{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 "" 261 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Sy stem" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 273 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 " System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 283 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 258 284 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Norma l" -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 O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "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 66 "Derivation of 11 stage, combined \+ order 6 and 7 Runge-Kutta schemes" }}{PARA 0 "" 0 "" {TEXT -1 46 "by P eter Stone, Gabriola Island, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.12.2012" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "load proc edures for constructing Runge-Kutta schemes etc. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 9 "butcher.m" }{TEXT -1 2 ", " }{TEXT 262 7 "roots.m" }{TEXT -1 6 " and " }{TEXT 262 6 "intg.m " }{TEXT -1 33 " are required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read into a Maple session by commands simila r to those that follow, where each file path gives the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Ma ple/procdrs/butcher.m\";\nread \"C:\\\\Maple/procdrs/roots.m\";\nread \+ \"C:\\\\Maple/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 29 "#============================" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 27 "Relations between the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "The Runge-Kutta schemes c onsidered in this worksheet have the property that the nodes " } {XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]; " "6#&%\"cG6#\"\"(" }{TEXT -1 31 " are related by the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p(x)*``((1-x)^3/3 !),x = 0 .. 1)*Int(q(x)*``((1-x)/1!),x = 0 .. 1) = Int(p(x)*``((1-x)^2 /2!),x = 0 .. 1)*Int(q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/*&-%$IntG6$ *&-%\"pG6#%\"xG\"\"\"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorialG6#F4F3F-/F ,;\"\"!F-F--F&6$*&-%\"qG6#F,F--F/6#*&,&F-F-F,F3F--F66#F-F3F-/F,;F:F-F- *&-F&6$*&-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F66#FSF3F-/F,;F:F-F--F&6$*&-F ?6#F,F--F/6#*&,&F-F-F,F3FS-F66#FSF3F-/F,;F:F-F-" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " } {XPPEDIT 18 0 "p(x) = x*(x-c[5])*(x-c[6]);" "6#/-%\"pG6#%\"xG*(F'\"\" \",&F'F)&%\"cG6#\"\"&!\"\"F),&F'F)&F,6#\"\"'F/F)" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "q(x) = (x-c[7])*p(x);" "6#/-%\"qG6#%\"xG*&,&F'\"\"\" &%\"cG6#\"\"(!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "See: J.H. Verner, SIAM J ournal of Numerical Analysis 1978, 772-790, \"Explicit Runge-Kutta met hods with estimates of the Local Truncation Error.\" (pages 780,781)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "p := x -> x*(x-c[5])*(x-c[6]):\n'p(x)'=p(x);\nq := x -> (x-c[7]) *p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG *(F'\"\"\",&F'F)&%\"cG6#\"\"&!\"\"F),&F'F)&F,6#\"\"'F/F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG**,&F'\"\"\"&%\"cG6#\"\"(!\"\"F*F' F*,&F'F*&F,6#\"\"&F/F*,&F'F*&F,6#\"\"'F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)' *(1-x)^3/3!,x=0..1)*Int('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2! ,x=0..1)*Int('q(x)'*(1-x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify( lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq \+ := expand(L*Req);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG 6$,$*&#\"\"\"\"\"'F-*&-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\" !F-F--F(6$*&-%\"qGF2F-F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8 F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$CeqG/,>**\"#_\"\"\"&%\"cG6#\"\"(F)&F+6#\"\"&F)&F+6#\"\"'F)F)**\"$0\" F))F.\"\"#F))F1F7F)F*F)F)**\"#qF)F*F)F.F)F8F)!\"\"**F:F)F*F)F6F)F1F)F; *&\"\"$F)F*F)F)*(F-F)F6F)F1F)F;F1F;F.F;*(\"#7F)F*F)F.F)F;*(\"#9F)F8F)F *F)F)*(F-F)F.F)F8F)F;*(FAF)F.F)F1F)F)*(FAF)F1F)F*F)F;*(FCF)F*F)F6F)F) \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The previous equation \+ involving integrals is equivalent to:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c [5]*c[6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]* c[5]+14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[ 5]^2=0" "6#/,>**\"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\"'F'F'** \"$0\"F'*$&F)6#F.\"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F)6#F.F' &F)6#F1F7!\"\"**F=F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6#F+F'F'* (F+F'*$&F)6#F.F7F'&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F'&F)6#F .F'FD*(\"#9F'*$&F)6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD*(FenF' &F)6#F.F'&F)6#F1F'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F'&F)6#F. F7F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 265 21 "_____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "#=================== =======" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Scheme of Sharp and Sm art" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "S ee: Explicit Runge-Kutta Pairs with One More Derivative Evaluation tha n the Minimum, by P.W.Sharp and E.Smart," }}{PARA 0 "" 0 "" {TEXT -1 89 " Siam Journal of Scientific Computing, Vol. 14, No. 2, page s. 338-348, March 1993." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 48 "#----------------------------------- ------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking 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 3193 "ee := \{c[2]=1/50,\nc[3]=27/125, \nc[4]=41/100,\nc[5]=57/100,\nc[6]=43/50,\nc[7]=2272510/11977321,\nc[8 ]=18/25,\nc[9]=5/6,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/50,\na[3,1]=-594/6 25,\na[3,2]=729/625,\na[4,1]=451/21600,\na[4,2]=0,\na[4,3]=1681/4320, \na[5,1]=19/160,\na[5,2]=0,\na[5,3]=361/3104,\na[5,4]=3249/9700,\na[6, 1]=-31/200,\na[6,2]=0,\na[6,3]=520921/412056,\na[6,4]=-17371/11640,\na [6,5]=132023/106200,\na[7,1]=25959766877768976976598957736980/48759451 4129628295945513157189933,\na[7,2]=0,\na[7,3]=347890318302644246405985 993187156250/1321817402067092875750818220388519949,\na[7,4]=-171704697 2617147709491116450178750/7467894926932728111586543618014237,\na[7,5]= 29780304732725103577764751746216250/2589126870028326251470674864678544 23,\na[7,6]=-302662548054389051180423185000/25662869164717278733974376 694207,\na[8,1]=42409705291266846/416462256407406875,\na[8,2]=0,\na[8, 3]=3247095172038/883201854817,\na[8,4]=-518509279926/374238074075,\na[ 8,5]=435669225629732566638/393965828849029186615,\na[8,6]=-64686945591 14760/61945939006089637,\na[8,7]=-8593750881095206170491007194502/3213 504543545558150903880585625,\na[9,1]=-1401024812030113404025/198875646 77841032175639,\na[9,2]=0,\na[9,3]=13281373111234375/5150833217292744, \na[9,4]=-50491693720625/29100752640072,\na[9,5]=890977646878316458397 3193125/6271093223575470807674793192,\na[9,6]=-4792324941735635008750/ 159776107397443897190271,\na[9,7]=-15328062904658911411660965319021185 41769245/1203242011387872547807852011647420329982736,\na[9,8]=-7500029 126894375/132689679447323376,\na[10,1]=36393032615434450612/3243905860 94889663425,\na[10,2]=0,\na[10,3]=-1462401427649331250/154787214582248 211,\na[10,4]=4135780451822750/874504037187843,\na[10,5]=-234937873364 7002895234008950/1090914599757106529355865311,\na[10,6]=-7868660590842 2443750/52446632451499515953,\na[10,7]=2315079813491204524435067899365 885119542372444358703/\n 31616904203952715759523523157378830803 1260760584200,\na[10,8]=-33473047374792524975/32907430028856870472,\na [10,9]=5594658687556280397846/1893189870520997940175,\na[11,1]=2508607 706701842363083/197875357745688550590720,\na[11,2]=0,\na[11,3]=-512283 3329940625/508724268374592,\na[11,4]=13293920580875/2874148408896,\na[ 11,5]=-599188464780493707137440161875/277270064173229869784600732736, \na[11,6]=-3601465055348923762849875/2146128454918752594358208,\na[11, 7]=606030238246181777051198920509497430523044409408159/\n 74752 050141640998967813674460513197348653288024576,\na[11,8]=-1922750201834 125/1941504226023936,\na[11,9]=12539348439579/3975412795840,\na[11,10] =0,\n\nb[1]=771570009067/14036203465200,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb [5]=28304779228000000/53707434325074117,\nb[6]=-296881060859375/515060 733835389,\nb[7]=744858303758379680905615939985761920312207508379/2487 223884477764590764433396524922145673887618400,\nb[8]=-5118512171875/11 763620626464,\nb[9]=136801854099/127885521925,\nb[10]=103626500437/171 7635089268,\nb[11]=0,\n\n`b*`[1]=448234490819/8120946290580,\n`b*`[2]= 0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=7786773134600000/14452831163890377 ,\n`b*`[6]=-408698637296875/567617951573694,\n`b*`[7]=4426705150369152 638325381078278067803359/14828075230102658203818343670586143438076,\n` b*`[8]=-5004542378125/10330679593521,\n`b*`[9]=154806770859/1242316498 70,\n`b*`[10]=0,\n`b*`[11]=16/243\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(ee,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[ j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'++?!\"(F(%!GF+F+F+F+F+F+ F+F+F+7.$\"'+g@!\"'$!'+/&*F/$\"'Sm6!\"&F+F+F+F+F+F+F+F+F+7.$\"'++TF/$ \"''z3#F*$\"\"!F;$\"'?\"*QF/F+F+F+F+F+F+F+F+7.$\"'++dF/$\"'](=\"F/F:$ \"'-j6F/$\"'[\\LF/F+F+F+F+F+F+F+7.$\"'++')F/$!'+]:F/F:$\"'?k7F4$!'N#\\ \"F4$\"':V7F4F+F+F+F+F+F+7.$\"'M(*=F/$\"'0C`F*F:$\"'\">j#F/$!'C*H#F/$ \"'@]6F/$!'Qz6F*F+F+F+F+F+7.$\"'++sF/$\"'L=5F/F:$\"']wOF4$!'^&Q\"F4$\" ''e5\"F4$!'DW5F/$!'EuEF4F+F+F+F+7.$\"'LL$)F/$!'tWqF*F:$\"'\\yDF4$!'1N< F4$\"'x?9F4$!'S**HF*$!'!RF\"F4$!'J_cF*F+F+F+7.$\"\"\"F;$\"'*=7\"F/F:$! '#yW*F4$\"'HHZF4$!'f`@F4$!'K+:F4$\"'GAtF4$!'><5F4$\"':bHF4F+F+7.Fjp$\" 'xn7F*F:$!'+25!\"%$\"'MDYF4$!'.h@F4$!'7y;F4$\"'@2\")F4$!'S.**F/$\"'BaJ F4F:F+7.%\"bG$\"'+(\\&F*F:F:F:$\"'=q_F/$!'+kdF/$\"'u%*HF/$!'9^VF/$\"'s p5F4$\"'4LgF*F:7.%#b*G$\"'\\>bF*F:F:F:$\"'r(Q&F/$!'C+sF/$\"'N&)HF/$!'N W[F/$\"'6Y7F4F:$\"'O%e'F*Q)pprint296\"" }}}{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 0 454 "subs(ee,matrix([seq( [c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));print(``);\nfor ii fro m 7 to 11 do\n print(c[ii]=subs(ee,c[ii]));print(``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]));\n end do:\n pri nt(`_________________________________`);\nend do:print(``);\nfor ii to 11 do\n print(b[ii]=subs(ee,b[ii]));\nend do:\nprint(`_____________ ____________________`);print(``);\nfor ii to 11 do\n print(`b*`[ii]= subs(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7'7(#\"\"\"\"#]F(%!GF+F+F+7(#\"#F\"$D\"#!$%f\"$D'#\"$H(F2F+F+F+7 (#\"#T\"$+\"#\"$^%\"&+;#\"\"!#\"%\"o\"\"%?VF+F+7(#\"#dF8#\"#>\"$g\"F<# \"$h$\"%/J#\"%\\K\"%+(*F+7(#\"#VF*#!#J\"$+#F<#\"'@4_\"'c?T#!&rt\"\"&S; \"#\"'B?8\"'+i5Q)pprint336\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"(5DF#\")@t(>\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"\"#\"A!)ptd*)fwp(*ox(owff#\"BL**=dJ^XfHG'HT^%f([" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"E]i:(=$*f)fSYUk-$=.*yM\"F \\*>&)Q?#=3vvG4n?S<=K\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"(\"\"%#!C](y,X;6\\4x9'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!@-X>25\\qh?&4\")3v$f)\" @Dce!)Q!4:ebaVX]8K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B______________ ___________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"* #\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!7DSS8,.7[-,9\"8Rc" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#\"2vVB6JP\"G8\"1WFHtR ekJyokx4*)\"=#>$zuw!3ZvNA$4ri" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"*\"\"'#!7](3]jN<%\\K#z%\"9r->(*QW(R2hxf\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#!LX#p`'4m69\"*eY!H1G`\"\"LOF) *H.UZ;,_y!yasyQ6?C.7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"* \"\")#!1vV*o7H+](\"3wLKZ%z'*oK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B _________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"571XMahKIRO\"6DMm*)[4' e!RC$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$#!4]7L\\wU,CY\"\"36# [Ae9sya\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\"1]F# =X!yNT\"0Vy=PS]u)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" &#!=]*3SB&*G+ZOty$\\B\"=6`'eNHl5d(*f944\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!5]PWA%3fg'oy\"5`f^*\\^CjYC&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"U.(eVWsBa>^)eO**y 1NW_/7\\8)z]J#\"T+Ueg2EJ!3$)yt:BN_fdr_R?/phJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!5v\\_#zut/tM$\"5s/(o&)G+V2H$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"7YyR!GcvoeYf&\"7v ,%z*4_q)*=$*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___________________ ______________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#\"7$3jB%=q1xg3D\"9?2f]&)oXxNvy>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$#!1D1%*HL$G7&\"0#fu$oUs3&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#\"/v3e?RH8\".'*)3%[T(G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!?v=;Su82P\\!yk%)= *f\"?OFt+Yyp)HK^qx\"=Y#Q-.1' \"SwX-)G`'[t>80YuO\"y'*)*4kT,0_Z(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"\")#!1DT$=?]F#>\"1OR-EU]T>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"/z&R%[$RD\"\".Sez7a(R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"-n!4+dr (\"/+_Y.i.9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"&#\"2+++G#zZIG\"2w&)*Rfh04oz$ePIe[u\"R+% =w)QnX@#\\_'RLWw!fkxZ%)QA([#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\")#!.v=<7&=^\"/kki?Ow6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"*#\"-*4a=!o8\"-D>_&)y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#5#\"-P/]EO5\".o#*3Nwr\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________ ________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"->3\\M#[%\".!e!HY47)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&# \"1++gMJx'y(\"2x.*Q;JGX9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"'#!0voHP')p3%\"0%pt:&zhn&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b *G6#\"\"(#\"IfL!y1y#y5QD$QE:p.:0nU%\"Jw!QM9'eqOM=Q?eE5I_2G[\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!.D\"yBa/]\"/@Nfz1L5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"-f3x1[:\"-q)\\;BC\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"#;\"$V#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11 eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'e xpanded'))]:\n`RK6_11eqs*` := subs(b=`b*`,OrderConditions(6,11,'expand ed')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK7_11eqs ):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK6_11eqs*`):\nmap(u- >lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtio ns to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 5 do\n so||ct||_11 := \+ StageOrderConditions(ct,11,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Stages 5 to 10 have stage -order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(exp and(subs(ee,so||i||_11[j])),i=2..5)],j=1..9)]:\nmap(proc(L) local i; f or i to nops(L) do if not evalb(L[i]) then break end if end do; i end \+ proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#F$\" \"$F%F%F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The stage-orders of the successive stages are given as fo llows." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[st age, `|`, 2, 3, 4, 5, 6, 7, 8, 9, 10], [`stage-order`, `|`, 1, 2, 2, 3 , 3, 3, 3, 3, 3]]);" "6#-%'matrixG6#7$7-%&stageG%\"|grG\"\"#\"\"$\"\"% \"\"&\"\"'\"\"(\"\")\"\"*\"#57-%,stage-orderGF)\"\"\"F*F*F+F+F+F+F+F+ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 9 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "[Sum(b[i]*a[i,1],i=1+1..11) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=2..9)];\nmap(u->l hs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7+/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/ F,;\"\"#\"#6&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F- &%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEF RFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F& 6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F -&F/6$F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F ,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,FaqF-/ F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,;\"#5 F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\" \"!F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The simpl ifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*a[i,3],i = 3 .. 10) = 0" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+\"\"$F,/F+;F0\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "S um(b[i]*c[i]*a[i,3],i = 3 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+;F3\"#5\"\"!" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,3],i = 3 .. 10) = 0;" "6#/-%$Sum G6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+\"\"$F,/F+;F5\"#5 \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "[Sum(b[i]*a[i,3],i=3..10),Sum(b[i]*c[i]*a[i,3],i=3..10),Sum(b[i ]*c[i]^2*a[i,3],i=3..10)];\nsubs(ee,eval(subs(Sum=add,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\" \"$F,/F+;F0\"#5-F%6$*(F(F,&%\"cGF*F,F-F,F1-F%6$*(F(F,)F7\"\"#F,F-F,F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the prin cipal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errterms7_11 := Principal ErrorTerms(7,11,'expanded'):\nnrm8 := sqrt(add(subs(ee,errterms7_11[i] )^2,i=1.. nops(errterms7_11))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lDou7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "In addition the 2-norm of the order 9 error ter ms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errter ms8_11 := PrincipalErrorTerms(8,11,'expanded'):\nnrm9 := sqrt(add(subs (ee,errterms8_11[i])^2,i=1.. nops(errterms8_11))):\nevalf[10](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!R!eIO!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The 2-norm of the order 9 error terms is approximately 2.848 times the principal error norm." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf[10](nrm9/nrm8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[JA[G!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm \+ of the order 6 embedded scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms6_11*` := subs(b=`b*`,PrincipalEr rorTerms(6,11,'expanded')):\nsqrt(add(subs(ee,`errterms6_11*`[i])^2,i= 1.. nops(`errterms6_11*`))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+a,:=>!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#-----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 7 scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The schem e will be constructed so that stage 4 has stage-order 2 and stages 5 t o 10 have " }{TEXT 260 13 "stage-order 3" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 60 "We start by determining the nodes and weights of t he scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We set up a system of equations that consists of the 7 order 7 \+ quadrature conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 10) = 1" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\" \"#5F-" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 1 .. 10) = 1/k;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF, F,!\"\"F,/F+;F,\"#5*&F,F,F2F3" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "k \+ = 2;" "6#/%\"kG\"\"#" }{TEXT -1 7 " . . 7," }}{PARA 0 "" 0 "" {TEXT -1 35 "and the relation between the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c [7]*c[5]*c[6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12* c[7]*c[5]+14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[ 7]*c[5]^2 = 0;" "6#/,>**\"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\" 'F'F'**\"$0\"F'*$&F)6#F.\"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F )6#F.F'&F)6#F1F7!\"\"**F=F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6# F+F'F'*(F+F'*$&F)6#F.F7F'&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F '&F)6#F.F'FD*(\"#9F'*$&F)6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD *(FenF'&F)6#F.F'&F)6#F1F'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F' &F)6#F.F7F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=1/50" "6#/&%\"cG6#\"\"#*&\"\"\" F)\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=27/125" "6#/&%\"cG 6#\"\"$*&\"#F\"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] =41/100" "6#/&%\"cG6#\"\"%*&\"#T\"\"\"\"$+\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5]=57/100" "6#/&%\"cG6#\"\"&*&\"#d\"\"\"\"$+\"!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "c[6]=43/50" "6#/&%\"cG6#\"\"'*&\"#V \"\"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=18/25" "6#/&% \"cG6#\"\")*&\"#=\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 9]=5/6" "6#/&%\"cG6#\"\"**&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[10]=1" "6#/&%\"cG6#\"#5\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "and the w eights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" " 6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/& %\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"b G6#\"\"%\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 45 ": The paper of Sharp and Smart has the node " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 24 " given incorrectly as " }{XPPEDIT 18 0 "43/100" "6#*&\"#V\"\"\"\"$+\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "Qeqs := QuadratureConditions(7,10,'expanded'):\nnode_eq := 52*c[7 ]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c[5]^2* c[6]+3*c[7]-\n 7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^ 2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-\n 12*c[6]*c[7]+14*c[7]*c[ 5]^2=0:\ncdns1 := [op(Qeqs),node_eq]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "e1 := \{c[2]=1/50,c[3]= 27/125,c[4]=41/100,c[5]=57/100,c[6]=43/50,c[8]=18/25,c[9]=5/6,c[10]=1, b[2]=0,b[3]=0,b[4]=0\}:\neqns1 := subs(e1,cdns1):\nnops(%);\nindets(eq ns1);\nnops(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"cG6#\"\"(&%\"bG6#\"\"'&F)6#\"\"&&F)F&& F)6#\"\")&F)6#\"\"*&F)6#\"#5&F)6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We have 8 equations and 8 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] : = 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e2 := solve(\{op(eq ns1)\}):\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 0 507 "e3 := \{c[10] = 1, c[9] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] = 27/125, c[2 ] = 1/50, b[3] = 0, b[2] = 0, b[1] = 771570009067/14036203465200, b[4] = 0, b[10] = 103626500437/1717635089268, b[8] = -5118512171875/117636 20626464, b[9] = 136801854099/127885521925, b[6] = -296881060859375/51 5060733835389, c[7] = 2272510/11977321, b[5] = 28304779228000000/53707 434325074117, c[6] = 43/50, b[7] = 74485830375837968090561593998576192 0312207508379/2487223884477764590764433396524922145673887618400\}:" }} }{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 38 "We now have all the nodes and weights." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 97 "seq(c[i]=subs(e3,c[i]),i=2..10);\nseq(b[i]=subs(e3, b[i]),i=1..6);\nseq(b[i]=subs(e3,b[i]),i=7..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%\"cG6#\"\"##\"\"\"\"#]/&F%6#\"\"$#\"#F\"$D\"/&F%6#\" \"%#\"#T\"$+\"/&F%6#\"\"&#\"#dF8/&F%6#\"\"'#\"#VF*/&F%6#\"\"(#\"(5DF# \")@t(>\"/&F%6#\"\")#\"#=\"#D/&F%6#\"\"*#Fw&)*Rfh04oz$ePIe[u\"R+%=w)QnX@#\\_'RLWw!fkxZ% )QA([#/&F%6#\"\")#!.v=<7&=^\"/kki?Ow6/&F%6#\"\"*#\"-*4a=!o8\"-D>_&)y7/ &F%6#\"#5#\"-P/]EO5\".o#*3Nwr\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 159 "We can determine the linking coefficient s using a system of equations that consists in part of the simple orde r conditions given in abreviated form as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO 7[i]],i=[45,50,51,54,55,59,61])]:\nlinalg[augment](linalg[delcols](%,2 ..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"#X%#~~G/*(%\"bG\"\" \"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\"#F-F3F-F-F-F-#F-\"$?% 7%\"#]F)/*(F,F-F.F--F06#*(F3F-F.F-F7F-F-#F-\"$0\"7%\"#^F)/*(F,F-F:F-F4 F-#F-\"#%)7%\"#aF)/*&F,F--F06#*(F3F-F.F--F06#*&)F.\"\"$F-F3F-F-F-#F-\" $o\"7%\"#bF)/*(F,F-F.F--F06#*&F3F-FTF-F-#F-\"$S\"7%\"#fF)/*(F,F-F:F-FT F-#F-\"#G7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"%F-F3F-F-#F-\"#NQ)pprint346 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We also incorporate the r ow-sum conditions," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j = 1 .. i-1) = c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+; \"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i= 2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . 10, " }}{PARA 0 "" 0 "" {TEXT -1 29 " the stage order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j],j = 2 .. i-1) = 1/2" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"&%\"cG6#F,F-/F, ;\"\"#,&F+F-F-!\"\"*&F-F-F3F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^2 " "6#*$&%\"cG6#%\"iG\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i=5" " 6#/%\"iG\"\"&" }{TEXT -1 9 " . . 10, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j]^2,j = 2 .. i-1) = 1/3" "6#/-%$SumG 6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2,&F+F-F-!\"\"*&F- F-\"\"$F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^3" "6#*$&%\"cG6#%\"iG \"\"$" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i = 2;" "6#/%\"iG\"\"#" } {TEXT -1 8 " . . 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 49 "together with the column simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,1],i=2.. 10)=b[1]" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"# 5&F)6#F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 10) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+ %\"jGF,/F+;,&F0F,F,F,\"#5*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 8 " . . \+ 9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "and the further simplifying condit ions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i ]*a[i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6# F+F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5\"\"!" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$ *(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5 \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 489 "SO7_10 := SimpleOrderConditions(7,10,'ex panded'):\nSO_eqs := [op(RowSumConditions(10,'expanded')),op(StageOrde rConditions(2,3..10,'expanded')),\n op(StageOrderConditions(3,5..10,' expanded'))]:\nord_cdns := [seq(SO7_10[i],i=[45,50,51,54,55,59,61])]: \nsimp_eqs := [add(b[i]*a[i,1],i=2..10)=b[1],seq(add(b[i]*a[i,j],i=j+1 ..10)=b[j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5. .10)=0,add(b[i]*c[i]^2*a[i,3],i=5..10)=0]:\ncdns2 := [op(simp_eqs),op( simp_eqs2),op(SO_eqs),op(ord_cdns)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "We specify the zero linking coefficie nts: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] \+ = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[ 6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "and the linking coeffici ent " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[6,1]=-31/20 0" "6#/&%\"aG6$\"\"'\"\"\",$*&\"#JF(\"$+#!\"\"F-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "e4 := \{seq(a[i,2]=0,i=4..10),a[6,1]=-31/200\}:\ne5 := `union`(e3 ,e4):\neqns2 := subs(e5,cdns2):\nnops(eqns2);\nindets(eqns2);\nnops(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e6 := solve(\{op(eqns2 )\}):\ninfolevel[solve] := 0:\ne7 := `union`(e5,e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2443 "e7 := \{c[10] = 1, c[9 ] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] = 27/125, c[ 2] = 1/50, b[3] = 0, b[2] = 0, a[3,2] = 729/625, a[6,2] = 0, a[7,2] = \+ 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[4,2] = 0, a[5,2] = 0, a[6,1] = -31/200, a[10,5] = -2349378733647002895234008950/109091459975710652 9355865311, a[5,4] = 3249/9700, a[2,1] = 1/50, a[10,1] = 3639303261543 4450612/324390586094889663425, a[4,3] = 1681/4320, a[10,6] = -78686605 908422443750/52446632451499515953, a[9,6] = -4792324941735635008750/15 9776107397443897190271, a[7,4] = -1717046972617147709491116450178750/7 467894926932728111586543618014237, a[9,5] = 89097764687831645839731931 25/6271093223575470807674793192, a[6,5] = 132023/106200, a[4,1] = 451/ 21600, a[5,1] = 19/160, b[1] = 771570009067/14036203465200, b[4] = 0, \+ b[10] = 103626500437/1717635089268, b[8] = -5118512171875/117636206264 64, a[9,1] = -1401024812030113404025/19887564677841032175639, b[9] = 1 36801854099/127885521925, a[8,7] = -8593750881095206170491007194502/32 13504543545558150903880585625, a[5,3] = 361/3104, b[6] = -296881060859 375/515060733835389, a[9,7] = -153280629046589114116609653190211854176 9245/1203242011387872547807852011647420329982736, a[9,4] = -5049169372 0625/29100752640072, a[10,8] = -33473047374792524975/32907430028856870 472, a[10,4] = 4135780451822750/874504037187843, a[7,3] = 347890318302 644246405985993187156250/1321817402067092875750818220388519949, a[8,3] = 3247095172038/883201854817, a[8,6] = -6468694559114760/619459390060 89637, c[7] = 2272510/11977321, a[7,1] = 25959766877768976976598957736 980/487594514129628295945513157189933, a[8,5] = 435669225629732566638/ 393965828849029186615, a[7,6] = -302662548054389051180423185000/256628 69164717278733974376694207, b[5] = 28304779228000000/53707434325074117 , c[6] = 43/50, a[10,7] = 23150798134912045244350678993658851195423724 44358703/316169042039527157595235231573788308031260760584200, b[7] = 7 44858303758379680905615939985761920312207508379/2487223884477764590764 433396524922145673887618400, a[6,3] = 520921/412056, a[8,4] = -5185092 79926/374238074075, a[8,1] = 42409705291266846/416462256407406875, a[3 ,1] = -594/625, a[9,3] = 13281373111234375/5150833217292744, a[9,8] = \+ -7500029126894375/132689679447323376, a[6,4] = -17371/11640, a[10,3] = -1462401427649331250/154787214582248211, a[7,5] = 2978030473272510357 7764751746216250/258912687002832625147067486467854423, a[10,9] = 55946 58687556280397846/1893189870520997940175\}:" }}}{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 0 338 "subs (e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));print( ``);\nfor ii from 7 to 10 do\n print(c[ii]=subs(e7,c[ii]));print(``) ; \n for jj to ii-1 do\n print(a[ii,jj]=subs(e7,a[ii,jj]));\n \+ end do:\n print(`_________________________________`);\nend do:print (``);\nfor ii to 10 do\n print(b[ii]=subs(e7,b[ii]));\nend do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"#]F(%!GF+F+F+ 7(#\"#F\"$D\"#!$%f\"$D'#\"$H(F2F+F+F+7(#\"#T\"$+\"#\"$^%\"&+;#\"\"!#\" %\"o\"\"%?VF+F+7(#\"#dF8#\"#>\"$g\"F<#\"$h$\"%/J#\"%\\K\"%+(*F+7(#\"#V F*#!#J\"$+#F<#\"'@4_\"'c?T#!&rt\"\"&S;\"#\"'B?8\"'+i5Q)pprint116\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"(#\"(5DF#\")@t(>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\"#\"A!)ptd*)fw p(*ox(owff#\"BL**=dJ^XfHG'HT^%f([" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"(\"\"$#\"E]i:(=$*f)fSYUk-$=.*yM\"F\\*>&)Q?#=3vvG4n?S<=K\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%#!C](y,X;6\\4x9'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"(#!@-X>25\\qh?&4\")3v$f)\"@Dce!)Q!4:ebaVX]8K" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"\"#!7DSS8,.7[-,9\"8Rc" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"$#\"2vVB6JP\"G8\"1WFHtRekJyokx4*)\"=#>$zuw!3ZvNA$4 ri" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!7](3]jN<%\\ K#z%\"9r->(*QW(R2hxf\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"(#!LX#p`'4m69\"*eY!H1G`\"\"LOF)*H.UZ;,_y!yasyQ6?C.7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!1vV*o7H+](\"3wLKZ %z'*oK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B________________________ _________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#5\"\"\"#\"571XMahKIRO\"6DMm*)[4'e!RC$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$#!4]7L\\wU,CY\"\"36#[Ae9sya\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\"1]F#=X!yNT\"0Vy=PS]u)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!=]*3SB&*G+ZOty$\\B \"=6`'eNHl5d(*f944\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"'#!5]PWA%3fg'oy\"5`f^*\\^CjYC&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#5\"\"(#\"U.(eVWsBa>^)eO**y1NW_/7\\8)z]J#\"T+Ueg2EJ!3$)yt:B N_fdr_R?/phJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!5v \\_#zut/tM$\"5s/(o&)G+V2H$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#5\"\"*#\"7YyR!GcvoeYf&\"7v,%z*4_q)*=$*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \"#\"-n!4+dr(\"/+_Y.i.9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"&#\"2+++G#zZIG\"2w&)*Rfh04oz$ePIe[ u\"R+%=w)QnX@#\\_'RLWw!fkxZ%)QA([#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\")#!.v=<7&=^\"/kki?Ow6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"*#\"-*4a=!o8\"-D>_&)y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"#5#\"-P/]EO5\".o#*3Nwr\"" }}}{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 52 "#------------------------ ---------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#------------------------------ -------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the embedded order 6 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2443 "e7 := \{c[10] = 1, c[9] = \+ 5/6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] = 27/125, c[2] = 1/50, b[3] = 0, b[2] = 0, a[3,2] = 729/625, a[6,2] = 0, a[7,2] = 0, a [8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[4,2] = 0, a[5,2] = 0, a[6,1] = - 31/200, a[10,5] = -2349378733647002895234008950/1090914599757106529355 865311, a[5,4] = 3249/9700, a[2,1] = 1/50, a[10,1] = 36393032615434450 612/324390586094889663425, a[4,3] = 1681/4320, a[10,6] = -786866059084 22443750/52446632451499515953, a[9,6] = -4792324941735635008750/159776 107397443897190271, a[7,4] = -1717046972617147709491116450178750/74678 94926932728111586543618014237, a[9,5] = 8909776468783164583973193125/6 271093223575470807674793192, a[6,5] = 132023/106200, a[4,1] = 451/2160 0, a[5,1] = 19/160, b[1] = 771570009067/14036203465200, b[4] = 0, b[10 ] = 103626500437/1717635089268, b[8] = -5118512171875/11763620626464, \+ a[9,1] = -1401024812030113404025/19887564677841032175639, b[9] = 13680 1854099/127885521925, a[8,7] = -8593750881095206170491007194502/321350 4543545558150903880585625, a[5,3] = 361/3104, b[6] = -296881060859375/ 515060733835389, a[9,7] = -1532806290465891141166096531902118541769245 /1203242011387872547807852011647420329982736, a[9,4] = -50491693720625 /29100752640072, a[10,8] = -33473047374792524975/32907430028856870472, a[10,4] = 4135780451822750/874504037187843, a[7,3] = 3478903183026442 46405985993187156250/1321817402067092875750818220388519949, a[8,3] = 3 247095172038/883201854817, a[8,6] = -6468694559114760/6194593900608963 7, c[7] = 2272510/11977321, a[7,1] = 25959766877768976976598957736980/ 487594514129628295945513157189933, a[8,5] = 435669225629732566638/3939 65828849029186615, a[7,6] = -302662548054389051180423185000/2566286916 4717278733974376694207, b[5] = 28304779228000000/53707434325074117, c[ 6] = 43/50, a[10,7] = 231507981349120452443506789936588511954237244435 8703/316169042039527157595235231573788308031260760584200, b[7] = 74485 8303758379680905615939985761920312207508379/24872238844777645907644333 96524922145673887618400, a[6,3] = 520921/412056, a[8,4] = -51850927992 6/374238074075, a[8,1] = 42409705291266846/416462256407406875, a[3,1] \+ = -594/625, a[9,3] = 13281373111234375/5150833217292744, a[9,8] = -750 0029126894375/132689679447323376, a[6,4] = -17371/11640, a[10,3] = -14 62401427649331250/154787214582248211, a[7,5] = 29780304732725103577764 751746216250/258912687002832625147067486467854423, a[10,9] = 559465868 7556280397846/1893189870520997940175\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We now turn our attent ion to the embedded order 6 scheme and introduce a new row correspondi ng to the node " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The linking coefficient s and weights can be chosen so as to form an 11 stage order 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We us e the order 6 quadrature conditions which are given in abreviated form as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "subs(b=`b*` ,QuadratureConditions(6)):\nListTools[Enumerate](%):\nlinalg[augment]( linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[de lcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7%\" \"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"$F)/*&F, F()F2F/F(#F(F57%\"\"%F)/*&F,F()F2F5F(#F(F;7%\"\"&F)/*&F,F()F2F;F(#F(FA 7%\"\"'F)/*&F,F()F2FAF(#F(FGQ(pprint86\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We incorporate the row sum condition for the new tenth ro w" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j],j = \+ 1 .. 10) = c[11];" "6#/-%$SumG6$&%\"aG6$\"#6%\"jG/F+;\"\"\"\"#5&%\"cG6 #F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "together with the stage-order equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j],j = 2 .. 10) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"# 6%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"#5*&F-F-F3!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[11]^2;" "6#*$&%\"cG6#\"#6\"\"#" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j]^2,j = 2 .. 10) = 1/3;" "6#/-%$SumG6 $*&&%\"aG6$\"#6%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"#5*&F-F-\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^3;" "6#*$&%\"cG6#\"#6\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 50 "which ensure that the \+ tenth row has stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 53 "We also i ncorporate the column simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,1],i=2..11)=`b*`[1]" "6 #/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#6&F)6#F," } {TEXT -1 2 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b *`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F +;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 6 " , \+ " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" }{TEXT -1 10 ", 6, 7, 8." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(6,11,'expanded')): \nSO_eqs2 := [add(a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2 *c[11]^2,\n add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3]:\n`simp_eqs*` : = [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..1 1)=`b*`[j]*(1-c[j]),j=[$5..8])]:\n`cdns*` := [op(SO_eqs2),op(`Qeqs*`), op(`simp_eqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We specify that " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"c G6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&% \"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,10] = 0; " "6#/&%\"aG6$\"#6\"#5\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] =0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] \+ = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4 ]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[10 ]=0" "6#/&%#b*G6#\"#5\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*` [11] = 16/243;" "6#/&%#b*G6#\"#6*&\"#;\"\"\"\"$V#!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We also set " }{XPPEDIT 18 0 "b[11] \+ = 0;" "6#/&%\"bG6#\"#6\"\"!" }{TEXT -1 66 ", so that the order 7 schem e can be regarded as a 11 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 14 equations for the 14 unkn own coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "e8 := \+ \{c[11]=1,b[11]=0,a[11,2]=0,a[11,10]=0,`b*`[2]=0,`b*`[3]=0,`b*`[4]=0,` b*`[10]=0,`b*`[11]=16/243\}:\ne9 := `union`(e7,e8):\n`eqns*` := subs(e 9,`cdns*`):\nnops(%);\nindets(`eqns*`);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<0&%#b*G6#\"\"& &F%6#\"\"'&F%6#\"\"(&F%6#\"\")&F%6#\"\"*&F%6#\"\"\"&%\"aG6$\"#6F6&F86$ F:\"\"$&F86$F:\"\"%&F86$F:F'&F86$F:F*&F86$F:F-&F86$F:F0&F86$F:F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e10 := solve(\{op(`eqns*` )\}):\ninfolevel[solve] := 0:\ne11 := `union`(e9,e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3347 "e11 := \{c[10] = 1, c [9] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] = 27/125, \+ c[2] = 1/50, b[3] = 0, b[2] = 0, a[3,2] = 729/625, a[6,2] = 0, a[7,2] \+ = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[4,2] = 0, a[5,2] = 0, a[6, 1] = -31/200, a[10,5] = -2349378733647002895234008950/1090914599757106 529355865311, a[5,4] = 3249/9700, a[2,1] = 1/50, a[10,1] = 36393032615 434450612/324390586094889663425, a[4,3] = 1681/4320, a[10,6] = -786866 05908422443750/52446632451499515953, a[9,6] = -4792324941735635008750/ 159776107397443897190271, a[7,4] = -1717046972617147709491116450178750 /7467894926932728111586543618014237, a[11,3] = -5122833329940625/50872 4268374592, a[9,5] = 8909776468783164583973193125/62710932235754708076 74793192, a[6,5] = 132023/106200, a[4,1] = 451/21600, a[5,1] = 19/160, b[1] = 771570009067/14036203465200, `b*`[4] = 0, b[4] = 0, a[11,1] = \+ 2508607706701842363083/197875357745688550590720, b[10] = 103626500437/ 1717635089268, b[8] = -5118512171875/11763620626464, a[9,1] = -1401024 812030113404025/19887564677841032175639, b[9] = 136801854099/127885521 925, a[8,7] = -8593750881095206170491007194502/32135045435455581509038 80585625, a[5,3] = 361/3104, b[6] = -296881060859375/515060733835389, \+ a[9,7] = -1532806290465891141166096531902118541769245/1203242011387872 547807852011647420329982736, a[9,4] = -50491693720625/29100752640072, \+ a[10,8] = -33473047374792524975/32907430028856870472, a[10,4] = 413578 0451822750/874504037187843, a[7,3] = 347890318302644246405985993187156 250/1321817402067092875750818220388519949, a[8,3] = 3247095172038/8832 01854817, `b*`[9] = 154806770859/124231649870, a[8,6] = -6468694559114 760/61945939006089637, c[7] = 2272510/11977321, a[7,1] = 2595976687776 8976976598957736980/487594514129628295945513157189933, a[8,5] = 435669 225629732566638/393965828849029186615, a[7,6] = -302662548054389051180 423185000/25662869164717278733974376694207, b[5] = 28304779228000000/5 3707434325074117, c[6] = 43/50, a[11,7] = 6060302382461817770511989205 09497430523044409408159/7475205014164099896781367446051319734865328802 4576, a[10,7] = 2315079813491204524435067899365885119542372444358703/3 16169042039527157595235231573788308031260760584200, b[7] = 74485830375 8379680905615939985761920312207508379/24872238844777645907644333965249 22145673887618400, a[6,3] = 520921/412056, a[8,4] = -518509279926/3742 38074075, a[8,1] = 42409705291266846/416462256407406875, a[3,1] = -594 /625, a[9,3] = 13281373111234375/5150833217292744, a[9,8] = -750002912 6894375/132689679447323376, a[6,4] = -17371/11640, a[10,3] = -14624014 27649331250/154787214582248211, a[11,8] = -1922750201834125/1941504226 023936, a[7,5] = 29780304732725103577764751746216250/25891268700283262 5147067486467854423, `b*`[3] = 0, `b*`[11] = 16/243, `b*`[5] = 7786773 134600000/14452831163890377, a[11,9] = 12539348439579/3975412795840, c [11] = 1, a[11,2] = 0, a[11,10] = 0, `b*`[2] = 0, `b*`[10] = 0, b[11] \+ = 0, a[11,4] = 13293920580875/2874148408896, a[10,9] = 559465868755628 0397846/1893189870520997940175, a[11,5] = -599188464780493707137440161 875/277270064173229869784600732736, a[11,6] = -36014650553489237628498 75/2146128454918752594358208, `b*`[6] = -408698637296875/5676179515736 94, `b*`[1] = 448234490819/8120946290580, `b*`[7] = 442670515036915263 8325381078278067803359/14828075230102658203818343670586143438076, `b*` [8] = -5004542378125/10330679593521\}:" }}}{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 0 137 "subs(e11,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[ j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'++?!\"(F(%!GF+F+F+F+F+F+ F+F+F+7.$\"'+g@!\"'$!'+/&*F/$\"'Sm6!\"&F+F+F+F+F+F+F+F+F+7.$\"'++TF/$ \"''z3#F*$\"\"!F;$\"'?\"*QF/F+F+F+F+F+F+F+F+7.$\"'++dF/$\"'](=\"F/F:$ \"'-j6F/$\"'[\\LF/F+F+F+F+F+F+F+7.$\"'++')F/$!'+]:F/F:$\"'?k7F4$!'N#\\ \"F4$\"':V7F4F+F+F+F+F+F+7.$\"'M(*=F/$\"'0C`F*F:$\"'\">j#F/$!'C*H#F/$ \"'@]6F/$!'Qz6F*F+F+F+F+F+7.$\"'++sF/$\"'L=5F/F:$\"']wOF4$!'^&Q\"F4$\" ''e5\"F4$!'DW5F/$!'EuEF4F+F+F+F+7.$\"'LL$)F/$!'tWqF*F:$\"'\\yDF4$!'1N< F4$\"'x?9F4$!'S**HF*$!'!RF\"F4$!'J_cF*F+F+F+7.$\"\"\"F;$\"'*=7\"F/F:$! '#yW*F4$\"'HHZF4$!'f`@F4$!'K+:F4$\"'GAtF4$!'><5F4$\"':bHF4F+F+7.Fjp$\" 'xn7F*F:$!'+25!\"%$\"'MDYF4$!'.h@F4$!'7y;F4$\"'@2\")F4$!'S.**F/$\"'BaJ F4F:F+7.%\"bG$\"'+(\\&F*F:F:F:$\"'=q_F/$!'+kdF/$\"'u%*HF/$!'9^VF/$\"'s p5F4$\"'4LgF*F:7.%#b*G$\"'\\>bF*F:F:F:$\"'r(Q&F/$!'C+sF/$\"'N&)HF/$!'N W[F/$\"'6Y7F4F:$\"'O%e'F*Q)pprint266\"" }}}{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 0 459 "subs(e11,matrix([seq ([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..5)]));print(``);\nfor ii fr om 6 to 11 do\n print(c[ii]=subs(e11,c[ii]));print(``); \n for jj \+ to ii-1 do\n print(a[ii,jj]=subs(e11,a[ii,jj]));\n end do:\n \+ print(`_________________________________`);\nend do:print(``);\nfor ii to 11 do\n print(b[ii]=subs(e11,b[ii]));\nend do:\nprint(`_________ ________________________`);print(``);\nfor ii to 11 do\n print(`b*`[ ii]=subs(e11,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7(#\"\"\"\"#]F(%!GF+F+F+7(#\"#F\"$D\"#!$%f\"$D'#\"$H(F2F+ F+F+7(#\"#T\"$+\"#\"$^%\"&+;#\"\"!#\"%\"o\"\"%?VF+F+7(#\"#dF8#\"#>\"$g \"F<#\"$h$\"%/J#\"%\\K\"%+(*F+Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'# \"#V\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\"#!#J\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#\"'@4_\"'c?T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%#!&rt\"\"&S;\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&#\"'B?8\"'+i5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"(5DF#\")@t(>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"\"#\"A!)ptd*)fwp(*ox(owff#\"BL**=dJ^XfHG'HT^%f([" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"E]i:(=$*f)fSYUk-$=.*yM\"F\\*>&)Q? #=3vvG4n?S<=K\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"% #!C](y,X;6\\4x9'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!@-X>25\\qh?&4\")3v$f)\"@Dce!)Q!4:e baVX]8K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________ ________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"\"&\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!7DSS8,.7[-,9\"8Rc" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#\"2vVB6JP\"G8\"1WFHtRekJyokx4 *)\"=#>$zuw!3ZvNA$4ri" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"'#!7](3]jN<%\\K#z%\"9r->(*QW(R2hxf\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#!LX#p`'4m69\"*eY!H1G`\"\"LOF) *H.UZ;,_y!yasyQ6?C.7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"* \"\")#!1vV*o7H+](\"3wLKZ%z'*oK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B _________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"571XMahKIRO\"6DMm*)[4' e!RC$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$#!4]7L\\wU,CY\"\"36# [Ae9sya\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\"1]F# =X!yNT\"0Vy=PS]u)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" &#!=]*3SB&*G+ZOty$\\B\"=6`'eNHl5d(*f944\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!5]PWA%3fg'oy\"5`f^*\\^CjYC&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"U.(eVWsBa>^)eO**y 1NW_/7\\8)z]J#\"T+Ueg2EJ!3$)yt:BN_fdr_R?/phJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!5v\\_#zut/tM$\"5s/(o&)G+V2H$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"7YyR!GcvoeYf&\"7v ,%z*4_q)*=$*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___________________ ______________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#\"7$3jB%=q1xg3D\"9?2f]&)oXxNvy>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$#!1D1%*HL$G7&\"0#fu$oUs3&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#\"/v3e?RH8\".'*)3%[T(G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!?v=;Su82P\\!yk%)= *f\"?OFt+Yyp)HK^qx\"=Y#Q-.1' \"SwX-)G`'[t>80YuO\"y'*)*4kT,0_Z(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"\")#!1DT$=?]F#>\"1OR-EU]T>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"/z&R%[$RD\"\".Sez7a(R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"-n!4+dr (\"/+_Y.i.9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"&#\"2+++G#zZIG\"2w&)*Rfh04oz$ePIe[u\"R+% =w)QnX@#\\_'RLWw!fkxZ%)QA([#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\")#!.v=<7&=^\"/kki?Ow6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"*#\"-*4a=!o8\"-D>_&)y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#5#\"-P/]EO5\".o#*3Nwr\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________ ________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"->3\\M#[%\".!e!HY47)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&# \"1++gMJx'y(\"2x.*Q;JGX9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"'#!0voHP')p3%\"0%pt:&zhn&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b *G6#\"\"(#\"IfL!y1y#y5QD$QE:p.:0nU%\"Jw!QM9'eqOM=Q?eE5I_2G[\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!.D\"yBa/]\"/@Nfz1L5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"-f3x1[:\"-q)\\;BC\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"#;\"$V#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11 eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'e xpanded'))]:\n`RK6_11eqs*` := subs(b=`b*`,OrderConditions(6,11,'expand ed')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e11,RK7_11eq s):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(e11,`RK6_11eqs*`):\nmap( u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q \"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------- ------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 62 "#---------------------------------------- ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "longer c onstruction of the order 7 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "The scheme will be constructed so that st age 4 has stage-order 2 and stages 5 to 10 have " }{TEXT 260 13 "stage -order 3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "We start by \+ determining the nodes and weights of the scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We set up a system of equ ations that consists of the 7 order 7 quadrature conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 10) = 1" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"#5F-" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 1 .. 10) = 1/k;" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;F,\"#5*&F,F,F2 F3" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 7 " . . 7," }}{PARA 0 "" 0 "" {TEXT -1 35 "and the relation b etween the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c [5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^2*c[7]- 7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2 = 0;" "6#/,>** \"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\"'F'F'**\"$0\"F'*$&F)6#F. \"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F)6#F.F'&F)6#F1F7!\"\"**F =F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6#F+F'F'*(F+F'*$&F)6#F.F7F '&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F'&F)6#F.F'FD*(\"#9F'*$&F )6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD*(FenF'&F)6#F.F'&F)6#F1F 'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F'&F)6#F.F7F'\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2]=1/50" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#]!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=27/125" "6#/&%\"cG6#\"\"$*&\"#F \"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]=41/100" "6#/ &%\"cG6#\"\"%*&\"#T\"\"\"\"$+\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=57/100" "6#/&%\"cG6#\"\"&*&\"#d\"\"\"\"$+\"!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "c[6]=43/50" "6#/&%\"cG6#\"\"'*&\"#V\"\"\"\"#]! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=18/25" "6#/&%\"cG6#\"\")* &\"#=\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9]=5/6" "6#/ &%\"cG6#\"\"**&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]=1" "6#/&%\"cG6#\"#5\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6# \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\" $\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "Qeqs := QuadratureConditions(7,10,'expande d'):\nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5] *c[6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5 ]+14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^ 2=0:\ncdns1 := [op(Qeqs),node_eq]:\ne1 := \{c[2]=1/50,c[3]=27/125,c[4] =41/100,c[5]=57/100,c[6]=43/50,c[8]=18/25,c[9]=5/6,c[10]=1,b[2]=0,b[3] =0,b[4]=0\}:\neqns1 := subs(e1,cdns1):\nnops(%);\nindets(eqns1);\nnops (%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"cG6#\"\"(&%\"bG6#\"\"'&F)6#\"\"&&F)F&&F)6#\"\")&F )6#\"\"*&F)6#\"#5&F)6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\") " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We h ave 8 equations and 8 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e2 := solve(\{op(eqns1)\}): \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 0 507 "e3 := \{c[10] = 1, c[9] = 5 /6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] = 27/125, c[2] = \+ 1/50, b[3] = 0, b[2] = 0, b[1] = 771570009067/14036203465200, b[4] = 0 , b[10] = 103626500437/1717635089268, b[8] = -5118512171875/1176362062 6464, b[9] = 136801854099/127885521925, b[6] = -296881060859375/515060 733835389, c[7] = 2272510/11977321, b[5] = 28304779228000000/537074343 25074117, c[6] = 43/50, b[7] = 744858303758379680905615939985761920312 207508379/2487223884477764590764433396524922145673887618400\}:" }}} {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 38 "We now have all the nodes and weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "seq(c[i]=subs(e3,c[i]),i=2..10);\nseq(b[i]=subs(e3,b[ i]),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%\"cG6#\"\"##\"\"\" \"#]/&F%6#\"\"$#\"#F\"$D\"/&F%6#\"\"%#\"#T\"$+\"/&F%6#\"\"&#\"#dF8/&F% 6#\"\"'#\"#VF*/&F%6#\"\"(#\"(5DF#\")@t(>\"/&F%6#\"\")#\"#=\"#D/&F%6#\" \"*#Fw&)*Rfh04oz$ePIe[u\"R+%=w)QnX@#\\_'RLWw!fkxZ%)QA([ #/&F%6#\"\")#!.v=<7&=^\"/kki?Ow6/&F%6#\"\"*#\"-*4a=!o8\"-D>_&)y7/&F%6# \"#5#\"-P/]EO5\".o#*3Nwr\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We construct a system of equations that incorpo rate the row-sum conditions," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[i,j],j = 1 .. i-1) = c[i]" "6#/-%$SumG6$&%\"aG6$% \"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . 10, " }}{PARA 0 "" 0 "" {TEXT -1 29 " the stage order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( a[i,j]*c[j],j = 2 .. i-1) = 1/2" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\" \"&%\"cG6#F,F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-F3F5" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG\"\"#" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "i = 3;" "6#/%\"iG\"\"$" }{TEXT -1 9 " . . 10, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j]^2,j = 2 .. i-1) = 1/3" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$&%\"cG6#F, \"\"#F-/F,;F2,&F+F-F-!\"\"*&F-F-\"\"$F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^3" "6#*$&%\"cG6#%\"iG\"\"$" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i = 5;" "6#/%\"iG\"\"&" }{TEXT -1 8 " . . 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "together with the column \+ simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,1],i=2..10)=b[1]" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#5&F)6#F," }{TEXT -1 6 ", " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 10) = b[j]*(1-c[j]);" "6#/-% $SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#5*&&F)6# F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j = 6; " "6#/%\"jG\"\"'" }{TEXT -1 8 " . . 9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "and the further simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 5 .. 10) = 0; " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+; \"\"&\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a [i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F +\"\"#F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "S O_eqs := [op(RowSumConditions(10,'expanded')),op(StageOrderConditions( 2,3..10,'expanded')),\n op(StageOrderConditions(3,5..10,'expanded'))] :\nsimp_eqs := [add(b[i]*a[i,1],i=2..10)=b[1],seq(add(b[i]*a[i,j],i=j+ 1..10)=b[j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5 ..10)=0,add(b[i]*c[i]^2*a[i,3],i=5..10)=0]:\ncdns2 := [op(simp_eqs),op (simp_eqs2),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 42 "We specify the zero linking coefficients: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"a G6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] = 0" "6#/ &%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2] = 0 " "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2 ] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "and the linking coefficient " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[6,1]=-31/200" "6#/&%\"aG6 $\"\"'\"\"\",$*&\"#JF(\"$+#!\"\"F-" }{TEXT -1 1 "." }}{PARA 257 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "e4 := \{se q(a[i,2]=0,i=4..10),a[6,1]=-31/200\}:\ne5 := `union`(e3,e4):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eqns2 := subs(e5,cdns2):\nnops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 141 "e6 := solve(\{op(eqns2)\},indets(eqns2) min us \{a[7,6],a[8,6],a[8,7],a[9,5],a[9,6],a[9,7],a[9,8]\}):\ninfolevel[s olve] := 0:\ne7 := `union`(e5,e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6565 "e7 := \{c[10] = 1, c[9] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] = 27/125, c[2] = 1/50, b[3] = 0, b [2] = 0, a[9,4] = 477500/322137-18144/3977*a[9,8]-10089/3977*a[9,5]-27 692/3977*a[9,6]+35745597848668000/570525380326412057*a[9,7], a[9,3] = \+ -374707300977568750/375711835824710379*a[9,7]-2046875/1697112+1550/291 *a[9,8]+1900/873*a[9,5]+5375/582*a[9,6], a[7,3] = -3312812295985745483 84950631227868490906250/1000479626172376396377167058557409583838253+33 280452084295878204957164858184257700/397880917806591363849742571756825 83397*a[9,8]-893937517979914757475868254239234807113284736162500/57078 49181700463494779734948110893595352446136708277*a[9,7]+135984642925079 93245036260909795718200/39788091780659136384974257175682583397*a[9,5]+ 57704009662287208177949922939593672625/3978809178065913638497425717568 2583397*a[9,6], a[8,3] = -21359605601871742/7013505929101797+959493276 /101419805*a[9,8]-5154539580732407777100/2909860337956024499401*a[9,7] +392051016/101419805*a[9,5]+332727507/20283961*a[9,6], a[7,4] = 392160 01615592693655754811926735764541250/7724983930897821516283210809953633 3215383-1590472805108500019414902908572625675483/163131176300702459178 3944544202985919277*a[9,8]+4272127398426012625977174387009303143194387 7541205875/234021816449719003285969132872546637409450291605039357*a[9, 7]-649870608538956997180282908879137372778/163131176300702459178394454 4202985919277*a[9,5]-11030698487042822715296907269132726458995/6525247 052028098367135778176811943677108*a[9,6]+1247/328*a[7,6], a[10,8] = -3 3567952125337682387076/1893189870520997940175*a[9,8]-19935909549337551 4375/98722290086570611416, a[8,5] = -549684895162665/457926503961973+2 158859871/496643375*a[9,8]-24416240119258773681/29998560185113654633*a [9,7]+46427094/26139125*a[9,5]+2994547563/397314700*a[9,6]-645/152*a[8 ,6]+394428737871125/860737310022246*a[8,7], a[7,5] = -7154519545019244 12208879890745740891250/3438074316743561499577893672018589635183-645/1 52*a[7,6]+2995240687586629038446144837236583193/7793543750850758673345 473055030609119*a[9,8]-42344408746417014827804285727121648757997698028 75/58843805996911994791543659258875191704664393161941*a[9,7]+644137782 27669441687013867467453402/410186513202671509123445950264768901*a[9,5] +20773443478423394944061972258253722145/311741750034030346933818922201 22436476*a[9,6], a[9,1] = -805/246*a[9,6]-10374402790173637/1588060336 99104387*a[9,7]+399655/717336-217/123*a[9,8]-236/369*a[9,5], a[8,4] = \+ 158620405107309993/40614935465137525+1247/328*a[8,6]-12937862368693375 /11763409903637362*a[8,7]-1146354591501/103955300125*a[9,8]+2463354465 63201767667609/119304273856197004475441*a[9,7]-468402951366/1039553001 25*a[9,5]-1590104755953/83164240100*a[9,6], a[3,2] = 729/625, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[4,2] = 0, a[5,2 ] = 0, a[6,1] = -31/200, a[5,4] = 3249/9700, a[2,1] = 1/50, a[10,7] = \+ 711996769619198265625/98722290086570611416*a[8,7]-33567952125337682387 076/1893189870520997940175*a[9,7]+127164413385140007828858885945081413 4722628077964123/316169042039527157595235231573788308031260760584200, \+ a[4,3] = 1681/4320, a[6,5] = 132023/106200, a[4,1] = 451/21600, a[5,1] = 19/160, b[1] = 771570009067/14036203465200, b[4] = 0, b[10] = 10362 6500437/1717635089268, b[8] = -5118512171875/11763620626464, b[9] = 13 6801854099/127885521925, a[5,3] = 361/3104, b[6] = -296881060859375/51 5060733835389, a[10,6] = -15694164460189059876681327855500003660978212 20154553/316169042039527157595235231573788308031260760584200*a[7,6]+71 1996769619198265625/98722290086570611416*a[8,6]-3356795212533768238707 6/1893189870520997940175*a[9,6]-1613457767667049893750/120627254638448 8866919, c[7] = 2272510/11977321, a[10,3] = -24111630789068970/7940898 72848731*a[9,8]+5631063244194925000/464361643746744633+647656207536894 626051087216250/113917130178620680194167144971*a[9,7]-9852064193383020 /794089872848731*a[9,5]-83612913220158525/1588179745697462*a[9,6], b[5 ] = 28304779228000000/53707434325074117, c[6] = 43/50, a[7,1] = 158618 4331355201100589962194783846582560/71675108637196282109844224009879072 05551-435/779*a[7,6]-78541866918938272563698909065314848172/3195352937 84881105607164395256254973879*a[9,8]+111036449601715727770686793684452 323409860630386500/241259604587339178645329002961388285989124011963958 1*a[9,7]-1689072406858887582015030302479889208/16817647041309531874061 283960855524941*a[9,5]-136181462802997811299961818137441067395/3195352 93784881105607164395256254973879*a[9,6], a[8,1] = 1012325975628712/954 660338768181-435/779*a[8,6]-6324110428774043/17645114855456043*a[8,7]- 56610103284/20362378375*a[9,8]+640248074238341176524/12299409675896598 39953*a[9,7]-1217421576/1071704125*a[9,5]-19630922913/4072475675*a[9,6 ], b[7] = 744858303758379680905615939985761920312207508379/24872238844 77764590764433396524922145673887618400, a[6,3] = 520921/412056, a[3,1] = -594/625, a[6,4] = -17371/11640, a[10,1] = -70905971989378447839230 29/1436401515228171429645900+13653923080364482092712755234285003185051 0446153446111/49259136749758331153337649079196218391270426499018360*a[ 7,6]+734495786468105866555554/58991796365434295815853*a[9,8]+127216548 4813667689771460512492276866/271589859418934661233563424904522175*a[9, 7]+6118650393785244/1678231174577215*a[9,5]+14967478017193243081115079 /589917963654342958158530*a[9,6]-103239531594783748515625/256348879924 79502097688*a[8,6]-4502746196002201456184929520619171875/1741966147371 187940840844106021986888*a[8,7], a[10,9] = 5594658687556280397846/1893 189870520997940175, a[10,5] = 69897131246445279371461624525/8727316798 056852234846922488+280831987209218528797583354767578125/84973958408350 631260528980781560336*a[8,7]+20245472153643887240918912933595004722661 8937399937337/9611538878001625590895151039843164564150327121759680*a[7 ,6]-153079305468127627109375/5001929364386244311744*a[8,6]+21181393888 38796385345283/71941215079797921726650*a[9,8]-119778501186351617200588 069359352065/21727188753514772898685073992361774*a[9,7]-23333836247486 1/40932467672615*a[9,5]+2938064313550588534511199/57552972063838337381 320*a[9,6], a[10,4] = -5111252825043479425/286837324197612504-92117162 12287471691295267097427734375/1161310764914125293893896070681324592*a[ 8,7]-1957062308185575766622161583580850456523983061532727591/103703445 788964907691237155956202565034253529471617600*a[7,6]+88785997171514023 7234375/32380911148395160544448*a[8,6]+18676366250170587568773723/3011 686446024803523230390*a[9,8]+27132994563098745846596421378798580545/21 07537309090932971172452177259092078*a[9,7]+2354150739008872629/1627884 23933989855*a[9,5]-364050910438003753823964609/60233728920496070464607 800*a[9,6]\}:" }}}{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 20 "indets(map(rhs,e7));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<)&%\"aG6$\"\"*\"\")&F%6$\"\"(\"\"'&F%6$F(F,&F%6 $F(F+&F%6$F'\"\"&&F%6$F'F,&F%6$F'F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We have numerical values for the linkin g coefficients in rows 2 to 6 of the Butcher tableau." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(6-i)],i=2..6)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7'7(#\"\"\"\"#]F(%!GF+F+F+7(#\"#F\"$D\"#!$%f\"$D'#\"$H(F2F+F+F+7( #\"#T\"$+\"#\"$^%\"&+;#\"\"!#\"%\"o\"\"%?VF+F+7(#\"#dF8#\"#>\"$g\"F<# \"$h$\"%/J#\"%\\K\"%+(*F+7(#\"#VF*#!#J\"$+#F<#\"'@4_\"'c?T#!&rt\"\"&S; \"#\"'B?8\"'+i5Q)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "Other examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "a[7,4]=subs(e7,a[7,4]);\n``;\na[8,5]=subs(e7,a[8,5 ]);\n``;\na[10,9]=subs(e7,a[10,9]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"%,.#\"J]7akdtE>\"[vbOp#f:;+;#R\"J$Q:KLO&*43@$G;:#y* 3$R)\\s(\"\"\"*&#\"I$[vcis&3H!\\T>+]3^!GZ!f\"\"Ix#>f)H?WX%Ry\"fCq+jVJI4qQur(fi7gU)RF@F%\"Wd$R]g\" H]%4ujYD(G8pfG.!>(\\k\"=-M#F-&F%6$F4F'F-F-*&#\"HyFPP\"z)3HG!=(*p&*Q&31 ()\\'F1F-&F%6$F4\"\"&F-F6*&#\"J&**eksK\"ps!pH:F#G/([)pI5\"\"I3rnV>\"o< yd8n$)4G?0Z__'F-&F%6$F4\"\"'F-F6*&#\"%Z7\"$G$F-&F%6$F'FIF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"&,0#\"0lE;&*[o\\&\"0t>'R]EzX!\"\"*&#\"+r)f)e@\"*vLk'\\\" \"\"&F%6$\"\"*F'F2F2*&#\"5\"ot(e#>,C;W#\"5LYl8^=g&)**HF2&F%6$F5\"\"(F2 F-*&#\")%4Fk%\")D\"Rh#F2&F%6$F5F(F2F2*&#\"+jva%*H\"*+ZJ(RF2&F%6$F5\"\" 'F2F2*&#\"$X'\"$_\"F2&F%6$F'FIF2F-*&#\"0D6(ytGWR\"0YA-5ttg)F2&F%6$F'F< F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"7YyR!GcvoeYf&\"7v,%z*4_q)*=$*=" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We can \+ find some simple order conditions that are not yet satisfied and deter mine which paramers are related by them." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_10 := SimpleOrderConditions(7,10,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 54 do\n eq := simplify(subs(e7,SO7_10 [ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets(lhs(eq))) \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#h<(&%\"aG6 $\"\"*\"\")&F&6$\"\"(\"\"'&F&6$F)F-&F&6$F)F,&F&6$F(F-&F&6$F(F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#f<(&%\"aG6$\"\"*\"\")&F&6$\"\"(\"\" '&F&6$F)F-&F&6$F)F,&F&6$F(F-&F&6$F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#e<(&%\"aG6$\"\"*\"\")&F&6$\"\"(\"\"'&F&6$F)F-&F&6$F)F,&F&6$F(F-& F&6$F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<)&%\"aG6$\"\"*\"\")&F &6$\"\"(\"\"'&F&6$F)F-&F&6$F)F,&F&6$F(\"\"&&F&6$F(F-&F&6$F(F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#a<(&%\"aG6$\"\"*\"\")&F&6$\"\"(\"\" '&F&6$F)F-&F&6$F)F,&F&6$F(F-&F&6$F(F," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations t hat consists of the simple order comditions given in abreviated form a s follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 167 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]], i=[54,59,61])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` ` ]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"#a%#~~G/*&%\"bG\"\"\"-%!G6#*(%\"aGF-% \"cGF--F/6#*&)F3\"\"$F-F2F-F-F-#F-\"$o\"7%\"#fF)/*(F,F-)F3\"\"#F-F4F-# F-\"#G7%\"#hF)/*(F,F-F3F--F/6#*&)F3\"\"%F-F2F-F-#F-\"#NQ)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "cdns3 := [seq(SO7_10[i],i=[54,59,61])]:\neqns3 := simplify(su bs(e7,cdns3)):\nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"aG6$\" \"*\"\")&F%6$\"\"(\"\"'&F%6$F(F,&F%6$F(F+&F%6$F'F,&F%6$F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 3 equations f or the linking coefficients " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\" \")\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,7]" "6#&%\"aG6$\"\"*\" \"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,8]" "6#&%\"aG6$\"\"*\" \")" }{TEXT -1 30 " in terms of the parameters " }{XPPEDIT 18 0 "a[7 ,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,6]" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,6]" " 6#&%\"aG6$\"\"*\"\"'" }{TEXT -1 89 ", and subsitute back into the expr essions previously obtained for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolev el[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "e8 := s olve(\{op(eqns3)\},\{a[8,7],a[9,7],a[9,8]\}):\ninfolevel[solve] := 0: \ne9 := `union`(map(u_->lhs(u_)=simplify(subs(e8,rhs(u_))),e7),e8):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8747 "e9 := \{c[ 10] = 1, c[9] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4] = 41/100, c[3] \+ = 27/125, c[2] = 1/50, b[3] = 0, b[2] = 0, a[3,2] = 729/625, a[6,2] = \+ 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[4,2] = 0, a[5,2] = 0, a[6,1] = -31/200, a[5,4] = 3249/9700, a[2,1] = 1/50, a[4,3] = 16 81/4320, a[6,5] = 132023/106200, a[4,1] = 451/21600, a[5,1] = 19/160, \+ b[1] = 771570009067/14036203465200, b[4] = 0, b[10] = 103626500437/171 7635089268, b[8] = -5118512171875/11763620626464, b[9] = 136801854099/ 127885521925, a[5,3] = 361/3104, b[6] = -296881060859375/5150607338353 89, a[10,6] = -1569416446018905987668132785550000366097821220154553/31 6169042039527157595235231573788308031260760584200*a[7,6]+7119967696191 98265625/98722290086570611416*a[8,6]-33567952125337682387076/189318987 0520997940175*a[9,6]-1613457767667049893750/1206272546384488866919, a[ 9,8] = -500544930841/118131596832*a[9,6]-61264306165084209987404481432 71551631263493587/1062340411043989126646573817622099058958961152*a[7,6 ]+15128656645422356219471828125/5224947905412840545053714944*a[8,6]+18 44671514591501860625/36361398317159597882328, a[9,7] = 161676075737718 8560240385731735041/867125550783556173144390510975200*a[9,6]+337716392 454169828806068331564017262170866354781837988249037167052895206622893/ 5912384053820196728892078008668942882938599255002909444504651138862303 500800*a[7,6]-1039775128787621089590899413439698394136310448125/170457 202035635253163492429938569724137172616704*a[8,6]-79407055271411247898 57645945416756356103865/6722048374749067114522615922086864221082872, a [8,7] = 6307871900509319105179508023367807990144609420098865978073291/ 78971543310578674083969005793484472022820379910423327762500*a[7,6]-192 83596263509636971887903/2730604796193727087006700*a[8,6]-2347211178153 53366134620879503433126/95041110424105351204486631545800625, c[7] = 22 72510/11977321, a[9,4] = 36624875320876084735690931247500/311217216817 85205036652159166151+184948767410650984744640/14813931446213043347479* a[9,6]+754744224048126221819480857955683912218009580033374488557682613 763/25251721615696537088671112473685301358183359988823543895935423604* a[7,6]-247380901456813373317766063045863796875/18200516805116965867987 545691385623788*a[8,6]-10089/3977*a[9,5], a[9,3] = -6041059510150/3976 13347281*a[9,6]-213947890191982352506726367714435353149182463152748230 6425/24399679270589811331307776595456830026253466521029581616*a[7,6]+3 78216416135558905486795703125/17586395864897719215799113552*a[8,6]+161 29378516717539598390625/66578553582001665554298456+1900/873*a[9,5], a[ 7,3] = -933564523142497901779801260084060142765625/9004316635551387567 394503527016686254544277-94929822175661026661424476625029531100/397880 91780659136384974257175682583397*a[9,6]-5215868472161871373925/3787959 64806621737208*a[7,6]+918786055467195218349440809037832877528066796875 /272052967865373355280205682463294096555598010008*a[8,6]+1359846429250 7993245036260909795718200/39788091780659136384974257175682583397*a[9,5 ], a[8,3] = -231170378781777897772699847/488734852236806096642853477-1 246528167721457796/46192288827762085*a[9,6]-75829631966778555318839725 26587582136933049326932848681/4868942014815299598094614671954479894395 8293100162280*a[7,6]+14469360245821499590625/378795964806621737208*a[8 ,6]+392051016/101419805*a[9,5], a[7,4] = 18279352373800939823324163626 40800174375/7536569688680801479300693473125495923452+45366962017748404 64149475737910161291269/1631311763007024591783944544202985919277*a[9,6 ]+45675729673903406851433/2300834749195776477856*a[7,6]-21683350909025 80715304680309329285590966237640625/5508232929619904971105399002960522 44878000909152*a[8,6]-649870608538956997180282908879137372778/16313117 63007024591783944544202985919277*a[9,5], a[10,8] = 1400189022920929099 517515017684243/18637128542567734204921296293800*a[9,6]+15857402349126 95897625870307735165225999117213221261381/1550806288809669457288322488 7113185172979869300700800*a[7,6]-99788184666255961616477176339375/1943 703628473084974215774772352*a[8,6]-2977668064676298338975/102013033089 4562984632, a[8,5] = -14392290221012124614830493/125139305875605630836 38860-2804688377373280041/226199352507082375*a[9,6]-162854484742628912 6247460898844336509576371713/46587529314490667823913696958146556691452 000*a[7,6]+332124638004118320545/32976510109167528096*a[8,6]+46427094/ 26139125*a[9,5], b[5] = 28304779228000000/53707434325074117, c[6] = 43 /50, a[7,5] = -81204383896651844636234438251358672189375/7838809442175 32021903759757220238436821724-348567956126034431417/329765101091675280 96*a[7,6]-8543683995809492399528202896252657799/7793543750850758673345 473055030609119*a[9,6]+36751442218687808733977632361513315101122671875 /23683878073045103094267619550069130971621361696*a[8,6]+64413778227669 441687013867467453402/410186513202671509123445950264768901*a[9,5], a[9 ,1] = 62348456339814459899/15277027884953137200*a[9,6]+603984506520864 9020514689264893435082469000463210977523134283043/93748004977630108787 1719443723136343472747164268825697008339200*a[7,6]-1272958519756546221 34840982213541875/27028052439496698584724476454602496*a[8,6]+849788415 08450205510291621325/155979823505808871122472973448-236/369*a[9,5], a[ 8,4] = 1202372500826294111503886894011502763/3318545113252667587681169 71680538900-20004059734792706416558424553961/6070042931324751751315049 98816*a[8,6]+164463032280796326890604730024740239100251077532164484765 7645995853/17555109363185032769083765365435388584388642636391765620377 676000*a[7,6]+1489289528385211701771/47347096048456137125*a[9,6]-46840 2951366/103955300125*a[9,5], b[7] = 7448583037583796809056159399857619 20312207508379/2487223884477764590764433396524922145673887618400, a[10 ,7] = -598654837029686893097733291010590776545275707919805426736260578 96247446006718134427629/1370791273596171564584624257136578847564380053 55946693248927095711213014831855920000*a[7,6]+326580946475274054311999 13048454080839357827356672675/5706974677728769924147929845708861532919 44816085888*a[8,6]+596874960261527554921938345637469786009760975827712 037302933/83411451405041783553489736391437541016416775629545513856200- 13567836925440539447282219361086670637795856153008682529/4104083273033 42433851067821543001392574473732627165000*a[9,6], a[6,3] = 520921/4120 56, a[10,3] = 31324687415813042985540870/361673232972883862409107*a[9, 6]+1630337087285738939355054093216330159440605605290562466415/32616345 53707477669996409840042363472130786191962245872*a[7,6]-866224433102384 65047814617045390625/706559403826210033283013200878128*a[8,6]+14732914 021779067062794440625/3806935825135059294314472018-9852064193383020/79 4089872848731*a[9,5], a[7,1] = 461753314468176748423154532852115889273 5/29893764821855034733617956843144686149981+10610725422231083909623/30 42083057570704466856*a[7,6]+224034380334560022920961764835069693396/31 9535293784881105607164395256254973879*a[9,6]-2168335090902580715304680 309329285590966237640625/218483775223841076044618790349387733213207061 6456*a[8,6]-1689072406858887582015030302479889208/16817647041309531874 061283960855524941*a[9,5], a[8,1] = 1124007309260005001892521011127434 1/9448717014952182306801086558169375-159909083595241495342578160099/17 282910422900004019937321400*a[8,6]+95186012520783916517745806022000731 7805953171925812476380576067/55537476976098498359857012770484480068619 137015931559856225000*a[7,6]+73545161895566009964/9274173452790377375* a[9,6]-1217421576/1071704125*a[9,5], a[10,1] = -5084648519680908164437 867361433268/1471990759044777156212364679272675-1092823622584592395239 815159883507162141443551362020130235590604884777217/137658513240508147 128096384344277909177204802986241460323908000438840000*a[7,6]-28469306 4162428945021968130844334496970321/15264554845159499772347747827624421 477500*a[9,6]+111954503464106025826805691400276624499975/5157986470534 883869572489387176763221184*a[8,6]+6118650393785244/1678231174577215*a [9,5], a[10,5] = 205603286081902609731080523575/2618195039417055670454 0767464-1419972689151596332115078111308072867970095137734526307/711256 8251259067525427810949846986080704140280897600*a[7,6]+1736585295452198 81212287424151875/2674362004621343818392798608832*a[8,6]-2751786257799 552616982702356686693/32766079421925320937915855215050*a[9,6]-23333836 2474861/40932467672615*a[9,5], a[3,1] = -594/625, a[10,4] = -140212437 09452004039648514944588163375/1068440647945340048996072245298432808+17 6714173858749817354587827562399097869695728372652829443044642965417047 3701/37469652113507240367651059838931628633330765974243592666918380381 097854400*a[7,6]+89152249316697832524199015786173885196253125/38999073 00148305519748706366120982325920704*a[8,6]-466445684455342443744708685 786458190834949127/56091169913126022593377574451969091558137650*a[9,6] +2354150739008872629/162788423933989855*a[9,5], a[6,4] = -17371/11640, a[10,9] = 5594658687556280397846/1893189870520997940175\}:" }}}{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 29 "indets(map(rhs,e8));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"(\"\"'&F%6$\"\")F(&F%6$\"\"*F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a[7,4]=subs(e9,a[7,4]);\n``;\na[8,5]=subs(e9,a[8,7]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,,#\"IvV<+3kijTKB)R4!QP_$z#= \"I_M#f\\DJZ$p+$z9!3o)opl`(\"\"\"*&#\"Ip7Hh,\"ztv%\\TYS[x,ipOX\"Ix#>f) H?WX%Ry\"fCq+j\\Z$ 3I#F-&F%6$F'F5F-F-*&#\"RD1kPi'4f&GH$4.o/`r!e-44N$o@\"Q_\"44+y[C_gH+*R0 6(\\!*>'HHB3bF-&F%6$\"\")F5F-!\"\"*&#\"HyFPP\"z)3HG!=(*p&*Q&31()\\'F1F -&F%6$F4\"\"&F-FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,(*&#\"hn\"Ht!yf'))4?%4Y9!*z! yOB!3&z^5>$40!>(yI'\"fn+DwFLU5*z.#G-sW[$z0!pR3u'y0JV:(*y\"\"\"&F%6$\" \"(\"\"'F.F.*&#\";.z)=(pj4NE'f$G>\":+n+(3FP>'z/1t#F.&F%6$F'F2F.!\"\"# \"EEJV.&z3iMhO``\"y6@ZB\"DD1!eaJm[/7N0TU56/&*F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We can find some simple \+ order conditions that are not yet satisfied and determine which parame rs are related by them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 47 do\n \+ eq := simplify(subs(e9,SO7_10[ii]));\n if lhs(eq)<>rhs(eq) then\n \+ print(ii,indets(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<&&%\"aG6$\"\"(\"\"'&F&6$\"\")F)&F&6$\"\"*\"\"&&F&6 $F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#^<%&%\"aG6$\"\"(\"\"'&F&6$ \"\")F)&F&6$\"\"*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#]<%&%\"aG6$ \"\"(\"\"'&F&6$\"\")F)&F&6$\"\"*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#[<&&%\"aG6$\"\"(\"\"'&F&6$\"\")F)&F&6$\"\"*\"\"&&F&6$F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Z<%&%\"aG6$\"\"(\"\"'&F&6$\"\")F)&F&6$\" \"*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple or der comditions given in abreviated form as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SO7 := Si mpleOrderConditions(7):\n[seq([i,SO7[i]],i=[50,51])]:\nlinalg[augment] (linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7% \"#]%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*(%\"aGF-F.F--F06#*&)F.\"\"#F-F3F- F-F-#F-\"$0\"7%\"#^F)/*(F,F-F7F--F06#*&F3F-F4F-F-#F-\"#%)Q)pprint386\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "cdns4 := [seq(SO7_10[i],i=[50,51])]:\neqns4 := simpl ify(subs(e9,cdns4)):\nnops(%);\nindets(eqns4);\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"a G6$\"\"(\"\"'&F%6$\"\")F(&F%6$\"\"*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 2 equations for the linking coefficien ts " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "a[9,6];" "6#&%\"aG6$\"\"*\"\"'" }{TEXT -1 29 " \+ in terms of the parameter " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"( \"\"'" }{TEXT -1 88 " and subsitute back into the expressions previous ly obtained for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "e10 := solve(\{op(eqns 4)\},\{a[8,6],a[9,6]\}):\ninfolevel[solve] := 0:\ne11 := `union`(map(u _->lhs(u_)=simplify(subs(e10,rhs(u_))),e9),e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[8,6]=subs( e10,a[8,6]);\n``;\na[9,6]=subs(e10,a[9,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"E(**y)4@I\\3\\%opKpj26F$\"Dvy ]%\\WZ]YUJ,5>)*3#*G\"\"\"&F%6$\"\"(F(F.F.#\"5SQ'>=_&**=v!)\"7j`dF]L4`] (y#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&*&#\"E,*oZE]fWB\"4:n-K^'=5%\"D-MXn u'Q,(oT%pwAyPTk\"\"\"&F%6$\"\"(F(F.F.#\"8]7)pB&>'[/9%3%\"9p[u]([)=>%zR 0*F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4334 "e11 := \{c[10] = 1, c[9] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4 ] = 41/100, c[3] = 27/125, c[2] = 1/50, b[3] = 0, b[2] = 0, a[3,2] = 7 29/625, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a [4,2] = 0, a[5,2] = 0, a[6,1] = -31/200, a[5,4] = 3249/9700, a[2,1] = \+ 1/50, a[9,1] = -330200143128150564844373989963138965305/15845789444006 682655005411396993536892*a[7,6]+1181196160700901160059807675289975/199 3695109095372538469668928368974-236/369*a[9,5], a[4,3] = 1681/4320, a[ 10,4] = -57824654117330032806972125/4504891914497692665956472+94084324 7662664339192236114935147524038/3722182981036603604663856136743935915* a[7,6]+2354150739008872629/162788423933989855*a[9,5], a[7,1] = 1928444 29333630775732683776601286031302135/1225644357696056424078336230568932 132149221-805/246*a[7,6]-1689072406858887582015030302479889208/1681764 7041309531874061283960855524941*a[9,5], a[8,1] = 120874462197201049046 99332896696341/9448717014952182306801086558169375-17554944318205473067 675579213230172839/474302730332421553642677809709929150*a[7,6]-1217421 576/1071704125*a[9,5], a[10,3] = 554677798642862769978653125/131273649 142588251528085242-1680077227969043462843278776669906292925/5011329281 785866316523045335323542988*a[7,6]-9852064193383020/794089872848731*a[ 9,5], a[9,3] = 347399897509328826223328125/193077805387804830107465522 4+2204752508464359361538521982673132842875/374888189285036150618420708 66057879964*a[7,6]+1900/873*a[9,5], a[8,5] = -177403572899995302408104 93/12513930587560563083638860+46427094/26139125*a[9,5], a[7,4] = 78470 915134262162154819489000191323849375/308999357235912860651328432398145 332861532-27692/3977*a[7,6]-649870608538956997180282908879137372778/16 31311763007024591783944544202985919277*a[9,5], a[8,4] = 51414638354943 21193947876483/1257886311469494661792654900-90583512681940241029205988 74026769184924/115018412105612226758349368854657818875*a[7,6]-46840295 1366/103955300125*a[9,5], a[7,3] = -1021703783184455623759561713382023 060265625/9004316635551387567394503527016686254544277+5375/582*a[7,6]+ 13598464292507993245036260909795718200/3978809178065913638497425717568 2583397*a[9,5], a[8,3] = -285280064336336028436159847/4887348522368060 96642853477+4688587612936865477702173330241598957/44885233992434039710 575363455476222*a[7,6]+392051016/101419805*a[9,5], a[6,5] = 132023/106 200, a[8,6] = 327110763693269684490849302109878997/2892089819100131424 6504744494507875*a[7,6]+80751899552181963840/2787505309335027575363, a [10,5] = 155667590349511247612945781575/26181950394170556704540767464- 233338362474861/40932467672615*a[9,5], a[10,6] = -15628625376456218259 00724443413866319/43052657060016033647105200475288170*a[7,6]-232623177 7585087107750/1206272546384488866919, a[10,1] = -372240362676250518801 412815177859/101516604072053596980163081329150+10940037763519352781305 071103897064233/92095249015338645888590254929746868*a[7,6]+61186503937 85244/1678231174577215*a[9,5], a[4,1] = 451/21600, a[7,5] = -847299542 98330153515424856383277188889375/7838809442175320219037597572202384368 21724+64413778227669441687013867467453402/4101865132026715091234459502 64768901*a[9,5], a[5,1] = 19/160, a[9,4] = 18273630606703730738657500/ 13573714027606306643357487-5679442461804189715323232627365990203246/12 8086798005720684794627075459031089877*a[7,6]-10089/3977*a[9,5], b[1] = 771570009067/14036203465200, b[4] = 0, b[10] = 103626500437/171763508 9268, b[8] = -5118512171875/11763620626464, b[9] = 136801854099/127885 521925, a[8,7] = -8593750881095206170491007194502/32135045435455581509 03880585625, a[5,3] = 361/3104, b[6] = -296881060859375/51506073383538 9, a[9,7] = -1532806290465891141166096531902118541769245/1203242011387 872547807852011647420329982736, a[10,8] = -33473047374792524975/329074 30028856870472, c[7] = 2272510/11977321, b[5] = 28304779228000000/5370 7434325074117, c[6] = 43/50, a[10,7] = 2315079813491204524435067899365 885119542372444358703/316169042039527157595235231573788308031260760584 200, b[7] = 744858303758379680905615939985761920312207508379/248722388 4477764590764433396524922145673887618400, a[9,6] = 4101865132026715091 23445950264768901/64413778227669441687013867467453402*a[7,6]+408414044 86195236981250/905397941918848750744869, a[6,3] = 520921/412056, a[3,1 ] = -594/625, a[9,8] = -7500029126894375/132689679447323376, a[6,4] = \+ -17371/11640, a[10,9] = 5594658687556280397846/1893189870520997940175 \}:" }}}{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 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a [7,4]=subs(e11,a[7,4]);\n``;\na[8,5]=subs(e11,a[8,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,(#\"Jv$\\QK\">+!*[>[:i@EM^\"4Z y\"KK:'GLX\")RK%G8lgG\"fBd$***3$\"\"\"*&#\"&#pF\"%xRF-&F%6$F'\"\"'F-! \"\"*&#\"HyFPP\"z)3HG!=(*p&*Q&31()\\'\"Ix#>f)H?WX%Ry\"fCq+j25\\qh?&4\")3v$f)\"@Dce!) Q!4:ebaVX]8K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We can find some simple order conditions that are not ye t satisfied and determine which paramers are related by them." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "for ii from 64 by -1 to 43 do\n eq := simplify(subs(e11,SO7_10[ ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<$&%\"aG6$ \"\"(\"\"'&F&6$\"\"*\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<$&% \"aG6$\"\"(\"\"'&F&6$\"\"*\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"# X<$&%\"aG6$\"\"(\"\"'&F&6$\"\"*\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations t hat consists of the simple order comditions given in abreviated form a s follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 164 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]], i=[45,55])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$( linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7%\"#X%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&% \"aGF--F06#*&F3F--F06#*&)F.\"\"#F-F3F-F-F-F-#F-\"$?%7%\"#bF)/*(F,F-F.F --F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-#F-\"$S\"Q)pprint396\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "We can solve th is system of two equations to obtain numerical values for the remainin g parameters " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,5]" "6#&%\"aG6$\"\"*\"\"&" }{TEXT -1 103 " and substitute these values back into the expressions obtain ed previously for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "cdns5 := [seq(SO7_10 [i],i=[45,55])]:\neqns5 := simplify(subs(e11,cdns5)):\nnops(%);\nindet s(eqns5);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"aG6$\"\"(\"\"'&F%6$\"\"*\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "e12 := solve(\{op(eqns5) \},\{a[7,6],a[9,5]\}):\ninfolevel[solve] := 0:\ne13 := `union`(map(u_- >lhs(u_)=simplify(subs(e12,rhs(u_))),e11),e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[7,6]=subs( e12,a[7,6]);\n``;\na[9,5]=subs(e12,a[9,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!?+]=B/=^!