloatG6$\"&F#R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions f or the two schemes can be displayed in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p_||(1..6),p||(1.. 7)],view=[-5.39..2.19,-6.19..6.19],font=[HELVETICA,9],\n lab els=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 425 574 574 {PLOTDATA 2 "65-%'CURVESG6$7av7$$\"\"!F)F(7 $F($\"3++++/^R%4#!#=7$$!3#)*****H19_o*!#F$\"35+++q.z)=%F-7$$\"3(****** H**fI-\"!#E$\"3I+++8')=$G'F-7$$\"39+++t%\\)[M!#C$\"3%******4L4wP)F-7$$ \"3I+++[o\")fV!#B$\"3!******p98s/\"!#<7$$\"3:+++'*f=DH!#A$\"3++++nrpc7 FF7$$\"3!*******e*H#e8!#@$\"3)*******GXDm9FF7$$\"3m********)fi)[FP$\"3 -+++vD$fn\"FF7$$\"37+++7TrZ9!#?$\"31+++k8r&)=FF7$$\"3%)*****>bWtl$Fen$ \"3)******zx5`4#FF7$$\"3p*****f+EM,)Fen$\"3$)******yI#QI#FF7$$\"3,+++R /PF:!#>$\"3#******>5[#4DFF7$$\"3&******z;%G:DFeo$\"3'******Hgp$3FFF7$$ \"3m*****z%p6VNFeo$\"3))******yoe(*GFF7$$\"3o*****\\%))3BUFeo$\"3A+++5 tLuIFF7$$\"3#)*****\\M;#[TFeo$\"3$******Ry`wB$FF7$$\"3%)*****H73+-$Feo $\"39+++I?[(Q$FF7$$\"3$)******G9d1pFen$\"3<+++Z&)zBNFF7$$!3,+++-L\"R%G Feo$\"3?+++?tEYOFF7$$!3g+++'4#=ouFeo$\"3#******49@Wv$FF7$$!3++++_Rb)H \"F-$\"3#)*****>$=/[QFF7$$!3/+++Y!zn\">F-$\"3%)*****fQ:u#RFF7$$!3=+++N ;**zDF-$\"31+++egK$*RFF7$$!3;+++_#31F$F-$\"3!)*****pVpo/%FF7$$!33+++%o &fvRF-$\"37+++xeB*3%FF7$$!3#)******)>z+hF-$\"3i******GD%)fTFF7$$!3y*** **\\\"p>*z'F-$\"3%)*****>g#HnTFF7$$!3!******fE-&*[(F-$\"3I+++!Hxw;%FF7 $$!3s*****>e_6<)F-$\"3g*****f'fPhTFF7$$!3@+++*z#>W))F-$\"3-+++TBl[TFF7 $$!3u******oAB4&*F-$\"3D+++7pmHTFF7$$!3/+++S3v;5FF$\"3@+++Ll[/TFF7$$!3 #******ft0@3\"FF$\"3m*****zu#4tSFF7$$!3)*******>9FF$ \"3#******HP#>>QFF7$$!3)******\\'R$p\\\"FF$\"3'******z0;)\\PFF7$$!31++ +*=>3e\"FF$\"3!)*****4O=pn$FF7$$!3-+++%\\E@n\"FF$\"3'******4SgOg$FF7$$ !3%******HA$epFF$\"3A+++#ff)3MFF7$$!3=+++M_;p?FF$\"3!)*****f[*f` LFF7$$!3-+++k,uj@FF$\"3')*****4nj6I$FF7$$!3/+++fT3aAFF$\"3-+++XH;]KFF7 $$!37+++$p+,M#FF$\"35+++@x[*>$FF7$$!3-+++zH\">U#FF$\"33+++;MI[JFF7$$!3 9+++e*R(*\\#FF$\"35+++,d*f4$FF7$$!35+++gT$Qd#FF$\"3<+++QN6UIFF7$$!3)** ****fm`Wk#FF$\"3#)*****p]Bj)HFF7$$!3&*******\\=&=r#FF$\"3++++-nPGHFF7$ $!3?+++'3wix#FF$\"3#)*****f<'3oGFF7$$!3>+++#yqz$GFF$\"3-+++aFJ0GFF7$$! 35+++e@=(*GFF$\"3)******f_d*RFFF7$$!3#******43jT&HFF$\"35+++\\*f>n#FF7 $$!3++++]k<4IFF$\"34+++J3I,EFF7$$!3'******\\Z&\\iIFF$\"38+++`%4!GDFF7$ $!3.+++')))R9JFF$\"3-+++iN<_CFF7$$!3'******fFF7$$!3#******pavWX$FF$\"3)******4')=:(= FF7$$!3')*****>[O(*\\$FF$\"3!******>]yby\"FF7$$!33+++)QENa$FF$\"3/+++z 6q*p\"FF7$$!31+++Hd`&e$FF$\"3!******f4cRh\"FF7$$!3%)*****4v#[DOFF$\"3) ******zgM$G:FF7$$!3#)*****fyHJm$FF$\"3-+++s.xU9FF7$$!37+++Z**G)p$FF$\" 3/+++_$frN\"FF7$$!3y*****f@F3t$FF$\"35+++klPr7FF7$$!35+++eHlgPFF$\"3)* *****fl)G&=\"FF7$$!37+++'>@xy$FF$\"3'******fri()4\"FF7$$!3++++US-7QFF$ \"3++++*Hs;,\"FF7$$!3&)*****z&zeLQFF$\"3x*****>8=!R#*F-7$$!3!******R#4 Z_QFF$\"3#)*****4x-NN)F-7$$!3=+++W-woQFF$\"3[*****Hy_$fuF-7$$!33+++t.d #)QFF$\"3O+++(*p(fb'F-7$$!31+++P*RS*QFF$\"3F+++#fuIk&F-7$$!3'******f%p K.RFF$\"3,+++&e+2s%F-7$$!3y*****pC,1\"RFF$\"3C+++%eJ$*y$F-7$$!30+++1O. ;RFF$\"3-+++[u\"*\\GF-7$$!3>+++?;y>RFF$\"3)******fY**Q!>F-7$$!3'****** H%Q(>#RFF$\"3U+++v%pARFFF(7$Fegl$!3U+++F-7$F[gl$!3-+++[u\"*\\GF-7$Fffl$!3C+++%eJ$*y$F- 7$Fafl$!3,+++&e+2s%F-7$F\\fl$!3F+++#fuIk&F-7$Fgel$!3O+++(*p(fb'F-7$Fbe l$!3[*****Hy_$fuF-7$F]el$!3#)*****4x-NN)F-7$Fhdl$!3x*****>8=!R#*F-7$Fc dl$!3++++*Hs;,\"FF7$F^dl$!3'******fri()4\"FF7$Ficl$!3)******fl)G&=\"FF 7$Fdcl$!35+++klPr7FF7$F_cl$!3/+++_$frN\"FF7$Fjbl$!3-+++s.xU9FF7$Febl$! 3)******zgM$G:FF7$F`bl$!3!******f4cRh\"FF7$F[bl$!3/+++z6q*p\"FF7$Ffal$ !3!******>]yby\"FF7$Faal$!3)******4')=:(=FF7$F\\al$!31+++u.Od>FF7$Fg`l $!3<+++Hd%G/#FF7$Fb`l$!3$)*****z[Ow7#FF7$F]`l$!35+++TTM6AFF7$Fh_l$!3;+ ++k4d$H#FF7$Fc_l$!3-+++U)\\RP#FF7$F^_l$!3-+++iN<_CFF7$Fi^l$!38+++`%4!G DFF7$Fd^l$!34+++J3I,EFF7$F_^l$!35+++\\*f>n#FF7$Fj]l$!3)******f_d*RFFF7 $Fe]l$!3-+++aFJ0GFF7$F`]l$!3#)*****f<'3oGFF7$F[]l$!3++++-nPGHFF7$Ff\\l $!3#)*****p]Bj)HFF7$Fa\\l$!3<+++QN6UIFF7$F\\\\l$!35+++,d*f4$FF7$Fg[l$! 33+++;MI[JFF7$Fb[l$!35+++@x[*>$FF7$F][l$!3-+++XH;]KFF7$Fhz$!3')*****4n j6I$FF7$Fcz$!3!)*****f[*f`LFF7$F^z$!3A+++#ff)3MFF7$Fiy$!3)******fNZ%oM FF7$Fdy$!3(******RxaM`$FF7$F_y$!3'******4SgOg$FF7$Fjx$!3!)*****4O=pn$F F7$Fex$!3'******z0;)\\PFF7$F`x$!3#******HP#>>QFF7$F[x$!3')******Rx/$)Q FF7$Ffw$!38+++XtZSRFF7$Faw$!3!******\\!*Q7*RFF7$F\\w$!3$******f^#RNSFF 7$Fgv$!3m*****zu#4tSFF7$Fbv$!3@+++Ll[/TFF7$F]v$!3D+++7pmHTFF7$Fhu$!3-+ ++TBl[TFF7$Fcu$!3g*****f'fPhTFF7$F^u$!3I+++!Hxw;%FF7$Fit$!3%)*****>g#H nTFF7$Fdt$!3i******GD%)fTFF7$F_t$!3V+++]k\"[9%FF7$Fjs$!3l*****p8b:7%FF 7$Fes$!37+++xeB*3%FF7$F`s$!3!)*****pVpo/%FF7$F[s$!31+++egK$*RFF7$Ffr$! 3%)*****fQ:u#RFF7$Far$!3#)*****>$=/[QFF7$F\\r$!3#******49@Wv$FF7$Fgq$! 3?+++?tEYOFF7$Fbq$!3<+++Z&)zBNFF7$F]q$!39+++I?[(Q$FF7$Fhp$!3$******Ry` wB$FF7$Fcp$!3A+++5tLuIFF7$F^p$!3))******yoe(*GFF7$Fio$!3'******Hgp$3FF F7$Fco$!3#******>5[#4DFF7$F^o$!3$)******yI#QI#FF7$Fin$!3)******zx5`4#F F7$FY$!31+++k8r&)=FF7$FT$!3-+++vD$fn\"FF7$FN$!3)*******GXDm9FF7$FH$!3+ +++nrpc7FF7$FA$!3!******p98s/\"FF7$F;$!3%******4L4wP)F-7$$\"3#******4+ gI-\"F7$!3I+++8')=$G'F-7$$!3+,++2T@&o*F1$!35+++q.z)=%F-7$F($!3++++/^R% 4#F-F'-%&COLORG6&%$RGBG$\"#D!\"#F(F_im-%)POLYGONSG6bv7%F'7$F($\"+/^R%4 #!#57$$!$'>FaimF(7%Ffim7$$!+jS@&o*Feo$\"+q.z)=%FiimFjim7%F^jm7$$\"+$** fI-\"F-$\"+8')=$G'FiimFjim7%Fdjm7$$\"+t%\\)[M!#;$\"+J$4wP)FiimFjim7%Fj jm7$$\"+[o\")fV!#:$\"+ZJ@Z5!\"*Fjim7%Fa[n7$$\"+'*f=DH!#9$\"+nrpc7Fg[nF jim7%Fi[n7$$\"+f*H#e8!#8$\"+HXDm9Fg[nFjim7%F`\\n7$$\"++*fi)[Fc\\n$\"+v D$fn\"Fg[nFjim7%Fg\\n7$$\"+7TrZ9!#7$\"+k8r&)=Fg[nFjim7%F]]n7$$\"+_XMdO F`]n$\"+y2J&4#Fg[nFjim7%Fd]n7$$\"+1gU8!)F`]n$\"+zI#QI#Fg[nFjim7%Fj]n7$ $\"+R/PF:!#6$\"+-\"[#4DFg[nFjim7%F`^n7$$\"+oTG:DFc^n$\"+.'p$3FFg[nFjim 7%Fg^n7$$\"+[p6VNFc^n$\"+zoe(*GFg[nFjim7%F]_n7$$\"+X))3BUFc^n$\"+5tLuI Fg[nFjim7%Fc_n7$$\"+Xj@[TFc^n$\"+%y`wB$Fg[nFjim7%Fi_n7$$\"+B\"3+-$Fc^n $\"+I?[(Q$Fg[nFjim7%F_`n7$$\"+H9d1pF`]n$\"+Z&)zBNFg[nFjim7%Fe`n7$$!+-L \"R%GFc^n$\"+?tEYOFg[nFjim7%F[an7$$!+'4#=ouFc^n$\"+T6UaPFg[nFjim7%Faan 7$$!+_Rb)H\"Fiim$\"+K=/[QFg[nFjim7%Fgan7$$!+Y!zn\">Fiim$\"+'Q:u#RFg[nF jim7%F]bn7$$!+N;**zDFiim$\"+egK$*RFg[nFjim7%Fcbn7$$!+_#31F$Fiim$\"+P%p o/%Fg[nFjim7%Fibn7$$!+%o&fvRFiim$\"+xeB*3%Fg[nFjim7%F_cn7$$!+*>z+hFiim$\"+HD%)fTFg[nFjim7%Fadn7$$!+:p>*z'Fiim$\"+-EHnTFg[nFjim7%Fgd n7$$!+mA]*[(Fiim$\"+!Hxw;%Fg[nFjim7%F]en7$$!+#e_6<)Fiim$\"+mfPhTFg[nFj im7%Fcen7$$!+*z#>W))Fiim$\"+TBl[TFg[nFjim7%Fien7$$!+pAB4&*Fiim$\"+7pmH TFg[nFjim7%F_fn7$$!+S3v;5Fg[n$\"+Ll[/TFg[nFjim7%Fefn7$$!+Od5#3\"Fg[n$ \"+[F4tSFg[nFjim7%F[gn7$$!+?9Fg[n$\"+tB>>QFg[nFjim7 %Fihn7$$!+lR$p\\\"Fg[n$\"+eg\")\\PFg[nFjim7%F_in7$$!+*=>3e\"Fg[n$\"+h$ =pn$Fg[nFjim7%Fein7$$!+%\\E@n\"Fg[n$\"+,/m.OFg[nFjim7%F[jn7$$!+BKepFg[n$\"+#ff)3MFg[nFjim7%F][o7$$!+M_;p?Fg[n$\"+'[*f`LFg[nFjim7%Fc[o7 $$!+k,uj@Fg[n$\"+rO;,LFg[nFjim7%Fi[o7$$!+fT3aAFg[n$\"+XH;]KFg[nFjim7%F _\\o7$$!+$p+,M#Fg[n$\"+@x[*>$Fg[nFjim7%Fe\\o7$$!+zH\">U#Fg[n$\"+;MI[JF g[nFjim7%F[]o7$$!+e*R(*\\#Fg[n$\"+,d*f4$Fg[nFjim7%Fa]o7$$!+gT$Qd#Fg[n$ \"+QN6UIFg[nFjim7%Fg]o7$$!+mOXWEFg[n$\"+2NK')HFg[nFjim7%F]^o7$$!+]=&=r #Fg[n$\"+-nPGHFg[nFjim7%Fc^o7$$!+'3wix#Fg[n$\"+wh3oGFg[nFjim7%Fi^o7$$! +#yqz$GFg[n$\"+aFJ0GFg[nFjim7%F__o7$$!+e@=(*GFg[n$\"+Ev&*RFFg[nFjim7%F e_o7$$!+\"3jT&HFg[n$\"+\\*f>n#Fg[nFjim7%F[`o7$$!+]k<4IFg[n$\"+J3I,EFg[ nFjim7%Fa`o7$$!+va\\iIFg[n$\"+`%4!GDFg[nFjim7%Fg`o7$$!+')))R9JFg[n$\"+ iN<_CFg[nFjim7%F]ao7$$!+w\"f^;$Fg[n$\"+U)\\RP#Fg[nFjim7%Fcao7$$!+VG-:K Fg[n$\"+k4d$H#Fg[nFjim7%Fiao7$$!+oQ=kKFg[n$\"+TTM6AFg[nFjim7%F_bo7$$!+ a^v7LFg[n$\"+)[Ow7#Fg[nFjim7%Febo7$$!+AfugLFg[n$\"+Hd%G/#Fg[nFjim7%F[c o7$$!+!p`!3MFg[n$\"+u.Od>Fg[nFjim7%Faco7$$!+ZbZaMFg[n$\"+h)=:(=Fg[nFji m7%Fgco7$$!+#[O(*\\$Fg[n$\"+-&yby\"Fg[nFjim7%F]do7$$!+)QENa$Fg[n$\"+z6 q*p\"Fg[nFjim7%Fcdo7$$!+Hd`&e$Fg[n$\"+'4cRh\"Fg[nFjim7%Fido7$$!+^F[DOF g[n$\"+3YLG:Fg[nFjim7%F_eo7$$!+'yHJm$Fg[n$\"+s.xU9Fg[nFjim7%Feeo7$$!+Z **G)p$Fg[n$\"+_$frN\"Fg[nFjim7%F[fo7$$!+;s#3t$Fg[n$\"+klPr7Fg[nFjim7%F afo7$$!+eHlgPFg[n$\"+c')G&=\"Fg[nFjim7%Fgfo7$$!+'>@xy$Fg[n$\"+;Fw)4\"F g[nFjim7%F]go7$$!+US-7QFg[n$\"+*Hs;,\"Fg[nFjim7%Fcgo7$$!+ezeLQFg[n$\"+ K\"=!R#*FiimFjim7%Figo7$$!+C4Z_QFg[n$\"+rF]`$)FiimFjim7%F_ho7$$!+W-woQ Fg[n$\"+$y_$fuFiimFjim7%Feho7$$!+t.d#)QFg[n$\"+(*p(fb'FiimFjim7%F[io7$ $!+P*RS*QFg[n$\"+#fuIk&FiimFjim7%Faio7$$!+YpK.RFg[n$\"+&e+2s%FiimFjim7 %Fgio7$$!+Z7g5RFg[n$\"+%eJ$*y$FiimFjim7%F]jo7$$!+1O.;RFg[n$\"+[u\"*\\G FiimFjim7%Fcjo7$$!+?;y>RFg[n$\"+m%**Q!>FiimFjim7%Fijo7$$!+VQ(>#RFg[n$ \"+FiimFjim7%F]\\p7$Fdjo$!+[u\"*\\GFiimFjim7%Fa\\ p7$F^jo$!+%eJ$*y$FiimFjim7%Fe\\p7$Fhio$!+&e+2s%FiimFjim7%Fi\\p7$Fbio$! +#fuIk&FiimFjim7%F]]p7$F\\io$!+(*p(fb'FiimFjim7%Fa]p7$Ffho$!+$y_$fuFii mFjim7%Fe]p7$F`ho$!+rF]`$)FiimFjim7%Fi]p7$Fjgo$!+K\"=!R#*FiimFjim7%F]^ p7$Fdgo$!+*Hs;,\"Fg[nFjim7%Fa^p7$F^go$!+;Fw)4\"Fg[nFjim7%Fe^p7$Fhfo$!+ c')G&=\"Fg[nFjim7%Fi^p7$Fbfo$!+klPr7Fg[nFjim7%F]_p7$F\\fo$!+_$frN\"Fg[ nFjim7%Fa_p7$Ffeo$!+s.xU9Fg[nFjim7%Fe_p7$F`eo$!+3YLG:Fg[nFjim7%Fi_p7$F jdo$!+'4cRh\"Fg[nFjim7%F]`p7$Fddo$!+z6q*p\"Fg[nFjim7%Fa`p7$F^do$!+-&yb y\"Fg[nFjim7%Fe`p7$Fhco$!+h)=:(=Fg[nFjim7%Fi`p7$Fbco$!+u.Od>Fg[nFjim7% F]ap7$F\\co$!+Hd%G/#Fg[nFjim7%Faap7$Ffbo$!+)[Ow7#Fg[nFjim7%Feap7$F`bo$ !+TTM6AFg[nFjim7%Fiap7$Fjao$!+k4d$H#Fg[nFjim7%F]bp7$Fdao$!+U)\\RP#Fg[n Fjim7%Fabp7$F^ao$!+iN<_CFg[nFjim7%Febp7$Fh`o$!+`%4!GDFg[nFjim7%Fibp7$F b`o$!+J3I,EFg[nFjim7%F]cp7$F\\`o$!+\\*f>n#Fg[nFjim7%Facp7$Ff_o$!+Ev&*R FFg[nFjim7%Fecp7$F`_o$!+aFJ0GFg[nFjim7%Ficp7$Fj^o$!+wh3oGFg[nFjim7%F]d p7$Fd^o$!+-nPGHFg[nFjim7%Fadp7$F^^o$!+2NK')HFg[nFjim7%Fedp7$Fh]o$!+QN6 UIFg[nFjim7%Fidp7$Fb]o$!+,d*f4$Fg[nFjim7%F]ep7$F\\]o$!+;MI[JFg[nFjim7% Faep7$Ff\\o$!+@x[*>$Fg[nFjim7%Feep7$F`\\o$!+XH;]KFg[nFjim7%Fiep7$Fj[o$ !+rO;,LFg[nFjim7%F]fp7$Fd[o$!+'[*f`LFg[nFjim7%Fafp7$F^[o$!+#ff)3MFg[nF jim7%Fefp7$Fhjn$!+ctWoMFg[nFjim7%Fifp7$Fbjn$!+uZXLNFg[nFjim7%F]gp7$F\\ jn$!+,/m.OFg[nFjim7%Fagp7$Ffin$!+h$=pn$Fg[nFjim7%Fegp7$F`in$!+eg\")\\P Fg[nFjim7%Figp7$Fjhn$!+tB>>QFg[nFjim7%F]hp7$Fdhn$!+Sx/$)QFg[nFjim7%Fah p7$F^hn$!+XtZSRFg[nFjim7%Fehp7$Fhgn$!+0*Q7*RFg[nFjim7%Fihp7$Fbgn$!+;DR NSFg[nFjim7%F]ip7$F\\gn$!+[F4tSFg[nFjim7%Faip7$Fffn$!+Ll[/TFg[nFjim7%F eip7$F`fn$!+7pmHTFg[nFjim7%Fiip7$Fjen$!+TBl[TFg[nFjim7%F]jp7$Fden$!+mf PhTFg[nFjim7%Fajp7$F^en$!+!Hxw;%Fg[nFjim7%Fejp7$Fhdn$!+-EHnTFg[nFjim7% Fijp7$Fbdn$!+HD%)fTFg[nFjim7%F][q7$F\\dn$!+]k\"[9%Fg[nFjim7%Fa[q7$Ffcn $!+P^b@TFg[nFjim7%Fe[q7$F`cn$!+xeB*3%Fg[nFjim7%Fi[q7$Fjbn$!+P%po/%Fg[n Fjim7%F]\\q7$Fdbn$!+egK$*RFg[nFjim7%Fa\\q7$F^bn$!+'Q:u#RFg[nFjim7%Fe\\ q7$Fhan$!+K=/[QFg[nFjim7%Fi\\q7$Fban$!+T6UaPFg[nFjim7%F]]q7$F\\an$!+?t EYOFg[nFjim7%Fa]q7$Ff`n$!+Z&)zBNFg[nFjim7%Fe]q7$F``n$!+I?[(Q$Fg[nFjim7 %Fi]q7$Fj_n$!+%y`wB$Fg[nFjim7%F]^q7$Fd_n$!+5tLuIFg[nFjim7%Fa^q7$F^_n$! +zoe(*GFg[nFjim7%Fe^q7$Fh^n$!+.'p$3FFg[nFjim7%Fi^q7$Fa^n$!+-\"[#4DFg[n Fjim7%F]_q7$F[^n$!+zI#QI#Fg[nFjim7%Fa_q7$Fe]n$!+y2J&4#Fg[nFjim7%Fe_q7$ F^]n$!+k8r&)=Fg[nFjim7%Fi_q7$Fh\\n$!+vD$fn\"Fg[nFjim7%F]`q7$Fa\\n$!+HX Dm9Fg[nFjim7%Fa`q7$Fj[n$!+nrpc7Fg[nFjim7%Fe`q7$Fb[n$!+ZJ@Z5Fg[nFjim7%F i`q7$F[[n$!+J$4wP)FiimFjim7%F]aq7$$\"+,+1B5F-$!+8')=$G'FiimFjim7%Faaq7 $$!+2T@&o*Feo$!+q.z)=%FiimFjim7%Fgaq7$F($!+/^R%4#FiimFjim7%F]bqF'Fjim- F\\im6&F^im$\"\"'!\"\"F(Fcbq-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3\"** ****43G]#=FF$\"3M+++Ca'3E&FF7$$\"3!*******pz?@=FF$\"3G+++61Ru_FF7$$\"3 )******HlCc\"=FF$\"3;+++#4oqG&FF7$$\"36+++Q^V3=FF$\"3')*****z!er)H&FF7 $$\"3#******4e.)*z\"FF$\"3H+++LR<4`FF7$$\"3-+++_)**)*y\"FF$\"3V+++%H-$ =`FF7$$\"3%*******y()*)y4)p^;FF$\"3/+++bo&eI&F F7$$\"3(******43$QU;FF$\"3#)******)\\TNH&FF7$$\"30+++5t%[j\"FF$\"3!)** ****ppbz_FF7$$\"3)******H6Y%H;FF$\"3i*****\\W2TE&FF7$$\"3++++l'Qli\"FF $\"3q*****\\<'[Z_FF7$$\"31+++AQYE;FF$\"3g*****4&[4I_FF7$$\"3.+++6?]H;F F$\"3#)*****>:dC@&FF7$$\"3-+++rK#ej\"FF$\"3E+++qr@&>&FF7$$\"3/+++0$Gak \"FF$\"3-+++/;6z^FF7$$\"3-+++t')4e;FF$\"3I+++q%3\\;&FF7$$\"3%******fNy Ln\"FF$\"3O+++atJ`^FF7$$\"31+++6Pg!p\"FF$\"3*)*****\\/$*[9&FF7$$\"35++ +p+)*3Jw\"FF$\"3%)*****zM de9&FF7$$\"35+++>Vry=FF$\"3w*****4)e(G?&FF7$$\"3-+++J<]C=FF$ \"3Q+++(G?u@&FF7$$\"3*******p3,o#=FF$\"3A+++a*G@B&FF7$$\"3\"******4SPp #=FF$\"3%******Hr*pY_FFF]cqF[im-Fcim6L7%7$$\"+\"3G]#=Fg[n$\"+Ca'3E&Fg[ n7$$\"+qz?@=Fg[n$\"+61Ru_Fg[n7$$\"$t\"Faim$\"$C&Faim7%F]`r7$$\"+`Yi:=F g[n$\"+#4oqG&Fg[nFb`r7%Fh`r7$$\"+Q^V3=Fg[n$\"+3er)H&Fg[nFb`r7%F^ar7$$ \"+\"e.)*z\"Fg[n$\"+LR<4`Fg[nFb`r7%Fdar7$$\"+_)**)*y\"Fg[n$\"+%H-$=`Fg [nFb`r7%Fjar7$$\"+z()*)y&Fg[nFb`r7%Ffhr7$$\"+0$Gak\"Fg[n$\"+/;6z^Fg[nFb`r 7%F\\ir7$$\"+t')4e;Fg[n$\"+q%3\\;&Fg[nFb`r7%Fbir7$$\"+c$yLn\"Fg[n$\"+a tJ`^Fg[nFb`r7%Fhir7$$\"+6Pg!p\"Fg[n$\"+XI*[9&Fg[nFb`r7%F^jr7$$\"+p+)*3 Jw\"Fg[n$\"+[t&e9&Fg[n Fb`r7%Ff[s7$$\"+>Vry=Fg[n$\"+\")e(G?&Fg[nF b`r7%Fd]s7$$\"+J<]C=Fg[n$\"+(G?u@&Fg[nFb`r7%Fj]s7$$\"+(3,o#=Fg[n$\"+a* G@B&Fg[nFb`r7%F`^s7$$\"+,u$p#=Fg[n$\"+8(*pY_Fg[nFb`r7%Ff^sFh_rFb`rFabq Ffbq-F$6$7K7$F^cq$!3M+++Ca'3E&FF7$Fa_r$!3%******Hr*pY_FF7$F\\_r$!3A+++ a*G@B&FF7$Fg^r$!3Q+++(G?u@&FF7$Fb^r$!3w*****4)e(G?&FF7$F]^r$!37+++qe$) )=&FF7$Fh]r$!3M+++#4!ov^FF7$Fc]r$!3q*****zAFQ;&FF7$F^]r$!3K+++i\"HP:&F F7$Fi\\r$!3%)*****zMde9&FF7$Fd\\r$!3g*****HGx19&FF7$F_\\r$!3&*******Qw gQ^FF7$Fj[r$!31+++SQ'*R^FF7$Fe[r$!3*)*****\\/$*[9&FF7$F`[r$!3O+++atJ`^ FF7$F[[r$!3I+++q%3\\;&FF7$Ffjq$!3-+++/;6z^FF7$Fajq$!3E+++qr@&>&FF7$F\\ jq$!3#)*****>:dC@&FF7$Fgiq$!3g*****4&[4I_FF7$Fbiq$!3q*****\\<'[Z_FF7$F ]iq$!3i*****\\W2TE&FF7$Fhhq$!3!)******ppbz_FF7$Fchq$!3#)******)\\TNH&F F7$F^hq$!3/+++bo&eI&FF7$Figq$!3g*****R_ojJ&FF7$Fdgq$!3M+++.\\*\\K&FF7$ F_gq$!3))*****\\/%pJ`FF7$Fjfq$!3<+++OPXO`FF7$Fefq$!3h*****HU%GR`FF7$F` fq$!3,+++\"Q9-M&FF7$F[fq$!31+++HmGR`FF7$Ffeq$!3\"*******frbO`FF7$Faeq$ !3V+++\"=%4K`FF7$F\\eq$!37+++H\"yfK&FF7$Fgdq$!3V+++%H-$=`FF7$Fbdq$!3H+ ++LR<4`FF7$F]dq$!3')*****z!er)H&FF7$Fhcq$!3;+++#4oqG&FF7$Fccq$!3G+++61 Ru_FFF__sF[im-Fcim6L7%7$Fi_r$!+Ca'3E&Fg[n7$Fg^s$!+8(*pY_Fg[n7$Fc`r$!$C &Faim7%F]gs7$Fa^s$!+a*G@B&Fg[nF`gs7%Fdgs7$F[^s$!+(G?u@&Fg[nF`gs7%Fhgs7 $Fe]s$!+\")e(G?&Fg[nF`gs7%F\\hs7$F_]s$!+qe$))=&Fg[nF`gs7%F`hs7$Fi\\s$! +#4!ov^Fg[nF`gs7%Fdhs7$Fc\\s$!+Gs#Q;&Fg[nF`gs7%Fhhs7$F]\\s$!+i\"HP:&Fg [nF`gs7%F\\is7$Fg[s$!+[t&e9&Fg[nF`gs7%F`is7$Fa[s$!+$Gx19&Fg[nF`gs7%Fdi s7$F[[s$!+RwgQ^Fg[nF`gs7%Fhis7$Fejr$!+SQ'*R^Fg[nF`gs7%F\\js7$F_jr$!+XI *[9&Fg[nF`gs7%F`js7$Fiir$!+atJ`^Fg[nF`gs7%Fdjs7$Fcir$!+q%3\\;&Fg[nF`gs 7%Fhjs7$F]ir$!+/;6z^Fg[nF`gs7%F\\[t7$Fghr$!+qr@&>&Fg[nF`gs7%F`[t7$Fahr $!+_rX7_Fg[nF`gs7%Fd[t7$F[hr$!+^[4I_Fg[nF`gs7%Fh[t7$Fegr$!+vh[Z_Fg[nF` gs7%F\\\\t7$F_gr$!+Xu5k_Fg[nF`gs7%F`\\t7$Fifr$!+qpbz_Fg[nF`gs7%Fd\\t7$ Fcfr$!+*\\TNH&Fg[nF`gs7%Fh\\t7$F]fr$!+bo&eI&Fg[nF`gs7%F\\]t7$Fger$!+C& ojJ&Fg[nF`gs7%F`]t7$Faer$!+.\\*\\K&Fg[nF`gs7%Fd]t7$F[er$!+XSpJ`Fg[nF`g s7%Fh]t7$Fedr$!+OPXO`Fg[nF`gs7%F\\^t7$F_dr$!+BWGR`Fg[nF`gs7%F`^t7$Ficr $!+\"Q9-M&Fg[nF`gs7%Fd^t7$Fccr$!+HmGR`Fg[nF`gs7%Fh^t7$F]cr$!+grbO`Fg[n F`gs7%F\\_t7$Fgbr$!+\"=%4K`Fg[nF`gs7%F`_t7$Fabr$!+H\"yfK&Fg[nF`gs7%Fd_ t7$F[br$!+%H-$=`Fg[nF`gs7%Fh_t7$Fear$!+LR<4`Fg[nF`gs7%F\\`t7$F_ar$!+3e r)H&Fg[nF`gs7%F``t7$Fi`r$!+#4oqG&Fg[nF`gs7%Fd`t7$F^`r$!+61Ru_Fg[nF`gs7 %Fh`tFjfsF`gsFabqFfbq-F$6$7e\\lF'7$F($\"3++++Fjzq:F-7$F($\"3)******Rl# fTJF-7$$\"3)******R15([WF1$\"34+++z*)Q7ZF-7$$\"3s*****pxN!*4%F7$\"3C++ +)H&=$G'F-7$$\"3-+++\"=G&fA!#D$\"3g*****pe\")R&yF-7$$\"3Z+++UmYN#*Fbbt $\"3H+++_yxC%*F-7$$\"3;+++Jas'=$F=$\"3%******zVd&*4\"FF7$$\"3$*******) [y_+\"FC$\"3'*******[sjc7FF7$$\"3.+++'R\")p+$FC$\"30+++&f#FF7$$\"31+++I(Hr+'FP$\"3#** ****>iJhN#FF7$$\"3!******RcS(R6Fen$\"3!)*****RE')H^#FF7$$\"3$)*****H.m (=?Fen$\"3;+++a9]pEFF7$$\"33+++x%G2L$Fen$\"3,+++%>^`#GFF7$$\"3++++4)Q) *3&Fen$\"3#******HF>+)HFF7$$\"3v*****z\\]m7(Fen$\"3$)*****pquF8$FF7$$ \"3O******zLUk*)Fen$\"3!******RT/FG$FF7$$\"3_*****fX30M#GFen$\" 3++++.@&fq$FF7$$!3(******pz1y$yFen$\"3.+++*oga$QFF7$$!37+++lUn1DFeo$\" 31+++u*pz&RFF7$$!3K+++FCLm\\Feo$\"3#******pkRE2%FF7$$!3=+++#\\Z$y\")Fe o$\"31+++g5`yTFF7$$!3.+++sQv37F-$\"3')******o@yuUFF7$$!35+++cH-e;F-$\" 3/+++eY&3O%FF7$$!3')*****R%pG^@F-$\"3;+++6fpOWFF7$$!35+++TpNuEF-$\"3U+ ++`Is-XFF7$$!3,+++GiK:KF-$\"37+++T=nfXFF7$$!3w*****pN'HlPF-$\"3$****** RP;%3YFF7$$!3#******\\#*[\"=VF-$\"3;+++.B$)\\YFF7$$!3E+++g\\$*p[F-$\"3 C+++i+s%o%FF7$$!3u*****H8W#=aF-$\"3P+++JUx8ZFF7$$!3!)*****HE(phfF-$\"3 7+++/)zvt%FF7$$!3#)*****4G$e*\\'F-$\"3i*****>4@mv%FF7$$!3C+++WhhJqF-$ \"35+++()\\HrZFF7$$!3s*****H)*yxb(F-$\"3m******)[C>y%FF7$$!39+++%*zAy! )F-$\"3K+++/3x)y%FF7$$!3o*****>COKf)F-$\"37++++N/#z%FF7$$!3b+++yT;.\"* F-$\"3M+++j(3>z%FF7$$!3]*****4b[%3'*F-$\"3S+++,g\\)y%FF7$$!30+++![g4, \"FF$\"3C+++-J!>y%FF7$$!31+++LSsg5FF$\"33+++!o+Ax%FF7$$!3)******zw2-6 \"FF$\"3#******RjN%fZFF7$$!3)******py)\\f6FF$\"3S+++zVjVZFF7$$!3-+++&[ .(37FF$\"38+++%>1[s%FF7$$!31+++dK&zD\"FF$\"3H+++ss%Hq%FF7$$!36+++p:T28 FF$\"3E+++-j/yYFF7$$!3!******4M#Gd8FF$\"3))*****\\!G4]YFF7$$!3.+++r!>y S\"FF$\"3k*****H%**3>YFF7$$!3%*******QJLf9FF$\"3e*****\\3u]e%FF7$$!3** *****\\t(>7:FF$\"3%)*****RgV\"[XFF7$$!3%******znSoc\"FF$\"3-+++'G)\\3X FF7$$!32+++*f:Pi\"FF$\"3#******f\\'[mWFF7$$!3#******H[RKo\"FF$\"3!**** ***ewkAWFF7$$!3'******RV#pX)G%FF7$$!3%****** H01#\\>FF$\"3;+++>%[_C%FF7$$!36+++B7V??FF$\"3%******f^)>/UFF7$$!3%**** ***[n&>4#FF$\"3%******4Ie^;%FF7$$!3%******f0DI;#FF$\"3L+++7L'z7%FF7$$! 38+++D&eIB#FF$\"3O+++k7G#4%FF7$$!39+++!)[m,BFF$\"3E+++p*4x0%FF7$$!3?++ +)z2'oBFF$\"3y******GX%Q-%FF7$$!36+++8wwLCFF$\"3++++*y8.*RFF7$$!3)**** **pX0r\\#FF$\"33+++a]zcRFF7$$!3+++++ajeDFF$\"3,+++Ti,BRFF7$$!3z*****fi /%=EFF$\"3/+++R>v))QFF7$$!3++++K,[wEFF$\"3=+++.s\"Q&QFF7$$!33+++(=SHt# FF$\"3/+++Z51=QFF7$$!3;+++-\"pyy#FF$\"3%)******o0O\"y$FF7$$!30++++UNTG FF$\"3!)*****4-;Ou$FF7$$!33+++QU[$*GFF$\"3#******\\!pu/PFF7$$!3A+++D([ V%HFF$\"31+++K!*okOFF7$$!3%)*****4yPS*HFF$\"3))******\\CRBOFF7$$!3#)** ***>DUE/$FF$\"3)******R:?3e$FF7$$!3y*****>*QD!4$FF$\"3/+++su%p`$FF7$$! 3:+++:a'p8$FF$\"3#)*****\\.i<\\$FF7$$!39+++h-(G=$FF$\"33+++WUEXMFF7$$! 3&)*****R$>1GKFF$\"37+++$3ouR$FF7$$!3%*******eFjsKFF$\"3%******p7-%[LF F7$$!3$*******\\>n;LFF$\"3'******4^5\")H$FF7$$!3;+++nHEgLFF$\"3!****** 4e`mC$FF7$$!31+++/-[.MFF$\"3)******p(y5%>$FF7$$!3')*****fA&QYMFF$\"3.+ ++]^cSJFF7$$!33+++EH-*[$FF$\"35++++,8'3$FF7$$!3!)*****46=9`$FF$\"32+++ pn\"4.$FF7$$!37+++'3tNd$FF$\"37+++zO/vHFF7$$!39+++jnY:OFF$\"3=+++J\"G' =HFF7$$!3%******Hfbql$FF$\"3#)*****pF!yhGFF7$$!3!)*****pLw#)p$FF$\"3*) ******)p(f/GFF7$$!3<+++V.0RPFF$\"3-+++f6;ZFFF7$$!3;+++n()GzPFF$\"3?+++ ^>`*o#FF7$$!3-+++R!)*)=QFF$\"3')*****\\,^)=*>G#FF7$$!3/+++ yAjrSFF$\"3(******>HbHA#FF7$$!3M+++1=`.TFF$\"39+++\"e-O;#FF7$$!3I+++Jw GMTFF$\"3@+++27*Q5#FF7$$!3-+++F[*Q;%FF$\"32+++s.yV?FF7$$!3S+++irN#>%FF $\"3/+++r@B$)>FF7$$!32+++mio>UFF$\"30+++XD@A>FF7$$!3O+++73!fC%FF$\"3#* *****\\7#pg=FF7$$!3))*****zpD5F%FF$\"33+++*)pk)z\"FF7$$!3O+++&>\"4&H%F F$\"3-+++\"Qfgt\"FF7$$!3'******z-K\"=VFF$\"3%******\\R=Hn\"FF7$$!3I+++ Yj=SVFF$\"3-+++61A4;FF7$$!35+++QXHhVFF$\"3&******Rkq\\a\"FF7$$!3E+++*) y\\\"Q%FF$\"3'******zb\"=![\"FF7$$!3K+++nq$3S%FF$\"34+++v]([T\"FF7$$!3 6+++n/N>WFF$\"3!******H\\\"3\\8FF7$$!3i*****H\\sqV%FF$\"33+++R#RGG\"FF 7$$!3K+++n=.aWFF$\"3/+++9S>;7FF7$$!3U+++,,DqWFF$\"3)******R](>\\6FF7$$ !3k*****fESd[%FF$\"3!******po0>3\"FF7$$!38+++)=10]%FF$\"3*)*****R$oP95 FF7$$!38+++y@a9XFF$\"3A+++_Kpm%*F-7$$!3*)*****zJLy_%FF$\"3#******pf%R) y)F-7$$!3$*******)Hc.a%FF$\"3a+++CUR4\")F-7$$!3O+++A23_XFF$\"3i*****po b,V(F-7$$!3#*******R2(Hc%FF$\"3I+++<82^nF-7$$!3%******p(o)Hd%FF$\"35++ +.TXsgF-7$$!3K++++z3#e%FF$\"3Y*****fMMXR&F-7$$!3=+++*eK-f%FF$\"31+++X1 Y^[&3YFF$\"3))******Q))o\"p#F-7$$!32+++bn]7YFF $\"3)******z!*e!=?F-7$$!3t*****\\;]`h%FF$\"3/+++^b-X8F-7$$!38+++-F13\"FF7$F_gv$!3 )******R](>\\6FF7$Fjfv$!3/+++9S>;7FF7$Fefv$!33+++R#RGG\"FF7$F`fv$!3!** ****H\\\"3\\8FF7$F[fv$!34+++v]([T\"FF7$Ffev$!3'******zb\"=![\"FF7$Faev $!3&******Rkq\\a\"FF7$F\\ev$!3-+++61A4;FF7$Fgdv$!3%******\\R=Hn\"FF7$F bdv$!3-+++\"Qfgt\"FF7$F]dv$!33+++*)pk)z\"FF7$Fhcv$!3#******\\7#pg=FF7$ Fccv$!30+++XD@A>FF7$F^cv$!3/+++r@B$)>FF7$Fibv$!32+++s.yV?FF7$Fdbv$!3@+ ++27*Q5#FF7$F_bv$!39+++\"e-O;#FF7$Fjav$!3(******>HbHA#FF7$Feav$!3!)*** **>)=*>G#FF7$F`av$!3!******RCb2M#FF7$F[av$!3%******f!pG*R#FF7$Ff`v$!3! ******p#\\idCFF7$Fa`v$!3=+++39!e^#FF7$F\\`v$!3-+++R'RQd#FF7$Fg_v$!3')* ****\\,^`*o#FF7$F]_v$!3-+++f6;ZFFF7$Fh^v$!3*)***** *)p(f/GFF7$Fc^v$!3#)*****pF!yhGFF7$F^^v$!3=+++J\"G'=HFF7$Fi]v$!37+++zO /vHFF7$Fd]v$!32+++pn\"4.$FF7$F_]v$!35++++,8'3$FF7$Fj\\v$!3.+++]^cSJFF7 $Fe\\v$!3)******p(y5%>$FF7$F`\\v$!3!******4e`mC$FF7$F[\\v$!3'******4^5 \")H$FF7$Ff[v$!3%******p7-%[LFF7$Fa[v$!37+++$3ouR$FF7$F\\[v$!33+++WUEX MFF7$Fgju$!3#)*****\\.i<\\$FF7$Fbju$!3/+++su%p`$FF7$F]ju$!3)******R:?3 e$FF7$Fhiu$!3))******\\CRBOFF7$Fciu$!31+++K!*okOFF7$F^iu$!3#******\\!p u/PFF7$Fihu$!3!)*****4-;Ou$FF7$Fdhu$!3%)******o0O\"y$FF7$F_hu$!3/+++Z5 1=QFF7$Fjgu$!3=+++.s\"Q&QFF7$Fegu$!3/+++R>v))QFF7$F`gu$!3,+++Ti,BRFF7$ F[gu$!33+++a]zcRFF7$Fffu$!3++++*y8.*RFF7$Fafu$!3y******GX%Q-%FF7$F\\fu $!3E+++p*4x0%FF7$Fgeu$!3O+++k7G#4%FF7$Fbeu$!3L+++7L'z7%FF7$F]eu$!3%*** ***4Ie^;%FF7$Fhdu$!3%******f^)>/UFF7$Fcdu$!3;+++>%[_C%FF7$F^du$!3%**** ***Q*)>)G%FF7$Ficu$!3s*******[0EL%FF7$Fdcu$!3c******=PsxVFF7$F_cu$!3!* ******ewkAWFF7$Fjbu$!3#******f\\'[mWFF7$Febu$!3-+++'G)\\3XFF7$F`bu$!3% )*****RgV\"[XFF7$F[bu$!3e*****\\3u]e%FF7$Ffau$!3k*****H%**3>YFF7$Faau$ !3))*****\\!G4]YFF7$F\\au$!3E+++-j/yYFF7$Fg`u$!3H+++ss%Hq%FF7$Fb`u$!38 +++%>1[s%FF7$F]`u$!3S+++zVjVZFF7$Fh_u$!3#******RjN%fZFF7$Fc_u$!33+++!o +Ax%FF7$F^_u$!3C+++-J!>y%FF7$Fi^u$!3S+++,g\\)y%FF7$Fd^u$!3M+++j(3>z%FF 7$F_^u$!37++++N/#z%FF7$Fj]u$!3K+++/3x)y%FF7$Fe]u$!3m******)[C>y%FF7$F` ]u$!35+++()\\HrZFF7$F[]u$!3i*****>4@mv%FF7$Ff\\u$!37+++/)zvt%FF7$Fa\\u $!3P+++JUx8ZFF7$F\\\\u$!3C+++i+s%o%FF7$Fg[u$!3;+++.B$)\\YFF7$Fb[u$!3$* *****RP;%3YFF7$F][u$!37+++T=nfXFF7$Fhjt$!3U+++`Is-XFF7$Fcjt$!3;+++6fpO WFF7$F^jt$!3/+++eY&3O%FF7$Fiit$!3')******o@yuUFF7$Fdit$!31+++g5`yTFF7$ F_it$!3#******pkRE2%FF7$Fjht$!31+++u*pz&RFF7$Feht$!3.+++*oga$QFF7$F`ht $!3++++.@&fq$FF7$F[ht$!3;+++hw;qNFF7$Ffgt$!3'******\\a:)GMFF7$Fagt$!3! ******RT/FG$FF7$F\\gt$!3$)*****pquF8$FF7$Fgft$!3#******HF>+)HFF7$Fbft$ !3,+++%>^`#GFF7$F]ft$!3;+++a9]pEFF7$Fhet$!3!)*****RE')H^#FF7$Fcet$!3#* *****>iJhN#FF7$F^et$!3!******R&H7*>#FF7$Fidt$!3A+++p[0U?FF7$Fddt$!3)** ****4tp\\)=FF7$F_dt$!3(*******=K)ys\"FF7$$\"3_+++'H\\bc)FC$!3/+++W#*zq :FF7$Fect$!30+++&fK)ys\"Fg[nFggx7%F^[y7$$ \"+;ev)y&F\\\\n$\"+J(p\\)=Fg[nFggx7%Fd[y7$$\"+-Yfb8Fc\\n$\"+p[0U?Fg[nF ggx7%Fj[y7$$\"+mB_bHFc\\n$\"+aH7*>#Fg[nFggx7%F`\\y7$$\"+I(Hr+'Fc\\n$\" +A;8cBFg[nFggx7%Ff\\y7$$\"+k0uR6F`]n$\"+ki)H^#Fg[nFggx7%F\\]y7$$\"+Lgw =?F`]n$\"+a9]pEFg[nFggx7%Fb]y7$$\"+x%G2L$F`]n$\"+%>^`#GFg[nFggx7%Fh]y7 $$\"+4)Q)*3&F`]n$\"+t#>+)HFg[nFggx7%F^^y7$$\"+)\\]m7(F`]n$\"+2ZxKJFg[n Fggx7%Fd^y7$$\"+!QBW'*)F`]n$\"+9Wq#G$Fg[nFggx7%Fj^y7$$\"+cu@K(*F`]n$\" +Xb\")GMFg[nFggx7%F`_y7$$\"+..n!=)F`]n$\"+hw;qNFg[nFggx7%Ff_y7$$\"+#30 M#GF`]n$\"+.@&fq$Fg[nFggx7%F\\`y7$$!+(z1y$yF`]n$\"+*oga$QFg[nFggx7%Fb` y7$$!+lUn1DFc^n$\"+u*pz&RFg[nFggx7%Fh`y7$$!+FCLm\\Fc^n$\"+Z'RE2%Fg[nFg gx7%F^ay7$$!+#\\Z$y\")Fc^n$\"+g5`yTFg[nFggx7%Fday7$$!+sQv37Fiim$\"+p@y uUFg[nFggx7%Fjay7$$!+cH-e;Fiim$\"+eY&3O%Fg[nFggx7%F`by7$$!+WpG^@Fiim$ \"+6fpOWFg[nFggx7%Ffby7$$!+TpNuEFiim$\"+`Is-XFg[nFggx7%F\\cy7$$!+GiK:K Fiim$\"+T=nfXFg[nFggx7%Fbcy7$$!+djHlPFiim$\"+ujT3YFg[nFggx7%Fhcy7$$!+D *[\"=VFiim$\"+.B$)\\YFg[nFggx7%F^dy7$$!+g\\$*p[Fiim$\"+i+s%o%Fg[nFggx7 %Fddy7$$!+LTC=aFiim$\"+JUx8ZFg[nFggx7%Fjdy7$$!+jsphfFiim$\"+/)zvt%Fg[n Fggx7%F`ey7$$!+\"G$e*\\'Fiim$\"+#4@mv%Fg[nFggx7%Ffey7$$!+WhhJqFiim$\"+ ()\\HrZFg[nFggx7%F\\fy7$$!+$)*yxb(Fiim$\"+*[C>y%Fg[nFggx7%Fbfy7$$!+%*z Ay!)Fiim$\"+/3x)y%Fg[nFggx7%Fhfy7$$!+UiB$f)Fiim$\"++N/#z%Fg[nFggx7%F^g y7$$!+yT;.\"*Fiim$\"+j(3>z%Fg[nFggx7%Fdgy7$$!+^&[%3'*Fiim$\"+,g\\)y%Fg [nFggx7%Fjgy7$$!+![g4,\"Fg[n$\"+-J!>y%Fg[nFggx7%F`hy7$$!+LSsg5Fg[n$\"+ !o+Ax%Fg[nFggx7%Ffhy7$$!+ox?56Fg[n$\"+McVfZFg[nFggx7%F\\iy7$$!+(y)\\f6 Fg[n$\"+zVjVZFg[nFggx7%Fbiy7$$!+&[.(37Fg[n$\"+%>1[s%Fg[nFggx7%Fhiy7$$! +dK&zD\"Fg[n$\"+ss%Hq%Fg[nFggx7%F^jy7$$!+p:T28Fg[n$\"+-j/yYFg[nFggx7%F djy7$$!+TBGd8Fg[n$\"+0G4]YFg[nFggx7%Fjjy7$$!+r!>yS\"Fg[n$\"+V**3>YFg[n Fggx7%F`[z7$$!+RJLf9Fg[n$\"+&3u]e%Fg[nFggx7%Ff[z7$$!+Nx>7:Fg[n$\"+/O9[ XFg[nFggx7%F\\\\z7$$!+y1%oc\"Fg[n$\"+'G)\\3XFg[nFggx7%Fb\\z7$$!+*f:Pi \"Fg[n$\"+'\\'[mWFg[nFggx7%Fh\\z7$$!+$[RKo\"Fg[n$\"+fwkAWFg[nFggx7%F^] z7$$!+MCpXPsxVFg[nFggx7%Fd]z7$$!+x(*46=Fg[n$\"+!\\0EL%Fg[nFg gx7%Fj]z7$$!+%e\\\"z=Fg[n$\"+R*)>)G%Fg[nFggx7%F`^z7$$!+`g?\\>Fg[n$\"+> %[_C%Fg[nFggx7%Ff^z7$$!+B7V??Fg[n$\"+;&)>/UFg[nFggx7%F\\_z7$$!+\\n&>4# Fg[n$\"+,$e^;%Fg[nFggx7%Fb_z7$$!+c]-j@Fg[n$\"+7L'z7%Fg[nFggx7%Fh_z7$$! +D&eIB#Fg[n$\"+k7G#4%Fg[nFggx7%F^`z7$$!+!)[m,BFg[n$\"+p*4x0%Fg[nFggx7% Fd`z7$$!+)z2'oBFg[n$\"+HX%Q-%Fg[nFggx7%Fj`z7$$!+8wwLCFg[n$\"+*y8.*RFg[ nFggx7%F`az7$$!+da5(\\#Fg[n$\"+a]zcRFg[nFggx7%Ffaz7$$!++ajeDFg[n$\"+Ti ,BRFg[nFggx7%F\\bz7$$!+EYS=EFg[n$\"+R>v))QFg[nFggx7%Fbbz7$$!+K,[wEFg[n $\"+.s\"Q&QFg[nFggx7%Fhbz7$$!+(=SHt#Fg[n$\"+Z51=QFg[nFggx7%F^cz7$$!+- \"pyy#Fg[n$\"+p0O\"y$Fg[nFggx7%Fdcz7$$!++UNTGFg[n$\"+@ghVPFg[nFggx7%Fj cz7$$!+QU[$*GFg[n$\"+0pu/PFg[nFggx7%F`dz7$$!+D([V%HFg[n$\"+K!*okOFg[nF ggx7%Ffdz7$$!+\"yPS*HFg[n$\"+]CRBOFg[nFggx7%F\\ez7$$!+_AkUIFg[n$\"+a,# 3e$Fg[nFggx7%Fbez7$$!+#*QD!4$Fg[n$\"+su%p`$Fg[nFggx7%Fhez7$$!+:a'p8$Fg [n$\"+N?w\"\\$Fg[nFggx7%F^fz7$$!+h-(G=$Fg[n$\"+WUEXMFg[nFggx7%Fdfz7$$! +M>1GKFg[n$\"+$3ouR$Fg[nFggx7%Fjfz7$$!+fFjsKFg[n$\"+F@S[LFg[nFggx7%F`g z7$$!+]>n;LFg[n$\"+606)H$Fg[nFggx7%Ffgz7$$!+nHEgLFg[n$\"+\"e`mC$Fg[nFg gx7%F\\hz7$$!+/-[.MFg[n$\"+xy5%>$Fg[nFggx7%Fbhz7$$!+E_QYMFg[n$\"+]^cSJ Fg[nFggx7%Fhhz7$$!+EH-*[$Fg[n$\"++,8'3$Fg[nFggx7%F^iz7$$!+6\"=9`$Fg[n$ \"+pn\"4.$Fg[nFggx7%Fdiz7$$!+'3tNd$Fg[n$\"+zO/vHFg[nFggx7%Fjiz7$$!+jnY :OFg[n$\"+J\"G'=HFg[nFggx7%F`jz7$$!+$fbql$Fg[n$\"+x-yhGFg[nFggx7%Ffjz7 $$!+PjF)p$Fg[n$\"+*p(f/GFg[nFggx7%F\\[[l7$$!+V.0RPFg[n$\"+f6;ZFFg[nFgg x7%Fb[[l7$$!+n()GzPFg[n$\"+^>`*o#Fg[nFggx7%Fh[[l7$$!+R!)*)=QFg[n$\"+:5 vJEFg[nFggx7%F^\\[l7$$!+x\\ydQFg[n$\"+R'RQd#Fg[nFggx7%Fd\\[l7$$!+!3he* QFg[n$\"+39!e^#Fg[nFggx7%Fj\\[l7$$!+ud/LRFg[n$\"+F\\idCFg[nFggx7%F`][l 7$$!+\"eo#pRFg[n$\"+1pG*R#Fg[nFggx7%Ff][l7$$!+9.Z/SFg[n$\"+W_vSBFg[nFg gx7%F\\^[l7$$!+4NgQSFg[n$\"+#)=*>G#Fg[nFggx7%Fb^[l7$$!+yAjrSFg[n$\"+#H bHA#Fg[nFggx7%Fh^[l7$$!+1=`.TFg[n$\"+\"e-O;#Fg[nFggx7%F^_[l7$$!+JwGMTF g[n$\"+27*Q5#Fg[nFggx7%Fd_[l7$$!+F[*Q;%Fg[n$\"+s.yV?Fg[nFggx7%Fj_[l7$$ !+irN#>%Fg[n$\"+r@B$)>Fg[nFggx7%F``[l7$$!+mio>UFg[n$\"+XD@A>Fg[nFggx7% Ff`[l7$$!+73!fC%Fg[n$\"+D@pg=Fg[nFggx7%F\\a[l7$$!+)pD5F%Fg[n$\"+*)pk)z \"Fg[nFggx7%Fba[l7$$!+&>\"4&H%Fg[n$\"+\"Qfgt\"Fg[nFggx7%Fha[l7$$!+G?8= VFg[n$\"+&R=Hn\"Fg[nFggx7%F^b[l7$$!+Yj=SVFg[n$\"+61A4;Fg[nFggx7%Fdb[l7 $$!+QXHhVFg[n$\"+W1(\\a\"Fg[nFggx7%Fjb[l7$$!+*)y\\\"Q%Fg[n$\"+e:=![\"F g[nFggx7%F`c[l7$$!+nq$3S%Fg[n$\"+v]([T\"Fg[nFggx7%Ffc[l7$$!+n/N>WFg[n$ \"+$\\\"3\\8Fg[nFggx7%F\\d[l7$$!+$\\sqV%Fg[n$\"+R#RGG\"Fg[nFggx7%Fbd[l 7$$!+n=.aWFg[n$\"+9S>;7Fg[nFggx7%Fhd[l7$$!+,,DqWFg[n$\"+/v>\\6Fg[nFggx 7%F^e[l7$$!+m-u&[%Fg[n$\"+(o0>3\"Fg[nFggx7%Fde[l7$$!+)=10]%Fg[n$\"+MoP 95Fg[nFggx7%Fje[l7$$!+y@a9XFg[n$\"+_Kpm%*FiimFggx7%F`f[l7$$!+=L$y_%Fg[ n$\"+(f%R)y)FiimFggx7%Fff[l7$$!+*Hc.a%Fg[n$\"+CUR4\")FiimFggx7%F\\g[l7 $$!+A23_XFg[n$\"+(ob,V(FiimFggx7%Fbg[l7$$!+S2(Hc%Fg[n$\"+<82^nFiimFggx 7%Fhg[l7$$!+xo)Hd%Fg[n$\"+.TXsgFiimFggx7%F^h[l7$$!++z3#e%Fg[n$\"+YV`%R &FiimFggx7%Fdh[l7$$!+*eK-f%Fg[n$\"+X1Y3\"Fg[nFggx7% Fd_\\l7$F_e[l$!+/v>\\6Fg[nFggx7%Fh_\\l7$Fid[l$!+9S>;7Fg[nFggx7%F\\`\\l 7$Fcd[l$!+R#RGG\"Fg[nFggx7%F``\\l7$F]d[l$!+$\\\"3\\8Fg[nFggx7%Fd`\\l7$ Fgc[l$!+v]([T\"Fg[nFggx7%Fh`\\l7$Fac[l$!+e:=![\"Fg[nFggx7%F\\a\\l7$F[c [l$!+W1(\\a\"Fg[nFggx7%F`a\\l7$Feb[l$!+61A4;Fg[nFggx7%Fda\\l7$F_b[l$!+ &R=Hn\"Fg[nFggx7%Fha\\l7$Fia[l$!+\"Qfgt\"Fg[nFggx7%F\\b\\l7$Fca[l$!+*) pk)z\"Fg[nFggx7%F`b\\l7$F]a[l$!+D@pg=Fg[nFggx7%Fdb\\l7$Fg`[l$!+XD@A>Fg [nFggx7%Fhb\\l7$Fa`[l$!+r@B$)>Fg[nFggx7%F\\c\\l7$F[`[l$!+s.yV?Fg[nFggx 7%F`c\\l7$Fe_[l$!+27*Q5#Fg[nFggx7%Fdc\\l7$F__[l$!+\"e-O;#Fg[nFggx7%Fhc \\l7$Fi^[l$!+#HbHA#Fg[nFggx7%F\\d\\l7$Fc^[l$!+#)=*>G#Fg[nFggx7%F`d\\l7 $F]^[l$!+W_vSBFg[nFggx7%Fdd\\l7$Fg][l$!+1pG*R#Fg[nFggx7%Fhd\\l7$Fa][l$ !+F\\idCFg[nFggx7%F\\e\\l7$F[][l$!+39!e^#Fg[nFggx7%F`e\\l7$Fe\\[l$!+R' RQd#Fg[nFggx7%Fde\\l7$F_\\[l$!+:5vJEFg[nFggx7%Fhe\\l7$Fi[[l$!+^>`*o#Fg [nFggx7%F\\f\\l7$Fc[[l$!+f6;ZFFg[nFggx7%F`f\\l7$F][[l$!+*p(f/GFg[nFggx 7%Fdf\\l7$Fgjz$!+x-yhGFg[nFggx7%Fhf\\l7$Fajz$!+J\"G'=HFg[nFggx7%F\\g\\ l7$F[jz$!+zO/vHFg[nFggx7%F`g\\l7$Feiz$!+pn\"4.$Fg[nFggx7%Fdg\\l7$F_iz$ !++,8'3$Fg[nFggx7%Fhg\\l7$Fihz$!+]^cSJFg[nFggx7%F\\h\\l7$Fchz$!+xy5%>$ Fg[nFggx7%F`h\\l7$F]hz$!+\"e`mC$Fg[nFggx7%Fdh\\l7$Fggz$!+606)H$Fg[nFgg x7%Fhh\\l7$Fagz$!+F@S[LFg[nFggx7%F\\i\\l7$F[gz$!+$3ouR$Fg[nFggx7%F`i\\ l7$Fefz$!+WUEXMFg[nFggx7%Fdi\\l7$F_fz$!+N?w\"\\$Fg[nFggx7%Fhi\\l7$Fiez $!+su%p`$Fg[nFggx7%F\\j\\l7$Fcez$!+a,#3e$Fg[nFggx7%F`j\\l7$F]ez$!+]CRB OFg[nFggx7%Fdj\\l7$Fgdz$!+K!*okOFg[nFggx7%Fhj\\l7$Fadz$!+0pu/PFg[nFggx 7%F\\[]l7$F[dz$!+@ghVPFg[nFggx7%F`[]l7$Fecz$!+p0O\"y$Fg[nFggx7%Fd[]l7$ F_cz$!+Z51=QFg[nFggx7%Fh[]l7$Fibz$!+.s\"Q&QFg[nFggx7%F\\\\]l7$Fcbz$!+R >v))QFg[nFggx7%F`\\]l7$F]bz$!+Ti,BRFg[nFggx7%Fd\\]l7$Fgaz$!+a]zcRFg[nF ggx7%Fh\\]l7$Faaz$!+*y8.*RFg[nFggx7%F\\]]l7$F[az$!+HX%Q-%Fg[nFggx7%F`] ]l7$Fe`z$!+p*4x0%Fg[nFggx7%Fd]]l7$F_`z$!+k7G#4%Fg[nFggx7%Fh]]l7$Fi_z$! +7L'z7%Fg[nFggx7%F\\^]l7$Fc_z$!+,$e^;%Fg[nFggx7%F`^]l7$F]_z$!+;&)>/UFg [nFggx7%Fd^]l7$Fg^z$!+>%[_C%Fg[nFggx7%Fh^]l7$Fa^z$!+R*)>)G%Fg[nFggx7%F \\_]l7$F[^z$!+!\\0EL%Fg[nFggx7%F`_]l7$Fe]z$!+>PsxVFg[nFggx7%Fd_]l7$F_] z$!+fwkAWFg[nFggx7%Fh_]l7$Fi\\z$!+'\\'[mWFg[nFggx7%F\\`]l7$Fc\\z$!+'G) \\3XFg[nFggx7%F``]l7$F]\\z$!+/O9[XFg[nFggx7%Fd`]l7$Fg[z$!+&3u]e%Fg[nFg gx7%Fh`]l7$Fa[z$!+V**3>YFg[nFggx7%F\\a]l7$F[[z$!+0G4]YFg[nFggx7%F`a]l7 $Fejy$!+-j/yYFg[nFggx7%Fda]l7$F_jy$!+ss%Hq%Fg[nFggx7%Fha]l7$Fiiy$!+%>1 [s%Fg[nFggx7%F\\b]l7$Fciy$!+zVjVZFg[nFggx7%F`b]l7$F]iy$!+McVfZFg[nFggx 7%Fdb]l7$Fghy$!+!o+Ax%Fg[nFggx7%Fhb]l7$Fahy$!+-J!>y%Fg[nFggx7%F\\c]l7$ F[hy$!+,g\\)y%Fg[nFggx7%F`c]l7$Fegy$!+j(3>z%Fg[nFggx7%Fdc]l7$F_gy$!++N /#z%Fg[nFggx7%Fhc]l7$Fify$!+/3x)y%Fg[nFggx7%F\\d]l7$Fcfy$!+*[C>y%Fg[nF ggx7%F`d]l7$F]fy$!+()\\HrZFg[nFggx7%Fdd]l7$Fgey$!+#4@mv%Fg[nFggx7%Fhd] l7$Faey$!+/)zvt%Fg[nFggx7%F\\e]l7$F[ey$!+JUx8ZFg[nFggx7%F`e]l7$Fedy$!+ i+s%o%Fg[nFggx7%Fde]l7$F_dy$!+.B$)\\YFg[nFggx7%Fhe]l7$Ficy$!+ujT3YFg[n Fggx7%F\\f]l7$Fccy$!+T=nfXFg[nFggx7%F`f]l7$F]cy$!+`Is-XFg[nFggx7%Fdf]l 7$Fgby$!+6fpOWFg[nFggx7%Fhf]l7$Faby$!+eY&3O%Fg[nFggx7%F\\g]l7$F[by$!+p @yuUFg[nFggx7%F`g]l7$Feay$!+g5`yTFg[nFggx7%Fdg]l7$F_ay$!+Z'RE2%Fg[nFgg x7%Fhg]l7$Fi`y$!+u*pz&RFg[nFggx7%F\\h]l7$Fc`y$!+*oga$QFg[nFggx7%F`h]l7 $F]`y$!+.@&fq$Fg[nFggx7%Fdh]l7$Fg_y$!+hw;qNFg[nFggx7%Fhh]l7$Fa_y$!+Xb \")GMFg[nFggx7%F\\i]l7$F[_y$!+9Wq#G$Fg[nFggx7%F`i]l7$Fe^y$!+2ZxKJFg[nF ggx7%Fdi]l7$F_^y$!+t#>+)HFg[nFggx7%Fhi]l7$Fi]y$!+%>^`#GFg[nFggx7%F\\j] l7$Fc]y$!+a9]pEFg[nFggx7%F`j]l7$F]]y$!+ki)H^#Fg[nFggx7%Fdj]l7$Fg\\y$!+ A;8cBFg[nFggx7%Fhj]l7$Fa\\y$!+aH7*>#Fg[nFggx7%F\\[^l7$F[\\y$!+p[0U?Fg[ nFggx7%F`[^l7$Fe[y$!+J(p\\)=Fg[nFggx7%Fd[^l7$F_[y$!+>K)ys\"Fg[nFggx7%F h[^l7$$\"+'H\\bc)Fd[n$!+W#*zq:Fg[nFggx7%F\\\\^l7$Fcjx$!+&fn<2bFF7$$\"3%******z[[!R:FF$\"3 5+++h_5AbFF7$$\"3/+++@QQK:FF$\"3<+++OH-ObFF7$$\"3.+++#)*yS_\"FF$\"3q** ****Hfx[bFF7$$\"3(******zA+V^\"FF$\"3I+++'oG-c&FF7$$\"3/+++[D@.:FF$\"3 -+++_.EqbFF7$$\"3++++>/)4\\\"FF$\"3-+++:=wybFF7$$\"31+++')*oxZ\"FF$\"3 %)*****zALce&FF7$$\"3/+++Tcuj9FF$\"3A+++]>y!f&FF7$$\"3'******z/#3\\9FF $\"37+++!*37%f&FF7$$\"3)******fncRV\"FF$\"3!******pDnbf&FF7$$\"3/+++Rz b=9FF$\"3U+++(yT]f&FF7$$\"3/+++$4)3.9FF$\"3m*****HdoCf&FF7$$\"35+++^!o xQ\"FF$\"3U+++gdx(e&FF7$$\"3#******fKVGP\"FF$\"3)******4\\(*3e&FF7$$\" 3/+++l7fe8FF$\"3;+++vxxrbFF7$$\"33+++s'H`M\"FF$\"3g*****pHx.c&FF7$$\"3 /+++clUL8FF$\"33+++`ZoYbFF7$$\"3%*******G+JB8FF$\"3R+++t\\tIbFF7$$\"3 \"*******4eZ:8FF$\"3d*******)oj7bFF7$$\"3'*******zn[58FF$\"31+++?Xh#\\ &FF7$$\"3.+++sU&*38FF$\"3A+++PB1raFF7$$\"3'******R)[[68FF$\"3G+++L$4'[ aFF7$$\"35+++j\\d=8FF$\"3C+++:.,Y\"FF$ \"3'******za$*3O&FF7$$\"3-+++ZR_![\"FF$\"3C+++:=1p`FF7$$\"3!******>U/% )\\\"FF$\"3/+++VU>!Q&FF7$$\"3*******fME`YU&FF 7$$\"36+++#f#)=a\"FF$\"3I+++t]LTaFFFc_^lF]gx-Fcim6L7%7$$\"+Zw$fa\"Fg[n $\"+gSBeaFg[n7$$\"+yz^Z:Fg[n$\"++,-vaFg[n7$$\"$V\"Faim$\"$Z&Faim7%Fc\\ _l7$$\"+[ZyY:Fg[n$\"+_OT\"\\&Fg[nFh\\_l7%F^]_l7$$\"+h^!Ra\"Fg[n$\"+>n< 2bFg[nFh\\_l7%Fd]_l7$$\"+)[[!R:Fg[n$\"+h_5AbFg[nFh\\_l7%Fj]_l7$$\"+@QQ K:Fg[n$\"+OH-ObFg[nFh\\_l7%F`^_l7$$\"+#)*yS_\"Fg[n$\"+Ifx[bFg[nFh\\_l7 %Ff^_l7$$\"+G-I9:Fg[n$\"+'oG-c&Fg[nFh\\_l7%F\\__l7$$\"+[D@.:Fg[n$\"+_. EqbFg[nFh\\_l7%Fb__l7$$\"+>/)4\\\"Fg[n$\"+:=wybFg[nFh\\_l7%Fh__l7$$\"+ ')*oxZ\"Fg[n$\"+GKj&e&Fg[nFh\\_l7%F^`_l7$$\"+Tcuj9Fg[n$\"+]>y!f&Fg[nFh \\_l7%Fd`_l7$$\"+[?3\\9Fg[n$\"+!*37%f&Fg[nFh\\_l7%Fj`_l7$$\"+wm&RV\"Fg [n$\"+dsc&f&Fg[nFh\\_l7%F`a_l7$$\"+Rzb=9Fg[n$\"+(yT]f&Fg[nFh\\_l7%Ffa_ l7$$\"+$4)3.9Fg[n$\"+t&oCf&Fg[nFh\\_l7%F\\b_l7$$\"+^!oxQ\"Fg[n$\"+gdx( e&Fg[nFh\\_l7%Fbb_l7$$\"+EL%GP\"Fg[n$\"+\"\\(*3e&Fg[nFh\\_l7%Fhb_l7$$ \"+l7fe8Fg[n$\"+vxxrbFg[nFh\\_l7%F^c_l7$$\"+s'H`M\"Fg[n$\"+(Hx.c&Fg[nF h\\_l7%Fdc_l7$$\"+clUL8Fg[n$\"+`ZoYbFg[nFh\\_l7%Fjc_l7$$\"+H+JB8Fg[n$ \"+t\\tIbFg[nFh\\_l7%F`d_l7$$\"+5eZ:8Fg[n$\"+!*oj7bFg[nFh\\_l7%Ffd_l7$ $\"+!y'[58Fg[n$\"+?Xh#\\&Fg[nFh\\_l7%F\\e_l7$$\"+sU&*38Fg[n$\"+PB1raFg [nFh\\_l7%Fbe_l7$$\"+%)[[68Fg[n$\"+L$4'[aFg[nFh\\_l7%Fhe_l7$$\"+j\\d=8 Fg[n$\"+:.)z8O&Fg[nFh\\_l7%Ffg_l 7$$\"+^H!RT\"Fg[n$\"+F.Vc`Fg[nFh\\_l7%F\\h_l7$$\"+9#ewV\"Fg[n$\"+W[Wc` Fg[nFh\\_l7%Fbh_l7$$\"+R!>,Y\"Fg[n$\"+[N*3O&Fg[nFh\\_l7%Fhh_l7$$\"+ZR_ ![\"Fg[n$\"+:=1p`Fg[nFh\\_l7%F^i_l7$$\"+AWS)\\\"Fg[n$\"+VU>!Q&Fg[nFh\\ _l7%Fdi_l7$$\"+Yt^8:Fg[n$\"+5\"yNR&Fg[nFh\\_l7%Fji_l7$$\"+'exd_\"Fg[n$ \"+)yz&3aFg[nFh\\_l7%F`j_l7$$\"+/V?N:Fg[n$\"+iKlCaFg[nFh\\_l7%Ffj_l7$$ \"+#f#)=a\"Fg[n$\"+t]LTaFg[nFh\\_l7%F\\[`lF^\\_lFh\\_lFj^^lFfbq-F$6$7K 7$Fd_^l$!3m******fSBeaFF7$Fg[_l$!3I+++t]LTaFF7$Fb[_l$!3k*****>E`YU&FF7 $F][_l$!3%******zyz&3aFF7$Fhj^l$!3)*******4\"yNR&FF7$Fcj^l$!3/+++VU>!Q &FF7$F^j^l$!3C+++:=1p`FF7$Fii^l$!3'******za$*3O&FF7$Fdi^l$!3K+++W[Wc`F F7$F_i^l$!3)******pKIkN&FF7$Fjh^l$!3s******=)z8O&FF7$Feh^l$!3?+++77Xr` FF7$F`h^l$!3=+++W\\F'Q&FF7$F[h^l$!3A+++S*R\\S&FF7$Ffg^l$!3C+++:.y!f&FF7$Ffb^ l$!3%)*****zALce&FF7$Fab^l$!3-+++:=wybFF7$F\\b^l$!3-+++_.EqbFF7$Fga^l$ !3I+++'oG-c&FF7$Fba^l$!3q******Hfx[bFF7$F]a^l$!3<+++OH-ObFF7$Fh`^l$!35 +++h_5AbFF7$Fc`^l$!3G+++>n<2bFF7$F^`^l$!3K+++_OT\"\\&FF7$Fi_^l$!31++++ ,-vaFFFe[`lF]gx-Fcim6L7%7$F_\\_l$!+gSBeaFg[n7$F][`l$!+t]LTaFg[n7$Fi\\_ l$!$Z&Faim7%Fcc`l7$Fgj_l$!+iKlCaFg[nFfc`l7%Fjc`l7$Faj_l$!+)yz&3aFg[nFf c`l7%F^d`l7$F[j_l$!+5\"yNR&Fg[nFfc`l7%Fbd`l7$Fei_l$!+VU>!Q&Fg[nFfc`l7% Ffd`l7$F_i_l$!+:=1p`Fg[nFfc`l7%Fjd`l7$Fih_l$!+[N*3O&Fg[nFfc`l7%F^e`l7$ Fch_l$!+W[Wc`Fg[nFfc`l7%Fbe`l7$F]h_l$!+F.Vc`Fg[nFfc`l7%Ffe`l7$Fgg_l$!+ >)z8O&Fg[nFfc`l7%Fje`l7$Fag_l$!+77Xr`Fg[nFfc`l7%F^f`l7$F[g_l$!+W\\F'Q& Fg[nFfc`l7%Fbf`l7$Fef_l$!+S*R\\S&Fg[nFfc`l7%Fff`l7$F_f_l$!+:.y!f&Fg[nFfc`l7%Ffj `l7$F_`_l$!+GKj&e&Fg[nFfc`l7%Fjj`l7$Fi__l$!+:=wybFg[nFfc`l7%F^[al7$Fc_ _l$!+_.EqbFg[nFfc`l7%Fb[al7$F]__l$!+'oG-c&Fg[nFfc`l7%Ff[al7$Fg^_l$!+If x[bFg[nFfc`l7%Fj[al7$Fa^_l$!+OH-ObFg[nFfc`l7%F^\\al7$F[^_l$!+h_5AbFg[n Ffc`l7%Fb\\al7$Fe]_l$!+>n<2bFg[nFfc`l7%Ff\\al7$F_]_l$!+_OT\"\\&Fg[nFfc `l7%Fj\\al7$Fd\\_l$!++,-vaFg[nFfc`l7%F^]alF`c`lFfc`lFj^^lFfbq-F$6%7$7$ $!3o*************Q&FFF(7$$\"3%**************=#FFF(-%'COLOURG6&F^imF)F) F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>'FF7$F($\"3Q++++++!>'FFF[ ^alF^^al-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HE LVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fd_al6#%(DEFAULTG-%%VIEW G6$;$!$R&Faim$\"$>#Faim;$!$>'Faim$\"$>'Faim" 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 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined sch eme" }}{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 4124 "ee := \{c[2]=1/100,\nc[3]=4/27,\n c[4]=2/9,\nc[5]=11417/21528,\nc[6]=49/78,\nc[7]=47/283,\nc[8]=30/73,\n c[9]=37/46,\nc[10]=32/33,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/100,\na[3,1] =-692/729,\na[3,2]=800/729,\na[4,1]=1/18,\na[4,2]=0,\na[4,3]=1/6,\na[5 ,1]=451814999377/985407860736,\na[5,2]=0,\na[5,3]=-552805397249/328469 286912,\na[5,4]=96066394193/54744881152,\na[6,1]=131173/1890096,\na[6, 2]=0,\na[6,3]=0,\na[6,4]=5764801/17935632,\na[6,5]=4769464/20091357,\n a[7,1]=1176539682457387/14353367828755172,\na[7,2]=0,\na[7,3]=0,\na[7, 4]=17717342174781/163306752068660,\na[7,5]=-528062841353883264/1033801 3178660912633,\na[7,6]=1679528554701504/62911488820241285,\na[8,1]=110 903909289926737793/4320958290769994040132,\na[8,2]=0,\na[8,3]=0,\na[8, 4]=-11607238985710021183/180261023052662924020,\na[8,5]=83329879774618 79955626624/57056453830329950469093005,\na[8,6]=-397914630242458427722 /6312988060727977701695,\na[8,7]=11/30,\na[9,1]=-199116158962708264981 792850380273725319682574083/212209570379395954037673565209665038844452 4173920,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-902399027478634049318468354365 831042971264299/376720159407433656819366352813041281670901920,\na[9,5] =-1932841049216489668013274675445061103749096456/122503447422553804621 2066937516775761976271555,\na[9,6]=61212379964740003623393766579069964 177763490780547/67292203490497755527349011795764425282435991108860,\na [9,7]=1767352010455801522759919865162797424677/85363129890431148674570 5433404624991760,\na[9,8]=48803138404357581919875739936453/25803619795 436169827228351014320,\na[10,1]=14681288291210349822347771070232766916 0817645499822332/\n 3471363275502643502810190506910949216726536 36486171345,\na[10,2]=0,\na[10,3]=0,\na[10,4]=838090727939994034141552 0095001429689250499016988/16655291405980027768151637904621488308989061 31435,\na[10,5]=645991781238548549725468179173866591806130183669466521 6/\n 2108701812611736155789874398208067694599974390240225435,\n a[10,6]=-329623101936975913713940238539594299201957668540750512/\n \+ 216369193273123311533898096852817531450154842258383315,\na[10,7]= -239812294208127777232571199557076199094738401624/62716593235272943126 739970926235455172660665115,\na[10,8]=-1920990949078220462934578748765 4324166296/6932257046340021610507379175381008191035,\na[10,9]=21064239 9729505665392/369807601102457012649,\na[11,1]=221939233494467949726864 332982580511239150575217/255387155540276753091910917079234490443699161 270,\na[11,2]=0,\na[11,3]=0,\na[11,4]=18116326319248729754632413293825 7205801916729/17154547141864477699072019121010975163450940,\na[11,5]=5 035138254118716342864368937874683246617259598976/775682801056824859328 708920179990438628643868105,\na[11,6]=-7722480175483523961899743026009 0636543172193252536784/\n 202652428901707058125367179822844135 27106230272155465,\na[11,7]=-20417460493227662758342875761671647513570 72803/242398642004974788232613894826395428361025315,\na[11,8]=-6988057 9855847721894868113496992143638/12013145933881997013005874021379315455 ,\na[11,9]=667711875409843102380/547572557872664725429,\na[11,10]=-180 8859196027797903/20206200178110899692,\na[12,1]=2354923/46428480,\na[1 2,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=31816481086962/14 3829479014165,\na[12,7]=7273629826007016071/29687570322557889840,\na[1 2,8]=131953911267751793/602866646471570040,\na[12,9]=29769129062566/20 5676274838545,\na[12,10]=30049446213369/166412182583360,\na[12,11]=-67 996657/1112423760,\n\nb[1]=2354923/46428480,\nb[2]=0,\nb[3]=0,\nb[4]=0 ,\nb[5]=0,\nb[6]=31816481086962/143829479014165,\nb[7]=727362982600701 6071/29687570322557889840,\nb[8]=131953911267751793/602866646471570040 ,\nb[9]=29769129062566/205676274838545,\nb[10]=30049446213369/16641218 2583360,\nb[11]=-67996657/1112423760,\n\n`b*`[1]=182127300329983916501 /3507184399433109696000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]= 0,\n`b*`[6]=285550402807929509723935449/1482899423367801818133032950, \n`b*`[7]=6443295529403196642395281317251/2684933869122948857487008798 8000,\n`b*`[8]=143483908435216987345410922193/615408757915355446585175 892000,\n`b*`[9]=139711798081481043314568703/7768342000936296608559979 50,\n`b*`[10]=2259083734163473774459617/17881499156003594882624000,\n` b*`[11]=-2987435988158249671/113556675995246584800,\n`b*`[12]=1/400\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-/&%\"cG6#\"\"##\"\"\"\"$+\"/&F%6#\"\"$#\"\"%\"#F/&F%6#F 0#F'\"\"*/&F%6#\"\"&#\"&<9\"\"&G:#/&F%6#\"\"'#\"#\\\"#y/&F%6#\"\"(#\"# Z\"$$G/&F%6#\"\")#\"#I\"#t/&F%6#F6#\"#P\"#Y/&F%6#\"#5#\"#K\"#L/&F%6#\" #6F)/&F%6#\"#7F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1 ..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6^o/&%\"aG6$\"\"#\"\"\"#F(\"$+ \"/&F%6$\"\"$F(#!$#p\"$H(/&F%6$F.F'#\"$+)F1/&F%6$\"\"%F(#F(\"#=/&F%6$F :F'\"\"!/&F%6$F:F.#F(\"\"'/&F%6$\"\"&F(#\"-x$**\\\"=X\"-O2'yS&)*/&F%6$ FIF'F@/&F%6$FIF.#!-\\sR0Gb\"-7pGp%G$/&F%6$FIF:#\",$>%Rmg*\",_6)[ua/&F% 6$FEF(#\"'t68\"('4!*=/&F%6$FEF'F@/&F%6$FEF.F@/&F%6$FEF:#\"(,[w&\")Kc$z \"/&F%6$FEFI#\"(k%pZ\")d84?/&F%6$\"\"(F(#\"1(QdCoRl<\"\"2s^vGyO`V\"/&F %6$FapF'F@/&F%6$FapF.F@/&F%6$FapF:#\"/\"yu@M`st-Q]Gz\")\\E3F'*eh6*>\"R?R!4n\"GTI\"GNm$>olLuSf,sw$/&F%6$FetFI#!Ock4\\P5h]WvYF8! o'*[;#\\5%G$>\"Ob:Fw>wvn^Pp17i/QbAuW.D7/&F%6$FetFE#\"SZ0y!\\jx*fF_,eX5?Nn<\"Hg<*\\i/MV0du'[6V!*)HJO&)/&F%6$FetF`r#\"A`k$*Rd()>>ed VSQJ!)[\"A?V,^$Gs#)phV&z>O!e#/&F%6$\"#5F(#\"WKB#)*\\Xw\"3;pwK-2rxMA)\\ .@\"H)G\"o9\"WX8<'[OOls;#\\4\"p]!>5G]VE]vKOrM/&F%6$F`wF'F@/&F%6$F`wF.F @/&F%6$F`wF:#\"R))p,*\\]#*oH9+&4?bTT.%**Rzs!4Q)\"RN981*)*3$)[@Y!zj^\"o x-!)fS\"Hbm\"/&F%6$F`wFI#\"X;_m%pO=Ih!=fmQ?*H%fR&Q-%Rr8f(p$>5 B'H$\"W:LQeA%[:]9`pj@/&F%6$F`wFap#!QC;SQZ4*>wqb*>rDBxx7 3UH7)R#\"P:^mgE\"IN5>35Qv\"zt]5;-Sj/dA$p/&F%6$F`wFet#\"6#Rlc]H(*RU1@\"6\\E,dC5,w !)p$/&F%6$F`tF(#\"Q<_d]\"R7^!e#)HLkos\\zY%\\L#R>A\"Qq7;*pV/\\M#zq\"4\" >4`nFSb:(Qb#/&F%6$F`tF'F@/&F%6$F`tF.F@/&F%6$F`tF:#\"NHn\">!e?d#QH8Cja( H([#>jK;\"=\"MS4Xj^(4,@\">?2*pxW'=9ZX:sJaO 14g-V(**='R_$[v,[Ax\"Vla:s-B1r_8WG#)zrOD\"eqq,*GCl-#/&F%6$F`tFap#!O.G2 d8vkr;wvGMeFmFK\\guT?\"N:`-h$GaRE[*QhK#)yu\\+U')RU#/&F%6$F`tF`r#!GQO9# *p\\8\"o[*=sZe&)z0))p\"GbaJz8-ue+8q*>)Q$f98?\"/&F%6$F`tFet#\"6!Q-J%)4a (=rn'\"6HaskE(ybsva/&F%6$F`tF`w#!4.zzFg>f)3=\"5#p**36y,?1-#/&F%6$\"#7F (#\"(B\\N#\")![Gk%/&F%6$Fg]lF'F@/&F%6$Fg]lF.F@/&F%6$Fg]lF:F@/&F%6$Fg]l FIF@/&F%6$Fg]lFE#\"/ip3\"[;=$\"0lT,z%HQ9/&F%6$Fg]lFap#\"4rg,2g#)HOF(\" 5S)*)ybA.d(oH/&F%6$Fg]lF`r#\"3$z^xE6R&>8\"3S+drkkmGg/&F%6$Fg]lFet#\"/m D1H\"p(H\"0X&Q[Fwc?/&F%6$Fg]lF`w#\"/pL@Y%\\+$\"0gLe#=7k;/&F%6$Fg]lF`t# !)dm*z'\"+gPU76" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 11 stage order 7 scheme" }{TEXT -1 1 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i =1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"(B\\N#\" )![Gk%/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\" \"'#\"/ip3\"[;=$\"0lT,z%HQ9/&F%6#\"\"(#\"4rg,2g#)HOF(\"5S)*)ybA.d(oH/& F%6#\"\")#\"3$z^xE6R&>8\"3S+drkkmGg/&F%6#\"\"*#\"/mD1H\"p(H\"0X&Q[Fwc? /&F%6#\"#5#\"/pL@Y%\\+$\"0gLe#=7k;/&F%6#\"#6#!)dm*z'\"+gPU76" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights \+ for the 12 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..12);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"#\"6,l\"R)*H.IF@=\"7+gp 4JV*R%=2N/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6# \"\"'#\"<\\a$Rs4&Hz!GS]bG\"=]H.L\"==!yOB%**G[\"/&F%6#\"\"(#\"@^sJ\"G&R Um>.%HbHVk\"A+!))z3q[d)[H7pQ$\\o#/&F%6#\"\")#\"?$>A4TXt)p@N%3R[V\"\"?+ ?*e<&eYaN:zv3ah/&F%6#\"\"*#\"<.(oXJV5[\"3)z6(R\"\"<]z*f&3mHO4+U$ox/&F% 6#\"#5#\":<'fWxtM;MP3fA\";+Si#)[f.g:*\\\")y\"/&F%6#\"#6#!4r'\\#e\"))fV ()H\"6+[eY_*fncN6/&F%6#\"#7#F'\"$+%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#=================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[10] = 81/82;" "6#/&%\"cG 6#\"#5*&\"#\")\"\"\"\"##)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "As for the previous sche me, one of the weights for the order 7 scheme is negative and all the \+ other weights are positive." }}{PARA 0 "" 0 "" {TEXT -1 37 "The nodes \+ of the order 7 scheme are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[2] = 1/100" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$+\"!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 2/15" "6#/&%\"cG6#\"\"$*&\"\"# \"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/5" "6#/&% \"cG6#\"\"%*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5 ] = 5120/9991" "6#/&%\"cG6#\"\"&*&\"%?^\"\"\"\"%\"***!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 60/97" "6#/&%\"cG6#\"\"'*&\"#g\"\"\"\"# (*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 52/313" "6#/&%\"cG6# \"\"(*&\"#_\"\"\"\"$8$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = \+ 63/157" "6#/&%\"cG6#\"\")*&\"#j\"\"\"\"$d\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9] = 19/23" "6#/&%\"cG6#\"\"**&\"#>\"\"\"\"#B!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 81/82" "6#/&%\"cG6#\"#5*&\"# \")\"\"\"\"##)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&% \"cG6#\"#6\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "The 2-norm of the principal error is appr oximately: 0.124631 " }{TEXT 270 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-4)" "6#)\"#5,$\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------ ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "ch ecking 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 4506 "ee := \{c[2]=1/1 00,\nc[3]=2/15,\nc[4]=1/5,\nc[5]=5120/9991,\nc[6]=60/97,\nc[7]=52/313, \nc[8]=63/157,\nc[9]=19/23,\nc[10]=81/82,\nc[11]=1,\nc[12]=1,\n\na[2,1 ]=1/100,\na[3,1]=-34/45,\na[3,2]=8/9,\na[4,1]=1/20,\na[4,2]=0,\na[4,3] =3/20,\na[5,1]=551874974720/997302429271,\na[5,2]=0,\na[5,3]=-20866990 08000/997302429271,\na[5,4]=2045902848000/997302429271,\na[6,1]=70365/ 1204352,\na[6,2]=0,\na[6,3]=0,\na[6,4]=15000000/48955027,\na[6,5]=1639 0905/64600448,\na[7,1]=21278500033609/276420238876800,\na[7,2]=0,\na[7 ,3]=0,\na[7,4]=9893050830728000/91236732526195641,\na[7,5]=-2901059421 298252903/76225368652427491200,\na[7,6]=436379361176824/23234416797464 775,\na[8,1]=627197176791399348709/22893533288085131673600,\na[8,2]=0, \na[8,3]=0,\na[8,4]=-26470464436388981652125/236136288092530359764076, \na[8,5]=825734282216068905765330677/6313097845989067825180262400,\na[ 8,6]=-49138310819605862695843/962154393150421608110400,\na[8,7]=13/32, \na[9,1]=-2805292064095175105994189803671936576349564929/\n 195 439700408341869784184509846462968849917952000,\na[9,2]=0,\na[9,3]=0,\n a[9,4]=-44489197025459151705444370840723674742716548825/\n 1493 2376340320772420636844398178733063223595932,\na[9,5]=-5381256054551232 24555096530708947439830683725706539/\n 221787063147752862838752 824077163711222570059776000,\na[9,6]=590395729067075691947185196568765 8329936372357281397/\n 47010380988313571014619878123036578848270 97474792000,\na[9,7]=80638533780709324566519854325500435393/3166476993 5063948010841554250731537120,\na[9,8]=38023609700214727511244770779309 498/15560366705131162570848900057690585,\na[10,1]=18745022701169741445 732083919370592367818346398255185593/\n 43542944158840438871835 940146729085397585013246181376000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=17 4955774658845813020015026443375837215431521574722425/\n 1298801 2824390651292175460014153260035238111030958552,\na[10,5]=7270389917310 0547975701530898752038618015765157471787669321/\n 6944684059165 807161861542521065278114605347407607709696000,\na[10,6]=-1851345468344 294261477635009838184638595117141582640612437867/\n 4207034078 64684992323487936970444634267986659475521049568000,\na[10,7]=-14238311 1687918201861880767560889949537519404224643/\n 136084445659098 41005179111919047260723369140257280,\na[10,8]=-15762011599975752003382 750329296637334838001/1705382686127610134765519769050554722692480,\na[ 10,9]=1570980729807122133129202485/2170749583944689600835012992,\na[11 ,1]=413401339654426004756492280722047075004416104055883/\n 7198 68089323382472727694115370573342900039059456000,\na[11,2]=0,\na[11,3]= 0,\na[11,4]=4068247489923915770015231221964449550938389658325/\n \+ 222675331628898775507843272599152181481922934868,\na[11,5]=422701288 179451309495857528449967681683868959323295591/\n 29766097532422 646168378497759855140870411136751616000,\na[11,6]=-6475633629566782288 4474260277213322605529059566403398599943/\n 105920343337637039 70644991697561788735926914730422752072000,\na[11,7]=-29332041896524497 98399861612546549582225310986953/\n 20501358149999954267743997 8824334676268210882720,\na[11,8]=-624797475608500160493593996249709038 784226/49773602893036152650616792845999901516465,\na[11,9]=38655300339 474326088/40800863136986779255,\na[11,10]=-6830025351434171965696/3696 66731905701279737985,\na[12,1]=6470068411/127052452800,\na[12,2]=0,\na [12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=14036525372223254273/576294 20394152745600,\na[12,7]=7354322880723039179131/3021642886516109069760 0,\na[12,8]=223707573986265289483/1101575429077660009200,\na[12,9]=531 5253908682019/32851416704215200,\na[12,10]=347047251949199216/12132172 05998680575,\na[12,11]=-95980357/508344480,\n\nb[1]=6470068411/1270524 52800,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=14036525372223254273/ 57629420394152745600,\nb[7]=7354322880723039179131/3021642886516109069 7600,\nb[8]=223707573986265289483/1101575429077660009200,\nb[9]=531525 3908682019/32851416704215200,\nb[10]=347047251949199216/12132172059986 80575,\nb[11]=-95980357/508344480,\n\n`b*`[1]=457593782868427426344821 /8836699182375187992960000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[ 5]=0,\n`b*`[6]=911082200138648538937372713121003/400821740041027803911 6332273920000,\n`b*`[7]=503904010603015936847352266508232049/210160045 2117104186799675302680320000,\n`b*`[8]=1628324543099503870464071215455 6497/76616314592355159097734873145440000,\n`b*`[9]=4185379964312305500 55107156803/2284868026806051955976984640000,\n`b*`[10]=213782918136340 8719740436246579/10547647109544872417769030948750,\n`b*`[11]=-43664669 29950751198793/35356163157685532736000,\n`b*`[12]=1/160\}:" }}}{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 1011 "subs(ee,matrix([[c[2],a[2,1],``],\n[c[3],a[3,1],a[3,2]],[c[4 ],a[4,1],a[4,2]],[``$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]],[``$2,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]],[``$2,a[8,7]],\n[c[9],a[9,1], a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6]],[``,a[9,7],a[9,8]],\n[c[ 10],a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[``,a[10,5],a[10,6]],\n[``,a [10,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[11,1],a[11,2]],[``,a[11,3],a [11,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7],a[11,8]],[``,a[11,9],a[11,1 0]],\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a[12,4]],[``,a[12,5],a[12,6] ],\n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,10]],[``$2,a[12,11]],\n[``,` ________________________________`$2],[`b`,b[1],b[2]],[``,b[3],b[4]],[` `,b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10]],[``$2,b[11]],\n[`b*`,`b* `[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[6]],\n[``,`b*`[7], `b*`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11],`b*`[12]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\"\"\"\"$+\"F(%!G7%#\"\"#\"#:# !#M\"#X#\"\")\"\"*7%#F)\"\"&#F)\"#?\"\"!7%F+F+#\"\"$F:7%#\"%?^\"%\"*** #\"-?Z(\\(=b\"-r#HCI(**F;7%F+#!.+!3!*p'3#FE#\".+![G!f/#FE7%#\"#g\"#(*# \"&l.(\"(_V?\"F;7%F+F;#\")+++:\")F]&*[7%F+F+#\")04R;\")[/gk7%#\"#_\"$8 $#\"/4O.+&y7#\"0+o()Q-Uw#F;7%F+F;#\"1+!G2$30$*)*\"2Tc>EDtO7*7%F+#!4.HD )H@%f5!H\"5+7\\FClo`Aw#\"0CoF'\"8+OnJ^3)GLN*G#F;7%F+F;#!8D@l\")*)QOWYqk#\"9wSwf.`#4)GOhB7%F+ #\"3JQ\"\\\"9+/63;U]JRa@ '*7%F+F+#\"#8\"#K7%#\"#>\"#B#!OH\\c\\jdO>n.)*=%*f5v^4k?H0G\"Q+?&z\"*\\ )oHYY)4X=%yp=M3/qRa>F;7%F+F;#!PD)[lrUZnB2%3PWaq^\"fa-(>*[W\"PKffBK1L(y \")RWoj?Cx?.MwB$\\\"7%F+#!TRlqDPoI)Ru%*32`'4bXAB^X0c7Q&\"T+gxf+dA7rjr2 CGvQG'GvZJ1(y@##\"U(R\"GdBPO*H$ewol>&=Z>pvq1HdR!f\"U+?zuu4F[)yl.B\"y)> Y,rNJ))4Q5q%7%F+#\"G$RN/]DV&)>lcC$42yL&Q1)\"G?r`J2Da:%3,[R1N*pZmJ#\"D) \\4$z2xW7^FZ@+(4O-Q\"D&e!pd+!*[3di680nOgb\"7%#\"#\")\"##)#\"Y$f&=b#)RY $=yO#fq$>R3KdWT(p6qA]u=\"Y+gP\"=YK,&e(R&3Hn9Sf$=()Q/%)eT%HaVF;7%F+F;# \"WDCsu:_Ja@PePVk-:+-8e%)eYxb\\<\"V_&e4.6\"Q_.gK:9+Yv@H^1RCG,))H\"7%F+ #\"fn@$pwyru:ld,='Q?v)*3`,d(za+J<**Qqs\"en+gp4xg2uM0Y6y_1@Dah=;2e;fSoW p#!hnnyV71k#eTr6&fQY=Q)4]jx9E%HW$oaM^=\"gn+!o&\\5_v%fm)zEMYWqp$z[BB*\\ okyS.2U7%F+#!TVYA/%>v`\\**)3cn2)='=?=zo6JQU\"\"S!Gd-9pLsgs/>>6z^+T)4fc W%3O\"#!M,!Q[LPmHH.v#Q.?vv**f6?w:\"L![#pAZb]!p(>bwM,hFho#Q0<7%F+F+#\"= &[-#HJ8Ar!)H2)4d\"\"=#*H,N3g*oWRe\\2<#7%F)#\"T$)e0/hT/]2Z?s!G#\\cZ+EWl R8S8%\"T+gXf!R+!HMt0P:TpFFZ#QB$*3o)>(F;7%F+F;#\"RD$e'*QQ4b\\W'>AJ_,qd \"R#**[Z#oS\"Qo[$H#>[\"=_\"*fsK%y]v()*)G;LvEA7%F+#\"W\"f&HB$f*oQo\"on* \\%Gv&e\\48Xz\")G,FU\"V+gh^n86/(39b)fx\\y$ohkAC`(4m(H#!fnV**f)R.kcf!Hb gAL@x-EuW)G#ym&HOjvk\"fn+?2_FUIZ\"p#ft)yhvp\"*\\kqRqjPLM?f57%F+#!R`p)4 JDAe\\la7;')*R)z\\Cl*=/K$H\"Q?F)3@oinMV#)y*RunU&****\\\"e8]?#!KEUyQ!4( \\i*Rf$\\g,]3cZ(zC'\"Jlk^,***f%Gz;1l_h.$*Ggt(\\7%F+#\"5)3EVZR.Ib'Q\"5b #zn)p8j3!3%#!7'pl>tm'p$7%F)#\"+6%o+Z'\"-+GX_q7F;7% F+F;F;7%F+F;#\"5tUDBAPDl.9\"5+cu_TR?%Hw&7%F+#\"7J\"z\"RIs!)GKat\"8+wp! 4h^')Gk@I#\"6$[*Gli)Rd2PA\"7+#4+mx!Had,67%F+#\"1>?o3RD:`\"2+_@/nT^G$# \"3;#*>\\>DZqM\"4v0o)*f?01o-o[G##\"@zlCO/u>(3MO\"=Hy8#\"A]([4.pxTs[a4rkZ0\"7%F+#!7$z)>^2&*Hp YmV\"8+gtKbodJ;c`$#F)\"$g\"Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(ee,matrix([seq( [c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[j],j=1..11 ),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[6](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'++5!\"(F(%!GF+F+F+F+F+F+F+F+F+F+7 /$\"'LL8!\"'$!'cbvF/$\"'*)))))F/F+F+F+F+F+F+F+F+F+F+7/$\"'++?F/$\"'++] F*$\"\"!F:$\"'++:F/F+F+F+F+F+F+F+F+F+7/$\"'hC^F/$\"'oLbF/F9$!'M#4#!\"& $\"'W^?FDF+F+F+F+F+F+F+F+7/$\"'d&='F/$\"'cUeF*F9F9$\"'/kIF/$\"'FPDF/F+ F+F+F+F+F+F+7/$\"'Mh;F/$\"')yp(F*F9F9$\"'L%3\"F/$!'!f!QF*$\"';y=F*F+F+ F+F+F+F+7/$\"'u7SF/$\"'jRFF*F9F9$!')47\"F/$\"'(zI\"F/$!'62^F*$\"']iSF/ F+F+F+F+F+7/$\"'(3E)F/$!'PN9F*F9F9$!'QzHFD$!'KECFD$\"')eD\"FD$\"'jYDFD $\"'iVCFDF+F+F+F+7/$\"'0y)*F/$\"'&\\I%F/F9F9$\"'1Z8!\"%$\"'!p/\"Fhp$!' g+WFD$!'GY5Fhp$!'^U#*FD$\"'/PsF/F+F+F+7/$\"\"\"F:$\"'uUdF/F9F9$\"'*p#= Fhp$\"'3?9Fhp$!'o8hFD$!'uI9Fhp$!'Gb7Fhp$\"'9u%*F/$!'iZ=F*F+F+7/Fdq$\"' W#4&F*F9F9F9F9$\"'lNCF/$\"')QV#F/$\"'!3.#F/$\"'(zh\"F/$\"'bgGF/$!'5))= F/F+7/%\"bGFgrF9F9F9F9FirF[sF]sF_sFasFcsF+7/%#b*G$\"'Ly^F*F9F9F9F9$\"' /tAF/$\"'s(R#F/$\"'ID@F/$\"'yJ=F/$\"'$o-#F/$!'*\\B\"F/$\"'+]i!\")Q)ppr int146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op (OrderConditions(7,11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,Orde rConditions(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(ee,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u), %);\nnops(%);\nsimplify(subs(ee,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u), %);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 #\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F $F$F$F$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#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for st age-orders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_11 := StageOrderCo nditions(ct,11,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 110 "Stages 6 to 11 have stage-order 4, w hile stages 4 and 5 have stage-order 3 has and stage 3 has stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(su bs(ee,so||i||_11[j])),i=2..6)],j=1..9)]:\nmap(proc(L) local i; for i t o 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&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 .. 11) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 10 " }}{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 0 151 "[Sum(b[i]*a[i,1],i=1+1..1 1)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=2..10)];\nmap(u ->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F -F-/F,;\"\"#\"#6&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,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F /6$F,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,Fa qF-/F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,; \"#5F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-/-F&6$*&F)F-&F/6$F,F]sF-/F,;F4F4*&&F *6#F]sF-,&F-F-&FEF]tFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F $F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the princi pal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "RK7_11err_eqs := PrincipalErrorConditions(7,11,'expa nded'):\nexpand(subs(ee,RK7_11err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0, 1),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"\"F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 93 "We c an calculate the principal error norm, that is, the 2-norm of the prin cipal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errt erms7_11 := PrincipalErrorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errterms7_11) do\n sm := sm+(evalf(subs(ee,errterms7_11[ct]))) ^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IMJY7!# 9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm of the order 6 embedded scheme is given as follo ws." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms6_12*` := s ubs(b=`b*`,PrincipalErrorTerms(6,12,'expanded')):\nsm := 0:\nfor ct to nops(`errterms6_12*`) do\n sm := sm+(evalf(subs(ee,`errterms6_12*`[ ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4 6MB#)!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "We incorporate the stage-order equations to ensure that stage 3 has stage-order 2, stages 4 and 5 ha ve stage order 3 and stages 6 to 11 have stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 57 "These conditions lead to relationships between the n odes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6 #/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c [4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0 " "6#/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/ F'!\"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" } {TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 68 "which we include expli citly to make the computation more efficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the s implifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and \+ " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 " . . 10. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "The simp lifying conditions allow us to manage with just 10 \"simple\" order co nditions which are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 7 order 7 quadrature conditions and 3 additional order 7 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO7 := SimpleOrde rConditions(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,32,38,48,61,64])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7 %\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#KF)/*&F,F()F2FGF(#F(\"\"'7%\"#QF)/*&F,F(-%!G6#* &%\"aGF(-FT6#*&FWF(-FT6#*&FWF(-FT6#*&F8F(FWF(F(F(F(F(#F(\"%?D7%\"#[F)/ *&F,F(-FT6#*&FWF(-FT6#*&FWF(-FT6#*&F?F(FWF(F(F(F(#F(\"$S)7%\"#hF)/*(F, F(F2F(-FT6#*&FEF(FWF(F(#F(\"#N7%\"#kF)/*&F,F()F2FNF(#F(\"\"(Q)pprint16 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "SO7_11 := SimpleOrderConditions(7,11,'expanded'):\nS O_eqs := [op(RowSumConditions(11,'expanded')),op(StageOrderConditions( 2,11,'expanded')),\n op(StageOrderConditions(3,4..11,'expanded')),op( StageOrderConditions(4,6..11,'expanded'))]:\nord_cdns := [seq(SO7_11[i ],i=[1,2,4,8,16,32,38,48,61,64])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2.. 11)=b[1],seq(add(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=[$8..10])]:\nn ode_eqs := [3*c[3]=2*c[4],3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] =0]:\ncdns := [op(node_eqs),op(simp_eqs),op(SO_eqs),op(ord_cdns)]:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The scheme can be obtained by s pecifying the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/100;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$+\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/5;" "6#/&%\"cG6#\"\"%*&\"\"\"F)\" \"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 60/97;" "6#/&%\"cG6 #\"\"'*&\"#g\"\"\"\"#(*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 52/313;" "6#/&%\"cG6#\"\"(*&\"#_\"\"\"\"$8$!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 63/157;" "6#/&%\"cG6#\"\")*&\"#j\"\"\"\"$d\"!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 19/23;" "6#/&%\"cG6#\"\"* *&\"#>\"\"\"\"#B!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 81/82 ;" "6#/&%\"cG6#\"#5*&\"#\")\"\"\"\"##)!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT -1 45 "together with the zero linking coeffici ents: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0 " "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2 ] = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10, 2] = 0;" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG6$\"#6\"\"# \"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[6,3] = 0;" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[7,3] = 0;" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,3] = 0;" "6#/&%\"aG6$\"\"*\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,3] = 0;" "6#/&%\"aG6$\"#5\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/&%\"aG6$\"#6 \"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the ze ro weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]= 0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" " 6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4] = 0;" "6 #/&%\"bG6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5] = 0;" "6# /&%\"bG6#\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "e1 := \{c[2]=1/100,c[4]=1/ 5,c[6]=60/97,c[7]=52/313,c[8]=63/157,c[9]=19/23,c[10]=81/82,c[11]=1,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),seq(b[i]=0,i=2..5)\}:\n eqns := subs(e1,cdns):\nnops(eqns);\nindets(eqns) minus \{a[8,7],a[9,7 ],a[9,8]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\\" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#F/&F%6$F2F>&F-6#F>&F-6#F3&F-6#FF&F%6$FCF/&F%6$F2F/&F%6$F6F/& F%6$F(F/&F%6$\"\"#F/&F%6$F+F/&F%6$F>F+&F%6$F>F'&F%6$F2FF&F%6$F(Fao&F%6 $FFF+&F%6$FFF'&F%6$FFF/&F%6$F3F+&F%6$F3F'&F%6$F'F/&F%6$F2F'&F%6$F2F6&F %6$FCF+&F%6$FCF'&F%6$FCF6&F%6$FCF>&F%6$FCF3&F%6$FCFF&F%6$FCF2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Z" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e2 := solve(\{op(eqns)\} ,indets(eqns) minus \{a[8,7],a[9,7],a[9,8]\}):\ne3 := `union`(e1,e2): \ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,e3));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"*\"\")&F%6$F'\"\"(&F%6$F(F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5025 "e3 := \{b[ 3] = 0, b[4] = 0, b[5] = 0, c[7] = 52/313, c[8] = 63/157, c[9] = 19/23 , c[10] = 81/82, c[6] = 60/97, a[2,1] = 1/100, a[8,5] = 22949265488410 3/1531644837993600*a[8,7]+4798798166112399099/68625931605835366400, c[ 2] = 1/100, c[4] = 1/5, b[9] = 5315253908682019/32851416704215200, a[8 ,1] = -24639157757/294377251200*a[8,7]+15279897640203/248861983129600, a[11,5] = -682897043632974173369/97569597601246003200-269902057628763 26378218209909261797/3482157897755973394651477999368000*a[8,7]-8945226 901314411093455855153671/5253672249638433106018454838400*a[9,7]+857996 41641507627056573953843509759/7309171042857638428261721329510400*a[9,8 ], a[7,6] = 436379361176824/23234416797464775, a[4,3] = 3/20, a[3,2] = 8/9, a[8,6] = -33118183969/466863921825*a[8,7]-33248810216853/1494144 015899200, a[7,4] = 9893050830728000/91236732526195641, b[7] = 7354322 880723039179131/30216428865161090697600, c[11] = 1, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[1 1,2] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, b[2] = 0, a[10,5] = 449211858724020391834087022291293850 46261297/5391202680992466919336266851087340545638400*a[9,8]-1121622145 204066232720966487/239020089819184019618201600-85542548146599598801639 718742505842927/16379966602917892859584925403676467200*a[8,7]-46369853 749783586918559003200470095093/38367114344037927338433637118493286400* a[9,7], a[11,10] = -6830025351434171965696/369666731905701279737985, a [10,9] = 1570980729807122133129202485/2170749583944689600835012992, a[ 7,1] = 21278500033609/276420238876800, a[11,1] = -73190759073874148857 /49552415603686195200+37670953757871488061140945206459/870037529710314 3211698868728000*a[8,7]+12485092996829826277940408537/1312660758126074 0030413206400*a[9,7]-23102218848351866228188759857/2650564820316884826 3381145600*a[9,8], a[6,4] = 15000000/48955027, a[11,8] = -152015257014 4099623/133773321760612391*a[9,8]+59055505597430365931240/388123882007 0407632837, a[5,4] = 2045902848000/997302429271, b[10] = 3470472519491 99216/1213217205998680575, a[10,4] = 172244301822334366894345753899223 975125/57615299877941310679817120681579037376*a[9,8]-50726221685668758 9619125/17880703948000573401568+563179376132055352396068874604243250/1 6236002052696934816873729770245717*a[8,7]+3052813584812045032648961595 29751750/38029903378123531492685685229756529*a[9,7], b[6] = 1403652537 2223254273/57629420394152745600, a[10,7] = 110965279287544351171358548 329/4578100819981168626783210496-16150726351202110361229287961/4633760 74748208035021852608*a[8,7]-8754787355347066021424092899/1085374791972 344800417506496*a[9,7], a[11,9] = 38655300339474326088/408008631369867 79255, c[5] = 5120/9991, a[9,4] = -96697071952000/97163719410219*a[9,7 ]-216417286125/583916545091*a[9,8]+4902133662625/10640540979684, a[11, 4] = -152318222968784875/3649511407918068+1568601848083132879579063430 84800/3046888160536476720320043249447*a[8,7]+1299684371823534290481760 16000/11492408046084072419415369959*a[9,7]+113443894388458531238377803 75/2693532960958740576452580089*a[9,8], a[7,5] = -2901059421298252903/ 76225368652427491200, a[10,1] = -1330491540298742798499435201230450083 41/215054556663307946840171799875836313600*a[9,8]-14419309680224390918 26791/1498648222895242475929600+1313334305139953081787632615577095259/ 450189356951231727641868985724723200*a[8,7]+71191612797816890161373784 4023381081/1054487286411190391965406516998758400*a[9,7], a[8,4] = -966 97071952000/97163719410219*a[8,7]+107149861263375/366699590317148, a[3 ,1] = -34/45, a[9,5] = 229492654884103/1531644837993600*a[9,7]-5644146 7341251433/54638480577894400*a[9,8]-80686691351360197/2844746010095616 00, a[9,6] = -33118183969/466863921825*a[9,7]+388915924731/11896051082 00*a[9,8]+27645587752973/43355654193600, a[9,1] = -24639157757/2943772 51200*a[9,7]+15197302759/198138521600*a[9,8]+11730347851/1031605862400 , a[10,8] = 9701308340069201361112803471/926752149496416070043705216-8 754787355347066021424092899/1085374791972344800417506496*a[9,8], c[3] \+ = 2/15, b[11] = -95980357/508344480, a[11,7] = 70733041463228208909104 685/1984252620240330013454452-3057811807481052937774/59110378003985813 505*a[8,7]-1520152570144099623/133773321760612391*a[9,7], a[11,6] = -7 391616554512663080198465473/12765159656751868946742254400+268618498627 952046883361382878/73200272932358176059966443625*a[8,7]+44513432784244 445759838677/55220104007707440031786325*a[9,7]-20386604915510305062152 878497/5487497479562300617028127800*a[9,8], a[6,5] = 16390905/64600448 , a[4,1] = 1/20, b[1] = 6470068411/127052452800, a[6,1] = 70365/120435 2, a[5,3] = -2086699008000/997302429271, a[5,1] = 551874974720/9973024 29271, a[10,6] = -3404876220128070079169860147337435585169/12911673968 41813727708375220348282867200*a[9,8]-53439960127126754966646106989909/ 127768395765882882949576116147200+137149417059509639806213488123916640 31/5547014654790326990289206619179086400*a[8,7]+7434427134457925854920 820615334065029/12992880308466728729847794361794196800*a[9,7], b[8] = \+ 223707573986265289483/1101575429077660009200\}:" }{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 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7)]));\nfor ii from 8 to 11 do\n print(``);\n print(c[ii ]=subs(e3,c[ii])); \n for jj to ii-1 do\n coeff_val := subs(e3, a[ii,jj]);\n if coeff_val=a[ii,jj] then \n print(a[ii,jj] *` is a parameter.`)\n else\n print(a[ii,jj]=coeff_val); \n end if;\n end do:\nend do:\nprint(``);\nseq(b[i]=subs(e3,b[i ]),i=1..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\" \"\"$+\"F(%!GF+F+F+F+F+7*#\"\"#\"#:#!#M\"#X#\"\")\"\"*F+F+F+F+F+7*#F) \"\"&#F)\"#?\"\"!#\"\"$F:F+F+F+F+7*#\"%?^\"%\"***#\"-?Z(\\(=b\"-r#HCI( **F;#!.+!3!*p'3#FD#\".+![G!f/#FDF+F+F+7*#\"#g\"#(*#\"&l.(\"(_V?\"F;F;# \")+++:\")F]&*[#\")04R;\")[/gkF+F+7*#\"#_\"$8$#\"/4O.+&y7#\"0+o()Q-Uw# F;F;#\"1+!G2$30$*)*\"2Tc>EDtO7*#!4.HD)H@%f5!H\"5+7\\FClo`Aw#\"0Co')[#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"%,&*&#\"/+?&>2(p'*\"/>-T>P;(*\"\"\"&F%6$F'\"\"(F.!\"\"# \"0vLEh)\\r5\"0[rJ!f*pm$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"&,&*&#\"0.T)[l#\\H#\"1+O*z$[kJ:\"\"\"&F%6$F'\"\"(F.F.#\"4*4* R7h;)z)z%\"5+kONegJfioF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"',&*&#\",pR==J$\"-D=#R'oY\"\"\"&F%6$F'\"\"(F.!\"\"#\"/`o@5)[K $\"1+#**e,WT\\\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\") \"\"(\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#>\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\",(*&#\",dx:RY#\"-+7DxVHF (&F%6$F'\"\"(F(!\"\"*&#\",fFI(>:\"-+;_Q\")>F(&F%6$F'\"\")F(F(#\",^yMI< \"\".+C'egJ5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,(*&#\"/+?&>2(p'*\" />-T>P;(*\"\"\"&F%6$F'\"\"(F.!\"\"*&#\"-DhGg8N\"p'o!)\"3+; c45guWGF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',(*&# \",pR==J$\"-D=#R'oY\"\"\"&F%6$F'\"\"(F.!\"\"*&#\"-JZ#f\"*)Q\".+#3^g*= \"F.&F%6$F'\"\")F.F.#\"/tHv(eXw#\"/+O>acNVF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\"(\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\")\"\"\"%1~is~a~para meter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#\")\"##)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",**&#\"HT$3]/B,_V*\\)zU()HS:\\I8\"H+OJOe()*zr ,%o%zIjmba]@F(&F%6$\"\"*\"\")F(!\"\"#\":\"zE=4RC-o4$>W\"\":+'HfZU_*GA[ ')\\\"F2*&#\"Ff_4xbhKwy\"3`*R^IMLJ\"\"E+KsCd)*o=kFr\"F+%e()*p^1a'>R!>6kG([a5F(&F%6$F 0F'*e(oco@is] \"8o:St0+[Rq!)y\"!\"\"*&#\"E]KC/Y()ogR_`0KhPzJc\"DpY#*4o-7R&\"\"\"&F% 6$\"\"*\"\")F.F.#\"=([m4sKi1/_9A;7\"\"<+;?='>S=>)*3?!R#!\"\"*&#\"GFH%e ]U(=(R;!))f*fY\"[Da&)\"G+sYwOSD\\efG*y\"Hgm*zj\"F.&F%6$F2\"\"(F.F6*&# \"G$4&4q/?.!f&=pe$y\\P&)pj%\"G+kG$\\=rjL%Qt#z.WV6n$QF.&F%6$F1F=F.F6" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',**&#\"Ip^eNuLZ,')p \"z+2G,Aw[S$\"I+s'GG[.Av$3xs8=%oRn6H\"\"\"\"&F%6$\"\"*\"\")F.!\"\"#\"A 4**)p5Ym'\\vEr7g*RM&\"B+s9;hd\\H)G)ew&Rox7F3*&#\"GJSm\"R7)[8i!)R'4&fqT \\r8\"F+k3z\">m?*G!*pK!zaY,ZbF.&F%6$F2\"\"(F.F.*&#\"FH]1M`h?3#\\&e#zXM rUMu\"G+o>%zhVzZ)H(GnY3.)G*H\"F.&F%6$F1F=F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,(#\"?H$[&e8<^Va(Gz_'46\"='\\5KyE'o6) *>35yX\"\"\"*&#\">hzGH7O5@?^js]h\"\"<3E&=-N!3#[Z2wLYF-&F%6$\"\")F(F-! \"\"*&#\"=**G4C9-mqMbtya()\"='\\1vT+[Ms>zu`3\"F-&F%6$\"\"*F(F-F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"),&#\"=rM!G6h8?p+M38q *\"<;_qV+2;k\\\\@vE*\"\"\"*&#\"=**G4C9-mqMbtya()\"='\\1vT+[Ms>zu`3\"F- &F%6$\"\"*F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"*#\"=&[-#HJ8Ar!)H2)4d\"\"=#*H,N3g*oWRe\\2<#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",*#\"5d)[T( Q2f2>t\"5+_>'o.cT_&\\!\"\"*&#\"Afk?X49h!)[ryv`4nP\"@+!G(o))p6K9.rHv.q) F(&F%6$\"\")\"\"(F(F(*&#\">P&3/%zFE)Ho*H4&[7\">+k?8/.S2E\"e2m78F(&F%6$ \"\"*F5F(F(*&#\">d)f()=Gi'=N[)=A5B\">+c9\"Qj#[)oJ?[c]EF(&F%6$Ff*p`T>C2%3Y!3C\\6F2&F%6$\"\"*F6F2F2*&#\">v .yPQ7`e%)Q%*QW8\"\"=*3!e_kdS(e4'H`$p#F2&F%6$F=F5F2F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&,*#\"6pL!3jE^alh \"R(\">+WDUn%*o=vc'f^w7!\"\"*&#\"?yGQhL)o/_zi)\\='o#\">DOWm*fg(\\yG:i]I5b\"\\g'Q?\"=+y7Gqh+Bczu\\([&F2&F%6$F=F5F2F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,(#\";&o/\"4*3#G KYTItq\":_WX8+LS-i_U)>\"\"\"*&#\"7ux$H0\"[2=\"y0$\"50N\"e)R+y.6fF-&F%6 $\"\")F(F-!\"\"*&#\"4B'*4W,d_,_\"\"3\"R71w@LxL\"F-&F%6$\"\"*F(F-F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&*&#\"4B'*4W,d_,_\" \"3\"R71w@LxL\"\"\"\"&F%6$\"\"*F(F.!\"\"#\"8S7$fOIuf0b0f\"7PGj2/2?)Q7) QF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"5)3EVZR.Ib 'Q\"5b#zn)p8j3!3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5 #!7'pl>tm'p$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"+6%o+Z'\"-+GX_q 7/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\" 5tUDBAPDl.9\"5+cu_TR?%Hw&/&F%6#\"\"(#\"7J\"z\"RIs!)GKat\"8+wp!4h^')Gk@ I/&F%6#\"\")#\"6$[*Gli)Rd2PA\"7+#4+mx!Had,6/&F%6#\"\"*#\"1>?o3RD:`\"2+ _@/nT^G$/&F%6#\"#5#\"3;#*>\\>DZqM\"4v0o)*f? " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK7_11eqs := [op(RowSumConditions(11,'expan ded')),op(OrderConditions(7,11,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simplify(subs(e3,RK7_11eqs)):\nmap(u->lhs(u )-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ 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#\"#&*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We now \+ turn our attention to the embedded 12 stage, order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 210 "The first 11 stages of the embedded order 6 sc heme coincide with those of the order 7 scheme and the weights of the \+ order 7 scheme provide the linking coefficients for the 12th stage of \+ the order 6 scheme. Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[12,j] = b[j];" "6#/&%\"aG6$\"#7%\"jG&%\"bG6#F(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 8 " . . 11." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplif ying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*& &%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and \+ " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The simplifying condit ions allow us to manage with just 7 \"simple\" order conditions which \+ are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 6 order 6 quadrature conditions and 1 additio nal order 6 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO6*` := subs(b=`b*`,Simpl eOrderConditions(6)):\n[seq([i,`SO6*`[i]],i=[1,2,4,8,16,31,32])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#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()F2F:F(#F(F57%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#JF)/*&F,F(-%!G6#*&FEF(%\"aGF(F(#F(\"#I7%\"#KF)/ *&F,F()F2FGF(#F(\"\"'Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "`SO6_12*` := subs(b=`b*`,Si mpleOrderConditions(6,12,'expanded')):\n`cdns*` := [seq(`SO6_12*`[i],i =[1,2,4,8,16,31,32])]:\n`simp_eqs*` := [seq(add(`b*`[i]*a[i,j],i=j+1.. 12)=`b*`[j]*(1-c[j]),j=[4,5])]:\nwt_eqns := [seq(b[i]=a[12,i],i=1..11) ]:\n`cdns*` := [op(`cdns*`),op(wt_eqns),op(`simp_eqs*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] = 0;" "6#/&%#b*G6#\"\"%\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5] = 0;" "6#/&%#b*G6#\"\"& \"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 1/160;" "6#/&% #b*G6#\"#7*&\"\"\"F)\"$g\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := \{c[12 ]=1,seq(`b*`[i]=0,i=2..5),`b*`[12]=1/160\}:\ne5 := `union`(e3,e4):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqns2 := [op(remove(_U->type(rhs(_U),realcons),e5)),op(subs(e5, `cdns*`))]:\nnops(eqns2);\nindets(eqns2) minus \{a[8,7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F%6$FCF'&F%6$FCF1&F%6$FC\"#5&F%6$FC\"#6&F%6$FQF+&F%6$FQF.&F%6$FQ F>&F%6$FQF'&F%6$FCF2&F%6$FC\"\"#&F%6$FC\"\"$&F%6$FCF(&F%6$F1F(&F%6$F1F +&F%6$F1F.&F%6$F1F>&F%6$F1F'&F%6$FNF2&F%6$FNF(&F%6$FNF+&F%6$FNF.&F%6$F NF>&F%6$FNF'&F%6$FQF2&F%6$FQF(&F66#FN&F66#FQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "e6 := solve(\{op(eqns2)\},indets(eqns2) \+ minus \{a[8,7]\}):\ne7 := simplify(subs(select(u_->member(lhs(u_),\{b[ 8],a[8,7],a[9,7],a[9,8]\}),e6),e5)):\ne8 := `union`(e7,e6):\ninfolevel [solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a[1 0,4]=subs(e8,a[10,4]);\na[11,1]=subs(e8,a[11,1]);\n`b*`[11]=subs(e8,`b *`[11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"IDm /?0dv<=H[?6.7(HKG[_N%=QR,H7F-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"Fn=AUKLQOO/Q !od*)[tWY\"F+78=()))G/T#G_<3Cc$>%=F(*&#\"Ozax+Vir%e['*fO,bL5Yi0:C/%\"N +!3A2D=.o\"[z$)QpR4T3i-IV)F(&F%6$\"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#6#!7$z)>^2&*HpYmV\"8+gtKbodJ;c`$" }}} {PARA 0 "" 0 "" {TEXT -1 46 "Note that some of the coefficients depend on " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5734 "e8 := \{b[ 6] = 14036525372223254273/57629420394152745600, a[8,2] = 0, a[8,3] = 0 , a[9,2] = 0, a[9,3] = 0, c[2] = 1/100, a[7,2] = 0, a[7,3] = 0, a[2,1] = 1/100, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, b[1] = 6 470068411/127052452800, c[4] = 1/5, a[10,1] = 418069560625981524207763 19337059511221001/22282776862342999774994722427318395863040-2915184381 892113538813906617406019858095831443/819180495983073515555532775492264 144666816000*a[8,7], a[10,4] = 509459332729389044326390177557052004662 5/166162966952470080766543674627304548952-5200328914524525009204910603 9489891122121178400/1229013938184355248283229712031120482918171961*a[8 ,7], c[9] = 19/23, c[10] = 81/82, `b*`[9] = 41853799643123055005510715 6803/2284868026806051955976984640000, a[5,1] = 551874974720/9973024292 71, b[9] = 5315253908682019/32851416704215200, a[11,6] = -190601505341 01620482439102399109584780668583993/4270496569979935758204201076691764 383218817600-187362883255490797590782763630711778335743167/46117983082 724732645092700290439240433636375*a[8,7], a[12,7] = 735432288072303917 9131/30216428865161090697600, a[12,10] = 347047251949199216/1213217205 998680575, `b*`[10] = 2137829181363408719740436246579/1054764710954487 2417769030948750, a[10,9] = 1570980729807122133129202485/2170749583944 689600835012992, a[12,6] = 14036525372223254273/57629420394152745600, \+ a[9,6] = 371284252549915362276863582572148504767/379072922987225956585 003219342595718720+183079091855206703689576672223282986182667549/26905 8289814543996624006181497070195017722875*a[8,7], a[12,11] = -95980357/ 508344480, a[12,9] = 5315253908682019/32851416704215200, c[11] = 1, a[ 6,2] = 0, c[12] = 1, a[5,3] = -2086699008000/997302429271, a[10,7] = - 342190921194606256966591144300536624466016118993/123376650642881604761 37000833225077718376373760+35849474905949822003106618003299146497/8431 75514653597238336215672137358540*a[8,7], `b*`[12] = 1/160, a[12,5] = 0 , b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[11,8] = -62479747560850016 0493593996249709038784226/49773602893036152650616792845999901516465, a [8,1] = -24639157757/294377251200*a[8,7]+15279897640203/24886198312960 0, b[10] = 347047251949199216/1213217205998680575, a[12,2] = 0, a[12,3 ] = 0, a[12,4] = 0, c[3] = 2/15, c[7] = 52/313, c[8] = 63/157, b[8] = \+ 223707573986265289483/1101575429077660009200, `b*`[11] = -436646692995 0751198793/35356163157685532736000, a[7,1] = 21278500033609/2764202388 76800, a[7,5] = -2901059421298252903/76225368652427491200, a[9,5] = -7 3325384643072596965000824955217641/39797956215568109384209145764823040 -1268647667485331387337591578226041520263736371563/8827020496741343302 83951392995787176151148528000*a[8,7], a[11,5] = 1165798427416910893328 79948864411149377/10880413121717968589105615747146137600+3765163454655 1307044855402849659963860092800873941/43876912636236211922727738443182 69585850698224000*a[8,7], a[10,6] = -754032904875554729107262519338979 478515193995847/237466889278601402574279455598118415631285372160-39575 6501709098392593896354851556194209442635531/13121607585266372101121162 8874358529360185378500*a[8,7], a[3,1] = -34/45, a[11,7] = -15116478448 577786983682178151791132383419817/402313999010960942171566681236012520 493520+164064660655766560463021267097383747/28646923586876314882134409 58198715*a[8,7], a[7,4] = 9893050830728000/91236732526195641, a[10,5] \+ = 28008111944344344024791046858546156433772697/35538902630322980190588 60417486925115555840+2468404192129448951423789226808842538884523958427 /387472374599993772857767002807840940427403968000*a[8,7], c[5] = 5120/ 9991, `b*`[8] = 16283245430995038704640712154556497/766163145923551590 97734873145440000, `b*`[6] = 911082200138648538937372713121003/4008217 400410278039116332273920000, a[11,1] = 4644734889576803804363638333242 221867/1841935624081752282410428888718131200-4042415056246103355013659 964858471624300775479/843300262084109396938837948168031825072208000*a[ 8,7], a[7,6] = 436379361176824/23234416797464775, a[9,8] = 38023609700 214727511244770779309498/15560366705131162570848900057690585, b[7] = 7 354322880723039179131/30216428865161090697600, `b*`[7] = 5039040106030 15936847352266508232049/2101600452117104186799675302680320000, a[10,8] = -15762011599975752003382750329296637334838001/170538268612761013476 5519769050554722692480, a[11,4] = 16858688135325572370611625320307875/ 406972999651523368143506780046924-109410843422145673109022937240151216 376650787200/1919614927835334271619338556889242943809680473*a[8,7], a[ 10,2] = 0, a[10,3] = 0, a[11,2] = 0, a[11,3] = 0, a[3,2] = 8/9, a[4,2] = 0, a[5,2] = 0, a[6,3] = 0, a[11,10] = -6830025351434171965696/36966 6731905701279737985, a[9,1] = -34056451860157223362655063910701/100014 808790968562415939055534080+136206581570146956207683042823082441734255 497/169652517715574539743215720352832438237776000*a[8,7], a[8,6] = -33 118183969/466863921825*a[8,7]-33248810216853/1494144015899200, b[11] = -95980357/508344480, `b*`[1] = 457593782868427426344821/8836699182375 187992960000, a[12,8] = 223707573986265289483/1101575429077660009200, \+ a[12,1] = 6470068411/127052452800, a[11,9] = 38655300339474326088/4080 0863136986779255, c[6] = 60/97, a[4,1] = 1/20, a[6,1] = 70365/1204352, a[5,4] = 2045902848000/997302429271, a[4,3] = 3/20, a[9,4] = -5104054 1528362486522669337380375/7443050847361739207136697182948+106909316855 058742985653105310193010870660918400/111992822552405793154776332986340 49797417696949*a[8,7], a[6,4] = 15000000/48955027, a[9,7] = 4404226898 827779641771638032167509/683520483854939947563820624503120-55280534713 67160750781463606141821/576309877967820835957366668803355*a[8,7], a[6, 5] = 16390905/64600448, a[8,4] = -96697071952000/97163719410219*a[8,7] +107149861263375/366699590317148, a[8,5] = 229492654884103/15316448379 93600*a[8,7]+4798798166112399099/68625931605835366400\}:" }}}{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 65 "The coef ficients for the embedded order 6 scheme are as follows. " }}{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 0 477 "subs(e8,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(8-i)],i=2..7)]));\nfor ii from 8 to 11 do\n print(` `);\n print(c[ii]=subs(e8,c[ii])); \n for jj to ii-1 do\n coe ff_val := subs(e8,a[ii,jj]);\n if coeff_val=a[ii,jj] then \n \+ print(a[ii,jj]*` is a parameter.`)\n else\n print(a[i i,jj]=coeff_val);\n end if;\n end do:\nend do:\nprint(``);\nc[1 2]=subs(e8,c[12]);\nseq(a[12,i]=subs(e8,a[12,i]),i=1..11);\nprint(``); \nseq(`b*`[i]=subs(e8,`b*`[i]),i=1..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\"$+\"F(%!GF+F+F+F+F+7*#\"\"#\"#:#!#M\"#X #\"\")\"\"*F+F+F+F+F+7*#F)\"\"&#F)\"#?\"\"!#\"\"$F:F+F+F+F+7*#\"%?^\"% \"***#\"-?Z(\\(=b\"-r#HCI(**F;#!.+!3!*p'3#FD#\".+![G!f/#FDF+F+F+7*#\"# g\"#(*#\"&l.(\"(_V?\"F;F;#\")+++:\")F]&*[#\")04R;\")[/gkF+F+7*#\"#_\"$ 8$#\"/4O.+&y7#\"0+o()Q-Uw#F;F;#\"1+!G2$30$*)*\"2Tc>EDtO7*#!4.HD)H@%f5! H\"5+7\\FClo`Aw#\"0Co')[ #F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\"/+?&>2(p'*\"/>-T>P;(* \"\"\"&F%6$F'\"\"(F.!\"\"#\"0vLEh)\\r5\"0[rJ!f*pm$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\"0.T)[l#\\H#\"1+O*z$[kJ:\" \"\"&F%6$F'\"\"(F.F.#\"4*4*R7h;)z)z%\"5+kONegJfioF." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\",pR==J$\"-D=#R'oY\"\"\"&F%6 $F'\"\"(F.!\"\"#\"/`o@5)[K$\"1+#**e,WT\\\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"(\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\"#>\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"\",&#\"A,2\"R1bEOBs:g=XcS$\"B!3Mb0RfTi&o4z3[,+\"!\"\"*&#\"N( \\bUtTC3BG/$o2i&p9q:e1i8\"N+gxP#QC$GN?d@V(RXd:x^_'p\"F(&F%6$\"\")\"\"( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,&#\"Av.QP$pE_'[i$G:aS5&\"@[H =(pOr?R5`5`c)Hue]&oJ4p5\"P\\pp6\"\"\"&F%6$\"\")\"\"(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"*\"\"&,&#\"DTw@b\\#3+lpfsIk%QDL(\"DSI#[wX\"4UQ4\"ob@czz R!\"\"*&#\"Rj:POPE?:/E#y:fPtQJ`[nwko7\"Q+!G&[6:wry&*HR^RGIV8u'\\?q#)) \"\"\"&F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"',&#\"HnZ][@d#ejoFi`\"*\\DD%Gr$\"H?(=dfU$>K+&ecfA()H#H2z$\"\" \"*&#\"N\\vm#=')HGBAnw&*o.n?b=4zI=\"NvGs<]>qq\\\"=1Si'*Ra9)*Ge!p#F-&F% 6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" (,&#\"C4v;K!Q;xT'zx#))*oA/W\"B?J]C1#QcZ*R\\&Q[?No\"\"\"*&#\"C@=91OY\"y ]2;n8Z`!Gb\"BbL!)omOdf$3#y'z()4jdF-&F%6$\"\")F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"D)\\4$z2xW7^FZ@+(4O-Q\"D&e!p d+!*[3di680nOgb\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#\")\"##)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",&#\"J,5A6&fqL>jx?C:)fig&p!=%\"JSI'e R=tUAZ*\\x**HMiox#GAF(*&#\"OV9$e4e)>gS)F(&F%6$\"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"%,&#\"IDm/?0dv<=H[?6.7(HKG[_N%=QR,H7F-&F %6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"&,&#\"M(psPVchaeo/\"zCSMWV%>63!G\"LSeb:^#p[!)HKIE!*Qb$\"\" \"*&#\"RF%eR_%))QD%)3oA*yB9&*[%H@>/%oC\"Q+!oRSF/%4%y!G+nx&Gx$***fuBZ(Q F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"',&#\"QZe*R>:&y%z*Q$>DE2\"HZbv[!H.a(\"Qg@P&GJcT=\")fb%zUd-9gy#*)oY P#!\"\"*&#\"QJbjU%4U>c:&[N'*Qf#R)44<]cdR\"Q+&y`=g$H&eV()G;@65sjE&e2;78 \"\"\"&F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"(,&#\"Q$**=h,mWiO0IW6fmpD1Y>@4>U$\"PgPPw$=x2DK$3+PhZg\")Gk]mPB \"!\"\"*&#\"G(\\Y\"*H.!=m5.?#)\\f!\\Z\\e$\"ES&et8sc@O$Qsf`Y^vJ%)\"\"\" &F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\") #!M,!Q[LPmHH.v#Q.?vv**f6?w:\"L![#pAZb]!p(>bwM,hFho#Q0<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"=&[-#HJ8Ar!)H2)4d\"\"=#*H,N3g *oWRe\\2<#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#6\"\"\",&#\"Fn=AUKLQOO/Q!od*)[tWY\"F+78=()))G/T#G_<3Cc$>%= F(*&#\"Ozax+Vir%e['*fO,bL5Yi0:C/%\"N+!3A2D=.o\"[z$)QpR4T3i-IV)F(&F%6$ \"\")\"\"(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\" #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,&#\"DvyI?`i61PsbKN \")oeo\"\"BCp/!y1N9oL_^'**H(pS\"\"\"*&#\"Q+sy]mP;7:Ss$H-4JnX@UV3T4\"\" Ot/o4Q%HC*)obQ$>;FM`$y#\\h>>F-&F%6$\"\")\"\"(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&,&#\"Hx$\\6Tk)[*zGL*3\"pTF%)zl6\"G +w8Yru:c5*eozr@JT!)3\"\"\"\"*&#\"STR(3!G4gQ'*f'\\GSb[/28bYXj^w$\"R+SA) p]eep#=V%QxsA>@Oij7p(Q%F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',&#\"P$*Reo1y%e4\"*R-\"RC[?;5M0:g!>\" O+w\")=KQkMQK6z^\"y@o$)pyx&[%yk6:\"K?N \\?D,O7om:e4W8#)[Jwo eBpkG\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#6\"\")#!KEUyQ!4(\\i*Rf$\\g,]3cZ(zC'\"Jlk^,***f%Gz;1l_h.$*Ggt(\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"5)3EVZR.Ib'Q\"5 b#zn)p8j3!3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!7'p l>tm'p$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 12 "" 1 " " {XPPMATH 20 "6-/&%\"aG6$\"#7\"\"\"#\"+6%o+Z'\"-+GX_q7/&F%6$F'\"\"#\" \"!/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'#\"5tUDBA PDl.9\"5+cu_TR?%Hw&/&F%6$F'\"\"(#\"7J\"z\"RIs!)GKat\"8+wp!4h^')Gk@I/&F %6$F'\"\")#\"6$[*Gli)Rd2PA\"7+#4+mx!Had,6/&F%6$F'\"\"*#\"1>?o3RD:`\"2+ _@/nT^G$/&F%6$F'\"#5#\"3;#*>\\>DZqM\"4v0o)*f?01o-o[G#/&F%6#\" #5#\"@zlCO/u>(3MO\"=Hy8#\"A]([4.pxTs[a4rkZ0\"/&F%6#\"#6#!7$z)>^2&*HpYm V\"8+gtKbodJ;c`$/&F%6#\"#7#F'\"$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 0 151 "RK7_11eqs : = [op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'expand ed'))]:\n`RK6_12eqs*` := subs(b=`b*`,OrderConditions(6,12,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Chec k: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(e8,RK7 _11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(e8,`RK6_1 2eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#P" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "We choos e a value for " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" } {TEXT -1 85 " that makes the principal error norm for the order 7 sch eme approximately a minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "errterms := PrincipalErrorT erms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm : = sm+(evalf(subs(e8,errterms[ct])))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(sqrt(sm ),a[8,7]=0.2..0.615,color=COLOR(RGB,.5,0,1),thickness=2,font=[HELVETIC A,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 277 277 {PLOTDATA 2 "6(-%'CUR VESG6#7S7$$\"\"#!\"\"$\"gnY:#>tJAO6L)=Dx_CV&=[H\\l='Q\"\\f(>V8!#k7$$\" gnnmmmmmmmmmmmmmmmmmmmmmm;z*=e/4#!#g$\"gnqs&*o*RrF&)\\#>#eun\"=Q:`#zx' y$p;bY^L\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLL$3_qPl\"p@F1$\"gn/K\"o([w)[fI -\"H&yH\"426L\"Gq-@juP)RG8F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmmmT&3%)zwD#F1$ \"gn#G3S/6'4k*)*\\p(yt\")\\\"[(z94!**RQg#)*4@8F-7$$\"gnnmmmmmmmmmmmmmm mmmmmmmm\"zf.\"yYBF1$\"gn6?.C!)[?gDw^(>^m(>/vm&p&[m@tg%oSJ\"F-7$$\"gnL LLLLLLLLLLLLLLLLLLLLL$3x'[(eaV#F1$\"gnm!p_y0fVvrbv)f$o*HV/PTH:?%f9#**Q 28F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmm;a))RVn4.Ky -&y8e&)\\;l__F\"F-7$$\"gn+++++++++++++++++++++++]([Q%*y.$F1$\"gntZ#Q=? NUdd6n**3z))[lh,htZ6\"3Nw\"3F\"F-7$$\"gn+++++++++++++++++++++++]i&yrx7 $F1$\"gnI)Hz)=EcvZ%fM)))H\\(ek2T.pXOS(pbsm7F-7$$\"gn++++++++++++++++++ +++++]7:`Q9KF1$\"gnHAco*G4gV'p\"z@_)HsIo8qPBAT'*=y7j7F-7$$\"gnnmmmmmmm mmmmmmmmmmmmmm;aj(pQIH$F1$\"gn!=\"\\\\V&[IjgH1+_(\\xc*=X+C$='z8t$[\\c1E*[*pD\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLLL$GP%zlMF1$\"gn3\\PLGP I8i0P'pij(4]oj)=B,9ZwtKYD\"F-7$$\"gn++++++++++++++++++++++](=Fwczb$F1$ \"gnTO))f5)zD;qpw&H?\\\\)4n,s,gM0kz_AD\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLL LLL3%4O&ROF1$\"gnu]%=kVk(z)o\"=i&*=)G>zFX^\"o,Oz-zZ]7F-7$$\"gn++++++++ ++++++++++++++](=-bT!HPF1$\"gn:e34)3Vvm0\"z]bz;VtnY^7OvGpL1*)[7F-7$$\" gn++++++++++++++++++++++]il5>F9QF1$\"gn_Fy][$G[;QV1%3<*oU*>1*f-@D2,kIx C\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLLL3_]3?.RF1$\"gnI\\0%QDNJ#*yFF&\\,a@8 5K(=-y$eoko)oC\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLL$3FzPl[)RF1$\"gn%ei'=be '*>A)QtJ]()poH*>%RLJQ99QUkC\"F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmmm\"HKg^H2% F1$\"gn(f(Qo;)\\%3TSJQjhXe6ti(zCX&zZJzJY7F-7$$\"gnnmmmmmmmmmmmmmmmmmmm mm;a8N%[W;%F1$\"gn&y#Q6%G(HNiP9y,L%F1$\"gn[T*41[7F-7$$\"gn++++++++++++++++++++++++vd%))*=WF1$\"gnv&)Q.,hq!=Dghv[o(* )QWN&4O\\uf'pGR\\7F-7$$\"gn+++++++++++++++++++++++v$R))Hf]%F1$\"gnh'3Y uXPWPw!H(zq'e'4Ijqk!yx-I\">b5D\"F-7$$\"gn++++++++++++++++++++++](=U=]+ f%F1$\"gnQ(R!\\#o$4t[x,gD9nh<'R%=;x[B)\\bD\"F-7$$\"gnLLLLL LLLLLLLLLLLLLLLLLLL39mPnZF1$\"gn\\+0eL%pyNyUkgKJxW/wB@h?SGsS&=e7F-7$$ \"gn+++++++++++++++++++++++]ibY)p&[F1$\"gnp+(*zd[*)H'eG\"3*)o!oC7jy`41 wt8Ke8E\"F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmm;ajfG=Q\\F1$\"gn*>K%Q%)42Gk')) )\\M$>(f%Rorj:Rz$oZ7bk7F-7$$\"gn+++++++++++++++++++++++](Q%R&p-&F1$\"g n,L577\"oG-(*4gSsl/RQ^'e4<4C$)3\\Qo7F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLL$3_ gh![5^F1$\"gnE/gK`L;T0*[D;'*)e0tW+HTR8,9yjJs7F-7$$\"gn++++++++++++++++ ++++++]Pfb+z(>&F1$\"gns*fO/!)yv)RH7c_CmDRit,*3@8*p%yenF\"F-7$$\"gnLLLL LLLLLLLLLLLLLLLLLLL$e9*4:$G&F1$\"gn'[X!o&GH/Q?q,m&4[F//#er&yd;0bwU\"G \"F-7$$\"gn++++++++++++++++++++++](o/)f^s`F1$\"gnYaGzZnM)>.@Ap^r.2#>X9 -Rq&yHicmG\"F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmmmm;'3&eeaF1$\"gnB\\MRM`e)3Q #oV&**4$o(pB>VLy*yC*o=?H\"F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmmm\"zz<.ma&F1$ \"gnRE(e*HN2L=:NH6:.q+V]4F(*RD$[wGyH\"F-7$$\"gnLLLLLLLLLLLLLLLLLLLLLL$ 3_bV#*Qj&F1$\"gn-Tr&F1$\"gn-nc'3z+h>u;#3v_;JQ:zI1&Q%G:w;y48F-7$$\"gnL LLLLLLLLLLLLLLLLLLLLLLe*ojJg!eF1$\"gnfO=E'*4aiJ1_h++!>\\$**o())3mVn $oJ\"F-7$$\"gnnmmmmmmmmmmmmmmmmmmmmmmm;BWD))eF1$\"gnj6rP07!)42IZevG^!Q Wtgc$[dGQvhUB8F-7$$\"gn++++++++++++++++++++++](=F1@f(fF1$\"gnIulc6Gar/ jn[0-#e`9Z]bl(H.1`BvI8F-7$$\"gn++++++++++++++++++++++]7yA#H)fgF1$\"gn# z-.`Hv[ioB!GtvVMpbj7^KCDQk`/Q8F-7$$\"$:'!\"$$\"gn&o)HYjQ0p([nuKWjiM:!e sm**ed6@Q=Y8F--%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\"&F*$\" \"!Ff[l$\"\"\"Ff[l-%*THICKNESSG6#F)-%+AXESLABELSG6$Q'a[8,7]6\"Q!F`\\l- %%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "evalf[12](minimize(sqrt(sm),a[8,7] =0..1,'location'));\nconvert(subs(op(1,op(op(2,[%]))),a[8,7]),rational ,4);\nconvert(subs(op(1,op(op(2,[%%]))),a[8,7]),rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-:xIJY7!#;<#7$<#/&%\"aG6$\"\")\"\"($\"-bN ZleS!#7F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#8\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#p\"$q\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "evalf[10](eval(sqrt(sm),a[8, 7]=13/32));\nevalf[10](eval(sqrt(sm),a[8,7]=69/170));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+HMJY7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+yIJY7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e9 := `union`(subs(a[8,7]=13/32,e8),\{a[8,7]=13/32 \}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4574 " e9 := \{b[6] = 14036525372223254273/57629420394152745600, a[9,6] = 590 3957290670756919471851965687658329936372357281397/47010380988313571014 61987812303657884827097474792000, a[10,7] = -1423831116879182018618807 67560889949537519404224643/1360844456590984100517911191904726072336914 0257280, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[9,3] = 0, c[2] = 1/100, a[7,2] = 0, a[7,3] = 0, a[2,1] = 1/100, `b*`[2] = 0, `b*`[3] = 0, `b* `[4] = 0, `b*`[5] = 0, b[1] = 6470068411/127052452800, c[4] = 1/5, a[1 1,7] = -2933204189652449798399861612546549582225310986953/205013581499 999542677439978824334676268210882720, a[11,5] = 4227012881794513094958 57528449967681683868959323295591/2976609753242264616837849775985514087 0411136751616000, c[9] = 19/23, c[10] = 81/82, `b*`[9] = 4185379964312 30550055107156803/2284868026806051955976984640000, a[11,1] = 413401339 654426004756492280722047075004416104055883/719868089323382472727694115 370573342900039059456000, a[11,4] = 4068247489923915770015231221964449 550938389658325/222675331628898775507843272599152181481922934868, a[5, 1] = 551874974720/997302429271, b[9] = 5315253908682019/32851416704215 200, a[10,6] = -185134546834429426147763500983818463859511714158264061 2437867/420703407864684992323487936970444634267986659475521049568000, \+ a[12,7] = 7354322880723039179131/30216428865161090697600, a[12,10] = 3 47047251949199216/1213217205998680575, `b*`[10] = 21378291813634087197 40436246579/10547647109544872417769030948750, a[10,5] = 72703899173100 547975701530898752038618015765157471787669321/694468405916580716186154 2521065278114605347407607709696000, a[10,9] = 157098072980712213312920 2485/2170749583944689600835012992, a[12,6] = 14036525372223254273/5762 9420394152745600, a[12,11] = -95980357/508344480, a[8,6] = -4913831081 9605862695843/962154393150421608110400, a[9,1] = -28052920640951751059 94189803671936576349564929/1954397004083418697841845098464629688499179 52000, a[9,7] = 80638533780709324566519854325500435393/316647699350639 48010841554250731537120, a[8,4] = -26470464436388981652125/23613628809 2530359764076, a[12,9] = 5315253908682019/32851416704215200, c[11] = 1 , a[6,2] = 0, c[12] = 1, a[8,5] = 825734282216068905765330677/63130978 45989067825180262400, a[5,3] = -2086699008000/997302429271, `b*`[12] = 1/160, a[9,4] = -44489197025459151705444370840723674742716548825/1493 2376340320772420636844398178733063223595932, a[12,5] = 0, b[2] = 0, b[ 3] = 0, b[4] = 0, b[5] = 0, a[11,8] = -6247974756085001604935939962497 09038784226/49773602893036152650616792845999901516465, b[10] = 3470472 51949199216/1213217205998680575, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0 , c[3] = 2/15, c[7] = 52/313, c[8] = 63/157, b[8] = 223707573986265289 483/1101575429077660009200, `b*`[11] = -4366466929950751198793/3535616 3157685532736000, a[7,1] = 21278500033609/276420238876800, a[7,5] = -2 901059421298252903/76225368652427491200, a[3,1] = -34/45, a[7,4] = 989 3050830728000/91236732526195641, c[5] = 5120/9991, `b*`[8] = 162832454 30995038704640712154556497/76616314592355159097734873145440000, `b*`[6 ] = 911082200138648538937372713121003/40082174004102780391163322739200 00, a[10,1] = 18745022701169741445732083919370592367818346398255185593 /43542944158840438871835940146729085397585013246181376000, a[7,6] = 43 6379361176824/23234416797464775, a[9,8] = 3802360970021472751124477077 9309498/15560366705131162570848900057690585, b[7] = 735432288072303917 9131/30216428865161090697600, `b*`[7] = 503904010603015936847352266508 232049/2101600452117104186799675302680320000, a[10,8] = -1576201159997 5752003382750329296637334838001/17053826861276101347655197690505547226 92480, a[10,2] = 0, a[10,3] = 0, a[11,2] = 0, a[11,3] = 0, a[3,2] = 8/ 9, a[4,2] = 0, a[5,2] = 0, a[6,3] = 0, a[11,10] = -6830025351434171965 696/369666731905701279737985, a[9,5] = -538125605455123224555096530708 947439830683725706539/221787063147752862838752824077163711222570059776 000, a[10,4] = 174955774658845813020015026443375837215431521574722425/ 12988012824390651292175460014153260035238111030958552, b[11] = -959803 57/508344480, a[11,6] = -647563362956678228844742602772133226055290595 66403398599943/1059203433376370397064499169756178873592691473042275207 2000, `b*`[1] = 457593782868427426344821/8836699182375187992960000, a[ 8,1] = 627197176791399348709/22893533288085131673600, a[12,8] = 223707 573986265289483/1101575429077660009200, a[12,1] = 6470068411/127052452 800, a[11,9] = 38655300339474326088/40800863136986779255, c[6] = 60/97 , a[4,1] = 1/20, a[6,1] = 70365/1204352, a[5,4] = 2045902848000/997302 429271, a[4,3] = 3/20, a[6,4] = 15000000/48955027, a[8,7] = 13/32, a[6 ,5] = 16390905/64600448\}:" }}}{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 62 "The coefficients for the combined sch eme are given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1011 "subs(e9,matrix([[c[2],a[2,1],``],\n[c[3],a[3,1],a[3,2]],[c[4],a[ 4,1],a[4,2]],[``$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]],[``$2,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]],[``$2,a[8,7]],\n[c[9],a[9,1],a[9 ,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6]],[``,a[9,7],a[9,8]],\n[c[10] ,a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[``,a[10,5],a[10,6]],\n[``,a[10 ,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[11,1],a[11,2]],[``,a[11,3],a[11 ,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7],a[11,8]],[``,a[11,9],a[11,10]] ,\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a[12,4]],[``,a[12,5],a[12,6]], \n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,10]],[``$2,a[12,11]],\n[``,`__ ______________________________`$2],[`b`,b[1],b[2]],[``,b[3],b[4]],[``, b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10]],[``$2,b[11]],\n[`b*`,`b*`[ 1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[6]],\n[``,`b*`[7],`b *`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11],`b*`[12]]]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\"\"\"\"$+\"F(%!G7%#\"\"#\"#:#! #M\"#X#\"\")\"\"*7%#F)\"\"&#F)\"#?\"\"!7%F+F+#\"\"$F:7%#\"%?^\"%\"***# \"-?Z(\\(=b\"-r#HCI(**F;7%F+#!.+!3!*p'3#FE#\".+![G!f/#FE7%#\"#g\"#(*# \"&l.(\"(_V?\"F;7%F+F;#\")+++:\")F]&*[7%F+F+#\")04R;\")[/gk7%#\"#_\"$8 $#\"/4O.+&y7#\"0+o()Q-Uw#F;7%F+F;#\"1+!G2$30$*)*\"2Tc>EDtO7*7%F+#!4.HD )H@%f5!H\"5+7\\FClo`Aw#\"0CoF'\"8+OnJ^3)GLN*G#F;7%F+F;#!8D@l\")*)QOWYqk#\"9wSwf.`#4)GOhB7%F+ #\"3JQ\"\\\"9+/63;U]JRa@ '*7%F+F+#\"#8\"#K7%#\"#>\"#B#!OH\\c\\jdO>n.)*=%*f5v^4k?H0G\"Q+?&z\"*\\ )oHYY)4X=%yp=M3/qRa>F;7%F+F;#!PD)[lrUZnB2%3PWaq^\"fa-(>*[W\"PKffBK1L(y \")RWoj?Cx?.MwB$\\\"7%F+#!TRlqDPoI)Ru%*32`'4bXAB^X0c7Q&\"T+gxf+dA7rjr2 CGvQG'GvZJ1(y@##\"U(R\"GdBPO*H$ewol>&=Z>pvq1HdR!f\"U+?zuu4F[)yl.B\"y)> Y,rNJ))4Q5q%7%F+#\"G$RN/]DV&)>lcC$42yL&Q1)\"G?r`J2Da:%3,[R1N*pZmJ#\"D) \\4$z2xW7^FZ@+(4O-Q\"D&e!pd+!*[3di680nOgb\"7%#\"#\")\"##)#\"Y$f&=b#)RY $=yO#fq$>R3KdWT(p6qA]u=\"Y+gP\"=YK,&e(R&3Hn9Sf$=()Q/%)eT%HaVF;7%F+F;# \"WDCsu:_Ja@PePVk-:+-8e%)eYxb\\<\"V_&e4.6\"Q_.gK:9+Yv@H^1RCG,))H\"7%F+ #\"fn@$pwyru:ld,='Q?v)*3`,d(za+J<**Qqs\"en+gp4xg2uM0Y6y_1@Dah=;2e;fSoW p#!hnnyV71k#eTr6&fQY=Q)4]jx9E%HW$oaM^=\"gn+!o&\\5_v%fm)zEMYWqp$z[BB*\\ okyS.2U7%F+#!TVYA/%>v`\\**)3cn2)='=?=zo6JQU\"\"S!Gd-9pLsgs/>>6z^+T)4fc W%3O\"#!M,!Q[LPmHH.v#Q.?vv**f6?w:\"L![#pAZb]!p(>bwM,hFho#Q0<7%F+F+#\"= &[-#HJ8Ar!)H2)4d\"\"=#*H,N3g*oWRe\\2<#7%F)#\"T$)e0/hT/]2Z?s!G#\\cZ+EWl R8S8%\"T+gXf!R+!HMt0P:TpFFZ#QB$*3o)>(F;7%F+F;#\"RD$e'*QQ4b\\W'>AJ_,qd \"R#**[Z#oS\"Qo[$H#>[\"=_\"*fsK%y]v()*)G;LvEA7%F+#\"W\"f&HB$f*oQo\"on* \\%Gv&e\\48Xz\")G,FU\"V+gh^n86/(39b)fx\\y$ohkAC`(4m(H#!fnV**f)R.kcf!Hb gAL@x-EuW)G#ym&HOjvk\"fn+?2_FUIZ\"p#ft)yhvp\"*\\kqRqjPLM?f57%F+#!R`p)4 JDAe\\la7;')*R)z\\Cl*=/K$H\"Q?F)3@oinMV#)y*RunU&****\\\"e8]?#!KEUyQ!4( \\i*Rf$\\g,]3cZ(zC'\"Jlk^,***f%Gz;1l_h.$*Ggt(\\7%F+#\"5)3EVZR.Ib'Q\"5b #zn)p8j3!3%#!7'pl>tm'p$7%F)#\"+6%o+Z'\"-+GX_q7F;7% F+F;F;7%F+F;#\"5tUDBAPDl.9\"5+cu_TR?%Hw&7%F+#\"7J\"z\"RIs!)GKat\"8+wp! 4h^')Gk@I#\"6$[*Gli)Rd2PA\"7+#4+mx!Had,67%F+#\"1>?o3RD:`\"2+_@/nT^G$# \"3;#*>\\>DZqM\"4v0o)*f?01o-o[G##\"@zlCO/u>(3MO\"=Hy8#\"A]([4.pxTs[a4rkZ0\"7%F+#!7$z)>^2&*Hp YmV\"8+gtKbodJ;c`$#F)\"$g\"Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "subs(e9,matrix([seq( [c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12),\n[`b`,seq(b[j],j=1..11 ),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[6](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'++5!\"(F(%!GF+F+F+F+F+F+F+F+F+F+7 /$\"'LL8!\"'$!'cbvF/$\"'*)))))F/F+F+F+F+F+F+F+F+F+F+7/$\"'++?F/$\"'++] F*$\"\"!F:$\"'++:F/F+F+F+F+F+F+F+F+F+7/$\"'hC^F/$\"'oLbF/F9$!'M#4#!\"& $\"'W^?FDF+F+F+F+F+F+F+F+7/$\"'d&='F/$\"'cUeF*F9F9$\"'/kIF/$\"'FPDF/F+ F+F+F+F+F+F+7/$\"'Mh;F/$\"')yp(F*F9F9$\"'L%3\"F/$!'!f!QF*$\"';y=F*F+F+ F+F+F+F+7/$\"'u7SF/$\"'jRFF*F9F9$!')47\"F/$\"'(zI\"F/$!'62^F*$\"']iSF/ F+F+F+F+F+7/$\"'(3E)F/$!'PN9F*F9F9$!'QzHFD$!'KECFD$\"')eD\"FD$\"'jYDFD $\"'iVCFDF+F+F+F+7/$\"'0y)*F/$\"'&\\I%F/F9F9$\"'1Z8!\"%$\"'!p/\"Fhp$!' g+WFD$!'GY5Fhp$!'^U#*FD$\"'/PsF/F+F+F+7/$\"\"\"F:$\"'uUdF/F9F9$\"'*p#= Fhp$\"'3?9Fhp$!'o8hFD$!'uI9Fhp$!'Gb7Fhp$\"'9u%*F/$!'iZ=F*F+F+7/Fdq$\"' W#4&F*F9F9F9F9$\"'lNCF/$\"')QV#F/$\"'!3.#F/$\"'(zh\"F/$\"'bgGF/$!'5))= F/F+7/%\"bGFgrF9F9F9F9FirF[sF]sF_sFasFcsF+7/%#b*G$\"'Ly^F*F9F9F9F9$\"' /tAF/$\"'s(R#F/$\"'ID@F/$\"'yJ=F/$\"'$o-#F/$!'*\\B\"F/$\"'+]i!\")Q)ppr int126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded 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 4506 "ee := \{c[2]=1/100,\nc[3]=2/15,\nc[4]=1/5,\nc[5]=5120/9991,\nc[6 ]=60/97,\nc[7]=52/313,\nc[8]=63/157,\nc[9]=19/23,\nc[10]=81/82,\nc[11] =1,\nc[12]=1,\n\na[2,1]=1/100,\na[3,1]=-34/45,\na[3,2]=8/9,\na[4,1]=1/ 20,\na[4,2]=0,\na[4,3]=3/20,\na[5,1]=551874974720/997302429271,\na[5,2 ]=0,\na[5,3]=-2086699008000/997302429271,\na[5,4]=2045902848000/997302 429271,\na[6,1]=70365/1204352,\na[6,2]=0,\na[6,3]=0,\na[6,4]=15000000/ 48955027,\na[6,5]=16390905/64600448,\na[7,1]=21278500033609/2764202388 76800,\na[7,2]=0,\na[7,3]=0,\na[7,4]=9893050830728000/9123673252619564 1,\na[7,5]=-2901059421298252903/76225368652427491200,\na[7,6]=43637936 1176824/23234416797464775,\na[8,1]=627197176791399348709/2289353328808 5131673600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-26470464436388981652125/236 136288092530359764076,\na[8,5]=825734282216068905765330677/63130978459 89067825180262400,\na[8,6]=-49138310819605862695843/962154393150421608 110400,\na[8,7]=13/32,\na[9,1]=-28052920640951751059941898036719365763 49564929/\n 195439700408341869784184509846462968849917952000,\n a[9,2]=0,\na[9,3]=0,\na[9,4]=-4448919702545915170544437084072367474271 6548825/\n 14932376340320772420636844398178733063223595932,\na[ 9,5]=-538125605455123224555096530708947439830683725706539/\n 22 1787063147752862838752824077163711222570059776000,\na[9,6]=59039572906 70756919471851965687658329936372357281397/\n 4701038098831357101 461987812303657884827097474792000,\na[9,7]=806385337807093245665198543 25500435393/31664769935063948010841554250731537120,\na[9,8]=3802360970 0214727511244770779309498/15560366705131162570848900057690585,\na[10,1 ]=18745022701169741445732083919370592367818346398255185593/\n 4 3542944158840438871835940146729085397585013246181376000,\na[10,2]=0,\n a[10,3]=0,\na[10,4]=17495577465884581302001502644337583721543152157472 2425/\n 12988012824390651292175460014153260035238111030958552, \na[10,5]=72703899173100547975701530898752038618015765157471787669321/ \n 6944684059165807161861542521065278114605347407607709696000, \na[10,6]=-18513454683442942614776350098381846385951171415826406124378 67/\n 42070340786468499232348793697044463426798665947552104956 8000,\na[10,7]=-142383111687918201861880767560889949537519404224643/\n 13608444565909841005179111919047260723369140257280,\na[10,8]= -15762011599975752003382750329296637334838001/170538268612761013476551 9769050554722692480,\na[10,9]=1570980729807122133129202485/21707495839 44689600835012992,\na[11,1]=413401339654426004756492280722047075004416 104055883/\n 71986808932338247272769411537057334290003905945600 0,\na[11,2]=0,\na[11,3]=0,\na[11,4]=4068247489923915770015231221964449 550938389658325/\n 22267533162889877550784327259915218148192293 4868,\na[11,5]=422701288179451309495857528449967681683868959323295591/ \n 29766097532422646168378497759855140870411136751616000,\na[11 ,6]=-64756336295667822884474260277213322605529059566403398599943/\n \+ 10592034333763703970644991697561788735926914730422752072000,\na[ 11,7]=-2933204189652449798399861612546549582225310986953/\n 20 5013581499999542677439978824334676268210882720,\na[11,8]=-624797475608 500160493593996249709038784226/497736028930361526506167928459999015164 65,\na[11,9]=38655300339474326088/40800863136986779255,\na[11,10]=-683 0025351434171965696/369666731905701279737985,\na[12,1]=6470068411/1270 52452800,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=140 36525372223254273/57629420394152745600,\na[12,7]=735432288072303917913 1/30216428865161090697600,\na[12,8]=223707573986265289483/110157542907 7660009200,\na[12,9]=5315253908682019/32851416704215200,\na[12,10]=347 047251949199216/1213217205998680575,\na[12,11]=-95980357/508344480,\n \nb[1]=6470068411/127052452800,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb [6]=14036525372223254273/57629420394152745600,\nb[7]=73543228807230391 79131/30216428865161090697600,\nb[8]=223707573986265289483/11015754290 77660009200,\nb[9]=5315253908682019/32851416704215200,\nb[10]=34704725 1949199216/1213217205998680575,\nb[11]=-95980357/508344480,\n\n`b*`[1] =457593782868427426344821/8836699182375187992960000,\n`b*`[2]=0,\n`b*` [3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=911082200138648538937372713121 003/4008217400410278039116332273920000,\n`b*`[7]=503904010603015936847 352266508232049/2101600452117104186799675302680320000,\n`b*`[8]=162832 45430995038704640712154556497/76616314592355159097734873145440000,\n`b *`[9]=418537996431230550055107156803/2284868026806051955976984640000, \n`b*`[10]=2137829181363408719740436246579/105476471095448724177690309 48750,\n`b*`[11]=-4366466929950751198793/35356163157685532736000,\n`b* `[12]=1/160\}:" }}}{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[7,11];" "6#&%\"TG6$\"\"( \"#6" }{TEXT -1 128 " denote the vector whose components are the prin cipal error terms of the 11 stage, order 7 scheme (the error terms of \+ order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6 , 12];" "6#&%#T*G6$\"\"'\"#7" }{TEXT -1 146 " denote the vector whose components are the principal error terms of the embedded 12 stage, or der 6 scheme (the error terms of order 7) and let " }{XPPEDIT 18 0 "` T*`[7,12];" "6#&%#T*G6$\"\"(\"#7" }{TEXT -1 99 " denote the vector wh ose components are the error terms of order 8 of the embedded order 6 \+ scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T [7, 11]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "abs(abs(`T*`[6,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" #7" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[7,12]));" "6#-% $absG6#-F$6#&%#T*G6$\"\"(\"#7" }{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[8] = abs(abs(T[7, 1 1]));" "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,12]))/abs(abs(`T*`[6, 12 ]));" "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\"#7\"\"\"-F*6#-F* 6#&F/6$\"\"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(a bs(`T*`[7,12]-T[7,11]))/abs(abs(`T*`[6,12]));" "6#/&%\"CG6#\"\")*&-%$a bsG6#-F*6#,&&%#T*G6$\"\"(\"#7\"\"\"&%\"TG6$F2\"#6!\"\"F4-F*6#-F*6#&F06 $\"\"'F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[8];" "6#&%\"AG6#\"\") " }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used f or error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\")" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \+ \"similar in magnitude\" and also not differ too much from 1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := PrincipalErrorTerms(7,11,'expanded'):\n`errterms7 _12*` :=subs(b=`b*`,PrincipalErrorTerms(7,12,'expanded')):\n`errterms6 _12*` := subs(b=`b*`,PrincipalErrorTerms(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "s nmB := sqrt(add(evalf(subs(ee,`errterms7_12*`[i]))^2,i=1..nops(`errter ms7_12*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms6_12*`[i]))^2,i= 1..nops(`errterms6_12*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms 7_12*`[i])-subs(ee,errterms7_11[i])))^2,i=1..nops(errterms7_11))):\n'B [8]'= evalf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")<6l:!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")Qdo:!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" } }{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4506 "ee := \{c[2]=1/100,\nc[3]=2/15,\nc[4]=1/5,\nc[ 5]=5120/9991,\nc[6]=60/97,\nc[7]=52/313,\nc[8]=63/157,\nc[9]=19/23,\nc [10]=81/82,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/100,\na[3,1]=-34/45,\na[3, 2]=8/9,\na[4,1]=1/20,\na[4,2]=0,\na[4,3]=3/20,\na[5,1]=551874974720/99 7302429271,\na[5,2]=0,\na[5,3]=-2086699008000/997302429271,\na[5,4]=20 45902848000/997302429271,\na[6,1]=70365/1204352,\na[6,2]=0,\na[6,3]=0, \na[6,4]=15000000/48955027,\na[6,5]=16390905/64600448,\na[7,1]=2127850 0033609/276420238876800,\na[7,2]=0,\na[7,3]=0,\na[7,4]=989305083072800 0/91236732526195641,\na[7,5]=-2901059421298252903/76225368652427491200 ,\na[7,6]=436379361176824/23234416797464775,\na[8,1]=62719717679139934 8709/22893533288085131673600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-264704644 36388981652125/236136288092530359764076,\na[8,5]=825734282216068905765 330677/6313097845989067825180262400,\na[8,6]=-49138310819605862695843/ 962154393150421608110400,\na[8,7]=13/32,\na[9,1]=-28052920640951751059 94189803671936576349564929/\n 195439700408341869784184509846462 968849917952000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-4448919702545915170544 4370840723674742716548825/\n 1493237634032077242063684439817873 3063223595932,\na[9,5]=-5381256054551232245550965307089474398306837257 06539/\n 221787063147752862838752824077163711222570059776000,\n a[9,6]=5903957290670756919471851965687658329936372357281397/\n 4 701038098831357101461987812303657884827097474792000,\na[9,7]=806385337 80709324566519854325500435393/31664769935063948010841554250731537120, \na[9,8]=38023609700214727511244770779309498/1556036670513116257084890 0057690585,\na[10,1]=1874502270116974144573208391937059236781834639825 5185593/\n 4354294415884043887183594014672908539758501324618137 6000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=1749557746588458130200150264433 75837215431521574722425/\n 129880128243906512921754600141532600 35238111030958552,\na[10,5]=727038991731005479757015308987520386180157 65157471787669321/\n 694468405916580716186154252106527811460534 7407607709696000,\na[10,6]=-185134546834429426147763500983818463859511 7141582640612437867/\n 420703407864684992323487936970444634267 986659475521049568000,\na[10,7]=-1423831116879182018618807675608899495 37519404224643/\n 13608444565909841005179111919047260723369140 257280,\na[10,8]=-15762011599975752003382750329296637334838001/1705382 686127610134765519769050554722692480,\na[10,9]=15709807298071221331292 02485/2170749583944689600835012992,\na[11,1]=4134013396544260047564922 80722047075004416104055883/\n 719868089323382472727694115370573 342900039059456000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=40682474899239157 70015231221964449550938389658325/\n 222675331628898775507843272 599152181481922934868,\na[11,5]=42270128817945130949585752844996768168 3868959323295591/\n 2976609753242264616837849775985514087041113 6751616000,\na[11,6]=-647563362956678228844742602772133226055290595664 03398599943/\n 10592034333763703970644991697561788735926914730 422752072000,\na[11,7]=-2933204189652449798399861612546549582225310986 953/\n 205013581499999542677439978824334676268210882720,\na[11 ,8]=-624797475608500160493593996249709038784226/4977360289303615265061 6792845999901516465,\na[11,9]=38655300339474326088/4080086313698677925 5,\na[11,10]=-6830025351434171965696/369666731905701279737985,\na[12,1 ]=6470068411/127052452800,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5 ]=0,\na[12,6]=14036525372223254273/57629420394152745600,\na[12,7]=7354 322880723039179131/30216428865161090697600,\na[12,8]=22370757398626528 9483/1101575429077660009200,\na[12,9]=5315253908682019/328514167042152 00,\na[12,10]=347047251949199216/1213217205998680575,\na[12,11]=-95980 357/508344480,\n\nb[1]=6470068411/127052452800,\nb[2]=0,\nb[3]=0,\nb[4 ]=0,\nb[5]=0,\nb[6]=14036525372223254273/57629420394152745600,\nb[7]=7 354322880723039179131/30216428865161090697600,\nb[8]=22370757398626528 9483/1101575429077660009200,\nb[9]=5315253908682019/32851416704215200, \nb[10]=347047251949199216/1213217205998680575,\nb[11]=-95980357/50834 4480,\n\n`b*`[1]=457593782868427426344821/8836699182375187992960000,\n `b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=91108220013864 8538937372713121003/4008217400410278039116332273920000,\n`b*`[7]=50390 4010603015936847352266508232049/2101600452117104186799675302680320000, \n`b*`[8]=16283245430995038704640712154556497/766163145923551590977348 73145440000,\n`b*`[9]=418537996431230550055107156803/22848680268060519 55976984640000,\n`b*`[10]=2137829181363408719740436246579/105476471095 44872417769030948750,\n`b*`[11]=-4366466929950751198793/35356163157685 532736000,\n`b*`[12]=1/160\}:" }}}{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 78 "The stability function R for th e 11 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(7,11,'expanded')):\n R := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/- %\"RG6#%\"zG,:\"\"\"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)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\"M>\\63rDpN(H([:4! >xk@!HkW5\"Q+?>j%*)z&z\\H#39Zi$)GUBLTM%GUF)*$)F'\"\")F)F)F)*&#\"N42H)) oX:(['>a.u!RNdc)3>Ra'\"T++BGu@^u:Qo:xN7O%*34W*=S?@DF)*$)F'\"\"*F)F)F)* &#\"MTg-nU#3)QQ,/SZw/dc'fT\\g$\"TD\"Q,TH*)p!Rk@XO&eMUQ`\"o>yJXD%F)*$)F '\"#5F)F)F)*&#\"Kk[3gZCZ'H(Qe9@!HXL6odE*\"Sveng>E8FHW,V-RchD-@J@XNOGF) *$)F'\"#6F)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,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=-1,z=-4.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+Lxz=Y!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-4.6 ):\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)):\np lots[display]([p1,p2,p3],view=[-5.19..0.49,-1.47..1.47],font=[HELVETIC A,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CUR VESG6$7V7$$!3Q++++++!>&!#<$!3[%**f**3q/W$F*7$$!3Ynm;#3'4G^F*$!3`AN&z2c 2.$F*7$$!3kLLLk@>m]F*$!3;(*ys([Ncm#F*7$$!3E+]U'*)HB,&F*$!3s.n)Row2Q#F* 7$$!3!pm;&GwYe\\F*$!3;QTJ5'eO7#F*7$$!3s+](\\(Q*y*[F*$!37myT#3#fk=F*7$$ !3nLLV@,KP[F*$!3jcB\"y&GIM;F*7$$!3NLL.q&p`r%F*$!3q(G27[)RY7F*7$$!3;nmJ G')*Rf%F*$!3NRA\"RtL=W*!#=7$$!3uLLyGAZ\"[%F*$!30gXXKmGSsFU7$$!3%3+])fw &\\O%F*$!31#z?44L([aFU7$$!3$QL$)f7eWC%F*$!3\")Gk?lK^7SFU7$$!3A++lN]MCT F*$!3?`*p$fiE8HFU7$$!3ummYeRz+SF*$!336#)[eo<_?FU7$$!3_LLV-,(>*QF*$!3;c g-4F**p9FU7$$!35++S:-YpPF*$!3MNlj=bKn'*!#>7$$!3K+++\"HZkk$F*$!3kj/S)fy 4&eFcp7$$!3;++gW:!z_$F*$!3e(pK!o)e)[IFcp7$$!3hLL)*\\1D?MF*$!3i:VF/:od5 Fcp7$$!3'ommSKVAH$F*$\"3qUV,%Guc7)!#?7$$!3/nmEGV!Q=$F*$\"3/mV!\\7ih4#F cp7$$!39++0(*RmdIF*$\"3%)>:h'=\\\"eLFcp7$$!39nmEI%3g%HF*$\"33<\\#=+n^M %Fcp7$$!3-++0xX]BGF*$\"3'\\DO#3pwb`Fcp7$$!3*)***\\\"R>&oq#F*$\"3mijpk- i)H'Fcp7$$!3gmm;\\r8&e#F*$\"3?A>'['=#eI(Fcp7$$!3ymmrw\\OtCF*$\"33Lu?O# HKG)Fcp7$$!3SLL$))e.GN#F*$\"30^&y!)[dHU*Fcp7$$!3nLL)**=uvA#F*$\"3uOKFI w&H2\"FU7$$!3K++:I;c=@F*$\"3G^V2EN6*>\"FU7$$!31LL.z]#3+#F*$\"3Cxm%G!*= 1N\"FU7$$!3M++?,<>z=F*$\"3ZVBRg6HE:FU7$$!3;++!4<(>gFU7$$!3H++q9zA<:F*$\"3%>7Ees(4$>#FU7$$! 3EnmEY;O-9F*$\"3!*=$RKd6,Y#FU7$$!3#)*****pQ<(z7F*$\"3#>iZ1qU6y#FU7$$!3 )RL$efMeo6F*$\"3D+tm[D13JFU7$$!3I****fAZ3Z5F*$\"3=(f8%z=f4NFU7$$!3xqm; (zQwK*FU$\"3qM$y$ykkMRFU7$$!3&z***\\)ecE8)FU$\"3'p)*>WiwSV%FU7$$!3'3nm m0VV'pFU$\"3+j]N88f$)\\FU7$$!3P)***\\iqATdFU$\"3M&o;1U()>j&FU7$$!3aFLL *)4AjXFU$\"37j[#4>(4OjFU7$$!33LLLO'R&eLFU$\"3%4a\")Quus9(FU7$$!3Uim;`O $Q;#FU$\"35VY0,ZEa!)FU7$$!3?*****>$H-m5FU$\"3]^1IQ4$)))*)FU7$$\"3v*QLL U?>#>Fcp$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\"FU$\"3Y/+-(HH39\"F*7$$\"3E0+] HcUiJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fa \\lF`\\l-F$6$7S7$F($!\"\"Fa\\l7$F3Ff\\l7$F=Ff\\l7$FGFf\\l7$FLFf\\l7$FQ Ff\\l7$FWFf\\l7$FfnFf\\l7$F[oFf\\l7$F`oFf\\l7$FeoFf\\l7$FjoFf\\l7$F_pF f\\l7$FepFf\\l7$FjpFf\\l7$F_qFf\\l7$FdqFf\\l7$FjqFf\\l7$F_rFf\\l7$FdrF f\\l7$FirFf\\l7$F^sFf\\l7$FcsFf\\l7$FhsFf\\l7$F]tFf\\l7$FbtFf\\l7$FgtF f\\l7$F\\uFf\\l7$FauFf\\l7$FfuFf\\l7$F[vFf\\l7$F`vFf\\l7$FevFf\\l7$Fjv Ff\\l7$F_wFf\\l7$FdwFf\\l7$FiwFf\\l7$F^xFf\\l7$FcxFf\\l7$FhxFf\\l7$F]y Ff\\l7$FbyFf\\l7$FgyFf\\l7$F\\zFf\\l7$FazFf\\l7$FfzFf\\l7$F[[lFf\\l7$F `[lFf\\l7$Fe[lFf\\l-Fj[l6&F\\\\lF`\\lF`\\lF]\\l-F$6&7#7$$!32+++Lxz=YF* Ff\\l-%'SYMBOLG6#%'CIRCLEG-Fj[l6&F\\\\lFa\\lFa\\lFa\\l-%&STYLEG6#%&POI NTG-F$6&F\\`l-Fa`l6#%&CROSSGFd`lFf`l-F$6&F\\`l-Fa`l6#%(DIAMONDGFd`lFf` l-F$6%7$7$F^`lF`\\lF]`l-%&COLORG6&F\\\\lF`\\l$\"\"&Fg\\lF`\\l-%*LINEST YLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fjbl-Fb bl6#%(DEFAULTG-%%VIEWG6$;$!$>&!\"#$\"#\\Fecl;$!$Z\"Fecl$\"$Z\"Fecl" 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 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1694 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/ 120*z^5+1/720*z^6+1/5040*z^7+\n 1044642902164771900915487297356925710 8114919/422843441332342288362471408229497957989463192000*z^8+\n 65439 1908856573539074035419648715456888290709/25212040189440908943612357715 6838157451217428230000*z^9+\n 360494159656570476474004013838808242670 26041/425453178196815338423458536452164390698929410138125*z^10+\n 926 576811334529021145838729647244760084864/283635452131210225615639024301 44292713261960675875*z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 280 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(RG B,.2,0,.45)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.3,0] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.55,. 