{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 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "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 "Map le 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 -1 100 "Derivation of 8 stage, order 6 R unge-Kutta schemes, each with a 9 stage FSAL order 5 embedded scheme" }}{PARA 3 "" 0 "" {TEXT 265 24 "(stage-order 3 examples)" }}{PARA 0 " " 0 "" {TEXT -1 46 "by Peter Stone, Gabriola Island, B.C., Canada " }} {PARA 0 "" 0 "" {TEXT -1 18 "Version: 4.12.2011" }}{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 et c. " }}{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 sessi on by commands similar to those that follow, where each file path give s the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/butcher.m\";\nread \"C:\\\\Maple/pro cdrs/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 between the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "For the Runge-Kutta schemes considered in this worksheet the stag e order conditions for stages 3 and 4 together with the condition that " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 24 " imply that the nodes " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"# " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$" } {TEXT -1 22 " satisfy the relation" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 2/3;" "6#/&%\"cG6#\"\"#*&F'\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 17 "while the nodes " }{XPPEDIT 18 0 " c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]" "6 #&%\"cG6#\"\"%" }{TEXT -1 22 " satisfy the relation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"# \"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "[op(StageOrderConditions(2,3..3,'expanded')) ,op(StageOrderConditions(3,3..3,'expanded'))];\nsol := solve(\{op(%)\} ,indets(%) minus \{c[3]\}):\nc[2]=subs(sol,c[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/*&&%\"aG6$\"\"$\"\"#\"\"\"&%\"cG6#F*F+,$*&#F+F*F+*$) &F-6#F)F*F+F+F+/*&F&F+)F,F*F+,$*&#F+F)F+*$)F4F)F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"#,$*&#F'\"\"$\"\"\"&F%6#F+F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[op(StageOrderConditions(2,4..4,'expanded')),op(StageOrderConditi ons(3,4..4,'expanded'))];\nsubs(a[4,2]=0,%);\nsol := solve(\{op(%)\},i ndets(%) minus \{c[4]\}):\nc[3]=subs(sol,c[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,&*&&%\"aG6$\"\"%\"\"#\"\"\"&%\"cG6#F+F,F,*&&F(6$F* \"\"$F,&F.6#F3F,F,,$*&#F,F+F,*$)&F.6#F*F+F,F,F,/,&*&F'F,)F-F+F,F,*&F1F ,)F4F+F,F,,$*&#F,F3F,*$)F;F3F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7$/*&&%\"aG6$\"\"%\"\"$\"\"\"&%\"cG6#F*F+,$*&#F+\"\"#F+*$)&F-6#F)F2F+F +F+/*&F&F+)F,F2F+,$*&#F+F*F+*$)F5F*F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$*&#\"\"#F'\"\"\"&F%6#\"\"%F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "#==========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "#======================================== ======" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Dormand, Lockyer, McCor rigan and Prince scheme" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 115 "See: Global Error estimation with Runge-Kutta tr iples, by J.R.Dormand, M.A.Lockyer, N.E.McCorrigan and P.J.Prince, " } }{PARA 0 "" 0 "" {TEXT -1 81 " Computers and Mathematics with A pplications, 18 (1989), pages 835 to 846." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[1/9, 1/9, ``, ``, ``, ``, ``, ``, ``, ``], [1/6, 1/24, 1/8, ``, ``, ``, `` , ``, ``, ``], [1/4, 1/16, 0, 3/16, ``, ``, ``, ``, ``, ``], [5/9, 280 /729, 0, -325/243, 1100/729, ``, ``, ``, ``, ``], [1/2, 6127/14680, 0, -1077/734, 6494/4037, -9477/161480, ``, ``, ``, ``], [48/49, -1342627 3320/14809773769, 0, 4192558704/2115681967, 14334750144/14809773769, 1 17092732328/14809773769, -361966176/40353607, ``, ``, ``], [1, -234068 9/1901060, 0, 31647/13579, 253549596/149518369, 10559024082/977620105, -152952/12173, -5764801/186010396, ``, ``], [1, 203/2880, 0, 0, 30208 /70785, 177147/164560, -536/705, 1977326743/3619661760, -259/720, ``], [b, 203/2880, 0, 0, 30208/70785, 177147/164560, -536/705, 1977326743/ 3619661760, -259/720, ``], [`b*`, 36567/458800, 0, 0, 9925984/27063465 , 85382667/117968950, -310378/808635, 262119736669/345979336560, -1/2, -101/2294]])" "6#-%'matrixG6#7,7,*&\"\"\"F)\"\"*!\"\"*&F)F)F*F+%!GF-F -F-F-F-F-F-7,*&F)F)\"\"'F+*&F)F)\"#CF+*&F)F)\"\")F+F-F-F-F-F-F-F-7,*&F )F)\"\"%F+*&F)F)\"#;F+\"\"!*&\"\"$F)F9F+F-F-F-F-F-F-7,*&\"\"&F)F*F+*& \"$!GF)\"$H(F+F:,$*&\"$D$F)\"$V#F+F+*&\"%+6F)FBF+F-F-F-F-F-7,*&F)F)\" \"#F+*&\"%FhF)\"&!o9F+F:,$*&\"%x5F)\"$M(F+F+*&\"%%\\'F)\"%PSF+,$*&\"%x %*F)\"'![h\"F+F+F-F-F-F-7,*&\"#[F)\"#\\F+,$*&\",?LFEM\"F)\",pPx4[\"F+F +F:*&\"+/(eD>%F)\"+n>o:@F+*&\",W,vMV\"F)\",pPx4[\"F+*&\"-GBt#4<\"F)\", pPx4[\"F+,$*&\"*wh'>OF)\")2ONSF+F+F-F-F-7,F),$*&\"(*oSBF)\"(g5!>F+F+F: *&\"&Z;$F)\"&zN\"F+*&\"*'f\\NDF)\"*p$=&\\\"F+*&\",#3C!f0\"F)\"*0,ix*F+ ,$*&\"'_H:F)\"&t@\"F+F+,$*&\"(,[w&F)\"*'R5g=F+F+F-F-7,F)*&\"$.#F)\"%!) GF+F:F:*&\"&3-$F)\"&&yqF+*&\"'ZrF)\"+gOF+,$*&\"$f#F)\"$?(F+F+F-7,%\"bG*&FaqF)FbqF+F:F:*&FdqF)Feq F+*&FgqF)FhqF+,$*&F[rF)F\\rF+F+*&\"+VnKx>F)\"+gOF+,$*&FbrF)FcrF+F+F -7,%#b*G*&\"&nl$F)\"'+)e%F+F:F:*&\"(%)f#**F)\")lM1FF+*&\")nEQ&)F)\"*]* oz6F+,$*&\"'y.JF)\"'N'3)F+F+*&\"-pmt>@EF)\"-glLzfMF+,$*&F)F)FKF+F+,$*& \"$,\"F)\"%%H#F+F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined s cheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1153 "# coefficients by Dormand, Lockyer, McCorrigan and \+ Prince\nee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/4,\nc[5]=5/9,\nc[6]=1/2, \nc[7]=48/49,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1 /8,\na[4,1]=1/16,\na[4,2]=0,\na[4,3]=3/16,\na[5,1]=280/729,\na[5,2]=0, \na[5,3]=-325/243,\na[5,4]=1100/729,\na[6,1]=6127/14680,\na[6,2]=0,\na [6,3]=-1077/734,\na[6,4]=6494/4037,\na[6,5]=-9477/161480,\na[7,1]=-134 26273320/14809773769,\na[7,2]=0,\na[7,3]=4192558704/2115681967,\na[7,4 ]=14334750144/14809773769,\na[7,5]=117092732328/14809773769,\na[7,6]=- 361966176/40353607,\na[8,1]=-2340689/1901060,\na[8,2]=0,\na[8,3]=31647 /13579,\na[8,4]=253549596/149518369,\na[8,5]=10559024082/977620105,\na [8,6]=-152952/12173,\na[8,7]=-5764801/186010396,\na[9,1]=203/2880,\na[ 9,2]=0,\na[9,3]=0,\na[9,4]=30208/70785,\na[9,5]=177147/164560,\na[9,6] =-536/705,\na[9,7]=1977326743/3619661760,\na[9,8]=-259/720,\n\nb[1]=20 3/2880,\nb[2]=0,\nb[3]=0,\nb[4]=30208/70785,\nb[5]=177147/164560,\nb[6 ]=-536/705,\nb[7]=1977326743/3619661760,\nb[8]=-259/720,\n\n`b*`[1]=36 567/458800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=9925984/27063465,\n`b*`[5 ]=85382667/117968950,\n`b*`[6]=-310378/808635,\n`b*`[7]=262119736669/3 45979336560,\n`b*`[8]=-1/2,\n`b*`[9]=-101/2294\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "s ubs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b `,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"\"*F(%!GF+F+F+F+F+F+F+7,# F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+F+7,#F)\"\"%#F)\"#;\"\"!#\"\"$F7F+F+ F+F+F+F+7,#\"\"&F*#\"$!G\"$H(F8#!$D$\"$V##\"%+6F@F+F+F+F+F+7,#F)\"\"## \"%Fh\"&!o9F8#!%x5\"$M(#\"%%\\'\"%PS#!%x%*\"'![h\"F+F+F+F+7,#\"#[\"#\\ #!,?LFEM\"\",pPx4[\"F8#\"+/(eD>%\"+n>o:@#\",W,vMV\"Fen#\"-GBt#4<\"Fen# !*wh'>O\")2ONSF+F+F+7,F)#!(*oSB\"(g5!>F8#\"&Z;$\"&zN\"#\"*'f\\ND\"*p$= &\\\"#\",#3C!f0\"\"*0,ix*#!'_H:\"&t@\"#!(,[w&\"*'R5g=F+F+7,F)#\"$.#\"% !)GF8F8#\"&3-$\"&&yq#\"'Zr<\"'gX;#!$O&\"$0(#\"+VnKx>\"+gO#!$f#\"$?( F+7,%\"bGFdpF8F8FgpFjpF]qF`qFcqF+7,%#b*G#\"&nl$\"'+)e%F8F8#\"(%)f#**\" )lM1F#\")nEQ&)\"*]*oz6#!'y.J\"'N'3)#\"-pmt>@E\"-glLzfM#!\"\"FH#!$,\"\" %%H#Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expande d')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*` ,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{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 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderConditions(c t,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8 [j])),i=2..5)],j=1..6)]:\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$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can calculate th e 2 norm of the principal error, that is, the 2-norm of the principal \+ error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(sub s(ee,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+lf'eP%!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the \+ order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5, 9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[i])^2 ,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+(pTK,#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "18 of the principal error conditions are satisfied." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops( errterms6_8) do\n if simplify(subs(ee,errterms6_8[ct]))=0 then ind : = [op(ind),ct] end if:\nend do:\nerr6 := PrincipalErrorConditions(6): \n[seq([i,err6[i]],i=ind)]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747%\"\"'%#~~G/*&%\"bG\"\" \"-%!G6#*(%\"aGF--F/6#*&F2F--F/6#*&F2F-%\"cGF-F-F-F6F-F-#F-\"$/&7%\"#5 F)/*&F,F-)F3\"\"#F-#F-\"$_#7%\"#:F)/**F,F-F9F-F6F-F3F-#F-\"#%)7%\"#?F) /*&F,F--F/6#*(F2F-F6F--F/6#*&)F9FAF-F2F-F-F-FB7%\"#@F)/*&F,F--F/6#*(F2 F-FTF-F3F-F-FB7%\"#CF)/*(F,F-F3F-FQF-#F-\"$E\"7%\"#EF)/*&F,F--F/6#*(F2 F-F9F-)F6FAF-F-#F-\"$o\"7%\"#GF)/*&F,F-)F6\"\"$F-#F-\"#c7%\"#HF)/*&F,F --F/6#*(F2F-FTF-FQF-F-Fjn7%\"#MF)/**F,F-F9F-F6F-FQF-#F-\"#U7%\"#PF)/*& F,F--F/6#*(F2F-)F9F[pF-F6F-F-FH7%\"#RF)/*(F,F-FbqF-F3F-Fip7%\"#TF)/*&F ,F-)FQFAF-#F-\"#j7%FjpF)/*(F,F-FTF-FcoF-#F-Fgo7%\"#WF)/*(F,F-FbqF-FQF- #F-FV7%\"#YF)/*(F,F-)F9\"\"%F-F6F-#F-\"#97%\"#ZF)/*&F,F--F/6#*&)F9\"\" &F-F2F-F-Fip7%\"#[F)/*&F,F-)F9F(F-#F-\"\"(Q*pprint1476\"" }}}{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 44 "simultaneous cons truction of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equations to en sure that stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifyin g conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6 #F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " } {XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 9 simple order condit ions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 97 "These include the 6 order 6 quadrature conditions, 2 a dditional order 6 order conditions and the " }{TEXT 260 28 "order 7 qu adrature condition" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "SO7 := SimpleOrderCondition s(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,20,29,32,64])]:\nlinalg[augment]( linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[de lcols](%,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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F( \"\"&7%\"#?F)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&FPF(-FM6#*&FPF(F2F(F(F(F (#F(\"$W\"7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(\"#C7%\"#KF)/*&F,F()F 2FGF(#F(\"\"'7%\"#kF)/*&F,F()F2FboF(#F(\"\"(Q)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection of 7 \"simple\" order conditions as given (in abreviated form) in the following table. " }}{PARA 0 "" 0 " " {TEXT -1 78 "These include the 5 quadrature conditions and two addit ional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "`SO5*` := subs(b=`b*`,SimpleOrderC onditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,9,15,16])]:\nlinalg[augm ent](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),lina lg[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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"\"*F)/*&F,F(-%!G 6#*&%\"aGF(-FF6#*&FIF(-FF6#*&FIF(F2F(F(F(F(#F(\"$?\"7%\"#:F)/*&F,F(-FF 6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint366\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO7_8 := SimpleOrderConditions(7,8,'expanded'):\nSO_eqs := [op(Ro wSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')) ,\n op(StageOrderConditions(3,3..8,'expanded'))]:\n`SO5_ 9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns : = [seq(SO7_8[i],i=[1,2,4,8,16,20,29,32,64])]:\n`ord_cdns*` := [seq(`SO 5_9*`[i],i=[1,2,4,8,9,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..8)= b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns : = [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO _eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[4] = 1/4;" "6#/&%\"cG6#\"\"%*&\"\"\"F)F'!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 1/2;" "6#/&%\"cG6#\"\"'*&\"\" \"F)\"\"#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 48/49;" "6#/& %\"cG6#\"\"(*&\"#[\"\"\"\"#\\!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and 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$\"\" )\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provide the \+ linking coefficients for the 9th stage of the embedded order 5 scheme \+ so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b [i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&% \"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"b G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6# \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G 6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b *G6#\"\"$\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8] = -1/2; " "6#/&%#b*G6#\"\"),$*&\"\"\"F*\"\"#!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We have 45 equa tions and 45 unknowns including " }{TEXT 260 18 "three of the nodes" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 11 ": Ta king " }{XPPEDIT 18 0 "c[7] = 48/49;" "6#/&%\"cG6#\"\"(*&\"#[\"\"\" \"#\\!\"\"" }{TEXT -1 87 " makes the principal error norm close to it s minimum value with repect to the values " }{XPPEDIT 18 0 "c[4]=1/4 " "6#/&%\"cG6#\"\"%*&\"\"\"F)F'!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]=1/2" "6#/&%\"cG6#\"\"'*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "preliminary calculat ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "e1 := \{c[4]=1/4,c[6]=1/2,c[8]=1,c[9]=1,\n seq (a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=-5/8\}:\ne qns := subs(e1,cdns):\ne2 := solve(\{op(eqns)\},indets(eqns) minus \{c [7]\}):\ne3 := `union`(e1,e2):\nerrterms6_8 := PrincipalErrorTerms(6,8 ,'expanded'):\nsm := 0:\nfor ct to nops(errterms6_8) do\n sm := sm+( simplify(subs(e3,errterms6_8[ct])))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(sm,c[7] =0.96..1,color=COLOR(RGB,.4,0,1),font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 272 272 {PLOTDATA 2 "6'-%'CURVESG6#7U7$$\"3k******* ******f*!#=$\"3i&eZ:v\"e_F!#E7$$\"3Lmmmh)=(3'*F*$\"3)[qnzI<8p#F-7$$\"3 *HL$e'40jh*F*$\"33:7q!fP'REF-7$$\"36mm;6m$[i*F*$\"3k>`(>k*F*$\"3A!ed:cHiZ#F-7$$\"3om m\">K'*)\\'*F*$\"3b#3'[\\MUHCF-7$$\"3u***\\Kd,\"e'*F*$\"3)fI\"3@I\"GQ# F-7$$\"3Ymm\"fX(em'*F*$\"3u@!pBwAmL#F-7$$\"3m***\\U7Y]n*F*$\"3kegU)Q^E H#F-7$$\"3_LLL/pu$o*F*$\"3-zy![(\\h\\AF-7$$\"3Omm;c0T\"p*F*$\"3l'*fA?C e8AF-7$$\"3o*****H,Q+q*F*$\"33d+*>\"e:v@F-7$$\"3y*****\\*3q3(*F*$\"3\" H$*o*=7*)Q@F-7$$\"3v*****p=\\qr*F*$\"3\"G(fg\")e=1@F-7$$\"3)pm;fBIYs*F *$\"3#RLa*=(H%y?F-7$$\"3xKLLO[kL(*F*$\"3w=!z]$3)y/#F-7$$\"3aLLL&Q\"GT( *F*$\"3POeUG58C?F-7$$\"3%)***\\s]k,v*F*$\"3w9XYUW,**>F-7$$\"3ULLLvv-e( *F*$\"31E\\:K<3z>F-7$$\"3M***\\sgamw*F*$\"3=l?ZQDvf>F-7$$\"3w***\\F-7$$\"3=LL$e/TMy*F*$\"3>A)**zN*)*H>F-7$$\"3ULLeDBJ \"z*F*$\"3\"=b5\"*f8(>>F-7$$\"3Smm;kD!)*z*F*$\"3m0Sg!QY8\">F-7$$\"3>mm \"f`@'3)*F*$\"3B&>&G)e7d!>F-7$$\"3X***\\nZ)H;)*F*$\"3R(GS#\\kS.>F-7$$ \"3?mm;$y*eC)*F*$\"36*yyPQ)p.>F-7$$\"3K*****R^bJ$)*F*$\"3:>a(Q-4r!>F-7 $$\"3q****\\5a`T)*F*$\"3'yGGx8\"e8>F-7$$\"3K++D\"RV'\\)*F*$\"3fy)HNWqG #>F-7$$\"3'*****\\@fke)*F*$\"3_F(z`Skn$>F-7$$\"3ELLL&4Nn')*F*$\"3#f`!e 1ga_>F-7$$\"3=+++:?Pv)*F*$\"3oZicnW$H(>F-7$$\"3Imm\"zM)>$))*F*$\"3o$>6 'Rrk%*>F-7$$\"3#)*****pfa<*)*F*$\"3\"o%=PD5+A?F-7$$\"3gLLeg`!)**)*F*$ \"3G3o*3Yy70#F-7$$\"36++DG2A3**F*$\"3y_6m$R\\c3#F-7$$\"3pKLL)G[k\"**F* $\"3p+:0bh2B@F-7$$\"3s***\\7yh]#**F*$\"3Y.(eM6?k;#F-7$$\"3ommm)fdL$**F *$\"3ciS&*f4I7AF-7$$\"3(pmm,FT=%**F*$\"3KAs]5]`jAF-7$$\"3]LLe#pa-&**F* $\"33*4j/pt(=BF-7$$\"3_+++ad)z&**F*$\"3!y!4$\\FFNP#F-7$$\"37LL$GUYo'** F*$\"34;U%)>\"=6W#F-7$$\"3ymmm5:xu**F*$\"3L02*[Vpg]#F-7$$\"3y***\\sI@K )**F*$\"3'32PmeJ,e#F-7$$\"3=+++u]E()**F*$\"3#[XU@%=P " 0 "" {MPLTEXT 1 0 68 "evalf[12](findmin(sm,c[7]=0.97..0.99));\nconvert(op(1 ,%),rational,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-d%eB'>)*!#7$ \"-B5F<.>!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#a\"#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "48/4 9=48/49.;\n54/55=54/55.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"#[\"# \\$\"+n$=fz*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"#a\"#b$\"+====) *!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "evalf(eval(sqrt(sm),c[7]=48/49));\nevalf(eval(sqrt(sm ),c[7]=54/55));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lf'eP%!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MseiV!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "#------------------------ ----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Taking " }{XPPEDIT 18 0 "c[6] = 1/2;" "6 #/&%\"cG6#\"\"'*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 87 " makes the princip al error norm close to its minimum value with repect to the values " }{XPPEDIT 18 0 "c[4]=1/4" "6#/&%\"cG6#\"\"%*&\"\"\"F)F'!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 48/49;" "6#/&%\"cG6#\"\"(*&\"#[ \"\"\"\"#\\!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "e1 := \{c[4]=1/4,c[7]=48/49 ,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,` b*`[3]=0,`b*`[8]=-5/8\}:\neqns := subs(e1,cdns):\ne2 := solve(\{op(eqn s)\},indets(eqns) minus \{c[6]\}):\ne3 := `union`(e1,e2):\nerrterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsm := 0:\nfor ct to nops(err terms6_8) do\n sm := sm+(simplify(subs(e3,errterms6_8[ct])))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "plot(sm,c[6]=0.4..0.6,color=COLOR(RGB,.4,0,1),font=[ HELVETICA,9]);\nevalf[12](findmin(sm,c[6]=0.45..0.55));\nsqrt(op(2,%)) ;\nconvert(op(1,%%),rational,4);" }}{PARA 13 "" 1 "" {GLPLOT2D 493 293 293 {PLOTDATA 2 "6'-%'CURVESG6#7U7$$\"3A+++++++S!#=$\"3K(oO&3IF-7$$ \"3±zjf)4UF*$\"3i7Ifw/.$p#F-7$$\"3?LLe4;[\\UF*$\"3Vfhbf5??CF-7$$\"3 7++Dmy]!H%F*$\"3k!Rc/-8r:#F-7$$\"3>LLezs$HL%F*$\"3jT%fOwN_!>F-7$$\"3G+ +D@1BvVF*$\"3Or')>t[.u;F-7$$\"3!omm;_M(=WF*$\"3u*3Rb5)Hc9F-7$$\"3CLL$3 y_qX%F*$\"3]f79fe)3G\"F-7$$\"3T+++l+>+XF*$\"3Lb$3,6F-7$$\"3K+++vW] VXF*$\"3k;16(\\Q8R*!#E7$$\"3?+++NfC&e%F*$\"3Mz+`.&3y*zFio7$$\"3bLLez6: BYF*$\"3O$3'['p52(oFio7$$\"3)pmm;=C#oYF*$\"3'G*4%)HV%fp&Fio7$$\"3$pmmm #pS1ZF*$\"3bO^%Q0%[O[Fio7$$\"31++DOD#3v%F*$\"3GVvF6`O()RFio7$$\"3!pmmm (y8!z%F*$\"3_Gd]M:xlLFio7$$\"3#)***\\i.tK$[F*$\"3CcsR(f[%=GFio7$$\"3C+ +v3zMu[F*$\"3fK[\\PkrACFio7$$\"3!omm\"H_?<\\F*$\"3&H.F)\\YMM@Fio7$$\"3 )om;zihl&\\F*$\"3)RmFio7$$\"3ULL$3#G,**\\F*$\"3,*)3Xh([\\\">Fio 7$$\"3&HL$ezw5V]F*$\"3]*z>EKhY'>Fio7$$\"3M++v$Q#\\\"3&F*$\"3=3**f_p\"p 4#Fio7$$\"3+LL$e\"*[H7&F*$\"3$>=[+.FuK#Fio7$$\"3q******pvxl^F*$\"3[EB6 zBVbEFio7$$\"3N****\\_qn2_F*$\"3#R`'QiG&*eIFio7$$\"3G++Dcp@[_F*$\"33bj *e6/>_$Fio7$$\"3e****\\2'HKH&F*$\"3ck38fM_8TFio7$$\"33mmmwanL`F*$\"3h/ Qcb$=(4ZFio7$$\"3i+++v+'oP&F*$\"3O!e\"R\\z!)3aFio7$$\"3YLLeR<*fT&F*$\" 3)o[/g'HJ%4'Fio7$$\"3'******\\)HxeaF*$\"31M)oLFsp*oFio7$$\"3mmm\"H!o-* \\&F*$\"3(3n>$H]$3q(Fio7$$\"3K++DTO5TbF*$\"3Wou#R^B2f)Fio7$$\"3emmmT9C #e&F*$\"33m6C*z>6^*Fio7$$\"3Y***\\i!*3`i&F*$\"3)fus7^tK0\"F-7$$\"3;LLL $*zymcF*$\"3)f0Vpz%=e6F-7$$\"3]LL$3N1#4dF*$\"3S(yBn\"ept7F-7$$\"3?mm\" HYt7v&F*$\"3#Qzn_=u!*R\"F-7$$\"3=+++q(G**y&F*$\"30S))4@OJF:F-7$$\"3Vmm ;9@BMeF*$\"3YbMw^#*H'p\"F-7$$\"3gKLL`v&Q(eF*$\"3a\"erxV)ow=F-7$$\"3$)* **\\i`1h\"fF*$\"3$)y'>f(3b:@F-7$$\"3g******p`KOfF*$\"3Q\">c/z?UD#F-7$$ \"3[++v.UacfF*$\"3c*pOS\"HM9CF-7$$\"3e**\\(=5s#yfF*$\"3_L:rwS5!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!GodP%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\" \"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "e1 := \{c [4]=1/4,c[6]=1/2,c[7]=48/49,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8 ),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=-1/2\}:\neqns := subs(e1,c dns):\nnops(%);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Y" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[sol ve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1016 "e3 := \{b[5] = 27/112, a[8,6] = 1152/425, `b *`[5] = 69/320, c[3] = 2/15, a[7,5] = -671175/876544, c[9] = 1, `b*`[2 ] = 0, `b*`[3] = 0, b[6] = 27/136, b[3] = 0, `b*`[8] = 0, a[8,4] = 109 56/2675, a[4,3] = 3/20, `b*`[7] = 10304/43605, b[4] = 5/16, a[6,5] = 2 17/428, a[5,1] = 88/135, a[3,2] = 8/75, a[9,5] = 27/112, a[6,4] = -845 6/2889, a[5,4] = 64/27, `b*`[6] = 435/1904, a[8,2] = 0, a[5,3] = -112/ 45, `b*`[9] = -1/18, a[7,4] = 819261/383488, a[8,3] = -6783/2140, a[8, 7] = -7168/40375, `b*`[4] = 241/756, a[9,1] = 53/912, a[9,3] = 0, a[9, 2] = 0, b[1] = 53/912, `b*`[1] = 617/10944, b[8] = -25/336, a[9,7] = 2 56/969, a[8,5] = -38493/13375, b[7] = 256/969, c[5] = 8/15, c[6] = 2/3 , a[9,8] = -25/336, c[4] = 1/5, a[7,1] = 1718911/4382720, a[9,6] = 27/ 136, a[6,2] = 0, a[9,4] = 5/16, a[4,1] = 1/20, a[3,1] = 2/75, a[6,3] = 3880/963, b[2] = 0, a[5,2] = 0, c[2] = 1/12, a[7,3] = -1000749/547840 , c[7] = 19/20, a[2,1] = 1/12, a[4,2] = 0, a[7,2] = 0, a[8,1] = 85153/ 203300, a[7,6] = 14535/14336, c[8] = 1, a[6,1] = -10891/11556\}:" }}} {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 0 123 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(1 0-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"\"*F(%! GF+F+F+F+F+F+F+7,#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+F+7,#F)\"\"%#F)\"# ;\"\"!#\"\"$F7F+F+F+F+F+F+7,#\"\"&F*#\"$!G\"$H(F8#!$D$\"$V##\"%+6F@F+F +F+F+F+7,#F)\"\"##\"%Fh\"&!o9F8#!%x5\"$M(#\"%%\\'\"%PS#!%x%*\"'![h\"F+ F+F+F+7,#\"#[\"#\\#!,?LFEM\"\",pPx4[\"F8#\"+/(eD>%\"+n>o:@#\",W,vMV\"F en#\"-GBt#4<\"Fen#!*wh'>O\")2ONSF+F+F+7,F)#!(*oSB\"(g5!>F8#\"&Z;$\"&zN \"#\"*'f\\ND\"*p$=&\\\"#\",#3C!f0\"\"*0,ix*#!'_H:\"&t@\"#!(,[w&\"*'R5g =F+F+7,F)#\"$.#\"%!)GF8F8#\"&3-$\"&&yq#\"'Zr<\"'gX;#!$O&\"$0(#\"+VnKx> \"+gO#!$f#\"$?(F+7,%\"bGFdpF8F8FgpFjpF]qF`qFcqF+7,%#b*G#\"&nl$\"'+) e%F8F8#\"(%)f#**\")lM1F#\")nEQ&)\"*]*oz6#!'y.J\"'N'3)#\"-pmt>@E\"-glLz fM#!\"\"FH#!$,\"\"%%H#Q)pprint506\"" }}}{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 s cheme" }}{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 1153 "# coefficients by Dormand, Lockye r, McCorrigan and Prince\nee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/4,\nc[5 ]=5/9,\nc[6]=1/2,\nc[7]=48/49,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/9,\na[3,1 ]=1/24,\na[3,2]=1/8,\na[4,1]=1/16,\na[4,2]=0,\na[4,3]=3/16,\na[5,1]=28 0/729,\na[5,2]=0,\na[5,3]=-325/243,\na[5,4]=1100/729,\na[6,1]=6127/146 80,\na[6,2]=0,\na[6,3]=-1077/734,\na[6,4]=6494/4037,\na[6,5]=-9477/161 480,\na[7,1]=-13426273320/14809773769,\na[7,2]=0,\na[7,3]=4192558704/2 115681967,\na[7,4]=14334750144/14809773769,\na[7,5]=117092732328/14809 773769,\na[7,6]=-361966176/40353607,\na[8,1]=-2340689/1901060,\na[8,2] =0,\na[8,3]=31647/13579,\na[8,4]=253549596/149518369,\na[8,5]=10559024 082/977620105,\na[8,6]=-152952/12173,\na[8,7]=-5764801/186010396,\na[9 ,1]=203/2880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=30208/70785,\na[9,5]=17714 7/164560,\na[9,6]=-536/705,\na[9,7]=1977326743/3619661760,\na[9,8]=-25 9/720,\n\nb[1]=203/2880,\nb[2]=0,\nb[3]=0,\nb[4]=30208/70785,\nb[5]=17 7147/164560,\nb[6]=-536/705,\nb[7]=1977326743/3619661760,\nb[8]=-259/7 20,\n\n`b*`[1]=36567/458800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=9925984/ 27063465,\n`b*`[5]=85382667/117968950,\n`b*`[6]=-310378/808635,\n`b*`[ 7]=262119736669/345979336560,\n`b*`[8]=-1/2,\n`b*`[9]=-101/2294\}:" }} }{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[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error term s of the 8 stage, order 6 scheme (the error terms of order 7)." }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T *G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose components ar e the principal error terms of the embedded 9 stage, order 5 scheme (t he error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#& %#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components \+ are the error terms of order 7 of the embedded order 5 scheme." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote th e 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6# -%$absG6#-F$6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ab s(abs(`T*`[5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G 6$\"\"'\"\"*" }{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[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6# \"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\" \"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\" " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8 ]))/abs(abs(`T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$ \"\"'\"\"*\"\"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*` ,PrincipalErrorTerms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`, PrincipalErrorTerms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf( subs(ee,`errterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sq rt(add(evalf(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms 6_8[i])))^2,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n' C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG 6#\"\"($\")7]!z\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" ($\")=rs " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "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 1153 "# c oefficients by Dormand, Lockyer, McCorrigan and Prince\nee := \{c[2]=1 /9,\nc[3]=1/6,\nc[4]=1/4,\nc[5]=5/9,\nc[6]=1/2,\nc[7]=48/49,\nc[8]=1, \nc[9]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1/8,\na[4,1]=1/16,\na[4 ,2]=0,\na[4,3]=3/16,\na[5,1]=280/729,\na[5,2]=0,\na[5,3]=-325/243,\na[ 5,4]=1100/729,\na[6,1]=6127/14680,\na[6,2]=0,\na[6,3]=-1077/734,\na[6, 4]=6494/4037,\na[6,5]=-9477/161480,\na[7,1]=-13426273320/14809773769, \na[7,2]=0,\na[7,3]=4192558704/2115681967,\na[7,4]=14334750144/1480977 3769,\na[7,5]=117092732328/14809773769,\na[7,6]=-361966176/40353607,\n a[8,1]=-2340689/1901060,\na[8,2]=0,\na[8,3]=31647/13579,\na[8,4]=25354 9596/149518369,\na[8,5]=10559024082/977620105,\na[8,6]=-152952/12173, \na[8,7]=-5764801/186010396,\na[9,1]=203/2880,\na[9,2]=0,\na[9,3]=0,\n a[9,4]=30208/70785,\na[9,5]=177147/164560,\na[9,6]=-536/705,\na[9,7]=1 977326743/3619661760,\na[9,8]=-259/720,\n\nb[1]=203/2880,\nb[2]=0,\nb[ 3]=0,\nb[4]=30208/70785,\nb[5]=177147/164560,\nb[6]=-536/705,\nb[7]=19 77326743/3619661760,\nb[8]=-259/720,\n\n`b*`[1]=36567/458800,\n`b*`[2] =0,\n`b*`[3]=0,\n`b*`[4]=9925984/27063465,\n`b*`[5]=85382667/117968950 ,\n`b*`[6]=-310378/808635,\n`b*`[7]=262119736669/345979336560,\n`b*`[8 ]=-1/2,\n`b*`[9]=-101/2294\}:" }}}{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 77 "The stability function R for th e 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR : = unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$ F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$ ?(F)*$)F'F1F)F)F)*&#\"$t(\"(SyA%F)*$)F'\"\"(F)F)F)*&#\"#8\"'7PcF)*$)F' \"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability regi on intersects the negative real axis by solving the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG 6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #z0G$!+C[M1U!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-4):\np1 := plot([R(z) ,1],z=-4.79..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=po int,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z 0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3] ,view=[-4.79..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3/++++++!z%! #<$\"3S(4*fWu$[(GF*7$$!3n*****H^bCt%F*$\"3n\"o'**[TO.EF*7$$!3=+++E5\" \\n%F*$\"3szEL;F,bBF*7$$!3[++bv=%[i%F*$\"35*fX1\">Rc@F*7$$!3))****4DFx uXF*$\"35!ph[j/H(>F*7$$!3h++&)GZY=XF*$\"37-i[fXQ$y\"F*7$$!3Y++gKn:iWF* $\"3^)y_Ed!R5;F*7$$!3k++S!\\vaS%F*$\"377:Ab!p;X\"F*7$$!3%*****>[Uz[VF* $\"35>JLILA28F*7$$!3K++0-CQ#H%F*$\"392mSq4`w6F*7$$!3$)*****ebqfB%F*$\" 3QCmmwf%y0\"F*7$$!3U++q]&o88%F*$\"35!H'4>kWj')!#=7$$!3p****4L#fI-%F*$ \"30mdH@fwAqF^o7$$!3&)*****=eX5\"RF*$\"3Z'e)y)fJjj&F^o7$$!3')*****)f6R *z$F*$\"3A:%>(RA]=XF^o7$$!3/++!G'3a%o$F*$\"3I!Q)QDsS)f$F^o7$$!3K++ge1Q $e$F*$\"3en&o!)Rp$\\HF^o7$$!3+++SG#)\\pMF*$\"3Y*[>rmk#pBF^o7$$!33+++'= [^N$F*$\"3$fuj3+')*>>F^o7$$!3#*****fJ2&\\C$F*$\"3AP2pv5+\"f\"F^o7$$!3: ++!f)3)[9$F*$\"3kVcp3;Yl8F^o7$$!33++Sg\"))e-$F*$\"3s8*pNhj@<\"F^o7$$!3 p****f8d3DHF*$\"33.-uQ*481\"F^o7$$!3v****H/&Gy!GF*$\"3=^d7;bmF)*!#>7$$ !3N++g0g./FF*$\"3'))p2N^\"z6&*F[s7$$!3-++I%=f,f#F*$\"3r!>rnu\\/^*F[s7$ $!3))*****3_@<[#F*$\"3;ah^.%\\!)z*F[s7$$!3++++&>y&oBF*$\"3>(3:-]Qh.\"F ^o7$$!3)*****H-tnkAF*$\"3+E[K<(y#46F^o7$$!3C+++`hg_@F*$\"3/(>qwx$e47F^ o7$$!3')*****es&>O?F*$\"3/8Kt7^WO8F^o7$$!3*******o5g[$>F*$\"3qx?RwNfl9 F^o7$$!3r****>i[TD=F*$\"3>Z+w$p9^i\"F^o7$$!3L++?:sM7YAF^o7$$!3E++?OQ(eP\"F*$\"3l&o%ot`%y_#F^o7$$!3I++gTn4p7F*$\"3^UeJ B9x6GF^o7$$!3w*****>S*3b6F*$\"37+Gckiv]JF^o7$$!3G++]2=y^5F*$\"3E[R24GQ $\\$F^o7$$!3c&****f>$R)Q*F^o$\"3P[s^K)=4\"RF^o7$$!3N-++.CpD$)F^o$\"3+C )\\E:[$\\VF^o7$$!3#o****4(Q'[@(F^o$\"3wH%Qk%)3.'[F^o7$$!3D-++Sf#)GhF^o $\"3P%*3(RntyT&F^o7$$!3'******\\([%=*\\F^o$\"3sL*fW8c-2'F^o7$$!3'4++g( pz'*QF^o$\"3U)=F0$ztsnF^o7$$!3u-++QB&px#F^o$\"3?Rp5rlFvvF^o7$$!3/$**** *y0Qm;F^o$\"3/8***3Uf]Y)F^o7$$!3%)******>Z!)ekF[s$\"3_3Hm/d`u$*F^o7$$ \"3:!*****R\"ysB&F[s$\"3!e:)f%\\oP0\"F*7$$\"3++++3%R)p:F^o$\"3m;3#[#o( *p6F*7$$\"3;0++dD@&o#F^o$\"3Y!)o;y(G!38F*7$$\"3+/++zpw_PF^o$\"3eo\\&f/ %Rb9F*7$$\"3!***************[F^o$\"3\"\\z2Q5;Bj\"F*-%'COLOURG6&%$RGBG$ \"*++++\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$ FGF_]l7$FQF_]l7$FenF_]l7$FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$ FdpF_]l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FgrF_]l 7$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$Fh wF_]l7$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$$!3K+++C[M1UF*F_]l-%'SYMBOLG6#%'CIRC LEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CR OSSGF]alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-% &COLORG6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6 %Q\"z6\"Q!F_cl-%%FONTG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$ !$z%!\"#$\"#\\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 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1304 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+773 /4227840*z^7+13/563712*z^8:\npts := []:\nz0 := 0:\nfor ct from 0 to 24 0 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pt s := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(R GB,.33,.13,.5)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0 ]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.65, .25,1)):\npts := []: z0 := 2+4.8*I:\nfor ct from 0 to 60 do\n zz := \+ newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.33,.13,.5)) :\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.89,4.82]],i=2..no ps(pts))],\n style=patchnogrid,color=COLOR(RGB,.65,.25,1)):\n pts := []: z0 := 2-4.8*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 := 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"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 "[-4.2063, 0];" "6#7$, $-%&FloatG6$\"&j?%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the bound ary curve horizontally by taking the 11th root of the real part of poi nts along the curve. In this way we see that there is " }{TEXT 260 53 "no largest interval on the nonnegative imaginary axis" }{TEXT -1 64 " that contains the origin and lies inside the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 119 "However the stability region intersects \+ the nonnegative imaginary axis in an interval that does not contain th e origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "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+773/4227840*z^7+13/563712*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 120 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,.65,.13,1),thickness=2,font=[HELVETICA, 9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7er7$$\"\"!F)F(7$$!:'y*Q3;IIAWC$>H!#E$\":b# Qv@Y(*e`EfTJF-7$$!:v]5a0ZwM@lD$[F-$\":Ta'z'ydtrI&=$G'F-7$$!:)**)\\l\"* f\"RK`5*['F-$\":Y9#oN.qmgzxC%*F-7$$!:,Y?ZpU*Q1Cs(*zF-$\":R_)RO@&f81PmD \"!#D7$$!:%=>/Pxp3\"o\"e/%*F-$\":,@*)R^WKkK'zq:F?7$$!:)RBF$pS_>#G[t5F? $\":Vj:'>YJ'3fb\\)=F?7$$!:/5k2'p:k(=3/?\"F?$\":\"o$pN9OVP&[6*>#F?7$$!: ?;5S.&[0L\")GA8F?$\":Wg!z'*HQN8TF8DF?7$$!:/9b([9X%fTv)R9F?$\":h`NWcRJn OLu#GF?7$$!:AyukX!['HiIPb\"F?$\":_gA*>qN74EfTJF?7$$!:\"o%*[>?-S8ZGk;F? 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=0)>r%4x'4=;rF?$\":m%>II$ep@n#fDOFfu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HE LVETICAG\"\"*-%&COLORG6&%$RGBG$\"#l!\"#$\"#8F^bm$\"\"\"F)-%+AXESLABELS G6$Q!6\"Ffbm-%%VIEWG6$%(DEFAULTGF[cm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 108 "The relevant intersection points of the \+ boundary curve with the imaginary axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundar y curve either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "Digits := 1 5:\nz0 := 2.5*I:\nfor ct from 77 to 80 do\n newton(R(z)=exp(ct*Pi/10 0*I),z=z0);\nend do;``;\nz0 := 3.4*I:\nfor ct from 109 to 112 do\n n ewton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0MIZ30\\h#!#=$\"0-D8)f(*=C!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0$*=9)RjZ7!#=$\"0Wj%o7F]C!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0dBP:2;S$!#>$\"0N'4]I`\"[#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0uLw$>fe@!#=$\"0]G:&ev7D!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\" 05gxOYz6#!#<$\"0kiQ@P3O$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0% f*4MAtf)!#=$\"0$48r$flQ$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0b= Q_65&f!#=$\"0KAoSj>T$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0gMIl# HbA!#<$\"0-QY`WqV$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to calculate the parameter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "Digits := 15:\nreal_part := proc(u )\n Re(newton(R(z)=exp(u*Pi*I),z=2.5*I))\nend proc:\nu0 := bisect('r eal_part'(u),u=0.77..0.8);\nnewton(R(z)=exp(u0*Pi*I),z=2.5*I);``;\nrea l_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.4*I))\nend proc: \nu0 := bisect('real_part'(u),u=1.09..1.12);\nnewton(R(z)=exp(u0*Pi*I) ,z=3.4*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\" 0!z6%[J(zy!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"09VU,M=$H!#H$\"0 B^`r*>vC!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0\"=#zdug5\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-ri'e]bW!#H$\"04iC\\H?S$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stabi lity region intersects the nonegative imaginary axis in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[2.4752, 3.4020];" "6#7$-%&FloatG6$ \"&_Z#!\"%-F%6$\"&?S$F(" }{TEXT -1 18 " (approximately)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#----------------- -------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 80 "The stability function R* for the 9 stage, order 5 sc heme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := una pply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# R*G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\" $F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\")6 2mB\",!ojE(p\"F)*$)F'F1F)F)F)*&#\"(zj@$FDF)*$)F'\"\"(F)F)F)*&#\"(*oH; \",?Zl!*y'F)*$)F'\"\")F)F)F)*&#\"%88\"+G`:$H\"F)*$)F'\"\"*F)F)!\"\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects \+ the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z_0 := ne wton(`R*`(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+lFI XS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-4):\np_1 := plot([`R*`(z) ,1],z=-4.59..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style= point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_0,0 ],[z_0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1, p_2,p_3],view=[-4.59..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3')* ************e%!#<$\"37'*G*\\`F*7$$!3n*\\(yb,vGVF*$\"3_[#3Z%)>:w\"F*7$$!3SLL=Q] duUF*$\"3_!)R'e)HL%e\"F*7$$!34LLt73/?UF*$\"3>'3t$*4UCU\"F*7$$!3yKLG(e1 b;%F*$\"3gg-*\\,TdF\"F*7$$!3Z*\\([`:B6TF*$\"3RK49YceV6F*7$$!3/n;p>l&p0 %F*$\"3g/zqt[5C5F*7$$!3KL$e;r;j&RF*$\"3Fo0AXafD$)!#=7$$!3++]s>+6_QF*$ \"3WG7)Qeb#*p'Fho7$$!3IL$e)4$RVu$F*$\"3\"o(f\\.3UQ`Fho7$$!39+]-AU\"pj$ F*$\"3k%y4.,LF*$\"3\"41e5HlA@#Fho7$$!3(*****\\L' )\\4KF*$\"3I*HBVgq$*z\"Fho7$$!3z****4D`Z.JF*$\"3-l\"z_xkK]\"Fho7$$!3>L $eQ+'>2IF*$\"3-,c<:9Q08Fho7$$!3ommcy0r#*GF*$\"3R)f$HuM1U6Fho7$$!3ommE1 ks&z#F*$\"3i*[k))HST0\"Fho7$$!3y**\\#zv5Ho#F*$\"3sk*R-)eg&***!#>7$$!3] mmE$z\\Ie#F*$\"31c$yKUw%p)*Fes7$$!3#)**\\#z['[tCF*$\"3?(Rbl#R[05Fho7$$ !3')**\\x6j:pBF*$\"3gVP\"=\\8-0\"Fho7$$!3Emm\"zr)HgAF*$\"3PY0al7j@6Fho 7$$!37m;4lMLg@F*$\"31$RA?!yX27Fho7$$!3ALL3Nu]_?F*$\"30qIA')*[/K\"Fho7$ $!3SL$eQ\\10%>F*$\"3o%G*e'4Q'f9Fho7$$!3/+]FX$4I%=F*$\"3G\"QFh o7$$!39++l'G+D_\"F*$\"3aXR:B]D&=#Fho7$$!3*)**\\7J*G&>9F*$\"3UinY)>1/U# Fho7$$!3z***\\pz'>08F*$\"3c&G;LxLBr#Fho7$$!3/nmE*GkC?\"F*$\"3yYs@dk:0I Fho7$$!3u****\\4ax#4\"F*$\"3)=2VhkhJN$Fho7$$!3BMLe9)4Q$**Fho$\"3s_bY4a Q.PFho7$$!3'o****4=mr%))Fho$\"3jys4xFQGTFho7$$!3Oqm\"f?>Z#yFho$\"3QG8( 3KtFd%Fho7$$!3E'**\\A^nfv'Fho$\"3_W4(H0Q&)3&Fho7$$!3%zmm;Qn5r&Fho$\"3c w@$Gm-!\\cFho7$$!3J'**\\7y`rh%Fho$\"3ot5u_x,-jFho7$$!3!)GLLp\\ejNFho$ \"3X;`d^c@-qFho7$$!39LL$))oeh[#Fho$\"3o\\3xN\\z)z(Fho7$$!3yim\">/awT\" Fho$\"31]rC2#[#y')Fho7$$!3Mf****>C4eVFes$\"3-)3Cwx]Nd*Fho7$$\"3wMLL)*p &\\*oFes$\"3%yNlz<#Qr5F*7$$\"31mmma=)fp\"Fho$\"3OSVOu'G[=\"F*7$$\"370+ v?g5pFFho$\"3L$H>nW[!>8F*7$$\"3g1+DvFA'z$Fho$\"3'[5A.OKA;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$FdoFi]l7$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$FasFi]l7$FgsFi]l7$F\\tFi ]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$Fdv Fi]l7$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$F`zFi]l7$FezFi]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]lF c]lF`]l-F$6&7#7$$!3u*****\\w-`/%F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]l Fd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_ al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\" \"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6 #%(DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$f%!\"#$\"#\\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 "T he following picture shows the stability region for the 7 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1398 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z ^5+\n 23660711/16972663680*z^6+3216379/16972663680*z^7+\n \+ 1629689/67890654720*z^8-1313/1293155328*z^9:\npts := []: z0 := 0: \nfor ct from 0 to 180 do\n zz := newton(`R*`(z)=exp(ct*Pi/15*I),z=z 0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 \+ := plot(pts,color=COLOR(RGB,.21,.1,.38)):\np_2 := plots[polygonplot]([ seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.45,.2,.75)):\npts := []: z0 := 2+4.6*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z 0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(p ts,color=COLOR(RGB,.21,.1,.38)):\np_4 := plots[polygonplot]([seq([pts[ i-1],pts[i],[1.91,4.58]],i=2..nops(pts))],\n style=patchnogri d,color=COLOR(RGB,.45,.2,.75)):\npts := []: z0 := 2-4.6*I:\nfor ct fro m 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,c olor=COLOR(RGB,.21,.1,.38)):\np_6 := plots[polygonplot]([seq([pts[i-1] ,pts[i],[1.91,-4.58]],i=2..nops(pts))],\n style=patchnogrid,c olor=COLOR(RGB,.45,.2,.75)):\np_7 := plot([[[-4.69,0],[2.19,0]],[[0,-5 .19],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)] ,view=[-4.69..2.19,-5.19..5.19],font=[HELVETICA,9],\n label s=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7av7$$\"\"!F)F(7$$ \"3++++qbWmR!#F$\"3%********4&R%4#!#=7$$\"3')*****f.)o^=!#D$\"3!****** z\"**y)=%F07$$\"3*******Rq0k,\"!#C$\"3_+++@5=$G'F07$$!3q*****HhG#*)eF4 $\"3I+++0Rbx$)F07$$!3%)*****H<3M)>!#B$\"3%*******>x=Z5!#<7$$!3!******R Z\"e4sFE$\"3%*******f4hc7FH7$$!3!)*****za+(fuFE$\"33+++\"p>gY\"FH7$$\" 3u*****>tr>C%!#A$\"33+++&R%Rv;FH7$$\"3C+++xm^8F!#@$\"3#******z#)\\Y)=F H7$$\"33+++&45ER*Ffn$\"3-+++:I]$4#FH7$$\"3&******HpXvT#!#?$\"3$******* 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]bl7$F]]`l$!+_,T%)[F_\\nFheal7%F\\^bl7$Fg\\`l$!+V)[A*[F_\\nFheal7%F`^b l7$Fa\\`l$!+%>'>**[F_\\nFheal7%Fd^bl7$F[\\`l$!+(HN_!\\F_\\nFheal7%Fh^b l7$Fe[`l$!+Q7O5\\F_\\nFheal7%F\\_bl7$F_[`l$!+C%yX\"\\F_\\nFheal7%F`_bl 7$Fij_l$!+J%)*y\"\\F_\\nFheal7%Fd_bl7$Fcj_l$!++$Q.#\\F_\\nFheal7%Fh_bl 7$F]j_l$!+?#>>#\\F_\\nFheal7%F\\`bl7$Fgi_l$!+>bmA\\F_\\nFheal7%F``bl7$ Fai_l$!+8RgA\\F_\\nFheal7%Fd`bl7$F[i_l$!+YHw@\\F_\\nFheal7%Fh`bl7$Feh_ l$!++E&FH7$F($\"3Q++++++!>&FHF]eb lF`ebl-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#% (DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fgfbl6$%*HELVETICAG\"\"*-%%VIEWG6 $;$!$z%Fcim$\"$>#Fcim;$!$>&Fcim$\"$>&Fcim" 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 35 "coefficients of the combined sche me" }}{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 1153 "# coefficients by Dormand, Lockye r, McCorrigan and Prince\nee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/4,\nc[5 ]=5/9,\nc[6]=1/2,\nc[7]=48/49,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/9,\na[3,1 ]=1/24,\na[3,2]=1/8,\na[4,1]=1/16,\na[4,2]=0,\na[4,3]=3/16,\na[5,1]=28 0/729,\na[5,2]=0,\na[5,3]=-325/243,\na[5,4]=1100/729,\na[6,1]=6127/146 80,\na[6,2]=0,\na[6,3]=-1077/734,\na[6,4]=6494/4037,\na[6,5]=-9477/161 480,\na[7,1]=-13426273320/14809773769,\na[7,2]=0,\na[7,3]=4192558704/2 115681967,\na[7,4]=14334750144/14809773769,\na[7,5]=117092732328/14809 773769,\na[7,6]=-361966176/40353607,\na[8,1]=-2340689/1901060,\na[8,2] =0,\na[8,3]=31647/13579,\na[8,4]=253549596/149518369,\na[8,5]=10559024 082/977620105,\na[8,6]=-152952/12173,\na[8,7]=-5764801/186010396,\na[9 ,1]=203/2880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=30208/70785,\na[9,5]=17714 7/164560,\na[9,6]=-536/705,\na[9,7]=1977326743/3619661760,\na[9,8]=-25 9/720,\n\nb[1]=203/2880,\nb[2]=0,\nb[3]=0,\nb[4]=30208/70785,\nb[5]=17 7147/164560,\nb[6]=-536/705,\nb[7]=1977326743/3619661760,\nb[8]=-259/7 20,\n\n`b*`[1]=36567/458800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=9925984/ 27063465,\n`b*`[5]=85382667/117968950,\n`b*`[6]=-310378/808635,\n`b*`[ 7]=262119736669/345979336560,\n`b*`[8]=-1/2,\n`b*`[9]=-101/2294\}:" }} }{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 31 "seq(c[i]=subs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/& %\"cG6#\"\"##\"\"\"\"\"*/&F%6#\"\"$#F)\"\"'/&F%6#\"\"%#F)F4/&F%6#\"\"& #F9F*/&F%6#F0#F)F'/&F%6#\"\"(#\"#[\"#\\/&F%6#\"\")F)/&F%6#F*F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking \+ coefficients" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#F(\"\"*/&F%6$\"\"$F(#F(\"#C /&F%6$F.F'#F(\"\")/&F%6$\"\"%F(#F(\"#;/&F%6$F9F'\"\"!/&F%6$F9F.#F.F;/& F%6$\"\"&F(#\"$!G\"$H(/&F%6$FGF'F?/&F%6$FGF.#!$D$\"$V#/&F%6$FGF9#\"%+6 FJ/&F%6$\"\"'F(#\"%Fh\"&!o9/&F%6$FfnF'F?/&F%6$FfnF.#!%x5\"$M(/&F%6$Ffn F9#\"%%\\'\"%PS/&F%6$FfnFG#!%x%*\"'![h\"/&F%6$\"\"(F(#!,?LFEM\"\",pPx4 [\"/&F%6$FbpF'F?/&F%6$FbpF.#\"+/(eD>%\"+n>o:@/&F%6$FbpF9#\",W,vMV\"Fep /&F%6$FbpFG#\"-GBt#4<\"Fep/&F%6$FbpFfn#!*wh'>O\")2ONS/&F%6$F5F(#!(*oSB \"(g5!>/&F%6$F5F'F?/&F%6$F5F.#\"&Z;$\"&zN\"/&F%6$F5F9#\"*'f\\ND\"*p$=& \\\"/&F%6$F5FG#\",#3C!f0\"\"*0,ix*/&F%6$F5Ffn#!'_H:\"&t@\"/&F%6$F5Fbp# !(,[w&\"*'R5g=/&F%6$F*F(#\"$.#\"%!)G/&F%6$F*F'F?/&F%6$F*F.F?/&F%6$F*F9 #\"&3-$\"&&yq/&F%6$F*FG#\"'Zr<\"'gX;/&F%6$F*Ffn#!$O&\"$0(/&F%6$F*Fbp# \"+VnKx>\"+gO/&F%6$F*F5#!$f#\"$?(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 6 scheme" } {TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=s ubs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\" \"#\"$.#\"%!)G/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"&3-$\"&&yq/&F %6#\"\"&#\"'Zr<\"'gX;/&F%6#\"\"'#!$O&\"$0(/&F%6#\"\"(#\"+VnKx>\"+gO /&F%6#\"\")#!$f#\"$?(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]) ,i=1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"&nl$\"' +)e%/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"(%)f#**\")lM1F/&F%6#\" \"&#\")nEQ&)\"*]*oz6/&F%6#\"\"'#!'y.J\"'N'3)/&F%6#\"\"(#\"-pmt>@E\"-gl LzfM/&F%6#\"\")#!\"\"F./&F%6#\"\"*#!$,\"\"%%H#" }}}{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 47 "#==============================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[ 4]=6/25" "6#/&%\"cG6#\"\"%*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]=63/64" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combi ned scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficie nts of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2060 "ee := \{c[2]=8/75,\nc[3]=4/25,\nc [4]=6/25,\nc[5]=1807/3162,\nc[6]=11/21,\nc[7]=63/64,\nc[8]=1,\nc[9]=1, \n\na[2,1]=8/75,\na[3,1]=1/25,\na[3,2]=3/25,\na[4,1]=3/50,\na[4,2]=0, \na[4,3]=9/50,\na[5,1]=1116733125001/2276240222016,\na[5,2]=0,\na[5,3] =-1364629188325/758746740672,\na[5,4]=2138982988675/1138120111008,\na[ 6,1]=373184883131711592359/629428679970118345044,\na[6,2]=0,\na[6,3]=- 128555397896225575/58054665188168082,\na[6,4]=20224502633443247773975/ 9127238351553409515876,\na[6,5]=-3590778379490959900760/50904073152027 811713543,\na[7,1]=-9955638050099787690037550073/717061083125790391982 1488128,\na[7,2]=0,\na[7,3]=421770368617928768410875/90187287207047138 902016,\na[7,4]=-3031979901406817815170871063875/140845378206500868366 7117047808,\na[7,5]=15692549351917507935540245739734013/22525630564781 89244473772594954240,\na[7,6]=-14873540135675332293/208966554719617024 0,\na[8,1]=-2691778286453656380245383/1521258163484980574735172,\na[8, 2]=0,\na[8,3]=2141335633391996675/371194659287217497,\na[8,4]=-1616792 33545622750791763569225/62137427083607735703094426884,\na[8,5]=1363639 349516531201308323118018680/154935659262427604083414890417653,\na[8,6] =-444161905271490/48398415269957,\na[8,7]=-311127404078694400/14477515 022855683323,\na[9,1]=31138111/450810360,\na[9,2]=0,\na[9,3]=0,\na[9,4 ]=2474072265625/6361142786136,\na[9,5]=13881573506953684872/1413933451 4114522375,\na[9,6]=-132955185321/214068151000,\na[9,7]=19335942766592 /29106652419495,\na[9,8]=-2486977/5149000,\n\nb[1]=31138111/450810360, \nb[2]=0,\nb[3]=0,\nb[4]=2474072265625/6361142786136,\nb[5]=1388157350 6953684872/14139334514114522375,\nb[6]=-132955185321/214068151000,\nb[ 7]=19335942766592/29106652419495,\nb[8]=-2486977/5149000,\n\n`b*`[1]=2 4359367024535936274621/339039904141177334687592,\n`b*`[2]=0,\n`b*`[3]= 0,\n`b*`[4]=470295508478676388272578125/1258949982543885938072825784, \n`b*`[5]=6740104862258656670972654206056/7847776077910531141899035067 695,\n`b*`[6]=-15929717413567933460447487/32198747781536819606222440, \n`b*`[7] = 16516196509757640754673483776/2189017272401670657866526198 9,\n`b*`[8]=-6/11,\n`b*`[9]=-559011373156103/30082707428567288\}:" }}} {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 0 588 "subs(ee,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3 ,1],a[3,2],``],\n [c[4],a[4,1],a[4,2],a[4,3]],\n [c[5],a[5,1 ],a[5,2],a[5,3]],[``$3,a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3]],[`` $2,a[6,4],a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5 ],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6] ],[``$3,a[8,7]],\n [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 [`b`,b[1],b[2],b[3]],[``,b[4],b[5 ],b[6]],[``$2,b[7],b[8]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3]],[``,`b *`[4],`b*`[5],`b*`[6]],[``,`b*`[7],`b*`[8],`b*`[9]]]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#777&#\"\")\"#vF(%!GF+7&#\"\"%\"#D#\" \"\"F/#\"\"$F/F+7&#\"\"'F/#F3\"#]\"\"!#\"\"*F87&#\"%2=\"%iJ#\".,]7Ln6 \"\".;?ASiF#F9#!.D$)=HYO\"\"-s1uY(e(7&F+F+F+#\".v'))H)*Q@\".356?\"Q67& #\"#6\"#@#\"6fBf6<8$)[=t$\"6W]M=,(*z'G%H'F9#!3vbA'*yRb&G\"\"2#3o\")=lY 0e7&F+F+#\"8vRxZKWLE]C-#\"7we^4Mb^$Qs7*#!7g2!*f4\\z$y2f$\"8VNr6y-_J2/4 &7&#\"#j\"#k#!=t+bP+p(y*4]!Qc&**\"=G\")[@)>R!zDJ3hqrF9#\"9v3To(Gzho.x@ %\"8;?!