*Qa![Dm-$\"A2UpwV(Rtysrk \"pGmD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#\"=DJ>tRekJyokx4*)\"=#>$zuw!3ZvNA$4 ri" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e13" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2444 "e13 := \{c[10] = 1, c[9] = 5/6, c[8] = 18/25, c[5] = 57/100, c[4 ] = 41/100, c[3] = 27/125, c[2] = 1/50, b[3] = 0, b[2] = 0, a[3,2] = 7 29/625, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a [4,2] = 0, a[5,2] = 0, a[6,1] = -31/200, a[10,5] = -234937873364700289 5234008950/1090914599757106529355865311, a[5,4] = 3249/9700, a[2,1] = \+ 1/50, a[10,1] = 36393032615434450612/324390586094889663425, a[4,3] = 1 681/4320, a[10,6] = -78686605908422443750/52446632451499515953, a[9,6] = -4792324941735635008750/159776107397443897190271, a[7,4] = -1717046 972617147709491116450178750/7467894926932728111586543618014237, a[9,5] = 8909776468783164583973193125/6271093223575470807674793192, a[6,5] = 132023/106200, a[4,1] = 451/21600, a[5,1] = 19/160, b[1] = 7715700090 67/14036203465200, b[4] = 0, b[10] = 103626500437/1717635089268, b[8] \+ = -5118512171875/11763620626464, a[9,1] = -1401024812030113404025/1988 7564677841032175639, b[9] = 136801854099/127885521925, a[8,7] = -85937 50881095206170491007194502/3213504543545558150903880585625, a[5,3] = 3 61/3104, b[6] = -296881060859375/515060733835389, a[9,7] = -1532806290 465891141166096531902118541769245/120324201138787254780785201164742032 9982736, a[9,4] = -50491693720625/29100752640072, a[10,8] = -334730473 74792524975/32907430028856870472, a[10,4] = 4135780451822750/874504037 187843, a[7,3] = 347890318302644246405985993187156250/1321817402067092 875750818220388519949, a[8,3] = 3247095172038/883201854817, a[8,6] = - 6468694559114760/61945939006089637, c[7] = 2272510/11977321, a[7,1] = \+ 25959766877768976976598957736980/487594514129628295945513157189933, a[ 8,5] = 435669225629732566638/393965828849029186615, a[7,6] = -30266254 8054389051180423185000/25662869164717278733974376694207, b[5] = 283047 79228000000/53707434325074117, c[6] = 43/50, a[10,7] = 231507981349120 4524435067899365885119542372444358703/31616904203952715759523523157378 8308031260760584200, b[7] = 744858303758379680905615939985761920312207 508379/2487223884477764590764433396524922145673887618400, a[6,3] = 520 921/412056, a[8,4] = -518509279926/374238074075, a[8,1] = 424097052912 66846/416462256407406875, a[3,1] = -594/625, a[9,3] = 1328137311123437 5/5150833217292744, a[9,8] = -7500029126894375/132689679447323376, a[6 ,4] = -17371/11640, a[10,3] = -1462401427649331250/154787214582248211, a[7,5] = 29780304732725103577764751746216250/258912687002832625147067 486467854423, a[10,9] = 5594658687556280397846/1893189870520997940175 \}:" }}}{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 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(6-i)],i=2..6)]));print(``);\nfor ii from 7 to 10 do\n prin t(c[ii]=subs(e13,c[ii]));print(``); \n for jj to ii-1 do\n prin t(a[ii,jj]=subs(e13,a[ii,jj]));\n end do:\n print(`_______________ __________________`);\nend do:print(``);\nfor ii to 10 do\n print(b[ ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7'7(#\"\"\"\"#]F(%!GF+F+F+7(#\"#F\"$D\"#!$%f\"$D'#\"$H(F2F+F+F +7(#\"#T\"$+\"#\"$^%\"&+;#\"\"!#\"%\"o\"\"%?VF+F+7(#\"#dF8#\"#>\"$g\"F <#\"$h$\"%/J#\"%\\K\"%+(*F+7(#\"#VF*#!#J\"$+#F<#\"'@4_\"'c?T#!&rt\"\"& S;\"#\"'B?8\"'+i5Q)pprint406\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"(5DF#\")@t(>\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"(\"\"\"#\"A!)ptd*)fwp(*ox(owff#\"BL**=dJ^XfHG'HT^%f([" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"E]i:(=$*f)fSYUk-$=.*yM \"F\\*>&)Q?#=3vvG4n?S<=K\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"(\"\"%#!C](y,X;6\\4x9'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!@-X>25\\qh?&4\")3v$f)\" @Dce!)Q!4:ebaVX]8K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B______________ ___________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"* #\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!7DSS8,.7[-,9\"8Rc" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#\"2vVB6JP\"G8\"1WFHtR ekJyokx4*)\"=#>$zuw!3ZvNA$4ri" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"*\"\"'#!7](3]jN<%\\K#z%\"9r->(*QW(R2hxf\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#!LX#p`'4m69\"*eY!H1G`\"\"LOF) *H.UZ;,_y!yasyQ6?C.7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"* \"\")#!1vV*o7H+](\"3wLKZ%z'*oK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B _________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"571XMahKIRO\"6DMm*)[4' e!RC$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$#!4]7L\\wU,CY\"\"36# [Ae9sya\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\"1]F# =X!yNT\"0Vy=PS]u)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" &#!=]*3SB&*G+ZOty$\\B\"=6`'eNHl5d(*f944\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!5]PWA%3fg'oy\"5`f^*\\^CjYC&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"U.(eVWsBa>^)eO**y 1NW_/7\\8)z]J#\"T+Ueg2EJ!3$)yt:BN_fdr_R?/phJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!5v\\_#zut/tM$\"5s/(o&)G+V2H$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"7YyR!GcvoeYf&\"7v ,%z*4_q)*=$*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___________________ ______________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"-n!4+dr(\"/+_Y.i.9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&# \"2+++G#zZIG\"2w&)*Rfh04oz$ePIe[u\"R+%=w)QnX@#\\_'RLWw!fkxZ%)QA ([#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!.v=<7&=^\"/kki? Ow6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"-*4a=!o8\"-D>_ &)y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"-P/]EO5\".o#*3 Nwr\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "sub s(e13,OrderConditions(7,10,'expanded')):\nmap(u->lhs(u)-rhs(u),%);\nno ps(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 57 " #--------------------------------------------------------" }}{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3193 "ee := \{c[2]=1/50,\nc[3]=27/125,\nc[4]=41/100, \nc[5]=57/100,\nc[6]=43/50,\nc[7]=2272510/11977321,\nc[8]=18/25,\nc[9] =5/6,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/50,\na[3,1]=-594/625,\na[3,2]=72 9/625,\na[4,1]=451/21600,\na[4,2]=0,\na[4,3]=1681/4320,\na[5,1]=19/160 ,\na[5,2]=0,\na[5,3]=361/3104,\na[5,4]=3249/9700,\na[6,1]=-31/200,\na[ 6,2]=0,\na[6,3]=520921/412056,\na[6,4]=-17371/11640,\na[6,5]=132023/10 6200,\na[7,1]=25959766877768976976598957736980/48759451412962829594551 3157189933,\na[7,2]=0,\na[7,3]=347890318302644246405985993187156250/13 21817402067092875750818220388519949,\na[7,4]=-171704697261714770949111 6450178750/7467894926932728111586543618014237,\na[7,5]=297803047327251 03577764751746216250/258912687002832625147067486467854423,\na[7,6]=-30 2662548054389051180423185000/25662869164717278733974376694207,\na[8,1] =42409705291266846/416462256407406875,\na[8,2]=0,\na[8,3]=324709517203 8/883201854817,\na[8,4]=-518509279926/374238074075,\na[8,5]=4356692256 29732566638/393965828849029186615,\na[8,6]=-6468694559114760/619459390 06089637,\na[8,7]=-8593750881095206170491007194502/3213504543545558150 903880585625,\na[9,1]=-1401024812030113404025/19887564677841032175639, \na[9,2]=0,\na[9,3]=13281373111234375/5150833217292744,\na[9,4]=-50491 693720625/29100752640072,\na[9,5]=8909776468783164583973193125/6271093 223575470807674793192,\na[9,6]=-4792324941735635008750/159776107397443 897190271,\na[9,7]=-1532806290465891141166096531902118541769245/120324 2011387872547807852011647420329982736,\na[9,8]=-7500029126894375/13268 9679447323376,\na[10,1]=36393032615434450612/324390586094889663425,\na [10,2]=0,\na[10,3]=-1462401427649331250/154787214582248211,\na[10,4]=4 135780451822750/874504037187843,\na[10,5]=-234937873364700289523400895 0/1090914599757106529355865311,\na[10,6]=-78686605908422443750/5244663 2451499515953,\na[10,7]=2315079813491204524435067899365885119542372444 358703/\n 316169042039527157595235231573788308031260760584200, \na[10,8]=-33473047374792524975/32907430028856870472,\na[10,9]=5594658 687556280397846/1893189870520997940175,\na[11,1]=250860770670184236308 3/197875357745688550590720,\na[11,2]=0,\na[11,3]=-5122833329940625/508 724268374592,\na[11,4]=13293920580875/2874148408896,\na[11,5]=-5991884 64780493707137440161875/277270064173229869784600732736,\na[11,6]=-3601 465055348923762849875/2146128454918752594358208,\na[11,7]=606030238246 181777051198920509497430523044409408159/\n 74752050141640998967 813674460513197348653288024576,\na[11,8]=-1922750201834125/19415042260 23936,\na[11,9]=12539348439579/3975412795840,\na[11,10]=0,\n\nb[1]=771 570009067/14036203465200,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=28304779228 000000/53707434325074117,\nb[6]=-296881060859375/515060733835389,\nb[7 ]=744858303758379680905615939985761920312207508379/2487223884477764590 764433396524922145673887618400,\nb[8]=-5118512171875/11763620626464,\n b[9]=136801854099/127885521925,\nb[10]=103626500437/1717635089268,\nb[ 11]=0,\n\n`b*`[1]=448234490819/8120946290580,\n`b*`[2]=0,\n`b*`[3]=0, \n`b*`[4]=0,\n`b*`[5]=7786773134600000/14452831163890377,\n`b*`[6]=-40 8698637296875/567617951573694,\n`b*`[7]=442670515036915263832538107827 8067803359/14828075230102658203818343670586143438076,\n`b*`[8]=-500454 2378125/10330679593521,\n`b*`[9]=154806770859/124231649870,\n`b*`[10]= 0,\n`b*`[11]=16/243\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[7,11];" "6#&% \"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose components a re the principal error terms of the 11 stage, order 7 scheme (the erro r terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6,11];" "6#&%#T*G6$\"\"'\"#6" }{TEXT -1 146 " denote the v ector whose components are the principal error terms of the embedded 1 1 stage, order 6 scheme (the error terms of order 7) and let " } {XPPEDIT 18 0 "`T*`[7,11];" "6#&%#T*G6$\"\"(\"#6" }{TEXT -1 99 " deno te the vector whose components are the error terms of order 8 of the e mbedded order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[7,11]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,11]));" "6#-%$absG6#-F$6#&%# T*G6$\"\"'\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[7,1 1]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\"#6" }{TEXT -1 15 " respectivel y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Defi ne: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[8] = abs(ab s(T[7,11]));" "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\"#6" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,11]))/abs(ab s(`T*`[6,11]));" "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\"#6\" \"\"-F*6#-F*6#&F/6$\"\"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 " C[8] = abs(abs(`T*`[7,11]-T[7,11]))/abs(abs(`T*`[6,11]));" "6#/&%\"CG6 #\"\")*&-%$absG6#-F*6#,&&%#T*G6$\"\"(\"#6\"\"\"&%\"TG6$F2F3!\"\"F4-F*6 #-F*6#&F06$\"\"'F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that \+ as well as attempting to ensure that " }{XPPEDIT 18 0 "A[8];" "6#&%\" AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embedded scheme is to \+ be used for error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\") " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\")" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := PrincipalErrorTerms(7,11,'expanded') :\n`errterms7_11*` :=subs(b=`b*`,PrincipalErrorTerms(7,11,'expanded')) :\n`errterms6_11*` := subs(b=`b*`,PrincipalErrorTerms(6,11,'expanded') ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms7_11*`[i]))^ 2,i=1..nops(`errterms7_11*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errte rms6_11*`[i]))^2,i=1..nops(`errterms6_11*`))):\nsnmC := sqrt(add((eval f(subs(ee,`errterms7_11*`[i])-subs(ee,errterms7_11[i])))^2,i=1..nops(e rrterms7_11))):\n'B[8]'= evalf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC/s dnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")ma;>!\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")/I;>!\"(" }}}{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 stabi lity regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffi cients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3193 "ee := \{c[2]=1/50,\nc[3]=2 7/125,\nc[4]=41/100,\nc[5]=57/100,\nc[6]=43/50,\nc[7]=2272510/11977321 ,\nc[8]=18/25,\nc[9]=5/6,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/50,\na[3,1]= -594/625,\na[3,2]=729/625,\na[4,1]=451/21600,\na[4,2]=0,\na[4,3]=1681/ 4320,\na[5,1]=19/160,\na[5,2]=0,\na[5,3]=361/3104,\na[5,4]=3249/9700, \na[6,1]=-31/200,\na[6,2]=0,\na[6,3]=520921/412056,\na[6,4]=-17371/116 40,\na[6,5]=132023/106200,\na[7,1]=25959766877768976976598957736980/48 7594514129628295945513157189933,\na[7,2]=0,\na[7,3]=347890318302644246 405985993187156250/1321817402067092875750818220388519949,\na[7,4]=-171 7046972617147709491116450178750/7467894926932728111586543618014237,\na [7,5]=29780304732725103577764751746216250/2589126870028326251470674864 67854423,\na[7,6]=-302662548054389051180423185000/25662869164717278733 974376694207,\na[8,1]=42409705291266846/416462256407406875,\na[8,2]=0, \na[8,3]=3247095172038/883201854817,\na[8,4]=-518509279926/37423807407 5,\na[8,5]=435669225629732566638/393965828849029186615,\na[8,6]=-64686 94559114760/61945939006089637,\na[8,7]=-859375088109520617049100719450 2/3213504543545558150903880585625,\na[9,1]=-1401024812030113404025/198 87564677841032175639,\na[9,2]=0,\na[9,3]=13281373111234375/51508332172 92744,\na[9,4]=-50491693720625/29100752640072,\na[9,5]=890977646878316 4583973193125/6271093223575470807674793192,\na[9,6]=-47923249417356350 08750/159776107397443897190271,\na[9,7]=-15328062904658911411660965319 02118541769245/1203242011387872547807852011647420329982736,\na[9,8]=-7 500029126894375/132689679447323376,\na[10,1]=36393032615434450612/3243 90586094889663425,\na[10,2]=0,\na[10,3]=-1462401427649331250/154787214 582248211,\na[10,4]=4135780451822750/874504037187843,\na[10,5]=-234937 8733647002895234008950/1090914599757106529355865311,\na[10,6]=-7868660 5908422443750/52446632451499515953,\na[10,7]=2315079813491204524435067 899365885119542372444358703/\n 31616904203952715759523523157378 8308031260760584200,\na[10,8]=-33473047374792524975/329074300288568704 72,\na[10,9]=5594658687556280397846/1893189870520997940175,\na[11,1]=2 508607706701842363083/197875357745688550590720,\na[11,2]=0,\na[11,3]=- 5122833329940625/508724268374592,\na[11,4]=13293920580875/287414840889 6,\na[11,5]=-599188464780493707137440161875/27727006417322986978460073 2736,\na[11,6]=-3601465055348923762849875/2146128454918752594358208,\n a[11,7]=606030238246181777051198920509497430523044409408159/\n \+ 74752050141640998967813674460513197348653288024576,\na[11,8]=-19227502 01834125/1941504226023936,\na[11,9]=12539348439579/3975412795840,\na[1 1,10]=0,\n\nb[1]=771570009067/14036203465200,\nb[2]=0,\nb[3]=0,\nb[4]= 0,\nb[5]=28304779228000000/53707434325074117,\nb[6]=-296881060859375/5 15060733835389,\nb[7]=744858303758379680905615939985761920312207508379 /2487223884477764590764433396524922145673887618400,\nb[8]=-51185121718 75/11763620626464,\nb[9]=136801854099/127885521925,\nb[10]=10362650043 7/1717635089268,\nb[11]=0,\n\n`b*`[1]=448234490819/8120946290580,\n`b* `[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=7786773134600000/144528311638 90377,\n`b*`[6]=-408698637296875/567617951573694,\n`b*`[7]=44267051503 69152638325381078278067803359/1482807523010265820381834367058614343807 6,\n`b*`[8]=-5004542378125/10330679593521,\n`b*`[9]=154806770859/12423 1649870,\n`b*`[10]=0,\n`b*`[11]=16/243\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability functi on R for the 11 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(7,11,'exp anded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\" 5RNZ3Qjfnn5\"9+++[?-32jlp^F)*$)F'\"\")F)F)F)*&#\"4(zj^7f(o\")e*\":++++ 3qL^%Q4;')F)*$)F'\"\"*F)F)F)*&#\"48DJ/R*3\\]d\":++++g0_G!)=ny%F)*$)F' \"#5F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-% \"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=-1,z=-3.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+&[`%**Q!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-3 .9):\np1 := plot([R(z),-1],z=-4.49..0.49,color=[red,blue]):\np2 := plo t([[[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)):\n plots[display]([p1,p2,p3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETI CA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CU RVESG6$7W7$$!3A++++++!\\%!#<$!3%\\>ib,2I5%F*7$$!3;+]7'3DdV%F*$!3BdvemS 3NOF*7$$!35++Ds,X\"Q%F*$!3a%G)e]'zb@$F*7$$!3(*\\P*)feAMVF*$!3IVg/[gU') GF*7$$!3%)*\\Pva,qG%F*$!3G#zO)*\\Qxe#F*7$$!33]iDpG*QB%F*$!3AfZH;J1&G#F *7$$!3I+](4>%y!=%F*$!3f@XgBRN9?F*7$$!3l++!RZBt7%F*$!39@XpFW.r\"F*7$$!3? +v8#z!zoQF*$!3%eG)R-Q,I#*!#=7$$!3;+v.8ajmPF*$!3d!3q+%**=@qFin7$$!3C+v$ Q<')4m$F*$!3aQDi6h%eA&Fin7$$!3G+v3`dnbNF*$!3;zJ1Tz]JQFin7$$!37++0T5NZM F*$!3v_#GcR%GAFFin7$$!3?+]sv&Q>N$F*$!3?(z]^>$og>Fin7$$!3E++:Qn_WKF*$!3 ['HsJs\"*=H\"Fin7$$!3#****\\s&QnOJF*$!3oCy6yVyfx!#>7$$!3&****\\=iPF.$F *$!35wOBI(Q!3RF]q7$$!3Q+v$Gc`$QHF*$!3Mm4*y/B28\"F]q7$$!3+++l(y@h#GF*$ \"3T=1+C/y5:F]q7$$!3=++g_n/JFF*$\"3l4;f(H`)QLF]q7$$!3C+vt%)=X?EF*$\"3] OOW1P&*Q^F]q7$$!3D++5(ocD_#F*$\"3H#G71Id\"QlF]q7$$!3#**\\P(R,::CF*$\"3 Crg'e)=9^zF]q7$$!3&)*\\7sqtGJ#F*$\"3U^]MV#*F]q7$$!30+]Pz*eh?#F*$\"3 i@9Wr'>#f5Fin7$$!3l*\\([Y:;3@F*$\"3m@C$*f_5(=\"Fin7$$!3$***\\7w!eC+#F* $\"39Z&3o)R%HL\"Fin7$$!3%**\\Py(=m#*=F*$\"3)*op/J0_'\\\"Fin7$$!3%)*\\i W'R3(z\"F*$\"3]q-liNY^;Fin7$$!3-+]d*>dQp\"F*$\"3I@G#yfmV$=Fin7$$!3)*** **p]Q@(e\"F*$\"397E$4%='G/#Fin7$$!3I+]FRT)G[\"F*$\"3%zBeY>y'oAFin7$$!3 ')*\\P*y(R>Q\"F*$\"3@5y'*=IH5DFin7$$!3++]Kx#e)p7F*$\"3)G/D5*oT3GFin7$$ !3=++5j![\"p6F*$\"3uIq<$onh5$Fin7$$!3s***\\KT=;1\"F*$\"377InT.*)eMFin7 $$!3'***\\P%o0=k*Fin$\"38tL(p\"f#H\"QFin7$$!3](***\\tEbw&)Fin$\"3^k_%f KV:C%Fin7$$!39-]P2EBuvFin$\"3I>Si+js)o%Fin7$$!3>)*\\(GL>l_'Fin$\"3Xg1. &eIm?&Fin7$$!3z++]-\")=-bFin$\"3aN>z1bBodFin7$$!3$*)*\\PM#3)HWFin$\"3c t]:^^>@kFin7$$!3;(****f'*ypR$Fin$\"3Ht/9GQ&)>rFin7$$!3L)**\\U'=wSBFin$ \"3go\"*Hr_,8zFin7$$!3?$*\\Pt2H$H\"Fin$\"31zh?Fw%oy)Fin7$$!33R****pit2 LF]q$\"3+r)*)36PYn*Fin7$$\"3Z^+]FkzBxF]q$\"3eoWS=\"*H!3\"F*7$$\"35**** *z2`!f\"F*7$$\"3l+]i_F06GFin$\"3%oP\"HVIfC8F*7$$\"3 \\.]Pt1&z\"QFin$\"37\"\\IQ%=\"\\Y\"F*7$$\"3!***************[Fin$\"3C[N I-iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fe\\lFd\\l-F$6$7S7$F($! \"\"Fe\\l7$F3Fj\\l7$F=Fj\\l7$FGFj\\l7$FQFj\\l7$FVFj\\l7$FenFj\\l7$F[oF j\\l7$F`oFj\\l7$FeoFj\\l7$FjoFj\\l7$F_pFj\\l7$FdpFj\\l7$FipFj\\l7$F_qF j\\l7$FdqFj\\l7$FiqFj\\l7$F^rFj\\l7$FcrFj\\l7$FhrFj\\l7$F]sFj\\l7$FbsF j\\l7$FgsFj\\l7$F\\tFj\\l7$FatFj\\l7$FftFj\\l7$F[uFj\\l7$F`uFj\\l7$Feu Fj\\l7$FjuFj\\l7$F_vFj\\l7$FdvFj\\l7$FivFj\\l7$F^wFj\\l7$FcwFj\\l7$Fhw Fj\\l7$F]xFj\\l7$FbxFj\\l7$FgxFj\\l7$F\\yFj\\l7$FayFj\\l7$FfyFj\\l7$F[ zFj\\l7$F`zFj\\l7$FezFj\\l7$FjzFj\\l7$F_[lFj\\l7$Fd[lFj\\l7$Fi[lFj\\l- F^\\l6&F`\\lFd\\lFd\\lFa\\l-F$6&7#7$$!3)******\\[`%**QF*Fj\\l-%'SYMBOL G6#%'CIRCLEG-F^\\l6&F`\\lFe\\lFe\\lFe\\l-%&STYLEG6#%&POINTG-F$6&F``l-F e`l6#%&CROSSGFh`lFj`l-F$6&F``l-Fe`l6#%(DIAMONDGFh`lFj`l-F$6%7$7$Fb`lFd \\lFa`l-%&COLORG6&F`\\lFd\\l$\"\"&F[]lFd\\l-%*LINESTYLEG6#\"\"$-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^cl-Ffbl6#%(DEFAULTG-%%V IEWG6$;$!$\\%!\"#$\"#\\Ficl;$!$Z\"Ficl$\"$Z\"Ficl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability regio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1000 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/ 720*z^6+1/5040*z^7+\n 10676759633808473539/51696563070802204800000 0*z^8+\n 9588168759125163797/8616093845133700800000000*z^9-\n \+ 5750490893904312513/4786718802852056000000000*z^10:\npts := []: z0 := \+ 0: tt := 0: \nwhile tt<=281/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z =z0):\n z0 := zz:\n if (3/4<=tt and tt<=33/20) or (53/20<=tt and t t<=71/20) or\n (209/20<=tt and tt<=227/20) or (247/20<=tt and tt< =53/4) then\n hh := 1/40\n else \n hh := 1/20\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 : = plot(pts,color=COLOR(RGB,.28,.12,.48)):\np2 := plots[polygonplot]([s eq([pts[i-1],pts[i],[-1.95,0]],i=2..nops(pts))],\n style=patc hnogrid,color=COLOR(RGB,.55,.23,.95)):\np3 := plot([[[-4.59,0],[1.19,0 ]],[[0,-4.59],[0,4.59]]],color=black,linestyle=3):\nplots[display]([p| |(1..3)],view=[-4.59..1.19,-4.59..4.59],font=[HELVETICA,9],\n 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" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the largest interval on the nonnegative imaginary axis th at contains the origin and lies inside the stability region is " } {XPPEDIT 18 0 "[0, 3.9];" "6#7$\"\"!-%&FloatG6$\"#R!\"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 491 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24 *z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 10676759633808473539/516965 630708022048000000*z^8+\n 9588168759125163797/86160938451337008000 00000*z^9-\n 5750490893904312513/4786718802852056000000000*z^10:\n Digits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 170 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,.5,0,.95 ),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 265 310 310 {PLOTDATA 2 "6(-%'CURVESG6#7gu7$$\"\"!F)F(7$$\"5 yc]4^8.];E!#@$\"55B$z*e`EfTJF-7$$\"5IM<=V%Q(4JVF-$\"5.m#ezrI&=$G'F-7$$ \"5W[$=uD*eL:eF-$\"5BPL#p2'zxC%*F-7$$\"5(*G$e\"*[R;m;(F-$\"5B5BdVhqjc7 !#?7$$\"5T+0X]^HBE%)F-$\"5>ccMzEjzq:F?7$$\"5?SvE&3D\\mh*F-$\"5.G)[Y@fb \\)=F?7$$\"5L.%yuM0!=v5F?$\"5zjuM[d[6*>#F?7$$\"55lC]I^w5%=\"F?$\"5%HJ) [xATF8DF?7$$\"5/ZO?e*o<\"*G\"F?$\"5M`oM&zQLu#GF?7$$\"5I_>)32R\"F?$ \"5'Qa5\")Gl#fTJF?7$$\"52=dg*o[k#*[\"F?$\"5jUXjH<>vbMF?7$$\"5a6*4@?P(3 &e\"F?$\"5Wacht!=6*pPF?7$$\"5)4\"4KS_;Uy;F?$\"5Pd0YUU/2%3%F?7$$\"5:p.L ??!o%pF?$\"5g2X<*4?[l-&F?7$$\"5w'4yk$*4N/.#F?$\"5muKYFNuqS`F?7$$ \"5*)RPV%e9zP6#F?$\"52TL@3_m'[l&F?7$$\"5_nRB1t!pa>#F?$\"5#)HR$eO%e-pfF ?7$$\"5/&=BZ*G(*evAF?$\"59P ju$HF?$\"5Ao'p9p-716*F?7$$\"5W#QDw*Hcd0IF?$\"5DiA.s`#pZU*F?7$$\"5!H23e CpLG2$F?$\"5Sz&fsL*f#*Q(*F?7$$\"5E&>WE7M1$RJF?$\"5*)3\">Bm@3`+\"!#>7$$ \"5@;%fE#[a10KF?$\"5kAXo&)oPsO5Ffu7$$\"5Bd@oDRi=qKF?$\"5*QhzquCR\"o5Ff u7$$\"52g:ePZuuMLF?$\"5S5f%*)Rka&*4\"Ffu7$$\"5wm\\]^T8$))R$F?$\"5Op*)e R]*p48\"Ffu7$$\"5gdM\"zbhBDY$F?$\"5)4CE(ef^Qi6Ffu7$$\"5;%>0/W78f_$F?$ \"5^7Eh#fE+Q>\"Ffu7$$\"5XiQNkb94*e$F?$\"53[mH*eE:_A\"Ffu7$$\"5WS?t8JB: _OF?$\"5nUlM%)e,jc7Ffu7$$\"5OT,diP3>:PF?$\"56'p0ry%\\/)G\"Ffu7$$\"5b(z &[n%\\-$yPF?$\"5$QH&=xS'f%>8Ffu7$$\"5Grf%)*p<#eTQF?$\"5TklY5^U(3N\"Ffu 7$$\"5Lnc_Q=H70RF?$\"5?9\"fL$*z)G#Q\"Ffu7$$\"5;,*)fJMY,pRF?$\"5w\")QV. 9Lq89Ffu7$$\"5&)4XS04GMLSF?$\"5T`*3QJ$y6X9Ffu7$$\"5#y!QaW*>(=)4%F?$\"5 aU=!y^SKlZ\"Ffu7$$\"5&3\"e_)3h?O;%F?$\"5Qm'f$\\!4Zz]\"Ffu7$$\"5OR:Ffu7$$\"53Gf#*4bX]'H%F?$\"5MIVm!y5x2d\"Ffu7$$\"5 Ki4WE\"*p0kVF?$\"5%[@)*Q&GE>-;Ffu7$$\"5NVj\">\\4,CV%F?$\"5/c!**H5k3Oj \"Ffu7$$\"5P/Y/FDDc,XF?$\"5MXF7\\w_-l;Ffu7$$\"5$*))RW-vmbrXF?$\"5pD813 !oUkp\"Ffu7$$\"5IXN?+t*)QUYF?$\"5d_ZRM45'ys\"Ffu7$$\"5d<2Z:+a09ZF?$\"5 WGO$o;V!Gf_S^IH*Q$\\F?$\"5=kjeq))pa`=Ffu7$$\"5Res DpNLp3]F?$\"5NJhyt)[s\\)=Ffu7$$\"5!p9CJ(\\Q>%3&F?$\"5J*\\B+d()*R;>Ffu7 $$\"5w+*)e1c7Ng^F?$\"5.1&=2IBHy%>Ffu7$$\"51/wcO=*=rB&F?$\"5:6ePm$eg#z> Ffu7$$\"507-$HD^ZWJ&F?$\"5g6Y4nnQp5?Ffu7$$\"5%=M5\\ok&G#R&F?$\"5-Sj%HX \"*G@/#Ffu7$$\"5LiyEVf.eqaF?$\"5c-@405act?Ffu7$$\"5rQM*[p[x#\\bF?$\"5* =9@!fcG+0@Ffu7$$\"5&)fr^p->KGcF?$\"5br@krD0WO@Ffu7$$\"5+lhz*[idwq&F?$ \"51s7oC-u(y;#Ffu7$$\"5$>(3[>)zFsy&F?$\"5hu*[([=@J*>#Ffu7$$\"5-NDif_X( p'eF?$\"5(f?Pnu(GuIAFfu7$$\"53iGZ9X(Qo%fF?$\"59QPg$ROn@E#Ffu7$$\"5))[] UC!ffn-'F?$\"5y#RaW>k#e$H#Ffu7$$\"5t@-t_&=um5'F?$\"5h'*4H(z.&)\\K#Ffu7 $$\"5RP62ZTp^'='F?$\"5I#>G\\)3+PcBFfu7$$\"5/[#[)3D*=iE'F?$\"5G4DY<%*>t (Q#Ffu7$$\"5\"QTl]782dM'F?$\"5OUZJo`U1>CFfu7$$\"5E<>S\")*p.\\U'F?$\"5A !)=%**4pe.X#Ffu7$$\"5IFM$pI5DP]'F?$\"5]!=Olmm0;[#Ffu7$$\"5>7^t;O83#e'F ?$\"5#[`gf^!Qz7DFfu7$$\"5*>T\\UJ:v)fmF?$\"5u$o\\VH!)4Ra#Ffu7$$\"5g/l)[ DS,qt'F?$\"5pqO/JZ#Q\\d#Ffu7$$\"5RsfGtGlM8oF?$\"5n-*4k)e9'eg#Ffu7$$\"5 Op$[]fJ)y))oF?$\"5\"34p^NPfmj#Ffu7$$\"5'zI]T2'f>jpF?$\"5Bf.:\"H[4tm#Ff u7$$\"5(\\r'3Ng0VOqF?$\"5Z7)\\jH&oy(p#Ffu7$$\"5=$R%GC)4Y$3rF?$\"53&HgM ACk!GFFfu7$$\"5(o=j#o))4zyrF?$\"5+NOI(pL7\"eFFfu7$$\"5!**p#)Q(3-hZsF?$ \"5%4!3)**f6+zy#Ffu7$$\"5e>H/w(*zk9tF?$\"5$*[jHd[`R)\\rwF?$\"5lHz7tG!Rk)HFfu7$ $\"5AD[#H%)z^Bs(F?$\"5NYj#G&o&HI,$Ffu7$$\"5w(4!*)o\\!\\0x(F?$\"5C8;'>w '46RIFfu7$$\"5y:$4o;*[1;yF?$\"5nO%o&>f\\*)eyF?$\"5 qS,1jz1q*3$Ffu7$$\"55%e#Q!f%R0**yF?$\"5@@k2aNk>9JFfu7$$\"5+jl82IMdOzF? $\"5I`#>6qVd\"QJFfu7$$\"5nmy%GuN*\\rzF?$\"5JeF\"HFO'ehJFfu7$$\"5@!G]15 P*)Q+)F?$\"5\"*GSQ9*p)[%=$Ffu7$$\"5'Hs&R&=:5Q.)F?$\"5\"=P7rWDso?$Ffu7$ $\"5+z!*y2)*[Lh!)F?$\"5NYLg`\"oY(GKFfu7$$\"5wLV))RI5a'3)F?$\"5&pq^>6.B ,D$Ffu7$$\"5aQ(*zqv\"3&4\")F?$\"5)=Y7#4]L,rKFfu7$$\"5')e*)>I\\kJI\")F? $\"5M)4s%3K.V\"H$Ffu7$$\"5c%3MSH.X!\\\")F?$\"5erj9>BqQ6LFfu7$$\"5!pPn \"fH5xl\")F?$\"5\\zVvL\"e'*3L$Ffu7$$\"5oTpd!eco0=)F?$\"5kS7cP&3s*\\LFf u7$$\"5:0'4px53N>)F?$\"5'QOV*[)QE'oLFfu7$$\"5%f!Q'4f&fl/#)F?$\"5u6]7$) 4?(oQ$Ffu7$$\"5`3Gw=%)Q29#)F?$\"5Myo*f#e5s/MFfu7$$\"5i=dB)F?$\"5$H(\\)o#plZ>NFfu7$$\"5q\"*>i6^b#HB)F?$\"5*Gtq/m /JX`$Ffu7$$\"5v7;_B#>L(G#)F?$\"5%>Ou@gMw#\\NFfu7$$\"5w]afD!p^JA)F?$\"5 I3*evf(*>Pc$Ffu7$$\"5j;-yEwg=;#)F?$\"5lw,RQ(3pyd$Ffu7$$\"5p'[@%GWc$y?) 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Digi ts := 15:\nz0 := 3.9*I:\nfor ct from 157 to 160 do\n newton(R(z)=exp (ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0X7Q:%=tf!#<$\"0c+(yzl)*Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0:m'*GT!fu!#=$\"0gaCa&)e!R!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0gFMN0Ab%!#<$\"0$z9L/&H\"R!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0iPN)[y=**!#<$\"0Yss)[&)>R!#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 0 176 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.9*I))\ne nd proc:\nu0 := bisect('real_part'(u),u=1.57..1.60);\nnewton(R(z)=exp( u0*Pi*I),z=3.9*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#u0G$\"06Y!))eT\"e\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0#4>& o&*o!R!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imagin ary axis that contains the origin and lies inside the stability region " }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.9069];" "6#7$\"\"!-%&Float G6$\"&p!R!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------ ------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 11 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(e e,subs(b=`b*`,StabilityFunction(6,11,'expanded'))):\n`R*` := unapply(% ,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#% \"zG,8\"\"\"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)*&#\"?x!Q@_t2!*Gjn=JTH#\"C+++/:;ZGqa$Q`:pD7\"F)*$)F'\"\"(F )F)F)*&#\"A$H3'4@$*4Yb\"\\-)o$QU'\"F++++_$=EgA\\/G6Y8biGF)*$)F'\"\")F) F)F)*&#\"@\\*fWDUrP(3\"ySlB^>\"E++++%ys3?u\\,w.#y$=a*F)*$)F'\"\"*F)F)F )*&#\">j408-Blc/r&pF\"*)*\"D++++%)zRLDWM$4:!G!oqF)*$)F'\"#5F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We ca n 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 33 "z_0 := newton(`R*`(z)=-1,z=-3.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+!Qugy$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-3.8): \np_1 := plot([`R*`(z),-1],z=-4.29..0.49,color=[red,blue]):\np_2 := pl ot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=blac k):\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=[-4.29..0.49,-1.47..1.47],font= [HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 263 263 {PLOTDATA 2 "6+-%'CURVESG6$7V7$$!3/++++++!H%!#<$!3?1uCTg(oV$F*7$$!3;n;a,[!zB%F*$!3 )RB!fDH)z/$F*7$$!3QLL3.'4e=%F*$!3Y[RQ&GW*)p#F*7$$!3`]iX\\=[STF*$!3P64d 0LyCCF*7$$!3ym\"He4a^4%F*$!3?8P:Vwtv@F*7$$!3-]P>'HyT/%F*$!3!oKemGuI#>F *7$$!3EL$el\\-K*RF*$!3q]7x7S!op\"F*7$$!3QL$3f4v0*QF*$!3/p!3Acb8J\"F*7$ $!3am\"z`YN%)y$F*$!3M.,v7,D15F*7$$!35Le4`*QPp$F*$!3tf.7?8$R!y!#=7$$!3- +Dm*>'o&f$F*$!3!oMiFxa`$fFZ7$$!3ELez,*zU\\$F*$!3RC@go2\"*4WFZ7$$!35+D@ :))>$R$F*$!3:Z\"4Imy*=KFZ7$$!3amm@$\\C#*G$F*$!3'\\)H\"H0VbE#FZ7$$!39L$ 3Q&Qk(>$F*$!3&>XDrOpgg\"FZ7$$!3a***\\YuXX4$F*$!3X;)f'[Qy@5FZ7$$!3!)*** \\ZIC5*HF*$!31s\")*))4'[dc!#>7$$!3#)***\\`@i7*GF*$!3XF()Q0]m.AFhp7$$!3 )H$ez!oo1!GF*$\"3AXIp*=,uF$!#?7$$!3gmm\"e?WHp#F*$\"3C]&4()>pRy#Fhp7$$! 3HmmEXuo,EF*$\"3\"Rr2i')\\K_%Fhp7$$!3#)*\\i$QT`&\\#F*$\"35cC%=k)[xiFhp 7$$!3Tmmwu/d,CF*$\"3!\\Z^0l^Rn(Fhp7$$!3q*\\iLWx%)H#F*$\"3%zn[+EgL6*Fhp 7$$!3%)*\\(3)\\3.?#F*$\"3w9([u+R^/\"FZ7$$!3Im;H-&zy4#F*$\"3?3Y0[pL'=\" FZ7$$!3Dm\"z#4x\"Q+#F*$\"3OK#=\\`25K\"FZ7$$!3NL$3#e$fB!>F*$\"37)Gc;UA \\Z\"FZ7$$!3/LezXE(pz\"F*$\"3+!*))Go#yuk\"FZ7$$!3\"**\\PG?L_q\"F*$\"37 BfQ1!z0\"=FZ7$$!38L$e64_hg\"F*$\"3ur+X)>GD+#FZ7$$!3w****p2;z.:F*$\"3;. m>\\P^?AFZ7$$!3p**\\_W=l.9F*$\"3+C,ym%[cX#FZ7$$!3!)*\\iXZhnI\"F*$\"3v* p([!)\\?1FFZ7$$!3)***\\2Q7=*>\"F*$\"3/6Q*H7JS,$FZ7$$!3$pmm2h:D5\"F*$\" 3j[o(Rc^,K$FZ7$$!3+(***\\2U/$***FZ$\"3om8K^@E\"o$FZ7$$!3WJ$eRU(zd!*FZ$ \"3y'=X!)*>AUSFZ7$$!3O%***\\ecKN!)FZ$\"3CETvhpUxWFZ7$$!3sl;H5%fK2(FZ$ \"3(okCzY*fH\\FZ7$$!3i(*\\7uHingFZ$\"3o$***z;q7^aFZ7$$!3[mm;W&HW3&FZ$ \"3P6^oHCJ9gFZ7$$!3u**\\iSr6bSFZ$\"3/7U2VeNmmFZ7$$!3)\\KL$fpwjIFZ$\"3/ sfkhF4htFZ7$$!3sGL3:#o*\\?FZ$\"3WVcnc!*\\Y\")FZ7$$!3&Hm\"HOUcW5FZ$\"3g c!zv]S\"3!*FZ7$$!3e)*)***pR-27Fhp$\"3#fn&QMJ-!))*FZ7$$\"33'RLeGv9Q*Fhp $\"3)*fGrfiN)4\"F*7$$\"31ummCb>&)=FZ$\"3IEqj\"pgu?\"F*7$$\"3[4]P;i%\\* GFZ$\"3/B&3!eAvN8F*7$$\"351]ipkShQFZ$\"3Iqx9&f\"Hr9F*7$$\"3!********** *****[FZ$\"3c`:(*\\iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fa\\lF` \\l-F$6$7S7$F($!\"\"Fa\\l7$F3Ff\\l7$F=Ff\\l7$FGFf\\l7$FLFf\\l7$FQFf\\l 7$FVFf\\l7$FfnFf\\l7$F[oFf\\l7$F`oFf\\l7$FeoFf\\l7$FjoFf\\l7$F_pFf\\l7 $FdpFf\\l7$FjpFf\\l7$F_qFf\\l7$FeqFf\\l7$FjqFf\\l7$F_rFf\\l7$FdrFf\\l7 $FirFf\\l7$F^sFf\\l7$FcsFf\\l7$FhsFf\\l7$F]tFf\\l7$FbtFf\\l7$FgtFf\\l7 $F\\uFf\\l7$FauFf\\l7$FfuFf\\l7$F[vFf\\l7$F`vFf\\l7$FevFf\\l7$FjvFf\\l 7$F_wFf\\l7$FdwFf\\l7$FiwFf\\l7$F^xFf\\l7$FcxFf\\l7$FhxFf\\l7$F]yFf\\l 7$FbyFf\\l7$FgyFf\\l7$F\\zFf\\l7$FazFf\\l7$FfzFf\\l7$F[[lFf\\l7$F`[lFf \\l7$Fe[lFf\\l-Fj[l6&F\\\\lF`\\lF`\\lF]\\l-F$6&7#7$$!33+++!Qugy$F*Ff\\ l-%'SYMBOLG6#%'CIRCLEG-Fj[l6&F\\\\lFa\\lFa\\lFa\\l-%&STYLEG6#%&POINTG- F$6&F\\`l-Fa`l6#%&CROSSGFd`lFf`l-F$6&F\\`l-Fa`l6#%(DIAMONDGFd`lFf`l-F$ 6%7$7$F^`lF`\\lF]`l-%&COLORG6&F\\\\lF`\\l$\"\"&Fg\\lF`\\l-%*LINESTYLEG 6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fjbl-Fbbl6# %(DEFAULTG-%%VIEWG6$;$!$H%!\"#$\"#\\Fecl;$!$Z\"Fecl$\"$Z\"Fecl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following picture shows the st ability region for the 11 stage, order 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1134 "`R*` := z \+ -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 22941311867 6328900773522138077/1122569155338354702847161504000000*z^7+\n 6423 8368802491554609932109608293/2862551346112804492260261835200000000*z^8 +\n 1951236540781087377142254459949/954183782037601497420087278400 000000*z^9-\n 98912769571045665230213050963/7068028015093344425333 9798400000000*z^10:\npts := []: z0 := 0: tt := 0: \nwhile tt<=281/20 d o\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (3 /4<=tt and tt<=33/20) or (53/20<=tt and tt<=71/20) or\n (209/20<= tt and tt<=227/20) or (247/20<=tt and tt<=53/4) then\n hh := 1/40 \n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [ op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.2 3,0,.4)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.9,0]],i =2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.45,0,.8 )):\np_3 := plot([[[-4.49,0],[1.19,0]],[[0,-4.49],[0,4.49]]],color=bla ck,linestyle=3):\nplots[display]([p_||(1..3)],view=[-4.49..1.19,-4.49. .4.49],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes= boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 595 560 560 {PLOTDATA 2 "6+-%'CURVESG6$7^al7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3G +++Ou8,w!#F$\"3>+++qEfTJF-7$$\"3')*****\\#Q'>'=!#D$\"3&*******Q#*Q7ZF- 7$$\"35+++)pN$\\f?\"!#A$\"3'** *****fFc*4\"!#<7$$\"3C+++c+)pC$FN$\"3/+++J%\\mD\"FQ7$$\"3e+++n;0myFN$ \"3/+++L\\u89FQ7$$\"3'******H%=6%y\"!#@$\"31+++C9'3d\"FQ7$$\"37+++hrk* *QFin$\"3++++OY-G t;w\"!#?$\"3)******R4*oU?FQ7$$\"3-+++:q%fk$Fio$\"3%)*****\\(QH+AFQ7$$ \"33+++_hPXtFio$\"33+++o`*zN#FQ7$$\"3++++#\\'zG5!#>$\"3<+++'=VnV#FQ7$$ \"3++++.!z]U\"Fip$\"3&*******H`C:DFQ7$$\"37+++^Z<[>Fip$\"3!)*****H'3E$ f#FQ7$$\"3(*********)4;i#Fip$\"3/+++a&H/n#FQ7$$\"3'******4C-?Y$Fip$\"3 ++++V'eiu#FQ7$$\"3!)*****>*QurWFip$\"30+++LS9?GFQ7$$\"3&******zR^Ej&Fi 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53288024576,\na[11,8]=-1922750201834125/1941504226023936,\na[11,9]=125 39348439579/3975412795840,\na[11,10]=0,\n\nb[1]=771570009067/140362034 65200,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=28304779228000000/537074343250 74117,\nb[6]=-296881060859375/515060733835389,\nb[7]=74485830375837968 0905615939985761920312207508379/24872238844777645907644333965249221456 73887618400,\nb[8]=-5118512171875/11763620626464,\nb[9]=136801854099/1 27885521925,\nb[10]=103626500437/1717635089268,\nb[11]=0,\n\n`b*`[1]=4 48234490819/8120946290580,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5 ]=7786773134600000/14452831163890377,\n`b*`[6]=-408698637296875/567617 951573694,\n`b*`[7]=4426705150369152638325381078278067803359/148280752 30102658203818343670586143438076,\n`b*`[8]=-5004542378125/103306795935 21,\n`b*`[9]=154806770859/124231649870,\n`b*`[10]=0,\n`b*`[11]=16/243 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,/&%\"cG6#\"\"##\"\"\"\"#]/&F%6#\"\"$#\"#F\"$D\"/&F%6#\" \"%#\"#T\"$+\"/&F%6#\"\"&#\"#dF8/&F%6#\"\"'#\"#VF*/&F%6#\"\"(#\"(5DF# \")@t(>\"/&F%6#\"\")#\"#=\"#D/&F%6#\"\"*#F " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..11);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6Y/&%\"aG6$\"\"#\"\"\"#F(\"#]/&F%6$\"\" $F(#!$%f\"$D'/&F%6$F.F'#\"$H(F1/&F%6$\"\"%F(#\"$^%\"&+;#/&F%6$F:F'\"\" !/&F%6$F:F.#\"%\"o\"\"%?V/&F%6$\"\"&F(#\"#>\"$g\"/&F%6$FKF'FA/&F%6$FKF .#\"$h$\"%/J/&F%6$FKF:#\"%\\K\"%+(*/&F%6$\"\"'F(#!#J\"$+#/&F%6$F[oF'FA /&F%6$F[oF.#\"'@4_\"'c?T/&F%6$F[oF:#!&rt\"\"&S;\"/&F%6$F[oFK#\"'B?8\"' +i5/&F%6$\"\"(F(#\"A!)ptd*)fwp(*ox(owff#\"BL**=dJ^XfHG'HT^%f([/&F%6$Fg pF'FA/&F%6$FgpF.#\"E]i:(=$*f)fSYUk-$=.*yM\"F\\*>&)Q?#=3vvG4n?S<=K\"/&F %6$FgpF:#!C](y,X;6\\4x9'/&F%6$FirFgp#!@-X>25\\qh?&4\")3v$f)\"@Dce!)Q!4:ebaVX]8K/&F% 6$\"\"*F(#!7DSS8,.7[-,9\"8Rc/&F%6$FauF'FA/&F%6$FauF.#\"2vVB 6JP\"G8\"1WFHtRekJyokx4*)\"=#>$zuw!3ZvNA$4ri/&F%6$FauF[o#!7](3]jN<%\\K#z%\"9r->(*Q W(R2hxf\"/&F%6$FauFgp#!LX#p`'4m69\"*eY!H1G`\"\"LOF)*H.UZ;,_y!yas yQ6?C.7/&F%6$FauFir#!1vV*o7H+](\"3wLKZ%z'*oK\"/&F%6$\"#5F(#\"571XMahKI RO\"6DMm*)[4'e!RC$/&F%6$F_xF'FA/&F%6$F_xF.#!4]7L\\wU,CY\"\"36#[Ae9sya \"/&F%6$F_xF:#\"1]F#=X!yNT\"0Vy=PS]u)/&F%6$F_xFK#!=]*3SB&*G+ZOty$\\B\" =6`'eNHl5d(*f944\"/&F%6$F_xF[o#!5]PWA%3fg'oy\"5`f^*\\^CjYC&/&F%6$F_xFg p#\"U.(eVWsBa>^)eO**y1NW_/7\\8)z]J#\"T+Ueg2EJ!3$)yt:BN_fdr_R?/phJ/&F%6 $F_xFir#!5v\\_#zut/tM$\"5s/(o&)G+V2H$/&F%6$F_xFau#\"7YyR!GcvoeYf&\"7v, %z*4_q)*=$*=/&F%6$\"#6F(#\"7$3jB%=q1xg3D\"9?2f]&)oXxNvy>/&F%6$Fc[lF'FA /&F%6$Fc[lF.#!1D1%*HL$G7&\"0#fu$oUs3&/&F%6$Fc[lF:#\"/v3e?RH8\".'*)3%[T (G/&F%6$Fc[lFK#!?v=;Su82P\\!yk%)=*f\"?OFt+Yyp)HK ^qx\"=Y#Q-.1'\"SwX-)G`'[t>80YuO\"y'*)*4kT,0_Z(/&F%6$Fc[lFir#!1DT$=?]F# >\"1OR-EU]T>/&F%6$Fc[lFau#\"/z&R%[$RD\"\".Sez7a(R/&F%6$Fc[lF_xFA" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights \+ for the 11 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..11);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"-n!4+dr(\"/+_Y.i.9/&F%6#\"\"# \"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\"2+++G#zZIG\"2w&)*Rfh04oz$eP Ie[u\"R+%=w)QnX@#\\_'RLWw!fkxZ%)QA([#/&F%6#\"\")#!.v=<7&=^\"/kki?Ow6/& F%6#\"\"*#\"-*4a=!o8\"-D>_&)y7/&F%6#\"#5#\"-P/]EO5\".o#*3Nwr\"/&F%6#\" #6F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 11 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..1 1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%#b*G6#\"\"\"#\"->3\\M#[%\".! e!HY47)/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\"1++gMJx 'y(\"2x.*Q;JGX9/&F%6#\"\"'#!0voHP')p3%\"0%pt:&zhn&/&F%6#\"\"(#\"IfL!y1 y#y5QD$QE:p.:0nU%\"Jw!QM9'eqOM=Q?eE5I_2G[\"/&F%6#\"\")#!.D\"yBa/]\"/@N fz1L5/&F%6#\"\"*#\"-f3x1[:\"-q)\\;BC\"/&F%6#\"#5F//&F%6#\"#6#\"#;\"$V# " }}}{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 38 "#================= ====================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Scheme wi th a small principal error norm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 48 "#--------------------------- --------------------" }}{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 "c oefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4251 "ee := \{c[2]=1/1440,\nc[3] =40/183,\nc[4]=49/120,\nc[5]=49/89,\nc[6]=269/315,\nc[7]=10809269/5706 4323,\nc[8]=269/280,\nc[9]=65/66,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/1440 ,\na[3,1]=-381560/11163,\na[3,2]=128000/3721,\na[4,1]=10339/384000,\na [4,2]=0,\na[4,3]=146461/384000,\na[5,1]=6163073/56397520,\na[5,2]=0,\n a[5,3]=3939947361/26112051760,\na[5,4]=94785600/326400647,\na[6,1]=-84 /515,\na[6,2]=0,\na[6,3]=4431643925510749/3223786245394050,\na[6,4]=-1 66218606939232/90566512053885,\na[6,5]=5209733339403491/35255085001825 50,\na[7,1]=8990648437774041845694826394521064409811/20408843232331890 3399600031639662643622130,\na[7,2]=0,\na[7,3]=547930339691225609501822 503013360248045868971/1737834520761638771583667016353839075551118195, \na[7,4]=-10831375873150718357864250769796161780597600/325475696570732 91338827325045823088043427021,\na[7,5]=1221450428943196975560168771911 668179455657701/6841738152147548926226151700669066667202112842,\na[7,6 ]=-2346622810543340886158743517271000/14976878991063184099068756036931 5577,\na[8,1]=562714525421474554187476828603/1576430076296951412495515 648000,\na[8,2]=0,\na[8,3]=-232434073319953744988674545591/74510875741 29649589321728000,\na[8,4]=2895719528678335306735611/16353514577606195 8545920,\na[8,5]=-3877135960439326710451592840782413717/40032047661641 7483052489638612992000,\na[8,6]=775651514059488014593497/5870401979381 64259487744,\na[8,7]=188187525132648765731051207041043965977201/838058 3499025736698082807492981506048000,\na[9,1]=31049434346235536518435948 28643316170473/5493198556137918605753979473772031976448,\na[9,2]=0,\na [9,3]=-80379088492007830915669557451/1850963041401006078449519616,\na[ 9,4]=8519139775589074767837125/338538398802108406103202,\na[9,5]=-2470 48879110939373809210763186554743125624445065/1766762789425993194337457 4719133235934016725376,\na[9,6]=17478588815033183487684605738930625/94 70320455773783765475210993539584,\na[9,7]=5680115883329072612844775910 212680982448198390650839995/\n 184203510561647602575374593408831 494422662678761353728,\na[9,8]=-373168103571100370594887280000/2087620 5037612721992947009240147,\na[10,1]=1169280695264980001334049232853059 /1626820675759937201701331837062960,\na[10,2]=0,\na[10,3]=-14685135128 66349156603823997571/28584615782549766069481480240,\na[10,4]=975093321 34466915877268800/3238039054339140321353321,\na[10,5]=-507655973661702 4493905835117377217502141590/30139735361305596755059429070170956786361 9,\na[10,6]=18947515655188164002766115308564375/8666094601554176945344 432830508061,\na[10,7]=45727568927147685551465920458209704007893276117 767183866475/\n 12624643823865838736175575303422784678965331666 68617282491,\na[10,8]=653479766201588970500527390720000/84524048323760 095514312876121590233,\na[10,9]=-638152078395004810819432233984/214262 32114086837877217243378141,\na[11,1]=110627591724088312853370407698744 217/153961713538018628577114230979964800,\na[11,2]=0,\na[11,3]=-293711 08267043584744126857/570660576895293065417600,\na[11,4]=15110782064016 68443300/50099027666910380431,\na[11,5]=-14207562098752542925511026308 53070453627869219/84214113659151179140822400721380572464133840,\na[11, 6]=189778934355414696052914575473965/86668623987144355187734174907968, \na[11,7]=22452213418020632097025222753496035518609830065148090541733/ \n 618662354654601543316945726201272636347870721298003014592,\n a[11,8]=24095734745599243154606080/3211814700096820264958983359,\na[11 ,9]=-1660488920478680996/55817684633385330175,\na[11,10]=0,\n\nb[1]=15 7555978800757/2874472964350575,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=21050 87478471774562042359/5315246555379439663510400,\nb[6]=-116455597328635 12771875/409009954655655860540672,\nb[7]=89543313432661343622256182806 845122951561468606052473/\n 29957181581476569403363555963384114988 4537293619715840,\nb[8]=54185546099841228800000/2235820226763235235663 1,\nb[9]=-18745074956372100336/4293668048721948475,\nb[10]=12475482426 0719/56172137577600,\nb[11]=0,\n\n`b*`[1]=1641717283464529/29894518829 245980,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=95166601700256909 068767/239186094992074784857968,\n`b*`[6]=-476838119396904213375/88915 20753383823055232,\n`b*`[7]=386665692883208129148336334007158171015085 889/1295304144774171894961513004109091083904203392,\n`b*`[8]=165297038 86583383552000/6097691527536096097263,\n`b*`[9]=-54796134375796472868/ 11163536926677066035,\n`b*`[10]=0,\n`b*`[11]=5/2\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11 ),\n[`b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11)]])):\nevalf[6]( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'WWp!\"*F(%! GF+F+F+F+F+F+F+F+F+7.$\"'z&=#!\"'$!'3=M!\"%$\"'%*RMF2F+F+F+F+F+F+F+F+F +7.$\"'L$3%F/$\"'X#p#!\"($\"\"!F<$\"'49QF/F+F+F+F+F+F+F+F+7.$\"'i0bF/$ \"'z#4\"F/F;$\"'')3:F/$\"''R!HF/F+F+F+F+F+F+F+7.$\"'oR&)F/$!'2J;F/F;$ \"'nu8!\"&$!'KN=FO$\"'tx9FOF+F+F+F+F+F+7.$\"'B%*=F/$\"'F0WF:F;$\"'&H:$ F/$!''yK$F/$\"'H&y\"F/$!'$oc\"F:F+F+F+F+F+7.$\"'92'*F/$\"'bpNF/F;$!'Y> JF2$\"'qq " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6 -i)],i=2..6)]));print(``);\nfor ii from 7 to 11 do\n print(c[ii]=sub s(ee,c[ii]));print(``); \n for jj to ii-1 do\n print(a[ii,jj]=s ubs(ee,a[ii,jj]));\n end do:\n print(`____________________________ _____`);\nend do:print(``);\nfor ii to 11 do\n print(b[ii]=subs(ee,b [ii]));\nend do:\nprint(`_________________________________`);print(``) ;\nfor ii to 11 do\n print(`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"%S9F(%!GF+F+F +7(#\"#S\"$$=#!'g:Q\"&j6\"#\"'+!G\"\"%@PF+F+F+7(#\"#\\\"$?\"#\"&R.\"\" '+SQ\"\"!#\"'hk9F=6bv!RQN;qm$er(Q;w?X$yt\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%#!M+wf!yhhzp2Dk yN=2:tePJ3\"\"M@qUV!)3Be/Dt#)Q8HtqlpvaK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#\"O,xlb%z\"o;\">xo,cvp>V*G/X@7\"OUG6-smm!p1q ^hAE*[v9_\"Q<%o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"' #!C+5FL2MCB\"=+!Gd*G \"9?fae>1wd9NN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"& #!FSqe" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"K,s(f'R/Tq?^5t l([E8Dv=)=\"I+![g]\")H\\2G3)pOd-*\\$e!Q)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#l\"#m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"It /<;LkG[fV=l`NiMM%\\5$\"I[k(>.sPZzRv0'=z8c&)>$\\&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#!>^ubpc\"4$y+#\\)3z.)\"=;'>&\\%yg+,9/j4&=" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#\":Dr$ywu!*ev(R\" >&)\"9-K51%3@!))RQ&Q$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"&#!Ql]WCc7VZb'=j2@4QPR46z)[qC\"Pw`s;S$fBL\">ZduL%>$*fU*yinw\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"DD1$*Qdg%o([$=L] \"))eyu\"\"C%eRN*4@vaw$ytdX?.Z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"(#\"X&**R3l!R)>[C)4o7-\"fxWGhs!HL)e6!o&\"WGPNh(yEmAW\\J )3Mfu`d-wkh0^.U=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" )#!?++G()[fq.5rN5oJP\"AZ,C4q%H*>s7w.0i(3#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"CfI&GB \\SL,+)\\E&p!Gp6\"CgH1P=L,S-[\"[pgw\\Dy:YeG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\";+)os(e\"pYM@L4v*\":@LN@.9R V0R!QK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!L!fT@]O'yc43\"[+&Ry?:Q'\"AT\"yLC# pyi`/2`3j-6b#Ha_()4iv?9\"MSQ8kCd!Q@2SA39z6:fO69U)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"BlRZvX\"H0'p9aNM*y(*=\"Aoz!\\OHPX))\\6%QjfbjLSplZ\"e\"=d*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"8++!)G7%)*4Yb=a\"8JmN_BjnA?eB#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#!5O.5sj&\\2X(=\"4v%[>s[ !oOH%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"0>2EC[vC\"\"/ +wdP@\"QoZ\"7K_0BQQ`2_\"*))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %#b*G6#\"\"(#\"N*)e3:5 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expa nded')),op(OrderConditions(7,11,'expanded'))]:\n`RK6_11eqs*` := subs(b =`b*`,OrderConditions(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK7_11eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops( %);\nsubs(ee,`RK6_11eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#P" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-orders fro m 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for \+ ct from 2 to 5 do\n so||ct||_11 := StageOrderConditions(ct,11,'expan ded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Stages 5 to 10 have stage-order 3. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(subs(ee,so||i||_11[j])),i=2. .5)],j=1..9)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb( L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#F$\"\"$F%F%F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The stage-order s of the successive stages are given as follows." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[stage, `|`, 2, 3, 4, 5, 6, 7, \+ 8, 9, 10], [`stage-order`, `|`, 1, 2, 2, 3, 3, 3, 3, 3, 3]]);" "6#-%'m atrixG6#7$7-%&stageG%\"|grG\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"# 57-%,stage-orderGF)\"\"\"F*F*F+F+F+F+F+F+" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying co nditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i] *a[i,j],i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG \"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0! \"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 7 " . . 9 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "[Sum(b[i]*a[i,1],i=1+1..11)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..11)=b[j]*(1-c[j]),j=2..9)];\nmap(u->lhs(u)-rhs(u),subs(ee,eval (subs(Sum=add,%))));\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"#6&F*6#F-/-F&6$* &F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F -&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FO F-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\" \"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;\"\"(F4*& &F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F*6#Fc pF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF-,&F- F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&FEF`s FFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"!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 26 "The simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,3],i = 3 .. 10 ) = 0" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;F0\"#5\" \"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 3 . . 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"$F,/F+;F3\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[ i]^2*a[i,3],i = 3 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&% \"cG6#F+\"\"#F,&%\"aG6$F+\"\"$F,/F+;F5\"#5\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 14 "are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "[Sum(b[i]*a[i,3], i=3..10),Sum(b[i]*c[i]*a[i,3],i=3..10),Sum(b[i]*c[i]^2*a[i,3],i=3..10) ];\nsubs(ee,eval(subs(Sum=add,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7%-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;F0\"#5-F%6$*(F(F ,&%\"cGF*F,F-F,F1-F%6$*(F(F,)F7\"\"#F,F-F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errterms7_11 := PrincipalErrorTerms(7,11,'expan ded'):\nnrm8 := sqrt(add(subs(ee,errterms7_11[i])^2,i=1.. nops(errterm s7_11))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W\"3U5 \"!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "In addition the 2-norm of the order 9 error terms is as follows" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errterms8_11 := PrincipalEr rorTerms(8,11,'expanded'):\nnrm9 := sqrt(add(subs(ee,errterms8_11[i])^ 2,i=1.. nops(errterms8_11))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+A9mF\")!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The 2-norm of the order 9 error terms is \+ approximately 73.61 times the principal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf[10](nrm9/nrm8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6Wigt!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm of the order 6 e mbedded scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms6_11*` := subs(b=`b*`,PrincipalErrorTerms(6 ,11,'expanded')):\nsqrt(add(subs(ee,`errterms6_11*`[i])^2,i=1.. nops(` errterms6_11*`))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+r8=L5!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#-------------- ---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 7 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 89 "The scheme will be constructed so tha t stage 4 has stage-order 2 and stages 5 to 10 have " }{TEXT 260 13 "s tage-order 3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "We start by determining the nodes and weights of the scheme." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We set up a system of e quations that consists of the 7 order 7 quadrature conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 10) = 1" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"#5F-" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 1 .. 10) = 1/k;" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;F,\"#5*&F,F,F2 F3" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 7 " . . 7," }}{PARA 0 "" 0 "" {TEXT -1 35 "and the relation b etween the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c [5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^2*c[7]- 7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2 = 0;" "6#/,>** \"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\"'F'F'**\"$0\"F'*$&F)6#F. \"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F)6#F.F'&F)6#F1F7!\"\"**F =F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6#F+F'F'*(F+F'*$&F)6#F.F7F '&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F'&F)6#F.F'FD*(\"#9F'*$&F )6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD*(FenF'&F)6#F.F'&F)6#F1F 'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F'&F)6#F.F7F'\"\"!" } {TEXT -1 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[2] = 1/1440;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"%S9!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 40/183;" "6#/&%\"cG6#\"\" $*&\"#S\"\"\"\"$$=!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 49/1 20;" "6#/&%\"cG6#\"\"%*&\"#\\\"\"\"\"$?\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5] = 49/89;" "6#/&%\"cG6#\"\"&*&\"#\\\"\"\"\"#*)!\"\" " }{TEXT -1 4 ", " }{XPPEDIT 18 0 "c[6] = 269/315;" "6#/&%\"cG6#\"\" '*&\"$p#\"\"\"\"$:$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 269 /280;" "6#/&%\"cG6#\"\")*&\"$p#\"\"\"\"$!G!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9] = 65/66;" "6#/&%\"cG6#\"\"**&\"#l\"\"\"\"#m!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]=1" "6#/&%\"cG6#\"#5\"\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "Qeqs := QuadratureConditions(7,10,'expanded'):\nnode_eq := 52*c[7 ]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c[5]^2* c[6]+3*c[7]-\n 7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^ 2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-\n 12*c[6]*c[7]+14*c[7]*c[ 5]^2=0:\ncdns1 := [op(Qeqs),node_eq]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "e1 := \{c[2]=1/1440,c[3 ]=40/183,c[4]=49/120,c[5]=49/89,c[6]=269/315,\n c[8]=269/280,c [9]=65/66,c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\neqns1 := subs(e1,cdns1):\nn ops(%);\nindets(eqns1);\nnops(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"bG6#\"\"&&F%6#\"\"'&F% 6#\"\"(&F%6#\"\")&F%6#\"\"*&F%6#\"#5&%\"cGF,&F%6#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We have 8 equations and 8 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "info level[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e2 := solve(\{op(eqns1)\}):\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 0 601 "e 3 := \{c[9] = 65/66, b[5] = 2105087478471774562042359/5315246555379439 663510400, b[8] = 54185546099841228800000/22358202267632352356631, b[6 ] = -11645559732863512771875/409009954655655860540672, b[1] = 15755597 8800757/2874472964350575, c[3] = 40/183, c[6] = 269/315, b[7] = 895433 13432661343622256182806845122951561468606052473/2995718158147656940336 35559633841149884537293619715840, c[7] = 10809269/57064323, c[10] = 1, b[2] = 0, b[3] = 0, b[4] = 0, c[8] = 269/280, c[2] = 1/1440, c[4] = 4 9/120, b[10] = 124754824260719/56172137577600, b[9] = -187450749563721 00336/4293668048721948475, c[5] = 49/89\}:" }}}{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 38 "We now have all the \+ nodes and weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "seq(c[ i]=subs(e3,c[i]),i=2..10);\nseq(b[i]=subs(e3,b[i]),i=1..6);\nseq(b[i]= subs(e3,b[i]),i=7..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%\"cG6#\" \"##\"\"\"\"%S9/&F%6#\"\"$#\"#S\"$$=/&F%6#\"\"%#\"#\\\"$?\"/&F%6#\"\"& #F7\"#*)/&F%6#\"\"'#\"$p#\"$:$/&F%6#\"\"(#\")p#43\"\")BV1d/&F%6#\"\")# FD\"$!G/&F%6#\"\"*#\"#l\"#m/&F%6#\"#5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(/&%\"bG6#\"\"\"#\"0d2!)yfbd\"\"1v0NkHZuG/&F%6#\"\"#\"\"!/&F%6#\" \"$F//&F%6#\"\"%F//&F%6#\"\"&#\":fB/iXxr%yu30@\":+/^j'R%z`bY_J&/&F%6# \"\"'#!8v=x7N'Gtfbk6\"9s1agelbY&*4!4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"bG6#\"\"(#\"VtC01'o9c^H7Xo!G=cAiV8mKMJV&*)\"WSer>OHPX))\\6%Qjf bjLSplZ\"e\"=d*H/&F%6#\"\")#\"8++!)G7%)*4Yb=a\"8JmN_BjnA?eB#/&F%6#\"\" *#!5O.5sj&\\2X(=\"4v%[>s[!oOH%/&F%6#\"#5#\"0>2EC[vC\"\"/+wdP@ " 0 "" {MPLTEXT 1 0 179 "SO7 := Simpl eOrderConditions(7):\n[seq([i,SO7[i]],i=[45,50,51,54,55,59,61])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7)7%\"#X%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F0 6#*&)F.\"\"#F-F3F-F-F-F-#F-\"$?%7%\"#]F)/*(F,F-F.F--F06#*(F3F-F.F-F7F- F-#F-\"$0\"7%\"#^F)/*(F,F-F:F-F4F-#F-\"#%)7%\"#aF)/*&F,F--F06#*(F3F-F. F--F06#*&)F.\"\"$F-F3F-F-F-#F-\"$o\"7%\"#bF)/*(F,F-F.F--F06#*&F3F-FTF- F-#F-\"$S\"7%\"#fF)/*(F,F-F:F-FTF-#F-\"#G7%\"#hF)/*(F,F-F.F--F06#*&)F. \"\"%F-F3F-F-#F-\"#NQ)pprint346\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We also incorporate the row-sum conditions," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j = 1 .. i-1) = c[i] " "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . 10, " }}{PARA 0 "" 0 "" {TEXT -1 29 " the stage order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j],j = 2 .. i-1) = 1/2" "6#/-%$SumG6$*& &%\"aG6$%\"iG%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-F3F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG\"\"#" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "i=5" "6#/%\"iG\"\"&" }{TEXT -1 9 " . . 10, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i, j]*c[j]^2,j = 2 .. i-1) = 1/3" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\" *$&%\"cG6#F,\"\"#F-/F,;F2,&F+F-F-!\"\"*&F-F-\"\"$F6" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[i]^3" "6#*$&%\"cG6#%\"iG\"\"$" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "i = 2;" "6#/%\"iG\"\"#" }{TEXT -1 8 " . . 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "together with t he column simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(b[i]*a[i,1],i=2..10)=b[1]" "6#/-%$SumG6$*&&%\"bG 6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#5&F)6#F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 10) = b[j]*(1-c[j]);" "6#/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#5*&&F)6 #F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j = 6 ;" "6#/%\"jG\"\"'" }{TEXT -1 8 " . . 9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "and the further simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 5 .. 10) = 0; " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+; \"\"&\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a [i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F +\"\"#F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 489 "S O7_10 := SimpleOrderConditions(7,10,'expanded'):\nSO_eqs := [op(RowSum Conditions(10,'expanded')),op(StageOrderConditions(2,3..10,'expanded') ),\n op(StageOrderConditions(3,5..10,'expanded'))]:\nord_cdns := [seq (SO7_10[i],i=[45,50,51,54,55,59,61])]:\nsimp_eqs := [add(b[i]*a[i,1],i =2..10)=b[1],seq(add(b[i]*a[i,j],i=j+1..10)=b[j]*(1-c[j]),j=6..9)]:\ns imp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5..10)=0,add(b[i]*c[i]^2*a[i,3],i= 5..10)=0]:\ncdns2 := [op(simp_eqs),op(simp_eqs2),op(SO_eqs),op(ord_cdn s)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " We specify 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#/&%\"a G6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/ &%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "and the linking coefficient " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[6,1] = -84/515;" "6#/&%\"aG6$\"\"'\"\"\",$*&\"#%)F (\"$:&!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "e4 := \{seq(a[i,2]=0,i=4..1 0),a[6,1]=-84/515\}:\ne5 := `union`(e3,e4):\neqns2 := subs(e5,cdns2): \nnops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e6 := solve(\{op(eqns2)\}):\ninfolevel[solve] := 0:\ne7 := `un ion`(e5,e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3258 "e7 := \{a[6,5] = 5209733339403491/3525508500182550, a[2,1] = 1/1 440, a[10,2] = 0, a[9,2] = 0, c[9] = 65/66, a[9,4] = 85191397755890747 67837125/338538398802108406103202, a[9,3] = -8037908849200783091566955 7451/1850963041401006078449519616, a[5,2] = 0, b[5] = 2105087478471774 562042359/5315246555379439663510400, a[10,8] = 65347976620158897050052 7390720000/84524048323760095514312876121590233, a[6,1] = -84/515, a[5, 1] = 6163073/56397520, a[9,7] = 56801158833290726128447759102126809824 48198390650839995/1842035105616476025753745934088314944226626787613537 28, a[7,3] = 547930339691225609501822503013360248045868971/17378345207 61638771583667016353839075551118195, a[7,4] = -10831375873150718357864 250769796161780597600/32547569657073291338827325045823088043427021, a[ 10,5] = -5076559736617024493905835117377217502141590/30139735361305596 7550594290701709567863619, b[8] = 54185546099841228800000/223582022676 32352356631, b[6] = -11645559732863512771875/409009954655655860540672, b[1] = 157555978800757/2874472964350575, a[7,2] = 0, a[8,2] = 0, a[6, 2] = 0, a[7,6] = -2346622810543340886158743517271000/14976878991063184 0990687560369315577, a[6,3] = 4431643925510749/3223786245394050, a[4,2 ] = 0, c[3] = 40/183, c[6] = 269/315, a[10,3] = -146851351286634915660 3823997571/28584615782549766069481480240, b[7] = 895433134326613436222 56182806845122951561468606052473/2995718158147656940336355596338411498 84537293619715840, a[4,3] = 146461/384000, a[10,6] = 18947515655188164 002766115308564375/8666094601554176945344432830508061, c[7] = 10809269 /57064323, c[10] = 1, b[2] = 0, b[3] = 0, b[4] = 0, a[9,6] = 174785888 15033183487684605738930625/9470320455773783765475210993539584, a[10,4] = 97509332134466915877268800/3238039054339140321353321, a[9,8] = -373 168103571100370594887280000/20876205037612721992947009240147, c[8] = 2 69/280, a[7,5] = 1221450428943196975560168771911668179455657701/684173 8152147548926226151700669066667202112842, a[9,5] = -247048879110939373 809210763186554743125624445065/176676278942599319433745747191332359340 16725376, a[10,9] = -638152078395004810819432233984/214262321140868378 77217243378141, a[8,4] = 2895719528678335306735611/1635351457760619585 45920, a[3,2] = 128000/3721, a[8,1] = 562714525421474554187476828603/1 576430076296951412495515648000, c[2] = 1/1440, a[8,3] = -2324340733199 53744988674545591/7451087574129649589321728000, a[5,3] = 3939947361/26 112051760, c[4] = 49/120, a[9,1] = 31049434346235536518435948286433161 70473/5493198556137918605753979473772031976448, a[8,7] = 1881875251326 48765731051207041043965977201/8380583499025736698082807492981506048000 , a[10,1] = 1169280695264980001334049232853059/16268206757599372017013 31837062960, b[10] = 124754824260719/56172137577600, b[9] = -187450749 56372100336/4293668048721948475, a[8,6] = 775651514059488014593497/587 040197938164259487744, a[10,7] = 4572756892714768555146592045820970400 7893276117767183866475/12624643823865838736175575303422784678965331666 68617282491, a[8,5] = -3877135960439326710451592840782413717/400320476 616417483052489638612992000, a[3,1] = -381560/11163, a[6,4] = -1662186 06939232/90566512053885, c[5] = 49/89, a[4,1] = 10339/384000, a[5,4] = 94785600/326400647, a[7,1] = 8990648437774041845694826394521064409811 /204088432323318903399600031639662643622130\}:" }}}{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 0 338 "s ubs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));pri nt(``);\nfor ii from 7 to 10 do\n print(c[ii]=subs(e7,c[ii]));print( ``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e7,a[ii,jj])); \n end do:\n print(`_________________________________`);\nend do:p rint(``);\nfor ii to 10 do\n print(b[ii]=subs(e7,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"%S9F(%!GF+F +F+7(#\"#S\"$$=#!'g:Q\"&j6\"#\"'+!G\"\"%@PF+F+F+7(#\"#\\\"$?\"#\"&R.\" \"'+SQ\"\"!#\"'hk9F=6bv!RQN;qm$er(Q;w?X $yt\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%#!M+wf!yhh zp2DkyN=2:tePJ3\"\"M@qUV!)3Be/Dt#)Q8HtqlpvaK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#\"O,xlb%z\"o;\">xo,cvp>V*G/X@7\"OUG 6-smm!p1q^hAE*[v9_\"Q<%o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"(\"\"'#!C+5FL2MCB\"=+!Gd*G\"9?fae>1wd9NN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"&#!FSqe" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"K,s(f'R /Tq?^5tl([E8Dv=)=\"I+![g]\")H\\2G3)pOd-*\\$e!Q)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#l\"#m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"It /<;LkG[fV=l`NiMM%\\5$\"I[k(>.sPZzRv0'=z8c&)>$\\&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#!>^ubpc\"4$y+#\\)3z.)\"=;'>&\\%yg+,9/j4&=" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#\":Dr$ywu!*ev(R\" >&)\"9-K51%3@!))RQ&Q$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"&#!Ql]WCc7VZb'=j2@4QPR46z)[qC\"Pw`s;S$fBL\">ZduL%>$*fU*yinw\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"DD1$*Qdg%o([$=L] \"))eyu\"\"C%eRN*4@vaw$ytdX?.Z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"(#\"X&**R3l!R)>[C)4o7-\"fxWGhs!HL)e6!o&\"WGPNh(yEmAW\\J )3Mfu`d-wkh0^.U=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" )#!?++G()[fq.5rN5oJP\"AZ,C4q%H*>s7w.0i(3#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"CfI&GB \\SL,+)\\E&p!Gp6\"CgH1P=L,S-[\"[pgw\\Dy:YeG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\";+)os(e\"pYM@L4v*\":@LN@.9R V0R!QK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!L!fT@]O'yc43\"[+&Ry?:Q'\"AT\"yLCOHPX))\\6%QjfbjLSpl Z\"e\"=d*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"8++!)G7 %)*4Yb=a\"8JmN_BjnA?eB#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"*#!5O.5sj&\\2X(=\"4v%[>s[!oOH%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"bG6#\"#5#\"0>2EC[vC\"\"/+wdP@ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------ ---------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#------------------------------ -------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the embedded order 6 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3258 "e7 := \{a[6,5] = 520973333 9403491/3525508500182550, a[2,1] = 1/1440, a[10,2] = 0, a[9,2] = 0, c[ 9] = 65/66, a[9,4] = 8519139775589074767837125/33853839880210840610320 2, a[9,3] = -80379088492007830915669557451/185096304140100607844951961 6, a[5,2] = 0, b[5] = 2105087478471774562042359/5315246555379439663510 400, a[10,8] = 653479766201588970500527390720000/845240483237600955143 12876121590233, a[6,1] = -84/515, a[5,1] = 6163073/56397520, a[9,7] = \+ 5680115883329072612844775910212680982448198390650839995/18420351056164 7602575374593408831494422662678761353728, a[7,3] = 5479303396912256095 01822503013360248045868971/1737834520761638771583667016353839075551118 195, a[7,4] = -10831375873150718357864250769796161780597600/3254756965 7073291338827325045823088043427021, a[10,5] = -50765597366170244939058 35117377217502141590/301397353613055967550594290701709567863619, b[8] \+ = 54185546099841228800000/22358202267632352356631, b[6] = -11645559732 863512771875/409009954655655860540672, b[1] = 157555978800757/28744729 64350575, a[7,2] = 0, a[8,2] = 0, a[6,2] = 0, a[7,6] = -23466228105433 40886158743517271000/149768789910631840990687560369315577, a[6,3] = 44 31643925510749/3223786245394050, a[4,2] = 0, c[3] = 40/183, c[6] = 269 /315, a[10,3] = -1468513512866349156603823997571/285846157825497660694 81480240, b[7] = 89543313432661343622256182806845122951561468606052473 /299571815814765694033635559633841149884537293619715840, a[4,3] = 1464 61/384000, a[10,6] = 18947515655188164002766115308564375/8666094601554 176945344432830508061, c[7] = 10809269/57064323, c[10] = 1, b[2] = 0, \+ b[3] = 0, b[4] = 0, a[9,6] = 17478588815033183487684605738930625/94703 20455773783765475210993539584, a[10,4] = 97509332134466915877268800/32 38039054339140321353321, a[9,8] = -373168103571100370594887280000/2087 6205037612721992947009240147, c[8] = 269/280, a[7,5] = 122145042894319 6975560168771911668179455657701/68417381521475489262261517006690666672 02112842, a[9,5] = -247048879110939373809210763186554743125624445065/1 7667627894259931943374574719133235934016725376, a[10,9] = -63815207839 5004810819432233984/21426232114086837877217243378141, a[8,4] = 2895719 528678335306735611/163535145776061958545920, a[3,2] = 128000/3721, a[8 ,1] = 562714525421474554187476828603/1576430076296951412495515648000, \+ c[2] = 1/1440, a[8,3] = -232434073319953744988674545591/74510875741296 49589321728000, a[5,3] = 3939947361/26112051760, c[4] = 49/120, a[9,1] = 3104943434623553651843594828643316170473/54931985561379186057539794 73772031976448, a[8,7] = 188187525132648765731051207041043965977201/83 80583499025736698082807492981506048000, a[10,1] = 11692806952649800013 34049232853059/1626820675759937201701331837062960, b[10] = 12475482426 0719/56172137577600, b[9] = -18745074956372100336/4293668048721948475, a[8,6] = 775651514059488014593497/587040197938164259487744, a[10,7] = 45727568927147685551465920458209704007893276117767183866475/126246438 2386583873617557530342278467896533166668617282491, a[8,5] = -387713596 0439326710451592840782413717/400320476616417483052489638612992000, a[3 ,1] = -381560/11163, a[6,4] = -166218606939232/90566512053885, c[5] = \+ 49/89, a[4,1] = 10339/384000, a[5,4] = 94785600/326400647, a[7,1] = 89 90648437774041845694826394521064409811/2040884323233189033996000316396 62643622130\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We now turn our attention to the embedded order \+ 6 scheme and introduce a new row corresponding to the node " } {XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 92 "The linking coefficients and weights can \+ be chosen so as to form an 11 stage order 6 scheme." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We use the order 6 quad rature conditions which are given in abreviated form as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "subs(b=`b*`,QuadratureCondi tions(6)):\nListTools[Enumerate](%):\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(F57% \"\"%F)/*&F,F()F2F5F(#F(F;7%\"\"&F)/*&F,F()F2F;F(#F(FA7%\"\"'F)/*&F,F( )F2FAF(#F(FGQ(pprint86\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We \+ incorporate the row sum condition for the new tenth row" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j],j = 1 .. 10) = c[11] ;" "6#/-%$SumG6$&%\"aG6$\"#6%\"jG/F+;\"\"\"\"#5&%\"cG6#F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "together with the stage-order eq uations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j ]*c[j],j = 2 .. 10) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"#6%\"jG\"\"\"&%\" cG6#F,F-/F,;\"\"#\"#5*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[ 11]^2;" "6#*$&%\"cG6#\"#6\"\"#" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j]^2,j = 2 .. 10) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"#6% \"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"#5*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^3;" "6#*$&%\"cG6#\"#6\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 50 "which ensure that the tenth row has sta ge-order 3." }}{PARA 0 "" 0 "" {TEXT -1 53 "We also incorporate the co lumn simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,1],i=2..11)=`b*`[1]" "6#/-%$SumG6$*&&%# b*G6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#6&F)6#F," }{TEXT -1 2 ", " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6* &&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 6 " , " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" }{TEXT -1 10 ", 6, 7, 8." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "`Qeqs*` : = subs(b=`b*`,QuadratureConditions(6,11,'expanded')):\nSO_eqs2 := [add (a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n add (a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3]:\n`simp_eqs*` := [add(`b*`[i]*a[ i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j ]),j=[$5..8])]:\n`cdns*` := [op(SO_eqs2),op(`Qeqs*`),op(`simp_eqs*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We s pecify that " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG6$\"#6\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,10] = 0;" "6#/&%\"aG6$\"#6 \"#5\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\" \"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6# \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6# \"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[10]=0" "6#/&%#b*G6# \"#5\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[11] = 5/2;" "6#/ &%#b*G6#\"#6*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 13 "We also set " }{XPPEDIT 18 0 "b[11] = 0;" "6#/&%\"bG6# \"#6\"\"!" }{TEXT -1 66 ", so that the order 7 scheme can be regarded \+ as a 11 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 14 equations for the 14 unknown coefficients. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "e8 := \{c[11]=1,b[11]=0 ,a[11,2]=0,a[11,10]=0,`b*`[2]=0,`b*`[3]=0,`b*`[4]=0,`b*`[10]=0,`b*`[11 ]=5/2\}:\ne9 := `union`(e7,e8):\n`eqns*` := subs(e9,`cdns*`):\nnops(%) ;\nindets(`eqns*`);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<0&%#b*G6#\"\"&&F%6#\"\"'&F%6#\"\" (&F%6#\"\")&F%6#\"\"*&%\"aG6$\"#6\"\"\"&F56$F7\"\"$&F56$F7\"\"%&F56$F7 F'&F56$F7F*&F56$F7F-&F56$F7F0&F56$F7F3&F%6#F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e10 := solve(\{op(`eqns*`)\}):\ninfolevel [solve] := 0:\ne11 := `union`(e9,e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4391 "e11 := \{c[11] = 1, b[11] = 0, a[ 6,5] = 5209733339403491/3525508500182550, a[2,1] = 1/1440, a[10,2] = 0 , a[9,2] = 0, c[9] = 65/66, a[9,4] = 8519139775589074767837125/3385383 98802108406103202, a[9,3] = -80379088492007830915669557451/18509630414 01006078449519616, a[5,2] = 0, b[5] = 2105087478471774562042359/531524 6555379439663510400, a[11,9] = -1660488920478680996/558176846333853301 75, `b*`[8] = 16529703886583383552000/6097691527536096097263, a[11,8] \+ = 24095734745599243154606080/3211814700096820264958983359, `b*`[7] = 3 86665692883208129148336334007158171015085889/1295304144774171894961513 004109091083904203392, `b*`[1] = 1641717283464529/29894518829245980, a [11,6] = 189778934355414696052914575473965/866686239871443551877341749 07968, a[11,2] = 0, a[11,10] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = \+ 0, `b*`[6] = -476838119396904213375/8891520753383823055232, a[10,8] = \+ 653479766201588970500527390720000/84524048323760095514312876121590233, a[6,1] = -84/515, a[5,1] = 6163073/56397520, `b*`[9] = -5479613437579 6472868/11163536926677066035, a[11,7] = 224522134180206320970252227534 96035518609830065148090541733/6186623546546015433169457262012726363478 70721298003014592, a[9,7] = 568011588332907261284477591021268098244819 8390650839995/184203510561647602575374593408831494422662678761353728, \+ a[7,3] = 547930339691225609501822503013360248045868971/173783452076163 8771583667016353839075551118195, a[7,4] = -108313758731507183578642507 69796161780597600/32547569657073291338827325045823088043427021, a[10,5 ] = -5076559736617024493905835117377217502141590/301397353613055967550 594290701709567863619, b[8] = 54185546099841228800000/2235820226763235 2356631, b[6] = -11645559732863512771875/409009954655655860540672, b[1 ] = 157555978800757/2874472964350575, a[7,2] = 0, a[8,2] = 0, a[6,2] = 0, a[11,3] = -29371108267043584744126857/570660576895293065417600, a[ 7,6] = -2346622810543340886158743517271000/149768789910631840990687560 369315577, a[6,3] = 4431643925510749/3223786245394050, a[4,2] = 0, c[3 ] = 40/183, c[6] = 269/315, a[10,3] = -1468513512866349156603823997571 /28584615782549766069481480240, b[7] = 8954331343266134362225618280684 5122951561468606052473/29957181581476569403363555963384114988453729361 9715840, `b*`[10] = 0, a[4,3] = 146461/384000, a[10,6] = 1894751565518 8164002766115308564375/8666094601554176945344432830508061, c[7] = 1080 9269/57064323, c[10] = 1, b[2] = 0, b[3] = 0, b[4] = 0, a[9,6] = 17478 588815033183487684605738930625/9470320455773783765475210993539584, a[1 0,4] = 97509332134466915877268800/3238039054339140321353321, a[9,8] = \+ -373168103571100370594887280000/20876205037612721992947009240147, c[8] = 269/280, a[7,5] = 1221450428943196975560168771911668179455657701/68 41738152147548926226151700669066667202112842, a[9,5] = -24704887911093 9373809210763186554743125624445065/17667627894259931943374574719133235 934016725376, a[10,9] = -638152078395004810819432233984/21426232114086 837877217243378141, a[11,5] = -142075620987525429255110263085307045362 7869219/84214113659151179140822400721380572464133840, a[8,4] = 2895719 528678335306735611/163535145776061958545920, `b*`[5] = 951666017002569 09068767/239186094992074784857968, a[3,2] = 128000/3721, a[8,1] = 5627 14525421474554187476828603/1576430076296951412495515648000, `b*`[11] = 5/2, c[2] = 1/1440, a[8,3] = -232434073319953744988674545591/74510875 74129649589321728000, a[5,3] = 3939947361/26112051760, c[4] = 49/120, \+ a[9,1] = 3104943434623553651843594828643316170473/54931985561379186057 53979473772031976448, a[8,7] = 188187525132648765731051207041043965977 201/8380583499025736698082807492981506048000, a[10,1] = 11692806952649 80001334049232853059/1626820675759937201701331837062960, a[11,1] = 110 627591724088312853370407698744217/153961713538018628577114230979964800 , b[10] = 124754824260719/56172137577600, b[9] = -18745074956372100336 /4293668048721948475, a[8,6] = 775651514059488014593497/58704019793816 4259487744, a[10,7] = 457275689271476855514659204582097040078932761177 67183866475/1262464382386583873617557530342278467896533166668617282491 , a[8,5] = -3877135960439326710451592840782413717/40032047661641748305 2489638612992000, a[3,1] = -381560/11163, a[6,4] = -166218606939232/90 566512053885, c[5] = 49/89, a[11,4] = 1511078206401668443300/500990276 66910380431, a[4,1] = 10339/384000, a[5,4] = 94785600/326400647, a[7,1 ] = 8990648437774041845694826394521064409811/2040884323233189033996000 31639662643622130\}:" }}}{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 0 137 "subs(e11,matrix([seq([c[i], seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[j],j=1..11)],[`b *`,seq(`b*`[j],j=1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'WWp!\"*F(%!GF+F+F+F+F+F+F+F+F+7.$\" 'z&=#!\"'$!'3=M!\"%$\"'%*RMF2F+F+F+F+F+F+F+F+F+7.$\"'L$3%F/$\"'X#p#!\" ($\"\"!F<$\"'49QF/F+F+F+F+F+F+F+F+7.$\"'i0bF/$\"'z#4\"F/F;$\"'')3:F/$ \"''R!HF/F+F+F+F+F+F+F+7.$\"'oR&)F/$!'2J;F/F;$\"'nu8!\"&$!'KN=FO$\"'tx 9FOF+F+F+F+F+F+7.$\"'B%*=F/$\"'F0WF:F;$\"'&H:$F/$!''yK$F/$\"'H&y\"F/$! '$oc\"F:F+F+F+F+F+7.$\"'92'*F/$\"'bpNF/F;$!'Y>JF2$\"'qq " 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 0 459 "subs(e11, matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..5)]));print(``); \nfor ii from 6 to 11 do\n print(c[ii]=subs(e11,c[ii]));print(``); \+ \n for jj to ii-1 do\n print(a[ii,jj]=subs(e11,a[ii,jj]));\n \+ end do:\n print(`_________________________________`);\nend do:print( ``);\nfor ii to 11 do\n print(b[ii]=subs(e11,b[ii]));\nend do:\nprin t(`_________________________________`);print(``);\nfor ii to 11 do\n \+ print(`b*`[ii]=subs(e11,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7(#\"\"\"\"%S9F(%!GF+F+F+7(#\"#S\"$$=#!'g :Q\"&j6\"#\"'+!G\"\"%@PF+F+F+7(#\"#\\\"$?\"#\"&R.\"\"'+SQ\"\"!#\"'hk9F =6bv!RQN;qm $er(Q;w?X$yt\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%# !M+wf!yhhzp2DkyN=2:tePJ3\"\"M@qUV!)3Be/Dt#)Q8HtqlpvaK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#\"O,xlb%z\"o;\">xo,cvp>V*G/X@7 \"OUG6-smm!p1q^hAE*[v9_\"Q<%o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"(\"\"'#!C+5FL2MCB\"=+!Gd*G\"9?fae>1wd9NN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"&#!FSqe" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"K,s(f'R /Tq?^5tl([E8Dv=)=\"I+![g]\")H\\2G3)pOd-*\\$e!Q)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#l\"#m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"It /<;LkG[fV=l`NiMM%\\5$\"I[k(>.sPZzRv0'=z8c&)>$\\&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#!>^ubpc\"4$y+#\\)3z.)\"=;'>&\\%yg+,9/j4&=" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#\":Dr$ywu!*ev(R\" >&)\"9-K51%3@!))RQ&Q$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"&#!Ql]WCc7VZb'=j2@4QPR46z)[qC\"Pw`s;S$fBL\">ZduL%>$*fU*yinw\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"DD1$*Qdg%o([$=L] \"))eyu\"\"C%eRN*4@vaw$ytdX?.Z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"(#\"X&**R3l!R)>[C)4o7-\"fxWGhs!HL)e6!o&\"WGPNh(yEmAW\\J )3Mfu`d-wkh0^.U=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" )#!?++G()[fq.5rN5oJP\"AZ,C4q%H*>s7w.0i(3#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"CfI&GB \\SL,+)\\E&p!Gp6\"CgH1P=L,S-[\"[pgw\\Dy:YeG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#\";+)os(e\"pYM@L4v*\":@LN@.9R V0R!QK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!L!fT@]O'yc43\"[+&Ry?:Q'\"AT\"yLC# pyi`/2`3j-6b#Ha_()4iv?9\"MSQ8kCd!Q@2SA39z6:fO69U)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"BlRZvX\"H0'p9aNM*y(*=\"Aoz!\\OHPX))\\6%QjfbjLSplZ\"e\"=d*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"8++!)G7%)*4Yb=a\"8JmN_BjnA?eB#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#!5O.5sj&\\2X(=\"4v%[>s[ !oOH%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"0>2EC[vC\"\"/ +wdP@\"QoZ\"7K_0BQQ`2_\"*))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %#b*G6#\"\"(#\"N*)e3:5 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expa nded')),op(OrderConditions(7,11,'expanded'))]:\n`RK6_11eqs*` := subs(b =`b*`,OrderConditions(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e11,RK7_11eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops (%);\nsubs(e11,`RK6_11eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#----- --------------------------------------------------------" }}{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4251 "ee := \{c[2]=1/1440,\nc[3]=40/183,\nc[4]=49/12 0,\nc[5]=49/89,\nc[6]=269/315,\nc[7]=10809269/57064323,\nc[8]=269/280, \nc[9]=65/66,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/1440,\na[3,1]=-381560/11 163,\na[3,2]=128000/3721,\na[4,1]=10339/384000,\na[4,2]=0,\na[4,3]=146 461/384000,\na[5,1]=6163073/56397520,\na[5,2]=0,\na[5,3]=3939947361/26 112051760,\na[5,4]=94785600/326400647,\na[6,1]=-84/515,\na[6,2]=0,\na[ 6,3]=4431643925510749/3223786245394050,\na[6,4]=-166218606939232/90566 512053885,\na[6,5]=5209733339403491/3525508500182550,\na[7,1]=89906484 37774041845694826394521064409811/2040884323233189033996000316396626436 22130,\na[7,2]=0,\na[7,3]=54793033969122560950182250301336024804586897 1/1737834520761638771583667016353839075551118195,\na[7,4]=-10831375873 150718357864250769796161780597600/325475696570732913388273250458230880 43427021,\na[7,5]=1221450428943196975560168771911668179455657701/68417 38152147548926226151700669066667202112842,\na[7,6]=-234662281054334088 6158743517271000/149768789910631840990687560369315577,\na[8,1]=5627145 25421474554187476828603/1576430076296951412495515648000,\na[8,2]=0,\na [8,3]=-232434073319953744988674545591/7451087574129649589321728000,\na [8,4]=2895719528678335306735611/163535145776061958545920,\na[8,5]=-387 7135960439326710451592840782413717/40032047661641748305248963861299200 0,\na[8,6]=775651514059488014593497/587040197938164259487744,\na[8,7]= 188187525132648765731051207041043965977201/838058349902573669808280749 2981506048000,\na[9,1]=3104943434623553651843594828643316170473/549319 8556137918605753979473772031976448,\na[9,2]=0,\na[9,3]=-80379088492007 830915669557451/1850963041401006078449519616,\na[9,4]=8519139775589074 767837125/338538398802108406103202,\na[9,5]=-2470488791109393738092107 63186554743125624445065/1766762789425993194337457471913323593401672537 6,\na[9,6]=17478588815033183487684605738930625/94703204557737837654752 10993539584,\na[9,7]=5680115883329072612844775910212680982448198390650 839995/\n 184203510561647602575374593408831494422662678761353728 ,\na[9,8]=-373168103571100370594887280000/2087620503761272199294700924 0147,\na[10,1]=1169280695264980001334049232853059/16268206757599372017 01331837062960,\na[10,2]=0,\na[10,3]=-1468513512866349156603823997571/ 28584615782549766069481480240,\na[10,4]=97509332134466915877268800/323 8039054339140321353321,\na[10,5]=-507655973661702449390583511737721750 2141590/301397353613055967550594290701709567863619,\na[10,6]=189475156 55188164002766115308564375/8666094601554176945344432830508061,\na[10,7 ]=45727568927147685551465920458209704007893276117767183866475/\n \+ 1262464382386583873617557530342278467896533166668617282491,\na[10,8] =653479766201588970500527390720000/84524048323760095514312876121590233 ,\na[10,9]=-638152078395004810819432233984/214262321140868378772172433 78141,\na[11,1]=110627591724088312853370407698744217/15396171353801862 8577114230979964800,\na[11,2]=0,\na[11,3]=-29371108267043584744126857/ 570660576895293065417600,\na[11,4]=1511078206401668443300/500990276669 10380431,\na[11,5]=-1420756209875254292551102630853070453627869219/842 14113659151179140822400721380572464133840,\na[11,6]=189778934355414696 052914575473965/86668623987144355187734174907968,\na[11,7]=22452213418 020632097025222753496035518609830065148090541733/\n 61866235465 4601543316945726201272636347870721298003014592,\na[11,8]=2409573474559 9243154606080/3211814700096820264958983359,\na[11,9]=-1660488920478680 996/55817684633385330175,\na[11,10]=0,\n\nb[1]=157555978800757/2874472 964350575,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=2105087478471774562042359/ 5315246555379439663510400,\nb[6]=-11645559732863512771875/409009954655 655860540672,\nb[7]=89543313432661343622256182806845122951561468606052 473/\n 299571815814765694033635559633841149884537293619715840,\nb[ 8]=54185546099841228800000/22358202267632352356631,\nb[9]=-18745074956 372100336/4293668048721948475,\nb[10]=124754824260719/56172137577600, \nb[11]=0,\n\n`b*`[1]=1641717283464529/29894518829245980,\n`b*`[2]=0, \n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=95166601700256909068767/239186094992 074784857968,\n`b*`[6]=-476838119396904213375/8891520753383823055232, \n`b*`[7]=386665692883208129148336334007158171015085889/12953041447741 71894961513004109091083904203392,\n`b*`[8]=16529703886583383552000/609 7691527536096097263,\n`b*`[9]=-54796134375796472868/111635369266770660 35,\n`b*`[10]=0,\n`b*`[11]=5/2\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[7,11]; " "6#&%\"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose compo nents are the principal error terms of the 11 stage, order 7 scheme (t he error terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "`T*`[6,11];" "6#&%#T*G6$\"\"'\"#6" }{TEXT -1 146 " den ote the vector whose components are the principal error terms of the e mbedded 11 stage, order 6 scheme (the error terms of order 7) and let \+ " }{XPPEDIT 18 0 "`T*`[7,11];" "6#&%#T*G6$\"\"(\"#6" }{TEXT -1 99 " \+ denote the vector whose components are the error terms of order 8 of t he embedded order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " } {XPPEDIT 18 0 "abs(abs(T[7,11]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6 " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,11]));" "6#-%$absG6 #-F$6#&%#T*G6$\"\"'\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[7,11]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\"#6" }{TEXT -1 15 " re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[8 ] = abs(abs(T[7,11]));" "6#/&%\"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\" #6" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,11]))/a bs(abs(`T*`[6,11]));" "6#/&%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\" #6\"\"\"-F*6#-F*6#&F/6$\"\"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7,11]-T[7,11]))/abs(abs(`T*`[6,11]));" "6#/& %\"CG6#\"\")*&-%$absG6#-F*6#,&&%#T*G6$\"\"(\"#6\"\"\"&%\"TG6$F2F3!\"\" F4-F*6#-F*6#&F06$\"\"'F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggest ed that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[8]; " "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embedded sche me is to be used for error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"B G6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\" )" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much \+ from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := PrincipalErrorTerms(7,11,'expanded') :\n`errterms7_11*` :=subs(b=`b*`,PrincipalErrorTerms(7,11,'expanded')) :\n`errterms6_11*` := subs(b=`b*`,PrincipalErrorTerms(6,11,'expanded') ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms7_11*`[i]))^ 2,i=1..nops(`errterms7_11*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errte rms6_11*`[i]))^2,i=1..nops(`errterms6_11*`))):\nsnmC := sqrt(add((eval f(subs(ee,`errterms7_11*`[i])-subs(ee,errterms7_11[i])))^2,i=1..nops(e rrterms7_11))):\n'B[8]'= evalf[8](snmB/sdnB);\n'C[8]'= evalf[8](snmC/s dnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")g5bC!\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\")$\")?7*[#!