15,1)):\npts := []: z0 := 2+5.6*I:\nfor ct from 0 to 40 do\n zz := n ewton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[R e(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.2,0,.45)):\n p4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.92,5.64]],i=2..nops( pts))],\n style=patchnogrid,color=COLOR(RGB,.55,.15,1)):\npts := []: z0 := 2-5.6*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=e xp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.2,0,.45)):\np6 := plots[ polygonplot]([seq([pts[i-1],pts[i],[1.92,-5.64]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.55,.15,1)):\np7 := plot([[[ -5.39,0],[2.39,0]],[[0,-6.19],[0,6.19]]],color=black,linestyle=3):\npl ots[display]([p||(1..7)],view=[-5.39..2.39,-6.19..6.19],font=[HELVETIC A,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=const rained);" }}{PARA 13 "" 1 "" {GLPLOT2D 386 493 493 {PLOTDATA 2 "6/-%'C URVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$ $\"3$*******[zw9?!#F$\"3D+++\")*)Q7ZF-7$$\"3#)*******QYU)=!#E$\"3=+++C `=$G'F-7$$\"32+++^4)f6\"!#D$\"3e*****zy\")R&yF-7$$\"3s******pek9`F@$\" 3;+++(*)yZU*F-7$$\"3++++YT-xA!#C$\"3++++gyb*4\"!#<7$$\"3a*****\\F0#))* )FK$\"3#*******f'QmD\"FN7$$\"31+++-\"y6B$!#B$\"32+++x;s89FN7$$\"3/+++' \\%yZ5!#A$\"35+++V(43d\"FN7$$\"3q*****pm]g1$Fgn$\"32+++6x!zs\"FN7$$\"3 _*****zsv%\\\")Fgn$\"3++++eA-&)=FN7$$\"37+++[Xb%)>!#@$\"3>+++!Qf@/#FN7 $$\"3))*****R./^Y%Fgo$\"3')*****\\!pJ*>#FN7$$\"3%******H_]([$*Fgo$\"3# )*****>%yYcBFN7$$\"31+++y:\\J=!#?$\"3&)*****R))GN^#FN7$$\"31+++jM')oLF gp$\"3#******\\*zIqEFN7$$\"3Q+++PS$[#eFgp$\"3%)*****\\nGk#GFN7$$\"3'3+ +ghHOX*Fgp$\"3!)******z=C\")HFN7$$\"3$******Ha-YV\"!#>$\"3.+++_VwLJFN7 $$\"39+++n[^B?F\\r$\"3'******>!>r#G$FN7$$\"3;+++>/&Rj#F\\r$\"3)******H t!oEMFN7$$\"3M+++]3sPJF\\r$\"3;+++8xUkNFN7$$\"3M+++-@nzLF\\r$\"33+++p! )3&p$FN7$$\"36+++S9l5KF\\r$\"3.+++7.!4MKF\\r$\"3C+++#)>YKUFN7$$!3N*****f%4Fd#>=F-$\"38+++OM()>XFN7$$!3/+++ x4/tXFN7$$!3w*****>jH(\\FF-$\"3')******QS_>YFN7$$!3 ))******)o,xB$F-$\"3')********QyfYFN7$$!39+++$y`\\t$F-$\"3'******pQtUp %FN7$$!3@+++p'*HQUF-$\"3m*****pp7Ms%FN7$$!3#******HnF`u%F-$\"3)******p 7!eZZFN7$$!3W+++K+Ca_F-$\"3m*****\\=3rw%FN7$$!3[*****f:=Pw&F-$\"3i**** *po$G#y%FN7$$!33+++9p\"GF'F-$\"3%******z,]Lz%FN7$$!3!)*****>\"f)3y'F-$ \"3n*****4P50![FN7$$!3=+++BS^(G(F-$\"3i*****f@JR![FN7$$!3W+++rM\\#z(F- $\"3l*****\\fXP![FN7$$!3M+++$e+eH)F-$\"3N+++ig0+[FN7$$!3Q+++l8f(z)F-$ \"3i******Gs$Hz%FN7$$!3!)*****zD6#)H*F-$\"3E+++#)zV#y%FN7$$!3!******Hd >#)z*F-$\"3#******4a$eoZFN7$$!35+++II%)H5FN$\"3-+++pwP^ZFN7$$!35+++/s* *z5FN$\"3J+++Gb!3t%FN7$$!3-+++jqVI6FN$\"3e*****\\/Qoq%FN7$$!3'******4v l8=\"FN$\"3'******\\8R%zYFN7$$!3-+++&R]IB\"FN$\"3k*****>.y&[YFN7$$!3++ ++Ar$eG\"FN$\"3M+++?,D9YFN7$$!3,+++UV;S8FN$\"3#*******)*4^wXFN7$$!3-++ +%>mlR\"FN$\"3U+++/*Gb`%FN7$$!3))*****4;]cX\"FN$\"35+++(3d;\\%FN7$$!3$ *******yW-=:FN$\"3q******[Y^XWFN7$$!35+++ST:%e\"FN$\"39+++Do.)R%FN7$$! 3#*******H7;a;FN$\"3y*****f@E/N%FN7$$!3&******\\cfws\"FN$\"3=+++-C&RI% FN7$$!3/+++C^v.=FN$\"3u*****fP]'fUFN7$$!3!******4]a7)=FN$\"3;+++v@3=UF N7$$!35+++:9'*e>FN$\"3)******>s&HzTFN7$$!3(******>\"f*e.#FN$\"3-+++;y& H9%FN7$$!30+++DBP6@FN$\"3u******)pS&3TFN7$$!3))******zV(\\=#FN$\"39+++ ++ZvSFN7$$!31+++B(*[cAFN$\"3V+++M%4K/%FN7$$!3$)*****>%Q%eK#FN$\"3#)*** **H.'H6SFN7$$!3))******H%[IR#FN$\"3A+++[nMzRFN7$$!3)******\\ol\"eCFN$ \"3)*******>=0ZRFN7$$!3%)*****\\N&G@DFN$\"3>+++VO;9RFN7$$!3%******fp5D e#FN$\"3!******\\E&[!)QFN7$$!3!)*****RJ]>k#FN$\"3#******fDge%QFN7$$!3- +++9Pr*p#FN$\"38+++%Qk,\"QFN7$$!3%******4;4fv#FN$\"39+++'4*HtPFN7$$!3) )******4Hk5GFN$\"3<+++\"f'=NPFN7$$!37+++]#>S'GFN$\"3))*****4Bmdp$FN7$$ !3.+++n49;HFN$\"3/+++'3#*\\l$FN7$$!37+++a*4r'HFN$\"3()******G9$Gh$FN7$ $!3!******pzFq,$FN$\"3')*****f.k#pNFN7$$!3#******>9'*f1$FN$\"36+++e@GC NFN7$$!3$******4q;T6$FN$\"3#)*****f0\"*yZ$FN7$$!35+++N2\\hJFN$\"3\"*** ****R*4,V$FN7$$!3$)******oy@3KFN$\"3))*****z8t4Q$FN7$$!3++++MTRaKFN$\" 3&)*****>AJ0L$FN7$$!3)******><4,I$FN$\"3++++6=&)yKFN7$$!33+++&oUaM$FN$ \"3********Q'>gA$FN7$$!31+++g.Y!R$FN$\"35+++Rb8sJFN7$$!38+++t'4_V$FN$ \"30+++BSJG4g#FN7$$!3')******eiV iQFN$\"3#******4AXCa#FN7$$!3()*****R_c9!RFN$\"35+++AW#Q[#FN7$$!3-+++3# e%RRFN$\"3))*****HFR]U#FN7$$!3#*******>/QwRFN$\"3')******)QagO#FN7$$!3 =+++0\\<7SFN$\"31+++)fGoI#FN7$$!3e*****\\s0o/%FN$\"35+++yvJZAFN7$$!3!* *****ze[-3%FN$\"35+++JgZ(=#FN7$$!3Q+++q**[7TFN$\"3#*******Q%fs7#FN7$$! 3()*****\\FN7$$!3i*****fc%\\HUFN$\"3++++L 1&=)=FN7$$!39+++Us$eD%FN$\"3(*******GG?>=FN7$$!3U+++3;2\"G%FN$\"3!**** **4O#*fv\"FN7$$!3%******\\'QB0VFN$\"3)******Ru3Ap\"FN7$$!3P+++ZNOGVFN$ \"33+++,#\\yi\"FN7$$!3)******fK-0N%FN$\"3)*******)>>Hc\"FN7$$!3w*****p j#prVFN$\"3'******H$GV(\\\"FN7$$!3a******[h(>R%FN$\"3)******z,89V\"FN7 $$!38+++&3#R6WFN$\"3'******pP\"*[O\"FN7$$!3N+++!\\v*HWFN$\"33+++kx!zH \"FN7$$!3Q+++@bvZWFN$\"32+++\"Q40B\"FN7$$!3i*****z!RvkWFN$\"3'******RV \\F;\"FN7$$!33+++HP)4[%FN$\"3%******4T&o%4\"FN7$$!3E+++:'[k\\%FN$\"36+ ++FqPE5FN7$$!3*)*****z\\U6^%FN$\"3Q+++x+%)y&*F-7$$!3O+++P*\\]_%FN$\"3S +++w&QE*))F-7$$!3))*****R*p9QXFN$\"3!)*****H5)p0#)F-7$$!3o*****\\d-/b% FN$\"3!******H*4\\=vF-7$$!3k*****p6!yhXFN$\"3L+++tmTJoF-7$$!3G+++Z%RAd %FN$\"38+++lOzWhF-7$$!3i*****fnQ#zc*3%F-7$$!3z* ****\\\\nSg%FN$\"3D+++)=$H1MF-7$$!3M+++,#zc*3%F-7$F`em$!3%)* ****4mcPx%F-7$F[em$!3!)*****RWc)eaF-7$Ffdm$!38+++lOzWhF-7$Fadm$!3L+++t mTJoF-7$F\\dm$!3!******H*4\\=vF-7$Fgcm$!3!)*****H5)p0#)F-7$Fbcm$!3S+++ w&QE*))F-7$F]cm$!3Q+++x+%)y&*F-7$Fhbm$!36+++FqPE5FN7$Fcbm$!3%******4T& o%4\"FN7$F^bm$!3'******RV\\F;\"FN7$Fiam$!32+++\"Q40B\"FN7$Fdam$!33+++k x!zH\"FN7$F_am$!3'******pP\"*[O\"FN7$Fj`m$!3)******z,89V\"FN7$Fe`m$!3' ******H$GV(\\\"FN7$F``m$!3)*******)>>Hc\"FN7$F[`m$!33+++,#\\yi\"FN7$Ff _m$!3)******Ru3Ap\"FN7$Fa_m$!3!******4O#*fv\"FN7$F\\_m$!3(*******GG?>= FN7$Fg^m$!3++++L1&=)=FN7$Fb^m$!3********eL&R%>FN7$F]^m$!3%******f.Nb+# FN7$Fh]m$!3y*****pXDm1#FN7$Fc]m$!3#*******Q%fs7#FN7$F^]m$!35+++JgZ(=#F N7$Fi\\m$!35+++yvJZAFN7$Fd\\m$!31+++)fGoI#FN7$F_\\m$!3')******)QagO#FN 7$Fj[m$!3))*****HFR]U#FN7$Fe[m$!35+++AW#Q[#FN7$F`[m$!3#******4AXCa#FN7 $F[[m$!3\"******f>G4g#FN7$Ffjl$!3?+++JwGfEFN7$Fajl$!32+++6@_gA$FN7$F_gl$!3++++6=&) yKFN7$Fjfl$!3&)*****>AJ0L$FN7$Fefl$!3))*****z8t4Q$FN7$F`fl$!3\"******* R*4,V$FN7$F[fl$!3#)*****f0\"*yZ$FN7$Ffel$!36+++e@GCNFN7$Fael$!3')***** f.k#pNFN7$F\\el$!3()******G9$Gh$FN7$Fgdl$!3/+++'3#*\\l$FN7$Fbdl$!3))** ***4Bmdp$FN7$F]dl$!3<+++\"f'=NPFN7$Fhcl$!39+++'4*HtPFN7$Fccl$!38+++%Qk ,\"QFN7$F^cl$!3#******fDge%QFN7$Fibl$!3!******\\E&[!)QFN7$Fdbl$!3>+++V O;9RFN7$F_bl$!3)*******>=0ZRFN7$Fjal$!3A+++[nMzRFN7$Feal$!3#)*****H.'H 6SFN7$F`al$!3V+++M%4K/%FN7$F[al$!39+++++ZvSFN7$Ff`l$!3u******)pS&3TFN7 $Fa`l$!3-+++;y&H9%FN7$F\\`l$!3)******>s&HzTFN7$Fg_l$!3;+++v@3=UFN7$Fb_ l$!3u*****fP]'fUFN7$F]_l$!3=+++-C&RI%FN7$Fh^l$!3y*****f@E/N%FN7$Fc^l$! 39+++Do.)R%FN7$F^^l$!3q******[Y^XWFN7$Fi]l$!35+++(3d;\\%FN7$Fd]l$!3U++ +/*Gb`%FN7$F_]l$!3#*******)*4^wXFN7$Fj\\l$!3M+++?,D9YFN7$Fe\\l$!3k**** *>.y&[YFN7$F`\\l$!3'******\\8R%zYFN7$F[\\l$!3e*****\\/Qoq%FN7$Ff[l$!3J +++Gb!3t%FN7$Fa[l$!3-+++pwP^ZFN7$F\\[l$!3#******4a$eoZFN7$Fgz$!3E+++#) zV#y%FN7$Fbz$!3i******Gs$Hz%FN7$F]z$!3N+++ig0+[FN7$Fhy$!3l*****\\fXP![ FN7$Fcy$!3i*****f@JR![FN7$F^y$!3n*****4P50![FN7$Fix$!3%******z,]Lz%FN7 $Fdx$!3i*****po$G#y%FN7$F_x$!3m*****\\=3rw%FN7$Fjw$!3)******p7!eZZFN7$ Few$!3m*****pp7Ms%FN7$F`w$!3'******pQtUp%FN7$F[w$!3')********QyfYFN7$F fv$!3')******QS_>YFN7$Fav$!3w*****\\(>/tXFN7$F\\v$!38+++OM()>XFN7$Fgu$ !3Q+++(Rr&fWFN7$Fbu$!3S+++!oWYKUFN7$Fct$!3:+++i9zSTFN7$F^t$!3x*****pBN6/%FN7$Fis$!3%**********yN$ RFN7$Fds$!3.+++7!>r#G$FN7$Fjq$!3.+++_VwLJFN7$Feq$!3!)** ****z=C\")HFN7$F`q$!3%)*****\\nGk#GFN7$F[q$!3#******\\*zIqEFN7$Fep$!3& )*****R))GN^#FN7$F`p$!3#)*****>%yYcBFN7$F[p$!3')*****\\!pJ*>#FN7$Feo$! 3>+++!Qf@/#FN7$F`o$!3++++eA-&)=FN7$F[o$!32+++6x!zs\"FN7$Fen$!35+++V(43 d\"FN7$FU$!32+++x;s89FN7$FP$!3#*******f'QmD\"FN7$$\"35+++\"RCqF#FK$!3+ +++gyb*4\"FN7$FD$!3;+++(*)yZU*F-7$$\"30+++_4)f6\"F@$!3e*****zy\")R&yF- 7$$\"3#*******)RYU)=F:$!3=+++C`=$G'F-7$$\"3%******fnnZ,#F4$!3D+++\")*) Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"\"# !\"\"F($\"#X!\"#-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$FWFdboF(7%F\\c o7$F($\"+aEfTJF_coF`co7%Fcco7$$\"+\\zw9?F\\r$\"+\")*)Q7ZF_coF`co7%Fgco 7$$\"+!RYU)=F-$\"+C`=$G'F_coF`co7%F]do7$$\"+^4)f6\"FN$\"+)y\")R&yF_coF `co7%Fcdo7$$\"+qek9`FN$\"+(*)yZU*F_coF`co7%Fido7$$\"+YT-xA!#;$\"+gyb*4 \"!\"*F`co7%F_eo7$$\"+v_?))*)Fbeo$\"+g'QmD\"FeeoF`co7%Fgeo7$$\"+-\"y6B $!#:$\"+x;s89FeeoF`co7%F]fo7$$\"+'\\%yZ5!#9$\"+V(43d\"FeeoF`co7%Fdfo7$ $\"+n10mIFgfo$\"+6x!zs\"FeeoF`co7%F[go7$$\"+GdZ\\\")Fgfo$\"+eA-&)=Feeo F`co7%Fago7$$\"+[Xb%)>!#8$\"+!Qf@/#FeeoF`co7%Fggo7$$\"+MS5lWFjgo$\"+0p J*>#FeeoF`co7%F^ho7$$\"+B0v[$*Fjgo$\"+UyYcBFeeoF`co7%Fdho7$$\"+y:\\J=! #7$\"+%))GN^#FeeoF`co7%Fjho7$$\"+jM')oLF]io$\"+&*zIqEFeeoF`co7%Faio7$$ \"+PS$[#eF]io$\"+v'Gk#GFeeoF`co7%Fgio7$$\"+;'HOX*F]io$\"+!)=C\")HFeeoF `co7%F]jo7$$\"+VDgM9!#6$\"+_VwLJFeeoF`co7%Fcjo7$$\"+n[^B?Ffjo$\"+->r#G $FeeoF`co7%Fjjo7$$\"+>/&Rj#Ffjo$\"+L2oEMFeeoF`co7%F`[p7$$\"+]3sPJFfjo$ \"+8xUkNFeeoF`co7%Ff[p7$$\"+-@nzLFfjo$\"+p!)3&p$FeeoF`co7%F\\\\p7$$\"+ S9l5KFfjo$\"+7YKUFeeoF`co7%Fj]p7$$!+Y4=F_co$\"+OM()>XFeeoF`co7%Fb_p7$$!+ x4/tXFeeoF`co7%Fh_p7$$!+K'H(\\FF_co$\"+RS_>YFeeoF`co7%F^` p7$$!+*o,xB$F_co$\"++RyfYFeeoF`co7%Fd`p7$$!+$y`\\t$F_co$\"+(QtUp%FeeoF `co7%Fj`p7$$!+p'*HQUF_co$\"+(p7Ms%FeeoF`co7%F`ap7$$!+twKXZF_co$\"+F,eZ ZFeeoF`co7%Ffap7$$!+K+Ca_F_co$\"+&=3rw%FeeoF`co7%F\\bp7$$!+c\"=Pw&F_co $\"+(o$G#y%FeeoF`co7%Fbbp7$$!+9p\"GF'F_co$\"+=+N$z%FeeoF`co7%Fhbp7$$!+ 7f)3y'F_co$\"+r.^+[FeeoF`co7%F^cp7$$!+BS^(G(F_co$\"+;7$R![FeeoF`co7%Fd cp7$$!+rM\\#z(F_co$\"+&fXP![FeeoF`co7%Fjcp7$$!+$e+eH)F_co$\"+ig0+[Feeo F`co7%F`dp7$$!+l8f(z)F_co$\"+Hs$Hz%FeeoF`co7%Ffdp7$$!+e7@)H*F_co$\"+#) zV#y%FeeoF`co7%F\\ep7$$!+t&>#)z*F_co$\"+TNeoZFeeoF`co7%Fbep7$$!+II%)H5 Feeo$\"+pwP^ZFeeoF`co7%Fhep7$$!+/s**z5Feeo$\"+Gb!3t%FeeoF`co7%F^fp7$$! +jqVI6Feeo$\"+X!Qoq%FeeoF`co7%Fdfp7$$!+^dO\"=\"Feeo$\"+N\"R%zYFeeoF`co 7%Fjfp7$$!+&R]IB\"Feeo$\"+K!y&[YFeeoF`co7%F`gp7$$!+Ar$eG\"Feeo$\"+?,D9 YFeeoF`co7%Ffgp7$$!+UV;S8Feeo$\"+**4^wXFeeoF`co7%F\\hp7$$!+%>mlR\"Feeo $\"+/*Gb`%FeeoF`co7%Fbhp7$$!+h,lb9Feeo$\"+(3d;\\%FeeoF`co7%Fhhp7$$!+zW -=:Feeo$\"+\\Y^XWFeeoF`co7%F^ip7$$!+ST:%e\"Feeo$\"+Do.)R%FeeoF`co7%Fdi p7$$!+I7;a;Feeo$\"+;iU]VFeeoF`co7%Fjip7$$!+l&fws\"Feeo$\"+-C&RI%FeeoF` co7%F`jp7$$!+C^v.=Feeo$\"+w.lfUFeeoF`co7%Ffjp7$$!+,XD\")=Feeo$\"+v@3=U FeeoF`co7%F\\[q7$$!+:9'*e>Feeo$\"+AdHzTFeeoF`co7%Fb[q7$$!+7f*e.#Feeo$ \"+;y&H9%FeeoF`co7%Fh[q7$$!+DBP6@Feeo$\"+*pS&3TFeeoF`co7%F^\\q7$$!+!Qu \\=#Feeo$\"+++ZvSFeeoF`co7%Fd\\q7$$!+B(*[cAFeeo$\"+M%4K/%FeeoF`co7%Fj \\q7$$!+UQ%eK#Feeo$\"+LgH6SFeeoF`co7%F`]q7$$!+I%[IR#Feeo$\"+[nMzRFeeoF `co7%Ff]q7$$!+&ol\"eCFeeo$\"+?=0ZRFeeoF`co7%F\\^q7$$!+b`G@DFeeo$\"+VO; 9RFeeoF`co7%Fb^q7$$!+'p5De#Feeo$\"+l_[!)QFeeoF`co7%Fh^q7$$!+9.&>k#Feeo $\"+c-'e%QFeeoF`co7%F^_q7$$!+9Pr*p#Feeo$\"+%Qk,\"QFeeoF`co7%Fd_q7$$!+h \"4fv#Feeo$\"+'4*HtPFeeoF`co7%Fj_q7$$!+5Hk5GFeeo$\"+\"f'=NPFeeoF`co7%F ``q7$$!+]#>S'GFeeo$\"+Jiw&p$FeeoF`co7%Ff`q7$$!+n49;HFeeo$\"+'3#*\\l$Fe eoF`co7%F\\aq7$$!+a*4r'HFeeo$\"+H9$Gh$FeeoF`co7%Fbaq7$$!+(zFq,$Feeo$\" +OSEpNFeeoF`co7%Fhaq7$$!+Uh*f1$Feeo$\"+e@GCNFeeoF`co7%F^bq7$$!+,n69JFe eo$\"+c5*yZ$FeeoF`co7%Fdbq7$$!+N2\\hJFeeo$\"+S*4,V$FeeoF`co7%Fjbq7$$!+ py@3KFeeo$\"+QJ(4Q$FeeoF`co7%F`cq7$$!+MTRaKFeeo$\"+A7`ILFeeoF`co7%Ffcq 7$$!+s\"4,I$Feeo$\"+6=&)yKFeeoF`co7%F\\dq7$$!+&oUaM$Feeo$\"+R'>gA$Feeo F`co7%Fbdq7$$!+g.Y!R$Feeo$\"+Rb8sJFeeoF`co7%Fhdq7$$!+t'4_V$Feeo$\"+BSJ G4g#Fe eoF`co7%Fdhq7$$!+fiViQFeeo$\"+@_WUDFeeoF`co7%Fjhq7$$!+ClX,RFeeo$\"+AW# Q[#FeeoF`co7%F`iq7$$!+3#e%RRFeeo$\"+t#R]U#FeeoF`co7%Ffiq7$$!+?/QwRFeeo $\"+*QagO#FeeoF`co7%F\\jq7$$!+0\\<7SFeeo$\"+)fGoI#FeeoF`co7%Fbjq7$$!+D d!o/%Feeo$\"+yvJZAFeeoF`co7%Fhjq7$$!+)e[-3%Feeo$\"+JgZ(=#FeeoF`co7%F^[ r7$$!+q**[7TFeeo$\"+R%fs7#FeeoF`co7%Fd[r7$$!+vh_VTFeeo$\"+daim?FeeoF`c o7%Fj[r7$$!+\"RiL<%Feeo$\"+O]`0?FeeoF`co7%F`\\r7$$!+t<,-UFeeo$\"+fL&R% >FeeoF`co7%Ff\\r7$$!+mX\\HUFeeo$\"+L1&=)=FeeoF`co7%F\\]r7$$!+Us$eD%Fee o$\"+HG?>=FeeoF`co7%Fb]r7$$!+3;2\"G%Feeo$\"+hB*fv\"FeeoF`co7%Fh]r7$$!+ lQB0VFeeo$\"+W(3Ap\"FeeoF`co7%F^^r7$$!+ZNOGVFeeo$\"+,#\\yi\"FeeoF`co7% Fd^r7$$!+EB]]VFeeo$\"+*>>Hc\"FeeoF`co7%Fj^r7$$!+PEprVFeeo$\"+LGV(\\\"F eeoF`co7%F`_r7$$!+\\h(>R%Feeo$\"+=ITJ9FeeoF`co7%Ff_r7$$!+&3#R6WFeeo$\" +x8*[O\"FeeoF`co7%F\\`r7$$!+!\\v*HWFeeo$\"+kx!zH\"FeeoF`co7%Fb`r7$$!+@ bvZWFeeo$\"+\"Q40B\"FeeoF`co7%Fh`r7$$!+3RvkWFeeo$\"+M%\\F;\"FeeoF`co7% F^ar7$$!+HP)4[%Feeo$\"+6ao%4\"FeeoF`co7%Fdar7$$!+:'[k\\%Feeo$\"+FqPE5F eeoF`co7%Fjar7$$!+)\\U6^%Feeo$\"+x+%)y&*F_coF`co7%F`br7$$!+P*\\]_%Feeo $\"+w&QE*))F_coF`co7%Ffbr7$$!+%*p9QXFeeo$\"+.\")p0#)F_coF`co7%F\\cr7$$ !+vDS]XFeeo$\"+$*4\\=vF_coF`co7%Fbcr7$$!+<,yhXFeeo$\"+tmTJoF_coF`co7%F hcr7$$!+Z%RAd%Feeo$\"+lOzWhF_coF`co7%F^dr7$$!+w'Q> Hc\"FeeoF`co7%F`]s7$Fe^r$!+,#\\yi\"FeeoF`co7%Fd]s7$F_^r$!+W(3Ap\"FeeoF `co7%Fh]s7$Fi]r$!+hB*fv\"FeeoF`co7%F\\^s7$Fc]r$!+HG?>=FeeoF`co7%F`^s7$ F]]r$!+L1&=)=FeeoF`co7%Fd^s7$Fg\\r$!+fL&R%>FeeoF`co7%Fh^s7$Fa\\r$!+O]` 0?FeeoF`co7%F\\_s7$F[\\r$!+daim?FeeoF`co7%F`_s7$Fe[r$!+R%fs7#FeeoF`co7 %Fd_s7$F_[r$!+JgZ(=#FeeoF`co7%Fh_s7$Fijq$!+yvJZAFeeoF`co7%F\\`s7$Fcjq$ !+)fGoI#FeeoF`co7%F``s7$F]jq$!+*QagO#FeeoF`co7%Fd`s7$Fgiq$!+t#R]U#Feeo F`co7%Fh`s7$Faiq$!+AW#Q[#FeeoF`co7%F\\as7$F[iq$!+@_WUDFeeoF`co7%F`as7$ Fehq$!+'>G4g#FeeoF`co7%Fdas7$F_hq$!+JwGfEFeeoF`co7%Fhas7$Figq$!+6@_gA$FeeoF`co7%F`ds7$F]dq$! +6=&)yKFeeoF`co7%Fdds7$Fgcq$!+A7`ILFeeoF`co7%Fhds7$Facq$!+QJ(4Q$FeeoF` co7%F\\es7$F[cq$!+S*4,V$FeeoF`co7%F`es7$Febq$!+c5*yZ$FeeoF`co7%Fdes7$F _bq$!+e@GCNFeeoF`co7%Fhes7$Fiaq$!+OSEpNFeeoF`co7%F\\fs7$Fcaq$!+H9$Gh$F eeoF`co7%F`fs7$F]aq$!+'3#*\\l$FeeoF`co7%Fdfs7$Fg`q$!+Jiw&p$FeeoF`co7%F hfs7$Fa`q$!+\"f'=NPFeeoF`co7%F\\gs7$F[`q$!+'4*HtPFeeoF`co7%F`gs7$Fe_q$ !+%Qk,\"QFeeoF`co7%Fdgs7$F__q$!+c-'e%QFeeoF`co7%Fhgs7$Fi^q$!+l_[!)QFee oF`co7%F\\hs7$Fc^q$!+VO;9RFeeoF`co7%F`hs7$F]^q$!+?=0ZRFeeoF`co7%Fdhs7$ Fg]q$!+[nMzRFeeoF`co7%Fhhs7$Fa]q$!+LgH6SFeeoF`co7%F\\is7$F[]q$!+M%4K/% FeeoF`co7%F`is7$Fe\\q$!+++ZvSFeeoF`co7%Fdis7$F_\\q$!+*pS&3TFeeoF`co7%F his7$Fi[q$!+;y&H9%FeeoF`co7%F\\js7$Fc[q$!+AdHzTFeeoF`co7%F`js7$F][q$!+ v@3=UFeeoF`co7%Fdjs7$Fgjp$!+w.lfUFeeoF`co7%Fhjs7$Fajp$!+-C&RI%FeeoF`co 7%F\\[t7$F[jp$!+;iU]VFeeoF`co7%F`[t7$Feip$!+Do.)R%FeeoF`co7%Fd[t7$F_ip $!+\\Y^XWFeeoF`co7%Fh[t7$Fihp$!+(3d;\\%FeeoF`co7%F\\\\t7$Fchp$!+/*Gb`% FeeoF`co7%F`\\t7$F]hp$!+**4^wXFeeoF`co7%Fd\\t7$Fggp$!+?,D9YFeeoF`co7%F h\\t7$Fagp$!+K!y&[YFeeoF`co7%F\\]t7$F[gp$!+N\"R%zYFeeoF`co7%F`]t7$Fefp $!+X!Qoq%FeeoF`co7%Fd]t7$F_fp$!+Gb!3t%FeeoF`co7%Fh]t7$Fiep$!+pwP^ZFeeo F`co7%F\\^t7$Fcep$!+TNeoZFeeoF`co7%F`^t7$F]ep$!+#)zV#y%FeeoF`co7%Fd^t7 $Fgdp$!+Hs$Hz%FeeoF`co7%Fh^t7$Fadp$!+ig0+[FeeoF`co7%F\\_t7$F[dp$!+&fXP ![FeeoF`co7%F`_t7$Fecp$!+;7$R![FeeoF`co7%Fd_t7$F_cp$!+r.^+[FeeoF`co7%F h_t7$Fibp$!+=+N$z%FeeoF`co7%F\\`t7$Fcbp$!+(o$G#y%FeeoF`co7%F``t7$F]bp$ !+&=3rw%FeeoF`co7%Fd`t7$Fgap$!+F,eZZFeeoF`co7%Fh`t7$Faap$!+(p7Ms%FeeoF `co7%F\\at7$F[ap$!+(QtUp%FeeoF`co7%F`at7$Fe`p$!++RyfYFeeoF`co7%Fdat7$F _`p$!+RS_>YFeeoF`co7%Fhat7$Fi_p$!+v>/tXFeeoF`co7%F\\bt7$Fc_p$!+OM()>XF eeoF`co7%F`bt7$F]_p$!+(Rr&fWFeeoF`co7%Fdbt7$Fg^p$!+!oWYKUFeeoF`co7%F`ct7$Fe]p$!+i 9zSTFeeoF`co7%Fdct7$F_]p$!+P_8TSFeeoF`co7%Fhct7$Fi\\p$!++!zN$RFeeoF`co 7%F\\dt7$Fc\\p$!+7r#G$F eeoF`co7%F`et7$Fdjo$!+_VwLJFeeoF`co7%Fdet7$F^jo$!+!)=C\")HFeeoF`co7%Fh et7$Fhio$!+v'Gk#GFeeoF`co7%F\\ft7$Fbio$!+&*zIqEFeeoF`co7%F`ft7$F[io$!+ %))GN^#FeeoF`co7%Fdft7$Feho$!+UyYcBFeeoF`co7%Fhft7$F_ho$!+0pJ*>#FeeoF` co7%F\\gt7$Fhgo$!+!Qf@/#FeeoF`co7%F`gt7$Fbgo$!+eA-&)=FeeoF`co7%Fdgt7$F \\go$!+6x!zs\"FeeoF`co7%Fhgt7$Fefo$!+V(43d\"FeeoF`co7%F\\ht7$F^fo$!+x; s89FeeoF`co7%F`ht7$Fheo$!+g'QmD\"FeeoF`co7%Fdht7$$\"+\"RCqF#Fbeo$!+gyb *4\"FeeoF`co7%Fhht7$Fjdo$!+(*)yZU*F_coF`co7%F^it7$$\"+_4)f6\"FN$!+)y\" )R&yF_coF`co7%Fbit7$$\"+*RYU)=F-$!+C`=$G'F_coF`co7%Fhit7$$\"+www9?F\\r $!+\")*)Q7ZF_coF`co7%F^jt7$F($!+aEfTJF_coF`co7%Fdjt7$F($!+Fjzq:F_coF`c o7%FhjtF'F`co-F_bo6&Fabo$\"#bFgbo$\"#:Fgbo$\"\"\"F)-%&STYLEG6#%,PATCHN OGRIDG-F$6$7K7$$\"37+++M.72?FN$\"3D+++V#[ih&FN7$$\"3A+++]/*3,#FN$\"3r* ******\\&QHcFN7$$\"31+++m1o7?FN$\"3w******y8mUcFN7$$\"35+++],c7?FN$\"3 ?+++g,#el&FN7$$\"3#*******3xh5?FN$\"3W+++o4jocFN7$$\"35+++Mv&p+#FN$\"3 m******R`)3o&FN7$$\"3y*****H@'p,?FN$\"3'******p'[R#p&FN7$$\"33+++92'\\ *>FN$\"3%******z1*)Hq&FN7$$\"3#*******=r)o)>FN$\"3H+++9N^7dFN7$$\"3%** ****p&*>w(>FN$\"39+++f#G3s&FN7$$\"3!******p.7t'>FN$\"3-+++nl!ys&FN7$$ \"3-+++FN$\"3K+++fRLLdFN7$$\"3++++&fGU%>FN$\"33+++>wIPdFN7$$\"3** *****Re/=$>FN$\"3K+++>ijRdFN7$$\"3!******>RX!>>FN$\"3t*****\\ES-u&FN7$ $\"32+++yt:1>FN$\"3$******\\6`!RdFN7$$\"3!******\\uiL*=FN$\"3-+++*)G-O dFN7$$\"3++++n>!4)=FN$\"3U+++`b6JdFN7$$\"3!******zAO!p=FN$\"3v*****>d> Vs&FN7$$\"3#*******f%\\!e=FN$\"3o*****4d_cr&FN7$$\"35+++P#\\#[=FN$\"3 \"******f^q^q&FN7$$\"31+++Q`'*R=FN$\"3))*****4!)zHp&FN7$$\"3)******>:W N$=FN$\"35+++qADzcFN7$$\"3'******zUO$H=FN$\"3C+++j&FN7$$\"3'**** **4&*GI$=FN$\"3n******>^g:cFN7$$\"3-+++y=FS=FN$\"3Q+++]n2+cFN7$$\"3!** ****H[E/&=FN$\"3G+++w#3ge&FN7$$\"3#*******H49j=FN$\"3s*****p>aSd&FN7$$ \"3)*******>$yy(=FN$\"3I+++r-vkbFN7$$\"3/+++\"yuR*=FN$\"33+++n2XebFN7$ $\"34+++$y=2\">FN$\"3O+++0;IbbFN7$$\"35+++gyUF>FN$\"3))*****Rr__b&FN7$ $\"3%******\\G/N%>FN$\"3m*****z?)4ebFN7$$\"33+++^MYe>FN$\"3#******R;FN c&FN7$$\"3!******f(*Q>(>FN$\"3)******R+r6d&FN7$$\"3++++\")Hn$)>FN$\"3' )*****4IS1e&FN7$$\"3!******f_*\\$*>FN$\"3e*****Rl\\:f&FN7$$\"32+++-nK, ?FN$\"3f*****4WKNg&FNF[\\uF^bo-Fibo6L7%7$$\"+M.72?Feeo$\"+V#[ih&Feeo7$ $\"+]/*3,#Feeo$\"++bQHcFeeo7$$\"$#>Fgbo$\"$k&Fgbo7%F[iu7$$\"+m1o7?Feeo $\"+z8mUcFeeoF`iu7%Ffiu7$$\"+],c7?Feeo$\"+g,#el&FeeoF`iu7%F\\ju7$$\"+4 xh5?Feeo$\"+o4jocFeeoF`iu7%Fbju7$$\"+Mv&p+#Feeo$\"+S`)3o&FeeoF`iu7%Fhj u7$$\"+8ip,?Feeo$\"+n[R#p&FeeoF`iu7%F^[v7$$\"+92'\\*>Feeo$\"+o!*)Hq&Fe eoF`iu7%Fd[v7$$\"+>r)o)>Feeo$\"+9N^7dFeeoF`iu7%Fj[v7$$\"+d*>w(>Feeo$\" +f#G3s&FeeoF`iu7%F`\\v7$$\"+P?Jn>Feeo$\"+nl!ys&FeeoF`iu7%Ff\\v7$$\"+Feeo$\"+fRLLdFeeoF`iu7%F\\]v7$$\"+&fGU%>Feeo$\"+>wIPdFeeoF`iu7%Fb]v 7$$\"+%e/=$>Feeo$\"+>ijRdFeeoF`iu7%Fh]v7$$\"+#RX!>>Feeo$\"+l-CSdFeeoF` iu7%F^^v7$$\"+yt:1>Feeo$\"+:J0RdFeeoF`iu7%Fd^v7$$\"+XFO$*=Feeo$\"+*)G- OdFeeoF`iu7%Fj^v7$$\"+n>!4)=Feeo$\"+`b6JdFeeoF`iu7%F`_v7$$\"+Gi.p=Feeo $\"+s&>Vs&FeeoF`iu7%Ff_v7$$\"+g%\\!e=Feeo$\"+rDl:dFeeoF`iu7%F\\`v7$$\" +P#\\#[=Feeo$\"+;0<0dFeeoF`iu7%Fb`v7$$\"+Q`'*R=Feeo$\"+,)zHp&FeeoF`iu7 %Fh`v7$$\"+_TaL=Feeo$\"+qADzcFeeoF`iu7%F^av7$$\"+GkLH=Feeo$\"+j&FeeoF`iu7%F`bv7$$\"+^*GI$=Feeo$\"+?^g:cFeeoF`iu7%Ffbv7$$\"+y=FS=F eeo$\"+]n2+cFeeoF`iu7%F\\cv7$$\"+$[E/&=Feeo$\"+w#3ge&FeeoF`iu7%Fbcv7$$ \"+I49j=Feeo$\"+(>aSd&FeeoF`iu7%Fhcv7$$\"+?$yy(=Feeo$\"+r-vkbFeeoF`iu7 %F^dv7$$\"+\"yuR*=Feeo$\"+n2XebFeeoF`iu7%Fddv7$$\"+$y=2\">Feeo$\"+0;Ib bFeeoF`iu7%Fjdv7$$\"+gyUF>Feeo$\"+9FDbbFeeoF`iu7%F`ev7$$\"+&G/N%>Feeo$ \"+3#)4ebFeeoF`iu7%Ffev7$$\"+^MYe>Feeo$\"+kr_jbFeeoF`iu7%F\\fv7$$\"+w* Q>(>Feeo$\"+/5Feeo$\"+,.k!e&FeeoF`iu7%F hfv7$$\"+E&*\\$*>Feeo$\"+a'\\:f&FeeoF`iu7%F^gv7$$\"+-nK,?Feeo$\"+TC`.c FeeoF`iu7%FdgvFfhuF`iuF\\[uFd[u-F$6$7K7$F\\\\u$!3D+++V#[ih&FN7$F_hu$!3 f*****4WKNg&FN7$Fjgu$!3e*****Rl\\:f&FN7$Fegu$!3')*****4IS1e&FN7$F`gu$! 3)******R+r6d&FN7$F[gu$!3#******R;FNc&FN7$Fffu$!3m*****z?)4ebFN7$Fafu$ !3))*****Rr__b&FN7$F\\fu$!3O+++0;IbbFN7$Fgeu$!33+++n2XebFN7$Fbeu$!3I++ +r-vkbFN7$F]eu$!3s*****p>aSd&FN7$Fhdu$!3G+++w#3ge&FN7$Fcdu$!3Q+++]n2+c FN7$F^du$!3n******>^g:cFN7$Ficu$!3M+++x,!>j&FN7$Fdcu$!3A+++oNI[cFN7$F_ cu$!3C+++d>Vs&FN7$Faau$! 3U+++`b6JdFN7$F\\au$!3-+++*)G-OdFN7$Fg`u$!3$******\\6`!RdFN7$Fb`u$!3t* ****\\ES-u&FN7$F]`u$!3K+++>ijRdFN7$Fh_u$!33+++>wIPdFN7$Fc_u$!3K+++fRLL dFN7$F^_u$!3-+++nl!ys&FN7$Fi^u$!39+++f#G3s&FN7$Fd^u$!3H+++9N^7dFN7$F_^ u$!3%******z1*)Hq&FN7$Fj]u$!3'******p'[R#p&FN7$Fe]u$!3m******R`)3o&FN7 $F`]u$!3W+++o4jocFN7$F[]u$!3?+++g,#el&FN7$Ff\\u$!3w******y8mUcFN7$Fa\\ u$!3r*******\\&QHcFNF]hvF^bo-Fibo6L7%7$Fghu$!+V#[ih&Feeo7$Fegv$!+TC`.c Feeo7$Faiu$!$k&Fgbo7%F[`w7$F_gv$!+a'\\:f&FeeoF^`w7%Fb`w7$Fifv$!+,.k!e& FeeoF^`w7%Ff`w7$Fcfv$!+/5aSd&FeeoF^`w7%Ffbw7$Fccv$!+w#3ge&FeeoF^`w7%Fjbw7$F]cv$!+]n 2+cFeeoF^`w7%F^cw7$Fgbv$!+?^g:cFeeoF^`w7%Fbcw7$Fabv$!+x,!>j&FeeoF^`w7% Ffcw7$F[bv$!+oNI[cFeeoF^`w7%Fjcw7$Feav$!+Vs&FeeoF^`w7%Fbew7$Fa_v$! +`b6JdFeeoF^`w7%Ffew7$F[_v$!+*)G-OdFeeoF^`w7%Fjew7$Fe^v$!+:J0RdFeeoF^` w7%F^fw7$F_^v$!+l-CSdFeeoF^`w7%Fbfw7$Fi]v$!+>ijRdFeeoF^`w7%Fffw7$Fc]v$ !+>wIPdFeeoF^`w7%Fjfw7$F]]v$!+fRLLdFeeoF^`w7%F^gw7$Fg\\v$!+nl!ys&FeeoF ^`w7%Fbgw7$Fa\\v$!+f#G3s&FeeoF^`w7%Ffgw7$F[\\v$!+9N^7dFeeoF^`w7%Fjgw7$ Fe[v$!+o!*)Hq&FeeoF^`w7%F^hw7$F_[v$!+n[R#p&FeeoF^`w7%Fbhw7$Fiju$!+S`)3 o&FeeoF^`w7%Ffhw7$Fcju$!+o4jocFeeoF^`w7%Fjhw7$F]ju$!+g,#el&FeeoF^`w7%F ^iw7$Fgiu$!+z8mUcFeeoF^`w7%Fbiw7$F\\iu$!++bQHcFeeoF^`w7%FfiwFh_wF^`wF \\[uFd[u-F$6%7$7$$!3o*************Q&FNF(7$$\"37++++++!R#FNF(-%'COLOURG 6&FaboF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>'FN7$F($\"3Q+++ +++!>'FNFcjwFfjw-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%FON TG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F\\\\x6#%(DEFAUL TG-%%VIEWG6$;$!$R&Fgbo$\"$R#Fgbo;$!$>'Fgbo$\"$>'Fgbo" 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 "int erval of absolute stability" }{TEXT -1 89 " (or stability interval) i s 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 "[-4.6188, 0];" "6#7$,$-%&FloatG6$ \"&)=Y!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve ho rizontally by taking the 11th root of the real part of points along th e curve. In this way we see that the largest interval on the nonnegati ve imaginary axis that contains the origin and lies inside the stabili ty region is " }{XPPEDIT 18 0 "[0, 4.1];" "6#7$\"\"!-%&FloatG6$\"#T! \"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 735 "R := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 1044642902164771900 9154872973569257108114919/42284344133234228836247140822949795798946319 2000*z^8+\n 654391908856573539074035419648715456888290709/25212040189 4409089436123577156838157451217428230000*z^9+\n 360494159656570476474 00401383880824267026041/4254531781968153384234585364521643906989294101 38125*z^10+\n 926576811334529021145838729647244760084864/283635452131 21022561563902430144292713261960675875*z^11:\nDigits := 18:\npts := [] : z0 := 0:\nfor ct from 0 to 155 do\n zz := newton(R(z)=exp(ct*Pi/10 0*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)] ]:\nend do:\nplot(pts,color=COLOR(RGB,.55,0,1),thickness=2,font=[HELVE TICA,9]);\nDigits := 10:\n" }}{PARA 13 "" 1 "" {GLPLOT2D 265 310 310 {PLOTDATA 2 "6(-%'CURVESG6#7ht7$$\"\"!F)F(7$$\"3j=$f7kU&e=!#>$\"3C$z*e `EfTJF-7$$\"3OQ!4'*>ah2$F-$\"3e'ezrI&=$G'F-7$$\"3n&))e>#*z'HTF-$\"3`$Q p2'zxC%*F-7$$\"3W$ojBxm\")3&F-$\"3WAfVhqjc7!#=7$$\"3)4JtIoP3)fF-$\"3oN \\zEjzq:F?7$$\"33^!*y'**pN#oF-$\"3z\"3a@fb\\)=F?7$$\"3>!H7>e:ii(F-$\"3 ')*p8v&[6*>#F?7$$\"38&))*4!pLaR)F-$\"3F@Z(G7uK^#F?7$$\"3GU0?tO,O\"*F-$ \"35)RR#)QLu#GF?7$$\"3B(f%p!Qu:&)*F-$\"3jqEh`EfTJF?7$$\"3*HdS)4p\\a5F? $\"34vX+>>vbMF?7$$\"3Mpy$4s`=7\"F?$\"3$)pUV%=6*pPF?7$$\"3hR(HQ*pU(=\"F ?$\"3;([O*\\/2%3%F?7$$\"3Q***Rb\\#R^7F?$\"331:d:(H#)R%F?7$$\"3&pEQn)3 \"RJ\"F?$\"3Q&4S9)*)Q7ZF?7$$\"3!G#*\\9%Q8v8F?$\"3E,`qZ#[l-&F?7$$\"3Zif 8i'3_V\"F?$\"3baMi9vqS`F?7$$\"3\"4fiZu\"G%\\\"F?$\"3*)[te#ym[l&F?7$$\" 3UbZ>\"3,Db\"F?$\"3'R>&=_g-pfF?7$$\"3[vJy!)y,5;F?$\"3Lr'zUK&=$G'F?7$$ \"3a6D!*Ht)pm\"F?$\"3!z?7,gWtf'F?7$$\"3I>65z%oNs\"F?$\"3T?%Q9)Q]6pF?7$ $\"3(ff'R\\K#*zwy\")R&yF?7$$\"3KFAfktC\\>F?$\"3e(Q^e7T\"o\")F? 7$$\"35s>P=KH1?F?$\"3y3]#R\\+B[)F?7$$\"3BL`UVq(Q1#F?$\"32h/C-*fkz)F?7$ $\"3FmTzk(=@7#F?$\"3VCV@k$>16*F?7$$\"3I1eH@'>6=#F?$\"3Wx#yp*)yZU*F?7$$ \"3\"4a*H!3g4C#F?$\"3:Yp>A&Q*Q(*F?7$$\"3'y@53x)p,BF?$\"3;X:qE)4`+\"!#< 7$$\"3d)*Q;*[sLO#F?$\"3EF?$\"3wt1&o-!Ri6Ffu7$$\"3uL+gp_r&o#F?$\"3N\"Ffu7$$\"3PqM@6@w_FF?$\"3y&QGLQA_A\"Ffu7$$\"3y[WPz1b?GF?$\"3QYl# *f'QmD\"Ffu7$$\"38J!e![.-*)GF?$\"3&4aw0-b!)G\"Ffu7$$\"33c5\"f<4\"eHF?$ \"3^gB'>[r%>8Ffu7$$\"31z1S)Qbx-$F?$\"3kC7Jj!))3N\"Ffu7$$\"3\"R2J$F?$\"3/*RfZ2clZ\"Ffu7$$\"3n %GG269BQ$F?$\"3+:W4\\O(z]\"Ffu7$$\"32!psc:BTX$F?$\"3ml`CC:RR:Ffu7$$\"3 %[Hg=&H7ENF?$\"3gQcpU(43d\"Ffu7$$\"3a\\/*e=t#)f$F?$\"3f^?a]$GAg\"Ffu7$ $\"3UC,Gkj`qOF?$\"37XDL(RZOj\"Ffu7$$\"3,+$*>=w(Gu$F?$\"3?mYxNp1l;Ffu7$ $\"3_>I.&\\k_\"QF?$\"3^m$)G@q['p\"Ffu7$$\"3Jnk$fumw)QF?$\"3CRMP6x!zs\" Ffu7$$\"3yx&Qd6c+'RF?$\"33;&eZ1H$fmBu%>p/TF?$\"3$*[]b&*R3!*o<%F?$\"3mn.4 &o(f`=Ffu7$$\"380\\^B$y*[UF?$\"3^p6IeA-&)=Ffu7$$\"3kQ-EGb$4K%F?$\"3fJW 'ovZk\">Ffu7$$\"3i^@&eZTFR%F?$\"3P1Hm6U(y%>Ffu7$$\"3yJ[4EnPkWF?$\"30$[ `\"R;Iz>Ffu7$$\"3()R%[R[Ae`%F?$\"355&3r.I2,#Ffu7$$\"3FH!e:Zgqg%F?$\"3$ GFf'z$f@/#Ffu7$$\"3@bm:1G2yYF?$\"3Xxkc6'*et?Ffu7$$\"3(>`qE)=%)[ZF?$\"3 G*\\Q;k?]5#Ffu7$$\"33&)H))f-N>[F?$\"3naW@NBXO@Ffu7$$\"3]!=i!\\0e*)[F?$ \"3VRgi0X)y;#Ffu7$$\"38c)>SI:&f\\F?$\"3Cu(\\X!pJ*>#Ffu7$$\"3X\\8k6p8H] F?$\"3e&4g6@\\2B#Ffu7$$\"3y(f4?[F%)4&F?$\"3E#)R**>5=iAFfu7$$\"3%HGPtto t;&F?$\"3?P*Ru#=h$H#Ffu7$$\"3dHF-**=%fB&F?$\"34h'on,T]K#Ffu7$$\"3&**=e AdFTI&F?$\"3dg-jTyYcBFfu7$$\"3vSguFc!>P&F?$\"3yDS&zS\"*yQ#Ffu7$$\"3AO> A!3b#RaF?$\"3_\"\\-Ul5$>CFfu7$$\"37!=ZO'R:1bF?$\"3+dD&*HVs]CFfu7$$\"3c \"p=J?zDd&F?$\"3BiQ!G(48#[#Ffu7$$\"32Tqv$[1&QcF?$\"3&RW\\O))GN^#Ffu7$$ \"3=>Q2<,\"Rq&F?$\"3U!)pQ+h\"\\a#Ffu7$$\"3E**4f0HwodF?$\"3t*p,-P!HwDFf u7$$\"3Ugwy1g.LeF?$\"3=h^h?\"\\wg#Ffu7$$\"37BBy(z)p'*eF?$\"3fo#o&H%*)* QEFfu7$$\"3H,'=Du=(ffF?$\"3$3H#*[*zIqEFfu7$$\"3+#*4gj71AgF?$\"3GB!=O5, ;q#Ffu7$$\"3s(Q:8i*o$3'F?$\"3B/%z;gkGt#Ffu7$$\"3]8$\\F![cWhF?$\"34_HFfu7$$\"3!*[<_xu]\"\\' F?$\"3bA)4Y3E/&HFfu7$$\"39/'[Gp6ea'F?$\"32))RX!)=C\")HFfu7$$\"3=3&zjeh *)f'F?$\"3kZ4@!3n>,$Ffu7$$\"30WG*\\4'*3l'F?$\"3)HgZd:#fUIFfu7$$\"3(3A* G\"3_:q'F?$\"3o/Cp\"42J2$Ffu7$$\"3:Hr'\\`k3v'F?$\"3m&)=MM9].JFfu7$$\"3 G0&H@!ow)z'F?$\"3cD!zANkP8$Ffu7$$\"3%4=$*[$3>XoF?$\"37-(p#3Z)Q;$Ffu7$$ \"3NyK2^u1!*oF?$\"3^Ve\\T6&Q>$Ffu7$$\"3p\"**46fEL$pF?$\"3zPfle@lBKFfu7 $$\"38b7$pb(*[(pF?$\"3#*z*\\>BwKD$Ffu7$$\"3*eUK_@4Z,(F?$\"3%Rv0:!>r#G$ Ffu7$$\"3`]%4e9!p_qF?$\"3dBiVzy%>J$Ffu7$$\"3B)GkTro()3(F?$\"3\"4ozC:t4 M$Ffu7$$\"30Z9gYI(G7(F?$\"3A,H4!3x(pLFfu7$$\"3o&=c`)3$\\:(F?$\"3;d%e\\ [\\$)R$Ffu7$$\"3=$GT2Ip[=(F?$\"3yv88L2oEMFfu7$$\"3KGNcOUh7sF?$\"3?1FZ, =waMFfu7$$\"3`.0sj)*3QsF?$\"3Mz\"p0L%e#[$Ffu7$$\"3-]hR9x@hsF?$\"3bS'y> mS,^$Ffu7$$\"30Y?%\\c:>G(F?$\"3RMLKfQUPNFfu7$$\"3weO\\Gg4+tF?$\"3E_!HM rFWc$Ffu7$$\"3>Z\"4v![m:tF?$\"3)>+LBtY6f$Ffu7$$\"3S$HM-a=&GtF?$\"3n1/_ %z$Ffu7$$\"3Op5Y[1O:tF?$\"36)piAr ,#=QFfu7$$\"3zIT:]Af&H(F?$\"3Blbs+%**=%QFfu7$$\"3mtX,L]WqsF?$\"3ikaf)e 'GlQFfu7$$\"3>-eKjy?RsF?$\"3-bY#Hji$))QFfu7$$\"3`F&Hj9`4?(F?$\"3d_VQ`p 76RFfu7$$\"37P=_u6WarF?$\"3OI/r***yN$RFfu7$$\"39*zwi!['z4(F?$\"3X0wwD# =d&RFfu7$$\"3(*f,/eG3HqF?$\"3a_\"f,2Wv(RFfu7$$\"3s#Qo\"z#HT%pF?$\"3uq) eJ%f0**RFfu7$$\"3o+_$>wAs$oF?$\"3z=vh>KD?SFfu7$$\"3tD)py>**zp'F?$\"3=h _RP_8TSFfu7$$\"3l(4:JHY`]'F?$\"3G&p#z+8qhSFfu7$$\"3O]GW.\"QN?'F?$\"3=( o$R)o]>3%Ffu7$$\"3W_(*3.:k*\\&F?$\"3gm;pkE)=5%Ffu7$$!3#p')o3JQ7%eF?$\" 3*ehX[\\'\\@TFfu7$$!3:d5h[R%GP'F?$\"3O$Q,>Y\"zSTFfu7$$!3TJ9w3/3qmF?$\" 37+q\"e)owfTFfu7$$!3t**>&HBtR)oF?$\"3eVjkV@UyTFfu7$$!3#3'3R]&pR0(F?$\" 3$pvi-pcn>%Ffu7$$!3)[6K3OTl>(F?$\"3WO?/z+x9UFfu7$$!3#[y.a_*>?tF?$\"3Q \\8(>)>YKUFfu7$$!3uW3@L^$*HuF?$\"3L(z0(3A$)\\UFfu7$$!3?_0QW]$*GvF?$\"3 tnf:C2)oE%Ffu7$$!3.Od&HwF?$\"3wNiDkwg$G%Ffu7$$!3A()RlX\"fFq(F?$\"3 I!)Hj\\L,+VFfu7$$!31f&4Z_f-y(F?$\"3B\"RipH)4;VFfu7$$!3e&QdcxGF&yF?$\"3 %[IxyAj=L%Ffu7$$!3k0pI04%3#zF?$\"3OK(\\`24tM%Ffu7$$!3x\\q*)QD8&)zF?$\" 3i@)Qv)pViVFfu7$$!3!yh5*f'Qg/)F?$\"33>::H$[sP%Ffu7$$!3\\APQ+u\"R5)F?$ \"3!z09/oW " 0 "" {MPLTEXT 1 0 113 "Digits := 15:\nz0 := 4.1* I:\nfor ct from 137 to 140 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0); \nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0sg'Q UTO_!#<$\"0p$R)o]>3%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0lji&[ 7#R\"!#<$\"0n\"pkE)=5%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0[Lk. %=,F!#<$\"0iX[\\'\\@T!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0,!of_ ^Tq!#<$\"0Q,>Y\"zST!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "real_part := proc(u)\n Re(newton (R(z)=exp(u*Pi*I),z=4.1*I))\nend proc:\nDigits := 15:\nu0 := bisect('r eal_part'(u),u=1.37..1.4);\nnewton(R(z)=exp(u0*Pi*I),z=4.1*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0\\\"oz'pMQ\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0dQ+@h`u(!#H$\"0!4[:Ts3T!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 112 "largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 4.1087];" "6#7$\"\"!-%&FloatG6$\"&(3T!\" %" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stab ility function R* for the 12 stage, order 6 scheme is given as follo ws." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,S tabilityFunction(6,12,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)' =`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,<\"\"\"F)F 'F)*&#F)\"\"#F)*$)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)*$)F'F1F)F)F)*&# \"ZfkL*4?Q'R#yRPm/\"3TkulD(GN#o^6>R%\"hn+++sq(*)GvR5;Y\"=2Hbz)GJy+D?& \\j5J<#F)*$)F'\"\"(F)F)F)*&#\"hn6AdRy`X*Gg())zRkz:f(QO4\")Gbv)f`_6L`L&RI*eZC/@F)*$)F'\"\")F)F)F)*&#\"en\\\"\\ \")3$*z^qadL;%eLmmcz)3h$*>cqrJz*\"[o+++c-I`eH=sZzVMy`-Ao#[T-,u73>AR^m.r'H8:7R8e)49V=xCi&f?\"]o+++WZ'[AJ /9QQ\"\\'z)*R%>T'=!***3!zn=:95p%*F)*$)F'\"#5F)F)F)*&#\"ZpP=F()4RmX'4;< Lr17>*3B?V/Wtm=f#\"\\o++I43J!Rbn(HUh&\\)**H\\,Lx[7wtM)*onPO=\"F)*$)F' \"#6F)F)F)*&#\"J_Ev)Rw9Igu!3\">./UNDb&*G\"Tv$zL!)4jcj9s]@^>y!G60c1Ex\" =9F)*$)F'\"#7F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 124 "We can find the point where the boundary of the stabil ity region intersects the negative real axis by solving the equation: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6# /-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+comFW!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z= -4.4):\np_1 := plot([`R*`(z),1],z=-4.89..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[display]([p_1,p_2,p_3],view=[-4.89..0.49,-0.07..1.47],fo nt=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 268 268 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3o*************)[!#<$\"3QRiEd]ooAF*7$ $!3)\\%F*$\"3*3'4P\"G7!R6F*7$$!3iK$e'y!Q/W%F*$\"3YLehT,*R-\"F*7$$!3' )\\7LEy&HQ%F*$\"3E=@s'og#*>*!#=7$$!3AmT+uvZDVF*$\"35j:P$)Rba#)FZ7$$!3_ L3AqW*)=UF*$\"3l=h?!*[zJnFZ7$$!3`*\\(yRQ`3TF*$\"3ul;t`A=IaFZ7$$!3!H$3# zr)R%*RF*$\"3G*[=\\r@VL%FZ7$$!3u*\\P)G'H1)QF*$\"3[^5zl^VbMFZ7$$!3QmmrO TgjPF*$\"32#*=BsuBPFFZ7$$!3'HLe&>!G0m$F*$\"3))Rst*\\>_B#FZ7$$!3\"**** \\^s)[WNF*$\"3aUMtfby\"z\"FZ7$$!3q***\\A'H(zU$F*$\"3?k8\\DS0a9FZ7$$!3w ***\\[V)o:LF*$\"31?N:\\XK77FZ7$$!3SL3#pKBP@$F*$\"3wy#4$4kh^5FZ7$$!3Kmm J^pZ#4$F*$\"3)*\\1Y#oU^?*!#>7$$!3?mmEn`w*)HF*$\"3%*4/^bE?#F*$\"3)Gr&)[\")*QH6FZ7$$!3KL3#>MSS3#F*$\"3OrUy!eb\"f7FZ7 $$!3u*\\7x[&y!)>F*$\"3-)*34od3*Q\"FZ7$$!3%HL3kTn#p=F*$\"3ARWx1'4![:FZ7 $$!3+++qO$eSv\"F*$\"3)f/o*=9)Qt\"FZ7$$!3;+]xG([8k\"F*$\"3M!p\\y!G#*Q>F Z7$$!3)**\\(o(Q'HK:F*$\"3!\\t'G)R68;#FZ7$$!3/+]#eN77T\"F*$\"3t!G!*Gd+* QCFZ7$$!3:nmwnHT-8F*$\"3UGO^Xo%*=FFZ7$$!3y***\\#)RYi=\"F*$\"39_^&*G7t` IFZ7$$!3qL3_?A)43\"F*$\"3!4:Cep]ER$FZ7$$!3!\\***\\.n+f'*FZ$\"3:75?&45k !QFZ7$$!37m;a,!zhd)FZ$\"3uI\"e_qPjFZ$\"3<;3V\"*R(eI&FZ7$$!3A1](=U!>z^FZ$\"3G,&4:%**ed fFZ7$$!3iKLLzHSjSFZ$\"3])yCXeM3m'FZ7$$!3mGLei\"\\B#HFZ$\"3Y6lb!GJfY(FZ 7$$!3gi;aZQu!z\"FZ$\"3mwxg\"3P/O)FZ7$$!3FJ****p3;4vFar$\"3mdbY3]ew#*FZ 7$$\"3DiL$3rQ%3WFar$\"3W%\\[;aq]/\"F*7$$\"3&pmmY=on]\"FZ$\"3/\")[xf3ii 6F*7$$\"3u+]7DeEVEFZ$\"3UNDY=Nb-8F*7$$\"3D)*\\(33R5t$FZ$\"3M](ffBNAX\" F*7$$\"3!***************[FZ$\"3)=pGdD;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++ \"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7 $FQF_]l7$FfnF_]l7$F[oF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l 7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]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$F`uF_]l7$Fe uF_]l7$FjuF_]l7$F_vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7$ F]xF_]l7$FbxF_]l7$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_] l7$FezF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe \\lFi\\lFi\\lFf\\l-F$6&7#7$$!34+++comFWF*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc \\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF] alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLOR G6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICA G\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%VIEWG6$;$!$*[! \"#$\"#\\F_dl;$!\"(F_dl$\"$Z\"F_dl" 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 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2045 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+1/720*z^6+ 4391911516823528725657464410810466 37397823963820099336459/2173110634952025007831288795529071814616103975 288977072000000*z^7+60210263520285767180173638759157964397988760289455 37839572211/2104244758930395335333115253598755528810919640312069388588 32000000*z^8+979317170561993610887956666633584163357547051799308814914 9/6099260170812740102414826822025378344379477218295853300256000000*z^9 +205956224771843140985813391215132967103665139221919056774803/94691014 1518677900899901864119439987964913838140431224864744000000*z^10+259186 673440443202308919120671331716096456639098727183769/118363767689834737 61248773301492999849561422976755390310809300000*z^11+28955525354204031 910807460301476398752652/141817726065605112807819512150721463566309803 379375*z^12:\npts := []: z0 := 0:\nfor ct from 0 to 210 do\n zz := n ewton(`R*`(z)=exp(ct*Pi/15*I),z=z0):\n z0 := zz:\n pts := [op(pts) ,[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.1,0,.35) ):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.6,0]],i=2..nop s(pts))],\n style=patchnogrid,color=COLOR(RGB,.35,0,.75)):\np ts := []: z0 := 2.1+5.6*I:\nfor ct from 0 to 40 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_3 := plot(pts,color=COLOR(RGB,.1,0,.35)):\np_4 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[2.00,5.36]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,.35,0,.75)):\npts := []: z0 := 2.1-5.6*I:\nfor ct from 0 to 40 do\n zz := newton(`R*`(z) =exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.1,0,.35)):\np_6 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[2.00,-5.36]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.35,0,.75)):\np_7 := plo t([[[-5.09,0],[2.39,0]],[[0,-6.19],[0,6.19]]],color=black,linestyle=3) :\nplots[display]([p_||(1..7)],view=[-5.09..2.39,-6.19..6.19],font=[HE LVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=c onstrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 481 481 {PLOTDATA 2 "6/ -%'CURVESG6$7_x7$$\"\"!F)F(7$F($\"3#******H5&R%4#!#=7$$\"3')*****H^:$> F!#F$\"3')*******H!z)=%F-7$$\"3/+++)*RRz=!#D$\"3O+++^r=$G'F-7$$\"3$)** ***>2\\?U$!#C$\"3[+++ngfx$)F-7$$\"3&)******G=f;J!#B$\"3)*******=e?Z5!# <7$$\"38+++DRLR=!#A$\"3/+++L&omD\"FF7$$\"3\"******4/sS.)FJ$\"33+++<$ph Y\"FF7$$\"3$)*****>1Ad!G!#@$\"3#******z!Qtv;FF7$$\"3'******H`5)>#)FU$ \"3-+++9jN&)=FF7$$\"3))*****>e.C3#!#?$\"37+++XI!\\4#FF7$$\"3>+++yv?VYF jn$\"3%******4%\\\"RI#FF7$$\"3%3+++'oBw\"*Fjn$\"3/+++t1I6DFF7$$\"3$*** ***4#)GOg\"!#>$\"3)******R5T]r#FF7$$\"3/+++m!ydX#Fjo$\"3')******)p`A\" HFF7$$\"3#*******)eKSD$Fjo$\"3')******\\P#**4$FF7$$\"3))*****fc&phOFjo $\"3'******Rf\"*eF$FF7$$\"3>+++'e_lL$Fjo$\"3()*****>\"G=RMFF7$$\"3$*** ***R48n-#Fjo$\"3!******\\dO&*e$FF7$$!3g*****f*H@ERFjn$\"3/+++0k!os$FF7 $$!37+++i!*=BRFjo$\"37+++'y?2&QFF7$$!3#******Hl%4n%)Fjo$\"3!)******o\" )*4'RFF7$$!3!*******>!*>'Q\"F-$\"3m******>NedSFF7$$!3/+++z!y?*>F-$\"32 +++#3$zSTFF7$$!3%)******)y%[YEF-$\"3))*****4c%H6UFF7$$!3x*****\\pPVL$F -$\"3#)*****zWy*pUFF7$$!3/+++%f?R/%F-$\"3g*****zr5yJ%FF7$$!3'******41E mw%F-$\"3#******Rz@dN%FF7$$!3e*****HBmj\\&F-$\"3#*******zKa%Q%FF7$$!3n *****HEi*GiF-$\"31+++#z\")\\S%FF7$$!3[+++0UkhpF-$\"3D+++H]hDg(R%FF7 $$!3#******47#Hf5FF$\"3/+++!3wfP%FF7$$!3$******\\^V;8\"FF$\"3W+++Hm\"y M%FF7$$!3\"******RBLW?\"FF$\"3t******y@28VFF7$$!3!******4Oa\"y7FF$\"37 +++7hmrUFF7$$!31+++9l^`8FF$\"34+++HdbBUFF7$$!3(*******\\?]J9FF$\"3-+++ ``%)oTFF7$$!33+++d4Q8:FF$\"3G+++kj)z5%FF7$$!3)******p?#e+;FF$\"3,+++z! p?/%FF7$$!3%******RxrUp\"FF$\"3#******zi$3tRFF7$$!3#******4^MYz\"FF$\" 35+++->!Q!RFF7$$!3(******zjO.!>FF$\"3&)*****fFuPFF7$$!37+++#zju6#FF$\"35+++>['er$FF7$$!3>++++^7CAFF$\" 35+++2?)3m$FF7$$!3\"******H$zgFBFF$\"3\"******f:i!3OFF7$$!3))*****\\2U tU#FF$\"3++++%z\"=cNFF7$$!3#******z(R;BDFF$\"32+++!3?U]$FF7$$!3<+++'pJ ^h#FF$\"3A+++=*z8X$FF7$$!3y*****4A7Mq#FF$\"39+++:f0(R$FF7$$!35+++s+A)y #FF$\"3*)*****f<#zSLFF7$$!3?+++JmypGFF$\"3=+++M`C#G$FF7$$!3)******zhY$ [HFF$\"3++++gk:@KFF7$$!3y*****zlLT-$FF$\"37+++5:LdJFF7$$!3)*******=.Q( 4$FF$\"3=+++/!H14$FF7$$!39+++N)>$oJFF$\"3++++GO&4-$FF7$$!3A+++9w=PKFF$ \"3;+++1ZD[HFF7$$!3:+++e4A/LFF$\"31+++F&GD(GFF7$$!3'******>_b'pLFF$\"3 ')*****pHBQz#FF7$$!3*******fD OFF$\"3')******flC_CFF7$$!3%)*****fYj&zOFF$\"35+++oFThBFF7$$!3z*****zs 0(QPFF$\"3y*****\\!Q-pAFF7$$!3%******p;Wnz$FF$\"3')*****p_\"Qv@FF7$$!3 !******4Z`M&QFF$\"3')*****f%*Q23#FF7$$!3,+++\"4g&3RFF$\"3/+++WBG&)>FF7 $$!3'******zop<'RFF$\"31+++T^7*)=FF7$$!3r*****4B#z7SFF$\"33+++jDJ#z\"F F7$$!3i*****z*4OhSFF$\"3/+++'fL[p\"FF7$$!3o*****zSWs5%FF$\"3/+++Ihj'f \"FF7$$!3s*****ze^-:%FF$\"31+++E?k(\\\"FF7$$!3'******H]L->%FF$\"3%**** **\\CgxR\"FF7$$!3f******>:3FUFF$\"3%******HT(*oH\"FF7$$!3))******Q[sgU FF$\"3'******>*f'\\>\"FF7$$!3k******)oF6H%FF$\"3%*******p1*=4\"FF7$$!3 $)*****zm'G=VFF$\"3m*****z]Oh()*F-7$$!3w*****f>GAM%FF$\"3L+++OW*4#))F- 7$$!3W+++Vd+jVFF$\"3'********)eR`xF-7$$!3E+++5jp!Q%FF$\"35+++0WdtmF-7$ $!3S+++XbR&R%FF$\"3y*****H+G@e&F-7$$!34+++x4@2WFF$\"37+++()z0![%F-7$$! 3B+++sJD;WFF$\"33+++f/zoLF-7$$!3))******p^iAWFF$\"3!*******o)o,D#F-7$$ !3o*****fE6kU%FF$\"31+++)G2k7\"F-7$$!34+++comFWFFF(7$Fegl$!31+++)G2k7 \"F-7$F`gl$!3!*******o)o,D#F-7$F[gl$!33+++f/zoLF-7$Fffl$!37+++()z0![%F -7$Fafl$!3y*****H+G@e&F-7$F\\fl$!35+++0WdtmF-7$Fgel$!3'********)eR`xF- 7$Fbel$!3L+++OW*4#))F-7$F]el$!3m*****z]Oh()*F-7$Fhdl$!3%*******p1*=4\" FF7$Fcdl$!3'******>*f'\\>\"FF7$F^dl$!3%******HT(*oH\"FF7$Ficl$!3%***** *\\CgxR\"FF7$Fdcl$!31+++E?k(\\\"FF7$F_cl$!3/+++Ihj'f\"FF7$Fjbl$!3/+++' fL[p\"FF7$Febl$!33+++jDJ#z\"FF7$F`bl$!31+++T^7*)=FF7$F[bl$!3/+++WBG&)> FF7$Ffal$!3')*****f%*Q23#FF7$Faal$!3')*****p_\"Qv@FF7$F\\al$!3y*****\\ !Q-pAFF7$Fg`l$!35+++oFThBFF7$Fb`l$!3')******flC_CFF7$F]`l$!33+++g8?TDF F7$Fh_l$!3\"******4Khzi#FF7$Fc_l$!3z*****p&\\C7FFF7$F^_l$!3')*****pHBQ z#FF7$Fi^l$!31+++F&GD(GFF7$Fd^l$!3;+++1ZD[HFF7$F_^l$!3++++GO&4-$FF7$Fj ]l$!3=+++/!H14$FF7$Fe]l$!37+++5:LdJFF7$F`]l$!3++++gk:@KFF7$F[]l$!3=+++ M`C#G$FF7$Ff\\l$!3*)*****f<#zSLFF7$Fa\\l$!39+++:f0(R$FF7$F\\\\l$!3A+++ =*z8X$FF7$Fg[l$!32+++!3?U]$FF7$Fb[l$!3++++%z\"=cNFF7$F][l$!3\"******f: i!3OFF7$Fhz$!35+++2?)3m$FF7$Fcz$!35+++>['er$FF7$F^z$!3#)*****H!>FuPFF7 $Fiy$!3&)*****f!Q!RFF7$F_y$!3#******zi$3tRFF7$Fjx $!3,+++z!p?/%FF7$Fex$!3G+++kj)z5%FF7$F`x$!3-+++``%)oTFF7$F[x$!34+++Hdb BUFF7$Ffw$!37+++7hmrUFF7$Faw$!3t******y@28VFF7$F\\w$!3W+++Hm\"yM%FF7$F gv$!3/+++!3wfP%FF7$Fbv$!3K+++>Dg(R%FF7$F]v$!3\"******Rl$p7WFF7$Fhu$!3k *****f(o<@WFF7$Fcu$!3(******Hj)*GU%FF7$F^u$!3D+++H]hNedSFF7$Far$!3!)******o\")*4'RFF7$F\\r$!37+++'y?2& QFF7$Fgq$!3/+++0k!os$FF7$Fbq$!3!******\\dO&*e$FF7$F]q$!3()*****>\"G=RM FF7$Fhp$!3'******Rf\"*eF$FF7$Fcp$!3')******\\P#**4$FF7$F^p$!3')******) p`A\"HFF7$Fho$!3)******R5T]r#FF7$Fco$!3/+++t1I6DFF7$F^o$!3%******4%\\ \"RI#FF7$Fhn$!37+++XI!\\4#FF7$FY$!3-+++9jN&)=FF7$FS$!3#******z!Qtv;FF7 $FN$!33+++<$phY\"FF7$FH$!3/+++L&omD\"FF7$FA$!3)*******=e?Z5FF7$F;$!3[+ ++ngfx$)F-7$$\"3!******p*RRz=F7$!3O+++^r=$G'F-7$$\"3B+++IbJ>FF1$!3')** *****H!z)=%F-7$F($!3#******H5&R%4#F-F'F*F.F4F:F@FGFMFRFXFgnF]oFboFgoF] pFbpFgpF\\qFaqFfqF[rF`rFerFjrF_sFdsFisF^tFctFhtF]u-%&COLORG6&%$RGBG$\" \"\"!\"\"F($\"#N!\"#-%)POLYGONSG6`x7%F'7$F($\"+.^R%4#!#57$$!#EFaimF(7% Fiim7$$\"+8bJ>FFjo$\"++.z)=%F\\jmF]jm7%Fajm7$$\"+)*RRz=FF$\"+^r=$G'F\\ jmF]jm7%Fgjm7$$\"+s!\\?U$!#;$\"+ngfx$)F\\jmF]jm7%F][n7$$\"+H=f;J!#:$\" +>e?Z5!\"*F]jm7%Fd[n7$$\"+DRLR=!#9$\"+L&omD\"Fj[nF]jm7%F\\\\n7$$\"+T?2 M!)F_\\n$\"+<$phY\"Fj[nF]jm7%Fc\\n7$$\"+i?s0G!#8$\"+3Qtv;Fj[nF]jm7%Fi \\n7$$\"+L0\")>#)F\\]n$\"+9jN&)=Fj[nF]jm7%F`]n7$$\"+#e.C3#!#7$\"+XI!\\ 4#Fj[nF]jm7%Ff]n7$$\"+yv?VYFi]n$\"+T\\\"RI#Fj[nF]jm7%F]^n7$$\"+goBw\"* Fi]n$\"+t1I6DFj[nF]jm7%Fc^n7$$\"+@)GOg\"!#6$\"+/6/:FFj[nF]jm7%Fi^n7$$ \"+m!ydX#F\\_n$\"+*p`A\"HFj[nF]jm7%F`_n7$$\"+*eKSD$F\\_n$\"+]P#**4$Fj[ nF]jm7%Ff_n7$$\"+mbphOF\\_n$\"+%f\"*eF$Fj[nF]jm7%F\\`n7$$\"+'e_lL$F\\_ n$\"+7G=RMFj[nF]jm7%Fb`n7$$\"+%48n-#F\\_n$\"+vl`*e$Fj[nF]jm7%Fh`n7$$!+ '*H@ERFi]n$\"+0k!os$Fj[nF]jm7%F^an7$$!+i!*=BRF\\_n$\"+'y?2&QFj[nF]jm7% Fdan7$$!+`Y4n%)F\\_n$\"+p\")*4'RFj[nF]jm7%Fjan7$$!+?!*>'Q\"F\\jm$\"+?N edSFj[nF]jm7%F`bn7$$!+z!y?*>F\\jm$\"+#3$zSTFj[nF]jm7%Ffbn7$$!+*y%[YEF \\jm$\"+hXH6UFj[nF]jm7%F\\cn7$$!+&pPVL$F\\jm$\"+[%y*pUFj[nF]jm7%Fbcn7$ $!+%f?R/%F\\jm$\"+=2\"yJ%Fj[nF]jm7%Fhcn7$$!+hgimZF\\jm$\"+%z@dN%Fj[nF] jm7%F^dn7$$!+LiO'\\&F\\jm$\"+!GVXQ%Fj[nF]jm7%Fddn7$$!+jA'*GiF\\jm$\"+# z\")\\S%Fj[nF]jm7%Fjdn7$$!+0UkhpF\\jm$\"+H]hDg(R%Fj[n F]jm7%Fhfn7$$!+@@Hf5Fj[n$\"+!3wfP%Fj[nF]jm7%F^gn7$$!+:NkJ6Fj[n$\"+Hm\" yM%Fj[nF]jm7%Fdgn7$$!+MKV/7Fj[n$\"+z@28VFj[nF]jm7%Fjgn7$$!+hV:y7Fj[n$ \"+7hmrUFj[nF]jm7%F`hn7$$!+9l^`8Fj[n$\"+HdbBUFj[nF]jm7%Ffhn7$$!+]?]J9F j[n$\"+``%)oTFj[nF]jm7%F\\in7$$!+d4Q8:Fj[n$\"+kj)z5%Fj[nF]jm7%Fbin7$$! +2Ae+;Fj[n$\"+z!p?/%Fj[nF]jm7%Fhin7$$!+u!Q!RFj[nF]jm7%Fdjn7$$!+QmL+>Fj[n$\"+wh(p$QFj[ nF]jm7%Fjjn7$$!+&oZ)3?Fj[n$\"+.>FuPFj[nF]jm7%F`[o7$$!+#zju6#Fj[n$\"+>[ 'er$Fj[nF]jm7%Ff[o7$$!++^7CAFj[n$\"+2?)3m$Fj[nF]jm7%F\\\\o7$$!+LzgFBFj [n$\"+c@13OFj[nF]jm7%Fb\\o7$$!+v?MFCFj[n$\"+%z\"=cNFj[nF]jm7%Fh\\o7$$! +yR;BDFj[n$\"+!3?U]$Fj[nF]jm7%F^]o7$$!+'pJ^h#Fj[n$\"+=*z8X$Fj[nF]jm7%F d]o7$$!+@AT.FFj[n$\"+:f0(R$Fj[nF]jm7%Fj]o7$$!+s+A)y#Fj[n$\"+w@zSLFj[nF ]jm7%F`^o7$$!+JmypGFj[n$\"+M`C#G$Fj[nF]jm7%Ff^o7$$!+=mM[HFj[n$\"+gk:@K Fj[nF]jm7%F\\_o7$$!+eO8CIFj[n$\"+5:LdJFj[nF]jm7%Fb_o7$$!+>.Q(4$Fj[n$\" +/!H14$Fj[nF]jm7%Fh_o7$$!+N)>$oJFj[n$\"+GO&4-$Fj[nF]jm7%F^`o7$$!+9w=PK Fj[n$\"+1ZD[HFj[nF]jm7%Fd`o7$$!+e4A/LFj[n$\"+F&GD(GFj[nF]jm7%Fj`o7$$!+ AblpLFj[n$\"+(HBQz#Fj[nF]jm7%F`ao7$$!+csrLMFj[n$\"+d\\C7FFj[nF]jm7%Ffa o7$$!+B\"4m\\$Fj[n$\"+@8'zi#Fj[nF]jm7%F\\bo7$$!+zJ\\eNFj[n$\"+g8?TDFj[ nF]jm7%Fbbo7$$!+/0Z>OFj[n$\"+glC_CFj[nF]jm7%Fhbo7$$!+mMczOFj[n$\"+oFTh BFj[nF]jm7%F^co7$$!+GdqQPFj[n$\"+0Q-pAFj[nF]jm7%Fdco7$$!+nTu'z$Fj[n$\" +F:Qv@Fj[nF]jm7%Fjco7$$!+rMX`QFj[n$\"+Y*Q23#Fj[nF]jm7%F`do7$$!+\"4g&3R Fj[n$\"+WBG&)>Fj[nF]jm7%Ffdo7$$!+)op<'RFj[n$\"+T^7*)=Fj[nF]jm7%F\\eo7$ $!+JAz7SFj[n$\"+jDJ#z\"Fj[nF]jm7%Fbeo7$$!+)*4OhSFj[n$\"+'fL[p\"Fj[nF]j m7%Fheo7$$!+3WC2TFj[n$\"+Ihj'f\"Fj[nF]jm7%F^fo7$$!+)e^-:%Fj[n$\"+E?k( \\\"Fj[nF]jm7%Fdfo7$$!+.NB!>%Fj[n$\"+X-w(R\"Fj[nF]jm7%Fjfo7$$!+?:3FUFj [n$\"+8u*oH\"Fj[nF]jm7%F`go7$$!+R[sgUFj[n$\"+#*f'\\>\"Fj[nF]jm7%Ffgo7$ $!+*oF6H%Fj[n$\"+q1*=4\"Fj[nF]jm7%F\\ho7$$!+omG=VFj[n$\"+3l8w)*F\\jmF] jm7%Fbho7$$!+'>GAM%Fj[n$\"+OW*4#))F\\jmF]jm7%Fhho7$$!+Vd+jVFj[n$\"+!*e R`xF\\jmF]jm7%F^io7$$!+5jp!Q%Fj[n$\"+0WdtmF\\jmF]jm7%Fdio7$$!+XbR&R%Fj [n$\"+.!G@e&F\\jmF]jm7%Fjio7$$!+x4@2WFj[n$\"+()z0![%F\\jmF]jm7%F`jo7$$ !+sJD;WFj[n$\"+f/zoLF\\jmF]jm7%Ffjo7$$!+q^iAWFj[n$\"+p)o,D#F\\jmF]jm7% F\\[p7$$!+m7TEWFj[n$\"+)G2k7\"F\\jmF]jm7%Fb[p7$$!+comFWFj[nF(F]jm7%Fh[ p7$Fc[p$!+)G2k7\"F\\jmF]jm7%F\\\\p7$F][p$!+p)o,D#F\\jmF]jm7%F`\\p7$Fgj o$!+f/zoLF\\jmF]jm7%Fd\\p7$Fajo$!+()z0![%F\\jmF]jm7%Fh\\p7$F[jo$!+.!G@ e&F\\jmF]jm7%F\\]p7$Feio$!+0WdtmF\\jmF]jm7%F`]p7$F_io$!+!*eR`xF\\jmF]j m7%Fd]p7$Fiho$!+OW*4#))F\\jmF]jm7%Fh]p7$Fcho$!+3l8w)*F\\jmF]jm7%F\\^p7 $F]ho$!+q1*=4\"Fj[nF]jm7%F`^p7$Fggo$!+#*f'\\>\"Fj[nF]jm7%Fd^p7$Fago$!+ 8u*oH\"Fj[nF]jm7%Fh^p7$F[go$!+X-w(R\"Fj[nF]jm7%F\\_p7$Fefo$!+E?k(\\\"F j[nF]jm7%F`_p7$F_fo$!+Ihj'f\"Fj[nF]jm7%Fd_p7$Fieo$!+'fL[p\"Fj[nF]jm7%F h_p7$Fceo$!+jDJ#z\"Fj[nF]jm7%F\\`p7$F]eo$!+T^7*)=Fj[nF]jm7%F``p7$Fgdo$ !+WBG&)>Fj[nF]jm7%Fd`p7$Fado$!+Y*Q23#Fj[nF]jm7%Fh`p7$F[do$!+F:Qv@Fj[nF ]jm7%F\\ap7$Feco$!+0Q-pAFj[nF]jm7%F`ap7$F_co$!+oFThBFj[nF]jm7%Fdap7$Fi bo$!+glC_CFj[nF]jm7%Fhap7$Fcbo$!+g8?TDFj[nF]jm7%F\\bp7$F]bo$!+@8'zi#Fj [nF]jm7%F`bp7$Fgao$!+d\\C7FFj[nF]jm7%Fdbp7$Faao$!+(HBQz#Fj[nF]jm7%Fhbp 7$F[ao$!+F&GD(GFj[nF]jm7%F\\cp7$Fe`o$!+1ZD[HFj[nF]jm7%F`cp7$F_`o$!+GO& 4-$Fj[nF]jm7%Fdcp7$Fi_o$!+/!H14$Fj[nF]jm7%Fhcp7$Fc_o$!+5:LdJFj[nF]jm7% F\\dp7$F]_o$!+gk:@KFj[nF]jm7%F`dp7$Fg^o$!+M`C#G$Fj[nF]jm7%Fddp7$Fa^o$! +w@zSLFj[nF]jm7%Fhdp7$F[^o$!+:f0(R$Fj[nF]jm7%F\\ep7$Fe]o$!+=*z8X$Fj[nF ]jm7%F`ep7$F_]o$!+!3?U]$Fj[nF]jm7%Fdep7$Fi\\o$!+%z\"=cNFj[nF]jm7%Fhep7 $Fc\\o$!+c@13OFj[nF]jm7%F\\fp7$F]\\o$!+2?)3m$Fj[nF]jm7%F`fp7$Fg[o$!+>[ 'er$Fj[nF]jm7%Fdfp7$Fa[o$!+.>FuPFj[nF]jm7%Fhfp7$F[[o$!+wh(p$QFj[nF]jm7 %F\\gp7$Fejn$!+->!Q!RFj[nF]jm7%F`gp7$F_jn$!+GO3tRFj[nF]jm7%Fdgp7$Fiin$ !+z!p?/%Fj[nF]jm7%Fhgp7$Fcin$!+kj)z5%Fj[nF]jm7%F\\hp7$F]in$!+``%)oTFj[ nF]jm7%F`hp7$Fghn$!+HdbBUFj[nF]jm7%Fdhp7$Fahn$!+7hmrUFj[nF]jm7%Fhhp7$F [hn$!+z@28VFj[nF]jm7%F\\ip7$Fegn$!+Hm\"yM%Fj[nF]jm7%F`ip7$F_gn$!+!3wfP %Fj[nF]jm7%Fdip7$Fifn$!+>Dg(R%Fj[nF]jm7%Fhip7$Fcfn$!+aOp7WFj[nF]jm7%F \\jp7$F]fn$!+wo<@WFj[nF]jm7%F`jp7$Fgen$!+L')*GU%Fj[nF]jm7%Fdjp7$Faen$! +H]he?Z5Fj[nF]jm7%F\\aq7$ F^[n$!+ngfx$)F\\jmF]jm7%F`aq7$$\"+(*RRz=FF$!+^r=$G'F\\jmF]jm7%Fdaq7$$ \"+IbJ>FFjo$!++.z)=%F\\jmF]jm7%Fjaq7$F($!+.^R%4#F\\jmF]jm7%F`bqF'F]jmF himF`jmFfjmF\\[nFc[nF[\\nFb\\nFh\\nF_]nFe]nF\\^nFb^nFh^nF__nFe_nF[`nFa `nFg`nF]anFcanFianF_bnFebnF[cnFacnFgcnF]dnFcdnFidnF_en-F\\im6&F^imFbim F($\"#vFdim-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3%******RhQY4#FF$\"3K+ ++U[%[N&FF7$$\"3!)*****H-,\\4#FF$\"3)******z6MzO&FF7$$\"3?+++(GPL4#FF$ \"3e*****z&pu!Q&FF7$$\"3)******4%*Q+4#FF$\"3K+++N,2$R&FF7$$\"3=+++X:6& 3#FF$\"3;+++@'3ZS&FF7$$\"3!*******=Gny?FF$\"3r*****4!Q[:aFF7$$\"3/+++( \\^32#FF$\"3B+++N>BDaFF7$$\"33+++nnyh?FF$\"3f*****z\"H!QV&FF7$$\"3y*** ***o\"G;0#FF$\"3@+++X$f5W&FF7$$\"31+++;f`S?FF$\"3#)*****f!f(oW&FF7$$\" 3'******pd\"oG?FF$\"3;+++e\"R6X&FF7$$\"3$)******H\"\\i,#FF$\"31+++Kxu` aFF7$$\"3@+++QjV.?FF$\"3M+++)38YX&FF7$$\"3)******zWc/*>FF$\"3S+++e4m`a FF7$$\"35+++H,ax>FF$\"3I+++4P$3X&FF7$$\"3!******4`P\\'>FF$\"3u*****\\* Q4YaFF7$$\"3)******pFF$\"3=+++h$H%RaFF7$$\"3$******>C#yT>FF$\"3&) ******>.'3V&FF7$$\"3'******>SS=$>FF$\"33+++!y[/U&FF7$$\"3)******R/JM#> FF$\"3=+++p0J3aFF7$$\"3.+++NI!p\">FF$\"3O+++L(HYR&FF7$$\"36+++&z.E\">F F$\"3')*****47u'z`FF7$$\"3-+++@j&3\">FF$\"3/+++*Q4QO&FF7$$\"3.+++:e#> \">FF$\"3;+++%[1vM&FF7$$\"3$*******>u(f\">FF$\"3(******RnN8L&FF7$$\"31 +++d:.B>FF$\"3.+++$4SfJ&FF7$$\"3.+++0*GH$>FF$\"3r******fc)>I&FF7$$\"3' ******z_>`%>FF$\"3q*****4(G4!H&FF7$$\"3%*******=eof>FF$\"31+++1fw!G&FF 7$$\"3%******\\)pRv>FF$\"3/+++]!RVF&FF7$$\"3-+++o,y\"*>FF$\"3;+++=\\&4 F&FF7$$\"3=+++\"H)=3?FF$\"3s*****4Bw0F&FF7$$\"37++++'\\S-#FF$\"3m***** 4@@IF&FF7$$\"3&******4!Q*)Q?FF$\"31+++#G3!y_FF7$$\"3')*****R*zN_?FF$\" 3M+++7$)>&G&FF7$$\"3))*****zJzT1#FF$\"3!******H7GUH&FF7$$\"3%******42 \"=u?FF$\"31+++)yLZI&FF7$$\"3!)*****z6dA3#FF$\"3)******p*\\O;`FF7$$\"3 ++++YtN)3#FF$\"33+++UFzG`FF7$$\"3')******[`Z#4#FF$\"3K+++FDrT`FFF_cqF[ im-Ffim6L7%7$$\"+9'QY4#Fj[n$\"+U[%[N&Fj[n7$$\"+B5!\\4#Fj[n$\"+=T$zO&Fj [n7$$\"$+#Fdim$\"$O&Fdim7%F_`r7$$\"+(GPL4#Fj[n$\"+epu!Q&Fj[nFd`r7%Fj`r 7$$\"+T*Q+4#Fj[n$\"+N,2$R&Fj[nFd`r7%F`ar7$$\"+X:6&3#Fj[n$\"+@'3ZS&Fj[n Fd`r7%Ffar7$$\"+>Gny?Fj[n$\"+,Q[:aFj[nFd`r7%F\\br7$$\"+(\\^32#Fj[n$\"+ N>BDaFj[nFd`r7%Fbbr7$$\"+nnyh?Fj[n$\"+=H!QV&Fj[nFd`r7%Fhbr7$$\"+p\"G;0 #Fj[n$\"+X$f5W&Fj[nFd`r7%F^cr7$$\"+;f`S?Fj[n$\"+1f(oW&Fj[nFd`r7%Fdcr7$ $\"+x:oG?Fj[n$\"+e\"R6X&Fj[nFd`r7%Fjcr7$$\"+I\"\\i,#Fj[n$\"+Kxu`aFj[nF d`r7%F`dr7$$\"+QjV.?Fj[n$\"+)38YX&Fj[nFd`r7%Ffdr7$$\"+[kX!*>Fj[n$\"+e4 m`aFj[nFd`r7%F\\er7$$\"+H,ax>Fj[n$\"+4P$3X&Fj[nFd`r7%Fber7$$\"+Jv$\\'> Fj[n$\"+&*Q4YaFj[nFd`r7%Fher7$$\"+x,#H&>Fj[n$\"+h$H%RaFj[nFd`r7%F^fr7$ $\"+UAyT>Fj[n$\"+?.'3V&Fj[nFd`r7%Fdfr7$$\"+-/%=$>Fj[n$\"+!y[/U&Fj[nFd` r7%Fjfr7$$\"+W5VB>Fj[n$\"+p0J3aFj[nFd`r7%F`gr7$$\"+NI!p\">Fj[n$\"+L(HY R&Fj[nFd`r7%Ffgr7$$\"+&z.E\">Fj[n$\"+@Tnz`Fj[nFd`r7%F\\hr7$$\"+@j&3\"> Fj[n$\"+*Q4QO&Fj[nFd`r7%Fbhr7$$\"+:e#>\">Fj[n$\"+%[1vM&Fj[nFd`r7%Fhhr7 $$\"+?u(f\">Fj[n$\"+ucLJ`Fj[nFd`r7%F^ir7$$\"+d:.B>Fj[n$\"+$4SfJ&Fj[nFd `r7%Fdir7$$\"+0*GH$>Fj[n$\"+gc)>I&Fj[nFd`r7%Fjir7$$\"+G&>`%>Fj[n$\"+rG 4!H&Fj[nFd`r7%F`jr7$$\"+>eof>Fj[n$\"+1fw!G&Fj[nFd`r7%Ffjr7$$\"+&)pRv>F j[n$\"+]!RVF&Fj[nFd`r7%F\\[s7$$\"+o,y\"*>Fj[n$\"+=\\&4F&Fj[nFd`r7%Fb[s 7$$\"+\"H)=3?Fj[n$\"+Jidq_Fj[nFd`r7%Fh[s7$$\"++'\\S-#Fj[n$\"+67-t_Fj[n Fd`r7%F^\\s7$$\"+,Q*)Q?Fj[n$\"+#G3!y_Fj[nFd`r7%Fd\\s7$$\"+%*zN_?Fj[n$ \"+7$)>&G&Fj[nFd`r7%Fj\\s7$$\"+=$zT1#Fj[n$\"+B\"GUH&Fj[nFd`r7%F`]s7$$ \"+r5=u?Fj[n$\"+)yLZI&Fj[nFd`r7%Ff]s7$$\"+=rD#3#Fj[n$\"+(*\\O;`Fj[nFd` r7%F\\^s7$$\"+YtN)3#Fj[n$\"+UFzG`Fj[nFd`r7%Fb^s7$$\"+\\`Z#4#Fj[n$\"+FD rT`Fj[nFd`r7%Fh^sFj_rFd`rFdbqFhbq-F$6$7K7$F`cq$!3K+++U[%[N&FF7$Fc_r$!3 K+++FDrT`FF7$F^_r$!33+++UFzG`FF7$Fi^r$!3)******p*\\O;`FF7$Fd^r$!31+++) yLZI&FF7$F_^r$!3!******H7GUH&FF7$Fj]r$!3M+++7$)>&G&FF7$Fe]r$!31+++#G3! y_FF7$F`]r$!3m*****4@@IF&FF7$F[]r$!3s*****4Bw0F&FF7$Ff\\r$!3;+++=\\&4F &FF7$Fa\\r$!3/+++]!RVF&FF7$F\\\\r$!31+++1fw!G&FF7$Fg[r$!3q*****4(G4!H& FF7$Fb[r$!3r******fc)>I&FF7$F][r$!3.+++$4SfJ&FF7$Fhjq$!3(******RnN8L&F F7$Fcjq$!3;+++%[1vM&FF7$F^jq$!3/+++*Q4QO&FF7$Fiiq$!3')*****47u'z`FF7$F diq$!3O+++L(HYR&FF7$F_iq$!3=+++p0J3aFF7$Fjhq$!33+++!y[/U&FF7$Fehq$!3&) ******>.'3V&FF7$F`hq$!3=+++h$H%RaFF7$F[hq$!3u*****\\*Q4YaFF7$Ffgq$!3I+ ++4P$3X&FF7$Fagq$!3S+++e4m`aFF7$F\\gq$!3M+++)38YX&FF7$Fgfq$!31+++Kxu`a FF7$Fbfq$!3;+++e\"R6X&FF7$F]fq$!3#)*****f!f(oW&FF7$Fheq$!3@+++X$f5W&FF 7$Fceq$!3f*****z\"H!QV&FF7$F^eq$!3B+++N>BDaFF7$Fidq$!3r*****4!Q[:aFF7$ Fddq$!3;+++@'3ZS&FF7$F_dq$!3K+++N,2$R&FF7$Fjcq$!3e*****z&pu!Q&FF7$Fecq $!3)******z6MzO&FFFa_sF[im-Ffim6L7%7$F[`r$!+U[%[N&Fj[n7$Fi^s$!+FDrT`Fj [n7$Fe`r$!$O&Fdim7%F_gs7$Fc^s$!+UFzG`Fj[nFbgs7%Ffgs7$F]^s$!+(*\\O;`Fj[ nFbgs7%Fjgs7$Fg]s$!+)yLZI&Fj[nFbgs7%F^hs7$Fa]s$!+B\"GUH&Fj[nFbgs7%Fbhs 7$F[]s$!+7$)>&G&Fj[nFbgs7%Ffhs7$Fe\\s$!+#G3!y_Fj[nFbgs7%Fjhs7$F_\\s$!+ 67-t_Fj[nFbgs7%F^is7$Fi[s$!+Jidq_Fj[nFbgs7%Fbis7$Fc[s$!+=\\&4F&Fj[nFbg s7%Ffis7$F][s$!+]!RVF&Fj[nFbgs7%Fjis7$Fgjr$!+1fw!G&Fj[nFbgs7%F^js7$Faj r$!+rG4!H&Fj[nFbgs7%Fbjs7$F[jr$!+gc)>I&Fj[nFbgs7%Ffjs7$Feir$!+$4SfJ&Fj [nFbgs7%Fjjs7$F_ir$!+ucLJ`Fj[nFbgs7%F^[t7$Fihr$!+%[1vM&Fj[nFbgs7%Fb[t7 $Fchr$!+*Q4QO&Fj[nFbgs7%Ff[t7$F]hr$!+@Tnz`Fj[nFbgs7%Fj[t7$Fggr$!+L(HYR &Fj[nFbgs7%F^\\t7$Fagr$!+p0J3aFj[nFbgs7%Fb\\t7$F[gr$!+!y[/U&Fj[nFbgs7% Ff\\t7$Fefr$!+?.'3V&Fj[nFbgs7%Fj\\t7$F_fr$!+h$H%RaFj[nFbgs7%F^]t7$Fier $!+&*Q4YaFj[nFbgs7%Fb]t7$Fcer$!+4P$3X&Fj[nFbgs7%Ff]t7$F]er$!+e4m`aFj[n Fbgs7%Fj]t7$Fgdr$!+)38YX&Fj[nFbgs7%F^^t7$Fadr$!+Kxu`aFj[nFbgs7%Fb^t7$F [dr$!+e\"R6X&Fj[nFbgs7%Ff^t7$Fecr$!+1f(oW&Fj[nFbgs7%Fj^t7$F_cr$!+X$f5W &Fj[nFbgs7%F^_t7$Fibr$!+=H!QV&Fj[nFbgs7%Fb_t7$Fcbr$!+N>BDaFj[nFbgs7%Ff _t7$F]br$!+,Q[:aFj[nFbgs7%Fj_t7$Fgar$!+@'3ZS&Fj[nFbgs7%F^`t7$Faar$!+N, 2$R&Fj[nFbgs7%Fb`t7$F[ar$!+epu!Q&Fj[nFbgs7%Ff`t7$F``r$!+=T$zO&Fj[nFbgs 7%Fj`tF\\gsFbgsFdbqFhbq-F$6%7$7$$!3')*************3&FFF(7$$\"37++++++! R#FFF(-%'COLOURG6&F^imF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++! >'FF7$F($\"3Q++++++!>'FFFgatFjat-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYL EG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G- F`ct6#%(DEFAULTG-%%VIEWG6$;$!$4&Fdim$\"$R#Fdim;$!$>'Fdim$\"$>'Fdim" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stabi lity interval for the embedded order 6 scheme is (approximately) " } {XPPEDIT 18 0 "[-4.4277, 0];" "6#7$,$-%&FloatG6$\"&xU%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two schemes can be displaye d in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "p lots[display]([p_||(1..6),p||(1..7)],view=[-5.39..2.39,-6.19..6.19],fo nt=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scal ing=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 401 509 509 {PLOTDATA 2 "65-%'CURVESG6$7_x7$$\"\"!F)F(7$F($\"3#******H5&R%4#!#=7$$\"3')***** H^:$>F!#F$\"3')*******H!z)=%F-7$$\"3/+++)*RRz=!#D$\"3O+++^r=$G'F-7$$\" 3$)*****>2\\?U$!#C$\"3[+++ngfx$)F-7$$\"3&)******G=f;J!#B$\"3)*******=e ?Z5!#<7$$\"38+++DRLR=!#A$\"3/+++L&omD\"FF7$$\"3\"******4/sS.)FJ$\"33++ +<$phY\"FF7$$\"3$)*****>1Ad!G!#@$\"3#******z!Qtv;FF7$$\"3'******H`5)># )FU$\"3-+++9jN&)=FF7$$\"3))*****>e.C3#!#?$\"37+++XI!\\4#FF7$$\"3>+++yv ?VYFjn$\"3%******4%\\\"RI#FF7$$\"3%3+++'oBw\"*Fjn$\"3/+++t1I6DFF7$$\"3 $******4#)GOg\"!#>$\"3)******R5T]r#FF7$$\"3/+++m!ydX#Fjo$\"3')******)p `A\"HFF7$$\"3#*******)eKSD$Fjo$\"3')******\\P#**4$FF7$$\"3))*****fc&ph OFjo$\"3'******Rf\"*eF$FF7$$\"3>+++'e_lL$Fjo$\"3()*****>\"G=RMFF7$$\"3 $******R48n-#Fjo$\"3!******\\dO&*e$FF7$$!3g*****f*H@ERFjn$\"3/+++0k!os $FF7$$!37+++i!*=BRFjo$\"37+++'y?2&QFF7$$!3#******Hl%4n%)Fjo$\"3!)***** *o\")*4'RFF7$$!3!*******>!*>'Q\"F-$\"3m******>NedSFF7$$!3/+++z!y?*>F-$ \"32+++#3$zSTFF7$$!3%)******)y%[YEF-$\"3))*****4c%H6UFF7$$!3x*****\\pP VL$F-$\"3#)*****zWy*pUFF7$$!3/+++%f?R/%F-$\"3g*****zr5yJ%FF7$$!3'***** *41Emw%F-$\"3#******Rz@dN%FF7$$!3e*****HBmj\\&F-$\"3#*******zKa%Q%FF7$ $!3n*****HEi*GiF-$\"31+++#z\")\\S%FF7$$!3[+++0UkhpF-$\"3D+++H]hDg(R %FF7$$!3#******47#Hf5FF$\"3/+++!3wfP%FF7$$!3$******\\^V;8\"FF$\"3W+++H m\"yM%FF7$$!3\"******RBLW?\"FF$\"3t******y@28VFF7$$!3!******4Oa\"y7FF$ \"37+++7hmrUFF7$$!31+++9l^`8FF$\"34+++HdbBUFF7$$!3(*******\\?]J9FF$\"3 -+++``%)oTFF7$$!33+++d4Q8:FF$\"3G+++kj)z5%FF7$$!3)******p?#e+;FF$\"3,+ ++z!p?/%FF7$$!3%******RxrUp\"FF$\"3#******zi$3tRFF7$$!3#******4^MYz\"F F$\"35+++->!Q!RFF7$$!3(******zjO.!>FF$\"3&)*****fFuPFF7$$!37+++#zju6#FF$\"35+++>['er$FF7$$!3>++++^7CAF F$\"35+++2?)3m$FF7$$!3\"******H$zgFBFF$\"3\"******f:i!3OFF7$$!3))***** \\2UtU#FF$\"3++++%z\"=cNFF7$$!3#******z(R;BDFF$\"32+++!3?U]$FF7$$!3<++ +'pJ^h#FF$\"3A+++=*z8X$FF7$$!3y*****4A7Mq#FF$\"39+++:f0(R$FF7$$!35+++s +A)y#FF$\"3*)*****f<#zSLFF7$$!3?+++JmypGFF$\"3=+++M`C#G$FF7$$!3)****** zhY$[HFF$\"3++++gk:@KFF7$$!3y*****zlLT-$FF$\"37+++5:LdJFF7$$!3)******* =.Q(4$FF$\"3=+++/!H14$FF7$$!39+++N)>$oJFF$\"3++++GO&4-$FF7$$!3A+++9w=P KFF$\"3;+++1ZD[HFF7$$!3:+++e4A/LFF$\"31+++F&GD(GFF7$$!3'******>_b'pLFF $\"3')*****pHBQz#FF7$$!3*******fDOFF$\"3')******flC_CFF7$$!3%)*****fYj&zOFF$\"35+++oFThBFF7$$!3z*** **zs0(QPFF$\"3y*****\\!Q-pAFF7$$!3%******p;Wnz$FF$\"3')*****p_\"Qv@FF7 $$!3!******4Z`M&QFF$\"3')*****f%*Q23#FF7$$!3,+++\"4g&3RFF$\"3/+++WBG&) >FF7$$!3'******zop<'RFF$\"31+++T^7*)=FF7$$!3r*****4B#z7SFF$\"33+++jDJ# z\"FF7$$!3i*****z*4OhSFF$\"3/+++'fL[p\"FF7$$!3o*****zSWs5%FF$\"3/+++Ih j'f\"FF7$$!3s*****ze^-:%FF$\"31+++E?k(\\\"FF7$$!3'******H]L->%FF$\"3%* *****\\CgxR\"FF7$$!3f******>:3FUFF$\"3%******HT(*oH\"FF7$$!3))******Q[ sgUFF$\"3'******>*f'\\>\"FF7$$!3k******)oF6H%FF$\"3%*******p1*=4\"FF7$ $!3$)*****zm'G=VFF$\"3m*****z]Oh()*F-7$$!3w*****f>GAM%FF$\"3L+++OW*4#) )F-7$$!3W+++Vd+jVFF$\"3'********)eR`xF-7$$!3E+++5jp!Q%FF$\"35+++0WdtmF -7$$!3S+++XbR&R%FF$\"3y*****H+G@e&F-7$$!34+++x4@2WFF$\"37+++()z0![%F-7 $$!3B+++sJD;WFF$\"33+++f/zoLF-7$$!3))******p^iAWFF$\"3!*******o)o,D#F- 7$$!3o*****fE6kU%FF$\"31+++)G2k7\"F-7$$!34+++comFWFFF(7$Fegl$!31+++)G2 k7\"F-7$F`gl$!3!*******o)o,D#F-7$F[gl$!33+++f/zoLF-7$Fffl$!37+++()z0![ %F-7$Fafl$!3y*****H+G@e&F-7$F\\fl$!35+++0WdtmF-7$Fgel$!3'********)eR`x F-7$Fbel$!3L+++OW*4#))F-7$F]el$!3m*****z]Oh()*F-7$Fhdl$!3%*******p1*=4 \"FF7$Fcdl$!3'******>*f'\\>\"FF7$F^dl$!3%******HT(*oH\"FF7$Ficl$!3%*** ***\\CgxR\"FF7$Fdcl$!31+++E?k(\\\"FF7$F_cl$!3/+++Ihj'f\"FF7$Fjbl$!3/++ +'fL[p\"FF7$Febl$!33+++jDJ#z\"FF7$F`bl$!31+++T^7*)=FF7$F[bl$!3/+++WBG& )>FF7$Ffal$!3')*****f%*Q23#FF7$Faal$!3')*****p_\"Qv@FF7$F\\al$!3y***** \\!Q-pAFF7$Fg`l$!35+++oFThBFF7$Fb`l$!3')******flC_CFF7$F]`l$!33+++g8?T DFF7$Fh_l$!3\"******4Khzi#FF7$Fc_l$!3z*****p&\\C7FFF7$F^_l$!3')*****pH BQz#FF7$Fi^l$!31+++F&GD(GFF7$Fd^l$!3;+++1ZD[HFF7$F_^l$!3++++GO&4-$FF7$ Fj]l$!3=+++/!H14$FF7$Fe]l$!37+++5:LdJFF7$F`]l$!3++++gk:@KFF7$F[]l$!3=+ ++M`C#G$FF7$Ff\\l$!3*)*****f<#zSLFF7$Fa\\l$!39+++:f0(R$FF7$F\\\\l$!3A+ ++=*z8X$FF7$Fg[l$!32+++!3?U]$FF7$Fb[l$!3++++%z\"=cNFF7$F][l$!3\"****** f:i!3OFF7$Fhz$!35+++2?)3m$FF7$Fcz$!35+++>['er$FF7$F^z$!3#)*****H!>FuPF F7$Fiy$!3&)*****f!Q!RFF7$F_y$!3#******zi$3tRFF7$F jx$!3,+++z!p?/%FF7$Fex$!3G+++kj)z5%FF7$F`x$!3-+++``%)oTFF7$F[x$!34+++H dbBUFF7$Ffw$!37+++7hmrUFF7$Faw$!3t******y@28VFF7$F\\w$!3W+++Hm\"yM%FF7 $Fgv$!3/+++!3wfP%FF7$Fbv$!3K+++>Dg(R%FF7$F]v$!3\"******Rl$p7WFF7$Fhu$! 3k*****f(o<@WFF7$Fcu$!3(******Hj)*GU%FF7$F^u$!3D+++H]hNedSFF7$Far$!3!)******o\")*4'RFF7$F\\r$!37+++'y? 2&QFF7$Fgq$!3/+++0k!os$FF7$Fbq$!3!******\\dO&*e$FF7$F]q$!3()*****>\"G= RMFF7$Fhp$!3'******Rf\"*eF$FF7$Fcp$!3')******\\P#**4$FF7$F^p$!3')***** *)p`A\"HFF7$Fho$!3)******R5T]r#FF7$Fco$!3/+++t1I6DFF7$F^o$!3%******4% \\\"RI#FF7$Fhn$!37+++XI!\\4#FF7$FY$!3-+++9jN&)=FF7$FS$!3#******z!Qtv;F F7$FN$!33+++<$phY\"FF7$FH$!3/+++L&omD\"FF7$FA$!3)*******=e?Z5FF7$F;$!3 [+++ngfx$)F-7$$\"3!******p*RRz=F7$!3O+++^r=$G'F-7$$\"3B+++IbJ>FF1$!3') *******H!z)=%F-7$F($!3#******H5&R%4#F-F'F*F.F4F:F@FGFMFRFXFgnF]oFboFgo F]pFbpFgpF\\qFaqFfqF[rF`rFerFjrF_sFdsFisF^tFctFhtF]u-%&COLORG6&%$RGBG$ \"\"\"!\"\"F($\"#N!\"#-%)POLYGONSG6`x7%F'7$F($\"+.^R%4#!#57$$!#EFaimF( 7%Fiim7$$\"+8bJ>FFjo$\"++.z)=%F\\jmF]jm7%Fajm7$$\"+)*RRz=FF$\"+^r=$G'F \\jmF]jm7%Fgjm7$$\"+s!\\?U$!#;$\"+ngfx$)F\\jmF]jm7%F][n7$$\"+H=f;J!#:$ \"+>e?Z5!\"*F]jm7%Fd[n7$$\"+DRLR=!#9$\"+L&omD\"Fj[nF]jm7%F\\\\n7$$\"+T ?2M!)F_\\n$\"+<$phY\"Fj[nF]jm7%Fc\\n7$$\"+i?s0G!#8$\"+3Qtv;Fj[nF]jm7%F i\\n7$$\"+L0\")>#)F\\]n$\"+9jN&)=Fj[nF]jm7%F`]n7$$\"+#e.C3#!#7$\"+XI! \\4#Fj[nF]jm7%Ff]n7$$\"+yv?VYFi]n$\"+T\\\"RI#Fj[nF]jm7%F]^n7$$\"+goBw \"*Fi]n$\"+t1I6DFj[nF]jm7%Fc^n7$$\"+@)GOg\"!#6$\"+/6/:FFj[nF]jm7%Fi^n7 $$\"+m!ydX#F\\_n$\"+*p`A\"HFj[nF]jm7%F`_n7$$\"+*eKSD$F\\_n$\"+]P#**4$F j[nF]jm7%Ff_n7$$\"+mbphOF\\_n$\"+%f\"*eF$Fj[nF]jm7%F\\`n7$$\"+'e_lL$F \\_n$\"+7G=RMFj[nF]jm7%Fb`n7$$\"+%48n-#F\\_n$\"+vl`*e$Fj[nF]jm7%Fh`n7$ $!+'*H@ERFi]n$\"+0k!os$Fj[nF]jm7%F^an7$$!+i!*=BRF\\_n$\"+'y?2&QFj[nF]j m7%Fdan7$$!+`Y4n%)F\\_n$\"+p\")*4'RFj[nF]jm7%Fjan7$$!+?!*>'Q\"F\\jm$\" +?NedSFj[nF]jm7%F`bn7$$!+z!y?*>F\\jm$\"+#3$zSTFj[nF]jm7%Ffbn7$$!+*y%[Y EF\\jm$\"+hXH6UFj[nF]jm7%F\\cn7$$!+&pPVL$F\\jm$\"+[%y*pUFj[nF]jm7%Fbcn 7$$!+%f?R/%F\\jm$\"+=2\"yJ%Fj[nF]jm7%Fhcn7$$!+hgimZF\\jm$\"+%z@dN%Fj[n F]jm7%F^dn7$$!+LiO'\\&F\\jm$\"+!GVXQ%Fj[nF]jm7%Fddn7$$!+jA'*GiF\\jm$\" +#z\")\\S%Fj[nF]jm7%Fjdn7$$!+0UkhpF\\jm$\"+H]hDg(R%F j[nF]jm7%Fhfn7$$!+@@Hf5Fj[n$\"+!3wfP%Fj[nF]jm7%F^gn7$$!+:NkJ6Fj[n$\"+H m\"yM%Fj[nF]jm7%Fdgn7$$!+MKV/7Fj[n$\"+z@28VFj[nF]jm7%Fjgn7$$!+hV:y7Fj[ n$\"+7hmrUFj[nF]jm7%F`hn7$$!+9l^`8Fj[n$\"+HdbBUFj[nF]jm7%Ffhn7$$!+]?]J 9Fj[n$\"+``%)oTFj[nF]jm7%F\\in7$$!+d4Q8:Fj[n$\"+kj)z5%Fj[nF]jm7%Fbin7$ $!+2Ae+;Fj[n$\"+z!p?/%Fj[nF]jm7%Fhin7$$!+u!Q!RFj[nF]jm7%Fdjn7$$!+QmL+>Fj[n$\"+wh(p$QF j[nF]jm7%Fjjn7$$!+&oZ)3?Fj[n$\"+.>FuPFj[nF]jm7%F`[o7$$!+#zju6#Fj[n$\"+ >['er$Fj[nF]jm7%Ff[o7$$!++^7CAFj[n$\"+2?)3m$Fj[nF]jm7%F\\\\o7$$!+LzgFB Fj[n$\"+c@13OFj[nF]jm7%Fb\\o7$$!+v?MFCFj[n$\"+%z\"=cNFj[nF]jm7%Fh\\o7$ $!+yR;BDFj[n$\"+!3?U]$Fj[nF]jm7%F^]o7$$!+'pJ^h#Fj[n$\"+=*z8X$Fj[nF]jm7 %Fd]o7$$!+@AT.FFj[n$\"+:f0(R$Fj[nF]jm7%Fj]o7$$!+s+A)y#Fj[n$\"+w@zSLFj[ nF]jm7%F`^o7$$!+JmypGFj[n$\"+M`C#G$Fj[nF]jm7%Ff^o7$$!+=mM[HFj[n$\"+gk: @KFj[nF]jm7%F\\_o7$$!+eO8CIFj[n$\"+5:LdJFj[nF]jm7%Fb_o7$$!+>.Q(4$Fj[n$ \"+/!H14$Fj[nF]jm7%Fh_o7$$!+N)>$oJFj[n$\"+GO&4-$Fj[nF]jm7%F^`o7$$!+9w= PKFj[n$\"+1ZD[HFj[nF]jm7%Fd`o7$$!+e4A/LFj[n$\"+F&GD(GFj[nF]jm7%Fj`o7$$ !+AblpLFj[n$\"+(HBQz#Fj[nF]jm7%F`ao7$$!+csrLMFj[n$\"+d\\C7FFj[nF]jm7%F fao7$$!+B\"4m\\$Fj[n$\"+@8'zi#Fj[nF]jm7%F\\bo7$$!+zJ\\eNFj[n$\"+g8?TDF j[nF]jm7%Fbbo7$$!+/0Z>OFj[n$\"+glC_CFj[nF]jm7%Fhbo7$$!+mMczOFj[n$\"+oF ThBFj[nF]jm7%F^co7$$!+GdqQPFj[n$\"+0Q-pAFj[nF]jm7%Fdco7$$!+nTu'z$Fj[n$ \"+F:Qv@Fj[nF]jm7%Fjco7$$!+rMX`QFj[n$\"+Y*Q23#Fj[nF]jm7%F`do7$$!+\"4g& 3RFj[n$\"+WBG&)>Fj[nF]jm7%Ffdo7$$!+)op<'RFj[n$\"+T^7*)=Fj[nF]jm7%F\\eo 7$$!+JAz7SFj[n$\"+jDJ#z\"Fj[nF]jm7%Fbeo7$$!+)*4OhSFj[n$\"+'fL[p\"Fj[nF ]jm7%Fheo7$$!+3WC2TFj[n$\"+Ihj'f\"Fj[nF]jm7%F^fo7$$!+)e^-:%Fj[n$\"+E?k (\\\"Fj[nF]jm7%Fdfo7$$!+.NB!>%Fj[n$\"+X-w(R\"Fj[nF]jm7%Fjfo7$$!+?:3FUF j[n$\"+8u*oH\"Fj[nF]jm7%F`go7$$!+R[sgUFj[n$\"+#*f'\\>\"Fj[nF]jm7%Ffgo7 $$!+*oF6H%Fj[n$\"+q1*=4\"Fj[nF]jm7%F\\ho7$$!+omG=VFj[n$\"+3l8w)*F\\jmF ]jm7%Fbho7$$!+'>GAM%Fj[n$\"+OW*4#))F\\jmF]jm7%Fhho7$$!+Vd+jVFj[n$\"+!* eR`xF\\jmF]jm7%F^io7$$!+5jp!Q%Fj[n$\"+0WdtmF\\jmF]jm7%Fdio7$$!+XbR&R%F j[n$\"+.!G@e&F\\jmF]jm7%Fjio7$$!+x4@2WFj[n$\"+()z0![%F\\jmF]jm7%F`jo7$ $!+sJD;WFj[n$\"+f/zoLF\\jmF]jm7%Ffjo7$$!+q^iAWFj[n$\"+p)o,D#F\\jmF]jm7 %F\\[p7$$!+m7TEWFj[n$\"+)G2k7\"F\\jmF]jm7%Fb[p7$$!+comFWFj[nF(F]jm7%Fh [p7$Fc[p$!+)G2k7\"F\\jmF]jm7%F\\\\p7$F][p$!+p)o,D#F\\jmF]jm7%F`\\p7$Fg jo$!+f/zoLF\\jmF]jm7%Fd\\p7$Fajo$!+()z0![%F\\jmF]jm7%Fh\\p7$F[jo$!+.!G @e&F\\jmF]jm7%F\\]p7$Feio$!+0WdtmF\\jmF]jm7%F`]p7$F_io$!+!*eR`xF\\jmF] jm7%Fd]p7$Fiho$!+OW*4#))F\\jmF]jm7%Fh]p7$Fcho$!+3l8w)*F\\jmF]jm7%F\\^p 7$F]ho$!+q1*=4\"Fj[nF]jm7%F`^p7$Fggo$!+#*f'\\>\"Fj[nF]jm7%Fd^p7$Fago$! +8u*oH\"Fj[nF]jm7%Fh^p7$F[go$!+X-w(R\"Fj[nF]jm7%F\\_p7$Fefo$!+E?k(\\\" Fj[nF]jm7%F`_p7$F_fo$!+Ihj'f\"Fj[nF]jm7%Fd_p7$Fieo$!+'fL[p\"Fj[nF]jm7% Fh_p7$Fceo$!+jDJ#z\"Fj[nF]jm7%F\\`p7$F]eo$!+T^7*)=Fj[nF]jm7%F``p7$Fgdo $!+WBG&)>Fj[nF]jm7%Fd`p7$Fado$!+Y*Q23#Fj[nF]jm7%Fh`p7$F[do$!+F:Qv@Fj[n F]jm7%F\\ap7$Feco$!+0Q-pAFj[nF]jm7%F`ap7$F_co$!+oFThBFj[nF]jm7%Fdap7$F ibo$!+glC_CFj[nF]jm7%Fhap7$Fcbo$!+g8?TDFj[nF]jm7%F\\bp7$F]bo$!+@8'zi#F j[nF]jm7%F`bp7$Fgao$!+d\\C7FFj[nF]jm7%Fdbp7$Faao$!+(HBQz#Fj[nF]jm7%Fhb p7$F[ao$!+F&GD(GFj[nF]jm7%F\\cp7$Fe`o$!+1ZD[HFj[nF]jm7%F`cp7$F_`o$!+GO &4-$Fj[nF]jm7%Fdcp7$Fi_o$!+/!H14$Fj[nF]jm7%Fhcp7$Fc_o$!+5:LdJFj[nF]jm7 %F\\dp7$F]_o$!+gk:@KFj[nF]jm7%F`dp7$Fg^o$!+M`C#G$Fj[nF]jm7%Fddp7$Fa^o$ !+w@zSLFj[nF]jm7%Fhdp7$F[^o$!+:f0(R$Fj[nF]jm7%F\\ep7$Fe]o$!+=*z8X$Fj[n F]jm7%F`ep7$F_]o$!+!3?U]$Fj[nF]jm7%Fdep7$Fi\\o$!+%z\"=cNFj[nF]jm7%Fhep 7$Fc\\o$!+c@13OFj[nF]jm7%F\\fp7$F]\\o$!+2?)3m$Fj[nF]jm7%F`fp7$Fg[o$!+> ['er$Fj[nF]jm7%Fdfp7$Fa[o$!+.>FuPFj[nF]jm7%Fhfp7$F[[o$!+wh(p$QFj[nF]jm 7%F\\gp7$Fejn$!+->!Q!RFj[nF]jm7%F`gp7$F_jn$!+GO3tRFj[nF]jm7%Fdgp7$Fiin $!+z!p?/%Fj[nF]jm7%Fhgp7$Fcin$!+kj)z5%Fj[nF]jm7%F\\hp7$F]in$!+``%)oTFj [nF]jm7%F`hp7$Fghn$!+HdbBUFj[nF]jm7%Fdhp7$Fahn$!+7hmrUFj[nF]jm7%Fhhp7$ F[hn$!+z@28VFj[nF]jm7%F\\ip7$Fegn$!+Hm\"yM%Fj[nF]jm7%F`ip7$F_gn$!+!3wf P%Fj[nF]jm7%Fdip7$Fifn$!+>Dg(R%Fj[nF]jm7%Fhip7$Fcfn$!+aOp7WFj[nF]jm7%F \\jp7$F]fn$!+wo<@WFj[nF]jm7%F`jp7$Fgen$!+L')*GU%Fj[nF]jm7%Fdjp7$Faen$! +H]he?Z5Fj[nF]jm7%F\\aq7$ F^[n$!+ngfx$)F\\jmF]jm7%F`aq7$$\"+(*RRz=FF$!+^r=$G'F\\jmF]jm7%Fdaq7$$ \"+IbJ>FFjo$!++.z)=%F\\jmF]jm7%Fjaq7$F($!+.^R%4#F\\jmF]jm7%F`bqF'F]jmF himF`jmFfjmF\\[nFc[nF[\\nFb\\nFh\\nF_]nFe]nF\\^nFb^nFh^nF__nFe_nF[`nFa `nFg`nF]anFcanFianF_bnFebnF[cnFacnFgcnF]dnFcdnFidnF_en-F\\im6&F^imFbim F($\"#vFdim-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3%******RhQY4#FF$\"3K+ ++U[%[N&FF7$$\"3!)*****H-,\\4#FF$\"3)******z6MzO&FF7$$\"3?+++(GPL4#FF$ \"3e*****z&pu!Q&FF7$$\"3)******4%*Q+4#FF$\"3K+++N,2$R&FF7$$\"3=+++X:6& 3#FF$\"3;+++@'3ZS&FF7$$\"3!*******=Gny?FF$\"3r*****4!Q[:aFF7$$\"3/+++( \\^32#FF$\"3B+++N>BDaFF7$$\"33+++nnyh?FF$\"3f*****z\"H!QV&FF7$$\"3y*** ***o\"G;0#FF$\"3@+++X$f5W&FF7$$\"31+++;f`S?FF$\"3#)*****f!f(oW&FF7$$\" 3'******pd\"oG?FF$\"3;+++e\"R6X&FF7$$\"3$)******H\"\\i,#FF$\"31+++Kxu` aFF7$$\"3@+++QjV.?FF$\"3M+++)38YX&FF7$$\"3)******zWc/*>FF$\"3S+++e4m`a FF7$$\"35+++H,ax>FF$\"3I+++4P$3X&FF7$$\"3!******4`P\\'>FF$\"3u*****\\* Q4YaFF7$$\"3)******pFF$\"3=+++h$H%RaFF7$$\"3$******>C#yT>FF$\"3&) ******>.'3V&FF7$$\"3'******>SS=$>FF$\"33+++!y[/U&FF7$$\"3)******R/JM#> FF$\"3=+++p0J3aFF7$$\"3.+++NI!p\">FF$\"3O+++L(HYR&FF7$$\"36+++&z.E\">F F$\"3')*****47u'z`FF7$$\"3-+++@j&3\">FF$\"3/+++*Q4QO&FF7$$\"3.+++:e#> \">FF$\"3;+++%[1vM&FF7$$\"3$*******>u(f\">FF$\"3(******RnN8L&FF7$$\"31 +++d:.B>FF$\"3.+++$4SfJ&FF7$$\"3.+++0*GH$>FF$\"3r******fc)>I&FF7$$\"3' ******z_>`%>FF$\"3q*****4(G4!H&FF7$$\"3%*******=eof>FF$\"31+++1fw!G&FF 7$$\"3%******\\)pRv>FF$\"3/+++]!RVF&FF7$$\"3-+++o,y\"*>FF$\"3;+++=\\&4 F&FF7$$\"3=+++\"H)=3?FF$\"3s*****4Bw0F&FF7$$\"37++++'\\S-#FF$\"3m***** 4@@IF&FF7$$\"3&******4!Q*)Q?FF$\"31+++#G3!y_FF7$$\"3')*****R*zN_?FF$\" 3M+++7$)>&G&FF7$$\"3))*****zJzT1#FF$\"3!******H7GUH&FF7$$\"3%******42 \"=u?FF$\"31+++)yLZI&FF7$$\"3!)*****z6dA3#FF$\"3)******p*\\O;`FF7$$\"3 ++++YtN)3#FF$\"33+++UFzG`FF7$$\"3')******[`Z#4#FF$\"3K+++FDrT`FFF_cqF[ im-Ffim6L7%7$$\"+9'QY4#Fj[n$\"+U[%[N&Fj[n7$$\"+B5!\\4#Fj[n$\"+=T$zO&Fj [n7$$\"$+#Fdim$\"$O&Fdim7%F_`r7$$\"+(GPL4#Fj[n$\"+epu!Q&Fj[nFd`r7%Fj`r 7$$\"+T*Q+4#Fj[n$\"+N,2$R&Fj[nFd`r7%F`ar7$$\"+X:6&3#Fj[n$\"+@'3ZS&Fj[n Fd`r7%Ffar7$$\"+>Gny?Fj[n$\"+,Q[:aFj[nFd`r7%F\\br7$$\"+(\\^32#Fj[n$\"+ N>BDaFj[nFd`r7%Fbbr7$$\"+nnyh?Fj[n$\"+=H!QV&Fj[nFd`r7%Fhbr7$$\"+p\"G;0 #Fj[n$\"+X$f5W&Fj[nFd`r7%F^cr7$$\"+;f`S?Fj[n$\"+1f(oW&Fj[nFd`r7%Fdcr7$ $\"+x:oG?Fj[n$\"+e\"R6X&Fj[nFd`r7%Fjcr7$$\"+I\"\\i,#Fj[n$\"+Kxu`aFj[nF d`r7%F`dr7$$\"+QjV.?Fj[n$\"+)38YX&Fj[nFd`r7%Ffdr7$$\"+[kX!*>Fj[n$\"+e4 m`aFj[nFd`r7%F\\er7$$\"+H,ax>Fj[n$\"+4P$3X&Fj[nFd`r7%Fber7$$\"+Jv$\\'> Fj[n$\"+&*Q4YaFj[nFd`r7%Fher7$$\"+x,#H&>Fj[n$\"+h$H%RaFj[nFd`r7%F^fr7$ $\"+UAyT>Fj[n$\"+?.'3V&Fj[nFd`r7%Fdfr7$$\"+-/%=$>Fj[n$\"+!y[/U&Fj[nFd` r7%Fjfr7$$\"+W5VB>Fj[n$\"+p0J3aFj[nFd`r7%F`gr7$$\"+NI!p\">Fj[n$\"+L(HY R&Fj[nFd`r7%Ffgr7$$\"+&z.E\">Fj[n$\"+@Tnz`Fj[nFd`r7%F\\hr7$$\"+@j&3\"> Fj[n$\"+*Q4QO&Fj[nFd`r7%Fbhr7$$\"+:e#>\">Fj[n$\"+%[1vM&Fj[nFd`r7%Fhhr7 $$\"+?u(f\">Fj[n$\"+ucLJ`Fj[nFd`r7%F^ir7$$\"+d:.B>Fj[n$\"+$4SfJ&Fj[nFd `r7%Fdir7$$\"+0*GH$>Fj[n$\"+gc)>I&Fj[nFd`r7%Fjir7$$\"+G&>`%>Fj[n$\"+rG 4!H&Fj[nFd`r7%F`jr7$$\"+>eof>Fj[n$\"+1fw!G&Fj[nFd`r7%Ffjr7$$\"+&)pRv>F j[n$\"+]!RVF&Fj[nFd`r7%F\\[s7$$\"+o,y\"*>Fj[n$\"+=\\&4F&Fj[nFd`r7%Fb[s 7$$\"+\"H)=3?Fj[n$\"+Jidq_Fj[nFd`r7%Fh[s7$$\"++'\\S-#Fj[n$\"+67-t_Fj[n Fd`r7%F^\\s7$$\"+,Q*)Q?Fj[n$\"+#G3!y_Fj[nFd`r7%Fd\\s7$$\"+%*zN_?Fj[n$ \"+7$)>&G&Fj[nFd`r7%Fj\\s7$$\"+=$zT1#Fj[n$\"+B\"GUH&Fj[nFd`r7%F`]s7$$ \"+r5=u?Fj[n$\"+)yLZI&Fj[nFd`r7%Ff]s7$$\"+=rD#3#Fj[n$\"+(*\\O;`Fj[nFd` r7%F\\^s7$$\"+YtN)3#Fj[n$\"+UFzG`Fj[nFd`r7%Fb^s7$$\"+\\`Z#4#Fj[n$\"+FD rT`Fj[nFd`r7%Fh^sFj_rFd`rFdbqFhbq-F$6$7K7$F`cq$!3K+++U[%[N&FF7$Fc_r$!3 K+++FDrT`FF7$F^_r$!33+++UFzG`FF7$Fi^r$!3)******p*\\O;`FF7$Fd^r$!31+++) yLZI&FF7$F_^r$!3!******H7GUH&FF7$Fj]r$!3M+++7$)>&G&FF7$Fe]r$!31+++#G3! y_FF7$F`]r$!3m*****4@@IF&FF7$F[]r$!3s*****4Bw0F&FF7$Ff\\r$!3;+++=\\&4F &FF7$Fa\\r$!3/+++]!RVF&FF7$F\\\\r$!31+++1fw!G&FF7$Fg[r$!3q*****4(G4!H& FF7$Fb[r$!3r******fc)>I&FF7$F][r$!3.+++$4SfJ&FF7$Fhjq$!3(******RnN8L&F F7$Fcjq$!3;+++%[1vM&FF7$F^jq$!3/+++*Q4QO&FF7$Fiiq$!3')*****47u'z`FF7$F diq$!3O+++L(HYR&FF7$F_iq$!3=+++p0J3aFF7$Fjhq$!33+++!y[/U&FF7$Fehq$!3&) ******>.'3V&FF7$F`hq$!3=+++h$H%RaFF7$F[hq$!3u*****\\*Q4YaFF7$Ffgq$!3I+ ++4P$3X&FF7$Fagq$!3S+++e4m`aFF7$F\\gq$!3M+++)38YX&FF7$Fgfq$!31+++Kxu`a FF7$Fbfq$!3;+++e\"R6X&FF7$F]fq$!3#)*****f!f(oW&FF7$Fheq$!3@+++X$f5W&FF 7$Fceq$!3f*****z\"H!QV&FF7$F^eq$!3B+++N>BDaFF7$Fidq$!3r*****4!Q[:aFF7$ Fddq$!3;+++@'3ZS&FF7$F_dq$!3K+++N,2$R&FF7$Fjcq$!3e*****z&pu!Q&FF7$Fecq $!3)******z6MzO&FFFa_sF[im-Ffim6L7%7$F[`r$!+U[%[N&Fj[n7$Fi^s$!+FDrT`Fj [n7$Fe`r$!$O&Fdim7%F_gs7$Fc^s$!+UFzG`Fj[nFbgs7%Ffgs7$F]^s$!+(*\\O;`Fj[ nFbgs7%Fjgs7$Fg]s$!+)yLZI&Fj[nFbgs7%F^hs7$Fa]s$!+B\"GUH&Fj[nFbgs7%Fbhs 7$F[]s$!+7$)>&G&Fj[nFbgs7%Ffhs7$Fe\\s$!+#G3!y_Fj[nFbgs7%Fjhs7$F_\\s$!+ 67-t_Fj[nFbgs7%F^is7$Fi[s$!+Jidq_Fj[nFbgs7%Fbis7$Fc[s$!+=\\&4F&Fj[nFbg s7%Ffis7$F][s$!+]!RVF&Fj[nFbgs7%Fjis7$Fgjr$!+1fw!G&Fj[nFbgs7%F^js7$Faj r$!+rG4!H&Fj[nFbgs7%Fbjs7$F[jr$!+gc)>I&Fj[nFbgs7%Ffjs7$Feir$!+$4SfJ&Fj [nFbgs7%Fjjs7$F_ir$!+ucLJ`Fj[nFbgs7%F^[t7$Fihr$!+%[1vM&Fj[nFbgs7%Fb[t7 $Fchr$!+*Q4QO&Fj[nFbgs7%Ff[t7$F]hr$!+@Tnz`Fj[nFbgs7%Fj[t7$Fggr$!+L(HYR &Fj[nFbgs7%F^\\t7$Fagr$!+p0J3aFj[nFbgs7%Fb\\t7$F[gr$!+!y[/U&Fj[nFbgs7% Ff\\t7$Fefr$!+?.'3V&Fj[nFbgs7%Fj\\t7$F_fr$!+h$H%RaFj[nFbgs7%F^]t7$Fier $!+&*Q4YaFj[nFbgs7%Fb]t7$Fcer$!+4P$3X&Fj[nFbgs7%Ff]t7$F]er$!+e4m`aFj[n Fbgs7%Fj]t7$Fgdr$!+)38YX&Fj[nFbgs7%F^^t7$Fadr$!+Kxu`aFj[nFbgs7%Fb^t7$F [dr$!+e\"R6X&Fj[nFbgs7%Ff^t7$Fecr$!+1f(oW&Fj[nFbgs7%Fj^t7$F_cr$!+X$f5W &Fj[nFbgs7%F^_t7$Fibr$!+=H!QV&Fj[nFbgs7%Fb_t7$Fcbr$!+N>BDaFj[nFbgs7%Ff _t7$F]br$!+,Q[:aFj[nFbgs7%Fj_t7$Fgar$!+@'3ZS&Fj[nFbgs7%F^`t7$Faar$!+N, 2$R&Fj[nFbgs7%Fb`t7$F[ar$!+epu!Q&Fj[nFbgs7%Ff`t7$F``r$!+=T$zO&Fj[nFbgs 7%Fj`tF\\gsFbgsFdbqFhbq-F$6$7e\\lF'7$F($\"3++++Fjzq:F-7$F($\"3)******R l#fTJF-7$$\"3$*******[zw9?F1$\"3D+++\")*)Q7ZF-7$$\"3#)*******QYU)=F_jm $\"3=+++C`=$G'F-7$$\"32+++^4)f6\"F7$\"3e*****zy\")R&yF-7$$\"3s******pe k9`F7$\"3;+++(*)yZU*F-7$$\"3++++YT-xAF=$\"3++++gyb*4\"FF7$$\"3a*****\\ F0#))*)F=$\"3#*******f'QmD\"FF7$$\"31+++-\"y6B$FC$\"32+++x;s89FF7$$\"3 /+++'\\%yZ5FJ$\"35+++V(43d\"FF7$$\"3q*****pm]g1$FJ$\"32+++6x!zs\"FF7$$ \"3_*****zsv%\\\")FJ$\"3++++eA-&)=FF7$$\"37+++[Xb%)>FU$\"3>+++!Qf@/#FF 7$$\"3))*****R./^Y%FU$\"3')*****\\!pJ*>#FF7$$\"3%******H_]([$*FU$\"3#) *****>%yYcBFF7$$\"31+++y:\\J=Fjn$\"3&)*****R))GN^#FF7$$\"31+++jM')oLFj n$\"3#******\\*zIqEFF7$$\"3Q+++PS$[#eFjn$\"3%)*****\\nGk#GFF7$$\"3'3++ ghHOX*Fjn$\"3!)******z=C\")HFF7$$\"3$******Ha-YV\"Fjo$\"3.+++_VwLJFF7$ $\"39+++n[^B?Fjo$\"3'******>!>r#G$FF7$$\"3;+++>/&Rj#Fjo$\"3)******Ht!o EMFF7$$\"3M+++]3sPJFjo$\"3;+++8xUkNFF7$$\"3M+++-@nzLFjo$\"33+++p!)3&p$ FF7$$\"36+++S9l5KFjo$\"3.+++7.!4MKFjo$\"3C+++#)>YKUFF7$$!3N*****f%4Fd#>=F-$\"38+++OM()>XFF7$$!3/+++x4/tXFF7$$!3w*****>jH(\\FF-$\"3')******QS_>YFF7$$!3))******)o ,xB$F-$\"3')********QyfYFF7$$!39+++$y`\\t$F-$\"3'******pQtUp%FF7$$!3@+ ++p'*HQUF-$\"3m*****pp7Ms%FF7$$!3#******HnF`u%F-$\"3)******p7!eZZFF7$$ !3W+++K+Ca_F-$\"3m*****\\=3rw%FF7$$!3[*****f:=Pw&F-$\"3i*****po$G#y%FF 7$$!33+++9p\"GF'F-$\"3%******z,]Lz%FF7$$!3!)*****>\"f)3y'F-$\"3n*****4 P50![FF7$$!3=+++BS^(G(F-$\"3i*****f@JR![FF7$$!3W+++rM\\#z(F-$\"3l***** \\fXP![FF7$$!3M+++$e+eH)F-$\"3N+++ig0+[FF7$$!3Q+++l8f(z)F-$\"3i******G s$Hz%FF7$$!3!)*****zD6#)H*F-$\"3E+++#)zV#y%FF7$$!3!******Hd>#)z*F-$\"3 #******4a$eoZFF7$$!35+++II%)H5FF$\"3-+++pwP^ZFF7$$!35+++/s**z5FF$\"3J+ ++Gb!3t%FF7$$!3-+++jqVI6FF$\"3e*****\\/Qoq%FF7$$!3'******4vl8=\"FF$\"3 '******\\8R%zYFF7$$!3-+++&R]IB\"FF$\"3k*****>.y&[YFF7$$!3++++Ar$eG\"FF $\"3M+++?,D9YFF7$$!3,+++UV;S8FF$\"3#*******)*4^wXFF7$$!3-+++%>mlR\"FF$ \"3U+++/*Gb`%FF7$$!3))*****4;]cX\"FF$\"35+++(3d;\\%FF7$$!3$*******yW-= :FF$\"3q******[Y^XWFF7$$!35+++ST:%e\"FF$\"39+++Do.)R%FF7$$!3#*******H7 ;a;FF$\"3y*****f@E/N%FF7$$!3&******\\cfws\"FF$\"3=+++-C&RI%FF7$$!3/+++ C^v.=FF$\"3u*****fP]'fUFF7$$!3!******4]a7)=FF$\"3;+++v@3=UFF7$$!35+++: 9'*e>FF$\"3)******>s&HzTFF7$$!3(******>\"f*e.#FF$\"3-+++;y&H9%FF7$$!30 +++DBP6@FF$\"3u******)pS&3TFF7$$!3))******zV(\\=#FF$\"39+++++ZvSFF7$$! 31+++B(*[cAFF$\"3V+++M%4K/%FF7$$!3$)*****>%Q%eK#FF$\"3#)*****H.'H6SFF7 $$!3))******H%[IR#FF$\"3A+++[nMzRFF7$$!3)******\\ol\"eCFF$\"3)*******> =0ZRFF7$$!3%)*****\\N&G@DFF$\"3>+++VO;9RFF7$$!3%******fp5De#FF$\"3!*** ***\\E&[!)QFF7$$!3!)*****RJ]>k#FF$\"3#******fDge%QFF7$$!3-+++9Pr*p#FF$ \"38+++%Qk,\"QFF7$$!3%******4;4fv#FF$\"39+++'4*HtPFF7$$!3))******4Hk5G FF$\"3<+++\"f'=NPFF7$$!37+++]#>S'GFF$\"3))*****4Bmdp$FF7$$!3.+++n49;HF F$\"3/+++'3#*\\l$FF7$$!37+++a*4r'HFF$\"3()******G9$Gh$FF7$$!3!******pz Fq,$FF$\"3')*****f.k#pNFF7$$!3#******>9'*f1$FF$\"36+++e@GCNFF7$$!3$*** ***4q;T6$FF$\"3#)*****f0\"*yZ$FF7$$!35+++N2\\hJFF$\"3\"*******R*4,V$FF 7$$!3$)******oy@3KFF$\"3))*****z8t4Q$FF7$$!3++++MTRaKFF$\"3&)*****>AJ0 L$FF7$$!3)******><4,I$FF$\"3++++6=&)yKFF7$$!33+++&oUaM$FF$\"3********Q '>gA$FF7$$!31+++g.Y!R$FF$\"35+++Rb8sJFF7$$!38+++t'4_V$FF$\"30+++BSJG4g#FF7$$!3')******eiViQFF$\"3#**** **4AXCa#FF7$$!3()*****R_c9!RFF$\"35+++AW#Q[#FF7$$!3-+++3#e%RRFF$\"3))* ****HFR]U#FF7$$!3#*******>/QwRFF$\"3')******)QagO#FF7$$!3=+++0\\<7SFF$ \"31+++)fGoI#FF7$$!3e*****\\s0o/%FF$\"35+++yvJZAFF7$$!3!******ze[-3%FF $\"35+++JgZ(=#FF7$$!3Q+++q**[7TFF$\"3#*******Q%fs7#FF7$$!3()*****\\FF7$$!3i*****fc%\\HUFF$\"3++++L1&=)=FF7$$!39 +++Us$eD%FF$\"3(*******GG?>=FF7$$!3U+++3;2\"G%FF$\"3!******4O#*fv\"FF7 $$!3%******\\'QB0VFF$\"3)******Ru3Ap\"FF7$$!3P+++ZNOGVFF$\"33+++,#\\yi \"FF7$$!3)******fK-0N%FF$\"3)*******)>>Hc\"FF7$$!3w*****pj#prVFF$\"3'* *****H$GV(\\\"FF7$$!3a******[h(>R%FF$\"3)******z,89V\"FF7$$!38+++&3#R6 WFF$\"3'******pP\"*[O\"FF7$$!3N+++!\\v*HWFF$\"33+++kx!zH\"FF7$$!3Q+++@ bvZWFF$\"32+++\"Q40B\"FF7$$!3i*****z!RvkWFF$\"3'******RV\\F;\"FF7$$!33 +++HP)4[%FF$\"3%******4T&o%4\"FF7$$!3E+++:'[k\\%FF$\"36+++FqPE5FF7$$!3 *)*****z\\U6^%FF$\"3Q+++x+%)y&*F-7$$!3O+++P*\\]_%FF$\"3S+++w&QE*))F-7$ $!3))*****R*p9QXFF$\"3!)*****H5)p0#)F-7$$!3o*****\\d-/b%FF$\"3!******H *4\\=vF-7$$!3k*****p6!yhXFF$\"3L+++tmTJoF-7$$!3G+++Z%RAd%FF$\"38+++lOz WhF-7$$!3i*****fnQ#zc*3%F-7$$!3z*****\\\\nSg%FF $\"3D+++)=$H1MF-7$$!3M+++,#zc*3%F-7$Fbjv$!3%)*****4mcPx%F-7 $F]jv$!3!)*****RWc)eaF-7$Fhiv$!38+++lOzWhF-7$Fciv$!3L+++tmTJoF-7$F^iv$ !3!******H*4\\=vF-7$Fihv$!3!)*****H5)p0#)F-7$Fdhv$!3S+++w&QE*))F-7$F_h v$!3Q+++x+%)y&*F-7$Fjgv$!36+++FqPE5FF7$Fegv$!3%******4T&o%4\"FF7$F`gv$ !3'******RV\\F;\"FF7$F[gv$!32+++\"Q40B\"FF7$Fffv$!33+++kx!zH\"FF7$Fafv $!3'******pP\"*[O\"FF7$F\\fv$!3)******z,89V\"FF7$Fgev$!3'******H$GV(\\ \"FF7$Fbev$!3)*******)>>Hc\"FF7$F]ev$!33+++,#\\yi\"FF7$Fhdv$!3)******R u3Ap\"FF7$Fcdv$!3!******4O#*fv\"FF7$F^dv$!3(*******GG?>=FF7$Ficv$!3+++ +L1&=)=FF7$Fdcv$!3********eL&R%>FF7$F_cv$!3%******f.Nb+#FF7$Fjbv$!3y** ***pXDm1#FF7$Febv$!3#*******Q%fs7#FF7$F`bv$!35+++JgZ(=#FF7$F[bv$!35+++ yvJZAFF7$Ffav$!31+++)fGoI#FF7$Faav$!3')******)QagO#FF7$F\\av$!3))***** HFR]U#FF7$Fg`v$!35+++AW#Q[#FF7$Fb`v$!3#******4AXCa#FF7$F]`v$!3\"****** f>G4g#FF7$Fh_v$!3?+++JwGfEFF7$Fc_v$!32+++6@_gA$FF7$Fa\\v$!3++++6=&)yKFF7$F\\\\v$!3 &)*****>AJ0L$FF7$Fg[v$!3))*****z8t4Q$FF7$Fb[v$!3\"*******R*4,V$FF7$F][ v$!3#)*****f0\"*yZ$FF7$Fhju$!36+++e@GCNFF7$Fcju$!3')*****f.k#pNFF7$F^j u$!3()******G9$Gh$FF7$Fiiu$!3/+++'3#*\\l$FF7$Fdiu$!3))*****4Bmdp$FF7$F _iu$!3<+++\"f'=NPFF7$Fjhu$!39+++'4*HtPFF7$Fehu$!38+++%Qk,\"QFF7$F`hu$! 3#******fDge%QFF7$F[hu$!3!******\\E&[!)QFF7$Ffgu$!3>+++VO;9RFF7$Fagu$! 3)*******>=0ZRFF7$F\\gu$!3A+++[nMzRFF7$Fgfu$!3#)*****H.'H6SFF7$Fbfu$!3 V+++M%4K/%FF7$F]fu$!39+++++ZvSFF7$Fheu$!3u******)pS&3TFF7$Fceu$!3-+++; y&H9%FF7$F^eu$!3)******>s&HzTFF7$Fidu$!3;+++v@3=UFF7$Fddu$!3u*****fP]' fUFF7$F_du$!3=+++-C&RI%FF7$Fjcu$!3y*****f@E/N%FF7$Fecu$!39+++Do.)R%FF7 $F`cu$!3q******[Y^XWFF7$F[cu$!35+++(3d;\\%FF7$Ffbu$!3U+++/*Gb`%FF7$Fab u$!3#*******)*4^wXFF7$F\\bu$!3M+++?,D9YFF7$Fgau$!3k*****>.y&[YFF7$Fbau $!3'******\\8R%zYFF7$F]au$!3e*****\\/Qoq%FF7$Fh`u$!3J+++Gb!3t%FF7$Fc`u $!3-+++pwP^ZFF7$F^`u$!3#******4a$eoZFF7$Fi_u$!3E+++#)zV#y%FF7$Fd_u$!3i ******Gs$Hz%FF7$F__u$!3N+++ig0+[FF7$Fj^u$!3l*****\\fXP![FF7$Fe^u$!3i** ***f@JR![FF7$F`^u$!3n*****4P50![FF7$F[^u$!3%******z,]Lz%FF7$Ff]u$!3i** ***po$G#y%FF7$Fa]u$!3m*****\\=3rw%FF7$F\\]u$!3)******p7!eZZFF7$Fg\\u$! 3m*****pp7Ms%FF7$Fb\\u$!3'******pQtUp%FF7$F]\\u$!3')********QyfYFF7$Fh [u$!3')******QS_>YFF7$Fc[u$!3w*****\\(>/tXFF7$F^[u$!38+++OM()>XFF7$Fij t$!3Q+++(Rr&fWFF7$Fdjt$!3S+++!oWYKUFF7$Feit$!3:+++i9zSTFF7$F`it$!3x*****pBN6/%FF7$F[it$!3%***** *****yN$RFF7$Ffht$!3.+++7!>r#G$FF7$F]gt$!3.+++_VwL JFF7$Fhft$!3!)******z=C\")HFF7$Fcft$!3%)*****\\nGk#GFF7$F^ft$!3#****** \\*zIqEFF7$Fiet$!3&)*****R))GN^#FF7$Fdet$!3#)*****>%yYcBFF7$F_et$!3')* ****\\!pJ*>#FF7$Fjdt$!3>+++!Qf@/#FF7$Fedt$!3++++eA-&)=FF7$F`dt$!32+++6 x!zs\"FF7$F[dt$!35+++V(43d\"FF7$Ffct$!32+++x;s89FF7$Fact$!3#*******f'Q mD\"FF7$$\"35+++\"RCqF#F=$!3++++gyb*4\"FF7$Fgbt$!3;+++(*)yZU*F-7$$\"30 +++_4)f6\"F7$!3e*****zy\")R&yF-7$$\"3#*******)RYU)=F_jm$!3=+++C`=$G'F- 7$$\"3%******fnnZ,#F1$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3+++ +Fjzq:F-F'-F\\im6&F^im$\"\"#FaimF($\"#XFdim-Ffim6f\\l7%F'7$F($\"+Fjzq: F\\jm7$$FCFaimF(7%Figx7$F($\"+aEfTJF\\jmF\\hx7%F_hx7$$\"+\\zw9?Fjo$\"+ \")*)Q7ZF\\jmF\\hx7%Fchx7$$\"+!RYU)=F-$\"+C`=$G'F\\jmF\\hx7%Fihx7$$\"+ ^4)f6\"FF$\"+)y\")R&yF\\jmF\\hx7%F_ix7$$\"+qek9`FF$\"+(*)yZU*F\\jmF\\h x7%Feix7$$\"+YT-xAF`[n$\"+gyb*4\"Fj[nF\\hx7%F[jx7$$\"+v_?))*)F`[n$\"+g 'QmD\"Fj[nF\\hx7%Fajx7$$\"+-\"y6B$Fg[n$\"+x;s89Fj[nF\\hx7%Fgjx7$$\"+' \\%yZ5F_\\n$\"+V(43d\"Fj[nF\\hx7%F][y7$$\"+n10mIF_\\n$\"+6x!zs\"Fj[nF \\hx7%Fc[y7$$\"+GdZ\\\")F_\\n$\"+eA-&)=Fj[nF\\hx7%Fi[y7$$\"+[Xb%)>F\\] n$\"+!Qf@/#Fj[nF\\hx7%F_\\y7$$\"+MS5lWF\\]n$\"+0pJ*>#Fj[nF\\hx7%Fe\\y7 $$\"+B0v[$*F\\]n$\"+UyYcBFj[nF\\hx7%F[]y7$$\"+y:\\J=Fi]n$\"+%))GN^#Fj[ nF\\hx7%Fa]y7$$\"+jM')oLFi]n$\"+&*zIqEFj[nF\\hx7%Fg]y7$$\"+PS$[#eFi]n$ \"+v'Gk#GFj[nF\\hx7%F]^y7$$\"+;'HOX*Fi]n$\"+!)=C\")HFj[nF\\hx7%Fc^y7$$ \"+VDgM9F\\_n$\"+_VwLJFj[nF\\hx7%Fi^y7$$\"+n[^B?F\\_n$\"+->r#G$Fj[nF\\ hx7%F__y7$$\"+>/&Rj#F\\_n$\"+L2oEMFj[nF\\hx7%Fe_y7$$\"+]3sPJF\\_n$\"+8 xUkNFj[nF\\hx7%F[`y7$$\"+-@nzLF\\_n$\"+p!)3&p$Fj[nF\\hx7%Fa`y7$$\"+S9l 5KF\\_n$\"+7YKUFj[nF\\hx7%F_by7$$!+Y4=F\\jm$\"+OM()>XFj[nF\\hx 7%Fgcy7$$!+x4/tXFj[nF\\hx7%F]dy7$$!+K'H(\\FF\\jm$\"+RS_> YFj[nF\\hx7%Fcdy7$$!+*o,xB$F\\jm$\"++RyfYFj[nF\\hx7%Fidy7$$!+$y`\\t$F \\jm$\"+(QtUp%Fj[nF\\hx7%F_ey7$$!+p'*HQUF\\jm$\"+(p7Ms%Fj[nF\\hx7%Feey 7$$!+twKXZF\\jm$\"+F,eZZFj[nF\\hx7%F[fy7$$!+K+Ca_F\\jm$\"+&=3rw%Fj[nF \\hx7%Fafy7$$!+c\"=Pw&F\\jm$\"+(o$G#y%Fj[nF\\hx7%Fgfy7$$!+9p\"GF'F\\jm $\"+=+N$z%Fj[nF\\hx7%F]gy7$$!+7f)3y'F\\jm$\"+r.^+[Fj[nF\\hx7%Fcgy7$$!+ BS^(G(F\\jm$\"+;7$R![Fj[nF\\hx7%Figy7$$!+rM\\#z(F\\jm$\"+&fXP![Fj[nF\\ hx7%F_hy7$$!+$e+eH)F\\jm$\"+ig0+[Fj[nF\\hx7%Fehy7$$!+l8f(z)F\\jm$\"+Hs $Hz%Fj[nF\\hx7%F[iy7$$!+e7@)H*F\\jm$\"+#)zV#y%Fj[nF\\hx7%Faiy7$$!+t&># )z*F\\jm$\"+TNeoZFj[nF\\hx7%Fgiy7$$!+II%)H5Fj[n$\"+pwP^ZFj[nF\\hx7%F]j y7$$!+/s**z5Fj[n$\"+Gb!3t%Fj[nF\\hx7%Fcjy7$$!+jqVI6Fj[n$\"+X!Qoq%Fj[nF \\hx7%Fijy7$$!+^dO\"=\"Fj[n$\"+N\"R%zYFj[nF\\hx7%F_[z7$$!+&R]IB\"Fj[n$ \"+K!y&[YFj[nF\\hx7%Fe[z7$$!+Ar$eG\"Fj[n$\"+?,D9YFj[nF\\hx7%F[\\z7$$!+ UV;S8Fj[n$\"+**4^wXFj[nF\\hx7%Fa\\z7$$!+%>mlR\"Fj[n$\"+/*Gb`%Fj[nF\\hx 7%Fg\\z7$$!+h,lb9Fj[n$\"+(3d;\\%Fj[nF\\hx7%F]]z7$$!+zW-=:Fj[n$\"+\\Y^X WFj[nF\\hx7%Fc]z7$$!+ST:%e\"Fj[n$\"+Do.)R%Fj[nF\\hx7%Fi]z7$$!+I7;a;Fj[ n$\"+;iU]VFj[nF\\hx7%F_^z7$$!+l&fws\"Fj[n$\"+-C&RI%Fj[nF\\hx7%Fe^z7$$! +C^v.=Fj[n$\"+w.lfUFj[nF\\hx7%F[_z7$$!+,XD\")=Fj[n$\"+v@3=UFj[nF\\hx7% Fa_z7$$!+:9'*e>Fj[n$\"+AdHzTFj[nF\\hx7%Fg_z7$$!+7f*e.#Fj[n$\"+;y&H9%Fj [nF\\hx7%F]`z7$$!+DBP6@Fj[n$\"+*pS&3TFj[nF\\hx7%Fc`z7$$!+!Qu\\=#Fj[n$ \"+++ZvSFj[nF\\hx7%Fi`z7$$!+B(*[cAFj[n$\"+M%4K/%Fj[nF\\hx7%F_az7$$!+UQ %eK#Fj[n$\"+LgH6SFj[nF\\hx7%Feaz7$$!+I%[IR#Fj[n$\"+[nMzRFj[nF\\hx7%F[b z7$$!+&ol\"eCFj[n$\"+?=0ZRFj[nF\\hx7%Fabz7$$!+b`G@DFj[n$\"+VO;9RFj[nF \\hx7%Fgbz7$$!+'p5De#Fj[n$\"+l_[!)QFj[nF\\hx7%F]cz7$$!+9.&>k#Fj[n$\"+c -'e%QFj[nF\\hx7%Fccz7$$!+9Pr*p#Fj[n$\"+%Qk,\"QFj[nF\\hx7%Ficz7$$!+h\"4 fv#Fj[n$\"+'4*HtPFj[nF\\hx7%F_dz7$$!+5Hk5GFj[n$\"+\"f'=NPFj[nF\\hx7%Fe dz7$$!+]#>S'GFj[n$\"+Jiw&p$Fj[nF\\hx7%F[ez7$$!+n49;HFj[n$\"+'3#*\\l$Fj [nF\\hx7%Faez7$$!+a*4r'HFj[n$\"+H9$Gh$Fj[nF\\hx7%Fgez7$$!+(zFq,$Fj[n$ \"+OSEpNFj[nF\\hx7%F]fz7$$!+Uh*f1$Fj[n$\"+e@GCNFj[nF\\hx7%Fcfz7$$!+,n6 9JFj[n$\"+c5*yZ$Fj[nF\\hx7%Fifz7$$!+N2\\hJFj[n$\"+S*4,V$Fj[nF\\hx7%F_g z7$$!+py@3KFj[n$\"+QJ(4Q$Fj[nF\\hx7%Fegz7$$!+MTRaKFj[n$\"+A7`ILFj[nF\\ hx7%F[hz7$$!+s\"4,I$Fj[n$\"+6=&)yKFj[nF\\hx7%Fahz7$$!+&oUaM$Fj[n$\"+R' >gA$Fj[nF\\hx7%Fghz7$$!+g.Y!R$Fj[n$\"+Rb8sJFj[nF\\hx7%F]iz7$$!+t'4_V$F j[n$\"+BSJG4g#Fj[nF\\hx7%Fi\\[l7$$!+fiViQFj[n$\"+@_WUDFj[nF\\ hx7%F_][l7$$!+ClX,RFj[n$\"+AW#Q[#Fj[nF\\hx7%Fe][l7$$!+3#e%RRFj[n$\"+t# R]U#Fj[nF\\hx7%F[^[l7$$!+?/QwRFj[n$\"+*QagO#Fj[nF\\hx7%Fa^[l7$$!+0\\<7 SFj[n$\"+)fGoI#Fj[nF\\hx7%Fg^[l7$$!+Dd!o/%Fj[n$\"+yvJZAFj[nF\\hx7%F]_[ l7$$!+)e[-3%Fj[n$\"+JgZ(=#Fj[nF\\hx7%Fc_[l7$$!+q**[7TFj[n$\"+R%fs7#Fj[ nF\\hx7%Fi_[l7$$!+vh_VTFj[n$\"+daim?Fj[nF\\hx7%F_`[l7$$!+\"RiL<%Fj[n$ \"+O]`0?Fj[nF\\hx7%Fe`[l7$$!+t<,-UFj[n$\"+fL&R%>Fj[nF\\hx7%F[a[l7$$!+m X\\HUFj[n$\"+L1&=)=Fj[nF\\hx7%Faa[l7$$!+Us$eD%Fj[n$\"+HG?>=Fj[nF\\hx7% Fga[l7$$!+3;2\"G%Fj[n$\"+hB*fv\"Fj[nF\\hx7%F]b[l7$$!+lQB0VFj[n$\"+W(3A p\"Fj[nF\\hx7%Fcb[l7$$!+ZNOGVFj[n$\"+,#\\yi\"Fj[nF\\hx7%Fib[l7$$!+EB]] VFj[n$\"+*>>Hc\"Fj[nF\\hx7%F_c[l7$$!+PEprVFj[n$\"+LGV(\\\"Fj[nF\\hx7%F ec[l7$$!+\\h(>R%Fj[n$\"+=ITJ9Fj[nF\\hx7%F[d[l7$$!+&3#R6WFj[n$\"+x8*[O \"Fj[nF\\hx7%Fad[l7$$!+!\\v*HWFj[n$\"+kx!zH\"Fj[nF\\hx7%Fgd[l7$$!+@bvZ WFj[n$\"+\"Q40B\"Fj[nF\\hx7%F]e[l7$$!+3RvkWFj[n$\"+M%\\F;\"Fj[nF\\hx7% Fce[l7$$!+HP)4[%Fj[n$\"+6ao%4\"Fj[nF\\hx7%Fie[l7$$!+:'[k\\%Fj[n$\"+FqP E5Fj[nF\\hx7%F_f[l7$$!+)\\U6^%Fj[n$\"+x+%)y&*F\\jmF\\hx7%Fef[l7$$!+P* \\]_%Fj[n$\"+w&QE*))F\\jmF\\hx7%F[g[l7$$!+%*p9QXFj[n$\"+.\")p0#)F\\jmF \\hx7%Fag[l7$$!+vDS]XFj[n$\"+$*4\\=vF\\jmF\\hx7%Fgg[l7$$!+<,yhXFj[n$\" +tmTJoF\\jmF\\hx7%F]h[l7$$!+Z%RAd%Fj[n$\"+lOzWhF\\jmF\\hx7%Fch[l7$$!+w 'Q>Hc\"Fj[nF\\hx7%Fea\\l7$Fjb[l$!+,#\\yi\"Fj[nF\\hx7%Fia\\l 7$Fdb[l$!+W(3Ap\"Fj[nF\\hx7%F]b\\l7$F^b[l$!+hB*fv\"Fj[nF\\hx7%Fab\\l7$ Fha[l$!+HG?>=Fj[nF\\hx7%Feb\\l7$Fba[l$!+L1&=)=Fj[nF\\hx7%Fib\\l7$F\\a[ l$!+fL&R%>Fj[nF\\hx7%F]c\\l7$Ff`[l$!+O]`0?Fj[nF\\hx7%Fac\\l7$F``[l$!+d aim?Fj[nF\\hx7%Fec\\l7$Fj_[l$!+R%fs7#Fj[nF\\hx7%Fic\\l7$Fd_[l$!+JgZ(=# Fj[nF\\hx7%F]d\\l7$F^_[l$!+yvJZAFj[nF\\hx7%Fad\\l7$Fh^[l$!+)fGoI#Fj[nF \\hx7%Fed\\l7$Fb^[l$!+*QagO#Fj[nF\\hx7%Fid\\l7$F\\^[l$!+t#R]U#Fj[nF\\h x7%F]e\\l7$Ff][l$!+AW#Q[#Fj[nF\\hx7%Fae\\l7$F`][l$!+@_WUDFj[nF\\hx7%Fe e\\l7$Fj\\[l$!+'>G4g#Fj[nF\\hx7%Fie\\l7$Fd\\[l$!+JwGfEFj[nF\\hx7%F]f\\ l7$F^\\[l$!+6@_gA$Fj[nF\\hx7%Feh\\l7$Fbhz$!+6=&)yKFj[nF\\hx7%Fih \\l7$F\\hz$!+A7`ILFj[nF\\hx7%F]i\\l7$Ffgz$!+QJ(4Q$Fj[nF\\hx7%Fai\\l7$F `gz$!+S*4,V$Fj[nF\\hx7%Fei\\l7$Fjfz$!+c5*yZ$Fj[nF\\hx7%Fii\\l7$Fdfz$!+ e@GCNFj[nF\\hx7%F]j\\l7$F^fz$!+OSEpNFj[nF\\hx7%Faj\\l7$Fhez$!+H9$Gh$Fj [nF\\hx7%Fej\\l7$Fbez$!+'3#*\\l$Fj[nF\\hx7%Fij\\l7$F\\ez$!+Jiw&p$Fj[nF \\hx7%F][]l7$Ffdz$!+\"f'=NPFj[nF\\hx7%Fa[]l7$F`dz$!+'4*HtPFj[nF\\hx7%F e[]l7$Fjcz$!+%Qk,\"QFj[nF\\hx7%Fi[]l7$Fdcz$!+c-'e%QFj[nF\\hx7%F]\\]l7$ F^cz$!+l_[!)QFj[nF\\hx7%Fa\\]l7$Fhbz$!+VO;9RFj[nF\\hx7%Fe\\]l7$Fbbz$!+ ?=0ZRFj[nF\\hx7%Fi\\]l7$F\\bz$!+[nMzRFj[nF\\hx7%F]]]l7$Ffaz$!+LgH6SFj[ nF\\hx7%Fa]]l7$F`az$!+M%4K/%Fj[nF\\hx7%Fe]]l7$Fj`z$!+++ZvSFj[nF\\hx7%F i]]l7$Fd`z$!+*pS&3TFj[nF\\hx7%F]^]l7$F^`z$!+;y&H9%Fj[nF\\hx7%Fa^]l7$Fh _z$!+AdHzTFj[nF\\hx7%Fe^]l7$Fb_z$!+v@3=UFj[nF\\hx7%Fi^]l7$F\\_z$!+w.lf UFj[nF\\hx7%F]_]l7$Ff^z$!+-C&RI%Fj[nF\\hx7%Fa_]l7$F`^z$!+;iU]VFj[nF\\h x7%Fe_]l7$Fj]z$!+Do.)R%Fj[nF\\hx7%Fi_]l7$Fd]z$!+\\Y^XWFj[nF\\hx7%F]`]l 7$F^]z$!+(3d;\\%Fj[nF\\hx7%Fa`]l7$Fh\\z$!+/*Gb`%Fj[nF\\hx7%Fe`]l7$Fb\\ z$!+**4^wXFj[nF\\hx7%Fi`]l7$F\\\\z$!+?,D9YFj[nF\\hx7%F]a]l7$Ff[z$!+K!y &[YFj[nF\\hx7%Faa]l7$F`[z$!+N\"R%zYFj[nF\\hx7%Fea]l7$Fjjy$!+X!Qoq%Fj[n F\\hx7%Fia]l7$Fdjy$!+Gb!3t%Fj[nF\\hx7%F]b]l7$F^jy$!+pwP^ZFj[nF\\hx7%Fa b]l7$Fhiy$!+TNeoZFj[nF\\hx7%Feb]l7$Fbiy$!+#)zV#y%Fj[nF\\hx7%Fib]l7$F\\ iy$!+Hs$Hz%Fj[nF\\hx7%F]c]l7$Ffhy$!+ig0+[Fj[nF\\hx7%Fac]l7$F`hy$!+&fXP ![Fj[nF\\hx7%Fec]l7$Fjgy$!+;7$R![Fj[nF\\hx7%Fic]l7$Fdgy$!+r.^+[Fj[nF\\ hx7%F]d]l7$F^gy$!+=+N$z%Fj[nF\\hx7%Fad]l7$Fhfy$!+(o$G#y%Fj[nF\\hx7%Fed ]l7$Fbfy$!+&=3rw%Fj[nF\\hx7%Fid]l7$F\\fy$!+F,eZZFj[nF\\hx7%F]e]l7$Ffey $!+(p7Ms%Fj[nF\\hx7%Fae]l7$F`ey$!+(QtUp%Fj[nF\\hx7%Fee]l7$Fjdy$!++RyfY Fj[nF\\hx7%Fie]l7$Fddy$!+RS_>YFj[nF\\hx7%F]f]l7$F^dy$!+v>/tXFj[nF\\hx7 %Faf]l7$Fhcy$!+OM()>XFj[nF\\hx7%Fef]l7$Fbcy$!+(Rr&fWFj[nF\\hx7%Fif]l7$ F\\cy$!+!oWYKUFj[nF\\hx7%Feg]l7$Fjay$!+i9zSTFj[nF\\hx7%Fig]l7$Fday$!+P_8TSFj[n F\\hx7%F]h]l7$F^ay$!++!zN$RFj[nF\\hx7%Fah]l7$Fh`y$!+7r#G$Fj[nF\\hx7%Fei]l7$Fj^y$!+_VwL JFj[nF\\hx7%Fii]l7$Fd^y$!+!)=C\")HFj[nF\\hx7%F]j]l7$F^^y$!+v'Gk#GFj[nF \\hx7%Faj]l7$Fh]y$!+&*zIqEFj[nF\\hx7%Fej]l7$Fb]y$!+%))GN^#Fj[nF\\hx7%F ij]l7$F\\]y$!+UyYcBFj[nF\\hx7%F][^l7$Ff\\y$!+0pJ*>#Fj[nF\\hx7%Fa[^l7$F `\\y$!+!Qf@/#Fj[nF\\hx7%Fe[^l7$Fj[y$!+eA-&)=Fj[nF\\hx7%Fi[^l7$Fd[y$!+6 x!zs\"Fj[nF\\hx7%F]\\^l7$F^[y$!+V(43d\"Fj[nF\\hx7%Fa\\^l7$Fhjx$!+x;s89 Fj[nF\\hx7%Fe\\^l7$Fbjx$!+g'QmD\"Fj[nF\\hx7%Fi\\^l7$$\"+\"RCqF#F`[n$!+ gyb*4\"Fj[nF\\hx7%F]]^l7$Ffix$!+(*)yZU*F\\jmF\\hx7%Fc]^l7$$\"+_4)f6\"F F$!+)y\")R&yF\\jmF\\hx7%Fg]^l7$$\"+*RYU)=F-$!+C`=$G'F\\jmF\\hx7%F]^^l7 $$\"+www9?Fjo$!+\")*)Q7ZF\\jmF\\hx7%Fc^^l7$F($!+aEfTJF\\jmF\\hx7%Fi^^l 7$F($!+Fjzq:F\\jmF\\hx7%F]_^lF'F\\hx-F\\im6&F^im$\"#bFdim$\"#:Fdim$F`i mF)Fhbq-F$6$7K7$$\"37+++M.72?FF$\"3D+++V#[ih&FF7$$\"3A+++]/*3,#FF$\"3r *******\\&QHcFF7$$\"31+++m1o7?FF$\"3w******y8mUcFF7$$\"35+++],c7?FF$\" 3?+++g,#el&FF7$$\"3#*******3xh5?FF$\"3W+++o4jocFF7$$\"35+++Mv&p+#FF$\" 3m******R`)3o&FF7$$\"3y*****H@'p,?FF$\"3'******p'[R#p&FF7$$\"33+++92' \\*>FF$\"3%******z1*)Hq&FF7$$\"3#*******=r)o)>FF$\"3H+++9N^7dFF7$$\"3% ******p&*>w(>FF$\"39+++f#G3s&FF7$$\"3!******p.7t'>FF$\"3-+++nl!ys&FF7$ $\"3-+++FF$\"3K+++fRLLdFF7$$\"3++++&fGU%>FF$\"33+++>wIPdFF7$$\"3* ******Re/=$>FF$\"3K+++>ijRdFF7$$\"3!******>RX!>>FF$\"3t*****\\ES-u&FF7 $$\"32+++yt:1>FF$\"3$******\\6`!RdFF7$$\"3!******\\uiL*=FF$\"3-+++*)G- OdFF7$$\"3++++n>!4)=FF$\"3U+++`b6JdFF7$$\"3!******zAO!p=FF$\"3v*****>d >Vs&FF7$$\"3#*******f%\\!e=FF$\"3o*****4d_cr&FF7$$\"35+++P#\\#[=FF$\"3 \"******f^q^q&FF7$$\"31+++Q`'*R=FF$\"3))*****4!)zHp&FF7$$\"3)******>:W N$=FF$\"35+++qADzcFF7$$\"3'******zUO$H=FF$\"3C+++j&FF7$$\"3'**** **4&*GI$=FF$\"3n******>^g:cFF7$$\"3-+++y=FS=FF$\"3Q+++]n2+cFF7$$\"3!** ****H[E/&=FF$\"3G+++w#3ge&FF7$$\"3#*******H49j=FF$\"3s*****p>aSd&FF7$$ \"3)*******>$yy(=FF$\"3I+++r-vkbFF7$$\"3/+++\"yuR*=FF$\"33+++n2XebFF7$ $\"34+++$y=2\">FF$\"3O+++0;IbbFF7$$\"35+++gyUF>FF$\"3))*****Rr__b&FF7$ $\"3%******\\G/N%>FF$\"3m*****z?)4ebFF7$$\"33+++^MYe>FF$\"3#******R;FN c&FF7$$\"3!******f(*Q>(>FF$\"3)******R+r6d&FF7$$\"3++++\")Hn$)>FF$\"3' )*****4IS1e&FF7$$\"3!******f_*\\$*>FF$\"3e*****Rl\\:f&FF7$$\"32+++-nK, ?FF$\"3f*****4WKNg&FFF[`^lF`gx-Ffim6L7%7$$\"+M.72?Fj[n$\"+V#[ih&Fj[n7$ $\"+]/*3,#Fj[n$\"++bQHcFj[n7$$\"$#>Fdim$\"$k&Fdim7%F[]_l7$$\"+m1o7?Fj[ n$\"+z8mUcFj[nF`]_l7%Ff]_l7$$\"+],c7?Fj[n$\"+g,#el&Fj[nF`]_l7%F\\^_l7$ $\"+4xh5?Fj[n$\"+o4jocFj[nF`]_l7%Fb^_l7$$\"+Mv&p+#Fj[n$\"+S`)3o&Fj[nF` ]_l7%Fh^_l7$$\"+8ip,?Fj[n$\"+n[R#p&Fj[nF`]_l7%F^__l7$$\"+92'\\*>Fj[n$ \"+o!*)Hq&Fj[nF`]_l7%Fd__l7$$\"+>r)o)>Fj[n$\"+9N^7dFj[nF`]_l7%Fj__l7$$ \"+d*>w(>Fj[n$\"+f#G3s&Fj[nF`]_l7%F``_l7$$\"+P?Jn>Fj[n$\"+nl!ys&Fj[nF` ]_l7%Ff`_l7$$\"+Fj[n$\"+fRLLdFj[nF`]_l7%F\\a_l7$$\"+&fGU%>Fj[n$\" +>wIPdFj[nF`]_l7%Fba_l7$$\"+%e/=$>Fj[n$\"+>ijRdFj[nF`]_l7%Fha_l7$$\"+# RX!>>Fj[n$\"+l-CSdFj[nF`]_l7%F^b_l7$$\"+yt:1>Fj[n$\"+:J0RdFj[nF`]_l7%F db_l7$$\"+XFO$*=Fj[n$\"+*)G-OdFj[nF`]_l7%Fjb_l7$$\"+n>!4)=Fj[n$\"+`b6J dFj[nF`]_l7%F`c_l7$$\"+Gi.p=Fj[n$\"+s&>Vs&Fj[nF`]_l7%Ffc_l7$$\"+g%\\!e =Fj[n$\"+rDl:dFj[nF`]_l7%F\\d_l7$$\"+P#\\#[=Fj[n$\"+;0<0dFj[nF`]_l7%Fb d_l7$$\"+Q`'*R=Fj[n$\"+,)zHp&Fj[nF`]_l7%Fhd_l7$$\"+_TaL=Fj[n$\"+qADzcF j[nF`]_l7%F^e_l7$$\"+GkLH=Fj[n$\"+j&Fj[nF`]_l7%F`f_l7$$ \"+^*GI$=Fj[n$\"+?^g:cFj[nF`]_l7%Fff_l7$$\"+y=FS=Fj[n$\"+]n2+cFj[nF`]_ l7%F\\g_l7$$\"+$[E/&=Fj[n$\"+w#3ge&Fj[nF`]_l7%Fbg_l7$$\"+I49j=Fj[n$\"+ (>aSd&Fj[nF`]_l7%Fhg_l7$$\"+?$yy(=Fj[n$\"+r-vkbFj[nF`]_l7%F^h_l7$$\"+ \"yuR*=Fj[n$\"+n2XebFj[nF`]_l7%Fdh_l7$$\"+$y=2\">Fj[n$\"+0;IbbFj[nF`]_ l7%Fjh_l7$$\"+gyUF>Fj[n$\"+9FDbbFj[nF`]_l7%F`i_l7$$\"+&G/N%>Fj[n$\"+3# )4ebFj[nF`]_l7%Ffi_l7$$\"+^MYe>Fj[n$\"+kr_jbFj[nF`]_l7%F\\j_l7$$\"+w*Q >(>Fj[n$\"+/5Fj[n$\"+,.k!e&Fj[nF`]_l7 %Fhj_l7$$\"+E&*\\$*>Fj[n$\"+a'\\:f&Fj[nF`]_l7%F^[`l7$$\"+-nK,?Fj[n$\"+ TC`.cFj[nF`]_l7%Fd[`lFf\\_lF`]_lFa_^lFhbq-F$6$7K7$F\\`^l$!3D+++V#[ih&F F7$F_\\_l$!3f*****4WKNg&FF7$Fj[_l$!3e*****Rl\\:f&FF7$Fe[_l$!3')*****4I S1e&FF7$F`[_l$!3)******R+r6d&FF7$F[[_l$!3#******R;FNc&FF7$Ffj^l$!3m*** **z?)4ebFF7$Faj^l$!3))*****Rr__b&FF7$F\\j^l$!3O+++0;IbbFF7$Fgi^l$!33++ +n2XebFF7$Fbi^l$!3I+++r-vkbFF7$F]i^l$!3s*****p>aSd&FF7$Fhh^l$!3G+++w#3 ge&FF7$Fch^l$!3Q+++]n2+cFF7$F^h^l$!3n******>^g:cFF7$Fig^l$!3M+++x,!>j& FF7$Fdg^l$!3A+++oNI[cFF7$F_g^l$!3C+++d>Vs&FF7$Fae^l$!3U+++`b6JdFF7$F\\e^l$!3-+++*)G-OdFF7 $Fgd^l$!3$******\\6`!RdFF7$Fbd^l$!3t*****\\ES-u&FF7$F]d^l$!3K+++>ijRdF F7$Fhc^l$!33+++>wIPdFF7$Fcc^l$!3K+++fRLLdFF7$F^c^l$!3-+++nl!ys&FF7$Fib ^l$!39+++f#G3s&FF7$Fdb^l$!3H+++9N^7dFF7$F_b^l$!3%******z1*)Hq&FF7$Fja^ l$!3'******p'[R#p&FF7$Fea^l$!3m******R`)3o&FF7$F`a^l$!3W+++o4jocFF7$F[ a^l$!3?+++g,#el&FF7$Ff`^l$!3w******y8mUcFF7$Fa`^l$!3r*******\\&QHcFFF] \\`lF`gx-Ffim6L7%7$Fg\\_l$!+V#[ih&Fj[n7$Fe[`l$!+TC`.cFj[n7$Fa]_l$!$k&F dim7%F[d`l7$F_[`l$!+a'\\:f&Fj[nF^d`l7%Fbd`l7$Fij_l$!+,.k!e&Fj[nF^d`l7% Ffd`l7$Fcj_l$!+/5aSd&Fj[nF^d`l7%Fff`l7$Fcg_l$!+w#3ge&Fj[nF^d`l 7%Fjf`l7$F]g_l$!+]n2+cFj[nF^d`l7%F^g`l7$Fgf_l$!+?^g:cFj[nF^d`l7%Fbg`l7 $Faf_l$!+x,!>j&Fj[nF^d`l7%Ffg`l7$F[f_l$!+oNI[cFj[nF^d`l7%Fjg`l7$Fee_l$ !+Vs&Fj[nF^d`l7%Fbi`l7$Fac_l$!+`b6JdFj[nF^d`l7%Ffi` l7$F[c_l$!+*)G-OdFj[nF^d`l7%Fji`l7$Feb_l$!+:J0RdFj[nF^d`l7%F^j`l7$F_b_ l$!+l-CSdFj[nF^d`l7%Fbj`l7$Fia_l$!+>ijRdFj[nF^d`l7%Ffj`l7$Fca_l$!+>wIP dFj[nF^d`l7%Fjj`l7$F]a_l$!+fRLLdFj[nF^d`l7%F^[al7$Fg`_l$!+nl!ys&Fj[nF^ d`l7%Fb[al7$Fa`_l$!+f#G3s&Fj[nF^d`l7%Ff[al7$F[`_l$!+9N^7dFj[nF^d`l7%Fj [al7$Fe__l$!+o!*)Hq&Fj[nF^d`l7%F^\\al7$F___l$!+n[R#p&Fj[nF^d`l7%Fb\\al 7$Fi^_l$!+S`)3o&Fj[nF^d`l7%Ff\\al7$Fc^_l$!+o4jocFj[nF^d`l7%Fj\\al7$F]^ _l$!+g,#el&Fj[nF^d`l7%F^]al7$Fg]_l$!+z8mUcFj[nF^d`l7%Fb]al7$F\\]_l$!++ bQHcFj[nF^d`l7%Ff]alFhc`lF^d`lFa_^lFhbq-F$6%7$7$$!3o*************Q&FFF (7$$\"37++++++!R#FFF(-%'COLOURG6&F^imF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$ 7$F($!3Q++++++!>'FF7$F($\"3Q++++++!>'FFFc^alFf^al-%(SCALINGG6#%,CONSTR AINEDG-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6% %&Re(z)G%&Im(z)G-F\\`al6#%(DEFAULTG-%%VIEWG6$;$!$R&Fdim$\"$R#Fdim;$!$> 'Fdim$\"$>'Fdim" 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" "C urve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for 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 4506 "ee := \{c[2]=1/100,\nc[3]=2/15,\nc[4]=1/5,\nc[5]=5120/9991,\nc[6 ]=60/97,\nc[7]=52/313,\nc[8]=63/157,\nc[9]=19/23,\nc[10]=81/82,\nc[11] =1,\nc[12]=1,\n\na[2,1]=1/100,\na[3,1]=-34/45,\na[3,2]=8/9,\na[4,1]=1/ 20,\na[4,2]=0,\na[4,3]=3/20,\na[5,1]=551874974720/997302429271,\na[5,2 ]=0,\na[5,3]=-2086699008000/997302429271,\na[5,4]=2045902848000/997302 429271,\na[6,1]=70365/1204352,\na[6,2]=0,\na[6,3]=0,\na[6,4]=15000000/ 48955027,\na[6,5]=16390905/64600448,\na[7,1]=21278500033609/2764202388 76800,\na[7,2]=0,\na[7,3]=0,\na[7,4]=9893050830728000/9123673252619564 1,\na[7,5]=-2901059421298252903/76225368652427491200,\na[7,6]=43637936 1176824/23234416797464775,\na[8,1]=627197176791399348709/2289353328808 5131673600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-26470464436388981652125/236 136288092530359764076,\na[8,5]=825734282216068905765330677/63130978459 89067825180262400,\na[8,6]=-49138310819605862695843/962154393150421608 110400,\na[8,7]=13/32,\na[9,1]=-28052920640951751059941898036719365763 49564929/\n 195439700408341869784184509846462968849917952000,\n a[9,2]=0,\na[9,3]=0,\na[9,4]=-4448919702545915170544437084072367474271 6548825/\n 14932376340320772420636844398178733063223595932,\na[ 9,5]=-538125605455123224555096530708947439830683725706539/\n 22 1787063147752862838752824077163711222570059776000,\na[9,6]=59039572906 70756919471851965687658329936372357281397/\n 4701038098831357101 461987812303657884827097474792000,\na[9,7]=806385337807093245665198543 25500435393/31664769935063948010841554250731537120,\na[9,8]=3802360970 0214727511244770779309498/15560366705131162570848900057690585,\na[10,1 ]=18745022701169741445732083919370592367818346398255185593/\n 4 3542944158840438871835940146729085397585013246181376000,\na[10,2]=0,\n a[10,3]=0,\na[10,4]=17495577465884581302001502644337583721543152157472 2425/\n 12988012824390651292175460014153260035238111030958552, \na[10,5]=72703899173100547975701530898752038618015765157471787669321/ \n 6944684059165807161861542521065278114605347407607709696000, \na[10,6]=-18513454683442942614776350098381846385951171415826406124378 67/\n 42070340786468499232348793697044463426798665947552104956 8000,\na[10,7]=-142383111687918201861880767560889949537519404224643/\n 13608444565909841005179111919047260723369140257280,\na[10,8]= -15762011599975752003382750329296637334838001/170538268612761013476551 9769050554722692480,\na[10,9]=1570980729807122133129202485/21707495839 44689600835012992,\na[11,1]=413401339654426004756492280722047075004416 104055883/\n 71986808932338247272769411537057334290003905945600 0,\na[11,2]=0,\na[11,3]=0,\na[11,4]=4068247489923915770015231221964449 550938389658325/\n 22267533162889877550784327259915218148192293 4868,\na[11,5]=422701288179451309495857528449967681683868959323295591/ \n 29766097532422646168378497759855140870411136751616000,\na[11 ,6]=-64756336295667822884474260277213322605529059566403398599943/\n \+ 10592034333763703970644991697561788735926914730422752072000,\na[ 11,7]=-2933204189652449798399861612546549582225310986953/\n 20 5013581499999542677439978824334676268210882720,\na[11,8]=-624797475608 500160493593996249709038784226/497736028930361526506167928459999015164 65,\na[11,9]=38655300339474326088/40800863136986779255,\na[11,10]=-683 0025351434171965696/369666731905701279737985,\na[12,1]=6470068411/1270 52452800,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=140 36525372223254273/57629420394152745600,\na[12,7]=735432288072303917913 1/30216428865161090697600,\na[12,8]=223707573986265289483/110157542907 7660009200,\na[12,9]=5315253908682019/32851416704215200,\na[12,10]=347 047251949199216/1213217205998680575,\na[12,11]=-95980357/508344480,\n \nb[1]=6470068411/127052452800,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb [6]=14036525372223254273/57629420394152745600,\nb[7]=73543228807230391 79131/30216428865161090697600,\nb[8]=223707573986265289483/11015754290 77660009200,\nb[9]=5315253908682019/32851416704215200,\nb[10]=34704725 1949199216/1213217205998680575,\nb[11]=-95980357/508344480,\n\n`b*`[1] =457593782868427426344821/8836699182375187992960000,\n`b*`[2]=0,\n`b*` [3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=911082200138648538937372713121 003/4008217400410278039116332273920000,\n`b*`[7]=503904010603015936847 352266508232049/2101600452117104186799675302680320000,\n`b*`[8]=162832 45430995038704640712154556497/76616314592355159097734873145440000,\n`b *`[9]=418537996431230550055107156803/2284868026806051955976984640000, \n`b*`[10]=2137829181363408719740436246579/105476471095448724177690309 48750,\n`b*`[11]=-4366466929950751198793/35356163157685532736000,\n`b* `[12]=1/160\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..12);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6-/&%\"cG6#\"\"##\"\"\"\"$+\"/&F%6#\"\"$#F'\"#:/&F%6#\" \"%#F)\"\"&/&F%6#F6#\"%?^\"%\"***/&F%6#\"\"'#\"#g\"#(*/&F%6#\"\"(#\"#_ \"$8$/&F%6#\"\")#\"#j\"$d\"/&F%6#\"\"*#\"#>\"#B/&F%6#\"#5#\"#\")\"##)/ &F%6#\"#6F)/&F%6#\"#7F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1. .i-1),i=1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6^o/&%\"aG6$\"\"#\"\" \"#F(\"$+\"/&F%6$\"\"$F(#!#M\"#X/&F%6$F.F'#\"\")\"\"*/&F%6$\"\"%F(#F( \"#?/&F%6$F;F'\"\"!/&F%6$F;F.#F.F=/&F%6$\"\"&F(#\"-?Z(\\(=b\"-r#HCI(** /&F%6$FIF'FA/&F%6$FIF.#!.+!3!*p'3#FL/&F%6$FIF;#\".+![G!f/#FL/&F%6$\"\" 'F(#\"&l.(\"(_V?\"/&F%6$FgnF'FA/&F%6$FgnF.FA/&F%6$FgnF;#\")+++:\")F]&* [/&F%6$FgnFI#\")04R;\")[/gk/&F%6$\"\"(F(#\"/4O.+&y7#\"0+o()Q-Uw#/&F%6$ F`pF'FA/&F%6$F`pF.FA/&F%6$F`pF;#\"1+!G2$30$*)*\"2Tc>EDtO7*/&F%6$F`pFI# !4.HD)H@%f5!H\"5+7\\FClo`Aw/&F%6$F`pFgn#\"0CoF'\"8+OnJ^3)GLN*G#/&F%6$F6F'FA/&F%6$F6F.FA/&F%6$F6F ;#!8D@l\")*)QOWYqk#\"9wSwf.`#4)GOhB/&F%6$F6FI#\"3JQ\"\\\"9+/63;U]JRa@'*/&F%6$F6F`p# \"#8\"#K/&F%6$F7F(#!OH\\c\\jdO>n.)*=%*f5v^4k?H0G\"Q+?&z\"*\\)oHYY)4X=% yp=M3/qRa>/&F%6$F7F'FA/&F%6$F7F.FA/&F%6$F7F;#!PD)[lrUZnB2%3PWaq^\"fa-( >*[W\"PKffBK1L(y\")RWoj?Cx?.MwB$\\\"/&F%6$F7FI#!TRlqDPoI)Ru%*32`'4bXAB ^X0c7Q&\"T+gxf+dA7rjr2CGvQG'GvZJ1(y@#/&F%6$F7Fgn#\"U(R\"GdBPO*H$ewol>& =Z>pvq1HdR!f\"U+?zuu4F[)yl.B\"y)>Y,rNJ))4Q5q%/&F%6$F7F`p#\"G$RN/]DV&)> lcC$42yL&Q1)\"G?r`J2Da:%3,[R1N*pZmJ/&F%6$F7F6#\"D)\\4$z2xW7^FZ@+(4O-Q \"D&e!pd+!*[3di680nOgb\"/&F%6$\"#5F(#\"Y$f&=b#)RY$=yO#fq$>R3KdWT(p6qA] u=\"Y+gP\"=YK,&e(R&3Hn9Sf$=()Q/%)eT%HaV/&F%6$F]wF'FA/&F%6$F]wF.FA/&F%6 $F]wF;#\"WDCsu:_Ja@PePVk-:+-8e%)eYxb\\<\"V_&e4.6\"Q_.gK:9+Yv@H^1RCG,)) H\"/&F%6$F]wFI#\"fn@$pwyru:ld,='Q?v)*3`,d(za+J<**Qqs\"en+gp4xg2uM0Y6y_ 1@Dah=;2e;fSoWp/&F%6$F]wFgn#!hnnyV71k#eTr6&fQY=Q)4]jx9E%HW$oaM^=\"gn+! o&\\5_v%fm)zEMYWqp$z[BB*\\okyS.2U/&F%6$F]wF`p#!TVYA/%>v`\\**)3cn2)='=? =zo6JQU\"\"S!Gd-9pLsgs/>>6z^+T)4fcW%3O\"/&F%6$F]wF6#!M,!Q[LPmHH.v#Q.?v v**f6?w:\"L![#pAZb]!p(>bwM,hFho#Q0(/&F%6$F^zF'FA/&F%6$F^zF.FA/&F%6$F^zF;#\"RD$e' *QQ4b\\W'>AJ_,qd\"R#**[Z#oS\"Qo[$H#>[\"=_\"*fsK%y]v()*)G;LvEA/&F%6$F^z FI#\"W\"f&HB$f*oQo\"on*\\%Gv&e\\48Xz\")G,FU\"V+gh^n86/(39b)fx\\y$ohkAC `(4m(H/&F%6$F^zFgn#!fnV**f)R.kcf!HbgAL@x-EuW)G#ym&HOjvk\"fn+?2_FUIZ\"p #ft)yhvp\"*\\kqRqjPLM?f5/&F%6$F^zF`p#!R`p)4JDAe\\la7;')*R)z\\Cl*=/K$H \"Q?F)3@oinMV#)y*RunU&****\\\"e8]?/&F%6$F^zF6#!KEUyQ!4(\\i*Rf$\\g,]3cZ (zC'\"Jlk^,***f%Gz;1l_h.$*Ggt(\\/&F%6$F^zF7#\"5)3EVZR.Ib'Q\"5b#zn)p8j3 !3%/&F%6$F^zF]w#!7'pl>tm'p$/&F%6$\"#7F(#\"+6%o+Z' \"-+GX_q7/&F%6$Fe]lF'FA/&F%6$Fe]lF.FA/&F%6$Fe]lF;FA/&F%6$Fe]lFIFA/&F%6 $Fe]lFgn#\"5tUDBAPDl.9\"5+cu_TR?%Hw&/&F%6$Fe]lF`p#\"7J\"z\"RIs!)GKat\" 8+wp!4h^')Gk@I/&F%6$Fe]lF6#\"6$[*Gli)Rd2PA\"7+#4+mx!Had,6/&F%6$Fe]lF7# \"1>?o3RD:`\"2+_@/nT^G$/&F%6$Fe]lF]w#\"3;#*>\\>DZqM\"4v0o)*f? " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=su bs(ee,b[i]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\" \"#\"+6%o+Z'\"-+GX_q7/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6# \"\"&F//&F%6#\"\"'#\"5tUDBAPDl.9\"5+cu_TR?%Hw&/&F%6#\"\"(#\"7J\"z\"RIs !)GKat\"8+wp!4h^')Gk@I/&F%6#\"\")#\"6$[*Gli)Rd2PA\"7+#4+mx!Had,6/&F%6# \"\"*#\"1>?o3RD:`\"2+_@/nT^G$/&F%6#\"#5#\"3;#*>\\>DZqM\"4v0o)*f? " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i] =subs(ee,`b*`[i]),i=1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6./&%#b*G 6#\"\"\"#\"9@[MEuUoGy$fd%\":++'H*z=vB=*pO))/&F%6#\"\"#\"\"!/&F%6#\"\"$ F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"B.578FPP*Q&['Q,?#36*\"C++#RF Kj6R!y-T+u@3S/&F%6#\"\"(#\"E\\?B3lE_t%o$f,.1,/R]\"F++K!o-`n*z'=/r6_/g, @/&F%6#\"\")#\"D(\\cX:72k/(Q]*4VXKG;\"D++WXJ([t(4f^N#f9jhw/&F%6#\"\"*# \"?.o:2^0]0BJk*z`=%\"@++k%)p(f&>01o-o[G#/&F%6#\"#5#\"@zlCO/u>(3MO\"=Hy 8#\"A]([4.pxTs[a4rkZ0\"/&F%6#\"#6#!7$z)>^2&*HpYmV\"8+gtKbodJ;c`$/&F%6# \"#7#F'\"$g\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#================= ================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with \+ " }{XPPEDIT 18 0 "c[10] = 169/170;" "6#/&%\"cG6#\"#5*&\"$p\"\"\"\"\"$q \"!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "As for the previous scheme, one of the weights for \+ the order 7 scheme is negative and all the other weights are positive. " }}{PARA 0 "" 0 "" {TEXT -1 37 "The nodes of the order 7 scheme are: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/1000" "6 #/&%\"cG6#\"\"#*&\"\"\"F)\"%+5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 28/249" "6#/&%\"cG6#\"\"$*&\"#G\"\"\"\"$\\#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 14/83" "6#/&%\"cG6#\"\"%*&\"#9\"\"\"\"#$) !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 415483/780760" "6#/&% \"cG6#\"\"&*&\"'$[:%\"\"\"\"'g2y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 87/131" "6#/&%\"cG6#\"\"'*&\"#()\"\"\"\"$J\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 109/659" "6#/&%\"cG6#\"\"(*&\"$4\"\" \"\"\"$f'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 7/16" "6#/&% \"cG6#\"\")*&\"\"(\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [9] = 13/16" "6#/&%\"cG6#\"\"**&\"#8\"\"\"\"#;!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[10] = 169/170" "6#/&%\"cG6#\"#5*&\"$p\"\"\"\"\"$q \"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\"#6\" \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The 2-norm of the principal error is approximately: 0 .936742 " }{TEXT 267 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" " 6#)\"#5,$\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------------------------------- -------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the com bined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffici ents of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4468 "ee := \{c[2]=1/1000,\nc[3]=28/249 ,\nc[4]=14/83,\nc[5]=415483/780760,\nc[6]=87/131,\nc[7]=109/659,\nc[8] =7/16,\nc[9]=13/16,\nc[10]=169/170,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/10 00,\na[3,1]=-385028/62001,\na[3,2]=392000/62001,\na[4,1]=7/166,\na[4,2 ]=0,\na[4,3]=21/166,\na[5,1]=201849498968676950243/1865686775770065920 00,\na[5,2]=0,\na[5,3]=-777541397023211694969/186568677577006592000,\n a[5,4]=337487397017512409163/93284338788503296000,\na[6,1]=206935503/6 884238116,\na[6,2]=0,\na[6,3]=0,\na[6,4]=404414081147/1257566797788,\n a[6,5]=1534888336000/4911984599757,\na[7,1]=39743335520386666547/57265 5525899435427204,\na[7,2]=0,\na[7,3]=0,\na[7,4]=2128983481028416471477 19/2010207409308970807088412,\na[7,5]=-49544686747746827756272000/2729 832755347968158104608573,\na[7,6]=1077348693331815806/1307302406204678 40411,\na[8,1]=17125839160165511097181/583729762655220563902464,\na[8, 2]=0,\na[8,3]=0,\na[8,4]=174040744000019283021419611/20490815871909640 14926856192,\na[8,5]=128489749336355703667293557125/956526769655402159 824357169664,\na[8,6]=-125509550840850981113069/2931683770285010740150 272,\na[8,7]=19/82,\na[9,1]=304584927066618919841641919928442600454537 797/1336435146974329082026479192445476428829818880,\na[9,2]=0,\na[9,3] =0,\na[9,4]=-183130124969528341358670952614060593600746254959/43039657 262695209676619384887672437213535272960,\na[9,5]=-59929859974878159390 72736862205038623064898960325/2245491222915232082243948587229615621971 585089024,\na[9,6]=1322708054627964884973707202944501083084973588321/1 346859151282526737213165214426258933563303526400,\na[9,7]=873822912254 162442527647371013373/228741131636931363287619371663360,\na[9,8]=51299 867691487048524852/18956439276208492414825,\na[10,1]=-1256795279708492 037734149713024454589348072577632030104333/\n 1844322652469644 188488724709227695887423797701298885000000,\na[10,2]=0,\na[10,3]=0,\na [10,4]=1155321277999101211105955197933773620274343648077208821293051/ \n 59396084444375336903774156155177493077283524727141295000000, \na[10,5]=100171919666627080787492626771580340159478395878727610639533 656/\n 84288684853878235080393786127427579167929622742228674202 00625,\na[10,6]=-75691885617633891987993521638144638721175631419293321 69849602853/\n 19653979525281151679347336094614453722396051644 36067222345000000,\na[10,7]=-38688518491828194541031368152041107061501 2243139331/\n 236919787007865614920290105576254790303222850000 00,\na[10,8]=-145680407248255277894206069634089984/1419249572624472888 0776189465890625,\na[10,9]=6596608202574079685632/83961997098579545281 25,\na[11,1]=-1913917650773276648000042701921031830180297961856463/\n \+ 2420535999874014114485551535784938587087050773067132,\na[11,2] =0,\na[11,3]=0,\na[11,4]=794034598781907555787098179682463625970047118 793613/\n 35481535396506256458169246778950856594722743868452,\n a[11,5]=12538765114319773465378231128982652519968027341646160000/\n \+ 915484882132450383552157953702371732055721397872191177,\na[11,6]= -879765828850292172490106340518569223117087719534557891994/\n \+ 196096868766029731312926688640518320991505603079281962497,\na[11,7]=-2 6773940308698113601982120741972828286815906/14220463788522665086401376 32652542482384789,\na[11,8]=-918990495679234610154177376026624/7752935 5682339127064062513128033,\na[11,9]=819316668609331200/920300389324474 639,\na[11,10]=-6894374368544685000000/970688489039629537954049,\na[12 ,1]=3042259121/61252403940,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12, 5]=0,\na[12,6]=13566961978425697607/75147921323806716000,\na[12,7]=492 897387352157699403367/1970142834564896505822000,\na[12,8]=541082406092 8/20472512295375,\na[12,9]=986554630144/6438843573435,\na[12,10]=26906 4197119512500/452758788468179739,\na[12,11]=-135068333/274428000,\n\nb [1]=3042259121/61252403940,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]= 13566961978425697607/75147921323806716000,\nb[7]=492897387352157699403 367/1970142834564896505822000,\nb[8]=5410824060928/20472512295375,\nb[ 9]=986554630144/6438843573435,\nb[10]=269064197119512500/4527587884681 79739,\nb[11]=-135068333/274428000,\n\n`b*`[1]=783278906947440245917/1 5535938516834202740000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0 ,\n`b*`[6]=2991788146029459739848009433033/190603700468340356327490360 00000,\n`b*`[7]=123570026638743472603558985757717817/49970312964637074 5806507058262000000,\n`b*`[8]=177511756302554128853504/649075937937135 306609375,\n`b*`[9]=37241388453197849045504/204141942686827696516875, \n`b*`[10]=190299067731754576532365655/459347372389706921889026076,\n` b*`[11]=-22960167428223228083/69605374826988000000,\n`b*`[12]=1/200\}: " }}}{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 1011 "subs(ee,matrix([[c[2],a[2,1],``],\n[c[3], a[3,1],a[3,2]],[c[4],a[4,1],a[4,2]],[``$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]],[``$2, 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]],[``$2,a[8 ,7]],\n[c[9],a[9,1],a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6]],[``, a[9,7],a[9,8]],\n[c[10],a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[``,a[10 ,5],a[10,6]],\n[``,a[10,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[11,1],a[ 11,2]],[``,a[11,3],a[11,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7],a[11,8] ],[``,a[11,9],a[11,10]],\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a[12,4]] ,[``,a[12,5],a[12,6]],\n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,10]],[`` $2,a[12,11]],\n[``,`________________________________`$2],[`b`,b[1],b[2 ]],[``,b[3],b[4]],[``,b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10]],[``$ 2,b[11]],\n[`b*`,`b*`[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*`[5],`b* `[6]],\n[``,`b*`[7],`b*`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11],`b*`[1 2]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\"\"\"\"%+ 5F(%!G7%#\"#G\"$\\##!'G]Q\"&,?'#\"'+?RF27%#\"#9\"#$)#\"\"(\"$m\"\"\"!7 %F+F+#\"#@F;7%#\"'$[:%\"'g2y#\"6V-&pno*)\\\\=?\"6+?f1qdx'ol=F<7%F+#!6p \\p6K-(RTvxFF#\"6j\"4C^\" \\7%#\"$4\"\"$f'#\"5Zlm'Q?bLV(R\"6/sUN%**e_bEdF<7%F+F<#\"9>x9Z;%G5[$)* G@\":7%)323(*3$4u?5?7%F+#!;+?Fcx#ouZnoW&\\\"=t&3Y5e\"ozMbF$)HF#\"41e\" =L$p[t2\"\"66/%yY?1CI287%#F:\"#;#\"8\"=(46b;g\"Re7<\"9kC!Rc?_li(HPeF<7 %F+F<#\"<6'>9-$G>++W2/u\"\"=#>co#\\,k4>(e\"3\\?7%F+#\"?Drb$HnOqbjL\\(* [G\"\"?k'prNC)f@Sb'pn_c*#!9pI6\")4&3%3b4b7\":s-:S2,&GqPoJH7%F+F+#\"#> \"##)7%#\"#8Fjo#\"N(zPXX+EWG*>>kT)>*=m1F\\e/$\"O!))=)H)GkZXC>zk-#3HV(p 9NkL\"F<7%F+F<#!Qf\\DY2g$fgSh_4ne8MG&p\\7IJ=\"PgHFNN@PCn()[Q>mn4_pislR I%7%F+#!RD.'*)*[1B'Q]?iots!Rf\"y[(*f)H*f\"RC!*3&er>i:'Hse[RC#3K_\"HA\" \\XA#\"R@$)et\\3$3,X%H?2P(\\)['zia!3FK\"\"R+k_.LcL*eiU9_;8stEDG^\"foM \"7%F+#\"BtL,rtkFDWiTD7H#Q()\"BgLmr$>wGj8$pjJ6uG##\"8_[_[q[\"pn)*H^\"8 D[T#\\3iFRk&*=7%#\"$p\"\"$q\"#!enLV5I?jxD2[$*eaW-8(\\TtP?\\3(z_zc7\"en +++&)))H,xzBu)epF#4Zs)[)=W'pClAV%=F<7%F+F<#\"hn^IH@)3s2[OMu-itP$z>bf56 75**zF@`:\"\"fn+++&HTrsCNGxI\\x^:cTx.pLvVW%3'Rf7%F+#\"jncO`R1hF(yeRy%f ,M!erni#\\(y!3Fmm>><+\"\"hnD1??u'GAuA'Hz;zvUFhy$R!3N#yQ&[o)G%)#![o`Gg \\)p@LH>9jv6sQY9Q;_$*z)>*Qj7%F+#!TJ$RJC7]hq5T?:o8.TX>G=\\=&)oQ\"S+++&GA..zaid0,H?\\hly+(y>pB#!E% )**3M'pg?%*yFb#[sS!oX\"\"DD1*eY*=w2))GZCEd\\#>97%F+F+#\"7KcozSd-#3mf' \"7D\"GX&z&)4(*>'R)7%F)#!Ujk&='zH!=I=.@>qU++[mFt2l\"UKr1t20(3(eQ \\yN:b&[9T,u)**f`?CF<7%F+F<#\"T8Oz=r/qfijCoz\")4(ybv!>y)fMSz\"S_%oQuAZ fc3&*ynCp\"ekD1lRN:[N7%F+#\"Y++;Y;MF!o*>Dl#)*G6By`Yt(>V6l(QD\"\"Wx6>sy R@d0K*ybM&>x3Gz Ig0:*4K=0k)oEHJJ(Hgwoo4'>7%F+#!M1f\"oGGG(>u?@)>g8\")p3.%RxE\"L*y%Q#[UD lKw8S'3lE_)yj/A9#!BCm-wt)\"3RYZC$*Q+.#*#!7+++&oW&oVP%*o\"9\\S&z`H'R!*[)oq*7%F)#\"+@\"f A/$\",SRS_7'F<7%F+F'pc8\"5+gr1QK@z9v7%F+#\"9nLS*pd @N(Q(*G\\\":+?#e]'*[cMG9q>#\".G41C3T&\"/v`H7DZ?7%F+#\"-W,jal)*\".NMdV) Qk#\"3+D^>r>k!p#\"3R(z\"o%)yeFX7%F+F+#!*L$o]8\"*+!GWF7%F+%A___________ _____________________GF[x7%%\"bGF`vF7%F+#\"E`%)QTs$\"9vo^'pFooU>9/##\"\" " 0 "" {MPLTEXT 1 0 139 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),`` $(13-i)],i=2..12),\n[`b`,seq(b[j],j=1..11),``],[`b*`,seq(`b*`[j],j=1.. 12)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7/7/$\"'++5!\")F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"']C6!\"'$!'.5i!\"&$\"'[Aj F2F+F+F+F+F+F+F+F+F+F+7/$\"'v'o\"F/$\"'(o@%!\"($\"\"!F<$\"'1l7F/F+F+F+ F+F+F+F+F+F+7/$\"'_@`F/$\"'!>3\"F2F;$!'fnTF2$\"'%yh$F2F+F+F+F+F+F+F+F+ 7/$\"'ATmF/$\"'$f+$F:F;F;$\"'&e@$F/$\"'yCJF/F+F+F+F+F+F+F+7/$\"'-a;F/$ \"'=SpF:F;F;$\"'4f5F/$!'$\\\"=F:$\"',T#)F*F+F+F+F+F+F+7/$\"'+vVF/$\"'' Q$HF:F;F;$\"'g$\\)F:$\"'HV8F/$!'9\"G%F:$\"'2!\"%$\"'W)=\"Fip$!'A^QF2$!')Hj\"Fip$!'YE 5Fip$\"'mcyF/F+F+F+7/$\"\"\"F<$!'+2zF/F;F;$\"')yB#Fip$\"'jp8Fip$!'Q'[% F2$!'y#)=Fip$!'N&=\"Fip$\"'r-*)F/$!'c-rF*F+F+7/Feq$\"'wm\\F:F;F;F;F;$ \"'P0=F/$\"'%=]#F/$\"'(Hk#F/$\"'>K:F/$\"'xUfF/$!'\"=#\\F/F+7/%\"bGFhrF ;F;F;F;FjrF\\sF^sF`sFbsFdsF+7/%#b*G$\"'sT]F:F;F;F;F;$\"'kp:F/$\"'(GZ#F /$\"'%[t#F/$\"'HC=F/$\"'\"G9%F/$!'i)H$F/$\"'++]F*Q)pprint196\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditi ons(7,11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,OrderConditions(6 ,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simp lify(subs(ee,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplif y(subs(ee,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 #\"#P" }}}{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 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct \+ from 2 to 6 do\n so||ct||_11 := StageOrderConditions(ct,11,'expanded ');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Stages 6 to 11 have stage-order 4, while stages 4 and 5 \+ have stage-order 3 has and stage 3 has stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(subs(ee,so||i||_1 1[j])),i=2..6)],j=1..9)]:\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&F&F&" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sim plifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&% \"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,& %\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\" \"" }{TEXT -1 8 " . . 10 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1],i=1+1..11)=b[1],seq(Sum(b[i]*a[i ,j],i=j+1..11)=b[j]*(1-c[j]),j=2..10)];\nmap(u->lhs(u)-rhs(u),subs(ee, eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 7,/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"#6&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,;\"\"( F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F* 6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF- ,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&F EF`sFFF-/-F&6$*&F)F-&F/6$F,F]sF-/F,;F4F4*&&F*6#F]sF-,&F-F-&FEF]tFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "None of the principal error conditions ar e satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "RK7_11err_e qs := PrincipalErrorConditions(7,11,'expanded'):\nexpand(subs(ee,RK7_1 1err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$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 93 "We can calculate the principal \+ error norm, that is, the 2-norm of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms7_11 := PrincipalEr rorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errterms7_11) do \n sm := sm+(evalf(subs(ee,errterms7_11[ct])))^2;\nend do:\nsqrt(sm) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1)>uO*!#:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error nor m of the order 6 embedded scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms6_12*` := subs(b=`b*`,Principal ErrorTerms(6,12,'expanded')):\nsm := 0:\nfor ct to nops(`errterms6_12* `) do\n sm := sm+(evalf(subs(ee,`errterms6_12*`[ct])))^2;\nend do:\n sqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!o/a\"=!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction o f the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "We incorporate the stage-order equations to ensure th at stage 3 has stage-order 2, stages 4 and 5 have stage order 3 and st ages 6 to 11 have stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 57 "These conditions lead to relationships between the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"# \"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6#/,**&\"\"$\"\"\"* $&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\" &F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" }{TEXT -1 1 "," }} {PARA 257 "" 0 "" {TEXT -1 68 "which we include explicitly to make the computation more efficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a [i,j],i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\" \"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\" \"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 " c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 9 " ), and " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 " . . 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "The simplifying conditio ns allow us to manage with just 10 \"simple\" order conditions which a re given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 7 order 7 quadrature conditions and 3 additio nal order 7 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO7 := SimpleOrderCondition s(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,32,38,48,61,64])]:\nlinalg[augmen t](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()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F( #F(\"\"&7%\"#KF)/*&F,F()F2FGF(#F(\"\"'7%\"#QF)/*&F,F(-%!G6#*&%\"aGF(-F T6#*&FWF(-FT6#*&FWF(-FT6#*&F8F(FWF(F(F(F(F(#F(\"%?D7%\"#[F)/*&F,F(-FT6 #*&FWF(-FT6#*&FWF(-FT6#*&F?F(FWF(F(F(F(#F(\"$S)7%\"#hF)/*(F,F(F2F(-FT6 #*&FEF(FWF(F(#F(\"#N7%\"#kF)/*&F,F()F2FNF(#F(\"\"(Q)pprint166\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "SO7_11 := SimpleOrderConditions(7,11,'expanded'):\nSO_eqs := [op( RowSumConditions(11,'expanded')),op(StageOrderConditions(2,11,'expande d')),\n op(StageOrderConditions(3,4..11,'expanded')),op(StageOrderCon ditions(4,6..11,'expanded'))]:\nord_cdns := [seq(SO7_11[i],i=[1,2,4,8, 16,32,38,48,61,64])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..11)=b[1],seq( add(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=[$8..10])]:\nnode_eqs := [3 *c[3]=2*c[4],3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4]=0]:\ncdns := [op(node_eqs),op(simp_eqs),op(SO_eqs),op(ord_cdns)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The scheme can be obtained by specifyin g the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/1000;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"%+5!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 14/83;" "6#/&%\"cG6#\"\"%*&\"#9\"\"\"\"#$)!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 87/131;" "6#/&%\"cG6#\"\"'* &\"#()\"\"\"\"$J\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 109/ 659;" "6#/&%\"cG6#\"\"(*&\"$4\"\"\"\"\"$f'!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 7/16;" "6#/&%\"cG6#\"\")*&\"\"(\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 13/16;" "6#/&%\"cG6#\"\"**&\" #8\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 169/170; " "6#/&%\"cG6#\"#5*&\"$p\"\"\"\"\"$q\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[11]=1" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT -1 45 "together with the zero linking coeffici ents: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0 " "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2 ] = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10, 2] = 0;" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG6$\"#6\"\"# \"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[6,3] = 0;" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[7,3] = 0;" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,3] = 0;" "6#/&%\"aG6$\"\"*\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,3] = 0;" "6#/&%\"aG6$\"#5\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/&%\"aG6$\"#6 \"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the ze ro weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]= 0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" " 6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4] = 0;" "6 #/&%\"bG6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5] = 0;" "6# /&%\"bG6#\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "e1 := \{c[2]=1/1000,c[4]=1 4/83,c[6]=87/131,c[7]=109/659,c[8]=7/16,c[9]=13/16,c[10]=169/170,c[11] =1,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),seq(b[i]=0,i=2..5) \}:\neqns := subs(e1,cdns):\nnops(eqns);\nindets(eqns) minus \{a[8,7], a[9,7],a[9,8]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\\" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#F/&F%6$F2F>&F-6#F>&F-6#F3&F-6#FF&F%6$FCF/&F%6$F2F/&F%6$F6F/ &F%6$F(F/&F%6$\"\"#F/&F%6$F+F/&F%6$F>F+&F%6$F>F'&F%6$F2FF&F%6$F(Fao&F% 6$FFF+&F%6$FFF'&F%6$FFF/&F%6$F3F+&F%6$F3F'&F%6$F'F/&F%6$F2F'&F%6$F2F6& F%6$FCF+&F%6$FCF'&F%6$FCF6&F%6$FCF>&F%6$FCF3&F%6$FCFF&F%6$FCF2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Z" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e2 := solve(\{op(eqns)\} ,indets(eqns) minus \{a[8,7],a[9,7],a[9,8]\}):\ne3 := `union`(e1,e2): \ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,e3));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"*\"\")&F%6$F'\"\"(&F%6$F(F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5230 "e3 := \{b[ 3] = 0, b[4] = 0, b[5] = 0, a[7,4] = 212898348102841647147719/20102074 09308970807088412, a[6,1] = 206935503/6884238116, c[11] = 1, a[4,2] = \+ 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[9,3] = 0, a[1 0,3] = 0, a[11,3] = 0, b[2] = 0, a[7,6] = 1077348693331815806/13073024 0620467840411, a[5,4] = 337487397017512409163/93284338788503296000, b[ 9] = 986554630144/6438843573435, a[10,8] = -69001287597332612028416/83 96199709857954528125*a[9,8]+3398644031223170808832/2838033755646622656 25, a[8,6] = -302307405522583/66125564299680243*a[8,7]-10431711556175/ 249848658690048, a[10,4] = 160825862937293429815709452831256890935296/ 3796978731377822610978425715185338828125*a[8,7]+2495608135799325520096 895494996561722976256/304901056196153824894361528159495753353125*a[9,7 ]-565666566050025476029234833241/19846652993031770373692700000+3606627 0376434853691939347597234/14205006003866798652302815066125*a[9,8], c[5 ] = 415483/780760, a[4,3] = 21/166, a[10,5] = -48619709808542196755429 668503452693561344/108103291395145722159976495416885962023125*a[8,7]-8 060747212739235144406950606988439095083008/927474799733468925292621753 50311259887247525*a[9,7]-1043882202108477612120927405602104/9575857068 0562525252827036570525+23991902317928964313060252941999872/27959433963 71953108329027708729759*a[9,8], a[11,9] = 819316668609331200/920300389 324474639, a[11,8] = -35811933074227200/3725912507386537*a[9,8]+552350 992271671296/39014999760963737, c[3] = 28/249, b[1] = 3042259121/61252 403940, a[10,9] = 6596608202574079685632/8396199709857954528125, a[11, 10] = -6894374368544685000000/970688489039629537954049, c[2] = 1/1000, c[4] = 14/83, c[6] = 87/131, c[7] = 109/659, c[8] = 7/16, c[9] = 13/1 6, c[10] = 169/170, b[7] = 492897387352157699403367/197014283456489650 5822000, b[11] = -135068333/274428000, a[9,6] = -302307405522583/66125 564299680243*a[9,7]+6541275531/25515590144*a[9,8]+1451489880535747/474 7124515110912, a[2,1] = 1/1000, a[10,6] = 1258180765628756133986028370 95368704/647126150323012942669282326738984375*a[8,7]+20859600231267206 740659095612125718528/555203443787168772385380795971450334375*a[9,7]+2 4410302041726378200041502489172463/14689779900849077326184730127500000 -881555535484229679550378627908/418425762820131414448879728125*a[9,8], a[8,5] = 14602530426277943936000/1380795526225578228682149*a[8,7]+107 50605584303932375/81518659284419698176, a[4,1] = 7/166, a[5,3] = -7775 41397023211694969/186568677577006592000, a[8,4] = -2531729125482375639 37/254199217160972535039*a[8,7]+7876012888267951/24947159878139904, a[ 8,1] = -727125698073181/72414709901294313*a[8,7]+225038371369/71067935 04768, b[6] = 13566961978425697607/75147921323806716000, a[3,2] = 3920 00/62001, a[11,4] = 1836617889215934917576682087645184/362913175890122 93485592423726259*a[8,7]+155987772829111381785440000/52530322612470310 096568859*a[9,8]+3022203800128219774031866508954828800/315708014195978 001129685557849789981*a[9,7]-6982791670630751174599/205662310658254392 948, a[5,1] = 201849498968676950243/186568677577006592000, a[11,7] = - 2618838845489152/51538969301141*a[8,7]-35811933074227200/3725912507386 537*a[9,7]+517747858453411026004792800/17454014721013731559681381, b[1 0] = 269064197119512500/452758788468179739, a[11,1] = 1818917779795074 07104770048/356498186126071144518957657*a[8,7]-41739545045364867360000 /44893545882887936018759*a[9,8]+8679925611982497228296873574400/899369 57780000060338596430288027*a[9,7]+121631459235204339751643/98211397799 379639741876, a[9,4] = -253172912548237563937/254199217160972535039*a[ 9,7]-13380859311275/43311041455104*a[9,8]+67385194281814825/1746301191 46979328, a[9,5] = 14602530426277943936000/1380795526225578228682149*a [9,7]-8692556020815625/8325026479209528*a[9,8]+180243654497876705125/1 548854526403974265344, a[3,1] = -385028/62001, a[11,5] = -131867841113 0136045151851819040768000/2453957870251479617180035914756966621*a[8,7] +1179156191141741901975040000000/117493637436404873518791323199*a[9,8] -174314947446744038023735496042086400000/17149077737690856619581271395 05884942671*a[9,7]-481030552035035028991504000/37145099243765853798812 811, a[9,1] = -727125698073181/72414709901294313*a[9,7]+1193493075/123 38183168*a[9,8]+223466909887/49747554533376, b[8] = 5410824060928/2047 2512295375, a[10,7] = -12069582662111879370752/283803375564662265625*a [8,7]-69001287597332612028416/8396199709857954528125*a[9,7]+7627558244 064663464949691781661/306097163668736461779798700000, a[7,5] = -495446 86747746827756272000/2729832755347968158104608573, a[11,6] = 272998060 98675246798717845504/117518738912477424574464657147*a[8,7]-41593700556 862924377600/16880123659601962122779*a[9,8]+36087375248060015447149576 19200/82126022360723763580734901029497*a[9,7]+155960594970603414053176 45148/7956480542572262759657531571, a[6,4] = 404414081147/125756679778 8, a[10,1] = 15927592955789587242007002230720512/372986190753595227861 13143191484375*a[8,7]+7167515630308400240997939673602787328/8685833803 7527893970946949233808721875*a[9,7]+11059869897678565366265881447/1047 6477440622383205827700000-189229225444622842204908546/2380373390058269 72035434875*a[9,8], a[7,1] = 39743335520386666547/57265552589943542720 4, a[6,5] = 1534888336000/4911984599757\}:" }{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 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7 )]));\nfor ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e3,c [ii])); \n for jj to ii-1 do\n coeff_val := subs(e3,a[ii,jj]); \n if coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a pa rameter.