*Qr/2sG(=!*7&F+#!@vQ1r3<:y\"oS,*z>.$\"@3y/N\\Dp:\"CSU&\\fsPZW#*=yk0jDD##!5$HK`nN,at[\"\"4S-<'>Zbm*3#7& F1#!:$QX-QcOX'GyRLcLT@\"3(\\r$7 &F+#!?D#pNw\"z]FiXNBz;;\">%)oU%4.dt2O3Fu8i#\"C!o=!=JK38?Jl^\\$ROO\"\"B `wT!*[T$3/wUi#fc$\\:#!0!\\r_!>;W%\"/d*p_T)R[7&F+F+F+#!3+WpySSF6J\"5BLo bG-:vZ97&F1#\")6\"Q6$\"*g.\"3XF9F97&F+#\".DcEsSZ#\".OhyU6O'#\"5s[o`p]t :)Q\"\"5vB_9T^M$RT\"#!-@`=bH8\"-+5:oS@7&F+F+#\"/#fmF%fL>\"/&\\>Cl1\"H# !(xp[#\"(+!\\^7&%\"bGF_qF9F9FbqF\\r7&%#b*G#\"8@YFOf`CqOfV#\"9#f(oMto`\"yZ()>K7&F+ #\">wP[tYvSwv4l>;l\"\">*)>El'ylq;Sss,*=##!\"'FL#!0.h:t8,f&\"2)Gn&Guq#3 IQ(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')) ,op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,Ord erConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{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 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderConditions(c t,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8 [j])),i=2..5)],j=1..6)]:\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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can calculate th e 2 norm of the principal error, that is, the 2-norm of the principal \+ error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(sub s(ee,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+P[B')G!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the \+ order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5, 9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[i])^2 ,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+WcOu7!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "18 of the principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops(errterm s6_8) do\n if simplify(subs(ee,errterms6_8[ct]))=0 then ind := [op(i nd),ct] end if:\nend do:\nerr6 := PrincipalErrorConditions(6):\n[seq([ i,err6[i]],i=ind)]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[ ` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747%\"\"'%#~~G/*&%\"bG\"\"\"-%!G6#*(% \"aGF--F/6#*&F2F--F/6#*&F2F-%\"cGF-F-F-F6F-F-#F-\"$/&7%\"#5F)/*&F,F-)F 3\"\"#F-#F-\"$_#7%\"#:F)/**F,F-F9F-F6F-F3F-#F-\"#%)7%\"#?F)/*&F,F--F/6 #*(F2F-F6F--F/6#*&)F9FAF-F2F-F-F-FB7%\"#@F)/*&F,F--F/6#*(F2F-FTF-F3F-F -FB7%\"#CF)/*(F,F-F3F-FQF-#F-\"$E\"7%\"#EF)/*&F,F--F/6#*(F2F-F9F-)F6FA F-F-#F-\"$o\"7%\"#GF)/*&F,F-)F6\"\"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F -FTF-FQF-F-Fjn7%\"#MF)/**F,F-F9F-F6F-FQF-#F-\"#U7%\"#PF)/*&F,F--F/6#*( F2F-)F9F[pF-F6F-F-FH7%\"#RF)/*(F,F-FbqF-F3F-Fip7%\"#TF)/*&F,F-)FQFAF-# F-\"#j7%FjpF)/*(F,F-FTF-FcoF-#F-Fgo7%\"#WF)/*(F,F-FbqF-FQF-#F-FV7%\"#Y F)/*(F,F-)F9\"\"%F-F6F-#F-\"#97%\"#ZF)/*&F,F--F/6#*&)F9\"\"&F-F2F-F-Fi p7%\"#[F)/*&F,F-)F9F(F-#F-\"\"(Q(pprint66\"" }}}{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 44 "simultaneous construction of \+ the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equations to ensure that \+ stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }} {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 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\" F," }{TEXT -1 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 5 " ), " }{XPPEDIT 18 0 "j = \+ 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=7" "6#/%\" jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 9 simple order conditions used are given (in abr eviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 97 "These inclu de the 6 order 6 quadrature conditions, 2 additional order 6 order con ditions and the " }{TEXT 260 28 "order 7 quadrature condition" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]],i= [1,2,4,8,16,20,29,32,64])]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(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%\"#?F)/*(F,F (F2F(-%!G6#*&%\"aGF(-FM6#*&FPF(-FM6#*&FPF(F2F(F(F(F(#F(\"$W\"7%\"#HF)/ *(F,F(F2F(-FM6#*&F?F(FPF(F(#F(\"#C7%\"#KF)/*&F,F()F2FGF(#F(\"\"'7%\"#k F)/*&F,F()F2FboF(#F(\"\"(Q)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we \+ use a selection of 7 \"simple\" order conditions as given (in abreviat ed form) in the following table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "Thes e include the 5 quadrature conditions and two additional order conditi ons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "`SO5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[s eq([i,`SO5*`[i]],i=[1,2,4,8,9,15,16])]:\nlinalg[augment](linalg[delcol s](%,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(-%!G6#*&%\"aGF(-FF6#* &FIF(-FF6#*&FIF(F2F(F(F(F(#F(\"$?\"7%\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F( \"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO7_8 := Si mpleOrderConditions(7,8,'expanded'):\nSO_eqs := [op(RowSumConditions(8 ,'expanded')),op(StageOrderConditions(2,8,'expanded')),\n \+ op(StageOrderConditions(3,3..8,'expanded'))]:\n`SO5_9*` := subs(b=`b *`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns := [seq(SO7_8[i], i=[1,2,4,8,16,20,29,32,64])]:\n`ord_cdns*` := [seq(`SO5_9*`[i],i=[1,2, 4,8,9,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..8)=b[1],seq(add(b[i ]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns := [seq(b[i]=a[9, i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO_eqs),op(wt_eqns ),op(`ord_cdns*`)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We speci fy the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4 ] = 6/25;" "6#/&%\"cG6#\"\"%*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 11/21;" "6#/&%\"cG6#\"\"'*&\"#6\"\"\"\"#@!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 63/64;" "6#/&%\"cG6#\"\"(*& \"#j\"\"\"\"#k!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&% \"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and 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 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provide the linking co efficients for the 9th stage of the embedded order 5 scheme so that: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/ &%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" " 6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"* \"\"\"" }{TEXT -1 3 ", " }{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*`[2] = 0;" "6#/&%#b*G6#\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$ \"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8] = -6/11;" "6#/&%# b*G6#\"\"),$*&\"\"'\"\"\"\"#6!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We have 45 equations a nd 45 unknowns including " }{TEXT 260 18 "three of the nodes" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 93 " was taken to mak e the principal error norm a minimum with respect to the chosen values for " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "preliminary calculation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "e 1 := \{c[4]=6/25,c[7]=63/64,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8 ),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=-6/11\}:\neqns := subs(e1, cdns):\ne2 := solve(\{op(eqns)\},indets(eqns) minus \{c[6]\}):\ne3 := \+ `union`(e1,e2):\nerrterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n sm := 0:\nfor ct to nops(errterms6_8) do\n sm := sm+(simplify(subs(e 3,errterms6_8[ct])))^2;\nend do:\nplot(sm,c[6]=0.5..0.6,color=COLOR(RG B,.4,0,1),font=[HELVETICA,9]);\nevalf[12](findmin(sm,c[6]=0.52..0.54)) ;\nsqrt(op(2,%));\nconvert(op(1,%%),rational,5);" }}{PARA 13 "" 1 "" {GLPLOT2D 444 246 246 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"3++++++++]!#=$ \"3_O(fgyN&)3\"!#D7$$\"3nT*)!#E7$$\"3OL$e9ui2/&F *$\"3))y,#Gzt%\\uF37$$\"3smm\"z_\"4i]F*$\"3eUZTzkq!)fF37$$\"3CmmT&phN3 &F*$\"3kcK,'HrRr%F37$$\"3kL$e*=)H\\5&F*$\"3'*H>'fj3Al$F37$$\"3\\m;z/3u C^F*$\"3Ju9w[75NGF37$$\"3C+]7LRDX^F*$\"3aHAXda-\\@F37$$\"3/n;zR'ok;&F* $\"3q,ib5Q#3g\"F37$$\"3-+]i5`h(=&F*$\"3%*Q#=P'4!y?\"F37$$\"3,LL$3En$4_ F*$\"3`t8?@dsK&*!#F7$$\"3ymmT!RE&G_F*$\"3%yY3'yepp%)F[o7$$\"3a****\\K] 4]_F*$\"3\"*R7g:153&)F[o7$$\"3K++]PAvr_F*$\"3S**o\\%y[Wx*F[o7$$\"3E++] nHi#H&F*$\"3s#RP&Q)[v?\"F37$$\"36m;z*ev:J&F*$\"3$R[]>Z2=]\"F37$$\"3#GL L347TL&F*$\"3]$=ynf]$\\>F37$$\"3kLLLjM?``F*$\"3d;Q\\:A-0CF37$$\"3#*** \\7o7Tv`F*$\"3ks)o$\\dy:IF37$$\"3yKLLQ*o]R&F*$\"3dR*3^(GZAOF37$$\"3!)* *\\7=lj;aF*$\"3AUs#fV&\\_VF37$$\"3t**\\PaREw9nG%RfF37$$\"3wK$eR\"3GyaF*$\"3!o\"[!f!=\\ZnF37$$\"3 'om;/T1&*\\&F*$\"3ABkkTwmcwF37$$\"3Om;zRQb@bF*$\"3$*)He6`ccj)F37$$\"31 +](=>Y2a&F*$\"3b3$p/)R=6&*F37$$\"3Smm\"zXu9c&F*$\"3ocss4\\cZ5F-7$$\"3u *****\\y))Ge&F*$\"3q+*GXYx'[6F-7$$\"37++DE&QQg&F*$\"3%pF2_Qk%[7F-7$$\" 3[**\\7y%3Ti&F*$\"3I#*yCEiHX8F-7$$\"3o***\\P![hYcF*$\"39!4.*RC__9F-7$$ \"3[LLLQx$om&F*$\"31z0aUd2[:F-7$$\"3m****\\P+V)o&F*$\"3M\")G))QAr[;F-7 $$\"3im;zpe*zq&F*$\"30s2X7e?QF-7$$\"3]**\\i?=bqdF*$\"32sL#yyA)3? F-7$$\"3uLL$3s?6z&F*$\"3_TmHMKB\"4#F-7$$\"3=+]7`Wl7eF*$\"3][h>B'pL<#F- 7$$\"3.nmm'*RRLeF*$\"3w\"zs-9$=[AF-7$$\"33mmTvJgaeF*$\"3&H/6$Gz7?BF-7$ $\"3aL$e9tOc(eF*$\"3o*\\**HGSoQ#F-7$$\"3'******\\Qk\\*eF*$\"3:$3%[AJ9W CF-7$$\"3mLL3dg6P8zYGr#F--%%FONTG6$%*HELVETICAG\"\"*-% +AXESLABELSG6$Q%c[6]6\"Q!Fc[l-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"!F]\\l$ \"\"\"F]\\l-%%VIEWG6$;$\"\"&F[\\l$\"\"'F[\\l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-VI@fQ_!#7$\"-)yu9+L)!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$\\wh)G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#6 \"#@" }}}{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 0 186 "e1 := \{c[4]=6/25,c[6]=11/21,c[7]=63/64,c[ 8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*` [3]=0,`b*`[8]=-6/11\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Y" }}{PARA 12 "" 1 " " {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[solve] := 0:\ne3 := `union`( e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2177 " e3 := \{`b*`[1] = 24359367024535936274621/339039904141177334687592, a[ 8,2] = 0, a[7,2] = 0, a[5,2] = 0, a[6,2] = 0, a[6,3] = -12855539789622 5575/58054665188168082, a[4,2] = 0, c[8] = 1, c[9] = 1, a[9,5] = 13881 573506953684872/14139334514114522375, c[4] = 6/25, c[7] = 63/64, c[5] \+ = 1807/3162, a[9,2] = 0, a[9,3] = 0, `b*`[5] = 67401048622586566709726 54206056/7847776077910531141899035067695, a[7,5] = 1569254935191750793 5540245739734013/2252563056478189244473772594954240, c[2] = 8/75, a[7, 3] = 421770368617928768410875/90187287207047138902016, b[5] = 13881573 506953684872/14139334514114522375, b[6] = -132955185321/214068151000, \+ a[7,6] = -14873540135675332293/2089665547196170240, a[8,4] = -16167923 3545622750791763569225/62137427083607735703094426884, a[6,4] = 2022450 2633443247773975/9127238351553409515876, b[2] = 0, b[3] = 0, `b*`[2] = 0, `b*`[3] = 0, c[3] = 4/25, a[8,3] = 2141335633391996675/37119465928 7217497, `b*`[6] = -15929717413567933460447487/32198747781536819606222 440, a[4,1] = 3/50, a[8,1] = -2691778286453656380245383/15212581634849 80574735172, a[6,5] = -3590778379490959900760/50904073152027811713543, `b*`[8] = -6/11, a[3,2] = 3/25, c[6] = 11/21, a[9,4] = 2474072265625/ 6361142786136, a[4,3] = 9/50, a[3,1] = 1/25, `b*`[7] = 165161965097576 40754673483776/21890172724016706578665261989, a[5,3] = -1364629188325/ 758746740672, `b*`[4] = 470295508478676388272578125/125894998254388593 8072825784, b[7] = 19335942766592/29106652419495, a[7,4] = -3031979901 406817815170871063875/1408453782065008683667117047808, `b*`[9] = -5590 11373156103/30082707428567288, a[2,1] = 8/75, b[4] = 2474072265625/636 1142786136, a[9,1] = 31138111/450810360, b[8] = -2486977/5149000, a[5, 4] = 2138982988675/1138120111008, a[6,1] = 373184883131711592359/62942 8679970118345044, a[7,1] = -9955638050099787690037550073/7170610831257 903919821488128, a[5,1] = 1116733125001/2276240222016, a[8,6] = -44416 1905271490/48398415269957, a[9,6] = -132955185321/214068151000, a[9,8] = -2486977/5149000, a[9,7] = 19335942766592/29106652419495, b[1] = 31 138111/450810360, a[8,7] = -311127404078694400/14477515022855683323, a [8,5] = 1363639349516531201308323118018680/154935659262427604083414890 417653\}:" }}}{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 0 331 "for ii from 2 to 9 do\n print(c[ii]=su bs(e3,c[ii]));print(``); \n for jj to ii-1 do\n print(a[ii,jj]= subs(e3,a[ii,jj]))\n end do:\n print(`____________________________ ______`):\nend do:\nprint(``);\nfor ii from 1 to 8 do\n print(b[ii]= subs(e3,b[ii]))\nend do:\nfor ii from 1 to 9 do\n print(`b*`[ii]=sub s(e3,`b*`[ii]))\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"##\"\")\"#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#\"\")\"#v" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"\"%\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\" \"\"#F(\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##F' \"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C____________________________ ______G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"'\"#D" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"%\"\"\"#\"\"$\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"%\"\"$#\"\"*\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________ ____________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\" &#\"%2=\"%iJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\"#\".,]7Ln6\"\".;?ASiF#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#!.D$)=HYO\"\"-s1uY(e(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\".v'))H)*Q@\".356?\"Q6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C________________________________ __G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#6\"#@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"'\"\"\"#\"6fBf6<8$)[=t$\"6W]M=,(*z'G%H'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#!3vbA'*yRb&G\"\"2#3o\")=lY0e" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%#\"8vRxZKWLE]C-#\"7 we^4Mb^$Qs7*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&#!7 g2!*f4\\z$y2f$\"8VNr6y-_J2/4&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C___ _______________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"cG6#\"\"(#\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\"#!=t+bP+p(y*4]!Qc&**\"=G \")[@)>R!zDJ3hqr" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\" #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"9v3To( Gzho.x@%\"8;?!*Qr/2sG(=!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"(\"\"%#!@vQ1r3<:y\"oS,*z>.$\"@3y/N\\Dp:\"CSU&\\fsP ZW#*=yk0jDD#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!5 $HK`nN,at[\"\"4S-<'>Zbm*3#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______ ____________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!:$QX-QcOX'GyRLcLT@\"3( \\r$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!?D# pNw\"z]FiXNBz;;\">%)oU%4.dt2O3Fu8i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\")\"\"&#\"C!o=!=JK38?Jl^\\$ROO\"\"B`wT!*[T$3/wUi#fc$\\:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!0!\\r_!>;W%\"/d* p_T)R[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!3+WpySS F6J\"5BLobG-:vZ9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C________________ __________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"* \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\")6\"Q6$\"*g.\"3X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#\".DcEsSZ#\".OhyU6O'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#\"5s[o`p]t:)Q\"\"5vB_9T^M$RT\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!-@`=bH8\"-+5:oS@ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"/#fmF%fL>\"/ &\\>Cl1\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!(xp [#\"(+!\\^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________________ ____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\")6\"Q6$\"*g.\"3X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"%#\".DcEsSZ#\".OhyU6O'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"&#\"5s[o`p]t:)Q\"\"5vB_9T^M$RT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"'#!-@`=bH8\"-+5:oS@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"(#\"/#fmF%fL>\"/&\\>Cl1\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!(xp[#\"(+!\\^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"\"#\"766H96$G)G\\BB\"8s))pd(ztY4=#3$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\" %#\";vV[Ex&Guw4380%\"1B*))er#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"=;3(>^X aS*)RSs+t\"\"=**>%ophpL$=-d,!*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% #b*G6#\"\")#!\"&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#! /P>)f#o))G\"0-Q5'*yp$o" }}}{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 0 588 "subs(e3,matrix([[c[2],a [2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1],a[4,2 ],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a[5,4]],\n [c [6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n [c[7],a[7,1],a[ 7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8 ,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n [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 [`b` ,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],[``$2,b[7],b[8]],\n [`b*`,`b *`[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``,`b*`[7],`b*`[8 ],`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777&#\" \")\"#vF(%!GF+7&#\"\"%\"#D#\"\"\"F/#\"\"$F/F+7&#\"\"'F/#F3\"#]\"\"!#\" \"*F87&#\"%2=\"%iJ#\".,]7Ln6\"\".;?ASiF#F9#!.D$)=HYO\"\"-s1uY(e(7&F+F+ F+#\".v'))H)*Q@\".356?\"Q67&#\"#6\"#@#\"6fBf6<8$)[=t$\"6W]M=,(*z'G%H'F 9#!3vbA'*yRb&G\"\"2#3o\")=lY0e7&F+F+#\"8vRxZKWLE]C-#\"7we^4Mb^$Qs7*#!7 g2!*f4\\z$y2f$\"8VNr6y-_J2/4&7&#\"#j\"#k#!=t+bP+p(y*4]!Qc&**\"=G\")[@) >R!zDJ3hqrF9#\"9v3To(Gzho.x@%\"8;?!*Qr/2sG(=!*7&F+#!@vQ1r3<:y\"oS,*z>. $\"@3y/N\\Dp:\"CSU&\\fsPZW#*=yk0jDD##!5 $HK`nN,at[\"\"4S-<'>Zbm*3#7&F1#!:$QX-QcOX'GyRLcLT@\"3(\\r$7&F+#!?D#pNw\"z]FiXNBz;;\">%)oU%4.dt2O3Fu8i #\"C!o=!=JK38?Jl^\\$ROO\"\"B`wT!*[T$3/wUi#fc$\\:#!0!\\r_!>;W%\"/d*p_T) R[7&F+F+F+#!3+WpySSF6J\"5BLobG-:vZ97&F1#\")6\"Q6$\"*g.\"3XF9F97&F+#\". DcEsSZ#\".OhyU6O'#\"5s[o`p]t:)Q\"\"5vB_9T^M$RT\"#!-@`=bH8\"-+5:oS@7&F+ F+#\"/#fmF%fL>\"/&\\>Cl1\"H#!(xp[#\"(+!\\^7&%\"bGF_qF9F9FbqF\\r7&%#b*G #\"8@YFOf`CqOfV#\"9#f(oMto`\"yZ()>K7&F+#\">wP[tYvSwv4l>;l\"\">*)>El'ylq;Sss,*=##! \"'FL#!0.h:t8,f&\"2)Gn&Guq#3IQ(pprint46\"" }}}{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 embedd ed scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2060 "ee := \{c[2]=8/75,\nc[3]=4/25,\nc [4]=6/25,\nc[5]=1807/3162,\nc[6]=11/21,\nc[7]=63/64,\nc[8]=1,\nc[9]=1, \n\na[2,1]=8/75,\na[3,1]=1/25,\na[3,2]=3/25,\na[4,1]=3/50,\na[4,2]=0, \na[4,3]=9/50,\na[5,1]=1116733125001/2276240222016,\na[5,2]=0,\na[5,3] =-1364629188325/758746740672,\na[5,4]=2138982988675/1138120111008,\na[ 6,1]=373184883131711592359/629428679970118345044,\na[6,2]=0,\na[6,3]=- 128555397896225575/58054665188168082,\na[6,4]=20224502633443247773975/ 9127238351553409515876,\na[6,5]=-3590778379490959900760/50904073152027 811713543,\na[7,1]=-9955638050099787690037550073/717061083125790391982 1488128,\na[7,2]=0,\na[7,3]=421770368617928768410875/90187287207047138 902016,\na[7,4]=-3031979901406817815170871063875/140845378206500868366 7117047808,\na[7,5]=15692549351917507935540245739734013/22525630564781 89244473772594954240,\na[7,6]=-14873540135675332293/208966554719617024 0,\na[8,1]=-2691778286453656380245383/1521258163484980574735172,\na[8, 2]=0,\na[8,3]=2141335633391996675/371194659287217497,\na[8,4]=-1616792 33545622750791763569225/62137427083607735703094426884,\na[8,5]=1363639 349516531201308323118018680/154935659262427604083414890417653,\na[8,6] =-444161905271490/48398415269957,\na[8,7]=-311127404078694400/14477515 022855683323,\na[9,1]=31138111/450810360,\na[9,2]=0,\na[9,3]=0,\na[9,4 ]=2474072265625/6361142786136,\na[9,5]=13881573506953684872/1413933451 4114522375,\na[9,6]=-132955185321/214068151000,\na[9,7]=19335942766592 /29106652419495,\na[9,8]=-2486977/5149000,\n\nb[1]=31138111/450810360, \nb[2]=0,\nb[3]=0,\nb[4]=2474072265625/6361142786136,\nb[5]=1388157350 6953684872/14139334514114522375,\nb[6]=-132955185321/214068151000,\nb[ 7]=19335942766592/29106652419495,\nb[8]=-2486977/5149000,\n\n`b*`[1]=2 4359367024535936274621/339039904141177334687592,\n`b*`[2]=0,\n`b*`[3]= 0,\n`b*`[4]=470295508478676388272578125/1258949982543885938072825784, \n`b*`[5]=6740104862258656670972654206056/7847776077910531141899035067 695,\n`b*`[6]=-15929717413567933460447487/32198747781536819606222440, \n`b*`[7] = 16516196509757640754673483776/2189017272401670657866526198 9,\n`b*`[8]=-6/11,\n`b*`[9]=-559011373156103/30082707428567288\}:" }}} {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[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\" &\"\"*" }{TEXT -1 145 " denote the vector whose components are the pr incipal error terms of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$ \"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of order 7 of the embedded order 5 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[6,8]));" "6#-%$absG6#-F $6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[ 5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" \"*" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG 6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs( `T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\" \"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") l`Z6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\"))Qt9\"! \"(" }}}{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 27 "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 2060 "ee := \{c[ 2]=8/75,\nc[3]=4/25,\nc[4]=6/25,\nc[5]=1807/3162,\nc[6]=11/21,\nc[7]=6 3/64,\nc[8]=1,\nc[9]=1,\n\na[2,1]=8/75,\na[3,1]=1/25,\na[3,2]=3/25,\na [4,1]=3/50,\na[4,2]=0,\na[4,3]=9/50,\na[5,1]=1116733125001/22762402220 16,\na[5,2]=0,\na[5,3]=-1364629188325/758746740672,\na[5,4]=2138982988 675/1138120111008,\na[6,1]=373184883131711592359/629428679970118345044 ,\na[6,2]=0,\na[6,3]=-128555397896225575/58054665188168082,\na[6,4]=20 224502633443247773975/9127238351553409515876,\na[6,5]=-359077837949095 9900760/50904073152027811713543,\na[7,1]=-9955638050099787690037550073 /7170610831257903919821488128,\na[7,2]=0,\na[7,3]=42177036861792876841 0875/90187287207047138902016,\na[7,4]=-3031979901406817815170871063875 /1408453782065008683667117047808,\na[7,5]=1569254935191750793554024573 9734013/2252563056478189244473772594954240,\na[7,6]=-14873540135675332 293/2089665547196170240,\na[8,1]=-2691778286453656380245383/1521258163 484980574735172,\na[8,2]=0,\na[8,3]=2141335633391996675/37119465928721 7497,\na[8,4]=-161679233545622750791763569225/621374270836077357030944 26884,\na[8,5]=1363639349516531201308323118018680/15493565926242760408 3414890417653,\na[8,6]=-444161905271490/48398415269957,\na[8,7]=-31112 7404078694400/14477515022855683323,\na[9,1]=31138111/450810360,\na[9,2 ]=0,\na[9,3]=0,\na[9,4]=2474072265625/6361142786136,\na[9,5]=138815735 06953684872/14139334514114522375,\na[9,6]=-132955185321/214068151000, \na[9,7]=19335942766592/29106652419495,\na[9,8]=-2486977/5149000,\n\nb [1]=31138111/450810360,\nb[2]=0,\nb[3]=0,\nb[4]=2474072265625/63611427 86136,\nb[5]=13881573506953684872/14139334514114522375,\nb[6]=-1329551 85321/214068151000,\nb[7]=19335942766592/29106652419495,\nb[8]=-248697 7/5149000,\n\n`b*`[1]=24359367024535936274621/339039904141177334687592 ,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=470295508478676388272578125/1258949 982543885938072825784,\n`b*`[5]=6740104862258656670972654206056/784777 6077910531141899035067695,\n`b*`[6]=-15929717413567933460447487/321987 47781536819606222440,\n`b*`[7] = 16516196509757640754673483776/2189017 2724016706578665261989,\n`b*`[8]=-6/11,\n`b*`[9]=-559011373156103/3008 2707428567288\}:" }}}{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 77 "The stability function R for the 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR := unapply(%,z): \n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\" \"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#C F)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F) F)*&#\"/(yvzzo%G\"3+D^?o0@6:F)*$)F'\"\"(F)F)F)*&#\".x(yBQdz\"3+D'yC*z: ENF)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabi lity region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/- %\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+S\\+3V!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-4): \np1 := plot([R(z),1],z=-4.89..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[d isplay]([p1,p2,p3],view=[-4.89..0.49,-0.07..1.47],font=[HELVETICA,9]); " }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$ 7X7$$!3o*************)[!#<$\"3u7'*GT&>W(GF*7$$!3F*7$$!3J\\7Q:?K8YF*$\"3[xm!*zDik)\\%F*$\"3Cw3/DdgH9F* 7$$!3iK$e'y!Q/W%F*$\"3O3h3H[<%G\"F*7$$!3')\\7LEy&HQ%F*$\"3I=p7(4oG:\"F *7$$!3AmT+uvZDVF*$\"30K8Zb=)Q.\"F*7$$!3_L3AqW*)=UF*$\"3U:(*)p8V\\U)!#= 7$$!3`*\\(yRQ`3TF*$\"33K(**e$QP#z'F^o7$$!3!H$3#zr)R%*RF*$\"3L3dKblJ>aF ^o7$$!3u*\\P)G'H1)QF*$\"3'3LyV/tyJ%F^o7$$!3QmmrOTgjPF*$\"3f.[dWu;;MF^o 7$$!3'HLe&>!G0m$F*$\"31[%49'*)f$y#F^o7$$!3\"****\\^s)[WNF*$\"3:+ig!y# \\@AF^o7$$!3q***\\A'H(zU$F*$\"3SNH%4*=2*y\"F^o7$$!3w***\\[V)o:LF*$\"3u (*\\priwu9F^o7$$!3SL3#pKBP@$F*$\"3=)p/(=7-h7F^o7$$!3KmmJ^pZ#4$F*$\"37` +e=Asz5F^o7$$!3?mmEn`w*)HF*$\"3=Bb'\\z9Hx*!#>7$$!3u*\\([xtGqGF*$\"3I*f '*e\"Q'e1*Ffr7$$!3gmmw6\"HXw#F*$\"3^bE?#F *$\"3%z2k?X7a9\"F^o7$$!3KL3#>MSS3#F*$\"3=f'\\7l)Hq7F^o7$$!3u*\\7x[&y!) >F*$\"3Qb;AB&yqR\"F^o7$$!3%HL3kTn#p=F*$\"3S_$)z&)f[`:F^o7$$!3+++qO$eSv \"F*$\"3'[`*f2$*\\PF^o7$$!3)**\\( o(Q'HK:F*$\"3E%R*eE1\"G;#F^o7$$!3/+]#eN77T\"F*$\"38=gjjjxRCF^o7$$!3:nm wnHT-8F*$\"3`*)GC2[Y>FF^o7$$!3y***\\#)RYi=\"F*$\"3Q;;wu;,aIF^o7$$!3qL3 _?A)43\"F*$\"3o:Zw'p-GR$F^o7$$!3!\\***\\.n+f'*F^o$\"3D],&pC#[1QF^o7$$! 37m;a,!zhd)F^o$\"3vi`]HquTUF^o7$$!3)[*\\P]?JWuF^o$\"3!H%=dIk0]ZF^o7$$! 3'\\mm\">_qPjF^o$\"3d^c?&HyeI&F^o7$$!3A1](=U!>z^F^o$\"3-!\\(f 3ii6F*7$$\"3u+]7DeEVEF^o$\"3tr62_ B_9F*7$$\"3!***************[F^o$\"3OWTzRhJK;F*-%'COLOURG6&%$RGBG$\"*++ ++\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_] l7$FQF_]l7$FenF_]l7$FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_ ]l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FhrF_]l7$F]s F_]l7$FbsF_]l7$FgsF_]l7$F\\tF_]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$ FeuF_]l7$FjuF_]l7$F_vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l 7$F]xF_]l7$FbxF_]l7$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF _]l7$FezF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6& Fe\\lFi\\lFi\\lFf\\l-F$6&7#7$$!3,+++S\\+3VF*F_]l-%'SYMBOLG6#%'CIRCLEG- Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSG F]alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COL ORG6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\" z6\"Q!F_cl-%%FONTG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$*[ !\"#$\"#\\F_dl;$!\"(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 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1346 "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 \+ 28468797975787/151121056820512500*z^7+7957382378777/352615799247862 500*z^8:\npts := []:\nz0 := 0:\nfor ct from 0 to 240 do\n zz := newt on(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(z z),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.1,.4,0)):\np2 : = plots[polygonplot]([seq([pts[i-1],pts[i],[-2.1,0]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.1,.85,0)):\npts := []: \+ z0 := 2+4.8*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi /30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.1,.4,0)):\np4 := plots[polygonpl ot]([seq([pts[i-1],pts[i],[1.89,4.78]],i=2..nops(pts))],\n st yle=patchnogrid,color=COLOR(RGB,.1,.85,0)):\npts := []: z0 := 2-4.8*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 := p lot(pts,color=COLOR(RGB,.1,.4,0)):\np6 := plots[polygonplot]([seq([pts [i-1],pts[i],[1.89,-4.78]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.1,.85,0)):\np7 := plot([[[-4.99,0],[2.29,0]],[[0, -5.19],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p||(1..7) ],view=[-4.99..2.29,-5.19..5.19],font=[HELVETICA,9],\n la bels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F (7$F($\"3++++Fjzq:!#=7$$!3%)******y]39r!#F$\"3!)*****\\i#fTJF-7$$!35++ +B9UNFyxs\"FP7$$!3%******p()>.2\"F^o$ \"31+++-ex%)=FP7$$!3\"********)G\"Rt\"FT$\"33+++xSmT?FP7$$\"3'******\\ NsxS#F^o$\"3'*******Q!)H)>#FP7$$\"3k*****>)[J&e(F^o$\"39+++2.SaBFP7$$ \"3!******H$4`!f\"!#?$\"3@+++Zu\\4DFP7$$\"3'******R`0rm#Fhp$\"3!****** RFnGm#FP7$$\"3A+++>lVuOFhp$\"3&)*****p8LN\"GFP7$$\"3#******44U!\\RFhp$ \"3++++$*zLgHFP7$$\"3)******HrPJY#Fhp$\"3/+++*o*3-JFP7$$!3!******4?;u/ 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(approximately) " } {XPPEDIT 18 0 "[-4.3080, 0];" "6#7$,$-%&FloatG6$\"&!3V!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that there is " }{TEXT 260 53 "no largest interval on the nonn egative imaginary axis" }{TEXT -1 65 " that contains the origin and li es inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "How ever the stability region intersects the nonnegative imaginary axis in an interval that does not contain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "R := z -> 1 +z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 28468797975787 /151121056820512500*z^7+7957382378777/352615799247862500*z^8:\nDigits \+ := 25:\npts := []: z0 := 0:\nfor ct from 0 to 112 do\n zz := newton( R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(R e(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,0,.8,0),thickne ss=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7]r7$$\"\"!F)F(7$$!:J !4%[C3s6Ir3x#!#E$\":9__lHw*e`EfTJF-7$$!:Zn->V`q$[Zr'e%F-$\":g&p-(yrvrI &=$G'F-7$$!:K$GB19oLeUuehF-$\":&y3+)3Y.2'zxC%*F-7$$!:a%*Q'pkf(H*)[,f(F -$\"::5+w@s'Qhqjc7!#D7$$!:p41wabhlX$pC*)F-$\":mG#>Bm9cEjzq:F?7$$!:\"3= %p#))G%*4%>'=5F?$\"::41[)oCK\"fb\\)=F?7$$!:'))*p%3U*z%zF%*Q6F?$\":14l^ IH%3b[6*>#F?7$$!:zii])z\"o@(\\Va7F?$\":K'yg*f=Qn6uK^#F?7$$!:/38\">AR;d d!eO\"F?$\":i\"\\eaR>PuLVFGF?7$$!:4q:oO)*[N^#et9F?$\":J*Q%fX1;\\i#fTJF ?7$$!:&z*oX%*Rb(e&p\"y:F?$\":#)*yI)p8;P'=vbMF?7$$!:l^72q/H%3X))z;F?$\" :)3TX^#zMS36*pPF?7$$!:*H7#e;Jf0t\")*yF?$\":RB95;N]L_)Q 7ZF?7$$!:^zY\"*)\\\"o'36Yi?F?$\":/Kv0[D#3SvaE]F?7$$!:G?OV?9Mv8GG:#F?$ \":#)>D@buoAX12M&F?7$$!:#)RH68=v=<:8C#F?$\":WalDsF?7$$!: pk,JFOIF?$\":L\\DFr,/?HLkz)F?7$$!: 38')oq[KU6\\v5$F?$\":y]0o4hc;R'e5\"*F?7$$!:!\\tA'Qld\\p%RxJF?$\":K-Qf> *od?%QZU*F?7$$!:JIccArOaw5eC$F?$\"::AxF#e7Gn#*))Q(*F?7$$!:hZ.%y4CYhTz7 LF?$\":s-6v$[g+#)QI05!#C7$$!:mYO%)zF9tlP$yLF?$\":08d3yGTop=n.\"Ffu7$$! :,,AJL8S$H*HCW$F?$\":!R&Rid*oVgL8o5Ffu7$$!:Cngk6wksXb]]$F?$\":#))H3)*p *eB'ya*4\"Ffu7$$!:qPA8Xks*fV>mNF?$\":_zO6Pt4E>i48\"Ffu7$$!:#>yA.:D\"Ffu 7$$!:s?tA<*3(Qe@/u$F?$\":W(=kp=RyfS?D7Ffu7$$!:qY\\)[9S#*pzJ&z$F?$\":UU EX%Gs56whc7Ffu7$$!:vxflq-%\\G;b[QF?$\":&=UO6Cj)\\%4.)G\"Ffu7$$!:n_H],S y>Io+!RF?$\":(pp%>=M!4^SW>8Ffu7$$!:/&4XeC()*R\\0)\\RF?$\":5FoFU)=Fb@Y)*)4sUF?$\":D)=B[#*[f=! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "Digits := 15:\nz0 := 2.05*I:\nfor ct fro m 64 to 67 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 \+ := 3.2*I:\nfor ct from 102 to 105 do\n newton(R(z)=exp(ct*Pi/100*I), z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$! 0Q40*zl$e%!#>$\"0)f*pg+.,#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0 %p\"**)G\"Rt\"!#>$\"03\"RxSmT?!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^ $$\"0J^O,>]y\"!#>$\"0kaNz$\"0B@)z+O/@!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0?\\y#)fa4\"!#<$\"0=%*p!y7dJ! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0#)G+?En.#!#=$\"0N.V(*eU=$! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0L0.7pfS)!#=$\"0\"ex#*>76K!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0k-4?;u/#!#<$\"0N^W55xB$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then \+ we apply the bisection method to calculate the parameter value associa ted with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R( z)=exp(u*Pi*I),z=2.05*I))\nend proc:\nu0 := bisect('real_part'(u),u=0. 64..0.67);\nnewton(R(z)=exp(u0*Pi*I),z=2.05*I);``;\nreal_part := proc( u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.2*I))\nend proc:\nu0 := bisect(' real_part'(u),u=1.02..1.05);\nnewton(R(z)=exp(u0*Pi*I),z=3.2*I);\nDigi ts := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0%HH&>#y^l!#:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0^kNP+Hw%!#H$\"0;w$Q5!z0#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#u0G$\"0tmiht?.\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0]B')f ^)*=$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the nonegative ima ginary axis in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[2.0579 , 3.1899];" "6#7$-%&FloatG6$\"&z0#!\"%-F%6$\"&**=$F(" }{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 80 "The stability function R * for the 9 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFuncti on(5,9,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"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)*&#\"@By80#H(=?OX)=paHD\"C++/ezPRMC[4X:AX ==F)*$)F'F1F)F)F)*&#\">>5.[)))HkvWC7moh\"B++wrw2XX%yWON74i%oP::F)*$)F'\"\"*F)F)!\"\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can \+ find the point where the boundary of the stability region intersects t he 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 := ne wton(`R*`(z)=1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+W k#yL%!\"*" }}}{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=-4.79..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],st yle=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=[-4.79..0.49,-0.07..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$ $!3/++++++!z%!#<$\"3Mr6qUq)=Q#F*7$$!3I++]cxAhZF*$\"3ipq53P)*eAF*7$$!3n *****H^bCt%F*$\"3i9_P'[u<9#F*7$$!3$*****\\pKo.ZF*$\"3m^0i1+.I?F*7$$!3= +++E5\"\\n%F*$\"3(f9d0\")GN#>F*7$$!3[++bv=%[i%F*$\"3%Q?M\"=k1][Uz[VF*$\"3XH.V()*z@-\"F*7$$!3K++0-CQ#H%F*$\"3PPPDGN.F\"*!#=7$$!3$) *****ebqfB%F*$\"3ec$o'=8nS\")F^o7$$!3U++q]&o88%F*$\"3cK?$oS#[nlF^o7$$! 3p****4L#fI-%F*$\"3.1eMq_gT_F^o7$$!3&)*****=eX5\"RF*$\"3h1@F2g]TTF^o7$ $!3')*****)f6R*z$F*$\"3!4C;&z.MsKF^o7$$!3/++!G'3a%o$F*$\"3/G]*pn)*Hd#F ^o7$$!3K++ge1Q$e$F*$\"3!)z4#)pIl\"4#F^o7$$!3+++SG#)\\pMF*$\"3#RXM9mQLn \"F^o7$$!33+++'=[^N$F*$\"3%ff&4bK$3O\"F^o7$$!3#*****fJ2&\\C$F*$\"3PNDB Ms7$$!3p****f8d3DHF*$\"3%pqWQdnsO)F[s7$$!3v****H/&Gy!GF*$\"3Bb) QL8QR7)F[s7$$!3N++g0g./FF*$\"3U')\\OSu]#>)F[s7$$!3-++I%=f,f#F*$\"3M=)e 2w=o_)F[s7$$!3))*****3_@<[#F*$\"3IlJo#)oSk!*F[s7$$!3++++&>y&oBF*$\"3\\ d@PCYQH)*F[s7$$!3)*****H-tnkAF*$\"3=*Gyx)zEq5F^o7$$!3C+++`hg_@F*$\"3aD Z@5!e@=\"F^o7$$!3')*****es&>O?F*$\"3'4_omEHyJ\"F^o7$$!3*******o5g[$>F* $\"39h%)R;pc_9F^o7$$!3r****>i[TD=F*$\"31)y;.\"pX;;F^o7$$!3L++?:sM7m1*zD'3WAF^o7$$!3E++?OQ(eP\"F*$\"3`[w#4stm_#F^o7$$!3I++gTn4 p7F*$\"3E$RBPj76\"GF^o7$$!3w*****>S*3b6F*$\"3m_,**z,U]JF^o7$$!3G++]2=y ^5F*$\"3w&>vG@6K\\$F^o7$$!3c&****f>$R)Q*F^o$\"3iE\"Gu6V3\"RF^o7$$!3N-+ +.CpD$)F^o$\"3'H%*pfS;$\\VF^o7$$!3#o****4(Q'[@(F^o$\"3'=*QS\\wHg[F^o7$ $!3D-++Sf#)GhF^o$\"3\"eyD-HqyT&F^o7$$!3'******\\([%=*\\F^o$\"3YCr2(Rb- 2'F^o7$$!3'4++g(pz'*QF^o$\"3ww.(y\"ytsnF^o7$$!3u-++QB&px#F^o$\"3;Gjajl FvvF^o7$$!3/$*****y0Qm;F^o$\"3QXh(3Uf]Y)F^o7$$!3%)******>Z!)ekF[s$\"37 TQm/d`u$*F^o7$$\"3:!*****R\"ysB&F[s$\"3k9#)f%\\oP0\"F*7$$\"3++++3%R)p: F^o$\"3]0U)[#o(*p6F*7$$\"3;0++dD@&o#F^o$\"3yyWH!yG!38F*7$$\"3+/++zpw_P F^o$\"3+WuclSRb9F*7$$\"3!***************[F^o$\"3*)GKY?iJK;F*-%'COLOURG 6&%$RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FG Fi]l7$FQFi]l7$FenFi]l7$F`oFi]l7$FeoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi]l7$Fi pFi]l7$F^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l7$FbrFi]l7$FgrFi]l7$F]sFi]l7$ FbsFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi] l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xF i]l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$F ezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$F h\\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!31+++Wk#yL%F*Fi]l-%'SYMBOLG6 #%'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`al- %&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVE TICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^dl-Ffcl6#%(DEFAULTG-%%VIEWG6$;$!$ z%!\"#$\"#\\Fidl;$!\"(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 7 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1582 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 2529546918845362018729205137823/18 18452215450948243439377958040000*z^6+\n 61686612244475642988848 031019/314300382917447844545077671760000*z^7+\n 699594166582026 779647021309573/31822913770391594260189114265700000*z^8-\n 6354 66750041186870327889433/1515376846209123536199481631700000*z^9:\npts : = []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(`R*`(z)=exp(c t*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,0,.35,0)):\np_2 := plots[po lygonplot]([seq([pts[i-1],pts[i],[-2.1,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,0,.65,0)):\npts := []: z0 := 2+4.7 *I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z =z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_ 3 := plot(pts,color=COLOR(RGB,0,.35,0)):\np_4 := plots[polygonplot]([s eq([pts[i-1],pts[i],[1.89,4.63]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,0,.65,0)):\npts := []: z0 := 2-4.7*I:\nfor c t from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot( pts,color=COLOR(RGB,0,.35,0)):\np_6 := plots[polygonplot]([seq([pts[i- 1],pts[i],[1.89,-4.63]],i=2..nops(pts))],\n style=patchnogrid ,color=COLOR(RGB,0,.65,0)):\np_7 := plot([[[-4.99,0],[2.29,0]],[[0,-5. 19],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)], view=[-4.99..2.29,-5.19..5.19],font=[HELVETICA,9],\n labels =[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$$ \"3v*****RA_RA$!#G$\"3++++Fjzq:!#=7$$\"37+++h?:^?!#E$\"3,+++TEfTJF07$$ \"3/+++)pm?O#!#D$\"3&******fx)Q7ZF07$$\"3++++YuN,9!#C$\"39+++hQ=$G'F07 $$\"32+++v0\"4.'F@$\"3U+++*3vR&yF07$$\"3)******f$=Wz@!#B$\"3s*****fJbZ U*F07$$\"3B+++'Hs)3qFK$\"3#******4w]&*4\"!#<7$$\"3:+++Bv/T?!#A$\"3)*** ****e\">mD\"FS7$$\"3n******>^20aFW$\"3%******43rOT\"FS7$$\"3*)*****4`< ^I\"!#@$\"3++++BUoq:FS7$$\"3E+++%RIK)GF\\o$\"3%******z15ws\"FS7$$\"3%) *****Hdrn%eF\\o$\"3-+++S0N%)=FS7$$\"3/+++#o'=!4\"!#?$\"3!)******H@sS?F S7$$\"3/+++Aehm=F\\p$\"3/+++YyS'>#FS7$$\"39+++:\"\\n\"HF\\p$\"3'****** f![!4N#FS7$$\"3O+++*f235%F\\p$\"3-+++bs[.DFS7$$\"3M+++5U\"H.&F\\p$\"3# )*****Rq9Kl#FS7$$\"3L+++-s)>+&F\\p$\"3#)*****Rl1!*z#FS7$$\"3A+++3!4'zH F\\p$\"3&*******p!o(RHFS7$$!39+++'fR/D#F\\p$\"3!)*****4*p]uIFS7$$!3-++ +(e&H(=\"!#>$\"39+++D\")Q-KFS7$$!3'******Ho-8o#Fer$\"32+++&*GsALFS7$$! 3E+++!3]uv%Fer$\"3/+++vz%\\V$FS7$$!3K*****H))>&=uFer$\"3=+++7'Q'QNFS7$ $!3'******468D1\"F0$\"3y******z`bLOFS7$$!37+++&e-4V\"F0$\"3#)*****>f&o >PFS7$$!3#******zY$*)Q=F0$\"3=+++)4Wsz$FS7$$!3!********)4SyAF0$\"3-+++ p!Hm'QFS7$$!3')******RfAUFF0$\"30+++jPNGRFS7$$!3?+++J)3WA$F0$\"3'***** *HCzH)RFS7$$!3&******fDZ-s$F0$\"3C+++DG1JSFS7$$!3z*****RNThA%F0$\"3/++ +6c7tSFS7$$!3*)*****p!RQRZF0$\"3z*****H%fj4TFS7$$!3]+++3z&zD&F0$\"3m** ***>u.59%FS7$$!3w******GqO!y&F0$\"3u*****\\H\"enTFS7$$!3E+++$=/bI'F0$ \"3u*****zIn'*=%FS7$$!3F+++sXbKoF0$\"3h******pA^2UFS7$$!3_+++,q#4O(F0$ \"3x*****p+C8A%FS7$$!3;+++it?!*yF0$\"3I+++mGFJUFS7$$!33+++p$H,U)F0$\"3 W+++/j\\PUFS7$$!3C+++M*e0&*)F0$\"3s******>G5SUFS7$$!3-+++P)*[\"[*F0$\" 3)******fHv\"RUFS7$$!3.+++S\\I,5FS$\"3U+++dQxMUFS7$$!35+++u8ba5FS$\"3T +++G%QpA%FS7$$!3!******fHLz5\"FS$\"3#)*****p@\"p:UFS7$$!3/+++XR5?%FS7$$!32+++ZMR:7FS$\"3W+++R)yH=%FS7$$!3%******p=(op7FS $\"3-+++_%)\\hTFS7$$!3#******4'3cC8FS$\"3:+++f!)eOTFS7$$!3!*******HZA! 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7$Fceal$!+O3L6[F^\\oF^gcl7%Fbedl7$Fjdal$!+!R*H.[F^\\oF^gcl7%FfedlFhfcl F^gclF^a`lFbes-F$6%7$7$$!3A++++++!*\\FSF(7$$\"3/++++++!H#FSF(-%'COLOUR G6&FghnF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>&FS7$F($\"3Q++ ++++!>&FSFcfdlFffdl-%(SCALINGG6#%,CONSTRAINEDG-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fhgdl6#%(DEFAULTG-%*AXESSTYLEG6#%$ BOXG-%%VIEWG6$;$!$*\\Fjhn$\"$H#Fjhn;$!$>&Fjhn$\"$>&Fjhn" 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 35 "coefficients of the comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficie nts of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2060 "ee := \{c[2]=8/75,\nc[3]=4/25,\nc [4]=6/25,\nc[5]=1807/3162,\nc[6]=11/21,\nc[7]=63/64,\nc[8]=1,\nc[9]=1, \n\na[2,1]=8/75,\na[3,1]=1/25,\na[3,2]=3/25,\na[4,1]=3/50,\na[4,2]=0, \na[4,3]=9/50,\na[5,1]=1116733125001/2276240222016,\na[5,2]=0,\na[5,3] =-1364629188325/758746740672,\na[5,4]=2138982988675/1138120111008,\na[ 6,1]=373184883131711592359/629428679970118345044,\na[6,2]=0,\na[6,3]=- 128555397896225575/58054665188168082,\na[6,4]=20224502633443247773975/ 9127238351553409515876,\na[6,5]=-3590778379490959900760/50904073152027 811713543,\na[7,1]=-9955638050099787690037550073/717061083125790391982 1488128,\na[7,2]=0,\na[7,3]=421770368617928768410875/90187287207047138 902016,\na[7,4]=-3031979901406817815170871063875/140845378206500868366 7117047808,\na[7,5]=15692549351917507935540245739734013/22525630564781 89244473772594954240,\na[7,6]=-14873540135675332293/208966554719617024 0,\na[8,1]=-2691778286453656380245383/1521258163484980574735172,\na[8, 2]=0,\na[8,3]=2141335633391996675/371194659287217497,\na[8,4]=-1616792 33545622750791763569225/62137427083607735703094426884,\na[8,5]=1363639 349516531201308323118018680/154935659262427604083414890417653,\na[8,6] =-444161905271490/48398415269957,\na[8,7]=-311127404078694400/14477515 022855683323,\na[9,1]=31138111/450810360,\na[9,2]=0,\na[9,3]=0,\na[9,4 ]=2474072265625/6361142786136,\na[9,5]=13881573506953684872/1413933451 4114522375,\na[9,6]=-132955185321/214068151000,\na[9,7]=19335942766592 /29106652419495,\na[9,8]=-2486977/5149000,\n\nb[1]=31138111/450810360, \nb[2]=0,\nb[3]=0,\nb[4]=2474072265625/6361142786136,\nb[5]=1388157350 6953684872/14139334514114522375,\nb[6]=-132955185321/214068151000,\nb[ 7]=19335942766592/29106652419495,\nb[8]=-2486977/5149000,\n\n`b*`[1]=2 4359367024535936274621/339039904141177334687592,\n`b*`[2]=0,\n`b*`[3]= 0,\n`b*`[4]=470295508478676388272578125/1258949982543885938072825784, \n`b*`[5]=6740104862258656670972654206056/7847776077910531141899035067 695,\n`b*`[6]=-15929717413567933460447487/32198747781536819606222440, \n`b*`[7] = 16516196509757640754673483776/2189017272401670657866526198 9,\n`b*`[8]=-6/11,\n`b*`[9]=-559011373156103/30082707428567288\}:" }}} {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 31 "s eq(c[i]=subs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&% \"cG6#\"\"##\"\")\"#v/&F%6#\"\"$#\"\"%\"#D/&F%6#F0#\"\"'F1/&F%6#\"\"&# \"%2=\"%iJ/&F%6#F6#\"#6\"#@/&F%6#\"\"(#\"#j\"#k/&F%6#F)\"\"\"/&F%6#\" \"*FN" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#\"\")\"#v/&F%6$\" \"$F(#F(\"#D/&F%6$F/F'#F/F1/&F%6$\"\"%F(#F/\"#]/&F%6$F9F'\"\"!/&F%6$F9 F/#\"\"*F;/&F%6$\"\"&F(#\".,]7Ln6\"\".;?ASiF#/&F%6$FHF'F?/&F%6$FHF/#!. D$)=HYO\"\"-s1uY(e(/&F%6$FHF9#\".v'))H)*Q@\".356?\"Q6/&F%6$\"\"'F(#\"6 fBf6<8$)[=t$\"6W]M=,(*z'G%H'/&F%6$FhnF'F?/&F%6$FhnF/#!3vbA'*yRb&G\"\"2 #3o\")=lY0e/&F%6$FhnF9#\"8vRxZKWLE]C-#\"7we^4Mb^$Qs7*/&F%6$FhnFH#!7g2! *f4\\z$y2f$\"8VNr6y-_J2/4&/&F%6$\"\"(F(#!=t+bP+p(y*4]!Qc&**\"=G\")[@)> R!zDJ3hqr/&F%6$FdpF'F?/&F%6$FdpF/#\"9v3To(Gzho.x@%\"8;?!*Qr/2sG(=!*/&F %6$FdpF9#!@vQ1r3<:y\"oS,*z>.$\"@3y/N\\Dp:\"CSU&\\fsPZW#*=yk0jDD#/&F%6$FdpFhn#!5$HK`nN,at[\"\"4S-<' >Zbm*3#/&F%6$F*F(#!:$QX-QcOX'GyRLcLT@\"3(\\r$/&F%6$F*F9#!?D#pNw\"z]FiXNBz;;\">%)o U%4.dt2O3Fu8i/&F%6$F*FH#\"C!o=!=JK38?Jl^\\$ROO\"\"B`wT!*[T$3/wUi#fc$\\ :/&F%6$F*Fhn#!0!\\r_!>;W%\"/d*p_T)R[/&F%6$F*Fdp#!3+WpySSF6J\"5BLobG-:v Z9/&F%6$FDF(#\")6\"Q6$\"*g.\"3X/&F%6$FDF'F?/&F%6$FDF/F?/&F%6$FDF9#\".D cEsSZ#\".OhyU6O'/&F%6$FDFH#\"5s[o`p]t:)Q\"\"5vB_9T^M$RT\"/&F%6$FDFhn#! -@`=bH8\"-+5:oS@/&F%6$FDFdp#\"/#fmF%fL>\"/&\\>Cl1\"H/&F%6$FDF*#!(xp[# \"(+!\\^" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 6 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]),i=1..8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\")6\"Q6$\"*g.\"3X/&F% 6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\".DcEsSZ#\".OhyU6O'/&F%6#\"\"&# \"5s[o`p]t:)Q\"\"5vB_9T^M$RT\"/&F%6#\"\"'#!-@`=bH8\"-+5:oS@/&F%6#\"\"( #\"/#fmF%fL>\"/&\\>Cl1\"H/&F%6#\"\")#!(xp[#\"(+!\\^" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order \+ 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"8@YFOf`CqOfV#\"9#f(oMto`\"yZ()>K/&F%6#\"\"(#\">wP[tYvSwv4l>;l\"\">*)>El'y lq;Sss,*=#/&F%6#\"\")#!\"'\"#6/&F%6#\"\"*#!0.h:t8,f&\"2)Gn&Guq#3I" }}} {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 35 "#==================================" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c [4]=9/38" "6#/&%\"cG6#\"\"%*&\"\"*\"\"\"\"#Q!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]=63/64" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\" " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combi ned scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficie nts of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2970 "ee := \{c[2]=2/19,\nc[3]=3/19,\nc [4]=9/38,\nc[5]=483979/844361,\nc[6]=330/613,\nc[7]=63/64,\nc[8]=1,\nc [9]=1,\n\na[2,1]=2/19,\na[3,1]=3/76,\na[3,2]=9/76,\na[4,1]=9/152,\na[4 ,2]=0,\na[4,3]=27/152,\na[5,1]=50496032797210018219/975213065730664967 22,\na[5,2]=0,\na[5,3]=-62238405188783017003/32507102191022165574,\na[ 5,4]=96058693980453356374/48760653286533248361,\na[6,1]=77026832972677 20889544548006160849795/12377955859939284746771758057563701031,\na[6,2 ]=0,\na[6,3]=-39922904315579295139013477837035/17050265762136077180720 317834126,\na[6,4]=639564226981491742693206700036358264060/27600850011 3722586808059264316424382117,\na[6,5]=-5313180967312747463264830899685 275868548285/89054878584359562560518476456399170688402362,\na[7,1]=-58 0440564877918386354914362593379258757943/42222747040520847093663842561 3284692459520,\na[7,2]=0,\na[7,3]=837083378582818460922126567546504549 /174481731812830090122355897926680576,\na[7,4]=-6809200678936021971721 50986392871771764900095/275506239843907734397585145149495475886358528, \na[7,5]=138939023470913107918594667116771850303891585832494354565/\n \+ 16440422995450808437828180734862212664987734041599934464,\na[7,6 ]=-5136972107040184160774194317/610237065866248643642654720,\na[8,1]=- 73912817561044808721210574412370874448581/4209820296536336488098719314 7226166573410,\na[8,2]=0,\na[8,3]=821199866821133946316318884532576/13 8069098844093769649611890045033,\na[8,4]=-9626283444389131976638088363 590116538104533086/\n 31707487687247989961352808387944912113028 69081, a[8,5]=22948818964225036611486944476353885456440719164455040674 8303/\n 21412703379882516143374345447138204592119497840386204840 294,\na[8,6]=-1608754695857040236707195633/148246171293910549234824945 ,\na[8,7]=-2136870666574914369093632/99262060255996938363115017,\na[9, 1]=74489238253/1086687728280,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1022143270 70044952/269723237135952465,\na[9,5]=517691403516342385674740758661900 23/45271316149543535439664321662847560,\na[9,6]=-100866038423285938709 /130767668626607626680,\na[9,7]=11559809987903488/17579279732593635,\n a[9,8]=-84698060813/177459304440,\n\nb[1]=74489238253/1086687728280,\n b[2]=0,\nb[3]=0,\nb[4]=102214327070044952/269723237135952465,\nb[5]=51 769140351634238567474075866190023/45271316149543535439664321662847560, \nb[6]=-100866038423285938709/130767668626607626680,\nb[7]=11559809987 903488/17579279732593635,\nb[8]=-84698060813/177459304440,\n\n`b*`[1]= 116199385772037258134667598271864434567/163100992796274153354439957260 2609517000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1018492330418071023333443 489476512190862024/2791915070320383197568190938763984505340675,\n`b*`[ 5]=73834772250747223839914017280548444801785295958509475023/\n \+ 75417453255544184981554000541305420593198255418756449800,\n`b*`[6]=-13 9813632869510160446859308498783581490009423/23117695163548596511469056 0680615525273273000,\n`b*`[7]=3955511412530100531582592349148563244056 576/5276949214412532313091563386220545755915925,\n`b*`[8]=-7/13,\n`b*` [9]=-770493302287576267044727411/36021606991976398621216598600\}:" }}} {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 0 342 "for ii from 2 to 9 do\n print(c[ii]=subs(ee,c[ii]) );print(``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii ,jj]))\n end do:\n print(`__________________________________`):\ne nd do:\nprint(``);\nfor ii from 1 to 8 do\n print(b[ii]=subs(ee,b[ii ]))\nend do:\nprint(``);\nfor ii from 1 to 9 do\n print(`b*`[ii]=sub s(ee,`b*`[ii]))\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"##F'\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#F'\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#F'\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#F'\" #w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\"*\"#w" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"*\"#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"%\"\"\"#\"\"*\"$_\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\" \"$#\"#F\"$_\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________ ________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\" 'zR[\"'hV%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\"#\"5>#=+@(zKg\\]\"5An\\mId18_(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#!5.q,$y)=0%QA'\"5ub;A5>- r]K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\"5ujN`/)Rp eg*\"5h$[K`'G`1w[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_______________ ___________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"' #\"$I$\"$8'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\"#\"F&z\\3;1![Xa*)3snsH$o-x\"GJ5qj v0e-O*y1?4o\"NG&ej)eZ&\\\\^9&e(RMx!R%)Ri]v#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#\"ZlXN%\\Kee\"*QI] =x;rm%f=z584ZB!R*Q\"\"YkW$**fTSt()\\m7A'[t!