\"(" }}} {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 "c oefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4251 "ee := \{c[2]=1/1440,\nc[3] =40/183,\nc[4]=49/120,\nc[5]=49/89,\nc[6]=269/315,\nc[7]=10809269/5706 4323,\nc[8]=269/280,\nc[9]=65/66,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/1440 ,\na[3,1]=-381560/11163,\na[3,2]=128000/3721,\na[4,1]=10339/384000,\na [4,2]=0,\na[4,3]=146461/384000,\na[5,1]=6163073/56397520,\na[5,2]=0,\n a[5,3]=3939947361/26112051760,\na[5,4]=94785600/326400647,\na[6,1]=-84 /515,\na[6,2]=0,\na[6,3]=4431643925510749/3223786245394050,\na[6,4]=-1 66218606939232/90566512053885,\na[6,5]=5209733339403491/35255085001825 50,\na[7,1]=8990648437774041845694826394521064409811/20408843232331890 3399600031639662643622130,\na[7,2]=0,\na[7,3]=547930339691225609501822 503013360248045868971/1737834520761638771583667016353839075551118195, \na[7,4]=-10831375873150718357864250769796161780597600/325475696570732 91338827325045823088043427021,\na[7,5]=1221450428943196975560168771911 668179455657701/6841738152147548926226151700669066667202112842,\na[7,6 ]=-2346622810543340886158743517271000/14976878991063184099068756036931 5577,\na[8,1]=562714525421474554187476828603/1576430076296951412495515 648000,\na[8,2]=0,\na[8,3]=-232434073319953744988674545591/74510875741 29649589321728000,\na[8,4]=2895719528678335306735611/16353514577606195 8545920,\na[8,5]=-3877135960439326710451592840782413717/40032047661641 7483052489638612992000,\na[8,6]=775651514059488014593497/5870401979381 64259487744,\na[8,7]=188187525132648765731051207041043965977201/838058 3499025736698082807492981506048000,\na[9,1]=31049434346235536518435948 28643316170473/5493198556137918605753979473772031976448,\na[9,2]=0,\na [9,3]=-80379088492007830915669557451/1850963041401006078449519616,\na[ 9,4]=8519139775589074767837125/338538398802108406103202,\na[9,5]=-2470 48879110939373809210763186554743125624445065/1766762789425993194337457 4719133235934016725376,\na[9,6]=17478588815033183487684605738930625/94 70320455773783765475210993539584,\na[9,7]=5680115883329072612844775910 212680982448198390650839995/\n 184203510561647602575374593408831 494422662678761353728,\na[9,8]=-373168103571100370594887280000/2087620 5037612721992947009240147,\na[10,1]=1169280695264980001334049232853059 /1626820675759937201701331837062960,\na[10,2]=0,\na[10,3]=-14685135128 66349156603823997571/28584615782549766069481480240,\na[10,4]=975093321 34466915877268800/3238039054339140321353321,\na[10,5]=-507655973661702 4493905835117377217502141590/30139735361305596755059429070170956786361 9,\na[10,6]=18947515655188164002766115308564375/8666094601554176945344 432830508061,\na[10,7]=45727568927147685551465920458209704007893276117 767183866475/\n 12624643823865838736175575303422784678965331666 68617282491,\na[10,8]=653479766201588970500527390720000/84524048323760 095514312876121590233,\na[10,9]=-638152078395004810819432233984/214262 32114086837877217243378141,\na[11,1]=110627591724088312853370407698744 217/153961713538018628577114230979964800,\na[11,2]=0,\na[11,3]=-293711 08267043584744126857/570660576895293065417600,\na[11,4]=15110782064016 68443300/50099027666910380431,\na[11,5]=-14207562098752542925511026308 53070453627869219/84214113659151179140822400721380572464133840,\na[11, 6]=189778934355414696052914575473965/86668623987144355187734174907968, \na[11,7]=22452213418020632097025222753496035518609830065148090541733/ \n 618662354654601543316945726201272636347870721298003014592,\n a[11,8]=24095734745599243154606080/3211814700096820264958983359,\na[11 ,9]=-1660488920478680996/55817684633385330175,\na[11,10]=0,\n\nb[1]=15 7555978800757/2874472964350575,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=21050 87478471774562042359/5315246555379439663510400,\nb[6]=-116455597328635 12771875/409009954655655860540672,\nb[7]=89543313432661343622256182806 845122951561468606052473/\n 29957181581476569403363555963384114988 4537293619715840,\nb[8]=54185546099841228800000/2235820226763235235663 1,\nb[9]=-18745074956372100336/4293668048721948475,\nb[10]=12475482426 0719/56172137577600,\nb[11]=0,\n\n`b*`[1]=1641717283464529/29894518829 245980,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=95166601700256909 068767/239186094992074784857968,\n`b*`[6]=-476838119396904213375/88915 20753383823055232,\n`b*`[7]=386665692883208129148336334007158171015085 889/1295304144774171894961513004109091083904203392,\n`b*`[8]=165297038 86583383552000/6097691527536096097263,\n`b*`[9]=-54796134375796472868/ 11163536926677066035,\n`b*`[10]=0,\n`b*`[11]=5/2\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stabi lity function R for the 11 stage, order 7 scheme is given as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunctio n(7,11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F) *&#\"E7\"@%yq`ekw`M0')\\Iju\\F)*$)F'\"\")F)F)F)*&#\":8s[ *)=*fkLo*4a$\"@C-#efvkH\"eJ&GPK@9F)*$)F'\"\"*F)F)F)*&#\"80*))R@')G'Ha= U'\">%ys())4z8GV\"o97[RF)*$)F'\"#5F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=- 1,z=-3.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!/v+'QHkw$!#8" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-3.8):\np1 := plot([R(z),-1],z=-4.29..0. 49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],style=point,symbol=[ci rcle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,-1]],linesty le=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.29. .0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7W7$$!3/++++++!H%!#<$!3x*H!oDgQ RPF*7$$!3;n;a,[!zB%F*$!3RS+P$=KwI$F*7$$!3QLL3.'4e=%F*$!3ZuT+5(**3#HF*7 $$!3`]iX\\=[STF*$!3t2qmQqs'HyT/%F*$!35'o&f $F*$!3g7SX1G@jhFin7$$!3ELez,*zU\\$F*$!3I*Q<\\)p7MXFin7$$!35+D@:))>$R$F *$!3!4=A3]hDF$Fin7$$!3amm@$\\C#*G$F*$!3sAPM4.urAFin7$$!39L$3Q&Qk(>$F*$ !3q_j0JB8'e\"Fin7$$!3a***\\YuXX4$F*$!3[uNTQNp\\)*!#>7$$!3!)***\\ZIC5*H F*$!36Krrm`<8_Fhp7$$!3#)***\\`@i7*GF*$!3J/H1F()fW[P\"zFhp7$$!3q*\\iLWx%)H#F*$\"3N7rLS!))HI*Fhp7$$!3%)*\\(3 )\\3.?#F*$\"3WuIM2\"R*f5Fin7$$!3Im;H-&zy4#F*$\"352`fsl[(>\"Fin7$$!3Dm \"z#4x\"Q+#F*$\"30vC^8`TH8Fin7$$!3NL$3#e$fB!>F*$\"3[kyStW(4[\"Fin7$$!3 /LezXE(pz\"F*$\"3CS+G:8m^;Fin7$$!3\"**\\PG?L_q\"F*$\"3$=rd6ZQN\"=Fin7$ $!38L$e64_hg\"F*$\"3jV$)pe%4X+#Fin7$$!3w****p2;z.:F*$\"3b(po8Z!y@AFin7 $$!3p**\\_W=l.9F*$\"3!H/b6gPkX#Fin7$$!3!)*\\iXZhnI\"F*$\"3\"F*$\"3Rv![X1&H9IFin7$$!3$pmm2h:D5\"F*$\"3NA%=C&yH?L Fin7$$!3+(***\\2U/$***Fin$\"3_C0&3GN8o$Fin7$$!3WJ$eRU(zd!*Fin$\"3x8Exv %eA/%Fin7$$!3O%***\\ecKN!)Fin$\"3'*o$))\\bUuZ%Fin7$$!3sl;H5%fK2(Fin$\" 3l6sRodgH\\Fin7$$!3i(*\\7uHingFin$\"3'HM&=M\"H6X&Fin7$$!3[mm;W&HW3&Fin $\"3o`E@KIJ9gFin7$$!3u**\\iSr6bSFin$\"31[Z$R'fNmmFin7$$!3)\\KL$fpwjIFi n$\"3QZ=@yF4htFin7$$!3sGL3:#o*\\?Fin$\"3CKDkd!*\\Y\")Fin7$$!3&Hm\"HOUc W5Fin$\"3!)=ue2093!*Fin7$$!3e)*)***pR-27Fhp$\"3#fn&QMJ-!))*Fin7$$\"33' RLeGv9Q*Fhp$\"3%=\\7(fiN)4\"F*7$$\"31ummCb>&)=Fin$\"3%e3!f\"pgu?\"F*7$ $\"3[4]P;i%\\*GFin$\"3\"\\#>5dAvN8F*7$$\"351]ipkShQFin$\"3'y`@')e\"Hr9 F*7$$\"3!***************[Fin$\"3FAR. " 0 "" {MPLTEXT 1 0 1029 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/504 0*z^7+\n 121937360638597471726198777/49746330498605345376645853707 84*z^8+\n 3540996833645991889487213/142132372853158129647559582022 4*z^9-\n 64218542962886213988905/39481214681432813790988772784*z^1 0:\npts := []: z0 := 0: tt := 0: \nwhile tt<=281/20 do\n zz := newto n(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (3/4<=tt and tt<=33/20 ) or (51/20<=tt and tt<=71/20) or\n (209/20<=tt and tt<=229/20) o r (247/20<=tt and tt<=53/4) then\n hh := 1/40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.5,.25,.12)):\np2 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[-1.9,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,1,.5,.23)):\np3 := plot([[[- 4.39,0],[1.19,0]],[[0,-4.59],[0,4.59]]],color=black,linestyle=3):\nplo ts[display]([p||(1..3)],view=[-4.39..1.19,-4.59..4.59],font=[HELVETICA ,9],\n 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HW=9O$F_[qFihp7%F_`w7$Fhcq$!/m6=3[ALF_[qFihp7%Fc`w7$Fbcq$!//VKQI\"G$F_ [qFihp7%Fg`w7$F\\cq$!/qk\"*RjPKF_[qFihp7%F[aw7$Ffbq$!/5M\"og6>$F_[qFih p7%F_aw7$F`bq$!/-oY@^TJF_[qFihp7%Fcaw7$Fjaq$!/?b.bE)3$F_[qFihp7%Fgaw7$ Fdaq$!/X1rj)4.$F_[qFihp7%F[bw7$F^aq$!/)4?z;$pHF_[qFihp7%F_bw7$Fh`q$!/P n#*\\6.HF_[qFihp7%Fcbw7$Fb`q$!/$fGX,E$GF_[qFihp7%Fgbw7$F\\`q$!/=L;)3%e FF_[qFihp7%F[cw7$Ff_q$!/Z4s&e9o#F_[qFihp7%F_cw7$F`_q$!/')RwFb]q$!/>8ApQB@F_[qFihp7%Fidw7$$\"/Q0!*R`9nFU$!/kBCc!\\ '>F_[qFihp7%F_ew7$$\"/YqY&eI%>FU$!/&4,!*eq!=F_[qFihp7%Feew7$$\"/!4Owd_ )QF0$!/'='))4g\\;F_[qFihp7%F[fw7$$\"/N%Gx2C2(Fho$!/r]cON#\\\"F_[qFihp7 %Fafw7$$!/XU#R2,T%F^q$!/&4&pz?N8F_[qFihp7%Fgfw7$$!/s2f$oLS#F^q$!/:qtc5 y6F_[qFihp7%F]gw7$$!/d)**eMJ)zF^p$!/rWl%>5-\"F_[qFihp7%Fcgw7$$!/Tw.L8u !\\%>cBFhhpFihp7%Faiw7$F($!/X(Rj \")R&yFe^qFihp7%Fgiw7$F($\"/X(Rj\")R&yFe^qFihp-Fdgp6&Ffgp$\"\"\"F)Fggp $\"#BF\\hp-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3o*************Q%FUF(7$$ \"3%**************=\"FUF(-%'COLOURG6&FfgpF)F)F)-%*LINESTYLEG6#\"\"$-F$ 6%7$7$F($!3')*************e%FU7$F($\"3')*************e%FUFa[xFd[x-%%FO NTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&I m(z)G-Fb\\x6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$R%F\\h p$\"$>\"F\\hp;$!$f%F\\hp$\"$f%F\\hp" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stab ility interval) is the intersection of the stability region with the r eal line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stabilit y interval is (approximately) " }{XPPEDIT 18 0 "[-3.7664, 0];" "6#7$, $-%&FloatG6$\"&kw$!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can distort the bound ary curve horizontally by taking the 11th root of the real part of poi nts along the curve." }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way we s ee that the intersection of the stability region with the non-negative imaginary axis consists of the union of two disjoint intervals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 524 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040 *z^7+\n 121937360638597471726198777/497463304986053453766458537078 4*z^8+\n 3540996833645991889487213/1421323728531581296475595820224 *z^9-\n 64218542962886213988905/39481214681432813790988772784*z^10 :\nDigits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 190 do\n zz \+ := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pt s),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.9,.3 ,0),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 " " 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$\"5tbZx?D1\"f1(F?$\"5KgxMoZpRwQFfu7$$\"5[iK(>HLUk(oF?$\"55Ik_?Ywh!)QF fu7$$\"5;;EGj')4K6mF?$\"5/apK\"y$\\r%)QFfu7$$\"59TJ,,$pcX:'F?$\"5\"eON #H&z*o))QFfu7$$!5qZM$>CO)pJ`F?$\"5F'HF%HbJa#*QFfu7$$!58(Q'e3MT8ajF?$\" 5E=g]')=fF'*QFfu7$$!5C'e$y_&*)))**QFfu7$$!5i9*)*[X[]1 'pF?$\"5+s>x'e3$Q.RFfu7$$!5oY()**=xHIQrF?$\"5[Wb(yx6fn!RFfu7$$!5zOfcO2 89\"G(F?$\"54)Q416\"y,5RFfu7$$!5sA*)oH!*e4,uF?$\"50L**)\\f*)fJ\"RFfu7$ $!5'H3o+R1+[](F?$\"5t)oyq%pg=;RFfu7$$!57yT'>R:Ojf(F?$\"5++n)\\i*p4>RFf u7$$!5,rI02q7SywF?$\"5T7kSi3L*=#RFfu7$$!5xW\"RB\"GN(Gv(F?$\"5=1*Q'*pgv X#RFfu7$$!5.Z[#RM$\\6@yF?$\"59\"=\"37gW9FRFfu7$$!5f&*Ho(\\nXT)yF?$\"5& =%\\8O0/gHRFfu7$$!5ex-\")eW.vUzF?$\"51&3&\\#*[R%>$RFfu-%%FONTG6$%*HELV ETICAG\"\"*-%+AXESLABELSG6$Q!6\"Fegn-%&COLORG6&%$RGBG$Fagn!\"\"$\"\"$F \\hnF(-%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTGFfhn" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The relevant intersectio n points of the boundary curve with the imaginary axis can be determin ed as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for point s on the boundary curve either side of each intersection point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Digits := 15:\nz0 := 0.41*I:\nfor ct from 12 to 15 do\n newton( R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 1.5*I:\nfor ct from 46 to 49 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3 .9*I:\nfor ct from 175 to 178 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0 );\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0*) 3Bo9_@#!#D$\"0a^Y=6*pP!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0:Sc >6nJ\"!#D$\"0010XqS3%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0(eY\" )=#>4$!#D$\"0\"z$prH#)R%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0=C XDv^_\"!#C$\"0&3\\%)*)Q7Z!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0U)>`_bcH!#?$\"0b2(=F?X9!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0kCM?`m=\"!#?$\"0H+>!ejw9!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0n`v'*R(H;!#?$\"0\")He[s!3:!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0(GwThCUe!#?$\"0 " 0 "" {MPLTEXT 1 0 482 "Digits := 15:\nreal_par t := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.41*I))\nend proc:\nu0 \+ := bisect('real_part'(u),u=0.12..0.15);\nnewton(R(z)=exp(u0*Pi*I),z=0. 41*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.5*I ))\nend proc:\nu0 := bisect('real_part'(u),u=0.46..0.49);\nnewton(R(z) =exp(u0*Pi*I),z=1.5*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=ex p(u*Pi*I),z=3.9*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.75..1. 78);\nnewton(R(z)=exp(u0*Pi*I),z=3.9*I);\nDigits := 10:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#u0G$\"0j_s,[SM\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0g)Gy8XAU!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"03v*H?YZZ!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0&fp\")[-N:!#H$\"0J2[!eb\"\\\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#u0G$\"0nTL2&Ho*QF," } {TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stabi lity function R* for the 11 stage, order 6 scheme is given as follow s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,St abilityFunction(6,11,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'= `R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,8\"\"\"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)*&# \"JXfogU2S2GZVoB,vyhFtQ)\"N%Q+s*)e[&*o5BP%\\?m(Q*fxE<=%F)*$)F'\"\"(F)F )F)*&#\"I`+g1F:r#4!o(3T!))G*pZo*>\"Ms]AVg.Ha#3[pUj'=&))\\qP2%yF)*$)F' \"\")F)F)F)*&#\"E0vX;4E`>F!>y:$4%yxB#\"JkOr8C3Z`*>IB_'y\"pS\"=0H'F)*$) F'\"\"*F)F)F)*&#\"DDkbJDDb)H)H(Q7O2@H9\"I[S&p5#*eYbaH\"R9!*H6J9:yF)*$) F'\"#5F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/ -%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-3.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+T*z2o$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)= -1,z=-3.7):\np_1 := plot([`R*`(z),-1],z=-4.19..0.49,color=[red,blue]): \np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond] ,color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR (RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.19..0.49,-1.47.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 263 263 {PLOTDATA 2 "6+-%'CURVESG6$7V7$$!3Q++++++!>%!#<$!3AL=Dl_:qOF*7$$!3l++D fY**QTF*$!3#)pmk'Rt#\\KF*7$$!3-++]=$*)z3%F*$!3_%>A!*p[>(GF*7$$!3O+vBW) 4O/%F*$!39-3&ei+fd#F*7$$!3E+](*p.B**RF*$!3()flBsvE2BF*7$$!3W+Dm45K\\RF *$!3uMPRmn1N?F*7$$!3;++N\\;T**QF*$!3mwq6**ej\"z\"F*7$$!3E++Xl7$*)z$F*$ !3y*\\dN5oyP\"F*7$$!37+]FZ%G*)p$F*$!31.vt/sU^5F*7$$!3X+]dLI@1OF*$!3FIV SAkQ1\")!#=7$$!3=+](Hf6-^$F*$!31A-ori-AhFZ7$$!3?+]xln#4T$F*$!3E&*[N**> L4XFZ7$$!3D+]FY.'>J$F*$!3)eK'39EocKFZ7$$!3?++I>7;5KF*$!3%eO&yWpbfAFZ7$ $!33++&G\\'\\?JF*$!3uD?2u(**Rd\"FZ7$$!32++!zCb&>IF*$!3pkY)y(yz/(*!#>7$ $!3>++]G&*>=HF*$!3coN-x\\/G]Fcp7$$!3******47X_?GF*$!3&)*RD#z2Q6:Fcp7$$ !3<+]xRi#=t#F*$\"3;!QC^()>'\\5Fcp7$$!3\"******[Tbji#F*$\"35%y!>gY[?NFc p7$$!3y****f\"z2q`#F*$\"3'o\"ofwy\"GE&Fcp7$$!3$)**\\F*$\"3S'[Rb\\['*R\"FZ7$$!31++D***4B&=F*$ \"3;uHgl\"zhb\"FZ7$$!3/+]xH!G\"\\t\"FZ7$$!3;+]-AyIf;F *$\"3oBh&QQ0$)*=FZ7$$!3!****\\p`*Hi:F*$\"3czt!eyUS4#FZ7$$!33++?'[!3i9F *$\"3#3wsGT(>;BFZ7$$!3%)***\\rpNSO\"F*$\"3!e=%f9Ly-\"GFZ7$$!3w***\\%=F%Q;\"F*$\"3Qi(*HzBnAJFZ7$$!33++g%Q*>p 5F*$\"3]#\\Om5aFV$FZ7$$!3)o****\\Cu9o*FZ$\"353)==dEyz$FZ7$$!3g,+v$H$zl ()FZ$\"3m;6!eZF?;%FZ7$$!3+&****4:7Zw(FZ$\"3a)p&*Hdf-g%FZ7$$!3%>+]<\"Gx AoFZ$\"3MPs/rSka]FZ7$$!37&**\\Zzu\"QeFZ$\"3q&R9w$)\\wd&FZ7$$!3M*****\\ E]b([FZ$\"3licTs1EThFZ7$$!3!z**\\Pfrx'QFZ$\"3'[a(*4QCCz'FZ7$$!3M$****f &4;(*GFZ$\"3!GA$*pKgZ[(FZ7$$!3N)***\\!RrX!>FZ$\"3]t!e+&=\"eE)FZ7$$!3#Q $**\\x'4??*Fcp$\"3Wg>d$>o37*FZ7$$!3_r'****>ymc\"!#?$\"3;bk*R[XV)**FZ7$ $\"3S-+]r9.@5FZ$\"3)>Po5q(\\26F*7$$\"352++[nE[>FZ$\"3ov$[2O+^@\"F*7$$ \"3-0+D[H*o$HFZ$\"3%*z0npqOT8F*7$$\"3)H+]xOMJ)QFZ$\"3glu0v=\\u9F*7$$\" 3!***************[FZ$\"3dbBUOiJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\" \"!Fa\\lF`\\l-F$6$7S7$F($!\"\"Fa\\l7$F3Ff\\l7$F=Ff\\l7$FGFf\\l7$FLFf\\ l7$FQFf\\l7$FVFf\\l7$FfnFf\\l7$F[oFf\\l7$F`oFf\\l7$FeoFf\\l7$FjoFf\\l7 $F_pFf\\l7$FepFf\\l7$FjpFf\\l7$F_qFf\\l7$FdqFf\\l7$FiqFf\\l7$F^rFf\\l7 $FcrFf\\l7$FhrFf\\l7$F]sFf\\l7$FbsFf\\l7$FgsFf\\l7$F\\tFf\\l7$FatFf\\l 7$FftFf\\l7$F[uFf\\l7$F`uFf\\l7$FeuFf\\l7$FjuFf\\l7$F_vFf\\l7$FdvFf\\l 7$FivFf\\l7$F^wFf\\l7$FcwFf\\l7$FhwFf\\l7$F]xFf\\l7$FbxFf\\l7$FgxFf\\l 7$F\\yFf\\l7$FayFf\\l7$FfyFf\\l7$F[zFf\\l7$FazFf\\l7$FfzFf\\l7$F[[lFf \\l7$F`[lFf\\l7$Fe[lFf\\l-Fj[l6&F\\\\lF`\\lF`\\lF]\\l-F$6&7#7$$!3%**** **4%*z2o$F*Ff\\l-%'SYMBOLG6#%'CIRCLEG-Fj[l6&F\\\\lFa\\lFa\\lFa\\l-%&ST YLEG6#%&POINTG-F$6&F\\`l-Fa`l6#%&CROSSGFd`lFf`l-F$6&F\\`l-Fa`l6#%(DIAM ONDGFd`lFf`l-F$6%7$7$F^`lF`\\lF]`l-%&COLORG6&F\\\\lF`\\l$\"\"&Fg\\lF` \\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z 6\"Q!Fjbl-Fbbl6#%(DEFAULTG-%%VIEWG6$;$!$>%!\"#$\"#\\Fecl;$!$Z\"Fecl$\" $Z\"Fecl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following pict ure shows the stability region for the 11 stage, order 6 scheme. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1197 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 83873276178750123684347280740074260685945/41817267759938766204943 7231068954858897200384*z^7+\n 199684769928880410876800927115270660 0053/78407377049885186634269480825429036043225072*z^8+\n 223777840 931578190271953260916457505/62905181406917865223301995347082413713664* z^9-\n 14292107361238729829855252531556425/78151431112990143912954 55465892106954048*z^10:\npts := []: z0 := 0: tt := 0: \nwhile tt<=281/ 20 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n i f (13/20<=tt and tt<=33/20) or (51/20<=tt and tt<=73/20) or\n (20 7/20<=tt and tt<=229/20) or (247/20<=tt and tt<=267/20) then\n hh := 1/40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLO R(RGB,.45,.15,0)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[- 1.85,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RG B,.9,.3,0)):\np_3 := plot([[[-4.29,0],[1.19,0]],[[0,-4.39],[0,4.39]]], color=black,linestyle=3):\nplots[display]([p_||(1..3)],view=[-4.29..1. 19,-4.39..4.39],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z )`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 595 560 560 {PLOTDATA 2 "6+-%'CURVESG6$7jal7$$\"\"!F)F(7$F($\"3++++Fjz q:!#=7$$\"3++++jc\"=:\"!#F$\"3#*******fEfTJF-7$$\"3!******\\-74=#!#E$ \"3*)*****R3*Q7ZF-7$$\"3%******fp\"\\N7!#D$\"3m*****f4'=$G'F-7$$\"3'** ****p'fJj9F=$\"3e*******z&)R&yF-7$$!35+++2e4[;!#C$\"3:+++gizC%*F-7$$!3 .+++!49b6\"!#B$\"3#******>bk&*4\"!#<7$$!3!)*****4Dr'[PFN$\"33+++(*=mc7 FQ7$$!3')*****4'e^\\qFN$\"3!******R=%z89FQ7$$\"3!******z8AC?(FH$\"3#** *****)G85d\"FQ7$$\"37+++*oS-g'!#A$\"32+++MZUG%y=C$!#@ $\"33+++$[>i)=FQ7$$\"32+++IuY46!#?$\"30+++1JqW?FQ7$$\"3/+++u$3n!>Fjo$ \"3-+++D-LC@FQ7$$\"3y*****H+9E;$Fjo$\"3)*******z!)G/AFQ7$$\"3O+++h(eM4 &Fjo$\"3')*******4@YG#FQ7$$\"3c******>WU(*zFjo$\"3/+++B5MlBFQ7$$\"3*** 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]l7%F\\hdl7$$!/XU#R2,T%Fdq$!/&4&pz?N8F`^qFgj]l7%Fbhdl7$$!/s2f$oLS#Fdq$ !/:qtc5y6F`^qFgj]l7%Fhhdl7$$!/d)**eMJ)zFjo$!/rWl%>5-\"F`^qFgj]l7%F^idl 7$$!/Tw.L8u!\\%>cBFi]qFgj ]l7%F\\[el7$F($!/X(Rj\")R&yF_\\qFgj]l7%Fb[el7$F($\"/X(Rj\")R&yF_\\qFgj ]l-Fihp6&F[ip$\"\"\"F)Fii]l$\"#BF^ipF\\^x-F$6%7$7$$!3o*************Q%F QF(7$$\"3%**************=\"FQF(-%'COLOURG6&F[ipF)F)F)-%*LINESTYLEG6#F[ ^x-F$6%7$7$F($!3')*************e%FQ7$F($\"3')*************e%FQFh\\elF[ ]el-%%FONTG6$%*HELVETICAGFh]x-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re (z)G%&Im(z)G-Fh]el6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$! $R%F^ip$\"$>\"F^ip;$!$f%F^ip$\"$f%F^ip" 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" }}}}{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 "coefficients of the combined sch eme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4251 "ee := \{c[2]=1/1440,\nc[3]=40/183,\nc[4]=49/120,\nc [5]=49/89,\nc[6]=269/315,\nc[7]=10809269/57064323,\nc[8]=269/280,\nc[9 ]=65/66,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/1440,\na[3,1]=-381560/11163, \na[3,2]=128000/3721,\na[4,1]=10339/384000,\na[4,2]=0,\na[4,3]=146461/ 384000,\na[5,1]=6163073/56397520,\na[5,2]=0,\na[5,3]=3939947361/261120 51760,\na[5,4]=94785600/326400647,\na[6,1]=-84/515,\na[6,2]=0,\na[6,3] =4431643925510749/3223786245394050,\na[6,4]=-166218606939232/905665120 53885,\na[6,5]=5209733339403491/3525508500182550,\na[7,1]=899064843777 4041845694826394521064409811/20408843232331890339960003163966264362213 0,\na[7,2]=0,\na[7,3]=547930339691225609501822503013360248045868971/17 37834520761638771583667016353839075551118195,\na[7,4]=-108313758731507 18357864250769796161780597600/3254756965707329133882732504582308804342 7021,\na[7,5]=1221450428943196975560168771911668179455657701/684173815 2147548926226151700669066667202112842,\na[7,6]=-2346622810543340886158 743517271000/149768789910631840990687560369315577,\na[8,1]=56271452542 1474554187476828603/1576430076296951412495515648000,\na[8,2]=0,\na[8,3 ]=-232434073319953744988674545591/7451087574129649589321728000,\na[8,4 ]=2895719528678335306735611/163535145776061958545920,\na[8,5]=-3877135 960439326710451592840782413717/400320476616417483052489638612992000,\n a[8,6]=775651514059488014593497/587040197938164259487744,\na[8,7]=1881 87525132648765731051207041043965977201/8380583499025736698082807492981 506048000,\na[9,1]=3104943434623553651843594828643316170473/5493198556 137918605753979473772031976448,\na[9,2]=0,\na[9,3]=-803790884920078309 15669557451/1850963041401006078449519616,\na[9,4]=85191397755890747678 37125/338538398802108406103202,\na[9,5]=-24704887911093937380921076318 6554743125624445065/17667627894259931943374574719133235934016725376,\n a[9,6]=17478588815033183487684605738930625/947032045577378376547521099 3539584,\na[9,7]=56801158833290726128447759102126809824481983906508399 95/\n 184203510561647602575374593408831494422662678761353728,\na [9,8]=-373168103571100370594887280000/20876205037612721992947009240147 ,\na[10,1]=1169280695264980001334049232853059/162682067575993720170133 1837062960,\na[10,2]=0,\na[10,3]=-1468513512866349156603823997571/2858 4615782549766069481480240,\na[10,4]=97509332134466915877268800/3238039 054339140321353321,\na[10,5]=-5076559736617024493905835117377217502141 590/301397353613055967550594290701709567863619,\na[10,6]=1894751565518 8164002766115308564375/8666094601554176945344432830508061,\na[10,7]=45 727568927147685551465920458209704007893276117767183866475/\n 12 62464382386583873617557530342278467896533166668617282491,\na[10,8]=653 479766201588970500527390720000/84524048323760095514312876121590233,\na [10,9]=-638152078395004810819432233984/2142623211408683787721724337814 1,\na[11,1]=110627591724088312853370407698744217/153961713538018628577 114230979964800,\na[11,2]=0,\na[11,3]=-29371108267043584744126857/5706 60576895293065417600,\na[11,4]=1511078206401668443300/5009902766691038 0431,\na[11,5]=-1420756209875254292551102630853070453627869219/8421411 3659151179140822400721380572464133840,\na[11,6]=1897789343554146960529 14575473965/86668623987144355187734174907968,\na[11,7]=224522134180206 32097025222753496035518609830065148090541733/\n 618662354654601 543316945726201272636347870721298003014592,\na[11,8]=24095734745599243 154606080/3211814700096820264958983359,\na[11,9]=-1660488920478680996/ 55817684633385330175,\na[11,10]=0,\n\nb[1]=157555978800757/28744729643 50575,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=2105087478471774562042359/5315 246555379439663510400,\nb[6]=-11645559732863512771875/4090099546556558 60540672,\nb[7]=89543313432661343622256182806845122951561468606052473/ \n 299571815814765694033635559633841149884537293619715840,\nb[8]=5 4185546099841228800000/22358202267632352356631,\nb[9]=-187450749563721 00336/4293668048721948475,\nb[10]=124754824260719/56172137577600,\nb[1 1]=0,\n\n`b*`[1]=1641717283464529/29894518829245980,\n`b*`[2]=0,\n`b*` [3]=0,\n`b*`[4]=0,\n`b*`[5]=95166601700256909068767/239186094992074784 857968,\n`b*`[6]=-476838119396904213375/8891520753383823055232,\n`b*`[ 7]=386665692883208129148336334007158171015085889/129530414477417189496 1513004109091083904203392,\n`b*`[8]=16529703886583383552000/6097691527 536096097263,\n`b*`[9]=-54796134375796472868/11163536926677066035,\n`b *`[10]=0,\n`b*`[11]=5/2\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..11);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6,/&%\"cG6#\"\"##\"\"\"\"%S9/&F%6#\"\"$ #\"#S\"$$=/&F%6#\"\"%#\"#\\\"$?\"/&F%6#\"\"&#F7\"#*)/&F%6#\"\"'#\"$p# \"$:$/&F%6#\"\"(#\")p#43\"\")BV1d/&F%6#\"\")#FD\"$!G/&F%6#\"\"*#\"#l\" #m/&F%6#\"#5F)/&F%6#\"#6F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j ]),j=1..i-1),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6Y/&%\"aG6$\" \"#\"\"\"#F(\"%S9/&F%6$\"\"$F(#!'g:Q\"&j6\"/&F%6$F.F'#\"'+!G\"\"%@P/&F %6$\"\"%F(#\"&R.\"\"'+SQ/&F%6$F;F'\"\"!/&F%6$F;F.#\"'hk9F>/&F%6$\"\"&F (#\"(tI;'\")?vRc/&F%6$FKF'FB/&F%6$FKF.#\"+ht%*RR\",g<07h#/&F%6$FKF;#\" )+cy%*\"*Z1SE$/&F%6$\"\"'F(#!#%)\"$:&/&F%6$F[oF'FB/&F%6$F[oF.#\"1\\2^D RkJW\"1]SRXiyBK/&F%6$F[oF;#!0K#Rpg=i;\"/&)Q07lc!*/&F%6$F[oFK#\"1\"\\.% RLt4_\"1]D=+&3b_$/&F%6$\"\"(F(#\"I6)4W1@XRE[pX=/uxV[1**)\"KI@iVEmR;.+' *R.*=LKK%)3/#/&F%6$FgpF'FB/&F%6$FgpF.#\"Nr*oe/[-O8I]A=]4cA\"pR.$za\"O& >=6bv!RQN;qm$er(Q;w?X$yt\"/&F%6$FgpF;#!M+wf!yhhzp2DkyN=2:tePJ3\"\"M@qU V!)3Be/Dt#)Q8HtqlpvaK/&F%6$FgpFK#\"O,xlb%z\"o;\">xo,cvp>V*G/X@7\"OUG6- smm!p1q^hAE*[v9_\"Q<%o/&F%6$FgpF[o#!C+5FL2MCB\"=+!Gd*G\"9?fae>1wd9NN;/&F%6$FirFK#!FSqe /&F%6$FirFgp#\"K,s(f'R/Tq?^5tl([E8Dv=)=\"I+![g]\")H\\2G3)pOd-*\\$e!Q)/ &F%6$\"\"*F(#\"It/<;LkG[fV=l`NiMM%\\5$\"I[k(>.sPZzRv0'=z8c&)>$\\&/&F%6 $FauF'FB/&F%6$FauF.#!>^ubpc\"4$y+#\\)3z.)\"=;'>&\\%yg+,9/j4&=/&F%6$Fau F;#\":Dr$ywu!*ev(R\">&)\"9-K51%3@!))RQ&Q$/&F%6$FauFK#!Ql]WCc7VZb'=j2@4 QPR46z)[qC\"Pw`s;S$fBL\">ZduL%>$*fU*yinw\"/&F%6$FauF[o#\"DD1$*Qdg%o([$ =L]\"))eyu\"\"C%eRN*4@vaw$ytdX?.Z*/&F%6$FauFgp#\"X&**R3l!R)>[C)4o7-\"f xWGhs!HL)e6!o&\"WGPNh(yEmAW\\J)3Mfu`d-wkh0^.U=/&F%6$FauFir#!?++G()[fq. 5rN5oJP\"AZ,C4q%H*>s7w.0i(3#/&F%6$\"#5F(#\"CfI&GB\\SL,+)\\E&p!Gp6\"CgH 1P=L,S-[ \"[pgw\\Dy:YeG/&F%6$F_xF;#\";+)os(e\"pYM@L4v*\":@LN@.9RV0R!QK/&F%6$F_x FK#!L!fT@]O'yc43\"[+&Ry?:Q'\"AT\"yLC#pyi`/2`3j-6b#Ha_()4iv?9\"MSQ8kCd!Q@2SA39z6:fO69U)/&F%6$Fc[lF[o#\"B lRZvX\"H0'p9aNM*y(*=\"Aoz!\\ " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=su bs(ee,b[i]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\" \"#\"0d2!)yfbd\"\"1v0NkHZuG/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F// &F%6#\"\"&#\":fB/iXxr%yu30@\":+/^j'R%z`bY_J&/&F%6#\"\"'#!8v=x7N'Gtfbk6 \"9s1agelbY&*4!4%/&F%6#\"\"(#\"VtC01'o9c^H7Xo!G=cAiV8mKMJV&*)\"WSer>OH PX))\\6%QjfbjLSplZ\"e\"=d*H/&F%6#\"\")#\"8++!)G7%)*4Yb=a\"8JmN_BjnA?eB #/&F%6#\"\"*#!5O.5sj&\\2X(=\"4v%[>s[!oOH%/&F%6#\"#5#\"0>2EC[vC\"\"/+wd P@ " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*` [i]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%#b*G6#\"\"\"#\"1H XY$G<\"QoZ\"7 K_0BQQ`2_\"*))/&F%6#\"\"(#\"N*)e3:5 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 38 "#===================================== " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Scheme with a moderately larg e stability region" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 48 "#---------------------------------------- -------" }}{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 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4903 "ee := \{c[2]=1/200,\nc[3]=93/479, \nc[4]=110/241,\nc[5]=509/845,\nc[6]=244/273,\nc[7]=222785849/11091452 21,\nc[8]=17/20,\nc[9]=171/181,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/200,\n a[3,1]=-820353/229441,\na[3,2]=864900/229441,\na[4,1]=-432520/5401533, \na[4,2]=0,\na[4,3]=2897950/5401533,\na[5,1]=311712996187/336669927750 0,\na[5,2]=0,\na[5,3]=996040375149076/5096677701243375,\na[5,4]=344546 810496217/1096059720697500,\na[6,1]=-228/1231,\na[6,2]=0,\na[6,3]=1508 9501252616323812/14457231345452578137,\na[6,4]=-6255616188978805360/52 00809295819122531,\na[6,5]=103959712518200500/83969153915827131,\na[7, 1]=20784021508544698439004341302945788046687497048823/\n 3338090 07739010923825478541340748044990946025336204,\na[7,2]=0,\na[7,3]=59087 07677638153633449865101135634541623011524075277164/\n 3154555408 7781255456304827404537875875692253868011534003,\na[7,4]=-4859760982689 9455036511090495924959702383070843517685/\n 4880912267236813758 28622960423688756917868034622078292,\na[7,5]=4083724255290790494045580 4285850036846494300434360648500/\n 73437256266214599098958725391 4642887902719996408150162943,\na[7,6]=-2267236012274582026887011527795 707923286/477373186945391376596029499088911158349111,\na[8,1]=-1485037 5572183825804482268314325533/131869870061161226458875010296840000,\na[ 8,2]=0,\na[8,3]=25892910773607576734414005208753/105292913205082732017 47374752000,\na[8,4]=-613522852101531567576566644397/81457670752223606 9963552640000,\na[8,5]=8122715759499790031095209746537365648167969/836 2965435593003722836700426272322940782592,\na[8,6]=-4293717856259525168 0361031143/2273295839722617724238904704000,\na[8,7]=-74242059958860599 56357359508852528825590423/4378197291051651257486258563957705093888000 ,\na[9,1]=-153831689800158592252487409542055462465307763/5177138697340 60023599533102763919424014608068,\na[9,2]=0,\na[9,3]=-1334984216509015 564184248020843701584/667067642557939663406130018977478253,\na[9,4]=-5 7633497085298645064371869898744017/30963779845911010423286154735455876 ,\na[9,5]=489177972756706421411944299196898256388191463578294580375/\n 319291201100986622665402847780351230046376607122311276442,\na[9 ,6]=-1185679300358377756233489606885369481439862/184386017579686381663 15261704657797344684897,\na[9,7]=1194448204582865834629880690823812141 7524848379605193796762013/\n 34123763173112983923507228128510968 67170775526794634954003706,\na[9,8]=103973108798709992837929472000/762 442644104164301247937014633,\na[10,1]=-4073447364725702015608831305484 6628592379/112331307637222010640347265020564130793284,\na[10,2]=0,\na[ 10,3]=-1111042984674821389643230968704444252/1475199643357862920568353 17593798019,\na[10,4]=-5213063483277854003640774248205317/228251499905 5829721733809658986516,\na[10,5]=4344125602637174263526484423723747582 563035478367000/\n 28196971747190084940997556852310139333529159 92886809,\na[10,6]=142812438169955950969368011788034519784/28198419677 1808918111529577339005012981,\na[10,7]=3501182975985292177489446628570 0128707491070752921128030483969446296/\n 3718044938427279904428 695460050071040231420419464687750808112491477,\na[10,8]=-2632810508797 5815319887872000/966150763091150023719018517747,\na[10,9]=-11985757874 06124991100974435169141280/4635648233480921649154730958571956313,\na[1 1,1]=-4549513437278902549066809605929617243853/11637130531712243439602 356681247846961408,\na[11,2]=0,\na[11,3]=-8070051946566656273895465364 457/1073987714691843866578232224704,\na[11,4]=-10059066143857143308915 950403/4154341208363403956814118464,\na[11,5]=581599188224381204095672 2134361657936380202840699902125/\n 3585951737451764935284196541 594543916113759295056049408,\na[11,6]=28274787710189875324117065188869 812987/61207302919331402809078602199860439808,\na[11,7]=57191704729097 59253773842353503580127446876084974327902851679027589/\n 602884 877866360860548455552363583579732315378471940789964794317568,\na[11,8] =1203167959926676490500/125196265101038882839767,\na[11,9]=-4684487619 570658562655565/18486842815356938821237824,\na[11,10]=0,\n\nb[1]=17499 592155841793/301627899585331605,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=8992 9293580433773826711625625/175034206955063764470139052544,\nb[6]=439656 79317071687650068867/11554641620922992752054730,\nb[7]=639711292154532 559948032664549872463482448777204372657657897547/\n 20116014255015 88622427604443016911932371008211435797382991660160,\nb[8]=-43614217354 789120000/21595629876813903267,\nb[9]=-210996982649291857501475159/990 36657939412172256631200,\nb[10]=7077278529846113/15546034297382400,\nb [11]=0,\n\n`b*`[1]=94288552637982437/1608682131121768560,\n`b*`[2]=0, \n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=532400445239427668047625/99415103005 1934322008696,\n`b*`[6]=15979368227021270883046053/3187487343702894552 290960,\n`b*`[7]=683482552868822360583731464535267340427333118635079/ \n 2165562941739167798045518397620231220712261142369860,\n`b*`[ 8]=-132403130312946592000/50389803045899107623,\n`b*`[9]=-169578451828 457839968131453/57771383797990433816368200,\n`b*`[10]=0,\n`b*`[11]=16/ 25\}:" }}}{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 0 136 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j =1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7.7.$\"'++]!\")F(%!GF+F+F+F+F+F+F+F+F+7.$\"'aT>!\"'$!'WvN!\"&$\"'g pPF2F+F+F+F+F+F+F+F+F+7.$\"'KkXF/$!'O2!)!\"($\"\"!F<$\"'0l`F/F+F+F+F+F 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));print(``);\nfor ii from 7 to 11 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,j j]));\n end do:\n print(`_________________________________`);\nend do:print(``);\nfor ii to 11 do\n print(b[ii]=subs(ee,b[ii]));\nend \+ do:\nprint(`_________________________________`);print(``);\nfor ii to \+ 11 do\n print(`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"$+#F(%!GF+F+F+7(#\"#$*\"$ z%#!'`.#)\"'T%H##\"'+\\')F2F+F+F+7(#\"$5\"\"$T##!'?DV\"(L:S&\"\"!#\"(] z*GF;F+F+7(#\"$4&\"$X)#\"-(='*Hr6$\".+vF*pmLF<#\"0w!\\^PSg**\"1vLC,xn' 4&#\"0eH43?&#\"3+0?=DrfR5\"2Jr#e\"R:pR )Q(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"*\\eyA#\"+@_946" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"( \"\"\"#\"SB)[q\\(oY!)yXHITV+R%)pW&3:-%y?\"T/iLDg%4*\\/[2MT&ya#Q#4,Rx+4 QL" }}{PARA 11 "" 1 "" {XPPMATH 20 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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"'#!>V6.h.o^_fi&yr$H%\"@+Sq/*QUsyV" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_____________ ____________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\" *#\"$r\"\"$\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!NjxIlCYb?a4u[_Afe,!)*oJQ:\"N o!3Y,C%>Rw-J`*fB+1M(pQr<&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$ #!F%e,P%3-[U=kb,4l@%)\\L\"\"E`#yu(*=+81MmRzbUw1n'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!D)Qc#)*o>*HW>T@kqcF(z<*[\"ZUkF6B72mPY+B^.yZGSlEi')45,7H>$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!Li)R9[p`)og*[Li vx$e.Izc=\"\"M(*[oWtzdYqh_Jm\"Q'ozv,'Q%=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"in8?w'z$>0'z$[[_<97Q#3p!))HY$e'Ge/#[W>\"\" hn1P+a\\j%zEbxqr'o4^G\"Gs]BR)H6tJwBT$" }}{PARA 11 "" 1 "" {XPPMATH 20 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{XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!>+?(y))>`\"e (z30\"Gj#\"?Zx^=!>P-]64j2:m*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#5\"\"*#!F!GT\"p^Vu45\"*\\71uyv&)>\"\"F8j&>de4ta\"\\;#4[L#[cj%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"\"#!I`QC<'Hfg4o1\\D!*ysV8&\\X\"J39'p%yC\"ocBgRMC7<`Irj6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$#!@dWOla*QFcmcY>0q!)\"@/ZAK#yl 'Q%=p9x)R2\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!>. /&f\"*3L9dQ9m!f+\"\"=k%=T\"o&RSj$37MaT" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#\"XD@!**pSG?!QOzlhV8An&4/7QC#)=*f\"e\"X3%\\g0 &HfP6;Ra%fTl>%GN\\w^ut^fe$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"'#\"G()H\")p))=lq6C`()*=5xyu#G\"G3)R/')*>-'y!4GSJ$>HI27'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"^o*eF!z;&G!zKu\\3 woWF,e.NNUQx`#f(4HZq\">d\"]oovJ%zk**yS>Zy`JK(zNejBbb%[0'3Omy([)Gg" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"7+0\\wm#*fz;.7\"9 n(RG))Q55li>D\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*# !:lbli&e1d>w[%o%\";CyB@)QpN:G%o[=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___________ ______________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"2$zTe:#f*\\<\"30;L&e **yi,$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"&#\">Dci6n#QxL/e$HH**)\"?WD0R,ZkP1bp?M]<" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\";n)o+l(orqJzc'R%\";IZ0_F*H#4iTY b6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"jnZv*yldEP/sx[C [jC()\\XmK![*fD`a@H6(R'\"[og,m\"*HQ(zN9@35PK>\"p,VWgFCi)e,bU,;,#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!5++7*yat@9O%\"5nK!R\"o ()Hcf@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#!0I5:%**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"'#\";`g/$)3F@qAo$zf\"\":g4H_X*GqVt[(=$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"Tz]j=JLF/Mn_`k9t$egB#)oGb#[$o\"Ug)pB 9hAr?7B?wR=b/)zn\"R<%Hcl@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\")#!6+?fYHJIJSK\"\"5Bw5**e/.)*Q]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"*#!<`98o*RyXG=Xy&p\"\";+#oj\"QV!*zz$Qrx&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"#;\"#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op( RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'expanded'))] :\n`RK6_11eqs*` := subs(b=`b*`,OrderConditions(6,11,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK7_11eqs):\nmap(u- >lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK6_11eqs*`):\nmap(u->lhs(u)-rh s(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 5 do\n so||ct||_11 := StageOr derConditions(ct,11,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Stages 5 to 10 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(su bs(ee,so||i||_11[j])),i=2..5)],j=1..9)]:\nmap(proc(L) local i; for i t o nops(L) do if not evalb(L[i]) then break end if end do; i end proc,% ):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#F$\"\"$F%F %F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The stage-orders of the successive stages are given as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[stage, ` |`, 2, 3, 4, 5, 6, 7, 8, 9, 10], [`stage-order`, `|`, 1, 2, 2, 3, 3, 3 , 3, 3, 3]]);" "6#-%'matrixG6#7$7-%&stageG%\"|grG\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#57-%,stage-orderGF)\"\"\"F*F*F+F+F+F+F+F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 11) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 9 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "[Sum(b[i]*a[i,1],i=1+1..11) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j=2..9)];\nmap(u->l hs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7+/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/ F,;\"\"#\"#6&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F- &%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEF RFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F& 6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F -&F/6$F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F ,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*&F)F-&F/6$F,FaqF-/ F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/6$F,F_rF-/F,;\"#5 F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\" \"!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 26 "The simpl ifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*a[i,3],i = 3 .. 10) = 0" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+\"\"$F,/F+;F0\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "S um(b[i]*c[i]*a[i,3],i = 3 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+;F3\"#5\"\"!" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,3],i = 3 .. 10) = 0;" "6#/-%$Sum G6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+\"\"$F,/F+;F5\"#5 \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "[Sum(b[i]*a[i,3],i=3..10),Sum(b[i]*c[i]*a[i,3],i=3..10),Sum(b[i ]*c[i]^2*a[i,3],i=3..10)];\nsubs(ee,eval(subs(Sum=add,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\" \"$F,/F+;F0\"#5-F%6$*(F(F,&%\"cGF*F,F-F,F1-F%6$*(F(F,)F7\"\"#F,F-F,F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the prin cipal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errterms7_11 := Principal ErrorTerms(7,11,'expanded'):\nnrm8 := sqrt(add(subs(ee,errterms7_11[i] )^2,i=1.. nops(errterms7_11))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V(47H)!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "In addition the 2-norm of the order 9 err or terms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 " errterms8_11 := PrincipalErrorTerms(8,11,'expanded'):\nnrm9 := sqrt(ad d(subs(ee,errterms8_11[i])^2,i=1.. nops(errterms8_11))):\nevalf[10](%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G.Jtc!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The 2-norm of the order 9 error terms is approximately 6.843 times the principal error norm." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf[10](nrm9/nrm8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z.cUo!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm \+ of the order 6 embedded scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms6_11*` := subs(b=`b*`,PrincipalEr rorTerms(6,11,'expanded')):\nsqrt(add(subs(ee,`errterms6_11*`[i])^2,i= 1.. nops(`errterms6_11*`))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)3D-9'!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#-----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 7 scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We first \+ construct a 10 stage, order 7 scheme with " }{TEXT 260 10 "parameters " }{TEXT -1 2 " " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 2 ".\n" }} {PARA 0 "" 0 "" {TEXT -1 89 "The scheme will be constructed so that st age 4 has stage-order 2 and stages 5 to 10 have " }{TEXT 260 13 "stage -order 3" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 60 "We start b y determining the nodes and weights of the scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We set up a system of equ ations that consists of the 7 order 7 quadrature conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 10) = 1" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"#5F-" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 1 .. 10) = 1/k;" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;F,\"#5*&F,F,F2 F3" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 7 " . . 7," }}{PARA 0 "" 0 "" {TEXT -1 35 "and the relation b etween the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c [5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^2*c[7]- 7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2 = 0;" "6#/,>** \"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\"'F'F'**\"$0\"F'*$&F)6#F. \"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F)6#F.F'&F)6#F1F7!\"\"**F =F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6#F+F'F'*(F+F'*$&F)6#F.F7F '&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F'&F)6#F.F'FD*(\"#9F'*$&F )6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD*(FenF'&F)6#F.F'&F)6#F1F 'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F'&F)6#F.F7F'\"\"!" } {TEXT -1 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[2] = 1/200;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$+#!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 509/845;" "6#/&%\"cG6#\"\"& *&\"$4&\"\"\"\"$X)!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "c[6] = 244 /273;" "6#/&%\"cG6#\"\"'*&\"$W#\"\"\"\"$t#!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 17/20;" "6#/&%\"cG6#\"\")*&\"#<\"\"\"\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 171/181;" "6#/&%\"cG6#\"\"**& \"$r\"\"\"\"\"$\"=!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]=1" "6 #/&%\"cG6#\"#5\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 442 "Qeqs := QuadratureConditions(7,10,'expanded'): \nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6 ]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14 *c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2=0: \ncdns1 := [op(Qeqs),node_eq]:\ne1 := \{c[2]=1/200,c[5]=509/845,c[6]=2 44/273,c[8]=17/20,c[9]=171/181,c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\neqns1 \+ := subs(e1,cdns1):\nnops(%);\nindets(eqns1);\nnops(%);\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"b G6#\"#5&%\"cG6#\"\"(&F%6#\"\"&&F%6#\"\"'&F%F*&F%6#\"\")&F%6#\"\"*&F%6# \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We have 8 equations and 8 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "e2 := solve(\{op(eqns1)\}):\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 0 625 "e3 := \{c[10] = 1, b[5] = 89929293580433773826711625 625/175034206955063764470139052544, c[6] = 244/273, b[2] = 0, b[3] = 0 , b[6] = 43965679317071687650068867/11554641620922992752054730, b[1] = 17499592155841793/301627899585331605, b[7] = 639711292154532559948032 664549872463482448777204372657657897547/201160142550158862242760444301 6911932371008211435797382991660160, c[2] = 1/200, b[4] = 0, b[8] = -43 614217354789120000/21595629876813903267, b[10] = 7077278529846113/1554 6034297382400, c[5] = 509/845, c[9] = 171/181, c[8] = 17/20, c[7] = 22 2785849/1109145221, b[9] = -210996982649291857501475159/99036657939412 172256631200\}:" }}}{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 53 "We now have all the weights and the nodes excludi ng " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "seq(c[i]= subs(e3,c[i]),i=[2,$5..10]);``;\nfor ii to 10 do b[ii]=subs(e3,b[ii]) \+ end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"\"\"\"$+#/ &F%6#\"\"&#\"$4&\"$X)/&F%6#\"\"'#\"$W#\"$t#/&F%6#\"\"(#\"*\\eyA#\"+@_9 46/&F%6#\"\")#\"#<\"#?/&F%6#\"\"*#\"$r\"\"$\"=/&F%6#\"#5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\"#\"2$zTe:#f*\\<\"30;L&e**yi,$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&#\">Dci6n#QxL/e$HH**)\"? WD0R,ZkP1bp?M]<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\";n )o+l(orqJzc'R%\";IZ0_F*H#4iTYb6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"\"(#\"jnZv*yldEP/sx[C[jC()\\XmK![*fD`a@H6(R'\"[og,m\"*HQ(zN9@3 5PK>\"p,VWgFCi)e,bU,;,#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \")#!5++7*yat@9O%\"5nK!R\"o()Hcf@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"\"*#! " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We construct a system of equations that incorporate the row-sum conditions," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j = 1 .. i-1) = c[i]" "6#/-%$SumG6$&%\"a G6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . 10, " }}{PARA 0 "" 0 "" {TEXT -1 29 " the stage order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( a[i,j]*c[j],j = 2 .. i-1) = 1/2" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\" \"&%\"cG6#F,F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-F3F5" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG\"\"#" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "i = 3;" "6#/%\"iG\"\"$" }{TEXT -1 9 " . . 10, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j]^2,j = 2 .. i-1) = 1/3" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$&%\"cG6#F, \"\"#F-/F,;F2,&F+F-F-!\"\"*&F-F-\"\"$F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^3" "6#*$&%\"cG6#%\"iG\"\"$" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i = 5;" "6#/%\"iG\"\"&" }{TEXT -1 8 " . . 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "together with the column \+ simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,1],i=2..10)=b[1]" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#5&F)6#F," }{TEXT -1 6 ", " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 10) = b[j]*(1-c[j]);" "6#/-% $SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#5*&&F)6# F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j = 6; " "6#/%\"jG\"\"'" }{TEXT -1 8 " . . 9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "and the further simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 5 .. 10) = 0; " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+; \"\"&\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a [i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F +\"\"#F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "S O_eqs := [op(RowSumConditions(10,'expanded')),op(StageOrderConditions( 2,3..10,'expanded')),\n op(StageOrderConditions(3,5..10,'expanded'))] :\nsimp_eqs := [add(b[i]*a[i,1],i=2..10)=b[1],seq(add(b[i]*a[i,j],i=j+ 1..10)=b[j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5 ..10)=0,add(b[i]*c[i]^2*a[i,3],i=5..10)=0]:\ncdcns2 := [op(simp_eqs),o p(simp_eqs2),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We specify the zero linking coefficients: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"a G6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] = 0" "6#/ &%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2] = 0 " "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2 ] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e4 := \{seq(a[i,2]=0,i=4..10)\}:\ne5 := `union`(e3,e4):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "eqns2 := subs(e5,cdcns2):\nnops(eqns2);\nindets(eqns2);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "e6 := solve(\{op(eqns2)\},indets(eqns2) minus \+ \{c[3],c[4],a[6,1],a[7,6],a[8,6],a[8,7],a[9,5],a[9,6],a[9,7],a[9,8]\}) :\ninfolevel[solve] := 0:\ne7 := `union`(e5,e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "indets(map(r hs,e6));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"\"( \"\"'&F%6$\"\"*F(&F%6$F+F'&F%6$F+\"\")&F%6$F+\"\"&&F%6$F0F(&F%6$F0F'&F %6$F(\"\"\"&%\"cG6#\"\"$&F<6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34747 "e7 := \{a[10,1] = -1/546002366288148993514714650048651048704452557246 78191985490545689453477013251250274377912079917117829700436883650*(-14 2801336664425700104606854688782114744822829809758576689553028669296064 309103420706865414685982617045649331583650*c[3]*c[4]+12002268185720125 3572857643646172929489778241047756632841927191798565800664231480111303 013984523095626579748729600*a[9,6]+60618801671802161417445794429172175 9297150971732088369821095194744991137433889125181047397888628650245895 1268400*a[9,7]+1085542470867675353465384648126900041026321994388666282 47739249983769683518232783801868716959233858440797233812179*a[9,8]+545 1689526923908230679027575582894612461531020476371021204545288796922534 2782537777487023153653677627185498328496*a[9,5]+7047958530457565416260 7618736353185016590063269884427445876817672579190284092824519564899646 036540742143405793825*c[3]+7632501836862474516072049070536133439065868 7380438014637614241896723426065561249143256270762389274367843714613825 *c[4]-9050447250001380068612531043944098128742620259926391970369038838 9654214960022091202311462799287539479315807637680*c[3]*a[9,5]-88769484 6220291004657247370672406704657015940544513430593024717917939283955855 54977696018855865985930890482019200*c[3]*a[9,6]-3017923017948333773821 9894261625540641324455002173495666537974781220113929774432245848320516 203789840525959743600*c[4]*a[9,7]-127710878925608865113574664485517651 8854496463986666214679285293926702159038032750610220199520398334597614 51543740*c[4]*a[9,8]-1342876727336718943663530193254311875029076221558 91642482975915413969113038258992091744765646618053713345374603200*c[4] *a[9,6]-90504472500013800686125310439440981287426202599263919703690388 389654214960022091202311462799287539479315807637680*c[4]*a[9,5]-909245 0878734605023167275551644509217680748870497753099010469888203058001498 4863404857327728494818347422451775465*c[3]*a[9,8]-44273693681061225493 9521390605510832175354512483447183023650149499173911912065705658210599 47870073384054991271600*c[3]*a[9,7]-4595556704353413573560857843288027 4720457097983558444441837089696298823250758266234875374804975895712089 843774550+115471476309041235661931117476319584753922779816526383128860 838578946803370063041078375525090758828481455453776000*c[4]*c[3]*a[6,1 ]-32429353409455616438301974630770547255098947710766822713949809837015 605431194472818490457295390565108940800000000*c[3]*a[8,7]+104730868873 8164308328455316001519602271080596555900026101028497485243693944204249 30918860813034816939622400000000*c[3]*a[8,6]+2204177508705544240604814 1523449123367392490512348487836702496005163428842607622249328634235801 3057745170749916400*c[4]*c[3]*a[9,7]+106970010338054176743144418254641 2849138911631823265070471819986847418588411586628292439149746997862910 85237382900*c[4]*c[3]*a[9,8]+99319956154975182078454316472773373103018 5867904312157999572737670481248032576086430779227362762875374307442264 00*c[4]*c[3]*a[9,6]+15024809285365748836891136998296194339464664282196 0731138739446344317901063297970660025905825929215835013472404400*c[4]* c[3]*a[9,5]-1649128817511334354817473744043406040232256688932799076248 7028553700684554329573187405753060692477682544421228000*c[3]*a[7,6]-11 7178390174392973841667336585415922713116804450721601280975729431750626 412609737730085446729338135346380800000000*c[3]*c[4]*a[8,6]+1614503906 5841993036169759829127384134504730475734081143785609836729193543361451 9422385266235141490602803200000000*c[3]*c[4]*a[8,7]+184513183270735360 1906435787392827249931992115076451425474983112770609378414743229574496 1416266583636617323751000*c[3]*c[4]*a[7,6])/c[3]/c[4], a[9,4] = -1/367 52301379872038764413534375032400*(10330370339106511106316839309446800- 26553537746957548007288778585960909*a[9,8]-133354196194791886490291276 90792016*a[9,5]-29358840383636727983444352970681600*a[9,6]-14828011630 72820160135703313576400*a[9,7]+22138368523496884888859750292179280*c[3 ]*a[9,5]+32848210757101748932296345741787200*c[3]*a[9,6]-1640172834542 3495703889016798332200*c[3]+31239456172891232949751504218777540*c[3]*a [9,8]+7382164671129808409002539661875600*c[3]*a[9,7])/c[4]/(-c[4]+c[3] ), c[10] = 1, a[9,1] = -1/36752301379872038764413534375032400*(-103303 70339106511106316839309446800+26553537746957548007288778585960909*a[9, 8]+13335419619479188649029127690792016*a[9,5]+293588403836367279834443 52970681600*a[9,6]+1482801163072820160135703313576400*a[9,7]-221383685 23496884888859750292179280*c[3]*a[9,5]-3284821075710174893229634574178 7200*c[3]*a[9,6]+16401728345423495703889016798332200*c[3]-312394561728 91232949751504218777540*c[3]*a[9,8]-7382164671129808409002539661875600 *c[3]*a[9,7]+16401728345423495703889016798332200*c[4]-3123945617289123 2949751504218777540*c[4]*a[9,8]-22138368523496884888859750292179280*c[ 4]*a[9,5]-32848210757101748932296345741787200*c[4]*a[9,6]-738216467112 9808409002539661875600*c[4]*a[9,7]+36752301379872038764413534375032400 *c[4]*c[3]*a[9,5]+36752301379872038764413534375032400*c[4]*c[3]*a[9,7] -34721787491481318390689029713428400*c[3]*c[4]+36752301379872038764413 534375032400*c[4]*c[3]*a[9,6]+36752301379872038764413534375032400*c[4] *c[3]*a[9,8])/c[3]/c[4], a[6,3] = -1/4695327*(10329496-551347524*c[4]+ 2389921443*c[4]*a[6,1])/(-509*c[3]+509*c[4]-845*c[3]*c[4]+845*c[3]^2), a[10,6] = -2597227846147402193389012618072482201738742035449752990091 064040820/371804493842727990442869546005007104023142041946468775080811 2491477*a[7,6]+1195816788097076461091569664000000/26955606290243085661 7606166451413*a[8,6]+21694221752050862338927637276561457168/4635648233 480921649154730958571956313*a[9,6]+25036335478788002548955683302246074 4960/281984196771808918111529577339005012981, a[10,7] = 11958167880970 76461091569664000000/269556062902430856617606166451413*a[8,7]+21694221 752050862338927637276561457168/4635648233480921649154730958571956313*a [9,7]+2075541776735585851263121275130843461312924593677931735732770872 240/371804493842727990442869546005007104023142041946468775080811249147 7, b[5] = 89929293580433773826711625625/175034206955063764470139052544 , c[6] = 244/273, b[2] = 0, a[8,1] = 1/5438764166087580851252075985112 166511416112375373262983833600000*(-1536662149216198095420665938957277 375681490937262747719138269600*c[3]*c[4]+19345243215358820242531869201 72404558800387236314557413121478400*a[9,6]+977053206626254530864036044 08527160319314577650878204390063600*a[9,7]+174967621074505761851772486 1611038752849496247518229755479514991*a[9,8]+8787027435234710247212093 24198445710992886759687464023456649584*a[9,5]+772679427544778383716889 719828243679277497202689430966753018400*c[3]+6691701402128034170807522 25282599041515947748362767715859760800*c[4]-14587501341401434496845223 55496437378760293343685475638154948720*c[3]*a[9,5]-3211538411980000610 007746458832380849089051502329668004101472000*c[3]*a[9,6]-486428514044 550354738842761538648855192183224335334542959684400*c[4]*a[9,7]-205844 2600876538374726735131307104415117054408844976182917076460*c[4]*a[9,8] -2164447294177441773037377168881419854723384079974894154844932800*c[4] *a[9,6]-14587501341401434496845223554964373787602933436854756381549487 20*c[4]*a[9,5]-2904668758506038679071665045307127202667631294995882403 497426655*c[3]*a[9,8]-162202349626558954534402840324568664970178424587 410771531638000*c[3]*a[9,7]+774604120444054491759266132950209267108083 3891499349583756539200*c[4]*c[3]*a[6,1]+728162736913159741818724733604 261765839942841915817476710400000*c[3]*a[8,7]-235160766715180710646876 1771880668524538417664219337076531200000*c[3]*a[8,6]+80752867262798634 5293363720039603698698221659260034752064702000*c[4]*c[3]*a[9,7]+341725 7362948280798907841229773090826667801523524567533526384300*c[4]*c[3]*a [9,8]+3593237649469426912016863865824753982792258443180325266884024000 *c[4]*c[3]*a[9,6]+2421697177501809852619688389773064017784769892758795 509314207600*c[4]*c[3]*a[9,5]+2631102021034603852729393293948452898356 508288245405827430400000*c[3]*c[4]*a[8,6]-3625177377210845288687219925 676989667393074207438047102361600000*c[3]*c[4]*a[8,7]-5154118012981479 72474140375293369067095992903322990125242253600)/c[3]/c[4], b[3] = 0, \+ a[7,4] = -1/7033365301668079730099181801955057627788799706345871709456 10557329973096518040393800*(598549928289783767396568775633191387120600 31336129203498135135922521213112054897814700*c[3]*c[4]-193340327375060 3508596183160992845035192212445297515684366963284102282675766812972160 0*a[9,6]-9764870088683901988085123396355029501865594615803438829590292 75779105178157189736400*a[9,7]-174866228157436982035388950210077646568 65541851141410206099495476538673400002926110509*a[9,8]-878193539398416 1040896861119447418716262624823693582104387199933622779180012637262416 *a[9,5]-19121857598927897318440445802429805343072976152933058638183810 198606761243048519898700*c[3]-3633550099915896021582505318392888653393 5834991585145301510804844342881303363359169800*c[4]+145790479526849038 89111684962540410246054848675876378935574035253263749326347109011280*c [3]*a[9,5]+32096773405093516007146852083280040368122190889516714210021 296170263828310863594528000*c[3]*a[9,6]+486146630998524330408302259640 4886466244197966290810173138917185128241964817974515600*c[4]*a[9,7]+20 5724974302867037688692882600091348904300492366369531836464652665160863 52944618953540*c[4]*a[9,8]+2163193007106208023962122962914125797571614 7441238597616072990842619802069030325467200*c[4]*a[9,6]+14579047952684 9038891116849625404102460548486758763789355740352532637493263471090112 80*c[4]*a[9,5]+2902985516562558935558028741208558179774338480199310731 6609182077947306528492087550845*c[3]*a[9,8]+16210835412451664400652120 37312377196282205785924146524755166479436824902834627362000*c[3]*a[9,7 ]+14801554640081753462612421244043592073320995957857785218749928595999 592595011277485800-383953618649405044120057194940224057420555363774073 20556175759755153958749437684278400*c[4]*c[3]*a[6,1]-80706071354371917 32711501166919703465572391515748005100790540317157886955346146298000*c [4]*c[3]*a[9,7]-341527707830889286536238675436300962326392762376389497 84246096562290948857049514765700*c[4]*c[3]*a[9,8]-35911553850780860122 750371388260045165972779150975667948097597764270594790433447976000*c[4 ]*c[3]*a[9,6]-24202938153278475022199162658834276341682410866631709627 819370901783631003464257592400*c[4]*c[3]*a[9,5]-1547909611352146560156 92657594452173911812236537406918484943229519728410122296877346400*c[4] *a[7,6]+15479096113521465601569265759445217391181223653740691848494322 9519728410122296877346400*c[3]*a[7,6])/(-509+845*c[4])/c[4]/(-c[4]+c[3 ]), a[8,5] = -1/761705005579297762858734075853389798874845050995870310 4000*(-615961169039064181898493052630261891038518645633002125539682960 0*c[3]*c[4]+1934524321535882024253186920172404558800387236314557413121 478400*a[9,6]+97705320662625453086403604408527160319314577650878204390 063600*a[9,7]+17496762107450576185177248616110387528494962475182297554 79514991*a[9,8]-109244458643580875898877128313905918770145139346630163 7478400000*c[4]*a[8,7]+87870274352347102472120932419844571099288675968 7464023456649584*a[9,5]+4344647900712343028353306710805699062427641138 083465295462400000*a[8,6]+21943131527458976300538898664285430500296787 3064137796249600000*a[8,7]+2737432982543916966231702169450013831526567 798293022219662906400*c[3]+2633923695211941999595564674904369193765018 343966358968769648800*c[4]-1458750134140143449684522355496437378760293 343685475638154948720*c[3]*a[9,5]-321153841198000061000774645883238084 9089051502329668004101472000*c[3]*a[9,6]-48642851404455035473884276153 8648855192183224335334542959684400*c[4]*a[9,7]-20584426008765383747267 35131307104415117054408844976182917076460*c[4]*a[9,8]-2164447294177441 773037377168881419854723384079974894154844932800*c[4]*a[9,6]-145875013 4140143449684522355496437378760293343685475638154948720*c[4]*a[9,5]-29 04668758506038679071665045307127202667631294995882403497426655*c[3]*a[ 9,8]-162202349626558954534402840324568664970178424587410771531638000*c [3]*a[9,7]+77460412044405449175926613295020926710808338914993495837565 39200*c[4]*c[3]*a[6,1]-36428184952264901717004654953479742186150855155 0484160768000000*c[3]*a[8,7]-72126276544242236914706172311017990329103 27626091410952192000000*c[3]*a[8,6]-1628772149130993169232534096745705 486703799574165025168557856800+807528672627986345293363720039603698698 221659260034752064702000*c[4]*c[3]*a[9,7]+3417257362948280798907841229 773090826667801523524567533526384300*c[4]*c[3]*a[9,8]+3593237649469426 912016863865824753982792258443180325266884024000*c[4]*c[3]*a[9,6]+2421 697177501809852619688389773064017784769892758795509314207600*c[4]*c[3] *a[9,5]+80698661871221847039814692790606194097726206636186688112640000 00*c[3]*c[4]*a[8,6]+18135867888767355625648560594351768440230381679352 15881472000000*c[3]*c[4]*a[8,7]-48610199872724165850018554592211305083 71909961872073875660800000*c[4]*a[8,6])/(-509+845*c[4])/(845*c[3]-509) , b[6] = 43965679317071687650068867/11554641620922992752054730, b[1] = 17499592155841793/301627899585331605, a[7,3] = -81997/100476647166686 853287131168599357966111268567233512452992230079618567585216862913400* (-10428080614269577238828751150010564622061091317997558011379359858552 5277252399300*c[3]*c[4]+3368421621262456481153810785747989099239192453 5523349885047419576365732470470400*a[9,6]+1701259120748999874226256256 126971457468930852139788882448712018695989188031600*a[9,7]+30465614274 640184054710877960705469462934256917311278297811410306890449650602071* a[9,8]+153000987736209182581538020022464562575680030518426320551574185 35561020838763504*a[9,5]+430996157388032265419852665813762405080362286 76239043575174999788751885342742200*c[3]+53519526421601348423763478558 008113060663690910571979022680697097283392161159300*c[4]-2539996751219 9756243889907056774568836237647502567827282137561223082637738222320*c[ 3]*a[9,5]-559197695474808590682705326907082669716526055648668578641749 89277070813236832000*c[3]*a[9,6]-8469763371108077654553603174340675296 908419935730767455148911693856816999956400*c[4]*a[9,7]-358418991466355 10652601032894947611132863831667425033291542835655165234883061260*c[4] *a[9,8]-37687668139534861448975014119229550167717194254909321797614530 919458381001796800*c[4]*a[9,6]-253999675121997562438899070567745688362 37647502567827282137561223082637738222320*c[4]*a[9,5]-5057651092744784 9756838294453430494491511683880408703657466879586095147258858055*c[3]* a[9,8]-282429068179352631379407963934634750797887341858177132744432545 3434402483078000*c[3]*a[9,7]-25787624007292520218705599410507339246420 158155364194889969717700472652475110200+668933216458102202554548502541 42408941887310123553859211183265860169028395529600*c[4]*c[3]*a[6,1]+14 0608055964367890335909522245930660626475733707121777988228494721198631 92462000*c[4]*c[3]*a[9,7]+59501777561703352655103875827565287637072569 271069063126431623042464879128068300*c[4]*c[3]*a[9,8]+6256597166582899 3957532194362964577390414595570527263102130213412460377105144000*c[4]* c[3]*a[9,6]+4216694017251236547364827399405601309748686078520592151946 2159594312041038895600*c[4]*c[3]*a[9,5])/(-c[4]+c[3])/c[3]/(845*c[3]-5 09), a[10,8] = 21694221752050862338927637276561457168/4635648233480921 649154730958571956313*a[9,8]-59790839404853823054578483200000/89852020 967476952205868722150471, b[7] = 6397112921545325599480326645498724634 82448777204372657657897547/2011601425501588622427604443016911932371008 211435797382991660160, a[4,3] = 1/2*1/c[3]*c[4]^2, a[4,2] = 0, a[5,2] \+ = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[5, 3] = -259081/3620106750*(-1018+2535*c[4])/c[3]/(-c[4]+c[3]), a[4,1] = \+ 1/2*c[4]*(-c[4]+2*c[3])/c[3], c[2] = 1/200, a[6,5] = -325/27783*(24379 992*c[3]-54555228*c[3]*c[4]+61039251*c[4]*c[3]*a[6,1]-14526784+2437999 2*c[4])/(-509+845*c[4])/(845*c[3]-509), a[10,9] = -1198575787406124991 100974435169141280/4635648233480921649154730958571956313, a[3,1] = -10 0*c[3]^2+c[3], b[4] = 0, b[8] = -43614217354789120000/2159562987681390 3267, a[3,2] = 100*c[3]^2, a[7,1] = 1/35799829385490525826204835371951 2433254449905053004870011315773680956306127682560444200*(-162108354124 5166440065212037312377196282205785924146524755166479436824902834627362 000*c[3]*a[9,7]-290298551656255893555802874120855817977433848019931073 16609182077947306528492087550845*c[3]*a[9,8]-1457904795268490388911168 4962540410246054848675876378935574035253263749326347109011280*c[3]*a[9 ,5]+191218575989278973184404458024298053430729761529330586381838101986 06761243048519898700*c[3]+64346899217236936595671075817911983051467649 541508042827141284881956301324885994600*c[3]*c[4]+80706071354371917327 11501166919703465572391515748005100790540317157886955346146298000*c[4] *c[3]*a[9,7]+242029381532784750221991626588342763416824108666317096278 19370901783631003464257592400*c[4]*c[3]*a[9,5]+35911553850780860122750 371388260045165972779150975667948097597764270594790433447976000*c[4]*c [3]*a[9,6]+34152770783088928653623867543630096232639276237638949784246 096562290948857049514765700*c[4]*c[3]*a[9,8]-3209677340509351600714685 2083280040368122190889516714210021296170263828310863594528000*c[3]*a[9 ,6]+173188247499645906115918424275760014253789920388164298140940580569 204327718799375063800*c[3]*c[4]*a[7,6]-1547909611352146560156926575944 52173911812236537406918484943229519728410122296877346400*c[3]*a[7,6]+3 8395361864940504412005719494022405742055536377407320556175759755153958 749437684278400*c[4]*c[3]*a[6,1]+1933403273750603508596183160992845035 1922124452975156843669632841022826757668129721600*a[9,6]+9764870088683 9019880851233963550295018655946158034388295902927577910517815718973640 0*a[9,7]+1748662281574369820353889502100776465686554185114141020609949 5476538673400002926110509*a[9,8]-1480155464008175346261242124404359207 3320995957857785218749928595999592595011277485800+87819353939841610408 96861119447418716262624823693582104387199933622779180012637262416*a[9, 5]+3071908425594432036692333765924693872644668464515919394745924383920 1624149270053854700*c[4]-486146630998524330408302259640488646624419796 6290810173138917185128241964817974515600*c[4]*a[9,7]-20572497430286703 768869288260009134890430049236636953183646465266516086352944618953540* c[4]*a[9,8]-2163193007106208023962122962914125797571614744123859761607 2990842619802069030325467200*c[4]*a[9,6]-14579047952684903889111684962 540410246054848675876378935574035253263749326347109011280*c[4]*a[9,5]) /c[3]/c[4], a[2,1] = 1/200, a[7,5] = -13/65179479991789760266189959712 24623272725533091543101866387178401109809852119846344*(-71844139317860 8698954353952713127360321090783338262370602332849625874954852411777752 00*c[3]*c[4]+285979771940394201664980219606393380110251439670942826832 423565348648373674914582492800*a[7,6]+19334032737506035085961831609928 450351922124452975156843669632841022826757668129721600*a[9,6]+97648700 8868390198808512339635502950186559461580343882959029275779105178157189 736400*a[9,7]+17486622815743698203538895021007764656865541851141410206 099495476538673400002926110509*a[9,8]+87819353939841610408968611194474 18716262624823693582104387199933622779180012637262416*a[9,5]+263437221 7033758846198040498984843378793092280598369778979086092214964612819859 8596800*c[3]+379409488273540115104632968466655671713046312982098330990 66294562744509034420132552800*c[4]-14579047952684903889111684962540410 246054848675876378935574035253263749326347109011280*c[3]*a[9,5]-320967 7340509351600714685208328004036812219088951671421002129617026382831086 3594528000*c[3]*a[9,6]-48614663099852433040830225964048864662441979662 90810173138917185128241964817974515600*c[4]*a[9,7]-2057249743028670376 8869288260009134890430049236636953183646465266516086352944618953540*c[ 4]*a[9,8]-216319300710620802396212296291412579757161474412385976160729 90842619802069030325467200*c[4]*a[9,6]-1457904795268490388911168496254 0410246054848675876378935574035253263749326347109011280*c[4]*a[9,5]-29 0298551656255893555802874120855817977433848019931073166091820779473065 28492087550845*c[3]*a[9,8]-1621083541245166440065212037312377196282205 785924146524755166479436824902834627362000*c[3]*a[9,7]-157686232135152 2223062528044443763993654077976859383976189282121927347249454725103840 0+38395361864940504412005719494022405742055536377407320556175759755153 958749437684278400*c[4]*c[3]*a[6,1]+8070607135437191732711501166919703 465572391515748005100790540317157886955346146298000*c[4]*c[3]*a[9,7]+3 4152770783088928653623867543630096232639276237638949784246096562290948 857049514765700*c[4]*c[3]*a[9,8]+3591155385078086012275037138826004516 5972779150975667948097597764270594790433447976000*c[4]*c[3]*a[9,6]+242 0293815327847502219916265883427634168241086663170962781937090178363100 3464257592400*c[4]*c[3]*a[9,5]-474760132199672102960527083629474275428 609953874158523916302382553257123291361143824000*c[3]*a[7,6]-319969171 0644574469448344260350221015167977173367516054313591530335287131690642 66477600*c[4]*a[7,6]+5311865413545511643779667779952724475082398254411 69168152256354250160633846481935508000*c[3]*c[4]*a[7,6])/(-509+845*c[4 ])/(845*c[3]-509), b[10] = 7077278529846113/15546034297382400, c[5] = \+ 509/845, a[5,4] = 259081/3620106750*(-1018+2535*c[3])/c[4]/(-c[4]+c[3] ), c[9] = 171/181, c[8] = 17/20, a[10,3] = 1/1326953905737796039551805 36895613170497005965524693635150*(-18969488914082577220434568520078846 9717784176378764992022875*c[3]*c[4]-5684826619530605670738901314313303 1261693217700990381950450+14847126924207330962963015485935158936325780 2124925342470400*a[9,6]+7498707981591382339723824304429132690395419328 044238531600*a[9,7]+13428450853785848937570369102421835613981773936317 1614038321*a[9,8]+6743885831005920122827581294015737515526616021038656 3983504*a[9,5]+9499219845259853646180758353002446628763957204515816009 5250*c[3]+94416090165330373312356979328258620785764978938040867973675* c[4]-111956454365422446046941182582383068774459539052606378322320*c[3] *a[9,5]-246479808466703235043295640188903915544111675433323996832000*c [3]*a[9,6]-37332515322625530389448021840477047789184096463825664456400 *c[4]*a[9,7]-157981774750421752206710224734374536635079693368437192986 260*c[4]*a[9,8]-166117444684778743970856689658208950394136803197150075 796800*c[4]*a[9,6]-111956454365422446046941182582383068774459539052606 378322320*c[4]*a[9,5]-222928113387996902794635793547081553905984262793 477433914305*c[3]*a[9,8]-124487391835849078134904352401623126196151853 28482085578000*c[3]*a[9,7]+1428413061895826518181956898796770813593793 99529385018224000*c[4]*c[3]*a[6,1]+61976376124987373632777167888414745 347466722027372664962000*c[4]*c[3]*a[9,7]+2622683686917610621113362277 02448886948216779757032275193300*c[4]*c[3]*a[9,8]+27577453980086058674 9261105621191675998124948333186275144000*c[4]*c[3]*a[9,6]+185860911471 084414360835558511028866629505521609926109395600*c[4]*c[3]*a[9,5])/(-c [4]+c[3])/c[3]/(845*c[3]-509), a[5,1] = 509/3620106750*(4284150*c[3]*c [4]-1290315*c[3]+518162-1290315*c[4])/c[3]/c[4], c[7] = 222785849/1109 145221, a[10,5] = 1/76468242188739749100481726837106690760751032141281 036357957418422959247944051329119257605938051353705683186*(-1167081440 1834428805266102943141070812860981753317831775496864340965297284984362 0910040311861080511469379130928000*c[3]*c[4]+3046802472660215722757255 2417359364588412500444594545627803564230329057300470310334008373728125 239252358373456000*a[7,6]-84095873298031751502944381288223219046350800 560618827294565059638809175905292387959722091492855775706376803168000* a[9,6]-424735640462988619605105505428809372917043696400060013593627379 6338578818595426625069020668007305191683614422000*a[9,7]-7606032516281 6759181146288723867349858752829276819999977206596274447973559365762499 592134243807650123553014335695*a[9,8]+48652958710792229327201686904011 3070323230369014357299447470714131739174497183053400537422172313158688 76800000000*c[4]*a[8,7]+5451689526923908230679027575582894612461531020 4763710212045452887969225342782537777487023153653677627185498328496*a[ 9,5]-19349262887185197690844032016739022138800578041720327810666488444 9782553312250853052086322204240693284044800000000*a[8,6]-9772562246603 9478821892941385556765995349839030183121662435575648101548135250068088 46996479546677084979200000000*a[8,7]+484092863627010719742637272154338 0847609607647223427019139874937473063458343799004915725960393167336896 6495772000*c[3]+542547194267501629723765991844419578501647005827878573 83136173598874870364906414672848630720284406994666804592000*c[4]-90504 4725000138006861253104394409812874262025992639197036903883896542149600 22091202311462799287539479315807637680*c[3]*a[9,5]+1396090627442766798 0351277443722715146201655495819824963439582592297397571703746134767223 4403660374208032217440000*c[3]*a[9,6]+21145575803959525939746275291800 6687081344132099836138241691842903202617441691432279949839591047371525 12635038000*c[4]*a[9,7]+8948273548566677550723092791043217630441509326 6847058796713642675821145363959720587755452051538411910062369806700*c[ 4]*a[9,8]+940908746326338859028844921790366344248105268567579502107223 82300798791074364024233623487612908306425577324856000*c[4]*a[9,6]-9050 4472500013800686125310439440981287426202599263919703690388389654214960 022091202311462799287539479315807637680*c[4]*a[9,5]+126269105623929590 3891328368795047360130572509605361492745374731864607812527781322439201 44275083427022401369574975*c[3]*a[9,8]+7051112302381638184014030492875 1261319234169638123911883421441216229844827370049080222445274384536089 83603510000*c[3]*a[9,7]+1154714763090412356619311174763195847539227798 16526383128860838578946803370063041078375525090758828481455453776000*c [4]*c[3]*a[6,1]+162236053013366128888997122732407597772240891906689072 30797261576158312018523832521563284921840750759936000000000*c[3]*a[8,7 ]+32122057248863442139023982424645331448499977299123137524583856062881 3865518373223632638393443189363113984000000000*c[3]*a[8,6]-35104148436 8286825522310464078026818435630238947665101796128894407084895359978900 34687154116784877984033745790000*c[4]*c[3]*a[9,7]-14855188896932892986 9568043387652630603596765835924881499455850807600919120915449698729581 500098149438119258323500*c[4]*c[3]*a[9,8]-1562019431524079245342581451 6952054241446934222782017274668057572529465315881650388489557373852164 8191773751480000*c[4]*c[3]*a[9,6]+150248092853657488368911369982961943 3946466428219607311387394463443179010632979706600259058259292158350134 72404400*c[4]*c[3]*a[9,5]-50580512561844445692139109612315644552472618 616271888124742655745831146206085289257833154813881782255879814480000* c[3]*a[7,6]-3408922438673110214396437217188158415015005172694389736225 5627192130461651755716070427401753189304573335393252000*c[4]*a[7,6]-35 9398427415562282948915868931482601862315319781172809188991504310107316 748015942834878202499961869385728000000000*c[3]*c[4]*a[8,6]-8076964658 2749378745550934054792837804151210573110396470159676511064754901791685 682407489535482243436544000000000*c[3]*c[4]*a[8,7]-2885952071571339742 1496047110110475326956132828426136295972658334072160387503891018164346 825500630061945701340000+565921308581292363686638398531236514869878068 94435350237929282863163536533857721177821521574548059655144218660000*c [3]*c[4]*a[7,6]+216489703614817990557394292646301354257891713335641372 635735710880289496123952798701719532630154546174361600000000*c[4]*a[8, 6])/(-509+845*c[4])/(845*c[3]-509), a[9,3] = 1/36752301379872038764413 534375032400*(-16401728345423495703889016798332200*c[4]+10330370339106 511106316839309446800-1482801163072820160135703313576400*a[9,7]-265535 37746957548007288778585960909*a[9,8]-293588403836367279834443529706816 00*a[9,6]-13335419619479188649029127690792016*a[9,5]+22138368523496884 888859750292179280*c[4]*a[9,5]+31239456172891232949751504218777540*c[4 ]*a[9,8]+32848210757101748932296345741787200*c[4]*a[9,6]+7382164671129 808409002539661875600*c[4]*a[9,7])/c[3]/(-c[4]+c[3]), a[8,4] = 1/10685 194825319412281438263232047478411426546906430772070400000*(-2897889187 494232973749141685574925828557161597223862888581789600*c[3]*c[4]+19345 24321535882024253186920172404558800387236314557413121478400*a[9,6]+977 05320662625453086403604408527160319314577650878204390063600*a[9,7]+174 9676210745057618517724861611038752849496247518229755479514991*a[9,8]-7 28162736913159741818724733604261765839942841915817476710400000*c[4]*a[ 8,7]+878702743523471024721209324198445710992886759687464023456649584*a [9,5]+772679427544778383716889719828243679277497202689430966753018400* c[3]+785614276903976987295284331825343114062470923806202560907792800*c [4]-1458750134140143449684522355496437378760293343685475638154948720*c [3]*a[9,5]-32115384119800006100077464588323808490890515023296680041014 72000*c[3]*a[9,6]-4864285140445503547388427615386488551921832243353345 42959684400*c[4]*a[9,7]-2058442600876538374726735131307104415117054408 844976182917076460*c[4]*a[9,8]-216444729417744177303737716888141985472 3384079974894154844932800*c[4]*a[9,6]-14587501341401434496845223554964 37378760293343685475638154948720*c[4]*a[9,5]-2904668758506038679071665 045307127202667631294995882403497426655*c[3]*a[9,8]-162202349626558954 534402840324568664970178424587410771531638000*c[3]*a[9,7]+774604120444 0544917592661329502092671080833891499349583756539200*c[4]*c[3]*a[6,1]+ 728162736913159741818724733604261765839942841915817476710400000*c[3]*a [8,7]-2351607667151807106468761771880668524538417664219337076531200000 *c[3]*a[8,6]+807528672627986345293363720039603698698221659260034752064 702000*c[4]*c[3]*a[9,7]+3417257362948280798907841229773090826667801523 524567533526384300*c[4]*c[3]*a[9,8]+3593237649469426912016863865824753 982792258443180325266884024000*c[4]*c[3]*a[9,6]+2421697177501809852619 688389773064017784769892758795509314207600*c[4]*c[3]*a[9,5]-5154118012 98147972474140375293369067095992903322990125242253600+2351607667151807 106468761771880668524538417664219337076531200000*c[4]*a[8,6])/c[4]/(-5 09*c[3]+509*c[4]+845*c[3]*c[4]-845*c[4]^2), a[10,4] = 1/10726962009590 3535071653172897573879902642938555359905668940168348631585487723477945 732636699247775696857439850*1/c[4]*(1533474418688040561256484328404026 0042078549306169380474914146799056069302087178422535358462253161742288 5003164625*c[3]*c[4]-1200226818572012535728576436461729294897782410477 56632841927191798565800664231480111303013984523095626579748729600*a[9, 6]-6061880167180216141744579442917217592971509717320883698210951947449 911374338891251810473978886286502458951268400*a[9,7]-10855424708676753 5346538464812690004102632199438866628247739249983769683518232783801868 716959233858440797233812179*a[9,8]-32429353409455616438301974630770547 2550989477107668227139498098370156054311944728184904572953905651089408 00000000*c[4]*a[8,7]-5451689526923908230679027575582894612461531020476 3710212045452887969225342782537777487023153653677627185498328496*a[9,5 ]-70479585304575654162607618736353185016590063269884427445876817672579 190284092824519564899646036540742143405793825*c[3]-8263617158197319612 6453394798863334053520919749952494678333510536166678065201006142634129 035916233073982718849750*c[4]+9050447250001380068612531043944098128742 6202599263919703690388389654214960022091202311462799287539479315807637 680*c[3]*a[9,5]+887694846220291004657247370672406704657015940544513430 59302471791793928395585554977696018855865985930890482019200*c[3]*a[9,6 ]-40310278605812617443881001441808493226531854187917595219011965624199 19478885255986417040979487674320389908106000*c[4]*a[9,7]+2169992164807 3544232360132501371510090063470882500991733310804708487417925405751246 6732601517361538457080969456470*c[4]*a[9,8]+24476998805139733077501867 2987102354795285819231802489505065343562819784959521375609237443877042 865040725791752000*c[4]*a[9,6]+905044725000138006861253104394409812874 2620259926391970369038838965421496002209120231146279928753947931580763 7680*c[4]*a[9,5]+90924508787346050231672755516445092176807488704977530 990104698882030580014984863404857327728494818347422451775465*c[3]*a[9, 8]+4427369368106122549395213906055108321753545124834471830236501494991 7391191206570565821059947870073384054991271600*c[3]*a[9,7]+45955567043 5341357356085784328802747204570979835584444418370896962988232507582662 34875374804975895712089843774550-1154714763090412356619311174763195847 5392277981652638312886083857894680337006304107837552509075882848145545 3776000*c[4]*c[3]*a[6,1]+324293534094556164383019746307705472550989477 10766822713949809837015605431194472818490457295390565108940800000000*c [3]*a[8,7]-10473086887381643083284553160015196022710805965559000261010 2849748524369394420424930918860813034816939622400000000*c[3]*a[8,6]-50 1010795710479771882039501985728523416879459269874338668459502753064759 73790560408137192212558354450381996038000*c[4]*c[3]*a[9,7]-21201511334 4085444049058136523108871990579471919200972770922607734393580429693059 777138716816254733346755258456700*c[4]*c[3]*a[9,8]-2229333663260368383 8814990438111857257358927450241343398450815034342613068237494757863325 5346546670702901456856000*c[4]*c[3]*a[9,6]-150248092853657488368911369 9829619433946466428219607311387394463443179010632979706600259058259292 15835013472404400*c[4]*c[3]*a[9,5]+16491288175113343548174737440434060 4023225668893279907624870285537006845543295731874057530606924776825444 21228000*c[3]*a[7,6]-1649128817511334354817473744043406040232256688932 7990762487028553700684554329573187405753060692477682544421228000*c[4]* a[7,6]+104730868873816430832845531600151960227108059655590002610102849 748524369394420424930918860813034816939622400000000*c[4]*a[8,6])/(-c[4 ]+c[3])/(-509+845*c[4]), a[8,3] = -17/35617316084398040938127544106824 92803808848968810257356800000*(-40361619625030164210328139829551066963 079498212646591821903460*c[4]*a[9,8]+131209831414275179819755338290705 69441489171536524857173720800*c[4]+15833895541725222456732621961560856 837220032534510485334602000*c[4]*c[3]*a[9,7]-4244014302308709358896817 9781988624602419295685782238330292800*c[4]*a[9,6]-95378140008735363674 28289441934291278278102437947736136464400*c[4]*a[9,7]-5682135661753397 9877434150697547565265826697984781625266309600*c[3]*c[4]+4748425838238 8428482738988034765961133034703779584225672827600*c[4]*c[3]*a[9,5]+704 55640185675037490526742467152038878279577317261279742824000*c[4]*c[3]* a[9,6]+67005046332319231351134141760256682875839245559305245755419300* c[4]*c[3]*a[9,8]-28602943806669479405578869715616419191378300856577953 689312720*c[4]*a[9,5]+151883160871383233678287477049060640609428115519 595089877579200*c[4]*c[3]*a[6,1]-2860294380666947940557886971561641919 1378300856577953689312720*c[3]*a[9,5]-10106113750944077891649811280262 138570509664771039022063573600+174337953771755285084592514974703480749 80791728095408074530400*c[3]+17229465559283745582768810278400896293978 171758577725950130384*a[9,5]+19157906012279500605177177335005325552806 77993154474595883600*a[9,7]+343073766812756395787789188551184069186175 73480749603048617941*a[9,8]+379318494418800396912389592190667560549095 53653226615943558400*a[9,6]-629713414113725609805440482123996244919421 86320189568707872000*c[3]*a[9,6]-3180438227971744206556918437736640489 611341658576681794738000*c[3]*a[9,7]-569542893824713466484640204962181 80444463358725409458892106405*c[3]*a[9,8])/(-c[4]+c[3])/c[3]/(845*c[3] -509), a[6,4] = 1/4695327*(2389921443*c[3]*a[6,1]-551347524*c[3]+10329 496)/(-509+845*c[4])/(-c[4]+c[3]), b[9] = -210996982649291857501475159 /99036657939412172256631200\}:" }}}{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 33 "We have expressions in terms of " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 69 " for the linking coefficients in rows 2 to 6 of the Butcher tableau." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "su bs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(5-i)],i=2..5)]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"\"\"$+#F(%!GF+F+7 '&%\"cG6#\"\"$,&*&\"$+\"F))F-\"\"#F)!\"\"F-F),$*&F3F)F4F)F)F+F+7'&F.6# \"\"%,$*&#F)F5F)*(F:F),&F:F6*&F5F)F-F)F)F)F-F6F)F)\"\"!,$*&F?F)*&F-F6F :F5F)F)F+7'#\"$4&\"$X),$*&#FI\"+]n5?OF)*(,**(\"(]TG%F)F-F)F:F)F)*&\"(: .H\"F)F-F)F6\"'i\"=&F)*&FTF)F:F)F6F)F-F6F:F6F)F)FC,$*&#\"'\"3f#FNF)*(, &\"%=5F6*&\"%NDF)F:F)F)F)F-F6,&F:F6F-F)F6F)F6,$*&#FZFNF)*(,&FgnF6*&Fin F)F-F)F)F)F:F6FjnF6F)F)Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "c[6]=subs(e7,c[6]);``;\nfo r ii from 2 to 5 do a[6,ii]=subs(e7,a[6,ii]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$W#\"$t#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\" \"\"\"(F`p%F,*&,(\")'\\H.\"F,*&\"*CvM^&F,&%\"cG6#\"\"%F,!\"\"*(\"+V9#* *Q#F,F3F,&F%6$F'F,F,F,F,,**&\"$4&F,&F46#F(F,F7*&F>F,F3F,F,*(\"$X)F,F?F ,F3F,F7*&FCF,)F?\"\"#F,F,F7F,F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"'\"\"%,$*&#\"\"\"\"(F`p%F,*(,(*(\"+V9#**Q#F,&%\"cG6#\"\"$F,& F%6$F'F,F,F,*&\"*CvM^&F,F2F,!\"\"\")'\\H.\"F,F,,&\"$4&F:*&\"$X)F,&F36# F(F,F,F:,&F@F:F2F,F:F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"'\"\"&,$*&#\"$D$\"&$yF\"\"\"*(,,*&\")#**zV#F.&%\"cG6#\"\"$F.F.*(\" )G_baF.F3F.&F46#\"\"%F.!\"\"**\")^#R5'F.F9F.F3F.&F%6$F'F.F.F.\")%yEX\" F<*&F2F.F9F.F.F.,&\"$4&F<*&\"$X)F.F9F.F.F<,&*&FFF.F3F.F.FDF " 0 " " {MPLTEXT 1 0 49 "SO7_10 := SimpleOrderConditions(7,10,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 54 do\n eq := simplify(subs(e7,SO7_ 10[ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets(lhs(eq)) )\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#h<)&%\"aG 6$\"\"(\"\"'&F&6$\"\"*F)&F&6$F,F(&F&6$F,\"\")&F&6$F1F)&F&6$F1F(&%\"cG6 #\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#f<)&%\"aG6$\"\"(\"\"'&F&6 $\"\"*F)&F&6$F,F(&F&6$F,\"\")&F&6$F1F)&F&6$F1F(&%\"cG6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#e<)&%\"aG6$\"\"(\"\"'&F&6$\"\"*F)&F&6$F,F (&F&6$F,\"\")&F&6$F1F)&F&6$F1F(&%\"cG6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<,&%\"aG6$\"\"(\"\"'&F&6$\"\"*F)&F&6$F,F(&F&6$F,\" \")&F&6$F,\"\"&&F&6$F1F)&F&6$F1F(&F&6$F)\"\"\"&%\"cG6#\"\"$&F=6#\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#a<)&%\"aG6$\"\"(\"\"'&F&6$\"\"* F)&F&6$F,F(&F&6$F,\"\")&F&6$F1F)&F&6$F1F(&%\"cG6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a syst em of equations that consists of the simple order comditions given in \+ abreviated form as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "SO7 := SimpleOrderConditions(7):\n [seq([i,SO7[i]],i=[54,59,61])]:\nlinalg[augment](linalg[delcols](%,2.. 