`)\n else\n print(a[ii,jj]=coeff_val);\n end if;\n end do:\nend do:\nprint(``);\nseq(b[i]=subs(e3,b[i]),i=1..11) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\"%+5F(%!G F+F+F+F+F+7*#\"#G\"$\\##!'G]Q\"&,?'#\"'+?RF2F+F+F+F+F+7*#\"#9\"#$)#\" \"(\"$m\"\"\"!#\"#@F;F+F+F+F+7*#\"'$[:%\"'g2y#\"6V-&pno*)\\\\=?\"6+?f1 qdx'ol=F<#!6p\\p6K-(RTvxFE#\"6j\"4C^\"\\F+F+7*#\"$4\"\"$f'#\"5Zlm'Q?bLV(R\"6/sUN%**e_bEdFx9Z ;%G5[$)*G@\":7%)323(*3$4u?5?#!;+?Fcx#ouZnoW&\\\"=t&3Y5e\"ozMbF$)HF#\"4 1e\"=L$p[t2\"\"66/%yY?1CI28F+Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")# \"\"(\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&*&# \"0\"=t!)pDrs\"28VH,*4ZTsF(&F%6$F'\"\"(F(!\"\"#\"-p8PQ]A\".oZ]$z1rF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\"6PRcP#[D\"H<`#\"6R]`s4;<#*> a#\"\"\"&F%6$F'\"\"(F.!\"\"#\"1^zE))G,wy\"2/*R\"y)fr%\\#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\"8+g$R%zFE/`-Y\"\": \\@oG#ybAEbz!Q\"\"\"\"&F%6$F'\"\"(F.F.#\"5vB$RI%e01v5\"5w\")p>WGf'=:)F ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"0$eAbS2 BI\"2V-o*Hkb7m\"\"\"&F%6$F'\"\"(F.!\"\"#\"/vhb6\"\",oJ=QB\"F(&F%6$F'\"\")F(F(#\"-())4pYB#\"/wL`avu \\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,(*&#\"6PRcP#[D\"H<`#\"6R]`s4 ;<#*>a#\"\"\"&F%6$F'\"\"(F.!\"\"*&#\"/v7Jf3Q8\"//^XT5JVF.&F%6$F'\"\")F .F2#\"2D[\"=G%>&Qn\"3G$zp9>,ju\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\"*\"\"&,(*&#\"8+g$R%zFE/`-Y\"\":\\@oG#ybAEbz!Q\"\"\"\"&F%6 $F'\"\"(F.F.*&#\"1Dc\"3-cDp)\"1G&4#zk-D$)F.&F%6$F'\"\")F.!\"\"#\"6D^qw y\\aOC!=\"7W`EuRSEX&)[:F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"',(*&#\"0$eAbS2BI\"2V-o*Hkb7m\"\"\"&F%6$F'\"\"(F.!\"\"*&#\"+J bFTl\",W,f:b#F.&F%6$F'\"\")F.F.#\"1Zd`!))*[^9\"1746:X7ZZF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\"(\"\"\"%1~is~a~parameter. GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"*\"\")\"\"\"%1~is ~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"$p\"\"$q\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\",**&#\"D70sIA+2?C(e*ybHfFf\"\"DvV[\" >VJ6'yA&f`2>')HPF(&F%6$\"\")\"\"(F(F(*&#\"FGty-OnRz*4C+%3.j:v;(\"Gv=s3 QB\\p%4(R*y_P!Q$eo)F(&F%6$\"\"*F1F(F(#\">Z9)eEm`cyw*)p)f5\"\">++qFe?$Q A1WxkZ5F(*&#\"c5!\\IF.&F%6$\"\"*F2F.F.#\"?TK$[BH gZD+0mlml&\">++q#pt.xJI*HlY)>!\"\"*&#\"AMsfZ$R>p`[Vw.Fmg$\"ADh1:GI_')z mQ+1]?9F.&F%6$F9F1F.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 5\"\"&,**&#\"JW8c$p_M]o'Hav'>U&3)4(>'[\"KDJ-if)oT&\\w*f@sX^R\"H.\"3\" \"\"\"&F%6$\"\")\"\"(F.!\"\"*&#\"L3I3&4R%))pg]pSW^BRF@Z21)\"MDvC())f7J ]`%HDgIJk*GzJ->*R#\"Cf(H(3x-H$3J&>P'RVfz#F.&F%6$F :F1F.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',**&#\"E/ (o`4PGg)R8c(Gcw!=e7\"EvV)*QnK#GpE%H,B.:Erk\"\"\"&F%6$\"\")\"\"(F.F.*&# \"GG&=d77c4f1u1sEJ-gf3#\"HvVL]9(fz!Q&Qs(oryVM?b&F.&F%6$\"\"*F2F.F.#\"D jC<*[-:/+#yjsT?I5W#\"D++]F,t%=Et2\\3!*z(*o9F.*&#\"?3ziy.bz'HU[Nbb\")) \"?D\"G(z)[WTJ,#GwD%=%F.&F%6$F9F1F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,(*&#\"8_2Pz=6iEep?\"\"6DcEiYcvL!QG\"\"\"&F%6$ \"\")F(F.!\"\"*&#\"8;%G?hKtf(G,!p\"7D\"GX&z&)4(*>'R)F.&F%6$\"\"*F(F.F2 #\"@h;y\"p\\\\YjY1W#evi(\"?++q)zz 'R)\"\"\"&F%6$\"\"*F(F.!\"\"#\"7K)33'R)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",**&#\"<[+x/rS2&zzx\"*==\"hD*z')\"AF!)GIkfQ.1++ydp$**)F(&F%6$F8F1F(F(# \"9V;vRV?N#f9j@\"\"8w=uR'z$*z(R6#)*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"$\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%, **&#\"C%=Xw3#owv\"\\$f@*)yhO=\"AfisBCf&[$H7!*e<8HO\"\"\"&F%6$\"\")\"\" (F.F.*&#\"<++W&y\"Q6\"HGx()f:\";f)ol45.Z7EKID&F.&F%6$\"\"*F1F.F.*&#\"F +)G[&*3l'=.u(>#G,!Q?AI\"E\")**y\\yb&oH6+yf>9!3dJF.&F%6$F9F2F.F.#\"7*fu 6vI1n\"z#)p\"6[HRa#e1Jic?!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"&,**&#\"F+!o2/>=&=:Xg8I6Ty'=8\"F@m'pv9f.!=<'z9Dqy&RX#\" \"\"&F%6$\"\")\"\"(F.!\"\"*&#\"@+++S](>!>uT6>c\"z6\"?*>B8z=N([SOuj$\\< \"F.&F%6$\"\"*F1F.F.*&#\"H++S'3Ug\\NP-QSuYu%\\Ju\"\"IrE%\\)e]Rr7e>m&3p Px2\\r\"F.&F%6$F:F2F.F3#\"<+S]\"**G].N?bI5[\";6G\"))z`ewV#*4Xr$F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',**&#\">/b%yr)zY_n)4 1)*HF\"?ZrlkWdCuZ7*Q(=v6\"\"\"&F%6$\"\")\"\"(F.F.*&#\"8+wPCH'ob+PfT\"8 zF7i>gfO7!)o\"F.&F%6$\"\"*F1F.!\"\"*&#\"@+#>w&\\rW:+1[_P(3O\"A(\\H5!\\ t!ejPsgB-E@)F.&F%6$F9F2F.F.#\">[^k<`ST.1(\\fgf:\"=r:`d'fFEsDa![czF." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,(*&#\"1_\"*[X)Q)=E \"/T6Ip*Q:&\"\"\"&F%6$\"\")F(F.!\"\"*&#\"2+sAuI$>\"e$\"1PlQ2D\"fs$F.&F %6$\"\"*F(F.F2#\"<+Gz/g-6MXeyu<&\";\"Q\"of:t85s9SX\"e$\"1PlQ2D\"fs$ \"\"\"&F%6$\"\"*F(F.!\"\"#\"3'Hr;F#*4N_&\"2PP'4w**\\,RF." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"3+7L4'omJ>)\"3RYZC$*Q+.#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!7+++&oW&oVP%*o\"9 \\S&z`H'R!*[)oq*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"+@\"fA/$\",SRS_7'/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"52wpD%y>'pc8\" 5+gr1QK@z9v/&F%6#\"\"(#\"9nLS*pd@N(Q(*G\\\":+?#e]'*[cMG9q>/&F%6#\"\")# \".G41C3T&\"/v`H7DZ?/&F%6#\"\"*#\"-W,jal)*\".NMdV)Qk/&F%6#\"#5#\"3+D^> r>k!p#\"3R(z\"o%)yeFX/&F%6#\"#6#!*L$o]8\"*+!GWF" }}}{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 88 "RK 7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7,1 1,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simpl ify(subs(e3,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#&*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We now turn our attention to the embedd ed 12 stage, order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 210 "The firs t 11 stages of the embedded order 6 scheme coincide with those of the \+ order 7 scheme and the weights of the order 7 scheme provide the linki ng coefficients for the 12th stage of the order 6 scheme. Thus" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[12,j] = b[j];" "6#/ &%\"aG6$\"#7%\"jG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6 #/%\"jG\"\"\"" }{TEXT -1 8 " . . 11." }}{PARA 0 "" 0 "" {TEXT -1 48 "W e also incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b* `[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+ ;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\" \"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\" \"\"\"!" }{TEXT -1 9 " ), and " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\" %" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The simplifying conditions allow us to manage with just 7 \"simpl e\" order conditions which are given (in abreviated form) as follows. \+ " }}{PARA 0 "" 0 "" {TEXT -1 92 "These include the 6 order 6 quadratur e conditions and 1 additional order 6 order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO6*` := subs(b=`b*`,SimpleOrderConditions(6)):\n[seq([i,`SO6*`[i]],i=[1,2, 4,8,16,31,32])]:\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()F 2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#JF)/*&F,F(-%!G6#*&FEF(%\" aGF(F(#F(\"#I7%\"#KF)/*&F,F()F2FGF(#F(\"\"'Q(pprint16\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "`SO6 _12*` := subs(b=`b*`,SimpleOrderConditions(6,12,'expanded')):\n`cdns*` := [seq(`SO6_12*`[i],i=[1,2,4,8,16,31,32])]:\n`simp_eqs*` := [seq(add (`b*`[i]*a[i,j],i=j+1..12)=`b*`[j]*(1-c[j]),j=[4,5])]:\nwt_eqns := [se q(b[i]=a[12,i],i=1..11)]:\n`cdns*` := [op(`cdns*`),op(wt_eqns),op(`sim p_eqs*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6 #\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b* G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%# b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] = 0;" "6#/& %#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5] = 0;" "6# /&%#b*G6#\"\"&\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = \+ 1/200;" "6#/&%#b*G6#\"#7*&\"\"\"F)\"$+#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := \{c[12]=1,seq(`b*`[i]=0,i=2..5),`b*`[12]=1/200\}:\ne5 := `un ion`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqns2 := [op(remove(_U->type(rhs(_U),realcons), e5)),op(subs(e5,`cdns*`))]:\nnops(eqns2);\nindets(eqns2) minus \{a[8,7 ]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F%6$FCF'&F%6$FCF1&F%6$FC\"#5&F%6$FC\"#6&F%6$FQF+& F%6$FQF.&F%6$FQF>&F%6$FQF'&F%6$FCF2&F%6$FC\"\"#&F%6$FC\"\"$&F%6$FCF(&F %6$F1F(&F%6$F1F+&F%6$F1F.&F%6$F1F>&F%6$F1F'&F%6$FNF2&F%6$FNF(&F%6$FNF+ &F%6$FNF.&F%6$FNF>&F%6$FNF'&F%6$FQF2&F%6$FQF(&F66#FN&F66#FQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "e6 := solve(\{op(eqns2)\},i ndets(eqns2) minus \{a[8,7]\}):\ne7 := simplify(subs(select(u_->member (lhs(u_),\{b[8],a[8,7],a[9,7],a[9,8]\}),e6),e5)):\ne8 := `union`(e7,e6 ):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a[10,4]=subs(e8,a[10,4]);\na[11,1]=subs(e8,a[11,1]);\n`b*`[11] =subs(e8,`b*`[11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"%,&#\"PVB&H'**o3A]j)G#[@JNJ+\"QS'R;%\"O++?E,PrQ'Rrbj$piHh#))[84M\"\" \"\"*&#\"Y)o!4J[1H:WvnSL_#oQ5G\"\"H%3)[b,!ya#*z>,gZx%QD;& >!\"\"*&#\"M7ll#)f%f(e'z7g^Sn!=(Q*>!3>\"\"Mx?ME*HS$[ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5786 "e8 := \{`b*`[1] = 783278906947440245917/1553 5938516834202740000, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[9,3] = 0, a [7,2] = 0, a[7,3] = 0, `b*`[6] = 2991788146029459739848009433033/19060 370046834035632749036000000, a[8,4] = -253172912548237563937/254199217 160972535039*a[8,7]+7876012888267951/24947159878139904, `b*`[2] = 0, ` b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, `b*`[11] = -22960167428223228083 /69605374826988000000, c[2] = 1/1000, c[4] = 14/83, a[9,4] = -47617241 39996116747086356113622401/694038739107811551598008835375104+300250865 90039944956245190732022651186753/2669646550787651848311967610160544618 035*a[8,7], a[9,6] = 1790916883668045363269998501493715739/18461033697 26031126867668128127385600+12942644388738147623013490841884464847/2507 01048850381734245786042367682560495*a[8,7], c[9] = 13/16, c[10] = 169/ 170, `b*`[12] = 1/200, a[7,6] = 1077348693331815806/130730240620467840 411, c[6] = 87/131, a[12,8] = 5410824060928/20472512295375, `b*`[9] = \+ 37241388453197849045504/204141942686827696516875, a[12,11] = -13506833 3/274428000, c[11] = 1, a[6,2] = 0, c[12] = 1, a[12,5] = 0, b[2] = 0, \+ b[3] = 0, b[4] = 0, b[5] = 0, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a [9,1] = 620771422319655632507313631709/3078681460960999177583749103616 +86233601481085784882201501464225789/760512493600077294141169017568093 845*a[8,7], a[6,1] = 206935503/6884238116, a[10,1] = -4700572051925188 224587395966163951657754269/832740792790871017773628916184255307572000 0-361691966733284749279380471633614704014040984682496/7164799842082044 31643577324822424808955442159140625*a[8,7], a[9,5] = -8129256670626170 039243233666796078375/3077834018031233667278986266887440896-1250352158 30282365041244045476846643464524800/1046998661830220982545725057850560 452895218357*a[8,7], a[11,7] = -11169360155507336613291984525239171359 14450/34684058020786988015613112991525426399629+5912038605248928275523 669327872/102416586106128305236542289469*a[8,7], a[8,6] = -30230740552 2583/66125564299680243*a[8,7]-10431711556175/249848658690048, `b*`[10] = 190299067731754576532365655/459347372389706921889026076, a[6,5] = 1 534888336000/4911984599757, a[11,1] = -1281038682523340677544193120326 40053413/195162538477476001197992547800155488084-119080199387180674051 60127965875945982656512/20544230947247121758159673174834029926342077*a [8,7], a[4,3] = 21/166, a[12,6] = 13566961978425697607/751479213238067 16000, a[12,10] = 269064197119512500/452758788468179739, a[11,8] = -91 8990495679234610154177376026624/77529355682339127064062513128033, `b*` [7] = 123570026638743472603558985757717817/499703129646370745806507058 262000000, a[11,4] = 102131112234674213277965249569591677263/286079881 3742493089370316757610678924-41461718359792002748937321548907806608312 22759424/72116941862817594427173265810875724786021837131*a[8,7], a[5,1 ] = 201849498968676950243/186568677577006592000, a[10,2] = 0, a[10,3] \+ = 0, a[11,2] = 0, a[11,3] = 0, a[11,5] = 10005322971671238729605060327 88876708464000/73813549372544508766486030006814584598499+8633072361447 7293726952929828094590947623218184192000/14141636390663867713407317984 0203763488282167150988881*a[8,7], a[4,2] = 0, a[5,2] = 0, a[6,3] = 0, \+ a[10,9] = 6596608202574079685632/8396199709857954528125, a[10,5] = 447 603542747118336047981504642936865756484986328/380576718257673274671757 09031929081814732802945+2330838422756427270818646082651699503189186782 087801208832/438390685232730137835668866557478307243344784457678358062 5*a[8,7], `b*`[8] = 177511756302554128853504/649075937937135306609375, a[9,8] = 51299867691487048524852/18956439276208492414825, a[9,7] = 10 26016039498904723223766355271/159401485461276211350257401856-428128592 03250000515209/3791287855241698482965*a[8,7], c[3] = 28/249, a[7,5] = \+ -49544686747746827756272000/2729832755347968158104608573, a[7,4] = 212 898348102841647147719/2010207409308970807088412, a[12,9] = 98655463014 4/6438843573435, a[8,5] = 14602530426277943936000/13807955262255782286 82149*a[8,7]+10750605584303932375/81518659284419698176, a[10,6] = -842 587005151655641450739461021034364290614996174637/221852050527362280821 968872873583346136193521000000-542856893750284084473879372409360788717 61241262686208/236185841827547771600283855523522378427726749041171875* a[8,7], a[6,4] = 404414081147/1257566797788, a[10,7] = -85656608866934 7914171585647905094924281714609/30614735843368194465551943863835217613 080000+538713459048622145964825786579960496128/10715334273314770304986 023046747421875*a[8,7], b[8] = 5410824060928/20472512295375, c[5] = 41 5483/780760, a[10,4] = 41639640381003135312148228863502208689962952343 /1340913488826129626936355713963871370126200000-1399278295437317869400 7941818127058928171529064831090688/27945312238583698859424004514442888 5676770573279609375*a[8,7], a[10,8] = -1456804072482552778942060696340 89984/14192495726244728880776189465890625, a[2,1] = 1/1000, a[11,6] = \+ -69966788792214045964953909937771565820803614/158108628410627996642557 80307674707851179339-1787253052102153756451486175595064180527333376/67 72354549914525451645466603454051303846260967*a[8,7], b[6] = 1356696197 8425697607/75147921323806716000, c[7] = 109/659, c[8] = 7/16, a[12,1] \+ = 3042259121/61252403940, b[9] = 986554630144/6438843573435, a[3,2] = \+ 392000/62001, a[7,1] = 39743335520386666547/572655525899435427204, b[1 1] = -135068333/274428000, a[5,4] = 337487397017512409163/932843387885 03296000, a[5,3] = -777541397023211694969/186568677577006592000, a[3,1 ] = -385028/62001, a[11,10] = -6894374368544685000000/9706884890396295 37954049, a[11,9] = 819316668609331200/920300389324474639, a[4,1] = 7/ 166, b[7] = 492897387352157699403367/1970142834564896505822000, a[8,1] = -727125698073181/72414709901294313*a[8,7]+225038371369/710679350476 8, b[10] = 269064197119512500/452758788468179739, b[1] = 3042259121/61 252403940, a[12,7] = 492897387352157699403367/197014283456489650582200 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 65 "The coefficients for the embedded order 6 scheme are as follows. " }}{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 0 477 "subs(e8,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7)]));\nfor ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e8,c[ii])); \n for j j to ii-1 do\n coeff_val := subs(e8,a[ii,jj]);\n if coeff_va l=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`)\n e lse\n print(a[ii,jj]=coeff_val);\n end if;\n end do:\ne nd do:\nprint(``);\nc[12]=subs(e8,c[12]);\nseq(a[12,i]=subs(e8,a[12,i] ),i=1..11);\nprint(``);\nseq(`b*`[i]=subs(e8,`b*`[i]),i=1..12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"\"\"%+5F(%!GF+F+F +F+F+7*#\"#G\"$\\##!'G]Q\"&,?'#\"'+?RF2F+F+F+F+F+7*#\"#9\"#$)#\"\"(\"$ m\"\"\"!#\"#@F;F+F+F+F+7*#\"'$[:%\"'g2y#\"6V-&pno*)\\\\=?\"6+?f1qdx'ol =F<#!6p\\p6K-(RTvxFE#\"6j\"4C^ \"\\F+F+7*#\"$4\"\"$f'#\"5Zlm'Q?bLV(R\"6/sUN%**e_bEdFx9Z;%G5[$ )*G@\":7%)323(*3$4u?5?#!;+?Fcx#ouZnoW&\\\"=t&3Y5e\"ozMbF$)HF#\"41e\"=L $p[t2\"\"66/%yY?1CI28F+Q)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"\"(\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&*&#\"0\"=t!)pDrs \"28VH,*4ZTsF(&F%6$F'\"\"(F(!\"\"#\"-p8PQ]A\".oZ]$z1rF(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&*&#\"6PRcP#[D\"H<`#\"6R]`s4;<#*>a#\"\"\"&F% 6$F'\"\"(F.!\"\"#\"1^zE))G,wy\"2/*R\"y)fr%\\#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&*&#\"8+g$R%zFE/`-Y\"\":\\@oG#ybAEb z!Q\"\"\"\"&F%6$F'\"\"(F.F.#\"5vB$RI%e01v5\"5w\")p>WGf'=:)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"0$eAbS2BI\"2V-o*Hk b7m\"\"\"&F%6$F'\"\"(F.!\"\"#\"/vhb6BUr2i\"@;O5\\Pex\"**4'4Y\"oyIF(*&#\"D*yDUY, :?#)[y&3\"[,OB')\"EXQ4ov,p69%Hx+g$\\70wF(&F%6$\"\")\"\"(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,&#\"C,Ci8hN'3Zn6'**RTshZ\"B/^PN)3!) f^:\"y5R(QSp!\"\"*&#\"J`n=^E-K2>Xi&\\%*R+f'3D+$\"IN!=Yag,hn>J[=l(y]lkp E\"\"\"&F%6$\"\")\"\"(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"&,&#\"Fv$ygzmOBV#R+0s0q%\"L++svIbU=;*GOx<5(3z#zSF$)!\"\"*&#\"T'\\#o%) 4/9Sq9Ojr/Qz#\\ZGLn'>ph$\"TD19f@Wb*3[UA[KxNkJW?3U)*zkrF(&F%6$\"\")\"\" (F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,&#\"PVB&H'**o3A]j)G#[@JNJ+\"Q S'R;%\"O++?E,PrQ'Rrbj$piHh#))[84M\"\"\"\"*&#\"Y)o!4J[1H:.4dOhM$etG()o>#3Git_]?&=A!\"\"*& #\"V3ioi7Ch<()yg$4CPzQZ%3%G]P*o&Ga\"Wv=blW>oL%et91$! \"\"*&#\"HGh\\g*zlyD['f9A'[!fMrQ&\"Gv=UZn/Bg)\\IqZJtUL:2\"\"\"\"&F%6$ \"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!E%) **3M'pg?%*yFb#[sS!oX\"\"DD1*eY*=w2))GZCEd\\#>9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"7KcozSd-#3mf'\"7D\"GX&z&)4(*>'R)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"\",&#\"H8M0SE.7$>WvnSL_#oQ5G\"\"H%3)[b,!ya#*z>,gZx%QD;&>!\"\"*&# \"M7ll#)f%f(e'z7g^Sn!=(Q*>!3>\"\"Mx?ME*HS$[%==KiZ4f%4G)HH&ps$HxWhB 2L')\"W\"))))4:n@G)[jP?S)zJ2MrnQm!RO;99F-&F%6$\"\")\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"',&#\"M9O!3#ecrx$*4R&\\'f/9 Az)ym*p\"MR$z6&yquwI!ybUm*zi5%G'3\"e\"!\"\"*&#\"OwLLF0=k]fvh[^kv`@5_ID (y\"\"On4EYQI^SX.mYX;XDX\"*\\XNsn\"\"\"&F%6$\"\")\"\"(F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"L]W\"f8H8mL2b: g$p6\"\"JH'*REa_\"*H68c,))py?!eSoM!\"\"*&#\"@syKpO_v#G*[_gQ?\"f\"?p%*G UlB0$Gh5'e;C5\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#6\"\")#!BCm-wt)\"3RYZ C$*Q+.#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!7+++&oW &oVP%*o\"9\\S&z`H'R!*[)oq*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 12 "" 1 " " {XPPMATH 20 "6-/&%\"aG6$\"#7\"\"\"#\"+@\"fA/$\",SRS_7'/&F%6$F'\"\"# \"\"!/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'#\"52wp D%y>'pc8\"5+gr1QK@z9v/&F%6$F'\"\"(#\"9nLS*pd@N(Q(*G\\\":+?#e]'*[cMG9q> /&F%6$F'\"\")#\".G41C3T&\"/v`H7DZ?/&F%6$F'\"\"*#\"-W,jal)*\".NMdV)Qk/& F%6$F'\"#5#\"3+D^>r>k!p#\"3R(z\"o%)yeFX/&F%6$F'\"#6#!*L$o]8\"*+!GWF" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6./&%#b*G6#\"\"\"#\"6/&F%6#\"\"(#\"E`%)QTs$\"9vo^'pFooU>9/#/&F%6#\"#5#\"\" " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op (OrderConditions(7,11,'expanded'))]:\n`RK6_12eqs*` := subs(b=`b*`,Orde rConditions(6,12,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "simplify(subs(e8,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u), %);\nnops(%);\nsimplify(subs(e8,`RK6_12eqs*`)):\nmap(u->lhs(u)-rhs(u), %);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 #\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F $F$F$F$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#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "We choose a value for " }{XPPEDIT 18 0 "a[8,7]" "6# &%\"aG6$\"\")\"\"(" }{TEXT -1 85 " that makes the principal error nor m for the order 7 scheme approximately a minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "errterms := PrincipalErrorTerms(7,11,'expanded'):\nsm := 0:\nfor ct to nops(errte rms) do\n sm := sm+(evalf(subs(e8,errterms[ct])))^2;\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot(sqrt(sm),a[8,7]=0..0.47,color=COLOR(RGB,.5,0,1),thickness=2,f ont=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 277 277 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$\"\"!F)$\"3Q'Q2rm:GQ*!#B7$$\"38LL$eCmW -\"!#>$\"3Y\"e!o+[[\"Q*F,7$$\"3lm;a[)[e\">F0$\"3S^l*Hlv.Q*F,7$$\"35LL3 7=I=HF0$\"3;1hA2E=z$*F,7$$\"3DLLe%o*RFRF0$\"3U\\ZCR(R!y$*F,7$$\"3\"emT 5\\,<$\\F0$\"3!)zRK(=gpP*F,7$$\"3EK$3_#y\"G'eF0$\"3qC(\\%44,w$*F,7$$\" 3_)*\\(o&[$p#oF0$\"3ky\")>G.3v$*F,7$$\"3!GL3-2ES#yF0$\"3'f*))*=&Rz\"))F0$\"3IH61*>FLP*F,7$$\"31mm;f7ES)*F0$\"3iSX&eI:DP *F,7$$\"3;L$e\\.uS2\"!#=$\"3UOmZU)\\=P*F,7$$\"3-++v_mWv6F]o$\"33r!HyMc 6P*F,7$$\"3')***\\i^NsF\"F]o$\"3]'=\"GM$>0P*F,7$$\"3!****\\s%zKv8F]o$ \"39DS0;<'*p$*F,7$$\"35L3-s_Sk9F]o$\"3M9%=h>.&p$*F,7$$\"3[mm\"p#oKq:F] o$\"3Q)*>Sgs,p$*F,7$$\"3Ommmxi0g;F]o$\"3=OB_4glo$*F,7$$\"3u*\\(=gHVke >F]o$\"37'y95w(yn$*F,7$$\"3'**\\ibezC#F]o$\"3a:)Q#eJVn$*F,7$$\"3'HLe *G,oZBF]o$\"3EPj>EIsuO*F,7$$\"3h*\\7 =52:a#F]o$\"3!=fs;tlvO*F,7$$\"3:L$3A&*H*QEF]o$\"3!H!z#)R\">xO*F,7$$\"3 z****\\*Gx&RFF]o$\"3oYpd\")[$zO*F,7$$\"3W**\\Pt5/QGF]o$\"3WUlFZA?o$*F, 7$$\"3l*\\(=Z)4L$HF]o$\"3S[EsQR^o$*F,7$$\"3J**\\ix&*3RIF]o$\"3'y6YQ3@* o$*F,7$$\"3$emm,PPT8$F]o$\"39OQTCQF] o$\"3'\\N(fw%4RP*F,7$$\"3\"HLLVy^p\"RF]o$\"3osHrxzxu$*F,7$$\"3qK$eW#\\ j;SF]o$\"3UGnxkCsv$*F,7$$\"3>mT&yj#\\:TF]o$\"3AG$H&e`rw$*F,7$$\"3w**** \\4EL1UF]o$\"3wq&Qa2xwP*F,7$$\"3`m;HoaW5VF]o$\"3wtff?u$)y$*F,7$$\"3zKL L]_c.WF]o$\"3GhpM5y#*z$*F,7$$\"3t*\\(=g.&G]%F]o$\"3)y?rX/X6Q*F,7$$\"3m *\\7)y)yyf%F]o$\"3)*oYS+HO#Q*F,7$$\"3u*************p%F]o$\"3!3)eZ1#HPQ *F,-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\"&!\"\"F($\"\"\"F)- %*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q'a[8,7]6\"Q!F_\\l-%%VIEWG6$;F($\"# Z!\"#%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "evalf[12](minimize(sqrt(sm),a[8,7]=0..1,'locati on'));\nconvert(subs(op(1,op(op(2,[%]))),a[8,7]),rational,4);\nconvert (subs(op(1,op(op(2,[%%]))),a[8,7]),rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-_-)>uO*!#<<#7$<#/&%\"aG6$\"\")\"\"($\"-x/&3hJ#!#7F# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#>\"##)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#j\"$s#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "evalf(eval(sqrt(sm),a[8,7]=19/82)); \nevalf(eval(sqrt(sm),a[8,7]=63/272));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1)>uO*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,)>uO*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e9 := `union`(subs(a[8,7]=19/82,e8),\{a[8,7]=19/82\}):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4579 "e9 := \{`b *`[1] = 783278906947440245917/15535938516834202740000, a[8,2] = 0, a[8 ,3] = 0, a[9,2] = 0, a[9,3] = 0, a[7,2] = 0, a[7,3] = 0, `b*`[6] = 299 1788146029459739848009433033/19060370046834035632749036000000, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[8,7] = 19/82, `b*`[11] \+ = -22960167428223228083/69605374826988000000, c[2] = 1/1000, c[4] = 14 /83, a[11,6] = -879765828850292172490106340518569223117087719534557891 994/196096868766029731312926688640518320991505603079281962497, c[9] = \+ 13/16, c[10] = 169/170, `b*`[12] = 1/200, a[7,6] = 1077348693331815806 /130730240620467840411, c[6] = 87/131, a[12,8] = 5410824060928/2047251 2295375, `b*`[9] = 37241388453197849045504/204141942686827696516875, a [9,5] = -5992985997487815939072736862205038623064898960325/22454912229 15232082243948587229615621971585089024, a[11,7] = -2677394030869811360 1982120741972828286815906/1422046378852266508640137632652542482384789, a[8,4] = 174040744000019283021419611/2049081587190964014926856192, a[ 9,4] = -183130124969528341358670952614060593600746254959/4303965726269 5209676619384887672437213535272960, a[12,11] = -135068333/274428000, c [11] = 1, a[6,2] = 0, c[12] = 1, a[9,1] = 3045849270666189198416419199 28442600454537797/1336435146974329082026479192445476428829818880, a[10 ,1] = -1256795279708492037734149713024454589348072577632030104333/1844 322652469644188488724709227695887423797701298885000000, a[12,5] = 0, b [2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[9,6] = 1322708054627964884973 707202944501083084973588321/134685915128252673721316521442625893356330 3526400, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[8,6] = -125509550840 850981113069/2931683770285010740150272, a[6,1] = 206935503/6884238116, `b*`[10] = 190299067731754576532365655/459347372389706921889026076, a [6,5] = 1534888336000/4911984599757, a[4,3] = 21/166, a[12,6] = 135669 61978425697607/75147921323806716000, a[12,10] = 269064197119512500/452 758788468179739, a[11,4] = 7940345987819075557870981796824636259700471 18793613/35481535396506256458169246778950856594722743868452, a[11,1] = -1913917650773276648000042701921031830180297961856463/242053599987401 4114485551535784938587087050773067132, a[11,8] = -91899049567923461015 4177376026624/77529355682339127064062513128033, `b*`[7] = 123570026638 743472603558985757717817/499703129646370745806507058262000000, a[5,1] \+ = 201849498968676950243/186568677577006592000, a[11,5] = 1253876511431 9773465378231128982652519968027341646160000/91548488213245038355215795 3702371732055721397872191177, a[10,2] = 0, a[10,3] = 0, a[11,2] = 0, a [11,3] = 0, a[4,2] = 0, a[5,2] = 0, a[6,3] = 0, a[10,9] = 659660820257 4079685632/8396199709857954528125, a[10,5] = 1001719196666270807874926 26771580340159478395878727610639533656/8428868485387823508039378612742 757916792962274222867420200625, `b*`[8] = 177511756302554128853504/649 075937937135306609375, a[8,5] = 128489749336355703667293557125/9565267 69655402159824357169664, a[9,7] = 873822912254162442527647371013373/22 8741131636931363287619371663360, a[9,8] = 51299867691487048524852/1895 6439276208492414825, c[3] = 28/249, a[7,5] = -495446867477468277562720 00/2729832755347968158104608573, a[7,4] = 212898348102841647147719/201 0207409308970807088412, a[12,9] = 986554630144/6438843573435, a[10,7] \+ = -386885184918281945410313681520411070615012243139331/236919787007865 61492029010557625479030322285000000, a[10,4] = 11553212779991012111059 55197933773620274343648077208821293051/5939608444437533690377415615517 7493077283524727141295000000, a[10,6] = -75691885617633891987993521638 14463872117563141929332169849602853/1965397952528115167934733609461445 372239605164436067222345000000, a[6,4] = 404414081147/1257566797788, b [8] = 5410824060928/20472512295375, c[5] = 415483/780760, a[10,8] = -1 45680407248255277894206069634089984/1419249572624472888077618946589062 5, a[2,1] = 1/1000, b[6] = 13566961978425697607/75147921323806716000, \+ c[7] = 109/659, c[8] = 7/16, a[12,1] = 3042259121/61252403940, b[9] = \+ 986554630144/6438843573435, a[3,2] = 392000/62001, a[7,1] = 3974333552 0386666547/572655525899435427204, b[11] = -135068333/274428000, a[5,4] = 337487397017512409163/93284338788503296000, a[5,3] = -7775413970232 11694969/186568677577006592000, a[8,1] = 17125839160165511097181/58372 9762655220563902464, a[3,1] = -385028/62001, a[11,10] = -6894374368544 685000000/970688489039629537954049, a[11,9] = 819316668609331200/92030 0389324474639, a[4,1] = 7/166, b[7] = 492897387352157699403367/1970142 834564896505822000, b[10] = 269064197119512500/452758788468179739, b[1 ] = 3042259121/61252403940, a[12,7] = 492897387352157699403367/1970142 834564896505822000\}:" }{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 62 "The coefficients for the \+ combined scheme are given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1011 "subs(e9,matrix([[c[2],a[2,1],``],\n[c[3],a[3,1],a[3 ,2]],[c[4],a[4,1],a[4,2]],[``$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]],[``$2,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]],[``$2,a[8,7]],\n[c[9 ],a[9,1],a[9,2]],[``,a[9,3],a[9,4]],[``,a[9,5],a[9,6]],[``,a[9,7],a[9, 8]],\n[c[10],a[10,1],a[10,2]],[``,a[10,3],a[10,4]],[``,a[10,5],a[10,6] ],\n[``,a[10,7],a[10,8]],[``$2,a[10,9]],\n[c[11],a[11,1],a[11,2]],[``, a[11,3],a[11,4]],[``,a[11,5],a[11,6]],\n[``,a[11,7],a[11,8]],[``,a[11, 9],a[11,10]],\n[c[12],a[12,1],a[12,2]],[``,a[12,3],a[12,4]],[``,a[12,5 ],a[12,6]],\n[``,a[12,7],a[12,8]],[``,a[12,9],a[12,10]],[``$2,a[12,11] ],\n[``,`________________________________`$2],[`b`,b[1],b[2]],[``,b[3] ,b[4]],[``,b[5],b[6]],\n[``,b[7],b[8]],[``,b[9],b[10]],[``$2,b[11]],\n [`b*`,`b*`[1],`b*`[2]],[``,`b*`[3],`b*`[4]],[``,`b*`[5],`b*`[6]],\n[`` ,`b*`[7],`b*`[8]],[``,`b*`[9],`b*`[10]],[``,`b*`[11],`b*`[12]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7S7%#\"\"\"\"%+5F(%!G7%#\" #G\"$\\##!'G]Q\"&,?'#\"'+?RF27%#\"#9\"#$)#\"\"(\"$m\"\"\"!7%F+F+#\"#@F ;7%#\"'$[:%\"'g2y#\"6V-&pno*)\\\\=?\"6+?f1qdx'ol=F<7%F+#!6p\\p6K-(RTvx FF#\"6j\"4C^\"\\7%#\"$4\" \"$f'#\"5Zlm'Q?bLV(R\"6/sUN%**e_bEdF<7%F+F<#\"9>x9Z;%G5[$)*G@\":7%)323 (*3$4u?5?7%F+#!;+?Fcx#ouZnoW&\\\"=t&3Y5e\"ozMbF$)HF#\"41e\"=L$p[t2\"\" 66/%yY?1CI287%#F:\"#;#\"8\"=(46b;g\"Re7<\"9kC!Rc?_li(HPeF<7%F+F<#\"<6' >9-$G>++W2/u\"\"=#>co#\\,k4>(e\"3\\?7%F+#\"?Drb$HnOqbjL\\(*[G\"\"?k'pr NC)f@Sb'pn_c*#!9pI6\")4&3%3b4b7\":s-:S2,&GqPoJH7%F+F+#\"#>\"##)7%#\"#8 Fjo#\"N(zPXX+EWG*>>kT)>*=m1F\\e/$\"O!))=)H)GkZXC>zk-#3HV(p9NkL\"F<7%F+ F<#!Qf\\DY2g$fgSh_4ne8MG&p\\7IJ=\"PgHFNN@PCn()[Q>mn4_pislRI%7%F+#!RD.' *)*[1B'Q]?iots!Rf\"y[(*f)H*f\"RC!*3&er>i:'Hse[RC#3K_\"HA\"\\XA#\"R@$)e t\\3$3,X%H?2P(\\)['zia!3FK\"\"R+k_.LcL*eiU9_;8stEDG^\"foM\"7%F+#\"BtL, rtkFDWiTD7H#Q()\"BgLmr$>wGj8$pjJ6uG##\"8_[_[q[\"pn)*H^\"8D[T#\\3iFRk&* =7%#\"$p\"\"$q\"#!enLV5I?jxD2[$*eaW-8(\\TtP?\\3(z_zc7\"en+++&)))H,xzBu )epF#4Zs)[)=W'pClAV%=F<7%F+F<#\"hn^IH@)3s2[OMu-itP$z>bf5675**zF@`:\"\" fn+++&HTrsCNGxI\\x^:cTx.pLvVW%3'Rf7%F+#\"jncO`R1hF(yeRy%f,M!erni#\\(y! 3Fmm>><+\"\"hnD1??u'GAuA'Hz;zvUFhy$R!3N#yQ&[o)G%)#![o`Gg\\)p@LH>9jv6sQ Y9Q;_$*z)>*Qj7%F+#!TJ$RJC7 ]hq5T?:o8.TX>G=\\=&)oQ\"S+++&GA..zaid0,H?\\hly+(y>pB#!E%)**3M'pg?%*yFb #[sS!oX\"\"DD1*eY*=w2))GZCEd\\#>97%F+F+#\"7KcozSd-#3mf'\"7D\"GX&z&)4(* >'R)7%F)#!Ujk&='zH!=I=.@>qU++[mFt2l\"UKr1t20(3(eQ\\yN:b&[9T,u)**f `?CF<7%F+F<#\"T8Oz=r/qfijCoz\")4(ybv!>y)fMSz\"S_%oQuAZfc3&*ynCp\"ekD1l RN:[N7%F+#\"Y++;Y;MF!o*>Dl#)*G6By`Yt(>V6l(QD\"\"Wx6>syR@d0K*ybM&>x3GzIg0:*4K=0k)oEHJJ (Hgwoo4'>7%F+#!M1f\"oGGG(>u?@)>g8\")p3.%RxE\"L*y%Q#[UDlKw8S'3lE_)yj/A9 #!BCm-wt)\"3RYZC $*Q+.#*#!7+++&oW&oVP%*o\"9\\S&z`H'R!*[)oq*7%F)#\"+@\"fA/$\",SRS_7'F<7% F+F'pc8\"5+gr1QK@z9v7%F+#\"9nLS*pd@N(Q(*G\\\":+?#e ]'*[cMG9q>#\".G41C3T&\"/v`H7DZ?7%F+#\"-W,jal)*\".NMdV)Qk#\"3+D^>r>k!p# \"3R(z\"o%)yeFX7%F+F+#!*L$o]8\"*+!GWF7%F+%A___________________________ _____GF[x7%%\"bGF`vF7%F+#\"E`%)QTs$\"9vo^'pFooU>9/##\"\" " 0 "" {MPLTEXT 1 0 139 "subs(e9,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..12 ),\n[`b`,seq(b[j],j=1..11),``],[`b*`,seq(`b*`[j],j=1..12)]])):\nevalf[ 6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7/7/$\"'++5!\")F (%!GF+F+F+F+F+F+F+F+F+F+7/$\"']C6!\"'$!'.5i!\"&$\"'[AjF2F+F+F+F+F+F+F+ F+F+F+7/$\"'v'o\"F/$\"'(o@%!\"($\"\"!F<$\"'1l7F/F+F+F+F+F+F+F+F+F+7/$ \"'_@`F/$\"'!>3\"F2F;$!'fnTF2$\"'%yh$F2F+F+F+F+F+F+F+F+7/$\"'ATmF/$\"' $f+$F:F;F;$\"'&e@$F/$\"'yCJF/F+F+F+F+F+F+F+7/$\"'-a;F/$\"'=SpF:F;F;$\" '4f5F/$!'$\\\"=F:$\"',T#)F*F+F+F+F+F+F+7/$\"'+vVF/$\"''Q$HF:F;F;$\"'g$ \\)F:$\"'HV8F/$!'9\"G%F:$\"'2!\"%$\"'W)=\"Fip$!'A^QF2$!')Hj\"Fip$!'YE5Fip$\"'mcyF/F+ F+F+7/$\"\"\"F<$!'+2zF/F;F;$\"')yB#Fip$\"'jp8Fip$!'Q'[%F2$!'y#)=Fip$!' N&=\"Fip$\"'r-*)F/$!'c-rF*F+F+7/Feq$\"'wm\\F:F;F;F;F;$\"'P0=F/$\"'%=]# F/$\"'(Hk#F/$\"'>K:F/$\"'xUfF/$!'\"=#\\F/F+7/%\"bGFhrF;F;F;F;FjrF\\sF^ sF`sFbsFdsF+7/%#b*G$\"'sT]F:F;F;F;F;$\"'kp:F/$\"'(GZ#F/$\"'%[t#F/$\"'H C=F/$\"'\"G9%F/$!'i)H$F/$\"'++]F*Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the e mbedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffi cients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4468 "ee := \{c[2]=1/1000,\nc[3] =28/249,\nc[4]=14/83,\nc[5]=415483/780760,\nc[6]=87/131,\nc[7]=109/659 ,\nc[8]=7/16,\nc[9]=13/16,\nc[10]=169/170,\nc[11]=1,\nc[12]=1,\n\na[2, 1]=1/1000,\na[3,1]=-385028/62001,\na[3,2]=392000/62001,\na[4,1]=7/166, \na[4,2]=0,\na[4,3]=21/166,\na[5,1]=201849498968676950243/186568677577 006592000,\na[5,2]=0,\na[5,3]=-777541397023211694969/18656867757700659 2000,\na[5,4]=337487397017512409163/93284338788503296000,\na[6,1]=2069 35503/6884238116,\na[6,2]=0,\na[6,3]=0,\na[6,4]=404414081147/125756679 7788,\na[6,5]=1534888336000/4911984599757,\na[7,1]=3974333552038666654 7/572655525899435427204,\na[7,2]=0,\na[7,3]=0,\na[7,4]=212898348102841 647147719/2010207409308970807088412,\na[7,5]=-495446867477468277562720 00/2729832755347968158104608573,\na[7,6]=1077348693331815806/130730240 620467840411,\na[8,1]=17125839160165511097181/583729762655220563902464 ,\na[8,2]=0,\na[8,3]=0,\na[8,4]=174040744000019283021419611/2049081587 190964014926856192,\na[8,5]=128489749336355703667293557125/95652676965 5402159824357169664,\na[8,6]=-125509550840850981113069/293168377028501 0740150272,\na[8,7]=19/82,\na[9,1]=30458492706661891984164191992844260 0454537797/1336435146974329082026479192445476428829818880,\na[9,2]=0, \na[9,3]=0,\na[9,4]=-183130124969528341358670952614060593600746254959/ 43039657262695209676619384887672437213535272960,\na[9,5]=-599298599748 7815939072736862205038623064898960325/22454912229152320822439485872296 15621971585089024,\na[9,6]=1322708054627964884973707202944501083084973 588321/1346859151282526737213165214426258933563303526400,\na[9,7]=8738 22912254162442527647371013373/228741131636931363287619371663360,\na[9, 8]=51299867691487048524852/18956439276208492414825,\na[10,1]=-12567952 79708492037734149713024454589348072577632030104333/\n 18443226 52469644188488724709227695887423797701298885000000,\na[10,2]=0,\na[10, 3]=0,\na[10,4]=1155321277999101211105955197933773620274343648077208821 293051/\n 59396084444375336903774156155177493077283524727141295 000000,\na[10,5]=10017191966662708078749262677158034015947839587872761 0639533656/\n 8428868485387823508039378612742757916792962274222 867420200625,\na[10,6]=-7569188561763389198799352163814463872117563141 929332169849602853/\n 1965397952528115167934733609461445372239 605164436067222345000000,\na[10,7]=-3868851849182819454103136815204110 70615012243139331/\n 23691978700786561492029010557625479030322 285000000,\na[10,8]=-145680407248255277894206069634089984/141924957262 44728880776189465890625,\na[10,9]=6596608202574079685632/8396199709857 954528125,\na[11,1]=-1913917650773276648000042701921031830180297961856 463/\n 2420535999874014114485551535784938587087050773067132,\n a[11,2]=0,\na[11,3]=0,\na[11,4]=79403459878190755578709817968246362597 0047118793613/\n 3548153539650625645816924677895085659472274386 8452,\na[11,5]=1253876511431977346537823112898265251996802734164616000 0/\n 915484882132450383552157953702371732055721397872191177,\na [11,6]=-879765828850292172490106340518569223117087719534557891994/\n \+ 196096868766029731312926688640518320991505603079281962497,\na[1 1,7]=-26773940308698113601982120741972828286815906/1422046378852266508 640137632652542482384789,\na[11,8]=-918990495679234610154177376026624/ 77529355682339127064062513128033,\na[11,9]=819316668609331200/92030038 9324474639,\na[11,10]=-6894374368544685000000/970688489039629537954049 ,\na[12,1]=3042259121/61252403940,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0, \na[12,5]=0,\na[12,6]=13566961978425697607/75147921323806716000,\na[12 ,7]=492897387352157699403367/1970142834564896505822000,\na[12,8]=54108 24060928/20472512295375,\na[12,9]=986554630144/6438843573435,\na[12,10 ]=269064197119512500/452758788468179739,\na[12,11]=-135068333/27442800 0,\n\nb[1]=3042259121/61252403940,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0, \nb[6]=13566961978425697607/75147921323806716000,\nb[7]=49289738735215 7699403367/1970142834564896505822000,\nb[8]=5410824060928/204725122953 75,\nb[9]=986554630144/6438843573435,\nb[10]=269064197119512500/452758 788468179739,\nb[11]=-135068333/274428000,\n\n`b*`[1]=7832789069474402 45917/15535938516834202740000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b *`[5]=0,\n`b*`[6]=2991788146029459739848009433033/19060370046834035632 749036000000,\n`b*`[7]=123570026638743472603558985757717817/4997031296 46370745806507058262000000,\n`b*`[8]=177511756302554128853504/64907593 7937135306609375,\n`b*`[9]=37241388453197849045504/2041419426868276965 16875,\n`b*`[10]=190299067731754576532365655/4593473723897069218890260 76,\n`b*`[11]=-22960167428223228083/69605374826988000000,\n`b*`[12]=1/ 200\}:" }}}{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[7,11];" "6#&%\"TG6$\"\"(\"#6" } {TEXT -1 128 " denote the vector whose components are the principal e rror terms of the 11 stage, order 7 scheme (the error terms of order 8 )." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6, 12]; " "6#&%#T*G6$\"\"'\"#7" }{TEXT -1 146 " denote the vector whose compo nents are the principal error terms of the embedded 12 stage, order 6 \+ scheme (the error terms of order 7) and let " }{XPPEDIT 18 0 "`T*`[7, 12];" "6#&%#T*G6$\"\"(\"#7" }{TEXT -1 99 " denote the vector whose co mponents are the error terms of order 8 of the embedded order 6 scheme ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Deno te the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[7, 11]) );" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,12]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"#7" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[7,12]));" "6#-%$absG 6#-F$6#&%#T*G6$\"\"(\"#7" }{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[8] = abs(abs(T[7, 11])); " "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,12]))/abs(abs(`T*`[6, 12])); " "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\"#7\"\"\"-F*6#-F*6#&F /6$\"\"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(` T*`[7,12]-T[7,11]))/abs(abs(`T*`[6,12]));" "6#/&%\"CG6#\"\")*&-%$absG6 #-F*6#,&&%#T*G6$\"\"(\"#7\"\"\"&%\"TG6$F2\"#6!\"\"F4-F*6#-F*6#&F06$\" \"'F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as at tempting to ensure that " }{XPPEDIT 18 0 "A[8];" "6#&%\"AG6#\"\")" } {TEXT -1 73 " is a minimum, if the embedded scheme is to be used for \+ error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\")" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errt erms7_11 := PrincipalErrorTerms(7,11,'expanded'):\n`errterms7_12*` :=s ubs(b=`b*`,PrincipalErrorTerms(7,12,'expanded')):\n`errterms6_12*` := \+ subs(b=`b*`,PrincipalErrorTerms(6,12,'expanded')):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := s qrt(add(evalf(subs(ee,`errterms7_12*`[i]))^2,i=1..nops(`errterms7_12*` ))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms6_12*`[i]))^2,i=1..nops( `errterms6_12*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms7_12*`[i ])-subs(ee,errterms7_11[i])))^2,i=1..nops(errterms7_11))):\n'B[8]'= ev alf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")?6V:!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")Bd[:!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" } }{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4468 "ee := \{c[2]=1/1000,\nc[3]=28/249,\nc[4]=14/83 ,\nc[5]=415483/780760,\nc[6]=87/131,\nc[7]=109/659,\nc[8]=7/16,\nc[9]= 13/16,\nc[10]=169/170,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/1000,\na[3,1]=- 385028/62001,\na[3,2]=392000/62001,\na[4,1]=7/166,\na[4,2]=0,\na[4,3]= 21/166,\na[5,1]=201849498968676950243/186568677577006592000,\na[5,2]=0 ,\na[5,3]=-777541397023211694969/186568677577006592000,\na[5,4]=337487 397017512409163/93284338788503296000,\na[6,1]=206935503/6884238116,\na [6,2]=0,\na[6,3]=0,\na[6,4]=404414081147/1257566797788,\na[6,5]=153488 8336000/4911984599757,\na[7,1]=39743335520386666547/572655525899435427 204,\na[7,2]=0,\na[7,3]=0,\na[7,4]=212898348102841647147719/2010207409 308970807088412,\na[7,5]=-49544686747746827756272000/27298327553479681 58104608573,\na[7,6]=1077348693331815806/130730240620467840411,\na[8,1 ]=17125839160165511097181/583729762655220563902464,\na[8,2]=0,\na[8,3] =0,\na[8,4]=174040744000019283021419611/2049081587190964014926856192, \na[8,5]=128489749336355703667293557125/956526769655402159824357169664 ,\na[8,6]=-125509550840850981113069/2931683770285010740150272,\na[8,7] =19/82,\na[9,1]=304584927066618919841641919928442600454537797/13364351 46974329082026479192445476428829818880,\na[9,2]=0,\na[9,3]=0,\na[9,4]= -183130124969528341358670952614060593600746254959/43039657262695209676 619384887672437213535272960,\na[9,5]=-59929859974878159390727368622050 38623064898960325/2245491222915232082243948587229615621971585089024,\n a[9,6]=1322708054627964884973707202944501083084973588321/1346859151282 526737213165214426258933563303526400,\na[9,7]=873822912254162442527647 371013373/228741131636931363287619371663360,\na[9,8]=51299867691487048 524852/18956439276208492414825,\na[10,1]=-1256795279708492037734149713 024454589348072577632030104333/\n 1844322652469644188488724709 227695887423797701298885000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=11553 21277999101211105955197933773620274343648077208821293051/\n 593 96084444375336903774156155177493077283524727141295000000,\na[10,5]=100 171919666627080787492626771580340159478395878727610639533656/\n \+ 8428868485387823508039378612742757916792962274222867420200625,\na[10, 6]=-7569188561763389198799352163814463872117563141929332169849602853/ \n 19653979525281151679347336094614453722396051644360672223450 00000,\na[10,7]=-386885184918281945410313681520411070615012243139331/ \n 23691978700786561492029010557625479030322285000000,\na[10,8 ]=-145680407248255277894206069634089984/141924957262447288807761894658 90625,\na[10,9]=6596608202574079685632/8396199709857954528125,\na[11,1 ]=-1913917650773276648000042701921031830180297961856463/\n 242 0535999874014114485551535784938587087050773067132,\na[11,2]=0,\na[11,3 ]=0,\na[11,4]=794034598781907555787098179682463625970047118793613/\n \+ 35481535396506256458169246778950856594722743868452,\na[11,5]=125 38765114319773465378231128982652519968027341646160000/\n 915484 882132450383552157953702371732055721397872191177,\na[11,6]=-8797658288 50292172490106340518569223117087719534557891994/\n 19609686876 6029731312926688640518320991505603079281962497,\na[11,7]=-267739403086 98113601982120741972828286815906/1422046378852266508640137632652542482 384789,\na[11,8]=-918990495679234610154177376026624/775293556823391270 64062513128033,\na[11,9]=819316668609331200/920300389324474639,\na[11, 10]=-6894374368544685000000/970688489039629537954049,\na[12,1]=3042259 121/61252403940,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12 ,6]=13566961978425697607/75147921323806716000,\na[12,7]=49289738735215 7699403367/1970142834564896505822000,\na[12,8]=5410824060928/204725122 95375,\na[12,9]=986554630144/6438843573435,\na[12,10]=2690641971195125 00/452758788468179739,\na[12,11]=-135068333/274428000,\n\nb[1]=3042259 121/61252403940,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=13566961978 425697607/75147921323806716000,\nb[7]=492897387352157699403367/1970142 834564896505822000,\nb[8]=5410824060928/20472512295375,\nb[9]=98655463 0144/6438843573435,\nb[10]=269064197119512500/452758788468179739,\nb[1 1]=-135068333/274428000,\n\n`b*`[1]=783278906947440245917/155359385168 34202740000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]= 2991788146029459739848009433033/19060370046834035632749036000000,\n`b* `[7]=123570026638743472603558985757717817/4997031296463707458065070582 62000000,\n`b*`[8]=177511756302554128853504/649075937937135306609375, \n`b*`[9]=37241388453197849045504/204141942686827696516875,\n`b*`[10]= 190299067731754576532365655/459347372389706921889026076,\n`b*`[11]=-22 960167428223228083/69605374826988000000,\n`b*`[12]=1/200\}:" }}}{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 78 "The s tability function R for the 11 stage, order 7 scheme is given as follo ws." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunc tion(7,11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,:\"\"\"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)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F )F)*&#\"N\"e:*Qj,@1uYH'*[8$ye/)4PLu(\"S+g<7j%QtUO,$=5v)pz$y()[+-(z7$F) *$)F'\"\")F)F)F)*&#\"P\"oe;nOHo\"=!p!4A%QiH8EW&HV)=\"USi!Hxv%o1\"yz>W* )HdXmOIpy\\U4*))F)*$)F'\"\"*F)F)F)*&#\"P^Pah&)\\&>0vIrs**[%GzQkB-@D\" \"W+!G8;ZfNLEZZ-oBmpIeHmLA\"yO66F)*$)F'\"#5F)F)!\"\"*&#\"Iw42:T'y9Cv&4 $p?Z(eSnQD\"Qv$oRGBJ;w>M4Nt#)yAu1J'\\Dy>F)*$)F'\"#6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the \+ point where the boundary of the stability region intersects the negati ve real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newt on(R(z)=-1,z=-4.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+U?9=Y! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-4.6):\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],v iew=[-5.19..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7V7$$!3Q++++++!>&!# <$!3'>PP'Rh]_MF*7$$!3Ynm;#3'4G^F*$!3U@M!Q#zPUIF*7$$!3kLLLk@>m]F*$!3q`q Q2nWwEF*7$$!3E+]U'*)HB,&F*$!35'pI\"o\\l!R#F*7$$!3!pm;&GwYe\\F*$!3\\ho? v?YK@F*7$$!3s+](\\(Q*y*[F*$!3[U_4t`3s=F*7$$!3nLLV@,KP[F*$!3oUgKp%R/k\" F*7$$!3NLL.q&p`r%F*$!3)[f![rX&)\\7F*7$$!3;nmJG')*Rf%F*$!3cE'f,'o^_%*!# =7$$!3uLLyGAZ\"[%F*$!3:sZ(RG!eKsFU7$$!3%3+])fw&\\O%F*$!3&4C:g,ZiU&FU7$ $!3$QL$)f7eWC%F*$!3U)[I%[qBzRFU7$$!3A++lN]MCTF*$!3+vdYf)fL(GFU7$$!3umm YeRz+SF*$!3o'eQ4.)34?FU7$$!3_LLV-,(>*QF*$!3]L6h2DbE9FU7$$!35++S:-YpPF* $!3'Rckv3d'\\#*!#>7$$!3K+++\"HZkk$F*$!3uR![z13bY&Fcp7$$!3;++gW:!z_$F*$ !3&GvSa`#Q.FFcp7$$!3hLL)*\\1D?MF*$!3sj`$Fcp7$$!39nmEI%3g%HF*$\"37QbF\"o`^[%Fcp7$$!3-++0xX]BGF*$\"3iD ?yjErjaFcp7$$!3*)***\\\"R>&oq#F*$\"3%[m@48O5Q'Fcp7$$!3gmm;\\r8&e#F*$\" 3ibT)Q2WlO(Fcp7$$!3ymmrw\\OtCF*$\"3]tMZhJ3G$)Fcp7$$!3SLL$))e.GN#F*$\"3 5oHw'f,XX*Fcp7$$!3nLL)**=uvA#F*$\"3'f;/*o?3v5FU7$$!3K++:I;c=@F*$\"3&G/ .x6!e+7FU7$$!31LL.z]#3+#F*$\"3s%>a0ns:N\"FU7$$!3M++?,<>z=F*$\"3'*)p$)* o5)o_\"FU7$$!3;++!4<(>gFU7$$!3H++q9zA<:F*$\"3oo]d![1K>#FU7$$!3EnmEY;O-9F*$\"3[oLSY!p,Y#FU7$$ !3#)*****pQ<(z7F*$\"3#RezJ')p6y#FU7$$!3)RL$efMeo6F*$\"3+\\lUj`23JFU7$$ !3I****fAZ3Z5F*$\"3B='Rv+(f4NFU7$$!3xqm;(zQwK*FU$\"3nng(yT[Y$RFU7$$!3& z***\\)ecE8)FU$\"3fX-mJs2MWFU7$$!3'3nmm0VV'pFU$\"392lCv9f$)\\FU7$$!3P) ***\\iqATdFU$\"3<)*oa^u)>j&FU7$$!3aFLL*)4AjXFU$\"3Uz=@&>(4OjFU7$$!33LL LO'R&eLFU$\"3!fZ'=WZFZrFU7$$!3Uim;`O$Q;#FU$\"3+i:1,ZEa!)FU7$$!3?*****> $H-m5FU$\"3Ep1IQ4$)))*)FU7$$\"3v*QLLU?>#>Fcp$\"3[-s.#30%>5F*7$$\"3%ymm Y^avJ\"FU$\"3\"H+?qHH39\"F*7$$\"3E0+]HcU/U\"=YF*Ff\\l-%'SYMBOLG6#%'CIRCLEG- Fj[l6&F\\\\lFa\\lFa\\lFa\\l-%&STYLEG6#%&POINTG-F$6&F\\`l-Fa`l6#%&CROSS GFd`lFf`l-F$6&F\\`l-Fa`l6#%(DIAMONDGFd`lFf`l-F$6%7$7$F^`lF`\\lF]`l-%&C OLORG6&F\\\\lF`\\l$\"\"&Fg\\lF`\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELV ETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fjbl-Fbbl6#%(DEFAULTG-%%VIEWG6$;$! $>&!\"#$\"#\\Fecl;$!$Z\"Fecl$\"$Z\"Fecl" 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 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1699 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5 040*z^7+\n 774333709804587831348962946740621016338915581/312797020048 87783796987510183013642733846312176000*z^8+\n 18843295442613296238422 090690181682936671658681/889094249786930366645572989441979781066847577 2906240*z^9-\n 12521022364387928448997271307505195498561543751/111136 781223366295830696623680247472633355947161328000*z^10+\n 253867405874 7206930957524147864115070976/19782549631067422788273350934197616312328 3968375*z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 280 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,.45,0,0)): \np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.3,0]],i=2..nops(p ts))],\n style=patchnogrid,color=COLOR(RGB,1,.15,.15)):\npts \+ := []: z0 := 2.4+5*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=ex p(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)] ]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.45,0,0)):\np4 := plots[po lygonplot]([seq([pts[i-1],pts[i],[2.34,4.99]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,1,.15,.15)):\npts := []: z0 := \+ 2.4-5*I:\nfor ct from 0 to 40 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:\n p5 := plot(pts,color=COLOR(RGB,.45,0,0)):\np6 := plots[polygonplot]([s eq([pts[i-1],pts[i],[2.34,-4.99]],i=2..nops(pts))],\n style=p atchnogrid,color=COLOR(RGB,1,.15,.15)):\np7 := plot([[[-5.39,0],[3.19, 0]],[[0,-5.69],[0,5.69]]],color=black,linestyle=3):\nplots[display]([p ||(1..7)],view=[-5.39..3.19,-5.69..5.69],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 386 493 493 {PLOTDATA 2 "6/-%'CURVESG6$7e\\ l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$\"3$)****** y7%pV#!#F$\"3?+++()*)Q7ZF-7$$\"3))*****HxtDd$!#E$\"3!)*******R&=$G'F-7 $$\"3o******oPM)3$!#D$\"3_+++OB)R&yF-7$$\"3#*******)*)=P)=!#C$\"3]**** *Hh\"yC%*F-7$$\"3()*****4,?N%*)FF$\"3/+++\"*)e&*4\"!#<7$$\"37+++f;65N! #B$\"3'******H(=kc7FN7$$\"3#*******>e)[=\"!#A$\"34+++A-t89FN7$$\"3**** ***>0+k`$FX$\"3))*****\\kH3d\"FN7$$\"3\"******f#)31_*FX$\"3)******4!*[ zs\"FN7$$\"33+++-@!oM#!#@$\"37++++$)4&)=FN7$$\"3o*****z0wqN&Fbo$\"32++ +9LGU?FN7$$\"3&******H%=BU6!#?$\"3#)*****R5)[*>#FN7$$\"3/+++U06*G#F]p$ \"3%)*****fjSmN#FN7$$\"3D+++Kf$)GVF]p$\"3*)*****foZN^#FN7$$\"3d******* >>^t(F]p$\"3-+++()yypEFN7$$\"3&*******y-1/8!#>$\"3&******f)\\cCGFN7$$ \"3;+++v20l?Fbq$\"3++++a7dwHFN7$$\"3/+++-l*=0$Fbq$\"3,+++SM*R7$FN7$$\" 3/+++8-n$=%Fbq$\"35+++^S\"[E$FN7$$\"3y*******=#*GI&Fbq$\"3')*****p!GN( R$FN7$$\"3!******\\,$==iFbq$\"3&******\\!zo?NFN7$$\"3!)*****HTO0w'Fbq$ \"3))*****H/nYj$FN7$$\"3i+++4z(o\"oFbq$\"3%******z%3iRPFN7$$\"3C+++bw1 MjFbq$\"37+++sH1OQFN7$$\"3/+++4`\"oI&Fbq$\"3(******4,=X#RFN7$$\"3-+++k @2iPFbq$\"3A+++eVY0SFN7$$\"39+++_(*\\XKl9%FN7$$!3=+++X*oX[$Fbq$\"3\")*****H8Uu?%FN7$$!3M* ****ROa!*e'Fbq$\"3-++++mViUFN7$$!3Y*****4-MS&**Fbq$\"3l*****fJs=J%FN7$ $!3!********GWPN\"F-$\"3g*****\\>'3cVFN7$$!3\"******f-K.t\"F-$\"3e**** *fs!R&R%FN7$$!3&******RTR@7#F-$\"3W+++c02IWFN7$$!3/+++zRkEDF-$\"3u**** *>i!QgWFN7$$!3.+++;\"e<%HF-$\"3?+++%oXl[%FN7$$!3++++\"3jdO$F-$\"3<+++l 3w3XFN7$$!3')*****p\\^sz$F-$\"3()*****4#G>FXFN7$$!3E+++0-3NUF-$\"3$*** ***f@\")>a%FN7$$!3/+++^PLyYF-$\"3O+++L-C`XFN7$$!3[+++J#)HE^F-$\"3!**** **R)*f5c%FN7$$!3]+++/^WybF-$\"3!******Rp2bc%FN7$$!3D+++5\\UMgF-$\"3s** ***HiGmc%FN7$$!3M+++682%\\'F-$\"3C+++)pYWc%FN7$$!3%******zk;u&pF-$\"37 +++>\\'*eXFN7$$!3,+++l6sCuF-$\"33+++Ec;]XFN7$$!3++++n$>l*yF-$\"3/+++]3 ,QXFN7$$!3/+++h&)ot$)F-$\"30+++^JWAXFN7$$!3]+++wlad))F-$\"3V+++CxQ.XFN 7$$!3W+++ft)*\\$*F-$\"3')*****>9d2[%FN7$$!37+++EKm`)*F-$\"3K+++\"HhWX% FN7$$!3)******>o@s.\"FN$\"3d*****H\"pUCWFN7$$!3'******zSa54\"FN$\"37++ +ETj!R%FN7$$!3++++Z5]Z6FN$\"3?+++Y1=`VFN7$$!3$******z7Tt?\"FN$\"3.+++% *[Q7VFN7$$!3#******H&\\Ur7FN$\"3q*****>rD*oUFN7$$!3#*******G[ZS8FN$\"3 c*****fNpRA%FN7$$!33+++khu99FN$\"3S+++sG:zTFN7$$!33+++vnp$\\\"FN$\"30+ ++'Qti8%FN7$$!31+++dM'fd\"FN$\"3A+++k:u'4%FN7$$!3++++Gdwf;FN$\"3M+++EG ;hSFN7$$!31+++>vWVFN$\"3A+++JI-uRFN7$$!3)******4]nR)>FN$\"3++++:,' )[RFN7$$!3)*******4\"=$f?FN$\"30+++PxRCRFN7$$!3\")*****4')o@8#FN$\"3)* *****Rw#3+RFN7$$!35+++T/i-AFN$\"3))******>\\[vQFN7$$!3%)*****RX23F#FN$ \"3A+++\"zt-&QFN7$$!3z*****HDwoL#FN$\"3y*****Hw'>CQFN7$$!3#)*****>Gt4S #FN$\"3=+++)ogqz$FN7$$!33+++9&RKY#FN$\"33+++uvroPFN7$$!36+++?$3Q_#FN$ \"3))*****fz`!RPFN7$$!3#*******=_!Ge#FN$\"31+++/@)zq$FN7$$!3z*****4i[. k#FN$\"35+++_eVvOFN7$$!3++++W5b'p#FN$\"3:+++6\\OTOFN7$$!3-+++(\\?:v#FN $\"3(******p$Ht0OFN7$$!3\"******p%=O0GFN$\"3))*****\\p:&oNFN7$$!3))*** **z+y\"eGFN$\"3++++t-qHNFN7$$!3@+++a325HFN$\"3!)*****>-&G*[$FN7$$!3'** ****HrT6'HFN$\"3))******)4!GZMFN7$$!3.+++=8\\6IFN$\"34+++!\\3PS$FN7$$! 3%*******p!>71$FN$\"35+++QtgeLFN7$$!3=+++%[@/6$FN$\"3)*******\\$H?J$FN 7$$!3?+++K(*=fJFN$\"3))*****4*R/kKFN7$$!3))*****pH1w?$FN$\"3&)******)4 QZ@$FN7$$!3?+++&*3ubKFN$\"3++++$[:U;$FN7$$!3\"******R,YOI$FN$\"3#***** *>:&f7JFN7$$!3!)*****Hk_8N$FN$\"3!******z)y+gIFN7$$!33+++#)o'))R$FN$\" 3>+++)>\"f1IFN7$$!3++++Q\"ohW$FN$\"3!******p'H[_HFN7$$!3++++[%4K\\$FN$ \"3)******\\\\9y*GFN7$$!35+++$4?*RNFN$\"3&)*****p\"QqUGFN7$$!3/+++Q-@' e$FN$\"31+++--D(y#FN7$$!3&******p'p(>j$FN$\"38+++90`JFFN7$$!3'*******[ 36xOFN$\"3%)*****ft(fvEFN7$$!34+++/A]@PFN$\"33+++\\:[>EFN7$$!3=+++Ak/l PFN$\"3'******fa!>jDFN7$$!3!*******))zk2QFN$\"3%)*****\\]:n]#FN7$$!3-+ ++JJA\\QFN$\"3-+++dJ.]CFN7$$!3))*****HA,(*)QFN$\"3++++^*4JR#FN7$$!3+++ +'>D!HRFN$\"3%)******>b!fL#FN7$$!3?+++S7:nRFN$\"3')*****zlv$yAFN7$$!3A +++X![S+%FN$\"3$******Rxu/A#FN7$$!36+++1epRSFN$\"3')******Gy:i@FN7$$!3 1+++Q^3uSFN$\"3.+++T=Q.@FN7$$!35+++))f@2TFN$\"3\")*****z02T/#FN7$$!34+ ++4m4RTFN$\"3'******>(yH%)>FN7$$!3-+++)fU(pTFN$\"3(******zcBR#>FN7$$!3 ))*****H*f<*>%FN$\"3!******p(*eH'=FN7$$!3\")*****RRCuA%FN$\"3&*******o ]Q,=FN7$$!37+++t+_aUFN$\"3-+++<&*=RaF-7$$!3S++ +[3r(e%FN$\"38+++5z4SZF-7$$!3A+++\\qm&f%FN$\"3=+++FL1hSF-7$$!3&******p XkCg%FN$\"3E+++S:z#Q$F-7$$!37+++e823YFN$\"3u*****HHU_q#F-7$$!3#******* [4Y7YFN$\"3(******fcF$G?F-7$$!3))*****f*=h:YFN$\"31+++>9#>N\"F-7$$!35+ ++@(3vh%FN$\"3W*****fy(penFbq7$$!3n*****>/U\"=YFNF(7$F^gm$!3W*****fy(p enFbq7$Fifm$!31+++>9#>N\"F-7$Fdfm$!3(******fcF$G?F-7$F_fm$!3u*****HHU_ q#F-7$Fjem$!3E+++S:z#Q$F-7$Feem$!3=+++FL1hSF-7$F`em$!38+++5z4SZF-7$F[e m$!3z*****H=y)>aF-7$Ffdm$!3%*******G3K+hF-7$Fadm$!3k*****z\"*o7y'F-7$F \\dm$!3=+++*y'[iuF-7$Fgcm$!3()******)3eO9)F-7$Fbcm$!3x*****Hx(QC))F-7$ F]cm$!3A+++7'3U]*F-7$Fhbm$!34+++P#f#=5FN7$Fcbm$!3%******H_'*e3\"FN7$F^ bm$!3!******fmsK:\"FN7$Fiam$!3,+++DqK?7FN7$Fdam$!3++++#\\+qG\"FN7$F_am $!3!******H]PKN\"FN7$Fj`m$!34+++ux)*=9FN7$Fe`m$!31+++&p2U[\"FN7$F``m$! 3#******pIh)[:FN7$F[`m$!32+++,4#Hh\"FN7$Ff_m$!3!******4Cnjn\"FN7$Fa_m$ !3-+++<&*=RFN7$F]^m$!3'******>(yH%)>FN7$Fh]m$!3\")*****z02T/#FN7$ Fc]m$!3.+++T=Q.@FN7$F^]m$!3')******Gy:i@FN7$Fi\\m$!3$******Rxu/A#FN7$F d\\m$!3')*****zlv$yAFN7$F_\\m$!3%)******>b!fL#FN7$Fj[m$!3++++^*4JR#FN7 $Fe[m$!3-+++dJ.]CFN7$F`[m$!3%)*****\\]:n]#FN7$F[[m$!3'******fa!>jDFN7$ Ffjl$!33+++\\:[>EFN7$Fajl$!3%)*****ft(fvEFN7$F\\jl$!38+++90`JFFN7$Fgil $!31+++--D(y#FN7$Fbil$!3&)*****p\"QqUGFN7$F]il$!3)******\\\\9y*GFN7$Fh hl$!3!******p'H[_HFN7$Fchl$!3>+++)>\"f1IFN7$F^hl$!3!******z)y+gIFN7$Fi gl$!3#******>:&f7JFN7$Fdgl$!3++++$[:U;$FN7$F_gl$!3&)******)4QZ@$FN7$Fj fl$!3))*****4*R/kKFN7$Fefl$!3)*******\\$H?J$FN7$F`fl$!35+++QtgeLFN7$F[ fl$!34+++!\\3PS$FN7$Ffel$!3))******)4!GZMFN7$Fael$!3!)*****>-&G*[$FN7$ F\\el$!3++++t-qHNFN7$Fgdl$!3))*****\\p:&oNFN7$Fbdl$!3(******p$Ht0OFN7$ F]dl$!3:+++6\\OTOFN7$Fhcl$!35+++_eVvOFN7$Fccl$!31+++/@)zq$FN7$F^cl$!3) )*****fz`!RPFN7$Fibl$!33+++uvroPFN7$Fdbl$!3=+++)ogqz$FN7$F_bl$!3y***** Hw'>CQFN7$Fjal$!3A+++\"zt-&QFN7$Feal$!3))******>\\[vQFN7$F`al$!3)***** *Rw#3+RFN7$F[al$!30+++PxRCRFN7$Ff`l$!3++++:,')[RFN7$Fa`l$!3A+++JI-uRFN 7$F\\`l$!3@+++&fx0+%FN7$Fg_l$!3o******G*Q$HSFN7$Fb_l$!3M+++EG;hSFN7$F] _l$!3A+++k:u'4%FN7$Fh^l$!30+++'Qti8%FN7$Fc^l$!3S+++sG:zTFN7$F^^l$!3c** ***fNpRA%FN7$Fi]l$!3q*****>rD*oUFN7$Fd]l$!3.+++%*[Q7VFN7$F_]l$!3?+++Y1 =`VFN7$Fj\\l$!37+++ETj!R%FN7$Fe\\l$!3d*****H\"pUCWFN7$F`\\l$!3K+++\"Hh WX%FN7$F[\\l$!3')*****>9d2[%FN7$Ff[l$!3V+++CxQ.XFN7$Fa[l$!30+++^JWAXFN 7$F\\[l$!3/+++]3,QXFN7$Fgz$!33+++Ec;]XFN7$Fbz$!37+++>\\'*eXFN7$F]z$!3C +++)pYWc%FN7$Fhy$!3s*****HiGmc%FN7$Fcy$!3!******Rp2bc%FN7$F^y$!3!***** *R)*f5c%FN7$Fix$!3O+++L-C`XFN7$Fdx$!3$******f@\")>a%FN7$F_x$!3()*****4 #G>FXFN7$Fjw$!3<+++l3w3XFN7$Few$!3?+++%oXl[%FN7$F`w$!3u*****>i!QgWFN7$ F[w$!3W+++c02IWFN7$Ffv$!3e*****fs!R&R%FN7$Fav$!3g*****\\>'3cVFN7$F\\v$ !3l*****fJs=J%FN7$Fgu$!3-++++mViUFN7$Fbu$!3\")*****H8Uu?%FN7$$!3M+++%) Hn')oF]p$!3G+++!>Kl9%FN7$Fht$!37+++]ZLzSFN7$Fct$!3A+++eVY0SFN7$F^t$!3( ******4,=X#RFN7$Fis$!37+++sH1OQFN7$Fds$!3%******z%3iRPFN7$F_s$!3))**** *H/nYj$FN7$Fjr$!3&******\\!zo?NFN7$Fer$!3')*****p!GN(R$FN7$F`r$!35+++^ S\"[E$FN7$F[r$!3,+++SM*R7$FN7$Ffq$!3++++a7dwHFN7$F`q$!3&******f)\\cCGF N7$F[q$!3-+++()yypEFN7$Ffp$!3*)*****foZN^#FN7$Fap$!3%)*****fjSmN#FN7$F [p$!3#)*****R5)[*>#FN7$Ffo$!32+++9LGU?FN7$F`o$!37++++$)4&)=FN7$F[o$!3) ******4!*[zs\"FN7$$\"3J+++`+SONFX$!3))*****\\kH3d\"FN7$FV$!34+++A-t89F N7$FP$!3'******H(=kc7FN7$$\"3Q+++1+_V*)FF$!3/+++\"*)e&*4\"FN7$$\"3'*** ***z*)=P)=FF$!3]*****Hh\"yC%*F-7$F>$!3_+++OB)R&yF-7$$\"3++++gPdsNF:$!3 !)*******R&=$G'F-7$$\"3u*****4CTpV#F4$!3?+++()*)Q7ZF-7$F($!3)******Rl# fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#X!\"#F(F(-%)POLYGONSG6f \\l7%F'7$F($\"+Fjzq:!#57$$FR!\"\"F(7%F]co7$F($\"+aEfTJF`coFaco7%Feco7$ $\"+z7%pV#Fbq$\"+()*)Q7ZF`coFaco7%Fico7$$\"+tPdsNF-$\"++a=$G'F`coFaco7 %F_do7$$\"+pPM)3$FN$\"+OB)R&yF`coFaco7%Fedo7$$\"+**)=P)=!#;$\"+8;yC%*F `coFaco7%F[eo7$$\"+6+_V*)F^eo$\"+\"*)e&*4\"!\"*Faco7%Fbeo7$$\"+f;65N!# :$\"+t=kc7FgeoFaco7%Fieo7$$\"+?e)[=\"!#9$\"+A-t89FgeoFaco7%F`fo7$$\"+_ +SONFcfo$\"+X'H3d\"FgeoFaco7%Fgfo7$$\"+E)31_*Fcfo$\"+,*[zs\"FgeoFaco7% F]go7$$\"+-@!oM#!#8$\"++$)4&)=FgeoFaco7%Fcgo7$$\"+eg2d`Ffgo$\"+9LGU?Fg eoFaco7%Fjgo7$$\"+V=BU6!#7$\"+/\")[*>#FgeoFaco7%F`ho7$$\"+U06*G#Fcho$ \"+O1kcBFgeoFaco7%Fgho7$$\"+Kf$)GVFcho$\"+'oZN^#FgeoFaco7%F]io7$$\"++# >^t(Fcho$\"+()yypEFgeoFaco7%Fcio7$$\"+z-1/8!#6$\"+')\\cCGFgeoFaco7%Fii o7$$\"+v20l?F\\jo$\"+a7dwHFgeoFaco7%F`jo7$$\"+-l*=0$F\\jo$\"+SM*R7$Fge oFaco7%Ffjo7$$\"+8-n$=%F\\jo$\"+^S\"[E$FgeoFaco7%F\\[p7$$\"+!>#*GI&F\\ jo$\"+2GN(R$FgeoFaco7%Fb[p7$$\"+:I==iF\\jo$\"+0zo?NFgeoFaco7%Fh[p7$$\" +8k`gnF\\jo$\"+VqmMOFgeoFaco7%F^\\p7$$\"+4z(o\"oF\\jo$\"+[3iRPFgeoFaco 7%Fd\\p7$$\"+bw1MjF\\jo$\"+sH1OQFgeoFaco7%Fj\\p7$$\"+4`\"oI&F\\jo$\"+6 !=X#RFgeoFaco7%F`]p7$$\"+k@2iPF\\jo$\"+eVY0SFgeoFaco7%Ff]p7$$\"+_(*\\X Kl9%FgeoFaco7%Fb^p 7$$!+X*oX[$F\\jo$\"+L@W2UFgeoFaco7%Fh^p7$$!+kV0*e'F\\jo$\"++mViUFgeoFa co7%F^_p7$$!+@S.a**F\\jo$\"+;B(=J%FgeoFaco7%Fd_p7$$!+!HWPN\"F`co$\"+&> '3cVFgeoFaco7%Fj_p7$$!+E?LIFXFgeoFaco7%F^bp7$$!+0-3NUF`co$\"+;7)>a%FgeoFa co7%Fdbp7$$!+^PLyYF`co$\"+L-C`XFgeoFaco7%Fjbp7$$!+J#)HE^F`co$\"+%)*f5c %FgeoFaco7%F`cp7$$!+/^WybF`co$\"+%p2bc%FgeoFaco7%Ffcp7$$!+5\\UMgF`co$ \"+B'Gmc%FgeoFaco7%F\\dp7$$!+682%\\'F`co$\"+)pYWc%FgeoFaco7%Fbdp7$$!+[ mTdpF`co$\"+>\\'*eXFgeoFaco7%Fhdp7$$!+l6sCuF`co$\"+Ec;]XFgeoFaco7%F^ep 7$$!+n$>l*yF`co$\"+]3,QXFgeoFaco7%Fdep7$$!+h&)ot$)F`co$\"+^JWAXFgeoFac o7%Fjep7$$!+wlad))F`co$\"+CxQ.XFgeoFaco7%F`fp7$$!+ft)*\\$*F`co$\"+Urv! [%FgeoFaco7%Fffp7$$!+EKm`)*F`co$\"+\"HhWX%FgeoFaco7%F\\gp7$$!+#o@s.\"F geo$\"+8pUCWFgeoFaco7%Fbgp7$$!+3W0\"4\"Fgeo$\"+ETj!R%FgeoFaco7%Fhgp7$$ !+Z5]Z6Fgeo$\"+Y1=`VFgeoFaco7%F^hp7$$!+G6M27Fgeo$\"+%*[Q7VFgeoFaco7%Fd hp7$$!+`\\Ur7Fgeo$\"+7d#*oUFgeoFaco7%Fjhp7$$!+H[ZS8Fgeo$\"+c$pRA%FgeoF aco7%F`ip7$$!+khu99Fgeo$\"+sG:zTFgeoFaco7%Ffip7$$!+vnp$\\\"Fgeo$\"+'Qt i8%FgeoFaco7%F\\jp7$$!+dM'fd\"Fgeo$\"+k:u'4%FgeoFaco7%Fbjp7$$!+Gdwf;Fg eo$\"+EG;hSFgeoFaco7%Fhjp7$$!+>vWVFgeo$\"+JI-uRFgeoFaco7%Fj[ q7$$!+,v'R)>Fgeo$\"+:,')[RFgeoFaco7%F`\\q7$$!+5\"=$f?Fgeo$\"+PxRCRFgeo Faco7%Ff\\q7$$!+h)o@8#Fgeo$\"+kF3+RFgeoFaco7%F\\]q7$$!+T/i-AFgeo$\"+? \\[vQFgeoFaco7%Fb]q7$$!+au!3F#Fgeo$\"+\"zt-&QFgeoFaco7%Fh]q7$$!+`i(oL# Fgeo$\"+jn>CQFgeoFaco7%F^^q7$$!+#Gt4S#Fgeo$\"+)ogqz$FgeoFaco7%Fd^q7$$! +9&RKY#Fgeo$\"+uvroPFgeoFaco7%Fj^q7$$!+?$3Q_#Fgeo$\"+'z`!RPFgeoFaco7%F `_q7$$!+>_!Ge#Fgeo$\"+/@)zq$FgeoFaco7%Ff_q7$$!+@'[.k#Fgeo$\"+_eVvOFgeo Faco7%F\\`q7$$!+W5b'p#Fgeo$\"+6\\OTOFgeoFaco7%Fb`q7$$!+(\\?:v#Fgeo$\"+ PHt0OFgeoFaco7%Fh`q7$$!+Z=O0GFgeo$\"+&p:&oNFgeoFaco7%F^aq7$$!+3!y\"eGF geo$\"+t-qHNFgeoFaco7%Fdaq7$$!+a325HFgeo$\"+A]G*[$FgeoFaco7%Fjaq7$$!+8 <9hHFgeo$\"+*4!GZMFgeoFaco7%F`bq7$$!+=8\\6IFgeo$\"+!\\3PS$FgeoFaco7%Ff bq7$$!+q!>71$Fgeo$\"+QtgeLFgeoFaco7%F\\cq7$$!+%[@/6$Fgeo$\"+]$H?J$Fgeo Faco7%Fbcq7$$!+K(*=fJFgeo$\"+\"*R/kKFgeoFaco7%Fhcq7$$!+(H1w?$Fgeo$\"+* 4QZ@$FgeoFaco7%F^dq7$$!+&*3ubKFgeo$\"+$[:U;$FgeoFaco7%Fddq7$$!+9gk.LFg eo$\"+_^f7JFgeoFaco7%Fjdq7$$!+VEN^LFgeo$\"+))y+gIFgeoFaco7%F`eq7$$!+#) o'))R$Fgeo$\"+)>\"f1IFgeoFaco7%Ffeq7$$!+Q\"ohW$Fgeo$\"+nH[_HFgeoFaco7% F\\fq7$$!+[%4K\\$Fgeo$\"+&\\9y*GFgeoFaco7%Fbfq7$$!+$4?*RNFgeo$\"+j$Fgeo$ \"+90`JFFgeoFaco7%Fdgq7$$!+\\36xOFgeo$\"+OxfvEFgeoFaco7%Fjgq7$$!+/A]@P Fgeo$\"+\\:[>EFgeoFaco7%F`hq7$$!+Ak/lPFgeo$\"+Y0>jDFgeoFaco7%Ffhq7$$!+ *)zk2QFgeo$\"+0br1DFgeoFaco7%F\\iq7$$!+JJA\\QFgeo$\"+dJ.]CFgeoFaco7%Fb iq7$$!+B7q*)QFgeo$\"+^*4JR#FgeoFaco7%Fhiq7$$!+'>D!HRFgeo$\"+?b!fL#Fgeo Faco7%F^jq7$$!+S7:nRFgeo$\"+ecPyAFgeoFaco7%Fdjq7$$!+X![S+%Fgeo$\"+uZZ? AFgeoFaco7%Fjjq7$$!+1epRSFgeo$\"+Hy:i@FgeoFaco7%F`[r7$$!+Q^3uSFgeo$\"+ T=Q.@FgeoFaco7%Ff[r7$$!+))f@2TFgeo$\"+eq5W?FgeoFaco7%F\\\\r7$$!+4m4RTF geo$\"+syH%)>FgeoFaco7%Fb\\r7$$!+)fU(pTFgeo$\"+oN#R#>FgeoFaco7%Fh\\r7$ $!+$*f<*>%Fgeo$\"+x*eH'=FgeoFaco7%F^]r7$$!+%RCuA%Fgeo$\"+p]Q,=FgeoFaco 7%Fd]r7$$!+t+_aUFgeo$\"+<&*=RaF`coFaco7%Ffdr7$$!+[3r(e%Fgeo$\"+5z4SZ F`coFaco7%F\\er7$$!+\\qm&f%Fgeo$\"+FL1hSF`coFaco7%Fber7$$!+dWY-YFgeo$ \"+S:z#Q$F`coFaco7%Fher7$$!+e823YFgeo$\"+$HU_q#F`coFaco7%F^fr7$$!+\\4Y 7YFgeo$\"+mvKG?F`coFaco7%Fdfr7$$!+'*=h:YFgeo$\"+>9#>N\"F`coFaco7%Fjfr7 $$!+@(3vh%Fgeo$\"+'y(penF\\joFaco7%F`gr7$$!+U?9=YFgeoF(Faco7%Ffgr7$Fag r$!+'y(penF\\joFaco7%Fjgr7$F[gr$!+>9#>N\"F`coFaco7%F^hr7$Fefr$!+mvKG?F `coFaco7%Fbhr7$F_fr$!+$HU_q#F`coFaco7%Ffhr7$Fier$!+S:z#Q$F`coFaco7%Fjh r7$Fcer$!+FL1hSF`coFaco7%F^ir7$F]er$!+5z4SZF`coFaco7%Fbir7$Fgdr$!+$=y) >aF`coFaco7%Ffir7$Fadr$!+H3K+hF`coFaco7%Fjir7$F[dr$!+=*o7y'F`coFaco7%F ^jr7$Fecr$!+*y'[iuF`coFaco7%Fbjr7$F_cr$!+*3eO9)F`coFaco7%Ffjr7$Fibr$!+ txQC))F`coFaco7%Fjjr7$Fcbr$!+7'3U]*F`coFaco7%F^[s7$F]br$!+P#f#=5FgeoFa co7%Fb[s7$Fgar$!+Bl*e3\"FgeoFaco7%Ff[s7$Faar$!+mEF`6FgeoFaco7%Fj[s7$F[ ar$!+DqK?7FgeoFaco7%F^\\s7$Fe`r$!+#\\+qG\"FgeoFaco7%Fb\\s7$F_`r$!+.vB` 8FgeoFaco7%Ff\\s7$Fi_r$!+ux)*=9FgeoFaco7%Fj\\s7$Fc_r$!+&p2U[\"FgeoFaco 7%F^]s7$F]_r$!+28')[:FgeoFaco7%Fb]s7$Fg^r$!+,4#Hh\"FgeoFaco7%Ff]s7$Fa^ r$!+TsOw;FgeoFaco7%Fj]s7$F[^r$!+<&*=RFgeoFaco7%Fj^s7 $Fc\\r$!+syH%)>FgeoFaco7%F^_s7$F]\\r$!+eq5W?FgeoFaco7%Fb_s7$Fg[r$!+T=Q .@FgeoFaco7%Ff_s7$Fa[r$!+Hy:i@FgeoFaco7%Fj_s7$F[[r$!+uZZ?AFgeoFaco7%F^ `s7$Fejq$!+ecPyAFgeoFaco7%Fb`s7$F_jq$!+?b!fL#FgeoFaco7%Ff`s7$Fiiq$!+^* 4JR#FgeoFaco7%Fj`s7$Fciq$!+dJ.]CFgeoFaco7%F^as7$F]iq$!+0br1DFgeoFaco7% Fbas7$Fghq$!+Y0>jDFgeoFaco7%Ffas7$Fahq$!+\\:[>EFgeoFaco7%Fjas7$F[hq$!+ OxfvEFgeoFaco7%F^bs7$Fegq$!+90`JFFgeoFaco7%Fbbs7$F_gq$!+--D(y#FgeoFaco 7%Ffbs7$Fifq$!+\"f1IFgeoFaco7%Ffcs7$Faeq$!+))y+gIFge oFaco7%Fjcs7$F[eq$!+_^f7JFgeoFaco7%F^ds7$Fedq$!+$[:U;$FgeoFaco7%Fbds7$ F_dq$!+*4QZ@$FgeoFaco7%Ffds7$Ficq$!+\"*R/kKFgeoFaco7%Fjds7$Fccq$!+]$H? J$FgeoFaco7%F^es7$F]cq$!+QtgeLFgeoFaco7%Fbes7$Fgbq$!+!\\3PS$FgeoFaco7% Ffes7$Fabq$!+*4!GZMFgeoFaco7%Fjes7$F[bq$!+A]G*[$FgeoFaco7%F^fs7$Feaq$! +t-qHNFgeoFaco7%Fbfs7$F_aq$!+&p:&oNFgeoFaco7%Fffs7$Fi`q$!+PHt0OFgeoFac o7%Fjfs7$Fc`q$!+6\\OTOFgeoFaco7%F^gs7$F]`q$!+_eVvOFgeoFaco7%Fbgs7$Fg_q $!+/@)zq$FgeoFaco7%Ffgs7$Fa_q$!+'z`!RPFgeoFaco7%Fjgs7$F[_q$!+uvroPFgeo Faco7%F^hs7$Fe^q$!+)ogqz$FgeoFaco7%Fbhs7$F_^q$!+jn>CQFgeoFaco7%Ffhs7$F i]q$!+\"zt-&QFgeoFaco7%Fjhs7$Fc]q$!+?\\[vQFgeoFaco7%F^is7$F]]q$!+kF3+R FgeoFaco7%Fbis7$Fg\\q$!+PxRCRFgeoFaco7%Ffis7$Fa\\q$!+:,')[RFgeoFaco7%F jis7$F[\\q$!+JI-uRFgeoFaco7%F^js7$Fe[q$!+&fx0+%FgeoFaco7%Fbjs7$F_[q$!+ H*Q$HSFgeoFaco7%Ffjs7$Fijp$!+EG;hSFgeoFaco7%Fjjs7$Fcjp$!+k:u'4%FgeoFac o7%F^[t7$F]jp$!+'Qti8%FgeoFaco7%Fb[t7$Fgip$!+sG:zTFgeoFaco7%Ff[t7$Faip $!+c$pRA%FgeoFaco7%Fj[t7$F[ip$!+7d#*oUFgeoFaco7%F^\\t7$Fehp$!+%*[Q7VFg eoFaco7%Fb\\t7$F_hp$!+Y1=`VFgeoFaco7%Ff\\t7$Figp$!+ETj!R%FgeoFaco7%Fj \\t7$Fcgp$!+8pUCWFgeoFaco7%F^]t7$F]gp$!+\"HhWX%FgeoFaco7%Fb]t7$Fgfp$!+ Urv![%FgeoFaco7%Ff]t7$Fafp$!+CxQ.XFgeoFaco7%Fj]t7$F[fp$!+^JWAXFgeoFaco 7%F^^t7$Feep$!+]3,QXFgeoFaco7%Fb^t7$F_ep$!+Ec;]XFgeoFaco7%Ff^t7$Fidp$! +>\\'*eXFgeoFaco7%Fj^t7$Fcdp$!+)pYWc%FgeoFaco7%F^_t7$F]dp$!+B'Gmc%Fgeo Faco7%Fb_t7$Fgcp$!+%p2bc%FgeoFaco7%Ff_t7$Facp$!+%)*f5c%FgeoFaco7%Fj_t7 $F[cp$!+L-C`XFgeoFaco7%F^`t7$Febp$!+;7)>a%FgeoFaco7%Fb`t7$F_bp$!+@G>FX FgeoFaco7%Ff`t7$Fiap$!+l3w3XFgeoFaco7%Fj`t7$Fcap$!+%oXl[%FgeoFaco7%F^a t7$F]ap$!+A1QgWFgeoFaco7%Fbat7$Fg`p$!+c02IWFgeoFaco7%Ffat7$Fa`p$!+E2R& R%FgeoFaco7%Fjat7$F[`p$!+&>'3cVFgeoFaco7%F^bt7$Fe_p$!+;B(=J%FgeoFaco7% Fbbt7$F__p$!++mViUFgeoFaco7%Ffbt7$Fi^p$!+L@W2UFgeoFaco7%Fjbt7$$!+%)Hn' )oFcho$!+!>Kl9%FgeoFaco7%F^ct7$F]^p$!+]ZLzSFgeoFaco7%Fdct7$Fg]p$!+eVY0 SFgeoFaco7%Fhct7$Fa]p$!+6!=X#RFgeoFaco7%F\\dt7$F[]p$!+sH1OQFgeoFaco7%F `dt7$Fe\\p$!+[3iRPFgeoFaco7%Fddt7$F_\\p$!+VqmMOFgeoFaco7%Fhdt7$Fi[p$!+ 0zo?NFgeoFaco7%F\\et7$Fc[p$!+2GN(R$FgeoFaco7%F`et7$F][p$!+^S\"[E$FgeoF aco7%Fdet7$Fgjo$!+SM*R7$FgeoFaco7%Fhet7$Fajo$!+a7dwHFgeoFaco7%F\\ft7$F jio$!+')\\cCGFgeoFaco7%F`ft7$Fdio$!+()yypEFgeoFaco7%Fdft7$F^io$!+'oZN^ #FgeoFaco7%Fhft7$Fhho$!+O1kcBFgeoFaco7%F\\gt7$Faho$!+/\")[*>#FgeoFaco7 %F`gt7$F[ho$!+9LGU?FgeoFaco7%Fdgt7$Fdgo$!++$)4&)=FgeoFaco7%Fhgt7$F^go$ !+,*[zs\"FgeoFaco7%F\\ht7$$\"+`+SONFcfo$!+X'H3d\"FgeoFaco7%F`ht7$Fafo$ !+A-t89FgeoFaco7%Ffht7$Fjeo$!+t=kc7FgeoFaco7%Fjht7$$\"+1+_V*)F^eo$!+\" *)e&*4\"FgeoFaco7%F^it7$$\"+)*)=P)=F^eo$!+8;yC%*F`coFaco7%Fdit7$Ffdo$! +OB)R&yF`coFaco7%Fjit7$$\"+gPdsNF-$!++a=$G'F`coFaco7%F^jt7$$\"+T7%pV#F bq$!+()*)Q7ZF`coFaco7%Fdjt7$F($!+aEfTJF`coFaco7%Fjjt7$F($!+Fjzq:F`coFa co7%F^[uF'Faco-Fcbo6&Febo$\"\"\"F)$\"#:FhboFf[u-%&STYLEG6#%,PATCHNOGRI DG-F$6$7K7$$\"3&******pgoxS#FN$\"3s*****\\u\\`)\\FN7$$\"3#*******4Lv2C FN$\"39+++E3?%*\\FN7$$\"3>+++ar\\1CFN$\"3P+++Dw'G+&FN7$$\"3%)*****pNbS S#FN$\"3%)*****\\1,7,&FN7$$\"3<+++0X\\+CFN$\"3j*****zui!>]FN7$$\"3++++ %Q\"*eR#FN$\"3\"******49Cj-&FN7$$\"3;+++?BL!R#FN$\"3T+++ie'G.&FN7$$\"3 =+++TH\"RQ#FN$\"3&)*****f7x&Q]FN7$$\"3%)******>$QnP#FN$\"3x*****pVbL/& FN7$$\"3++++GM#*oBFN$\"3Q+++fl5Z]FN7$$\"3y*****\\'QfgBFN$\"3-+++v[u\\] FN7$$\"3A+++9o)=N#FN$\"3')*****R=%>^]FN7$$\"32+++O@&HM#FN$\"3W+++(3*Q^ ]FN7$$\"3$)*****>_`RL#FN$\"3o*****p1x-0&FN7$$\"3()*****H^p]K#FN$\"3k** ***pV@y/&FN7$$\"3?+++_T\\;BFN$\"3d*****RK0S/&FN7$$\"3,+++loV3BFN$\"3e* ****H#o$)Q]FN7$$\"3'******R-@6I#FN$\"3z*****\\QbB.&FN7$$\"35+++\\/y%H# FN$\"3v*****>IRY-&FN7$$\"37+++FHl*G#FN$\"37+++6Q\"e,&FN7$$\"3++++T)pfG #FN$\"3D+++h(eg+&FN7$$\"37+++V7%RG#FN$\"3!)*****p49c*\\FN7$$\"3#****** *flt$G#FN$\"3u*****fn!y%)\\FN7$$\"3#)*****f`iaG#FN$\"3#)*****Rg8R(\\FN 7$$\"39+++0>9*G#FN$\"3')*****4*eSj\\FN7$$\"3-+++yyp%H#FN$\"3Q+++YAm`\\ FN7$$\"3&*******p'\\>I#FN$\"3G+++Uf1X\\FN7$$\"3!******pDA1J#FN$\"3!*** ***f\"[%z$\\FN7$$\"3()*****pap.K#FN$\"3d*****HoVD$\\FN7$$\"3%)*****R21 3L#FN$\"36+++(*)3!H\\FN7$$\"3z*****R>S:M#FN$\"3K+++cpQF\\FN7$$\"3'**** **fS-AN#FN$\"3T+++*)\\jF\\FN7$$\"3))*****\\8lCO#FN$\"3L+++w&Q'H\\FN7$$ \"3/+++JL0sBFN$\"3\"******R(>BL\\FN7$$\"3?+++otu!Q#FN$\"3i*****H3<#Q\\ FN7$$\"39+++L1Q)Q#FN$\"3n*****p4zV%\\FN7$$\"3#******H6M[R#FN$\"3M+++I# )\\^\\FN7$$\"3)******HxH+S#FN$\"3e*****4yd$f\\FN7$$\"3'******>)Q#RS#FN $\"3q*****f?\\x'\\FN7$$\"3;+++f4]1CFN$\"3g*****\\&\\Zw\\FNF_\\uFbbo-Fj bo6L7%7$$\"+2'oxS#Fgeo$\"+X(\\`)\\Fgeo7$$\"+5Lv2CFgeo$\"+E3?%*\\Fgeo7$ $\"$M#Fhbo$\"$*\\Fhbo7%F_iu7$$\"+ar\\1CFgeo$\"+Dw'G+&FgeoFdiu7%Fjiu7$$ \"+d`0/CFgeo$\"+l5?6]FgeoFdiu7%F`ju7$$\"+0X\\+CFgeo$\"+[F1>]FgeoFdiu7% Ffju7$$\"+%Q\"*eR#Fgeo$\"+TTKE]FgeoFdiu7%F\\[v7$$\"+?BL!R#Fgeo$\"+ie'G .&FgeoFdiu7%Fb[v7$$\"+TH\"RQ#Fgeo$\"+ErdQ]FgeoFdiu7%Fh[v7$$\"+?$QnP#Fg eo$\"+PaNV]FgeoFdiu7%F^\\v7$$\"+GM#*oBFgeo$\"+fl5Z]FgeoFdiu7%Fd\\v7$$ \"+lQfgBFgeo$\"+v[u\\]FgeoFdiu7%Fj\\v7$$\"+9o)=N#Fgeo$\"+%=%>^]FgeoFdi u7%F`]v7$$\"+O@&HM#Fgeo$\"+(3*Q^]FgeoFdiu7%Ff]v7$$\"+AN&RL#Fgeo$\"+nqF ]]FgeoFdiu7%F\\^v7$$\"+8&p]K#Fgeo$\"+P9#y/&FgeoFdiu7%Fb^v7$$\"+_T\\;BF geo$\"+C`+W]FgeoFdiu7%Fh^v7$$\"+loV3BFgeo$\"+Bo$)Q]FgeoFdiu7%F^_v7$$\" +C57,BFgeo$\"+&QbB.&FgeoFdiu7%Fd_v7$$\"+\\/y%H#Fgeo$\"+-$RY-&FgeoFdiu7 %Fj_v7$$\"+FHl*G#Fgeo$\"+6Q\"e,&FgeoFdiu7%F``v7$$\"+T)pfG#Fgeo$\"+h(eg +&FgeoFdiu7%Ff`v7$$\"+V7%RG#Fgeo$\"+(49c*\\FgeoFdiu7%F\\av7$$\"+glt$G# Fgeo$\"+w1y%)\\FgeoFdiu7%Fbav7$$\"+ODY&G#Fgeo$\"+/O\"R(\\FgeoFdiu7%Fha v7$$\"+0>9*G#Fgeo$\"+\"*eSj\\FgeoFdiu7%F^bv7$$\"+yyp%H#Fgeo$\"+YAm`\\F geoFdiu7%Fdbv7$$\"+q'\\>I#Fgeo$\"+Uf1X\\FgeoFdiu7%Fjbv7$$\"+dAi5BFgeo$ \"+;[%z$\\FgeoFdiu7%F`cv7$$\"+Z&p.K#Fgeo$\"+$oVD$\\FgeoFdiu7%Ffcv7$$\" +ug!3L#Fgeo$\"+(*)3!H\\FgeoFdiu7%F\\dv7$$\"+%>S:M#Fgeo$\"+cpQF\\FgeoFd iu7%Fbdv7$$\"+1C?_BFgeo$\"+*)\\jF\\FgeoFdiu7%Fhdv7$$\"+N^YiBFgeo$\"+w& Q'H\\FgeoFdiu7%F^ev7$$\"+JL0sBFgeo$\"+u>BL\\FgeoFdiu7%Fdev7$$\"+otu!Q# Fgeo$\"+$3<#Q\\FgeoFdiu7%Fjev7$$\"+L1Q)Q#Fgeo$\"+(4zV%\\FgeoFdiu7%F`fv 7$$\"+8T$[R#Fgeo$\"+I#)\\^\\FgeoFdiu7%Fffv7$$\"+t(H+S#Fgeo$\"+\"yd$f\\ FgeoFdiu7%F\\gv7$$\"+#)Q#RS#Fgeo$\"+1#\\x'\\FgeoFdiu7%Fbgv7$$\"+f4]1CF geo$\"+b\\Zw\\FgeoFdiu7%FhgvFjhuFdiuFb[uFh[u-F$6$7K7$F`\\u$!3s*****\\u \\`)\\FN7$Fchu$!3g*****\\&\\Zw\\FN7$F^hu$!3q*****f?\\x'\\FN7$Figu$!3e* ****4yd$f\\FN7$Fdgu$!3M+++I#)\\^\\FN7$F_gu$!3n*****p4zV%\\FN7$Fjfu$!3i *****H3<#Q\\FN7$Fefu$!3\"******R(>BL\\FN7$F`fu$!3L+++w&Q'H\\FN7$F[fu$! 3T+++*)\\jF\\FN7$Ffeu$!3K+++cpQF\\FN7$Faeu$!36+++(*)3!H\\FN7$F\\eu$!3d *****HoVD$\\FN7$Fgdu$!3!******f\"[%z$\\FN7$Fbdu$!3G+++Uf1X\\FN7$F]du$! 3Q+++YAm`\\FN7$Fhcu$!3')*****4*eSj\\FN7$Fccu$!3#)*****Rg8R(\\FN7$F^cu$ !3u*****fn!y%)\\FN7$Fibu$!3!)*****p49c*\\FN7$Fdbu$!3D+++h(eg+&FN7$F_bu $!37+++6Q\"e,&FN7$Fjau$!3v*****>IRY-&FN7$Feau$!3z*****\\QbB.&FN7$F`au$ !3e*****H#o$)Q]FN7$F[au$!3d*****RK0S/&FN7$Ff`u$!3k*****pV@y/&FN7$Fa`u$ !3o*****p1x-0&FN7$F\\`u$!3W+++(3*Q^]FN7$Fg_u$!3')*****R=%>^]FN7$Fb_u$! 3-+++v[u\\]FN7$F]_u$!3Q+++fl5Z]FN7$Fh^u$!3x*****pVbL/&FN7$Fc^u$!3&)*** **f7x&Q]FN7$F^^u$!3T+++ie'G.&FN7$Fi]u$!3\"******49Cj-&FN7$Fd]u$!3j**** *zui!>]FN7$F_]u$!3%)*****\\1,7,&FN7$Fj\\u$!3P+++Dw'G+&FN7$Fe\\u$!39+++ E3?%*\\FNFahvFbbo-Fjbo6L7%7$F[iu$!+X(\\`)\\Fgeo7$Figv$!+b\\Zw\\Fgeo7$F eiu$!$*\\Fhbo7%F_`w7$Fcgv$!+1#\\x'\\FgeoFb`w7%Ff`w7$F]gv$!+\"yd$f\\Fge oFb`w7%Fj`w7$Fgfv$!+I#)\\^\\FgeoFb`w7%F^aw7$Fafv$!+(4zV%\\FgeoFb`w7%Fb aw7$F[fv$!+$3<#Q\\FgeoFb`w7%Ffaw7$Feev$!+u>BL\\FgeoFb`w7%Fjaw7$F_ev$!+ w&Q'H\\FgeoFb`w7%F^bw7$Fidv$!+*)\\jF\\FgeoFb`w7%Fbbw7$Fcdv$!+cpQF\\Fge oFb`w7%Ffbw7$F]dv$!+(*)3!H\\FgeoFb`w7%Fjbw7$Fgcv$!+$oVD$\\FgeoFb`w7%F^ cw7$Facv$!+;[%z$\\FgeoFb`w7%Fbcw7$F[cv$!+Uf1X\\FgeoFb`w7%Ffcw7$Febv$!+ YAm`\\FgeoFb`w7%Fjcw7$F_bv$!+\"*eSj\\FgeoFb`w7%F^dw7$Fiav$!+/O\"R(\\Fg eoFb`w7%Fbdw7$Fcav$!+w1y%)\\FgeoFb`w7%Ffdw7$F]av$!+(49c*\\FgeoFb`w7%Fj dw7$Fg`v$!+h(eg+&FgeoFb`w7%F^ew7$Fa`v$!+6Q\"e,&FgeoFb`w7%Fbew7$F[`v$!+ -$RY-&FgeoFb`w7%Ffew7$Fe_v$!+&QbB.&FgeoFb`w7%Fjew7$F__v$!+Bo$)Q]FgeoFb `w7%F^fw7$Fi^v$!+C`+W]FgeoFb`w7%Fbfw7$Fc^v$!+P9#y/&FgeoFb`w7%Fffw7$F]^ v$!+nqF]]FgeoFb`w7%Fjfw7$Fg]v$!+(3*Q^]FgeoFb`w7%F^gw7$Fa]v$!+%=%>^]Fge oFb`w7%Fbgw7$F[]v$!+v[u\\]FgeoFb`w7%Ffgw7$Fe\\v$!+fl5Z]FgeoFb`w7%Fjgw7 $F_\\v$!+PaNV]FgeoFb`w7%F^hw7$Fi[v$!+ErdQ]FgeoFb`w7%Fbhw7$Fc[v$!+ie'G. &FgeoFb`w7%Ffhw7$F][v$!+TTKE]FgeoFb`w7%Fjhw7$Fgju$!+[F1>]FgeoFb`w7%F^i w7$Faju$!+l5?6]FgeoFb`w7%Fbiw7$F[ju$!+Dw'G+&FgeoFb`w7%Ffiw7$F`iu$!+E3? %*\\FgeoFb`w7%FjiwF\\`wFb`wFb[uFh[u-F$6%7$7$$!3o*************Q&FNF(7$$ \"3%**************=$FNF(-%'COLOURG6&FeboF)F)F)-%*LINESTYLEG6#\"\"$-F$6 %7$7$F($!3Q++++++!p&FN7$F($\"3Q++++++!p&FNFgjwFjjw-%(SCALINGG6#%,CONST RAINEDG-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6 %%&Re(z)G%&Im(z)G-F`\\x6#%(DEFAULTG-%%VIEWG6$;$!$R&Fhbo$\"$>$Fhbo;$!$p &Fhbo$\"$p&Fhbo" 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 stab ility region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For t his scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.6181, 0];" "6#7$,$-%&FloatG6$\"&\"=Y!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th roo t of the real part of points along the curve. In this way we see that \+ the largest interval on the nonnegative imaginary axis that contains t he origin and lies inside the stability region is " }{XPPEDIT 18 0 "[ 0,4.14]" "6#7$\"\"!-%&FloatG6$\"$9%!\"#" }{TEXT -1 18 " approximately . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 728 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/7 20*z^6+1/5040*z^7+\n 774333709804587831348962946740621016338915581/31 279702004887783796987510183013642733846312176000*z^8+\n 1884329544261 3296238422090690181682936671658681/88909424978693036664557298944197978 10668475772906240*z^9-\n 12521022364387928448997271307505195498561543 751/111136781223366295830696623680247472633355947161328000*z^10+\n 25 38674058747206930957524147864115070976/1978254963106742278827335093419 76163123283968375*z^11:\nDigits := 18:\npts := []: z0 := 0:\nfor ct fr om 0 to 160 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=red,thickness=2,font=[HELVETICA,9]);\nDigits := 10:\n" }}{PARA 13 "" 1 "" {GLPLOT2D 243 279 279 {PLOTDATA 2 "6(-%'CURVESG6#7]u7$$\"\" !F)F(7$$\"3Kjfp!\\N+u\"!#>$\"3C$z*e`EfTJF-7$$\"3p\")R/2*fV)GF-$\"3Q(ez rI&=$G'F-7$$\"31ArZ-t\"=)QF-$\"3O9%p2'zxC%*F-7$$\"3&)y%)*p.n!*z%F-$\"3 ZjfVhqjc7!#=7$$\"36#Gym2m]m&F-$\"3+T_zEjzq:F?7$$\"31(**[u:+f\\'F-$\"3& \\lb@fb\\)=F?7$$\"3^UVRBIn,tF-$\"3b))*>v&[6*>#F?7$$\"3sK`*Ry&>*3)F-$\" 3w'f&*G7uK^#F?7$$\"3sEpqtNEj))F-$\"3>@&*H)QLu#GF?7$$\"39#4\"o1vMF'*F-$ \"3Eruw`EfTJF?7$$\"3z1#H@u+%Q5F?$\"3Dv&o$>>vbMF?7$$\"3MRtz*e:N6\"F?$\" 3\"[OG_=6*pPF?7$$\"3a=9?#eB#)=\"F?$\"3L%4j:XqS3%F?7$$\"3N$yrlSMEE\"F?$ \"397#H(=(H#)R%F?7$$\"3ZM$**Q(>$oL\"F?$\"3T\"f#H()*)Q7ZF?7$$\"3`#*\\\" *>3)3T\"F?$\"3X!*H7e#[l-&F?7$$\"3vN7\\+3$[[\"F?$\"3eP,_KvqS`F?7$$\"3.J Vuj0se:F?$\"3)R\")yB\"o'[l&F?7$$\"3eWoML,eK;F?$\"3^D8S+h-pfF?7$$\"3Tp; ZSGV1F?$\"3PId\\LMmDsF?7$$\"3J<0jz% z?+#F?$\"3#4M\"3aG#)RvF?7$$\"3$*)\\,(*o(4w?F?$\"31>#fkL#)R&yF?7$$\"3pU &ftQp,:#F?$\"3FEI9,>9o\")F?7$$\"3#pH2V,(HCAF?$\"3Y`;>u:I#[)F?7$$\"3#R< U,t\"[)H#F?$\"33*Qz$)Qhkz)F?7$$\"3H,i2)fBFP#F?$\"3cS8a%Q@16*F?7$$\"3G, /kM<-ZCF?$\"3y7o58;yC%*F?7$$\"3WhF!=Vu8_#F?$\"3vo;$e8U*Q(*F?7$$\"3&R!y !yDzdf#F?$\"3SKQx-.J05!#<7$$\"3#plo99L-n#F?$\"3'\\)o#zVEn.\"Ffu7$$\"39 %RoMWKZu#F?$\"3hc_')HE9o5Ffu7$$\"3kGF?$\"3JQ%[9*)e&*4\"Ffu7$$\" 3cnmBi-&Q*GF?$\"3P%Q]yBv48\"Ffu7$$\"3R$Hy(o\"f%oHF?$\"3\\Cd&oo\"Ri6Ffu 7$$\"351dcfV4VIF?$\"3y<;\"Ffu7$$\"3,hh$[7]x6$F?$\"3QA:vy\\AD7Ffu 7$$\"3(z)p>a/U#>$F?$\"3]?f7t=kc7Ffu7$$\"3Noy([1*4nKF?$\"3X;Ott*e!)G\"F fu7$$\"3E%=:pUz8Ffu7$$\"37kPd&yakT$F?$\"3>.;(3%R*3N \"Ffu7$$\"3!e5N7==6\\$F?$\"3k\\uj#*=J#Q\"Ffu7$$\"3q\"R>pYidc$F?$\"3nP) =;AIPT\"Ffu7$$\"32)HdNK!QSOF?$\"3?EZ6$)*[^W\"Ffu7$$\"3\\TP3tU'\\r$F?$ \"37m/uP#olZ\"Ffu7$$\"3uaqQ*o1&*y$F?$\"3q`wT^!))z]\"Ffu7$$\"35n>62)**R 'QF?$\"3<2TB&\\3%R:Ffu7$$\"3EG>GGdVQRF?$\"3+aM9X'H3d\"Ffu7$$\"3%)G[nEk !G,%F?$\"35@&p9e^Ag\"Ffu7$$\"3eS`&)\\P5(3%F?$\"3^?g<)QuOj\"Ffu7$$\"3\" eTO(=%>8;%F?$\"3S%*o&=:)4l;Ffu7$$\"3[z:^C]WNUF?$\"3***4(RgH_'p\"Ffu7$$ \"3dWF?$\"3$yW^R^/3z\"Ffu7$$\"3.a(zLot3`%F?$\"3]()=IOVBA=Ff u7$$\"3%Hm/;rFfu7$$\"3l3>aW![T#[F?$\"3c$>PdFfu7$$\"3&*>tB)pdq*[F?$\"3k?%>cF0%z>Ffu7$$\"3'>YhI.$yp\\F?$\"3%p`*p0P %3,#Ffu7$$\"3qlp$H+8B/&F?$\"3A?IJ9LGU?Ffu7$$\"3<(Qaz3OY6&F?$\"35$\\u6) Qst?Ffu7$$\"3G!f0yASn=&F?$\"3S&H:P5l^5#Ffu7$$\"3)[@%*pu7'e_F?$\"3FDk_ \"e1m8#Ffu7$$\"38UpDr-CI`F?$\"3Y`Ps)zZ!o@Ffu7$$\"3In*[Yj3;S&F?$\"312p/ /\")[*>#Ffu7$$\"3F]yywFqsaF?$\"3JR5])oE4B#Ffu7$$\"3/)o<)Hm]VbF?$\"3r^J NgDOiAFfu7$$\"3rkuv%*H+9cF?$\"3bJLIo c6@_K#Ffu7$$\"3&QRf7*z*Rv&F?$\"3Sc8WO1kcBFfu7$$\"3$zT_`IbM#eF?$\"3o2n8 [50)Q#Ffu7$$\"3sRz([CFfu7$$\"3=y*\\$=Nd#ehd#Ffu7$$\"3dH1bRF;(H'F?$\"3yOGnEDT2EFfu7$$\"3HZiQEFfu7$$\"3.;x%*ya]FkF?$\"3\"4hDm)yypEFfu7$$\"3UVYM9+k\"\\ 'F?$\"3zm^]7U*3q#Ffu7$$\"3p(edg]<]b'F?$\"344@]o]$>t#Ffu7$$\"3-$pQ?M%e< mF?$\"3d`b%)G4!Hw#Ffu7$$\"3w2M!H_$GzmF?$\"3wn]Qb8y$z#Ffu7$$\"38h5I[Z0S nF?$\"3d/J?')\\cCGFfu7$$\"3Im4Z#eM)*z'F?$\"3(RI%yS&R_&GFfu7$$\"3`\"Qk$ 3nbeoF?$\"3Z$=&oW=z&)GFfu7$$\"3c&evvF_h\"pF?$\"3br\"zj(y?;HFfu7$$\"3l_ E;g.bspF?$\"3lP&R%QGZYHFfu7$$\"3qY]B%\\yw-(F?$\"3#>!GWa7dwHFfu7$$\"3`+ %Q+Lj93(F?$\"3AGq#>4([1IFfu7$$\"3&[f5uNJQ8(F?$\"3]XKX5R?OIFfu7$$\"3:Hc H*p4Z=(F?$\"3<7.OK]qlIFfu7$$\"3zX.*z%p-MsF?$\"3JxjFKP(\\4$Ffu7$$\"3A-% eb,9P(F?$\"3!4!HKI;A\"=$Ffu7$$\"3$))>+<7bVT(F?$\"3$4:hBi*R4KFfu7$$ \"3\"[]ljw/\\X(F?$\"3Wn[RC!osB$Ffu7$$\"3X*o=#z,a$\\(F?$\"3*z8c809[E$Ff u7$$\"3n>H>52AIvF?$\"3/+6Elh-#H$Ffu7$$\"3'RjRLq6\\c(F?$\"3(HxP@<%*)=LF fu7$$\"3xUQQ:We(f(F?$\"3/dhD$G4aM$Ffu7$$\"3RE)HSw:#GwF?$\"3K[SVSTcrLFf u7$$\"3V*o2*=zycwF?$\"3%*[T/2GN(R$Ffu7$$\"3%z#[Q`xG$o(F?$\"3BWu#zsqFU$ Ffu7$$\"3E>s8,!y-v(F?$\"3&3WP$ROx'\\$Ffu7$$\"39*yb4AE%oxF?$\"3dNZ&[!z o?NFfu7$$\"3!*ei:n3[%y(F?$\"3$\\1ehREUa$Ffu7$$\"3BXp\\Z/W)z(F?$\"3vSa# =)3RnNFfu7$$\"3YMua$=-.\"yF?$\"3+!px#yP=!f$Ffu7$$\"3Unne?51?yF?$\"35\" 4!\\P!3Eh$Ffu7$$\"3@%zZ;h3x#yF?$\"3.nd#G/nYj$Ffu7$$\"3tP>ojABLyF?$\"3( e#)zW^kjl$Ffu7$$\"3,l1@FQhOyF?$\"31d2QDWqxOFfu7$$\"3moIb.&Gy$yF?$\"3d2 i&y#4p)p$Ffu7$$\"3-=bKNN%o$yF?$\"3[u'Q4HG$>PFfu7$$\"3y>huNnhLyF?$\"3y9 )ey%3iRPFfu7$$\"3UPgT*z%4GyF?$\"3rb-/`HdfPFfu7$$\"3S7LP?8@?yF?$\"3eMrv Z*)=zPFfu7$$\"3+lMU?W))4yF?$\"3?2,`LJZ)z$Ffu7$$\"3zlWDfQ,(z(F?$\"3JzWI _(Hu\"QFfu7$$\"3oS@X`uZ\"y(F?$\"3k-GYsH1OQFfu7$$\"3oeD\"ycEJw(F?$\"3k$ pi(zoPaQFfu7$$\"3Z.9>\\.yTxF?$\"3S9&\\BZvB(QFfu7$$\"3&*eLEM#=sr(F?$\"3 :(Ql!fE1!*QFfu7$$\"36')e5^+<*o(F?$\"3,$*[NgAW2RFfu7$$\"3E3/![q-tl(F?$ \"3[O#\\6,=X#RFfu7$$\"3q2x?4>?@wF?$\"3>E8>lNHTRFfu7$$\"3v#p5abY.e(F?$ \"3@yf !y^$3Y'F?$\"3MS%)y\"e#*p5%Ffu7$$\"3(f#QKl9%Ffu7$$!3 0fC4zVo*p'F?$\"3g)[7=30#fTFfu7$$!38@dC\"*poGpF?$\"3bHh+L&H;<%Ffu7$$!3q s7Fwu>/rF?$\"3IO)QJa3Q=%Ffu7$$!3u4'\\)*>%yZsF?$\"3Lirs!4Xd>%Ffu7$$!3x$ *\\v.E!*=UFfu7$$!3G;p )fF)3svF?$\"3A-z?9%H,B%Ffu7$$!3d())z'*)3 " 0 " " {MPLTEXT 1 0 113 "Digits := 15:\nz0 := 4.1*I:\nfor ct from 147 to 15 0 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0El,(=&z=)!#<$\"0W)y\"e#*p5%!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0,iL)46:L!#<$\"0CT'*)*G/7%!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0e4_ce7r\"!#<$\"0w#RRygLT!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0*3a$)Hn')o!#<$\"0yv.>Kl9%!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then w e apply the bisection method to calculate the parameter value associat ed with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=4.1*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=1.47..1.5); \nnewton(R(z)=exp(u0*Pi*I),z=4.1*I);\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0[GC,Hm[\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0G%oJG3@d!#H$\"0\"R+`Q>HT!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interva l on the nonnegative imaginary axis that contains the origin and lies \+ inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0,4. 1292]" "6#7$\"\"!-%&FloatG6$\"&#HT!\"%" }{TEXT -1 18 " (approximately )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-- ----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 12 st age, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,StabilityFunction(6,12,'expanded' ))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"#F)*$)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)*$)F'F1F)F)F)*&#\"XjRG+RBV[w3+T2Ad[=!*zdjrT9sP :'\"fn+++_:ax,tL-:mPI)*\\w`n]F^f*p^71HF)*$)F'\"\"(F)F)F)*&#\"ZF(3>Vbr7 *QV5+E!fSlao,d5gE,'>Db&\"in+++cEo@@vhb#pV=)fZ!4._dU*>q^>5M4,b$\"jn+!3-VEoFDk(ph$)>HNi! H3v:szL(o*>@;5XF)*$)F'\"\"*F)F)F)*&#\"Zp0t*H3d:5<-8Fw>#Ga\"e`Zrc+/Sk# \\$\"\\o+++wPI.'fJ0A@&z\\6%zKO&)o>lC<4'*\\EqPcF)*$)F'\"#5F)F)!\"\"*&# \"WdSOejRwmm&)Hg\"=JvDPOjMKtSLu8&\"in+++ofAA+x2On>$))[b&ff/kwo?&py6`>R 'F)*$)F'\"#6F)F)F)*&#\"HsQQ9I[=0ppj'3SVtDMtJ\"Rv$4#*4#3ySS\\NtP$oqpbow 2ujX\\F)*$)F'\"#7F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 124 "We can find the point where the boundary of the s tability region intersects the negative real axis by solving the equat ion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1; " "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-4.5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!+30$=[%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "z_0 := newton(`R*`(z)=1 ,z=-4.5):\np_1 := plot([`R*`(z),1],z=-5.09..0.49,color=[red,blue]):\np _2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,cross,diamond],col or=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=[-5.09..0.49,-.07..1.47], font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 268 268 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3')*************3&!#<$\"3+j#R_')49'GF *7$$!32+]()Rf=H]F*$\"3=sJ,L#\\Nf#F*7$$!3G++vz=Po\\F*$\"3w0n'ei;tM#F*7$ $!35]i?\"*yX:\\F*$\"31')o)*G[c\\@F*7$$!3#**\\iE!Rai[F*$\"3'Ry.C+Di'>F* 7$$!3P\\PW)eOI![F*$\"3%o6GQ3)>wL$F^o7$$!3++]ZTF#[\"QF*$\"3/jH!yn5% zEF^o7$$!3s***\\'=(pWp$F*$\"3WpnG6&*y+@F^o7$$!3#)***\\Z^AOd$F*$\"3o<], *o@vl\"F^o7$$!3))***\\8%Q;dMF*$\"34SM*\\YTtL\"F^o7$$!3#**\\i*3#39N$F*$ \"3<*[A\\Og87\"F^o7$$!3t***\\J`acA$F*$\"3\\49u]PG-%*!#>7$$!3l****fuY7> JF*$\"3E<1Ode\\&R)Far7$$!3q*\\iQ70_*HF*$\"3DqE-bCz=xFar7$$!3+++5C`^&)G F*$\"3yB#pPpF^[(Far7$$!3q*\\i)G#o^w#F*$\"37vBf]^TavFar7$$!3W*\\(eM$p0l #F*$\"3!R/&\\UWa()yFar7$$!3]**\\i5u*4`#F*$\"3imO]'G%Gu%)Far7$$!3T*\\7 \"eI>@CF*$\"3&yPa\"3$3_?*Far7$$!3n**\\()HUv-BF*$\"3ux>&yIv%=5F^o7$$!3y *\\iRdH(z@F*$\"3$)Q`^/_^S6F^o7$$!3o*\\P$\\ijs?F*$\"35'H2/IiOE\"F^o7$$! 3S**\\#[_sp&>F*$\"32p+n\\+.:9F^o7$$!3y****pz0[P=F*$\"3CV(odTdFf\"F^o7$ $!3')**\\_B5e?!z0OF^o7$$!3c-]i)>_r2*F^o$\"3K0 ?(3'=TMSF^o7$$!3*)**\\74%3K!zF^o$\"3'GWHj-wp`%F^o7$$!3E****\\xPYbnF^o$ \"3Oui!\\:x()3&F^o7$$!3)4+Dc^\")Qb&F^o$\"3,M*G$RJ\\QdF^o7$$!3)e****f) \\h'R%F^o$\"3;Lk.4VaUkF^o7$$!3F)**\\<\"G98KF^o$\"3)))\\_&eH&>D(F^o7$$! 3'G*\\i%Qq%R?F^o$\"3)\\Q@Hgb]:)F^o7$$!3xr****pJ()4'*Far$\"3!RB()HEVP3* F^o7$$\"3k<+]_)f2v#Far$\"3@w5_D%*)y-\"F*7$$\"3)3++!Qdi!Q\"F^o$\"33Bvs& QZ![6F*7$$\"3o4]PhBPfDF^o$\"3!fVMhlr;H\"F*7$$\"3]/]i%G$e(o$F^o$\"3%*\\ z:L\"QfW\"F*7$$\"3!***************[F^o$\"3F8MWBjJK;F*-%'COLOURG6&%$RGB G$\"*++++\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l 7$FGF_]l7$FQF_]l7$FenF_]l7$FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l 7$FdpF_]l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]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$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_] l7$F`zF_]l7$FezF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l- Fc\\l6&Fe\\lFi\\lFi\\lFf\\l-F$6&7#7$$!3#)*****z]I=[%F*F_]l-%'SYMBOLG6# %'CIRCLEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l 6#%&CROSSGF]alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\l Ff`l-%&COLORG6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%%FONTG6 $%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%VI EWG6$;$!$4&!\"#$\"#\\F_dl;$!\"(F_dl$\"$Z\"F_dl" 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" "Curv e 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 82 "The following picture shows the stability region f or the 8 stage, order 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2000 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n61537721441716357799018485722074 10008764843233900283963/2906125169959512750675376499830376615023373017 7541552000000*z^7+5552519601266010570168546540590260010433891271554319 08727/18661710195170199425752030904759818436925561752121682656000000*z ^8+355010934192663373796797154107855000718233029713972960543/451016211 996873379721575082906235291983616976425276826430208000*z^9-34926440040 0567147535815428219762713021710155708299730569/56377026499609172465196 885363279411497952122053159603303776000000*z^10+5137433407332346336372 57531181602985666676396358364057/6391953117869520687664045959555488831 9673607770022225968000000*z^11+317334257343400866369690518483014383872 /4945637407766855697068337733549404078082099209375*z^12:\npts := []: z 0 := 0:\nfor ct from 0 to 280 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,.4,0,0)):\np_2 := plots[polygonplot ]([seq([pts[i-1],pts[i],[-2.25,0]],i=2..nops(pts))],\n style= patchnogrid,color=COLOR(RGB,.78,0,0)):\npts := []: z0 := 2.4+4.7*I:\nf or ct from 0 to 40 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_3 := \+ plot(pts,color=COLOR(RGB,.4,0,0)):\np_4 := plots[polygonplot]([seq([pt s[i-1],pts[i],[2.30,4.70]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.78,0,0)):\npts := []: z0 := 2.4-4.7*I:\nfor ct fr om 0 to 40 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_5 := plot(pts, color=COLOR(RGB,.4,0,0)):\np_6 := plots[polygonplot]([seq([pts[i-1],pt s[i],[2.30,-4.70]],i=2..nops(pts))],\n style=patchnogrid,colo r=COLOR(RGB,.78,0,0)):\np_7 := plot([[[-5.29,0],[3.09,0]],[[0,-5.69],[ 0,5.69]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view= [-5.29..3.09,-5.39..5.39],font=[HELVETICA,9],\n labels=[`Re (z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 481 481 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F(7$F( $\"3++++Fjzq:!#=7$$\"38+++L*oq8)!#F$\"3'******Rp#fTJF-7$$\"3.+++![,39# !#D$\"3++++p'*Q7ZF-7$$\"3%******\\(3A7A!#C$\"3q*****RW!>$G'F-7$$\"3'** ****zr\"fr8!#B$\"3Q+++Ve+ayF-7$$\"3#******47`L:'FC$\"3T+++$oi[U*F-7$$ \"3/+++zy+0A!#A$\"3!******fM\"e*4\"!#<7$$\"3_+++'Gp%*o'FN$\"3++++FSpc7 FQ7$$\"3/+++^mw#y\"!#@$\"31+++[P$QT\"FQ7$$\"3G+++K+*zF%FZ$\"33+++^W+r: FQ7$$\"3[+++()4\">S*FZ$\"34+++,$)=G!#?$\"3)******f; 8`)=FQ7$$\"3'*******p'e:k$Fdo$\"3'******fn&>U?FQ7$$\"3u*****R>4T\\'Fdo $\"3=+++>PX)>#FQ7$$\"3%******>;%*o3\"!#>$\"3)******H5&R`BFQ7$$\"3!**** **H=!e.'G#f]#FQ7$$\"3++++`_-!\\#Fdp$\"3;+++&4YXl#FQ7$ $\"3M+++.*4!yLFdp$\"3!)*****f-*e(z#FQ7$$\"39+++r*y$QUFdp$\"3z*****RclN $HFQ7$$\"3%)*****f5a#3\\Fdp$\"3))*****\\)R]hIFQ7$$\"3D+++6NOK_Fdp$\"33 +++&Q<5=$FQ7$$\"3++++\"\\\"3%4&Fdp$\"39+++=H9#H$FQ7$$\"3N+++lqCEWFdp$ \"3))*****f_B^R$FQ7$$\"36+++1/62KFdp$\"3y*****H)zD!\\$FQ7$$\"3++++@[(* \\9Fdp$\"31+++z(Ryd$FQ7$$!3!3++ge=83)Fdo$\"3#******>(=:eOFQ7$$!3;+++Iu ];NFdp$\"3()******o3[JPFQ7$$!3t*****4=t(=mFdp$\"3))*****f')G\")z$FQ7$$ !3)******4yje+\"F-$\"3#******Rk:%eQFQ7$$!3)******H5d$y8F-$\"3()*****fU sE\"RFQ7$$!3'******R=vYx\"F-$\"3?+++.9BhRFQ7$$!3')*****p+w2>#F-$\"3')* ****H*[T/SFQ7$$!3@+++%eYKi#F-$\"3r******3k_USFQ7$$!3=+++&Q[#pIF-$\"3k* ******\\We2%FQ7$$!3%*******[\\WENF-$\"32+++5#>Y5%FQ7$$!3y*****R`FH*RF- $\"3Q+++V12HTFQ7$$!3!)*****zXZrY%F-$\"3;+++n*)Q\\TFQ7$$!3))*****>0fy% \\F-$\"3q*******HNd;%FQ7$$!3\"******R_uSV&F-$\"3m*****p.V#yTFQ7$$!3U++ +;2.DfF-$\"3g*****RJ>q=%FQ7$$!3e*****RIn,U'F-$\"3w*****RTY@>%FQ7$$!37+ ++Fh6>pF-$\"3S+++:Io$>%FQ7$$!3g******R%FQ7$$!3.++ +UR&z#zF-$\"3?+++Jv5'=%FQ7$$!31+++GR8Q%)F-$\"3q*****H`.q<%FQ7$$!3M+++G nx_*)F-$\"3%******f6FV;%FQ7$$!3)******H\"Qvs%*F-$\"3E+++*3M![TFQ7$$!3M +++64O****F-$\"3l*****RJl!GTFQ7$$!31+++aOW`5FQ$\"3%******\\f_V5%FQ7$$! 3\"******>]^!36FQ$\"3++++L'Ho2%FQ7$$!33+++&*45k6FQ$\"3U+++zK)GrRFQ7 $$!3'******RoefM\"FQ$\"3$)*****HOW*GRFQ7$$!3'******4\"*=LT\"FQ$\"3)*** ****yf\"Q)QFQ7$$!3'*******zW5&[\"FQ$\"3*)*****>1/p$QFQ7$$!31+++NX`h:FQ $\"3=+++$[p&*y$FQ7$$!3,+++X;AU;FQ$\"3$******zC6Lu$FQ7$$!3'******RRKhs \"FQ$\"3!******4H\"Q*p$FQ7$$!3,+++8w$=\"=FQ$\"3#******Rca%eOFQ7$$!3+++ +17!z*=FQ$\"3%******fKi0i$FQ7$$!3-+++fs;$)>FQ$\"3'******>Uo_e$FQ7$$!3= +++7>&o1#FQ$\"3)******\\<:>b$FQ7$$!3/+++Y&*\\[@FQ$\"34+++\\&4)>NFQ7$$! 3-+++sD*yA#FQ$\"3!*******\\Y$fLFQ7$$!3y*****4nH7f#FQ$\"3/+++v#zZK $FQ7$$!33+++,(*zdEFQ$\"3))*****4Vq))G$FQ7$$!3)******f\\'fAFFQ$\"3))*** **H/'[^KFQ7$$!33+++V*Gdy#FQ$\"34+++C,_7KFQ7$$!3))*****R+-t%GFQ$\"3%)** ***Ht))=<$FQ7$$!3z*****f94DNJFQ$\"31+++@]&>%HFQ7$$!3?+++o 9`*=$FQ$\"3!)*****H^Z/*GFQ7$$!3&******z&*HHC$FQ$\"3()*****>f3r$GFQ7$$! 3%)*****p1QbH$FQ$\"3;+++KD(>y#FQ7$$!39+++[IWZLFQ$\"3-+++(3\"4DFFQ7$$!3 -+++lYs)R$FQ$\"37+++$QMlm#FQ7$$!3!)*****\\H^%\\MFQ$\"3?+++45R1EFQ7$$!3 9+++\"[v'*\\$FQ$\"3)*******yowWDFQ7$$!3=++++%H%\\NFQ$\"3++++fDy\"[#FQ7 $$!3/+++14s)f$FQ$\"34+++L*ovT#FQ7$$!3/+++_3`ZOFQ$\"3%******zShAN#FQ7$$ !3')*****RJ7ep$FQ$\"3'******z=$*fG#FQ7$$!3!)*****z/#\\VPFQ$\"3*)***** \\T)))=AFQ7$$!3?+++.TZ!z$FQ$\"3%)*****H3c5:#FQ7$$!3%)*****R8Xm$QFQ$\"3 1+++,be#3#FQ7$$!3%******fk!)=)QFQ$\"3!)*****fvVN,#FQ7$$!3>+++\\80ERFQ$ \"31+++p_(R%>FQ7$$!35+++6*G!pRFQ$\"3/+++ZJ!R(=FQ7$$!3&*******G1p5SFQ$ \"3)*******o=L.=FQ7$$!3O+++KG#40%FQ$\"30+++>2DK%FQ$\"3'******4?$GU9FQ7$$!3m*** **4`)[FUFQ$\"3)*******)o1#o8FQ7$$!3s*****4#QUdUFQ$\"3))******HeU$H\"FQ 7$$!3S+++4tY&G%FQ$\"3.+++W-\"z@\"FQ7$$!3%)*****4<(f6VFQ$\"3#*******)eM ;9\"FQ7$$!3I+++y2!eL%FQ$\"31+++n'zX1\"FQ7$$!3i*****H9u!eVFQ$\"3]*****4 DMt')*F-7$$!3A+++&Q@%yVFQ$\"3u*****p<>43*F-7$$!3')*****p;aoR%FQ$\"3[++ +hDg'G)F-7$$!3!)*****4-\"R8WFQ$\"3a*****\\]NX[(F-7$$!3c*****>]c!GWFQ$ \"3/+++)>y\\n'F-7$$!3\"******p5!)3W%FQ$\"3a*****p63$eeF-7$$!3++++))[*= X%FQ$\"3K+++Wk-N]F-7$$!3A+++!yN6Y%FQ$\"3$)*****p_hd?%F-7$$!3?+++WwjoWF Q$\"3?+++;\"o7P$F-7$$!3!)*****H8LWZ%FQ$\"3C+++M6UKDF-7$$!3-+++B0byWFQ$ \"3')*****R3.-p\"F-7$$!3e*****pp65[%FQ$\"3))*****H\"z%oX)Fdp7$$!3#)*** **z]I=[%FQF(7$F[am$!3))*****H\"z%oX)Fdp7$Ff`m$!3')*****R3.-p\"F-7$Fa`m $!3C+++M6UKDF-7$F\\`m$!3?+++;\"o7P$F-7$Fg_m$!3$)*****p_hd?%F-7$Fb_m$!3 K+++Wk-N]F-7$F]_m$!3a*****p63$eeF-7$Fh^m$!3/+++)>y\\n'F-7$Fc^m$!3a**** *\\]NX[(F-7$F^^m$!3[+++hDg'G)F-7$Fi]m$!3u*****p<>43*F-7$Fd]m$!3]*****4 DMt')*F-7$F_]m$!31+++n'zX1\"FQ7$Fj\\m$!3#*******)eM;9\"FQ7$Fe\\m$!3.++ +W-\"z@\"FQ7$F`\\m$!3))******HeU$H\"FQ7$F[\\m$!3)*******)o1#o8FQ7$Ff[m $!3'******4?$GU9FQ7$Fa[m$!3*******p^*o::FQ7$F\\[m$!35+++D?Y)e\"FQ7$Fgj l$!31+++#[P1m\"FQ7$Fbjl$!30+++>2DKFQ7$F^il$!3!)*****fvVN,#FQ7$Fihl$!31+++ ,be#3#FQ7$Fdhl$!3%)*****H3c5:#FQ7$F_hl$!3*)*****\\T)))=AFQ7$Fjgl$!3'** ****z=$*fG#FQ7$Fegl$!3%******zShAN#FQ7$F`gl$!34+++L*ovT#FQ7$F[gl$!3+++ +fDy\"[#FQ7$Fffl$!3)*******yowWDFQ7$Fafl$!3?+++45R1EFQ7$F\\fl$!37+++$Q Mlm#FQ7$Fgel$!3-+++(3\"4DFFQ7$Fbel$!3;+++KD(>y#FQ7$F]el$!3()*****>f3r$ GFQ7$Fhdl$!3!)*****H^Z/*GFQ7$Fcdl$!31+++@]&>%HFQ7$F^dl$!3')*****\\.8;* HFQ7$Ficl$!3#******pK=%RIFQ7$Fdcl$!3')*****RV\"Q&3$FQ7$F_cl$!3<+++**f_ HJFQ7$Fjbl$!3%)*****Ht))=<$FQ7$Febl$!34+++C,_7KFQ7$F`bl$!3))*****H/'[^ KFQ7$F[bl$!3))*****4Vq))G$FQ7$Ffal$!3/+++v#zZK$FQ7$Faal$!3?+++\">Y$fLF Q7$F\\al$!3-+++V*RFR$FQ7$Fg`l$!31+++%Hu^U$FQ7$Fb`l$!3#)******H/#pX$FQ7 $F]`l$!3!*******\\NFQ7$Fc_l$!3)******\\<:>b$ FQ7$F^_l$!3'******>Uo_e$FQ7$Fi^l$!3%******fKi0i$FQ7$Fd^l$!3#******Rca% eOFQ7$F_^l$!3!******4H\"Q*p$FQ7$Fj]l$!3$******zC6Lu$FQ7$Fe]l$!3=+++$[p &*y$FQ7$F`]l$!3*)*****>1/p$QFQ7$F[]l$!3)*******yf\"Q)QFQ7$Ff\\l$!3$)** ***HOW*GRFQ7$Fa\\l$!3%******>K)GrRFQ7$F\\\\l$!3G+++$GI-,%FQ7$Fg[l$!3U+ ++z%FQ7$Fjx$!3S+++:Io$>%FQ7$Fex$ !3w*****RTY@>%FQ7$F`x$!3g*****RJ>q=%FQ7$F[x$!3m*****p.V#yTFQ7$Ffw$!3q* ******HNd;%FQ7$Faw$!3;+++n*)Q\\TFQ7$F\\w$!3Q+++V12HTFQ7$Fgv$!32+++5#>Y 5%FQ7$Fbv$!3k*******\\We2%FQ7$F]v$!3r******3k_USFQ7$Fhu$!3')*****H*[T/ SFQ7$Fcu$!3?+++.9BhRFQ7$F^u$!3()*****fUsE\"RFQ7$Fit$!3#******Rk:%eQFQ7 $Fdt$!3))*****f')G\")z$FQ7$F_t$!3()******o3[JPFQ7$Fjs$!3#******>(=:eOF Q7$Fes$!31+++z(Ryd$FQ7$F`s$!3y*****H)zD!\\$FQ7$F[s$!3))*****f_B^R$FQ7$ Ffr$!39+++=H9#H$FQ7$Far$!33+++&Q<5=$FQ7$F\\r$!3))*****\\)R]hIFQ7$Fgq$! 3z*****RclN$HFQ7$Fbq$!3!)*****f-*e(z#FQ7$F]q$!3;+++&4YXl#FQ7$Fhp$!3\") *****>'G#f]#FQ7$Fbp$!3)******H5&R`BFQ7$F]p$!3=+++>PX)>#FQ7$Fho$!3'**** **fn&>U?FQ7$Fbo$!3)******f;8`)=FQ7$F]o$!34+++,$)=G$G' F-7$F5$!3++++p'*Q7ZF-7$$\"3U+++G*oq8)F1$!3'******Rp#fTJF-7$F($!3++++Fj zq:F-F'F*F.F4F:F@FFFKFRFWFgnF\\oFaoFgoF\\pFapFgpF\\qFaqFfqF[rF`rFerFjr F_sFdsFisF^tFctFhtF]uFbuFguF\\vFavFfvF[wF`wFewFjwF_x-%&COLORG6&%$RGBG$ \"\"%!\"\"F(F(-%)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$!$D#!\"#F(7%Ffhn 7$$\"+L*oq8)Fdp$\"+%p#fTJFihnFjhn7%F_in7$$\"+![,39#FQ$\"+p'*Q7ZFihnFjh n7%Fein7$$\"+v3A7A!#;$\"+W/>$G'FihnFjhn7%F[jn7$$\"+=S*F`\\o$\"+,$)=G!#7$\"+mJJ& )=Fe[oFjhn7%F`]o7$$\"+q'e:k$Fc]o$\"+wc>U?Fe[oFjhn7%Fg]o7$$\"+%>4T\\'Fc ]o$\"+>PX)>#Fe[oFjhn7%F]^o7$$\"+iT*o3\"!#6$\"+.^R`BFe[oFjhn7%Fc^o7$$\" +$=!e.#Fihn$\"+$*[T/SFe[oFjhn7%Fjdo7$$!+%eYKi#Fihn$\"+4k_USFe [oFjhn7%F`eo7$$!+&Q[#pIFihn$\"++X%e2%Fe[oFjhn7%Ffeo7$$!+\\\\WENFihn$\" +5#>Y5%Fe[oFjhn7%F\\fo7$$!+Mv#H*RFihn$\"+V12HTFe[oFjhn7%Fbfo7$$!+eu9nW Fihn$\"+n*)Q\\TFe[oFjhn7%Fhfo7$$!+_!fy%\\Fihn$\"++`tlTFe[oFjhn7%F^go7$ $!+CX2MaFihn$\"+PICyTFe[oFjhn7%Fdgo7$$!+;2.DfFihn$\"+9$>q=%Fe[oFjhn7%F jgo7$$!+/t;?kFihn$\"+9k9#>%Fe[oFjhn7%F`ho7$$!+Fh6>pFihn$\"+:Io$>%Fe[oF jhn7%Ffho7$$!+S%Fe[oFjhn7%F\\io7$$!+UR&z#zFihn$\"+Jv 5'=%Fe[oFjhn7%Fbio7$$!+GR8Q%)Fihn$\"+LN+xTFe[oFjhn7%Fhio7$$!+Gnx_*)Fih n$\"+;rKkTFe[oFjhn7%F^jo7$$!+8Qvs%*Fihn$\"+*3M![TFe[oFjhn7%Fdjo7$$!+64 O****Fihn$\"+9`1GTFe[oFjhn7%Fjjo7$$!+aOW`5Fe[o$\"+&f_V5%Fe[oFjhn7%F`[p 7$$!+-:036Fe[o$\"+L'Ho2%Fe[oFjhn7%Ff[p7$$!+&*45k6Fe[o$\"+zFe[ o$\"+A%o_e$Fe[oFjhn7%F^`p7$$!+7>&o1#Fe[o$\"+v^\">b$Fe[oFjhn7%Fd`p7$$!+ Y&*\\[@Fe[o$\"+\\&4)>NFe[oFjhn7%Fj`p7$$!+sD*yA#Fe[o$\"+]Y$fLFe[oFjhn7%Fhbp7$$!+r'H7f#Fe[o$\"+v#zZK$Fe[oFjhn7%F^cp7$$!+,( *zdEFe[o$\"+J/())G$Fe[oFjhn7%Fdcp7$$!+'\\'fAFFe[o$\"+Vg[^KFe[oFjhn7%Fj cp7$$!+V*Gdy#Fe[o$\"+C,_7KFe[oFjhn7%F`dp7$$!+/?IZGFe[o$\"+L())=<$Fe[oF jhn7%Ffdp7$$!+YrT2HFe[o$\"+**f_HJFe[oFjhn7%F\\ep7$$!+JJ%HFe[oFjhn7%Fdfp7$$!+o9` *=$Fe[o$\"+8vW!*GFe[oFjhn7%Fjfp7$$!+e*HHC$Fe[o$\"+#f3r$GFe[oFjhn7%F`gp 7$$!+n!QbH$Fe[o$\"+KD(>y#Fe[oFjhn7%Ffgp7$$!+[IWZLFe[o$\"+(3\"4DFFe[oFj hn7%F\\hp7$$!+lYs)R$Fe[o$\"+$QMlm#Fe[oFjhn7%Fbhp7$$!+&H^%\\MFe[o$\"+45 R1EFe[oFjhn7%Fhhp7$$!+\"[v'*\\$Fe[o$\"+zowWDFe[oFjhn7%F^ip7$$!++%H%\\N Fe[o$\"+fDy\"[#Fe[oFjhn7%Fdip7$$!+14s)f$Fe[o$\"+L*ovT#Fe[oFjhn7%Fjip7$ $!+_3`ZOFe[o$\"+39E_BFe[oFjhn7%F`jp7$$!+9B\"ep$Fe[o$\"+)=$*fG#Fe[oFjhn 7%Ffjp7$$!+[?\\VPFe[o$\"+:%)))=AFe[oFjhn7%F\\[q7$$!+.TZ!z$Fe[o$\"+$3c5 :#Fe[oFjhn7%Fb[q7$$!+M^kOQFe[o$\"+,be#3#Fe[oFjhn7%Fh[q7$$!+Y1)=)QFe[o$ \"+cPa8?Fe[oFjhn7%F^\\q7$$!+\\80ERFe[o$\"+p_(R%>Fe[oFjhn7%Fd\\q7$$!+6* G!pRFe[o$\"+ZJ!R(=Fe[oFjhn7%Fj\\q7$$!+H1p5SFe[o$\"+p=L.=Fe[oFjhn7%F`]q 7$$!+KG#40%Fe[o$\"+>2DK%Fe[o$\"+,KGU9Fe[oFjhn7%F^_q7$$!+J&)[FUFe [o$\"+*o1#o8Fe[oFjhn7%Fd_q7$$!+@QUdUFe[o$\"+IeU$H\"Fe[oFjhn7%Fj_q7$$!+ 4tY&G%Fe[o$\"+W-\"z@\"Fe[oFjhn7%F``q7$$!+rrf6VFe[o$\"+*eM;9\"Fe[oFjhn7 %Ff`q7$$!+y2!eL%Fe[o$\"+n'zX1\"Fe[oFjhn7%F\\aq7$$!+VT2eVFe[o$\"+^ULn)* FihnFjhn7%Fbaq7$$!+&Q@%yVFe[o$\"+x\">43*FihnFjhn7%Fhaq7$$!+nT&oR%Fe[o$ \"+hDg'G)FihnFjhn7%F^bq7$$!+@5R8WFe[o$\"+0b`%[(FihnFjhn7%Fdbq7$$!+-l0G WFe[o$\"+)>y\\n'FihnFjhn7%Fjbq7$$!+2,)3W%Fe[o$\"+<\"3$eeFihnFjhn7%F`cq 7$$!+))[*=X%Fe[o$\"+Wk-N]FihnFjhn7%Ffcq7$$!+!yN6Y%Fe[o$\"+F:w0UFihnFjh n7%F\\dq7$$!+WwjoWFe[o$\"+;\"o7P$FihnFjhn7%Fbdq7$$!+LJVuWFe[o$\"+M6UKD FihnFjhn7%Fhdq7$$!+B0byWFe[o$\"+%3.-p\"FihnFjhn7%F^eq7$$!+(p65[%Fe[o$ \"+8z%oX)Ff^oFjhn7%Fdeq7$$!+30$=[%Fe[oF(Fjhn7%Fjeq7$Feeq$!+8z%oX)Ff^oF jhn7%F^fq7$F_eq$!+%3.-p\"FihnFjhn7%Fbfq7$Fidq$!+M6UKDFihnFjhn7%Fffq7$F cdq$!+;\"o7P$FihnFjhn7%Fjfq7$F]dq$!+F:w0UFihnFjhn7%F^gq7$Fgcq$!+Wk-N]F ihnFjhn7%Fbgq7$Facq$!+<\"3$eeFihnFjhn7%Ffgq7$F[cq$!+)>y\\n'FihnFjhn7%F jgq7$Febq$!+0b`%[(FihnFjhn7%F^hq7$F_bq$!+hDg'G)FihnFjhn7%Fbhq7$Fiaq$!+ x\">43*FihnFjhn7%Ffhq7$Fcaq$!+^ULn)*FihnFjhn7%Fjhq7$F]aq$!+n'zX1\"Fe[o Fjhn7%F^iq7$Fg`q$!+*eM;9\"Fe[oFjhn7%Fbiq7$Fa`q$!+W-\"z@\"Fe[oFjhn7%Ffi q7$F[`q$!+IeU$H\"Fe[oFjhn7%Fjiq7$Fe_q$!+*o1#o8Fe[oFjhn7%F^jq7$F__q$!+, KGU9Fe[oFjhn7%Fbjq7$Fi^q$!+<&*o::Fe[oFjhn7%Ffjq7$Fc^q$!+D?Y)e\"Fe[oFjh n7%Fjjq7$F]^q$!+#[P1m\"Fe[oFjhn7%F^[r7$Fg]q$!+>2DKFe [oFjhn7%F^\\r7$F_\\q$!+cPa8?Fe[oFjhn7%Fb\\r7$Fi[q$!+,be#3#Fe[oFjhn7%Ff \\r7$Fc[q$!+$3c5:#Fe[oFjhn7%Fj\\r7$F][q$!+:%)))=AFe[oFjhn7%F^]r7$Fgjp$ !+)=$*fG#Fe[oFjhn7%Fb]r7$Fajp$!+39E_BFe[oFjhn7%Ff]r7$F[jp$!+L*ovT#Fe[o Fjhn7%Fj]r7$Feip$!+fDy\"[#Fe[oFjhn7%F^^r7$F_ip$!+zowWDFe[oFjhn7%Fb^r7$ Fihp$!+45R1EFe[oFjhn7%Ff^r7$Fchp$!+$QMlm#Fe[oFjhn7%Fj^r7$F]hp$!+(3\"4D FFe[oFjhn7%F^_r7$Fggp$!+KD(>y#Fe[oFjhn7%Fb_r7$Fagp$!+#f3r$GFe[oFjhn7%F f_r7$F[gp$!+8vW!*GFe[oFjhn7%Fj_r7$Fefp$!+@]&>%HFe[oFjhn7%F^`r7$F_fp$!+ NIh\"*HFe[oFjhn7%Fb`r7$Fiep$!+F$=%RIFe[oFjhn7%Ff`r7$Fcep$!+M9Q&3$Fe[oF jhn7%Fj`r7$F]ep$!+**f_HJFe[oFjhn7%F^ar7$Fgdp$!+L())=<$Fe[oFjhn7%Fbar7$ Fadp$!+C,_7KFe[oFjhn7%Ffar7$F[dp$!+Vg[^KFe[oFjhn7%Fjar7$Fecp$!+J/())G$ Fe[oFjhn7%F^br7$F_cp$!+v#zZK$Fe[oFjhn7%Fbbr7$Fibp$!+\">Y$fLFe[oFjhn7%F fbr7$Fcbp$!+V*RFR$Fe[oFjhn7%Fjbr7$F]bp$!+%Hu^U$Fe[oFjhn7%F^cr7$Fgap$!+ I/#pX$Fe[oFjhn7%Fbcr7$Faap$!+]NFe[oF jhn7%Fjcr7$Fe`p$!+v^\">b$Fe[oFjhn7%F^dr7$F_`p$!+A%o_e$Fe[oFjhn7%Fbdr7$ Fi_p$!+EBc?OFe[oFjhn7%Ffdr7$Fc_p$!+kXXeOFe[oFjhn7%Fjdr7$F]_p$!+\"H\"Q* p$Fe[oFjhn7%F^er7$Fg^p$!+[7JVPFe[oFjhn7%Fber7$Fa^p$!+$[p&*y$Fe[oFjhn7% Ffer7$F[^p$!+iS!p$QFe[oFjhn7%Fjer7$Fe]p$!+zf\"Q)QFe[oFjhn7%F^fr7$F_]p$ !+jV%*GRFe[oFjhn7%Fbfr7$Fi\\p$!+A$)GrRFe[oFjhn7%Fffr7$Fc\\p$!+$GI-,%Fe [oFjhn7%Fjfr7$F]\\p$!+z%Fe[oFjhn7%F^ir7$Fgho$!+ :Io$>%Fe[oFjhn7%Fbir7$Faho$!+9k9#>%Fe[oFjhn7%Ffir7$F[ho$!+9$>q=%Fe[oFj hn7%Fjir7$Fego$!+PICyTFe[oFjhn7%F^jr7$F_go$!++`tlTFe[oFjhn7%Fbjr7$Fifo $!+n*)Q\\TFe[oFjhn7%Ffjr7$Fcfo$!+V12HTFe[oFjhn7%Fjjr7$F]fo$!+5#>Y5%Fe[ oFjhn7%F^[s7$Fgeo$!++X%e2%Fe[oFjhn7%Fb[s7$Faeo$!+4k_USFe[oFjhn7%Ff[s7$ F[eo$!+$*[T/SFe[oFjhn7%Fj[s7$Fedo$!+.9BhRFe[oFjhn7%F^\\s7$F_do$!+ECn7R Fe[oFjhn7%Fb\\s7$Fico$!+WcTeQFe[oFjhn7%Ff\\s7$Fcco$!+m)G\")z$Fe[oFjhn7 %Fj\\s7$F]co$!+p3[JPFe[oFjhn7%F^]s7$Fgbo$!+s=:eOFe[oFjhn7%Fb]s7$Fabo$! +z(Ryd$Fe[oFjhn7%Ff]s7$F[bo$!+$)zD!\\$Fe[oFjhn7%Fj]s7$Feao$!+EN7&R$Fe[ oFjhn7%F^^s7$F_ao$!+=H9#H$Fe[oFjhn7%Fb^s7$Fi`o$!+&Q<5=$Fe[oFjhn7%Ff^s7 $Fc`o$!+&)R]hIFe[oFjhn7%Fj^s7$F]`o$!+kbcLHFe[oFjhn7%F^_s7$Fg_o$!+E!*e( z#Fe[oFjhn7%Fb_s7$Fa_o$!+&4YXl#Fe[oFjhn7%Ff_s7$F[_o$!+iG#f]#Fe[oFjhn7% Fj_s7$Fd^o$!+.^R`BFe[oFjhn7%F^`s7$F^^o$!+>PX)>#Fe[oFjhn7%Fb`s7$Fh]o$!+ wc>U?Fe[oFjhn7%Ff`s7$Fa]o$!+mJJ&)=Fe[oFjhn7%Fj`s7$F[]o$!+,$)=G$G'Fih nFjhn7%F\\cs7$Ffin$!+p'*Q7ZFihnFjhn7%F`cs7$$\"+G*oq8)Fdp$!+%p#fTJFihnF jhn7%Fdcs7$F($!+Fjzq:FihnFjhn7%FjcsF'FjhnFehnF^inFdinFjinFajnFhjnF^[oF f[oF\\\\oFc\\oFi\\oF_]oFf]oF\\^oFb^oFi^oF__oFe_oF[`oFa`oFg`oF]aoFcaoFi aoF_boFeboF[coFacoFgcoF]doFcdoFidoF_eoFeeoF[foFafoFgfoF]goFcgoFigo-F\\ hn6&F^hn$\"#yF]inF(F(-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3)*******pua mBFQ$\"3c******>*[)>ZFQ7$$\"3#)******pb(QO#FQ$\"3u*****4:t'GZFQ7$$\"3? +++r^-gBFQ$\"3\")*****4P))pt%FQ7$$\"3-+++Sx2bBFQ$\"3W+++K1mWZFQ7$$\"37 +++!QC\"\\BFQ$\"37+++u`c^ZFQ7$$\"34+++&olAM#FQ$\"3#)*****Hy'edZFQ7$$\" 31+++v>hMBFQ$\"3++++7uhiZFQ7$$\"3.+++uQGEBFQ$\"3%)*****z6elw%FQ7$$\"3! ********=8uJ#FQ$\"3C+++)R=$pZFQ7$$\"3-+++AS93BFQ$\"3O+++=s\"3x%FQ7$$\" 3!)*****R1M')H#FQ$\"3s*****RX%)4x%FQ7$$\"33+++pf0*G#FQ$\"31+++&4j(pZFQ 7$$\"3/+++B')fzAFQ$\"3i*****HS7rw%FQ7$$\"3A+++\\!o/F#FQ$\"3A+++4A,jZFQ 7$$\"3#******z_()=E#FQ$\"39+++Y&ouv%FQ7$$\"3;+++Gj4aAFQ$\"3e*****\\q?0 v%FQ7$$\"3%*******)=YtC#FQ$\"3U+++:'\\Au%FQ7$$\"3<+++9X*=C#FQ$\"3))*** **>1(yKZFQ7$$\"3,+++dJ*zB#FQ$\"3o*****HgCBs%FQ7$$\"3(******4'>(eB#FQ$ \"3*)*****4@?6r%FQ7$$\"3*)******>rrNAFQ$\"3'******zv*\\*p%FQ7$$\"3,+++ UhkPAFQ$\"3m*****\\**\\yo%FQ7$$\"3<+++TMoTAFQ$\"3u*****RL+mn%FQ7$$\"3) ******4bUxC#FQ$\"33++++J>mYFQ7$$\"3A+++V7ibAFQ$\"3-+++Hd/dYFQ7$$\"3'** *****QK,lAFQ$\"3/+++b@^\\YFQ7$$\"3'******4SNbF#FQ$\"3%******f2_Qk%FQ7$ $\"3%******HJknG#FQ$\"3;+++2Z@SYFQ7$$\"31+++6QF)H#FQ$\"3E+++[)Q'QYFQ7$ $\"3!******H8m'4BFQ$\"3\")*****>Im!RYFQ7$$\"3(******zx#f?BFQ$\"3<+++zH OTYFQ7$$\"3>+++/[wIBFQ$\"33+++l>MXYFQ7$$\"3#)*****Rha*RBFQ$\"3E+++5Xy] YFQ7$$\"35+++s@*zM#FQ$\"3e*****\\%oXdYFQ7$$\"3'*******o#eZN#FQ$\"3E+++ IC7lYFQ7$$\"3/+++@oZFe[o7$$\"+qb(QO#Fe[o$\"+^JnGZFe[o7$$\"$I#F]i n$\"$q%F]in7%Fiat7$$\"+r^-gBFe[o$\"+r$))pt%Fe[oF^bt7%Fdbt7$$\"+Sx2bBFe [o$\"+K1mWZFe[oF^bt7%Fjbt7$$\"+!QC\"\\BFe[o$\"+u`c^ZFe[oF^bt7%F`ct7$$ \"+&olAM#Fe[o$\"+$y'edZFe[oF^bt7%Ffct7$$\"+v>hMBFe[o$\"+7uhiZFe[oF^bt7 %F\\dt7$$\"+uQGEBFe[o$\"+=\"elw%Fe[oF^bt7%Fbdt7$$\"+!>8uJ#Fe[o$\"+)R=$ pZFe[oF^bt7%Fhdt7$$\"+AS93BFe[o$\"+=s\"3x%Fe[oF^bt7%F^et7$$\"+kSj)H#Fe [o$\"+aW)4x%Fe[oF^bt7%Fdet7$$\"+pf0*G#Fe[o$\"+&4j(pZFe[oF^bt7%Fjet7$$ \"+B')fzAFe[o$\"+.C6nZFe[oF^bt7%F`ft7$$\"+\\!o/F#Fe[o$\"+4A,jZFe[oF^bt 7%Ffft7$$\"+Gv)=E#Fe[o$\"+Y&ouv%Fe[oF^bt7%F\\gt7$$\"+Gj4aAFe[o$\"+02_] ZFe[oF^bt7%Fbgt7$$\"+*=YtC#Fe[o$\"+:'\\Au%Fe[oF^bt7%Fhgt7$$\"+9X*=C#Fe [o$\"+iqyKZFe[oF^bt7%F^ht7$$\"+dJ*zB#Fe[o$\"+.YKAZFe[oF^bt7%Fdht7$$\"+ h>(eB#Fe[o$\"+6-76ZFe[oF^bt7%Fjht7$$\"+?rrNAFe[o$\"+e(*\\*p%Fe[oF^bt7% F`it7$$\"+UhkPAFe[o$\"+&**\\yo%Fe[oF^bt7%Ffit7$$\"+TMoTAFe[o$\"+M.gwYF e[oF^bt7%F\\jt7$$\"+^DuZAFe[o$\"++J>mYFe[oF^bt7%Fbjt7$$\"+V7ibAFe[o$\" +Hd/dYFe[oF^bt7%Fhjt7$$\"+RK,lAFe[o$\"+b@^\\YFe[oF^bt7%F^[u7$$\"+,a`vA Fe[o$\"+w?&Qk%Fe[oF^bt7%Fd[u7$$\"+8Vw'G#Fe[o$\"+2Z@SYFe[oF^bt7%Fj[u7$$ \"+6QF)H#Fe[o$\"+[)Q'QYFe[oF^bt7%F`\\u7$$\"+Lhm4BFe[o$\"+-j1RYFe[oF^bt 7%Ff\\u7$$\"+yFf?BFe[o$\"+zHOTYFe[oF^bt7%F\\]u7$$\"+/[wIBFe[o$\"+l>MXY Fe[oF^bt7%Fb]u7$$\"+9Y&*RBFe[o$\"+5Xy]YFe[oF^bt7%Fh]u7$$\"+s@*zM#Fe[o$ \"+XoXdYFe[oF^bt7%F^^u7$$\"+p#eZN#Fe[o$\"+IC7lYFe[oF^bt7%Fd^u7$$\"+@o< gBFe[o$\"+)3]Nn%Fe[oF^bt7%Fj^u7$$\"+yt?kBFe[o$\"+)*)=Do%Fe[oF^bt7%F`_u 7$$\"+.&QoO#Fe[o$\"+W2#=p%Fe[oF^bt7%Ff_u7$$\"+JA3oBFe[o$\"+r7E,ZFe[oF^ bt7%F\\`u7$$\"+V'pzO#Fe[o$\"+U)f1r%Fe[oF^bt7%Fb`uFdatF^btF^dsFbds-F$6$ 7K7$Fjds$!3c******>*[)>ZFQ7$F]at$!35+++U)f1r%FQ7$Fh`t$!3\"******4Fh7q% FQ7$Fc`t$!39+++W2#=p%FQ7$F^`t$!3k*****z*)=Do%FQ7$Fi_t$!3#)*****z3]Nn%F Q7$Fd_t$!3E+++IC7lYFQ7$F__t$!3e*****\\%oXdYFQ7$Fj^t$!3E+++5Xy]YFQ7$Fe^ t$!33+++l>MXYFQ7$F`^t$!3<+++zHOTYFQ7$F[^t$!3\")*****>Im!RYFQ7$Ff]t$!3E +++[)Q'QYFQ7$Fa]t$!3;+++2Z@SYFQ7$F\\]t$!3%******f2_Qk%FQ7$Fg\\t$!3/+++ b@^\\YFQ7$Fb\\t$!3-+++Hd/dYFQ7$F]\\t$!33++++J>mYFQ7$Fh[t$!3u*****RL+mn %FQ7$Fc[t$!3m*****\\**\\yo%FQ7$F^[t$!3'******zv*\\*p%FQ7$Fijs$!3*)**** *4@?6r%FQ7$Fdjs$!3o*****HgCBs%FQ7$F_js$!3))*****>1(yKZFQ7$Fjis$!3U+++: '\\Au%FQ7$Feis$!3e*****\\q?0v%FQ7$F`is$!39+++Y&ouv%FQ7$F[is$!3A+++4A,j ZFQ7$Ffhs$!3i*****HS7rw%FQ7$Fahs$!31+++&4j(pZFQ7$F\\hs$!3s*****RX%)4x% FQ7$Fggs$!3O+++=s\"3x%FQ7$Fbgs$!3C+++)R=$pZFQ7$F]gs$!3%)*****z6elw%FQ7 $Fhfs$!3++++7uhiZFQ7$Fcfs$!3#)*****Hy'edZFQ7$F^fs$!37+++u`c^ZFQ7$Fies$ !3W+++K1mWZFQ7$Fdes$!3\")*****4P))pt%FQ7$F_es$!3u*****4:t'GZFQF[auF[hn -Fchn6L7%7$Feat$!+?*[)>ZFe[o7$Fc`u$!+U)f1r%Fe[o7$F_bt$!$q%F]in7%Fihu7$ F]`u$!+r7E,ZFe[oF\\iu7%F`iu7$Fg_u$!+W2#=p%Fe[oF\\iu7%Fdiu7$Fa_u$!+)*)= Do%Fe[oF\\iu7%Fhiu7$F[_u$!+)3]Nn%Fe[oF\\iu7%F\\ju7$Fe^u$!+IC7lYFe[oF\\ iu7%F`ju7$F_^u$!+XoXdYFe[oF\\iu7%Fdju7$Fi]u$!+5Xy]YFe[oF\\iu7%Fhju7$Fc ]u$!+l>MXYFe[oF\\iu7%F\\[v7$F]]u$!+zHOTYFe[oF\\iu7%F`[v7$Fg\\u$!+-j1RY Fe[oF\\iu7%Fd[v7$Fa\\u$!+[)Q'QYFe[oF\\iu7%Fh[v7$F[\\u$!+2Z@SYFe[oF\\iu 7%F\\\\v7$Fe[u$!+w?&Qk%Fe[oF\\iu7%F`\\v7$F_[u$!+b@^\\YFe[oF\\iu7%Fd\\v 7$Fijt$!+Hd/dYFe[oF\\iu7%Fh\\v7$Fcjt$!++J>mYFe[oF\\iu7%F\\]v7$F]jt$!+M .gwYFe[oF\\iu7%F`]v7$Fgit$!+&**\\yo%Fe[oF\\iu7%Fd]v7$Fait$!+e(*\\*p%Fe [oF\\iu7%Fh]v7$F[it$!+6-76ZFe[oF\\iu7%F\\^v7$Feht$!+.YKAZFe[oF\\iu7%F` ^v7$F_ht$!+iqyKZFe[oF\\iu7%Fd^v7$Figt$!+:'\\Au%Fe[oF\\iu7%Fh^v7$Fcgt$! +02_]ZFe[oF\\iu7%F\\_v7$F]gt$!+Y&ouv%Fe[oF\\iu7%F`_v7$Fgft$!+4A,jZFe[o F\\iu7%Fd_v7$Faft$!+.C6nZFe[oF\\iu7%Fh_v7$F[ft$!+&4j(pZFe[oF\\iu7%F\\` v7$Feet$!+aW)4x%Fe[oF\\iu7%F``v7$F_et$!+=s\"3x%Fe[oF\\iu7%Fd`v7$Fidt$! +)R=$pZFe[oF\\iu7%Fh`v7$Fcdt$!+=\"elw%Fe[oF\\iu7%F\\av7$F]dt$!+7uhiZFe [oF\\iu7%F`av7$Fgct$!+$y'edZFe[oF\\iu7%Fdav7$Fact$!+u`c^ZFe[oF\\iu7%Fh av7$F[ct$!+K1mWZFe[oF\\iu7%F\\bv7$Febt$!+r$))pt%Fe[oF\\iu7%F`bv7$Fjat$ !+^JnGZFe[oF\\iu7%FdbvFfhuF\\iuF^dsFbds-F$6%7$7$$!3/++++++!H&FQF(7$$\" 3')*************3$FQF(-%'COLOURG6&F^hnF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7 $7$F($!3Q++++++!p&FQ7$F($\"3Q++++++!p&FQFacvFdcv-%(SCALINGG6#%,CONSTRA INEDG-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%% &Re(z)G%&Im(z)G-Fjdv6#%(DEFAULTG-%%VIEWG6$;$!$H&F]in$\"$4$F]in;$!$R&F] in$\"$R&F]in" 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 75 "The stability interval for the embedded order 6 scheme is (appr oximately) " }{XPPEDIT 18 0 "[-4.4818, 0];" "6#7$,$-%&FloatG6$\"&=[%! \"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two schemes can be displayed in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plots[display]([p_||(1..6),p||(1..7)],view=[-5.39..3 .19,-5.69..5.69],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z )`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 401 509 509 {PLOTDATA 2 "65-%'CURVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fj zq:!#=7$$\"38+++L*oq8)!#F$\"3'******Rp#fTJF-7$$\"3.+++![,39#!#D$\"3+++ +p'*Q7ZF-7$$\"3%******\\(3A7A!#C$\"3q*****RW!>$G'F-7$$\"3'******zr\"fr 8!#B$\"3Q+++Ve+ayF-7$$\"3#******47`L:'FC$\"3T+++$oi[U*F-7$$\"3/+++zy+0 A!#A$\"3!******fM\"e*4\"!#<7$$\"3_+++'Gp%*o'FN$\"3++++FSpc7FQ7$$\"3/++ +^mw#y\"!#@$\"31+++[P$QT\"FQ7$$\"3G+++K+*zF%FZ$\"33+++^W+r:FQ7$$\"3[++ +()4\">S*FZ$\"34+++,$)=G!#?$\"3)******f;8`)=FQ7$$\" 3'*******p'e:k$Fdo$\"3'******fn&>U?FQ7$$\"3u*****R>4T\\'Fdo$\"3=+++>PX )>#FQ7$$\"3%******>;%*o3\"!#>$\"3)******H5&R`BFQ7$$\"3!******H=!e.'G#f]#FQ7$$\"3++++`_-!\\#Fdp$\"3;+++&4YXl#FQ7$$\"3M+++.*4 !yLFdp$\"3!)*****f-*e(z#FQ7$$\"39+++r*y$QUFdp$\"3z*****RclN$HFQ7$$\"3% )*****f5a#3\\Fdp$\"3))*****\\)R]hIFQ7$$\"3D+++6NOK_Fdp$\"33+++&Q<5=$FQ 7$$\"3++++\"\\\"3%4&Fdp$\"39+++=H9#H$FQ7$$\"3N+++lqCEWFdp$\"3))*****f_ B^R$FQ7$$\"36+++1/62KFdp$\"3y*****H)zD!\\$FQ7$$\"3++++@[(*\\9Fdp$\"31+ ++z(Ryd$FQ7$$!3!3++ge=83)Fdo$\"3#******>(=:eOFQ7$$!3;+++Iu];NFdp$\"3() ******o3[JPFQ7$$!3t*****4=t(=mFdp$\"3))*****f')G\")z$FQ7$$!3)******4yj e+\"F-$\"3#******Rk:%eQFQ7$$!3)******H5d$y8F-$\"3()*****fUsE\"RFQ7$$!3 '******R=vYx\"F-$\"3?+++.9BhRFQ7$$!3')*****p+w2>#F-$\"3')*****H*[T/SFQ 7$$!3@+++%eYKi#F-$\"3r******3k_USFQ7$$!3=+++&Q[#pIF-$\"3k*******\\We2% FQ7$$!3%*******[\\WENF-$\"32+++5#>Y5%FQ7$$!3y*****R`FH*RF-$\"3Q+++V12H TFQ7$$!3!)*****zXZrY%F-$\"3;+++n*)Q\\TFQ7$$!3))*****>0fy%\\F-$\"3q**** ***HNd;%FQ7$$!3\"******R_uSV&F-$\"3m*****p.V#yTFQ7$$!3U+++;2.DfF-$\"3g *****RJ>q=%FQ7$$!3e*****RIn,U'F-$\"3w*****RTY@>%FQ7$$!37+++Fh6>pF-$\"3 S+++:Io$>%FQ7$$!3g******R%FQ7$$!3.+++UR&z#zF-$\"3 ?+++Jv5'=%FQ7$$!31+++GR8Q%)F-$\"3q*****H`.q<%FQ7$$!3M+++Gnx_*)F-$\"3%* *****f6FV;%FQ7$$!3)******H\"Qvs%*F-$\"3E+++*3M![TFQ7$$!3M+++64O****F-$ \"3l*****RJl!GTFQ7$$!31+++aOW`5FQ$\"3%******\\f_V5%FQ7$$!3\"******>]^! 36FQ$\"3++++L'Ho2%FQ7$$!33+++&*45k6FQ$\"3U+++zK)GrRFQ7$$!3'******R oefM\"FQ$\"3$)*****HOW*GRFQ7$$!3'******4\"*=LT\"FQ$\"3)*******yf\"Q)QF Q7$$!3'*******zW5&[\"FQ$\"3*)*****>1/p$QFQ7$$!31+++NX`h:FQ$\"3=+++$[p& *y$FQ7$$!3,+++X;AU;FQ$\"3$******zC6Lu$FQ7$$!3'******RRKhs\"FQ$\"3!**** **4H\"Q*p$FQ7$$!3,+++8w$=\"=FQ$\"3#******Rca%eOFQ7$$!3++++17!z*=FQ$\"3 %******fKi0i$FQ7$$!3-+++fs;$)>FQ$\"3'******>Uo_e$FQ7$$!3=+++7>&o1#FQ$ \"3)******\\<:>b$FQ7$$!3/+++Y&*\\[@FQ$\"34+++\\&4)>NFQ7$$!3-+++sD*yA#F Q$\"3!*******\\Y$fLFQ7$$!3y*****4nH7f#FQ$\"3/+++v#zZK$FQ7$$!33+++ ,(*zdEFQ$\"3))*****4Vq))G$FQ7$$!3)******f\\'fAFFQ$\"3))*****H/'[^KFQ7$ $!33+++V*Gdy#FQ$\"34+++C,_7KFQ7$$!3))*****R+-t%GFQ$\"3%)*****Ht))=<$FQ 7$$!3z*****f94DNJFQ$\"31+++@]&>%HFQ7$$!3?+++o9`*=$FQ$\"3! )*****H^Z/*GFQ7$$!3&******z&*HHC$FQ$\"3()*****>f3r$GFQ7$$!3%)*****p1Qb H$FQ$\"3;+++KD(>y#FQ7$$!39+++[IWZLFQ$\"3-+++(3\"4DFFQ7$$!3-+++lYs)R$FQ $\"37+++$QMlm#FQ7$$!3!)*****\\H^%\\MFQ$\"3?+++45R1EFQ7$$!39+++\"[v'*\\ $FQ$\"3)*******yowWDFQ7$$!3=++++%H%\\NFQ$\"3++++fDy\"[#FQ7$$!3/+++14s) f$FQ$\"34+++L*ovT#FQ7$$!3/+++_3`ZOFQ$\"3%******zShAN#FQ7$$!3')*****RJ7 ep$FQ$\"3'******z=$*fG#FQ7$$!3!)*****z/#\\VPFQ$\"3*)*****\\T)))=AFQ7$$ !3?+++.TZ!z$FQ$\"3%)*****H3c5:#FQ7$$!3%)*****R8Xm$QFQ$\"31+++,be#3#FQ7 $$!3%******fk!)=)QFQ$\"3!)*****fvVN,#FQ7$$!3>+++\\80ERFQ$\"31+++p_(R%> FQ7$$!35+++6*G!pRFQ$\"3/+++ZJ!R(=FQ7$$!3&*******G1p5SFQ$\"3)*******o=L .=FQ7$$!3O+++KG#40%FQ$\"30+++>2DK%FQ$\"3'******4?$GU9FQ7$$!3m*****4`)[FUFQ$\"3) *******)o1#o8FQ7$$!3s*****4#QUdUFQ$\"3))******HeU$H\"FQ7$$!3S+++4tY&G% FQ$\"3.+++W-\"z@\"FQ7$$!3%)*****4<(f6VFQ$\"3#*******)eM;9\"FQ7$$!3I+++ y2!eL%FQ$\"31+++n'zX1\"FQ7$$!3i*****H9u!eVFQ$\"3]*****4DMt')*F-7$$!3A+ ++&Q@%yVFQ$\"3u*****p<>43*F-7$$!3')*****p;aoR%FQ$\"3[+++hDg'G)F-7$$!3! )*****4-\"R8WFQ$\"3a*****\\]NX[(F-7$$!3c*****>]c!GWFQ$\"3/+++)>y\\n'F- 7$$!3\"******p5!)3W%FQ$\"3a*****p63$eeF-7$$!3++++))[*=X%FQ$\"3K+++Wk-N ]F-7$$!3A+++!yN6Y%FQ$\"3$)*****p_hd?%F-7$$!3?+++WwjoWFQ$\"3?+++;\"o7P$ F-7$$!3!)*****H8LWZ%FQ$\"3C+++M6UKDF-7$$!3-+++B0byWFQ$\"3')*****R3.-p \"F-7$$!3e*****pp65[%FQ$\"3))*****H\"z%oX)Fdp7$$!3#)*****z]I=[%FQF(7$F [am$!3))*****H\"z%oX)Fdp7$Ff`m$!3')*****R3.-p\"F-7$Fa`m$!3C+++M6UKDF-7 $F\\`m$!3?+++;\"o7P$F-7$Fg_m$!3$)*****p_hd?%F-7$Fb_m$!3K+++Wk-N]F-7$F] _m$!3a*****p63$eeF-7$Fh^m$!3/+++)>y\\n'F-7$Fc^m$!3a*****\\]NX[(F-7$F^^ m$!3[+++hDg'G)F-7$Fi]m$!3u*****p<>43*F-7$Fd]m$!3]*****4DMt')*F-7$F_]m$ !31+++n'zX1\"FQ7$Fj\\m$!3#*******)eM;9\"FQ7$Fe\\m$!3.+++W-\"z@\"FQ7$F` \\m$!3))******HeU$H\"FQ7$F[\\m$!3)*******)o1#o8FQ7$Ff[m$!3'******4?$GU 9FQ7$Fa[m$!3*******p^*o::FQ7$F\\[m$!35+++D?Y)e\"FQ7$Fgjl$!31+++#[P1m\" FQ7$Fbjl$!30+++>2DKFQ7$F^il$!3!)*****fvVN,#FQ7$Fihl$!31+++,be#3#FQ7$Fdhl$ !3%)*****H3c5:#FQ7$F_hl$!3*)*****\\T)))=AFQ7$Fjgl$!3'******z=$*fG#FQ7$ Fegl$!3%******zShAN#FQ7$F`gl$!34+++L*ovT#FQ7$F[gl$!3++++fDy\"[#FQ7$Fff l$!3)*******yowWDFQ7$Fafl$!3?+++45R1EFQ7$F\\fl$!37+++$QMlm#FQ7$Fgel$!3 -+++(3\"4DFFQ7$Fbel$!3;+++KD(>y#FQ7$F]el$!3()*****>f3r$GFQ7$Fhdl$!3!)* ****H^Z/*GFQ7$Fcdl$!31+++@]&>%HFQ7$F^dl$!3')*****\\.8;*HFQ7$Ficl$!3#** ****pK=%RIFQ7$Fdcl$!3')*****RV\"Q&3$FQ7$F_cl$!3<+++**f_HJFQ7$Fjbl$!3%) *****Ht))=<$FQ7$Febl$!34+++C,_7KFQ7$F`bl$!3))*****H/'[^KFQ7$F[bl$!3))* ****4Vq))G$FQ7$Ffal$!3/+++v#zZK$FQ7$Faal$!3?+++\">Y$fLFQ7$F\\al$!3-+++ V*RFR$FQ7$Fg`l$!31+++%Hu^U$FQ7$Fb`l$!3#)******H/#pX$FQ7$F]`l$!3!****** *\\NFQ7$Fc_l$!3)******\\<:>b$FQ7$F^_l$!3'*** ***>Uo_e$FQ7$Fi^l$!3%******fKi0i$FQ7$Fd^l$!3#******Rca%eOFQ7$F_^l$!3!* *****4H\"Q*p$FQ7$Fj]l$!3$******zC6Lu$FQ7$Fe]l$!3=+++$[p&*y$FQ7$F`]l$!3 *)*****>1/p$QFQ7$F[]l$!3)*******yf\"Q)QFQ7$Ff\\l$!3$)*****HOW*GRFQ7$Fa \\l$!3%******>K)GrRFQ7$F\\\\l$!3G+++$GI-,%FQ7$Fg[l$!3U+++z%FQ7$Fjx$!3S+++:Io$>%FQ7$Fex$!3w*****RTY@>%F Q7$F`x$!3g*****RJ>q=%FQ7$F[x$!3m*****p.V#yTFQ7$Ffw$!3q*******HNd;%FQ7$ Faw$!3;+++n*)Q\\TFQ7$F\\w$!3Q+++V12HTFQ7$Fgv$!32+++5#>Y5%FQ7$Fbv$!3k** *****\\We2%FQ7$F]v$!3r******3k_USFQ7$Fhu$!3')*****H*[T/SFQ7$Fcu$!3?+++ .9BhRFQ7$F^u$!3()*****fUsE\"RFQ7$Fit$!3#******Rk:%eQFQ7$Fdt$!3))*****f ')G\")z$FQ7$F_t$!3()******o3[JPFQ7$Fjs$!3#******>(=:eOFQ7$Fes$!31+++z( Ryd$FQ7$F`s$!3y*****H)zD!\\$FQ7$F[s$!3))*****f_B^R$FQ7$Ffr$!39+++=H9#H $FQ7$Far$!33+++&Q<5=$FQ7$F\\r$!3))*****\\)R]hIFQ7$Fgq$!3z*****RclN$HFQ 7$Fbq$!3!)*****f-*e(z#FQ7$F]q$!3;+++&4YXl#FQ7$Fhp$!3\")*****>'G#f]#FQ7 $Fbp$!3)******H5&R`BFQ7$F]p$!3=+++>PX)>#FQ7$Fho$!3'******fn&>U?FQ7$Fbo $!3)******f;8`)=FQ7$F]o$!34+++,$)=G$G'F-7$F5$!3++++p' *Q7ZF-7$$\"3U+++G*oq8)F1$!3'******Rp#fTJF-7$F($!3++++Fjzq:F-F'F*F.F4F: F@FFFKFRFWFgnF\\oFaoFgoF\\pFapFgpF\\qFaqFfqF[rF`rFerFjrF_sFdsFisF^tFct FhtF]uFbuFguF\\vFavFfvF[wF`wFewFjwF_x-%&COLORG6&%$RGBG$\"\"%!\"\"F(F(- %)POLYGONSG6f\\l7%F'7$F($\"+Fjzq:!#57$$!$D#!\"#F(7%Ffhn7$$\"+L*oq8)Fdp $\"+%p#fTJFihnFjhn7%F_in7$$\"+![,39#FQ$\"+p'*Q7ZFihnFjhn7%Fein7$$\"+v3 A7A!#;$\"+W/>$G'FihnFjhn7%F[jn7$$\"+= S*F`\\o$\"+,$)=G!#7$\"+mJJ&)=Fe[oFjhn7%F`] o7$$\"+q'e:k$Fc]o$\"+wc>U?Fe[oFjhn7%Fg]o7$$\"+%>4T\\'Fc]o$\"+>PX)>#Fe[ oFjhn7%F]^o7$$\"+iT*o3\"!#6$\"+.^R`BFe[oFjhn7%Fc^o7$$\"+$=!e.#Fihn$\"+$*[T/SFe[oFjhn7%Fjdo7$$!+%eYKi#Fihn$\"+4k_USFe[oFjhn7%F`eo7$ $!+&Q[#pIFihn$\"++X%e2%Fe[oFjhn7%Ffeo7$$!+\\\\WENFihn$\"+5#>Y5%Fe[oFjh n7%F\\fo7$$!+Mv#H*RFihn$\"+V12HTFe[oFjhn7%Fbfo7$$!+eu9nWFihn$\"+n*)Q\\ TFe[oFjhn7%Fhfo7$$!+_!fy%\\Fihn$\"++`tlTFe[oFjhn7%F^go7$$!+CX2MaFihn$ \"+PICyTFe[oFjhn7%Fdgo7$$!+;2.DfFihn$\"+9$>q=%Fe[oFjhn7%Fjgo7$$!+/t;?k Fihn$\"+9k9#>%Fe[oFjhn7%F`ho7$$!+Fh6>pFihn$\"+:Io$>%Fe[oFjhn7%Ffho7$$! +S%Fe[oFjhn7%F\\io7$$!+UR&z#zFihn$\"+Jv5'=%Fe[oFjhn7 %Fbio7$$!+GR8Q%)Fihn$\"+LN+xTFe[oFjhn7%Fhio7$$!+Gnx_*)Fihn$\"+;rKkTFe[ oFjhn7%F^jo7$$!+8Qvs%*Fihn$\"+*3M![TFe[oFjhn7%Fdjo7$$!+64O****Fihn$\"+ 9`1GTFe[oFjhn7%Fjjo7$$!+aOW`5Fe[o$\"+&f_V5%Fe[oFjhn7%F`[p7$$!+-:036Fe[ o$\"+L'Ho2%Fe[oFjhn7%Ff[p7$$!+&*45k6Fe[o$\"+zFe[o$\"+A%o_e$Fe [oFjhn7%F^`p7$$!+7>&o1#Fe[o$\"+v^\">b$Fe[oFjhn7%Fd`p7$$!+Y&*\\[@Fe[o$ \"+\\&4)>NFe[oFjhn7%Fj`p7$$!+sD*yA#Fe[o$\"+]Y$fLFe [oFjhn7%Fhbp7$$!+r'H7f#Fe[o$\"+v#zZK$Fe[oFjhn7%F^cp7$$!+,(*zdEFe[o$\"+ J/())G$Fe[oFjhn7%Fdcp7$$!+'\\'fAFFe[o$\"+Vg[^KFe[oFjhn7%Fjcp7$$!+V*Gdy #Fe[o$\"+C,_7KFe[oFjhn7%F`dp7$$!+/?IZGFe[o$\"+L())=<$Fe[oFjhn7%Ffdp7$$ !+YrT2HFe[o$\"+**f_HJFe[oFjhn7%F\\ep7$$!+JJ%HFe[oFjhn7%Fdfp7$$!+o9`*=$Fe[o$\"+8 vW!*GFe[oFjhn7%Fjfp7$$!+e*HHC$Fe[o$\"+#f3r$GFe[oFjhn7%F`gp7$$!+n!QbH$F e[o$\"+KD(>y#Fe[oFjhn7%Ffgp7$$!+[IWZLFe[o$\"+(3\"4DFFe[oFjhn7%F\\hp7$$ !+lYs)R$Fe[o$\"+$QMlm#Fe[oFjhn7%Fbhp7$$!+&H^%\\MFe[o$\"+45R1EFe[oFjhn7 %Fhhp7$$!+\"[v'*\\$Fe[o$\"+zowWDFe[oFjhn7%F^ip7$$!++%H%\\NFe[o$\"+fDy \"[#Fe[oFjhn7%Fdip7$$!+14s)f$Fe[o$\"+L*ovT#Fe[oFjhn7%Fjip7$$!+_3`ZOFe[ o$\"+39E_BFe[oFjhn7%F`jp7$$!+9B\"ep$Fe[o$\"+)=$*fG#Fe[oFjhn7%Ffjp7$$!+ [?\\VPFe[o$\"+:%)))=AFe[oFjhn7%F\\[q7$$!+.TZ!z$Fe[o$\"+$3c5:#Fe[oFjhn7 %Fb[q7$$!+M^kOQFe[o$\"+,be#3#Fe[oFjhn7%Fh[q7$$!+Y1)=)QFe[o$\"+cPa8?Fe[ oFjhn7%F^\\q7$$!+\\80ERFe[o$\"+p_(R%>Fe[oFjhn7%Fd\\q7$$!+6*G!pRFe[o$\" +ZJ!R(=Fe[oFjhn7%Fj\\q7$$!+H1p5SFe[o$\"+p=L.=Fe[oFjhn7%F`]q7$$!+KG#40% Fe[o$\"+>2DK%Fe[o$\"+,KGU9Fe[oFjhn7%F^_q7$$!+J&)[FUFe[o$\"+*o1#o 8Fe[oFjhn7%Fd_q7$$!+@QUdUFe[o$\"+IeU$H\"Fe[oFjhn7%Fj_q7$$!+4tY&G%Fe[o$ \"+W-\"z@\"Fe[oFjhn7%F``q7$$!+rrf6VFe[o$\"+*eM;9\"Fe[oFjhn7%Ff`q7$$!+y 2!eL%Fe[o$\"+n'zX1\"Fe[oFjhn7%F\\aq7$$!+VT2eVFe[o$\"+^ULn)*FihnFjhn7%F baq7$$!+&Q@%yVFe[o$\"+x\">43*FihnFjhn7%Fhaq7$$!+nT&oR%Fe[o$\"+hDg'G)Fi hnFjhn7%F^bq7$$!+@5R8WFe[o$\"+0b`%[(FihnFjhn7%Fdbq7$$!+-l0GWFe[o$\"+)> y\\n'FihnFjhn7%Fjbq7$$!+2,)3W%Fe[o$\"+<\"3$eeFihnFjhn7%F`cq7$$!+))[*=X %Fe[o$\"+Wk-N]FihnFjhn7%Ffcq7$$!+!yN6Y%Fe[o$\"+F:w0UFihnFjhn7%F\\dq7$$ !+WwjoWFe[o$\"+;\"o7P$FihnFjhn7%Fbdq7$$!+LJVuWFe[o$\"+M6UKDFihnFjhn7%F hdq7$$!+B0byWFe[o$\"+%3.-p\"FihnFjhn7%F^eq7$$!+(p65[%Fe[o$\"+8z%oX)Ff^ oFjhn7%Fdeq7$$!+30$=[%Fe[oF(Fjhn7%Fjeq7$Feeq$!+8z%oX)Ff^oFjhn7%F^fq7$F _eq$!+%3.-p\"FihnFjhn7%Fbfq7$Fidq$!+M6UKDFihnFjhn7%Fffq7$Fcdq$!+;\"o7P $FihnFjhn7%Fjfq7$F]dq$!+F:w0UFihnFjhn7%F^gq7$Fgcq$!+Wk-N]FihnFjhn7%Fbg q7$Facq$!+<\"3$eeFihnFjhn7%Ffgq7$F[cq$!+)>y\\n'FihnFjhn7%Fjgq7$Febq$!+ 0b`%[(FihnFjhn7%F^hq7$F_bq$!+hDg'G)FihnFjhn7%Fbhq7$Fiaq$!+x\">43*FihnF jhn7%Ffhq7$Fcaq$!+^ULn)*FihnFjhn7%Fjhq7$F]aq$!+n'zX1\"Fe[oFjhn7%F^iq7$ Fg`q$!+*eM;9\"Fe[oFjhn7%Fbiq7$Fa`q$!+W-\"z@\"Fe[oFjhn7%Ffiq7$F[`q$!+Ie U$H\"Fe[oFjhn7%Fjiq7$Fe_q$!+*o1#o8Fe[oFjhn7%F^jq7$F__q$!+,KGU9Fe[oFjhn 7%Fbjq7$Fi^q$!+<&*o::Fe[oFjhn7%Ffjq7$Fc^q$!+D?Y)e\"Fe[oFjhn7%Fjjq7$F]^ q$!+#[P1m\"Fe[oFjhn7%F^[r7$Fg]q$!+>2DKFe[oFjhn7%F^\\ r7$F_\\q$!+cPa8?Fe[oFjhn7%Fb\\r7$Fi[q$!+,be#3#Fe[oFjhn7%Ff\\r7$Fc[q$!+ $3c5:#Fe[oFjhn7%Fj\\r7$F][q$!+:%)))=AFe[oFjhn7%F^]r7$Fgjp$!+)=$*fG#Fe[ oFjhn7%Fb]r7$Fajp$!+39E_BFe[oFjhn7%Ff]r7$F[jp$!+L*ovT#Fe[oFjhn7%Fj]r7$ Feip$!+fDy\"[#Fe[oFjhn7%F^^r7$F_ip$!+zowWDFe[oFjhn7%Fb^r7$Fihp$!+45R1E Fe[oFjhn7%Ff^r7$Fchp$!+$QMlm#Fe[oFjhn7%Fj^r7$F]hp$!+(3\"4DFFe[oFjhn7%F ^_r7$Fggp$!+KD(>y#Fe[oFjhn7%Fb_r7$Fagp$!+#f3r$GFe[oFjhn7%Ff_r7$F[gp$!+ 8vW!*GFe[oFjhn7%Fj_r7$Fefp$!+@]&>%HFe[oFjhn7%F^`r7$F_fp$!+NIh\"*HFe[oF jhn7%Fb`r7$Fiep$!+F$=%RIFe[oFjhn7%Ff`r7$Fcep$!+M9Q&3$Fe[oFjhn7%Fj`r7$F ]ep$!+**f_HJFe[oFjhn7%F^ar7$Fgdp$!+L())=<$Fe[oFjhn7%Fbar7$Fadp$!+C,_7K Fe[oFjhn7%Ffar7$F[dp$!+Vg[^KFe[oFjhn7%Fjar7$Fecp$!+J/())G$Fe[oFjhn7%F^ br7$F_cp$!+v#zZK$Fe[oFjhn7%Fbbr7$Fibp$!+\">Y$fLFe[oFjhn7%Ffbr7$Fcbp$!+ V*RFR$Fe[oFjhn7%Fjbr7$F]bp$!+%Hu^U$Fe[oFjhn7%F^cr7$Fgap$!+I/#pX$Fe[oFj hn7%Fbcr7$Faap$!+]NFe[oFjhn7%Fjcr7$F e`p$!+v^\">b$Fe[oFjhn7%F^dr7$F_`p$!+A%o_e$Fe[oFjhn7%Fbdr7$Fi_p$!+EBc?O Fe[oFjhn7%Ffdr7$Fc_p$!+kXXeOFe[oFjhn7%Fjdr7$F]_p$!+\"H\"Q*p$Fe[oFjhn7% F^er7$Fg^p$!+[7JVPFe[oFjhn7%Fber7$Fa^p$!+$[p&*y$Fe[oFjhn7%Ffer7$F[^p$! +iS!p$QFe[oFjhn7%Fjer7$Fe]p$!+zf\"Q)QFe[oFjhn7%F^fr7$F_]p$!+jV%*GRFe[o Fjhn7%Fbfr7$Fi\\p$!+A$)GrRFe[oFjhn7%Fffr7$Fc\\p$!+$GI-,%Fe[oFjhn7%Fjfr 7$F]\\p$!+z%Fe[oFjhn7%F^ir7$Fgho$!+:Io$>%Fe[oFj hn7%Fbir7$Faho$!+9k9#>%Fe[oFjhn7%Ffir7$F[ho$!+9$>q=%Fe[oFjhn7%Fjir7$Fe go$!+PICyTFe[oFjhn7%F^jr7$F_go$!++`tlTFe[oFjhn7%Fbjr7$Fifo$!+n*)Q\\TFe [oFjhn7%Ffjr7$Fcfo$!+V12HTFe[oFjhn7%Fjjr7$F]fo$!+5#>Y5%Fe[oFjhn7%F^[s7 $Fgeo$!++X%e2%Fe[oFjhn7%Fb[s7$Faeo$!+4k_USFe[oFjhn7%Ff[s7$F[eo$!+$*[T/ SFe[oFjhn7%Fj[s7$Fedo$!+.9BhRFe[oFjhn7%F^\\s7$F_do$!+ECn7RFe[oFjhn7%Fb \\s7$Fico$!+WcTeQFe[oFjhn7%Ff\\s7$Fcco$!+m)G\")z$Fe[oFjhn7%Fj\\s7$F]co $!+p3[JPFe[oFjhn7%F^]s7$Fgbo$!+s=:eOFe[oFjhn7%Fb]s7$Fabo$!+z(Ryd$Fe[oF jhn7%Ff]s7$F[bo$!+$)zD!\\$Fe[oFjhn7%Fj]s7$Feao$!+EN7&R$Fe[oFjhn7%F^^s7 $F_ao$!+=H9#H$Fe[oFjhn7%Fb^s7$Fi`o$!+&Q<5=$Fe[oFjhn7%Ff^s7$Fc`o$!+&)R] hIFe[oFjhn7%Fj^s7$F]`o$!+kbcLHFe[oFjhn7%F^_s7$Fg_o$!+E!*e(z#Fe[oFjhn7% Fb_s7$Fa_o$!+&4YXl#Fe[oFjhn7%Ff_s7$F[_o$!+iG#f]#Fe[oFjhn7%Fj_s7$Fd^o$! +.^R`BFe[oFjhn7%F^`s7$F^^o$!+>PX)>#Fe[oFjhn7%Fb`s7$Fh]o$!+wc>U?Fe[oFjh n7%Ff`s7$Fa]o$!+mJJ&)=Fe[oFjhn7%Fj`s7$F[]o$!+,$)=G$G'FihnFjhn7%F\\c s7$Ffin$!+p'*Q7ZFihnFjhn7%F`cs7$$\"+G*oq8)Fdp$!+%p#fTJFihnFjhn7%Fdcs7$ F($!