=GyV33X&*HUSk\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!=u2;%=Sq5sp8&\"()4 )[Oj`'H?)4U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\"BwD`%))=jJY R8@o')*>@)\"BL]/!*=h\\'pP4W))4p!Q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"%#!O'3LX5Ql6!fj$)3Qm(>8*QWMGE'*\"O\"3pGI67\\%zQ3GNh **)zC(o([2<$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#\" gn.$[nS]Xk\">2Wca)QNwW%p[6m.DU'*=)[H#\"fn%HS[?'QSy\\>@f/#QrWXVPVh^#))z Lq79#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!=Lc>2nBS q&epa(3;\"me2uucQUj^.9p<&\"Dgv%Gm@VmRa`V&\\hJr_%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!64(QfGB%Qg'35\"6!oEwgE'owwI\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"2)[.z)*4)f:\"\" 2NOfK(z#zv\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!, 831)p%)\"-SWIfu<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C________________ __________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\",`#Q#*[u\".!GGxo'3\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"%#\"3_\\/qqK9A5\"3lC&f8PKsp#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&#\"DB+>me2uucQUj^.9p<&\"Dgv%Gm@VmRa`V&\\h Jr_%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#!64(QfGB%Qg'35 \"6!oEwgE'owwI\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"2 )[.z)*4)f:\"\"2NOfK(z#zv\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6 #\"\")#!,831)p%)\"-SWIfu<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"HnXVk=F)fnY8es.sdQ*> ;\"\"I+q^4Egs&*RWN`TF'z#*45j\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%# b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%#\"LC?'3>7lZ*[VMLB52 =/L#\\=5\"Lv1M0X)RwQ4>ov>$Q?.2:>z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"&#\"YB]Z4&efH&y,[W[0G$f?aIT 0+a:)\\=WbD`uTv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#!NB% 4+\\\"e$y)\\3$foWg,^pGj8)R\"\"N+IFt__:1og0p9^'f[N;&p+')f;7i)Rw>*pg@g$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(R owSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n `RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs) ):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)) :\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order cond tions to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_ 8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have sta ge-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(e xpand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; \+ for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$F$F $F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can calculate the 2 norm of the principal error, that is, the \+ 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'): \nevalf(evalf[14](sqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops(errter ms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3!\\:e#!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 nor m of the principal error of the order 5 embedded scheme is as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs( b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add (subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)eq?s\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "18 of the principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops(errterms6_8) do\n if simplify(subs(e e,errterms6_8[ct]))=0 then ind := [op(ind),ct] end if:\nend do:\nerr6 \+ := PrincipalErrorConditions(6):\n[seq([i,err6[i]],i=ind)]:\nlinalg[aug ment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),lin alg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 747%\"\"'%#~~G/*&%\"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F--F/6#*&F2F-%\"cG F-F-F-F6F-F-#F-\"$/&7%\"#5F)/*&F,F-)F3\"\"#F-#F-\"$_#7%\"#:F)/**F,F-F9 F-F6F-F3F-#F-\"#%)7%\"#?F)/*&F,F--F/6#*(F2F-F6F--F/6#*&)F9FAF-F2F-F-F- FB7%\"#@F)/*&F,F--F/6#*(F2F-FTF-F3F-F-FB7%\"#CF)/*(F,F-F3F-FQF-#F-\"$E \"7%\"#EF)/*&F,F--F/6#*(F2F-F9F-)F6FAF-F-#F-\"$o\"7%\"#GF)/*&F,F-)F6\" \"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F-FTF-FQF-F-Fjn7%\"#MF)/**F,F-F9F- F6F-FQF-#F-\"#U7%\"#PF)/*&F,F--F/6#*(F2F-)F9F[pF-F6F-F-FH7%\"#RF)/*(F, F-FbqF-F3F-Fip7%\"#TF)/*&F,F-)FQFAF-#F-\"#j7%FjpF)/*(F,F-FTF-FcoF-#F-F go7%\"#WF)/*(F,F-FbqF-FQF-#F-FV7%\"#YF)/*(F,F-)F9\"\"%F-F6F-#F-\"#97% \"#ZF)/*&F,F--F/6#*&)F9\"\"&F-F2F-F-Fip7%\"#[F)/*&F,F-)F9F(F-#F-\"\"(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 44 "simultaneous construction of the two schemes" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate th e stage-order equations to ensure that stage 2 has stage-order 2 and s tages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We al so incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1 -c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F, F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 9 simple order conditions used are given (in abreviated form) as foll ows. " }}{PARA 0 "" 0 "" {TEXT -1 97 "These include the 6 order 6 quad rature conditions, 2 additional order 6 order conditions and the " } {TEXT 260 28 "order 7 quadrature condition" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,20,29,32, 64])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F5 7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#?F)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#* &FPF(-FM6#*&FPF(F2F(F(F(F(#F(\"$W\"7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F (#F(\"#C7%\"#KF)/*&F,F()F2FGF(#F(\"\"'7%\"#kF)/*&F,F()F2FboF(#F(\"\"(Q )pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection of 7 \+ \"simple\" order conditions as given (in abreviated form) in the follo wing table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 quadr ature conditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "`SO5*` : = subs(b=`b*`,SimpleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4, 8,9,15,16])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$ (linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"\"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F) /*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F :F(#F(F57%\"\"*F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&FIF(-FF6#*&FIF(F2F(F(F(F (#F(\"$?\"7%\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F (#F(\"\"&Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO7_8 := SimpleOrderConditions(7,8 ,'expanded'):\nSO_eqs := [op(RowSumConditions(8,'expanded')),op(StageO rderConditions(2,8,'expanded')),\n op(StageOrderConditio ns(3,3..8,'expanded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderCondition s(5,9,'expanded')):\nord_cdns := [seq(SO7_8[i],i=[1,2,4,8,16,20,29,32, 64])]:\n`ord_cdns*` := [seq(`SO5_9*`[i],i=[1,2,4,8,9,15,16])]:\nsimp_e qs := [add(b[i]*a[i,1],i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j] *(1-c[j]),j=[6,7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [ op(ord_cdns),op(simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*`)]:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4] = 9/38;" "6#/&% \"cG6#\"\"%*&\"\"*\"\"\"\"#Q!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [6] = 330/613;" "6#/&%\"cG6#\"\"'*&\"$I$\"\"\"\"$8'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\" \"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and the zero linki ng coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{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 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Th e weights of the order 6 scheme provide the linking coefficients for t he 9th stage of the embedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"i G&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" } {TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify \+ that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "`b*`[8] = -7/13;" "6#/&%#b*G6#\"\"),$*&\"\"(\" \"\"\"#8!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "We have 45 equations and 45 unknowns incl uding " }{TEXT 260 18 "three of the nodes" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "c[6]" "6#&% \"cG6#\"\"'" }{TEXT -1 93 " was taken to make the principal error norm a minimum with respect to the chosen values for " }{XPPEDIT 18 0 "c[ 4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#& %\"cG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "preliminary calculation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "e1 := \{c[4]=9/38,c[7]= 63/64,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2 ]=0,`b*`[3]=0,`b*`[8]=-7/13\}:\neqns := subs(e1,cdns):\ne2 := solve(\{ op(eqns)\},indets(eqns) minus \{c[6]\}):\ne3 := `union`(e1,e2):\nerrte rms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsm := 0:\nfor ct to no ps(errterms6_8) do\n sm := sm+(simplify(subs(e3,errterms6_8[ct])))^2 ;\nend do:\nplot(sm,c[6]=0.5..0.6,color=COLOR(RGB,.4,0,1),font=[HELVET ICA,9]);\nevalf[12](findmin(sm,c[6]=0.53..0.55));\nsqrt(op(2,%));\ncon vert(op(1,%%),rational,5);" }}{PARA 13 "" 1 "" {GLPLOT2D 435 263 263 {PLOTDATA 2 "6'-%'CURVESG6#7X7$$\"3++++++++]!#=$\"3ye%)3pF-7$$\"3OL$e9ui2/&F*$\"3X(*zzl#Q=V'F-7$$\"3 g+voMrU^]F*$\"3tsy::9#[#fF-7$$\"3smm\"z_\"4i]F*$\"3wDai\"R2wW&F-7$$\"3 [mmm6m#G2&F*$\"3k&\\$>T)yf*\\F-7$$\"3CmmT&phN3&F*$\"3!)3YE0&)*=d%F-7$$ \"3%**\\(=ddC%4&F*$\"3'HXN)o!ef<%F-7$$\"3kL$e*=)H\\5&F*$\"32%*Qv2F\"\\ !QF-7$$\"3\\m;z/3uC^F*$\"3?3;dOJ.zJF-7$$\"3C+]7LRDX^F*$\"3FEP!=-'45EF- 7$$\"3/n;zR'ok;&F*$\"3m]o26_E*4#F-7$$\"3-+]i5`h(=&F*$\"3u0F1E_\"=m\"F- 7$$\"3,LL$3En$4_F*$\"3Y!**zX\"p+!G\"F-7$$\"3ymmT!RE&G_F*$\"3abRoO]dg** !#E7$$\"3a****\\K]4]_F*$\"3-Vg(*QJp*H(Fhp7$$\"3K++]PAvr_F*$\"3qU*pkuq) [^Fhp7$$\"3E++]nHi#H&F*$\"3X$p2R1J__$Fhp7$$\"36m;z*ev:J&F*$\"3_D;$[lV) )R#Fhp7$$\"3#GLL347TL&F*$\"3aF4)zS\"H^9Fhp7$$\"3kLLLjM?``F*$\"36B\"zh$ Q&[^*!#F7$$\"3#***\\7o7Tv`F*$\"3)*=tUq4'f&oFgr7$$\"3yKLLQ*o]R&F*$\"3M! H#Q1E:iqFgr7$$\"3!)**\\7=lj;aF*$\"3=t`tsidp(*Fgr7$$\"3t**\\PaR%HFhp7$$\"3'om;/T1&*\\&F*$\"3GCIrmL*4&RFhp7$$\"3Om;zRQb@bF *$\"3l7zsd&o=9&Fhp7$$\"31+](=>Y2a&F*$\"3mg!*)[)y!*ziFhp7$$\"3Smm\"zXu9 c&F*$\"3kQH6UXI)f(Fhp7$$\"3u*****\\y))Ge&F*$\"3,\"yBg-$fS!*Fhp7$$\"37+ +DE&QQg&F*$\"3]))*Q@V-90\"F-7$$\"3[**\\7y%3Ti&F*$\"3S1m]o-P)>\"F-7$$\" 3o***\\P![hYcF*$\"31(HWmVy\\O\"F-7$$\"3[LLLQx$om&F*$\"3t@$G,^]j^\"F-7$ $\"3m****\\P+V)o&F*$\"3aqi]f%o#y;F-7$$\"3im;zpe*zq&F*$\"3sQ=#)f\\1C=F- 7$$\"3K****\\#\\'QHdF*$\"3C-b.0^<\")>F-7$$\"3oK$e9S8&\\dF*$\"3EU&*H&>q c7#F-7$$\"3]**\\i?=bqdF*$\"3)4ccccj?F#F-7$$\"3uLL$3s?6z&F*$\"3wTLyW=Z4 CF-7$$\"3=+]7`Wl7eF*$\"3mf.-%z&3YDF-7$$\"3.nmm'*RRLeF*$\"3a+_MSacpEF-7 $$\"33mmTvJgaeF*$\"3OG&eZddly#F-7$$\"3aL$e9tOc(eF*$\"3YOy.f=K#*GF-7$$ \"3'******\\Qk\\*eF*$\"3Gzpl\"=L(zHF-7$$\"3mLL3dg6$F- 7$$\"3e**\\(=5s#yfF*$\"3SD@)4!G5QKF-7$$\"3w**************fF*$\"3%Hg^2q 5PF$F--%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q%c[6]6\"Q!F\\]l-%&CO LORG6&%$RGBG$\"\"%!\"\"$\"\"!Ff]l$\"\"\"Ff]l-%%VIEWG6$;$\"\"&Fd]l$\"\" 'Fd]l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-ZM]Q$Q&!#7$\"-@ &\\$Rkm!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i'[:e#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$I$\"$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 0 188 "e1 := \{c[4 ]=9/38,c[6]=330/613,c[7]=63/64,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4 ..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=-7/13\}:\neqns := subs( e1,cdns):\nnops(%);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Y" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}) :\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3055 "e3 := \{a[8,2] = 0, a[7,2] = 0, a[5,2] = 0, a[6,2] = 0, a[4,2] = 0, `b*`[1] = 116199385772037258 134667598271864434567/1631009927962741533544399572602609517000, c[8] = 1, c[9] = 1, `b*`[6] = -139813632869510160446859308498783581490009423 /231176951635485965114690560680615525273273000, c[7] = 63/64, c[4] = 9 /38, a[7,3] = 837083378582818460922126567546504549/1744817318128300901 22355897926680576, a[4,1] = 9/152, c[3] = 3/19, a[9,2] = 0, a[9,3] = 0 , a[9,8] = -84698060813/177459304440, a[8,7] = -2136870666574914369093 632/99262060255996938363115017, a[5,4] = 96058693980453356374/48760653 286533248361, a[7,1] = -580440564877918386354914362593379258757943/422 227470405208470936638425613284692459520, a[6,4] = 63956422698149174269 3206700036358264060/276008500113722586808059264316424382117, a[8,1] = \+ -73912817561044808721210574412370874448581/420982029653633648809871931 47226166573410, `b*`[7] = 3955511412530100531582592349148563244056576/ 5276949214412532313091563386220545755915925, b[2] = 0, b[3] = 0, `b*`[ 2] = 0, `b*`[3] = 0, a[9,5] = 51769140351634238567474075866190023/4527 1316149543535439664321662847560, c[2] = 2/19, b[6] = -1008660384232859 38709/130767668626607626680, a[6,1] = 77026832972677208895445480061608 49795/12377955859939284746771758057563701031, a[9,1] = 74489238253/108 6687728280, b[7] = 11559809987903488/17579279732593635, c[6] = 330/613 , `b*`[4] = 1018492330418071023333443489476512190862024/27919150703203 83197568190938763984505340675, a[9,6] = -100866038423285938709/1307676 68626607626680, a[9,7] = 11559809987903488/17579279732593635, a[8,5] = 229488189642250366114869444763538854564407191644550406748303/21412703 379882516143374345447138204592119497840386204840294, a[7,4] = -6809200 67893602197172150986392871771764900095/2755062398439077343975851451494 95475886358528, `b*`[8] = -7/13, a[7,5] = 1389390234709131079185946671 16771850303891585832494354565/1644042299545080843782818073486221266498 7734041599934464, a[6,5] = -531318096731274746326483089968527586854828 5/89054878584359562560518476456399170688402362, a[6,3] = -399229043155 79295139013477837035/17050265762136077180720317834126, a[8,4] = -96262 83444389131976638088363590116538104533086/3170748768724798996135280838 794491211302869081, a[8,3] = 821199866821133946316318884532576/1380690 98844093769649611890045033, a[4,3] = 27/152, a[3,2] = 9/76, a[5,3] = - 62238405188783017003/32507102191022165574, a[3,1] = 3/76, b[1] = 74489 238253/1086687728280, `b*`[5] = 73834772250747223839914017280548444801 785295958509475023/754174532555441849815540005413054205931982554187564 49800, b[5] = 51769140351634238567474075866190023/45271316149543535439 664321662847560, a[2,1] = 2/19, c[5] = 483979/844361, b[8] = -84698060 813/177459304440, `b*`[9] = -770493302287576267044727411/3602160699197 6398621216598600, b[4] = 102214327070044952/269723237135952465, a[8,6] = -1608754695857040236707195633/148246171293910549234824945, a[7,6] = -5136972107040184160774194317/610237065866248643642654720, a[9,4] = 1 02214327070044952/269723237135952465, a[5,1] = 50496032797210018219/97 521306573066496722\}:" }}}{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 0 342 "for ii from 2 to 9 do\n print(c[ii]=subs(e3,c[ii]));print(``); \n for jj to ii-1 do\n \+ print(a[ii,jj]=subs(e3,a[ii,jj]))\n end do:\n print(`___________ _______________________`):\nend do:\nprint(``);\nfor ii from 1 to 8 do \n print(b[ii]=subs(e3,b[ii]))\nend do:\nprint(``);\nfor ii from 1 t o 9 do\n print(`b*`[ii]=subs(e3,`b*`[ii]))\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##F'\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\" \"\"#F'\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_____________________ _____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#F'\"# >" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#F'\"#w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\"$\"\"##\"\"*\"#w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C____ ______________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"%#\"\"*\"#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\"#\"\"*\"$_\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$#\"#F\"$_\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"'zR[\"'hV%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\" \"\"#\"5>#=+@(zKg\\]\"5An\\mId18_(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"&\"\"$#!5.q,$y)=0%QA'\"5ub;A5>-r]K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\"5ujN`/)Rpeg*\"5h$[K`'G`1w[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$I$\"$8'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"\"#\"F&z\\3;1![Xa*)3snsH$o-x\"GJ5qjv0e-O*y1?4o\"NG& ej)eZ&\\\\^9&e(RMx!R%)Ri]v#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"(\"\"&#\"ZlXN%\\Kee\"*QI]=x;rm%f=z584ZB!R*Q\"\"YkW$**fTSt()\\m7A '[t!=GyV33X&*HUSk\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"( \"\"'#!=u2;%=Sq5sp8&\"()4)[Oj`'H?)4U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\"BwD`%))=jJYR8@o')*>@)\"BL]/!*=h\\'pP4W))4p !Q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!O'3LX5Ql6 !fj$)3Qm(>8*QWMGE'*\"O\"3pGI67\\%zQ3GNh**)zC(o([2<$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#\"gn.$[nS]Xk\">2Wca)QNwW%p[6m.DU' *=)[H#\"fn%HS[?'QSy\\>@f/#QrWXVPVh^#))zLq79#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!=Lc>2nBSq&epa(3;\"me2uucQUj^.9 p<&\"Dgv%Gm@VmRa`V&\\hJr_%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"'#!64(QfGB%Qg'35\"6!oEwgE'owwI\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"2)[.z)*4)f:\"\"2NOfK(z#zv\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!,831)p%)\"-SWIfu< " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C________________________________ __G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\",`#Q#*[u\".!GGxo'3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"%#\"3_\\/qqK9A5\"3lC&f8PKsp#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"bG6#\"\"&#\"DB+>me2uucQUj^.9p<&\"Dgv%Gm@VmRa`V&\\hJr_%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#!64(QfGB%Qg'35\"6!oEwgE'owwI\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"2)[.z)*4)f:\"\"2NOf K(z#zv\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!,831)p%)\" -SWIfu<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"HnXVk=F)fnY8es.sdQ*>;\"\"I+q^4Egs&*RW N`TF'z#*45j\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%#\"LC?'3>7lZ*[VMLB52=/L#\\=5\"Lv1M0X )RwQ4>ov>$Q?.2:>z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&# \"YB]Z4&efH&y,[W[0G$f?aIT0+a:)\\=WbD`uTv " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#!NB%4+\\\"e$y)\\3$f oWg,^pGj8)R\"\"N+IFt__:1og0p9^'f[N;&p+') f;7i)Rw>*pg@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 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 comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 2970 "ee := \{c[2]=2/19,\nc[3]=3/19,\nc[4]=9/38,\nc[5]= 483979/844361,\nc[6]=330/613,\nc[7]=63/64,\nc[8]=1,\nc[9]=1,\n\na[2,1] =2/19,\na[3,1]=3/76,\na[3,2]=9/76,\na[4,1]=9/152,\na[4,2]=0,\na[4,3]=2 7/152,\na[5,1]=50496032797210018219/97521306573066496722,\na[5,2]=0,\n a[5,3]=-62238405188783017003/32507102191022165574,\na[5,4]=96058693980 453356374/48760653286533248361,\na[6,1]=770268329726772088954454800616 0849795/12377955859939284746771758057563701031,\na[6,2]=0,\na[6,3]=-39 922904315579295139013477837035/17050265762136077180720317834126,\na[6, 4]=639564226981491742693206700036358264060/276008500113722586808059264 316424382117,\na[6,5]=-5313180967312747463264830899685275868548285/890 54878584359562560518476456399170688402362,\na[7,1]=-580440564877918386 354914362593379258757943/422227470405208470936638425613284692459520,\n a[7,2]=0,\na[7,3]=837083378582818460922126567546504549/174481731812830 090122355897926680576,\na[7,4]=-68092006789360219717215098639287177176 4900095/275506239843907734397585145149495475886358528,\na[7,5]=1389390 23470913107918594667116771850303891585832494354565/\n 1644042299 5450808437828180734862212664987734041599934464,\na[7,6]=-5136972107040 184160774194317/610237065866248643642654720,\na[8,1]=-7391281756104480 8721210574412370874448581/42098202965363364880987193147226166573410,\n a[8,2]=0,\na[8,3]=821199866821133946316318884532576/138069098844093769 649611890045033,\na[8,4]=-96262834443891319766380883635901165381045330 86/\n 3170748768724798996135280838794491211302869081, a[8,5]=22 9488189642250366114869444763538854564407191644550406748303/\n 21 412703379882516143374345447138204592119497840386204840294,\na[8,6]=-16 08754695857040236707195633/148246171293910549234824945,\na[8,7]=-21368 70666574914369093632/99262060255996938363115017,\na[9,1]=74489238253/1 086687728280,\na[9,2]=0,\na[9,3]=0,\na[9,4]=102214327070044952/2697232 37135952465,\na[9,5]=51769140351634238567474075866190023/4527131614954 3535439664321662847560,\na[9,6]=-100866038423285938709/130767668626607 626680,\na[9,7]=11559809987903488/17579279732593635,\na[9,8]=-84698060 813/177459304440,\n\nb[1]=74489238253/1086687728280,\nb[2]=0,\nb[3]=0, \nb[4]=102214327070044952/269723237135952465,\nb[5]=517691403516342385 67474075866190023/45271316149543535439664321662847560,\nb[6]=-10086603 8423285938709/130767668626607626680,\nb[7]=11559809987903488/175792797 32593635,\nb[8]=-84698060813/177459304440,\n\n`b*`[1]=1161993857720372 58134667598271864434567/1631009927962741533544399572602609517000,\n`b* `[2]=0,\n`b*`[3]=0,\n`b*`[4]=10184923304180710233334434894765121908620 24/2791915070320383197568190938763984505340675,\n`b*`[5]=7383477225074 7223839914017280548444801785295958509475023/\n 7541745325554418 4981554000541305420593198255418756449800,\n`b*`[6]=-139813632869510160 446859308498783581490009423/231176951635485965114690560680615525273273 000,\n`b*`[7]=3955511412530100531582592349148563244056576/527694921441 2532313091563386220545755915925,\n`b*`[8]=-7/13,\n`b*`[9]=-77049330228 7576267044727411/36021606991976398621216598600\}:" }}}{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[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denot e the vector whose components are the principal error terms of the 8 s tage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose components are the principal \+ error terms of the embedded 9 stage, order 5 scheme (the error terms o f order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\" *" }{TEXT -1 99 " denote the vector whose components are the error te rms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&% \"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]) );" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" \"*" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG 6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs( `T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\" \"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") e@*3\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")po.6! \"(" }}}{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 2970 "ee := \{c[ 2]=2/19,\nc[3]=3/19,\nc[4]=9/38,\nc[5]=483979/844361,\nc[6]=330/613,\n c[7]=63/64,\nc[8]=1,\nc[9]=1,\n\na[2,1]=2/19,\na[3,1]=3/76,\na[3,2]=9/ 76,\na[4,1]=9/152,\na[4,2]=0,\na[4,3]=27/152,\na[5,1]=5049603279721001 8219/97521306573066496722,\na[5,2]=0,\na[5,3]=-62238405188783017003/32 507102191022165574,\na[5,4]=96058693980453356374/48760653286533248361, \na[6,1]=7702683297267720889544548006160849795/12377955859939284746771 758057563701031,\na[6,2]=0,\na[6,3]=-39922904315579295139013477837035/ 17050265762136077180720317834126,\na[6,4]=6395642269814917426932067000 36358264060/276008500113722586808059264316424382117,\na[6,5]=-53131809 67312747463264830899685275868548285/8905487858435956256051847645639917 0688402362,\na[7,1]=-580440564877918386354914362593379258757943/422227 470405208470936638425613284692459520,\na[7,2]=0,\na[7,3]=8370833785828 18460922126567546504549/174481731812830090122355897926680576,\na[7,4]= -680920067893602197172150986392871771764900095/27550623984390773439758 5145149495475886358528,\na[7,5]=13893902347091310791859466711677185030 3891585832494354565/\n 16440422995450808437828180734862212664987 734041599934464,\na[7,6]=-5136972107040184160774194317/610237065866248 643642654720,\na[8,1]=-73912817561044808721210574412370874448581/42098 202965363364880987193147226166573410,\na[8,2]=0,\na[8,3]=8211998668211 33946316318884532576/138069098844093769649611890045033,\na[8,4]=-96262 83444389131976638088363590116538104533086/\n 317074876872479899 6135280838794491211302869081, a[8,5]=229488189642250366114869444763538 854564407191644550406748303/\n 214127033798825161433743454471382 04592119497840386204840294,\na[8,6]=-1608754695857040236707195633/1482 46171293910549234824945,\na[8,7]=-2136870666574914369093632/9926206025 5996938363115017,\na[9,1]=74489238253/1086687728280,\na[9,2]=0,\na[9,3 ]=0,\na[9,4]=102214327070044952/269723237135952465,\na[9,5]=5176914035 1634238567474075866190023/45271316149543535439664321662847560,\na[9,6] =-100866038423285938709/130767668626607626680,\na[9,7]=115598099879034 88/17579279732593635,\na[9,8]=-84698060813/177459304440,\n\nb[1]=74489 238253/1086687728280,\nb[2]=0,\nb[3]=0,\nb[4]=102214327070044952/26972 3237135952465,\nb[5]=51769140351634238567474075866190023/4527131614954 3535439664321662847560,\nb[6]=-100866038423285938709/13076766862660762 6680,\nb[7]=11559809987903488/17579279732593635,\nb[8]=-84698060813/17 7459304440,\n\n`b*`[1]=116199385772037258134667598271864434567/1631009 927962741533544399572602609517000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=10 18492330418071023333443489476512190862024/2791915070320383197568190938 763984505340675,\n`b*`[5]=73834772250747223839914017280548444801785295 958509475023/\n 75417453255544184981554000541305420593198255418 756449800,\n`b*`[6]=-139813632869510160446859308498783581490009423/231 176951635485965114690560680615525273273000,\n`b*`[7]=39555114125301005 31582592349148563244056576/5276949214412532313091563386220545755915925 ,\n`b*`[8]=-7/13,\n`b*`[9]=-770493302287576267044727411/36021606991976 398621216598600\}:" }}}{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 77 "The stability function R for the 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR := una pply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#% \"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F )*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)* $)F'F1F)F)F)*&#\"9**\\HjWBr:B&o-#\"=?&\\'*)[#***H!eR])o5F)*$)F'\"\"(F) F)F)*&#\"9^*o^?^q'zM]\\Q\">?BRM#)z)*z%GL1;5 " 0 "" {MPLTEXT 1 0 26 "z0 := newt on(R(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+I*z)HV!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-4):\np1 := plot([R(z),1],z=-4 .89..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symb ol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],li nestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[- 4.89..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o************* )[!#<$\"3wjWSpj\\sFF*7$$!3OLekFGog[F*$\"3]01;\")pANEF*7$$!3)\\%F*$\"3?lKeQejt8F*7$$!3iK$e'y!Q/W%F*$\"326?6gz:L 7F*7$$!3')\\7LEy&HQ%F*$\"3)QzF;M+k5\"F*7$$!3AmT+uvZDVF*$\"3m_=/tq3;**! #=7$$!3_L3AqW*)=UF*$\"3sH&)>74Sr!)Fco7$$!3`*\\(yRQ`3TF*$\"3[8hR'3]**\\ 'Fco7$$!3!H$3#zr)R%*RF*$\"3Qzv`o&*G!=&Fco7$$!3u*\\P)G'H1)QF*$\"3m6$=8< zM7%Fco7$$!3QmmrOTgjPF*$\"3;9z)))f#**fKFco7$$!3'HLe&>!G0m$F*$\"3QV$)[Z E`bEFco7$$!3\"****\\^s)[WNF*$\"3XN%==;Y(>@Fco7$$!3q***\\A'H(zU$F*$\"3M 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1401 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+\n 202685231571234463294999/10688503958 02999924889649520*z^7+\n 384950347967051205168951/171016063328479 98798234392320*z^8:\npts := []:\nz0 := 0:\nfor ct from 0 to 240 do\n \+ zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.4,.1 ,0)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.1,0]],i=2..n ops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95,.35,.1)): \npts := []: z0 := 2+4.8*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),I m(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.4,.1,0)):\np4 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[1.90,4.77]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.95,.35,.1)):\npts := [] : z0 := 2-4.8*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)]]:\ne nd do:\np5 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" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 148 "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 there is " }{TEXT 260 53 "no largest \+ interval on the nonnegative imaginary axis" }{TEXT -1 65 " that contai ns the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability region intersects the nonnegati ve imaginary axis in an interval that does not contain the origin." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "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 \+ 202685231571234463294999/1068850395802999924889649520*z^7+\n 38 4950347967051205168951/17101606332847998798234392320*z^8:\nDigits := 2 5:\npts := []: z0 := 0:\nfor ct from 0 to 112 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,.95,.35,0),thicknes s=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7]r7$$\"\"!F)F(7$$!:/\")4%p(e'= ;z=DF!#E$\":EJ:Cnw*e`EfTJF-7$$!:$yALA8^Q*zU5^%F-$\":$*e(p>)>wrI&=$G'F- 7$$!:R*3L$3*e,%4Bq0'F-$\":hQi#eX;rgzxC%*F-7$$!:w_=PCwuC,*ekuF-$\":*QeX fIGRhqjc7!#D7$$!:Gt\"[T&eCb6enx)F-$\":oL2.:X!fEjzq:F?7$$!:w!y2%y4/7$Hp ,5F?$\":9%HO\"yqD9fb\\)=F?7$$!:i\"pqkaQ)z>h*>6F?$\":Z#3(p(eeQb[6*>#F?7 $$!:qN`=#3b`YtXL7F?$\":j?%)GN\")*\\F4wLVFGF?7$$!:3*og\\R3$emO([9F?$\":nutN/\"fZGEfTJF?7$$!:V!R^?GJ/O8 V^:F?$\":rDt/*z>dq=vbMF?7$$!:7ye9[z)*p&4F^;F?$\":_:zYU$yZ'46*pPF?7$$!: SG+vZK@I6.&[6>.BltRYH# )R%F?7$$!:Fo*f*R&yq3@\"f$>F?$\":Ng[0hoq*z&)Q7ZF?7$$!:%R#[R:4)Rk+RE?F?$ \":?,!fi]KUFwaE]F?7$$!:oKxkO\\Kh4w[6#F?$\":)[r%e6 UfXL3EF?$\":*zGDS=IfwjlDsF?7$$!:,lMrqp7bOB[o#F?$\":9e*4])=oca8)RvF?7$$ !:+rHR8StN(evfFF?$\":Q]U$f3%=[@qR&yF?7$$!:/N#oSd!ydb\\J$GF?$\":#>y8(oi a=IE\"o\")F?7$$!:A*)G>X5`'4h,0HF?$\":.#zAid:1=`OM:!f5\"*F?7$$!:, cwn4c&\\&*))[6JF?$\":`F,zlhUs'HuC%*F?7$$!:Fw#puDk'4rcs<$F?$\":4g?tjMT2 q%*)Q(*F?7$$!:k!3rE-yq#=w9C$F?$\":t!zL^h+4CXI05!#C7$$!:/W<=Vn)\\hr7/LF ?$\":Q=Qd6VSrW>n.\"Ffu7$$!:;m,R/JZ>w$=lLF?$\":%G)ftM$)ysAM\"o5Ffu7$$!: CY#H@KadjRhCMF?$\":]r,YAe=A&)[&*4\"Ffu7$$!:v%*H&=_M*43zB[$F?$\":wmV\"Ffu7$$!:xJR`)RR%pQ\"=XOF?$\":A*G+CCFsNb?D7F fu7$$!:`[euW5UY%yt&p$F?$\":?>`=X?(Hy\">mD\"Ffu7$$!:xDUCWUxLM/Vu$F?$\": N%)f`AQWTdK!)G\"Ffu7$$!:#e8Ar&=.TYz2z$F?$\":zNa=Q`33qX%>8Ffu7$$!:hK-Lr Z$ybU/NQF?$\":9diig[V:`e3N\"Ffu7$$!:6?@;NpUQbep(QF?$\":5S.JH7ZR.rAQ\"F fu7$$!:sb%)e30S_;cj\"RF?$\":C6/Qk6WSF?$\":B0bs4<[ ([oKR:Ffu7$$!:R*e\"3'GyjnA#p1%F?$\":\"Gc8?VdqChtq:Ffu7$$!:=T$*ph$3Gq*y ]3%F?$\":6rU%[>#)>'\\W@g\"Ffu7$$!:#4NZ\\ZAR'[=y4%F?$\":@a)4_Mrq*y^Nj\" Ffu7$$!:W;3U6VK-<,T5%F?$\":O&41$=*3A%zd\\m\"Ffu7$$!:;xaV'*>fb(pY-TF?$ \":$e]l>:SeaAO'p\"Ffu7$$!:H\"zWm-!3m>m24%F?$\":zGpi3)G-k[wF#=Ffu7$$!:5\"y;v4? 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "Digits := 15:\nz0 := 1.9*I:\nfor ct from 59 to 62 do\n newton(R (z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.15*I:\nfor ct from 10 1 to 104 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits : = 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!00'HS=i![#!#>$\"05RkN[L&= !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0E`M'Qx)))*!#?$\"0Dt_mMZ)=! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"05)od\\aI\"*!#?$\"0w(fjc6;> !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0)*>?5$Q$G$!#>$\"0mHZR!\\Z >!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0kR)=]Sh8!#<$\"0mi(f1bEJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0')y'\\`p4]!#=$\"0Ns>k(y`J!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0coenN$y]!#=$\"0h[ " 0 "" {MPLTEXT 1 0 330 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.9*I))\ne nd proc:\nu0 := bisect('real_part'(u),u=0.59..0.62);\nnewton(R(z)=exp( u0*Pi*I),z=1.9*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*P i*I),z=3.15*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.01..1.04); \nnewton(R(z)=exp(u0*Pi*I),z=3.15*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0sW3pa[0'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0MVT^[>G#!#H$\"0dv.L\\>!>!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0fi1(f:D5!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o==Bttb$!#H$\"0\"*H&H!Hx;$!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 73 "stability region intersects the nonegative imaginary axi s in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[1.9019, 3.1677]; " "6#7$-%&FloatG6$\"&>!>!\"%-F%6$\"&x;$F(" }{TEXT -1 18 " (approximat ely)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " #------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded '))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"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)*&#\"WnsIrQcv_^)\\l[)fCt@RbP#3;\"Z+g,o:[Xv4W?e70g;!zx h\\A]sm70b6F)*$)F'F1F)F)F)*&#\"TPW^[48elmQA/7R\\KALL;Ton#HI&paSaBWgB e@)\"Y+?nA0;:D.[t_Po')Qj#f?$3M3*)3<]QF)*$)F'\"\")F)F)F)*&#\"The\"e*Rb( fY`lu3)zjRV&[)=#[m,mH\"Z+?vi$oDC?&ovV+%p=U@[HJ`M`AMFghF)*$)F'\"\"*F)F) !\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region \+ intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#% \"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z_0 := newton(`R*`(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$z_0G$!+*y`KV%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-4):\np_1 := p lot([`R*`(z),1],z=-5.19..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[di splay]([p_1,p_2,p_3],view=[-5.19..0.49,-0.07..1.47],font=[HELVETICA,9] );" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG 6$7Z7$$!3Q++++++!>&!#<$\"3%QL(Qc$H(>OF*7$$!3QML3T![!f^F*$\"3DlSThz$eS$ F*7$$!3Ynm;#3'4G^F*$\"3*Gb;MjvH?$F*7$$!3a++DBT9(4&F*$\"3fvLND[l5IF*7$$ !3kLLLk@>m]F*$\"389i%*zhSGGF*7$$!3E+]U'*)HB,&F*$\"3xpe(*4+%R`#F*7$$!3! pm;&GwYe\\F*$\"3iq!y?KdjE#F*7$$!3s+](\\(Q*y*[F*$\"3ydPoA&H\\*>F*7$$!3n LLV@,KP[F*$\"3'Qw'4fy/_%y^*QF *$\"3L.gn^qIR;Fho7$$!35++S:-YpPF*$\"3_P*R_d$\\j6Fho7$$!3K+++\"HZkk$F*$ \"31\"=sgCGjN)!#>7$$!3;++gW:!z_$F*$\"3I1fXx(oTG'F\\r7$$!3hLL)*\\1D?MF* $\"3_p=Fm)=06&F\\r7$$!3'ommSKVAH$F*$\"3y+rTz%3\"zVF\\r7$$!3/nmEGV!Q=$F *$\"3s&yLioBv=%F\\r7$$!39++0(*RmdIF*$\"3XRfgoswKVF\\r7$$!39nmEI%3g%HF* $\"3\\Rlu)3lAr%F\\r7$$!3-++0xX]BGF*$\"31/YKPD6O`F\\r7$$!3*)***\\\"R>&o q#F*$\"3-4?t>ef\"4'F\\r7$$!3gmm;\\r8&e#F*$\"3%)3z)y\")o8-(F\\r7$$!3ymm rw\\OtCF*$\"3ez=F*$\"34LbzM`D<:Fho7$$ !3;++!4<(>gFho7$$! 3H++q9zA<:F*$\"3Ul@*>Dh2>#Fho7$$!3EnmEY;O-9F*$\"3ul!R\"z?ueCFho7$$!3#) *****pQ<(z7F*$\"39Is\\w$>/y#Fho7$$!3)RL$efMeo6F*$\"3j,Mf%z%o2JFho7$$!3 I****fAZ3Z5F*$\"3!on,A*GU4NFho7$$!3xqm;(zQwK*Fho$\"3CS9XscdMRFho7$$!3& z***\\)ecE8)Fho$\"3^LcL7A0MWFho7$$!3'3nmm0VV'pFho$\"3?!>\\GR%e$)\\Fho7 $$!3P)***\\iqATdFho$\"3f#=Ae;')>j&Fho7$$!3aFLL*)4AjXFho$\"3Kq)*y+r4OjF ho7$$!33LLLO'R&eLFho$\"3%zyTjwus9(Fho7$$!3Uim;`O$Q;#Fho$\"3kFFB0ZEa!)F ho7$$!3?*****>$H-m5Fho$\"37CKRQ4$)))*)Fho7$$\"3v*QLLU?>#>F\\r$\"3$H?P? 30%>5F*7$$\"3%ymmY^avJ\"Fho$\"3dWl2(HH39\"F*7$$\"3E0+]HcU&!\"#$\"#\\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 7 stage, order 5 scheme. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1787 "`R*` := \+ z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 160823755392173245 984865498515275563819900391771307267/\n 1155051266725022496177790 16600512582044097545481568016000*z^6+\n 4275518429762946566465717 12042238666558130948514437/\n 21258612271012683978732334343652622 46210384272666896000*z^7+\n 8215823604423544054695302926768411633 33223249391911/\n 38501708890834083205926338866837527348032515160 522672000*z^8-\n 296601664821884854339637980874655346597553995815 861/\n 616027342253345331294821421869400437568520242568362752000* z^9:\npts := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(`R *`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.35,0,0)):\np_2 : = plots[polygonplot]([seq([pts[i-1],pts[i],[-2.3,0]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.75,.25,0)):\npts := []: z0 := 2+4.6*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(c t*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,.35,0,0)):\np_4 := plots[po lygonplot]([seq([pts[i-1],pts[i],[1.89,4.58]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.25,0)):\npts := []: z0 := \+ 2-4.6*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30 *I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np_5 := plot(pts,color=COLOR(RGB,.35,0,0)):\np_6 := plots[polygonplo t]([seq([pts[i-1],pts[i],[1.89,-4.58]],i=2..nops(pts))],\n st yle=patchnogrid,color=COLOR(RGB,.75,.25,0)):\np_7 := plot([[[-5.29,0], [2.29,0]],[[0,-5.09],[0,5.09]]],color=black,linestyle=3):\nplots[displ ay]([p_||(1..7)],view=[-5.29..2.29,-5.09..5.09],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7] z7$$\"\"!F)F(7$$\"3%******>l6qO&!#G$\"3++++Fjzq:!#=7$$\"3o*****\\s#eeP 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YFd]oFihcl7%F]]dl7$Fhebl$!+c-KyYFd]oFihcl7%Fa]dl7$Fbebl$!+Ic%>o%Fd]oFi hcl7%Fe]dl7$F\\ebl$!+&=\"4(o%Fd]oFihcl7%Fi]dl7$Ffdbl$!+;Nr$p%Fd]oFihcl 7%F]^dl7$F`dbl$!+$y8X'[Fd]oFihcl7%Ficdl7$Ff[bl$!+R9Vj[Fd]oFihcl 7%F]ddl7$F`[bl$!+g-gh[Fd]oFihcl7%Faddl7$Fjjal$!+Mu0f[Fd]oFihcl7%Feddl7 $Fdjal$!+hc$e&[Fd]oFihcl7%Fiddl7$F^jal$!+R&o>&[Fd]oFihcl7%F]edl7$Fhial $!+X0\\Z[Fd]oFihcl7%Faedl7$Fbial$!+tqVU[Fd]oFihcl7%Feedl7$F\\ial$!+-X% o$[Fd]oFihcl7%Fiedl7$Ffhal$!+:.vI[Fd]oFihcl7%F]fdl7$F`hal$!+UK>C[Fd]oF ihcl7%Fafdl7$Fjgal$!+QM@<[Fd]oFihcl7%Fefdl7$Fdgal$!+$p_)4[Fd]oFihcl7%F ifdl7$F^gal$!+qY:-[Fd]oFihcl7%F]gdl7$Fcfal$!+#=lTz%Fd]oFihcl7%FagdlFch clFihclFgb`lFefs-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&I m(z)G-Ffgdl6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOX G-%%VIEWG6$;$!$H&F_jn$\"$H#F_jn;$!$>&F_jn$\"$>&F_jn" 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 35 "coefficients of 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 2970 "ee := \{c[2]=2/19,\nc[3]=3/19,\nc [4]=9/38,\nc[5]=483979/844361,\nc[6]=330/613,\nc[7]=63/64,\nc[8]=1,\nc [9]=1,\n\na[2,1]=2/19,\na[3,1]=3/76,\na[3,2]=9/76,\na[4,1]=9/152,\na[4 ,2]=0,\na[4,3]=27/152,\na[5,1]=50496032797210018219/975213065730664967 22,\na[5,2]=0,\na[5,3]=-62238405188783017003/32507102191022165574,\na[ 5,4]=96058693980453356374/48760653286533248361,\na[6,1]=77026832972677 20889544548006160849795/12377955859939284746771758057563701031,\na[6,2 ]=0,\na[6,3]=-39922904315579295139013477837035/17050265762136077180720 317834126,\na[6,4]=639564226981491742693206700036358264060/27600850011 3722586808059264316424382117,\na[6,5]=-5313180967312747463264830899685 275868548285/89054878584359562560518476456399170688402362,\na[7,1]=-58 0440564877918386354914362593379258757943/42222747040520847093663842561 3284692459520,\na[7,2]=0,\na[7,3]=837083378582818460922126567546504549 /174481731812830090122355897926680576,\na[7,4]=-6809200678936021971721 50986392871771764900095/275506239843907734397585145149495475886358528, \na[7,5]=138939023470913107918594667116771850303891585832494354565/\n \+ 16440422995450808437828180734862212664987734041599934464,\na[7,6 ]=-5136972107040184160774194317/610237065866248643642654720,\na[8,1]=- 73912817561044808721210574412370874448581/4209820296536336488098719314 7226166573410,\na[8,2]=0,\na[8,3]=821199866821133946316318884532576/13 8069098844093769649611890045033,\na[8,4]=-9626283444389131976638088363 590116538104533086/\n 31707487687247989961352808387944912113028 69081, a[8,5]=22948818964225036611486944476353885456440719164455040674 8303/\n 21412703379882516143374345447138204592119497840386204840 294,\na[8,6]=-1608754695857040236707195633/148246171293910549234824945 ,\na[8,7]=-2136870666574914369093632/99262060255996938363115017,\na[9, 1]=74489238253/1086687728280,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1022143270 70044952/269723237135952465,\na[9,5]=517691403516342385674740758661900 23/45271316149543535439664321662847560,\na[9,6]=-100866038423285938709 /130767668626607626680,\na[9,7]=11559809987903488/17579279732593635,\n a[9,8]=-84698060813/177459304440,\n\nb[1]=74489238253/1086687728280,\n b[2]=0,\nb[3]=0,\nb[4]=102214327070044952/269723237135952465,\nb[5]=51 769140351634238567474075866190023/45271316149543535439664321662847560, \nb[6]=-100866038423285938709/130767668626607626680,\nb[7]=11559809987 903488/17579279732593635,\nb[8]=-84698060813/177459304440,\n\n`b*`[1]= 116199385772037258134667598271864434567/163100992796274153354439957260 2609517000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1018492330418071023333443 489476512190862024/2791915070320383197568190938763984505340675,\n`b*`[ 5]=73834772250747223839914017280548444801785295958509475023/\n \+ 75417453255544184981554000541305420593198255418756449800,\n`b*`[6]=-13 9813632869510160446859308498783581490009423/23117695163548596511469056 0680615525273273000,\n`b*`[7]=3955511412530100531582592349148563244056 576/5276949214412532313091563386220545755915925,\n`b*`[8]=-7/13,\n`b*` [9]=-770493302287576267044727411/36021606991976398621216598600\}:" }}} {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 31 "s eq(c[i]=subs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&% \"cG6#\"\"##F'\"#>/&F%6#\"\"$#F-F)/&F%6#\"\"%#\"\"*\"#Q/&F%6#\"\"&#\"' zR[\"'hV%)/&F%6#\"\"'#\"$I$\"$8'/&F%6#\"\"(#\"#j\"#k/&F%6#\"\")\"\"\"/ &F%6#F4FO" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#F'\"#>/&F%6$ \"\"$F(#F.\"#w/&F%6$F.F'#\"\"*F0/&F%6$\"\"%F(#F5\"$_\"/&F%6$F9F'\"\"!/ &F%6$F9F.#\"#FF;/&F%6$\"\"&F(#\"5>#=+@(zKg\\]\"5An\\mId18_(*/&F%6$FHF' F?/&F%6$FHF.#!5.q,$y)=0%QA'\"5ub;A5>-r]K/&F%6$FHF9#\"5ujN`/)Rpeg*\"5h$ [K`'G`1w[/&F%6$\"\"'F(#\"F&z\\3;1![Xa*)3snsH$o-x\"GJ5qjv0e-O*y1?4o\"NG&ej)eZ&\\\\^9&e(RMx!R%)Ri]v#/&F%6$Fdp FH#\"ZlXN%\\Kee\"*QI]=x;rm%f=z584ZB!R*Q\"\"YkW$**fTSt()\\m7A'[t!=GyV33 X&*HUSk\"/&F%6$FdpFhn#!=u2;%=Sq5sp8&\"()4)[Oj`'H?)4U/&F%6$FfrF'F?/&F %6$FfrF.#\"BwD`%))=jJYR8@o')*>@)\"BL]/!*=h\\'pP4W))4p!Q\"/&F%6$FfrF9#! O'3LX5Ql6!fj$)3Qm(>8*QWMGE'*\"O\"3pGI67\\%zQ3GNh**)zC(o([2<$/&F%6$FfrF H#\"gn.$[nS]Xk\">2Wca)QNwW%p[6m.DU'*=)[H#\"fn%HS[?'QSy\\>@f/#QrWXVPVh^ #))zLq79#/&F%6$FfrFhn#!=Lc>2nBSq&epa(3;\"me2uucQUj^.9p<&\"Dgv%Gm@VmRa`V&\\hJr_%/&F%6$F5Fhn#!64(QfGB% Qg'35\"6!oEwgE'owwI\"/&F%6$F5Fdp#\"2)[.z)*4)f:\"\"2NOfK(z#zv\"/&F%6$F5 Ffr#!,831)p%)\"-SWIfu<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 6 scheme" }{TEXT -1 2 ": " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]),i=1.. 8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\",`#Q#*[u\".!G Gxo'3\"/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"3_\\/qqK9A5\"3lC&f8P Ksp#/&F%6#\"\"&#\"DB+>me2uucQUj^.9p<&\"Dgv%Gm@VmRa`V&\\hJr_%/&F%6#\"\" '#!64(QfGB%Qg'35\"6!oEwgE'owwI\"/&F%6#\"\"(#\"2)[.z)*4)f:\"\"2NOfK(z#z v\"/&F%6#\"\")#!,831)p%)\"-SWIfu<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs (ee,`b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\" \"#\"HnXVk=F)fnY8es.sdQ*>;\"\"I+q^4Egs&*RWN`TF'z#*45j\"/&F%6#\"\"#\"\" !/&F%6#\"\"$F//&F%6#\"\"%#\"LC?'3>7lZ*[VMLB52=/L#\\=5\"Lv1M0X)RwQ4>ov> $Q?.2:>z#/&F%6#\"\"&#\"YB]Z4&efH&y,[W[0G $f?aIT0+a:)\\=WbD`uTv/&F%6#\"\"'#!NB%4+\\\"e$y)\\3$foWg,^pGj8)R\"\"N+I Ft__:1og0p9^'f[N;&p+')f;7i)Rw>*pg@g$" }}}{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 35 "#============ ======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme \+ with " }{XPPEDIT 18 0 "c[4]=7/30" "6#/&%\"cG6#\"\"%*&\"\"(\"\"\"\"#I! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=81/82" "6#/&%\"cG6#\" \"(*&\"#\")\"\"\"\"##)!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------- -----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2433 "ee := \{c[ 2]=14/135,\nc[3]=7/45,\nc[4]=7/30,\nc[5]=51612/89803,\nc[6]=49/87,\nc[ 7]=81/82,\nc[8]=1,\nc[9]=1,\n\na[2,1]=14/135,\na[3,1]=7/180,\na[3,2]=7 /60,\na[4,1]=7/120,\na[4,2]=0,\na[4,3]=7/40,\na[5,1]=19468520817039492 /35486945168446723,\na[5,2]=0,\na[5,3]=-72573280055824680/354869451684 46723,\na[5,4]=73499982271800480/35486945168446723,\na[6,1]=2025265834 0120203773069749/33784135725639504675502296,\na[6,2]=0,\na[6,3]=-79221 6000912879280500/351533212217208184283,\na[6,4]=2249034659172295455085 05100/100340318815275363580633077,\na[6,5]=-21679358721707162238705728 210467/901105029036754619331512380401576,\na[7,1]=-9721587211241934408 14630918467125/799434474429485902833274628345432,\na[7,2]=0,\na[7,3]=3 058612407219548732633706208815/709347359742223516267324426216,\na[7,4] =-53478378574860120460721550328853039040/23405367722957560812286413128 409759761,\na[7,5]=524767583662593596626716546857270370413091/24088433 982273003693314369668047287199799,\na[7,6]=-696206010458633764414572/3 2219569565392536806647,\na[8,1]=-114361999204933693744433721507907/778 77427759375157832377305006776,\na[8,2]=0,\na[8,3]=74051210740135739810 98320/1457324938194042260233261,\na[8,4]=-2993009538247695784606055118 2764365/11155804472406678900783685305189917,\na[8,5]=26499197252515606 41426119276806860337581199555/1007124822780127331605758150395071698262 41912,\na[8,6]=-42066702439513241792550/1604007647963024581459,\na[8,7 ]=-425342741234074038059/27197351224216978505337,\na[9,1]=5847536753/8 6036171760,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1815811171875/4929812076868, \na[9,5]=74928498684378188697374218447/29744854556648749370324659440, \na[9,6]=-6206299092285813/2910288487182440,\na[9,7]=1044913680622457/ 1298591690996340,\na[9,8]=-418874561/667578680,\n\nb[1]=5847536753/860 36171760,\nb[2]=0,\nb[3]=0,\nb[4]=1815811171875/4929812076868,\nb[5]=7 4928498684378188697374218447/29744854556648749370324659440,\nb[6]=-620 6299092285813/2910288487182440,\nb[7]=1044913680622457/129859169099634 0,\nb[8]=-418874561/667578680,\n\n`b*`[1]=1360833350266810598272988918 9/197441635892684338840372330104,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=715 4586848463452050019495500125/19675246518315548228466935287856,\n`b*`[5 ]=85316751507451678656433588557034105747933/36476436347097667766853358 930957710182664,\n`b*`[6]=-6519743523284530424551776811925577/33393635 95999139623414387108668538,\n`b*`[7]=202517037851021117714407375979242 69/23840769535138099800229313089019088,\n`b*`[8]=-25/38,\n`b*`[9]=-245 172123941262584/22948677498512207029\}:" }}}{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 0 342 "for \+ ii from 2 to 9 do\n print(c[ii]=subs(ee,c[ii]));print(``); \n for \+ jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]))\n end do:\n \+ print(`__________________________________`):\nend do:\nprint(``);\nfo r ii from 1 to 8 do\n print(b[ii]=subs(ee,b[ii]))\nend do:\nprint(`` );\nfor ii from 1 to 9 do\n print(`b*`[ii]=subs(ee,`b*`[ii]))\nend d o:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"#9\"$N\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"#\"\"\"#\"#9\"$N\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C __________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"$#\"\"(\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"\"(\"$!