2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"#a%#~~G/*&%\"bG\"\" \"-%!G6#*(%\"aGF-%\"cGF--F/6#*&)F3\"\"$F-F2F-F-F-#F-\"$o\"7%\"#fF)/*(F ,F-)F3\"\"#F-F4F-#F-\"#G7%\"#hF)/*(F,F-F3F--F/6#*&)F3\"\"%F-F2F-F-#F- \"#NQ)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 109 "cdcns3 := [seq(SO7_10[i],i=[54,59,61])]:\ne qns3 := simplify(subs(e7,cdcns3)):\nnops(%);\nindets(eqns3);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"aG6$\"\"(\"\"'&F%6$\"\"*F(&F%6$F+F'&F%6$F+\"\")&F %6$F0F(&F%6$F0F'&%\"cG6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We c an solve this system of 3 equations for the linking coefficients " } {XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[9,7]" "6#&%\"aG6$\"\"*\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[9,8]" "6#&%\"aG6$\"\"*\"\")" }{TEXT -1 30 " in terms of the parameters " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,6]" "6#&%\"aG6$\"\")\"\"'" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,6]" "6#&%\"aG6$\"\"*\"\"'" } {TEXT -1 89 ", and subsitute back into the expressions previously obta ined for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "e8 := solve(\{op(eqns3)\}, \{a[8,7],a[9,7],a[9,8]\}):\ninfolevel[solve] := 0:\ne9 := `union`(map( u_->lhs(u_)=simplify(subs(e8,rhs(u_))),e7),e8):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "indets(map(r hs,e8));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"aG6$\"\"( \"\"'&F%6$\"\"*F(&F%6$\"\")F(&%\"cG6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48039 "e9 := \{a[8,5] = -1/290504670534382988161194474231 2728058034435028864593217053713752323841870091904000*(-418057121536677 5413405388145551300500885381921365892831768253075542886254504045619081 25*c[4]*c[3]*a[9,6]-24458900527164860270139277795858018978249580369803 69402416242303905340427605403124107600*c[3]*c[4]+600884109756960149173 1755961603182571500014820289982040779342756087463543038577712013750*c[ 3]*c[4]*a[7,6]+1115123243161362189511212914515683792213704990358042619 777564103732605098899532144000000*c[4]*a[8,6]+251823757233335939103354 149832616799402444899168667390694679386443944272608586889954125*c[4]*a [9,6]-1851236032360218173157121636082029085305659561596357590789865751 776132236876433520000000*c[3]*c[4]*a[8,6]+2954242300574577064107143730 162328488133511412355909998276310550638677860163743138591700*c[4]*c[3] *a[6,1]-36195267676484345080372352478769466614124349627545572292978526 18755643720007853320017750*c[4]*a[7,6]+9236047229979358215908190580613 47998462133513464478514742159914574078428262555062661975*c[4]*c[3]*a[9 ,5]+106278925506214240708007082039425662323283331611462669149603309003 1346972840361122066400*c[4]-556348880480413412058848403021569386055888 708110555697045869108305569136077681096917095*c[4]*a[9,5]-996666927953 7449605887763778088895432239707606130490814129144370357349601885928320 00000*a[8,6]-225073248223201352165635210839408421443943426363204554320 519305100082060499982421790500*a[9,6]-53705392959962738607419357312497 30943025654271614489443040877774671579137367813046635000*c[3]*a[7,6]-5 5634888048041341205884840302156938605588870811055569704586910830556913 6077681096917095*c[3]*a[9,5]+10724959618505683736247600122291887349207 79874699060719828106653828383855323736501443275*c[3]+16545845856992426 1630160322052752782715963711732421704085248074517720243881996256000000 0*c[3]*a[8,6]+32350349132095898167072725292380036094675242890553551792 99179629950099149018008095547000*a[7,6]+373648123278202637681653739016 306711434444391506695183498701007484419137765196751302500*c[3]*a[9,6]- 6383446770191975264667907903853648170426633978459970248356201057346255 38793152982215675+3351261303722253570863359019384364704170974584949974 55380292752813650521021940447728759*a[9,5])/(845*c[3]-509)/(-509+845*c [4]), a[9,7] = 1364473912812730509838628861/90264152844692051935675458 5825478353489977447025280359665552150313136912864878648280455586603621 2633286*(-362076707906572194444829219813868023273107437039122591716485 310749073596692629778533132*a[7,6]+14281398663884623923493304585516560 08780451067160020363281207599923654306635238469096*a[9,6]-568766986489 5827928869411810620641118085864927191363567081221760531431935017018955 343+581827493474785420489169296092705175416308076811942946257368102665 92958419755827200000*a[8,6]+262749210086339168249281885332173582430067 95789147534721633627421956082123043752251605*c[4]-15978778013280747258 66259078625418403266652218584776881867908503193268957833688942882*c[4] *a[9,6]+40511041499382872575179662708682774735064889472000191614180528 6206955294660196432539119*c[4]*a[7,6]-65097912179760827784239023702175 620036332829905598534560762906568769990363087462400000*c[4]*a[8,6])/(- 222785849+1109145221*c[4]), c[10] = 1, a[6,3] = -1/4695327*(10329496-5 51347524*c[4]+2389921443*c[4]*a[6,1])/(-509*c[3]+509*c[4]-845*c[3]*c[4 ]+845*c[3]^2), a[10,6] = -25972278461474021933890126180724822017387420 35449752990091064040820/3718044938427279904428695460050071040231420419 464687750808112491477*a[7,6]+1195816788097076461091569664000000/269556 062902430856617606166451413*a[8,6]+21694221752050862338927637276561457 168/4635648233480921649154730958571956313*a[9,6]+250363354787880025489 556833022460744960/281984196771808918111529577339005012981, a[10,8] = \+ 2193443992576000/79367777649092228181944332547726338039416024435398527 1346644512231918791511207881555305031*(-863194527821852072241047842272 45561428060980738151247846013292402809712624628*a[7,6]+399803747839367 75835221330601070730225688099111869347008169121692188888719064*a[9,6]+ 4272028469345799926507418752686236293262833813309311471486205622574208 172528+102896033709136642564411419844181624259141375407271438508795575 039300116480000*a[8,6]-47894278524765653848395735121202118377431678029 95594526588531459508829795562*c[4]-44732140639404671323833701861034054 719724799416148900546025287794949043525838*c[4]*a[9,6]+965787320062973 83492543467598516550286314130088177420745744380434291194862801*c[4]*a[ 7,6]-11512548033850124352493572794041632550305571920567665046270980322 0200540160000*c[4]*a[8,6])/(-17+20*c[4]), b[5] = 899292935804337738267 11625625/175034206955063764470139052544, a[9,4] = -1/58875982795076623 9606546530820174734575292137846034148888518195161195047921006065090883 50*(-38367037769486314341906274966158694357163923747232173949250117031 1483883248619433254817266344241150*c[3]*c[4]+2895773739967825282023622 5442372677951605792012844502249081487320586333229856917607638151660155 375-356096030040774730384851199544973131359757455412094582973594986988 525824115901239424847414468945000*c[3]*c[4]^2*a[9,6]+17649816068585183 8156337655746357986920068985849399110299910915571053033903285217391434 12547652809500*c[3]*c[4]^2*a[7,6]-610115672716142057256093587605228747 4477879193364441891100911758492352754722216409845072588800000000*c[3]* c[4]^2*a[8,6]+78671539902859803634414362255958092964429947536032715529 3807646695156613807524213176928515097305400*c[3]*a[9,5]*c[4]^2+3574204 7466356040484777592656507829008259606313297901368471170477424847093542 5975478407275691199000*c[3]*c[4]^2-47389128769888331420020012293825644 1643725956163794700644435612032940492814236478706575874774589880*a[9,5 ]*c[4]^2-2693304633925506743110036270704816184427555850488143591797002 121380127335990154091710960626583705900*a[7,6]+54339098158555194097888 7493233581960464661423304775028849232947703535832200902939653819406644 29000*a[9,6]-809090011254208245649751028453453170868587065455107051559 44335723468148188850522231211677305589262*a[9,5]+931016709846475480449 8725763338110276838693058759457892289892921652135626843047990936454553 60000000*a[8,6]+177678619550082642020135876053364424793551475021479151 834558387987366518759577164631216554533068500*c[4]^2+72160721180334956 7343779477799605913342878611580933362082556865920629712489787983261924 84026346500*c[3]-18937194605425657342455671267667895960653690929180094 7320422042810765182542793334464737784685688525*c[4]+134318479275010995 5941138740752785715882035501590501883237189856312978098616477235469034 72147785710*c[3]*a[9,5]-1073371822568851648238861337106246289994701904 47771986966430825489661582089361282134254657039316400*c[3]*a[9,6]-2879 1723445931346892521693131986171953180764200717609069337373567942796257 5669073644927144843288850*c[4]*a[9,6]+49799465469160472832308199019792 4230793471423381002795731234194608550181467219268349073819800271118*c[ 4]*a[9,5]+214500448864798032859040545051350679126765141780776500276402 187428591295236678971440529389307329000*c[4]^2*a[9,6]+3675134643935104 2265485400720835672479399295969497052338110817574823758013652167486522 38991360000000*c[4]^2*a[8,6]-24456692784625469610575970177903741530302 10062895849152981148261856821497692948879007449640960000000*c[3]*a[8,6 ]+47797654836565791992496720425399440668836435657379920753615089714954 1509580432941512698305289939250*c[4]*c[3]*a[9,6]-826729829497850678650 3031074995009332425999071845724212041117769041746627500987853732168521 24222190*c[4]*c[3]*a[9,5]+11571982667615172087021135443702282022325647 609015537688276116740967771474198806184259238769148103100*c[3]*a[7,6]+ 8018709636755707118740714909879399823021068311665108688869927713396087 110856571459709596777779079175*c[4]*a[7,6]+818939158954568993077297673 7758178441297417003448480062918331250548366105306846561881980062720000 000*c[3]*c[4]*a[8,6]-5600191233232616745100588019979967340800156080280 928618482344358714066185238641221665034198178631500*c[4]^2*a[7,6]-2872 2268445535018677132799956992798411235676827077727007912693381543991815 996542610465066589546878075*c[3]*c[4]*a[7,6]-4326406060911972862460854 4252259331915511130724437941290701961578882472726142151071790368512000 00000*c[4]*a[8,6])/(-17+20*c[4])/(-222785849+1109145221*c[4])/c[4]/(-c [4]+c[3]), a[10,4] = -1/1293580563118200021439089890808937427422636520 373737997591420407969652611253487792200651960019101124843501350*(10995 9099284315237475249964895010470989719036646310521769801165280809086743 18308435150950339357940221747989082508725664850*c[3]*c[4]+353778808855 9086074309029834920958035363579091820847289129929315553826951987052525 2758934546354306930611562943149330790000*c[3]*c[4]^2*a[9,6]-2010548729 4460086075795721977441782039287038218157645989507182309524053570582585 86639214500766571097494288826347271105257000*c[3]*c[4]^2*a[7,6]-311701 3226736897512859472630512902962582975075595814774226605535166156191930 04914542118240032469602085987406647296000000000*c[3]*c[4]^3*a[8,6]+901 7095051883577351385050099340022028021321923745160539486112604236553541 34868274257143753057166179892916524153590048740000*c[3]*c[4]^3*a[7,6]- 1819255077785888048280070694438476508647726545470946622486505556332965 5145535514598467879255489060119117850121821269400000*c[3]*c[4]^3*a[9,6 ]+60614547130200439756273741369570628867273815906195699769458135482881 3371771756611321217315174568778055782732258713600000000*c[3]*c[4]^2*a[ 8,6]-66447288712114422827271139876632009215655157797966695152243410866 783167406027240194725574398560231510231439421832346944400*c[3]*a[9,5]* c[4]^2+308894279204487372219068094852246313687219083319702149606404749 83438869943657334830349469583779109483773907960553302720000*c[4]^3*c[3 ]*a[6,1]-3246054610026824313238306876386679842186476577838699158490564 5859463406381209249122873688606323035473670934032848386992000*c[4]^2*c [3]*a[6,1]+40192416194340739544941814230683440654795818711728255485210 348463462673646971505182916748225095746961643519914015085768000*c[4]^3 *c[3]*a[9,5]+701619345599099665955607897937827621088981295180874115236 1136753000871137084472978969184848875422149986704070104184182500*c[3]* c[4]^3-163042176459101902093602191278739976614272816418767230391156395 61113531553218474101790208293192668757784162081700863906125*c[3]*c[4]^ 2+40025644916528096117255633369474192533453816945757453056203427374192 464153453094981201559016410837678944145166523863425680*a[9,5]*c[4]^2+8 2884441521208852315748730808407423404521686673095252041659014638492373 155682684019429724138864423351661529722525577261600*a[7,6]-16722452213 1890655277359902096673844182706493467562431207335729944706281241222569 6270436884045095780260254615877005496000*a[9,6]-2421057969576264666079 9270347240084370758664762449327860321973216452663770779285370538017569 909745802930830338738081249600*c[4]^3*a[9,5]+1877585482140924063959137 9513977131455085613177257629823447836892302644990442544556442388659944 0269185523775128371200000000*c[4]^3*a[8,6]-286107551843918525714797818 0787417833784119956660006048318638313572160839942573979649943479891648 99725483832033530668980000*c[4]^3*a[7,6]+10958589758497242799698887378 3335448864105658182806133827885364280885141645888484386037284509395640 24415367706517190680000*c[4]^3*a[9,6]-10612226950265591986388985794867 7193513406397919262849750408409829856118417062141302266938528844883420 0225697419040525250+24899141700674196249595258784177027076692121578868 4924885574116450813511191199026673605952204117427119031711911284299988 8*a[9,5]-2865134492048432677985903591293668009049910742036076075331962 8111865370256622127213230342082204468672590856588656640000000*a[8,6]-1 8045444565355359199413582494102396944844386028815134263061233267664869 894588164290408847243201266432955755857081220592250*c[4]^2-17835194259 5139813123986708812862801249565205381225098992460099689944724165503796 9659895689532060975325797715964779179625*c[3]+864544041256136303023351 0800267150583466339981902800261269176896516552695176817773436428916771 313563529343368586974626100*c[4]-4133551028893849868547739424878111567 7416193976707025840532441729064325531741292247386449825634425523689940 38605800265040*c[3]*a[9,5]+3672541331878058167613904292273021313081541 3777360084907486200831130695864319314754894631986959727368883927025659 44801600*c[3]*a[9,6]+1126724276200904144780724773531608190353553840445 6885498089015844043974520545553054910482592172892828531236679556215604 400*c[4]*a[9,6]-194589651371749808107216106892517529398793292987734301 29108806636244939205355320897026501025031978650591253906057423738272*c [4]*a[9,5]-21310463160669524400275694508577131834320257488009600830380 284279489916195993014619709228028514014470628740281731372038000*c[4]^2 *a[9,6]-33182909775674630929621682155710965818740002683621190594164527 2049003954035189929065248407846122205090398173772554240000000*c[4]^2*a [8,6]+5273852512703794019697938596211982494683472141780462538107055805 009311889735437339515143541094173550593645426421367728000*c[4]*c[3]*a[ 6,1]+78170607605745443272144571503977651267139143067529825951325701997 825295669723689085528056995575263555325100897239040000000*c[3]*a[8,6]- 2019310927296512954923960287277120214549704169637415333289910280843516 6727775913830686812187120452327706321163250384830000*c[4]*c[3]*a[9,6]+ 3230417591534942786848676037803090615755998675336649991964035679298030 1824214628994081322919748569665519861592570772217760*c[4]*c[3]*a[9,5]- 3698020464470545621260376083301644195145508859268911500386227386000095 28373693322580923339028394913727768293198774986350400*c[3]*a[7,6]+1086 4578804223497676840633310270776358176391882077902339567641654827795817 528472493417993526888759110958721732950983485000*c[4]^3-37068484224779 7548387332751539884012249618915888843090563514695085854277778456292439 484763652370251121259510315938176172200*c[4]*a[7,6]-379270658468299107 7873555938693284235600799929218922692747907549987739517425967028721815 56728868864298095674919034880000000*c[3]*c[4]*a[8,6]+56684923092180247 4011522742596329568043264199332663590175677696624165653875017974493561 142315201432114441082327175527597000*c[4]^2*a[7,6]+1490265585997037356 2678915552673704228543477432771193681033863258656092760290045662732676 13584597852134525327024576078085800*c[3]*c[4]*a[7,6]+17779957048447744 0991200653102631897302915552394066698264702010200903067577989469473096 293573979495730096978056192000000000*c[4]*a[8,6])/(-17+20*c[4])/(-2227 85849+1109145221*c[4])/(-509+845*c[4])/c[4]/(-c[4]+c[3]), a[9,3] = 1/5 8875982795076623960654653082017473457529213784603414888851819516119504 792100606509088350*(-1605272345309842571033586951482582871624514962464 4804874184955217353175798243882052702250*c[4]*a[9,6]+16112429098387667 5265310423478387006493983740986149465492179993251870338688240164846009 50*c[4]+35464941115614203072157654933428276911103396232382412045474054 596100388093703205577663870*c[4]*a[9,5]-275038678959489497388409875141 691994653506206351153808519223280253053209284044528640000000*c[4]*a[8, 6]+7956494665627371357367894016574317770065430676791380697835189769575 40503631152630605050975*c[4]*a[7,6]-2136290535839956137719319095989939 9938167607908026802048693838804041535549934830342048414*a[9,5]+7645890 9992444471589280566290452417040518562297344283375497324551361991709480 60795696375+2458221159930968401566740275991679366133901624530458947937 38023376355249323468369920000000*a[8,6]-711129926158636853918595655693 821808020500031185749776656331979405274296285718834679972300*a[7,6]+14 3474890936117797557580665260714366548125146828327193747294105239347065 74254605204613000*a[9,6])/c[3]/(-c[4]+c[3]), c[6] = 244/273, b[2] = 0, b[3] = 0, b[6] = 43965679317071687650068867/1155464162092299275205473 0, b[1] = 17499592155841793/301627899585331605, a[10,1] = -1/249372056 4968223873295672920441096105726040529312396604940034257254219247236665 282466214862029895286226685252715403885950*(-4109031900011106064584409 3106169459773412327709915176966162839454502786766226118557730469395245 92447420209868034848077200*c[4]*c[3]*a[9,6]+16732440860609186168870919 1818463450989391964612160648665216614307518060375265742049860415194230 21315863118290168730665050*c[3]*c[4]+413753929016581538772165028992356 0923257065240083659178710819985155844313361404797729183260440648010150 85016570760538006800*c[3]*c[4]*a[7,6]+32056627718410742667629167230457 8428881403947776987200231813871907346150822042652836552597886959833918 74196101242880000000*c[4]*a[8,6]+1870995677951071675849149724278358994 3392980621575632671968300932338038843792525208271691366570129016846291 39895174182000*c[4]*a[9,6]-8746137654249387710366995090404056883577453 3022277223297999658382812728351781012788316227704065766190998985839943 680000000*c[3]*c[4]*a[8,6]+5273852512703794019697938596211982494683472 1417804625381070558050093118897354373395151435410941735505936454264213 67728000*c[4]*c[3]*a[6,1]-92735461210204986402456571765144371268173854 3514549336367742253947066306209072653168209618438933917008344164518421 41772200*c[4]*a[7,6]+6862181963487825420280628318314350245072040060966 0976100687452379291463800238491059020727117212356714180745827542263732 00*c[4]*c[3]*a[9,5]-14544503816596039216067654632396818165141138044498 0392524013710664559446032976763114815874689258665300124395702934296802 5*c[4]-413355102889384986854773942487811156774161939767070258405324417 2906432553174129224738644982563442552368994038605800265040*c[4]*a[9,5] -286513449204843267798590359129366800904991074203607607533196281118653 70256622127213230342082204468672590856588656640000000*a[8,6]-167224522 1318906552773599020966738441827064934675624312073357299447062812412225 696270436884045095780260254615877005496000*a[9,6]-36980204644705456212 6037608330164419514550885926891150038622738600009528373693322580923339 028394913727768293198774986350400*c[3]*a[7,6]-413355102889384986854773 9424878111567741619397670702584053244172906432553174129224738644982563 442552368994038605800265040*c[3]*a[9,5]-178351942595139813123986708812 8628012495652053812250989924600996899447241655037969659895689532060975 325797715964779179625*c[3]+7817060760574544327214457150397765126713914 3067529825951325701997825295669723689085528056995575263555325100897239 040000000*c[3]*a[8,6]+828844415212088523157487308084074234045216866730 9525204165901463849237315568268401942972413886442335166152972252557726 1600*a[7,6]+3672541331878058167613904292273021313081541377736008490748 620083113069586431931475489463198695972736888392702565944801600*c[3]*a [9,6]-1061222695026559198638898579486771935134063979192628497504084098 298561184170621413022669385288448834200225697419040525250+248991417006 7419624959525878417702707669212157886849248855741164508135111911990266 736059522041174271190317119112842999888*a[9,5])/c[3]/c[4], a[9,1] = -1 /222984508816079157715179221205521400770200392700191373543305585183588 702029650935947216962602905550*(12947341263356319589167090066109520459 405335234677220446308933894607383657607680689765459770399312075*c[3]*c [4]*a[7,6]-13431847927501099559411387407527857158820355015905018832371 8985631297809861647723546903472147785710*c[4]*a[9,5]-27363430861486693 4577345895842939403187396453758429024083546506347095192159907804905341 7016320000000*c[3]*c[4]*a[8,6]-301341051254780057733213074550170007519 9683390095340985904022865314650666907016668184804307612097175*c[4]*a[7 ,6]+607974335954326556914902810052327357405133477713949110146887683291 25115651986271526841269677824250*c[4]*a[9,6]-1200944703120067622824627 6435655952342973509013213832967965416130605578651801487714201443185136 6300*c[4]*c[3]*a[9,6]+104167033519708117279842300548823938753154229714 8086887129156052299603699232849221936742661120000000*c[4]*a[8,6]-61023 5603343222176974751210034093700669621579172521900479715885176347472123 81123449900907223261350*c[4]+22298450881607915771517922120552140077020 0392700191373543305585183588702029650935947216962602905550*c[4]*c[3]*a [9,5]+5880126668797714524031267218298644291149761750179244967831640081 79882107078914705651340394909879575*c[3]*c[4]-134318479275010995594113 8740752785715882035501590501883237189856312978098616477235469034721477 85710*c[3]*a[9,5]-2895773739967825282023622544237267795160579201284450 2249081487320586333229856917607638151660155375+10733718225688516482388 6133710624628999470190447771986966430825489661582089361282134254657039 316400*c[3]*a[9,6]-931016709846475480449872576333811027683869305875945 789228989292165213562684304799093645455360000000*a[8,6]+24456692784625 4696105759701779037415303021006289584915298114826185682149769294887900 7449640960000000*c[3]*a[8,6]+80909001125420824564975102845345317086858 706545510705155944335723468148188850522231211677305589262*a[9,5]-11571 9826676151720870211354437022820223256476090155376882761167409677714741 98806184259238769148103100*c[3]*a[7,6]-7216072118033495673437794777996 0591334287861158093336208255686592062971248978798326192484026346500*c[ 3]+2693304633925506743110036270704816184427555850488143591797002121380 127335990154091710960626583705900*a[7,6]-54339098158555194097888749323 358196046466142330477502884923294770353583220090293965381940664429000* a[9,6])/c[3]/c[4], b[7] = 63971129215453255994803266454987246348244877 7204372657657897547/20116014255015886224276044430169119323710082114357 97382991660160, a[4,3] = 1/2*1/c[3]*c[4]^2, a[4,2] = 0, a[5,2] = 0, a[ 6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[5,3] = -2 59081/3620106750*(-1018+2535*c[4])/c[3]/(-c[4]+c[3]), a[4,1] = 1/2*c[4 ]*(-c[4]+2*c[3])/c[3], c[2] = 1/200, a[8,1] = 1/3638186567739131728876 3124455333206479420954793341379343724410804208032492188874354217224632 3776275200000*(-733253348805093924195565061061471681987518398268897900 85754649257857481358200547347582320871969162653750*c[4]*c[3]*a[9,6]+87 9237143955863956501966224061822963051143141304601550931755615914703624 2654240485675802250125711807024925*c[3]*c[4]*a[7,6]+756763501313525838 5124094924229767722555073141667174714980305102762642256852166661931533 72121850880000000*c[4]*a[8,6]+1619961056089846804606500395219733930853 83012468485788056119718716663733238853600035177358411654860524450*c[4] *c[3]*a[9,5]+441687520167802138953304871100933829741593922744223705960 34457363608826048904235029490415767848880225750*c[4]*a[9,6]-1863472517 0847874095106341755353992644954688565959382999797064358192805741788696 04447354091991232940800000*c[3]*c[4]*a[8,6]+51816078437212433584537034 6478440842492755354251287271226415598598164090341937300757426916234481 977829400*c[4]*c[3]*a[6,1]-2189213816806261671988243115795622167501984 016712091505814787985259703065281411366077195985724820067057825*c[4]*a [7,6]-7186504471846966953792530464247651203619115652992531979092306304 9112340861353297939062077047907695765775*c[4]+511743964900691578933203 5903632183157231836331807284978543965322645213274108798221806751030951 96105169475*c[3]*c[4]-975810861005600027863560593096857480242129625401 88480616053179676664899666954417062609793410097424860290*c[4]*a[9,5]-3 9476833304374989708647028772391155478735867087762118774477683504470892 146273382224159932041593870971000*a[9,6]+58779612810869871500893768270 5681014725732519916638303355870632608549513970175127631578518884492180 51938*a[9,5]+166552122406112867370181223014885502028166447256193752818 6990367545437582782579426685547247054435302400000*c[3]*a[8,6]-97581086 1005600027863560593096857480242129625401884806160531796766648996669544 17062609793410097424860290*c[3]*a[9,5]-8025189676096224377023827404108 8687868065561042296147571579912787128879776522321386206699937180003566 425*c[3]-9981269766973329890367329534394347232571442517094631082636338 190734217819055370028127699198005801903675-676374704470697086435999692 8615616572540065372039526118883496135802508097699372401140271897352806 40000000*a[8,6]+655361967430193837009955585710619378772727066584655999 30125034501528298356779976383919730010111632555000*c[3]*a[9,6]+1956659 9681345342416305176566085414244340076925924920418271365142980496261123 23711805259415812659693634100*a[7,6]-785838326466046906177581533593717 2270493733570634534008327779131252296128965694793058226186925544618732 900*c[3]*a[7,6])/c[3]/c[4], a[6,5] = -325/27783*(24379992*c[3]-5455522 8*c[3]*c[4]+61039251*c[4]*c[3]*a[6,1]-14526784+24379992*c[4])/(-509+84 5*c[4])/(845*c[3]-509), a[7,4] = 1/54939588728766269160473018724022762 9522624310776471198703323866265057661701591403845876734632850*(-855735 9973492949289200451680213433902471731207214756904194668489488731015914 813472004410579377500*c[4]*c[3]*a[9,6]+4241435366396227463117791580046 07063221486997984055958827905944770972790035483998167205299620930250*c [3]*c[4]*a[7,6]+883173286026424048204365386016837904217300182667107113 20577606646743841741113607808285829120000000*c[4]*a[8,6]+1890559371135 6423234838856518087503036245171587090482077517915810861295163366473967 180075780769300*c[4]*c[3]*a[9,5]+5154670090541906731601218822755784445 394214419493859484301877232129898327929751547041710041305500*c[4]*a[9, 6]-1466171761674515364897227409006341903857796963366808468877958302878 16397389471510015720089600000000*c[3]*c[4]*a[8,6]+29991693868818419116 4876687298513986820902682365426698181555796061612880902405030488042956 89408800*c[4]*c[3]*a[6,1]-13457845148591943624149023177120143865662152 6164645854595693757858641887994054505700873842094479250*c[4]*a[7,6]+14 7716939596836871584592749756081881264007373410658396621005553205148077 76922611689266192694798100*c[4]-24158574160874749494375830482547609903 543252819122094405511467703471894650490500916898305149058525*c[3]*c[4] -113881031941780111556603289558657266810044879737621957129664131925779 87264087023963662317837173460*c[4]*a[9,5]-4607104403268224331541016090 668173643502521312661178440181897599412802901153331053033616300654000* a[9,6]+685981600690722802157527507519012411909027737117746463656793410 0617982860852420352075881395409812*a[9,5]+1310424578199933146648071383 8738000898948808024230815619275524758324982037740310785287802880000000 0*c[3]*a[8,6]-11388103194178011155660328955865726681004487973762195712 966413192577987264087023963662317837173460*c[3]*a[9,5]+101906095777948 1183875824597169190659634007431926591324951707233483740432909260991040 6219347646025*c[3]-870307047386674611478936787864535885197802654467034 6187642031507806299303315362074036979312754000-78935634355475262916434 1224132265379593484412347158005942158828637564006770392685172957593600 00000*a[8,6]+764833638656512683723410333323105447693444107897582668360 2560847748169845726060392560718613070000*c[3]*a[9,6]+22835004736543159 1134827684104038749046777203553017378080892154742509207929631188141293 232727403400*a[7,6]-49999955664286863317701779110083648309413971979076 1395385826384588501341234013382005022663943056800*c[3]*a[7,6])/(-c[4]+ c[3])/c[4]/(-509+845*c[4]), a[8,7] = -1364473912812730509838628861/293 5845327576564009232166859753935522600757340596428075052317089665931008 000*(50760423613309354212766144628409176211205387593470507739117*c[4]* a[7,6]+4142305456187345504854547971426874427643170894706755582927*c[4] -898215739262789460547388309012805441974911273950333625599-77071314071 89655728939089923350304685588155618327787264000*c[4]*a[8,6]-4536829070 2005430138882561499384025624667086347277669920676*a[7,6]+6888425140491 853472018820297792946312393809417113480192000*a[8,6])/(-222785849+1109 145221*c[4]), a[10,9] = -1198575787406124991100974435169141280/4635648 233480921649154730958571956313, a[3,1] = -100*c[3]^2+c[3], a[10,7] = 2 3382008649107441653912266705822682312/14428188201453250203881915418750 4073707208909220139045693875301534068313543226568285048801886447197104 21177075928771*(112261899465211438672788008261303011478603670915781409 234699110140931184822435375572522947*c[4]*a[7,6]+658172867760155823470 5521630858460934258611191977963457787891266199412441094398108110275*c[ 4]-1416808941501262828395045425682351443176265029018429068956062948153 261449384999905103549-186116395671981956771714028005744106931043028441 99632556897333016915854503484114739200000*c[4]*a[8,6]-1003366427454637 0342915851287823419340944796960970939140390689697577732269844040893661 3916*a[7,6]+1663457895383281957959641862029361248760970657137256536220 8605333800250911538915737600000*a[8,6]+6234250557922811419779047842889 31720068219257023206536229990682296084624090594392891848*a[9,6]-697520 6566856260318031475660281899982730485949480958376671617060116028785931 65037948666*c[4]*a[9,6])/(-222785849+1109145221*c[4]), b[4] = 0, b[8] \+ = -43614217354789120000/21595629876813903267, a[3,2] = 100*c[3]^2, a[2 ,1] = 1/200, a[7,1] = 1/2796425066294203100268076653052758634270157741 85223840139991847928914349806110024557551257928120650*(-85573599734929 4928920045168021343390247173120721475690419466848948873101591481347200 4410579377500*c[4]*c[3]*a[9,6]+559425733456980069087401053157903114281 557965175728938280043454887954369495433005276111423182190600*c[3]*c[4] *a[7,6]+88317328602642404820436538601683790421730018266710711320577606 646743841741113607808285829120000000*c[4]*a[8,6]+189055937113564232348 3885651808750303624517158709048207751791581086129516336647396718007578 0769300*c[4]*c[3]*a[9,5]+515467009054190673160121882275578444539421441 9493859484301877232129898327929751547041710041305500*c[4]*a[9,6]-14661 7176167451536489722740900634190385779696336680846887795830287816397389 471510015720089600000000*c[3]*c[4]*a[8,6]+2999169386881841911648766872 9851398682090268236542669818155579606161288090240503048804295689408800 *c[4]*c[3]*a[6,1]-2554900120113230507369178596737810593843040023359579 68098703107560266449855693911322020707108939050*c[4]*a[7,6]+1038455334 1118230569978768236911329431808457669230283548383692419687714924952255 769030606964480525*c[4]+2264610241102964654804345614742652131132706277 9263574151340077696879421509849381426339164707292200*c[3]*c[4]-1138810 3194178011155660328955865726681004487973762195712966413192577987264087 023963662317837173460*c[4]*a[9,5]-460710440326822433154101609066817364 3502521312661178440181897599412802901153331053033616300654000*a[9,6]+6 8598160069072280215752750751901241190902773711774646365679341006179828 60852420352075881395409812*a[9,5]+131042457819993314664807138387380008 989488080242308156192755247583249820377403107852878028800000000*c[3]*a [8,6]-1138810319417801115566032895586572668100448797376219571296641319 2577987264087023963662317837173460*c[3]*a[9,5]+10190609577794811838758 2459716919065963400743192659132495170723348374043290926099104062193476 46025*c[3]-87030704738667461147893678786453588519780265446703461876420 31507806299303315362074036979312754000-7893563435547526291643412241322 6537959348441234715800594215882863756400677039268517295759360000000*a[ 8,6]+76483363865651268372341033332310544769344410789758266836025608477 48169845726060392560718613070000*c[3]*a[9,6]+2283500473654315911348276 8410403874904677720355301737808089215474250920792963118814129323272740 3400*a[7,6]-4999995566428686331770177911008364830941397197907613953858 26384588501341234013382005022663943056800*c[3]*a[7,6])/c[3]/c[4], a[8, 3] = -1/41708342804150190696525524088253674761829840625107088853995136 052832838313141663903422073600000*(-4278679986746474644600225840106716 951235865603607378452097334244744365507957406736002205289688750*c[4]*c [3]*a[9,6]+21207176831981137315588957900230353161074349899202797941395 2972385486395017741999083602649810465125*c[3]*c[4]*a[7,6]+441586643013 2120241021826930084189521086500913335535566028880332337192087055680390 4142914560000000*c[4]*a[8,6]+94527968556782116174194282590437515181225 85793545241038758957905430647581683236983590037890384650*c[4]*c[3]*a[9 ,5]+257733504527095336580060941137789222269710720974692974215093861606 4949163964875773520855020652750*c[4]*a[9,6]-73308588083725768244861370 4503170951928898481683404234438979151439081986947357550078600448000000 00*c[3]*c[4]*a[8,6]+30235718413325329141483373588148010928673569411793 001834436094645204028081896641962419030981847800*c[4]*c[3]*a[6,1]-1277 4500600566152536845892983689052969215200116797898404935155378013322492 7846955661010353554469525*c[4]*a[7,6]-41934691342990357704138962630081 35029574903722403971705052055705919007590098744885757115577043675*c[4] -136341323868685229015753478976963925092662322340337536951096176221418 63090906004544796706949525*c[3]*c[4]-569405159708900557783016447793286 3340502243986881097856483206596288993632043511981831158918586730*c[4]* a[9,5]-230355220163411216577050804533408682175126065633058922009094879 9706401450576665526516808150327000*a[9,6]+6552122890999665733240356919 3690004494744040121154078096377623791624910188701553926439014400000000 *c[3]*a[8,6]-394678171777376314582170612066132689796742206173579002971 07941431878200338519634258647879680000000*a[8,6]-569405159708900557783 0164477932863340502243986881097856483206596288993632043511981831158918 586730*c[3]*a[9,5]+109757958220869939624338975181954868361315306771686 8180684422041878434083529577126519077391063975*c[3]-582427059676726992 5424001411648300607433497839543220003810582981845711019451680271023860 65705975-1895439980587325093407950815991284311832286218097246409414085 17443438389686186988191937899464298500*c[3]*a[7,6]+3824168193282563418 6170516666155272384672205394879133418012804238740849228630301962803593 06535000*c[3]*a[9,6]+3429908003453614010787637537595062059545138685588 732318283967050308991430426210176037940697704906*a[9,5]+11417502368271 5795567413842052019374523388601776508689040446077371254603964815594070 646616363701700*a[7,6])/(845*c[3]-509)/c[3]/(-c[4]+c[3]), b[10] = 7077 278529846113/15546034297382400, c[5] = 509/845, a[5,4] = 259081/362010 6750*(-1018+2535*c[3])/c[4]/(-c[4]+c[3]), c[9] = 171/181, c[8] = 17/20 , a[5,1] = 509/3620106750*(4284150*c[3]*c[4]-1290315*c[3]+518162-12903 15*c[4])/c[3]/c[4], a[8,4] = 1/417083428041501906965255240882536747618 29840625107088853995136052832838313141663903422073600000*(-47556846570 7605829089418570858248207339529735192866139019721395090300269433529484 5156825895764277718717700*c[3]*c[4]-4745677459488195690656013946075075 2364629503780394041704821357043978573698281951527900546425191976399637 50*c[3]*c[4]^2*a[9,6]+235218388340937984157302614554106910077478584163 826150465272320040107204794446748494564258500213949180917625*c[3]*c[4] ^2*a[7,6]-813098701313219837233415468424798240677958482753303841639157 34193729305844984600524794786126765900800000000*c[3]*c[4]^2*a[8,6]+104 8452445755931512928743920607072702623716092507319674543257393174857271 2159169436159345949329056399257650*c[3]*a[9,5]*c[4]^2+3353580258164149 1535528316666251988562194061382101989025909027605631738316985457048565 129812967425119363800*c[4]^2*c[3]*a[6,1]-15122232779376545331611350492 1423506605928396990546914636219234567360312313146835010655479296031419 70025*c[3]*c[4]^2-6315530117038688048292670480343195332964159657825156 382751692463029613621880493778704268743441999653517330*a[9,5]*c[4]^2-2 5436579585748945141196729535911038517642100003702396543752774685874645 139460208253468372405564576017243300*a[7,6]+51319883295687486621241137 4041085021223566272140907544068209885558121597901553968914079116540720 322623000*a[9,6]-76413496654130832951161898751738531914345227589162979 4362631822391114368161227665921052074549839834675194*a[9,5]+8792871158 1190621236679960072003015443020849836513839545485449765432605270091841 21482353466558648320000000*a[8,6]+132493104739041804152650397986713715 3975383082404129642324217908663695275592039577452387633259538387975725 *c[4]^2+10432746578925091690130975625341529422848522935498499184305388 66232675437094790178020687099183340046363525*c[3]-99955046544135740389 8045355121672369240582072823396459421214402622225511644057982858023151 300206847493700*c[4]+1268554119307280036222628771025914724314768513022 450248008691335796643695670407421813927314331266523183770*c[3]*a[9,5]- 8519705576592519881129422614238051924045451865600527990916254485198678 78638139692990956490131451223215000*c[3]*a[9,6]-3129167691984646679881 6151136555677254085617072618872473264409922769561106503313182349419410 91077080201750*c[4]*a[9,6]+5072820189807507511466379391729685640857676 519925414270281604310568328114057592431707977953872068681337996*c[4]*a [9,5]+2858638848378096575791610767517412183857564192215451742929475826 672792190819587376059334689990853963007750*c[4]^2*a[9,6]+4897837147555 3715639267274961919799349713712156382444425364625685926883639168238659 314255785235317760000000*c[4]^2*a[8,6]-6736090196837616365989814504219 7309524058196052667345259434027817761331744451849098465499110482657117 82200*c[4]*c[3]*a[6,1]-21651775912794672758123558991935115263661638143 305187866430874778090688576173532546912114211707658931200000*c[3]*a[8, 6]+5194787229326181619842759864516610467525018944275628141435840154172 942855598290695301622672341768433733750*c[4]*c[3]*a[9,6]-8421479489955 4888942811209941288494430741388199154716274812488063462421539855905791 61574402793512840335180*c[4]*c[3]*a[9,5]+10215898244058609780308559936 7183239516418536418248942108261128706279849676554032309756940430032080 043527700*c[3]*a[7,6]+129756506970653288574775283947126514023428752722 230204074272396479544831647719810365660089574075424747775+951651995835 1042284496106996849207651432653925627439056986881334108730805935719549 6074098191441105764726225*c[4]*a[7,6]+89004673954793959967818707517601 4982125756553062751228832263197344822141846199337428657201236562176000 00000*c[3]*c[4]*a[8,6]-74633523157593945749131546243605220557158212875 621704391137309354221921809511415705820738741500145310582125*c[4]^2*a[ 7,6]-32453264833033875439793633422401532240875400960027065844723242819 9996298882875113979256781171239394883792525*c[3]*c[4]*a[7,6]-465588930 2050751455724008340098965675382853189158569817268906795683610402322254 4565367052664558637388800000*c[4]*a[8,6])/(-509*c[3]+509*c[4]+845*c[3] *c[4]-845*c[4]^2)/c[4]/(-222785849+1109145221*c[4]), c[7] = 222785849/ 1109145221, a[7,3] = -1/5493958872876626916047301872402276295226243107 76471198703323866265057661701591403845876734632850*(-85573599734929492 8920045168021343390247173120721475690419466848948873101591481347200441 0579377500*c[4]*c[3]*a[9,6]+424143536639622746311779158004607063221486 997984055958827905944770972790035483998167205299620930250*c[3]*c[4]*a[ 7,6]+88317328602642404820436538601683790421730018266710711320577606646 743841741113607808285829120000000*c[4]*a[8,6]+189055937113564232348388 5651808750303624517158709048207751791581086129516336647396718007578076 9300*c[4]*c[3]*a[9,5]+515467009054190673160121882275578444539421441949 3859484301877232129898327929751547041710041305500*c[4]*a[9,6]-14661717 6167451536489722740900634190385779696336680846887795830287816397389471 510015720089600000000*c[3]*c[4]*a[8,6]+2999169386881841911648766872985 1398682090268236542669818155579606161288090240503048804295689408800*c[ 4]*c[3]*a[6,1]-2554900120113230507369178596737810593843040023359579680 98703107560266449855693911322020707108939050*c[4]*a[7,6]+1038455334111 8230569978768236911329431808457669230283548383692419687714924952255769 030606964480525*c[4]-2415857416087474949437583048254760990354325281912 2094405511467703471894650490500916898305149058525*c[3]*c[4]-1138810319 4178011155660328955865726681004487973762195712966413192577987264087023 963662317837173460*c[4]*a[9,5]-460710440326822433154101609066817364350 2521312661178440181897599412802901153331053033616300654000*a[9,6]+6859 8160069072280215752750751901241190902773711774646365679341006179828608 52420352075881395409812*a[9,5]+131042457819993314664807138387380008989 488080242308156192755247583249820377403107852878028800000000*c[3]*a[8, 6]-1138810319417801115566032895586572668100448797376219571296641319257 7987264087023963662317837173460*c[3]*a[9,5]+14577750196360268427238752 7103887652909323539911014693632339352356644971810629658306418050779636 00*c[3]-87030704738667461147893678786453588519780265446703461876420315 07806299303315362074036979312754000-7893563435547526291643412241322653 7959348441234715800594215882863756400677039268517295759360000000*a[8,6 ]+76483363865651268372341033332310544769344410789758266836025608477481 69845726060392560718613070000*c[3]*a[9,6]+2283500473654315911348276841 0403874904677720355301737808089215474250920792963118814129323272740340 0*a[7,6]-3790879961174650186815901631982568623664572436194492818828170 34886876779372373976383875798928597000*c[3]*a[7,6])/(845*c[3]-509)/c[3 ]/(-c[4]+c[3]), a[10,5] = -4/27930927653853408443454157133507556037035 90268364344846025364651720022470248192118083563060494459*(-43512503255 5864737520312437133109360106222562252141816541137579870512747485621644 493584413248143068750*c[4]*c[3]*a[9,6]+5362322029889928077015979932408 6604099367662207786464077285829145256415201366929985430505557657984656 25*c[3]*c[4]*a[7,6]+10790828513853056132333894935837561012923400864100 10127497743574284324955680131594119453816652800000000*c[4]*a[8,6]-1371 9972146302710257458458162321421473651947896179816052566684228195327633 45373606750599605789815219650*c[4]*c[3]*a[9,5]+26210490126737887739389 2343787872975495937614421704360496377548111350282213232446209744930583 792688750*c[4]*a[9,6]-179140473363572346401220849131291533515133079178 0861606553228527053545358643833392988091306624000000000*c[3]*c[4]*a[8, 6]-1054432977192983570087915213369750549165588605526248914131804790003 615486032630485938337025700389886000*c[4]*c[3]*a[6,1]-3230085104395234 7824865488586935007676423834394986165935311818976255047736681381494182 39920573717655625*c[4]*a[7,6]-4163706291249207608469055723162355589258 93593335813825076702341293534198506220930370409589688651416375*c[4]+93 4478769104569528526378945225209918938024548888527393363577403026202212 937329291789889572724024342375*c[3]*c[4]+82644565946367804982797103013 2734145572643962030239807190111511497209676382006113415449940055640173 730*c[4]*a[9,5]-234262256077803831809925757817732622787577941094856644 546212900143477907912193102106878252975990535000*a[9,6]+16011089927000 6053193765154534927231420118942562098986080215296923467057695639321571 0968054272000000000*c[3]*a[8,6]-96445500270335007190090489536423622240 0479784190631762305675575550825235113377688517020993638400000000*a[8,6 ]+82644565946367804982797103013273414557264396203023980719011151149720 9676382006113415449940055640173730*c[3]*a[9,5]-40577861802692494049438 4084430793288921078701898344582294136140868202895872088359414201024394 005077625*c[3]+2416823836626978198576213511719213506566522060267365583 83771093120962835378195491059631534559793019625-4792698077996858794109 5205256692789011889046075823799395083305170119286846642970389908583721 86281412500*c[3]*a[7,6]+3889029595004798386628433504046838236846824365 91657887311492928528956448302167330609650537848157175000*c[3]*a[9,6]-4 9782348007930429273661213531072388177097725050105569450859971520956180 5063243919205282863299787986306*a[9,5]+2886962510888048670061237807770 0153381126064440940016440351955422000848526557718258536649839559967325 00*a[7,6])/(845*c[3]-509)/(-509+845*c[4]), a[6,4] = 1/4695327*(2389921 443*c[3]*a[6,1]-551347524*c[3]+10329496)/(-509+845*c[4])/(-c[4]+c[3]), b[9] = -210996982649291857501475159/99036657939412172256631200, a[9,8 ] = 64000/598759850831288931162665329527551680250286891568777298786362 217634181969198479*(-4769030540452221393596949404820196763981269654041 50540585708797805578522788*a[7,6]+220886048530037435553709008845694642 130873475756184237614194042498281153144*a[9,6]+98745963650506523950948 9981854496178079935716163895563789633515903542288+56848637408362785947 1886297481666432370946825454538334302737983642542080000*a[8,6]+1448462 82309408781557124346235894851795078427932343365876049500248044798*c[4] -247138898560246802894108850060961628285772372464911052740471203287011 290198*c[4]*a[9,6]+533584154730924770677035732588489228101183039161201 219589747958200503839021*c[4]*a[7,6]-636052377560780350966495734477438 262447821653070036742887899465304975360000*c[4]*a[8,6])/(-17+20*c[4]), a[10,3] = 1/156572715649228503454610889461873437511248746217064730037 9725498554025395348727830946306384205950*(-992653756925182117547252394 9047583326867208200369118008865815447806927978461183627525116272077900 00*c[4]*c[3]*a[9,6]+49200650250196238572166382328534419333692491766150 491224037089593432843644116143787395814756027909000*c[3]*c[4]*a[7,6]+1 0244810117906518959170638477795319688920682118938442513187002371022285 641969178505761156177920000000*c[4]*a[8,6]+219304887051734509524130735 6098150352204439904102495920992078234059910238950510980192888790569238 800*c[4]*c[3]*a[9,5]+5979417305028611808657413834396709956657288726612 87700179017758927068206039851179456838364791438000*c[4]*a[9,6]-1700759 2435424378232807837944473566084750444775054978238984316313386702097178 695161823530393600000000*c[3]*c[4]*a[8,6]+1685442962282158289169568813 4267929124088590296019889069225018356160379502211772429196847591047520 00*c[4]*c[3]*a[6,1]-29636841393313473885482471722158602888579264270971 124299449560476990908183260493713354402024636929800*c[4]*a[7,6]-464820 1962136405050875340521872636660765969031967098918325633928638087889383 38585832756669373684725*c[4]+38282981196235664085281490042364390680792 3323179679402419005285365485813550391825530425759210045125*c[3]*c[4]-1 3210199705246492940565981588804242949965206049564147027041039303390465 22634094779784828869112121360*c[4]*a[9,5]-5344241107791140224587578665 1750814264629247226869669906110012153188513653378640215189949087586400 0*a[9,6]+1520092510711922450111762805293608104278061730810774611835960 8719656979163778760510933851340800000000*c[3]*a[8,6]-91565335852351304 9830635819993427840328441918322703286892904241219574247853655514800630 8085760000000*a[8,6]-1321019970524649294056598158880424294996520604956 414702704103930339046522634094779784828869112121360*c[3]*a[9,5]+570314 6639349531648442577541954588359679302317464892470856862254768390329339 73737856020098652380250*c[3]-33915061493246537151408247325737070690464 9186993075189897803858301412286365256296591827034344952250-43974207549 6259421670644589309977960345090402598561166984067760468777064071953812 60529592675717252000*c[3]*a[7,6]+8872070208415547131191559866548023193 24395165161195895297897058338787702104223005537043359116120000*c[3]*a[ 9,6]+79573865680123845050273190872205439781447217505658589784188035567 1686011858880760840802241867538192*a[9,5]+2648860549439006457164001135 6068494889426155612150015857383489950131068119837217824390014996378794 400*a[7,6])/(845*c[3]-509)/c[3]/(-c[4]+c[3]), a[7,5] = -13/25456759820 6117715090402972512768196110164564574623432080101818779166454079299066 5066465707129*(-427867998674647464460022584010671695123586560360737845 2097334244744365507957406736002205289688750*c[4]*c[3]*a[9,6]+419534120 0432001895571043592315894888542868696804763892100176514084343596507715 14916831340555155625*c[3]*c[4]*a[7,6]-56940515970890055778301644779328 63340502243986881097856483206596288993632043511981831158918586730*c[4] *a[9,5]+44158664301321202410218269300841895210865009133355355660288803 323371920870556803904142914560000000*c[4]*a[8,6]+945279685567821161741 9428259043751518122585793545241038758957905430647581683236983590037890 384650*c[4]*c[3]*a[9,5]+2577335045270953365800609411377892222697107209 746929742150938616064949163964875773520855020652750*c[4]*a[9,6]-167618 1634670426172673560669315750937707132488228978507733676867668691784812 8726404905394393936325*c[3]*c[4]+1499584693440920955824383436492569934 1045134118271334909077789803080644045120251524402147844704400*c[4]*c[3 ]*a[6,1]-2527134521917028360764096081051823074873751676536834107785786 79960820223742299054547535091529673625*c[4]*a[7,6]+8012877139825204601 6636739989394582691320945418564761862175819801397675170887778924990041 75823925*c[4]-73308588083725768244861370450317095192889848168340423443 897915143908198694735755007860044800000000*c[3]*c[4]*a[8,6]-2303552201 6341121657705080453340868217512606563305892200909487997064014505766655 26516808150327000*a[9,6]-394678171777376314582170612066132689796742206 17357900297107941431878200338519634258647879680000000*a[8,6]+655212289 0999665733240356919369000449474404012115407809637762379162491018870155 3926439014400000000*c[3]*a[8,6]-37496822450747562729645957381871001934 2293026381085124422140318474937669431458789889036069946732500*c[3]*a[7 ,6]-569405159708900557783016447793286334050224398688109785648320659628 8993632043511981831158918586730*c[3]*a[9,5]+34299080034536140107876375 3759506205954513868558873231828396705030899143042621017603794069770490 6*a[9,5]-4729237427976604726529505932420786385538419743937173312495758 829429922535647483726206356028765475+382416819328256341861705166661552 7238467220539487913341801280423874084922863030196280359306535000*c[3]* a[9,6]+791590525816349523605341286632974685139790286687429103678427193 7714612219158954963186810367406675*c[3]+225868433460716087921772690028 075029402635680979848909267301091247033460048062158643218177044836500* a[7,6])/(845*c[3]-509)/(-509+845*c[4])\}:" }}}{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 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a[8,5]=subs(e9,a[8,7]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,$*&#\"=h)G'Q)40t7G\"RZk8 \"go+!35$fm*3W0G,4$)QZ0Y*yi#Rd@) *)!\"\"*(\"en+SE(yF$=c:)e&o/.NB**3R*Gdl*=2982xF.F3F.&F%6$F'F:F.F>*&\"f nw1#*pwFZj3nYiDSQ*\\hD))Q,V0?q!Ho`%F.F7F.F>*&\"en+?>![8rT4QR7j%Hz(H?)= ?Z`=\\S^U))oF.FAF.F.F.,&\"*\\eyA#F>*&\"+@_946F.F3F.F.F>F.F>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can find so me simple order conditions that are not yet satisfied and determine wh ich parameters are related by them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 47 do\n eq := simplify(subs(e9,SO7_10[ii]));\n if lhs(eq)<>rhs(eq) t hen\n print(ii,indets(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<)&%\"aG6$\"\"(\"\"'&F&6$\"\"*F)&F&6$F,\"\" &&F&6$\"\")F)&F&6$F)\"\"\"&%\"cG6#\"\"$&F76#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#^<'&%\"aG6$\"\"(\"\"'&F&6$\"\"*F)&F&6$\"\")F)&%\"cG6 #\"\"$&F16#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#]<'&%\"aG6$\"\" (\"\"'&F&6$\"\"*F)&F&6$\"\")F)&%\"cG6#\"\"$&F16#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<)&%\"aG6$\"\"(\"\"'&F&6$\"\"*F)&F&6$F,\"\"&&F &6$\"\")F)&F&6$F)\"\"\"&%\"cG6#\"\"$&F76#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Z<'&%\"aG6$\"\"(\"\"'&F&6$\"\"*F)&F&6$\"\")F)&%\"cG6 #\"\"$&F16#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the \+ simple order comditions given in abreviated form as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "S O7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]],i=[50,51])]:\nlinalg[ augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]), linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7$7%\"#]%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*(%\"aGF-F.F--F06#*&)F.\"\" #F-F3F-F-F-#F-\"$0\"7%\"#^F)/*(F,F-F7F--F06#*&F3F-F4F-F-#F-\"#%)Q)ppri nt386\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "cdcns4 := [seq(SO7_10[i],i=[50,51])]:\neqns4 := simp lify(subs(e9,cdcns4)):\nnops(%);\nindets(eqns4);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'& %\"aG6$\"\"(\"\"'&F%6$\"\"*F(&F%6$\"\")F(&%\"cG6#\"\"$&F06#\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 2 equatio ns for the linking coefficients " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG 6$\"\")\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,6];" "6#&%\"aG 6$\"\"*\"\"'" }{TEXT -1 29 " in terms of the parameter " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 88 " and subsitute back into the expressions previously obtained for the other coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "e10 := solve(\{op(eqns4)\},\{a[8,6],a[9,6]\}):\ninfolevel[sol ve] := 0:\ne11 := `union`(map(u_->lhs(u_)=simplify(subs(e10,rhs(u_))), e9),e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "a[8,6]=subs(e10,a[8,6]);\n``;\na[9,6]=subs(e10,a[9, 6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"B\\Q$e Q/b7l<8tU9zv8\"D+S)>7\"p?jl.aGL-k,6\"\"\"\"*&#\"TfS)=fmIS]PFX5\\6`tywM \")p?e0]*)F-&F%6$\"\"(F(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"',&#\"Lj8d*fI:\"zGlhdDwe#3An+\"[#\"M(*[oWtzdYqh_Jm\"Q'ozv,'Q %=!\"\"*&#\"U2&3fTr%pmK(*oA,G$>hf\"G?47yp`$*\"Taon)>RC=lz=RjHdIiHP7O2& )oBj\"\"\"&F%6$\"\"(F(F2F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17583 "e11 := \{c[10] = 1, a[6,3] = -1/4695327*(10329496- 551347524*c[4]+2389921443*c[4]*a[6,1])/(-509*c[3]+509*c[4]-845*c[3]*c[ 4]+845*c[3]^2), a[9,1] = 1/1486657730083357734178180449788807945201309 7304682037672876441028650*(2198993062517721175657278489176196173278532 10391507263118737490262325*c[3]*c[4]*a[7,6]+89551335457092199609076195 14112464427307297666370600207685335483530*c[4]*a[9,5]-3801983535650827 6354053698820247395307496597630191479555113788577825*c[3]*c[4]-1965400 39287298156359112070094868815487165506723544953117113361260100*c[4]*a[ 7,6]+37532426710840788123178418881310611295147681836467269056642588103 00*c[4]-14866577300833577341781804497888079452013097304682037672876441 028650*c[4]*c[3]*a[9,5]+8955133545709219960907619514112464427307297666 370600207685335483530*c[3]*a[9,5]+221240279603962951247803659659927771 7204980157493690702023365273125+37532426710840788123178418881310611295 14768183646726905664258810300*c[3]-19654003928729815635911207009486881 5487165506723544953117113361260100*c[3]*a[7,6]+17566215965604670385210 0165212996303951898841174157393994782637902800*a[7,6]-5394275709782240 189469796843412123542602857410867024267114598533866*a[9,5])/c[3]/c[4], b[5] = 89929293580433773826711625625/175034206955063764470139052544, \+ a[10,3] = -1/737285365636180172365717201591044746279068994997784860204 10234182150*(-14391653814477288355745613816196928352831197371050876700 850505560434625*c[3]*c[4]+12202113593801907743815767240983710318277547 26857288137767993655568579200*a[7,6]-374705429421626576173005511555794 61464640366231439127156178159509833424*a[9,5]-301047924636924401582455 60368229172952172493866504041760786289722899250*c[3]+19698063725830830 659874586320581669010798396620374360439809791849899825*c[4]+6220551824 3865315690803469010736041134815539225080672783832111563475920*c[3]*a[9 ,5]+622055182438653156908034690107360411348155392250806727838321115634 75920*c[4]*a[9,5]-7936583494464354640728753500084406412881973757556609 9547659523773744000*c[4]*c[3]*a[6,1]-103268492958872675360960572326271 030960548390265605439100860774599483600*c[4]*c[3]*a[9,5]-2025694692880 670342539159787550340907454720519045989148161011078497936000*c[3]*a[7, 6]-1365236479962262628713813301962521687249905903409998613158451917910 746400*c[4]*a[7,6]+226645348834599591603766648361165191694728976106375 0153475229608319412000*c[3]*c[4]*a[7,6]+179275009506577751615407870332 58774025336146046125210387419394848831250)/(845*c[3]-509)/c[3]/(-c[4]+ c[3]), c[6] = 244/273, b[2] = 0, b[3] = 0, b[6] = 43965679317071687650 068867/11554641620922992752054730, b[1] = 17499592155841793/3016278995 85331605, a[10,1] = 1/121214875108147473598130467961978893601502896260 60081551157585371184735050*(-78752078883226201395621284301140510424585 693024572180917193014729976059150*c[3]*c[4]+57905828070624613771776742 11742584010183575172898442955136464536172493750+3941282690798016201252 49281883773843280364776774904068499061950748651081600*a[7,6]-121029853 70318538410388078023252166053078838292754838071445545521676195952*a[9, 5]+6362474583443358303139491381547879090487882108380918422058562767517 643475*c[3]+6362474583443358303139491381547879090487882108380918422058 562767517643475*c[4]+2009238239276849696812952049046774128654541916970 1057309177772035002722160*c[3]*a[9,5]+20092382392768496968129520490467 741286545419169701057309177772035002722160*c[4]*a[9,5]-256351646871198 65489553873805272632713608775236907850153894026178919312000*c[4]*c[3]* a[6,1]-333557232257158741415902648613855430002571300557905568295780301 95633202800*c[4]*c[3]*a[9,5]-44097138302781082907456169653389450498171 9606801429552050179969485171087200*c[3]*a[7,6]-44097138302781082907456 1696533894504981719606801429552050179969485171087200*c[4]*a[7,6]+49338 1916256526050562931734236693442049219068265533884056143982251851257400 *c[3]*c[4]*a[7,6])/c[3]/c[4], a[7,5] = -1065961/2446394966882199569115 703998829974634621308530824849490327307237*(-2003475994948271232492399 22850787236826351282712333040698783675*c[3]*c[4]-667339581138892567176 31184566706828007241012611968975143921770*c[4]*a[9,5]+9625973542205699 4004163284998229123412582366622716791166512075*c[4]+110786236947419296 515517388917224498361726239012011363451107850*c[4]*c[3]*a[9,5]+1757504 65927419314691974871738642967869984924000470927766675600*c[4]*c[3]*a[6 ,1]-57526109664919419205354063686725920390214989872864487873301275+962 59735422056994004163284998229123412582366622716791166512075*c[3]-66733 958113889256717631184566706828007241012611968975143921770*c[3]*a[9,5]+ 40198325065052818543519849638406834858799615881055867867758794*a[9,5]) /(845*c[3]-509)/(-509+845*c[4]), b[7] = 639711292154532559948032664549 872463482448777204372657657897547/201160142550158862242760444301691193 2371008211435797382991660160, a[4,3] = 1/2*1/c[3]*c[4]^2, a[4,2] = 0, \+ a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = \+ 0, a[5,3] = -259081/3620106750*(-1018+2535*c[4])/c[3]/(-c[4]+c[3]), a[ 8,6] = 137579144273131765125504385833849/11101640233285403656320691121 984000+589465263817347678735311491045273750403066591884059/89500558206 981193102100325516008077529644572928000*a[7,6], a[4,1] = 1/2*c[4]*(-c[ 4]+2*c[3])/c[3], a[9,6] = -2481006722082587625576165287911530599571363 /18438601757968638166315261704657797344684897-935369781209202815961193 2801226897326669471415908507/63236885073612372962305729633918796518243 9198676854*a[7,6], c[2] = 1/200, a[6,5] = -325/27783*(24379992*c[3]-54 555228*c[3]*c[4]+61039251*c[4]*c[3]*a[6,1]-14526784+24379992*c[4])/(-5 09+845*c[4])/(845*c[3]-509), a[8,5] = -527/149418793283586102823792373 90488213399874673024000*(-23546330767866054403409954828915233039370818 359667175*c[3]*c[4]+10176795721024446782279896655886340123474106606013 575*c[4]+28832852001085293333668622206708160754717432979870100*c[4]*c[ 3]*a[6,1]+9014209254441825577264388741859206904813883534427175*c[4]*c[ 3]*a[9,5]-5429860959184484282636182094208682029053570081684535*c[4]*a[ 9,5]-6055098366205546798084480665958835446643447969590775-542986095918 4484282636182094208682029053570081684535*c[3]*a[9,5]+10176795721024446 782279896655886340123474106606013575*c[3]+3270768317425920118179664717 103217932293807303641927*a[9,5])/(845*c[3]-509)/(-509+845*c[4]), a[10, 9] = -1198575787406124991100974435169141280/46356482334809216491547309 58571956313, a[3,1] = -100*c[3]^2+c[3], b[4] = 0, b[8] = -436142173547 89120000/21595629876813903267, a[8,1] = -17/12591247547810421671513219 2748871873137035480179879846285627954569798400000*(-101488000842807638 57516998348994540790015563189844638426393860189749314425*c[3]*c[4]+389 6787052434410968703923253812427144315446022105882677897872842287760040 0*a[7,6]-1196634709734841078821464584102104439618622730210053849569499 930208430338*a[9,5]+13930939304371514597536539590501620023901217518779 80729096726907625965025*c[3]+13930939304371514597536539590501620023901 21751877980729096726907625965025*c[4]+19865546752965436377291504392264 79865378656595338890968342293597300832290*c[3]*a[9,5]+1986554675296543 637729150439226479865378656595338890968342293597300832290*c[4]*a[9,5]- 1054871153707993536185860449579968667352175337051242920612829645357774 9400*c[4]*c[3]*a[6,1]-329791493246675711960929689812647443270130613568 0477147837402926756784450*c[4]*c[3]*a[9,5]-435992977587948440350889773 88966910262217900165364998814185216637072069300*c[3]*a[7,6]-4359929775 8794844035088977388966910262217900165364998814185216637072069300*c[4]* a[7,6]+487811815088155427113905361769998627114159292833796912961990333 68527356225*c[3]*c[4]*a[7,6]+26570356640131036835058866253572243426138 0040315260820262812841943524675)/c[3]/c[4], a[9,8] = 10397310879870999 2837929472000/762442644104164301247937014633, a[3,2] = 100*c[3]^2, a[2 ,1] = 1/200, a[9,3] = -1/143361401165222539457876610394291990858371237 26790778855232826450*(-36193275516722071478474849451601360940354563005 27219774025321900*c[4]+18952752100993071972913410809534119140517406627 1499472629810377300*c[4]*a[7,6]-86356157624968369921963544012656358990 42717132469238387353264690*c[4]*a[9,5]+5201808784746615418967981526916 223281198512450209280874748889618*a[9,5]-21334646056312724324764094470 58127017555429274342999712655125625-1693945609026487018824495325101217 97446382681942292568943859824400*a[7,6])/c[3]/(-c[4]+c[3]), a[10,6] = \+ -422243741906977924863571623920353258161809980804/10373753223221056460 708245576806198484379171923*a[7,6]+88300604865507531890572886199765841 296/281984196771808918111529577339005012981, b[10] = 7077278529846113/ 15546034297382400, c[5] = 509/845, a[5,4] = 259081/3620106750*(-1018+2 535*c[3])/c[4]/(-c[4]+c[3]), c[9] = 171/181, c[8] = 17/20, a[5,1] = 50 9/3620106750*(4284150*c[3]*c[4]-1290315*c[3]+518162-1290315*c[4])/c[3] /c[4], a[10,8] = -26328105087975815319887872000/9661507630911500237190 18517747, a[7,4] = -1/258705066178555432430268057550140726268062600460 59678013969116501450*(116894310335474118198750342129772949525541696556 2717764021842577341925*c[3]*c[4]+1051906344293267908949635106981354337 7825471783252483946275807375591200*a[7,6]-3230219219151953242870737168 58443633315865226133095923760156547498564*a[9,5]-507876384694593323148 794033652187789881040276510442190360463991851925*c[3]-7144624786793729 98771964163217749603417457110876283304451942036699200*c[4]+53625446761 9528583541409215609793458058754648492074765377863030719620*c[3]*a[9,5] +536254467619528583541409215609793458058754648492074765377863030719620 *c[4]*a[9,5]-141227907355575895158619062867944372877064507370012823708 0241718973600*c[4]*c[3]*a[6,1]-890245628955798925525522175226474404832 313709186253785352248056892100*c[4]*c[3]*a[9,5]+4266916625812024144400 90504827527317345753186965359466214789518856000-1176927999967467783373 9769844504497303878499167327574251365964809575400*c[3]*a[7,6]-17462885 283454054677061722306468456098747590681430940932422509297396000*c[4]*a [7,6]+1953839214091375789687643520354872342195939449192888063340715179 5857000*c[3]*c[4]*a[7,6])/(-c[4]+c[3])/(-509+845*c[4])/c[4], a[8,7] = \+ -7424205995886059956357359508852528825590423/4378197291051651257486258 563957705093888000, c[7] = 222785849/1109145221, a[7,1] = -1/131680878 68488471510700644129302162967044386363444376109110280299238050*(-10350 42625366297663189795634265558605637170465679917512429564073005900*c[3] *c[4]+1051906344293267908949635106981354337782547178325248394627580737 5591200*a[7,6]-3230219219151953242870737168584436333158652261330959237 60156547498564*a[9,5]-507876384694593323148794033652187789881040276510 442190360463991851925*c[3]-5078763846945933231487940336521877898810402 76510442190360463991851925*c[4]+53625446761952858354140921560979345805 8754648492074765377863030719620*c[3]*a[9,5]+53625446761952858354140921 5609793458058754648492074765377863030719620*c[4]*a[9,5]-14122790735557 58951586190628679443728770645073700128237080241718973600*c[4]*c[3]*a[6 ,1]-890245628955798925525522175226474404832313709186253785352248056892 100*c[4]*c[3]*a[9,5]+4266916625812024144400905048275273173457531869653 59466214789518856000-1176927999967467783373976984450449730387849916732 7574251365964809575400*c[3]*a[7,6]-11769279999674677833739769844504497 303878499167327574251365964809575400*c[4]*a[7,6]+131680878684884715107 00644129302162967044386363444376109110280299238050*c[3]*c[4]*a[7,6])/c [3]/c[4], a[9,7] = 119444820458286583462988069082381214175248483796051 93796762013/3412376317311298392350722812851096867170775526794634954003 706, a[9,4] = 1/143361401165222539457876610394291990858371237267907788 55232826450*(-36193275516722071478474849451601360940354563005272197740 25321900*c[3]-21334646056312724324764094470581270175554292743429997126 55125625+1895275210099307197291341080953411914051740662714994726298103 77300*c[3]*a[7,6]-8635615762496836992196354401265635899042717132469238 387353264690*c[3]*a[9,5]-169394560902648701882449532510121797446382681 942292568943859824400*a[7,6]+52018087847466154189679815269162232811985 12450209280874748889618*a[9,5])/(-c[4]+c[3])/c[4], a[6,4] = 1/4695327* (2389921443*c[3]*a[6,1]-551347524*c[3]+10329496)/(-509+845*c[4])/(-c[4 ]+c[3]), b[9] = -210996982649291857501475159/9903665793941217225663120 0, a[8,3] = 17/2473722504481418795975092195459172360256099807070330968 28345686777600000*(120855157218997240839855624313087199609704653081811 063005615398995686575*c[3]*c[4]+38967870524344109687039232538124271443 154460221058826778978728422877600400*a[7,6]-11966347097348410788214645 84102104439618622730210053849569499930208430338*a[9,5]-486691899982726 676457159783210766253970187793332109425923372139205482675*c[3]+1393093 930437151459753653959050162002390121751877980729096726907625965025*c[4 ]+19865546752965436377291504392264798653786565953388909683422935973008 32290*c[3]*a[9,5]+1986554675296543637729150439226479865378656595338890 968342293597300832290*c[4]*a[9,5]+265703566401310368350588662535722434 261380040315260820262812841943524675-105487115370799353618586044957996 86673521753370512429206128296453577749400*c[4]*c[3]*a[6,1]-32979149324 66757119609296898126474432701306135680477147837402926756784450*c[4]*c[ 3]*a[9,5]-646912585325555455511751502843123956178104496793609206841591 85692203482000*c[3]*a[7,6]-4359929775879484403508897738896691026221790 0165364998814185216637072069300*c[4]*a[7,6]+72379973686015016128978754 211546245916648576895350538306456793827752256500*c[3]*c[4]*a[7,6])/(84 5*c[3]-509)/c[3]/(-c[4]+c[3]), a[10,7] = 35011829759852921774894466285 700128707491070752921128030483969446296/371804493842727990442869546005 0071040231420419464687750808112491477, a[8,4] = -17/247372250448141879 597509219545917236025609980707033096828345686777600000*(-4866918999827 26676457159783210766253970187793332109425923372139205482675*c[4]+19865 54675296543637729150439226479865378656595338890968342293597300832290*c [4]*a[9,5]+72379973686015016128978754211546245916648576895350538306456 793827752256500*c[3]*c[4]*a[7,6]-1054871153707993536185860449579968667 3521753370512429206128296453577749400*c[4]*c[3]*a[6,1]+120855157218997 240839855624313087199609704653081811063005615398995686575*c[3]*c[4]-64 6912585325555455511751502843123956178104496793609206841591856922034820 00*c[4]*a[7,6]-3297914932466757119609296898126474432701306135680477147 837402926756784450*c[4]*c[3]*a[9,5]+3896787052434410968703923253812427 1443154460221058826778978728422877600400*a[7,6]+1986554675296543637729 150439226479865378656595338890968342293597300832290*c[3]*a[9,5]-119663 4709734841078821464584102104439618622730210053849569499930208430338*a[ 9,5]+13930939304371514597536539590501620023901217518779807290967269076 25965025*c[3]+26570356640131036835058866253572243426138004031526082026 2812841943524675-43599297758794844035088977388966910262217900165364998 814185216637072069300*c[3]*a[7,6])/c[4]/(-509*c[3]+509*c[4]+845*c[3]*c [4]-845*c[4]^2), a[7,3] = 1/258705066178555432430268057550140726268062 60046059678013969116501450*(116894310335474118198750342129772949525541 6965562717764021842577341925*c[3]*c[4]+1051906344293267908949635106981 3543377825471783252483946275807375591200*a[7,6]-3230219219151953242870 73716858443633315865226133095923760156547498564*a[9,5]-714462478679372 998771964163217749603417457110876283304451942036699200*c[3]-5078763846 94593323148794033652187789881040276510442190360463991851925*c[4]+53625 4467619528583541409215609793458058754648492074765377863030719620*c[3]* a[9,5]+536254467619528583541409215609793458058754648492074765377863030 719620*c[4]*a[9,5]-141227907355575895158619062867944372877064507370012 8237080241718973600*c[4]*c[3]*a[6,1]-890245628955798925525522175226474 404832313709186253785352248056892100*c[4]*c[3]*a[9,5]+4266916625812024 14440090504827527317345753186965359466214789518856000-1746288528345405 4677061722306468456098747590681430940932422509297396000*c[3]*a[7,6]-11 769279999674677833739769844504497303878499167327574251365964809575400* c[4]*a[7,6]+1953839214091375789687643520354872342195939449192888063340 7151795857000*c[3]*c[4]*a[7,6])/(845*c[3]-509)/c[3]/(-c[4]+c[3]), a[10 ,4] = 1/73728536563618017236571720159104474627906899499778486020410234 182150*(22664534883459959160376664836116519169472897610637501534752296 08319412000*c[3]*c[4]*a[7,6]-14391653814477288355745613816196928352831 197371050876700850505560434625*c[3]*c[4]-79365834944643546407287535000 844064128819737575566099547659523773744000*c[4]*c[3]*a[6,1]-1032684929 58872675360960572326271030960548390265605439100860774599483600*c[4]*c[ 3]*a[9,5]+622055182438653156908034690107360411348155392250806727838321 11563475920*c[4]*a[9,5]-2025694692880670342539159787550340907454720519 045989148161011078497936000*c[4]*a[7,6]-301047924636924401582455603682 29172952172493866504041760786289722899250*c[4]-13652364799622626287138 13301962521687249905903409998613158451917910746400*c[3]*a[7,6]-3747054 2942162657617300551155579461464640366231439127156178159509833424*a[9,5 ]+12202113593801907743815767240983710318277547268572881377679936555685 79200*a[7,6]+622055182438653156908034690107360411348155392250806727838 32111563475920*c[3]*a[9,5]+1969806372583083065987458632058166901079839 6620374360439809791849899825*c[3]+179275009506577751615407870332587740 25336146046125210387419394848831250)/(-c[4]+c[3])/(-509+845*c[4])/c[4] , a[10,5] = 28/218215672232116429655758316664889129307628406359*(-1083 5382771363120659540908813131963488582760363965875*c[3]*c[4]+4891474028 957959404985550538206650817888180098487875*c[4]+1531284082553715976723 5739871108108800646178723754950*c[4]*c[3]*a[9,5]+117685110208511401361 91274370084963573354054277498000*c[4]*c[3]*a[6,1]-92239479055602536349 38451590998848969856692272652390*c[4]*a[9,5]-2915822622994453282544692 459152118213748507795513875+489147402895795940498555053820665081788818 0098487875*c[3]-9223947905560253634938451590998848969856692272652390*c [3]*a[9,5]+5556200572698424970631564331146052219712492741751558*a[9,5] )/(845*c[3]-509)/(-509+845*c[4])\}:" }}}{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 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[7,4]=subs(e11,a[7,4]);\n``;\na[8,5]=sub s(e11,a[8,7]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%, $*&#\"\"\"\"_o]9];\"pR,y'fg/gi!oisS,bd!o-VKabyh10(e#F,**,<*(\"aoD>MxD% =-kxrib'pTb_\\HxH@M]()>=TZN.J%*o6F,&%\"cG6#\"\"$F,&F36#F(F,F,*&\"bo+7f vt!eFYR[_Kyra#yPVN\")p5N'\\*3zE$HWj!>0\"F,&F%6$F'\"\"'F,F,*&\"`ok&)\\Z l:gP#f4LhAleJLOWeortqGC`>:>#>-B$F,&F%6$\"\"*\"\"&F,!\"\"*&\"`oD>&=*RYg .>U/^w-/\"))*y(=_O.%z[JK$f%p%Qwy]F,F2F,FC*&\"`o+#*pO?%>X/LGw36duT.'\\x @jT'>x)*HPz'yCY9(F,F6F,FC*(\"`o?'>2.jyPlZ2#\\[Yve!eMz4c@49a$eG&>wYai`F ,F2F,F?F,F,*(FIF,F6F,F?F,F,**\"ao+O(*='e^*edbt!z AT\"F,F6F,F2F,&F%6$F\\%Rf> UB([N?Nk(o*yv849#RQ&>F,F2F,F6F,F:F,F,F,,&F6FCF2F,FC,&\"$4&FC*&\"$X)F,F 6F,F,FCF6FCF,FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!LB/fD)GD&)3&ftNc*fg)e*f?Cu\"L +!))Q40x&Rcei[d7l^5H(>yV" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 117 "We can find some simple order conditions that are not yet satisfied and determine which paramers are related by them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "for ii from 64 by -1 to 43 do\n eq := simplify(subs(e11,SO7_1 0[ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets(lhs(eq))) \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<'&%\"aG6 $\"\"(\"\"'&F&6$\"\"*\"\"&&F&6$F)\"\"\"&%\"cG6#\"\"$&F26#\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<'&%\"aG6$\"\"(\"\"'&F&6$\"\"*\"\" &&F&6$F)\"\"\"&%\"cG6#\"\"$&F26#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#X<'&%\"aG6$\"\"(\"\"'&F&6$\"\"*\"\"&&F&6$F)\"\"\"&%\"cG6#\"\"$&F 26#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple or der comditions given in abreviated form as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SO7 := Si mpleOrderConditions(7):\n[seq([i,SO7[i]],i=[45,55])]:\nlinalg[augment] (linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7% \"#X%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\" #F-F3F-F-F-F-#F-\"$?%7%\"#bF)/*(F,F-F.F--F06#*&F3F--F06#*&)F.\"\"$F-F3 F-F-F-#F-\"$S\"Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "We can solve this system of two equations to ob tain numerical values for the remaining parameters " }{XPPEDIT 18 0 " a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a [9,5]" "6#&%\"aG6$\"\"*\"\"&" }{TEXT -1 103 " and substitute these va lues back into the expressions obtained previously for the other coeff icients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "cdcns5 := [seq(SO7_10[i],i=[45,55])]:\neqns5 := simp lify(subs(e11,cdcns5)):\nnops(%);\nindets(eqns5);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'& %\"aG6$\"\"(\"\"'&F%6$\"\"*\"\"&&F%6$F(\"\"\"&%\"cG6#\"\"$&F16#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "e12 := solve(\{op(eqn s5)\},\{a[7,6],a[9,5]\}):\ninfolevel[solve] := 0:\ne13 := `union`(map( u_->lhs(u_)=simplify(subs(e12,rhs(u_))),e11),e12):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[7,6]=sub s(e12,a[7,6]);\n``;\na[9,5]=subs(e12,a[9,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!I'GBzq&zF:,()o-#euA,OsE#\"K6\"\\$e 6\"*)3*\\Hgfw8RXp=ttZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,$*&#\"-D\\,6_N\"N1j]ijm) 47>SN*o%f&Qy)*eVk0\"\"\"\"*&,,*&\"H(ouO3Dm&y;a&\\[atVtO1BF.&%\"cG6#\" \"$F.!\"\"*&\"H(ouO3Dm&y;a&\\[atVtO1BF.&F46#\"\"%F.F7*(\"H$=mT\\*[NXM7 \"\\N$REp7!)f$F.F3F.F:F.F.\"HREZF0!=$QKVP#p=-[U)=R\"F.**\"H9!R/+%R0KVe ()>:fVPs0+\"F.F:F.F3F.&F%6$\"\"'F.F.F.F.,**(\"'DSrF.F3F.F:F.F.*&\"'0,V F.F3F.F7*&FIF.F:F.F7\"'\"3f#F.F7F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 85 "This completes the construction of th e 10 stage, order 7 scheme with the parameters " }{XPPEDIT 18 0 "c[3] " "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6 #\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"' \"\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6781 "e13 := \{c[10] = 1, a[6,3] = -1/4695327*(10329496-551347524*c[4]+2389921443*c[4]*a[6,1 ])/(-509*c[3]+509*c[4]-845*c[3]*c[4]+845*c[3]^2), b[5] = 8992929358043 3773826711625625/175034206955063764470139052544, a[8,4] = 221/32504925 206004096000*(58731650087224319790*c[3]*c[4]+122329965152952406395*c[4 ]*c[3]*a[6,1]-52377483446440770930*c[3]-34389599109076936180*c[4]+2103 3640295257082764)/(-509*c[3]+509*c[4]+845*c[3]*c[4]-845*c[4]^2)/c[4], \+ a[10,1] = -1/490602568229440225362487290779260374*(5569591608419108059 282789108812768378*c[3]*c[4]+1673643642371217202490685432960840432*c[4 ]*c[3]*a[6,1]-1808500732512691780061201251224743925*c[3]-1808500732512 691780061201251224743925*c[4]+726253943076102655661657938361652590)/c[ 3]/c[4], c[6] = 244/273, b[2] = 0, b[3] = 0, b[6] = 439656793170716876 50068867/11554641620922992752054730, b[1] = 17499592155841793/30162789 9585331605, a[9,5] = 355211014925/105644358987838559468935401912098666 362506306*(-230636734373544849554167856625083674687*c[3]-2306367343735 44849554167856625083674687*c[4]+35980126926393354911234453548949416618 3*c[3]*c[4]+139188424802186923743323831800527472639+100057237435915198 758433205394000439014*c[4]*c[3]*a[6,1])/(714025*c[3]*c[4]-430105*c[3]- 430105*c[4]+259081), b[7] = 639711292154532559948032664549872463482448 777204372657657897547/201160142550158862242760444301691193237100821143 5797382991660160, a[4,3] = 1/2*1/c[3]*c[4]^2, a[7,6] = -22672360122745 82026887011527795707923286/477373186945391376596029499088911158349111, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = \+ 0, a[10,2] = 0, a[5,3] = -259081/3620106750*(-1018+2535*c[4])/c[3]/(-c [4]+c[3]), a[4,1] = 1/2*c[4]*(-c[4]+2*c[3])/c[3], c[2] = 1/200, a[6,5] = -325/27783*(24379992*c[3]-54555228*c[3]*c[4]+61039251*c[4]*c[3]*a[6 ,1]-14526784+24379992*c[4])/(-509+845*c[4])/(845*c[3]-509), a[10,9] = \+ -1198575787406124991100974435169141280/4635648233480921649154730958571 956313, a[10,5] = 119956200/932957436771935526461936536220191972637*(1 5078219258418020620716653129620669339*c[3]*c[4]+1894461238868833279120 7935967526969237*c[4]*c[3]*a[6,1]-117152422164933797790558390815721648 71*c[3]-11715242216493379779055839081572164871*c[4]+710619589831993973 3488135949957226687)/(-509+845*c[4])/(845*c[3]-509), a[3,1] = -100*c[3 ]^2+c[3], a[8,1] = 17/3685993415578871314242289536000*(467454951648432 586433783320359*c[3]*c[4]+354294006881631917298940356615*c[4]*c[3]*a[6 ,1]-151696507535283791338109404410*c[3]-151696507535283791338109404410 *c[4]+60917966339612978138933086268)/c[3]/c[4], b[4] = 0, b[8] = -4361 4217354789120000/21595629876813903267, a[9,8] = 1039731087987099928379 29472000/762442644104164301247937014633, a[10,6] = 1428124381699559509 69368011788034519784/281984196771808918111529577339005012981, a[3,2] = 100*c[3]^2, a[2,1] = 1/200, a[9,1] = -57/6783284894710044595261302150 938385052994*(56071271465125681631559259475028420846*c[4]*c[3]*a[6,1]+ 514333727908302766076402226015376773674*c[3]*c[4]-16676330830254467136 3096221106292016775*c[3]-166763308302544671363096221106292016775*c[4]+ 66968460691120503135160533761816675770)/c[3]/c[4], b[10] = 70772785298 46113/15546034297382400, a[8,6] = -42937178562595251680361031143/22732 95839722617724238904704000, a[9,6] = -11856793003583777562334896068853 69481439862/18438601757968638166315261704657797344684897, c[5] = 509/8 45, a[5,4] = 259081/3620106750*(-1018+2535*c[3])/c[4]/(-c[4]+c[3]), c[ 9] = 171/181, c[8] = 17/20, a[8,3] = -221/32504925206004096000*(587316 50087224319790*c[3]*c[4]+122329965152952406395*c[4]*c[3]*a[6,1]-343895 99109076936180*c[3]-52377483446440770930*c[4]+21033640295257082764)/(- c[4]+c[3])/c[3]/(845*c[3]-509), a[5,1] = 509/3620106750*(4284150*c[3]* c[4]-1290315*c[3]+518162-1290315*c[4])/c[3]/c[4], a[10,8] = -263281050 87975815319887872000/966150763091150023719018517747, a[7,3] = -2896216 037/2864239121672348259576176994533466950094666*(700221973143731731555 67487266470956*c[4]*c[3]*a[6,1]+16521542580547372792200192592168317*c[ 3]*c[4]-12819094696534439520794423926674718*c[3]-196826071798410550727 15404651048125*c[4]+7904100240267532175157507666574750)/(-c[4]+c[3])/c [3]/(845*c[3]-509), a[8,7] = -7424205995886059956357359508852528825590 423/4378197291051651257486258563957705093888000, c[7] = 222785849/1109 145221, a[7,1] = 222785849/1457897712931225264124274090217534677598184 994*(910288565086851251022377334464122428*c[4]*c[3]*a[6,1]+12950255336 48294623396802830976646486*c[3]*c[4]-255873893337933715945300260463625 625*c[3]-255873893337933715945300260463625625*c[4]+1027533031234779182 77047599665471750)/c[3]/c[4], a[10,3] = 13/13394358263687335618218*(27 96667131730390002417099*c[3]*c[4]+1789078990693805760982032*c[4]*c[3]* a[6,1]-1281092414945294383977746*c[3]-1933237508439185394627675*c[4]+7 76345476761771491807090)/(-c[4]+c[3])/c[3]/(845*c[3]-509), a[8,5] = -1 35464823/2767066927100306195780163168256*(-596063379425572264321345930 7*c[3]*c[4]+23832232105482509503159986894*c[4]*c[3]*a[6,1]+27866163994 6832881883660243*c[3]+278661639946832881883660243*c[4]-105809465112319 063822465715)/(-509+845*c[4])/(845*c[3]-509), a[9,3] = 741/18170306027 5893979654498*(276792658456904830262607*c[3]*c[4]+58808082428981301250 926*c[4]*c[3]*a[6,1]-174902942710783538168775*c[4]+7023715805900498692 5370-116347786987344190639978*c[3])/(-c[4]+c[3])/c[3]/(845*c[3]-509), \+ a[10,4] = -13/13394358263687335618218*(2796667131730390002417099*c[3]* c[4]+1789078990693805760982032*c[4]*c[3]*a[6,1]-1933237508439185394627 675*c[3]-1281092414945294383977746*c[4]+776345476761771491807090)/c[4] /(-509+845*c[4])/(-c[4]+c[3]), a[9,7] = 119444820458286583462988069082 38121417524848379605193796762013/3412376317311298392350722812851096867 170775526794634954003706, a[7,4] = 2896216037/286423912167234825957617 6994533466950094666*(70022197314373173155567487266470956*c[4]*c[3]*a[6 ,1]+16521542580547372792200192592168317*c[3]*c[4]-19682607179841055072 715404651048125*c[3]-12819094696534439520794423926674718*c[4]+79041002 40267532175157507666574750)/c[4]/(-509+845*c[4])/(-c[4]+c[3]), a[6,4] \+ = 1/4695327*(2389921443*c[3]*a[6,1]-551347524*c[3]+10329496)/(-509+845 *c[4])/(-c[4]+c[3]), b[9] = -210996982649291857501475159/9903665793941 2172256631200, a[9,4] = -741/181703060275893979654498*(-17490294271078 3538168775*c[3]+58808082428981301250926*c[4]*c[3]*a[6,1]+2767926584569 04830262607*c[3]*c[4]+70237158059004986925370-116347786987344190639978 *c[4])/c[4]/(-509+845*c[4])/(-c[4]+c[3]), a[7,5] = -299299679781805713 13390950/242982952155204210687379015036255779599697499*(-4472292987448 2783894542*c[3]*c[4]+806348169208705234768839*c[4]*c[3]*a[6,1]-8511390 2924867643878662*c[3]-85113902924867643878662*c[4]+5336911739124166663 2414)/(-509+845*c[4])/(845*c[3]-509), a[10,7] = 3501182975985292177489 4466285700128707491070752921128030483969446296/37180449384272799044286 95460050071040231420419464687750808112491477\}:" }}}{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 0 443 "s ubs(e13,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(5-i)],i=2..5)]));pr int(``);\nfor ii from 6 to 10 do\n print(c[ii]=subs(e13,c[ii]));prin t(``); \n for jj to ii-1 do\n if ii=6 and jj=1 then\n \+ print(a[ii,jj]*` is a parameter`);\n else\n print(a[ii,jj ]=subs(e13,a[ii,jj]));\n end if;\n end do:\n print(`_________ ________________________`);\nend do:print(``);\nfor ii to 10 do\n pr int(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7&7'#\"\"\"\"$+#F(%!GF+F+7'&%\"cG6#\"\"$,&*&\"$+\"F))F- \"\"#F)!\"\"F-F),$*&F3F)F4F)F)F+F+7'&F.6#\"\"%,$*&#F)F5F)*(F:F),&F:F6* &F5F)F-F)F)F)F-F6F)F)\"\"!,$*&F?F)*&F-F6F:F5F)F)F+7'#\"$4&\"$X),$*&#FI \"+]n5?OF)*(,**(\"(]TG%F)F-F)F:F)F)*&\"(:.H\"F)F-F)F6\"'i\"=&F)*&FTF)F :F)F6F)F-F6F:F6F)F)FC,$*&#\"'\"3f#FNF)*(,&\"%=5F6*&\"%NDF)F:F)F)F)F-F6 ,&F:F6F-F)F6F)F6,$*&#FZFNF)*(,&FgnF6*&FinF)F-F)F)F)F:F6FjnF6F)F)Q(ppri nt66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$W#\"$t#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"'\"\"\"F(%0 ~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\" \"\"\"(F`p%F,*&,(\")'\\H.\"F,*&\"*CvM^&F,&%\"cG6#\"\"%F,!\"\"*(\"+V9#* *Q#F,F3F,&F%6$F'F,F,F,F,,**&\"$4&F,&F46#F(F,F7*&F>F,F3F,F,*(\"$X)F,F?F ,F3F,F7*&FCF,)F?\"\"#F,F,F7F,F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"'\"\"%,$*&#\"\"\"\"(F`p%F,*(,(*(\"+V9#**Q#F,&%\"cG6#\"\"$F,& F%6$F'F,F,F,*&\"*CvM^&F,F2F,!\"\"\")'\\H.\"F,F,,&\"$4&F:*&\"$X)F,&F36# F(F,F,F:,&F@F:F2F,F:F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"'\"\"&,$*&#\"$D$\"&$yF\"\"\"*(,,*&\")#**zV#F.&%\"cG6#\"\"$F.F.*(\" )G_baF.F3F.&F46#\"\"%F.!\"\"**\")^#R5'F.F9F.F3F.&F%6$F'F.F.F.\")%yEX\" F<*&F2F.F9F.F.F.,&\"$4&F<*&\"$X)F.F9F.F.F<,&*&FFF.F3F.F.FDFA+(F.&%\"cG6#\" 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"6#/&%\"aG6$\"#5\"\"$,$*&#\"#8\"8=#=cL(oj#eVR8\"\"\"**,,*(\":*4F.F 6F.F@\"9!42=\\rF.F3F .!\"\"*&\":Yx(RQ%HX\\T#4\"G\"F.F7F.F@\"9!42=\\r\"\"HPE(>>?i`O>YEb$>xOu&H$*\"\"\"*(,,*( \"GR$p1iHJl;2i?!=%e#>#y]\"F.&%\"cG6#\"\"$F.&F46#\"\"%F.F.**\"GP#pp_nf$ z?\"zK$)o)Q7Y%*=F.F7F.F3F.&F%6$\"\"'F.F.F.*&\"Gr[;s:3Re0z(zL\\;AC:<\"F .F3F.!\"\"*&\"Gr[;s:3Re0z(zL\\;AC:<\"F.F7F.FA\"F(oEs&*\\f8)[L(R*>$)*e> 1rF.F.,&\"$4&FA*&\"$X)F.F7F.F.FA,&*&FHF.F3F.F.FFFAFAF.F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"H%y>X.)y6!o$p4&f&*p\"QC\"G9 \"H\")H,0!RtdH:6=*3=x'>%)>G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\"(#\"_o'HY%pR[I!G6#Hvq5\\2(G,q&GmW*[x@H&)f(H=,N\"^ox9\\7\"33v (ok%>/UJ-/r+0gapGW!*zsUQ\\/=P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\")#!>+?(y))>`\"e(z30\"Gj#\"?Zx^=!>P-]64j2:m*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!F!GT\"p^Vu45\"*\\71uyv&)>\" \"F8j&>de4ta\"\\;#4[L#[cj%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B______ ___________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"2$zTe:#f*\\<\"30;L& e**yi,$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"&#\">Dci6n#QxL/e$HH**)\"?WD0R,ZkP1bp?M]<" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\";n)o+l(orqJzc'R%\";IZ0_F*H#4iTY b6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"jnZv*yldEP/sx[C [jC()\\XmK![*fD`a@H6(R'\"[og,m\"*HQ(zN9@35PK>\"p,VWgFCi)e,bU,;,#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!5++7*yat@9O%\"5nK!R\"o ()Hcf@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#! " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "subs(e13,OrderConditions(7,10,'expanded')):\nmap (u->simplify(lhs(u)-rhs(u)),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#&)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "The \+ stability function of a general 10 stage, order 7 Runge-Kutta scheme h as the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z^5/120+z^6/720+z^7/5040+t[8]*z^8+t[9]*z^9+t [10]*z^10;" "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$ \"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F,F)*&F'F/\"$?(F,F)*&F'\"\"( \"%S]F,F)*&&%\"tG6#\"\")F)*$F'F?F)F)*&&F=6#\"\"*F)*$F'FDF)F)*&&F=6#\"# 5F)*$F'FIF)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 20 "where, \+ for example, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "t[9 ]=b[9]*a[9,8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[10]*(a[ 10,8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[10,9]*(a[9,7]*a [7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[9,8]*(a[8,6]*a[6,5]*a[5,4]*a[ 4,3]*a[3,2]*c[2]+a[8,7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6, 4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a [4,3]*c[3])))))))" "6#/&%\"tG6#\"\"*,&*4&%\"bG6#F'\"\"\"&%\"aG6$F'\"\" )F-&F/6$F1\"\"(F-&F/6$F4\"\"'F-&F/6$F7\"\"&F-&F/6$F:\"\"%F-&F/6$F=\"\" $F-&F/6$F@\"\"#F-&%\"cG6#FCF-F-*&&F+6#\"#5F-,&*2&F/6$FJF1F-&F/6$F1F4F- &F/6$F4F7F-&F/6$F7F:F-&F/6$F:F=F-&F/6$F=F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F /6$FJF'F-,&*0&F/6$F'F4F-&F/6$F4F7F-&F/6$F7F:F-&F/6$F:F=F-&F/6$F=F@F-&F /6$F@FCF-&FE6#FCF-F-*&&F/6$F'F1F-,&*.&F/6$F1F7F-&F/6$F7F:F-&F/6$F:F=F- &F/6$F=F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F/6$F1F4F-,&*,&F/6$F4F:F-&F/6$F:F= F-&F/6$F=F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F/6$F4F7F-,&**&F/6$F7F=F-&F/6$F= F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F/6$F7F:F-,&*(&F/6$F:F@F-&F/6$F@FCF-&FE6# FCF-F-*&&F/6$F:F=F-,&*&&F/6$F=FCF-&FE6#FCF-F-*&&F/6$F=F@F-&FE6#F@F-F-F -F-F-F-F-F-F-F-F-F-F-F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[10]=b[ 10]*a[10,9]*a[9,8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^10 " "6#/&%\"tG6#\"#5*8&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"*F,&F.6$F0\"\")F,&F. 6$F3\"\"(F,&F.6$F6\"\"'F,&F.6$F9\"\"&F,&F.6$F<\"\"%F,&F.6$F?\"\"$F,&F. 6$FB\"\"#F,&%\"cG6#FEF,%\"zGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "t[8] = 28/31;" "6#/&%\"tG6#\"\")* &\"#G\"\"\"\"#J!\"\"" }{TEXT -1 1 " " }{TEXT 268 1 "x" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "1/8! = 1/44640;" "6#/*&\"\"\"F%-%*factorialG6#\"\")! \"\"*&F%F%\"&SY%F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "t[8] = -9/161;" "6#/&%\"tG6#\"\"),$*&\"\"*\"\"\"\"$h\"!\"\"F-" }{TEXT -1 1 " " }{TEXT 267 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/9! = -1/6491520;" "6#/*&\" \"\"F%-%*factorialG6#\"\"*!\"\",$*&F%F%\"(?:\\'F*F*" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "t[9] = -24/13;" "6#/&%\"tG6#\"\"*,$*&\"#C\"\"\"\" #8!\"\"F-" }{TEXT -1 1 " " }{TEXT 266 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/10! = -1/1965600;" "6#/*&\"\"\"F%-%*factorialG6#\"#5!\"\",$*&F %F%\"(+c'>F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This give s three equations which we can solve for " }{XPPEDIT 18 0 "c[6]" "6#& %\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"( " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"( " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 204 "Rz := StabilityFunction(7,10,'expanded'):\neq A := simplify(subs(e13,coeff(Rz,z^10)))=-24/13*1/10!:\neqB := simplify (subs(e13,coeff(Rz,z^9)))=-9/161*1/9!:\neqC := simplify(subs(e13,coeff (Rz,z^8)))=28/31*1/8!:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The first two of these equations are as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eqA;``;eqB;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\".Z0H\"HL%*\"9+#fB(fn$4#*\\l[$\"\"\"*0,,*& \")#**zV#F)&%\"cG6#\"\"$F)F)*(\")G_baF)F.F)&F/6#\"\"%F)!