+Fjzq:FihnFjhn7%FjcsF'FjhnFehnF^inFdinFjinFajnFhjnF^[oFf[oF\\\\oFc \\oFi\\oF_]oFf]oF\\^oFb^oFi^oF__oFe_oF[`oFa`oFg`oF]aoFcaoFiaoF_boFeboF [coFacoFgcoF]doFcdoFidoF_eoFeeoF[foFafoFgfoF]goFcgoFigo-F\\hn6&F^hn$\" #yF]inF(F(-%&STYLEG6#%,PATCHNOGRIDG-F$6$7K7$$\"3)*******puamBFQ$\"3c** ****>*[)>ZFQ7$$\"3#)******pb(QO#FQ$\"3u*****4:t'GZFQ7$$\"3?+++r^-gBFQ$ \"3\")*****4P))pt%FQ7$$\"3-+++Sx2bBFQ$\"3W+++K1mWZFQ7$$\"37+++!QC\"\\B FQ$\"37+++u`c^ZFQ7$$\"34+++&olAM#FQ$\"3#)*****Hy'edZFQ7$$\"31+++v>hMBF Q$\"3++++7uhiZFQ7$$\"3.+++uQGEBFQ$\"3%)*****z6elw%FQ7$$\"3!********=8u J#FQ$\"3C+++)R=$pZFQ7$$\"3-+++AS93BFQ$\"3O+++=s\"3x%FQ7$$\"3!)*****R1M ')H#FQ$\"3s*****RX%)4x%FQ7$$\"33+++pf0*G#FQ$\"31+++&4j(pZFQ7$$\"3/+++B ')fzAFQ$\"3i*****HS7rw%FQ7$$\"3A+++\\!o/F#FQ$\"3A+++4A,jZFQ7$$\"3#**** **z_()=E#FQ$\"39+++Y&ouv%FQ7$$\"3;+++Gj4aAFQ$\"3e*****\\q?0v%FQ7$$\"3% *******)=YtC#FQ$\"3U+++:'\\Au%FQ7$$\"3<+++9X*=C#FQ$\"3))*****>1(yKZFQ7 $$\"3,+++dJ*zB#FQ$\"3o*****HgCBs%FQ7$$\"3(******4'>(eB#FQ$\"3*)*****4@ ?6r%FQ7$$\"3*)******>rrNAFQ$\"3'******zv*\\*p%FQ7$$\"3,+++UhkPAFQ$\"3m *****\\**\\yo%FQ7$$\"3<+++TMoTAFQ$\"3u*****RL+mn%FQ7$$\"3)******4bUxC# FQ$\"33++++J>mYFQ7$$\"3A+++V7ibAFQ$\"3-+++Hd/dYFQ7$$\"3'*******QK,lAFQ $\"3/+++b@^\\YFQ7$$\"3'******4SNbF#FQ$\"3%******f2_Qk%FQ7$$\"3%******H JknG#FQ$\"3;+++2Z@SYFQ7$$\"31+++6QF)H#FQ$\"3E+++[)Q'QYFQ7$$\"3!******H 8m'4BFQ$\"3\")*****>Im!RYFQ7$$\"3(******zx#f?BFQ$\"3<+++zHOTYFQ7$$\"3> +++/[wIBFQ$\"33+++l>MXYFQ7$$\"3#)*****Rha*RBFQ$\"3E+++5Xy]YFQ7$$\"35++ +s@*zM#FQ$\"3e*****\\%oXdYFQ7$$\"3'*******o#eZN#FQ$\"3E+++IC7lYFQ7$$\" 3/+++@oZFe[o7$$\"+qb(QO#Fe[o$\"+^JnGZFe[o7$$\"$I#F]in$\"$q%F]in7 %Fiat7$$\"+r^-gBFe[o$\"+r$))pt%Fe[oF^bt7%Fdbt7$$\"+Sx2bBFe[o$\"+K1mWZF e[oF^bt7%Fjbt7$$\"+!QC\"\\BFe[o$\"+u`c^ZFe[oF^bt7%F`ct7$$\"+&olAM#Fe[o $\"+$y'edZFe[oF^bt7%Ffct7$$\"+v>hMBFe[o$\"+7uhiZFe[oF^bt7%F\\dt7$$\"+u QGEBFe[o$\"+=\"elw%Fe[oF^bt7%Fbdt7$$\"+!>8uJ#Fe[o$\"+)R=$pZFe[oF^bt7%F hdt7$$\"+AS93BFe[o$\"+=s\"3x%Fe[oF^bt7%F^et7$$\"+kSj)H#Fe[o$\"+aW)4x%F e[oF^bt7%Fdet7$$\"+pf0*G#Fe[o$\"+&4j(pZFe[oF^bt7%Fjet7$$\"+B')fzAFe[o$ \"+.C6nZFe[oF^bt7%F`ft7$$\"+\\!o/F#Fe[o$\"+4A,jZFe[oF^bt7%Ffft7$$\"+Gv )=E#Fe[o$\"+Y&ouv%Fe[oF^bt7%F\\gt7$$\"+Gj4aAFe[o$\"+02_]ZFe[oF^bt7%Fbg t7$$\"+*=YtC#Fe[o$\"+:'\\Au%Fe[oF^bt7%Fhgt7$$\"+9X*=C#Fe[o$\"+iqyKZFe[ oF^bt7%F^ht7$$\"+dJ*zB#Fe[o$\"+.YKAZFe[oF^bt7%Fdht7$$\"+h>(eB#Fe[o$\"+ 6-76ZFe[oF^bt7%Fjht7$$\"+?rrNAFe[o$\"+e(*\\*p%Fe[oF^bt7%F`it7$$\"+UhkP AFe[o$\"+&**\\yo%Fe[oF^bt7%Ffit7$$\"+TMoTAFe[o$\"+M.gwYFe[oF^bt7%F\\jt 7$$\"+^DuZAFe[o$\"++J>mYFe[oF^bt7%Fbjt7$$\"+V7ibAFe[o$\"+Hd/dYFe[oF^bt 7%Fhjt7$$\"+RK,lAFe[o$\"+b@^\\YFe[oF^bt7%F^[u7$$\"+,a`vAFe[o$\"+w?&Qk% Fe[oF^bt7%Fd[u7$$\"+8Vw'G#Fe[o$\"+2Z@SYFe[oF^bt7%Fj[u7$$\"+6QF)H#Fe[o$ \"+[)Q'QYFe[oF^bt7%F`\\u7$$\"+Lhm4BFe[o$\"+-j1RYFe[oF^bt7%Ff\\u7$$\"+y Ff?BFe[o$\"+zHOTYFe[oF^bt7%F\\]u7$$\"+/[wIBFe[o$\"+l>MXYFe[oF^bt7%Fb]u 7$$\"+9Y&*RBFe[o$\"+5Xy]YFe[oF^bt7%Fh]u7$$\"+s@*zM#Fe[o$\"+XoXdYFe[oF^ bt7%F^^u7$$\"+p#eZN#Fe[o$\"+IC7lYFe[oF^bt7%Fd^u7$$\"+@o*[)>ZFQ7$F]at$!35+++U)f1r%FQ7$Fh`t$!3\"******4Fh7q%FQ7$Fc`t$!39++ +W2#=p%FQ7$F^`t$!3k*****z*)=Do%FQ7$Fi_t$!3#)*****z3]Nn%FQ7$Fd_t$!3E+++ IC7lYFQ7$F__t$!3e*****\\%oXdYFQ7$Fj^t$!3E+++5Xy]YFQ7$Fe^t$!33+++l>MXYF Q7$F`^t$!3<+++zHOTYFQ7$F[^t$!3\")*****>Im!RYFQ7$Ff]t$!3E+++[)Q'QYFQ7$F a]t$!3;+++2Z@SYFQ7$F\\]t$!3%******f2_Qk%FQ7$Fg\\t$!3/+++b@^\\YFQ7$Fb\\ t$!3-+++Hd/dYFQ7$F]\\t$!33++++J>mYFQ7$Fh[t$!3u*****RL+mn%FQ7$Fc[t$!3m* ****\\**\\yo%FQ7$F^[t$!3'******zv*\\*p%FQ7$Fijs$!3*)*****4@?6r%FQ7$Fdj s$!3o*****HgCBs%FQ7$F_js$!3))*****>1(yKZFQ7$Fjis$!3U+++:'\\Au%FQ7$Feis $!3e*****\\q?0v%FQ7$F`is$!39+++Y&ouv%FQ7$F[is$!3A+++4A,jZFQ7$Ffhs$!3i* ****HS7rw%FQ7$Fahs$!31+++&4j(pZFQ7$F\\hs$!3s*****RX%)4x%FQ7$Fggs$!3O++ +=s\"3x%FQ7$Fbgs$!3C+++)R=$pZFQ7$F]gs$!3%)*****z6elw%FQ7$Fhfs$!3++++7u hiZFQ7$Fcfs$!3#)*****Hy'edZFQ7$F^fs$!37+++u`c^ZFQ7$Fies$!3W+++K1mWZFQ7 $Fdes$!3\")*****4P))pt%FQ7$F_es$!3u*****4:t'GZFQF[auF[hn-Fchn6L7%7$Fea t$!+?*[)>ZFe[o7$Fc`u$!+U)f1r%Fe[o7$F_bt$!$q%F]in7%Fihu7$F]`u$!+r7E,ZFe [oF\\iu7%F`iu7$Fg_u$!+W2#=p%Fe[oF\\iu7%Fdiu7$Fa_u$!+)*)=Do%Fe[oF\\iu7% Fhiu7$F[_u$!+)3]Nn%Fe[oF\\iu7%F\\ju7$Fe^u$!+IC7lYFe[oF\\iu7%F`ju7$F_^u $!+XoXdYFe[oF\\iu7%Fdju7$Fi]u$!+5Xy]YFe[oF\\iu7%Fhju7$Fc]u$!+l>MXYFe[o F\\iu7%F\\[v7$F]]u$!+zHOTYFe[oF\\iu7%F`[v7$Fg\\u$!+-j1RYFe[oF\\iu7%Fd[ v7$Fa\\u$!+[)Q'QYFe[oF\\iu7%Fh[v7$F[\\u$!+2Z@SYFe[oF\\iu7%F\\\\v7$Fe[u $!+w?&Qk%Fe[oF\\iu7%F`\\v7$F_[u$!+b@^\\YFe[oF\\iu7%Fd\\v7$Fijt$!+Hd/dY Fe[oF\\iu7%Fh\\v7$Fcjt$!++J>mYFe[oF\\iu7%F\\]v7$F]jt$!+M.gwYFe[oF\\iu7 %F`]v7$Fgit$!+&**\\yo%Fe[oF\\iu7%Fd]v7$Fait$!+e(*\\*p%Fe[oF\\iu7%Fh]v7 $F[it$!+6-76ZFe[oF\\iu7%F\\^v7$Feht$!+.YKAZFe[oF\\iu7%F`^v7$F_ht$!+iqy KZFe[oF\\iu7%Fd^v7$Figt$!+:'\\Au%Fe[oF\\iu7%Fh^v7$Fcgt$!+02_]ZFe[oF\\i u7%F\\_v7$F]gt$!+Y&ouv%Fe[oF\\iu7%F`_v7$Fgft$!+4A,jZFe[oF\\iu7%Fd_v7$F aft$!+.C6nZFe[oF\\iu7%Fh_v7$F[ft$!+&4j(pZFe[oF\\iu7%F\\`v7$Feet$!+aW)4 x%Fe[oF\\iu7%F``v7$F_et$!+=s\"3x%Fe[oF\\iu7%Fd`v7$Fidt$!+)R=$pZFe[oF\\ iu7%Fh`v7$Fcdt$!+=\"elw%Fe[oF\\iu7%F\\av7$F]dt$!+7uhiZFe[oF\\iu7%F`av7 $Fgct$!+$y'edZFe[oF\\iu7%Fdav7$Fact$!+u`c^ZFe[oF\\iu7%Fhav7$F[ct$!+K1m WZFe[oF\\iu7%F\\bv7$Febt$!+r$))pt%Fe[oF\\iu7%F`bv7$Fjat$!+^JnGZFe[oF\\ iu7%FdbvFfhuF\\iuF^dsFbds-F$6$7e\\lF'F*7$F($\"3)******Rl#fTJF-7$$\"3$) ******y7%pV#F1$\"3?+++()*)Q7ZF-7$$\"3))*****HxtDd$!#E$\"3!)*******R&=$ G'F-7$$\"3o******oPM)3$F7$\"3_+++OB)R&yF-7$$\"3#*******)*)=P)=F=$\"3]* ****Hh\"yC%*F-7$$\"3()*****4,?N%*)F=$\"3/+++\"*)e&*4\"FQ7$$\"37+++f;65 NFC$\"3'******H(=kc7FQ7$$\"3#*******>e)[=\"FN$\"34+++A-t89FQ7$$\"3**** ***>0+k`$FN$\"3))*****\\kH3d\"FQ7$$\"3\"******f#)31_*FN$\"3)******4!*[ zs\"FQ7$$\"33+++-@!oM#FZ$\"37++++$)4&)=FQ7$$\"3o*****z0wqN&FZ$\"32+++9 LGU?FQ7$$\"3&******H%=BU6Fdo$\"3#)*****R5)[*>#FQ7$$\"3/+++U06*G#Fdo$\" 3%)*****fjSmN#FQ7$$\"3D+++Kf$)GVFdo$\"3*)*****foZN^#FQ7$$\"3d*******>> ^t(Fdo$\"3-+++()yypEFQ7$$\"3&*******y-1/8Fdp$\"3&******f)\\cCGFQ7$$\"3 ;+++v20l?Fdp$\"3++++a7dwHFQ7$$\"3/+++-l*=0$Fdp$\"3,+++SM*R7$FQ7$$\"3/+ ++8-n$=%Fdp$\"35+++^S\"[E$FQ7$$\"3y*******=#*GI&Fdp$\"3')*****p!GN(R$F Q7$$\"3!******\\,$==iFdp$\"3&******\\!zo?NFQ7$$\"3!)*****HTO0w'Fdp$\"3 ))*****H/nYj$FQ7$$\"3i+++4z(o\"oFdp$\"3%******z%3iRPFQ7$$\"3C+++bw1MjF dp$\"37+++sH1OQFQ7$$\"3/+++4`\"oI&Fdp$\"3(******4,=X#RFQ7$$\"3-+++k@2i PFdp$\"3A+++eVY0SFQ7$$\"39+++_(*\\XKl9%FQ7$$!3=+++X*oX[$Fdp$\"3\")*****H8Uu?%FQ7$$!3M**** *ROa!*e'Fdp$\"3-++++mViUFQ7$$!3Y*****4-MS&**Fdp$\"3l*****fJs=J%FQ7$$!3 !********GWPN\"F-$\"3g*****\\>'3cVFQ7$$!3\"******f-K.t\"F-$\"3e*****fs !R&R%FQ7$$!3&******RTR@7#F-$\"3W+++c02IWFQ7$$!3/+++zRkEDF-$\"3u*****>i !QgWFQ7$$!3.+++;\"e<%HF-$\"3?+++%oXl[%FQ7$$!3++++\"3jdO$F-$\"3<+++l3w3 XFQ7$$!3')*****p\\^sz$F-$\"3()*****4#G>FXFQ7$$!3E+++0-3NUF-$\"3$****** f@\")>a%FQ7$$!3/+++^PLyYF-$\"3O+++L-C`XFQ7$$!3[+++J#)HE^F-$\"3!******R )*f5c%FQ7$$!3]+++/^WybF-$\"3!******Rp2bc%FQ7$$!3D+++5\\UMgF-$\"3s***** HiGmc%FQ7$$!3M+++682%\\'F-$\"3C+++)pYWc%FQ7$$!3%******zk;u&pF-$\"37+++ >\\'*eXFQ7$$!3,+++l6sCuF-$\"33+++Ec;]XFQ7$$!3++++n$>l*yF-$\"3/+++]3,QX FQ7$$!3/+++h&)ot$)F-$\"30+++^JWAXFQ7$$!3]+++wlad))F-$\"3V+++CxQ.XFQ7$$ !3W+++ft)*\\$*F-$\"3')*****>9d2[%FQ7$$!37+++EKm`)*F-$\"3K+++\"HhWX%FQ7 $$!3)******>o@s.\"FQ$\"3d*****H\"pUCWFQ7$$!3'******zSa54\"FQ$\"37+++ET j!R%FQ7$$!3++++Z5]Z6FQ$\"3?+++Y1=`VFQ7$$!3$******z7Tt?\"FQ$\"3.+++%*[Q 7VFQ7$$!3#******H&\\Ur7FQ$\"3q*****>rD*oUFQ7$$!3#*******G[ZS8FQ$\"3c** ***fNpRA%FQ7$$!33+++khu99FQ$\"3S+++sG:zTFQ7$$!33+++vnp$\\\"FQ$\"30+++' Qti8%FQ7$$!31+++dM'fd\"FQ$\"3A+++k:u'4%FQ7$$!3++++Gdwf;FQ$\"3M+++EG;hS FQ7$$!31+++>vWVFQ$\"3A+++JI-uRFQ7$$!3)******4]nR)>FQ$\"3++++:,')[R FQ7$$!3)*******4\"=$f?FQ$\"30+++PxRCRFQ7$$!3\")*****4')o@8#FQ$\"3)**** **Rw#3+RFQ7$$!35+++T/i-AFQ$\"3))******>\\[vQFQ7$$!3%)*****RX23F#FQ$\"3 A+++\"zt-&QFQ7$$!3z*****HDwoL#FQ$\"3y*****Hw'>CQFQ7$$!3#)*****>Gt4S#FQ $\"3=+++)ogqz$FQ7$$!33+++9&RKY#FQ$\"33+++uvroPFQ7$$!36+++?$3Q_#FQ$\"3) )*****fz`!RPFQ7$$!3#*******=_!Ge#FQ$\"31+++/@)zq$FQ7$$!3z*****4i[.k#FQ $\"35+++_eVvOFQ7$$!3++++W5b'p#FQ$\"3:+++6\\OTOFQ7$$!3-+++(\\?:v#FQ$\"3 (******p$Ht0OFQ7$$!3\"******p%=O0GFQ$\"3))*****\\p:&oNFQ7$$!3))*****z+ y\"eGFQ$\"3++++t-qHNFQ7$$!3@+++a325HFQ$\"3!)*****>-&G*[$FQ7$$!3'****** HrT6'HFQ$\"3))******)4!GZMFQ7$$!3.+++=8\\6IFQ$\"34+++!\\3PS$FQ7$$!3%** *****p!>71$FQ$\"35+++QtgeLFQ7$$!3=+++%[@/6$FQ$\"3)*******\\$H?J$FQ7$$! 3?+++K(*=fJFQ$\"3))*****4*R/kKFQ7$$!3))*****pH1w?$FQ$\"3&)******)4QZ@$ FQ7$$!3?+++&*3ubKFQ$\"3++++$[:U;$FQ7$$!3\"******R,YOI$FQ$\"3#******>:& f7JFQ7$$!3!)*****Hk_8N$FQ$\"3!******z)y+gIFQ7$$!33+++#)o'))R$FQ$\"3>++ +)>\"f1IFQ7$$!3++++Q\"ohW$FQ$\"3!******p'H[_HFQ7$$!3++++[%4K\\$FQ$\"3) ******\\\\9y*GFQ7$$!35+++$4?*RNFQ$\"3&)*****p\"QqUGFQ7$$!3/+++Q-@'e$FQ $\"31+++--D(y#FQ7$$!3&******p'p(>j$FQ$\"38+++90`JFFQ7$$!3'*******[36xO FQ$\"3%)*****ft(fvEFQ7$$!34+++/A]@PFQ$\"33+++\\:[>EFQ7$$!3=+++Ak/lPFQ$ \"3'******fa!>jDFQ7$$!3!*******))zk2QFQ$\"3%)*****\\]:n]#FQ7$$!3-+++JJ A\\QFQ$\"3-+++dJ.]CFQ7$$!3))*****HA,(*)QFQ$\"3++++^*4JR#FQ7$$!3++++'>D !HRFQ$\"3%)******>b!fL#FQ7$$!3?+++S7:nRFQ$\"3')*****zlv$yAFQ7$$!3A+++X ![S+%FQ$\"3$******Rxu/A#FQ7$$!36+++1epRSFQ$\"3')******Gy:i@FQ7$$!31+++ Q^3uSFQ$\"3.+++T=Q.@FQ7$$!35+++))f@2TFQ$\"3\")*****z02T/#FQ7$$!34+++4m 4RTFQ$\"3'******>(yH%)>FQ7$$!3-+++)fU(pTFQ$\"3(******zcBR#>FQ7$$!3))** ***H*f<*>%FQ$\"3!******p(*eH'=FQ7$$!3\")*****RRCuA%FQ$\"3&*******o]Q,= FQ7$$!37+++t+_aUFQ$\"3-+++<&*=RaF-7$$!3S+++[3r( e%FQ$\"38+++5z4SZF-7$$!3A+++\\qm&f%FQ$\"3=+++FL1hSF-7$$!3&******pXkCg% FQ$\"3E+++S:z#Q$F-7$$!37+++e823YFQ$\"3u*****HHU_q#F-7$$!3#*******[4Y7Y FQ$\"3(******fcF$G?F-7$$!3))*****f*=h:YFQ$\"31+++>9#>N\"F-7$$!35+++@(3 vh%FQ$\"3W*****fy(penFdp7$$!3n*****>/U\"=YFQF(7$Fh]y$!3W*****fy(penFdp 7$Fc]y$!31+++>9#>N\"F-7$F^]y$!3(******fcF$G?F-7$Fi\\y$!3u*****HHU_q#F- 7$Fd\\y$!3E+++S:z#Q$F-7$F_\\y$!3=+++FL1hSF-7$Fj[y$!38+++5z4SZF-7$Fe[y$ !3z*****H=y)>aF-7$F`[y$!3%*******G3K+hF-7$F[[y$!3k*****z\"*o7y'F-7$Ffj x$!3=+++*y'[iuF-7$Fajx$!3()******)3eO9)F-7$F\\jx$!3x*****Hx(QC))F-7$Fg ix$!3A+++7'3U]*F-7$Fbix$!34+++P#f#=5FQ7$F]ix$!3%******H_'*e3\"FQ7$Fhhx $!3!******fmsK:\"FQ7$Fchx$!3,+++DqK?7FQ7$F^hx$!3++++#\\+qG\"FQ7$Figx$! 3!******H]PKN\"FQ7$Fdgx$!34+++ux)*=9FQ7$F_gx$!31+++&p2U[\"FQ7$Fjfx$!3# ******pIh)[:FQ7$Fefx$!32+++,4#Hh\"FQ7$F`fx$!3!******4Cnjn\"FQ7$F[fx$!3 -+++<&*=RFQ7$Fgdx$!3'******>(yH%)>FQ7$Fbdx$!3\")*****z02T/#FQ7$F] dx$!3.+++T=Q.@FQ7$Fhcx$!3')******Gy:i@FQ7$Fccx$!3$******Rxu/A#FQ7$F^cx $!3')*****zlv$yAFQ7$Fibx$!3%)******>b!fL#FQ7$Fdbx$!3++++^*4JR#FQ7$F_bx $!3-+++dJ.]CFQ7$Fjax$!3%)*****\\]:n]#FQ7$Feax$!3'******fa!>jDFQ7$F`ax$ !33+++\\:[>EFQ7$F[ax$!3%)*****ft(fvEFQ7$Ff`x$!38+++90`JFFQ7$Fa`x$!31++ +--D(y#FQ7$F\\`x$!3&)*****p\"QqUGFQ7$Fg_x$!3)******\\\\9y*GFQ7$Fb_x$!3 !******p'H[_HFQ7$F]_x$!3>+++)>\"f1IFQ7$Fh^x$!3!******z)y+gIFQ7$Fc^x$!3 #******>:&f7JFQ7$F^^x$!3++++$[:U;$FQ7$Fi]x$!3&)******)4QZ@$FQ7$Fd]x$!3 ))*****4*R/kKFQ7$F_]x$!3)*******\\$H?J$FQ7$Fj\\x$!35+++QtgeLFQ7$Fe\\x$ !34+++!\\3PS$FQ7$F`\\x$!3))******)4!GZMFQ7$F[\\x$!3!)*****>-&G*[$FQ7$F f[x$!3++++t-qHNFQ7$Fa[x$!3))*****\\p:&oNFQ7$F\\[x$!3(******p$Ht0OFQ7$F gjw$!3:+++6\\OTOFQ7$Fbjw$!35+++_eVvOFQ7$F]jw$!31+++/@)zq$FQ7$Fhiw$!3)) *****fz`!RPFQ7$Fciw$!33+++uvroPFQ7$F^iw$!3=+++)ogqz$FQ7$Fihw$!3y*****H w'>CQFQ7$Fdhw$!3A+++\"zt-&QFQ7$F_hw$!3))******>\\[vQFQ7$Fjgw$!3)****** Rw#3+RFQ7$Fegw$!30+++PxRCRFQ7$F`gw$!3++++:,')[RFQ7$F[gw$!3A+++JI-uRFQ7 $Fffw$!3@+++&fx0+%FQ7$Fafw$!3o******G*Q$HSFQ7$F\\fw$!3M+++EG;hSFQ7$Fge w$!3A+++k:u'4%FQ7$Fbew$!30+++'Qti8%FQ7$F]ew$!3S+++sG:zTFQ7$Fhdw$!3c*** **fNpRA%FQ7$Fcdw$!3q*****>rD*oUFQ7$F^dw$!3.+++%*[Q7VFQ7$Ficw$!3?+++Y1= `VFQ7$Fdcw$!37+++ETj!R%FQ7$F_cw$!3d*****H\"pUCWFQ7$Fjbw$!3K+++\"HhWX%F Q7$Febw$!3')*****>9d2[%FQ7$F`bw$!3V+++CxQ.XFQ7$F[bw$!30+++^JWAXFQ7$Ffa w$!3/+++]3,QXFQ7$Faaw$!33+++Ec;]XFQ7$F\\aw$!37+++>\\'*eXFQ7$Fg`w$!3C++ +)pYWc%FQ7$Fb`w$!3s*****HiGmc%FQ7$F]`w$!3!******Rp2bc%FQ7$Fh_w$!3!**** **R)*f5c%FQ7$Fc_w$!3O+++L-C`XFQ7$F^_w$!3$******f@\")>a%FQ7$Fi^w$!3()** ***4#G>FXFQ7$Fd^w$!3<+++l3w3XFQ7$F_^w$!3?+++%oXl[%FQ7$Fj]w$!3u*****>i! QgWFQ7$Fe]w$!3W+++c02IWFQ7$F`]w$!3e*****fs!R&R%FQ7$F[]w$!3g*****\\>'3c VFQ7$Ff\\w$!3l*****fJs=J%FQ7$Fa\\w$!3-++++mViUFQ7$F\\\\w$!3\")*****H8U u?%FQ7$$!3M+++%)Hn')oFdo$!3G+++!>Kl9%FQ7$Fb[w$!37+++]ZLzSFQ7$F][w$!3A+ ++eVY0SFQ7$Fhjv$!3(******4,=X#RFQ7$Fcjv$!37+++sH1OQFQ7$F^jv$!3%******z %3iRPFQ7$Fiiv$!3))*****H/nYj$FQ7$Fdiv$!3&******\\!zo?NFQ7$F_iv$!3')*** **p!GN(R$FQ7$Fjhv$!35+++^S\"[E$FQ7$Fehv$!3,+++SM*R7$FQ7$F`hv$!3++++a7d wHFQ7$F[hv$!3&******f)\\cCGFQ7$Ffgv$!3-+++()yypEFQ7$Fagv$!3*)*****foZN ^#FQ7$F\\gv$!3%)*****fjSmN#FQ7$Fgfv$!3#)*****R5)[*>#FQ7$Fbfv$!32+++9LG U?FQ7$F]fv$!37++++$)4&)=FQ7$Fhev$!3)******4!*[zs\"FQ7$$\"3J+++`+SONFN$ !3))*****\\kH3d\"FQ7$F^ev$!34+++A-t89FQ7$Fidv$!3'******H(=kc7FQ7$$\"3Q +++1+_V*)F=$!3/+++\"*)e&*4\"FQ7$$\"3'******z*)=P)=F=$!3]*****Hh\"yC%*F -7$Fjcv$!3_+++OB)R&yF-7$$\"3++++gPdsNFfcv$!3!)*******R&=$G'F-7$$\"3u** ***4CTpV#F1$!3?+++()*)Q7ZF-7$F($!3)******Rl#fTJF-FhgnF'-F\\hn6&F^hn$\" #XF]inF(F(-Fchn6f\\l7%F'Ffhn7$$FCFahnF(7%Ffhn7$F($\"+aEfTJFihnF`iz7%Fc iz7$$\"+z7%pV#Fdp$\"+()*)Q7ZFihnF`iz7%Fgiz7$$\"+tPdsNF-$\"++a=$G'FihnF `iz7%F]jz7$$\"+pPM)3$FQ$\"+OB)R&yFihnF`iz7%Fcjz7$$\"+**)=P)=F^jn$\"+8; yC%*FihnF`iz7%Fijz7$$\"+6+_V*)F^jn$\"+\"*)e&*4\"Fe[oF`iz7%F_[[l7$$\"+f ;65NFejn$\"+t=kc7Fe[oF`iz7%Fe[[l7$$\"+?e)[=\"Fb[o$\"+A-t89Fe[oF`iz7%F[ \\[l7$$\"+_+SONFb[o$\"+X'H3d\"Fe[oF`iz7%Fa\\[l7$$\"+E)31_*Fb[o$\"+,*[z s\"Fe[oF`iz7%Fg\\[l7$$\"+-@!oM#F`\\o$\"++$)4&)=Fe[oF`iz7%F]][l7$$\"+eg 2d`F`\\o$\"+9LGU?Fe[oF`iz7%Fc][l7$$\"+V=BU6Fc]o$\"+/\")[*>#Fe[oF`iz7%F i][l7$$\"+U06*G#Fc]o$\"+O1kcBFe[oF`iz7%F_^[l7$$\"+Kf$)GVFc]o$\"+'oZN^# Fe[oF`iz7%Fe^[l7$$\"++#>^t(Fc]o$\"+()yypEFe[oF`iz7%F[_[l7$$\"+z-1/8Ff^ o$\"+')\\cCGFe[oF`iz7%Fa_[l7$$\"+v20l?Ff^o$\"+a7dwHFe[oF`iz7%Fg_[l7$$ \"+-l*=0$Ff^o$\"+SM*R7$Fe[oF`iz7%F]`[l7$$\"+8-n$=%Ff^o$\"+^S\"[E$Fe[oF `iz7%Fc`[l7$$\"+!>#*GI&Ff^o$\"+2GN(R$Fe[oF`iz7%Fi`[l7$$\"+:I==iFf^o$\" +0zo?NFe[oF`iz7%F_a[l7$$\"+8k`gnFf^o$\"+VqmMOFe[oF`iz7%Fea[l7$$\"+4z(o \"oFf^o$\"+[3iRPFe[oF`iz7%F[b[l7$$\"+bw1MjFf^o$\"+sH1OQFe[oF`iz7%Fab[l 7$$\"+4`\"oI&Ff^o$\"+6!=X#RFe[oF`iz7%Fgb[l7$$\"+k@2iPFf^o$\"+eVY0SFe[o F`iz7%F]c[l7$$\"+_(*\\XKl9%Fe[oF`iz7%Fic[l7$$!+X*oX[$Ff^o$\"+L@W2UFe[oF`iz7%F_d[l7$$!+kV 0*e'Ff^o$\"++mViUFe[oF`iz7%Fed[l7$$!+@S.a**Ff^o$\"+;B(=J%Fe[oF`iz7%F[e [l7$$!+!HWPN\"Fihn$\"+&>'3cVFe[oF`iz7%Fae[l7$$!+E?LIFXFe[oF`iz7%Feg[l 7$$!+0-3NUFihn$\"+;7)>a%Fe[oF`iz7%F[h[l7$$!+^PLyYFihn$\"+L-C`XFe[oF`iz 7%Fah[l7$$!+J#)HE^Fihn$\"+%)*f5c%Fe[oF`iz7%Fgh[l7$$!+/^WybFihn$\"+%p2b c%Fe[oF`iz7%F]i[l7$$!+5\\UMgFihn$\"+B'Gmc%Fe[oF`iz7%Fci[l7$$!+682%\\'F ihn$\"+)pYWc%Fe[oF`iz7%Fii[l7$$!+[mTdpFihn$\"+>\\'*eXFe[oF`iz7%F_j[l7$ $!+l6sCuFihn$\"+Ec;]XFe[oF`iz7%Fej[l7$$!+n$>l*yFihn$\"+]3,QXFe[oF`iz7% F[[\\l7$$!+h&)ot$)Fihn$\"+^JWAXFe[oF`iz7%Fa[\\l7$$!+wlad))Fihn$\"+CxQ. XFe[oF`iz7%Fg[\\l7$$!+ft)*\\$*Fihn$\"+Urv![%Fe[oF`iz7%F]\\\\l7$$!+EKm` )*Fihn$\"+\"HhWX%Fe[oF`iz7%Fc\\\\l7$$!+#o@s.\"Fe[o$\"+8pUCWFe[oF`iz7%F i\\\\l7$$!+3W0\"4\"Fe[o$\"+ETj!R%Fe[oF`iz7%F_]\\l7$$!+Z5]Z6Fe[o$\"+Y1= `VFe[oF`iz7%Fe]\\l7$$!+G6M27Fe[o$\"+%*[Q7VFe[oF`iz7%F[^\\l7$$!+`\\Ur7F e[o$\"+7d#*oUFe[oF`iz7%Fa^\\l7$$!+H[ZS8Fe[o$\"+c$pRA%Fe[oF`iz7%Fg^\\l7 $$!+khu99Fe[o$\"+sG:zTFe[oF`iz7%F]_\\l7$$!+vnp$\\\"Fe[o$\"+'Qti8%Fe[oF `iz7%Fc_\\l7$$!+dM'fd\"Fe[o$\"+k:u'4%Fe[oF`iz7%Fi_\\l7$$!+Gdwf;Fe[o$\" +EG;hSFe[oF`iz7%F_`\\l7$$!+>vWVFe[o$\"+JI-uRFe[oF`iz7%Fa a\\l7$$!+,v'R)>Fe[o$\"+:,')[RFe[oF`iz7%Fga\\l7$$!+5\"=$f?Fe[o$\"+PxRCR Fe[oF`iz7%F]b\\l7$$!+h)o@8#Fe[o$\"+kF3+RFe[oF`iz7%Fcb\\l7$$!+T/i-AFe[o $\"+?\\[vQFe[oF`iz7%Fib\\l7$$!+au!3F#Fe[o$\"+\"zt-&QFe[oF`iz7%F_c\\l7$ $!+`i(oL#Fe[o$\"+jn>CQFe[oF`iz7%Fec\\l7$$!+#Gt4S#Fe[o$\"+)ogqz$Fe[oF`i z7%F[d\\l7$$!+9&RKY#Fe[o$\"+uvroPFe[oF`iz7%Fad\\l7$$!+?$3Q_#Fe[o$\"+'z `!RPFe[oF`iz7%Fgd\\l7$$!+>_!Ge#Fe[o$\"+/@)zq$Fe[oF`iz7%F]e\\l7$$!+@'[. k#Fe[o$\"+_eVvOFe[oF`iz7%Fce\\l7$$!+W5b'p#Fe[o$\"+6\\OTOFe[oF`iz7%Fie \\l7$$!+(\\?:v#Fe[o$\"+PHt0OFe[oF`iz7%F_f\\l7$$!+Z=O0GFe[o$\"+&p:&oNFe [oF`iz7%Fef\\l7$$!+3!y\"eGFe[o$\"+t-qHNFe[oF`iz7%F[g\\l7$$!+a325HFe[o$ \"+A]G*[$Fe[oF`iz7%Fag\\l7$$!+8<9hHFe[o$\"+*4!GZMFe[oF`iz7%Fgg\\l7$$!+ =8\\6IFe[o$\"+!\\3PS$Fe[oF`iz7%F]h\\l7$$!+q!>71$Fe[o$\"+QtgeLFe[oF`iz7 %Fch\\l7$$!+%[@/6$Fe[o$\"+]$H?J$Fe[oF`iz7%Fih\\l7$$!+K(*=fJFe[o$\"+\"* R/kKFe[oF`iz7%F_i\\l7$$!+(H1w?$Fe[o$\"+*4QZ@$Fe[oF`iz7%Fei\\l7$$!+&*3u bKFe[o$\"+$[:U;$Fe[oF`iz7%F[j\\l7$$!+9gk.LFe[o$\"+_^f7JFe[oF`iz7%Faj\\ l7$$!+VEN^LFe[o$\"+))y+gIFe[oF`iz7%Fgj\\l7$$!+#)o'))R$Fe[o$\"+)>\"f1IF e[oF`iz7%F][]l7$$!+Q\"ohW$Fe[o$\"+nH[_HFe[oF`iz7%Fc[]l7$$!+[%4K\\$Fe[o $\"+&\\9y*GFe[oF`iz7%Fi[]l7$$!+$4?*RNFe[o$\"+j$Fe[o$\"+90`JFFe[oF`iz7 %F[]]l7$$!+\\36xOFe[o$\"+OxfvEFe[oF`iz7%Fa]]l7$$!+/A]@PFe[o$\"+\\:[>EF e[oF`iz7%Fg]]l7$$!+Ak/lPFe[o$\"+Y0>jDFe[oF`iz7%F]^]l7$$!+*)zk2QFe[o$\" +0br1DFe[oF`iz7%Fc^]l7$$!+JJA\\QFe[o$\"+dJ.]CFe[oF`iz7%Fi^]l7$$!+B7q*) QFe[o$\"+^*4JR#Fe[oF`iz7%F__]l7$$!+'>D!HRFe[o$\"+?b!fL#Fe[oF`iz7%Fe_]l 7$$!+S7:nRFe[o$\"+ecPyAFe[oF`iz7%F[`]l7$$!+X![S+%Fe[o$\"+uZZ?AFe[oF`iz 7%Fa`]l7$$!+1epRSFe[o$\"+Hy:i@Fe[oF`iz7%Fg`]l7$$!+Q^3uSFe[o$\"+T=Q.@Fe [oF`iz7%F]a]l7$$!+))f@2TFe[o$\"+eq5W?Fe[oF`iz7%Fca]l7$$!+4m4RTFe[o$\"+ syH%)>Fe[oF`iz7%Fia]l7$$!+)fU(pTFe[o$\"+oN#R#>Fe[oF`iz7%F_b]l7$$!+$*f< *>%Fe[o$\"+x*eH'=Fe[oF`iz7%Feb]l7$$!+%RCuA%Fe[o$\"+p]Q,=Fe[oF`iz7%F[c] l7$$!+t+_aUFe[o$\"+<&*=RaFihnF`iz7%F]j]l7$$!+[3r(e%Fe [o$\"+5z4SZFihnF`iz7%Fcj]l7$$!+\\qm&f%Fe[o$\"+FL1hSFihnF`iz7%Fij]l7$$! +dWY-YFe[o$\"+S:z#Q$FihnF`iz7%F_[^l7$$!+e823YFe[o$\"+$HU_q#FihnF`iz7%F e[^l7$$!+\\4Y7YFe[o$\"+mvKG?FihnF`iz7%F[\\^l7$$!+'*=h:YFe[o$\"+>9#>N\" FihnF`iz7%Fa\\^l7$$!+@(3vh%Fe[o$\"+'y(penFf^oF`iz7%Fg\\^l7$$!+U?9=YFe[ oF(F`iz7%F]]^l7$Fh\\^l$!+'y(penFf^oF`iz7%Fa]^l7$Fb\\^l$!+>9#>N\"FihnF` iz7%Fe]^l7$F\\\\^l$!+mvKG?FihnF`iz7%Fi]^l7$Ff[^l$!+$HU_q#FihnF`iz7%F]^ ^l7$F`[^l$!+S:z#Q$FihnF`iz7%Fa^^l7$Fjj]l$!+FL1hSFihnF`iz7%Fe^^l7$Fdj]l $!+5z4SZFihnF`iz7%Fi^^l7$F^j]l$!+$=y)>aFihnF`iz7%F]_^l7$Fhi]l$!+H3K+hF ihnF`iz7%Fa_^l7$Fbi]l$!+=*o7y'FihnF`iz7%Fe_^l7$F\\i]l$!+*y'[iuFihnF`iz 7%Fi_^l7$Ffh]l$!+*3eO9)FihnF`iz7%F]`^l7$F`h]l$!+txQC))FihnF`iz7%Fa`^l7 $Fjg]l$!+7'3U]*FihnF`iz7%Fe`^l7$Fdg]l$!+P#f#=5Fe[oF`iz7%Fi`^l7$F^g]l$! +Bl*e3\"Fe[oF`iz7%F]a^l7$Fhf]l$!+mEF`6Fe[oF`iz7%Faa^l7$Fbf]l$!+DqK?7Fe [oF`iz7%Fea^l7$F\\f]l$!+#\\+qG\"Fe[oF`iz7%Fia^l7$Ffe]l$!+.vB`8Fe[oF`iz 7%F]b^l7$F`e]l$!+ux)*=9Fe[oF`iz7%Fab^l7$Fjd]l$!+&p2U[\"Fe[oF`iz7%Feb^l 7$Fdd]l$!+28')[:Fe[oF`iz7%Fib^l7$F^d]l$!+,4#Hh\"Fe[oF`iz7%F]c^l7$Fhc]l $!+TsOw;Fe[oF`iz7%Fac^l7$Fbc]l$!+<&*=RFe[oF`iz 7%Fad^l7$Fja]l$!+syH%)>Fe[oF`iz7%Fed^l7$Fda]l$!+eq5W?Fe[oF`iz7%Fid^l7$ F^a]l$!+T=Q.@Fe[oF`iz7%F]e^l7$Fh`]l$!+Hy:i@Fe[oF`iz7%Fae^l7$Fb`]l$!+uZ Z?AFe[oF`iz7%Fee^l7$F\\`]l$!+ecPyAFe[oF`iz7%Fie^l7$Ff_]l$!+?b!fL#Fe[oF `iz7%F]f^l7$F`_]l$!+^*4JR#Fe[oF`iz7%Faf^l7$Fj^]l$!+dJ.]CFe[oF`iz7%Fef^ l7$Fd^]l$!+0br1DFe[oF`iz7%Fif^l7$F^^]l$!+Y0>jDFe[oF`iz7%F]g^l7$Fh]]l$! +\\:[>EFe[oF`iz7%Fag^l7$Fb]]l$!+OxfvEFe[oF`iz7%Feg^l7$F\\]]l$!+90`JFFe [oF`iz7%Fig^l7$Ff\\]l$!+--D(y#Fe[oF`iz7%F]h^l7$F`\\]l$!+\"f1IFe[oF`iz7%F]i^l7$Fhj\\l$!+))y+gIFe[oF`iz7%Fai^l7$Fbj\\l $!+_^f7JFe[oF`iz7%Fei^l7$F\\j\\l$!+$[:U;$Fe[oF`iz7%Fii^l7$Ffi\\l$!+*4Q Z@$Fe[oF`iz7%F]j^l7$F`i\\l$!+\"*R/kKFe[oF`iz7%Faj^l7$Fjh\\l$!+]$H?J$Fe [oF`iz7%Fej^l7$Fdh\\l$!+QtgeLFe[oF`iz7%Fij^l7$F^h\\l$!+!\\3PS$Fe[oF`iz 7%F][_l7$Fhg\\l$!+*4!GZMFe[oF`iz7%Fa[_l7$Fbg\\l$!+A]G*[$Fe[oF`iz7%Fe[_ l7$F\\g\\l$!+t-qHNFe[oF`iz7%Fi[_l7$Fff\\l$!+&p:&oNFe[oF`iz7%F]\\_l7$F` f\\l$!+PHt0OFe[oF`iz7%Fa\\_l7$Fje\\l$!+6\\OTOFe[oF`iz7%Fe\\_l7$Fde\\l$ !+_eVvOFe[oF`iz7%Fi\\_l7$F^e\\l$!+/@)zq$Fe[oF`iz7%F]]_l7$Fhd\\l$!+'z`! RPFe[oF`iz7%Fa]_l7$Fbd\\l$!+uvroPFe[oF`iz7%Fe]_l7$F\\d\\l$!+)ogqz$Fe[o F`iz7%Fi]_l7$Ffc\\l$!+jn>CQFe[oF`iz7%F]^_l7$F`c\\l$!+\"zt-&QFe[oF`iz7% Fa^_l7$Fjb\\l$!+?\\[vQFe[oF`iz7%Fe^_l7$Fdb\\l$!+kF3+RFe[oF`iz7%Fi^_l7$ F^b\\l$!+PxRCRFe[oF`iz7%F]__l7$Fha\\l$!+:,')[RFe[oF`iz7%Fa__l7$Fba\\l$ !+JI-uRFe[oF`iz7%Fe__l7$F\\a\\l$!+&fx0+%Fe[oF`iz7%Fi__l7$Ff`\\l$!+H*Q$ HSFe[oF`iz7%F]`_l7$F``\\l$!+EG;hSFe[oF`iz7%Fa`_l7$Fj_\\l$!+k:u'4%Fe[oF `iz7%Fe`_l7$Fd_\\l$!+'Qti8%Fe[oF`iz7%Fi`_l7$F^_\\l$!+sG:zTFe[oF`iz7%F] a_l7$Fh^\\l$!+c$pRA%Fe[oF`iz7%Faa_l7$Fb^\\l$!+7d#*oUFe[oF`iz7%Fea_l7$F \\^\\l$!+%*[Q7VFe[oF`iz7%Fia_l7$Ff]\\l$!+Y1=`VFe[oF`iz7%F]b_l7$F`]\\l$ !+ETj!R%Fe[oF`iz7%Fab_l7$Fj\\\\l$!+8pUCWFe[oF`iz7%Feb_l7$Fd\\\\l$!+\"H hWX%Fe[oF`iz7%Fib_l7$F^\\\\l$!+Urv![%Fe[oF`iz7%F]c_l7$Fh[\\l$!+CxQ.XFe [oF`iz7%Fac_l7$Fb[\\l$!+^JWAXFe[oF`iz7%Fec_l7$F\\[\\l$!+]3,QXFe[oF`iz7 %Fic_l7$Ffj[l$!+Ec;]XFe[oF`iz7%F]d_l7$F`j[l$!+>\\'*eXFe[oF`iz7%Fad_l7$ Fji[l$!+)pYWc%Fe[oF`iz7%Fed_l7$Fdi[l$!+B'Gmc%Fe[oF`iz7%Fid_l7$F^i[l$!+ %p2bc%Fe[oF`iz7%F]e_l7$Fhh[l$!+%)*f5c%Fe[oF`iz7%Fae_l7$Fbh[l$!+L-C`XFe [oF`iz7%Fee_l7$F\\h[l$!+;7)>a%Fe[oF`iz7%Fie_l7$Ffg[l$!+@G>FXFe[oF`iz7% F]f_l7$F`g[l$!+l3w3XFe[oF`iz7%Faf_l7$Fjf[l$!+%oXl[%Fe[oF`iz7%Fef_l7$Fd f[l$!+A1QgWFe[oF`iz7%Fif_l7$F^f[l$!+c02IWFe[oF`iz7%F]g_l7$Fhe[l$!+E2R& R%Fe[oF`iz7%Fag_l7$Fbe[l$!+&>'3cVFe[oF`iz7%Feg_l7$F\\e[l$!+;B(=J%Fe[oF `iz7%Fig_l7$Ffd[l$!++mViUFe[oF`iz7%F]h_l7$F`d[l$!+L@W2UFe[oF`iz7%Fah_l 7$$!+%)Hn')oFc]o$!+!>Kl9%Fe[oF`iz7%Feh_l7$Fdc[l$!+]ZLzSFe[oF`iz7%F[i_l 7$F^c[l$!+eVY0SFe[oF`iz7%F_i_l7$Fhb[l$!+6!=X#RFe[oF`iz7%Fci_l7$Fbb[l$! +sH1OQFe[oF`iz7%Fgi_l7$F\\b[l$!+[3iRPFe[oF`iz7%F[j_l7$Ffa[l$!+VqmMOFe[ oF`iz7%F_j_l7$F`a[l$!+0zo?NFe[oF`iz7%Fcj_l7$Fj`[l$!+2GN(R$Fe[oF`iz7%Fg j_l7$Fd`[l$!+^S\"[E$Fe[oF`iz7%F[[`l7$F^`[l$!+SM*R7$Fe[oF`iz7%F_[`l7$Fh _[l$!+a7dwHFe[oF`iz7%Fc[`l7$Fb_[l$!+')\\cCGFe[oF`iz7%Fg[`l7$F\\_[l$!+( )yypEFe[oF`iz7%F[\\`l7$Ff^[l$!+'oZN^#Fe[oF`iz7%F_\\`l7$F`^[l$!+O1kcBFe [oF`iz7%Fc\\`l7$Fj][l$!+/\")[*>#Fe[oF`iz7%Fg\\`l7$Fd][l$!+9LGU?Fe[oF`i z7%F[]`l7$F^][l$!++$)4&)=Fe[oF`iz7%F_]`l7$Fh\\[l$!+,*[zs\"Fe[oF`iz7%Fc ]`l7$$\"+`+SONFb[o$!+X'H3d\"Fe[oF`iz7%Fg]`l7$F\\\\[l$!+A-t89Fe[oF`iz7% F]^`l7$Ff[[l$!+t=kc7Fe[oF`iz7%Fa^`l7$$\"+1+_V*)F^jn$!+\"*)e&*4\"Fe[oF` iz7%Fe^`l7$$\"+)*)=P)=F^jn$!+8;yC%*FihnF`iz7%F[_`l7$Fdjz$!+OB)R&yFihnF `iz7%Fa_`l7$$\"+gPdsNF-$!++a=$G'FihnF`iz7%Fe_`l7$$\"+T7%pV#Fdp$!+()*)Q 7ZFihnF`iz7%F[``l7$F($!+aEfTJFihnF`iz7%Fa``lFjcsF`iz7%FjcsF'F`iz-F\\hn 6&F^hn$\"\"\"F)$\"#:F]inFj``lFbds-F$6$7K7$$\"3&******pgoxS#FQ$\"3s**** *\\u\\`)\\FQ7$$\"3#*******4Lv2CFQ$\"39+++E3?%*\\FQ7$$\"3>+++ar\\1CFQ$ \"3P+++Dw'G+&FQ7$$\"3%)*****pNbSS#FQ$\"3%)*****\\1,7,&FQ7$$\"3<+++0X\\ +CFQ$\"3j*****zui!>]FQ7$$\"3++++%Q\"*eR#FQ$\"3\"******49Cj-&FQ7$$\"3;+ ++?BL!R#FQ$\"3T+++ie'G.&FQ7$$\"3=+++TH\"RQ#FQ$\"3&)*****f7x&Q]FQ7$$\"3 %)******>$QnP#FQ$\"3x*****pVbL/&FQ7$$\"3++++GM#*oBFQ$\"3Q+++fl5Z]FQ7$$ \"3y*****\\'QfgBFQ$\"3-+++v[u\\]FQ7$$\"3A+++9o)=N#FQ$\"3')*****R=%>^]F Q7$$\"32+++O@&HM#FQ$\"3W+++(3*Q^]FQ7$$\"3$)*****>_`RL#FQ$\"3o*****p1x- 0&FQ7$$\"3()*****H^p]K#FQ$\"3k*****pV@y/&FQ7$$\"3?+++_T\\;BFQ$\"3d**** *RK0S/&FQ7$$\"3,+++loV3BFQ$\"3e*****H#o$)Q]FQ7$$\"3'******R-@6I#FQ$\"3 z*****\\QbB.&FQ7$$\"35+++\\/y%H#FQ$\"3v*****>IRY-&FQ7$$\"37+++FHl*G#FQ $\"37+++6Q\"e,&FQ7$$\"3++++T)pfG#FQ$\"3D+++h(eg+&FQ7$$\"37+++V7%RG#FQ$ \"3!)*****p49c*\\FQ7$$\"3#*******flt$G#FQ$\"3u*****fn!y%)\\FQ7$$\"3#)* ****f`iaG#FQ$\"3#)*****Rg8R(\\FQ7$$\"39+++0>9*G#FQ$\"3')*****4*eSj\\FQ 7$$\"3-+++yyp%H#FQ$\"3Q+++YAm`\\FQ7$$\"3&*******p'\\>I#FQ$\"3G+++Uf1X \\FQ7$$\"3!******pDA1J#FQ$\"3!******f\"[%z$\\FQ7$$\"3()*****pap.K#FQ$ \"3d*****HoVD$\\FQ7$$\"3%)*****R213L#FQ$\"36+++(*)3!H\\FQ7$$\"3z*****R >S:M#FQ$\"3K+++cpQF\\FQ7$$\"3'******fS-AN#FQ$\"3T+++*)\\jF\\FQ7$$\"3)) *****\\8lCO#FQ$\"3L+++w&Q'H\\FQ7$$\"3/+++JL0sBFQ$\"3\"******R(>BL\\FQ7 $$\"3?+++otu!Q#FQ$\"3i*****H3<#Q\\FQ7$$\"39+++L1Q)Q#FQ$\"3n*****p4zV% \\FQ7$$\"3#******H6M[R#FQ$\"3M+++I#)\\^\\FQ7$$\"3)******HxH+S#FQ$\"3e* ****4yd$f\\FQ7$$\"3'******>)Q#RS#FQ$\"3q*****f?\\x'\\FQ7$$\"3;+++f4]1C FQ$\"3g*****\\&\\Zw\\FQF_a`lFihz-Fchn6L7%7$$\"+2'oxS#Fe[o$\"+X(\\`)\\F e[o7$$\"+5Lv2CFe[o$\"+E3?%*\\Fe[o7$$\"$M#F]in$\"$*\\F]in7%F_^al7$$\"+a r\\1CFe[o$\"+Dw'G+&Fe[oFd^al7%Fj^al7$$\"+d`0/CFe[o$\"+l5?6]Fe[oFd^al7% F`_al7$$\"+0X\\+CFe[o$\"+[F1>]Fe[oFd^al7%Ff_al7$$\"+%Q\"*eR#Fe[o$\"+TT KE]Fe[oFd^al7%F\\`al7$$\"+?BL!R#Fe[o$\"+ie'G.&Fe[oFd^al7%Fb`al7$$\"+TH \"RQ#Fe[o$\"+ErdQ]Fe[oFd^al7%Fh`al7$$\"+?$QnP#Fe[o$\"+PaNV]Fe[oFd^al7% F^aal7$$\"+GM#*oBFe[o$\"+fl5Z]Fe[oFd^al7%Fdaal7$$\"+lQfgBFe[o$\"+v[u\\ ]Fe[oFd^al7%Fjaal7$$\"+9o)=N#Fe[o$\"+%=%>^]Fe[oFd^al7%F`bal7$$\"+O@&HM #Fe[o$\"+(3*Q^]Fe[oFd^al7%Ffbal7$$\"+AN&RL#Fe[o$\"+nqF]]Fe[oFd^al7%F\\ cal7$$\"+8&p]K#Fe[o$\"+P9#y/&Fe[oFd^al7%Fbcal7$$\"+_T\\;BFe[o$\"+C`+W] Fe[oFd^al7%Fhcal7$$\"+loV3BFe[o$\"+Bo$)Q]Fe[oFd^al7%F^dal7$$\"+C57,BFe [o$\"+&QbB.&Fe[oFd^al7%Fddal7$$\"+\\/y%H#Fe[o$\"+-$RY-&Fe[oFd^al7%Fjda l7$$\"+FHl*G#Fe[o$\"+6Q\"e,&Fe[oFd^al7%F`eal7$$\"+T)pfG#Fe[o$\"+h(eg+& Fe[oFd^al7%Ffeal7$$\"+V7%RG#Fe[o$\"+(49c*\\Fe[oFd^al7%F\\fal7$$\"+glt$ G#Fe[o$\"+w1y%)\\Fe[oFd^al7%Fbfal7$$\"+ODY&G#Fe[o$\"+/O\"R(\\Fe[oFd^al 7%Fhfal7$$\"+0>9*G#Fe[o$\"+\"*eSj\\Fe[oFd^al7%F^gal7$$\"+yyp%H#Fe[o$\" +YAm`\\Fe[oFd^al7%Fdgal7$$\"+q'\\>I#Fe[o$\"+Uf1X\\Fe[oFd^al7%Fjgal7$$ \"+dAi5BFe[o$\"+;[%z$\\Fe[oFd^al7%F`hal7$$\"+Z&p.K#Fe[o$\"+$oVD$\\Fe[o Fd^al7%Ffhal7$$\"+ug!3L#Fe[o$\"+(*)3!H\\Fe[oFd^al7%F\\ial7$$\"+%>S:M#F e[o$\"+cpQF\\Fe[oFd^al7%Fbial7$$\"+1C?_BFe[o$\"+*)\\jF\\Fe[oFd^al7%Fhi al7$$\"+N^YiBFe[o$\"+w&Q'H\\Fe[oFd^al7%F^jal7$$\"+JL0sBFe[o$\"+u>BL\\F e[oFd^al7%Fdjal7$$\"+otu!Q#Fe[o$\"+$3<#Q\\Fe[oFd^al7%Fjjal7$$\"+L1Q)Q# Fe[o$\"+(4zV%\\Fe[oFd^al7%F`[bl7$$\"+8T$[R#Fe[o$\"+I#)\\^\\Fe[oFd^al7% Ff[bl7$$\"+t(H+S#Fe[o$\"+\"yd$f\\Fe[oFd^al7%F\\\\bl7$$\"+#)Q#RS#Fe[o$ \"+1#\\x'\\Fe[oFd^al7%Fb\\bl7$$\"+f4]1CFe[o$\"+b\\Zw\\Fe[oFd^al7%Fh\\b lFj]alFd^alFf``lFbds-F$6$7K7$F`a`l$!3s*****\\u\\`)\\FQ7$Fc]al$!3g***** \\&\\Zw\\FQ7$F^]al$!3q*****f?\\x'\\FQ7$Fi\\al$!3e*****4yd$f\\FQ7$Fd\\a l$!3M+++I#)\\^\\FQ7$F_\\al$!3n*****p4zV%\\FQ7$Fj[al$!3i*****H3<#Q\\FQ7 $Fe[al$!3\"******R(>BL\\FQ7$F`[al$!3L+++w&Q'H\\FQ7$F[[al$!3T+++*)\\jF \\FQ7$Ffj`l$!3K+++cpQF\\FQ7$Faj`l$!36+++(*)3!H\\FQ7$F\\j`l$!3d*****HoV D$\\FQ7$Fgi`l$!3!******f\"[%z$\\FQ7$Fbi`l$!3G+++Uf1X\\FQ7$F]i`l$!3Q+++ YAm`\\FQ7$Fhh`l$!3')*****4*eSj\\FQ7$Fch`l$!3#)*****Rg8R(\\FQ7$F^h`l$!3 u*****fn!y%)\\FQ7$Fig`l$!3!)*****p49c*\\FQ7$Fdg`l$!3D+++h(eg+&FQ7$F_g` l$!37+++6Q\"e,&FQ7$Fjf`l$!3v*****>IRY-&FQ7$Fef`l$!3z*****\\QbB.&FQ7$F` f`l$!3e*****H#o$)Q]FQ7$F[f`l$!3d*****RK0S/&FQ7$Ffe`l$!3k*****pV@y/&FQ7 $Fae`l$!3o*****p1x-0&FQ7$F\\e`l$!3W+++(3*Q^]FQ7$Fgd`l$!3')*****R=%>^]F Q7$Fbd`l$!3-+++v[u\\]FQ7$F]d`l$!3Q+++fl5Z]FQ7$Fhc`l$!3x*****pVbL/&FQ7$ Fcc`l$!3&)*****f7x&Q]FQ7$F^c`l$!3T+++ie'G.&FQ7$Fib`l$!3\"******49Cj-&F Q7$Fdb`l$!3j*****zui!>]FQ7$F_b`l$!3%)*****\\1,7,&FQ7$Fja`l$!3P+++Dw'G+ &FQ7$Fea`l$!39+++E3?%*\\FQFa]blFihz-Fchn6L7%7$F[^al$!+X(\\`)\\Fe[o7$Fi \\bl$!+b\\Zw\\Fe[o7$Fe^al$!$*\\F]in7%F_ebl7$Fc\\bl$!+1#\\x'\\Fe[oFbebl 7%Ffebl7$F]\\bl$!+\"yd$f\\Fe[oFbebl7%Fjebl7$Fg[bl$!+I#)\\^\\Fe[oFbebl7 %F^fbl7$Fa[bl$!+(4zV%\\Fe[oFbebl7%Fbfbl7$F[[bl$!+$3<#Q\\Fe[oFbebl7%Fff bl7$Fejal$!+u>BL\\Fe[oFbebl7%Fjfbl7$F_jal$!+w&Q'H\\Fe[oFbebl7%F^gbl7$F iial$!+*)\\jF\\Fe[oFbebl7%Fbgbl7$Fcial$!+cpQF\\Fe[oFbebl7%Ffgbl7$F]ial $!+(*)3!H\\Fe[oFbebl7%Fjgbl7$Fghal$!+$oVD$\\Fe[oFbebl7%F^hbl7$Fahal$!+ ;[%z$\\Fe[oFbebl7%Fbhbl7$F[hal$!+Uf1X\\Fe[oFbebl7%Ffhbl7$Fegal$!+YAm` \\Fe[oFbebl7%Fjhbl7$F_gal$!+\"*eSj\\Fe[oFbebl7%F^ibl7$Fifal$!+/O\"R(\\ Fe[oFbebl7%Fbibl7$Fcfal$!+w1y%)\\Fe[oFbebl7%Ffibl7$F]fal$!+(49c*\\Fe[o Fbebl7%Fjibl7$Fgeal$!+h(eg+&Fe[oFbebl7%F^jbl7$Faeal$!+6Q\"e,&Fe[oFbebl 7%Fbjbl7$F[eal$!+-$RY-&Fe[oFbebl7%Ffjbl7$Fedal$!+&QbB.&Fe[oFbebl7%Fjjb l7$F_dal$!+Bo$)Q]Fe[oFbebl7%F^[cl7$Fical$!+C`+W]Fe[oFbebl7%Fb[cl7$Fcca l$!+P9#y/&Fe[oFbebl7%Ff[cl7$F]cal$!+nqF]]Fe[oFbebl7%Fj[cl7$Fgbal$!+(3* Q^]Fe[oFbebl7%F^\\cl7$Fabal$!+%=%>^]Fe[oFbebl7%Fb\\cl7$F[bal$!+v[u\\]F e[oFbebl7%Ff\\cl7$Feaal$!+fl5Z]Fe[oFbebl7%Fj\\cl7$F_aal$!+PaNV]Fe[oFbe bl7%F^]cl7$Fi`al$!+ErdQ]Fe[oFbebl7%Fb]cl7$Fc`al$!+ie'G.&Fe[oFbebl7%Ff] cl7$F]`al$!+TTKE]Fe[oFbebl7%Fj]cl7$Fg_al$!+[F1>]Fe[oFbebl7%F^^cl7$Fa_a l$!+l5?6]Fe[oFbebl7%Fb^cl7$F[_al$!+Dw'G+&Fe[oFbebl7%Ff^cl7$F`^al$!+E3? %*\\Fe[oFbebl7%Fj^clF\\eblFbeblFf``lFbds-F$6%7$7$$!3o*************Q&FQ F(7$$\"3%**************=$FQF(-%'COLOURG6&F^hnF)F)F)-%*LINESTYLEG6#\"\" $-F$6%7$7$F($!3Q++++++!p&FQ7$F($\"3Q++++++!p&FQFg_clFj_cl-%(SCALINGG6# %,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESL ABELSG6%%&Re(z)G%&Im(z)G-F`acl6#%(DEFAULTG-%%VIEWG6$;$!$R&F]in$\"$>$F] in;$!$p&F]in$\"$p&F]in" 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 52 "#------- --------------------------------------------" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 36 "coefficients for 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 sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4468 "ee := \{c[2]=1/1000,\nc[3]=28/249,\nc[4]=14/83,\nc[ 5]=415483/780760,\nc[6]=87/131,\nc[7]=109/659,\nc[8]=7/16,\nc[9]=13/16 ,\nc[10]=169/170,\nc[11]=1,\nc[12]=1,\n\na[2,1]=1/1000,\na[3,1]=-38502 8/62001,\na[3,2]=392000/62001,\na[4,1]=7/166,\na[4,2]=0,\na[4,3]=21/16 6,\na[5,1]=201849498968676950243/186568677577006592000,\na[5,2]=0,\na[ 5,3]=-777541397023211694969/186568677577006592000,\na[5,4]=33748739701 7512409163/93284338788503296000,\na[6,1]=206935503/6884238116,\na[6,2] =0,\na[6,3]=0,\na[6,4]=404414081147/1257566797788,\na[6,5]=15348883360 00/4911984599757,\na[7,1]=39743335520386666547/572655525899435427204, \na[7,2]=0,\na[7,3]=0,\na[7,4]=212898348102841647147719/20102074093089 70807088412,\na[7,5]=-49544686747746827756272000/272983275534796815810 4608573,\na[7,6]=1077348693331815806/130730240620467840411,\na[8,1]=17 125839160165511097181/583729762655220563902464,\na[8,2]=0,\na[8,3]=0, \na[8,4]=174040744000019283021419611/2049081587190964014926856192,\na[ 8,5]=128489749336355703667293557125/956526769655402159824357169664,\na [8,6]=-125509550840850981113069/2931683770285010740150272,\na[8,7]=19/ 82,\na[9,1]=304584927066618919841641919928442600454537797/133643514697 4329082026479192445476428829818880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-183 130124969528341358670952614060593600746254959/430396572626952096766193 84887672437213535272960,\na[9,5]=-599298599748781593907273686220503862 3064898960325/2245491222915232082243948587229615621971585089024,\na[9, 6]=1322708054627964884973707202944501083084973588321/13468591512825267 37213165214426258933563303526400,\na[9,7]=8738229122541624425276473710 13373/228741131636931363287619371663360,\na[9,8]=512998676914870485248 52/18956439276208492414825,\na[10,1]=-12567952797084920377341497130244 54589348072577632030104333/\n 18443226524696441884887247092276 95887423797701298885000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=115532127 7999101211105955197933773620274343648077208821293051/\n 5939608 4444375336903774156155177493077283524727141295000000,\na[10,5]=1001719 19666627080787492626771580340159478395878727610639533656/\n 842 8868485387823508039378612742757916792962274222867420200625,\na[10,6]=- 7569188561763389198799352163814463872117563141929332169849602853/\n \+ 1965397952528115167934733609461445372239605164436067222345000000 ,\na[10,7]=-386885184918281945410313681520411070615012243139331/\n \+ 23691978700786561492029010557625479030322285000000,\na[10,8]=-145 680407248255277894206069634089984/14192495726244728880776189465890625, \na[10,9]=6596608202574079685632/8396199709857954528125,\na[11,1]=-191 3917650773276648000042701921031830180297961856463/\n 242053599 9874014114485551535784938587087050773067132,\na[11,2]=0,\na[11,3]=0,\n a[11,4]=794034598781907555787098179682463625970047118793613/\n \+ 35481535396506256458169246778950856594722743868452,\na[11,5]=125387651 14319773465378231128982652519968027341646160000/\n 915484882132 450383552157953702371732055721397872191177,\na[11,6]=-8797658288502921 72490106340518569223117087719534557891994/\n 19609686876602973 1312926688640518320991505603079281962497,\na[11,7]=-267739403086981136 01982120741972828286815906/1422046378852266508640137632652542482384789 ,\na[11,8]=-918990495679234610154177376026624/775293556823391270640625 13128033,\na[11,9]=819316668609331200/920300389324474639,\na[11,10]=-6 894374368544685000000/970688489039629537954049,\na[12,1]=3042259121/61 252403940,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=13 566961978425697607/75147921323806716000,\na[12,7]=49289738735215769940 3367/1970142834564896505822000,\na[12,8]=5410824060928/20472512295375, \na[12,9]=986554630144/6438843573435,\na[12,10]=269064197119512500/452 758788468179739,\na[12,11]=-135068333/274428000,\n\nb[1]=3042259121/61 252403940,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=13566961978425697 607/75147921323806716000,\nb[7]=492897387352157699403367/1970142834564 896505822000,\nb[8]=5410824060928/20472512295375,\nb[9]=986554630144/6 438843573435,\nb[10]=269064197119512500/452758788468179739,\nb[11]=-13 5068333/274428000,\n\n`b*`[1]=783278906947440245917/155359385168342027 40000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=299178 8146029459739848009433033/19060370046834035632749036000000,\n`b*`[7]=1 23570026638743472603558985757717817/4997031296463707458065070582620000 00,\n`b*`[8]=177511756302554128853504/649075937937135306609375,\n`b*`[ 9]=37241388453197849045504/204141942686827696516875,\n`b*`[10]=1902990 67731754576532365655/459347372389706921889026076,\n`b*`[11]=-229601674 28223228083/69605374826988000000,\n`b*`[12]=1/200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" } {TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=s ubs(ee,c[i]),i=2..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-/&%\"cG6#\" \"##\"\"\"\"%+5/&F%6#\"\"$#\"#G\"$\\#/&F%6#\"\"%#\"#9\"#$)/&F%6#\"\"&# \"'$[:%\"'g2y/&F%6#\"\"'#\"#()\"$J\"/&F%6#\"\"(#\"$4\"\"$f'/&F%6#\"\") #FJ\"#;/&F%6#\"\"*#\"#8FS/&F%6#\"#5#\"$p\"\"$q\"/&F%6#\"#6F)/&F%6#\"#7 F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "l inking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..12);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6^o/&%\"aG6$\"\"#\"\"\"#F(\"%+5/&F%6$\" \"$F(#!'G]Q\"&,?'/&F%6$F.F'#\"'+?RF1/&F%6$\"\"%F(#\"\"(\"$m\"/&F%6$F:F '\"\"!/&F%6$F:F.#\"#@F=/&F%6$\"\"&F(#\"6V-&pno*)\\\\=?\"6+?f1qdx'ol=/& F%6$FJF'FA/&F%6$FJF.#!6p\\p6K-(RTvxFM/&F%6$FJF:#\"6j\"4C^\"\\/& F%6$Fx9Z;%G5[$)*G@\":7%)323(*3$4u?5?/&F%6$F9-$G>++W2/u\"\"=#>co#\\,k4>(e\"3\\?/&F%6$F`rFJ#\"?Drb$HnOqbj L\\(*[G\"\"?k'prNC)f@Sb'pn_c*/&F%6$F`rFin#!9pI6\")4&3%3b4b7\":s-:S2,&G qPoJH/&F%6$F`rF<#\"#>\"##)/&F%6$\"\"*F(#\"N(zPXX+EWG*>>kT)>*=m1F\\e/$ \"O!))=)H)GkZXC>zk-#3HV(p9NkL\"/&F%6$FetF'FA/&F%6$FetF.FA/&F%6$FetF:#! Qf\\DY2g$fgSh_4ne8MG&p\\7IJ=\"PgHFNN@PCn()[Q>mn4_pislRI%/&F%6$FetFJ#!R D.'*)*[1B'Q]?iots!Rf\"y[(*f)H*f\"RC!*3&er>i:'Hse[RC#3K_\"HA\"\\XA/&F%6 $FetFin#\"R@$)et\\3$3,X%H?2P(\\)['zia!3FK\"\"R+k_.LcL*eiU9_;8stEDG^\"f oM\"/&F%6$FetF<#\"BtL,rtkFDWiTD7H#Q()\"BgLmr$>wGj8$pjJ6uG#/&F%6$FetF`r #\"8_[_[q[\"pn)*H^\"8D[T#\\3iFRk&*=/&F%6$\"#5F(#!enLV5I?jxD2[$*eaW-8( \\TtP?\\3(z_zc7\"en+++&)))H,xzBu)epF#4Zs)[)=W'pClAV%=/&F%6$F`wF'FA/&F% 6$F`wF.FA/&F%6$F`wF:#\"hn^IH@)3s2[OMu-itP$z>bf5675**zF@`:\"\"fn+++&HTr sCNGxI\\x^:cTx.pLvVW%3'Rf/&F%6$F`wFJ#\"jncO`R1hF(yeRy%f,M!erni#\\(y!3F mm>><+\"\"hnD1??u'GAuA'Hz;zvUFhy$R!3N#yQ&[o)G%)/&F%6$F`wFin#![o`Gg\\)p @LH>9jv6sQY9Q;_$*z)>*Qj/&F %6$F`wF<#!TJ$RJC7]hq5T?:o8.TX>G=\\=&)oQ\"S+++&GA..zaid0,H?\\hly+(y>pB/ &F%6$F`wF`r#!E%)**3M'pg?%*yFb#[sS!oX\"\"DD1*eY*=w2))GZCEd\\#>9/&F%6$F` wFet#\"7KcozSd-#3mf'\"7D\"GX&z&)4(*>'R)/&F%6$\"#6F(#!Ujk&='zH!=I=.@>qU ++[mFt2l\"UKr1t20(3(eQ\\yN:b&[9T,u)**f`?C/&F%6$FazF'FA/&F%6$FazF. FA/&F%6$FazF:#\"T8Oz=r/qfijCoz\")4(ybv!>y)fMSz\"S_%oQuAZfc3&*ynCp\"ekD 1lRN:[N/&F%6$FazFJ#\"Y++;Y;MF!o*>Dl#)*G6By`Yt(>V6l(QD\"\"Wx6>syR@d0K

*ybM&>x3GzIg0:*4K=0k)oEHJJ(Hgwoo4'>/&F%6$FazF<#!M1f\"oGGG(>u?@)>g8\")p3.%R xE\"L*y%Q#[UDlKw8S'3lE_)yj/A9/&F%6$FazF`r#!BCm-wt)\"3RYZC$*Q+.#*/&F%6$FazF`w#! 7+++&oW&oVP%*o\"9\\S&z`H'R!*[)oq*/&F%6$\"#7F(#\"+@\"fA/$\",SRS_7'/&F%6 $Fh]lF'FA/&F%6$Fh]lF.FA/&F%6$Fh]lF:FA/&F%6$Fh]lFJFA/&F%6$Fh]lFin#\"52w pD%y>'pc8\"5+gr1QK@z9v/&F%6$Fh]lF<#\"9nLS*pd@N(Q(*G\\\":+?#e]'*[cMG9q> /&F%6$Fh]lF`r#\".G41C3T&\"/v`H7DZ?/&F%6$Fh]lFet#\"-W,jal)*\".NMdV)Qk/& F%6$Fh]lF`w#\"3+D^>r>k!p#\"3R(z\"o%)yeFX/&F%6$Fh]lFaz#!*L$o]8\"*+!GWF " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "wei ghts for the 11 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..11);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"+@\"fA/$\",SRS_7'/&F %6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"52wp D%y>'pc8\"5+gr1QK@z9v/&F%6#\"\"(#\"9nLS*pd@N(Q(*G\\\":+?#e]'*[cMG9q>/& F%6#\"\")#\".G41C3T&\"/v`H7DZ?/&F%6#\"\"*#\"-W,jal)*\".NMdV)Qk/&F%6#\" #5#\"3+D^>r>k!p#\"3R(z\"o%)yeFX/&F%6#\"#6#!*L$o]8\"*+!GWF" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the \+ 12 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..12);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6./&%#b*G6#\"\"\"#\"6/&F%6#\"\"(#\"E`%)QTs$\"9vo^'pFooU>9/#/&F%6#\"#5#\"\" " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#================= ================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Abreviated calculations" }}{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 1114 "SO7_11 := Simple OrderConditions(7,11,'expanded'):\nSO_eqs := [op(RowSumConditions(11,' expanded')),op(StageOrderConditions(2,11,'expanded')),\n op(StageOrde rConditions(3,4..11,'expanded')),op(StageOrderConditions(4,6..11,'expa nded'))]:\nord_cdns := [seq(SO7_11[i],i=[1,2,4,8,16,32,38,48,61,64])]: \nsimp_eqs := [add(b[i]*a[i,1],i=2..11)=b[1],seq(add(b[i]*a[i,j],i=j+1 ..11)=b[j]*(1-c[j]),j=[$8..10])]:\nnode_eqs := [3*c[3]=2*c[4],3*c[6]^2 -4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4]=0]:\ncdns := [op(node_eqs),op(sim p_eqs),op(SO_eqs),op(ord_cdns)]:\n\n`SO6_12*` := subs(b=`b*`,SimpleOrd erConditions(6,12,'expanded')):\n`cdns*` := [seq(`SO6_12*`[i],i=[1,2,4 ,8,16,29,30,31,32])]:\n`simp_eqs*` := [seq(add(`b*`[i]*a[i,j],i=j+1..1 2)=`b*`[j]*(1-c[j]),j=[4,5])]:\nwt_eqns := [seq(b[i]=a[12,i],i=1..11)] :\n`cdns*` := [op(`cdns*`),op(wt_eqns),op(`simp_eqs*`)]:\n\nRK6_12eqs \+ := [op(RowSumConditions(12,'expanded')),op(OrderConditions(6,12,'expan ded'))]:\nerrterms7_11 := PrincipalErrorTerms(7,11,'expanded'):\n`errt erms7_12*` := subs(b=`b*`,PrincipalErrorTerms(7,12,'expanded')):\n`err terms6_12*` := subs(b=`b*`,PrincipalErrorTerms(6,12,'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 13 "calc_RKcoeff s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2825 "calc_RKcoeffs := proc()\n local eqns,eqns2,ct,mn, sm,e_temp,sm2,Rz,stb7,stb6,\n nmB,snmB,dnB,sdnB,nmC, snmC,B_8,C_8;\n global cdns,e1,e2,e3,e4,e5,e6,e7,e8,e9;\n\n e1 := \+ \{c[2]=c_2,c[4]=c_4,c[6]=c_6,c[7]=c_7,c[8]=c_8,c[9]=c_9,c[10]=c_10,c[1 1]=1,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),seq(b[i]=0,i= 2..5)\};\n eqns := subs(e1,cdns);\n e2 := solve(\{op(eqns)\},indet s(eqns) minus \{a[8,7],a[9,7],a[9,8]\});\n e3 := `union`(e1,e2);\n \+ e4 := \{c[12]=1,seq(`b*`[i]=0,i=2..5),`b*`[bs[1]]=bs[2]\}:\n e5 := \+ `union`(e3,e4):\n eqns2 := [op(remove(_U->type(rhs(_U),realcons),e5) ),op(subs(e5,`cdns*`))]:\n e6 := solve(\{op(eqns2)\},indets(eqns2) m inus \{a[8,7]\}):\n e7 := simplify(subs(select(u_->member(lhs(u_),\{ b[8],a[8,7],a[9,7],a[9,8]\}),e6),e5)):\n e8 := `union`(e7,e6):\n D igits := 16:\n sm := 0;\n for ct to nops(errterms7_11) do\n s m := sm+(evalf(subs(e8,errterms7_11[ct])))^2;\n end do;\n mn := mi nimize(sqrt(sm),a[8,7]=0..1,location);\n e_temp := \{a[8,7]=convert( subs(op(1,op(1,op(2,[mn]))),a[8,7]),rational,4)\};\n ##e_temp := \{a [8,7]= a8_7\};\n e9 := `union`(subs(e_temp,e8),e_temp): ##combined s cheme\n sm2 := 0:\n for ct to nops(errterms7_11) do\n sm2 := \+ sm2+(evalf(subs(e9,errterms7_11[ct])))^2;\n end do:\n Rz := subs(e 9,StabilityFunction(7,11,'expanded'));\n stb7 := max(fsolve(Rz=1,z=- 8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n stb7 := evalf[8](stb7):\n \+ Rz := subs(e9,subs(b=`b*`,StabilityFunction(6,12,'expanded')));\n st b6 := max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n stb 6 := evalf[8](stb6);\n nmB := 0;\n for ct to nops(`errterms7_12*`) do\n nmB := nmB+evalf(subs(e9,`errterms7_12*`[ct]))^2;\n end d o:\n snmB := sqrt(simplify(nmB));\n dnB := 0;\n for ct to nops(` errterms6_12*`) do\n dnB := dnB+evalf(subs(e9,`errterms6_12*`[ct] ))^2;\n end do;\n sdnB := sqrt(simplify(dnB));\n nmC := 0;\n f or ct to nops(errterms7_11) do\n nmC := nmC+(evalf(subs(e9,`errte rms7_12*`[ct]))-evalf(subs(e9,errterms7_11[ct])))^2;\n end do;\n s nmC := sqrt(simplify(nmC));\n B_8 := evalf[8](snmB/sdnB);\n C_8 := evalf[8](snmC/sdnB);\n print(`nodes:`,c[2]=c_2,c[3]=subs(e9,c[3]),c [4]=c_4,c[5]=subs(e9,c[5]),\n c[6]=c_6, c[7]=c_7,c[8]=c_8,c[9]=c_9,c[10]=c_10);\n print(`order 7 weights:`,s eq(b[i]=evalf[6](subs(e9,b[i])),i=[1,$6..11]));\n print(`order 6 wei ghts:`,seq(`b*`[i]=evalf[6](subs(e9,`b*`[i])),i=[1,$6..12]));\n prin t(infinity*`-norm of linking coeffs`=evalf(max(seq(seq(subs(e9,abs(a[i ,j])),j=1..i-1),i=2..12))));\n print(`2-norm of principal error of o rder 7 scheme` = evalf[10](sqrt(sm2)));\n print(`2-norm of principal error of order 6 scheme` = evalf[10](sdnB));\n print(`order 7 stabi lity interval` = [stb7,0]);\n print(`order 6 stability interval` = [ stb6,0]);\n print('B[8]'=B_8,'C[8]'=C_8);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#-------- ------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 35 "Sharp-Verner (1994) scheme ( with " }{XPPEDIT 18 0 "a[8,7]=9/5 3" "6#/&%\"aG6$\"\")\"\"(*&\"\"*\"\"\"\"#`!\"\"" }{TEXT -1 14 " inste ad of " }{XPPEDIT 18 0 "a[8,7]=1/6" "6#/&%\"aG6$\"\")\"\"(*&\"\"\"F* \"\"'!\"\"" }{TEXT -1 3 " ) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "c_2 := 1/12: c_4 := 2/9: c_6 := 2/3: c_7 := 1/6: c_8 := 4/9: c_9 \+ := 3/4: c_10 := 11/12: bs := [11,0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#7/&F&6#\"\"$#\"\" %\"#F/&F&6#F1#F(\"\"*/&F&6#\"\"&#F;F7/&F&6#\"\"'#F(F//&F&6#\"\"(#F*F@/ &F&6#\"\")#F1F7/&F&6#F7#F/F1/&F&6#\"#5#\"#6F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'Xx\\!\"(/&F&6#\" \"'$\"'9#[\"!\"'/&F&6#\"\"($\"'0ODF2/&F&6#\"\")$\"'2%p#F2/&F&6#\"\"*$ \"'%=3\"F2/&F&6#\"#5$\"''>`\"F2/&F&6#\"#6$\"'!>w\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'ROX!\"(/&F&6# \"\"'$\"'=$R%!\"'/&F&6#\"\"($\"'\\8FF2/&F&6#\"\")$\"'<8?F2/&F&6#\"\"*$ !'CW@F2/&F&6#\"#5$\"'NgGF2/&F&6#\"#6$\"\"!FP/&F&6#\"#7$!'v&*GF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"1wZj_fns " 0 "" {MPLTEXT 1 0 129 "c_2 := 1/16: c_4 := 6/25: c_6 := 5/8: c_7 := 35/209: c_8 := 29/72 : c_9 := 11/13: c_10 := 15/16:\nbs := [12,1/50]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6,%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#;/&F& 6#\"\"$#\"\"%\"#D/&F&6#F1#\"\"'F2/&F&6#\"\"&#\"$:*\"%'p\"/&F&6#F7#F;\" \")/&F&6#\"\"(#\"#N\"$4#/&F&6#FC#\"#H\"#s/&F&6#\"\"*#\"#6\"#8/&F&6#\"# 5#\"#:F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG 6#\"\"\"$\"'v?^!\"(/&F&6#\"\"'$\"'-ME!\"'/&F&6#\"\"($\"'.eCF2/&F&6#\" \")$\"'Z)*>F2/&F&6#\"\"*$\"'i'G\"F2/&F&6#\"#5$\"'^n**F+/&F&6#\"#6$\"'M S6F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\" \"\"$\"'@a^!\"(/&F&6#\"\"'$\"'1mD!\"'/&F&6#\"\"($\"'wVCF2/&F&6#\"\")$ \"'+O?F2/&F&6#\"\"*$\"'/G9F2/&F&6#\"#5$\"'aU$)F+/&F&6#\"#6$!'TaB!\")/& F&6#\"#7$\"'++?F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\" \"\"%8-norm~of~linking~coeffsGF&$\"1\\Dow\"yp.\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~schemeG$\"+O_ (yp\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~err or~of~order~6~schemeG$\"+sbxF8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %;order~7~stability~intervalG7$$!).()>Y!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)GS0O!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")c*4f\"!\"(/&%\"CGF&$ \")Wie:F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#--------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 38 " This scheme has one negative weight. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "c_2 := 1/20: c_4 := 4/19: c_6 := 3 7/56: c_7 := 61/367: c_8 := 25/57: c_9 := 39/50:\nc_10 := 19/20: bs := [11,0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%'nodes: G/&%\"cG6#\"\"##\"\"\"\"#?/&F&6#\"\"$#\"\")\"#d/&F&6#\"\"%#F6\"#>/&F&6 #\"\"&#\"&\")[%\"&3A)/&F&6#\"\"'#\"#P\"#c/&F&6#\"\"(#\"#h\"$n$/&F&6#F1 #\"#DF2/&F&6#\"\"*#\"#R\"#]/&F&6#\"#5#F8F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'p*)\\!\"(/&F&6#\" \"'$\"'cZ;!\"'/&F&6#\"\"($\"'!R^#F2/&F&6#\"\")$\"'9GEF2/&F&6#\"\"*$\"' .U8F2/&F&6#\"#5$\"'t0:F2/&F&6#\"#6$!'Hj8F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'EY]!\"(/&F&6#\"\" '$\"'%*49!\"'/&F&6#\"\"($\"'T\"\\#F2/&F&6#\"\")$\"'x0FF2/&F&6#\"\"*$\" '\"*=;F2/&F&6#\"#5$\"'6!H\"F2/&F&6#\"#6$\"\"!FP/&F&6#\"#7$!'.w?!\")" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linkin g~coeffsGF&$\"1Qh9)ezE,\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-n orm~of~principal~error~of~order~7~schemeG$\"+k+A:;!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\" +R&=Cr#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~int ervalG7$$!)4E>Y!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6 ~stability~intervalG7$$!)SX'o$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")c$o]\"!\"(/&%\"CGF&$\")8z,:F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#------------------- -------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "c_2 := 1/100: c_4 := 2/9: c_6 := 49/78: c_7 := 47/283: c_8 := 30/73: c_9 := 37/46:\nc_10 := 32/33: bs := [12,1/400]:\ncalc_RKcoeffs ();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%'nodes:G/&%\"cG6#\"\"##\"\"\" \"$+\"/&F&6#\"\"$#\"\"%\"#F/&F&6#F1#F(\"\"*/&F&6#\"\"&#\"&<9\"\"&G:#/& F&6#\"\"'#\"#\\\"#y/&F&6#\"\"(#\"#Z\"$$G/&F&6#\"\")#\"#I\"#t/&F&6#F7# \"#P\"#Y/&F&6#\"#5#\"#K\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order ~7~weights:G/&%\"bG6#\"\"\"$\"':s]!\"(/&F&6#\"\"'$\"'57A!\"'/&F&6#\"\" ($\"'1]CF2/&F&6#\"\")$\"'x)=#F2/&F&6#\"\"*$\"'QZ9F2/&F&6#\"#5$\"'s0=F2 /&F&6#\"#6$!'[7hF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weight s:G/&%#b*G6#\"\"\"$\"')H>&!\"(/&F&6#\"\"'$\"'iD>!\"'/&F&6#\"\"($\"'!)* R#F2/&F&6#\"\")$\"'_JBF2/&F&6#\"\"*$\"'[)z\"F2/&F&6#\"#5$\"'Oj7F2/&F&6 #\"#6$!'zIEF+/&F&6#\"#7$\"'++D!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"1IaJ`b1c5!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 7~schemeG$\"+Ee_E9!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~ principal~error~of~order~6~schemeG$\"+r'oS3\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)Yjx:F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 51 "#------------------------------------------------- -" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "c_2 := 1/100: c_4 := 1/5: c_6 := 60/97: c_7 := 52/31 3: c_8 := 63/157: c_9 := 19/23:\nc_10 := 81/82: bs := [12,1/160]:\ncal c_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%'nodes:G/&%\"cG6#\" \"##\"\"\"\"$+\"/&F&6#\"\"$#F(\"#:/&F&6#\"\"%#F*\"\"&/&F&6#F7#\"%?^\"% \"***/&F&6#\"\"'#\"#g\"#(*/&F&6#\"\"(#\"#_\"$8$/&F&6#\"\")#\"#j\"$d\"/ &F&6#\"\"*#\"#>\"#B/&F&6#\"#5#\"#\")\"##)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'W#4&!\"(/&F&6#\" \"'$\"'lNC!\"'/&F&6#\"\"($\"')QV#F2/&F&6#\"\")$\"'!3.#F2/&F&6#\"\"*$\" '(zh\"F2/&F&6#\"#5$\"'bgGF2/&F&6#\"#6$!'5))=F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'Ly^!\"(/&F&6#\"\" '$\"'/tA!\"'/&F&6#\"\"($\"'s(R#F2/&F&6#\"\")$\"'ID@F2/&F&6#\"\"*$\"'yJ =F2/&F&6#\"#5$\"'$o-#F2/&F&6#\"#6$!'*\\B\"F2/&F&6#\"#7$\"'+]i!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"1Q!*pfh)p#=!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-no rm~of~principal~error~of~order~7~schemeG$\"+GMJY7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+5 6MB#)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~inter valG7$$!)xz=Y!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~s tability~intervalG7$$!)pmFW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$/&%\"BG6#\"\")$\")<6l:!\"(/&%\"CGF&$\")Qdo:F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#------------------------ --------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "c_2 := 1/1000: c_4 := 14/83: c_6 := 87/131: c_7 := 109/659:\nc_8 \+ := 7/16: c_9 := 13/16: c_10 := 169/170: bs := [12,1/200]:\ncalc_RKcoef fs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%'nodes:G/&%\"cG6#\"\"##\"\" \"\"%+5/&F&6#\"\"$#\"#G\"$\\#/&F&6#\"\"%#\"#9\"#$)/&F&6#\"\"&#\"'$[:% \"'g2y/&F&6#\"\"'#\"#()\"$J\"/&F&6#\"\"(#\"$4\"\"$f'/&F&6#\"\")#FK\"#; /&F&6#\"\"*#\"#8FT/&F&6#\"#5#\"$p\"\"$q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'wm\\!\"(/&F&6#\" \"'$\"'P0=!\"'/&F&6#\"\"($\"'%=]#F2/&F&6#\"\")$\"'(Hk#F2/&F&6#\"\"*$\" '>K:F2/&F&6#\"#5$\"'xUfF2/&F&6#\"#6$!'\"=#\\F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'sT]!\"(/&F&6#\"\" '$\"'kp:!\"'/&F&6#\"\"($\"'(GZ#F2/&F&6#\"\")$\"'%[t#F2/&F&6#\"\"*$\"'H C=F2/&F&6#\"#5$\"'\"G9%F2/&F&6#\"#6$!'i)H$F2/&F&6#\"#7$\"'++]!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"1aR2o6)yB#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-nor m~of~principal~error~of~order~7~schemeG$\"+6)>uO*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+! o/a\"=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~inte rvalG7$$!)?9=Y!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~ stability~intervalG7$$!)0$=[%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")?6V:!\"(/&%\"CGF&$\")Bd[:F*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "#====================== ===========" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedure s for the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "RK7_11step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4075 "rk7_11step := proc(x_rk7step::realcons)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB, a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a 72,a73,a74,a75,a76,a81,a82,a83,a84,\n a85,a86,a87,a91,a92,a93,a94,a9 5,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3,aB4 ,aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,b1,b2,b 3,b4,b5,b6,b7,b8,b9,bA,bB,\n xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,save Digits;\n options `Copyright 2004 by Peter Stone`;\n \n data := \+ SOLN_;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digi ts)),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 error \"evaluation of slope field failed at %1\",evalf([X_,Y_],saveDigits); \n end if;\n val;\n end proc;\n\n xx := evalf(x_rk7st ep);\n n := nops(data);\n\n if (data[1,1]data[n ,1] or xxdata[1,1])) then\n error \"independent variable is outside 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 := c 6_; c7 := c7_; c8 := c8_;\n c9 := c9_; cA := cA_; cB := cB_;\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_; a65 := a65_;\n a71 := a 71_; a72 := a72_; a73 := a73_; a74 := a74_; a75 := a75_; a76 := a76_; \n a81 := a81_; a82 := a82_; a83 := a83_; a84 := a84_; a85 := a85_; \+ a86 := a86_;\n a87 := a87_;\n a91 := a91_; a92 := a92_; a93 := a93 _; a94 := a94_; a95 := a95_; a96 := a96_; \n a97 := a97_; a98 := a98 _; \n aA1 := aA1_; aA2 := aA2_; aA3 := aA3_; aA4 := aA4_; aA5 := a A5_; aA6 := aA6_; \n aA7 := aA7_; aA8 := aA8_; aA9 := aA9_;\n aB1 \+ := aB1_; aB2 := aB2_; aB3 := aB3_; aB4 := aB4_; aB5 := aB5_; aB6 := aB 6_; \n aB7 := aB7_; aB8 := aB8_; aB9 := aB9_; aBA := aBA_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_; b5 := b5_; b6 := b6_; b7 := b7_ ; \n b8 := b8_; b9 := b9_; bA := bA_; bB := bB_;\n # Perform a bin ary search for the 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] t hen jF := jM else jS := jM end if;\n end do;\n if jM = n the n jF := n-1; jS := n end if;\n else\n while jS-jF> 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 end if;\n \n # Get the data needed from the list. \n xk := data[jF,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(x k + 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 t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7 ;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 \+ + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,y k + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t : = aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f 8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 \+ + b9*f9 + bA*fA\n \+ + bB*fB; \n ys := yk + t*h;\n\n evalf[ saveDigits](ys);\nend proc: # of rk7_11step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "RK7_1 Sharp-Verner scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5633 "RK7_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA ,cB,a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a 71,a72,a73,a74,a75,a76,a81,a82,a83,a84,\n a85,a86,a87,a91,a92,a93,a9 4,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3 ,aB4,aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,b1, b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,\n t,k,fn,xk,yk,soln,eqns,A,saveDigits ;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)), Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/12,1/12 ,0,0,0,0,0,0,0,0,0,0],\n[4/27,4/243,32/243,0,0,0,0,0,0,0,0,0],\n[2/9,1 /18,0,1/6,0,0,0,0,0,0,0,0],\n[5/9,5/9,0,-25/12, 25/12,0,0,0,0,0,0,0], \n[2/3, 1/15,0,0,1/3,4/15,0,0,0,0,0,0],\n[1/6,319/3840,0,0,161/1536,-4 1/960,11/512,0,0,0,0,0],\n[4/9,245/5184,0,0,1627/10368,151/1296,-445/1 0368,1/6,0,0,0,0],\n[3/4,-556349853/7539261440,0,0,-4356175383/3015704 576,-814787343/1884815360,831004641/3015704576,355452237/235601920,107 943759/117800960,0,0,0],\n[11/12,-68998698967/1063035863040,0,0,-76738 7292485/425214345216,-205995991597/265758965760,-22181208863/141738115 072,26226796959/15502606336,1614200643/1107329024,187/329,0,0],\n[1,24 511479161/17979371520,0,0,3889847115/217931776,22028391/3681280,614528 179/217931776,-148401247/10215552,-3122234829/318384704,-4160/1221,150 40/20757,0],\n[0,5519/110880,0,0,0,0,83/560,932/3675,282123/1047200,26 24/24255,3008/19635,37/2100]]);\n\n c2 := evalf(A[1,1]);\n c3 := e valf(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 c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10, 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 a81 := evalf(A[ 7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := \+ evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n \+ a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8, 3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := ev alf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a 98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3] );\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := eval f(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2] );\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := ev alf(A[10,5]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n \+ aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A [10,10]);\n aBA := evalf(A[10,11]);\n b1 := evalf(A[11,2]);\n b2 := evalf(A[11,3]);\n b3 := evalf(A[11,4]);\n b4 := evalf(A[11,5]) ;\n b5 := evalf(A[11,6]);\n b6 := evalf(A[11,7]);\n b7 := evalf( A[11,8]);\n b8 := evalf(A[11,9]);\n b9 := evalf(A[11,10]);\n bA \+ := evalf(A[11,11]);\n bB := evalf(A[11,12]);\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,y k + 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 + a 72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,y k + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86 *f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 \+ + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n \+ f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + \+ aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(x k + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB 5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n \+ fB := fn(xk + cB*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + b B*fB;\n yk := yk + t*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,c8_=c8,c9_=c9,cA_=cA,cB_=cB,\n a31_=a31,a32_=a32,a41_=a41,a 42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a 61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a7 3_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a8 3,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93 _=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1 ,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_ =aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n \+ aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,b1_=b1,b2_=b2,\n \+ b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB\};\n \+ return subs(eqns,eval(rk7_11step)); \n else\n return evalf[s aveDigits]([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 19 "RK7_2 scheme with " }{XPPEDIT 18 0 "c[10 ]=15/16" "6#/&%\"cG6#\"#5*&\"#:\"\"\"\"#;!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6937 "RK7_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6, c7,c8,c9,cA,cB,a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62,a6 3,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,\n a85,a86,a87,a91 ,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9, aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f3,f4,f5,f6,f7,f8,f 9,fA,fB,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,\n t,k,fn,xk,yk,soln,eqns,A ,saveDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalh f(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix( [[1/16,1/16,0,0,0,0,0,0,0,0,0,0],\n[4/25,-28/625,128/625,0,0,0,0,0,0,0 ,0,0],\n[6/25,3/50,0,9/50,0,0,0,0,0,0,0,0],\n[915/1696,15090136995/390 27212288,0,-53100995625/39027212288,29533111875/19513606144,0,0,0,0,0, 0,0],\n[5/8,32755/421632,0,0,9765625/29258496,1488770/6971751,0,0,0,0, 0,0],\n[35/209,1072887507055/12570100065468,0,0,1075426748046875/95951 07694235844,-26411117095731200/385766943434163819,23430395520/60866149 3759,0,0,0,0,0],\n[29/72,232418455423819/11076629098233600,0,0,-232407 582514046875/2367427911414660864,881809586743097626/531145999301508607 5,-14411638756714/175988393502075,19/48,0,0,0,0],\n[11/13,-80469422945 92811005033672093485229/31693413191384942205414737020389900,0,0,-53623 1038652840856428143497005609375/138242660274515884207093807951326924,- 312405090743882815734153055386047660032/138949582969101476796789174014 714391225,46374757254514641197506707162751264/335428141562998715027252 37873062925,2051784331177468089852258884/674022310351068617436737421,1 762562147235459382311840/629339225351137831406851,0,0,0],\n[15/16,2157 20375695479264063749229806841635/440720278308021223336256911460990976, 0,0,12320199720948281040487332306693759765625/235489849560647123861381 8261942735405056,18664883699983476632333388794677364275/66041869145822 46254039171103542098296,-78458209082510636098652961448851403235/657093 91590090207335818975096879372288,-284269998655149797778753043937075/78 560914257412947734286452523008,-40415483223195085710169801875/13000342 692048285899311310848,1620538162595625/5204697537445888,0,0],\n[1,-975 767605592872353548579494188313/1054764737809245464809448652511500,0,0, -5137868354683248477586346693453125/495465427355479410622167542061972, -405677779726268282752892649892020224/71142722483104145892095323206788 025,245921960510918251629683797176094768/59908811922681865929060892977 735375,367953427734636456312437471451631/44380048256994921706877045210 625,808591598711928461147452272/133856729103497890946741285,-304924046 7/3093372235,3687441170432/6749831213125,0],\n[0,18009539/351697500,0, 0,0,0,1351744/5131875,898538649982788221/3655523545713900000,986955378 624/4938547627475,397569776903/3090028048800,2957049856/29666896875,71 669/6284880]]);\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 c8 := evalf(A[7,1]);\n c9 := evalf(A [8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n a21 := \+ c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := eva lf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a5 1 := 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 a81 := evalf(A[7,2]);\n a82 : = evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[ 7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := \+ evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n \+ a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8, 9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := ev alf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n a A6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9] );\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := ev alf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n \+ aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A [10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aB A := evalf(A[10,11]);\n b1 := evalf(A[11,2]);\n b2 := evalf(A[11,3 ]);\n b3 := evalf(A[11,4]);\n b4 := evalf(A[11,5]);\n b5 := eval f(A[11,6]);\n b6 := evalf(A[11,7]);\n b7 := evalf(A[11,8]);\n b8 := evalf(A[11,9]);\n b9 := evalf(A[11,10]);\n bA := evalf(A[11,11 ]);\n bB := evalf(A[11,12]);\n xk := evalf(xx);\n yk := evalf(yy );\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(x k,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 \+ t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n \+ f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f 3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c 9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 \+ + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t* h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + \+ aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB *h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB;\n yk \+ := yk + t*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,c8_=c8,c9_=c9 ,cA_=cA,cB_=cB,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43 ,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_ =a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74, a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_= a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a 95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=a A3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,aB 1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6,aB7_=aB 7,aB8_=aB8,aB9_=aB9,aBA_=aBA,b1_=b1,b2_=b2,\n b3_=b3,b4_=b4,b5 _=b5,b6_=b6,b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB\};\n return subs(e qns,eval(rk7_11step)); \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 19 "RK7_3 scheme with " }{XPPEDIT 18 0 "c[10]=19/20" "6#/&% \"cG6#\"#5*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7397 "RK7_3 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,a 21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a7 2,a73,a74,a75,a76,a81,a82,a83,a84,\n a85,a86,a87,a91,a92,a93,a94,a95 ,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3,aB4, aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,b1,b2,b3 ,b4,b5,b6,b7,b8,b9,bA,bB,\n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digit s+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/20,1/20,0,0, 0,0,0,0,0,0,0,0],\n[8/57,-184/3249,640/3249,0,0,0,0,0,0,0,0,0],[4/19,1 /19,0,3/19,0,0,0,0,0,0,0,0],\n[44881/82208,10879382999084569/177783816 94173184,0,-41275080377096907/17778381694173184,20050852951463513/8889 190847086592,0,0,0,0,0,0,0],\n[37/56,5533979/91295232,0,0,12854870299/ 39431336448,3657883862/13345483011,0,0,0,0,0,0],\n[61/367,158475499801 357/1970033827325064,0,0,31153019468948131/297713925429454536,-3016545 448677376/81418799560147215,150256844634176/8265656149993815,0,0,0,0,0 ],\n[25/57,2924917590401981135/66605023970341440552,0,0,44587661735110 8665/3098009485599691752,196990339933642293248/1651616643670523695197, -7149897627279951488/167672519638445079477,35/201,0,0,0,0],\n[39/50,-4 2211185912602462564070166033950198162012423/34220582419886119007473321 0927141821875000000,0,0,-231885975739312376352234586354522482324560602 3/847779727566936317150916383049577334375000000,-311125308777988754259 23566022361029283298688/28981348290292110337421437961130838134765625,1 5292107243270980188039355727714753150306881614/29901512151935859805913 259351031993989013671875,40769840728265525109900718286661374039/155493 75265121654840821192853125000000,945612121233663016296797907/598971783 761117752865000000,0,0,0],\n[19/20,53421576995301575930555089451778526 350776610941/205195002329815636484038978294464618799958400000,0,0,5028 96807923589267114219976720637605496180343553/1955190424455206666383767 28307225007182256000000,363012698911770750340098104804203047936446648/ 417738629670365256911027966408581558444453125,-11581344894509944294868 3847145831524761051670902703/19193952100161669036556601046617781331541 0464375000,-28699281021898330563484694969577298696532859/1346330884497 0886630335729997542445904000000,-16493845602934685528675361413563797/2 8001674400224052367433014064000000,1285791152889375/2245412845835068,0 ,0],\n[1,89462997577799701289200380185298019046438429/8410858488785640 6378631238816963629525265400,0,0,6492689767645528199626511179748230475 508923/641139590808912277141769373352511460044088,69407198604727672359 4566813393158946881536/175338845029991694153905751089388426975345,-187 824233463716665067125996081330354271085997696/302112836272077066552318 02682148684012056060045,-197008013321007326029676937491820383240505/20 203428941091626061502271171655020032789,-17297582435764003600121633126 256/9677847661048496307295108424975,165638979000000000/398275890687341 03,-845624320000/1531259401089,0],\n[0,125174203/2508655500,0,0,0,0,56 446289600512/342606012403635,62143252047047649977/24719883468262273368 0,9636489994384107/36666550775008000,5373710937500/40041548453907,1321 2880000/87750675837,-98471/7223040]]);\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 c8 := evalf(A[7 ,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := eval f(A[10,1]);\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 a81 := ev alf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a 84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7] );\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := eval f(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]); \n a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf( A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 : = evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A [10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7 ]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := e valf(A[10,10]);\n aBA := evalf(A[10,11]);\n b1 := evalf(A[11,2]); \n b2 := evalf(A[11,3]);\n b3 := evalf(A[11,4]);\n b4 := evalf(A [11,5]);\n b5 := evalf(A[11,6]);\n b6 := evalf(A[11,7]);\n b7 := evalf(A[11,8]);\n b8 := evalf(A[11,9]);\n b9 := evalf(A[11,10]); \n bA := evalf(A[11,11]);\n bB := evalf(A[11,12]);\n xk := evalf (xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to s tps 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 := a7 1*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk \+ + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85* f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8 ;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + a A3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4 *f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA; \n fB := fn(xk + cB*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ \+ + bB*fB;\n yk := yk + t*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,c8_=c8,c9_=c9,cA_=cA,cB_=cB,\n a31_=a31,a32_=a32,a4 1_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a5 4,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72 _=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82 ,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n a92_ =a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n \+ aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n \+ aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n \+ aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,b1_=b1,b2_=b2,\n \+ b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB \};\n return subs(eqns,eval(rk7_11step)); \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 19 "RK7_4 scheme with " }{XPPEDIT 18 0 "c[10]=32/33" "6#/&%\"cG6#\"#5*&\"#K\"\"\"\"#L!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7561 "RK7_4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,c8,c9,cA,cB,a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a 54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,\n a8 5,a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n \+ aA7,aA8,aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f3,f4, f5,f6,f7,f8,f9,fA,fB,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,\n t,k,fn,xk,y k,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits := ma x(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n \+ A := matrix([[1/100,1/100,0,0,0,0,0,0,0,0,0,0],\n[4/27,-692/729,800/7 29,0,0,0,0,0,0,0,0,0],\n[2/9,1/18,0,1/6,0,0,0,0,0,0,0,0],\n[11417/2152 8,451814999377/985407860736,0,-552805397249/328469286912,96066394193/5 4744881152,0,0,0,0,0,0,0],\n[49/78,131173/1890096,0,0,5764801/17935632 ,4769464/20091357,0,0,0,0,0,0],\n[47/283,1176539682457387/143533678287 55172,0,0,17717342174781/163306752068660,-528062841353883264/103380131 78660912633,1679528554701504/62911488820241285,0,0,0,0,0],\n[30/73,110 903909289926737793/4320958290769994040132,0,0,-11607238985710021183/18 0261023052662924020,8332987977461879955626624/570564538303299504690930 05,-397914630242458427722/6312988060727977701695,11/30,0,0,0,0],\n[37/ 46,-199116158962708264981792850380273725319682574083/21220957037939595 40376735652096650388444524173920,0,0,-90239902747863404931846835436583 1042971264299/376720159407433656819366352813041281670901920,-193284104 9216489668013274675445061103749096456/12250344742255380462120669375167 75761976271555,61212379964740003623393766579069964177763490780547/6729 2203490497755527349011795764425282435991108860,17673520104558015227599 19865162797424677/853631298904311486745705433404624991760,488031384043 57581919875739936453/25803619795436169827228351014320,0,0,0],\n[32/33, 146812882912103498223477710702327669160817645499822332/347136327550264 350281019050691094921672653636486171345,0,0,83809072793999403414155200 95001429689250499016988/1665529140598002776815163790462148830898906131 435,6459917812385485497254681791738665918061301836694665216/2108701812 611736155789874398208067694599974390240225435,-32962310193697591371394 0238539594299201957668540750512/21636919327312331153389809685281753145 0154842258383315,-239812294208127777232571199557076199094738401624/627 16593235272943126739970926235455172660665115,-192099094907822046293457 87487654324166296/6932257046340021610507379175381008191035,21064239972 9505665392/369807601102457012649,0,0],\n[1,221939233494467949726864332 982580511239150575217/255387155540276753091910917079234490443699161270 ,0,0,181163263192487297546324132938257205801916729/1715454714186447769 9072019121010975163450940,50351382541187163428643689378746832466172595 98976/775682801056824859328708920179990438628643868105,-77224801754835 239618997430260090636543172193252536784/202652428901707058125367179822 84413527106230272155465,-204174604932276627583428757616716475135707280 3/242398642004974788232613894826395428361025315,-698805798558477218948 68113496992143638/12013145933881997013005874021379315455,6677118754098 43102380/547572557872664725429,-1808859196027797903/202062001781108996 92,0],\n[0,2354923/46428480,0,0,0,0,31816481086962/143829479014165,727 3629826007016071/29687570322557889840,131953911267751793/6028666464715 70040,29769129062566/205676274838545,30049446213369/166412182583360,-6 7996657/1112423760]]);\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 c8 := evalf(A[7,1]);\n c9 := \+ evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\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 a81 := evalf(A[7,2]);\n \+ a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7, 5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := ev alf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a 93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6] );\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := eval f(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]); \n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf( A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5 ]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := e valf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]); \n aBA := evalf(A[10,11]);\n b1 := evalf(A[11,2]);\n b2 := evalf (A[11,3]);\n b3 := evalf(A[11,4]);\n b4 := evalf(A[11,5]);\n b5 \+ := evalf(A[11,6]);\n b6 := evalf(A[11,7]);\n b7 := evalf(A[11,8]); \n b8 := evalf(A[11,9]);\n b9 := evalf(A[11,10]);\n bA := evalf( A[11,11]);\n bB := evalf(A[11,12]);\n xk := evalf(xx);\n yk := e valf(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 + a 73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h); \n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87 *f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 \+ + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn (xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + \+ aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h, yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB 6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn( xk + cB*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + \+ b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB;\n \+ yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk,yk]; \n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fx y,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c 8,c9_=c9,cA_=cA,cB_=cB,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42, a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_= a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a 74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84_=a 84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93,a9 4_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA 2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9 _=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6 ,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,b1_=b1,b2_=b2,\n b3_=b3,b 4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB\};\n retur n subs(eqns,eval(rk7_11step)); \n else\n return evalf[saveDigit s]([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 19 "RK7_5 scheme with " }{XPPEDIT 18 0 "c[10]=81/82" "6#/ &%\"cG6#\"#5*&\"#\")\"\"\"\"##)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7748 "RK7_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB ,a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71, a72,a73,a74,a75,a76,a81,a82,a83,a84,\n a85,a86,a87,a91,a92,a93,a94,a 95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3,aB 4,aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,b1,b2, b3,b4,b5,b6,b7,b8,b9,bA,bB,\n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n \n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Dig its+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/100,1/100, 0,0,0,0,0,0,0,0,0,0],\n[2/15,-34/45,8/9,0,0,0,0,0,0,0,0,0],\n[1/5,1/20 ,0,3/20,0,0,0,0,0,0,0,0],\n[5120/9991,551874974720/997302429271,0,-208 6699008000/997302429271,2045902848000/997302429271,0,0,0,0,0,0,0],\n[6 0/97,70365/1204352,0,0,15000000/48955027,16390905/64600448,0,0,0,0,0,0 ],\n[52/313,21278500033609/276420238876800,0,0,9893050830728000/912367 32526195641,-2901059421298252903/76225368652427491200,436379361176824/ 23234416797464775,0,0,0,0,0],\n[63/157,627197176791399348709/228935332 88085131673600,0,0,-26470464436388981652125/236136288092530359764076,8 25734282216068905765330677/6313097845989067825180262400,-4913831081960 5862695843/962154393150421608110400,13/32,0,0,0,0],\n[19/23,-280529206 4095175105994189803671936576349564929/19543970040834186978418450984646 2968849917952000,0,0,-44489197025459151705444370840723674742716548825/ 14932376340320772420636844398178733063223595932,-538125605455123224555 096530708947439830683725706539/221787063147752862838752824077163711222 570059776000,5903957290670756919471851965687658329936372357281397/4701 038098831357101461987812303657884827097474792000,806385337807093245665 19854325500435393/31664769935063948010841554250731537120,3802360970021 4727511244770779309498/15560366705131162570848900057690585,0,0,0],\n[8 1/82,18745022701169741445732083919370592367818346398255185593/43542944 158840438871835940146729085397585013246181376000,0,0,17495577465884581 3020015026443375837215431521574722425/12988012824390651292175460014153 260035238111030958552,727038991731005479757015308987520386180157651574 71787669321/6944684059165807161861542521065278114605347407607709696000 ,-1851345468344294261477635009838184638595117141582640612437867/420703 407864684992323487936970444634267986659475521049568000,-14238311168791 8201861880767560889949537519404224643/13608444565909841005179111919047 260723369140257280,-15762011599975752003382750329296637334838001/17053 82686127610134765519769050554722692480,1570980729807122133129202485/21 70749583944689600835012992,0,0],\n[1,413401339654426004756492280722047 075004416104055883/719868089323382472727694115370573342900039059456000 ,0,0,4068247489923915770015231221964449550938389658325/222675331628898 775507843272599152181481922934868,422701288179451309495857528449967681 683868959323295591/297660975324226461683784977598551408704111367516160 00,-64756336295667822884474260277213322605529059566403398599943/105920 34333763703970644991697561788735926914730422752072000,-293320418965244 9798399861612546549582225310986953/20501358149999954267743997882433467 6268210882720,-624797475608500160493593996249709038784226/497736028930 36152650616792845999901516465,38655300339474326088/4080086313698677925 5,-6830025351434171965696/369666731905701279737985,0],\n[0,6470068411/ 127052452800,0,0,0,0,14036525372223254273/57629420394152745600,7354322 880723039179131/30216428865161090697600,223707573986265289483/11015754 29077660009200,5315253908682019/32851416704215200,347047251949199216/1 213217205998680575,-95980357/508344480]]);\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 c8 := eval f(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := e valf(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 := eva lf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a6 4 := 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 a81 \+ := evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]); \n a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf( A[7,7]);\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 : = evalf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[ 8,8]);\n a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := \+ evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n \+ aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9, 8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := e valf(A[10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf( A[10,7]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB 9 := evalf(A[10,10]);\n aBA := evalf(A[10,11]);\n b1 := evalf(A[11 ,2]);\n b2 := evalf(A[11,3]);\n b3 := evalf(A[11,4]);\n b4 := ev alf(A[11,5]);\n b5 := evalf(A[11,6]);\n b6 := evalf(A[11,7]);\n \+ b7 := evalf(A[11,8]);\n b8 := evalf(A[11,9]);\n b9 := evalf(A[11,1 0]);\n bA := evalf(A[11,11]);\n bB := evalf(A[11,12]);\n xk := e valf(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 := f n(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 + a6 3*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 := f n(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n \+ t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a 98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f 2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n \+ fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 \+ + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aB A*fA;\n fB := fn(xk + cB*h,yk + t*h);\n \n t := b1*f1 + b2 *f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ \+ + bB*fB;\n yk := yk + t*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,c8_=c8,c9_=c9,cA_=cA,cB_=cB,\n a31_=a31,a32_= a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a 54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a 71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a8 2_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n \+ a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n \+ aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7, \n aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_= aB5,\n aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,b1_=b1,b2_ =b2,\n b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8,b9_=b9,bA_=bA ,bB_=bB\};\n return subs(eqns,eval(rk7_11step)); \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 19 "RK7_6 scheme with " }{XPPEDIT 18 0 "c[10]=169/170" "6#/&%\"cG6#\"#5*&\"$p\"\"\"\"\"$q\"!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7834 "RK7_6 := proc(fxy,x,y,xx,yy,h,stps,bb)\n loc al c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,a21,a31,a32,a41,a42,a43,a51,a52,\n \+ a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,\n a85,a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6 ,\n aA7,aA8,aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,\n f1,f2,f 3,f4,f5,f6,f7,f8,f9,fA,fB,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,\n t,k,fn ,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits \+ := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y); \n\n A := matrix([[1/1000,1/1000,0,0,0,0,0,0,0,0,0,0],\n[28/249,-385 028/62001,392000/62001,0,0,0,0,0,0,0,0,0],\n[14/83,7/166,0,21/166,0,0, 0,0,0,0,0,0],\n[415483/780760,201849498968676950243/186568677577006592 000,0,-777541397023211694969/186568677577006592000,3374873970175124091 63/93284338788503296000,0,0,0,0,0,0,0],\n[87/131,206935503/6884238116, 0,0,404414081147/1257566797788,1534888336000/4911984599757,0,0,0,0,0,0 ],\n[109/659,39743335520386666547/572655525899435427204,0,0,2128983481 02841647147719/2010207409308970807088412,-49544686747746827756272000/2 729832755347968158104608573,1077348693331815806/130730240620467840411, 0,0,0,0,0],\n[7/16,17125839160165511097181/583729762655220563902464,0, 0,174040744000019283021419611/2049081587190964014926856192,12848974933 6355703667293557125/956526769655402159824357169664,-125509550840850981 113069/2931683770285010740150272,19/82,0,0,0,0],\n[13/16,3045849270666 18919841641919928442600454537797/1336435146974329082026479192445476428 829818880,0,0,-183130124969528341358670952614060593600746254959/430396 57262695209676619384887672437213535272960,-599298599748781593907273686 2205038623064898960325/22454912229152320822439485872296156219715850890 24,1322708054627964884973707202944501083084973588321/13468591512825267 37213165214426258933563303526400,873822912254162442527647371013373/228 741131636931363287619371663360,51299867691487048524852/189564392762084 92414825,0,0,0],\n[169/170,-125679527970849203773414971302445458934807 2577632030104333/18443226524696441884887247092276958874237977012988850 00000,0,0,115532127799910121110595519793377362027434364807720882129305 1/59396084444375336903774156155177493077283524727141295000000,10017191 9666627080787492626771580340159478395878727610639533656/84288684853878 23508039378612742757916792962274222867420200625,-756918856176338919879 9352163814463872117563141929332169849602853/19653979525281151679347336 09461445372239605164436067222345000000,-386885184918281945410313681520 411070615012243139331/236919787007865614920290105576254790303222850000 00,-145680407248255277894206069634089984/14192495726244728880776189465 890625,6596608202574079685632/8396199709857954528125,0,0],\n[1,-191391 7650773276648000042701921031830180297961856463/24205359998740141144855 51535784938587087050773067132,0,0,794034598781907555787098179682463625 970047118793613/35481535396506256458169246778950856594722743868452,125 38765114319773465378231128982652519968027341646160000/9154848821324503 83552157953702371732055721397872191177,-879765828850292172490106340518 569223117087719534557891994/196096868766029731312926688640518320991505 603079281962497,-26773940308698113601982120741972828286815906/14220463 78852266508640137632652542482384789,-918990495679234610154177376026624 /77529355682339127064062513128033,819316668609331200/92030038932447463 9,-6894374368544685000000/970688489039629537954049,0],\n[0,3042259121/ 61252403940,0,0,0,0,13566961978425697607/75147921323806716000,49289738 7352157699403367/1970142834564896505822000,5410824060928/2047251229537 5,986554630144/6438843573435,269064197119512500/452758788468179739,-13 5068333/274428000]]);\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 c8 := evalf(A[7,1]);\n c9 := e valf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n a 21 := 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 a81 := evalf(A[7,2]);\n \+ a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7, 5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := ev alf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a 93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6] );\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := eval f(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]); \n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf( A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5 ]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := e valf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]); \n aBA := evalf(A[10,11]);\n b1 := evalf(A[11,2]);\n b2 := evalf (A[11,3]);\n b3 := evalf(A[11,4]);\n b4 := evalf(A[11,5]);\n b5 \+ := evalf(A[11,6]);\n b6 := evalf(A[11,7]);\n b7 := evalf(A[11,8]); \n b8 := evalf(A[11,9]);\n b9 := evalf(A[11,10]);\n bA := evalf( A[11,11]);\n bB := evalf(A[11,12]);\n xk := evalf(xx);\n yk := e valf(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 + a 73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h); \n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87 *f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 \+ + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn (xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + \+ aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h, yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB 6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn( xk + cB*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + \+ b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB;\n \+ yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk,yk]; \n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fx y,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c 8,c9_=c9,cA_=cA,cB_=cB,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42, a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_= a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a 74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84_=a 84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93,a9 4_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA 2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9 _=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6 ,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,b1_=b1,b2_=b2,\n b3_=b3,b 4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB\};\n retur n subs(eqns,eval(rk7_11step)); \n else\n return evalf[saveDigit s]([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 "Testi ng the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 80 "These tests do not make use of the embedd ed order 4 method for error correction." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 1 of 11 stage, order 7 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;\n dsolve(\{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=[H ELVETICA,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-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG6#,$*&\"\"%F0F'F0F0F0-F) 6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0*&#\"#[F/F0*(F8F0-%$sinG F4F0F'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$$\"3ALL$3FWYs#F/$\"3!H*fm:2P/5 F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3WmmmT&)G\\aF/$\"34$Q7t`D r,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$\"3GLLL3x&)*3\"!#=$\"3 U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u7\"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$$\"3r LLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"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$$\"31 nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$3_DG1qFJ$\"31le1yn9(R(FJ7$$\"3] ***\\il'pisFJ$\"3E!)4GzFfsmFJ7$$\"3+MLe*[!)y_(FJ$\"3CJpN=**=vfFJ7$$\"3 Qnm\"HKkIz(FJ$\"3'oU:>LtrL&FJ7$$\"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$$\"3%o;HKR'\\5:F2$\"3XwZW,h_FQFJ7$$\"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$$\"3pTg-gl[Q>F2$\"3>kb3F 2$\"3OF\\_#G6DA\"F27$$\"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+))GPMfE5F27$$\"3?+](=xpe=#F2 $\"3ES3-I\\16#*FJ7$$\"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$41oWW#F2$\"3QVRl9U0BZFJ7$$\"3GLL3_?`(\\#F2$ \"3/\\cKWs=$>%FJ7$$\"3AL3_D1l_DF2$\"3o6E$fFc$yPFJ7$$\"3fL$e*)>pxg#F2$ \"3ym)*p(*f`&[$FJ7$$\"3%omm\"z+vbEF2$\"3jG&[,$f<=LFJ7$$\"33+]Pf4t.FF2$ \"3%R>3YHT'HFF2$\"3/C%)f*f*e+KFJ7$$\"3om\"zWi^bv#F2$ \"3-Gu$[oUh>$FJ7$$\"3)*\\7.d>Y\"y#F2$\"3#p*R$)o?n4KFJ7$$\"3uLLe*Gst!GF 2$\"3>.X!=mk1C$FJ7$$\"3)om\"H2\"34'GF2$\"3'[>IF2$\"3a-&\\&*p%H,TFJ7$$\"3F+]i!RU07$F2$ \"3'fkDHe#=P[FJ7$$\"3+++v=S2LKF2$\"3K%\\5FaXpw&FJ7$$\"3Jmmm\"p)=MLF2$ \"3))zmB`6`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3#o,C;(=8foFJ7$$\"3B++](=]@W$F2$ \"3#G%=QV$\\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3u,muc\"4C(FJ7$$\"3hTgx.2vFNF2$\"3/^M\"Q[;lC(FJ7$$\"35L$e*[$z*RN F2$\"3=wJ%fi2nC(FJ7$$\"3)*\\PMFwrmNF2$\"3R[i&\\xl(GsFJ7$$\"3%o;Hd!fX$f $F2$\"3IEKi0hy'=(FJ7$$\"3r$e9T=%>?OF2$\"3(>gS`&3dArFJ7$$\"3e++]iC$pk$F 2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3MoFJ7$$\"3[m;H2qcZPF2$ \"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+'FJ7$$\"3Ymm;/OgbRF2$ \"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\FJ7$$\"3ym\"zW7@^6%F 2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p%FJ7$$\"3)QL3-$H**>U F2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))oXb%FJ7$$\"3h+v=n(*fD VF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOhXFJ7$$\"3MMe9\"z-lU% F2$\"3kC\">#=Lu2YFJ7$$\"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(-%%FONTG6$%*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 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 738 "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 := [`Sharp-Verner scheme with `*c[10]=11/ 12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with \+ `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170] : errs := []:\nDigits := 25:\nfor ct to 6 do\n Fn_RK7_||ct := RK7_|| ct(F(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Fn _RK7_||ct):\n for ii to numpts do\n sm := sm+(Fn_RK7_||ct[ii,2] -f(Fn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nu mpts)];\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(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-V erner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+D[#e*G!#D7$/*&%-sc heme~with~GF+F,F+#\"#:\"#;$\"+Q')ybPF57$/F8#\"#>\"#?$\"+&f`)R " 0 "" {MPLTEXT 1 0 669 "F := (x,y) -> 1 2*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)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`S harp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`sche me with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]= 81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 25:\nfor c t to 6 do\n fn_RK7_||ct := RK7_||ct(F(x,y),x,y,x0,y0,hh,numsteps,tru e);\nend do:\nxx := 4.999: fxx := evalf(f(xx)):\nfor ct to 6 do\n er rs := [op(errs),abs(fn_RK7_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nli nalg[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(pprint06 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+&Q0sj#!#D7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+!)=G5TF57$/F8#\"#>\"#?$\"+y1D1:F57$/F8#\"#K\"#L$\"+cFx@IF57$/F8#\"# \")\"##)$\"+$p*>Z6F57$/F8#\"$p\"\"$q\"$\"*]&HXkF5Q(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, 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 integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((f(x)-'fn_RK7_||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$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+B5O&*G!#D7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+-Vf_PF57$/F8#\"#>\"#?$\"+._`R " 0 "" {MPLTEXT 1 0 518 "evalf[25]( plot(['fn_RK7_1'(x)-f(x),'fn_RK7_2'(x)-f(x),'fn_RK7_3'(x)-f(x),\n'fn_R K7_4'(x)-f(x),'fn_RK7_5'(x)-f(x),'fn_RK7_6'(x)-f(x)],x=0..5,-0.83e-15. .0.93e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7 ,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp -Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme wit h c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`sc heme with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 R unge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 915 582 582 {PLOTDATA 2 "6,-%'CURVESG6%7at7$$\"\"!F)F(7$$\":MLLLLLL$3x&)*3\"!#D$\" )X&yr&!#C7$$\":nmmmm;z%\\v#pK\"F-$\"*0I1-\"F07$$\":++++++D1R(*Rc\"F-$ \"*$*=)48F07$$\":nmmmm\"z>6B`#o\"F-$\"*)G[;9F07$$\":MLLLL$3xJs1,=F-$\" *#y\"GY\"F07$$\":nmmm;Hd?pM.'=F-$\"*+;QZ\"F07$$\":+++++vVB:-'>>F-$\"*) 3GJ9F07$$\":nmmm\"zp[#)eB\\>F-$\"*BjnV\"F07$$\":MLLL$3-j7'p)y>F-$\"*Te wU\"F07$$\":++++vVtFM.&3?F-$\"*[iEM\"F07$$\":nmmmmm;H2P\"Q?F-$\"*!G0[8 F07$$\":MLLLL$ek.pu/BF-$\")aJ.xF07$$\":++++++vVtc8d#F-$\")i>\"H\"F07$$ \":MLLLLeR(\\;m/FF-$!)=5cgF07$$\":nmmmm;/^cmz$GF-$!*OI<-\"F07$$\":++++ +vo/[r7(HF-$!*NKvZ\"F07$$\":MLLLLLLeRwX5$F-$!*c#fzAF07$$\":MLLLLL$e*[` HP$F-$!*S-()3$F07$$\":MLLLLLLLeI8k$F-$!*R7<$RF07$$\":MLLLLL3-8>bx$F-$! *^/w8%F07$$\":MLLLLL$3xwq4RF-$!*=jnJ%F07$$\":MLLLL$3_]>!o(RF-$!*l=sJ%F 07$$\":MLLLLLeRA'*Q/%F-$!*y!4;VF07$$\":MLLLL3x1OVu2%F-$!*+RmI%F07$$\": MLLLL$eR(\\!*46%F-$!*M14D%F07$$\":MLLLLe9TjPX9%F-$!*7()pC%F07$$\":NLLL LLL3x%3yTF-$!*JZkA%F07$$\":ommmm;/^J'=XWF-$!*(*=ni$F07$$\":-+++++v$fyG 7ZF-$!*WH[V#F07$$\":ommmmT5:jQe%[F-$!*R'39>F07$$\":NLLLL$ek.%*Qz\\F-$! *g!Rj7F07$$\":+++++D\"yv,%H6&F-$!)6Q]:F07$$\":ommmmm;z%4\\Y_F-$\")<&Rx %F07$$\":++++++v$4'4.P&F-$\"*6F:9\"F07$$\":NLLLLLL3FGT\\&F-$\"*9%y/@F0 7$$\":qmmmmm\"HKp%zh&F-$\"*b>n*GF07$$\":++++++]PflVB$)F-$!+gy^K^F-7$$\":0++++++]Plxe)F-$!+Q))\\I \\F-7$$\":qmmmmm;/w)4_))F-$!+jAz\\WF-7$$\":NLLLLLLe9Kk6*F-$!+x;jGPF-7$ $\":.+++++]7`l2Q*F-$!+l_9!3$F-7$$\":qmmmm;a)3\\m_'*F-$!+W\"3DT#F-7$$\" :NLLLLLekGkX#**F-$!+\\()\\q=F-7$$\":+++++D1kOY'>5F0$!+96`::F-7$$\":nmm mmmmTIOo/\"F0$!+UG=08F-7$$\":MLLLL$e9;_yq5F0$!+)Q/m;\"F-7$$\":++++++D \"GTt%4\"F0$!+'\\i\")4\"F-7$$\":nmmmm;/,/$o=6F0$!+\"zmM0\"F-7$$\":MLLL LL$3_>jU6F0$!+W?ZC5F-7$$\":nmmmmm\"HdNb'>\"F0$!*1G!3**F-7$$\":+++++++D ;v/D\"F0$!*$>A/'*F-7$$\":++++++Dc\"4,k7F0$!*6d`e*F-7$$\":++++++](omax7 F0$!*dPT`*F-7$$\":++++++v=U#3\"H\"F0$!*4/]@*F-7$$\":+++++++]<=YI\"F0$! *6R.5*F-7$$\":++++++]7o*oJ8F0$!*hdS>)F-7$$\":+++++++v=h(e8F0$!*mn\"prF -7$$\":+++++++D,Q4T\"F0$!*lWh%GF-7$$\":+++++++v$[6j9F0$\"*J;&R8F-7$$\" :nmmm;HKkAg\\Z\"F0$\"*QMo\">F-7$$\":MLLLLek`h0o[\"F0$\"*68dJ#F-7$$\":n mmm\"H2$)4$GF\\\"F0$\"*8=gU#F-7$$\":++++](oH/5l)\\\"F0$\"**H)oT#F-7$$ \":MLLL3-j()ptX]\"F0$\"*2D5K#F-7$$\":nmmmm;HKR'\\5:F0$\"*-Q!f=F-7$$\": +++++v$4rr=M:F0$!(-,T&F-7$$\":MLLLLLe*[z(yb\"F0$!*Tj7(RF-7$$\":nmmmmTg _d[ge\"F0$!+&R6l@\"F-7$$\":+++++]i:?>Uh\"F0$!++WGuBF-7$$\":nmmm;a8Z^/$ G;F0$!+lH))QHF-7$$\":MLLLLeky#)*QU;F0$!+=s)pr$F-7$$\":++++]i:59vkl\"F0 $!+Bb.`UF-7$$\":nmmmmmmTXg0n\"F0$!+.0k]]F-7$$\":nmmmmm\"zpYU%p\"F0$!+v +\"p$fF-7$$\":nmmmmm;a)))G=F0$!*mgMF&F07$$\":nmmmm mm;pW`(>F0$!*')3FT%F07$$\":MLLLL$e9TOEH?F0$!*.pop$F07$$\":,+++++D1f#=$ 3#F0$!*2?/E$F07$$\":,++++D\"G)[=g4#F0$!*S$z*>$F07$$\":,++++]PfQa)3@F0$ !*Db&eJF07$$\":,++++v$f$G!p@@F0$!*$[YZJF07$$\":,+++++]7=EX8#F0$!*_%RFJ F07$$\":,+++]7y+8W49#F0$!*f&QbJF07$$\":,++++D1*y?OZ@F0$!*5a>7$F07$$\": ,+++]PMx-!y`@F0$!+ypp_JF-7$$\":,++++]il(z>g@F0$!+'\\C)*>$F-7$$\":,++++ v=U(Q.t@F0$!+o$4.@$F-7$$\":,+++++v=xpe=#F0$!+Q))*eA$F-7$$\":ommmmTNrfb E@#F0$!+\\7DoLF-7$$\":MLLLL$eRA9WRAF0$!+eR!)fMF-7$$\":+++++DcwCFiE#F0$ !+q#**p[$F-7$$\":nmmmmm;H28IH#F0$!+qse7NF-7$$\":nmmmm;a8d3AM#F0$!+0QW> LF-7$$\":nmmmmm\"zpSS\"R#F0$!+&HGC$HF-7$$\":MLLLLL$3_?`(\\#F0$!+.]m\\? F-7$$\":MLLLL$3_D1l_DF0$!+cfO\\pxg#F0$!+gBn)e\"F-7$ $\":ommmmmm\"z+vbEF0$!+rQ1?:F-7$$\":,+++++v$f4t.FF0$!+gNF$\\\"F-7$$\": ommmm;zWi^bv#F0$!+&oZ#*[\"F-7$$\":MLLLLL$e*Gst!GF0$!+>I6-:F-7$$\":ommm mm\"H2\"34'GF0$!+3EUC:F-7$$\":,++++++]#RW9HF0$!+[SLo:F-7$$\":,+++++D\" y#=o'HF0$!+=XuM;F-7$$\":,+++++]7j#>>IF0$!+cw-N#[Z%F0$!+o'QN6#F-7$$\":ommmmm;aG!e&e%F0$!+H'eo@#F-7$ $\":NLLLLLLL)Qk%o%F0$!+FV[MBF-7$$\":-+++++D1Mm-z%F0$!+vWLjCF-7$$\":-++ +++v$40O\"*[F0$!+GXgmDF-7$$\"\"&F)$!+'[Iwj#F--%&COLORG6&%$RGBGF($\"\"# !\"\"$\"\"\"F)-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6% 7itF'7$$\":ommmmm;a8ABO\"!#E$!).2&>\"F07$$\":NLLLLLL3FWYs#F][n$!)vK3@F 07$$\":ommmmmT&Q`!eS$F][n$!)]M\"f#F07$$\":-+++++]iSmp3%F][n$!)rIpFF07$ $\":OLLLLLeRZF\"oZF][n$!)oLiFF07$$\":qmmmmmm;a)G\\aF][n$!),O:EF07$$\": 0++++++D\"G$R<)F][n$!(:ox#F07$F+$\")>H!>$F07$F7$\"*cp:3\"F07$Fin$\"*=J nX\"F07$F^o$\"*+CH@\"F07$Fco$\")=b:$)F07$F]p$\"(Bd&HF07$Fgp$!))3(f#*F0 7$F\\q$!*;Aeb\"F07$Faq$!*)R\"f?#F07$Ffq$!*-))*[BF07$F[r$!*pyiV#F07$F`r $!*!G\">V#F07$Fer$!*Xb\\R#F07$Fjr$!*&\\%=Q#F07$F_s$!*10\\H#F07$Fds$!*3 jEH#F07$Fis$!*)H$*pAF07$F^t$!*;p*R;F07$Fct$!)wEEdF07$Fht$!)o)[`\"F07$F ]u$\")'*yeMF07$Fbu$\"*&*\\N:\"F07$Fgu$\"*htwc\"F07$Fav$\"*5kBa#F07$F[w $\"*y@2<$F07$$\":NLLLLL3_D%eleF-$\"*$eXGLF07$F`w$\"*'e6qMF07$$\":NLLLL $eRZAJ^gF-$\"*:$R%\\$F07$$\":++++++D\"y:A8hF-$\"*6&*3_$F07$$\":qmmmm;a )348vhF-$\"+g:DeMF-7$Few$\"+Z'H(>MF-7$Fjw$\"+SD%e&HF-7$F_x$\"+sVePAF-7 $Fix$\"+$*zOv6F-7$Fcy$\"*T7*zSF-7$F]z$!*-a=[&F-7$Fgz$!+*ys=7\"F-7$$\": SLLLLL$eRilDzF-$!+%[]hL\"F-7$F\\[l$!+u@>39F-7$$\":qmmmmm;a8'R\"4)F-$!+ a8^S9F-7$$\":SLLLLL$e9TaC\")F-$!+2E\\w9F-7$$\":0+++++]P4#pd\")F-$!+Z_[ c9F-7$$\":qmmmmm;H2S3>)F-$!+ej#*o9F-7$$\":0+++++]7.OrD)F-$!++$*ym9F-7$ Fa[l$!+,j0j9F-7$F[\\l$!*=/eh*F-7$Fe\\l$!*u!yzFF-7$Fj\\l$\"(ngF)F-7$F_] l$\"*&*oW6#F-7$$\":nmmm;/EvR^g+\"F0$\"*>Zra#F-7$Fd]l$\"*lS%=JF-7$$\":M LLL$ekGN8CL5F0$\"*\"fqhLF-7$Fi]l$\"*]DsY$F-7$Fc^l$\"*o'f%e$F-7$F]_l$\" *(Rz#f$F-7$$\":+++++](oaFfp6F0$\"*HNds$F-7$Fb_l$\"*n#GBRF-7$$\":MLLLL$ e*)fV^B7F0$\"*wZtE%F-7$Fg_l$\"*'**R;ZF-7$Fa`l$\"*[nh=&F-7$F[al$\"*.Yz( fF-7$F`al$\"*.kI=(F-7$Feal$\"*\\`dm)F-7$Fjal$\"+'G'=)Q\"F-7$F_bl$\"+BH (Q4#F-7$F]dl$\"+1`\"[!GF-7$Fgdl$\"+af)\\L$F-7$$\":++++]P4@Ej>d\"F0$\"+ )Gw;S$F-7$F\\el$\"+v3)z\\$F-7$$\":++++v$f$=B\"4$f\"F0$\"+@Jn'[$F-7$$\" :MLLL$e9T))Q8+;F0$\"+\\'[%eMF-7$$\":+++](=#*p;_l.;F0$\"+\"[Kb]$F-7$$\" :nmmm\"zp)\\awrg\"F0$\"+#*o[VNF-7$$\":MLL$eRZFtyp5;F0$\"+'f4NY$F-7$Fae l$\"+M^25NF-7$F[fl$\"+qR4:MF-7$Fefl$\"+DR%4H$F-7$$\":nmmmm;z>c#\\#o\"F 0$\"+A3O(G$F-7$Fjfl$\"+BDh#H$F-7$$\":nmmmm;/wxcjq\"F0$\"+-[63LF-7$F_gl $\"+HyPGLF-7$Fdgl$\"+jz9[MF-7$Figl$\"+(y'[:PF-7$$\":+++++DcE2xPz\"F0$ \"*Mg(>UF07$F^hl$\"*%H7;\\F07$$\":ommmmTNYe'H\\=F0$\"*a#3_cF07$Fchl$\" *4&)>T'F07$$\":ommmmT&QG%G;!>F0$\"*%)3mA(F07$Fhhl$\"*lb#yxF07$$\":++++ +D1Rgs2&>F0$\"*iANT)F07$F]il$\"*\\vwt)F07$$\":MLLL$ek.HW#)))>F0$\"*(QR $*))F07$$\":+++++D1k;/B+#F0$\"*L<)H!*F07$$\":nmmm;/wP!Ry:?F0$\"*_U<0*F 07$Fbil$\"*LLy1*F07$$\":ommmmT&)e6Bi0#F0$\"*#)ft#*)F07$Fgil$\"*))[2i)F 07$Fajl$\"*A7B?)F07$F[[m$\"*N`co(F07$F_\\m$\"+'>'z!3(F-7$Fi\\m$\"+Q;rL lF-7$F^]m$\"+C0C2fF-7$Fc]m$\"+I$>IN&F-7$Fh]m$\"+V.uq[F-7$F]^m$\"+YZ3PW F-7$Fb^m$\"++w%H&QF-7$Fg^m$\"+(*)[BF-7$F``m$\"+kE\\NAF-7$$\":MLLLL3F>HT'HFF0$\"+^<+8 AF-7$Fe`m$\"+W^92AF-7$$\":,++++DJq&>Y\"y#F0$\"+#R8q@#F-7$Fj`m$\"+\"RV< C#F-7$F_am$\"+gqUXBF-7$Fdam$\"+K&pv]#F-7$F^bm$\"+\\m.uHF-7$Fhbm$\"+m3; =NF-7$F]cm$\"+s7XGTF-7$Fgcm$\"+OxSSYF-7$F\\dm$\"+r^Ru[F-7$Fadm$\"+qZ2e ]F-7$$\":MLLLLekyZ2mY$F0$\"+,Og;^F-7$Ffdm$\"+#Q+>;&F-7$$\":,++++v$fe?_ :NF0$\"+Kg)p=&F-7$F[em$\"+J*=i>&F-7$$\":,++++vVti?OF0$\"+%Q2C5&F-7$Feem$\"+%ydq.&F-7$$ \":MLLLL$e*[t\\sp$F0$\"+y\")[v[F-7$Fjem$\"+*GL@o%F-7$F_fm$\"+:zdZUF-7$ Fdfm$\"+uc9hQF-7$Fifm$\"+[lCTNF-7$$\":ommmm;zW7@^6%F0$\"+9qcBMF-7$F^gm $\"++?cJLF-7$$\":NLLLLL3-$H**>UF0$\"+$\\MWE$F-7$Fcgm$\"+s+pDKF-7$$\":- ++++](=n(*fDVF0$\"+Wy:9KF-7$Fhgm$\"+p'**yA$F-7$$\":MLLLL$e9\"z-lU%F0$ \"+(pA4E$F-7$F]hm$\"+Yz76LF-7$Fbhm$\"+oQKvMF-7$Fghm$\"+aFkaOF-7$F\\im$ \"+yUJYQF-7$Faim$\"++Bf,SF-7$Ffim$\"+Sw$46%F--F[jm6&F]jmF($\"\"(F`jmF( -Fdjm6#%8scheme~with~c[10]=15/16G-F$6%7itF'7$F[[n$!)@PT5F07$Fa[n$!)*Q) Q=F07$$\":-++++](o/[AlIF][n$!)A2fAF07$Ff[n$!)j6iAF07$$\":NLLLL$eRseQYP F][n$!)%oKG#F07$F[\\n$!)I5@CF07$F`\\n$!)fu:CF07$Fe\\n$!)>Q$H#F07$Fj\\n $!(0&4JF07$F+$\")$H`e#F07$F7$\")Il^')F07$Fin$\"***4$y5F07$F^o$\")$[l\" zF07$Fco$\")W2OTF07$Fho$!(LF07$F^t$! *AK9m\"F07$$\":NLLLLeRs3P(yXF-$!*7&)fJ\"F07$Fct$!)eV$p'F07$Fht$!)'e)GF F07$F]u$\")=:c?F07$Fbu$\")GEY**F07$Fgu$\"*KK4T\"F07$F\\v$\"*(**GL=F07$ Fav$\"*9piU#F07$Ffv$\"*`f8)GF07$F[w$\"*K7z7$F07$Fjan$\"*/9?K$F07$F`w$ \"*^1\"4NF07$Fbbn$\"*F19b$F07$Fgbn$\"*GJDg$F07$F\\cn$\"+,37SNF-7$Few$ \"+THV:NF-7$Fjw$\"+>f8HIF-7$F_x$\"+GU`+AF-7$$\":NLLLLLeR(>(R\"oF-$\"+J VL2=F-7$Fdx$\"+7^Y5\"F-7$Fhy$!*)*>.&*)F-7$F]z$!+#zyrX\"F- 7$Fbz$!+PeM?=F-7$Fgz$!+KB$H&F-7$F`al$!*wahz%F-7$ Feal$!*dT\\@%F-7$Fjal$!*1p0g\"F-7$F_bl$\"*XN/<\"F-7$Fdbl$\"*YVDg\"F-7$ Fibl$\"*sY1%>F-7$Fccl$\"*PBn7#F-7$F]dl$\"*Jsk&>F-7$Fbdl$\"*[Cf.\"F-7$F gdl$!*;*z(4\"F-7$F\\el$!*?R/%eF-7$Fael$!+T%pDG\"F-7$Ffel$!+dC6I;F-7$F[ fl$!+:b&>7#F-7$F`fl$!+QP\\hCF-7$Fefl$!+Qx4*)HF-7$Fjfl$!+mji#e$F-7$F_gl $!+3D/HTF-7$Fdgl$!+\\TZhXF-7$Figl$!+(yCF\"[F-7$$\":MLLL$e9m%>(*)z$F07$F\\ao$!* `=Jq#F07$F]il$!*%*o(3CF07$Fbil$!*!QC2=F07$Fgil$!*>%[b:F07$F\\jl$!*BIz` \"F07$Fajl$!*Uj0a\"F07$Ffjl$!*XH\"p:F07$F[[m$!*38Ie\"F07$F_\\m$!+H*Hdp \"F-7$Fi\\m$!+,#))ev\"F-7$F^]m$!+ze(z)=F-7$Fc]m$!+)3]$p>F-7$Fh]m$!+S8& *)*>F-7$F]^m$!+K#Hu,#F-7$Fb^m$!+QlS$*=F-7$Fg^m$!+/4V`;F-7$$\":+++++]P4 1oWW#F0$!+`(QGP\"F-7$F\\_m$!+L!379\"F-7$$\":MLLLL3-)Q84DDF0$!+QAa\\5F- 7$Fa_m$!*VG$4)*F-7$$\":MLLLLeRA\"*4-e#F0$!*Y,VL*F-7$Ff_m$!*(z:4!*F-7$F [`m$!*m5uq)F-7$F``m$!*->eh)F-7$Fj`m$!*B*[8()F-7$Fdam$!*aI%\\!*F-7$Fiam $!*SYZP*F-7$F^bm$!*))o\\()*F-7$Fcbm$!+.XJj5F-7$Fhbm$!+x/@o6F-7$F]cm$!+ u,Hd9F-7$Fbcm$!+RyK!e\"F-7$Fgcm$!+Vq;!o\"F-7$F\\dm$!+j)>[v\"F-7$Fadm$! +;%f\\z\"F-7$Ffdm$!+<544=F-7$F[em$!+c&3t!=F-7$F`em$!+8GU\"z\"F-7$Feem$ !+5mAil7F-7$Fghm$!+q`(=L\"F-7$F\\im$!+hL!eS\"F-7$Faim$!+)) [Vl9F-7$Ffim$!+5:^1:F--F[jm6&F]jm$\"\"*F`jm$\"\"%F`jmF(-Fdjm6#%8scheme ~with~c[10]=19/20G-F$6%7euF'7$Fa[n$!)K!pP#F07$Fe\\n$!)0'))e$F07$Fj\\n$ !)p`UBF07$F+$\"(,X^\"F07$F7$\")Nk#e'F07$Fin$\"*.(RK6F07$$\":MLLL$e*)fI &*y/@F-$\"*a))[9\"F07$$\":+++++D\"G))>Wr@F-$\"*f$[`6F07$$\":nmmm;ajfW% 4QAF-$\"*0jR8\"F07$F^o$\"*$*zz2\"F07$$\":nmmmmT5!>=0QCF-$\")\">B'**F07 $Fco$\")_D*o)F07$F]p$\")uSVMF07$Fgp$!)F9#>$F07$F\\q$!)-q\"[(F07$Faq$!* w&RP6F07$$\":MLLLL3_+sx[n$F-$!*-WO9\"F07$$\":MLLLL$3xc[U3PF-$!*=\")\\= \"F07$$\":MLLLLe*[$*>(>u$F-$!*wxf=\"F07$Ffq$!*y6p=\"F07$$\":MLLLL3FpEm !4QF-$!*@0C=\"F07$$\":MLLLL$ek.MhUQF-$!*$3'G=\"F07$$\":MLLLLek.agh(QF- $!*H#Qx6F07$F[r$!*!HSD6F07$Fer$!**\\L75F07$Fis$!)0*H:)F07$Ffcp$!)G^>OF 07$F^t$!(>\"4bF07$F^dp$\")Kt_LF07$Fct$\"*byB-\"F07$Fht$\"*pRsR\"F07$F] u$\"*tv*==F07$Fbu$\"*Xr!fCF07$Fgu$\"*C1_t#F07$Fav$\"*NstF$F07$F[w$\"*W ]tV$F07$F`w$\"*L&ojKF07$Few$\"+yiG!z#F-7$$\":qmmmmmm;HZ_O'F-$\"+xy>0DF -7$Fjw$\"+NfX!3#F-7$$\":NLLLLLLeRh;i'F-$\"+`?Rb;F-7$F_x$\"+n]3F8F-7$Fd x$\"*%Hi`(*F-7$Fix$\"*U#y(e%F-7$F^y$\"*E/K(>F-7$Fcy$!)D**4PF-7$F]z$!*V -YR&F-7$Fgz$!*())QktF-7$$\":0+++++]7Gg$fyF-$!*kRIZ(F-7$Ffdn$!*f*44xF-7 $$\":qmmmmm;z>_>*zF-$!*uT'*f(F-7$F\\[l$!*\"\\K%[(F-7$F]fn$!*]#4YpF-7$F a[l$!*JvJ0'F-7$F[\\l$!*&*yGg\"F-7$Fe\\l$\"*%eg%=#F-7$Fj\\l$\"*IUfL$F-7 $F_]l$\"*li7*QF-7$Ffgn$\"*<*f/RF-7$Fd]l$\"*%zRERF-7$F^hn$\"*Su#\\QF-7$ Fi]l$\"*)fAVPF-7$F^^l$\"*9vD`$F-7$Fc^l$\"*jHMP$F-7$Fh^l$\"*=kfH$F-7$F] _l$\"*=;nL$F-7$Fb_l$\"*>XTp$F-7$Fg_l$\"*PfSK%F-7$Fa`l$\"*uH[i%F-7$F[al $\"*-N.0&F-7$F`al$\"*yj@m&F-7$Feal$\"*sf)okF-7$Fjal$\"*.#43&*F-7$F_bl$ \"+xn7G9F-7$F]dl$\"+bD*3+#F-7$Fgdl$\"+[%>=_#F-7$F\\el$\"+m]I$HF-7$Ffel$\"+`^K]HF-7$$\":++++D 1*GrrMN;F0$\"+N+:^HF-7$F[fl$\"+A)=/#HF-7$$\":nmmmT5SW[K%\\;F0$\"+Ve)H# HF-7$F`fl$\"+&\\$[\\HF-7$$\":MLLLe9\"f(zE$F07$F\\ `o$\"*W?au$F07$Fchl$\"*%zt=VF07$Fd`o$\"*^+9-&F07$Fhhl$\"*8GG`&F07$F\\a o$\"*Q$\\xhF07$F]il$\"*8;'GlF07$Fdao$\"*w2'3nF07$Fiao$\"*[)HtoF07$F^bo $\"*7W%=pF07$Fbil$\"*T$=fpF07$Ffbo$\"*YQ\\'oF07$Fgil$\"**Hs)f'F07$Fajl $\"*?L0B'F07$F[[m$\"*f'fwdF07$F_\\m$\"+%=$\\d_F-7$Fi\\m$\"+T\")4>[F-7$ F^]m$\"+\\d$GL%F-7$Fc]m$\"+(f@%GRF-7$Fh]m$\"+.J[\"f$F-7$F]^m$\"+t+w0LF -7$Fb^m$\"+OjyGHF-7$Fg^m$\"+Re^iEF-7$F\\_m$\"+dLb)>#F-7$Ff_m$\"+_X:>=F -7$F[`m$\"+O\\c4]tUGF-7$Fifm$\"+/*>pg#F-7$Fa[p $\"+'R9#>DF-7$F^gm$\"+?VG]CF-7$Fi[p$\"+q*3'*R#F-7$Fcgm$\"+%)oHqBF-7$Fa \\p$\"+@wGhBF-7$Fhgm$\"+_g9rBF-7$Fi\\p$\"+cGX&R#F-7$F]hm$\"+>phKCF-7$F bhm$\"+l,QaDF-7$Fghm$\"+&y'4(o#F-7$F\\im$\"+'Rq#GGF-7$Faim$\"+(ob@%HF- 7$Ffim$\"+6dCAIF--%'COLOURG6&F]jm$\"*++++\"!\")F(Fjat-Fdjm6#%8scheme~w ith~c[10]=32/33G-F$6%7iwF'7$Fa[n$!)X*GP#F07$Fe\\n$!)m`KRF07$Fj\\n$!)&z sV$F07$F+$!)GQ$o\"F07$F2$\")GnT6F07$F7$\")'zEw$F07$FA$\")_\"yO(F07$Fin $\")[yO#*F07$F_`r$\")?\"R!**F07$F^o$\"*F,P-\"F07$$\":nmmm\"zp[KJ2QBF-$ \"*.9#G5F07$$\":++++]7G8O*RrBF-$\"*x13.\"F07$$\":MLLL3Fp,fDZS#F-$\"*DX U+\"F07$F\\ar$\"*Dc&35F07$$\":++++Dc^y/y8Z#F-$\"*#y645F07$$\":MLLL$3Fp wUq/DF-$\")zdo&*F07$$\":nmmmT&Q`00.QDF-$\")UH3'*F07$Fco$\")yM$f*F07$Fh o$\")k()[yF07$F]p$\")*=?p'F07$Fbp$\")1mw_F07$Fgp$\")4z\"[#F07$F\\q$!(: 1G#F07$Faq$!)B(4J#F07$F[r$!(1u**)F07$Fis$\")J\"Gn#F07$$\":ommm;/,p:g[C %F-$\")V<\\YF07$Ffcp$\")\"fKI(F07$$\":NLLL$ek.H4TyVF-$\")b**)p(F07$F^t $\"*mmL-\"F07$$\":-+++](=<,<'>^%F-$\"*584M\"F07$F^dp$\"*D;KQ\"F07$$\": ommm;H2LZ7bk%F-$\"*//Jm\"F07$Fct$\"*3Ea*>F07$Fht$\"*EiOI#F07$F]u$\"*#4 ZKEF07$$\":qmmm;a8(*ykh/&F-$\"*q%zfGF07$Fbu$\"*()\\()4$F07$$\":qmmmT5: )oyKY^F-$\"*%*o;3$F07$$\":NLLL$e*[=c:(z^F-$\"*La95$F07$$\":++++D\"G)[D .J@&F-$\"*WB]F$F07$Fgu$\"*)GEbKF07$F\\v$\"*yGiO$F07$Fav$\"*<#fZMF07$Ff v$\"**H(yR$F07$F[w$\"*Xo'eKF07$Fjan$\"*`F&fIF07$F`w$\"*_tbu#F07$Fbbn$ \"*Vr\\e#F07$Fgbn$\"*9aGJ#F07$F\\cn$\"+^o5VAF-7$Few$\"+[drp>F-7$$\":++ ++++DcwV6I'F-$\"+MZCZ;F-7$F^gr$\"+vU#Hg\"F-7$$\":NLLLLL3x\"3NHkF-$\"+7 q3$G\"F-7$Fjw$\"+XeTx5F-7$$\":qmmmmm\"zpybdlF-$\"**)[=C*F-7$Ffgr$\"*p' QXgF-7$$\":++++++v=#\\w&o'F-$\"*3nd.&F-7$F_x$\"*.`q%GF-7$Ffgp$!'dSzF-7 $Fdx$!)3A0PF-7$F^hp$!*M$=qDF-7$Fix$!*J1;%[F-7$F^y$!*$zu>mF-7$Fcy$!*,!> **zF-7$Fhy$!*ULNf*F-7$F]z$!+g'4+,\"F-7$$\":qmmmmm;zWwTf(F-$!+vSe25F-7$ Fbz$!*DyV%**F-7$$\":NLLLLL$ek$ons(F-$!*\">5w(*F-7$Fgz$!*U`VU*F-7$F\\[l $!*>PRW(F-7$Fa[l$!*<(G\\VF-7$Ff[l$!*$3O5=F-7$F[\\l$\")nG6^F-7$F`\\l$\" *spAE#F-7$Fe\\l$\"*0nH&GF-7$$\":qmmm;/^cPS([%*F-$\"*u\"[#*HF-7$$\":NLL LL3_+A:n^*F-$\"*&3p'4$F-7$$\":++++]7`W1!p%e*F-$\"*WdR0$F-7$Fj\\l$\"*3i r0$F-7$$\":+++++Dcwf9')y*F-$\"*\\v_&HF-7$F_]l$\"*D,4w#F-7$Fd]l$\"*$f% \\D#F-7$Fi]l$\"*n=<\"=F-7$$\":+++++Dc,w5)e5F0$\"*j4#4;F-7$F^^l$\"*DM6T \"F-7$$\":nmmmmTN@nfF3\"F0$\"*=5/J\"F-7$Fc^l$\"*gApB\"F-7$$\":MLLLLe9T e3n5\"F0$\"*%*[+>\"F-7$Fh^l$\"*w\"3u6F-7$$\":+++++v$4'\\d18\"F0$\"*X#) p?\"F-7$F]_l$\"*-*\\Z7F-7$Fb_l$\"*0(\\C;F-7$Fg_l$\"*-=,-#F-7$F[al$\"*E g_7#F-7$Feal$\"*A&3cAF-7$Fjal$\"*+=nA$F-7$F_bl$\"*\"3I%o&F-7$F]dl$\"*D aPa*F-7$Fgdl$\"+UxbZ8F-7$F\\el$\"+0#GXb\"F-7$Fael$\"+%4$4l;F-7$F^_s$\" +N\\-\"o\"F-7$Fc_s$\"+a1!zl\"F-7$Fh_s$\"+1q\")z;F-7$Ffel$\"+j/B(o\"F-7 $$\":MLL$3-8+Ve#=j\"F0$\"+x\\dc;F-7$F``s$\"+ke!yn\"F-7$$\":nmm;H#od*\\ o)Q;F0$\"+61ro;F-7$F[fl$\"+>`PQ;F-7$Fh`s$\"+z#*e>;F-7$F`fl$\"+^B3?;F-7 $F`as$\"+,[3^:F-7$Fefl$\"+J\\>l9F-7$Fjfl$\"+\"eq&F-7$Fj`q$\"*jBuo$F-7$Fd_o$\")?a$=$F07 $Fbaq$\")!)[\"=$F07$Fgaq$\")C_4HF07$F\\bq$\");4JHF07$Fabq$\")G\\SHF07$ $\":MLL$eR(\\,#p76=F0$\")RA)*GF07$Ffbq$\")dv=HF07$$\":nmm;a8Z\"fo1==F0 $\")&Hk'HF07$F^hl$\")[ZPJF07$F\\`o$\")38V\\F07$Fchl$\");xy\")F07$$\":M LLL$3_]%QU$*)=F0$\"*@*[p5F07$Fd`o$\"*JJ/L\"F07$$\":,+++]ilAZ9R\">F0$\" *4%))G:F07$Fhhl$\"*DtUt\"F07$$\":nmmm;/E+c'[Q>F0$\"*31/)>F07$F\\ao$\"* pqZI#F07$$\":MLLL$ekyZ'eI'>F0$\"*PV*yCF07$F]il$\"*ERqj#F07$Fiao$\"*NeW *HF07$Fbil$\"*^>/8$F07$$\":ommm\"H2$)4N+O?F0$\"*`se8$F07$$\":,+++]i:&y LuU?F0$\"*@Pu9$F07$$\":MLLL3_+sC$[\\?F0$\"*^s[8$F07$Ffbo$\"*3ef6$F07$$ \":MLLL$3_D`Gqp?F0$\"*1td/$F07$Fgil$\"*k%o^HF07$Fajl$\"*ET#4FF07$F[[m$ \"*[&=.CF07$F_\\m$\"+S+0d?F-7$Fi\\m$\"+Dy22=F-7$F^]m$\"+p,/Q:F-7$Fc]m$ \"+**G.`8F-7$Fh]m$\"+_H7F7F-7$F]^m$\"+!R/*[6F-7$$\":nmmm;/Ev%>J0BF0$\" +(p*GB6F-7$$\":nmmmmTN@#3hlP5\"F-7$$\":nmmm;zWnp4*HBF0$\"+z7 z-6F-7$Fb^m$\"+8.-!4\"F-7$$\":nmmm;ajfW2XN#F0$\"+g`-x5F-7$$\":nmmmm\"H d?j!oO#F0$\"+)4:c1\"F-7$$\":nmmm;H#=&>0\"zBF0$\"+T?]i5F-7$Fg^m$\"+=v(* f5F-7$F\\_m$\"*rf&e!*F-7$Ff_m$\"*#eOmqF-7$F[`m$\"*UjcZ'F-7$F``m$\"*%\\ X8hF-7$Fheo$\"*R'z1gF-7$Fe`m$\"*J0D'fF-7$F`fo$\"*w(fgfF-7$Fj`m$\"*`-?, 'F-7$F_am$\"*&f6riF-7$Fdam$\"*Vm9t'F-7$Fiam$\"*x+?Q(F-7$F^bm$\"*wtO@)F -7$Fhbm$\"*j>(f**F-7$F]cm$\"+81?a6F-7$Fgcm$\"+)>#[o7F-7$Fadm$\"+Z1G'Q \"F-7$F`ho$\"++1n19F-7$Ffdm$\"+\\M$fU\"F-7$Fhho$\"+*G*HP9F-7$F[em$\"+p ?[X9F-7$F`io$\"+rGAX9F-7$F`em$\"+')==R9F-7$Fhio$\"+=+)fU\"F-7$Feem$\"+ ]/g29F-7$F`jo$\"+WPtg8F-7$Fjem$\"+IEA08F-7$F_fm$\"+I8^(=\"F-7$Fdfm$\"+ km+'3\"F-7$Fifm$\"*.GQ(**F-7$F^gm$\"*p=VM*F-7$Fi[p$\"*&))*z7*F-7$Fcgm$ \"*)))G+!*F-7$Fa\\p$\"*u=_&*)F-7$Fhgm$\"*+!)y)*)F-7$F]hm$\"*,#yF#*F-7$ Fbhm$\"*sK&3(*F-7$Fghm$\"+szDA5F-7$F\\im$\"+*zma2\"F-7$Faim$\"+Z!4x6\" F-7$Ffim$\"+%Rat9\"F--F[jm6&F]jm$FgimF`jmF(Fajm-Fdjm6#%8scheme~with~c[ 10]=81/82G-F$6%7gtF'7$Fa[n$!)Sl7=F07$Fe\\n$!)]H:JF07$$\":0+++++v$4'\\/ 8'F][n$!)sgGKF07$$\":SLLLLL$3x1h6oF][n$!)anNKF07$$\":qmmmmm\"zW$F07$Fj\\n$!)t!H+$F07$$\":qmmmmm;z%\\DO&*F][n$!)L,xEF07$F+$! )D6n>F07$F7$\")0)fe\"F07$Fin$\")i%>]&F07$F_`r$\")8`YgF07$F^o$\"),\\%[' F07$F\\ar$\")VL.lF07$Fco$\")jq4jF07$F]p$\")NzfZF07$Fgp$\")^R_BF07$F\\q $\"(4-Y)F07$Faq$!'\"R\"eF07$F[r$\")uq>:F07$Fis$\")QpVWF07$Ffcp$\")5\"e /)F07$F^t$\"*mxm-\"F07$F^dp$\"*&=s&H\"F07$Fct$\"*w;%[r-5F-7$Fc^l$!*e?!>5F-7$Fffu$!*+)R=5F-7$Fh^l$!). 8n**F-7$F^gu$!)JY$R*F-7$F]_l$!)y&['))F-7$Fb_l$!)k*Hv&F-7$Fg_l$!)M0dRF- 7$F[al$!)?O2dF-7$Feal$!)h@z\"*F-7$$\":++++++]PR0=P\"F0$!)YDX)*F-7$$\": ++++++++g\\[Q\"F0$!*&\\V>5F-7$$\":++++++]i!Q*yR\"F0$!*aX(H5F-7$Fjal$!) -r*e*F-7$$\":++++++](=A)RU\"F0$!)pLI'=#F07$Fd`o$!*w+!**=F07$Fhhl$!*?]Pl\"F07$F \\ao$!*xXRD\"F07$F]il$!*(>)G+\"F07$Fiao$!)y$4-(F07$Fbil$!)=4CaF07$Ffav $!)?B0_F07$F[bv$!)\"z3#\\F07$F`bv$!)v?6[F07$Ffbo$!)\"*fQZF07$Fhbv$!)y[ (z%F07$Fgil$!)')G)*\\F07$F[[m$!)[)H*oF07$Fi\\m$!*M40%))F-7$$\":MLLL$3_ ]%oi#*>#F0$!*54xC*F-7$F^]m$!*PC(*\\*F-7$$\":,+++]il(4&[gA#F0$!*;ej\\*F -7$Fc]m$!*n5*Q&*F-7$Fh]m$!*p>H<*F-7$F]^m$!*0$3=&)F-7$Fb^m$!*<#p(*oF-7$ Fg^m$!*kbMQ&F-7$$\":MLLLLek`1OzT#F0$!*:$)4\"[F-7$Fjfq$!*C)>AVF-7$$\":n mmmmT5l0+5Z#F0$!*tdi*RF-7$F\\_m$!*U4>z$F-7$Fa_m$!*oq0k$F-7$Ff_m$!*9,;h $F-7$F``m$!*xf;m$F-7$Fj`m$!*SK&*y$F-7$Fdam$!*enI3%F-7$F^bm$!*-N)pWF-7$ Fhbm$!*FE%*4&F-7$F]cm$!*\")fAJ'F-7$Fbcm$!*`#3?pF-7$Fgcm$!*U^WW(F-7$F\\ dm$!*VpK$yF-7$Fadm$!*'eT7!)F-7$Ffdm$!*@?B.)F-7$F[em$!*9K!pzF-7$Feem$!* fs=p(F-7$Fjem$!*c]+E(F-7$F_fm$!*%HS7mF-7$Fdfm$!*(**)G&fF-7$Fifm$!*FnSV &F-7$F^gm$!*$)o0:&F-7$Fcgm$!*`[K/&F-7$Fhgm$!*[#>q]F-7$F]hm$!*'z2\">&F- 7$Fbhm$!*BIPU&F-7$Fghm$!*gG[p&F-7$F\\im$!**Rt0gF-7$Faim$!*u4\\E'F-7$Ff im$!*zRmW'F--Fhat6&F]jmFjatF(F(-Fdjm6#%:scheme~with~c[10]=169/170G-%%F ONTG6$%*HELVETICAGF^^r-%&TITLEG6#%Verror~curves~for~11~stage~order~7~R unge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fc_y-%%VIEWG6$;F(Ffim;$!#$ )!#<$\"#$*F\\`y" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" " scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme with c[10]= 81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 2 of 11 stage, order 7 Runge-Kutta methods " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dy G\"\"\"%#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)*$% \"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of \+ each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 741 "G := (x,y ) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`sl ope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: ` ,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner sc heme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10 ]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme \+ with `*c[10]=169/170]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^ 2):\nfor ct to 6 do\n Gn_RK7_||ct := RK7_||ct(G(x,y),x,y,x0,y0,hh,nu msteps,false);\n sm := 0: numpts := nops(Gn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK7_||ct[ii,2]-g(Gn_RK7_||ct[ii,1]))^2; \n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits \+ := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\" \"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no .~of~steps:~~~G\"$+#Q(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner ~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+KUZhQ!#D7$/*&%-scheme~w ith~GF+F,F+#\"#:\"#;$\"+N\"H[y#F57$/F8#\"#>\"#?$\"+s!)HbAF57$/F8#\"#K \"#L$\"+C/^k8F57$/F8#\"#\")\"##)$\"+$zac>'!#E7$/F8#\"$p\"\"$q\"$\"+,fs QMFTQ(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical \+ procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Ku tta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value ob tained by each of the methods at the point where " }{XPPEDIT 18 0 "9. 99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 671 "G := (x,y) -> x/y: hh := 0. 05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]= 11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme wi th `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/1 70]: errs := []:\nDigits := 30:\nfor ct to 6 do\n gn_RK7_||ct := RK7 _||ct(G(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\ng := x -> sqrt(1+x ^2):\nxx := 9.99: gxx := evalf(g(xx)):\nfor ct to 6 do\n errs := [op (errs),abs(gn_RK7_||ct(xx)-gxx)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~ G-%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q (pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\" &%\"cG6#\"#5F+#\"#6\"#7$\"+()f>=8!#D7$/*&%-scheme~with~GF+F,F+#\"#:\"# ;$\"+cd'=O*!#E7$/F8#\"#>\"#?$\"+\\))zbyF?7$/F8#\"#K\"#L$\"+RW>PZF?7$/F 8#\"#\")\"##)$\"++NBg=F?7$/F8#\"$p\"\"$q\"$\"+'e/J)f!#FQ(pprint76\"" } }}{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$\"\"!\"#5" }{TEXT -1 82 " of each R unge-Kutta method is estimated as follows using the special procedure \+ " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 100 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2 ):\nfor ct to 6 do\n sm := NCint((g(x)-'gn_RK7_||ct'(x))^2,x=0..10,a daptive=false,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/1 0)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~ scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+eV'*\\Q!#D7$/*&%-scheme~ with~GF+F,F+#\"#:\"#;$\"+0xuwFF57$/F8#\"#>\"#?$\"+16v[AF57$/F8#\"#K\"# L$\"+<^#3O\"F57$/F8#\"#\")\"##)$\"+\"RwU='!#E7$/F8#\"$p\"\"$q\"$\"+,q? MMFTQ(pprint86\"" }}}{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 0 515 "evalf[20](plot(['gn_RK7_1'(x)-g(x),'gn_RK7_2'(x)-g(x ),'gn_RK7_3'(x)-g(x),\n'gn_RK7_4'(x)-g(x),'gn_RK7_5'(x)-g(x),'gn_RK7_6 '(x)-g(x)],x=0..10,-9e-17..8.9e-16,font=[HELVETICA,9],\ncolor=[COLOR(R GB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0 ,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`sch eme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error curv es for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 935 587 587 {PLOTDATA 2 "6,-%'CURVESG6%7_q7$$\"\"!F)F(7$$\"5 lmmmmT&)G\\a!#@$!$4%!#>7$$\"5LLLLL3x&)*3\"!#?$!$*pF07$$\"5+++](=#**3E7 F4$!$(pF07$$\"5mmmmTN@Ki8F4$!$/(F07$$\"5+++v=U#Q/V\"F4$!$@(F07$$\"5LLL $e*[Vb)\\\"F4$!$l(F07$$\"5mmm\"HdXqmc\"F4FI7$$\"5++++]ilyM;F4FI7$$\"5m mm;/w(=5x\"F4$!$i(F07$$\"5LLLLe*)4D2>F4$!$D(F07$$\"5mmmTN'4n`(>F4$!$C' F07$$\"5+++]7.K[V?F4$!$]&F07$$\"5LLLe*)4$*f6@F4$!$\\&F07$$\"5mmmmm;arz @F4$!$[&F07$$\"5LLL$e*)4bQl#F4$!#SF07$$\"5++++D\"y%*z7$F4$\"$C(F07$$\" 5LLLeRAY1lLF4$\"$&zF07$$\"5mmm;ajW8-OF4$\"%x;F07$$\"5LL$e9TQp1s$F4$\"% s;F07$$\"5+++vo/V?RQF4$\"%8#F07$$\"5LLLL$e 9ui2%F4$\"%MFF07$$\"5mm;z>h!z&4UF4$\"%AFF07$$\"5+++DcwR)GM%F4$\"%jFF07 $$\"5LL$3F>*))=wWF4$\"%gMF07$$\"5mmm;H2Q\\4YF4$\"%1QF07$$\"5LLeR(\\EYh n%F4$\"%(z$F07$$\"5++]ilA()zUZF4$\"%\"z$F07$$\"5mmT&Q.=^%4[F4$\"%-QF07 $$\"5LLL3-QO5w[F4$\"%#)QF07$$\"5***\\7.d4cF%\\F4$\"%xTF07$$\"5lm;aQ`&3 %4]F4$\"%`[F07$$\"5IL3x1651w]F4$\"%S[F07$$\"5++++voMrU^F4$\"%F[F07$$\" 5qmm\"z%*HB$4aF4$\"%%)\\F07$$\"5NLL$3-8Lfn&F4$\"%XdF07$$\"5+++v$4'HaUf F4$\"%JgF07$$\"5mmmmm\"z_\"4iF4$\"%JlF07$$\"5lmmm;zp!fu'F4$\"%trF07$$ \"5lmmmmm6m#G(F4$\"%swF07$$\"5lmmm;a`T>yF4$\"%V!)F07$$\"5lmmmmT&phN)F4 $\"%-$)F07$$\"5ILL$3-js.*))F4$\"%n%)F07$$\"5++++v=ddC%*F4$\"%d&)F07$$ \"5IL3x\"[5^8\\*F4$\"%n')F07$$\"5lm;a)3\\E\"e&*F4$\"%p')F07$$\"5++DJ&p (=!\\i*F4$\"%R')F07$$\"5ILL3-jsn\"p*F4$\"%6')F07$$\"5lmT&)3\\EXe(*F4$ \"%$e)F07$$\"5++]i:N!G_#)*F4$\"%f&)F07$$\"5ILeRA@M+#*)*F4$\"%_&)F07$$ \"5lmm;H2)y(e**F4$\"%*e)F07$$\"5++]i:N!)eA5F0$\"%j&)F07$$\"5LLLLe*=)H \\5F0$\"%w&)F07$$\"5++++v=JN[6F0$\"%\\$)F07$$\"5mmmm\"z/3uC\"F0$\"%n!) F07$$\"5++++DJ$RDX\"F0$\"%LuF07$$\"5mmmm\"zR'ok;F0$\"%)y'F07$$\"5++++D 1J:w=F0$\"%9iF07$$\"5LLLLL3En$4#F0$\"%+dF07$$\"5mmmm;/RE&G#F0$\"%/`F07 $$\"5+++++D.&4]#F0$\"%8\\F07$$\"5+++++vB_GF07$$\"5LLLLeR\"3Gy%F0$\"%3FF07$$\"5mmmm;/T 1&*\\F0$\"%*f#F07$$\"5mmmm\"zRQb@&F0$\"%#\\#F07$$\"5*******\\(=>Y2aF0$ \"%2CF07$$\"5mmmm;zXu9cF0$\"%?BF07$$\"5**********\\y))GeF0$\"%QAF07$$ \"5********\\i_QQgF0$\"%i@F07$$\"5*******\\7y%3TiF0$\"%&4#F07$$\"5**** ****\\P![hY'F0$\"%B?F07$$\"5KLLLL$Qx$omF0$\"%j>F07$$\"5*********\\P+V) oF0$\"%->F07$$\"5mmmm\"zpe*zqF0$\"%^=F07$$\"5*********\\#\\'QH(F0$\"%) z\"F07$$\"5KLLLe9S8&\\(F0$\"%]F-F(7$$\"5KLL L$3FWYs#F-$\"\"&F07$$\"5)******\\iSmp3%F-$\"$E\"F07$F+$\"$I'F07$$\"5IL LL3x1h6oF-F[\\m7$$\"5++++]7G$R<)F-$\"$X'F07$$\"5lmmm\"z%\\DO&*F-$\"$2* F07$F2$\"%K7F07$FO$\"%'y\"F07$Feo$\"%%G#F07$Fjo$\"%LFF07$F_p$\"%QJF07$ Fip$\"%5NF07$F]r$\"%fQF07$Fas$\"%!=%F07$Fiu$\"%#[%F07$Fcv$\"%iZF07$F]w $\"%9]F07$Fgw$\"%AaF07$Fax$\"%*o&F07$Ffx$\"%tdF07$F[y$\"%IeF07$F`y$\"% 5fF07$Fey$\"%7fF07$Fjy$\"%\"*eF07$F_z$\"%seF07$Fdz$\"%`eF07$Fiz$\"%PeF 07$F^[l$\"%MeF07$Fc[l$\"%keF07$$\"5+]PfL>ab-5F0$\"%8fF07$$\"5LL3F%z&HB 45F0$\"%$*eF07$$\"5m;z%\\l\\5f,\"F0$\"%teF07$Fh[l$\"%aeF07$$\"5L$3-jPd l#H5F0$\"%OeF07$$\"5mm\"zpB6Vf.\"F0$\"%CeF07$$\"5+]il(4l?E/\"F0$\"%HeF 07$F]\\l$\"%!)eF07$Fb\\l$\"%jdF07$Fg\\l$\"%/cF07$F\\]l$\"%>_F07$Fa]l$ \"%#z%F07$Ff]l$\"%+WF07$F[^l$\"%USF07$F`^l$\"%kPF07$Fe^l$\"%)[$F07$Fj^ l$\"%XKF07$F__l$\"%RIF07$Fd_l$\"%sGF07$Fi_l$\"%%p#F07$F^`l$\"%hDF07$Fc `l$\"%>CF07$Fh`l$\"%1BF07$F]al$\"%%>#F07$Fbal$\"%'4#F07$Fgal$\"%-?F07$ F\\bl$\"%B>F07$Fabl$\"%Y=F07$Ffbl$\"%p5F07$Fjhl$\"$(**F07$F_il$\"$v*F07$Fdi l$\"$c*F07$Fiil$\"#%*F]jl-F_jl6&FajlF($\"\"(FdjlF(-Fhjl6#%8scheme~with ~c[10]=15/16G-F$6%7[oF'7$F+$\"$S#F07$F2$\"$w%F07$FO$\"$9(F07$Feo$\"$f* F07$Fjo$\"%D7F07$F_p$\"%::F07$Fip$\"%L=F07$F]r$\"%v@F07$Fas$\"%FDF07$F iu$\"%#)GF07$Fcv$\"%FKF07$F]w$\"%[NF07$Fbw$\"%PQF07$Fgw$\"%)3%F07$F\\x $\"%)H%F07$Fax$\"%oWF07$F[y$\"%-ZF07$F]\\lF[v7$Fb\\l$\"%mZF07$Fg\\l$\" %eYF07$F\\]l$\"%jVF07$Fa]l$\"%:SF07$Ff]l$\"%*o$F07$F[^l$\"%!R$F07$F`^l $\"%eJF07$Fe^l$\"%EHF07$Fj^lFdr7$F__l$\"%]DF07$Fd_l$\"%5CF07$Fi_l$\"%g AF07$F^`l$\"%\\@F07$Fc`l$\"%I?F07$Fh`l$\"%N>F07$F]al$\"%T=F07$Fbal$\"% e8F07$F2$\"%<8F07$$\"5mmmT5:Q(z:\"F4F`[o7$F8F`[o7$$\"5 LLLekGg?