=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\"(\"#g" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"(\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"%\"\"\"#\"\"(\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\" \"$#\"\"(\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C___________________ _______________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"& 7;&\"&.)*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\"#\"2#\\Rq\"3_o%>\"2BnWo^%p[N" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#!2!oCe0!GtD(\"2BnWo^%p[N" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\"2![+=F#)**\\t\"2 BnWo^%p[N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_______________________ ___________G" }}{PARA 11 "" 1 "" {XPPMATH 20 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"6#/&%\"aG6$\"\"(\"\"&#\"K\"48/Pqs&oa;ni'f$fiOenZ_\"J*z*>( GZ!o'pVJ$p.IF#)RV)3C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"( \"\"'#!9sXTkPje/,1ip\"8Zm!o`#Rl&p&>A$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!B2z]@PVWPpL\\?** >O9\"\"Awn+0tPKy:v$fxUxy(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$ #\":?$)4\")Rd8S2@^S(\":hKBgA/%>Q\\Kd9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!DlVw#=^01Yy&pZ#Q&4I*H\"D<**=0`o$y+*ymSsW!e:6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#\"Ob&*>\"eP.'o!o F>hUT1c^_s>*\\E\"N7>CE)pr]R]\"edgJt7!yA[725" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!8]DzTK^RCqm?%\"7f9eCI'zk2Sg\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!6f!QS2M7uU`U\"8P` ]yp@C7N(>F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________________ ____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"+`n`Ze\",g<Z%=UP(p)=yVo)\\G\\(\">S%fYKq$\\([mba[uH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!18eG#4*H1i\"1SC =([)G5H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"1dCi! o8\\/\"\"1Sj*4p\"f)H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\")#!*hX()=%\"*!oyvm" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_______ ___________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"+`n`Ze\",g<Z%=UP(p)=yVo)\\G\\(\">S%fYKq$\\([mba[uH" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#!18eG#4*H1i\"1SC=([)G5H " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"1dCi!o8\\/\"\"1Sj *4p\"f)H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!*hX()=% \"*!oyvm" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\">*=*))HF)f5oE]L$3O\"\"?/,Ls.%)QVo#*ej Tu>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%#\"@D,]&\\>+0_MY[oear\"AcyGNpYG#[bJ=lCv'> " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&#\"JLzu0T.d&)eLkly;X 2:v;`)\"JkE=5x&4$*eL&ownw4ZjVwk$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %#b*G6#\"\"'#!Cxb#>\"ox^XUIXGBNu>l\"CQ&o'3rQ9MiR\"**ffj$RL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"DpU#zfP2Wr<6-^y.!* 38$H-!)*4Q^`p2%Q#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!# D\"#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!3%ei7%R7s^C\" 5Hq?7&)\\x'[H#" }}}{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 0 146 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := su bs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs (u),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->lhs(u)-rhs( u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for \+ stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderCon ditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have stage-order 2. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so ||i||_8[j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimpl ify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$F$F$F$F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can c alculate the 2 norm of the principal error, that is, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 " errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](s qrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5z-SA!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the princip al error of the order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs(b=`b*`,Prin cipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`e rrterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+ck8W5!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "18 of the principal error conditions are \+ satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nf or ct to nops(errterms6_8) do\n if simplify(subs(ee,errterms6_8[ct]) )=0 then ind := [op(ind),ct] end if:\nend do:\nerr6 := PrincipalErrorC onditions(6):\n[seq([i,err6[i]],i=ind)]:\nlinalg[augment](linalg[delco ls](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747%\"\"'%#~~G/*&% \"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F--F/6#*&F2F-%\"cGF-F-F-F6F-F-#F-\"$ /&7%\"#5F)/*&F,F-)F3\"\"#F-#F-\"$_#7%\"#:F)/**F,F-F9F-F6F-F3F-#F-\"#%) 7%\"#?F)/*&F,F--F/6#*(F2F-F6F--F/6#*&)F9FAF-F2F-F-F-FB7%\"#@F)/*&F,F-- F/6#*(F2F-FTF-F3F-F-FB7%\"#CF)/*(F,F-F3F-FQF-#F-\"$E\"7%\"#EF)/*&F,F-- F/6#*(F2F-F9F-)F6FAF-F-#F-\"$o\"7%\"#GF)/*&F,F-)F6\"\"$F-#F-\"#c7%\"#H F)/*&F,F--F/6#*(F2F-FTF-FQF-F-Fjn7%\"#MF)/**F,F-F9F-F6F-FQF-#F-\"#U7% \"#PF)/*&F,F--F/6#*(F2F-)F9F[pF-F6F-F-FH7%\"#RF)/*(F,F-FbqF-F3F-Fip7% \"#TF)/*&F,F-)FQFAF-#F-\"#j7%FjpF)/*(F,F-FTF-FcoF-#F-Fgo7%\"#WF)/*(F,F -FbqF-FQF-#F-FV7%\"#YF)/*(F,F-)F9\"\"%F-F6F-#F-\"#97%\"#ZF)/*&F,F--F/6 #*&)F9\"\"&F-F2F-F-Fip7%\"#[F)/*&F,F-)F9F(F-#F-\"\"(Q(pprint16\"" }}} {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 44 "simultan eous construction of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equatio ns to ensure that stage 2 has stage-order 2 and stages 3 to 8 have sta ge-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the si mplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&& %\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " } {XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 9 simple order condit ions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 97 "These include the 6 order 6 quadrature conditions, 2 a dditional order 6 order conditions and the " }{TEXT 260 28 "order 7 qu adrature condition" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "SO7 := SimpleOrderCondition s(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,20,29,32,64])]:\nlinalg[augment]( linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[de lcols](%,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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F( \"\"&7%\"#?F)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&FPF(-FM6#*&FPF(F2F(F(F(F (#F(\"$W\"7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(\"#C7%\"#KF)/*&F,F()F 2FGF(#F(\"\"'7%\"#kF)/*&F,F()F2FboF(#F(\"\"(Q)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection of 7 \"simple\" order conditions as given (in abreviated form) in the following table. " }}{PARA 0 "" 0 " " {TEXT -1 78 "These include the 5 quadrature conditions and two addit ional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "`SO5*` := subs(b=`b*`,SimpleOrderC onditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,9,15,16])]:\nlinalg[augm ent](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),lina lg[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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"\"*F)/*&F,F(-%!G 6#*&%\"aGF(-FF6#*&FIF(-FF6#*&FIF(F2F(F(F(F(#F(\"$?\"7%\"#:F)/*&F,F(-FF 6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint366\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO7_8 := SimpleOrderConditions(7,8,'expanded'):\nSO_eqs := [op(Ro wSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')) ,\n op(StageOrderConditions(3,3..8,'expanded'))]:\n`SO5_ 9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns : = [seq(SO7_8[i],i=[1,2,4,8,16,20,29,32,64])]:\n`ord_cdns*` := [seq(`SO 5_9*`[i],i=[1,2,4,8,9,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..8)= b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns : = [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO _eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[4] = 7/30;" "6#/&%\"cG6#\"\"%*&\"\"(\"\"\"\"#I!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 49/87;" "6#/&%\"cG6#\"\"' *&\"#\\\"\"\"\"#()!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 81/8 2;" "6#/&%\"cG6#\"\"(*&\"#\")\"\"\"\"##)!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 35 "and 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$\"\")\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provide the linking coefficients for the 9th stage of the em bedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8 ." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "` b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[8] = -25/38;" "6#/&%#b*G6#\"\"),$*&\"#D\"\"\"\"#Q! \"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We have 45 equations and 45 unknowns including " } {TEXT 260 18 "three of the nodes" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 93 " was taken to make the principal error norm a min imum with respect to the chosen values for " }{XPPEDIT 18 0 "c[4]" "6 #&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6 #\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "preliminary calculation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "e1 := \{c[4]=7/30,c[7]=81/8 2,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0, `b*`[3]=0,`b*`[8]=-25/38\}:\neqns := subs(e1,cdns):\ne2 := solve(\{op( eqns)\},indets(eqns) minus \{c[6]\}):\ne3 := `union`(e1,e2):\nerrterms 6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsm := 0:\nfor ct to nops( errterms6_8) do\n sm := sm+(simplify(subs(e3,errterms6_8[ct])))^2;\n end do:\nplot(sm,c[6]=0.55..0.6,color=COLOR(RGB,.4,0,1),font=[HELVETIC A,9]);\nevalf[12](findmin(sm,c[6]=0.56..0.57));\nsqrt(op(2,%));\nconve rt(op(1,%%),rational,5);" }}{PARA 13 "" 1 "" {GLPLOT2D 482 273 273 {PLOTDATA 2 "6'-%'CURVESG6#7W7$$\"3U+++++++b!#=$\"3yIvE%pEpF$!#D7$$\"3 En;a)G\\a]&F*$\"3)e]q'=*))4&HF-7$$\"33ML3x&)*3^&F*$\"3gfr#4)zq[EF-7$$ \"30]i!R(*Rc^&F*$\"3.'pI2HyRS#F-7$$\"37n\"H2P\"Q?bF*$\"3!\\?.>U\\a<#F- 7$$\"3u]PMnNrDbF*$\"3'\\eTF-7$$\"3DL$eRwX5`&F*$\"3s%pP8o$)or\"F -7$$\"3CML$eI8k`&F*$\"3diz,,z@8:F-7$$\"37M$3x%3yTbF*$\"3yJ)G:G$\\E8F-7 $$\"3qm\"z%4\\Y_bF*$\"3f=i54Z&=+\"F-7$$\"3CMeR-/PibF*$\"3b:(>\\+hG_(!# E7$$\"3c+DcmpisbF*$\"31JKId+/5aFjn7$$\"3ULe*)>VB$e&F*$\"35WR8&H-5o$Fjn 7$$\"3Y+DJbw!Qf&F*$\"35U+K0wftBFjn7$$\"3±/j$o/cF*$\"3!*G)ocyb%=9Fjn 7$$\"3GL$3_>jUh&F*$\"3MA.>n%f9s)!#F7$$\"3@++D;v/DcF*$\"3#HzG$\\^yWbFdp 7$$\"3h++v=h(ej&F*$\"3T.xWGh_,_Fdp7$$\"3e++v$[6jk&F*$\"3Gj.x!>oEG(Fdp7 $$\"3]Le*[z(ybcF*$\"3iE0MGx%45\"Fjn7$$\"3SnmTXg0ncF*$\"3))[x[L<0ZpxgdF*$\"3#=G,\"*pjtE\"F-7$$\"3#**\\Pf4t.x&F*$\"3@)p= J&pN69F-7$$\"3kL$e*Gst!y&F*$\"3YP^S)Gf'p:F-7$$\"3K++]#RW9z&F*$\"3lb](3 ^:`t\"F-7$$\"3]+]7j#>>!eF*$\"3Ait2^2e)*=F-7$$\"3j*\\i!RU07eF*$\"39zv`K 6\"p0#F-7$$\"3u**\\(=S2L#eF*$\"33&=P&H#zAB#F-7$$\"3jmm;p)=M$eF*$\"3uOR Jcsv)Q#F-7$$\"3r***\\(=]@WeF*$\"3y***e;?lRb#F-7$$\"3wLe*[$z*R&eF*$\"3o Dlp>JU,FF-7$$\"35++DYKpkeF*$\"3a\"eRm)pdfGF-7$$\"3ym\"H2qcZ(eF*$\"3#Q' )*enZ#\\+$F-7$$\"3k*\\7.\"fF&)eF*$\"3o!eInu3F:$F-7$$\"3?mmTg.c&*eF*$\" 33geECTd#H$F-7$$\"3)**\\ilAFj!fF*$\"3^\"**p\\XeNV$F-7$$\"3&GLL$)*pp;fF *$\"3Y(H&3K;djNF-7$$\"3[L$3xe,t#fF*$\"3m[&*pjf;!p$F-7$$\"35m\"HdO=y$fF *$\"3v6Ufb1()3QF-7$$\"3))****\\#>#[ZfF*$\"3d#>&p=[`6RF-7$$\"3;m;aG!e&e fF*$\"3,%)\\6OF<@SF-7$$\"3ELLL)Qk%ofF*$\"3f9mARHc6TF-7$$\"3y*\\iSjE!zf F*$\"3o(>@S8N&*>%F-7$$\"3C+v$40O\"*)fF*$\"3M\"*H*GaI_F%F-7$$\"3w****** ********fF*$\"3]*>&R#y*)oM%F--%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG 6$Q%c[6]6\"Q!Fg\\l-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"!Fa]l$\"\"\"Fa]l-%% VIEWG6$;$\"#b!\"#$\"\"'F_]l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-3\\**zJc!#7$\"-ZWz0;]!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ldlRA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#\\ \"#()" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "e1 := \{c[4]=7/30,c[6]=49/87,c[7]=81/82,c[ 8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*` [3]=0,`b*`[8]=-25/38\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns );\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Y" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolev el[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := so lve(\{op(eqns)\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2552 "e3 := \{a[ 7,3] = 3058612407219548732633706208815/709347359742223516267324426216, a[9,4] = 1815811171875/4929812076868, a[5,2] = 0, a[4,2] = 0, c[8] = \+ 1, c[9] = 1, a[6,2] = 0, a[7,5] = 524767583662593596626716546857270370 413091/24088433982273003693314369668047287199799, a[7,4] = -5347837857 4860120460721550328853039040/23405367722957560812286413128409759761, a [8,4] = -29930095382476957846060551182764365/1115580447240667890078368 5305189917, `b*`[8] = -25/38, `b*`[2] = 0, `b*`[3] = 0, b[2] = 0, b[3] = 0, a[9,3] = 0, a[9,2] = 0, a[8,5] = 2649919725251560641426119276806 860337581199555/100712482278012733160575815039507169826241912, a[8,2] \+ = 0, a[7,2] = 0, a[5,1] = 19468520817039492/35486945168446723, a[4,1] \+ = 7/120, a[9,6] = -6206299092285813/2910288487182440, b[4] = 181581117 1875/4929812076868, `b*`[6] = -6519743523284530424551776811925577/3339 363595999139623414387108668538, a[6,1] = 20252658340120203773069749/33 784135725639504675502296, a[3,2] = 7/60, c[6] = 49/87, a[9,1] = 584753 6753/86036171760, a[2,1] = 14/135, a[6,5] = -2167935872170716223870572 8210467/901105029036754619331512380401576, c[4] = 7/30, c[7] = 81/82, \+ a[8,3] = 7405121074013573981098320/1457324938194042260233261, b[1] = 5 847536753/86036171760, a[8,1] = -114361999204933693744433721507907/778 77427759375157832377305006776, a[7,6] = -696206010458633764414572/3221 9569565392536806647, a[9,7] = 1044913680622457/1298591690996340, b[8] \+ = -418874561/667578680, `b*`[1] = 13608333502668105982729889189/197441 635892684338840372330104, c[5] = 51612/89803, a[6,3] = -79221600091287 9280500/351533212217208184283, a[8,7] = -425342741234074038059/2719735 1224216978505337, b[7] = 1044913680622457/1298591690996340, a[6,4] = 2 24903465917229545508505100/100340318815275363580633077, a[5,4] = 73499 982271800480/35486945168446723, b[5] = 74928498684378188697374218447/2 9744854556648749370324659440, a[5,3] = -72573280055824680/354869451684 46723, a[9,8] = -418874561/667578680, b[6] = -6206299092285813/2910288 487182440, a[8,6] = -42066702439513241792550/1604007647963024581459, c [3] = 7/45, a[9,5] = 74928498684378188697374218447/2974485455664874937 0324659440, a[3,1] = 7/180, c[2] = 14/135, `b*`[7] = 20251703785102111 771440737597924269/23840769535138099800229313089019088, a[4,3] = 7/40, a[7,1] = -972158721124193440814630918467125/7994344744294859028332746 28345432, `b*`[9] = -245172123941262584/22948677498512207029, `b*`[4] \+ = 7154586848463452050019495500125/19675246518315548228466935287856, `b *`[5] = 85316751507451678656433588557034105747933/36476436347097667766 853358930957710182664\}:" }}}{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 0 342 "for ii from 2 to 9 do\n print(c[ii]=subs(e3,c[ii]));print(``); \n for jj to ii-1 do\n \+ print(a[ii,jj]=subs(e3,a[ii,jj]))\n end do:\n print(`___________ _______________________`):\nend do:\nprint(``);\nfor ii from 1 to 8 do \n print(b[ii]=subs(e3,b[ii]))\nend do:\nprint(``);\nfor ii from 1 t o 9 do\n print(`b*`[ii]=subs(e3,`b*`[ii]))\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"#9\"$N\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\" \"\"#\"#9\"$N\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_________________ _________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$# \"\"(\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#\"\"(\"$!=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\"(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"(\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\" \"\"#\"\"(\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\" #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$#\"\"(\"# S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_______________________________ ___G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"&7;&\"&.)*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\"#\"2#\\Rq\"3_o%>\"2BnWo^%p[N" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#!2!oCe0!GtD(\"2BnWo^%p[N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\"2![+=F#)**\\t\"2BnWo^% p[N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_____________________________ _____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#\\\"#()" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"'\"\"\"#\";\\(pIx.-7S$eED?\";'H-bn/&RcsNTyL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#!6+0GzG\"4+;Az\"6$G%=3s@7K`^$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%#\"<+^]3baHs\"fY.\\ A\"YvO!H]5,*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"#\")\"##)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"\"#!BDrY=4j93W$>C6se@(*\"BKaMGYFLG!f[HWZM%*z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"@:)3iqLEt[&>sS7'eI\"?;iUCtE;NAU(ftM4(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%#!GS!RI&)G.b@2Y?,'[ dy$yM&\"Gh(f(4%GJT'G73cdHsn`SB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"(\"\"&#\"K\"48/Pqs&oa;ni'f$fiOenZ_\"J*z*>(GZ!o'pVJ$p.IF#)RV) 3C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!9sXTkPje/,1 ip\"8Zm!o`#Rl&p&>A$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_____________ _____________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!B2z]@PVWPpL\\?**>O9\"\"Awn+0tPKy: v$fxUxy(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\":?$)4\")Rd8S2@ ^S(\":hKBgA/%>Q\\Kd9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"%#!DlVw#=^01Yy&pZ#Q&4I*H\"D<**=0`o$y+*ymSsW!e:6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#\"Ob&*>\"eP.'o!oF>hUT1c^_s>*\\E\" N7>CE)pr]R]\"edgJt7!yA[725" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"'#!8]DzTK^RCqm?%\"7f9eCI'zk2Sg\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!6f!QS2M7uU`U\"8P`]yp@C7N(>F" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"\"#\"+`n`Ze\",g<Z%=UP(p)=yVo)\\G\\(\">S%fYKq$\\([mba[uH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!18eG#4*H1i\"1SC=([)G5H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"1dCi!o8\\/\"\"1Sj*4p\" f)H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!*hX()=% \"*!oyvm" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C________________________ __________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"+`n`Ze\",g<Z%=UP(p)=yVo)\\G\\(\">S%fYKq$\\([mba[uH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#!18eG#4*H1i\"1SC=([)G5H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"1dCi!o8\\/\"\"1Sj*4p\"f)H\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!*hX()=%\"*!oyvm" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"\"#\">*=*))HF)f5oE]L$3O\"\"?/,Ls.%)QVo#*ejTu>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"%#\"@D,]&\\>+0_MY[oear\"AcyGNpYG#[bJ=lCv'>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&#\"JLzu0T.d&)eLkly;X2:v;`)\"JkE=5x &4$*eL&ownw4ZjVwk$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#! Cxb#>\"ox^XUIXGBNu>l\"CQ&o'3rQ9MiR\"**ffj$RL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"DpU#zfP2Wr<6-^y.!*38$H-!)*4Q^ `p2%Q#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!#D\"#Q" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!3%ei7%R7s^C\"5Hq?7&)\\ x'[H#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 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 2433 "ee := \{c[2]=14/135,\nc[3]=7/45,\nc[4]=7/30,\nc[5]= 51612/89803,\nc[6]=49/87,\nc[7]=81/82,\nc[8]=1,\nc[9]=1,\n\na[2,1]=14/ 135,\na[3,1]=7/180,\na[3,2]=7/60,\na[4,1]=7/120,\na[4,2]=0,\na[4,3]=7/ 40,\na[5,1]=19468520817039492/35486945168446723,\na[5,2]=0,\na[5,3]=-7 2573280055824680/35486945168446723,\na[5,4]=73499982271800480/35486945 168446723,\na[6,1]=20252658340120203773069749/337841357256395046755022 96,\na[6,2]=0,\na[6,3]=-792216000912879280500/351533212217208184283,\n a[6,4]=224903465917229545508505100/100340318815275363580633077,\na[6,5 ]=-21679358721707162238705728210467/901105029036754619331512380401576, \na[7,1]=-972158721124193440814630918467125/79943447442948590283327462 8345432,\na[7,2]=0,\na[7,3]=3058612407219548732633706208815/7093473597 42223516267324426216,\na[7,4]=-53478378574860120460721550328853039040/ 23405367722957560812286413128409759761,\na[7,5]=5247675836625935966267 16546857270370413091/24088433982273003693314369668047287199799,\na[7,6 ]=-696206010458633764414572/32219569565392536806647,\na[8,1]=-11436199 9204933693744433721507907/77877427759375157832377305006776,\na[8,2]=0, \na[8,3]=7405121074013573981098320/1457324938194042260233261,\na[8,4]= -29930095382476957846060551182764365/111558044724066789007836853051899 17,\na[8,5]=2649919725251560641426119276806860337581199555/10071248227 8012733160575815039507169826241912,\na[8,6]=-42066702439513241792550/1 604007647963024581459,\na[8,7]=-425342741234074038059/2719735122421697 8505337,\na[9,1]=5847536753/86036171760,\na[9,2]=0,\na[9,3]=0,\na[9,4] =1815811171875/4929812076868,\na[9,5]=74928498684378188697374218447/29 744854556648749370324659440,\na[9,6]=-6206299092285813/291028848718244 0,\na[9,7]=1044913680622457/1298591690996340,\na[9,8]=-418874561/66757 8680,\n\nb[1]=5847536753/86036171760,\nb[2]=0,\nb[3]=0,\nb[4]=18158111 71875/4929812076868,\nb[5]=74928498684378188697374218447/2974485455664 8749370324659440,\nb[6]=-6206299092285813/2910288487182440,\nb[7]=1044 913680622457/1298591690996340,\nb[8]=-418874561/667578680,\n\n`b*`[1]= 13608333502668105982729889189/197441635892684338840372330104,\n`b*`[2] =0,\n`b*`[3]=0,\n`b*`[4]=7154586848463452050019495500125/1967524651831 5548228466935287856,\n`b*`[5]=8531675150745167865643358855703410574793 3/36476436347097667766853358930957710182664,\n`b*`[6]=-651974352328453 0424551776811925577/3339363595999139623414387108668538,\n`b*`[7]=20251 703785102111771440737597924269/23840769535138099800229313089019088,\n` b*`[8]=-25/38,\n`b*`[9]=-245172123941262584/22948677498512207029\}:" } }}{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[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error term s of the 8 stage, order 6 scheme (the error terms of order 7)." }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T *G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose components ar e the principal error terms of the embedded 9 stage, order 5 scheme (t he error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#& %#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components \+ are the error terms of order 7 of the embedded order 5 scheme." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote th e 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6# -%$absG6#-F$6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ab s(abs(`T*`[5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G 6$\"\"'\"\"*" }{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[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6# \"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\" \"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\" " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8 ]))/abs(abs(`T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$ \"\"'\"\"*\"\"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*` ,PrincipalErrorTerms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`, PrincipalErrorTerms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf( subs(ee,`errterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sq rt(add(evalf(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms 6_8[i])))^2,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n' C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG 6#\"\"($\")\"\\95\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\" \"($\")!\\S7\"!\"(" }}}{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 27 "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 2433 "ee := \{c[2]=14/135,\nc[3]=7/45,\nc[4]=7/30,\nc[5]=51612/89803, \nc[6]=49/87,\nc[7]=81/82,\nc[8]=1,\nc[9]=1,\n\na[2,1]=14/135,\na[3,1] =7/180,\na[3,2]=7/60,\na[4,1]=7/120,\na[4,2]=0,\na[4,3]=7/40,\na[5,1]= 19468520817039492/35486945168446723,\na[5,2]=0,\na[5,3]=-7257328005582 4680/35486945168446723,\na[5,4]=73499982271800480/35486945168446723,\n a[6,1]=20252658340120203773069749/33784135725639504675502296,\na[6,2]= 0,\na[6,3]=-792216000912879280500/351533212217208184283,\na[6,4]=22490 3465917229545508505100/100340318815275363580633077,\na[6,5]=-216793587 21707162238705728210467/901105029036754619331512380401576,\na[7,1]=-97 2158721124193440814630918467125/799434474429485902833274628345432,\na[ 7,2]=0,\na[7,3]=3058612407219548732633706208815/7093473597422235162673 24426216,\na[7,4]=-53478378574860120460721550328853039040/234053677229 57560812286413128409759761,\na[7,5]=5247675836625935966267165468572703 70413091/24088433982273003693314369668047287199799,\na[7,6]=-696206010 458633764414572/32219569565392536806647,\na[8,1]=-11436199920493369374 4433721507907/77877427759375157832377305006776,\na[8,2]=0,\na[8,3]=740 5121074013573981098320/1457324938194042260233261,\na[8,4]=-29930095382 476957846060551182764365/11155804472406678900783685305189917,\na[8,5]= 2649919725251560641426119276806860337581199555/10071248227801273316057 5815039507169826241912,\na[8,6]=-42066702439513241792550/1604007647963 024581459,\na[8,7]=-425342741234074038059/27197351224216978505337,\na[ 9,1]=5847536753/86036171760,\na[9,2]=0,\na[9,3]=0,\na[9,4]=18158111718 75/4929812076868,\na[9,5]=74928498684378188697374218447/29744854556648 749370324659440,\na[9,6]=-6206299092285813/2910288487182440,\na[9,7]=1 044913680622457/1298591690996340,\na[9,8]=-418874561/667578680,\n\nb[1 ]=5847536753/86036171760,\nb[2]=0,\nb[3]=0,\nb[4]=1815811171875/492981 2076868,\nb[5]=74928498684378188697374218447/2974485455664874937032465 9440,\nb[6]=-6206299092285813/2910288487182440,\nb[7]=1044913680622457 /1298591690996340,\nb[8]=-418874561/667578680,\n\n`b*`[1]=136083335026 68105982729889189/197441635892684338840372330104,\n`b*`[2]=0,\n`b*`[3] =0,\n`b*`[4]=7154586848463452050019495500125/1967524651831554822846693 5287856,\n`b*`[5]=85316751507451678656433588557034105747933/3647643634 7097667766853358930957710182664,\n`b*`[6]=-651974352328453042455177681 1925577/3339363595999139623414387108668538,\n`b*`[7]=20251703785102111 771440737597924269/23840769535138099800229313089019088,\n`b*`[8]=-25/3 8,\n`b*`[9]=-245172123941262584/22948677498512207029\}:" }}}{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 77 "The stab ility function R for the 8 stage, order 6 scheme is given as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunctio n(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"4hn#[6&*[,P8\"7+?%[B]?/ 3]+(F)*$)F'\"\"(F)F)F)*&#\"2\\9=Z6*y6E\"7+q!eqT.M,v;\"F)*$)F'\"\")F)F) F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "W e can find the point where the boundary of the stability region inters ects 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 26 "z0 := n ewton(R(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+x1\"zN %!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-4):\np1 := plot([R(z),1],z=-4 .89..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symb ol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],li nestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[- 4.89..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o************* )[!#<$\"3esuC!pQWk#F*7$$!3OLekFGog[F*$\"3?k**Hq(3F^#F*7$$!3:#F*7$$!3w\\(o2)Qr@ZF*$\"3aY\"\\#*4F*7$$!3'f;a4X'pqYF*$\" 31pC(=/o=z\"F*7$$!3J\\7Q:?K8YF*$\"39\\$pV'o09;F*7$$!3kK$3)zv%fb%F*$\"3 UnLA*\\gAX\"F*7$$!3=KLBHG>)\\%F*$\"3ua^p(z$H/8F*7$$!3iK$e'y!Q/W%F*$\"3 SYKiIh3q6F*7$$!3')\\7LEy&HQ%F*$\"3Mv&opSu!\\5F*7$$!3AmT+uvZDVF*$\"3[V( 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ot]([seq([pts[i-1],pts[i],[1.9,4.76]],i=2..nops(pts))],\n sty le=patchnogrid,color=COLOR(RGB,.2,.35,.95)):\npts := []: z0 := 2-4.8*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 := p lot(pts,color=COLOR(RGB,0,.1,.4)):\np6 := plots[polygonplot]([seq([pts [i-1],pts[i],[1.9,-4.76]],i=2..nops(pts))],\n style=patchnogr id,color=COLOR(RGB,.2,.35,.95)):\np7 := plot([[[-5.19,0],[2.29,0]],[[0 ,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p||(1..7 )],view=[-5.19..2.29,-5.19..5.19],font=[HELVETICA,9],\n l abels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F (7$F($\"3++++Fjzq:!#=7$$!3*)*****HT?7i%!#F$\"3$)*****>j#fTJF-7$$!3%*** ***H?$f66!#D$\"3%******\\j)Q7ZF-7$$!3++++SdE75!#C$\"3[*****R&H=$G'F-7$ $!3E+++e8$)=`F=$\"3c*****RorR&yF-7$$!3,++++E!\\$>!#B$\"31+++oquC%*F-7$ 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary cu rve horizontally by taking the 11th root of the real part of points al ong the curve. In this way we see that there is " }{TEXT 260 53 "no la rgest interval on the nonnegative imaginary axis" }{TEXT -1 64 " that \+ contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability region intersects the nonn egative imaginary axis in an interval that does not contain the origin ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 418 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/7 20*z^6+\n 1337014895114826761/7005008042050234842000*z^7+\n \+ 26117891147181449/1167501340341705807000*z^8:\nDigits := 25:\npts := [ ]: z0 := 0:\nfor ct from 0 to 110 do\n zz := newton(R(z)=exp(ct*Pi/1 00*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz) ]]:\nend do:\nplot(pts,color=COLOR(RGB,0,.35,.95),thickness=2,font=[HE LVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7[r7$$\"\"!F)F(7$$!:X=e!>&*Q76p#om#!#E$\":e T:c/x*e`EfTJF-7$$!:N2l$*zk=0>jVT%F-$\":WB(p-vm<2`=$G'F-7$$!:m92]2z*>n[ /FfF-$\":1ixC!o(>2'zxC%*F-7$$!:`-X=q_Dg4>TI(F-$\":8Q3P(*)))Rhqjc7!#D7$ $!:_xLAjl]_cWwe)F-$\":N'3rq&=>mK'zq:F?7$$!:GM+@LpoN\"H[+)*F-$\":Cff()o (z_\"fb\\)=F?7$$!:H@01,EIMtzc4\"F?$\":.=bRyK%ob[6*>#F?7$$!:[XdNi0#yq5h 17F?$\":UfS**Q%GD=TF8DF?7$$!:>(=YG/1\"Ra>NJ\"F?$\":;frFE1#zxLVFGF?7$$! :F\\Lss)4=in!pT\"F?$\":+j!3p*o')>j#fTJF?7$$!:0Z%\\j]K][d:<:F?$\":jB&Qa >JKx=vbMF?7$$!:Fdw7!*eU,)ec9;F?$\":$*pZCW>5(36\"*pPF?7$$!:'Rkem')G_3aP 4.2%3%F?7$$!:_!Gke`M\"pdw1&4]QNw)\\H#)R%F?7 $$!:I-cv4*=F?$\":>8\\*yQpGN')Q7ZF?7$$!:(Q+YhTN3+F&*z>F?$\":f>l1G ()QDrZl-&F?7$$!:aXH$oeqr^C'f1#F?$\":u;f.ax#Q5nqS`F?7$$!:#=,%pZv))R%H/] @F?$\":o@5?9f\"4/c'[l&F?7$$!:j&[\"QnNXvFmAB#F?$\":sTJ*[@=$QOC!pfF?7$$! :F'o&)*\\$*4&*R\"p7BF?$\":zB>nB,\"=aH=$G'F?7$$!:1:%eYXyc$)eO\"R#F?$\": 68)3[.@pL8M(f'F?7$$!:[@-_QF:ZHF$oCF?$\":8IVet/;XX*\\6pF?7$$!:0v/5utz*G UgVDF?$\":,JG@W1\"4islDsF?7$$!:A#*>uio\\>4H;9rE!\\p9)RvF?7 $$!::Id=^s15my\"*o#F?$\":E)QLLTJJ%orR&yF?7$$!:5+tbtl;@Y%\\fFF?$\":K?3< #RAXa\"G\"o\")F?7$$!:)\\_Mm9&p6)H;GGF?$\":2J^]m#G:ASG#[)F?7$$!:qBDggxh 4sv^*GF?$\":RTsvZB0p>Rkz)F?7$$!:`'H=B(4(42r^gHF?$\":nY8#[bIz!e$f5\"*F? 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$\":FlLnT,L\\/j!HAFfu7$$\":Pr;%puxcjjE%G&F?$\":^'48B(ptWPX-E#Ffu7$$\": 6UL[jO*eY(=$\\`F?$\":@jg/>37#oOR\"H#Ffu7$$\"::@#p&=Hac\"o/7aF?$\":#*3( ye\"36Uw.DK#Ffu7$$\":[Vdj!f&4i`mCZ&F?$\":x+>#4k)3<8rNN#Ffu7$$\":u@w\"* y9\"y')=dIbF?$\":&=:F2d6Mf3f%Q#Ffu7$$\":sIBVA+X%*eOje&F?$\":,0HITfXikd bT#Ffu7$$\":QQ\"=5X/yn]rRcF?$\":9\\>Su8mlzlkW#Ffu7$$\":qigbtS#HKFk!p&F ?$\":4S8A=$Q'y@4tZ#Ffu7$$\":&o^acc=)QjN!RdF?$\":$3l0z&H2RU\"33DFfu7$$ \":N&Q@.&*Q\\t%*y%y&F?$\":`6([vc&fAav(QDFfu7$$\":Yz%4r:bH.zxFeF?$\":U( [P&y0&*pK%QpDFfu7$$\":ww])=fC]%**\\y'eF?$\"::-&4H46xw,!**f#Ffu7$$\":&= +$HOS#zqm#[!fF?$\":\\ASBGbU$e^JIEFfu7$$\":%[H^el_L-e\\QfF?$\":;oZlE;&*>!e!y#Ffu7$$\":]r[Zx')Qyh&RYgF?$ \":X*RG2^xoH!4-\"GFfu7$$\":;BBiA`jPoZE0'F?$\":>(3*>mV;y;v'RGFfu7$$\":, C`A-Wc549@0'F?$\":9B])3V-e]fF?$\":$=LSd*=9*pkC%)HFfu 7$$\":,;L0d')\\.KoZ)eF?$\":A,NJ09UDxYD,$Ffu7$$\":RM\"*HY!Qr;&)e)y&F?$ \":un&F?$\":7,%ymGgqPa_]JFfu7$$!:8r9M%GU$pl k'fcF?$\":XOoQ1(yApWOxJFfu7$$!:W-kX'zYZ![&oPfF?$\":!QO>=qHQ,_$R?$Ffu7$ $!:m(ec+\">>,V_p9'F?$\":2AK?fu)[%HJ-B$Ffu7$$!:'4\"QcrN9UW\"H>jF?$\":N$ R-3+'=PmYiD$Ffu7$$!:H\"e-@\\D3(pK#okF?$\":#3WfDc;\\ma(>G$Ffu7$$!:tic'p IAO+f#3g'F?$\":JWKkgR6**37uI$Ffu7$$!:`a2Q)f'4imA7s'F?$\":#zACt%><#R6bK LFfu7$$!:z9*z[([$)[Bz?$oF?$\":aO_$>tt4GuQdLFfu-%*THICKNESSG6#\"\"#-%%F ONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBGF($\"#N!\"#$\"#&*F\\_m-%+AXESL ABELSG6$Q!6\"Fb_m-%%VIEWG6$%(DEFAULTGFg_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 -1 108 "The relevant intersection poin ts 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 t he boundary curve either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "Dig its := 15:\nz0 := 1.7*I:\nfor ct from 53 to 56 do\n newton(R(z)=exp( ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.15*I:\nfor ct from 100 to 103 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0*[58[Nx7!#>$\"0Z@L^X\\m\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0BVH(y+Iq!#?$\"0j,HbYjp\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0TOdfzs%o!#@$\"0P6>4Xxs\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0%p!pjeB2\"!#>$\"0&p#RnS\"f6!#<$\"0:Js!#<$\"0(yApWOxJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to ca lculate the parameter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "Digits := 15:\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.7*I))\nend proc:\nu0 := \+ bisect('real_part'(u),u=0.53..0.56);\nnewton(R(z)=exp(u0*Pi*I),z=1.7*I );``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.15*I)) \nend proc:\nu0 := bisect('real_part'(u),u=1.0..1.03);\nnewton(R(z)=ex p(u0*Pi*I),z=3.15*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#u0G$\"0MV#\\9:#\\&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0- \"=i3GD%#u0G$\"0=\"e:cr75!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0fQWs438$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects th e nonegative imaginary axis in the interval" }{TEXT -1 3 " " } {XPPEDIT 18 0 "[1.7253, 3.1308];" "6#7$-%&FloatG6$\"&`s\"!\"%-F%6$\"&3 8$F(" }{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 80 "The s tability function R* for the 9 stage, order 5 scheme is given as fol lows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*` ,StabilityFunction(5,9,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z) '=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"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)*&#\"Ft:D-%\\!)QiuE7W.X5%)* y\"\"IW`$[Q&QX^[#p6A'>Xj`/'G\"F)*$)F'F1F)F)F)*&#\"Fz*H3equ)ei*p5cX?c\" *zj\"J+O)3iMYjG@J#Hb!*H'3M6:KF)*$)F'\"\"(F)F)F)*&#\"E'Gp=NWApR*p*[*=_G \"=:#\"JDh->LK)>Sv%)yZXo>SHs/5F)*$)F'\"\")F)F)F)*&#\"Bh$*=2')3.7T#4630 YV6\"HDO&*>\\#='47X*z6$=L&\\R%y%F)*$)F'\"\"*F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the poi nt where the boundary of the stability region intersects the negative \+ real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)= 1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+wK)eY%!\"*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-4.5):\np_1 := plot([`R*`(z),1],z=-4. 99..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,sym bol=[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=[-4.99..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++++++!*\\! #<$\"3'*\\?g(>gRp#F*7$$!3Wm;z)*y8g\\F*$\"3Ws_c)orIb#F*7$$!3cLLe(zv-$\\ F*$\"3Ox(e98p(=CF*7$$!3o+]P'p8/!\\F*$\"3WF%eE?&y!H#F*7$$!3#pmm^f^0([F* $\"3mh#GN$=')o@F*7$$!3#3]()f)ee=[F*$\"3Ct')4hiSq>F*7$$!3%QL3oIz\"3ZF*$\"3<&Hxlt)e,;F*7$$!3imm,F%Q(\\YF*$ \"3f))))QxB^K9F*7$$!3Gnm1o,\"4f%F*$\"3eI[H)Rg'y7F*7$$!33nm64>3KXF*$\"3 Xe'H\"R_$)R6F*7$$!3I+Dh]K`tWF*$\"3L#)RhjYK:5F*7$$!3\\L$3@f%)\\T%F*$\"3 XV7Z,RoK!*!#=7$$!3jm;u*Q?kI%F*$\"3=_oC7uK[sFco7$$!3E+]ZY%3S>%F*$\"3G%* RFZuyZdFco7$$!3%omTR&=vxSF*$\"3/kV\"[a4c]%Fco7$$!3/+]x(4o='RF*$\"3!*3O W)H%QENFco7$$!3#Fco7$$!3#)*****=Az%>OF*$\"3PrcK/kLI7$$!3[LLBUd1fJF*$\"34h3Jf]sR%)Ffr7$$!3eLL$4-XW0$F*$\"37#*3d2]k )y(Ffr7$$!3;+]n]iuKHF*$\"3J&G*\\5zUOuFfr7$$!3ILL$z@A]#GF*$\"3&*ztbDw>C uFfr7$$!3!)**\\n!)=$oq#F*$\"3E+qzR0$*zwFfr7$$!3')**\\-InG%f#F*$\"3Qx5R :YwZ\")Ffr7$$!3ILL3sw&oZ#F*$\"3,0v`o8gT))Ffr7$$!3#HL3&R6-pBF*$\"3Y)Qot W`)['*Ffr7$$!3#om;4([q_AF*$\"3c=$\\:u?%p5Fco7$$!3xm;%z&\\)=8#F*$\"3'RY $>D05(>\"Fco7$$!3$***\\_o3rE?F*$\"30<^%fDaXK\"Fco7$$!3hmmhq*>J\">F*$\" 3A(z)*4_O-[\"Fco7$$!35++?e%pdz\"F*$\"3S+u%GAqAm\"Fco7$$!3+++:w['4o\"F* $\"3aPNr`')>j=Fco7$$!3-+]()Rb))p:F*$\"3!='*3%Q$y83#Fco7$$!3#)***\\a(3b Y9F*$\"3]I([uy\\TN#Fco7$$!3cLL$R>HdL\"F*$\"3/wl`[<%)HEFco7$$!3z****\\% R.u@\"F*$\"3++WO+W1gHFco7$$!3pm;aLE=56F*$\"3#4aU7?K]H$Fco7$$!3E%****4@ ?'H**Fco$\"3Et$)4$*yz/PFco7$$!3MML3+cmE))Fco$\"3C@x[ULzOTFco7$$!3%H** \\(H-wtwFco$\"36[hp\"H+Bk%Fco7$$!3cOLL)\\%eYlFco$\"3iHK1*z'>'>&Fco7$$! 3g.+vof`m`Fco$\"3!yH5f%o,ZeFco7$$!3Dkmm#)*3+B%Fco$\"3O=!pte!y]lFco7$$! 3Yjm;()funIFco$\"3=;Ih$Qk\"etFco7$$!3HBL3;r5:>Fco$\"3#Q\"*R]!y5d#)Fco7 $$!3_^****>q^f&)Ffr$\"3!*fZ6fxlz\"*Fco7$$\"3OMnm\"G*fzNFfr$\"3g)QAGQWk .\"F*7$$\"3\"RLL8'ppV9Fco$\"3%4;rO96`:\"F*7$$\"3@0+D$4>8g#Fco$\"3!>_>f <,rH\"F*7$$\"3Q,+v#=6$4PFco$\"3;A!=/E$3\\9F*7$$\"3!***************[Fco $\"3*[\"*=KC;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$ 7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi]l7$FeoFi]l7$ FjoFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l 7$FbrFi]l7$FhrFi]l7$F]sFi]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftF i]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^ wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7 $FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[l Fi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$ !3e*****fF$)eY%F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&ST YLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMON DGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LI NESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^d l-Ffcl6#%(DEFAULTG-%%VIEWG6$;$!$*\\!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture sho ws the stability region for the 7 stage, order 5 scheme. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1628 "`R* ` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 1789841045034 412267462388049402251573/1286045363451962211692485145385384835344*z^6+ \n 6379915620455610699625887470580829979/321511340862990552923121 28634634620883600*z^7+\n 215181285218948969939692244351869286/100 47229401968454778847540198323319026125*z^8-\n 1143460508110924112 03088607189361/478439495331831179945120961824919953625*z^9:\npts := [] : z0 := 0:\nfor ct from 0 to 240 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,0,0,.35)):\np_2 := plots[polygon plot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n sty le=patchnogrid,color=COLOR(RGB,0,.2,.75)):\npts := []: z0 := 2+4.7*I: \nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 : = plot(pts,color=COLOR(RGB,0,0,.35)):\np_4 := plots[polygonplot]([seq( [pts[i-1],pts[i],[1.88,4.64]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,0,.2,.75)):\npts := []: z0 := 2-4.7*I:\nfor ct \+ from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pt s,color=COLOR(RGB,0,0,.35)):\np_6 := plots[polygonplot]([seq([pts[i-1] ,pts[i],[1.88,-4.64]],i=2..nops(pts))],\n style=patchnogrid,c olor=COLOR(RGB,0,.2,.75)):\np_7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.1 9],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],v iew=[-5.19..2.29,-5.19..5.19],font=[HELVETICA,9],\n labels= [`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$$ \"39+++08ldV!#G$\"3++++Fjzq:!#=7$$\"3\"******pn.(RH!#E$\"3z*****\\k#fT JF07$$\"3A+++h\"3Im$!#D$\"3A+++L))Q7ZF07$$\"36+++`9,OB!#C$\"3A+++%=%=$ 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538,\n`b*`[7]=20251703785102111771440737597924269/23840769535138099800 229313089019088,\n`b*`[8]=-25/38,\n`b*`[9]=-245172123941262584/2294867 7498512207029\}:" }}}{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 31 "seq(c[i]=subs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"cG6#\"\"##\"#9\"$N\"/&F%6#\"\"$#\"\"(\"#X/&F% 6#\"\"%#F0\"#I/&F%6#\"\"&#\"&7;&\"&.)*)/&F%6#\"\"'#\"#\\\"#()/&F%6#F0# \"#\")\"##)/&F%6#\"\")\"\"\"/&F%6#\"\"*FP" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=s ubs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/ &%\"aG6$\"\"#\"\"\"#\"#9\"$N\"/&F%6$\"\"$F(#\"\"(\"$!=/&F%6$F/F'#F1\"# g/&F%6$\"\"%F(#F1\"$?\"/&F%6$F;F'\"\"!/&F%6$F;F/#F1\"#S/&F%6$\"\"&F(# 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4p\"f)H\"/&F%6$F]uFer#!*hX()=%\"*!oyvm" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 6 scheme" } {TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=s ubs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\" \"#\"+`n`Ze\",g<Z%=UP(p)=yVo)\\G\\(\">S%fYKq$\\([mba[uH/& F%6#\"\"'#!18eG#4*H1i\"1SC=([)G5H/&F%6#\"\"(#\"1dCi!o8\\/\"\"1Sj*4p\"f )H\"/&F%6#\"\")#!*hX()=%\"*!oyvm" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs (ee,`b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\" \"#\">*=*))HF)f5oE]L$3O\"\"?/,Ls.%)QVo#*ejTu>/&F%6#\"\"#\"\"!/&F%6#\" \"$F//&F%6#\"\"%#\"@D,]&\\>+0_MY[oear\"AcyGNpYG#[bJ=lCv'>/&F%6#\"\"&# \"JLzu0T.d&)eLkly;X2:v;`)\"JkE=5x&4$*eL&ownw4ZjVwk$/&F%6#\"\"'#!Cxb#> \"ox^XUIXGBNu>l\"CQ&o'3rQ9MiR\"**ffj$RL/&F%6#\"\"(#\"DpU#zfP2Wr<6-^y.< D?\"D)3>!*38$H-!)*4Q^`p2%Q#/&F%6#\"\")#!#D\"#Q/&F%6#\"\"*#!3%ei7%R7s^C \"5Hq?7&)\\x'[H#" }}}{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 35 "#=============== ===================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme wit h " }{XPPEDIT 18 0 "c[4] = 3/13;" "6#/&%\"cG6#\"\"%*&\"\"$\"\"\"\"#8! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 86/87;" "6#/&%\"cG6 #\"\"(*&\"#')\"\"\"\"#()!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------- --------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2141 "ee \+ := \{c[2]=4/39,\nc[3]=2/13,\nc[4]=3/13,\nc[5]=13021/22659,\nc[6]=39/67 ,\nc[7]=86/87,\nc[8]=1,\nc[9]=1,\n\na[2,1]=4/39,\na[3,1]=1/26,\na[3,2] =3/26,\na[4,1]=3/52,\na[4,2]=0,\na[4,3]=9/52,\n\na[5,1]=1406640413621/ 2478209482476,\na[5,2]=0,\na[5,3]=-1755653396555/826069827492,\na[5,4] =1321105868272/619552370619,\na[6,1]=1463083752990765495750771/2783511 518115684243554356,\na[6,2]=0,\na[6,3]=-417501847634533557363/21377094 8323146013636,\na[6,4]=414679390177938970209021/2082129036667442172814 64,\na[6,5]=48490458547529962724706855/2711140218644676453221942744,\n a[7,1]=-1146771707244809451668952985850/117842899561081747475116169888 1,\na[7,2]=0,\na[7,3]=883524649813655720289029/25784099269201941696881 1,\na[7,4]=-550972740958654450507278066587/325473716439106878117398000 544,\na[7,5]=-968282586950392419883943203143069455/3282846083555917612 7341032228568032,\na[7,6]=3230428272165469542719/108684219530393291772 ,\na[8,1]=-368234904360842614256649493/316816365518493517722015828,\na [8,2]=0,\na[8,3]=19247613365107236707/4836252322132133628,\na[8,4]=-36 27331815384766429266963559/1852940251413900055822878936,\na[8,5]=-1113 629335962635330712822690622675431/31397585101055611361934971740947112, \na[8,6]=7515696221383336/210977455127283,\na[8,7]=-246623972988792974 4/169565652664150899277,\na[9,1]=265211783/3930519060,\na[9,2]=0,\na[9 ,3]=0,\na[9,4]=53198489747/147124055808,\na[9,5]=-16525725503573410691 1939/60068315060067285425920,\na[9,6]=1687862952891371/536423949472320 ,\na[9,7]=397538864947251/474823057340620,\na[9,8]=-7142267/10794560, \n\nb[1]=265211783/3930519060,\nb[2]=0,\nb[3]=0,\nb[4]=53198489747/147 124055808,\nb[5]=-165257255035734106911939/60068315060067285425920,\nb [6]=1687862952891371/536423949472320,\nb[7]=397538864947251/4748230573 40620,\nb[8]=-7142267/10794560,\n\n`b*`[1]=61909109615135759874805/909 927606074377568473224,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=19102617167037 3376608531013/532182573462887006324068800,\n`b*`[5]=-16134045022121298 883366692111903/6240618480477179290700503465280,\n`b*`[6]=123307988643 675607258315033/41372548832224521099482580,\n`b*`[7]=16485988876547280 61640361417/1901782716664438059984219160,\n`b*`[8]=-17/25,\n`b*`[9]=-2 59237562821839/28937895739220050\}:" }}}{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 0 342 "for ii fr om 2 to 9 do\n print(c[ii]=subs(ee,c[ii]));print(``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]))\n end do:\n prin t(`__________________________________`):\nend do:\nprint(``);\nfor ii \+ from 1 to 8 do\n print(b[ii]=subs(ee,b[ii]))\nend do:\nprint(``);\nf or ii from 1 to 9 do\n print(`b*`[ii]=subs(ee,`b*`[ii]))\nend do:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"%\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"#\"\"\"#\"\"%\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________ ________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"$#\"\"#\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\"#F(\"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##F'\"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"$\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\" \"\"#\"\"$\"#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$#\"\"*\"#_ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C________________________________ __G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"&@I\"\"&fE#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"&\"\"\"#\".@OTSmS\"\".wC[4#yC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#!.blR`cv\"\"-#\\F)pg#)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\".s#oe5@8\"->1P_&>'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#R\"#n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"\"#\":r2v&\\l2*Hv$3j9\":cVbVUo:\"=:^$y#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#!6jtbLXjZ=]<%\"6OO,YJK[4x8#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%#\"9@!4-(*QzA`knW'=-96F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_______ ___________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"c G6#\"\"(#\"#')\"#()" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\"#!@]e)H&*o;X4[C2?p#*4%yD" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"(\"\"%#!?(em!ys]]Wle4us4b\"?W0+)R<\"yo5RkrtaK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#!Eb%pI9.K%R))>CR]pe#Go* \"DK!o&GA.Tt7w\"fb$3YGG$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"(\"\"'#\"7>Fapa;s#G/B$\"6sUo3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!<$ \\\\mD9E%3O/\\Bo$\"\"4GO8K@K_i$[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!=fN'pEHkw%Q:=LFO\"=O*yG#e0+RT^-%H&=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!FJanA1pAGrI`jifLHO6\"\" D7r%4ur\\$>O6c0,^e(RJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" )\"\"'#\"1OLQ@ip:v\"0$GF^Xx4@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"(#!4W(Hz))H(RiY#\"6x#**3:kEll&p\"" }}{PARA 11 "" 1 "" {XPPMATH 20 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20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%#\",Z(*[)>`\"-3e0Cr9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&#!9R>\"p5Md.bsDl\"\"8?fU&Gn+1: $o+'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"1r8*G&H'yo\" \"0?BZ\\RUO&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"0^s% \\')QvR\"0?1MdI#[Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#! (nA9(\")gXz5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"80[()fd8:'4\"4>'\"9CKZovPuggF*4*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%#\"<85`3mPt.nrh-\">\"<+)oSK1q)GYtD=K&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&#!A.>6#pmL)))H@@-XS8;\"@!GlM]+2HzrZ! [=1C'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"U)*f!QWm;Fy,>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!