\"\"**\")^#R5' F)F4F)F.F)&%\"aG6$\"\"'F)F)F)\")%yEX\"F7*&F-F)F4F)F)F),&\"$4&F7*&\"$X) F)F4F)F)F7,&*&FCF)F.F)F)FAF7F7,&\"%=5F7*&\"%NDF)F.F)F)F)F4F),&F4F7F.F) F7F.F)F)F)#F7\"(+c'>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\":+oV*)Qqu$o*>YR\"F'**,>*(\":ob *pR'oP?706K&F'&%\"cG6#\"\"$F'&F.6#\"\"%F'!\"\"**\":o(ey/JatFOK*o%F')F1 \"\"#F'F-F'&%\"aG6$\"\"'F'F'F'**\":o(ey/JatFOK*o%F'F1F')F-F8F'F9F'F4** \";QJX@hZ5xMY!*3xF'F7F'F?F'F9F'F4*(\":+o\\Ite*y&H%G4VF'F-F'F7F'F'**\"< NbD,E`mo\\aDF#=F')F-F0F'F7F'F9F'F'*(\";;%f:YFy$>)*[4i " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sol := solve(\{eqA,eqB,eqC\},\{c[3],c[4],a[6,1]\}):\ninfolevel[sol ve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c[3]=subs( sol,c[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"gn59L+P q-Irna4))>XA03[_]$HY,'3KXbQ\"hnJ@n&ycNc9'pVW7^57sGF-,G\"y&)>" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Since the rational expressions for the three parameters " }{XPPEDIT 18 0 "c[3] " "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6 #\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"' \"\"\"" }{TEXT -1 164 " involve a large number of digits, we replace \+ them by simpler rational approximations and substitute these values in the expressions obtained for the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "[c[3]=subs( sol,c[3]),c[4]=subs(sol,c[4]),a[6,1]=subs(sol,a[6,1])]:\nevalf[10](%); \nconvert(%,rational,6);\ne14 := \{op(%)\}:\ne15 := `union`(map(u_->lh s(u_)=simplify(subs(e14,rhs(u_))),e13),e14):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"$$\"+P'H:%>!#5/&F&6#\"\"%$\"+ZDMkXF+/&% \"aG6$\"\"'\"\"\"$!+c;:_=F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\" cG6#\"\"$#\"#$*\"$z%/&F&6#\"\"%#\"$5\"\"$T#/&%\"aG6$\"\"'\"\"\"#!$G#\" %J7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3719 "e15 := \{c[10] = 1, b[5] = 89929293580433773826711625625/1750342 06955063764470139052544, c[6] = 244/273, b[2] = 0, b[3] = 0, b[6] = 43 965679317071687650068867/11554641620922992752054730, b[1] = 1749959215 5841793/301627899585331605, a[8,4] = -613522852101531567576566644397/8 14576707522236069963552640000, a[6,5] = 103959712518200500/83969153915 827131, b[7] = 6397112921545325599480326645498724634824487772043726576 57897547/2011601425501588622427604443016911932371008211435797382991660 160, a[7,6] = -2267236012274582026887011527795707923286/47737318694539 1376596029499088911158349111, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7, 2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, c[2] = 1/200, a[10,9] = - 1198575787406124991100974435169141280/46356482334809216491547309585719 56313, b[4] = 0, b[8] = -43614217354789120000/21595629876813903267, a[ 8,3] = 25892910773607576734414005208753/105292913205082732017473747520 00, a[5,1] = 311712996187/3366699277500, a[9,8] = 10397310879870999283 7929472000/762442644104164301247937014633, a[10,6] = 14281243816995595 0969368011788034519784/281984196771808918111529577339005012981, a[9,3] = -1334984216509015564184248020843701584/6670676425579396634061300189 77478253, a[2,1] = 1/200, b[10] = 7077278529846113/15546034297382400, \+ a[8,6] = -42937178562595251680361031143/227329583972261772423890470400 0, a[10,4] = -5213063483277854003640774248205317/228251499905582972173 3809658986516, a[10,5] = 434412560263717426352648442372374758256303547 8367000/2819697174719008494099755685231013933352915992886809, a[9,6] = -1185679300358377756233489606885369481439862/184386017579686381663152 61704657797344684897, c[5] = 509/845, c[9] = 171/181, c[8] = 17/20, a[ 10,8] = -26328105087975815319887872000/966150763091150023719018517747, a[10,1] = -40734473647257020156088313054846628592379/1123313076372220 10640347265020564130793284, a[7,4] = -48597609826899455036511090495924 959702383070843517685/488091226723681375828622960423688756917868034622 078292, a[7,3] = 59087076776381536334498651011356345416230115240752771 64/31545554087781255456304827404537875875692253868011534003, a[3,1] = \+ -820353/229441, a[7,1] = 207840215085446984390043413029457880466874970 48823/333809007739010923825478541340748044990946025336204, a[3,2] = 86 4900/229441, a[8,1] = -14850375572183825804482268314325533/13186987006 1161226458875010296840000, a[4,3] = 2897950/5401533, a[6,4] = -6255616 188978805360/5200809295819122531, a[8,7] = -74242059958860599563573595 08852528825590423/4378197291051651257486258563957705093888000, c[7] = \+ 222785849/1109145221, a[9,4] = -57633497085298645064371869898744017/30 963779845911010423286154735455876, a[10,3] = -111104298467482138964323 0968704444252/147519964335786292056835317593798019, a[7,5] = 408372425 52907904940455804285850036846494300434360648500/7343725626621459909895 87253914642887902719996408150162943, a[9,1] = -15383168980015859225248 7409542055462465307763/517713869734060023599533102763919424014608068, \+ a[9,7] = 1194448204582865834629880690823812141752484837960519379676201 3/3412376317311298392350722812851096867170775526794634954003706, a[9,5 ] = 489177972756706421411944299196898256388191463578294580375/31929120 1100986622665402847780351230046376607122311276442, a[6,1] = -228/1231, c[4] = 110/241, b[9] = -210996982649291857501475159/99036657939412172 256631200, a[5,3] = 996040375149076/5096677701243375, a[5,4] = 3445468 10496217/1096059720697500, c[3] = 93/479, a[4,1] = -432520/5401533, a[ 8,5] = 8122715759499790031095209746537365648167969/8362965435593003722 836700426272322940782592, a[6,3] = 15089501252616323812/14457231345452 578137, a[10,7] = 3501182975985292177489446628570012870749107075292112 8030483969446296/37180449384272799044286954600500710402314204194646877 50808112491477\}:" }}}{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 45 "subs(e15,StabilityFunction(7 ,10,'expanded'));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"zGF$ *&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F$\"#CF$*$)F% \"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F$*&#F$\"$?(F$*$)F%F-F$F$F$*&#F$ \"%S]F$*$)F%\"\"(F$F$F$*&#\"FRuM#flozinR#eC]wHL0\"\"J;3Ve&3^ocJ?+\\k;9 sX?-ZF$*$)F%\"\")F$F$F$*&#\"Ezc!=2+4ZdxZ)[A'\\)>aB\"L?:+\\\"GgE#f-mDf4 /Yf[;#G:F$*$)F%\"\"*F$F$!\"\"*&#\"C*f\\-x_Bj^r(Q8sS:^=\"Ic4$3f46z`!\\e /2R#3o5'QOF$*$)F%\"#5F$F$FV" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We may compare the coefficients of " } {XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "z^9" "6#*$%\"zG\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^10" "6# *$%\"zG\"#5" }{TEXT -1 58 " with those of the original specified stab ility function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 284 "[1053329765024582396762796865592347439/470220 45721416644900203156685108558430816*8!,-235419849622488477757470900071 805679/1528216485946040959256602592266028149001520*9!,-185115407213387 7151632352770249599/3638610680823907045849053791109590830956*10!]:\nev alf(%);\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$ \"+#=))>.*!#5$!+$G@,f&!#6$!+fG;Y=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#G\"#J#!\"*\"$h\"#!#C\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#---------------------------------------------- ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction o f the embedded order 6 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3719 "e15 := \{c[10] = 1, b[5] = 8992929358043377 3826711625625/175034206955063764470139052544, c[6] = 244/273, b[2] = 0 , b[3] = 0, b[6] = 43965679317071687650068867/115546416209229927520547 30, b[1] = 17499592155841793/301627899585331605, a[8,4] = -61352285210 1531567576566644397/814576707522236069963552640000, a[6,5] = 103959712 518200500/83969153915827131, b[7] = 6397112921545325599480326645498724 63482448777204372657657897547/2011601425501588622427604443016911932371 008211435797382991660160, a[7,6] = -2267236012274582026887011527795707 923286/477373186945391376596029499088911158349111, a[4,2] = 0, a[5,2] \+ = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, c[2] = 1/200, a[10,9] = -1198575787406124991100974435169141280/46356482334 80921649154730958571956313, b[4] = 0, b[8] = -43614217354789120000/215 95629876813903267, a[8,3] = 25892910773607576734414005208753/105292913 20508273201747374752000, a[5,1] = 311712996187/3366699277500, a[9,8] = 103973108798709992837929472000/762442644104164301247937014633, a[10,6 ] = 142812438169955950969368011788034519784/28198419677180891811152957 7339005012981, a[9,3] = -1334984216509015564184248020843701584/6670676 42557939663406130018977478253, a[2,1] = 1/200, b[10] = 707727852984611 3/15546034297382400, a[8,6] = -42937178562595251680361031143/227329583 9722617724238904704000, a[10,4] = -5213063483277854003640774248205317/ 2282514999055829721733809658986516, a[10,5] = 434412560263717426352648 4423723747582563035478367000/28196971747190084940997556852310139333529 15992886809, a[9,6] = -1185679300358377756233489606885369481439862/184 38601757968638166315261704657797344684897, c[5] = 509/845, c[9] = 171/ 181, c[8] = 17/20, a[10,8] = -26328105087975815319887872000/9661507630 91150023719018517747, a[10,1] = -4073447364725702015608831305484662859 2379/112331307637222010640347265020564130793284, a[7,4] = -48597609826 899455036511090495924959702383070843517685/488091226723681375828622960 423688756917868034622078292, a[7,3] = 59087076776381536334498651011356 34541623011524075277164/3154555408778125545630482740453787587569225386 8011534003, a[3,1] = -820353/229441, a[7,1] = 207840215085446984390043 41302945788046687497048823/3338090077390109238254785413407480449909460 25336204, a[3,2] = 864900/229441, a[8,1] = -14850375572183825804482268 314325533/131869870061161226458875010296840000, a[4,3] = 2897950/54015 33, a[6,4] = -6255616188978805360/5200809295819122531, a[8,7] = -74242 05995886059956357359508852528825590423/4378197291051651257486258563957 705093888000, c[7] = 222785849/1109145221, a[9,4] = -57633497085298645 064371869898744017/30963779845911010423286154735455876, a[10,3] = -111 1042984674821389643230968704444252/14751996433578629205683531759379801 9, a[7,5] = 40837242552907904940455804285850036846494300434360648500/7 34372562662145990989587253914642887902719996408150162943, a[9,1] = -15 3831689800158592252487409542055462465307763/51771386973406002359953310 2763919424014608068, a[9,7] = 1194448204582865834629880690823812141752 4848379605193796762013/34123763173112983923507228128510968671707755267 94634954003706, a[9,5] = 489177972756706421411944299196898256388191463 578294580375/319291201100986622665402847780351230046376607122311276442 , a[6,1] = -228/1231, c[4] = 110/241, b[9] = -210996982649291857501475 159/99036657939412172256631200, a[5,3] = 996040375149076/5096677701243 375, a[5,4] = 344546810496217/1096059720697500, c[3] = 93/479, a[4,1] \+ = -432520/5401533, a[8,5] = 812271575949979003109520974653736564816796 9/8362965435593003722836700426272322940782592, a[6,3] = 15089501252616 323812/14457231345452578137, a[10,7] = 3501182975985292177489446628570 0128707491070752921128030483969446296/37180449384272799044286954600500 71040231420419464687750808112491477\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We now turn our attentio n to the embedded order 6 scheme and introduce a new row corresponding to the node " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The linking coefficients and weights can be chosen so as to form an 11 stage order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We use \+ the order 6 quadrature conditions which are given in abreviated form a s follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "subs(b=`b*`,Q uadratureConditions(6)):\nListTools[Enumerate](%):\nlinalg[augment](li nalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delc ols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7%\"\" \"%#~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"$F)/*&F,F( )F2F/F(#F(F57%\"\"%F)/*&F,F()F2F5F(#F(F;7%\"\"&F)/*&F,F()F2F;F(#F(FA7% \"\"'F)/*&F,F()F2FAF(#F(FGQ(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We incorporate the row sum condition for the new tenth ro w" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j],j = \+ 1 .. 10) = c[11];" "6#/-%$SumG6$&%\"aG6$\"#6%\"jG/F+;\"\"\"\"#5&%\"cG6 #F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "together with the stage-order equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j],j = 2 .. 10) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"# 6%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"#5*&F-F-F3!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[11]^2;" "6#*$&%\"cG6#\"#6\"\"#" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j]^2,j = 2 .. 10) = 1/3;" "6#/-%$SumG6 $*&&%\"aG6$\"#6%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"#5*&F-F-\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^3;" "6#*$&%\"cG6#\"#6\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 50 "which ensure that the \+ tenth row has stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 53 "We also i ncorporate the column simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,1],i=2..11)=`b*`[1]" "6 #/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#6&F)6#F," } {TEXT -1 2 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 11) = `b *`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F +;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 6 " , \+ " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" }{TEXT -1 10 ", 6, 7, 8." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(6,11,'expanded')): \nSO_eqs2 := [add(a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2 *c[11]^2,\n add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3]:\n`simp_eqs*` : = [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..1 1)=`b*`[j]*(1-c[j]),j=[$5..8])]:\n`cdns*` := [op(SO_eqs2),op(`Qeqs*`), op(`simp_eqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We specify that " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"c G6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0;" "6#/&% \"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,10] = 0; " "6#/&%\"aG6$\"#6\"#5\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] =0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] \+ = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4 ]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[10 ]=0" "6#/&%#b*G6#\"#5\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*` [11] = 16/25;" "6#/&%#b*G6#\"#6*&\"#;\"\"\"\"#D!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We also set " }{XPPEDIT 18 0 "b[11] = \+ 0;" "6#/&%\"bG6#\"#6\"\"!" }{TEXT -1 66 ", so that the order 7 scheme \+ can be regarded as a 11 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 14 equations for the 14 unkno wn coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "e16 := \+ \{c[11]=1,b[11]=0,a[11,2]=0,a[11,10]=0,`b*`[2]=0,`b*`[3]=0,`b*`[4]=0,` b*`[10]=0,`b*`[11]=16/25\}:\ne17 := `union`(e15,e16):\n`eqns*` := subs (e17,`cdns*`):\nnops(%);\nindets(`eqns*`);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<0&%\"aG6$\" #6\"\"&&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"\"$&F% 6$F'\"\"%&F%6$F'\"\"\"&%#b*G6#F=&F?6#F(&F?6#F+&F?6#F.&F?6#F1&F?6#F4" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e18 := solve(\{op(`eqns*` )\}):\ninfolevel[solve] := 0:\ne19 := `union`(e17,e18):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4975 "e19 := \{a[8,2] = 0, \+ a[9,2] = 0, a[4,2] = 0, a[11,5] = 581599188224381204095672213436165793 6380202840699902125/35859517374517649352841965415945439161137592950560 49408, b[7] = 63971129215453255994803266454987246348244877720437265765 7897547/20116014255015886224276044430169119323710082114357973829916601 60, a[5,1] = 311712996187/3366699277500, a[7,6] = -2267236012274582026 887011527795707923286/477373186945391376596029499088911158349111, c[6] = 244/273, b[2] = 0, b[3] = 0, c[10] = 1, b[5] = 89929293580433773826 711625625/175034206955063764470139052544, a[11,7] = 571917047290975925 3773842353503580127446876084974327902851679027589/60288487786636086054 8455552363583579732315378471940789964794317568, a[8,3] = 2589291077360 7576734414005208753/10529291320508273201747374752000, b[6] = 439656793 17071687650068867/11554641620922992752054730, b[1] = 17499592155841793 /301627899585331605, a[8,4] = -613522852101531567576566644397/81457670 7522236069963552640000, a[6,2] = 0, a[7,2] = 0, a[6,5] = 1039597125182 00500/83969153915827131, `b*`[7] = 68348255286882236058373146453526734 0427333118635079/2165562941739167798045518397620231220712261142369860, a[5,2] = 0, `b*`[6] = 15979368227021270883046053/31874873437028945522 90960, a[6,4] = -6255616188978805360/5200809295819122531, a[8,7] = -74 24205995886059956357359508852528825590423/4378197291051651257486258563 957705093888000, a[11,8] = 1203167959926676490500/12519626510103888283 9767, c[11] = 1, `b*`[9] = -169578451828457839968131453/57771383797990 433816368200, a[11,6] = 28274787710189875324117065188869812987/6120730 2919331402809078602199860439808, a[6,3] = 15089501252616323812/1445723 1345452578137, a[10,7] = 350118297598529217748944662857001287074910707 52921128030483969446296/3718044938427279904428695460050071040231420419 464687750808112491477, c[3] = 93/479, a[4,1] = -432520/5401533, a[8,5] = 8122715759499790031095209746537365648167969/83629654355930037228367 00426272322940782592, a[5,3] = 996040375149076/5096677701243375, a[5,4 ] = 344546810496217/1096059720697500, c[4] = 110/241, b[9] = -21099698 2649291857501475159/99036657939412172256631200, a[6,1] = -228/1231, a[ 9,5] = 489177972756706421411944299196898256388191463578294580375/31929 1201100986622665402847780351230046376607122311276442, a[9,7] = 1194448 2045828658346298806908238121417524848379605193796762013/34123763173112 98392350722812851096867170775526794634954003706, a[9,1] = -15383168980 0158592252487409542055462465307763/51771386973406002359953310276391942 4014608068, a[10,3] = -1111042984674821389643230968704444252/147519964 335786292056835317593798019, a[7,5] = 40837242552907904940455804285850 036846494300434360648500/734372562662145990989587253914642887902719996 408150162943, c[7] = 222785849/1109145221, a[9,4] = -57633497085298645 064371869898744017/30963779845911010423286154735455876, a[8,1] = -1485 0375572183825804482268314325533/131869870061161226458875010296840000, \+ a[4,3] = 2897950/5401533, a[7,1] = 20784021508544698439004341302945788 046687497048823/333809007739010923825478541340748044990946025336204, a [3,2] = 864900/229441, a[7,3] = 59087076776381536334498651011356345416 23011524075277164/3154555408778125545630482740453787587569225386801153 4003, a[3,1] = -820353/229441, a[10,1] = -4073447364725702015608831305 4846628592379/112331307637222010640347265020564130793284, a[7,4] = -48 597609826899455036511090495924959702383070843517685/488091226723681375 828622960423688756917868034622078292, c[8] = 17/20, a[10,8] = -2632810 5087975815319887872000/966150763091150023719018517747, a[9,6] = -11856 79300358377756233489606885369481439862/1843860175796863816631526170465 7797344684897, c[5] = 509/845, c[9] = 171/181, a[10,4] = -521306348327 7854003640774248205317/2282514999055829721733809658986516, a[10,5] = 4 344125602637174263526484423723747582563035478367000/281969717471900849 4099755685231013933352915992886809, a[8,6] = -429371785625952516803610 31143/2273295839722617724238904704000, a[2,1] = 1/200, b[10] = 7077278 529846113/15546034297382400, a[9,8] = 103973108798709992837929472000/7 62442644104164301247937014633, a[10,6] = 14281243816995595096936801178 8034519784/281984196771808918111529577339005012981, a[9,3] = -13349842 16509015564184248020843701584/667067642557939663406130018977478253, b[ 4] = 0, b[8] = -43614217354789120000/21595629876813903267, a[10,2] = 0 , c[2] = 1/200, a[10,9] = -1198575787406124991100974435169141280/46356 48233480921649154730958571956313, b[11] = 0, a[11,2] = 0, a[11,10] = 0 , `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[10] = 0, `b*`[11] = 16/2 5, `b*`[8] = -132403130312946592000/50389803045899107623, a[11,3] = -8 070051946566656273895465364457/1073987714691843866578232224704, a[11,9 ] = -4684487619570658562655565/18486842815356938821237824, a[11,1] = - 4549513437278902549066809605929617243853/11637130531712243439602356681 247846961408, a[11,4] = -10059066143857143308915950403/415434120836340 3956814118464, `b*`[5] = 532400445239427668047625/99415103005193432200 8696, `b*`[1] = 94288552637982437/1608682131121768560\}:" }}}{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 0 137 "subs(e19,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(12-i)],i=2. .11),\n[`b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11)]])):\nevalf[ 6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'++]!\")F (%!GF+F+F+F+F+F+F+F+F+7.$\"'aT>!\"'$!'WvN!\"&$\"'gpPF2F+F+F+F+F+F+F+F+ F+7.$\"'KkXF/$!'O2!)!\"($\"\"!F<$\"'0l`F/F+F+F+F+F+F+F+F+7.$\"'nBgF/$ \"'re#*F:F;$\"'Ha>F/$\"']VJF/F+F+F+F+F+F+F+7.$\"'tP*)F/$!':_=F/F;$\"'t V5F2$!'#G?\"F2$\"'2Q7F2F+F+F+F+F+F+7.$\"'j3?F/$\"'KEiF:F;$\"'2t=F/$!'n c**F:$\"'$3c&F:$!'S\\ZF*F+F+F+F+F+7.$\"'++&)F/$!'9E6F/F;$\"'8fCF2$!'!= `(F/$\"'s7(*F/$!'w))=F:$!'s&p\"F2F+F+F+F+7.$\"'^Z%*F/$!'OrHF/F;$!'F,?F 2$!'Kh=F2$\"'2K:F2$!'UIkF:$\"'M+NF2$\"'oj8F/F+F+F+7.$\"\"\"F<$!'GEOF/F ;$!'[JvF2$!'\"RG#F2$\"'kS:F2$\"'bk]F/$\"'t;%*F2$!'0DFF:$!'c&e#F/F+F+7. F[q$!'[4RF/F;$!'59vF2$!'M@CF2$\"')=i\"F2$\"'^>YF/$\"'M'[*F2$\"'D5'*F*$ !''R`#F/F;F+7.%\"bG$\"'s,eF:F;F;F;$\"'\"y8&F/$\"'-0QF2$\"'6!=$F/$!'f>? F2$!'\\I@F2$\"'Z_XF/F;7.%#b*G$\"'BheF:F;F;F;$\"'Lb`F/$\"';8]F2$\"'9cJF /$!'eFEF2$!'MNHF2F;$\"'++kF/Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "subs(e19,matrix([seq ([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..5)]));print(``);\nfor ii fr om 6 to 11 do\n print(c[ii]=subs(e19,c[ii]));print(``); \n for jj \+ to ii-1 do\n print(a[ii,jj]=subs(e19,a[ii,jj]));\n end do:\n \+ print(`_________________________________`);\nend do:print(``);\nfor ii to 11 do\n print(b[ii]=subs(e19,b[ii]));\nend do:\nprint(`_________ ________________________`);print(``);\nfor ii to 11 do\n print(`b*`[ ii]=subs(e19,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7(#\"\"\"\"$+#F(%!GF+F+F+7(#\"#$*\"$z%#!'`.#)\"'T%H##\"'+ \\')F2F+F+F+7(#\"$5\"\"$T##!'?DV\"(L:S&\"\"!#\"(]z*GF;F+F+7(#\"$4&\"$X )#\"-(='*Hr6$\".+vF*pmLF<#\"0w!\\^PSg**\"1vLC,xn'4&#\"0eH43?&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&#\"3+0?=DrfR5\"2Jr#e\"R:pR)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"*\\eyA#\"+@_946" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"\"#\"SB)[q\\(oY!)yXHITV+R%)pW&3:-%y?\"T/iLDg%4*\\/[ 2MT&ya#Q#4,Rx+4QL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"XkrF vS_6IiTXjN65l)\\Mj`\"Qwn2(3f\"Y.S`6!oQD#pve(y`/u#[IcaD\"y(3abaJ" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%#!V&oF!z)Gk9RD(e*)4*f9iEcsVt" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!I'GBzq&zF:,()o-#euA,OsE#\"K6\"\\$e 6\"*)3*\\Hgfw8RXp=ttZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___________ ______________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \")#\"#<\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!DLbK9$oA[/e#Q=sbP][\"\"E++%oH5] ()ekAh61q)p=8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\"A`(3_+9Wt wvgt2\"H*e#\"A+?vutu,KF30K\"HH0\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\")\"\"%#!?(RWmcwvcJ:5_G_8'\"?++k_N'*pgBAvqwX\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#\"Lpz;[cOPlu4_4J+z*\\fdr A\")\"L#f#ySHKsiU+n$Gs.IfNa'HO)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"'#!>V6.h.o^_fi&yr$H%\"@+Sq/*QUsyV" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B________ _________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"*#\"$r\"\"$\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!NjxIlCYb?a4u[_Afe,!)*o JQ:\"No!3Y,C%>Rw-J`*fB+1M(pQr<&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"$#!F%e,P%3-[U=kb,4l@%)\\L\"\"E`#yu(*=+81MmRzbUw1n'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!D)Qc#)*o>*HW>T@kqcF(z<*[\"ZUkF6B72mPY+B^.yZGSlEi')4 5,7H>$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!Li)R9[p `)og*[Livx$e.Izc=\"\"M(*[oWtzdYqh_Jm\"Q'ozv,'Q%=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"in8?w'z$>0'z$[[_<97Q#3p!))HY$e'Ge /#[W>\"\"hn1P+a\\j%zEbxqr'o4^G\"Gs]BR)H6tJwBT$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"?+?ZHz$G**4()z3J(R5\"?LY,PzC,V;/T kUCw" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B____________________________ _____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#5\"\"\"#!JzBfGm%[08$)3c,-dsktWtS\"K%G$zITc?]EZ.k5?APwIJB6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$#!F_UW/(o4BV'*Q@[n%)H/66\" E>!)z$f<`$o0#H'yNV'*>v9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"%#!C<`?[UxSO+ayF$[jI@&\"C;l)*e'4Qt@(He0**\\^#G#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#\"U+qOya.jDeZPsBW[ENEurj-c7WV\" U4o)G*f\"HNLR,J_ob(*4%\\3!>Z<(p>G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#5\"\"'#\"H%y>X.)y6!o$p4&f&*p\"QC\"G9\"H\")H,0!RtdH:6=*3=x'> %)>G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"_o'HY%pR[ I!G6#Hvq5\\2(G,q&GmW*[x@H&)f(H=,N\"^ox9\\7\"33v(ok%>/UJ-/r+0gapGW!*zsU Q\\/=P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!>+?(y))> `\"e(z30\"Gj#\"?Zx^=!>P-]64j2:m*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#5\"\"*#!F!GT\"p^Vu45\"*\\71uyv&)>\"\"F8j&>de4ta\"\\;#4[L#[cj %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_______________________________ __G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"\"#!I`QC<'Hfg4o1\\D!*ysV8&\\X\"J39'p%yC\"ocBgRMC7<`Irj6" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$#!@dWOla*QFcmcY>0q!)\"@/ZAK#yl 'Q%=p9x)R2\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!>. /&f\"*3L9dQ9m!f+\"\"=k%=T\"o&RSj$37MaT" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#\"XD@!**pSG?!QOzlhV8An&4/7QC#)=*f\"e\"X3%\\g0 &HfP6;Ra%fTl>%GN\\w^ut^fe$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"'#\"G()H\")p))=lq6C`()*=5xyu#G\"G3)R/')*>-'y!4GSJ$>HI27'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"^o*eF!z;&G!zKu\\3 woWF,e.NNUQx`#f(4HZq\">d\"]oovJ%zk**yS>Zy`JK(zNejBbb%[0'3Omy([)Gg" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"7+0\\wm#*fz;.7\"9 n(RG))Q55li>D\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*# !:lbli&e1d>w[%o%\";CyB@)QpN:G%o[=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___________ ______________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"2$zTe:#f*\\<\"30;L&e **yi,$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"&#\">Dci6n#QxL/e$HH**)\"?WD0R,ZkP1bp?M]<" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\";n)o+l(orqJzc'R%\";IZ0_F*H#4iTY b6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"jnZv*yldEP/sx[C [jC()\\XmK![*fD`a@H6(R'\"[og,m\"*HQ(zN9@35PK>\"p,VWgFCi)e,bU,;,#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!5++7*yat@9O%\"5nK!R\"o ()Hcf@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#!0I5:%**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"'#\";`g/$)3F@qAo$zf\"\":g4H_X*GqVt[(=$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"Tz]j=JLF/Mn_`k9t$egB#)oGb#[$o\"Ug)pB 9hAr?7B?wR=b/)zn\"R<%Hcl@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\")#!6+?fYHJIJSK\"\"5Bw5**e/.)*Q]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"*#!<`98o*RyXG=Xy&p\"\";+#oj\"QV!*zz$Qrx&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"#;\"#D" }}}{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 "Check" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 120 "`RK6_11eqs*` := subs(b=`b*`,OrderConditions(6,11,' expanded')):\nsubs(e19,`RK6_11eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 52 "#---------------------------------------------------" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#-------------------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the e mbedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffi cients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4903 "ee := \{c[2]=1/200,\nc[3]= 93/479,\nc[4]=110/241,\nc[5]=509/845,\nc[6]=244/273,\nc[7]=222785849/1 109145221,\nc[8]=17/20,\nc[9]=171/181,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1 /200,\na[3,1]=-820353/229441,\na[3,2]=864900/229441,\na[4,1]=-432520/5 401533,\na[4,2]=0,\na[4,3]=2897950/5401533,\na[5,1]=311712996187/33666 99277500,\na[5,2]=0,\na[5,3]=996040375149076/5096677701243375,\na[5,4] =344546810496217/1096059720697500,\na[6,1]=-228/1231,\na[6,2]=0,\na[6, 3]=15089501252616323812/14457231345452578137,\na[6,4]=-625561618897880 5360/5200809295819122531,\na[6,5]=103959712518200500/83969153915827131 ,\na[7,1]=20784021508544698439004341302945788046687497048823/\n \+ 333809007739010923825478541340748044990946025336204,\na[7,2]=0,\na[7,3 ]=5908707677638153633449865101135634541623011524075277164/\n 315 45554087781255456304827404537875875692253868011534003,\na[7,4]=-485976 09826899455036511090495924959702383070843517685/\n 488091226723 681375828622960423688756917868034622078292,\na[7,5]=408372425529079049 40455804285850036846494300434360648500/\n 7343725626621459909895 87253914642887902719996408150162943,\na[7,6]=-226723601227458202688701 1527795707923286/477373186945391376596029499088911158349111,\na[8,1]=- 14850375572183825804482268314325533/1318698700611612264588750102968400 00,\na[8,2]=0,\na[8,3]=25892910773607576734414005208753/10529291320508 273201747374752000,\na[8,4]=-613522852101531567576566644397/8145767075 22236069963552640000,\na[8,5]=8122715759499790031095209746537365648167 969/8362965435593003722836700426272322940782592,\na[8,6]=-429371785625 95251680361031143/2273295839722617724238904704000,\na[8,7]=-7424205995 886059956357359508852528825590423/437819729105165125748625856395770509 3888000,\na[9,1]=-153831689800158592252487409542055462465307763/517713 869734060023599533102763919424014608068,\na[9,2]=0,\na[9,3]=-133498421 6509015564184248020843701584/667067642557939663406130018977478253,\na[ 9,4]=-57633497085298645064371869898744017/3096377984591101042328615473 5455876,\na[9,5]=48917797275670642141194429919689825638819146357829458 0375/\n 31929120110098662266540284778035123004637660712231127644 2,\na[9,6]=-1185679300358377756233489606885369481439862/18438601757968 638166315261704657797344684897,\na[9,7]=119444820458286583462988069082 38121417524848379605193796762013/\n 3412376317311298392350722812 851096867170775526794634954003706,\na[9,8]=103973108798709992837929472 000/762442644104164301247937014633,\na[10,1]=-407344736472570201560883 13054846628592379/112331307637222010640347265020564130793284,\na[10,2] =0,\na[10,3]=-1111042984674821389643230968704444252/147519964335786292 056835317593798019,\na[10,4]=-5213063483277854003640774248205317/22825 14999055829721733809658986516,\na[10,5]=434412560263717426352648442372 3747582563035478367000/\n 2819697174719008494099755685231013933 352915992886809,\na[10,6]=142812438169955950969368011788034519784/2819 84196771808918111529577339005012981,\na[10,7]=350118297598529217748944 66285700128707491070752921128030483969446296/\n 371804493842727 9904428695460050071040231420419464687750808112491477,\na[10,8]=-263281 05087975815319887872000/966150763091150023719018517747,\na[10,9]=-1198 575787406124991100974435169141280/463564823348092164915473095857195631 3,\na[11,1]=-4549513437278902549066809605929617243853/1163713053171224 3439602356681247846961408,\na[11,2]=0,\na[11,3]=-807005194656665627389 5465364457/1073987714691843866578232224704,\na[11,4]=-1005906614385714 3308915950403/4154341208363403956814118464,\na[11,5]=58159918822438120 40956722134361657936380202840699902125/\n 358595173745176493528 4196541594543916113759295056049408,\na[11,6]=2827478771018987532411706 5188869812987/61207302919331402809078602199860439808,\na[11,7]=5719170 472909759253773842353503580127446876084974327902851679027589/\n \+ 602884877866360860548455552363583579732315378471940789964794317568,\n a[11,8]=1203167959926676490500/125196265101038882839767,\na[11,9]=-468 4487619570658562655565/18486842815356938821237824,\na[11,10]=0,\n\nb[1 ]=17499592155841793/301627899585331605,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[ 5]=89929293580433773826711625625/175034206955063764470139052544,\nb[6] =43965679317071687650068867/11554641620922992752054730,\nb[7]=63971129 2154532559948032664549872463482448777204372657657897547/\n 2011601 425501588622427604443016911932371008211435797382991660160,\nb[8]=-4361 4217354789120000/21595629876813903267,\nb[9]=-210996982649291857501475 159/99036657939412172256631200,\nb[10]=7077278529846113/15546034297382 400,\nb[11]=0,\n\n`b*`[1]=94288552637982437/1608682131121768560,\n`b*` [2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=532400445239427668047625/99415 1030051934322008696,\n`b*`[6]=15979368227021270883046053/3187487343702 894552290960,\n`b*`[7]=68348255286882236058373146453526734042733311863 5079/\n 2165562941739167798045518397620231220712261142369860,\n `b*`[8]=-132403130312946592000/50389803045899107623,\n`b*`[9]=-1695784 51828457839968131453/57771383797990433816368200,\n`b*`[10]=0,\n`b*`[11 ]=16/25\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[7,11];" "6#&%\"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose components are the principal \+ error terms of the 11 stage, order 7 scheme (the error terms of order \+ 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6,11]; " "6#&%#T*G6$\"\"'\"#6" }{TEXT -1 146 " denote the vector whose compo nents are the principal error terms of the embedded 11 stage, order 6 \+ scheme (the error terms of order 7) and let " }{XPPEDIT 18 0 "`T*`[7, 11];" "6#&%#T*G6$\"\"(\"#6" }{TEXT -1 99 " denote the vector whose co mponents are the error terms of order 8 of the embedded order 6 scheme ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Deno te the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[7,11])) ;" "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,11]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[7,11]));" "6#-%$absG6#-F$6 #&%#T*G6$\"\"(\"#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[8] = abs(abs(T[7,11]));" "6#/&% \"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,11]))/abs(abs(`T*`[6,11]));" "6#/ &%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\"#6\"\"\"-F*6#-F*6#&F/6$\" \"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7 ,11]-T[7,11]))/abs(abs(`T*`[6,11]));" "6#/&%\"CG6#\"\")*&-%$absG6#-F*6 #,&&%#T*G6$\"\"(\"#6\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"'F3F8 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[8];" "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[8];" "6#&%\"CG6#\"\")" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errterms7_1 1 := PrincipalErrorTerms(7,11,'expanded'):\n`errterms7_11*` :=subs(b=` b*`,PrincipalErrorTerms(7,11,'expanded')):\n`errterms6_11*` := subs(b= `b*`,PrincipalErrorTerms(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add( evalf(subs(ee,`errterms7_11*`[i]))^2,i=1..nops(`errterms7_11*`))):\nsd nB := sqrt(add(evalf(subs(ee,`errterms6_11*`[i]))^2,i=1..nops(`errterm s6_11*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms7_11*`[i])-subs( ee,errterms7_11[i])))^2,i=1..nops(errterms7_11))):\n'B[8]'= evalf[8](s nmB/sdnB);\n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")4D$4#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"CG6#\"\")$\")(=J8#!\"(" }}}{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 4903 "ee := \{c[2]=1/200,\nc[3]=93/479,\nc[4]=110/241,\nc[5]=509/845, \nc[6]=244/273,\nc[7]=222785849/1109145221,\nc[8]=17/20,\nc[9]=171/181 ,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/200,\na[3,1]=-820353/229441,\na[3,2] =864900/229441,\na[4,1]=-432520/5401533,\na[4,2]=0,\na[4,3]=2897950/54 01533,\na[5,1]=311712996187/3366699277500,\na[5,2]=0,\na[5,3]=99604037 5149076/5096677701243375,\na[5,4]=344546810496217/1096059720697500,\na [6,1]=-228/1231,\na[6,2]=0,\na[6,3]=15089501252616323812/1445723134545 2578137,\na[6,4]=-6255616188978805360/5200809295819122531,\na[6,5]=103 959712518200500/83969153915827131,\na[7,1]=207840215085446984390043413 02945788046687497048823/\n 3338090077390109238254785413407480449 90946025336204,\na[7,2]=0,\na[7,3]=59087076776381536334498651011356345 41623011524075277164/\n 3154555408778125545630482740453787587569 2253868011534003,\na[7,4]=-4859760982689945503651109049592495970238307 0843517685/\n 4880912267236813758286229604236887569178680346220 78292,\na[7,5]=4083724255290790494045580428585003684649430043436064850 0/\n 734372562662145990989587253914642887902719996408150162943, \na[7,6]=-2267236012274582026887011527795707923286/4773731869453913765 96029499088911158349111,\na[8,1]=-14850375572183825804482268314325533/ 131869870061161226458875010296840000,\na[8,2]=0,\na[8,3]=2589291077360 7576734414005208753/10529291320508273201747374752000,\na[8,4]=-6135228 52101531567576566644397/814576707522236069963552640000,\na[8,5]=812271 5759499790031095209746537365648167969/83629654355930037228367004262723 22940782592,\na[8,6]=-42937178562595251680361031143/227329583972261772 4238904704000,\na[8,7]=-7424205995886059956357359508852528825590423/43 78197291051651257486258563957705093888000,\na[9,1]=-153831689800158592 252487409542055462465307763/517713869734060023599533102763919424014608 068,\na[9,2]=0,\na[9,3]=-1334984216509015564184248020843701584/6670676 42557939663406130018977478253,\na[9,4]=-576334970852986450643718698987 44017/30963779845911010423286154735455876,\na[9,5]=4891779727567064214 11944299196898256388191463578294580375/\n 3192912011009866226654 02847780351230046376607122311276442,\na[9,6]=-118567930035837775623348 9606885369481439862/18438601757968638166315261704657797344684897,\na[9 ,7]=11944482045828658346298806908238121417524848379605193796762013/\n \+ 3412376317311298392350722812851096867170775526794634954003706,\n a[9,8]=103973108798709992837929472000/762442644104164301247937014633, \na[10,1]=-40734473647257020156088313054846628592379/11233130763722201 0640347265020564130793284,\na[10,2]=0,\na[10,3]=-111104298467482138964 3230968704444252/147519964335786292056835317593798019,\na[10,4]=-52130 63483277854003640774248205317/2282514999055829721733809658986516,\na[1 0,5]=4344125602637174263526484423723747582563035478367000/\n 28 19697174719008494099755685231013933352915992886809,\na[10,6]=142812438 169955950969368011788034519784/281984196771808918111529577339005012981 ,\na[10,7]=35011829759852921774894466285700128707491070752921128030483 969446296/\n 37180449384272799044286954600500710402314204194646 87750808112491477,\na[10,8]=-26328105087975815319887872000/96615076309 1150023719018517747,\na[10,9]=-1198575787406124991100974435169141280/4 635648233480921649154730958571956313,\na[11,1]=-4549513437278902549066 809605929617243853/11637130531712243439602356681247846961408,\na[11,2] =0,\na[11,3]=-8070051946566656273895465364457/107398771469184386657823 2224704,\na[11,4]=-10059066143857143308915950403/415434120836340395681 4118464,\na[11,5]=5815991882243812040956722134361657936380202840699902 125/\n 3585951737451764935284196541594543916113759295056049408, \na[11,6]=28274787710189875324117065188869812987/612073029193314028090 78602199860439808,\na[11,7]=571917047290975925377384235350358012744687 6084974327902851679027589/\n 6028848778663608605484555523635835 79732315378471940789964794317568,\na[11,8]=1203167959926676490500/1251 96265101038882839767,\na[11,9]=-4684487619570658562655565/184868428153 56938821237824,\na[11,10]=0,\n\nb[1]=17499592155841793/301627899585331 605,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=89929293580433773826711625625/17 5034206955063764470139052544,\nb[6]=43965679317071687650068867/1155464 1620922992752054730,\nb[7]=6397112921545325599480326645498724634824487 77204372657657897547/\n 201160142550158862242760444301691193237100 8211435797382991660160,\nb[8]=-43614217354789120000/215956298768139032 67,\nb[9]=-210996982649291857501475159/99036657939412172256631200,\nb[ 10]=7077278529846113/15546034297382400,\nb[11]=0,\n\n`b*`[1]=942885526 37982437/1608682131121768560,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b* `[5]=532400445239427668047625/994151030051934322008696,\n`b*`[6]=15979 368227021270883046053/3187487343702894552290960,\n`b*`[7]=683482552868 822360583731464535267340427333118635079/\n 21655629417391677980 45518397620231220712261142369860,\n`b*`[8]=-132403130312946592000/5038 9803045899107623,\n`b*`[9]=-169578451828457839968131453/57771383797990 433816368200,\n`b*`[10]=0,\n`b*`[11]=16/25\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability f unction R for the 11 stage, order 7 scheme is given as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(7, 11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F) *&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$ )F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&# \"FRuM#flozinR#eC]wHL0\"\"J;3Ve&3^ocJ?+\\k;9sX?-ZF)*$)F'\"\")F)F)F)*&# \"Ezc!=2+4ZdxZ)[A'\\)>aB\"L?:+\\\"GgE#f-mDf4/Yf[;#G:F)*$)F'\"\"*F)F)! \"\"*&#\"C*f\\-x_Bj^r(Q8sS:^=\"Ic4$3f46z`!\\e/2R#3o5'QOF)*$)F'\"#5F)F) FZ" }}}{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,$\" \"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=-1,z=-4.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#z0G$!+F\"[Tq%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-4.7):\np1 := plo t([R(z),-1],z=-5.29..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3] ,style=point,symbol=[circle,cross,diamond],color=black):\np3 := plot([ [z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]( [p1,p2,p3],view=[-5.29..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7W7$ $!3/++++++!H&!#<$!3v-[T,`^@RF*7$$!3'RLeWA1qA&F*$!3VF5o4%z*>MF*7$$!3+nm \"*[C,k^F*$!37tPv,#Rh(HF*7$$!3V]Pk,>?4^F*$!3.;3R'HnAj#F*7$$!3)H$3Pa8Ra ]F*$!3Q8BteG+CBF*7$$!3K]i]h6v#*\\F*$!3*[QQ:J%y:?F*7$$!3wm;ko46J\\F*$!3 l+&ziYpTu\"F*7$$!3Gnmc%=i!p[F*$!3(G@y'e.z.:F*7$$!3#pm\"\\+M,2[F*$!3)fO .QarHH\"F*7$$!3cL3UYc]$o%F*$!3u*)G4H-\\([*!#=7$$!3%o;/$[\")**oXF*$!3u$ Gc?&=lOqFZ7$$!3!)*\\PlEK/X%F*$!3sU)\\AW)y\"4&FZ7$$!3WmT+i7\"yK%F*$!3qc i*Q+(yxNFZ7$$!3j*\\(e/Ne0UF*$!3Oa.(f'pZdCFZ7$$!3aLLQKs&)zSF*$!3gV=e2;8 8;FZ7$$!3gm;Rju6pRF*$!3swT1=4al5FZ7$$!3c***\\@r]W%QF*$!3p;8*H5fC8'!#>7 $$!3#****\\s1s#>PF*$!3QF_6F'p#zGFhp7$$!3a***\\yCR')f$F*$!3!>(pQk<.zh!# ?7$$!3UmT+\"4$4*[$F*$\"3v9@]KyF-\"*Fcq7$$!3cLL)\\6K)eLF*$\"3IuON!)**[$ G#Fhp7$$!35LL$>)R[[KF*$\"355/&yi2!*>$Fhp7$$!37+vBqG7?JF*$\"35dwd7is&4% Fhp7$$!3SLLVO:]1IF*$\"3::!*\\EgF8[Fhp7$$!3!**\\P_#4%=)GF*$\"3![N))RJO2 e&Fhp7$$!3!**\\7PaMJw#F*$\"3g*)pvQ+NOjFhp7$$!3CL$3x)oFREF*$\"3sb_o'y# \\'=(Fhp7$$!3DL3K&*o`DDF*$\"3ARBM*e;v/)Fhp7$$!3pm;zZH&GS#F*$\"31N!z'=? +&3*Fhp7$$!3omT+1)=aF#F*$\"3+#pC\"3CYI5FZ7$$!31+D'4,([k@F*$\"35%*pm@k> ]6FZ7$$!3Hm;CLwnW?F*$\"3!=H._)4g&H\"FZ7$$!3,++qAG!4#>F*$\"3W$[5n)edl9F Z7$$!3-+]F=L\")*z\"F*$\"3?*z\"yvDw`;FZ7$$!35+vV'*Hl#o\"F*$\"3!=yC%*\\X !f=FZ7$$!33+]KMkc_:F*$\"39xGB%)Q9<@FZ7$$!3nLLVsynN9F*$\"3i)p'*p*3ezBFZ 7$$!3$)***\\KQu3J\"F*$\"3o\"QG'*\\_ep#FZ7$$!3'p;/E(Qy(>\"F*$\"3PE(yqHC '=IFZ7$$!3o***\\L2YT2\"F*$\"3(pMt2a!*eT$FZ7$$!3)*Q$3dRD\"y&*FZ$\"3Y`rg S9JPQFZ7$$!3-'*\\(yw/@O)FZ$\"3Ba!*zS[\\LVFZ7$$!3eLL$eLAK<(FZ$\"3W@$\\ \"=Od!)[FZ7$$!3j/]P4EdGfFZ$\"3LX%o=tcu_&FZ7$$!3=fmm#*p#)HZFZ$\"3f\\yy) [39B'FZ7$$!3)ym;4YOR]$FZ$\"3*4d>EY2T/(FZ7$$!3+K$3<#p>)G#FZ$\"3sYI#Q]>Z &zFZ7$$!3M#****pae5<\"FZ$\"35STt8.\"\\*))FZ7$$\"3/tm;%*43$4\"Fhp$\"3#p S9&p2*4,\"F*7$$\"3![LL8H$[a7FZ$\"3qP%G/fcO8\"F*7$$\"3%3+Dw*)yaZ#FZ$\"3 ?`G%>q!)3G\"F*7$$\"3*=+v$)[FTk$FZ$\"3.*>c=Jo'R9F*7$$\"3!************** *[FZ$\"3?sBg1iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Ff\\lFe\\l-F$ 6$7S7$F($!\"\"Ff\\l7$F3F[]l7$F=F[]l7$FGF[]l7$FQF[]l7$FVF[]l7$FfnF[]l7$ F[oF[]l7$F`oF[]l7$FeoF[]l7$FjoF[]l7$F_pF[]l7$FdpF[]l7$FjpF[]l7$F_qF[]l 7$FeqF[]l7$FjqF[]l7$F_rF[]l7$FdrF[]l7$FirF[]l7$F^sF[]l7$FcsF[]l7$FhsF[ ]l7$F]tF[]l7$FbtF[]l7$FgtF[]l7$F\\uF[]l7$FauF[]l7$FfuF[]l7$F[vF[]l7$F` vF[]l7$FevF[]l7$FjvF[]l7$F_wF[]l7$FdwF[]l7$FiwF[]l7$F^xF[]l7$FcxF[]l7$ FhxF[]l7$F]yF[]l7$FbyF[]l7$FgyF[]l7$F\\zF[]l7$FazF[]l7$FfzF[]l7$F[[lF[ ]l7$F`[lF[]l7$Fe[lF[]l7$Fj[lF[]l-F_\\l6&Fa\\lFe\\lFe\\lFb\\l-F$6&7#7$$ !3&)*****p7[Tq%F*F[]l-%'SYMBOLG6#%'CIRCLEG-F_\\l6&Fa\\lFf\\lFf\\lFf\\l -%&STYLEG6#%&POINTG-F$6&Fa`l-Ff`l6#%&CROSSGFi`lF[al-F$6&Fa`l-Ff`l6#%(D IAMONDGFi`lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLORG6&Fa\\lFe\\l$\"\"&F\\]lF e\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\" z6\"Q!F_cl-Fgbl6#%(DEFAULTG-%%VIEWG6$;$!$H&!\"#$\"#\\Fjcl;$!$Z\"Fjcl$ \"$Z\"Fjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following pic ture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1096 "R := z -> 1+z+1/2*z^2+1/6* z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 1053329765024582396 762796865592347439/47022045721416644900203156685108558430816*z^8-\n \+ 235419849622488477757470900071805679/1528216485946040959256602592266 028149001520*z^9-\n 1851154072133877151632352770249599/36386106808 23907045849053791109590830956*z^10:\npts := []: z0 := 0: tt := 0: \nwh ile tt<=281/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := \+ zz:\n if (3/4<=tt and tt<=33/20) or (53/20<=tt and tt<=71/20) or\n \+ (209/20<=tt and tt<=227/20) or (247/20<=tt and tt<=53/4) then\n \+ hh := 1/40\n else \n hh := 1/20\n end if;\n tt := tt+hh; \n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color= COLOR(RGB,.5,.12,.12)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i ],[-2.35,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLO R(RGB,1,.23,.23)):\np3 := plot([[[-5.39,0],[1.19,0]],[[0,-4.59],[0,4.5 9]]],color=black,linestyle=3):\nplots[display]([p||(1..3)],view=[-5.39 ..1.19,-4.59..4.59],font=[HELVETICA,9],\n labels=[`Re(z)` ,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 484 535 535 {PLOTDATA 2 "6+-%'CURVESG6$7^al7$$\"\"!F)F(7$F($ \"3++++Fjzq:!#=7$$\"35+++dO]kB!#F$\"3)******Rl#fTJF-7$$\"3=+++LNxNj!#E $\"3.+++&)*)Q7ZF-7$$\"30+++73j6n!#D$\"3S+++n`=$G'F-7$$\"3A+++F)R&yF-7$$\"3!******>t')[*>!#B$\"3H+++G$zZU*F-7$$\"3l*****fI ..W(FI$\"3/+++Hxb*4\"!#<7$$\"3'******>ONxN#!#A$\"3%******\\3PmD\"FQ7$$ \"3#)*****H/-Ae'FU$\"3********)\\8PT\"FQ7$$\"3))******Q(o$e;!#@$\"35++ +_%y2d\"FQ7$$\"3-+++,koLQFjn$\"3#******pA3ys\"FQ7$$\"38+++JtlD#)Fjn$\" 37+++-^u%)=FQ7$$\"31+++L8\"3l\"!#?$\"37+++ZAYT?FQ7$$\"3<+++!*4(G6$Fjo$ \"3))*****>)Qq(>#FQ7$$\"33+++Mj8BbFjo$\"3;+++yU*HN#FQ7$$\"35+++fK!z=(F jo$\"38+++fc/ICFQ7$$\"3c+++5e)[?*Fjo$\"39+++,I_1DFQ7$$\"31+++$*)H\"f6! 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3')*************e%FQ7$F($\"3')*************e%FQFihwF\\iw-%*AXESSTYLEG6 #%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F^j w6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$R&Fgfp$\"$>\"Fgf p;$!$f%Fgfp$\"$f%Fgfp" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 " interval of absolute stability" }{TEXT -1 89 " (or stability interval ) is the intersection of the stability region with the real line." }} {PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.70415, 0];" "6#7$,$-%&FloatG6$ \"':/Z!\"&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve ho rizontally by taking the 11th root of the real part of points along th e curve. In this way we see that the largest interval on the nonnegati ve imaginary axis that contains the origin and lies inside the stabili ty region is " }{XPPEDIT 18 0 "[0, 3.65];" "6#7$\"\"!-%&FloatG6$\"$l$ !\"#" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 587 "R := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7+\n 1053329765024582 396762796865592347439/47022045721416644900203156685108558430816*z^8-\n 235419849622488477757470900071805679/1528216485946040959256602592 266028149001520*z^9-\n 1851154072133877151632352770249599/36386106 80823907045849053791109590830956*z^10:\nDigits := 20:\npts := []: z0 : = 0:\nfor ct from 0 to 140 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z =z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nen d do:\nplot(pts,color=COLOR(RGB,1,0,0),thickness=2,font=[HELVETICA,9]) ;\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 265 310 310 {PLOTDATA 2 "6(-%'CURVESG6#7is7$$\"\"!F)F(7$$\"5&3]+dpxb(*[#!#@$\"5+C$z*e`EfTJF- 7$$\"5i!*o_:$3MA7%F-$\"5>D(ezrI&=$G'F-7$$\"5kzKs.(*=*p`&F-$\"5NU4%p2'z xC%*F-7$$\"5JV%)p#[c0s#oF-$\"5=qcfVhqjc7!#?7$$\"5K;Du+NahK!)F-$\"5wo)= &zEjzq:F?7$$\"5q#*zC*fW2]<*F-$\"5#yKPb@fb\\)=F?7$$\"5S*4Fk5@-o-\"F?$\" 5M70)=v&[6*>#F?7$$\"5YcSyO,[4K6F?$\"5F!yW\"*G7uK^#F?7$$\"5IDjIla*[SB\" F?$\"5Z))\\oG)QLu#GF?7$$\"5Y&\\?@/wpJL\"F?$\"5ughFt`EfTJF?7$$\"5L::?J> /&)H9F?$\"5p.@:G>>vbMF?7$$\"5$eNn:m.0W_\"F?$\"5qt.a-&=6*pPF?7$$\"5HVs! 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YJdU(F?$\"5&p?^>PM\")\\E$Ffu7$$\"5UOQ(f3)ecAuF?$\"55!pgTOl%>(G$Ffu7$$ \"5\"RO\"f!z'*RhT(F?$\"5.s1t/rN.4LFfu7$$\"51[**)pM?viS(F?$\"5![i2[/;+0 L$Ffu7$$\"5#ps$GYSYv#R(F?$\"5'RJd#Gyof^LFfu7$$\"5YA5Z$HN7`P(F?$\"5?LB1 =:lKsLFfu7$$\"5^yD*3JM>ON(F?$\"5$*)H%=iW@p#R$Ffu7$$\"5::/geuUEFtF?$\"5 /(\\X,z2(p7MFfu7$$\"5'eKfw.YEdH(F?$\"5U,o?6D[MKMFfu7$$\"5A#>oui\"pLesF ?$\"5C)))[0<1R;X$Ffu7$$\"5ZkVR&zQ?U@(F?$\"5C*z'e5+OeqMFfu7$$\"55o#o.ez 0A;(F?$\"5d@)R'zlB=*[$Ffu7$$\"5Zqp-;T;o+rF?$\"5@tBzGx$Ru]$Ffu7$$\"5b%H DM')f\\t-(F?$\"5,6p5RH(e`_$Ffu7$$\"5Aj.6U7^xQpF?$\"56V\\HoF?$\"50TgWhM6?gNFfu7$$\"5A`&R@ge))**o'F?$\"5NUk'yRkKrd$ Ffu7$$\"5=?\")3T]wH,lF?$\"5FsE(y!)QVPf$Ffu7$$\"5***QOXc\\Ia@'F?$\"5%[: eUDnP+h$Ffu7$$\"5ubQKeDQe:cF?$\"5'4EmZ8#)>gi$Ffu7$$!5]8Kn1s,#zo&F?$\"5 xV&e\"*3<%pTOFfu7$$!5<_$)yq*yrfF'F?$\"53+]1]k]1dOFfu7$$!5j5ap&>QZ?e'F? $\"5\"RQDYs%o8sOFfu7$$!5(R>!Q8ctu'z'F?$\"5Xo<_(4'Q\"po$Ffu7$$!5&)>%Qp \\m&zkpF?$\"57$z7!\\R/S,PFfu7$$!5tH!Rn#o^>/rF?$\"51wz.%Q!4g:PFfu7$$!56 $Ge*f7#oSA(F?$\"5-j&4/xb>&HPFfu7$$!5t%*>PK!))*pHtF?$\"5g&=-cGogJu$Ffu7 $$!5&zNgqB$pVCuF?$\"5qnUusM&Glv$Ffu7$$!5L/X+zz#Q0^(F?$\"5$p(y;2QtipPFf u7$$!5633a;?Qg*e(F?$\"5ocAMv#GhCy$Ffu7$$!5/2K=#*)H5Gm(F?$\"5F$)QDo>X.& z$Ffu7$$!5&p&R%eU)*\\5t(F?$\"5iS7N\"p:^t!QFfu-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6$Q!6\"F[hm-%&COLORG6&%$RGBG$\"\"\"F)F(F(-%*THICKNESS G6#\"\"#-%%VIEWG6$%(DEFAULTGFjhm" 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 -1 107 "The relevant intersection point of the b oundary curve with the imaginary axis can be determined as follows." } }{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Digits := 1 5:\nz0 := 3.65*I:\nfor ct from 126 to 129 do\n newton(R(z)=exp(ct*Pi /100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0@Y%y4!zM&!#<$\"0eUDnP+h$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"099:^L7v\"!#<$\"0mZ8#)>gi$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0N>!4-*f,#!#<$\"0f\"*3<%pTO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0F2_nT'\\f!#<$\"0l+X1lql$!#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 poi nt." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "Digits := 15:\nreal_ part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.65*I))\nend proc:\n u0 := bisect('real_part'(u),u=1.26..1.29);\nnewton(R(z)=exp(u0*Pi*I),z =3.65*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0 ?:wL/ZF\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Kh)R;VLO!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 112 "largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.6334];" "6#7$\"\"!-%&FloatG6$\"&Mj$!\" %" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stab ility function R* for the 11 stage, order 6 scheme is given as follo ws." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,S tabilityFunction(6,11,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)' =`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,8\"\"\"F)F 'F)*&#F)\"\"#F)*$)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)*$)F'F1F)F)F)*&# \"VVJSolH_XjD(*eZZ*G4()*\\G/)*))o1G\"\"YoJFnqi-G+)Q`F/Xsf\"3$\\+&*>bxF %efF)*$)F'\"\"(F)F)F)*&#\"W*)e\"QW\"))*fDcK>+6X#oQ&4Y,XL$*>lL#\"enS=^X (=:Mk.WSzbb=k215k]$f<3h&fu(F)*$)F'\"\")F)F)F)*&#\"U<=.17aQkzXJxVU`J7(y mwHKN6.#\"en?fvs$f2<#=?-(*yx#4#QI]?`nz3a!yH(QF)*$)F'\"\"*F)F)F)*&#\"P* 3vI//TVj_)zrhCdLSg`DZd-(\"W.gA,LQCNNzOi7&elhd;XsiH^BB+\"F)*$)F'\"#5F)F )!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region \+ intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6# %\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!++,#fZ%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "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,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=[-4.99..0.49,-1.47.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 263 263 {PLOTDATA 2 "6+-%'CURVESG6$7W7$$!3A++++++!*\\!#<$!3!)G?[VzQKMF*7$$!3cL Le(zv-$\\F*$!3)R]>v\"f-**HF*7$$!3#pmm^f^0([F*$!3'44>M'=;:EF*7$$!3#3]() f)ee=[F*$!3h%Rgk;uuJ#F*7$$!3%QL3oIz\" 3ZF*$!3]ma@:<(Hy\"F*7$$!3imm,F%Q(\\YF*$!3#[`^@WLqa\"F*7$$!3Gnm1o,\"4f% F*$!3btk)=A@zL\"F*7$$!33nm64>3KXF*$!3KJf9%zAU:\"F*7$$!3\\L$3@f%)\\T%F* $!3m:@bW&[W`)!#=7$$!3jm;u*Q?kI%F*$!3k[WzB:W$Q'FZ7$$!3E+]ZY%3S>%F*$!3#R \"HX%[7sm%FZ7$$!3%omTR&=vxSF*$!3#QUfE64;K$FZ7$$!3/+]x(4o='RF*$!3Kxj%*e C_;BFZ7$$!3OF*$!3EG-%\\PWl='!#>7$$!3?++]Qxz+NF*$!3s_2O'Hy&QIFhp7 $$!3/++5QhU'Q$F*$!3\"*>r@pE([s(!#?7$$!3mm;%zwlDG$F*$\"3pt!o7$*pB@)Fcq7 $$!3[LLBUd1fJF*$\"3%z[u+U%o@BFhp7$$!3eLL$4-XW0$F*$\"3a9'Q1m.2P$Fhp7$$! 3;+]n]iuKHF*$\"3eS3gJzrQWFhp7$$!3ILL$z@A]#GF*$\"3%p2p6\"zK9`Fhp7$$!3!) **\\n!)=$oq#F*$\"35$Q>#GsVciFhp7$$!3')**\\-InG%f#F*$\"3a;Pm/=SwrFhp7$$ !3ILL3sw&oZ#F*$\"3=)GM!=%=V>)Fhp7$$!3#HL3&R6-pBF*$\"3Q:ruA8j0#*Fhp7$$! 3#om;4([q_AF*$\"3'fcT;vQ,/\"FZ7$$!3xm;%z&\\)=8#F*$\"3uVGIy\")=y6FZ7$$! 3$***\\_o3rE?F*$\"35d$*=NZq68FZ7$$!3hmmhq*>J\">F*$\"3mMaHdL\"F*$\"3\"HU99\\%3HEFZ7$$!3z****\\%R.u@\"F*$\"35g/MzZkfHF Z7$$!3pm;aLE=56F*$\"3+9!*yQ#)z%H$FZ7$$!3E%****4@?'H**FZ$\"3Gcq\"f]#o/P FZ7$$!3MML3+cmE))FZ$\"3()R>nA(Qn8%FZ7$$!3%H**\\(H-wtwFZ$\"3dvG'Q$zFUYF Z7$$!3cOLL)\\%eYlFZ$\"3)[(R*Ht)='>&FZ7$$!3g.+vof`m`FZ$\"3s\"H+&4Y,ZeFZ 7$$!3Dkmm#)*3+B%FZ$\"3Q!z%H4,y]lFZ7$$!3Yjm;()funIFZ$\"3-c[GCV;etFZ7$$! 3HBL3;r5:>FZ$\"3Y<&HA!y5d#)FZ7$$!3_^****>q^f&)Fhp$\"3+U#)4fxlz\"*FZ7$$ \"3OMnm\"G*fzNFhp$\"3%RQAGQWk.\"F*7$$\"3\"RLL8'ppV9FZ$\"3oz&oO96`:\"F* 7$$\"3@0+D$4>8g#FZ$\"3erCbw65(H\"F*7$$\"3Q,+v#=6$4PFZ$\"3GrO8sK3\\9F*7 $$\"3!***************[FZ$\"3!Q81aM;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"! \")$\"\"!Ff\\lFe\\l-F$6$7S7$F($!\"\"Ff\\l7$F3F[]l7$F=F[]l7$FGF[]l7$FQF []l7$FVF[]l7$FfnF[]l7$F[oF[]l7$F`oF[]l7$FeoF[]l7$FjoF[]l7$F_pF[]l7$Fdp F[]l7$FjpF[]l7$F_qF[]l7$FeqF[]l7$FjqF[]l7$F_rF[]l7$FdrF[]l7$FirF[]l7$F ^sF[]l7$FcsF[]l7$FhsF[]l7$F]tF[]l7$FbtF[]l7$FgtF[]l7$F\\uF[]l7$FauF[]l 7$FfuF[]l7$F[vF[]l7$F`vF[]l7$FevF[]l7$FjvF[]l7$F_wF[]l7$FdwF[]l7$FiwF[ ]l7$F^xF[]l7$FcxF[]l7$FhxF[]l7$F]yF[]l7$FbyF[]l7$FgyF[]l7$F\\zF[]l7$Fa zF[]l7$FfzF[]l7$F[[lF[]l7$F`[lF[]l7$Fe[lF[]l7$Fj[lF[]l-F_\\l6&Fa\\lFe \\lFe\\lFb\\l-F$6&7#7$$!3u*******4?fZ%F*F[]l-%'SYMBOLG6#%'CIRCLEG-F_\\ l6&Fa\\lFf\\lFf\\lFf\\l-%&STYLEG6#%&POINTG-F$6&Fa`l-Ff`l6#%&CROSSGFi`l F[al-F$6&Fa`l-Ff`l6#%(DIAMONDGFi`lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLORG6 &Fa\\lFe\\l$\"\"&F\\]lFe\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG \"\"*-%+AXESLABELSG6%Q\"z6\"Q!F_cl-Fgbl6#%(DEFAULTG-%%VIEWG6$;$!$*\\! \"#$\"#\\Fjcl;$!$Z\"Fjcl$\"$Z\"Fjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following picture shows the stability region for the \+ 11 stage, order 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1329 "`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 1280668889804284998709289474758972 5634552296568403143/\n 5958427775519950049308159724504275338800280 2627067273168*z^7+\n 233651993334501460953868245110019325625599881 443815889/\n 77459561081759350641006076418555579404403643415187455 11840*z^8+\n 2031135322976667871231534243773145796438541206031817/ \n 3872978054087967532050303820927778970220182170759372755920*z^9- \n 70257472553604033572461717985263434104043075089/\n 10023235 1296272451657616558512623679353524383301226003*z^10:\npts := []: z0 := 0: tt := 0: \nwhile tt<=281/20 do\n zz := newton(`R*`(z)=exp(tt*Pi* I),z=z0):\n z0 := zz:\n if (13/20<=tt and tt<=33/20) or (53/20<=tt and tt<=71/20) or\n (209/20<=tt and tt<=227/20) or (247/20<=tt a nd tt<=267/20) then\n hh := 1/40\n else \n hh := 1/20\n \+ end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np_1 := plot(pts,color=COLOR(RGB,.43,0,0)):\np_2 := plots[polygonplo t]([seq([pts[i-1],pts[i],[-2.25,0]],i=2..nops(pts))],\n style =patchnogrid,color=COLOR(RGB,.85,0,0)):\np_3 := plot([[[-5.09,0],[1.19 ,0]],[[0,-4.49],[0,4.49]]],color=black,linestyle=3):\nplots[display]([ p_||(1..3)],view=[-5.09..1.19,-4.49..4.49],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 595 560 560 {PLOTDATA 2 "6+-%'CURVESG6$7bal 7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"35+++Y>')p5!#E$\"3:+++.FfTJF-7$$ \"3/+++=SYxF!#D$\"31+++C)*Q7ZF-7$$\"3>+++hB+CG!#C$\"3_******o:>$G'F-7$ $\"3&******>UA7s\"!#B$\"3++++e5,ayF-7$$\"3?+++_nZ*f(FC$\"3u*****z4\")[ U*F-7$$\"3!******zyyzo#!#A$\"33+++/oe*4\"!#<7$$\"3]+++#fVb3)FN$\"3#*** ***HT3nD\"FQ7$$\"3'*******)pG)[@!#@$\"3\"******4coQT\"FQ7$$\"3a+++2PJw ^FZ$\"3%******p'H3r:FQ7$$\"35+++R#FQ7$$\"3++++%f[f9\"!#>$\"3y*****RmCsF#FQ7$$\"3%******zrVE]\"Fip$ \"3?+++$*=eaBFQ7$$\"3++++t@\\S>Fip$\"34+++-sAJCFQ7$$\"31+++YFalCFip$\" 34+++A`*o]#FQ7$$\"3++++'G<(yIFip$\"3!******4lv7e#FQ7$$\"3s*****HJnQx$F ip$\"3!)*****p\\HSl#FQ7$$\"3G+++U2hOXFip$\"36+++\"*z\"[s#FQ7$$\"30+++) RlXM&Fip$\"3?+++bdL$z#FQ7$$\"3A+++g8>phFip$\"33+++1;MfGFQ7$$\"3#****** RgN)ypFip$\"3#******RhxE#HFQ7$$\"3c+++Uo6UxFip$\"3%********GsK)HFQ7$$ \"3[+++zhxI%)Fip$\"3@+++;)H6/$FQ7$$\"33+++R#)[@!*Fip$\"3y*****fF6j4$FQ 7$$\"3!)*****p4$e'\\*Fip$\"39+++(H9*[JFQ7$$\"30+++m&GR%)*Fip$\"33+++za 0*>$FQ7$$\"31+++R5j05F-$\"3#)*****H\\doC$FQ7$$\"3'******>tfI,\"F-$\"3# )*****z!3W#H$FQ7$$\"3,+++x!)o15F-$\"3!)*****H;>fL$FQ7$$\"3I+++1+pn)*Fi p$\"37+++FwRxLFQ7$$\"3q+++wPKP&*Fip$\"33+++wA(pT$FQ7$$\"3Q+++EVI\"3*Fi p$\"3!******RuIZX$FQ7$$\"3U+++(yVf])Fip$\"3%)*****4:`2\\$FQ7$$\"3w**** *z!p+=yFip$\"3$******f78^_$FQ7$$\"3]*****za`W-(Fip$\"3#*******f(yyb$FQ 7$$\"3/+++seCKhFip$\"3')******3M6*e$FQ7$$\"3;+++2u?[^Fip$\"3(******fMw )=OFQ7$$\"31+++uK#*ySFip$\"3++++fKAZOFQ7$$\"3:+++,GnIHFip$\"3)******Ri 1Un$FQ7$$\"3%)*****fJ(Q4gg+,\"F-$\"3;+++6*=I 'QFQ7$$!3'******>#>Dx6F-$\"3;+++!)[nyQFQ7$$!3'******f%[vZ8F-$\"3-+++!= yL*QFQ7$$!33+++upv(p\"F-$\"3*******p\"G-?RFQ7$$!3!******p?\"Re?F-$\"3' ******p*o7VRFQ7$$!30+++&))o#GCF-$\"3')*****f$F-$\"3/+++`$QD*RFQ7$$!3/+++=\"\\He $F-$\"3A+++bEq-SFQ7$$!3!******\\0_.)RF-$\"33+++wC#)4SFQ7$$!3%******z=$ =$Q%F-$\"3P+++M>$R,%FQ7$$!3)******p.$>\"z%F-$\"3k*****f5X],%FQ7$$!3m** ****pvI/_F-$\"3h*****>,cJ,%FQ7$$!3;+++[7lAcF-$\"3*******RFQ#3SFQ7$$!3I +++$>&eYgF-$\"3K+++ZXC+SFQ7$$!3g******>bwwkF-$\"3;+++@f5*)RFQ7$$!3\")* ****HVHU\"pF-$\"3++++F85FQ$\"3)******f \"[#\\z$FQ7$$!32+++HyNV5FQ$\"3A+++bH%\\x$FQ7$$!31+++M0)[2\"FQ$\"3#**** **\\*ePaPFQ7$$!3%******Rn*)y5\"FQ$\"3,+++*46Mt$FQ7$$!3(******fJzC9\"FQ $\"3))******=eG7PFQ7$$!3++++'o*oy6FQ$\"33+++T1G\"p$FQ7$$!3!******f/([; 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22785849/1109145221,\nc[8]=17/20,\nc[9]=171/181,\nc[10]=1,\nc[11]=1,\n \na[2,1]=1/200,\na[3,1]=-820353/229441,\na[3,2]=864900/229441,\na[4,1] =-432520/5401533,\na[4,2]=0,\na[4,3]=2897950/5401533,\na[5,1]=31171299 6187/3366699277500,\na[5,2]=0,\na[5,3]=996040375149076/509667770124337 5,\na[5,4]=344546810496217/1096059720697500,\na[6,1]=-228/1231,\na[6,2 ]=0,\na[6,3]=15089501252616323812/14457231345452578137,\na[6,4]=-62556 16188978805360/5200809295819122531,\na[6,5]=103959712518200500/8396915 3915827131,\na[7,1]=20784021508544698439004341302945788046687497048823 /\n 333809007739010923825478541340748044990946025336204,\na[7,2] =0,\na[7,3]=5908707677638153633449865101135634541623011524075277164/\n 31545554087781255456304827404537875875692253868011534003,\na[7, 4]=-48597609826899455036511090495924959702383070843517685/\n 48 8091226723681375828622960423688756917868034622078292,\na[7,5]=40837242 552907904940455804285850036846494300434360648500/\n 734372562662 145990989587253914642887902719996408150162943,\na[7,6]=-22672360122745 82026887011527795707923286/477373186945391376596029499088911158349111, \na[8,1]=-14850375572183825804482268314325533/131869870061161226458875 010296840000,\na[8,2]=0,\na[8,3]=25892910773607576734414005208753/1052 9291320508273201747374752000,\na[8,4]=-613522852101531567576566644397/ 814576707522236069963552640000,\na[8,5]=812271575949979003109520974653 7365648167969/8362965435593003722836700426272322940782592,\na[8,6]=-42 937178562595251680361031143/2273295839722617724238904704000,\na[8,7]=- 7424205995886059956357359508852528825590423/43781972910516512574862585 63957705093888000,\na[9,1]=-153831689800158592252487409542055462465307 763/517713869734060023599533102763919424014608068,\na[9,2]=0,\na[9,3]= -1334984216509015564184248020843701584/6670676425579396634061300189774 78253,\na[9,4]=-57633497085298645064371869898744017/309637798459110104 23286154735455876,\na[9,5]=4891779727567064214119442991968982563881914 63578294580375/\n 3192912011009866226654028477803512300463766071 22311276442,\na[9,6]=-1185679300358377756233489606885369481439862/1843 8601757968638166315261704657797344684897,\na[9,7]=11944482045828658346 298806908238121417524848379605193796762013/\n 341237631731129839 2350722812851096867170775526794634954003706,\na[9,8]=10397310879870999 2837929472000/762442644104164301247937014633,\na[10,1]=-40734473647257 020156088313054846628592379/112331307637222010640347265020564130793284 ,\na[10,2]=0,\na[10,3]=-1111042984674821389643230968704444252/14751996 4335786292056835317593798019,\na[10,4]=-521306348327785400364077424820 5317/2282514999055829721733809658986516,\na[10,5]=43441256026371742635 26484423723747582563035478367000/\n 281969717471900849409975568 5231013933352915992886809,\na[10,6]=1428124381699559509693680117880345 19784/281984196771808918111529577339005012981,\na[10,7]=35011829759852 921774894466285700128707491070752921128030483969446296/\n 37180 44938427279904428695460050071040231420419464687750808112491477,\na[10, 8]=-26328105087975815319887872000/966150763091150023719018517747,\na[1 0,9]=-1198575787406124991100974435169141280/46356482334809216491547309 58571956313,\na[11,1]=-4549513437278902549066809605929617243853/116371 30531712243439602356681247846961408,\na[11,2]=0,\na[11,3]=-80700519465 66656273895465364457/1073987714691843866578232224704,\na[11,4]=-100590 66143857143308915950403/4154341208363403956814118464,\na[11,5]=5815991 882243812040956722134361657936380202840699902125/\n 35859517374 51764935284196541594543916113759295056049408,\na[11,6]=282747877101898 75324117065188869812987/61207302919331402809078602199860439808,\na[11, 7]=5719170472909759253773842353503580127446876084974327902851679027589 /\n 60288487786636086054845555236358357973231537847194078996479 4317568,\na[11,8]=1203167959926676490500/125196265101038882839767,\na[ 11,9]=-4684487619570658562655565/18486842815356938821237824,\na[11,10] =0,\n\nb[1]=17499592155841793/301627899585331605,\nb[2]=0,\nb[3]=0,\nb [4]=0,\nb[5]=89929293580433773826711625625/175034206955063764470139052 544,\nb[6]=43965679317071687650068867/11554641620922992752054730,\nb[7 ]=639711292154532559948032664549872463482448777204372657657897547/\n \+ 2011601425501588622427604443016911932371008211435797382991660160,\n b[8]=-43614217354789120000/21595629876813903267,\nb[9]=-21099698264929 1857501475159/99036657939412172256631200,\nb[10]=7077278529846113/1554 6034297382400,\nb[11]=0,\n\n`b*`[1]=94288552637982437/1608682131121768 560,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=53240044523942766804 7625/994151030051934322008696,\n`b*`[6]=15979368227021270883046053/318 7487343702894552290960,\n`b*`[7]=6834825528688223605837314645352673404 27333118635079/\n 216556294173916779804551839762023122071226114 2369860,\n`b*`[8]=-132403130312946592000/50389803045899107623,\n`b*`[9 ]=-169578451828457839968131453/57771383797990433816368200,\n`b*`[10]=0 ,\n`b*`[11]=16/25\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,/&%\"cG6#\"\"##\"\"\"\"$+#/&F%6#\"\"$#\"#$* \"$z%/&F%6#\"\"%#\"$5\"\"$T#/&F%6#\"\"&#\"$4&\"$X)/&F%6#\"\"'#\"$W#\"$ t#/&F%6#\"\"(#\"*\\eyA#\"+@_946/&F%6#\"\")#\"#<\"#?/&F%6#\"\"*#\"$r\" \"$\"=/&F%6#\"#5F)/&F%6#\"#6F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j ]),j=1..i-1),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6Y/&%\"aG6$\" \"#\"\"\"#F(\"$+#/&F%6$\"\"$F(#!'`.#)\"'T%H#/&F%6$F.F'#\"'+\\')F1/&F%6 $\"\"%F(#!'?DV\"(L:S&/&F%6$F:F'\"\"!/&F%6$F:F.#\"(]z*GF=/&F%6$\"\"&F(# \"-(='*Hr6$\".+vF*pmL/&F%6$FJF'FA/&F%6$FJF.#\"0w!\\^PSg**\"1vLC,xn'4&/ &F%6$FJF:#\"0eH43?&/&F%6$FjnFJ#\"3+0?=DrfR5\"2Jr#e\"R:pR)/&F%6$\"\"(F(#\"SB) [q\\(oY!)yXHITV+R%)pW&3:-%y?\"T/iLDg%4*\\/[2MT&ya#Q#4,Rx+4QL/&F%6$FfpF 'FA/&F%6$FfpF.#\"XkrFvS_6IiTXjN65l)\\Mj`\"Qwn2(3f\"Y.S`6!oQD#pve(y`/u# [IcaD\"y(3abaJ/&F%6$FfpF:#!V&oF!z)Gk9RD(e*)4*f9iEcsVt/&F%6$FfpFjn#!I'GBzq&zF:,( )o-#euA,OsE#\"K6\"\\$e6\"*)3*\\Hgfw8RXp=ttZ/&F%6$\"\")F(#!DLbK9$oA[/e# Q=sbP][\"\"E++%oH5]()ekAh61q)p=8/&F%6$FhrF'FA/&F%6$FhrF.#\"A`(3_+9Wtwv gt2\"H*e#\"A+?vutu,KF30K\"HH0\"/&F%6$FhrF:#!?(RWmcwvcJ:5_G_8'\"?++k_N' *pgBAvqwX\")/&F%6$FhrFJ#\"Lpz;[cOPlu4_4J+z*\\fdrA\")\"L#f#ySHKsiU+n$Gs .IfNa'HO)/&F%6$FhrFjn#!>V6.h.o^_fi&yr$H%\"@+Sq/*QUsyV/&F%6$\"\"*F(# !NjxIlCYb?a4u[_Afe,!)*oJQ:\"No!3Y,C%>Rw-J`*fB+1M(pQr<&/&F%6$F`uF'FA/&F %6$F`uF.#!F%e,P%3-[U=kb,4l@%)\\L\"\"E`#yu(*=+81MmRzbUw1n'/&F%6$F`uF:#! D)Qc#)*o>*HW>T@kqcF(z<*[\"ZUkF6B72mPY+B^.yZGSlEi')45,7H>$/&F%6$F`uFj n#!Li)R9[p`)og*[Livx$e.Izc=\"\"M(*[oWtzdYqh_Jm\"Q'ozv,'Q%=/&F%6$F`uFfp #\"in8?w'z$>0'z$[[_<97Q#3p!))HY$e'Ge/#[W>\"\"hn1P+a\\j%zEbxqr'o4^G\"Gs ]BR)H6tJwBT$/&F%6$F`uFhr#\"?+?ZHz$G**4()z3J(R5\"?LY,PzC,V;/TkUCw/&F%6$ \"#5F(#!JzBfGm%[08$)3c,-dsktWtS\"K%G$zITc?]EZ.k5?APwIJB6/&F%6$F^xF'FA/ &F%6$F^xF.#!F_UW/(o4BV'*Q@[n%)H/66\"E>!)z$f<`$o0#H'yNV'*>v9/&F%6$F^xF: #!C<`?[UxSO+ayF$[jI@&\"C;l)*e'4Qt@(He0**\\^#G#/&F%6$F^xFJ#\"U+qOya.jDe ZPsBW[ENEurj-c7WV\"U4o)G*f\"HNLR,J_ob(*4%\\3!>Z<(p>G/&F%6$F^xFjn#\"H%y >X.)y6!o$p4&f&*p\"QC\"G9\"H\")H,0!RtdH:6=*3=x'>%)>G/&F%6$F^xFfp#\"_o'H Y%pR[I!G6#Hvq5\\2(G,q&GmW*[x@H&)f(H=,N\"^ox9\\7\"33v(ok%>/UJ-/r+0gapGW !*zsUQ\\/=P/&F%6$F^xFhr#!>+?(y))>`\"e(z30\"Gj#\"?Zx^=!>P-]64j2:m*/&F%6 $F^xF`u#!F!GT\"p^Vu45\"*\\71uyv&)>\"\"F8j&>de4ta\"\\;#4[L#[cj%/&F%6$\" #6F(#!I`QC<'Hfg4o1\\D!*ysV8&\\X\"J39'p%yC\"ocBgRMC7<`Irj6/&F%6$Fb[lF'F A/&F%6$Fb[lF.#!@dWOla*QFcmcY>0q!)\"@/ZAK#yl'Q%=p9x)R2\"/&F%6$Fb[lF:#!> ./&f\"*3L9dQ9m!f+\"\"=k%=T\"o&RSj$37MaT/&F%6$Fb[lFJ#\"XD@!**pSG?!QOzlh V8An&4/7QC#)=*f\"e\"X3%\\g0&HfP6;Ra%fTl>%GN\\w^ut^fe$/&F%6$Fb[lFjn#\"G ()H\")p))=lq6C`()*=5xyu#G\"G3)R/')*>-'y!4GSJ$>HI27'/&F%6$Fb[lFfp#\"^o* eF!z;&G!zKu\\3woWF,e.NNUQx`#f(4HZq\">d\"]oovJ%zk**yS>Zy`JK(zNejBbb%[0' 3Omy([)Gg/&F%6$Fb[lFhr#\"7+0\\wm#*fz;.7\"9n(RG))Q55li>D\"/&F%6$Fb[lF`u #!:lbli&e1d>w[%o%\";CyB@)QpN:G%o[=/&F%6$Fb[lF^xFA" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 11 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"2$zTe:#f*\\<\"30;L&e**yi,$/&F%6#\"\"#\"\"!/&F% 6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\">Dci6n#QxL/e$HH**)\"?WD0R,ZkP1bp?M ]\"p,VWgFCi)e,bU,; ,#/&F%6#\"\")#!5++7*yat@9O%\"5nK!R\"o()Hcf@/&F%6#\"\"*#! " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..11);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%#b*G6#\"\"\"#\"2PC)zj_&)G%*\"4g&o< 7J@o3;/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\"9Dw/owUR _W+C`\"9'p3?KM>0I5:%**/&F%6#\"\"'#\";`g/$)3F@qAo$zf\"\":g4H_X*GqVt[(=$ /&F%6#\"\"(#\"Tz]j=JLF/Mn_`k9t$egB#)oGb#[$o\"Ug)pB9hAr?7B?wR=b/)zn\"R< %Hcl@/&F%6#\"\")#!6+?fYHJIJSK\"\"5Bw5**e/.)*Q]/&F%6#\"\"*#!<`98o*RyXG= Xy&p\"\";+#oj\"QV!*zz$Qrx&/&F%6#\"#5F//&F%6#\"#6#\"#;\"#D" }}}{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 43 "#======================================== ==" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Scheme with a larger stabil ity region" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 48 "#-----------------------------------------------" }}{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4911 "ee := \{c[2]=1/24,\nc[3]=897/1072,\nc[4]=160/4 99,\nc[5]=141/763,\nc[6]=169/192,\nc[7]=998622639/2007664595,\nc[8]=10 4/105,\nc[9]=20/21,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/24,\na[3,1]=-21734 31/287296,\na[3,2]=2413827/287296,\na[4,1]=57894880/223353897,\na[4,2] =0,\na[4,3]=13721600/223353897,\na[5,1]=5056664940861/42500572528960, \na[5,2]=0,\na[5,3]=-286249149686016/36667767392227699,\na[5,4]=288925 9817673483/39243095536832320,\na[6,1]=174/407,\na[6,2]=0,\na[6,3]=8617 65466269918263/4970009789268581376,\na[6,4]=10826192777269994995/77126 91245665173504,\na[6,5]=-16750236560040896731/14897194035750715392,\na [7,1]=-3439778790612913527970381936467065474966136664036779/\n \+ 39381209389424561778044106217782104388290028625000000,\na[7,2]=0,\na[7 ,3]=2631414486943314085034368959766533330714638432739014413056/\n \+ 42832809612378095870766530128571820142439373914153488671875,\na[7,4] =-3226817813200049695169462352322166717564113587934800370011/\n \+ 40015346463976160761929086193516183832403838724337375000000,\na[7,5]= 786190577550706940215874040419538587608361105183350570182/\n 126 1335096542994096899051116924589799398264209359351953125,\na[7,6]=-5411 006332774526240948446773664264724938752/279717068193641240266628706086 067991948046875,\na[8,1]=15322017632690457879761200795014874391/293049 23702961208406725757182921875000,\na[8,2]=0,\na[8,3]=33468248159108461 26629531019335170048/2298048540174087438681694595766796875,\na[8,4]=18 9748010561998005455394521375650921/29817877073894091150172096921875000 ,\na[8,5]=-382845331583835193195089397869250404738354182/1102254632855 88517977748940118009136362890625,\na[8,6]=-120391896911431755652090822 656/233667703961223196028992578125,\na[8,7]=-7969512806056250503723894 1490242415433886906/23692035580167289837348760854832233004866581,\na[9 ,1]=1032897956298204654305923433045759171807509/2004961136769885546067 510633666861543410000,\na[9,2]=0,\na[9,3]=8472900696201518374537737273 138176/9559881927124203744915849518389875,\na[9,4]=1046253623595072271 778442420598129/248084737254798838369431846390000,\na[9,5]=-2255808953 526127614059281040689393645809880661461199/\n 89151923156535278 3145488093784087901162293552521625,\na[9,6]=-1788698788744957883891433 528082038784/5923143676314255446643283231146667875,\na[9,7]=-574800468 754997103944423838666252602754508075193305936911875/\n 31154518 4693907215706597063235165448890282539301416442871559,\na[9,8]=88873364 366227887565000/8176290045932637014386689,\na[10,1]=862802087344743900 54682197174277603231/163235672733343926284895054973116600000,\na[10,2] =0,\na[10,3]=871191425901232070238397371934719232/53336244680174479037 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998773730568797417283222822299497058344239378524000,\na[11,6]=-7360838 5714443424401508261029604294656/12376818362185003161183187069164749312 5,\na[11,7]=-105278338515332628649993716344997524414442824414296086168 50996735536625/\n 27463985814499522354606501995948603403618171 26809954433857815662665184,\na[11,8]=-5616863065079321503840875/276176 908218169072485950384,\na[11,9]=40097947059612729/1419769450975237280, \na[11,10]=0,\n\nb[1]=164949402359214517/2969763122318428800,\nb[2]=0, \nb[3]=0,\nb[4]=0,\nb[5]=166164647343343629239462104853/59415021855318 2839761586950000,\nb[6]=8835157365899998531682304/69998514485057912349 78125,\nb[7]=110567843372870117205263375074449005180061856568412069340 896875/\n 34071775464939293118333187684633880466821185714939883454 1328544,\nb[8]=758177409913942921875/78785859799058312576,\nb[9]=-2973 333145929506703/747247079460651200,\nb[10]=-1895689785077201/288707084 450720,\nb[11]=0,\n\n`b*`[1]=20506743773621213/371220390289803600,\n`b *`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=268231112188987944945827753/ 955225431757528681288725000,\n`b*`[6]=415490030471198546067456/3043413 67326338749346875,\n`b*`[7]=108419211385790972003069732530101032173894 775381027875/\n 33766460613793638050005115629338489535336215211 7883624,\n`b*`[8]=54584684081980124625/4924116237441144536,\n`b*`[9]=- 842056888251867309/186811769865162800,\n`b*`[10]=0,\n`b*`[11]=-38/5\}: " }}}{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 0 136 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j =1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7.7.$\"'nmT!\"(F(%!GF+F+F+F+F+F+F+F+F+7.$\"'an$)!\"'$!'8lv!\"&$\"' )=S)F2F+F+F+F+F+F+F+F+F+7.$\"'T1KF/$\"'2#f#F/$\"\"!F;$\"'VVhF*F+F+F+F+ F+F+F+F+7.$\"'(z%=F/$\"'z*=\"F/F:$!'c1y!\")$\"'ZitF*F+F+F+F+F+F+F+7.$ 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#/&%\"aG6$\"#6\"\"%#\"B$[7IM')[ZN30\")f`^Ed\"A++gH\\c\"y0\"\\<*4(eV\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!YFl%>\"e)\\1U) yVz4bEy,I-[H'*G%om-!>\"X+S_y$RUMeq\\*HAGA$G]1YNA&*\\9e)RYF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\")#!:v3%Q]@$z]1joh&\"<%Q]f[s!p\"=#3p9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \"#\"3r/.!\\: %\"9voM\\(QjKn8M/$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(# \"Wvy-\"QvZ*Q%3\"\"WCO)y6_@O``*[Q$Hc60+0QOz81YmP$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"5DY7!)>3%o%ea\"4OX9T uB;T#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!34t'=D))o0U )\"3+G;l)p<\"o=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!#Q\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderCon ditions(7,11,'expanded'))]:\n`RK6_11eqs*` := subs(b=`b*`,OrderConditio ns(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "s ubs(ee,RK7_11eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK6_1 1eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}} {PARA 257 "" 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 in clusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 t o 5 do\n so||ct||_11 := StageOrderConditions(ct,11,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " Stages 5 to 10 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(subs(ee,so||i||_11[j])),i=2..5)],j= 1..9)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) t hen break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#F$\"\"$F%F%F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The stage-orders of the s uccessive stages are given as follows." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[stage, `|`, 2, 3, 4, 5, 6, 7, 8, 9, 10] , [`stage-order`, `|`, 1, 2, 2, 3, 3, 3, 3, 3, 3]]);" "6#-%'matrixG6#7 $7-%&stageG%\"|grG\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57-%,stag e-orderGF)\"\"\"F*F*F+F+F+F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j] ,i = j+1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 9 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0 " "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "[Su m(b[i]*a[i,1],i=1+1..11)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c [j]),j=2..9)];\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%)))); \nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"#6&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F, ;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\" \"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F* 6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-, &F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&F EFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF -/-F&6$*&F)F-&F/6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$ *&F)F-&F/6$F,F_rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7+\"\"!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 26 "The simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,3],i = 3 .. 10) = 0" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;F0\"#5\"\"!" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 3 .. 10) = 0;" "6#/-% $SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+;F3\"#5\" \"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,3],i = 3 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&% \"aG6$F+\"\"$F,/F+;F5\"#5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "[Sum(b[i]*a[i,3],i=3..10),Sum(b[i] *c[i]*a[i,3],i=3..10),Sum(b[i]*c[i]^2*a[i,3],i=3..10)];\nsubs(ee,eval( subs(Sum=add,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%$SumG6$*&&% \"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;F0\"#5-F%6$*(F(F,&%\"cGF*F,F-F,F 1-F%6$*(F(F,)F7\"\"#F,F-F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\" !F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errterms7_11 := PrincipalErrorTerms(7,11,'expanded'):\nnrm8 := sqrt(a dd(subs(ee,errterms7_11[i])^2,i=1.. nops(errterms7_11))):\nevalf[10](% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U(f6v$!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "In addition the 2-norm \+ of the order 9 error terms is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "errterms8_11 := PrincipalErrorTerms(8,11,'expanded') :\nnrm9 := sqrt(add(subs(ee,errterms8_11[i])^2,i=1.. nops(errterms8_11 ))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+63ax=!#8" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The 2-n orm of the order 9 error terms is approximately 5.005 times the princi pal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf[10] (nrm9/nrm8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bvA0]!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The princ ipal error norm of the order 6 embedded scheme is given as follows." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms6_11*` := subs(b= `b*`,PrincipalErrorTerms(6,11,'expanded')):\nsqrt(add(subs(ee,`errterm s6_11*`[i])^2,i=1.. nops(`errterms6_11*`))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V@8[e!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#-----------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 7 schem e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We f irst construct a 10 stage, order 7 scheme with " }{TEXT 260 10 "parame ters" }{TEXT -1 2 " " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 2 ".\n" } }{PARA 0 "" 0 "" {TEXT -1 89 "The scheme will be constructed so that s tage 4 has stage-order 2 and stages 5 to 10 have " }{TEXT 260 13 "stag e-order 3" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 60 "We start \+ by determining the nodes and weights of the scheme." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We set up a system of e quations that consists of the 7 order 7 quadrature conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 10) = 1" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"#5F-" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 1 .. 10) = 1/k;" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;F,\"#5*&F,F,F2 F3" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 7 " . . 7," }}{PARA 0 "" 0 "" {TEXT -1 35 "and the relation b etween the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c [5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^2*c[7]- 7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2 = 0;" "6#/,>** \"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\"'F'F'**\"$0\"F'*$&F)6#F. \"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F)6#F.F'&F)6#F1F7!\"\"**F =F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6#F+F'F'*(F+F'*$&F)6#F.F7F '&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F'&F)6#F.F'FD*(\"#9F'*$&F )6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD*(FenF'&F)6#F.F'&F)6#F1F 'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F'&F)6#F.F7F'\"\"!" } {TEXT -1 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[2] = 1/24;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#C!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 141/763;" "6#/&%\"cG6#\"\"&*& \"$T\"\"\"\"\"$j(!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "c[6] = 169/ 192;" "6#/&%\"cG6#\"\"'*&\"$p\"\"\"\"\"$#>!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 104/105;" "6#/&%\"cG6#\"\")*&\"$/\"\"\"\"\"$0\"! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 20/21;" "6#/&%\"cG6#\" \"**&\"#?\"\"\"\"#@!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]=1" " 6#/&%\"cG6#\"#5\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "Qeqs := QuadratureConditions(7,10,'expanded'): \nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6 ]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14 *c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2=0: \ncdns1 := [op(Qeqs),node_eq]:\ne1 := \{c[2]=1/24,c[5]=141/763,c[6]=16 9/192,c[8]=104/105,c[9]=20/21,c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\neqns1 : = subs(e1,cdns1):\nnops(%);\nindets(eqns1);\nnops(%);\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"b G6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\")&F%6#\"\"*&F%6#\"#5&%\"cGF,&F%6# \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We have 8 equations and 8 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "e2 := solve(\{op(eqns1)\}):\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 0 607 "e3 := \{b[8] = 758177409913942921875/787858597990583 12576, b[10] = -1895689785077201/288707084450720, b[9] = -297333314592 9506703/747247079460651200, b[5] = 166164647343343629239462104853/5941 50218553182839761586950000, b[6] = 8835157365899998531682304/699985144 8505791234978125, b[1] = 164949402359214517/2969763122318428800, c[7] \+ = 998622639/2007664595, b[7] = 110567843372870117205263375074449005180 061856568412069340896875/340717754649392931183331876846338804668211857 149398834541328544, b[4] = 0, b[3] = 0, b[2] = 0, c[10] = 1, c[9] = 20 /21, c[2] = 1/24, c[5] = 141/763, c[6] = 169/192, c[8] = 104/105\}:" } }}{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 53 "We now have all the weights and the nodes excluding " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[ 4]" "6#&%\"cG6#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "seq(c[i]=subs(e3,c[i]),i=[ 2,$5..10]);``;\nfor ii to 10 do b[ii]=subs(e3,b[ii]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"\"\"\"#C/&F%6#\"\"&#\"$T\" \"$j(/&F%6#\"\"'#\"$p\"\"$#>/&F%6#\"\"(#\"*REi)**\"+&fkw+#/&F%6#\"\")# \"$/\"\"$0\"/&F%6#\"\"*#\"#?\"#@/&F%6#\"#5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"3 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We construct a system of equations that incorporate \+ the row-sum conditions," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j = 1 .. i-1) = c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG /F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . 10, " }}{PARA 0 "" 0 "" {TEXT -1 29 " the stage order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j] *c[j],j = 2 .. i-1) = 1/2" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"&%\" cG6#F,F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-F3F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i = 3;" "6#/%\"iG\"\"$" }{TEXT -1 9 " . . 10, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j]^2,j = 2 .. i-1) = \+ 1/3" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2, &F+F-F-!\"\"*&F-F-\"\"$F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^3" "6# *$&%\"cG6#%\"iG\"\"$" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i = 5;" "6# /%\"iG\"\"&" }{TEXT -1 8 " . . 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "together with the column simplifying cond itions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]* a[i,1],i=2..10)=b[1]" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+F,F, /F+;\"\"#\"#5&F)6#F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*a[ i,j],i = j+1 .. 10) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#5*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F ," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 8 " . . 9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "and the further simplif ying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S um(b[i]*c[i]*a[i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5\"\"!" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,3],i = 5 .. 