%H\"F4$\"%A8F07$F=$\"%_8F07$FB$\"%j9F07$FG$\"%(y\"F07$FL$\"%(z \"F07$FO$\"%&z\"F07$FW$\"%`=F07$Feo$\"%-@F07$$\"5+++D\"yD&y;CF4$\"%O@F 07$Fjo$\"%^AF07$$\"5++]7`>+RsFF4$\"%XAF07$$\"5mmmT5S\\#4*GF4$\"%ZAF07$ $\"5++D1R+C>]HF4$\"%eAF07$$\"5LL$3x1')f%4IF4$\"%zAF07$$\"5mmTN'4KF(oIF 4$\"%wAF07$F_p$\"%rAF07$Fip$\"%4AF07$F]r$\"%6@F07$Fas$\"%8?F07$Fiu$\"% Y>F07$$\"5NL$e9TQ=gF&F4$\"%O>F07$F^v$\"%G>F07$$\"5++]P%[@GEa&F4$\"%L>F 07$FcvFjfm7$$\"5qm;HdX!Q#4eF4$\"%8>F07$Fhv$\"%C>F07$$\"5qm\"z>'=a>4gF4 $\"%f>F07$$\"5NL$3-j(y%e2'F4$\"%`>F07$$\"5++vV)RL+D9'F4$\"%Z>F07$F]w$ \"%T>F07$Fgw$\"%y?F07$FaxF__o7$F[y$\"%uCF07$F]\\l$\"%ZEF07$$\"5++]ilA \"*[b5F0$\"%]EF07$$\"5mmm\"Hd0!oh5F0$\"%UEF07$$\"5LL$3-)))4(y1\"F0$\"% MEF07$$\"5+++](=#>1u5F0$\"%EEF07$$\"5mm;z%\\&GD!3\"F0$\"%>EF07$$\"5LLL 3-)yVk3\"F0$\"%.yH6F0$\"%WEF 07$$\"5mmmTg_7(f8\"F0$\"%SEF07$$\"5LL$3xc=i@9\"F0F`co7$Fb\\l$\"%xEF07$ $\"5mm;H#=0WX:\"F0$\"%(o#F07$$\"5LLLe*[)\\tg6F0$\"%zEF07$$\"5++](oz\"f #p;\"F0$\"%rEF07$$\"5mmm;/^o6t6F0$\"%jEF07$$\"5LL$e9Ty2$z6F0$\"%bEF07$ $\"5+++v=<()\\&=\"F0Feco7$$\"5mm;/E]'*o\">\"F0$\"%`EF07$$\"5LLLLL$e!)y >\"F0$\"%wEF07$$\"5++]iS;:2/7F0Faho7$$\"5mmm\"z%\\CE57F0$\"%!o#F07$$\" 5LL$3_DQ`k@\"F0F[io7$$\"5+++]i:VkA7F0F`io7$$\"5mm;zp[_$)G7F0$\"%cEF07$ $\"5LLL3x\"=E]B\"F0Feco7$$\"5++]P%[6<7C\"F0F\\go7$Fg\\l$\"%mEF07$$\"5L LL3_+%=QD\"F0F\\ho7$$\"5+++]7`(G-E\"F0Fheo7$$\"5mmm\"Hd5RmE\"F0$\"%gEF 07$$\"5LLLLLe%\\IF\"F0F\\go7$$\"5+++v$4\")f%z7F0Fago7$$\"5mmm;aj,(eG\" F0$\"%QEF07$$\"5LLLe9;0G#H\"F0$\"%REF07$$\"5++++vo3p)H\"F0Fe[p7$$\"5LL L$eRd6:J\"F0$\"%[EF07$$\"5mmmm;zALC8F0$\"%KEF07$$\"5LLL3xJEuI8F0$\"%DE F07$$\"5+++]P%)H:P8F0Fido7$$\"5mmm\"zpLjNM\"F0$\"%?EF07$$\"5LLLLe*ot* \\8F0$\"%PEF07$$\"5mmmmT5lD,9F0$\"%.EF07$F\\]l$\"%nDF07$Fa]l$\"%\"R#F0 7$Ff]l$\"%6AF07$F[^l$\"%Q?F07$F`^l$\"%,>F07$Fe^l$\"%jF07$F^q$\"%))>F07$Fcq$\"%k>F07$Fhq$\"%yA)F4$\"$L#F07$Fax$\"$Q#F07$F[y$\"$E%F07$F] \\l$\"$^'F07$Fb\\l$\"$l(F07$Fg\\l$\"$T)F07$F\\^pFdfn7$Ff_p$\"$,*F07$F[ `p$\"$6*F07$F\\]l$\"$;*F07$$\"5mmmm\"z4wb]\"F0Fefq7$$\"5LLLLekGhe:F0$ \"$8*F07$$\"5++++DJ'\\;h\"F0$\"$0*F07$Fa]l$\"$(*)F07$$\"5LLLL3_(>/x\"F 0$\"$u)F07$Ff]l$\"$\\)F07$F[^l$\"$#zF07$F`^l$\"$U(F07$Fe^l$\"$!pF07$Fj ^l$\"$V'F07$F__l$\"$.'F07$Fd_l$\"$q&F07$Fi_l$\"$M&F07$F^`l$\"$4&F07$Fc `l$\"$\"[F07$Fh`l$\"$e%F07$F]al$\"$O%F07$Fbal$\"$;%F07$Fgal$\"$(RF07$F \\bl$\"$#QF07$Fabl$\"$n$F07$Ffbl$\"$^$F07$F[cl$\"$S$F07$F`cl$\"$F$F07$ Fecl$\"$;$F07$Fjcl$\"$0$F07$F_dl$\"$'HF07$Fddl$\"$&GF07$Fidl$\"$x#F07$ F^el$\"$o#F07$Fcel$\"$h#F07$Fhel$\"$a#F07$F]fl$\"$Z#F07$FbflFb\\n7$Fgf l$\"$M#F07$F\\gl$\"$G#F07$Fagl$\"$B#F07$Ffgl$\"$<#F07$F[hl$\"$7#F07$F` hl$\"$2#F07$Fehl$\"$.#F07$Fjhl$\"$)>F07$F_il$\"$$>F07$Fdil$\"$!>F07$Fi il$\"#>F]jl-F_jl6&Fajl$Fd[mFdjlF(Fejl-Fhjl6#%8scheme~with~c[10]=81/82G -F$6%7frF'F^[mFahn7$Fehn$\"#MF07$Ff[m$\"$X\"F07$F]in$\"$(\\F07$F+$\"$D (F07$FeinFe`r7$F^\\mFe`r7$F[jn$\"$E(F07$Fa\\m$\"$T(F07$Fcjn$\"$.)F07$F f\\mF]cp7$F[[o$\"%Z8F07$F2Ffil7$Fc[o$\"%X8F07$F8Fhar7$Fg[o$\"%]8F07$F= $\"%w8F07$FB$\"%w9F07$FG$\"%lF07$Fjo$\"%**=F07$F^^o$\"%e=F07$F_pFfgm7$$\"5LLek`T AE(=$F4$\"%X;F07$F\\]q$\"%U;F07$$\"5++v$4@;(z0LF4$\"%L;F07$Fdp$\"%3;F0 7$$\"5mm\"H#o#3KVU$F4$\"%R:F07$Fd]q$\"%e8F07$$\"5LL3_D.q'Ga$F4$\"%q7F0 7$Fip$\"%o7F07$$\"5++D\"GQ#>ShOF4$\"%m7F07$F^q$\"%j7F07$$\"5mmT5SWo$*z PF4$\"%c7F07$Fcq$\"%S7F07$Fhq$\"%[5F07$F]r$\"$K)F07$$\"5++Dc^.m#H9%F4$ \"$H)F07$Fbr$\"$F)F07$$\"5LL3-))=:BwUF4$\"$@)F07$FgrF`ar7$F^_q$\"$P(F0 7$F\\s$\"$R&F07$Ff_q$\"$/%F07$Fas$\"$.%F07$Ffs$\"$-%F07$F[t$\"$+%F07$F `t$\"$#RF07$Fet$\"$g$F07$Fjt$\"$e#F07$F_u$\"#JF07$FduF[jr7$FiuF[jr7$F^ v$!#HF07$Fcv$!$d#F07$Fhv$!$L$F07$F]w$!$b%F07$Fbw$!$n&F07$Fgw$!$3'F07$F ecq$!$,'F07$F\\x$!$#fF07$F`dq$!$t&F07$FedqFgo7$Fjdq$!$X&F07$Fax$!$P&F0 7$F[y$!$d$F07$F]\\l$!$O\"F07$Fb\\l$!\"'F07$Fg\\l$\"#\"*F07$Ff_p$\"$u\" F07$F\\]l$\"$:#F07$Fhfq$\"$H#F07$F[gqFb\\n7$F`gqF^]r7$Fa]l$\"$_#F07$Fh gq$\"$c#F07$Ff]lF\\^s7$F[^lF^]r7$F`^lFgdq7$Fe^l$\"$?#F07$Fj^l$\"$1#F07 $F__lFj^r7$Fd_l$\"$$=F07$Fi_lFbcq7$F^`l$\"$k\"F07$Fc`l$\"$b\"F07$Fh`l$ \"$Z\"F07$F]alFjhn7$Fbal$\"$M\"F07$Fgal$\"$F\"F07$F\\bl$\"$A\"F07$Fabl $\"$>\"F07$Ffbl$\"$8\"F07$F[cl$\"$5\"F07$F`cl$\"$0\"F07$Fecl$\"$-\"F07 $Fjcl$\"#)*F07$F_dl$\"#'*F07$Fddl$\"##*F07$Fidl$\"#*)F07$F^el$\"#')F07 $Fcel$\"#%)F07$Fhel$\"##)F07$F]fl$\"#!)F07$Fbfl$\"#xF07$Fgfl$\"#vF07$F \\gl$\"#tF07$Fagl$\"#sF07$Ffgl$\"#qF07$F[hl$\"#oF07$F`hl$\"#nF07$Fehl$ \"#mF07$Fjhl$\"#kF07$F_il$\"#iF07$Fdil$\"#hF07$Fiil$FchnF]jl-Fjgp6&Faj lF\\hpF(F(-Fhjl6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGFhg n-%&TITLEG6#%Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-% +AXESLABELSG6$Q\"x6\"Q!Fjes-%%VIEWG6$;F(Fiil;$!\"*!#<$FiasFcfs" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20 " "scheme with c[10]=32/33" "scheme with c[10]=81/82" "scheme with c[1 0]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 3 \+ of 11 stage, order 7 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$* &%\"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 "Th e 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 0 741 "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 st eps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]= 11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme wi th `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/1 70]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 6 d o\n Hn_RK7_||ct := RK7_||ct(H(x,y),x,y,x0,y0,hh,numsteps,false);\n \+ sm := 0: numpts := nops(Hn_RK7_||ct):\n for ii to numpts do\n \+ sm := sm+(Hn_RK7_||ct[ii,2]-h(Hn_RK7_||ct[ii,1]))^2;\n end do:\n e rrs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~po int:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\" Q(pprint96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\" &%\"cG6#\"#5F+#\"#6\"#7$\"+V;bn8!#@7$/*&%-scheme~with~GF+F,F+#\"#:\"#; $\"+]1l+k!#A7$/F8#\"#>\"#?$\"+5]My()F?7$/F8#\"#K\"#L$\"+^1\\?MF?7$/F8# \"#\")\"##)$\"+WcDPDF?7$/F8#\"$p\"\"$q\"$\"+3B`VKF?Q)pprint106\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$ ***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 671 "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 := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[1 0]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n hn_RK7_||ct := RK7_||ct(H(x,y),x,y,x0,y0,h h,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(errs),abs(hn_RK7_||ct(x x)-hxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field: ~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/st ep~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pprint666\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"# 7$\"+t2R+F!#O7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+N)y3\">F57$/F8#\"# >\"#?$\"+H$Ggm\"F57$/F8#\"#K\"#L$\"+(\\o[L\"F57$/F8#\"#\")\"##)$\"+QqK \\t!#P7$/F8#\"$p\"\"$q\"$\"+lQd,$*FTQ)pprint676\"" }}}{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 interval [0, 0.5] of eac h Runge-Kutta method is estimated as follows using the special procedu re " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integrat ion by the 7 point Newton-Cotes method over 50 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 437 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2 ):\nfor ct to 6 do\n sm := NCint((h(x)-'hn_RK7_||ct'(x))^2,x=0..10,a daptive=false,numpoints=7,factor=50);\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$/*&%:Sharp-Verner~sc heme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+`7xu7!#@7$/*&%-scheme~with ~GF+F,F+#\"#:\"#;$\"+f+/cf!#A7$/F8#\"#>\"#?$\"+y>V&=)F?7$/F8#\"#K\"#L$ \"+-[D%=$F?7$/F8#\"#\")\"##)$\"+r$**=P#F?7$/F8#\"$p\"\"$q\"$\"+9poVIF? Q)pprint136\"" }}}{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_RK7_1'(x),h(x)-'hn_RK7_2'(x ),h(x)-'hn_RK7_3'(x),\nh(x)-'hn_RK7_4'(x),h(x)-'hn_RK7_5'(x),h(x)-'hn_ RK7_6'(x)],x=0..5,numpoints=100,font=[HELVETICA,9],\ncolor=[COLOR(RGB, 0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1) ,red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[ 10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error curves \+ for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 835 507 507 {PLOTDATA 2 "6,-%'CURVESG6%7g_l7$$\"\"!F)F(7$$\" 5#HHHHHHz?k#!#@$!#g!#?7$$\"5&eeeeeeeTG&F-$!&E`\"F07$$\"5qoooo=\")eLkF- $!&JR(F07$$\"5]^^^^^w,$e(F-$!'x]FF07$$\"5NMMMM%=ZCt)F-$!']'\\)F07$$\"5 ;<<<<Qm#F0$ !(x$*>&F07$$\"5[[[[[[j)Qy#F0$!(*=5bF07$$\"5HHHHHHH&R!HF0$!(ZJW'F07$$\" 5555555&>S-$F0$!(/Y;)F07$$\"5!**)*)*)*)*[jEF$F0$!(a55)F07$$\"5ppppppuI @NF0$!(bL0)F07$$\"5[\\\\\\\\\\9))\\OF0$!(#R;\")F07$$\"5GHHHHHaXyPF0$!( k8Z)F07$$\"5344444%Hq!RF0$!(rBn*F07$$\"5)))))))))))Q.c.%F0$!)c_r6F07$$ \"5rrrrrrm#>H%F0$!)K6f6F07$$\"5aaaaaa*\\#[XF0$!)'\\)\\6F07$$\"5!3333e& *z+o%F0$!)zig6F07$$\"512222d*4>\"[F0$!)!e9A\"F07$$\"5KLLLLe*RP%\\F0$!) lxC9F07$$\"5ffffff*pb2&F0$!)$3Of\"F07$$\"5NLLLLL=!yI&F0$!)fbu:F07$$\"5 222222P.SbF0$!)a'*e:F07$$\"5+++++]FvqcF0$!)#*=n:F07$$\"5!HHHHHzr9!eF0$ !)(*>J;F07$$\"5&eeeee$3>KfF0$!)\\#[&=F07$$\"5yyyyyy)4H1'F0$!)m#[4#F07$ $\"5baaaaa9UDjF0$!)_7h?F07$$\"5IIIIIII$ze'F0$!)hFO?F07$$\"5+.....BU9nF 0$!)!Rm0#F07$$\"5vvvvvv:\"4%oF0$!)\\?n@F07$$\"5577777i:/pF0$!)v'pH#F07 $$\"5][[[[[3SnpF0$!)Zd9DF07$$\"5&[[[[[[X1.(F0$!),:pEF07$$\"5@@@@@@,*Q4 (F0$!)\\DdEF07$$\"5DEEEEw)=OK(F0$!)lq8EF07$$\"5JJJJJJwM`vF0$!)SKwDF07$ $\"556666'[K**o(F0$!)\\x$e#F07$$\"5!44444M-J'zF0$!)*>i3$F07$$\"5]]]]]]qo*4)F0$!)#>JE$F07$$\"5DEEEEEw4J$)F0$!)Tz ,KF07$$\"5,-----#3Dc)F0$!)BT[JF07$$\"5NLLLLe45(p)F0$!)Q0_JF07$$\"5lkkk k9PpJ))F0$!)v/nKF07$$\"5IIII!G4!**)*))F0$!)4'=U$F07$$\"5&ffff4Z'Gm*)F0 $!)uu&p$F07$$\"5ghhh6\\GeL!*F0$!))e^)QF07$$\"5EFFFFF#z35*F0$!)S_hQF07$ $\"5lnnnn$!)u[3WF07$$\"5ts sssAz-55Fi_l$!)ddzVF07$$\"5&[[[[[y@\\.\"Fi_l$!)'*opUF07$$\"5(pppppk:)f 5Fi_l$!)i%H<%F07$$\"5EEEEw8F!G2\"Fi_l$!)F2mTF07$$\"5cbbbb!y*y&3\"Fi_l$ !)@+!G%F07$$\"5????&RJ$G#4\"Fi_l$!)ukNWF07$$\"5&[[[[t%ox)4\"Fi_l$!)Re1 ZF07$$\"5]\\\\\\u!Qq_5\"Fi_l$!)m:ZZF07$$\"5999999Rw66Fi_l$!)$3Jr%F07$$ \"5-----xihN6Fi_l$!))H')e%F07$$\"5!**)*)*)*)R'o%f6Fi_l$!)CNuWF07$$\"5D DDDvPELs6Fi_l$!)Y*RW%F07$$\"5gggggNm>&=\"Fi_l$!)i4)\\%F07$$\"5GGGG`M'G ;>\"Fi_l$!)]'Gf%F07$$\"5'ffffMjg!)>\"Fi_l$!)jalZF07$$\"5kjjjQKE\\/7Fi_ l$!)=17[F07$$\"5JJJJJJY#4@\"Fi_l$!)l#[x%F07$$\"588888)y[wB\"Fi_l$!)gE@ YF07$$\"5&\\\\\\\\\\%HPk7Fi_l$!)^uxWF07$$\"5------kj(G\"Fi_l$!)\\JDWF0 7$$\"544444f)**3J\"Fi_l$!)'QGV%F07$$\"500000I]-O8Fi_l$!)T#yG%F07$$\"5, ,,,,,-:h8Fi_l$!),KTTF07$$\"566666hp5(Q\"Fi_l$!)3J:RF07$$\"5@@@@@@P189F i_l$!)-ksNF07$$\"5(pppppMd%Q9Fi_l$!)@CXMF07$$\"5tsssss4&QY\"Fi_l$!)5l( H$F07$$\"5OOOOO61U)[\"Fi_l$!)-^bGF07$$\"5+++++]-*H^\"Fi_l$!)9&Q8#F07$$ \"5qppppW1FS:Fi_l$!)(Qd/#F07$$\"5RRRRRR5bn:Fi_l$!)-!)*)=F07$$\"5PPPPP( oj?f\"Fi_l$!(O>>>>>)[Fk\"Fi_l$!'lOYF07$$\"566666h]$el\"Fi_l$!'^3CF07$$\" 5......8#*o;Fi_l$\"'\\#e)F07$$\"5QQQQQjuj#p\"Fi_l$\")RBU9F07$$\"5utttt BON;lEF07$$\"5++++]PwJHR%F07$$\"5NNNNN&=-q\"=Fi_l$\"))*\\ufF07 $$\"5UUUUUUFvH=Fi_l$\")DZPeF07$$\"5\\\\\\\\\\*H.D%=Fi_l$\")n)pq&F07$$ \"5ccccccQDb=Fi_l$\")0q4cF07$$\"5jjjjj8W+o=Fi_l$\")&4dl&F07$$\"5)))))) )))QTOH*=Fi_l$\")cCOzF07$$\"5999999%oy\">Fi_l$\")/ns(*F07$$\"5[[[[)4;> 4$>Fi_l$\")uuI&*F07$$\"5#GGGGy!*pR%>Fi_l$\")\"Q-I*F07$$\"5;<<Fi _l$\")6=B\"*F07$$\"5^^^^^,92q>Fi_l$\")qsq\"*F07$$\"5<<<<Fi_l$\" *pp3D\"F07$$\"5$GGGGG3\\.-#Fi_l$\"*u\"z'Q\"F07$$\"5cbbbbIJ?L?Fi_l$\"*3 D7N\"F07$$\"5GGGGGyr0Y?Fi_l$\"*cDvJ\"F07$$\"5kkkk9-U[_?Fi_l$\"*7XGI\"F 07$$\"5,,,,,E7\"*e?Fi_l$\"*62=H\"F07$$\"5QPPP()\\#Q`1#Fi_l$\"*_,zG\"F0 7$$\"5uttttt_wr?Fi_l$\"*4\">(H\"F07$$\"5'ffffM)G^%3#Fi_l$\"*(4`*R\"F07 $$\"5=====$\\gs4#Fi_l$\"*$H5fQ;@Fi_l$\"*uAP$=F07$$\"5iiiii7dvA@Fi_l $\"*B@\"4=F07$$\"5YYYYYrJ=Y@Fi_l$\"+JNYA>>>>9hk>#Fi_l$\"+2?ba@F -7$$\"5---_*))HR\")>#Fi_l$\"+i<'3C#F-7$$\"57777(eX<)*>#Fi_l$\"+')\\8RB F-7$$\"5AAAs%Gh&\\,AFi_l$\"+Zn(GM#F-7$$\"5KKKK#)pP<.AFi_l$\"+wPBMBF-7$ $\"5____x$3Il?#Fi_l$\"+\"zBqJ#F-7$$\"5sssss(R'))4AFi_l$\"+)z9**H#F-7$$ \"57888jD!*f;AFi_l$\"+U()*fE#F-7$$\"5``````;JBAFi_l$\"+K&)[KAF-7$$\"5@ @@@@rqKZAFi_l$\"+kb$z6#F-7$$\"5*))))))))))[U8F#Fi_l$\"+qIli?F-7$$\"5oo oo$*\\QuxAFi_l$\"+\"G%[#4#F-7$$\"5[[[[)4@XTG#Fi_l$\"+uX@p@F-7$$\"5GGGG .sla!H#Fi_l$\"+l.p@BF-7$$\"533333Lz%pH#Fi_l$\"+\\p_#f#F-7$$\"5...GMt#[ &)H#Fi_l$\"+.?U&o#F-7$$\"5)zzz/Oh[,I#Fi_l$\"+VNLzFF-7$$\"5$HHzmQ&*[R#Fi_l$\"+v!p@t#F-7 $$\"5$GGGGyN\"y)R#Fi_l$\"+F-=\"3$F-7$$\"5999R&p9#[+CFi_l$\"+PX\"G;$F-7 $$\"5YXX&zg$H=-CFi_l$\"+PM#*\\JF-7$$\"5xww^?DP)QS#Fi_l$\"+.e2PJF-7$$\" 53333L9Xe0CFi_l$\"+O:FCJF-7$$\"5rqq?e#4')*3CFi_l$\"+'o#z)4$F-7$$\"5MLL L$3n(Q7CFi_l$\"+0h[tIF-7$$\"5feeeLF3>>CFi_l$\"+kyQBIF-7$$\"5%QQQQQ)R*f U#Fi_l$\"+)z%)R(HF-7$$\"5WVVVV=\"e&\\CFi_l$\"+&pJ9\"GF-7$$\"5.....`A7t CFi_l$\"+!Hf+t#F-7$$\"5UTTT\"*G'3)zCFi_l$\"+-\"p\"pFF-7$$\"5!)zzzz/]\\ '[#Fi_l$\"+=vwvGF-7$$\"5====o!Q\"=$\\#Fi_l$\"+RL8#4$F-7$$\"5ccccccx')* \\#Fi_l$\"+r;5\"[$F-7$$\"5LLLLL30C8DFi_l$\"+1H8xLF-7$$\"555555gKhEDFi_ l$\"+t#\\_E$F-7$$\"533333$))))4b#Fi_l$\"+]3xtIF-7$$\"5111111XOvDFi_l$ \"+%\\A#))HF-7$$\"5++++D\"GI?e#Fi_l$\"+@W$F-7$$\"5#====ogF?g#Fi_l$\"+ q(o.s$F-7$$\"5qppp>d\"f`h#Fi_l$\"+%R1Kf$F-7$$\"5ddddd22pGEFi_l$\"+1t\" )pMF-7$$\"5HHHHHakp^EFi_l$\"+M9JqKF-7$$\"5,,,,,,AquEFi_l$\"+``))oJF-7$ $\"5a```.;CD\"o#Fi_l$\"+qKW5KF-7$$\"511111JE!yo#Fi_l$\"+*G$)eK$F-7$$\" 5eeee3YGN%p#Fi_l$\"++'3$fNF-7$$\"566666hI!4q#Fi_l$\"+bOe%*QF-7$$\"5;;; ;;\"\\.Sr#Fi_l$\"+I'o)ePF-7$$\"5@@@@@@R5FFFi_l$\"+$[(GFOF-7$$\"5eeeeee ET_FFi_l$\"+$yi(*Q$F-7$$\"5'fffffR@xx#Fi_l$\"+sKK(G$F-7$$\"5mlll!pWWSy #Fi_l$\"+3sd[LF-7$$\"5NNNN&y\\n.z#Fi_l$\"+PFa#\\$F-7$$\"5/000!)[0p'z#F i_l$\"+r*zZw$F-7$$\"5uuuuu*f8I!GFi_l$\"+%>ze%RF-7$$\"59999k,(fc\"GFi_l $\"+=Y:3QF-7$$\"5`````.eIGGFi_l$\"+;houOF-7$$\"5'oooo='\\]`GFi_l$\"+bh ,FMF-7$$\"5??????TqyGFi_l$\"+U!GEK$F-7$$\"5%QQQQ8P33*GFi_l$\"+dHH>NF-7 $$\"5ZZZZZAE\"H!HFi_l$\"+Y.oLRF-7$$\"55666hto,:HFi_l$\"+*\\#e(z$F-7$$ \"5uuuuuC67FHFi_l$\"+`3olOF-7$$\"5#====o:\"H`HFi_l$\"+BH()*R$F-7$$\"5* ))))))))))=h%zHFi_l$\"+/1c$G$F-7$$\"5YXXX&z:K>*HFi_l$\"+9O&3\\$F-7$$\" 5-----FJS/IFi_l$\"+]+UAQF-7$$\"5eeee3'4uo,$Fi_l$\"+cjc\"o$F-7$$\"5:::: :l]MHIFi_l$\"+(=%RXNF-7$$\"5cbbbb!G&fbIFi_l$\",+))G1G$!#A7$$\"5'fffff \\X=3$Fi_l$\",d>+\")=$Fbao7$$\"5/////zos$4$Fi_l$\",Oo$H:MFbao7$$\"5777 77i#3c5$Fi_l$\",(QP(Qk$Fbao7$$\"5?????X'*[$Fi_l$\",&=RDWKFbao7$$\"5D DDDD].02KFi_l$\",vUQbS$Fbao7$$\"5oooo=1xi>KFi_l$\",[:[1F$Fbao7$$\"5777 77i]?KKFi_l$\",a7w19$Fbao7$$\"5srrrrY&)HdKFi_l$\",v@MF!HFbao7$$\"5JJJJ JJ?R#G$Fi_l$\",V=J>\"GFbao7$$\"5FFFFF_r\\&H$Fi_l$\",]vXS.$Fbao7$$\"5BB BBBtAg3LFi_l$\",co4R7$Fbao7$$\"5>>>>>%R2>>>pr5.[$Fi_l$\",$*zg6I#Fbao7$$\"5KKKK#[*Gs$[$Fi_l $\",FoH7J#Fbao7$$\"5ssssss]8([$Fi_l$\",UxdPL#Fbao7$$\"5#HHHHz'RJ$\\$Fi _l$\",u0#pXNf$Fi_l$\",`$4EJ@Fbao7$$\"5gffffM:(**f$Fi_l$\",*fmPzAFbao7$$\"5OOO O'Q(yR1OFi_l$\",&=\\GGAFbao7$$\"5888888U#Gh$Fi_l$\",JB\">x@Fbao7$$\"5m mmmm\"*onDOFi_l$\",yvS#y?Fbao7$$\"5?????q&H&QOFi_l$\",sZ?O)>Fbao7$$\"5 XXXXX&zd@m$Fi_l$\",Lv@x#=Fbao7$$\"5qqqqq?gy&o$Fi_l$\",MP4$z#[z$)p$Fi_l$\",4!onI>Fbao7$$\" 5122K)[m`**p$Fi_l$\",c#)4\"o>Fbao7$$\"5WWW%pv%y_,PFi_l$\",w*3>e>Fbao7$ $\"5\"==ob-.-Jq$Fi_l$\",&=:\"o%>Fbao7$$\"5=>>>%H@wYq$Fi_l$\",-Z$\\N>Fb ao7$$\"5$RRR9$yX#yq$Fi_l$\",q;SI\">Fbao7$$\"5oooooVH(4r$Fi_l$\",LgG3*= Fbao7$$\"5nnnn<0kcBPFi_l$\"-)zNbV!=!#B7$$\"5mmmmmm)fht$Fi_l$\"-*=oW;s \"Fidp7$$\"59::::l]:hPFi_l$\"-))zu_t:Fidp7$$\"5jjjjjj-:'y$Fi_l$\"-_Y&y >_\"Fidp7$$\"5\\\\\\\\\\\\WiRQFi_l$\"-BnsoR9Fidp7$$\"5------J\"o)QFi_l $\"-goA!HG\"Fidp7$$\"5666666!f7%RFi_l$\"-7X6o$>\"Fidp7$$\"5YYYYYYAR!*R Fi_l$\"->]onv5Fidp7$$\"5kkkkk9q+RSFi_l$\",%pl%H!**Fidp7$$\"5333333NG\" 4%Fi_l$\",&y&*[8))Fidp7$$\"5qqqqqq9xVTFi_l$\",=$[%Fi_l$\",o\"G7)4(Fidp7$$\"5'oooooyUAC%Fi_l$\",@W6\"GjFidp7$$\" 5666666dt\"H%Fi_l$\",,(eqNcFidp7$$\"588888jlQXVFi_l$\"-A8B))=\\!#C7$$ \"5mmmmmm%4=R%Fi_l$\"-21vh5WF`ip7$$\"544444fy7XWFi_l$\"-0\\\")3EQF`ip7 $$\"5,,,,,^*\\_\\%Fi_l$\"-*3H9?Y$F`ip7$$\"5======z)[a%Fi_l$\"-VY#[h$HF `ip7$$\"5ssssss`c%f%Fi_l$\"-\\5%\\Ui#F`ip7$$\"5********\\PBB2YFi_l$\"- ')z+>VEF`ip7$$\"5EFFFF-$**)>YFi_l$\"-))e*\\J\\#F`ip7$$\"5aaaa/nicKYFi_ l$\"-**=[G^BF`ip7$$\"5\"=====BL_k%Fi_l$\"-[>leF`ip7$$\"5=====VN^&o%F i_l$\"-v_eDH>F`ip7$$\"5jjjjj8.%*)p%Fi_l$\"-/$*3.I?F`ip7$$\"511111\"QiN s%Fi_l$\"-?\"y*3M=F`ip7$$\"5[[[[[[W=[ZFi_l$\"-XGa%Gj\"F`ip7$$\"51222d> qLgZFi_l$\"-)plsWa\"F`ip7$$\"5lllll!f*[sZFi_l$\"-KMo?r9F`ip7$$\"5CCCCu h@k%y%Fi_l$\"-\\+'HMV\"F`ip7$$\"5#GGGGGt%z'z%Fi_l$\"-(>!)*zz9F`ip7$$\" 5zzzzzHlxA[Fi_l$\".,(3pvf8!#D7$$\"5wwwwwE$e([[Fi_l$\".@)=Sq*>\"Fe_q7$$ \"5?????X4sh[Fi_l$\".$o`4TH6Fe_q7$$\"5jjjjjjNou[Fi_l$\".gWBhI2\"Fe_q7$ $\"512222#=Yw)[Fi_l$\".\\zp980\"Fe_q7$$\"5]]]]]+)31!\\Fi_l$\"..\"[)Gc5 \"Fe_q7$$\"59::::S#yQ#\\Fi_l$\"-&eUi?')*Fe_q7$$\"5zzzzzzw9Z\\Fi_l$\"-$ Q%eq%z)Fe_q7$$\"5%[[[[)f2Og\\Fi_l$\"-n9Sxa#)Fe_q7$$\"5!**)*)*)*)RQdt\\ Fi_l$\"-gjkS6yFe_q7$$\"5&\\\\\\\\*>py')\\Fi_l$\"-`'*Rs1wFe_q7$$\"\"&F) $\"-5s*)*G+)Fe_q-%&COLORG6&%$RGBGF($\"\"#!\"\"$\"\"\"F)-%'LEGENDG6#%ES harp-Verner~scheme~with~c[10]=11/12G-F$6%7g`lF'7$F+$!#kF07$F2$!&0k\"F0 7$F7$!&W\"zF07$F<$!'%\\%HF07$$\"5!HHHHzTKx:)F-$!'3!G&F07$FA$!'*p4*F07$ $\"5vvvvv]>;2$*F-$!(_P^\"F07$FF$!(!)HW#F07$FK$!(+)yEF07$FP$!(95o#F07$F U$!(Ptt#F07$FZ$!()4%)HF07$Fin$!(ecz$F07$F^o$!(BUS&F07$Fco$!(oUP&F07$Fh o$!(bLO&F07$F]p$!(,gV&F07$Fbp$!(L*HdF07$Fgp$!(P7h'F07$F\\q$!(O?B)F07$F aq$!([z;)F07$Ffq$!(gr6)F07$F[r$!(@p;)F07$F`r$!(0$o%)F07$Fer$!(EN\\*F07 $Fjr$!):4B6F07$F_s$!)()=66F07$Fds$!)-r,6F07$Fis$!)_746F07$F^t$!)`\"e: \"F07$Fct$!)fG88F07$Fht$!)j,V9F07$F]u$!)uvD9F07$Fbu$!)(y4T\"F07$Fgu$!) Z9:9F07$F\\v$!)O&*f9F07$Fav$!)/\"*>;F07$Ffv$!)r]!z\"F07$F[w$!)'*ohJDF07$Fa[l$!)wh$[#F07$Ff[l$!)2STCF07$F[\\l$!)hFSCF07 $F`\\l$!)k\"Q^#F07$Fj\\l$!)o;'z#F07$Fd]l$!)RS-HF07$Fi]l$!)bWRGF07$F^^l $!)^q$y#F07$Fc^l$!)7A!y#F07$Fh^l$!)gscGF07$F]_l$!)B@iHF07$Fb_l$!);\"y9 $F07$$\"5&\\\\\\u!)p0f)**F0$!)`@6KF07$$\"5CCCuOxWA+5Fi_l$!)9zrKF07$$\" 5**)*)RF\\Qe=+\"Fi_l$!)VWmKF07$Fg_l$!)t4hKF07$$\"5CBBtg2,w15Fi_l$!)OS] KF07$F]`l$!)0rRKF07$Fb`l$!)bUeJF07$Fg`l$!)!*)p3$F07$F\\al$!)O\\#3$F07$ Faal$!)7:pJF07$Ffal$!)O<(G$F07$F[bl$!)nr#\\$F07$F`bl$!)#zR_$F07$Febl$! )Tq)\\$F07$Fjbl$!)&>jS$F07$F_cl$!)x%GK$F07$Fdcl$!)M)fI$F07$Ficl$!)$QpO $F07$F^dl$!)()*3Y$F07$Fcdl$!)4-HOF07$Fhdl$!)W\\zOF07$F]el$!)F-^OF07$Fb el$!)(zO`$F07$Fgel$!)*3,V$F07$$\"5[[[[[tY+w7Fi_l$!)_4:MF07$F\\fl$!)mol MF07$$\"5zyyyGmAX$H\"Fi_l$!)?TRNF07$$\"5cbbbbI\"o#*H\"Fi_l$!)n0mOF07$$ \"5KKKK#[*R308Fi_l$!)CYiOF07$Fafl$!)tqMOF07$Fffl$!)W!f^$F07$F[gl$!)3_. MF07$$\"511111\"eGTP\"Fi_l$!)70hLF07$F`gl$!)wzeLF07$$\"5kjjj8^hf$R\"Fi _l$!)ih*Q$F07$$\"5;;;;;T`3+9Fi_l$!)KXbMF07$$\"5oooo=JXd19Fi_l$!)*GTU$F 07$Fegl$!)f%HR$F07$Fjgl$!)SHsKF07$F_hl$!)$zL:$F07$Fihl$!)Xl.HF07$F^il$ !)&f\\y#F07$Fcil$!)7'Ql#F07$Fhil$!)p**yBF07$F]jl$!)Vs5@F07$Fgjl$!)ur@? F07$Fa[m$!)d3(*=F07$Ff[m$!)))=Q9F07$F[\\m$!)]xG5F07$F`\\m$!)M)f+\"F07$ Fe\\m$!(*pB)*F07$Fj\\m$!(4k]*F07$F_]m$!(v#3))F07$Fd]m$!($)4@#F07$Fi]m$ \"(@va$F07$F^^m$\"(+rY$F07$Fc^m$\"(4]S$F07$Fh^m$\"())HY$F07$F]_m$\"(/w /%F07$Fb_m$\")N#oH\"F07$Fg_m$\")603?F07$F\\`m$\")RWe>F07$Fa`m$\")'eE\" >F07$Ff`m$\")-n()=F07$F[am$\")kR]>F07$F`am$\")?B%H$F07$Feam$\")uDnQF07 $Fjam$\")]BoPF07$F_bm$\")fBwOF07$Fibm$\")a+5'F07$Ffem$\"),`B#Fi_l$\"*/6Js(F-7$Ffjm$\"*1\"3FvF-7$$\"58888jD4 L`AFi_l$\"*#y5UuF-7$$\"500000!yM$fAFi_l$\"*tIqP(F-7$$\"5(ppppWjQ`E#Fi_ l$\"*8f$[tF-7$F[[n$\"*2TKQ(F-7$Fe[n$\"*!y+&*yF-7$F_\\n$\"*,^-u*F-7$Fd \\n$\"+o+m85F-7$Fi\\n$\"+]Kg`5F-7$F^]n$\"+$pI(\\5F-7$Fc]n$\"+-(pe/\"F- 7$Fh]n$\"+(G#=Q5F-7$F]^n$\"+?3aI5F-7$Fb^n$\"+1_R:5F-7$Fg^n$\"+cPV+5F-7 $F\\_n$\"*#y]e%*F-7$Fa_n$\"*Pue@*F-7$Ff_n$\"*t5)*Q*F-7$F[`n$\"*%*ot#)* F-7$F``n$\"+6q_p5F-7$Fe`n$\"+u51C7F-7$Fj`n$\"+vM'*f7F-7$F_an$\"+v!G[D \"F-7$Fdan$\"+!**4(\\7F-7$Fian$\"+#=4YC\"F-7$F^bn$\"+H#fWB\"F-7$Fcbn$ \"+6zPC7F-7$Fhbn$\"+\"e?W?\"F-7$F]cn$\"+>?u%=\"F-7$Fbcn$\"+\"e:.7\"F-7 $Fgcn$\"+-Os$4\"F-7$F\\dn$\"+E\\![6\"F-7$Fadn$\"+[hRm6F-7$Ffdn$\"+it'p E\"F-7$F[en$\"+>![NW\"F-7$F`en$\"+Qcx+9F-7$Feen$\"+L4Pa8F-7$Fjen$\"+0M Nv7F-7$F_fn$\"+g$HpC\"F-7$Fdfn$\"+!=4cF\"F-7$Fifn$\"+iyrT8F-7$F^gn$\"+ Z\"Fbao7$ F]ho$\",!y@369Fbao7$Fbho$\",4;-EN\"Fbao7$Fgho$\",[b%=\"H\"Fbao7$F\\io$ \",(o)=iB\"Fbao7$Faio$\",^?+W@\"Fbao7$Ffio$\",y1b.?\"Fbao7$F[jo$\",F(> (y>\"Fbao7$F`jo$\",J))Q'*>\"Fbao7$Fejo$\",7sgo?\"Fbao7$Fjjo$\",-NC5A\" Fbao7$F_[p$\",g&QZq7Fbao7$Fd[p$\",%\\Y;j8Fbao7$Fi[p$\",tQViM\"Fbao7$F^ \\p$\",yfqtJ\"Fbao7$Fc\\p$\",l>G8E\"Fbao7$Fh\\p$\",?(pc27Fbao7$F]]p$\" ,mlSb:\"Fbao7$Fb]p$\",39S'46Fbao7$Fg]p$\",kv#z\"4\"Fbao7$F\\^p$\",OsD0 3\"Fbao7$Fa^p$\",,**)ey5Fbao7$Ff^p$\",(Gd,!3\"Fbao7$F[_p$\",Jnrc3\"Fba o7$F`_p$\",Tb2m4\"Fbao7$Fe_p$\",%Q=mS6Fbao7$Fj_p$\",Y1chA\"Fbao7$F_`p$ \",__*p)>\"Fbao7$Fd`p$\",j#Q@r6Fbao7$Fi`p$\",Xg%)z6\"Fbao7$F^ap$\",t\" R5n5Fbao7$$\"5#GGGGGoV.l$Fi_l$\",0Z')H-\"Fbao7$Fcap$\"+@9qR)*Fbao7$$\" 5wwwww^[1oOFi_l$\"+PGl%o*Fbao7$$\"5333333>(Rn$Fi_l$\"+N$4)y&*Fbao7$$\" 5RRRRRk*y)zOFi_l$\"+W7w]&*Fbao7$Fhap$\"+I&3Hk*Fbao7$F]bp$\"+\"emE%**Fb ao7$Fbbp$\",3M-a0\"Fbao7$Fgbp$\",JWvr2\"Fbao7$F\\cp$\",)f\\yr5Fbao7$Fa cp$\",gjcb1\"Fbao7$Ffcp$\",.)=Of5Fbao7$F[dp$\",$=C2Z5Fbao7$F`dp$\",&p_ \"\\.\"Fbao7$Fedp$\",u'*ee()*Fidp7$F[ep$\",b+hKU*Fidp7$F`ep$\",fgS\"=' )Fidp7$Feep$\",?#4,#R)Fidp7$$\"5'ooooo`MGz$Fi_l$\",4$\\@r')Fidp7$$\"55 55555)=&*z$Fi_l$\",wQ&ei#*Fidp7$$\"5LLLLL$3.i!QFi_l$\",*ops.\"*Fidp7$$ \"5cccccct)G\"QFi_l$\",I.O[())Fidp7$$\"5-.....fDEQFi_l$\",!*4\"3L%)Fid p7$Fjep$\",u\"f\"G,)Fidp7$F_fp$\",1*H!R>(Fidp7$Fdfp$\",%Hqr^nFidp7$Fif p$\",4Q.49'Fidp7$F^gp$\",hCu\"*o&Fidp7$Fcgp$\",f7Q<6&Fidp7$Fhgp$\",;$4 a!f%Fidp7$F]hp$\",8T&eyTFidp7$Fbhp$\",Qu^Wu$Fidp7$Fghp$\",\"p(ebO$Fidp 7$F\\ip$\"-8skf^HF`ip7$Fbip$\"-*yPi.n#F`ip7$Fgip$\"-6$QCpK#F`ip7$F\\jp $\"-z#4.w7#F`ip7$Fajp$\"-M=)4*3=F`ip7$Ffjp$\"-abZfK;F`ip7$$\"5OOOO6b)) *3g%Fi_l$\"-407X(p\"F`ip7$F[[q$\"-bA'o'[;F`ip7$$\"5ijjj))>ec8YFi_l$\"- C%yB7g\"F`ip7$F`[q$\"-uRC3b:F`ip7$Fe[q$\"-oz9gm9F`ip7$Fj[q$\"-.9(zKQ\" F`ip7$F_\\q$\"-rixK-8F`ip7$Fd\\q$\"-zJ;pO7F`ip7$Fi\\q$\"-v^X+57F`ip7$F ^]q$\"-#[=/3G\"F`ip7$Fc]q$\"-'pf#pd6F`ip7$Fh]q$\"-I&3e2.\"F`ip7$F]^q$ \",`i8gwFe_q7$F\\`q$\"-y4:s9sFe_q7$F a`q$\"-%)=0(f'oFe_q7$Ff`q$\"-dM0]_nFe_q7$F[aq$\"-#zdag8(Fe_q7$F`aq$\"- %[?o_O'Fe_q7$Feaq$\"-cWWywcFe_q7$Fjaq$\"-SafTI`Fe_q7$F_bq$\"-ea&o:0&Fe _q7$Fdbq$\"-\\)R_o$\\Fe_q7$Fibq$\"-1jjD>_Fe_q-F^cq6&F`cqF($\"\"(FccqF( -Fgcq6#%8scheme~with~c[10]=15/16G-F$6%7\\`lF'7$F+$!#dF07$F2$!&,X\"F07$ F7$!&w*pF07$F<$!'l/EF07$Fjdq$!''3n%F07$FA$!'1\\!)F07$Fbeq$!(v'R8F07$FF $!(kD;#F07$FK$!(S9P#F07$FP$!(aMP#F07$FU$!(#oBCF07$FZ$!(pRk#F07$$\"5!)z zzzHZbI=F0$!(I#4HF07$Fin$!()fpLF07$$\"532222dpng>F0$!(wh8%F07$F^o$!(^, \"[F07$Fco$!(\"\\$y%F07$Fho$!()3uZF07$F]p$!(7-%[F07$Fbp$!(Nx5&F07$Fgp$ !(O<\"fF07$F\\q$!(Bl\"F07$F`w$!),GJ;F07$Few$!)-HV;F07$Fjw$!)B.<F07$Fix$!)JBd?F07$F^y$!)]1[?F07$Fcy$!)3\\9?F07$ Fhy$!)f'\\)>F07$F]z$!)_8()>F07$Fbz$!)ZUb?F07$Fgz$!))\\XJ#F07$F\\[l$!)? pGCF07$Fa[l$!)w.$Q#F07$Ff[l$!)A_UBF07$F[\\l$!)C\\TBF07$F`\\l$!)]`7CF07 $Fe\\l$!),&3^#F07$Fj\\l$!)Kz&o#F07$F_]l$!)\"Gg!GF07$Fd]l$!)!e*)y#F07$F i]l$!)jXGFF07$F^^l$!)4euEF07$Fc^l$!)l&)pEF07$Fh^l$!)H=QFF07$F]_l$!)ccL GF07$Fb_l$!)Cp,IF07$Fg_l$!)P!Q5$F07$F]`l$!)%[M3$F07$Fb`l$!);21IF07$Fg` l$!)U,PHF07$F\\al$!)D=GHF07$Faal$!)Zh$*HF07$Ffal$!)+L'3$F07$F[bl$!)(o% [KF07$F`bl$!)@DqKF07$Febl$!)nzYKF07$Fjbl$!)r.hJF07$F_cl$!)@m\"3$F07$Fd cl$!)bwdIF07$Ficl$!)*)H$3$F07$F^dl$!)yoMJF07$Fcdl$!)bZHKF07$Fhdl$!)$)p ^KF07$F]el$!)p`EKF07$Fbel$!)hwAJF07$Fgel$!)$\\\\-$F07$Fafl$!)S%*RHF07$ Fffl$!)8xVGF07$F[gl$!)NQYFF07$F`gl$!)Dz$e#F07$Fegl$!)!4nJ#F07$Fjgl$!)' 3TB#F07$F_hl$!)F8Q@F07$Fdhl$!)^xM=F07$Fihl$!)')o@8F07$F^il$!)46n7F07$F cil$!).Eo6F07$Fhil$!(\"[)\\&F07$F]jl$\"']D$)F07$Fbjl$\"'Df\")F07$Fgjl$ \"'^\\\")F07$F\\[m$\"'1H$*F07$Fa[m$\"(x:i\"F07$Ff[m$\")'>K1\"F07$F[\\m $\")-Y*)=F07$F`\\m$\")*Ry%=F07$Fe\\m$\")e#)3=F07$Fj\\m$\")Kl&y\"F07$F_ ]m$\")\\yN=F07$Fd]m$\").&)4IF07$Fi]m$\")l)o1%F07$F^^m$\")GftRF07$Fc^m$ \");]%)QF07$Fh^m$\")'4l\"QF07$F]_m$\"))H1%QF07$Fb_m$\")\\IH`F07$Fg_m$ \")=8YlF07$F\\`m$\")l0%Q'F07$Fa`m$\")\"*HHiF07$$\"5*******\\7G&\\]>Fi_ l$\")v*3;'F07$Ff`m$\")U[3hF07$$\"5MMMM4Ggaj>Fi_l$\");w)3'F07$F[am$\")e zJhF07$F`am$\")Sc0$)F07$Feam$\")y-,#*F07$Fjam$\")f+l*)F07$F_bm$\")k!4u )F07$Fdbm$\")!)fU')F07$Fibm$\")!\\uc)F07$F^cm$\")y!z`)F07$Fccm$\")IG$f )F07$Fhcm$\")sS[#*F07$F]dm$\"*97&f6F07$Fbdm$\"*7D.@\"F07$Fgdm$\"*s_(\\ 7F07$F\\em$\"*DvbC\"F07$Faem$\"*h39C\"F07$Ffem$\"*o2JB\"F07$F[fm$\"*$) \\[A\"F07$F`fm$\"*0j%37F07$Fefm$\"*))\\A>\"F07$Fjfm$\"+A=3N6F-7$F_gm$ \"+6uv06F-7$Fdgm$\"+\")RGj6F-7$Figm$\"+r\"\\FT\"F-7$F^hm$\"+8F;p9F-7$F chm$\"+?7]L:F-7$Fhhm$\"+A1&f`\"F-7$F]im$\"+NWGI:F-7$Fbim$\"+d=+>:F-7$F gim$\"+*[&y2:F-7$F\\jm$\"+q1b&[\"F-7$Fajm$\"+P;ej9F-7$Ffjm$\"+K0T)Q\"F -7$F[[n$\"+b'>4N\"F-7$F`[n$\"+\"[5&p8F-7$Fe[n$\"+VrW=9F-7$Fj[n$\"+![#p ;:F-7$F_\\n$\"+$3fCp\"F-7$Fd\\n$\"+N\"oHv\"F-7$Fi\\n$\"+5hA9=F-7$F^]n$ \"+hyb2=F-7$Fc]n$\"+?&44!=F-7$Fh]n$\"+lBn(y\"F-7$F]^n$\"+HV^uif\"F-7$F[`n$\"+>4#[l\"F-7$F``n$\"+bWqxF-7$Fhbn$\"+n6=m>F-7$F]cn$\"+#\\_S$>F-7$ Fbcn$\"+i$R#G=F-7$Fgcn$\"+^!zQx\"F-7$F\\dn$\"+Yc>)z\"F-7$Fadn$\"+8s)f' =F-7$Ffdn$\"+o3!\\+#F-7$F[en$\"+oyIcAF-7$F`en$\"+2%>*)=#F-7$Feen$\"+L, S;@F-7$Fjen$\"+%>q@*>F-7$F_fn$\"+l@,N>F-7$Fdfn$\"+E%)zo>F-7$Fifn$\"+HM oa?F-7$F^gn$\"+%y9YA#F-7$Fcgn$\"+Nc-/CF-7$Fhgn$\"+U#F-7$Fejn$ \"+0o;PBF-7$Fjjn$\"+>w)R=#F-7$F_[o$\"+6Q2;@F-7$Fd[o$\"+2FLa@F-7$Fi[o$ \"+/:eXAF-7$F^\\o$\"+X4U>CF-7$Fc\\o$\"+XV]NDF-7$Fh\\o$\"+Fm+ZCF-7$F]]o $\"+uDChBF-7$Fb]o$\"+q6&>?#F-7$Fg]o$\"+H3%H8#F-7$F\\^o$\"+c$fnD#F-7$Fa ^o$\"+!e%*4_#F-7$Ff^o$\"+iHxLCF-7$F[_o$\"+D)R#\\BF-7$F`_o$\"+\"))f(y@F -7$Fe_o$\"+`uH-@F-7$Fj_o$\"+0GeKAF-7$F_`o$\"+ULPVCF-7$Fd`o$\"+2mL`BF-7 $Fi`o$\"+i6HmAF-7$F^ao$\",'op)o4#Fbao7$Fdao$\",pHXc.#Fbao7$Fiao$\",N[O %y@Fbao7$F^bo$\",%3ZKBBFbao7$Fcbo$\",7B,!RAFbao7$Fhbo$\",>Ddu:#Fbao7$F ]co$\",$\\er!*>Fbao7$Fbco$\",h*31C>Fbao7$Fgco$\",dbbS1#Fbao7$F\\do$\", (4N*e;#Fbao7$Fado$\",2S/,3#Fbao7$Ffdo$\",ALTu*>Fbao7$F[eo$\",f(e&f%=Fb ao7$F`eo$\",1s`ky\"Fbao7$Feeo$\",z_h`#>Fbao7$Fjeo$\",)ykz\")>Fbao7$F_f o$\",PbTv*=Fbao7$Fdfo$\",9_Gm\"=Fbao7$Fifo$\",=\"=\"3o\"Fbao7$F^go$\", tH0(H;Fbao7$Fcgo$\",>bb\\u\"Fbao7$Fhgo$\",3Gnez\"Fbao7$F]ho$\",l&[l@Jz9Fbao7$Ffio$\",uf5)f9Fbao7$F[jo$\",edC]X\"Fbao7$F`jo$\",/Z O\\X\"Fbao7$Fejo$\",?DL4Y\"Fbao7$Fjjo$\",KJZZZ\"Fbao7$F_[p$\",j!H\"p_ \"Fbao7$Fd[p$\",0*o>H;Fbao7$Fi[p$\",@$fB3;Fbao7$F^\\p$\",I?WPd\"Fbao7$ Fc\\p$\",2Z%z1:Fbao7$Fh\\p$\",e6\\DW\"Fbao7$F]]p$\",u1/-Q\"Fbao7$Fb]p$ \",@H3WK\"Fbao7$Fg]p$\",Mc()=I\"Fbao7$F\\^p$\",!\\6O'G\"Fbao7$Fa^p$\", 6opDG\"Fbao7$Ff^p$\",`xGCG\"Fbao7$F[_p$\",Bg6$3P\"Fbao7$Fi`p$\",`[G&38Fbao7$F^ap$\",[h[*[7Fbao7$Fcap$\" ,piT1:\"Fbao7$Fhap$\",DyB*=6Fbao7$F]bp$\",A=(\\[6Fbao7$Fbbp$\",ab.F@\" Fbao7$Fgbp$\",vsogB\"Fbao7$F\\cp$\",6;N)H7Fbao7$Facp$\",2Q)oA7Fbao7$Ff cp$\",(=,e:7Fbao7$F[dp$\",wRy9?\"Fbao7$F`dp$\",,]Gv=\"Fbao7$Fedp$\"-L/ #>K8\"Fidp7$F[ep$\"-`M6F\"3\"Fidp7$F`ep$\",w+r8))*Fidp7$Feep$\",mFkna* Fidp7$Fjep$\",k-[!>!*Fidp7$F_fp$\",ePxk-)Fidp7$Fdfp$\",yz4!fuFidp7$Fif p$\",*3`!3r'Fidp7$F^gp$\",^8nC<'Fidp7$Fcgp$\",iMQT[&Fidp7$Fhgp$\",GxmO *[Fidp7$F]hp$\",#pab0WFidp7$Fbhp$\",^^@W#RFidp7$Fghp$\",-J^!*[$Fidp7$F \\ip$\"-qI@$G/$F`ip7$Fbip$\"-qz6yBFF`ip7$Fgip$\"-@^%\\4O#F`ip7$F\\jp$ \"-p6;'>8#F`ip7$Fajp$\"-DMJM2=F`ip7$Ffjp$\"-X#f;Ah\"F`ip7$F[[q$\"-2vX2 B;F`ip7$F`[q$\"-I!3T4`\"F`ip7$Fe[q$\"-*RzEQW\"F`ip7$Fj[q$\"-!)p\"=5$)Fe_q7$Fg_q$\"-%HUA>L(Fe_q7$F\\`q$\"-^Z \"f>>>>WIX<6F0$!(l\\#GF07$FK$!(\"p?GF07$$\"599999*y$)fP\"F0$!(6q\"G F07$FP$!(4F#GF07$FU$!(&H!)GF07$FZ$!(SG8$F07$Fin$!(uI'RF07$F^o$!(0xg&F0 7$Fco$!(9md&F07$Fho$!(wOc&F07$F]p$!(wCj&F07$Fbp$!([R\"fF07$Fgp$!(X)enF 07$F\\q$!(m8J)F07$Faq$!([mC)F07$Ffq$!(qJ>)F07$F[r$!(UCB)F07$F`r$!(cH\\ )F07$Fer$!(^!)Q*F07$Fjr$!)KY!4\"F07$F_s$!)I!*y5F07$Fds$!)>=p5F07$Fis$! )*HR2\"F07$F^t$!)#o(46F07$Fct$!)3&HB\"F07$Fht$!)Y!RL\"F07$F]u$!)R%zJ\" F07$Fbu$!)kl.8F07$Fgu$!)$)\\/8F07$F\\v$!)J?M8F07$Fav$!)OFX9F07$Ffv$!)) *Qj:F07$F[w$!)W@Q:F07$F`w$!)#Ro^\"F07$Few$!)*e4_\"F07$Fjw$!),tk:F07$Fd x$!)*34r\"F07$F^y$!)s6nu\"F07$Fgz$!)`\\')=F07$F\\[l$!)y8Y>F07$Fa[l$!)k`4>F07$Ff[l$!) Vwv=F07$F[\\l$!)2no=F07$F`\\l$!)Z6,>F07$Fe\\l$!)]O^>F07$Fj\\l$!)#4F/#F 07$$\"5SPP()\\l06$)*)F0$!)(=V2#F07$$\"5!)yyy.gY$****)F0$!)D@5@F07$$\"5 ???qda(en,*F0$!)r<2@F07$F_]l$!)+)R5#F07$$\"5XWWW>Q5Bn!*F0$!)De(4#F07$F d]l$!)3=\"4#F07$Fi]l$!)szX?F07$F^^l$!)0-/?F07$Fc^l$!)pa%*>F07$Fh^l$!)s 0C?F07$F]_l$!)08r?F07$Fb_l$!)$ei:#F07$Fg_l$!)H@1AF07$F]`l$!)Vu\">#F07$ Fb`l$!)usO@F07$Fg`l$!)>R'3#F07$F\\al$!)qAv?F07$Faal$!)g&[5#F07$Ffal$!) 2S_@F07$F[bl$!)agPAF07$F`bl$!)b8YAF07$Febl$!)a-IAF07$Fjbl$!)e6r@F07$F_ cl$!)`D;@F07$Fdcl$!)qm)4#F07$Ficl$!)(4F6#F07$F^dl$!)flW@F07$Fcdl$!)2&[ ?#F07$Fhdl$!)SN=AF07$F]el$!)&)=,AF07$Fbel$!)NTI@F07$Fgel$!)NPl?F07$F\\ er$!)&)=[?F07$F\\fl$!)t8`?F07$Fder$!)a)32#F07$Fier$!)%RV5#F07$F^fr$!)$ **f4#F07$Fafl$!)`6!3#F07$Fffl$!)H67?F07$F[gl$!)]cY>F07$Fegl$!)/2Y=F07$ F_hl$!)O[9eO\"F07$Fh il$!)z5*=\"F07$F]jl$!)=625F07$Fgjl$!(=jk*F07$Fa[m$!(_d-*F07$Ff[m$!(pxO 'F07$F[\\m$!()e&*QF07$F`\\m$!(I#4QF07$Fe\\m$!(1'=PF07$Fj\\m$!(#)pe$F07 $F_]m$!(0`D$F07$Fd]m$\"'ttMF07$Fi]m$\"(6hh$F07$F^^m$\"(:N`$F07$Fc^m$\" (2.Y$F07$Fh^m$\"(jmW$F07$F]_m$\"(o>q$F07$Fb_m$\"(T$>%)F07$Fg_m$\")tTJ7 F07$F\\`m$\")d'4?\"F07$Fa`m$\")jWs6F07$Ff`m$\")#yT:\"F07$F[am$\");E!= \"F07$F`am$\")A2$)=F07$Feam$\")Ly*=#F07$Fjam$\")$oO8#F07$F_bm$\").1\"3 #F07$Fibm$\")(zX/#F07$Fccm$\")X:s?F07$Fhcm$\")K*4I#F07$F]dm$\")I2oIF07 $Fbdm$\");1KKF07$Fgdm$\")cofLF07$F\\em$\")`X[LF07$Faem$\")UDPLF07$Ffem $\")!R\\J$F07$F[fm$\")'RFH$F07$F`fm$\")wo[KF07$Fefm$\")X50KF07$Fg`s$\" )&)[EJF07$Fjfm$\"*Lj@0$F-7$F_as$\"*\\G&>IF-7$Fdas$\"*v4R*HF-7$Fias$\"* Q28)HF-7$F_gm$\"*\\\\<*HF-7$Figm$\"*l\\80%F-7$Fajm$\"*_<\\D%F-7$Ffjm$ \"*5js.%F-7$F[[n$\"*z2'\\RF-7$Fe[n$\"*&>g/UF-7$F_\\n$\"*WY,&F-7$Fa_n$\"*X?i([F-7$Ff_n$\"*[03'\\F-7$F[`n$\"*gaD=&F-7$F``n$ \"*pU:j&F-7$Fe`n$\"*W1PW'F-7$Fj`n$\"*$)\\Rj'F-7$F_an$\"*'4\"pg'F-7$Fda n$\"*Cj*zlF-7$Fian$\"*^1Jb'F-7$F^bn$\"*1l'*\\'F-7$Fcbn$\"**\\eYkF-7$Fh bn$\"*D0:M'F-7$F]cn$\"*a))yB'F-7$Fbcn$\"*O9z*eF-7$Fgcn$\"*O)F-7$Fhgn$ \"*(G7w!)F-7$F]hn$\"*M1))z(F-7$Fbhn$\"*+P=N(F-7$Fghn$\"*H2*[rF-7$F\\in $\"*vLtE(F-7$Fain$\"*O#fqvF-7$Ffin$\"*(H+p\")F-7$F[jn$\"*@K?-*F-7$F`jn $\"*USwq)F-7$Fejn$\"*8MGS)F-7$Fjjn$\"*#3CayF-7$F_[o$\"**zo^wF-7$Fd[o$ \"*\"GcCyF-7$Fi[o$\"*'Q95#)F-7$F^\\o$\"*Oua#*)F-7$Fc\\o$\"*%QP0%*F-7$F h\\o$\"*\\%4x!*F-7$F]]o$\"*$H(*e()F-7$Fb]o$\"*sw0<)F-7$Fg]o$\"*8>(fzF- 7$F__t$\"*r5T8)F-7$F\\^o$\"*p25^)F-7$Fg_t$\"*fN]>*F-7$Fa^o$\"*5vdj*F-7 $Ff^o$\"*s&R-$*F-7$F[_o$\"*a/$z*)F-7$F`_o$\"*l2s\")Fbao7$Fjeo$\"+v-?m%)Fbao7$F_fo$\"+PkE1\")Fbao 7$Fdfo$\"+\"Hw1w(Fbao7$Fifo$\"+#Q&[%=(Fbao7$F^go$\"+9O'e,(Fbao7$Fcgo$ \"+55`.wFbao7$Fhgo$\"+oXhvyFbao7$F]ho$\"+VN;]vFbao7$Fbho$\"+)R]sB(Fbao 7$Fgho$\"+E8_3pFbao7$F\\io$\"+jO#Rh'Fbao7$Faio$\"+^2u'\\'Fbao7$Ffio$\" +I>G@kFbao7$F[jo$\"+jK53kFbao7$F`jo$\"+\"Ri!=kFbao7$Fejo$\"+#e\")yX'Fb ao7$Fjjo$\"+D`\"f`'Fbao7$F_[p$\"+'=i#4oFbao7$Fd[p$\"+t*H[K(Fbao7$Fi[p$ \"+i:(fB(Fbao7$F^\\p$\"+x@y!3(Fbao7$Fc\\p$\"+.rbznFbao7$Fh\\p$\"+Zrd! \\'Fbao7$F]]p$\"+<8\"3@'Fbao7$Fb]p$\"+upljfFbao7$Fg]p$\"+v'>u'eFbao7$F \\^p$\"+/^w1eFbao7$Fa^p$\"+)zqlz&Fbao7$Ff^p$\"+-w![!eFbao7$F[_p$\"+1qL OeFbao7$F`_p$\"+W&)4(*eFbao7$Fe_p$\"+#RiB9'Fbao7$Fj_p$\"+O36@mFbao7$F_ `p$\"+KY&HZ'Fbao7$Fd`p$\"+uB`CjFbao7$Fi`p$\"+8Y4PgFbao7$F^ap$\"+&**>Bw &Fbao7$Fc[u$\"+*=rR_&Fbao7$Fcap$\"+q>%HJ&Fbao7$F[\\u$\"+9>+H_Fbao7$F` \\u$\"+vu!=<&Fbao7$Fe\\u$\"+nJDd^Fbao7$Fhap$\"+\\T=4_Fbao7$F]bp$\"+Wx9 x`Fbao7$Fbbp$\"+7d8@dFbao7$Fgbp$\"+Ff4WeFbao7$F\\cp$\"+([0]\"eFbao7$Fa cp$\"+1N@\"y&Fbao7$Ffcp$\"+\\OgZdFbao7$F[dp$\"+en#4o&Fbao7$F`dp$\"+Nx' \\h&Fbao7$Fedp$\",^A!=e`Fidp7$F[ep$\",,k6E6&Fidp7$F`ep$\",-i`an%Fidp7$ Feep$\",dzv_b%Fidp7$Fa_u$\",R=\"*Gr%Fidp7$Ff_u$\",6#>0[]Fidp7$F[`u$\", !*4?G'\\Fidp7$F``u$\",,8U!Q[Fidp7$Fe`u$\",JUAsf%Fidp7$Fjep$\",D(H5oVFi dp7$$\"5iiiii7;U^QFi_l$\",;VE\"yTFidp7$$\"5wvvvvv(=K'QFi_l$\",'fEZ5SFi dp7$$\"5KKKKKdt6pQFi_l$\",2k?Q%RFidp7$$\"5*)))))))))Qf,vQFi_l$\",R#p6) *QFidp7$$\"5YXXXX?X\"4)QFi_l$\",]+![&)QFidp7$F_fp$\",R?&*R#RFidp7$Fdfp $\",T_D#)p$Fidp7$Fifp$\",;Q;xO$Fidp7$F^gp$\",1;+68$Fidp7$Fcgp$\",u&oC< GFidp7$Fhgp$\",C()R%QDFidp7$F]hp$\",/F&*RJ#Fidp7$Fbhp$\",mwI.3#Fidp7$F ghp$\",.D&fs=Fidp7$F\\ip$\"-l9*zuk\"F`ip7$Fbip$\"-*)HRq#\\\"F`ip7$Fgip $\"-S6nz/8F`ip7$F\\jp$\"-J-a$f>\"F`ip7$Fajp$\"-SlM%*=5F`ip7$Ffjp$\",$* =hh@*F`ip7$Fjcu$\",_-3Yg*F`ip7$F[[q$\",B%QeG$*F`ip7$Fbdu$\",!\\*G,1*F` ip7$F`[q$\",0[\\!*z)F`ip7$Fe[q$\",G&yR)H)F`ip7$Fj[q$\",nF(*o#yF`ip7$F_ \\q$\",j\"QiotF`ip7$Fd\\q$\",Tr%)p*pF`ip7$Fi\\q$\",oO6#\\oF`ip7$F^]q$ \",cJ!>vsF`ip7$Fc]q$\",d`W&zlF`ip7$Fh]q$\",U=x!eeF`ip7$F]^q$\",zHbIa&F `ip7$Fb^q$\",6DCqG&F`ip7$Fg^q$\",Ps$yp^F`ip7$F\\_q$\",-QRyP&F`ip7$Fa_q $\"-EK,Rb\\Fe_q7$Fg_q$\"-LDY[sVFe_q7$F\\`q$\"-'R4w!=TFe_q7$Fa`q$\"-^B% H*=RFe_q7$Ff`q$\"-VcfncQFe_q7$F[aq$\"-Af&R44%Fe_q7$F`aq$\"-*G%\\1\\OFe _q7$Feaq$\"->hiMaKFe_q7$Fjaq$\"-yN2nbIFe_q7$F_bq$\"-M=sq&*GFe_q7$Fdbq$ \"-AjaYJGFe_q7$Fibq$\"-+#*)3[+$Fe_q-%'COLOURG6&F`cq$\"*++++\"!\")F(Fj[ ^l-Fgcq6#%8scheme~with~c[10]=32/33G-F$6%7e^lF'F]dq7$F2$!&yj\"F07$F7$!& z!zF07$F<$!'[XHF07$FA$!'^4\"*F07$FF$!((p\\CF07$F_[z$!(V1p#F07$FK$!(rlo #F07$Fg[z$!(_Io#F07$FP$!(V$)o#F07$FU$!($[UFF07$FZ$!(+0)HF07$Fin$!(BXw$ F07$F^o$!(W/K&F07$Fco$!(T4H&F07$Fho$!(:yF&F07$F]p$!(,*R`F07$Fbp$!(Ykf& F07$Fgp$!(&ppjF07$F\\q$!(%\\%z(F07$Faq$!($zLxF07$Ffq$!(uBo(F07$F[r$!(i Kr(F07$F`r$!(L_$zF07$Fer$!(%o2()F07$Fjr$!)AF-5F07$F_s$!(Yk\"**F07$Fds$ !(()Q#)*F07$Fis$!(WI&)*F07$F^t$!)V*G,\"F07$Fct$!)@?56F07$Fht$!)T1!>\"F 07$F]u$!)&>e<\"F07$Fbu$!)Cki6F07$Fgu$!)OSh6F07$F\\v$!)-W!=\"F07$Fav$!) Q*pD\"F07$Ffv$!)6#)Q8F07$F[w$!)7D<8F07$F`w$!)0w(H\"F07$Few$!)%foH\"F07 $Fjw$!)0]>8F07$F_x$!)%*3]8F07$Fdx$!)\"F07$F_hl$!(S2/\"F07$Fdhl$\" '6JkF07$Fihl$\"(()Q#QF07$$\"5&[[[[tWIm_\"Fi_l$\"(Tbu$F07$F^il$\"(^6n$F 07$$\"5aaaa/U3\"Rb\"Fi_l$\"(oui$F07$Fcil$\"(mAv$F07$Fhil$\"(hFe'F07$F] jl$\"(Bw`*F07$Fbjl$\"(-vL*F07$Fgjl$\"())f9*F07$F\\[m$\"(:W+*F07$Fa[m$ \"(v%*4*F07$Ff[m$\")f>f7F07$F[\\m$\")CZ&e\"F07$F`\\m$\")%*[]:F07$Fe\\m $\"))zm^\"F07$Fj\\m$\")@f)[\"F07$F_]m$\")\"pn[\"F07$Fd]m$\")ei')=F07$F i]m$\")OAcAF07$Fc^m$\")5Da@F07$F]_m$\");O(4#F07$Fb_m$\")\\WfDF07$Fg_m$ \")'*pYHF07$Fa`m$\")4?.GF07$F[am$\").&es#F07$F`am$\")jEoLF07$Feam$\")Y ]BOF07$F_bm$\")a9TMF07$Fccm$\")G&yM$F07$Fhcm$\")Vu:NF07$F]dm$\")\\8.UF 07$Fbdm$\")*3mN%F07$Fgdm$\")qrvWF07$F\\em$\")jvgWF07$Faem$\")V$eW%F07$ Ffem$\")k5;WF07$F[fm$\")F`'Q%F07$F`fm$\")o%yK%F07$Fefm$\"))y(pUF07$Fjf m$\"*XTR1%F-7$F_gm$\"*:8S$RF-7$Fdgm$\"*\"yFoSF-7$Figm$\"*[%\\mZF-7$F^h m$\"*&o&F-7$F h]n$\"*p/Mk&F-7$F]^n$\"*zm=g&F-7$Fb^n$\"*#p`>bF-7$Fg^n$\"*\"4?QaF-7$F \\_n$\"*vR*Q^F-7$Fa_n$\"*1Y'\\\\F-7$Ff_n$\"*gy\"))\\F-7$F[`n$\"*-R@8&F -7$F``n$\"*v`bX&F-7$Fe`n$\"*=sN2'F-7$Fj`n$\"*5*Q?iF-7$F_an$\"*$e.&>'F- 7$Fdan$\"*/o(phF-7$Fian$\"*c&eWhF-7$F^bn$\"*kvW4'F-7$Fcbn$\"*a/Z/'F-7$ Fhbn$\"*)QfF-7$F^gn$\"*OugO'F-7$Fcgn$\"*@%eZoF-7$Fh gn$\"*ILNh'F-7$F]hn$\"*00kQ'F-7$Fbhn$\"*#p<#fF-7$Fe_o$\"*.ydp&F-7$Fj _o$\"*e_$GgF-7$F_`o$\"*-+Af'F-7$Fd`o$\"*a\"G\\jF-7$Fi`o$\"*4AW6'F-7$F^ ao$\"+6F$fl&Fbao7$Fdao$\"+)eMVZ&Fbao7$Fiao$\"+zq*p%eFbao7$F^bo$\"+ia,R iFbao7$Fcbo$\"++Xd7gFbao7$Fhbo$\"+3$)e$z&Fbao7$F]co$\"+t'oYM&Fbao7$Fbc o$\"+.*=O:&Fbao7$Fgco$\"+]S!Q_&Fbao7$F\\do$\"+Yr\\.eFbao7$Fado$\"+zait bFbao7$Ffdo$\"+O_6_`Fbao7$F[eo$\"+8&3^%\\Fbao7$F`eo$\"+@m=xZFbao7$Feeo $\"+1kB]^Fbao7$Fjeo$\"+8[u4`Fbao7$F_fo$\"+WN+%3&Fbao7$Fdfo$\"+hb>n[Fba o7$Fifo$\"+(H)H-XFbao7$F^go$\"+P\"[)fVFbao7$Fcgo$\"+P.htYFbao7$Fhgo$\" +_jw>[Fbao7$F]ho$\"+!3%f?YFbao7$Fbho$\"+ON1HWFbao7$Fgho$\"+$*HVFUFbao7 $F\\io$\"+r5EWSFbao7$Faio$\"+_MaoRFbao7$Ffio$\"+@_2:RFbao7$F[jo$\"+,ck ,RFbao7$F`jo$\"+v!\\4!RFbao7$Fejo$\"+#Q+p\"RFbao7$Fjjo$\"+\"yZW&RFbao7 $F_[p$\"+s#R%)4%Fbao7$Fd[p$\"+2P.&Q%Fbao7$Fi[p$\"+Eg1IVFbao7$F^\\p$\"+ #R*>PUFbao7$Fc\\p$\"+$4Tp0%Fbao7$Fh\\p$\"+0&[R)QFbao7$F]]p$\"+\\0&fr$F bao7$Fb]p$\"+!zY^c$Fbao7$Fg]p$\"+:U&R]$Fbao7$F\\^p$\"+![7:Y$Fbao7$Fa^p $\"+`i1^MFbao7$Ff^p$\"+$)pi]MFbao7$F[_p$\"+2z)HY$Fbao7$F`_p$\"+bfh\"\\ $Fbao7$Fe_p$\"+UOz=OFbao7$Fj_p$\"+)*QQ\")QFbao7$F_`p$\"+6qX%z$Fbao7$Fd `p$\"+9;X2PFbao7$Fi`p$\"+C;&*QNFbao7$F^ap$\"+#H*zxLFbao7$Fcap$\"+S:V6J Fbao7$Fhap$\"+Ss8EIFbao7$F]bp$\"+G.95JFbao7$Fbbp$\"+4')4%H$Fbao7$Fgbp$ \"+BL[hLFbao7$F\\cp$\"+'[bYM$Fbao7$Facp$\"+G\">_K$Fbao7$Ffcp$\"+Av)eI$ Fbao7$F[dp$\"+Hl`nKFbao7$F`dp$\"+N%)fHKFbao7$Fedp$\",[\"))*=3$Fidp7$F[ ep$\",#)391%HFidp7$F`ep$\",L&e.(o#Fidp7$Feep$\",QU@uf#Fidp7$Fa_u$\",%R ArvEFidp7$Ff_u$\",ja,Q&GFidp7$F[`u$\",@Y#)[!GFidp7$F``u$\",fagVt#Fidp7 $Fe`u$\",%*f^#)f#Fidp7$Fjep$\",jf%poCFidp7$F_fp$\",N8:*)>#Fidp7$Fdfp$ \",'QuWc?Fidp7$Fifp$\",%Qg3a=Fidp7$F^gp$\",Y\\C^r\"Fidp7$Fcgp$\",!\\<0 G:Fidp7$Fhgp$\",U8t6P\"Fidp7$F]hp$\",^B^#Q7Fidp7$Fbhp$\",I!RE46Fidp7$F ghp$\"+Rsv&*)*Fidp7$F\\ip$\",,\"[Dz')F`ip7$Fbip$\",iOqtz(F`ip7$Fgip$\" ,ut'[(z'F`ip7$F\\jp$\",x4YH<'F`ip7$Fajp$\",M)GX`_F`ip7$Ffjp$\",#RiI6ZF `ip7$F[[q$\",e;)pjZF`ip7$F`[q$\",Z\"zG$\\%F`ip7$Fe[q$\",Q97wB%F`ip7$Fj [q$\",$4Jl'*RF`ip7$F_\\q$\",Nd,9w$F`ip7$Fd\\q$\",![z$oc$F`ip7$Fi\\q$\" ,(H%>&zMF`ip7$F^]q$\",*)\\ZAo$F`ip7$Fc]q$\",)=O**HLF`ip7$Fh]q$\",n$fhk HF`ip7$F]^q$\",+mRT!GF`ip7$Fb^q$\",smo6n#F`ip7$Fg^q$\",8BWUg#F`ip7$F\\ _q$\",'>4\"**p#F`ip7$Fa_q$\"-b&pSt[#Fe_q7$Fg_q$\"-9;/d%>#Fe_q7$F\\`q$ \"-\\\"*z(f1#Fe_q7$Fa`q$\"-gU*GI'>Fe_q7$Ff`q$\"-bC[TD>Fe_q7$F[aq$\"-yv _#y.#Fe_q7$F`aq$\"-H*4i\"Fe_q7$Fjaq$\"-t!zx9_\"F e_q7$F_bq$\"-[I(Q)R9Fe_q7$Fdbq$\"-F>=\\.9Fe_q7$Fibq$\"-p0jJ'[\"Fe_q-F^ cq6&F`cq$FjbqFccqF(Fdcq-Fgcq6#%8scheme~with~c[10]=81/82G-F$6%7^\\lF'7$ F+$!#[F07$F2$!&`B\"F07$F7$!&l(fF07$F<$!'`JAF07$FA$!'7@pF07$FF$!((Gn=F0 7$F_[z$!('o^?F07$FK$!(\"e[?F07$Fg[z$!(!*e/#F07$FP$!(3*\\?F07$FU$!(094# F07$FZ$!(8_F#F07$Fin$!(`d)GF07$F^o$!(#>2TF07$Fco$!(9W3%F07$Fho$!(.T2%F 07$F]p$!(C@7%F07$Fbp$!(\">BVF07$Fgp$!(=t$\\F07$F\\q$!(>J3'F07$Faq$!(Ud .'F07$Ffq$!()H&*fF07$F[r$!(g*=gF07$F`r$!(zY>'F07$Fer$!(I'>oF07$Fjr$!(i R!zF07$F_s$!(W,#yF07$Fds$!(zku(F07$Fis$!(#eoxF07$F^t$!(&3\"*zF07$Fct$! (i5!))F07$Fht$!(Q@[*F07$F]u$!(B'o$*F07$Fbu$!(pFE*F07$Fgu$!(y7D*F07$F\\ v$!(`fS*F07$Fav$!)Ya05F07$Ffv$!)]Mx5F07$F[w$!)W)*f5F07$F`w$!)#3T/\"F07 $Few$!)!4J/\"F07$Fjw$!)Q$=1\"F07$F_x$!)U5)3\"F07$Fdx$!)T$\\8\"F07$Fix$ !)l0o6F07$F^y$!)5&G;\"F07$Fcy$!)MwV6F07$Fhy$!)CFD6F07$F]z$!):E=6F07$Fb z$!)FsD6F07$Fge^l$!)M*H9\"F07$Fgz$!)Nrx6F07$F_f^l$!)wA/7F07$F\\[l$!)Hh (>\"F07$Fa[l$!)v1v6F07$Ff[l$!)9!H:\"F07$F[\\l$!)bVU6F07$F`\\l$!)nSS6F0 7$Fj\\l$!)lcj6F07$Fd]l$!)J[m6F07$Fi]l$!)p8T6F07$F^^l$!)Vr:6F07$F]`l$!) fL`5F07$Fb`l$!)-&o-\"F07$Fg`l$!(j>***F07$Faal$!(Q#\\%*F07$Febl$!(9]R)F 07$Fjbl$!(TE<)F07$F_cl$!(*oDzF07$$\"5eddd#)Q1!f;\"Fi_l$!(j*GyF07$Fdcl$ !(77p(F07$$\"5#HHHzmjk(y6Fi_l$!(srZ(F07$Ficl$!(l!GrF07$F^dl$!(g&\\lF07 $Fcdl$!(a]f&F07$Fhdl$!(bX<&F07$F]el$!(9X8&F07$Fbel$!(5$o\\F07$Fgel$!(i zr%F07$F\\fl$!((4/MF07$Fafl$!'s>fF07$Fffl$!'q;dF07$F[gl$!'X,ZF07$F`gl$ \"(*)QB\"F07$Fegl$\"(*>z_F07$Fjgl$\"(\"p$4&F07$F_hl$\"($*>2&F07$Fdhl$ \"(eh[(F07$Fihl$\")l!yC\"F07$Fh[_l$\")\"=AA\"F07$F^il$\")bK(>\"F07$F` \\_l$\");!p<\"F07$Fcil$\")&[<=\"F07$Fhil$\")'f*4;F07$F]jl$\")#[v2#F07$ Fbjl$\")X#R.#F07$Fgjl$\")bl\"*>F07$F\\[m$\")cgc>F07$Fa[m$\")r4d>F07$Ff [m$\")$4G\\#F07$F[\\m$\")rlF-7$F_gm$\"*v%zUjF-7$$\"5]]]]DWp'H<#Fi_l$\"*! QgYjF-7$$\"5qqqq?eKKw@Fi_l$\"*&=RrjF-7$$\"5!444f@dz'z@Fi_l$\"*XRWU'F-7 $Fdgm$\"*4E^^'F-7$$\"5_^^^,9&[(*=#Fi_l$\"*oI/'oF-7$Figm$\"*\"=0XvF-7$F ^hm$\"*-y?z(F-7$Fchm$\"*43t2)F-7$Fhhm$\"*BzT3)F-7$F]im$\"*NcV0)F-7$Fbi m$\"*ns\\*zF-7$Fgim$\"*\\Pf$zF-7$F\\jm$\"*#z!*=yF-7$Fajm$\"*0mKq(F-7$F fjm$\"*'*QWI(F-7$F[[n$\"*M<4/(F-7$F`[n$\"*;'yyqF-7$Fe[n$\"*gX&QsF-7$Fj [n$\"*B+Wg(F-7$F_\\n$\"*R*)oI)F-7$Fd\\n$\"*'*fjb)F-7$Fi\\n$\"*\"yK6))F -7$F^]n$\"*WT*y()F-7$Fc]n$\"*\"=lY()F-7$Fh]n$\"*lhBo)F-7$F]^n$\"*rb%=' )F-7$Fb^n$\"*7\"z\"\\)F-7$Fg^n$\"*6_mO)F-7$F\\_n$\"*iM]!zF-7$Fa_n$\"*x F#)e(F-7$Ff_n$\"*uILi(F-7$F[`n$\"*?P^!yF-7$F``n$\"*M^9C)F-7$Fe`n$\"*#* \\H5*F-7$Fj`n$\"*2*p4$*F-7$F_an$\"*Ia*F-7$F^bn$\"*l^77*F-7$Fcbn$\"*/in/*F-7$Fhbn$\"*v&H**))F-7$F]cn$\"*2 ]Qv)F-7$Fbcn$\"*@L3F)F-7$Fgcn$\"*s%yazF-7$F\\dn$\"*!Rc+!)F-7$Fadn$\"*_ iL?)F-7$Ffdn$\"*r#Rt')F-7$F[en$\"*#Q`#e*F-7$F`en$\"*KLHH*F-7$Feen$\"*` D])*)F-7$Fjen$\"*ekGX)F-7$F_fn$\"*\")='G\")F-7$Fdfn$\"*-I4?)F-7$Fifn$ \"*qSGX)F-7$F^gn$\"**Rp0!*F-7$Fcgn$\"*$yy7'*F-7$Fhgn$\"*)=A%G*F-7$F]hn $\"*H[`'*)F-7$Fbhn$\"*a(\\W%)F-7$Fghn$\"*@[d5)F-7$F\\in$\"*EDm9)F-7$Fa in$\"*W0/M)F-7$Ffin$\"*S)p)y)F-7$F[jn$\"*$p#QZ*F-7$F`jn$\"*<\"pV\"*F-7 $Fejn$\"*HyN#))F-7$Fjjn$\"*%eNS#)F-7$F_[o$\"*Xoj*yF-7$Fi[o$\"*N;+@)F-7 $Fc\\o$\"*Js![!*F-7$Fh\\o$\"*[jAt)F-7$F]]o$\"*\\thU)F-7$Fb]o$\"*)Qe_yF -7$Fg]o$\"*j*z=vF-7$F\\^o$\"*3;lz(F-7$Fa^o$\"*fdR])F-7$Ff^o$\"*hO(4#)F -7$F[_o$\"*:iX#zF-7$F`_o$\"*Bv]M(F-7$Fe_o$\"*\">Z,qF-7$Fj_o$\"*[/IF(F- 7$F_`o$\"*+6Uy(F-7$Fd`o$\"*Rnt\\(F-7$Fi`o$\"*`=+A(F-7$F^ao$\"+[o,vmFba o7$Fdao$\"+[o]%Q'Fbao7$Fiao$\"+j-pomFbao7$F^bo$\"+E&Q%ypFbao7$Fcbo$\"+ &Rg^s'Fbao7$Fhbo$\"+')>?!['Fbao7$F]co$\"+43![(fFbao7$Fbco$\"+Y^**)o&Fb ao7$Fgco$\"+(*3)y$fFbao7$F\\do$\"+y0&y6'Fbao7$Fado$\"+=q_veFbao7$Ffdo$ \"+t])>k&Fbao7$F[eo$\"+MzOY_Fbao7$F_fo$\"+!)*>L-&Fbao7$Fdfo$\"+;![!4[Fbao7$Fifo$ \"+_UYWWFbao7$F^go$\"+5cfOUFbao7$$\"5<<<<#4^u#*Q$Fi_l$\"+1HasUFbao7$Fc go$\"+o&*3\"R%Fbao7$$\"5EEEwjNda)R$Fi_l$\"+P@c%\\%Fbao7$$\"5iiii(Q9O;S $Fi_l$\"+A]*3`%Fbao7$$\"5)*)*)*[6_ls/MFi_l$\"+7B\\$[%Fbao7$Fhgo$\"+tJa OWFbao7$F]ho$\"+pn?`UFbao7$Fbho$\"+<6)o2%Fbao7$F\\io$\"+DGs=PFbao7$Fjj o$\"+T5AXNFbao7$Fh\\p$\"+dB$zI$Fbao7$F`_p$\"+8lb!*GFbao7$F^ap$\"+;2DTE Fbao7$Fhap$\"+z!=LI#Fbao7$F[ep$\",DU&=#4#Fidp7$Feep$\",!f:_#z\"Fidp7$F jep$\",NK\\6e\"Fidp7$F_fp$\",rdg(e8Fidp7$Fdfp$\",s`i/<\"Fidp7$Fifp$\"+ n/kn**Fidp7$F^gp$\"+#H,o`)Fidp7$Fcgp$\"+d;+'3(Fidp7$Fhgp$\"+B*Hb%eFidp 7$F]hp$\"+Xvh_[Fidp7$Fbhp$\"+hFCRRFidp7$Fghp$\"+H6oqJFidp7$F\\ip$\",E< &4nCF`ip7$Fbip$\",K\"zQZ>F`ip7$Fgip$\",K0,RX\"F`ip7$F\\jp$\",I[o:.\"F` ip7$Fajp$\"+^'GCd(F`ip7$Ffjp$\"+u.xXZF`ip7$Fj[q$\"+W+\"H+$F`ip7$F^]q$ \"*\\i'QYF`ip7$Fh]q$\"*J;z1#F`ip7$F\\_q$!+uuu)>\"F`ip7$Fg_q$!, " 0 "" {MPLTEXT 1 0 520 "evalf[20](plot([1-'hn_RK7_1'(x)/h(x),1-'hn_RK7_2'(x) /h(x),1-'hn_RK7_3'(x)/h(x),\n1-'hn_RK7_4'(x)/h(x),1-'hn_RK7_5'(x)/h(x) ,1-'hn_RK7_6'(x)/h(x)],x=0..10,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0 ,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1), red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[1 0]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme \+ with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`relative error curves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 845 666 666 {PLOTDATA 2 "6,-%'CURVESG6%7Y7$$\"\"!F)F(7$$ \"5mmmmm;arz@!#?$!'vl_!#>7$$\"5LLLL$e9ui2%F-$!(:4F\"F07$$\"5mmmmm\"z_ \"4iF-$!(+v^#F07$$\"5lmmmmT&phN)F-$!(c-`%F07$$\"5LLLLe*=)H\\5F0$!(%y)H (F07$$\"5mmmm\"z/3uC\"F0$!(m8%**F07$$\"5++++DJ$RDX\"F0$!(SAo*F07$$\"5m mmm\"zR'ok;F0$\"(XgF\"F-7$$\"5++++D1J:w=F0$\"*]L#HMF-7$$\"5LLLLL3En$4# F0$\"+]a$pW\"F-7$$\"5mmmm;/RE&G#F0$\"+MX*4)HF-7$$\"5+++++D.&4]#F0$\"+? +qXzF-7$$\"5+++++vB_*z_%F-7$$\"5LLLLL347TLF0$\",x)ptWuF-7$$\"5 LLLLLLY.KNF0$\"-Zw&>\")=\"F-7$$\"5*******\\7o7Tv$F0$\"-p@Y0_=F-7$$\"5L LLLL$Q*o]RF0$\"-O>;y>GF-7$$\"5*******\\7=lj;%F0$\"-sWG^SUF-7$$\"5***** **\\PaRY2aF0$\".G99-o6%F-7$$\"5mmmm;zXu 9cF0$\".v%3jzCcF-7$$\"5**********\\y))GeF0$\".FgP\\Wi(F-7$$\"5******** \\i_QQgF0$\"/Tf&Qbf-\"F-7$$\"5*******\\7y%3TiF0$\"/Yt**))4r8F-7$$\"5** ******\\P![hY'F0$\"/8vp$3t#=F-7$$\"5KLLLL$Qx$omF0$\"/G$G*zR9CF-7$$\"5* ********\\P+V)oF0$\"/'H\"yOT[KF-7$$\"5mmmm\"zpe*zqF0$\"/1-6G>\">%F-7$$ \"5*********\\#\\'QH(F0$\"/'z8[FZv&F-7$$\"5KLLLe9S8&\\(F0$\"/$=INF]\\( F-7$$\"5*******\\i?=bq(F0$\"0uT[*)\\n+\"F-7$$\"5KLLLL3s?6zF0$\"0Jx#o;y 'G\"F-7$$\"5*******\\7`Wl7)F0$\"0Bkb#=!zj\"F-7$$\"5lmmmmm*RRL)F0$\"0w \"4A*[l2#F-7$$\"5lmmm;a<.Y&)F0$\"0V#=0@\"Gi#F-7$$\"5KLLLe9tOc()F0$\"0# 3[z)==I$F-7$$\"5**********\\Qk\\*)F0$\"0DaCK0\"QTF-7$$\"5KLLL$3dg6<*F0 $\"0KA)3O1$>&F-7$$\"5++++voTAq#*F0$\"0S,v-bSz&F-7$$\"5lmmmmmxGp$*F0$\" 0U#RO9sfkF-7$$\"5ILL$eRA5\\Z*F0$\"0ov`0gcA(F-7$$\"5)******\\7oK0e*F0$ \"0\"RZ@O:,\")F-7$$\"5+++++]oi\"o*F0$\"0(ykg%)>I!*F-7$$\"5)******\\(=5 s#y*F0$\"1L')e8B*f+\"F-7$$\"5+++D1k2/P)*F0$\"1-m*GgW&)4\"F-7$$\"5+++]P 40O\"*)*F0$\"1(pEEa\"G[6F-7$$\"5+++voa-oX**F0$\"1/'p2HJ(>7F-7$$\"#5F)$ \"1h')[Qw5`8F--%&COLORG6&%$RGBGF($\"\"#!\"\"$\"\"\"F)-%'LEGENDG6#%ESha rp-Verner~scheme~with~c[10]=11/12G-F$6%7YF'7$F+$!'E;bF07$F2$!(s$=7F07$ F7$!(y<:#F07$F<$!((39NF07$FA$!(%=*R&F07$FF$!(_Eg(F07$FK$!(Ax?*F07$FP$! (')4q(F07$FU$\")du@IF-7$FZ$\"*(oC,XF-7$Fin$\"+*ROt3\"F-7$F^o$\"+:tt&H$ F-7$Fco$\"+My_8mF-7$Fho$\",T4x*R7F-7$F]p$\",E`l/@#F-7$Fbp$\",%[9dZ#F-7$F[r$\"-!y(floPF-7$F`r$\"-m(Qp='eF-7$Fer$\"-;+.MS%)F-7$Fjr$\". sGD'Q?8F-7$F_s$\".'R,M?')>F-7$Fds$\"._blx^y#F-7$Fis$\".!y4L&['QF-7$F^t $\".+Dal5J&F-7$Fct$\".x@jIEB(F-7$Fht$\".3ti_dw*F-7$F]u$\"/$[C5KQJ\"F-7 $Fbu$\"/6E()Rw[l-A JB%F-7$Ffv$\"/cF$fA<_&F-7$F[w$\"/@$zGrtS(F-7$F`w$\"/$zz())ej%*F-7$Few$ \"03c8qGI?\"F-7$Fjw$\"0M'[rt0A:F-7$F_x$\"0XprM)><>F-7$Fdx$\"0kDpGscS#F -7$Fix$\"08%HI/o-IF-7$F^y$\"0jP'y\"oUv$F-7$Fcy$\"0R\"**oCQyTF-7$Fhy$\" 0!)G2&RWYYF-7$F]z$\"0iFAK*y&=&F-7$Fbz$\"07o>R/1!eF-7$Fgz$\"0w?CC?\"[kF -7$F\\[l$\"0r&y^q4jrF-7$Fa[l$\"0Q\"psIt(y(F-7$Ff[l$\"0.jL#4X[\")F-7$F[ \\l$\"0ALtx88i)F-7$F`\\l$\"0a3Q^]Y`*F--Fe\\l6&Fg\\lF($\"\"(Fj\\lF(-F^] l6#%8scheme~with~c[10]=15/16G-F$6%7[oF'7$F+$!'()4\\F07$F2$!(+$36F07$F7 $!((==?F07$F<$!(vCa*F-7$Fin$\"+&4O*[>F-7$F^o$\"+Sx2]^F-7 $Fco$\"+;g6I'*F-7$Fho$\",1+LRq\"F-7$F]p$\",xAOq)GF-7$Fbp$\",<)G)Gs%F-7 $Fgp$\",:_M'*\\(F-7$F\\q$\"-*RQDJ;\"F-7$Faq$\"-)R0v>w\"F-7$Ffq$\"-f^:$ fj#F-7$F[r$\"-E&Glp(QF-7$F`r$\"-x!==i\"eF-7$Fer$\"-cIW4[\")F-7$Fjr$\". Q*fuCM7F-7$F_s$\".bP#fj4=F-7$Fds$\".Y(4@$R[#F-7$Fis$\".]@tY9Q$F-7$F^t$ \".=GjR'oXF-7$Fct$\".9FrR,8'F-7$Fht$\".v,fSF<)F-7$F]u$\"/\"y=\\Cp3\"F- 7$Fbu$\"/R*3S$)RV\"F-7$Fgu$\"/y?.NIE>F-7$F\\v$\"/I;!\\/Z[#F-7$Fav$\"/, ]Y[>6MF-7$Ffv$\"/MT)z>gW%F-7$F[w$\"/)euBh])fF-7$F`w$\"/hCaLXiwF-7$Few$ \"/$*z7l_t(*F-7$Fjw$\"0lGNso@C\"F-7$F_x$\"0o<\\=2Md\"F-7$Fdx$\"0.YhG9q )>F-7$Fix$\"0$y(H_&>*\\#F-7$F^y$\"0es'o(zu9$F-7$Fcy$\"0[@E^V*=NF-7$Fhy $\"0+%)G!G`JRF-7$F]z$\"0F/+fjkS%F-7$Fbz$\"0umrF([]\\F-7$Fgz$\"0])e'zH1 `&F-7$F\\[l$\"0-obTLb<'F-7$Fa[l$\"0K1zGQRw'F-7$Ff[l$\"0/ex+C#oqF-7$$\" 5+]i!RvZ]\")*)*F0$\"01i#Q%)o%R(F-7$$\"5++DJqX/%\\!**F0$\"0#G4C#)zFvF-7 $$\"5+](=nQTI<\"**F0$\"05x'G#)zFvF-7$$\"5++]7.#Q?&=**F0$\"01W3U)zFvF-7 $$\"5++v$f$=.5K**F0$\"0*GcM(=y_(F-7$F[\\l$\"0(f.#4c#GvF-7$$\"5++Dc,\"> g#f**F0$\"0!*)[%*4yKvF-7$$\"5++]PMF,%G(**F0$\"0*[()=\\PjvF-7$$\"5++v=n j+U')**F0$\"0&p&H&4h>xF-7$F`\\l$\"0CF3Kv@P)F--Fe\\l6&Fg\\l$\"\"*Fj\\l$ \"\"%Fj\\lF(-F^]l6#%8scheme~with~c[10]=19/20G-F$6%7YF'7$F+$!''Rs&F07$F 2$!(wH=\"F07$F7$!(M)y=F07$F<$!(&z,FF07$FA$!(a>l$F07$FF$!(wJe%F07$FK$!( &G4]F07$FP$!(K!pOF07$FU$\")v`%[#F-7$FZ$\"*],_Z#F-7$Fin$\"*0z!)y&F-7$F^ o$\"+Q@ZGS=e +G#F-7$F^t$\".QeE]y:$F-7$Fct$\".#*)4ilLVF-7$Fht$\".)H)4ci*eF-7$F]u$\". G`=_=*zF-7$Fbu$\"/-%H4g<2\"F-7$Fgu$\"/O$fTkTY\"F-7$F\\v$\"/%yx`o-\">F- 7$Fav$\"/Qy'4Opl#F-7$Ffv$\"/fn!ekM\\$F-7$F[w$\"/g*yveQt%F-7$F`w$\"/`?F-7$F\\q$\",h*H2jJF-7$Faq$\" ,on,x&[F-7$Ffq$\",ASjeQ(F-7$F[r$\"-Cpq116F-7$F`r$\"-%zG\"z$p\"F-7$Fer$ \"-Mo7;=CF-7$Fjr$\"-m`!yDv$F-7$F_s$\"-W>,pLcF-7$Fds$\"-/:/#R)yF-7$Fis$ \".oWs@Q4\"F-7$F^t$\".E8k5b]\"F-7$Fct$\".pTn5p0#F-7$Fht$\"..^t24z#F-7$ F]u$\".`@-$>wPF-7$Fbu$\".&)>6M`1&F-7$Fgu$\".%*>w@-#pF-7$F\\v$\".r%4[mi !*F-7$Fav$\"/Fw9vqm7F-7$Ffv$\"/dk#G)Qw;F-7$F[w$\"/H1:bV(H#F-7$F`w$\"/) Hy!p)=F-7$Fbz$\"0?eeJ[B8#F-7 $Fgz$\"0C<+b_iR#F-7$F\\[l$\"0K5/w48p#F-7$Fa[l$\"0C3#GSalHF-7$Ff[l$\"0c $=]xA,JF-7$F]bm$\"0!*f=?'HbKF-7$Fbbm$\"0@koBC%>LF-7$Fgbm$\"0(p+SUU>LF- 7$F\\cm$\"0=S\"eVU>LF-7$Facm$\"0*yJzYV>LF-7$F[\\l$\"0HQ$G&G'>LF-7$Ficm $\"0eSXt?:K$F-7$F^dm$\"0jlWAIWL$F-7$Fcdm$\"07=mZ$o.MF-7$F`\\l$\"0:P[\" H#Gr$F--Fe\\l6&Fg\\l$\"\"&Fj\\lF(F[]l-F^]l6#%8scheme~with~c[10]=81/82G -F$6%7YF'7$F+$!'M#>%F07$F2$!'_u&)F07$F7$!(?ZH\"F07$F<$!(uDm\"F07$FA$!( NQv\"F07$FF$!(@u1\"F07$FK$\")biT9F-7$FP$\")G]\"z(F-7$FU$\"*Xvr=#F-7$FZ $\"*F*=6dF-7$Fin$\"*1^%=**F-7$F^o$\"+cjV'=#F-7$Fco$\"+-&*pNOF-7$Fho$\" +@yyZdF-7$F]p$\"+Qlir')F-7$Fbp$\",eV.-D\"F-7$Fgp$\",2%ex>sl+@F07$F[w$!.*\\iW;eHF07$F`w$!.+d )Qw;RF07$Few$!.;hspm8&F07$Fjw$!.`$)eYpn'F07$F_x$!.]j5;\"3')F07$Fdx$!/> `r%3<5\"F07$Fix$!/0?#oG\")R\"F07$F^y$!/_FKc\"Gx\"F07$Fcy$!/[@X3F%)>F07 $Fhy$!/ " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 4 of 11 stage, order 7 Rung e-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathema tics 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)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=- 32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^si mplify(log[2](rhs(%)));\nk := unapply(rhs(%),x):\nplot(k(x),x=-1..1,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"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$7eo7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p 0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p \"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"Q LL3i.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$$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3 hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn7$$!3Q+++]$*4)*\\F-$\"3e;N4:OFap7$$!3]++]PYx\"\\#F-$\"37]-4,Tp9TFap7$$ !3QnmTNz>&H#F-$\"3y(*QMk^JnXFap7$$!3EMLLL7i)4#F-$\"3yCPsPtXE]Fap7$$!3# pm;aVXH)=F-$\"3_cYryDpGbFap7$$!3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3 s*****\\F&*=Y\"F-$\"3K`b3X@JjkFap7$$!3')****\\74_c7F-$\"3co2Qfx9woFap7 $$!3ZmmT5VBU5F-$\"3E!>K?nVAE(Fap7$$!3)3LLL3x%z#)!#>$\"3C'Q/NU&H#f(Fap7 $$!3gKL$e9d;J'Fft$\"3@erx_1&z$yFap7$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap 7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o%4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")F ap7$$!3=o;/^wj!p\"Fft$\"3]3j^OK2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4% yz\")Fap7$$!3S<+D\"GyNH'!#?$\"3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#) *eHY6>>)Fap7$$\"3o'H$3x\"yY_%Fjv$\"3Q>*>/AS,>)Fap7$$\"3&yK$3_Nl.5Fft$ \"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b\"Fft$\"3-l$f@nl+<)Fap7$$\"3@CL$3-Dg5#F ft$\"3kJX?)*G!=:)Fap7$$\"3e?Le*['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ez w5VFft$\"3U_-I6(**[-)Fap7$$\"3L`mmmJ+IiFft$\"3%pB(\\hv&o%yFap7$$\"3s*) ***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H (Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9 -Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"3 1))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%z AF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\ (=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3 uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i ******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap 7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"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$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 748 "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 poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,` scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\n Digits := 20:\nfor ct to 6 do\n Kn_RK7_||ct := RK7_||ct(evalf(K(x,y) ),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_ RK7_||ct):\n for ii to numpts do\n sm := sm+(Kn_RK7_||ct[ii,2]- k(Kn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/num pts)];\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~poin t:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ +#Q)pprint146\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\" \"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+/S8&=\"!#;7$/*&%-scheme~with~GF+F,F+#\" #:\"#;$\"+%4+%G5F57$/F8#\"#>\"#?$\"+h0)[J(!#<7$/F8#\"#K\"#L$\"+%pIFk&F F7$/F8#\"#\")\"##)$\"+#ea(yCFF7$/F8#\"$p\"\"$q\"$\"+V`5WGFFQ)pprint156 \"" }}}{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 20 ".995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: num steps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)] ,[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/1 2,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with ` *c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n kn_RK7_||ct := RK7_||c t(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 _RK7_||ct(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~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)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~sc heme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+Q>_F6!#?7$/*&%-scheme~with ~GF+F,F+#\"#:\"#;$\"*P*)Hu)F57$/F8#\"#>\"#?$\"*Jp&\\oF57$/F8#\"#K\"#L$ \"*]&*4w%F57$/F8#\"#\")\"##)$\"*u$e)3#F57$/F8#\"$p\"\"$q\"$\"*HAiS\"F5 Q)pprint176\"" }}}{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 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" } {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 0 414 "mthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/1 6,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with ` *c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20: \nfor ct to 6 do\n sm := NCint((k(x)-'kn_RK7_||ct'(x))^2,x=-1..1,ada ptive=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$/*&%:Sharp-Verner~sche me~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+/A4)=\"!#;7$/*&%-scheme~with ~GF+F,F+#\"#:\"#;$\"+%*z'4.\"F57$/F8#\"#>\"#?$\"+L79Lt!#<7$/F8#\"#K\"# L$\"+eL#ol&FF7$/F8#\"#\")\"##)$\"+oU%\\[#FF7$/F8#\"$p\"\"$q\"$\"+.e?^G FFQ)pprint186\"" }}}{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 0 499 "evalf[20](plot([k(x)-'kn_RK7_1'(x),k(x)-'kn_RK7_2'(x ),k(x)-'kn_RK7_3'(x),\nk(x)-'kn_RK7_4'(x),k(x)-'kn_RK7_5'(x),k(x)-'kn_ RK7_6'(x)],x=-1..1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR (RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegen d=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`s cheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=8 1/82`,`scheme with c[10]=169/170`],\ntitle=`error curves for 11 stage \+ order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 840 605 605 {PLOTDATA 2 "6,-%'CURVESG6%7eo7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!# ?$\"*G$zCFF/7$$!5LLLL$3L !#>7$$!5nmmmm\"4m(G$)F/$\"*\"[>K*)F<7$$!5LLLL$3i.9!zF/$\"+38`%>#F<7$$! 5mmmm;/R=0vF/$\"*+,s%HaF/$\"+`RzG5Ffn7$$!5+++++]$*4)*\\F/$\"+QYU0&H#F/$\"+FO?F: F`p7$$!5LLLLLL7i)4#F/$\"+D')p!o\"F`p7$$!5mmmmTNa%H)=F/$\"+7ff[=F`p7$$! 5********\\P'psm\"F/$\"+kwR7?F`p7$$!5*********\\F&*=Y\"F/$\"+&fE5;#F`p 7$$!5********\\74_c7F/$\"+`*G!*H#F`p7$$!5mmmmT5VBU5F/$\"+cJ4GCF`p7$$!4 LLLLL3x%z#)F/$\"+B=UQDF`p7$$!5ILLL$e9d;J'!#@$\"+0q`?EF`p7$$!4LLLL$3s$Q M%F/$\"+za<#o#F`p7$$!5immmT&QdDG$Fjt$\"+DKN1FF`p7$$!5&*******\\ivF@AFj t$\"+5d$Rs#F`p7$$!5imm;/^wj!p\"Fjt$\"+C\"4-t#F`p7$$!5GLLLeRx**f6Fjt$\" +[=zMFF`p7$$!5S*****\\7GyNH'!#A$\"+M$pwt#F`p7$$!2mmmmm\"zr)*F/$\"+'RS) QFF`p7$$\"5]LLL3x\"yY_%Fiv$\"+C`CQFF`p7$$\"5ILLL3_Nl.5Fjt$\"+-T!et#F`p 7$$\"5DLL$ekGR[b\"Fjt$\"+!pG:t#F`p7$$\"5?LLL$3-Dg5#Fjt$\"+PyTDFF`p7$$ \"55LLLe*['R3KFjt$\"+ZAw2FF`p7$$\"3LLLLezw5VF<$\"+&4zHo#F`p7$$\"5]mmmm mJ+IiFjt$\"+6:WBEF`p7$$\"3++++v$Q#\\\")F<$\"+B#\\Va#F`p7$$\"5lmmm\"z\\ 1A-\"F/$\"+!y8\"RCF`p7$$\"4LLLL$e\"*[H7F<$\"+,$[gJ#F`p7$$\"5lmmm;HCjV9 F/$\"+Q(eN<#F`p7$$\"4+++++qvxl\"F<$\"+(*o=>?F`p7$$\"5++++]7JFn=F/$\"+b :Pg=F`p7$$\"4++++]_qn2#F<$\"+5xV(p\"F`p7$$\"5++++vV+ZzAF/$\"+$ed!R:F`p 7$$\"4++++Dcp@[#F<$\"+Bxz#Q\"F`p7$$\"5++++v=GB2FF/$\"+Z2y97F`p7$$\"4++ ++]2'HKHF<$\"+PFfn7$$\"4LLLLL$*zym'F<$\"+'fF35#FL7$$\"4LLLL$3N1# 4(F<$\"+9G=U6FL7$$\"4nmmm\"HYt7vF<$\"*#>u1hFL7$$\"4+++++q(G**yF<$\"+F% \\3P$F<7$$\"4nmmm;9@BM)F<$\"+/7^e;F<7$$\"4LLLLL`v&Q()F<$\"*Tq!Q()F<7$$ \"4++++DOl5;*F<$\"+^^:;VF/7$$\"4++++v.Uac*F<$\"+wr_x@F/7$$\"\"\"F*$\"+ >G\\d5F/-%&COLORG6&%$RGBGF+$\"\"#F)Fabl-%'LEGENDG6#%ESharp-Verner~sche me~with~c[10]=11/12G-F$6%7eoF'7$F-$\"*hBJh#F/7$F3$\"+uR>.5F/7$F8$\"*m9 53$F<7$F>$\"*lQ_<)F<7$FC$\"+%3Q2)>F<7$FH$\"*J:[7%FL7$FN$\"*J-kO)FL7$FS $\"+TA+S;FL7$FX$\"*rmd0$Ffn7$Fhn$\"*@K(GbFfn7$F]o$\"*0Ar'*)Ffn7$Fbo$\" +_%[L[\"Ffn7$Fgo$\"+))pv_BFfn7$F\\p$\"*)e9DNF`p7$Fbp$\"*Gv$>\\F`p7$Fgp $\"*(>*=,(F`p7$F\\q$\"*%**HP!)F`p7$Faq$\"*j&HQ\"*F`p7$Ffq$\"+t&e-0\"F` p7$F[r$\"+lr!R>\"F`p7$F`r$\"+)3<_K\"F`p7$Fer$\"+\"p8%e9F`p7$Fjr$\"+D+6 /;F`p7$F_s$\"+EODYF`p7$F^t$\"+Pe) p5#F`p7$Fct$\"+Ccs-AF`p7$Fht$\"+5=)RF#F`p7$F^u$\"+P#ouK#F`p7$Fcu$\"+Pz W[BF`p7$Fhu$\"+uVqjBF`p7$F]v$\"+$QZ\"pBF`p7$Fbv$\"+:Q7tBF`p7$Fgv$\"+T+ ivBF`p7$F]w$\"+&zNmP#F`p7$Fbw$\"+V)=hP#F`p7$Fgw$\"+c%**RP#F`p7$F\\x$\" +\\%*GqBF`p7$Fax$\"+\"y&)\\O#F`p7$Ffx$\"+7Um\\BF`p7$F[y$\"+\"F`p7$Fb\\l$\"+\"oRT0\"F`p7$F g\\l$\"*g]r:*F`p7$F\\]l$\"*W;A*zF`p7$Fa]l$\"*-3E\"pF`p7$Ff]l$\"*Ia$>\\ F`p7$F[^l$\"*HM%)[$F`p7$F`^l$\"+Pyd0BFfn7$Fe^l$\"+Gr21:Ffn7$Fj^l$\"*P^ VI*Ffn7$F__l$\"*zDhg&Ffn7$Fd_l$\"*$flzJFfn7$Fi_l$\"+CX?#y\"FL7$F^`l$\" *!)\\Ke*FL7$Fc`l$\"*T&3`]FL7$Fh`l$\"++Ue]FF<7$F]al$\"+P.7L8F<7$Fbal$\" *E*)e#pF<7$Fgal$\"+vlj\"Q$F/7$F\\bl$\"+2#RJp\"F/7$Fabl$\"*O\")3?)F/-Ff bl6&FhblF+$\"\"(F)F+-F\\cl6#%8scheme~with~c[10]=15/16G-F$6%7eoF'7$F-$ \"*)*oRr\"F/7$F3$\"*27(pmF/7$F8$\"*D$zt?F<7$F>$\"*a(fkbF<7$FC$\"+bRfj8 F<7$FH$\"*@t3'GFL7$FN$\"*\")>Q%eFL7$FS$\"+L*z9:\"FL7$FX$\"*b0U:#Ffn7$F hn$\"*jI-\"RFfn7$F]o$\"*(>sajFfn7$Fbo$\"+Su1`5Ffn7$Fgo$\"+l3wr;Ffn7$F \\p$\"**4:1DF`p7$Fbp$\"*&=S)\\$F`p7$Fgp$\"*Wbs)\\F`p7$F\\q$\"*U#y;dF`p 7$Faq$\"*[T+]'F`p7$Ffq$\"*M\\0Z(F`p7$F[r$\"*-gB\\)F`p7$F`r$\"*OgjU*F`p 7$Fer$\"+vdPP5F`p7$Fjr$\"+h]+T6F`p7$F_s$\"+Zj5U7F`p7$Fds$\"+^B%QL\"F`p 7$Fis$\"+f!>!>9F`p7$F^t$\"+L&y')\\\"F`p7$Fct$\"+SNxm:F`p7$Fht$\"+RUX<; F`p7$F^u$\"+Df\\b;F`p7$Fcu$\"+\\tTq;F`p7$Fhu$\"+FzE\"o\"F`p7$F]v$\"+*f Q^o\"F`p7$Fbv$\"+Nn'zo\"F`p7$Fgv$\"+i:u*o\"F`p7$F]w$\"+XOY!p\"F`p7$Fbw $\"+6a4!p\"F`p7$Fgw$\"+1qe)o\"F`p7$F\\x$\"+I\"[fo\"F`p7$Fax$\"+*zu@o\" F`p7$Ffx$\"+ZeFr;F`p7$F[y$\"+*oxfl\"F`p7$F`y$\"+9rA>;F`p7$Fey$\"+ltSq: F`p7$Fjy$\"+^?X0:F`p7$F_z$\"+SA\\H9F`p7$Fdz$\"+vSaT8F`p7$Fiz$\"+$[iiC \"F`p7$F^[l$\"+W(Q#[6F`p7$Fc[l$\"+*\\*F`p7$F]\\l$ \"*_PZ`)F`p7$Fb\\l$\"*l@x\\(F`p7$Fg\\l$\"*keK^'F`p7$F\\]l$\"*OV[o&F`p7 $Fa]l$\"*fyr\"\\F`p7$Ff]l$\"*Q]+]$F`p7$F[^l$\"*H]I[#F`p7$F`^l$\"+Q/fU; Ffn7$Fe^l$\"+04hu5Ffn7$Fj^l$\"*)*)QdmFfn7$F__l$\"*@>l-%Ffn7$Fd_l$\"*E# >'H#Ffn7$Fi_l$\"+'eZmH\"FL7$F^`l$\"*l=x/(FL7$Fc`l$\"*aPfw$FL7$Fh`l$\"+ ;l%o2#F<7$F]al$\"+n\\L?5F<7$Fbal$\"*x_WO&F<7$Fgal$\"+@qEUEF/7$F\\bl$\" +u>KG8F/7$Fabl$\"*x/pT'F/-Ffbl6&Fhbl$\"\"*F)$\"\"%F)F+-F\\cl6#%8scheme ~with~c[10]=19/20G-F$6%7eoF'7$F-$\"*Jl2]\"F/7$F3$\"*[=er&F/7$F8$\"*5%) Gu\"F<7$F>$\"*$y)[f%F<7$FC$\"+&Q'416F<7$FH$\"*P2QH#FL7$FN$\"*GTVj%FL7$ FS$\"*NS(e!*FL7$FX$\"*7^Uo\"Ffn7$Fhn$\"*KL@/$Ffn7$F]o$\"*!o-H\\Ffn7$Fb o$\"*lZk9)Ffn7$Fgo$\"+`)p:H\"Ffn7$F\\p$\"*m%oM>F`p7$Fbp$\"*_:&*p#F`p7$ Fgp$\"*MWv%QF`p7$F\\q$\"*bV,T%F`p7$Faq$\"*nIU,&F`p7$Ffq$\"*@+Gw&F`p7$F [r$\"*%o*4b'F`p7$F`r$\"*L+:F(F`p7$Fer$\"*ndB+)F`p7$Fjr$\"*g*z,))F`p7$F _s$\"*FUl%4\"F`p7$F^t$\"+6I5c6F`p7$ Fct$\"+*yJ'37F`p7$Fht$\"+mdsZ7F`p7$F^u$\"+m)oqF\"F`p7$Fcu$\"+Bxd)G\"F` p7$Fhu$\"+og%pH\"F`p7$F]v$\"+9.$**H\"F`p7$Fbv$\"+y96-8F`p7$Fgv$\"+T\"z MI\"F`p7$F]w$\"++`./8F`p7$Fbw$\"+9,v.8F`p7$Fgw$\"+ZXe-8F`p7$F\\x$\"+') )[0I\"F`p7$Fax$\"++ij(H\"F`p7$Ffx$\"+EoA*G\"F`p7$F[y$\"+7SUx7F`p7$F`y$ \"+&[r!\\7F`p7$Fey$\"+9(49@\"F`p7$Fjy$\"+y@Ih6F`p7$F_z$\"+%=2F5\"F`p7$ Fdz$\"+)zl[.\"F`p7$Fiz$\"*u'o8'*F`p7$F^[l$\"*Srv&))F`p7$Fc[l$\"*0`=3)F `p7$Fh[l$\"*O5yK(F`p7$F]\\l$\"*HYQe'F`p7$Fb\\l$\"*3lQy&F`p7$Fg\\l$\"*' GPC]F`p7$F\\]l$\"*H/_Q%F`p7$Fa]l$\"*GTGz$F`p7$Ff]l$\"*UP\"*p#F`p7$F[^l $\"*IKR\">F`p7$F`^l$\"+Wgzk7Ffn7$Fe^l$\"*l7,E)Ffn7$Fj^l$\"*6c15&Ffn7$F __l$\"*#3JrIFfn7$Fd_l$\"*ne.u\"Ffn7$Fi_l$\"*fbCu*FL7$F^`l$\"**pQH_FL7$ Fc`l$\"*P/qR>*F/7$Fabl$\"*`z)pWF/-%'COLOURG6&Fhb l$\"*++++\"!\")F+Fc]o-F\\cl6#%8scheme~with~c[10]=32/33G-F$6%7eoF'7$F-$ \")B.9pF/7$F3$\"*S7xg#F/7$F8$\")cZ$)yF<7$F>$\"*_cG1#F<7$FC$\"*'*[,$\\F <7$FH$\"*F,z,\"FL7$FN$\"*H*[[?FL7$FS$\"*^oP*RFL7$FX$\")T27uFfn7$Fhn$\" *)f:P8Ffn7$F]o$\"*g0_;#Ffn7$Fbo$\"*dSsd$Ffn7$Fgo$\"*Dd6n&Ffn7$F\\p$\") /X&\\)F`p7$Fbp$\"*f5b=\"F`p7$Fgp$\"*hn)*o\"F`p7$F\\q$\"*?`q$>F`p7$Faq$ \"*nxC?#F`p7$Ffq$\"*K*QJDF`p7$F[r$\"*?Ex(GF`p7$F`r$\"*H'H%>$F`p7$Fer$ \"*$pS:NF`p7$Fjr$\"*COm'QF`p7$F_s$\"*m'G4UF`p7$Fds$\"*b$=?XF`p7$Fis$\" *PL)3[F`p7$F^t$\"*(owy]F`p7$Fct$\"*a\"\\4`F`p7$Fht$\"*X)=\"[&F`p7$F^u$ \"*BU+h&F`p7$Fcu$\"*^s0m&F`p7$Fhu$\"*l/tp&F`p7$F]v$\"*')*Q5dF`p7$Fbv$ \"*Ek*>dF`p7$Fgv$\"*7^fs&F`p7$F]w$\"*u\"QGdF`p7$Fbw$\"*e7rs&F`p7$Fgw$ \"**Q'>s&F`p7$F\\x$\"*T@Ir&F`p7$Fax$\"*a)>+dF`p7$Ffx$\"*#*HKm&F`p7$F[y $\"*gb8h&F`p7$F`y$\"*fen[&F`p7$Fey$\"*?v7K&F`p7$Fjy$\"*gI65&F`p7$F_z$ \"*C:P%[F`p7$Fdz$\"*M(pXXF`p7$Fiz$\"*M^GA%F`p7$F^[l$\"*F`p7$Fa]l$\"*^Mlm\"F`p7$Ff]l$\"*jsi=\" F`p7$F[^l$\")F([T)F`p7$F`^l$\"*B4Tc&Ffn7$Fe^l$\"*z+mj$Ffn7$Fj^l$\"*fP# [AFfn7$F__l$\"*+JbN\"Ffn7$Fd_l$\")^N$p(Ffn7$Fi_l$\"*FYYJ%FL7$F^`l$\"*K g4K#FL7$Fc`l$\"*#=(QA\"FL7$Fh`l$\"*#[RemF<7$F]al$\"*dfKA$F<7$Fbal$\"*] $Hr;F<7$Fgal$\"*,ex8)F/7$F\\bl$\"*VX/1%F/7$Fabl$\"*waq&>F/-Ffbl6&Fhbl$ \"\"&F)F+Fabl-F\\cl6#%8scheme~with~c[10]=81/82G-F$6%7eoF'7$F-$\")`X$p) F/7$F3$\"*I:!GKF/7$F8$\")D\\?'*F<7$F>$\"*UfZ[#F<7$FC$\"*\"pGieF<7$FH$ \"*g'=+7FL7$FN$\"*%y['R#FL7$FS$\"*wjdk%FL7$FX$\")$eZe)Ffn7$Fhn$\"*&GVV :Ffn7$F]o$\"*`+S\\#Ffn7$Fbo$\"*FoG6%Ffn7$Fgo$\"*3&G9lFfn7$F\\p$\")p&Gv *F`p7$Fbp$\"*\"3_g8F`p7$Fgp$\"*Us*Q>F`p7$F\\q$\"*^!\\AAF`p7$Faq$\"*`Xp _#F`p7$Ffq$\"*[^U!HF`p7$F[r$\"*()p:I$F`p7$F`r$\"*PnZm$F`p7$Fer$\"*(3>L SF`p7$Fjr$\"*'R=OWF`p7$F_s$\"*5Z$H[F`p7$Fds$\"*y)3'=&F`p7$Fis$\"*t7t^& F`p7$F^t$\"*^nq#eF`p7$Fct$\"*#*R=4'F`p7$Fht$\"*`&)))G'F`p7$F^u$\"*RunV 'F`p7$Fcu$\"**[x%\\'F`p7$Fhu$\"*AWp`'F`p7$F]v$\"*'p(>b'F`p7$Fbv$\"*!y' Hc'F`p7$Fgv$\"*D`)plF`p7$F]w$\"*Q^Ed'F`p7$Fbw$\"*H27d'F`p7$Fgw$\"*u?`c 'F`p7$F\\x$\"*Vg]b'F`p7$Fax$\"*:o.a'F`p7$Ffx$\"*>rz\\'F`p7$F[y$\"*7r%Q kF`p7$F`y$\"*M[bH'F`p7$Fey$\"*N5d5'F`p7$Fjy$\"*i]J&eF`p7$F_z$\"*)=#yb& F`p7$Fdz$\"*8+f@&F`p7$Fiz$\"*g%[X[F`p7$F^[l$\"*iDWY%F`p7$Fc[l$\"*<-N2% F`p7$Fh[l$\"*iF`p7$Ff]l$\"*4d8O\"F`p7$F[^ l$\")Cod'*F`p7$F`^l$\"*'ej'Q'Ffn7$Fe^l$\"*2BX<%Ffn7$Fj^l$\"*FP/e#Ffn7$ F__l$\"*E:Zb\"Ffn7$Fd_l$\")!Qo!))Ffn7$Fi_l$\"*#G,=\\FL7$F^`l$\"*xk,i#F L7$Fc`l$\"*-OmN\"FL7$Fh`l$\"*q\\K=(F<7$F]al$\"*\"Q5ILF<7$Fbal$\"*fe8i \"F<7$Fgal$\"*6;PA(F/7$F\\bl$\"*%*Qm>$F/7$Fabl$\"*\\,%e7F/-Fa]o6&FhblF c]oF+F+-F\\cl6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGF__n- %&TITLEG6#%Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%+A XESLABELSG6$Q\"x6\"Q!Fgjp-%%VIEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[1 0]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "scheme \+ with c[10]=32/33" "scheme with c[10]=81/82" "scheme with c[10]=169/170 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 5 of \+ 11 stage, order 7 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&\"#;F&*&,&*&F*F&%\"xGF&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 ", \+ " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-%%sqrtG6#,&* &\"\"#\"\"\"-%#lnG6#,&*&\"#;F+%\"xGF+F+F+F+F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve( \{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"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$$\"3umm\"H2P\"Q?F6$\"3]_v ibZz]7F27$$\"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$$\"3`KLe*)>VB$)F6$\"3 K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/ \"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za=F27$$\" 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'>F 27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"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#Fco$\"3ab\\ mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q

>IFco $\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S 2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\ (=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"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$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F2 7$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF27$$ \"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\"\"*-%+A XESLABELSG6$%\"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 following cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 731 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numste ps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`ini tial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,n umsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`sch eme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10] =32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs \+ := []:\nDigits := 20:\nfor ct to 6 do\n Sn_RK7_||ct := RK7_||ct(S(x, y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Sn_RK7_|| ct):\n for ii to numpts do\n sm := sm+(Sn_RK7_||ct[ii,2]-s(Sn_R K7_||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,!\"\"%\"yGF0F,7$%0initial~point:~G-%!G6 $\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\"Q)ppri nt196\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+>T]Uf!#A7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+\"#?$\"+zdpD@F57$/F8#\"#K\"#L$\"+;r%G&zF57$/F8#\" #\")\"##)$\"+Lp(R<\"!#@7$/F8#\"$p\"\"$q\"$\"+Df*z3\"FTQ)pprint206\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fol lowing 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 o f 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 0 663 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numste ps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`ini tial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,n umsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`sch eme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10] =32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs \+ := []:\nDigits := 25:\nfor ct to 6 do\n sn_RK7_||ct := RK7_||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_RK7_||ct(xx)-sxx)] ;\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,!\"\"%\"yGF0F,7$%0initial~point:~G-%!G6 $\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\"Q)ppri nt216\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+j3&y([!#A7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+J&*=)*HF57$/F8#\"#>\"#?$\"+tc$[v\"F57$/F8#\"#K\"#L$\"+Zs!za'F57$/F8 #\"#\")\"##)$\"+U3'*f'*F57$/F8#\"$p\"\"$q\"$\"+'z)Q\\*)F5Q)pprint226\" " }}}{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 inte rval [0, 0.5] of each Runge-Kutta method is estimated as follows usi ng the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perf orm numerical integration by the 7 point Newton-Cotes method over 50 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 416 "mthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/1 6,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with ` *c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20: \nfor ct to 6 do\n sm := NCint((s(x)-'sn_RK7_||ct'(x))^2,x=0..0.5,ad aptive=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$/*&%:Sharp-Verner~sc heme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+clQSf!#A7$/*&%-scheme~with ~GF+F,F+#\"#:\"#;$\"+bj[MOF57$/F8#\"#>\"#?$\"+2()*[7#F57$/F8#\"#K\"#L$ \"+E1\"*\\zF57$/F8#\"#\")\"##)$\"+Fmat6!#@7$/F8#\"$p\"\"$q\"$\"+3!)f(3 \"FTQ)pprint236\"" }}}{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 0 500 "evalf[20](plot([s(x)-'sn_RK7_1'(x),s(x)-'sn_RK7_2'(x ),s(x)-'sn_RK7_3'(x),\ns(x)-'sn_RK7_4'(x),s(x)-'sn_RK7_5'(x),s(x)-'sn_ RK7_6'(x)],x=0..0.5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLO R(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlege nd=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,` scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]= 81/82`,`scheme with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 764 485 485 {PLOTDATA 2 "6,-%'CURVESG6%7\\p7$$\"\"!F)F(7$$\"5SLLL3x1h6o!#B $\"\"\"!#>7$$\"5ommmTN@Ki8!#A$\"$9$F07$$\"5NLL3FpE!Hq\"F4$\"%,=F07$$\" 5-++]7.K[V?F4$\"%quF07$$\"5omm\"zptjSQ#F4$\"&XZ#F07$$\"5NLLL$3FWYs#F4$ \"&D&pF07$$\"5-++vo/[AlIF4$\"'wA6B`#o\"Fhq$ \"(p6%))F07$$\"5MLL3xJs1,=Fhq$\"(h'e()F07$$\"5nm\"Hd?pM.'=Fhq$\"(s>Fhq$\"(ujt)F07$$\"5ML3-j7'p)y>Fhq$\"(^I\"))F07$$\"5nm mm\"H2P\"Q?Fhq$\"(!pW))F07$$\"5+++]PMnNrDFhq$\"(Jzh)F07$$\"5MLLL$eRwX5 $Fhq$\"(D0Q)F07$$\"5NLLL$3x%3yTFhq$\"(A0&zF07$$\"5ommm\"z%4\\Y_Fhq$\"( =8g(F07$$\"5NLLLeR-/PiFhq$\"($QNtF07$$\"5-+++DcmpisFhq$\"(\"e/rF07$$\" 5OLLLe*)>VB$)Fhq$\"('\\-pF07$$\"5.+++DJbw!Q*Fhq$\"(*pHnF07$$\"5nmmm;/j $o/\"!#?$\"(f`d'F07$$\"5MLLL3_>jU6Fbx$\"(b_X'F07$$\"5++++]i^Z]7Fbx$\"( SXL'F07$$\"5++++](=h(e8Fbx$\"(phA'F07$$\"5++++]P[6j9Fbx$\"(/>8'F07$$\" 5MLL$e*[z(yb\"Fbx$\"(nO0'F07$$\"5nmmm;a/cq;Fbx$\"(n%ofF07$$\"5nmmmm;t, mFbx$\"(@ ?x&F07$$\"5,++]i!f#=$3#Fbx$\"(n?r&F07$$\"5,++](=xpe=#Fbx$\"(k(ecF07$$ \"5nmmm\"H28IH#Fbx$\"(hmg&F07$$\"5nmm;zpSS\"R#Fbx$\"(c;c&F07$$\"5MLLL3 _?`(\\#Fbx$\"(me^&F07$$\"5MLL$e*)>pxg#Fbx$\"(S5Z&F07$$\"5,++]Pf4t.FFbx $\"($3MaF07$$\"5MLLLe*Gst!GFbx$\"(ShR&F07$$\"5,++++DRW9HFbx$\"(6*e`F07 $$\"5,+++DJE>>IFbx$\"(pUK&F07$$\"5,++]i!RU07$Fbx$\"(\"H#H&F07$$\"5,+++ v=S2LKFbx$\"(A%e_F07$$\"5ommmm\"p)=MLFbx$\"([$H_F07$$\"5,+++](=]@W$Fbx $\"(>'*>&F07$$\"5MLL$e*[$z*RNFbx$\"(tP<&F07$$\"5,+++]iC$pk$Fbx$\"(@m9& F07$$\"5omm;H2qcZPFbx$\"(g?7&F07$$\"5,++]7.\"fF&QFbx$\"(Ut4&F07$$\"5om mm;/OgbRFbx$\"(jS2&F07$$\"5,++]ilAFjSFbx$\"(s00&F07$$\"5NLLLL$)*pp;%Fb x$\"([(G]F07$$\"5NLLL3xe,tUFbx$\"(%>2]F07$$\"5omm;HdO=yVFbx$\"(Pl)\\F0 7$$\"5,++++D>#[Z%Fbx$\"(_\"o\\F07$$\"5ommmT&G!e&e%Fbx$\"(Xx%\\F07$$\"5 NLLLL$)Qk%o%Fbx$\"(h+$\\F07$$\"5-++]iSjE!z%Fbx$\"(o<\"\\F07$$\"5-++]P4 0O\"*[Fbx$\"(uZ*[F07$$\"\"&!\"\"$\"(Zq([F0-%&COLORG6&%$RGBGF($\"\"#Fad l$F/F)-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7`pF'7$F +$FadlF07$F2$!$<#F07$F8$!%E7F07$F=$!%!*\\F07$FB$!&Bi\"F07$FG$!&OZ%F07$ FL$!'&y3\"F07$FQ$!'t&R#F07$FV$!'up[F07$Fen$!'>l#*F07$Fjn$!(bwm\"F07$F_ o$!(/K'GF07$Fdo$!(BZ.%F07$Fio$!(9t,%F07$$\"50++]P4'\\/8'F4$!(JM)RF07$F ^p$!(_7&RF07$$\"50++v$4@\">_rF4$!(()o$RF07$$\"5qmm;zW&F07$Fdt$!(VG=&F07$Fit$!(Lp>&F07$F^u$!(;,F&F07$Fcu$!(t_I&F07$Fhu$!( ^(3_F07$F]v$!(D54&F07$Fbv$!(Tx&[F07$Fgv$!(hsl%F07$F\\w$!(:1]%F07$Faw$! (cAO%F07$Ffw$!(S*RUF07$F[x$!(1[8%F07$F`x$!(N1/%F07$Ffx$!()4nRF07$F[y$! (rJ*QF07$F`y$!(!oEQF07$Fey$!(<)oPF07$Fjy$!(!y?PF07$F_z$!(Z%oOF07$Fdz$! (lwi$F07$Fiz$!($y$e$F07$F^[l$!(cxa$F07$Fc[l$!(:4^$F07$Fh[l$!(f\"yMF07$ F]\\l$!(RhW$F07$Fb\\l$!(\"[=MF07$Fg\\l$!(Q.R$F07$F\\]l$!((yiLF07$Fa]l$ !(t+M$F07$Ff]l$!(`nJ$F07$F[^l$!(sQH$F07$F`^l$!(zDF$F07$Fe^l$!(CHD$F07$ Fj^l$!(2@B$F07$F__l$!(PU@$F07$Fd_l$!(lf>$F07$Fi_l$!(x+=$F07$F^`l$!(*Qj JF07$Fc`l$!(#H[JF07$Fh`l$!(,J8$F07$F]al$!(#z=JF07$Fbal$!(aV5$F07$Fgal$ !(R44$F07$F\\bl$!(!pxIF07$Fabl$!(%*\\1$F07$Ffbl$!($p`IF07$F[cl$!(\\6/$ F07$F`cl$!(\"GIIF07$Fecl$!(P!>IF07$Fjcl$!(#f3IF07$F_dl$!(&p(*HF0-Fedl6 &FgdlF($\"\"(FadlF(-F\\el6#%8scheme~with~c[10]=15/16G-F$6%7^pF'Fbel7$F 2$!$Z\"F07$F8$!$@)F07$F=$!%$H$F07$FB$!&[0\"F07$FG$!&Z'GF07$FL$!&#eoF07 $FQ$!'Q'[\"F07$FV$!'7sHF07$Fen$!'?gbF07$Fjn$!''f$)*F07$F_o$!(<*e;F07$F do$!(?)3BF07$Fio$!(e))H#F07$F\\hl$!(r%zAF07$F^p$!(26E#F07$Fihl$!(`pC#F 07$Fbp$!(c=D#F07$Fgp$!(q:K#F07$F\\q$!(r\\c#F07$Faq$!(\"*R\"GF07$Ffq$!( R$GHF07$F\\r$!(#*y\"HF07$Far$!(1w!HF07$Ffr$!(yL(GF07$F[s$!(sC&GF07$F`s $!(AZ$HF07$Fes$!(1F2$F07$Fjs$!(x7/$F07$F_t$!(rR,$F07$Fdt$!(,y+$F07$Fit $!(Fz,$F07$F^u$!(nW1$F07$Fcu$!(8q3$F07$Fhu$!(wl.$F07$F]v$!(u:(HF07$Fbv $!(o#RGF07$Fgv$!(]Qs#F07$F\\w$!(\"4LEF07$Faw$!((e_DF07$Ffw$!(c7[#F07$F [x$!(p)>CF07$F`x$!(Z[O#F07$Ffx$!(X=K#F07$F[y$!(9'yAF07$F`y$!(8(RAF07$F ey$!(de?#F07$Fjy$!([x<#F07$F_z$!(Br9#F07$Fdz$!(dK7#F07$Fiz$!(vv4#F07$F ^[l$!(\"\\w?F07$Fc[l$!(H\\0#F07$Fh[l$!(ed.#F07$F]\\l$!(F07$F\\]l$!(L#o>F07$Fa]l$!(R\\&>F07$Ff]l$!(!HT>F0 7$F[^l$!()*y#>F07$F`^l$!(Na\">F07$Fe^l$!(JR!>F07$Fj^l$!(Z<*=F07$F__l$! ((G\")=F07$Fd_l$!($fq=F07$Fi_l$!(%Hh=F07$F^`l$!(F:&=F07$Fc`l$!(!pU=F07 $Fh`l$!(*zL=F07$F]al$!(Ca#=F07$Fbal$!(tp\"=F07$Fgal$!(A\"4=F07$F\\bl$! (n8!=F07$Fabl$!(ORz\"F07$Ffbl$!(Aty\"F07$F[cl$!(!)*zOx\"F 07$Fecl$!(Qqw\"F07$Fjcl$!(D4w\"F07$F_dl$!(YXv\"F0-Fedl6&Fgdl$\"\"*Fadl $\"\"%FadlF(-F\\el6#%8scheme~with~c[10]=19/20G-F$6%7`pF'7$F+$!\"#F07$F 2$!$j%F07$F8$!%HEF07$F=$!&[2\"F07$FB$!&%4NF07$FG$!&,s*F07$FL$!'SuBF07$ FQ$!'P`_F07$FV$!(3H2\"F07$Fen$!(670#F07$Fjn$!(N-r$F07$F_o$!(U@S'F07$Fd o$!(fH0*F07$Fio$!(+R,*F07$F\\hl$!(sy$*)F07$F^p$!((fl))F07$Fdhl$!(lJ$)) F07$Fihl$!(rv!))F07$F^il$!(Umz)F07$Fbp$!(A[\"))F07$Fgp$!(n=0*F07$F\\q$ !(P*H**F07$Faq$!)=Z&3\"F07$Ffq$!),VG6F07$F\\r$!)XSC6F07$Far$!)/W?6F07$ Ffr$!)AA26F07$F[s$!))H%)4\"F07$F`s$!)!eY7\"F07$Fes$!)=Gr6F07$Fjs$!)oHf 6F07$F_t$!)(y'[6F07$Fdt$!).uX6F07$Fit$!)!4![6F07$F^u$!)t3i6F07$Fcu$!)9 go6F07$Fhu$!)&GV9\"F07$F]v$!)()[;6F07$Fbv$!)#pJ1\"F07$Fgv$!)1H=5F07$F \\w$!(&\\N)*F07$Faw$!(&eI&*F07$Ffw$!(k>E*F07$F[x$!(6:.*F07$F`x$!()HD)) F07$Ffx$!(rWm)F07$F[y$!(0G])F07$F`y$!('\\d$)F07$Fey$!(m5B)F07$Fjy$!(8h 7)F07$F_z$!($z6!)F07$Fdz$!(1F#zF07$Fiz$!(bo#yF07$F^[l$!(m\"[xF07$Fc[l$ !(*pnwF07$Fh[l$!(chf(F07$F]\\l$!(@i_(F07$Fb\\l$!(9eY(F07$Fg\\l$!(\\VS( F07$F\\]l$!(yTM(F07$Fa]l$!(rXH(F07$Ff]l$!(ROC(F07$F[^l$!(mO>(F07$F`^l$ !(jr9(F07$Fe^l$!(RU5(F07$Fj^l$!(u(eqF07$F__l$!(Y(>qF07$Fd_l$!(S)zpF07$ Fi_l$!(V^%pF07$F^`l$!('p3pF07$Fc`l$!(Ed(oF07$Fh`l$!(ZD%oF07$F]al$!()H6 oF07$Fbal$!(l(znF07$Fgal$!(p/v'F07$F\\bl$!(M:s'F07$Fabl$!(0Qp'F07$Ffbl $!(C\"pmF07$F[cl$!(I^J\"F07$Fgp$!)ZN\\8F07$F\\q$!)8xx9F07$Faq$!)fv8;F07$Ffq $!)o.x;F07$F\\r$!)U0r;F07$Far$!)G;l;F07$Ffr$!)$4bk\"F07$F[s$!)>?K;F07$ F`s$!)=Ip;F07$Fes$!)m?O19F07$F fw$!)U\\m8F07$F[x$!)eXK8F07$F`x$!)!3?I\"F07$Ffx$!)6Fy7F07$F[y$!)4Ta7F0 7$F`y$!)'pHB\"F07$Fey$!)ZJ97F07$Fjy$!)&H))>\"F07$F_z$!)G'>=\"F07$Fdz$! )#>)o6F07$Fiz$!)zna6F07$F^[l$!)'oI9\"F07$Fc[l$!)r>J6F07$Fh[l$!)Bk?6F07 $F]\\l$!)[K56F07$Fb\\l$!)IT,6F07$Fg\\l$!)^M#4\"F07$F\\]l$!)#oM3\"F07$F a]l$!)(\\h2\"F07$Ff]l$!)ejo5F07$F[^l$!)MEh5F07$F`^l$!)HSa5F07$Fe^l$!). 2[5F07$Fj^l$!)JOT5F07$F__l$!)`gN5F07$Fd_l$!)!=(H5F07$Fi_l$!)$*fC5F07$F ^`l$!)CA>5F07$Fc`l$!)%eV,\"F07$Fh`l$!)NY45F07$F]al$!)M&[+\"F07$Fbal$!) 9?+5F07$Fgal$!(&ze**F07$F\\bl$!(3h\"**F07$Fabl$!(+_()*F07$Ffbl$!(!zQ)* F07$F[cl$!(v$)z*F07$F`cl$!(cLw*F07$Fecl$!(Irs*F07$Fjcl$!(xMp*F07$F_dl$ !(o$e'*F0-Fedl6&FgdlF_dlF(Fjdl-F\\el6#%8scheme~with~c[10]=81/82G-F$6%7 \\pF'F]fo7$F2$!$)fF07$F8$!%CMF07$F=$!&BT\"F07$FB$!&+l%F07$FG$!'C)H\"F0 7$FL$!'F&>$F07$FQ$!'z>rF07$FV$!(JPY\"F07$Fen$!(Vb\"GF07$Fjn$!(&H@^F07$ F_o$!(c;)))F07$Fdo$!)^#)f7F07$Fio$!)'*Qa7F07$F^p$!)#QPB\"F07$Fbp$!)w+E 7F07$Fgp$!)qKd7F07$F\\q$!)j!eP\"F07$Faq$!)2m,:F07$Ffq$!)kFg:F07$F\\r$! ).ra:F07$Far$!)\"H#\\:F07$Ffr$!)u$4`\"F07$F[s$!)dS=:F07$F`s$!)F)=b\"F0 7$Fes$!)h!Hh\"F07$Fjs$!)'*R'f\"F07$F_t$!))o;e\"F07$Fdt$!)3Kx:F07$Fit$! )Jjz:F07$F^u$!)3;(f\"F07$Fcu$!)b20;F07$Fhu$!)],\"F07$F\\]l$!)+x.5F07$Fa]l$ !()*)p**F07$Ff]l$!('G+**F07$F[^l$!(&)>$)*F07$F`^l$!(F%o(*F07$Fe^l$!(f( 4(*F07$Fj^l$!(?wk*F07$F__l$!(yUf*F07$Fd_l$!(O(R&*F07$Fi_l$!(9B\\*F07$F ^`l$!(*\\U%*F07$Fc`l$!(PuR*F07$Fh`l$!(!4_$*F07$F]al$!(z$4$*F07$Fbal$!( \"Gm#*F07$Fgal$!(TiA*F07$F\\bl$!(%p'=*F07$Fabl$!(&z[\"*F07$Ffbl$!(j]6* F07$F[cl$!(Aw2*F07$F`cl$!(z^/*F07$Fecl$!(<;,*F07$Fjcl$!(S/)*)F07$F_dl$ !(8z%*)F0-Fbeo6&FgdlFdeoF(F(-F\\el6#%:scheme~with~c[10]=169/170G-%%FON TG6$%*HELVETICAGF_en-%&TITLEG6#%Verror~curves~for~11~stage~order~7~Run ge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fieq-%%VIEWG6$;F(F_dl%(DEFAU LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Ve rner scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c [10]=19/20" "scheme with c[10]=32/33" "scheme with c[10]=81/82" "schem e with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 6 of 11 stage, order 7 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-%#l nG6#,,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$$\"3 GLLL3x&)*3\"!#=$\"3QWK+t!=.P\"F-7$$\"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\"F I7$$\"3Y++DJbw!Q*F-$\"3/$y^LV#)3O\"FI7$$\"3+N$ekGkX#**F-$\"3'R4-T\"[w( Q\"FI7$$\"3%ommTIOo/\"FI$\"3i$3slO!o09FI7$$\"3E+]7GTt%4\"FI$\"3qC'f&H6 %RT\"FI7$$\"3YLL3_>jU6FI$\"3#[%e$e^!3:9FI7$$\"3ym;HdNb'>\"FI$\"3CNMY.K v29FI7$$\"37++]i^Z]7FI$\"3=<^j^\"=7R\"FI7$$\"35+++v\"=YI\"FI$\"3![4*)z 8y_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$$\"3%ommmJFI$\"3yKBC%\\'\\`dF-7$$\"3gmmm\"pW`(>FI$\"3$)>nU%[* 3T`F-7$$\"3_ek.HW#)))>FI$\"3kKE$Q%*GSE&F-7$$\"3?]iSmTI-?FI$\"3NxPe(47T ?&F-7$$\"3*=/wP!Ry:?FI$\"3MId=-(\\?;&F-7$$\"3dLe9TOEH?FI$\"3+V!)\\%**R %Q^F-7$$\"3EDc^yLuU?FI$\"3=k(\\#*\\cP8&F-7$$\"3'pT&)e6Bi0#FI$\"3[P(R22 Q$[^F-7$$\"3k3_D`Gqp?FI$\"3m1F&*)f\"Q#=&F-7$$\"3K+]i!f#=$3#FI$\"3z7KO0 x$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$p;a8d3AM#FI$\"3NM9xI K\")Q#*F-7$$\"3um;zpSS\"R#FI$\"3r*)ek&)o0L5FI7$$\"3-+v$41oWW#FI$\"3if$ >8HM6:\"FI7$$\"3GLL3_?`(\\#FI$\"31,>EjcGm7FI7$$\"3AL3_D1l_DFI$\"3+C:%e =o-Q\"FI7$$\"3fL$e*)>pxg#FI$\"3%R*>K#Rqn[\"FI7$$\"33+]Pf4t.FFI$\"3A89r RoB^;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-#FI 7$$\"3;L$3_NJOG$FI$\"3:>!e))R4?FI7$$\"3Jmmm\"p)=MLFI$\"3OrPOD8'y)>FI 7$$\"3GLLeR%p\")Q$FI$\"3S(z'\\,LGb>FI7$$\"3B++](=]@W$FI$\"3K:aBe2s7>FI 7$$\"35L$e*[$z*RNFI$\"3!>**fc?u*4=FI7$$\"3e++]iC$pk$FI$\"3'e\\^`F#Qg;F I7$$\"3[m;H2qcZPFI$\"351&p,mkr[\"FI7$$\"3O+]7.\"fF&QFI$\"3o;2PnR[#G\"F I7$$\"3Ymm;/OgbRFI$\"3/lp%G(QG#3\"FI7$$\"3*G$e*[$zV4SFI$\"3P9Az61+7**F -7$$\"3w**\\ilAFjSFI$\"3!p>ERLl'*>*F-7$$\"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$$\"3)RL$3xe,tUFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9 \"FI7$$\"3MMe9\"z-lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'Gmo O\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p 9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_eaMDR_K " 0 "" {MPLTEXT 1 0 743 "U := (x,y) -> (1+2*(x+1)*si n(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatri x([[`slope field: `,U(x,y)],[`initial point: `,``(x0,y0)],[`step wid th: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Ve rner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,` scheme with `*c[10]=169/170]: errs := []:\nDigits := 25:\nfor ct to 6 \+ do\n Un_RK7_||ct := RK7_||ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n \+ sm := 0: numpts := nops(Un_RK7_||ct):\n for ii to numpts do\n \+ sm := sm+(Un_RK7_||ct[ii,2]-u(Un_RK7_||ct[ii,1]))^2;\n end do:\n \+ errs := [op(errs),sqrt(sm/numpts)];\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+,&%\"xGF+F+F+F+-%$s inG6#,$*&\"\"$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)pp rint246\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+t)3O;*!#E7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+>/kAeF57$/F8#\"#>\"#?$\"+6)\\MQ&F57$/F8#\"#K\"#L$\"+'y>I'HF57$/F8# \"#\")\"##)$\"+Lu3'G\"F57$/F8#\"$p\"\"$q\"$\"+h11#R(!#FQ)pprint256\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fo llowing 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 o f 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 0 674 "U := (x,y) -> (1+2*(x+1)*sin(3*x)) *exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`sl ope field: `,U(x,y)],[`initial point: `,``(x0,y0)],[`step width: ` ,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner sc heme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10 ]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme \+ with `*c[10]=169/170]: errs := []:\nDigits := 30:\nfor ct to 6 do\n \+ un_RK7_||ct := RK7_||ct(U(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\n xx := 4.999: uxx := evalf(u(xx)):\nfor ct to 6 do\n errs := [op(errs ),abs(un_RK7_||ct(xx)-uxx)];\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+-%$sinG6#,$ *&\"\"$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)pprint266 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+g[#y())!#E7$/*&%-scheme~with~GF+F,F+#\"#:\"#; $\"+**e]5bF57$/F8#\"#>\"#?$\"+iTZ2_F57$/F8#\"#K\"#L$\"+1,(Qu#F57$/F8# \"#\")\"##)$\"+NyBE6F57$/F8#\"$p\"\"$q\"$\"+k1JLg!#FQ)pprint276\"" }}} {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 integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((u(x)-'un_RK7_||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$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+T!3\"R\"*!#E7$/*&%-scheme~with~GF+F,F+#\"#:\" #;$\"+=`V0eF57$/F8#\"#>\"#?$\"+)pp1P&F57$/F8#\"#K\"#L$\"+&R%=dHF57$/F8 #\"#\")\"##)$\"+(Q9`G\"F57$/F8#\"$p\"\"$q\"$\"+.To.u!#FQ)pprint286\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The fo llowing error graphs are constructed using the numerical procedures fo r the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 515 "evalf[2 0](plot(['un_RK7_1'(x)-u(x),'un_RK7_2'(x)-u(x),'un_RK7_3'(x)-u(x),\n'u n_RK7_4'(x)-u(x),'un_RK7_5'(x)-u(x),'un_RK7_6'(x)-u(x)],x=0..5,-9e-18. .2.83e-16,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7 ,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp -Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme wit h c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`sc heme with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 R unge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 974 839 839 {PLOTDATA 2 "6,-%'CURVESG6%7br7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$!%_DF- 7$$\"5MLLLL3x&)*3\"!#?$!$K%F37$$\"5nmm\"z%\\v#pK\"F3$!$Y%F37$$\"5+++]i !R(*Rc\"F3$!$:%F37$$\"5MLL3xJs1,=F3$!$H$F37$$\"5nmmm\"H2P\"Q?F3$!$Z#F3 7$$\"5+++]PMnNrDF3$!\"%F37$$\"5MLLL$eRwX5$F3$\"$m#F37$$\"5MLLLL$eI8k$F 3$\"$<%F37$$\"5NLLL$3x%3yTF3$\"$/&F37$$\"5-++]PfyG7ZF3$\"$I&F37$$\"5om mm\"z%4\\Y_F3$\"$:&F37$$\"5NLLLeR-/PiF3$\"#W!#>7$$\"5-+++DcmpisF3$\"#P Fgo7$$\"5OLLLe*)>VB$)F3$\"#KFgo7$$\"5.+++DJbw!Q*F3$\"#HFgo7$$\"5nmmm;/ j$o/\"Fgo$\"#GFgo7$$\"5MLLL3_>jU6FgoFep7$$\"5++++]i^Z]7Fgo$\"#JFgo7$$ \"5++++](=h(e8FgoF`p7$$\"5++++]P[6j9Fgo$\"#MFgo7$$\"5MLL$e*[z(yb\"FgoF `p7$$\"5+++Dc,#>Uh\"FgoFep7$$\"5nmmm;a/cq;Fgo$\"$<$F37$$\"5nmm;zpYU%p \"Fgo$\"$i$F37$$\"5nmmmT&)))G=Fgo$\"%RhF37$$\"5MLLe9;0?E>Fgo$\"%!Q(F37$$ \"5++]i!Rgs2&>Fgo$\"%r*)F37$$\"5nmmmm\"pW`(>Fgo$\"%n)*F37$$\"5MLek.HW# )))>Fgo$\"&Z-\"F37$$\"5++]iSmTI-?Fgo$\"&*\\5F37$$\"5M$e9\"4NS/4?Fgo$\" &70\"F37$$\"5nmTgx.Ry:?Fgo$\"&;0\"F37$$\"5+]P4YsP_A?Fgo$\"&B/\"F37$$\" 5MLLe9TOEH?Fgo$\"&E.\"F37$$\"5om;a)e6Bi0#Fgo$\"%l&*F37$$\"5,++]i!f#=$3 #Fgo$\"%C\")F37$$\"5,+++D\"=EX8#Fgo$\"%XcF37$$\"5,++](=xpe=#Fgo$\"%#o% F37$$\"5MLLeRA9WRAFgo$\"%'e&F37$$\"5nmmm\"H28IH#Fgo$\"%WqF37$$\"5nm;a8 A3hpxg#Fgo$\"$1'Fgo7$$\"5,++]Pf4t.F Fgo$\"$5&Fgo7$$\"5MLLLe*Gst!GFgo$\"$Q%Fgo7$$\"5,++++DRW9HFgo$\"$!RFgo7 $$\"5,+++DJE>>IFgo$F[sFgo7$$\"5,++]i!RU07$Fgo$\"$\\$Fgo7$$\"5,+++v=S2L KFgoF^]l7$$\"5ommmm\"p)=MLFgo$\"$h$Fgo7$$\"5,+++](=]@W$Fgo$\"$*QFgo7$$ \"5MLL$e*[$z*RNFgo$\"$H%Fgo7$$\"5,+++]iC$pk$Fgo$\"$!\\Fgo7$$\"5omm;H2q cZPFgo$\"$t&Fgo7$$\"5MLL$3_0j,!QFgo$\"$Y'Fgo7$$\"5,++]7.\"fF&QFgo$\"$l (Fgo7$$\"5om;aQG-ZyQFgo$\"$\\)Fgo7$$\"5MLLek`8=/RFgo$\"$x*Fgo7$$\"5,+] i!*yC*)HRFgo$\"%c6Fgo7$$\"5ommm;/OgbRFgo$\"%Q8Fgo7$$\"5MLe*[o=i!pRFgo$ \"%v9Fgo7$$\"5,+]7`p2_#)RFgo$\"%J;Fgo7$$\"5omTN@_$zf*RFgo$\"%bF37$$\"5,+D\"yv^'*G-%Fgo$\"&=5#F37$$\"5om;/E+^ NOSFgo$\"&IB#F37$$\"5ML3F%Ho8)\\SFgo$\"&_U#F37$$\"5,++]ilAFjSFgo$\"&Ta #F37$$\"5ML$3_]p'>*3%Fgo$\"&+t#F37$$\"5omm\"zW7@^6%Fgo$\"&wz#F37$$\"5- +]i!RbX59%Fgo$\"&zp#F37$$\"5NLLLL$)*pp;%Fgo$\"&``#F37$$\"5NL$3_+sD-=%F go$\"&LM#F37$$\"5NLL3xc9[$>%Fgo$\"&YB#F37$$\"5NL$e*[$>Pn?%Fgo$\"&@7#F3 7$$\"5NLL$3-$H**>UFgo$\"&r%>F37$$\"5NLLek.W]YUFgo$\"&Cz\"F37$$\"5NLLL3 xe,tUFgo$\"&)*p\"F37$$\"5-++v=n(*fDVFgo$\"%rFgo7$$\"5,]PM-\\NA%Q%Fgo$\"%**>Fgo7$$\"5ML3_vSME!R%Fgo$\"%U?Fgo7$$ \"5,v$4@mQ$G$R%Fgo$\"%H?Fgo7$$\"5o;zp[KLI'R%Fgo$\"%=?Fgo7$$\"5MekGNyKK *R%Fgo$\"%S?Fgo7$$\"5,+](=UAVBS%Fgo$\"%b?Fgo7$$\"5om\"H#o2IU9WFgo$\"%i ?Fgo7$$\"5MLLe9\"z-lU%Fgo$\"%j?Fgo7$$\"5,v=<,PF_HWFgo$\"%$3#Fgo7$$\"5o ;/w(GoUDV%Fgo$\"%'3#Fgo7$$\"5Me*[V(GEcNWFgo$\"%r?Fgo7$$\"5,+v$4Yd#eQWF go$\"%q?Fgo7$$\"5oTg_Z?DgTWFgo$\"%!4#Fgo7$$\"5M$e9TjYAYW%Fgo$\"%u?Fgo7 $$\"5,DJq?7CkZWFgo$\"%k?Fgo7$$\"5om;H2eBm]WFgo$\"%*3#Fgo7$$\"5MLek`T@u iWFgo$\"%n?Fgo7$$\"5,++++D>#[Z%Fgo$\"%V?Fgo7$$\"5MLL$3_5,-`%Fgo$\"%8>F go7$$\"5ommmT&G!e&e%Fgo$\"%E6but%Fgo$\"%x7Fgo7$$\"5-++]iSjE!z%Fg o$\"%f6Fgo7$$\"5-++++DM\"3%[Fgo$\"%r5Fgo7$$\"5-++]P40O\"*[Fgo$\"$)**Fg o7$$\"\"&F)$\"$())Fgo-%&COLORG6&%$RGBGF($\"\"#!\"\"$\"\"\"F)-%'LEGENDG 6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7asF'7$$\"5NLLL$3FWYs#F- $!$G$F-7$F+$!$$oF-7$$\"50+++]7G$R<)F-$!$v(F-7$F1$!#lF37$F<$\"#8F37$Fat$\"%CEF37$Ff t$\"%[OF37$F[u$\"%bWF37$F`u$\"%CbF37$Feu$\"%WhF37$Fju$\"%6kF37$F_v$\"% +mF37$Fdv$\"%5mF37$Fiv$\"%;mF37$F^w$\"%ilF37$Fcw$\"%,lF37$Fhw$\"%:gF37 $F]x$\"%=^F37$$\"5,++v$fQa)3@Fgo$\"%tVF37$Fbx$\"%iPF37$$\"5,+]i!*y?OZ@ Fgo$\"%4OF37$$\"5,++Dcwz>g@Fgo$\"%>NF37$$\"5,+](=U(Q.t@Fgo$\"%ENF37$Fg x$\"%#e$F37$F\\y$\"%\"\\%F37$Fay$\"%f`F37$Ffy$\"%8bF37$F[z$\"%(o&F37$F `z$\"%OcF37$Fez$\"$f&Fgo7$Fjz$\"$?&Fgo7$F_[l$\"$r%Fgo7$Fd[l$\"$z$Fgo7$ Fi[l$\"$?$Fgo7$F^\\l$\"$w#Fgo7$Fc\\l$\"$Y#Fgo7$Fh\\l$\"$H#Fgo7$F\\]l$ \"$@#Fgo7$Fa]lFi[n7$Fd]l$\"$G#Fgo7$Fi]l$\"$Z#Fgo7$F^^l$\"$v#Fgo7$Fc^l$ FfrFgo7$Fh^l$\"$t$Fgo7$F]_l$\"$:%Fgo7$Fb_l$\"$w%Fgo7$Fg_l$\"$=&Fgo7$F \\`l$\"$$eFgo7$Fa`l$\"$y'Fgo7$Ff`l$\"$z(Fgo7$F[al$\"$e)Fgo7$F`al$\"$_* Fgo7$Feal$\"%F5Fgo7$Fjal$\"&N9\"F37$F_bl$\"&)\\7F37$Fdbl$\"&kL\"F37$Fi bl$\"&)p9F37$F^cl$\"&Eb\"F37$Fccl$\"&!*o\"F37$Fhcl$\"&^u\"F37$F]dl$\"& mo\"F37$Fbdl$\"&3e\"F37$Fgdl$\"&tX\"F37$F\\el$\"&%*Q\"F37$Fael$\"&5K\" F37$Ffel$\"&=A\"F37$$\"5NL$3Fpm[KB%Fgo$\"&#y6F37$F[fl$\"&b9\"F37$$\"5N $3_DyL=)\\UFgo$\"&r8\"F37$$\"5NL3_+sA8`UFgo$\"&A8\"F37$$\"5N$e*[=1iWcU Fgo$\"&!G6F37$$\"5NL$ek.9g(fUFgo$\"&*G6F37$$\"5N$3FWX2uIE%Fgo$\"&b7\"F 37$$\"5NLeRs3!)QmUFgo$\"&27\"F37$$\"5N$ek.H%>qpUFgo$\"&)H6F37$F`fl$\"& &G6F37$Fefl$\"%I7Fgo7$Fjfl$\"%e8Fgo7$F_gl$\"%t8Fgo7$Fdgl$\"%%R\"Fgo7$F igl$\"%&Q\"Fgo7$F^hl$\"%y8Fgo7$Fchl$\"%(Q\"Fgo7$FhhlFddn7$$\"5oTNY3qJO 0WFgoFgdn7$$\"5M$3_]f6$Q3WFgo$\"%$Q\"Fgo7$$\"5,D1k\"=1.9T%Fgo$\"%)R\"F go7$F]il$\"%!R\"Fgo7$$\"5M3x\"[N&HWF37$F<$!$u\" F37$FA$!$;\"F37$FF$!#pF37$FK$\"#kF37$FP$\"$2#F37$FU$\"$!GF37$FZ$\"$@$F 37$Fin$\"$F$F37$F^o$\"$;$F37$Fco$\"#FFgo7$FioF_cm7$F^p$\"#?Fgo7$Fcp$\" #>Fgo7$Fhp$\"#=Fgo7$F]qFh]o7$F`qFb]o7$Feq$\"#AFgo7$FhqF\\cm7$F]rFjp7$F `r$\"#IFgo7$Fcr$Fg]lF37$Fhr$\"$.%F37$F]s$\"$s%F37$Fbs$\"$4'F37$Fgs$Fe_ lF37$F\\t$\"%#f\"F37$Fat$\"%BHF37$Fft$\"%\"*QF37$F[u$\"%FYF37$F`u$\"%_ bF37$Feu$\"%jgF37$Fju$\"%eiF37$F_v$\"%pjF37$Fdv$\"%fjF37$Fiv$\"%TjF37$ F^w$\"%^iF37$Fcw$\"%rhF37$Fhw$\"%7cF37$F]x$\"%JYF37$Fhgm$\"%(z$F37$Fbx $\"%XIF37$F`hm$\"%BGF37$Fehm$\"%\"f#F37$Fjhm$\"%GDF37$Fgx$\"%;DF37$F\\ y$\"%WJF37$Fay$\"%USF37$Ffy$\"%%G%F37$F[z$\"%*e%F37$F`z$\"%dYF37$FezF^ ]n7$Fjz$\"$f%Fgo7$F_[l$\"$F%Fgo7$Fd[l$\"$_$Fgo7$Fi[l$\"$)HFgo7$F^\\l$ \"$d#Fgo7$Fc\\lFf[n7$Fh\\l$\"$8#Fgo7$F\\]l$\"$0#Fgo7$Fa]lFaeo7$Fd]l$\" $7#Fgo7$Fi]l$\"$I#Fgo7$F^^l$\"$b#Fgo7$Fc^l$\"$&HFgo7$Fh^lFddo7$F]_l$\" $-%Fgo7$Fb_l$\"$y%Fgo7$Fg_l$F\\oFgo7$F\\`l$\"$3'Fgo7$Fa`lFdin7$Ff`l$\" $D)Fgo7$F[al$\"$2*Fgo7$F`al$\"%-5Fgo7$Feal$\"%w5Fgo7$Fjal$\"&o=\"F37$F _bl$\"&cG\"F37$Fdbl$\"&YO\"F37$Fibl$\"&1[\"F37$F^cl$\"&7b\"F37$Fccl$\" &!f;F37$Fhcl$\"&\"*o\"F37$F]dl$\"F37$Fbdl$\"&>\\\"F37$Fgdl$\"&#f8F 37$F\\el$\"&gG\"F37$Fael$\"&8@\"F37$Ffel$\"&f4\"F37$F[fl$\"%*)**F37$F` fl$\"%]%*F37$Fefl$\"%+5Fgo7$Fjfl$\"%G6Fgo7$F_gl$\"%Z6Fgo7$Fdgl$\"%t6Fg o7$Figl$\"%l6Fgo7$F^hlFd^m7$Fchl$\"%s6Fgo7$Fhhl$\"%\"=\"Fgo7$F]il$\"%( =\"Fgo7$FbilF]\\p7$Fgil$\"%*>\"Fgo7$F\\jl$\"%,7Fgo7$Fajl$\"%#>\"Fgo7$F fjlFg\\p7$F[[m$\"%/7Fgo7$F`[m$\"%%>\"Fgo7$Fe[m$\"%)=\"Fgo7$Fj[m$\"%.7F go7$F_\\m$\"%!>\"Fgo7$Fd\\m$\"%x6Fgo7$Fi\\m$\"%/6Fgo7$F^]m$\"$***Fgo7$ Fc]m$\"$-*Fgo7$Fh]m$\"$C)Fgo7$F]^m$\"$Y(Fgo7$Fb^m$\"$w'Fgo7$F\\_m$\"$% eFgo7$Fa_mFdjm-Ff_m6&Fh_m$\"\"*F[`m$\"\"%F[`mF(-F_`m6#%8scheme~with~c[ 10]=19/20G-F$6%7fsF'7$Ff`m$!$1#F-7$F+$!$#HF-7$F^am$!##)F-7$F1$\"#CF37$ F<$\"$>\"F37$FF$\"$J#F37$FK$\"$3$F37$FPF_dm7$FU$\"$T$F37$FZ$F^[nF37$F^ o$F\\foF37$FcoFbcm7$FioFh]o7$F^p$\"#;Fgo7$Fcp$\"#:Fgo7$FhpFabp7$F]qF^b p7$F`qFh]o7$FeqFb]o7$FhqF^]o7$F]rFjq7$F`r$\"#RFgo7$Fcr$\"$Z%F37$Fhr$Fh jmF37$F]s$\"$.&F37$Fbs$FbjmF37$Fgs$\"$F'F37$F\\t$\"%M5F37$Fat$\"%H=F37 $Fft$\"%lCF37$F[u$\"%mHF37$F`u$\"%#f$F37$Feu$\"% Fgo$\"%SRF37$$\"5+]i:NgX3#)>Fgo$\"%*)RF37$$\"5nT5Sp%\\aa)>Fgo$\"%+SF37 $Fju$\"%3SF37$$\"5n;a8s(Hkb*>Fgo$\"%GSF37$F_v$\"%8SF37$Fiv$\"%HRF37$Fc w$\"%*p$F37$Fhw$\"%)4$F37$F]x$\"%%=#F37$Fhgm$\"%0:F37$Fbx$\"%,5F37$$\" 5,]iSmb^tP@Fgo$\"$$**F37$$\"5,+D\"y+8W49#Fgo$\"$3*F37$$\"5,](=#\\/J:W@ Fgo$Fd^pF37$F`hm$\"$,*F37$$\"5,+vVtF+y`@Fgo$\"$d)F37$Fehm$\"$l)F37$Fjh m$\"$9*F37$Fgx$\"%05F37$F\\y$\"%5>F37$Fay$\"%cEF37$Ffy$\"%'y#F37$F[z$ \"%,HF37$F`z$\"%vGF37$Fez$\"$$GFgo7$Fjz$\"$f#Fgo7$F_[l$\"$K#Fgo7$Fd[l$ \"$(=Fgo7$Fi[l$\"$f\"Fgo7$F^\\l$\"$R\"Fgo7$Fc\\l$\"$D\"Fgo7$Fh\\l$\"$; \"Fgo7$F\\]l$\"$8\"Fgo7$Fa]lF^\\q7$Fd]l$\"$<\"Fgo7$Fi]l$\"$G\"Fgo7$F^^ l$\"$X\"Fgo7$Fc^l$\"$s\"Fgo7$Fh^l$\"$4#Fgo7$F]_l$\"$N#Fgo7$Fb_l$\"$o#F go7$Fg_l$\"$!HFgo7$F\\`l$\"$E$Fgo7$Fa`l$\"$y$Fgo7$Ff`l$\"$P%Fgo7$F[al$ \"$'[Fgo7$F`al$\"$X&Fgo7$Feal$\"$#fFgo7$Fjal$\"%tmF37$F_bl$\"%gtF37$Fd bl$\"%;zF37$Fibl$\"%y()F37$F^cl$\"%(G*F37$$\"5omT&Q.[Mi2%Fgo$\"%(o*F37 $Fccl$\"&R+\"F37$$\"5o;a)3C!yn&4%Fgo$\"&C,\"F37$$\"5,+Dcw4*e@5%Fgo$\"& m,\"F37$$\"5oT5SWj%*R0TFgo$\"&$=5F37$$\"5M$eRAr,S'3TFgo$\"&p,\"F37$$\" 5,D\"y+3d!)=6%Fgo$\"&,,\"F37$Fhcl$\"&:,\"F37$$\"5NL3F>RL3GTFgo$\"%q)*F 37$F]dl$\"%x\"*F37$$\"5om\"z>'ox+aTFgo$\"%\\')F37$Fbdl$\"%7!)F37$$\"5N L3Fp^yftTFgo$\"%1uF37$Fgdl$\"%inF37$$\"5NLe9T)e`o=%Fgo$\"%=nF37$F\\el$ \"%>hF37$Fael$\"%'\\&F37$Ffel$\"%)e%F37$F]an$\"%VUF37$F[fl$\"%7SF37$Fe an$\"%URF37$Fjan$\"%?RF37$F_bn$\"%2RF37$Fdbn$\"%BRF37$Fibn$\"%:RF37$F^ cn$\"%)*QF37$FccnFcep7$F`flFhep7$$\"5om;a8AyI*H%Fgo$\"%SXF37$FeflFa]n7 $$\"5omTNrRduQVFgo$\"$j&Fgo7$$\"5NL$eRAr\"*=N%Fgo$F\\emFgo7$$\"5-+Dcw% oP]O%Fgo$\"$B'Fgo7$Fjfl$\"$\\'Fgo7$FdglFg]n7$Fhhl$\"$%oFgo7$F]il$\"$&o Fgo7$Fbil$\"$#oFgo7$Ffjl$\"$\"oFgo7$Fj[m$\"$!oFgo7$F_\\m$\"$n'Fgo7$Fd \\m$\"$b'Fgo7$Fi\\m$\"$!fFgo7$F^]m$\"$C&Fgo7$Fc]m$\"$n%Fgo7$Fh]m$\"$G% Fgo7$F]^m$\"$)QFgo7$Fb^mFddo7$F\\_m$\"$1$Fgo7$Fa_m$\"$u#Fgo-%'COLOURG6 &Fh_m$\"*++++\"!\")F(Fdjq-F_`m6#%8scheme~with~c[10]=32/33G-F$6%7^tF'7$ F1$Ff\\qF37$F<$\"$P#F37$FF$\"$J$F37$$\"5++]7G))>Wr@F3$\"$Y$F37$$\"5MLL ek.pu/BF3$\"$m$F37$$\"5nm;/,>=0QCF3$\"$q$F37$FK$\"$o$F37$$\"5ML$eR(\\; m/FF3$Fi\\nF37$$\"5nmmT5ll'z$GF3Fb\\r7$$\"5++](o/[r7(HF3$\"$j$F37$FPFj r7$FU$\"$=$F37$FZ$Fa[nF37$F^o$\"$+#F37$FcoF^bp7$Fio$\"#9Fgo7$F^pF^^r7$ Fcp$\"#8Fgo7$FhpFb^r7$F]qFb^r7$F`qFabp7$$\"5+++++v\"=YI\"Fgo$FgamFgo7$ FeqFh]o7$$\"5++++]7!Q4T\"FgoFbcm7$FhqFhcm7$F]r$\"#NFgo7$Fcr$\"$1&F37$F ]s$F^jnF37$Fgs$\"$.'F37$F\\t$\"$X(F37$Fat$\"%&4\"F37$Fft$F\\fnF37$F[u$ \"%W;F37$F`u$\"%D>F37$Feu$\"%Y?F37$F_v$\"%`>F37$Fcw$\"%+;F37$$\"5,+Dc^ yLuU?Fgo$\"%18F37$Fhw$\"%s5F37$$\"5ML3_D`Gqp?Fgo$\"$V'F37$F]x$\"$d$F37 $Fhgm$!$=\"F37$Fbx$!$6%F37$Fhgp$!$3%F37$F]hp$!$T%F37$Fbhp$!$V%F37$F`hm $!$N%F37$$\"5,]7.K`5d]@Fgo$!$E%F37$Fihp$!$I%F37$$\"5,]P%[@+*)p:#Fgo$!$ ?%F37$Fehm$!$!QF37$Fjhm$Fc`pF37$FgxFc[o7$$\"5om;a8(fbE@#Fgo$FjjpF37$F \\y$FghqF37$$\"5++]ilZsAmAFgo$F\\inF37$Fay$\"%?7F37$Ffy$\"%%H\"F37$F[z $\"%I8F37$F`z$\"%&H\"F37$Fez$\"$B\"Fgo7$Fjz$\"$2\"Fgo7$F_[l$\"##*Fgo7$ $\"5MLL3_D1l_DFgo$\"#\")Fgo7$Fd[l$\"#tFgo7$$\"5ommm;z+vbEFgo$\"#oFgo7$ Fi[l$\"#lFgo7$F^\\l$\"#eFgo7$Fc\\l$\"#aFgo7$Fh\\l$\"#]Fgo7$F\\]l$\"#\\ Fgo7$Fa]lF`hr7$$\"5MLL$3_NJOG$FgoF]hr7$Fd]l$\"#^Fgo7$$\"5MLLLeR%p\")Q$ FgoFjgr7$Fi]l$\"#dFgo7$$\"5omm\"H#oZ1\"\\$Fgo$\"#hFgo7$F^^lFagr7$Fc^l$ \"#')Fgo7$Fh^l$\"$7\"Fgo7$F]_lFe\\q7$Fb_l$\"$U\"Fgo7$F\\`l$\"$j\"Fgo7$ Ff`l$\"$3#Fgo7$F[alF\\[q7$F`al$\"$h#Fgo7$Feal$\"$&GFgo7$Fjal$\"%iKF37$ F_bl$\"%ROF37$Fdbl$\"%WRF37$$\"5,]i:g\"R%3VSFgo$\"%7UF37$Fibl$\"%LWF37 $$\"5o;aQGuHacSFgo$\"%*\\%F37$F^cl$\"%0ZF37$F[`q$\"%7\\F37$Fccl$\"%g]F 37$$\"5,vo/t[sV#4%Fgo$\"%s]F37$Fc`q$\"%z]F37$$\"5MeRs3c$=*)4%Fgo$\"%w] F37$Fh`q$\"%V]F37$Fbaq$\"%;]F37$Fhcl$\"%F\\F37$$\"5-]Pf$=B-;7%Fgo$\"%F ZF37$F_bq$\"%jYF37$$\"5o;z%\\lWkX8%Fgo$\"%6WF37$F]dl$\"%SSF37$$\"5N$3- j7mEv9%Fgo$\"%\"*RF37$Fgbq$\"%\"f$F37$$\"5-]il(f())[gTFgo$\"%1JF37$Fbd l$\"%rIF37$$\"5N$3-8v\"RGqTFgo$\"%*e#F37$F_cq$\"%'e#F37$$\"5N$eRsey6p< %Fgo$\"%kDF37$Fgdl$FjilF37$Fgcq$\"%d?F37$F\\el$\"%,;F37$$\"5NL3-8D$4,? %Fgo$\"%r6F37$Fael$\"%^6F37$$\"5NLe*[=1lL@%Fgo$\"$+)F37$Ffel$Fb]nF37$F ]anFer7$F[fl$\"$;#F37$Fdbn$F[eoF37$F`fl$\"$/%F37$$\"5-+v$4'\\=;'G%Fgo$ \"$g&F37$F_fq$\"$!)*F37$$\"5NLe9m%z`CJ%Fgo$\"$P\"Fgo7$Fefl$\"$e\"Fgo7$ Fefq$\"$'>Fgo7$Fjfq$\"$M#Fgo7$F^gq$\"$W#Fgo7$Fjfl$\"$l#Fgo7$FdglFfjp7$ Fhhl$\"$)GFgo7$F]ilFg]q7$FbilF\\es7$Fj[mFfjp7$Fd\\m$\"$p#Fgo7$Fi\\m$F_ apFgo7$F^]m$\"$.#Fgo7$Fc]m$\"$x\"Fgo7$Fh]m$\"$n\"Fgo7$Fb^mFe[q7$F\\_mF ^fr7$Fa_mF^\\q-Ff_m6&Fh_m$Fb_mF[`mF(F\\`m-F_`m6#%8scheme~with~c[10]=81 /82G-F$6%7ftF'7$F1$\"$1\"F37$F<$\"$$=F37$FF$\"$X#F37$Ff[r$\"$a#F37$F[ \\rFR7$F`\\r$\"$n#F37$FK$\"$k#F37$Fh\\r$FidsF37$F\\]r$\"$j#F37$F_]r$\" $c#F37$FPFjap7$FU$\"$A#F37$FZ$\"$%>F37$F^o$Fi\\qF37$Fco$\"#7Fgo7$Fio$ \"#6Fgo7$F^pFcis7$Fcp$\"#5Fgo7$FhpFgis7$F]qFgis7$F`qF`is7$Fh^rFb^r7$Fe qF^^r7$F]_rFj^r7$FhqFb]o7$F]rFb^o7$Fcr$\"$j%F37$Fgs$\"$'fF37$F\\t$\"$+ (F37$Fat$\"$*))F37$F[u$\"%T6F37$Feu$\"%X7F37$F_v$\"%#3\"F37$Fcw$\"$s(F 37$Fcar$\"$T&F37$Fhw$\"$l$F37$F[br$FhhrF37$F]x$!$^\"F37$Fhgm$!$$[F37$F bx$F\\amF37$Fhgp$!$x'F37$F]hp$!$-(F37$Fbhp$!$.(F37$F`hm$!$&pF37$Fecr$! $*oF37$Fihp$!$#pF37$F]dr$!$#oF37$Fehm$!$f'F37$Fjhm$!$(fF37$Fgx$!$8&F37 $Fhdr$!$C#F37$F\\y$\"#`F37$F^erFegs7$Fay$\"$L%F37$Ffy$\"$$\\F37$F[z$\" $K&F37$$\"5nmTN'fW2XN#Fgo$\"$M&F37$F`zF[o7$$\"5n;/w(yvbHP#Fgo$F`iqF37$ $\"5nm\"H#=&>0\"zBFgo$F\\_tFgo7$$\"5nTNY$Q\"*z@Q#Fgo$\"#_Fgo7$$\"5n;zp [KYD&Q#FgoFghr7$$\"5n\"HKR6NH$)Q#FgoFghr7$FezFghr7$Fjz$\"#XFgo7$F_[l$ \"#SFgo7$FgfrF[p7$Fd[lFa_r7$Fi[lF\\dm7$F^\\lFb^o7$Fc\\lFep7$Fh\\lF^]o7 $F\\]lF\\cm7$Fa]lF^]o7$Fd]lFjp7$FjhrFb^o7$Fi]lFbq7$F`irFjq7$F^^lFjbp7$ Fc^lFh`t7$Fh^l$\"#uFgo7$F]_l$\"#\"*Fgo7$Fb_lFafr7$F\\`l$\"$E\"Fgo7$Ff` l$\"$d\"Fgo7$F[alF[]q7$F`al$\"$*=Fgo7$FealFges7$Fjal$\"%jAF37$F_bl$\"% mCF37$Fdbl$\"%FEF37$Fibl$\"%uGF37$F^cl$\"%(*HF37$F[`q$\"%$3$F37$FcclF \\`s7$Fh`q$\"%:IF37$Fhcl$\"%&)GF37$Fb^s$\"%4FF37$F_bq$\"%]EF37$Fj^s$\" %XCF37$F]dl$\"%f@F37$Fb_s$\"%?@F37$Fgbq$\"%==F37$$\"5N3x\")HA$[s:%Fgo$ \"%'y\"F37$Fj_s$\"%g9F37$$\"5o\"z%\\lH%HP;%Fgo$\"%e9F37$Fbdl$\"%M9F37$ Fb`sFd[t7$F_cq$\"%\"3\"F37$Fj`s$\"%n5F37$Fgdl$\"$>(F37$Fgcq$\"$-(F37$F \\elFhbm7$$\"5N$3_]4R&z'>%FgoFe^o7$Fgas$\"#rF37$$\"5N$e*)4$fKU.UFgo$\" #qF37$Fael$\"#fF37$F_bs$!$#>F37$Ffel$!$$RF37$F]an$!$O&F37$F[fl$!$3'F37 $Fean$!$F'F37$Fjan$!$E'F37$F_bn$!$@'F37$Fdbn$!$'fF37$Fibn$!$(eF37$F^cn $!$'eF37$Fccn$!$A&F37$F`fl$!$.&F37$F_cs$!$7%F37$F_fq$!$E\"F37$FgcsFb^r 7$FeflFhcm7$FefqFd`t7$Fjfq$\"#yFgo7$F^gq$\"#%)Fgo7$Fjfl$\"$,\"Fgo7$Fhh l$\"$=\"Fgo7$Fbil$\"$A\"Fgo7$FfjlF^fr7$Fj[m$\"$C\"Fgo7$$\"5,](o/)\\Aqc WFgoFh\\u7$F_\\m$\"$?\"Fgo7$$\"5o;H#oK.#yoWFgoFb]u7$Fd\\m$F\\apFgo7$Fi \\m$\"$0\"Fgo7$F^]m$\"#'*Fgo7$Fh]mF_\\u7$F]^m$\"#zFgo7$Fb^m$FehtFgo7$F g^mFagr7$F\\_mFdgr7$Fa_m$\"#gFgo-Fbjq6&Fh_mFdjqF(F(-F_`m6#%:scheme~wit h~c[10]=169/170G-%%FONTG6$%*HELVETICAGFe_p-%&TITLEG6#%Verror~curves~fo r~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F[`u-% %VIEWG6$;F(Fa_m;$!\"*!#=$FgjpFd`u" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33 " "scheme with c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "eval f[20](plot(['un_RK7_1'(x)-u(x),'un_RK7_2'(x)-u(x),'un_RK7_3'(x)-u(x), \n'un_RK7_4'(x)-u(x),'un_RK7_5'(x)-u(x),'un_RK7_6'(x)-u(x)],x=0..1,-4. 5e-18..5.3e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB ,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[` Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`schem e with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82 `,`scheme with c[10]=169/170`],\ntitle=`error curves for 11 stage orde r 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 778 570 570 {PLOTDATA 2 "6,-%'CURVESG6%7hx7$$\"\"!F)F(7$$\"5LLLLL3x&)*3\"!#@$!$!fF -7$$\"5mmmmm;arz@F-$!%B5F-7$$\"5++++D\"y%*z7$F-$!%t:F-7$$\"5LLLL$e9ui2 %F-$!%j?F-7$$\"5++++voMrU^F-$!%aDF-7$$\"5mmmmm\"z_\"4iF-$!%KIF-7$$\"5l mmmmm6m#G(F-$!%RMF-7$$\"5lmmmmT&phN)F-$!$x$!#?7$$\"5++++v=ddC%*F-$!$,% FS7$$\"5LLLLe*=)H\\5FS$!$A%FS7$$\"5mmm\"Hd0!oh5FS$!$E%FS7$$\"5+++](=#> 1u5FS$!$D%FS7$$\"5LLL3-)yVk3\"FS$!$M%FS7$$\"5mmmm;ac#))4\"FS$!$T%FS7$$ \"5LL$eRse;]5\"FS$!$Y%FS7$$\"5+++DJ?v?66FS$!$Q%FS7$$\"5mm;aQ`%)R<6FS$! $P%FS7$$\"5LLL$ekQ*eB6FS$!$V%FS7$$\"5++]7`>.yH6FS$!$O%FS7$$\"5mmmTg_7( f8\"FSFip7$$\"5LL$3xc=i@9\"FS$!$U%FS7$$\"5++++v=JN[6FSFip7$$\"5LLLe*[) \\tg6FSFdp7$$\"5mmm;/^o6t6FS$!$R%FS7$$\"5LL$e9Ty2$z6FSF[r7$$\"5+++v=<( )\\&=\"FSFdp7$$\"5mm;/E]'*o\">\"FS$!$X%FS7$$\"5LLLLL$e!)y>\"FSFas7$$\" 5+++]i:VkA7FS$!$W%FS7$$\"5mmmm\"z/3uC\"FSFjo7$$\"5LLLLe*ot*\\8FSFip7$$ \"5++++DJ$RDX\"FS$!$H%FS7$$\"5LLLLekGhe:FS$!$9%FS7$$\"5mmmm\"zR'ok;FS$ !$!RFS7$$\"5LLLL3_(>/x\"FS$!$e$FS7$$\"5++++D1J:w=FS$!$B$FS7$$\"5mmm;Hd G\"\\)>FS$!$x#FS7$$\"5LLLLL3En$4#FS$!$<#FS7$$\"5++++Dc#o%*=#FS$!$%=FS7 $$\"5mmmm;/RE&G#FS$!$R\"FS7$$\"5LLLLe9r5$R#FS$!#vFS7$$\"5+++++D.&4]#FS $!\"*FS7$$\"5+++++]jB4EFS$\"#YFS7$$\"5+++++vB_T%FS$\"$B&FS7$$\"5KLL3F%>6`U%FS$\" $C&FS7$$\"5****\\7`>q4_WFS$\"$?&FS7$$\"5mmm;zWG))yWFSF\\\\l7$$\"5LL$3_ +noc]%FS$\"$F&FS7$$\"5+++DJ&\\aC`%FS$\"$D&FS7$$\"5mm;Hd?.CfXFSFc]l7$$ \"5LLLL$e9Ege%FSF\\\\l7$$\"5LLL$3F9&FS7$$\"5mmmm\"zRQb@&FS$\"$8&FS7$$\"5*******\\(=>Y2aFS$ \"$,&FS7$$\"5mmmm;zXu9cFS$\"$#\\FS7$$\"5**********\\y))GeFS$\"$u%FS7$$ \"5********\\i_QQgFS$F\\x!#>7$$\"5+++](=-N(RhFS$\"#XF\\bl7$$\"5******* \\7y%3TiFS$\"#WF\\bl7$$\"5+]7y+3!=\"[iFSF`bl7$$\"5++DcwM7:biFSF`bl7$$ \"5+]PM_hW=iiFSFebl7$$\"5++]7G)o<#piFSFebl7$$\"5+]i!R]\"4DwiFSF`bl7$$ \"5++vozTTG$G'FSFebl7$$\"5+](oa&otJ!H'FSFebl7$$\"5+++DJ&f]tH'FS$\"#VF \\bl7$$\"5++]PM-N[DjFSF_dl7$$\"5+++]P4kh`jFSF_dl7$$\"5+++vVBA))4kFSF_d l7$$\"5********\\P![hY'FSF_dl7$$\"5lm\"HdXqnCZ'FSF_dl7$$\"5IL$e9;P(yyk FS$\"#UF\\bl7$$\"5++v=nQq5&['FSF_dl7$$\"5lmm\"HdqE9\\'FSFcel7$$\"5ILek ysju(\\'FSF_dl7$$\"5++]P%)Rg1/lFSF_dl7$$\"5lmT5!pq&Q5lFSFcel7$$\"5ILL$ eRP0n^'FSFcel7$$\"5++Dc,T]-BlFSFcel7$$\"5lm;H23ZMHlFSFcel7$$\"5IL3-8vV mNlFSFcel7$$\"5+++v=US)>a'FSF_dl7$$\"5lm\"zW#4PI[lFSFcel7$$\"5IL$3-jPB Yb'FSF_dl7$$\"5++v$fL/V4c'FSF_dl7$$\"5lmmmT5FEnlFSFcel7$$\"5+++](o/?yh 'FS$\"#TF\\bl7$$\"5KLLLL$Qx$omFS$\"#SF\\bl7$$\"5++++DJ\"eBs'FSFghl7$$ \"5lmmm;z)Qjx'FSFghl7$$\"5+++]7`#HL!oFSFghl7$$\"5ILLL3F'>.$oFSFghl7$$ \"5+++D19[\"Q%oFSFghl7$$\"5lmm;/,+JdoFSFghl7$$\"5++]7`%fdS'oFSFghl7$$ \"5ILL3-)=03(oFSFghl7$$\"5lm;/^\"y_v(oFSFghl7$$\"5*********\\P+V)oFS$ \"#RF\\bl7$$\"5mmmm\"zpe*zqFS$\"#QF\\bl7$$\"5*********\\#\\'QH(FS$\"#P F\\bl7$$\"5KLLLe9S8&\\(FS$\"#OF\\bl7$$\"5*******\\i?=bq(FS$\"#MF\\bl7$ $\"5ILL3xc/%pv(FSF[\\m7$$\"5lmm;H2FO3yFSF[\\m7$$\"5IL$3_D$Q2MyFSF[\\m7 $$\"5+++D\"y&\\yfyFSF[\\m7$$\"5l;/w7RF@myFSF[\\m7$$\"5IL3FW?0ksyFS$\"# NF\\bl7$$\"5+]7yv,$o!zyFSF_]m7$$\"5lm;H2$3'\\&)yFSF[\\m7$$\"5I$3-)QkQ# >*yFSF[\\m7$$\"5++DJqX;N)*yFS$\"#LF\\bl7$$\"5l;H#=qUzZ!zFSF[\\m7$$\"5K LLLL3s?6zFSF]^m7$$\"5lmm;zpe()=!)FSF]^m7$$\"5*******\\7`Wl7)FSF]^m7$$ \"5I$3x1'Qc-L\")FS$\"#KF\\bl7$$\"5lmTN'fu1&R\")FSF^_m7$$\"5+]7.K`y)f9) FSF]^m7$$\"5IL$3x1'*oC:)FSF^_m7$$\"5l;aQ.o+&*e\")FSF^_m7$$\"5++D1Rv6Vl \")FSF]^m7$$\"5I$eRZFG7><)FSF]^m7$$\"5lmmT5!R$Ry\")FSF^_m7$$\"5+]P4Y( \\u[=)FSF]^m7$$\"5IL3x\"[gb8>)FSF]^m7$$\"5l;zW<7n$y>)FSF^_m7$$\"5++]7` >yJ/#)FSF^_m7$$\"5I$3-))o#*)z5#)FSF^_m7$$\"5lm\"zWU.!G<#)FSF^_m7$$\"5+ ]i:gT6wB#)FSF^_m7$$\"5ILL$e*[ACI#)FSF]^m7$$\"5l;/^JcLsO#)FSF^_m7$$\"5+ +v=njW?V#)FSF^_m7$$\"5I$ekG5d&o\\#)FSF^_m7$$\"5lm;aQym;c#)FSF^_m7$$\"5 ILe*)4$*)G\"p#)FSF^_m7$$\"5+++D\"y5\"4#G)FSF^_m7$$\"5I$3Fp^@s&)G)FSF]^ m7$$\"5lmTg_AL0&H)FSF]^m7$$\"5+]7G))HW`,$)FSF^_m7$$\"5IL$eRs`:!3$)FSF] ^m7$$\"5l;ajfWm\\9$)FSF^_m7$$\"5++DJ&>vx4K)FSF^_m7$$\"5I$e*)4$f)euK)FS F^_m7$$\"5lmmmmm*RRL)FSF^_m7$$\"5lmm;a8H'pQ)FSF^_m7$$\"5lmmmTge)*R%)FS F^_m7$$\"5lm;a8(fTKX)FSF^_m7$$\"5lmmT&QL(\\m%)FSF^_m7$$\"5lmTN@-_7t%)F S$\"#JF\\bl7$$\"5lm;HdqIvz%)FSFfem7$$\"5lm\"HK*Q4Q'[)FSFfem7$$\"5lmm;H 2)3I\\)FSFfem7$$\"5lmm\"H2G?&>&)FSFfem7$$\"5lmmm;a<.Y&)FSFfem7$$\"5+++ ]PM&*>^')FSFfem7$$\"5KLLLe9tOc()FSFfem7$$\"5lm;/^\")o_!y)FSFfem7$$\"5+ ++vV[ko/))FSFfem7$$\"5I$3Fp,ME2\"))FSFfem7$$\"5lmT5!>Bmn\"))FSFfem7$$ \"5+]7GjBh!G#))FS$\"#IF\\bl7$$\"5IL$ek`,Y)G))FSF\\hm7$$\"5l;aj42f)[$)) FSF\\hm7$$\"5++D\"G))zD4%))FSFfem7$$\"5I$e*)f0plp%))FSFfem7$$\"5lmm;H# e0I&))FSF\\hm7$$\"5ILLe9;ZK,*)FSF\\hm7$$\"5**********\\Qk\\*)FSF\\hm7$ $\"5lmmmT5ASg!*FSF\\hm7$$\"5KLLL$3dg6<*FSF\\hm7$$\"5++DJqX?Nx\"*FSF\\h m7$$\"5lm;Hd?Na$=*FS$\"#HF\\bl7$$\"5IL3FW&*\\t*=*FSF_jm7$$\"5+++DJqk#f >*FSF_jm7$$\"5IL$3_+U4$3#*FSF_jm7$$\"5lmm;zpBp?#*FSF_jm7$$\"5ILe9mWQ)o A*FSF\\hm7$$\"5++]7`>`2L#*FSF\\hm7$$\"5lmT5S%zm#R#*FSF\\hm7$$\"5ILL3Fp #eaC*FSF\\hm7$$\"5lm;/,>7%yD*FSF\\hm7$$\"5++++voTAq#*FSF\\hm7$$\"5IL$e *[=rg#G*FSF\\hm7$$\"5lmm\"H#o+*\\H*FSF\\hm7$$\"5ILe*)4V:=,$*FSF\\hm7$$ \"5++](oz,ttI*FSF_jm7$$\"5lmT&QG\\kNJ*FSF_jm7$$\"5ILL$3x'fv>$*FSF_jm7$ $\"5+++v=n=_W$*FSF_jm7$$\"5lmmmmmxGp$*FSF_jm7$$\"5ILL$eRA5\\Z*FSF_jm7$ $\"5)******\\7oK0e*FSF_jm7$$\"5)******\\(=5s#y*FSF_jm7$$\"\"\"F)F_jm-% &COLORG6&%$RGBGF($\"\"#!\"\"Fa^n-%'LEGENDG6#%ESharp-Verner~scheme~with ~c[10]=11/12G-F$6%7c[lF'7$F1$!$=$F-7$F;$!$!eF-7$F@$!$z'F-7$FE$!$t(F-7$ $\"5lmmT5lqCwiF-$!$U(F-7$$\"5lmm;aQ8MVjF-$!$[(F-7$$\"5lmm\"z>hN/T'F-$! $v(F-7$$\"5lmmmT&))HvZ'F-$!$A)F-7$$\"5lmmT&)eTiWlF-Fe`n7$$\"5lmm;HK%=< h'F-$!$W(F-7$$\"5lmm\"Hdq7)ymF-$!$!zF-7$$\"5lmmm;zp!fu'F-$!$B)F-7$$\"5 lmmTg_7+8oF-$!$&zF-7$$\"5lmm;/Eb4!)oF-$!$$zF-7$$\"5lmm\"z%*z*=ZpF-$!$e (F-7$$\"5lmmm\"H2%G9qF-F[cn7$$\"5lmmTNY$y83(F-$!$)zF-7$$\"5lmm;z>EZ[rF -$!$F)F-7$$\"5lmm\"HK*oc:sF-$!$b)F-7$FJ$!$7)F-7$$\"5lmmT5Sav\\tF-$!$K) F-7$$\"5lmm;a8(\\oT(F-$!$k)F-7$$\"5lmm\"zp)R%R[(F-$!$(zF-7$$\"5lmmmTg# Q5b(F-$!$V)F-7$$\"5lmmT&Q`K\"=wF-$!$(yF-7$$\"5lmm;H2oA&o(F-$!$n(F-7$$ \"5lmm\"H23@Bv(F-$!$H)F-7$$\"5lmmm;a`T>yF-$!$?)F-7$$\"5lmmTgF'4l)yF-$! $y(F-7$$\"5lmm;/,Rg`zF-$!$Y)F-7$$\"5lmm\"zW<)p?!)F-$!$_)F-7$$\"5lmmm\" zW#z(3)F-$!$)yF-7$$\"5lmmTN@n)[:)F-Fbdn7$$\"5lmm;z%*4)>A)F-$!$c)F-7$$ \"5lmm\"H#o_2*G)F-Fahn7$FO$!#!)FS7$FU$!#tFS7$FZ$!#mFS7$F\\t$!#SFS7$Fbt $!\"#FS7$Fgt$\"#=FS7$F\\u$F`dlFS7$Fau$\"#rFS7$Ffu$\"#**FS7$F[v$\"$J\"F S7$F`v$\"$r\"FS7$Fev$\"$)=FS7$Fjv$\"$9#FS7$F_w$\"$\\#FS7$Fdw$\"$\"GFS7 $Fiw$\"$3$FS7$F^x$\"$B$FS7$Fhx$\"$i$FS7$F]y$\"$*QFS7$Fby$\"$1%FS7$$\"5 LLLL$3xxlV$FS$\"$;%FS7$Fgy$\"$B%FS7$$\"5mmm;HdO2VOFS$\"$D%FS7$F\\z$\"$ G%FS7$$\"5mm;/,pZoyPFSFg]o7$$\"5KLL3xcoD.QFS$\"$I%FS7$$\"5****\\7`W*Gy #QFSFg]o7$$\"5mmm;HK5S_QFS$\"$F%FS7$$\"5L$3xJU0W&eQFSFg^o7$$\"5++v=6)4I3(QFSFg]o7$$\"5LL$3_+7tp(QFSFg]o7$$\"5+](=# *>9;J)QFS$\"$H%FS7$$\"5mm\"HKR;f#*)QFSF__o7$$\"5L$eRse=-a*QFSF__o7$$\" 5+++D\"y?X:!RFS$\"$K%FS7$$\"5m;/EvH#)o2RFS$\"$M%FS7$$\"5LL3Fp^7$Q\"RFS $\"$L%FS7$$\"5+]7GjtU(*>RFSFe`o7$$\"5mm;Hd&Hq+sSFS$\"$E%FS7$$\"5****\\7.dn[&3%FSFd]o7$$ \"5K$e9;diEA4%FSFbdo7$$\"5mmT5S%\\m*)4%FSF_^o7$$\"5**\\Pf3jjq0TFSFe`o7 $$\"5KLL3xJiW7TFSFj_o7$$\"5***\\iS\"pf#f7%FSF__o7$$\"5mm;/^1dSRTFSF_^o 7$$\"5**\\7`>vb9YTFSFg]o7$$\"5KL3-)QW&)G:%FSFg^o7$$\"5m;/^c7`ifTFSFd]o 7$FfzFg^o7$$\"5********\\iB0pUFS$\"$@%FS7$F[[l$\"$<%FS7$Fi]l$\"$6%FS7$ F_^l$\"$-%FS7$F]`l$\"$\"RFS7$Fe`l$\"$u$FS7$Fj`l$\"$g$FS7$F_al$\"$^$FS7 $Fdal$\"$N$FS7$FialF^_m7$FcblFfem7$FhblF^_m7$F[clF^_m7$F^clFfem7$FaclF fem7$FdclF^_m7$FgclFfem7$FjclFfem7$F]dlF\\hm7$FbdlF\\hm7$FedlF\\hm7$Fh dlF\\hm7$F[elF\\hm7$F]hlF_jm7$Fehl$\"#GF\\bl7$FjhlF`io7$F]ilF`io7$$\"5 ++]ilsk3$y'FSF`io7$$\"5ILLe9mS$)*y'FSF_jm7$$\"5lm;ajf;e'z'FSF`io7$F`il F_jm7$$\"5IL$e9m%o25oFSF`io7$$\"5lmmT5SW#o\"oFSF`io7$$\"5++]PfL?dBoFSF `io7$FcilF`io7$FfilF`io7$FiilF`io7$F\\jlF`io7$F_jlF`io7$FbjlF`io7$Fejl $\"#FF\\bl7$$\"5ILL$ek`H@)pFSF^[p7$FjjlF^[p7$$\"5NL$3-j(pp1rFSF^[p7$$ \"5+++voa_VLrFSF^[p7$$\"5l;aQGC)>,9(FSF^[p7$$\"5IL3-)QR/o9(FSF^[p7$$\" 5+]ilZj*)[`rFSF^[p7$$\"5lm;H2LN'=CF-=(FSF^[p7$$\"5ILL$e9\"=\"p=(FSFf \\p7$$\"5lmm\"H#o$)QSsFSFf\\p7$F_[mFf\\p7$Fd[m$\"#DF\\bl7$$\"5+++DcEIG 3vFSFi]p7$$\"5lmm;aQ?V@vFSFi]p7$$\"5++]7`Wl+GvFSFi]p7$$\"5ILL3_]5eMvFS Ff\\p7$$\"5lm;/^cb:TvFSFi]p7$$\"5++++]i+tZvFSFf\\p7$$\"5IL$e*[oXIavFSF i]p7$$\"5lmm\"zW2z3c(FSFf\\p7$$\"5++](o/e`uc(FSFi]p7$$\"5ILL$ek3GSd(FS Fi]p7$$\"5+++vV)4xre(FSFi]p7$$\"5lmmmT5hK+wFSFi]p7$$\"5+++]PMTiEwFSFi] p7$$\"5ILLLLe@#Hl(FSFi]p7$$\"5+++DJq62mwFSFi]p7$$\"5lmm;H#=?#zwFSFi]p7 $$\"5++]7G)o%z&o(FSFi]p7$$\"5ILL3F%>pBp(FSFi]p7$$\"5lm;/E+P%*)p(FSFi]p 7$Fi[m$\"#CF\\bl7$Fa\\mFeap7$Fc^mFeap7$Ff^mFeap7$Fi^mFeap7$F\\_m$\"#BF \\bl7$Fa_mF\\bp7$Fd_mF\\bp7$Fg_mF\\bp7$F]`mF\\bp7$Fc`mF\\bp7$F_amF\\bp 7$F[bmF\\bp7$F]cmF\\bp7$FedmF\\bp7$F[emF\\bp7$FefmF\\bp7$$\"5IL3FpEx6[liW')FSFbcp7$FhfmF\\bp7$$\"5I$3-Q1_ sxl)FSF\\bp7$$\"5lmT5!p]XVm)FSF\\bp7$$\"5+]iS;$\\=4n)FSFbcp7$$\"5IL$3F %z9\\x')FSFbcp7$$\"5++DJ&>XP1p)FSFbcp7$$\"5lmm\"zWU$y.()FSFbcp7$$\"5++ ]7`p`2I()FSFbcp7$F[gmFbcp7$F[imFbcp7$FaimFbcp7$$\"5lmTg_Zicc*)FSFbcp7$ $\"5IL$3_]k)[j*)FSFbcp7$$\"5++D\"yD/6/(*)FSFbcp7$$\"5lmmT5SMLx*)FS$\"# @F\\bl7$$\"5IL3-jPeD%)*)FSFbcp7$$\"5++]i:N#y6**)FSFbcp7$$\"5lm\"H#oK15 )**)FSFbcp7$$\"5ILL$3-.B]+*FSFbcp7$$\"5lm;/EDy')=!*FSFbcp7$$\"5+++DJ?E rK!*FSFbcp7$$\"5lmT&Qy,N'R!*FSFbcp7$$\"5IL$ek`Tdl/*FSFagp7$$\"5++D1*G \")zM0*FSFbcp7$FdimFagp7$$\"5IL3F%zgCt1*FSFbcp7$$\"5++](oa+ZU2*FSFbcp7 $$\"5lm\"z%*HSp63*FSFbcp7$$\"5ILL3_+=4)3*FSFbcp7$$\"5lm;Hd&fO>5*FSFbcp 7$$\"5+++]i!R\"y:\"*FSFbcp7$$\"5lmT5:)y.F7*FSFbcp7$$\"5IL$3xc=E'H\"*FS Fagp7$$\"5++DJ?$e[l8*FSFbcp7$$\"5lmm\"H2)4ZV\"*FSFbcp7$$\"5IL3_DyLR]\" *FSFbcp7$$\"5++]7yvdJd\"*FSFbcp7$$\"5lm\"H2LFS7$FZ$!$(>FS 7$$\"5++]ilA\"*[b5FS$!$0#FS7$Fin$!$,#FS7$$\"5LL$3-)))4(y1\"FSFj^q7$F^o $!$*>FS7$$\"5mm;z%\\&GD!3\"FS$!$-#FS7$Fco$!$1#FS7$$\"5++]P4@Zj#4\"FS$! $/#FS7$FhoF__q7$F]p$!$4#FS7$FbpFb_q7$Fgp$!$+#FS7$F\\qF``q7$FaqF]aq7$Ff qF]aq7$FiqF``q7$F^rFb_q7$Fdr$!$.#FS7$FdsF]`q7$$\"5++]iS;:2/7FSF__q7$$ \"5mmm\"z%\\CE57FS$!$2#FS7$$\"5LL$3_DQ`k@\"FSFb_q7$FgsFb_q7$$\"5LLL3x \"=E]B\"FSFh_q7$F\\t$!$)>FS7$F_t$!$#>FS7$Fbt$!$$=FS7$F\\u$!$c\"FS7$Fau $!$O\"FS7$Ffu$!$:\"FS7$F[v$!#()FS7$F`v$!#^FS7$Fev$!#NFS7$FjvFfw7$F_w$F _[pFS7$Fdw$\"#hFS7$Fiw$\"#$*FS7$F^x$\"$8\"FS7$Fcx$\"$U\"FS7$Fhx$\"$l\" FS7$$\"5LLL$eky#*4-$FS$\"$#=FS7$F]y$\"$0#FS7$Fby$\"$Q#FS7$Fgy$\"$p#FS7 $F\\z$\"$!HFS7$Fe^o$\"$'HFS7$FazF[\\o7$F]do$\"$6$FS7$Ffz$\"$;$FS7$Fdfo $\"$<$FS7$F[[l$\"$>$FS7$F`[lF^\\o7$Fe[l$\"$H$FS7$$\"5K$3_D`#=A0WFS$\"$ G$FS7$Fj[l$\"$D$FS7$$\"5m;Hd&zt9'=WFSFggq7$F_\\l$\"$F$FS7$$\"5**\\Pfe] w+KWFSF^\\o7$$\"5mmT5!p5/(QWFSF\\hq7$$\"5K$e9;Kc+aW%FS$\"$C$FS7$Fd\\lF `iq7$$\"5m;aj%eZ$zeWFSF^\\o7$$\"5KLe9;K**[lWFSF`iq7$$\"5**\\ilZ)Q'=sWF S$\"$A$FS7$Fi\\lFehq7$Fa]lF_hq7$Fi]lF`iq7$F\\^lF_hq7$F_^lF_hq7$Fb^l$\" $E$FS7$Fg^lFdjq7$F\\_lF`iq7$F__lFdjq7$Fb_lF^\\o7$Fe_l$\"$@$FS7$Fh_l$\" $?$FS7$F]`lF\\jq7$F``lF`gq7$Fe`l$\"$8$FS7$$\"5NL3F%z+O:A&FSF]gq7$$\"5+ +](ozhLvA&FSF`gq7$$\"5qm\"z%*zAJNB&FS$\"$:$FS7$$\"5NLL3-Q)G&R_FSF^\\r7 $$\"5++vo/[k_X_FS$\"$7$FS7$$\"5qm;H2eS_^_FSF^\\r7$$\"5NLe*)4o;_d_FSF`g q7$$\"5+++]7y#>NE&FSFf\\r7$$\"5IL$3x\")\\9bF&FSFc[r7$$\"5lmm\"H#=(4vG& FSFjfq7$$\"5IL3_DGt]$H&FSFjfq7$$\"5++]7GQ\\]*H&FSF^\\r7$$\"5lm\"H2$[D] 0`FSFf\\r7$$\"5ILLLLe,]6`FS$\"$9$FS7$$\"5+++vV)f!\\N`FSFjfq7$$\"5lmm;a Q5[f`FS$\"$2$FS7$$\"5IL3xc['yaO&FS$\"$4$FS7$$\"5++]PfeiZr`FSF[_r7$$\"5 lm\"z>'oQZx`FS$\"$0$FS7$$\"5ILLeky9Z$Q&FS$\"$5$FS7$$\"5++v=n)3p%*Q&FSF ]`r7$$\"5lm;zp)pmaR&FSFh_r7$$\"5ILeRs3VY,aFSF]`r7$Fj`lFh_r7$$\"5lmTg-& 3&[&FSF[_r7$$\"5I$3FW&*\\q;\\&FSF\\ar7$$\"5lm\"H#o#3[\") \\&FSFebr7$$\"5+]7.#emDY]&FSFebr7$$\"5ILL$e*[K56bFSFh_r7$$\"5++vVB:%eS _&FSF]br7$$\"5lm;/^\"e8q`&FS$\"$,$FS7$$\"5ILekyZ(o*\\bFS$\"$-$FS7$$\"5 +++D19R#Hc&FSF_dr7$$\"5NL$e9mCM))e&FSF]br7$F_alF_dr7$$\"5ILLLe9i\"=s&F S$\"$&HFS7$Fdal$\"$*GFS7$FialF`io7$FcblF^[p7$F[elFf\\p7$FehlFi]p7$Fejl Feap7$$\"5lmm\"Hd&\\@LpFSFeap7$Fa[pFeap7$$\"5++D1k\"=eV*pFSFeap7$$\"5l m;H#o#oe1qFSFeap7$$\"5+]iST\\6q7qFSFeap7$$\"5IL3_+sa\")=qFSFeap7$$\"5l ;ajf%zH\\-(FSF\\bp7$$\"5+++v=hN->tFSF\\bp7$$\"5ILLeR(>#=WtFS F\\bp7$$\"5+]7`Wc=Z]tFSF\\bp7$$\"5lm\"z%\\::wctFSFbcp7$$\"5I$3FWX<^IO( FSFbcp7$$\"5++]PfL3MptFSF\\bp7$$\"5l;HKk#\\IcP(FSF\\bp7$$\"5IL3Fp^,#>Q (FSF\\bp7$$\"5+](=U2\")4#)Q(FSF\\bp7$$\"5lmm;zp%*\\%R(FSFbcp7$$\"5++D1 *yyyqS(FSFbcp7$$\"5IL$e*)f5e'>uFSFbcp7$$\"5+]i!R]wZfU(FSF\\bp7$$\"5lmT &)3CuBKuFSFbcp7$$\"5I$3-QJ3F&QuFSFbcp7$$\"5+++v=Un\"[W(FSF\\bp7$$\"5l; zpB,k5^uFSF\\bp7$$\"5ILekGggRduFSFbcp7$$\"5+]PfL>dojuFSFbcp7$$\"5lm;aQ y`(*puFSFbcp7$$\"5I$e*[VP]EwuFSF\\bp7$$\"5++vV['paD[(FSF\\bp7$$\"5l;aQ `bV%))[(FSFbcp7$Fd[mFbcp7$F[_pFbcp7$F]`pFbcp7$F``pFbcp7$Fc`pFbcp7$Ff`p Fbcp7$Fi`pFbcp7$F\\apFbcp7$F_apFbcp7$FbapFbcp7$Fi[mFagp7$Fa\\mFagp7$Fc ^mFagp7$Ff^mFagp7$Fi^mFagp7$F\\_m$\"#?F\\bl7$Fa_mF]^s7$Fd_mFagp7$Fg_mF ]^s7$Fj_mF]^s7$F]`mFagp7$F``mF]^s7$Fc`mF]^s7$Ff`mFagp7$Fi`mFagp7$F\\am F]^s7$F_amF]^s7$FbamF]^s7$FeamF]^s7$FhamF]^s7$F[bmFagp7$F^bmF]^s7$Fabm F]^s7$FdbmF]^s7$FgbmF]^s7$FjbmF]^s7$F]cmF]^s7$F`cmFagp7$FccmFagp7$Ffcm F]^s7$FicmFagp7$F\\dmF]^s7$F_dmF]^s7$FbdmF]^s7$FedmF]^s7$F[emF]^s7$Fef mF]^s7$FjbpF]^s7$F]cpF]^s7$F`cpF]^s7$FecpF]^s7$FhcpF]^s7$F[dp$\"#>F\\b l7$$\"5l;aQ`I')=0')FSF]^s7$F^dpFd`s7$$\"5I$e*)fIgM$=')FSF]^s7$FadpF]^s 7$FddpF]^s7$FgdpFd`s7$FjdpF]^s7$FhfmF]^s7$F^epF]^s7$FaepF]^s7$FdepF]^s 7$FgepFd`s7$$\"5l;/,plW1%o)FSF]^s7$FjepF]^s7$$\"5I$e9;#Q/@(p)FSF]^s7$F ]fpF]^s7$$\"5+](=U2Tc.r)FSFd`s7$$\"5IL3_+(RHpr)FSFd`s7$$\"5l;H#oKQ-Ns) FSF]^s7$F`fpFd`s7$$\"5lm\"Hd?M@Ku)FSFd`s7$F[gmFd`s7$F[imFd`s7$FaimFd`s 7$FgimFd`s7$F]\\nFd`s7$Fe]nFd`s7$$\"5+++DJ&**)4A%*FSFd`s7$Fh]nFd`s7$$ \"5++]7GQeJ,&*FSFd`s7$$\"5lmmTg_9sF&*FSFd`s7$$\"5I$e*[=cGKM&*FS$F_jnF \\bl7$$\"5++DcwfU#4a*FSF^ds7$$\"5l;ajMjc_Z&*FSF^ds7$$\"5IL$3Fp1FTb*FSF d`s7$$\"5+]7y]q%G2c*FSF^ds7$$\"5lmT&)3u)Htc*FSFd`s7$$\"5I$3FpwFJRd*FSF d`s7$F[^nF^ds7$$\"5+++++]oi\"o*FSF^ds7$F^^nF^ds7$Fa^nF^ds-Fd^n6&Ff^n$ \"\"*Fi^n$\"\"%Fi^nF(-F[_n6#%8scheme~with~c[10]=19/20G-F$6%7c^lF'7$F1$ Fc_qF-7$$\"5LLL$e*)4bQl#F-$!$Z#F-7$F6$!$&GF-7$$\"5LLLeRAY1lLF-$!$%HF-7 $$\"5mmm;ajW8-OF-$!$\"GF-7$$\"5+++vo/V?RQF-$!$#HF-7$F;$!$(GF-7$F@F]hs7 $FE$!$z#F-7$FJ$!$;#F-7$FO$!#6FS7$FZ$F^^sFS7$F^r$Fg[mFS7$F\\t$\"#dFS7$F _t$\"#zFS7$Fbt$\"$+\"FS7$Fgt$\"$?\"FS7$F\\u$\"$W\"FS7$Fau$\"$o\"FS7$Ff u$\"$!>FS7$F[v$\"$;#FS7$F`v$\"$X#FS7$Fev$\"$a#FS7$Fjv$\"$r#FS7$F_w$\"$ $HFS7$FdwF`_r7$$\"5++++DJ=-GDFSF]`r7$$\"5++++]PL4bDFSFjfq7$$\"5+++D197 '=c#FSFf\\r7$$\"5+++]i!4H'oDFSF[_r7$$\"5+++v=npRvDFSF[\\o7$$\"5++++vV[ ;#e#FSFjfq7$$\"5+++](of+df#FSF^[r7$FiwF_hq7$$\"5+++DcEU+;EFSF`iq7$$\"5 +++]7.@xAEFSF^\\o7$$\"5+++voz*R&HEFS$\"$=$FS7$$\"5++++DcyIOEFSF`iq7$$ \"5+++D\"GtvIk#FSF[[r7$$\"5+++]P4O%)\\EFSF^[r7$$\"5+++v$f[6ml#FSF\\jq7 $$\"5++++]i$zLm#FSFcgq7$$\"5+++]i:^\"pn#FSF^[r7$$\"5++++vo3X!p#FSF[[r7 $$\"5+++DJX(=sp#FS$\"$I$FS7$$\"5+++](=i')Rq#FS$\"$J$FS7$$\"5+++vV)\\a2 r#FS$\"$L$FS7$F^xF\\hq7$$\"5+++]7yW/CFFSF\\hq7$$\"5++++D\"em0t#FSFdy7$ $\"5+++]P%o)3PFFSF`_t7$$\"5++++](y5Ou#FSFggq7$$\"5+++]i!*G8]FFSF[_t7$$ \"5++++v$*\\lcFFSF`_t7$$\"5+++](o4xJw#FSFehq7$$\"5++++++#*ppFFSF`iq7$$ \"5++++D1Mu#y#FSFggq7$$\"5++++]7wy&z#FS$\"$P$FS7$$\"5+++]i:(4B!GFS$\"$ S$FS7$$\"5++++v==$)3GFSFfat7$$\"5+++](=#RN:GFSFfat7$Fcx$\"$R$FS7$$\"5+ +++]PW'z%GFS$\"$M$FS7$$\"5+++++]G0uGFSFdy7$$\"5++++Dcq4()GFS$\"$Q$FS7$ $\"5++++]i79+HFS$\"$[$FS7$$\"5+++]ilLm1HFS$\"$Y$FS7$$\"5++++voa=8HFS$ \"$W$FS7$$\"5+++](=d2(>HFS$\"$V$FS7$FhxF`dt7$$\"5L$e9m%pB:KHFSF`dt7$$ \"5mm\"HKR1v!QHFS$\"$T$FS7$$\"5+]P%)Rex*R%HFSFedt7$$\"5LL$ekGX?*\\HFSF `dt7$$\"5++vozTewhHFSF[bt7$$\"5mmm\"H2B6O(HFSFact7$$\"5LLe9m>mX&)HFSF^ et7$$\"5++]Pf3?I(*HFS$\"$U$FS7$$\"5L$e*)fIqCK+$FS$\"$X$FS7$$\"5mmTg_(R Z\"4IFSFgft7$$\"5+](=#*>4q],$FSFfct7$FeeqFedt7$$\"5mm;HKkNoWIFSF`dt7$$ \"5+++v=UVPoIFSFedt7$$\"5L$ek`m.(HuIFSF^et7$$\"5mm\"z>6t>-3$FSF^et7$$ \"5+]PfeDC9'3$FS$\"$Z$FS7$$\"5LL$3_+7l?4$FS$\"$\\$FS7$$\"5++vV)*30\"R5 $FSFfct7$F]yFdht7$$\"5++](=U_ER9$FSF[dt7$$\"5LLL3_]r4sJFSF`dt7$$\"5mm; H#oxn-?$FS$\"$]$FS7$$\"5+++]7.%Q%GKFS$\"$_$FS7$$\"5mmm\"HdlzZG$FSFcit7 $FbyF_ht7$Fj\\oF_ht7$FgyF[dt7$Fb]oF^et7$F\\zFfat7$FazF[_tF\\gq7$F[[l$ \"$+$FS7$Fi]lFcer7$F_^l$\"$z#FS7$F]`l$\"$n#FS7$Fe`l$\"$_#FS7$Ff[r$\"$b #FS7$Fi[r$\"$c#FS7$F\\\\rFijs7$Fa\\rFijs7$Fd\\r$\"$^#FS7$Fi\\rFijs7$F \\]r$\"$d#FS7$F_]rFi[u7$$\"5lmT5:))o^p_FSFe[o7$Fb]rF^[u7$$\"5++DJ?3@^ \"G&FSF^[u7$Fe]rFe[o7$Fh]rFe[o7$F[^rF^[u7$F^^rFe[o7$Fa^rFi[u7$Fi^r$\"$ W#FS7$Fj`l$\"$T#FS7$Fj`r$\"$U#FS7$F_ar$\"$V#FS7$FbarFc]u7$FearFf]u7$Fh arFc]u7$F[br$\"$R#FS7$Fcbr$\"$S#FS7$F[crFf]u7$F^crFc]u7$FacrF_^u7$Fdcr F_^u7$FgcrF_^u7$Fgdr$\"$P#FS7$F_al$\"$N#FS7$Fdal$\"$C#FS7$FialFagp7$$ \"5+++v=U,1*3'FSFagp7$F^blFagp7$$\"5++](=' FSFagp7$$\"5+]PM_hUu'>'FSFagp7$$\"5++vV[@'yI?'FSFagp7$$\"5+]7`W\")HT4i FSFagp7$$\"5++]iSTtu:iFSFbcp7$$\"5++D\"G81;%GiFSFagp7$FcblF]^s7$FedlF] ^s7$F[elF]^s7$FeflF]^s7$F]hlF]^s7$$\"5ILLeky8a#f'FSF]^s7$F`hlF]^s7$$\" 5lm\"HKRrRTi'FSF]^s7$$\"5IL$e*)4Qf/j'FSFd`s7$$\"5++vo/[!znj'FSF]^s7$$ \"5lmmT5:()4VmFSFd`s7$$\"5ILe9;#Q=%\\mFSF]^s7$$\"5++](=#\\!Qdl'FSF]^s7 $$\"5lmTgF;x0imFSF]^s7$FehlFd`s7$FjhlFd`s7$F]ilFd`s7$F`ilFd`s7$FcilFd` s7$FfilFd`s7$FiilFd`s7$F\\jlFd`s7$F_jlFd`s7$FbjlFd`s7$FejlF^ds7$Fa[pF^ ds7$FjjlF^ds7$Fb]pF^ds7$F_[mF^ds7$FihrF^ds7$F\\irF^ds7$F_irF^ds7$Fbir$ \"#LwFSFjcu7$$\"5lmmTNYJxRwFSFjcu7$$\"5++]PM_wMYwFSF^ds7$Fc`p Fjcu7$Fi`pFjcu7$Fi[mFjcu7$Fc^mFjcu7$Ff^mFjcu7$Fi^mFjcu7$F\\_m$\"#;F\\b l7$Fa_mF`hu7$Fd_mFjcu7$Fg_mF`hu7$Fj_mF`hu7$F]`mFjcu7$F``mF`hu7$Fc`mF`h u7$Ff`mFjcu7$Fi`mFjcu7$F\\amF`hu7$F_amF`hu7$FbamF`hu7$FeamF`hu7$FhamF` hu7$F[bmFjcu7$F^bmF`hu7$FabmF`hu7$FdbmF`hu7$FgbmF`hu7$FjbmF`hu7$F]cmF` hu7$F`cmFjcu7$FccmFjcu7$FfcmF`hu7$FicmFjcu7$F\\dmFjcu7$F_dmFjcu7$FbdmF `hu7$FedmF`hu7$F[emF`hu7$FefmF`hu7$F[gmF`hu7$FaimF`hu7$FffpF`hu7$Fifp$ \"#:F\\bl7$F\\gpF`hu7$F_gpFeju7$FdgpF`hu7$FggpFeju7$FjgpF`hu7$F]hpF`hu 7$F`hpF`hu7$FchpF`hu7$FfhpF`hu7$FihpFeju7$F\\ipF`hu7$FdimFeju7$F`ipF`h u7$FcipF`hu7$FfipF`hu7$FiipF`hu7$F\\jpF`hu7$F_jpF`hu7$F[[qF`hu7$FgimF` hu7$FjimF`hu7$F]jmFeju7$FbjmFeju7$FejmFeju7$FhjmFeju7$F[[nFeju7$F^[nF` hu7$Fa[nF`hu7$Fd[nF`hu7$Fg[nF`hu7$Fj[nF`hu7$F]\\nF`hu7$F`\\nF`hu7$Fc\\ nF`hu7$Ff\\nF`hu7$Fi\\nFeju7$F\\]nFeju7$F_]nFeju7$Fb]nFeju7$Fe]nFeju7$ Fh]nFeju7$F[^nFeju7$F^^nFeju7$Fa^nFeju-%'COLOURG6&Ff^n$\"*++++\"!\")F( Ff]v-F[_n6#%8scheme~with~c[10]=32/33G-F$6%7`alF'7$$\"5ILLL3x1h6o!#B$!% ,;Fb^v7$$\"5mmmmTN@Ki8!#A$FRFh^v7$$\"5******\\7.K[V?Fh^v$\"#vFh^v7$$\" 5KLLL$3FWYs#Fh^v$!$9#Fh^v7$$\"5lmm;aQ`!eS$Fh^v$!$Q&Fh^v7$$\"5)******\\ iSmp3%Fh^v$F\\oFh^v7$$\"5KLL$eRZF\"oZFh^v$!$X'Fh^v7$$\"5lmmmmT&)G\\aFh ^v$\"$B\"Fh^v7$$\"5++++]7G$R<)Fh^v$!$\"yFh^v7$F+$!$4\"F-7$$\"5+++](=#* *3E7F-$!#\"*F-7$$Fg^vF-$F-F-7$$\"5LLL$e*[Vb)\\\"F-$!#pF-7$$\"5++++]ily M;F-$!$1\"F-7$$\"5LLLLe*)4D2>F-$!#nF-7$F1$!\"'F-7$$\"5++DJ&p(G)*QAF-$! #:F-7$$\"5LL$eRsL]#)H#F-$!#GF-7$$\"5mmTg_(zShOF-F\\cv7$$\"5LL$e9TQp1s$F -$\"#7F-7$$\"5mmT5SWo$*zPF-$F\\fsF-7$Fhgs$FchlF-7$$\"5LLeR(\\wr%)*QF-$ FgemF-7$$\"5mm;/ED#Rx&RF-$FdelF-7$$\"5++voa&o1q,%F-$\"#')F-7$F;$\"$N\" F-7$F@$Fj[uF-7$FE$\"$s$F-7$FJ$\"$T&F-7$FO$F^_vFS7$FUFfjn7$FZ$\"$A\"FS7 $F^rFjis7$F\\t$\"$q\"FS7$F_t$\"$&>FS7$Fbt$\"$=#FS7$FgtF\\^u7$F\\u$\"$h #FS7$Fau$\"$$GFS7$FfuF_dr7$F[vF^[r7$F`vFbft7$$\"5mm;HKkqk0@FSF`dt7$$\" 5+++DJ?:i<@FSF[dt7$$\"5LL$3-j(ffH@FSFgft7$$\"5mmm;HK/dT@FSFbft7$$\"5LL ekGgwbZ@FSF`dt7$$\"5++]7G))[a`@FSFgft7$$\"5mmTgF;@`f@FSF^et7$$\"5LLL3F W$>b;#FSFgft7$$\"5++DcEsl]r@FSFedt7$$\"5mm;/E+Q\\x@FSFedt7$$\"5LL3_DG5 [$=#FSFdht7$FevFgft7$$\"5LL$eRArU9?#FS$\"$c$FS7$$\"5mmm\"H#orT8AFS$\"$ e$FS7$$\"5LLeRA'R/%>AFS$\"$f$FS7$$\"5++](=Ui\"RDAFS$\"$`$FS7$$\"5mmTN@ _)y8B#FSF_\\w7$$\"5LLL$3-3mtB#FSF^]w7$$\"5mm;z>O0M\\AFS$\"$a$FS7$$\"5+ ++v=#*\\JhAFS$\"$b$FS7$$\"5mm\"H#=?AInAFSFi]w7$$\"5LL$3x\"[%*GtAFSFcit 7$$\"5++v=(GqW#FS$\"$p$FS7$Fdw$\"$r$FS7$Fc[t$F\\hvFS7$Ff[tF[`w7$Fi[ t$\"$t$FS7$F\\\\t$\"$o$FS7$F_\\tFa`w7$Fb\\t$\"$q$FS7$Fe\\t$\"$v$FS7$Fi w$\"$y$FS7$Fd]t$\"$w$FS7$F`^tFh_w7$Fc^tFh_w7$Ff^tFe`w7$Fi^tF^aw7$F^_tF h`w7$Fc_t$\"$x$FS7$F^xF[`w7$Fi_tF[`w7$F\\`tFh`w7$F_`tFego7$Fb`tF^`w7$F e`tF^`w7$Fh`tFego7$F[atFe`w7$F^atF`_w7$FaatFe`w7$FdatFego7$FiatFh`w7$F ^btF[`w7$FabtF[`w7$FcxFego7$FgbtFe_w7$F\\ct$\"$m$FS7$F_ctFe`w7$FdctFeg o7$FictF^`w7$F^dtFe`w7$FcdtFe_w7$FhxFe`w7$Fjet$\"$k$FS7$FeeqF]_w7$Fagt Fa\\o7$FdgtF]_w7$FggtFi\\w7$FjgtFd\\w7$F]htFa\\o7$Fbht$\"$j$FS7$$\"5m; H#=X\"y)z4$FSF]_w7$FghtFi\\w7$$\"5L$3_]M?L)4JFSF]_w7$F]yFhgo7$FbyFbft7 $FgyF\\hq7$F\\zF`_r7$FazF_[t7$Ffz$\"$s#FS7$F[[lFi[u7$Fi]l$\"$O#FS7$F_^ l$\"$D#FS7$F]`l$\"$6#FS7$Fe`l$\"$'>FS7$Ff[r$\"$+#FS7$Fi[rFjew7$F\\\\r$ \"$*>FS7$Fa\\rF^fw7$Fd\\rFgew7$Fi\\rF^fw7$F\\]r$\"$,#FS7$F_]rFgew7$Fa \\u$\"$%>FS7$Fb]r$\"$(>FS7$Fe\\uF[gw7$Fe]rFhfw7$Fh]rFhfw7$F[^rFgew7$F^ ^r$\"$$>FS7$Fa^rF[iv7$Fi^r$\"$*=FS7$Fj`l$\"$'=FS7$Fj`r$\"$(=FS7$F_arF_ [o7$FbarF\\hw7$FearFfgw7$FharF\\hw7$F[br$\"$&=FS7$FcbrFigw7$F[crFfgw7$ F^crF_[o7$FacrFchw7$FdcrFchw7$FgcrFigw7$Fjcr$\"$%=FS7$F]drFgeq7$$\"5+] P%[Y;\"\\VbFSFchw7$FbdrF\\iw7$$\"5l;zW#4LYkb&FS$\"$\"=FS7$FgdrFjx7$$\" 5N$3_+s\\,%pbFSF\\hw7$$\"5qmT&Q.3zed&FSFgeq7$$\"5+]ilZjmN#e&FSFchw7$Fj drF\\iw7$$\"5q;/EvH=J&f&FSFfiw7$$\"5++D1*GT*y,cFS$\"$!=FS7$$\"5N$ekGg* pE3cFSFchw7$F_alFgeq7$F^er$\"$x\"FS7$Fdal$\"$s\"FS7$$\"5++++DcljLfFSF \\[o7$FialFjcu7$$\"5+]P4YA'>Z/'FSF`hu7$$\"5++v=U#)R0^gFSFjcu7$$\"5+]7G QU$)QdgFSFjcu7$$\"5++]PM-FsjgFSF`hu7$$\"5++DcEA9RwgFSF`hu7$Fa_uF`hu7$$ \"5+]P%[@]%R&4'FSFjcu7$$\"5++v$4@')G<5'FSFjcu7$$\"5+]7.2AK13hFSFjcu7$$ \"5++]7.#e(R9hFSFjcu7$$\"5+](=#*>%>t?hFSF`hu7$$\"5++DJ&>Imq7'FSFjcu7$$ \"5+]iS\">m+M8'FSF`hu7$F^blF`hu7$$\"5+]Pf$=Qpg9'FSFjcu7$$\"5++vozTPS_h FSFjcu7$$\"5+]7yv,\"Q(ehFSFjcu7$Fe_uF`hu7$$\"5+](oz;#oSrhFSF`hu7$$\"5+ +D1k\"=Tx<'FSFjcu7$$\"5+]i:gTb2%='FSFjcu7$Fh_uFjcu7$F[`uF`hu7$F^`uF`hu 7$Fa`uF`hu7$Fd`uFjcu7$$\"5+](=n8q\"3AiFSF`hu7$Fg`uF`hu7$$\"5+]i!*G@/vM iFSFjcu7$FcblF`hu7$FaclF`hu7$F]dlF`hu7$$\"5+]7.2AQQ/jFSF`hu7$$\"5++D\" G)[qT6jFSF`hu7$$\"5+]Pfev-X=jFSF`hu7$FbdlFeju7$$\"5+]i:5Hn^KjFSF`hu7$$ \"5++v$fe&*\\&RjFSF`hu7$$\"5+](=*zFSF[cx7$Ff^mF[cx7$$\"5+]7G8h^gD!)FSF[cx7$$\"5ILeRZ_WLK! )FSFgex7$$\"5l;/^\"Quj!R!)FSFgex7$$\"5++]i:NIzX!)FSF[cx7$$\"5I$eR(\\EB __!)FSF[cx7$$\"5lmT&Qyh^#f!)FSFgex7$$\"5+](oz\"44)f1)FSF[cx7$$\"5ILL3_ +-rs!)FSFgex7$$\"5++DJ?$yoh3)FSFgex7$$\"5lm;a)eOF'*4)FSFgex7$$\"5+]ilA dmN1\")FSFgex7$$\"5IL3xc[f38\")FSF[cx7$$\"5l;a)3*R_\")>\")FSF[cx7$Fi^m F[cx7$F\\_mFgex7$Fa_mFgex7$Fd_mF[cx7$Fg_mFgex7$Fj_mFgex7$F]`mF[cx7$F`` mFgex7$Fc`mFgex7$Ff`mF[cx7$Fi`mF[cx7$F\\amFgex7$F_amFgex7$FbamFgex7$Fe amFgex7$FhamFgex7$F[bmF[cx7$F^bmFgex7$FabmFgex7$FdbmFgex7$FgbmFgex7$Fj bmFgex7$F]cmFgex7$F`cmF[cx7$FccmF[cx7$FfcmFgex7$FicmF[cx7$F\\dmF[cx7$F _dmF[cx7$FbdmFgex7$FedmFgex7$$\"5lmTg-NycS$)FSFgex7$$\"5lm;aQ.d>Z$)FSF gex7$$\"5lm\"zWy!f$R)FSF[cx7$$\"5lm;/E]'=-S)FSFgex7$$\"5lm\" z>'=l%oS)FSFgex7$$\"5lmm\"zpQuMT)FSFgex7$$\"5lmT&Q`D-,U)FSFgex7$$\"5lm ;zpB,tE%)FSFgex7$$\"5lm\"Hd?*zNL%)FSF[cx7$F[emF[cx7$$\"5lmTgxGPhY%)FSF gex7$F^emFgex7$$\"5lm\"z%\\l%p)f%)FSFgex7$FaemF[cx7$FdemFgex7$FiemFgex 7$F\\fmFgex7$F_fmFgex7$FbfmFgex7$FefmFgex7$F[gmFgex7$FaimFgex7$FgimFge x7$Fe]nFgex7$F[^nFgex7$F^^nFgex7$Fa^nFgex-Fd^n6&Ff^n$FjdvFi^nF(Fa^n-F[ _n6#%8scheme~with~c[10]=81/82G-F$6%7`clF'F_^vFe^vFj^vF__v7$Fe_v$!$P&Fh ^vFi_v7$F^`v$!$W'Fh^v7$Fc`v$\"$C\"Fh^v7$$Fa^vFh^v$!$k%Fh^v7$Fh`v$!$X(F h^v7$$\"5ILL$3-)Q4b))Fh^v$!$*[Fh^v7$$\"5lmmm\"z%\\DO&*Fh^v$F__mFh^v7$$ \"5+++Dc,;u@5F-$!#8F-7$F+Fbav7$F`av$!#sF-7$Feav$!\"$F-7$Fhav$!#]F-7$F] bv$F[dqF-7$Fbbv$!#WF-7$F1$FfapF-7$Fjbv$F\\cxF-7$F_cv$Fb^nF-7$Fdcv$Fg[m F-7$Ficv$FbgpF-7$F]dv$!#ZF-7$Fbdv$!#KF-7$Fgdv$F`]mF-7$Fffs$F\\blF-7$F_ ev$F]bpF-7$F6$\"#aF-7$$\"5LLek`TAE(=$F-$\"#!)F-7$$\"5mm;H#=qHlC$F-$F^^ mF-7$$\"5++v$4@;(z0LF-Fjdy7$F^gs$FfblF-7$$\"5mm\"H#o#3KVU$F-Fiev7$$\"5 ++](oHa*f$[$F-$F^_vF-7$$\"5LL3_D.q'Ga$F-$\"#`F-7$Fcgs$\"#fF-7$F[fv$F[d uF-7$F^fvFdey7$FcfvFecy7$Fhgs$\"#sF-7$Fifv$FgdqF-7$F]gv$\"#qF-7$Fagv$F ]eqF-7$F;$\"$h\"F-7$F@$\"$i#F-7$FE$Fe\\wF-7$FJ$\"$$\\F-7$FO$\"#mFS7$FU $\"#&)FS7$FZ$\"$-\"FS7$F^r$\"$=\"FS7$F\\t$\"$P\"FS7$F_t$\"$b\"FS7$FbtF \\[o7$FgtF\\iw7$F\\uF^fw7$FauFb[o7$FfuFaew7$F[vF]fq7$F`vF^[u7$$\"5++D \"Gj$)f'*4#FSFd[u7$F[jvF^[u7$$\"5LL3xJ#HM;6#FSF]\\u7$F^jvFijs7$Fajv$\" $`#FS7$FdjvFi[u7$FjjvF_jy7$F`[wFijs7$Fc[wF^[u7$Ff[wFi[u7$Fi[wFd[u7$Fev F^[u7$F]\\w$\"$f#FS7$Fb\\w$FggyFS7$Fg\\w$\"$j#FS7$F\\]wF]\\u7$Fa]w$\"$ g#FS7$Fd]wF]\\u7$Fg]wFijy7$F\\^wFb[z7$Fa^w$\"$e#FS7$Fd^wFijs7$Fg^wF\\[ z7$FjvFijy7$$\"5LL3x\"[5/?H#FSFijy7$$\"5++](oaIW()H#FSF^[z7$$\"5mm\"z> h]%[0BFSF_y7$$\"5LLL3x1ZA7BFSF\\[z7$$\"5++v=U2\\'*=BFSFbiv7$$\"5mm;H23 ^qDBFSF_y7$$\"5LLeRs3`WKBFS$\"$l#FS7$F[_wFbiv7$$\"5mmTg-5d#fM#FSF_y7$$ \"5LL$3x1\"fm_BFSF_y7$$\"5++D\"G861%fBFSF^[z7$$\"5mmm\"z>JYhO#FSF\\[z7 $$\"5LL3-j7l)GP#FSFbiv7$$\"5++]7G8nizBFSFijy7$$\"5mm\"HKR\"pO'Q#FSF_y7 $F_w$\"$m#FS7$$\"5+++v=%eFS$\"$G\"FS7$$\"5+++DcE]2beFSF`hz 7$$\"5++DJq?