#<\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!0R=#GcP#f#\"2]+ARd*y$*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 0 588 "subs(e e,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [ c[4],a[4,1],a[4,2],a[4,3]],\n [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a [5,4]],\n [c[6],a[6,1],a[6,2],a[6,3]],[``$2,a[6,4],a[6,5]],\n \+ [c[7],a[7,1],a[7,2],a[7,3]],[``,a[7,4],a[7,5],a[7,6]],\n [c[8], a[8,1],a[8,2],a[8,3]],[``,a[8,4],a[8,5],a[8,6]],[``$3,a[8,7]],\n \+ [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 [`b`,b[1],b[2],b[3]],[``,b[4],b[5],b[6]],[``$2,b[7],b[8]] ,\n [`b*`,`b*`[1],`b*`[2],`b*`[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[ ``,`b*`[7],`b*`[8],`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#777&#\"\"%\"#RF(%!GF+7&#\"\"#\"#8#\"\"\"\"#E#\"\"$F2F+7&#F4F/ #F4\"#_\"\"!#\"\"*F87&#\"&@I\"\"&fE##\".@OTSmS\"\".wC[4#yCF9#!.blR`cv \"\"-#\\F)pg#)7&F+F+F+#\".s#oe5@8\"->1P_&>'7&#F*\"#n#\":r2v&\\l2*Hv$3j 9\":cVbVUo:\"=:^$y#F9#!6jtbLXjZ=]<%\"6OO,YJK[4x8#7&F+F+#\"9@!4-(*QzA`knW'=-96F7&#\"#')\"#()# !@]e)H&*o;X4[C2?p#*4%yD7&F+#!?(em!ys]]Wle4us4b\"?W0+)R<\"yo5RkrtaK#!Eb%pI9.K%R)) >CR]pe#Go*\"DK!o&GA.Tt7w\"fb$3YGG$#\"7>Fapa;s#G/B$\"6sUo3\"7&F1 #!<$\\\\mD9E%3O/\\Bo$\"\"4GO8K@K_i$[7& F+#!=fN'pEHkw%Q:=LFO\"=O*yG#e0+RT^-%H&=#!FJanA1pAGrI`jifLHO6\"\"D7r%4u r\\$>O6c0,^e(RJ#\"1OLQ@ip:v\"0$GF^Xx4@7&F+F+F+#!4W(Hz))H(RiY#\"6x#**3: kEll&p\"7&F1#\"*$y6_E\"+g!>0$RF9F97&F+#\",Z(*[)>`\"-3e0Cr9#!9R>\"p5Md. bsDl\"\"8?fU&Gn+1:$o+'#\"1r8*G&H'yo\"\"0?BZ\\RUO&7&F+F+#\"0^s%\\')QvR \"0?1MdI#[Z#!(nA9(\")gXz57&%\"bGF^qF9F9FaqF[r7&%#b*G#\"80[()fd8:'4\"4> '\"9CKZovPuggF*4*F9F97&F+#\"<85`3mPt.nrh-\">\"<+)oSK1q)GYtD=K&#!A.>6#p mL)))H@@-XS8;\"@!GlM]+2HzrZ![=1C'#\"U)*f!QWm;Fy,>#!#<\"#D#!0R=#GcP#f#\"2]+ ARd*y$*GQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expa nded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=` b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{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 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderConditions(c t,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have stage-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8 [j])),i=2..5)],j=1..6)]:\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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can calculate th e 2 norm of the principal error, that is, the 2-norm of the principal \+ error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(sub s(ee,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+m**R`?!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the \+ order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5, 9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[i])^2 ,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+r!Rx-\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "18 of the principal error conditions are satisfied." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops( errterms6_8) do\n if simplify(subs(ee,errterms6_8[ct]))=0 then ind : = [op(ind),ct] end if:\nend do:\nerr6 := PrincipalErrorConditions(6): \n[seq([i,err6[i]],i=ind)]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747%\"\"'%#~~G/*&%\"bG\"\" \"-%!G6#*(%\"aGF--F/6#*&F2F--F/6#*&F2F-%\"cGF-F-F-F6F-F-#F-\"$/&7%\"#5 F)/*&F,F-)F3\"\"#F-#F-\"$_#7%\"#:F)/**F,F-F9F-F6F-F3F-#F-\"#%)7%\"#?F) /*&F,F--F/6#*(F2F-F6F--F/6#*&)F9FAF-F2F-F-F-FB7%\"#@F)/*&F,F--F/6#*(F2 F-FTF-F3F-F-FB7%\"#CF)/*(F,F-F3F-FQF-#F-\"$E\"7%\"#EF)/*&F,F--F/6#*(F2 F-F9F-)F6FAF-F-#F-\"$o\"7%\"#GF)/*&F,F-)F6\"\"$F-#F-\"#c7%\"#HF)/*&F,F --F/6#*(F2F-FTF-FQF-F-Fjn7%\"#MF)/**F,F-F9F-F6F-FQF-#F-\"#U7%\"#PF)/*& F,F--F/6#*(F2F-)F9F[pF-F6F-F-FH7%\"#RF)/*(F,F-FbqF-F3F-Fip7%\"#TF)/*&F ,F-)FQFAF-#F-\"#j7%FjpF)/*(F,F-FTF-FcoF-#F-Fgo7%\"#WF)/*(F,F-FbqF-FQF- #F-FV7%\"#YF)/*(F,F-)F9\"\"%F-F6F-#F-\"#97%\"#ZF)/*&F,F--F/6#*&)F9\"\" &F-F2F-F-Fip7%\"#[F)/*&F,F-)F9F(F-#F-\"\"(Q(pprint46\"" }}}{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 44 "simultaneous const ruction of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equations to ensu re that stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3 ." }}{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 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"i G\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&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 5 " ), " } {XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 9 simple order condit ions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 97 "These include the 6 order 6 quadrature conditions, 2 a dditional order 6 order conditions and the " }{TEXT 260 28 "order 7 qu adrature condition" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "SO7 := SimpleOrderCondition s(7):\n[seq([i,SO7[i]],i=[1,2,4,8,16,20,29,32,64])]:\nlinalg[augment]( linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[de lcols](%,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%\"#?F)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&FPF(-FM6#*&FPF(F2F(F(F(F (#F(\"$W\"7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(\"#C7%\"#KF)/*&F,F()F 2FGF(#F(\"\"'7%\"#kF)/*&F,F()F2FboF(#F(\"\"(Q)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection of 7 \"simple\" order conditions as given (in abreviated form) in the following table. " }}{PARA 0 "" 0 " " {TEXT -1 78 "These include the 5 quadrature conditions and two addit ional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "`SO5*` := subs(b=`b*`,SimpleOrderC onditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,9,15,16])]:\nlinalg[augm ent](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),lina lg[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(-%!G 6#*&%\"aGF(-FF6#*&FIF(-FF6#*&FIF(F2F(F(F(F(#F(\"$?\"7%\"#:F)/*&F,F(-FF 6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint366\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO7_8 := SimpleOrderConditions(7,8,'expanded'):\nSO_eqs := [op(Ro wSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')) ,\n op(StageOrderConditions(3,3..8,'expanded'))]:\n`SO5_ 9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns : = [seq(SO7_8[i],i=[1,2,4,8,16,20,29,32,64])]:\n`ord_cdns*` := [seq(`SO 5_9*`[i],i=[1,2,4,8,9,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..8)= b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns : = [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO _eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[4] = 3/13;" "6#/&%\"cG6#\"\"%*&\"\"$\"\"\"\"#8!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 39/67;" "6#/&%\"cG6#\"\"' *&\"#R\"\"\"\"#n!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 86/87; " "6#/&%\"cG6#\"\"(*&\"#')\"\"\"\"#()!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 35 "and 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 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provide the linking coefficients for the 9th stage of the em bedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8 ." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{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*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[8] = -17/25;" "6#/&%#b*G6#\"\"),$*&\"#<\"\"\"\"#D! \"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We have 45 equations and 45 unknowns including " } {TEXT 260 18 "three of the nodes" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 93 " was taken to make the principal error norm a min imum with respect to the chosen values for " }{XPPEDIT 18 0 "c[4]" "6 #&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6 #\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "preliminary calculation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 525 "e1 := \{c[4]=3/13,c[7]=86/8 7,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0, `b*`[3]=0,`b*`[8]=-17/25\}:\neqns := subs(e1,cdns):\ne2 := solve(\{op( eqns)\},indets(eqns) minus \{c[6]\}):\ne3 := `union`(e1,e2):\nerrterms 6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsm := 0:\nfor ct to nops( errterms6_8) do\n sm := sm+(simplify(subs(e3,errterms6_8[ct])))^2;\n end do:\nplot(sm,c[6]=0.57..0.625,color=COLOR(RGB,.4,0,1),font=[HELVET ICA,9]);\nevalf[12](findmin(sm,c[6]=0.58..0.585));\nsqrt(op(2,%));\nco nvert(op(1,%%),rational,5);" }}{PARA 13 "" 1 "" {GLPLOT2D 546 336 336 {PLOTDATA 2 "6'-%'CURVESG6#7_o7$$\"3]*************p&!#=$\"3W:2NCh]8r&F*$\"3@4\"f?4% ))H7F-7$$\"3'4FWI?'f9dF*$\"3FdF*$\"3^fbowZg:)*!#D7$$\"3T$3-y]>Cs&F*$\"3'y$y&y (R!p3*FQ7$$\"3(3GjFQ7$$\"3.mmTOY0SdF*$\"39k1?x#Qm@&FQ7$$\"3+m\"zC$*efu&F*$\"3%)og lZ)38E%FQ7$$\"3?CJXm^$=v&F*$\"3=^/)zOHrW$FQ7$$\"3S#3F/S6xv&F*$\"3]WOX \"zZKv#FQ7$$\"39;ajUugodF*$\"3*e`inMP(Q'e0=/Q)[Fip7$$\"3O$ekj3ttz&F*$\"3 =W63efAoJFip7$$\"33]P%3U)=.eF*$\"3v[OeU[O8>Fip7$$\"3/U5:2-<4eF*$\"3E/s fI+r[5Fip7$$\"3*GLeM*>::eF*$\"3wj4+gq[kc!#F7$$\"3:\\P4a2U?eF*$\"3kR_1M :(pA%Fcr7$$\"3_m\"HZ^*oDeF*$\"3wuZ\"z%3mL^Fcr7$$\"3:**\\(yE_v$eF*$\"33 I`g-ZR]9Fip7$$\"3q**\\iIPY\\eF*$\"3?9ydI:3>KFip7$$\"35**\\7KE%4'eF*$\" 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++++]iF*$\"3-A_H!HHTG(FQ-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q%c [6]6\"Q!F^al-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"!Fhal$\"\"\"Fhal-%%VIEWG6 $;$\"#d!\"#$\"$D'!\"$%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 $$\"-\\RE#4#e!#7$\"-'=s?k@%!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ \"\\#R`?!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#R\"#n" }}}{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 0 187 "e1 := \{c[4]=3/13,c[6]=39/67,c[7]=86/87,c[8]=1,c[9]=1,\n \+ seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=-17/25 \}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Y" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}) :\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2259 "e3 := \{a[5,2] = 0, a[4,2] = 0, c[8] = 1, c[9] = 1, a[6,2] = 0, a[4,1] = 3/52, a[6,4] = 41467939 0177938970209021/208212903666744217281464, a[6,5] = 484904585475299627 24706855/2711140218644676453221942744, `b*`[2] = 0, `b*`[3] = 0, b[2] \+ = 0, b[3] = 0, a[9,3] = 0, a[9,2] = 0, a[8,2] = 0, a[7,2] = 0, b[4] = \+ 53198489747/147124055808, `b*`[6] = 123307988643675607258315033/413725 48832224521099482580, a[8,3] = 19247613365107236707/483625232213213362 8, a[7,3] = 883524649813655720289029/257840992692019416968811, a[2,1] \+ = 4/39, b[5] = -165257255035734106911939/60068315060067285425920, a[9, 1] = 265211783/3930519060, a[8,6] = 7515696221383336/210977455127283, \+ a[9,4] = 53198489747/147124055808, a[9,5] = -165257255035734106911939/ 60068315060067285425920, a[7,4] = -550972740958654450507278066587/3254 73716439106878117398000544, a[7,1] = -1146771707244809451668952985850/ 1178428995610817474751161698881, b[8] = -7142267/10794560, a[9,8] = -7 142267/10794560, `b*`[5] = -16134045022121298883366692111903/624061848 0477179290700503465280, a[7,5] = -968282586950392419883943203143069455 /32828460835559176127341032228568032, c[5] = 13021/22659, a[5,3] = -17 55653396555/826069827492, a[5,1] = 1406640413621/2478209482476, c[2] = 4/39, a[9,7] = 397538864947251/474823057340620, a[6,1] = 146308375299 0765495750771/2783511518115684243554356, `b*`[9] = -259237562821839/28 937895739220050, b[6] = 1687862952891371/536423949472320, c[3] = 2/13, c[4] = 3/13, a[3,1] = 1/26, a[9,6] = 1687862952891371/536423949472320 , c[6] = 39/67, a[3,2] = 3/26, c[7] = 86/87, a[8,4] = -362733181538476 6429266963559/1852940251413900055822878936, a[4,3] = 9/52, `b*`[8] = - 17/25, `b*`[7] = 1648598887654728061640361417/190178271666443805998421 9160, a[8,1] = -368234904360842614256649493/31681636551849351772201582 8, a[6,3] = -417501847634533557363/213770948323146013636, b[7] = 39753 8864947251/474823057340620, a[8,5] = -11136293359626353307128226906226 75431/31397585101055611361934971740947112, b[1] = 265211783/3930519060 , a[8,7] = -2466239729887929744/169565652664150899277, a[7,6] = 323042 8272165469542719/108684219530393291772, a[5,4] = 1321105868272/6195523 70619, `b*`[1] = 61909109615135759874805/909927606074377568473224, `b* `[4] = 191026171670373376608531013/532182573462887006324068800\}:" }}} {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 0 342 "for ii from 2 to 9 do\n print(c[ii]=subs(e3,c[ii]) );print(``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e3,a[ii ,jj]))\n end do:\n print(`__________________________________`):\ne nd do:\nprint(``);\nfor ii from 1 to 8 do\n print(b[ii]=subs(e3,b[ii ]))\nend do:\nprint(``);\nfor ii from 1 to 9 do\n print(`b*`[ii]=sub s(e3,`b*`[ii]))\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"##\"\"%\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#\"\"%\"#R" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"\"#\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\" \"\"#F(\"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##F' \"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C____________________________ ______G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"$\"#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"%\"\"\"#\"\"$\"#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"%\"\"$#\"\"*\"#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________ ____________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\" &#\"&@I\"\"&fE#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\"#\".@OTSmS\"\".wC[4#yC" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#!.blR`cv\"\"-#\\F)pg#)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%#\".s#oe5@8\"->1P_& >'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________________________ ____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#R\"#n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"'\"\"\"#\":r2v&\\l2*Hv$3j9\":cVbVUo:\"=:^$y#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#!6jtbLXjZ=]<%\"6OO,YJK[4x8#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%#\"9@!4-(*QzA`knW'=-96F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"#')\"#()" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\"#!@]e )H&*o;X4[C2?p#*4%yD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%#!?(em!ys]]Wle4us4b\"?W0+)R<\"y o5RkrtaK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#!Eb%pI 9.K%R))>CR]pe#Go*\"DK!o&GA.Tt7w\"fb$3YGG$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#\"7>Fapa;s#G/B$\"6sUo3\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"\"#!<$\\\\mD9E%3O/\\Bo$\"\"4GO8K@K_i$[" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!=fN'pEHkw%Q:=LFO \"=O*yG#e0+RT^-%H&=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"&#!FJanA1pAGrI`jifLHO6\"\"D7r%4ur\\$>O6c0,^e(RJ" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"1OLQ@ip:v\"0$GF^Xx4@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!4W(Hz))H(RiY#\"6x#**3:k Ell&p\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_________________________ _________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\"\"#\"*$y6_E\"+g!>0$R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"%#\",Z(*[)>`\"-3e0Cr9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"&#!9R>\"p5Md.bsDl\"\"8?fU&Gn+1:$o+'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"1r8*G&H'yo\"\"0?BZ\\RUO&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"0^s%\\')QvR\"0?1 MdI#[Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!(nA9(\" )gXz5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C___________________________ _______G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"*$y6_E\"+g!>0$R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%# \",Z(*[)>`\"-3e0Cr9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&# !9R>\"p5Md.bsDl\"\"8?fU&Gn+1:$o+'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"\"'#\"1r8*G&H'yo\"\"0?BZ\\RUO&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"0^s%\\')QvR\"0?1MdI#[Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!(nA9(\")gXz5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"# \"80[()fd8:'4\"4>'\"9CKZovPuggF*4*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%#\"<85`3mPt.nrh -\">\"<+)oSK1q)GYtD=K&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\" \"&#!A.>6#pmL)))H@@-XS8;\"@!GlM]+2HzrZ![=1C'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"U)*f!QWm;Fy,>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!# <\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!0R=#GcP#f#\"2 ]+ARd*y$*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 0 588 "subs(e3,matrix([[c[2],a[2,1],``$2],\n \+ [c[3],a[3,1],a[3,2],``],\n [c[4],a[4,1],a[4,2],a[4,3]],\n \+ [c[5],a[5,1],a[5,2],a[5,3]],[``$3,a[5,4]],\n [c[6],a[6,1],a[6,2] ,a[6,3]],[``$2,a[6,4],a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3]],[``, a[7,4],a[7,5],a[7,6]],\n [c[8],a[8,1],a[8,2],a[8,3]],[``,a[8,4],a [8,5],a[8,6]],[``$3,a[8,7]],\n [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 [`b`,b[1],b[2],b[3]], [``,b[4],b[5],b[6]],[``$2,b[7],b[8]],\n [`b*`,`b*`[1],`b*`[2],`b* `[3]],[``,`b*`[4],`b*`[5],`b*`[6]],[``,`b*`[7],`b*`[8],`b*`[9]]]));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777&#\"\"%\"#RF(%!GF+7&# \"\"#\"#8#\"\"\"\"#E#\"\"$F2F+7&#F4F/#F4\"#_\"\"!#\"\"*F87&#\"&@I\"\"& fE##\".@OTSmS\"\".wC[4#yCF9#!.blR`cv\"\"-#\\F)pg#)7&F+F+F+#\".s#oe5@8 \"->1P_&>'7&#F*\"#n#\":r2v&\\l2*Hv$3j9\":cVbVUo:\"=:^$y#F9#!6jtbLXjZ=] <%\"6OO,YJK[4x8#7&F+F+#\"9@!4-(*QzA`knW'=-96F7&#\"#')\"#()#!@]e)H&*o;X4[C2?p#*4%yD7&F+#!?(em!ys]]Wle4u s4b\"?W0+)R<\"yo5RkrtaK#!Eb%pI9.K%R))>CR]pe#Go*\"DK!o&GA.Tt7w\"fb$3YGG $#\"7>Fapa;s#G/B$\"6sUo3\"7&F1#!<$\\\\mD9E%3O/\\Bo$\"\"4GO8K@K_i$[7&F+#!=fN'pEHkw%Q:=LFO\"=O*yG#e0+RT ^-%H&=#!FJanA1pAGrI`jifLHO6\"\"D7r%4ur\\$>O6c0,^e(RJ#\"1OLQ@ip:v\"0$GF ^Xx4@7&F+F+F+#!4W(Hz))H(RiY#\"6x#**3:kEll&p\"7&F1#\"*$y6_E\"+g!>0$RF9F 97&F+#\",Z(*[)>`\"-3e0Cr9#!9R>\"p5Md.bsDl\"\"8?fU&Gn+1:$o+'#\"1r8*G&H' yo\"\"0?BZ\\RUO&7&F+F+#\"0^s%\\')QvR\"0?1MdI#[Z#!(nA9(\")gXz57&%\"bGF^ qF9F9FaqF[r7&%#b*G#\"80[()fd8:'4\"4>'\"9CKZovPuggF*4*F9F97&F+#\"<85`3m Pt.nrh-\">\"<+)oSK1q)GYtD=K&#!A.>6#pmL)))H@@-XS8;\"@!GlM]+2HzrZ![=1C'# \"U) *f!QWm;Fy,>#!#<\"#D#!0R=#GcP#f#\"2]+ARd*y$*GQ(pprint26\"" }}}{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 o f 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 2141 "ee := \{c[2]=4/39,\nc[3] =2/13,\nc[4]=3/13,\nc[5]=13021/22659,\nc[6]=39/67,\nc[7]=86/87,\nc[8]= 1,\nc[9]=1,\n\na[2,1]=4/39,\na[3,1]=1/26,\na[3,2]=3/26,\na[4,1]=3/52, \na[4,2]=0,\na[4,3]=9/52,\n\na[5,1]=1406640413621/2478209482476,\na[5, 2]=0,\na[5,3]=-1755653396555/826069827492,\na[5,4]=1321105868272/61955 2370619,\na[6,1]=1463083752990765495750771/2783511518115684243554356, \na[6,2]=0,\na[6,3]=-417501847634533557363/213770948323146013636,\na[6 ,4]=414679390177938970209021/208212903666744217281464,\na[6,5]=4849045 8547529962724706855/2711140218644676453221942744,\na[7,1]=-11467717072 44809451668952985850/1178428995610817474751161698881,\na[7,2]=0,\na[7, 3]=883524649813655720289029/257840992692019416968811,\na[7,4]=-5509727 40958654450507278066587/325473716439106878117398000544,\na[7,5]=-96828 2586950392419883943203143069455/32828460835559176127341032228568032,\n a[7,6]=3230428272165469542719/108684219530393291772,\na[8,1]=-36823490 4360842614256649493/316816365518493517722015828,\na[8,2]=0,\na[8,3]=19 247613365107236707/4836252322132133628,\na[8,4]=-362733181538476642926 6963559/1852940251413900055822878936,\na[8,5]=-11136293359626353307128 22690622675431/31397585101055611361934971740947112,\na[8,6]=7515696221 383336/210977455127283,\na[8,7]=-2466239729887929744/16956565266415089 9277,\na[9,1]=265211783/3930519060,\na[9,2]=0,\na[9,3]=0,\na[9,4]=5319 8489747/147124055808,\na[9,5]=-165257255035734106911939/60068315060067 285425920,\na[9,6]=1687862952891371/536423949472320,\na[9,7]=397538864 947251/474823057340620,\na[9,8]=-7142267/10794560,\n\nb[1]=265211783/3 930519060,\nb[2]=0,\nb[3]=0,\nb[4]=53198489747/147124055808,\nb[5]=-16 5257255035734106911939/60068315060067285425920,\nb[6]=1687862952891371 /536423949472320,\nb[7]=397538864947251/474823057340620,\nb[8]=-714226 7/10794560,\n\n`b*`[1]=61909109615135759874805/90992760607437756847322 4,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=191026171670373376608531013/532182 573462887006324068800,\n`b*`[5]=-16134045022121298883366692111903/6240 618480477179290700503465280,\n`b*`[6]=123307988643675607258315033/4137 2548832224521099482580,\n`b*`[7]=1648598887654728061640361417/19017827 16664438059984219160,\n`b*`[8]=-17/25,\n`b*`[9]=-259237562821839/28937 895739220050\}:" }}}{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[6, 8];" "6#&%\"TG6$\"\"' \"\")" }{TEXT -1 127 " denote the vector whose components are the pri ncipal error terms of the 8 stage, order 6 scheme (the error terms of \+ order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5 ,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose \+ components are the principal error terms of the embedded 9 stage, orde r 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T* `[6,9];" "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whos e components are the error terms of order 7 of the embedded order 5 sc heme." }}{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[6,8 ]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "abs(abs(`T*`[5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\" \"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-% $absG6#-F$6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 15 " respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9 ]));" "6#/&%\"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F *6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs( abs(`T*`[6,9]-T[6, 8]))/abs(abs(`T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$a bsG6#-F*6#,&&%#T*G6$\"\"'\"\"*\"\"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F 06$\"\"&F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well \+ as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\" (" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used \+ for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \+ \"similar in magnitude\" and also not differ too much from 1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms6_9 *` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB : = sqrt(add(evalf(subs(ee,`errterms6_9*`[i]))^2,i=1..nops(`errterms6_9* `))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms5_9*`[i]))^2,i=1..nops( `errterms5_9*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms6_9*`[i]) -subs(ee,errterms6_8[i])))^2,i=1..nops(errterms6_8))):\n'B[7]'= evalf[ 8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\"))pK8\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")NCk6!\"(" }}}{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 27 "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 2141 "ee := \{c[2]=4/39,\nc[3]=2/13,\nc[4]=3/13,\nc[ 5]=13021/22659,\nc[6]=39/67,\nc[7]=86/87,\nc[8]=1,\nc[9]=1,\n\na[2,1]= 4/39,\na[3,1]=1/26,\na[3,2]=3/26,\na[4,1]=3/52,\na[4,2]=0,\na[4,3]=9/5 2,\n\na[5,1]=1406640413621/2478209482476,\na[5,2]=0,\na[5,3]=-17556533 96555/826069827492,\na[5,4]=1321105868272/619552370619,\na[6,1]=146308 3752990765495750771/2783511518115684243554356,\na[6,2]=0,\na[6,3]=-417 501847634533557363/213770948323146013636,\na[6,4]=41467939017793897020 9021/208212903666744217281464,\na[6,5]=48490458547529962724706855/2711 140218644676453221942744,\na[7,1]=-1146771707244809451668952985850/117 8428995610817474751161698881,\na[7,2]=0,\na[7,3]=883524649813655720289 029/257840992692019416968811,\na[7,4]=-550972740958654450507278066587/ 325473716439106878117398000544,\na[7,5]=-96828258695039241988394320314 3069455/32828460835559176127341032228568032,\na[7,6]=32304282721654695 42719/108684219530393291772,\na[8,1]=-368234904360842614256649493/3168 16365518493517722015828,\na[8,2]=0,\na[8,3]=19247613365107236707/48362 52322132133628,\na[8,4]=-3627331815384766429266963559/1852940251413900 055822878936,\na[8,5]=-1113629335962635330712822690622675431/313975851 01055611361934971740947112,\na[8,6]=7515696221383336/210977455127283, \na[8,7]=-2466239729887929744/169565652664150899277,\na[9,1]=265211783 /3930519060,\na[9,2]=0,\na[9,3]=0,\na[9,4]=53198489747/147124055808,\n a[9,5]=-165257255035734106911939/60068315060067285425920,\na[9,6]=1687 862952891371/536423949472320,\na[9,7]=397538864947251/474823057340620, \na[9,8]=-7142267/10794560,\n\nb[1]=265211783/3930519060,\nb[2]=0,\nb[ 3]=0,\nb[4]=53198489747/147124055808,\nb[5]=-165257255035734106911939/ 60068315060067285425920,\nb[6]=1687862952891371/536423949472320,\nb[7] =397538864947251/474823057340620,\nb[8]=-7142267/10794560,\n\n`b*`[1]= 61909109615135759874805/909927606074377568473224,\n`b*`[2]=0,\n`b*`[3] =0,\n`b*`[4]=191026171670373376608531013/532182573462887006324068800, \n`b*`[5]=-16134045022121298883366692111903/62406184804771792907005034 65280,\n`b*`[6]=123307988643675607258315033/41372548832224521099482580 ,\n`b*`[7]=1648598887654728061640361417/1901782716664438059984219160, \n`b*`[8]=-17/25,\n`b*`[9]=-259237562821839/28937895739220050\}:" }}} {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 77 "The stability function R for the 8 stage, order 6 scheme is given as \+ follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,Stabilit yFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"1ZxsMX*e.\" \"4g?^vM%fM0aF)*$)F'\"\"(F)F)F)*&#\"/Vihwx?K\"4;K!o#\\eD9W\"F)*$)F'\" \")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability regi on intersects the negative real axis by solving the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG 6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #z0G$!+c&\\>P%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-4):\np1 := plot([R( z),1],z=-4.89..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,p 3],view=[-4.89..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o******** *****)[!#<$\"3eGxs4n'Qe#F*7$$!3OLekFGog[F*$\"3yAP5@JoaCF*7$$!3F*7$$!3'f;a4X'pqYF *$\"3%)zAMDk@[)\\%F*$\"3M1;E(z?4F\"F*7$$!3iK$e'y!Q/W%F*$ \"3c*\\!G0m$F *$\"3NIyqt?'\\U#Fco7$$!3\"****\\^s)[WNF*$\"3C&ztUGqq$>Fco7$$!3q***\\A' H(zU$F*$\"3I:!>6!eUl:Fco7$$!3w***\\[V)o:LF*$\"3-DS3]ls)H\"Fco7$$!3SL3# pKBP@$F*$\"3k9zH)=d.7\"Fco7$$!3KmmJ^pZ#4$F*$\"3e'ywo')e)H(*!#>7$$!3?mm En`w*)HF*$\"3K.N+M2=N*)F[s7$$!3u*\\([xtGqGF*$\"3!pNsI@T/W)F[s7$$!3gmmw 6\"HXw#F*$\"3EEQpappC$)F[s7$$!3#**\\([Kb\\[EF*$\"3s'*4Vr?y!\\)F[s7$$!3 ()*\\ia7/!QDF*$\"3[c_sj+<&*))F[s7$$!3?m;aLzrACF*$\"3PD>$4\"HgV&*F[s7$$ !3XmT!4A\\oJ#F*$\"3#y(G.Q[eK5Fco7$$!34L$e>^bE?#F*$\"32;d5xO)f8\"Fco7$$ !3KL3#>MSS3#F*$\"3iLiw&zVRE\"Fco7$$!3u*\\7x[&y!)>F*$\"33Z*)>sKl#R\"Fco 7$$!3%HL3kTn#p=F*$\"3C(=262c0b\"Fco7$$!3+++qO$eSv\"F*$\"3L\"R;!)RNct\" Fco7$$!3;+]xG([8k\"F*$\"3O[7Noq5S>Fco7$$!3)**\\(o(Q'HK:F*$\"3?C^Awn4i@ Fco7$$!3/+]#eN77T\"F*$\"3\\R**[q\"y$RCFco7$$!3:nmwnHT-8F*$\"3N=y1g#R#> FFco7$$!3y***\\#)RYi=\"F*$\"3CvJK\\_*Q0$Fco7$$!3qL3_?A)43\"F*$\"3uPSU \\Bu#R$Fco7$$!3!\\***\\.n+f'*Fco$\"3x'HGJ+bk!QFco7$$!37m;a,!zhd)Fco$\" 3_qPjFco $\"3g(*\\0**o(eI&Fco7$$!3A1](=U!>z^Fco$\"37P%F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]l Fd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_ al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\" \"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6%Q\"z6\"Q!F^dl-Ffcl6#%(DEFAULTG-%%VIEWG6$;$!$*[!\"#$\"#\\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 50 "The follo wing picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1347 "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 1035894534727747/54053 45943475512060*z^7+32207776616243/1441425584926803216*z^8:\npts := []: \nz0 := 0:\nfor ct from 0 to 240 do\n zz := newton(R(z)=exp(ct*Pi/20 *I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np1 := plot(pts,color=COLOR(RGB,.4,.1,0)):\np2 := plots[polygonplot] ([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patc hnogrid,color=COLOR(RGB,1,.25,.2)):\npts := []: z0 := 2+4.8*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,c olor=COLOR(RGB,.4,.1,0)):\np4 := plots[polygonplot]([seq([pts[i-1],pts [i],[1.9,4.75]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,1,.25,.2)):\npts := []: z0 := 2-4.8*I:\nfor ct from 0 to 60 d o\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts : = [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB, .4,.1,0)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.9,-4.75] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,1,.25 ,.2)):\np7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.19],[0,5.19]]],color=b lack,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.19..2.29,-5.19 ..5.19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],ax es=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$! 3=+++si=sQ!#F$\"3*******Rj#fTJF-7$$!3!3++5$))RQ#*!#E$\"39+++r')Q7ZF-7$ $!3H+++H'z^I)!#D$\"3Q+++$>$=$G'F-7$$!3')*****zL2!zU!#C$\"35+++qE(R&yF- 7$$!34+++g*[+^\"!#B$\"3*******>&*\\ZU*F-7$$!3i*****\\K$e_RFI$\"3-+++9. b*4\"!#<7$$!39+++jDWwxFI$\"3#******4G@mD\"FQ7$$!3&*******)3_%R5!#A$\"3 !******H7%o89FQ7$$!37+++eG-eGFI$\"3!*******4+tq:FQ7$$\"3'******piu,c$F Z$\"3'******zUPxs\"FQ7$$\"3.+++E!\\XV\"!#@$\"3)******p.dY)=FQ7$$\"3/++ +LdD(z$Fdo$\"3')*****>\\&QT?FQ7$$\"3G+++Yb%=<)Fdo$\"35+++#pDx>#FQ7$$\" 3#*******4qv::!#?$\"35+++;\\L`BFQ7$$\"3/+++5:.lCFdp$\"3$)*****f2rw]#FQ 7$$\"3?+++X6\\#\\$Fdp$\"3?+++*Hg*fEFQ7$$\"3#)*****H\"=2dTFdp$\"3-+++jr @4GFQ7$$\"3=+++\">N*zOFdp$\"3!)*****\\lMV&HFQ7$$\"3e*****fZ*zd(*Fdo$\" 3;+++5u@%4$FQ7$$!3/+++Ag'>>&Fdp$\"3*)*****4(G*yA$FQ7$$!3')*****ftFzf\" !#>$\"37+++B4baLFQ7$$!3%******fOKRA$Fhr$\"3?+++ki_tMFQ7$$!3&******fkj! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary cu rve horizontally by taking the 11th root of the real part of points al ong the curve. In this way we see that there is " }{TEXT 260 53 "no la rgest interval on the nonnegative imaginary axis" }{TEXT -1 65 " that \+ contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 397 "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 1035 894534727747/5405345943475512060*z^7+32207776616243/144142558492680321 6*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 110 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, 1,.25,0),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 " " 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7[r7$$\"\"!F)F (7$$!:JT0()3%R_r)>di#!#E$\":O*[(F-$\":\"*G?6Cq-91PmD\"!#D7$$!:Qw&o=USLVH>a%)F-$\":xU$*3$zsjEjzq:F ?7$$!:U'[oKCR@)*=hZ'*F-$\":kFMT2V#f\"fb\\)=F?7$$!:NmS4W1[Ev7&y5F?$\":9 yI_-jse&[6*>#F?7$$!:$>?(zlUn(*=3w=\"F?$\":mtc)*Q]G(=TF8DF?7$$!:([9u@)) >YR3r#H\"F?$\":jI=k.(p')yLVFGF?7$$!:)33\\\"R:\">85J%R\"F?$\":**ps?t#4@ MEfTJF?7$$!:\"=:g*[#fb?@y#\\\"F?$\":Kg:<<*yg\")=vbMF?7$$!:(*[Bas*3)3.: %)e\"F?$\":!f,v2$>*[;6\"*pPF?7$$!:xI\"zb!)H].-W\"o\"F?$\":[K]E$3N[K.2% 3%F?7$$!:jli*)Rdbp_U?x\"F?$\":OGhsNi!*4_H#)R%F?7$$!:MI\"Gng;/)fr.'=F?$ \":Vi%4q:Ksq')Q7ZF?7$$!:)yl&eO+'pJ!\\l%>F?$\":w,W]B1EswZl-&F?7$$!:!z1w c+uKpPnI?F?$\":\">P(*e1LQ#z12M&F?7$$!:_Bj+Vn4e*f#G6#F?$\":\"4uyH8^#Rsl [l&F?7$$!:Uz#F?$\":i/W%RZ^%\\`C!pfF?7$$!:V-z7t*)*3tmXrAF?$ \":S:8C\"p'RM>$=$G'F?7$$!:@%p+E>2)>`C![BF?$\":#[0%3&pv(=mTtf'F?7$$!:7J Y<(GdsU:!GU#F?$\":.?1GPD3o*)*\\6pF?7$$!:]dEP7l9`n0e\\#F?$\":.?$R,?3^[y lDsF?7$$!:->B0ih-y`Xqc#F?$\":r2'[6\"Q$oga\")RvF?7$$!:#*>\")H4lH_p?lj#F ?$\":J!QmxYb;qE(R&yF?7$$!:k^WF`+RXv@Uq#F?$\":2i0V.&*zmSH\"o\")F?7$$!:/ (yDqDf&G#[)F?7$$!:n*z_APR+?\"=U$GF?$\":O!Q7#zj-D9 Tkz)F?7$$!:-e(G*=Y$\\etW'*GF?$\":_vcn/3)3jff5\"*F?7$$!:4%R$*HTui&zon&H F?$\":Z[ci**RmB&*\\ZU*F?7$$!:Y&3#*\\u_3e47:IF?$\":'*e\"e=$*G3**H!*Q(*F ?7$$!:gz'>$pPW/KG92$F?$\":otfXe!p;)\\0`+\"!#C7$$!:Dhc3;(Q/H!*fDJF?$\": y%*)4y?Bow0sO5Ffu7$$!:KoXD/Bfv*=_xJF?$\":%Ra5bXnv?b8o5Ffu7$$!:!>\"pDBY \\iUiqA$F?$\":.z%[\"QujSJ]&*4\"Ffu7$$!:q$H)yzwi@[dSF$F?$\":pRX,]'RIQ\\ '48\"Ffu7$$!:*f0)Ros!***43$=LF?$\":D#fcnj)eFPzB;\"Ffu7$$!:j`ZU%GbH!ep& fLF?$\":=JoU8E@Pf$z$>\"Ffu7$$!:\"epKd?;p'fPvR$F?$\":)Q)zd<:zPd2_A\"Ffu 7$$!:'=bUf')HjJ#G=V$F?$\":h%>6%eiE4G@mD\"Ffu7$$!:)[=)Q;1CoX[>Y$F?$\":* )47T#>KZxY.)G\"Ffu7$$!:@AF>ln;s'4D([$F?$\":&)zRv*)Q?(=xW>8Ffu7$$!:\\4' p)RRm&f@'o]$F?$\":.SKUNBr8Ng3N\"Ffu7$$!:Qop3xROcIh&>NF?$\":LVE2d,P;^sA Q\"Ffu7$$!:]0a\"f\">xX\\kN_$F?$\":\\WPuec,I7%o89Ffu7$$!:L/o![T2I)*\\5; NF?$\":N98E#zAg$4&4X9Ffu7$$!:%yQwdY8YA=_#\\$F?$\":qg/R4A%=8`]w9Ffu7$$! :C\\-w$**[uYM$RW$F?$\":L9B&*y(=e\\Y\"z]\"Ffu7$$!:jZmw8D?$G;a\\LF?$\":E D!*)z39oWHKR:Ffu7$$!:#[VFolVF;)GL8$F?$\":;:'ywC\\\")4+tq:Ffu7$$\":\"Qs V]jbT%=N)RHF?$\":w&*3H$pu`?c8-;Ffu7$$\":'*)o@GoBFUb8/MF?$\":N\"=\"e]1P 2^RNj\"Ffu7$$\":,?DWcsT.\"G%=j$F?$\":D]!oVv&GgOT\\m\"Ffu7$$\":K'3*=`9W (z_R+QF?$\":Zj*3NNLQ;3M'p\"Ffu7$$\":!=#\\n9!4ORa$3%RF?$\":D,kmLp[zUPxs \"Ffu7$$\":jme_sKu@Q$fkSF?$\":<9^st'\\,%pI\"f#=Ff u7$$\":***\\[*4r>\"4a()zVF?$\":6Wi`Y3+W/%G`=Ffu7$$\":/Hr#y,Vx&R>JZ%F?$ \":)H!\\o@M/u.dY)=Ffu7$$\":!*fu$GP^VUd@iXF?$\":S]p'3U>^yE-;>Ffu7$$\":= J+!\\'3()=fmxk%F?$\":=>U(H\"GREyzt%>Ffu7$$\":$*oxm]u?W76-t%F?$\":Eks+& fmi-qsy>Ffu7$$\":SXH6Je>pTv)4[F?$\":/n!)ezPFG\"G15?Ffu7$$\":Pj\"*Ryo_< @/q)[F?$\":s6D4>@C>\\&QT?Ffu7$$\":xv*o;&zdR*=yh\\F?$\":]%))HB#Ffu7$$\":XKel4'f%pM6TI&F?$ \":ikoV$>RQ\"=?*GAFfu7$$\"::0R'y$yx%>yim`F?$\":K3fEeKVDZ#3gAFfu7$$\":< B;G)y)p[O)=FaF?$\":_i?6<#>:?'37H#Ffu7$$\":$yxF,dt>sCx&[&F?$\":y%p)=i\" *>QK%HABFfu7$$\":.)yUQ*f.9aXBa&F?$\":g_RdP0J$ >#fof&F?$\":$[E6V@A\\\\`K%Q#Ffu7$$\":/p_h8mS$f8D\\cF?$\":3XXY:)43&>g_T #Ffu7$$\":kcO9&H&pYhX%*p&F?$\":4$R9P-MDLO8YCFfu7$$\":y#H_%=R&z\\1NZdF? $\":YB7yL\"=-d%RpZ#Ffu7$$\":chUuHe4p#*eGz&F?$\":3]\\dKNbj2rw]#Ffu7$$\" :IKVN,F[I*\\%e$eF?$\":wKZ%=Kr*y_@$QDFfu7$$\":$=efw8xr`J;weF?$\":]=&zZw &p(*[$))oDFfu7$$\":U!*[o#H()zlVk8fF?$\":(z*>X*)HE;I\\$*f#Ffu7$$\":R;Ig %)))p%3@4[fF?$\":LKy*>dz'**)4rHEFfu7$$\":6f\"e]F'>R)oFzfF?$\":?S*p\\/% y&)Hg*fEFfu7$$\":a>%G5&GD/%)Gp+'F?$\":`zEUiihPs)3!p#Ffu7$$\":lI*zq .Kix+'F?$\":3b`$yX&=ZlMV&HFfu7$$\":ZH&4Q:=v'*zkdfF?$\":\"e*)3nmfRkLw#) HFfu7$$\":#3dh7[&*4pO5jh&F?$\":N](zamARU'3n1$Ffu7$$ \":5_M9%yr[st\"[K&F?$\":!RtZ@JFfu7$$!:C#[@]S<9B6&)R`F?$\":0>&*\\%f)HmV\"[[JFfu7$$!:agBe kI-qir/u&F?$\":-a=oRomV)HAvJFfu7$$!:!3=n9$e'[)o5#**fF?$\":br?u9z$eGbp, KFfu7$$!:sgQ$G]=[#\\K()>'F?$\":8Bx%\\kVaqG*yA$Ffu7$$!:F_o')fR>]O'F? $\":$)4Z!>!4GX04QD$Ffu7$$!:8H2>/DNf$3t4lF?$\":v#\\***poo`NQ%zKFfu7$$!: .g*Hm>5^B78RmF?$\":)o!z8a>9kEvZI$Ffu7$$!:8)QK\"\\Jaf![)pv'F?$\":'*zb>K [:))[9)HLFfu7$$!:/r\"GZdi68YtloF?$\":Y9`IXFDH#4baLFfu-%*THICKNESSG6#\" \"#-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\"\"F)$\"#D!\"#F(-%+ AXESLABELSG6$Q!6\"Fb_m-%%VIEWG6$%(DEFAULTGFg_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 -1 108 "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 0 201 "Digits := 15:\nz0 := 1.6*I:\nfor ct from 49 to 52 do\n newton(R (z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.15*I:\nfor ct from 10 0 to 103 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits : = 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0+]h.;W&f!#?$\"0T\"oWHKR: !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0*=xwG-eG!#?$\"0#\\\")4+tq: !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"09/25(f<9!#?$\"0ZP0iN@g\"! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0&R\"H7oL6(!#?$\"02P2^RNj\" !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0!oKv%*zd(*!#=$\"02y.T$\"0_>!RtZ@J!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!03tf*e^15!#<$\"0')HmV\"[[J!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0<7?M31B#!#<$\"0omV)HAvJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisect ion method to calculate the parameter value associated with each inter section point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.6*I))\ne nd proc:\nu0 := bisect('real_part'(u),u=0.49..0.52);\nnewton(R(z)=exp( u0*Pi*I),z=1.7*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*P i*I),z=3.15*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.0..1.03); \nnewton(R(z)=exp(u0*Pi*I),z=3.15*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0Wv\\ze+2&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0PM(pDt#f\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0p')*fQf55!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0abr@)3BJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region \+ intersects the nonegative imaginary axis in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[1.5927, 3.1231];" "6#7$-%&FloatG6$\"&Ff\"!\"%-F %6$\"&J7$F(" }{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 80 "The stability function R* for the 9 stage, order 5 scheme is \+ given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := unapply(%,z ):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\" zG,6\"\"\"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)*&#\"A.#)ymBggU GO+S\"ROB#\"D+!3\"*[%y:NFkpje*3I/;F)*$)F'F1F)F)F)*&#\"A$zd*GCO2&fqBA7P $>J\"E+I!)o()*)ynm,/rYtL>k:F)*$)F'\"\"(F)F)F)*&#\"?*H`DV1Yr7OVF4r+\"\" C+5pi**H$**)*[G(f>*z*RZF)*$)F'\"\")F)F)F)*&#\"=fp<%p[Ro,8 " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-4.5);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+'z!H5X!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-4.5):\np_1 := plot([`R*`(z),1],z=-5.09..0.49,colo r=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,c ross,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=[-5.09.. 0.49,-0.37..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3')*************3&!#<$\"3 %f^G^4FR(HF*7$$!3_*\\P*pHff]F*$\"3a\"G,DS\"[%>pKgPoEF*7$$!3i+D\")4*y()*\\F*$\"3'3yQ/37j_#F*7$$!3G++vz=Po\\F*$\"3! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1587 "`R*` := z -> 1+z+1/ 2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 2233639140003628426060236678 8203/16043008958636964273515784489108000*z^6+\n 3119337122223705 9507362428957793/156419337346710401666778898768803000*z^7+\n 100 710927433612714606432553299/4739979919597284898993299962691000*z^8-\n \+ 2783155171301683948694176959/13903941097485369037047013223893600 *z^9:\npts := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(` R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz ),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.35,0,0)):\np_2 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.4,0]],i=2..nops(pts))] ,\n style=patchnogrid,color=COLOR(RGB,.75,0,0)):\npts := []: \+ z0 := 2+4.7*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct *Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\n end do:\np_3 := plot(pts,color=COLOR(RGB,.35,0,0)):\np_4 := plots[poly gonplot]([seq([pts[i-1],pts[i],[1.88,4.65]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,0,0)):\npts := []: z0 := 2-4. 7*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I), z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np _5 := plot(pts,color=COLOR(RGB,.35,0,0)):\np_6 := plots[polygonplot]([ seq([pts[i-1],pts[i],[1.88,-4.65]],i=2..nops(pts))],\n style= patchnogrid,color=COLOR(RGB,.75,0,0)):\np_7 := plot([[[-5.19,0],[2.29, 0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p _||(1..7)],view=[-5.19..2.29,-5.19..5.19],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7]z7 $$\"\"!F)F(7$$\"3v*****H;pR?&!#G$\"3++++Fjzq:!#=7$$\"3!******f6N?a$!#E $\"3()*****fk#fTJF07$$\"3!)*****\\;BuX%!#D$\"3'*******\\))Q7ZF07$$\"3E +++y6ofG!#C$\"3U+++nU=$G'F07$$\"3)******f*zix7!#B$\"3S+++;i(R&yF07$$\" 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22121298883366692111903/6240618480477179290700503465280,\n`b*`[6]=1233 07988643675607258315033/41372548832224521099482580,\n`b*`[7]=164859888 7654728061640361417/1901782716664438059984219160,\n`b*`[8]=-17/25,\n`b *`[9]=-259237562821839/28937895739220050\}:" }}}{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 31 "seq(c[i]=subs(ee,c[i]) ,i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"cG6#\"\"##\"\"%\"#R/ &F%6#\"\"$#F'\"#8/&F%6#F)#F.F0/&F%6#\"\"&#\"&@I\"\"&fE#/&F%6#\"\"'#F* \"#n/&F%6#\"\"(#\"#')\"#()/&F%6#\"\")\"\"\"/&F%6#\"\"*FM" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficie nts" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq( seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#\"\"%\"#R/&F%6$\"\"$F(#F(\"#E/&F%6 $F/F'#F/F1/&F%6$F*F(#F/\"#_/&F%6$F*F'\"\"!/&F%6$F*F/#\"\"*F:/&F%6$\"\" &F(#\".@OTSmS\"\".wC[4#yC/&F%6$FGF'F>/&F%6$FGF/#!.blR`cv\"\"-#\\F)pg#) /&F%6$FGF*#\".s#oe5@8\"->1P_&>'/&F%6$\"\"'F(#\":r2v&\\l2*Hv$3j9\":cVbV Uo:\"=:^$y#/&F%6$FgnF'F>/&F%6$FgnF/#!6jtbLXjZ=]<%\"6OO,YJK[4x8#/&F%6$F gnF*#\"9@!4-(*QzA`knW'=-96F/&F%6$\"\"(F(#!@]e)H&*o;X4[C2/&F%6$FcpF/#\"9H!*G?dl8)\\Y_$))\"96)opT>?p#*4%yD/&F%6 $FcpF*#!?(em!ys]]Wle4us4b\"?W0+)R<\"yo5RkrtaK/&F%6$FcpFG#!Eb%pI9.K%R)) >CR]pe#Go*\"DK!o&GA.Tt7w\"fb$3YGG$/&F%6$FcpFgn#\"7>Fapa;s#G/B$\"6sUo3\"/&F%6$\"\")F(#!<$\\\\mD9E%3O/\\Bo$\"/&F%6$FerF/#\"52nB2^O8wC>\"4GO8K@K_i$[/&F%6$FerF*#!=fN'pEHkw%Q:=LFO \"=O*yG#e0+RT^-%H&=/&F%6$FerFG#!FJanA1pAGrI`jifLHO6\"\"D7r%4ur\\$>O6c0 ,^e(RJ/&F%6$FerFgn#\"1OLQ@ip:v\"0$GF^Xx4@/&F%6$FerFcp#!4W(Hz))H(RiY#\" 6x#**3:kEll&p\"/&F%6$FCF(#\"*$y6_E\"+g!>0$R/&F%6$FCF'F>/&F%6$FCF/F>/&F %6$FCF*#\",Z(*[)>`\"-3e0Cr9/&F%6$FCFG#!9R>\"p5Md.bsDl\"\"8?fU&Gn+1:$o+ '/&F%6$FCFgn#\"1r8*G&H'yo\"\"0?BZ\\RUO&/&F%6$FCFcp#\"0^s%\\')QvR\"0?1M dI#[Z/&F%6$FCFer#!(nA9(\")gXz5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 6 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee ,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"* $y6_E\"+g!>0$R/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\",Z(*[)>`\"-3e 0Cr9/&F%6#\"\"&#!9R>\"p5Md.bsDl\"\"8?fU&Gn+1:$o+'/&F%6#\"\"'#\"1r8*G&H 'yo\"\"0?BZ\\RUO&/&F%6#\"\"(#\"0^s%\\')QvR\"0?1MdI#[Z/&F%6#\"\")#!(nA9 (\")gXz5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"80[()fd8:'4\"4 >'\"9CKZovPuggF*4*/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"<85`3mPt. nrh-\">\"<+)oSK1q)GYtD=K&/&F%6#\"\"&#!A.>6#pmL)))H@@-XS8;\"@!GlM]+2Hzr Z![=1C'/&F%6#\"\"'#\"U)*f!QWm;Fy,>/&F%6#\"\")#!#<\"#D/&F%6#\"\"*#!0 R=#GcP#f#\"2]+ARd*y$*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 35 "#===== =============================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Abreviated calculations" }}{PARA 257 "" 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 0 770 "S O7_8 := SimpleOrderConditions(7,8,'expanded'):\nSO_eqs := [op(RowSumCo nditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')),\n \+ op(StageOrderConditions(3,3..8,'expanded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns := [seq (SO7_8[i],i=[1,2,4,8,16,20,29,32,64])]:\n`ord_cdns*` := [seq(`SO5_9*`[ i],i=[1,2,4,8,9,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..8)=b[1],s eq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns := [seq (b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO_eqs), op(wt_eqns),op(`ord_cdns*`)]:\n\nerrterms6_8 := PrincipalErrorTerms(6, 8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,' expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'e xpanded')):" }}}{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_RKcoeffs" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1985 "calc_RKcoeffs := proc()\n local eqns,sm,ct,Rz,stb6,stb5,nmB,snmB,dnB,sdnB,nmC,snmC,B_7,C_7,nr m;\n global e1,e2,e3;\n\n e1 := \{c[4]=c_4,c[6]=c_6,c[7]=c_7,c[8]= 1,c[9]=1,seq(a[i,2]=0,i=4..8),\n b[2]=0,b[3]=0,`b*`[2]=0,`b*` [3]=0,`b*`[bs[1]]=bs[2]\};\n eqns := subs(e1,cdns):\n e2 := solve( \{op(eqns)\});\n e3 := `union`(e1,e2);\n Digits := 14;\n sm := 0 ;\n for ct to nops(errterms6_8) do\n sm := sm+(evalf(subs(e3,er rterms6_8[ct])))^2;\n end do;\n Rz := subs(e3,StabilityFunction(6, 8,'expanded'));\n stb6 := max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1, z=-8..-1e-7));\n stb6 := evalf[8](stb6);\n Rz := subs(e3,subs(b=`b *`,StabilityFunction(5,9,'expanded')));\n stb5 := max(fsolve(Rz=1,z= -8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n stb5 := evalf[8](stb5);\n \+ nmB := 0;\n for ct to nops(`errterms6_9*`) do\n nmB := nmB+eva lf(subs(e3,`errterms6_9*`[ct]))^2;\n end do:\n snmB := sqrt(nmB); \n dnB := 0;\n for ct to nops(`errterms5_9*`) do\n dnB := dnB +evalf(subs(e3,`errterms5_9*`[ct]))^2;\n end do;\n sdnB := sqrt(dn B);\n nmC := 0;\n for ct to nops(errterms6_8) do\n nmC := nmC +(evalf[14](subs(e3,`errterms6_9*`[ct]))-evalf[14](subs(e3,errterms6_8 [ct])))^2;\n end do;\n snmC := sqrt(simplify(nmC));\n B_7 := eva lf[8](snmB/sdnB);\n C_7 := evalf[8](snmC/sdnB);\n print(`nodes:`,c [2]=subs(e3,c[2]),c[3]=subs(e3,c[3]),c[4]=c_4,c[5]=subs(e3,c[5]),c[6]= c_6,c[7]=c_7);\n print(`order 6 weights:`,seq(b[i]=evalf[6](subs(e3, b[i])),i=[1,$4..8]));\n print(`order 5 weights:`,seq(`b*`[i]=evalf[6 ](subs(e3,`b*`[i])),i=[1,$4..9]));\n nrm := evalf(max(seq(seq(subs(e 3,abs(a[i,j])),j=1..i-1),i=2..9)));\n print(infinity*`-norm of linki ng coeffs`=evalf[10](nrm));\n print(`2-norm of principal error of or der 6 scheme` = evalf[10](sqrt(sm)));\n print(`2-norm of principal e rror of order 5 scheme` = evalf[10](sdnB));\n print(`order 6 stabili ty interval` = [stb6,0]);\n print(`order 5 stability interval` = [st b5,0]);\n print('B[7]'=B_7,'C[7]'=C_7);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------- ------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Dormand, Lockyer, McCorri gan and Prince scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "c_4 := 1/4: c_6 := 1/2: c_7 := 48/49: bs := [8,-1/2]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/ &F&6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F5/&F&6#\"\"&#F:F+/&F&6#F1#F*F(/&F&6# \"\"(#\"#[\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G /&%\"bG6#\"\"\"$\"'h[q!\"(/&F&6#\"\"%$\"'dnU!\"'/&F&6#\"\"&$\"'\\w5!\" &/&F&6#\"\"'$!'%Gg(F2/&F&6#\"\"($\"'uiaF2/&F&6#\"\")$!'A(f$F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"' 9qz!\"(/&F&6#\"\"%$\"'nnO!\"'/&F&6#\"\"&$\"'sPsF2/&F&6#\"\"'$!'IQQF2/& F&6#\"\"($\"' " 0 "" {MPLTEXT 1 0 61 "evalf(sqrt(subs(e3,add( (b[i]-`b*`[i])^2,i=1..8)+`b*`[9]^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.#[tz&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 60 "#------------------------------------------------------ -----" }}{PARA 0 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c [4]=6/25" "6#/&%\"cG6#\"\"%*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]=63/64" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\" " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "c_4 := 6 /25: c_6 := 11/21: c_7 := 63/64: bs := [8,-6/11]:\ncalc_RKcoeffs();" } }{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\")\"#v/&F& 6#\"\"$#\"\"%\"#D/&F&6#F1#\"\"'F2/&F&6#\"\"&#\"%2=\"%iJ/&F&6#F7#\"#6\" #@/&F&6#\"\"(#\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~we ights:G/&%\"bG6#\"\"\"$\"'92p!\"(/&F&6#\"\"%$\"'N*)Q!\"'/&F&6#\"\"&$\" 'q<)*F2/&F&6#\"\"'$!')3@'F2/&F&6#\"\"($\"'9VmF2/&F&6#\"\")$!'-I[F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"' \"[=(!\"(/&F&6#\"\"%$\"'iNP!\"'/&F&6#\"\"&$\"'b)e)F2/&F&6#\"\"'$!'JZ\\ F2/&F&6#\"\"($\"'.XvF2/&F&6#\"\")$!'baaF2/&F&6#\"\"*$!'De=F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeff sGF&$\"+_$*>x\"*!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~p rincipal~error~of~order~6~schemeG$\"+P[B')G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+WcOu 7!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG 7$$!)\\+3V!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stab ility~intervalG7$$!)k#yL%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /&%\"BG6#\"\"($\")l`Z6!\"(/&%\"CGF&$\"))Qt9\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#------------------------ --------------------------" }}{PARA 0 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[4]=9/38" "6#/&%\"cG6#\"\"%*&\"\"*\"\"\"\"#Q!\"\" " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=63/64" "6#/&%\"cG6#\"\"(* &\"#j\"\"\"\"#k!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "c_4 := 9/38: c_6 := 330/613: c_7 := 63/64: bs := [8,- 7/13]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/ &%\"cG6#\"\"##F(\"#>/&F&6#\"\"$#F.F*/&F&6#\"\"%#\"\"*\"#Q/&F&6#\"\"&# \"'zR[\"'hV%)/&F&6#\"\"'#\"$I$\"$8'/&F&6#\"\"(#\"#j\"#k" }}{PARA 11 " " 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'rao!\"(/&F &6#\"\"%$\"'g*y$!\"'/&F&6#\"\"&$\"'`V6!\"&/&F&6#\"\"'$!'Q8xF2/&F&6#\" \"($\"'#ed'F2/&F&6#\"\")$!'#Gx%F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*% 1order~5~weights:G/&%#b*G6#\"\"\"$\"'QCr!\"(/&F&6#\"\"%$\"',[O!