10) = 0;" "6#/-%$S umG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+\"\"$F,/F+;\"\" &\"#5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "SO_eqs := [op(RowSumConditions(10, 'expanded')),op(StageOrderConditions(2,3..10,'expanded')),\n op(Stage OrderConditions(3,5..10,'expanded'))]:\nsimp_eqs := [add(b[i]*a[i,1],i =2..10)=b[1],seq(add(b[i]*a[i,j],i=j+1..10)=b[j]*(1-c[j]),j=6..9)]:\ns imp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5..10)=0,add(b[i]*c[i]^2*a[i,3],i= 5..10)=0]:\ncdcns2 := [op(simp_eqs),op(simp_eqs2),op(SO_eqs)]:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We specif y the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[5,2] = 0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\" \"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\" \"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG 6$\"#5\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e4 := \{seq(a[i,2]=0,i=4..10 )\}:\ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "eqns2 := subs(e5,cdcns2):\nn ops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "e6 := solve(\{op(eqns2)\},indets(eqns2) minus \+ \{c[3],c[4],a[6,1],a[7,6],a[8,6],a[8,7],a[9,5],a[9,6],a[9,7],a[9,8]\}) :\ninfolevel[solve] := 0:\ne7 := `union`(e5,e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "indets(map(r hs,e6));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"\"* \"\"(&F%6$F'\"\")&%\"cG6#\"\"$&F-6#\"\"%&F%6$F'\"\"&&F%6$F'\"\"'&F%6$F +F8&F%6$F+F(&F%6$F(F8&F%6$F8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33449 "e7 := \{b[8] = 758177409913942921875/78785859799058312576, b[10] = -1 895689785077201/288707084450720, b[9] = -2973333145929506703/747247079 460651200, b[5] = 166164647343343629239462104853/594150218553182839761 586950000, b[6] = 8835157365899998531682304/6999851448505791234978125, b[1] = 164949402359214517/2969763122318428800, c[7] = 998622639/20076 64595, a[7,1] = -1/572444230332379121732271745082032217713201845921217 5011645023681558308860592000000000*(-949504081843347292164889981986170 37813412248264331834224708786992683868338312704000*c[3]*a[9,5]-2154163 3786427712576469281471437646455082761994399786290553257397996171789355 72938375*c[3]*a[9,6]+1382986477628671268221543740478226134049588242665 862203369004475545779007474802995200*c[4]*c[3]*a[9,7]-2154172427789270 2063562050838590437418562514056303203653639804504977567788593750000000 *c[3]*c[4]*a[7,6]+1896120522377013879553118016521762460279721289330854 9049297536256985463313918457031250*c[3]*a[7,6]-91092377134984561557893 4938114011840594235599418929966910993044742010542860366811136*c[3]+237 5022066796538545365752756544932307890132470177463302738117271771972905 999556100096*c[3]*c[4]+11867863045103298054617006870231389060680856444 99675190370702968309206456006367772672*c[4]*c[3]*a[6,1]+27539225798049 1560844949118378074828938221709486634578517009872586334049233477550080 0*c[4]*c[3]*a[9,8]+244733354260007148797757517308640717122833745735192 8383305458828648085789086568072000*c[4]*c[3]*a[9,6]-272769474571153545 9797591267744741162816672170153332968168478737997975344788730019840*c[ 3]*a[9,8]-687904548115447622332989762550600392503991717677501658955493 977903384204349775114240*c[3]*a[9,7]+513809655635797151717596493798190 069869741456919753117116686556563246748525763072000*c[4]*c[3]*a[9,5]+1 2712259670285467201697451706374135693717278137945967747408211125082198 2717324103680*a[9,7]+5040694091026559630818615579973899134431858138815 46459386311273994383386127406202880*a[9,8]+976529401698929770596177578 1462306088398070606288717182137408083162177737326628864*c[4]+175465367 6800943226674305209174967540195429489550562074139441542066635050550732 8000*a[9,5]+3980826164988607435494323312546144364569684719803892355135 00562662838823368172718625*a[9,6]-508916230344027655034571765285826354 918601062091945944572718113167406303301708185600*c[4]*a[9,8]-255571550 9117203785311109664579683943656513004140059903997767117325751507915428 86400*c[4]*a[9,7]-1204448236825108573591643463977445870746484397241862 71602369949343574055122221637632-4522595406377589512514260804786152177 49928678226241024962083479474941148441947704000*c[4]*a[9,6]-9495040818 4334729216488998198617037813412248264331834224708786992683868338312704 000*c[4]*a[9,5])/c[3]/c[4], b[7] = 11056784337287011720526337507444900 5180061856568412069340896875/34071775464939293118333187684633880466821 1857149398834541328544, a[6,5] = -582169/21233664*(21233664*c[4]*c[3]* a[6,1]-18690048*c[3]*c[4]+8225568*c[3]-4826809+8225568*c[4])/(763*c[4] -141)/(763*c[3]-141), a[6,4] = 1/21233664*(2993946624*c[3]*a[6,1]+3640 811616*c[3]-2523050179)/(-c[4]+c[3])/(763*c[4]-141), a[4,2] = 0, a[5,2 ] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[ 5,4] = 19881/888389894*(-94+763*c[3])/c[4]/(-c[4]+c[3]), a[2,1] = 1/24 , a[6,3] = -1/21233664*(2993946624*c[4]*a[6,1]-2523050179+3640811616*c [4])/(763*c[3]^2-763*c[3]*c[4]+141*c[4]-141*c[3]), a[10,8] = -53651396 47887469158218320707351/8853429096639550069183960833070*a[9,8]-5011245 181150477887211158796875/359022955840383035894248406127836, a[4,1] = 1 /2*c[4]*(2*c[3]-c[4])/c[3], a[8,3] = 1/1370107016129978621204929098930 5315707264373460000000000*(9118002190033072431697968169725813650098727 42134051246080*a[9,7]+361549094758499642805329817666511911797188232821 9122401280*a[9,8]+3691944705140515708478636642198520451099172043770998 882304*c[3]+1723385697507514971218348570393544981609973574213505122304 *c[4]+2474334822799195044239620406956274280531443105926628769792*c[4]* c[3]*a[6,1]-1067591590963413044062012716306391758764300821455936356352 -3243879207122769011986296581846756044567057508839178424000*c[4]*a[9,6 ]-681041807062030157293252429787086009279156404408743424000*c[3]*a[9,5 ]-15450963804700786635217087224416136660711745145991634908375*c[3]*a[9 ,6]-19564677964591150883721039069471531113564157563341775831040*c[3]*a [9,8]-4934067851769669691762801215248791358173992214526816317440*c[3]* a[9,7]+125854383742786700102684918217534898176095744458234368000*a[9,5 ]+2855289510436187307425438137146363393394962078092818508625*a[9,6]+19 752799868096835026833741368216449681963812924527754444800*c[4]*c[3]*a[ 9,8]-6316905192329236916518749655307024702475941128369347231744*c[3]*c [4]-3650255283619467547553810659133052955644688889067383193600*c[4]*a[ 9,8]-681041807062030157293252429787086009279156404408743424000*c[4]*a[ 9,5]-1833113876968881876128843342574718749478593915432052838400*c[4]*a [9,7]+3685353892115808581664904992393947695602810897616108032000*c[4]* c[3]*a[9,5]+9919616227852885613378067165847591530866433740955009331200 *c[4]*c[3]*a[9,7]+1755375769528136706486201625495797774471393531378931 3032000*c[4]*c[3]*a[9,6])/c[3]/(763*c[3]^2-763*c[3]*c[4]+141*c[4]-141* c[3]), a[5,1] = 141/888389894*(13254-107583*c[3]-107583*c[4]+1164338*c [3]*c[4])/c[3]/c[4], a[7,4] = -1/4059888158385667530016111667248455444 7744811767462234125142011925945452912000000000*(1382986477628671268221 543740478226134049588242665862203369004475545779007474802995200*c[4]*c [3]*a[9,7]-27276947457115354597975912677447411628166721701533329681684 78737997975344788730019840*c[3]*a[9,8]+2753922579804915608449491183780 748289382217094866345785170098725863340492334775500800*c[4]*c[3]*a[9,8 ]+24473335426000714879775751730864071712283374573519283833054588286480 85789086568072000*c[4]*c[3]*a[9,6]+51380965563579715171759649379819006 9869741456919753117116686556563246748525763072000*c[4]*c[3]*a[9,5]+118 6786304510329805461700687023138906068085644499675190370702968309206456 006367772672*c[4]*c[3]*a[6,1]+1896120522377013879553118016521762460279 7212893308549049297536256985463313918457031250*c[3]*a[7,6]-21541633786 4277125764692814714376464550827619943997862905532573979961717893557293 8375*c[3]*a[9,6]-94950408184334729216488998198617037813412248264331834 224708786992683868338312704000*c[3]*a[9,5]-910923771349845615578934938 114011840594235599418929966910993044742010542860366811136*c[3]-6879045 4811544762233298976255060039250399171767750165895549397790338420434977 5114240*c[3]*a[9,7]+13903571333412047077612199360704480476451771633745 32163604127231253666696453636100096*c[3]*c[4]+127122596702854672016974 517063741356937172781379459677474082111250821982717324103680*a[9,7]+50 4069409102655963081861557997389913443185813881546459386311273994383386 127406202880*a[9,8]+57233268576649517122203076374495360810449735065589 9570438973563253396405124142628864*c[4]-189612052237701387955311801652 17624602797212893308549049297536256985463313918457031250*c[4]*a[7,6]+1 7546536768009432266743052091749675401954294895505620741394415420666350 505507328000*a[9,5]+39808261649886074354943233125461443645696847198038 9235513500562662838823368172718625*a[9,6]-1204448236825108573591643463 97744587074648439724186271602369949343574055122221637632-5089162303440 2765503457176528582635491860106209194594457271811316740630330170818560 0*c[4]*a[9,8]-25557155091172037853111096645796839436565130041400599039 9776711732575150791542886400*c[4]*a[9,7]-45225954063775895125142608047 8615217749928678226241024962083479474941148441947704000*c[4]*a[9,6]-94 9504081843347292164889981986170378134122482643318342247087869926838683 38312704000*c[4]*a[9,5])/(763*c[4]-141)/c[4]/(-c[4]+c[3]), a[5,3] = -1 9881/888389894*(-94+763*c[4])/c[3]/(-c[4]+c[3]), a[3,1] = -12*c[3]^2+c [3], a[9,1] = -1/5449722686217665780077639987200*(-1569225119308257792 197427200000+5346412750488006628328322367488*a[9,8]+186107361822243048 622863052800*a[9,5]+4222263716391676224631007912175*a[9,6]+13483259638 73108493477827309568*a[9,7]-1007091610428166284391805030400*c[3]*a[9,5 ]-4796891322764507900172506030400*c[3]*a[9,6]+247152956291050602271094 7840000*c[3]-5397820565396545153600710082560*c[3]*a[9,8]-2710719940114 725353176398704640*c[3]*a[9,7]+2471529562910506022710947840000*c[4]-53 97820565396545153600710082560*c[4]*a[9,8]-1007091610428166284391805030 400*c[4]*a[9,5]-4796891322764507900172506030400*c[4]*a[9,6]-2710719940 114725353176398704640*c[4]*a[9,7]+5449722686217665780077639987200*c[4] *c[3]*a[9,5]+5449722686217665780077639987200*c[4]*c[3]*a[9,7]-51902120 82112062647692990464000*c[3]*c[4]+5449722686217665780077639987200*c[4] *c[3]*a[9,6]+5449722686217665780077639987200*c[4]*c[3]*a[9,8])/c[3]/c[ 4], a[10,1] = -1/10175207200619880846993794744106420240803299863344846 056317132656925942778855792097549034157654220000976168960000*(55241254 5161303983252601207547058059203367050899689487704421232639933758344584 3867528610741436646172821177671680*a[9,7]+2190438776759976033974516790 5590804046834226022714177620218259137129415289959767231609572574304108 834189677690880*a[9,8]+68074673516320611483271860274225274748555164889 32058292407701340130141596268344597803517337469776388329425739776*c[3] +105152946197585262845718431636582148838815038771763641027093373701531 27207640414664683729431979350431194321739776*c[4]+17248631676234255856 2354972594969520762141017522056645536139909558074814451116558206514597 68623009871841143554048*c[4]*c[3]*a[6,1]-41260797064913318846772265371 6722576195367273084578317121930295067281813259816862205473358313760871 5386543104000*c[3]*a[9,5]-11403361270617289774606355576545211828238942 3865406883364403181215627041935483535320755335354053279066983873416500 *c[3]*a[9,6]-145159391155122674199258400003292938844388296292673778830 155789438081709442343823001693476688616382910582061465600*c[3]*a[9,8]- 3508130819714689508726798628220201879650724973279760686199439092518048 8095543013222970288608376656379888659046400*c[3]*a[9,7]+76248655126510 8513420037931507967014987507018413178803593606443047008331187866023210 638840396333982790960128000*a[9,5]+17298720846509324586241696875209565 0016626573270889382241275218882149398819119293637236471641361546760487 29034250*a[9,6]-178621368788168522338097862674290968294378362930151113 64898282523126582778080459283018315255812131875782412251136*c[3]*c[4]- 2211500688074975803531964067391379254728455511908642932810497316729412 1206209380378067356925980109880672270745600*c[4]*a[9,8]-25226782886266 2653258426059582822388971083776463361028998447083126168498153481667194 60199247962716396560000000000*c[3]*c[4]*a[8,7]+16657133357387798273230 6179271783194733347738928221991204456945102014576496763118905713565705 6899383520507812500*c[3]*a[7,6]-18924080500700930582605199065196670644 2619916415494806575477712778620110576204253431343222576878509843750000 0000*c[3]*c[4]*a[7,6]-561181408289523809008352041155022081443875732535 45588832051844759650231194696728530072249275345525884765625000000*c[3] *c[4]*a[8,6]+493956552088174602720893202891660061270911452075479401698 78967522817130582832016258240677747569759763153076171875*c[3]*a[8,6]+1 2547930845671587258511742625364638317118312062574758787743863812735451 241281489527487193062975911681872000000000*c[3]*a[8,7]-111058879044117 9424542087099676172334274522623636283776416423034584792133651706531433 3477608170921865827367526400*c[4]*a[9,7]-19652984630353788879043821302 0132336113563917562193854380620367014039553687993517031653269557049804 60363053104000*c[4]*a[9,6]-4126079706491331884677226537167225761953672 730845783171219302950672818132598168622054733583137608715386543104000* c[4]*a[9,5]+1465551545316142383742512692340938324871227991416417959342 91902798063264340827898222863606272160771207799196672000*c[4]*c[3]*a[9 ,8]+129552980115888735900853270455424891776442422379633855656600063866 274509181141057879201327739516151366040850272000*c[4]*c[3]*a[9,6]+2232 7651177680044170274637218855271321777675841385337302413674832364256987 038316727856466127191457091063350272000*c[4]*c[3]*a[9,5]+7052864381707 1626873950102122331989086091723331500564546639713972150555195094755855 394030553286538738666959872000*c[4]*c[3]*a[9,7]-6477125130039841728928 2852050864955182194000016662014074872352852122129425327615170671386974 64195476466827378688)/c[3]/c[4], a[4,3] = 1/2*c[4]^2/c[3], a[3,2] = 12 *c[3]^2, b[4] = 0, b[3] = 0, b[2] = 0, c[10] = 1, a[10,4] = -1/7216459 0075318303879388615206428512346122694066275503945511579127134345949331 858847865490479817163127490560000*(55241254516130398325260120754705805 9203367050899689487704421232639933758344584386752861074143664617282117 7671680*a[9,7]+2190438776759976033974516790559080404683422602271417762 0218259137129415289959767231609572574304108834189677690880*a[9,8]+6807 4673516320611483271860274225274748555164889320582924077013401301415962 68344597803517337469776388329425739776*c[3]+24222084461427954704294183 0000736055731655590369433740426988837612362460342647425056729283579583 24613060044619776*c[4]+17248631676234255856235497259496952076214101752 205664553613990955807481445111655820651459768623009871841143554048*c[4 ]*c[3]*a[6,1]-49395655208817460272089320289166006127091145207547940169 878967522817130582832016258240677747569759763153076171875*c[4]*a[8,6]- 1254793084567158725851174262536463831711831206257475878774386381273545 1241281489527487193062975911681872000000000*c[4]*a[8,7]-16657133357387 7982732306179271783194733347738928221991204456945102014576496763118905 7135657056899383520507812500*c[4]*a[7,6]-41260797064913318846772265371 6722576195367273084578317121930295067281813259816862205473358313760871 5386543104000*c[3]*a[9,5]-11403361270617289774606355576545211828238942 3865406883364403181215627041935483535320755335354053279066983873416500 *c[3]*a[9,6]-145159391155122674199258400003292938844388296292673778830 155789438081709442343823001693476688616382910582061465600*c[3]*a[9,8]- 3508130819714689508726798628220201879650724973279760686199439092518048 8095543013222970288608376656379888659046400*c[3]*a[9,7]+76248655126510 8513420037931507967014987507018413178803593606443047008331187866023210 638840396333982790960128000*a[9,5]+17298720846509324586241696875209565 0016626573270889382241275218882149398819119293637236471641361546760487 29034250*a[9,6]-359265833966990205585007521714657635530787399212543153 67323023710418594411102616838315670672939573686708338139136*c[3]*c[4]+ 4512129900765702410565971301253022270617823050673040819593144486179475 709614567462504815690338152175394017116160*c[4]*a[9,8]+166571333573877 9827323061792717831947333477389282219912044569451020145764967631189057 135657056899383520507812500*c[3]*a[7,6]+493956552088174602720893202891 6600612709114520754794016987896752281713058283201625824067774756975976 3153076171875*c[3]*a[8,6]+12547930845671587258511742625364638317118312 062574758787743863812735451241281489527487193062975911681872000000000* c[3]*a[8,7]-5917542257482979954020595449198378246391810534520059208444 548676289380589691632736216593224731577754467492618240*c[4]*a[9,7]+771 2379631480829145203393160282598748017729562168605858496608212585537305 22466564248570198093787841903229864750*c[4]*a[9,6]-4126079706491331884 6772265371672257619536727308457831712193029506728181325981686220547335 83137608715386543104000*c[4]*a[9,5]+1196719875887380523471552186822427 2137289443656640386934286591862869088283927487396074747045760867970888 6117580800*c[4]*c[3]*a[9,8]+106349129595460573863194579102383668407552 6731205346885761796737813561556481837258830861309730702134131702802720 00*c[4]*c[3]*a[9,6]+22327651177680044170274637218855271321777675841385 337302413674832364256987038316727856466127191457091063350272000*c[4]*c [3]*a[9,5]+60097818943731907867064713266164502911451117860601739106789 416694198326097606530743520875283932009812952350515200*c[4]*c[3]*a[9,7 ]-64771251300398417289282852050864955182194000016662014074872352852122 12942532761517067138697464195476466827378688)/(-c[4]+c[3])/(763*c[4]-1 41)/c[4], a[7,5] = 49/481814855931638011726514388588529768296609583302 093679963388913522288432000000000*(12712259670285467201697451706374135 6937172781379459677474082111250821982717324103680*a[9,7]+5040694091026 5596308186155799738991344318581388154645938631127399438338612740620288 0*a[9,8]-1619071197436763496371173347362177374282378276382651173858883 936661335652526606811136*c[3]-6983821320699285830862766334667032275997 44606357432489765753483836162931928913371136*c[4]+11867863045103298054 61700687023138906068085644499675190370702968309206456006367772672*c[4] *c[3]*a[6,1]+503870181907146206108093358952413774966307874795238321337 5046886371636445000250000000*c[4]*a[7,6]-94950408184334729216488998198 617037813412248264331834224708786992683868338312704000*c[3]*a[9,5]-215 4163378642771257646928147143764645508276199439978629055325739799617178 935572938375*c[3]*a[9,6]-272769474571153545979759126774474116281667217 0153332968168478737997975344788730019840*c[3]*a[9,8]-68790454811544762 2332989762550600392503991717677501658955493977903384204349775114240*c[ 3]*a[9,7]+175465367680094322667430520917496754019542948955056207413944 15420666350505507328000*a[9,5]+398082616498860743549432331254614436456 968471980389235513500562662838823368172718625*a[9,6]-44351073303285265 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6407171228337457351928383305458828648085789086568072000*c[4]*c[3]*a[9, 6]+5138096556357971517175964937981900698697414569197531171166865565632 46748525763072000*c[4]*c[3]*a[9,5]+13829864776286712682215437404782261 34049588242665862203369004475545779007474802995200*c[4]*c[3]*a[9,7])/( 763*c[4]-141)/(763*c[3]-141), a[10,5] = 2177/1223466904014799240924139 9526382363486482979863822003987539688406370246346334233647419089491810 7181946880000*(2266784383155410673236749671092572352348019170683421176 9520608568923296764529028782310563551510425064048640000*a[9,7]+8988304 9451161634787124034736173740608402813462205331028910171729836445303904 299583032888579821384308490240000*a[9,8]+49095732885674286998010506572 8869286736086890605214164016446959634810053128184645839638042830019614 3438888960*c[3]+168318474626075095237388549538664658867370341415643789 72324070979058665198073068538397062404215225220174888960*c[4]+55461838 1872484111132974188408262124637109381099860596579228005009886863186869 96207882507294607748784055123968*c[4]*c[3]*a[6,1]+42206669254156788021 5730638587040299172702024518963276595652546514179191332429595926251475 50382373465625000000*c[4]*a[8,6]+2385095935155223459574947723723598971 4371816993342819901472069268656072915747512127692346945809224920000000 000*c[4]*a[8,7]+142328736275604907929031397163181177608622570793279569 0414710801981098333867120479755541049307616597500000000*c[4]*a[7,6]-13 2671373199078195648785419201518513246098801634912642161392377835138846 70733661164163130492403886544651264000*c[3]*a[9,5]-3841192700686137062 2612490718188480024226449857403271599969736450162426665659480953780218 3747399224679634187500*c[3]*a[9,6]-48638841653359097406082013123191889 4214264870011791968617435893828831260758006954481234709123430611541688 320000*c[3]*a[9,8]-122663580450182861253875177237137071265357349449039 032483292371192116847030749283410659290707818824991979520000*c[3]*a[9, 7]+2451725245225429303601408139897000048191340895219224448854040009797 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000000*c[3]*c[4]*a[8,7]+6779278795999714118914306874261496494200300914 628036565156091673428512369132815621418356666853916866054687500*c[3]*a [7,6]-7701902537467130833322762839397676490452412873423568168698044978 096298076174559759244523550508591942500000000*c[3]*c[4]*a[7,6]-2283949 5490015339901035636683823528246012173383543899293793112978036788864300 9774249453812630792559959375000000*c[3]*c[4]*a[8,6]+201035142594405856 4205740937274050142487529844697353635755747965254279853159825617091546 57992728867880908203125*c[3]*a[8,6]+6419800387139656661668755641847276 8869093077644708603784892853232435658900575452730545057759236593672000 000000*c[3]*a[8,7]-455721968722564919874857390202872466551900713254545 65930783816274648786712337875344302985240919357101107200000*c[4]*a[9,7 ]-80644581721928468711591998939316039242158286004297995979307832301550 033323205151603429166389366110188892000000*c[4]*a[9,6]-132671373199078 1956487854192015185132460988016349126421613923778351388467073366116416 3130492403886544651264000*c[4]*a[9,5]+49106522823102934881140494018607 1960504786647608059199084911238961800792111449329043554273634232828960 358400000*c[4]*c[3]*a[9,8]+4363958571193717845882602495794194180267146 96604818233561786354936756563302166884208627333014796752298756000000*c [4]*c[3]*a[9,6]+717930906034728108368959396104671103594137486861264865 02937861197312723430991372115294103302866421514673152000*c[4]*c[3]*a[9 ,5]+246606994422210662315259708315455100694397336321431445426865615727 354781996551765161015444956180634525849600000*c[4]*c[3]*a[9,7])/(763*c [4]-141)/(763*c[3]-141), a[8,5] = 1/6636735699576136331199187966010040 090503880000000000*(18236004380066144863395936339451627300197454842681 02492160*a[9,7]+723098189516999285610659635333023823594376465643824480 2560*a[9,8]+611843559476031966327948461524019479761461009848557764608* c[3]+1551542437562766290608756685550263881473210066212610244608*c[4]+4 948669645598390088479240813912548561062886211853257539584*c[4]*c[3]*a[ 6,1]+3400862509100131308822109947751628833212528699774375000000*c[4]*a [8,6]+1921825031402508836877665012347820295065398125864000000000*c[4]* a[8,7]-1362083614124060314586504859574172018558312808817486848000*c[3] *a[9,5]-30901927609401573270434174448832273321423490291983269816750*c[ 3]*a[9,6]-39129355929182301767442078138943062227128315126683551662080* c[3]*a[9,8]-9868135703539339383525602430497582716347984429053632634880 *c[3]*a[9,7]+251708767485573400205369836435069796352191488916468736000 *a[9,5]+5710579020872374614850876274292726786789924156185637017250*a[9 ,6]-2993467521030844745786128026927214962567277865947236328125*a[8,6]- 955925600986867652514236002049310377441789052436800000000*a[8,7]-23780 67870218210241940204506743848693088333365039494463488*c[3]*c[4]-730051 0567238935095107621318266105911289377778134766387200*c[4]*a[9,8]-88373 0410021839295805893322488149152438692712433792712704-10399663113192299 592465662442704871525779423900952000000000*c[3]*c[4]*a[8,7]-1840324889 6761703465469999220812005671923116297360625000000*c[3]*c[4]*a[8,6]+161 98693039337124404502238897485567492473992990905966796875*c[3]*a[8,6]+5 172845628035319282754340918890949063745284021342400000000*c[3]*a[8,7]- 3666227753937763752257686685149437498957187830864105676800*c[4]*a[9,7] -6487758414245538023972593163693512089134115017678356848000*c[4]*a[9,6 ]-1362083614124060314586504859574172018558312808817486848000*c[4]*a[9, 5]+39505599736193670053667482736432899363927625849055508889600*c[4]*c[ 3]*a[9,8]+35107515390562734129724032509915955489427870627578626064000* c[4]*c[3]*a[9,6]+73707077842316171633298099847878953912056217952322160 64000*c[4]*c[3]*a[9,5]+19839232455705771226756134331695183061732867481 910018662400*c[4]*c[3]*a[9,7])/(763*c[4]-141)/(763*c[3]-141), a[10,9] \+ = 255482840375593769438967652731/8853429096639550069183960833070, a[10 ,7] = 526180744020800178157171673671875/359022955840383035894248406127 836*a[8,7]-5365139647887469158218320707351/885342909663955006918396083 3070*a[9,7]-1987490824767545062357006754936745522836709757055434291947 233012500/800134180797252928440343744342240823139585255602078766676541 96707053, a[9,4] = -1/5449722686217665780077639987200*(156922511930825 7792197427200000-5346412750488006628328322367488*a[9,8]-18610736182224 3048622863052800*a[9,5]-4222263716391676224631007912175*a[9,6]-1348325 963873108493477827309568*a[9,7]+1007091610428166284391805030400*c[3]*a [9,5]+4796891322764507900172506030400*c[3]*a[9,6]-24715295629105060227 10947840000*c[3]+5397820565396545153600710082560*c[3]*a[9,8]+271071994 0114725353176398704640*c[3]*a[9,7])/c[4]/(-c[4]+c[3]), a[9,3] = 1/5449 722686217665780077639987200*(1569225119308257792197427200000+539782056 5396545153600710082560*c[4]*a[9,8]+1007091610428166284391805030400*c[4 ]*a[9,5]+4796891322764507900172506030400*c[4]*a[9,6]+27107199401147253 53176398704640*c[4]*a[9,7]-5346412750488006628328322367488*a[9,8]-2471 529562910506022710947840000*c[4]-1348325963873108493477827309568*a[9,7 ]-4222263716391676224631007912175*a[9,6]-18610736182224304862286305280 0*a[9,5])/c[3]/(-c[4]+c[3]), a[8,4] = -1/27402140322599572424098581978 610631414528746920000000000*(18236004380066144863395936339451627300197 45484268102492160*a[9,7]+723098189516999285610659635333023823594376465 6438244802560*a[9,8]+2507072516928295618155888916760845561508198092062 957764608*c[3]+8323588288367765741238081508423285303910093143906050244 608*c[4]+4948669645598390088479240813912548561062886211853257539584*c[ 4]*c[3]*a[6,1]-1279783053023699309568012894973393865926146429113159179 6875*c[4]*a[8,6]-32510205966328104458766759065431287686798858954784000 00000*c[4]*a[8,7]-1362083614124060314586504859574172018558312808817486 848000*c[3]*a[9,5]-309019276094015732704341744488322733214234902919832 69816750*c[3]*a[9,6]-3912935592918230176744207813894306222712831512668 3551662080*c[3]*a[9,8]-98681357035393393835256024304975827163479844290 53632634880*c[3]*a[9,7]+2517087674855734002053698364350697963521914889 16468736000*a[9,5]+571057902087237461485087627429272678678992415618563 7017250*a[9,6]-1263381038465847383303749931061404940495188225673869446 3488*c[3]*c[4]-7300510567238935095107621318266105911289377778134766387 200*c[4]*a[9,8]+127978305302369930956801289497339386592614642911315917 96875*c[3]*a[8,6]+3251020596632810445876675906543128768679885895478400 000000*c[3]*a[8,7]-366622775393776375225768668514943749895718783086410 5676800*c[4]*a[9,7]-64877584142455380239725931636935120891341150176783 56848000*c[4]*a[9,6]-1362083614124060314586504859574172018558312808817 486848000*c[4]*a[9,5]-213518318192682608812402543261278351752860164291 1872712704+39505599736193670053667482736432899363927625849055508889600 *c[4]*c[3]*a[9,8]+3510751539056273412972403250991595548942787062757862 6064000*c[4]*c[3]*a[9,6]+737070778423161716332980998478789539120562179 5232216064000*c[4]*c[3]*a[9,5]+198392324557057712267561343316951830617 32867481910018662400*c[4]*c[3]*a[9,7])/(763*c[4]-141)/c[4]/(-c[4]+c[3] ), a[10,3] = 1/1626468745308242723424216539530693409488664110080000*(1 24504516450975770684415923553512030283835665905930240*a[9,7]+493688137 796243958682840977289935377085158724595875840*a[9,8]+46235692773000895 6846492787106928491971152477935443968*c[3]+236997092741674919107422917 167696771827109097611603968*c[4]+3887552093270170129274801845925220489 37686191900196864*c[4]*c[3]*a[6,1]-92994911718538689814352428980225013 966894630927872000*c[3]*a[9,5]-210979854699818037403291910552414397407 4910306567168875*c[3]*a[9,6]-26715180789966960317376430189519198064962 84445862789120*c[3]*a[9,8]-6737372060432234966823358132718416957912525 75079608320*c[3]*a[9,7]+1718516717210216941523419722963529091655588854 6304000*a[9,5]+389884135159558889565716374677463041080684604490132125* a[9,6]-809724893600225476582798675432315703643912612250472448*c[3]*c[4 ]-498435139121207842900945217456184755710977558486220800*c[4]*a[9,8]-1 45983529768064337515597167144263275907172091577360384-2503080741755736 50304497821396389042915964442938675200*c[4]*a[9,7]-4429452896487296260 15488425669070437204091384982872000*c[4]*a[9,6]-9299491171853868981435 2428980225013966894630927872000*c[4]*a[9,5]+26972057528332027243505049 71057226727712594873226854400*c[4]*c[3]*a[9,8]+23969308936310688273036 71409826246408416466147105896000*c[4]*c[3]*a[9,6]+50322778468968099523 6531229162494224515890804240896000*c[4]*c[3]*a[9,5]+135450397585789145 5193842820747835742871495531646873600*c[4]*c[3]*a[9,7])/(-c[4]+c[3])/c [3]/(763*c[3]-141), a[8,1] = -1/38637017854865397117979000589840990294 48553315720000000000*(182360043800661448633959363394516273001974548426 8102492160*a[9,7]+7230981895169992856106596353330238235943764656438244 802560*a[9,8]+25070725169282956181558889167608455615081980920629577646 08*c[3]+3446771395015029942436697140787089963219947148427010244608*c[4 ]+4948669645598390088479240813912548561062886211853257539584*c[4]*c[3] *a[6,1]-1362083614124060314586504859574172018558312808817486848000*c[3 ]*a[9,5]-30901927609401573270434174448832273321423490291983269816750*c [3]*a[9,6]-39129355929182301767442078138943062227128315126683551662080 *c[3]*a[9,8]-986813570353933938352560243049758271634798442905363263488 0*c[3]*a[9,7]+25170876748557340020536983643506979635219148891646873600 0*a[9,5]+5710579020872374614850876274292726786789924156185637017250*a[ 9,6]-6204972495842973385054315041356670588923090934895494463488*c[3]*c [4]-7300510567238935095107621318266105911289377778134766387200*c[4]*a[ 9,8]-6535961327705759880667762383720772496330870585232000000000*c[3]*c [4]*a[8,7]-14539547111275163753672099161827906642474562981640625000000 *c[3]*c[4]*a[8,6]+1279783053023699309568012894973393865926146429113159 1796875*c[3]*a[8,6]+32510205966328104458766759065431287686798858954784 00000000*c[3]*a[8,7]-3666227753937763752257686685149437498957187830864 105676800*c[4]*a[9,7]-648775841424553802397259316369351208913411501767 8356848000*c[4]*a[9,6]-13620836141240603145865048595741720185583128088 17486848000*c[4]*a[9,5]-2135183181926826088124025432612783517528601642 911872712704+395055997361936700536674827364328993639276258490555088896 00*c[4]*c[3]*a[9,8]+35107515390562734129724032509915955489427870627578 626064000*c[4]*c[3]*a[9,6]+7370707784231617163329809984787895391205621 795232216064000*c[4]*c[3]*a[9,5]+1983923245570577122675613433169518306 1732867481910018662400*c[4]*c[3]*a[9,7])/c[3]/c[4], a[10,6] = 39544589 18230699741846244609537668670264912300169257659977176734375/8001341807 9725292844034374434224082313958525560207876667654196707053*a[7,6]+5261 80744020800178157171673671875/359022955840383035894248406127836*a[8,6] -5365139647887469158218320707351/8853429096639550069183960833070*a[9,6 ]-204388824020215684266404402278957056/8875951095645670561315492876310 816875, a[7,3] = 2995867917/451098684265074170001790185249828382752720 1307495803791682445769549494768000000000*1/c[3]*(190562345600601438782 61806330132012880494626400586075308261052735681024000*c[4]*c[3]*a[9,5] +362176403838556281643150505551710768725680943232938084393503063593996 288*c[4]-3521532205725400113807227644231472891415127552401882855130810 531757568000*c[4]*a[9,5]-947866854511635439704982184551764530225305658 7608119707359979112359628800*c[4]*a[9,7]+51292369503005520602475277078 935910394461575718758832175288397608016998400*c[4]*c[3]*a[9,7]+5156573 3272212388756182935850277824427315370910835050661552291928350380032*c[ 3]*c[4]+44015673787661247085231772549957808587864227946041142372814687 162517684224*c[4]*c[3]*a[6,1]+1021378132259299248245748636209817957509 63160698365839099029453338855833600*c[4]*c[3]*a[9,8]+90767001987886944 912463990131927181770875465073743079668082814978866024000*c[4]*c[3]*a[ 9,6]-18874746611869094888945027222225993710204201387247160305325233185 817395200*c[4]*a[9,8]-167734564617196058095116941135016155041853742796 82535036958947460052568000*c[4]*a[9,6]+6507680747146545426563815174792 10586749060268530361051865588840075776000*a[9,5]-129198834769716415098 20346556817568453945380565888988002187269161898483712*c[3]+18694987120 327484461431265058204793770107018516892425445274516679285800960*a[9,8] -352153220572540011380722764423147289141512755240188285513081053175756 8000*c[3]*a[9,5]-25513086957462878113668548868753244189169331707204443 076531222191224238080*c[3]*a[9,7]-446707216663505890392861648230240597 1400375463815374415415374244305772544+14764136156409444696913939089488 401146913167985762231360656573545567104125*a[9,6]-79893871541421321303 158407980706738121239341653450939916177061101189364875*c[3]*a[9,6]+471 4738218875839861110439568144439620803244784686535352281654428522434560 *a[9,7]-10116507214758773506434081730078196912476351154885759301237202 9973723873280*c[3]*a[9,8])/(-c[4]+c[3])/(763*c[3]-141), c[9] = 20/21, \+ c[2] = 1/24, c[5] = 141/763, c[6] = 169/192, c[8] = 104/105\}:" }}} {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 33 "We have expressions in terms of " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6# \"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" } {TEXT -1 69 " for the linking coefficients in rows 2 to 6 of the Butc her tableau." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$ (5-i)],i=2..5)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7' #\"\"\"\"#CF(%!GF+F+7'&%\"cG6#\"\"$,&*&\"#7F))F-\"\"#F)!\"\"F-F),$*&F3 F)F4F)F)F+F+7'&F.6#\"\"%,$*&#F)F5F)*(F:F),&*&F5F)F-F)F)F:F6F)F-F6F)F) \"\"!,$*&F?F)*&F:F5F-F6F)F)F+7'#\"$T\"\"$j(,$*&#FI\"*%*)*Q)))F)*(,*\"& aK\"F)*&\"'$e2\"F)F-F)F6*&FSF)F:F)F6*(\"(QV;\"F)F-F)F:F)F)F)F-F6F:F6F) F)FC,$*&#\"&\"))>FNF)*(,&\"#%*F6*&FJF)F:F)F)F)F-F6,&F:F6F-F)F6F)F6,$*& #FZFNF)*(,&FgnF6*&FJF)F-F)F)F)F:F6FinF6F)F)Q(pprint06\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "c[6]= subs(e7,c[6]);``;\nfor ii from 2 to 5 do a[6,ii]=subs(e7,a[6,ii]) end \+ do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$p\"\"$#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"'\"\"$,$*&#\"\"\"\")kOB@F,*&,(*(\"+Cm%R*HF,&%\"cG6#\"\"%F,&F%6$F'F ,F,F,\"+z,0BD!\"\"*&\"+;;\"3k$F,F2F,F,F,,**&\"$j(F,)&F36#F(\"\"#F,F,*( F>F,F@F,F2F,F9*&\"$T\"F,F2F,F,*&FEF,F@F,F9F9F,F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,$*&#\"\"\"\")kOB@F,*(,(*(\"+Cm%R*HF ,&%\"cG6#\"\"$F,&F%6$F'F,F,F,*&\"+;;\"3k$F,F2F,F,\"+z,0BD!\"\"F,,&&F36 #F(F;F2F,F;,&*&\"$j(F,F=F,F,\"$T\"F;F;F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,$*&#\"'p@e\")kOB@\"\"\"*(,,**F-F.&% \"cG6#\"\"%F.&F36#\"\"$F.&F%6$F'F.F.F.*(\")[+p=F.F6F.F2F.!\"\"*&\"(obA )F.F6F.F.\"(4o#[F=*&F?F.F2F.F.F.,&*&\"$j(F.F2F.F.\"$T\"F=F=,&*&FDF.F6F .F.FEF=F=F.F=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can find some simple order conditions that are not ye t satisfied and determine which parameters are related by them." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_10 := SimpleOrderConditi ons(7,10,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 54 do\n \+ eq := simplify(subs(e7,SO7_10[ii]));\n if lhs(eq)<>rhs(eq) then\n \+ print(ii,indets(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#h<)&%\"aG6$\"\"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F (\"\"'&F&6$F,F3&F&6$F,F)&F&6$F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#f<)&%\"aG6$\"\"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F(\"\"'&F&6$F,F3 &F&6$F,F)&F&6$F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#e<)&%\"aG6$\" \"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F(\"\"'&F&6$F,F3&F&6$F,F)&F&6$F) F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<,&%\"aG6$\"\"*\"\"(&F&6$F( \"\")&%\"cG6#\"\"$&F.6#\"\"%&F&6$F(\"\"&&F&6$F(\"\"'&F&6$F,F9&F&6$F,F) &F&6$F)F9&F&6$F9\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#a<)&%\"aG 6$\"\"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F(\"\"'&F&6$F,F3&F&6$F,F)&F& 6$F)F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple or der comditions given in abreviated form as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "SO7 := Si mpleOrderConditions(7):\n[seq([i,SO7[i]],i=[54,59,61])]:\nlinalg[augme nt](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linal g[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7% 7%\"#a%#~~G/*&%\"bG\"\"\"-%!G6#*(%\"aGF-%\"cGF--F/6#*&)F3\"\"$F-F2F-F- F-#F-\"$o\"7%\"#fF)/*(F,F-)F3\"\"#F-F4F-#F-\"#G7%\"#hF)/*(F,F-F3F--F/6 #*&)F3\"\"%F-F2F-F-#F-\"#NQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cdcns3 := [seq(SO7_10[ i],i=[54,59,61])]:\neqns3 := simplify(subs(e7,cdcns3)):\nnops(%);\nind ets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"aG6$\"\"*\"\"(&F%6$F'\"\")&%\"cG 6#\"\"%&F%6$F'\"\"'&F%6$F+F2&F%6$F+F(&F%6$F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 3 equations for the linkin g coefficients " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,7]" "6#&%\"aG6$\"\"*\"\"(" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,8]" "6#&%\"aG6$\"\"*\"\")" } {TEXT -1 30 " in terms of the parameters " }{XPPEDIT 18 0 "a[7,6]" " 6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,6]" "6#&% \"aG6$\"\")\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,6]" "6#&% \"aG6$\"\"*\"\"'" }{TEXT -1 89 ", and subsitute back into the expressi ons previously obtained for the other coefficients." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[ solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "e8 := solv e(\{op(eqns3)\},\{a[8,7],a[9,7],a[9,8]\}):\ninfolevel[solve] := 0:\ne9 := `union`(map(u_->lhs(u_)=simplify(subs(e8,rhs(u_))),e7),e8):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "indets(map(rhs,e8));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <&&%\"cG6#\"\"%&%\"aG6$\"\"*\"\"'&F)6$\"\")F,&F)6$\"\"(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49793 "e9 := \{b[8] = 758177409913942921875/787858 59799058312576, b[10] = -1895689785077201/288707084450720, b[9] = -297 3333145929506703/747247079460651200, b[5] = 16616464734334362923946210 4853/594150218553182839761586950000, b[6] = 8835157365899998531682304/ 6999851448505791234978125, a[10,4] = -1/558579568234474740277628136405 8527772618879293485280604090126926504515905449107640036555531591211121 48297646080000*(286987632426880009270808380248913623175778584514032659 7956756921744083530278315013457458456153225067225893807910956713525431 5008*c[3]+116227259283209442335497205637192821156785467405676243942480 731083355640357492648237543640827391575992232679975381905736265105408* c[4]+13865971146295113617509156502781555190938965432232433674733540972 095082093788844499844689152841195151698362691107415859904118784*c[4]*c [3]*a[6,1]-41875913300654540471045092813209028179576574970541289203295 867185595786240465781383329161645027556504977031155997519651432038400* c[4]^2*c[3]*a[6,1]-846631153254761989061389996540837603426896658855490 3199224414667724367000701189716376348316229707029538070797590416616699 218750000*c[3]*c[4]^2*a[7,6]+11261373328717687744302536150712601553546 0776121102185163714212770897976076124385985571073681838046140893638967 56512332185600000*c[4]^2*a[9,5]+54676233015852369789083183054208270257 8839121447347965771451412849403210453878984583063293513400167722046441 03427574005126953125*c[4]*a[8,6]-1272711149431224338721087607050242496 0310191770039805565534450436499693106715329371710475127198787507409105 938401816091271431250*c[4]^2*a[9,6]+6887082319262582769107729391342801 5919976427805250862741154508390420325109388626316419095901082800483355 658375890678564823418750*c[4]^2*c[3]*a[9,6]-60939204608592877651793156 6169767020237989873619865016169602442157412452099878769553125739143563 32769859465625710772407500800000*c[4]^2*c[3]*a[9,5]-264175378051325756 7293957591688559100443176548399706246109179707761358685318879048828492 00853574209141395528000380993272597708800*c[3]*c[4]^3+5374421341609747 8028935909013991491289864280143403712008077343773436411347031771375826 04887400313092661237144822274261666800000*c[4]^3*a[9,6]-67325404236274 9932322635703941657059004578540496293084409819815089071867385516274418 4788549565495284775390286829341324083200000*c[4]^3*a[9,5]+433800230208 0495594325496813034190383090602322812175703990292487849158494596004462 406341134228426469575954790619467321582031250000*c[4]^2*a[7,6]+4142905 7198939911934675060089214069323580195512277689205642396767366742487979 46689487109503456446465940431901253830156250000000000*c[3]*c[4]^3*a[7, 6]+2814465988861353127873279984070598573787667388941829667493098931669 3574857790710252881540981289454556848625328590389676238438400*c[4]^3*c [3]*a[6,1]-25606140632762048392089742174848955739820681273012795934714 8106200379456172800252813646605552105105902068516808688477880859375000 0*c[4]*a[7,6]+36432115909416893500863194475708108937623647262317136411 6803204902809811925637530057659125058047723566214382187999108530176000 00*c[4]^3*c[3]*a[9,5]-290828615861577132879986514735287289745861317371 7520018593121510576736301970584507784062077366268716099662369857727135 9232400000*c[4]^3*c[3]*a[9,6]-3316906710713254742166285099837481214468 7144513081504388560024395105489031669065980981308677613009134409748363 59923113132032000*c[3]*a[9,5]+1401852703393677581542266996489114785667 3304109538955348789415007173917129983560529439669568779616854982844814 236558427327030500*c[3]*a[9,6]+456375071348254484124833846957087477374 1819706513340152498721616451083510573203429336388603904237135741207010 39796235858975129600*c[3]*c[4]^2-1535257595453608402742289508883842201 4933015019473710751188526922865776790473061450493069788392547215174703 005235153145539067904+612953926881479578827583485028944759161322067672 9347468921315124128275168368726478792089808051683208324737246746384783 11424000*a[9,5]-233062002836920662334326683384801585506737979330174835 9314610457547547115054746655223190463975135582470750728069678847474826 000*a[9,6]-71094762153552038795245185229390043044214620148399578691267 03444172448194490040449616681315178659593333430692244233154296875000*a [8,6]+3320009971448591955329471739845285053004218692495671850749127061 11709393743849505746222280900171015954849833837302303906250000000*a[7, 6]-2217846281534803590754078930326982531744181650896316704630723366839 76755844539940256171561125885026046114752504056095523825778688*c[3]*c[ 4]-2371857404455728500000248012326490665430008651162492834082878617001 60406068113357875346186621909007639382200276637243178955571200*c[4]^2- 8871626872010264348124390518768969025083197421249521226175409330847946 1974397961144023803048491569882209238227706201171875000000*c[3]*c[4]^3 *a[8,6]-61580257683103911932175852213151233751260605696694292242157156 2593583946673084953227291780827307925145318612016354824414062500000*c[ 3]*a[7,6]-211005652341471948179363999038240385952292131976976288285458 8204084200862623230751916674016739720238976383102780986593750000000000 *c[4]^3*a[7,6]-6350648679385234976530533768497714961965260592549318062 6761611095523057554052184241793991634867050789322509058872436614990234 375*c[4]^2*a[8,6]+4941812216047246444683718128458699745668359066731280 9337119150243609134493060358373028561250151069009925481882183591342773 43750000*c[3]*c[4]*a[7,6]-13521165311206770562393590048506034126404566 3825785267159471663133957923816916465607913238773170480071919533334488 600616455078125*c[3]*c[4]*a[8,6]+1334361857226692885447962320144786754 8471594617901968541949913616529973046371388526544170807029087104475659 981066692565917968750*c[3]*a[8,6]+827965972918631053877182049229742049 6321445302110912038493691704912236965167196812664217652511006332081182 194722152333364161750*c[4]*a[9,6]-516805764911996600049545848832342425 3404912373402225545114271666857407492450867941335973317011272382277268 939639729718034432000*c[4]*a[9,5]-524164325356570770352688273960692222 1173546094597063238932905859596432571773538160107303101987668665673229 4004084709763687367750*c[4]*c[3]*a[9,6]+279661559310534330381420909687 2888443509183078656665312710772540292341784921994495914430950978440303 3174157453511445211774976000*c[4]*c[3]*a[9,5]+136741145962431251161539 9225816149318740605161419825642064184819121681136245671315532944265923 57202249030491077274324813270220800*c[4]^3+163944874043702132776610624 2655864524949843822275468535898732261663906178032780146960335023569765 5771155311389392626953125000000*c[4]^3*a[8,6]+210088076760636573197333 9894386266752738362007933393380236666332846612044756113860273958596823 47539184500131855926438140869140625*c[3]*c[4]^2*a[8,6])/(-998622639+20 07664595*c[4])/(-104+105*c[4])/(763*c[4]-141)/c[4]/(-c[4]+c[3]), b[1] \+ = 164949402359214517/2969763122318428800, c[7] = 998622639/2007664595, b[7] = 11056784337287011720526337507444900518006185656841206934089687 5/340717754649392931183331876846338804668211857149398834541328544, a[6 ,5] = -582169/21233664*(21233664*c[4]*c[3]*a[6,1]-18690048*c[3]*c[4]+8 225568*c[3]-4826809+8225568*c[4])/(763*c[4]-141)/(763*c[3]-141), a[8,1 ] = -1/285716178370373365337725245044195133105412493724021611868511178 0461374996855130704800758492385516912640000000000*(-435104093300697124 2823334733433584009870327112687647196444084136862781659926475281725714 001330442283496448432000*c[4]*c[3]*a[9,6]+3659482686963620315830937990 6558337472587777280683017628500373675356574939409708648989430409709555 52188486647808*c[4]*c[3]*a[6,1]+54505512508911947668802771929831361108 2874557751982145177781049013998571510378012885344030329658473024638156 8000*c[4]*c[3]*a[9,5]+215429672127812353366309067913099359199382664474 823680437744588733807837943705006654413442400393030368750000000000*c[3 ]*c[4]*a[7,6]-17102678109023838526269647012107568874498915681454912258 671221488072411703418515594819411334274112117818005651456*c[3]*c[4]+80 4058678314525485239960940254436887800414315712920386236718038397971446 985102247343808223050579766674966224000*c[4]*a[9,6]-100724472657360217 8414310726357303003442795709607201605112282148243431174088640888818263 542286786955392843776000*c[4]*a[9,5]-114539598782299340764953487358712 5950019234077248055253561776622772138478737815838940073943980535674812 50000000000*c[4]*a[7,6]+8231827414329509252493856187179835264272582647 734011152639464902018804986082400231481029372163033221715141328896*c[4 ]+24527533358695942739908376153716640328679799934507957468317403929728 17499185050911768213129902682787109375000000*c[4]*a[8,6]-4998092345389 2721896636480888501220376287243716616648133645343298375858724265194707 29146193871363297265625000000*c[3]*c[4]*a[8,6]+18613565720429607753134 7067387129388578550714357294136724550174183910610152684620345183695232 551717837996032000*a[9,5]-18962299265416816520263662748600933179528994 9459610427051973101541737107356698677732270165446179281939160156250000 *c[3]*a[7,6]-465988386546803866469217785815386729036335509887287928219 0799937252629994205442190530904486106626889537445429248+38298224879071 7781269345609349101925868794417731360612603671989130109427358111626360 2321178254399718285936380250*c[3]*a[9,6]+86960513601712071954501559506 9898054199151177616287925365574421694278105687531522817687659018918803 9453647568896*c[3]-100724472657360217841431072635730300344279570960720 1605112282148243431174088640888818263542286786955392843776000*c[3]*a[9 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134426746647605637932373130625417216*c[4]*a[9,5]-116741947881750706808 1724343782649251873881523460963905538226054291812858446404636703125000 000*c[4]*a[8,6]+221341996695852779303438640368487789173791638768700316 1286677937507106234350168137239887872000-88593658703863486237952044084 577425801362244426840363529765339674020068669657227537900634112*a[9,5] -479859606741394244816054933750276123849696931957354464998443022323231 14068159804086669531250000*a[7,6]+336857545578062061687013345689534846 085420437529680706739822779501449271751498300373725337138*a[9,6]+10275 7235375082653388443444843368606024315613262970260435395939153810610977 8345747931396484375*a[8,6])/c[3]/(-c[4]+c[3]), a[9,4] = -1/25942599312 8964850458539829750386577140753767294045709953020260714675264611164823 1905441087488*(-426192836369824465682329065436358183408066954610075436 441424906539244003269965417061514557020291307929600*c[3]-6711638378551 0462543415708279971943984519174052679911874484692033181020565834594089 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695591756886743160446958321309946157264081914866438709541076992000*c[4 ]^2+989882330929596544575624944420297699763772183810632013052542482690 5087559868464590409202061129664062500000*c[3]*a[7,6]+45478064101936494 2685881404059972512255436518700950522172667351969897825970465579553482 92287890625000000*c[4]^2*a[8,6]-21361670447829921153330440363693498440 999859783987477480247160897037509149578159358695062296785964843750000* c[3]*c[4]*a[7,6]+53730313462135549817190830791208511013201475614745698 1007995484405360941315747404523014539453057861328125*c[3]*c[4]*a[8,6]- 2822704615148880234286207236482014497919936742612883138941346702992428 35894827365699917757685925292968750*c[3]*a[8,6]-3254976201415121157339 8645061615182443308064763420491496809674455692986120773487231307230013 320328819050*c[4]*a[9,6]+277876220914304071303352351615288995211602067 18244268877146896465034747726458818495303810262863393587200*c[4]*a[9,5 ]+17613807387799556333690188781569066811520605258503429086571476318931 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7516384765625000000*c[3]*c[4]*a[8,6]+125955438051753357214622116663609 9705887480634873149851578368407736136864385462239005773146852687872000 *c[4]*c[3]*a[9,5]+1432311826339691060004670465334114189929510985534501 21897739264583963040965031200186920070368750000000000*c[3]*c[4]*a[7,6] -369821309875231010217706861025284378055496042787591900571670710454138 4796857787115708040589383254933504*c[3]*c[4]+1858079272804944087300630 6537558190482194922997787164039203232659076340670395245049974390616262 3696000*c[4]*a[9,6]-72950221826661544517971529333767763970807769714258 072350804738077586876332898632563984562441250000000000*c[4]*a[7,6]-100 5471266063952013198851906961482222547143705483092635596600462331577867 483090210860316314908382128000*c[4]*c[3]*a[9,6]+2716283826491652395026 8492044146993800410814214918688363632320035779835115975202980699286258 29486526464*c[4]+56680068975012590947838296628043376134164297750948291 8940809168697287025304857287792974849609375000000*c[4]*a[8,6]-23276168 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668160781247297488000*c[4]*a[9,6]-189330410960275390755188794919190504 7024609021249657271202733726071315283292587321967529962411812500000000 00*c[4]*a[7,6]-7192135966155448750411387690495482337879718925320561622 422483107057776486106544278283842600539657361584000*c[4]*c[3]*a[9,6]+1 3636894014467885771207330017297646582946066801390250472664613322054535 285690257777241544707532005077680128*c[4]+4054325333782650630498873357 8039426948767722181253313191836079836916940920056441795831490992558593 75000000*c[4]*a[8,6]-1664944351535662368886278795147668399300201843192 552248041873866442540977357586902727089960734804672512000*c[4]*a[9,5]+ 3076764791173373447089977852107748942350307469071426827967289844933136 01320340436808020556308790902784000*a[9,5]+633057801187641061885169020 6738211016154544262391536011403123151524813677875031161614423955683344 240144250*c[3]*a[9,6]-770633208780554012780687757919237158042827693115 4877812810410757940343422629640383925768007557974761406464-16649443515 3566236888627879514766839930020184319255224804187386644254097735758690 2727089960734804672512000*c[3]*a[9,5]-35686509448399372737203624868170 1205955299221282906767157307161064529323723413472057058436340750122070 3125*a[8,6]+1666502054806590679043068038611624754933119398912458743923 23958180235563998149613235683626899789746093750000*a[7,6]-901802175757 0416227729510024543756652581348236668127812861218446207073427701287581 47706434925812597656250000*c[3]*a[7,6]-1169870903898524111740613786566 3011183195160432466665499447449074246379142599992054883797873543270483 09750*a[9,6]+358362280870838766976014808443815325049881255506514963287 74700409768541783464824377498897626526932038320128*c[3]+19311210432006 1853889974225350452496555952699176494938539734300632791400000684027786 90467158013641357421875*c[3]*a[8,6])/(763*c[3]-141)/c[3]/(-c[4]+c[3]), a[10,8] = 5951113579723375/147354657844158824229032633629544910874858 120201834698294687588746407806910337300545142784*(-8258421418348454772 00288395154454989632689232828293774654585373743159479750*a[9,6]-195259 13695778558326720483527682296387493485659482909134441543071629836288*c [4]-453821195943256774535150294752266254962609168689738602254157157335 625000000*c[4]*a[8,6]-807613651683840126392809024202501389332454108832 440662351027260754000000000*c[4]*a[7,6]+230767902310443330293738756777 41870610109141153344284662679530197596766208+3994571985125541400856270 82360067693170213278690446998859127914529794921875*a[8,6]+710868266325 880111252003776511576743735337210378554541340227120142843750000*a[7,6] +938234859362664684156540661950623420174416169840428430376807051826548 048000*c[4]*a[9,6])/(-104+105*c[4]), a[9,8] = -625/7213928131542586288 13749797158802780672179835691844523770590627841521483776*(-63526318602 680421323099107319573460740976094832945674973429644134089190750*a[9,6] +128948978135826320656351616360457366155556979559082339073796945847150 1824*c[4]-349093227648659057334730995963281734586622437453645078657043 96718125000000*c[4]*a[8,6]-6212412705260308664560069416942318379480416 2217880050950079020058000000000*c[4]*a[7,6]-98975987875288421477549244 9750509890594134103387447534562860451688873984+30727476808658010775817 467873851361013093329130034384527625224194599609375*a[8,6]+54682174332 760008557846444347044364902718246952196503180017470780218750000*a[7,6] +721719122586665141658877432269710323211089361415714177212928501405036 96000*c[4]*a[9,6])/(-104+105*c[4]), a[9,7] = -404616403318175429665672 24375/4915909273589333322681838105120622073719568013311053801291630085 85256453213721465323506175172865450573824*(506133475657806036856445202 71257356920483298003019050033251318130640359284492935888896*c[4]+36058 4814337750604220206449054854911163806068732349285008513648510361654741 75625000000*c[4]*a[8,6]-1005622214475735417551140728108093624683008020 686425251680530292986661219692686000000000*c[4]*a[7,6]+214737343618843 7812411256302130788517902661838641913853600789350892279280620995038400 0*c[4]*a[9,6]+88515705336666294565699366172014490922618935154169722673 9633434972634094416999656250000*a[7,6]-2732275356106012978649401497293 8509471236834656068757587559645611284638072689326555136-31738975845354 0896422994218178492083263975133415453276908535451032557914850816699218 75*a[8,6]-189013599331169786613282455760470447669557213922126792322152 81265666416584632716744250*a[9,6])/(-998622639+2007664595*c[4]), a[7,3 ] = 1/9952420820076276841624648082009505162177273976923488429803831493 1265817450160018664029683712000000000*(-100547126606395201319885190696 1482222547143705483092635596600462331577867483090210860316314908382128 000*c[4]*c[3]*a[9,6]+2909291159066449198603797432257719526636550925798 350946508235285457771791582283197051246697214191337472*c[4]*c[3]*a[6,1 ]+12595543805175335721462211666360997058874806348731498515783684077361 36864385462239005773146852687872000*c[4]*c[3]*a[9,5]+14323118263396910 6000467046533411418992951098553450121897739264583963040965031200186920 070368750000000000*c[3]*c[4]*a[7,6]-3698213098752310102177068610252843 780554960427875919005716707104541384796857787115708040589383254933504* c[3]*c[4]+185807927280494408730063065375581904821949229977871640392032 326590763406703952450499743906162623696000*c[4]*a[9,6]-232761687618574 3560584759954071953584929682431416961062549802693195220155679556693313 42088736866304000*c[4]*a[9,5]-2646867202016991342865773730171823077065 0203009222106405742118356931571135084402655774220081250000000000*c[4]* a[7,6]+133720459969588723320181052956067972997763494034267103511768698 6471508281418046245751824876828270526464*c[4]+566800689750125909478382 9662804337613416429775094829189408091686972870253048572877929748496093 75000000*c[4]*a[8,6]-3067155505527277084624157470013978439033146041416 563596821541813588865250408553975787516384765625000000*c[3]*c[4]*a[8,6 ]+43013627725057646401369744891762182893195966294861272584472107436504 068407708714778137922033942528000*a[9,5]-12607328054760822351082776491 7429842759420498205901409378739248514009135016095170997861936939160156 250000*c[3]*a[7,6]-100722582167970397717037372779916183153347332212074 8915013775918447738412534560585012540779540808466432+88502418731670776 1617739438940054664637850449097097163624091031948107602107511696017674 256351648852250*c[3]*a[9,6]+299873938344234824538806281361589829633980 8354396384887721984454344968599613162240479659005373489086464*c[3]-232 7616876185743560584759954071953584929682431416961062549802693195220155 67955669331342088736866304000*c[3]*a[9,5]-4989026904571420765721183401 1142346701425866249532611094269140369709118373187959185944140408325195 3125*a[8,6]+2699735835594322017195221939751887271857300421871871082618 961283836032433953362614104636817840576171875*c[3]*a[8,6]+232979456844 2039254918311252078323437624939744040904157588759376209081000952741692 0967933300683593750000*a[7,6]-1635496859916851826842742606691319891401 53228470105766803403454134578206942541479866962084070226065750*a[9,6]) /(763*c[3]-141)/c[3]/(-c[4]+c[3]), a[8,5] = 1/142737995874540232530271 00389489432668614738423156007539071311722918219284480000000000*(-68696 0892307812045141193352460760086565507834738234451830851669129168925739 11331250000000000*c[3]*c[4]*a[7,6]+10643232146535506503442680672723568 998259155523361021188679215157225276477289325724029681664*c[4]*c[3]*a[ 6,1]+15852372384895282122180420102714278610301673145292539296145306683 213433501830396975775744000*c[4]*c[3]*a[9,5]-1265455876975214894394931 3805242716267055558005806178710944780826489209110452207577614256000*c[ 4]*c[3]*a[9,6]-1015374190552976287240128917174398767546523462424054698 1329816385395203055316320611937026048*c[3]*c[4]+2338522656009243776011 603206473424631264526446682399997697528304764060923425637311197392000* c[4]*a[9,6]+1269482120778525535582021791572308941097465330250210454890 5646834497092860971818750000000000*c[4]*a[7,6]+42681652617331774178502 45223824304738922595895408028042600169438194304901233736895397101568*c [4]-182314284558901415262277020610207517722037720912386556373129289828 6410183800338550625000000*c[4]*a[8,6]+98656595119462255209303096968502 36597298920642280208688843804832571141632905378114375000000*c[3]*c[4]* a[8,6]-292946855343412159793897671622898202365994221951015470610286794 5390686924977832206532608000*c[4]*a[9,5]+54135657409464108166369032370 6797464398495220119176688808000498427374647997214077485056000*a[9,5]-2 4548818406201213366264611126345243422531115081543484697425673926979048 87472243171796189184+1113864808379225610170538558898968254756452761969 3980219529520623316022602429286878212548250*c[3]*a[9,6]+16047455255444 9683225650085849609742161585285594756916807598176984585063053258966174 8046875*a[8,6]-1117408741726931330798758764456876099195164795897320660 8151324557447961945334569628906250000*a[7,6]-2929468553434121597938976 716228982023659942219510154706102867945390686924977832206532608000*c[3 ]*a[9,5]+6046687020834387272336545654472315345290147087018834497886142 2959807056484328203027343750000*c[3]*a[7,6]-20583871295081364486768799 05697962305644296716090237497973345226589199458640274508293537750*a[9, 6]+4314984447846162359578592202576759001941035590335906238735122240473 022277419634918391021568*c[3]-8683835716244333922068866347748385338247 487440340392022992724045336056958130254694423828125*c[3]*a[8,6])/(763* c[4]-141)/(763*c[3]-141), a[10,6] = 3954458918230699741846244609537668 670264912300169257659977176734375/800134180797252928440343744342240823 13958525560207876667654196707053*a[7,6]+526180744020800178157171673671 875/359022955840383035894248406127836*a[8,6]-5365139647887469158218320 707351/8853429096639550069183960833070*a[9,6]-204388824020215684266404 402278957056/8875951095645670561315492876310816875, a[8,7] = -10940827 545723463618159777471/206951964789026015539767126898726574758728833804 67582825560417874963451084800*(143303498169329948767120521680712953011 17028639505422221312*c[4]+14949517491349841279071895663858475762527155 369900875000000*c[4]*a[8,6]-317734001205852985031485251874998941407013 002412595300000000*c[4]*a[7,6]+279672115644735179532921914410806359884 297903165253154687500*a[7,6]-77236183450999614428178945805573394070249 94110604217155584-1315868987519855820918307482912542918680775654954816 6015625*a[8,6])/(-998622639+2007664595*c[4]), c[9] = 20/21, c[2] = 1/2 4, c[5] = 141/763, c[6] = 169/192, c[8] = 104/105\}:" }}}{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 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a[8,5]=subs(e9,a[8,7]);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,$*&#\">ruxf\"=OYB daF3%4\"\"ho+[3^M'\\(yTgb#Gen/Q$)G(eZdE()*o7n(Rb,E!*yk>&p?\"\"\"*&,.*& \"fn78AAa]R'Gq6,`Hr!o@07n([*H$p\")\\.L9F.&%\"cG6#\"\"%F.F.*(\"fn+++v3! *p`:FDwv%eQm&*=2z7%)\\8\\<&\\\\\"F.F3F.&F%6$F'\"\"'F.F.*(\"gn++++`f7C+ 8qST*)*\\(=D&[J])H&e?,St<$F.F3F.&F%6$\"\"(F;F.!\"\"*&\"gn+voaJDlJ!zH%) )fj!3T9>#H`z^tWc6s'z#F.F>F.F.\"en%ebr@/16%*\\-2%Rtb!e%*y\"GWh**4X$=Os( FA*&\"fnDc,m\"[&\\lv2o=Ha7H[2$=4#e&)>v)*oeJ\"F.F9F.FAF.,&\"*REi)**FA*& \"+&fkw+#F.F3F.F.FAF.FA" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 119 "We can find some simple order conditions that are not yet satisfied and determine which parameters are related by them. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 47 do\n eq := simplify(subs(e9,SO7_ 10[ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets(lhs(eq)) )\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<)&%\"cG 6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$F/\"\"'&F-6$\"\")F3&F-6$\"\"(F 3&F-6$F3\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#^<'&%\"cG6#\"\"$& F&6#\"\"%&%\"aG6$\"\"*\"\"'&F-6$\"\")F0&F-6$\"\"(F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#]<'&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"'&F-6$ \"\")F0&F-6$\"\"(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<)&%\"cG6# \"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$F/\"\"'&F-6$\"\")F3&F-6$\"\"(F3& F-6$F3\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Z<'&%\"cG6#\"\"$&F& 6#\"\"%&%\"aG6$\"\"*\"\"'&F-6$\"\")F0&F-6$\"\"(F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple order comditions given in abrev iated form as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SO7 := SimpleOrderConditions(7):\n [seq([i,SO7[i]],i=[50,51])]:\nlinalg[augment](linalg[delcols](%,2..2), matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7%\"#]%#~~G/*(%\"bG\"\" \"%\"cGF--%!G6#*(%\"aGF-F.F--F06#*&)F.\"\"#F-F3F-F-F-#F-\"$0\"7%\"#^F) /*(F,F-F7F--F06#*&F3F-F4F-F-#F-\"#%)Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "cdcns4 := [ seq(SO7_10[i],i=[50,51])]:\neqns4 := simplify(subs(e9,cdcns4)):\nnops( %);\nindets(eqns4);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%\"cG6#\"\"$&F%6#\"\"%&%\"aG6$ \"\"*\"\"'&F,6$\"\")F/&F,6$\"\"(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 2 equations for the linking coefficients \+ " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,6];" "6#&%\"aG6$\"\"*\"\"'" }{TEXT -1 29 " in terms of the parameter " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\" \"'" }{TEXT -1 88 " and subsitute back into the expressions previously obtained for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "e10 := solve(\{op(eqns4) \},\{a[8,6],a[9,6]\}):\ninfolevel[solve] := 0:\ne11 := `union`(map(u_- >lhs(u_)=simplify(subs(e10,rhs(u_))),e9),e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[8,6]=subs( e10,a[8,6]);\n``;\na[9,6]=subs(e10,a[9,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"C_F/%Gx>r>x4)H$f+`D\"\"Dv=#*>>X OIEN&e*R&e217!\"\"*&#\"S3\"f%[w-.K\"3/?omc`wV?3Y8^!*f#\"R:$y&)=Ylql@_ \"HcZ&\\mM.Z;W'GA\"\"\"\"&F%6$\"\"(F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"',&#\"EC-UQD=[V$>+eq=ed87&\"FvymY6B$GVmWbUJwO9Bf!\"\"*&#\"T]ilHWnBu` ,'G&pk$z-Jen:Y^np;\"S(y'\\63CQt " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17529 "e11 := \{b[8] = 758177409913942921875/78785859799058312576, b[1 0] = -1895689785077201/288707084450720, b[9] = -2973333145929506703/74 7247079460651200, b[5] = 166164647343343629239462104853/59415021855318 2839761586950000, b[6] = 8835157365899998531682304/6999851448505791234 978125, b[1] = 164949402359214517/2969763122318428800, a[7,1] = -1/545 65437560151192492509904202869302049306342340566333403829352000000000*( 4907714065767855824374499178853327688084299558769041316371524647792640 *c[3]+4907714065767855824374499178853327688084299558769041316371524647 792640*c[4]+1131245815132082500066285374092674826344914601904927411828 6799726968832*c[4]*c[3]*a[6,1]-480289528524247475585113219285672502413 16520081019324714828960875000000*c[4]*a[7,6]-9050681786215100400085466 83267691052266694512297174993469981805824000*c[3]*a[9,5]+1672537525368 71449071042047628760731808130964919923557115684711168000*a[9,5]+422754 84541978033007231319822540965056158811946313884775031741603515625*a[7, 6]-3414910001308436716549630937945704623290124889532955837243326317996 032*c[3]*c[4]-36595901815956698821189962313880297312387323402040745895 21985061341184-4802895285242474755851132192856725024131652008101932471 4828960875000000*c[3]*a[7,6]+54565437560151192492509904202869302049306 342340566333403829352000000000*c[3]*c[4]*a[7,6]-9050681786215100400085 46683267691052266694512297174993469981805824000*c[4]*a[9,5]+4897638441 760369932812206520093959382124027750941450496578695871232000*c[4]*c[3] *a[9,5])/c[3]/c[4], c[7] = 998622639/2007664595, b[7] = 11056784337287 0117205263375074449005180061856568412069340896875/34071775464939293118 3331876846338804668211857149398834541328544, a[8,1] = -1/6341296781358 19226004322779116016777815003406170008881577327170000000000*(175491492 2922390842033381441862127428813186920743846994540045511218104320*c[3]+ 1754914922922390842033381441862127428813186920743846994540045511218104 320*c[4]+8121999221968002349524198861543381752651788436182668901170275 57774721024*c[4]*c[3]*a[6,1]-11863151354548912646952296516356110809605 180460011773204562753336125000000*c[4]*a[7,6]-223551840119512979882111 030767119689909873544537402223387085506038528000*c[3]*a[9,5]+413116768 76607247920547385764303900756608348335221118607574123658496000*a[9,5]+ 1044204468186857415278613599616761836887122655073952953943284017606835 9375*a[7,6]-3596881587881899883822784903186053233350496281081873041327 276607933548544*c[3]*c[4]-11863151354548912646952296516356110809605180 460011773204562753336125000000*c[3]*a[7,6]+134776630773573445456499463 38108717606178666558119884350745849944000000000*c[3]*c[4]*a[7,6]-22355 1840119512979882111030767119689909873544537402223387085506038528000*c[ 4]*a[9,5]+120971669511481137340461501046320796738463485448253827265493 7880194304000*c[4]*c[3]*a[9,5]-970779453197127397476853056833607978015 047233239347467128640950756845568)/c[3]/c[4], a[6,5] = -582169/2123366 4*(21233664*c[4]*c[3]*a[6,1]-18690048*c[3]*c[4]+8225568*c[3]-4826809+8 225568*c[4])/(763*c[4]-141)/(763*c[3]-141), a[7,4] = -1/11609667565989 61542393827748997219192538432815756730497953816000000000*(147231421973 03567473123497536559983064252898676307123949114573943377920*c[3]+30810 335112180858328728380604710120913822532600182989555977391831377920*c[4 ]+33937374453962475001988561222780244790347438057147822354860399180906 496*c[4]*c[3]*a[6,1]-6863020150112604081812233409489097093159398620220 74342198919549861328125*c[4]*a[7,6]-2715204535864530120025640049803073 156800083536891524980409945417472000*c[3]*a[9,5]+501761257610614347213 126142886282195424392894759770671347054133504000*a[9,5]+12682645362593 4099021693959467622895168476435838941654325095224810546875*a[7,6]-3840 2236061553434236945397862120059384540110954396948727083529513988096*c[ 3]*c[4]-10978770544787009646356988694164089193716197020612223768565955 184023552-144086858557274242675533965785701750723949560243057974144486 882625000000*c[3]*a[7,6]+779704064391491114620088055989293870938819251 528037122498180790375000000*c[3]*c[4]*a[7,6]-2715204535864530120025640 049803073156800083536891524980409945417472000*c[4]*a[9,5]+146929153252 81109798436619560281878146372083252824351489736087613696000*c[4]*c[3]* a[9,5])/(-c[4]+c[3])/c[4]/(763*c[4]-141), a[8,4] = -1/1349212081140040 9063921761257787591017340498003617210246326110000000000*(-255263178147 52576898322324489372275903426207230554933401900640154283845632*c[3]*c[ 4]+1370631053158115386200486996258028459181349830820221561652059354207 6712960*c[4]-169516597707781320820762165214380698201037145919452362523 133128815748046875*c[4]*a[7,6]+243659976659040070485725965846301452579 5536530854800670351082673324163072*c[4]*c[3]*a[6,1]+192586903904698305 311161749829355586121888355127425169257050655222625000000*c[3]*c[4]*a[ 7,6]-67065552035853893964633309230135906972962063361220667016125651811 5584000*c[4]*a[9,5]+36291500853444341202138450313896239021539045634476 14817964813640582912000*c[4]*c[3]*a[9,5]-29123383595913821924305591705 00823934045141699718042401385922852270536704-6706555203585389396463330 92301359069729620633612206670161256518115584000*c[3]*a[9,5]-3558945406 3646737940856889549068332428815541380035319613688260008375000000*c[3]* a[7,6]+123935030629821743761642157292911702269825045005663355822722370 975488000*a[9,5]+31326134045605722458358407988502855106613679652218588 618298520528205078125*a[7,6]+52647447687671725261001443255863822864395 60762231540983620136533654312960*c[3])/(-c[4]+c[3])/c[4]/(763*c[4]-141 ), a[10,1] = -1/159652296196468703969722380827685773647051788230955552 45680781858209280000*(509378947095227520766022185009033561115424406623 28554280638120112190324736*c[3]+50937894709522752076602218500903356111 542440662328554280638120112190324736*c[4]+2706366169320052211466456263 5071339911514709324851536017113811121564811264*c[4]*c[3]*a[6,1]-343551 099753394219286031485755041540976137068139531229685171557138875000000* c[4]*a[7,6]-6473952681679661316181134425413794096863665846461692728290 779857059072000*c[3]*a[9,5]+119636609189624147520516376668852551462356 0792072213204048492738984704000*a[9,5]+3023965409287688701007256306906 35523046703981851983217795802047689947265625*a[7,6]-104964250941271348 019520032246395454518614207133702768427306741853727337472*c[3]*c[4]-28 1276429381884482399243543972590187342813102490533216384516304475923353 60-3435510997533942192860314857550415409761370681395312296851715571388 75000000*c[3]*a[7,6]+3903065748677614798989233447631241175586882667620 70982837591354856000000000*c[3]*c[4]*a[7,6]-64739526816796613161811344 25413794096863665846461692728290779857059072000*c[4]*a[9,5]+3503280777 3911926129405713238232091460333170502484195402027411566922496000*c[4]* c[3]*a[9,5])/c[3]/c[4], a[6,4] = 1/21233664*(2993946624*c[3]*a[6,1]+36 40811616*c[3]-2523050179)/(-c[4]+c[3])/(763*c[4]-141), a[4,2] = 0, a[5 ,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, \+ a[9,1] = -1/1354665951976272534607631590664286637608773633239124605436 660560128000*(-3891919432850628622623936856789927411671341175869017545 302687300216832*c[3]*c[4]+15092567835786500051119760736963849502999626 604837496473399608000000000*c[3]*c[4]*a[7,6]+1354665951976272534607631 590664286637608773633239124605436660560128000*c[4]*c[3]*a[9,5]-1328460 3980457908899162706065348388364619463001133004708356946625000000*c[4]* a[7,6]+194606747583746497018797269758412813536221963961487297989123946 6625024*c[4]-250338006852758096172576742180425184669511248082197338619 356669696000*c[4]*a[9,5]+116932191286322218956171735679368626751077564 95788946852668354060546875*a[7,6]+462616762336027412324158855143380747 55440479658702260478806409472000*a[9,5]-132846039804579088991627060653 48388364619463001133004708356946625000000*c[3]*a[7,6]+1946067475837464 970187972697584128135362219639614872979891239466625024*c[3]-1084743161 962272384530357463553882450889497180488189291710271957485568-250338006 852758096172576742180425184669511248082197338619356669696000*c[3]*a[9, 5])/c[3]/c[4], a[9,3] = 1/61659806644345586463706490244164161930303761 1852127722092244224000*(6046701857286258033301186192693849961137670915 399638010176125000000*c[4]*a[7,6]-885784012670671356480643012100194872 718352134553879371821228705792*c[4]+1139453831828666800967577342651002 20605148497078833563322419968000*c[4]*a[9,5]+4937383531917489233183238 34116469026349338725757027442744775583744-2105674839945504835339821825 8688245223231897887438443549752576000*a[9,5]-5322357363965508373061981 596694065851209720753659056373540443359375*a[7,6])/c[3]/(-c[4]+c[3]), \+ a[5,4] = 19881/888389894*(-94+763*c[3])/c[4]/(-c[4]+c[3]), a[2,1] = 1/ 24, a[10,6] = 4905403672098941222092370996583097700708683596875/200652 259134124242003874071522321513808126072209*a[7,6]-10932641078500713592 43261837492355072/8875951095645670561315492876310816875, a[6,3] = -1/2 1233664*(2993946624*c[4]*a[6,1]-2523050179+3640811616*c[4])/(763*c[3]^ 2-763*c[3]*c[4]+141*c[4]-141*c[3]), a[8,7] = -796951280605625050372389 41490242415433886906/23692035580167289837348760854832233004866581, a[4 ,1] = 1/2*c[4]*(2*c[3]-c[4])/c[3], a[9,6] = -5121357581870580019343481 82538420224/5923143676314255446643283231146667875+16696751461567583102 7936469528601537423674429656250/14986529104718765832944052478963177338 240811496787*a[7,6], a[10,3] = 1/3396857365882312850419625123993314332 91599549427565005227250677834240000*(388204316481235054416485397284572 737153059131552095049340315645070012254208*c[3]+1528136841285682562298 06655502710068334627321986985662841914360336570974208*c[4]+81190985079 601566343993687905214019734544127974554608051341433364694433792*c[4]*c [3]*a[6,1]-10306532992601826578580944572651246229284112044185936890555 14671416625000000*c[4]*a[7,6]-1942185804503898394854340327624138229059 0997539385078184872339571177216000*c[3]*a[9,5]+35890982756887244256154 91300065576543870682376216639612145478216954112000*a[9,5]+907189622786 306610302176892071906569140111945555949653387406143069841796875*a[7,6] -728195033931527853113207066706226108443793786948968513398271056992326 844416*c[3]*c[4]-49091183133755456997202905578075511507369178330438977 69748871540158080078125*c[3]*a[7,6]+5577223172592335942877489864491419 058825374106180049537229487193552375000000*c[3]*c[4]*a[7,6]-8438292881 4565344719773063191777056202843930747159964915354891342777006080-19421 858045038983948543403276241382290590997539385078184872339571177216000* c[4]*a[9,5]+1050984233217357783882171397146962743809995115074525862060 82234700767488000*c[4]*c[3]*a[9,5])/(763*c[3]-141)/c[3]/(-c[4]+c[3]), \+ a[8,6] = -1255300593298097719711977284042752/1206075853995853526303645 1919921875+25990511346082043765356666820040813203027648459108/12228644 16470334664954756291522165706546188578315*a[7,6], a[5,1] = 141/8883898 94*(13254-107583*c[3]-107583*c[4]+1164338*c[3]*c[4])/c[3]/c[4], a[5,3] = -19881/888389894*(-94+763*c[4])/c[3]/(-c[4]+c[3]), a[3,1] = -12*c[3 ]^2+c[3], a[4,3] = 1/2*c[4]^2/c[3], a[10,8] = -73761140646847191787348 72354875/359022955840383035894248406127836, a[3,2] = 12*c[3]^2, b[4] = 0, b[3] = 0, b[2] = 0, c[10] = 1, a[10,5] = 15239/4947437709096129785 40055609786712733534648750*(320393966128966686257654769214341718251027 82464*c[4]*c[3]*a[6,1]-30674915571600453437583163126552732810047337489 *c[3]*c[4]+41473693968533951656010824381216280725533124125*c[4]*c[3]*a [9,5]+12868637218712264623488801609526569904424075655*c[4]-76642081907 77571669066220495087150173394718875*c[4]*a[9,5]+1416321566054570911321 542712722527096262982125*a[9,5]+12868637218712264623488801609526569904 424075655*c[3]-7367626312315685971706359410434021064958210593-76642081 90777571669066220495087150173394718875*c[3]*a[9,5])/(763*c[4]-141)/(76 3*c[3]-141), a[9,8] = 88873364366227887565000/817629004593263701438668 9, a[10,9] = 255482840375593769438967652731/88534290966395500691839608 33070, a[9,7] = -57480046875499710394442383866625260275450807519330593 6911875/311545184693907215706597063235165448890282539301416442871559, \+ c[9] = 20/21, c[2] = 1/24, c[5] = 141/763, c[6] = 169/192, c[8] = 104/ 105, a[10,7] = -306989390668037647170793389323371972068207220184067211 641263783356000/800134180797252928440343744342240823139585255602078766 67654196707053, a[9,4] = -1/616598066443455864637064902441641619303037 611852127722092244224000*(-5322357363965508373061981596694065851209720 753659056373540443359375*a[7,6]-88578401267067135648064301210019487271 8352134553879371821228705792*c[3]+493738353191748923318323834116469026 349338725757027442744775583744+604670185728625803330118619269384996113 7670915399638010176125000000*c[3]*a[7,6]-21056748399455048353398218258 688245223231897887438443549752576000*a[9,5]+11394538318286668009675773 4265100220605148497078833563322419968000*c[3]*a[9,5])/(-c[4]+c[3])/c[4 ], a[7,3] = 1/11609667565989615423938277489972191925384328157567304979 53816000000000*(308103351121808583287283806047101209138225326001829895 55977391831377920*c[3]+14723142197303567473123497536559983064252898676 307123949114573943377920*c[4]+3393737445396247500198856122278024479034 7438057147822354860399180906496*c[4]*c[3]*a[6,1]-144086858557274242675 533965785701750723949560243057974144486882625000000*c[4]*a[7,6]-271520 4535864530120025640049803073156800083536891524980409945417472000*c[3]* a[9,5]+501761257610614347213126142886282195424392894759770671347054133 504000*a[9,5]+12682645362593409902169395946762289516847643583894165432 5095224810546875*a[7,6]-3840223606155343423694539786212005938454011095 4396948727083529513988096*c[3]*c[4]-1097877054478700964635698869416408 9193716197020612223768565955184023552-68630201501126040818122334094890 9709315939862022074342198919549861328125*c[3]*a[7,6]+77970406439149111 4620088055989293870938819251528037122498180790375000000*c[3]*c[4]*a[7, 6]-2715204535864530120025640049803073156800083536891524980409945417472 000*c[4]*a[9,5]+146929153252811097984366195602818781463720832528243514 89736087613696000*c[4]*c[3]*a[9,5])/(763*c[3]-141)/c[3]/(-c[4]+c[3]), \+ a[10,4] = -1/339685736588231285041962512399331433291599549427565005227 250677834240000*(-1942185804503898394854340327624138229059099753938507 8184872339571177216000*c[4]*a[9,5]-72819503393152785311320706670622610 8443793786948968513398271056992326844416*c[3]*c[4]-4909118313375545699 720290557807551150736917833043897769748871540158080078125*c[4]*a[7,6]+ 5577223172592335942877489864491419058825374106180049537229487193552375 000000*c[3]*c[4]*a[7,6]+8119098507960156634399368790521401973454412797 4554608051341433364694433792*c[4]*c[3]*a[6,1]+388204316481235054416485 397284572737153059131552095049340315645070012254208*c[4]+1050984233217 35778388217139714696274380999511507452586206082234700767488000*c[4]*c[ 3]*a[9,5]-194218580450389839485434032762413822905909975393850781848723 39571177216000*c[3]*a[9,5]+1528136841285682562298066555027100683346273 21986985662841914360336570974208*c[3]-10306532992601826578580944572651 24622928411204418593689055514671416625000000*c[3]*a[7,6]+3589098275688 724425615491300065576543870682376216639612145478216954112000*a[9,5]-84 3829288145653447197730631917770562028439307471599649153548913427770060 80+9071896227863066103021768920719065691401119455559496533874061430698 41796875*a[7,6])/(-c[4]+c[3])/c[4]/(763*c[4]-141), a[7,5] = 1631083643 7/249167953115315531420226053377075221794488515472693609613281250*(386 7015363048067540487032589634949193268117784490374858873*c[3]*c[4]+1843 759925247606162189462457929479138820217329899695767552*c[4]*c[3]*a[6,1 ]+798241139677989892922225320819986138445217685230940748875*c[4]*c[3]* a[9,5]-300275863823318584589754094639121943736266137904851482335*c[4]- 147512451762249770513805727700678958742825286523673192125*c[4]*a[9,5]+ 27259837088436720370179040112445259741465747575147994875*a[9,5]-231641 425619579554735960449842760766703307711492397502199-300275863823318584 589754094639121943736266137904851482335*c[3]-1475124517622497705138057 27700678958742825286523673192125*c[3]*a[9,5])/(763*c[4]-141)/(763*c[3] -141), a[8,3] = 1/1349212081140040906392176125778759101734049800361721 0246326110000000000*(1370631053158115386200486996258028459181349830820 2215616520593542076712960*c[3]+526474476876717252610014432558638228643 9560762231540983620136533654312960*c[4]+243659976659040070485725965846 3014525795536530854800670351082673324163072*c[4]*c[3]*a[6,1]-355894540 63646737940856889549068332428815541380035319613688260008375000000*c[4] *a[7,6]-67065552035853893964633309230135906972962063361220667016125651 8115584000*c[3]*a[9,5]+12393503062982174376164215729291170226982504500 5663355822722370975488000*a[9,5]+3132613404560572245835840798850285510 6613679652218588618298520528205078125*a[7,6]-2552631781475257689832232 4489372275903426207230554933401900640154283845632*c[3]*c[4]-1695165977 07781320820762165214380698201037145919452362523133128815748046875*c[3] *a[7,6]+19258690390469830531116174982935558612188835512742516925705065 5222625000000*c[3]*c[4]*a[7,6]-670655520358538939646333092301359069729 620633612206670161256518115584000*c[4]*a[9,5]+362915008534443412021384 5031389623902153904563447614817964813640582912000*c[4]*c[3]*a[9,5]-291 2338359591382192430559170500823934045141699718042401385922852270536704 )/(763*c[3]^2-763*c[3]*c[4]+141*c[4]-141*c[3])/c[3], a[8,5] = 2/240623 7082129327550433554304098001826171875*(8971031051611067215214333538001 56811102877908992*c[4]*c[3]*a[6,1]+13361742274210286163958269941948375 65983480216375*c[4]*c[3]*a[9,5]-85016985597900090645497103750735248594 0442679072*c[3]*c[4]+358708975828216722221215394299911075781980972000* c[4]-246920794320268722033829103776503403412412464625*c[4]*a[9,5]+4563 0186106366828056054919570756199057863902375*a[9,5]-2059954841266249701 16896781502475740149882408224+3587089758282167222212153942999110757819 80972000*c[3]-246920794320268722033829103776503403412412464625*c[3]*a[ 9,5])/(763*c[4]-141)/(763*c[3]-141)\}:" }}}{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 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[7,4]=subs(e11,a[7,4]);\n``;\na[8,5]=sub s(e11,a[8,7]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%, $*&#\"\"\"\"ao++++g\"Q&z\\Inv:GVQD>>s**[x#QRU:'*)fcn'4;\"F,**,<*&\"bo? zPVRd9\"\\R72jn)*GDkI)*fl`(\\BJZnNI(>UJs9F,&%\"cG6#\"\"$F,F,*&\"bo?zPJ =Rxfb*)H=+E`AQ\"475Zg!QG(G$e3=7^L53$F,&F36#F(F,F,**\"bo'\\14=*Rg[NAy9d !QuM!zW-yA7c))>+vC'RXut$R$F,F8F,F2F,&F%6$\"\"'F,F,F,*(\"coD\"G8')\\&>* )>UV2A?')RfJ4(4*[4MB7=3/E6],-joF,F8F,&F%6$F'F>F,!\"\"*(\"ao+?Z#G')G9EJ@ZVh5wDh<]F,FFF,F,*&\"covoa5[A&4DVlT*QeVw%o^*Gin% fRp@!*4Mfi`k#o7F,FAF,F,*(\"bo'4))R^HN3F([pRa46SXQf+7iyRXpBMMbhgB-%QF,F 2F,F8F,FC\"bo_N-%=bfcoPA71-(>;P>*3kTp))pNY'4qyW0xy4\"FC*(\"co+++DE)o[W T(z0V-c\\Rs])Q4 (QH*)f0)3?Y6\"\\\"RkSqz(F,F2F,F8F,FAF,F,*(\"ao+?ZmV)z46G D`\"Hp9F,F8F,F2F,FFF,F,F,,&F8FCF2F,FCF8FC,&*&\"$j(F,F8F,F,\"$T\"FCFCF, FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!M1p)QV:CC!\\T*Qs.0Dcg!G^pz\"M\"em[ +LA$[&3w[t$)*Gn,eN?pB" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We can find some simple order conditions that are n ot yet satisfied and determine which paramers are related by them." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "for ii from 64 by -1 to 43 do\n eq := simplify(subs(e11,SO7_10[ ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<'&%\"cG6# \"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$\"\"(\"\"'&F-6$F4\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<'&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$ \"\"*\"\"&&F-6$\"\"(\"\"'&F-6$F4\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#X<'&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$\"\"(\"\"'&F -6$F4\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple order comditions given in abreviated form as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]],i=[45,55])]:\nlinalg[augme nt](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linal g[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$ 7%\"#X%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\" \"#F-F3F-F-F-F-#F-\"$?%7%\"#bF)/*(F,F-F.F--F06#*&F3F--F06#*&)F.\"\"$F- F3F-F-F-#F-\"$S\"Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 99 "We can solve this system of two equations to obtain numerical values for the remaining parameters " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,5]" "6#&%\"aG6$\"\"*\"\"&" }{TEXT -1 103 " and substitute t hese values back into the expressions obtained previously for the othe r coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 107 "cdcns5 := [seq(SO7_10[i],i=[45,55])]:\neqns5 \+ := simplify(subs(e11,cdcns5)):\nnops(%);\nindets(eqns5);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%\"cG6#\"\"$&F%6#\"\"%&%\"aG6$\"\"*\"\"&&F,6$\"\"(\"\"'&F,6$F3\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "e12 := solve (\{op(eqns5)\},\{a[7,6],a[9,5]\}):\ninfolevel[solve] := 0:\ne13 := `un ion`(map(u_->lhs(u_)=simplify(subs(e12,rhs(u_))),e11),e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[ 7,6]=subs(e12,a[7,6]);\n``;\na[9,5]=subs(e12,a[9,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!L_(Q\\skUmtnW[4CEXxKj+6a\"Nvo /[>*z1'31(GmES7k$>oqrz#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,$*&#\"(J@)H\"ID'Rt bE_4[?ufO]$Gz\"o@H*\"\"\"*&,,*&\"IDv#[WSTdw.A;&*))H)*Q='>OF.&%\"cG6#\" \"$F.F.*(\"Jrfb;N>Z5DMS6)*yQ^_lB*3R*G(Ga\")*)z+6[z$=F<**\"IcuU`J(\\&R3c**oI#ymXdo](F.F9F.F3F.& F%6$\"\"'F.F.F.F.,**(\"'p@eF.F3F.F9F.F.*&\"'$e2\"F.F9F.F<*&FHF.F3F.F< \"&\"))>F.F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6795 "e13 := \{b[8] = 75817740991394292 1875/78785859799058312576, b[10] = -1895689785077201/288707084450720, \+ b[9] = -2973333145929506703/747247079460651200, b[5] = 166164647343343 629239462104853/594150218553182839761586950000, b[6] = 883515736589999 8531682304/6999851448505791234978125, a[8,3] = -4/16674740851991167968 75*(257336995614031430107569*c[3]*c[4]+388742192102923837636608*c[4]*c [3]*a[6,1]-301063389367092128361754*c[3]+39804486757208501540121*c[4]- 4903829299053209888298)/c[3]/(763*c[3]^2-763*c[3]*c[4]+141*c[4]-141*c[ 3]), a[10,1] = 1/55147186734237812934086167220647500*(-309009262239356 643946328558242779763*c[3]*c[4]+407290680260571184072790357556854784*c [4]*c[3]*a[6,1]+42554636263023389936757844914108258*c[3]+4255463626302 3389936757844914108258*c[4]-5242641951145738734017349176836404)/c[3]/c [4], b[1] = 164949402359214517/2969763122318428800, c[7] = 998622639/2 007664595, b[7] = 1105678433728701172052633750744490051800618565684120 69340896875/3407177546493929311833318768463388046682118571493988345413 28544, a[6,5] = -582169/21233664*(21233664*c[4]*c[3]*a[6,1]-18690048*c [3]*c[4]+8225568*c[3]-4826809+8225568*c[4])/(763*c[4]-141)/(763*c[3]-1 41), a[7,1] = 2995867917/262934044102022765850630983720903912431164062 50*(-3822589202129136873319699094318709954*c[3]*c[4]+14404099284968865 12730498702950531072*c[4]*c[3]*a[6,1]+63304992217353824270853475076830 1539*c[3]+633049922173538242708534750768301539*c[4]-779904229152196524 43777544655596782)/c[3]/c[4], a[6,4] = 1/21233664*(2993946624*c[3]*a[6 ,1]+3640811616*c[3]-2523050179)/(-c[4]+c[3])/(763*c[4]-141), a[8,1] = \+ 4/78263336457005684239733354296875*(-96165719468899273963950510504928* c[3]*c[4]+129402251256155587452225667989504*c[4]*c[3]*a[6,1]+132498872 03174702027077065799773*c[3]+13249887203174702027077065799773*c[4]-163 2358318608678886691014659474)/c[3]/c[4], a[4,2] = 0, a[5,2] = 0, a[6,2 ] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[10,4] = 1/19 5827124410706209801250*(132688265310778621266519879*c[3]*c[4]+20392622 0152801474828566528*c[4]*c[3]*a[6,1]+21306665101061958768970911*c[3]-1 57647680526575230922872214*c[4]-2624936460681289809021318)/(763*c[4]-1 41)/c[4]/(-c[4]+c[3]), a[5,4] = 19881/888389894*(-94+763*c[3])/c[4]/(- c[4]+c[3]), a[2,1] = 1/24, a[6,3] = -1/21233664*(2993946624*c[4]*a[6,1 ]-2523050179+3640811616*c[4])/(763*c[3]^2-763*c[3]*c[4]+141*c[4]-141*c [3]), a[8,7] = -79695128060562505037238941490242415433886906/236920355 80167289837348760854832233004866581, a[4,1] = 1/2*c[4]*(2*c[3]-c[4])/c [3], a[5,1] = 141/888389894*(13254-107583*c[3]-107583*c[4]+1164338*c[3 ]*c[4])/c[3]/c[4], a[5,3] = -19881/888389894*(-94+763*c[4])/c[3]/(-c[4 ]+c[3]), a[10,5] = -362109118/7650676869736283173299829017320300003113 125*(-135644048482565065653950914902152861699627*c[3]*c[4]+90842815360 260750668593357724505669107712*c[4]*c[3]*a[6,1]+4548112015002394144556 7321574695351841245*c[3]+45481120150023941445567321574695351841245*c[4 ]-22551853969095703273086939368535328690779)/(763*c[4]-141)/(763*c[3]- 141), a[3,1] = -12*c[3]^2+c[3], a[9,1] = 2/267727958654241740474777084 9357522625*(-3583230875138748043884382563418927772*c[3]*c[4]+553965196 2001052310278126563120644096*c[4]*c[3]*a[6,1]+494091673878574353523863 469261185927*c[3]+494091673878574353523863469261185927*c[4]-6087105811 8723445912507426095087126)/c[3]/c[4], a[7,3] = -2995867917/55943413638 7282480533257412172135983896093750*(-502209495199304144990667134841341 5487*c[3]*c[4]+4321229785490659538191496108851593216*c[4]*c[3]*a[6,1]- 2375474623408812321858511238314227258*c[3]+189914976652061472812560425 2304904617*c[4]-233971268745658957331332633966790346)/(-c[4]+c[3])/c[3 ]/(763*c[3]-141), a[4,3] = 1/2*c[4]^2/c[3], a[10,8] = -737611406468471 9178734872354875/359022955840383035894248406127836, a[3,2] = 12*c[3]^2 , a[7,5] = -92799444559123870397599014/1314670220510113829253154918604 5195621558203125*(-78540214845344325328516313*c[3]*c[4]+13535754589416 905351626752*c[4]*c[3]*a[6,1]+17555788477868847229879167*c[3]+17555788 477868847229879167*c[4]-5352197575816551620174313)/(763*c[4]-141)/(763 *c[3]-141), b[4] = 0, b[3] = 0, b[2] = 0, c[10] = 1, a[9,8] = 88873364 366227887565000/8176290045932637014386689, a[10,9] = 25548284037559376 9438967652731/8853429096639550069183960833070, a[7,4] = 2995867917/559 434136387282480533257412172135983896093750*(-5022094951993041449906671 348413415487*c[3]*c[4]+4321229785490659538191496108851593216*c[4]*c[3] *a[6,1]+1899149766520614728125604252304904617*c[3]-2375474623408812321 858511238314227258*c[4]-233971268745658957331332633966790346)/(763*c[4 ]-141)/c[4]/(-c[4]+c[3]), a[7,6] = -5411006332774526240948446773664264 724938752/279717068193641240266628706086067991948046875, a[9,4] = 2/69 36692194428325875*(1484249510163638917881*c[3]*c[4]+180503382739792768 929*c[3]+2023765975437297057792*c[4]*c[3]*a[6,1]-158512671522661286794 6*c[4]-22237638240551140602)/(763*c[4]-141)/c[4]/(-c[4]+c[3]), a[9,5] \+ = -2982131/9292168179283503659742048095226557339625*(36196183898298895 16220376574140444827525*c[3]-10458241751387898114034251047193516555971 *c[3]*c[4]+3619618389829889516220376574140444827525*c[4]-1837948110079 898154287289390892365521907+7506857456678230689956083954973153427456*c [4]*c[3]*a[6,1])/(582169*c[3]*c[4]-107583*c[4]-107583*c[3]+19881), a[9 ,3] = -2/6936692194428325875*(1484249510163638917881*c[3]*c[4]+2023765 975437297057792*c[4]*c[3]*a[6,1]+180503382739792768929*c[4]-2223763824 0551140602-1585126715226612867946*c[3])/(-c[4]+c[3])/c[3]/(763*c[3]-14 1), a[9,7] = -57480046875499710394442383866625260275450807519330593691 1875/311545184693907215706597063235165448890282539301416442871559, a[8 ,5] = -219109402/382954364574110830665030948046875*(-99557227157605776 64704774878003*c[3]*c[4]+6729434197165541735195153006592*c[4]*c[3]*a[6 ,1]+3352049080673811535943913280917*c[3]+33520490806738115359439132809 17*c[4]-1667432869088799425276846771043)/(763*c[4]-141)/(763*c[3]-141) , a[10,6] = -5290899117464226058433745200964698112/8875951095645670561 315492876310816875, a[10,3] = -1/195827124410706209801250*(13268826531 0778621266519879*c[3]*c[4]+203926220152801474828566528*c[4]*c[3]*a[6,1 ]-157647680526575230922872214*c[3]+21306665101061958768970911*c[4]-262 4936460681289809021318)/(-c[4]+c[3])/c[3]/(763*c[3]-141), a[9,6] = -17 88698788744957883891433528082038784/5923143676314255446643283231146667 875, a[8,6] = -120391896911431755652090822656/233667703961223196028992 578125, c[9] = 20/21, c[2] = 1/24, c[5] = 141/763, c[6] = 169/192, c[8 ] = 104/105, a[10,7] = -3069893906680376471707933893233719720682072201 84067211641263783356000/8001341807972529284403437443422408231395852556 0207876667654196707053, a[8,4] = 4/1667474085199116796875*(25733699561 4031430107569*c[3]*c[4]+388742192102923837636608*c[4]*c[3]*a[6,1]+3980 4486757208501540121*c[3]-301063389367092128361754*c[4]-490382929905320 9888298)/(763*c[4]-141)/c[4]/(-c[4]+c[3])\}:" }}}{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 0 443 "s ubs(e13,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(5-i)],i=2..5)]));pr int(``);\nfor ii from 6 to 10 do\n print(c[ii]=subs(e13,c[ii]));prin t(``); \n for jj to ii-1 do\n if ii=6 and jj=1 then\n \+ print(a[ii,jj]*` is a parameter`);\n else\n print(a[ii,jj ]=subs(e13,a[ii,jj]));\n end if;\n end do:\n print(`_________ ________________________`);\nend do:print(``);\nfor ii to 10 do\n pr int(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7&7'#\"\"\"\"#CF(%!GF+F+7'&%\"cG6#\"\"$,&*&\"#7F))F-\"\" #F)!