=iheFSFehz7$$\"5++]P%[ho\"oeFS$\"$D\"FS7$$\"5++vV)*3arueFS Fehz7$$\"5+++]7.AE\")eFS$\"$H\"FS7$$\"5++DcE(**3y)eFSFehz7$$\"5++]iS\" zbV*eFS$\"$E\"FS7$$\"5++voa&e-4!fFSFehz7$$\"5+++voz$\\u!fFSF`jz7$$\"5+ +](oz'Ha?fFSF`hz7$Ff[xF`hz7$$\"5++]7`W,tYfFSFehz7$$\"5+++D\"GtB)ffFSF` hz7$$\"5++]P4@t\"H(fFSF`jz7$$\"5+++]P44,')fFS$FafvF\\bl7$$\"5+++v$f3)> 7gFSFh[[l7$FialFh[[l7$F^blFh[[l7$FcblFh[[l7$FaclFh[[l7$F]dlFh[[l7$Fb`x Fh[[l7$Fe`xFh[[l7$Fh`xFh[[l7$Fbdl$FgevF\\bl7$F\\axFh[[l7$F_axFh[[l7$Fb axFh[[l7$FedlFe\\[l7$FfaxFh[[l7$FiaxFh[[l7$F\\bxFh[[l7$F_bxFh[[l7$Fbbx Fh[[l7$FhdlFh[[l7$FfbxFh[[l7$F[elFh[[l7$FeflFh[[l7$F]hlFh[[l7$F_auFh[[ l7$F`hlFh[[l7$FcauFh[[l7$FfauFe\\[l7$FiauFh[[l7$F\\buFe\\[l7$F_buFh[[l 7$FbbuFh[[l7$FebuFh[[l7$FehlFe\\[l7$F]ilFe\\[l7$FejlFe\\[l7$FjjlFe\\[l 7$F_[mFe\\[l7$FihrFe\\[l7$F\\irFe\\[l7$F_irFe\\[l7$Fbir$\"#5F\\bl7$Fei rFe\\[l7$FhirFe\\[l7$F[jrFe\\[l7$F^jrFe\\[l7$FajrFe\\[l7$FdjrFf^[l7$$ \"5I$e9T)G\"*y+uFSFe\\[l7$FgjrFe\\[l7$$\"5l;/,%pWoLT(FSFe\\[l7$FjjrFf^ [l7$F][sFe\\[l7$F`[sFe\\[l7$Fc[sFe\\[l7$Ff[sFe\\[l7$Fi[sFe\\[l7$F\\\\s Fe\\[l7$F_\\sFe\\[l7$Fb\\sFf^[l7$Fe\\sFe\\[l7$Fh\\sFe\\[l7$F[]sFe\\[l7 $Fd[mFe\\[l7$F[_pFe\\[l7$F]`pFe\\[l7$FjfuFe\\[l7$F``pFe\\[l7$FaguFe\\[ l7$FdguFe\\[l7$FgguFe\\[l7$Fc`pFf^[l7$$\"5lm;HKkm\\fwFSFe\\[l7$Ff`pFe \\[l7$$\"5IL$3-jnXEn(FSFe\\[l7$Fi`pFe\\[l7$F\\apFe\\[l7$F_apFe\\[l7$Fb apFe\\[l7$Fi[mFf^[l7$Fa\\mFf^[l7$Fc^mFf^[l7$F\\fxFf^[l7$F_fxFe\\[l7$Fb fxFe\\[l7$FefxFe\\[l7$FhfxFe\\[l7$F[gxFf^[l7$F^gxFe\\[l7$FagxFe\\[l7$F dgxFe\\[l7$Ff^mFe\\[l7$F[hxFe\\[l7$FahxFe\\[l7$FdhxFe\\[l7$FghxFf^[l7$ FjhxFe\\[l7$F]ixFf^[l7$F`ixFf^[l7$FcixFf^[l7$FfixFf^[l7$FiixFe\\[l7$F \\jxFe\\[l7$Fi^mFe\\[l7$F\\_mFf^[l7$Fa_mFf^[l7$Fd_mFe\\[l7$Fg_mFf^[l7$ Fj_mFf^[l7$F]`mFe\\[l7$F``mFe\\[l7$Fc`mFf^[l7$Ff`mFe\\[l7$Fi`mFe\\[l7$ F\\amFf^[l7$F_amFf^[l7$FbamFf^[l7$FeamFe\\[l7$FhamFf^[l7$F[bmFe\\[l7$F ^bmFf^[l7$FabmFf^[l7$FdbmFf^[l7$FgbmFf^[l7$$\"5+](=UdyZEE)FSFe\\[l7$Fj bmFf^[l7$$\"5l;HdX++hv#)FSFf^[l7$F]cmFe\\[l7$FccmFe\\[l7$FicmFe\\[l7$F \\dmFe\\[l7$F_dmFe\\[l7$FbdmFf^[l7$FedmFf^[l7$F^\\yFf^[l7$Fa\\yFe\\[l7 $Fd\\yFe\\[l7$Fg\\yFe\\[l7$Fj\\yFe\\[l7$FhdmFe\\[l7$F^]yFe\\[l7$Fa]yFf ^[l7$Fd]yFf^[l7$Fg]yFf^[l7$Fj]yFf^[l7$F]^yFe\\[l7$F`^yFe\\[l7$F[emFe\\ [l7$Fd^yFf^[l7$F^emFf^[l7$Fh^yFf^[l7$FaemFe\\[l7$FdemFf^[l7$FiemFf^[l7 $F\\fmFf^[l7$F_fmFf^[l7$FbfmFf^[l7$FefmFf^[l7$F[gmFf^[l7$FaimFf^[l7$Fg imFf^[l7$Fe]nFf^[l7$F[^nFf^[l7$F^^nFf^[l7$Fa^nFf^[l-Fd]v6&Ff^nFf]vF(F( -F[_n6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGFjes-%&TITLEG 6#%Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABEL SG6$Q\"x6\"Q!F`h[l-%%VIEWG6$;F(Fa^n;$!#XF\\bl$F`fyF\\bl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c [10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "schem e with c[10]=32/33" "scheme with c[10]=81/82" "scheme with c[10]=169/1 70" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 7 of \+ 11 stage, order 7 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#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF &*&F&F&F2F(F(F&F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-% \"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Sol ution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" " 6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3 *sin(3*x)+8/3*sin(3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\" \"\"\"$!\"\"-%$sinG6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F 1F*F*-%$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#-%$expG6#,&*(\"\"%\"\"\"\"\"$! \"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := dif f(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(dsolve( \{de,ic\},y(x)));\nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1.1,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0*&\"\"%F0-%$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$$\"3ALL$3FWYs#F/$\"3K tP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$ \"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$zx(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(e8Fgt$\"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$e*[z(yb\"Fg t$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$\"3wmm;a/c q;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF27$$\"3%om mmJFgt$\"3%RlX>.MR=&F27$$\"3gmmm \"pW`(>Fgt$\"3+6YS9:C2[F27$$\"3dLe9TOEH?Fgt$\"3!eWte3T%oWF27$$\"3K+]i! f#=$3#Fgt$\"3:XZ<;2j,UF27$$\"3?+](=xpe=#Fgt$\"3E#Q(H44MbQF27$$\"37nm\" H28IH#Fgt$\"3MH4)f2==l$F27$$\"3um;zpSS\"R#Fgt$\"3wpxg#Fgt$\"37l*=e[EHY$F27$$ \"33+]Pf4t.FFgt$\"35!4Ne]qiX$F27$$\"3uLLe*Gst!GFgt$\"3U+pq))z7kMF27$$ \"30+++DRW9HFgt$\"37'z:1TS%*[$F27$$\"3:++DJE>>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$$\"35L$e*[$z*RNFgt$\"3UAxt;S)>+% F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$pk$Fgt$\"3eIRs#H!Q \"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m;H2qcZPFgt$\"3w))) [$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ymm;/OgbRFgt$\"3W-T ml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e%Fgt$\"3 5gRzc#\\LF27$$\"\"&F)$\"3Ii# 4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fiel;F($\"#6Fcfl" 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 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 0 738 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: n umsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12 ,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `* c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: \+ errs := []:\nDigits := 30:\nfor ct to 6 do\n Vn_RK7_||ct := RK7_||ct (V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Vn_R K7_||ct):\n for ii to numpts do\n sm := sm+(Vn_RK7_||ct[ii,2]-v (Vn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nump ts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F, F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#! \"#7$%1no.~of~steps:~~~G\"$]#Q)pprint296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ /*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+0TO)o* !#C7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+4AwHmF57$/F8#\"#>\"#?$\"+-00 .iF57$/F8#\"#K\"#L$\"+JrHC[F57$/F8#\"#\")\"##)$\"+_:X)3$F57$/F8#\"$p\" \"$q\"$\"+6[\"3<#F5Q)pprint306\"" }}}{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 0 669 " 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 := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[ 10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`schem e with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigit s := 30:\nfor ct to 6 do\n vn_RK7_||ct := RK7_||ct(V(x,y),x,y,x0,y0, hh,numsteps,true);\nend do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(vn_RK7_||ct(xx)-vxx)];\nend do:\nDi gits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\" \"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0ini tial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~step s:~~~G\"$]#Q)pprint316\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~sche me~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+UP'fb)!#D7$/*&%-scheme~with~ GF+F,F+#\"#:\"#;$\"+!e&R`?F57$/F8#\"#>\"#?$\"+L&)\\?cF57$/F8#\"#K\"#L$ \"+2Q^]_!#E7$/F8#\"#\")\"##)$\"+K]qhpFM7$/F8#\"$p\"\"$q\"$\"+R(p*)\\\" F5Q)pprint326\"" }}}{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$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "mthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/1 6,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with ` *c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20: \nfor ct to 6 do\n sm := NCint((v(x)-'vn_RK7_||ct'(x))^2,x=0..5,adap tive=false,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/5)]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~sche me~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+wcFH&*!#C7$/*&%-scheme~with~ GF+F,F+#\"#:\"#;$\"+yqjalF57$/F8#\"#>\"#?$\"+E\")3)4'F57$/F8#\"#K\"#L$ \"+0*[Ty%F57$/F8#\"#\")\"##)$\"+\"yl_1$F57$/F8#\"$p\"\"$q\"$\"+9?+Y@F5 Q)pprint336\"" }}}{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 0 498 "evalf[20](plot([v(x)-'vn_RK7_1'(x),v(x)-'vn_RK7_2'(x ),v(x)-'vn_RK7_3'(x),\nv(x)-'vn_RK7_4'(x),v(x)-'vn_RK7_5'(x),v(x)-'vn_ RK7_6'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR( RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend =[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`sc heme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81 /82`,`scheme with c[10]=169/170`],\ntitle=`error curves for 11 stage o rder 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 929 671 671 {PLOTDATA 2 "6,-%'CURVESG6%7ju7$$\"\"!F)F(7$$\"5qmm;aQ`!eS$!#A$\" \"\"!#?7$$\"5SLLL3x1h6oF-$\"$Q#F07$$\"5,++Dc,;u@5!#@$\"%pgF07$$\"5ommm TN@Ki8F9$\"&-.'F07$$\"5NLL3FpE!Hq\"F9$\"')[d$F07$$\"5-++]7.K[V?F9$\"(] ^G\"F07$$\"5omm\"zptjSQ#F9$\"(!\\j7F07$$\"5NLLL$3FWYs#F9$\"(!\\U7F07$$ \"5-++vo/[AlIF9$\"(uuA\"F07$$\"5omm;aQ`!eS$F9$\"(#=d7F07$$\"5NLLeRseQY PF9$\"(i\")\\\"F07$$\"5-+++D1k'p3%F9$\"(444#F07$$\"5OLL$eRZF\"oZF9$\"( !p@?F07$$\"5qmmmmT&)G\\aF9$\"('f-?F07$$F3F9$\"(\\pU#F07$$\"50+++]7G$R< )F9$\"(L1j#F07$$\"5SLL$3-)Q4b))F9$\"(*)\\a#F07$$\"5qmmm\"z%\\DO&*F9$\" (%z\"[#F07$$\"5NLL3x\"[No()*F9$\"(;p^#F07$$\"5+++Dc,;u@5F0$\"(Ks`#F07$ $\"5nm;z%\\l*zb5F0$\"(Ig\\#F07$$\"5MLLLL3x&)*3\"F0$\"(#F07$$\"5+++]i!R(*Rc\"F0$\"(]&H=F07$$\"5nm;z>6B`# o\"F0$\"(0rf\"F07$$\"5MLL3xJs1,=F0$\"(0DI\"F07$$\"5++]PM_@g>>F0$\"(8FB \"F07$$\"5nmmm\"H2P\"Q?F0$\"'%\\c*F07$$\"5MLLek.pu/BF0$\"'p3lF07$$\"5+ ++]PMnNrDF0$\"'i'f$F07$$\"5nmmT5ll'z$GF0$\"&%R*)F07$$\"5MLLL$eRwX5$F0$ !&cv\"F07$$\"5MLLLe*[`HP$F0$!')*G5F07$$\"5MLLLL$eI8k$F0$!'X%f\"F07$$\" 5MLL$3-8>bx$F0$!'!Hg\"F07$$\"5MLLL3xwq4RF0$!'pe;F07$$\"5MLL$eRA'*Q/%F0 $!'R!o\"F07$$\"5NLLL$3x%3yTF0$!'7N;F07$$\"5-++]PfyG7ZF0$!'\"\\T\"F07$$ \"5ommm\"z%4\\Y_F0$!'&fC\"F07$$\"5NLLL$3FGT\\&F0$!'YG7F07$$\"5++++v$fl F07$$\"5qmmm\"HK kIz(F0$!'DuCF07$$\"5OLLLe*)>VB$)F0$!'etJF07$$\"5.+++DJbw!Q*F0$!'J+cF07 $$\"5NLL$ekGkX#**F0$!')R\"pF07$$\"5nmmm;/j$o/\"!#>$!'U$o(F07$$\"5++]i: g2\")e5Fc[l$!'Y@xF07$$\"5MLLe9;_yq5Fc[l$!'uSyF07$$\"5n;HK9I)yP2\"Fc[l$ !'L))yF07$$\"5++D19WCxw5Fc[l$!'[dyF07$$\"5M$3-Q\"egwz5Fc[l$!'FyvF07$$ \"5nm;a8s'fF3\"Fc[l$!'n)f(F07$$\"5ML3-8+pu)3\"Fc[l$!'KMxF07$$\"5+++]7G Tt%4\"Fc[l$!'*y\"yF07$$\"5nmmT5SIo=6Fc[l$!'0TtF07$$\"5MLLL3_>jU6Fc[l$! '?LnF07$$\"5nmm;HdNb'>\"Fc[l$!'F!*eF07$$\"5++++]i^Z]7Fc[l$!'C[[F07$$\" 5+++D1RICd7Fc[l$!'l6[F07$$\"5+++]i:4,k7Fc[l$!'yLYF07$$\"5+++v=#zy2F\"F c[l$!'O+ZF07$$\"5++++vomax7Fc[l$!'L(p%F07$$\"5+++](=U#3\"H\"Fc[l$!'m*p %F07$$\"5+++++v\"=YI\"Fc[l$!'%=z%F07$$\"5++++D\"o*oJ8Fc[l$!'#e:&F07$$ \"5++++](=h(e8Fc[l$!'PueF07$$\"5++++]7!Q4T\"Fc[l$!'9#[(F07$$\"5++++]P[ 6j9Fc[l$!'P8)*F07$$\"5nmm\"HKR'\\5:Fc[l$!(RR:\"F07$$\"5MLL$e*[z(yb\"Fc [l$!(OlQ\"F07$$\"5+++Dc,#>Uh\"Fc[l$!(\"f&o\"F07$$\"5nmmm;a/cq;Fc[l$!(p 23#F07$$\"5nmm;zpYU%p\"Fc[l$!(.&)>#F07$$\"5nmmmT&)))G=\"[Fc[ l$!'n-hF07$$\"5,+DcEsW\"R\">Fc[l$!'.zbF07$$\"5MLLe9;0?E>Fc[l$!'%Rx\"F0 7$$\"5++]i!Rgs2&>Fc[l$\"'(>N#F07$$\"5nmmmm\"pW`(>Fc[l$\"'Y@lF07$$\"5ML ek.HW#)))>Fc[l$\"'&yp*F07$$\"5++]iSmTI-?Fc[l$\"('p38F07$$\"5n\"zp[25uc +#Fc[l$\"(vuG\"F07$$\"5M$e9\"4NS/4?Fc[l$\"(zrE\"F07$$\"5+v$fL%pRT7?Fc[ l$\"(*4a7F07$$\"5nmTgx.Ry:?Fc[l$\"('y\"G\"F07$$\"5Me*[=\"QQ:>?Fc[l$\"( F3Z\"F07$$\"5+]P4YsP_A?Fc[l$\"(&Qc:F07$$\"5nT&Q.oq$*e-#Fc[l$\"(]3`\"F0 7$$\"5MLLe9TOEH?Fc[l$\"(sj]\"F07$$\"5om;a)e6Bi0#Fc[l$\"(V=o\"F07$$\"5, ++]i!f#=$3#Fc[l$\"(C(=>F07$$\"5,+++D\"=EX8#Fc[l$\"($y!z\"F07$$\"5,++]( =xpe=#Fc[l$\"(vFc\"F07$$\"5MLLeRA9WRAFc[l$\"(=g?\"F07$$\"5nmmm\"H28IH# Fc[l$\"''y**)F07$$\"5nmmTNr&3AM#Fc[l$\"'\"Gm'F07$$\"5nmm;zpSS\"R#Fc[l$ \"'ZF_F07$$\"5+++v$41oWW#Fc[l$\"',mTF07$$\"5MLLL3_?`(\\#Fc[l$\"'pdNF07 $$\"5MLL3_D1l_DFc[l$\"'NpJF07$$\"5MLL$e*)>pxg#Fc[l$\"'PPHF07$$\"5ommm; z+vbEFc[l$\"'[?GF07$$\"5,++]Pf4t.FFc[l$\"'+pFF07$$\"5MLLLe*Gst!GFc[l$ \"'([*GF07$$\"5,++++DRW9HFc[l$\"'ihLF07$$\"5,+++DJE>>IFc[l$\"'*=;%F07$ $\"5,++]i!RU07$Fc[l$\"'%Ha&F07$$\"5,+++v=S2LKFc[l$\"'/?\")F07$$\"5ommm m\"p)=MLFc[l$\"(_E6\"F07$$\"5MLLLeR%p\")Q$Fc[l$\"(*Gi7F07$$\"5,+++](=] @W$Fc[l$\"(6$z8F07$$\"5ML$ekyZ2mY$Fc[l$\"(()3U\"F07$$\"5omm\"H#oZ1\"\\ $Fc[l$\"(&*QX\"F07$$\"5MLe9T8MH.NFc[l$\"(6LY\"F07$$\"5,+]Pfe?_:NFc[l$ \"(qmZ\"F07$$\"5omTgx.2vFNFc[l$\"(f<[\"F07$$\"5MLL$e*[$z*RNFc[l$\"(%y# [\"F07$$\"5omT5:)[[Lb$Fc[l$\"(O,\\\"F07$$\"5,+]PMFwrmNFc[l$\"(yd[\"F07 $$\"5MLek`mn3!e$Fc[l$\"(PrZ\"F07$$\"5omm\"Hd!fX$f$Fc[l$\"(C`Z\"F07$$\" 5ML$e9T=%>?OFc[l$\"(m'R9F07$$\"5,+++]iC$pk$Fc[l$\"(+:S\"F07$$\"5MLLe*[ t\\sp$Fc[l$\"($['G\"F07$$\"5omm;H2qcZPFc[l$\"(w37\"F07$$\"5,++]7.\"fF& QFc[l$\"'XquF07$$\"5MLLek`8=/RFc[l$\"'n'3'F07$$\"5ommm;/OgbRFc[l$\"'&4 H&F07$$\"5MLe*[o=i!pRFc[l$\"'oS_F07$$\"5,+]7`p2_#)RFc[l$\"'o3`F07$$\"5 M$eRs31]#*)RFc[l$\"'pe^F07$$\"5omTN@_$zf*RFc[l$\"'wz]F07$$\"5,](oaNk3F +%Fc[l$\"'=*G&F07$$\"5MLLe*[$zV4SFc[l$\"'XN^F07$$\"5,D1kc!e-G,%Fc[l$\" 'Or]F07$$\"5o;zpBEs;;SFc[l$\"'3p]F07$$\"5M3_v!>(=`>SFc[l$\"':0`F07$$\" 5,+D\"yv^'*G-%Fc[l$\"''*3`F07$$\"5M$3F>*3eiHSFc[l$\"'w]^F07$$\"5om;/E+ ^NOSFc[l$\"'w%4&F07$$\"5ML3F%Ho8)\\SFc[l$\"'p$=&F07$$\"5,++]ilAFjSFc[l $\"'=\"Q&F07$$\"5omm\"zW7@^6%Fc[l$\"'MH^F07$$\"5NLLLL$)*pp;%Fc[l$\"'&R '\\F07$$\"5NLL$3-$H**>UFc[l$\"')>S%F07$$\"5NLLL3xe,tUFc[l$\"'HaNF07$$ \"5omm;HdO=yVFc[l$\"'?P@F07$$\"5MLLe9\"z-lU%Fc[l$\"'0q;F07$$\"5,++++D> #[Z%Fc[l$\"'KU8F07$$\"5MLL$3_5,-`%Fc[l$\"'^!3\"F07$$\"5ommmT&G!e&e%Fc[ l$\"&d:*F07$$\"5NLLLL$)Qk%o%Fc[l$\"&he(F07$$\"5-++]iSjE!z%Fc[l$\"&>.(F 07$$\"5-++]P40O\"*[Fc[l$\"&jL(F07$$\"\"&F)$\"&Sc)F0-%&COLORG6&%$RGBGF( $\"\"#!\"\"$F/F)-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$ 6%7^wF'7$F+F(7$F2$\"$,\"F07$F7$\"%pDF07$F=$\"&Kb#F07$FB$\"'59:F07$FG$ \"'9YaF07$FL$\"'Na`F07$FQ$\"'Nl_F07$FV$\"'d,_F07$Fen$\"'fE`F07$Fjn$\"' 2YjF07$F_o$\"':g))F07$$\"5pmmT5SpaFWF9$\"'`6()F07$Fdo$\"'\"oc)F07$$\"5 0++D\"y+3(3^F9$\"')fW)F07$Fio$\"'>$[)F07$F^p$\"(kq-\"F07$Fbp$\"(P*46F0 7$Fgp$\"(!zt5F07$F\\q$\"((=Y5F07$Faq$\"(e\"e5F07$Ffq$\"(M[1\"F07$F[r$ \"(Uv/\"F07$F`r$\"(g0.\"F07$Fer$\"'O<\"*F07$Fjr$\"'2\"[(F07$Fds$\"'\\t ]F07$F^t$\"'WONF07$Fht$\"&a3*F07$Fbu$!&#\\rF07$Fgu$!'tZ5F07$F\\v$!'W]7 F07$Ffv$!'3N7F07$F`w$!'O(=\"F07$Few$!'!e0\"F07$Fjw$!'7=5F07$F_x$!'RM5F 07$Fdx$!'hs5F07$Fix$!'.h6F07$F^y$!'r^7F07$Fhy$!'3*p\"F07$F]z$!'Zh?F07$ Fbz$!'cYDF07$$\"5qmmmTg()4_))F0$!'&GQ$F07$Fgz$!'p2WF07$Fa[l$!'dIpF07$F \\\\l$!'*eR(F07$Fj]l$!'&oz(F07$$\"5ML$e9Te3n5\"Fc[l$!'#4*yF07$F_^l$!'( R,)F07$$\"5MLe*)4o-nC6Fc[l$!'%*Q!)F07$$\"5++]P4'\\d18\"Fc[l$!'ww\")F07 $$\"5nmT&)3CZkO6Fc[l$!'=V#)F07$Fd^l$!'CD\")F07$$\"5++vVt_@P\\6Fc[l$!'d !G)F07$$\"5nm;aQ`B6c6Fc[l$!''3P)F07$$\"5+]P4r`C[f6Fc[l$!']1#)F07$$\"5M Lek.aD&G;\"Fc[l$!'[5#)F07$$\"5n;z>OaEAm6Fc[l$!')pG)F07$$\"5+++voaFfp6F c[l$!'xi$)F07$$\"5M$3-8]&G'H<\"Fc[l$!'6G%)F07$$\"5nmT&Q`&HLw6Fc[l$!'*f V)F07$$\"5+]iSmbIqz6Fc[l$!'rE#)F07$$\"5ML$e*)f:tI=\"Fc[l$!'Na#)F07$$\" 5++D1kcL\")*=\"Fc[l$!'*>S)F07$Fi^l$!'ye%)F07$$\"5++]PfeR.57Fc[l$!'Z9%) F07$$\"5MLLe*)fV^B7Fc[l$!'&RG)F07$$\"5++voagXDI7Fc[l$!'.>%)F07$$\"5nm; z>hZ*pB\"Fc[l$!'r_%)F07$$\"5MLe*[='\\tV7Fc[l$!'j2$)F07$F^_l$!'gN%)F07$ Fb`l$!'M<&)F07$F\\al$!'WS')F07$Faal$!'`()*)F07$Ffal$!'w([*F07$F[bl$!(Y #z5F07$F`bl$!(0&)H\"F07$Febl$!(jyZ\"F07$Fjbl$!(]Uq\"F07$F_cl$!(^4$>F07 $Fdcl$!(\"oM@F07$Ficl$!(5D;#F07$F^dl$!(Ed?#F07$$\"5nm\"zpB\">F@#F07$$\"5nm\"H#=?S?L9#F07$Fgel$!(e&R@F07$F\\fl$!(zW<#F07$Fafl$!(SS:#F07$Fffl$!(+O8#F0 7$F[gl$!(.K*>F07$F`gl$!(Nx!=F07$Fegl$!(s*z:F07$Fjgl$!(4%38F07$F_hl$!(q l;\"F07$Fdhl$!'gV(*F07$Fihl$!'#*[\"*F07$F^il$!''*[sF07$Fcil$!'&)z[F07$ Fhil$!'\\ODF07$F]jl$!&p1*F07$Fbjl$\"&N/)F07$Fgjl$\"&L\"zF07$F\\[m$\"&= \"yF07$Fa[m$\"&C+)F07$Ff[m$\"&U))*F07$F[\\m$\"'FX=F07$F`\\m$\"'opAF07$ Fe\\m$\"'XKAF07$Fj\\m$\"'r)>#F07$F_]m$\"'jXMF07$Fd]m$\"'/**[F07$$\"5,+ DJXR0g*3#Fc[l$\"'qYZF07$$\"5,+]7G)[=g4#Fc[l$\"'g@Fc[l$\"'2H^F07$$\"5,+](=U(Q.t@F c[l$\"'-@[F07$F^^m$\"'RrYF07$Fc^m$\"'+qNF07$Fh^m$\"'(>d#F07$F]_m$\"'-x % Fc[l$\"'GA7F07$Fb^n$\"'H@7F07$$\"5NLLek.W]YUFc[l$\"'v<6F07$Fg^n$\"'$*4 5F07$F\\_n$\"&>&fF07$Ff_n$\"&*HNF07$F``n$\"&IL#F07$Fe`n$\"&*H>F07$Fj`n $\"&5w\"F07$F_an$\"&+!=F07$Fdan$\"&>0#F0-Fian6&F[bnF($\"\"(F^bnF(-Fabn 6#%8scheme~with~c[10]=15/16G-F$6%7\\vF'F*7$F2$\"$[\"F07$F7$\"%\"y$F07$ F=$\"&bw$F07$FB$\"'=PAF07$FG$\"'+d!)F07$FL$\"'?@zF07$FQ$\"'a*y(F07$FV$ \"'B&p(F07$Fen$\"'(3)yF07$Fjn$\"'`#R*F07$F_o$\"(7=J\"F07$Fdo$\"(%Qo7F0 7$Fio$\"(^kD\"F07$F^p$\"(zZ_\"F07$Fbp$\"(Nll\"F07$Fgp$\"(3Eg\"F07$F\\q $\"(!Qj:F07$Faq$\"(u\")e\"F07$Ffq$\"(+Lg\"F07$F[r$\"(ksd\"F07$F`r$\"(O [&)F07$F_cl$!(/U6\"F07$Fdcl$!(udR\"F07$Ficl$!(6qY\"F07$F^dl$! (/mb\"F07$F`eo$!(kZg\"F07$Feeo$!()f\"f\"F07$Fjeo$!(A%y:F07$Fcdl$!(Lcc \"F07$Fbfo$!([bb\"F07$Fgfo$!(()pb\"F07$F\\go$!(K_f\"F07$Fhdl$!($)*3;F0 7$F]el$!(A0e\"F07$Fbel$!(3jb\"F07$Fgel$!(Cvb\"F07$F\\fl$!(M=f\"F07$Faf l$!(sod\"F07$Fffl$!(2>c\"F07$F[gl$!(.VX\"F07$F`gl$!(ILD\"F07$Fegl$!(E& G5F07$Fjgl$!'+5tF07$F_hl$!'8=cF07$Fdhl$!'JAJF07$Fihl$!'3EGF07$F^il$!&< x$F07$Fcil$\"'Q7AF07$Fhil$\"'w4[F07$F]jl$\"'g'z'F07$Fbjl$\"'!p!*)F07$F gjl$\"'Yi()F07$F\\[m$\"'*Ri)F07$Fa[m$\"'.J&)F07$Ff[m$\"'F&p)F07$F[\\m$ \"'R!))*F07$F`\\m$\"(h7/\"F07$Fe\\m$\"(vT-\"F07$Fj\\m$\"(qx+\"F07$F_]m $\"(-36\"F07$Fd]m$\"(:FD\"F07$Fi]m$\"([7;\"F07$F^^m$\"(R2,\"F07$Fc^m$ \"'hDyF07$Fh^m$\"';teF07$F]_m$\"'C'R%F07$Fb_m$\"'euMF07$Fg_m$\"'>!z#F0 7$F\\`m$\"'N(Q#F07$Fa`m$\"'-C@F07$Ff`m$\"'$3'>F07$F[am$\"'uw=F07$F`am$ \"'\"e$=F07$$\"5omm\"zWi^bv#Fc[l$\"'E]=F07$Feam$\"'-:>F07$$\"5omm;H2\" 34'GFc[l$\"'$*Q?F07$Fjam$\"'nIAF07$F_bm$\"''>x#F07$Fdbm$\"'\">o$F07$Fi bm$\"'`n`F07$F^cm$\"'KptF07$Fccm$\"')pR)F07$Fhcm$\"'67#*F07$F]dm$\"'+' \\*F07$Fbdm$\"'x<(*F07$Fgdm$\"'4w(*F07$F\\em$\"''R')*F07$Faem$\"'?)))* F07$Ffem$\"'ix)*F07$F[fm$\"'aD**F07$F`fm$\"'pt)*F07$Fefm$\"'W(y*F07$Fj fm$\"'4u(*F07$F_gm$\"'Ms%*F07$Fdgm$\"']'=*F07$Figm$\"'>l$)F07$F^hm$\"' 2YsF07$Fchm$\"'lu[F07$Fhhm$\"'vOSF07$F]im$\"'hTNF07$Fbim$\"',:NF07$Fgi m$\"'giNF07$F\\jm$\"'\">Y$F07$Fajm$\"'<3MF07$Ffjm$\"'tXNF07$F[[n$\"'iU MF07$F`[n$\"'U*R$F07$Fe[n$\"'X'R$F07$Fj[n$\"'F\\NF07$F_\\n$\"'g]NF07$F d\\n$\"'sWMF07$Fi\\n$\"'![S$F07$F^]n$\"'^bMF07$Fc]n$\"'#Rd$F07$Fh]n$\" 'KzLF07$F]^n$\"'QTKF07$Fb^n$\"'4fGF07$Fg^n$\"'`0BF07$F\\_n$\"'n#R\"F07 $Fa_n$\"'K$4\"F07$Ff_n$\"&<#))F07$F[`n$\"&!HrF07$F``n$\"&(\\gF07$Fe`n$ \"&B+&F07$Fj`n$\"&0i%F07$F_an$\"&J\"[F07$Fdan$\"&li&F0-Fian6&F[bn$\"\" *F^bn$\"\"%F^bnF(-Fabn6#%8scheme~with~c[10]=19/20G-F$6%7iuF'7$FQ$\"'xs $F07$$\"50++v$4@\">_rF 9$\"'*e9$F07$$\"5qmm;zW 5(F07$F^`o$!'AYpF07$Fc`o$!'zspF07$Fh`o$!'`(4(F07$Fi^l$!'wWrF07$$\"5+]( =\"Fc[l$!'7JpF07$$\"5ML3F%zv$H.7Fc[l$!'T!)pF07$$\"5n;H#o#eQm17F c[l$!'zSqF07$F`ao$!'()*4(F07$$\"5M$3F>*eSS87Fc[l$!''e9(F07$$\"5nm\"zW# fTx;7Fc[l$!'cCrF07$$\"5+]7.dfU9?7Fc[l$!'=!)oF07$Feao$!'aPpF07$$\"5n;a8 AgW)oA\"Fc[l$!'y%*pF07$Fjao$!'D]qF07$$\"5M$eRs3mCOB\"Fc[l$!'?!4(F07$F_ bo$!'5`qF07$$\"5+]PM_h[OS7Fc[l$!'e/oF07$Fdbo$!'OeoF07$$\"5n;zW\"pF07$F^_l$!'EjpF07$Fb`l$!';DoF07$F\\al$!'>4mF07$Faal$!'#[n'F0 7$Ffal$!'s&z'F07$$\"5++++++'\\[Q\"Fc[l$!'tXrF07$F[bl$!'HBwF07$$\"5++++ +Dk-P9Fc[l$!'_#Q)F07$F`bl$!'e)p*F07$$\"5ML$ek`h0o[\"Fc[l$!([W1\"F07$Fe bl$!(**\\;\"F07$$\"5++]P4rr=M:Fc[l$!(_IF\"F07$Fjbl$!(?IT\"F07$$\"5nm;/ Ev&[ge\"Fc[l$!(lhc\"F07$F_cl$!(p;j\"F07$$\"5nmTNr9XIG;Fc[l$!(=%)p\"F07 $$\"5ML$eky#)*QU;Fc[l$!(Pyv\"F07$$\"5+vVB:c6\"fk\"Fc[l$!('GYbs\"F07$$\"5++Dc,T^Z c;Fc[l$!(^)Hh(*)zk8F07$$ \"5nmT5l]pl2=Fc[l$!(8$p7F07$F`gl$!($[f6F07$Fegl$!'VL)*F07$Fjgl$!'q!*yF 07$F_hl$!'\"z#pF07$Fdhl$!'6;dF07$Fihl$!'tn`F07$F^il$!'))fUF07$Fcil$!'z _HF07$Fhil$!')Qt\"F07$Fbjl$!&IP\"F07$Fj\\m$\"&*[^F07$F_]m$\"'$[1\"F07$ Fd]m$\"'lZ;F07$Fh\\p$\"'W'f\"F07$F]]p$\"'l$f\"F07$Fb]p$\"'&)H=F07$Fj_p$\"'\">w\"F07$Fi]m$\"'_; F07$Fccm$\"&mr$F07$Fhcm$\"&;!\\F07$F]d m$\"&1/&F07$Fbdm$\"&K'\\F07$Fgdm$\"&Zh%F07$F\\em$\"&@e%F07$Faem$\"&N3% F07$Ffem$\"&fI$F07$Fjfm$\"&**4\"F07$Fdgm$!&,/$F07$Figm$!&)[iF07$F^hm$! &(R\"*F07$F\\hp$!&aq*F07$Fahp$!'a_5F07$Ffhp$!'(>-\"F07$Fchm$!&-k*F07$F hhm$!&7U(F07$F]im$!&mn%F07$F[[n$!&w9\"F07$Fc]n$\"&+-#F07$Fh]n$\"&fJ$F0 7$F]^n$\"&,C%F07$Fb^n$\"&d7%F07$Fg^n$\"&@O$F07$F\\_n$\"&/'=F07$Ff_n$\" &n1\"F07$F``n$\"%=rF07$Fe`n$\"%odF07$Fj`n$\"%=\\F07$F_an$\"%TZF07$Fdan $\"%O_F0-%'COLOURG6&F[bn$\"*++++\"!\")F(Feeu-Fabn6#%8scheme~with~c[10] =32/33G-F$6%7_uF'7$Fio$!&!)f%F07$F`r$!'!)\\:F07$Fer$!'T\"z\"F07$Fjr$!' Jv>F07$F_s$!'gq?F07$Fds$!'6S@F07$$\"5nm\"Hd?pM.'=F0$!'*[3#F07$Fis$!'lL ?F07$$\"5ML3-j7'p)y>F0$!'=B?F07$F^t$!'yD?F07$Fht$!'`9;F07$Fbu$!'u86F07 $Fgu$!&w(*)F07$F\\v$!&K%pF07$Ffv$!&9!fF07$F`w$!&p8&F07$$\"5-+](oHaN;J% F0$!&Q\"\\F07$Fces$!&N&[F07$$\"5NL$eRs3P(yXF0$!&Z$[F07$Few$!&Y)[F07$F[ fs$!&GK&F07$Fjw$!&#*4'F07$Fdx$!&WI(F07$F^y$!&)Q\"*F07$Fhy$!'RZ6F07$F]z $!')\\C\"F07$Fbz$!'z#Q\"F07$F[[o$!'qV;F07$Fgz$!'[x?F07$F\\[l$!'IZFF07$ Fa[l$!'AWOF07$Fj]l$!'XyVF07$Fd^l$!''Q)[F07$Ff]o$!'Px\\F07$F[^o$!'eS]F0 7$F`^o$!'@p\\F07$Fe^o$!'^z\\F07$Fj^o$!'!f-&F07$F__o$!'!>2&F07$Fd_o$!'g 6^F07$Fi_o$!'h:^F07$F^`o$!'`!)\\F07$Fc`o$!'^&*\\F07$$\"5n;/^JcKW'=\"Fc [l$!'[S]F07$Fh`o$!'n%3&F07$$\"5M$e9mpX$=$>\"Fc[l$!']>^F07$Fi^l$!'$Q5&F 07$Feao$!')\\y%F07$F^_l$!'3UYF07$F\\al$!'uZPF07$Ffal$!'o7LF07$$\"5++++ v$R0=P\"Fc[l$!'e4LF07$Fh_t$!'UfLF07$$\"5++++D1Q*yR\"Fc[l$!'QrMF07$F[bl $!'_>OF07$F``t$!'*>=%F07$F`bl$!'TH`F07$Fh`t$!'-5iF07$Febl$!'K$=(F07$F` at$!'m^#)F07$Fjbl$!'n='*F07$$\"5++v$4@Ej>d\"Fc[l$!(Lt,\"F07$Fhat$!(it5 \"F07$$\"5M3x\"[N!*p&*e\"Fc[l$!(`H5\"F07$$\"5+]Pf$=B\"4$f\"Fc[l$!(I75 \"F07$$\"5n\"zpB,c7mf\"Fc[l$!(Ja6\"F07$$\"5MLe9T))Q8+;Fc[l$!(vk=\"F07$ $\"5n;zp)\\awrg\"Fc[l$!(1\\<\"F07$F_cl$!(%Ro6F07$F`bt$!(q1A\"F07$Febt$ !(**yD\"F07$F_ct$!(l8C\"F07$Fict$!(*[M7F07$F^dt$!(oFE\"F07$Fcdt$!(BOD \"F07$Fhdt$!(5VC\"F07$Fdcl$!(i]B\"F07$Ficl$!([O>\"F07$F^dl$!(=d6\"F07$ Fhdl$!'\\I**F07$Fffl$!'#*\\')F07$Ffet$!'tbvF07$F[gl$!'sMrF07$F^ft$!'U( *fF07$F`gl$!'eoZF07$$\"5,+D1k1nTN=Fc[l$!'_/WF07$Fegl$!'!=M$F07$$\"5ML3 -jik#F07$Fjgl$!'A7=F07$F_hl$!'td5F07$Fdhl$!&VD\"F07$$\"5M $3_D.Xrx!>Fc[l$!&r@\"F07$Fihl$!%tvF07$$\"5M3xcB$)f)p\">Fc[l$\"%o(*F07$ $\"5o;Hd?%\\d+#>Fc[l$\"&(poF07$$\"5,D\"yv^+HJ#>Fc[l$\"&kx'F07$F^il$\"& Po'F07$Fcil$\"'Vr7F07$Fhil$\"'UTH#F07$F \\[m$\"'d=AF07$Ff[m$\"'S(=#F07$F[\\m$\"'Q3BF07$F`\\m$\"'<`BF07$Fe\\m$ \"'^9BF07$Fj\\m$\"''oF#F07$F_]m$\"'r*=#F07$Fd]m$\"'4a?F07$Fi]m$\"';q:F 07$F^^m$\"'xx5F07$Fc^m$\"&E6(F07$Fh^m$\"&v4&F07$F]_m$\"&a6%F07$Fb_m$\" &eh$F07$F\\`m$\"&/2$F07$Ff`m$\"&$zBF07$F`am$\"&2&=F07$Feam$\"&#HgrgPFc[l$!&+@#F07$F\\hp$!&2;#F07$$\"5omm\"HK/9qy$Fc [l$!&QJ#F07$Fahp$!&x>#F07$Ffhp$!&'R\"F07$Fbu$!&y&zF07$Fgu$!&e L'F07$F\\v$!&'p[F07$Ffv$!&!3UF07$F`w$!&R%QF07$Fces$!&3!RF07$Few$!&)*3% F07$F[fs$!&\"GYF07$Fjw$!&\"4aF07$F^y$!&.*yF07$Fhy$!&]]*F07$F]z$!'Y75F0 7$Fbz$!'q-6F07$F[[o$!'Vb7F07$Fgz$!'J3:F07$Fa[l$!'$4N#F07$Fj]l$!'!os#F0 7$Fd^l$!':(*GF07$Ff]o$!']_HF07$F[^o$!'7!)HF07$F`^o$!'4,HF07$Fe^o$!'m&* GF07$Fj^o$!'hAHF07$F__o$!'?\\HF07$Fd_o$!'TqHF07$Fi_o$!']hHF07$F^`o$!'' f$GF07$Fc`o$!'QOGF07$Fh`o$!'u')GF07$Fi^l$!''*zGF07$Feao$!'$p`#F07$F^_l $!'\\QBF07$F\\al$!'vQ9F07$Ffal$!&Ru)F07$Fd`v$!&QD)F07$Fh_t$!&6&zF07$F \\av$!&aS)F07$F[bl$!&>2*F07$F``t$!'-f7F07$F`bl$!'Xp?F07$Fh`t$!'*)>FF07 $Febl$!';gMF07$F`at$!'T!H%F07$Fjbl$!'Jj`F07$Ffbv$!'j7eF07$Fhat$!'hFlF0 7$F^cv$!'.-lF07$Fccv$!']'\\'F07$Fhcv$!'(3h'F07$F]dv$!'JgrF07$Fbdv$!'\\ !4(F07$F_cl$!'CeqF07$F`bt$!'9\"\\(F07$Febt$!'L/yF07$F_ct$!'x,xF07$Fict $!'KlwF07$F^dt$!']xyF07$Fcdt$!']?yF07$Fhdt$!'QixF07$Fdcl$!']/xF07$Ficl $!'SSuF07$F^dl$!'&>'oF07$Fhdl$!'L.fF07$Fffl$!'6>\\F07$Ffet$!'#Q+%F07$F [gl$!'))[PF07$F^ft$!'&)yFF07$F`gl$!'#**p\"F07$Fjgv$!'B$\\\"F07$Fegl$!& <`&F07$Fbhv$\"&&z\\F07$Fjgl$\"&kZ(F07$F_hl$\"'XQ9F07$Fdhl$\"'!zM#F07$$ \"5,voaNR**p/>Fc[l$\"'s;BF07$F`iv$\"'7'G#F07$$\"5o\"Hd&HhH%3\">Fc[l$\" '1hAF07$Fihl$\"';rAF07$Fhiv$\"'yMCF07$F]jv$\"'\"=4$F07$Fbjv$\"'m\\IF07 $F^il$\"')y+$F07$Fcil$\"'D#\\$F07$Fhil$\"'ziQF07$F]jl$\"'-cTF07$Fbjl$ \"')pR%F07$F\\[m$\"'(eD%F07$Ff[m$\"'8tTF07$F[\\m$\"'3FVF07$F`\\m$\"'jv VF07$Fe\\m$\"'w.VF07$Fj\\m$\"'`LUF07$F_]m$\"'#*3SF07$Fd]m$\"'osPF07$Fi ]m$\"'#4)HF07$F^^m$\"'*4A#F07$Fc^m$\"'x5;F07$Fh^m$\"'q?7F07$F]_m$\"&;# **F07$Fb_m$\"&QU)F07$Fg_m$\"&UM(F07$F\\`m$\"&(\\kF07$Ff`m$\"&*)3&F07$F [am$\"&3t%F07$F`am$\"&G[%F07$F_er$\"&(fWF07$Feam$\"&:h%F07$Fjam$\"&ta& F07$F_bm$\"&R6(F07$Fdbm$\"&;A*F07$Fibm$\"'*))H\"F07$F^cm$\"'8e=F07$Fcc m$\"'YJAF07$Fhcm$\"'#Gc#F07$F]dm$\"'&em#F07$Fbdm$\"'ySFF07$Fgdm$\"'bcF F07$F\\em$\"'SzFF07$Faem$\"'xsFF07$Ffem$\"')*RFF07$F`fm$\"'N%p#F07$Fjf m$\"':3EF07$F_gm$\"'+%R#F07$Fdgm$\"'x[AF07$Figm$\"'u/>F07$F^hm$\"'6n:F 07$F\\hp$\"'cY9F07$Fahp$\"'^b8F07$Ffhp$\"'-&H\"F07$Fchm$\"'i_7F07$Fcip $\"'0r7F07$Fhhm$\"'w48F07$F`jp$\"'#[I\"F07$F]im$\"'#GJ\"F07$Fbim$\"'m \\8F07$Fgim$\"'y*R\"F07$F\\jm$\"'w*F07$ Fg^n$\"&-\"eF07$$\"5-++v=n(*fDVFc[l$\"&]O%F07$F\\_n$\"&*pLF07$Fa_n$\"& et#F07$Ff_n$\"&>H#F07$F``n$\"&gn\"F07$Fe`n$\"&aO\"F07$Fj`n$\"&[A\"F07$ F_an$\"&YE\"F07$Fdan$\"&C]\"F0-Fceu6&F[bnFeeuF(F(-Fabn6#%:scheme~with~ c[10]=169/170G-%%FONTG6$%*HELVETICAGFg_s-%&TITLEG6#%Verror~curves~for~ 11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fbiy-%%V IEWG6$;F(Fdan%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10 ]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme w ith c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 8 of 11 stage, order 7 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/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 "Solu tion: " }}{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 := diff( 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.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6# \"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\" \",(*&\"\"$\"\"\")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&-%'CURVESG 6$7io7$$\"\"!F)F(7$$\"3()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZb cF-$\"3ZF^'eEW*Q9F-7$$\"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3 c?q**e5wz?F>7$$\"3$4\"3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F >$\"3L\\!fpl0?S%F>7$$\"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$$\"3oKCVK;BK aF>$\"3y!44&)e3J')*F>7$$\"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$$\"3ynvcnz>k\"*F>$\"31*=-!RxBm$\"3+Ep.yT!)o=Fho7$$\"3\">*=*=tV]-\"Fho$\"3eT(\\)>LNc >Fho7$$\"3!z$y$y&G+\"3\"Fho$\"3!4zI&*RE\"\\?Fho7$$\"37>*=*y\"*GM6Fho$ \"3tVizMbXM@Fho7$$\"3'\\'['['y))*=\"Fho$\"3fa-R]1_?AFho7$$\"3s%f%f9[%4 C\"Fho$\"3#y2[?x6qH#Fho7$$\"3[KCVKm,'H\"Fho$\"3)*e,ywZ!pP#Fho7$$\"3c8N ^t2A`8Fho$\"3WUCG,38dCFho7$$\"3Y'['[Yr,.9Fho$\"3@#)=I[fvCDFho7$$\"3q8N ^$f)zc9Fho$\"3%3DX;Fho$\"3iG:8:1'ez#Fho7 $$\"3c%f%fuKqx;Fho$\"3[Z4\"[C.I'GFho7$$\"3=^8N\"ft,t\"Fho$\"3!\\dJru-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$)******>MoW>F ho$\"3Mca,!\\@%QJFho7$$\"3!*['[')ep#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31Yf% faPE0#Fho$\"3!)Qt2wy;NKFho7$$\"3C^8N^)3&3@Fho$\"3Ik#>&=s*=G$Fho7$$\"3E >*=*e&>B;#Fho$\"3HYm&4R?ZK$Fho7$$\"3_f%f%z([t@#Fho$\"3-!>y6]piO$Fho7$$ \"3oq-FIB#>F#Fho$\"3%>b4%)\\*>0MFho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%*QMFh o7$$\"3eCVKkVazBFho$\"3))))QO[-#QDFho$\"3]n/[U d8hNFho7$$\"3AdnvO#pkf#Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\" *)HR3Wug$Fho7$$\"3$ovcn0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\" 3)*3G:eW%Hk$Fho7$$\"3[aS09&4M\"GFho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUliGFho $\"3?xe_?\\\"fm$Fho7$$\"3163\"3DQ(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=*=HE \"H(HFho$\"3IP\"\\xLemn$Fho7$$\"3%pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\"3-d nvO9*43$Fho$\"3y,I^j&GHn$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$ \"3'>*=*=tG))=$Fho$\"3#R\\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOFho 7$$\"3!>*=*=h2%)H$Fho$\"3%>QOHhJ`h$Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f0f $Fho7$$\"3)\\'['[I)[0MFho$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59(y GT$>NFho7$$\"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$%*HELVETICAG \"\"*-%+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 following c ode constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " ba sed on each of the methods and gives the " }{TEXT 260 22 "root mean sq uare error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: nums teps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`i nitial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: ` ,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`s cheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[1 0]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: err s := []:\nDigits := 30:\nfor ct to 6 do\n Wn_RK7_||ct := RK7_||ct(W( x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Wn_RK7_ ||ct):\n for ii to numpts do\n sm := sm+(Wn_RK7_||ct[ii,2]-w(Wn _RK7_||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*(% \"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initia l~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G \"$+%Q)pprint346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G \"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+D0$\\V#!#E7$/*&%-scheme~with~GF+F,F+# \"#:\"#;$\"+l8`^9F57$/F8#\"#>\"#?$\"+a<*GY\"F57$/F8#\"#K\"#L$\"+FJU65F 57$/F8#\"#\")\"##)$\"+GPW,5F57$/F8#\"$p\"\"$q\"$\"+C>W%=)!#FQ)pprint35 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "T he 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 = 3.499;" "6 #/%\"xG-%&FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 663 "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 := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19 /20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 30:\nfor ct to 6 do\n wn_R K7_||ct := RK7_||ct(W(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx : = 3.499: wxx := evalf(w(xx)):\nfor ct to 6 do\n errs := [op(errs),ab s(wn_RK7_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,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)pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Shar p-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+Ja(*f%*!#F7$/*& %-scheme~with~GF+F,F+#\"#:\"#;$\"+f$)*e+&F57$/F8#\"#>\"#?$\"+*RPi)fF57 $/F8#\"#K\"#L$\"+*Hlfk#F57$/F8#\"#\")\"##)$\"+Pw1c6F57$/F8#\"$p\"\"$q \"$\"+tVkG6F5Q)pprint376\"" }}}{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 "[0,4]" "6#7$\"\"!\" \"%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follow s 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 0 413 "mthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10 ]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme \+ with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits \+ := 20:\nfor ct to 6 do\n sm := NCint((w(x)-'wn_RK7_||ct'(x))^2,x=0.. 4,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(s m/4)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner ~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+'*\\1)R#!#E7$/*&%-schem e~with~GF+F,F+#\"#:\"#;$\"+%o][T\"F57$/F8#\"#>\"#?$\"+(pGYW\"F57$/F8# \"#K\"#L$\"+_;a.)*!#F7$/F8#\"#\")\"##)$\"+63%H))*FM7$/F8#\"$p\"\"$q\"$ \"+DK'\\:)FMQ)pprint386\"" }}}{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 0 512 "evalf[20](plot(['wn_RK7_1'(x)-w(x),'wn_RK7_2'(x)-w (x),'wn_RK7_3'(x)-w(x),\n'wn_RK7_4'(x)-w(x),'wn_RK7_5'(x)-w(x),'wn_RK7 _6'(x)-w(x)],x=0..4,-6e-17..7e-17,font=[HELVETICA,9],\ncolor=[COLOR(RG B,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0, 1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with \+ c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`sche me with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error curve s for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1000 633 633 {PLOTDATA 2 "6,-%'CURVESG6%7dq7$$\"\"!F)F(7$$\" 5LLLLL3x&)*3\"!#@$!#8!#?7$$\"5mmmmm;arz@F-$!#7F07$$\"5+++++DJdpKF-$!\" $F07$$\"5LLLLLL3VfVF-$\"#?F07$$\"5+++++]i9RlF-$\"$l\"F07$$\"5mmmmmm;') =()F-$\"$b%F07$$\"5MLLLeR?ah5F0$\"$0*F07$$\"5++++]7z>^7F0$\"%M:F07$$\" 5mmmmT&y`3W\"F0$\"%#H#F07$$\"5LLLLLe'40j\"F0$\"%sIF07$$\"5mmmm\"H_(zV= F0$\"%.PF07$$\"5++++](Q&3d?F0$\"%SSF07$$\"5MLLek`tS5@F0$\"%kSF07$$\"5n mm;z>$HP;#F0$\"%WSF07$$\"5ML$ekGI!R!>#F0$\"%4SF07$$\"5+++v$fG^q@#F0$\" %gRF07$$\"5nm;/,pArVAF0$\"%eRF07$$\"5MLLL3_KPqAF0$\"%XRF07$$\"5+++]P%= F07$$\"5nmmmT&))Rn'HF0$\"%q>F07$$\"5nmmm ;/`T?IF0$\"%)*=F07$$\"5nmmm\"Hs!4uIF0$\"%))=F07$$\"5nmmmmThwFJF0$\"%C> F07$$\"5nmmm;zp6NKF0$\"%x?F07$$\"5nmmmm;yYULF0$\"%]BF07$$\"5MLLL3_!\\h b$F0$\"%SJF07$$\"5++++](GI)pPF0$\"%=TF07$$\"5mmmm\"H_6N)RF0$\"%F^F07$$ \"5LLLLLeF>(>%F0$\"%(4'F07$$\"5mmmmm;EI&R%F0$\"%8lF07$$\"5+++++vCT$f%F 0$\"%QnF07$$\"5mmmm;/uY#p%F0$\"%(y'F07$$\"5LLLLLLB_\"z%F0$\"%8oF07$$\" 5++++]isd!*[F0$\"%!z'F07$$\"5mmmmm\">K'*)\\F0$\"%[nF07$$\"5NLLLLeZ*)*R &F0$\"%/kF07$$\"5+++++Dt:5eF0$\"%%*eF07$$\"5mmmmm\"fX(emF0$\"%%*[F07$$ \"5+++++DCh/vF0$\"%'4%F07$$\"5LLLLLL/pu$)F0$\"%#\\$F07$$\"5mmmmm;c0T\" *F0$\"%yIF07$$\"5+++++I,Q+5!#>$\"%5FF07$$\"5+++++]*3q3\"Fey$\"%5CF07$$ \"5+++++q=\\q6Fey$\"%w@F07$$\"5nmmm;fBIY7Fey$\"%#*>F07$$\"5LLLLLj$[kL \"Fey$\"%A=F07$$\"5LLLLL`Q\"GT\"Fey$\"%)p\"F07$$\"5++++]s]k,:Fey$\"%s: F07$$\"5LLLLL`dF!e\"Fey$\"%![\"F07$$\"5++++]sgam;Fey$\"%)Q\"F07$$\"5++ ++]Fey$\" %u6F07$$\"5nmmmmTc-)*>Fey$\"%A6F07$$\"5nmmm;f`@'3#Fey$\"%!4\"F07$$\"5+ +++]nZ)H;#Fey$\"%[5F07$$\"5nmmmmJy*eC#Fey$\"%75F07$$\"5+++++S^bJBFey$ \"$y*F07$$\"5+++++0TN:CFey$\"$[*F07$$\"5++++]7RV'\\#Fey$\"$S*F07$$\"5+ ++++:#fke#Fey$\"$9*F07$$\"5LLLLL`4NnEFey$\"$1*F07$$\"5+++++],s`FFey$\" $!*)F07$$\"5nmmm;zM)>$GFey$\"$q)F07$$\"5+++++qfaQFeyF[bl7$$\"5MLL3xE3l@QFe y$\"$O*F07$$\"5nm;aj\\>$p#QFeyFaal7$$\"5++++]sI@KQFey$\"$G)F07$$\"5+++ ++S2lsQFey$\"$[&F07$$\"5++++]2%)38RFey$\"\"#F07$$\"5+++]i0j\"[$RFey$!$ S&F07$$\"5++++v.UacRFey$!%W:F07$$\"5+++D\"G:3u'RFey$!%UBF07$$\"5+++](= 5s#yRFey$!%/OF07$$\"5++]iSwSq$)RFey$!%%[%F07$$\"5+++v$40O\"*)RFey$!%w_ F07$$\"5++DJ?Q?&=*RFey$!%GhF07$$\"5++](oa-oX*RFey$!%#G'F07$$\"5++vVt7S G(*RFey$!%oiF07$$\"\"%F)$!%y$)F0-%&COLORG6&%$RGBGF($F]hl!\"\"$\"\"\"F) -%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7`qF'7$F2$!#LF 07$F<$!#*)F07$FA$!$m\"F07$FF$!$/$F07$FP$!$l'F07$FZ$!$/*F07$Fin$!$h(F07 $F^o$!$\"HF07$Fho$\"#pF07$F\\q$\"$H&F07$FaqF\\^l7$Ffq$\"%d;F07$$\"5nmm m;a>,\"f#F0$\"%KBF07$F[r$\"%\\IF07$$\"5nmmm;HOr0GF0$\"%UNF07$F`r$\"%qQ F07$Fjr$\"%*4%F07$Fds$\"%BUF07$Fis$\"%lUF07$F^t$\"%TUF07$Fht$\"%lPF07$ Fbu$\"%4MF07$$\"5++++](oZiH%F0$\"%!Q$F07$Fgu$\"%`LF07$$\"5LLLL$eadV\\% F0$\"%TLF07$F\\v$\"%JLF07$Fav$\"%DLF07$Ffv$\"%ILF07$F[w$\"%:LF07$F`w$ \"%/LF07$$\"5+++++vMw%>&F0$\"%pKF07$Few$\"%=KF07$$\"5qmmmmTg-0cF0$\"%K JF07$Fjw$\"%MIF07$F_x$\"%uDF07$Fdx$\"%m@F07$Fix$\"%U=F07$F^y$\"%G;F07$ Fcy$\"%S9F07$Fiy$\"%q7F07$F^z$\"%Y6F07$Fcz$\"%_5F07$Fhz$\"$i*F07$F][lF `bl7$Fb[l$\"$K)F07$Fg[l$\"$!yF07$F\\\\l$\"$Q(F07$Fa\\l$\"$%oF07$Ff\\l$ \"$c'F07$F[]l$\"$9'F07$F`]l$\"$#fF07$Fe]l$\"$!eF07$Fj]l$\"$e&F07$F_^l$ \"$K&F07$Fd^l$\"$3&F07$Fi^l$\"$)\\F07$F^_l$\"$+&F07$Fc_l$\"$%[F07$Fh_l $\"$w%F07$F]`l$\"$q%F07$Fb`l$\"$g%F07$Fg`l$\"$e%F07$F\\alF\\hm7$F_alFi gm7$Fdal$\"$[%F07$FialFfgm7$F^bl$\"$y%F07$Fcbl$\"$!\\F07$FhblF[im7$F[c l$\"$/&F07$F`cl$\"$?&F07$Fecl$\"$G&F07$Fjcl$\"$]&F07$$\"5++]7`Hi*Hv$Fe y$\"$9&F07$F_dl$\"$_&F07$$\"5nm;/Ev%eNw$Fey$\"$7&F07$Fddl$\"$%\\F07$Fi dl$\"$;&F07$F^el$\"$A&F07$FcelF]jm7$FfelF[[n7$F[flFdhm7$F`fl$\"$O%F07$ Fefl$\"$Y%F07$Fhfl$\"$'\\F07$F]gl$\"$I%F07$F`gl$\"$)RF07$Fegl$\"$)>F07 $Fjgl$!$o\"F0F^hl7$Fdhl$!%%H\"F07$Fihl$!%7>F07$F^il$!%CHF07$FcilF`il7$ Fhil$!%YUF07$F]jl$!%Q\\F07$Fbjl$!%s]F07$Fgjl$!%Q]F07$F\\[m$!%)z'F0-Fa[ m6&Fc[mF($\"\"(Fe[mF(-Fi[m6#%8scheme~with~c[10]=15/16G-F$6%7[qF'7$FFFf \\m7$FK$!$I#F07$FPFe]m7$$\"5LLL$e*[e-Y8F0$!$<$F07$FU$!$C$F07$$\"5LLLek `xE)[\"F0$!$G$F07$$\"5+++](=s\"oN:F0$!$=$F07$$\"5mmmT5!p&4$e\"F0$!$5$F 07$FZ$!$!GF07$Fin$!$>\"F07$F^o$\"$A#F07$F\\q$\"$I(F07$Ffq$\"%f8F07$F[r $\"%!3#F07$F`r$\"%!\\#F07$Fds$\"%vEF07$F^t$\"%dFF07$Fct$\"%2GF07$Fht$ \"%+HF07$F]u$\"%wIF07$Fbu$\"%kLF07$F\\v$\"%'o$F07$F`w$\"%RQF07$$\"5NLL LLLy>#4&F0$\"%\\QF07$F`bm$\"%XQF07$$\"5qmmmm;\"HtH&F0$\"%EQF07$Few$\"% 5QF07$Fhbm$\"%APF07$Fjw$\"%/OF07$F_x$\"%kIF07$Fdx$\"%'e#F07$Fix$\"%-AF 07$F^y$\"%Q>F07$Fcy$\"%5F07$FU$!%#o#F07 $FZ$!%ALF07$$\"5+++]i!f`rt\"F0$!%BNF07$Fin$!%BOF07$$\"5LLL$3_XT/&>F0$! %cNF07$F^o$!%PLF07$Fho$!%SHF07$F\\q$!%]BF07$$\"5nmm\"H#=_pBBF0$!%:NF07$Fct$\"%iIF07$Fht$\"%XCF07$F]u $\"%s=F07$Fbu$\"%l9F07$Fd`m$\"%%Q\"F07$Fgu$\"%E8F07$F\\am$\"%,8F07$F\\ v$\"%&H\"F07$Ffv$\"%Q8F07$F`wF^\\l7$F`bm$\"%\\9F07$Few$\"%-:F07$$\"5++ ++++/Y-bF0$\"%9:F07$Fhbm$\"%7:F07$$\"5NLLLL$o\"f2dF0$\"%'\\\"F07$Fjw$ \"%/:F07$$\"5NLLLLe9XMiF0$\"%O9F07$F_x$\"%W8F07$FdxFbdm7$Fix$\"$s*F07$ F^yFfal7$Fcy$\"$g(F07$Fiy$\"$q'F07$F^zFh\\o7$FczF`jm7$Fhz$\"$-&F07$F][ lFhhm7$Fb[l$\"$U%F07$Fg[l$\"$5%F07$F\\\\l$\"$)QF07$Fa\\l$\"$a$F07$Ff\\ l$\"$Y$F07$F[]l$\"$C$F07$F`]l$\"$7$F07$Fe]l$\"$5$F07$Fj]l$\"$)HF07$F_^ l$\"$#GF07$Fd^l$\"$o#F07$Fi^l$\"$e#F07$F^_l$\"$g#F07$Fc_l$\"$a#F07$Fh_ l$\"$c#F07$F]`l$\"$]#F07$Fb`l$\"$S#F07$Fg`l$\"$Q#F07$F\\alF__p7$F_alF \\_p7$Fdal$\"$G#F07$FialFi^p7$F^bl$\"$[#F07$Fcbl$\"$q#F07$FhblF\\_p7$F [cl$\"$k#F07$F`clF^`p7$Fecl$\"$y#F07$Fjcl$\"$+$F07$F[jmFb`p7$F_dl$\"$# HF07$Fcjm$\"$i#F07$Fddl$\"$W#F07$Fidl$\"$m#F07$F^el$\"$s#F07$Fcel$\"$u #F07$Ffel$\"$'GF07$F[fl$\"$=#F07$F`fl$\"$'>F07$Fefl$\"$;#F07$FhflFfap7 $$\"5++DJ?)Q\"HCQFeyF__p7$F]gl$\"$+#F07$$\"5ML3x16DdHQFey$\"$9#F07$F`g l$\"$y\"F07$Fegl$\"#GF07$Fjgl$!$=#F07$F_hl$!$![F07$Fdhl$!%W5F07$Fihl$! %7:F07$F^il$!%CBF07$Fcil$!%MGF07$Fhil$!%YLF07$F]jl$!%))QF07$Fbjl$!%ASF 07$Fgjl$!%[RF07$F\\[m$!%)Q&F0-%'COLOURG6&Fc[m$\"*++++\"!\")F(F`fp-Fi[m 6#%8scheme~with~c[10]=32/33G-F$6%7\\tF'F\\`o7$F2$!#YF07$F7$!#&*F07$F<$ !$!=F07$FA$!$k%F07$FF$!%55F07$$\"50+++DJ59n'*F-$!%a8F07$FK$!%ur*=F0$!%=bF07$Fb bo$!%2bF07$$\"5++]7GQC5x>F0$!%([&F07$$\"5mmmTN@Mw.?F0$!%A`F07$$\"5LL$3 FWSC/.#F0$!%3`F07$F^o$!%%H&F07$Fho$!%O[F07$F\\q$!%9TF07$F`co$!%8LF07$F aq$!%sJF07$$\"5nm;zWn\"yOS#F0$!%xAF07$Fhco$!%lAF07$$\"5++]PfL,+dCF0$!% XAF07$Ffq$!%*)>F07$$\"5nmm;/E))\\5DF0$!%Q6F07$F`do$!%U6F07$$\"5nmm;zWU f()yFF0$\"%97F07$Fj^m$\"%J?F07$$\"5nmm;a Q8bKGF0$\"%=?F07$$\"5nmmm\"z/*QfGF0$\"%K?F07$$\"5nmm;HdnA')GF0$\"%]AF0 7$F`r$\"%zFF07$Fjr$\"%*H$F07$Fds$\"%tNF07$Fjeo$\"%nNF07$F_fo$\"%pNF07$ Fdfo$\"%aOF07$Fis$\"%EOF07$F\\go$\"%uNF07$F^t$\"%!\\$F07$$\"5+++]PM%3$ \\MF0$\"%(>$F07$Fct$\"%%z#F07$$\"5nmm;zp'*)Hm$F0$\"%PBF07$Fht$\"%e=F07 $$\"5LLL$3_!4nwQF0$\"%\"Q\"F07$F]u$\"$q*F07$$\"5+++]iS@N!4%F0$\"$*eF07 $Fbu$\"$'HF07$Fd`m$\"$e\"F07$Fgu$\"#hF07$$\"5++++D\"3I[W%F0$\"#QF07$F \\am$\"#8F07$$\"5mmmmT5])Qa%F0$\"#5F07$F\\vF97$Ffv$\"#aF07$F`w$\"$Y\"F 07$Few$\"$*QF07$FjwF]jm7$$\"5qmmmm\"R/B-'F0$\"$U&F07$FfjoF^jn7$$\"5+++ ++D&)fYkF0Fdin7$F_x$\"$a&F07$$\"5NLLLL3!z;3(F0F^[n7$Fdx$\"$'[F07$Fix$ \"$A%F07$F^y$\"$o$F07$Fcy$\"$I$F07$Fiy$\"$!HF07$F^zFi^p7$Fcz$\"$K#F07$ Fhz$\"$7#F07$F][l$\"$3#F07$Fb[l$\"$#>F07$Fg[l$\"$!=F07$F\\\\l$\"$o\"F0 7$Fa\\l$\"$a\"F07$Ff\\lFafq7$F[]l$\"$M\"F07$F`]l$\"$K\"F07$Fe]l$\"$S\" F07$Fj]l$\"$G\"F07$F_^l$\"$A\"F07$Fd^l$\"$3\"F07$Fi^lF\\[r7$F^_l$\"$? \"F07$Fc_l$\"$/\"F07$Fh_l$\"$;\"F07$F]`l$\"$5\"F07$Fb`l$\"$+\"F07$Fg`l $\"#)*F07$F\\alFi[r7$F_alFi[r7$Fdal$\"#))F07$FialFf[r7$F^blF\\[r7$Fcbl F`[r7$FhblF\\\\r7$F[cl$\"$9\"F07$F`clFi[r7$Fecl$\"$=\"F07$Fjcl$\"$]\"F 07$F[jmF[]r7$F_dl$\"$U\"F07$Fcjm$\"$7\"F07$FddlFc[r7$Fidl$\"$E\"F07$F^ elFf]r7$Fcel$\"$W\"F07$Ffel$\"$c\"F07$F[flFd\\r7$F`fl$\"#wF07$Fefl$\"# '*F07$FhflFafq7$F\\cpF`[r7$F]glFh^q7$FbcpFc[r7$F`gl$\"#oF07$Fegl$!#AF0 7$Fjgl$!$3\"F07$F_hlF]_nFf]o7$Fihl$!$_)F07$F^il$!%99F07$Fcil$!%9%F07$Fjao$!%2XF07$Fin$!%4ZF07$F[jp$!%JZF07$Fbbo$!%DZF07$Fhjp$!%#f% F07$F^o$!%nXF07$Fho$!%/UF07$F\\q$!%![%3QFeyFfjq7$FfelFebp7$F[flFfjq7$F`flFg_s7$FeflF] ^r7$Fhfl$\"$'=F07$F\\cpFfbs7$F]glFb_s7$FbcpFa^r7$F`glF_]r7$$\"5+++v$4y ns$QFeyFf]r7$$\"5+++]P*[AB%QFeyFiiq7$$\"5++]PfV)\\[%QFeyFfiq7$$\"5+++D \"y>xt%QFey$\"#%)F07$$\"5++]7._X!*\\QFeyF\\\\r7$$\"5++++D1>V_QFey$\"$O \"F07$$\"5++](o/Ef\\&QFeyF[_r7$$\"5+++vo9m[dQFey$\"$5#F07$$\"5++]i!*oR ,gQFeyFcjq7$$\"5+++]7B8aiQFey$\"#!*F07$$\"5++]PMx'o]'QFeyFb_p7$$\"5+++ DcJgfnQFeyFfiq7$$\"5++]7y&QB,(QFeyF_cp7$FeglFd\\r7$$\"5+++vV[aqxQFeyFj es7$$\"5+++](o:gF)QFey$\"$O#F07$$\"5+++DJl[\"y)QFeyFj\\s7$$\"5++++vt&p G*QFeyF]as7$$\"5++](oz#pR&*QFeyF_\\r7$$\"5+++v=#GCz*QFeyFd\\r7$$\"5++] iSO;X+RFeyF]jq7$$\"5+++]i!**yH!RFeyF^_s7$$\"5++]P%[M1b!RFeyF\\gs7$$\"5 +++D1*pL!3RFey$\"#sF07$$\"5++]7G`5c5RFey$\"#')F07$FjglFijq7$$\"5++]7.# Q?&=RFeyFh^r7$$\"5+++DccB&R#RFey$\"#SF07$$\"5++D\"GQMom#RFeyFd^r7$$\"5 ++]P4JVQHRFeyFjeq7$$\"5++v$f$=.5KRFey$\"#eF07$F_hlFcjq7$$\"5++]i:!G[-% RFeyFi\\t7$$\"5+++voa-oXRFeyF47$$\"5++DJ&>C'R[RFeyF^fq7$$\"5++](=#HA6^ RFey$!$!>F07$$\"5++vV[;#GQ&RFeyFdhs7$Fdhl$\"#EF07$Fihl$!##*F07$F^il$!$ %RF07$Fhil$!$;'F07$F\\[m$!%)G\"F0-F^fp6&Fc[mF`fpF(F(-Fi[m6#%:scheme~wi th~c[10]=169/170G-%%FONTG6$%*HELVETICAGFd_o-%&TITLEG6#%Verror~curves~f or~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F]`t- %%VIEWG6$;F(F\\[m;$!\"'F]br$Fd^nF]br" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33 " "scheme with c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 9 of 11 stage, order 7 Rung e-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x))*y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-%$co sG6#*&\"\"#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 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" "6#/%\"yG *&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\"xGF&F&*&F/F&F0F&F&*&F&F&F/! \"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x)=-(1+cos(2*x))*y (x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x));\nm := unapply(rhs (%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9],labels=[`x`,`y(x) `]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\"\"$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&-%'CURVESG6$7Y7$$\"\"!F)$\"3 :&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H8F,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%o FJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY%G?7 \"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"33++]7[X$ocbFJ7$$\"3))***\\PpnsM*FJ$\" 3!\\;$Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aFJ7$$\"3-+++!R5'f5F,$\"3 q`:6QhHm`FJ7$$\"3)***\\P/QBE6F,$\"3@Igj*yDKK&FJ7$$\"3!******\\\"o?&=\" F,$\"3i/K.-M\\%H&FJ7$$\"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&QAi\"F,$\"3:*4?^OZ,C&FJ7$ $\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"3!******\\Nm'[F,$\"3[h+0^h(R>&FJ7$$\"3z*****\\@80+#F,$\" 3!zBIi>A%o^FJ7$$\"31++]7,Hl?F,$\"3<)30`]&>L^FJ7$$\"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@O tBF,$\"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%>5p i#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%FJ7$$\"33++Dr\"[8v #F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ7$$\"31+]P/)fT(GF ,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4%FJ7$$\"\"$F)$\"3n tdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l;F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 745 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numst eps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y) ],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps : `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/ 12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with \+ `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170] : errs := []:\nDigits := 20:\nfor ct to 6 do\n Mn_RK7_||ct := RK7_|| ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := \+ nops(Mn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Mn_RK7_||c t[ii,2]-m(Mn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqr t(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf (errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~f ield:~~~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)pprint396\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Shar p-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+q;_^7!#B7$/*&%- scheme~with~GF+F,F+#\"#:\"#;$\"+tsvU$*!#C7$/F8#\"#>\"#?$\"+htxz6F57$/F 8#\"#K\"#L$\"+b9M]wF?7$/F8#\"#\")\"##)$\"+De*)=sF?7$/F8#\"$p\"\"$q\"$ \"+.UUCsF?Q)pprint406\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "num erical procedures" }{TEXT -1 56 " for solutions based on each of the R unge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the v alue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is a lso given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 676 "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 width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`sc heme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10 ]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 25:\nfor ct to 6 do\n mn_RK7_||ct := RK7_||ct(M(x,y),x,y,x0,evalf(y0),hh,num steps,true);\nend do:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 6 \+ do\n errs := [op(errs),abs(mn_RK7_||ct(xx)-mxx)];\nend do:\nDigits : = 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG 6#,$*&\"\"#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)pp rint416\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+%QxdK#!#C7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+a@4V\"#?$\"+Fad(>#F57$/F8#\"#K\"#L$\"+m&Q@V\"F57$/F8# \"#\")\"##)$\"+?$oAN\"F57$/F8#\"$p\"\"$q\"$\"+6v!yM\"F5Q)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, 3];" "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 integratio n by the 7 point Newton-Cotes method over 150 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((m(x)-'mn_RK7_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7 ,factor=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 1 0:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+hI;M7!#B7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+\"z*e8#*!#C7$/F8#\"#>\"#?$\"+UqWj6F57$/F8#\"#K\"#L$\"+-G)[a(F?7$/F8 #\"#\")\"##)$\"+=#>%>rF?7$/F8#\"$p\"\"$q\"$\"+MXWCrF?Q)pprint436\"" }} }{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 0 500 "evalf[20] (plot(['mn_RK7_1'(x)-m(x),'mn_RK7_2'(x)-m(x),'mn_RK7_3'(x)-m(x),'mn_RK 7_4'(x)-m(x),\n'mn_RK7_5'(x)-m(x),'mn_RK7_6'(x)-m(x)],x=0..0.5,font=[H ELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9, .4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme wi th c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/20`,`s cheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10]=1 69/170`],\ntitle=`error curves for 11 stage order 7 Runge-Kutta method s`));" }}{PARA 13 "" 1 "" {GLPLOT2D 873 562 562 {PLOTDATA 2 "6,-%'CURV ESG6%7\\q7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#AF(7$$\"5NLLL$3FWYs#F-$\"#8!#>7 $$\"5omm;aQ`!eS$F-$\"#wF37$$\"5-+++D1k'p3%F-$\"$C$F37$$\"5OLL$eRZF\"oZ F-$\"%&3\"F37$$\"5qmmmmT&)G\\aF-$\"%'3$F37$$\"50++]P4'\\/8'F-$\"%RxF37 $$\"5SLLL3x1h6oF-$\"&rv\"F37$$\"5qmm;zW*F-$\"':*y\"F37$$\"5qmmm\"z%\\DO&*F-$\"'1mBF37$$\"5N LL3x\"[No()*F-$\"'@(4$F37$$\"5+++Dc,;u@5!#@$\"'l(R$F37$$\"5nm;z%\\l*zb 5Fbp$\"''[Q$F37$$\"5MLLLL3x&)*3\"Fbp$\"':sLF37$$\"5nmm\"z%\\v#pK\"Fbp$ \"'5'G$F37$$\"5+++]i!R(*Rc\"Fbp$\"'MCKF37$$\"5nm;z>6B`#o\"Fbp$\"'BdKF3 7$$\"5MLL3xJs1,=Fbp$\"'i]MF37$$\"5nm\"Hd?pM.'=Fbp$\"'MsOF37$$\"5++]PM_ @g>>Fbp$\"'=NSF37$$\"5n;zp[#)eB\\>Fbp$\"'s)G%F37$$\"5ML3-j7'p)y>Fbp$\" '7.YF37$$\"5+]PMxUL]3?Fbp$\"'Vm[F37$$\"5nmmm\"H2P\"Q?Fbp$\"'c^[F37$$\" 5MLLek.pu/BFbp$\"'D@ZF37$$\"5+++]PMnNrDFbp$\"'W4YF37$$\"5nm\"Hd?>4!QEF bp$\"'(of%F37$$\"5ML$eR(\\;m/FFbp$\"'\"Gg%F37$$\"5++v=U2TJrFFbp$\"'7TY F37$$\"5nmmT5ll'z$GFbp$\"'JLZF37$$\"5++](o/[r7(HFbp$\"'t@_F37$$\"5MLLL $eRwX5$Fbp$\"'Pd`F37$$\"5MLLLe*[`HP$Fbp$\"'XA_F37$$\"5MLLLL$eI8k$Fbp$ \"'!46&F37$$\"5MLLL3xwq4RFbp$\"'?Y_F37$$\"5NLLL$3x%3yTFbp$\"'m&F37$$\"5NL3FWXK1z ZFbp$\"'J(=&F37$$\"5om;/^J'Qe%[Fbp$\"'+-_F37$$\"5-+D\"yv,9E\"\\Fbp$\"' /[_F37$$\"5NLLek.%*Qz\\Fbp$\"'j`F37$$\"5N$eRZF[_&z]Fbp$\"'&zM&F37$$\"5++]7yv,%H6 &Fbp$\"'zK`F37$$\"5NLe*[=c:(z^Fbp$\"'o-`F37$$\"5ommm\"z%4\\Y_Fbp$\"'&G F&F37$$\"5NLLLeR-/PiFbp$\"'An]F37$$\"5-+++DcmpisFbp$\"'A3[F37$$\"5OLLL e*)>VB$)Fbp$\"'nHXF37$$\"5.+++DJbw!Q*Fbp$\"'7hUF37$$\"5nmmm;/j$o/\"!#? $\"'6+SF37$$\"5MLLL3_>jU6F`]l$\"'x)y$F37$$\"5++++]i^Z]7F`]l$\"'slNF37$ $\"5++++](=h(e8F`]l$\"(01O$F`]l7$$\"5++++]P[6j9F`]l$\"(*yzJF`]l7$$\"5M LL$e*[z(yb\"F`]l$\"(\"zGIF`]l7$$\"5nmmm;a/cq;F`]l$\"(oI'GF`]l7$$\"5nmm mm;t,mF`]l$\"(fK[#F`]l7$$\"5,++]i!f#=$3#F`]l$\"(T)oBF`]l7$$\"5,++](=xpe=# F`]l$\"(i!oAF`]l7$$\"5nmmm\"H28IH#F`]l$\"(r1<#F`]l7$$\"5nmm;zpSS\"R#F` ]l$\"(\"Q(3#F`]l7$$\"5MLLL3_?`(\\#F`]l$\"($z.?F`]l7$$\"5MLL$e*)>pxg#F` ]l$\"(=I#>F`]l7$$\"5,++]Pf4t.FF`]l$\"(At&=F`]l7$$\"5MLLLe*Gst!GF`]l$\" (%y!z\"F`]l7$$\"5,++++DRW9HF`]l$\"(sks\"F`]l7$$\"5,+++DJE>>IF`]l$\"(Sv m\"F`]l7$$\"5,++]i!RU07$F`]l$\"(sRh\"F`]l7$$\"5,+++v=S2LKF`]l$\"(P\"e: F`]l7$$\"5ommmm\"p)=MLF`]l$\"(&)4^\"F`]l7$$\"5,+++](=]@W$F`]l$\"(MNY\" F`]l7$$\"5MLL$e*[$z*RNF`]l$\"(MHU\"F`]l7$$\"5,+++]iC$pk$F`]l$\"(e4Q\"F `]l7$$\"5omm;H2qcZPF`]l$\"(.OM\"F`]l7$$\"5,++]7.\"fF&QF`]l$\"(7mI\"F`] l7$$\"5ommm;/OgbRF`]l$\"(RBF\"F`]l7$$\"5,++]ilAFjSF`]l$\"(H$Q7F`]l7$$ \"5NLLLL$)*pp;%F`]l$\"(ds?\"F`]l7$$\"5NLLL3xe,tUF`]l$\"(wq<\"F`]l7$$\" 5omm;HdO=yVF`]l$\"(O'[6F`]l7$$\"5,++++D>#[Z%F`]l$\"(HP7\"F`]l7$$\"5omm mT&G!e&e%F`]l$\"(Sl4\"F`]l7$$\"5NLLLL$)Qk%o%F`]l$\"(pL2\"F`]l7$$\"5-++ ]iSjE!z%F`]l$\"((z\\5F`]l7$$\"5-++]P40O\"*[F`]l$\"(b#G5F`]l7$$\"\"&!\" \"$\"(jh+\"F`]l-%&COLORG6&%$RGBGF($\"\"#F_il$\"\"\"F)-%'LEGENDG6#%ESha rp-Verner~scheme~with~c[10]=11/12G-F$6%7\\qF'F*7$F/$\"#5F37$F5$\"#cF37 $F:$\"$S#F37$F?$\"$.)F37$FD$\"%%G#F37$FI$\"%GdF37$FN$\"&0I\"F37$FS$\"& Xs#F37$FX$\"&4M&F37$Fgn$\"&\">tF37$F\\o$\"&A!**F37$Fao$\"'\"RK\"F37$Ff o$\"'s] FF37$Fir$\"'\"*))HF37$F^s$\"'XxJF37$Fcs$\"'B6MF37$Fhs$\"'-2OF37$F]t$\" ','f$F37$Fbt$\"'U*\\$F37$Fgt$\"'i;MF37$F\\u$\"'S2MF37$Fau$\"'-7MF37$Ff u$\"'#3W$F37$F[v$\"'!*4NF37$F`v$\"'OvQF37$Fev$\"'8xRF37$Fjv$\"'(p(QF37 $F_w$\"'K%z$F37$Fdw$\"'>(*QF37$Fiw$\"'JFSF37$F^x$\"'4LRF37$Fcx$\"'8gQF 37$Fhx$\"'rcQF37$F]y$\"'@oQF37$Fby$\"'Y.RF37$Fgy$\"'7vRF37$F\\z$\"'#G+ %F37$Faz$\"'W\"*RF37$Ffz$\"'5!)RF37$F[[l$\"'\")oRF37$F`[l$\"'TYRF37$Fe [l$\"'@CRF37$Fj[l$\"'*[x$F37$F_\\l$\"''\\e$F37$Fd\\l$\"'pzLF37$Fi\\l$ \"'D\"=$F37$F^]l$\"'$z)HF37$Fd]l$\"'KJGF37$Fi]l$\"'klEF37$F^^l$\"(F`]l7$F\\`l$\"(G#f=F`]l7$Fa`l$\"(&ytAF37$F[p$\"'8/HF37$F `p$\"'j&=$F37$Ffp$\"'jtJF37$F[q$\"'rhJF37$F`q$\"'/\"3$F37$Feq$\"'EBIF3 7$Fjq$\"'WaIF37$F_r$\"'iOKF37$Fdr$\"'CXMF37$Fir$\"'X'y$F37$F^s$\"'![-% F37$Fcs$\"'I?VF37$Fhs$\"'vnXF37$F]t$\"'!Qb%F37$Fbt$\"'\\JWF37$Fgt$\"'k EVF37$F\\u$\"'(\\J%F37$Fau$\"'y?VF37$Ffu$\"'8dVF37$F[v$\"'IWWF37$F`v$ \"'E0\\F37$Fev$\"'RL]F37$Fjv$\"'j1\\F37$F_w$\"'+-[F37$Fdw$\"'9J\\F37$F iw$\"'B%4&F37$F^x$\"'/v\\F37$Fcx$\"'o#)[F37$Fhx$\"'Ay[F37$F]y$\"']#*[F 37$Fby$\"'fO\\F37$Fgy$\"'LE]F37$F\\z$\"'(41&F37$Faz$\"'dY]F37$Ffz$\"'C K]F37$F[[l$\"'(z,&F37$F`[l$\"'k*)\\F37$Fe[l$\"'dh\\F37$Fj[l$\"'%4x%F37 $F_\\l$\"'RHXF37$Fd\\l$\"'#)oUF37$Fi\\l$\"'?F`]l7$Feal$\"($y#*=F`]l7$Fjal$\"(]l\"=F`]l7$F_bl$\"(QXv\"F`]l 7$Fdbl$\"(AF37$F?$\"$c'F37$FD$\"%m=F37$FI $\"%uYF37$FN$\"&+1\"F37$FS$\"&)=AF37$FX$\"&iM%F37$Fgn$\"&S&fF37$F\\o$ \"&H0)F37$Fao$\"'Qw5F37$Ffo$\"'1B9F37$F[p$\"'Mi=F37$F`p$\"'$G/#F37$Ffp $\"'9N?F37$F[q$\"']F?F37$F`q$\"'xv>F37$Feq$\"'!)Q>F37$Fjq$\"'3f>F37$F_ r$\"'ww?F37$Fdr$\"'R6AF37$Fir$\"'aJCF37$F^s$\"'J&e#F37$Fcs$\"'%fx#F37$ Fhs$\"'eNHF37$F]t$\"'iEHF37$Fbt$\"',[GF37$Fgt$\"'r!y#F37$F\\u$\"'LtFF3 7$Fau$\"'HxFF37$Ffu$\"'3,GF37$F[v$\"'(y&GF37$F`v$\"'adJF37$Fev$\"'4TKF 37$Fjv$\"'ZfJF37$F_w$\"'G#4$F37$Fdw$\"'$z<$F37$Fiw$\"'y&G$F37$F^x$\"' \"*3KF37$Fcx$\"'m\\JF37$Fhx$\"'7ZJF37$F]y$\"'%p:$F37$Fby$\"'T'=$F37$Fg y$\"'+YKF37$F\\z$\"'0pKF37$Faz$\"'vfKF37$Ffz$\"'\\]KF37$F[[l$\"'GTKF37 $F`[l$\"'(HA$F37$Fe[l$\"'&[?$F37$Fj[l$\"'_&3$F37$F_\\l$\"'SKHF37$Fd\\l $\"'>mFF37$Fi\\l$\"'80EF37$F^]l$\"'\"zW#F37$Fd]l$\"'Z?BF37$Fi]l$\"'R&= #F37$F^^l$\"(<41#F`]l7$Fc^l$\"(M5&>F`]l7$Fh^l$\"(*=f=F`]l7$F]_l$\"(]\" e5 \"F`]l7$Fibl$\"(GC1\"F`]l7$F^cl$\"(Bi-\"F`]l7$Fccl$\"'6L**F`]l7$Fhcl$ \"'%**e*F`]l7$F]dl$\"':+$*F`]l7$Fbdl$\"'V3!*F`]l7$Fgdl$\"'$)e()F`]l7$F \\el$\"'s+&)F`]l7$Fael$\"'-r#)F`]l7$Ffel$\"'^V!)F`]l7$F[fl$\"'rKyF`]l7 $F`fl$\"'^BwF`]l7$Fefl$\"'OKuF`]l7$Fjfl$\"'nYsF`]l7$F_gl$\"'qrqF`]l7$F dgl$\"'X=pF`]l7$Figl$\"';^nF`]l7$F^hl$\"'d3mF`]l7$Fchl$\"'`jkF`]l7$Fhh l$\"'&4L'F`]l7$F]il$\"'-&>'F`]l-%'COLOURG6&Feil$\"*++++\"!\")F(Fh`p-F[ jl6#%8scheme~with~c[10]=32/33G-F$6%7jpF'F*7$F/$F^\\nF37$F5$\"#VF37$F:$ \"$$=F37$F?$\"$6'F37$FD$\"%QF37$Ffp$\"'%=#>F37$F[q$\"'i9>F37$F`q$\"'xl=F37$Feq$\"'rI=F37$Fjq$ \"'V\\=F37$F_r$\"'df>F37$Fdr$\"'2'3#F37$Fir$\"'Z$H#F37$F^s$\"'jQCF37$F cs$\"'()=EF37$Fhs$\"'-qFF37$F]t$\"'chFF37$Fbt$\"'R(o#F37$Fgt$\"'\"Qi#F 37$F\\u$\"'v;EF37$Fau$\"'K?EF37$Ffu$\"'^UEF37$F[v$\"'v&p#F37$F`v$\"'1y HF37$Fev$\"'2dIF37$Fjv$\"'3!)HF37$F_w$\"'i;HF37$Fdw$\"')o*HF37$Fiw$\"' l)4$F37$F^x$\"';EIF37$Fcx$\"'@qHF37$Fhx$\"'wnHF37$F]y$\"''p(HF37$Fby$ \"'q/IF37$Fgy$\"'$41$F37$F\\z$\"'r#3$F37$Faz$\"'%R2$F37$Ffz$\"'@lIF37$ F[[l$\"'_cIF37$F`[l$\"'ERIF37$Fe[l$\"'1#F37$F^^l$\"(VY%>F`]l7$Fc^l$\"(\"4T=F`]l7$Fh^l$\"(KXv\"F`]l 7$F]_l$\"(z#f;F`]l7$Fb_l$\"(I_e\"F`]l7$Fg_l$\"(3c]\"F`]l7$F\\`l$\"('fS 9F`]l7$Fa`l$\"($fu8F`]l7$Ff`l$\"(7kJ\"F`]l7$F[al$\"(u,E\"F`]l7$F`al$\" ()*>@\"F`]l7$Feal$\"(vO;\"F`]l7$Fjal$\"(Dp6\"F`]l7$F_bl$\"(z)y5F`]l7$F dbl$\"(C./\"F`]l7$Fibl$\"(YI+\"F`]l7$F^cl$\"'w)o*F`]l7$Fccl$\"'7y$*F`] l7$Fhcl$\"'Aa!*F`]l7$F]dl$\"'p!y)F`]l7$Fbdl$\"'L0&)F`]l7$Fgdl$\"'tp#)F `]l7$F\\el$\"'4E!)F`]l7$Fael$\"'E4yF`]l7$Ffel$\"'\\%f(F`]l7$F[fl$\"'\\ &R(F`]l7$F`fl$\"'+)>(F`]l7$Fefl$\"'bF37$Ffp $\"''o&>F37$F[q$\"'^\\>F37$F`q$\"'w**=F37$Feq$\"'Rj=F37$$\"5MLe9\"4&[E B;Fbp$\"'dl=F37$Fjq$\"'X!)=F37$$\"5++vV[r(*zTF37$F_r$\"'y() >F37$Fdr$\"'08@F37$Fir$\"'o?BF37$F^s$\"'?nCF37$Fcs$\"'A]EF37$Fhs$\"'a/ GF37$F]t$\"')fz#F37$Fbt$\"')3s#F37$Fgt$\"'6cEF37$F\\u$\"'Y[EF37$Fau$\" ';^EF37$Ffu$\"'5sEF37$F[v$\"'tBFF37$F`v$\"'l.IF37$Fev$\"'$G3$F37$Fjv$ \"'>0IF37$F_w$\"'nSHF37$Fdw$\"'_;IF37$Fiw$\"')e6$F37$F^x$\"''H/$F37$Fc x$\"'.')HF37$Fhx$\"''H)HF37$F]y$\"'@\"*HF37$Fby$\"'kF`]l7$Fc^l$\"($)o$=F`]l7$Fh^l$\"(^-v\"F`]l7$F]_l$\"(4]l\"F`]l7 $Fb_l$\"(*)4e\"F`]l7$Fg_l$\"(S9]\"F`]l7$F\\`l$\"(3lV\"F`]l7$Fa`l$\"(,1 P\"F`]l7$Ff`l$\"(?DJ\"F`]l7$F[al$\"($Qc7F`]l7$F`al$\"(9$37F`]l7$Feal$ \"($4g6F`]l7$Fjal$\"(bM6\"F`]l7$F_bl$\"(1b2\"F`]l7$Fdbl$\"(_q.\"F`]l7$ Fibl$\"'v)***F`]l7$F^cl$\"',e'*F`]l7$Fccl$\"'C[$*F`]l7$Fhcl$\"'GD!*F`] l7$F]dl$\"'a_()F`]l7$Fbdl$\"'*zZ)F`]l7$Fgdl$\"'4V#)F`]l7$F\\el$\"'<+!) F`]l7$Fael$\"'+%y(F`]l7$Ffel$\"'*)pvF`]l7$F[fl$\"'\\rtF`]l7$F`fl$\"'hu rF`]l7$Fefl$\"'s%*pF`]l7$Fjfl$\"'(*>oF`]l7$F_gl$\"'HbmF`]l7$Fdgl$\"'26 lF`]l7$Figl$\"'j`jF`]l7$F^hl$\"'V>iF`]l7$Fchl$\"'%H3'F`]l7$Fhhl$\"' " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "Test 10 of 11 stage, order 7 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!\"\",$*&,&*&\"\"#F&-%$sin G6#*&\"\"&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#\"\"!*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }{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))*sinh(y(x));\nic := y(0)=sqrt(5)/2;\ndsolve(\{de,ic\}, y(x));\nsimplify(convert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x), x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$s inG6#,$*&\"\"&F2F,F2F2F2F2*&\"\"$F2-%$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#,*#\"\"\"\"\"&F0*&#F0\"\"#F0-F)6#,$*&,& -%$expG6#,$*&F4!\"\"F1F3F0F0F0F0F0,&F:F0F0F?F?F?F0F0*&#\"\"$\"#9F0-%$s inG6#,$*&\"\"(F0F'F0F0F0F0*&#F0F1F0-%$cosG6#,$*&F1F0F'F0F0F0F?" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#*&,*-%$expG6#,4# \"\"#\"\"&!\"\"*&#\"\"$\"\"(\"\"\"-%$sinGF&F9F9*&#\"$#>F8F9*&)-%$cosGF &\"\"'F9F:F9F9F4*&#\"$S#F8F9*&)FA\"\"%F9F:F9F9F9*&#\"#sF8F9*&)FAF2F9F: F9F9F4*&#\"#KF3F9*$)FAF3F9F9F9*&\"\")F9)FAF7F9F4*&F2F9FAF9F9*&F2F4F3#F 9F2F9F9-F.6#,2#F2F3F4*&F6F9F:F9F9*&#F>F8F9F?F9F4*&FEF9FGF9F9*&#FLF8F9F MF9F4*&FPF9FRF9F9*&FUF9FVF9F4*&F2F9FAF9F9F4-F.6#,$*&F2F4F3FYF9F9F9F9F9 ,*F-F4FZF9FaoF9F9F9F4" }}{PARA 13 "" 1 "" {GLPLOT2D 552 388 388 {PLOTDATA 2 "6&-%'CURVESG6$7av7$$\"\"!F)$\"3!\\*)\\())R.=6!#<7$$\"3ALL $3FWYs#!#>$\"33uw,WSt45F,7$$\"3WmmmT&)G\\aF0$\"3yUB%H69F5*!#=7$$\"3m** **\\7G$R<)F0$\"3[G6@7@G;#)F87$$\"3GLLL3x&)*3\"F8$\"3u_\"Hlv:eW(F87$$\" 3))**\\i!R(*Rc\"F8$\"3aT]N\"zi(yjF87$$\"3umm\"H2P\"Q?F8$\"3@V:-NK+KcF8 7$$\"3YLek.pu/BF8$\"3:Vt%)yf.P`F87$$\"3!***\\PMnNrDF8$\"3\"zkU]7kD7&F8 7$$\"37$eR(\\;m/FF8$\"3#f>&4'\\tL/&F87$$\"3MmT5ll'z$GF8$\"3OEk>%*z$=) \\F87$$\"37](o/[r7(HF8$\"3oQ2*>TRs$\\F87$$\"3MLL$eRwX5$F8$\"359>\\xg!* 3\\F87$$\"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$\"38&yxl/-Si&F87$$\"3emm\"z%4\\Y_F8$\"3[*Q!R (Q;g:'F87$$\"32++v$flMLe*)>VB$)F8$\"3R-PW ?nJGpF87$$\"3wmmTg()4_))F8$\"3XS!**oz*pDlF87$$\"3Y++DJbw!Q*F8$\"3wsIZl ^+ohF87$$\"3=nT&)3\\m_'*F8$\"3:%=[TB`#HgF87$$\"3+N$ekGkX#**F8$\"3;==#[ laH$fF87$$\"3nTg_(R^g+\"F,$\"3PhYLpC*H!fF87$$\"31]iSmjk>5F,$\"3?Z/nW4A ')eF87$$\"3XekGN8CL5F,$\"3S5jkI7P$)eF87$$\"3%ommTIOo/\"F,$\"3i#*RMEK8& *eF87$$\"3cLe9;_yq5F,$\"3_[.xr\")Q`fF87$$\"3E+]7GTt%4\"F,$\"3U$yw2JYC1 'F87$$\"3(p;/,/$o=6F,$\"3q*3')fymbA'F87$$\"3YLL3_>jU6F,$\"3?OGM!H;eW'F 87$$\"3ym;HdNb'>\"F,$\"33P8#3QfS;(F87$$\"37++]i^Z]7F,$\"3[rQ!Q)F8 7$$\"35+++v\"=YI\"F,$\"3&441r_RP_*F87$$\"33++](=h(e8F,$\"3G(\\BrLl\"*4 \"F,7$$\"3&*****\\7!Q4T\"F,$\"39o3wy`oD7F,7$$\"3/++]P[6j9F,$\"3%z>\\NC [mH\"F,7$$\"3'=HKkAg\\Z\"F,$\"3=cb1EJ%4I\"F,7$$\"3W$ek`h0o[\"F,$\"3:bt 7ruA+8F,7$$\"3/voH/5l)\\\"F,$\"39tx0vuR%H\"F,7$$\"3%o;HKR'\\5:F,$\"37z 4l5FZ$G\"F,7$$\"3-]P4rr=M:F,$\"3i,%*Q\"yBqC\"F,7$$\"3UL$e*[z(yb\"F,$\" 3W\")o!f;2L>\"F,7$$\"3w;/Ev&[ge\"F,$\"3r>R!QDA@6\"F,7$$\"34+Dc,#>Uh\"F ,$\"3#4HI%*[n*=5F,7$$\"3V$eky#)*QU;F,$\"3mms=Lk!y?*F87$$\"3wmm;a/cq;F, $\"3Pe)[!HQiL#)F87$$\"3\"pm;a)))G=F,$\"35Bp0?YWZMF87$$\"3KLe9;0?E>F,$\"34: XWhm;,LF87$$\"3pTg-gl[Q>F,$\"3-rZpoe$*\\KF87$$\"31]i!Rgs2&>F,$\"35MI#4 &[)H@$F87$$\"3WekyZ'eI'>F,$\"3=g(Q?ez,>$F87$$\"3gmmm\"pW`(>F,$\"3Q*HR/ (>[\"=$F87$$\"3_ek.HW#)))>F,$\"3SdFyRyC)=$F87$$\"3?]iSmTI-?F,$\"3]%GeC R0B@$F87$$\"3*=/wP!Ry:?F,$\"3OG\"o!ej/aKF87$$\"3dLe9TOEH?F,$\"32!yA?() GSJ$F87$$\"3'pT&)e6Bi0#F,$\"3EaW;d/\"=\\$F87$$\"3K+]i!f#=$3#F,$\"35[ni L?f`PF87$$\"3/++D\"=EX8#F,$\"3)=7zi$ePDXF87$$\"3?+](=xpe=#F,$\"3rT'oxR DBu&F87$$\"3$pTNrfbE@#F,$\"3S'**Rb6vOf'F87$$\"3mLeRA9WRAF,$\"3Y7cI=h\\ :wF87$$\"3S]ilZsAmAF,$\"3.'zb%4IS@))F87$$\"37nm\"H28IH#F,$\"3I'yDR:j=- \"F,7$$\"3!oTN@#3hF,7$$\"3WeRseStdCF,$\"3!>Ien#[n??F,7$$\"3)oT5l0+5Z#F,$\"3=]_;#*)R!\\ ?F,7$$\"35YOSbIjxCF,$\"3'f>URy8d0#F,7$$\"3IvoHagE%[#F,$\"3+2\\y.l?d?F, 7$$\"3_/,>`!**3\\#F,$\"3WHUb4&*\\`?F,7$$\"3GLL3_?`(\\#F,$\"3&3)GRRomW? F,7$$\"3.$3-)Q84DDF,$\"3%H:bmLUz&>F,7$$\"3AL3_D1l_DF,$\"3aG*H8v%))4=F, 7$$\"3I3-))o-VmDF,$\"3Y]!f;TM@s\"F,7$$\"3S$eRA\"*4-e#F,$\"34g!)pZ(f(H; F,7$$\"3[e*)fb&*)Rf#F,$\"3\"z'>0Ar[N:F,7$$\"3fL$e*)>pxg#F,$\"3')[_^!fj 9W\"F,7$$\"3V+D1R'f:&H/%HG\"F,7$$\"3%omm\"z+vbEF,$\"3;#=@: ,\\a8\"F,7$$\"3AL3F>0uzEF,$\"3C!3Cbi?=+\"F,7$$\"33+]Pf4t.FF,$\"3l3u3() HVJ))F87$$\"3Q$3F>HT'HFF,$\"3A'y5tq@xr(F87$$\"3om\"zWi^bv#F,$\"3w5fooR nqnF87$$\"3)*\\7.d>Y\"y#F,$\"3Op_f3vnxfF87$$\"3uLLe*Gst!GF,$\"3eVA%fwp RK&F87$$\"3)om\"H2\"34'GF,$\"3!H2v@w6,N%F87$$\"30+++DRW9HF,$\"3UKzOA[p (y$F87$$\"3S+Dc,6jSHF,$\"3l.`HJu\"\\j$F87$$\"3K+]7y#=o'HF,$\"3w)4GNSsE b$F87$$\"3G]iSm=\"*zHF,$\"354>3R$*[ONF87$$\"3C+voaa+$*HF,$\"3>.\"*[foP ONF87$$\"3>](oH/*41IF,$\"3UC#)e$)41_NF87$$\"3:++DJE>>IF,$\"37lJ^K5P$e$ F87$$\"3A+v$4^n)pIF,$\"3q;[s.$y8&QF87$$\"3F+]i!RU07$F,$\"3'\\(yiwjedVF 87$$\"39+vo/#3o<$F,$\"3\"=P&zNU*o?&F87$$\"3+++v=S2LKF,$\"3rC&z7e?,M'F8 7$$\"3;L$3_NJOG$F,$\"3'oVq=V<'GvF87$$\"3Jmmm\"p)=MLF,$\"35IlY8#))ft)F8 7$$\"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$$\"3C$ekyZ2mY$F,$\"3i=$4cd g.0\"F,7$$\"3=vo/Bh$)yMF,$\"3i,C<-Qm\\5F,7$$\"3mm\"H#oZ1\"\\$F,$\"3%43 uAU\"*f/\"F,7$$\"36]Pfe?_:NF,$\"3]?:\"y&yYI5F,7$$\"35L$e*[$z*RNF,$\"3W ZcwFIo05F,7$$\"3%o;Hd!fX$f$F,$\"3\\R'zK#G91$*F87$$\"3e++]iC$pk$F,$\"3! e-txpVQY)F87$$\"3ILe*[t\\sp$F,$\"3o)z!\\!\\E.v(F87$$\"3[m;H2qcZPF,$\"3 -m*ep\"3v-sF87$$\"3s***\\7.lQx$F,$\"3Il\"\\:&*zo*pF87$$\"3UL$3_0j,!QF, $\"35]vs*pf#\\oF87$$\"3F+v=n?J8QF,$\"3+R^fg!Quz'F87$$\"36nm;z5YEQF,$\" 3-)RYLJ)=gnF87$$\"3^Le9\"45'RQF,$\"3qjisVpNPnF87$$\"3O+]7.\"fF&QF,$\"3 VEh1S\\sGnF87$$\"3i3_vlYhlQF,$\"3(\\vU\")GTPt'F87$$\"3)oT&QG-ZyQF,$\"3 aR=Q3Gt^nF87$$\"39Dc,\"zD8*QF,$\"3o0VIG#3By'F87$$\"3TLek`8=/RF,$\"33OP 7dc+DoF87$$\"3$*\\i!*yC*)HRF,$\"3\\cG3V'=X%pF87$$\"3Ymm;/OgbRF,$\"33DC (y#>%[5(F87$$\"3*G$e*[$zV4SF,$\"39]/NuwfQvF87$$\"3w**\\ilAFjSF,$\"3D8x FXpB@!)F87$$\"3#G3_]p'>*3%F,$\"3\"o(e-u54K#)F87$$\"3ym\"zW7@^6%F,$\"3% eL`Rp%o0%)F87$$\"3w3F>RL3GTF,$\"3-T$yF^SJZ)F87$$\"3t]i!RbX59%F,$\"3i)p QA'3/D&)F87$$\"3#=z>'ox+aTF,$\"37$4T=0@'f&)F87$$\"3yLLL$)*pp;%F,$\"3![ sp'zRPv&)F87$$\"3!Q3_+sD-=%F,$\"3se;6C(p2d)F87$$\"3#Q$3xc9[$>%F,$\"3X[ ZFn2MW&)F87$$\"3'Qe*[$>Pn?%F,$\"3'>&G!HhJc\\)F87$$\"3)QL3-$H**>UF,$\"3 [:#3'R6kC%)F87$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$\"3)RL$3xe,tUF,$\"3% p?zi!3VLzF87$$\"3Cn;HdO=yVF,$\"3;#>X)HduejF87$$\"3a+++D>#[Z%F,$\"3gxAO ;Hx\")\\F87$$\"3TM$3_5,-`%F,$\"3'RSzCT^zW%F87$$\"3SnmT&G!e&e%F,$\"3%=E R;G9D:%F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe*[=Y.h%F,$\"3v*>WD ]()H5%F87$$\"386but%F,$\"3')>+X\\xqwZF87$$\"37+]iSjE!z%F,$\"3<\"z* *[W(=JcF87$$\"3y*\\7G))Rb\"[F,$\"3[YJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$\"3 !**fZR-9n(oF87$$\"3)3](=np3m[F,$\"3gVt^2I*Ho(F87$$\"3a+]P40O\"*[F,$\"3 `9x`tb,B')F87$$\"3>]7.#Q?&=\\F,$\"3Ik*[g6ply*F87$$\"3s+voa-oX\\F,$\"3) pQ=b([U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3ftg')yo>49F ,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLAB ELSG6$%\"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 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "P := (x,y) -> -(2*sin(5 *x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := s qrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0 ,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmth ds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15 /16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 3 0:\nfor ct to 6 do\n Pn_RK7_||ct := RK7_||ct(P(x,y),x,y,x0,evalf(y0) ,hh,numsteps,false);\n sm := 0: numpts := nops(Pn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Pn_RK7_||ct[ii,2]-p(Pn_RK7_||ct[ii,1 ]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nD igits := 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-FBF 4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint44 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+g2#G*o!#>7$/*&%-scheme~with~GF+F,F+#\"#:\"#;F 37$/F8#\"#>\"#?F37$/F8#\"#K\"#LF37$/F8#\"#\")\"##)F37$/F8#\"$p\"\"$q\" F3Q)pprint456\"" }}}{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 \+ = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also giv en." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "P := (x,y) -> -(2*si n(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 : = sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,`` (x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\n mthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10] =15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme w ith `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits : = 30:\nfor ct to 6 do\n pn_RK7_||ct := RK7_||ct(P(x,y),x,y,x0,evalf( y0),hh,numsteps,true);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfo r ct to 6 do\n errs := [op(errs),abs(pn_RK7_||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.F5 F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-F BF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint 466\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+O\"z3F$!#A7$/*&%-scheme~with~GF+F,F+#\"#:\"#; $\"+Dl7d:F57$/F8#\"#>\"#?$\"+!e)4;>F57$/F8#\"#K\"#L$\"+;R.+n!#B7$/F8# \"#\")\"##)$\"+^)Hg(GFM7$/F8#\"$p\"\"$q\"$\"+0TcfDFMQ)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 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((p(x)-'pn_RK7_||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$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+ZpX86!#A7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+I'*4Ra!#B7$/F8#\"#>\"#?$\"+5b>.lF?7$/F8#\"#K\"#L$\"+0[YWDF?7$/F8#\" #\")\"##)$\"+=(*\\L8F?7$/F8#\"$p\"\"$q\"$\"+K7!\\2\"F?Q)pprint486\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The fol lowing error graphs are constructed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "evalf[20 ](plot([p(x)-'pn_RK7_1'(x),p(x)-'pn_RK7_2'(x),p(x)-'pn_RK7_3'(x),\np(x )-'pn_RK7_4'(x),p(x)-'pn_RK7_5'(x),p(x)-'pn_RK7_6'(x)],x=0..2.2,font=[ HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9 ,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme w ith c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/20`,` scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10]= 169/170`],\ntitle=`error curves for 11 stage order 7 Runge-Kutta metho ds`));" }}{PARA 13 "" 1 "" {GLPLOT2D 860 651 651 {PLOTDATA 2 "6,-%'CUR VESG6%7]v7$$\"\"!F)F(7$$\"5mmmm;zM%))>\"!#@$\"%6()!#>7$$\"5LLLLLepo(R# F-$\"&=Z\"F07$$\"5++++]P/`'f$F-$\"',o=!#?7$$\"5mmmmm;RP&z%F-$\"'DJ@F;7 $$\"5++voHHBwD\\F-$\"'+3AF;7$$\"5NL$3F>u]h0&F-$\"')=I#F;7$$\"5qm\"HdX: Rl=&F-$\"'X(G#F;7$$\"5+++v=nv#pJ&F-$\"'7tAF;7$$\"5qm;zW#R/xd&F-$\"'tYA F;7$$\"5NLL$3x@\"[QeF-$\"':aAF;7$$\"5qmm\"H#o[.gjF-$\"')zI#F;7$$\"5+++ +v=&)e\")oF-$\"'L\"G#F;7$$\"5ILL3Fp@9.uF-$\"'LuAF;7$$\"5lmm;z>epCzF-$ \"'gLAF;7$$\"5+++DJq%\\iW)F-$\"'g\">#F;7$$\"5LLLL$37.y'*)F-$\"'HG@F;7$ $\"5+++]7jpRJ6F;$\"'f6=F;7$$\"5nmmm;9O,m8F;$\"'n%\\\"F;7$$\"5nmmmmca=- ;F;$\"'V^6F;7$$\"5nmmm;*Hd$Q=F;$\"&@g*F;7$$\"5+++]7emSt?F;$\"&TO)F;7$$ \"5LLLL3$F;$\"&>H)F;7$$\"5mmmmTv+JiOF;$\"&m8)F;7$$\"5+ +++vLo`FTF;$\"&s7(F;7$$\"5mmm;/'G3oO%F;$\"&Qr'F;7$$\"5LLLLLQ(zgg%F;$\" &BF'F;7$$\"5++++vj,$o\"[F;$\"&\"fjF;7$$\"5mmmm;*e!eF]F;$\"&l!oF;7$$\"5 +++]i?\"3i9&F;$\"&F>(F;7$$\"5NLLL3_c$[E&F;$\"&so(F;7$$\"5qmm;a$=jMQ&F; $\"&1N)F;7$$\"5+++++:24-bF;$\"&\")R*F;7$$\"5+++++D#\\&yfF;$\"'#\\C\"F; 7$$\"5+++++S#QL4'F;$\"'kE8F;7$$\"5+++++bs73iF;$\"'.$R\"F;7$$\"5+++++qi \"HK'F;$\"'(GV\"F;7$$\"5+++++&G0xV'F;$\"'Gh9F;7$$\"5NLL$3-8%=YmF;$\"'` #[\"F;7$$\"5mmmmTvHmaoF;$\"'1p9F;7$$\"5+++](o[kD5(F;$\"'$=U\"F;7$$\"5L LLLL)*fY]tF;$\"'%>S\"F;7$$\"5NLLL$e4r/c(F;$\"'T39F;7$$\"5LLLLL$>w/x(F; $\"'\"pW\"F;7$$\"5lmm;/Y?w9!)F;$\"'n`:F;7$$\"5*******\\()*y/f#)F;$\"'; B;F;7$$\"5lmm;/rAGv%)F;$\"'F)o\"F;7$$\"5LLLLLVm^\"p)F;$\"';O:>R*F;$\"'HG;F;7$$\"5*******\\i*p#yh*F;$\"'4]:F;7$$\"5lmm;aeAa`)*F;$ \"'d([\"F;7$$\"5LLLL3_d#*35F0$\"'DU9F;7$$\"5LLLeRrF;7$$\"5nmmT5Zudq7F0$\"'UU>F;7$$\"5ML$3_qQmkF\"F0$\"'g#)>F;7$$\" 5+++++F`N#G\"F0$\"'OR?F;7$$\"5+++]PU+S08F0$\"'`lBF;7$$\"5++++vdZWG8F0$ \"&#GIF07$$\"5++]7GJKfR8F0$\"&Sl$F07$$\"5+++D\"[qT2N\"F0$\"&5S%F07$$\" 5++]PMy,*=O\"F0$\"&%p^F07$$\"5+++](=lQIP\"F0$\"&M2'F07$$\"5++](o489F07$$\"5+++]())y=*y9F0$ \"'\"4O\"F07$$\"5mmmmT`Xz!\\\"F0$\"'G>7F07$$\"5LLL$ezJqE]\"F0$\"'A(4\" F07$$\"5++++]#3YX^\"F0$\"&ml*F07$$\"5nm;/E!H2`_\"F0$\"&uR)F07$$\"5MLL3 -)\\og`\"F0$\"&!)G(F07$$\"5++]7y0(Hoa\"F0$\"&CS'F07$$\"5nmm;a84fd:F0$ \"&Qy&F07$$\"5+]iS;C@`g:F0$\"&Ke&F07$$\"5MLekyMLZj:F0$\"&,`&F07$$\"5n; a)3aa9kc\"F0$\"&wZ&F07$$\"5++]7.cdNp:F0$\"&+Y&F07$$\"5M$ek`m'pHs:F0$\" &cV&F07$$\"5nmTgFx\"Q_d\"F0$\"&-Q&F07$$\"5MeRse#y3nd\"F0$\"&1O&F07$$\" 5+]P%)*yQz\"y:F0$\"&QP&F07$$\"5nTN'4K**\\'z:F0$\"&^\\&F07$$\"5MLL3_)f? 6e\"F0$\"&O`&F07$$\"5nm;/,Ta)Gf\"F0$\"&'4dF07$$\"5++++]$G]Yg\"F0$\"&G( fF07$$\"5mmmTN$))*yE;F0$\"'PlmF;7$$\"5LLL$3K[H*[;F0$\"'wBvF;7$$\"5mmm \"z%[]ra;F0$\"'\"Rq(F;7$$\"5++++v81]g;F0$\"'E>\")F;7$$\"5mm;aQ'R$Rj;F0 $\"'_.!)F;7$$\"5LLL3-zhGm;F0$\"'e.zF;7$$\"5++]ilh*y\"p;F0$\"'?w!)F;7$$ \"5mmm;HW<2s;F0$\"'u_#)F;7$$\"5LL$3Fp_k\\n\"F0$\"'yN\")F;7$$\"5+++Dc4t &yn\"F0$\"'J'4)F;7$$\"5LL3-)3q.$z;F0$\"'#fE)F;7$$\"5mm;z>#4]2o\"F0$\"' WX%)F;7$$\"5++Dc^$['>#o\"F0$\"'9%Q)F;7$$\"5LLLL$[(Gk$o\"F0$\"'QB$)F;7$ $\"5mmT5:m#*3&o\"F0$\"'+l#)F;7$$\"5++](oulNlo\"F0$\"'B>#)F;7$$\"5LLeky [?)zo\"F0$\"'&[A)F;7$$\"5mmmT5S%G%*o\"F0$\"'g#R)F;7$$\"5++v=UJ[(3p\"F0 $\"'[;&)F;7$$\"5LL$eRFA@Bp\"F0$\"']a%)F;7$$\"5mm\"HdShnPp\"F0$\"'>$R)F ;7$$\"5+++]P0S@&p\"F0$\"'LM$)F;7$$\"5m\"HKks6Gmp\"F0$\"'2*G)F;7$$\"5L$ ek`\"HA/)p\"F0$\"'l!H)F;7$$\"5+voH/TjX*p\"F0$\"'0N%)F;7$$\"5mm\"HKHXq3 q\"F0$\"':K&)F;7$$\"5+]P4rw')p.F0$\"''Q^$F;7$$\"5LLL$3#43SE>F0$\"'=rKF;7$$\" 5++]il=5.P>F0$\"'0#>$F;7$$\"5mmmT5G7mZ>F0$\"'.MJF;7$$\"5LL$3_vV\"He>F0 $\"'`&4$F;7$$\"5+++++Z;#*o>F0$\"'dvIF;7$$\"5LLLekm]5\")>F0$\"'buIF;7$$ \"5mmm;H'[)G$*>F0$\"'\"=4$F;7$$\"5+++v$f!>Z0?F0$\"')[7$F;7$$\"5LLLLeD` l_@F0$\"')Hd$F;7 $$\"5+++D17$*4w@F0$\"'$\\i$F;7$$\"#A!\"\"$\"'1BPF;-%&COLORG6&%$RGBGF($ \"\"#Febn$\"\"\"F)-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G- F$6%7huF'7$F2$\"%xiF07$F=$\"&'e')F;7$FB$\"&Z())F;7$FG$\"&`9*F;7$FL$\"& \")3*F;7$FQ$\"&5.*F;7$FV$\"&:#*)F;7$Fen$\"&-)))F;7$Fjn$\"&#y))F;7$F_o$ \"&!\\')F;7$Fio$\"&$f!)F;7$Fcp$\"&q>(F;7$Fhp$\"&%)>&F;7$F]q$\"&9V$F;7$ $\"5nmmmTN&*4%[\"F;$\"&7c#F;7$Fbq$\"&Pe\"F;7$$\"5nmmm\"zPr-s\"F;$\"&i9 \"F;7$Fgq$\"%;#)F;7$$\"5MLLeky>)e&>F;$\"%)3'F;7$F\\r$\"%@\\F;7$$\"5mmm TgP8$4>#F;$\"%$*[F;7$Far$\"%zhF;7$Ffr$\"%r$*F;7$F[s$\"&#H8F;7$$\"5LLLL 3<@%*pHF;$\"&!4hF;7$Fhu$!%.[F;7$$\"5+++D\"yT\\T K&F;$!%oGF;7$F]v$!%+BF;7$$\"5NLL3F\\pxUaF;$!$z#F;7$Fbv$\"%JHF;7$Fgv$\" &G'>F;7$F\\w$\"&3^#F;7$Faw$\"&:)HF;7$Ffw$\"&GD$F;7$F[x$\"&%eMF;7$F`x$ \"&im$F;7$Fex$\"&Em$F;7$Fjx$\"&q^$F;7$F_y$\"&;`$F;7$Fdy$\"&Yv$F;7$Fiy$ \"&&4UF;7$Fcz$\"&+x&F;7$Fhz$\"&MG'F;7$F][l$\"&=g'F;7$$\"5+++v=K$Q,\")) F;$\"&ek'F;7$Fb[l$\"&Mc'F;7$$\"5ILLe*)4&F;7$F_^l$\"&&=eF;7$$\"5nmm\"H_FmJ 7\"F0$\"&s`'F;7$Fd^l$\"&54(F;7$$\"5+++v=TU(z:\"F0$\"&pH(F;7$$\"5MLLL$[ .I&o6F0$\"&XW(F;7$$\"5nmm\"z%Ge3z6F0$\"&=^(F;7$Fi^l$\"&=Z(F;7$F^_l$\"& ;B(F;7$Fc_l$\"&9&oF;7$F]`l$\"&?E'F;7$Faal$\"&g*eF;7$$\"5++]P%e];\")G\" F0$\"&`+'F;7$$\"5+++vo%oxQH\"F0$\"&#)3'F;7$$\"5++]7`j)Q'*H\"F0$\"&:N'F ;7$Ffal$\"&r`'F;7$$\"5+++D1+C#pJ\"F0$\"&7O(F;7$F[bl$\"%#)))F07$F`bl$\" &u;\"F07$Febl$\"&G`\"F07$Fjbl$\"&4$>F07$F_cl$\"&aU#F07$$\"5++v=U\"*yAz 8F0$\"&`t#F07$Fdcl$\"&].$F07$$\"5++Dc^qjg\"R\"F0$\"&(oOF07$Ficl$\"&(=Q F07$F^dl$\"&7W%F07$Fcdl$\"&*y^F07$Fhdl$\"&CF&F07$F]el$\"&2)fF07$Fbel$ \"&')o'F07$Fgel$\"&KC(F07$F\\fl$\"&mH(F07$Fafl$\"&i_(F07$Fffl$\"&%\\vF 07$F[gl$\"&3d(F07$F`gl$\"&Nf(F07$Fegl$\"&^g(F07$$\"5mmmTg14)HZ\"F0$\"& \\O(F07$Fjgl$\"&9<(F07$$\"5LLLe9rm&[[\"F0$\"&^&oF07$F_hl$\"&Y:'F07$$\" 5++vV)R-zA\\\"F0$\"&(\\hF07$$\"5LL$3_X\\jP\\\"F0$\"&U9'F07$$\"5mm\"z>^ 'zC&\\\"F0$\"&U8'F07$$\"5+++voNCt'\\\"F0$\"&$*4'F07$$\"5mm;H#oP,(*\\\" F0$\"&X]&F07$Fdhl$\"&3K&F07$$\"5mmm\"H-?3'3:F0$\"&&\\]F07$Fihl$\"&sY%F 07$F^il$\"&:q$F07$Fcil$\"&\"*3$F07$Fhil$\"&&pEF07$F]jl$\"&ZX#F07$F_]m$ \"&Vt#F07$Fi]m$\"&&)=$F07$F^^m$\"'7gOF;7$Fc^m$\"'eDTF;7$Fh^m$\"'18UF;7 $F]_m$\"'O8WF;7$Fb_m$\"'V]VF;7$Fg_m$\"'P&H%F;7$F\\`m$\"'=uVF;7$Fa`m$\" 'yaWF;7$Ff`m$\"'[\"R%F;7$F[am$\"'NlVF;7$F`am$\"'jUWF;7$Feam$\"'fCXF;7$ Fjam$\"'w\"\\%F;7$F_bm$\"'?fWF;7$Fdbm$\"'xFWF;7$Fibm$\"'=-WF;7$F^cm$\" 'T+WF;7$Fccm$\"'btWF;7$Fhcm$\"'XEXF;7$F]dm$\"']$\\%F;7$Fbdm$\"'#4Y%F;7 $Fgdm$\"'WHWF;7$Figm$\"'\\*H%F;7$Fchm$\"'YySF;7$Fhhm$\"'8cOF;7$F]im$\" 'e!>$F;7$Fbim$\"'-nEF;7$Fgim$\"']_AF;7$F\\jm$\"'\"*)*=F;7$Fajm$\"'gp:F ;7$Ffjm$\"':$R\"F;7$F[[n$\"'dy7F;7$F`[n$\"'0U7F;7$Fe[n$\"'y:7F;7$Fj[n$ \"'i)>\"F;7$F_\\n$\"',!>\"F;7$Fd\\n$\"'H!>\"F;7$Fi\\n$\"'h)>\"F;7$F^]n $\"'z87F;7$Fc]n$\"'RO7F;7$Fh]n$\"'3#H\"F;7$F]^n$\"'=\\8F;7$$\"5++](o*y <'G2#F0$\"'4x8F;7$Fb^n$\"'P+9F;7$$\"5++v=U%[*G!4#F0$\"'J)Q\"F;7$$\"5nm ;Hd>()4'4#F0$\"'V:9F;7$$\"5MLeRsaz!>5#F0$\"'c\"R\"F;7$Fg^n$\"'Y89F;7$F _an$\"'%3G\"F;7$Fian$\"'@P6F;7$Fcbn$\"&I.'F;-Fibn6&F[cnF($\"\"(FebnF(- Facn6#%8scheme~with~c[10]=15/16G-F$6%7ivF'7$F+$\"%*p&F07$F2$\"%#p*F07$ F7$\"'lP7F;7$F=$\"'*3U\"F;7$FB$\"'\"[Z\"F;7$FG$\"'gS:F;7$FL$\"'&4`\"F; 7$FQ$\"'M@:F;7$FV$\"'v.:F;7$Fen$\"'W5:F;7$$\"5qmT&Q.jp)ofF-$\"'Ad:F;7$ $\"5++](oH/e#*4'F-$\"'pq:F;7$$\"5NLe*)fbkkHiF-$\"'\"3c\"F;7$Fjn$\"'X^: F;7$$\"5NL$e*[$p63i'F-$\"'#Q`\"F;7$F_o$\"'$f`\"F;7$Fdo$\"''\\`\"F;7$Fi o$\"'l5:F;7$F^p$\"'z%[\"F;7$Fcp$\"'PY9F;7$Fhp$\"'jV7F;7$F]q$\"'=S5F;7$ Fbq$\"&HF)F;7$Fgq$\"&(4rF;7$Fbgn$\"&rq'F;7$F\\r$\"&&4kF;7$Fjgn$\"&0B'F ;7$Far$\"&#)='F;7$FfrFi_o7$F[s$\"&rV'F;7$F`s$\"&k#pF;7$Fes$\"&/'pF;7$F js$\"&4O'F;7$F_t$\"&o3'F;7$Fdt$\"&ks&F;7$Fit$\"&7t&F;7$F^u$\"&4(fF;7$F cu$\"&O?'F;7$Fhu$\"&/^'F;7$F]v$\"&d#pF;7$Fbv$\"&ue(F;7$Fgv$\"&]n*F;7$F \\w$\"'yD5F;7$Faw$\"'.v5F;7$Ffw$\"'D16F;7$F[x$\"'/I6F;7$F`x$\"'r`6F;7$ Fex$\"'q`6F;7$F_y$\"'SB6F;7$Fiy$\"'/_6F;7$Fcz$\"'qZ7F;7$Fhz$\"'!eF\"F; 7$F][l$\"'j)G\"F;7$Fb[l$\"'dp7F;7$Fg[l$\"',E7F;7$Fa\\l$\"'$o5\"F;7$Ff \\l$\"'6i5F;7$F[]l$\"'RL5F;7$F`]l$\"'cK5F;7$Fe]l$\"'*\\0\"F;7$Fj]l$\"' N)4\"F;7$F_^l$\"'2m6F;7$Fd^l$\"''[K\"F;7$Fi^l$\"'nG9F;7$F^_l$\"'-k9F;7 $Fc_l$\"'j/:F;7$Fh_l$\"'_I:F;7$F]`l$\"'Xg:F;7$Fg`l$\"'f2;F;7$Faal$\"'F 'o\"F;7$Ffal$\"'MH>F;7$F[bl$\"&sQ#F07$F`bl$\"&2!GF07$Febl$\"&#*G$F07$F jbl$\"&+z$F07$F_cl$\"&YP%F07$Fdcl$\"&C1&F07$Ficl$\"&B*eF07$F^dl$\"&)3l F07$Fcdl$\"&w?(F07$Fhdl$\"&&GtF07$F]el$\"&S(zF07$Fbel$\"&zh)F07$Fgel$ \"&d4*F07$Fafl$\"&6H*F07$F[gl$\"&[M*F07$$\"5m;/^1P2Jd9F0$\"&@N*F07$$\" 5LL3x\"Q1,(e9F0$\"&?M*F07$$\"5+]7.d!R\"4g9F0$\"&VG*F07$F`gl$\"&RH*F07$ $\"5++D\"G3PiUY\"F0$\"&5J*F07$Fegl$\"&vI*F07$Fdio$\"&W,*F07$Fjgl$\"&c! ))F07$F\\jo$\"&UY)F07$F_hl$\"&3t(F07$Fdhl$\"&8&oF07$Fihl$\"&u#fF07$F^i l$\"&61&F07$Fcil$\"&SJ%F07$Fhil$\"&7t$F07$F]jl$\"&cL$F07$Fbjl$\"&!3KF0 7$Fgjl$\"&v<$F07$F\\[m$\"&v9$F07$Fa[m$\"&/9$F07$Ff[m$\"&(GJF07$F[\\m$ \"&q4$F07$F`\\m$\"&l3$F07$Fe\\m$\"&r4$F07$Fj\\m$\"&s<$F07$F_]m$\"&R?$F 07$Fd]m$\"&;L$F07$Fi]m$\"&N^$F07$F^^m$\"'ryRF;7$Fc^m$\"'J]XF;7$Fh^m$\" '!=n%F;7$F]_m$\"'rV\\F;7$Fb_m$\"'At[F;7$Fg_m$\"'%G\"[F;7$F\\`m$\"'\\E \\F;7$Fa`m$\"'KU]F;7$$\"5++v$4c8=Nn\"F0$\"'\"f+&F;7$Ff`m$\"'*3(\\F;7$$ \"5mm\"zW#=4Tw;F0$\"'yV\\F;7$F[am$\"'(*[\\F;7$F`am$\"'ne]F;7$Feam$\"' \\u^F;7$Fjam$\"'%p8&F;7$F_bm$\"'s*4&F;7$Fdbm$\"',k]F;7$Fibm$\"'JO]F;7$ F^cm$\"'LT]F;7$Fccm$\"'C\\^F;7$Fhcm$\"'7H_F;7$F]dm$\"'1\">&F;7$Fbdm$\" 'V`^F;7$Fgdm$\"'M<^F;7$F\\em$\"'$)*3&F;7$Faem$\"'&>4&F;7$Ffem$\"':%=&F ;7$F[fm$\"'KY_F;7$F`fm$\"'1s^F;7$Fefm$\"'p1^F;7$Fjfm$\"'bl^F;7$F_gm$\" '2*=&F;7$Fdgm$\"'p;^F;7$Figm$\"'As]F;7$$\"5Le*[$\\2RD>!o1s\"F0$\"'rn^F;7$$\"5mTN@FJ@3At\"F0$\"'TQ]F;7$$\"5+v=nQE]RLF;7$F`[n$\"'m@>F;7$Fe[n$\"'R ))=F;7$Fj[n$\"'Um=F;7$F_\\n$\"'Db=F;7$Fd\\n$\"'Vb=F;7$Fi\\n$\"'\\m=F;7 $F^]n$\"'P()=F;7$Fc]n$\"'h=>F;7$Fh]n$\"'P'*>F;7$F]^n$\"'h%3#F;7$Fb^n$ \"'X#=#F;7$Fg^n$\"'%*fAF;7$F_an$\"'-?AF;7$Fian$\"'n4AF;7$F^bn$\"'1CAF; 7$Fcbn$\"')\\C#F;-Fibn6&F[cn$\"\"*Febn$\"\"%FebnF(-Facn6#%8scheme~with ~c[10]=19/20G-F$6%7dwF'7$F+$\"%K>F07$F2$\"%IJF07$$\"5mmm\"HKRUva#F-$\" %8JF07$$\"5+++]7GyR(p#F-$\"%MJF07$$\"5LLL3-jKDZGF-$\"%(H$F07$$\"5mmmm \"zp3r*HF-$\"&%))QF;7$$\"5LLL$3xc>oH$F-$\"&;&QF;7$F7$\"&H!QF;7$$\"5mmm ;H28C'*QF-$\"&W&RF;7$$\"5LLLL3x@&f>%F-$\"&e=%F;7$$\"5+++](o/jc\\%F-$\" &W7%F;7$F=$\"&L5%F;7$FQ$\"&3=%F;7$Fen$\"&(ySF;7$Fjn$\"&&yRF;7$F_o$\"&8 #QF;7$Fio$\"&7Q$F;7$Fcp$\"&.\"GF;7$Fhp$\"&#Q;F;7$F]q$\"%:rF;7$Fbfn$\"% DHF;7$Fbq$!%w7F;7$Fjfn$!%$f#F;7$Fgq$!%+KF;7$Fbgn$!%%4$F;7$F\\r$!%WBF;7 $Fjgn$!$i&F;7$Far$\"%2@F;7$F[s$\"&61\"F;7$Fhhn$\"&wX\"F;7$F`s$\"&g$=F; 7$F`in$\"&-+#F;7$Fes$\"&&)*>F;7$Fhin$\"&bx\"F;7$Fjs$\"&(39F;7$Fdt$\"%` PF;7$Fit$\"$#QF;7$F^u$!%Y:F;7$Fcu$!%;=F;7$Fhu$!%;:F;7$Fb[o$!$=(F;7$F]v $!$N%F;7$Fj[o$\"$'fF;7$Fbv$\"%VCF;7$$\"5+++++q*>.u&F;$\"%)H(F;7$Fgv$\" &zM\"F;7$F\\w$\"&=u\"F;7$Faw$\"&#*3#F;7$Ffw$\"&GH#F;7$F[x$\"&\"[CF;7$$ \"5qmmTg2Z%>a'F;$\"&3b#F;7$F`x$\"&Sg#F;7$$\"5+++D\"GbB/v'F;$\"&fh#F;7$ Fex$\"&8f#F;7$Fjx$\"&EW#F;7$F_y$\"&MT#F;7$Fdy$\"&.`#F;7$Fiy$\"&]!GF;7$ F^z$\"&UP$F;7$Fcz$\"&0r$F;7$$\"5ILLe*[3lrO)F;$\"&V%QF;7$Fhz$\"&9%RF;7$ $\"5+++v=d%*R$e)F;$\"&V*RF;7$F][l$\"&0*RF;7$Fb[l$\"&0y$F;7$Fg[l$\"&nV$ F;7$F\\\\l$\"&Z'HF;7$Fa\\l$\"&__#F;7$Ff\\l$\"&$fAF;7$F[]l$\"&_8#F;7$F` ]l$\"&cC#F;7$Fe]l$\"&'4DF;7$Fj]l$\"&\\\"HF;7$F_^l$\"&+_$F;7$Fd^l$\"&l# [F;7$Fabo$\"&UI&F;7$Fi^l$\"&+f&F;7$F^_l$\"&`l&F;7$Fc_l$\"&Ed&F;7$F]`l$ \"&\")H&F;7$Faal$\"&U0&F;7$Fjco$\"&a8&F;7$F_do$\"&4:&F;7$Fddo$\"&aF&F; 7$Ffal$\"&XR&F;7$F\\eo$\"&-'eF;7$F[bl$\"%\\nF07$F`bl$\"%W%)F07$Febl$\" &g2\"F07$Fjbl$\"&oL\"F07$F_cl$\"&rm\"F07$F`fo$\"&t(=F07$Fdcl$\"&93#F07 $Fhfo$\"&p^#F07$Ficl$\"&+i#F07$$\"5+]Pf$)H-*3S\"F0$\"&]-$F07$F^dl$\"&* \\IF07$$\"5+]7GQp%zqS\"F0$\"&E5$F07$Fcdl$\"&fb$F07$Fhdl$\"&)>OF07$F]el $\"&O4%F07$Fbel$\"&Ca%F07$Fgel$\"&x%[F07$Fafl$\"&&>\\F07$F[gl$\"&j%\\F 07$Fbiq$\"&\\!\\F07$F`gl$\"&ry%F07$F_jq$\"&dz%F07$Fegl$\"&/y%F07$Fdio$ \"&0S%F07$Fjgl$\"&=;%F07$$\"5LL3FWfKS![\"F0$\"&#*y$F07$$\"5mm;/,Ix)=[ \"F0$\"&*)y$F07$$\"5++D\"y0?sL[\"F0$\"&zy$F07$F\\jo$\"&Xy$F07$$\"5mmTN rT6M'[\"F0$\"&jw$F07$$\"5++]7G7c#y[\"F0$\"&*yOF07$$\"5LLe*[G35$*[\"F0$ \"&wN$F07$F_hl$\"&`-$F07$Fdjo$\"&G-$F07$Fijo$\"&+-$F07$F^[p$\"&G,$F07$ Fc[p$\"&,)HF07$$\"5LL3_D1p@)\\\"F0$\"&J%GF07$Fh[p$\"&TP#F07$$\"5++D1RZ e=,:F0$\"&s>#F07$Fdhl$\"&T>#F07$$\"5mmTg_)yaT]\"F0$\"&->#F07$$\"5++]P4 f#Rc]\"F0$\"&$z@F07$$\"5LLe9mHP72:F0$\"&?8#F07$F`\\p$\"&n%>F07$$\"5++v ozqE45:F0$\"&!49F07$$\"5LL$ek89x:^\"F0$\"&jS\"F07$$\"5mm\"HK>hhI^\"F0$ \"&MS\"F07$Fihl$\"&#*R\"F07$$\"5M3_+ZL7*e^\"F0$\"&wQ\"F07$$\"5n;/,W%QO s^\"F0$\"&YM\"F07$$\"5+Dc,TN:e=:F0$\"&&)>\"F07$$\"5ML3-Q'oE*>:F0$\"%*z (F07$$\"5nTg-NP=F@:F0$\"%>uF07$$\"5+]7.K))phA:F0$\"%,uF07$$\"5Mek.HR@' R_\"F0$\"%!Q(F07$F^il$\"%JtF07$$\"5M$3_+Af(*z_\"F0$\"%6lF07$$\"5++D19% *yoI:F0$\"%>EF07$$\"5n;H23'>yL`\"F0$\"%-EF07$Fcil$\"%LDF07$Fhil$!$S\"F 07$F]jl$!$5(F07$Fa[m$\"%q8F07$F_]m$\"%+^F07$Fd]m$\"%t\")F07$Fi]m$\"&v6 \"F07$$\"5LL$3FM3?dh\"F0$\"&cR\"F07$F^^m$\"''Gk\"F;7$$\"5++]7G$ofyj\"F 0$\"'6k=F;7$Fc^m$\"'3u?F;7$Fh^m$\"'\"4:#F;7$F]_m$\"')=H#F;7$Fb_m$\"'=f AF;7$Fg_m$\"'DJAF;7$F\\`m$\"'m$G#F;7$Fa`m$\"'VOBF;7$Ff`m$\"'D.BF;7$F[a m$\"'P\"H#F;7$F`am$\"'dOBF;7$Feam$\"'+%Q#F;7$Fjam$\"'rmBF;7$F_bm$\"'c \\BF;7$Fdbm$\"'-LBF;7$Fibm$\"'j>BF;7$F^cm$\"'0>BF;7$Fccm$\"'>eBF;7$Fhc m$\"'E'Q#F;7$F]dm$\"'))oBF;7$Fbdm$\"'t^BF;7$Fgdm$\"'6NBF;7$Figm$\"'iXA F;7$Fchm$\"'5p?F;7$Fhhm$\"'Q+=F;7$F]im$\"'H6:F;7$Fbim$\"'F\">\"F;7$Fgi m$\"&,m*F;7$F\\jm$\"&V'yF;7$Fajm$\"&WK'F;7$Ffjm$\"&ng&F;7$F[[n$\"&r=&F ;7$Fe[n$\"&n)\\F;7$F_\\n$\"&y#\\F;7$Fi\\n$\"&7,&F;7$Fc]n$\"&^B&F;7$Fh] n$\"&th&F;7$F]^n$\"&;2'F;7$Fb^n$\"&)*\\'F;7$Fg^n$\"&\"HnF;7$F_an$\"&@5 'F;7$Fian$\"&t8&F;7$F^bn$\"&*=OF;7$Fcbn$\"%SbF;-%'COLOURG6&F[cn$\"*+++ +\"!\")F(F\\ju-Facn6#%8scheme~with~c[10]=32/33G-F$6%7jwF'7$F+$\"$&oF07 $F2$\"%K5F07$Fh`s$\"&y<\"F;7$F7$\"&'[6F;7$Fjas$\"&28\"F;7$F=$\"&Q3\"F; 7$F_o$\"%?gF;7$Fcp$!%,;F;7$Fhp$!%\">)F;7$F]q$!&U?\"F;7$Fbfn$!&xK\"F;7$ Fbq$!&5Q\"F;7$Fjfn$!&!38F;7$Fgq$!&A>\"F;7$F\\r$!%I$)F;7$Far$!%b>F;7$F[ s$\"%'*pF;7$Fhhn$\"&,3\"F;7$F`s$\"&WX\"F;7$F`in$\"&_l\"F;7$Fes$\"&Ts\" F;7$Fhin$\"&*Q;F;7$Fjs$\"&@T\"F;7$F_t$\"&r6\"F;7$Fdt$\"%*z&F;7$Fit$\"% CBF;7$F^u$!$7%F;7$Fcu$!%^8F;7$Fhu$!%8>F;7$Fb[o$!%_?F;7$F]v$!%J?F;7$Fj[ o$!%H=F;7$Fbv$!%?6F;7$Fghs$\"%&H\"F;7$Fgv$\"%`]F;7$Faw$\"&_-\"F;7$F[x$ \"&yI\"F;7$F`x$\"&@Y\"F;7$Fex$\"&/]\"F;7$Fjx$\"&\">E*yF;$\"&h&=F;7$F^z$\"&`+# F;7$$\"5ILLeRs\\!p8)F;$\"&+3#F;7$Fcz$\"&I8#F;7$F]\\t$\"&%f@F;7$Fhz$\"& <:#F;7$Fe\\t$\"&K5#F;7$F][l$\"&C+#F;7$Fb[l$\"&Nn\"F;7$Fg[l$\"&.K\"F;7$ F\\\\l$\"%n*)F;7$Fa\\l$\"%kbF;7$$\"5ILLeRFYoN(*F;$\"%\"f%F;7$Ff\\l$\"% \"*QF;7$$\"5+++vo*))*Rr**F;$\"%HNF;7$F[]l$\"%mNF;7$F`]l$\"%1_F;7$Fe]l$ \"%pyF;7$Fj]l$\"&3;\"F;7$F_^l$\"&5r\"F;7$Fd^l$\"&T%HF;7$Fabo$\"&FY$F;7 $Fi^l$\"&/)QF;7$$\"5nm;zWo@/,7F0$\"&&>SF;7$F^_l$\"&/6%F;7$$\"5++]P4hK% QA\"F0$\"&];%F;7$Fc_l$\"&S=%F;7$F]`l$\"&H1%F;7$Faal$\"&4\"QF;7$F_do$\" &jy$F;7$Ffal$\"&*=QF;7$$\"5++](=7Ah6J\"F0$\"&B'QF;7$F\\eo$\"&f%RF;7$$ \"5++]i!*yNoA8F0$\"%9TF07$F[bl$\"%jUF07$Febl$\"%)4'F07$F_cl$\"%i#*F07$ Fdcl$\"&4;\"F07$Ficl$\"&[Z\"F07$F^dl$\"&+t\"F07$Fcdl$\"&6.#F07$Fhdl$\" &y1#F07$F]el$\"&QM#F07$Fbel$\"&%)e#F07$Fgel$\"&8s#F07$$\"5L$3FWI7)=Y9F 0$\"&ds#F07$F\\fl$\"&os#F07$$\"5m;z%\\lxo*[9F0$\"&Yr#F07$Fafl$\"&4o#F0 7$Fffl$\"&$*o#F07$F[gl$\"&Pp#F07$Fbiq$\"&3j#F07$F`gl$\"&kZ#F07$$\"5L$3 _vS/sGY\"F0$\"&!zCF07$F_jq$\"&2[#F07$$\"5m;H2e(p_cY\"F0$\"&$zCF07$Fegl $\"&;Y#F07$$\"5mmT5!\\\\F&o9F0$\"&fP#F07$$\"5++](oa'>,q9F0$\"&*z?F07$$ \"5LLek.Ok\\r9F0$\"&53#F07$Fdio$\"&>3#F07$$\"5++v=F07$Fdhl$\"%%)>F07$F^jt$\"%w>F07$Fcjt$\"%8>F07$Fhjt$\"%h:F0 7$F`\\p$\"$7\"F07$F`[u$!%NTF07$Fe[u$!%ETF07$Fj[u$!%;TF07$Fihl$!%\"F0 7$Fb_u$!&x=\"F07$Fcil$!&M=\"F07$$\"5+]P4'**ze(Q:F0$!&bA\"F07$$\"5nmT5! >5\\9a\"F0$!&mI\"F07$$\"5M$e9TQSRTa\"F0$!&wH\"F07$Fhil$!&#)G\"F07$$\"5 MLe9m4.@_:F0$!&uC\"F07$F]jl$!&<@\"F07$Fgjl$!&K0\"F07$Fa[m$!%.*)F07$F[ \\m$!%,xF07$F_]m$!%;YF07$Fd]m$!%@8F07$Fi]m$\"%X\"F;7$Fg_m$\"'Jy6F;7$F\\`m$\"'F77F;7$Fa`m$\"'$eC\"F;7 $Ff`m$\"'6G7F;7$F[am$\"'GA7F;7$F`am$\"'9Z7F;7$Feam$\"'zs7F;7$Fjam$\"'c j7F;7$F_bm$\"'Sa7F;7$Fccm$\"'za7F;7$Fgdm$\"'eR7F;7$Figm$\"'([:\"F;7$Fc hm$\"&)>)*F;7$Fhhm$\"&:'yF;7$F]im$\"&;)eF;7$Fbim$\"&x#QF;7$Fgim$\"&ip# F;7$F\\jm$\"&_'>F;7$Fajm$\"&Ib\"F;7$Ffjm$\"&'*[\"F;7$F[[n$\"&._\"F;7$F _\\n$\"&Wj\"F;7$Fi\\n$\"&'3F;7$F^u$\"&ax\"F;7$Fcu$ \"&\"QF;7$Fgv$\"&QE#F;7$Faw$\"&Mi#F;7$F[x$\"&A%GF;7$F `x$\"&3(HF;7$Fex$\"&(>IF;7$Fjx$\"&I+$F;7$F_y$\"&I*HF;7$Fiy$\"&E2$F;7$F gbv$\"&R7$F;7$F^z$\"&l;$F;7$F_cv$\"&V;$F;7$Fcz$\"&V9$F;7$F]\\t$\"&s5$F ;7$Fhz$\"&a/$F;7$Fe\\t$\"&[&HF;7$F][l$\"&q#GF;7$Fg[l$\"&==#F;7$F\\\\l$ \"&0%=F;7$Fa\\l$\"&Ze\"F;7$F_ev$\"&J^\"F;7$Ff\\l$\"&QY\"F;7$Fgev$\"&FW \"F;7$F[]l$\"&YX\"F;7$F`]l$\"&_f\"F;7$Fe]l$\"&z!=F;7$Fj]l$\"&l5#F;7$F_ ^l$\"&ua#F;7$Fd^l$\"&$*f$F;7$Fi^l$\"&,f%F;7$Fc_l$\"&jR&F;7$Faal$\"&HD' F;7$Ffal$\"pF;7$F[bl$\"%czF07$Febl$\"%P**F07$F_cl$\"&ID\"F07$Fdcl$ \"&bU\"F07$Ficl$\"&!Q;F07$F^dl$\"&nz\"F07$Fcdl$\"&I(>F07$Fhdl$\"&b+#F0 7$F]el$\"&d:#F07$Fbel$\"&-G#F07$Fgel$\"&2K#F07$F[gl$\"&mC#F07$Fegl$\"& ,.#F07$Fj^w$\"&*f>F07$F__w$\"&)=F07$F \\au$\"%XQF07$F^^m$\"&%[bF;7$Fdau$\"&Z/(F;7$Fc^m$\"&'3%)F;7$Fh^m$\"&D \"*)F;7$F]_m$\"&Is*F;7$Fb_m$\"&Te*F;7$Fg_m$\"&*p%*F;7$F\\`m$\"&xv*F;7$ Fa`m$\"'!Q+\"F;7$Fa`r$\"&^'**F;7$Ff`m$\"&`*)*F;7$Fi`r$\"&K%)*F;7$F[am$ \"&[&)*F;7$F`am$\"'\"o+\"F;7$Feam$\"'XG5F;7$Fjam$\"'+@5F;7$F_bm$\"'f85 F;7$Fccm$\"'-;5F;7$Fgdm$\"'T/5F;7$Figm$\"&=T*F;7$Fchm$\"&82)F;7$Fhhm$ \"&Ia'F;7$F]im$\"&W.&F;7$Fbim$\"&+b$F;7$Fgim$\"&dw#F;7$$\"5nmm\"z4%=8X =F0$\"&d\\#F;7$F\\jm$\"&+I#F;7$$\"5nm;zW8%HE'=F0$\"&K@#F;7$$\"5nmmTgP> Yo=F0$\"&:<#F;7$$\"5nm;/whWHu=F0$\"&H8#F;7$Fajm$\"&q7#F;7$Ffjm$\"&i7#F ;7$F[[n$\"&S<#F;7$F_\\n$\"&QF#F;7$Fi\\n$\"&AM#F;7$Fc]n$\"&m[#F;7$Fh]n$ \"&uz#F;7$F]^n$\"&#eKF;7$Fb^n$FaftF;7$Fg^n$\"&>O%F;7$Fa_n$\"&0h%F;7$F_ an$\"&`w%F;7$Fdan$\"&N'[F;7$Fian$\"&d!\\F;7$$\"5++v$4hzt\"e@F0$\"&f*\\ F;7$$\"5++]P4o*[T;#F0$\"&v!\\F;7$$\"5+vV)*3hFkl@F0$\"&4%\\F;7$$\"5+]Pf 3al8n@F0$\"&#f\\F;7$$\"5+DJ?3Z.jo@F0$\"&G\"\\F;7$$\"5++D\"y+9C,<#F0$\" &Eq%F;7$$\"5+]7.2E<6t@F0$\"&)yZF;7$F^bn$\"&f%[F;7$$\"5+](oa!)*o3z@F0$ \"&Mu%F;7$$\"5++vo/%[u?=#F0$\"&\\g%F;7$$\"5+]i!R+2i]=#F0$\"&,o%F;7$$\" 5++]7.c'\\!)=#F0$\"&Ho%F;7$$\"5++Dc,G[-%>#F0$\"&)*[%F;7$Fcbn$\"&!QUF;- Fjiu6&F[cnF\\juF(F(-Facn6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELV ETICAGFi^s-%&TITLEG6#%Verror~curves~for~11~stage~order~7~Runge-Kutta~m ethodsG-%+AXESLABELSG6$Q\"x6\"Q!Fa^[l-%%VIEWG6$;F(Fcbn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner sche me with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/2 0" "scheme with c[10]=32/33" "scheme with c[10]=81/82" "scheme with c[ 10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 502 "evalf[20](plot([p(x)-'pn_RK7_1'(x),p(x)-'pn_RK7 _2'(x),p(x)-'pn_RK7_3'(x),\np(x)-'pn_RK7_4'(x),p(x)-'pn_RK7_5'(x),p(x) -'pn_RK7_6'(x)],x=2.2..2.8,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2, 1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red] ,\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=1 5/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error curves for 1 1 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1027 617 617 {PLOTDATA 2 "6,-%'CURVESG6%7ft7$$\"#A!\"\"$\"'1BP!#?7$$\" 5++++]#HyI@#!#>$\"'K+SF-7$$\"5+++]([kdWA#F1$\"'k`VF-7$$\"5++++v;\\DPAF 1$\"'o<\\F-7$$\"5++++DF17$$\"5++++]c.iDBF 1$\"'_iDF17$$\"5++](=7HdqK#F1$\"'&Qh#F17$$\"5+++v$fA%\\GBF1$\"'4QFF17$ $\"5++voH$p7#HBF1$\"'@nGF17$$\"5++]ilg6$*HBF1$\"'B\"3$F17$$\"5++Dc,G' \\1L#F1$\"'FBJF17$$\"5+++]P&4o8L#F1$\"'`SJF17$$\"5+++D\"['>CMBF1$\"''4 @$F17$$\"5++++DMe6PBF1$\"'+@LF17$$\"5+++](oU'eVBF1$\"'&33%F17$$\"5++++ ]>q0]BF1$\"')o5&F17$$\"5++++]U80jBF1$\"'@hnF17$$\"5++++]!ytbP#F1$\"'') p*)F17$$\"5++]P%Qn;%yBF1$\"'[='*F17$$\"5+++v=n&f7Q#F1$\"(Mr7\"F17$$\"5 ++]7`gC5%Q#F1$\"(/<:\"F17$$\"5+++](QNXpQ#F1$\"(4\"*=\"F17$$\"5+++v=/jq $R#F1$\"(g+\\\"F17$$\"5++++]asY+CF1$\"(Bc(=F17$$\"5+++]Pg4L.CF1$\"(BQ \">F17$$\"5++++DmY>1CF1$\"(T)f>F17$$\"5+++]7s$e!4CF1$\"(_w:#F17$$\"5++ +++y?#>T#F1$\"(*31CF17$$\"5+++vV>We=CF1$\"(/Ej#F17$$\"5+++](3wY_U#F1$ \"(\"3XIF17$$\"5++]iS6a>GCF1$\"(,y<$F17$$\"5+++v$>1W6V#F1$\"(anb$F17$$ \"5++](oCr#4MCF1$\"(SKh$F17$$\"5+++++j8/PCF1$\"(Rxo$F17$$\"5+++v$pj6NW #F1$\"(&z2TF17$$\"5+++](3\">)*\\CF1$\"((G^VF17$$\"5+++DJED1`CF1$\"(j=S %F17$$\"5++++vTJ9cCF1$\"(G]W%F17$$\"5+++v=dPAfCF1$\"(;6U%F17$$\"5+++]i sVIiCF1$\"(.lQ%F17$$\"5++D19A:\"RY#F1$\"(\"*HS%F17$$\"5++]ilr'=bY#F1$ \"(6aT%F17$$\"5+]iSTYAKmCF1$\"(\"3;WF17$$\"5++v=<@e7nCF1$\"(ImS%F17$$ \"5+](oHfRHzY#F1$\"(ojP%F17$$\"5+++voqHtoCF1$\"(=gI%F17$$\"5+]7`WXl`pC F1$\"($oiTF17$$\"5++DJ??,MqCF1$\"(+#HSF17$$\"5+]P4'\\pV6Z#F1$\"(tR.%F1 7$$\"5++](=(ps%>Z#F1$\"(;%QSF17$$\"5++vVB>WbtCF1$\"(\\i/%F17$$\"5++++v o:;vCF1$\"(&4]SF17$$\"5+]ilA,&**eZ#F1$\"(go/%F17$$\"5++DJqLujwCF1$\"([ ].%F17$$\"5+](ozhOvtZ#F1$\"(4g+%F17$$\"5++]il)H8\"yCF1$\"(QZ%RF17$$\"5 ++v$4O;*ezCF1$\"(66h$F17$$\"5+++DcG]1\"[#F1$\"(@#HMF17$$\"5++Dc^$*3a#[ #F1$\"(m)HMF17$$\"5++](o%en,%[#F1$\"(F\"HMF17$$\"5+]7`%4paZ[#F1$\"(BtU $F17$$\"5++v=UBE\\&[#F1$\"(FFU$F17$$\"5+]P%)*ebIi[#F1$\"(e=T$F17$$\"5+ ++]P)[op[#F1$\"(7\")Q$F17$$\"5+](=U#[Ww([#F1$\"(bTL$F17$$\"5++v$4\"3/c )[#F1$\"(AbA$F17$$\"5+]il(zOc$*[#F1$\"(%p>IF17$$\"5++]P%yK_,\\#F1$\"(, qt#F17$$\"5++D\"yvCW<\\#F1$\"(0Ft#F17$$\"5+++DJnhL$\\#F1$\"(*\\FFF17$$ \"5+](ozr7KT\\#F1$\"(EUs#F17$$\"5++vo/(3G\\\\#F1$\"(C'>FF17$$\"5+]iS\" p/Cd\\#F1$\"(*o6FF17$$\"5++]7y1+_'\\#F1$\"(Nhp#F17$$\"5++Dc^E>6)\\#F1$ \"(8Ag#F17$$\"5++++DYQq*\\#F1$\"(#\\pAF17$$\"5+++D1v!=j]#F1$\"(?H6#F17 $$\"5+++](QIKH^#F1$\"(qlz\"F17$$\"5++++]4+p=DF1$\"(lJv\"F17$$\"5+++]7: xWCDF1$\"(CKr\"F17$$\"5++D\"y]L-g_#F1$\"(bRq\"F17$$\"5++]7.bpbFDF1$\"( O+r\"F17$$\"5++vV)\\d6\"HDF1$\"(rvx\"F17$$\"5+++v$\\>m1`#F1$\"(I;)=F17 $$\"5++D1*[\"3AKDF1$\"(km'=F17$$\"5++]P%[VvP`#F1$\"(:;&=F17$$\"5++voza +LNDF1$\"(1$Q=F17$$\"5++++vuY)o`#F1$\"(sy$=F17$$\"5+++](G(*3La#F1$\"(] c4#F17$$\"5+++++rKt\\DF1$\"(\"fFBF17$$\"5+]P%['yg7]DF1$\"(nsR#F17$$\"5 ++voH'))=0b#F1$\"(.8R#F17$$\"5+]7`%Rp64b#F1$\"(@`Q#F17$$\"5++]Pf,XI^DF 1$\"(b$zBF17$$\"5++D1*o6!4_DF1$\"()QnBF17$$\"5+++v=Kd(Gb#F1$\"(TaN#F17 $$\"5++]7yipWaDF1$\"(d?L#F17$$\"5+++]P$>=gb#F1$\"(jSJ#F17$$\"5+++Dca1; fDF1$\"(.QW#F17$$\"5++++v:JIiDF1$\"(_Wd#F17$$\"5++D19iL#Qc#F1$\"(rxa#F 17$$\"5++]7`3OMlDF1$\"(!4BDF17$$\"5+]ilsJP5mDF1$\"(OR^#F17$$\"5++v=#\\ &Q'oc#F1$\"(![5DF17$$\"5+DJ&>l\"RCnDF1$\"(2D^#F17$$\"5+](=<\"yRinDF1$ \"(0$=DF17$$\"5+vV[rRS+oDF1$\"(A#HDF17$$\"5+++DJ,TQoDF1$\"(9pa#F17$$\" 5+]7y]CU9pDF1$\"('o6EF17$$\"5++DJqZV!*pDF1$\"(_jt#F17$$\"5+]P%)*3Zk1d# F1$\"(\"*[u#F17$$\"5++]P4%fC9d#F1$\"(F+t#F17$$\"5++vV[S[%Hd#F1$\"(U/q# F17$$\"5+++](o3lWd#F1$\"(U9n#F17$$\"5+]Pf3u!4`d#F1$\"(*[cEF17$$\"5++vo HhI:wDF1$\"(`Tk#F17$$\"5+]7y][q*pd#F1$\"())yj#F17$$\"5++](=d.Tyd#F1$\" (OWk#F17$$\"5++D195!H&zDF1$\"(H;v#F17$$\"5+++Dc%)p@\"e#F1$\"(O%*z#F17$ $\"5++]iSLHf%e#F1$\"(v3t#F17$$\"5++++D#))oze#F1$\"()p'p#F17$$\"5+++]ii d.%f#F1$\"(.:u#F17$$\"5+++++VE5+EF1$\"(XSx#F17$$\"5+++](y[TLg#F1$\"(Ip q#F17$$\"5++++vK.e1EF1$\"(Xck#F17$$\"5++](oR/!R2EF1$\"(')fj#F17$$\"5++ +v=b(*>3EF1$\"()pLEF17$$\"5++voz5Yg3EF1$\"(jqj#F17$$\"5++]iSm%4!4EF1$ \"(#yWEF17$$\"5++Dc,AVT4EF1$\"()=eEF17$$\"5+++]ix\">)4EF1$\"(G*yEF17$$ \"5+++D1+'Q9h#F1$\"(>>m#F17$$\"5++++]A!eIh#F1$\"(\"[HEF17$$\"5+++](=_( zCEF1$\"(w9\\#F17$$\"5++++]&*=jPEF1$\"(&RJBF17$$\"5+++](3/3(\\EF1$\"(@ H?#F17$$\"5+++]P#4JBm#F1$\"(-b/#F17$$\"5++++]KCnuEF1$\"(X^*=F17$$\"5++ +](=n#f(o#F1$\")q8]w7FF1 $\")]^0:F-7$$\"5+++]()Q?QDFF1$\")))o%R\"F-7$$\"5+++++J'ypt#F1$\")#4AI \"F-7$$\"5++++DM'p-v#F1$\")1A17F-7$$\"5+++++ms:iFF1$\")U]F6F-7$$\"5+++ ](3'>$[x#F1$\")ul^5F-7$$\"5+++]7hK'py#F1$\"(r8')*F-7$$\"#GF*$\"(!Q;#*F --%&COLORG6&%$RGBG$\"\"!F[\\n$\"\"#F*$\"\"\"F[\\n-%'LEGENDG6#%ESharp-V erner~scheme~with~c[10]=11/12G-F$6%7fv7$F($\"&I.'F-7$F/$\"&#*Q&F-7$F5$ \"&7.&F-7$F:$\"&,D&F-7$F?$\"&]J(F-7$FD$\"'>;5F-7$FI$\"'UW9F-7$FN$\"'U4 @F-7$FS$\"&wr$F17$FX$\"&NH&F17$Fgn$\"&DN(F17$F\\o$\"&r_(F17$Fao$\"&M*z F17$Ffo$\"&^\\)F17$F[p$\"&iL*F17$F`p$\"&x[*F17$Fep$\"&1a*F17$Fjp$\"&\\ v*F17$F_q$\"']75F17$Fdq$\"'#oH\"F17$Fiq$\"'@#o\"F17$F^r$\"':(H#F17$Fcr $\"'HVJF17$Fhr$\"'k-MF17$F]s$\"'6)3%F17$Fbs$\"'LxTF17$Fgs$\"'5F17$$\"5++v$fLm(e8CF1$\"(f3.\"F17$$\"5++](=([KD:CF1$\"(1M /\"F17$$\"5++D\"yS$)=pT#F1$\"(di1\"F17$Fju$\"(&[R6F17$$\"5+](=<@@<%>CF 1$\"(=#H7F17$$\"5++voz/+D?CF1$\"(u_L\"F17$$\"5+]ilZ(z#3@CF1$\"(;@M\"F1 7$$\"5++]i:!f:>U#F1$\"(W*[8F17$$\"5++Dc^v6eBCF1$\"(#ei8F17$F_v$\"(!ex8 F17$$\"5++D19'3@nU#F1$\"(y&*R\"F17$Fdv$\"(#pd9F17$$\"5++v=nO(p'HCF1$\" (N;j\"F17$Fiv$\"(5Mr\"F17$F^w$\"(-2u\"F17$Fcw$\"(+Iy\"F17$$\"5++vV[J*e 'QCF1$\"()*['=F17$$\"5++](o**\\w-W#F1$\"(3'e?F17$$\"5++DJXoS*=W#F1$\"( [S2#F17$Fhw$\"(U\"*3#F17$$\"5++]i!RxYnW#F1$\"(aa7#F17$F]x$\"([kH#F17$F bx$\"(2VK#F17$Fgx$\"(dzM#F17$$\"5++](o%\\ModCF1$\"(>L#F17$$\"5+] il(z)*y.Y#F1$\"(iSL#F17$$\"5++]i!\\1k2Y#F1$\"(KkL#F17$$\"5++Dcw=U`hCF1 $\"(86M#F17$Fay$\"(4cM#F17$F_[l$\"(+%)G#F17$F]]l$\"((*z6#F17$Fb]l$\"(Z _6#F17$Fg]l$\"()\\1@F17$F\\^l$\"(*f&3#F17$Fa^l$\"(!*=/#F17$$\"5+]7G8J7 &)yCF1$\"(vv&>F17$Ff^l$\"($R/=F17$$\"5+Dcw%)H\"e*zCF1$\"(%>*o\"F17$$\" 5+]Pf3'4F.[#F1$\"([Un\"F17$$\"5+v=UKigp![#F1$\"(EWn\"F17$F[_l$\"(wXn\" F17$F`_l$\"($*[n\"F17$Fe_l$\"(;Wn\"F17$Fj_l$\"(FKn\"F17$F_`l$\"(z+n\"F 17$Fd`l$\"(aCm\"F17$Fi`l$\"(2ck\"F17$F^al$\"(wqg\"F17$Fcal$\"(o#H:F17$ Fhal$\"(m;Q\"F17$F]bl$\"(:!z6F17$Fbbl$\"(ir<\"F17$Fgbl$\"('*[<\"F17$F \\cl$\"(xL<\"F17$Facl$\"(%)4<\"F17$Ffcl$\"(@j;\"F17$F[dl$\"(Lk:\"F17$F `dl$\"(MO4\"F17$Fedl$\"'M%o)F17$Fjdl$\"'GLxF17$F_el$\"',lgF17$$\"5++]7 GIDF1$\"'\"oC'F17$$\"5++]i!fVH,_#F 1$\"'GzkF17$$\"5++v$4\"\\\"\\3_#F1$\"'qfkF17$$\"5+++DJi)o:_#F1$\"'\")R kF17$$\"5++](=()G3I_#F1$\"'b*R'F17$Fiel$\"'5hjF17$$\"5+]i:5D]ADDF1$\"' %fM'F17$F^fl$\"'tWjF17$$\"5+](oa]kzn_#F1$\"'/wjF17$Fcfl$\"'TvkF17$Fhfl $\"'$*yrF17$F]gl$\"'&>@)F17$Fbgl$\"'jY\")F17$Fggl$\"'X\"3)F17$F\\hl$\" 'dI!)F17$Fahl$\"'(*z!)F17$Ffhl$\"(2e-\"F17$F[il$\"(U`>\"F17$F`il$\"(EC C\"F17$Feil$\"(P$R7F17$Fjil$\"(KiB\"F17$F_jl$\"(XJB\"F17$Fdjl$\"(QpA\" F17$Fijl$\"(^2A\"F17$F^[m$\"(*o37F17$Fc[m$\"(#))*>\"F17$Fh[m$\"(z2G\"F 17$F]\\m$\"(q7O\"F17$Fb\\m$\"(mrM\"F17$Fg\\m$\"(XTL\"F17$F\\]m$\"(b$H8 F17$Fa]m$\"(@wK\"F17$Ff]m$\"(l(G8F17$F[^m$\"(6>L\"F17$F`^m$\"(#yP8F17$ Fe^m$\"(QsM\"F17$Fj^m$\"(x;Q\"F17$F__m$\"()fZ9F17$Fd_m$\"(%3_9F17$Fi_m $\"(Df#F1$\"(+(Q9F17$F`cm$\"( YCU\"F17$$\"5++](=F[_bf#F1$\"(voS\"F17$$\"5+++D\"G?pqf#F1$\"(P8 F17$F\\gm$\"(HxD\"F17$Fagm$\"(o)p6F17$Ffgm$\"(G*)4\"F17$F[hm$\"(gw,\"F 17$F`hm$\"'!RT*F17$Fehm$\"(j?o)F-7$Fjhm$\"($\\k!)F-7$F_im$\"(ivW(F-7$F dim$\"(4%fF-7$Fcjm$\"(!QYbF-7$Fhjm$\" (>o;&F-7$F][n$\"(w'Q[F-7$Fb[n$\"(oA^%F--Fg[n6&Fi[nFj[n$\"\"(F*Fj[n-Fa \\n6#%8scheme~with~c[10]=15/16G-F$6%7`t7$F($\"')\\C#F-7$F/$\"'v,CF-7$F 5$\"''*3EF-7$F:$\"']^HF-7$F?$\"')3q$F-7$FD$\"'ovWF-7$FI$\"'mwaF-7$FN$ \"'E$)oF-7$FS$\"&k&)*F17$FX$\"'/!G\"F17$Fgn$\"'Ig;F17$F\\o$\"'&Rp\"F17 $Fao$\"'!fx\"F17$Ffo$\"'Hh=F17$F[p$\"'#H+#F17$F`p$\"'aI?F17$Fep$\"'zT? F17$Fjp$\"'d(3#F17$F_q$\"'Of@F17$Fdq$\"'1cEF17$Fiq$\"'))>LF17$F^r$\"'( 4Q%F17$Fcr$\"'p$y&F17$Fhr$\"'L#>'F17$F]s$\"'IFsF17$Fbs$\"'y%Q(F17$Fgs$ \"',AwF17$F\\t$\"'#G]*F17$Fat$\"(a7>\"F17$Fft$\"(;b@\"F17$F[u$\"(_YC\" F17$F`u$\"(W)o8F17$Feu$\"(c]_\"F17$Fju$\"(o&o;F17$F_v$\"(01$>F17$Fdv$ \"(7a,#F17$Fiv$\"(&HfAF17$F^w$\"(v^H#F17$Fcw$\"(NFM#F17$Fhw$\"(VTh#F17 $F]x$\"(Hmw#F17$Fbx$\"(Y()z#F17$Fgx$\"(ae#GF17$F\\y$\"(O2!GF17$Fay$\"( Q&oFF17$F_[l$\"(SBq#F17$F]]l$\"(C&3DF17$Fb]l$\"(F17$Fcal$\"(\\')*=F17$Fhal$\"(Dou \"F17$F]bl$\"(w#Q:F17$Fbbl$\"(ge`\"F17$Fgbl$\"(>H`\"F17$F\\cl$\"(25`\" F17$Facl$\"(Y\"G:F17$Ffcl$\"(VG_\"F17$F[dl$\"(?>^\"F17$F`dl$\"('eV9F17 $Fedl$\"([#)>\"F17$Fjdl$\"(:'*3\"F17$F_el$\"'\"4f)F17$Fdel$\"'Z=$)F17$ Fiel$\"'$R-)F17$F^fl$\"']%)zF17$Fcfl$\"'?Q!)F17$Fhfl$\"'-k%)F17$F]gl$ \"'=6\"*F17$Fbgl$\"'pQ!*F17$Fggl$\"'/m*)F17$F\\hl$\"'c/*)F17$Fahl$\"'% Q#*)F17$Ffhl$\"(W;1\"F17$F[il$\"(?W@\"F17$F`il$\"($*zD\"F17$Feil$\"(k[ D\"F17$Fjil$\"(@\"*\\\"F17$$\"5++D\"Gy!4G'e#F1$\"(J>\\\"F17$$\"5+]i!R]*[7(e#F 1$\"($3)[\"F17$F[cm$\"(<6\\\"F17$F`cm$\"(sl_\"F17$Fecm$\"(%Qb:F17$Fjcm $\"(cx^\"F17$F_dm$\"(5P[\"F17$Fddm$\"(w(y9F17$Fidm$\"(C&y9F17$Fcem$\"( in[\"F17$F]fm$\"(k'4:F17$$\"5++]P%))))G1h#F1$\"(_/^\"F17$Fbfm$\"(87]\" F17$$\"5++]7G6$[Ah#F1$\"(Q?\\\"F17$Fgfm$\"(=H[\"F17$F\\gm$\"(-@T\"F17$ Fagm$\"(VvK\"F17$Ffgm$\"(#Rh7F17$F[hm$\"(C\\<\"F17$F`hm$\"(#y!4\"F17$F ehm$\")]545F-7$Fjhm$\"(G.T*F-7$F_im$\"(.Zq)F-7$Fdim$\"(Jw1)F-7$Fiim$\" (sY`(F-7$F^jm$\"(7(zpF-7$Fcjm$\"(2M_'F-7$Fhjm$\"(=M3'F-7$F][n$\"(YIq&F -7$Fb[n$\"(CwK&F--Fg[n6&Fi[n$\"\"*F*$\"\"%F*Fj[n-Fa\\n6#%8scheme~with~ c[10]=19/20G-F$6%7fv7$F($\"%SbF-7$F/$!%(G%F-7$F5$!&^J\"F-7$F:$!&V0#F-7 $F?$!&(=BF-7$FD$!&c'=F-7$FI$!%9wF-7$FN$\"&=K\"F-7$FS$\"%(H(F17$FX$\"&F I\"F17$Fgn$\"&M0#F17$F_q$\"&v3$F17$Fiq$\"&%>dF17$F^r$\"&WC)F17$Fcr$\"' Q.7F17$$\"5+]Pf$Q]%GwBF1$\"'(e@\"F17$$\"5++v=DkF17$F_v$ \"'r1#)F17$Fien$\"'*4O)F17$Fdv$\"'s9))F17$Fafn$\"(_K-\"F17$Fiv$\"(.')3 \"F17$F^w$\"(&)f5\"F17$Fcw$\"(ci8\"F17$F_gn$\"(7F?\"F17$Fdgn$\"(FSO\"F 17$Fign$\"(gUP\"F17$Fhw$\"(mUQ\"F17$Fahn$\"(I&49F17$F]x$\"(tHa\"F17$Fb x$\"(\\>c\"F17$Fgx$\"(xtd\"F17$F\\y$\"(BCc\"F17$Fay$\"(!oR:F17$Ffy$\"( 2aa\"F17$F[z$\"(V)[:F17$F`z$\"(8qa\"F17$Fez$\"(V%Q:F17$Fjz$\"(&y::F17$ F_[l$\"(fcY\"F17$Fd[l$\"(A]O\"F17$Fi[l$\"(l2F\"F17$F^\\l$\"(xAF\"F17$F c\\l$\"(qOF\"F17$Fh\\l$\"(xgF\"F17$F]]l$\"(0hF\"F17$Fg]l$\"(SIE\"F17$F a^l$\"(=_>\"F17$Fe\\o$\"(av5\"F17$Ff^l$\"'*o[*F17$F]]o$\"'V$H)F17$Fb]o $\"'lP\")F17$Fg]o$\"'_Q\")F17$F[_l$\"'ER\")F17$F`_l$\"'!39)F17$Fe_l$\" 'eO\")F17$Fj_l$\"'eD\")F17$F_`l$\"')\\4)F17$Fd`l$\"'$)=!)F17$Fi`l$\"'? [yF17$F^al$\"'0buF17$Fcal$\"'4emF17$$\"5+voH/)Qe*)[#F1$\"'V;gF17$Fhal$ \"'[V^F17$$\"5+Dc,\"zMa(*[#F1$\"''>(RF17$F]bl$\"'mjIF17$Fbbl$\"'\")eIF 17$Fgbl$\"'W_IF17$F\\cl$\"'EYIF17$Facl$\"'nJIF17$Ffcl$\"':%*HF17$F[dl$ \"'%H!HF17$$\"5+]P%[m'fJ(\\#F1$\"'x)p#F17$F`dl$\"'WvAF17$$\"5+]7GQ')y! *)\\#F1$\"'\"pX\"F17$Fedl$!%$\\$F17$Fjdl$!&X#))F17$F_el$!''3a#F17$F[bo $!'5GDF17$F`bo$!'x9DF17$$\"5++D1*)p3`;DF1$!':1DF17$Febo$!'p#\\#F17$$\" 5++voH'Hqz^#F1$!'dmCF17$Fdel$!']8CF17$F]co$!'41BF17$Fbco$!'pW@F17$Fgco $!'@Q@F17$F\\do$!'sJ@F17$Fado$!'F=@F17$Fiel$!'H-@F17$Fido$!'w)3#F17$F^ fl$!'ij?F17$Faeo$!'o6?F17$Fcfl$!'#Q!>F17$Fhfl$!'Os7F17$F]gl$!&Qm$F17$F bgl$!&[j$F17$Fggl$!&hf$F17$$\"5+]7.#[u_X`#F1$!&\"\\NF17$F\\hl$!&@V$F17 $$\"5+]PMxkt5ODF1$!&U8$F17$Fahl$!&tV#F17$$\"5++]7G\\2\\QDF1$\"&()Q#F17 $$\"5+++D\"Q#o4SDF1$\"'lA=F17$$\"5++]PM)*GqTDF1$\"'q0=F17$Ffhl$\"'.*y \"F17$$\"5++]iSZ]\"\\a#F1$\"'.z7@la#F1$\"':C=F17$$\"5++ ](ok>F\"[DF1$\"'\"\\8#F17$F[il$\"'\"fO$F17$F`il$\"'i\\PF17$Feil$\"'LSP F17$FjilFbhr7$F_jl$\"'k@PF17$Fdjl$\"'&Gq$F17$Fijl$\"'F%o$F17$F^[m$\"'^ \\OF17$Fc[m$\"'9'F17$Fc^p$\"'7vgF17$Fh^p$\"'aCgF17$F]_p$\"'l;gF17$Fb_p$\"'^LgF17$F g_p$\"'A#4'F17$Fecm$\"'_\">'F17$Fjcm$\"'uTgF17$F_dm$\"'W.fF17$Fidm$\"' [peF17$F]fm$\"'3TfF17$Fbfm$\"'^(*eF17$Fgfm$\"'hDeF17$F\\gm$\"'[yaF17$F agm$\"'s\"4&F17$Ffgm$\"']yZF17$F[hm$\"'xBWF17$F`hm$\"'p\"4%F17$Fehm$\" ((*Hx$F-7$Fjhm$\"(XA]$F-7$F_im$\"(<@B$F-7$Fdim$\"(R$))HF-7$Fiim$\"(\"[ $y#F-7$F^jm$\"(?[c#F-7$Fcjm$\"(M.R#F-7$Fhjm$\"('zAAF-7$F][n$\"(cy2#F-7 $Fb[n$\"(2?$>F--%'COLOURG6&Fi[n$\"*++++\"!\")Fj[nF^ht-Fa\\n6#%8scheme~ with~c[10]=32/33G-F$6%7_v7$F($\"%'4&F-7$F/$!%aLF-7$F5$!&Y=\"F-7$F:$!&2 .#F-7$F?$!&&>HF-7$FD$!&e6$F-7$FI$!&%fHF-7$FN$!&@K#F-7$FS$\"#nF17$FX$\" %:AF17$Fgn$\"%`[F17$F_q$\"%i#)F17$Fiq$\"&8i\"F17$F^r$\"&;S#F17$$\"5+++ +]hDJpBF1$\"&A4$F17$Fcr$\"&jt$F17$Fjer$\"&4'QF17$Fhr$\"&CE%F17$Fgfr$\" &Vs%F17$F\\gr$\"&e_&F17$Fagr$\"&$*z&F17$F]s$\"&5$eF17$Fbs$\"&)efF17$Fg s$\"&>E'F17$F\\t$\"'Q45F17$Fat$\"'=*p\"F17$Fft$\"'\"Rt\"F17$F[u$\"'\") )y\"F17$F`u$\"'-iAF17$Feu$\"'')))GF17$Fju$\"'8ZMF17$F_v$\"'O.ZF17$Fien $\"'`3[F17$Fdv$\"'RY^F17$Fafn$\"'UOiF17$Fiv$\"'AKnF17$F^w$\"'/SoF17$Fc w$\"'%Q0(F17$F_gn$\"'BzvF17$Fdgn$\"'!Q)))F17$Fign$\"'Y]*)F17$Fhw$\"'o: !*F17$$\"5++v=U0#H^W#F1$\"''\\3*F17$Fahn$\"'O$>*F17$$\"5++D1RUVO[CF1$ \"'*zZ*F17$F]x$\"(*=K5F17$Fbx$\"(R_/\"F17$Fgx$\"(ec0\"F17$F\\y$\"(#3X5 F17$Fay$\"(xz-\"F17$Ffy$\"($yJ5F17$F[z$\"(1Q.\"F17$$\"5+Dc^.f/#fY#F1$ \"(@L.\"F17$F`z$\"(v>.\"F17$$\"5+voHzLSsmCF1$\"(/$H5F17$Fez$\"(#pC5F17 $Fjz$\"(uf+\"F17$F_[l$\"'%*['*F17$Fd[l$\"'wD))F17$Fi[l$\"'f`!)F17$F^\\ l$\"'@j!)F17$Fc\\l$\"'+s!)F17$Fh\\l$\"'-(3)F17$F]]l$\"'7$3)F17$Fg]l$\" 'erzF17$Fa^l$\"'Y0uF17$Fe\\o$\"'YvmF17$Ff^l$\"''GN&F17$F]]o$\"'@fVF17$ Fb]o$\"'BHUF17$Fg]o$\"'pHUF17$F[_l$\"'3IUF17$F`_l$\"'*3B%F17$Fe_l$\"'a FUF17$Fj_l$\"'q=UF17$F_`l$\"'n$>%F17$Fd`l$\"'pITF17$Fi`l$\"'%*))RF17$F ^al$\"'VhOF17$Fcal$\"'v'*HF17$F[bs$\"'OhCF17$Fhal$\"'%Gt\"F17$Fcbs$\"& 7b(F17$F]bl$!$i#F17$Fbbl$!$l#F17$Fgbl$!$=$F17$F\\cl$!$l&F17$Facl$!%%\\ \"F17$$\"5+vo/)p1E`\\#F1$!%bDF17$Ffcl$!%F17$Fhjs$!&@h'F17$F][t$!&6b'F17$Ffh l$!&k['F17$Fe[t$!&.P'F17$Fj[t$!&?%eF17$F_\\t$!&0M$F17$F[il$\"&m&fF17$F `il$\"&[$))F17$Feil$\"&K\"))F17$Fjil$\"&-z)F17$F_jl$\"&\"p()F17$Fdjl$ \"&Vs)F17$Fijl$\"&9o)F17$F^[m$\"&Ch)F17$Fc[m$\"&Pr)F17$Fh[m$\"'<%R\"F1 7$F]\\m$\"'02>F17$Fb\\m$\"'f()=F17$Fg\\m$\"'zs=F17$F\\]m$\"'%H(=F17$Fa ]m$\"'L')=F17$F[^m$\"'iD>F17$Fe^m$\"'!=,#F17$Fj^m$\"'Ly@F17$F__m$\"']v CF17$Fd_m$\"'m8DF17$Fi_m$\"'***\\#F17$F^`m$\"')GZ#F17$Fc`m$\"'uYCF17$F ]am$\"'?ECF17$Fgam$\"'&>X#F17$F\\bm$\"'8^EF17$Fabm$\"'%*\\FF17$F][r$\" '%fr#F17$Ffbm$\"'l#o#F17$Fj[r$\"'JbEF17$F[cm$\"'=dEF17$Fa]p$\"'WXFF17$ Ff]p$\"'+%z#F17$F[^p$\"'_iFF17$F`cm$\"'NJFF17$Fc^p$\"'d,FF17$Fh^p$\"'m yEF17$Fb_p$\"'I!o#F17$Fecm$\"'jUFF17$Fjcm$\"'IwEF17$F_dm$\"'S9EF17$Fid m$\"'`'f#F17$F]fm$\"'%ph#F17$Fbfm$\"']&f#F17$Fgfm$\"'$Qc#F17$F\\gm$\"' O(R#F17$Fagm$\"'+>AF17$Ffgm$\"'Nx?F17$F[hm$\"'mA>F17$F`hm$\"'Wy:F-7$F_im$\"(>$*R\"F-7$Fdim$\"(m.H\"F-7$Fiim$ \"(;y>\"F-7$F^jm$\"(Tb4\"F-7$Fcjm$\"(Tp,\"F-7$Fhjm$\"'e?%*F-7$F][n$\"' 7v()F-7$Fb[n$\"'%37)F--Fg[n6&Fi[n$\"\"&F*Fj[nF^\\n-Fa\\n6#%8scheme~wit h~c[10]=81/82G-F$6%7hu7$F($\"&!QUF-7$F/$\"&z@%F-7$F5$\"&FE%F-7$F:$\"&q ^%F-7$F?$\"&sQ&F-7$FD$\"&wb'F-7$FI$\"&?=)F-7$FN$\"'?b5F-7$FS$\"&/c\"F1 7$FX$\"&W.#F17$Fgn$\"&\\h#F17$F_q$\"&@L$F17$Fiq$\"&$f[F17$F^r$\"&!yiF1 7$Fcr$\"&iE)F17$Fjer$\"&YW)F17$Fhr$\"&&4*)F17$Fgfr$\"%*F17$F\\gr$\" 'OD5F17$Fagr$\"'ub5F17$F]s$\"'Zh5F17$Fbs$\"'f%3\"F17$Fgs$\"'lB6F17$F\\ t$\"'a!\\\"F17$Fat$\"'F17$F_el$!'GCIF17$F`bo$!'+&*HF17$Fdel$!',lHF17$ F]co$!'KTHF17$Fbco$!'=,HF17$Fgco$!'T#*GF17$F\\do$!'g$)GF17$Fado$!'\\lG F17$Fiel$!'%e%GF17$Fido$!'(G$GF17$F^fl$!'C9GF17$Faeo$!'i#y#F17$Fcfl$!' 2CFF17$Fhfl$!'P1CF17$F]gl$!'Lc>F17$Fbgl$!'zS>F17$Fggl$!'`C>F17$Fcis$!' #\\\">F17$F\\hl$!'$=!>F17$F[js$!'fz=F17$Fahl$!'[P=F17$Ffhl$!&qm(F17$F[ il$\"%=rF17$F`il$\"&zi#F17$Feil$\"&+\"F17$Fb\\m$\"&y\"**F17$Fg\\m$\"&/&)*F17$F\\]m$\"& E()*F17$Fa]m$\"&G***F17$F[^m$\"'bI5F17$Fe^m$\"'v'4\"F17$Fj^m$\"'+B7F17 $F__m$\"'1Z9F17$Fd_m$\"'xx9F17$Fi_m$\"'rp9F17$F^`m$\"'w`9F17$Fc`m$\"'e Q9F17$F]am$\"'gG9F17$Fgam$\"'gc9F17$F\\bm$\"'^G;F17$Fabm$\"'t< " 0 "" {MPLTEXT 1 0 500 "evalf[20](plot([p(x)-'pn_RK7_1'(x),p(x)-'pn_RK7_2'(x),p(x)-'pn_RK 7_3'(x),\np(x)-'pn_RK7_4'(x),p(x)-'pn_RK7_5'(x),p(x)-'pn_RK7_6'(x)],x= 2.8..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0) ,COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Ve rner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c [10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`schem e with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 848 571 571 {PLOTDATA 2 "6,-%'CURVESG6%7dp7$$\"#G!\"\"$\"(!Q;#*!#?7$$\"5nmmm;RP&z% G!#>$\"(tcW(F-7$$\"5LLL$37.y'*)GF1$\"(T.['F-7$$\"5+++DJ'pRJ\"HF1$\"(J# GhF-7$$\"5nmmmTh8gOHF1$\"(Rx)eF-7$$\"5nmmmmX&=-'HF1$\"(+mu&F-7$$\"5nmm m\"*Hd$Q)HF1$\"(pXp&F-7$$\"5+++D\"emSt+$F1$\"('RFdF-7$$\"5LLL$3JF1$\"(#G/qF-7$$\"5nmm;a25BmJF1$\"(Qz9)F-7$$\"5+++]P$o` F@$F1$\"(nAs*F-7$$\"5LLLL$Q(zggKF1$\")N.#=\"F-7$$\"5nmmm\"*e!eFI$F1$\" )X/(R\"F-7$$\"5++++]r!4-N$F1$\")5mX;F-7$$\"5++++]A\\&yR$F1$\")3]]=F-7$ $\"5++++]DF\"3U$F1$\"(hb\">F17$$\"5++++]G0xVMF1$\"(ah&>F17$$\"5ML3-j* \\#)*[MF1$\"(Q<'>F17$$\"5nm;/wqW>aMF1$\"(tb'>F17$$\"5++D1*=W1%fMF1$\"( H\"o>F17$$\"5MLL3-8%=YY$F1$\"(S'p>F17$$\"5nmT5:%QI)pMF1$\"(&Rp>F17$$\" 5++]7GbB/vMF1$\"(x&o>F17$$\"5MLe9TEVD![$F1$\"(jg'>F17$$\"5nmm;a(Hma[$F 1$\"(aE'>F17$$\"5+++vo[kD5NF1$\"(')=$>F17$$\"5LLLL$)*fY]`$F1$\"(+'y=F1 7$$\"5LLLLL>w/xNF1$\")'R]v\"F-7$$\"5+++]()*y/fi$F1$\")!Rqe\"F-7$$\"5LL LLLk;:pOF1$\")cgV9F-7$$\"5+++]()R.g;PF1$\")p488F-7$$\"5+++]i*p#yhPF1$ \")&)4C7F-7$$\"5mmmT&eAa`y$F1$\")%*H#>\"F-7$$\"5LLLL3_d#*3QF1$\")FLq6F -7$$\"5LLLeRr\"F-7$$\"5nmm;aZ%=u%RF1$\"),A=7F-7$$\"5+++]7A;k*)RF1$\")e$>G\"F -7$$\"5nmmmT2QCNSF1$\")TGm8F-7$$\"5+++++F`N#3%F1$\")vgc9F-7$$\"5+++]PU +S0TF1$\")a#[\\\"F-7$$\"5++++vdZWGTF1$\")%o__\"F-7$$\"5++]7GJKfRTF1$\" )DTO:F-7$$\"5+++D\"[qT2:%F1$\")XQW:F-7$$\"5++]PMy,*=;%F1$\")W%*[:F-7$$ \"5+++](=lQI<%F1$\")H2]:F-7$$\"5++](o48%F1$\")+tR:F-7$$\"5++]i:*3u,@%F1$\")sXF:F-7$$\"5++++DoDbAUF1$\")dA6:F -7$$\"5mmm;H'z(zWUF1$\")r!>Z\"F-7$$\"5LLLLLCI/nUF1$\")9W@9F-7$$\"5++++ ]#3YXJ%F1$\")#z&)G\"F-7$$\"5nmm;a84fdVF1$\")x]b6F-7$$\"5++++]$G]YS%F1$ \")wO95F-7$$\"5LLL$3K[H*[WF1$\"(h@'*)F-7$$\"5+++]P0S@&\\%F1$\"(#QdzF-7 $$\"5LLLL$eel/a%F1$\"(!GZsF-7$$\"5mmmTN\"p_Tc%F1$\"(Ns*pF-7$$\"5+++](o zRye%F1$\"(z3$oF-7$$\"5MLL3x%H`1h%F1$\"(g;v'F-7$$\"5nmmmm#zmMj%F1$\"(] lv'F-7$$\"5nmm;H*)ozcYF1$\"(8+&oF-7$$\"5nmmm\"f)p7!o%F1$\"(g'RqF-7$$\" 5+++Dc(*QE.ZF1$\"(qdK(F-7$$\"5LLL$3#43SEZF1$\"(\\,s(F-7$$\"5+++++Z;#*o ZF1$\"(I\\w)F-7$$\"5LLLLeD`l<[F1$\")neg5F-7$$\"5+++]7<$\\%R[F1$\")tvr6 F-7$$\"5nmmmm3LCh[F1$\"))ypI\"F-7$$\"5MLL3F\\-[%)[F1$\")kj$[\"F-7$$\"5 +++]()*=:q\"F-7$$\"5+++vV[v$)=\\F1$\")&fY#=F-7$$\"5+++++2 z&*H\\F1$\"()*['>F17$$\"5+++Dcl#y5%\\F1$\"(Ox6#F17$$\"5+++]7C')>_\\F1$ \"(xyG#F17$$\"5++]P4o*[T'\\F1$\"(a8\\#F17$$\"5+++D17$*4w\\F1$\"(X1s#F1 7$$\"5++vo/%[u?)\\F1$\"(oL'GF17$$\"5++]7.c'\\!))\\F1$\"(u])HF17$$\"5++ Dc,G[-%*\\F1$\"(zB9$F17$$\"\"&\"\"!$\"(#eELF1-%&COLORG6&%$RGBG$FjflFjf l$\"\"#F*$\"\"\"Fjfl-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12 G-F$6%7dp7$F($\"(oA^%F-7$F/$\"(XTj$F-7$F5$\"($phJF-7$F:$\"(\\9*HF-7$F? $\"(Wd(GF-7$FD$\"(6*3GF-7$FI$\"(8Wy#F-7$FN$\"(o4!GF-7$FS$\"(2c&GF-7$FX $\"(K4%HF-7$Fgn$\"(*ffIF-7$F\\o$\"(1>T$F-7$Fao$\"($Q`RF-7$Ffo$\"(F))p% F-7$F[p$\"(Cnq&F-7$F`p$\"(uOv'F-7$Fep$\"(X)yzF-7$Fjp$\"(+a)*)F-7$F_q$ \"'=)H*F17$Fdq$\"'*z[*F17$Fiq$\"'U8&*F17$F^r$\"'!)H&*F17$Fcr$\"'7S&*F1 7$Fhr$\"'Q'F-7$F_v$\"(F!\\fF-7$Fdv$\"(^fz&F-7$Fiv$\"(q