\"'/&F& 6#\"\"&$\"'9!z*F2/&F&6#\"\"'$!'\"z/'F2/&F&6#\"\"($\"'$e\\(F2/&F&6#\"\" )$!'i%Q&F2/&F&6#\"\"*$!')*Q@F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&% )infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+'R\">&3\"!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 6~schemeG$\"+3!\\:e#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~o f~principal~error~of~order~5~schemeG$\"+)eq?s\"!#8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)*z)HV!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)QDL W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")e@*3\" !\"(/&%\"CGF&$\")po.6F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 52 "#------------------------------------------------- --" }}{PARA 0 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[4] =7/30" "6#/&%\"cG6#\"\"%*&\"\"(\"\"\"\"#I!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=81/82" "6#/&%\"cG6#\"\"(*&\"#\")\"\"\"\"##)!\"\" " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "c_4 := 7 /30: c_6 := 49/87: c_7 := 81/82: bs := [8,-25/38]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#9\"$N\"/& F&6#\"\"$#\"\"(\"#X/&F&6#\"\"%#F1\"#I/&F&6#\"\"&#\"&7;&\"&.)*)/&F&6#\" \"'#\"#\\\"#()/&F&6#F1#\"#\")\"##)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %1order~6~weights:G/&%\"bG6#\"\"\"$\"'g'z'!\"(/&F&6#\"\"%$\"'L$o$!\"'/ &F&6#\"\"&$\"'/>D!\"&/&F&6#\"\"'$!'aK@F9/&F&6#\"\"($\"'^Y!)F2/&F&6#\" \")$!'`uiF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%# b*G6#\"\"\"$\"'L#*o!\"(/&F&6#\"\"%$\"'MOO!\"'/&F&6#\"\"&$\"''*QB!\"&/& F&6#\"\"'$!'R_>F9/&F&6#\"\"($\"'d%\\)F2/&F&6#\"\")$!'&*ylF2/&F&6#\"\"* $!'No5F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-nor m~of~linking~coeffsGF&$\"+$3t6j#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+5z-SA!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+ck8W5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~st ability~intervalG7$$!)2\"zN%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)L)eY%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")\"\\95\"!\"(/&%\"CGF&$\")!\\S7\"F* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "#--- --------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[4] = 3/13;" "6#/&%\"cG6# \"\"%*&\"\"$\"\"\"\"#8!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7 ] = 86/87;" "6#/&%\"cG6#\"\"(*&\"#')\"\"\"\"#()!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "c_4 := 3/13: c_6 := 39/67: c_7 := 86/87: bs := [8,-17/25]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 " " {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"%\"#R/&F&6#\"\"$#F(\"#8/& F&6#F*#F/F1/&F&6#\"\"&#\"&@I\"\"&fE#/&F&6#\"\"'#F+\"#n/&F&6#\"\"(#\"#' )\"#()" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6# \"\"\"$\"']Zn!\"(/&F&6#\"\"%$\"'*eh$!\"'/&F&6#\"\"&$!';^F!\"&/&F&6#\" \"'$\"'^YJF9/&F&6#\"\"($\"'Os$)F2/&F&6#\"\")$!'a;mF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'u.o!\"(/&F&6# \"\"%$\"'\\*e$!\"'/&F&6#\"\"&$!'L&e#!\"&/&F&6#\"\"'$\"'V!)HF9/&F&6#\" \"($\"'qo')F2/&F&6#\"\")$!'++oF2/&F&6#\"\"*$!'Te*)!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+j>KiN!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal ~error~of~order~6~schemeG$\"+m**R`?!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+r!Rx-\"!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)' \\>P%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability ~intervalG7$$!)3H5X!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"B G6#\"\"($\"))pK8\"!\"(/&%\"CGF&$\")NCk6F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "#----------------------------------------------------" }}{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 35 "#==================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Test-bed procedures for the examples " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "RK6_8step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2923 "rk6step := proc(x_rk6step::realco ns)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,a52,a53 ,a54,\n a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84, a85,a86,a87,\n f1,f2,f3,f4,f5,f6,f7,f8,b1,b2,b3,b4,b5,b6,b7,b8,\n \+ xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits;\n options `Copyright \+ 2004 by Peter Stone`;\n \n data := SOLN_;\n\n saveDigits := Digi ts;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n # procedu re 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 fa iled at %1\",evalf([X_,Y_],saveDigits);\n end if;\n val;\n \+ end proc;\n\n xx := evalf(x_rk6step);\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 := c6_; c7 := c7_; c8 := c8_;\n a2 1 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a4 3_;\n a51 := a51_; a52 := a52_; a53 := a53_; a54 := a54_;\n a61 := a61_; a62 := a62_; a63 := a63_; a64 := a64_; a65 := a65_;\n a71 := \+ a71_; a72 := a72_; a73 := a73_; a74 := a74_; a75 := a75_; a76 := a76_; \n a81 := a81_; a82 := a82_; a83 := a83_; a84 := a84_; a85 := a85_; \+ a86 := a86_; a87 := a87_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b 4_; b5 := b5_; b6 := b6_; b7 := b7_; b8 := b8_;\n # Perform a binary 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] then jF := jM else jS := jM end if;\n end do;\n if jM = n then j F := n-1; jS := n end if;\n else\n while jS-jF> 1 do\n j M := trunc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM else j S := 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(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 + a 65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a7 3*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 \n ys := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n\n evalf[saveD igits](ys);\nend proc: # of rk6step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "RK6_1 scheme of Dormand, Lockyer, McCorrigan and Pr ince" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3491 "RK6_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a6 2,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87 ,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,sol n,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := max(trunc (evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := m atrix([[1/9,1/9,0,0,0,0,0,0,0],\n [1/6,1/24,1/8,0,0,0,0,0,0], \n [1/4,1/16,0,3/16,0,0,0,0,0],\n [5/9,280/729,0,-32 5/243,1100/729,0,0,0,0],\n [1/2,6127/14680,0,-1077/734,6494/4 037,-9477/161480,0,0,0],\n [48/49,-13426273320/14809773769,0, 4192558704/2115681967,14334750144/14809773769,\n 1 17092732328/14809773769, -361966176/40353607,0,0],\n [1,-2340 689/1901060,0,31647/13579,253549596/149518369,10559024082/977620105,\n -152952/12173,-5764801/186010396,0],\n [0 ,203/2880,0,0,30208/70785,177147/164560,-536/705,1977326743/3619661760 ,-259/720]]);\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 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 := eva lf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a5 4 := 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 b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := evalf(A[8,4 ]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n b6 := evalf( A[8,7]);\n b7 := evalf(A[8,8]);\n b8 := evalf(A[8,9]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from \+ 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := \+ fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(x k + 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 + a 63*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\n \+ yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 \+ + b8*f8)*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,a31_=a3 1,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53 _=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65 ,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n \+ a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87, \n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\}; \n return subs(eqns,eval(rk6step)); \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 20 "RK6_2 scheme with " }{XPPEDIT 18 0 "c [4]=6/25" "6#/&%\"cG6#\"\"%*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]=63/64" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4286 "RK6_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n l ocal c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54 ,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85, a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk ,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := ma x(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n \+ A := matrix([[8/75,8/75,0,0,0,0,0,0,0],\n [4/25,1/25,3/25,0, 0,0,0,0,0],\n [6/25,3/50,0,9/50,0,0,0,0,0],\n [1807/ 3162,1116733125001/2276240222016,0,-1364629188325/758746740672,\n \+ 2138982988675/1138120111008,0,0,0,0],\n [11/21,3731 84883131711592359/629428679970118345044,0,\n -12855539789 6225575/58054665188168082,20224502633443247773975/91272383515534095158 76,\n -3590778379490959900760/50904073152027811713543,0,0 ,0],\n [63/64,-9955638050099787690037550073/71706108312579039 19821488128,0,\n 421770368617928768410875/90187287207047 138902016,\n -3031979901406817815170871063875/14084537820 65008683667117047808,\n 15692549351917507935540245739734 013/2252563056478189244473772594954240,\n -14873540135675 332293/2089665547196170240,0,0],\n [1,-2691778286453656380245 383/1521258163484980574735172,0,\n 2141335633391996675/3 71194659287217497,\n -161679233545622750791763569225/6213 7427083607735703094426884,\n 136363934951653120130832311 8018680/154935659262427604083414890417653,\n -44416190527 1490/48398415269957,-311127404078694400/14477515022855683323,0],\n \+ [0,31138111/450810360,0,0,2474072265625/6361142786136,1388157350 6953684872/14139334514114522375, -132955185321/214068151 000,19335942766592/29106652419495,-2486977/5149000]]);\n\n c2 := eva lf(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 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 := ev alf(A[7,7]);\n a87 := evalf(A[7,8]);\n b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := evalf(A[8,4]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n b6 := evalf(A[8,7]);\n b7 := evalf(A[8,8 ]);\n b8 := evalf(A[8,9]);\n\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\n yk := yk + (b1*f1 + b2*f2 \+ + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*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,a31_=a31,a32_=a32,a41_=a41,\n \+ a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_ =a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72, a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n a81_=a81,a82_=a82,a83_= a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,\n b1_=b1,b2_=b2,b3_=b 3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\};\n return subs(eqns,eval(r k6step)); \n else\n return evalf[saveDigits]([soln]);\n end i f;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "RK6 _3 scheme with " }{XPPEDIT 18 0 "c[4]=9/38" "6#/&%\"cG6#\"\"%*&\"\"* \"\"\"\"#Q!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=63/64" "6# /&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4749 "RK6_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31, a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a 74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n sa veDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5); \n\n fn := unapply(fxy,x,y);\n\n A :=matrix([[2/19,2/19,0,0,0,0,0, 0,0],\n [3/19,3/76,9/76,0,0,0,0,0,0],\n [9/38,9/152,0, 27/152,0,0,0,0,0],\n [483979/844361,50496032797210018219/97521 306573066496722,0,\n -62238405188783017003/325071021910221 65574,\n 96058693980453356374/48760653286533248361,0,0,0, 0],\n [330/613,7702683297267720889544548006160849795/123779558 59939284746771758057563701031,\n 0,-399229043155792951390 13477837035/17050265762136077180720317834126,\n 639564226 981491742693206700036358264060/276008500113722586808059264316424382117 ,\n -5313180967312747463264830899685275868548285/890548785 84359562560518476456399170688402362,\n 0,0,0],\n[63/64,-5 80440564877918386354914362593379258757943/4222274704052084709366384256 13284692459520,0,837083378582818460922126567546504549/1744817318128300 90122355897926680576,-680920067893602197172150986392871771764900095/27 5506239843907734397585145149495475886358528,13893902347091310791859466 7116771850303891585832494354565/16440422995450808437828180734862212664 987734041599934464,-5136972107040184160774194317/610237065866248643642 654720,0,0],\n[1,-73912817561044808721210574412370874448581/4209820296 5363364880987193147226166573410,0,821199866821133946316318884532576/13 8069098844093769649611890045033,-9626283444389131976638088363590116538 104533086/3170748768724798996135280838794491211302869081,2294881896422 50366114869444763538854564407191644550406748303/2141270337988251614337 4345447138204592119497840386204840294,-1608754695857040236707195633/14 8246171293910549234824945,-2136870666574914369093632/99262060255996938 363115017,0],\n[0,74489238253/1086687728280,0,0,102214327070044952/269 723237135952465,51769140351634238567474075866190023/452713161495435354 39664321662847560,-100866038423285938709/130767668626607626680,1155980 9987903488/17579279732593635,-84698060813/177459304440]]);\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 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 := ev alf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a 61 := 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 := eval f(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 b1 := evalf(A[8,2]);\n \+ b2 := evalf(A[8,3]);\n b3 := evalf(A[8,4]);\n b4 := evalf(A[8,5] );\n b5 := evalf(A[8,6]);\n b6 := evalf(A[8,7]);\n b7 := evalf(A [8,8]);\n b8 := evalf(A[8,9]);\n\n xk := evalf(xx);\n yk := eval f(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := \+ fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n \+ t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n \+ t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5* h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73* f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ 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\n yk := yk + (b1*f1 + b2 *f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n xk := x k + 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,a31_=a31,a32_=a32,a41_=a41,\n \+ a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n \+ a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_= a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n a81_=a81,a82_=a82,a 83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,\n b1_=b1,b2_=b2,b 3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\};\n return subs(eqns,ev al(rk6step)); \n else\n return evalf[saveDigits]([soln]);\n e nd if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 " RK6_4 scheme with " }{XPPEDIT 18 0 "c[4]=7/30" "6#/&%\"cG6#\"\"%*&\" \"(\"\"\"\"#I!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=81/82" "6#/&%\"cG6#\"\"(*&\"#\")\"\"\"\"##)!\"\"" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4587 "RK6 _4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21 ,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72, a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7, b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digit s+5);\n\n fn := unapply(fxy,x,y);\n\n A :=matrix([[14/135,14/135,0 ,0,0,0,0,0,0],\n [7/45,7/180,7/60,0,0,0,0,0,0],\n [7/3 0,7/120,0,7/40,0,0,0,0,0],\n [51612/89803,19468520817039492/35 486945168446723,0,-72573280055824680/35486945168446723,\n \+ 73499982271800480/35486945168446723,0,0,0,0],\n [49/87,2025265 8340120203773069749/33784135725639504675502296,\n 0,-79221 6000912879280500/351533212217208184283,\n 2249034659172295 45508505100/100340318815275363580633077,\n -216793587217071 62238705728210467/901105029036754619331512380401576,0,0,0],\n \+ [81/82,-972158721124193440814630918467125/7994344744294859028332746283 45432,\n 0,3058612407219548732633706208815/709347359742223 516267324426216,\n -53478378574860120460721550328853039040 /23405367722957560812286413128409759761,\n 52476758366259 3596626716546857270370413091/24088433982273003693314369668047287199799 ,\n -696206010458633764414572/32219569565392536806647,0,0] ,\n [1,-114361999204933693744433721507907/77877427759375157832 377305006776,0,\n 7405121074013573981098320/1457324938194 042260233261,\n -29930095382476957846060551182764365/11155 804472406678900783685305189917,\n 26499197252515606414261 19276806860337581199555/100712482278012733160575815039507169826241912, \n -42066702439513241792550/1604007647963024581459,\n \+ -425342741234074038059/27197351224216978505337,0],\n \+ [0,5847536753/86036171760,0,0,1815811171875/4929812076868,\n \+ 74928498684378188697374218447/29744854556648749370324659440,\n \+ -6206299092285813/2910288487182440,1044913680622457/129859 1690996340,\n -418874561/667578680]]);\n\n c2 := evalf( A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := e valf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\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 := e valf(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 := eva lf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a8 1 := 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 b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := evalf(A[8,4]);\n b4 := evalf(A[8,5]);\n \+ b5 := evalf(A[8,6]);\n b6 := evalf(A[8,7]);\n b7 := evalf(A[8,8]); \n b8 := evalf(A[8,9]);\n\n xk := evalf(xx);\n yk := evalf(yy); \n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk, yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := \+ a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n \+ t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n \+ f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a 74*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\n yk := yk + (b1*f1 + b2*f2 + \+ b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n xk := xk + h: \n soln := soln,[xk,yk];\n end do;\n if bb=true then\n e qns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_= c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,a31_=a31,a32_=a32,a41_=a41,\n a 42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n 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,\n a81_=a81,a82_=a82,a83_=a8 3,a84_=a84,a85_=a85,a86_=a86,a87_=a87,\n b1_=b1,b2_=b2,b3_=b3, b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\};\n return subs(eqns,eval(rk6 step)); \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 20 "RK6_5 scheme with " }{XPPEDIT 18 0 "c[4] = 3/13;" "6#/&%\"cG6#\"\"%*&\"\" $\"\"\"\"#8!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 86/87; " "6#/&%\"cG6#\"\"(*&\"#')\"\"\"\"#()!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4349 "RK 6_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a2 1,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72 ,a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7 ,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits; \n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Dig its+5);\n\n fn := unapply(fxy,x,y);\n\n A :=matrix([[4/39,4/39,0,0 ,0,0,0,0,0],\n [2/13,1/26,3/26,0,0,0,0,0,0],\n [3/13,3 /52,0,9/52,0,0,0,0,0],\n [13021/22659,1406640413621/2478209482 476,0,-1755653396555/826069827492,\n 1321105868272/6195523 70619,0,0,0,0],\n [39/67,1463083752990765495750771/27835115181 15684243554356,0,\n -417501847634533557363/2137709483231460 13636,\n 414679390177938970209021/208212903666744217281464 ,\n 48490458547529962724706855/271114021864467645322194274 4,0,0,0],\n [86/87,-1146771707244809451668952985850/1178428995 610817474751161698881,\n 0,883524649813655720289029/257840 992692019416968811,\n -550972740958654450507278066587/32547 3716439106878117398000544,\n -96828258695039241988394320314 3069455/32828460835559176127341032228568032,\n 32304282721 65469542719/108684219530393291772,0,0],\n [1,-3682349043608426 14256649493/316816365518493517722015828,0,\n 1924761336510 7236707/4836252322132133628,\n -362733181538476642926696355 9/1852940251413900055822878936,\n -111362933596263533071282 2690622675431/31397585101055611361934971740947112,\n 75156 96221383336/210977455127283,-2466239729887929744/169565652664150899277 ,0],\n [0,265211783/3930519060,0,0,53198489747/147124055808,\n -165257255035734106911939/60068315060067285425920,16878629 52891371/536423949472320,\n 397538864947251/47482305734062 0,-7142267/10794560]]);\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 := eval f(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n a21 : = c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := e valf(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 := eva lf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a6 5 := 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 b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := e valf(A[8,4]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n b6 := evalf(A[8,7]);\n b7 := evalf(A[8,8]);\n b8 := evalf(A[8,9]);\n \n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n f or 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*f 3 + 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\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f 6 + b7*f7 + b8*f8)*h;\n xk := xk + h:\n soln := soln,[xk,yk] ;\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=f xy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_= c8,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a5 2_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a6 4,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76 _=a76,\n a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86 ,a87_=a87,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7, b8_=b8\};\n return subs(eqns,eval(rk6step)); \n else\n ret urn evalf[saveDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Testing the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 80 "These tests do not \+ make use of the embedded order 4 method for error correction." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 1 of 8 stage, order 6 Runge- Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%# dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F +F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6 #\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x* cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/ 289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 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ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXES LABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 765 "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 := [`scheme of Dor mand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=6 3/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30, c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigit s := 25:\nfor ct to 5 do\n Fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0, hh,numsteps,false);\n sm := 0: numpts := nops(Fn_RK6_||ct):\n for \+ ii to numpts do\n sm := sm+(Fn_RK6_||ct[ii,2]-f(Fn_RK6_||ct[ii,1] ))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDi gits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"x GF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initia l~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G \"$+&Q)pprint566\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~M cCorrigan~and~PrinceG$\"+o'*\\y'*!#C7$*&%-scheme~with~G\"\"\"6$/&%\"cG 6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+n$>G%\\F+7$*&F.F/6$/F2#\"\" *\"#QF9F/$\"+?a'\\C'F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+Dt6%R&F +7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+)QG#QcF+Q)pprint576\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%& FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 696 "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 := [`scheme of Dormand, Lockyer, \+ McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme w ith `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`sc heme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 25:\nfor c t to 5 do\n fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,numsteps,tru e);\nend do:\nxx := 4.999: fxx := evalf(f(xx)):\nfor ct to 5 do\n er rs := [op(errs),abs(fn_RK6_||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)pprint58 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorriga n~and~PrinceG$\"+Esv0)*!#C7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\" \"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+'yCgO\"F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$ \"+V@Y,UF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+i%oAw$F+7$*&F.F/6$/ F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+$*zqsUF+Q)pprint596\"" }}}{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 me thod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 po int Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "mthds := [`scheme of Dormand, Lockyer, M cCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme wi th `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`sch eme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((f(x)-'fn_RK6_||ct'(x))^2,x=0..5,adaptive=fal se,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$%Rscheme~of~Dormand,~Lockyer,~M cCorrigan~and~PrinceG$\"+NZgc'*!#C7$*&%-scheme~with~G\"\"\"6$/&%\"cG6# \"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+>p#>\"\\F+7$*&F.F/6$/F2#\"\"* \"#QF9F/$\"+>)G4A'F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+CgOu`F+7$ *&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+/>j?cF+Q)pprint606\"" }}} {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 0 513 "evalf[20]( plot([f(x)-'fn_RK6_1'(x),f(x)-'fn_RK6_2'(x),f(x)-'fn_RK6_3'(x),f(x)-'f n_RK6_4'(x),\nf(x)-'fn_RK6_5'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[C OLOR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,. 35,.95),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrigan and Pri nce`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]= 3/13 and c[7]=86/87`],\ntitle=`error curves for 8 stage order 6 Runge- Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 834 555 555 {PLOTDATA 2 "6+-%'CURVESG6%7fv7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$\"%(o#!#>7$$\"5qm mmmT&)G\\aF-$\"%QhF07$$\"5SLLL3x1h6oF-$\"%MnF07$$\"50+++]7G$R<)F-$\"%& ='F07$$\"5SLL$3-)Q4b))F-$\"%$z&F07$$\"5qmmm\"z%\\DO&*F-$\"%6\\F07$$\"5 +++Dc,;u@5!#?$\"%9HF07$$\"5MLLLL3x&)*3\"FM$\"%S;F07$$\"5+++]i!R(*Rc\"F M$!&bv\"F07$$\"5nmmm\"H2P\"Q?FM$!&=J%F07$$\"5++]7G))>Wr@FM$!&&QZF07$$ \"5MLLek.pu/BFM$!&b:&F07$$\"5nm;/,>=0QCFM$!&'[_F07$$\"5+++]PMnNrDFM$!& 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'$R+$FM7$Fc]n$!'xFOFM7$Fh]n$!'LqTFM7$F]^n$!'NCXFM7$Fg^n$!'OC[FM7$Fa_n$ !'yD\\FM7$Fe`n$!'o!*\\FM7$Fj`n$!'l.]FM7$F_an$!'#o&\\FM7$$\"5MLLe*[t\\s p$F0$!'Xj[FM7$Fdan$!'&ps%FM7$Fian$!'q^VFM7$F^bn$!'HbRFM7$Fcbn$!'K7OFM7 $Fhbn$!'S)R$FM7$F]cn$!'8(H$FM7$Fbcn$!'1,LFM7$Fgcn$!'M&Q$FM7$F\\dn$!'Sa NFM7$Fadn$!'kRPFM7$Ffdn$!'4NRFM7$F[en$!'V\"4%FM7$F`en$!';-UFM-Feen6&Fg en$\"\"*Fien$\"\"$FienFihp-F]fn6#%Escheme~with~c[4]=9/38~and~c[7]=63/6 4G-F$6%7`tF'7$F2$\"%8aF07$FQ$\"%q()F07$$\"5nmm\"z%\\v#pK\"FM$\"%4()F07 $FV$\"%L$)F07$$\"5nm;z>6B`#o\"FM$\"%R\")F07$$\"5MLL3xJs1,=FM$\"%v!)F07 $$\"5++]PM_@g>>FM$\"%!Q)F07$Fen$\"%h*)F07$Fio$\"&&[(F07$F\\[o$\"&%)4 (F07$Far$\"&a*oF07$F[s$\"&WY'F07$Fes$\"&<'eF07$Fjs$\"&[<&F07$F_t$\"&zK %F07$Fdt$\"&2-$F07$Fit$\"'&*f5FM7$F^u$\"%J6FM7$Fcu$!'&4#=FM7$Fhu$!'uCO FM7$F]v$!',4]FM7$Fbv$!'qalFM7$Fgv$!'(oq'FM7$F\\w$!'c'*zFM7$Faw$!'kP'*F M7$Ffw$!(]:5\"FM7$F[x$!('yN7FM7$F`x$!(onY\"FM7$Fex$!(<4g\"FM7$Fjx$!(Hi m\"FM7$F_y$!(eDx\"FM7$F`_o$!(%*Rz\"FM7$Fe_o$!(*4r$FM7$F`]l$!']f@FM7$Fe]l$!'J37FM7$F_^l$!%!f#FM7$Fg`l$ \"&xR&FM7$F\\al$\"&&ftFM7$Faal$\"&6#))FM7$Ffal$\"&iB*FM7$F[bl$\"&iT*FM 7$$\"5ML$e9Te3n5\"F0$\"&IU*FM7$F`bl$\"&_F*FM7$$\"5++]P4'\\d18\"F0$\"&q %*)FM7$Febl$\"&+o)FM7$Ficl$\"&^\"yFM7$Fcdl$\"&(GvFM7$Fbfo$\"&SP(FM7$Fh dl$\"&9.(FM7$Fjfo$\"&^*eFM7$F]el$\"&)>]FM7$Fafl$\"%,VFM7$F_hl$!&6!eFM7 $Fdhl$!''eb\"FM7$Fihl$!'dPIFM7$F^il$!'ltVFM7$Fcil$!'lxiFM7$Fhil$!'slxF M7$F]jl$!'%zQ*FM7$Fbjl$!(S80\"FM7$Fgjl$!('3M6FM7$F[\\m$!(*3\\6FM7$Fe\\ m$!(m;<\"FM7$Ffio$!(Nm:\"FM7$Fj\\m$!(AB8\"FM7$F^jo$!($pZ6FM7$Fcjo$!(/q :\"FM7$Fhjo$!(qe6\"FM7$F_]m$!(Z28\"FM7$Fd]m$!(ml-\"FM7$Fi]m$!'=f))FM7$ Fc^m$!'.^pFM7$F]_m$!'uP^FM7$Fb_m$!&MR%F07$Fg_m$!&T&RF07$F^\\r$!&$**QF0 7$F\\`m$!&q\"QF07$F[]r$!&[!QF07$Fa`m$!&Q%QF07$F_bm$!&jG%F07$Fibm$!&_9& F07$F^cm$!&=@'F07$Fccm$!&\\(pF07$Fhcm$!&'ztF07$F]dm$!'*\\j(FM7$Ff^p$!' SNxFM7$Fbdm$!'I)y(FM7$F^_p$!'_WxFM7$Fgdm$!'O+yFM7$Faem$!'&=m(FM7$F[fm$ !'XwuFM7$F_gm$!'k!p'FM7$Figm$!'IbbFM7$Fchm$!'EALFM7$Fgim$!'^@EFM7$Fajm $!'%\\G#FM7$Ffjm$!'2W@FM7$F[[n$!'yy?FM7$Fe[n$!'`[?FM7$F_\\n$!'.`?FM7$F d\\n$!'k4@FM7$Fi\\n$!'8TAFM7$F]dp$!'OUCFM7$F^]n$!'Z.FFM7$Fc]n$!'XOKFM7 $Fh]n$!'&Hs$FM7$F]^n$!'jrSFM7$Fg^n$!'IlVFM7$Fa_n$!'UaWFM7$Fe`n$!'M.XFM 7$Fj`n$!'P,XFM7$F_an$!'CXWFM7$Fger$!'Z]VFM7$Fdan$!'o@UFM7$Fian$!''f)QF M7$F^bn$!'MQNFM7$Fcbn$!'oNKFM7$Fhbn$!'8WIFM7$F]cn$!'W_HFM7$Fbcn$!'YbHF M7$Fgcn$!'SIIFM7$F\\dn$!'y!=$FM7$Fadn$!'9YLFM7$Ffdn$!')=_$FM7$F[en$!'8 jOFM7$F`en$!'*Gw$FM-Feen6&FgenF($\"#NF\\ip$\"#&*F\\ip-F]fn6#%Escheme~w ith~c[4]=7/30~and~c[7]=81/82G-F$6%7ftF'7$F2$\"%v[F07$FQ$\"%6yF07$Feir$ \"%ByF07$FV$\"%YwF07$F]jr$\"%?wF07$Fbjr$\"%OyF07$Fgjr$\"%-$)F07$Fen$\" %N!*F07$Fio$\"&?y\"F07$Fcp$\"&%[FM7$ F[]l$!'?rRFM7$F`]l$!'u%)HFM7$Fe]l$!'n\"3#FM7$F_^l$!&oT*FM7$Fg`l$!&fv$F M7$F\\al$!&Dz\"FM7$Faal$!%)3$FM7$Ffal$\"%(>#FM7$F[bl$\"%m^FM7$Faes$\"% (Q'FM7$F`bl$\"%WgFM7$Fies$\"%xRFM7$Febl$\"%5BFM7$Ficl$!%)z$FM7$Fcdl$!% uvFM7$Fbfo$!&r+\"FM7$Fhdl$!&DX\"FM7$Fjfo$!&oj#FM7$F]el$!&Tj$FM7$Fafl$! &(4$)FM7$F_hl$!'gi9FM7$Fdhl$!'&eU#FM7$Fihl$!'\"y&QFM7$F^il$!'Tk^FM7$Fc il$!'xxpFM7$Fhil$!'\\.%)FM7$F]jl$!'d0**FM7$Fbjl$!(ok4\"FM7$Fgjl$!(4?< \"FM7$Fa[m$!(&)H=\"FM7$F[\\m$!(q3=\"FM7$F`\\m$!(UE>\"FM7$Fe\\m$!(g6?\" FM7$Ffio$!(5N=\"FM7$Fj\\m$!(mo:\"FM7$F^jo$!(pD<\"FM7$Fcjo$!(z>=\"FM7$F hjo$!(v(Q6FM7$F_]m$!(dR:\"FM7$Fd]m$!(#pY5FM7$Fi]m$!'dk!*FM7$Fc^m$!'sIs FM7$F]_m$!'(3e&FM7$Fb_m$!&;(\\F07$Fg_m$!&%zYF07$F^\\r$!&'zYF07$F\\`m$! &'oYF07$F[]r$!&\\r%F07$Fa`m$!&uz%F07$F_bm$!&8L&F07$Fibm$!&]6'F07$F^cm$ !&N&pF07$Fccm$!&$ouF07$Fhcm$!&bq(F07$F]dm$!'IuyFM7$Ff^p$!'FjzFM7$Fbdm$ !',4!)FM7$F^_p$!'^gzFM7$Fgdm$!'d;!)FM7$Faem$!'TzyFM7$F[fm$!'p1xFM7$F_g m$!'u]pFM7$Figm$!'d\\eFM7$Fchm$!''[l$FM7$Fgim$!'.aHFM7$Fajm$!'T.EFM7$F fjm$!'()\\CFM7$F[[n$!'yvBFM7$Fe[n$!'^VBFM7$F_\\n$!')RN#FM7$Fd\\n$!'dBC FM7$Fi\\n$!'%>d#FM7$F]dp$!'#>z#FM7$F^]n$!'ssIFM7$Fc]n$!'N`OFM7$Fh]n$!' '*=UFM7$F]^n$!'o]YFM7$Fg^n$!'Z3]FM7$Fa_n$!'44^FM7$Fe`n$!'!z:&FM7$Fj`n$ !'AX^FM7$F_an$!'.q]FM7$Fger$!'O_\\FM7$Fdan$!'Z)z%FM7$Fian$!'j4WFM7$F^b n$!'4:SFM7$Fcbn$!'*Hn$FM7$Fhbn$!'fbMFM7$F]cn$!'F^LFM7$Fbcn$!''\\N$FM7$ Fgcn$!'5SMFM7$F\\dn$!'L5OFM7$Fadn$!'$yz$FM7$Ffdn$!'2)*RFM7$F[en$!'_fTF M7$F`en$!'UtUFM-%'COLOURG6&Fgen$\"*++++\"!\")F(F(-F]fn6#%Escheme~with~ c[4]=3/13~and~c[7]=86/87G-%&TITLEG6#%Uerror~curves~for~8~stage~order~6 ~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F_dv-%%FONTG6$%*HELVETIC AGFehr-%%VIEWG6$;F(F`en%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, McCorrigan and Pr ince" "scheme with c[4]=6/25 and c[7]=63/64" "scheme with c[4]=9/38 an d c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "scheme with c[4] =3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 2 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&%\"xGF&%\"yGF(" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code 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 soluti on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 768 "G := (x,y) -> x/y: h h := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no . of steps: `,numsteps]]);``;\nmthds := [`scheme of Dormand, Lockyer , McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),` scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\ng : = x -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK6_||ct := RK6_||ct(G(x,y) ,x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn_RK6_||ct ):\n for ii to numpts do\n sm := sm+(Gn_RK6_||ct[ii,2]-g(Gn_RK6 _||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\n end 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)pprint616\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+!G$=![\"!#B7 $*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#k F/$\"+wav`))!#C7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+EhX9!)FB7$*&F.F/6$/F2#F< \"#I/F:#\"#\")\"##)F/$\"+/\"3U!fFB7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#( )F/$\"+Ex+#H&FBQ)pprint626\"" }}}{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 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is al so given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 698 "G := (x,y) -> \+ x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Dormand, L ockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),` scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81 /82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 25 :\nfor ct to 5 do\n gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,nums teps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := evalf (g(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(gn_RK6_||ct(xx)-gxx )];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*& %\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~ ~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint636\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 '7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"*hv-k%!#B7$* &%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/ $\"*te@)GF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"*IAhk#F+7$*&F.F/6$/F2#F<\"#I/ F:#\"#\")\"##)F/$\"*K#p%*>F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\" *yEM!=F+Q)pprint646\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows usi ng the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perf orm numerical integration by the 7 point Newton-Cotes method over 100 \+ equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 465 "mthd s := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`schem e with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)] : errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n sm := NCint((g(x)-'gn_RK6_||ct'(x))^2,x=0..10,adaptive=false,numpoi nts=7,factor=100);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorriga n~and~PrinceG$\"+O*=eZ\"!#B7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%# \"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+ac#z#))!#C7$*&F.F/6$/F2#\"\"*\"#QF9 F/$\"+YV@\"*zFB7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+RVD()eFB7$*&F. F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+H*zoF&FBQ)pprint656\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following e rror graphs are constructed using the numerical procedures for the sol utions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 514 "evalf[20](plot([ g(x)-'gn_RK6_1'(x),g(x)-'gn_RK6_2'(x),g(x)-'gn_RK6_3'(x),g(x)-'gn_RK6_ 4'(x),\ng(x)-'gn_RK6_5'(x)],x=0..10,font=[HELVETICA,9],\ncolor=[COLOR( RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.9 5),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrigan and Prince`, `scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7] =63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 \+ and c[7]=86/87`],\ntitle=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 818 563 563 {PLOTDATA 2 "6+ -%'CURVESG6%7fq7$$\"\"!F)F(7$$\"5mmmmTN@Ki8!#@F(7$$\"5KLLL$3FWYs#F-$!# F!#>7$$\"5lmm;aQ`!eS$F-$!$h\"F37$$\"5)******\\iSmp3%F-$!$%oF37$$\"5KLL $eRZF\"oZF-$!%ABF37$$\"5lmmmmT&)G\\aF-$!%\"Q$F37$$\"5ILLL3x1h6oF-$!%pL F37$$\"5++++]7G$R<)F-$!%(*GF37$$\"5lmmm\"z%\\DO&*F-$\"%7@F37$$\"5LLLLL 3x&)*3\"!#?$\"%esF37$$\"5++++]ilyM;FZ$\"&$yIF37$$\"5mmmmm;arz@FZ$\"&dY 'F37$$\"5+++D\"yD&y;CFZ$\"&;f(F37$$\"5LLL$e*)4bQl#FZ$\"'xa5F37$$\"5mm \"zW#fD78FFZ$\"'Ba5F37$$\"5++]7`>+RsFFZ$\"'#z0\"F37$$\"5LL3x\")zulJGFZ $\"'Uv5F37$$\"5mmmT5S\\#4*GFZ$\"':H6F37$$\"5++D1R+C>]HFZ$\"'[k7F37$$\" 5LL$3x1')f%4IFZ$\"'5+:F37$$\"5mmTN'4KF(oIFZ$\"'j(\\\"F37$$\"5++++D\"y% *z7$FZ$\"';&\\\"F37$$\"5LLek`TAE(=$FZ$\"'1$\\\"F37$$\"5mm;H#=qHlC$FZ$ \"'B$\\\"F37$$\"5++v$4@;(z0LFZ$\"'+-:F37$$\"5LLLeRAY1lLFZ$\"'MN:F37$$ \"5++](oHa*f$[$FZ$\"'NV=F37$$\"5mmm;ajW8-OFZ$\"'VK>F37$$\"5++D\"GQ#>Sh OFZ$\"'!*G>F37$$\"5LL$e9TQp1s$FZ$\"'XE>F37$$\"5mmT5SWo$*zPFZ$\"'yG>F37 $$\"5+++vo/V?RQFZ$\"'EY>F37$$\"5mm;/ED#Rx&RFZ$\"'-X@F37$$\"5LLLL$e9ui2 %FZ$\"'&yL#F37$$\"5mm;z>h!z&4UFZ$\"'vFBF37$$\"5+++DcwR)GM%FZ$\"'xVBF37 $$\"5LL$3F>*))=wWFZ$\"']%e#F37$$\"5mmm;H2Q\\4YFZ$\"'\\&o#F37$$\"5LLeR( \\EYhn%FZ$\"'))yEF37$$\"5++]ilA()zUZFZ$\"''Rn#F37$$\"5mmT&Q.=^%4[FZ$\" 'wwEF37$$\"5LLL3-QO5w[FZ$\"'Y/FF37$$\"5***\\7.d4cF%\\FZ$\"'`'z#F37$$\" 5lm;aQ`&3%4]FZ$\"'H%)HF37$$\"5IL3x1651w]FZ$\"'JwHF37$$\"5++++voMrU^FZ$ \"'KoHF37$$\"5NLL$3-8Lfn&FZ$\"'&Q=$F37$$\"5mmmmm\"z_\"4iFZ$\"'iNLF37$$ \"5lmmm;zp!fu'FZ$\"'\\IMF37$$\"5lmmmmm6m#G(FZ$\"';yMF37$$\"5lmmT5Sav\\ tFZ$\"'mtMF37$$\"5lmm;a8(\\oT(FZ$\"'a#[$F37$$\"5lmm\"zp)R%R[(FZ$\"'*=_ $F37$$\"5lmmmTg#Q5b(FZ$\"'?INF37$$\"5lmm;H2oA&o(FZ$\"'_2NF37$$\"5lmmm; a`T>yFZ$\"'5)[$F37$$\"5lmm;/,Rg`zFZ$\"'v)\\$F37$$\"5lmmm\"zW#z(3)FZ$\" '))4NF37$$\"5lmm;z%*4)>A)FZ$\"'4([$F37$$\"5lmmmmT&phN)FZ$\"'1pMF37$$\" 5++++v=ddC%*FZ$\"'jsLF37$$\"5LLLLe*=)H\\5F3$\"'YJKF37$$\"5mmmm\"z/3uC \"F3$\"'QNHF37$$\"5++++DJ$RDX\"F3$\"'\"[l#F37$$\"5mmmm\"zR'ok;F3$\"',1 CF37$$\"5++++D1J:w=F3$\"'%\\>#F37$$\"5LLLLL3En$4#F3$\"'o4?F37$$\"5mmmm ;/RE&G#F3$\"'io=F37$$\"5+++++D.&4]#F3$\"'8I*>\"F37$$\"5LLLLL$Q*o]RF3$\"'?V6F37$$\"5*******\\7=lj;%F3$\"'K(3\" F37$$\"5*******\\PaRY2aF3$\"&=Z)F37$$\"5mmmm;zXu9cF3$\"&*o\") F37$$\"5**********\\y))GeF3$\"&v(yF37$$\"5********\\i_QQgF3$\"&;h(F37$ $\"5*******\\7y%3TiF3$\"&2P(F37$$\"5********\\P![hY'F3$\"&.7(F37$$\"5K LLLL$Qx$omF3$\"&\"4pF37$$\"5*********\\P+V)oF3$\"&qp'F37$$\"5mmmm\"zpe *zqF3$\"&c^'F37$$\"5*********\\#\\'QH(F3$\"&\"GjF37$$\"5KLLLe9S8&\\(F3 $\"&7;'F37$$\"5*******\\i?=bq(F3$\"&e*fF37$$\"5KLLLL3s?6zF3$\"&C%eF37$ $\"5*******\\7`Wl7)F3$\"&**o&F37$$\"5lmmmmm*RRL)F3$\"&.b&F37$$\"5lmmm; a<.Y&)F3$\"&XT&F37$$\"5KLLLe9tOc()F3$\"&hG&F37$$\"5**********\\Qk\\*)F 3$\"&M<&F37$$\"5KLLL$3dg6<*F3$\"&*\\]F37$$\"5lmmmmmxGp$*F3$\"&W%\\F37$ $\"5)******\\7oK0e*F3$\"&m$[F37$$\"5)******\\(=5s#y*F3$\"&wt%F37$$\"#5 F)$\"%NY!#=-%&COLORG6&%$RGBG$\"\"&!\"\"F($\"\"\"F)-%'LEGENDG6#%Rscheme ~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG-F$6%7\\qF'F*7$F/$!#`F37$F :$!%Z8F37$FD$!%/nF37$FN$!%rlF37$FX$!%PdF37$Fhn$\"%\"*GF37$F]o$\"&7$=F3 7$Fbo$\"&kS#F37$Fgo$\"&g*QF37$F[q$\"&nI%F37$F_r$\"&5H'F37$Fdr$\"&GG'F3 7$Fir$\"&uG'F37$F^s$\"&1M'F37$Fcs$\"&J`'F37$Fhs$\"&[H)F37$F]t$\"&A\")) F37$Fbt$\"&jz)F37$Fgt$\"&oy)F37$F\\u$\"&b!))F37$Fau$\"&S\"*)F37$Ffu$\" '&4,\"F37$F[v$\"'kF6F37$$\"5++Dc^.m#H9%FZ$\"'/D6F37$F`v$\"')G7\"F37$$ \"5LL3-))=:BwUFZ$\"'XB6F37$Fev$\"'HM6F37$Fjv$\"'1'G\"F37$F_w$\"'.^8F37 $Fcx$\"'En8F37$Fgy$\"'kV:F37$F\\z$\"'\"4q\"F37$Faz$\"'KA=F37$F[[l$\"') z'>F37$F]^l$\"')[,#F37$$\"5++vVtF\\%HU)FZ$\"'I>?F37$$\"5IL$3-QJ?(*[)FZ $\"'XR?F37$$\"5lm\"zp)*p&\\c&)FZ$\"'3R?F37$$\"5+++v$f3rKi)FZ$\"'PK?F37 $$\"5lm;H2e=#ov)FZ$\"'S>?F37$$\"5ILL$3-js.*))FZ$\"'$G,#F37$$\"5lmTgF;! [r&*)FZ$\"'a>?F37$$\"5++]PM-M#R-*FZ$\"'TH?F37$$\"5ILe9T)y)p!4*FZ$\"'oA ?F37$$\"5lmm\"zWF37$Fb^l$\"'$))*>F 37$$\"5lmm;H2)y(e**FZ$\"'!f(>F37$Fg^l$\"'RY>F37$F\\_l$\"'a$z\"F37$Fa_l $\"'rN;F37$Ff_l$\"'5)[\"F37$F[`l$\"'Ag8F37$F``l$\"'wY7F37$Fe`l$\"'\")f 6F37$Fj`l$\"'Au5F37$F_al$\"&D***F37$Fdal$\"&qN*F37$Fial$\"&F%))F37$F^b l$\"&rH)F37$Fcbl$\"&F)yF37$Fhbl$\"&#[uF37$F]cl$\"&05(F37$Fbcl$\"&Nv'F3 7$Fgcl$\"&CX'F37$F\\dl$\"&\\;'F37$Fadl$\"&@#fF37$Ffdl$\"&.o&F37$F[el$ \"&!\\aF37$F`el$\"&?E&F37$Feel$\"&R2&F37$Fjel$\"&H*[F37$F_fl$\"&xs%F37 $Fdfl$\"&!yXF37$Fifl$\"&DU%F37$F^gl$\"&9H%F37$Fcgl$\"&(fTF37$Fhgl$\"&p /%F37$F]hl$\"&0$RF37$Fbhl$\"&o#QF37$Fghl$\"&Ts$F37$F\\il$\"&)GOF37$Fai l$\"&T`$F37$Ffil$\"&uW$F37$F[jl$\"&IO$F37$F`jl$\"&LG$F37$Fejl$\"&L@$F3 7$Fjjl$\"&l8$F37$F_[m$\"&52$F37$Fd[m$\"&T+$F37$Fi[m$\"&E%HF37$F^\\m$\" %zGFb\\m-Fd\\m6&Ff\\m$F[]mFi\\m$\"#&)!\"#F(-F]]m6#%Escheme~with~c[4]=6 /25~and~c[7]=63/64G-F$6%7]qF'F*Fc]m7$F:$!%O8F37$FD$!%^mF37$FN$!%nlF37$ FX$!%joF37$Fhn$!$x%F37$F]o$\"&J>\"F37$Fbo$\"&Qn\"F37$Fgo$\"&_\"HF37$F[ q$\"&sE$F37$F_r$\"&<'\\F37$Fdr$\"&a&\\F37$Fir$\"&-'\\F37$F^s$\"&s+&F37 $Fcs$\"&]<&F37$Fhs$\"&vq'F37$F]t$\"&&frF37$Fbt$\"&m9(F37$Fgt$\"&%RrF37 $F\\u$\"&q:(F37$Fau$\"&RD(F37$Ffu$\"&FJ)F37$F[v$\"&wM*F37$Fiam$\"&iK*F 37$F`v$\"&'3$*F37$Fabm$\"&bJ*F37$Fev$\"&XT*F37$Fjv$\"'?y5F37$F_w$\"'-P 6F37$Fcx$\"'!G:\"F37$Fgy$\"''\\J\"F37$F\\z$\"'mj9F37$Faz$\"'w\"e\"F37$ F[[l$\"'`KlF37$Fbcl$\"&/?'F37$Fgcl$\" &R#fF37$F\\dl$\"&,m&F37$Fadl$\"&rV&F37$Ffdl$\"&^@&F37$F[el$\"&F+&F37$F `el$\"&5$[F37$Feel$\"&$eYF37$Fjel$\"&A\\%F37$F_fl$\"&0M%F37$Fdfl$\"&J? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code c onstructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based o n each of the methods and gives the " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 768 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`schem e with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64), `scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=8 6/87)]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to \+ 5 do\n Hn_RK6_||ct := RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,false); \n sm := 0: numpts := nops(Hn_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Hn_RK6_||ct[ii,2]-h(Hn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initia l~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\" $+\"Q)pprint666\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~Mc Corrigan~and~PrinceG$\"+%oXF+7$* &F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+Y6\"*eUF+Q)pprint676\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$ ***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 698 "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 := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`schem e with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64), `scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=8 6/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n hn_RK6_||ct := \+ RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nh := x -> exp(- x^2/2):\nxx := 9.99: hxx := evalf(h(xx)):\nfor ct to 5 do\n errs := \+ [op(errs),abs(hn_RK6_||ct(xx)-hxx)];\nend do:\nDigits := 10:\nlinalg[t ranspose]([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$%/step~width:~~~G$F,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$%Rscheme~of~Dormand,~Lockyer,~McCo rrigan~and~PrinceG$\"+!>5i-\"!#M7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\" \"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+)[[d^'!#N7$*&F.F/6$/F2#\"\"*\"# QF9F/$\"+(*p&*4dFB7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+g:Li[FB7$*& F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+Sw2oVFBQ)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interval [0, 0.5] of each Runge-Kutta method is estimated as follows using the sp ecial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numer ical integration by the 7 point Newton-Cotes method over 50 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 464 "mthds := [`sch eme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6 /25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*( c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := \+ []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n sm := N Cint((h(x)-'hn_RK6_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,fact or=50);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nl inalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~Pri nceG$\"++;f]z!#?7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F 36#\"\"(#\"#j\"#kF/$\"+K)RlO&F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+%\\;L%[F +7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+#***oqUF+7$*&F.F/6$/F2#\"\"$ \"#8/F:#\"#')\"#()F/$\"+@9M#)RF+Q)pprint706\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 527 "evalf[20](plot(['hn_RK6_1' (x)-h(x),'hn_RK6_2'(x)-h(x),'hn_RK6_3'(x)-h(x),\n'hn_RK6_4'(x)-h(x),'h n_RK6_5'(x)-h(x)],x=0..6,numpoints=100,font=[HELVETICA,9],\ncolor=[COL OR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35 ,.95),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrigan and Princ e`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c [7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/ 13 and c[7]=86/87`],\ntitle=`error curves for 8 stage order 6 Runge-Ku tta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 925 518 518 {PLOTDATA 2 "6+-%'CURVESG6%7f`l7$$\"\"!F)F(7$$\"5.......*4M'!#@$\"'7Fs!#?7$$\"5111 111_#e=\"F0$\")X@dFF07$$\"5%RRRRR**)H1=F0$\")$R\")=$F07$$\"5&[[[[[o&)3 V#F0$\")>&QA&F07$$\"5CCCCCCQ]_IF0$\"))zu3(F07$$\"5777777M#)GOF0$\")/^m pF07$$\"5kjjjjj*obA%F0$\")m\"=q(F07$$\"5nmmmmmSsU[F0$\")*yrI(F07$$\"5Y XXXXX***yX&F0$\")5kpnF07$$\"5_^^^^^Ro!4'F0$\")RkVXF07$$\"5\\[[[[[//[mF 0$\"))zIC%F07$$\"5baaaaa=\\vsF0$\")\\YG6F07$$\"5POOOOO'>b!zF0$!(W4c%F0 7$$\"5YXXXXX\"oE^)F0$!)#*>#*>F07$$\"5edddddr,k!*F0$!)(p%[HF07$$\"5hggg ggWi>(*F0$!)x:rBF07$$\"5CCCCC%)4]F5!#>$\")(40\\\"F07$$\"5FFFFF2b5#4\"F ip$\")7$f6*F07$$\"5qpppp%*>S16Fip$\"*$[(R_\"F07$$\"577777#[)p?6Fip$\"* 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th~c[4]=7/30~and~c[7]=81/82G-F$6%7YF'7$F+$\"(\\&\\NF07$F2$\"(.Ve&F07$F 7$\"(os(RF07$F<$!'I\"Q*F-7$FA$\"(6q7'F07$FF$\")h`_iF07$FK$\"*l7)[EF07$ FP$\"**o&*G#)F07$FU$\"+!f2w=#F07$FZ$\"+g\"e2#fF07$Fin$\",W'[(*z5F07$F^ o$\",O\"=\"4b#F07$Fco$\",pveef%F07$Fho$\",@uqT&zF07$F]p$\"-3)z(=I8F07$ Fbp$\"-iCF@g@F07$Fgp$\"-.X\"F07$F[r$\".*)*3=Pj@R_q>F07$Fct$\"/\\]')3T5EF07$Fht$\"/\"f Uc//V$F07$F]u$\"/!HZ^V(*\\%F07$Fbu$\"/dF&ehb$eF07$Fgu$\"/%[C4LDr(F07$F \\v$\"/yTYhpU(*F07$Fav$\"07#4%=ZGI\"F07$Ffv$\"07Jlh$Gc;F07$F[w$\"0ex2s #[`@F07$F`w$\"0>#ytMS(o#F07$Few$\"0-\\ZT/#QLF07$Fjw$\"05\\JuT%GTF07$F_ x$\"06@p[ia3&F07$Fdx$\"0RET7uNC'F07$Fix$\"0LyIOZLi(F07$F^y$\"0Y115N%R$ *F07$Fcy$\"1JNN#G#QG5F07$Fhy$\"1MSNzz[J6F07$F]z$\"1q?6iE4]7F07$Fbz$\"1 Y*[A&)[RQ\"F07$Fgz$\"1;IQmo_A:F07$F\\[l$\"1Kwpg0#Rn\"F07$Fa[l$\"1)y,38 ]))z\"F07$Ff[l$\"1@+m;ZN!)=F07$F[\\l$\"1A=%H2T9(>F07$F`\\l$\"1hD&p\"p@ e@F0-%'COLOURG6&Fg\\l$\"*++++\"!\")F(F(-F^]l6#%Escheme~with~c[4]=3/13~ and~c[7]=86/87G-%&TITLEG6#%hnrelative~error~curves~for~8~stage~order~6 ~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fain-%%FONTG6$%*HELVETIC AGFgbm-%%VIEWG6$;F(F`\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, McCorrigan and Pr ince" "scheme with c[4]=6/25 and c[7]=63/64" "scheme with c[4]=9/38 an d c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "scheme with c[4] =3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 4 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathematics of Computation, Vol. \+ 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2) ;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$**\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F (" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\" \"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2 ^(13-6*x^2);" "6#/%\"yG)\"\"#,&\"#8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=-32*x*y(x)*ln(2);\nic := y(- 1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^simplify(log[2](rhs(%)));\nk \+ := unapply(rhs(%),x):\nplot(k(x),x=-1..1,font=[HELVETICA,9],labels=[`x `,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG 6#%\"xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" } }{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo 7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3w KL$3$3(F-7$$!3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3 :mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$ $!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$ FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7 $$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn 7$$!3Q+++]$*4)*\\F-$\"3e;N4:OF ap7$$!3]++]PYx\"\\#F-$\"37]-4,Tp9TFap7$$!3QnmTNz>&H#F-$\"3y(*QMk^JnXFa p7$$!3EMLLL7i)4#F-$\"3yCPsPtXE]Fap7$$!3#pm;aVXH)=F-$\"3_cYryDpGbFap7$$ !3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3s*****\\F&*=Y\"F-$\"3K`b3X@Jjk Fap7$$!3')****\\74_c7F-$\"3co2Qfx9woFap7$$!3ZmmT5VBU5F-$\"3E!