\"\"F-F),$*&F3F)F4F)F)F+F+7'&F.6#\"\"%,$*&#F)F5F)*(F:F),&*&F5F)F-F )F)F:F6F)F-F6F)F)\"\"!,$*&F?F)*&F:F5F-F6F)F)F+7'#\"$T\"\"$j(,$*&#FI\"* %*)*Q)))F)*(,*\"&aK\"F)*&\"'$e2\"F)F-F)F6*&FSF)F:F)F6*(\"(QV;\"F)F-F)F :F)F)F)F-F6F:F6F)F)FC,$*&#\"&\"))>FNF)*(,&\"#%*F6*&FJF)F:F)F)F)F-F6,&F :F6F-F)F6F)F6,$*&#FZFNF)*(,&FgnF6*&FJF)F-F)F)F)F:F6FinF6F)F)Q(pprint46 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$p\"\"$#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"' 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"6#/&%\"aG6$\"\"(\"\"\",$*&#\"+L(o8H@?*eAQF(&%\"cG6#\"\"$F(&F36#\" \"%F(!\"\"**\"Fs5`]Hq)\\IF^')o\\G*4/W\"F(F6F(F2F(&F%6$\"\"'F(F(F(*&\"E R:Io2vM&3FCQN_\"HU!*z(F9F(F 2F9F6F9F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\" !" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,$*&#\"+&\\4A]F.& %\"cG6#F(F.&F46#\"\"%F.!\"\"**\"F;Kf^)3h\\\">Q&f1\\&yH7K%F.F6F.F3F.&F% 6$\"\"'F.F.F.*&\"FesA9$Q7^e=K7)3MiuaP#F.F3F.F9*&\"F&\\4A]F.&%\"cG6#\"\"$F.&F46#F(F.!\"\"**\"F;Kf^)3h\\\">Q&f1\\&y H7K%F.F7F.F3F.&F%6$\"\"'F.F.F.*&\"FXg=\\:`#HQ650AqY J\"\"\"\"*(,,*(\";8j^G`KW`%[@S&yF.&%\"cG6#\"\"$F.&F46#\"\"%F.!\"\"**\" ;_ni^`!pT*ead`8F.F7F.F3F.&F%6$\"\"'F.F.F.*&\";n\"z)Hs%)oyZ)ybv\"F.F3F. 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@v[Ns[ty\">ZokS6wt\"BOy71%[U*e.$QSe&H-f$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"?JFln*Q%pPfv.%G[b#\"@qI$3'R=p+bRm4HM&))" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"3 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "sub s(e13,OrderConditions(7,10,'expanded')):\nmap(u->simplify(lhs(u)-rhs(u )),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "The stability function of a gen eral 10 stage, order 7 Runge-Kutta scheme has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2+z^3/6+z^4/24+z ^5/120+z^6/720+z^7/5040+t[8]*z^8+t[9]*z^9+t[10]*z^10;" "6#/-%\"RG6#%\" zG,8\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)* &F'\"\"&\"$?\"F,F)*&F'F/\"$?(F,F)*&F'\"\"(\"%S]F,F)*&&%\"tG6#\"\")F)*$ F'F?F)F)*&&F=6#\"\"*F)*$F'FDF)F)*&&F=6#\"#5F)*$F'FIF)F)" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 20 "where, for example, " }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "t[9]=b[9]*a[9,8]*a[8,7]*a[7,6]* a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[10]*(a[10,8]*a[8,7]*a[7,6]*a[6,5]*a [5,4]*a[4,3]*a[3,2]*c[2]+a[10,9]*(a[9,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a [3,2]*c[2]+a[9,8]*(a[8,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[8,7]*(a[7 ,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5] *(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3])))))))" "6#/&%\"t G6#\"\"*,&*4&%\"bG6#F'\"\"\"&%\"aG6$F'\"\")F-&F/6$F1\"\"(F-&F/6$F4\"\" 'F-&F/6$F7\"\"&F-&F/6$F:\"\"%F-&F/6$F=\"\"$F-&F/6$F@\"\"#F-&%\"cG6#FCF -F-*&&F+6#\"#5F-,&*2&F/6$FJF1F-&F/6$F1F4F-&F/6$F4F7F-&F/6$F7F:F-&F/6$F :F=F-&F/6$F=F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F/6$FJF'F-,&*0&F/6$F'F4F-&F/6 $F4F7F-&F/6$F7F:F-&F/6$F:F=F-&F/6$F=F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F/6$F 'F1F-,&*.&F/6$F1F7F-&F/6$F7F:F-&F/6$F:F=F-&F/6$F=F@F-&F/6$F@FCF-&FE6#F CF-F-*&&F/6$F1F4F-,&*,&F/6$F4F:F-&F/6$F:F=F-&F/6$F=F@F-&F/6$F@FCF-&FE6 #FCF-F-*&&F/6$F4F7F-,&**&F/6$F7F=F-&F/6$F=F@F-&F/6$F@FCF-&FE6#FCF-F-*& &F/6$F7F:F-,&*(&F/6$F:F@F-&F/6$F@FCF-&FE6#FCF-F-*&&F/6$F:F=F-,&*&&F/6$ F=FCF-&FE6#FCF-F-*&&F/6$F=F@F-&FE6#F@F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F- " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[10]=b[10]*a[10,9]*a[9,8]*a[8,7]*a[ 7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^10" "6#/&%\"tG6#\"#5*8&%\"bG6# F'\"\"\"&%\"aG6$F'\"\"*F,&F.6$F0\"\")F,&F.6$F3\"\"(F,&F.6$F6\"\"'F,&F. 6$F9\"\"&F,&F.6$F<\"\"%F,&F.6$F?\"\"$F,&F.6$FB\"\"#F,&%\"cG6#FEF,%\"zG F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "t[8] = 1/8!;" "6#/&%\"tG6#\"\")*&\"\"\"F)-%*factorialG6#F'!\"\" " }{XPPEDIT 18 0 "`` = 1/44640;" "6#/%!G*&\"\"\"F&\"&SY%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "t[8] = 82/91;" "6#/&%\"tG6#\"\")*&\"##)\"\" \"\"#\"*!\"\"" }{TEXT -1 1 " " }{TEXT 272 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/9! = 41/16511040;" "6#/*&\"\"\"F%-%*factorialG6#\"\"* !\"\"*&\"#TF%\")S5^;F*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[9] = 7 1/82;" "6#/&%\"tG6#\"\"**&\"#r\"\"\"\"##)!\"\"" }{TEXT -1 1 " " } {TEXT 271 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/10! = 71/297561600; " "6#/*&\"\"\"F%-%*factorialG6#\"#5!\"\"*&\"#rF%\"*+;c(HF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "T his gives three equations which we can solve for " }{XPPEDIT 18 0 "c[ 6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"c G6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\" \")\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "Rz := StabilityFunction(7,10,'expa nded'):\neqA := simplify(subs(e13,coeff(Rz,z^10)))=71/82*1/10!:\neqB : = simplify(subs(e13,coeff(Rz,z^9)))=82/91*1/9!:\neqC := simplify(subs( e13,coeff(Rz,z^8)))=1/8!:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The first two of these equations are as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eqA;``;eqB;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\",hQ(*f]'\"7+]ifr%=WO[;$\"\"\"*0,,**\")k OB@F)&%\"cG6#\"\"%F)&F/6#\"\"$F)&%\"aG6$\"\"'F)F)F)*(\")[+p=F)F2F)F.F) !\"\"*&\"(obA)F)F2F)F)\"(4o#[F;*&F=F)F.F)F)F),&*&\"$j(F)F.F)F)\"$T\"F; F;,&*&FBF)F2F)F)FCF;F;,&\"#%*F;*&FBF)F2F)F)F)F.F),&F.F;F2F)F;F2F)F)F)# \"#r\"*+;c(H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/,$*&#\"\"\"\"9++]Yx\\1>6MR6F'**,>*&\"7KwmjR(>.p`7#F' &%\"cG6#\"\"$F'F'*&\"7KwmjR(>.p`7#F'&F.6#\"\"%F'!\"\"**\"7sEP@htwAt\\$ *F')F3\"\"#F'F-F'&%\"aG6$\"\"'F'F'F'*(\"6KteLl?$)>ES(F'F-F'F9F'F'*(\"8 #3np:*QZciix\"F'F-F'F3F'F'*&\"7k)G1ZQ!e1$>i$F')F-F:F'F6*&\"7k)G1ZQ!e1$ >i$F'F9F'F'*(\"9tPInzD\"p$RN4>F'FEF'F3F'F6*(\":'3)>ta^jP()pRV\"F')F-F0 F'F9F'F6*(\"9I?1t(=t*4CKhCF'FEF'F9F'F'*(\"9E#\\x.Ve2Q%=jWF'FLF'F3F'F'* *\"9s)>f6U]#QU.Z7F'F9F'FEF'F;F'F6**\"9[wUM&)elpT\")o!)F'F9F'FLF'F;F'F' **\"7sEP@htwAt\\$*F'F3F'FEF'F;F'F6F',&*&\"$j(F'F3F'F'\"$T\"F6F6,&*&FYF 'F-F'F'FZF6F6,&F3F6F-F'F6F'F6#\"#T\")S5^;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "These three equations can be so lved to obtain rational values for the three parameters " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&% \"aG6$\"\"'\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sol := solve(\{eqA,eqB,eqC\} ,\{c[3],c[4],a[6,1]\}):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c[3]=subs(sol,c[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$#\"in!RB9U\")3>4Y3W8D)3*e!ob&eJo1:%yedmp[ \"in$H'RV/kwEC[.L:]l!R[%G)4f_K&fe:Dr>e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Since the rational expressions for t he three parameters " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 164 " in volve a large number of digits, we replace them by simpler rational ap proximations and substitute these values in the expressions obtained f or the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "[c[3]=subs(sol,c[3]),c[4]=subs(sol,c[4]) ,a[6,1]=subs(sol,a[6,1])]:\nevalf[10](%);\nconvert(%,rational,6);\ne14 := \{op(%)\}:\ne15 := `union`(map(u_->lhs(u_)=simplify(subs(e14,rhs(u _))),e13),e14):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"$$ \"+>m`n$)!#5/&F&6#\"\"%$\"+'>4k?$F+/&%\"aG6$\"\"'\"\"\"$\"+UC>vUF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"$#\"$(*)\"%s5/&F&6#\"\" %#\"$g\"\"$*\\/&%\"aG6$\"\"'\"\"\"#\"$u\"\"$2%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3716 "e15 := \{b[8] = 7581774099 13942921875/78785859799058312576, b[10] = -1895689785077201/2887070844 50720, b[9] = -2973333145929506703/747247079460651200, b[5] = 16616464 7343343629239462104853/594150218553182839761586950000, b[6] = 88351573 65899998531682304/6999851448505791234978125, b[1] = 164949402359214517 /2969763122318428800, c[7] = 998622639/2007664595, b[7] = 110567843372 870117205263375074449005180061856568412069340896875/340717754649392931 183331876846338804668211857149398834541328544, a[6,3] = 86176546626991 8263/4970009789268581376, a[4,1] = 57894880/223353897, a[5,1] = 505666 4940861/42500572528960, a[5,3] = -286249149686016/36667767392227699, a [4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[2,1] = 1/24, a[8,7] = -7969512806056250503723894149024 2415433886906/23692035580167289837348760854832233004866581, a[7,1] = - 3439778790612913527970381936467065474966136664036779/39381209389424561 778044106217782104388290028625000000, a[10,8] = -737611406468471917873 4872354875/359022955840383035894248406127836, b[4] = 0, b[3] = 0, b[2] = 0, c[10] = 1, a[8,3] = 3346824815910846126629531019335170048/229804 8540174087438681694595766796875, a[10,1] = 862802087344743900546821971 74277603231/163235672733343926284895054973116600000, a[9,8] = 88873364 366227887565000/8176290045932637014386689, a[7,3] = 263141448694331408 5034368959766533330714638432739014413056/42832809612378095870766530128 571820142439373914153488671875, a[10,9] = 2554828403755937694389676527 31/8853429096639550069183960833070, a[7,6] = -541100633277452624094844 6773664264724938752/279717068193641240266628706086067991948046875, a[1 0,4] = 8964183417443891431743857714046506701/1273379282362081167514274 901917400000, a[5,4] = 2889259817673483/39243095536832320, a[9,7] = -5 74800468754997103944423838666252602754508075193305936911875/3115451846 93907215706597063235165448890282539301416442871559, a[8,4] = 189748010 561998005455394521375650921/29817877073894091150172096921875000, a[10, 6] = -5290899117464226058433745200964698112/88759510956456705613154928 76310816875, a[9,6] = -1788698788744957883891433528082038784/592314367 6314255446643283231146667875, a[8,6] = -120391896911431755652090822656 /233667703961223196028992578125, a[9,5] = -225580895352612761405928104 0689393645809880661461199/89151923156535278314548809378408790116229355 2521625, c[9] = 20/21, c[3] = 897/1072, c[4] = 160/499, a[6,1] = 174/4 07, c[2] = 1/24, c[5] = 141/763, c[6] = 169/192, c[8] = 104/105, a[10, 7] = -3069893906680376471707933893233719720682072201840672116412637833 56000/8001341807972529284403437443422408231395852556020787666765419670 7053, a[3,2] = 2413827/287296, a[6,4] = 10826192777269994995/771269124 5665173504, a[8,1] = 15322017632690457879761200795014874391/2930492370 2961208406725757182921875000, a[6,5] = -16750236560040896731/148971940 35750715392, a[9,3] = 8472900696201518374537737273138176/9559881927124 203744915849518389875, a[8,5] = -3828453315838351931950893978692504047 38354182/110225463285588517977748940118009136362890625, a[10,3] = 8711 91425901232070238397371934719232/533362446801744790377544382297593125, a[7,4] = -3226817813200049695169462352322166717564113587934800370011/ 40015346463976160761929086193516183832403838724337375000000, a[9,4] = \+ 1046253623595072271778442420598129/248084737254798838369431846390000, \+ a[7,5] = 786190577550706940215874040419538587608361105183350570182/126 1335096542994096899051116924589799398264209359351953125, a[10,5] = -25 2047840384081563219583820196041648831131662918986674/66729954558238813 530334990145293701430108220259249375, a[3,1] = -2173431/287296, a[9,1] = 1032897956298204654305923433045759171807509/20049611367698855460675 10633666861543410000, a[4,3] = 13721600/223353897\}:" }}}{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 45 "subs(e15,StabilityFunction(7,10,'expanded'));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8\"\"\"F$%\"zGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$* $)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F $*&#F$\"$?(F$*$)F%F-F$F$F$*&#F$\"%S]F$*$)F%\"\"(F$F$F$*&#\"EZ3r7A$H(*G z[!=7-L)*eX\"J++]$3Q(p?&))R/\"[$Gqi\\=Q=F$*$)F%\"\")F$F$F$*&#\"DP0]8u0 ,1,Iv)eUzB$z\"\"I+]Ph*RX)=pc\")*Q*R!\\#4W@sF$*$)F%\"\"*F$F$F$*&#\"?;Dd ]l^%GeIM;G;O$\"FD1WM)=dNm2<2DN'Gl)39F$*$)F%\"#5F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We may compare the c oefficients of " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "z^9" "6#*$%\"zG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "z^10" "6#*$%\"zG\"#5" }{TEXT -1 58 " with those of the original specified stability function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "[455898330212180487928 972932212710847/18381849627028348104398852069738083500000*8!,179323794 25887530010601057413500537/7221440924903993898156691884539961375000*9! ,336162816343058284516550572516/1408865286352507170766355718834440625* 10!]:\nevalf[10](%);\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"+\\U)*****!#5$\"+k]36!*F&$\"+(\\6&e')F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"#\"##)\"#\"*#\"#rF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------- --------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#-------------------------- -----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the embedded order 6 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3716 "e15 := \{b[8] = 7581774099 13942921875/78785859799058312576, b[10] = -1895689785077201/2887070844 50720, b[9] = -2973333145929506703/747247079460651200, b[5] = 16616464 7343343629239462104853/594150218553182839761586950000, b[6] = 88351573 65899998531682304/6999851448505791234978125, b[1] = 164949402359214517 /2969763122318428800, c[7] = 998622639/2007664595, b[7] = 110567843372 870117205263375074449005180061856568412069340896875/340717754649392931 183331876846338804668211857149398834541328544, a[6,3] = 86176546626991 8263/4970009789268581376, a[4,1] = 57894880/223353897, a[5,1] = 505666 4940861/42500572528960, a[5,3] = -286249149686016/36667767392227699, a [4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[2,1] = 1/24, a[8,7] = -7969512806056250503723894149024 2415433886906/23692035580167289837348760854832233004866581, a[7,1] = - 3439778790612913527970381936467065474966136664036779/39381209389424561 778044106217782104388290028625000000, a[10,8] = -737611406468471917873 4872354875/359022955840383035894248406127836, b[4] = 0, b[3] = 0, b[2] = 0, c[10] = 1, a[8,3] = 3346824815910846126629531019335170048/229804 8540174087438681694595766796875, a[10,1] = 862802087344743900546821971 74277603231/163235672733343926284895054973116600000, a[9,8] = 88873364 366227887565000/8176290045932637014386689, a[7,3] = 263141448694331408 5034368959766533330714638432739014413056/42832809612378095870766530128 571820142439373914153488671875, a[10,9] = 2554828403755937694389676527 31/8853429096639550069183960833070, a[7,6] = -541100633277452624094844 6773664264724938752/279717068193641240266628706086067991948046875, a[1 0,4] = 8964183417443891431743857714046506701/1273379282362081167514274 901917400000, a[5,4] = 2889259817673483/39243095536832320, a[9,7] = -5 74800468754997103944423838666252602754508075193305936911875/3115451846 93907215706597063235165448890282539301416442871559, a[8,4] = 189748010 561998005455394521375650921/29817877073894091150172096921875000, a[10, 6] = -5290899117464226058433745200964698112/88759510956456705613154928 76310816875, a[9,6] = -1788698788744957883891433528082038784/592314367 6314255446643283231146667875, a[8,6] = -120391896911431755652090822656 /233667703961223196028992578125, a[9,5] = -225580895352612761405928104 0689393645809880661461199/89151923156535278314548809378408790116229355 2521625, c[9] = 20/21, c[3] = 897/1072, c[4] = 160/499, a[6,1] = 174/4 07, c[2] = 1/24, c[5] = 141/763, c[6] = 169/192, c[8] = 104/105, a[10, 7] = -3069893906680376471707933893233719720682072201840672116412637833 56000/8001341807972529284403437443422408231395852556020787666765419670 7053, a[3,2] = 2413827/287296, a[6,4] = 10826192777269994995/771269124 5665173504, a[8,1] = 15322017632690457879761200795014874391/2930492370 2961208406725757182921875000, a[6,5] = -16750236560040896731/148971940 35750715392, a[9,3] = 8472900696201518374537737273138176/9559881927124 203744915849518389875, a[8,5] = -3828453315838351931950893978692504047 38354182/110225463285588517977748940118009136362890625, a[10,3] = 8711 91425901232070238397371934719232/533362446801744790377544382297593125, a[7,4] = -3226817813200049695169462352322166717564113587934800370011/ 40015346463976160761929086193516183832403838724337375000000, a[9,4] = \+ 1046253623595072271778442420598129/248084737254798838369431846390000, \+ a[7,5] = 786190577550706940215874040419538587608361105183350570182/126 1335096542994096899051116924589799398264209359351953125, a[10,5] = -25 2047840384081563219583820196041648831131662918986674/66729954558238813 530334990145293701430108220259249375, a[3,1] = -2173431/287296, a[9,1] = 1032897956298204654305923433045759171807509/20049611367698855460675 10633666861543410000, a[4,3] = 13721600/223353897\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We now t urn our attention to the embedded order 6 scheme and introduce a new r ow corresponding to the node " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG 6#\"#6\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The linki ng coefficients and weights can be chosen so as to form an 11 stage or der 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We use the order 6 quadrature conditions which are given \+ in abreviated form as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "subs(b=`b*`,QuadratureConditions(6)):\nListTools[Enumerate](%): \nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdi m](%))]),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%\"\"$F)/*&F,F()F2F/F(#F(F57%\"\"%F)/*&F,F()F2F5F(#F(F;7%\"\"&F)/ *&F,F()F2F;F(#F(FA7%\"\"'F)/*&F,F()F2FAF(#F(FGQ(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We incorporate the row sum condition fo r the new tenth row" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j],j = 1 .. 10) = c[11];" "6#/-%$SumG6$&%\"aG6$\"#6%\"jG/F +;\"\"\"\"#5&%\"cG6#F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "together with the stage-order equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j],j = 2 .. 10) = 1/2;" " 6#/-%$SumG6$*&&%\"aG6$\"#6%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"#5*&F-F-F3 !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^2;" "6#*$&%\"cG6#\"#6\"\" #" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j]^2,j = 2 .. \+ 10) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"#6%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/ F,;F2\"#5*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^3;" " 6#*$&%\"cG6#\"#6\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 50 "which ensure that the tenth row has stage-order 3." }}{PARA 0 "" 0 " " {TEXT -1 53 "We also incorporate the column simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,1], i=2..11)=`b*`[1]" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+; \"\"#\"#6&F)6#F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j], i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\" &%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F, " }{TEXT -1 6 " , " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" }{TEXT -1 10 ", 6, 7, 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "`Qeqs*` := subs(b=`b*`,QuadratureConditions( 6,11,'expanded')):\nSO_eqs2 := [add(a[11,j],j=1..10)=c[11],add(a[11,j] *c[j],j=2..10)=1/2*c[11]^2,\n add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^ 3]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*`[ i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[$5..8])]:\n`cdns*` := [op(SO_ eqs2),op(`Qeqs*`),op(`simp_eqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We specify that " }{XPPEDIT 18 0 "c[1 1] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[1 1,2] = 0;" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,10] = 0;" "6#/&%\"aG6$\"#6\"#5\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[10]=0" "6#/&%#b*G6#\"#5\"\"!" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "`b*`[11] = -38/5;" "6#/&%#b*G6#\"#6,$*&\"#Q\"\"\" \"\"&!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We also s et " }{XPPEDIT 18 0 "b[11] = 0;" "6#/&%\"bG6#\"#6\"\"!" }{TEXT -1 66 ", so that the order 7 scheme can be regarded as a 11 stage scheme." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have \+ 14 equations for the 14 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "e16 := \{c[11]=1,b[11]=0,a[11,2]=0,a[11,10]=0,` b*`[2]=0,`b*`[3]=0,`b*`[4]=0,`b*`[10]=0,`b*`[11]=-38/5\}:\ne17 := `uni on`(e15,e16):\n`eqns*` := subs(e17,`cdns*`):\nnops(%);\nindets(`eqns*` );\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<0&%#b*G6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\")&F%6#\" \"*&F%6#\"\"\"&%\"aG6$\"#6F6&F86$F:\"\"$&F86$F:\"\"%&F86$F:F'&F86$F:F* &F86$F:F-&F86$F:F0&F86$F:F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e18 := solve(\{op(`eqns*`)\}):\ninfolevel[solve] := 0:\ne19 := ` union`(e17,e18):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e19 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4976 "e19 := \{`b*`[5] = 268231112188987944945827753/9552254317575 28681288725000, b[8] = 758177409913942921875/78785859799058312576, `b* `[9] = -842056888251867309/186811769865162800, b[10] = -18956897850772 01/288707084450720, b[9] = -2973333145929506703/747247079460651200, b[ 5] = 166164647343343629239462104853/594150218553182839761586950000, b[ 6] = 8835157365899998531682304/6999851448505791234978125, a[11,7] = -1 0527833851533262864999371634499752441444282441429608616850996735536625 /274639858144995223546065019959486034036181712680995443385781566266518 4, `b*`[8] = 54584684081980124625/4924116237441144536, a[11,3] = 16004 6766004414315742455765106336/98065952173285752701328766250625, `b*`[7] = 108419211385790972003069732530101032173894775381027875/337664606137 936380500051156293384895353362152117883624, b[1] = 164949402359214517/ 2969763122318428800, c[7] = 998622639/2007664595, b[7] = 1105678433728 70117205263375074449005180061856568412069340896875/3407177546493929311 83331876846338804668211857149398834541328544, a[11,9] = 40097947059612 729/1419769450975237280, a[11,6] = -7360838571444342440150826102960429 4656/123768183621850031611831870691647493125, a[6,3] = 861765466269918 263/4970009789268581376, a[4,1] = 57894880/223353897, a[5,1] = 5056664 940861/42500572528960, a[5,3] = -286249149686016/36667767392227699, a[ 4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, \+ a[10,2] = 0, a[11,5] = -1900266684289629480230017826550979437884206498 5811946527/5038998773730568797417283222822299497058344239378524000, a[ 11,1] = 2429287816990748646637471052716009978490201/460701576400638855 6345602665417429396800000, a[2,1] = 1/24, `b*`[6] = 415490030471198546 067456/304341367326338749346875, a[8,7] = -796951280605625050372389414 90242415433886906/23692035580167289837348760854832233004866581, c[11] \+ = 1, a[11,2] = 0, a[11,10] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[10] = 0, b[11] = 0, a[7,1] = -3439778790612913527970381936467065 474966136664036779/393812093894245617780441062177821043882900286250000 00, `b*`[1] = 20506743773621213/371220390289803600, a[10,8] = -7376114 064684719178734872354875/359022955840383035894248406127836, b[4] = 0, \+ b[3] = 0, b[2] = 0, c[10] = 1, a[8,3] = 334682481591084612662953101933 5170048/2298048540174087438681694595766796875, a[10,1] = 8628020873447 4390054682197174277603231/163235672733343926284895054973116600000, a[1 1,4] = 572651535981050835474886343012483/81435870991749105781564929600 000, a[9,8] = 88873364366227887565000/8176290045932637014386689, a[7,3 ] = 2631414486943314085034368959766533330714638432739014413056/4283280 9612378095870766530128571820142439373914153488671875, a[10,9] = 255482 840375593769438967652731/8853429096639550069183960833070, `b*`[11] = - 38/5, a[7,6] = -5411006332774526240948446773664264724938752/2797170681 93641240266628706086067991948046875, a[10,4] = 89641834174438914317438 57714046506701/1273379282362081167514274901917400000, a[5,4] = 2889259 817673483/39243095536832320, a[9,7] = -5748004687549971039444238386662 52602754508075193305936911875/3115451846939072157065970632351654488902 82539301416442871559, a[8,4] = 189748010561998005455394521375650921/29 817877073894091150172096921875000, a[11,8] = -561686306507932150384087 5/276176908218169072485950384, a[10,6] = -5290899117464226058433745200 964698112/8875951095645670561315492876310816875, a[9,6] = -17886987887 44957883891433528082038784/5923143676314255446643283231146667875, a[8, 6] = -120391896911431755652090822656/233667703961223196028992578125, a [9,5] = -2255808953526127614059281040689393645809880661461199/89151923 1565352783145488093784087901162293552521625, c[9] = 20/21, c[3] = 897/ 1072, c[4] = 160/499, a[6,1] = 174/407, c[2] = 1/24, c[5] = 141/763, c [6] = 169/192, c[8] = 104/105, a[10,7] = -3069893906680376471707933893 23371972068207220184067211641263783356000/8001341807972529284403437443 4224082313958525560207876667654196707053, a[3,2] = 2413827/287296, a[6 ,4] = 10826192777269994995/7712691245665173504, a[8,1] = 1532201763269 0457879761200795014874391/29304923702961208406725757182921875000, a[6, 5] = -16750236560040896731/14897194035750715392, a[9,3] = 847290069620 1518374537737273138176/9559881927124203744915849518389875, a[8,5] = -3 82845331583835193195089397869250404738354182/1102254632855885179777489 40118009136362890625, a[10,3] = 871191425901232070238397371934719232/5 33362446801744790377544382297593125, a[7,4] = -32268178132000496951694 62352322166717564113587934800370011/4001534646397616076192908619351618 3832403838724337375000000, a[9,4] = 1046253623595072271778442420598129 /248084737254798838369431846390000, a[7,5] = 7861905775507069402158740 40419538587608361105183350570182/1261335096542994096899051116924589799 398264209359351953125, a[10,5] = -252047840384081563219583820196041648 831131662918986674/667299545582388135303349901452937014301082202592493 75, a[3,1] = -2173431/287296, a[9,1] = 1032897956298204654305923433045 759171807509/2004961136769885546067510633666861543410000, a[4,3] = 137 21600/223353897\}:" }}}{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 0 137 "subs(e19,matrix([seq([c[i], seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[j],j=1..11)],[`b *`,seq(`b*`[j],j=1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'nmT!\"(F(%!GF+F+F+F+F+F+F+F+F+7.$\" 'an$)!\"'$!'8lv!\"&$\"')=S)F2F+F+F+F+F+F+F+F+F+7.$\"'T1KF/$\"'2#f#F/$ \"\"!F;$\"'VVhF*F+F+F+F+F+F+F+F+7.$\"'(z%=F/$\"'z*=\"F/F:$!'c1y!\")$\" 'ZitF*F+F+F+F+F+F+F+7.$\"'3-))F/$\"'=vUF/F:$\"'$Rt\"F/$\"'p.9F2$!'RC6F 2F+F+F+F+F+F+7.$\"'0u\\F/$!'dM()F*F:$\"'YVhF*$!'&R1)F*$\"'+LiF/$!'YM>F *F+F+F+F+F+7.$\"'w/**F/$\"'[G_F/F:$\"'Qc9F2$\"'djjF2$!'HtMF2$!'F_^F/$! 'zjLF2F+F+F+F+7.$\"'\"Q_*F/$\"'r^^F/F:$\"')H'))F/$\"'KIF/$!'+X=F2$\"''p3\"F*F+F+F+7.$\"\"\"F;$\"'i&G&F/F:$\"'RL;F2$\"'oRqF 2$!'8xPF2$!'%4'fF/$!'sOQF2$!']a?F*$\"'p&)GF*F+F+7.F[q$\"'-t_F/F:$\"'.K ;F2$\"'$>.(F2$!'7rPF2$!'GZfF/$!'KLQF2$!'zL?F*$\"'ECGF*F:F+7.%\"bG$\"'H abF*F:F:F:$\"'o'z#F/$\"'>i7F2$\"'9XKF/$\"'FB'*F2$!'0zRF2$!'9mlF2F:7.%# b*G$\"'9CbF*F:F:F:$\"'/3GF/$\"'@l8F2$\"''3@$F/$\"'_36!\"%$!'_2XF2F:$!' ++wF2Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "pr inted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 459 "subs(e19,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(6-i)],i=2..5)]));print(``);\nfor ii from 6 to 11 do\n print (c[ii]=subs(e19,c[ii]));print(``); \n for jj to ii-1 do\n print (a[ii,jj]=subs(e19,a[ii,jj]));\n end do:\n print(`________________ _________________`);\nend do:print(``);\nfor ii to 11 do\n print(b[i i]=subs(e19,b[ii]));\nend do:\nprint(`________________________________ _`);print(``);\nfor ii to 11 do\n print(`b*`[ii]=subs(e19,`b*`[ii])) ;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7(#\"\"\" \"#CF(%!GF+F+F+7(#\"$(*)\"%s5#!(JM<#\"''H(G#\"(FQT#F2F+F+F+7(#\"$g\"\" $*\\#\")!)[*y&\"*(*QNB#\"\"!#\")+;s8F;F+F+7(#\"$T\"\"$j(#\".h3%\\mc]\" /g*GDd+D%F<#!0;go\\\"\\iG\"2*pFARnxmO#\"1$[tw\")f#*)G\"2?B$o`&4V#RF+Q( pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$p\"\"$#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"\"#\"$u\"\"$2%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\" #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#\"3j#=*p iYl<')\"4w8eo#*y4+(\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" '\"\"%#\"5&*\\**psx#>E3\"\"4/N(*[\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"*REi)**\"+&fkw+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"(\"\"\"#!Uzn.km8m\\ZlqYO>Qqz_8Hh!zy(RM\"V+++D'G+H)Q/@y'3H>wgh(RYY`,S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&#\"Z#=q0N$=06O3weQ&>//ue@Spq]vd!>'y \"enDJ&>Nf$4UE)R*z*eCp6^!**o4%*Ha'4N8E\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!L_(Q\\skUmtnW[4CEXxKj+6a\"Nvo/[>*z1'31(GmES 7k$>oqrz#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_______________________ __________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"$/\"\" $0\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#\"G\"Ru[,&z+7wzyX!pKw,A`\"\"G+](=# H=dds1%37'HqB\\IH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\"F[+< N$>5`Hm7Y3\"f\"[#oM$\"Fvozmdf%p\"oQu3u,a[!)H#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#\"E@4lv8_%Rba+)*>c5![(*=\"D+](=#p4s ,:\"4%*Q2xy\")H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"& #!N#=a$QZS]#pyR*3&>$>NQeJ`%GQ\"ND1*GOO\"4!=,%*[x(z^)e&GjaA5\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!?cE#34_cvJ9\"p*=R ?\"\"?D\"yD**Gg>B7'RqnOB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"(#!M1p)QV:CC!\\T*Qs.0Dcg!G^pz\"M\"em[+LA$[&3w[t$)*Gn,eN?pB" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#?\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"*\"\"\"#\"L4v!=))f&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"*\"\"%#\"CH\")f?CWyh9m!))4ek$R*oS 5GfShFh_`*3eD#\"TD;__NHi6!z3%y$4)[XJy_`cJ#>:*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#!F%yQ?3GNV\"*Q)y&\\u)y)p)y\"\"FvymY 6B$GVmWbUJwO9Bf" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"( #!gnv=\"p$fI$>v!3Xv-EDm'QQUWR5(*\\vo/![d\"gnf:(GW;9IRDG!*)[a;NK1(f1d@2 Rp%=X:J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"8+]c( )yAmVOt)))\":*o'Q9qjKf/!Hw\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___ ______________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"GJKgxU<(>#oa+RuWt3-G')\"H+ +g;J(\\0&*[GERMLFnNK;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$#\"EK# >Z$>P(RQ-2K7!fU\">r)\"EDJf(H#QWvP!zW!\\F9 v;\"3iBGzLF\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!W um)*=HmJ6$)[;/'>?Qe>Kc\"3%QSy/_#\"Vv$\\#f-A3,V,PHX,*\\LIN\")Q#eX&*Hn' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!F7\")pk4?XPVeg Aku6**3H&\"Fvo\"3JwG\\:8cqck&4^f())" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#5\"\"(#!`o+gN$yj7k6s1%=?s?o?(>PB$*Q$zqrkP!o1R*)pI\"_o`qq' >awmwy?gb_eRJ#3CUVuV.WGHD(z!=M,!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#5\"\")#!@v[Ns[ty\">ZokS6wt\"BOy71%[U*e.$QSe&H-f$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"?JFln*Q%pPfv.%G[b#\"@qI $3'R=p+bRm4HM&))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B________________ _________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#\"L,-\\y*4gr_5ZPmk[2*p\"yGHC\"L++!o RHuTlEgXjb)Q1Sw:qg%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$#\"BOj5l dXUdJ9W+mn/g\"\"AD1Dm(G8q_dGt@&f1)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#6\"\"%#\"B$[7IM')[ZN30\")f`^Ed\"A++gH\\c\"y0\"\\<*4(eV\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!YFl%>\"e)\\1U) yVz4bEy,I-[H'*G%om-!>\"X+S_y$RUMeq\\*HAGA$G]1YNA&*\\9e)RYF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\")#!:v3%Q]@$z]1joh&\"<%Q]f[s!p\"=#3p9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \"#\"3r/.!\\: %\"9voM\\(QjKn8M/$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(# \"Wvy-\"QvZ*Q%3\"\"WCO)y6_@O``*[Q$Hc60+0QOz81YmP$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"5DY7!)>3%o%ea\"4OX9T uB;T#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!34t'=D))o0U )\"3+G;l)p<\"o=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!#Q\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "`RK6_11eqs*` := subs(b= `b*`,OrderConditions(6,11,'expanded')):\nsubs(e19,`RK6_11eqs*`):\nmap( u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------ ---------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#------------------------------ -------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4911 "ee := \{c[2]=1/24,\nc[3]=897/1072,\nc[4]=160/499,\nc[5]=141/763, \nc[6]=169/192,\nc[7]=998622639/2007664595,\nc[8]=104/105,\nc[9]=20/21 ,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/24,\na[3,1]=-2173431/287296,\na[3,2] =2413827/287296,\na[4,1]=57894880/223353897,\na[4,2]=0,\na[4,3]=137216 00/223353897,\na[5,1]=5056664940861/42500572528960,\na[5,2]=0,\na[5,3] =-286249149686016/36667767392227699,\na[5,4]=2889259817673483/39243095 536832320,\na[6,1]=174/407,\na[6,2]=0,\na[6,3]=861765466269918263/4970 009789268581376,\na[6,4]=10826192777269994995/7712691245665173504,\na[ 6,5]=-16750236560040896731/14897194035750715392,\na[7,1]=-343977879061 2913527970381936467065474966136664036779/\n 3938120938942456177 8044106217782104388290028625000000,\na[7,2]=0,\na[7,3]=263141448694331 4085034368959766533330714638432739014413056/\n 42832809612378095 870766530128571820142439373914153488671875,\na[7,4]=-32268178132000496 95169462352322166717564113587934800370011/\n 400153464639761607 61929086193516183832403838724337375000000,\na[7,5]=7861905775507069402 15874040419538587608361105183350570182/\n 1261335096542994096899 051116924589799398264209359351953125,\na[7,6]=-54110063327745262409484 46773664264724938752/279717068193641240266628706086067991948046875,\na [8,1]=15322017632690457879761200795014874391/2930492370296120840672575 7182921875000,\na[8,2]=0,\na[8,3]=334682481591084612662953101933517004 8/2298048540174087438681694595766796875,\na[8,4]=189748010561998005455 394521375650921/29817877073894091150172096921875000,\na[8,5]=-38284533 1583835193195089397869250404738354182/11022546328558851797774894011800 9136362890625,\na[8,6]=-120391896911431755652090822656/233667703961223 196028992578125,\na[8,7]=-79695128060562505037238941490242415433886906 /23692035580167289837348760854832233004866581,\na[9,1]=103289795629820 4654305923433045759171807509/20049611367698855460675106336668615434100 00,\na[9,2]=0,\na[9,3]=8472900696201518374537737273138176/955988192712 4203744915849518389875,\na[9,4]=1046253623595072271778442420598129/248 084737254798838369431846390000,\na[9,5]=-22558089535261276140592810406 89393645809880661461199/\n 891519231565352783145488093784087901 162293552521625,\na[9,6]=-1788698788744957883891433528082038784/592314 3676314255446643283231146667875,\na[9,7]=-5748004687549971039444238386 66252602754508075193305936911875/\n 311545184693907215706597063 235165448890282539301416442871559,\na[9,8]=88873364366227887565000/817 6290045932637014386689,\na[10,1]=8628020873447439005468219717427760323 1/163235672733343926284895054973116600000,\na[10,2]=0,\na[10,3]=871191 425901232070238397371934719232/533362446801744790377544382297593125,\n a[10,4]=8964183417443891431743857714046506701/127337928236208116751427 4901917400000,\na[10,5]=-252047840384081563219583820196041648831131662 918986674/\n 6672995455823881353033499014529370143010822025924 9375,\na[10,6]=-5290899117464226058433745200964698112/8875951095645670 561315492876310816875,\na[10,7]=-3069893906680376471707933893233719720 68207220184067211641263783356000/\n 80013418079725292844034374 434224082313958525560207876667654196707053,\na[10,8]=-7376114064684719 178734872354875/359022955840383035894248406127836,\na[10,9]=2554828403 75593769438967652731/8853429096639550069183960833070,\na[11,1]=2429287 816990748646637471052716009978490201/460701576400638855634560266541742 9396800000,\na[11,2]=0,\na[11,3]=160046766004414315742455765106336/980 65952173285752701328766250625,\na[11,4]=572651535981050835474886343012 483/81435870991749105781564929600000,\na[11,5]=-1900266684289629480230 0178265509794378842064985811946527/\n 503899877373056879741728 3222822299497058344239378524000,\na[11,6]=-736083857144434244015082610 29604294656/123768183621850031611831870691647493125,\na[11,7]=-1052783 3851533262864999371634499752441444282441429608616850996735536625/\n \+ 2746398581449952235460650199594860340361817126809954433857815662 665184,\na[11,8]=-5616863065079321503840875/27617690821816907248595038 4,\na[11,9]=40097947059612729/1419769450975237280,\na[11,10]=0,\n\nb[1 ]=164949402359214517/2969763122318428800,\nb[2]=0,\nb[3]=0,\nb[4]=0,\n b[5]=166164647343343629239462104853/594150218553182839761586950000,\nb [6]=8835157365899998531682304/6999851448505791234978125,\nb[7]=1105678 43372870117205263375074449005180061856568412069340896875/\n 340717 754649392931183331876846338804668211857149398834541328544,\nb[8]=75817 7409913942921875/78785859799058312576,\nb[9]=-2973333145929506703/7472 47079460651200,\nb[10]=-1895689785077201/288707084450720,\nb[11]=0,\n \n`b*`[1]=20506743773621213/371220390289803600,\n`b*`[2]=0,\n`b*`[3]=0 ,\n`b*`[4]=0,\n`b*`[5]=268231112188987944945827753/9552254317575286812 88725000,\n`b*`[6]=415490030471198546067456/304341367326338749346875, \n`b*`[7]=108419211385790972003069732530101032173894775381027875/\n \+ 337664606137936380500051156293384895353362152117883624,\n`b*`[8]= 54584684081980124625/4924116237441144536,\n`b*`[9]=-842056888251867309 /186811769865162800,\n`b*`[10]=0,\n`b*`[11]=-38/5\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "T[7,11];" "6#&%\"TG6$\"\"(\"#6" }{TEXT -1 128 " denote the vector whose components are the principal error terms of the 11 s tage, order 7 scheme (the error terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6,11];" "6#&%#T*G6$\"\"'\"#6" }{TEXT -1 146 " denote the vector whose components are the principal \+ error terms of the embedded 11 stage, order 6 scheme (the error terms \+ of order 7) and let " }{XPPEDIT 18 0 "`T*`[7,11];" "6#&%#T*G6$\"\"(\" #6" }{TEXT -1 99 " denote the vector whose components are the error t erms of order 8 of the embedded order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[7,11]));" "6#-%$absG6#-F$6#& %\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,11] ));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"#6" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[7,11]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\" #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[8] = abs(abs(T[7,11]));" "6#/&%\"AG6#\"\")-%$absG 6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[8] = \+ abs(abs(`T*`[7,11]))/abs(abs(`T*`[6,11]));" "6#/&%\"BG6#\"\")*&-%$absG 6#-F*6#&%#T*G6$\"\"(\"#6\"\"\"-F*6#-F*6#&F/6$\"\"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7,11]-T[7,11]))/abs(abs (`T*`[6,11]));" "6#/&%\"CG6#\"\")*&-%$absG6#-F*6#,&&%#T*G6$\"\"(\"#6\" \"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"'F3F8" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[8];" "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [8];" "6#&%\"CG6#\"\")" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "errterms7_11 := PrincipalEr rorTerms(7,11,'expanded'):\n`errterms7_11*` :=subs(b=`b*`,PrincipalErr orTerms(7,11,'expanded')):\n`errterms6_11*` := subs(b=`b*`,PrincipalEr rorTerms(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs (ee,`errterms7_11*`[i]))^2,i=1..nops(`errterms7_11*`))):\nsdnB := sqrt (add(evalf(subs(ee,`errterms6_11*`[i]))^2,i=1..nops(`errterms6_11*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms7_11*`[i])-subs(ee,errterm s7_11[i])))^2,i=1..nops(errterms7_11))):\n'B[8]'= evalf[8](snmB/sdnB); \n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"BG6#\"\")$\")A)[>#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6# \"\")$\"))y()H#!\"(" }}}{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 4911 "ee := \{c[2]=1/24,\nc[3]=897/1072,\nc[4]=160/499,\nc[5]=141/763, \nc[6]=169/192,\nc[7]=998622639/2007664595,\nc[8]=104/105,\nc[9]=20/21 ,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/24,\na[3,1]=-2173431/287296,\na[3,2] =2413827/287296,\na[4,1]=57894880/223353897,\na[4,2]=0,\na[4,3]=137216 00/223353897,\na[5,1]=5056664940861/42500572528960,\na[5,2]=0,\na[5,3] =-286249149686016/36667767392227699,\na[5,4]=2889259817673483/39243095 536832320,\na[6,1]=174/407,\na[6,2]=0,\na[6,3]=861765466269918263/4970 009789268581376,\na[6,4]=10826192777269994995/7712691245665173504,\na[ 6,5]=-16750236560040896731/14897194035750715392,\na[7,1]=-343977879061 2913527970381936467065474966136664036779/\n 3938120938942456177 8044106217782104388290028625000000,\na[7,2]=0,\na[7,3]=263141448694331 4085034368959766533330714638432739014413056/\n 42832809612378095 870766530128571820142439373914153488671875,\na[7,4]=-32268178132000496 95169462352322166717564113587934800370011/\n 400153464639761607 61929086193516183832403838724337375000000,\na[7,5]=7861905775507069402 15874040419538587608361105183350570182/\n 1261335096542994096899 051116924589799398264209359351953125,\na[7,6]=-54110063327745262409484 46773664264724938752/279717068193641240266628706086067991948046875,\na [8,1]=15322017632690457879761200795014874391/2930492370296120840672575 7182921875000,\na[8,2]=0,\na[8,3]=334682481591084612662953101933517004 8/2298048540174087438681694595766796875,\na[8,4]=189748010561998005455 394521375650921/29817877073894091150172096921875000,\na[8,5]=-38284533 1583835193195089397869250404738354182/11022546328558851797774894011800 9136362890625,\na[8,6]=-120391896911431755652090822656/233667703961223 196028992578125,\na[8,7]=-79695128060562505037238941490242415433886906 /23692035580167289837348760854832233004866581,\na[9,1]=103289795629820 4654305923433045759171807509/20049611367698855460675106336668615434100 00,\na[9,2]=0,\na[9,3]=8472900696201518374537737273138176/955988192712 4203744915849518389875,\na[9,4]=1046253623595072271778442420598129/248 084737254798838369431846390000,\na[9,5]=-22558089535261276140592810406 89393645809880661461199/\n 891519231565352783145488093784087901 162293552521625,\na[9,6]=-1788698788744957883891433528082038784/592314 3676314255446643283231146667875,\na[9,7]=-5748004687549971039444238386 66252602754508075193305936911875/\n 311545184693907215706597063 235165448890282539301416442871559,\na[9,8]=88873364366227887565000/817 6290045932637014386689,\na[10,1]=8628020873447439005468219717427760323 1/163235672733343926284895054973116600000,\na[10,2]=0,\na[10,3]=871191 425901232070238397371934719232/533362446801744790377544382297593125,\n a[10,4]=8964183417443891431743857714046506701/127337928236208116751427 4901917400000,\na[10,5]=-252047840384081563219583820196041648831131662 918986674/\n 6672995455823881353033499014529370143010822025924 9375,\na[10,6]=-5290899117464226058433745200964698112/8875951095645670 561315492876310816875,\na[10,7]=-3069893906680376471707933893233719720 68207220184067211641263783356000/\n 80013418079725292844034374 434224082313958525560207876667654196707053,\na[10,8]=-7376114064684719 178734872354875/359022955840383035894248406127836,\na[10,9]=2554828403 75593769438967652731/8853429096639550069183960833070,\na[11,1]=2429287 816990748646637471052716009978490201/460701576400638855634560266541742 9396800000,\na[11,2]=0,\na[11,3]=160046766004414315742455765106336/980 65952173285752701328766250625,\na[11,4]=572651535981050835474886343012 483/81435870991749105781564929600000,\na[11,5]=-1900266684289629480230 0178265509794378842064985811946527/\n 503899877373056879741728 3222822299497058344239378524000,\na[11,6]=-736083857144434244015082610 29604294656/123768183621850031611831870691647493125,\na[11,7]=-1052783 3851533262864999371634499752441444282441429608616850996735536625/\n \+ 2746398581449952235460650199594860340361817126809954433857815662 665184,\na[11,8]=-5616863065079321503840875/27617690821816907248595038 4,\na[11,9]=40097947059612729/1419769450975237280,\na[11,10]=0,\n\nb[1 ]=164949402359214517/2969763122318428800,\nb[2]=0,\nb[3]=0,\nb[4]=0,\n b[5]=166164647343343629239462104853/594150218553182839761586950000,\nb [6]=8835157365899998531682304/6999851448505791234978125,\nb[7]=1105678 43372870117205263375074449005180061856568412069340896875/\n 340717 754649392931183331876846338804668211857149398834541328544,\nb[8]=75817 7409913942921875/78785859799058312576,\nb[9]=-2973333145929506703/7472 47079460651200,\nb[10]=-1895689785077201/288707084450720,\nb[11]=0,\n \n`b*`[1]=20506743773621213/371220390289803600,\n`b*`[2]=0,\n`b*`[3]=0 ,\n`b*`[4]=0,\n`b*`[5]=268231112188987944945827753/9552254317575286812 88725000,\n`b*`[6]=415490030471198546067456/304341367326338749346875, \n`b*`[7]=108419211385790972003069732530101032173894775381027875/\n \+ 337664606137936380500051156293384895353362152117883624,\n`b*`[8]= 54584684081980124625/4924116237441144536,\n`b*`[9]=-842056888251867309 /186811769865162800,\n`b*`[10]=0,\n`b*`[11]=-38/5\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stabi lity function R for the 11 stage, order 7 scheme is given as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunctio n(7,11,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F) *&#\"EZ3r7A$H(*Gz[!=7-L)*eX\"J++]$3Q(p?&))R/\"[$Gqi\\=Q=F)*$)F'\"\")F) F)F)*&#\"DP0]8u0,1,Iv)eUzB$z\"\"I+]Ph*RX)=pc\")*Q*R!\\#4W@sF)*$)F'\"\" *F)F)F)*&#\"?;Dd]l^%GeIM;G;O$\"FD1WM)=dNm2<2DN'Gl)39F)*$)F'\"#5F)F)F) " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We \+ can find the point where the boundary of the stability region intersec ts the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := new ton(R(z)=1,z=-4.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+W*GE) \\!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.9):\np1 := plot([R(z),1],z= -5.49..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,sy mbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]], linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view= [-5.49..0.49,-.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 429 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++++++!\\&!#< $\"3oW!zW@)>ZEF*7$$!3eL3_aKTdaF*$\"3q*eZ&[Ls#\\#F*7$$!3(pmT!4l#[U&F*$ \"37fHwKm]YBF*7$$!3M+Dcj(RAR&F*$\"3@T?$\\X]\"3AF*7$$!3sLL3=Ilf`F*$\"3w :+\\vUFx?F*7$$!3x]737f%HI&F*$\"33x%[HN&Gm=F*7$$!3%p;zg!)QiC&F*$\"3S%)G (e5Z]n\"F*7$$!3O](oXtlC=&F*$\"3I#o%pP:]\"[\"F*7$$!3#QLeIm#p=^F*$\"36Yr +)eS'38F*7$$!3%RLLA'o\\a]F*$\"3;#z!zsD]`6F*7$$!3=L$391,.*\\F*$\"392X') )eRa,\"F*7$$!3l](=?P5k#\\F*$\"36+@%HJ[I$*)!#=7$$!3An\"HEo>D'[F*$\"3i:l c`P!)[yF^o7$$!3%R$eM()*\\Su%F*$\"3K)>J&e'f\\:'F^o7$$!3P+D\"*z9Q@YF*$\" 3mY)>+&*Rjw%F^o7$$!3ULe/Mv^%\\%F*$\"3GlOt2jPYOF^o7$$!3C+DYU/1oVF*$\"3+ 9)z\\Gyiy#F^o7$$!3Amm@!y$)zB%F*$\"3)3V#ya$)R8@F^o7$$!3?L$3`=7M7%F*$\"3 )=L)*4%)RAm\"F^o7$$!3G++l0;#\\'QF*$ \"33Jlyq'>6+\"F^o7$$!38++NaY6SPF*$\"3'z*p<+\"Hg5)!#>7$$!3$Q$e/tzxEOF*$ \"3[Pp8#H'=)*oFar7$$!3'pm;op4?\\$F*$\"3E#[-&G`wqfFar7$$!3)pmm#*GVyP$F* $\"3.ZRMl:=HbFar7$$!35+Dh;1/XKF*$\"3z*=E%yB'3L&Far7$$!3!omm([x[FJF*$\" 3)4\"HF*\\*Q'Q&Far7$$!37+Dh@O^)*HF*$\"3Y!G%[O,)[l&Far7$$!3!**\\PGv*pvG F*$\"3I?e(e?r_3'Far7$$!3am;zkjbZFF*$\"3UUoTGP)yp'Far7$$!3mm\"HDt!))HEF *$\"3GLd*)3U[*R(Far7$$!3sL$3dm^H]#F*$\"3yx8L\"*3k.$)Far7$$!3eLe/Q!36P# F*$\"34\"H6,l5$4%*Far7$$!3-+vesxLcAF*$\"3#3eB5gB=0\"F^o7$$!3>L$e;u#QK@ F*$\"3'G!H/d->)=\"F^o7$$!3C++ql]K/?F*$\"3G>yRrt**[8F^o7$$!3;+]-8c/z=F* $\"3rv-Y,SF^o7$$!3PnmwC.J-:F*$\"3JU%)ocmDEAF^o7$$!3')***\\dP)=t8F*$\" 3_yn`fk-LDF^o7$$!3#Q$ek)p%=c7F*$\"3?:'=Ye.u%GF^o7$$!3b***\\[xo#G6F*$\" 3JtsgTv$fB$F^o7$$!3an\"z#f)4z+\"F*$\"3/%y;KA8)\\OF^o7$$!3-,]iE6+@))F^o $\"3*oe$=$=G\"RTF^o7$$!3)ymmT*3)4f(F^o$\"3]Vcn@V)3o%F^o7$$!3S**\\7.PE. jF^o$\"3A4Uy7,=C`F^o7$$!3MJLL***QI1&F^o$\"3zyE>z<>FgF^o7$$!3[PL35,t%z$ F^o$\"3k^Nm;*=A%oF^o7$$!3Ei;zeM#p`#F^o$\"3aR\"3a$[IfxF^o7$$!3S'****pxH 6Q\"F^o$\"3JAEyDG+5()F^o7$$!3s:nmT'yfk&!#?$\"3^?#=%*H*pV**F^o7$$\"3tom mW3MG6F^o$\"3%[.HS=Y%>6F*7$$\"3)3+vQV&e\"R#F^o$\"3'=HW#**)z,F\"F*7$$\" 3F**\\7#pr1g$F^o$\"3c!RsLpDMV\"F*7$$\"3!***************[F^o$\"3UoiJ K;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fe]lFd]l-F$6$7S7$F($\"\"\"Fe ]l7$F=Fj]l7$FGFj]l7$FQFj]l7$FenFj]l7$F`oFj]l7$FeoFj]l7$FjoFj]l7$F_pFj] l7$FdpFj]l7$FipFj]l7$F^qFj]l7$FcqFj]l7$FhqFj]l7$F]rFj]l7$FcrFj]l7$FhrF j]l7$F]sFj]l7$FbsFj]l7$FgsFj]l7$F\\tFj]l7$FatFj]l7$FftFj]l7$F[uFj]l7$F `uFj]l7$FeuFj]l7$FjuFj]l7$F_vFj]l7$FdvFj]l7$FivFj]l7$F^wFj]l7$FcwFj]l7 $FhwFj]l7$F]xFj]l7$FbxFj]l7$FgxFj]l7$F\\yFj]l7$FayFj]l7$FfyFj]l7$F[zFj ]l7$F`zFj]l7$FezFj]l7$FjzFj]l7$F_[lFj]l7$Fd[lFj]l7$Fj[lFj]l7$F_\\lFj]l 7$Fd\\lFj]l7$Fi\\lFj]l-F^]l6&F`]lFd]lFd]lFa]l-F$6&7#7$$!3++++W*GE)\\F* Fj]l-%'SYMBOLG6#%'CIRCLEG-F^]l6&F`]lFe]lFe]lFe]l-%&STYLEG6#%&POINTG-F$ 6&F`al-Feal6#%&CROSSGFhalFjal-F$6&F`al-Feal6#%(DIAMONDGFhalFjal-F$6%7$ 7$FbalFd]lFaal-%&COLORG6&F`]lFd]l$\"\"&!\"\"Fd]l-%*LINESTYLEG6#\"\"$-% +AXESLABELSG6%Q\"z6\"Q!Fjcl-%%FONTG6#%(DEFAULTG-F]dl6$%*HELVETICAG\"\" *-%%VIEWG6$;$!$\\&!\"#$\"#\\Fjdl;$!\"(Fjdl$\"$Z\"Fjdl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1086 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/ 720*z^6+1/5040*z^7+\n 455898330212180487928972932212710847/1838184 9627028348104398852069738083500000*z^8+\n 179323794258875300106010 57413500537/7221440924903993898156691884539961375000*z^9+\n 336162 816343058284516550572516/1408865286352507170766355718834440625*z^10:\n pts := []: z0 := 0: tt := 0: \nwhile tt<=321/20 do\n zz := newton(R( z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (23/20<=tt and tt<=7/4) or (59/20<=tt and tt<=79/20) or\n (241/20<=tt and tt<=261/20) or (5 7/4<=tt and tt<=297/20) then\n hh := 1/40\n else \n hh := \+ 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]] :\nend do:\np1 := plot(pts,color=COLOR(RGB,.38,.18,.18)):\np2 := plots [polygonplot]([seq([pts[i-1],pts[i],[-2.5,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.35,.35)):\np3 := plot([[[- 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th roo t of the real part of points along the curve. In this way we see that \+ the largest interval on the nonnegative imaginary axis that contains t he origin and lies inside the stability region is " }{XPPEDIT 18 0 "[ 0, 5.5];" "6#7$\"\"!-%&FloatG6$\"#b!\"\"" }{TEXT -1 18 " approximatel y. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 581 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/7 20*z^6+1/5040*z^7+\n 455898330212180487928972932212710847/18381849 627028348104398852069738083500000*z^8+\n 1793237942588753001060105 7413500537/7221440924903993898156691884539961375000*z^9+\n 3361628 16343058284516550572516/1408865286352507170766355718834440625*z^10:\nD igits := 20:\npts := []: z0 := 0:\nfor ct from 0 to 260 do\n zz := n ewton(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,.63,.15,. 15),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 " " {GLPLOT2D 265 310 310 {PLOTDATA 2 "6(-%'CURVESG6#7a[l7$$\"\"!F)F(7$F ($\"5$RKz*e`EfTJ!#@7$$\"5z6H1.P\\HD?F-$\"5?*oezrI&=$G'F-7$$\"5eqw=Od(* p@HF-$\"5zj&Rp2'zxC%*F-7$$\"5D>?q'e]d#F=7$$\"5c[bNCb2$)3rF-$\"5M`CC)G7uK^#F=7$$ \"5!H=n8#)[(H4zF-$\"5UHX6E)QLu#GF=7$$\"5REF&)=0q'3q)F-$\"5.Y/um`EfTJF= 7$$\"5z#R4R)=[K%[*F-$\"5.;9+8>>vbMF=7$$\"5eu15?2$Hg-\"F=$\"5)[a:+Z=6*p PF=7$$\"5y(*QHhya7F=$\"5*G!3'*>$)*)Q7ZF=7$$\"5IA61RR**zH8F=$ \"5Q8j@q]#[l-&F=7$$\"5&GG\"y&o(QC/9F=$\"5<0Xe_>vqS`F=7$$\"5-/()eBMH8y9 F=$\"5A)*3gJ!zm[l&F=7$$\"5dpE`.#3$[^:F=$\"5%['[%eR1E!pfF=7$$\"5'H9s;>h 2Vi\"F=$\"5k)>IU;M&=$G'F=7$$\"5OxEV1Uvh'p\"F=$\"5cE&zJ\\iWtf'F=7$$\"5D c\"p!=V>UoF=$\"5^4Q!H9`A)RvF=7$$\"52o-F=$\"5#[Q116*F=7$$\"5pH\"y/7gw_K#F=$\"5%yW;L7)*yZU*F=7$$\"5!>ULhw8>FR#F= $\"5(*>+@o]&Q*Q(*F=7$$\"5K%QuE6f!ofCF=$\"5&z4Pq\">)4`+\"!#>7$$\"5*[f!Q 5V\"fh_#F=$\"5SGJKh\"zDn.\"Fdu7$$\"5Ft<*o2X_@f#F=$\"5+$\\TdOxT\"o5Fdu7 $$\"5%GeU_Nodwl#F=$\"5A1vehmxb*4\"Fdu7$$\"5WI^#G\\brEs#F=$\"5A!oW>=xt4 8\"Fdu7$$\"5RXBf3].>(y#F=$\"5P%oGr0z*Qi6Fdu7$$\"5ilI+0]*47&GF=$\"5?%\\ g)3Ce!Q>\"Fdu7$$\"5k&R!\\s_es9HF=$\"5i'z/IM(=AD7Fdu7$$\"5@'z*=F\">Lx(H F=$\"5DJyqSRzjc7Fdu7$$\"5dU/(p,vE-/$F=$\"5$*[*pxC-a!)G\"Fdu7$$\"5ycSAE z4?-JF=$\"5^&p#4iA,Z>8Fdu7$$\"5eWr=11+ljJF=$\"5jA'[!>Ri)3N\"Fdu7$$\"5W XglaYwcCKF=$\"5()HZjuqBI#Q\"Fdu7$$\"58?Qom9u%\\G$F=$\"52()yG'[^=PT\"Fd u7$$\"5ozY8wIDyWLF=$\"5*[>j7zmM^W\"Fdu7$$\"55Ivu1Hf1/MF=$\"55)yp'z]\"Fdu7$$\"5n)Qx7*=z%4_$F=$\"5It kT6@JQR:Fdu7$$\"5z1mv*o+J&yNF=$\"5mFapgS#*zq:Fdu7$$\"5\\M$\\*\\\"QJbj$ F=$\"5VE#fONK:Ag\"Fdu7$$\"5$[7WL)R1%>p$F=$\"54y*\\=JNJOj\"Fdu7$$\"5L,I %=*G,vZPF=$\"5[\"R*>;4t/l;Fdu7$$\"5r\"3a+>&4&H!QF=$\"5X['\\qw;jkp\"Fdu 7$$\"5xM&[**[(R`dQF=$\"5K*[&)G/!*yys\"Fdu7$$\"5Fw14tM)*[6RF=$\"5[tqDeu WHf:<%F=$\"5M=KI(*Q%oj\">Fdu7$ $\"5()*4\")**e#R]@UF=$\"5sgoypf?yZ>Fdu7$$\"5;S'3xg_!zqUF=$\"5/yFq.G^>z >Fdu7$$\"5t%R'4fu#o$>VF=$\"5iU@aHgvg5?Fdu7$$\"5.Nlj[1kAnVF=$\"5l-!\\&H l#>?/#Fdu7$$\"5PR%zQd8aVT%F=$\"5#f8suV9IM2#Fdu7$$\"5H6y(=#Fdu7$$\"5s-U^7RV mQYF=$\"5=s(R6/?o/B#Fdu7$$\"5mK0eMb*)=\"o%F=$\"5W[C.8m;(=E#Fdu7$$\"5k% 4A+z<7Hs%F=$\"5\"=\")Rj\\VtKH#Fdu7$$\"5+\"RqB?fDQw%F=$\"5'=>LFkOtYK#Fd u7$$\"5#H1?mfg@R![F=$\"5ri9kV[F=$\"5<*=qz!erY (Q#Fdu7$$\"5cals5=jj\")[F=$\"59*)y4iY2')=CFdu7$$\"5EJWN-edC>\\F=$\"5#3 0ZS'f>D]CFdu7$$\"5j4S`g6)>g&\\F=$\"5]XtdV\"oS;[#Fdu7$$\"5;`EeA8%e>*\\F =$\"5#)*egP-\"o-8DFdu7$$\"5rif;?]R1F]F=$\"5\"y\"*pt?E5Wa#Fdu7$$\"56e<' *z\"pT81&F=$\"5U'4[J](4zvDFdu7$$\"5Vq9)\\)y-![4&F=$\"5(4)4r#R\"*org#Fd u7$$\"54Ei*e0G_u7&F=$\"5q\\4\\QvSaQEFdu7$$\"5cFu<2D[Jf^F=$\"5315tK$\\; *pEFdu7$$\"5A.kC!*4.T!>&F=$\"53e@5([C'G,FFdu7$$\"5#o)*>!Q(>n2A&F=$\"5; 5L.icMlKFFdu7$$\"5#[/\\,&)*3U]_F=$\"5^$37V8J=Sw#Fdu7$$\"5C!f*=yXZTz_F= $\"5f:L$4]0\"Q&z#Fdu7$$\"5*pzT9:0,yI&F=$\"56hU`;/?uEGFdu7$$\"5juBg)=CU cL&F=$\"5yqR0h`:5eGFdu7$$\"5Z&*Q(f#\\?,j`F=$\"5E0yZL%=g%*)GFdu7$$\"5\" \\ov#>Mn***Q&F=$\"5UM4mmr%=3#HFdu7$$\"5:;TY6&H'p;aF=$\"5pKcCd$4x@&HFdu 7$$\"5(GveDpgDKW&F=$\"5Y1wc.Ro`$)HFdu7$$\"5>?.TO5arpaF=$\"5V#y[Jkh)*[, $Fdu7$$\"5)yf56F08j\\&F=$\"5bW*[)zhMEYIFdu7$$\"5Iab2)*eG=BbF=$\"5HrL$* )oaKw2$Fdu7$$\"59*[&f/%*f]]bF=$\"5>l!Gho=2!4JFdu7$$\"5U`8!pUmz%ybF=$\" 5*47xSu%))QSJFdu7$$\"5**f'pZu]:tg&F=$\"5Vy8OO^\"z<<$Fdu7$$\"5^:m'R#QqB PcF=$\"5XbbA&Q))zJ?$Fdu7$$\"5EB(ej$egZocF=$\"5B)G'>%o*HfMKFdu7$$\"5p38 ')\\5!o7q&F=$\"5*>c81;h?gE$Fdu7$$\"5Q](yv:NYet&F=$\"5._]WN>]Y(H$Fdu7$$ \"5Mo5*\\'QhVsdF=$\"5-eFCGz'G*GLFdu7$$\"5A&e%o1XrC6eF=$\"5]::HV8UTgLFd u7$$\"55#Reye$pY_eF=$\"5[VfNe(RC>R$Fdu7$$\"5#[DU+cSai*eF=$\"55=#>$>Z@Y BMFdu7$$\"5,]Wso'HMF%fF=$\"5l&zt;s\\I]X$Fdu7$$\"5%4xvO/;$*>*fF=$\"5t%e /CVdKm[$Fdu7$$\"5$[)ePW%4xS/'F=$\"5@OgKsf:F=NFdu7$$\"5[O*>g#49**)4'F=$ \"5Q_!\\$QS1&*\\NFdu7$$\"5\"Rh5%\\r@qchF=$\"59#o7kf%Hn\"e$Fdu7$$\"5z)f `>mPRr@'F=$\"5^,i+=o9W8OFdu7$$\"57$3wMRN,-G'F=$\"5YCHzle*e_k$Fdu7$$\"5 4;h'3'=*fdM'F=$\"5U3#oq0!y7xOFdu7$$\"5Y*y=PnwlOT'F=$\"5recH\\[)\\!4PFd u7$$\"5Xf%\\U+l`P['F=$\"5J#f=Q+BE5u$Fdu7$$\"5+M@lU\\p%eb'F=$\"5X.-%**> 5dIx$Fdu7$$\"5X0U(=-Ph(HmF=$\"5S)*fs0`890QFdu7$$\"5ljrDC)e2`q'F=$\"5s3 9j'[Cws$QFdu7$$\"5l)plq@x#H#y'F=$\"5^]M5$Q)pXpQFdu7$$\"5DtD&y$))4_goF= $\"5t'f^Vw@w;!RFdu7$$\"5f_arHgDzRpF=$\"5+\\]PPaM#R$RFdu7$$\"5<*y3%[5D! 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Digi ts := 15:\nz0 := 5.5*I:\nfor ct from 240 to 243 do\n newton(R(z)=exp (ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0U72crrw'!#<$\"0P5&y^x,b!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"03?kfaK;\"!#=$\"0@$QK#eY]&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0H'Qq9vwl!#<$\"0\"fB[:U2b!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0\"Gc'*Q6J8!#;$\"0fHTCl+^&!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisect ion method to calculate the parameter value associated with each inter section point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=5.5*I))\ne nd proc:\nu0 := bisect('real_part'(u),u=2.40..2.43);\nnewton(R(z)=exp( u0*Pi*I),z=5.5*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#u0G$\"0HJgLu,T#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0[$HRbAu5 !#G$\"0u#4Kuq/b!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegativ e imaginary axis that contains the origin and lies inside the stabilit y region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 5.5047];" "6#7$\"\"! -%&FloatG6$\"&Z]&!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 11 stage, order 6 sch eme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 " subs(ee,subs(b=`b*`,StabilityFunction(6,11,'expanded'))):\n`R*` := una pply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%# R*G6#%\"zG,8\"\"\"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)*&#\"UZ<2Hg*H(eH9Uh)o&Q>&[3hkeT/[\" \"X++][\"p*\\3$)Gi#3X]Z&GHG?xW*)=H-h&F)*$)F'\"\")F)F)F)*&#\"Q$phxPdv^ \\;#)>eS8vjE%35?I)[(\"W]P4&p+@.Z7ME`Pj\"R8!pH@#*))>n)oGF)*$)F'\"\"*F)F )F)*&#\"Mo$[;\\=z`g\\(G&H(y)RVtF/$4&*\"TDJS**GV^[m2$R3oxA>FEJz5.7=NF)* $)F'\"#5F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/- %#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-5.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+8+.p^!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "z_0 := newton(`R*`(z)=1,z= -5.2):\np_1 := plot([`R*`(z),1],z=-5.79..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=[-5.79..0.49,-.17..1.47],fon t=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 263 263 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3/++++++!z&!#<$\"3O'3FO?0'pRF*7$$!3#GLezYydv&F* $\"3'4.ak*QC/PF*7$$!3Zmm\"f$pb@dF*$\"3g0m-H1LaMF*7$$!39+](QSNto&F*$\"3 cQ#RUq\"3>KF*7$$!3!QLL=(Q6`cF*$\"3)4]Ok'zu(*HF*7$$!3Q+vtF>c$f&F*$\"3+m 0Y[=lVEF*7$$!3'pmTO)*4S`&F*$\"386Wbbx*fK#F*7$$!3-+D;%fPqY&F*$\"3r4S'f^ Q#3?F*7$$!31LLo/_1+aF*$\"3jb2(z4r\"G&F*$\"3a0!H;H&Gu 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:= z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120* z^5+1/720*z^6+\n 5354575971726701663049056208895715107829625322171 747/\n 25246031350251474127317846377028718029738824861168250000*z^ 7+\n 148044158646108485193856886142142958729960290719057/\n 56 10229188944772028292854750450826228830849969148500000*z^8+\n 74883 0201008426637513405819821649517557377761693/\n 2868867198892212969 01339163375326341247032100695093750*z^9+\n 95093042773433987872952 874960537918491648368/\n 35181203107931262719227768083930766485143 2899403125*z^10:\npts := []: z0 := 0: tt := 0: \nwhile tt<=241/20 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (31/2 0<=tt and tt<=7/4) or (41/4<=tt and tt<=209/20) then\n hh := 1/80 \n elif (21/20<=tt and tt<=43/20) or (197/20<=tt and tt<=219/20) the n\n hh := 1/40\n else \n hh := 1/20\n end if;\n tt := \+ tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts ,color=COLOR(RGB,.32,.08,.08)):\np_2 := plots[polygonplot]([seq([pts[i -1],pts[i],[-2.6,0]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.63,.15,.15)):\npts := []: z0 := 0.065+5.7*I: tt := 0: \+ \nwhile tt<=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n \+ z0 := zz:\n if (3/2<=tt and tt<=53/30) then\n hh := 1/120\n e lif (0<=tt and tt<=7/30) or (11/10<=tt and tt<=61/30) then\n hh : = 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n pt s := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR( RGB,.32,.08,.08)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[- 0.18,5.65]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.63,.15,.15)):\npts := []: z0 := 0.065-5.7*I: tt := 0: \nwhile tt <=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz: \n if (7/30<=tt and tt<=1/2) then\n hh := 1/120\n elif (0<=tt and tt<=9/10) or (53/30<=tt and tt<=61/30) then\n hh := 1/60\n \+ else \n hh := 1/30\n end if;\n tt := tt+hh;\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.32,.0 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%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-Fgg^m6$%*HELVETICAG\"\"*-%%VIE WG6$;$!$4'Figp$\"$>\"Figp;$!$H'Figp$\"$H'Figp" 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" }}}} {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 "coeffici ents 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 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4911 "ee := \{c[2]=1/2 4,\nc[3]=897/1072,\nc[4]=160/499,\nc[5]=141/763,\nc[6]=169/192,\nc[7]= 998622639/2007664595,\nc[8]=104/105,\nc[9]=20/21,\nc[10]=1,\nc[11]=1, \n\na[2,1]=1/24,\na[3,1]=-2173431/287296,\na[3,2]=2413827/287296,\na[4 ,1]=57894880/223353897,\na[4,2]=0,\na[4,3]=13721600/223353897,\na[5,1] =5056664940861/42500572528960,\na[5,2]=0,\na[5,3]=-286249149686016/366 67767392227699,\na[5,4]=2889259817673483/39243095536832320,\na[6,1]=17 4/407,\na[6,2]=0,\na[6,3]=861765466269918263/4970009789268581376,\na[6 ,4]=10826192777269994995/7712691245665173504,\na[6,5]=-167502365600408 96731/14897194035750715392,\na[7,1]=-343977879061291352797038193646706 5474966136664036779/\n 3938120938942456177804410621778210438829 0028625000000,\na[7,2]=0,\na[7,3]=263141448694331408503436895976653333 0714638432739014413056/\n 42832809612378095870766530128571820142 439373914153488671875,\na[7,4]=-32268178132000496951694623523221667175 64113587934800370011/\n 400153464639761607619290861935161838324 03838724337375000000,\na[7,5]=7861905775507069402158740404195385876083 61105183350570182/\n 1261335096542994096899051116924589799398264 209359351953125,\na[7,6]=-5411006332774526240948446773664264724938752/ 279717068193641240266628706086067991948046875,\na[8,1]=153220176326904 57879761200795014874391/29304923702961208406725757182921875000,\na[8,2 ]=0,\na[8,3]=3346824815910846126629531019335170048/2298048540174087438 681694595766796875,\na[8,4]=189748010561998005455394521375650921/29817 877073894091150172096921875000,\na[8,5]=-38284533158383519319508939786 9250404738354182/110225463285588517977748940118009136362890625,\na[8,6 ]=-120391896911431755652090822656/233667703961223196028992578125,\na[8 ,7]=-79695128060562505037238941490242415433886906/23692035580167289837 348760854832233004866581,\na[9,1]=103289795629820465430592343304575917 1807509/2004961136769885546067510633666861543410000,\na[9,2]=0,\na[9,3 ]=8472900696201518374537737273138176/955988192712420374491584951838987 5,\na[9,4]=1046253623595072271778442420598129/248084737254798838369431 846390000,\na[9,5]=-22558089535261276140592810406893936458098806614611 99/\n 891519231565352783145488093784087901162293552521625,\na[9 ,6]=-1788698788744957883891433528082038784/592314367631425544664328323 1146667875,\na[9,7]=-5748004687549971039444238386662526027545080751933 05936911875/\n 311545184693907215706597063235165448890282539301 416442871559,\na[9,8]=88873364366227887565000/817629004593263701438668 9,\na[10,1]=86280208734474390054682197174277603231/1632356727333439262 84895054973116600000,\na[10,2]=0,\na[10,3]=871191425901232070238397371 934719232/533362446801744790377544382297593125,\na[10,4]=8964183417443 891431743857714046506701/1273379282362081167514274901917400000,\na[10, 5]=-252047840384081563219583820196041648831131662918986674/\n \+ 66729954558238813530334990145293701430108220259249375,\na[10,6]=-52908 99117464226058433745200964698112/8875951095645670561315492876310816875 ,\na[10,7]=-3069893906680376471707933893233719720682072201840672116412 63783356000/\n 80013418079725292844034374434224082313958525560 207876667654196707053,\na[10,8]=-7376114064684719178734872354875/35902 2955840383035894248406127836,\na[10,9]=255482840375593769438967652731/ 8853429096639550069183960833070,\na[11,1]=2429287816990748646637471052 716009978490201/4607015764006388556345602665417429396800000,\na[11,2]= 0,\na[11,3]=160046766004414315742455765106336/980659521732857527013287 66250625,\na[11,4]=572651535981050835474886343012483/81435870991749105 781564929600000,\na[11,5]=-1900266684289629480230017826550979437884206 4985811946527/\n 503899877373056879741728322282229949705834423 9378524000,\na[11,6]=-73608385714443424401508261029604294656/123768183 621850031611831870691647493125,\na[11,7]=-1052783385153326286499937163 4499752441444282441429608616850996735536625/\n 274639858144995 2235460650199594860340361817126809954433857815662665184,\na[11,8]=-561 6863065079321503840875/276176908218169072485950384,\na[11,9]=400979470 59612729/1419769450975237280,\na[11,10]=0,\n\nb[1]=164949402359214517/ 2969763122318428800,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=1661646473433436 29239462104853/594150218553182839761586950000,\nb[6]=88351573658999985 31682304/6999851448505791234978125,\nb[7]=1105678433728701172052633750 74449005180061856568412069340896875/\n 340717754649392931183331876 846338804668211857149398834541328544,\nb[8]=758177409913942921875/7878 5859799058312576,\nb[9]=-2973333145929506703/747247079460651200,\nb[10 ]=-1895689785077201/288707084450720,\nb[11]=0,\n\n`b*`[1]=205067437736 21213/371220390289803600,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5] =268231112188987944945827753/955225431757528681288725000,\n`b*`[6]=415 490030471198546067456/304341367326338749346875,\n`b*`[7]=1084192113857 90972003069732530101032173894775381027875/\n 337664606137936380 500051156293384895353362152117883624,\n`b*`[8]=54584684081980124625/49 24116237441144536,\n`b*`[9]=-842056888251867309/186811769865162800,\n` b*`[10]=0,\n`b*`[11]=-38/5\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2.. 11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,/&%\"cG6#\"\"##\"\"\"\"#C/&F%6 #\"\"$#\"$(*)\"%s5/&F%6#\"\"%#\"$g\"\"$*\\/&F%6#\"\"&#\"$T\"\"$j(/&F%6 #\"\"'#\"$p\"\"$#>/&F%6#\"\"(#\"*REi)**\"+&fkw+#/&F%6#\"\")#\"$/\"\"$0 \"/&F%6#\"\"*#\"#?\"#@/&F%6#\"#5F)/&F%6#\"#6F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i ,j]=subs(ee,a[i,j]),j=1..i-1),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6Y/&%\"aG6$\"\"#\"\"\"#F(\"#C/&F%6$\"\"$F(#!(JM<#\"''H(G/&F%6$F.F' #\"(FQT#F1/&F%6$\"\"%F(#\")!)[*y&\"*(*QNB#/&F%6$F:F'\"\"!/&F%6$F:F.#\" )+;s8F=/&F%6$\"\"&F(#\".h3%\\mc]\"/g*GDd+D%/&F%6$FJF'FA/&F%6$FJF.#!0;g o\\\"\\iG\"2*pFARnxmO/&F%6$FJF:#\"1$[tw\")f#*)G\"2?B$o`&4V#R/&F%6$\"\" 'F(#\"$u\"\"$2%/&F%6$FjnF'FA/&F%6$FjnF.#\"3j#=*piYl<')\"4w8eo#*y4+(\\/ &F%6$FjnF:#\"5&*\\**psx#>E3\"\"4/N(*[\"/&F%6$\"\"(F(#!Uzn.km8m\\ZlqYO>Qqz_8Hh!zy(RM\"V+++D'G+ H)Q/@y'3H>wgh(RYY`,S/&F%6$F fpFJ#\"Z#=q0N$=06O3weQ&>//ue@Spq]vd!>'y\"enDJ&>Nf$4UE)R*z*eCp6^!**o4%* Ha'4N8E\"/&F%6$FfpFjn#!L_(Q\\skUmtnW[4CEXxKj+6a\"Nvo/[>*z1'31(GmES7k$> oqrz#/&F%6$\"\")F(#\"G\"Ru[,&z+7wzyX!pKw,A`\"\"G+](=#H=dds1%37'HqB\\IH /&F%6$FhrF'FA/&F%6$FhrF.#\"F[+5`Hm7Y3\"f\"[#oM$\"Fvozmdf%p\"oQu3u, a[!)H#/&F%6$FhrF:#\"E@4lv8_%Rba+)*>c5![(*=\"D+](=#p4s,:\"4%*Q2xy\")H/& F%6$FhrFJ#!N#=a$QZS]#pyR*3&>$>NQeJ`%GQ\"ND1*GOO\"4!=,%*[x(z^)e&GjaA5\" /&F%6$FhrFjn#!?cE#34_cvJ9\"p*=R?\"\"?D\"yD**Gg>B7'RqnOB/&F%6$FhrFfp#!M 1p)QV:CC!\\T*Qs.0Dcg!G^pz\"M\"em[+LA$[&3w[t$)*Gn,eN?pB/&F%6$\"\"*F(#\" L4v!=))f&*/&F%6$F`uF: #\"CH\")f?CWyh9m!))4ek $R*oS5GfShFh_`*3eD#\"TD;__NHi6!z3%y$4)[XJy_`cJ#>:*)/&F%6$F`uFjn#!F%yQ? 3GNV\"*Q)y&\\u)y)p)y\"\"FvymY6B$GVmWbUJwO9Bf/&F%6$F`uFfp#!gnv=\"p$fI$> v!3Xv-EDm'QQUWR5(*\\vo/![d\"gnf:(GW;9IRDG!*)[a;NK1(f1d@2Rp%=X:J/&F%6$F 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scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i ]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"3 " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%#b*G6#\"\"\"#\"287itPu10#\"3+O!)*G!R?7P/&F%6#\"\"#\"\"!/&F%6# \"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\"<`x#e%\\%z)*)=76Bo#\"<+]s)G\"oGvvJaA b*/&F%6#\"\"'#\"9cu1Y&)>r/.!\\:%\"9voM\\(QjKn8M/$/&F%6#\"\"(#\"Wvy-\"Q vZ*Q%3\"\"WCO)y6_@O``*[Q$Hc60+0QOz81YmP$/&F%6#\"\")# \"5DY7!)>3%o%ea\"4OX9TuB;T#\\/&F%6#\"\"*#!34t'=D))o0U)\"3+G;l)p<\"o=/& F%6#\"#5F//&F%6#\"#6#!#QF;" }}}{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 40 "#===== ==================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Scheme that satisfies some error conditions" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 48 "#--------- --------------------------------------" }}{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 6302 "ee \+ := \{c[2]=1/200,\nc[3]=125788166632/804556341815,\nc[4]=20081493398174 7403/389065106257376584,\nc[5]=119/198,\nc[6]=17/19,\nc[7]=9526409/472 45149,\nc[8]=25/29,\nc[9]=44/45,\nc[10]=1,\nc[11]=1,\n\na[2,1]=1/200, \na[3,1]=-296212523854984450605064/129462181430947023498845,\na[3,2]=6 3290651458559192893696/25892436286189404699769,\na[4,1]=-1278937215707 1146745045111775696471200626449407/38081526404775332901721474741064649 138530038784,\na[4,2]=0,\na[4,3]=3244505211373170947213130728585983354 5001662335/38081526404775332901721474741064649138530038784,\na[5,1]=-4 319556318016342264550098836819533/588237381338985769780687033796930496 ,\na[5,2]=0,\na[5,3]=122312177918886060585665458413264940675777568875/ 329913188780101678448178322632546427763796661824,\na[5,4]=156435672850 20817980224090362461345189277853300810312/\n 6583637495330361087 7005093358513531407567316792726287,\na[6,1]=-3824213620692329526455584 82090899/7276891295937830443110867514470288,\na[6,2]=0,\na[6,3]=220573 40392597202004799056057490702918544557070119184193875/\n 4129997 9864570857330923564925248608645282955350725839567504,\na[6,4]=-9436081 2687526097028123187068745127462524044420500458282895614759616/\n \+ 11701787925765017413597259781168301649785274316799073782550336425080 9,\na[6,5]=1042940990325035750229825636422309760/855155063453934295147 878200275545499,\na[7,1]=345935972702034336631409026535217689699087841 70157140274555/\n 5873155060211325709024462330057232620393755812 89686893222448,\na[7,2]=0,\na[7,3]=51771404702431227976835698966059875 1616539775823173072267317364897221236919340917125/\n 33333078077 3975293533161697727890456259487743000308970714929289253541421782715374 8784,\na[7,4]=-3046404853655506630695951934475238619460577978474785952 003633121978548393408440668463488064/\n 51136263942302953858508 176386231981963730605396561411808693067571768322150482526166450770407, \na[7,5]=319578151553445472127225629704831153236060812834175249153952/ \n 6153982820746383354666623304676483805766376504613845749252065 ,\na[7,6]=-59190400148110099510923215866/11994398797423357814553515510 595,\na[8,1]=-246878890533156807885166723922846376763102826028025/\n \+ 945916869588539937065443533422779502568160417427184,\na[8,2]=0, \na[8,3]=4105216230872532336132293706540464473954640713998671441385245 340625/\n 194997865429882410899648851032266533908818489693684482 6392190722672,\na[8,4]=-4932471259935775148927656253720019927767958725 726761483449498264713532779200/\n 10260713295176800963125818770 711044624508389420487189702045571431972578553533,\na[8,5]=212696875762 29892809323240545664650122099097000742563680/\n 2098952683541446 3140588698214708112363244319596705442863,\na[8,6]=-7115819429493526581 990/378493240381452882505729,\na[8,7]=-2838085862362767273670332075312 6600/18970757838400100725353564886292693,\na[9,1]=55835332432633377407 3044906290708552756484723746581752261/\n 58431944020157116455094 5793240631864149041989001022312500,\na[9,2]=0,\na[9,3]=-36145100904073 14534068821036471754993634788628766244060214613687/\n 502464522 228026334461988984585059909779594952312786529837471612,\na[9,4]=-13436 3809278462440447578035839524277657822517625740761194787095930448569694 72/\n 660987287247208667958590882181936133876300215856141449861 4431157761209623125,\na[9,5]=98993186489712313901089693322164771791022 1123481001424643841888/\n 57587452730227615610167811388166529824 2482451696692517167915625,\na[9,6]=-372567259603706115805768838168/168 3402581390679277326651009375,\na[9,7]=23144360888445940796012547061348 8480291827060789/31418904104758208921880920386941521649080865000,\na[9 ,8]=4101332345819049706502423/10667097363909719917125000,\na[10,1]=269 3249580763133758720448819763251094287099002263195049/\n 1972459 096581551425883657161923834583484247606165156240,\na[10,2]=0,\na[10,3] =-76272016498630274835542997163916675561009406468590453432456541457037 5/\n 739302547895265811564113341331219106351003006692324719048 05722424944,\na[10,4]=-91410637813378364974443788253123984352292510395 0674595588223056094266864010432/\n 389018180564378499646810101 592305688081747704847671122805431872050870556393341,\na[10,5]=38681164 03444609936063780569978932749234663970855789282731320288/\n 222 4720532488974689873485313867737218754855271707412737609361885,\na[10,6 ]=-2147337295919765809044534716/15282699757843572625171655145,\na[10,7 ]=76963822828349512771032624512097499610054141220500895/\n 7457 737217451650694295122868750841403431006754989491,\na[10,8]=14194090907 0976808611971717/336796637787572286621096705,\na[10,9]=-54500989446984 4730599500000/14146180885244972240022216299,\na[11,1]=6936698309764946 835725063360649553583306425655914739761/\n 50785294555790876523 93405412856849782233648892058268000,\na[11,2]=0,\na[11,3]=-51260626059 4671504932297273531042063679229867086229919385168625/\n 496261 25652150749082665578724450361459713081709904842453083616,\na[11,4]=-15 4894885026891330281918205630253613274325644541693861447226781547973091 2/\n 652826950367613499218361365117961613704987867512238468998 956163729502185,\na[11,5]=11187030302114879079207593593591827587827107 0980030897149566532/\n 6364479960999970554842249969714404629465 0653180108535971446675,\na[11,6]=-87122197804121104293267367/582943250 254476957489125300,\na[11,7]=14114528561974414913592938450722541815279 2014862255637/\n 1365445940930576614823174588909837943526579257 5731520,\na[11,8]=60701635156866657458077/141408544478495930424000,\na [11,9]=-2472599248597125/64519956837334856,\na[11,10]=0,\n\nb[1]=18530 703372187/317986769215500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=3974063783 66413672608/803680368165950784145,\nb[6]=383280144717771439/1194918123 71674080,\nb[7]=32739939833109197146986610130768938398484878542073/\n \+ 102638217510541641713801705364029167418501950476800,\nb[8]=-160705 16558763250309/7373332958527296000,\nb[9]=-68588512868559375/322599784 18667428,\nb[10]=1754022361907/1430294620800,\nb[11]=0,\n\n`b*`[1]=106 44890436863/181706725266000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*` [5]=3667053975210116568/7266549440921797325,\n`b*`[6]=8397994718591749 49/224047148196888900,\n`b*`[7]=43232559208971791847448214852556880865 6853/\n 1360573252321546818820057421907905293476160,\n`b*`[8]=- 7030001847946787323/2764999859447736000,\n`b*`[9]=-333800898560611875/ 129039913674669712,\n`b*`[10]=0,\n`b*`[11]=3/2\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "s ubs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11),\n[` b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11)]])):\nevalf[6](%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'++]!\")F(%!GF+F+F +F+F+F+F+F+F+7.$\"'Xj:!\"'$!'-)G#!\"&$\"'PWCF2F+F+F+F+F+F+F+F+F+7.$\"' Zh^F/$!'UeLF/$\"\"!F;$\"'*)>&)F/F+F+F+F+F+F+F+F+7.$\"'55gF/$!'AVtF*F:$ \"'S2PF/$\"'8wBF/F+F+F+F+F+F+F+7.$\"'PZ*)F/$!'Gb_!\"(F:$\"'wS`F/$!'zj! )F/$\"'f>7F2F+F+F+F+F+F+7.$\"'Q;?F/$\"'7!*eFLF:$\"':`:F/$!'VdfFL$\"'.$ >&FL$!'%[$\\F*F+F+F+F+F+7.$\"'p?')F/$!'%*4EF/F:$\"'E0@F2$!'92[F/$\"'N8 5F2$!'/!)=FL$!'.'\\\"F2F+F+F+F+7.$\"'yx(*F/$\"'ib&*F/F:$!'c$>(F2$!'xK? F2$\"',> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "subs(ee,m atrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4)]));print(``); \nfor ii from 5 to 11 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:print(``) ;\nfor ii to 11 do\n print(b[ii]=subs(ee,b[ii]));\nend do:\nprint(`_ ________________________________`);print(``);\nfor ii to 11 do\n pri nt(`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"\"\"\"$+#F(%!GF+7&#\"-Km;)yD\"\"-:=McX!)#!9k] g]W)\\&Q_7iH\"9X))\\Bq%4V\"=i%H\"#\"8'p$*G>f&e9l!Hj\"8p(*p/%*='GOC*e#F +7&#\"3.uu\")R$\\\"3?\"3%ewtD1^1*Q#!P2%\\ki+7Z'pv<6X]uY62d@P*y7\"P%yQ+ `Q\"\\Y1TZZ@\"\"$)>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\" \"\"#!CL&>o$))4]XEUj,=jb>V\"E'\\IpzLqo!ypd)*Q8QP#)e" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$#\"Qv)ovxv1%\\E8%eam&egg))=z<7B7\"QC =m'zjxUYDjA$y\"[%y;5!y)=8*H$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"&\"\"%#\"V7.\"3I`yF*=X8Yi.4C-)z\"3-&GnNk:\"V(GEFz;tc29`8&eL40q( 3h.L&\\POe'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_____________________ ____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#<\" #>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\"#!B**34#[ebk_HBp?O@CQ\"C)GqW^n36V/$ y$fH\"*oF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$#\"fnvQ>%=>,2dX a=Hq!\\dg0*z/??(f#RSt0A\"fn/vcRes]`&HGX'3'[_#\\cB4Ld3dk)z**HT" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%#!_o;'fZh&*GGe/]?W/ CDYF^uoq=B\"Gq4Evo73O%*\"`o43DkL]Dyt!*z;VF&y\\;Io6yfsf8u,ld#zy,<\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&#\"Fg(4BUOc#)H-vN]K !*4%H/\"\"E*\\XbF+#yy9&HMRXj]:b)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B _________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"cG6#\"\"(#\"(4k_*\")\\^CZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\"#\"fnbXFSr:qTy 3*p*o<_`E!49jOV.-F(f$fM\"gn[CA$*oo*G\"ev$R?EBd+LiW-4dK6-1bJ(e" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$#\"_pDr\"4M>pB@s*[O7LO+_fyu%yzdg%>'Q_ZM>&fpIm]bO&[SYI\"gp2/x]k;ED[]@Ko)>B'Qw\"3&eQ&HIUREO6&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"(\"\"&#\"gn_R:\\_$\"hnl?D\\d%Qh/lPm d!Q[wYIBmmaLQY2#G)R:'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" (\"\"'#!>me@B4^*45\"[,S!>f\"A&f5b^`X\"yNBuz)R%*>\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#D\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"\"#!TD!Gg#G5jnPYG#Rsm^)y!o:L0*)yoC\"T%=FuTg\"oD]zFULNWlq$*R&)epo\"f% *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\"^oD1MX_QT9n)*RrS Y&RZk/a1PHKhLKD(3B;_5%\"^osEs!>#RE[%o$p*[=)3R`mA.^)['**3T#))Ha'y*\\>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!go+#zF`8ZE)\\\\ M[hnsD(ezwF*>+s`ilF*[^xN*f7ZK\\\"hoLNbyD(>Vrb/-(*=([?%*Q3XiW5rq(=e7j4! o<&H82E5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#\"Y!oj Du+q4*4A,lkcaSKK4G*)Hid(op7#\"YjGW0nf>VCjB63Z@)p)eSJY9a$o_*)4#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!7!*>eEN\\H%>e6(\" 9Hd]#)GX\"QSK\\y$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"(#!D+m7`2K.ntswiB'e3QG\"D$p#H')[c``s+,SQyvq*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#W\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"Zh Av\"eYPs%[cFb32H1\\/tSxLjKCLNe&\"Z+DJA5+*)>/\\T'=jSKzX4bk6d,-W>Ve" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$#![o(o8Y@gSCm(G')yMO*\\vrk.@) oS`9tS!45Xh$\"jn7;ZP)Hly7B&\\fz(4*f]e%)*))>YMj-GA_kC]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!hos%pp&[/$f4(y%>h2uDw^AylxU_R e.yvWSCYy#4QOM\"\"goDJi47wd6V9')\\99ce@+j(Q8O>=#)3fezm3sC(G()4m" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#\"jn))=%QkC9+\"[B6 A5zrZ;ALp*3,RJ7(*['=$**)*\"jnDc\"z;eMK85%\";+]7<*>(4RO(4n1\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B______ ___________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"c G6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"X\\]>jA+*4(G%4^Kw>)[/seP8j2e\\ Kp#\"XSi:lhgZU[$eMQ#>;dO)eU^:e'4fC(>" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"$#!`ov.d9acCV`/fokS45cvm\"R;(*HaN[FI')\\;?Fw\"_oW\\UAd![!>ZK#p 1I+^j5>7LTL6k:\"eE&*ya-$R(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#5\"\"%#!ioK/,koE%4cIA)e&fu1&R5DH_V)R7`#)yVW(\\OyL\"yj59*\"ioTLRc0( 30s=V0G7rw%[qZ<3)o0Bf,,\"ok*\\yVc!==!*Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#\"[o)G?8t#G*yb3(RmM#\\F$*y*p0yjg$*4YW.k6oQ\"[ o&)=O4wt7uqr_&[v=stnQJ&[t)*ou*)[K0sCA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!=;Z`W!4ew>fHPt9#\">X^lr^isN%yv*p#G:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"V&*3]?79a+h*\\(47XiK5x7 &\\$GG#Q'p(\"U\"\\*)\\v15V.9%3voG7&H%p];Xh3o(4244%>9\"<0n4@mGsvyPmzO$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!<++]*fIZ%)pW*) 4]a\">*H;A-SA(\\C&)3=YT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_______ __________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG 6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#\"Xh(RZ\"flDkI$e`&\\1Oj]sNo%\\w4$ )pOp\"X+!o#e?*)[OB#y\\o>0MR_w3zbXH&y]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"$#!jnD'o^Q>*Hi3n)H#zO1U5`tsHK\\]rYfgig7&\"in;O3`C%[!*4<38( f9O]Wsybm#3\\2:_c7E'\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"%#!do74tza\"yEsWhQpTXkDVF8ODIc?=>GI8*o-&)[*[:\"co&=-&HP;c*)**o%QA ^ny)\\q8;'z6l8O=#*\\8wO]p#Gl" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#6\"\"&#\"jnKlc\\r*3.!)42r#yeF=f$f$f2#z!z[6-..(=6\"invmWrf`3,=`1l %HYS9(p*\\A%[bq***4'*zWO'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"'#!;ntE$H/67/y>Ar)\"<+`7*[dpZa-DVHe" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"WPcDi[,#z_\"=aA2XQHf8\\Tu>cGX69\"V ?:tvDzl_Vz$)4*)euJ#[hw0$4%fWl8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\")#\"8x!eulmo:N;qg\"9+SUIf\\yWa399" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!1Drf[#*fsC\"2c[LPo&*>X'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \"#\"/(=sLqI&=\"0+b@pn)zJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"&#\"63En8kOyjS(R\"6XTy]f;o.o.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"3R9x\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"St?ay[[)RQ*o285m) p9(>4J$)R*RF$\"T+oZ]>]=u;HSO0OASv\"\".+3i%HI9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\" \"\"#\"/joV!*[k5\"0+gEDnq\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b* G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&#\"4ol65_(R0nO\"4Dtz@4W\\lE(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"3\\\\[r/C#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"K`ol3)ob_[@ [u%=zr*3#fDBV\"LghZ$H0z!>Ud+#)=oa@BDt0O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!4BtyYz%=+Iq\"4+gtZ%f)**\\w#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!3v=hg&)*3!QL\"37(pYn8*R!H\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"\"$\"\"#" }}}{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 0 151 "R K7_11eqs := [op(RowSumConditions(11,'expanded')),op(OrderConditions(7, 11,'expanded'))]:\n`RK6_11eqs*` := subs(b=`b*`,OrderConditions(6,11,'e xpanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK7_ 11eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK6_11eqs*`):\nm ap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 [q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"! F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-orde r condtions to check for stage-orders from 2 to 5 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 5 do\n so ||ct||_11 := StageOrderConditions(ct,11,'expanded');\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Stages 5 \+ to 10 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "[seq([seq(expand(subs(ee,so||i||_11[j])),i=2..5)],j=1..9)]:\nmap( proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#F$\"\"$F%F%F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The stage-orders of the success ive stages are given as follows." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[stage, `|`, 2, 3, 4, 5, 6, 7, 8, 9, 10], [`sta ge-order`, `|`, 1, 2, 2, 3, 3, 3, 3, 3, 3]]);" "6#-%'matrixG6#7$7-%&st ageG%\"|grG\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#57-%,stage-order GF)\"\"\"F*F*F+F+F+F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j +1 .. 11) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 9 \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/ &%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "[Sum(b[i] *a[i,1],i=1+1..11)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..11)=b[j]*(1-c[j]),j =2..9)];\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops( %);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+/-%$SumG6$*&&%\"bG6#%\"iG\"\" \"&%\"aG6$F,F-F-/F,;\"\"#\"#6&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4 *&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F *6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F -F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEF hoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/ -F&6$*&F)F-&F/6$F,FcpF-/F,;\"\")F4*&&F*6#FcpF-,&F-F-&FEFdqFFF-/-F&6$*& F)F-&F/6$F,FaqF-/F,;\"\"*F4*&&F*6#FaqF-,&F-F-&FEFbrFFF-/-F&6$*&F)F-&F/ 6$F,F_rF-/F,;\"#5F4*&&F*6#F_rF-,&F-F-&FEF`sFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"!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 26 "The simplifying conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,3],i = 3 .. 10) = 0" "6#/-%$SumG6$*&&%\"b G6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;F0\"#5\"\"!" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 3 .. 10) = 0;" "6#/-%$SumG6$*( &%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/F+;F3\"#5\"\"!" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,3],i = 3 .. 10 ) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F +\"\"$F,/F+;F5\"#5\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 121 "[Sum(b[i]*a[i,3],i=3..10),Sum(b[i]*c[i]*a[i ,3],i=3..10),Sum(b[i]*c[i]^2*a[i,3],i=3..10)];\nsubs(ee,eval(subs(Sum= add,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%$SumG6$*&&%\"bG6#%\"i G\"\"\"&%\"aG6$F+\"\"$F,/F+;F0\"#5-F%6$*(F(F,&%\"cGF*F,F-F,F1-F%6$*(F( F,)F7\"\"#F,F-F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can ca lculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "errterms7_ 11 := PrincipalErrorTerms(7,11,'expanded'):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$:\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "nrm8 := sqrt(add(subs(ee,errterms7_ 11[i])^2,i=1.. nops(errterms7_11))):\nevalf[10](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+(pT*o@!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "26 of the 115 principal error terms are z ero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "ind := []:\n for ct to nops(errterms7_11) do\n dd := subs(ee,errterms7_11[ct]);\n i nd := [op(ind),`if`(dd=0,0,1)];\n end do:\nind;\nnops(ind);\nListTools [Occurrences](0,ind);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"\"\"\" !F$F$F$F$F$F$F$F$F$F$F$F%F%F%F$F$F$F%F$F%F$F$F%F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F%F$F$F$F$F$F$F$F%F%F$F$F$F$F$F$F$F$F%F$F%F$F$F$F$F$F%F$F$F$F$F %F%F%F$F%F$F%F%F$F$F$F$F$F$F$F$F$F$F$F$F%F%F$F%F%F$F$F%F%F$F$F$F$F$F$F $F$F%F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#E" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 101 "An additional two error terms are (appro ximately) zero. They have a magnitude that is less than 0.2 " }{TEXT 270 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-27)" "6#)\"#5,$\"#F!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 54 "subs(ee,[seq(errterms7_11[i],i=[1,6])]);\nev alf[10](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$#!fn.Nj*RgUk5\"GbBaPo q]+H/c3s1*)\\!>G<\"bp+o6H(*Q8#*QyNi&)G&>TWf[Qy&4J^EOfdc5+z`:%4;W\"4C!* >\"#\"fn.Nj*RgUk5\"GbBaPoq]+H/c3s1*)\\!