u,\"F17$Fjdl$\"($y(4\"F17$F_el$\"(tO>\"F17$Fdel$\"(s7I\"F17$Fie l$\"(]pO\"F17$F^fl$\"(XXU\"F17$Fcfl$\"(Br\\\"F17$Fhfl$\"(V?e\"F1-F^gl6 &F`glFagl$\"\"(F*Fagl-Fggl6#%8scheme~with~c[10]=15/16G-F$6%7dp7$F($\"( CwK&F-7$F/$\"(M>I%F-7$F5$\"(=eu$F-7$F:$\"(TOa$F-7$F?$\"(0dS$F-7$FD$\"( B`K$F-7$FI$\"(=eH$F-7$FN$\"(4`J$F-7$FS$\"(L2Q$F-7$FX$\"([I[$F-7$Fgn$\" (9di$F-7$F\\o$\"(gD0%F-7$Fao$\"((z4ZF-7$Ffo$\"(Beh&F-7$F[p$\"(=.$oF-7$ F`p$\"(P(z!)F-7$Fep$\"(#)F-7$Fg]l$\"(\"\\` uF-7$F\\^l$\"(4lo'F-7$Fa^l$\"(.3(eF-7$Ff^l$\"(5]=&F-7$F[_l$\"(l-g%F-7$ F`_l$\"(ap=%F-7$Fe_l$\"(')>/%F-7$Fj_l$\"(dd%RF-7$F_`l$\"(#R+RF-7$Fd`l$ \"(YQ!RF-7$Fi`l$\"(x(eRF-7$F^al$\"()>qSF-7$Fcal$\"(2rB%F-7$Fhal$\"(*ym WF-7$F]bl$\"(wQ2&F-7$Fbbl$\"(+&QhF-7$Fgbl$\"(9+y'F-7$F\\cl$\"(i9c(F-7$ Facl$\"(]`e)F-7$Ffcl$\"(F1-F^gl6&F`g l$\"\"*F*$\"\"%F*Fagl-Fggl6#%8scheme~with~c[10]=19/20G-F$6%7ep7$F($\"( 2?$>F-7$F/$\"()o^:F-7$F5$\"((4`8F-7$F:$\"(iFG\"F-7$F?$\"(P_B\"F-7$FD$ \"(b)37F-7$FI$\"(L%*>\"F-7$FN$\"(\"[27F-7$FS$\"(l>B\"F-7$FX$\"(=!p7F-7 $Fgn$\"(0*>8F-7$F\\o$\"(vtY\"F-7$Fao$\"(a4p\"F-7$Ffo$\"(5\"**>F-7$F[p$ \"(q$HCF-7$$\"5+++]P;Io\"G$F1$\"(8@l#F-7$F`p$\"(c**)GF-7$$\"5MLL$3_c$[ ELF1$\"()=kJF-7$Fep$\"(,HW$F-7$$\"5+++++(*>.uLF1$\"(evo$F-7$Fjp$\"(Fr) QF-7$F_q$\"'c5SF17$Fdq$\"'`wSF17$Fiq$\"'z$3%F17$F^r$\"'(e3%F17$Fcr$\"' F&3%F17$Fhr$\"'q%3%F17$Fbs$\"'UuSF17$F\\t$\"'BcSF17$Fat$\"'l*)RF17$Fft $\"'.()QF17$F[u$\"(S(\\OF-7$F`u$\"(nqI$F-7$Feu$\"(OH+$F-7$Fju$\"(^ks#F -7$F_v$\"(AHa#F-7$Fdv$\"(W%zCF-7$Fiv$\"(dyV#F-7$F^w$\"(f%=CF-7$Fcw$\"( kNT#F-7$Fhw$\"(UYU#F-7$F]x$\"((4_CF-7$Fbx$\"(_d\\#F-7$Fgx$\"(@Nb#F-7$F \\y$\"(N@o#F-7$Fay$\"($p`GF-7$Ffy$\"(4O/$F-7$F[z$\"(?k7$F-7$F`z$\"(@Q> $F-7$Fez$\"(H!>KF-7$Fjz$\"(3nB$F-7$F_[l$\"(+kC$F-7$Fd[l$\"(-$[KF-7$Fi[ l$\"(57C$F-7$F^\\l$\"(vRA$F-7$Fc\\l$\"(&*[>$F-7$Fh\\l$\"(`\"fJF-7$F]]l $\"(GW2$F-7$Fb]l$\"(X)oHF-7$Fg]l$\"(05q#F-7$F\\^l$\"(yJV#F-7$Fa^l$\"(I 59#F-7$Ff^l$\"(;l)=F-7$F[_l$\"(:^m\"F-7$F`_l$\"(5!3:F-7$Fe_l$\"(FUX\"F -7$Fj_l$\"(\\*=9F-7$F_`l$\"()=.9F-7$Fd`l$\"(zcS\"F-7$Fi`l$\"('QF9F-7$F ^al$\"(S8Z\"F-7$Fcal$\"((pM:F-7$Fhal$\"('*4i\"F-7$F]bl$\"(qa%=F-7$Fbbl $\"(c_A#F-7$Fgbl$\"(*zYCF-7$F\\cl$\"(;Zr#F-7$Facl$\"((RoIF-7$Ffcl$\"(W f]$F-7$F[dl$\"(!Q`PF-7$F`dl$\"''[.%F17$Fedl$\"'fWVF17$Fjdl$\"'Y*o%F17$ F_el$\"'#>5&F17$Fdel$\"'xmbF17$Fiel$\"'-ceF17$F^fl$\"'K0hF17$Fcfl$\"'* )HkF17$Fhfl$\"'uCoF1-%'COLOURG6&F`gl$\"*++++\"!\")FaglFg[p-Fggl6#%8sch eme~with~c[10]=32/33G-F$6%7^q7$F($\"'%37)F-7$F/$\"'/FlF-7$$\"5+++v=&)e \")oGF1$\"'*R3'F-7$F5$\"'_gdF-7$F:$\"'%H\\&F-7$F?$\"'y5`F-7$FD$\"')o@& F-7$FI$\"'3&=&F-7$FN$\"'#*G_F-7$FS$\"'F]`F-7$FX$\"'hAbF-7$Fgn$\"'p`dF- 7$F\\o$\"'K\"R'F-7$Fao$\"'F.tF-7$Ffo$\"'uF&)F-7$F[p$\"(&GL5F-7$Fa]o$\" (,88\"F-7$F`p$\"(Y)Q7F-7$Fi]o$\"(t]O\"F-7$Fep$\"(&4+:F-7$Fa^o$\"(J_h\" F-7$Fjp$\"(Wtq\"F-7$$\"5+++++CQL4MF1$\"'\"yt\"F17$F_q$\"'TdUHMF1$\" '_p:>RPF1$\"(Pd 6\"F-7$F_v$\"(-H3\"F-7$Fdv$\"(Dz0\"F-7$Fiv$\"(uI/\"F-7$F^w$\"(\"4Q5F-7 $Fcw$\"(6%Q5F-7$Fhw$\"(B`/\"F-7$F]x$\"(d!f5F-7$Fbx$\"(F!z5F-7$Fgx$\"( \"*R5\"F-7$F\\y$\"(lm:\"F-7$Fay$\"(cfA\"F-7$Ffy$\"(SOI\"F-7$F[z$\"(4'Q 8F-7$F`z$\"(xsO\"F-7$Fez$\"(ZyP\"F-7$Fjz$\"(E]Q\"F-7$F_[l$\"(w')Q\"F-7 $Fd[l$\"(!z)Q\"F-7$Fi[l$\"(:\\Q\"F-7$F^\\l$\"(ZkP\"F-7$Fc\\l$\"(,AO\"F -7$Fh\\l$\"(YhM\"F-7$F]]l$\"(7#48F-7$Fb]l$\"(#)[E\"F-7$Fg]l$\"(9s:\"F- 7$F\\^l$\"(xy/\"F-7$Fa^l$\"''GA*F-7$Ff^l$\"'%e2)F-7$F[_l$\"'znqF-7$$\" 5mmmTg&zRy^%F1$\"':%o'F-7$F`_l$\"'zpjF-7$Fe_l$\"'%>9'F-7$Fj_l$\"'y(*fF -7$F_`l$\"'UVfF-7$Fd`l$\"'flfF-7$Fi`l$\"'`sgF-7$F^al$\"'5)G'F-7$Fcal$ \"';%e'F-7$Fhal$\"'t%)pF-7$F]bl$\"'I@!)F-7$Fbbl$\"'\\'p*F-7$Fgbl$\"(YK 1\"F-7$F\\cl$\"(C[<\"F-7$Facl$\"(\"GB8F-7$Ffcl$\"(1#3:F-7$F[dl$\"(&e8; F-7$F`dl$\"'NM#F17$Fdel$ \"'n\"R#F17$Fiel$\"'/9DF17$F^fl$\"'y?EF17$Fcfl$\"'2fFF17$Fhfl$\"'QJHF1 -F^gl6&F`gl$FiflF*FaglFdgl-Fggl6#%8scheme~with~c[10]=81/82G-F$6%7jp7$F ($\"'5pjF-7$F/$\"'+V^F-7$F5$\"',^XF-7$F:$\"'qQVF-7$F?$\"'w\">%F-7$FD$ \"'u7TF-7$FI$\"'y&3%F-7$FN$\"'`?TF-7$FS$\"'o?UF-7$FX$\"'wiVF-7$Fgn$\"' FaXF-7$F\\o$\"'X%4&F-7$Fao$\"'NueF-7$Ffo$\"''=$pF-7$F[p$\"'NX%)F-7$F`p $\"(>>,\"F-7$Fep$\"(N0A\"F-7$Fa^o$\"($p78F-7$Fjp$\"(FrQ\"F-7$Fe`p$\"'Y 79F17$F_q$\"'bH9F17$F]ap$\"'%[V\"F17$Fbap$\"'kR9F17$Fgap$\"'0S9F17$F\\ bp$\"'iT9F17$Fabp$\"',[9F17$Fdq$\"'bZ9F17$Fiq$\"'f[9F17$F^r$\"'CZ9F17$ Fcr$\"'hW9F17$Fhr$\"'*GW\"F17$Fbs$\"'JN9F17$F\\t$\"'7D9F17$Fat$\"'q#R \"F17$Fft$\"'r_8F17$F[u$\"(lOE\"F-7$F`u$\"(\\y8\"F-7$Feu$\"(Y*G5F-7$Fj u$\"'YO$*F-7$F]ep$\"'*z**)F-7$F_v$\"'\"zt)F-7$Fdv$\"'#y`)F-7$Fiv$\"'K< %)F-7$F^w$\"'tu$)F-7$Fcw$\"'6v$)F-7$Fhw$\"'9H%)F-7$F]x$\"'8R&)F-7$Fgx$ \"'t0*)F-7$F\\y$\"'sQ$*F-7$Fay$\"'0)*)*F-7$Ffy$\"(A,0\"F-7$F[z$\"(lm2 \"F-7$F`z$\"(F17$Fdel$\"'e\"4#F17$Fiel$\"'h :AF17$F^fl$\"'l7BF17$Fcfl$\"'Q\\CF17$Fhfl$\"'@ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 11 of 11 stage, or der 7 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that appears in a n article by F. G. Lether: Mathematics of Computation, Vol. 20, no. 9 5, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/ *&%#dyG\"\"\"%#dxG!\"\"*&-%$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,$*&-%$expG6#,$% \"xG!\"\"\"\"\"-%$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'!\"\"\"\"\"-%$sin G6#*&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(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"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$$\"39 pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"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$$\"3IVkw 'oSC/)F,$!3&[P'\\m/)z7%F,7$$\"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$$\"3Z,MUjZ 4H%)F,$\"3c]$R4W&G]NF07$$\"3=ryb([][Z)F,$\"37Ccp$Rpv:\"F,7$$\"3%QzY4r \"HF&)F,$\"3CcG)))y\"yp?F,7$$\"3g()F,$\"3y_cgRCTxTF,7$$\"3Wx*okV>:t)F,$\"37VUul#=X<%F,7$$\"3q% 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$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF07$$\"376: 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$yA*F,$\"3ML bSaq^*[\"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,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;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 0 785 "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 \+ := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16 ,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `* c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 30: \nfor ct to 6 do\n Qn_RK7_||ct := RK7_||ct(Q(x,y),x,y,x0,evalf[33](y 0),evalf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK7_| |ct):\n for ii to numpts do\n sm := sm+(Qn_RK7_||ct[ii,2]-q(Qn_ RK7_||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,&*( -%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF 07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$ +&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint496\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+# \"#6\"#7$\"+^\\T2#)!#@7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+bR**G7!#? 7$/F8#\"#>\"#?$\"+w@F9)*F57$/F8#\"#K\"#L$\"+'f=MI\"F?7$/F8#\"#\")\"##) $\"+G!\\(>8F?7$/F8#\"$p\"\"$q\"$\"+Nn,,5F?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-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 0 709 "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 := [`Sharp-Verner scheme with ` *c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`sc heme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10 ]=169/170]: errs := []:\nDigits := 30:\nfor ct to 6 do\n qn_RK7_||ct := RK7_||ct(Q(x,y),x,y,x0,evalf(y0),evalf(hh),numsteps,true);\nend do :\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 6 do\n errs := [op( errs),abs(qn_RK7_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0! \"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sin G6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~ of~steps:~~~GFFQ)pprint516\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~sch eme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+1Iq55!#>7$/*&%-scheme~with~ GF+F,F+#\"#:\"#;$\"+l>U&\\\"F57$/F8#\"#>\"#?$\"+'G*o(=\"F57$/F8#\"#K\" #L$\"+>#3 " 0 "" {MPLTEXT 1 0 433 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint((q(x)-'qn_RK7_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false,nu mpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~sche me~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+#[1;(o!#@7$/*&%-scheme~with~ GF+F,F+#\"#:\"#;$\"+PS7N5!#?7$/F8#\"#>\"#?$\"+29C)G)F57$/F8#\"#K\"#L$ \"+Sc$f5\"F?7$/F8#\"#\")\"##)$\"+e:W=6F?7$/F8#\"$p\"\"$q\"$\"+UH#R[)F5 Q)pprint536\"" }}}{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 0 518 "evalf[25](plot(['qn_RK7_1'(x)-q(x),'qn_RK7_2'(x)-q(x ),'qn_RK7_3'(x)-q(x),\n'qn_RK7_4'(x)-q(x),'qn_RK7_5'(x)-q(x),'qn_RK7_6 '(x)-q(x)],x=0..0.5,-2.5e-22..1.5e-22,font=[HELVETICA,9],\ncolor=[COLO R(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,. 5,0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme w ith c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,` scheme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error c urves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 " " {GLPLOT2D 783 536 536 {PLOTDATA 2 "6,-%'CURVESG6%7hn7$$\"\"!F)F(7$$ \":MLLLLLL$3x&)*3\"!#E$\"\"'!#D7$$\":nmmmmm;H2P\"Q?F-$\"#5F07$$\":MLLL LLLeRwX5$F-$\"#=F07$$\":NLLLLLL3x%3yTF-$\"#IF07$$\":ommmmm;z%4\\Y_F-$ \"#QF07$$\":NLLLLL$eR-/PiF-$\"#YF07$$\":-+++++]il'pisF-$\"#cF07$$\":OL LLLL$e*)>VB$)F-$\"#mF07$$\":.+++++]7`l2Q*F-$\"#!)F07$$\":nmmmmmmTIOo/ \"F0$\"#$*F07$$\":MLLLLL$3_>jU6F0$\"$2\"F07$$\":+++++++D;v/D\"F0$\"$B \"F07$$\":+++++++v=h(e8F0$\"$T\"F07$$\":+++++++v$[6j9F0$\"$h\"F07$$\": MLLLLLe*[z(yb\"F0$\"$!=F07$$\":nmmmmmmTXg0n\"F0$\"$.#F07$$\":nmmmmmmmJ F0$\"$! GF07$$\":,+++++D1f#=$3#F0$\"$;$F07$$\":,+++++v=xpe=#F0$\"$[$F07$$\":nm mmmm;H28IH#F0$\"$(QF07$$\":nmmmmm\"zpSS\"R#F0$\"$C%F07$$\":MLLLLL$3_?` (\\#F0$\"$n%F07$$\":MLLLLLe*)>pxg#F0$\"$9&F07$$\":,+++++v$f4t.FF0$\"$c &F07$$\":MLLLLL$e*Gst!GF0$\"$6'F07$$\":,++++++]#RW9HF0$\"$j'F07$$\":,+ ++++]7j#>>IF0$\"$>(F07$$\":,+++++D1RU07$F0$\"$q(F07$$\":,+++++](=S2LKF 0$\"$F)F07$$\":ommmmmm;p)=MLF0$\"$#))F07$$\":,++++++v=]@W$F0$\"$A*F07$ $\":MLLLLLe*[$z*RNF0$\"$b*F07$$\":,++++++DYKpk$F0$\"$t*F07$$\":ommmmm \"H2qcZPF0$\"$p*F07$$\":,+++++DJ5fF&QF0$\"$K*F07$$\":ommmmmmTg.c&RF0$ \"$`)F07$$\":,+++++DcEsK1%F0$\"$/(F07$$\":NLLLLLLL)*pp;%F0$\"$[%F07$$ \":NLLLLL$3xe,tUF0$\"$E\"F07$$\":ommmmm\"HdO=yVF0$!$H%F07$$\":,++++++] #>#[Z%F0$!%$3\"F07$$\":MLLLLL3_5,-`%F0$!%%e\"F07$$\":ommmmm;aG!e&e%F0$ !%y@F07$$\":-+++++vV37^j%F0$!%aEF07$$\":NLLLLLLL)Qk%o%F0$!%EMF07$$\":o mmmm;z>6but%F0$!%iVF07$$\":-+++++D1Mm-z%F0$!%(H&F07$$\":-++++]7G))Rb\" [F0$!%xeF07$$\":-++++++]U83%[F0$!%8jF07$$\":-++++](=np3m[F0$!%\\qF07$$ \":-+++++v$40O\"*[F0$!%fxF07$$\":-++++DJ?Q?&=\\F0$!%C$)F07$$\":,++++]( oa-oX\\F0$!%l%*F07$$\":++++]i:5>g#f\\F0$!%N'*F07$$\":+++++vVt7SG(\\F0$ !&%45F07$$\":++++](=nj+U')\\F0$!&d2\"F07$$\"\"&!\"\"$!&l9\"F0-%&COLORG 6&%$RGBGF($\"\"#F[^l$\"\"\"F)-%'LEGENDG6#%ESharp-Verner~scheme~with~c[ 10]=11/12G-F$6%7hnF'7$F+$\"\")F07$F2$\"#:F07$F7$\"#EF07$F<$\"#TF07$FA$ \"#_F07$FF$\"#kF07$FK$\"#yF07$FPFfn7$FU$\"$6\"F07$FZ$\"$I\"F07$Fin$\"$ \\\"F07$F^o$\"$s\"F07$Fco$\"$(>F07$Fho$\"$D#F07$F]p$\"$^#F07$Fbp$\"$$G F07$Fgp$\"$:$F07$F\\q$\"$e$F07$Faq$\"$#RF07$Ffq$\"$T%F07$F[r$\"$&[F07$ F`r$\"$S&F07$Fer$\"$#fF07$Fjr$\"$]'F07$F_s$\"$:(F07$Fds$\"$u(F07$Fis$ \"$])F07$F^t$\"$@*F07$Fct$\"$)**F07$Fht$\"%m5F07$F]u$\"%W6F07$Fbu$\"%; 7F07$Fgu$\"%m7F07$F\\v$\"%18F07$Fav$\"%A8F07$Ffv$\"%08F07$F[w$\"%P7F07 $F`w$\"%06F07$Few$\"$o)F07$Fjw$\"$r%F07$F_x$!#CF07$Fdx$!$o)F07$Fix$!%e =F07$F^y$!%6EF07$Fcy$!%0NF07$Fhy$!%?UF07$F]z$!%x`F07$Fbz$!%!y'F07$Fgz$ !%z\")F07$F\\[l$!%X!*F07$Fa[l$!%'p*F07$Ff[l$!&&z5F07$F[\\l$!&c=\"F07$F `\\l$!&*p7F07$Fe\\l$!&,W\"F07$Fj\\l$!&bY\"F07$F_]l$!&R`\"F07$Fd]l$!&Fj \"F07$Fi]l$!&#Q\"F07$Fbu$\"%l7F07$ Fgu$\"%;8F07$F\\v$\"%a8F07$Fav$\"%m8F07$Ffv$\"%V8F07$F[w$\"%k7F07$F`w$ \"%=6F07$Few$\"$c)F07$Fjw$\"$A%F07$F_x$!$;\"F07$Fdx$!%J5F07$Fix$!%-@F0 7$F^y$!%;HF07$Fcy$!%\")QF07$Fhy$!%`YF07$F]z$!%+fF07$Fbz$!%7uF07$Fgz$!% =*)F07$F\\[l$!%^)*F07$Fa[l$!&`0\"F07$Ff[l$!&O<\"F07$F[\\l$!&yG\"F07$F` \\l$!&&y8F07$Fe\\l$!&;c\"F07$Fj\\l$!&*)e\"F07$F_]l$!&Dm\"F07$Fd]l$!&)o F07$Fbp$\"$B#F07$Fgp$\"$[#F07$F\\q$\"$#GF07 $Faq$\"$4$F07$FfqF]r7$F[r$\"$#QF07$F`r$\"$E%F07$FerF\\s7$Fjr$\"$8&F07$ F_sFecn7$Fds$\"$5'F07$Fis$\"$q'F07$F^t$\"$E(F07$Fct$\"$'yF07$Fht$\"$R) F07$F]u$\"$**)F07$Fbu$\"$a*F07$Fgu$\"$#**F07$F\\v$\"%@5F07$Fav$\"%H5F0 7$Ffv$\"%65F07$F[w$\"$]*F07$F`w$\"$Q)F07$Few$\"$R'F07$Fjw$\"$3$F07$F_x $!$,\"F07$Fdx$!$(zF07$Fix$!%6;F07$F^y$!%IAF07$Fcy$!%jHF07$Fhy$!%\\NF07 $F]z$!%(\\%F07$Fbz$!%XcF07$Fgz$!%*y'F07$F\\[l$!%)\\(F07$Fa[l$!%I!)F07$ Ff[l$!%H*)F07$F[\\l$!%'z*F07$F`\\l$!&&[5F07$Fe\\l$!&v=\"F07$Fj\\l$! 7F07$F_]l$!&TE\"F07$Fd]l$!&[M\"F07$Fi]l$!&5V\"F0-Ff`n6&Fa^lFh`nF(F(-Fg ^l6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGFaem-%&TITLEG6#% Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6 $Q\"x6\"Q!F`fo-%%VIEWG6$;F(Fi]l;$F0!#B$Fb_lFhfo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/ 12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "scheme with c [10]=32/33" "scheme with c[10]=81/82" "scheme with c[10]=169/170" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "evalf[25](plot(['qn_RK7_1'(x)-q(x),'qn_RK7_2'(x)-q(x),'qn_RK7_3'( x)-q(x),\n'qn_RK7_4'(x)-q(x),'qn_RK7_5'(x)-q(x),'qn_RK7_6'(x)-q(x)],x= 0.5..0.76,-2.3e-19..4.3e-19,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2 ,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red ],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]= 15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme wit h c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error curves for \+ 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 854 449 449 {PLOTDATA 2 "6,-%'CURVESG6%7in7$$\"\"&!\"\"$!&l9 \"!#D7$$\":LLLLLLL3gsm0&F-$!&'z8F-7$$\":nmmmmm\"zFJ)f5&F-$!&%e:F-7$$\" :LLLLLL$eszVh^F-$!&:'=F-7$$\":LLLLLL$33/E<_F-$!&k@#F-7$$\":nmmmmm\"HHv \"GF&F-$!&6j#F-7$$\":LLLLLLeC4EVK&F-$!&j6$F-7$$\":++++++DhAgwP&F-$!&*3 OF-7$$\":LLLLLLeMY=GV&F-$!&B<%F-7$$\":++++++Dw!)*z([&F-$!&0'[F-7$$\":n mmmmmm\"y[NWbF-$!&kq&F-7$$\":LLLLLL$3:'oTf&F-$!&Gm'F-7$$\":+++++++X3Z- l&F-$!&k!yF-7$$\":+++++++v\"eb1dF-$!&W8*F-7$$\":+++++++br>3w&F-$!'sn5F -7$$\":LLLLLLeM`'45eF-$!'J\">\"F-7$$\":nmmmmmmhV\"poeF-$!'G/9F-7$$\":n mmmmmmY+H$=fF-$!'u2;F-7$$\":++++++DrHpg(fF-$!'&3(=F-7$$\":nmmmmmm'R#zr -'F-$!'Lu@F-7$$\":++++++Dr%\\D$3'F-$!'PBDF-7$$\":++++++v8G_m8'F-$!'%[# HF-7$$\":nmmmmm;zzmB>'F-$!'&=M$F-7$$\":nmmmmm\"H;,`ViF-$!'8FPF-7$$\":L LLLLL$3nmr)H'F-$!'9#H%F-7$$\":LLLLLLeM)*RgN'F-$!'[P\\F-7$$\":++++++v)) 4SfS'F-$!'GhcF-7$$\":LLLLLL$e!fL)fkF-$!':skF-7$$\":+++++++5%3^:lF-$!'c ntF-7$$\":++++++]#o,)*plF-$!'Z]\")F-7$$\":++++++DJ/#oAmF-$!(JA/*!#E7$$ \":++++++](*[)>\"o'F-$!)je:5F^u7$$\":nmmmmmm'>\"yPt'F-$!)@RA6F^u7$$\": +++++++v4=**y'F-$!)ICJ7F^u7$$\":LLLLLLe9E*ySoF-$!)sxI8F^u7$$\":+++++++ 0)[S'*oF-$!)_=39F^u7$$\":nmmmmm\"zV[t[pF-$!)wlT9F^u7$$\":++++++DOtMM+( F-$!)^LM9F^u7$$\":nmmmmmmT(Q\"p0(F-$!(j%[8F-7$$\":++++++D\"y:!H6(F-$!( v(H6F-7$$\":LLLLLLL8RCo;(F-$!'!R=(F-7$$\":LLLLLL$3c#o>A(F-$!&NY#F-7$$ \":nmmmmm\"z,blwsF-$\"(#oq5F-7$$\":+++++++5S2pK(F-$\"(BZI#F-7$$\":nmmm mm;%[<]%Q(F-$\"(6wo%F-7$$\":++++++vQGe-T(F-$\"(&39iF-7$$\":LLLLLLL$>[, OuF-$\"(`@T(F-7$$\":lmmmm;H#emZjuF-$\")t]=5F-7$$\":++++++Dr\\Q4\\(F-$ \")6h%=\"F-7$$\":+++++]i!R23/vF-$\")5()p8F-7$$\":+++++++5)HA6#F-7$$\" :++++D\"G)e^%GvvF-$\")Xf>BF-7$$\":++++]7G$*H:)yvF-$\")x5aBF-7$$\":++++ vVtF3YBe(F-$\")!fLN#F-7$$\":+++++v=i'o(ee(F-$\")\"[[N#F-7$$\":++++]P4J VQHf(F-$\")Hf5DF-7$$\"#w!\"#$\")*Q@n#F--%&COLORG6&%$RGBG$\"\"!F[_l$\" \"#F*$\"\"\"F[_l-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$ 6%7in7$F($!&#Q$!&0L$F-7$FC$ !&p%RF-7$FH$!&zm%F-7$FM$!&**R&F-7$FR$!&jB'F-7$FW$!&\"esF-7$Ffn$!&P^)F- 7$F[o$!&H$**F-7$F`o$!''H;\"F-7$Feo$!'%*f8F-7$Fjo$!'t)e\"F-7$F_p$!'(>x \"F-7$Fdp$!'p(3#F-7$Fip$!'D*Q#F-7$F^q$!':zFF-7$Fcq$!'#)GKF-7$Fhq$!'(eu $F-7$F]r$!'XSVF-7$Fbr$!')y&\\F-7$Fgr$!'BGbF-7$F\\s$!'RkjF-7$Fas$!'5>tF -7$Ffs$!'U*Q)F-7$F[t$!'(ye*F-7$F`t$!(I54\"F-7$Fet$!(kl?\"F-7$Fjt$!)J/Q 8F^u7$F`u$!):*>]\"F^u7$Feu$!)N*)e;F^u7$Fju$!)WD==F^u7$F_v$!)w4j>F^u7$F dv$!)c=u?F^u7$Fiv$!)/!*>@F^u7$F^w$!)DU.@F^u7$Fcw$!(N#o>F-7$Fhw$!(+Ij\" F-7$F]x$!(I#35F-7$Fbx$\"&+/%F-7$Fgx$\"(a8p\"F-7$F\\y$\"(-\"[NF-7$Fay$ \"(*=HrF-7$Ffy$\"(-;U*F-7$F[z$\")z)>7\"F-7$F`z$\")l1Q:F-7$Fez$\")1?(y \"F-7$Fjz$\")]-l?F-7$F_[l$\")+2@AF-7$Fd[l$\")?F\"Q#F-7$Fi[l$\")*[Hs#F- 7$F^\\l$\")%z46$F-7$Fc\\l$\")QHxJF-7$Fh\\l$\"),_)[$F-7$F]]l$\")rCSNF-7 $Fb]l$\")07RNF-7$Fg]l$\")UMTNF-7$F\\^l$\")lluPF-7$Fa^l$\")&Hn,%F--Fg^l 6&Fi^lFj^l$\"\"(F*Fj^l-Fa_l6#%8scheme~with~c[10]=15/16G-F$6%7in7$F($!& #)Q\"F-7$F/$!&mm\"F-7$F4$!&*z=F-7$F9$!&:C#F-7$F>$!&Zm#F-7$FC$!&!fJF-7$ FH$!&st$F-7$FM$!&TK%F-7$FR$!&]*\\F-7$FW$!&Y\"eF-7$Ffn$!&;#oF-7$F[o$!&+ 'zF-7$F`o$!&5K*F-7$Feo$!'7!4\"F-7$Fjo$!'mt7F-7$F_p$!'n?9F-7$Fdp$!'(Rn \"F-7$Fip$!'#f\">F-7$F^q$!'wGAF-7$Fcq$!'c*e#F-7$Fhq$!'W/IF-7$F]r$!'c\" [$F-7$Fbr$!'/xRF-7$Fgr$!'wMWF-7$F\\s$!'$e5&F-7$Fas$!'5seF-7$Ffs$!'CJnF -7$F[t$!'J$p(F-7$F`t$!'3b()F-7$Fet$!'%Go*F-7$Fjt$!)B)Q2\"F^u7$F`u$!)$) f07F^u7$Feu$!)qqJ8F^u7$Fju$!))z)f9F^u7$F_v$!)u^w:F^u7$Fdv$!)fAm;F^u7$F iv$!)#GNq\"F^u7$F^w$!)`?\"p\"F^u7$Fcw$!(.Se\"F-7$Fhw$!(6oJ\"F-7$F]x$!' ^z\")F-7$Fbx$\"&=,#F-7$Fgx$\"()GS8F-7$F\\y$\"(*[DGF-7$Fay$\"(%o!p&F-7$ Ffy$\"(K^_(F-7$F[z$\"(VU'*)F-7$F`z$\")XUH7F-7$Fez$\")w#)G9F-7$Fjz$\")7 ?^;F-7$F_[l$\")$*4wF-7$Fi[l$\")!Ry<#F-7$F^\\l$\")DX)[ #F-7$Fc\\l$\")E`TDF-7$Fh\\l$\")_n!z#F-7$F]]l$\")n3KGF-7$Fb]l$\")a=JGF- 7$Fg]l$\")b'H$GF-7$F\\^l$\")&H(>IF-7$Fa^l$\")k_8KF--Fg^l6&Fi^l$\"\"*F* $\"\"%F*Fj^l-Fa_l6#%8scheme~with~c[10]=19/20G-F$6%7in7$F($!&P&=F-7$F/$ !&HA#F-7$F4$!&g]#F-7$F9$!&b)HF-7$F>$!&na$F-7$FC$!&>?%F-7$FH$!&%o\\F-7$ FM$!&lu&F-7$FR$!&aj'F-7$FW$!&:s(F-7$Ffn$!&e0*F-7$F[o$!'Tc5F-7$F`o$!'sO 7F-7$Feo$!'/Y9F-7$Fjo$!';*o\"F-7$F_p$!'&Q)=F-7$Fdp$!'K>AF-7$Fip$!'vRDF -7$F^q$!',aHF-7$Fcq$!'wJMF-7$Fhq$!'2\")RF-7$F]r$!'u7YF-7$Fbr$!'ko_F-7$ Fgr$!'^ueF-7$F\\s$!'qinF-7$Fas$!'uwxF-7$Ffs$!'\\8*)F-7$F[t$!(>'=5F-7$F `t$!(P!f6F-7$Fet$!($p\"G\"F-7$Fjt$!)^D@9F^u7$F`u$!)PB&f\"F^u7$Feu$!)zm hF^u7$F_v$!)*oR3#F^u7$Fdv$!)\"F-7$F`z$\")TsR;F-7$Fez$\")e)\\!>F-7$Fjz$\")$)y+AF-7$F_[l$\" )/*oO#F-7$Fd[l$\")FZPDF-7$Fi[l$\")-B,HF-7$F^\\l$\")>J9LF-7$Fc\\l$\")#p [Q$F-7$Fh\\l$\")T>;PF-7$F]]l$\")9FrPF-7$Fb]l$\")52qPF-7$Fg]l$\")PVsPF- 7$F\\^l$\")lu?SF-7$Fa^l$\")+ZyUF--%'COLOURG6&Fi^l$\"*++++\"!\")Fj^lFec n-Fa_l6#%8scheme~with~c[10]=32/33G-F$6%7in7$F($!&A)=F-7$F/$!&cD#F-7$F4 $!&>a#F-7$F9$!&n-$F-7$F>$!&Sf$F-7$FC$!&kD%F-7$FH$!&6.&F-7$FM$!&v\"eF-7 $FR$!&fr'F-7$FW$!&L\"yF-7$Ffn$!&;;*F-7$F[o$!'do5F-7$F`o$!'u]7F-7$Feo$! 'Bi9F-7$Fjo$!'%yq\"F-7$F_p$!'`/>F-7$Fdp$!'VVAF-7$Fip$!'6nDF-7$F^q$!'c& )HF-7$Fcq$!'8oMF-7$Fhq$!'(H-%F-7$F]r$!''4m%F-7$Fbr$!'TB`F-7$Fgr$!'HNfF -7$F\\s$!'FKoF-7$Fas$!'GcyF-7$Ffs$!'8/!*F-7$F[t$!(:*G5F-7$F`t$!(x1<\"F -7$Fet$!(#\\%H\"F-7$Fjt$!)GNN9F^u7$F`u$!)?\"4h\"F^u7$Feu$!)S!)yF^u7$F_v$!)(>O5#F^u7$Fdv$!)(Q:A#F^u7$Fiv$!)4@pAF^u7$F^w$!) N^\\AF^u7$Fcw$!($f,@F-7$Fhw$!(Nyt\"F-7$F]x$!(T>1\"F-7$Fbx$\"&`2(F-7$Fg x$\"(9C&=F-7$F\\y$\"(E\\&QF-7$Fay$\"(Ddr(F-7$Ffy$\")/n=5F-7$F[z$\")?Y7 7F-7$F`z$\")$43m\"F-7$Fez$\")GAH>F-7$Fjz$\")L`GAF-7$F_[l$\")^g'R#F-7$F d[l$\")1?pDF-7$Fi[l$\")CDPHF-7$F^\\l$\")B?bLF-7$Fc\\l$\")IfEMF-7$Fh\\l $\")P\"=w$F-7$F]]l$\")i`F-7$F9$!&)*H#F-7$F >$!&1t#F-7$FC$!&MB$F-7$FH$!&:#QF-7$FM$!&%=WF-7$FR$!&/5&F-7$FW$!&M$fF-7 $Ffn$!&p&pF-7$F[o$!&O6)F-7$F`o$!&j\\*F-7$Feo$!':56F-7$Fjo$!'d'H\"F-7$F _p$!''eW\"F-7$Fdp$!'2.F-7$F^q$!'LmAF-7$Fcq$!'dKEF-7$Fhq$ !'n`IF-7$F]r$!''y`$F-7$Fbr$!'iSSF-7$Fgr$!'*\\]%F-7$F\\s$!'s&=&F-7$Fas$ !'&G'fF-7$Ffs$!''R$oF-7$F[t$!'84yF-7$F`t$!'$\\)))F-7$Fet$!']C)*F-7$Fjt $!)*Q$*3\"F^u7$F`u$!)NbA7F^u7$Feu$!)=%*\\8F^u7$Fju$!)Y;z9F^u7$F_v$!)KN 'f\"F^u7$Fdv$!)Zw&o\"F^u7$Fiv$!)r&=s\"F^u7$F^w$!)>y1=F-7$Fd[l$\")as]>F-7$Fi[l$ \")A9IAF-7$F^\\l$\")%Rua#F-7$Fc\\l$\")&Q;g#F-7$Fh\\l$\")\"Gh&GF-7$F]]l $\")/V)*GF-7$Fb]l$\")y](*GF-7$Fg]l$\")EK**GF-7$F\\^l$\")a0!4$F-7$Fa^l$ \")8+)G$F--Fccn6&Fi^lFecnFj^lFj^l-Fa_l6#%:scheme~with~c[10]=169/170G-% %FONTG6$%*HELVETICAGFbgm-%&TITLEG6#%Verror~curves~for~11~stage~order~7 ~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fe\\p-%%VIEWG6$;F(Fa^l;$ !#B!#?$\"#VF^]p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" " scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme with c[10]= 81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[25](plot(['qn_RK7_1' (x)-q(x),'qn_RK7_2'(x)-q(x),'qn_RK7_3'(x)-q(x),\n'qn_RK7_4'(x)-q(x),'q n_RK7_5'(x)-q(x),'qn_RK7_6'(x)-q(x)],x=0.76..0.86,-2.3e-15..1.3e-16,fo nt=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RG B,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner sche me with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/2 0`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[ 10]=169/170`],\ntitle=`error curves for 11 stage order 7 Runge-Kutta m ethods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 981 600 600 {PLOTDATA 2 "6,-% 'CURVESG6%7co7$$\"#w!\"#$\")*Q@n#!#D7$$\":nmmmmmmT:(z@wF-$\")0%\\-$F-7 $$\":LLLLLLe9ui2k(F-$\")&QdT$F-7$$\":nmmmmm;z_\"4iwF-$\")\"G5&QF-7$$\" :nmmmmm;aphNo(F-$\")lf^VF-7$$\":LLLLLLe*=)H\\q(F-$\")+'Q(\\F-7$$\":nmm mmm\"z/3uCxF-$\")H;RcF-7$$\":++++++DJ$RDXxF-$\")D*)[lF-7$$\":nmmmmm\"z R'okw(F-$\")J2/vF-7$$\":++++++D1J:wy(F-$\")*\\JK)F-7$$\":LLLLLLL3En$4y F-$\")OR6#*F-7$$\":nmmmmm;/RE&GyF-$\"*3Qu,\"F-7$$\":+++++++D.&4]yF-$\" *m>87\"F-7$$\":+++++++vB_<(yF-$\"*Px_B\"F-7$$\":+++++++v'Hi#*yF-$\"*rO AO\"F-7$$\":nmmmmm\"z*ev:\"zF-$\"*aT%*[\"F-7$$\":LLLLLLL347T$zF-$\"*#z h!p\"F-7$$\":LLLLLLLLY.K&zF-$\"*-#*)R=F-7$$\":++++++D\"o7TvzF-$\"*%=n[ ?F-7$$\":LLLLLLL$Q*o]*zF-$\"*'o$3?#F-7$$\":++++++D\"=lj;!)F-$\"*q`BN#F -7$$\":++++++vV&RY29)F-$\"*ITb*HF-7$$\":nm mmmm;zXu9;)F-$\"*trU%HF-7$$\":+++++++]y))G=)F-$\"*/+*\\FF-7$$\":++++++ ]i_QQ?)F-$\"*jc#yCF-7$$\":++++++D\"y%3TA)F-$\"*j,:2#F-7$$\":++++++]P![ hY#)F-$\"*YFh\\\"F-7$$\":LLLLLLL$Qx$oE)F-$\")[8QrF-7$$\":+++++++v.I%)G )F-$!)!o`Z$F-7$$\":nmmmmm\"zpe*zI)F-$!*>s>r\"F-7$$\":+++++++D\\'QH$)F- $!*=pUp$F-7$$\":LLLLLLe9S8&\\$)F-$!+a$R*yi!#E7$$\":++++++D1#=bq$)F-$!, NI`f2\"F\\w7$$\":LLLLLLL3s?6R)F-$!,hnK1`\"F\\w7$$\":++++++DJXaET)F-$!- )>p.f+#!#F7$$\":mmmmmmmm*RRL%)F-$!,cobRd#F\\w7$$\":mmmmmm;a<.YX)F-$!,n '\\DZKF\\w7$$\":LLLLLLe9tOcZ)F-$!+NmXTSF-7$$\":+++++++]Qk\\\\)F-$!+JOj j\\F-7$$\":LLLLLL$3dg6<&)F-$!+ob)H4'F-7$$\":lmmmm;zpBp?_)F-$!+?!G^+(F- 7$$\":++++++voTAq_)F-$!+7))*GA(F-7$$\":qmmmm\"H#o+*\\H&)F-$!+?*R:A(F-7 $$\":NLLLL$3x'fv>`)F-$!+hHRCsF-7$$\":qmmm;zW<*Q@L&)F-$!+-y3KsF-7$$\":+ ++++v=n=_W`)F-$!+Xw^]sF-7$$\":NLLL$3Fp\"[!pN&)F-$!+'yK'*G(F-7$$\":mmmm mmmmxGp`)F-$!+!3dhO(F-7$$\":NLLLL$eRA5\\Z&)F-$!+]])\\d)F-7$$\":++++++D \"oK0e&)F-$!+7Mh%3*F-7$$\":+++++++]oi\"o&)F-$!,a3Qp+\"F-7$$\":++++++v= 5s#y&)F-$!,+g6\"36F-7$$\":+++]iS\"p)4^*y&)F-$!,&yoyV6F-7$$\":++++D\"y] &4I'z&)F-$!,!y$)fp6F-7$$\":+++](=UK#44.e)F-$!,`'*=&p6F-7$$\":++++]iS\" *3))4e)F-$!,(e&R%p6F-7$$\":++++vVtF3YBe)F-$!,g#3Gp6F-7$$\":+++++D1k2/P e)F-$!,*4L7p6F-7$$\":++++](=nj+U'e)F-$!,1#*o)o6F-7$$\":+++++]P40O\"*e) F-$!,.Je$p6F-7$$\":++++DJqX/%\\!f)F-$!,lK'zq6F-7$$\":++++]7.#Q?&=f)F-$ !,a_rV<\"F-7$$\":++++v$f$=.5Kf)F-$!,LA$=#=\"F-7$$\":+++++voa-oXf)F-$!, y7Jy>\"F-7$$\":++++]PMF,%G(f)F-$!,9Gh$z7F-7$$\"#')F*$!,c0w/M\"F--%&COL ORG6&%$RGBG$\"\"!F]bl$\"\"#!\"\"$\"\"\"F]bl-%'LEGENDG6#%ESharp-Verner~ scheme~with~c[10]=11/12G-F$6%7co7$F($\")&Hn,%F-7$F/$\")1AXXF-7$F4$\")Y aI^F-7$F9$\")**R#y&F-7$F>$\")_)=`'F-7$FC$\")'>MY(F-7$FH$\")VCf%)F-7$FM $\")'>2#)*F-7$FR$\"*t'*\\7\"F-7$FW$\"*DFvC\"F-7$Ffn$\"*<#Q!Q\"F-7$F[o$ \"*D\"RC:F-7$F`o$\"*k3(z;F-7$Feo$\"*\"H0]=F-7$Fjo$\"*9%yR?F-7$F_p$\"*, M)HAF-7$Fdp$\"*.y-`#F-7$Fip$\"*Y3Jv#F-7$F^q$\"*m/Y1$F-7$Fcq$\"*+`9H$F- 7$Fhq$\"*\\^r^$F-7$F]r$\"*)[xPPF-7$Fbr$\"*_+p%RF-7$Fgr$\"*OCl8%F-7$F\\ s$\"*5g')H%F-7$Fas$\"*tL]U%F-7$Ffs$\"*8blY%F-7$F[t$\"*ztYQ%F-7$F`t$\"* qxp3%F-7$Fet$\"*EFYn$F-7$Fjt$\"*#33fIF-7$F_u$\"*)*\\+>#F-7$Fdu$\"*$R** 45F-7$Fiu$!)/U$*eF-7$F^v$!*0ERk#F-7$Fcv$!*D!zEcF-7$Fhv$!+&*z>9&*F\\w7$ F^w$!,c&R/D;F\\w7$Fcw$!,?KN$3BF\\w7$Fhw$!-b$*GHAIF\\x7$F^x$!,v'=RvQF\\ w7$Fcx$!,;oxi)[F\\w7$Fhx$!+N(e$ygF-7$F]y$!+zO:iuF-7$Fby$!+nBCc\"*F-7$F gy$!,(oiT_5F-7$F\\z$!,t#=3&3\"F-7$Faz$!,NZx[3\"F-7$Ffz$!,z*RI&3\"F-7$F [[l$!,&\\jX'3\"F-7$F`[l$!,1E<#*3\"F-7$Fe[l$!,L=y]4\"F-7$Fj[l$!,lHYl5\" F-7$F_\\l$!,z'[x(G\"F-7$Fd\\l$!,)oe7k8F-7$Fi\\l$!,Y'\\q6:F-7$F^]l$!,vw @Km\"F-7$Fc]l$!,2Tdmr\"F-7$Fh]l$!,%o)>`v\"F-7$F]^l$!,oo+_v\"F-7$Fb^l$! ,T^\"3b>F-7$Fdal$!,tTh5,#F--Fial6&F[blF\\bl$\" \"(F`blF\\bl-Fdbl6#%8scheme~with~c[10]=15/16G-F$6%7co7$F($\")k_8KF-7$F /$\")1hOOF-7$F4$\")A?0TF-7$F9$\")k1FYF-7$F>$\")&*4F_F-7$FC$\")9*G(fF-7 $FH$\")V:,8F\\w7$Fcw$!,B4T#[=F\\w7$Fhw$!-&zkh)>CF\\ x7$F^x$!,%\\#pG5$F\\w7$Fcx$!,%=K=7RF\\w7$Fhx$!+>:`m[F-7$F]y$!+[S\"F-7$F[`l$!,$30a19F-7$F``l$!,!*y?3T\"F-7$Fe`l$!,'GV $\")(Ha&pF-7 $FC$\")2vYzF-7$FH$\")pX1!*F-7$FM$\"**=_X5F-7$FR$\"*^-w>\"F-7$FW$\"*Knz K\"F-7$Ffn$\"*\"oIp9F-7$F[o$\"*R,Di\"F-7$F`o$\"*$4r(y\"F-7$Feo$\"*A&)) o>F-7$Fjo$\"*,b1<#F-7$F_p$\"*`TFP#F-7$Fdp$\"*?\\@p#F-7$Fip$\"*:%**GHF- 7$F^q$\"*H(**fKF-7$Fcq$\"*!Q&4]$F-7$Fhq$\"*=!fSPF-7$F]r$\"*O5Z(RF-7$Fb r$\"*Eck>%F-7$Fgr$\"*>'G(R%F-7$F\\s$\"*8X'oXF-7$Fas$\"*\"p[,ZF-7$Ffs$ \"*#o#Ru%F-7$F[t$\"*H&)Rl%F-7$F`t$\"*;*QLVF-7$Fet$\"*Pe8*QF-7$Fjt$\"*) )HDB$F-7$F_u$\"*(>J.BF-7$Fdu$\"*KoB/\"F-7$Fiu$!)))ecmF-7$F^v$!*Iw!fGF- 7$Fcv$!*:s@/'F-7$Fhv$!,6Q`*=5F\\w7$F^w$!,Ko)\\PNSOFKF\\x7$F^x$!,U;Hn8%F\\w7$Fcx$!,2_!49_F\\w7$Fhx$!+)y8V[ 'F-7$F]y$!+$[D&ezF-7$Fby$!+@L!Gw*F-7$Fgy$!,s&>%>7\"F-7$F\\z$!,Q?Mn:\"F -7$Faz$!,!ph^c6F-7$Ffz$!,g1pp:\"F-7$F[[l$!,=Y%>e6F-7$F`[l$!,rHJ6;\"F-7 $Fe[l$!,!zpOn6F-7$Fj[l$!,*30dz6F-7$F_\\l$!,im.DP\"F-7$Fd\\l$!,/YLPX\"F -7$Fi\\l$!,FF$z5;F-7$F^]l$!,!)3V>x\"F-7$Fc]l$!,PP&yG=F-7$Fh]l$!,:\"R\" *p=F-7$F]^l$!,#\\pyp=F-7$Fb^l$!,m**f'p=F-7$Fg^l$!,(>iSp=F-7$F\\_l$!,kP a\"p=F-7$Fa_l$!,Z5Z(o=F-7$Ff_l$!,jhA&p=F-7$F[`l$!,rL3=(=F-7$F``l$!,Pl$ \\x=F-7$Fe`l$!,dd=**)=F-7$Fj`l$!,#R2\"[\">F-7$F_al$!,A5DX/#F-7$Fdal$!, \"e\\wT@F--%'COLOURG6&F[bl$\"*++++\"!\")F\\blFb\\o-Fdbl6#%8scheme~with ~c[10]=32/33G-F$6%7co7$F($\")+pIVF-7$F/$\"))>)**[F-7$F4$\")w6IbF-7$F9$ \")1,KiF-7$F>$\")e'*QqF-7$FC$\")@&=/)F-7$FH$\")=!R6*F-7$FM$\"*<^z0\"F- 7$FR$\"*`(z67F-7$FW$\"*)\\nV8F-7$Ffn$\"**Hl'[\"F-7$F[o$\"*5A;k\"F-7$F` o$\"*uV(3=F-7$Feo$\"*V8?*>F-7$Fjo$\"*/:h>#F-7$F_p$\"*`M0S#F-7$Fdp$\"*A EOs#F-7$Fip$\"*;*>jHF-7$F^q$\"*$*4!)H$F-7$Fcq$\"*eMTy$ F-7$F]r$\"*^?4-%F-7$Fbr$\"*e)>XUF-7$Fgr$\"*88$[WF-7$F\\s$\"*u1;i%F-7$F as$\"*J;fv%F-7$Ffs$\"*(3y)z%F-7$F[t$\"*[*p2ZF-7$F`t$\"*L)G$Q%F-7$Fet$ \"*:Tg$RF-7$Fjt$\"*:s%pKF-7$F_u$\"*el$HBF-7$Fdu$\"*UqO0\"F-7$Fiu$!)IFV nF-7$F^v$!*ZjL*GF-7$Fcv$!*YfO6'F-7$Fhv$!,WvW4.\"F\\w7$F^w$!,T,oyv\"F\\ w7$Fcw$!,7*y/&\\#F\\w7$Fhw$!-kV$=^E$F\\x7$F^x$!,1[3^=%F\\w7$Fcx$!,odo] F&F\\w7$Fhx$!+:J;glF-7$F]y$!+gjl^!)F-7$Fby$!+IV8x)*F-7$Fgy$!,!4*)3N6F- 7$F\\z$!,W)*)Gq6F-7$Faz$!,`To+<\"F-7$Ffz$!,Z!o_q6F-7$F[[l$!,q&owr6F-7$ F`[l$!,WpQZ<\"F-7$Fe[l$!,%)\\[5=\"F-7$Fj[l$!,![nR$>\"F-7$F_\\l$!,W%Qg) Q\"F-7$Fd\\l$!,)G!)zq9F-7$Fi\\l$!,d+9(H;F-7$F^]l$!,&*R\"y#z\"F-7$Fc]l$ !,))[)H]=F-7$Fh]l$!,O0:>*=F-7$F]^l$!,vf'y\"*=F-7$Fb^l$!,6:e;*=F-7$Fg^l $!,%)Q,9*=F-7$F\\_l$!,EeY6*=F-7$Fa_l$!,baM2*=F-7$Ff_l$!,B\\>:*=F-7$F[` l$!,!)oKQ*=F-7$F``l$!,$*>'e**=F-7$Fe`l$!,&y*f@\">F-7$Fj`l$!,p5]t$>F-7$ F_al$!,ra:'o?F-7$Fdal$!,4]%F-7$F9$\")'=9t%F-7$F>$\")$HSM&F-7$FC$\")EQ0hF-7$FH$\")$R#>pF-7$FM $\")t'=.)F-7$FR$\")!*z*>*F-7$FW$\"*P&4?5F-7$Ffn$\"*$yjG6F-7$F[o$\"*L$G Y7F-7$F`o$\"*ZaJP\"F-7$Feo$\"*c%G7:F-7$Fjo$\"*VHsm\"F-7$F_p$\"*a:C#=F- 7$Fdp$\"*'Gpn?F-7$Fip$\"*]m&\\AF-7$F^q$\"*&4u.DF-7$Fcq$\"*\"yw)o#F-7$F hq$\"*yyF(GF-7$F]r$\"*_^D0$F-7$Fbr$\"*C>GA$F-7$Fgr$\"*fAqP$F-7$F\\s$\" *$**e3NF-7$Fas$\"*kn0h$F-7$Ffs$\"*/FJk$F-7$F[t$\"*![,uNF-7$F`t$\"*i&yF LF-7$Fet$\"*$))H))HF-7$Fjt$\"*cAB[#F-7$F_u$\"*<+(o#F-7$Fcv$!*&eWSYF-7$Fhv$!+,u_DyF\\w7$F^w$!, =DqVL\"F\\w7$Fcw$!,B,wR*=F\\w7$Fhw$!-]=PbyCF\\x7$F^x$!,(Q:%p<$F\\w7$Fc x$!,$G9O/SF\\w7$Fhx$!+s$=*z\\F-7$F]y$!+T\\;7hF-7$Fby$!+FU'z\\(F-7$Fgy$ !+&[fnh)F-7$F\\z$!+zW(R)))F-7$Faz$!+`,I#)))F-7$Ffz$!+c0y&)))F-7$F[[l$! +k^>&*))F-7$F`[l$!+Lsv<*)F-7$Fe[l$!+A.ml*)F-7$Fj[l$!+y^Sf!*F-7$F_\\l$! ,'*yJT0\"F-7$Fd\\l$!,WoJl6\"F-7$Fi\\l$!,5FtrB\"F-7$F^]l$!,)>0(4O\"F-7$ Fc]l$!,&>`j/9F-7$Fh]l$!,[))GiV\"F-7$F]^l$!,'o88O9F-7$Fb^l$!,)fQ.O9F-7$ Fg^l$!,3%*QeV\"F-7$F\\_l$!,y]XcV\"F-7$Fa_l$!,UsK`V\"F-7$Ff_l$!,psGfV\" F-7$F[`l$!,'y\\oP9F-7$F``l$!,K:`?W\"F-7$Fe`l$!,]G*f^9F-7$Fj`l$!,-lB2Z \"F-7$F_al$!,!*fy.d\"F-7$Fdal$!,)eH3X;F--F`\\o6&F[blFb\\oF\\blF\\bl-Fd bl6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGFa^n-%&TITLEG6#% Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6 $Q\"x6\"Q!F`ip-%%VIEWG6$;F(Fdal;$!#B!#;$\"#8!#<" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/ 12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "scheme with c [10]=32/33" "scheme with c[10]=81/82" "scheme with c[10]=169/170" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 503 "evalf[20](plot(['qn_RK7_1'(x)-q(x),'qn_RK7_2'(x)-q(x),'qn_RK7_3'( x)-q(x),\n'qn_RK7_4'(x)-q(x),'qn_RK7_5'(x)-q(x),'qn_RK7_6'(x)-q(x)],x= 0.86..0.9,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7 ,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp -Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme wit h c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`sc heme with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 R unge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 630 630 {PLOTDATA 2 "6,-%'CURVESG6%7[u7$$\"#')!\"#$!'YS8!#?7$$\"5nmmmmh)=(3')F -$!'zS8F-7$$\"5LLLLe'40jh)F-$!'&oX\"F-7$$\"5nmmm;6m$[i)F-$!'59:F-7$$\" 5nmmm;yYUL')F-$!')>c\"F-7$$\"5LLLLeF>(>k)F-$!'5$o\"F-7$$\"5nmmm\">K'*) \\')F-$!'r$p\"F-7$$\"5++++Dt:5e')F-$!'/M=F-7$$\"5nmmm\"fX(em')F-$!'xM= F-7$$\"5++++DCh/v')F-$!'9[>F-7$$\"5LLLLL/pu$o)F-$!'yY>F-7$$\"5nmmm;c0T \"p)F-$!'1u>F-7$$\"5+++++8!Q+q)F-$!'b)*>F-7$$\"5+++++&*3q3()F-$!'q*)>F -7$$\"5+++++(=\\qr)F-$!'Oa>F-7$$\"5nmmm\"fBIYs)F-$!'WY>F-7$$\"5LLLLLO[ kL()F-$!'0pv!))F-$\"'W*y \"F-7$$\"5nmmm\"f`@'3))F-$\"'D*y\"F-7$$\"5NLLLL1+Y7))F-$\"'s)y\"F-7$$ \"5++++vw%)H;))F-$\"''pz\"F-7$$\"5qmmTN.8P=))F-$\"'3J=F-7$$\"5NLL$e*HT W?))F-$\"'c\\>F-7$$\"5qm;/EV0[@))F-$\"'$*y?F-7$$\"5+++Dccp^A))F-$\"'c \"H#F-7$$\"5NL$ek)pLbB))F-$\"',IEF-7$$\"5nmmm;$y*eC))F-$\"'%R:$F-7$$\" 5+++++9b:L))F-$\"'G#)QF-7$$\"5+++]7)[]_$))F-$\"'C#*QF-7$$\"5++++DiaMP) )F-$\"'tRRF-7$$\"5+++DJ\\HRQ))F-$\"'g**RF-7$$\"5+++]PO/WR))F-$\"'g1TF- 7$$\"5+++vVBz[S))F-$\"'k*G%F-7$$\"5++++]5a`T))F-$\"'<\"f%F-7$$\"5+++vo 0CcV))F-$\"':fdF-7$$\"5+++](3S*eX))F-$\"')or'F-7$$\"5+++D1'R;w%))F-$\" '_:nF-7$$\"5++++D\"RV'\\))F-$\"'E9nF-7$$\"5+++D\"Q-%*=&))F-$\"'^9nF-7$ $\"5+++]PcY9a))F-$\"'4FnF-7$$\"5++]ils*p_&))F-$\"'A\\nF-7$$\"5+++v$*)G &Rc))F-$\"'E'z'F-7$$\"5++](=_g?v&))F-$\"'m))oF-7$$\"5++++]@fke))F-$\"' afqF-7$$\"5lmmmT30pi))F-$\"'^*R*F-7$$\"5LLLLL&4Nn'))F-$\"(aM/\"F-7$$\" 5lmmm;bN0r))F-$\"(4L/\"F-7$$\"5+++++:?Pv))F-$\"(\\R0\"F-7$$\"5NL$e9mI] j())F-$\"(oR1\"F-7$$\"5qmm\"H#)fGt())F-$\"(_63\"F-7$$\"5++]P%)*)oIy))F -$\"(#\\46F-7$$\"5NLL$e9=&Gz))F-$\"(1Z:\"F-7$$\"5+++vok$)))F-$\"(,y^\"F-7$$\"5NLL$eCZwu)))F-$\"(mr^\"F-7$ $\"5+++++(fa<*))F-$\"(Q&=:F-7$$\"5lm;zWU4w#*))F-$\"(q4_\"F-7$$\"5ILLe* yGnP*))F-$\"(Ef_\"F-7$$\"5++]PMLOx%*))F-$\"(!HN:F-7$$\"5lmm;zy*zd*))F- $\"()*>b\"F-7$$\"5+++vopEz(*))F-$\"(@si\"F-7$$\"5LLLLeg`!)**))F-$\"(_g \"=F-7$$\"5lmm;a\"Gd3!*)F-$\"(h\")*>F-7$$\"5++++]-#4>!*)F-$\"(FN5#F-7$ $\"5ILL$eM7hH!*)F-$\"(-L5#F-7$$\"5lmmmTWI,/*)F-$\"(%3.@F-7$$\"5ILLLL') o61*)F-$\"(`E5#F-7$$\"5++++DG2A3*)F-$\"(4C5#F-7$$\"5ILL3F=wF5*)F-$\"(m M5#F-7$$\"5lmm;H3XL7*)F-$\")rG5@!#@7$$\"5IL$3-L&HO8*)F-$\"):Z>@Fdhl7$$ \"5+++DJ)R\"R9*)F-$\")9RO@Fdhl7$$\"5lm;HKV)>a\"*)F-$\")P(f;#Fdhl7$$\"5 LLLLL)G[k\"*)F-$\")Ic:AFdhl7$$\"5+++Dch;g=*)F-$\")AQOCFdhl7$$\"5lmm;zM ]v?*)F-$\")-)=!GFdhl7$$\"5ILL3-3%3H#*)F-$\")VF,GFdhl7$$\"5++++D\"yh]#* )F-$\")po+GFdhl7$$\"5qmmTgNd8F*)F-$\")$G.!GFdhl7$$\"5NLL$e**o4#H*)F-$ \")bY,GFdhl7$$\"5qm;a8nmCI*)F-$\")!GR!GFdhl7$$\"5+++DJWOGJ*)F-$\")#G$4 GFdhl7$$\"5NL$e*[@1KK*)F-$\")$!#A7$$\"5nmmm;q7%=%*)F-$\")$*z(f$Fdhl7$$ \"5+++]P\")z/Y*)F-$\")z`'f$Fdhl7$$\"5LLLLe#pa-&*)F-$\")VP1OFdhl7$$\"5l m;/Ev5A^*)F-$\")87V_)*)F-$\"(mkG&F-7$$\"5+++++u]E()*)F -$\"(0(*G&F-7$$\"5+++vo:gF))*)F-$\"(x\\H&F-7$$\"5+++]PdpG*)*)F-$\"(`_I &F-7$$\"5+++D1**yH!**)F-$\"(3QK&F-7$$\"5++++vS)38**)F-$\"(QbN&F-7$$\"5 +++DcI;[$**)F-$\"(%>0bF-7$$\"5+++]P?Wl&**)F-$\"(RW(eF-7$$\"5+]i:5>g#f* *)F-$\"(He%fF-7$$\"5++D\"Gyh(>'**)F-$\"(m!pfF-7$$\"5+](oal@pk**)F-$\"( ,*ofF-7$$\"5++]7G:3u'**)F-$\"(S(ofF-7$$\"5++vVt7SG(**)F-$\"(9%ofF-7$$ \"5+++v=5s#y**)F-$\"(!4ofF-7$$\"5++]P40O\"*)**)F-$\"(Wu'fF-7$$\"\"*!\" \"$\"('zmfF--%&COLORG6&%$RGBG$\"\"!Ff]n$\"\"#F^]n$\"\"\"Ff]n-%'LEGENDG 6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7[u7$F($!'/6?F-7$F/$!'a6 ?F-7$F4$!'1&=#F-7$F9$!'lqAF-7$F>$!'HF-7$F_p$!'X2HF-7$Fdp$!'oOEF-7$Fip$!'VIEF-7$ F^q$!'\\`?F-7$Fcq$!'x]?F-7$Fhq$!'dq;F-7$F]r$!':e5F-7$Fbr$!'1Y5F-7$Fgr$ !&Lj)F-7$F\\s$!&2G(F-7$Fas$!&.;&F-7$Ffs$!&6#>F-7$F[t$\"&b\"HF-7$F`t$\" &t#[F-7$Fet$\"&p#[F-7$Fjt$\"&k#[F-7$F_u$\"&g#[F-7$Fdu$\"&A%[F-7$Fiu$\" &&=aF-7$F^v$\"&/.'F-7$Fcv$\"&&QrF-7$Fhv$\"&i0*F-7$F]w$\"'3D7F-7$Fbw$\" '*)R:5F-7$F`^l$\"(0],\"F-7$ Fe^l$\"(U],\"F-7$Fj^l$\"(Gp,\"F-7$F__l$\"(_--\"F-7$Fd_l$\"(;t-\"F-7$Fi _l$\"(+7/\"F-7$F^`l$\"($)o1\"F-7$Fc`l$\"(m)=9F-7$Fh`l$\"(HYd\"F-7$F]al $\"(3Wd\"F-7$Fbal$\"(x.f\"F-7$Fgal$\"(Cag\"F-7$F\\bl$\"(P7j\"F-7$Fabl$ \"(=Qn\"F-7$Ffbl$\"(o$Fdhl7$F[il$ \")b(p@$Fdhl7$F`il$\")TOhKFdhl7$Feil$\")]yNLFdhl7$Fjil$\")*fsm$Fdhl7$F _jl$\")G1;UFdhl7$Fdjl$\")D::UFdhl7$Fijl$\")sE9UFdhl7$F^[m$\")ks8UFdhl7 $Fc[m$\")nU:UFdhl7$Fh[m$\")e6>UFdhl7$F]\\m$\")#4sA%Fdhl7$Fb\\m$\")hEVU Fdhl7$Fg\\m$\")L'HF%Fdhl7$F\\]m$\"*g?@![F`]m7$Fb]m$\")Dt4aFdhl7$Fg]m$ \")R$yS&Fdhl7$F\\^m$\")cbAaFdhl7$Fa^m$\")%['QaFdhl7$Ff^m$\")BFnaFdhl7$ F[_m$\")p'Fdhl7$F^am$\") :\"=p'Fdhl7$Fcam$\")9W\"p'Fdhl7$Fham$\")'p5p'Fdhl7$F]bm$\")CL!p'Fdhl7$ Fbbm$\")df*o'Fdhl7$Fgbm$\")SN)o'Fdhl7$F\\cm$\"(8#*o'F-7$Facm$\"(w?p'F- 7$Ffcm$\"(y&)p'F-7$F[dm$\"(h:r'F-7$F`dm$\"(.bt'F-7$Fedm$\"(rm%oF-7$Fjd m$\"(^;8(F-7$F_em$\"(GcT(F-7$Fdem$\"(`c%yF-7$Fiem$\"(#)3%zF-7$F^fm$\"( u1%zF-7$Fcfm$\"(m/%zF-7$Fhfm$\"(a-%zF-7$F]gm$\"(O)RzF-7$Fbgm$\"(<%RzF- 7$Fggm$\"(Wx$zF-7$F\\hm$\"(,i$zF-7$Fahm$\"(,e$zF-7$Ffhm$\"(B1%zF-7$F[i m$\"(![[zF-7$F`im$\"(!zjzF-7$Feim$\"(G9*zF-7$Fjim$\"(q'Q!)F-7$F_jm$\"( ,8E)F-7$Fdjm$\"(P*4))F-7$Fijm$\"(Qf\"*)F-7$F^[n$\"(N/&*)F-7$Fc[n$\"(*= ]*)F-7$Fh[n$\"([*\\*)F-7$F]\\n$\"(g%\\*)F-7$Fb\\n$\"(u*[*)F-7$Fg\\n$\" (/![*)F-7$F\\]n$\"(Mq%*)F--Fb]n6&Fd]nFe]n$\"\"(F^]nFe]n-F\\^n6#%8schem e~with~c[10]=15/16G-F$6%7[u7$F($!'s4;F-7$F/$!'75;F-7$F4$!'%*[$!'hu=F-7$FC$!'@>?F-7$FH$!'&=.#F-7$FM$!'*))>#F-7$FR$!'u*>#F- 7$FW$!'xLBF-7$Ffn$!'8KBF-7$F[o$!'gjBF-7$F`o$!'\\\"R#F-7$Feo$!'=!Q#F-7$ Fjo$!'vMBF-7$F_p$!'7DBF-7$Fdp$!'^2@F-7$Fip$!'^-@F-7$F^q$!'aR;F-7$Fcq$! 'PP;F-7$Fhq$!'dK8F-7$F]r$!&OT)F-7$Fbr$!&pJ)F-7$Fgr$!&H&oF-7$F\\s$!&)od F-7$Fas$!&$pSF-7$Ffs$!&DZ\"F-7$F[t$\"&aS#F-7$F`t$\"&%QRF-7$Fet$\"&\"QR F-7$Fjt$\"&x$RF-7$F_u$\"&t$RF-7$Fdu$\"&.&RF-7$Fiu$\"&! \\F-7$Fcv$\"&&*y&F-7$Fhv$\"&gK(F-7$F]w$\"&g))*F-7$Fbw$\"'>,9F-7$Fgw$\" '0Z?F-7$F\\x$\"'+*>#F-7$Fax$\"'v)>#F-7$Ffx$\"'6)>#F-7$F[y$\"'*z?#F-7$F `y$\"'1\\AF-7$Fey$\"'\"=R#F-7$Fjy$\"'xZDF-7$F_z$\"'B/GF-7$Fdz$\"'j7KF- 7$Fiz$\"'8XQF-7$F^[l$\"'sCZF-7$Fc[l$\"'pOZF-7$Fh[l$\"'!Qz%F-7$F]\\l$\" ''e'[F-7$Fb\\l$\"'r%*\\F-7$Fg\\l$\"'A:_F-7$F\\]l$\"'jybF-7$Fa]l$\"'>() pF-7$Ff]l$\"'zU\")F-7$F[^l$\"'9T\")F-7$F`^l$\"'gR\")F-7$Fe^l$\"'*)R\") F-7$Fj^l$\"')\\:)F-7$F__l$\"'f\"=)F-7$Fd_l$\"':Q#)F-7$Fi_l$\"'J\\$)F-7 $F^`l$\"''\\b)F-7$Fc`l$\"(Du8\"F-7$Fh`l$\"(3AE\"F-7$F]al$\"(I?E\"F-7$F bal$\"(7[F\"F-7$Fgal$\"(eoG\"F-7$F\\bl$\"(DvI\"F-7$Fabl$\"(>;M\"F-7$Ff bl$\"(JgR\"F-7$F[cl$\"(8zg\"F-7$F`cl$\"(*GL=F-7$Fecl$\"(AD$=F-7$Fjcl$ \"(jT$=F-7$F_dl$\"($3P=F-7$Fddl$\"(NI%=F-7$Fidl$\"(%Ga=F-7$F^el$\"(iV( =F-7$Fcel$\"('zk>F-7$Fhel$\"(J>>#F-7$F]fl$\"(c5T#F-7$Fbfl$\"(hy`#F-7$F gfl$\"(\"fPDF-7$F\\gl$\"(Ft`#F-7$Fagl$\"(2o`#F-7$Ffgl$\"(6l`#F-7$F[hl$ \"(wx`#F-7$F`hl$\"),'fa#Fdhl7$Ffhl$\")Z)pb#Fdhl7$F[il$\")%*HxDFdhl7$F` il$\")p#Gh#Fdhl7$Feil$\")XRsEFdhl7$Fjil$\")YtPHFdhl7$F_jl$\")M2xLFdhl7 $Fdjl$\")PMwLFdhl7$Fijl$\")^jvLFdhl7$F^[m$\"):?vLFdhl7$Fc[m$\")6cwLFdh l7$Fh[m$\")B^zLFdhl7$F]\\m$\")u)fQ$Fdhl7$Fb\\m$\")d$))R$Fdhl7$Fg\\m$\" )#)fAMFdhl7$F\\]m$\"*qYg%QF`]m7$Fb]m$\")UHKVFdhl7$Fg]m$\")NxIVFdhl7$F \\^m$\")![DM%Fdhl7$Fa^m$\")4UbVFdhl7$Ff^m$\")\"=$yVFdhl7$F[_m$\"))HrT% Fdhl7$F`_m$\")yR![%Fdhl7$Fe_m$\")%\\Jt%Fdhl7$Fj_m$\")+S'H&Fdhl7$F_`m$ \")sBe`Fdhl7$Fd`m$\")r3e`Fdhl7$Fi`m$\")&QzN&Fdhl7$F^am$\")7zd`Fdhl7$Fc am$\")\\\\d`Fdhl7$Fham$\")q>d`Fdhl7$F]bm$\")ugc`Fdhl7$Fbbm$\")n,c`Fdhl 7$Fgbm$\")D-b`Fdhl7$F\\cm$\"(3dN&F-7$Facm$\"((*zN&F-7$Ffcm$\"('>j`F-7$ F[dm$\"(wNP&F-7$F`dm$\"(>FR&F-7$Fedm$\"(#f\"[&F-7$Fjdm$\"('Q4dF-7$F_em $\"(\\j$fF-7$Fdem$\"(++G'F-7$Fiem$\"((4cjF-7$F^fm$\"(JfN'F-7$Fcfm$\"(k dN'F-7$Fhfm$\"(%fbjF-7$F]gm$\"(g_N'F-7$Fbgm$\"(D\\N'F-7$Fggm$\"(&e`jF- 7$F\\hm$\"(\\BN'F-7$Fahm$\"(H?N'F-7$Ffhm$\"(!)eN'F-7$F[im$\"(c@O'F-7$F `im$\"(&QujF-7$Feim$\"(bkR'F-7$Fjim$\"(wTV'F-7$F_jm$\"(n=h'F-7$Fdjm$\" (0&\\qF-7$Fijm$\"(MS8(F-7$F^[n$\"(U:;(F-7$Fc[n$\"(W8;(F-7$Fh[n$\"(_6;( F-7$F]\\n$\"(h2;(F-7$Fb\\n$\"(r.;(F-7$Fg\\n$\"((ffrF-7$F\\]n$\"(?)erF- -Fb]n6&Fd]nF\\]n$\"\"%F^]nFe]n-F\\^n6#%8scheme~with~c[10]=19/20G-F$6%7 [u7$F($!'vT@F-7$F/$!'FU@F-7$F4$!'dEBF-7$F9$!'Y$!'M$\\#F-7$FC$! '6&o#F-7$FH$!'(=q#F-7$FM$!'-BHF-7$FR$!'8CHF-7$FW$!'\"45$F-7$Ffn$!'u)4$ F-7$F[o$!'lRJF-7$F`o$!'lvJF-7$Feo$!'5gJF-7$Fjo$!'X(4$F-7$F_p$!'b%3$F-7 $Fdp$!'h\"z#F-7$Fip$!''\\y#F-7$F^q$!'ck@F-7$Fcq$!'ph@F-7$Fhq$!'LaKF-7$Fjy$\"'vFMF-7$F_z$\"'!)pPF-7$Fdz$\"'l9VF-7$Fiz$\"'ne^F-7$ F^[l$\"'pKjF-7$Fc[l$\"'g[jF-7$Fh[l$\"'mCkF-7$F]\\l$\"'l?lF-7$Fb\\l$\"' N#p'F-7$Fg\\l$\"'E')pF-7$F\\]l$\"'vquF-7$Fa]l$\"'I\\$*F-7$Ff]l$\"(%4*3 \"F-7$F[^l$\"(u))3\"F-7$F`^l$\"(o')3\"F-7$Fe^l$\"(1()3\"F-7$Fj^l$\"(82 4\"F-7$F__l$\"(aU4\"F-7$Fd_l$\"(%y,6F-7$Fi_l$\"(*e;6F-7$F^`l$\"(%)R9\" F-7$Fc`l$\"((y>:F-7$Fh`l$\"(yho\"F-7$F]al$\"(Rfo\"F-7$Fbal$\"([Hq\"F-7 $Fgal$\"(')*=z\"F-7$Ffbl$\"($Rk=F-7$F [cl$\"(9n9#F-7$F`cl$\"(#4ZCF-7$Fecl$\"(ngW#F-7$Fjcl$\"(Z#[CF-7$F_dl$\" (J@X#F-7$Fddl$\"(\\+Y#F-7$Fidl$\"(>]Z#F-7$F^el$\"(U<]#F-7$Fcel$\"(R@i# F-7$Fhel$\"(BY#HF-7$F]fl$\"(+l@$F-7$Fbfl$\"(CaQ$F-7$Fgfl$\"(k]Q$F-7$F \\gl$\"(7ZQ$F-7$Fagl$\"(g(F-7$F_em$\"(RH !zF-7$Fdem$\"(b'e$)F-7$Fiem$\"(e&f%)F-7$F^fm$\"(P$f%)F-7$Fcfm$\"(:\"f% )F-7$Fhfm$\"(*))e%)F-7$F]gm$\"(W%e%)F-7$Fbgm$\"()*zX)F-7$Fggm$\"(:iX)F -7$F\\hm$\"(qXX)F-7$Fahm$\"(QTX)F-7$Ffhm$\"(L#f%)F-7$F[im$\"(UvY)F-7$F `im$\"(EP[)F-7$Feim$\"(LH^)F-7$Fjim$\"(IGc)F-7$F_jm$\"(Cxz)F-7$Fdjm$\" (%ov$*F-7$Fijm$\"(bs[*F-7$F^[n$\"(eN_*F-7$Fc[n$\"('HB&*F-7$Fh[n$\"(RI_ *F-7$F]\\n$\"(?D_*F-7$Fb\\n$\"(-?_*F-7$Fg\\n$\"(r4_*F-7$F\\]n$\"(Q*>&* F--%'COLOURG6&Fd]n$\"*++++\"!\")Fe]nF`is-F\\^n6#%8scheme~with~c[10]=32 /33G-F$6%7[u7$F($!'+n@F-7$F/$!'_n@F-7$F4$!'0aBF-7$F9$!'/YCF-7$F>$!'&G_ #F-7$FC$!''pr#F-7$FH$!'#Rt#F-7$FM$!'%y&HF-7$FR$!'(*eHF-7$FW$!'3QJF-7$F fn$!')e8$F-7$F[o$!'UxJF-7$F`o$!'-9KF-7$Feo$!'P)>$F-7$Fjo$!'ONJF-7$F_p$ !'KAJF-7$Fdp$!'_EGF-7$Fip$!'y>GF-7$F^q$!''G>#F-7$Fcq$!'%**=#F-7$Fhq$!' 4y.*F-7$F\\s$!&$ovF-7$Fas$!&IF &F-7$Ffs$!&`w\"F-7$F[t$\"&VZ$F-7$F`t$\"&ea&F-7$Fet$\"&aa&F-7$Fjt$\"&[a &F-7$F_u$\"&Va&F-7$Fdu$\"&F-7$Fgw$\"'4'y#F-7$F\\x$\"' G\"*HF-7$Fax$\"''4*HF-7$Ffx$\"'2!*HF-7$F[y$\"'P.IF-7$F`y$\"'seIF-7$Fey $\"'=^KF-7$Fjy$\"'^hMF-7$F_z$\"'W2QF-7$Fdz$\"'VeVF-7$Fiz$\"'$>@&F-7$F^ [l$\"'0*R'F-7$Fc[l$\"'::kF-7$Fh[l$\"'5#\\'F-7$F]\\l$\"'@*e'F-7$Fb\\l$ \"'*Gw'F-7$Fg\\l$\"'U$F-7$F[hl$\"(.OU$F-7$F`hl$\")_h MMFdhl7$Ffhl$\")TX\\MFdhl7$F[il$\")*)zwMFdhl7$F`il$\")(GY_$Fdhl7$Feil$ \")3$[g$Fdhl7$Fjil$\")#Q@'RFdhl7$F_jl$\"))[Qb%Fdhl7$Fdjl$\")d'Gb%Fdhl7 $Fijl$\")#4>b%Fdhl7$F^[m$\")QK^XFdhl7$Fc[m$\")/:`XFdhl7$Fh[m$\")(>rb%F dhl7$F]\\m$\")6$ec%Fdhl7$Fb\\m$\")k6$e%Fdhl7$Fg\\m$\")04:YFdhl7$F\\]m$ \"*:y\\=&F`]m7$Fb]m$\")OYReFdhl7$Fg]m$\")ETPeFdhl7$F\\^m$\")0C`eFdhl7$ Fa^m$\")-bqeFdhl7$Ff^m$\"),M,fFdhl7$F[_m$\")E``fFdhl7$F`_m$\")RhQgFdhl 7$Fe_m$\")U^yjFdhl7$Fj_m$\")b(f8(Fdhl7$F_`m$\")O8>sFdhl7$Fd`m$\")>$*=s Fdhl7$Fi`m$\")=t=sFdhl7$F^am$\")I`=sFdhl7$Fcam$\")P8=sFdhl7$Fham$\")Ft d@(F-7$Facm$\"($z=sF-7$Ffcm$\"(xdA(F-7$F[dm$\"(A(RsF-7$F`dm$\"(QaE (F-7$Fedm$\"(@[Q(F-7$Fjdm$\"(n2p(F-7$F_em$\"(eb*zF-7$Fdem$\"(-qX)F-7$F iem$\"(w\"f&)F-7$F^fm$\"(_*e&)F-7$Fcfm$\"(F(e&)F-7$Fhfm$\"(*\\e&)F-7$F ]gm$\"([!e&)F-7$Fbgm$\"((fd&)F-7$Fggm$\"($zb&)F-7$F\\hm$\"(HTb)F-7$Fah m$\"($p`&)F-7$Ffhm$\"(a)e&)F-7$F[im$\"(psc)F-7$F`im$\"(jOe)F-7$Feim$\" (\\Kh)F-7$Fjim$\"()zj')F-7$F_jm$\"(/=!*)F-7$Fdjm$\"(nv[*F-7$Fijm$\"(g1 g*F-7$F^[n$\"(fuj*F-7$Fc[n$\"(&>P'*F-7$Fh[n$\"(Mpj*F-7$F]\\n$\"(4kj*F- 7$Fb\\n$\"(&)ej*F-7$Fg\\n$\"(U[j*F-7$F\\]n$\"('zL'*F--Fb]n6&Fd]n$\"\"& F^]nFe]nFi]n-F\\^n6#%8scheme~with~c[10]=81/82G-F$6%7[u7$F($!'1X;F-7$F/ $!'ZX;F-7$F4$!'3(y\"F-7$F9$!'#p&=F-7$F>$!'B:>F-7$FC$!'hi?F-7$FH$!'\\v? F-7$FM$!'_XAF-7$FR$!'PYAF-7$FW$!'R#Q#F-7$Ffn$!'s!Q#F-7$F[o$!'H7CF-7$F` o$!'6SCF-7$Feo$!'DGCF-7$Fjo$!'\\!Q#F-7$F_p$!'fqBF-7$Fdp$!';Y@F-7$Fip$! '0T@F-7$F^q$!'Gl;F-7$Fcq$!'2j;F-7$Fhq$!'[]8F-7$F]r$!&LY)F-7$Fbr$!&WO)F -7$Fgr$!&Z'oF-7$F\\s$!&Qv&F-7$Fas$!&<,%F-7$Ffs$!&%\\8F-7$F[t$\"&ti#F-7 $F`t$\"&&*>%F-7$Fet$\"&#*>%F-7$Fjt$\"&()>%F-7$F_u$\"&$)>%F-7$Fdu$\"&:@ %F-7$Fiu$\"&Po%F-7$F^v$\"&b=&F-7$Fcv$\"&W4'F-7$Fhv$\"&!owF-7$F]w$\"'0H 5F-7$Fbw$\"'!=X\"F-7$Fgw$\"'p8@F-7$F\\x$\"'VpAF-7$Fax$\"'=pAF-7$Ffx$\" '^oAF-7$F[y$\"'hyAF-7$F`y$\"'i?BF-7$Fey$\"'rmCF-7$Fjy$\"'OEEF-7$F_z$\" '%*))GF-7$Fdz$\"';2LF-7$Fiz$\"')\\&RF-7$F^[l$\"'-c[F-7$Fc[l$\"'Do[F-7$ Fh[l$\"'mE\\F-7$F]\\l$\"'Q+]F-7$Fb\\l$\"'@K^F-7$Fg\\l$\"'(yN&F-7$F\\]l $\"'#)HdF-7$Fa]l$\"'(=<(F-7$Ff]l$\"'>b$)F-7$F[^l$\"']`$)F-7$F`^l$\"'#> N)F-7$Fe^l$\"'@_$)F-7$Fj^l$\"'jn$)F-7$F__l$\"'$[R)F-7$Fd_l$\"'m_%)F-7$ Fi_l$\"'Nm&)F-7$F^`l$\"'sw()F-7$Fc`l$\"(*=m6F-7$Fh`l$\"(>RH\"F-7$F]al$ \"(OPH\"F-7$Fbal$\"(,oI\"F-7$Fgal$\"(=\">8F-7$F\\bl$\"(_-M\"F-7$Fabl$ \"(A^P\"F-7$Ffbl$\"(x2V\"F-7$F[cl$\"(Wvk\"F-7$F`cl$\"(c\"y=F-7$Fecl$\" (qt(=F-7$Fjcl$\"(X!z=F-7$F_dl$\"(G?)=F-7$Fddl$\"(6\"))=F-7$Fidl$\"(4'* *=F-7$F^el$\"(M,#>F-7$Fcel$\"((f7?F-7$Fhel$\"(w[C#F-7$F]fl$\"(&**oCF-7 $Fbfl$\"()p)f#F-7$Fgfl$\"(@%)f#F-7$F\\gl$\"(^\")f#F-7$Fagl$\"(=wf#F-7$ Ffgl$\"(:tf#F-7$F[hl$\"(1')f#F-7$F`hl$\")o'pg#Fdhl7$Ffhl$\")XFdhl7$F`_m$\")?p$e%Fdhl7$Fe_m$\")#H<%[Fdhl7$Fj_m$\")+ u;aFdhl7$F_`m$\")o')zaFdhl7$Fd`m$\")LrzaFdhl7$Fi`m$\")8czaFdhl7$F^am$ \")1TzaFdhl7$Fcam$\")x5zaFdhl7$Fham$\")I!)yaFdhl7$F]bm$\")**>yaFdhl7$F bbm$\")ffxaFdhl7$Fgbm$\")(ylZ&Fdhl7$F\\cm$\"(wsZ&F-7$Facm$\"(5'zaF-7$F fcm$\"(7\\[&F-7$F[dm$\"(+b\\&F-7$F`dm$\"(D]^&F-7$Fedm$\"(kcg&F-7$Fjdm$ \"(Uz$eF-7$F_em$\"(P$pgF-7$Fdem$\"(_'>kF-7$Fiem$\"(>s\\'F-7$F^fm$\"(\\ q\\'F-7$Fcfm$\"(zo\\'F-7$Fhfm$\"(0n\\'F-7$F]gm$\"(kj\\'F-7$Fbgm$\"(@g \\'F-7$Fggm$\"(_Y\\'F-7$F\\hm$\"()Q$\\'F-7$Fahm$\"(eI\\'F-7$Ffhm$\"(yp \\'F-7$F[im$\"(oL]'F-7$F`im$\"(8tF--F^is6&Fd]nF`isFe]nFe]n-F\\^n6#%:scheme~with~ c[10]=169/170G-%%FONTG6$%*HELVETICAGF]]n-%&TITLEG6#%Verror~curves~for~ 11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F^gw-%%V IEWG6$;F(F\\]n%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10 ]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme w ith c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 504 "evalf[20](p lot(['qn_RK7_1'(x)-q(x),'qn_RK7_2'(x)-q(x),'qn_RK7_3'(x)-q(x),\n'qn_RK 7_4'(x)-q(x),'qn_RK7_5'(x)-q(x),'qn_RK7_6'(x)-q(x)],x=0.9..0.925,font= [HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,. 9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme \+ with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/20`, `scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10] =169/170`],\ntitle=`error curves for 11 stage order 7 Runge-Kutta meth ods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 617 617 {PLOTDATA 2 "6,-%'CU RVESG6%7ct7$$\"\"*!\"\"$\"('zmf!#?7$$\"5nmm;a)G\\a+*F-$\"(%>lfF-7$$\"5 LL$ek`o!>5!*F-$\"(6y*fF-7$$\"5nmm\"z>)G_:!*F-$\"(!puiF-7$$\"5nmmT&QU!* 3-*F-$\"(*HriF-7$$\"5LL$eRZXKi-*F-$\"(6XE'F-7$$\"5nm;z>,_=J!*F-$\"(f6< 'F-7$$\"5++]7G$[8j.*F-$\"(:<)eF-7$$\"5nm;z%*frhT!*F-$\"($pxeF-7$$\"5NL $3-Q\\gU/*F-$\"(0p'eF-7$$\"5++]ilFQ!p/*F-$\"(?!4eF-7$$\"5I$3FWXKj#[!*F -$\"(wDs&F-7$$\"5lm\"HK9#Gi\\!*F-$\"(1,b&F-7$$\"5+]7.K=B)40*F-$\"(9VA& F-7$$\"5LLL$3_\"=M_!*F-$\"(%HPYF-7$$\"5nmmTg(fJr0*F-$\"(iOL%F-7$$\"5++ +]7eP_i!*F-$\"(**yJ%F-7$$\"5++++v34Bl!*F-$\"(\"RHUF-7$$\"5+++]Pf!Qz1*F -$\"(@\"=QF-7$$\"5+++]i+paq!*F-$\"(WPU#F-7$$\"5+++](=ubJ2*F-$\"('=B5F- 7$$\"5NLe9m>[_v!*F-$\"(NH-\"F-7$$\"5nm;zW(*Q*y2*F-$\"(5@-\"F-7$$\"5++D \"y+'4r!3*F-$\"(!o45F-7$$\"5LLL$3F-GN3*F-$\"'PT!*F-7$$\"5NL$e*[L7s%3*F -$\"'EDwF-7$$\"5NLL3FWW\"f3*F-$\"'&*))[F-7$$\"5NL$3_]l2r3*F-$!&*=7F-7$ $\"5LLLL$e'3I)3*F-$!'T*)))F-7$$\"5+]7Gj`))o*3*F-$!(BCo#F-7$$\"5lm\"HK9 %o2\"4*F-$!(;l![F-7$$\"5I$3xJ#H[Y#4*F-$!(\\e![F-7$$\"5++]7.FW0\"* F-$!(S`)fF-7$$\"5+++D14js1\"*F-$!(\\>9(F-7$$\"5++D\"G))*)4!3\"*F-$!('> &>*F-7$$\"5++]Pf)[$H4\"*F-$!)b?p7F-7$$\"5SgF;++4Y4\"*F-$!)zJH8F-7$$\"5 !3_]49JG'4\"*F-$!)ZU$R\"F-7$$\"5?\"GQ62s>6*F- $!)1)*)R\"F-7$$\"5+]7GQi8J8\"*F-$!)hz)R\"F-7$$\"5LLLek`1l9\"*F-$!)\\j) R\"F-7$$\"5NL3xcG/6<\"*F-$!)5(*)R\"F-7$$\"5LL$e*[.-d>\"*F-$!)d(\\S\"F- 7$$\"5l;H#oK!o*37*F-$!)!)><9F-7$$\"5++vo/.MAA\"*F-$!)OaX9F-7$$\"5q\"z> OHq')G7*F-$!)KDq9F-7$$\"5N$3_DG+]N7*F-$!)+e0:F-7$$\"5+vV[r-L@C\"*F-$!) /Bb:F-7$$\"5nmmTg-m([7*F-$!)([Ri\"F-7$$\"5lmTgFJDjF\"*F-$!)4\"RF#F-7$$ \"5nm;z%*f%)QI\"*F-$!):)*HFF-7$$\"5NLL$eR](yK\"*F-$!)mLHFF-7$$\"5++](o za'=N\"*F-$!)P.HFF-7$$\"5NLeRA\"exx8*F-$!)%eJt#F-7$$\"5nmm\"zWho.9*F-$ !)stlFF-7$$\"5N$3_+A\"y.T\"*F-$!)K=(y#F-7$$\"5++v=#*4qqT\"*F-$!)*Q'=GF -7$$\"5q;HKk2iPU\"*F-$!)\"fQ'GF-7$$\"5NL$ek`SXI9*F-$!)`rFHF-7$$\"5qm\" H23!QQW\"*F-$!)U'z8$F-7$$\"5++++D'>Ad9*F-$!)kV@NF-7$$\"5++D\"y]bJq9*F- $!)**eqTF-7$$\"5++]i!R\"4M[\"*F-$!)-87XF-7$$\"5++vVts-l\\\"*F-$!)&Q:^% F-7$$\"5+++DcJ'f4:*F-$!)3&4^%F-7$$\"5++voavL\\`\"*F-$!)!4+^%F-7$$\"5++ ]7`>r-c\"*F-$!)197XF-7$$\"5+]7G)[wLu:*F-$!)g#*>XF-7$$\"5++vVB5/%)e\"*F -$!)v)>a%F-7$$\"5+Dc,\"HtV&f\"*F-$!)IEjXF-7$$\"5+]PfebqCg\"*F-$!)ja&f% F-7$$\"5+v=Z3%p\"*F-$!)>#oo'F-7$ $\"5+++]P4v5s\"*F-$!)R;&o'F-7$$\"5NLe9TQKbu\"*F-$!)'))fo'F-7$$\"5nm;zW n*)*p<*F-$!)fJ.nF-7$$\"5N3_vSCumx\"*F-$!)Ue;nF-7$$\"5+](=n8)eLy\"*F-$! )%[pt'F-7$$\"5q\"H#oKQV+z\"*F-$!)8MnnF-7$$\"5NLekG&zs'z\"*F-$!))f;\"oF -7$$\"5q;Hd?4(45=*F-$!)8xjpF-7$$\"5+++]7BmM#=*F-$!)e8`sF-7$$\"5I$e*[$ \\b/O=*F-$!)%pWt(F-7$$\"5lm\"zWn[i[=*F-$!)'yv`)F-7$$\"5]ilsppp<&=*F-$! )?-2))F-7$$\"5IeR(\\EX\"\\&=*F-$!)Q=*F-$!*`QJ,*!#@7$$\"5++]i:bzj#>*F -$!*]:<,*Fe\\m7$$\"5I$3F>9^BR>*F-$!*xeJ,*Fe\\m7$$\"5lm\"H#on!4_>*F-$!* iE$>!*Fe\\m7$$\"5+]7`%Ri%\\'>*F-$!*f*=O!*Fe\\m7$$\"5LLL$3-=!y(>*F-$!*Z D\\2*Fe\\m7$$\"5+]ilZQg7*>*F-$!*#>Cg\"*Fe\\m7$$\"5lm\"zWn*=Z+#*F-$!*Dd !G$*Fe\\m7$$\"5I$3-8]v<=?*F-$!*.:tj*Fe\\m7$$\"5++]7G8O;.#*F-$!,=vfx,\" !#A7$$\"5N3Fp,C<\"Q?*F-$!,^@&pb5Fc_m7$$\"5q;/EvM)fW?*F-$!,U;#*Q4\"Fc_m 7$$\"5+D\"G)[Xz50#*F-$!,t`@Q4\"Fc_m7$$\"5NLeRAcgv0#*F-$!,5k]P4\"Fc_m7$ $\"5+]7`pxA02#*F-$!+I'3O4\"Fe\\m7$$\"5nmmm;*\\[$3#*F-$!+NwY$4\"Fe\\m7$ $\"5lm;/^Y'**4@*F-$!+MD?$4\"Fe\\m7$$\"5nmmT&Qz]O@*F-$!+gu@$4\"Fe\\m7$$ \"5N$3x16Rl\\@*F-$!+c[j$4\"Fe\\m7$$\"5++v$f$))*zi@*F-$!+rYr%4\"Fe\\m7$ $\"5l;z>h&e%f<#*F-$!+nU(p4\"Fe\\m7$$\"5LL$ekG=4*=#*F-$!+_G:,6Fe\\m7$$ \"5lm\"Hd&R^K@#*F-$!*2*3<6F-7$$\"5++++D'4TPA*F-$!*&*G79\"F-7$$\"5lm;/E b+^E#*F-$!*w749\"F-7$$\"5LLL3F9!z#H#*F-$!*x#eS6F-7$$\"5++](=UgbKUB*F-$!*e/x8\"F-7$$\"5NeRZ_MB*[B*F-$!**y6O6F-7$ $\"5+]7G)[Z_bB*F-$!*1:P8\"F-7$$\"5qT&)3C:E@O#*F-$!*`M,8\"F-7$$\"5NLe*) fbF(oB*F-$!*?%)[7\"F-7$$\"5q;/^JOI>Q#*F-$!*DCl5\"F-7$$\"5++]7.U#*F-$!):Vt()F-7$$\"5+vVt_!fcBC*F-$!)wH_()F-7$$\"5 ]i:g;]X^U#*F-$!).;_()F-7$$\"5+](o/)4DnU#*F-$!)/-_()F-7$$\"5+DJ?3H%))HC *F-$!)Hu^()F-7$$\"5++v$f$[VIV#*F-$!)vY^()F-7$$\"5+]iS\"p=ORC*F-$!)f\"4 v)F-7$$\"5++](oa-oXC*F-$!))f.v)F-7$$\"5++vVt7SGZ#*F-$!)s#yu)F-7$$\"$D* !\"$$!)kSS()F--%&COLORG6&%$RGBG$\"\"!F^[n$\"\"#F*$\"\"\"F^[n-%'LEGENDG 6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7ct7$F($\"(Mq%*)F-7$F/$ \"(#fW*)F-7$F4$\"(9C**)F-7$F9$\"(2LR*F-7$F>$\"(O#)Q*F-7$FC$\"(NyP*F-7$ FH$\"(*3K#*F-7$FM$\"(]'z()F-7$FR$\"(LOx)F-7$FW$\"(Qtv)F-7$Ffn$\"(E&p') F-7$F[o$\"(%=Q&)F-7$F`o$\"(TeF)F-7$Feo$\"('yzxF-7$Fjo$\"(G^)oF-7$F_p$ \"(z@U'F-7$Fdp$\"(y%)R'F-7$Fip$\"(i[E'F-7$F^q$\"(wGk&F-7$Fcq$\"(\\3`$F -7$Fhq$\"(GuS\"F-7$F]r$\"(zqS\"F-7$Fbr$\"(reS\"F-7$Fgr$\"(3sQ\"F-7$F\\ s$\"(Q#G7F-7$Fas$\"(NZ,\"F-7$Ffs$\"'I>gF-7$F[t$!'eW:F-7$F`t$!(w'y9F-7$ Fet$!(\\\"*=%F-7$Fjt$!($*4S(F-7$F_u$!(m**R(F-7$Fdu$!(X*)R(F-7$Fiu$!(Hy R(F-7$F^v$!(X'3uF-7$Fcv$!(c8`(F-7$Fhv$!(u7C)F-7$F]w$!('Hw\"*F-7$Fbw$!) q*>4\"F-7$Fgw$!)[p,9F-7$F\\x$!)sRH>F-7$Fax$!)U8??F-7$Ffx$!)r!p6#F-7$F[ y$!)[vD@F-7$F`y$!)#>d7#F-7$Fey$!)\"[c7#F-7$Fjy$!)idD@F-7$F_z$!)TVD@F-7 $Fdz$!)9HD@F-7$Fiz$!)2,D@F-7$F^[l$!)`wC@F-7$Fc[l$!)PED@F-7$Fh[l$!)$zU8 #F-7$F]\\l$!)$\\E:#F-7$Fb\\l$!)iF&>#F-7$Fg\\l$!)oWKAF-7$F\\]l$!)7g&G#F -7$Fa]l$!)]KgBF-7$Ff]l$!)twjCF-7$F[^l$!)ZxUMF-7$F`^l$!);2ITF-7$Fe^l$!) d4HTF-7$Fj^l$!)HjGTF-7$F__l$!)b\"[8%F-7$Fd_l$!)jt$=%F-7$Fi_l$!)U&f@%F- 7$F^`l$!)x@jUF-7$Fc`l$!)W`F-7$Fgal$!)*)\\'H'F-7$F\\bl$!)SQ5oF-7$Fabl$!)9\\4oF-7$Ffbl$!)Sg3oF -7$F[cl$!)-=2oF-7$F`cl$!)]O5oF-7$Fecl$!)e.AoF-7$Fjcl$!)G7boF-7$F_dl$!) \"Rq)oF-7$Fddl$!)PZNpF-7$Fidl$!)n>2qF-7$F^el$!)(e66(F-7$Fcel$!)FC=uF-7 $Fhel$!)UOmzF-7$F]fl$!)Dg-*)F-7$Fbfl$!*[gx+\"F-7$Fgfl$!*c)[25F-7$F\\gl $!*bQs+\"F-7$Fagl$!*agt+\"F-7$Ffgl$!*r`*45F-7$F[hl$!*.S>,\"F-7$F`hl$!* r))\\,\"F-7$Fehl$!*2R&>5F-7$Fjhl$!*#R(F-7$F9$ \"(x<^(F-7$F>$\"(>x](F-7$FC$\"(_$*\\(F-7$FH$\"(R7Q(F-7$FM$\"(fT,(F-7$F R$\"(Z$4qF-7$FW$\"(xi*pF-7$Ffn$\"(ld#pF-7$F[o$\"(G-#oF-7$F`o$\"(J$4mF- 7$Feo$\"(z.@'F-7$Fjo$\"(W0\\&F-7$F_p$\"(\")z6&F-7$Fdp$\"(.!*4&F-7$Fip$ \"(m=*\\F-7$F^q$\"(+G\\%F-7$Fcq$\"(>rz#F-7$Fhq$\"(@;4\"F-7$F]r$\"(^84 \"F-7$Fbr$\"(#R!4\"F-7$Fgr$\"(aa2\"F-7$F\\s$\"'y![*F-7$Fas$\"'XpxF-7$F fs$\"'*)fWF-7$F[t$!'k0;F-7$F`t$!(%pA7F-7$Fet$!(EsR$F-7$Fjt$!(KX(fF-7$F _u$!(.P(fF-7$Fdu$!(zG(fF-7$Fiu$!(t>(fF-7$F^v$!(?1)fF-7$Fcv$!(*))ygF-7$ Fhv$!(#oZmF-7$F]w$!(DqR(F-7$Fbw$!('p%z)F-7$Fgw$!)#ex7\"F-7$F\\x$!))**3 b\"F-7$Fax$!)[mB;F-7$Ffx$!)F-7$F[^l$!)ugjFF-7$F`^l$!)kV9LF-7$Fe^l$!)Ll8LF-7$Fj^l$!)1G8LF-7$F__ l$!)nA=LF-7$Fd_l$!)wQdLF-7$Fi_l$!)5=$Q$F-7$F^`l$!)A-@MF-7$Fc`l$!)SVvMF -7$Fh`l$!)lG_NF-7$F]al$!)$4a!QF-7$Fbal$!):EnUF-7$Fgal$!)**Q\\]F-7$F\\b l$!)d)4Y&F-7$Fabl$!)(p-Y&F-7$Ffbl$!)%e&faF-7$F[cl$!)hTeaF-7$F`cl$!)8'4 Y&F-7$Fecl$!)\")HqaF-7$Fjcl$!)Nx'\\&F-7$F_dl$!)KJAbF-7$Fddl$!)D2hbF-7$ Fidl$!))o%=cF-7$F^el$!)am,dF-7$Fcel$!)xTZfF-7$Fhel$!):2'Q'F-7$F]fl$!)4 LNrF-7$Fbfl$!)Omv!)F-7$Fgfl$!)V[t!)F-7$F\\gl$!)3[r!)F-7$Fagl$!)GXs!)F- 7$Ffgl$!)y=$4)F-7$F[hl$!)<24\")F-7$F`hl$!)3XL\")F-7$Fehl$!)k$)p\")F-7$ Fjhl$!)&*)GA)F-7$F_il$!)*f\\S)F-7$Fdil$!)RC^()F-7$Fiil$!)R6F$*F-7$F^jl $!*(ywG5F-7$Fcjl$!*w))41\"F-7$Fhjl$!*d=h3\"F-7$F][m$!*I%3'3\"F-7$Fb[m$ !*9]g3\"F-7$Fg[m$!*u\")f3\"F-7$F\\\\m$!*K8f3\"F-7$Fa\\m$!+08j&3\"Fe\\m 7$Fg\\m$!+w*ea3\"Fe\\m7$F\\]m$!+=*Hc3\"Fe\\m7$Fa]m$!+ASO'3\"Fe\\m7$Ff] m$!+F:P)3\"Fe\\m7$F[^m$!+P/)H4\"Fe\\m7$F`^m$!+$=AJ5\"Fe\\m7$Fe^m$!+gv/ B6Fe\\m7$Fj^m$!+1yrf6Fe\\m7$F__m$!,CS&oB7Fc_m7$Fe_m$!,5GZ&o7Fc_m7$Fj_m $!,**o$o88Fc_m7$F_`m$!,&=))f88Fc_m7$Fd`m$!,(yO^88Fc_m7$Fi`m$!+(=VLJ\"F e\\m7$F^am$!+)ztJJ\"Fe\\m7$Fcam$!+U\\&GJ\"Fe\\m7$Fham$!+&HjGJ\"Fe\\m7$ F]bm$!+!\\TLJ\"Fe\\m7$Fbbm$!+3ed98Fe\\m7$Fgbm$!+(>PrJ\"Fe\\m7$F\\cm$!+ =T\"=K\"Fe\\m7$Facm$!*V@\"R8F-7$Ffcm$!*y+RO\"F-7$F[dm$!*#H_j8F-7$F`dm$ !*8GJO\"F-7$Fedm$!*\"[[i8F-7$Fjdm$!*d.'f8F-7$F_em$!*5ZwN\"F-7$Fdem$!*% *zYN\"F-7$Fiem$!*o_-N\"F-7$F^fm$!*:`PM\"F-7$Fcfm$!*V()4K\"F-7$Fhfm$!*k )>w7F-7$F]gm$!*c:r>\"F-7$Fbgm$!*_K(e5F-7$Fggm$!*jNc.\"F-7$F\\hm$!*j+I. \"F-7$Fahm$!*S%)H.\"F-7$Ffhm$!*#z'H.\"F-7$F[im$!*>NH.\"F-7$F`im$!*m-H. \"F-7$Feim$!*`PG.\"F-7$Fjim$!*&>xK5F-7$F_jm$!*jsC.\"F-7$Fdjm$!*D#eJ5F- -Fjjm6&F\\[nF($\"\"%F*F][n-Fd[n6#%8scheme~with~c[10]=19/20G-F$6%7ct7$F ($\"(Q*>&*F-7$F/$\"(3t^*F-7$F4$\"(upc*F-7$F9$\"(cp(**F-7$F>$\"(r:(**F- 7$FC$\"(t-'**F-7$FH$\"((f)z*F-7$FM$\"(abH*F-7$FR$\"(n\"*G*F-7$FW$\"($p r#*F-7$Ffn$\"(Br<*F-7$F[o$\"(,a.*F-7$F`o$\"(_>v)F-7$Feo$\"(J`@)F-7$Fjo $\"(vjC(F-7$F_p$\"(2Zu'F-7$Fdp$\"(A%>nF-7$Fip$\"(Oid'F-7$F^q$\"(>$3fF- 7$Fcq$\"(1ij$F-7$Fhq$\"(H#\\8F-7$F]r$\"($*)[8F-7$Fbr$\"(SwM\"F-7$Fgr$ \"(exK\"F-7$F\\s$\"(6y:\"F-7$Fas$\"'3$H*F-7$Ffs$\"'*=([F-7$F[t$!'oMKF- 7$F`t$!(OOu\"F-7$Fet$!($e_YF-7$Fjt$!(x<5)F-7$F_u$!(`15)F-7$Fdu$!(O&*4) F-7$Fiu$!($H)4)F-7$F^v$!(t(4\")F-7$Fcv$!(;2C)F-7$Fhv$!(Y%***)F-7$F]w$! ($f****F-7$Fbw$!)5e'=\"F-7$Fgw$!)XB=:F-7$F\\x$!)km$3#F-7$Fax$!)%>4=#F- 7$Ffx$!)\"\\YG#F-7$F[y$!)I8%H#F-7$F`y$!)X4%H#F-7$Fey$!)y,%H#F-7$Fjy$!) -%RH#F-7$F_z$!)py$H#F-7$Fdz$!)Hj$H#F-7$Fiz$!)*HLH#F-7$F^[l$!)W1$H#F-7$ Fc[l$!)!)e$H#F-7$Fh[l$!)Q>.BF-7$F]\\l$!)')yABF-7$Fb\\l$!)[GoBF-7$Fg\\l $!)A(zS#F-7$F\\]l$!),ukCF-7$Fa]l$!)OcWDF-7$Ff]l$!)>4bEF-7$F[^l$!)e)=q$ F-7$F`^l$!)NUWF-7$Fd_l$! )$GV\\%F-7$Fi_l$!)inGXF-7$F^`l$!)W2zXF-7$Fc`l$!):b^YF-7$Fh`l$!)F$Rv%F- 7$F]al$!)Z?\"4&F-7$Fbal$!)Vu1dF-7$Fgal$!)7P\\nF-7$F\\bl$!),8)H(F-7$Fab l$!)P<(H(F-7$Ffbl$!)FA'H(F-7$F[cl$!)Wp%H(F-7$F`cl$!)&p!)H(F-7$Fecl$!)D [5tF-7$Fjcl$!)NpXtF-7$F_dl$!)lmztF-7$Fddl$!)'G7V(F-7$Fidl$!)2f2vF-7$F^ el$!)wG=wF-7$Fcel$!)CIXzF-7$Fhel$!)\"p!H&)F-7$F]fl$!)**GE&*F-7$Fbfl$!* N\"zx5F-7$Fgfl$!*`+v2\"F-7$F\\gl$!**HBx5F-7$Fagl$!*)3Ox5F-7$Ffgl$!*09, 3\"F-7$F[hl$!*eBA3\"F-7$F`hl$!*Uha3\"F-7$Fehl$!*.%H!4\"F-7$Fjhl$!*0St4 \"F-7$F_il$!*,?:7\"F-7$Fdil$!*?/v;\"F-7$Fiil$!*8nRC\"F-7$F^jl$!*)=\\r8 F-7$Fcjl$!*jkUT\"F-7$Fhjl$!*uAwW\"F-7$F][m$!*4xvW\"F-7$Fb[m$!*dJvW\"F- 7$Fg[m$!*VSuW\"F-7$F\\\\m$!*G\\tW\"F-7$Fa\\m$!+sK(pW\"Fe\\m7$Fg\\m$!+- HuY9Fe\\m7$F\\]m$!+)\\opW\"Fe\\m7$Fa]m$!+)=SzW\"Fe\\m7$Ff]m$!+cxf]9Fe \\m7$F[^m$!+Wupc9Fe\\m7$F`^m$!+6G6q9Fe\\m7$Fe^m$!+kJX'\\\"Fe\\m7$Fj^m$ !+#3%*[a\"Fe\\m7$F__m$!,hTD$H;Fc_m7$Fe_m$!,PT,&)o\"Fc_m7$Fj_m$!,_p9![< Fc_m7$F_`m$!,_n,zu\"Fc_m7$Fd`m$!,()Q)yZ$\")\"f$45F-7 $FC$\")\">#35F-7$FH$\"(+'>**F-7$FM$\"(LRT*F-7$FR$\"(muS*F-7$FW$\"(/)*Q *F-7$Ffn$\"(vUH*F-7$F[o$\"(i6:*F-7$F`o$\"(%*\\'))F-7$Feo$\"(OLK)F-7$Fj o$\"(uaM(F-7$F_p$\"(U#RoF-7$Fdp$\"(kO\"oF-7$Fip$\"(>*omF-7$F^q$\"(kR*f F-7$Fcq$\"(%f)p$F-7$Fhq$\"(U')Q\"F-7$F]r$\"((H)Q\"F-7$Fbr$\"(CqQ\"F-7$ Fgr$\"(2pO\"F-7$F\\s$\"([]>\"F-7$Fas$\"'\"F-7$Fgw$!)g[%F-7$F__l$!)nf)[%F-7$Fd_l$!)(R8 a%F-7$Fi_l$!)$)3wXF-7$F^`l$!)G2FYF-7$Fc`l$!)3R+ZF-7$Fh`l$!)n&R![F-7$F] al$!)k6X^F-7$Fbal$!)#=xw&F-7$Fgal$!)TDAoF-7$F\\bl$!)TExtF-7$Fabl$!)tHw tF-7$Ffbl$!)gLvtF-7$F[cl$!):zttF-7$F`cl$!)#4sP(F-7$Fecl$!)=x*Q(F-7$Fjc l$!)PSDuF-7$F_dl$!):yfuF-7$Fddl$!)p&>^(F-7$Fidl$!)hA*e(F-7$F^el$!)iB,x F-7$Fcel$!)Q7K!)F-7$Fhel$!)BzA')F-7$F]fl$!)lyJ'*F-7$Fbfl$!*C<)*3\"F-7$ Fgfl$!*z(o8tF-7$F/$\"(!=6tF-7$F4$\" (.&\\tF-7$F9$\"(Qnm(F-7$F>$\"('fiwF-7$FC$\"(bRl(F-7$FH$\"($pIvF-7$FM$ \"(ks9(F-7$FR$\"(aB9(F-7$FW$\"(Z*GrF-7$Ffn$\"(ik0(F-7$F[o$\"(sy%pF-7$F `o$\"(U2t'F-7$Feo$\"(e(>jF-7$Fjo$\"(0yd&F-7$F_p$\"()p$>&F-7$Fdp$\"($Gu ^F-7$Fip$\"(JW1&F-7$F^q$\"(5Ab%F-7$Fcq$\"(K.\"GF-7$Fhq$\"(?u0\"F-7$F]r $\"(er0\"F-7$Fbr$\"(\">c5F-7$Fgr$\"(?4/\"F-7$F\\s$\"'x/\"*F-7$Fas$\"'H ^tF-7$Ffs$\"'1fRF-7$F[t$!')*fAF-7$F`t$!(a`J\"F-7$Fet$!(&QYNF-7$Fjt$!(V 9>'F-7$F_u$!(%e!>'F-7$Fdu$!(I(*='F-7$Fiu$!($y)='F-7$F^v$!(2w>'F-7$Fcv$ !([\")H'F-7$Fhv$!(I0)oF-7$F]w$!(\"3[wF-7$Fbw$!(8,3*F-7$Fgw$!)LZi6F-7$F \\x$!)tv\"F-7$Fiz$!)x3 drzF-7$Ff]l $!)F![.#F-7$F[^l$!)ClWeF-7$Fcel$!)U\"e4'F-7$Fhel$!)![Ta'F-7$F] fl$!)Q(*4tF-7$Fbfl$!)\\7r#)F-7$Fgfl$!)H*)o#)F-7$F\\gl$!).%oE)F-7$Fagl$ !)g#yE)F-7$Ffgl$!)j)*)G)F-7$F[hl$!)%)>0$)F-7$F`hl$!)93I$)F-7$Fehl$!)(= sO)F-7$Fjhl$!)iO@%)F-7$F_il$!)8=2')F-7$Fdil$!)9bg*)F-7$Fiil$!)X8[&*F-7 $F^jl$!*35G0\"F-7$Fcjl$!*!)yc3\"F-7$Fhjl$!*%HJ66F-7$F][m$!*(yF66F-7$Fb [m$!*#HC66F-7$Fg[m$!*$H<66F-7$F\\\\m$!*$H566F-7$Fa\\m$!+cV\"36\"Fe\\m7 $Fg\\m$!+Swj56Fe\\m7$F\\]m$!+b8\"36\"Fe\\m7$Fa]m$!+U)e:6\"Fe\\m7$Ff]m$ !+LJg86Fe\\m7$F[^m$!+@`H=6Fe\\m7$F`^m$!+?hhG6Fe\\m7$Fe^m$!+*)H))[6Fe\\ m7$Fj^m$!+J)fh=\"Fe\\m7$F__m$!,uXU6D\"Fc_m7$Fe_m$!,yp#p'H\"Fc_m7$Fj_m$ !,731DM\"Fc_m7$F_`m$!,>M>CM\"Fc_m7$Fd`m$!,ULKBM\"Fc_m7$Fi`m$!+3\"e@M\" Fe\\m7$F^am$!+$*\\)>M\"Fe\\m7$Fcam$!+O*e;M\"Fe\\m7$Fham$!+\"\\j;M\"Fe \\m7$F]bm$!+WD9U8Fe\\m7$Fbbm$!+PyPV8Fe\\m7$Fgbm$!+M;$fM\"Fe\\m7$F\\cm$ !+`xc]8Fe\\m7$Facm$!*F#[n8F-7$Ffcm$!*\\@5R\"F-7$F[dm$!*7O1R\"F-7$F`dm$ !*KL-R\"F-7$Fedm$!*'Qd*Q\"F-7$Fjdm$!*#Gg'Q\"F-7$F_em$!*.#e%Q\"F-7$Fdem $!*\\::Q\"F-7$Fiem$!*ePpP\"F-7$F^fm$!*v8-P\"F-7$Fcfm$!*\"pkY8F-7$Fhfm$ !*qZ-I\"F-7$F]gm$!*An#=7F-7$Fbgm$!*tFZ2\"F-7$Fggm$!*+j20\"F-7$F\\hm$!* AH![5F-7$Fahm$!*w7![5F-7$Ffhm$!*.'*z/\"F-7$F[im$!*$G'z/\"F-7$F`im$!*$) Hz/\"F-7$Feim$!*vjy/\"F-7$Fjim$!*A(zZ5F-7$F_jm$!*P$\\Z5F-7$Fdjm$!*y$eY 5F--F^bs6&F\\[nF`bsF][nF][n-Fd[n6#%:scheme~with~c[10]=169/170G-%%FONTG 6$%*HELVETICAGF)-%&TITLEG6#%Verror~curves~for~11~stage~order~7~Runge-K utta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F^]w-%%VIEWG6$;F(Fdjm%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verne r scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10 ]=19/20" "scheme with c[10]=32/33" "scheme with c[10]=81/82" "scheme w ith c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "evalf[20](plot(['qn_RK7_1'(x)-q(x),'qn_RK7 _2'(x)-q(x),'qn_RK7_3'(x)-q(x),\n'qn_RK7_4'(x)-q(x),'qn_RK7_5'(x)-q(x) ,'qn_RK7_6'(x)-q(x)],x=0.925..0.9469483523,font=[HELVETICA,9],\ncolor= [COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR( RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`sch eme with c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/ 33`,`scheme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`er ror curves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 986 617 617 {PLOTDATA 2 "6,-%'CURVESG6%7cs7$$\"$D*!\" $$!)kSS()!#?7$$\"5#HT(y=;Tya#*F-$!)m^t%)F-7$$\"5x:8C5]n%*e#*F-$!)FFL m..t#*F-$!(Nj9#F-7$$\"55GIvH^&ytF*F-$\")hUNSF-7$$\"501=hrCT]y#*F-$\")e dOuF-7$$\"5+%eqM\")pH'z#*F-$\"*!G*F-$\"*RT\\f \"F-7$$\"5&>OH`:Fb2G*F-$\"*h^[f\"F-7$$\"5&4vei#e!=8G*F-$\"*&=w%f\"F-7$ $\"5%*R\")=(\\%3)=G*F-$\"*9sYf\"F-7$$\"5D>(=^U)*3UG*F-$\"*Y7Vf\"F-7$$ \"5g)H\\IN7PlG*F-$\"*EdVf\"F-7$$\"5>L9ZMp%y6H*F-$\"*\"3pF;F-7$$\"5S7Pm nmbc$H*F-$\"*&HAhJ*F-$\"*s-&pWF-7$$\"5D]9w\")>fA9$*F-$\"*X4y#\\F-7$$\"5!ph %H(R\"eI;$*F-$\"*Al)ogF-7$$\"5g$yFG\"3dQ=$*F-$\"*%Q*[-)F-7$$\"55#pj]M! *e3K*F-$\"*j5H-)F-7$$\"5e+'*Hx)4KLK*F-$\"*k#4@!)F-7$$\"5LK,#o/MAvK*F-$ \"*vQX.)F-7$$\"5S'p3&3k%f*H$*F-$\"*?A*>\")F-7$$\"5Wgs>q(e'RK$*F-$\"*4@ -Z)F-7$$\"50*ozz8AvML*F-$\"*(\\tB))F-7$$\"5l<@w0bQbM$*F-$\"*)R'pR*F-7$ $\"5DYXat)[KcL*F-$\"+^^oH5F-7$$\"5$[(pKTA6rO$*F-$\"+B$**p;\"F-7$$\"5:0 **3h')z2R$*F-$\"+v:#))>\"F-7$$\"5WNG&33&[WT$*F-$\"+l7a)>\"F-7$$\"5JEm+ nzC&fM*F-$\"+o>y)>\"F-7$$\"5+s8ze)4/$[$*F-$\"+$3MC?\"F-7$$\"5r\">]8Fjx7$$\"5z*QL>nUL'f$*F-$\",\"H$)\\79Fjx7$$\"5ge_;>uCZk$*F-$\",! >P]69Fjx7$$\"55y0mrw'ylO*F-$\",0hK$49Fjx7$$\"5c(*e:Cz[oo$*F-$\",x&4K+9 Fjx7$$\"5?a&obuDA)p$*F-$\",/3dwQ\"Fjx7$$\"5!3@\")pcjf4P*F-$\",x0*[j8Fj x7$$\"5SnQR)Q,(4s$*F-$\",A=g$>8Fjx7$$\"5/Cl!)4#RMKP*F-$\",Q%>*=C\"Fjx7 $$\"5&z2$>f.>#QP*F-$\"+=GV#=\"F-7$$\"5!>jz&3:%4WP*F-$\"+=G'\\5\"F-7$$ \"5&)3HF$3<.ZP*F-$\"+Qy0e5F-7$$\"5!e=mzl#p*\\P*F-$\"+u[AV5F-7$$\"5!GYf EBo!Hv$*F-$\"+^U>V5F-7$$\"5vRFN2QWev$*F-$\"+3O;V5F-7$$\"5gZe71h%fnP*F- $\"+?5/V5F-7$$\"5]b*)*[S[MzP*F-$\"+)*y\"H/\"F-7$$\"5qwLnbXNB!Q*F-$\"+3 #GE/\"F-7$$\"5)yzZkqgKDQ*F-$\"+Xi`T5F-7$$\"5lqg()Gnqv%Q*F-$\"+!ft\\.\" F-7$$\"5WVVI^F:)pQ*F-$\"+jX`.5F-7$$\"5lX)yn%)**)f(Q*F-$\"*[g\"R)*F-7$$ \"5&yM`A%pk@)Q*F-$\"*0f(f&*F-7$$\"55]ysPSR$))Q*F-$\"*%R*y;*F-7$$\"5I_B ?L69X*Q*F-$\"*^yfi)F-7$$\"5qc8:C`jo!R*F-$\"*>F$))oF-7$$\"57h.5:&H@>R*F -$\"*xB$*z$F-7$$\"5buSHQX4.$R*F-$!)8[B\")F-7$$\"5+)y([h&fSTR*F-$!*)ej \"*\\F-7$$\"5S,:o%eC]_R*F-$!*?#3\"*\\F-7$$\"5$[@vyg*)fjR*F-$!*%=`!*\\F -7$$\"508PT-t%H()R*F-$!*As**)\\F-7$$\"5D6A&p*\\!*4,%*F-$!*B/()QiC.%*F-$!*^/H5&F-7$$\"5gFC8xFMR0%*F-$!*Xg_f&F-7$$\"5lZ@\"z+:nl S*F-$!*R.0K'F-7$$\"5qn=pQs3u2%*F-$!*S6(>xF-7$$\"5q(er%p%f9*3%*F-$!+J6# y-\"F-7$$\"5v28D+<$)35%*F-$!+*)y)[Z\"F-7$$\"5D*=$QA)p#>6%*F-$!+A$>2<#F -7$$\"5vq]^WzqH7%*F-$!+@APrKF-7$$\"55T![]Z)R6Ol`78%*F-$!+[v-hPF-7$$\"5D_pkmg9S8%*F-$ !+7P#4w$F-7$$\"5+$*GrF^O&RT*F-$!+XgrgPF-7$$\"5xL)y()=%e]9%*F-$!+;%30w$ F-7$$\"5+q8j1NY\"oT*F-$!+BMmfPF-7$$\"5>1R[CGM7>%*F-$!+Y_efPF-7$$\"5!z# >ee&p!Q@%*F-$!+)>[U*F-$!+IMn*)QF-7$$\"5X><=@D.TD%*F-$!+\\ (z:&RF-7$$\"5vUc,k76+E%*F-$!+z[ePSF-7$$\"5N*[$o\\(o#=F%*F-$!+,eK9VF-7$ $\"5%fL^`BEk$G%*F-$!+PjB1[F-7$$\"5Ne(*RPfA]H%*F-$!+6hW/cF-7$$\"5!3=[%R c-kI%*F-$!+dh=\"*oF-7$$\"5+#Rs/\\D47V*F-$!+gUd)y(F-7$$\"5D.m\\T`#yDZBV*F-$!+:(f*Q&)F-7$$\"5nD]aV]i\"HV*F-$!+fQZ Q&)F-7$$\"5]iV92$yV_V*F-$!+Fc\\O&)F-7$$\"5H*pV2dJrvV*F-$!+Dr&\\`)F-7$$ \"50Ye!)>v&z)R%*F-$!+(Rl'Q&)F-7$$\"5x#*z')oMy=U%*F-$!+E8Xt&)F-7$$\"5X. U81g$[KW*F-$!+Unf?')F-7$$\"5:9/SV&))3VW*F-$!+!e]*3()F-7$$\"5![im13Tp`W *F-$!+Iw5l))F-7$$\"5]NG$zh$*HkW*F-$!+*)f3G\"*F-7$$\"5D/Em$zTXwW*F-$!+& QuIj*F-7$$\"5+tBRp**3')[%*F-$!,0\\uo/\"F-7$$\"5vT@7X\"Qw+X*F-$!,bUG(z6 F-7$$\"5Y5>&3K'=H^%*F-$!,;Ej$[7F-7$$\"50l*pbf9mMX*F-$!,e)=4[7F-7$$\"5n >!)GqG/kb%*F-$!,)[9#yC\"F-7$$\"50S4jgn'ezX*F-$!,eFBvC\"F-7$$\"5WgQ(4l! pFg%*F-$!,;:@pC\"F-7$$\"5&H*Q`sJjQh%*F-$!,)e]/Y7F-7$$\"5]DR4%pv&\\i%*F -$!-n3(*)RC\"Fjx7$$\"5!=%R([&p/0j%*F-$!-5f#G@C\"Fjx7$$\"50eRl:#=0OY*F- $!-/0kPR7Fjx7$$\"5IuRVw%*)fTY*F-$!-,b1PN7Fjx7$$\"5c!*R@P2Yrk%*F-$!-q\" )oiH7Fjx7$$\"5&H\\5zi$o!fY*F-$!-AO.;37Fjx7$$\"5I&*pg=l!*4n%*F-$!-(\\$Q rl6Fjx7$$\"5]Y_&R'z^pn%*F-$!-\\y!p=8\"Fjx7$$\"5l(\\.$4%H\"Ho%*F-$!-vSP )f3\"Fjx7$$\"5!)[$!(XyH&F-7$FC$!&?(>F-7$FH$\")rQ@ kF-7$FM$\"*\"Qtc6F-7$FR$\"*]B;)>F-7$FW$\"*5QcW#F-7$Ffn$\"*W+bW#F-7$F[o $\"*\"GOXCF-7$F`o$\"*CD_W#F-7$Feo$\"*JtYW#F-7$Fjo$\"*\")GZW#F-7$F_p$\" *[^[\\#F-7$Fdp$\"*\"*Gfp#F-7$Fip$\"*\"z$4S$F-7$F^q$\"*W\\`4&F-7$Fcq$\" *%RZOlF-7$Fhq$\"*qL\\`'F-7$F]r$\"*E9Q`'F-7$Fbr$\"*^y@e'F-7$Fgr$\"*\\U, y'F-7$F\\s$\"*D\"zouF-7$Fas$\"*<1R=*F-7$Ffs$\"+PuY77F-7$F[t$\"+$zn@@\" F-7$F`t$\"+&*G*=@\"F-7$Fet$\"+q6!R@\"F-7$Fjt$\"+;inE7F-7$F_u$\"+x72z7F -7$Fdu$\"+%*z#>L\"F-7$Fiu$\"+U$4wT\"F-7$F^v$\"+S!p?b\"F-7$Fcv$\"+d*frv \"F-7$Fhv$\"+:Zn/=F-7$F]w$\"+0FD/=F-7$Fbw$\"+9**f/=F-7$Fgw$\"+$>***4=F -7$F\\x$\"+ze!>$=F-7$Ffx$\",\"Q>%**)=Fjx7$Fay$\",*)*)ob-#Fjx7$Ffy$\",V d7c6#Fjx7$F[z$\",-12T6#Fjx7$F`z$\",H.a26#Fjx7$Fez$\",&)eln4#Fjx7$Fjz$ \",>vvq2#Fjx7$F_[l$\",Svm%R?Fjx7$Fd[l$\",y'ouq>Fjx7$Fi[l$\",(pU.]=Fjx7 $F^\\l$\"+waMdoPR:F-7$F[_l$\"+MftP:F-7$F`_l$\"+\"H8y_\"F-7$Fe_l$ \"+#4_,[\"F-7$Fj_l$\"+=tP]9F-7$F_`l$\"+:,&zS\"F-7$Fd`l$\"+BWT[8F-7$Fi` l$\"+[./m7F-7$F^al$\"+lnq,5F-7$Fcal$\"*n$p8`F-7$Fhal$!*^lRr\"F-7$F]bl$ !*#ec'3)F-7$Fbbl$!*voc3)F-7$Fgbl$!*xwZ3)F-7$F\\cl$!*t'z$3)F-7$Facl$!*# fs-\")F-7$Ffcl$!*!zh`#)F-7$F[dl$!*#f)f**)F-7$F`dl$!+Or/45F-7$Fedl$!+2o L?7F-7$Fjdl$!+Q;\"pg\"F-7$F_el$!+C^x#G#F-7$Fdel$!+sUCNLF-7$Fiel$!+Ewz+ ]F-7$F^fl$!+`dfWbF-7$Fcfl$!++C8UdF-7$Fhfl$!+pQ(>u&F-7$F]gl$!+T`\"=u&F- 7$Fbgl$!+&H)\\TdF-7$Fggl$!+38=TdF-7$F\\hl$!+?1*)RdF-7$Fahl$!+f.vRdF-7$ Ffhl$!+&>%)3v&F-7$F[il$!+caWBeF-7$F`il$!+^iApeF-7$Feil$!+;.2NfF-7$Fjil $!+(3h!GgF-7$F_jl$!+b;EdhF-7$Fdjl$!+%RMId'F-7$Fijl$!+cv87tF-7$F^[m$!+z w]6&)F-7$Fc[m$!,)*)H\\W5F-7$Fh[m$!,:)*G$z6F-7$F]\\m$!,3SZ@H\"F-7$Fb\\m $!,+)Q2#H\"F-7$Fg\\m$!,IO+?H\"F-7$F\\]m$!,3,,(H\"F-7$F`^m$!,$fU@/8F-7$Fe^m$!,yJ^tJ\" F-7$Fj^m$!,1\\V0M\"F-7$F__m$!,@E\\&z8F-7$Fd_m$!,pFGVX\"F-7$Fi_m$!,HXIy d\"F-7$F^`m$!,,aFPx\"F-7$Fc`m$!,+gfZ(=F-7$Fh`m$!,Q0_V(=F-7$F]am$!,AvXR (=F-7$Fbam$!,F9$\\t=F-7$Fgam$!,8,MD(=F-7$F\\bm$!,m$o6r=F-7$Fabm$!-yZcz n=Fjx7$Ffbm$!-p?Uzk=Fjx7$F[cm$!-QYWOg=Fjx7$F`cm$!-ksk#R&=Fjx7$Fecm$!-h g(3Z%=Fjx7$Fjcm$!-&)oFL5=Fjx7$F_dm$!-WOP^U$!((4`KF-7$FC$\" 'Av(*F-7$FH$\")h6^_F-7$FM$\")1F\"Q*F-7$FR$\"*_B.g\"F-7$FW$\"*frG(>F-7$ Ffn$\"*bgF(>F-7$F[o$\"*_\\E(>F-7$F`o$\"*aQD(>F-7$Feo$\"*@$4s>F-7$Fjo$ \"*5M@(>F-7$F_p$\"*FtA,#F-7$Fdp$\"*(fPt@F-7$Fip$\"*w&QQFF-7$F^q$\"*%)G m4%F-7$Fcq$\"*A')>D&F-7$Fhq$\"*i[2D&F-7$F]r$\"*vZ)\\_F-7$Fbr$\"*$pa)G& F-7$Fgr$\"*()**pW&F-7$F\\s$\"*'GB)*fF-7$Fas$\"*(f4rtF-7$Ffs$\"*x@\\s*F -7$F[t$\"*S=Ds*F-7$F`t$\"*#HJ?(*F-7$Fet$\"*7ejt*F-7$Fjt$\"*7o%Q)*F-7$F _u$\"+Q@rD5F-7$Fdu$\"+EO$z1\"F-7$Fiu$\"+-!ej8\"F-7$F^v$\"+TgqV7F-7$Fcv $\"+=>R29F-7$Fhv$\"+=dIX9F-7$F]w$\"+Kx'\\W\"F-7$Fbw$\"+t=CX9F-7$Fgw$\" +t\"Q&\\9F-7$F\\x$\"+@i$pY\"F-7$Ffx$\",T]**G^\"Fjx7$Fay$\",%Ho!*>;Fjx7 $Ffy$\",Z/82p\"Fjx7$F[z$\",([^]*o\"Fjx7$F`z$\",\"GTz'o\"Fjx7$Fez$\",7_ ean\"Fjx7$Fjz$\",))y)[f;Fjx7$F_[l$\",1Ip*G;Fjx7$Fd[l$\",HIxJd\"Fjx7$Fi [l$\",`C)*R\"F-7$Fc\\l$\"+T#e;I\"F-7$Fh\\l$\"+v? ?U7F-7$F]]l$\"+#G*RB7F-7$Fb]l$\"+nLOB7F-7$Fg]l$\"+JuKB7F-7$F\\^l$\"+sO =B7F-7$Fa^l$\"+n#RIA\"F-7$Ff^l$\"+=ypA7F-7$F[_l$\"+EXQ@7F-7$F`_l$\"+IH U87F-7$Fe_l$\"+1=9v6F-7$Fj_l$\"+Ll@^6F-7$F_`l$\"+ft6<6F-7$Fd`l$\"+*>e# p5F-7$Fi`l$\"+/o-.5F-7$F^al$\"*&GI/zF-7$Fcal$\"*'*=,7%F-7$Fhal$!*Dwf` \"F-7$F]bl$!*:'3mmF-7$Fbbl$!*nY`m'F-7$Fgbl$!*D6Ym'F-7$F\\cl$!*;\"yjmF- 7$Facl$!*!z*)ymF-7$Ffcl$!*Tl(*z'F-7$F[dl$!*\"Hz%R(F-7$F`dl$!*)eHs#)F-7 $Fedl$!*Q=m'**F-7$Fjdl$!+/Zq18F-7$F_el$!+aD%)[=F-7$Fdel$!+S\\<$p#F-7$F iel$!+gl[HSF-7$F^fl$!+$37eY%F-7$Fcfl$!+Z3JCYF-7$Fhfl$!+xJ=CYF-7$F]gl$! +5b0CYF-7$Fbgl$!+$=+Qi%F-7$Fggl$!+.\\aBYF-7$F\\hl$!+#H0Di%F-7$Fahl$!+n iQAYF-7$Ffhl$!+**)*GJYF-7$F[il$!+yMN*o%F-7$F`il$!+\"))))fs%F-7$Feil$!+ i\"z'yZF-7$Fjil$!+)\\\"4`[F-7$F_jl$!+4iZc\\F-7$Fdjl$!+&**\\\"*G&F-7$Fi jl$!+-JY!)eF-7$F^[m$!+\"3o)RoF-7$Fc[m$!+oJ='Q)F-7$Fh[m$!+A]Tk%*F-7$F] \\m$!,zL\\m.\"F-7$Fb\\m$!,XN!fO5F-7$Fg\\m$!,SPJl.\"F-7$F\\]m$!,'47HO5F -7$Fa]m$!,Xd.h.\"F-7$Ff]m$!,L_Tl.\"F-7$F[^m$!,3Wu1/\"F-7$F`^m$!,+\")oi /\"F-7$Fe^m$!,j5Pn0\"F-7$Fj^m$!,o*[?v5F-7$F__m$!,-kRi5\"F-7$Fd_m$!,m=w c;\"F-7$Fi_m$!,D0;PE\"F-7$F^`m$!,^!y**=9F-7$Fc`m$!,zV%**)\\\"F-7$Fh`m$ !,udo')\\\"F-7$F]am$!,#oOM)\\\"F-7$Fbam$!,zO!)z\\\"F-7$Fgam$!,!pj>(\\ \"F-7$F\\bm$!,'e0.'\\\"F-7$Fabm$!-v5yH$\\\"Fjx7$Ffbm$!-gc\"H3\\\"Fjx7$ F[cm$!-k>q=([\"Fjx7$F`cm$!-l_c*=[\"Fjx7$Fecm$!-4a@Ku9Fjx7$Fjcm$!-j,:4Y 9Fjx7$F_dm$!-;@IU!R\"Fjx7$Fddm$!-c>h5Y8Fjx7$Fidm$!-&R%*zgG\"Fjx7$F^em$ !.Z`mNc?\"Fbem7$Fdem$!5A\"[.,5q!*))4\"Fiem-F[fm6&F]fm$\"\"*Fbfm$\"\"%F bfmF^fm-Fffm6#%8scheme~with~c[10]=19/20G-F$6%7as7$F($!*&))3f8F-7$F/$!* j&o:8F-7$F4$!)E_`))F-7$F9$!([Qx\"F-7$F>$!(gvn\"F-7$FC$\"(l5(RF-7$FH$\" )3^\"H(F-7$FM$\"**y;#G\"F-7$FR$\"*DA#p@F-7$FW$\"*V>%oEF-7$Ffn$\"*Cp#oE F-7$F[o$\"*3>\"oEF-7$F`o$\"*(*ozm#F-7$Feo$\"*Imtm#F-7$Fjo$\"*$4TnEF-7$ F_p$\"*g**3s#F-7$Fdp$\"*X@e$HF-7$Fip$\"*C@,p$F-7$F^q$\"*&\\\\/bF-7$Fcq $\"**eT[qF-7$Fhq$\"*#\\vYqF-7$F]r$\"*tTb/(F-7$Fbr$\"*+>q4(F-7$Fgr$\"*i wzI(F-7$F\\s$\"*H9@/)F-7$Fas$\"*,W4()*F-7$Ffs$\"+TJq+8F-7$F[t$\"+)o\"Q +8F-7$F`t$\"+El3+8F-7$Fet$\"+!\\8AI\"F-7$Fjt$\"+luw:8F-7$F_u$\"+')HKr8 F-7$Fdu$\"+RKLF9F-7$Fiu$\"+t*y!=:F-7$F^v$\"+-=Sg;F-7$Fcv$\"+E@Mx=F-7$F hv$\"+(pxv#>F-7$F]w$\"+**o7F>F-7$Fbw$\"+2,[F>F-7$Fgw$\"+%fGJ$>F-7$F\\x $\"+&yHf&>F-7$Ffx$\",@Y%)e,#Fjx7$Fay$\",XM#ea@Fjx7$Ffy$\",)yb%eC#Fjx7$ F[z$\",G)yAWAFjx7$F`z$\",)3=aSAFjx7$Fez$\",nnq]A#Fjx7$Fjz$\",dqZK?#Fjx 7$F_[l$\",XR9:;#Fjx7$Fd[l$\",(G/=&3#Fjx7$Fi[l$\",,rp4&>Fjx7$F^\\l$\"+B ?&y%=F-7$Fc\\l$\"+1TS8j\"F-7$F]]l$\"+*y.ig\"F-7$Fb]l $\"+Gm:1;F-7$Fg]l$\"+Z%4hg\"F-7$F\\^l$\"+32#fg\"F-7$Fa^l$\"+06t0;F-7$F f^l$\"+6:G0;F-7$F[_l$\"+hC`.;F-7$F`_l$\"+ax(Gf\"F-7$Fe_l$\"+SCaT:F-7$F j_l$\"+u1V4:F-7$F_`l$\"+K[kj9F-7$Fd`l$\"+HuN*R\"F-7$Fi`l$\"+koN58F-7$F ^al$\"+#41X-\"F-7$Fcal$\"*R]L:&F-7$Fhal$!*B9>Y#F-7$F]bl$!*'>>s$*F-7$Fb bl$!*C_6P*F-7$Fgbl$!*(z6q$*F-7$F\\cl$!*;&*)o$*F-7$Facl$!*NJ*)Q*F-7$Ffc l$!**)3,b*F-7$F[dl$!+B4ZM5F-7$F`dl$!+]Ht^6F-7$Fedl$!+4_Dy8F-7$Fjdl$!+B )\\Hz\"F-7$F_el$!+f?Q=DF-7$Fdel$!+Byg[OF-7$Fiel$!+FN-QaF-7$F^fl$!+F,TA gF-7$Fcfl$!+HSqMiF-7$Fhfl$!+)*=`MiF-7$F]gl$!+r(fVB'F-7$Fbgl$!+Gb,MiF-7 $Fggl$!+[8nLiF-7$F\\hl$!+!=pAB'F-7$Fahl$!+9;4KiF-7$Ffhl$!+:>#RC'F-7$F[ il$!+aq>@jF-7$F`il$!+Cr'*pjF-7$Feil$!+#z:,W'F-7$Fjil$!+/0>RlF-7$F_jl$! +Hr%on'F-7$Fdjl$!+ww#)>rF-7$Fijl$!+:qB2zF-7$F^[m$!+W-\"[=*F-7$Fc[m$!,. H'QC6F-7$Fh[m$!,t\"e&zE\"F-7$F]\\m$!,Zpl!)Q\"F-7$Fb\\m$!,jr')zQ\"F-7$F g\\m$!,@u2zQ\"F-7$F\\]m$!,6:'e(Q\"F-7$Fa]m$!,]BMtQ\"F-7$Ff]m$!,\"[7\"z Q\"F-7$F[^m$!,(GoP$R\"F-7$F`^m$!,b[q2S\"F-7$Fe^m$!,'zXf99F-7$Fj^m$!,t( )e*Q9F-7$F__m$!,YNc)z9F-7$Fd_m$!,f'42e:F-7$Fi_m$!,Y8joo\"F-7$F^`m$!,?A Y/*=F-7$Fc`m$!,$y'o^*>F-7$Fh`m$!,b&\\t%*>F-7$F]am$!,DO-V*>F-7$Fbam$!,x y9Q*>F-7$Fgam$!,+?DF*>F-7$F\\bm$!,P)z3\"*>F-7$Fabm$!-kyDD()>Fjx7$Ffbm$ !-))RDz$)>Fjx7$F[cm$!-9%>$py>Fjx7$F`cm$!--oBHr>Fjx7$Fecm$!-]8(42'>Fjx7 $Fjcm$!-HjsJ@>Fjx7$F_dm$!-d$4`P%=Fjx7$Fddm$!-R-(e?y\"Fjx7$Fidm$!-r6(Q& )p\"Fjx7$F^em$!.\"G?5m'e\"Fbem7$Fdem$!5A\"=LX`*pEQ9Fiem-%'COLOURG6&F]f m$\"*++++\"!\")F^fmFbcr-Fffm6#%8scheme~with~c[10]=32/33G-F$6%7as7$F($! *8_&y8F-7$F/$!*\"*)pM8F-7$F4$!)H!=+*F-7$F9$!(TYW#F-7$F>$!(TpM#F-7$FC$ \"($4kLF-7$FH$\")mj.tF-7$FM$\"*)z5*G\"F-7$FR$\"*(QD&=#F-7$FW$\"*'=`*o# F-7$Ffn$\"*[!Q*o#F-7$F[o$\"*8H#*o#F-7$F`o$\"*$y2*o#F-7$Feo$\"*Zq%)o#F- 7$Fjo$\"*,=&)o#F-7$F_p$\"*OAEu#F-7$Fdp$\"*Vh*fHF-7$Fip$\"**)>Es$F-7$F^ q$\"*$)\\ob&F-7$Fcq$\"*PRv6(F-7$Fhq$\"*8ie6(F-7$F]r$\"*.QY6(F-7$Fbr$\" *\")Hn;(F-7$Fgr$\"*2k,Q(F-7$F\\s$\"*SsG7)F-7$Fas$\"*#>)H(**F-7$Ffs$\"+ C-b98F-7$F[t$\"+\\`A98F-7$F`t$\"+(3FRJ\"F-7$Fet$\"+w63;8F-7$Fjt$\"+iS! )H8F-7$F_u$\"+kw0'Q\"F-7$Fdu$\"+LvxU9F-7$Fiu$\"+A4oM:F-7$F^v$\"+CN$)y; F-7$Fcv$\"+9re)*=F-7$Fhv$\"+D#y%\\>F-7$F]w$\"+8B-\\>F-7$Fbw$\"+F -7$Fgw$\"+2X6b>F-7$F\\x$\"+mHFy>F-7$Ffx$\",#oQCR?Fjx7$Fay$\",p,N0=#Fjx 7$Ffy$\",sdYOF#Fjx7$F[z$\",e17?F#Fjx7$F`z$\",LH,$oAFjx7$Fez$\",*[;u_AF jx7$Fjz$\",\"z8!3B#Fjx7$F_[l$\",e&>&))=#Fjx7$Fd[l$\",ybN@6#Fjx7$Fi[l$ \",#fFFx>Fjx7$F^\\l$\"+kYmt=F-7$Fc\\l$\"+aneQd'\\'F-7$Fjil$!+)H8of'F-7$F_jl$!+J.4OnF-7$Fdjl$!+mgG%=(F-7$Fijl $!+Qb(4)zF-7$F^[m$!+qljt#*F-7$Fc[m$!,C%yqN6F-7$Fh[m$!,+c#)4G\"F-7$F]\\ m$!,lk@DS\"F-7$Fb\\m$!,c%=W-9F-7$Fg\\m$!,*[?O-9F-7$F\\]m$!,:6P?S\"F-7$ Fa]m$!,ys#y,9F-7$Ff]m$!,*fzO-9F-7$F[^m$!,T'o!zS\"F-7$F`^m$!,?;,aT\"F-7 $Fe^m$!,im;%H9F-7$Fj^m$!,v?DTX\"F-7$F__m$!,3,9c\\\"F-7$Fd_m$!,*=>*\\d \"F-7$Fi_m$!,t&\\w0F-7$Fc`m$!,YFT!>?F-7$Fh`m$!,EO- '=?F-7$F]am$!,dik\"=?F-7$Fbam$!,D>sw,#F-7$Fgam$!,L*4e;?F-7$F\\bm$!,UiX \\,#F-7$Fabm$!-lkV66?Fjx7$Ffbm$!--arl2?Fjx7$F[cm$!-By3c-?Fjx7$F`cm$!-Q dH;&*>Fjx7$Fecm$!-v#R#e%)>Fjx7$Fjcm$!-iLl=X>Fjx7$F_dm$!-&3o#fn=Fjx7$Fd dm$!-@mT'e!=Fjx7$Fidm$!-\\b&*GAMK\"=Y\"Fiem-F[fm6&F]fm$\"\"&FbfmF^fmFcfm-Fffm6#%8scheme~with~c[10] =81/82G-F$6%7as7$F($!*y$eY5F-7$F/$!*'eI85F-7$F4$!)\\(f$oF-7$F9$!(g8\"> F-7$F>$!(ur$=F-7$FC$\"(?s\\#F-7$FH$\")y'o`&F-7$FM$\")#Qnx*F-7$FR$\"*!3 nd;F-7$FW$\"*A6./#F-7$Ffn$\"*P'>S?F-7$F[o$\"*b\"3S?F-7$F`o$\"*ym*R?F-7 $Feo$\"*01&R?F-7$Fjo$\"*XU&R?F-7$F_p$\"*4413#F-7$Fdp$\"*')ebC#F-7$Fip$ \"*IKV#GF-7$F^q$\"*Z,i@%F-7$Fcq$\"*+P/S&F-7$Fhq$\"*Qk\"*R&F-7$F]r$\"*z N#)R&F-7$Fbr$\"*oyxV&F-7$Fgr$\"*jx(*f&F-7$F\\s$\"*zYM;'F-7$Fas$\"*\"3U nvF-7$Ffs$\"*!Rou**F-7$F[t$\"*!)=A(**F-7$F`t$\"*sb*p**F-7$Fet$\"*M6j)* *F-7$Fjt$\"+Y%[!45F-7$F_u$\"+9ju^5F-7$Fdu$\"+**Rz%4\"F-7$Fiu$\"+!**QX; \"F-7$F^v$\"+`j#RF\"F-7$Fcv$\"+XimS9F-7$Fhv$\"+d!y#z9F-7$F]w$\"+4@$*y9 F-7$Fbw$\"+Xh?z9F-7$Fgw$\"+7%fN[\"F-7$F\\x$\"+BU9,:F-7$Ffx$\",ebIua\"F jx7$Fay$\",!\\Vla;Fjx7$Ffy$\",I#=HD0C8F-7$Fh\\l$\"+]NHd7F-7$F]]l$\"+-%fwB\"F-7$Fb]l$\"+oIiP7F-7$Fg]l$\"+8 neP7F-7$F\\^l$\"+y7WP7F-7$Fa^l$\"+$=&HP7F-7$Ff^l$\"+(**[pB\"F-7$F[_l$ \"+ChgN7F-7$F`_l$\"+biVF7F-7$Fe_l$\"+Nd4)=\"F-7$Fj_l$\"+%3$\\j6F-7$F_` l$\"+O!=%G6F-7$Fd`l$\"+a]%yF-7$F`dl$!*d59u)F-7$Fedl$!+sy'y/\"F-7$Fjdl$!+E%feO\"F-7$F_el$!+oF-7$Fdel$!+m&\\%)y#F-7$Fiel$!+It)*fTF-7$F^fl$!+&*o'yg%F-7$Fcfl$!+Juc qZF-7$Fhfl$!+BdVqZF-7$F]gl$!+>SIqZF-7$Fbgl$!+<1/qZF-7$Fggl$!+isxpZF-7$ F\\hl$!+\"\\/(oZF-7$Fahl$!+OCdoZF-7$Ffhl$!+cFmxZF-7$F[il$!+:w+P[F-7$F` il$!+XnXu[F-7$Feil$!+i!>$G\\F-7$Fjil$!+[zQ/]F-7$F_jl$!+\"yu+6&F-7$Fdjl $!+u=:]aF-7$Fijl$!+BWfagF-7$F^[m$!+c.ANqF-7$Fc[m$!+>Sb:')F-7$Fh[m$!+D4 R<(*F-7$F]\\m$!,tA:R1\"F-7$Fb\\m$!,DpaQ1\"F-7$Fg\\m$!,2;%zj5F-7$F\\]m$ !,KnZN1\"F-7$Fa]m$!,cvaL1\"F-7$Ff]m$!,RH*zj5F-7$F[^m$!,XD/!o5F-7$F`^m$ !,)fGpt5F-7$Fe^m$!,N@IV3\"F-7$Fj^m$!,fx!3.6F-7$F__m$!,5YgX8\"F-7$Fd_m$ !,^ZwZ>\"F-7$Fi_m$!,8sbRH\"F-7$F^`m$!,(HIx]9F-7$Fc`m$!,c7c9`\"F-7$Fh`m $!,z?B6`\"F-7$F]am$!,**=\"zI:F-7$Fbam$!,&QyTI:F-7$Fgam$!,$R8fH:F-7$F\\ bm$!,\"HCNG:F-7$Fabm$!-bT)[a_\"Fjx7$Ffbm$!-TX!GG_\"Fjx7$F[cm$!-9jS'*=: Fjx7$F`cm$!-B#3aL^\"Fjx7$Fecm$!-Mb%H`]\"Fjx7$Fjcm$!-vKPWv9Fjx7$F_dm$!- >Ogc;9Fjx7$Fddm$!-:R#>(p8Fjx7$Fidm$!-0ljG18Fjx7$F^em$!.Rd-+8A\"Fbem7$F dem$!5A\"ecBbAb&36Fiem-F`cr6&F]fmFbcrF^fmF^fm-Fffm6#%:scheme~with~c[10 ]=169/170G-%%FONTG6$%*HELVETICAGFgip-%&TITLEG6#%Verror~curves~for~11~s tage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fdgu-%%VIEWG 6$;F(Fdem%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10 ]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme w ith c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Test 12 of 11 stage, order 7 Runge-Kutt a methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(- x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$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-%$cosGF+F,F+%\"xGF+F+**F1F+\"#NF+F2F+\"#kF-F-*,F/F+F5F-F1F+\"# @F+F2F+F+*,\"#DF+F5F-F1F+\"\"(F+F2F+F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "de := \+ diff(y(x),x)=5*y(x)*sin(7*x)^7;\nic := y(0)=1;\ndsolve(\{de,ic\},y(x)) :\ny(x)=combine((numer(rhs(%))/convert(denom(rhs(%)),exp)));\nr := una pply(rhs(%),x):\nplot(r(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x) `]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,,$*(\"\"&\"\"\"F)F0)-%$sinG6#,$*&\"\"(F0F,F0F0F7F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#D\"#k\"\"\"-%$cosG6#,$*& \"\"(F0F'F0F0F0!\"\"*&#F0F/F0-F26#,$*&\"#NF0F'F0F0F0F7*&#\"\"&\"%OJF0- F26#,$*&\"#\\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&H SekZ\"F27$$\"3o;/,>=0QCFJ$\"3mjV0*4cU_\"F27$$\"3c3FpwUq/DFJ$\"3#pc5I%e Tr:F27$$\"3!***\\PMnNrDFJ$\"3U+AysC:<;F27$$\"37$eR(\\;m/FFJ$\"3)f:eA`w 9q\"F27$$\"3MmT5ll'z$GFJ$\"35TSzD!3Dx\"F27$$\"37](o/[r7(HFJ$\"3e:[6>4v F=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$$\"3Um;HKp%zh&FJ$\"3!*H'3:A\\o!>F2 7$$\"32++v$flV;F27$ $\"3I++vVVX$\\'FJ$\"3s/S7/D%eb\"F27$$\"3pLL$eRh;i'FJ$\"3)z%[YI??k9F27$ $\"31nm\"zWo)\\nFJ$\"31\")Q9\"48QP\"F27$$\"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<5ljQ 5F27$$\"3Qnm\"HKkIz(FJ$\"3z[=lFuU55F27$$\"35MLeRilDzFJ$\"3tR>*>I/Y+\"F 27$$\"3!3+]i:[#e!)FJ$\"3ZybuD9w,5F27$$\"3[nm\"H2S3>)FJ$\"36f+OZ/c+5F27 $$\"3>MLe*)>VB$)FJ$\"3q[OmV#R,+\"F27$$\"3wmmTg()4_))FJ$\"3!z[$G++++5F2 7$$\"3Y++DJbw!Q*FJ$\"3(f!Q*>W.++\"F27$$\"3E$3_+A:n^*FJ$\"3$o[9!4H.+5F2 7$$\"3=nT&)3\\m_'*FJ$\"3qlL!>[$=+5F27$$\"33^il(f9')y*FJ$\"39t!)=KMs+5F 27$$\"3+N$ekGkX#**FJ$\"3)[%4Vj`B-5F27$$\"31]iSmjk>5F2$\"3?UoC!*)fG,\"F 27$$\"3%ommTIOo/\"F2$\"3%pJ)[*fok/\"F27$$\"3?]i:g2\")e5F2$\"3'ob!*>J( \\t5F27$$\"3cLe9;_yq5F2$\"3!f)3IJ$>-6\"F27$$\"3!pTN@nfF3\"F2$\"3O\\W6v rpd6F27$$\"3E+]7GTt%4\"F2$\"3&p\"fsjqH;7F27$$\"3i$e9Te3n5\"F2$\"3e&f$* [!\\V&G\"F27$$\"3(p;/,/$o=6F2$\"3K1G\">I$Qj8F27$$\"3L]P4'\\d18\"F2$\"3 mAQZ9DAZ9F27$$\"3YLL3_>jU6F2$\"3A//J![AI`\"F27$$\"3)\\PMF:s$\\6F2$\"3% **=P`q)\\!e\"F27$$\"3u;aQ`B6c6F2$\"3vn0oImOE;F27$$\"3[ek.aD&G;\"F2$\"3 q#GT!4W()p;F27$$\"3C+voaFfp6F2$\"3Hrm)p6m.r\"F27$$\"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$\"3 4bF!Hpw)>>F27$$\"37++vomax7F2$\"3s-.5q=,@>F27$$\"35+](=U#3\"H\"F2$\"3E fS@r$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$$\"3V$ek y#)*QU;F2$\"3k!o/)p1f`5F27$$\"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\"pm;a)))G=F2$\"3:l7r\"p]]+\"F27$ $\"3%\\ilAZ9R\">F2$\"3^hCaoQo55F27$$\"3KLe9;0?E>F2$\"3:4Zh8k[?5F27$$\" 31]i!Rgs2&>F2$\"3wO#Q8N_'f5F27$$\"3gmmm\"pW`(>F2$\"3,$4k)\\,ZO6F27$$\" 3_ek.HW#)))>F2$\"3Q;r;:R*z>\"F27$$\"3?]iSmTI-?F2$\"3_fyny')>t7F27$$\"3 EY64NS/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$$\"3dLe9TOEH?F2$\"3%\\u#Gw;Va9F27$$\"3kH 2$)4N+O?F2$\"3\\;?B!3#z-:F27$$\"3EDc^yLuU?F2$\"37f,s,Z'3b\"F27$$\"3)3_ +sC$[\\?F2$\"3C:ye37$yf\"F27$$\"3'pT&)e6Bi0#F2$\"3VLP%*y0!Hk\"F27$$\"3 k3_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$$\"3?+](=xpe=#F2$\"3.\"p)[NLI@>F27$$\"3mLeR A9WRAF2$\"3HST.9PT@>F27$$\"37nm\"H28IH#F2$\"3`7Ny#=%Q@>F27$$\"3=Ug_Z>J 0BF2$\"3F27$$\"3!oTN@#3hF27$$\"3(=zWnp4*HB F2$\"3aFU&R3A$>>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??JCF 2$\"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()Q DF2$\"3A!)Gqgm(f0\"F27$$\"3AL3_D1l_DF2$\"3/w0[YW]J5F27$$\"3I3-))o-VmDF 2$\"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#3 Fp,aRzy#F2$\"3V/:%[T%G-5F27$$\"3k\"H2L7ST$GF2$\"3>]6Z^*QK.\"F27$$\"3)om\"H2\"34'GF2$\"3c89;GxR$4\"F27$$\"3=] (=<1#HuGF2$\"3H$eWMp\"*>9\"F27$$\"3YLe9;gn()GF2$\"3LgFbL%HW?\"F27$$\"3 w;Hdq*f5!HF2$\"337;\"\\lV-G\"F27$$\"30+++DRW9HF2$\"3/%p0(oZBn8F27$$\"3 -D19>2*4#HF2$\"3.aKDn)*p79F27$$\"3X]7G8v`FHF2$\"39c[>!pr#f9F27$$\"3Uv= U2V3MHF2$\"3FI^JzcD1:F27$$\"3S+Dc,6jSHF2$\"3)fI*>\"*G\"Hb\"F27$$\"3O]P %)*oCP&HF2$\"3kq@\"z(y\"F27$$\"3C+voaa+$*HF2$\"3] (oH/*41IF2$\"3/1YYU*QP(=F27$$\"3:++DJE>>IF2$\"3w5uUgAb'*=F27$$\"3Fv=<^ 8'=.$F2$\"3=[IQQI\\4>F27$$\"3S]P4r+`WIF2$\"3bz\"Rx1kj\">F27$$\"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$$\"3+++v=S2LKF2$\"39dX:D?6=>F27$$\"3Om\"zpo_$ eKF2$\"3#zYTO,TK!>F27$$\"3;L$3_NJOG$F2$\"3=J`$Q^]x&=F27$$\"3y;HK*oqiH$ F2$\"3h#pBbd4n\"=F27$$\"3'**\\PM-5*3LF2$\"3uG&fHPn7w\"F27$$\"39$3_vN\\ :K$F2$\"3Y(o1g2R=p\"F27$$\"3Jmmm\"p)=MLF2$\"3896'o\"f'3h\"F27$$\"3&)\\ i:&GO4M$F2$\"3=*)4prhIk:F27$$\"3%H$ekyQoZLF2$\"3IceZ)o3k^\"F27$$\"3/;a 8s9VaLF2$\"3G6b_$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$$\"3GLLeR%p\")Q$F2$\"3F&[%Q!RZWC\"F 27$$\"3!**\\ilik;S$F2$\"3_Z%zlm()R<\"F27$$\"3)pmTN\")f^T$F2$\"331)eDsV v6\"F27$$\"3hL3_+]lGMF2$\"3CD\\786*[2\"F27$$\"3B++](=]@W$F2$\"3.\">ZcV 5Y/\"F27$$\"3C$ekyZ2mY$F2$\"3bMcTbf795F27$$\"3mm\"H#oZ1\"\\$F2$\"38K^# 3O0J+\"F27$$\"35+5F27$$\"35L$e*[$z*RNF2$\"3:J>88\">++\" F27$$\"3%o;Hd!fX$f$F2$\"3+/++++++5F27$$\"3e++]iC$pk$F2$\"3>3i#3WY++\"F 27$$\"3)*e*)f!y6&fOF2$\"3/sYmoa@+5F27$$\"3%p\"zp)4\"4sOF2$\"3/b\">zh_2 +\"F27$$\"3!\\(oz;/n%o$F2$\"3FA+oZ!R@+\"F27$$\"3ILe*[t\\sp$F2$\"3]TEK8 J>05F27$$\"36]P4r$3Cs$F2$\"3cyY&>9K:-\"F27$$\"3[m;H2qcZPF2$\"3yA^kd)oL 1\"F27$$\"3KL3F>grgPF2$\"3kx1i!p\"\\+6F27$$\"3s***\\7.lQx$F2$\"3WMY`\\ )G.:\"F27$$\"3em\"HK/9qy$F2$\"3j!GcKQeN@\"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$$\"36nm;z5YEQF2$\"3%GU A'*pv(o9F27$$\"3J]i:&eNI$QF2$\"3eJ%\\YU_f^\"F27$$\"3^Le9\"45'RQF2$\"3] [!)RoHki:F27$$\"3s;a8(f%=YQF2$\"3Zk-ND-43;F27$$\"3O+]7.\"fF&QF2$\"3]5U W.;d^;F27$$\"3i3_vlYhlQF2$\"3S)o\\Z'>`GF27$$\"3Ymm;/OgbRF2$\"3e5rkQQ q>>F27$$\"3*G$e*[$zV4SF2$\"3^Ar&3889#>F27$$\"3w**\\ilAFjSF2$\"3Q')**40 OT@>F27$$\"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-([etep\">F27$$\"3t]i!RbX59%F2$\"38M:lS7 J9>F27$$\"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$$\"3\"R3Fpm[KB%F2$\"3x%eAK2 J4g\"F27$$\"3OfkGNl()RUF2$\"3/aZW5%e[b\"F27$$\"3#R$ek.W]YUF2$\"3_pHqN- l2:F27$$\"3]3_+sA8`UF2$\"3?puckU3g9F27$$\"3'Rek.9g(fUF2$\"30z2N6y#HT\" F27$$\"3SfRs3!)QmUF2$\"3E%zqA:-pO\"F27$$\"3)RL$3xe,tUF2$\"3Hzu(eD`EK\" F27$$\"3kv$4'\\=;'G%F2$\"32(*p'*eu>U7F27$$\"3G#[Z%F2$\"3S)\\h/++ ++\"F27$$\"3TM$3_5,-`%F2$\"3y/++\"F27$$\"3SnmT&G!e&e%F2$\"3$eA?a3 RF+\"F27$$\"3fLe*[=Y.h%F2$\"3M:+/vy285F27$$\"3m+]P%37^j%F2$\"356M^f6but%F2$\"3 %\\.QNQyVc\"F27$$\"3ID19>zl]ZF2$\"3S[]>-K'z\"F27$$\"37+]iSjE!z%F2$\"3&)4 L&Q*oyW=F27$$\"3y*\\7G))Rb\"[F2$\"3)fCVDB`!)*=F27$$\"3L+++DM\"3%[F2$\" 3$Gxb\\1an\">F27$$\"3i]P4'>]M&[F2$\"3p-qe54u>>F27$$\"3)3](=np3m[F2$\"3 zS9J5F#4#>F27$$\"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\"\"*-%+AXESL ABELSG6$%\"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 constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h 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 0 729 "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: `,numst eps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme \+ with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/ 33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := [ ]:\nDigits := 30:\nfor ct to 6 do\n Rn_RK7_||ct := RK7_||ct(R(x,y),x ,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Rn_RK7_||ct): \n for ii to numpts do\n sm := sm+(Rn_RK7_||ct[ii,2]-r(Rn_RK7_| |ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nen d 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 )pprint546\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\" &%\"cG6#\"#5F+#\"#6\"#7$\"+Q&fcD*!#@7$/*&%-scheme~with~GF+F,F+#\"#:\"# ;$\"+R**=>lF57$/F8#\"#>\"#?$\"+pDLReF57$/F8#\"#K\"#L$\"+uor7OF57$/F8# \"#\")\"##)$\"+e::@:F57$/F8#\"$p\"\"$q\"$\"+'H8))e'!#AQ)pprint556\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fol lowing 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 o f 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 0 660 "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. o f steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[ 10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`schem e with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=1 69/170]: errs := []:\nDigits := 25:\nfor ct to 6 do\n rn_RK7_||ct := RK7_||ct(R(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: r xx := evalf(r(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(rn_RK7_| |ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf (errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~f ield:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$% 0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~ste ps:~~~G\"$+&Q)pprint566\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~sche me~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+\\So1?!#?7$/*&%-scheme~with~ GF+F,F+#\"#:\"#;$\"+nnU;9F57$/F8#\"#>\"#?$\"+M#)*yE\"F57$/F8#\"#K\"#L$ \"+D#H?(y!#@7$/F8#\"#\")\"##)$\"+Ks(pL$FM7$/F8#\"$p\"\"$q\"$\"+!R7IX\" FMQ)pprint576\"" }}}{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 0 413 "mthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/1 6,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with ` *c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20: \nfor ct to 6 do\n sm := NCint((r(x)-'rn_RK7_||ct'(x))^2,x=0..5,adap tive=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$/*&%:Sharp-Verner~sche me~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+Y27K#*!#@7$/*&%-scheme~with~ GF+F,F+#\"#:\"#;$\"+xWG-lF57$/F8#\"#>\"#?$\"+&4$RCeF57$/F8#\"#K\"#L$\" +`HH.OF57$/F8#\"#\")\"##)$\"+'3:q^\"F57$/F8#\"$p\"\"$q\"$\"+cW=ql!#AQ) pprint586\"" }}}{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([r(x)-'rn_RK7_1'(x),r(x)-'rn_RK7_2'(x),r(x)-'rn_ RK7_3'(x),\nr(x)-'rn_RK7_4'(x),r(x)-'rn_RK7_5'(x),r(x)-'rn_RK7_6'(x)], x=0..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0) ,COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Ve rner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c [10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`schem e with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 959 559 559 {PLOTDATA 2 "6,-%'CURVESG6%7jw7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$!&*=\" )!#>7$$\"5MLLLL3x&)*3\"!#?$!'l\\>F07$$\"5nmm\"z%\\v#pK\"F4$\"'8z\"*F07 $$\"5+++]i!R(*Rc\"F4$\"(unw\"F07$$\"5nm;z>6B`#o\"F4$\"(]q/#F07$$\"5MLL 3xJs1,=F4$\"(o;\">F07$$\"5++]PM_@g>>F4$\"(PLZ\"F07$$\"5nmmm\"H2P\"Q?F4 $\"'9'>*F07$$\"5+++]PMnNrDF4$\"(m(p!*F07$$\"5MLLL$eRwX5$F4$\")'3G]\"F0 7$$\"5MLLLL$eI8k$F4$\")Has;F07$$\"5NLLL$3x%3yTF4$\")B(Hy\"F07$$\"5-+]( oHaN;J%F4$\")/Y(z\"F07$$\"5ommT5:j=XWF4$\")p.&y\"F07$$\"5NL$eRs3P(yXF4 $\")lSi=F07$$\"5ommm \"z%4\\Y_F4$\")[@X=F07$$\"5NLLLeR-/PiF4$\")VB$)F4$\")?vT>F07$$\"50++++v`w(e)F4$\")l'>!>F0 7$$\"5qmmmTg()4_))F4$\")H*e)=F07$$\"5NLLL$e9Kk6*F4$\")l18>F07$$\"5.+++ DJbw!Q*F4$\")$QM\">F07$$\"5qmmT&)3\\m_'*F4$\")j[q=F07$$\"5NLL$ekGkX#** F4$\")'Qw&=F07$$\"5++]iSmjk>5F0$\"))*eo>F07$$\"5nmmm;/j$o/\"F0$\")J&*Q @F07$$\"5MLLe9;_yq5F0$\"):AjU6F0$\")ME!e$F07$$\"5nmm;HdNb'>\"F0$ \");JO[F07$$\"5++++]i^Z]7F0$\")#QCK&F07$$\"5+++]i:4,k7F0$\")!Q1K&F07$$ \"5++++vomax7F0$\")J&)G`F07$$\"5+++](=U#3\"H\"F0$\")Xuy`F07$$\"5+++++v \"=YI\"F0$\")R87aF07$$\"5+++]7GR:=8F0$\")#32W&F07$$\"5++++D\"o*oJ8F0$ \")!HyW&F07$$\"5+++]PMaAX8F0$\")9!HV&F07$$\"5++++](=h(e8F0$\")/M1aF07$ $\"5++++v$R0=P\"F0$\")SB(Q&F07$$\"5++++++'\\[Q\"F0$\")/<%R&F07$$\"5+++ +D1Q*yR\"F0$\")14?aF07$$\"5++++]7!Q4T\"F0$\")nWyaF07$$\"5++++v=A)RU\"F 0$\")o#)*\\&F07$$\"5+++++Dk-P9F0$\")mr+bF07$$\"5++++DJ12]9F0$\")z-iaF0 7$$\"5++++]P[6j9F0$\")]vGaF07$$\"5nm\"HKkAg\\Z\"F0$\")lC3aF07$$\"5ML$e k`h0o[\"F0$\")'>.S&F07$$\"5++voH/5l)\\\"F0$\"),O%R&F07$$\"5nmm\"HKR'\\ 5:F0$\");`I`F07$$\"5++]P4rr=M:F0$\")AVh\\F07$$\"5MLL$e*[z(yb\"F0$\")F \\/[F07$$\"5+++Dc,#>Uh\"F0$\")A9WTF07$$\"5nmmm;a/cq;F0$\"))Q%zPF07$$\" 5nmm;zpYU%p\"F0$\"):+EQF07$$\"5nmmmT&)))G=F0 $\").q,QF07$$\"5,+DcEsW\"R\">F0$\").\"='QF07$$\"5MLLe9;0?E>F0$\")^QZRF 07$$\"5++]i!Rgs2&>F0$\")YQ;UF07$$\"5nmmmm\"pW`(>F0$\")RSFXF07$$\"5++]i SmTI-?F0$\")1D8\\F07$$\"5MLLe9TOEH?F0$\")s9oeF07$$\"5om;a)e6Bi0#F0$\") H)\\D(F07$$\"5,++]i!f#=$3#F0$\")c:/\")F07$$\"5,++v$fQa)3@F0$\")lKA')F0 7$$\"5,+++D\"=EX8#F0$\")E!3%*)F07$$\"5,++Dcwz>g@F0$\")0Jo*)F07$$\"5,++ ](=xpe=#F0$\"))Ho+*F07$$\"5ML3_]%oi#*>#F0$\")&po0*F07$$\"5om;a8(fbE@#F 0$\")'*o%4*F07$$\"5,+Dcw4&[gA#F0$\")fV.\"*F07$$\"5MLLeRA9WRAF0$\"))ep3 *F07$$\"5nmTg-NV$GD#F0$\")'=,1*F07$$\"5++]ilZsAmAF0$\"))[S/*F07$$\"5ML ekGg,izAF0$\")dLZ!*F07$$\"5nmmm\"H28IH#F0$\")-ku!*F07$$\"5nmTg_Z>J0BF0 $\")\"yl5*F07$$\"5nm;a8A3hpxg#F0$\")oy_dF07$ $\"5ommm;z+vbEF0$\")N4,dF07$$\"5,++]Pf4t.FF0$\")h75dF07$$\"5omT5lDho;F F0$\")I&zr&F07$$\"5ML$3F>HT'HFF0$\")&GXr&F07$$\"5,+DJ?ekfUFF0$\")hD*p& F07$$\"5omm\"zWi^bv#F0$\")yy\"o&F07$$\"5ML3_v!z1&oFF0$\")6*Gm&F07$$\"5 ,+]7.d>Y\"y#F0$\")N?ecF07$$\"5om\"H2L7>IF0$\"*[zLC\"F07$$\"5,++v$4^ n)pIF0$\"**Qaj7F07$$\"5,++]i!RU07$F0$\"*gibF\"F07$$\"5,+]il(Hv'[JF0$\" *;^@F\"F07$$\"5,++vo/#3o<$F0$\"*!ePp7F07$$\"5,+DJ?eY(3>$F0$\"*qBMF\"F0 7$$\"5,+](=<6T\\?$F0$\"*$Hhw7F07$$\"5,+vVBlv+>KF0$\"*cn(z7F07$$\"5,+++ v=S2LKF0$\"*O))*y7F07$$\"5omm\"zpo_$eKF0$\"*GBlE\"F07$$\"5MLL$3_NJOG$F 0$\"*K0&[7F07$$\"5,++vVB+\"*3LF0$\"*rtu>\"F07$$\"5ommmm\"p)=MLF0$\"*)* [F5\"F07$$\"5MLLLeR%p\")Q$F0$\")1Gk$*F07$$\"5,+++](=]@W$F0$\")>E]xF07$ $\"5omm\"H#oZ1\"\\$F0$\");N`wF07$$\"5MLL$e*[$z*RNF0$\")+'[h(F07$$\"5om m\"Hd!fX$f$F0$\"*BTHg(F47$$\"5,+++]iC$pk$F0$\")%[`f(F07$$\"5MLe*)f!y6& fOF0$\")!=#ovF07$$\"5om;zp)4\"4sOF0$\")'fpb(F07$$\"5,+voz;/n%o$F0$\")[ :pvF07$$\"5MLLe*[t\\sp$F0$\")#e6h(F07$$\"5,+]P4r$3Cs$F0$\")6'p&yF07$$ \"5omm;H2qcZPF0$\")y&HE)F07$$\"5,+++DJ]'Qx$F0$\")+PT*)F07$$\"5MLL$3_0j ,!QF0$\")\\'H()*F07$$\"5,++v=n?J8QF0$\"*L/&f5F07$$\"5ommm;z5YEQF0$\"*# 4_\\6F07$$\"5MLLe9\"45'RQF0$\"*8^pE\"F07$$\"5,++]7.\"fF&QF0$\"*_0UO\"F 07$$\"5om;aQG-ZyQF0$\"*2#f\"\\\"F07$$\"5MLLek`8=/RF0$\"*oGgd\"F07$$\"5 omTgF;p.oej\"F07$$\"5, ++]ilAFjSF0$\"*F$\\M;F07$$\"5ML$3_]p'>*3%F0$\"*Ruzj\"F07$$\"5omm\"zW7@ ^6%F0$\"*[GZk\"F07$$\"5NL3F>RL3GTF0$\"*//Vk\"F07$$\"5-+]i!RbX59%F0$\"* >F&Q;F07$$\"5om\"z>'ox+aTF0$\"*ZX(H;F07$$\"5NLLLL$)*pp;%F0$\"*E2th\"F0 7$$\"5NLL3xc9[$>%F0$\"*([7x:F07$$\"5NLL$3-$H**>UF0$\"*[5*p9F07$$\"5NLL ek.W]YUF0$\"*$>()R8F07$$\"5NLLL3xe,tUF0$\"*3F2C\"F07$$\"5om;a8AyI*H%F0 $\"*FX15\"F07$$\"5-++v=n(*fDVF0$\"*nY.+\"F07$$\"5omTNrRduQVF0$\")eEU(* F07$$\"5NL$eRAr\"*=N%F0$\")(*37'*F07$$\"5-+Dcw%oP]O%F0$\")/kS&*F07$$\" 5omm;HdO=yVF0$\")B,I&*F07$$\"5ML3_vSME!R%F0$\")N#[Z%F0$\")B3*[*F07$$\"5MLL$3_ 5,-`%F0$\")gr3&*F07$$\"5ommmT&G!e&e%F0$\")dU#[*F07$$\"5-++]P%37^j%F0$ \"*(y_05F07$$\"5NLLLL$)Qk%o%F0$\"*F#Gn6F07$$\"5om\"z%\\!pYyp%F0$\"*+#H M7F07$$\"5-+]il(\\\\5r%F0$\"*Al]K\"F07$$\"5NL3x\"[I_Us%F0$\"*'e#\\V\"F 07$$\"5omm\"z>6but%F0$\"*aHhb\"F07$$\"5-+D19>zl]ZF0$\"*T_Go\"F07$$\"5N L$3-jsgQw%F0$\"*BS\"pF07$$\"5-++++DM\"3%[F0$ \"*Cq'*)>F07$$\"5-+]P4'>]M&[F0$\"*p&p\"*>F07$$\"5-++v=np3m[F0$\"*k0K*> F07$$\"5-+]7GQPsy[F0$\"*z0l*>F07$$\"5-++]P40O\"*[F0$\"*?#*>+#F07$$\"5, ++voa-oX\\F0$\"*TVA+#F07$$\"\"&F)$\"****\\3?F0-%&COLORG6&%$RGBGF($\"\" #!\"\"$\"\"\"F)-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6 %7]]lF'7$F+$!'f_7F07$F2$!',#y\"F07$F8$\"(Eu+\"F07$F=$\"(N>r\"F07$FB$\" (Wf/#F07$FG$\"(DwB#F07$$\"5nm\"Hd?pM.'=F4$\"(l:G#F07$FL$\"(6!>>F07$$\" 5ML3-j7'p)y>F4$\"(?%[=F07$FQ$\"(6\\;\"F07$FV$\"((pWYF07$Fen$\")O&[B\"F 07$$\"5MLL$3F%\\wQKF4$\"):WR7F07$$\"5MLLLe*[`HP$F4$\")ss37F07$$\"5MLL3 -jx/SMF4$\"))3j;\"F07$$\"5MLL$ek.Ur]$F4$\")B)R7\"F07$$\"5MLLe*)4jBuNF4 $\")ksA6F07$Fjn$\")#**>5\"F07$$\"5MLLL3xwq4RF4$\")fZ%=\"F07$F_o$\")o\\ #G\"F07$Fdo$\")1].8F07$Fio$\")\\g%G\"F07$F^p$\")?j]7F07$Fcp$\")U$3@\"F 07$Fhp$\")R5;7F07$F]q$\")#z2D\"F07$Fbq$\")/gS8F07$Fgq$\")/l%Q\"F07$$\" 5+++]P4'4.P&F4$\")d*yS\"F07$$\"5NLLL$3FGT\\&F4$\")5?$Q\"F07$$\"5qmm;HK p%zh&F4$\")\"F07$F \\r$\")wb?7F07$$\"5++++vVVX$\\'F4$\")>s@:F07$$\"5qmmm\"zWo)\\nF4$\")!3 Wy\"F07$$\"5NLLL3_DG1qF4$\")E\\\"[\"F07$Far$\")(=BC\"F07$$\"5qmmm\"Hd) G&R(F4$\")F(f@\"F07$$\"5NLLLe*[!)y_(F4$\")<2>7F07$$\"5++++D1CZgwF4$\") x@I7F07$$\"5qmmm\"HKkIz(F4$\")mc\"G\"F07$$\"5SLLLeRilDzF4$\")Q+V8F07$$ \"50+++Dc\"[#e!)F4$\")66!Q\"F07$$\"5qmmm\"H2S3>)F4$\")&f_S\"F07$Ffr$\" )Qr%R\"F07$$\"5qmm;H#o)fb%)F4$\")96r8F07$F[s$\")pzP8F07$$\"5SLL$3x1K*> ()F4$\")&4MJ\"F07$F`s$\")*zVJ\"F07$$\"5+++]7`aE%)*)F4$\"*yt!H8F47$Fes$ \")2^!=F07$Fbv$\")i!oH#F0 7$$\"5+++voaFfp6F0$\")tUzJF07$Fgv$\")b.FOF07$$\"5ML3F%zv$H.7F0$\")!=&) p$F07$$\"5++]PfeR.57F0$\"))e)RPF07$$\"5nm\"zW#fTx;7F0$\")!ziw$F07$$\"5 MLLe*)fV^B7F0$\")b?pPF07$$\"5++voagXDI7F0$\")Z1WPF07$$\"5nm;z>hZ*pB\"F 0$\")Ht_PF07$$\"5MLe*[='\\tV7F0$\")k09PF07$F\\w$\")W+zOF07$Faw$\")ZDnO F07$Ffw$\")G\\$o$F07$F[x$\")+ylPF07$F`x$\")'zm\"QF07$Fex$\")gPfQF07$Fj x$\")OMpQF07$F_y$\"))Qn%QF07$Fdy$\")8(p!QF07$Fiy$\").$)yPF07$F^z$\")Dm *y$F07$$\"5+++]7.b \"F0$\")hOrOF07$$\"5nTg_s,m\"\\b\"F0$\")*))3i$F07$Fj]l$\")l7,OF07$$\"5 nm;/Ev&[ge\"F0$\")6%QU$F07$F_^l$\")MtGGF07$$\"5M$3-Q\"e=E@;F0$\")#\\#Q FF07$$\"5nmTNr9XIG;F0$\")#REo#F07$$\"5+]i!*GrrMN;F0$\")\\RTEF07$$\"5ML $eky#)*QU;F0$\")%[Si#F07$$\"5++Dc,T^Zc;F0$\")N#Gf#F07$Fd^l$\")==GEF07$ Fi^l$\")4)Qp#F07$F^_l$\"))\\Mu#F07$Fc_l$\")//-FF07$Fh_l$\")qTdEF07$F]` l$\"*\\*p#F07$Fg`l$\")G6gEF07$F\\al$\")O?3EF07$Faal$ \")i,?EF07$Ffal$\")ZvqEF07$F[bl$\")ALKFF07$F`bl$\")Tx2GF07$Febl$\")Zh: IF07$Fjbl$\")O#yE$F07$F_cl$\")\"3[Z$F07$Fdcl$\")(f'GRF07$Ficl$\")!oE( \\F07$F^dl$\")eA'*eF07$$\"5,+]7G)[=g4#F0$\")r^3hF07$Fcdl$\")+**[iF07$$ \"5,+]Pf$G!p@@F0$\")gL)H'F07$Fhdl$\")]u'H'F07$$\"5,+]i!*y?OZ@F0$\")&[o E'F07$F]el$\")\">LB'F07$$\"5,+](=U(Q.t@F0$\")'GXD'F07$Fbel$\")FX'H'F07 $Fgel$\")p(RP'F07$F\\fl$\")kpIkF07$Fafl$\")fRVkF07$Fffl$\")$R#=kF07$F[ gl$\")p(zP'F07$F`gl$\")69ajF07$Fegl$\")&e(fjF07$Fjgl$\")1_,kF07$F_hl$ \")lI^kF07$Fdhl$\")\"yN]'F07$Fihl$\")\\hTlF07$F^il$\")*)o8lF07$Fcil$\" )-j^jF07$Fhil$\")RK4hF07$F]jl$\")HfccF07$Fbjl$\")UUMZF07$Fgjl$\")v:ZTF 07$F\\[m$\")S;$)RF07$Fa[m$\")Rs:SF07$Ff[m$\")korSF07$$\"5om;/,>Ww>EF0$ \")vxySF07$$\"5,++D1R'f0uzEF0$\")p :\"*RF07$$\"5om;HKRdt\"p#F0$\"*s;j+%F47$F`\\m$\")4>ASF07$Fe\\m$\")(yP. %F07$Fj\\m$\")MKGSF07$F_]m$\");Q0SF07$Fd]m$\")*)ezRF07$Fi]m$\")\"eH&RF 07$F^^m$\")EX\\RF07$$\"5M$3Fp,aRzy#F0$\")uDcRF07$Fc^m$\")kh!)RF07$$\"5 ,]7`W1Z*3!GF0$\")sCFSF07$Fh^m$\")-ZUSF07$F]_m$\")a#**f%F07$Fb_m$\")5rj aF07$Fg_m$\")O(=f'F07$F\\`m$\")*o-y(F07$Fa`m$\")KOn&)F07$Ff`m$\")w`R)) F07$$\"5,+v=<^8'=.$F0$\")z,P))F07$$\"5,+]P4r+`WIF0$\")gC@))F07$$\"5,+D c,\"z)>dIF0$\")0h/))F07$F[am$\")Q$y#))F07$$\"5,+]7y]\\?&4$F0$\")$G)G*) F07$F`am$\")#f>,*F07$Feam$\")()Rg*)F07$Fjam$\")rd>*)F07$Fdbm$\")G1K!*F 07$F^cm$\")!365*F07$Fhcm$\")uyS')F07$Fbdm$\")HbSyF07$Fgdm$\")=&Gg'F07$ F\\em$\")k(yS&F07$Faem$\")s'[R&F07$Ffem$\")9yj`F07$F[fm$\"*&=9Z`F47$F` fm$\")r9N`F07$Fefm$\")D8'H&F07$Fjfm$\")VA\"G&F07$F_gm$\")Q(eH&F07$Fdgm $\")NoT`F07$Figm$\")V*>c&F07$F^hm$\")R6feF07$Fchm$\")<'*yjF07$Fhhm$\") q!H#pF07$F]im$\")[#GL(F07$Fbim$\")h-!)yF07$Fgim$\")?g:\"F07$Fc]n$\"*$Rm]6F07$Fh]n$\"* hP'[6F07$$\"5omT&Q.[Mi2%F0$\"*(=7\\6F07$F]^n$\"*F4S:\"F07$$\"5,+Dcw4*e @5%F0$\"*I\"Gh6F07$Fb^n$\"*G4a;\"F07$Fg^n$\"*yLo;\"F07$F\\_n$\"*\"y%4; \"F07$Fa_n$\"*(4f]6F07$Ff_n$\"*&=IN6F07$F[`n$\"*Qf,4\"F07$F``n$\"*c^G. \"F07$Fe`n$\")'y#H'*F07$Fj`n$\")\\sL))F07$F_an$\")!R!owF07$Fdan$\")3lr pF07$Fian$\")1c0oF07$F^bn$\")p8AnF07$Fcbn$\"))Rxn'F07$Fhbn$\")9'fo'F07 $F]cn$\")VOUnF07$Fbcn$\").9cnF07$Fgcn$\")*Qqv'F07$F\\dn$\")[$Hu'F07$Fa dn$\")3a7nF07$Ffdn$\")$oFn'F07$F[en$\")!*QhmF07$F`en$\")j)=m'F07$$\"5M L$3_+sm')[%F0$\"*&>FxmF47$$\"5ommT5::^-XF0$\")(*[-nF07$$\"5,+]i:5jN;XF 0$\")'4)4nF07$Feen$\")R)3p'F07$$\"5om;/E+f/WXF0$\")CInmF07$$\"5,++DJ&p !*yb%F0$\"))y$RmF07$$\"5ML$ek.\\ND49F07$$\"5++Dc,\">g#f\\F0$\"*lDfS\"F0 7$$\"5++]PMF,%G(\\F0$\"*5+kS\"F07$$\"5+]7y]&4I'z\\F0$\"*]I(39F07$$\"5+ +v=nj+U')\\F0$\"*!Gq49F07$$\"5+]Pf$=.5K*\\F0$\"**>G99F07$Fd[o$\"*:E$>9 F0-Fi[o6&F[\\oF($\"\"(F^\\oF(-Fb\\o6#%8scheme~with~c[10]=15/16G-F$6%7e \\lF'7$F+$!')G-\"F07$F2$!'tN9F07$F8$\"'$ea)F07$F=$\"(HDZ\"F07$FB$\"(w \"[-\"F07$F]ao$\")8@(3\"F07$F_o$\")@Fl 6F07$Fdo$\")N(==\"F07$Fio$\")Eqm6F07$F^p$\")B`R6F07$Fcp$\")/%y5\"F07$F hp$\")&f@6\"F07$F]q$\")<(*R6F07$$\"5qmTNr*ykh/&F4$\")iiq6F07$Fbq$\")! \\;@\"F07$$\"5NLe*[=c:(z^F4$\")xU<7F07$Fgq$\")#[kC\"F07$F]co$\")\\Lk7F 07$Fbco$\")&3XC\"F07$Fgco$\")WS%>\"F07$F\\do$\"))*R_6F07$Fado$\")Z.@6F 07$Ffdo$\")'o<6\"F07$F[eo$\")&4P7\"F07$F\\r$\")\\yE6F07$Fceo$\")LD78F0 7$Fheo$\")'fl`\"F07$F]fo$\")D%>L\"F07$Far$\")quD6F07$Fefo$\")y5'4\"F07 $Fjfo$\")Hi&4\"F07$F_go$\")cy/6F07$Fdgo$\")a7\\6F07$Figo$\")Rn+7F07$F^ ho$\")W$3B\"F07$Fcho$\")8o]7F07$Ffr$\")>nT7F07$F[io$\")7eA7F07$F[s$\") 0)e>\"F07$Fcio$\")RWw6F07$F`s$\")zEx6F07$F[jo$\"*bZ!*=\"F47$Fes$\")z)) 47F07$Fcjo$\")7U:7F07$Fjs$\")\"3*47F07$F[[p$\")S>!=\"F07$F_t$\")_Wf6F0 7$Fc[p$\")11U6F07$Fdt$\")$46:\"F07$Fit$\")_ec7F07$F^u$\")jHy8F07$Fhu$ \")+j0;F07$Fbv$\")t-]@F07$Fg\\p$\")9%H(GF07$Fgv$\")3LAKF07$F_]p$\")&e` G$F07$Fd]p$\")(>7L$F07$Fi]p$\")QNbLF07$F^^p$\")sQrLF07$Fh^p$\")gxrLF07 $F\\w$\")YrBLF07$Faw$\")tt:LF07$Ffw$\")[nGLF07$F[x$\")u(QR$F07$F`x$\") PUMMF07$Fex$\")5VoMF07$Fjx$\")5AwMF07$F_y$\")d4eMF07$Fdy$\")vJEMF07$Fi y$\")M&RS$F07$F^z$\")(=FT$F07$Fcap$\")c&)QMF07$Fcz$\")vXWMF07$F[bp$\") g.xMF07$Fhz$\")o[$F07$$\"5++v$4@Ej>d\"F0$\")w,6KF07$$\"5M$e*[o=f+z:F0$\")\\`=J F07$Fcfp$\")9T#F07$Fjgp$\")2A`BF07$F_hp$\")PcBBF07$Fd^l$\")&QPN#F07$Fi^l$ \")Uu2CF07$F^_l$\")23ZCF07$Fc_l$\")N48CF07$Fh_l$\")rRxBF07$F]`l$\"*^sS Q#F47$Fb`l$\")>$3T#F07$Fg`l$\")SYzBF07$F\\al$\")\\!zL#F07$Faal$\")y^ZB F07$Ffal$\")MI*Q#F07$F[bl$\")_GTCF07$F`bl$\")$Gk]#F07$Febl$\")\\\"**o# F07$Fjbl$\")Jx4HF07$F_cl$\")7#*=JF07$Fdcl$\")Fx&f$F07$Ficl$\")zYBXF07$ F^dl$\")&o4E&F07$F]\\q$\")rqRaF07$Fcdl$\")!*)3d&F07$$\"5,+DcwMBF:@F0$ \")M*[h&F07$Fe\\q$\")d@OcF07$$\"5,+v=UK#3\"G@F0$\")zr`cF07$Fhdl$\")**y [cF07$F]]q$\")98LcF07$F]el$\")s,5cF07$Fe]q$\")B0FcF07$Fbel$\")KIgcF07$ Fgel$\")S*>s&F07$F\\fl$\")w;ndF07$Fafl$\")m@xdF07$Fffl$\")b*pv&F07$F[g l$\")a![s&F07$F`gl$\")k\"eq&F07$Fegl$\")'y/r&F07$Fjgl$\")7&Ru&F07$F_hl $\")Rm$y&F07$Fdhl$\")-+DeF07$Fihl$\")^-aeF07$F^il$\")ZkIeF07$Fcil$\")8 )>q&F07$Fhil$\")h+@bF07$F]jl$\")1#4,&F07$Fbjl$\")&zNB%F07$Fgjl$\")jV@P F07$F\\[m$\")S\\mNF07$Fa[m$\")Ce$F47$F`\\m$\")-5&f$F07$Fe\\m$\")fK/OF07$Fj \\m$\")O()*f$F07$F_]m$\")kZ\"e$F07$Fd]m$\")&Q3c$F07$Fi]m$\")&R&RNF07$F ^^m$\")F$o`$F07$Fc^m$\")(zEc$F07$Fh^m$\")]r9OF07$F]_m$\")**G.TF07$Fb_m $\")OOL\\F07$Fg_m$\")*z`)fF07$F\\`m$\")O*H*pF07$Fa`m$\")&)fPwF07$Ff`m$ \")Lc/zF07$Ffgq$\")AG=zF07$F[am$\")\"Qs#zF07$Fchq$\")Fg2!)F07$F`am$\") ktt!)F07$Feam$\")jTK!)F07$Fjam$\")Y)***zF07$Fdbm$\")Z()*3)F07$F^cm$\") \\RU\")F07$Fhcm$\"))Rtx(F07$Fbdm$\")eB))pF07$Fgdm$\")\"\\0*eF07$F\\em$ \")>>T[F07$Faem$\")\"47#[F07$Ffem$\")//%z%F07$F[fm$\"*(f'3y%F47$F`fm$ \")Q3rZF07$Fefm$\")n&)RZF07$Fjfm$\")U,GZF07$F_gm$\")wHSZF07$Fdgm$\")pp yZF07$Figm$\"),Lo\\F07$F^hm$\")WlK_F07$Fchm$\")md!p&F07$Fhhm$\")T49iF0 7$F]im$\")-\"=h'F07$Fbim$\")CjIrF07$Fgim$\")yZ%)yF07$F\\jm$\")'R\\d)F0 7$Fajm$\")!zG`*F07$Ffjm$\"*`yc+\"F07$Fi]r$\"*T>E,\"F07$F[[n$\"*SPx,\"F 07$Fa^r$\"*Ss+-\"F07$F`[n$\"*%*e,-\"F07$Fe[n$\"*jN.-\"F07$Fj[n$\"*X<,- \"F07$F_\\n$\"*y;s-\"F07$Fd\\n$\"*<\"4O5F07$Fi\\n$\"*Fw'oF07$Fdan$ \")myWiF07$Fian$\")d#=4'F07$F^bn$\")z3:gF07$Fcbn$\")xpufF07$Fhbn$\")E' *zfF07$F]cn$\")6HDgF07$Fbcn$\")DHNgF07$Fgcn$\")VBNgF07$F\\dn$\")HWBgF0 7$Fadn$\")&p))*fF07$Ffdn$\")*Gq'fF07$F[en$\")u(z&fF07$F`en$\")NUefF07$ Fcer$\"*xd2(fF47$Fher$\")o*3*fF07$F]fr$\"):q'*fF07$Feen$\")yY\")fF07$F efr$\")ZfifF07$Fjfr$\")(4.%fF07$F_gr$\")I4DfF07$Fjen$\")hmXfF07$F_fn$ \")$>hN'F07$Fdfn$\")6z2uF07$Fifn$\")b%oz(F07$F^gn$\")jhp#)F07$Fcgn$\") QY?*)F07$Fhgn$\")?3&o*F07$F]hn$\"*i%yf5F07$Fbhn$\"*Yq>7\"F07$Fghn$\"*@ @1<\"F07$F\\in$\"*')H(37F07$Feir$\"*4)eJ7F07$Fain$\"*_&3X7F07$F]jr$\"* \\<.D\"F07$Ffin$\"*$3$*[7F07$F[jn$\"*ii%\\7F07$F`jn$\"*U<5D\"F07$Fejn$ \"*$HCb7F07$Fjjn$\"*Qz>E\"F07$Fa[s$\"*>S^E\"F07$Ff[s$\"*]6mE\"F07$F[\\ s$\"*.6YE\"F07$F_[o$\"*\")\\@E\"F07$Fc\\s$\"*L+&f7F07$Fh\\s$\"*\"[*)f7 F07$F]]s$\"*m7F07$Fd[o$\"*#R?q7 F0-Fi[o6&F[\\o$\"\"*F^\\o$\"\"%F^\\oF(-Fb\\o6#%8scheme~with~c[10]=19/2 0G-F$6%7i^lF'7$F+$!'N,9F07$F2$!'I:7F07$F8$\"'46(*F07$F=$\"(1A^\"F07$FB $\"(Bk%=F07$FG$\"(0sA#F07$F[^o$\"(cLF#F07$FL$\"(^\"R?F07$$\"5n;zp[#)eB \\>F4$\"(7D1#F07$Fc^o$\"(ub(>F07$$\"5+]PMxUL]3?F4$\"(`QG\"F07$FQ$\"(k4 I\"F07$$\"5++]7G))>Wr@F4$\"&$GzF07$$\"5MLLek.pu/BF4$!(&G$4\"F07$$\"5nm ;/,>=0QCF4$!'__F07$$\"5ML$eR(\\;m/FF4$\"(XR@'F07$$\"5nm mT5ll'z$GF4$\"(W#)y(F07$$\"5+]7`%Rz#HrGF4$\"(19!zF07$$\"5MLekyA!>Y!HF4 $\"(Ye_)F07$$\"5n;/wi^_%z$HF4$\"(c))e)F07$$\"5++](o/[r7(HF4$\"(mSl)F07 $$\"5nmT5:QR#z.$F4$\"(g\"*y)F07$Fen$\"(%3*[)F07$Ff_o$\"(He5(F07$Fjn$\" (!=)f&F07$F]ao$\"(YSa'F07$F_o$\"(Aye(F07$Fdo$\"(>C!yF07$Fio$\"(9&)f(F0 7$F^p$\"(dpB(F07$Fcp$\"(&e>oF07$$\"5NL3FWXK1zZF4$\"(b\\#oF07$Fhp$\"(1/ )oF07$$\"5-+D\"yv,9E\"\\F4$\"(i,=(F07$F]q$\"(\"[asF07$Fbq$\"(+QA)F07$F gq$\"(2Mr)F07$F\\do$\"(]%*3(F07$F\\r$\"(Kvl&F07$Fceo$\"(_*e%*F07$Fheo$ \")!ym7\"F07$F]fo$\"(Db\\(F07$Far$\"(?s'fF07$Fdgo$\"('ppnF07$Ffr$\"(4^ &zF07$F[io$\"(<4r(F07$F[s$\"(')4O(F07$Fcio$\"(4h5(F07$F`s$\"(i(=rF07$F [jo$\")%fkF(F47$Fes$\"(9Gb(F07$Fcjo$\"(!)[i(F07$Fjs$\"(R(\\vF07$F[[p$ \"(a_:(F07$F_t$\"(%p&)oF07$Fc[p$\"(8kn'F07$Fdt$\"(eh#oF07$Fit$\"(k2.)F 07$F^u$\"(Zg.*F07$Fcu$\")V0H5F07$Fhu$\")zK)3\"F07$$\"5ML$e9Te3n5\"F0$ \")pLZ5F07$F]v$\"(-Xg*F07$$\"5MLe*)4o-nC6F0$\"(h$4$*F07$$\"5++]P4'\\d1 8\"F0$\"(o)p'*F07$$\"5nmT&)3CZkO6F0$\"(fm***F07$Fbv$\")g**\\6F07$Fg\\p $\")9Eq=F07$Fgv$\")()=BAF07$F_]p$\")Y'3C#F07$Fd]p$\")Ik9AF07$$\"5M$3F> *eSS87F0$\")m7BAF07$Fi]p$\")AfFAF07$$\"5+]7.dfU9?7F0$\")2mk@F07$F^^p$ \")dFq@F07$Fh^p$\")j1*4#F07$F\\w$\")Fj')>F07$Faw$\")^vr>F07$Ffw$\")0L \"*>F07$F[x$\")\\8#3#F07$F`x$\")=VO@F07$$\"5++]7G8@+38F0$\")I`V@F07$$ \"5+++Dc^gQ68F0$\")lHx@F07$$\"5++]P%)*)*pZJ\"F0$\")`Ox@F07$Fex$\");W\" =#F07$$\"5+++vo/=#\\K\"F0$\")/H'>#F07$Fjx$\")P?\">#F07$F_y$\")I#p;#F07 $Fdy$\")KsC@F07$Fiy$\")FT&4#F07$F^z$\")ZT2@F07$Fcz$\")I5]@F07$Fhz$\")* f$\\AF07$$\"5+++D1k!*>99F0$\")tF\\AF07$$\"5+++]i:,Y<9F0$\")Rh`AF07$$\" 5+++v=n6s?9F0$\")TF%H#F07$F][l$\")1(RH#F07$$\"5+++](=K/0V\"F0$\")-d:BF 07$Fb[l$\")4y8BF07$Fg[l$\")xHdAF07$F\\\\l$\")dZw@F07$Ff\\l$\")I_o>F07$ F`]l$\")_DJ=F07$Fe]l$\")B)[#>F07$Fj]l$\")e!o8#F07$Fcfp$\")k\\Q>F07$F_^ l$\")-(*)z[\"F07$Fb`l$\")!o&)\\\"F 07$Fg`l$\")Vtc9F07$F\\al$\")'4MS\"F07$F`bl$\")*3;e\"F07$Fjbl$\")oa$*=F 07$F_cl$\")*[4&>F07$Fdcl$\")X%H/#F07$Ficl$\")e(pr#F07$F^dl$\")GwhMF07$ $\"5,+DJXR0g*3#F0$\")-'e`$F07$F]\\q$\")CT(e$F07$$\"5,+v$4rVOC5#F0$\")] :1OF07$Fcdl$\")HH;OF07$Fe\\q$\")WkZNF07$Fhdl$\")xj*[$F07$F]]q$\")w^GMF 07$F]el$\")F8&Q$F07$Fe]q$\")@!)3MF07$Fbel$\")k0bMF07$Fgel$\")lOQNF07$F \\fl$\")^G)f$F07$Fafl$\")uH6OF07$Fffl$\")7/%e$F07$F[gl$\")7BTNF07$F`gl $\"):?;NF07$$\"5n;a8(RqBHF#F0$\")XC5NF07$Fegl$\").'H_$F07$$\"5nT5SW)Qo HG#F0$\")P.GNF07$$\"5+]i:g;mJ'G#F0$\")x%*GNF07$$\"5Me9\"fZ%[m*G#F0$\") ;CeNF07$Fjgl$\")5,oNF07$F_hl$\")E$=i$F07$Fdhl$\")F&)zOF07$$\"5n;/,Wf-w BBF0$\")R%\\r$F07$Fihl$\")8(ys$F07$$\"5n;z%\\S8fgL#F0$\")***fs$F07$F^i l$\")fP0PF07$Fcil$\")zpSNF07$Fhil$\")g#4C$F07$Fbjl$\")%3Rc#F07$$\"5ML3 FW&p68^#F0$\"),M-BF07$Fgjl$\")98-AF07$$\"5M$3x\"[g6)>`#F0$\")\"HR=#F07 $$\"5MLe9;#)4()QDF0$\")!R'f@F07$$\"5M$e9TQ!3wXDF0$\"))3n9#F07$F\\[m$\" )74X@F07$Fa[m$\")9h(>#F07$Ff[m$\")$\\jE#F07$Fiaq$\")t9xAF07$F^bq$\")51 dAF07$Fcbq$\")]:IAF07$F[\\m$\"):?/AF07$F[cq$\")v9'=#F07$F`cq$\")D0(=#F 07$Fecq$\"*@%H.AF47$F`\\m$\")!>,A#F07$Fe\\m$\")9EKAF07$Fj\\m$\")!pgA#F 07$F_]m$\")^_,AF07$Fd]m$\")9Fu@F07$Fi]m$\")3&o9#F07$F^^m$\")8CX@F07$Fc ^m$\")Ymv@F07$Fh^m$\")JyIAF07$F]_m$\")v%>h#F07$Fb_m$\")OdaHF07$$\"5,+] 7G8v`FHF0$\")No4JF07$Fg_m$\")(=Ja$F07$$\"5,+]P%)*oCP&HF0$\")DpuRF07$F \\`m$\"):R*Q%F07$$\"5,+]iSm=\"*zHF0$\"):%yx%F07$Fa`m$\")LOO\\F07$$\"5, +D\"G)[Ab**HF0$\")])*p\\F07$$\"5,+](oH/*41IF0$\")C[-]F07$$\"5,+v$4r$ek 7IF0$\")agw\\F07$Ff`m$\")*GE'\\F07$Fagq$\")*)Rw[F07$Ffgq$\"),fH[F07$F[ hq$\")f(=![F07$F[am$\")>gG[F07$Fchq$\")SxP\\F07$F`am$\")\"Q^-&F07$Feam $\")g')p\\F07$Fjam$\")ISF\\F07$Fdbm$\")7?\\]F07$F^cm$\")+KP^F07$Fhcm$ \")\"R3n%F07$Fbdm$\")cI^VF07$Fgdm$\")v0POF07$F\\em$\")&Q]#HF07$$\"5om \"H#oK)yVX$F0$\")[K%*GF07$$\"5ML$ekyZ2mY$F0$\")MI))GF07$$\"5,+vo/Bh$)y MF0$\")wX?HF07$Faem$\")_K\")HF07$$\"5MLe9T8MH.NF0$\")'*42IF07$$\"5,+]P fe?_:NF0$\")**)o,$F07$$\"5M$e*[=\"QO;_$F0$\")'Q!4IF07$$\"5omTgx.2vFNF0 $\")_11IF07$$\"5,](=nj-lQ`$F0$\")=)p)HF07$Ffem$\");3gHF07$F[fm$\"*o^M% HF47$F`fm$\")j))HHF07$$\"5o;z%\\:7AKl$F0$\")i8/HF07$Fefm$\")VB*)GF07$$ \"5,]P%['R9!em$F0$\")AQ#)GF07$Fjfm$\")mauGF07$$\"5M$eRZxv!QyOF0$\")PAy GF07$F_gm$\")c!*))GF07$$\"5o;aj%e2g4p$F0$\")WFAHF07$Fdgm$\")frIHF07$Fi gm$\")1H.JF07$F^hm$\")29$G$F07$Fchm$\")ZM7OF07$Fhhm$\"))\\e\"QF07$F]im $\"):gmRF07$Fbim$\")gj7UF07$Fgim$\")'ROr%F07$F\\jm$\")EWm_F07$$\"5ML3_ vlYhlQF0$\")K=:dF07$Fajm$\");\\xgF07$$\"5,+Dc,\"zD8*QF0$\")^M%H'F07$Ff jm$\"),2ujF07$F`[n$\")L#=F'F07$Fj[n$\")H47iF07$F_\\n$\")\"G)3jF07$Fd\\ n$\")\">qU'F07$Fi\\n$\"))RvV'F07$F^]n$\")f1/U\\F07$$\"5-+v$4'\\=;'G%F0$\")BoP XF07$F_an$\")h$>9%F07$$\"5NLe9m%z`CJ%F0$\")*4z*QF07$Fdan$\")\">^w$F07$ Fian$\")s(4p$F07$F^bn$\")L\\bOF07$Fcbn$\")@HROF07$Fhbn$\")H*=m$F07$$\" 5,]PM-\\NA%Q%F0$\")19\"p$F07$F]cn$\")4UFPF07$$\"5o;zp[KLI'R%F0$\").SCP F07$Fbcn$\")gJ[PF07$Fgcn$\")]&Hv$F07$F\\dn$\")\\*)RPF07$Fadn$\")F^3PF0 7$Ffdn$\")6xmOF07$F[en$\")@%\\l$F07$F`en$\")nqbOF07$Fcer$\"*5>An$F47$F her$\")v#*)p$F07$F]fr$\")L\\1PF07$Feen$\")V&fo$F07$Fefr$\")_%3m$F07$Fj fr$\")gMJOF07$F_gr$\")\"yAh$F07$Fjen$\")lUMOF07$F_fn$\")^+oRF07$Fdfn$ \")W6iYF07$Fifn$\")8ZT[F07$F^gn$\")\"Q\"o\\F07$Fcgn$\")%3#3`F07$Fhgn$ \")(>6z&F07$F]hn$\")`9TlF07$Fbhn$\")2%)QqF07$Fghn$\")N1@uF07$F\\in$\") 58hwF07$$\"5-]P%)RLbU$z%F0$\")0z*p(F07$$\"5-+v=F07$F efv$\"(<%z>F07$Fc^o$\"(Ye*=F07$F]gv$\"(c&=7F07$FQ$\"(FZB\"F07$Fegv$!'i +4!QEF4$\" (-')e\"F07$Fghv$\"(NPp$F07$$\"5++v=U2TJrFF4$\"(\"fyQF07$F\\iv$\"(8`J&F 07$Faiv$\"(5.T&F07$Ffiv$\"(`W.'F07$F[jv$\"(@!zgF07$F`jv$\"(1q6'F07$$\" 5M$e*)4$4xf/IF4$\"()R;gF07$Fejv$\"(-*\\gF07$$\"5+](=#*p;]72$F4$\"(8q.' F07$Fen$\"(xFQ&F07$Ff_o$\"(m-?$F07$Fjn$\"('fn8F07$F]ao$\"(GCO#F07$F_o$ \"(=bU$F07$$\"5-](=nQYs9@%F4$\"(*fQOF07$$\"5omT5!p:g[C%F4$\"(<(QOF07$$ \"5N$e*[$*\\yCyUF4$\"(SIk$F07$Fdo$\"(,Wk$F07$$\"5o;/E+OK-XVF4$\"(QTk$F 07$$\"5NLek.H4TyVF4$\"(\"p(F4$\"(SdV\"F07$Far$\"(vmS\"F07$Fdgo$\"(8I[#F07$Ffr$\"(Jgm $F07$F[io$\"(k\\U$F07$F[s$\"(,I2$F07$$\"5qmmTN@([Ql)F4$\"(M*GHF07$Fcio $\"(I[\"GF07$$\"50++D19a,'y)F4$\"(qe\"GF07$F`s$\"(bs#GF07$F[jo$\")#>m) HF47$Fes$\"(6lE$F07$Fcjo$\"(4)RLF07$Fjs$\"(dUE$F07$F[[p$\"(!emGF07$F_t $\"(Ulf#F07$$\"5+](oa5L_mo*F4$\"(7]`#F07$$\"5NL3_D`(R1s*F4$\"('>3CF07$ $\"5q;HdXvria(*F4$\"('*zS#F07$Fc[p$\"()4%R#F07$$\"5N$3xc)>?gA)*F4$\"(B 6Q#F07$$\"5qm\"Hd?W*ec)*F4$\"(kKQ#F07$$\"5+]7yDkod!*)*F4$\"(*>dCF07$Fd t$\"(+wb#F07$Fit$\"((yqOF07$F^u$\"([UO%F07$$\"5++]i:g2\")e5F0$\"(J2r%F 07$Fcu$\"(YZI&F07$$\"5++D19WCxw5F0$\"(:!>aF07$$\"5nm;a8s'fF3\"F0$\"(x[ l&F07$$\"5ML3-8+pu)3\"F0$\"(V\"*o&F07$Fhu$\"(^xf&F07$Fcbw$\"(cy'[F07$F ]v$\"(-.K$F07$F[cw$\"(<[`#F07$F`cw$\"($\\8@F07$$\"5M$e9\"456lL6F0$\"() )[9#F07$Fecw$\"((R%>#F07$$\"5+]Pf3Q$Q'R6F0$\"(1#3FF07$Fbv$\"(,G%HF07$$ \"5++vVt_@P\\6F0$\"(.K0%F07$$\"5nm;aQ`B6c6F0$\"(V)[\\F07$$\"5MLek.aD&G ;\"F0$\"(S9D(F07$Fg\\p$\"(JW$))F07$$\"5ML$e*)f:tI=\"F0$\")$oL5\"F07$Fg v$\")1')z6F07$F^^p$\"(>O%**F07$F\\w$\"(W/V(F07$$\"5++]7y+\"fQD\"F0$\"( $QMuF07$$\"5+++D1RICd7F0$\"(k'>uF07$$\"5++]PMxpig7F0$\"(@=D(F07$Faw$\" (lQD(F07$$\"5+++v=#zy2F\"F0$\"(V*=uF07$Ffw$\"(6CY(F07$F[x$\"(rDS)F07$F `x$\"(kb&*)F07$F[gw$\"(rw-*F07$F`gw$\"(%pq$*F07$Fegw$\"(%Qr$*F07$Fex$ \"()z7%*F07$F]hw$\"(tQc*F07$Fjx$\"(;J^*F07$F_y$\"(wxE*F07$Fdy$\"(D0%)) F07$$\"5+++]i!H$Gl8F0$\"(Nzl)F07$Fiy$\"(aGa)F07$$\"5+++](o\\F$y8F0$\"( Uec)F07$F^z$\"(OOm)F07$Fcz$\"(uh4*F07$Fhz$\")M>65F07$F][l$\")!)[e5F07$ Fb[l$\").L#3\"F07$Fg[l$\")?8F5F07$F\\\\l$\"(o%3%*F07$Ff\\l$\"(ni+(F07$ F`]l$\"(#H=bF07$Fe]l$\"(Btt(F07$Fj]l$\")A<'3\"F07$$\"5+v$4YsF*Rh:F0$\" )I[Y6F07$F]at$\")1qE6F07$$\"5Me9;#Q$>Wo:F0$\")\\N.6F07$Fbat$\")]=k5F07 $Fgat$\"(Pfy*F07$Fcfp$\"([Fx)F07$F_bt$\"(%padF07$F_^l$\"(Qv&\\F07$F[gp $\"(MD([F07$F`gp$\"(7%H[F07$Fegp$\"(j/&\\F07$Fjgp$\"(n%G^F07$Fe]x$\"(4 5<&F07$F_hp$\"(uS=&F07$F]^x$\"(\\')G&F07$Fd^l$\"(]H\\&F07$Fi^l$\"(X+A' F07$F^_l$\"(#*=#oF07$Fc_l$\"(oHV'F07$Fh_l$\"(t0'fF07$Fa_x$\"(jX\"fF07$ F]`l$\")V'Q0'F47$Fi_x$\"(3IC'F07$Fb`l$\"(h=T'F07$Fg`l$\"(.,*fF07$F\\al $\"(P#eaF07$F`bl$\"()esUk2#F0$\")r5 Z:F07$F^dl$\"))f%y;F07$Fiax$\")!y9s\"F07$F]\\q$\")@)yu\"F07$Fabx$\")B< @t 9F07$F\\fl$\")!*4M:F07$$\"5M$3_]M0_$>AF0$\")cQU:F07$Fafl$\")!eta\"F07$ $\"5o;H23m\\uKAF0$\")J/P:F07$Fffl$\")l&)>:F07$F[gl$\")y_w9F07$F`gl$\") D9^9F07$F]ex$\")./X9F07$Fegl$\")i'yX\"F07$Feex$\")[)HY\"F07$Fjex$\")a \"RY\"F07$F_fx$\")Gf$\\\"F07$Fjgl$\")=].:F07$Fdhl$\")?*)=;F07$F^il$\") `$[l\"F07$Fcil$\")[*3\\\"F07$Fhil$\")(zM:\"F07$$\"5M$e*[ooq.)R#F0$\")K &>8\"F07$$\"5++D\"yv1qYS#F0$\")NC(3\"F07$$\"5MeRZ-nl)zS#F0$\")Wx#3\"F0 7$$\"5n;a8ZmII6CF0$\")S(R5\"F07$$\"5+voz\"fc>YT#F0$\")cc$4\"F07$$\"5ML $ek`1OzT#F0$\")#*Q*4\"F07$$\"5nmT5:j??JCF0$\")#o4L\"F07$F]jl$\")]Jm9F0 7$$\"5Me9TQgXyZCF0$\")c$zY\"F07$$\"5n;H2$)f55^CF0$\")0-!f\"F07$$\"5+vV tFfvTaCF0$\"),4l:F07$$\"5MLeRseStdCF0$\")+I]:F07$$\"5n\"Hdq\"e00hCF0$ \")Yx\"e\"F07$$\"5+](=)F07$Fcix$\"(jxG)F07$F\\[m$\"(0: R)F07$Fa[m$\"(X'4!*F07$Ff[m$\"($Rf(*F07$$\"5,+vV)*3ow8EF0$\"(>7\"**F07 $Fiaq$\"(?b*)*F07$$\"5,]PM-CKwAEF0$\"(p'*))*F07$$\"5MLek.H?wDEF0$\"(Jr ))*F07$$\"5o;z%\\S$3wGEF0$\"(wx#)*F07$F^bq$\"(!G0(*F07$Fcbq$\"('\\P%*F 07$F[\\m$\"(ff<*F07$$\"5,](ozT))[(eEF0$\"(`a6*F07$$\"5ML3F>*oZ;+*F07$Fecq$\")-!e;*F47$F`\\m$\"(xhL*F07$Fe \\m$\"(p$f%*F07$Fj\\m$\"(*R(R*F07$F_]m$\"(s'\\\"*F07$Fd]m$\"($zu))F07$ $\"5M3x\"[gT!zeFF0$\"(H$*y)F07$$\"5,](= '=**zElFF0$\"(M^j)F07$Fi]m$\"(s1g)F07$$\"5o;HK*QP%)\\x#F0$\"(%G>&)F07$ F^^m$\"(N9f)F07$Fc^m$\"(rs())F07$Fh^m$\"(X7O*F07$$\"5,++vV)>ST$GF0$\") /!p0\"F07$F]_m$\")^;!=\"F07$$\"5,+](=<1#HuGF0$\")#ooD\"F07$$\"5MLLe9;g n()GF0$\")8\\78F07$$\"5om;Hdq*f5!HF0$\")GN77F07$Fb_m$\")*Rq9\"F07$$\"5 ,]7.2Atr2*4#HF0$\")!)*e3\"F07$$\"5,]P4@;TECH F0$\")/p.6F07$F\\^y$\")\"o17\"F07$$\"5,+v=U2V3MHF0$\")Msj6F07$Fg_m$\") c,D8F07$Fd^y$\")T#*3;F07$F\\`m$\")6R:>F07$F\\_y$\")=JJAF07$Fa`m$\")k$f I#F07$Fd_y$\")Ui!H#F07$Fi_y$\")8o*G#F07$F^`y$\")EN2AF07$Ff`m$\")=zg@F0 7$Fagq$\")[S9?F07$Ffgq$\")!)4X>F07$$\"5,v=U2^spZIF0$\")v$4%>F07$$\"5,] (oa5Vk30$F0$\")3A3>F07$$\"5,Dc^.6;.aIF0$\")l\"*3>F07$F[hq$\")J04>F07$$ \"5,v$4'*4(fOgIF0$\")@x3>F07$$\"5,]il(4:LN1$F0$\"))G\"4>F07$$\"5,DJq&4 L+n1$F0$\")HW5>F07$F[am$\")QaO>F07$Fchq$\")vI[?F07$F`am$\")d9P@F07$Fea m$\"))y73#F07$Fjam$\")6:Q?F07$Fdbm$\")UAi@F07$F^cm$\")cehAF07$Fccm$\") ;-M@F07$Fhcm$\")P&fz\"F07$$\"5om;HK*oqiH$F0$\")&e4o\"F07$F]dm$\")`*3j \"F07$$\"5ML$3_vN\\:K$F0$\")6*et\"F07$Fbdm$\")A)z%=F07$$\"5MLLekyQoZLF 0$\")J2g>F07$$\"5,++]il!z6O$F0$\")S?'*>F07$$\"5ommTg_UnuLF0$\")A6$z\"F 07$Fgdm$\")x??:F07$$\"5,++DcEYm,MF0$\")0$*\\7F07$$\"5omm;a8)f^T$F0$\") 2Cp6F07$$\"5,+]7.2u!>U$F0$\")BRj6F07$$\"5MLL3_+]lGMF0$\")=W^6F07$$\"5o m;/,%f-aV$F0$\")w2`6F07$F\\em$\")uAf6F07$F`cy$\")ZL`6F07$Fecy$\")>.f6F 07$Fjcy$\")y`%>\"F07$Faem$\")*4wD\"F07$$\"5MeRZ_H>7%\\$F0$\")A'oD\"F07 $$\"5,]7.#34zr\\$F0$\")uee7F07$$\"5oT&)e6_iB+NF0$\")ub(G\"F07$Fbdy$\") z6(G\"F07$$\"5o;/E+OxS4NF0$\")l6&H\"F07$Fgdy$\")5a*H\"F07$F\\ey$\")F+$ H\"F07$Faey$\")BF!H\"F07$Ffey$\"),kr7F07$Ffem$\")+uW7F07$F[fm$\"*I-zA \"F47$F`fm$\")`L97F07$Fdfy$\")6U)=\"F07$Fefm$\")-bt6F07$F\\gy$\")&)[m6 F07$Fjfm$\")t,f6F07$Fdgy$\")T3i6F07$F_gm$\")VMs6F07$$\"5Me9;KY_\"yo$F0 $\")b'p<\"F07$F\\hy$\")SU/7F07$$\"5,v$4r`!\\5%p$F0$\")MW07F07$Fdgm$\") #H(47F07$Figm$\")t,V8F07$F^hm$\")!ybV\"F07$$\"5MLL3F>grgPF0$\")qfK:F07 $Fchm$\")Fz@;F07$$\"5ML$eRs`R/y$F0$\")^0Y;F07$$\"5omm\"HK/9qy$F0$\")%> %)o\"F07$$\"5,+](=#\\&)e$z$F0$\")'zKm\"F07$Fhhm$\")i(Ge\"F07$$\"5,+DJ? 3.X.QF0$\")+K3;F07$$\"5om;z>hvt1QF0$\")YXG;F07$$\"5ML3F>9[-5QF0$\")y\\ @:F07$F]im$\"),lY:F07$$\"5om\"H#=?$*f;QF0$\")e!*o:F07$$\"5ML$3xJd'))>Q F0$\").`E:F07$$\"5,+v=F07$F\\ jm$\")')Qy@F07$F\\jy$\")YK$\\#F07$Fajm$\")8!Rv#F07$$\"5M$3_+(4!)*[)QF0 $\")DoTGF07$Fdjy$\")^AlGF07$$\"5M3xJn\"oRX*QF0$\")(H/)GF07$$\"5o;H2LsN v(*QF0$\")o#f)GF07$$\"5,D\"G))HYn4!RF0$\")EMOGF07$Ffjm$\")lNZGF07$F`[n $\"),.)e#F07$Fj[n$\")\\m&\\#F07$F_\\n$\")()\\%f#F07$Fd\\n$\")%4Zr#F07$ $\"5o;zpBEs;;SF0$\")83CFF07$Fi\\n$\")jZDFF07$$\"5M$3F>*3eiHSF0$\")h26F F07$F^]n$\")e![q#F07$Fc]n$\")sCZEF07$Fh]n$\")%*zDEF07$F]^n$\")9$\\o#F0 7$Fb^n$\")I%o\"GF07$$\"5N3_v:y;O=TF0$\")U\\CGF07$Fd]z$\")YE[GF07$$\"5o \"HK9byU[7%F0$\")([v%GF07$Fg^n$\")UgZGF07$F\\^z$\")q&y%GF07$Fa^z$\")gN YGF07$Ff^z$\")lVSGF07$F\\_n$\")A.3GF07$Fa_n$\")'\\Is#F07$Ff_n$\")](Gf# F07$F``n$\")V$=H#F07$Fj`n$\")q&H;#F07$Fj_z$\")`y))=F07$F_an$\")X#zi\"F 07$Fb`z$\")x\"f^\"F07$Fdan$\")k8#[\"F07$Fian$\")F8u9F07$F^bn$\")\"oAZ \"F07$$\"5o;/E])pk%eVF0$\")iBn9F07$Fcbn$\")Bqu9F07$$\"5oTNr*y*3v\")>XF0$\")s**e:F07$$\"5om\"H# o2(yK_%F0$\")iKe:F07$$\"5,]7`W1*Rn_%F0$\")=qd:F07$Feen$\")3uT:F07$Fefr $\")IO;:F07$Fjfr$\"))=l[\"F07$$\"5omT&QG48[c%F0$\")PZq9F07$F_gr$\")l_n 9F07$$\"5,+D1*y)ylyXF0$\")*pLZ\"F07$Fjen$\")(yq[\"F07$F_fn$\")E]9?F07$$\"5N3FWq))fa%p%F0$\")_@^?F07$Fifn$\")!)Hk?F07$$\"5N$3_vS4[W q%F0$\")9#*3?F07$F^gn$\")M9W>F07$Fcgn$\")*G2-#F07$Fhgn$\")TTKAF07$$\"5 N$e*)fb^cSu%F0$\")\\wbCF07$F]hn$\")96TFF07$$\"5o;a8sA$fsv%F0$\")=aCGF0 7$Fbhn$\")qFrIF07$Fghn$\")yO?LF07$F\\in$\")O.EMF07$Fain$\")e%4L$F07$Ff in$\")MX*4$F07$$\"5-]PMx9;$F07$Fjjn$\")X2aKF07$Fa[s$\")?`' H$F07$Ff[s$\")b-;LF07$F[\\s$\")1x)G$F07$F_[o$\")KkbKF07$$\"5+DJ&p(Q_2 \\\\F0$\")eeUKF07$$\"5+]i:&GAqC&\\F0$\")vOFKF07$$\"5+v$fLp?le&\\F0$\") f8FKF07$Fc\\s$\")ZB?KF07$$\"5+](oz\"f,0m\\F0$\")*[]@$F07$Fh\\s$\")(\\f A$F07$Fb]s$\")c9iKF07$Fd[o$\")CUpLF0-Fi[o6&F[\\o$Fe[oF^\\oF(F_\\o-Fb\\ o6#%8scheme~with~c[10]=81/82G-F$6%7[[mF'7$Fijz$\"'fS7F07$F^[[l$\"'i!)> F07$Fc[[l$\"&.z(F07$F+$!':a5F07$F[\\[l$!'2^MF07$F`\\[l$!'I2`F07$Fe\\[l $!'&Q1&F07$Fj\\[l$!'yFVF07$F_][l$!'^e>F07$F2$!&j?'F07$F8$\"'?nkF07$F=$ \"'J,))F07$FB$\"(fy2\"F07$FG$\"(zET\"F07$$\"5+]iS\">'4qI=F4$\"(tzU\"F0 7$F[^o$\"(cKW\"F07$$\"5M$3_+AUo**)=F4$\"($o89F07$FL$\"(yjM\"F07$Fefv$ \"(>=O\"F07$Fc^o$\"(7@I\"F07$F]gv$\"'&e?)F07$FQ$\"'R9$)F07$$\"5MLe*)fI &*y/@F4$!'f!G\"F07$Fegv$!''H2#F07$$\"5nmTN'fW%4QAF4$!(OYA\"F07$Fjgv$!( Kk#>F07$$\"5++D\"G8O*RrBF4$!(#y!*>F07$F_hv$!(!y=>F07$$\"5ML3FpwUq/DF4$ !(%y15F07$FV$!'O'Q*F07$Fa`[l$\"'@,WF07$Fghv$\"(]g%>F07$Fi`[l$\"(8m1#F0 7$F\\iv$\"(W/7$F07$Faiv$\"(AQ=$F07$Ffiv$\"(c;j$F07$F[jv$\"(l%eOF07$F`j v$\"(]un$F07$Fejv$\"(:1c$F07$Fen$\"(c[%HF07$Fa_o$\"(=\"f?F07$Ff_o$\"(D ./\"F07$F``o$!'O[BF07$Fjn$!'FLXF07$F]ao$\"'')zJF07$F_o$\"(#yF6F07$Fic[ l$\"(d)*G\"F07$F^d[l$\"(Y**G\"F07$Fcd[l$\"(iKH\"F07$Fdo$\"(tXH\"F07$F[ e[l$\"(tVH\"F07$F`e[l$\"(CVF\"F07$Fee[l$\"(]*Q6F07$Fio$\"(^&Q6F07$F^p$ \"'_4')F07$Fcp$\"'6%R&F07$F]q$\"'MC()F07$Fgq$\"(LjF4$\"']K=F07$F_g[l$\"'RJBF07$$\"5NLL$3x\"3NHkF4$\"(A3 \"=F07$Fceo$\"(!4`EF07$Fgg[l$\"(Qp?$F07$F\\h[l$\"('RQRF07$Fah[l$\"(()R !QF07$Fheo$\"(M+f$F07$$\"5NLL$eR(>(R\"oF4$\"(Eif#F07$Fih[l$\"(r`N#F07$ $\"5qmm;/E!z@%pF4$\"(#Q%H\"F07$F]fo$\"'')zLF07$Fai[l$\"'aILF07$Ffi[l$ \"'9%)HF07$F[j[l$!&>#*)F07$F`j[l$!&[\"))F07$Fej[l$!&ui&F07$Far$!&&QVF0 7$Fdgo$\"'ER\"*F07$Ffr$\"(^$3=F07$F[s$\"(\\=O\"F07$F`s$\"(\"=v6F07$F[j o$\")S,'H\"F47$Fes$\"(N%3:F07$Fcjo$\"(eTc\"F07$Fjs$\"(Vp]\"F07$F[[p$\" (1`?\"F07$F_t$\"(R1+\"F07$F[^\\l$\"'hU&*F07$F`^\\l$\"'/(e)F07$Fe^\\l$ \"'G%e)F07$Fc[p$\"'%e[)F07$F]_\\l$\"'v)R)F07$Fb_\\l$\"'=7%)F07$Fg_\\l$ \"'b\")*)F07$Fdt$\"'P_(*F07$Fit$\"((e%z\"F07$F^u$\"(o#HAF07$Fe`\\l$\"( yYU#F07$Fcu$\"(o/!GF07$F]a\\l$\"())G'GF07$Fba\\l$\"(8c,$F07$$\"5+]7G8' G`d3\"F0$\"('[\\IF07$Fga\\l$\"(H*>IF07$$\"5n;/w790u\"4\"F0$\"(jy*GF07$ Fhu$\"(vU$HF07$Fcbw$\"(k%\\BF07$F]v$\"(z+6\"F07$F[cw$\"'N0XF07$F`cw$!& ='HF07$F[c\\l$!&U+$F07$Fecw$!&wK#F07$Fcc\\l$\"'iU@F07$Fbv$\"'%z=$F07$F [d\\l$\"'N@**F07$F`d\\l$\"(./`\"F07$Fed\\l$\"(i&)3$F07$Fg\\p$\"(0;>%F0 7$F]e\\l$\"(y`s&F07$Fgv$\"(=v>'F07$Fd]p$\"(bPH&F07$F^^p$\"(6aS%F07$Fh^ p$\"(LiW$F07$F\\w$\"(6;H#F07$F[f\\l$\"(dFH#F07$F`f\\l$\"(J*zAF07$Fef\\ l$\"((Q]@F07$Faw$\"(=5:#F07$$\"5++]i!R&[Rn7F0$\"(%Hg@F07$F]g\\l$\"(P=G #F07$$\"5++](o/tiTF\"F0$\"(qBG#F07$Ffw$\"(A[J#F07$F[x$\"(lx.$F07$F`x$ \"(s!fMF07$F[gw$\"(2R^$F07$F`gw$\"('zuPF07$Fegw$\"(>`x$F07$Fex$\"(:o!Q F07$$\"5++]iSmy`@8F0$\"(g<#RF07$F]hw$\"(\"y@RF07$$\"5++](oHu0$G8F0$\"( %R;RF07$Fjx$\"(bN)QF07$F_y$\"(bvp$F07$Fdy$\"(CLP$F07$Fci\\l$\"((pMKF07 $Fiy$\"(mr9$F07$F[j\\l$\"(qX;$F07$F^z$\"(*fQKF07$Fcz$\"(vnc$F07$Fhz$\" (J\"RVF07$F][l$\"('G-ZF07$Fb[l$\"(l?*[F07$Fg[l$\"(L,[%F07$F\\\\l$\"(Cc \"QF07$Fa\\l$\"(Dx\"HF07$Ff\\l$\"(\\m!>F07$F[]l$\"'5\"[*F07$F`]l$\"'hU sF07$Fecp$\"(5)y9F07$Fe]l$\"(5[z#F07$F\\ep$\"(DbJ%F07$Fj]l$\"(Z6_&F07$ Fbat$\"(5.K&F07$Fcfp$\"(=B'RF07$$\"5+]Pf$=B\"4$f\"F0$\"(.*=FF07$F_bt$ \"(3O#=F07$$\"5n;zp)\\awrg\"F0$\"(WYu\"F07$F_^l$\"('RQ9F07$Fjgp$\"(<5 \">F07$Fd^l$\"(S`@#F07$Fi^l$\"(y&fFF07$F^_l$\"(S]A$F07$Fc_l$\"(b/%HF07 $Fh_l$\"(XFe#F07$$\"5+]iSmbs&Hx\"F0$\"(](>DF07$Fa_x$\"(hva#F07$$\"5+Dc ,;krO$y\"F0$\"(6/b#F07$$\"5n;a)eO8Poy\"F0$\"(**[b#F07$$\"5M3_v:.rI!z\" F0$\")87`EF47$F]`l$\")Q<`EF47$Fi_x$\"(3nz#F07$Fb`l$\"(/]#HF07$Fg`l$\"( Z^g#F07$F\\al$\"(KB?#F07$F`bl$\"(%eFLF07$Fjbl$\"(iMX%F07$$\"5Me9\"4gi9 (y>F0$\"(^q_%F07$$\"5+]i:NgX3#)>F0$\"(dQg%F07$$\"5nT5Sp%\\aa)>F0$\"(7Y m%F07$$\"5MLek.HW#)))>F0$\"(<+i%F07$$\"5n;a8s(Hkb*>F0$\"(Zm_%F07$F_cl$ \"(pJ(QF07$$\"5nmTgx.Ry:?F0$\"(4G,$F07$Fdcl$\"(jL2#F07$F_b]l$\"('eUEF0 7$Ficl$\"(k-9%F07$Fgb]l$\"(n$ydF07$F\\c]l$\"(5\"opF07$Fac]l$\"(aUU(F07 $F^dl$\"(=1L)F07$Fiax$\"(.$o&)F07$F]\\q$\"(k:q)F07$Fabx$\"(YOP)F07$Fcd l$\"(ZL9)F07$Fe\\q$\"(gCv'F07$Fhdl$\"(+U)eF07$F]]q$\"(a:=&F07$F]el$\"( V3x%F07$Fe]q$\"(![f\\F07$Fbel$\"(fwK&F07$Fgel$\"(!exfF07$F\\fl$\"(54W' F07$F]f]l$\"(&*R]'F07$Fafl$\"(!*=a'F07$Fef]l$\"(!)QY'F07$Fffl$\"(RNL'F 07$F[gl$\"((z/gF07$F`gl$\"(??\"eF07$$\"5M3-Q\"eZv&pAF0$\"(=rx&F07$F]ex $\"(-bw&F07$$\"5+D1*G@$>FwAF0$\"(Izw&F07$Fegl$\"(\"oieF07$Feex$\"(h9!f F07$Fjex$\"(O&3fF07$F_fx$\"(!oLhF07$Fjgl$\"(j)3iF07$Fdhl$\"(7%)3(F07$F ^il$\"(5^Q(F07$$\"5nmTN'fW2XN#F0$\"(.n$pF07$Fcil$\"(.#[hF07$$\"5nm\"H# =&>0\"zBF0$\"(rKy%F07$Fhil$\"(-aY$F07$Fai]l$\"(7TN$F07$Ffi]l$\"(R@.$F0 7$F`j]l$\"()[RKF07$Fjj]l$\"(rIJ$F07$$\"5+]7yDk!pXU#F0$\"(A8/%F07$F_[^l $\"(S$RaF07$$\"5M$3FW?1NyV#F0$\"(7)GbF07$F]jl$\"(Qqx'F07$Fg[^l$\"(#*) \\oF07$F\\\\^l$\"(jG#yF07$Fa\\^l$\"(7/q(F07$Ff\\^l$\"(q,k(F07$F[]^l$\" (B/&yF07$F`]^l$\"(k;s(F07$Fe]^l$\"(DDa(F07$Fj]^l$\"(lG%pF07$F_^^l$\"(@ %HaF07$Fbjl$\"(&QkRF07$$\"5M$3-jP(=U/DF0$\"(&y]KF07$Ffhx$\"(2K*HF07$$ \"5M$eRAr^,#=DF0$\"(XA&HF07$Fgjl$\"(%*H.$F07$Fcix$\"(?$pKF07$F\\[m$\"( i8U$F07$Fa[m$\"(&)z*QF07$Ff[m$\"(T\"yWF07$$\"5o;a8(R+o2h#F0$\"(b\"*f%F 07$Fi_^l$\"(6$)f%F07$$\"5M$eR(*Rhlnh#F0$\"(a#)f%F07$Fiaq$\"(P6f%F07$Fa `^l$\"(0ye%F07$Ff`^l$\"(9he%F07$F[a^l$\"(VBa%F07$F^bq$\"(&p^WF07$Fcbq$ \"()H]UF07$F[\\m$\"(MB0%F07$Fia^l$\"(;k+%F07$F^b^l$\"(@-#RF07$Fcb^l$\" (]+#RF07$F[cq$\"(5N\"RF07$F[c^l$\"(I#zQF07$F`cq$\"(c(>RF07$Fecq$\")2DW SF47$F`\\m$\"(aN<%F07$$\"5oT5S%4vppq#F0$\"(Z\"zTF07$$\"5M$3-8Da3-r#F0$ \"(efE%F07$$\"5,DJ?3MtW8FF0$\"(mfE%F07$Fe\\m$\"(>rE%F07$$\"5,]i!*y3P;B FF0$\"(B-G%F07$Fj\\m$\"(n-A%F07$F_]m$\"(NB.%F07$Fd]m$\"('oBQF07$Fed^l$ \"(_*ePF07$Fjd^l$\"(-Bk$F07$F_e^l$\"(M>k$F07$Fi]m$\"(Efh$F07$Fge^l$\"( `Vb$F07$F^^m$\"(a+h$F07$Fc^m$\"((RAQF07$Fh^m$\"($QtTF07$Fef^l$\"(oJ'\\ F07$F]_m$\"(pRj&F07$F]g^l$\"(@$fgF07$Fbg^l$\"(k\"4jF07$Fgg^l$\"(1FR&F0 7$Fb_m$\"(8Gk%F07$F_h^l$\"(b#3YF07$Fdh^l$\"(gF(RF07$Fih^l$\"(lv.%F07$F \\^y$\"(8.3%F07$$\"5,]i:N54\"3$HF0$\"(O(fSF07$Fai^l$\"(mW7%F07$$\"5,]( =#\\/xNPHF0$\"(`PC%F07$Fg_m$\"('34]F07$$\"5,+DJq&*y?+\"F07$F\\_y$\")!z'z5F07$$\"5,+voag'ek)HF0$\"):d+6F07$Fa` m$\")'pV6\"F07$Fi_y$\")v_!3\"F07$Ff`m$\"(s)y&*F07$Fagq$\"(8EH)F07$Ffgq $\"((z-xF07$Fg[_l$\"(#)em(F07$F\\\\_l$\"(+bS(F07$Fa\\_l$\"(r\"3uF07$F[ hq$\"(YtS(F07$Fi\\_l$\"(mtS(F07$F^]_l$\"(q(3uF07$Fc]_l$\"(D!=uF07$F[am $\"(\"p@wF07$Fchq$\"(dpZ)F07$F`am$\"(6I:*F07$Feam$\"(%RH()F07$Fjam$\"( ;zF07$Fgdm$\"(^ YZ'F07$F\\em$\"(&42[F07$F`cy$\"(')4#[F07$Fecy$\"(9n)[F07$Fjcy$\"(Ll:&F 07$Faem$\"(e\\j&F07$Fbdy$\"(Uw'eF07$Fgdy$\"($ypfF07$F\\ey$\"(Ra#fF07$F aey$\"(2b!fF07$Ffey$\"(tow&F07$Ffem$\"(%4kbF07$F[fm$\"))3jV&F47$F`fm$ \"($fL`F07$Fefm$\"(8V-&F07$Fjfm$\"(IU\"\\F07$F_gm$\"(?F,&F07$Fdgm$\"(q eG&F07$Figm$\"(7&)>'F07$F^hm$\"(NRn'F07$Fai_l$\"(w^@(F07$Fchm$\"(3yo(F 07$Fii_l$\"(B4x(F07$F^j_l$\"(#\\hzF07$Fcj_l$\"(NEm(F07$Fhhm$\"(J8$pF07 $F]im$\"(Q^I'F07$Fbim$\"(%RAgF07$F_]`l$\"(H&=jF07$Fgim$\"('e(=(F07$Fg] `l$\"(h'*p(F07$F\\jm$\"(9JO*F07$$\"5o;/,W%)o=fQF0$\")lvN5F07$F\\jy$\") t>M6F07$$\"5,]7.2ZC/sQF0$\")z$fE\"F07$Fajm$\"),a,8F07$$\"5,voH/>To\")Q F0$\")%*yX8F07$Fe^`l$\")9=b8F07$$\"5o\"H2e.!>6))QF0$\")*o@O\"F07$Fdjy$ \")agd8F07$F]_`l$\")Rwk8F07$Fb_`l$\")3uj8F07$Fg_`l$\")&4RJ\"F07$Ffjm$ \")R(*=8F07$F[[n$\")4eO7F07$F`[n$\")eSy5F07$Fe[n$\")>\"4-\"F07$Fj[n$\" (pa***F07$F_\\n$\")2cv5F07$Fd\\n$\")')3n6F07$F[a`l$\")u@u6F07$Fi\\n$\" )sIv6F07$Fca`l$\")WRk6F07$F^]n$\")Fkf6F07$Fc]n$\")]'f6\"F07$Fh]n$\").n *4\"F07$F]^n$\")#>X9\"F07$Fb^n$\")_IX7F07$Fgb`l$\")XO^7F07$Fd]z$\")X#* p7F07$F_c`l$\")shp7F07$Fg^n$\")s/q7F07$F\\^z$\")C*4F\"F07$Fa^z$\")zIq7 F07$Ff^z$\"))fjE\"F07$F\\_n$\")\")HU7F07$Fa_n$\")L2!=\"F07$Ff_n$\")&RN 3\"F07$$\"5NL3Fp^yftTF0$\"(P$y)*F07$$\"5NL$3_+sD-=%F0$\"(61#*)F07$$\"5 NLe9T)e`o=%F0$\"(Wrz)F07$F[`n$\"(Fy8)F07$$\"5NL3-8D$4,?%F0$\"(BT%zF07$ $\"5NL$e*[$>Pn?%F0$\"(Bn#yF07$$\"5NLe*[=1lL@%F0$\"(YsB)F07$F``n$\"(KeH *F07$$\"5NL$3Fpm[KB%F0$\")(=c/\"F07$Fe`n$\")2dP6F07$$\"5N$3_DyL=)\\UF0 $\")rQ$>\"F07$$\"5NL3_+sA8`UF0$\")%Gc=\"F07$$\"5N$e*[=1iWcUF0$\")-pm6F 07$$\"5NL$ek.9g(fUF0$\")YlU6F07$$\"5NLeRs3!)QmUF0$\")!485\"F07$Fj`n$\" (1?#)*F07$$\"5o;ajMj))ezUF0$\"(9&y')F07$Fj_z$\"(!>o!)F07$$\"5N$eRse$[t #H%F0$\"(.u#pF07$F_an$\"(%*H['F07$$\"5-]P%)R33)eI%F0$\"(qf9'F07$Fb`z$ \"(_3.'F07$$\"5o;zW#4yE!>VF0$\"('\\8gF07$Fdan$\"(\"G\\gF07$F^bn$\"(\"4 SiF07$Fhbn$\"(0)QlF07$F]cn$\"(T`1(F07$Fbcn$\"(i#osF07$$\"5M$3_]f6$Q3WF 0$\"(0^G(F07$Fgcn$\"(]IL(F07$$\"5,]iST**GY?WF0$\"(&HcsF07$F\\dn$\"(!R] sF07$Fadn$\"(2$=qF07$Ffdn$\"(l,q'F07$$\"5M3-)QRI#o`WF0$\"(=,q'F07$$\"5 ,](o/)\\AqcWF0$\"(Iep'F07$$\"5o\"Hdqc>A(fWF0$\"(bji'F07$F[en$\"(%)*3mF 07$$\"5o;H#oK.#yoWF0$\"(g)3mF07$F`en$\"(;Wh'F07$Fcer$\")'Q4u'F47$Fher$ \"(Ei%pF07$Fii`l$\"(MX)pF07$F]fr$\"(1Z+(F07$Faj`l$\"(`'ypF07$Ffj`l$\"( vN(pF07$F[[al$\"(K)opF07$Feen$\"(#yZoF07$Fefr$\"(4^l'F07$Fjfr$\"(o#GkF 07$$\"5M$3_vS*=NhXF0$\"()f0jF07$Fi[al$\"(,jI'F07$$\"5,]i:g\"Hu#oXF0$\" ()[*H'F07$F_gr$\"(]Vr% F0$\"(O$*4)F07$$\"5o;zpB,4l5F07$$\"5oTg-N\"F07$Fg_al$\")mqS7F07$Fbhn$\").K-9F07$Fghn$\")(3_b \"F07$F\\in$\")YE+;F07$Feir$\")-yb:F07$Fain$\")(*)oZ\"F07$F]jr$\")&HlP \"F07$Ffin$\")3Nt7F07$F[aal$\")k*RF\"F07$F`aal$\")mAt7F07$Feaal$\")/Pf 7F07$F[jn$\")crf7F07$$\"5-](=n))Q4m&[F0$\")oFg7F07$F]bal$\"),(*p7F07$$ \"5-]iSTux#H'[F0$\")RUs7F07$F`jn$\")g4t7F07$Fejn$\")\\8>8F07$Fjjn$\")2 &)*Q\"F07$Fa[s$\")]8A9F07$Ff[s$\")R(pV\"F07$F[\\s$\")AL;9F07$F_[o$\")p >\"R\"F07$F\\dal$\")&H(p8F07$Fc\\s$\")kIk8F07$$\"5+Dcw4v^li\\F0$\")yFg 8F07$Fidal$\");Og8F07$$\"5+v=qx9F0-F]jz6&F[\\oF_jzF(F(-Fb\\o6#%:scheme~wit h~c[10]=169/170G-%%FONTG6$%*HELVETICAGFcdv-%&TITLEG6#%Verror~curves~fo r~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F^bhl- %%VIEWG6$;F(Fd[o%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10 ]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme w ith c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 13 of 11 stage, order 7 Runge-Kutt a methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathematica in Action\" by Stan Wagon, Springer-Verlag, page 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\" %#dxG!\"\",&*&%$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#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation " } {XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponenti al term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5 " "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this t erm disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$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 slight deviation of a nume rical solution from the correct solution tends to become rapidly magni fied." }}{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,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#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$$\"\"!F)$!3A+++++++S!#=7$$\"3E LLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLL LLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm ^&Q%R)F,$!39J7$$\"3))** ***\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LLL=\"\\g**FK7 $$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g (y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@E VOJ&[$F,7$$\"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$$\"3s******z-6jY FO$!3_=%f%oq`+=F,7$$\"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%)*******HSu]&FO$!3%*oc=HW4cUF,7$$\"3'fmm\" HOq&e&FO$!3oqc`'[F/N%F,7$$\"3'HLL$ep'Rm&FO$!3$e%**GFr7=WF,7$$\"3D***\\ P?[nq&FO$!3TlAsE+sVWF,7$$\"3Umm;\\%H&\\dFO$!3y[ey96=hWF,7$$\"3eLLe%p5B z&FO$!3=)zg%Q%y/Z%F,7$$\"3')******R>4NeFO$!3waa0%)\\frWF,7$$\"3HLL$ed* f:fFO$!3]_J$4k<:X%F,7$$\"3#emm;@2h*fFO$!3V5vHeMg-WF,7$$\"37LLL))3E!3'F O$!3=l`a'y%*4K%F,7$$\"3]*****\\c9W;'FO$!3>=$e-d.)3UF,7$$\"3Lmmmmd'*GjF O$!3Gy*y<4!G/RF,7$$\"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$$\"3=+++XhUkwFO$\"3Z X^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,- %'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABEL SG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]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 0 735 "E := (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := \+ 0: y0 := -2/5:\nmatrix([[`slope field: `,E(x,y)],[`initial point: `, ``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[1 0]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n En_RK7_||ct := RK7_||ct(E(x,y),x,y,x0,eval f(y0),hh,numsteps,false);\n sm := 0: numpts := nops(En_RK7_||ct):\n \+ for ii to numpts do\n sm := sm+(En_RK7_||ct[ii,2]-e(En_RK7_||ct [ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend d o:\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)pprint596\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+# \"#6\"#7$\"+y#3d0#!#?7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+tt#HE%!#@7 $/F8#\"#>\"#?$\"++=J\"[&F?7$/F8#\"#K\"#L$\"+8c![6%F?7$/F8#\"#\")\"##)$ \"+b3F#*>F?7$/F8#\"$p\"\"$q\"$\"+?\"G7)>F?Q)pprint606\"" }}}{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-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 = 7.999;" "6#/%\"xG-%&FloatG6$ \"%**z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 666 "E := (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := \+ 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope field: `,E(x,y)],[`initia l point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nums teps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32 /33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := \+ []:\nDigits := 20:\nfor ct to 6 do\n en_RK7_||ct := RK7_||ct(E(x,y), x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 7.999: exx := eval f(e(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(en_RK7_||ct(xx)-ex x)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\" !#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint6 16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+?I:R6!#>7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$ \"+-TEiB!#?7$/F8#\"#>\"#?$\"+I7UPIF?7$/F8#\"#K\"#L$\"+'f%=!G#F?7$/F8# \"#\")\"##)$\"+(****R5\"F?7$/F8#\"$p\"\"$q\"$\"+S2)y4\"F?Q)pprint626\" " }}}{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 inter val " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of ea ch Runge-Kutta method is estimated as follows using the special proced ure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integra tion by the 7 point Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "mthds := [`Sharp-Verner s cheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[1 0]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n \+ sm := NCint((e(x)-'en_RK7_||ct'(x))^2,x=0..8,adaptive=false,numpoints =7,factor=200);\n errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&% \"cG6#\"#5F+#\"#6\"#7$\"+@()y\"#?$\"+wO?!Q&F?7$/F8#\"#K\"#L$\"+AR!*QSF?7$/F8# \"#\")\"##)$\"+K9_b>F?7$/F8#\"$p\"\"$q\"$\"+$R#oW>F?Q)pprint636\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "evalf[20]( plot([e(x)-'en_RK7_1'(x),e(x)-'en_RK7_2'(x),e(x)-'en_RK7_3'(x),e(x)-'e n_RK7_4'(x),\ne(x)-'en_RK7_5'(x),e(x)-'en_RK7_6'(x)],x=0..2,-1.7e-16.. 7.3e-16,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0 ),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-V erner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with \+ c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`sche me with c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 587 587 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5MLLLLL3VfV!#@$\"$(>!#?7 $$\"5nmmmm\"H[D:)F-$\"$/%F07$$\"5LLLLLe0$=C\"F0$\"$F'F07$$\"5LLLLL3RBr ;F0$\"$j)F07$$\"5nmmm;zjf)4#F0$\"%<6F07$$\"5MLLL$e4;[\\#F0$\"%x8F07$$ \"5++++]i'y]!HF0$\"%e;F07$$\"5MLLL$ezs$HLF0$\"%h>F07$$\"5++++]7iI_PF0$ \"%*G#F07$$\"5nmmmm;_M(=%F0$\"%oEF07$$\"5MLLLL3y_qXF0$\"%!*HF07$$\"5++ +++]1!>+&F0$\"%AMF07$$\"5+++++]Z/NaF0$\"%SQF07$$\"5+++++]$fC&eF0$\"%uU F07$$\"5MLLL$ez6:B'F0$\"%.ZF07$$\"5nmmmm;=C#o'F0$\"%H_F07$$\"5nmmmmm#p S1(F0$\"%>dF07$$\"5,+++]i`A3vF0$\"%9jF07$$\"5nmmmmm(y8!zF0$\"%()oF07$$ \"5,+++]i.tK$)F0$\"%avF07$$\"5,+++](3zMu)F0$\"%Q#)F07$$\"5nmmmm\"H_?<* F0$\"&_+*F-7$$\"5nmmm;zihl&*F0$\"&!e(*F-7$$\"5MLLLL3#G,***F0$\"'!H1\"F -7$$\"5LLLLezw5V5!#>$\"'Cg6F-7$$\"5++++v$Q#\\\"3\"Fas$\"'@^7F-7$$\"5LL LL$e\"*[H7\"Fas$\"()yc8!#A7$$\"5+++++qvxl6Fas$\"'ou9F-7$$\"5++++]_qn27 Fas$\"'.*f\"F-7$$\"5++++Dcp@[7Fas$\"']Ge#F07$$\"5nmmm\"H!o-*\\ \"Fas$\"&py#F07$$\"5++++DTO5T:Fas$\"&x,$F07$$\"5nmmmmT9C#e\"Fas$\"&PE$ F07$$\"5++++D1*3`i\"Fas$\"&Na$F07$$\"5LLLLL$*zym;Fas$\"&b$QF07$$\"5LLL L$3N1#4Fas$\"&k='F07$$\"5++++v.Uac>Fas$\"&vo'F07$$\"\"#F)$\"&p E(F0-%&COLORG6&%$RGBGF($Fez!\"\"$\"\"\"F)-%'LEGENDG6#%ESharp-Verner~sc heme~with~c[10]=11/12G-F$6%7SF'7$F+$\"#iF07$F2$\"$D\"F07$F7$\"$%>F07$F <$\"$k#F07$FA$\"$Q$F07$FF$\"$8%F07$FK$\"$#\\F07$FP$\"$w&F07$FU$\"$n'F0 7$FZ$\"$n(F07$Fin$\"$_)F07$F^o$\"$j*F07$Fco$\"%p5F07$Fho$\"%y6F07$F]p$ \"%$G\"F07$Fbp$\"%89F07$Fgp$\"%I:F07$F\\q$\"%s;F07$Faq$\"%1=F07$FfqFR7 $F[r$\"%=@F07$F`r$\"&:H#F-7$Fer$\"&1Y#F-7$Fjr$\"&+l#F-7$F_s$\"&E'GF-7$ Fes$\"&,1$F-7$Fjs$\"''**G$F^t7$F`t$\"&oa$F-7$Fet$\"&V\"QF-7$Fjt$\"&(*3 %F-7$F_u$\"&eU%F-7$Fdu$\"%YZF07$Fiu$\"%6^F07$F^v$\"%raF07$Fcv$\"%$)eF0 7$Fhv$\"%5jF07$F]w$\"%&y'F07$Fbw$\"%(H(F07$Fgw$\"%zyF07$F\\x$\"%&[)F07 $Fax$\"%e\"*F07$Ffx$\"%z)*F07$F[y$\"&\"f5F07$F`y$\"&%[6F07$Fey$\"&XB\" F07$Fjy$\"&RL\"F07$F_z$\"&qV\"F07$Fdz$\"&[b\"F0-Fiz6&F[[lF($\"\"(F][lF (-Fa[l6#%8scheme~with~c[10]=15/16G-F$6%7SF'7$F+$\"#zF07$F2$\"$h\"F07$F 7$\"$\\#F07$F<$\"$R$F07$FA$\"$M%F07$FF$\"$I&F07$FK$\"$K'F07$FP$\"$S(F0 7$FU$\"$c)F07$FZ$\"$&)*F07$Fin$\"%&4\"F07$F^o$\"%O7F07$Fco$\"%t8F07$Fh o$\"%8:F07$F]p$\"%\\;F07$Fbp$\"%:=F07$Fgp$\"%l>F07$F\\q$\"%[@F07$Faq$ \"%?BF07$Ffq$\"%?DF07$F[r$\"%@FF07$F`r$\"&S%HF-7$Fer$\"&:;$F-7$Fjr$\"& ^S$F-7$F_s$\"&%yOF-7$Fes$\"&B$RF-7$Fjs$\"'\"yA%F^t7$F`t$\"&xb%F-7$Fet$ \"&F0-Fiz6&F[[l$\"\"*F][l$\"\"%F][lF(-Fa[l6#%8scheme~with~c[10]=19/20G- F$6%7SF'7$F+$!\"(F07$F2$!#;F07$F7$!#FF07$F<$!#UF07$FA$!#gF07$FF$!#!)F0 7$FK$!$.\"F07$FP$!$J\"F07$FU$!$i\"F07$FZ$!$/#F07$Fin$!$S#F07$F^o$!$&HF 07$Fco$!$\\$F07$Fho$!$1%F07$F]p$!$m%F07$Fbp$!$T&F07$Fgp$!$9'F07$F\\q$! $0(F07$Faq$!$'zF07$Ffq$!$/*F07$F[r$!%=5F07$F`r$!&!\\6F-7$Fer$!&'z7F-7$ Fjr$!&/W\"F-7$F_s$!&'=;F-7$Fes$!&ry\"F-7$Fjs$!'h\")>F^t7$F`t$!&')>#F-7 $Fet$!&EV#F-7$Fjt$!&6o#F-7$F_u$!&e(HF-7$Fdu$!%qKF07$Fiu$!%7OF07$F^v$!% ZRF07$Fcv$!%`VF07$Fhv$!%hZF07$F]w$!%G_F07$Fbw$!%1\" F07$Fjy$!&(f6F07$F_z$!&8E\"F07$Fdz$!&3Q\"F0-%'COLOURG6&F[[l$\"*++++\"! \")F(F^hm-Fa[l6#%8scheme~with~c[10]=32/33G-F$6%7SF'7$F+$!\"$F07$F2$!\" 'F07$F7$!\"*F07$F<$!#F07$F]p$!$E#F07$Fbp$!$i#F07$Fgp$!$(HF07$F\\q$!$U$F07$Faq $!$&QF07$Ffq$!$P%F07$F[r$!$#\\F07$F`r$!%`bF-7$Fer$!%%='F-7$Fjr$!%hpF-7 $F_s$!%>yF-7$Fes$!%I')F-7$Fjs$!&\"o&*F^t7$F`t$!&41\"F-7$Fet$!&Q<\"F-7$ Fjt$!&TH\"F-7$F_u$!&cV\"F-7$Fdu$!%x:F07$Fiu$!%UF07$Fcv$!% *4#F07$Fhv$!%&H#F07$F]w$!%?DF07$Fbw$!%bFF07$Fgw$!%CIF07$F\\x$!%0LF07$F ax$!% " 0 "" {MPLTEXT 1 0 498 "evalf[20](p lot([e(x)-'en_RK7_1'(x),e(x)-'en_RK7_2'(x),e(x)-'en_RK7_3'(x),e(x)-'en _RK7_4'(x),\ne(x)-'en_RK7_5'(x),e(x)-'en_RK7_6'(x)],x=2..8,font=[HELVE TICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0 ),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme with c [10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/20`,`schem e with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10]=169/1 70`],\ntitle=`error curves for 11 stage order 7 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 1076 686 686 {PLOTDATA 2 "6,-%'CURVESG 6%7Z7$$\"\"#\"\"!$\"&pE(!#?7$$\"5+++++DHyI@!#>$\"&.P*F-7$$\"5++++v[kdW AF1$\"'Gp6F-7$$\"5++++]n\"\\DP#F1$\"'n-:F-7$$\"5++++]s,P,DF1$\"':N>F-7 $$\"5++++v8*y&HEF1$\"'!H\\#F-7$$\"5++++vG[W[FF1$\"'7aJF-7$$\"5++++v)fB :(GF1$\"'uESF-7$$\"5++++vQ=\"))*HF1$\"'*f=&F-7$$\"5++++vj=pDJF1$\"'jwm F-7$$\"5+++++lN?cKF1$\"'Ih')F-7$$\"5++++]U$e6P$F1$\"(Z%*3\"F-7$$\"5+++ ++&>q0]$F1$\"(z1T\"F-7$$\"5+++++DM^IOF1$\"(9*G=F-7$$\"5+++++0ytbPF1$\" (p!\\BF-7$$\"5++++vQNXpQF1$\"(/([HF-7$$\"5+++++XDn/SF1$\"(*>kQF-7$$\"5 +++++!y?#>TF1$\")1.f[!#@7$$\"5++++v3wY_UF1$\"*C7HM'!#A7$$\"5+++++IOTqV F1$\")Y\\I!)F^q7$$\"5++++v3\">)*\\%F1$\")wGS5F-7$$\"5++++DEP/BYF1$\")O 0J8F-7$$\"5++++](o:;v%F1$\")6S@n(\\cF-7$$\"5++++vB4JBmF1$\"*#fKssF-7$$\"5+ ++++DVsYnF1$\"*%=F3$*F-7$$\"5++++v=n#f(oF1$\"+?RH07F-7$$\"5+++++!)RO+q F1$\"+#\\()ea\"F-7$$\"5++++]_!>w7(F1$\"+s;$R*>F-7$$\"5++++v)Q?QD(F1$\" ,D)yTmDF^q7$$\"5+++++5jyptF1$\"-S<'RjB$Fdq7$$\"5++++DE8COuF1$\",zIwjp$ F^q7$$\"5++++]Ujp-vF1$\",>!f!=A%F^q7$$\"5++++D,X8ivF1$\",J'psaZF^q7$$ \"5+++++gEd@wF1$\"+e*=\\N&F-7$$\"5+++]PMh%\\o(F1$\"+0w_ygF-7$$\"5++++v 3'>$[xF1$\"+xs\"***oF-7$$\"5+++++5h(*3yF1$\"+?i&)*y(F-7$$\"5++++D6Ejpy F1$\"+#*yd%z)F-7$$\"5+++vVeWA-zF1$\"+)>PpQ*F-7$$\"5+++]i0j\"[$zF1$\",k \\>>+\"F-7$$\"5+++D\"G:3u'zF1$\",P%QSp5F-7$$\"\")F*$\",GgL99\"F--%&COL ORG6&%$RGBG$F*F*$F)!\"\"$\"\"\"F*-%'LEGENDG6#%ESharp-Verner~scheme~wit h~c[10]=11/12G-F$6%7Z7$F($\"&[b\"F-7$F/$\"&!*)>F-7$F5$\"&kY#F-7$F:$\"& T:$F-7$F?$\"&X/%F-7$FD$\"&j>&F-7$FI$\"&(R?F-7$F]u$\")cV,EF-7$Fbu$ \")133MF-7$Fgu$\")N:WVF-7$F\\v$\")L1HcF-7$Fav$\")'z(=rF-7$Ffv$\")%[??* F-7$F[w$\"*G%fr6F-7$F`w$\"*Bl!3:F-7$Few$\"*Gg-$>F-7$Fjw$\"*.B%*\\#F-7$ F_x$\"*,8d?$F-7$Fdx$\"*^B[8%F-7$Fix$\"+$f*)>K&F^q7$F^y$\",Kv27r'Fdq7$F cy$\"+6o=lwF^q7$Fhy$\"+A6xa()F^q7$F]z$\"+J0*)f)*F^q7$Fbz$\"+z4X56F-7$F gz$\"+Pe]g7F-7$F\\[l$\"+jv$3V\"F-7$Fa[l$\"+fVQ:;F-7$Ff[l$\"+lOtB=F-7$F [\\l$\"+38dY>F-7$F`\\l$\"+?Fox?F-7$Fe\\l$\"+?^iF-7$F/$\"&sb#F-7$F5$\"&6<$F-7$F:$\"&`0%F-7$F?$\"&.?&F-7$FD$ \"&8o'F-7$FI$\"&pV)F-7$FN$\"'gv5F-7$FS$\"'!QQ\"F-7$FX$\"'S!y\"F-7$Fgn$ \"'t3BF-7$F\\o$\"'R.HF-7$Fao$\"'0fPF-7$Ffo$\"'Ut[F-7$F[p$\"'hfiF-7$F`p $\"''y&yF-7$Fep$\"(I)H5F-7$Fjp$\")D/&H\"F^q7$F`q$\"*y\\1p\"Fdq7$Ffq$\" )NfS@F^q7$F[r$\"(ZJx#F-7$F`r$\"(E%[NF-7$Fer$\"(H#*e%F-7$Fjr$\"(\"*=\"e F-7$F_s$\"(&>)\\(F-7$Fds$\"(&fp(*F-7$Fis$\")(=+B\"F-7$F^t$\")_Ux:F-7$F ct$\")vmR?F-7$Fht$\")2pAEF-7$F]u$\").&\\M$F-7$Fbu$\")?9#Q%F-7$Fgu$\")K v&e&F-7$F\\v$\")A!zB(F-7$Fav$\")LR`\"*F-7$Ffv$\"*(z?$=\"F-7$F[w$\"*&oW 1:F-7$F`w$\"*M%3R>F-7$Few$\"*oY>[#F-7$Fjw$\"*X\"y8KF-7$F_x$\"*pN>7%F-7 $Fdx$\"*k%f;`F-7$Fix$\"+rQ1VoF^q7$F^y$\",%[NLH')Fdq7$Fcy$\"+B(of&)*F^q 7$Fhy$\",,k'pD6F^q7$F]z$\",a#Rzn7F^q7$Fbz$\"+J#GyU\"F-7$Fgz$\"+p,x?;F- 7$F\\[l$\"+#H%yR=F-7$Fa[l$\"+%Gwq2#F-7$Ff[l$\"+nP(\\M#F-7$F[\\l$\"+b&> H]#F-7$F`\\l$\"+KQ]rEF-7$Fe\\l$\"+xIW^GF-7$Fj\\l$\"+M@]VIF--F_]l6&Fa]l $\"\"*Fd]l$\"\"%Fd]lFb]l-Fh]l6#%8scheme~with~c[10]=19/20G-F$6%7Z7$F($! &3Q\"F-7$F/$!&\\!=F-7$F5$!&jF#F-7$F:$!&$\\HF-7$F?$!&\\#QF-7$FD$!&([\\F -7$FI$!&7G'F-7$FN$!&r.)F-7$FS$!'$o.\"F-7$FX$!'DO8F-7$Fgn$!'aMF-7$F]u$!)i66D F-7$Fbu$!)tr*G$F-7$Fgu$!)-D$>%F-7$F\\v$!).]LaF-7$Fav$!)&R9(oF-7$Ffv$!) sI#)))F-7$F[w$!*'Q)38\"F-7$F`w$!*FicX\"F-7$Few$!*U'=j=F-7$Fjw$!*xrDT#F -7$F_x$!*0?V4$F-7$Fdx$!*!o9\"*RF-7$Fix$!+@I1P^F^q7$F^y$!,3D6!ykFdq7$Fc y$!+>L%))R(F^q7$Fhy$!+o#p0X)F^q7$F]z$!+k/H<&*F^q7$Fbz$!+_k'=2\"F-7$Fgz $!+[vq;7F-7$F\\[l$!+#)47\"Q\"F-7$Fa[l$!+'eb#f:F-7$Ff[l$!+ycOg]F-7$FX$!&(pkF-7$Fgn$!&%)R)F-7 $F\\o$!'wc5F-7$Fao$!'ho8F-7$Ffo$!'XuF-7$Fix$!+]%=s[#F^q7$F^y$!,D8nk8$Fdq7 $Fcy$!+:wI#e$F^q7$Fhy$!+rJ_\"4%F^q7$F]z$!+)**)*zg%F^q7$Fbz$!*()p'*=&F- 7$Fgz$!*%*[4*eF-7$F\\[l$!*m\"*po'F-7$Fa[l$!*No%\\vF-7$Ff[l$!*x&=B&)F-7 $F[\\l$!*zls4*F-7$F`\\l$!*$H,5(*F-7$Fe\\l$!+wJSO5F-7$Fj\\l$!+6-@16F--F _]l6&Fa]l$\"\"&Fd]lFb]lFe]l-Fh]l6#%8scheme~with~c[10]=81/82G-F$6%7Z7$F ($!%^mF-7$F/$!%$p)F-7$F5$!&i4\"F-7$F:$!&-U\"F-7$F?$!&=%=F-7$FD$!&HQ#F- 7$FI$!&W-$F-7$FN$!&)pQF-7$FS$!&B*\\F-7$FX$!&RV'F-7$Fgn$!&;N)F-7$F\\o$! '%30\"F-7$Fao$!'$4O\"F-7$Ffo$!'[kF-7$Fix$!+dCVtCF^q7$F^y$!,+[#3>JFdq7$Fcy$!+ac'*F-7$Fe\\l$!+(fe1.\"F-7$Fj\\l$!+-(y+5\"F--F]_n6&Fa]lF__nFb] lFb]l-Fh]l6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGF[dm-%&T ITLEG6#%Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%+AXES LABELSG6$Q\"x6\"Q!Fcfo-%%VIEWG6$;F(Fj\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10] =11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "scheme wi th c[10]=32/33" "scheme with c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 14 of 11 stage , order 7 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 "Solution: " }}{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/-%%diffG6$-%\"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*&-%$ex pG6#,$*&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$$\"3GLLL3x &)*3\"!#=$\"3(**[ro!GyVzF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\" 3umm\"H2P\"Q?F@$\"3/$)oqvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3 !***\\PMnNrDF@$\"3M:4%*3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3 MLL$eRwX5$F@$\"3)GTJ!oG0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33 ML$3x%3yTF@$\"3n$**Q]`\"yRLF@7$$\"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$$\"3GLL 3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"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$F,$!3D 1>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!y EF,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[gfMO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$ $\"\"&F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%* HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULT G" 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 fo llowing 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 0 744 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`slope fi eld: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme \+ with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/ 20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with \+ `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n Bn_RK 7_||ct := RK7_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm \+ := 0: numpts := nops(Bn_RK7_||ct):\n for ii to numpts do\n sm : = sm+(Bn_RK7_||ct[ii,2]-b(Bn_RK7_||ct[ii,1]))^2;\n end do:\n errs \+ := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\" yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width: ~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint646\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7( 7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+`cCD q!#?7$/*&%-scheme~with~GF+F,F+#\"#:\"#;F37$/F8#\"#>\"#?$\"+\\cCDqF57$/ F8#\"#K\"#LF37$/F8#\"#\")\"##)F37$/F8#\"$p\"\"$q\"$\"+bcCDqF5Q)pprint6 56\"" }}}{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-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 = 4.999;" "6 #/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 675 "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 := [`Sharp-Verner s cheme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[1 0]=19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []:\nDigits := 25:\nfor ct to 6 do\n \+ bn_RK7_||ct := RK7_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\ne nd do:\nxx := 4.999: bxx := evalf(b(xx)):\nfor ct to 6 do\n errs := \+ [op(errs),abs(bn_RK7_||ct(xx)-bxx)];\nend do:\nDigits := 10:\nlinalg[t ranspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F +F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~ width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint686\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\" +rsCRB!#C7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+<\\#=%)*!#D7$/F8#\"#> \"#?$\"+$p<=F\"F57$/F8#\"#K\"#L$\"+%f(QMKF?7$/F8#\"#\")\"##)$\"+w'\\k) >F?7$/F8#\"$p\"\"$q\"$\"+\\*QL*>F?Q)pprint696\"" }}}{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 0 413 "mthds := [`Sharp-Verner scheme with `*c[10]=11/ 12,`scheme with `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with \+ `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170] : errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((b(x)-'bn _RK7_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n er rs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose ]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"#7$\"+h .\"#?$ \"+Wd75>F57$/F8#\"#K\"#L$\"+nZq8NF57$/F8#\"#\")\"##)$\"+w.Qk9F57$/F8# \"$p\"\"$q\"$\"+Z5p>B!