>K?nVAE(F ap7$$!3)3LLL3x%z#)!#>$\"3C'Q/NU&H#f(Fap7$$!3gKL$e9d;J'Fft$\"3@erx_1&z$ yFap7$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o %4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")Fap7$$!3=o;/^wj!p\"Fft$\"3]3j^O K2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4%yz\")Fap7$$!3S<+D\"GyNH'!#?$\" 3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#)*eHY6>>)Fap7$$\"3o'H$3x\"yY_%F jv$\"3Q>*>/AS,>)Fap7$$\"3&yK$3_Nl.5Fft$\"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b \"Fft$\"3-l$f@nl+<)Fap7$$\"3@CL$3-Dg5#Fft$\"3kJX?)*G!=:)Fap7$$\"3e?Le* ['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ezw5VFft$\"3U_-I6(**[-)Fap7$$\"3L `mmmJ+IiFft$\"3%pB(\\hv&o%yFap7$$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wF ap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\ \\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$ \"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"31))p(30[[c&Fap7$$\"3#z****\\_q n2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%zAF-$\"3ZUhzOe!Rg%Fap7$$\"3U)** *\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\(=GB2FF-$\"3WV]5@%**Rj$Fap7$$ \"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7 $$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\" Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-? C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v^Fgn7$$\"3y)***\\7k.6aF-$\"3# pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9+3>Fgn7$$\"33****\\i!*3`iF-$ \"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4*yfd(\\\"fFM7$$\"3wKLL3N1#4( F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\"3%o[)olFVm:FM7$$\"3Y*******p (G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$\"35omE\\#[Ck$F=7$$\"3fKLLLb dQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 775 "K := (x,y) -> -32*x*y(x)*ln (2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slop e field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Dormand , Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64 ),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7] =81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Kn_RK6_||ct := RK6_||ct(evalf(K(x,y)),x,y,x0, evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK6_||ct) :\n for ii to numpts do\n sm := sm+(Kn_RK6_||ct[ii,2]-k(Kn_RK6_ ||ct[ii,1]))^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,$** \"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G- %!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)p print716\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorriga n~and~PrinceG$\"+#=0')p\"!#97$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%# \"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+H?@<6F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$ \"+L\"oP))*!#:7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+bIeu&)FK7$*&F.F /6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+ms;$y(FKQ)pprint726\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following cod e 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 20 ".995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 706 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 : = -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial point: \+ `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);` `;\nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`sch eme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64 ),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7] =86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n kn_RK6_||ct : = RK6_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,true);\nend do: \nxx := 0.995: kxx := evalf(k(xx)):\nfor ct to 5 do\n errs := [op(er rs),abs(kn_RK6_||ct(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\" #F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"# 7$%1no.~of~steps:~~~G\"$+#Q)pprint736\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~ of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+$y&G'G#!#>7$*&%-scheme~ with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+u#o9U \"F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+J=:W7F+7$*&F.F/6$/F2#F<\"#I/F:#\"# \")\"##)F/$\"+;W'>0\"F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+wU; \"[*!#?Q)pprint746\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F% " }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows u sing the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to pe rform numerical integration by the 7 point Newton-Cotes method over 10 0 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "mt hds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme wi th `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`sch eme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87 )]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((k(x)-' kn_RK6_||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7,factor=100);\n \+ errs := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+Dl%Gq \"!#97$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\" #j\"#kF/$\"+C<+?6F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+.$[%3**!#:7$*&F.F/6$ /F2#F<\"#I/F:#\"#\")\"##)F/$\"+WW*ff)FK7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#' )\"#()F/$\"+kEg-yFKQ)pprint756\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 514 "evalf[20](plot([k(x)-'kn_RK6_1'(x),k(x)-'k n_RK6_2'(x),k(x)-'kn_RK6_3'(x),k(x)-'kn_RK6_4'(x),\nk(x)-'kn_RK6_5'(x) ],x=-1..1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1,. 85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegend=[`scheme \+ of Dormand, Lockyer, McCorrigan and Prince`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[4]= 7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86/87`],\ntitle=` error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 742 610 610 {PLOTDATA 2 "6+-%'CURVESG6%7eo7$$!\"\"\" \"!$F*F*7$$!5nmmmm;p0k&*!#?$\",dkL^8%F/7$$!5LLLL$32W\\!#>7$$!5nmmmm\"4m(G$)F/$\"-;s\\n?8F<7$ $!5LLLL$3i.9!zF/$\"-\"ycMj@$F<7$$!5mmmm;/R=0vF/$\",[8SYs'!#=7$$!5++++] P8#\\4(F/$\"-JS63o8FL7$$!5mmmm;/siqmF/$\"-BD1m)o#FL7$$!5++++](y$pZiF/$ 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\"-pX\"4k!RF`p7$$\"55LLLe*['R3KFjt$\"-\\/+6\")QF`p7$$\"3LLLLezw5VF<$\" -)H;&fXQF`p7$$\"5]mmmmmJ+IiFjt$\"-HL`FgPF`p7$$\"3++++v$Q#\\\")F<$\"-;o 3$pk$F`p7$$\"5lmmm\"z\\1A-\"F/$\"-2(3>h\\$F`p7$$\"4LLLL$e\"*[H7F<$\"-/ Mcv>LF`p7$$\"5lmmm;HCjV9F/$\"-RQ\\b:JF`p7$$\"4+++++qvxl\"F<$\"-\"R;DV* GF`p7$$\"5++++]7JFn=F/$\"-X1dsmEF`p7$$\"4++++]_qn2#F<$\"-'HqF`p7$$\"5++ ++v=GB2FF/$\"-d1YPTBkE^\"F`p7$$\"5NLLL$3UD X8$F/$\"-Ng#[,K\"F`p7$$\"4nmmmmwanL$F<$\"-/![*oT6F`p7$$\"4+++++v+'oPF< $\",K$)Q27)F`p7$$\"4LLLLeR<*fTF<$\",+?.Av&F`p7$$\"4+++++&)Hxe%F<$\"-*Q &G*>z$Ffn7$$\"4nmmm\"H!o-*\\F<$\"--T-dlCFfn7$$\"4++++DTO5T&F<$\"-gv\"R )3:Ffn7$$\"4nmmmmT9C#eF<$\",@yy(f*)Ffn7$$\"4++++D1*3`iF<$\",,]AE'\\Ffn 7$$\"4LLLLL$*zym'F<$\"-qfK$zn#FL7$$\"4LLLL$3N1#4(F<$\"-/qsjY8FL7$$\"4n mmm\"HYt7vF<$\",[Qi\")Q'FL7$$\"4+++++q(G**yF<$\"-d\\o:7IF<7$$\"4nmmm;9 @BM)F<$\"-'H**ey;\"F<7$$\"4LLLLL`v&Q()F<$\",Z>*)RG%F<7$$\"4++++DOl5;*F <$\"-P1BT86F/7$$\"4++++v.Uac*F<$\"+cU\"4u%F/7$$\"\"\"F*$!,I!\\M8FF/-%& COLORG6&%$RGBG$\"\"&F)F+Fabl-%'LEGENDG6#%Rscheme~of~Dormand,~Lockyer,~ McCorrigan~and~PrinceG-F$6%7eoF'7$F-$\",rh?ls#F/7$F3$\"--kO*Q0\"F/7$F8 $\",c$)3qD$F<7$F>$\",f3Aqp)F<7$FC$\"-Cv.L<@F<7$FH$\",\\e+fU%FL7$FN$\", 2oYC+*FL7$FS$\"-h&3')*okt,(*=F`p7$Fds$\"-tSg:P?F`p7$Fi s$\"-n=aFn@F`p7$F^t$\"-Z\"fm*)G#F`p7$Fct$\"-3PM*HR#F`p7$Fht$\"-18)>/Z# F`p7$F^u$\"-ng,aGDF`p7$Fcu$\"-;\"pQ8b#F`p7$Fhu$\"-9T%>zc#F`p7$F]v$\"-% QgPQd#F`p7$Fbv$\"-ik*e\"yDF`p7$Fgv$\"-n3\\(3e#F`p7$F]w$\"-h[2)>e#F`p7$ Fbw$\"-tv@U\"e#F`p7$Fgw$\"-*G!\\7zDF`p7$F\\x$\"-Y$Q%4vDF`p7$Fax$\"-zM=#F`p7$Fdz$\"-()G8<\\?F`p7 $Fiz$\"-5^Jm.>F`p7$F^[l$\"-#RblRv\"F`p7$Fc[l$\"-fk 3^9F`p7$F]\\l$\"-fppw.8F`p7$Fb\\l$\"-Ns'Q`9\"F`p7$Fg\\l$\",hV%4\\**F`p 7$F\\]l$\",r4dGo)F`p7$Fa]l$\",*oR24vF`p7$Ff]l$\",%oW2T`F`p7$F[^l$\",3, \">$y$F`p7$F`^l$\"-X:1)Q\\#Ffn7$Fe^l$\"-7\\1X@;Ffn7$Fj^l$\",j/b>#**Ffn 7$F__l$\",U:39*eFfn7$Fd_l$\",*HN*GE$Ffn7$Fi_l$\"-H'ek1w\"FL7$F^`l$\",a mOW&))FL7$Fc`l$\",h(Gy,UFL7$Fh`l$\"-&yH*y#)>F<7$F]al$\",Cq\"z,xF<7$Fba 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\",qS\"f%3$FL7$FN$\",s6'GtiFL7$FS$\"-1*=*eK7FL7$FX$\",#4(R4I#Ffn7$Fhn$ \",l')3)pTFfn7$F]o$\",sR!=pnFfn7$Fbo$\"-h.7x?6Ffn7$Fgo$\"-v@vgy$>q.6F`p7$Fjr$\"-LOW)R@\"F`p7$F_s$\"-o&*zd@8F`p 7$Fds$\"-$p\"o?>9F`p7$Fis$\"-qWc&)4:F`p7$F^t$\"-iKHj%f\"F`p7$Fct$\"-4K U5n;F`p7$Fht$\"->#3W5s\"F`p7$F^u$\"-(Q@M:w\"F`p7$Fcu$\"-D[qTxk()z\"F`p7$Fbw$\"-%z1v$)z\"F`p7$Fgw$\"-;wYx'z\"F`p7 $F\\x$\"-\\sn'Rz\"F`p7$Fax$\"-1/j&**y\"F`p7$Ffx$\"--`QOyg\"F`p7$F _z$\"-@N^9@:F`p7$Fdz$\"-e]%yvU\"F`p7$Fiz$\"-7Y)3iK\"F`p7$F^[l$\"-OE/#> A\"F`p7$Fc[l$\"-Z(HC\\6\"F`p7$Fh[l$\"-h%=74,\"F`p7$F]\\l$\",HLAG3*F`p7 $Fb\\l$\",'=V3zzF`p7$Fg\\l$\",sGu5$pF`p7$F\\]l$\",ID,*[gF`p7$Fa]l$\",u eK6B&F`p7$Ff]l$\",\\n(o?PF`p7$F[^l$\",L(*)GNEF`p7$F`^l$\"-hK$4qt\"Ffn7 $Fe^l$\"-@MZ>H6Ffn7$Fj^l$\",dl3#3pFfn7$F__l$\",mNX35%Ffn7$Fd_l$\",Hkr/ F#Ffn7$Fi_l$\"-Q>joC7FL7$F^`l$\",X$fDchFL7$Fc`l$\",bwr-#HFL7$Fh`l$\"-% y-))yP\"F<7$F]al$\",831XN&F<7$Fbal$\",?5Vo(>F<7$Fgal$\",IfEYD&F/7$F\\b l$\"+P8EzMF/7$Fabl$!,I0'*>9\"F/-%'COLOURG6&Fhbl$\"*++++\"!\")F+F+-F\\c l6#%Escheme~with~c[4]=3/13~and~c[7]=86/87G-%&TITLEG6#%Uerror~curves~fo r~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fg\\p-% %FONTG6$%*HELVETICAGFa_n-%%VIEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, Mc Corrigan and Prince" "scheme with c[4]=6/25 and c[7]=63/64" "scheme wi th c[4]=9/38 and c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "s cheme with c[4]=3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 47 "Test 5 of 8 stage, order 6 Runge-Kutta methods " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=16/((16*x+1)*y )" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"#;F&*&,&*&F*F&%\"xGF&F&F&F&F&%\"yGF& F(" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\" !\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1) +1)" "6#/%\"yG-%%sqrtG6#,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F+%\"xGF+F+F+F+ F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y( x));\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));\ns := unapply(rhs(%),x): \nplot(s(x),x=0..0.5,0..2.6,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*( \"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+ " }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$ 7U7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3 x&)*3\"!#>$\"3?25A!pa&\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3 umm\"H2P\"Q?F6$\"3]_vibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3C LL$3x%3yTF6$\"31#\\\\E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$ $\"3)HL$eR-/PiF6$\"3.fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$ $\"3`KLe*)>VB$)F6$\"3K!\\`od36k\"F27$$\"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$$\"3Wmmm\"pW` (>Fco$\"3So#znsrC'>F27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=x pe=#Fco$\"3?*eB@.[<+#F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;z pSS\"R#Fco$\"3)4US+%ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#H Le*)>pxg#Fco$\"3ab\\mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"3 2LLe*Gst!GFco$\"3Q>IFco$\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f .9#F27$$\"3b***\\(=S2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,I A7m@F27$$\"3!*****\\(=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3 UyOr,.R*=#F27$$\"3#*****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$ \"3Qbx\"QY%\\6AF27$$\"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbR Fco$\"36qG(yA8CB#F27$$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp; %Fco$\"3IImU*yHDD#F27$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVF co$\"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$\"#5!\"\"F(F(-%+AXESLABE LSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F($\"\"&Fj[l;F( $\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 758 "S := (x,y) -> 16/((16*x+1)* y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Dormand, L ockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),` scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81 /82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20 :\nfor ct to 5 do\n Sn_RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Sn_RK6_||ct):\n for ii to n umpts do\n sm := sm+(Sn_RK6_||ct[ii,2]-s(Sn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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\"$+\"Q)pprint766\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+1dT g;!#?7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\" #j\"#kF/$\"+BbsO6F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+WCOn)*!#@7$*&F.F/6$/ F2#F<\"#I/F:#\"#\")\"##)F/$\"+*)3Zr!)FK7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#' )\"#()F/$\"+6k]*4(FKQ)pprint776\"" }}}{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 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".4995 is also g iven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 690 "S := (x,y) -> 16/( (16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([ [`slope field: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of D ormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7] =63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/3 0,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDig its := 20:\nfor ct to 5 do\n sn_RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y 0,hh,numsteps,true);\nend do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(sn_RK6_||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\"$+\"Q)pprint786\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG $\"*]=#o8!#>7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36# \"\"(#\"#j\"#kF/$\")!\\'f$*F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\")`vA\")F+7$ *&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\")O=TmF+7$*&F.F/6$/F2#\"\"$\"#8/F :#\"#')\"#()F/$\")CTSeF+Q)pprint796\"" }}}{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 each Runge-Kutt a method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Dormand, Lock yer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`sch eme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82 ),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\n for ct to 5 do\n sm := NCint((s(x)-'sn_RK6_||ct'(x))^2,x=0..0.5,adap tive=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$%Rscheme~of~Dormand,~L ockyer,~McCorrigan~and~PrinceG$\"+0(>)f;!#?7$*&%-scheme~with~G\"\"\"6$ /&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+k6JO6F+7$*&F.F/6$/F2 #\"\"*\"#QF9F/$\"+Vzuj)*!#@7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+5@ \\o!)FK7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+u&zo4(FKQ)pprint806 \"" }}}{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 0 515 "eva lf[20](plot(['sn_RK6_1'(x)-s(x),'sn_RK6_2'(x)-s(x),'sn_RK6_3'(x)-s(x), 'sn_RK6_4'(x)-s(x),\n'sn_RK6_5'(x)-s(x)],x=0..0.5,font=[HELVETICA,9], \ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COL OR(RGB,0,.35,.95),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrig an and Prince`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4 ]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme \+ with c[4]=3/13 and c[7]=86/87`],\ntitle=`error curves for 8 stage orde r 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 875 539 539 {PLOTDATA 2 "6+-%'CURVESG6%7^p7$$\"\"!F)F(7$$\"5SLLL3x1h6o!#B$\"$@\"!# >7$$\"5ommmTN@Ki8!#A$\"&?r\"F07$$\"5NLL3FpE!Hq\"F4$\"&?\\)F07$$\"5-++] 7.K[V?F4$\"'DWJF07$$\"5omm\"zptjSQ#F4$\"'70&*F07$$\"5NLLL$3FWYs#F4$\"( x`Z#F07$$\"5-++vo/[AlIF4$\"('*)\\dF07$$\"5omm;aQ`!eS$F4$\"):**>7F07$$ \"5NLLeRseQYPF4$\")O60CF07$$\"5-+++D1k'p3%F4$\")7^hWF07$$\"5pmmT5SpaFW F4$\")1xiyF07$$\"5OLL$eRZF\"oZF4$\"*-.kK\"F07$$\"50++D\"y+3(3^F4$\"*Yi *\\=F07$$\"5qmmmmT&)G\\aF4$\"*'4)>%=F07$$\"50++]P4'\\/8'F4$\"*k_k#=F07 $$F,F4$\"*]U=\"=F07$$\"5qmm;zWZ<'=F07$$\"5qmmm\"z%\\DO&*F4$\"*)f$ 30#F07$$\"5NLL3x\"[No()*F4$\"*$=ZUAF07$$\"5+++Dc,;u@5!#@$\"*y+,L#F07$$ \"5nm;z%\\l*zb5Fbr$\"*X)y@BF07$$\"5MLLLL3x&)*3\"Fbr$\"*X.OJ#F07$$\"5++ ]i!*GER37Fbr$\"*ctkG#F07$$\"5nmm\"z%\\v#pK\"Fbr$\"*J*\\qAF07$$\"5ML$3_ +ZiaW\"Fbr$\"*8-BL#F07$$\"5+++]i!R(*Rc\"Fbr$\"*GI=V#F07$$\"5nm;z>6B`#o \"Fbr$\"*PzpS#F07$$\"5MLL3xJs1,=Fbr$\"*m'f&Q#F07$$\"5nm\"Hd?pM.'=Fbr$ 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*F07$F`x$\")7p%R*F07$Fex$\")bjg\"*F07$Fjx$\")6G^*)F07$F`y$\").3)y)F07$ Fey$\")C.C')F07$Fjy$\")+iw%)F07$F_z$\")qO[$)F07$Fdz$\")b!>C)F07$Fiz$\" )t%f7)F07$F^[l$\")WeN!)F07$Fc[l$\")HOQzF07$Fh[l$\"),beyF07$F]\\l$\")O$ px(F07$Fb\\l$\")%pVq(F07$Fg\\l$\")yVLwF07$F\\]l$\")'o@d(F07$Fa]l$\")z# )4vF07$Ff]l$\"))*z[uF07$F[^l$\")_[)R(F07$F`^l$\")r#oM(F07$Fe^l$\")@9'H (F07$Fj^l$\")s(*[sF07$F__l$\")/W0sF07$Fd_l$\")$G$frF07$Fi_l$\")Vu>rF07 $F^`l$\")$p#zqF07$Fc`l$\")&yS/(F07$Fh`l$\")?62qF07$F]al$\")CntpF07$Fba l$\")0-SpF07$Fgal$\")gK3pF07$F\\bl$\")PMwoF07$Fabl$\").jYoF07$Ffbl$\") RGF07$FQ$\"(\"3UWF07$FV$\"(XtC*F07$Fen$\")!\\qz\"F0 7$Fjn$\")Z2(H$F07$F_o$\")*y7w&F07$Fdo$\")Gh5#)F07$Fio$\")()=v\")F07$Fc p$\")6VS!)F07$F\\q$\")8X')zF07$Faq$\")[3'=)F07$Ffq$\")T:d*)F07$F[r$\") \")=$y*F07$F`r$\"*=Vp,\"F07$Ffr$\"*O:L,\"F07$F[s$\"*xU(45F07$F`s$\")'* 3y**F07$Fes$\")qF&*)*F07$Fjs$\"*W&H65F07$F_t$\"*[u:0\"F07$Fdt$\"*$3\"3 /\"F07$Fit$\"*N\"=J5F07$F^u$\"*#QKG5F07$Fcu$\"*ZY)H5F07$Fhu$\"*C_9/\"F 07$F]v$\"*ngn/\"F07$Fbv$\"*9dN-\"F07$Fgv$\")R]x**F07$F\\w$\")'4<\\*F07 $Faw$\")9*p3*F07$Ffw$\")+2v()F07$F[x$\")I3-&)F07$F`x$\")y%>E)F07$Fex$ \")c4c!)F07$Fjx$\")\\(>(yF07$F`y$\")\"\\%GxF07$Fey$\")!zTe(F07$Fjy$\") +aauF07$F_z$\")1vTtF07$Fdz$\")b7[sF07$Fiz$\")*[h9(F07$F^[l$\")9omqF07$ Fc[l$\")D=\")pF07$Fh[l$\")J*4\"pF07$F]\\l$\")w@RoF07$Fb\\l$\")HSvnF07$ Fg\\l$\")S-8nF07$F\\]l$\")C9fmF07$Fa]l$\")&=Vg'F07$Ff]l$\")*[1b'F07$F[ ^l$\")6S1lF07$F`^l$\")<(4Y'F07$Fe^l$\")#)R;kF07$Fj^l$\")-#\\P'F07$F__l $\")IjOjF07$Fd_l$\")43'H'F07$Fi_l$\"))p7E'F07$F^`l$\")^nDiF07$Fc`l$\") ws%>'F07$Fh`l$\")$=A;'F07$F]al$\")3\"G8'F07$Fbal$\")l@.hF07$Fgal$\")OM vgF07$F\\bl$\")w@ZgF07$Fabl$\")q3@gF07$Ffbl$\")\"z_*fF07$F[cl$\")daqfF 07$F`cl$\")B`[fF07$Fecl$\")y4CfF07$Fjcl$\")`#H!fF07$F_dl$\")F-\")eF07$ Fddl$\")hngeF07$Fidl$\")%\\%ReF0-%'COLOURG6&Fael$\"*++++\"!\")F(F(-Fee l6#%Escheme~with~c[4]=3/13~and~c[7]=86/87G-%&TITLEG6#%Uerror~curves~fo r~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fiep-%% FONTG6$%*HELVETICAGFjen-%%VIEWG6$;F(Fidl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, Mc Corrigan and Prince" "scheme with c[4]=6/25 and c[7]=63/64" "scheme wi th c[4]=9/38 and c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "s cheme with c[4]=3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 46 "Test 6 of 8 stage, order 6 Runge-Kutta methods" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*s in(3*x))*exp(-y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xG F&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F&F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x+2/9*sin(3*x)-2/3*x*cos(3*x)-2/3 *cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\"xG\"\"\"*(\"\"#F*\"\"*!\"\"-%$si nG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$cosG6#*&F3F*F)F*F*F.*(F,F*F3F.-F 66#*&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,i c\},y(x));\nu := unapply(rhs(%),x):\nplot(u(x),x=0..5,font=[HELVETICA, 9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/*(\"\"#F/,&F,F/F/F/F/-%$sinG6#,$*&\" \"$F/F,F/F/F/F/F/-%$expG6#,$F)!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG-%#lnG6#,,F'\"\"\"*&#\"\"#\"\"*F,-%$sinG6#,$*&\"\"$F,F'F,F,F,F,*& #F/F6F,*&F'F,-%$cosGF3F,F,!\"\"*&#F/F6F,F:F,F<#\"\"&F6F," }}{PARA 13 " " 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F (7$$\"3GLLL3x&)*3\"!#=$\"3QWK+t!=.P\"F-7$$\"3umm\"H2P\"Q?F-$\"3pUCE&Gm M$HF-7$$\"3MLL$eRwX5$F-$\"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF-$\"3dz%)zau hMpF-7$$\"3emm\"z%4\\Y_F-$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF-$\"36YrjIBvP 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O9*z2=!e_*)F-7$$\"3ym\"zW7@^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3J eP:9JjA()F-7$$\"3t]i!RbX59%FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3o Lr_-o*=n)F-7$$\"3yLLL$)*pp;%FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32p cM,k23()F-7$$\"3#Q$3xc9[$>%FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3s e,X+?^M))F-7$$\"3)QL3-$H**>UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3 i#fiyx0s=*F-7$$\"3)RL$3xe,tUFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#) >Y<=$f\\9\"FI7$$\"3MMe9\"z-lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\" 3qZKS'GmoO\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI $\"3t\\(p9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_eaMDR_K " 0 "" {MPLTEXT 1 0 770 "U := (x,y) -> \+ (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with ` *(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme \+ with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: \+ errs := []:\nDigits := 25:\nfor ct to 5 do\n Un_RK6_||ct := RK6_||ct (U(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Un_R K6_||ct):\n for ii to numpts do\n sm := sm+(Un_RK6_||ct[ii,2]-u (Un_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nump ts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G *&,&\"\"\"F+*(\"\"#F+,&%\"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)pprint816\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"++Xa9:!#C7$*&%- scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\" +(o;qA\"F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+]%pbA\"F+7$*&F.F/6$/F2#F<\"#I /F:#\"#\")\"##)F/$\"+PHoA5F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\" +XLnS'*!#DQ)pprint826\"" }}}{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 = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "U := (x,y) - > (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: \+ y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial point: `,``(x0, y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthd s := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`schem e with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)] : errs := []:\nDigits := 25:\nfor ct to 5 do\n un_RK6_||ct := RK6_|| ct(U(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: uxx := e valf(u(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(un_RK6_||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)pprint856\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"*8a[s)!#C7$*& %-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$ \"*wrb!\\F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"*)=!>v$F+7$*&F.F/6$/F2#F<\"#I /F:#\"#\")\"##)F/$\"*H'>PMF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\" *tK@7$F+Q)pprint866\"" }}}{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$\"\"!\"\"& " }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows u sing the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to pe rform numerical integration by the 7 point Newton-Cotes method over 20 0 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "mt hds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme wi th `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`sch eme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87 )]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((u(x)-' un_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+/$oN^\" !#C7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j \"#kF/$\"+0.cE7F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+79DD7F+7$*&F.F/6$/F2#F <\"#I/F:#\"#\")\"##)F/$\"+\"= " 0 "" {MPLTEXT 1 0 529 "evalf[20](plot(['un_RK6_1'(x)-u(x),'un_RK6 _2'(x)-u(x),'un_RK6_3'(x)-u(x),'un_RK6_4'(x)-u(x),\n'un_RK6_5'(x)-u(x) ],x=0..5,-1e-15..7.3e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1) ,COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nl egend=[`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme wit h c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`sc heme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86 /87`],\ntitle=`error curves for 8 stage order 6 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 967 587 587 {PLOTDATA 2 "6+-%'CURVESG6 %7[u7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$\"&Wf$F-7$$\"5qmmmmT&)G\\aF-$\"&_ 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" }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin(3/2*x)*cos(3/2*x))+ 2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F+F*%\"xGF*F*F, **\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+F*F7F,F1F*F*F*F**&F7 F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)-x)" "6#-%$ expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/3);\nic : = y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x)));\nv := unapply(rhs(%),x): \nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,& \"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0F0F0,&F)F0#F0F8!\"\"F0F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+-%$exp G6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*&#\"\")F,F+*&-F56#,$*( F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F.6#,&*&#F3F,F+F4F+F9F 'F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CU RVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3W+7cSy5h&*!#=7$$\"3 ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F27$$ \"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$zx(F2 7$$\"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$$\"3Y++DJbw!Q*F2$\"3%Gs Cu$*)zK]F27$$\"3+N$ekGkX#**F2$\"3u>+\\,YW?`F27$$\"3%ommTIOo/\"!#<$\"3q ]2x8ZEqcF27$$\"3E+]7GTt%4\"Fgt$\"39b$=$pWlHgF27$$\"3YLL3_>jU6Fgt$\"3nw Ydkc=KkF27$$\"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$$\"3%o;HK R'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr=M:Fgt$\"3Erd.MaCV()F27$$\"3UL$e *[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$ \"3wmm;a/cq;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF 27$$\"3%ommmJFgt$\"3%RlX>.MR=&F2 7$$\"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$$\"3uLLe*Gst!GFgt$\"3U+pq))z 7kMF27$$\"30+++DRW9HFgt$\"37'z:1TS%*[$F27$$\"3:++DJE>>IFgt$\"3N!o4Joz] `$F27$$\"3F+]i!RU07$Fgt$\"3=,?;D0\"Qg$F27$$\"3+++v=S2LKFgt$\"3wRH=fZn5 PF27$$\"3Jmmm\"p)=MLFgt$\"3RsXuk([b#QF27$$\"3B++](=]@W$Fgt$\"3%4[=*QOM SRF27$$\"3mm\"H#oZ1\"\\$Fgt$\"3QK??D+QyRF27$$\"35L$e*[$z*RNFgt$\"3UAxt ;S)>+%F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$pk$Fgt$\"3eI Rs#H!Q\"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m;H2qcZPFgt$ \"3w)))[$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ymm;/OgbRFgt $\"3W-Tml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e% Fgt$\"35gRzc#\\LF27$$\"\"&F) $\"3Ii#4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG %%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%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 follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 765 "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 := [`scheme of Dormand, Lockyer, \+ McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme w ith `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`sc heme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 30:\nfor c t to 5 do\n Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,fal se);\n sm := 0: numpts := nops(Vn_RK6_||ct):\n for ii to numpts do \n sm := sm+(Vn_RK6_||ct[ii,2]-v(Vn_RK6_||ct[ii,1]))^2;\n end d o:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nli nalg[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)pp rint886\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorriga n~and~PrinceG$\"+u]\\Qj!#A7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\" \"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+C^;9WF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\" +33[aSF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+F/P(f$F+7$*&F.F/6$/F2 #\"\"$\"#8/F:#\"#')\"#()F/$\"+ " 0 "" {MPLTEXT 1 0 696 "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 := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4 ]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with \+ `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 30:\nfor ct to 5 do \n vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\nend \+ do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 5 do\n errs := [op (errs),abs(vn_RK6_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F ,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"! F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint906\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~Pri nceG$\"+Dxe)\\'!#B7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/ &F36#\"\"(#\"#j\"#kF/$\"+7-&oi%F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+(HE8I% F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+\\h$e#QF+7$*&F.F/6$/F2#\"\" $\"#8/F:#\"#')\"#()F/$\"+6,>#o$F+Q)pprint916\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "mthds := [`scheme of Dormand, Lockyer, McCorrig an and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c [4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme wit h `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 d o\n sm := NCint((v(x)-'vn_RK6_||ct'(x))^2,x=0..5,adaptive=false,nump oints=7,factor=100);\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$%Rscheme~of~Dormand,~Lockyer,~McCorri gan~and~PrinceG$\"+S$[bB'!#A7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%# \"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+YF#)RVF+7$*&F.F/6$/F2#\"\"*\"#QF9F/ $\"+w)\\_)RF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+hW9ONF+7$*&F.F/6 $/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+Y=#)zLF+Q)pprint926\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following erro r graphs are constructed using the numerical procedures for the soluti ons." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "evalf[20](plot(['vn _RK6_1'(x)-v(x),'vn_RK6_2'(x)-v(x),'vn_RK6_3'(x)-v(x),\n'vn_RK6_4'(x)- v(x),'vn_RK6_5'(x)-v(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB, .5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),r ed],\nlegend=[`scheme of Dormand, Lockyer, McCorrigan and Prince`,`sch eme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/ 64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and \+ c[7]=86/87`],\ntitle=`error curves for 8 stage order 6 Runge-Kutta met hods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 997 679 679 {PLOTDATA 2 "6+-%'C URVESG6%7_w7$$\"\"!F)F(7$$\"5qmm;aQ`!eS$!#A$\"$9$!#?7$$\"5SLLL3x1h6oF- $\"&.+%F07$$\"5,++Dc,;u@5!#@$\"'85oF07$$\"5ommmTN@Ki8F9$\"(W23&F07$$\" 5NLL3FpE!Hq\"F9$\")*R9T#F07$$\"5-++]7.K[V?F9$\")Mf$Q(F07$$\"5omm\"zptj SQ#F9$\")!*>fsF07$$\"5NLLL$3FWYs#F9$\")\"e<9(F07$$\"5-++vo/[AlIF9$\")p $))3(F07$$\"5omm;aQ`!eS$F9$\")uL(R(F07$$\"5NLLeRseQYPF9$\")f$*Q!*F07$$ \"5-+++D1k'p3%F9$\"*KdJC\"F07$$\"5pmmT5SpaFWF9$\"*%=JA7F07$$\"5OLL$eRZ F\"oZF9$\"*3yB?\"F07$$\"50++D\"y+3(3^F9$\"*7'*))=\"F07$$\"5qmmmmT&)G\\ aF9$\"*b-#37F07$$\"5SLL3_v!p)*y&F9$\"*fYAM\"F07$$\"50++]P4'\\/8'F9$\"* Na4b\"F07$$\"5qmm\"HK9I5Z'F9$\"*2]^_\"F07$$F3F9$\"*Hu.]\"F07$$\"50++v$ 4@\">_rF9$\"*.h:[\"F07$$\"5qmm;zW6B`#o\"F0$\"*9YKL\"F07$$\"5MLL3 xJs1,=F0$\"*Rv%*>\"F07$$\"5++]PM_@g>>F0$\"*.tk8\"F07$$\"5nmmm\"H2P\"Q? 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iOm[&F97$F`fm$\"*fIg([F97$Fefm$\"*+0pG%F97$Fjfm$\"*F'yhPF97$F_gm$\"*[( >.HF97$Fdgm$\"*\"Q[:BF97$Figm$\"*&3n^=F97$F^hm$\"*,8fc\"F97$Fchm$\"*6! [y8F97$Fhhm$\"*P?'p7F97$F]im$\"*N^0C\"F97$Fbim$\"*:#y?7F97$Fgim$\"*k\" ))47F97$F\\jm$\"*-6n?\"F97$Fajm$\"*NO@@\"F97$Ffjm$\"*\\(zE7F97$F[[n$\" *rW5D\"F97$F`[n$\"*df\\G\"F97$Fe[n$\"*\\\"***Q\"F97$Fj[n$\"*bh[a\"F97$ F_\\n$\"*a4Q.#F97$Fd\\n$\"*oTBO#F97$Fi\\n$\"*f<\\v#F97$F^]n$\"*b$3cKF9 7$Fc]n$\"*rly\"QF97$Fh]n$\"*(*fLN%F97$F]^n$\"*jCH\"\\F97$Fb^n$\"*`kJ[& F97$Fg^n$\"*Tjo)fF97$F\\_n$\"*CguM'F97$Fa_n$\"*I1;d'F97$Ff_n$\"*8Shg'F 97$F[`n$\"*Z%*)GmF97$F``n$\"*2y*RmF97$Fe`n$\"*5?Tj'F97$Fj`n$\"*8()*=mF 97$F_an$\"*qb6f'F97$Fdan$\"*s()Qa'F97$Fian$\"*>)z)['F97$F^bn$\"*E`c:'F 97$Fcbn$\"*Rlzl&F97$Fhbn$\"*P4%=]F97$F]cn$\"*Xd)RVF97$Fbcn$\"*l_rv$F97 $Fgcn$\"***RNC$F97$F\\dn$\"*L`\"yGF97$Fadn$\"*#)[*GEF97$Ffdn$\"*Lg[0#F 97$F[en$\"+)*\\.79F-7$F`en$\"*QQ)y()F-7$Feen$\"*$=f&*pF-7$Fjen$\"* " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 8 of 8 stage, order 6 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&*$%\"yGF.F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) =0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = rho(x)/2-2/rho(x);" "6#/%\"yG,&*&-%$rhoG6#%\"xG\"\"\"\"\"#!\"\"F+*&F, F+-F(6#F*F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where \+ " }{XPPEDIT 18 0 "rho(x) = (54*x^2-3*x^4+sqrt(64+9*x^8-324*x^6+2916*x^ 4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*&\"#a\"\"\"*$F'\"\"#F,F,*&\"\"$F,*$F '\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\"\"*F,*$F'\"\")F,F,*&\"$C$F,*$F'\"\" 'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,F0F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := di ff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x) );\nw := unapply(rhs(%),x):\nplot(w(x),x=0..4,0..3.7,numpoints=75,font =[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\" F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#! \"\",(*&\"\"$\"\"\")F'\"\"%F/F+*&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F' \"\")F/F/*&\"$C$F/)F'\"\"'F/F+*&\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+ F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVE SG6$7io7$$\"\"!F)F(7$$\"3()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$y ZbcF-$\"3ZF^'eEW*Q9F-7$$\"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$ \"3c?q**e5wz?F>7$$\"3$4\"3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=l 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0f$Fho7$$\"3)\\'['[I)[0MFho$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59 (yGT$>NFho7$$\"3['['[1m/8NFho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\" 3s<,a='3,U$Fho7$$\"37.Fq-Y#3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$ \"3?^'pWBqHH$Fho7$$\"32dnv'*p'o+i*\\/FFFho7$$\"\" %F)$\"3CxC=rRoRDFho-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\" xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 759 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.0 1: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x, y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of ste ps: `,numsteps]]);``;\nmthds := [`scheme of Dormand, Lockyer, McCorr igan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `* (c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme w ith `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Wn_RK6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Wn_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Wn_RK6_||ct[ii,2]-w(Wn_RK6_||ct[ii,1]))^2;\n end do:\n \+ errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[t ranspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\" F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~widt h:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint936\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 '7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+WSxJV!#D7$* &%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/ $\"+DYQ))RF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+\"G!3z_F+7$*&F.F/6$/F2#F<\" #I/F:#\"#\")\"##)F/$\"+L<5LYF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$ \"+!f7Bb%F+Q)pprint946\"" }}}{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 = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 690 "W := (x,y) - > x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nm atrix([[`slope field: `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`sche me of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/ 25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c [4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := [ ]:\nDigits := 30:\nfor ct to 5 do\n wn_RK6_||ct := RK6_||ct(W(x,y),x ,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 3.499: wxx := evalf(w(xx)) :\nfor ct to 5 do\n errs := [op(errs),abs(wn_RK6_||ct(xx)-wxx)];\nen d 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)pprint956\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~M cCorrigan~and~PrinceG$\"+1*pow)!#E7$*&%-scheme~with~G\"\"\"6$/&%\"cG6# \"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+SE#\\4\"!#D7$*&F.F/6$/F2#\"\" *\"#QF9F/$\"+LJ%ox\"FB7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+TY#*Q;F B7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+=j\"zm\"FBQ)pprint966\"" } }}{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 Ru nge-Kutta method is estimated as follows using the special procedure \+ " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration \+ by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "mthds := [`scheme of Dorman d, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/6 4),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7 ]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits : = 20:\nfor ct to 5 do\n sm := NCint((w(x)-'wn_RK6_||ct'(x))^2,x=0..4 ,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm /4)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dorman d,~Lockyer,~McCorrigan~and~PrinceG$\"+QZc0U!#D7$*&%-scheme~with~G\"\" \"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+3Q(R'RF+7$*&F.F/ 6$/F2#\"\"*\"#QF9F/$\"+u#ojD&F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$ \"+:e!>h%F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+CWbGXF+Q)pprint9 76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 " The following error graphs are constructed using the numerical procedu res for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 546 "e valf[20](plot([w(x)-'wn_RK6_1'(x),w(x)-'wn_RK6_2'(x),w(x)-'wn_RK6_3'(x ),\nw(x)-'wn_RK6_4'(x),w(x)-'wn_RK6_5'(x)],x=0..4,-2.19e-15..2.19e-15, font=[HELVETICA,9],numpoints=50,\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,. 1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegend=[`sche me of Dormand, Lockyer, McCorrigan and Prince`,`scheme with c[4]=6/25 \+ and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[ 4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86/87`],\ntitl e=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 906 673 673 {PLOTDATA 2 "6+-%'CURVESG6%7`q7$$\"\" !F)F(7$$\"59dG9dG9BN@!#@$\"&A[#!#A7$$\"5G9dG9dGYqUF-$\"&:@&F07$$\"5&G9 dG9dy!Q`F-$\"%l`F-7$$\"5Sr&G9dG%p0kF-$\"%2FF-7$$\"5+++++++JtuF-$!%)[%F -7$$\"5dG9dG9d#4a)F-$!&a'=F-7$$\"5BhIlK;z()R5!#?$!&yv(F-7$$\"5gz*[C7Ej cA\"FM$!'`$*>F-7$$\"5'z*[C71'[9T\"FM$!'FM7$$\"5'G9dG9<9O&GFM$!'XM\"FM7$$\"5yQpMn$ =%4pJFM$!&jk)FM7$$\"53/-^v(=aUF$FM$!&9C&FM7$$\"55bxQp%Q9*yLFM$!&%))=FM 7$$\"571`Ej\"euN[$FM$\"&>E\"FM7$$\"59dG9dyZB)e$FM$\"&e1%FM7$$\"5;3/-^v \\*Gp$FM$\"&SS'FM7$$\"5=fz*[C%*FM7$$\"5D71`Ejd`6TFM$\"&^\\*FM7$$\"5D71`Ej$p'*\\%FM$\"&*=qFM7$$\"5 D71`EjH!y)[FM$\"&ru$FM7$$\"5&=fz*[CGp*G&FM$\"%q^FM7$$\"5Vr&G9do#e\"p&F M$!%;;!#>7$$\"5bEj\"3/A+%**eFM$!%TCF_x7$$\"5l\"3/-^v>F_x7$$\"5#3/-^v\"4)RQ\"F_x$!%!z\"F _x7$$\"55bxQpa#**4Z\"F_x$!%g;F_x7$$\"53/-^vF_-[:F_x$!%i:F_x7$$\"5hIlK; 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vWFM$\"'BA;FM7$F\\w$\"'9P;FM7$Fdfn$\"'*ye\"FM7$Faw$\"'L@:FM7$Ffw$\"'J_ 8FM7$F[x$\"&b=\"F_x7$Ffx$\"&k.\"F_x7$F`y$\"%&>*F_x7$Fjy$\"%\\#)F_x7$Fd z$\"%tuF_x7$F^[l$\"%WiF_x7$Fc[l$\"%daF_x7$Fh[l$\"%mZF_x7$F]\\l$\"%EUF_ x7$Fb\\l$\"%1QF_x7$Fg\\l$\"%*[$F_x7$F\\]l$\"%vJF_x7$Fa]l$\"%]HF_x7$Ff] l$\"%FFF_x7$F[^l$\"%cDF_x7$F`^l$\"%&R#F_x7$Fe^l$\"%jAF_x7$Fj^l$\"%T@F_ x7$F__l$\"%V?F_x7$Fd_l$\"%]>F_x7$Fi_l$\"%m=F_x7$F^`l$\"%+=F_x7$Fc`l$\" %QF_x7$F\\gl$\"%,?F_x7$F`gl$\"%E?F_x7$$\"5k\"3/-^q@_&QF_x$ \"%1?F_x7$Fegl$\"%t>F_x7$$\"5%=fz*[%>1#**QF_x$\"%@=F_x7$Fjgl$\"%L:F_x7 $F_hl$\"%t5F_x7$Fdhl$\"$0$F_x7$Fihl$!$\\#F_x7$F^il$!%=5F_x7$Fhil$!%5?F _x7$Fbjl$!%RLF_x-Fgjl6&FijlF($\"#NFd]n$\"#&*Fd]n-F`[m6#%Escheme~with~c [4]=7/30~and~c[7]=81/82G-F$6%7\\qF'7$F+$\"&gT#F07$F2$\"&^[)F07$F<$\"&[ 7#F-7$FF$\"&#QUF-7$FQ$\"'xb5F-7$Fen$\"&M_\"FM7$F_o$\"&q,\"FM7$Fio$!%AK FM7$F^co$!%>KFM7$F^p$!%8OFM7$Ffco$!%M_FM7$Fcp$!%E_FM7$Fhp$!%e^FM7$F]q$ !%&f&FM7$Fbq$!%tQFM7$Fgq$\"$=$FM7$F\\r$\"%LrFM7$Far$\"&$f;FM7$Ffr$\"&x %GFM7$F[s$\"&nC%FM7$F`s$\"&Oz&FM7$Fes$\"&TV(FM7$Fjs$\"&h5*FM7$F_t$\"'w u5FM7$Fdt$\"'()H7FM7$Fit$\"'#)\\8FM7$F]v$\"'mF9FM7$Fgv$\"'@'[\"FM7$Fcd n$\"'SH:FM7$Fgen$\"'6c:FM7$Fdgo$\"'/Z:FM7$Figo$\"'$*Q:FM7$F^ho$\"'GQ:F M7$F\\fn$\"':o:FM7$Ffho$\"')*e:FM7$F[io$\"'<]:FM7$F`io$\"'-Y:FM7$F\\w$ \"'&oc\"FM7$Fdfn$\"'mJ:FM7$Faw$\"'hy9FM7$Ffw$\"'gL8FM7$F[x$\"&<=\"F_x7 $Ffx$\"&F_x7$Fi_l$\"%1>F_x7$F^`l$\"%R=F_x7$Fc`l$\"%wF_x7$F\\gl$\"%o>F_ x7$F`gl$\"%\\>F_x7$Fcbp$\"%&*=F_x7$Fegl$\"%8=F_x7$F[cp$\"%w:F_x7$Fjgl$ \"%h6F_x7$F_hl$\"$T&F_x7$Fdhl$!$o%F_x7$Fihl$!%(=\"F_x7$F^il$!%j@F_x7$F hil$!%;MF_x7$Fbjl$!%z]F_x-%'COLOURG6&Fijl$\"*++++\"!\")F(F(-F`[m6#%Esc heme~with~c[4]=3/13~and~c[7]=86/87G-%&TITLEG6#%Uerror~curves~for~8~sta ge~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fjgq-%%FONTG6$ %*HELVETICAGF\\ao-%%VIEWG6$;F(Fbjl;$!$>#!#<$\"$>#Fghq" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, \+ McCorrigan and Prince" "scheme with c[4]=6/25 and c[7]=63/64" "scheme \+ with c[4]=9/38 and c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "scheme with c[4]=3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 48 "Test 9 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x ))*y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-%$cosG6#*&\"\"#F&%\"xGF& F&F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2) ;" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" "6#/%\"yG*&\"\"\"F&-%%sqr tG6#,(-%$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$-%\"yG6#%\"xGF,,$*&,& \"\"\"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$$\"\"!F)$\"3:&4tB c8UT\"!#<7$$\"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$$\"3/++v VA)GA\"!#=$\"3V)o6<$fq15F,7$$\"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$$\"3 z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\P*zhdVFJ$\"3P$G(zQ8#4%oFJ7$$ \"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY%G?7\"*G' FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"33++]7[X$ocbFJ7$$\"3))***\\PpnsM*FJ$\"3!\\; $Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aFJ7$$\"3-+++!R5'f5F,$\"3q`:6Q hHm`FJ7$$\"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$$\"3z*****\\@80+#F,$\"3!zBI i>A%o^FJ7$$\"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$$\"3!)**\\P/-a[AF,$\"3 gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>;?IA\\FJ7$$\"3')****\\i@OtBF,$ \"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8tZFJ7$$\"3>+++!*>=+DF, $\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1YFJ7$$\"3=+]P%>5pi#F,$ \"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$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)$\"3ntdq; jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%F ONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(Fc\\l;F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code con structs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on \+ each of the methods and gives the " }{TEXT 260 22 "root mean square er ror" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 772 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numstep s := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4 ]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with \+ `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do \n Mn_RK6_||ct := RK6_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,false );\n sm := 0: numpts := nops(Mn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Mn_RK6_||ct[ii,2]-m(Mn_RK6_||ct[ii,1]))^2;\n end do: \n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlina lg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,% \"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F2 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint986\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$ \"+DRvIN!#B7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\" \"(#\"#j\"#kF/$\"+gY7B7F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+tX4lFF+7$*&F.F /6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+J5h2TF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\" #')\"#()F/$\"+@u1?]F+Q)pprint996\"" }}}{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 = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 703 "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]]);``;\n mthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme \+ with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`s cheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/ 87)]: errs := []:\nDigits := 25:\nfor ct to 5 do\n mn_RK6_||ct := RK 6_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 2.99 9: mxx := evalf(m(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(mn_R K6_||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,e valf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slo pe~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\" $F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F, !\"#7$%1no.~of~steps:~~~G\"$+$Q*pprint1026\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+W(G=&e!#C7$*&% -scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$ \"+v5(p\"HF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+AgVpeF+7$*&F.F/6$/F2#F<\"#I /F:#\"#\")\"##)F/$\"+%yBUQ)F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$ \"+(Rzu+\"!#BQ*pprint1036\"" }}}{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 foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 150 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "mthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`sc heme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/6 4),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7 ]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint( (m(x)-'mn_RK6_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7,factor=150 );\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 10:\nlinalg[ transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+ ))z^xM!#B7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\" (#\"#j\"#kF/$\"+^'p8@\"F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+R<1LFF+7$*&F.F /6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+FfUdSF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\" #')\"#()F/$\"+P5Qd\\F+Q*pprint1046\"" }}}{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(['mn_RK6_1'(x)-m(x) ,'mn_RK6_2'(x)-m(x),'mn_RK6_3'(x)-m(x),\n'mn_RK6_4'(x)-m(x),'mn_RK6_5' (x)-m(x)],x=0..0.5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR 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fl$!(*\\c`Fj]l7$F_gl$!((*GA&Fj]l7$Fdgl$!(]I4&Fj]l7$Figl$!(I1(\\Fj]l7$F ^hl$!(WL'[Fj]l7$Fchl$!(4iu%Fj]l7$Fhhl$!(=jk%Fj]l7$F]il$!(lYa%Fj]l7$Fbi l$!(J " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 10 of 8 stage, order 6 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 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z**[W(=JcF87$$\"3y*\\7G))Rb\"[F,$\"3OXJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$ \"3!**fZR-9n(oF87$$\"3)3](=np3m[F,$\"3#eM]7.#Q?&=\\F,$\"3'4(*[g6ply*F87$$\"3s+voa-oX\\F, $\"3?