#DQ)pprint686\"" }}}{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([b(x)-'bn_RK7_1'(x) ,b(x)-'bn_RK7_2'(x),b(x)-'bn_RK7_3'(x),\nb(x)-'bn_RK7_4'(x),b(x)-'bn_R K7_5'(x),b(x)-'bn_RK7_6'(x)],x=0..0.65,numpoints=100,font=[HELVETICA,9 ],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),mage nta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=1 1/12`,`scheme with c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme with c[10]=169/170`], \ntitle=`error curves for 11 stage order 7 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 1041 574 574 {PLOTDATA 2 "6,-%'CURVESG6%7]a l7$$\"\"!F)F(7$$\"5SSSSSS:N<7$$\"5+ ,,,,^)yLH%F-$\"#JF37$$\"5?@@@@@Y0_^F-$\"$P\"F37$$\"5STTTT\"RI2,'F-$\"$ \"[F37$$\"5ihhhhhhSpoF-$\"%?9F37$$\"5XXXX&zNNlh(F-$\"%%G$F37$$\"5IHHHH aXmj$)F-$\"%GqF37$$\"55888j]Pz5\"*F-$\"&9T\"F37$$\"5&ppppp%H#z&)*F-$\" &Zo#F37$$\"5HHHagC0Z/5!#@$\"&h,$F37$$\"5*)))))Q^a([J-\"Fin$\"&0,$F37$$ \"5[[[BU%)p#=/\"Fin$\"&\\+$F37$$\"53333L9_]g5Fin$\"&$**HF37$$\"5FFFx9u ;'y4\"Fin$\"&\"))HF37$$\"5YYYY'R8=_8\"Fin$\"&q(HF37$$\"5%[[[)f`5$*47Fi n$\"&[&HF37$$\"5BBBBBtRk%G\"Fin$\"&H$HF37$$\"5%\\\\\\\\u&*)o_9Fin$\"&$ ))GF37$$\"5mmmmmTRt?;Fin$\"&C*GF37$$\"5____xLkv/F in$\"&ok%F37$$\"5GGGG.s(oz(>Fin$\"&I,&F37$$\"5YYYY'Rj8\"**>Fin$\"&=W&F 37$$\"5kkkk*e\\e--#Fin$\"&,X&F37$$\"5$GGGGyN.9/#Fin$\"&'QaF37$$\"5?>>> p\"3$p$3#Fin$\"&dT&F37$$\"5cbbbb0G)f7#Fin$\"&HR&F37$$\"5GGGGG`Ac5AFin$ \"&uM&F37$$\"5,,,,,,<9&H#Fin$\"&DI&F37$$\"5utttt[6szBFin$\"&'e_F37$$\" 5YYYYY'f+VY#Fin$\"&$=_F37$$\"5>>>>>W+))[DFin$\"&%)=&F37$$\"5#>>>>>\\fM j#Fin$\"&i=&F37$$\"5aaaaH[pj]Nk4$Fin$\"&;M(F37 $$\"5onnnOFin$\"&I ,(F37$$\"5333333)Q^x$Fin$\"&/9(F37$$\"588888j`AJRFin$\"&M+)F37$$\"5%[[ [[)4+jrRFin$\"&Q\\)F37$$\"5ccccccY.7SFin$\"&A$*)F37$$\"5GGGGG.$RC0%Fin $\"&h*))F37$$\"5+++++]R%G4%Fin$\"&/'))F37$$\"5WVVVVVKltTFin$\"&\"*y)F3 7$$\"5(ooooo`iWD%Fin$\"&%=()F37$$\"5uttttB63;WFin$\"&-e)F37$$\"5hgggg5 (*pxXFin$\"&WY)F37$$\"533333e?*>\"\\Fin$\"&r6*F37$$\"5cbbbb0WGY_Fin$\" &$o)*F37$$\"5SRRRRRX*GT&Fin$\"&mq*F37$$\"5DBBBBtY]zbFin$\"&kc*F37$$\"5 ?::::S(4Gm&Fin$\"&*H&*F37$$\"5522222[6YdFin$\"&`b*F37$$\"5+**)*)*)R()> %HeFin$\"&tq*F37$$\"5\"44444%\\s7fFin$\"'+55F37$$\"5XWW%>8ppb&fFin$\"' WX5F37$$\"5+)zzHf7/'Fin$\"'b$4\"F37$ $\"50000bUR5%3'Fin$\"''))3\"F37$$\"55777PVMzphFin$\"'ez5F37$$\"5?>>>>W H[biFin$\"'Oq5F37$$\"5NLLL$e%>'oU'Fin$\"'M_5F37$$\"5[ZZZZZ4C)f'Fin$\"' $o.\"F37$$\"5NLLLL$QU,!pFin$\"'ht5F37$$\"5?>>>>>Q/-sFin$\"'sN6F37$$\"5 +++++v&y>P(Fin$\"'j;6F37$$\"5!33333L8>a(Fin$\"'!))4\"F37$$\"5?@@@r32)o i(Fin$\"'#=4\"F37$$\"5ghhhh'3[=r(Fin$\"'l)3\"F37$$\"5+---_ka\"oz(Fin$ \"'h$4\"F37$$\"5VUUUUUGy\")yFin$\"'j96F37$$\"5baaaa/\"*4nzFin$\"'yl6F3 7$$\"5lmmmmm`T_!)Fin$\"(r<>\"!#?7$$\"5!)yyyyG;tP\")Fin$\"(_;=\"Fcfl7$$ \"5!44444*y/B#)Fin$\"(1;<\"Fcfl7$$\"5::::::/o$R)Fin$\"(g=:\"Fcfl7$$\"5 SRRRRRHJk&)Fin$\"(\"fL6Fcfl7$$\"5][[[[[]=$*))Fin$\"(t$\\6Fcfl7$$\"5edd dddr0A#*Fin$\"(OA>\"Fcfl7$$\"5&eeee$[3Qr$*Fin$\"(/Y<\"Fcfl7$$\"5:9999R Xq?&*Fin$\"(jx:\"Fcfl7$$\"5IGGG`%Qm`f*Fin$\"(W.:\"Fcfl7$$\"5XUUU#*H#G+ n*Fin$\"()oW6Fcfl7$$\"5gcccJv+pW(*Fin$\"(lD9\"Fcfl7$$\"5rqqqq?>N>)*Fin $\"(>r9\"Fcfl7$$\"5gdddKw?83**Fin$\"(\"po6Fcfl7$$\"5XWWW%>B7p***Fin$\" (t@A\"Fcfl7$$\"5888jvQ#p&35Fcfl$\"(@W@\"Fcfl7$$\"5#====VDZu,\"Fcfl$\"( (o.7Fcfl7$$\"5?>>>W&G._.\"Fcfl$\"(DD=\"Fcfl7$$\"5dcccc;$fH0\"Fcfl$\"(> B;\"Fcfl7$$\"5yxxxFg,[g5Fcfl$\"(H[:\"Fcfl7$$\"5+**)*)*)R+,!o5Fcfl$\"(5 \"\\6Fcfl7$$\"5@???qZ=_v5Fcfl$\"(Sp9\"Fcfl7$$\"5UTTTT\"pUI3\"Fcfl$\"(J 9:\"Fcfl7$$\"5jiii7NNc!4\"Fcfl$\"(%zn6Fcfl7$$\"5%QQQQ)yV3)4\"Fcfl$\"(Q V?\"Fcfl7$$\"5999kw*ek**4\"Fcfl$\"(.!=7Fcfl7$$\"5WWWWp+[%=5\"Fcfl$\"(O g@\"Fcfl7$$\"5uuuCi6]s.6Fcfl$\"(^P@\"Fcfl7$$\"50000bA_g06Fcfl$\"(q9@\" Fcfl7$$\"5mlllSWcO46Fcfl$\"(Cp?\"Fcfl7$$\"5EEEEEmg786Fcfl$\"($R-7Fcfl7 $$\"5SSSS!H=?\"[6Fcfl$\"(28;\"Fcfl7$$\"5baaaa*H9J=\"Fcfl$\"(kI9\"Fcfl7 $$\"5'eeeL-Be3>\"Fcfl$\"(\"Fcfl$\"(%G#>\"Fcf l7$$\"5%GGyli7uC?\"Fcfl$\"(W&)>\"Fcfl7$$\"5\\[[)4;4Yj?\"Fcfl$\"(7R>\"F cfl7$$\"5999R&p0=-@\"Fcfl$\"((H*=\"Fcfl7$$\"5!)zzzHA+497Fcfl$\"(+Z=\"F cfl7$$\"5UUUUn$)ydH7Fcfl$\"(*\\m6Fcfl7$$\"500000Xd1X7Fcfl$\"(0([6Fcfl7 $$\"5UUUUn$Qe?E\"Fcfl$\"(d58\"Fcfl7$$\"5!)zzzHA50z7Fcfl$\"(OL7\"Fcfl7$ $\"5\\[[)4;MZvG\"Fcfl$\"(&)48\"Fcfl7$$\"5=<<<#4mVgH\"Fcfl$\"(Sd:\"Fcfl 7$$\"5NM%o\\2un\")H\"Fcfl$\"(*)f;\"Fcfl7$$\"5_^^wd?=H+8Fcfl$\"(wh<\"Fc fl7$$\"5po=cS+fT-8Fcfl$\"(#ot6Fcfl7$$\"5'eeeL-)*RXI\"Fcfl$\"(\">r6Fcfl 7$$\"5???&*))R\")y38Fcfl$\"(Ei;\"Fcfl7$$\"5baaaa*HOIJ\"Fcfl$\"($Gh6Fcf l7$$\"5IIIII?$)RX8Fcfl$\"(OW7\"Fcfl7$$\"511111T.wx8Fcfl$\"(^))4\"Fcfl7 $$\"59999*y_VYR\"Fcfl$\"(%e@6Fcfl7$$\"5AAAAs9n_69Fcfl$\"(m=8\"Fcfl7$$ \"5IIIIb,*4%G9Fcfl$\"(=H6\"Fcfl7$$\"5QQQQQ)3$HX9Fcfl$\"(+W4\"Fcfl7$$\" 5]]]]DCrzg9Fcfl$\"(6(y5Fcfl7$$\"5iiii7g6Iw9Fcfl$\"(\\)o5Fcfl7$$\"5uuuu *f>0=\\\"Fcfl$\"(M03\"Fcfl7$$\"5(oooo=B4t]\"Fcfl$\"(o-5\"Fcfl7$$\"5zyy yGEcvS:Fcfl$\"(DT1\"Fcfl7$$\"5rqqqq???u:Fcfl$\"((yM5Fcfl7$$\"5!*)))))Q Ess:f\"Fcfl$\"(-I/\"Fcfl7$$\"53222dCM%*3;Fcfl$\"()He5Fcfl7$$\"5EDDD]ET JE;Fcfl$\"(u+/\"Fcfl7$$\"5WVVVVG[oV;Fcfl$\"(MA-\"Fcfl7$$\"5/...`vg!)e; Fcfl$\"()f25Fcfl7$$\"5jiiiiAt#Rn\"Fcfl$\"'vm**Fcfl7$$\"5UUUU4mp\"F cfl$\"(%R85Fcfl7$$\"5#====o\")pTq\"Fcfl$\"(Z,-\"Fcfl7$$\"5cccc1RD$ot\" Fcfl$\"'!Q()*Fcfl7$$\"5JJJJJh_\\pFcfl$\"''fH*Fcfl7$$\"5GFFFx%zY\\$>Fcfl$\"'#Q+*Fcfl7$$\"5 +++++Dt)o'>Fcfl$\"'MO()Fcfl7$$\"5:::lF`Mvv>Fcfl$\"'w'o)Fcfl7$$\"5IIIIb \"e>Y)>Fcfl$\"'Vt')Fcfl7$$\"5XXX&H)4d[$*>Fcfl$\"'!3t)Fcfl7$$\"5gggg5Q= N-?Fcfl$\"'RI))Fcfl7$$\"5!444fY4%3??Fcfl$\"'>>p\")>8]=#Fcfl$\" 'x!y(Fcfl7$$\"5UTTmAuq'))=#Fcfl$\"'&))y(Fcfl7$$\"5\"444%G]4s#>#Fcfl$\" ''3\"yFcfl7$$\"5SSS:ME[d'>#Fcfl$\"'(3&yFcfl7$$\"5!**)*)*)R-(G/?#Fcfl$ \"'K-zFcfl7$$\"5)yyyGm?We@#Fcfl$\"'T\"y(Fcfl7$$\"5'eeee3rf7B#Fcfl$\"'U iwFcfl7$$\"5VUUUU#es')H#Fcfl$\"'B?uFcfl7$$\"5'ffff4%G5iBFcfl$\"'l(*pFc fl7$$\"5tssssPdSGCFcfl$\"'i)y'Fcfl7$$\"5RQQQQQ*GK\\#Fcfl$\"','['Fcfl7$ $\"5(pppp>#G4hDFcfl$\"'#[:'Fcfl7$$\"5onnnn2QXEEFcfl$\"'$p%fFcfl7$$\"5' eeeee&[H$p#Fcfl$\"'HacFcfl7$$\"5UTTTTECefFFcfl$\"'og`Fcfl7$$\"5SRRRR>Q \\?GFcfl$\"'kw^Fcfl7$$\"5gffff\\^I!*GFcfl$\"'&p([Fcfl7$$\"5cbbbbN_u_HF cfl$\"'LWYFcfl7$$\"5YXXXX!R>$>IFcfl$\"':OWFcfl7$$\"5POOOO^$RI3$Fcfl$\" '1yTFcfl7$$\"5**)*)*)*)*y@z`JFcfl$\"(^![RFin7$$\"5%RRRR*G*e]@$Fcfl$\"( Lyw$Fin7$$\"588888Qsf%G$Fcfl$\"(+U_$Fin7$$\"5)yyyyy&Q(zM$Fcfl$\"(UfL$F in7$$\"5&[[[[)>zHCFin7$$\"5EEEEEc`JUPFcfl$\"(%Q_AFin7$$\"5<<<<Fin7$$\"5qpppp%e[\"QRFcfl$\"( !4*y\"Fin7$$\"5vuuuuW\"*R1SFcfl$\"(J=i\"Fin7$$\"5xwwwwOB=oSFcfl$\"(;=_ \"Fin7$$\"5!**)*)*)*)HVYOTFcfl$\"(zpO\"Fin7$$\"5OOOOh#*[\"G:%Fcfl$\"() RW8Fin7$$\"5$GGGG`Xl\"pTFcfl$\"(n$>8Fin7$$\"5IHHH/=g^&=%Fcfl$\"(m=G\"F in7$$\"5wvvvv!em=?%Fcfl$\"(2C@\"Fin7$$\"5CBBBt5\")[MUFcfl$\"()[t6Fin7$ $\"5rqqqqS'4rE%Fcfl$\"(QE8\"Fin7$$\"5qpppppiDNVFcfl$\"'(*p)*Fin7$$\"5s rrr'HlZ4N%Fcfl$\"'k7(*Fin7$$\"5utttBO!RmO%Fcfl$\"'zJ&*Fin7$$\"5vuuC(ys %[uVFcfl$\"'T4%*Fin7$$\"5wvvv]>/L#Q%Fcfl$\"'>O#*Fin7$$\"5xwwE96h>>>p#=_\"z%Fcfl$\"')Gk#Fin7$$\"5n mmmmEy+d[Fcfl$\"'&f\">Fin7$$\"5qpppp%e,&*)[Fcfl$\"'ww8Fin7$$\"5tssssU` *>#\\Fcfl$\"&eT'Fin7$$\"598888`WPR\\Fcfl$\"&&)H'Fin7$$\"5a````jNvc\\Fc fl$\"&K1'Fin7$$\"5%RRRRRnKT(\\Fcfl$\"&)\\\\Fin7$$\"5NMMMM%y6:*\\Fcfl$! %%p\"Fin7$$\"5jiiiiKq&G0&Fcfl$!&cK'Fin7$$\"5b````toC)3&Fcfl$!',%3\"Fin 7$$\"5XWWWW9njB^Fcfl$!'#3'=Fin7$$\"5XVVV$f,0'R^Fcfl$!'AK=Fin7$$\"5XUUU U\"42D&Fcfl$!'CUHFin7$$\"5llllSz*)pn_Fcfl$!'z[HFi n7$$\"5IFFFxR))o%G&Fcfl$!'47KFin7$$\"5533e&*pP=$H&Fcfl$!'6>OFin7$$\"5! *)))))Q,qy;I&Fcfl$!'M0UFin7$$\"5qpp>KIO<5`Fcfl$!'!)pTFin7$$\"5^]]]]g&o '=`Fcfl$!'pMTFin7$$\"5&eeee$[rsN`Fcfl$!'/lSFin7$$\"5?@@@@Ody_`Fcfl$!'r /SFin7$$\"5bccc1CV%)p`Fcfl$!'36SFin7$$\"5#>>>>>\"H!pQ&Fcfl$!'jQVFin7$$ \"5&eee$[1Lm%R&Fcfl$!'\\yZFin7$$\"5!)zzz/,PU-aFcfl$!'un_Fin7$$\"5vttBh &4%=5aFcfl$!(,rA&F-7$$\"5qnnn[&Fcfl$!(TI<&F-7$$\"5!poo$**R?>!\\&Fcfl$!(#)*)[&F-7$$\" 5bbbbIMKV)\\&Fcfl$!(_L7'F-7$$\"5DCCuhGWn1bFcfl$!(qYD'F-7$$\"5$HHHHHi: \\^&Fcfl$!(DK?'F-7$$\"5!3333L#3+JbFcfl$!()\\/hF-7$$\"5qooooBg3ZbFcfl$! ()f5gF-7$$\"5gccc1C7UA(F-7$$\"5vvvvDy%HTh&Fcfl$!(`9;(F-7$$\"5STTT;0gcJcFcfl$!(_ z.(F-7$$\"522222KD+\\cFcfl$!(![@pF-7$$\"5&ppp>d(**3kcFcfl$!(B\"eoF-7$$ \"5&oooo$>u>>p'*e&\\z&Fcfl$!(mw`)F-7$$\"5bccccwbC6eFcfl$!(k>'*)F-7$$\"5&RRRR kDNv#eFcfl$!(Wt\"))F-7$$\"5KJJJJO\\#Q%eFcfl$!(A#*eFcfl$!(*G `\"*F-7$$\"5kjjjj$Ha$3fFcfl$!(@&>)*F-7$$\"554444RYkSfFcfl$!(7)3&*F-7$$ \"5baaaa%)\\$H(fFcfl$!((HW$*F-7$$\"5++++vQ?S*)fFcfl$!(:]w*F-7$$\"5XXXX &H4pe+'Fcfl$!)-1k5F-7$$\"5!444fr9OB-'Fcfl$!)6pY5F-7$$\"5POOOO,K!)QgFcf l$!)#p'H5F-7$$\"5XXXX&H+ei0'Fcfl$!)Fb85F-7$$\"5baaaa/GrtgFcfl$!)Et45F- 7$$\"5ljjj81w;\"4'Fcfl$!)E2i5F-7$$\"5tssss2Ci3hFcfl$!)-6P6F-7$$\"5!zyy y`4J19'Fcfl$!)(H95\"F-7$$\"5.....$yRE<'Fcfl$!)vk!3\"F-7$$\"5566hBa*Q0= 'Fcfl$!)'[.4\"F-7$$\"5?>>>WD\"Q%)='Fcfl$!)#3!=6F-7$$\"5IFFxk'HPj>'Fcfl $!)))*o<\"F-7$$\"5NNNN&yYOU?'Fcfl$!)_<:7F-7$$\"5]^^^E5[.?iFcfl$!)n7'> \"F-7$$\"5onnnn_J$eB'Fcfl$!)iUx6F-7$$\"5qqqqq?8s_iFcfl$!)rve6F-7$$\"5v tttt)[4'piFcfl$!)5Z[6F-7$$\"5IDDDvsN0yiFcfl$!)=Da6F-7$$\"5!onnnnl(\\'G 'Fcfl$!)\"Hu<\"F-7$$\"5IGGGyS<%\\H'Fcfl$!))GFB\"F-7$$\"5!)zzzzCeQ.jFcf l$!)8H'G\"F-7$$\"5vssssK')3PjFcfl$!)3rV7F-7$$\"5mllllS9zqjFcfl$!)#[V@ \"F-7$$\"5qmm;a3SNyjFcfl$!)BD?7F-7$$\"5qnnnUwl\"fQ'Fcfl$!)`oS7F-7$$\"5 qoo=JW\"zMR'Fcfl$!)UB'G\"F-7$$\"5qppp>7HJkFcfl$!))yiJ\"F-7$$\"#l!\"#$!)M*GU\" F--%&COLORG6&%$RGBGF($\"\"#!\"\"$\"\"\"F)-%'LEGENDG6#%ESharp-Verner~sc heme~with~c[10]=11/12G-F$6%7^\\lF'F*7$F/$\"#9F37$F5$\"#\")F37$F:$\"$\\ $F37$F?$\"%'>\"F37$FD$\"%![$F37$FI$\"%XzF37$FN$\"&\"z;F37$$\"5?@@@Y_\" Hst)F-$\"&9Q#F37$FS$\"&'GLF37$$\"5+000!)[$eV[*F-$\"&**e%F37$FX$\"&7D'F 37$Fgn$\"&f+(F37$F]o$\"&H*pF37$Fbo$\"&)zpF37$Fgo$\"&o'pF37$F\\p$\"&3%p F37$Fap$\"&]\"pF37$Ffp$\"&N'oF37$F[q$\"&E\"oF37$F`q$\"&,r'F37$Feq$\"&s s'F37$Fjq$\"&'=pF37$F_r$\"&(GuF37$Fdr$\"&.c)F37$Fir$\"'j#3\"F37$F^s$\" 'Sm6F37$Fcs$\"'4k7F37$Fhs$\"'%fE\"F37$F]t$\"'Ej7F37$Fbt$\"'$zD\"F37$Fg t$\"'j_7F37$F\\u$\"'3U7F37$Fau$\"'lJ7F37$Ffu$\"'[@7F37$F[v$\"'?77F37$F `v$\"'\\07F37$Fev$\"'b07F37$Fjv$\"'4@7F37$F_w$\"'\"*o7F37$Fdw$\"'**y8F 37$Fiw$\"'y,;F37$F^x$\"'r$o\"F37$Fcx$\"'r>F3 7$F_\\l$\"'#\\2#F37$Fd\\l$\"'bm?F37$Fi\\l$\"'Be?F37$F^]l$\"'nT?F37$Fc] l$\"'CD?F37$Fh]l$\"'>$*>F37$F]^l$\"'mm>F37$Fb^l$\"'Q@@F37$Fg^l$\"'m#H# F37$F\\_l$\"'8bAF37$Fa_l$\"'#HA#F37$Ff_l$\"')\\@#F37$F[`l$\"'w@AF37$F` `l$\"'6eAF37$Fe`l$\"'y\\BF37$Fj`l$\"'VJCF37$F_al$\"'bYDF37$Fdal$\"'3TD F37$Fial$\"'@IDF37$F^bl$\"'j3DF37$Fcbl$\"'A([#F37$Fhbl$\"'QXCF37$F]cl$ \"'u4CF37$Fbcl$\"'!z\\#F37$Fgcl$\"'#)REF37$F\\dl$\"'Y&f#F37$Fadl$\"'?a DF37$Ffdl$\"'LQDF37$F[el$\"'gJDF37$F`el$\"'.WDF37$Feel$\"'q$f#F37$Fjel $\"'%>r#F37$F_fl$\"(,6x#Fcfl7$Fefl$\"(ivu#Fcfl7$Fjfl$\"(;Us#Fcfl7$F_gl $\"(5$yEFcfl7$Fdgl$\"(jgj#Fcfl7$Figl$\"(]`n#Fcfl7$F^hl$\"(`Mx#Fcfl7$Fc hl$\"(ZCt#Fcfl7$Fhhl$\"(\"R$p#Fcfl7$F]il$\"(Jjn#Fcfl7$Fbil$\"([Nm#Fcfl 7$Fgil$\"(=#fEFcfl7$F\\jl$\"(,1n#Fcfl7$Fajl$\"()\\@FFcfl7$Ffjl$\"(T[%G Fcfl7$F[[m$\"()oEGFcfl7$F`[m$\"(/FFcfl7$Fhbm$\"(Vzn#Fcfl7$F]cm$\" (>rj#Fcfl7$Fbcm$\"(J.i#Fcfl7$Fgcm$\"(.\"REFcfl7$F\\dm$\"(qtp#Fcfl7$Fad m$\"(\">@FFcfl7$Ffdm$\"(fZu#Fcfl7$F[em$\"(Q*QFFcfl7$F`em$\"(CJt#Fcfl7$ Feem$\"(Q:s#Fcfl7$Fjem$\"(-+r#Fcfl7$F_fm$\"(^Si#Fcfl7$Fdfm$\"(,dc#Fcfl 7$Fifm$\"(r0i#Fcfl7$F^gm$\"(kXk#Fcfl7$Fcgm$\"('H+EFcfl7$Fhgm$\"(iqb#Fc fl7$F]hm$\"(^1_#Fcfl7$Fbhm$\"(Z&)\\#Fcfl7$Fghm$\"(:x_#Fcfl7$F\\im$\"(E Wd#Fcfl7$Faim$\"(n)*[#Fcfl7$Ffim$\"(;AU#Fcfl7$F[jm$\"(7OW#Fcfl7$F`jm$ \"(z.[#Fcfl7$Fejm$\"(owV#Fcfl7$Fjjm$\"(weR#Fcfl7$F_[n$\"(cP>#Fcfl7$Fcan$ \"(!yC@Fcfl7$Fhan$\"(U@1#Fcfl7$F]bn$\"(060#Fcfl7$Fbbn$\"(\">\\?Fcfl7$F gbn$\"(EZ1#Fcfl7$F\\cn$\"(s,4#Fcfl7$Facn$\"(MM0#Fcfl7$Ffcn$\"($QFcfl7$F`dn$\"(3$f>Fcfl7$Fedn$\"(45&>Fcfl7$Fjdn$\"(c.&>Fc fl7$F_en$\"(tK'>Fcfl7$Fden$\"(2n)>Fcfl7$Fien$\"(r-#>Fcfl7$F^fn$\"(\"pg =Fcfl7$Fcfn$\"(-E&=Fcfl7$Fhfn$\"(-3&=Fcfl7$Fbgn$\"($**f=Fcfl7$F\\hn$\" (]X)=Fcfl7$Fahn$\"(>d&=Fcfl7$Ffhn$\"(Tt#=Fcfl7$F[in$\"(tqx\"Fcfl7$$\"5 ?>>>p6xQIBFcfl$\"(F#H%G\"Fcfl7$Fh[ o$\"(gc@\"Fcfl7$F]\\o$\"(^S;\"Fcfl7$Fb\\o$\"(/[7\"Fcfl7$Fg\\o$\"(\"zi5 Fcfl7$F\\]o$\")\")e95Fin7$Fa]o$\"($GI)*Fin7$Ff]o$\"(g(Q#*Fin7$F[^o$\"( Au&))Fin7$F`^o$\"(eK\\)Fin7$Fe^o$\"(8h-)Fin7$Fj^o$\"(K@n(Fin7$F__o$\"( q_O(Fin7$Fd_o$\"(!=;pFin7$Fi_o$\"(k]h'Fin7$F^`o$\"(eNN'Fin7$Fc`o$\"(`g %fFin7$Fh`o$\"(+Lo&Fin7$F]ao$\"(6jT&Fin7$Fbao$\"('='4&Fin7$Fgao$\"(*=Y [Fin7$F[co$\"(x$>YFin7$Feco$\"(3/L%Fin7$Fjco$\"(Pz5%Fin7$Fheo$\"(9g!RF in7$Fbfo$\"(%zrOFin7$Fgfo$\"(bOY$Fin7$Feho$\"(?yF$Fin7$F_io$\"(vi2$Fin 7$F][p$\"(vM!HFin7$Fb[p$\"(Q+u#Fin7$Fg[p$\"(&*ed#Fin7$Fa\\p$\"([aU#Fin 7$Fe]p$\"('4lAFin7$Fj]p$\"(CG8#Fin7$Fd^p$\"(ef)>Fin7$Fh_p$\"(W1'=Fin7$ F]`p$\"(OAu\"Fin7$F[bp$\"(Jch\"Fin7$F_cp$\"((Q.:Fin7$F]ep$\")Hb,9F-7$F [gp$\")vT!H\"F-7$F_hp$\")T'e?\"F-7$F]jp$\")?>.6F-7$Fa[q$\")k[35F-7$Fe \\q$\"(i!f$*F-7$Fi]q$\"($[([)F-7$F]_q$\"(zHe(F-7$Fg_q$\"(.b2(F-7$F[aq$ \"(3+E'F-7$F_bq$\"()p*R&F-7$Fibq$\"(D#G]F-7$Fgdq$\"(Q(*H%F-7$Fefq$\"(( o@NF-7$F_gq$\"(=\"fKF-7$F]iq$\"(7-f#F-7$Fbiq$\"(m'o=F--Fhiq6&FjiqF($\" \"(F]jqF(-Fajq6#%8scheme~with~c[10]=15/16G-F$6%7[_lF'F*7$F/$\"\"$F37$F 5$\"#;F37$F:$\"#tF37$F?$\"$`#F37$FD$\"$Q(F37$FI$\"%$p\"F37$FN$\"%%f$F3 7$FS$\"%jrF37$FX$\"&FN\"F37$Fgn$\"&x^\"F37$F]o$\"&\\^\"F37$Fbo$\"&@^\" F37$Fgo$\"&$4:F37$F\\p$\"&O]\"F37$Fap$\"&\")\\\"F37$Ffp$\"&p[\"F37$F[q $\"&eZ\"F37$F`q$\"&OX\"F37$Feq$\"&mX\"F37$Fjq$\"&s\\\"F37$F_r$\"&kg\"F 37$Fdr$\"&0&=F37$Fir$\"&IM#F37$F^s$\"&f_#F37$Fcs$\"&'RFF37$Fhs$\"&Pu#F 37$F]t$\"&yt#F37$Fbt$\"&js#F37$Fgt$\"&\\r#F37$F\\u$\"&?p#F37$Fau$\"&%p EF37$Ffu$\"&tk#F37$F[v$\"&ri#F37$F`v$\"&Ch#F37$Fev$\"&@h#F37$Fjv$\"&\\ k#F37$F_w$\"&uu#F37$Fdw$\"&`)HF37$Fiw$\"&2Z$F37$F^x$\"&*\\OF37$Fcx$\"& *GPF37$Fhx$\"&5s$F37$F]y$\"&Jr$F37$Fby$\"&wp$F37$Fgy$\"&@o$F37$F\\z$\" &6l$F37$Faz$\"&1i$F37$Ffz$\"&oc$F37$F[[l$\"&J`$F37$F`[l$\"&)*f$F37$Fe[ l$\"&k.%F37$Fj[l$\"&@G%F37$F_\\l$\"&5]%F37$Fd\\l$\"&G[%F37$Fi\\l$\"&[Y %F37$F^]l$\"&*GWF37$Fc]l$\"&LR%F37$Fh]l$\"&PK%F37$F]^l$\"&eE%F37$Fb^l$ \"&'*f%F37$Fg^l$\"&b(\\F37$F\\_l$\"&R*[F37$Fa_l$\"&P#[F37$Ff_l$\"&h![F 37$F[`l$\"&,#[F37$F``l$\"&$)*[F37$Fe`l$\"&v4&F37$Fj`l$\"&fF&F37$F_al$ \"&%GbF37$Fdal$\"&n^&F37$Fial$\"&I\\&F37$F^bl$\"&iW&F37$Fcbl$\"&(*R&F3 7$Fhbl$\"&)3`F37$F]cl$\"&7B&F37$Fbcl$\"&7U&F37$Fgcl$\"&Jt&F37$F\\dl$\" &oj&F37$Fadl$\"&qa&F37$Ffdl$\"&B^&F37$F[el$\"&s\\&F37$F`el$\"&P_&F37$F eel$\"&8j&F37$Fjel$\"&**)eF37$F_fl$\"'J?gFcfl7$Fefl$\"'7pfFcfl7$Fjfl$ \"'Y=fFcfl7$F_gl$\"'p=eFcfl7$Fdgl$\"'tEdFcfl7$Figl$\"'\"3\"eFcfl7$F^hl $\"'=FgFcfl7$Fchl$\"'0QfFcfl7$Fhhl$\"'4`eFcfl7$F]il$\"'(e\"eFcfl7$Fbil $\"'%yy&Fcfl7$Fgil$\"'-ydFcfl7$F\\jl$\"'K-eFcfl7$Fajl$\"'+8fFcfl7$Ffjl $\"'l$='Fcfl7$F[[m$\"'PWhFcfl7$F`[m$\"'2!4'Fcfl7$Fe[m$\"'*H)fFcfl7$Fj[ m$\"''4)eFcfl7$F_\\m$\"'PVeFcfl7$Fd\\m$\"'*\\\"eFcfl7$Fi\\m$\"''\\!eFc fl7$F^]m$\"'**GeFcfl7$Fc]m$\"'A8fFcfl7$Fh]m$\"'$*)4'Fcfl7$F]^m$\"'*z;' Fcfl7$Fb^m$\"'/ehFcfl7$Fg^m$\"'YYhFcfl7$F\\_m$\"'\"\\8'Fcfl7$Fa_m$\"') =6'Fcfl7$Ff_m$\"'%*)3'Fcfl7$F[`m$\"'*4)eFcfl7$F``m$\"'t\"z&Fcfl7$Fe`m$ \"'+oeFcfl7$Fj`m$\"'jVgFcfl7$F_am$\"'NvgFcfl7$Fdam$\"'(=0'Fcfl7$Fiam$ \"'[GgFcfl7$F^bm$\"';0gFcfl7$Fcbm$\"'#H\"fFcfl7$Fhbm$\"'xAeFcfl7$F]cm$ \"'yLdFcfl7$Fbcm$\"'_'p&Fcfl7$Fgcm$\"'(pt&Fcfl7$F\\dm$\"'=keFcfl7$Fadm $\"'W;fFcfl7$Ffdm$\"'EofFcfl7$F[em$\"'ibfFcfl7$F`em$\"'(H%fFcfl7$Feem$ \"'y')\\Fcfl7$F[_n$\"'w-\\Fcfl7$F`_n$\"'#3 #[Fcfl7$Fe_n$\"'VVZFcfl7$Fj_n$\"'5yYFcfl7$F_`n$\"'(*fYFcfl7$Fd`n$\"'Ik YFcfl7$Fi`n$\"'b4ZFcfl7$F^an$\"'IeZFcfl7$Fcan$\"'x3YFcfl7$Fhan$\"'^sWF cfl7$F]bn$\"'/[WFcfl7$Fbbn$\"'!HW%Fcfl7$Fgbn$\"'/vWFcfl7$F\\cn$\"'))GX Fcfl7$Facn$\"'E\\WFcfl7$Ffcn$\"';rVFcfl7$F[dn$\"'L.VFcfl7$F`dn$\"'zWUF cfl7$Fedn$\"'$Fcfl7$Fdjn$\"'I,JFcfl7$Fijn$\"'vbHFcfl7$F^[o$\"'s1G Fcfl7$Fc[o$\"'$=s#Fcfl7$Fh[o$\"'9qDFcfl7$F]\\o$\"'\"QX#Fcfl7$Fb\\o$\"' NcBFcfl7$Fg\\o$\"'eAAFcfl7$F\\]o$\"(h)4@Fin7$Fa]o$\"(Lr-#Fin7$Ff]o$\"( ?,!>Fin7$F[^o$\"(A'3=Fin7$F`^o$\"(3Zr\"Fin7$Fe^o$\"($o<;Fin7$Fj^o$\"(7 'G:Fin7$F__o$\"(g_W\"Fin7$Fd_o$\"(?UN\"Fin7$Fi_o$\"(%=v7Fin7$F^`o$\"() y*>\"Fin7$Fc`o$\"(817\"Fin7$Fh`o$\"(!oZ5Fin7$F]ao$\"'@2(*Fin7$Fbao$\"' w>\"*Fin7$Fgao$\"'>-%)Fin7$F[co$\"'2,xFin7$Feco$\"'=1sFin7$Fjco$\"'(e` 'Fin7$Fheo$\"'WDfFin7$Fbfo$\"'%\\^&Fin7$Fgfo$\"'bl[Fin7$Feho$\"'S*H%Fi n7$F_io$\"'XhRFin7$F][p$\"':pLFin7$Fb[p$\"'yyHFin7$Fg[p$\"'b(f#Fin7$Fa \\p$\"'=I?Fin7$Fe]p$\"'Ov;Fin7$Fj]p$\"'9b8Fin7$F_^p$\"'\\P6Fin7$Fd^p$ \"&Q-)Fin7$Fi^p$\"&G*yFin7$F^_p$\"&cs(Fin7$Fc_p$\"&$4tFin7$Fh_p$\"&W\" eFin7$F]`p$\"&;G#Fin7$Fb`p$\"&\"R?Fin7$Fg`p$\"%\"[)Fin7$F\\ap$!%@xFin7 $Faap$!&C1$Fin7$Ffap$!&q.$Fin7$F[bp$!&H,$Fin7$F`bp$!&C'HFin7$Febp$!&T% HFin7$Fjbp$!&)oJFin7$F_cp$!&Vf%Fin7$F]ep$!'r4zF-7$Fbep$!'K`zF-7$Fgep$! 'T&z)F-7$F\\fp$!(#)z+\"F-7$Fafp$!(_vD\"F-7$Fffp$!(qjJ\"F-7$F[gp$!(DaI \"F-7$Fegp$!()\\l7F-7$F_hp$!(fiJ\"F-7$Fdhp$!(NoU\"F-7$Fihp$!(v#f;F-7$F ^ip$!(>uy\"F-7$Fcip$!(`=x\"F-7$Fhip$!(_9u\"F-7$F]jp$!(!Q8H$F-7$Feaq$!(E()G$F-7$Fjaq$!(Ep]$F-7$F_bq$!(-b\" QF-7$Fdbq$!((*ep$F-7$Fibq$!(vTj$F-7$F^cq$!('exOF-7$Fccq$!(#G#z$F-7$Fhc q$!()3JSF-7$F]dq$!(_%)=%F-7$Fbdq$!(nF7%F-7$Fgdq$!(i$eSF-7$F\\eq$!(rX*R F-7$Faeq$!(5O'RF-7$Ffeq$!(=;*RF-7$F[fq$!(\"z)3%F-7$F`fq$!()Q9VF-7$Fefq $!(8O`%F-7$Fjfq$!(3MQ%F-7$F_gq$!(#=&G%F-7$Fdgq$!(BHJ%F-7$Figq$!(`))R%F -7$F^hq$!(Ube%F-7$Fchq$!(?C([F-7$Fhhq$!((H*z%F-7$F]iq$!()GFZF-7$Fbiq$! (MK>&F--Fhiq6&Fjiq$\"\"*F]jq$\"\"%F]jqF(-Fajq6#%8scheme~with~c[10]=19/ 20G-F$6%7^\\lF'F*7$F/$\"#5F37$F5$\"#aF37$F:$\"$N#F37$F?$\"$/)F37$FD$\" %KBF37$FI$\"%6`F37$FN$\"&#>6F37$F]\\r$\"&`e\"F37$FS$\"&I@#F37$Fe\\r$\" &w/$F37$FX$\"&a9%F37$Fgn$\"&Ok%F37$F]o$\"&]j%F37$Fbo$\"&ki%F37$Fgo$\"& yh%F37$F\\p$\"&0g%F37$Fap$\"&Me%F37$Ffp$\"&$\\XF37$F[q$\"&b^%F37$F`q$ \"&xW%F37$Feq$\"&-Y%F37$Fjq$\"&&)e%F37$F_r$\"&(G\\F37$Fdr$\"&/o&F37$Fi r$\"&2=(F37$F^s$\"&Wt(F37$Fcs$\"&&z$)F37$Fhs$\"&;R)F37$F]t$\"&QP)F37$F bt$\"&'Q$)F37$Fgt$\"&MI)F37$F\\u$\"&MB)F37$Fau$\"&V;)F37$Ffu$\"&q4)F37 $F[v$\"&b.)F37$F`v$\"&9*zF37$Fev$\"&E*zF37$Fjv$\"&r4)F37$F_w$\"&gT)F37 $Fdw$\"&r9*F37$Fiw$\"'=i5F37$F^x$\"'I;6F37$Fcx$\"'2S6F37$Fhx$\"'nP6F37 $F]y$\"'FN6F37$Fby$\"'^I6F37$Fgy$\"'wD6F37$F\\z$\"'K;6F37$Faz$\"')p5\" F37$Ffz$\"'c!4\"F37$F[[l$\"'\\!3\"F37$F`[l$\"'i,6F37$Fe[l$\"'JN7F37$Fj [l$\"'t48F37$F_\\l$\"'rv8F37$Fd\\l$\"':q8F37$Fi\\l$\"'kk8F37$F^]l$\"'m `8F37$Fc]l$\"'xU8F37$Fh]l$\"'_@8F37$F]^l$\"'*RI\"F37$Fb^l$\"'129F37$Fg ^l$\"'F?:F37$F\\_l$\"'Q&\\\"F37$Fa_l$\"'3u9F37$Ff_l$\"'*)o9F37$F[`l$\" '^t9F37$F``l$\"'y(\\\"F37$Fe`l$\"'ne:F37$Fj`l$\"'v7;F37$F_al$\"')))o\" F37$Fdal$\"'C&o\"F37$Fial$\"'.y;F37$F^bl$\"'sj;F37$Fcbl$\"'_\\;F37$Fhb l$\"'y@;F37$F]cl$\"'?)f\"F37$Fbcl$\"'6d;F37$Fgcl$\"',^z\"Fcfl 7$Fadm$\"(Ay!=Fcfl7$Ffdm$\"('[B=Fcfl7$F[em$\"(='>=Fcfl7$F`em$\"(dd\"=F cfl7$Feem$\"(f!3=Fcfl7$Fjem$\"('R+=Fcfl7$F_fm$\"(+Lu\"Fcfl7$Fdfm$\"(_Z q\"Fcfl7$Fifm$\"(x:u\"Fcfl7$F^gm$\"(3wv\"Fcfl7$Fcgm$\"()=G `i\"Fcfl7$F`jm$\"(D+l\"Fcfl7$Fejm$\"(8;i\"Fcfl7$Fjjm$\"(;Qf\"Fcfl7$F_[ n$\"(Y6d\"Fcfl7$Fd[n$\"(]Zb\"Fcfl7$Fi[n$\"(?Jb\"Fcfl7$F^\\n$\"(N1c\"Fc fl7$Fc\\n$\"(_Ne\"Fcfl7$Fh\\n$\"(RXf\"Fcfl7$F]]n$\"(GLa\"Fcfl7$Fb]n$\" (K%)\\\"Fcfl7$Fg]n$\"(PH\\\"Fcfl7$F\\^n$\"(Na\\\"Fcfl7$Fa^n$\"(9D^\"Fc fl7$Ff^n$\"(O(H:Fcfl7$F[_n$\"(VT]\"Fcfl7$F`_n$\"(1!z9Fcfl7$Fe_n$\"()Hb 9Fcfl7$Fj_n$\"(%[N9Fcfl7$F_`n$\"(9-V\"Fcfl7$Fd`n$\"(M?V\"Fcfl7$Fi`n$\" (3mW\"Fcfl7$F^an$\"(a?Y\"Fcfl7$Fcan$\"(4hT\"Fcfl7$Fhan$\"(QWP\"Fcfl7$F ]bn$\"((=n8Fcfl7$Fbbn$\"(7hO\"Fcfl7$Fgbn$\"(/oP\"Fcfl7$F\\cn$\"(CTR\"F cfl7$Facn$\"(?'p8Fcfl7$Ffcn$\"(vbM\"Fcfl7$F[dn$\"(LZK\"Fcfl7$F`dn$\"(S pI\"Fcfl7$Fedn$\"(>:I\"Fcfl7$Fjdn$\"(*G,8Fcfl7$F_en$\"(T-J\"Fcfl7$Fden $\"(jiK\"Fcfl7$Fien$\"(8>G\"Fcfl7$F^fn$\"(QAC\"Fcfl7$Fcfn$\"(]pB\"Fcfl 7$Fhfn$\"(\\fB\"Fcfl7$Fbgn$\"(:CC\"Fcfl7$F\\hn$\"('Gf7Fcfl7$Fahn$\"(>+ C\"Fcfl7$Ffhn$\"(d5A\"Fcfl7$F[in$\"(W')=\"Fcfl7$Fb`t$\"(Xn:\"Fcfl7$F`i n$\"()p@6Fcfl7$Fj`t$\"($G96Fcfl7$F_at$\"(_#46Fcfl7$Fdat$\"(g(36Fcfl7$F iat$\"(/k6\"Fcfl7$F^bt$\"(n\\7\"Fcfl7$Fcbt$\"(qD7\"Fcfl7$Fhbt$\"(Ez6\" Fcfl7$F]ct$\"(/L6\"Fcfl7$Fbct$\"(:T5\"Fcfl7$Fein$\"(.]4\"Fcfl7$Fjin$\" (i40\"Fcfl7$F_jn$\"(3.+\"Fcfl7$Fdjn$\"':W(*Fcfl7$Fijn$\"'R=$*Fcfl7$F^[ o$\"'Gp))Fcfl7$Fc[o$\"'(el)Fcfl7$Fh[o$\"'#>?)Fcfl7$F]\\o$\"'[iyFcfl7$F b\\o$\"'#[h(Fcfl7$Fg\\o$\"'^*>(Fcfl7$F\\]o$\"(\"\\')oFin7$Fa]o$\"(L7p' Fin7$Ff]o$\"(STH'Fin7$F[^o$\"(7$[gFin7$F`^o$\"(Q,#eFin7$Fe^o$\"($)G]&F in7$Fj^o$\"(K#y_Fin7$F__o$\"(?$*3&Fin7$Fd_o$\"(]>y%Fin7$Fi_o$\"(%H$f%F in7$F^`o$\"(ycV%Fin7$Fc`o$\"(VL:%Fin7$Fh`o$\"(!y\"*RFin7$F]ao$\"(\"*)H QFin7$Fbao$\"(;[g$Fin7$Fgao$\"(*\\_MFin7$F[co$\"(Z%=LFin7$Feco$\"(o?6$ Fin7$Fjco$\"((fyHFin7$Fheo$\"(kt&GFin7$Fbfo$\"(%z!p#Fin7$Fgfo$\"(Drc#F in7$Feho$\"(5`X#Fin7$F_io$\"(N0J#Fin7$F][p$\"(&p6AFin7$Fb[p$\"(oR5#Fin 7$Fg[p$\"(0Y*>Fin7$Fa\\p$\"(o@\">Fin7$Fe]p$\"('z.=Fin7$Fj]p$\"(/kr\"Fi n7$Fd^p$\"(o^j\"Fin7$Fh_p$\"(%pX:Fin7$F]`p$\"(;CZ\"Fin7$F[bp$\"(\")fS \"Fin7$F_cp$\"(2DK\"Fin7$F]ep$\")HFh7F-7$F[gp$\")vB17F-7$F_hp$\")TxM6F -7$F]jp$\")?[x5F-7$Fa[q$\")kvN5F-7$Fe\\q$\"(iIp*F-7$Fi]q$\"($yV#*F-7$F ]_q$\"(z\\%))F-7$Fg_q$\"(.]I)F-7$F[aq$\"(3>#zF-7$F_bq$\"()fOvF-7$Fibq$ \"(D*zqF-7$Fgdq$\"(Q%onF-7$Fefq$\"((Q]kF-7$F_gq$\"(=t.'F-7$F]iq$\"(7fy &F-7$Fbiq$\"(mW]&F--%'COLOURG6&Fjiq$\"*++++\"!\")F(Fg_\\l-Fajq6#%8sche me~with~c[10]=32/33G-F$6%7^\\lF'F*7$F/$\"\"&F37$F5$\"#CF37$F:$\"$.\"F3 7$F?F^[r7$FD$\"%.5F37$FI$\"%kAF37$FN$\"%KZF37$F]\\r$\"%vmF37$FS$\"%\"G *F37$Fe\\r$\"&FF\"F37$FX$\"&Ts\"F37$Fgn$\"&%G>F37$F]o$\"&[#>F37$Fbo$\" &8#>F37$Fgo$\"&x\">F37$F\\p$\"&0\">F37$Fap$\"&M!>F37$Ffp$\"&#*)=F37$F[ q$\"&_(=F37$F`q$\"&t%=F37$Feq$\"&R&=F37$Fjq$\"&\"4>F37$F_r$\"&I0#F37$F dr$\"&uO#F37$Fir$\"&!))HF37$F^s$\"&d@$F37$Fcs$\"&-[$F37$Fhs$\"&^[$F37$ F]t$\"&xZ$F37$Fbt$\"&JY$F37$Fgt$\"&&[MF37$F\\u$\"&%>MF37$Fau$\"&2R$F37 $Ffu$\"&FO$F37$F[v$\"&uL$F37$F`v$\"&'>LF37$Fev$\"&6K$F37$Fjv$\"&kO$F37 $F_w$\"&8]$F37$Fdw$\"&l!QF37$Fiw$\"&cT%F37$F^x$\"&yj%F37$Fcx$\"&_t%F37 $Fhx$\"&_s%F37$F]y$\"&_r%F37$Fby$\"&bp%F37$Fgy$\"&dn%F37$F\\z$\"&lj%F3 7$Faz$\"&xf%F37$Ffz$\"&(HXF37$F[[l$\"&\"*[%F37$F`[l$\"&-e%F37$Fe[l$\"& r8&F37$Fj[l$\"&OW&F37$F_\\l$\"&Ur&F37$Fd\\l$\"&6p&F37$Fi\\l$\"&$ocF37$ F^]l$\"&Fi&F37$Fc]l$\"&vd&F37$Fh]l$\"&$*[&F37$F]^l$\"&sT&F37$Fb^l$\"&2 &eF37$Fg^l$\"&`J'F37$F\\_l$\"&>@'F37$Fa_l$\"&S7'F37$Ff_l$\"&M5'F37$F[` l$\"&U7'F37$F``l$\"&pA'F37$Fe`l$\"&/['F37$Fj`l$\"&Pq'F37$F_al$\"&k,(F3 7$Fdal$\"&6+(F37$Fial$\"&6(pF37$F^bl$\"&<\"pF37$Fcbl$\"&F&oF37$Fhbl$\" &wt'F37$F]cl$\"&.k'F37$Fbcl$\"&&*)oF37$Fgcl$\"&\\F(F37$F\\dl$\"&F:(F37 $Fadl$\"&&RqF37$Ffdl$\"&m*pF37$F[el$\"&&zpF37$F`el$\"&g,(F37$Feel$\"&Z :(F37$Fjel$\"&*yuF37$F_fl$\"'\"*QwFcfl7$Fefl$\"'-uvFcfl7$Fjfl$\"'m4vFc fl7$F_gl$\"'<$Q(Fcfl7$Fdgl$\"'EnsFcfl7$Figl$\"'P\"Q(Fcfl7$F^hl$\"'yZwF cfl7$Fchl$\"'tMvFcfl7$Fhhl$\"'LFuFcfl7$F]il$\"'w!Q(Fcfl7$Fbil$\"'SYtFc fl7$Fgil$\"'$eL(Fcfl7$F\\jl$\"'\"*otFcfl7$Fajl$\"'d5vFcfl7$Ffjl$\"'iZy Fcfl7$F[[m$\"'F(z(Fcfl7$F`[m$\"'OGxFcfl7$Fe[m$\"'^#f(Fcfl7$Fj[m$\"'Yju Fcfl7$F_\\m$\"'T;uFcfl7$Fd\\m$\"'h\"Q(Fcfl7$Fi\\m$\"'sqtFcfl7$F^]m$\"' M.uFcfl7$Fc]m$\"'G6vFcfl7$Fh]m$\"'[VxFcfl7$F]^m$\"'$)GyFcfl7$Fb^m$\"': ;yFcfl7$Fg^m$\"'Y,yFcfl7$F\\_m$\"'z'y(Fcfl7$Fa_m$\"'ddxFcfl7$Ff_m$\"'X GxFcfl7$F[`m$\"'rkuFcfl7$F``m$\"'9dtFcfl7$Fe`m$\"'4buFcfl7$Fj`m$\"'#[n (Fcfl7$Fgas$\"'_GxFcfl7$F_am$\"'d8xFcfl7$F_bs$\"'l)p(Fcfl7$Fdam$\"'w$o (Fcfl7$Fiam$\"'1awFcfl7$F^bm$\"'ZCwFcfl7$Fcbm$\"'N2vFcfl7$Fhbm$\"',$R( Fcfl7$F]cm$\"'.\"G(Fcfl7$Fbcm$\"'8PsFcfl7$Fgcm$\"'`!H(Fcfl7$F\\dm$\"'] ^uFcfl7$Fadm$\"'p;vFcfl7$Ffdm$\"'#4e(Fcfl7$F[em$\"''[c(Fcfl7$F`em$\"'z [vFcfl7$Feem$\"'z;vFcfl7$Fjem$\"'%\\[(Fcfl7$F_fm$\"'kZsFcfl7$Fdfm$\"'= *3(Fcfl7$Fifm$\"'*HC(Fcfl7$F^gm$\"'+3tFcfl7$Fcgm$\"'n&=(Fcfl7$Fhgm$\"' FmqFcfl7$F]hm$\"'=mpFcfl7$Fbhm$\"''o!pFcfl7$Fghm$\"'***)pFcfl7$F\\im$ \"'D=rFcfl7$Faim$\"']%)oFcfl7$Ffim$\"'K*p'Fcfl7$F[jm$\"'JhnFcfl7$F`jm$ \"'fioFcfl7$Fejm$\"'UWnFcfl7$Fjjm$\"'&)GmFcfl7$F_[n$\"'![`'Fcfl7$Fd[n$ \"'VnkFcfl7$Fi[n$\"'HhkFcfl7$F^\\n$\"'0$\\'Fcfl7$Fc\\n$\"')ze'Fcfl7$Fh \\n$\"'\"Gj'Fcfl7$F]]n$\"'!)>kFcfl7$Fb]n$\"'wLiFcfl7$Fg]n$\"'\\6iFcfl7 $F\\^n$\"'_AiFcfl7$Fa^n$\"'d$H'Fcfl7$Ff^n$\"'GkjFcfl7$F[_n$\"'!yD'Fcfl 7$F`_n$\"'B`hFcfl7$Fe_n$\"'ragFcfl7$Fj_n$\"'#G(fFcfl7$F_`n$\"'X^fFcfl7 $Fd`n$\"'fffFcfl7$Fi`n$\"'E?gFcfl7$F^an$\"'s$3'Fcfl7$Fcan$\"'a#*eFcfl7 $Fhan$\"'h>dFcfl7$F]bn$\"'#**o&Fcfl7$Fbbn$\"'/'o&Fcfl7$Fgbn$\"'wIdFcfl 7$F\\cn$\"'3-eFcfl7$Facn$\"'4+dFcfl7$Ffcn$\"'/+cFcfl7$F[dn$\"'S8bFcfl7 $F`dn$\"'zRaFcfl7$Fedn$\"'mFin7$Fi_o$\"(W2#>Fin7$F^`o$\"(eb&=Fin7$Fc`o$\"(.wt\"Fin7 $Fh`o$\"(S0n\"Fin7$F]ao$\"(hMg\"Fin7$Fbao$\"(1$4:Fin7$Fgao$\"(pgW\"Fin 7$F[co$\"(<0R\"Fin7$Feco$\"(3TI\"Fin7$Fjco$\"(n'[7Fin7$Fheo$\"(C$)>\"F in7$Fbfo$\"(M&G6Fin7$Fgfo$\"(Dr2\"Fin7$Feho$\"(?1.\"Fin7$F_io$\"':*p*F in7$F][p$\"':)G*Fin7$Fb[p$\"'=Q))Fin7$Fg[p$\"':!Q)Fin7$Fa\\p$\"')p.)Fi n7$Fe]p$\"'c$e(Fin7$Fj]p$\"'9'Fin7$F[bp$\"'\"z\"fFin7$F_cp$\"'ZnbFin7$F]ep$\"(H-J&F-7$F[g p$\"(v&z]F-7$F_hp$\"(T)yZF-7$F]jp$\"(?y`%F-7$Fa[q$\"(kBO%F-7$Fe\\q$\"( iD3%F-7$Fi]q$\"($)H*QF-7$F]_q$\"(zWs$F-7$Fg_q$\"(.r\\$F-7$F[aq$\"(3[L$ F-7$F_bq$\"()4rJF-7$Fibq$\"(D*yHF-7$Fgdq$\"(Qg%GF-7$Fefq$\"(())4FF-7$F _gq$\"(=j`#F-7$F]iq$\"(7\"GCF-7$Fbiq$\"(mnI#F--Fhiq6&Fjiq$Fb`\\lF]jqF( F^jq-Fajq6#%8scheme~with~c[10]=81/82G-F$6%7^_lF'F*7$F/$F\\jqF37$F:$\"# a F37$Fiw$!%zlF37$F^x$!%!4(F37$Fcx$!%HtF37$Fhx$!%8tF37$F]y$!%*H(F37$Fby$ !%nsF37$Fgy$!%PsF37$F\\z$!%xrF37$Faz$!%\\nFin$!%&***F37$$\"5ljjj)[MHpy'Fin$!%))**F 37$$\"5!poooV-nY#oFin$!&7+\"F37$$\"55555&Qq/C'oFin$!&!45F37$Fbcl$!&Z- \"F37$$\"5!)zzzHUxhvpFin$!&h4\"F37$$\"5IEEEE,J4^qFin$!&48\"F37$$\"5vss sAg%ol7(Fin$!&C7\"F37$Fgcl$!&R6\"F37$F\\dl$!&^4\"F37$Fadl$!&b2\"F37$Ff dl$!&_1\"F37$F[el$!&c0\"F37$$\"5!===obxJVv(Fin$!&B0\"F37$F`el$!&80\"F3 7$$\"5?AAAZ`\"*HRyFin$!&Y0\"F37$Feel$!&^1\"F37$Fjel$!&a7\"F37$F_fl$!'* e;\"Fcfl7$Fefl$!')f:\"Fcfl7$Fjfl$!'9Y6Fcfl7$F_gl$!'jE6Fcfl7$Fdgl$!'O16 Fcfl7$$\"5qmmmmm4`Y')Fin$!'8'4\"Fcfl7$$\"5&RRRRR**[(G()Fin$!'3(3\"Fcfl 7$$\"5gdddd2!e)p()Fin$!'D%3\"Fcfl7$$\"5?@@@@@q'4\"))Fin$!'&R3\"Fcfl7$$ \"5&[[[[[.w?&))Fin$!'!z3\"Fcfl7$Figl$!'n)4\"Fcfl7$$\"5!edddd2.a(*)Fin$ !'%p:\"Fcfl7$$\"50.....6id!*Fin$!'a#=\"Fcfl7$$\"5IIIIII\"R)R\"*Fin$!'& G<\"Fcfl7$F^hl$!'Cj6Fcfl7$Fhhl$!'BG6Fcfl7$F\\jl$!'r*4\"Fcfl7$Fajl$!'0< 6Fcfl7$Ffjl$!'r(=\"Fcfl7$F[[m$!'h\"=\"Fcfl7$F`[m$!'6Fcfl7$F_fm$!'+%3\"Fcfl7$Fdfm$!')* [5Fcfl7$Fifm$!'in5Fcfl7$F^gm$!'1)3\"Fcfl7$Fcgm$!'&)p5Fcfl7$Fhgm$!'v^5F cfl7$F]hm$!'fM5Fcfl7$Fbhm$!'g=5Fcfl7$$\"5lllS4pr 5Fcfl7$Ffim$!&Z%)*Fcfl7$$\"5EDD+>'pW&y:Fcfl$!&>\")*Fcfl7$$\"5!)zzHnrt) Ge\"Fcfl$!&]z*Fcfl7$$\"5NMMf:Z+B(e\"Fcfl$!&^!)*Fcfl7$F[jm$!&/')*Fcfl7$ $\"5WVV=7)R:ff\"Fcfl$!&`)**Fcfl7$$\"5*zzz/O2e-g\"Fcfl$!'V>5Fcfl7$$\"5a __x3\\2g/;Fcfl$!'-:5Fcfl7$F`jm$!'i55Fcfl7$Fejm$!&A$**Fcfl7$Fjjm$!&(f(* Fcfl7$F_[n$!&jg*Fcfl7$Fd[n$!&aX*Fcfl7$Fi[n$!&dS*Fcfl7$F^\\n$!&6U*Fcfl7 $Fc\\n$!&pe*Fcfl7$Fh\\n$!&$3(*Fcfl7$F]]n$!&gR*Fcfl7$Fb]n$!&'z!*Fcfl7$$ \"5YXX&H[B8Px\"Fcfl$!&9/*Fcfl7$Fg]n$!!*Fcfl7$$\"5uttB'==\\@y\"Fcfl$ !&x)*)Fcfl7$F\\^n$!&`)*)Fcfl7$Fa^n$!&\"*3*Fcfl7$Ff^n$!&RD*Fcfl7$F[_n$! &\"*4*Fcfl7$F`_n$!&l%*)Fcfl7$Fe_n$!&jz)Fcfl7$Fj_n$!&Ek)Fcfl7$$\"5777iC :57u=Fcfl$!&qg)Fcfl7$F_`n$!&nd)Fcfl7$$\"5CCCuhVRP#)=Fcfl$!&ib)Fcfl7$Fd `n$!&Ab)Fcfl7$Fi`n$!&!Q')Fcfl7$F^an$!&Ly)Fcfl7$Fcan$!&q])Fcfl7$Fhan$!& #G#)Fcfl7$F]bn$!&Z:)Fcfl7$Fbbn$!&76)Fcfl7$Fgbn$!&3;)Fcfl7$F\\cn$!&EJ)F cfl7$Facn$!&n;)Fcfl7$Ffcn$!&F-)Fcfl7$F[dn$!&E*yFcfl7$F`dn$!&!fxFcfl7$F edn$!&'*p(Fcfl7$Fjdn$!&sm(Fcfl7$F_en$!&%3xFcfl7$Fden$!&T%yFcfl7$Fien$! &9e(Fcfl7$F^fn$!&nJ(Fcfl7$Fcfn$!&7E(Fcfl7$Fhfn$!&qA(Fcfl7$F]gn$!&!GsFc fl7$Fbgn$!&%\\sFcfl7$Fggn$!&AI(Fcfl7$F\\hn$!&GQ(Fcfl7$Fahn$!&-F(Fcfl7$ Ffhn$!&'erFcfl7$F[in$!&F*oFcfl7$F`in$!&=]'Fcfl7$Fein$!&VH'Fcfl7$Fjin$! &t&fFcfl7$F_jn$!&mn&Fcfl7$Fdjn$!&DZ&Fcfl7$Fijn$!&r:&Fcfl7$F^[o$!\\F cfl7$Fc[o$!&'HZFcfl7$Fh[o$!&uT%Fcfl7$F]\\o$!&uA%Fcfl7$Fb\\o$!&v-%Fcfl7 $Fg\\o$!&yw$Fcfl7$F\\]o$!'>tNFin7$Fa]o$!'P/MFin7$Ff]o$!'+jJFin7$F[^o$! 'y3IFin7$F`^o$!'AEGFin7$Fe^o$!'P^EFin7$Fj^o$!'e%\\#Fin7$F__o$!'SNBFin7 $Fd_o$!'Sw@Fin7$Fi_o$!'cR?Fin7$F^`o$!'A**=Fin7$Fc`o$!'Pm)*Fin7$Fheo$!&m!))Fin7$Fbfo$!&Y>)Fin7$Fgf o$!&0=(Fin7$Feho$!&gG'Fin7$F_io$!&bz&Fin7$F][p$!&v\"\\Fin7$Fb[p$!&sI%F in7$Fg[p$!&&*y$Fin7$Fa\\p$!&_*HFin7$Fe]p$!&%pCFin7$Fj]p$!&w0#Fin7$Fd^p $!&#G8Fin7$Fh_p$!&'45Fin7$F]`p$!%9eFin7$F[bp$\"$r'Fin7$F_cp$\"%()GFin7 $F]ep$\"&Hk'F-7$Fbep$\"&ow'F-7$Fgep$\"&f#yF-7$Fafp$\"'[&>\"F-7$F[gp$\" 'vX7F-7$Fegp$\"'-37F-7$F_hp$\"'Ty7F-7$F]jp$\"'?+ \+ " 0 "" {MPLTEXT 1 0 519 "evalf[25](plot([b(x)-'bn_RK7_1'(x),b(x)-'bn_R K7_2'(x),b(x)-'bn_RK7_3'(x),\nb(x)-'bn_RK7_4'(x),b(x)-'bn_RK7_5'(x),b( x)-'bn_RK7_6'(x)],x=0.65..5,-3.5e-15..6.3e-15,font=[HELVETICA,9],\ncol or=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COL OR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,` scheme with c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]= 32/33`,`scheme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle= `error curves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 880 564 564 {PLOTDATA 2 "6,-%'CURVESG6%7\\x7$$\"# l!\"#$!.f$p$**GU\"!#F7$$\":++++++]i?w\"[u!#D$!.oZ3#=@=F-7$$\":+++++]PM DzJF)F1$!/WkIXF/@!#G7$$\":+++++D1*e.3P()F1$!/`v1&3)GBF97$$\":++++++vVY \")4?*F1$!/lDSDF97$$\":++++DJXkhVrp*F1$!/:+'4)zdEF97$$\":++++]i!RD'H js*F1$!/6&H%=\"*QEF97$$\":++++v$fLMc^b(*F1$!/+\"z#Q_lDF97$$\":+++++D\" GV;q%y*F1$!/lNq)QSc#F97$$\":++++](=EF97$$\":+++++] i!zcW,**F1$!/W=]I*fv#F97$$\":++++]P%[r*==+\"!#C$!0]08S8Ut#!#H7$$\":+++ ++]i]P$\\85Fio$!0.0cs!)=r#F\\p7$$\":+++]7GQAH7^-\"Fio$!0+e&eOo\"p#F\\p 7$$\":++++D19%47tO5Fio$!0o)3-2c!o#F\\p7$$\":++]7y+3(QfjR5Fio$!0,L2oY*4 GF\\p7$$\":+++DJ&>+o1aU5Fio$!0;kt'>+/GF\\p7$$\":++]P%)*eH(RXa/\"Fio$!0 Xriurfs#F\\p7$$\":+++]P%)*eE,N[5Fio$!0$QI#\\m^r#F\\p7$$\":+++vVtx^efT0 \"Fio$!0$))z3'G)zFF\\p7$$\":++++]ilP/p*f5Fio$!0c(>D#Q?\"HF\\p7$$\":+++ +v=<\"yo?$3\"Fio$!0`,H[Z_'GF\\p7$$\":+++++voCrWk5\"Fio$!0C-:2!G=GF\\p7 $$\":++++]i:g$p@<6Fio$!0e!oIFtEGF\\p7$$\":+++++]i&f\"*)z7\"Fio$!0_`Vrf %\\GF\\p7$$\":++++]P4JQh(Q6Fio$!0=)[1R.**GF\\p7$$\":+++++DcmgL&\\6Fio$ !0!)*RKo*e*HF\\p7$$\":+++](oHaimA_6Fio$!0eAUx8I,$F\\p7$$\":++++voH%=(> \\:\"Fio$!0V0OQ&*R$HF\\p7$$\":+++]iS;Vx7w:\"Fio$!0A%*3-l&*)GF\\p7$$\": ++++]7.-$eIg6Fio$!07&*3s!f'3$F\\p7$$\":+++]P%)*3')))*H;\"Fio$!0u95B'f/ IF\\p7$$\":++++Dcw>%>pl6Fio$!0nyRx0\"GHF\\p7$$\":+++]7Gjy*\\Qo6Fio$!0, p!)oOA#HF\\p7$$\":++++++]P0y5<\"Fio$!0Jzrj6t2$F\\p7$$\":++++](oHx-&== \"Fio$!0!*o.KLu1$F\\p7$$\":+++++vV3]AE>\"Fio$!0I(=!oup0$F\\p7$$\":++++ +D1z_I\\@\"Fio$!0%RwXdK7IF\\p7$$\":+++++vo\\bQsB\"Fio$!1%[aH9at)H!#I7$ $\":++++Dc^$eq-S7Fio$!1=\"zczLh>$Fdx7$$\":++++]PMIFdx7$$\":+++++ ]ibf+2F\"Fio$!1cc6w5%G?$Fdx7$$\":++++DJX*)4*[t7Fio$!10tl49$[6$Fdx7$$\" :++++]7GBgxiF\"Fio$!13`Vb%Gm.$Fdx7$$\":++++v$4r0h1z7Fio$!1[e>G**G*3$Fd x7$$\":+++++v$44Y&=G\"Fio$!1!3vtrKU<$Fdx7$$\":+++++vVe5!R$H\"Fio$!1jkx O5PLJFdx7$$\":+++++v$f-c#\\I\"Fio$!1u6d2MV$4$Fdx7$$\":+++++vV$*4hkJ\"F io$!1+7;kh&31$Fdx7$$\":+++++v$4'f'*zK\"Fio$!1OKVs?%G1$Fdx7$$\":+++++vV G4K&R8Fio$!1MyTuw=%=$Fdx7$$\":+++++v$f*en5N\"Fio$!1a'>WKH0C$Fdx7$$\":+ ++++vVj3.EO\"Fio$!1Sb&>\\gY>$Fdx7$$\":+++++v$4$eQTP\"Fio$!1Y?Et?7\\JFd x7$$\":++++]Pf`3PcQ\"Fio$!1?8>8-,+9Fio$!1s`;-X,(G$Fdx7$$\":++++D\"yvQ[)GS\"Fio$ !1S!p!f])Q>$Fdx7$$\":+++](=UKk%fdS\"Fio$!1Yv%y79r5$Fdx7$$\":++++]i!*)* 3M'39Fio$!1A\"p:JY=6$Fdx7$$\":++++vVB5M$Q99Fio$!1DS/)*>QZJFdx7$$\":+++ ++Dc@fK,U\"Fio$!10&p'R0k$G$Fdx7$$\":+++++](o'fHJW\"Fio$!1wYhHf-&=$Fdx7 $$\":+++++v=7gEhY\"Fio$!2$>^O&pLJ4$!#J7$$\":++++](=#H4PM^\"Fio$!2xL*)* p%*4^JFebl7$$\":++++++DYeZ2c\"Fio$!2xiV,!QF,KFebl7$$\":+++++vo9p=Cg\"F io$!2&[.$)R6f4JFebl7$$\":+++++]7$)z*3W;Fio$!26n3Qgk5,$Febl7$$\":++++DJ Xuu[&=Fio$!3x8mh,`B'f#Fcel7$$ \":++++++v32;v(=Fio$!3G4kx_ex+DFcel7$$\":++++]PMd=`.)=Fio$!3]**G(HEB[j #Fcel7$$\":+++++v$f+.>$)=Fio$!3o_tjGV9hDFcel7$$\":++++]7`aTFg)=Fio$!3[ lQbpuL$\\#Fcel7$$\":+++++]7.`k)))=Fio$!30=$)*ygo\\]#Fcel7$$\":++++](=< X;q\"*=Fio$!3'4QO+7sBd#Fcel7$$\":+++++DJ+wQX*=Fio$!3R+A?jy!3]#Fcel7$$ \":++++]i!*[(eP(*=Fio$!3@6'Q@gr'\\CFcel7$$\":++++++](*)H@+>Fio$!3b$R5; (*QGe#Fcel7$$\":+++++](=\\9c6>Fio$!3j$)\\#RRs*>DFcel7$$\":++++++D'3*4H #>Fio$!4,pCJy/7yX#!#L7$$\":+++++v$f:QN0?Fio$!40)*[#G?)p(f@F_jl7$$\":++ ++++D^f(Q.@Fio$!4D4crC)*o;&=F_jl7$$\":+++++++b1Nk=#Fio$!5DN?%)\\g?G'\\ \"!#M7$$\":+++++vVj,RIG#Fio$!5#*3$[uee$=;6F_[m7$$\":++++++]<))\\&oBFio $!4ql^y%4s[UrF_[m7$$\":+++++vVQ&)oBY#Fio$!558')Rb[*ewg#!#N7$$\":+++++D J:q1IwU?F_\\m7$$\":+++++]PMP@\\k#Fio$\"60=2k1>3fnk'!#O 7$$\":+++++v=2a@0t#Fio$\"7fWH#R!)f-J7>\"Fj\\m7$$\":+++++]7`)G&G#GFio$ \"7,>2Mtu([]Ll\"Fj\\m7$$\":+++++v$4.#f(=HFio$\"8&>O^_Cv!Q!*)4@!#P7$$\" :+++++DcY$fC-IFio$\"8o5?XmGo=Rvj#Fj]m7$$\":+++++]P>*QT#4$Fio$\"9JqbtHR M#*=P-J!#Q7$$\":++++++]Z@mb=$Fio$\"9\"=TWZsSiI-gW$Fe^m7$$\":+++++](=*e (pwKFio$\"92]&\\E)R.DL\"*GQFe^m7$$\":+++++vV)z=([O$Fio$\":L<>=6['y*fx \"pU!#R7$$\":+++++]7jRuFY$Fio$\":;_Zq?**3))o(oaZFe_m7$$\":++++v=#*phqP Z$Fio$\":_HtwKe?^;tVu%Fe_m7$$\":++++](=nPow%[$Fio$\":nSy1e$3Bx%\\Ut%Fe _m7$$\":++++Dc^$eIw&\\$Fio$\":c8)H*\\ws4Epqs%Fe_m7$$\":+++++DJ!z#fn]$F io$\":`zz6&p;E_FUIZFe_m7$$\":++++]i!R?Tv4L#f[Fe_m7$ $\":++++++]<;W2b$Fio$\":HFL;MO$4(*pxs^Fe_m7$$\":++++](=Uu\\[iNFio$\":w :'QA#oi??/e6&Fe_m7$$\":+++++v$4(yDUd$Fio$\":*fwlED_F'Q?#f]Fe_m7$$\":++ +](='4!e$Fio$\":@pK%= sRokr'QI&Fe_m7$$\":+++]il(f'RJIe$Fio$\":Kl&>4+am`Fe_m7$$\":++++]7`/&)HHj$Fio$\"::W!>C&zij \"p.-`Fe_m7$$\":++++++DJmqYk$Fio$\":+bG2cb/R8Q(Q_Fe_m7$$\":+++]PfLWe4` l$Fio$\":bh:5RZo!**4$QB&Fe_m7$$\":++++v=Ud][fm$Fio$\":\\\"olAMh&\\eN6B &Fe_m7$$\":+++]7y]qU(ewOFio$\":a81N]oc\"o#=NB&Fe_m7$$\":++++]Pf$[jA(o$ Fio$\":-!GW0)=Q\\]%)eC&Fe_m7$$\":++++Dcw4>/&3PFio$\":L9Cd=!e#=?vpL&Fe_ m7$$\":+++++v$fL?yHPFio$\":JKc$=n^7;p]LcFe_m7$$\":++]iS;o8#)*oKPFio$\" :(e+Kw\")>:Wx6ibFe_m7$$\":+++D\"yD94wfNPFio$\":'=#en7O6Dxy!3aFe_m7$$\" :++](=#*p\"pR0&QPFio$\":Tx\"evh.k(3p?S&Fe_m7$$\":+++]iS\"p%=89u$Fio$\" :Lyb#H$*=auy/bcFe_m7$$\":++]7.#eYs4KWPFio$\":iPDC0)e/\"QcO\\&Fe_m7$$\" :+++vVBS-wGsu$Fio$\":d$f#fh\"*[\"eQ[x`Fe_m7$$\":++]P%[Y,[l8]PFio$\":,l <4y:I%Gby[dFe_m7$$\":++++D1*yNV/`PFio$\":1?!*=JGTD6rSe&Fe_m7$$\":+++]( =n)o[vYw$Fio$\":N*)\\)R%yi#p$eW^&Fe_m7$$\":++++]P%)zj1jx$Fio$\"::J@\\# )[iC[RmX&Fe_m7$$\":++]i!*yedU9#zPFio$\":U3$[MzX+!3KCe&Fe_m7$$\":+++DJ? LN@A@y$Fio$\":^NNq?U^:%zs$p&Fe_m7$$\":++](=cFe_m7$$\":++++voz,%*o&*z$Fio$\":$R%)en/tkAd' Rq&Fe_m7$$\":++]i:5azswC!QFio$\":qHXS&)pnhE)p3dFe_m7$$\":+++Dc^Gd^%Q0Q Fio$\":<4?oYs+eW%3\\bFe_m7$$\":++](oHH].BH3QFio$\":m9sg^1>\"4H\"Q^&Fe_ m7$$\":+++]PMx74+7\"QFio$\":hUAtaB`78v()z&Fe_m7$$\":++]7yv^!zy59QFio$ \":=N/.dhGX0XIj&Fe_m7$$\":+++v=QFio$\":1yWt([L(QH&paeFe_m7$$\":++++++vBCJG#QFio$\":J* zN'QCT[%fv?dFe_m7$$\":++++v=neD>Z%QFio$\":17XoE-g;5^Sk&Fe_m7$$\":++++] Pf$psgmQFio$\":d7Wh+^)\\O2e$e&Fe_m7$$\":++]P%)*e/rKMpQFio$\":*3g-(H3x= Doxu&Fe_m7$$\":+++v=UKFFz?(QFio$\":ME$yh*f_1n>G$eFe_m7$$\":++]7`%*=WF: [(QFio$\":)o7w*=2=>aUqn&Fe_m7$$\":+++](oa5w7bxQFio$\":)GCqz)G**yldHe&F e_m7$$\":++](=#*>zxsG!)QFio$\":W$H[aR%Q%GF8]fFe_m7$$\":+++Dc^y%zK-$)QF io$\":m**H;Ln\"\\vMe*y&Fe_m7$$\":++]i!R];\"Gfd)QFio$\":$\\Q2O>-k]=@ScF e_m7$$\":++++Dc^GG&\\))QFio$\":SRXn:q9'4ujPcFe_m7$$\":+++]il(f*GR%**QF io$\":IMazy#4/,_T/eFe_m7$$\":+++++vVjH$Q5RFio$\":3FIQUMC=g5b(fFe_m7$$ \":+++++](=xDELRFio$\":Xat\\Q(Q38h]AeFe_m7$$\":+++++DJ!e=9cRFio$\":Kg3 V_CSH8)*>o&Fe_m7$$\":++++++v)Q6-zRFio$\":je'GRjr)>$*GGx&Fe_m7$$\":++++ +v=(>/!>+%Fio$\":M0&yf/\")3!o7u$fFe_m7$$\":++++DJ&p)4pU-%Fio$\":Y> X#)>2A!eFe_m7$$\":++++](=nxxjYSFio$\":8%ph$>Pbi&)*R(o&Fe_m7$$\":++++vV [mX1!pSFio$\":n)z[Ep%ey\"41ydFe_m7$$\":++++++Dc8v84%Fio$\":]#p0B$pdF e_m7$$\":+++](oHHq*>*\\TFio$\":wWS1O!\\GkoVofFe_m7$$\":++](=nj-7s%G:%F io$\":-j%G@I6TtN'3$eFe_m7$$\":+++DcwfPXud:%Fio$\":c5;Wxt_\\Z#[pcFe_m7$ $\":++]iS;$\\&p,(eTFio$\":h'z#yMz!)*QZ,$o&Fe_m7$$\":++++DcEs$*G;;%Fio$ \":1*yR%Fio$\":9 KmpvE5DtUTg&Fe_m7$$\":++++DJ&pC6g2UFio$\":/\\tSm-:WpH\"obFe_m7$$\":+++ ]P4'\\^#y)=UFio$\":\\u&yq&Fe_m7$$\":+++ ++++b)QEvUFio$\":s)ok=FHCj8FKbFe_m7$$\":+++++]iI\"Q_nVFio$\":'3'*pJC75 Mgbn_Fe_m7$$\":+++++vV=y>!fWFio$\":ak=4l\\1J:4'*4&Fe_m7$$\":++++++]Z2& 4VXFio$\":ayLTg!y7h%er'[Fe_m7$$\":+++++]7$[[XRYFio$\":fyd6FxEZ5%fdWFe_ m7$$\":+++++++&y,kDZFio$\":!yC\\$3.Gv[eV%RFe_m7$$\":+++++vVjrJv\"[Fio$ \":j(oS/$*G\"f\">LAMFe_m7$$\":+++++DcJk$[0\\Fio$\":**yk&>B&)=)RP3$HFe_ m7$$\"\"&\"\"!$\",$eJ26CF1-%&COLORG6&%$RGBG$F_]oF_]o$\"\"#!\"\"$\"\"\" F_]o-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7`x7$F($\" -TW.go=F-7$$\":++++++DJ5)3upF1$!,]2K4_)F-7$F/$!-oD3JKKF-7$$\":+++++vV) HxngyF1$!.`.D\"3ZYF97$F5$!.W%Ht$ex&F97$F;$!.`&4l`sqF97$F@$!.lNF_uF)F97 $FE$!.x&f(*z=$)F97$FJ$!.F**)fmp$)F97$Fhn$!.lvs'p!e)F97$Fbo$!.W[kfYI*F9 7$Fgo$!/]0?p;u#*F\\p7$F^p$!/.0=U-Q#*F\\p7$F`s$!/C-df')o)*F\\p7$Fes$!/e !eHe:#**F\\p7$Fjs$!0_`(4TX-5F\\p7$F_t$!0=)QYLHA5F\\p7$Fdt$!0!)*\\E6)*e 5F\\p7$Fit$!0eA&e?Yl5F\\p7$F^u$!0V0qfLv.\"F\\p7$Fcu$!0A%*=cO@-\"F\\p7$ Fhu$!07&fr%\\L4\"F\\p7$F]v$!0u9RC0V1\"F\\p7$Fbv$!0nyjw]s.\"F\\p7$Fgv$! 0,pEoWd.\"F\\p7$F\\w$!0JzK.^=4\"F\\p7$Faw$!0!*owqg+4\"F\\p7$Ffw$!0I() \\/\"*z3\"F\\p7$F[x$!0%Rwyw>v5F\\p7$F`x$!1%[aJX^$p5Fdx7$Ffx$!1=\"z9AV` 9\"Fdx7$F[y$!1Y%4J,^Q6\"Fdx7$F`y$!1!fa,vIV3\"Fdx7$Fey$!1Qw#4)za\"3\"Fd x7$Fjy$!1KTuB/)f5\"Fdx7$F_z$!1!H`,$*=[7\"Fdx7$Fdz$!1I\\6s2$)G6Fdx7$Fiz $!1r`@IRL)4\"Fdx7$F^[l$!1*Rq^kE_3\"Fdx7$Fc[l$!1cc@Ke(>:\"Fdx7$Fh[l$!10 t&>2A.7\"Fdx7$F]\\l$!13`8OrE#4\"Fdx7$Fb\\l$!1[e*3e[>6\"Fdx7$Fg\\l$!1!3 v])[-V6Fdx7$F\\]l$!1jk()Q'3'H6Fdx7$Fa]l$!1u6(yL\"[;6Fdx7$Ff]l$!1+7;q_- 16Fdx7$F[^l$!1OKLe!H#36Fdx7$F`^l$!1Myh\"G+S:\"Fdx7$Fe^l$!1a'>ms\")e<\" Fdx7$Fj^l$!1Sb0'[V/;\"Fdx7$F__l$!1Y?'Q(R3X6Fdx7$Fd_l$!1?84v,7J6Fdx7$Fi _l$!1b#eX&pmA6Fdx7$F^`l$!1s`EKH'))>\"Fdx7$Fc`l$!1S!pXK'*[;\"Fdx7$Fh`l$ !1Yva,kGL6Fdx7$F]al$!1A\"pg%o`N6Fdx7$Fbal$!1DSMV34\\6Fdx7$Fgal$!10&p+i G+?\"Fdx7$F\\bl$!1wY\"Rv+j;\"Fdx7$Fabl$!2$>^'[6[\\8\"Febl7$Fgbl$!2xL*G 4s(=;\"Febl7$F\\cl$!2xiVo'fG'=\"Febl7$Facl$!2&[.&G&4.d6Febl7$Ffcl$!26n G'G)*4D6Febl7$F[dl$!2s8$)Q6Febl7$F_el$!3jX;toGa!4\"Fcel7$Feel$ !3#yU)p'[b(e5Fcel7$Fjel$!3;eABF]'>,i*Fcel7$F^gl$!3]**Gdaz([,\"Fcel7$Fcgl$ !2o_t`<))\\')*Fcel7$Fhgl$!2[lQl-@Vg*Fcel7$F]hl$!20=$)\\8Qhl*Fcel7$Fbhl $!2'4QO]M$Q#**Fcel7$Fghl$!2R+Agc6yk*Fcel7$F\\il$!2@6'Q*3hEX*Fcel7$Fail $!2b$R5wsE!)**Fcel7$Ffil$!2j$)\\AO4Mv*Fcel7$F[jl$!3,pCJI1!*G&*F_jl7$Fa jl$!30)*[#yIHB])F_jl7$Ffjl$!3D4cr_Y4xuF_jl7$F[[m$!4DN?%)\\FdFA'F_[m7$F a[m$!4#*3$[uy&R'H\\F_[m7$Ff[m$!4ql^y%HNt*[$F_[m7$F[\\m$!558')Rb3V'>\"> F_\\m7$Fa\\m$!4)=%yJggl+b#F_\\m7$Ff\\m$\"60=2k1>O&oM9Fj\\m7$F\\]m$\"6f WH#R!)>'z5I$Fj\\m7$Fa]m$\"6,>2Mtu1*z7]Fj\\m7$Ff]m$\"7&>O^_CvEH\"=nFj]m 7$F\\^m$\"7o5?XmGyB4P')Fj]m7$Fa^m$\"9JqbtHRM$oos.\"Fe^m7$Fg^m$\"9\"=TW ZsS-8i.<\"Fe^m7$F\\_m$\"92]&\\E)R.?(onJ\"Fe^m7$Fa_m$\":L<>=6['y%Q4K[\" Fe_m7$Fg_m$\":;_Zq?**3e#z.o;Fe_m7$F\\`m$\":_HtwKe?@(*=fm\"Fe_m7$Fa`m$ \":nSy1e$3B&3_Qm\"Fe_m7$Ff`m$\":c8)H*\\wsH5FGm\"Fe_m7$F[am$\":`zz6&p;E W4cl;Fe_m7$F`am$\":*3s)))Q>TD2*[9?4t2>Fe_m7$Ffem $\"::W!>C&ziBDJi)=Fe_m7$F[fm$\":+bG2cb/f?F^'=Fe_m7$F`fm$\":bh:5RZo!Q6z k=Fe_m7$Fefm$\":\\\"olAMh&>H__'=Fe_m7$Fjfm$\":a81N]ocr\"*[v'=Fe_m7$F_g m$\":-!GW0)=Qf1`M(=Fe_m7$Fdgm$\":L9Cd=!e#Q5)=4>Fe_m7$Figm$\":JKc$=n^7 \"GT(=?Fe_m7$F^hm$\":(e+Kw\")>:ngA$*>Fe_m7$Fchm$\":'=#en7O6NDt!Q>Fe_m7 $Fhhm$\":Tx\"evh.k1wfO>Fe_m7$F]im$\":Lyb#H$*=aK]&z-#Fe_m7$Fbim$\":iPDC 0)e/oe3q>Fe_m7$Fgim$\":d$f#fh\"*[rH$pG>Fe_m7$F\\jm$\":,l<4y:IWT9I1#Fe_ m7$Fajm$\":1?!*=JGTvP0R+#Fe_m7$Ffjm$\":N*)\\)R%yiUH;.)>Fe_m7$F[[n$\":: J@\\#)[i%H6,h>Fe_m7$F`[n$\":U3$[MzX+Eo=2?Fe_m7$Fe[n$\":^NNq?U^:&G^Z?Fe _m7$Fj[n$\":_nzG-Ei5=&o*)>Fe_m7$F_\\n$\":#HM)3w$)4D7UX'>Fe_m7$Fd\\n$\" :heL#R\\L!RX7A-#Fe_m7$Fi\\n$\":$R%)en/tkvw'Q0#Fe_m7$F^]n$\":qHXS&)pnrU ;d0#Fe_m7$Fc]n$\":<4?oYs+)zhF)*>Fe_m7$Fh]n$\":m9sg^1>J6uh)>Fe_m7$F]^n$ \":hUAtaB`A+z&*3#Fe_m7$Fb^n$\":=N/.dhG&*ej)H?Fe_m7$Fg^n$\":Y$e'\\MaTJ \\-Q)>Fe_m7$F\\_n$\":1yWt([L()))e96@Fe_m7$Fa_n$\":J*zN'QCT)\\!oG1#Fe_m 7$Ff_n$\":17XoE-gY;\")z.#Fe_m7$F[`n$\":d7Wh+^)\\\"f0!>?Fe_m7$F``n$\":* 3g-(H3xGjr$z?Fe_m7$Fe`n$\":ME$yh*f_E$QQ5@Fe_m7$Fj`n$\":)o7w*=2=faQS0#F e_m7$F_an$\":)GCqz)G**Gv\\.-#Fe_m7$Fdan$\":W$H[aR%QWRsU:#Fe_m7$Fian$\" :m**H;Ln\"\\D`9'4#Fe_m7$F^bn$\":$\\Q2O>-kRZ7U?Fe_m7$Fcbn$\":SRXn:q9mMl =/#Fe_m7$Fhbn$\":IMazy#4/>(HT5#Fe_m7$F]cn$\":3FIQUMCGO-y;#Fe_m7$Fbcn$ \":Xat\\Q(Q3(p9^6#Fe_m7$Fgcn$\":Kg3V_CSf*)>p1#Fe_m7$F\\dn$\":je'GRjr)p ,pO5#Fe_m7$Fadn$\":M0&yf/\")3'\\[p;#Fe_m7$Ffdn$\":Y>X#Q%\\/7#Fe_m7 $F[en$\":8%ph$>Pb7'fW\"3#Fe_m7$F`en$\":n)z[Ep%eGMI$=@Fe_m7$Feen$\":]#p 0@Fe_m7$Fejn$\":s)ok=FHC]3(y0#Fe_m7$$\":+++D\"GjB3q9yUFi o$\":8\"=TvjJ\"ecgs.#Fe_m7$$\":+++]ils48I5G%Fio$\":f+fars\"ekF3S@Fe_m7 $$\":+++vV)*3PD8RG%Fio$\":j$[\"R(4ek.KPz?Fe_m7$$\":++++DJXkP'z'G%Fio$ \":t5V=dr(\\v5IH?Fe_m7$$\":+++D1k\"=*\\z'*G%Fio$\":U!\\?DKO,mh#o6#Fe_m 7$$\":+++](oz\">iiDH%Fio$\"::_-f'>FW*e#z,@Fe_m7$$\":+++voHaYuXaH%Fio$ \":3nzz;^'R3TgV?Fe_m7$$\":++++]i!Rn)G$)H%Fio$\":5mCyphQ3Q!>I?Fe_m7$$\" :+++]7GjG6&4/VFio$\":m7uFh#Gu;$QS1#Fe_m7$$\":++++v$fLe8')4VFio$\":)R.& p$QD`I=XD@Fe_m7$$\":+++DcEs5[WFJ%Fio$\":H9huZD3lWQc3#Fe_m7$$\":+++]Pf3 QgFcJ%Fio$\":H%*ziIk1;(GNG?Fe_m7$$\":+++v=#\\aE2^=VFio$\":$e'QJY/\"*H^ SJ-#Fe_m7$$\":+++++D\"G\\QR@VFio$\":kJp;XN/JZ,t5#Fe_m7$$\":++++](=Fe_m7$F_[o$\":a k=4l\\1Jo1q#>Fe_m7$Fd[o$\":ayLTg!y7$=e[&=Fe_m7$Fi[o$\":fyd6FxEO(3t=B&)[K!f)*=\"Fe_m7$F]]o$\",A2op,\"F1-Fc]o6&Fe]oFf]o$\"\"(Fi]oF f]o-F]^o6#%8scheme~with~c[10]=15/16G-F$6%7ay7$F($!-fK3F$>&F-7$F/$!-o)) 4h=sF-7$F5$!.WufK'Q')F97$F;$!.`vYTHq*F97$F@$!/l:x[b$3\"F97$FE$!/xX^!)z v5F97$FJ$!/F>mFqr5F97$FO$!/:Sk`@K6F97$FT$!/6&R;dY7\"F97$FY$!/+ri\"*Q$4 \"F97$Fhn$!/lDqFA$4\"F97$F]o$!/y7()*H(=6F97$Fbo$!/W=Fv+z6F97$Fgo$!0]0_ ZF=<\"F\\p7$F^p$!0.0u'GLk6F\\p7$Fcp$!0+e%Rbod6F\\p7$Fhp$!0o)e31#\\:\"F \\p7$F]q$!0,Ld*R377F\\p7$Fbq$!0;k0U'357F\\p7$Fgq$!0XrAL9Fdx7$F[y$!1Y%4\\*)*z!Q\"Fdx7$F `y$!1!fa1ynTM\"Fdx7$Fey$!1Qw#p.A/M\"Fdx7$Fjy$!1KT9#*=5r8Fdx7$F_z$!1!H` $f+D%R\"Fdx7$Fdz$!1I\\h:y_*R\"Fdx7$Fiz$!1r`JCyph8Fdx7$F^[l$!1*RqZ[-_M \"Fdx7$Fc[l$!1ccrF!\\$G9Fdx7$Fh[l$!10tv**)*4*Q\"Fdx7$F]\\l$!13`$HLRUN \"Fdx7$Fb\\l$!1[ef!e5%y8Fdx7$Fg\\l$!1!3v)*o,uT\"Fdx7$F\\]l$!1jk<=n$4S \"Fdx7$Fa]l$!1u6<'4M[Q\"Fdx7$Ff]l$!1+7O>(**>P\"Fdx7$F[^l$!1OK.#\\#zu8F dx7$F`^l$!1Myr)G%*>V\"Fdx7$Fe^l$!1a'>jV0)f9Fdx7$Fj^l$!1Sb:0V$4W\"Fdx7$ F__l$!1Y?1d/;A9Fdx7$Fd_l$!1?8zFW409Fdx7$Fi_l$!1b#e%fkz%R\"Fdx7$F^`l$!1 s`Oo'o**[\"Fdx7$Fc`l$!1S!p(fSvZ9Fdx7$Fh`l$!1YvM;rU39Fdx7$F]al$!1A\"p(H t*4T\"Fdx7$Fbal$!1DSM!RX%G9Fdx7$Fgal$!10&pcBP@\\\"Fdx7$F\\bl$!1wYhUz#4 X\"Fdx7$Fabl$!2$>^')3')f79Febl7$Fgbl$!2xL*)\\pd\"[9Febl7$F\\cl$!2xiVf! ft![\"Febl7$Facl$!2&[.#Qp!*fW\"Febl7$Ffcl$!26nG@=()yS\"Febl7$F[dl$!2%o'38Fcel7$Fdfl$!3x8mTm**zb7Fcel7$Fifl$!3G 4kZZ>c97Fcel7$F^gl$!3]**GxaU/#G\"Fcel7$Fcgl$!3o_t$f(Q>Y7Fcel7$Fhgl$!3[ lQlT#GK@\"Fcel7$F]hl$!30=$)fZ#\\(>7Fcel7$Fbhl$!3'4QOm=WUD\"Fcel7$Fghl$ !3R+A?cHN>7Fcel7$F\\il$!3@6'Qyb\"f%>\"Fcel7$Fail$!3b$R5`s5?E\"Fcel7$Ff il$!3j$)\\_k'eRB\"Fcel7$F[jl$!4,pCJ)p#*=17F_jl7$Fajl$!40)*[#yvF_\\m7$Ff\\m$\"60=2k1>AT_M\"Fj\\m7$F\\]m$\"6fWH#R!)fEsKOFj\\m 7$Fa]m$\"6,>2Mtu5WJw&Fj\\m7$Ff]m$\"7&>O^_Cv7ox)yFj]m7$F\\^m$\"8o5?XmGy 5aj-\"Fj]m7$Fa^m$\"9JqbtHRMi\"HGC\"Fe^m7$Fg^m$\"9\"=TWZsS7Ye2T\"Fe^m7$ F\\_m$\"92]&\\E)R.[ig%f\"Fe^m7$Fa_m$\":L<>=6['yy28.=Fe_m7$Fg_m$\":;_Zq ?**33`I].#Fe_m7$F\\`m$\":_HtwKe?h!)*4L?Fe_m7$Fa`m$\":nSy1e$3B@.@J?Fe_m 7$Ff`m$\":c8)H*\\ws>%4bI?Fe_m7$F[am$\":`zz6&p;E#3:W.#Fe_m7$F`am$\":*3s )))Q>T&QZC&4#Fe_m7$Feam$\":HFL;MO$4KtEPAFe_m7$Fjam$\":w:'QA#oi?:`]@#Fe _m7$F_bm$\":*fwlED_F*QIH>#Fe_m7$Fdbm$\":<+yT0r1Ji<_9#Fe_m7$Fibm$\":@pK %=sRoTNW,BFe_m7$F^cm$\":Kl&>#Fe_m7$F]dm$\":vQGs%*zzXM*\\xAFe_m7$Fbdm$ \":=o\\Kno$f'pSA@#Fe_m7$Fgdm$\":tm!fCG*o#f&fn<#Fe_m7$F\\em$\":+(Q$\\GI m,VibF#Fe_m7$Faem$\":gVNvDC!>MDUQBFe_m7$Ffem$\"::W!>C&zi`J%o7BFe_m7$F[ fm$\":+bG2cb/\\#fR(G#Fe_m7$F`fm$\":bh:5RZo!*RdvG#Fe_m7$Fefm$\":\\\"olA Mh&R%*p')G#Fe_m7$Fjfm$\":a81N]oc6f+?H#Fe_m7$F_gm$\":-!GW0)=Q*z`s*H#Fe_ m7$Fdgm$\":L9Cd=!e#Qi$eWBFe_m7$Figm$\":JKc$=n^7')Q2\"[#Fe_m7$F^hm$\":( e+Kw\")>:ee3]CFe_m7$Fchm$\":'=#en7O60RJAQ#Fe_m7$Fhhm$\":Tx\"evh.k,l+!Q #Fe_m7$F]im$\":Lyb#H$*=alGQ$\\#Fe_m7$Fbim$\":iPDC0)e/mAAACFe_m7$Fgim$ \":d$f#fh\"*[,@h5P#Fe_m7$F\\jm$\":,l<4y:IM'H5PDFe_m7$Fajm$\":1?!*=JGT& z'4WY#Fe_m7$Ffjm$\":N*)\\)R%yi7&*ofV#Fe_m7$F[[n$\"::J@\\#)[iCa&o7CFe_m 7$F`[n$\":U3$[MzX+cHIpCFe_m7$Fe[n$\":^NNq?U^:pZ)>DFe_m7$Fj[n$\":_nzG-E i+/]'[CFe_m7$F_\\n$\":#HM)3w$)4vY;tT#Fe_m7$Fd\\n$\":heL#R\\L!\\T'H*[#F e_m7$Fi\\n$\":$R%)en/tkHT@GDFe_m7$F^]n$\":qHXS&)pn@wQ6`#Fe_m7$Fc]n$\": <4?oYs+)emOgCFe_m7$Fh]n$\":m9sg^1>6-kXJ)*=DFe_m7$Fcbn$\":SRXn:q9OrT#=DFe_m7$Fhbn$\":I Mazy#4/vk%ff#Fe_m7$F]cn$\":3FIQUMC)yH\"fn#Fe_m7$Fbcn$\":Xat\\Q(Q3dg07E Fe_m7$Fgcn$\":Kg3V_CSHrvNb#Fe_m7$F\\dn$\":je'GRjr)*GB$)*f#Fe_m7$Fadn$ \":M0&yf/\")3!*HH!o#Fe_m7$Ffdn$\":Y>X#\\4&Ri#Fe_m7$F[en$\":8%ph$>P b_mumd#Fe_m7$F`en$\":n)z[Ep%eGrxKi#Fe_m7$Feen$\":]#p0r#Fe_m7 $Fjen$\":][QH+pM#R;g@EFe_m7$F_fn$\":4+HiTA:%\\;)[e#Fe_m7$Fdfn$\":c4%yw *\\.cYsxj#Fe_m7$Fifn$\":wWS1O!\\G'=p6t#Fe_m7$F^gn$\":-j%G@I6T0@SoEFe_m 7$Fcgn$\":c5;Wxt_4r\\Xf#Fe_m7$Fhgn$\":h'z#yMz!)4'[L,EFe_m7$F]hn$\":1*y R\"p[g#F e_m7$$\":++]7`%Rz_h'y=%Fio$\":TF&H=&4/)>[xoDFe_m7$$\":+++D19%\\Mao!>%F io$\":BZy;XnPP&*)*or#Fe_m7$$\":++]PfL%>;Z]$>%Fio$\":!3&GxM6n$Q\"G9k#Fe _m7$F\\in$\":9KmpvE5N9$RuDFe_m7$$\":++]ils%fzK9*>%Fio$\":Osx0\"G=&=_3: i#Fe_m7$$\":+++v=#\\HhD'>?%Fio$\":Z3\"=%\\A>pD9\"zEFe_m7$$\":++](=<^*H %=y/UFio$\":n7N)4*>f!Q'=`g#Fe_m7$Fain$\":/\\tSm-:/uM/c#Fe_m7$Ffin$\": \\FWY&4vh#Fe_m7$ F]dq$\":3nzz;^'RpU*\\a#Fe_m7$Fbdq$\":5mCyphQy&*)*y_#Fe_m7$Fgdq$\":m7uF h#GufS9rDFe_m7$F\\eq$\":)R.&p$QD`H(z![EFe_m7$Faeq$\":H9huZD3DKH()f#Fe_ m7$Ffeq$\":H%*ziIk1'z0HFDFe_m7$F[fq$\":$e'QJY/\"*z>>/_#Fe_m7$F`fq$\":k Jp;XN/\"zrSEEFe_m7$Fefq$\":*)z\"GP&*HchQgJDFe_m7$Fjjn$\":'3'*pJC75J+Wi CFe_m7$F_[o$\":ak=4l\\1\"4'=-T#Fe_m7$Fd[o$\":ayLTg!y7'3AqK#Fe_m7$Fi[o$ \":fyd6FxEP0`P;#Fe_m7$F^\\o$\":!yC\\$3.GT1Ld%>Fe_m7$Fc\\o$\":j(oS/$*G! HZ%)fs\"Fe_m7$Fh\\o$\":**yk&>B&)37k4A:Fe_m7$F]]o$\",%4AW:8F1-Fc]o6&Fe] o$\"\"*Fi]o$\"\"%Fi]oFf]o-F]^o6#%8scheme~with~c[10]=19/20G-F$6%7dv7$F( $\"-T&4%R/bF-7$$\":+++++D\"yD?_=mF1$\"-Z1AY!)\\F-7$$\":+++++]i:0Wqt'F1 $\"-&Gu)p/XF-7$$\":+++++vVt2mb&oF1$\"-')=/@tSF-7$Fg^o$\"-]?Lp&o$F-7$$ \":+++++](oa@86sF1$\"-m&Q\"GYIF-7$F/$\"-KMg#>Z#F-7$$\":++++](o/opUawF1 $\"-;E#)of?F-7$F__o$\".Z'4jO1F\\p7$F`s$!.C-qwM8&F\\p7$Ffw$!.I(3^q^zF\\p7$Fg\\l$!0!3vX8*)G5F dx7$F__l$!0Y?'zRM.7Fdx7$Fabl$!1$>^1)fHc8Febl7$Fgbl$!1xL*oNH3[\"Febl7$F \\cl$!1xiV\\$[zg\"Febl7$Facl$!1&[.ONm\\k\"Febl7$Ffcl$!16noxDHw;Febl7$F [dl$!1[(Qru\"Fcel7$Feel$!2#yU)H*3X`< Fcel7$Fjel$!2;eA$\\z*)H=Fcel7$F_fl$!2pz2Lh83$>Fcel7$$\":+++++]iDpqM%=F io$!2Ph)4(4TN\">Fcel7$Fdfl$!2x8mr!Qc'*=Fcel7$$\":+++++]P9hnh'=Fio$!2Vm N$)p*R#)=Fcel7$Fifl$!2G4k()Gd/)=Fcel7$F]hl$!20=$)zt*)y\">Fcel7$Fail$!2 b$R5iK9B?Fcel7$Ffil$!2j$)\\-FOD+#Fcel7$F[jl$!3,pCJ$*)3=)>F_jl7$$\":+++ +]P4@'=8k>Fio$!3`F'Q.Vf(y>F_jl7$Fajl$!30)*[#eG<9(>F_jl7$$\":++++]Pf`qq V0#Fio$!3*H3LIs\"4-?F_jl7$Ffjl$!3D4crgNsD?F_jl7$$\":+++++]7.L6\\9#Fio$ !3Wyn&>%H(H*>F_jl7$F[[m$!4DN?%)\\I-*f>F_[m7$$\":++++](=#4/PZB#Fio$!4\" G$pa#[udq>F_[m7$Fa[m$!4#*3$[uy%Hu(>F_[m7$$\":++++DJ&pKnT/BFio$!4'\\*[U )3=&y$>F_[m7$$\":++++](o/\\WzDBFio$!4^C.F*4DX**=F_[m7$$\":+++]ilAsI$[O BFio$!49(R2z]XM#)=F_[m7$$\":++++vV)Rl@F_[m7$$\":+++v$fe]nNTuBFio$!4wF_[m7$$\":++]P%['Rh44KQ#Fio$!5?oc)pMRnW=F_\\m7$$\":++]7y]&*=ys!*Q#Fio$!5AAyXQ%G$fo=F_\\m7$$ \":++++vVtZi/?R#Fio$!5@s3F_\\m7$$\":+++]i:&G'*>t.CFio$!5NErJ\"G 9$em=F_\\m7$$\":++++](ozn$faT#Fio$!5jW8vL*\\)zD=F_\\m7$$\":+++]Pf3$Rn= FCFio$!5L#**HG.*H+&z\"F_\\m7$$\":++++DJ?369*QCFio$!5z>%>+\")fb$4=F_\\m 7$$\":++](=U#)p`f%=W#Fio$!51PtA&)\\p?e=F_\\m7$$\":+++v=S_Jx\"F_\\m7$$\":+++ ]7.KB[T1X#Fio$!5;s-'R#y]yo=F_\\m7$$\":++]P4'*4_KtNX#Fio$!5q'G:[-pL[\"= F_\\m7$$\":+++D1*y3o^]cCFio$!5%Hx\"[J:'=ww\"F_\\m7$$\":++]7.#e'4,P%fCF io$!5(oNMdWt\"*f\"=F_\\m7$F[\\m$!558')Rb[&>Z#=F_\\m7$$\":++++]ildJ.Z[# Fio$!5^3k&GlHi%eMQ p>[GfL43_69:Fj\\m7$$\":++++D\"yD:9KmEFio$!6kW&yp*>Us3Y\"Fj\\m7$$\":+ +++]7yqX@xo#Fio$!68jR3qY5NtT\"Fj\\m7$$\":+++]7G))zZ@%)p#Fio$!6feh?0Ss: YS\"Fj\\m7$$\":++++vV)*))\\@\"4FFio$!6_>0`uCY)Q.9Fj\\m7$$\":+++]Pf3)>: #)>FFio$!6lTppW7tR-U\"Fj\\m7$F\\]m$!6Tbq2'>Smc09Fj\\m7$$\":++++v=noP/O v#Fio$!6*QJ!G(o3m'=K\"Fj\\m7$$\":++++]i:I@(owFFio$!68*z!fNO_\\\\C\"Fj \\m7$$\":+++]P%)*3J'G#)y#Fio$!6a*\\d8tSHdA7Fj\\m7$$\":++++D1k\"\\+x*z# Fio$!6mdGL?\"4<\"[C\"Fj\\m7$$\":+++]7GQsY68\"GFio$!6F>`b:Jiw)>7Fj\\m7$ Fa]m$!6*4Gfm_#QQ&y6Fj\\m7$Ff]m$!60Q'[ZvCxW;$*Fj]m7$F\\^m$!6K*)zaLrTm)Q tFj]m7$Fa^m$!7pHWEqgl`.;[Fe^m7$Fg^m$!7>)eb_Ff(G3,@Fe^m7$F\\_m$\"62]&\\ E)R.nLw&Fe^m7$Fa_m$\"8L<>=6['y][RLFe_m7$Fg_m$\"8;_Zq?**3e09g'Fe_m7$Fea m$\"9\"HFL;MO$4(Q[o*!#S7$F[fm$\":)*\\&G2cb/4u387F_av7$Figm$\":2BjN=n^7 J\"fM:F_av7$Fa_n$\":3$*zN'QCT[8)**y\"F_av7$F]cn$\":#3FIQUMCQ*\\A5#F_av 7$Fadn$\":R`]yf/\")3W1&>BF_av7$Ffdn$\":cu@#G>>X#y7sJ#F_av7$F[en$\":ITp h$>PbK`,ABF_av7$F`en$\":q')z[Ep%eG$f-U#F_av7$Feen$\":(\\#p0a#F_av7$Fdfn$\":b&4%yw*\\.Oyv7EF_av7$Fifn$\":gZ/kg.\\GNH(GFF_av7$F^g n$\":=IYG@I6T5Cum#F_av7$Fcgn$\":j0hTuPF&RP!Rf#F_av7$Fhgn$\":6mz#yMz!)f of3EF_av7$F]hn$\":e!*yRH&pWFio$\":btc ^.G(=j)HPB$F_av7$$\":++++DJq]gQ+[%Fio$\":6Mj9'H*H?0=fL$F_av7$$\":+++]P f$o;![0\\%Fio$\":n-#Gb\"4A-+*oILF_av7$$\":++++](oHGu0,XFio$\":\\3iuvX+ f;M]K$F_av7$$\":++++vVB:Dw?_%Fio$\":XBXI^`$eBVY7LF_av7$Fd[o$\":Q&yLTg! y7uk#)H$F_av7$$\":++++DJXk/S^b%Fio$\":2yQV2]9kng>C$F_av7$$\":++++]iS\" =]=nXFio$\":!3rH/X.Go6_1KF_av7$$\":++++v$f$)*)*H#zXFio$\":%*=b\\+&G+4K (>I$F_av7$$\":+++++DJ:'\\F\"f%Fio$\":[>wq.m*o77)4S$F_av7$$\":++++DcEK$ *>Lg%Fio$\":yg_*zG5gn!Q#RLF_av7$$\":++++](=#\\!\\O:YFio$\":kTN[+K5_?g. G$F_av7$$\":+++DJqXy9w$=YFio$\":Q]fw*)Q@\">vahKF_av7$$\":+++]7`p2R(Q@Y Fio$\":$>iF(3')>()yxuT$F_av7$$\":+++v$fLpL')RCYFio$\":sVk%f$*[t59c;LF_ av7$$\":++++v=04k(yC$F_av7$$\":+++Dc,T&>6UIYFio $\":6b[y[R3Z=`mX$F_av7$$\":+++]P%[YiBVLYFio$\":&GaTzC&))yEzTN$F_av7$$ \":+++v=n)Q0OWOYFio$\":OkH#)4n>*=qVkKF_av7$Fi[o$\":%fyd6FxE_&Q?P$F_av7 $$\":+++v$f3))R\"[@k%Fio$\":65'eltj%zIUHS$F_av7$$\":+++](=#\\9VT[k%Fio $\":Kf6Aj$Qp1G]8LF_av7$$\":+++D\"yv,BZ`ZYFio$\":$\\-F-f?`hf)*eKF_av7$$ \":++++v$fe9!G-l%Fio$\":)\\rs\"H7i0(G1uMF_av7$$\":+++voHahI@Hl%Fio$\": %fckYV@'*[Aw\"Q$F_av7$$\":+++]ilAxf9cl%Fio$\":*eMSOB_)*)3J^H$F_av7$$\" :+++Dc,\"H*)yIeYFio$\":4\"fn')fu&R&=1!G$F_av7$$\":++++]Pf3=,5m%Fio$\": SKLw]+w#\\'*)>X$F_av7$$\":++++D\"G8ZVxrYFio$\"::kry6*)QtZD'HMF_av7$$\" :+++++D1M^ZDo%Fio$\":!GAW^@x%4i#)pS$F_av7$$\":++++]7`f%Q4/ZFio$\":V.)R A;p_%4;6O$F_av7$F^\\o$\":'zZ#\\$3.Gss0-WQ-3)R$F_av7$$\":+++v=U#[4sxD[Fio$\":UC,GYx'*R]b*4LF_av7$$\" :++++DJ&>2d_G[Fio$\":PM7z6(ylN+$zI$F_av7$$\":+++]P4@EqAS$[Fio$\":W\">F DM)*G_G'>O$F_av7$$\":++++](o/)p>&R[Fio$\":%4+J)p#HOEj'QS$F_av7$$\":+++ Dcwfd>oA%[Fio$\":po`$y!)*[%4/0=MF_av7$$\":+++]ilsMp;]%[Fio$\":!GU\"\\, 86tWsmK$F_av7$$\":+++voa&=\">lx%[Fio$\":(zk[0u;_*3r#yKF_av7$$\":++++vV )*))o80&[Fio$\":RNn]n]kEOT[Z$F_av7$$\":+++D\"G8h'=iK&[Fio$\":sH^0M![L$ zw1Q$F_av7$$\":+++](=UK%o5g&[Fio$\":r1ntUrQ,&QZ%H$F_av7$$\":+++v$4r.#= f(e[Fio$\":O&4D:nIpn4)GJ$F_av7$$\":++++++](zw]h[Fio$\":W(zn!z2D'H`NOMF _av7$$\":++++]7`9m&\\$)[Fio$\":9Y\"3\"zla^,#RfLF_av7$Fh\\o$\":()*yk&>B &)y#)\\[G$F_av7$$\":++++]7y:#=u_\\Fio$\":4q_d'f(=V?+wL$F_av7$F]]o$\"+3 :*>Q$F1-%'COLOURG6&Fe]o$\"*++++\"!\")Ff]oFbew-F]^o6#%8scheme~with~c[10 ]=32/33G-F$6%7bv7$F($\"-T\\Io1BF-7$Ffet$\"-ZY*RN3#F-7$F[ft$\"-&e.p0)=F -7$F`ft$\"-'Q=Chp\"F-7$Fg^o$\"-]'fa(H:F-7$Fhft$\"-mHhf[7F-7$F/$\"-KfaI +5F-7$F__o$\"-Zc]C?mF97$F5$\"-cb44lSF97$F;$\"-ZC`:/=F97$F@$!+lvl'o$F97 $FJ$!-Ff!*)Q?\"F97$F^p$!..0G&ouCF\\p7$F`s$!.C-r\"p'=%F\\p7$Ffw$!.I(e57 ZeF\\p7$Fg\\l$!/!3vST_A(Fdx7$F__l$!/Y?cptD#)Fdx7$Fabl$!0$>^c`j)4*Febl7 $Fgbl$!0xL*)o5D&)*Febl7$F\\cl$!1xiVi6)=1\"Febl7$Facl$!1&[.9^;/3\"Febl7 $Ffcl$!16ny:TQ&4\"Febl7$F[dl$!1Nr'38\"Fcel7$Feel$!2#yU)*3!\\J8\"Fc el7$Fjel$!2;eAj\"Q&o<\"Fcel7$F_fl$!2pz2t?[`B\"Fcel7$Fdfl$!2x8m\"o*Q4@ \"Fcel7$Fifl$!2G4k(o\"Qx>\"Fcel7$F]hl$!20=$)RYE(>7Fcel7$Fail$!2b$R5L:L &G\"Fcel7$Ffil$!2j$)\\_/z4F\"Fcel7$F[jl$!3,pCJm,ec7F_jl7$Fe^u$!3`F'Q.8 6(\\7F_jl7$Fajl$!30)*[#eL`,C\"F_jl7$F]_u$!3*H3Lgmc)pM'4N6\"F_\\m7$F_du$!5AAyXQ %GJo7\"F_\\m7$Fddu$!5@s3rOF')GN6F_\\m7$$\":++]iSmIk*[DCCFio$!5]T?%3/K& e-6F_\\m7$Fceu$!5L#**HG.(=\\x5F_\\m7$$\":++]PM_'=#e=,V#Fio$!5Zwft@t.lS 6F_\\m7$$\":+++DJXk]U]IV#Fio$!5YZt&R*Q/q26F_\\m7$$\":++]7GQUzE#)fV#Fio $!5?G1*))[bAs2\"F_\\m7$Fheu$!5z>%>+\"e\\y$3\"F_\\m7$F]fu$!51PtA&)z&fE6 \"F_\\m7$Fbfu$!5\"=C:qcid23\"F_\\m7$Fgfu$!5QdB?(>N(=h5F_\\m7$F\\gu$!5; s-'R#=eP<6F_\\m7$Fagu$!5q'G:[-,:^3\"F_\\m7$Ffgu$!5%Hx\"[JXEqc5F_\\m7$F [hu$!5(oNMdW)yN%3\"F_\\m7$F[\\m$!558')Rb)HA%*3\"F_\\m7$Fchu$!5^3k&Gl=q m/\"F_\\m7$Fhhu$!5lhd#*pm2W45F_\\m7$F]iu$!54$\\'R!pxP9+\"F_\\m7$Fbiu$! 51XY))>/?f:5F_\\m7$Fgiu$!5evN!\\EO\">C5F_\\m7$F\\ju$!4e%RyBtP!H'**F_\\ m7$Faju$!4\"Q(p!z'ejpz*F_\\m7$Ffju$!5:q2D(*e(o..\"F_\\m7$F[[v$!5'Hd&[o mD,-5F_\\m7$F`[v$!4<1S.9#zQg(*F_\\m7$Fe[v$!4&z?ET.Tp>)*F_\\m7$Fa\\m$!5 )=%yJg5Lu25F_\\m7$Ff\\m$!5&>GfL4[C$[()Fj\\m7$F\\]m$!5Tbq2'>?+C$zFj\\m7 $Fa]m$!5*4Gfm_iLkV'Fj\\m7$Ff]m$!60Q'[ZvC7(Q#[Fj]m7$F\\^m$!6K*)zaLr,0A[ $Fj]m7$Fa^m$!7pHWEqglCUP=Fe^m7$Fg^m$!6>)eb_Ffd&3>\"Fe^m7$F\\_m$\"72]& \\E)R..)*)e\"Fe^m7$Fa_m$\"8L<>=6['y$*GnLFe_m7$Fg_m$\"8;_Zq?**3yNWX&Fe_ m7$Feam$\"9\"HFL;MO$45g8uF_av7$F[fm$\"9)*\\&G2cb/*>f!)))F_av7$Figm$\": 2BjN=n^7wh14\"F_av7$Fa_n$\":3$*zN'QCT3LqW7F_av7$F]cn$\":#3FIQUMCol^P9F _av7$Fadn$\":R`]yf/\")3d-Yc\"F_av7$Ffdn$\":cu@#G>>X_W!)e:F_av7$F[en$\" :ITph$>Pb-:hd:F_av7$F`en$\":q')z[Ep%eG(=zh\"F_av7$Feen$\":(\\#p0lNKTFio$\":;e)pr'\\amkw&[F_av7$F_[o$\":SX '=4l\\1\"4o4.#F_av7$F]fv$\":btc^.G(=t_;y?F_av7$Fbfv$\":6Mj9'H*H?FfB9#F _av7$Fgfv$\":n-#Gb\"4A-L;s8#F_av7$F\\gv$\":\\3iuvX+f=8=8#F_av7$Fagv$\" :XBXI^`$e$)yB?@F_av7$Fd[o$\":Q&yLTg!y73pw5#F_av7$Figv$\":2yQV2]9kW=*p? F_av7$F^hv$\":!3rH/X.Go7.X?F_av7$Fchv$\":%*=b\\+&G+HH%G5#F_av7$Fhhv$\" :[>wq.m*oZ*>W;#F_av7$F]iv$\":yg_*zG5g;qQB@F_av7$Fbiv$\":kTN[+K5_O^T3#F _av7$Fgiv$\":Q]fw*)Q@,'z#32#F_av7$F\\jv$\":$>iF(3')>()3%fp@F_av7$Fajv$ \":sVk%f$*[txT]0@F_av7$Ffjv$\":HC&y(p*>0uR9h?F_av7$F[[w$\":6b[y[R3FjsE >#F_av7$F`[w$\":&GaTzC&)))=jmF@F_av7$Fe[w$\":OkH#)4n>H`9/2#F_av7$Fi[o$ \":%fyd6FxE#>Op8#F_av7$F]\\w$\":65'eltj%pYQo:#F_av7$Fb\\w$\":Kf6Aj$Qp[ X5+@F_av7$Fg\\w$\":$\\-F-f?`&\\TZ1#F_av7$F\\]w$\":)\\rs\"H7il&f6+AF_av 7$Fa]w$\":%fckYV@'47h;9#F_av7$Ff]w$\":*eMSOB_)zgom3#F_av7$F[^w$\":4\"f n')fu&HkVe2#F_av7$F`^w$\":SKLw]+w#3%[V=#F_av7$Fe^w$\"::kry6*)Qj=>%o@F_ av7$Fj^w$\":!GAW^@x%4vNB:#F_av7$F__w$\":V.)RA;p_*)e&)>@F_av7$F^\\o$\": 'zZ#\\$3.GaL\\)3#F_av7$Fc\\o$\":MwoS/$*GK-0<0#F_av7$Fj_w$\":crHWMk1n$p y!=#F_av7$F_`w$\":?1C*e>-%Q1x;7#F_av7$Fd`w$\":UC,GYx'fI5]m?F_av7$Fi`w$ \":PM7z6(y0r%HQ1#F_av7$F^aw$\":W\">FDM)*GAFI(4#F_av7$Fcaw$\":%4+J)p#HO KTO@@F_av7$Fhaw$\":po`$y!)*[M<^/8#F_av7$F]bw$\":!GU\"\\,86BwQM2#F_av7$ Fbbw$\":(zk[0u;_8+MU?F_av7$Fgbw$\":RNn]n]kOLcR;#F_av7$F\\cw$\":sH^0M![ L2IJ0@F_av7$Facw$\":r1ntUrQh\"=V^?F_av7$Ffcw$\":O&4D:nIp![k81#F_av7$F[ dw$\":W(zn!z2DY'y5Q@F_av7$F`dw$\":9Y\"3\"zlaT#z]'3#F_av7$Fh\\o$\":()*y k&>B&))fwVO?F_av7$Fhdw$\":4q_d'f(=jV&pf?F_av7$F]]o$\"+=%>q2#F1-Fc]o6&F e]o$F^]oFi]oFf]oFj]o-F]^o6#%8scheme~with~c[10]=81/82G-F$6%7cu7$F($\",T 6nsl%F-7$Fg^o$\",]H$Rr[F-7$F/$\",K$fQ$>&F-7$F__o$\"-Zc#)z2^F97$F5$\"-c N2k,\\F97$F;$\"-ZM=dyYF97$F@$\"-N%4_dM%F97$FJ$\"-t![Dwj$F97$F^p$\".(\\ zk\"z7$F\\p7$F`s$\".w(4]?\\=F\\p7$Ffw$\"-q7@y()fF\\p7$Fg\\l$!.!3vZ#*Rx Fdx7$F__l$!/Y?YxY@@Fdx7$Fabl$!0$>^m-LBMFebl7$F\\cl$!0xiVp@E9&Febl7$Ffc l$!06n)ebNIhFebl7$Feel$!1#yU)*zPnA(Fcel7$F_fl$!1pz2VMWb))Fcel7$F[jl$!2 ,pCJI7=))*F_jl7$Fajl$!30)*[#y(Qu^5F_jl7$F]_u$!3*H3Lg/7>6\"F_jl7$Ffjl$! 3D4cr5L!*p6F_jl7$Fe_u$!3Wyn&RJLp=\"F_jl7$F[[m$!4DN?%)\\nnL?\"F_[m7$F]` u$!4\"G$pa#=;#pD\"F_[m7$Fa[m$!4#*3$[u)*y`58F_[m7$Fe`u$!4'\\*[U))>#H/8F _[m7$Fj`u$!4^C.F***eX)H\"F_[m7$F_au$!49(R2zq\\(oH\"F_[m7$Fdau$!47_8I4 \"3w(H\"F_[m7$Fiau$!4k@;jI`ZMI\"F_[m7$Ff[m$!4ql^y%\\R\"yJ\"F_[m7$Ffbu$ !4wc)pC#[CK\"F_\\m7$F_du$!5AAyXQ/hKV8F_\\m7$Fddu$!5@s3rOdY^+9F_\\m7$F`bx$!5]T?%3/5t,O\"F_\\m7$Fc eu$!5L#**HG.\"fxH8F_\\m7$Fhbx$!5Zwft@Vs(3U\"F_\\m7$F]cx$!5YZt&R*[B$)z8 F_\\m7$Fbcx$!5?G1*))[Bd>M\"F_\\m7$Fheu$!5z>%>+\"QZoa8F_\\m7$F]fu$!51Pt A&)p+o*R\"F_\\m7$Fbfu$!5\"=C:qcAd&f8F_\\m7$Fgfu$!5QdB?(>N)3O8F_\\m7$F \\gu$!5;s-'R#QKh>9F_\\m7$Fagu$!5q'G:[-YE'y8F_\\m7$Ffgu$!5%Hx\"[Jb.vU8F _\\m7$F[hu$!5(oNMdWl*)eQ\"F_\\m7$F[\\m$!558')RbQh-)R\"F_\\m7$Fchu$!5^3 k&GlI$yq8F_\\m7$Fhhu$!5lhd#*pw:O]8F_\\m7$Fbiu$!51XY))>%HNeR\"F_\\m7$Fa \\m$!5)=%yJg+4)GU\"F_\\m7$$\":+++]7`>b`eLc#Fio$!5rS%4bEr$G+9F_\\m7$$\" :++++D\"y]p.,vDFio$!561/\"*o0zFy8F_\\m7$$\":++]7G)[+GL#zd#Fio$!5^#\\m- ;c$Re8F_\\m7$$\":+++DJ&>]'GO3e#Fio$!5KwV\\cmS$oV\"F_\\m7$$\":++]PM-** \\C\\Pe#Fio$!5wQ&*R>#QFcR\"F_\\m7$$\":+++]P4'\\.Am'e#Fio$!5o#R\\$>STMg 8F_\\m7$$\":++]iS;$*>;v&*e#Fio$!5v_!G%e^'fGT\"F_\\m7$$\":+++vVB!\\?\") [#f#Fio$!56%[70e*>Q89F_\\m7$$\":++](o/t)*y5S&f#Fio$!5K,j,f2emt8F_\\m7$ $\":++++]P%[PSJ)f#Fio$!5EpYj$=gtKO\"F_\\m7$$\":+++Dc^yW&*RTg#Fio$!5'[j 6(3&40-R\"F_\\m7$$\":+++]ils9(e'*4EFio$!5)GGou2XI^W\"F_\\m7$$\":++]ils p*H)yGh#Fio$!5PR=A_\\!GuS\"F_\\m7$$\":+++vozm%)y\"z:EFio$!663@'H(Q9S&o 8Fj\\m7$$\":++](=nQ'pu/(=EFio$!6Fh_Wu(f%)Hr8Fj\\m7$$\":++++v$4Y0xh@EFi o$!6+*z#)\\ONguC9Fj\\m7$$\":+++](=#\\%R&pKj#Fio$!6s$Hx&4Xw$o+9Fj\\m7$F f\\m$!6&>GfL4[m>x8Fj\\m7$$\":+++]ilZVR@cl#Fio$!6'*f.&4*fMbsO\"Fj\\m7$F `\\v$!6kW&yp*>y=$e8Fj\\m7$$\":+++](oz;O9-xEFio$!6`mB@bt\\u:N\"Fj\\m7$F e\\v$!68jR3qYFF\"\\8Fj\\m7$F_]v$!6_>0`uCY(Rt8Fj\\m7$F\\]m$!6Tbq2'>+QY? 9Fj\\m7$F\\^v$!6*QJ!G(oys;r8Fj\\m7$Fa^v$!68*z!fNO-*H&*='Hv4N\"Fj\\m7$F[_v$!6mdGL?\"*Hq =Q\"Fj\\m7$$\":++](=F6E%3GFio$!6Sw\")HYJ%Gg>8Fj\\m7$F `_v$!6F>`b:J5G)y8Fj\\m7$$\":++]Pf$ov@o>9GFio$!6[2nc**[H:(R8Fj\\m7$$\": +++D1Rvi<#3f'Q*38Fj\\m7$$\":++]7`%RzIv'*>GFio$!63BXT5g.9 #*Q\"Fj\\m7$Fa]m$!6*4Gfm_U\"fIN\"Fj\\m7$$\":+++]i:N]]>K\"Fj\\m7$$\":++++D\"yvk%Ho%GFio$!7Zr\"[z%)*4/8'H\"Fj]m7$$\":++]il Z)=Pl#)\\GFio$!7@CTR'zLnN;O\"Fj]m7$$\":+++D19>'4O#G&GFio$!7:U`HQI==;Q8 Fj]m7$$\":++](o/)\\?o?e&GFio$!7XOnsyvW35+8Fj]m7$$\":+++](o/[ax\")eGFio $!7.e)*HfvNca/8Fj]m7$$\":++]7G86p#[\"='GFio$!7zad0^'=JfhM\"Fj]m7$$\":+ ++vozT$**=\"['GFio$!7s\"o\"yhDUur18Fj]m7$$\":++]P4Ys<(*3y'GFio$!7uWr%z I;]!)RG\"Fj]m7$$\":++++]7.U/13(GFio$!7ReQV*)zca(RN\"Fj]m7$$\":+++]7yDR L%z#)GFio$!7\\'y*R#p1?,9K\"Fj]m7$$\":++++vV[Oi#y%*GFio$!72lr7$H9dh'*G \"Fj]m7$$\":++]i:5zgpzx*GFio$!7s%e6ZUA6imE\"Fj]m7$$\":+++Dcw4&onx+HFio $!7?#obP$eoWhN8Fj]m7$$\":++](oH/%4%Qx.HFio$!7q;kv-\")f,A'H\"Fj]m7$$\": +++]P4rL\"4x1HFio$!77([VjvV\"exi7Fj]m7$$\":++]7yv,e)zw4HFio$!7I@\"ez[: $oo?8Fj]m7$$\":+++v=UK#e]w7HFio$!7QP8(4>F%fy-8Fj]m7$$\":++]Pf3j18id\"H Fio$!7Y\\h_-^\\[gl7Fj]m7$Ff]m$!70Q'[ZvCtmsE\"Fj]m7$$\":+++]iS;Q7xR#HFi o$!7_,pg'Qt**)[!G\"Fj]m7$$\":++++D1RX/&>HHFio$!7y9bM**fxe1v7Fj]m7$$\": +++D1R+\\+/=$HFio$!7I_B:L&QzW:I\"Fj]m7$$\":+++](=-PHFio$!7u`UQw36&)zT7Fj]m7$$\":++++]P%)f)3jRH Fio$!7)[NAfXBPQ0H\"Fj]m7$$\":+++DJqXj%)RA%HFio$!7r?eD\"Fj]m7$$\":+++v$f$oqwdu%HFio$!7o 56]%>m)RkL7Fj]m7$$\":++++voHusm+&HFio$!7>r;?L)3^g%48Fj]m7$$\":+++Dc,\" z(ovE&HFio$!7.@v\"Fj]m7$$\":++++](o H8*Ht/$Fio$!7'GU)pm17'>mZ,0$Fio$!7AhG)4iOD2HfJybIFio $!7SIZB3hL)Rf:\"Fj]m7$$\":+++]7.dz!4geIFio$!7US+F'e\\*RO`6Fj]m7$$\":++ +v$4rG0kBkIFio$!7-(=Wq)e'QoM;\"Fj]m7$$\":++++v=()pIFio$!7#HVw$pT0o X%>\"Fj]m7$$\":+++]PMxs*G9\"3$Fio$!8ME0#er@3$z*z6Fe^m7$Fa^m$!8pHWEqglE T\\:\"Fe^m7$Fg^m$!8>)eb_FfxjCB5Fe^m7$F\\_m$!7$*\\/NV,RlFe_m7$Feam$!94FnOeOm!Rr$>^F_ av7$F[fm$!9-]9FRWa4`mMLF_av7$Figm$!9$pnV;G$[(QwSv\"F_av7$Fa_n$\"8yI*zN 'QCTQ:B'!#T7$F]cn$\":;3FIQUMCo\"f:>F\\`\\l7$Fadn$\":!R`]yf/\")3jZQPF\\ `\\l7$Feen$\":m\\#p0m\"F_av7$Fc\\o$\":MwoS/$*G0fSZw\"F_av7$Fh \\o$\":()*yk&>B&)GoV$))=F_av7$F]]o$\"+[UR&4#F1-F`ew6&Fe]oFbewFf]oFf]o- F]^o6#%:scheme~with~c[10]=169/170G-%%FONTG6$%*HELVETICAGFidt-%&TITLEG6 #%Verror~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELS G6$Q\"x6\"Q!Fec\\l-%%VIEWG6$;F(F]]o;$F_\\m!#;$\"#jF]d\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]=19/20" "sch eme with c[10]=32/33" "scheme with c[10]=81/82" "scheme with c[10]=169 /170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 313 "evalf[25](plot(['bn_RK7_4'(x)-b(x),'bn_RK7_5'(x)-b(x ),'bn_RK7_6'(x)-b(x)],x=0.65..5,-5.3e-16..2.3e-16,\nfont=[HELVETICA,9] ,color=[magenta,COLOR(RGB,.5,0,1),red],legend=[`scheme with c[10]=32/3 3`,`scheme with c[10]=81/82`,`scheme with c[10]=169/170`],title=`error curves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 821 400 400 {PLOTDATA 2 "6)-%'CURVESG6%7dv7$$\"#l!\"#$!-s &4%R/b!#F7$$\":+++++D\"yD?_=m!#D$!-^1AY!)\\F-7$$\":+++++]i:0Wqt'F1$!-j U()p/XF-7$$\":+++++vVt2mb&oF1$!-j>/@tSF-7$$\":++++++DJ5)3upF1$!-@?Lp&o $F-7$$\":+++++](oa@86sF1$!-]&Q\"GYIF-7$$\":++++++]i?w\"[uF1$!-6Mg#>Z#F -7$$\":++++](o/opUawF1$!-&fA)of?F-7$$\":+++++vV)HxngyF1$!.m'4jO1!#H7$$\": +++++voCrWk5\"F^p$\".\"R+nZL^Fap7$$\":+++++vV3]AE>\"F^p$\".rL60<&zFap7 $$\":+++++v$44Y&=G\"F^p$\"0-_hM\"*)G5!#I7$$\":+++++v$4$eQTP\"F^p$\"0:x 'zRM.7Faq7$$\":+++++v=7gEhY\"F^p$\"1*>A.)fHc8!#J7$$\":++++](=#H4PM^\"F ^p$\"1gO`c$H3[\"F\\r7$$\":++++++DYeZ2c\"F^p$\"1rf?]$[zg\"F\\r7$$\":+++ ++vo9p=Cg\"F^p$\"18T0`j'\\k\"F\\r7$$\":+++++]7$)z*3W;F^p$\"1:N%yd#Hw;F \\r7$$\":++++DJXuuFjt7$$\":+++++] iDpqM%=F^p$\"2*ehc'4TN\">Fjt7$$\":+++++++?:>[&=F^p$\"2A()\\p!Qc'*=Fjt7 $$\":+++++]P9hnh'=F^p$\"2xf;zp*R#)=Fjt7$$\":++++++v32;v(=F^p$\"2;u\\)) Gd/)=Fjt7$$\":+++++]7.`k)))=F^p$\"2O\"[`P(*)y\">Fjt7$$\":++++++](*)H@+ >F^p$\"2A[uGEVJ-#Fjt7$$\":+++++](=\\9c6>F^p$\"2TL[%pi`-?Fjt7$$\":+++++ +D'3*4H#>F^p$\"3$o\\aH*)3=)>!#L7$$\":++++]P4@'=8k>F^p$\"3^W8&*H%f(y>F \\y7$$\":+++++v$f:QN0?F^p$\"3lh\"3bG<9(>F\\y7$$\":++++]Pf`qqV0#F^p$\"3 x>(>Fs\"4-?F\\y7$$\":++++++D^f(Q.@F^p$\"3JawYcNsD?F\\y7$$\":+++++]7.L6 \\9#F^p$\"3ofJ4QH(H*>F\\y7$$\":+++++++b1Nk=#F^p$\"4p*RDO(H-*f>!#M7$$\" :++++](=#4/PZB#F^p$\"4mHl=qWx0(>F[[l7$$\":+++++vVj,RIG#F^p$\"4'[!**oH[ Hu(>F[[l7$$\":++++DJ&pKnT/BF^p$\"4%f#f/7\"=&y$>F[[l7$$\":++++](o/\\WzD BF^p$\"4oUCC,^_%**=F[[l7$$\":+++]ilAsI$[OBF^p$\"4n1w%GVXM#)=F[[l7$$\": ++++vV)Rl@,`] h=F[[l7$$\":++++++]<))\\&oBF^p$\"4%o\")*e>NKa'=F[[l7$$\":++](oHziC<[rB F^p$\"4Zc(f$oCm%Q>F[[l7$$\":+++v$fe]nNTuBF^p$\"4Iez,V%zq#)=F[[l7$$\":+ +]i!*y$Q5aMxBF^p$\"40/=tByTH%=F[[l7$$\":+++](=F[[l7$$\":++]P%['Rh44KQ#F^p$\"54!Q_)o-o'f*=!#N7$$\":+++D\"yv,R49'Q#F^ p$\"52MO.'=$RnW=F]_l7$$\":++]7y]&*=ys!*Q#F^p$\"5g+hAT\"G$fo=F]_l7$$\": ++++vVtZi/?R#F^p$\"5$zmWMQD]\"4>F]_l7$$\":+++]i:&G'*>t.CF^p$\"5Iox3PVJ em=F]_l7$$\":++++](ozn$faT#F^p$\"5QL&Go#4&)zD=F]_l7$$\":+++]Pf3$Rn=FCF ^p$\"55A\"R(o$*H+&z\"F]_l7$$\":++++DJ?369*QCF^p$\"53fa4q,cN4=F]_l7$$\" :++](=U#)p`f%=W#F^p$\"5%o&ph7cp?e=F]_l7$$\":+++v=Z#=F]_l7$$\":++++]il dJ.Z[#F^p$\"5V#H,S;Hi%e[d84_69:!#O7$$\":++++D\" yD:9KmEF^p$\"6]7W5#3@C(3Y\"Fjgl7$$\":++++]7yqX@xo#F^p$\"6_TKf7\\4NtT\" Fjgl7$$\":+++]7G))zZ@%)p#F^p$\"6&)o)QS.Ddh/9Fjgl7$$\":++++vV)*))\\@\"4 FF^p$\"6@?K$))fo%)Q.9Fjgl7$$\":+++]Pf3)>:#)>FF^p$\"6ui+2WfsR-U\"Fjgl7$ $\":+++++v=2a@0t#F^p$\"62y/(44Umc09Fjgl7$$\":++++v=noP/Ov#F^p$\"6*3Jld 64m'=K\"Fjgl7$$\":++++]i:I@(owFF^p$\"6\"ed&==6_\\\\C\"Fjgl7$$\":+++]P% )*3J'G#)y#F^p$\"6sMS'Q&p%HdA7Fjgl7$$\":++++D1k\"\\+x*z#F^p$\"6(=r)3Mpq 6[C\"Fjgl7$$\":+++]7GQsY68\"GF^p$\"6$*ROtR(Qm()>7Fjgl7$$\":+++++]7`)G& G#GF^p$\"6p\\YcL/QQ&y6Fjgl7$$\":+++++v$4.#f(=HF^p$\"6(*>$z&>(*yZkJ*!#P 7$$\":+++++DcY$fC-IF^p$\"6to**=:eUm)QtF\\\\m7$$\":+++++]P>*QT#4$F^p$\" 7a],zj=#QNg\"[!#Q7$$\":++++++]Z@mb=$F^p$\"7S#oSK'**zG3,@Fg\\m7$$\":+++ ++](=*e(pwKF^p$!6Kn[[LPBnLw&Fg\\m7$$\":+++++vV)z=([O$F^p$!8)4M$e$yf/^[ RL!#R7$$\":+++++]7jRuFY$F^p$!8R*3k1n;icS,mFg]m7$$\":++++++]<;W2b$F^p$! 9Q=y=E>Km'Q[o*!#S7$$\":++++++DJmqYk$F^p$!:P5Pd$[Ur.u387Fb^m7$$\":+++++ v$fL?yHPF^p$!:@_&>Y19*yI\"fM:Fb^m7$$\":++++++vBCJG#QF^p$!:rzq7+\"HBY8) **y\"Fb^m7$$\":+++++vVjH$Q5RF^p$!:\"f]rbK/>V*\\A5#Fb^m7$$\":+++++v=(>/ !>+%F^p$!:>wO**3g&oWk]>BFb^m7$$\":++++DJ&p)4pU-%F^p$!:n&odC$\\Yqx7sJ#F b^m7$$\":++++](=nxxjYSF^p$!:\"4r7ybd]L`,ABFb^m7$$\":++++vV[mX1!pSF^p$! :rmU(Q)*zhL$f-U#Fb^m7$$\":++++++Dc8v84%F^p$!:rC/W@d&epozjDFb^m7$$\":++ +]PfeD5%3.TF^p$!:>'\\w(e5x!**o\\WDFb^m7$$\":++++v=#\\pIz9TF^p$!:n)[rBv W^ZlcDDFb^m7$$\":+++]7yDk.-l7%F^p$!:Hit6L`/*3md8DFb^m7$$\":++++]PfL+6# QTF^p$!:Vk%yh.L\\'yx>a#Fb^m7$$\":+++v=*\\TF^p$!:YXzwOO#\\`$H(GFFb^m7$$\":++](=nj-7s%G:%F^ p$!:+-*)e4+GF6Cum#Fb^m7$$\":+++DcwfPXud:%F^p$!:/vS'>x*Qzt.Rf#Fb^m7$$\" :++]iS;$\\&p,(eTF^p$!:*3-nt1>L]of3EFb^m7$$\":++++DcEs$*G;;%F^p$!:(eJ1 \\4J/*pSHs#Fb^m7$$\":+++]i:gT!zLtTF^p$!:*))RLudf-k!4!*p#Fb^m7$$\":++++ +v$4ro/&=%F^p$!:)f#f(Q>ZgbC$fn#Fb^m7$$\":+++++++b)QEvUF^p$!:Px'HEVO))o ?n]GFb^m7$$\":+++++]iI\"Q_nVF^p$!:T*Rojxj[-CboHFb^m7$$\":+++++vV=y>!fW F^p$!:D3:0mL/\")**eu:$Fb^m7$$\":+++]7.dM>H&pWF^p$!:M'f[]a%ov&)HPB$Fb^m 7$$\":++++DJq]gQ+[%F^p$!:k)3+levm[!=fL$Fb^m7$$\":+++]Pf$o;![0\\%F^p$!: #=ug;R#z()**)oILFb^m7$$\":++++](oHGu0,XF^p$!:k5YZ E)H$Fb^m7$$\":++++DJXk/S^b%F^p$!:@.'yzgP1p1'>C$Fb^m7$$\":++++]iS\"=]=n XF^p$!:>mOcg\\Y];@l?$Fb^m7$$\":++++v$f$)*)*H#zXF^p$!:lW>I$yYf,K(>I$Fb^ m7$$\":+++++DJ:'\\F\"f%F^p$!:xk0k*pq777)4S$Fb^m7$$\":++++DcEK$*>Lg%F^p $!:7m%f%HvahKFb^m7$$\":+++]7`p2R(Q@YF^p$!:Lj\\Qj 5t&)yxuT$Fb^m7$$\":+++v$fLpL')RCYF^p$!:$Q'**>m!G569c;LFb^m7$$\":++++v= 6UIYF^p$!:mQ@InI&=1!G$Fb^m7$$\":++++]Pf3=,5m%F^p$!:'\\'>r))G(R\\'*)>X $Fb^m7$$\":++++D\"G8ZVxrYF^p$!:76@;4(>.xaiHMFb^m7$$\":+++++D1M^ZDo%F^p $!:))[X9QDM,i#)pS$Fb^m7$$\":++++]7`f%Q4/ZF^p$!:7OB\"f\"*p]%4;6O$Fb^m7$ $\":+++++++&y,kDZF^p$!:+4$RWZL9ts0MJV[F\\$Fb^m7$$\":+++]7`p< rGI#[F^p$!:m\"R(fyh`&Q-3)R$Fb^m7$$\":+++v=U#[4sxD[F^p$!:\\kM#[#*f-/b&* 4LFb^m7$$\":++++DJ&>2d_G[F^p$!:b;&ely.hN+$zI$Fb^m7$$\":+++]P4@EqAS$[F^ p$!:s*)>a-OCE&G'>O$Fb^m7$$\":++++](o/)p>&R[F^p$!:u'*fzyFaiKmQS$Fb^m7$$ \":+++Dcwfd>oA%[F^p$!:4*[>@OUy4/0=MFb^m7$$\":+++]ilsMp;]%[F^p$!::B_'e+ ]FZCnELFb^m7$$\":+++voa&=\">lx%[F^p$!:O?f'*pI!))*3r#yKFb^m7$$\":++++vV )*))o80&[F^p$!:!p9f'**)fOi8%[Z$Fb^m7$$\":+++D\"G8h'=iK&[F^p$!:vr2\\Xl4 Pzw1Q$Fb^m7$$\":+++](=UK%o5g&[F^p$!:[L9CsN+0&QZ%H$Fb^m7$$\":+++v$4r.#= f(e[F^p$!:m]_OaHQv'4)GJ$Fb^m7$$\":++++++](zw]h[F^p$!:\"zO&fb]M-LbjV$Fb ^m7$$\":++++]7`9m&\\$)[F^p$!:)>LZC\\wP:?RfLFb^m7$$\":+++++DcJk$[0\\F^p $!:j$3rQ)4`t#)\\[G$Fb^m7$$\":++++]7y:#=u_\\F^p$!:U_/6=/'o.-gPLFb^m7$$ \"\"&\"\"!$!:pSx39h**z]\"*>Q$Fb^m-%'COLOURG6&%$RGBG$\"*++++\"!\")$Fhen FhenF_fn-%'LEGENDG6#%8scheme~with~c[10]=32/33G-F$6%7bv7$F($!-s\\Io1BF- 7$F/$!-^Y*RN3#F-7$F5$!-jN!p0)=F-7$F:$!-j%=Chp\"F-7$F?$!-@'fa(H:F-7$FD$ !-]Hhf[7F-7$FI$!-6faI+5F-7$FS$!-mc]C?mFW7$Fhn$!-Td44lSFW7$F]o$!-_?`:/= FW7$Fbo$\"+Pwl'o$FW7$Fgo$\"-G_!*)Q?\"FW7$F\\p$\".\"3$H&ouCFap7$Fcp$\". \"R5Aj`j)4*F\\r7$F^r$\"0gO`o5D&)*F\\r7$Fcr$\"1rf?j6)=1\"F\\r7$F hr$\"18T06lT!3\"F\\r7$F]s$\"1:N%e6%Q&4\"F\\r7$Fbs$\"1+ZTX6f&3\"F\\r7$F gs$\"1!eFh#=w%3\"F\\r7$F\\t$\"1<72k#ya7\"F\\r7$Fat$\"17E(\\pJy;\"F\\r7 $Fft$\"2))\\6b$Qya6Fjt7$F\\u$\"23K(p61#=9\"Fjt7$Fau$\"2WTK`8n38\"Fjt7$ Ffu$\"2'4&)R4!\\J8\"Fjt7$F[v$\"2V..n\"Q&o<\"Fjt7$F`v$\"2TXxs?[`B\"Fjt7 $Fjv$\"2A()\\z'*Q4@\"Fjt7$Fdw$\"2;u\\)o\"Qx>\"Fjt7$Fiw$\"2O\"[`jks>7Fj t7$F^x$\"2A[uG`J`G\"Fjt7$Fcx$\"2TL[W/z4F\"Fjt7$Fhx$\"3$o\\af;!ec7F\\y7 $F^y$\"3^W8&*H6r\\7F\\y7$Fcy$\"3lh\"3bL`,C\"F\\y7$Fhy$\"3x>(>dm6\"yu7\"F[[l7$F`]l$\"4%o\")*e>P'4G6 F[[l7$Fe]l$\"4Zc(f$o[v<<\"F[[l7$Fj]l$\"4Iez,V&z0Q6F[[l7$F_^l$\"40/=tB; :O6\"F[[l7$Fd^l$\"4C`26E.['y6F[[l7$Fi^l$\"54!Q_)o_SgW6F]_l7$F__l$\"52M O.'=j4N6\"F]_l7$Fd_l$\"5g+hAT\"GJo7\"F]_l7$Fi_l$\"5$zmWMQ``5:\"F]_l7$F ^`l$\"5Iox3Pt\"))Q7\"F]_l7$Fc`l$\"5QL&Go#*R#z(4\"F]_l7$$\":++](o/[n57R =CF^p$\"5a5o.\\%po(*3\"F]_l7$$\":+++vVt_N0B8U#F^p$\"5PGG.wC')GN6F]_l7$ $\":++]iSmIk*[DCCF^p$\"5\\/AXh?`e-6F]_l7$Fh`l$\"55A\"R(ot=\\x5F]_l7$$ \":++]PM_'=#e=,V#F^p$\"5&p4PKGP]19\"F]_l7$$\":+++DJXk]U]IV#F^p$\"55D-# 4>V+x5\"F]_l7$$\":++]7GQUzE#)fV#F^p$\"5B([(y+aDAx5F]_l7$F]al$\"53fa4qh \\y$3\"F]_l7$Fbal$\"5%o&ph7'efE6\"F]_l7$Fgal$\"5X08R*phd23\"F]_l7$F\\b l$\"5%ybm#pYt=h5F]_l7$Fabl$\"5&)4S2C:eP<6F]_l7$Ffbl$\"5BJK]78]6&3\"F]_ l7$F[cl$\"5BdNXI^Eqc5F]_l7$F`cl$\"5%Gm.uS)yN%3\"F]_l7$Fecl$\"5f:5F]_l7$F^el$\"5r,q$=#f8>C5F]_l7$ Fcel$\"4X%y,Z`P!H'**F]_l7$Fhel$\"4qzXc'zO'pz*F]_l7$F]fl$\"5(e)*F]_l7$Fagl$\"5ajz6H6Lu25F]_l7$Ffgl$\"57r(>d8\\C$[()Fjgl7$F eil$\"52y/(44A+C$zFjgl7$Fc[m$\"5p\\YcL/OVOkFjgl7$Fh[m$\"6(*>$z&>(*GrQ# [F\\\\m7$F^\\m$\"6to**=:e-0A[$F\\\\m7$Fc\\m$\"7a],zj=#[Au$=Fg\\m7$Fi\\ m$\"6S#oSK'**zb3>\"Fg\\m7$F^]m$!7Kn[[LPB.)*)e\"Fg\\m7$Fc]m$!8)4M$e$yf/ %*GnLFg]m7$Fi]m$!8R*3k1n;ieVaaFg]m7$F^^m$!9Q=y=E>Km4g8uFb^m7$Fd^m$!9P5 Pd$[Ur$>f!)))Fb^m7$Fi^m$!:@_&>Y19*yvh14\"Fb^m7$F^_m$!:qzq7+\"HB1LqW7Fb ^m7$Fc_m$!:\"f]rbK/>tl^P9Fb^m7$Fh_m$!:>wO**3g&ouDgk:Fb^m7$F]`m$!:n&odC $\\YqW/)e:Fb^m7$Fb`m$!:\"4r7ybd].:hd:Fb^m7$Fg`m$!:rmU(Q)*zhL(=zh\"Fb^m 7$F\\am$!:rC/W@d&eRIg4'\\w(e5x!*[MZp\"Fb^m7$Ffam$!:n)[r BvW^6!p'HlNKTF^p$!:=d;:%ednZmd[*)>e\"*p\"Fb^m7$F`bm$!:Vk%yh.L\\cu)fo\"Fb^m7$Fe bm$!:\")G-tE#\\,2sHKx*QzCvxr\"Fb^m7$Ficm$!:*3-nt1>L?:+EZg :3^mFb^m7$Fg em$!:D3:0mL/\")4o4.#Fb^m7$F\\fm$!:M'f[]a%ovEl\"y?Fb^m7$Fafm$!:k)3+levm o#fB9#Fb^m7$Fffm$!:#=ug;R#z(Gj@P@Fb^m7$F[gm$!:k5'3pw5#Fb^m7$Fjgm$!:@.'yzgP1R%=*p? Fb^m7$F_hm$!:>mOcg\\Y]EJ]/#Fb^m7$Fdhm$!:lW>I$yYf@H%G5#Fb^m7$Fihm$!:xk0 k*pq7Z*>W;#Fb^m7$F^im$!:7m%f%Hm! G5yT]0@Fb^m7$Fgjm$!:S.5;Oy&4uR9h?Fb^m7$F\\[n$!:m#Fb^m7$Fa[ n$!:m*4$oI\\7&=jmF@Fb^m7$Ff[n$!:#Qpe?RhLKXTq?Fb^m7$F[\\n$!:vH@c<\"RJ#> Op8#Fb^m7$F`\\n$!:.soO`M`pYQo:#Fb^m7$Fe\\n$!:\"H'[m+eD*[X5+@Fb^m7$Fj\\ n$!:Y@7ZpIBc\\TZ1#Fb^m7$F_]n$!:-`TG*)y@h&f6+AFb^m7$Fd]n$!:75@'omm(47h; 9#Fb^m7$Fi]n$!:c7<3F#=K3'om3#Fb^m7$F^^n$!:7a>Q@In?kVe2#Fb^m7$Fc^n$!:' \\'>r))G(R3%[V=#Fb^m7$Fh^n$!:76@;4(>.'=>%o@Fb^m7$F]_n$!:))[X9QDM,vNB:# Fb^m7$Fb_n$!:7OB\"f\"*p]*)e&)>@Fb^m7$Fg_n$!:+4$RWZL9bL\\)3#Fb^m7$F\\`n $!:6N[z5tBI-0<0#Fb^m7$Fa`n$!:R;qpK>Mr$py!=#Fb^m7$Ff`n$!:m\"R(fyh`N1x;7 #Fb^m7$F[an$!:\\kM#[#*f-J5]m?Fb^m7$F`an$!:b;&ely.hq%HQ1#Fb^m7$Fean$!:s *)>a-OCEs-t4#Fb^m7$Fjan$!:u'*fzyFaA8k87#Fb^m7$F_bn$!:4*[>@OUyt6XI@Fb^m 7$Fdbn$!::B_'e+]Fi(QM2#Fb^m7$Fibn$!:O?f'*pI!)Q,SB/#Fb^m7$F^cn$!:!p9f'* *)fOLj&R;#Fb^m7$Fccn$!:vr2\\Xl4x+8`5#Fb^m7$Fhcn$!:[L9CsN+l\"=V^?Fb^m7$ F]dn$!:m]_OaHQ0[k81#Fb^m7$Fbdn$!:\"zO&fb]MU'y5Q@Fb^m7$Fgdn$!:)>LZC\\wP Cz]'3#Fb^m7$F\\en$!:j$3rQ)4`$fwVO?Fb^m7$Faen$!:U_/6=/'oNapf?Fb^m7$Ffen $!:pSx39h**zT>q2#Fb^m-%&COLORG6&F^fn$Fgen!\"\"Fbfn$\"\"\"Fhen-Fdfn6#%8 scheme~with~c[10]=81/82G-F$6%7cu7$F($!,s6nsl%F-7$F?$!,@H$Rr[F-7$FI$!,6 $fQ$>&F-7$FS$!-mc#)z2^FW7$Fhn$!-TP2k,\\FW7$F]o$!-_I=dyYFW7$Fbo$!-j$4_d M%FW7$Fgo$!-s([Dwj$FW7$F\\p$!.>pY;z7$Fap7$Fcp$!.4'4]?\\=Fap7$Fhp$!-Hm? y()fFap7$F]q$\".-_\"[#*RxFaq7$Fcq$\"/:xYxY@@Faq7$Fhq$\"0*>Aj-LBMF\\r7$ Fcr$\"0rf?q@E9&F\\r7$F]s$\"0:N%ebNIhF\\r7$Ffu$\"1'4&)R\"ytEsFjt7$F`v$ \"1TXx_MWb))Fjt7$Fhx$\"2$o\\a*H7=))*F\\y7$Fcy$\"3lh\"3v(Qu^5F\\y7$Fhy$ \"3x>(>d/7>6\"F\\y7$F]z$\"3JawY1L!*p6F\\y7$Fbz$\"3ofJ45L$p=\"F\\y7$Fgz $\"4p*RDOnwO.7F[[l7$F][l$\"4mHl=qh@pD\"F[[l7$Fb[l$\"4'[!**oH!z`58F[[l7 $Fg[l$\"4%f#f/7?#H/8F[[l7$F\\\\l$\"4oUCC,!fX)H\"F[[l7$Fa\\l$\"4n1w%Gj \\(oH\"F[[l7$Ff\\l$\"4#)*=<,;3w(H\"F[[l7$F[]l$\"4NpH!>JvW.8F[[l7$F`]l$ \"4%o\")*e>&R\"yJ\"F[[l7$Fj]l$\"4Iez,VVZ*Q8F[[l7$Fd^l$\"4C`26Ex)p*R\"F [[l7$Fi^l$\"54!Q_)o#))p#f8F]_l7$F__l$\"52MO.'=A[CK\"F]_l7$Fd_l$\"5g+hA T,hKV8F]_l7$Fi_l$\"5$zmWMQ;R)z8F]_l7$F^`l$\"5Iox3P8,5g8F]_l7$Fc`l$\"5Q L&Go#*R/8M\"F]_l7$Febo$\"5a5o.\\kq)RL\"F]_l7$Fjbo$\"5PGG.waY^+9F]_l7$F _co$\"5\\/AXh+JMK)z8F]_l7$Fado$\"5B([(y+Ms&>M\"F]_l7$F]al$\"53fa 4qTZoa8F]_l7$Fbal$\"5%o&ph7w+o*R\"F]_l7$Fgal$\"5X08R*p@d&f8F]_l7$F\\bl $\"5%ybm#pY$)3O8F]_l7$Fabl$\"5&)4S2CNKh>9F]_l7$Ffbl$\"5BJK]7jkiy8F]_l7 $F[cl$\"5BdNXIh.vU8F]_l7$F`cl$\"5%Gm.uSl*)eQ\"F]_l7$Fecl$\"5b`eLc#F^p$\"5mn)f %*Hr$G+9F]_l7$$\":++++D\"y]p.,vDF^p$\"5^>OR&o!zFy8F]_l7$$\":++]7G)[+GL #zd#F^p$\"5&on)eKbNRe8F]_l7$$\":+++DJ&>]'GO3e#F^p$\"5-u6o5hS$oV\"F]_l7 $$\":++]PM-**\\C\\Pe#F^p$\"5\\t&=g+QFcR\"F]_l7$$\":+++]P4'\\.Am'e#F^p$ \"5>D'*oyLTMg8F]_l7$$\":++]iS;$*>;v&*e#F^p$\"5gh2SS`'fGT\"F]_l7$$\":++ +vVB!\\?\")[#f#F^p$\"5h'=m**o)>Q89F]_l7$$\":++](o/t)*y5S&f#F^p$\"5CbP* 3P!emt8F]_l7$$\":++++]P%[PSJ)f#F^p$\"5'*>9L_/OFj8F]_l7$$\":+++Dc^yW&*R Tg#F^p$\"5!o[!)Qp50-R\"F]_l7$$\":+++]ils9(e'*4EF^p$\"5n/2H?Q/8X9F]_l7$ $\":++]ilsp*H)yGh#F^p$\"5m'[e@!R!GuS\"F]_l7$$\":+++vozm%)y\"z:EF^p$\"6 aPP6d8\\m>x8Fjgl7$$\":+++]ilZVR@cl#F^p$\"6XCn%>5_`Dn8Fjgl 7$F\\hl$\"6]7W5#3\"y=$e8Fjgl7$$\":+++](oz;O9-xEF^p$\"6T3'pm`.Xd^8Fjgl7 $Fahl$\"6_TKf7\\EF\"\\8Fjgl7$F[il$\"6@?K$))fouRt8Fjgl7$Feil$\"62y/(44- QY?9Fjgl7$Fjil$\"6*3Jld6zs;r8Fjgl7$F_jl$\"6\"ed&==6-F6E%3GF^p$\"6cLh%)fl%Gg>8Fjgl7$ F^[m$\"6$*ROtR(=\"G)y8Fjgl7$$\":++]Pf$ov@o>9GF^p$\"6K69Zo1H:(R8Fjgl7$$ \":+++D1Rvi<#3GF^p$\"6EZy]Q \"HS@*Q\"Fjgl7$Fc[m$\"6p\\YcL/9fIN\"Fjgl7$$\":+++]i:N]ex( f+&>]>K\"Fjgl7$$\":++++D\"yvk%Ho%GF^p$\"7RZ'4.]6TIhH\"F\\\\m7$$\":++]i lZ)=Pl#)\\GF^p$\"72U%y@0*ycjh8F\\\\m7$$\":+++D19>'4O#G&GF^p$\"7.Byd*4W \"=;Q8F\\\\m7$$\":++](o/)\\?o?e&GF^p$\"7nR!H)=;]35+8F\\\\m7$$\":+++](o /[ax\")eGF^p$\"73azx\\\\Fca/8F\\\\m7$$\":++]7G86p#[\"='GF^p$\"7p71CWD9 $fhM\"F\\\\m7$$\":+++vozT$**=\"['GF^p$\"7-&='HhYSur18F\\\\m7$$\":++]P4 Ys<(*3y'GF^p$\"7$ep$RZ'e^!)RG\"F\\\\m7$$\":++++]7.U/13(GF^p$\"70%QP0!z da(RN\"F\\\\m7$$\":+++]7yDRL%z#)GF^p$\"7vo+'\\a%*>,9K\"F\\\\m7$$\":+++ +vV[Oi#y%*GF^p$\"7Aa!QRzadh'*G\"F\\\\m7$$\":++]i:5zgpzx*GF^p$\"77_eEHX 4@mm7F\\\\m7$$\":+++Dcw4&onx+HF^p$\"7[(*e*)>qiWhN8F\\\\m7$$\":++](oH/% 4%Qx.HF^p$\"7Th9uL>`,A'H\"F\\\\m7$$\":+++]P4rL\"4x1HF^p$\"72&R\\!*>b\" exi7F\\\\m7$$\":++]7yv,e)zw4HF^p$\"7\\#*QOR+Joo?8F\\\\m7$$\":+++v=UK#e ]w7HF^p$\"7oz7b!y:%fy-8F\\\\m7$$\":++]Pf3j18id\"HF^p$\"7jn_U=w[[gl7F\\ \\m7$Fh[m$\"7(*>$z&>(*QnEn7F\\\\m7$$\":+++]iS;Q7xR#HF^p$\"7Za!Gl:2+*[! G\"F\\\\m7$$\":++++D1RX/&>HHF^p$\"7@s@i#o*ye1v7F\\\\m7$$\":+++D1R+\\+/ =$HF^p$\"7>8&31&**)yW:I\"F\\\\m7$$\":+++](=-PHF^p$\"7dS4lM>5&)zT7F\\\\m7$$\":++++]P%)f)3jRHF ^p$\"7ciA_Imy$Q0H\"F\\\\m7$$\":+++DJqXj%)RA%HF^p$\"71=!R!3IK**p)G\"F\\ \\m7$$\":+++]7.2n!)[[%HF^p$\"7*>q=C$e&>?eD\"F\\\\m7$$\":+++v$f$oqwdu%H F^p$\"73n+R4=))RkL7F\\\\m7$$\":++++voHusm+&HF^p$\"7lSn[Rz60Y48F\\\\m7$ $\":+++Dc,\"z(ovE&HF^p$\"7.@86^%pVTdF\"F\\\\m7$$\":+++]PM_\"[YGbHF^p$ \"7-#>RX9\"4D]V7F\\\\m7$$\":+++v=n8&3O*y&HF^p$\"7)e\\s;0V^]uA\"F\\\\m7 $$\":++++++v)oD]gHF^p$\"7;>H9$\\QIUgH\"F\\\\m7$$\":++++]il\"F\\\\m7$$\":++++](oH8*Ht/$F^p$\"7/4^W\"R!3'>mZ,0$F^p$\"78*3;6G@C2HfJybIF^p$\"7JX1E!G:%)Rf:\"F\\\\m 7$$\":+++]7.dz!4geIF^p$\"7%4(3]`+&*RO`6F\\\\m7$$\":+++v$4rG0kBkIF^p$\" 7_vsqSR&RoM;\"F\\\\m7$$\":++++v=()pIF^p$\"7pvI+6*)3oX%>\"F\\\\m7$$ \":+++]PMxs*G9\"3$F^p$\"8n6(**fz?/$z*z6Fg\\m7$Fc\\m$\"8a],zj=#o7%\\:\" Fg\\m7$Fi\\m$\"8S#oSK'**zPYK-\"Fg\\m7$F^]m$\"7oK^^miwvxV!*Fg\\m7$Fc]m$ \"8-fmT;-a>SX$zFg]m7$Fi]m$\"8h5fLHLyB9!RlFg]m7$F^^m$\"9i\"=7Q2yOVr$>^F b^m7$Fd^m$\"9j*GEk^dGOlYL$Fb^m7$Fi^m$\"9zZ/QNf3@k2at\"f:>F_as7$Fh_m$!:*=wO**3g&oMw%QPF_as7$F \\am$!:-ZUS9sbe4'\\GcF_as7$Fhdm$!:xf#f(Q>ZgNN9A(F_as7$F]em$!:utnHEVO)) [Au'*)F_as7$Fbem$!:T*Rojxj[7QUb5Fb^m7$Fgem$!:D3:0mL/\"[NoY7Fb^m7$Fegm$ !:.9bM!p>5m(**zT\"Fb^m7$F[\\n$!:vH@c<\"RJb^\"pd\"Fb^m7$Fg_n$!:+4$RWZL9 n[#>m\"Fb^m7$F\\`n$!:6N[z5tB+fSZw\"Fb^m7$F\\en$!:j$3rQ)4`BoV$))=Fb^m7$ Ffen$!:pSx39h**zC%R&4#Fb^m-F\\fn6&F^fnF_fnFbfnFbfn-Fdfn6#%:scheme~with ~c[10]=169/170G-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Verror~curves~fo r~11~stage~order~7~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fids-% %VIEWG6$;F(Ffen;$!#`!#<$\"#BFbes" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with c[10]=32/33" "scheme with c[10]=8 1/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 49 "Test 15 of 11 stage, order 7 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 273 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$$\"3>+++#Qx/\"[F*$!3>+++#Qx/\"[F*F_]m-%'SYMBOL G6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fh\\n-Fjz6&F\\[lF`[lF][lF`[l-F b]n6$Fd]n\"#5Ff]n-F$6&Fh\\nF\\^n-Fb]n6$%(DIAMONDGF`^nFf]n-F$6&Fh\\nF\\ ^n-Fb]n6$%&CROSSGF`^nFf]n-%+AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAU LTG-Fa_n6$%*HELVETICAG\"\"*-%%VIEWG6$Fc_nFc_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "The following procedure uses " }{TEXT 0 6 "fsolve" }{TEXT -1 23 " to solve the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" " 6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F, F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 "for " }{TEXT 271 1 "y" }{TEXT -1 25 " numerically i n terms of " }{TEXT 272 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) lo cal y;\n fsolve(ln(x^2+y^2)+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/ 2-x);\n end proc:\nuu := evalf(exp(Pi/2)):\nplot('phi'(x),x=1..u u,numpoints=100,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 13 " " 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\" \"!$\"+++++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-$!+ " 0 " " {MPLTEXT 1 0 796 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme w ith `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/3 3,`scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]:\nerrs := [ ]: vals := []:\nDigits := 25:\nfor ct to 6 do\n Cn_RK7_||ct := RK7_ ||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops( Cn_RK7_||ct):\n for ii to numpts do\n if ct=1 then vals := [op( vals),phi(Cn_RK7_||ct[ii,1])] end if;\n sm := sm+(Cn_RK7_||ct[ii, 2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\ne nd do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,& %\"yG\"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~G-%!G6$F,F,7$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint706\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"# 7$\"+ycD?M!#C7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+VDIQNF57$/F8#\"#> \"#?$\"+3/s=VF57$/F8#\"#K\"#L$\"+0&)fvuF57$/F8#\"#\")\"##)$\"+3GMB5!#B 7$/F8#\"$p\"\"$q\"$\"+*))z#>!*F5Q)pprint716\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6#/%\"xG-%&FloatG6$\" %\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 659 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Sharp-Verner scheme with `*c[10]=11/12,`scheme wit h `*c[10]=15/16,`scheme with `*c[10]=19/20,`scheme with `*c[10]=32/33, `scheme with `*c[10]=81/82,`scheme with `*c[10]=169/170]: errs := []: \nDigits := 30:\nfor ct to 6 do\n cn_RK7_||ct := RK7_||ct(C(x,y),x,y ,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749: cxx := evalf(phi(xx)) :\nfor ct to 6 do\n errs := [op(errs),abs(cn_RK7_||ct(xx)-cxx)];\nen d do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,& %\"yG\"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~G-%!G6$F,F,7$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint726\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\"&%\"cG6#\"#5F+#\"#6\"# 7$\"+@t(3+%!#B7$/*&%-scheme~with~GF+F,F+#\"#:\"#;$\"+Z#yA+%F57$/F8#\"# >\"#?$\"+6&4&))[F57$/F8#\"#K\"#L$\"+NJkB&)F57$/F8#\"#\")\"##)$\"+Y*[@< \"!#A7$/F8#\"$p\"\"$q\"$\"+UvAP5FTQ)pprint736\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" }{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 integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 420 "mthds := [`Sharp-Verner sch eme with `*c[10]=11/12,`scheme with `*c[10]=15/16,`scheme with `*c[10] =19/20,`scheme with `*c[10]=32/33,`scheme with `*c[10]=81/82,`scheme w ith `*c[10]=169/170]: errs := []:\nDigits := 20:\nfor ct to 6 do\n s m := NCint(('phi'(x)-'cn_RK7_||ct'(x))^2,x=1..4.75,adaptive=false,nump oints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$/*&%:Sharp-Verner~scheme~with~G\"\"\" &%\"cG6#\"#5F+#\"#6\"#7$\"+7\"=9_\"!#C7$/*&%-scheme~with~GF+F,F+#\"#: \"#;$\"+)o?&y9F57$/F8#\"#>\"#?$\"+[\"[\"G=F57$/F8#\"#K\"#L$\"+hvG\"=$F 57$/F8#\"#\")\"##)$\"+gjs)Q%F57$/F8#\"$p\"\"$q\"$\"+X9c'*QF5Q(pprint46 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Th e following error graphs are constructed using the numerical procedure s for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 537 "eva lf[30](plot(['cn_RK7_1'(x)-'phi'(x),'cn_RK7_2'(x)-'phi'(x),'cn_RK7_3'( x)-'phi'(x),'cn_RK7_4'(x)-'phi'(x),\n'cn_RK7_5'(x)-'phi'(x),'cn_RK7_6' (x)-'phi'(x)],x=1..3.75,0..2.35e-20,font=[HELVETICA,9],\ncolor=[COLOR( RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5, 0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme wit h c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`sc heme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error cur ves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 887 616 616 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F* 7$$\"?LLLLLLLL$eRF27$$\"?LLLLLLLLeRiYzH7F/$\"+@&)*3P# F27$$\"?nmmmmmm;a8-qb)G\"F/$\"+(QfJr#F27$$\"?LLLLLLL$3xJ@PIM\"F/$\"+o( zY,$F27$$\"?+++++++]P4;$[%*R\"F/$\"+S>2zKF27$$\"?LLLLLLL$3F%f()yd9F/$ \"+'R7r^$F27$$\"?+++++++](=U5Uf^\"F/$\"+nhDVPF27$$\"?nmmmmmmm;Hn*fdd\" F/$\"+JI;eRF27$$\"?LLLLLLLLektvWG;F/$\"+]M#p8%F27$$\"?++++++++]PR8w(o \"F/$\"+Ml1IVF27$$\"?++++++++]7`'=tu\"F/$\"+UO'*=XF27$$\"?++++++++]igJ r/=F/$\"++?-*p%F27$$\"?LLLLLLL$3F>(G$o&=F/$\"+U,%)e[F27$$\"?nmmmmmmm;z \\#3)=>F/$\"+l%p00&F27$$\"?nmmmmmmmm;C&48(>F/$\"+8)G[@&F27$$\"?+++++++ ]PM()4QK?F/$\"+$Q?aS&F27$$\"?nmmmmmmmmT!eRk3#F/$\"+au0wbF27$$\"?++++++ +]P%[U]d9#F/$\"+`d6E#F/$\"+\"ffh:'F27$$\"?nmmmmmm;aQQAF:BF/$\"+9fNWjF27$$\"?LLLL LLLLekGEktBF/$\"+R(4Sb'F27$$\"?LLLLLLL$3F%fIFMCF/$\"+()HqynF27$$\"?+++ ++++]ilF?0([#F/$\"+ARr!)pF27$$\"?LLLLLLLL3FfZ0WDF/$\"+M591sF27$$\"?+++ ++++++veT%Hg#F/$\"+CgMZuF27$$\"?++++++++v=ZfbgEF/$\"+u]&>p(F27$$\"?+++ ++++]P%[J)H;FF/$\"+b8FQzF27$$\"?++++++++DJ52>yFF/$\"+m02A#)F27$$\"?nmm mmmmmmT!y.Q$GF/$\"+A4'o[)F27$$\"?++++++++]7.E=$*GF/$\"+!3m6y)F27$$\"?L LLLLLL$3F>k))p%HF/$\"+;6hf!*F27$$\"?++++++++]PaG\"e+$F/$\",nl%)[P*!#J7 $$\"?nmmmmmm;/,a=;hIF/$\",;BX:o*Fgv7$$\"?+++++++](=n]JF/$\"-c^)p=+ \"Fgv7$$\"?nmmmmmmm;H#)>evJF/$\"-\\S!=g.\"Fgv7$$\"?+++++++]P4Y(*zMKF/$ \",/+AI2\"F27$$\"?LLLLLLLLL$3\\L=H$F/$\",TV!p46F27$$\"?LLLLLLLLeRK(e,N $F/$\",)=![%[6F27$$\"?nmmmmmm;/^65+3MF/$\",>mB\"*=\"F27$$\"?+++++++++v e?:hMF/$\",+u^aA\"F27$$\"?nmmmmmmm\"zp:p?_$F/$\",l#)p'o7F27$$\"?LLLLLL LLL$e8ald$F/$\",a1U#38F27$$\"?+++++++]PM([YYj$F/$\",FP#RZ8F27$$\"?++++ +++]i:!G[-p$F/$\",@`#=!Q\"F27$$\"$v$!\"#$\",[B$R69F2-%&COLORG6&%$RGBGF +$\"\"#!\"\"F(-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6% 7SF'7$F-$\"+!G2/<\"F27$F4$\"+.pH\\>F27$F9$\"+R_c*o#F27$F>$\"+92*[F$F27 $FC$\"+n]bQPF27$FH$\"+)[T^9%F27$FM$\"+$H[2]%F27$FR$\"+')yc?[F27$FW$\"+ .b/C^F27$Ffn$\"+xlc7aF27$F[o$\"+Bm\"Gl&F27$F`o$\"+[**y7fF27$Feo$\"+n8^ nhF27$Fjo$\"+*zx2T'F27$F_p$\"+'4Wsi'F27$Fdp$\"+![Yu)oF27$Fip$\"+5%z36( F27$F^q$\"+#ps2P(F27$Fcq$\"+Sa$Rg(F27$Fhq$\"+')RFnyF27$F]r$\"+eT0D\")F 27$Fbr$\"+^Uc+%)F27$Fgr$\"+(H%)*f')F27$F\\s$\"+b4+]*)F27$Fas$\"+I!H:E* F27$Ffs$\"+pz?U&*F27$F[t$\"+)f%zc)*F27$F`t$\",[rR%>5F27$Fet$\",7[CQ0\" F27$Fjt$\",frW%)3\"F27$F_u$\",1P3'G6F27$Fdu$\",3iEj;\"F27$Fiu$\",!pZV3 7F27$F^v$\",jlz$[7F27$Fcv$\"-^ir0%H\"Fgv7$Fiv$\"-u0i.R8Fgv7$F^w$\"-:5v ])Q\"Fgv7$Fcw$\"-r(=h#R9Fgv7$Fhw$\",#3T2&\\\"F27$F]x$\",\"oq[^:F27$Fbx $\",SNd@h\"F27$Fgx$\",(fp.w;F27$F\\y$\",02Ylt\"F27$Fay$\",\\2%y4=F27$F fy$\",)3Uyy=F27$F[z$\",:+NQ&>F27$F`z$\",5LRh-#F27$Fez$\",Wi__5#F2-F[[l 6&F][lF+$\"\"(F`[lF+-Fb[l6#%8scheme~with~c[10]=15/16G-F$6%7SF'7$F-$\"* #35e%*F27$F4$\"+:cuv:F27$F9$\"+Ssvu@F27$F>$\"+u%3&[EF27$FC$\"+gkwBIF27 $FH$\"+_ft_LF27$FM$\"+'Q1.k$F27$FR$\"+stw)*QF27$FW$\"+G\\\"Q9%F27$Ffn$ \"+o%plP%F27$F[o$\"+Wh?qXF27$F`o$\"+m:bzZF27$Feo$\"+F[Y%)\\F27$Fjo$\"+ s9&*z^F27$F_p$\"+%3.QN&F27$Fdp$\"+0ZaibF27$Fip$\"+tXcTdF27$F^q$\"+:$Q( \\fF27$Fcq$\"+$GCk8'F27$Fhq$\"+eK'pM'F27$F]r$\"+bNw_lF27$Fbr$\"+L$yDx' F27$Fgr$\"+pkbzpF27$F\\s$\"+l*\\/@(F27$Fas$\"+h>FeuF27$Ffs$\"+UDP\"o(F 27$F[t$\"+[r!3$zF27$F`t$\"+VQ9)>)F27$Fet$\"+4!H)p%)F27$Fjt$\"+7?bV()F2 7$F_u$\"+-!**)f!*F27$Fdu$\"+3,(fN*F27$Fiu$\"+a6!eo*F27$F^v$\"+YsL)***F 27$Fcv$\"-W6lQN5Fgv7$Fiv$\"-tCD\"F27$F]x$\",D(pkK7F27$Fbx$\",\\];\"y7F27$Fgx$\",B! >&fK\"F27$F\\y$\",C3W+P\"F27$Fay$\",UC#3B9F27$Ffy$\",y$)HDZ\"F27$F[z$ \",/_7W_\"F27$F`z$\",2z$>s:F27$Fez$\",_55Gi\"F2-F[[l6&F][l$\"\"*F`[l$ \"\"%F`[lF+-Fb[l6#%8scheme~with~c[10]=19/20G-F$6%7SF'7$F-$\"+d_8D7F27$ F4$\"+e\"yu.#F27$F9$\"+Al-2GF27$F>$\"+\"=cOT$F27$FC$\"+Kd3$*QF27$FH$\" +vHf7VF27$FM$\"+0/,zYF27$FR$\"+h?J3]F27$FW$\"+Ng_?`F27$Ffn$\"+*G6th&F2 7$F[o$\"+tP[keF27$F`o$\"+j'\\?8'F27$Feo$\"+FAM%R'F27$Fjo$\"+0@(\\k'F27 $F_p$\"+B!4#ooF27$Fdp$\"+%p/n8(F27$Fip$\"+x/UntF27$F^q$\"+&)\\2OwF27$F cq$\"+`aIxyF27$Fhq$\"+5H&*\\\")F27$F]r$\"+5K4<%)F27$Fbr$\"+$RtGq)F27$F gr$\"+7]:s*)F27$F\\s$\"+-2at#*F27$Fas$\"+(>'e(f*F27$Ffs$\"+B-!)*))*F27 $F[t$\",&)o\"y@5F27$F`t$\",%[_-d5F27$Fet$\",!R'zH4\"F27$Fjt$\",@a)=H6F 27$F_u$\",?0,8<\"F27$Fdu$\",?5[4@\"F27$Fiu$\",3.\"Hb7F27$F^v$\",;A3uH \"F27$Fcv$\"-'HEadM\"Fgv7$Fiv$\"-_E-f$R\"Fgv7$F^w$\"-N1oBY9Fgv7$Fcw$\" -8\\z_+:Fgv7$Fhw$\",ct!fg:F27$F]x$\",d$**y@;F27$Fbx$\",]V$4)o\"F27$Fgx $\",#*ek!eF27$Ffy$\",C!o\\()>F27$ F[z$\",NMPj2#F27$F`z$\",v`]f;#F27$Fez$\",jU'=oAF2-%'COLOURG6&F][l$\"*+ +++\"!\")F+Fahm-Fb[l6#%8scheme~with~c[10]=32/33G-F$6%7SF'7$F-$\"+wmT=7 F27$F4$\"+-x$Q-#F27$F9$\"+#o`Zy#F27$F>$\"+h<2$Q$F27$FC$\"+JM/bQF27$FH$ \"+gsJnUF27$FM$\"+`a/FYF27$FR$\"+=6<]\\F27$FW$\"+;.Nc_F27$Ffn$\"+,MRZb F27$F[o$\"+.^$)*y&F27$F`o$\"+bsP_gF27$Feo$\"+S0')4jF27$Fjo$\"+WY+clF27 $F_p$\"+')=TvnF27$Fdp$\"+:KTRqF27$Fip$\"+7qPmsF27$F^q$\"+!\\f3`(F27$Fc q$\"+$Qx%oxF27$Fhq$\"+F(\\r.)F27$F]r$\"+sx_+$)F27$Fbr$\"+IrU#e)F27$Fgr $\"+.\\:[))F27$F\\s$\"+3erX\"*F27$Fas$\"+1nyl%*F27$Ffs$\"+Jg_a(*F27$F[ t$\",(\\V)y+\"F27$F`t$\",/gYF/\"F27$Fet$\",b!RLy5F27$Fjt$\",P\"[<96F27 $F_u$\",TO(*e:\"F27$Fdu$\",G?6_>\"F27$Fiu$\",7O2#R7F27$F^v$\",l=55G\"F 27$Fcv$\"-w8b0H8Fgv7$Fiv$\"-6vmmw8Fgv7$F^w$\"-fe)p!H9Fgv7$Fcw$\"-&eR&> $[\"Fgv7$Fhw$\",-(==V:F27$F]x$\",taZWg\"F27$Fbx$\",Oqg4n\"F27$Fgx$\",+ \\!>TF27$F[z$\" ,Kt@Z1#F27$F`z$\",=cvx:#F27$Fez$\",esZ^E#F2-F[[l6&F][l$\"\"&F`[lF+F(-F b[l6#%8scheme~with~c[10]=81/82G-F$6%7SF'7$F-$\"*dtE=*F27$F4$\"+$HzW_\" F27$F9$\"+@))['4#F27$F>$\"+'oDda#F27$FC$\"+3Sv**GF27$FH$\"+T7r3KF27$FM $\"+Z@6yMF27$FR$\"+V\"***>PF27$FW$\"+ZW3\\RF27$Ffn$\"+CUzmTF27$F[o$\"+ O(H\"[VF27$F`o$\"+\")z\\WXF27$Feo$\"+.c3PZF27$Fjo$\"+3D>@\\F27$F_p$\"+ %GZ`3&F27$Fdp$\"+@Z'GG&F27$Fip$\"+OMn_aF27$F^q$\"+H:i]cF27$Fcq$\"+FN]G eF27$Fhq$\"+`$>'HgF27$F]r$\"+i9xEiF27$Fbr$\"+_:#yV'F27$Fgr$\"+rz!oj'F2 7$F\\s$\"+!G-'foF27$Fas$\"+26F*4(F27$Ffs$\"+Z.\\:tF27$F[t$\"+%>>$evF27 $F`t$\"+,tM>yF27$Fet$\"+=>y&3)F27$Fjt$\"+#paTN)F27$F_u$\"+cs\\m')F27$F du$\"+mkvg*)F27$Fiu$\"+rg.!H*F27$F^v$\"+v`*Gg*F27$Fcv$\",@buB'**Fgv7$F iv$\"-`M\"\\=.\"Fgv7$F^w$\"-gZ([52\"Fgv7$Fcw$\"-9vg_66Fgv7$Fhw$\",*RbP c6F27$F]x$\",M8q@?\"F27$Fbx$\",e'e)=D\"F27$Fgx$\",&[!)R/8F27$F\\y$\",h @1cN\"F27$Fay$\",NL\"G=9F27$Ffy$\",_nH$y9F27$F[z$\",mwSna\"F27$F`z$\", I1@ph\"F27$Fez$\",&4kW)p\"F2-F_hm6&F][lFahmF+F+-Fb[l6#%:scheme~with~c[ 10]=169/170G-%&TITLEG6#%Verror~curves~for~11~stage~order~7~Runge-Kutta ~methodsG-%%FONTG6$%*HELVETICAGFe^m-%+AXESLABELSG6$Q\"x6\"Q!Fe\\o-%%VI EWG6$;F(Fez;F+$\"$N#!#A" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10 ]=15/16" "scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme w ith c[10]=81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 546 "evalf[30](p lot(['cn_RK7_1'(x)-'phi'(x),'cn_RK7_2'(x)-'phi'(x),'cn_RK7_3'(x)-'phi' (x),'cn_RK7_4'(x)-'phi'(x),\n'cn_RK7_5'(x)-'phi'(x),'cn_RK7_6'(x)-'phi '(x)],x=3.75..4.6,-1.95e-17..3.1e-17,font=[HELVETICA,9],\ncolor=[COLOR (RGB,0,.2,1),COLOR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5 ,0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme wi th c[10]=15/16`,`scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`s cheme with c[10]=81/82`,`scheme with c[10]=169/170`],\ntitle=`error cu rves for 11 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 900 623 623 {PLOTDATA 2 "6,-%'CURVESG6%7co7$$\"$v$!\"#$\",[B $R69!#I7$$\"?nmmmmmmm;/\"eF&oP!#H$\",[y=SU\"F-7$$\"?LLLLLLL$eR-L[Yy$F1 $\",lf_$H9F-7$$\"?nmmmmmmm\"H()zxF!QF1$\",=YkKV\"F-7$$\"?nmmmmmmmT5Tu- @QF1$\",\"zgQN9F-7$$\"?LLLLLLL$e9hM!>RQF1$\",*=\\BT9F-7$$\"?nmmmmmm;H2 %oHg&QF1$\",p_DTW\"F-7$$\"?+++++++]i:VeYtQF1$\"+^/'*Q9F17$$\"?nmmmmmm; H#QM)\\\"*QF1$\"+tz#3V\"F17$$\"?+++++++]7.9IZ4RF1$\"+])p^U\"F17$$\"?LL LLLLLL$3<8F17$$\"?nmmmmmm;HK^#R[,%F1$\"+Ieny7F1 7$$\"?LLLLLLLL$3sF&*R.%F1$\"+^AN<7F17$$\"?LLLLLLLLL$Q%HA]SF1$\"+>\\]O6 F17$$\"?+++++++]i!zd*4pSF1$\"+H!oH2\"F17$$\"?LLLLLLLLLe(f3e3%F1$\"*G84 !**F17$$\"?+++++++]iSS59/TF1$\"*835b)F17$$\"?+++++++](=7'yf@TF1$\"*!e' e&oF17$$\"?LLLLLLLLeRAA\")RTF1$\"*#eX?\\F17$$\"?LLLLLLL$ek=pQl:%F1$\"* ZkT]$F17$$\"?nmmmmmmmT&)[/euTF1$\")Vj?fF17$$\"?nmmmmmm;H#Qw?L>%F1$!*)) p^6$F17$$\"?+++++++]P4jUj4UF1$!*fAW:(F17$$\"?nmmmmmmm\"H#*G`sA%F1$!+l% R34\"F17$$\"?+++++++++DnaXXUF1$!+HZdZ2 ^%F1$!-uxLzB>F17$$\"?LLLLLLLL3_['[&HXF1$!-jdq&\\#GF17$$\"?+++++++]PMa! pz`%F1$!-a&)3rtJF17$$\"?mmmmmmmmm;g%*QYXF1$!-/e**3\")QF17$$\"?LLLLLLLe k.ptObXF1$!-ge-z>\\F17$$\"?+++++++]i!zFXVc%F1$!-YNK/?jF17$$\"?+++++++D \"y+yT'oXF1$!-1+\"4O#pF17$$\"?+++++++++D#GQHd%F1$!-!yBc[?)F17$$\"?++++ ++voHz2C,uXF1$!-4ZY1J#)F17$$\"?++++++]PfLLl3vXF1$!-X+hvk#)F17$$\"?++++ ++D1*y)e1;wXF1$!-L_pRF$)F17$$\"?+++++++v=U%yMsd%F1$!-**yk$)y%)F17$$\"? ++++++]7y]NIQzXF1$!-')=PEn(*F17$$\"?+++++++]Pf'GJ:e%F1$!.]rqew2\"F17$$ \"?++++++DJ&pd()RQe%F1$!.F&>s1&3\"F17$$\"?++++++]7`%\\Y[he%F1$!.()3J2) )4\"F17$$\"?+++++]7.K`fFI(e%F1$!.b(HH.A6F17$$\"?++++++v$4@T0d%)e%F1$!. DoP)f%=\"F17$$\"?+++++]P%)*3([8h*e%F1$!.I(o]^T8F17$$\"?+++++++voHVcw!f %F1$!.Z,-idV\"F17$$\"?++++++DcEZKU2$f%F1$!.[6uAeW\"F17$$\"?++++++]P%[; #GQ&f%F1$!.`)=Y7f9F17$$\"?+++++]7GjB;r`'f%F1$!.0V&e:w9F17$$\"?++++++v= U#3T\"p(f%F1$!.mgZt>_\"F17$$\"?+++++D1k\"=\"e&o#)f%F1$!.g\"=Qjo:F17$$ \"?+++++]P4@T0d%))f%F1$!.MzsELk\"F17$$\"?+++++voagq_GU*f%F1$!.0e\"4?g< F17$$\"#Y!\"\"$!.e_#>'*Q>F1-%&COLORG6&%$RGBG$\"\"!F]bl$\"\"#Feal$\"\" \"F]bl-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7_p7$F($ \",Wi__5#F-7$F/$\",KtvE8#F-7$F5$\",AgeJ:#F-7$F:$\",ri-_<#F-7$F?$\",,P5 m>#F-7$FD$\",!*)*f4A#F-7$FI$\",9/nAC#F-7$FN$\"+T\"e\"fAF17$FS$\"+D`=vA F17$FX$\"+r-W$H#F17$Fgn$\"+e(pLJ#F17$F\\o$\"+/t8@BF17$Fao$\"+Na7FBF17$ Ffo$\"++XRGBF17$F[p$\"+7_1RBF17$F`p$\"+Up+MBF17$Fep$\"+qR\"*>BF17$Fjp$ \"+'3P&)G#F17$F_q$\"+kq5vAF17$Fdq$\"+.\"GdC#F17$Fiq$\"+#o'f%=#F17$F^r$ \"+ivY+@F17$Fcr$\"+8$RX+#F17$Fhr$\"+FpfV>F17$F]s$\"+DLF*y\"F17$Fbs$\"+ l+3\"f\"F17$Fgs$\"+9B=u8F17$F\\t$\"+6K8&=\"F17$Fat$\"*gK'p$)F17$Fft$\" *yW-&QF17$F[u$!*?SS%=F17$F`u$!*/A8-)F17$Feu$!+MQGi8F17$Fju$!+6^CmBF17$ F_v$!+Aw&yj$F17$Fdv$!+,&)\\H_F17$Fiv$!+U#=pL'F17$F^w$!+OQzV&)F17$Fcw$! ,b5r;8\"F17$Fhw$!,G#QMo9F17$F]x$!,AX\"[GF17$$\"?++++++vVt_1n7TXF1$!,$R\"[*)) >F17$$\"?LLLLLLL3_Dd#z@a%F1$!,f#fB%*>F17$$\"?++++++]P4reVGWXF1$!,rW-S+ #F17$F[z$!,8=M-*>F17$F`z$!+z+,l\"*F17$Fez$\",*))z))35F17$Fjz$\",>1kJ'> F17$F_[l$\",3J&QrUF17$Fd[l$\",w.coG%F17$Fi[l$\",;*)RkJ%F17$F^\\l$\",W' =#QS%F17$Fc\\l$\",dp**pn%F17$Fh\\l$\",>%**4GuF17$F]]l$\",Ob9`k*F17$$\" ?+++++]iS;=\"e&o#e%F1$\",4J^yn*F17$Fb]l$\",kCoMr*F17$$\"?+++++](=Ud.<% *\\e%F1$\",D_h.x*F17$Fg]l$\",oMij#**F17$F\\^l$\"-C-T1V5F17$Fa^l$\"-<'o QS>\"F17$$\"?+++++D1R]T,U.*e%F1$\"-fu^F\\8F17$Ff^l$\"-2C[e'f\"F17$$\"? +++++voHH+'\\)=!f%F1$\"-AQQ&y$=F17$F[_l$\"-T]c.T=F17$F`_l$\"-'e'f(R&=F 17$Fe_l$\"-%*\\r6v=F17$Fj_l$\"-\"o'G'Q\">F17$F_`l$\"--TtbM?F17$Fd`l$\" -]WAnk@F17$Fi`l$\"-32SBzBF17$F^al$\"-5N_WBFF17$Fcal$\"-?<.phKF1-Fial6& F[blF\\bl$\"\"(FealF\\bl-Fcbl6#%8scheme~with~c[10]=15/16G-F$6%7eo7$F($ \",_55Gi\"F-7$F/$\",!*eQ8k\"F-7$F5$\",y5.Ml\"F-7$F:$\",#oXsl;F-7$F?$\" ,$\\#ern\"F-7$FD$\",_)zt\"p\"F-7$FI$\",O'eu.N&[%F1$\",uD\"[>:F17$Fgx$\",Eh69>#F17$F[[m$\",A,*fwNF17$Fh\\l$\"-=S<2-VF17$F]]l$\"-Pyap&)[F17$Fb]l$\"-!\\lv$> \\F17$Fg]l$\"-VCRt()\\F17$F\\^l$\"-N43u:^F17$Fa^l$\"-7)f#*)yaF17$Ff^l$ \"-J@cmEkF17$F[_l$\"-A%eET+(F17$F`_l$\"-0!=@K0(F17$Fe_l$\"-f1\\\\?rF17 $Fj_l$\"-ZWh>:sF17$F_`l$\"->3dK'[(F17$Fd`l$\"-h([w:x(F17$Fi`l$\"-tE=bP #)F17$F^al$\"-*\\Pr/)*)F17$Fcal$\".)=fVr85F1-Fial6&F[bl$\"\"*Feal$\"\" %FealF\\bl-Fcbl6#%8scheme~with~c[10]=19/20G-F$6%7eo7$F($\",jU'=oAF-7$F /$\",\\3DBI#F-7$F5$\",'GO;JBF-7$F:$\",?!fyjBF-7$F?$\",=vtpR#F-7$FD$\", V#F-7$FI$\",\">QdkCF-7$FN$\"+(R&*o\\#F17$FS$\"+:uaIDF17$FX$\"+UK#f c#F17$Fgn$\"+$HjPg#F17$F\\o$\"+ap)Rj#F17$Fao$\"+9Y-oEF17$Ffo$\"+I'H;q# F17$F[p$\"+b)z!RFF17$F`p$\"+LeboFF17$Fep$\"+V$\\B!GF17$Fjp$\"+X'=c#GF1 7$F_q$\"+$3pL'GF17$Fdq$\"+39s#*GF17$Fiq$\"+Y/C=HF17$F^r$\"+`X=QHF17$Fc r$\"+94ahHF17$Fhr$\"+33p%*HF17$F]s$\"+;b?8IF17$Fbs$\"+n)GO.$F17$Fgs$\" +F#HN0$F17$F\\t$\"+n\"\\I4$F17$Fat$\"+\"GY08$F17$Fft$\"+/T\\'=$F17$F[u $\"+r#['yKF17$F`u$\"+(G>[V$F17$Feu$\"+&o:&=OF17$Fju$\"+5AA')RF17$F_v$ \"+F'Rpf%F17$Fdv$\"+.*G!HcF17$Fiv$\"+d'\\$RlF17$F^w$\"+$HrSx)F17$Fcw$ \",/(pDf7F17$Fhw$\",'zWB3>F17$F]x$\",Y58(oDF17$Fbx$\",*yhaSRF17$Fc\\n$ \",)o4kN]F17$Fgx$\",%fdD4lF17$F[[m$\",#\\#=/Z)F17$F\\y$\"-W3bG56F17$Fa y$\"-;NSo;=F17$Ffy$\"-C')\\h*4#F17$F[z$\"-Bmy=1FF17$F`z$\"-RccxSOF17$F ez$\"-(\\L/L'\\F17$Fjz$\"-&ov@ia&F17$F_[l$\"-rUQfJoF17$Fd[l$\"--^]h`oF 17$Fi[l$\"-N\")Q/$)oF17$F^\\l$\"-A[*><%pF17$Fc\\l$\"-bVsz\"4(F17$Fh\\l $\"-:4vPN%)F17$F]]l$\"-AOA6-&*F17$Fb]l$\"-^U?jn&*F17$Fg]l$\"-p.:M*p*F1 7$F\\^l$\"-1sSoS**F17$Fa^l$\".D^)3Qh5F17$Ff^l$\".)[6,NM7F17$F[_l$\".*Q $3k*Q8F17$F`_l$\".^I[]$[8F17$Fe_l$\".bZd;7O\"F17$Fj_l$\".An)*>\"z8F17$ F_`l$\".nw_t&H9F17$Fd`l$\".$\\dD3#[\"F17$Fi`l$\".&G[\"4sc\"F17$F^al$\" .:f9I>q\"F17$Fcal$\".v3?w,\">F1-%'COLOURG6&F[bl$\"*++++\"!\")F\\blF[ao -Fcbl6#%8scheme~with~c[10]=32/33G-F$6%7co7$F($\",esZ^E#F-7$F/$\",RL)o+ BF-7$F5$\",S%fjJBF-7$F:$\",C\")fqO#F-7$F?$\",xj*[.CF-7$FD$\",&3\"H8W#F -7$FI$\",TyxqZ#F-7$FN$\"+Wu59DF17$FS$\"+\\pJ`DF17$FX$\"+%o.Tf#F17$Fgn$ \"+&z`wj#F17$F\\o$\"+g`mvEF17$Fao$\"+x*R(>FF17$Ffo$\"+%fT`w#F17$F[p$\" +%\\+E\"GF17$F`p$\"+h)\\b&GF17$Fep$\"+RBq3HF17$Fjp$\"+p(z^&HF17$F_q$\" +L>a9IF17$Fdq$\"+4*o'pIF17$Fiq$\"+^:#\\8$F17$F^r$\"+I2R/KF17$Fcr$\"+r$ [gG$F17$Fhr$\"+?9HlLF17$F]s$\"+xR$GZ$F17$Fbs$\"+F+O4OF17$Fgs$\"+q-%)eP F17$F\\t$\"+xR]CRF17$Fat$\"+*)3s#=%F17$Fft$\"+>(\\z`%F17$F[u$\"+v%Q3.& F17$F`u$\"+6l1lcF17$Feu$\"+W&)G9jF17$Fju$\"+9'yOc(F17$F_v$\"+>SD8%*F17 $Fdv$\",d>$e?7F17$Fiv$\",`kP![9F17$F^w$\",\"4h%z(>F17$Fcw$\",,FTC#GF17 $Fhw$\",Q3&ftTF17$F]x$\",#f?;'\\&F17$Fbx$\",@7a+9)F17$Fgx$\"-I(>J$*G\" F17$F\\y$\"-^s'R#3@F17$Fay$\"-GJK8KLF17$Ffy$\"-)=,$RqdH&*F17$F_[l$\". CtDK6;\"F17$Fd[l$\".biPm[;\"F17$Fi[l$\".l%>9F17$F]]l$\".c3g\"F17$Fg]l$\".r\\%eVA;F17$F\\^l$\".f!4h>h;F17$Fa^l$\".]w;a$ oo')>/#F17$F[_l$\".@.Cbr?#F17$F`_l$\".)*)=jiAAF17$Fe_l$ \".KirYOC#F17$Fj_l$\".*R=8JsAF17$F_`l$\".r;?x@N#F17$Fd`l$\".QD:Z[V#F17 $Fi`l$\".?r!*R%oDF17$F^al$\".\"z'o^#zFF17$Fcal$\".IL3QU5$F1-Fial6&F[bl $\"\"&FealF\\blF`bl-Fcbl6#%8scheme~with~c[10]=81/82G-F$6%7co7$F($\",&4 kW)p\"F-7$F/$\",y:Uas\"F-7$F5$\",h\"Q?\\F17 $FX$\"+2oZc>F17$Fgn$\"+/0w\"*>F17$F\\o$\"+[p.C?F17$Fao$\"+2/@i?F17$Ffo $\"+BU%G5#F17$F[p$\"+0*GP9#F17$F`p$\"+m#*o$=#F17$Fep$\"+^D8NAF17$Fjp$ \"+Q#\\WG#F17$F_q$\"+(y^FM#F17$Fdq$\"+&p\\4S#F17$Fiq$\"+?HnwCF17$F^r$ \"+()45jDF17$Fcr$\"+!fbcm#F17$Fhr$\"+FB0eFF17$F]s$\"+in&Q!HF17$Fbs$\"+ W]m#4$F17$Fgs$\"+M/j-LF17$F\\t$\"+=LhANF17$Fat$\"+#e/f)QF17$Fft$\"+k'H WQ%F17$F[u$\"+d:&p1&F17$F`u$\"+GD/s([F17$F]x$\",C@oGP'F17$Fbx$\",%fu:8 $*F17$Fgx$\"-ksFD]9F17$F\\y$\"-*Gv#*zK#F17$Fay$\"-n5MK?OF17$Ffy$\"-gJG !)HTF17$F[z$\"-z'[Ga>&F17$F`z$\"-j6.k.oF17$Fez$\"-dKQIK!*F17$Fjz$\".)f k[s+5F17$F_[l$\".:B))Q;@\"F17$Fd[l$\".0r@Kb@\"F17$Fi[l$\".BQZg1A\"F17$ F^\\l$\".e_U51B\"F17$Fc\\l$\".p5dqaD\"F17$Fh\\l$\".h'\\Fts9F17$F]]l$\" .vc#>(Qk\"F17$Fb]l$\".zR$e>b;F17$Fg]l$\".H^6&Gx;F17$F\\^l$\".*[8tY;!#>$\"#J!#=" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[10]=11/12" "scheme with c[10]=15/16" " scheme with c[10]=19/20" "scheme with c[10]=32/33" "scheme with c[10]= 81/82" "scheme with c[10]=169/170" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 545 "evalf[30](plot(['cn_RK7_1' (x)-'phi'(x),'cn_RK7_2'(x)-'phi'(x),'cn_RK7_3'(x)-'phi'(x),'cn_RK7_4'( x)-'phi'(x),\n'cn_RK7_5'(x)-'phi'(x),'cn_RK7_6'(x)-'phi'(x)],x=4.6..4. 67,-2.9e-16..5.9e-16,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COL OR(RGB,0,.7,0),COLOR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nleg end=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`, `scheme with c[10]=19/20`,`scheme with c[10]=32/33`,`scheme with c[10] =81/82`,`scheme with c[10]=169/170`],\ntitle=`error curves for 11 stag e order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 823 481 481 {PLOTDATA 2 "6,-%'CURVESG6%7^r7$$\"#Y!\"\"$!.e_#>'*Q>!#H7$$\"? nmmmmmmm;z+e_,YF-$!.'e)=E#[>F-7$$\"?LLLLLLL$3->R`Gg%F-$!.'4H$)Rc>F-7$$ \"?nmmmmmmmT&pSYVg%F-$!.A7$GVm>F-7$$\"?nmmmmmmm\"z'=$\\eg%F-$!.%3>N$Q) >F-7$$\"?LLLLLLL$3Ft3Xtg%F-$!.k\\`(\\R?F-7$$\"?+++++++]7$=ZQ!3YF-$!.,i 3*p-@F-7$$\"?nmmmmmm;aLc=t3YF-$!.jvP1^@#F-7$$\"?MLLLLLL$3Fe\")\\%4YF-$ !.YMffvT#F-7$$\"?+++++++](=`xn,h%F-$!.Mej\\cm#F-7$$\"?MLLLLLL$3_+H54h% F-$!.vvK-@n#F-7$$\"?nmmmmmm;ay/Gl6YF-$!.ks8#fyEF-7$$\"?+++++++]PurI88Y F-$!.a*eSs\"p#F-7$$\"?LLLLLLLL$e#3dl9YF-$!.'e))zD2FF-7$$\"?nmmmmmmm\"H t%o*fh%F-$!.#G%>_Ht#F-7$$\"?MLLLLLL$3-)\\YF-$!.WJoKk;$F-7$$\"?+++++++++S%*yQ>YF-$!.r`7X!HLF-7$$\"?++ ++++++]YF-$!.KGS$>UNF-7$$\"?+++++++](=;WW)>YF-$!.0&)op[g$F-7$$\"? ++++++++D1^d$*>YF-$!.w+e]L*4`XF-7$$\"?+++++++DJShR_HYF-$!.))=[%[<[F-7$$\"?++++++++v`mL ))HYF-$!.7/\"Q^h^F-7$$\"?++++++v$4@y@t*HYF-$!.))zZZAE&F-7$$\"?++++++]( o/\"pI1IYF-$!.Ee'zr%H&F-7$$\"?++++++D\"G)Q?H:IYF-$!.cb98kH&F-7$$\"?+++ ++++v=nrFCIYF-$!.I!4'4\")H&F-7$$\"?++++++]i!RUZA/j%F-$!.[OR1:I&F-7$$\" ?+++++++]i!o<-1j%F-$!.^GL3\\I&F-7$$\"?mmmmmmmTN\"*z@NJYF-$!.,vvi\">`F- 7$$\"?LLLLLLLL3-$=-@j%F-$!.\"zLa^L`F-7$$\"?LLLLLLL$3xplzMj%F-$!.w9\"p` g`F-7$$\"?nmmmmmmm\"H([a'\\j%F-$!.xBZVoR&F-7$$\"?nmmmmmm\"Hd(3rtNYF-$! .sITW!HaF-7$$\"?nmmmmmm;ayo(3lj%F-$!.3(3B&z[&F-7$$\"?MLLLLLLL$36]!=PYF -$!.Q,TvQe&F-7$$\"?+++++++]7VLA&yj%F-$!.X=v_pv&F-7$$\"?MLLLLLL3FuAxdQY F-$!.(RUY.&4'F-7$$\"?nmmmmmmmT07KIRYF-$!.MWs.!)p'F-7$$\"?+++++++D\"yF' znRYF-$!.kc8kd;(F-7$$\"?MLLLLLL$3-Nr_+k%F-$!.Rh8\\Uo(F-7$$\"?nmmmmmmTg AkuUSYF-$!.YNh]]p(F-7$$\"?+++++++++&\\@-3k%F-$!.!='H))eq(F-7$$\"?+++++ ++]P*eXN:k%F-$!.<&Q)*>FxF-7$$\"?++++++++v$opoAk%F-$!.E1Zs'[xF-7$$\"?++ +++++](oMf(oVYF-$!.y!H=m\"z(F-7$$\"?++++++++DEOIEXYF-$!.4+13p&yF-7$$\" ?mmmmmmm;H(*Q3(fk%F-$!.<:qL@\"zF-7$$\"?LLLLLLLLLoT'ymk%F-$!.;\"R8Z6!)F -7$$\"?+++++++](=p]cqk%F-$!.W#z-;'4)F-7$$\"?mmmmmmmmT:sVVZYF-$!.k)y^c9 #)F-7$$\"?LLLLLLL$e*QPA\"yk%F-$!.2Z#>Ez$)F-7$$\"?++++++++]i-,>[YF-$!.A z!4b1')F-7$$\"?MLLLLLL3_v1\\()[YF-$!.XvB#>\\#*F-7$$\"?nmmmmmm;a)3rf&\\ YF-$!/VB'*\\TM5F-7$$\"?MLLLLLe9T$)you\\YF-$!/kMxKQv5F-7$$\"?++++++]7Gy YS$*\\YF-$!/]cdIEA6F-7$$\"?nmmmm\"zp)*p(Q3)*\\YF-$!/,%pu!*\\8\"F-7$$\" ?MLLLL$e9;d2jF+l%F-$!/b(*['\\/9\"F-7$$\"?+++++v$fLWFUu+l%F-$!/QhT/mS6F -7$$\"?nmmmmmT5:t977]YF-$!/*pvKr39\"F-7$$\"?+++++]Pfeq)z9-l%F-$!/oqzLH T6F-7$$\"?MLLLLLL3-o#Q3.l%F-$!/M$f!erT6F-7$$\"?nmmmmm;/wd=Fo]YF-$!/*QH E4M9\"F-7$$\"?++++++++]Zaq0^YF-$!/0oc(3^9\"F-7$$\"?mmmmmmmT&)G'\\hH;\"F-7$$\"?LLLLLL LL$e/Xy`l%F-$!/-HP*>)o6F-7$$\"?mmmmmmmT&)3J@8cYF-$!/kJOZDz6F-7$$\"?+++ ++++](=<\"e)ol%F-$!/%*z;I7*>\"F-7$$\"?MLLLLLL3Fu&p6wl%F-$!/b[iA+O7F-7$ $\"?nmmmmmmmmwzvLeYF-$!/OITm-08F-7$$\"?nmmmmmm;H-,*z!fYF-$!/0z*\\hOV\" F-7$$\"?nmmmmmmm\"zAAA)fYF-$!/\"o$=\")fg;F-7$$\"?MLLLL$e*[o3WU\"*fYF-$ !/.M#>!H)p\"F-7$$\"?++++++DJX*eE1+m%F-$!/P+A1vNgYF-$!/b#yD(3Pdf^u\"F-7$$\" ?mmmmmmmTg2R5(>m%F-$!/bkYF-$!/mTi$Qww\"F-7$$\"?+++++++]7=p:*[m%F-$!/^E`])[x \"F-7$$\"?nmmmmmmmmO9]elYF-$!/P`e2`&y\"F-7$$\"?++++++](=nHpafm%F-$!/]= G?)Qz\"F-7$$\"?MLLLLLL3xcrVKmYF-$!/\"32,y_!=F-7$$\"?nmmmmm;H#o,0%pmYF- $!/c30)f5#=F-7$$\"?+++++++](o(GP1nYF-$!/Zt8U.V=F-7$$\"?+++++++++&zQrxm %F-$!/>q0+q6>F-7$$\"?+++++++]78Z!z%oYF-$!/'pYF-$!/K*HMx/Z#F-7$$\"$n%!\"#$!/yeO:C=FF--%&COLORG6&%$RGBG$\"\"! Fc_m$\"\"#F*$\"\"\"Fc_m-%'LEGENDG6#%ESharp-Verner~scheme~with~c[10]=11 /12G-F$6%7jq7$F($\"-?<.phKF-7$F/$\"-:$ousF$F-7$F4$\"-dF00\"H$F-7$F9$\" -X&Q\\)3LF-7$F>$\"-Q;Of[LF-7$FC$\"->]WC4NF-7$FM$\"-(H\"e?pSF-7$FR$\"-N W(>Bu%F-7$FW$\"-6)p:Se&F-7$Ffn$\"-JJG`(f&F-7$F[o$\"-O_\"G6h&F-7$F`o$\" -gk;sQcF-7$Feo$\"-s;H]tcF-7$Fjo$\"-93OsWdF-7$Fdp$\"-(R&p)F-7$Fbr$\"-$>ij#*=*F-7$F\\ s$\"-ffaG$Q*F-7$Fas$\"-rlL6*Q*F-7$Ffs$\"-%fy\\\\R*F-7$F[t$\"-W&)>A;%*F -7$F`t$\"-/^EhP%*F-7$Fet$\"-&\\YY(*[*F-7$Fjt$\"-/g/>X&*F-7$F_u$\"-W]U( 4r*F-7$Fdu$\"-(4f0^!**F-7$Fiu$\".JI(y!y-\"F-7$F^v$\".VhR[m5\"F-7$Fcv$ \".lnSEhD\"F-7$Fhv$\".CD(*)*fO\"F-7$F]w$\".w+Wi1^\"F-7$Fbw$\".?Z\\9Lb \"F-7$Fgw$\".nLVNpc\"F-7$F\\x$\".p,6Puc\"F-7$Fax$\".()o;Rzc\"F-7$Ffx$ \".!)>UW*o:F-7$F[y$\".]D?^*p:F-7$F`y$\".GlrpTd\"F-7$Fey$\".`y%yTy:F-7$ Fjy$\".S0\"HX'e\"F-7$F_z$\".-\"[g&zf\"F-7$Fiz$\".t/K2Bj\"F-7$F^[l$\".k *zFpr;F-7$Fc[l$\".zTj=bu\"F-7$Fh[l$\".zW^[U*=F-7$F]\\l$\".e#QLlm@F-7$F b\\l$\".nsD!p\"Q#F-7$Fg\\l$\".0l;pCi#F-7$F\\]l$\".D(ya:EEF-7$Fa]l$\".h s9a)HEF-7$Ff]l$\".6SPFrj#F-7$F[^l$\".-I\"pXWEF-7$F`^l$\".#z,d@fEF-7$Fe ^l$\".[9t&H$o#F-7$Fj^l$\".!yDj$fq#F-7$F__l$\"._C;%H\\FF-7$Fd_l$\".Nv7) e(y#F-7$Fi_l$\".GC5hA%GF-7$F^`l$\".$eA$*p>HF-7$Fc`l$\".&G=`HGIF-7$Fh`l $\".1x(3!=M$F-7$F]al$\".#*\\Lu%))QF-7$Fbal$\".GqJ&o&4%F-7$Fgal$\".dZ* \\;MVF-7$F\\bl$\".q*H$Q\"*R%F-7$Fabl$\".*ye-)oU%F-7$Ffbl$\".**>\\)pFWF -7$F[cl$\".Lu3<&GWF-7$F`cl$\".\\gOb,V%F-7$Fecl$\".kn4&zJWF-7$Fjcl$\".N Ueo$QWF-7$F_dl$\".Cu]l\\W%F-7$Fddl$\".**fwXuX%F-7$Fidl$\".4K'p.qWF-7$F ^el$\".\"e6*=z\\%F-7$Fcel$\".9!HC1:XF-7$Fhel$\".NnR$RSXF-7$F]fl$\".c&) Hx#)e%F-7$Fbfl$\".I-N#p%o%F-7$Fgfl$\".yJqt5([F-7$F\\gl$\".;@JK4B&F-7$F agl$\".uJ@5'>fF-7$Ffgl$\".I.XFT;(F-7$F[hl$\"..!*>\"HttF-7$F`hl$\".\"oH On\"e(F-7$Fehl$\".A`X\"f%e(F-7$Fjhl$\".D%*)>^(e(F-7$F_il$\".t'*>hLf(F- 7$Fdil$\".'4%H@#*f(F-7$Fiil$\".`!)Gu4h(F-7$F^jl$\".58[rFi(F-7$Fcjl$\". j'\\edWwF-7$Fhjl$\".h4l5mm(F-7$F][m$\".p\"*y;:s(F-7$Fb[m$\".)=A=zaxF-7 $Fg[m$\".(e#\\Xi!yF-7$F\\\\m$\".[Yyi#[yF-7$Fa\\m$\"..doMr!zF-7$Ff\\m$ \".'ex\"*e!*zF-7$F[]m$\".mW&R=4\")F-7$F`]m$\".(p-4q*[)F-7$Fe]m$\".,)QA 92#*F-7$Fj]m$\"._10AO\")*F-7$F_^m$\"/uT0**3k5F-7$$\"?++++++v=UnU'H%pYF -$\"/]OFq!e6\"F-7$Fd^m$\"/6&\\lOd<\"F-7$$\"?++++++D19*3))4)pYF-$\"/j9A s+X7F-7$Fi^m$\"/$z(\\s([K\"F--F__m6&Fa_mFb_m$\"\"(F*Fb_m-Fi_m6#%8schem e~with~c[10]=15/16G-F$6%7_r7$F($\".)=fVr85F-7$F/$\".0x$yb=5F-7$F4$\".= t:JG-\"F-7$F9$\".J+n=\"G5F-7$F>$\".2ThOy.\"F-7$FC$\".H_OC<2\"F-7$FH$\" .k(>i'=6\"F-7$FM$\".Nved]=\"F-7$FR$\"..#)>d*>8F-7$FW$\".%fA2G)[\"F-7$F fn$\".GyP$)=\\\"F-7$F[o$\".Hrv1b\\\"F-7$F`o$\".yVbSG]\"F-7$Feo$\".8\"> $4;^\"F-7$Fjo$\".$='p]p_\"F-7$F_p$\".7@o-qa\"F-7$Fdp$\".AE@Kse\"F-7$Fi p$\".h5qFqm\"F-7$F^q$\".U^-w!>=F-7$Fcq$\".G6_SH$>F-7$Fhq$\".frp()Q3#F- 7$F]r$\".R`1t&G@F-7$Fbr$\".7Uy(Qw@F-7$Fgr$\".Qw&R@7AF-7$F\\s$\".JiT+H@ #F-7$Fas$\".2j$[F9AF-7$Ffs$\".)Gk7l:AF-7$F[t$\".:0(zm?AF-7$F`t$\".s\\K 7dA#F-7$Fet$\".d+!)ezB#F-7$Fjt$\".uR1^.D#F-7$F_u$\".IL!*[EG#F-7$Fdu$\" .`>UR#=BF-7$Fiu$\".!*pzq`Q#F-7$F^v$\".n9F@f_#F-7$Fcv$\".#H$[Z6z#F-7$Fh v$\".6OVpd)HF-7$F]w$\".*=l3&=C$F-7$Fbw$\".h'\\@LG1MF-7$Fdz$\".\\#G1aFMF-7$Fiz$\".tVaEyY$F-7$F^[l$\".Jfa(\\NNF-7$F c[l$\".G\"zN&3m$F-7$Fh[l$\".PtiB>\"RF-7$F]\\l$\".[=&QfqVF-7$Fb\\l$\".[ ug4Ct%F-7$Fg\\l$\".wHQJw8&F-7$F\\]l$\".@)pJ&[9&F-7$Fa]l$\".=H;*4_^F-7$ Ff]l$\".f?'yMm^F-7$F[^l$\".eXv.2=&F-7$F`^l$\".mMwY%4_F-7$Fe^l$\".g9y`O D&F-7$Fj^l$\".))*e&[BH&F-7$F__l$\".3N[zSO&F-7$Fd_l$\".i#RNbEaF-7$Fi_l$ \".YvY+_^&F-7$F^`l$\".C#enDScF-7$Fc`l$\".XEt*=:eF-7$Fh`l$\".o/ec#>jF-7 $F]al$\".Ph]/w>(F-7$Fbal$\".>)F-7$ Fhel$\".Rs>nVE)F-7$F]fl$\".>#*f,9M)F-7$Fbfl$\".0p>rD\\)F-7$Fgfl$\".)=A CL\"y)F-7$F\\gl$\".3,qfiL*F-7$Fagl$\"/td()=pR5F-7$Ffgl$\"/O592XJ7F-7$F [hl$\"/\"pp(ypj7F-7$F`hl$\"/U5HK$eH\"F-7$Fehl$\"/ApL>L'H\"F-7$Fjhl$\"/ 5?,6$oH\"F-7$F_il$\"/s#y#3$yH\"F-7$Fdil$\"/,t9C$))H\"F-7$Fiil$\"/[%\\> T3I\"F-7$F^jl$\"/G_Av&GI\"F-7$Fcjl$\"/p$)=Se18F-7$Fhjl$\"/w%>!yM58F-7$ F][m$\"/mPp%G'>8F-7$Fb[m$\"/H!yVb]K\"F-7$Fg[m$\"/)3\"4=9L8F-7$F\\\\m$ \"/MUA&*eR8F-7$Fa\\m$\"/Oh'y4&[8F-7$Ff\\m$\"/G7\")4/h8F-7$F[]m$\"/eA\\ 4uy8F-7$F`]m$\"/Z`/YCN9F-7$Fe]m$\"/&yPtA:a\"F-7$Fj]m$\"/Us]([8j\"F-7$F _^m$\"/(H=!*HRv\"F-7$F^dn$\"/,!>$*)fI=F-7$Fd^m$\"/+QO!z%>>F-7$Ffdn$\"/ (\\i**eA-#F-7$Fi^m$\"/U^AS#39#F--F__m6&Fa_m$\"\"*F*$\"\"%F*Fb_m-Fi_m6# %8scheme~with~c[10]=19/20G-F$6%7`r7$F($\".v3?w,\">F-7$F/$\".T>%HI>>F-7 $F4$\".90]ct#>F-7$F9$\".m4:Rt$>F-7$F>$\".'4YIob>F-7$FC$\".orYC(=?F-7$F H$\".)e/:[#4#F-7$FM$\".\"R#\\udA#F-7$FR$\".r#>SGpCF-7$FW$\".1q#4*3x#F- 7$Ffn$\".cZP)fxFF-7$F[o$\".U-WWVy#F-7$F`o$\".')\\u-!)z#F-7$Feo$\".B/f' Q9GF-7$Fjo$\"._\"=,4VGF-7$F_p$\".FVZP.)GF-7$Fdp$\".qCH;W&HF-7$Fip$\".B @G%)**4$F-7$F^q$\".9.]wZP$F-7$Fcq$\".E3nH#zNF-7$Fhq$\".\"zg(z!\\QF-7$F ]r$\".WUNa(GRF-7$Fbr$\".8q@;R,%F-7$Fgr$\".6%4*zw2%F-7$F\\s$\".U$H_%*yS F-7$Fas$\".?%R'y93%F-7$Ffs$\".%e]d,%3%F-7$F[t$\".U^$GE$4%F-7$F`t$\".B> *4c-TF-7$Fet$\".?6LZ^7%F-7$Fjt$\"..;D1\"[TF-7$F_u$\".0`E#43UF-7$Fdu$\" .`J!fstUF-7$Fiu$\".&y?Od'R%F-7$F^v$\".FX))Q:l%F-7$Fcv$\".8I\"*4$G^F-7$ Fhv$\".;m\")efZ&F-7$F]w$\".w-3\\9$fF-7$Fbw$\".6bdk`1'F-7$Fgw$\".&***ym $3hF-7$F\\x$\".Q5ZA.6'F-7$Fax$\".+JjzA6'F-7$Ffx$\".@xS)>;hF-7$F[y$\".Q !HJ7?hF-7$F`y$\".6[1ol8'F-7$Fey$\".v=sEJ:'F-7$Fjy$\".\\_1MV='F-7$F_z$ \".')[k,qA'F-7$Fdz$\".lRkFjE'F-7$Fiz$\".kBwT1M'F-7$F^[l$\".o&pqokkF-7$ Fc[l$\".E;ilFp'F-7$Fh[l$\".4n$fzXrF-7$F]\\l$\".Ce%zImzF-7$Fb\\l$\".6bM _&4')F-7$Fg\\l$\".24'Q4F$*F-7$F\\]l$\".$3FY?S$*F-7$Fa]l$\".m*Q$fLN*F-7 $Ff]l$\".d\"zrAz$*F-7$F[^l$\".@*\\8H0%*F-7$F`^l$\".#>ZZad%*F-7$Fe^l$\" .+M00'Q&*F-7$Fj^l$\".4/Fu)4'*F-7$F__l$\".rR>e:u*F-7$Fd_l$\".>.oMd&)*F- 7$Fi_l$\"/Me9-r,5F-7$F^`l$\"/)Q)e\"yV-\"F-7$Fc`l$\"/HKcx%f0\"F-7$Fh`l$ \"/E5)Q@j9\"F-7$F]al$\"/V2E6`-8F-7$Fbal$\"/R^BGYh8F-7$Fgal$\"/P8w#\\\" H9F-7$F\\bl$\"/ru%onvW\"F-7$Fabl$\"/xLk5Xb9F-7$Ffbl$\"/$3'z+sb9F-7$F[c l$\"/2(R@*)fX\"F-7$F`cl$\"/2KSy_c9F-7$Fecl$\"/j3Wp1d9F-7$Fjcl$\"/ygZ\" G#f9F-7$F_dl$\"/FBbqRh9F-7$Fddl$\"/+X%>+bY\"F-7$Fidl$\"/JKP)Q'p9F-7$F^ el$\"/_CS9vy9F-7$Fcel$\"/TspV@%[\"F-7$Fhel$\"/MbV_*>\\\"F-7$F]fl$\"/@W gP81:F-7$Fbfl$\"/ip$zoP`\"F-7$Fgfl$\"/xK([%='e\"F-7$F\\gl$\"/K`FD4'o\" F-7$Fagl$\"/Sn2dSv=F-7$Ffgl$\"/R()*o'o9AF-7$F[hl$\"/lakg[rAF-7$F`hl$\" /3il5.GBF-7$Fehl$\"/629q#*GBF-7$Fjhl$\"/C=%zB)HBF-7$F_il$\"/=]a)>;L#F- 7$Fdil$\"/H\"pD>ML#F-7$Fiil$\"/:WM\"GqL#F-7$F^jl$\"/9fx0lSBF-7$Fcjl$\" /M17cMZBF-7$Fhjl$\"/o(e=4TN#F-7$F][m$\"/@kb9%3P#F-7$Fb[m$\"/\\'Gp52Q#F -7$Fg[m$\"/q^O%)\\&R#F-7$F\\\\m$\"/+fONI2CF-7$Fa\\m$\"/\"zdZT 1U9x#F-7$Fj]m$\"/F=,YbJHF-7$F_^m$\"/ZTMV1\\JF-7$F^dn$\"/Wmvul%G$F-7$Fd ^m$\"/`mcA\\TMF-7$Ffdn$\"/R3YnWAOF-7$Fi^m$\"/GYZ]sIQF--%'COLOURG6&Fa_m $\"*++++\"!\")Fb_mF[bq-Fi_m6#%8scheme~with~c[10]=32/33G-F$6%7`r7$F($\" .IL3QU5$F-7$F/$\".`o**p!>JF-7$F4$\".nr!p:KJF-7$F9$\".`*>IM[JF-7$F>$\". \"*e/zw<$F-7$FC$\".T?q'ywKF-7$FH$\".Lgx(*=R$F-7$FM$\".'*\\k,\"*f$F-7$F R$\".zeT$HwRF-7$FW$\".,+X?AW%F-7$Ffn$\".hhltHX%F-7$F[o$\".+Jw)yjWF-7$F `o$\".0rs#o&[%F-7$Feo$\".mTzi=^%F-7$Fjo$\".4v[@rb%F-7$F_p$\".78?M_h%F- 7$Fdp$\".G'4E5IZF-7$Fip$\".WdYQZ&\\F-7$F^q$\".]3esqP&F-7$Fcq$\".J%)['[ !p&F-7$Fhq$\".+3!4X.hF-7$F]r$\".!4K!f_A'F-7$Fbr$\".1'3JSbjF-7$Fgr$\".* 4dN$GX'F-7$F\\s$\".Uk*e$[X'F-7$Fas$\".K*e\\%)ekF-7$Ffs$\".-$y)fGY'F-7$ F[t$\".\"*R<$\\xkF-7$F`t$\".UrE2A\\'F-7$Fet$\".bU4Vz_'F-7$Fjt$\".<'Q.: klF-7$F_u$\".!)H8$[dmF-7$Fdu$\".g>(pjenF-7$Fiu$\".\"fM-.ZpF-7$F^v$\".$ *=jYjL(F-7$Fcv$\".2S#4_h!)F-7$Fhv$\".A\\6o*)e)F-7$F]w$\".rT'p*)y#*F-7$ Fbw$\".H?tS:[*F-7$Fgw$\".]6bkma*F-7$F\\x$\".FC0@(\\&*F-7$Fax$\".k$o)zF b*F-7$Ffx$\".TeX/*e&*F-7$F[y$\".7tLQ]c*F-7$F`y$\".%y^)R2f*F-7$Fey$\".! Q6!>mh*F-7$Fjy$\".S!\\CQl'*F-7$F_z$\".dy(f!=t*F-7$Fdz$\".w4l5Dz*F-7$Fi z$\".B8r)Q1**F-7$F^[l$\"/0&*3)R&45F-7$Fc[l$\"/m0.-9W5F-7$Fh[l$\"/-(>0( f76F-7$F]\\l$\"/ei\\\\8O7F-7$Fb\\l$\"/KJ9YuK8F-7$Fg\\l$\"/lGCX\"F-7$F`^l$\"/X$*oa\\g9F-7$Fe^l$\"/W:CK&HZ\"F-7$Fj^l$\"/0%GM;Q [\"F-7$F__l$\"/%[)p$oP]\"F-7$Fd_l$\"/%Hc$=+@:F-7$Fi_l$\"/xF$)zHX:F-7$F ^`l$\"/-+N/Nz:F-7$Fc`l$\"/d?k`nE;F-7$Fh`l$\"/SW\"zj\"\\%*>F-7$Fbal$\"/*Ho(Q7#3#F-7$Fgal$\"/%ynt*o#=#F-7$F\\bl$\"/$=E%4/5 AF-7$Fabl$\"/yz5Ev@AF-7$Ffbl$\"/5#QEj@A#F-7$F[cl$\"/%Q')4uDA#F-7$F`cl$ \"/)*=9jRBAF-7$Fecl$\"/&=&e#>UA#F-7$Fjcl$\"/&RaM=vA#F-7$F_dl$\"/dx#>H3 B#F-7$Fddl$\"/+#[l#4PAF-7$Fidl$\"/9f#H5MC#F-7$F^el$\"/ZMehJdAF-7$Fcel$ \"/J1f[jlAF-7$Fhel$\"/j&e)QVxAF-7$F]fl$\"/)\\n)*[()H#F-7$Fbfl$\"/'*f(G r,M#F-7$Fgfl$\"/9yYvN=CF-7$F\\gl$\"/]G:?xmDF-7$Fagl$\"/wJ\"[Tp%GF-7$Ff gl$\"/gv1\"zsM$F-7$F[hl$\"/G<(Q$*3V$F-7$F`hl$\"/$*yS?59NF-7$Fehl$\"/fb ^WX:NF-7$Fjhl$\"/Gp<\"3o^$F-7$F_il$\"/uzB#>&>NF-7$Fdil$\"/Oqu`BANF-7$F iil$\"/R2\")GoFNF-7$F^jl$\"/:9l3:LNF-7$Fcjl$\"/2qeoDVNF-7$Fhjl$\"/C!*> jY`NF-7$F][m$\"/J;pJryNF-7$Fb[m$\"/iV\"*zc$f$F-7$Fg[m$\"/w,uat:OF-7$F \\\\m$\"/8!QQmLj$F-7$Fa\\m$\"/'fYykwl$F-7$Ff\\m$\"/>$\\jI;p$F-7$F[]m$ \"/\"))fKO$RPF-7$F`]m$\"/tJI8@!*QF-7$Fe]m$\"/?`EwgqTF-7$Fj]m$\"/g\"oP> bS%F-7$F_^m$\"/XMAv/CZF-7$F^dn$\"/ZPV*\\B#\\F-7$Fd^m$\"/m#\\V7::&F-7$F fdn$\"/C;uQo:aF-7$Fi^m$\"/nZN%y%>dF--F__m6&Fa_m$\"\"&F*Fb_mFf_m-Fi_m6# %8scheme~with~c[10]=81/82G-F$6%7`r7$F($\".)y$*p/_JF-7$F/$\".YsL2r;$F-7 $F4$\".y')o&R!=$F-7$F9$\".(*)G(=o>$F-7$F>$\".ye7!QEKF-7$FC$\".^-n0^K$F -7$FH$\".v*RA1RMF-7$FM$\".&>rcSVOF-7$FR$\".P$pO09SF-7$FW$\".AH!RfqWF-7 $Ffn$\".rAz:9[%F-7$F[o$\".gA)**H#\\%F-7$F`o$\"./HeLV^%F-7$Feo$\".&4y7l SXF-7$Fjo$\".4>WGee%F-7$F_p$\".&3dwTVYF-7$Fdp$\".1#f(pmv%F-7$Fip$\".U* >k9x\\F-7$F^q$\".tMro**Q&F-7$Fcq$\".;=$[\\&p&F-7$Fhq$\".:!*pXt4'F-7$F] r$\".:J')\\d@'F-7$Fbr$\".LW*>?UjF-7$Fgr$\".Q#*H\\oV'F-7$F\\s$\"./'ym%) QkF-7$Fas$\".!G5e%GW'F-7$Ffs$\".HVy]oW'F-7$F[t$\".:F$yWhkF-7$F`t$\".Si \\DhZ'F-7$Fet$\".0ams<^'F-7$Fjt$\".#\\W,&ya'F-7$F_u$\".?r*Q9SmF-7$Fdu$ \".&oewYRnF-7$Fiu$\".27%)HO#pF-7$F^v$\".;M?&e-tF-7$Fcv$\".K(H5g0!)F-7$ Fhv$\".f8b`b^)F-7$F]w$\".DA)eT\"=*F-7$Fbw$\".X)\\Yyw$*F-7$Fgw$\".Sz^1' R%*F-7$F\\x$\".bIuGEW*F-7$Fax$\".uoD`cW*F-7$Ffx$\".8@;4IJ%*pb*F-7$F_z $\".\"=e3fA'*F-7$Fdz$\".5yEkAo*F-7$Fiz$\".\\pmAOz*F-7$F^[l$\".$*\\5fv( **F-7$Fc[l$\"/v&oE'GJ5F-7$Fh[l$\"/BNRvN(4\"F-7$F]\\l$\"/F(f[Eh@\"F-7$F b\\l$\"/]S[lv38F-7$Fg\\l$\"/9F-7$F[^l$\"/QX\"p'fB9F-7$F`^l$\"/=\"[L/:V\"F- 7$Fe^l$\"/Q@JapV9F-7$Fj^l$\"/)*=.QFa9F-7$F__l$\"/F-7$Fbal$\"/`kq<,E?F-7$Fgal$ \"/[y$yJ:7#F-7$F\\bl$\"/\">(>d\\Z@F-7$Fabl$\"/zU\"*phe@F-7$Ffbl$\"/,\" \\(f,f@F-7$F[cl$\"/1-N^Tf@F-7$F`cl$\"/[k&)R@g@F-7$Fecl$\"/lLWN,h@F-7$F jcl$\"/8+\"))=U;#F-7$F_dl$\"/T8WcVn@F-7$Fddl$\"/O$)H6_t@F-7$Fidl$\"/NN s$f'z@F-7$F^el$\"/Z)y'><$>#F-7$Fcel$\"/UB4>D,AF-7$Fhel$\"/9/\\&)o7AF-7 $F]fl$\"/xc4iELAF-7$Fbfl$\"/3%\\9fIF#F-7$Fgfl$\"/?+*z8yM#F-7$F\\gl$\"/ 1=*=#4*[#F-7$Fagl$\"/tbu'zYv#F-7$Ffgl$\"/%4Mq$4FKF-7$F[hl$\"/1(*pZ)eI$ F-7$F`hl$\"/nni,E%Q$F-7$Fehl$\"/)yMgibQ$F-7$Fjhl$\"/pE`i'oQ$F-7$F_il$ \"/aB(=x%*Q$F-7$Fdil$\"/$4'zH4#R$F-7$Fiil$\"/()H2#RtR$F-7$F^jl$\"//:e^ g-MF-7$Fcjl$\"/)yP!yL7MF-7$Fhjl$\"//Br/$4\\YMF-7$Fb[ m$\"/`QDKzgMF-7$Fg[m$\"/u&*Qw3#[$F-7$F\\\\m$\"/\"3(R&y*)\\$F-7$Fa\\m$ \"/l!>Z.A_$F-7$Ff\\m$\"/W+J1faNF-7$F[]m$\"/V*)[\"p**f$F-7$F`]m$\"/Sn(* y'Hu$F-7$Fe]m$\"/*>[c)e2SF-7$Fj]m$\"/)=[j='GUF-7$F_^m$\"/7nE*)oFXF-7$F ^dn$\"/#\\Me&f8ZF-7$Fd^m$\"/*HPr9#G\\F-7$Ffdn$\"/ " 0 "" {MPLTEXT 1 0 528 "evalf[30](plot(['phi'(x)-'c n_RK7_1'(x),'phi'(x)-'cn_RK7_2'(x),'phi'(x)-'cn_RK7_3'(x),'phi'(x)-'cn _RK7_4'(x),\n'phi'(x)-'cn_RK7_5'(x),'phi'(x)-'cn_RK7_6'(x)],x=4.67..4. 75,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.2,1),COLOR(RGB,0,.7,0),COL OR(RGB,.9,.4,0),magenta,COLOR(RGB,.5,0,1),red],\nlegend=[`Sharp-Verner scheme with c[10]=11/12`,`scheme with c[10]=15/16`,`scheme with c[10] =19/20`,`scheme with c[10]=32/33`,`scheme with c[10]=81/82`,`scheme wi th c[10]=169/170`],\ntitle=`error curves for 11 stage order 7 Runge-Ku tta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 886 462 462 {PLOTDATA 2 "6,-%'CURVESG6%7[p7$$\"$n%!\"#$\"/yeO:C=F!#H7$$\"?LLLLLLLLLLsPurYF-$\" /r[,d]RFF-7$$\"?nmmmmmmmmJ>5EtYF-$\"/03!))=&eFF-7$$\"?LLLLLLLLLBAt'\\n %F-$\"/4c?eF&y#F-7$$\"?LLLLLLLLLjN\\owYF-$\"/(fJU>7'GF-7$$\"?nmmmmmmmm ^&Q%RyYF-$\"/_qEV'))>$F-7$$\"?LLLLLLLLLQk#z*zYF-$\"/JnYnArVF-7$$\"?+++ ++++++l9.i\"o%F-$\"/X!=UE;V%F-7$$\"?LLLLLLLLL=\"\\ZF-$\"0SR2(*fPa$F-7$$\"?+++++++++Dy'G*>ZF-$\"04V^Gcud%F -7$$\"?++++++++]FJ*G3s%F-$\"0[;P@&RFZF-7$$\"?+++++++++I%=HqMBZF-$\"03+Hb3\"4[F-7$$\"?++++++++++.W2DZF- $\"0GG![XW&)[F-7$$\"?mmmmmmmm;HOq&es%F-$\"0*z$*e\"*za\\F-7$$\"?LLLLLLL LLep'Rms%F-$\"01H73Z>4&F-7$$\"?++++++++v.#[nqs%F-$\"0tVI_\\6A&F-7$$\"? mmmmmmmm;\\%H&\\FZF-$\"00![Aib7aF-7$$\"?LLLLLLLLe%p5Bzs%F-$\"0B0=zbMp& F-7$$\"?+++++++++S>4NGZF-$\"0Bc)4km+hF-7$$\"?MLLLLLLL$ed*f:HZF-$\"0t8: 0fGR(F-7$$\"?nmmmmmmmm6s5'*HZF-$\"0fRXEI`!)*F-7$$\"?MLLLLLLLL))3E!3t%F -$\"1HcA!og@+\"F-7$$\"?+++++++++lXTkJZF-$\"1:Ux&yr%35F-7$$\"?nmmmmmmmm md'*GLZF-$\"1CatVCD@5F-7$$\"?+++++++++DcB,NZF-$\"1$[`RkD\"R5F-7$$\"?mm mmmmmm;*z$>%et%F-$\"1G5n\"eTm0\"F-7$$\"?LLLLLLLLLt>:nOZF-$\"1XYt]<0$4 \"F-7$$\"?LLLLLLLL$3Lq&4PZF-$\"1=Co(3$)e7\"F-7$$\"?LLLLLLLLL)o))>vt%F- $\"13c&)GE\\u6F-7$$\"?LLLLLLLL$e/2Wzt%F-$\"1sI)Q3\\dC\"F-7$$\"?LLLLLLL LL.a#o$QZF-$\"1[>\"fw5*[8F-7$$\"?mmmmmmmmT:D*)yQZF-$\"1+f*eyJ[\\\"F-7$ $\"?++++++++]F'f4#RZF-$\"1%)*o+iA**p\"F-7$$\"?MLLLLLLLeRn-jRZF-$\"1M2) z`zW)>F-7$$\"?nmmmmmmmm^Q40SZF-$\"19h1xse@BF-7$$\"?MLLLLLLL$eY/C3u%F-$ \"1CEi!HOmL#F-7$$\"?+++++++++!3:(fTZF-$\"17prS/%>N#F-7$$\"?nmmmmmmmmc% GpLu%F-$\"1XCf\"R)\\)Q#F-7$$\"?+++++++++N$zhTu%F-$\"1')*)GO()H2CF-7$$ \"?LLLLLLLLL8-V&\\u%F-$\"1c&)[\"y%fKCF-7$$\"?++++++++D'>zw`u%F-$\"1b'4 1*GC_CF-7$$\"?mmmmmmmm;z\"G*zXZF-$\"1&G%*enC*zCF-7$$\"?LLLLLLLL3ir#4pd!e#F-7$$\"?++++++++ +![,`uu%F-$\"13\"Q*Q#=.z#F-7$$\"?+++++++++:o%F-7$$\"?+++++++]i0j\"[$\\ZF-$\"1kt0*R=)fXF-7$$\"?++++++++v.Uac\\ZF- $\"1wV-e2u$*\\F-7$$\"?+++++++](=5s#y\\ZF-$\"1`QoIkl0bF-7$$\"$v%F*$\"19 (=&flm2hF--%&COLORG6&%$RGBG$\"\"!Fcdl$\"\"#!\"\"$\"\"\"Fcdl-%'LEGENDG6 #%ESharp-Verner~scheme~with~c[10]=11/12G-F$6%7_p7$F($!/$z(\\s([K\"F-7$ F/$!/5(*z9CN8F-7$F4$!/..'f;XM\"F-7$F9$!/uO&fEzN\"F-7$F>$!/r\"o,\"G+9F- 7$FC$!/e/TmY0;F-7$FH$!/y20'z)eBF-7$FM$!/$y)*oYXR#F-7$FR$!/BH?0Y9CF-7$F W$!/R2>+USCF-7$Ffn$!/n3bb+FDF-7$F[o$!/WL\"f>r'GF-7$F`o$!/swA$o@S%F-7$F eo$!/$>:0([TWF-7$Fjo$!/*\\W/G2[%F-7$F_p$!/atiluEXF-7$Fdp$!/\"4wni'*p%F -7$Fip$!//JWYtm`F-7$F^q$!/f=94;!\\)F-7$Fcq$!/p)pS&ek&)F-7$Fhq$!/W4.Z. \\')F-7$F]r$!/\"p(*GKQv)F-7$Fbr$!/<07rs(4*F-7$Fgr$!/`q8?(yc*F-7$F\\s$! 0bS6(y6a5F-7$Fas$!03Xj!QBo7F-7$Ffs$!0_Ut&yX#p\"F-7$F[t$!06wrdk%GF-7$Fiu$!0='QiT(3?#F-7$$\"?++++++++v8\"[7(=ZF-$!0C *oT$3$)R#F-7$F^v$!0`ZF-$!0c4%*='*>1$F-7$F cv$!0g#fh^+#f$F-7$Fhv$!070GYLQs$F-7$F]w$!0MFk#eMYPF-7$Fbw$!0_%e&[z\")y $F-7$Fgw$!0Xr/q9'[QF-7$F\\x$!0U]L<%>0RF-7$Fax$!0\\qTnD4-%F-7$Ffx$!05+T SNF8%F-7$F[y$!0qt1a;9I%F-7$F`y$!0j3%z`M`XF-7$Fey$!0zSn\\+[#\\F-7$Fjy$! 0Tc0g>I8'F-7$F_z$!0%*3Re\">c%)F-7$Fdz$!0k)3z**og')F-7$Fiz$!0s*=8,B:()F -7$F^[l$!08(>sBgD))F-7$Fc[l$!0eFp>J+)*)F-7$Fh[l$!06OiW$3N\"*F-7$F]\\l$ !0OaP5L'o%*F-7$Fb\\l$!0D[R68tx*F-7$Fg\\l$!1cj)3w'GC5F-7$F\\]l$!1*HT0-t P4\"F-7$Fa]l$!1ym8P=:'>\"F-7$Ff]l$!1N6Bk6XV8F-7$F[^l$!1-F%HO#*)G#F-7$F^`l$!1AQ1pa'GJ #F-7$Fc`l$!1mU%[z[;L#F-7$Fh`l$!1U_6k^ZeBF-7$F]al$!11S/u!f$)R#F-7$Fbal$ !1j$yTS;)eCF-7$Fgal$!18U[kT'en#F-7$F\\bl$!1!f)=!e>f9$F-7$Fabl$!1_))z1M %Ge$F-7$Ffbl$!1g?+,r2>UF-7$F[cl$!1Cl\"R<7`j%F-7$F`cl$!1HNPhyLK^F-7$$\" ?+++++++D\"G:3u'\\ZF-$!1uH&[.[`T&F-7$Fecl$!1G!\\L,NQs&F-7$$\"?+++++++v $40O\"*)\\ZF-$!1ppo;!>)fgF-7$Fjcl$!1+<8!ReaU'F--F_dl6&FadlFbdl$\"\"(Ff dlFbdl-Fjdl6#%8scheme~with~c[10]=15/16G-F$6%7_p7$F($!/U^AS#39#F-7$F/$! /tC)3rv:#F-7$F4$!/8g&*o`s@F-7$F9$!/xzn*zN>#F-7$F>$!/m(>'z]bAF-7$FC$!/& \\G%**))[DF-7$FH$!/`\\9*[ti$F-7$FM$!/%y(3!*R!o$F-7$FR$!/^QF`'4r$F-7$FW $!/KNH#p(\\PF-7$Ffn$!/$zCA@9(QF-7$F[o$!/0wBGSTVF-7$F`o$!/i4(o2(okF-7$F eo$!/5bhO[ElF-7$Fjo$!/!yZ4cSe'F-7$F_p$!/%>whS+l'F-7$Fdp$!/N?Hoz%)oF-7$ Fip$!/Pa'o!*zx(F-7$F^q$!0)[`f<'z>\"F-7$Fcq$!0;Ga,j%37F-7$Fhq$!055'[[O? 7F-7$F]r$!0%Hz&\\>[B\"F-7$Fbr$!0$eUmn,!G\"F-7$Fgr$!0&>$43W6M\"F-7$F\\s $!0m\\#3vvn9F-7$Fas$!0<0zjoru\"F-7$Ffs$!0!)zCX0KI#F-7$F[t$!0qV\\l84N#F -7$F`t$!0%)puf[NO#F-7$Fet$!0b%4*zghQ#F-7$Fjt$!0t*4aM&oT#F-7$F_u$!0GE` \"p^5DF-7$Fdu$!01OAba)\\EF-7$Fiu$!0BSul*)Q&HF-7$Fg[m$!0XJ\"yFv/KF-7$F^ v$!0R:DSy%eNF-7$F_\\m$!0!o\">1U70%F-7$Fcv$!0K5x`J(HZF-7$Fhv$!0$eQ\")[V **[F-7$F]w$!0tX#=1&F-7$F\\x$! 0AsP)G_K^F-7$Fax$!0)RQjxbv_F-7$Ffx$!0rF#p2V8aF-7$F[y$!0(yVy@a@cF-7$F`y $!0Gk&=(fG$fF-7$Fey$!0^w+!y#HR'F-7$Fjy$!0o#*o7Gd*yF-7$F_z$!16c*))QL-3 \"F-7$Fdz$!1r?=[v+16F-7$Fiz$!11u%G]sH6\"F-7$F^[l$!1qo(HN`q7\"F-7$Fc[l$ !1.\"[t\"4RY6F-7$Fh[l$!1/QO)eq_;\"F-7$F]\\l$!1j[F-7$F`^l$!1fwWfs12BF-7$Fe^l$ !1oGN(fgeu#F-7$Fj^l$!1%eOt$)fOw#F-7$F__l$!1Udb50w\"y#F-7$Fd_l$!1e.5(f- \\#GF-7$Fi_l$!1Dp@EC!o%GF-7$F^`l$!1/<'>+:e(GF-7$Fc`l$!1l-Cn1E)*GF-7$Fh `l$!1E]\"G(*y+$HF-7$F]al$!1<@*))4%=xHF-7$Fbal$!1$eKTR!\\[IF-7$Fgal$!1j !=7!3)\\I$F-7$F\\bl$!1bHeDAijQF-7$Fabl$!1M>JY(GcQ%F-7$Ffbl$!1[fHUA**[^ F-7$F[cl$!1#4cGi$**\\cF-7$F`cl$!1$R&32vd\\iF-7$Fbdm$!1jr%3,b:f'F-7$Fec l$!1U\"oLMTZ'pF-7$Fjdm$!1:/oT*\\\"yF--F_dl6&Fadl$ \"\"*Ffdl$\"\"%FfdlFbdl-Fjdl6#%8scheme~with~c[10]=19/20G-F$6%7_p7$F($! /GYZ]sIQF-7$F/$!/;Nv=pgQF-7$F4$!/\"Q#=&)[()QF-7$F9$!/6'RC,b#RF-7$F>$!/ SWz$\\%QSF-7$FC$!/,S>^.mXF-7$FH$!/BAF]-pkF-7$FM$!/qQ,)GGc'F-7$FR$!/\\V ->PL:%F-7$F`t$!0I`$H=kvTF-7$Fet$!0!3\"*=8i:UF-7$Fjt$!0'))GaI YqUF-7$F_u$!0?G6F-7$Fjy$!1S\\rwG0\"R \"F-7$F_z$!1-exP^\"Q*=F-7$Fdz$!1tlfgKPQ>F-7$Fiz$!1t;vT+e]>F-7$F^[l$!1> 0Ya&z_(>F-7$Fc[l$!1c\"fOGM'4?F-7$Fh[l$!1v&445>O/#F-7$F]\\l$!1B.fx.)e6# F-7$Fb\\l$!1c&4=p.B=#F-7$Fg\\l$!1*)\\C#=v?G#F-7$F\\]l$!1J0If>XICF-7$Fa ]l$!1xIDJxN[EF-7$Ff]l$!1m-9.I&4'HF-7$F[^l$!1c^$)z%RkS$F-7$F`^l$!1WO7fe 2LSF-7$Fe^l$!1#p<@#Hu%y%F-7$Fj^l$!1([PIied\"[F-7$F__l$!1\"[d*=)*HZ[F-7 $Fd_l$!1T&F-7$Fbal$!1Dh1n,P@`F-7 $Fgal$!19\\^^8OvdF-7$F\\bl$!1hj^![zBv'F-7$Fabl$!1*zjUyRql(F-7$Ffbl$!1K Qx\\10r*)F-7$F[cl$!1g1uGOBH)*F-7$F`cl$!2:\\0Z[&G&3\"F-7$Fbdm$!2g#)31FD N9\"F-7$Fecl$!2a!\\dF-7$F/$!/[$!/7UvstFgF-7$FC$!/\\eKht)z'F-7$FH$!/l)*=3#pb*F-7$FM$!/28 n%*3%p*F-7$FR$!/D#H+fYx*F-7$FW$!/&*>K&[y()*F-7$Ffn$!0tqWOw,-\"F-7$F[o$ !0F^?o%[U6F-7$F`o$!0$yb?Dh!o\"F-7$Feo$!0>#p$=Bcp\"F-7$Fjo$!0Q7SY&f5zs\"F-7$Fdp$!0W*oxvv*y\"F-7$Fip$!081(R,u>?F-7$F^q$!0k%*G, 4/2$F-7$Fcq$!0FG+7Ct4$F-7$Fhq$!0_(R$=by7$F-7$F]r$!0`mnRJ`;$F-7$Fbr$!0R Jtg'4$G$F-7$Fgr$!0$oXF%Q\\eWF-7$ Ffs$!0GAs\"4ZDeF-7$F[t$!0(R:\\U'H%fF-7$F`t$!0#)p#*)\\!\\(fF-7$Fet$!0v) >\"4=@.'F-7$Fjt$!0i[,yJ16'F-7$F_u$!0Tb[Bj>N'F-7$Fdu$!0_5B\\oeq'F-7$Fiu $!08$>E0QmuF-7$Fg[m$!0lE]RHq3)F-7$F^v$!0a9Lj/b&*)F-7$F_\\m$!1$fH4CAc, \"F-7$Fcv$!1m5S6rrz6F-7$Fhv$!1tw'4:83A\"F-7$F]w$!1^MUmN>G7F-7$Fbw$!1Y* G'Hs!>C\"F-7$Fgw$!1`Oa**fdh7F-7$F\\x$!14`K(RJ'z7F-7$Fax$!16?t8A&fJ\"F- 7$Ffx$!1xTg/Cm]8F-7$F[y$!1Iz\\%[]ES\"F-7$F`y$!1u*)Q8Ryz9F-7$Fey$!1V\"3 I4>Gf\"F-7$Fjy$!1^v6omOd>F-7$F_z$!1*\\qFQz:l#F-7$Fdz$!1<*pOnIKr#F-7$Fi z$!1?8@0rJIFF-7$F^[l$!17y/Dj*[w#F-7$Fc[l$!1-@?#*Q08GF-7$Fh[l$!1q$ov@<1 'GF-7$F]\\l$!1Px0(R88'HF-7$Fb\\l$!1h(z'4j``IF-7$Fg\\l$!1X+5Qgo\">$F-7$ F\\]l$!1zW\\S/f'R$F-7$Fa]l$!1?4Wm/t'p$F-7$Ff]l$!17\\_lA@ETF-7$F[^l$!14 l?4,zOZF-7$F`^l$!1\"=)R:)GOf&F-7$Fe^l$!1I#\\c1z#>mF-7$Fj^l$!1[=v7m=imF -7$F__l$!1PQiG7#eq'F-7$Fd_l$!1.D74a&*4oF-7$Fi_l$!1lK#*o1DjoF-7$F^`l$!1 )>tsy:Z$pF-7$Fc`l$!1WXi3(y.*pF-7$Fh`l$!1JGN:%=$pqF-7$F]al$!19m\\$3,e=( F-7$Fbal$!13gK\"Rm5O(F-7$Fgal$!1$H:wuqO)zF-7$F\\bl$!1/?D1a_;$*F-7$Fabl $!2'Q*[79:Y0\"F-7$Ffbl$!2@#))fqNvK7F-7$F[cl$!2$G0]_I))[8F-7$F`cl$!2&f \\BC(Hs[\"F-7$Fbdm$!2^@eD,tec\"F-7$Fecl$!2^.3.R+:l\"F-7$Fjdm$!2wVY*pzl W$!/u]aqk_dF-7$FC$!/`6v+PwkF-7$FH$!/23PeKT!* F-7$FM$!/U$erv)p\"*F-7$FR$!/JX*p)4Y#*F-7$FW$!/uR%\\xPM*F-7$Ffn$!/qHe!o &['*F-7$F[o$!0S0h8\"yy5F-7$F`o$!0!zNT47v:F-7$Feo$!0^5(4%*=*e\"F-7$Fjo$ !0fexjCKg\"F-7$F_p$!0-%eH+[>;F-7$Fdp$!0Its`(>x;F-7$Fip$!0(*Ht\\a'*)=F- 7$F^q$!0Y+U_z/&GF-7$Fcq$!0z34wma(GF-7$Fhq$!0'=ydj\"Q!HF-7$F]r$!0!4Wt:m QHF-7$Fbr$!0%f0(>[w/$F-7$Fgr$!0%**>%)\\*>>$F-7$F\\s$!0$3Sp4I&[$F-7$Fas $!0^9T!Gh>TF-7$Ffs$!0yE+,mmN&F-7$F[t$!0Idg4^KY&F-7$F`t$!0bsYp8E\\&F-7$ Fet$!0`Eh%)>_a&F-7$Fjt$!0ge!>B`)F-7$Fcv$!1P!)>YJ$R2\"F-7$Fhv$!1W= &pL.36\"F-7$F]w$!1VTx9(=v6\"F-7$Fbw$!1\"fMH7++8\"F-7$Fgw$!15/N&zRz9\"F -7$F\\x$!1AI-e-Rk6F-7$Fax$!1_!*zAqL(>\"F-7$Ffx$!1F/zW8F-7$Fey$!1)**Gzj/gW\"F-7$Fjy$!1to&f;06x\"F-7$F_ z$!1EOX6#=uQ#F-7$Fdz$!1_i\"f>rAW#F-7$Fiz$!1%[#RA8ldCF-7$F^[l$!1!)=(=*f y)[#F-7$Fc[l$!1%ea^&QDKDF-7$Fh[l$!1txR$oa^d#F-7$F]\\l$!1<8f=ualEF-7$Fb \\l$!1u229c)zu#F-7$Fg\\l$!1@!\\c.'4rGF-7$F\\]l$!1LF-7$Ff]l$!1lXCJPe)p$F-7$F[^l$!1?A=VN%oB%F-7$F`^l$!1#4OA:n0*\\ F-7$Fe^l$!12s<,Y>\"*eF-7$Fj^l$!1%GWOb#QHfF-7$F__l$!1]`_?x@ofF-7$Fd_l$! 1kO[f+$41'F-7$Fi_l$!1+P,xmY3hF-7$F^`l$!1?L!3\"[JshF-7$Fc`l$!1&Q6v_G?A' F-7$Fh`l$!1yVQ_bT#H'F-7$F]al$!14+:+#=gR'F-7$Fbal$!1\\Gg7,W^lF-7$Fgal$! 1CME$>.45(F-7$F\\bl$!15r*)pgrq#)F-7$Fabl$!1M\"GeW(=Y$*F-7$Ffbl$!2!p[91 X3!4\"F-7$F[cl$!2[\"F-7$F`cl$!2$3q)RvhL*)e/hPi\"F--F jeo6&FadlF\\foFbdlFbdl-Fjdl6#%:scheme~with~c[10]=169/170G-%&TITLEG6#%V error~curves~for~11~stage~order~7~Runge-Kutta~methodsG-%%FONTG6$%*HELV ETICAGFgen-%+AXESLABELSG6$Q\"x6\"Q!F\\gq-%%VIEWG6$;F(Fjcl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner \+ scheme with c[10]=11/12" "scheme with c[10]=15/16" "scheme with c[10]= 19/20" "scheme with c[10]=32/33" "scheme with c[10]=81/82" "scheme wit h c[10]=169/170" }}}}{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 34 "#============================== ===" }}{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 }