(Q=b([U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3.ug')y o>49F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AX ESLABELSG6$%\"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 constr ucts 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 0 796 "P := (x,y) -> -(2*sin( 5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := \+ sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x 0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmt hds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme wi th `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`sch eme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87 )]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Pn_RK6_||ct := RK6_ ||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts : = nops(Pn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Pn_RK6_| |ct[ii,2]-evalf(p(Pn_RK6_||ct[ii,1])))^2;\n end do:\n errs := [op( errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,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$%0ini tial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F-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$%Rscheme~of~Do rmand,~Lockyer,~McCorrigan~and~PrinceG$\"+31#3$f!#A7$*&%-scheme~with~G \"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+R<*=/&F+7$*& F.F/6$/F2#\"\"*\"#QF9F/$\"+x)GtK&F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##) F/$\"+)4Sc_%F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+JH2cXF+Q(ppri nt16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999; " "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 720 "P := (x,y) -> -(2*sin(5*x)+ 3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt( 5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0,y0) ],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds : = [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `* (c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme w ith `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: e rrs := []:\nDigits := 30:\nfor ct to 5 do\n pn_RK6_||ct := RK6_||ct( P(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: pxx \+ := evalf(p(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(pn_RK6_||ct (xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(er rs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fiel d:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$ cosG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~ G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~step s:~~~G\"$+&Q*pprint1076\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~L ockyer,~McCorrigan~and~PrinceG$\"+Bp$zn&!#B7$*&%-scheme~with~G\"\"\"6$ /&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+rwbV " 0 "" {MPLTEXT 1 0 440 "mthds := [`scheme of Dorman d, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/6 4),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7 ]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits : = 20:\nfor ct to 5 do\n sm := NCint((p(x)-'pn_RK6_||ct'(x))^2,x=0..5 ,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm /5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dorman d,~Lockyer,~McCorrigan~and~PrinceG$\"+\\kqVf!#A7$*&%-scheme~with~G\"\" \"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+Qa8^]F+7$*&F.F/6 $/F2#\"\"*\"#QF9F/$\"+`jfM`F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+ ^9ZIXF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+pe+fXF+Q*pprint1096 \"" }}}{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 0 516 "eva lf[20](plot([p(x)-'pn_RK6_1'(x),p(x)-'pn_RK6_2'(x),p(x)-'pn_RK6_3'(x), \np(x)-'pn_RK6_4'(x),p(x)-'pn_RK6_5'(x)],x=0..2.2,font=[HELVETICA,9], \ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COL OR(RGB,0,.35,.95),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrig an and Prince`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4 ]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme \+ with c[4]=3/13 and c[7]=86/87`],\ntitle=`error curves for 8 stage orde r 6 Runge-Kutta methods`));\n" }}{PARA 13 "" 1 "" {GLPLOT2D 936 546 546 {PLOTDATA 2 "6+-%'CURVESG6%7\\s7$$\"\"!F)F(7$$\"5mmmmm;RP&z%!#@$\" (z+\"R!#?7$$\"5LLLL$37.y'*)F-$\"([j1)F07$$\"5+++]7jpRJ6F0$\"(N7\\*F07$ $\"5nmmm;9O,m8F0$\")^IP5F07$$\"5nmmmTN&*4%[\"F0$\")@7\"3\"F07$$\"5nmmm mca=-;F0$\")QUP6F07$$\"5nmmm\"zPr-s\"F0$\")F*39\"F07$$\"5nmmm;*Hd$Q=F0 $\")#G$Q6F07$$\"5+++]7emSt?F0$\")vu@6F07$$\"5LLLL3U-\"F07$$\"5++++vGle&>$F0$\")C#G-\"F07$$\"5LLLL3-$[*GMF0$\" 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s.#F17$F\\bs$\"(\"*>G#F17$Fhem$\"(#=zBF17$Fdbs$\"(rC`#F17$F\\do$\"(P_p #F17$F\\cs$\"(\\1v#F17$Fado$\"(.l'GF17$Ffdo$\"(XV;$F17$F[eo$\"(>qA$F17 $F`eo$\"(>OE$F17$Feeo$\"(o9K$F17$Fjeo$\"($**[MF17$F]fm$\"(?Gw$F1-%'COL OURG6&Fefm$\"*++++\"!\")FgfmFgfm-F[gm6#%Escheme~with~c[4]=3/13~and~c[7 ]=86/87G-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~meth odsG-%+AXESLABELSG6$Q\"x6\"Q!Fjbu-%%FONTG6$%*HELVETICAGFchq-%%VIEWG6$; F(F]fm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, McCorrigan and Prince" "scheme with c [4]=6/25 and c[7]=63/64" "scheme with c[4]=9/38 and c[7]=63/64" "schem e with c[4]=7/30 and c[7]=81/82" "scheme with c[4]=3/13 and c[7]=86/87 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 11 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that \+ appears in an article by F. G. Lether: Mathematics of Computation, Vo l. 20, no. 95, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x- 1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\" \"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6# *&\"\"\"F(,&%\"xGF(F(!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6#/-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$ *&-%$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'!\"\" \"\"\"-%$sinG6#*&F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#= 7$$\"3#>=\"*)>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$z TlU<)F,7$$\"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$$\"3kgswR?Pw:F,$\"3 T8>lD8j?zF,7$$\"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)QgTz pp+(F,7$$\"3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS% zr'F,7$$\"3\"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9 @F_jF,7$$\"3`G+(=Gs!HXF,$\"3S.Rv)o&))[hF,7$$\"3&\\%Rcec>> `F,7$$\"39pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"3QD(p)Qdl>bF,$\"34_#)R=svWX F,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&p Z&F07$$\"3.\"Q#ekL\"p!pF,$!3OST&zF\\gd%F07$$\"3%yB5rz/v4(F,$!3cn!**p^J 2Z\"F,7$$\"33-p_gxs'H(F,$!3A9e\"Q#e(4b#F,7$$\"3%324>i/:\\(F,$!3S!=W0GJ d`$F,7$$\"39FEN$GhMf(F,$!3U'\\`Wd^A(RF,7$$\"3c%='zWzT&p(F,$!3f\\'**z(4 Q8VF,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,MUjZ4H%)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&HR F,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<(fF0 7$$\"376:PheCZ*)F,$!3ocHHl.]EIF07$$\"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$$\"3 EJK:t[EZ!*F,$!3I(>(y]J<_NF,7$$\"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$$\"3E20FIC$y A*F,$\"3MLbSaq^*[\"F,7$$\"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$$\"3oE3CP5 fw#*F,$\"3Ma)HpV]]I!H!G$* F,$\"3a%4t07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[ PN*F,$\"33n1kl[ ]%*F,$!3m(=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY *F,$!3U$3!Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RG BG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\" *-%%VIEWG6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete solution based on each of the methods and gives the " } {TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 812 "Q := (x,y) -> exp(-x)/( x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]); ``;\nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`sc heme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/6 4),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7 ]=86/87)]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Qn_RK6_||ct \+ := RK6_||ct(Q(x,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,false); \n sm := 0: numpts := nops(Qn_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Qn_RK6_||ct[ii,2]-q(Qn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1 F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!- %$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$% 1no.~of~steps:~~~GFFQ*pprint1106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dor mand,~Lockyer,~McCorrigan~and~PrinceG$\"+'\\j*pi!#?7$*&%-scheme~with~G \"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+x5pQfF+7$*&F .F/6$/F2#\"\"*\"#QF9F/$\"+w(Hvy&F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F /$\"+UglgcF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+l \+ " 0 "" {MPLTEXT 1 0 736 "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):\nmat rix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0)],[`step w idth: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25 ,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4 ]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []: \nDigits := 30:\nfor ct to 5 do\n qn_RK6_||ct := RK6_||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 5 do\n errs := [op(errs),abs(qn_RK6_||ct(x x)-qxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field: ~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1 F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G ,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ*pprint1 126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorriga n~and~PrinceG$\"+w-u!R(!#>7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\" \"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+r[V))pF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$ \"+\\>N4oF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+(>:Pl'F+7$*&F.F/6$ /F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+!ok.['F+Q*pprint1136\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the interval " } {XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"F&*&F&F&*&\"\"'F&% #PiGF&!\"\"F+" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "mthds := [`scheme of Dormand, Lockyer, McCorrigan an d Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9 /38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*( c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n \+ sm := NCint((q(x)-'qn_RK6_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false, numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~L ockyer,~McCorrigan~and~PrinceG$\"+pTJf`!#?7$*&%-scheme~with~G\"\"\"6$/ &%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+Z/%*z]F+7$*&F.F/6$/F2 #\"\"*\"#QF9F/$\"+,_ W[F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+(Q#f>ZF+Q*pprint1146\" " }}}{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 0 533 "evalf [25](plot([q(x)-'qn_RK6_1'(x),q(x)-'qn_RK6_2'(x),q(x)-'qn_RK6_3'(x),\n q(x)-'qn_RK6_4'(x),q(x)-'qn_RK6_5'(x)],x=0..0.7,-0.9e-19..1.8e-18,font =[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RG B,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegend=[`scheme of Dormand, Lo ckyer, McCorrigan and Prince`,`scheme with c[4]=6/25 and c[7]=63/64`,` scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]= 81/82`,`scheme with c[4]=3/13 and c[7]=86/87`],\ntitle=`error curves f or 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 685 540 540 {PLOTDATA 2 "6+-%'CURVESG6%7gn7$$\"\"!F)F(7$$\": mmmmmmm;z+e_\"!#E$!$D(!#D7$$\":LLLLLL$3->R`GF-$!%69F07$$\":mmmmmmmT&pS YVF-$!%RAF07$$\":lmmmmmm\"z'=$\\eF-$!%bJF07$$\":KLLLLL$3Ft3XtF-$!%qTF0 7$$\":lmmmmm;aLc=t)F-$!%6`F07$$\":++++++v=`xn,\"F0$!%TkF07$$\":mmmmmmT &y/Gl6F0$!%cyF07$$\":++++++vV<2LJ\"F0$!%[%*F07$$\":LLLLLLLe#3dl9F0$!&I 7\"F07$$\":mmmmmm;Ht%o*f\"F0$!&oH\"F07$$\":+++++++vAl1v\"F0$!&&=:F07$$ \":+++++++DmlA!>F0$!&$oZ$F07$$\":LLLLLLL$oD[lFF0$!&l'RF0 7$$\":***********\\(oibk\"HF0$!&M_%F07$$\":***********\\i!o<-1$F0$!&72 &F07$$\":LLLLLL$3-$=-@$F0$!&Xw&F07$$\":LLLLLL3xplzM$F0$!&0X'F07$$\":mm mmmm;H([a'\\$F0$!&/G(F07$$\":mmmmmmT&yo(3l$F0$!&!=#)F07$$\":********** *\\7VLA&y$F0$!&S6*F07$$\":mmmmmm;a?@.$RF0$!'a65F07$$\":************** \\\\@-3%F0$!';>6F07$$\":************\\PopoA%F0$!',D7F07$$\":********** 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2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dorma nd, Lockyer, McCorrigan and Prince" "scheme with c[4]=6/25 and c[7]=63 /64" "scheme with c[4]=9/38 and c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "scheme with c[4]=3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 12 of 8 stage, order 6 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx =exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$, &F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "5*y*sin^7*7*x;" "6# *,\"\"&\"\"\"%\"yGF%%$sinG\"\"(F(F%%\"xGF%" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(16/49+5/3136*cos*49*x-cos*35*x/64+5/64*c os*21*x-25/64*cos*7*x);" 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F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q*pprint1176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~Prince G$\"+Dor0l!#?7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36# \"\"(#\"#j\"#kF/$\"+f_N'G)F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+zREp#)F+7$* &F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+;v-QtF+7$*&F.F/6$/F2#\"\"$\"#8/F :#\"#')\"#()F/$\"+.sS'\\'F+Q*pprint1186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5 ];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is est imated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "mthds := [`scheme of Dormand, Lockyer, McCorrigan an d Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9 /38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*( c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n \+ sm := NCint((r(x)-'rn_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints =7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorriga n~and~PrinceG$\"+$*Gs7I!#?7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\" \"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+pkxYQF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\" +d/x[QF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+fA87MF+7$*&F.F/6$/F2# \"\"$\"#8/F:#\"#')\"#()F/$\"+ip=AIF+Q*pprint1196\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error gra phs are constructed using the numerical procedures for the solutions. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "evalf[20](plot([r(x)-'r n_RK6_1'(x),r(x)-'rn_RK6_2'(x),r(x)-'rn_RK6_3'(x),\nr(x)-'rn_RK6_4'(x) ,r(x)-'rn_RK6_5'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0 ,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red], \nlegend=[`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme \+ with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`, `scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7] =86/87`],\ntitle=`error curves for 8 stage order 6 Runge-Kutta methods `));" }}{PARA 13 "" 1 "" {GLPLOT2D 1044 595 595 {PLOTDATA 2 "6+-%'CURV ESG6%7icl7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$!'**[r!#>7$$\"5MLLLL3x&)*3 \"!#?$!(Xf8\"F07$$\"5+++]i!R(*Rc\"F4$\"))3;-\"F07$$\"5nmmm\"H2P\"Q?F4$ \")s#e#=F07$$\"5++]7G))>Wr@F4$\"(SJ_\"F07$$\"5MLLek.pu/BF4$!)$)**=CF07 $$\"5n;zp[KJ2QBF4$!)K0fCF07$$\"5++D\"G8O*RrBF4$!)PNtCF07$$\"5M$3Fp,fDZ S#F4$!)n!))4#F07$$\"5nm;/,>=0QCF4$!)BxI@F07$$\"5+]i:&y/y8Z#F4$!)g%*>?F 07$$\"5ML3FpwUq/DF4$!(53`$F07$$\"5n;aQ`00.QDF4$!(jTb\"F07$$\"5ML$eR(\\;m/FF4$\")UR9XF07$$\"5nmmT5ll'z$GF4$\")bmge F07$$\"5MLekyA!>Y!HF4$\")tq\"y&F07$$\"5++](o/[r7(HF4$\")4GddF07$$\"5nm T5:QR#z.$F4$\")6tLZF07$$\"5MLLL$eRwX5$F4$\")1N\\JF07$$\"5MLLLe*[`HP$F4 $\"()Q:ZF07$$\"5MLLLL$eI8k$F4$!(M@(HF07$$\"5MLLL3xwq4RF4$\"(1do%F07$$ \"5NLLL$3x%3yTF4$\"(3z*)*F07$$\"5-+](oHaN;J%F4$\").n)3\"F07$$\"5ommT5: j=XWF4$\"(no))*F07$$\"5NL$eRs3P(yXF4$\"(#GU\")F07$$\"5-++]PfyG7ZF4$\"( PD;'F07$$\"5om;/^J'Qe%[F4$\"(H>['F07$$\"5NLLek.%*Qz\\F4$\"(8FJ)F07$$\" 5++]7yv,%H6&F4$\")![QI\"F07$$\"5ommm\"z%4\\Y_F4$\")a\"Ha\"F07$$\"5+++] P4'4.P&F4$\")j$Qn\"F07$$\"5NLLL$3FGT\\&F4$\")6C::F07$$\"5qmm;HKp%zh&F4 $\")eR'4\"F07$$\"5++++v$flLjF4$\")o*[!fF07$$\"5qmmmm\"HZ_O'F4$\" )$3X$eF07$$\"5+++voa!*H(R'F4$\")9*o4'F07$$\"5NLL$3x\"3NHkF4$\")2AVB$)F4$\")#3r*oF07$$\"5qm m;H#o)fb%)F4$\")\"*Q,oF07$$\"50++++v`w(e)F4$\");xTmF07$$\"5SLL$3x1K*>( )F4$\")&4D_'F07$$\"5qmmmTg()4_))F4$\")M&*HlF07$$\"5+++]7`aE%)*)F4$\"*v )y1mF47$$\"5NLLL$e9Kk6*F4$\")nPRnF07$$\"5qmm;aQ))f[#*F4$\")3#Hx'F07$$ \"5.+++DJbw!Q*F4$\")BENnF07$$\"5NL$3_+A:n^*F4$\")g!\\a'F07$$\"5qmmT&)3 \\m_'*F4$\")O%yT'F07$$\"5++]il(f9')y*F4$\")u_?jF07$$\"5NLL$ekGkX#**F4$ \")0\"fQ'F07$$\"5++]iSmjk>5F0$\")@R)*oF07$$\"5nmmm;/j$o/\"F0$\")'=&zwF 07$$\"5MLLe9;_yq5F0$\")aTt'*F07$$\"5+++]7GTt%4\"F0$\"*1H31\"F07$$\"5ML $e9Te3n5\"F0$\")h&o)**F07$$\"5nmmT5SIo=6F0$\")9\"fN)F07$$\"5MLe*)4o-nC 6F0$\")VQlvF07$$\"5++]P4'\\d18\"F0$\")]`PrF07$$\"5nmT&)3CZkO6F0$\")(3? 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$\"*,#*QT#F07$Fbfp$\"*f#y;DF07$Fgfp$\"*NDql#F07$F\\gp$\"*qG&F07$Fa \\q$\"*\\-zC&F07$Ff\\q$\"*`>)H_F07$F[]q$\"*?jmA&F07$F`]q$\"*wcqF&F07$F e]q$\"*sbwK&F07$Fj]q$\"*U\\tFH&F07$Fc_q$\"*&GZ:`F07$Fh_q$\"*l&R\\`F07$ F]`q$\"*c$3q`F07$Fg`q$\"*\"HS\"Q&F07$Faaq$\"*HeUO&F07$Ffaq$\"*:t#=`F07 $F[bq$\"**=4V_F07$Fcdq$\"*iW$H]F07$F[gq$\"*)\\g!*[F07$F`gq$\"*-!47XF07 $Fegq$\"*9;D*QF07$Ffcy$\"*HnTU$F07$Fjgq$\"*^Jl>$F07$F^dy$\"*'Rn=JF07$F cdy$\"*L#)f0$F07$Fhdy$\"*[oT.$F07$F]ey$\"*QW$=IF07$Fbey$\"*$3w8IF07$F_ hq$\"*Jzs+$F07$Fihq$\"*@m]-$F07$Fciq$\"*Q,$F07$F`\\r$\"*21p,$F07$Fe \\r$\"*]uz+$F07$Fj\\r$\"*zHu*HF07$F_]r$\"*M;d)HF07$Fd]r$\"*\"*o8)HF07$ Fi]r$\"*4Sj*HF07$F^^r$\"*q!Q!4$F07$Fc^r$\"*&z2;KF07$Fh^r$\"*lm`[$F07$F ]_r$\"*4(zyQF07$Fi_v$\"*+%f5TF07$Fb_r$\"*yZDM%F07$Fa`v$\"*/KJl%F07$Fg_ r$\"*Zzy.&F07$Fa`r$\"*8.bd&F07$F[ar$\"***p1'fF07$Fear$\"*,jcD'F07$Fibr $\"*A(e3kF07$Feav$\"*Q34W'F07$Fjav$\"*TywY'F07$F_bv$\"*'Q%fW'F07$Fdbv$ \"*w!eekF07$Fibv$\"*^#ofkF07$F^cr$\"*F;lX'F07$Facv$\"*q))=U'F07$Fccr$ \"*(o+yjF07$Ficv$\"*i%y!Q'F07$Fhcr$\"*z)p$R'F07$Fadv$\"*1/>U'F07$F]dr$ \"*#[vhkF07$F_gr$\"*PD,^'F0-%'COLOURG6&Ffgr$\"*++++\"!\")F(F(-F\\hr6#% Escheme~with~c[4]=3/13~and~c[7]=86/87G-%&TITLEG6#%Uerror~curves~for~8~ stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fcbal-%%FON TG6$%*HELVETICAGFe\\z-%%VIEWG6$;F(F_gr%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Dormand, Lockyer, McCo rrigan and Prince" "scheme with c[4]=6/25 and c[7]=63/64" "scheme with c[4]=9/38 and c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "sch eme with c[4]=3/13 and c[7]=86/87" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 13 of 8 stage, order 6 Runge-Kutta 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!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"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#,&*&%$sinG\"\"\"%\"x GF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x" "6# *&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general s olution of the differential equation " }{XPPEDIT 18 0 "dy/dx = cos* x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yG F&F&" }{TEXT -1 64 " contains an exponential term, but with the initi al condition " }{XPPEDIT 18 0 "y(0) = -2/5" "6#/-%\"yG6#\"\"!,$*&\"\" #\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this term disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de : = diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\" #F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"# \"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-*&-%$expG6#,$*&F+F-F' F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a numerical solution from the co rrect solution tends to become rapidly magnified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff( y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5;\ndsolve(\{de,ic\},y(x));\ne : = unapply(rhs(%),x):\nplot(e(x),x=0..8,font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6# %\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF &F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURV ESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3ELLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLLLLAKn\\F,$!3C&4%=OwYjDF,7$$\"3 =LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm^&Q%R)F,$!39J7$$\"3))*****\\YJ?;\"!#<$\"3m!=?Y3*>`CFK 7$$\"3?LLL=\"\\g**FK7$$\"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$$\"3&)******f-w+?FO$\"375@EVOJ&[$F,7$$\"3%*********y,u@FO 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mm;\\%H&\\dFO$!3y[ey96=hWF,7$$\"3eLLe%p5Bz&FO$!3=)zg%Q%y/Z%F,7$$\"3')* *****R>4NeFO$!3waa0%)\\frWF,7$$\"3HLL$ed*f:fFO$!3]_J$4k<:X%F,7$$\"3#em m;@2h*fFO$!3V5vHeMg-WF,7$$\"37LLL))3E!3'FO$!3=l`a'y%*4K%F,7$$\"3]***** \\c9W;'FO$!3>=$e-d.)3UF,7$$\"3Lmmmmd'*GjFO$!3Gy*y<4!G/RF,7$$\"3j***** \\iN7]'FO$!3;B6I^7jsMF,7$$\"3aLLLt>:nmFO$!37+2hu:afHF,7$$\"35LLL.a#o$o FO$!3;\"e/Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4`1I$pa!o\"F,7$$\"3y******z]rf rFO$!3pfL'*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3?j;%3XMsQ#FK7$$\"3/LLL8-V&\\( FO$\"3qi(R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3ZX^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,-%'COLOURG6&%$RGBG$\"#5!\"\"F (F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;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 the methods and gives the \+ " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each soluti on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "E := (x,y) -> cos(x) +2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slo pe field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `, hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Dorman d, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/6 4),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7 ]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits : = 20:\nfor ct to 5 do\n En_RK6_||ct := RK6_||ct(E(x,y),x,y,x0,evalf( y0),hh,numsteps,false);\n sm := 0: numpts := nops(En_RK6_||ct):\n \+ for ii to numpts do\n sm := sm+(En_RK6_||ct[ii,2]-e(En_RK6_||ct[i i,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\" xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$% /step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q*pprint1206\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$ \"+)yZEm'!#=7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36# \"\"(#\"#j\"#kF/$\"+!4X)QWF+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+\"\\f[+%F+7 $*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+tJpXMF+7$*&F.F/6$/F2#\"\"$\"#8 /F:#\"#')\"#()F/$\"+hv*\\=$F+Q*pprint1216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "x = 7.999;" "6#/%\"xG-%&FloatG6$\"%**z!\"$ " }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "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 := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`schem e with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64), `scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=8 6/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n en_RK6_||ct := \+ RK6_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 7. 999: exx := evalf(e(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(en _RK6_||ct(xx)-exx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~field:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~poi nt:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~ G\"$+%Q*pprint1226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~of~Dormand,~Lockyer ,~McCorrigan~and~PrinceG$\"+Z*[?p$!#<7$*&%-scheme~with~G\"\"\"6$/&%\"c G6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+;yufCF+7$*&F.F/6$/F2#\"\"* \"#QF9F/$\"+I%e#>AF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+D7S4>F+7$ *&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+&**Q\\w\"F+Q*pprint1236\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows 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 0 440 "mthds := [`scheme of Dorman d, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/6 4),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7 ]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits : = 20:\nfor ct to 5 do\n sm := NCint((e(x)-'en_RK6_||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$%Rscheme~of~Dorman d,~Lockyer,~McCorrigan~and~PrinceG$\"+(p[(Rl!#=7$*&%-scheme~with~G\"\" \"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+Fi'pN%F+7$*&F.F/ 6$/F2#\"\"*\"#QF9F/$\"+>f)4$RF+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$ \"+uR8#Q$F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+[rCEJF+Q*pprint1 246\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical proced ures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 " evalf[20](plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x),e(x)-'en_RK6_3'( x),\ne(x)-'en_RK6_4'(x),e(x)-'en_RK6_5'(x)],x=0..2,font=[HELVETICA,9], \ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COL OR(RGB,0,.35,.95),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrig an and Prince`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4 ]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme \+ with c[4]=3/13 and c[7]=86/87`],\ntitle=`error curves for 8 stage orde r 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5MLLLLL3VfV!#@$\"&zp&!#? 7$$\"5nmmmm\"H[D:)F-$\"'=F07$$\"5LLLLL 3RBr;F0$\"':=DF07$$\"5nmmm;zjf)4#F0$\"'QnKF07$$\"5MLLL$e4;[\\#F0$\"'=V SF07$$\"5++++]i'y]!HF0$\"'g\")[F07$$\"5MLLL$ezs$HLF0$\"'A)z&F07$$\"5++ ++]7iI_PF0$\"'u!z'F07$$\"5nmmmm;_M(=%F0$\"'zYzF07$$\"5MLLLL3y_qXF0$\"' (z#*)F07$$\"5+++++]1!>+&F0$\"(P_-\"F07$$\"5+++++]Z/NaF0$\"(`T:\"F07$$ \"5+++++]$fC&eF0$\"(n()G\"F07$$\"5MLLL$ez6:B'F0$\"(g?U\"F07$$\"5nmmmm; =C#o'F0$\"(\"4'e\"F07$$\"5nmmmmm#pS1(F0$\"(\"QRF07$$\"5nmmmmm(y8!zF0$\"(Mo5#F07$$\"5,+++]i.tK$)F0$\"(\\wJ#F07$ 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\"G5F07$F_z$\")BJ76F07$Fdz$\")D))47F0-Fiz6&F[[lF($\"#NF]el$\"#&*F]el-F b[l6#%Escheme~with~c[4]=7/30~and~c[7]=81/82G-F$6%7SF'7$F+$\"&es#F07$F2 $\"&zf&F07$F7$\"&hq)F07$F<$\"'f/7F07$FA$\"''Hc\"F07$FF$\"'-M>F07$FK$\" '-NBF07$FP$\"'VtFF07$FU$\"'9[KF07$FZ$\"'*4!QF07$Fin$\"'AqUF07$F^o$\"'X .\\F07$Fco$\"'!*>bF07$Fho$\"'fjhF07$F]p$\"'%4!oF07$Fbp$\"'F&e(F07$Fgp$ \"'?=$)F07$F\\q$\"'d7#*F07$Faq$\"(4v+\"F07$Ffq$\"(/$36F07$F[r$\"(T>@\" F07$F`r$\")\"z&G8F-7$Fer$\")&GJW\"F-7$Fjr$\")Hmw:F-7$F_s$\")(3ds\"F-7$ Fes$\")pCl=F-7$Fjs$\"*(e6F?F^t7$F`t$\")h!y?#F-7$Fet$\")I&*)R#F-7$Fjt$ \")2\\)f#F-7$F_u$\")5!)QGF-7$Fdu$\"(SB2$F07$Fiu$\"(s " 0 "" {MPLTEXT 1 0 513 "evalf[20](plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2 '(x),e(x)-'en_RK6_3'(x),\ne(x)-'en_RK6_4'(x),e(x)-'en_RK6_5'(x)],x=2.. 8,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1,.85,0),CO LOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegend=[`scheme of Dorma nd, Lockyer, McCorrigan and Prince`,`scheme with c[4]=6/25 and c[7]=63 /64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86/87`],\ntitle=`error cu rves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 813 648 648 {PLOTDATA 2 "6+-%'CURVESG6%7Z7$$\"\"#\"\"!$\")OP RB!#?7$$\"5+++++DHyI@!#>$\")\"H<-$F-7$$\"5++++v[kdWAF1$\")U)ex$F-7$$\" 5++++]n\"\\DP#F1$\")6ed[F-7$$\"5++++]s,P,DF1$\")YXhiF-7$$\"5++++v8*y&H EF1$\")Evq!)F-7$$\"5++++vG[W[FF1$\"*0r:-\"F-7$$\"5++++v)fB:(GF1$\"*%fe /8F-7$$\"5++++vQ=\"))*HF1$\"*JI0o\"F-7$$\"5++++vj=pDJF1$\"*x*)Q;#F-7$$ \"5+++++lN?cKF1$\"*iSt!GF-7$$\"5++++]U$e6P$F1$\"*xN8`$F-7$$\"5+++++&>q 0]$F1$\"*:!psXF-7$$\"5+++++DM^IOF1$\"*W<%GfF-7$$\"5+++++0ytbPF1$\"*w_W h(F-7$$\"5++++vQNXpQF1$\"*yd!e&*F-7$$\"5+++++XDn/SF1$\"+%QQDD\"F-7$$\" 5+++++!y?#>TF1$\",YD!)\\d\"!#@7$$\"5++++v3wY_UF1$\"-eYr#f0#!#A7$$\"5++ +++IOTqVF1$\",R`))Gg#F^q7$$\"5++++v3\">)*\\%F1$\"+$e#zrLF-7$$\"5++++DE P/BYF1$\"+YE=9VF-7$$\"5++++](o:;v%F1$\"+$G@$zbF-7$$\"5++++v$)[op[F1$\" 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*>P[O#F-7$Ffo$\"*Qsf1$F-7$F[p$\"*eJz$RF-7$F`p$\"*<(4V\\F-7$Fep$\"*;'px kF-7$Fjp$\"+KZDX\")F^q7$F`q$\"-3'paK1\"Fdq7$Ffq$\",tOChM\"F^q7$F[r$\"+ !>vPu\"F-7$F`r$\"+Mq9JAF-7$Fer$\"+&QLa)GF-7$Fjr$\"+PF-aOF-7$F_s$\"+&Hm Sr%F-7$Fds$\"+F/!>9'F-7$Fis$\"+)QkEt(F-7$F^t$\"+a1_;**F-7$Fct$\",+bDAG \"F-7$Fht$\",WBB([;F-7$F]u$\",xI\\F5#F-7$Fbu$\",:\\[Zv#F-7$Fgu$\",?=YU&Fdq7$F cy$\".71q7d>'F^q7$Fhy$\".z)3hTwqF^q7$F]z$\".$\\+knpzF^q7$Fbz$\"-(zX$pv *)F-7$Fgz$\".u;kd)=5F-7$F\\[l$\".T)*eNl:\"F-7$Fa[l$\".C\\[.dI\"F-7$Ff[ l$\".J#)y5TZ\"F-7$F[\\l$\".[Oh*Rt:F-7$F`\\l$\".L70w$z;F-7$Fe\\l$\".IE` !\\#z\"F-7$Fj\\l$\".Aj%QA8>F--F_]l6&Fa]lFe]l$\"#NF\\il$\"#&*F\\il-Fi]l 6#%Escheme~with~c[4]=7/30~and~c[7]=81/82G-F$6%7Z7$F($\")]N=6F-7$F/$\") ZaW9F-7$F5$\")a00=F-7$F:$\")N9ABF-7$F?$\")BC$*HF-7$FD$\")0:eQF-7$FI$\" )f^$)[F-7$FN$\")#GkB'F-7$FS$\")pdL!)F-7$FX$\"*C@W.\"F-7$Fgn$\"*L9?M\"F -7$F\\o$\"*)36)o\"F-7$Fao$\"*,>Mi?F-7$Fer$\"+8g7nEF -7$Fjr$\"+J^cxLF-7$F_s$\"+*\\2uN%F-7$Fds$\"+)*R@xcF-7$Fis$\"+>LiZrF-7$ F^t$\"+LDDm\"*F-7$Fct$\",dT9_=\"F-7$Fht$\",\\Z$)R_\"F-7$F]u$\",cmeO%>F -7$Fbu$\",'R'Gja#F-7$Fgu$\",fZ*pXKF-7$F\\v$\",KG+d?%F-7$Fav$\",Z[>(=`F -7$Ffv$\",)>Z?voF-7$F[w$\",H=FMv)F-7$F`w$\"-'[lKn7\"F-7$Few$\"-Ut/ei8F-7$F[\\l$\".FaseVX\"F-7$F`\\l$\".9f : " 0 "" {MPLTEXT 1 0 2 " ;" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 14 of 8 stage, order 6 Ru nge-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#/*&%#d yG\"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F&%\"yGF&F(" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5);" "6#/-%\"yG6#\" \"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Sol ution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101 " "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\"$!**F&\"&,-\"! \"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+10/101" "6#,&*&%$cosG\" \"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x* sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F&F&*&\"$+#F&\"&,-\"!\"\" F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990/10201+sqrt(5))*exp(-10* x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F&\"&,-\"!\"\"F&-%%sqrtG6# \"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := \+ diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic \},y(x));\nb := unapply(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9 ],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-%$cosGF+F0F0*&F/F0F)F0!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"&#\"\"\"\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&#\"$+\"\"$,\" \"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,-\"F-F/F-!\"\"*&#\"#5F,F-*&-%$ sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$expG6#,$*&F8F-F'F-F5F-,&#F3F4F- *$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\")*y*\\xz1OA!#<7$$\"3ALL$ 3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$ \"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x&)*3\"!#=$\"3(**[ro!GyV zF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\"3umm\"H2P\"Q?F@$\"3/$)o qvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3!***\\PMnNrDF@$\"3M:4%* 3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3MLL$eRwX5$F@$\"3)GTJ!oG 0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33ML$3x%3yTF@$\"3n$**Q]` \"yRLF@7$$\"3emm\"z%4\\Y_F@$\"3%)*)G8T#p2%RF@7$$\"3`LLeR-/PiF@$\"3PZ.% R2Cm^%F@7$$\"3]***\\il'pisF@$\"31e'*fKlL9]F@7$$\"3>MLe*)>VB$)F@$\"3%3) yy-pAk`F@7$$\"3Y++DJbw!Q*F@$\"3Kg$RQBm7^&F@7$$\"3%ommTIOo/\"F,$\"3xrTB 'zz$GaF@7$$\"3YLL3_>jU6F,$\"3!p\\3Dp!RX^F@7$$\"37++]i^Z]7F,$\"3Kvxv\"* =%>e%F@7$$\"33++](=h(e8F,$\"3C^UQ-(*\\]PF@7$$\"3/++]P[6j9F,$\"3icMTpV \"zp#F@7$$\"3UL$e*[z(yb\"F,$\"3C-jVJM+L:F@7$$\"3wmm;a/cq;F,$!33XIKjA5+ 5F07$$\"3%ommmJF,$!3!\\F1VDv$)p&F@7$$\"3K+]i!f#=$3#F,$!3V$*[LN4F4!)F @7$$\"3?+](=xpe=#F,$!3$[6ral`?.\"F,7$$\"37nm\"H28IH#F,$!3s6ToLL*4G\"F, 7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F,$!3[Tb]!*H%e)HF,7$$\"3+++v =S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$!3N`)=cl>V?$F,7$$\"3B++](=] @W$F,$!3Y_BiN[odKF,7$$\"3mm\"H#oZ1\"\\$F,$!3)o4&)z%=-oKF,7$$\"35L$e*[$ z*RNF,$!3%)Q61)ek$pKF,7$$\"3%o;Hd!fX$f$F,$!3#*y!45Ut-E$F,7$$\"3e++]iC$ pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D1>x`HA5KF,7$$\"3[m;H2qcZPF ,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX&>E0$F,7$$\"3Ymm;/OgbRF,$! 3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!yEF,7$$\"3yLLL$)*pp;%F,$!30 ,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F,7$$\"3Cn;HdO=yVF,$!35Q!)* 4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$$\"3SnmT&G!e&e%F,$!3]X/0\" RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@7$$\"37+]iSjE!z%F,$!3+r[gf MO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$$\"\"&F)$\"3&Q8`\">jC3#*F@ -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 778 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: \+ x0 := 0: y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Princ e`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7 ]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/ 13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Bn_RK6 _||ct := RK6_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm : = 0: numpts := nops(Bn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Bn_RK6_||ct[ii,2]-evalf(b(Bn_RK6_||ct[ii,1])))^2;\n end do:\n \+ errs := [op(errs),sqrt(sm/numpts)];\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(pprint26\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+$=^'QG! #@7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j \"#kF/$\"+%QK8\"=F+7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+QwL#e\"F+7$*&F.F/6$/ F2#F<\"#I/F:#\"#\")\"##)F/$\"+4Mc_8F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#') \"#()F/$\"+=3.57F+Q(pprint36\"" }}}{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 0 702 " B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: \+ y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial point: `, ``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`schem e with `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64), `scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=8 6/87)]: errs := []:\nDigits := 25:\nfor ct to 5 do\n bn_RK6_||ct := \+ RK6_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4. 999: bxx := evalf(b(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(bn _RK6_||ct(xx)-bxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~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)pprint186\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%Rscheme~ of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+f*Q$oF!#A7$*&%-scheme~w ith~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+x%f8(=F +7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+8\\+#p\"F+7$*&F.F/6$/F2#F<\"#I/F:#\"# \")\"##)F/$\"+G[dv9F+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+E%Q*o8 F+Q)pprint196\"" }}}{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 0 440 "mthds := [`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with \+ `*(c[4]=6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((b(x)-'bn_ RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n err s := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+J " 0 "" {MPLTEXT 1 0 530 "evalf[20](plot(['bn_RK6_1'(x)-b(x),'bn_RK6_2'(x )-b(x),'bn_RK6_3'(x)-b(x),\n'bn_RK6_4'(x)-b(x),'bn_RK6_5'(x)-b(x)],x=0 ..0.65,numpoints=100,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COL OR(RGB,.1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegen d=[`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with c[ 4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86/87` ],\ntitle=`error curves for 8 stage order 6 Runge-Kutta methods`));" } }{PARA 13 "" 1 "" {GLPLOT2D 938 575 575 {PLOTDATA 2 "6+-%'CURVESG6%7b \\l7$$\"\"!F)F(7$$\"5SSSSSS:N<7$$\"5\"333333.ZV$F-$\"&[/ #F07$$\"5+,,,,^)yLH%F-$\"&Du*F07$$\"5?@@@@@Y0_^F-$\"'2)[$F07$$\"5STTTT \"RI2,'F-$\"(T`-\"F07$$\"5ihhhhhhSpoF-$\"(L!4EF07$$\"5XXXX&zNNlh(F-$\" (W6P&F07$$\"5IHHHHaXmj$)F-$\"),SL5F07$$\"55888j]Pz5\"*F-$\")O%)z=F07$$ \"5&ppppp%H#z&)*F-$\")z*=E$F07$$\"5HHHagC0Z/5!#@$\")+b.OF07$$\"5*))))) Q^a([J-\"F[o$\")c#of$F07$$\"5[[[BU%)p#=/\"F[o$\")O6!f$F07$$\"53333L9_] g5F[o$\")TT$e$F07$$\"5FFFx9u;'y4\"F[o$\")I0qNF07$$\"5YYYY'R8=_8\"F[o$ \")RucNF07$$\"5%[[[)f`5$*47F[o$\")eKINF07$$\"5BBBBBtRk%G\"F[o$\")l[/NF 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"Test 15 of 8 stage, order 6 Runge-Kutta methods" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Differential Equati ons, Hull, Enright, Fellen and Sedgwick,\n Siam Journal on Nume rical Analysis, Vol. 9, No. 4 (Dec. 1972), page 617, Example A5." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx = (y-x)/(y+x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&% \"yGF&%\"xGF(F&,&F+F&F,F&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y( 1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/(x^2))+4*arctan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/, ,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"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\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\" \"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 18 "The solution (for " }{TEXT 268 1 "x" }{TEXT -1 47 " inc reasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&- %%sqrtG6#\"\"#\"\"\"-%$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-*$)% \"yGF,F-F-F-*&F,F--%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF -F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"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=red):\np2 := plot([sqrt(2)*exp(Pi/4-t ),t,t=Pi/4..2*Pi],coords=polar,color=black,linestyle=2):\np3 := plot([ sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi/4],coords=polar,color=black,linesty le=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[12,10$3],color=[black,gr een$3]):\nplots[display]([p1,p2,p3,p4],font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 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GL![F*$!3AMPAGam$=&F*7$$\"3B_c%y]m_![F*$!3C%yzJ*p\")G^F*7$$\"3'*o#foxs n![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* $!3>+++#Qx/\"[F*F_]m-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&F h\\n-Fjz6&F\\[lF`[lF][lF`[l-Fb]n6$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#%(DEFAULTG-Fa_n6$%*HELVETICAG\"\"*-%%VIEWG6$Fc_n Fc_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 follo wing 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#,&*$%\"xG\"\"#\"\"\"*$%\"y GF+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 266 1 "y " }{TEXT -1 25 " numerically in terms of " }{TEXT 267 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) local 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 := eval f(exp(Pi/2)):\nplot('phi'(x),x=1..uu,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$$\"+N$4`2\"F-$\"+]6*f)**F37$$\"+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$$\"+\"*o!oQ\"F-$\"+ig9X'*F37$$\"+jM?A9F-$\"+N0$)y &*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?]\"F-$\"+2v=4%*F37$$\"+d@iS:F- 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F37$$\"+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$$\"+zL#f h$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$\"+I6?&p$F-$!+q5-gsF37$$\"+aq sLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F37$$\"+a3#*3QF-$!+'*3rE!*F37$ $\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6MXI5F-7$$\"+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$$\"+`'4eU%F-$!+)[X5D#F-7$$\"+C&QOY%F-$!+d+zqBF- 7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$!+'GZ)QEF-7$$\"+d6/\"e%F-$!+$ pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhbYF-$!+9%4S9$F-7$$\"+(y;_p%F- $!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M`Y_ZF-$!+El5vPF-7$$\"+o))>qZ F-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q.z%F-$!+!p\\]>%F-7$$ \"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 823 "C := \+ (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\n matrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`sch eme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4]=6 /25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with `*( c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]:\nerrs := []: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn_RK6_||ct := RK 6_||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nop s(Cn_RK6_||ct):\n for ii to numpts do\n if ct=1 then vals := [o p(vals),phi(Cn_RK6_||ct[ii,1])] end if;\n sm := sm+(Cn_RK6_||ct[i i,2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,& %\"yG\"\"\"%\"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)pprint206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+&>Y &>y!#B7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(# \"#j\"#kF/$\"+;PbN8!#A7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+&z`JR\"FB7$*&F.F/ 6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+av3X9FB7$*&F.F/6$/F2#\"\"$\"#8/F:#\"# ')\"#()F/$\"+5A2y8FBQ)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{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 = 4.749;" "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 686 "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)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`s cheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with `*(c[4] =6/25,c[7]=63/64),`scheme with `*(c[4]=9/38,c[7]=63/64),`scheme with ` *(c[4]=7/30,c[7]=81/82),`scheme with `*(c[4]=3/13,c[7]=86/87)]: errs : = []:\nDigits := 30:\nfor ct to 5 do\n cn_RK6_||ct := RK6_||ct(C(x,y ),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749: cxx := evalf(phi (xx)):\nfor ct to 5 do\n errs := [op(errs),abs(cn_RK6_||ct(xx)-cxx)] ;\nend 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)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%Rscheme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG$\"+4x1 n()!#A7$*&%-scheme~with~G\"\"\"6$/&%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(# \"#j\"#kF/$\"+]dTb:!#@7$*&F.F/6$/F2#\"\"*\"#QF9F/$\"+#4j " 0 "" {MPLTEXT 1 0 447 "mthds := [`scheme of Dormand, Lockyer, M cCorrigan and Prince`,`scheme with `*(c[4]=6/25,c[7]=63/64),`scheme wi th `*(c[4]=9/38,c[7]=63/64),`scheme with `*(c[4]=7/30,c[7]=81/82),`sch eme with `*(c[4]=3/13,c[7]=86/87)]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint(('phi'(x)-'cn_RK6_||ct'(x))^2,x=1..4.75,adapt ive=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$%Rscheme~of~Dormand,~L ockyer,~McCorrigan~and~PrinceG$\"+GwnRL!#B7$*&%-scheme~with~G\"\"\"6$/ &%\"cG6#\"\"%#\"\"'\"#D/&F36#\"\"(#\"#j\"#kF/$\"+#3')\\(fF+7$*&F.F/6$/ F2#\"\"*\"#QF9F/$\"+#[:8G'F+7$*&F.F/6$/F2#F<\"#I/F:#\"#\")\"##)F/$\"+a ZWnlF+7$*&F.F/6$/F2#\"\"$\"#8/F:#\"#')\"#()F/$\"+;dp*G'F+Q)pprint246\" " }}}{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 0 548 "evalf [30](plot(['cn_RK6_1'(x)-'phi'(x),'cn_RK6_2'(x)-'phi'(x),'cn_RK6_3'(x) -'phi'(x),\n'cn_RK6_4'(x)-'phi'(x),'cn_RK6_5'(x)-'phi'(x)],x=1..3.75,0 ..2.65e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.1, .85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegend=[`scheme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with c[4]=6/25 an d c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c[4] =7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86/87`],\ntitle= `error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 927 604 604 {PLOTDATA 2 "6+-%'CURVESG6%7V7$$\"\"\"\" \"!$F*F*7$$\"?LLLLLLLL$eR(*\\F27$$\"?LLLLLLLLeRiYzH7F/$ \",a>G%)4\"F27$$\"?nmmmmmm;a8-qb)G\"F/$\",QVgu!=F27$$\"?LLLLLLL$3xJ@PI M\"F/$\",:4Wgm#F27$$\"?+++++++]P4;$[%*R\"F/$\",&yGFQOF27$$\"?LLLLLLL$3 F%f()yd9F/$\",jq:]o%F27$$\"?+++++++](=U5Uf^\"F/$\",m02M)eF27$$\"?nmmmm mmm;Hn*fdd\"F/$\",z&fE,sF27$$\"?LLLLLLLLektvWG;F/$\",2y.OV)F27$$\"?+++ +++++]PR8w(o\"F/$\",tJ/'z)*F27$$\"?++++++++]7`'=tu\"F/$\"-(G$o&=F/$\"-@e-?Z9F 27$$\"?nmmmmmmm;z\\#3)=>F/$\"-\"4A7`j\"F27$$\"?nmmmmmmmm;C&48(>F/$\"-b \"poq!=F27$$\"?+++++++]PM()4QK?F/$\"-AL2m/?F27$$\"?nmmmmmmmmT!eRk3#F/$ \"-\">e\"z\"=#F27$$\"?+++++++]P%[U]d9#F/$\"-*y*3G'R#F27$$\"?+++++++]7` u$GA?#F/$\"-$R&e@=EF27$$\"?nmmmmmmmT5!>d6E#F/$\"-9Uh#f&GF27$$\"?nmmmmm m;aQQAF:BF/$\"-)=+$HB=%F27$$\"?+++++++++veT%Hg#F/$\"-1Vo;yFF/$\"-jl%Q;k&F27$$\"?nmmmmmmmmT!y.Q$GF/$\"-lrfZlgF27 $$\"?++++++++]7.E=$*GF/$\"-7%[#G\\lF27$$\"?LLLLLLL$3F>k))p%HF/$\"-+$HP y+(F27$$\"?++++++++]PaG\"e+$F/$\".S!4*HCd(!#J7$$\"?nmmmmmm;/,a=;hIF/$ \".KthEi<)Fgv7$$\"?+++++++](=n]JF/$\".pFJ))Q#))Fgv7$$\"?nmmmmmmm;H# )>evJF/$\".G4'=YL&*Fgv7$$\"?+++++++]P4Y(*zMKF/$\".\"[\")=ZP5F27$$\"?LL LLLLLLL$3\\L=H$F/$\".'*=MX.8\"F27$$\"?LLLLLLLLeRK(e,N$F/$\".xRS)\\O7F2 7$$\"?nmmmmmm;/^65+3MF/$\".+kP(oZ8F27$$\"?+++++++++ve?:hMF/$\".I-9ynZ \"F27$$\"?nmmmmmmm\"zp:p?_$F/$\".GPVi'Q;F27$$\"?LLLLLLLLL$e8ald$F/$\". E?509!=F27$$\"?mmmmmmmT&)e6.g0OF/$\".F(>\"eU!>F27$$\"?+++++++]PM([YYj$ F/$\".FTaGr,#F27$$\"?+++++++++v$QZCm$F/$\".gB!y$*Q@F27$$\"?+++++++]i:! 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" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of D ormand, Lockyer, McCorrigan and Prince" "scheme with c[4]=6/25 and c[7 ]=63/64" "scheme with c[4]=9/38 and c[7]=63/64" "scheme with c[4]=7/30 and c[7]=81/82" "scheme with c[4]=3/13 and c[7]=86/87" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "eval f[30](plot(['cn_RK6_1'(x)-'phi'(x),'cn_RK6_2'(x)-'phi'(x),'cn_RK6_3'(x )-'phi'(x),\n'cn_RK6_4'(x)-'phi'(x),'cn_RK6_5'(x)-'phi'(x)],x=3.75..4. 6,0..1.55e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB, .1,.85,0),COLOR(RGB,.9,.3,.1),COLOR(RGB,0,.35,.95),red],\nlegend=[`sch eme of Dormand, Lockyer, McCorrigan and Prince`,`scheme with c[4]=6/25 and c[7]=63/64`,`scheme with c[4]=9/38 and c[7]=63/64`,`scheme with c [4]=7/30 and c[7]=81/82`,`scheme with c[4]=3/13 and c[7]=86/87`],\ntit le=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 958 662 662 {PLOTDATA 2 "6+-%'CURVESG6%7]o7$$\"$v $!\"#$\".*)z&=$\\f#!#I7$$\"?nmmmmmmm;/\"eF&oP!#H$\".D**f-')o#F-7$$\"?L LLLLLL$eR-L[Yy$F1$\".5.btgz#F-7$$\"?nmmmmmmm\"H()zxF!QF1$\"./>/41$HF-7 $$\"?nmmmmmmmT5Tu-@QF1$\".RGcQl2$F-7$$\"?LLLLLLL$e9hM!>RQF1$\".Zv(f;5K F-7$$\"?nmmmmmm;H2%oHg&QF1$\".C(y%=SM$F-7$$\"?+++++++]i:VeYtQF1$\"-%=V !Q?NF17$$\"?nmmmmmm;H#QM)\\\"*QF1$\"-tWBX;PF17$$\"?+++++++]7.9IZ4RF1$ \"-'>UBl!RF17$$\"?LLLLLLLL$3<_!\\F17$$\"?++++++++]PA&H()*RF1$\"-o\"zBI;&F17$$\"?nmm mmmm;HK^#R[,%F1$\"-;]oGjaF17$$\"?LLLLLLLL$3sF&*R.%F1$\"-/e(e%oeF17$$\" ?LLLLLLLLL$Q%HA]SF1$\"-X,ba-jF17$$\"?+++++++]i!zd*4pSF1$\"-cZ)ylr'F17$ $\"?LLLLLLLLLe(f3e3%F1$\"-B$y^4:(F17$$\"?+++++++]iSS59/TF1$\"-d.;0exF1 7$$\"?+++++++](=7'yf@TF1$\"-JW6kV%)F17$$\"?LLLLLLLLeRAA\")RTF1$\"-FA6r &>*F17$$\"?LLLLLLL$ek=pQl:%F1$\"-n72_\"y*F17$$\"?nmmmmmmmT&)[/euTF1$\" .&H&fv[2\"F17$$\"?nmmmmmm;H#Qw?L>%F1$\".kN'Qd*=\"F17$$\"?+++++++]P4jUj 4UF1$\".XAi.mI\"F17$$\"?nmmmmmmm\"H#*G`sA%F1$\".)))*e')fT\"F17$$\"?+++ ++++++DnaXXUF1$\".xv[X5e\"F17$$\"?++++++++DJZFEjUF1$\".&H\\i#3y\"F17$$ \"?+++++++]iS1A\\!G%F1$\".rk1=#=?F17$$\"?++++++++v=$eA'*H%F1$\".*y')GB !G#F17$$\"?LLLLLLLLLex?\"oJ%F1$\".7)pA$e^#F17$$\"?++++++++](=`l^L%F1$ \".ZA&R++HF17$$\"?nmmmmmm;HK*['z^VF1$\".(e2GgqLF17$$\"?++++++++]i=&y*p VF1$\".YT&R@hRF17$$\"?LLLLLLL$eR7R'3(Q%F1$\".?FB#z.WF17$$\"?+++++++]7` Z!p\\S%F1$\".2F2^%F1$\"/o['Rcdz\"F17$$\"?LLLLLLLL3_['[&HXF 1$\"/uY9E'[I#F17$$\"?+++++++]PMa!pz`%F1$\"/:>w3#f\\#F17$$\"?mmmmmmmmm; g%*QYXF1$\"/e+8X1VGF17$$\"?LLLLLLLek.ptObXF1$\"/qbDYI>LF17$$\"?+++++++ ]i!zFXVc%F1$\"/#*=!e^f\"RF17$$\"?+++++++D\"y+yT'oXF1$\"/k+39dpTF17$$\" ?+++++++++D#GQHd%F1$\"/)fW:0(fYF17$$\"?+++++++v=U%yMsd%F1$\"/kOV!4Hy%F 17$$\"?+++++++]Pf'GJ:e%F1$\"/\\%Qo))3g&F17$$\"?++++++DJ&pd()RQe%F1$\"/ v]bxERcF17$$\"?++++++]7`%\\Y[he%F1$\"/+9P6h+dF17$$\"?++++++v$4@T0d%)e% F1$\"/F!GW(Q**fF17$$\"?+++++++voHVcw!f%F1$\"/;na87?oF17$$\"?++++++DcEZ KU2$f%F1$\"/0=')y)y'oF17$$\"?++++++]P%[;#GQ&f%F1$\"/#4oN;n#pF17$$\"?++ +++]7GjB;r`'f%F1$\"/LjOJ%)))pF17$$\"?++++++v=U#3T\"p(f%F1$\"/621JmQrF1 7$$\"?+++++D1k\"=\"e&o#)f%F1$\"/N))\\@*\\G(F17$$\"?+++++]P4@T0d%))f%F1 $\"/^zI>1:vF17$$\"?+++++voagq_GU*f%F1$\"/CkiDuqyF17$$\"#Y!\"\"$\"/>!pL +/T)F1-%&COLORG6&%$RGBG$\"\"&Fg_l$\"\"!Fa`l$\"\"\"Fa`l-%'LEGENDG6#%Rsc heme~of~Dormand,~Lockyer,~McCorrigan~and~PrinceG-F$6%7co7$F($\".;!QSJM ;F-7$F/$\".(4()G2%p\"F-7$F5$\".9!R>(Hw\"F-7$F:$\".(RGMb\\=F-7$F?$\".Cg CjQ%>F-7$FD$\".:$)*3QI?F-7$FI$\".Wwuxt6#F-7$FN$\"-Kt&HGB#F17$FS$\"-idR !>O#F17$FX$\"-ltV\\([#F17$Fgn$\"-P]X*eh#F17$F\\o$\"-6#H\"fmFF17$Fao$\" -Z-*pA&HF17$Ffo$\"-_V2pfJF17$F[p$\"-V(F17$Fbs$\"-'o(Q!4L)F17$Fgs$\"-T$*)*Gm#*F17$F\\t$\".- sp(\\95F17$Fat$\".wgA?4:\"F17$Fft$\".sW&eD?8F17$F[u$\".0qw^p_\"F17$F`u $\"./1Y(efF17$Fju$\".()3R%)3L#F17$F_v$\".b\\=Mqy# F17$Fdv$\"._g/Y(yLF17$Fiv$\".2aO$zEQF17$F^w$\".k#GF&Rq%F17$Fcw$\".AL4? 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