>G " 0 "" {MPLTEXT 1 0 200 "ind := []:\n for ct \+ to nops(errterms7_11) do\n dd := evalf[30](subs(ee,errterms7_11[ct] ));\n ind := [op(ind),`if`(abs(dd)<0.2*10^(-27),0,1)];\n end do:\ni nd;\nnops(ind);\nListTools[Occurrences](0,ind);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_r\"\"!F$\"\"\"F%F%F$F%F%F%F%F%F%F%F$F$F$F%F%F%F$F%F$F %F%F$F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F$F%F%F%F%F%F%F%F$F$F%F%F%F%F%F%F%F %F$F%F$F%F%F%F%F%F$F%F%F%F%F$F$F$F%F$F%F$F$F%F%F%F%F%F%F%F%F%F%F%F%F$F $F%F$F$F%F%F$F$F%F%F%F%F%F%F%F%F$F%F%F%F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The 2-nor m of the order 9 error terms is as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 139 "errterms8_11 := PrincipalErrorTerms(8,11,'expanded '):\nnrm9 := sqrt(add(subs(ee,errterms8_11[i])^2,i=1.. nops(errterms8_ 11))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/>%)o*)!# 9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The 2-norm of the order 9 error terms is approximately 4.135 times the pr incipal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf [10](nrm9/nrm8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bN7NT!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The princ ipal error norm of the order 6 embedded scheme is given as follows." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms6_11*` := subs(b= `b*`,PrincipalErrorTerms(6,11,'expanded')):\nsqrt(add(subs(ee,`errterm s6_11*`[i])^2,i=1.. nops(`errterms6_11*`))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+d%\\k@$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 48 "#-----------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 7 schem e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We f irst construct a 10 stage, order 7 scheme with " }{TEXT 260 10 "parame ters" }{TEXT -1 2 " " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 2 ".\n" } }{PARA 0 "" 0 "" {TEXT -1 89 "The scheme will be constructed so that s tage 4 has stage-order 2 and stages 5 to 10 have " }{TEXT 260 13 "stag e-order 3" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 60 "We start \+ by determining the nodes and weights of the scheme." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We set up a system of e quations that consists of the 7 order 7 quadrature conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 10) = 1" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"#5F-" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 1 .. 10) = 1/k;" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;F,\"#5*&F,F,F2 F3" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 7 " . . 7," }}{PARA 0 "" 0 "" {TEXT -1 35 "and the relation b etween the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c [5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^2*c[7]- 7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2 = 0;" "6#/,>** \"#_\"\"\"&%\"cG6#\"\"(F'&F)6#\"\"&F'&F)6#\"\"'F'F'**\"$0\"F'*$&F)6#F. \"\"#F'&F)6#F1F7&F)6#F+F'F'**\"#qF'&F)6#F+F'&F)6#F.F'&F)6#F1F7!\"\"**F =F'&F)6#F+F'&F)6#F.F7&F)6#F1F'FD*&\"\"$F'&F)6#F+F'F'*(F+F'*$&F)6#F.F7F '&F)6#F1F'FD&F)6#F1FD&F)6#F.FD*(\"#7F'&F)6#F+F'&F)6#F.F'FD*(\"#9F'*$&F )6#F1F7F'&F)6#F+F'F'*(F+F'&F)6#F.F'&F)6#F1F7FD*(FenF'&F)6#F.F'&F)6#F1F 'F'*(FenF'&F)6#F1F'&F)6#F+F'FD*(F[oF'&F)6#F+F'&F)6#F.F7F'\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 1/200;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$+#!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 119/198;" "6#/&%\"cG6#\"\"& *&\"$>\"\"\"\"\"$)>!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "c[6] = 17 /19;" "6#/&%\"cG6#\"\"'*&\"#<\"\"\"\"#>!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 25/29;" "6#/&%\"cG6#\"\")*&\"#D\"\"\"\"#H!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 44/45;" "6#/&%\"cG6#\"\"**&\" #W\"\"\"\"#X!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]=1" "6#/&%\" cG6#\"#5\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "Qeqs := QuadratureCondition s(7,10,'expanded'):\nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[ 7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c [5]-12*c[7]*c[5]+14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7 ]+14*c[7]*c[5]^2=0:\ncdns1 := [op(Qeqs),node_eq]:\ne1 := \{c[2]=1/200, c[5]=119/198,c[6]=17/19,c[8]=25/29,c[9]=44/45,c[10]=1,b[2]=0,b[3]=0,b[ 4]=0\}:\neqns1 := subs(e1,cdns1):\nnops(%);\nindets(eqns1);\nnops(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"bG6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\")&F%6#\"\"* &F%6#\"#5&%\"cGF,&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\") " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We h ave 8 equations and 8 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e2 := solve(\{op(eqns1)\}): \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 0 525 "e3 := \{b[1] = 185307033721 87/317986769215500, b[5] = 397406378366413672608/803680368165950784145 , b[10] = 1754022361907/1430294620800, b[9] = -68588512868559375/32259 978418667428, c[7] = 9526409/47245149, c[10] = 1, b[2] = 0, b[3] = 0, \+ b[4] = 0, b[6] = 383280144717771439/119491812371674080, b[8] = -160705 16558763250309/7373332958527296000, b[7] = 327399398331091971469866101 30768938398484878542073/1026382175105416417138017053640291674185019504 76800, c[9] = 44/45, c[8] = 25/29, c[5] = 119/198, c[6] = 17/19, c[2] \+ = 1/200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We now have all the weights and the nodes excluding \+ " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "seq(c[i]= subs(e3,c[i]),i=[2,$5..10]);``;\nfor ii to 10 do b[ii]=subs(e3,b[ii]) \+ end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"\"\"\"$+#/ &F%6#\"\"&#\"$>\"\"$)>/&F%6#\"\"'#\"#<\"#>/&F%6#\"\"(#\"(4k_*\")\\^CZ/ &F%6#\"\")#\"#D\"#H/&F%6#\"\"*#\"#W\"#X/&F%6#\"#5F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \"#\"/(=sLqI&=\"0+b@pn)zJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"&#\"63En8kOyjS(R\"6XTy]f;o.o.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"3R9x\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"St?ay[[)RQ*o285m) p9(>4J$)R*RF$\"T+oZ]>]=u;HSO0OASv\"\".+3i%HI9" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We construct a system of \+ equations that incorporate the row-sum conditions," }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j = 1 .. i-1) = c[i]" "6#/ -%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . 1 0, " }}{PARA 0 "" 0 "" {TEXT -1 29 " the stage order conditions: " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[i,j]*c[j],j = 2 .. i-1) = 1/2" "6#/-%$SumG6$*&&% \"aG6$%\"iG%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-F3F5" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG\"\"#" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "i = 3;" "6#/%\"iG\"\"$" }{TEXT -1 9 " . . 10, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a [i,j]*c[j]^2,j = 2 .. i-1) = 1/3" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\" \"\"*$&%\"cG6#F,\"\"#F-/F,;F2,&F+F-F-!\"\"*&F-F-\"\"$F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^3" "6#*$&%\"cG6#%\"iG\"\"$" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "i = 5;" "6#/%\"iG\"\"&" }{TEXT -1 8 " . . 10." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "together \+ with the column simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,1],i=2..10)=b[1]" "6#/-%$SumG6$* &&%\"bG6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"#5&F)6#F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 10) = b[j]*(1-c[j]); " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"# 5*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 8 " . . 9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "and the further simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,3],i = 5 .. 10) \+ = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"$F,/ F+;\"\"&\"#5\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^ 2*a[i,3],i = 5 .. 10) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG 6#F+\"\"#F,&%\"aG6$F+\"\"$F,/F+;\"\"&\"#5\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "SO_eqs := [op(RowSumConditions(10,'expanded')),op(StageOrderCondi tions(2,3..10,'expanded')),\n op(StageOrderConditions(3,5..10,'expand ed'))]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..10)=b[1],seq(add(b[i]*a[i,j ],i=j+1..10)=b[j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 := [add(b[i]*c[i]*a[i, 3],i=5..10)=0,add(b[i]*c[i]^2*a[i,3],i=5..10)=0]:\ncdcns2 := [op(simp_ eqs),op(simp_eqs2),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "We specify the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] = 0" "6#/ &%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2] = 0 " "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2 ] = 0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[7,2] = 0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2] = 0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[9,2] = 0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "e4 := \{seq(a[i,2]=0,i=4..10)\}:\ne5 := `union`(e3,e4 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "eqns2 := subs(e5,cdcns2):\nnops(eqns2);\nindets(eqns2 );\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "in folevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "e6 := solve(\{op(eqns2)\},indets(eqns2) minus \{c[3],c[4],a[6,1],a[7,6], a[8,6],a[8,7],a[9,5],a[9,6],a[9,7],a[9,8]\}):\ninfolevel[solve] := 0: \ne7 := `union`(e5,e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "indets(map(rhs,e6));\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"\"*\"\"(&F%6$F'\"\")&%\"cG 6#\"\"$&F-6#\"\"%&F%6$F'\"\"&&F%6$F'\"\"'&F%6$F+F8&F%6$F+F(&F%6$F(F8&F %6$F8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27408 "e7 := \{a[6,2] = 0, a [7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[10,3] = 1/6858932212 89967670687406904062797711477706*(469212632107972952521482257041173128 6497500*a[9,7]+85764925883887023070224812882829580185937500*a[9,8]+637 17257853159865094333269095862072184584245*c[3]+60794305800016331114134 561171465321649436432*c[4]+4168590527850178538677924525136551311482437 5*a[9,5]+92388163861065294493939370842587685915177500*a[9,6]-103257359 609425917375579296824068590140492500*c[4]*a[9,6]-693597415558265000553 13366048490517619623750*c[4]*a[9,5]-2327007030311565064007576720122611 1116847500*c[4]*a[9,7]-111193901118837598207778938753878317922945756*c [3]*c[4]-99487314025308946761460782944082313015687500*c[4]*a[9,8]+1655 33514092530852594699453974187377958875000*c[4]*c[3]*a[9,8]+17180636304 7616232271972275387946057544685000*c[4]*c[3]*a[9,6]+115405284269358378 243294508215135483098197500*c[4]*c[3]*a[9,5]+3871826823543612459441178 0721367815135595000*c[4]*c[3]*a[9,7]+438961335427006987616390460527030 73355730658*c[4]*c[3]*a[6,1]-78070673241494659327103770499287629808950 00*c[3]*a[9,7]-69359741555826500055313366048490517619623750*c[3]*a[9,5 ]-153721482726814523611764667452372788329455000*c[3]*a[9,6]-1427013052 52181769478189184460506360309375000*c[3]*a[9,8]-3821531621074158554538 9622215790483432425424)/(-c[4]+c[3])/c[3]/(198*c[3]-119), a[5,2] = 0, \+ b[1] = 18530703372187/317986769215500, a[7,3] = -161948953/21022170836 38746340466587848749216797803827259696017357258748774*1/c[3]*(10943661 2017658709336508904621155195488468780136380884784*c[4]+502101293573169 98977430259940477086445780883482606282500*c[4]*c[3]*a[9,5]-10124261185 009891172352582170273193140122492161822832500*c[4]*a[9,7]+101303763234 243088277711366455695691273975513308018400146*c[4]*c[3]*a[6,1]-4492485 2582862578032437600999374235240961843116016147500*c[4]*a[9,6]-30176794 916771327668253540065236228722464268355707816250*c[4]*a[9,5]+720197450 93804446519350850972286751059581388942017125000*c[4]*c[3]*a[9,8]-43284 594273549137049508844776273350384293865071212312500*c[4]*a[9,8]+168454 09366655113043074044283311699510455911328074965000*c[4]*c[3]*a[9,7]+74 748914381569667650610462167026038468154999470346195000*c[4]*c[3]*a[9,6 ]-205258989532761046844008831008743148713490883408664579772*c[3]*c[4]+ 83293580192208089361292154523610183319429556758107461165*c[3]+20414339 86613713343423262141900115069777876463736382500*a[9,7]-498770447025138 10476988168376958984047762647173158971888+4019592073203493823954943247 3124315741913228051172342500*a[9,6]+1813655856108983834607157205940965 2616026504718834495625*a[9,5]-6688060760456233421370409772839171862940 1841631362385000*c[3]*a[9,6]-30176794916771327668253540065236228722464 268355707816250*c[3]*a[9,5]-620859871498314194132334922174885784996391 28398290625000*c[3]*a[9,8]+3731430540823201469785245239333909515887402 1613114062500*a[9,8]-3396671675206010436956352135262376334588399494284 065000*c[3]*a[9,7])/(-c[4]+c[3])/(198*c[3]-119), b[5] = 39740637836641 3672608/803680368165950784145, a[6,5] = -6534/6859*(16473*c[3]+41154*c [4]*c[3]*a[6,1]-36822*c[3]*c[4]-9826+16473*c[4])/(198*c[3]-119)/(198*c [4]-119), a[9,3] = 1/1106972373717205797850933500*(3449340088988622174 08811776+223207499752640271438223500*c[4]*a[9,7]-529160127288027265343 063520*c[4]-886191180067236774456841500*a[9,6]-39985296868200569593324 8375*a[9,5]-45007074366747156530113500*a[9,7]+665301578143169141132631 750*c[4]*a[9,5]-822660800919445450245937500*a[9,8]+9542865290665567222 85287500*c[4]*a[9,8]+990448965957499924392940500*c[4]*a[9,6])/c[3]/(-c [4]+c[3]), a[5,3] = -14161/23287176*(-119+297*c[4])/c[3]/(-c[4]+c[3]), a[4,3] = 1/2*c[4]^2/c[3], a[3,2] = 100*c[3]^2, a[8,1] = 1/88920559586 5720097212391938050047333230783792698*(6677370906928969517370587689601 758099803562500*a[9,7]+12205217458855694390721550111838120952410156250 0*a[9,8]+31356504583124416657156998313277100666912310500*c[3]+14773576 425475193289166825774331233490893287600*c[4]-5908786951943183263545852 77912918962527988307142*c[3]*c[4]*a[8,7]+43457416339302110014139455618 9872756842112379890*c[3]*c[4]*a[8,6]+593232645688045977482108436377818 71037140890625*a[9,5]+131477712937724882702292832493391399860934562500 *a[9,6]-388829514614808352758089866064622992963995287270*c[3]*a[8,6]-1 46945679165692515961386106904378623373985687500*c[4]*a[9,6]-9870593600 5237902135678546556981600549192406250*c[4]*a[9,5]-33115666504149076243 171115539213234712804812500*c[4]*a[9,7]-475145381276119940850891133350 33332804377729500*c[3]*c[4]-141580522522726054932369981297322203047957 812500*c[4]*a[9,8]+119143493860235489003310337380397137088027610222*c[ 3]*a[8,7]-25638802326226303682115745677457872187241017200+235570953441 174444341254254595544505911728125000*c[4]*c[3]*a[9,8]+2444978527294715 81179449152664428297714698875000*c[4]*c[3]*a[9,6]+16423340612636222372 1549178304893755535631062500*c[4]*c[3]*a[9,5]+551000165363152697155284 10729111096412902125000*c[4]*c[3]*a[9,7]+10480850138456498379744975142 38743454746442901650*c[4]*c[3]*a[6,1]-11110247391360806423860305567572 673140849625000*c[3]*a[9,7]-987059360052379021356785465569816005491924 06250*c[3]*a[9,5]-218761236652685098950033452383962161113151625000*c[3 ]*a[9,6]-203078408138943486501081253961676298199765625000*c[3]*a[9,8]) /c[3]/c[4], a[10,1] = -1/627240456844588883556949208769439959515954404 028307671282713911837587405437940363711220*(36057888046211693436057281 427330273213745686088224845368394664759512098006074626925000*a[9,7]+65 9083299168554790175312477589930058536490637505483201559497820432746122 948025578125000*a[9,8]+44106408674932037709493031050613366849869189699 7682394717427014493565039736667040257880*c[3]+467189952353991250246720 227809470406460304064451407470859251929982973080704252367803360*c[4]+7 4077544300412592218420504851312413991228476579087179362786276040926338 7750959442725713*c[3]*c[4]*a[8,7]-544818879109771109604861763259747091 444345970146668146213898601788090469856993294244835*c[3]*c[4]*a[8,6]+3 2034638515258792313717606758175658112700908620746614207268176317353605 1806278492731250*a[9,5]+7099813264470607122318870797374178871899634923 21640716480152402253112520493715983325000*a[9,6]-293675723815638581526 568844902042417837433189185979983345803884788712572671332312615520+487 4695234140057296464552618639842397133621838154399202966461173893441046 08888736955905*c[3]*a[8,6]-7935085413231855019062267361771141092123121 38477127859595464449577008111140035510775000*c[4]*a[9,6]-5330133131110 2864521983917127048574002645209301746467336463016057445494334153900471 2500*c[4]*a[9,5]-17882502140823371555103798646154615345552807855986552 3822435697162298851287993057425000*c[4]*a[9,7]-87582390569913945189048 4320512619192651108077494975872826253525129732119937117654759352*c[3]* c[4]-76453662703552355660336247400431886790232913950636051380901747170 1985502619709670625000*c[4]*a[9,8]-14936834778981207620339837709054240 5201804926118299535316580638819186503580319532780333*c[3]*a[8,7]-71342 8170293710359287678177834697956891510421275010124501337622128863555919 51924948325*c[3]*a[7,6]+7997312474667629846490263802002084499541243111 13422650651300930031866140945850116250000*c[4]*c[3]*a[9,8]+78882798778 5438605892565748358135758658171704923171691636920961717001999306880322 550000*c[4]*c[3]*a[9,6]+8868624873612073256599004698450098867667018018 27378196018460267174303183038863217925000*c[4]*c[3]*a[9,5]+10201573356 8142485021283601163451757718163594183239824254807256232359529758189540 1850000*c[4]*c[3]*a[9,7]+337331469920876497612014200217697910421197934 779754700387260393237865982599142253257340*c[4]*c[3]*a[6,1]+7973608962 1061746038034619875642712829051164730736425679561263649696515073358033 765775*c[3]*c[4]*a[7,6]-2057023044641376160310387188483451064557195336 81556070643234533779807936582903920850000*c[3]*a[9,7]-5330133131110286 4521983917127048574002645209301746467336463016057445494334153900471250 0*c[3]*a[9,5]-70579346275539243685124303800464778406257468335231151356 9876649957317578327208709650000*c[3]*a[9,6]-68942348919548533159398825 8793283146512176130270191940216638732786091500815388031250000*c[3]*a[9 ,8])/c[3]/c[4], a[2,1] = 1/200, a[8,5] = -2/12248010962337742385845618 98140561065056176023*(180289014487082176969005867619247468694696187500 *a[9,7]+3295408713891037485494818530196292657150742187500*a[9,8]+97677 57865262805528256386370983183708654359197875*c[3]+93200188050062764973 20651712431645294901845579575*c[4]-58193351579336540376430719243975131 18393281495650+8054826318127847813160779823702466008975478110012*c[3]* c[4]*a[8,7]+35742053499986012328552235344477842431968196659876*c[3]*c[ 4]*a[8,6]+1601728143357724139201692778220110518002804046875*a[9,5]+354 9898249318571832961906477321567796245233187500*a[9,6]+1922014200703660 3652488349841009748867589490123054*a[8,6]+9761351501879145215023429372 21045119458801872526*a[8,7]-319797320789348531360730526766380695443925 97011468*c[3]*a[8,6]-3967533337473697930957424886418222831097613562500 *c[4]*a[9,6]-2665060272141423357663320757038503214828194968750*c[4]*a[ 9,5]-894122995612025058565620119558757337245729937500*c[4]*a[9,7]-2197 9919329768319206447908066383208604020924905850*c[3]*c[4]-3822674108113 603483173989495027699482294860937500*c[4]*a[9,8]-162415764485047962401 2301693863587677754981266892*c[3]*a[8,7]+63604157429117099972138648740 79701659616659375000*c[4]*c[3]*a[9,8]+66014420236957326918451271219395 64038296869625000*c[4]*c[3]*a[9,6]+44343019654117800404818278142321313 99462038687500*c[4]*c[3]*a[9,5]+14877004464805122823192670896859996031 48357375000*c[4]*c[3]*a[9,7]+28298295373832545625311432884446073278153 958344550*c[4]*c[3]*a[6,1]-2148133518433502761160462629289324873436472 4255178*c[4]*a[8,6]-4841031979076837827101680803134310379131726742886* c[4]*a[8,7]-299976679566741773444228250324462174802939875000*c[3]*a[9, 7]-2665060272141423357663320757038503214828194968750*c[3]*a[9,5]-59065 53389622497671650903214366978350055093875000*c[3]*a[9,6]-5483117019751 474135529193856965260051393671875000*c[3]*a[9,8])/(198*c[4]-119)/(198* c[3]-119), a[5,1] = 119/23287176*(14161-35343*c[3]-35343*c[4]+117612*c [3]*c[4])/c[3]/c[4], a[8,4] = 1/74723159316447066992637977987398935565 61208342*(6677370906928969517370587689601758099803562500*a[9,7]+122052 174588556943907215501118381209524101562500*a[9,8]+31356504583124416657 156998313277100666912310500*c[3]+2923090997849026555008406135709554228 5851343225*c[4]+59323264568804597748210843637781871037140890625*a[9,5] +131477712937724882702292832493391399860934562500*a[9,6]-3888295146148 08352758089866064622992963995287270*c[3]*a[8,6]-1469456791656925159613 86106904378623373985687500*c[4]*a[9,6]-9870593600523790213567854655698 1600549192406250*c[4]*a[9,5]-33115666504149076243171115539213234712804 812500*c[4]*a[9,7]-264309435549884102838997158863277569165963322300*c[ 3]*c[4]-141580522522726054932369981297322203047957812500*c[4]*a[9,8]+1 19143493860235489003310337380397137088027610222*c[3]*a[8,7]-2563880232 6226303682115745677457872187241017200+23557095344117444434125425459554 4505911728125000*c[4]*c[3]*a[9,8]+244497852729471581179449152664428297 714698875000*c[4]*c[3]*a[9,6]+1642334061263622237215491783048937555356 31062500*c[4]*c[3]*a[9,5]+55100016536315269715528410729111096412902125 000*c[4]*c[3]*a[9,7]+1048085013845649837974497514238743454746442901650 *c[4]*c[3]*a[6,1]+388829514614808352758089866064622992963995287270*c[4 ]*a[8,6]-119143493860235489003310337380397137088027610222*c[4]*a[8,7]- 11110247391360806423860305567572673140849625000*c[3]*a[9,7]-9870593600 5237902135678546556981600549192406250*c[3]*a[9,5]-21876123665268509895 0033452383962161113151625000*c[3]*a[9,6]-20307840813894348650108125396 1676298199765625000*c[3]*a[9,8])/(-c[4]+c[3])/(198*c[4]-119)/c[4], a[7 ,5] = -2/27804267110822448924837818781236452789448527205976943964167*( 720279077888250308081757602887279011176577420186164736127500*a[9,7]+13 165604995175190329842736413036188747161255902272752704687500*a[9,8]+31 866604960480066053215919880886049013560385611053916981272451*c[3]+4969 2160670406273584543872658276819337443751195632437724454855*c[4]-190874 99488620669284435307851760686763130813206839135306617634+6399121285384 936512168938470774394421044016251557311430831875*a[9,5]+14182325223146 082368877328498402360726022364923175580153447500*a[9,6]+43631734484468 9016801467858625183705956411464693243368598999566*a[7,6]-1585083407292 7974412274661262920285517319113737666824877382500*c[4]*a[9,6]-10647277 432825356549659242161456555423249707712675190447938750*c[4]*a[9,5]-357 2142698934403420493340368893706651437080386400684738277500*c[4]*a[9,7] -90856561097128808091108854082163219292953891315895292051159258*c[3]*c [4]-15272101794403220782617574239121978946707056846636393137437500*c[4 ]*a[9,8]-4876487971793583128957581949340288478336363428924484707871171 62*c[4]*a[7,6]-7259733973046086161906776135108098636921804202459007309 46234572*c[3]*a[7,6]+2541072399404905642822083780963152799536132147591 6015472375000*c[4]*c[3]*a[9,8]+263736566927709154086586800845228280036 06592605529675006065000*c[4]*c[3]*a[9,6]+17715638081507736107836386117 381495578180185942098216039427500*c[4]*c[3]*a[9,5]+5943565162932872918 131776412108856445248251399221307379655000*c[4]*c[3]*a[9,7]+3574300302 9946757939132742992808797578282288731745417011720582*c[4]*c[3]*a[6,1]+ 811382032281621394566051450394434553538319293216006699292850404*c[3]*c [4]*a[7,6]-11984475413602820252116639106863970101929607495534505693550 00*c[3]*a[9,7]-1064727743282535654965924216145655542324970771267519044 7938750*c[3]*a[9,5]-23597482304058187470905134812467793476911161804947 603952795000*c[3]*a[9,6]-219057965465940141622593429393375241339321736 86134496096875000*c[3]*a[9,8])/(198*c[4]-119)/(198*c[3]-119), b[10] = \+ 1754022361907/1430294620800, b[9] = -68588512868559375/322599784186674 28, c[7] = 9526409/47245149, a[4,1] = 1/2*c[4]*(2*c[3]-c[4])/c[3], c[1 0] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[6] = 383280144717771439/119491 812371674080, b[8] = -16070516558763250309/7373332958527296000, a[6,4] = 17/13718*(96026*c[3]*a[6,1]-21964*c[3]+289)/(-c[4]+c[3])/(198*c[4]- 119), a[10,6] = -7759365740446879723835826600992238400440916214471301/ 29830948869806602777180491475003365613724027019957964*a[7,6]+239433056 1195458751842358107/1347186551150289146484386820*a[8,6]+24525445251143 012876977500000/14146180885244972240022216299*a[9,6]+12022007220378816 3018766520/436648564509816360719190147, a[9,4] = -1/110697237371720579 7850933500*(344934008898862217408811776-822660800919445450245937500*a[ 9,8]-399852968682005695933248375*a[9,5]-886191180067236774456841500*a[ 9,6]-45007074366747156530113500*a[9,7]+665301578143169141132631750*c[3 ]*a[9,5]+990448965957499924392940500*c[3]*a[9,6]-529160127288027265343 063520*c[3]+954286529066556722285287500*c[3]*a[9,8]+223207499752640271 438223500*c[3]*a[9,7])/c[4]/(-c[4]+c[3]), a[4,2] = 0, a[10,9] = -54500 9894469844730599500000/14146180885244972240022216299, b[7] = 327399398 33109197146986610130768938398484878542073/1026382175105416417138017053 64029167418501950476800, a[7,1] = 1/1471551958547122438326611494124451 7584626790817872121500811241418*(5612479214865244842058477325735636459 53257512526620028168335748*c[3]-32358083616727614680714925588532719275 209940237943165372585000*c[3]*a[9,7]-637132022209571061714438639936630 423876601368733585306725465000*c[3]*a[9,6]-591456506758038382381002259 362113151616168689525631394615625000*c[3]*a[9,8]+965061081808562464356 584060805837534613621795757126259316455714*c[4]*c[3]*a[6,1]-6434764203 381758188962824301573087428180690088543211024296170070*c[3]*a[7,6]-287 476490686284626840799538359326996427742108242230142094346250*c[3]*a[9, 5]+160476259399187568789557963126939124021702787778975299250685000*c[4 ]*c[3]*a[9,7]+71208873070481471603378436228211635609737800034930122516 3755000*c[4]*c[3]*a[9,6]+719179528613255327001727421940521536090783009 8960059380095719490*c[3]*c[4]*a[7,6]+478322228200708874911582425169300 380610865020436651833064542500*c[4]*c[3]*a[9,5]+6860895478393245235619 62620860051255874755679849732417754125000*c[4]*c[3]*a[9,8]+51407818650 0164978280225030941645162296648235019087405096812772*c[3]*c[4]+1944753 5102982758318207455277956533301767590345026447875442500*a[9,7]+3554713 34869730138905753883151977096173353909361364323026562500*a[9,8]+104253 7925654532087551702457563114444698108383310240088234260656*c[4]+172776 274705393285828561338710908649368188438792047408632460625*a[9,5]+38292 2781024944223959687869456863739602603852925740664143082500*a[9,6]-4279 72519969055309131415854098847708967616070917004271689327500*c[4]*a[9,6 ]-287476490686284626840799538359326996427742108242230142094346250*c[4] *a[9,5]-96447852871228892353320189960130079588801170432818487933492500 *c[4]*a[9,7]-412346748448886961130674504456293431561090534859182614710 812500*c[4]*a[9,8]-475149127547429886752274380119777458263462511894206 188224590192)/c[3]/c[4], c[9] = 44/45, a[7,4] = 289/210221708363874634 0466587848749216797803827259696017357258748774*(-190341668333691851063 0289740501924663247643543408421492505000*c[3]*a[9,7]-37478354247621827 159672861172742966110388315807857959219145000*c[3]*a[9,6]-347915592210 61081316529544668359597153892275854448905565625000*c[3]*a[9,8]+3301458 3616854381423873396033739037997250441913330589892255044*c[3]-378515541 375397540527224958916063966363570005208424177899774710*c[3]*a[7,6]-169 10381805075566284752914021136882142808359308366478946726250*c[3]*a[9,5 ]+9439779964658092281738703713349360236570752222292664661805000*c[4]*c [3]*a[9,7]+56768298929915439079799062400343384389036576221007427018615 042*c[4]*c[3]*a[6,1]+4188757239440086564904613895777155035866929413819 4189715515000*c[4]*c[3]*a[9,6]+281366016588652279359754367746647282712 27354143332460768502500*c[4]*c[3]*a[9,5]+40358208696430854327174271815 297132698515039991160730456125000*c[4]*c[3]*a[9,8]-1150224167774000371 47305136694186459446619880771897231218894044*c[3]*c[4]+114397265311663 2842247497369291560782456917079119202816202500*a[9,7]+2091007852174883 1700338463714822182127844347609492019001562500*a[9,8]+7498698330313414 5630004268086627150682983204420592105945607729*c[4]+101633102767878403 42856549335935802904011084634826318154850625*a[9,5]+225248694720555425 85863992320991984682506108995631803773122500*a[9,6]-251748541158267828 90083285535226335821624474759823780687607500*c[4]*a[9,6]-1691038180507 5566284752914021136882142808359308366478946726250*c[4]*a[9,5]-56734031 10072287785489422938831181152282421790165793407852500*c[4]*a[9,7]-2425 5691085228644772392617909193731268299443227010742041812500*c[4]*a[9,8] +378515541375397540527224958916063966363570005208424177899774710*c[4]* a[7,6]-27949948679260581573663198830575144603733088934953305189681776) /(-c[4]+c[3])/(198*c[4]-119)/c[4], a[9,1] = -1/11069723737172057978509 33500*(-344934008898862217408811776+822660800919445450245937500*a[9,8] +399852968682005695933248375*a[9,5]+886191180067236774456841500*a[9,6] +45007074366747156530113500*a[9,7]-665301578143169141132631750*c[3]*a[ 9,5]-990448965957499924392940500*c[3]*a[9,6]+5291601272880272653430635 20*c[3]-954286529066556722285287500*c[3]*a[9,8]-2232074997526402714382 23500*c[3]*a[9,7]+529160127288027265343063520*c[4]-9542865290665567222 85287500*c[4]*a[9,8]-665301578143169141132631750*c[4]*a[9,5]-990448965 957499924392940500*c[4]*a[9,6]-223207499752640271438223500*c[4]*a[9,7] +1106972373717205797850933500*c[4]*c[3]*a[9,5]+11069723737172057978509 33500*c[4]*c[3]*a[9,7]-1082372987634601224565357200*c[3]*c[4]+11069723 73717205797850933500*c[4]*c[3]*a[9,6]+1106972373717205797850933500*c[4 ]*c[3]*a[9,8])/c[3]/c[4], a[10,7] = 2394330561195458751842358107/13471 86551150289146484386820*a[8,7]+24525445251143012876977500000/141461808 85244972240022216299*a[9,7]+154869603082860073165731441680874994300603 6471889565/7457737217451650694295122868750841403431006754989491, c[8] \+ = 25/29, a[8,3] = -25/2490771977214902233087932599579964518853736114*( -341850697683017382428209942366104962496546896+89031612092386260231607 835861356774664047500*a[9,7]+16273623278474259187628733482450827936546 87500*a[9,8]+610851175148526518907656451947218792824938215*c[3]+196981 019006335910522224343657749779878577168*c[4]-1959275722209233546151814 758725048311653142500*c[4]*a[9,6]+790976860917394636642811248503758280 495211875*a[9,5]+1753036172502998436030571099911885331479127500*a[9,6] -148136631884810752318137407567635641877995000*c[3]*a[9,7]-18877403003 03014065764933083964296040639437500*c[4]*a[9,8]-1316079146736505361809 047287426421340655898750*c[4]*a[9,5]-441542220055321016575614873856176 462837397500*c[4]*a[9,7]-352412580733178803785329545151036758887951096 4*c[3]*c[4]+734666887150870262873712143054814618838695000*c[4]*c[3]*a[ 9,7]+3140946045882325924550056727940593412156375000*c[4]*c[3]*a[9,8]+3 259971369726287749059322035525710636195985000*c[4]*c[3]*a[9,6]+2189778 748351496316287322377398583407141747500*c[4]*c[3]*a[9,5]-1316079146736 505361809047287426421340655898750*c[3]*a[9,5]+139744668512753311729933 00189849912729952572022*c[4]*c[3]*a[6,1]-27077121085192464866810833861 55683975996875000*c[3]*a[9,8]-2916816488702467986000446031786162148175 355000*c[3]*a[9,6])/c[3]/(198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3]), c[5] = 119/198, c[6] = 17/19, a[10,5] = -79/4319837857056397269675958 73808154242090877688724729801158893878676024383910427247735*(278739100 3755323532313735029429194864968900694794110629452504742455779188043774 975000*a[9,7]+50949264041015279586205012935429056676299308964586016617 051217296080810793120609375000*a[9,8]-51963470066595125746938034555700 725410607373852152536703204884220947798233657297845110*c[3]-6089256337 4520614039321930342917078891158367792033258928891880654036622361819371 816350*c[4]+1278254512971659320657796945910845062961220282052006456158 15138169966043184718422132342*c[3]*c[4]*a[8,7]+56720578923482694293039 7465703331636000286963272461131384518388541213564139335180337466*c[3]* c[4]*a[8,6]-1094854734065806825911867572747775657016360168050833650121 82374755512321503411636756250*a[9,5]+548838456549793803352799653803572 15716865909117046722449895540443634991611998541775000*a[9,6]+305012576 1887439940319495143853907229209411288410922032936950292447833706150651 43493539*a[8,6]+154907022413362194338462734382711993067867952647917522 24388342005339511706006106985491*a[8,7]-446396243652130746716905547299 49065929745335609671032717094406619280206527155118765135*a[7,6]-507499 9166837925278850924693135072532634146513490441701861480318526647679141 42003459838*c[3]*a[8,6]-6134076867321224861001878483686982933061483960 1405160385177368731121461213410134925000*c[4]*a[9,6]+18216910701263004 3302983007902571075705243120398374002289177396652028904686348773762500 *c[4]*a[9,5]-138237507221955114280069988819398470447248899025594304224 14928330342515595354678475000*c[4]*a[9,7]+1276874024360534369224503945 24890474392321634235002344236370307347082327446331523554580*c[3]*c[4]- 5910114628757772431999781500509770574450719839891977927577941206345374 0520019906875000*c[4]*a[9,8]+49891344878767554044830619992296014862656 551563749977742634925045077877883291015090445*c[4]*a[7,6]-257744457460 8883569665178269561090304826710472629215916326799762232960771251436288 3422*c[3]*a[8,7]+74274332977413351134409494424621134908315768493402222 504073046307709923465350533743670*c[3]*a[7,6]+310220328243322062627140 96875411548098172260198546308363076654617852760598532920389940+9833636 1049919238784534179588313829726154834310807699971464904105578492629949 088750000*c[4]*c[3]*a[9,8]+1020627915739161783595270537621867748526196 49084690939128278310998000414455926106850000*c[4]*c[3]*a[9,6]-30310490 0743703769529333071972345151173429729738471029018967433084888429646193 758025000*c[4]*c[3]*a[9,5]+2300086254617404422475114099684109004080275 7989132497677631561423595109982186775950000*c[4]*c[3]*a[9,7]-115290502 3780210814623339671630106782452195473297895811450130457901567282300865 92885420*c[4]*c[3]*a[6,1]-34089640868153740509453181019543669032340479 1057691286034129738567699061275661042728073*c[4]*a[8,6]-76824387395771 4440193322406885810921678713199819135193347575830415452481766742032007 51*c[4]*a[8,7]-8301248979828551009139884670987068019164703537497895456 3375757638028737990685890654690*c[3]*c[4]*a[7,6]-463784385498784923863 9659964932609943393633088817091635559629739548271254056028950000*c[3]* a[9,7]+182169107012630043302983007902571075705243120398374002289177396 652028904686348773762500*c[3]*a[9,5]-913193398292934227427347323135355 35394449159707355050798985857208737212934249674550000*c[3]*a[9,6]-8477 2725043033826538391534127856749763926581302420431009883538022050424680 990593750000*c[3]*a[9,8])/(198*c[4]-119)/(198*c[3]-119), c[2] = 1/200, a[5,4] = 14161/23287176*(-119+297*c[3])/c[4]/(-c[4]+c[3]), a[10,4] = \+ -1/3100546005163563438244929356250321104873724191934293975693098921589 65598338082236140*(212105223801245255506219302513707489492621682871910 8551082039103500711647416154525000*a[9,7]+3876960583344439942207720456 4113532855087684559146070679970460025455654291060328125000*a[9,8]+2594 4946279371786887937077088596098146981876293981317336319236146680296455 098061191640*c[3]+3033988338485322090520731236474789529783576190431321 8409438404763307204537488411360990*c[4]+188439050089757601845397686812 79798889824063894556831886628339010208003047428146631250*a[9,5]+417636 0743806239483716982821984811101117432307774357155765602366194779532315 9763725000*a[9,6]+2867467784788268997920325069788142586549189316561411 2958626242199373182624052278644465*c[3]*a[8,6]-74648738005973996917180 772056615044170257799624091946139453842709273085745171603575000*c[4]*a [9,6]-3135372430064874383646112772179327882508541723632145137439000944 5556173137737588512500*c[4]*a[9,5]-19481290902112462398513369098281357 83143283581802938364442452824962834530453971025000*c[4]*a[9,7]-5026464 6917513852538837605726259409287334578696792390128594336974212376243282 202765640*c[3]*c[4]-68925831259682845887705092522451508865869181574488 252183031951743289727413951230625000*c[4]*a[9,8]+419663629584535505463 3401046086458569950061301617706614713750718405079740703054408725*c[4]* a[7,6]-878637339940071036490578688767896501187087800695879619509297875 4069794328254090163549*c[3]*a[8,7]-41966362958453550546334010460864585 69950061301617706614713750718405079740703054408725*c[3]*a[7,6]-1727504 2577390504795680520288355436343378422893292940196811993222865445451254 841918560+748285972086177282290982549939966069424919276432325169224841 61837366846030006186250000*c[4]*c[3]*a[9,8]+77664207208102189509358820 446400688889786358585502380721651982706993379142722210150000*c[4]*c[3] *a[9,6]+52168381609482783862347086461471169809805988342786952706968251 010253128414050777525000*c[4]*c[3]*a[9,5]+1750239952487902900598394528 8871052093027463942092621708770275846829052177470403050000*c[4]*c[3]*a [9,7]+1984302764240449985953024707162928884830576086939733531689767019 0462704858773073721020*c[4]*c[3]*a[6,1]-286746778478826899792032506978 81425865491893165614112958626242199373182624052278644465*c[4]*a[8,6]+8 7863733994007103649057868876789650118708780069587961950929787540697943 28254090163549*c[4]*a[8,7]-1210013555671397741359051287343206508563056 0804797415920190266692929878622523760050000*c[3]*a[9,7]-31353724300648 7438364611277217932788250854172363214513743900094455561731377375885125 00*c[3]*a[9,5]-4151726251502308452066135517674398729779851078543008903 3522155879842210489835806450000*c[3]*a[9,6]-40554322893852078329058132 870193126265422125310011290600978748987417147106787531250000*c[3]*a[9, 8])/(-c[4]+c[3])/(198*c[4]-119)/c[4], a[6,3] = -17/13718*(96026*c[4]*a [6,1]+289-21964*c[4])/(198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3]), a[ 10,8] = 24525445251143012876977500000/14146180885244972240022216299*a[ 9,8]-82563122799843405235943383/336796637787572286621096705, a[3,1] = \+ -100*c[3]^2+c[3]\}:" }}}{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 33 "We have expressions in terms of " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1 ]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 69 " for the linking coefficient s in rows 2 to 6 of the Butcher tableau." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "subs(e7,matrix([seq([ c[i],seq(a[i,j],j=1..i-1),``$(5-i)],i=2..5)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"\"\"$+#F(%!GF+F+7'&%\"cG6#\"\"$,&* &\"$+\"F))F-\"\"#F)!\"\"F-F),$*&F3F)F4F)F)F+F+7'&F.6#\"\"%,$*&#F)F5F)* (F:F),&*&F5F)F-F)F)F:F6F)F-F6F)F)\"\"!,$*&F?F)*&F:F5F-F6F)F)F+7'#\"$> \"\"$)>,$*&#FI\")wrGBF)*(,*\"&hT\"F)*&\"&V`$F)F-F)F6*&FSF)F:F)F6*(\"'7 w6F)F-F)F:F)F)F)F-F6F:F6F)F)FC,$*&#FQFNF)*(,&FIF6*&\"$(HF)F:F)F)F)F-F6 ,&F:F6F-F)F6F)F6,$*&#FQFNF)*(,&FIF6*&FgnF)F-F)F)F)F:F6FhnF6F)F)Q(pprin t06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "c[6]=subs(e7,c[6]);``;\nfor ii from 2 to 5 do a[6,ii] =subs(e7,a[6,ii]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"'#\"#<\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\"#<\"&=P\"\"\"\"*&,(*(\"&Eg*F. &%\"cG6#\"\"%F.&F%6$F'F.F.F.\"$*GF.*&\"&k>#F.F3F.!\"\"F.,**&\"$)>F.)&F 46#F(\"\"#F.F.*(F?F.FAF.F3F.F<*&\"$>\"F.F3F.F.*&FFF.FAF.F#F.F3F.!\"\"\"$*G F.F.,&&F46#F(F;F3F.F;,&*&\"$)>F.F>F.F.\"$>\"F;F;F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,$*&#\"%Ml\"%fo\"\"\"*(,,*&\"&tk \"F.&%\"cG6#\"\"$F.F.**\"&a6%F.&F46#\"\"%F.F3F.&F%6$F'F.F.F.*(\"&Ao$F. F3F.F9F.!\"\"\"%E)*F@*&F2F.F9F.F.F.,&*&\"$)>F.F3F.F.\"$>\"F@F@,&*&FEF. F9F.F.FFF@F@F.F@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can find some simple order conditions that are not ye t satisfied and determine which parameters are related by them." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_10 := SimpleOrderConditi ons(7,10,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 54 do\n \+ eq := simplify(subs(e7,SO7_10[ii]));\n if lhs(eq)<>rhs(eq) then\n \+ print(ii,indets(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#h<)&%\"aG6$\"\"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F (\"\"'&F&6$F,F3&F&6$F,F)&F&6$F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#f<)&%\"aG6$\"\"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F(\"\"'&F&6$F,F3 &F&6$F,F)&F&6$F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#e<)&%\"aG6$\" \"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F(\"\"'&F&6$F,F3&F&6$F,F)&F&6$F) F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#b<,&%\"aG6$\"\"*\"\"(&F&6$F( \"\")&%\"cG6#\"\"$&F.6#\"\"%&F&6$F(\"\"&&F&6$F(\"\"'&F&6$F,F9&F&6$F,F) &F&6$F)F9&F&6$F9\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#a<)&%\"aG 6$\"\"*\"\"(&F&6$F(\"\")&%\"cG6#\"\"%&F&6$F(\"\"'&F&6$F,F3&F&6$F,F)&F& 6$F)F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple or der comditions given in abreviated form as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "SO7 := Si mpleOrderConditions(7):\n[seq([i,SO7[i]],i=[54,59,61])]:\nlinalg[augme nt](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linal g[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7% 7%\"#a%#~~G/*&%\"bG\"\"\"-%!G6#*(%\"aGF-%\"cGF--F/6#*&)F3\"\"$F-F2F-F- F-#F-\"$o\"7%\"#fF)/*(F,F-)F3\"\"#F-F4F-#F-\"#G7%\"#hF)/*(F,F-F3F--F/6 #*&)F3\"\"%F-F2F-F-#F-\"#NQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cdcns3 := [seq(SO7_10[ i],i=[54,59,61])]:\neqns3 := simplify(subs(e7,cdcns3)):\nnops(%);\nind ets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"aG6$\"\"*\"\"(&F%6$F'\"\")&%\"cG 6#\"\"%&F%6$F'\"\"'&F%6$F+F2&F%6$F+F(&F%6$F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 3 equations for the linkin g coefficients " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[9,7]" "6#&%\"aG6$\"\"*\"\"(" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,8]" "6#&%\"aG6$\"\"*\"\")" } {TEXT -1 30 " in terms of the parameters " }{XPPEDIT 18 0 "a[7,6]" " 6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,6]" "6#&% \"aG6$\"\")\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[9,6]" "6#&% \"aG6$\"\"*\"\"'" }{TEXT -1 89 ", and subsitute back into the expressi ons previously obtained for the other coefficients." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[ solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "e8 := solv e(\{op(eqns3)\},\{a[8,7],a[9,7],a[9,8]\}):\ninfolevel[solve] := 0:\ne9 := `union`(map(u_->lhs(u_)=simplify(subs(e8,rhs(u_))),e7),e8):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "indets(map(rhs,e8));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <&&%\"cG6#\"\"%&%\"aG6$\"\"*\"\"'&F)6$\"\")F,&F)6$\"\"(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37915 "e9 := \{a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[5,2] = 0, a[8,1] = 1/151981378421081225402 76830245365390369450817266432870707865952898029887104661564*(-77451940 4427256144393658296444025138971392602802659084934475148176665714017767 74*c[3]*c[4]*a[8,6]-83262331915943825364235637516492335768655284703910 147386246918389013295717497175*c[4]*a[7,6]+325816620202841800471932804 627256351878948648915312323639249022690249341456177850*c[3]*c[4]*a[7,6 ]+31204742796965980424941103690755626281455857062997395860925986162599 646125669827*c[4]*a[8,6]+280704704985269535369062970857623196904081146 6152039312685915187044454765375000*c[4]*c[3]*a[9,5]+179136755152398219 61803764690379506872050439276156330533871783818451569331754700*c[4]*c[ 3]*a[6,1]-907534349424260381408960774653544922331904222715337327245444 851236263564000000*c[4]*c[3]*a[9,6]+2166838087686048828752104776066557 6948559672661342911534146137658146399395347640*c[3]*c[4]-3994970696893 359603638101784678990967112137632455663320711940022313312057552720*c[4 ]-16870636309720744802484087642453111329083664872327913040890096326176 26853937500*c[4]*a[9,5]+5454373110176110373114461221402618472600838510 25884555263676450995532142000000*c[4]*a[9,6]+4419447914105323607763024 79751274756266705760250758470160217046192454020292960+1013942283260994 258331114358309050630384321272629808915083798718593422200093750*a[9,5] +744978759247918437469476756726510372666915705245511845034840848743803 17220918525*a[7,6]+692991046066492339720641633660443545395456539349747 60230979355363175353359484482*c[3]*a[8,6]-4247504112696267118016845210 844112584067759745255416930907045597597733761052520*c[3]-2915201338657 00558316992509403334630628533001661068921150907020301802042355527550*c [3]*a[7,6]-27920033028864298274947303302255034041302608951102933138723 250777062841270336161*a[8,6]-16870636309720744802484087642453111329083 66487232791304089009632617626853937500*c[3]*a[9,5]-4880228572262835596 99714951388655337022180287760001970499078929838107706000000*a[9,6]+812 0044179059171833659122720584349305074932519031965559564506563692884520 00000*c[3]*a[9,6])/c[3]/c[4], a[9,1] = -1/1542333543875107337192653686 030896686286160146237384237739513839035414706250000*(19351486475437945 4009319979145869959719517127616398681266321082079597729531500575*c[3]* c[4]*a[7,6]+1542333543875107337192653686030896686286160146237384237739 513839035414706250000*c[4]*c[3]*a[9,5]+1006986682205670940428035952501 6334816017191828898350765432243303030015928175080*c[3]*c[4]-4574853401 9749354595733866767303481191568837749401179882553251862095217427196250 *c[4]*a[7,6]-926958038995645318817807013321599523576025542435599617631 323973965729040625000*c[4]*a[9,5]-456385248786818675435085607353044081 10763359600670846932710099077085259240065343*c[3]*c[4]*a[8,6]-15965055 7000386632479034722046270560393961481681308933289118069720977034923600 0*c[4]+171454630752560332005170899399756188359647566280205471763329594 29999805563554850*c[4]*a[8,6]-5968295810440114592809054140148743170205 42918027559161734946192174471165000000*c[4]*c[3]*a[9,6]+29969083022945 6613907387979197946069923122995069167338056965082964578100000000*c[4]* a[9,6]-153406774883869770741468699462939747479684664566499632630347531 74210352346338550*a[8,6]+557111144648897944137974922147830016694682017 928466436859230065161220989062500*a[9,5]-26814442704740854928555766559 8162273089110048219781302472021390020938300000000*a[9,6]+5340054146183 26042514494317802782283649959452972026618394425540366632095000000*c[3] *a[9,6]+40834469628294302538928712236851312520156690169021284097687983 384760495109532149*c[3]*a[8,6]+301044598004900088933497080168249755079 9218480834384486690190301488828608000-92695803899564531881780701332159 9523576025542435599617631323973965729040625000*c[3]*a[9,5]-17314487899 0760564113602086604199437643778482604146188501445178702797968528184725 *c[3]*a[7,6]+409328988597757383224987228970610094871931706178852662107 05541139769405066438750*a[7,6]-172796521322819911835628783004423081299 4219713278232404156340589380048585311640*c[3])/c[3]/c[4], b[1] = 18530 703372187/317986769215500, b[5] = 397406378366413672608/80368036816595 0784145, a[6,5] = -6534/6859*(16473*c[3]+41154*c[4]*c[3]*a[6,1]-36822* c[3]*c[4]-9826+16473*c[4])/(198*c[3]-119)/(198*c[4]-119), a[5,3] = -14 161/23287176*(-119+297*c[4])/c[3]/(-c[4]+c[3]), a[4,3] = 1/2*c[4]^2/c[ 3], a[3,2] = 100*c[3]^2, a[8,7] = 2789843676680233784205/2451345287320 98032152859876183406860406499163237177934671*(136426814567036613952595 3704001822575312800+13547861062143029882850308869543074941942683*c[4]* a[8,6]-12121770424022710947813434251696435474369769*a[8,6]-63356295873 75138659856595162111221617167440*c[4]+68877980492164986429617611298562 601836732225*a[7,6]-76981272314772631891925565568981731464583075*c[4]* a[7,6])/(47245149*c[4]-9526409), a[8,3] = -1/2279098607746726950653978 28670134485684567084833523174635088519234565828*(-92652796989314713357 637677545646513568278001837890622650950715159971610542*c[3]*c[4]*a[8,6 ]-14858270756284398782290429035540776257529356969309972997865172622897 3165775*c[4]*a[7,6]+24722164787767318982298360916277930243620274621204 8290216580183137283082550*c[3]*c[4]*a[7,6]+556852668774164186341357758 98646136942550920296510020684157248000184957851*c[4]*a[8,6]+5009211744 687407501897168707095815797295055768084770275521506391312375000*c[4]*c [3]*a[9,5]+31967185511253713057109347261539118971782280993253346468309 341422741421100*c[4]*c[3]*a[6,1]-1619506777444935073011491861110546868 147522512014835239928556759532000000*c[4]*c[3]*a[9,6]+4549979983544294 569782935119666389325889285482706075103849778383519129720*c[3]*c[4]-71 2907684807434923924090707627006634303716539482466860829752117291272136 0*c[4]-301058685665556309457456099062829333271773553738428112518716798 2657437500*c[4]*a[9,5]+97333993189872360448670470440482362277553120671 5986836118678052446000000*c[4]*a[9,6]+78865619290322829233944733590292 7415412669970842412287014308307072656480+18093931108182424659311755448 72560134310154186609744716652893888566843750*a[9,5]+132942422556228831 209966996633785892830525825514878705770372597152239148325*a[7,6]+82899 870990439480372623185172420564771617159539165293950850639879974598906* c[3]*a[8,6]-1216009375600489744312082395931676132351760719317045405562 419929260437010*c[3]-2211983165221286435258274397772235863902866676634 11628088519111228095389650*c[3]*a[7,6]-4982365983768837456738464159352 5490948598191844245807980561748210691804393*a[8,6]-3010586856655563094 574560990628293332717735537384281125187167982657437500*c[3]*a[9,5]-870 883096962015856645998946046421136167580553377461906000922467978000000* a[9,6]+144903237981915243374712429678312088202673066864485258309397183 7476000000*c[3]*a[9,6])/(198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3])/c [3], a[9,3] = 1/323801664168546057006927518235023645591147754885893359 7622176077125000*(-359956476259963921358343735608572313785073822214027 28302622655462304433*c[4]*a[8,6]+9604570624618228042850956066930042829 6893063796444557193698594847429325*c[4]*a[7,6]-62917901221636949223445 6822498269956545269041186804677516921818000000*c[4]*a[9,6]+19460807088 91766706253756296463020900270029435930369182409287642312500*c[4]*a[9,5 ]+33517468544629278982045537210562880597287284575186501711005284304080 80*c[4]-63202114879783156262448332875115850071085935546844240819183604 26240+3220663208641782454258865002813541754919081567178138848129395488 7325019*a[8,6]-8593563190447888248866644901990038321300958339681881433 1204005916120975*a[7,6]+5629496425093832298939876832879257505931354579 03983132515140574000000*a[9,6]-116961416342484968709190403676312872288 9563145836939054074268835531250*a[9,5])/(-c[4]+c[3])/c[3], a[2,1] = 1/ 200, a[10,8] = -105377105176463/55781254832038388478751220237857265073 4368718301544315590190257476920*(-359640354005295035279147639470165222 95247534385586263120+3783295947265530583617913211376369588688415132161 9900576*c[4]-291836171688416228135068249643347025208895840378777500000 *a[9,6]-1856916193213350460832858736046610174215135955020741598793*a[8 ,6]-1758076336047996976967140480534921175785426457259646060275*c[4]*a[ 7,6]+1573015669095576242549546745741771578334328935442841211825*a[7,6] +326169838945876960856840984895505498762883586305692500000*c[4]*a[9,6] +2075376921826685809166136234405034900593387243846711198651*c[4]*a[8,6 ])/(-25+29*c[4]), a[10,7] = -27710405580754394328019718781/13224541999 3550185298968146018572499491284783980701288898324137391615233182737379 501757128*(50773779611854110755239133395059371081084199531966438439755 0625681680+72110830744647671550885159230664311169738797787135256848505 92834981803*c[4]*a[8,6]-6452021698205317980868672141691017315187155591 480522981182109378667929*a[8,6]-23699684417771535587285625742326222909 95184845543862758831962615302720*c[4]+34924473037929676230269269920357 687992808123203228157166936230126936225*a[7,6]-39033234571803755786771 536969811533639020843580078528598340492494811075*c[4]*a[7,6]-564612187 84461061684901799018335575329950759226598915846798012500000*a[9,6]+631 03715112044716000772598902845643015827319135610553005244837500000*c[4] *a[9,6])/(47245149*c[4]-9526409), a[5,1] = 119/23287176*(14161-35343*c [3]-35343*c[4]+117612*c[3]*c[4])/c[3]/c[4], a[9,7] = -2686516133099484 38479/3799697632667533641792030544873193952696701432521911387456095794 930784652000000*(10769087321847282437463336378017127115075471955158054 67754203039160-5016640108121378302587813686873249123390891889327683909 638413250920*c[4]-9651490390506164390581504105698388945290728072922891 5977432500000*a[9,6]-1302932581661704743675204012003397808728750561766 5706488415747801611*a[8,6]-7942625645475576884950973012953860363235327 5660914127619758885784425*c[4]*a[7,6]+71065597880570951075877126958008 224302631878222923166817679003070275*a[7,6]+10786959848212771965944034 0004864347035602254932667611974777500000*c[4]*a[9,6]+14562187677395523 605781691898861504921086035690332260192935247542977*c[4]*a[8,6])/(4724 5149*c[4]-9526409), b[10] = 1754022361907/1430294620800, b[9] = -68588 512868559375/32259978418667428, a[10,4] = -1/5008736216204618313902734 665967779784161747857482155965040540568013429751736709100*(-3064806927 8111894137889964482059331240927406094450669659635431816291554672530048 77276350470760*c[3]*c[4]^2+7311003993238768949295617128429061601394285 97304292770878922577373059469193892393433076000000*c[3]*c[4]^2*a[9,6]- 2147732021039342921928756071563286694767716689965707095870800588995674 03296552298993320037000*c[3]+44505800532041840937739933470375075600632 305128044114901112848784029318227786546849273726190*c[4]+5698619663311 5412245117720212919707838563199526636635229878025690336533226251254900 844236288850*c[3]*c[4]^3*a[7,6]-12713758635815817644432674071707382606 9719395979047157593966992422902809842634170857621273746150*c[3]*c[4]^2 *a[7,6]-21357071895474761993051588912481529418839411785374133466950830 407291015299779658439468053095034*c[3]*c[4]^3*a[8,6]+41826621426204723 4456774369567829740335081229534591524523043163078811800745884081060928 03477506*c[3]*c[4]^2*a[8,6]-373306838055725544837175051725927059782718 030645002900787767457243917228939548592105764000000*c[3]*c[4]^3*a[9,6] +115465586412122408514149056979428911570808854618848907944587688056202 9380179226140155284125000*c[4]^3*c[3]*a[9,5]-1922175118797956990022125 024463303376783829002168706741287147030832458450978847896935953062500* c[4]^2*c[3]*a[9,5]+439190703685873076380355103371926794372423178854661 200541381352867924602613391039906245862300*c[4]^3*c[3]*a[6,1]-46717013 3041757232304745414115547393248268258853552398052235024997008168594860 287546553131800*c[4]^2*c[3]*a[6,1]-26329172717176400730324639030920113 155648554740594282470958376504494041535925289963158643234192*c[3]*c[4] *a[8,6]+72498683256207753221728115380681017895282777919748159321702408 514360234876312298699482031250*a[9,5]-34894505468345840015662493512101 954407078794872267880935527709912196522494472589644350000000*a[9,6]-19 9633220201889963041039351207222200471454528703895359783187332261797434 0704971815969117030475*a[8,6]+5326731115859286927838393425127460019296 695828063313551661447633646034066156600690931773149375*a[7,6]+11552466 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2406179148092250176982298572290942480904008135565186792419575233884000 +238514879426808086485554954348107822590783439994741792183280465095928 687734036922443246000000*c[4]*a[9,6]-564275561422116846568492975202651 522761409161599102935741011526802043822968599988624142125000*c[4]*a[9, 5]+1562696524766438181539321871900558499670204636292117056226302536581 052428951085691762713629720*c[3]*c[4]+37537727426314048923568937941814 850874930645672923849745243080753484485712577140581381767065575*a[7,6] *c[4]^2-18715947276065218363023352291320568843017941316027674666570232 836652639571772580436918432714425*a[7,6]*c[4]^3-2455456318173719731735 7169703502380173453568681540404889079108289947373231930997700746483779 525*c[4]*a[7,6]-228581357308200638348269175290562173497281513315070110 82360289202314045658657347315982783692500*c[3]*a[7,6]-4201804999520717 2648155580424662324821072119975292038828056431860199407288538314988313 2000000*c[4]*c[3]*a[9,6]+938878665223354080845055538572479004258479109 215314128375800691653820814687250401240169250000*c[4]*c[3]*a[9,5]+7634 2635733178740408107592896929385995517242431227393875751542901757485847 686454851185917500*c[4]*c[3]*a[6,1]+1238381741354728577244233040893970 4172051914137528867742045134177273897367313534328747043711901*c[4]*a[8 ,6]+936791315554689172638049070693835608165017330376116745590385890731 87852431760238171033665712800*c[3]*c[4]*a[7,6]+52160167556961659592643 605699177177441455033607660191942308497934898227519779840604342402830* c[4]^3-120628061216211219646236696179620517170302437211009542400815772 149943920214368362541995312500*c[3]*a[9,5]+720776106929745308932475301 43228941626479100701018144266829867436139495527282752877700000000*c[3] *a[9,6])/(-25+29*c[4])/(47245149*c[4]-9526409)/(-c[4]+c[3])/(198*c[4]- 119)/c[4], a[7,3] = -1/99617150321540375206332626186079950162170804272 007072621858865121630900*(77742817583948041228613386752813559538116486 8093400279343397111565218300*c[4]*c[3]*a[6,1]-124577444418841159462422 450854657451395963270154987326148350519964000000*c[4]*c[3]*a[9,6]-2315 83604358120238044196999279099487132133502875713932706705229435187500*c [4]*a[9,5]-11429439043295691370992637719646750967330274591776902306050 132786844089675*c[4]*a[7,6]+190170498367440915248448930125214848027848 26631696022324352321779791006350*c[3]*c[4]*a[7,6]-60508372923235082490 0087229745251403049564090395459827685879511808550760*c[3]*c[4]-7127138 229947285642895205965049731812944461679837740203919285781536277734*c[3 ]*c[4]*a[8,6]+74872302453747969575900361877294124828887015901229756624 513696342000000*c[4]*a[9,6]+256789259188754925372272530293164255622581 742150211360037067040621501680*c[4]+4283482067493570664164290453742010 534042378484346924668012096000014227527*c[4]*a[8,6]+385323980360569807 838243746699678138253465828314213098117038953177875000*c[4]*c[3]*a[9,5 ]+13918408544755711276393658037481231802385801435459574743483799142821 8750*a[9,5]+1114640292168578795190095612910092986174408206649886602379 97833652000000*c[3]*a[9,6]-2619482620030024931871944283026401336161599 72999304866387334411866743840+6376913153110729259432552705570812674739 781503012714919296203067690353762*c[3]*a[8,6]+102263401966329870161513 07433368145602348140424221438905413276704018396025*a[7,6]-231583604358 120238044196999279099487132133502875713932706705229435187500*c[3]*a[9, 5]-3832589218283721120568049353348114688353707064941985229273980631591 677261*a[8,6]+43818643709425672162984323047763900099575433607970560588 2782907743578230*c[3]-669910074586166043573845343112631643205831194905 73992769301728306000000*a[9,6]-170152551170868187327559569059402758761 75897512570125237578393171391953050*c[3]*a[7,6])/(198*c[3]-119)/c[3]/( -c[4]+c[3]), c[7] = 9526409/47245149, a[4,1] = 1/2*c[4]*(2*c[3]-c[4])/ c[3], c[10] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[6] = 3832801447177714 39/119491812371674080, b[8] = -16070516558763250309/737333295852729600 0, a[6,4] = 17/13718*(96026*c[3]*a[6,1]-21964*c[3]+289)/(-c[4]+c[3])/( 198*c[4]-119), a[10,6] = -77593657404468797238358266009922384004409162 14471301/29830948869806602777180491475003365613724027019957964*a[7,6]+ 2394330561195458751842358107/1347186551150289146484386820*a[8,6]+24525 445251143012876977500000/14146180885244972240022216299*a[9,6]+12022007 2203788163018766520/436648564509816360719190147, a[10,3] = 1/235194573 3799947400089876606361359814164276032948098955214929552777204700*(-731 9570962155862355253376526106074571893962145193359189425106497637757232 818*c[3]*c[4]*a[8,6]+1505209860827906621430777874367835337015708727563 12114827602263255424837100*c[4]*c[3]*a[6,1]-12794103541814987076790785 7027733202583654278449171983954355984003028000000*c[4]*c[3]*a[9,6]-237 8363616757894844713903182596351732847011074533582088897862706299375625 00*c[4]*a[9,5]+1953051018233618199601570512385956489246001695075181492 7109834467845363521450*c[3]*c[4]*a[7,6]+615027024258736837169432172179 916130788113143106740846349025504154303863480*c[3]*c[4]-11738033897464 675038009438938077213243448192005754878668313486372088880096225*c[4]*a [7,6]+7689385461999916475444967164798106619926696533056296005337556614 3234000000*c[4]*a[9,6]-39032940407370906900276335409539791354931850234 2343138568563844423634462640*c[4]+439913608331589707209672629599304481 8461522703424291634048422592014611670229*c[4]*a[8,6]+39572772783030519 2649876327860569447986309405678696851766199004913677625000*c[4]*c[3]*a [9,5]+1429420557546411548085628680449322506105021807421698326155786171 96780656250*a[9,5]+105024513819420776655873927340690855336115402156754 17755859435175026892717675*a[7,6]+114473558005713042266022819445866549 680111722822943354064423775160604000000*c[3]*a[9,6]-695963673362670896 8398907220108269463622944687314332009189169196276947680+12032907970745 182286140145009914640226401776989030058469186225593442389710*c[3]+6549 0898082447189494372316286212246169577556035940582221172005505179933135 74*c[3]*a[8,6]-3936069127177381590823386685888513784939257155695418830 464378108644652547047*a[8,6]-68799764659999252675033916737667269757238 863716819490574072874970262000000*a[9,6]-23783636167578948447139031825 9635173284701107453358208889786270629937562500*c[3]*a[9,5]-17474667005 248162838540367742400663324832646745409518618993009787019535782350*c[3 ]*a[7,6])/(198*c[3]-119)/c[3]/(-c[4]+c[3]), a[4,2] = 0, a[10,5] = 711/ 7477562635759367604690075838242203919934631422792700836478319371681236 925*(54564450790579780470022486711201167657605818153074792914861473867 4418077298*c[3]*c[4]*a[8,6]-327937860812070397774377571648128229861368 301021005068528712898496241167669*c[4]*a[8,6]+107516699740743476900995 9163790932223700227980229498477727144604494804745325*c[4]*a[7,6]-17889 3332341741247280648667588743344783735411836504788731071119067202806365 0*c[3]*c[4]*a[7,6]+582965779816222792734897174948880996908179083522294 481290485591768184875000*c[4]*c[3]*a[9,5]+2217397919311051353709450708 82747119260383357377306501897166709487546152900*c[4]*c[3]*a[6,1]+42630 200691083221868255126330551052925565340325393966979052782677828000000* c[4]*c[3]*a[9,6]-20234577063539514772482576579448889572082739849155705 8403590322464858398780*c[3]*c[4]+9112953307606397899991443250051169978 7605939480810682787966599168180747290*c[4]-350368322212780365330569514 236953730465016721914914359967514067779868687500*c[4]*a[9,5]-256211812 23428805062234141582502905546173108579403444800541823932634000000*c[4] *a[9,6]+16006245525313690546163301836887562428018431585371481096990573 81127604056950*c[3]*a[7,6]+2105749007238427448198877383545327976027120 70241791963818859464978809968750*a[9,5]-961991523996125845956279251812 939358047572403363235480071655698758509508975*a[7,6]-52450647243092276 779373685744868766853626787970314940210889402618578023420-350368322212 780365330569514236953730465016721914914359967514067779868687500*c[3]*a [9,5]+2934180859897471980086536167377989425075400588082676928941115407 59794728967*a[8,6]+229242147788573518977884424685552312781548866236767 66400484789834462000000*a[9,6]-488208243915713825258095933731799921146 999425580142883975076345129742490214*c[3]*a[8,6]+879381732497350658683 42152983550206285516186397677354605426093801678511490*c[3]-38142811144 653409040017744611545678933400567659563023086520910817004000000*c[3]*a [9,6])/(198*c[4]-119)/(198*c[3]-119), a[10,9] = -545009894469844730599 500000/14146180885244972240022216299, b[7] = 3273993983310919714698661 0130768938398484878542073/10263821751054164171380170536402916741850195 0476800, a[10,1] = -1/334006888380743352229894856525412157320995725716 741095473361237223917745154686458404353300*(-1416212692010297122552234 9025446980643509046789973120567480526465093920062997973362148133488*c[ 3]*c[4]*a[8,6]-1400801262302096216317362976850127894347920010148137819 8140831147304449447332237111100856725*c[4]*a[7,6]+17962973113689115390 1429630345716202342393511602887985590003630357076437288674011414555100 *c[4]*c[3]*a[6,1]+4722561763877374491340556765905064241170515584336112 58435225366004723602618782714450125000*c[4]*c[3]*a[9,5]-28383073227343 8163873498108657930628636005734614140099766625346235162165210278500098 812500*c[4]*a[9,5]-189546657877718489546256480653474033342990022604753 597379898613326872029760328346668000000*c[4]*c[3]*a[9,6]+6011136039938 8403164250717377448876075409671321610132950151625583940050867057383945 145036700*c[3]*c[4]*a[7,6]+5249870150637936744331830689186466171567662 346538424686339874481625122833687815156181761049*c[4]*a[8,6]-465813889 0348370854672578178166692494229292512822306069899962231855565160575556 48892073840*c[4]+91764097425407745369908287436669499478823128383818648 826993285582246910366087086954000000*c[4]*a[9,6]+300962513775160371482 9614381631064570118871837471453143250661448705387255495155956783484600 *c[3]*c[4]-53783848778400150199592747127191099646419179603545908429083 033417209519196840817214077138100*c[3]*a[7,6]+125334849784924398302079 84529711670633639284301325443650968112079167138979192001625721819175*a 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674104156138335578702144352168749863358295263210068301595714737*a[8,6] -126646389072146134369811048643506763353375669891737013655922944286446 5392755614925*a[7,6]-1422356246201337200820762256118213221199559904063 15432017355404228680050645687500*a[9,5]*c[4]^2-11780847783130369775250 90777222754027172276116894570923926649041173981007111236194*c[3]*a[8,6 ]-53955833312874709725131379000603767156778830863953932692428916676653 428632686290*c[4]^2+45985590110205049603791432278606848376749765415446 598754465526270841084454000000*a[9,6]*c[4]^2+2630858730728303433416121 218562145638125222669495740107216091121198690361229214799*a[8,6]*c[4]^ 2-50417435964851274121897444536190578898408057681260678430320075803992 385136000000*c[4]*a[9,6]+114165128846706643385268761354830578816385477 042308646159741916801029585945406250*c[4]*a[9,5]-383653979007546605292 069439179404882424255438574682851231762529267503751395336320*c[3]*c[4] -383606414551837723501282059309792792842620515467124425318777564492146 8322438614975*a[7,6]*c[4]^2+484772756224538255725610847371009469665900 7609935480521576312663332434526221475900*c[4]*a[7,6]+49558422757169094 91388872659856688720685061028238171659565419345130634720043968350*c[3] *a[7,6]-75130614539790501331971421557716708565339979242628939927236897 85271718344980320+8388783463059287627004784889214903043600668420915642 2934482142934373884512000000*c[4]*c[3]*a[9,6]-189955424467629541094816 930657617265593649785330900100333015962408435781657062500*c[4]*c[3]*a[ 9,5]-30453248375907697335066399973645161682485746769465761907582032491 3676678639829900*c[4]*c[3]*a[6,1]-249606015909444934147988678275301469 4802563021176702905157150714913179607440798976*c[4]*a[8,6]-15989429966 0854401685684604192268131272655927980316737192419039850531784749823313 00*c[3]*c[4]*a[7,6]+28680081726525266164222948992170289259442230282957 452169513163754499656516937500*c[3]*a[9,5]-138040751044005921172205086 24993393818627385282354341451259661158277903684000000*c[3]*a[9,6])/c[4 ]/(198*c[4]-119)/(-c[4]+c[3])/(47245149*c[4]-9526409), a[7,4] = 1/9961 7150321540375206332626186079950162170804272007072621858865121630900*(- 2315836043581202380441969992790994871321335028757139327067052294351875 00*c[4]*a[9,5]+7774281758394804122861338675281355953811648680934002793 43397111565218300*c[4]*c[3]*a[6,1]-71271382299472856428952059650497318 12944461679837740203919285781536277734*c[3]*c[4]*a[8,6]+42834820674935 70664164290453742010534042378484346924668012096000014227527*c[4]*a[8,6 ]-62457599053728771099096505093821751673403062863762018650484187111935 73175*c[4]*a[7,6]-1245774444188411594624224508546574513959632701549873 26148350519964000000*c[4]*c[3]*a[9,6]+19017049836744091524844893012521 484802784826631696022324352321779791006350*c[3]*c[4]*a[7,6]+3853239803 60569807838243746699678138253465828314213098117038953177875000*c[4]*c[ 3]*a[9,5]+443876416174302497201043397985378205054931232465776180570262 856021646830*c[4]+7487230245374796957590036187729412482888701590122975 6624513696342000000*c[4]*a[9,6]-60508372923235082490008722974525140304 9564090395459827685879511808550760*c[3]*c[4]+1391840854475571127639365 80374812318023858014354595747434837991428218750*a[9,5]-231583604358120 238044196999279099487132133502875713932706705229435187500*c[3]*a[9,5]- 2619482620030024931871944283026401336161599729993048663873344118667438 40+1022634019663298701615130743336814560234814042422143890541327670401 8396025*a[7,6]-2219893425500963299383894411620485167616586581797082567 8580107247042469550*c[3]*a[7,6]+25109928010870914980107236278542505156 3404845764140785349587092343433080*c[3]-383258921828372112056804935334 8114688353707064941985229273980631591677261*a[8,6]-6699100745861660435 7384534311263164320583119490573992769301728306000000*a[9,6]+6376913153 110729259432552705570812674739781503012714919296203067690353762*c[3]*a [8,6]+1114640292168578795190095612910092986174408206649886602379978336 52000000*c[3]*a[9,6])/c[4]/(-c[4]+c[3])/(198*c[4]-119), c[9] = 44/45, \+ a[7,5] = -1/3023783513994312990907453962897539554458301629519651474849 81250624275*(-71271382299472856428952059650497318129444616798377402039 19285781536277734*c[3]*c[4]*a[8,6]-22036044048583911320593211549880421 450386671278212181669945947741636684975*c[4]*a[7,6]-124577444418841159 462422450854657451395963270154987326148350519964000000*c[4]*c[3]*a[9,6 ]+42834820674935706641642904537420105340423784843469246680120960000142 27527*c[4]*a[8,6]+3853239803605698078382437466996781382534658283142130 98117038953177875000*c[4]*c[3]*a[9,5]+77742817583948041228613386752813 5595381164868093400279343397111565218300*c[4]*c[3]*a[6,1]+366650144673 91717995608872998960701236777822799042117400414265990286248950*c[3]*c[ 4]*a[7,6]-100605504822677333110477122118322089945526382558640238626832 9642402437860*c[3]*c[4]+4977770721197462296064007877735600640684315829 97292998781064846382474230*c[4]-23158360435812023804419699927909948713 2133502875713932706705229435187500*c[4]*a[9,5]+74872302453747969575900 361877294124828887015901229756624513696342000000*c[4]*a[9,6]-294343100 677284332360111243377355493326294022056125580362513385871483540+139184 085447557112763936580374812318023858014354595747434837991428218750*a[9 ,5]+197164604645224469710570840183140612977143900910319520204779532425 17033925*a[7,6]+637691315311072925943255270557081267473978150301271491 9296203067690353762*c[3]*a[8,6]+49208709303970045403520062026582086000 9254686611222424093584898104405630*c[3]-328055392602978529434395179464 38522159222262504406105042475922201835064850*c[3]*a[7,6]-3832589218283 721120568049353348114688353707064941985229273980631591677261*a[8,6]-23 1583604358120238044196999279099487132133502875713932706705229435187500 *c[3]*a[9,5]-669910074586166043573845343112631643205831194905739927693 01728306000000*a[9,6]+111464029216857879519009561291009298617440820664 988660237997833652000000*c[3]*a[9,6])/(198*c[4]-119)/(198*c[3]-119), c [8] = 25/29, c[5] = 119/198, c[6] = 17/19, a[7,1] = 1/1185444088826330 4649553582516143514069298325708368841642001204949474077100*(-231583604 358120238044196999279099487132133502875713932706705229435187500*c[4]*a [9,5]+7774281758394804122861338675281355953811648680934002793433971115 65218300*c[4]*c[3]*a[6,1]-71271382299472856428952059650497318129444616 79837740203919285781536277734*c[3]*c[4]*a[8,6]+42834820674935706641642 90453742010534042378484346924668012096000014227527*c[4]*a[8,6]-1142943 9043295691370992637719646750967330274591776902306050132786844089675*c[ 4]*a[7,6]-124577444418841159462422450854657451395963270154987326148350 519964000000*c[4]*c[3]*a[9,6]+2481057357912841334605529048281718716747 9497090673275758413061040812171850*c[3]*c[4]*a[7,6]+385323980360569807 838243746699678138253465828314213098117038953177875000*c[4]*c[3]*a[9,5 ]+25678925918875492537227253029316425562258174215021136003706704062150 1680*c[4]+748723024537479695759003618772941248288870159012297566245136 96342000000*c[4]*a[9,6]+1384248607454776954264190033739103881897213096 713838604355597839369593240*c[3]*c[4]+13918408544755711276393658037481 2318023858014354595747434837991428218750*a[9,5]-2315836043581202380441 96999279099487132133502875713932706705229435187500*c[3]*a[9,5]-2619482 62003002493187194428302640133616159972999304866387334411866743840+1022 6340196632987016151307433368145602348140424221438905413276704018396025 *a[7,6]-22198934255009632993838944116204851676165865817970825678580107 247042469550*c[3]*a[7,6]+251099280108709149801072362785425051563404845 764140785349587092343433080*c[3]-3832589218283721120568049353348114688 353707064941985229273980631591677261*a[8,6]-66991007458616604357384534 311263164320583119490573992769301728306000000*a[9,6]+63769131531107292 59432552705570812674739781503012714919296203067690353762*c[3]*a[8,6]+1 1146402921685787951900956129100929861744082066498866023799783365200000 0*c[3]*a[9,6])/c[3]/c[4], c[2] = 1/200, a[5,4] = 14161/23287176*(-119+ 297*c[3])/c[4]/(-c[4]+c[3]), a[8,5] = -135/600436864599601795403265969 22624312116947107350890982500419861*(-14517908884373180100835500962220 927352599734436958439423890833462*c[3]*c[4]*a[8,6]-3032738285833979228 4420934998081222959474516499816433095560801655*c[4]*a[7,6]+20552198442 095380399353667591588955981015413698224909051365858620*c[4]*c[3]*a[6,1 ]+3220499777155791161582542182445465906106903882046000644774475000*c[4 ]*c[3]*a[9,5]-19355528963714098395369824227828810243773816260781519026 67487500*c[4]*a[9,5]-1041205978444654479724644096446147332773044822106 321716834400000*c[4]*c[3]*a[9,6]+5046068744496873001945668176151329534 4335750142551712209420493510*c[3]*c[4]*a[7,6]+872540988505256783838093 2396486314924037214131303304502237420111*c[4]*a[8,6]+72616717434253539 86761151747164150707454277315578531740870729558*c[4]+62577531027734284 3874912360995411780808042090053799415673200000*c[4]*a[9,6]-16783343997 034071092664231996525809393535098325419126332379220516*c[3]*c[4]-45149 036134972021596355978418196106360721460653862058292639388930*c[3]*a[7, 6]+27135026767988235201850310261441094226898251605098913822343875165*a [7,6]+1163286841758574600529802567228095161115699058097475133421368750 *a[9,5]-19355528963714098395369824227828810243773816260781519026674875 00*c[3]*a[9,5]-4372066921315521826809143510611768452743145942346726011 885265684-559904224984990965572290007206421067038774501627083687707600 000*a[9,6]+12989707949176003248115974545145040262852393969910182642428 640466*c[3]*a[8,6]+733638544475948770681914275866142869909703559627088 2328141331358*c[3]+931605349134690850279944717872868666165355893463551 009799200000*c[3]*a[9,6]-780694568662598175013030793369828177413856001 2218746133580849573*a[8,6])/(198*c[4]-119)/(198*c[3]-119), a[6,3] = -1 7/13718*(96026*c[4]*a[6,1]+289-21964*c[4])/(198*c[3]^2-198*c[3]*c[4]+1 19*c[4]-119*c[3]), a[9,8] = -9194653/937589087293698923940281796670234 7070117654438557121300000000*(-391397289956719451706507952040835990249 894761618264280+22318457243248091068216550503428640289545860626057160* c[4]-32426241298712914237229805515927447245432871153197500000*a[9,6]-2 06324021468150051203650970671845574912792883891193510977*a[8,6]-195341 815116444108551904497837213463976158495251071784475*c[4]*a[7,6]+174779 518788397360283282971749085730926036548382537912425*a[7,6]+36241093216 208551206315664988389499862542620700632500000*c[4]*a[9,6]+230597435758 520645462904026045003877843709693760745688739*c[4]*a[8,6])/(-25+29*c[4 ]), a[3,1] = -100*c[3]^2+c[3], a[9,4] = -1/647603328337092114013855036 4700472911822955097717867195244352154250000*(9184519213864175924561996 435728908472590671182248510261526157661055220483427360*c[3]*c[4]^2-172 4088058430441420180932819753844347673876143565547981744667436140131156 000000*c[3]*c[4]^2*a[9,6]+17279652132281991183562878300442308129942197 13278232404156340589380048585311640*c[3]-11303755921442359100776377049 4141278251327224943799922074128214211880579402360*c[4]+263186234739844 526160273646421247030717730684095512668941608094530130982142476650*c[3 ]*c[4]^2*a[7,6]-986359446556183575553555466558529069238932868977071274 84507617989349735585839986*c[3]*c[4]^2*a[8,6]+533268663734179428879148 0986092848534555557494225142247580228471404940489625000*c[4]^2*c[3]*a[ 9,5]+13389173851791934535619510248001490377950998471967196099779586254 1240285816869541*c[3]*c[4]*a[8,6]-557111144648897944137974922147830016 694682017928466436859230065161220989062500*a[9,5]+26814442704740854928 5557665598162273089110048219781302472021390020938300000000*a[9,6]+1534 0677488386977074146869946293974747968466456649963263034753174210352346 338550*a[8,6]-40932898859775738322498722897061009487193170617885266210 705541139769405066438750*a[7,6]-32049985345640076786171022088133786647 07633039458545088192157515642363223562500*a[9,5]*c[4]^2-40834469628294 302538928712236851312520156690169021284097687983384760495109532149*c[3 ]*a[8,6]+6142763637507463655622873729983136774628364585140126988695558 5116524800553320*c[4]^2+1036194338147588530310762654296502410975713439 819698029432401135861998018000000*a[9,6]*c[4]^2+5928119906069992196508 7424505285332949208591620339132175032356266326356235934133*a[8,6]*c[4] ^2-1406564937978751718286275668544175122442007352324932569988312413751 010733000000*c[4]*a[9,6]+340917869896859030366307005522025753032068720 9295766832439601285623536367625000*c[4]*a[9,5]-98483818363894174890303 76161157629597783917281731972081963029603910209839829160*c[3]*c[4]-864 3811982908542332196720839148400149522917597938312614020628881330425901 3768325*a[7,6]*c[4]^2+123087904393141575462757158485999693055721258362 533450639579931326630607071094225*c[4]*a[7,6]-301044598004900088933497 0801682497550799218480834384486690190301488828608000+17314487899076056 4113602086604199437643778482604146188501445178702797968528184725*c[3]* a[7,6]+234033493882178857328304691068694684238249962823812309964441897 4140337186000000*c[4]*c[3]*a[9,6]-567241497811580571533855353725723521 8516773675971107838849084492045884040250000*c[4]*c[3]*a[9,5]-701865359 1904017390612162765523091673788823334095556017504612029987075587991696 9*c[4]*a[8,6]-42899728531108245110009113647014361878274984496501527979 3023061396030713553716525*c[3]*c[4]*a[7,6]+926958038995645318817807013 321599523576025542435599617631323973965729040625000*c[3]*a[9,5]-534005 4146183260425144943178027822836499594529720266183944255403666320950000 00*c[3]*a[9,6])/(-25+29*c[4])/(47245149*c[4]-9526409)/c[4]/(-c[4]+c[3] )\}:" }}}{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 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a[ 8,5]=subs(e9,a[8,7]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" )\"\"&,$*&#\"70UyL-owO%)*y#\"ZrY$z%\\2V&p)3.&G))HI9i5'ya8F.&% \"cG6#\"\"%F.&F%6$F'\"\"'F.F.*&\"Mp(pVZNkp^UV8y%4rASUq<77F.F8F.!\"\"*& \"LSu;<;A6@;&fc)f'Q^P(eHcL'F.F4F.F=*&\"MDAtO=gi&)H6whHk)\\;#\\!)z()oF. &F%6$\"\"(F:F.F.*(\"MvIek9t\")*obcD>*=jsZJs7)p(F.F4F.FBF.F=F.,&*&\")\\ ^CZF.F4F.F.\"(4k_*F=F=F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can find some simple order conditions that \+ are not yet satisfied and determine which parameters are related by th em." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for ii from 64 by -1 to 47 do\n eq := simplify(sub s(e9,SO7_10[ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indets (lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"# b<)&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$F/\"\"'&F-6$\"\")F3&F -6$\"\"(F3&F-6$F3\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#^<'&%\"c G6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"'&F-6$\"\")F0&F-6$\"\"(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#]<'&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\" \"'&F-6$\"\")F0&F-6$\"\"(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<)& %\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$F/\"\"'&F-6$\"\")F3&F-6$ \"\"(F3&F-6$F3\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Z<'&%\"cG6# \"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"'&F-6$\"\")F0&F-6$\"\"(F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a s ystem of equations that consists of the simple order comditions given \+ in abreviated form as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SO7 := SimpleOrderCondition s(7):\n[seq([i,SO7[i]],i=[50,51])]:\nlinalg[augment](linalg[delcols](% ,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7%\"#]%#~~G/*(%\"bG \"\"\"%\"cGF--%!G6#*(%\"aGF-F.F--F06#*&)F.\"\"#F-F3F-F-F-#F-\"$0\"7%\" #^F)/*(F,F-F7F--F06#*&F3F-F4F-F-#F-\"#%)Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "cdcns4 : = [seq(SO7_10[i],i=[50,51])]:\neqns4 := simplify(subs(e9,cdcns4)):\nno ps(%);\nindets(eqns4);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%\"cG6#\"\"$&F%6#\"\"%&%\"aG 6$\"\"*\"\"'&F,6$\"\")F/&F,6$\"\"(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We can solve this system of 2 equations for the linking coefficien ts " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "a[9,6];" "6#&%\"aG6$\"\"*\"\"'" }{TEXT -1 29 " \+ in terms of the parameter " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"( \"\"'" }{TEXT -1 88 " and subsitute back into the expressions previous ly obtained for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "e10 := solve(\{op(eqns 4)\},\{a[8,6],a[9,6]\}):\ninfolevel[solve] := 0:\ne11 := `union`(map(u _->lhs(u_)=simplify(subs(e10,rhs(u_))),e9),e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[8,6]=subs( e10,a[8,6]);\n``;\na[9,6]=subs(e10,a[9,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"L'G[vwQFf0$* 4KLA\"\"\"*&#\"JvY][WwSl%>!=*3:**GW\"eBX%\"IFSiZ%pq36$*\\.#R\\\"o%GmNy F-&F%6$\"\"(F(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&#\"=V]>$><&H(\\&Q\"H0Q\"\" =DcfO?3Gj;$*ey*p$!\"\"*&#\"Hh:G[[>%eM0/&Hy7B(GnZZ\"G]i6-?bW&e)3Yq;JhD@ V:\"\"\"&F%6$\"\"(F(F2F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13594 "e11 := \{a[10,8] = 141940909070976808611971717/336 796637787572286621096705, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[9,2] = 0, a[10,2] = 0, a[5,2] = 0, a[10,1] = -1/3745361515630432440818009997 0863755192597143108791910*(-506336259629333177335499911268480707618977 30337719420*c[3]-50633625962933317733549991126848070761897730337719420 *c[4]+19128438057991915449917418562618168523109679123584375*a[9,5]-130 4252401583771151835557813448421131676832989762218710*a[7,6]-3182714903 7667220664568477944524347626686692995543750*c[4]*a[9,5]+33073901176194 6767330311175992526000668348458170003844*c[3]*c[4]-2362976534417117173 262702309512223323373799399533780+145769386059362658146327049738352950 0109401576793067970*c[4]*a[7,6]+14576938605936265814632704973835295001 09401576793067970*c[3]*a[7,6]+5295609671813537555953410615979681369818 4581622837500*c[4]*c[3]*a[9,5]+201426469173963814602291731688602630009 67127777084770*c[4]*c[3]*a[6,1]-16291872559575826498707140853110035589 45801762298134790*c[3]*c[4]*a[7,6]-31827149037667220664568477944524347 626686692995543750*c[3]*a[9,5])/c[3]/c[4], b[1] = 18530703372187/31798 6769215500, a[10,3] = 1/6294725236353667967761361339640967259260024051 8978*(798459316770054075049774568847477462970448441170233*c[3]-1012672 5192586663546709998225369614152379546067543884*c[4]-472595306883423434 652540461902444664674759879906756+382568761159838308998348371252363370 4621935824716875*a[9,5]-2608504803167542303671115626896842263353665979 52443742*a[7,6]-6365429807533444132913695588904869525337338599108750*c [4]*a[9,5]+15928064749962572875208800997265417992371437750142572*c[3]* c[4]+291538772118725316292654099476705900021880315358613594*c[4]*a[7,6 ]+434020126913591072375530163130735099280694003315830764*c[3]*a[7,6]+1 0591219343627075111906821231959362739636916324567500*c[4]*c[3]*a[9,5]+ 4028529383479276292045834633772052600193425555416954*c[4]*c[3]*a[6,1]- 485081318315190022066769005851998052137246239000046148*c[3]*c[4]*a[7,6 ]-6365429807533444132913695588904869525337338599108750*c[3]*a[9,5])/(1 98*c[3]-119)/c[3]/(-c[4]+c[3]), b[5] = 397406378366413672608/803680368 165950784145, a[6,5] = -6534/6859*(16473*c[3]+41154*c[4]*c[3]*a[6,1]-3 6822*c[3]*c[4]-9826+16473*c[4])/(198*c[3]-119)/(198*c[4]-119), a[5,3] \+ = -14161/23287176*(-119+297*c[4])/c[3]/(-c[4]+c[3]), a[4,3] = 1/2*c[4] ^2/c[3], a[3,2] = 100*c[3]^2, a[7,3] = -17/135973341712971148966814362 385116446705728342937666*(-2186867715596276770977996123843471617047915 7533444+36574400926063413335365366306678910051623953724287*c[3]+215528 52194513171901667507376253657491254393170804*c[4]+11175328953062462775 024782022951218221875177676875*a[9,5]-76197803403141449961181566511105 4215195812846988638*a[7,6]-1859424481265855150802442723146505216748979 1428750*c[4]*a[9,5]-50138595520196552477358677496946871905756678714332 *c[3]*c[4]+851622508623345617213205743359413534630614358399066*c[4]*a[ 7,6]+1267828997800168663219659678083938946292192804233196*c[3]*a[7,6]+ 30938323301734396626796946149832607808092258007500*c[4]*c[3]*a[9,5]+62 421041704937023884333536978575385715399923247126*c[4]*c[3]*a[6,1]-1416 985350482541447127854934329108234091274310613572*c[3]*c[4]*a[7,6]-1859 4244812658551508024427231465052167489791428750*c[3]*a[9,5])/(198*c[3]- 119)/c[3]/(-c[4]+c[3]), a[10,7] = 769638228283495127710326245120974996 10054141220500895/7457737217451650694295122868750841403431006754989491 , a[8,1] = 25/18146779692042741190680225783425301833004760241157842*(- 186751126846457585584215693614464764754553196386484*c[3]-1867511268464 57585584215693614464764754553196386484*c[4]+48426425463270672025107388 766121945628125769933125*a[9,5]-33019048141361294983178678821479015991 81855670284098*a[7,6]-805750608548537232014391846696818927257890961912 50*c[4]*a[9,5]+1016885991268578686029743536046986485482775898156508*c[ 3]*c[4]+3690364204034497674590558221224125316732662219729286*c[4]*a[7, 6]+3690364204034497674590558221224125316732662219729286*c[3]*a[7,6]+13 4066067640849052049453433315941300501733118032500*c[4]*c[3]*a[9,5]+855 566718572954599295247580449479525498181798689114*c[4]*c[3]*a[6,1]-4124 524698626791518660035659015198883407093069109202*c[3]*c[4]*a[7,6]-8057 5060854853723201439184669681892725789096191250*c[3]*a[9,5]+17483111256 930402758680366094087418655584747596612)/c[3]/c[4], a[9,4] = -1/288873 23344289819446122265312635488149479232500*(-10434480815184372338958713 373437178545168233125*a[9,5]+72456964031873800889687968128665440789218 7916+711464084062945377788810144828248566942869138178*a[7,6]+290291137 70950012447882944102029026673648554500*c[3]-79516574101152718694043486 7749218986583206683846*c[3]*a[7,6]+17361573121063073303477523091937490 352464791250*c[3]*a[9,5])/(-c[4]+c[3])/c[4], a[2,1] = 1/200, a[8,4] = \+ 25/152493946991955808325043914146431107840376136480318*(-8057506085485 3723201439184669681892725789096191250*c[4]*a[9,5]+54939256571340642072 85191938363735433932835485010516*c[4]*a[7,6]-2651463152486899046477041 5444366517460623485057383*c[4]+115035230522899588205346181770481776521 353360585532*c[3]*c[4]-61402698520910129375540380487594690143955220126 58812*c[3]*c[4]*a[7,6]+13406606764084905204945343331594130050173311803 2500*c[4]*c[3]*a[9,5]+855566718572954599295247580449479525498181798689 114*c[4]*c[3]*a[6,1]+3690364204034497674590558221224125316732662219729 286*c[3]*a[7,6]-186751126846457585584215693614464764754553196386484*c[ 3]-3301904814136129498317867882147901599181855670284098*a[7,6]+1748311 1256930402758680366094087418655584747596612-80575060854853723201439184 669681892725789096191250*c[3]*a[9,5]+484264254632706720251073887661219 45628125769933125*a[9,5])/c[4]/(198*c[4]-119)/(-c[4]+c[3]), a[10,4] = \+ -1/62947252363536679677613613396409672592600240518978*(-48508131831519 0022066769005851998052137246239000046148*c[3]*c[4]*a[7,6]+105912193436 27075111906821231959362739636916324567500*c[4]*c[3]*a[9,5]+40285293834 79276292045834633772052600193425555416954*c[4]*c[3]*a[6,1]+15928064749 962572875208800997265417992371437750142572*c[3]*c[4]+43402012691359107 2375530163130735099280694003315830764*c[4]*a[7,6]+79845931677005407504 9774568847477462970448441170233*c[4]-636542980753344413291369558890486 9525337338599108750*c[4]*a[9,5]+29153877211872531629265409947670590002 1880315358613594*c[3]*a[7,6]-26085048031675423036711156268968422633536 6597952443742*a[7,6]-1012672519258666354670999822536961415237954606754 3884*c[3]-472595306883423434652540461902444664674759879906756+38256876 11598383089983483712523633704621935824716875*a[9,5]-636542980753344413 2913695588904869525337338599108750*c[3]*a[9,5])/c[4]/(198*c[4]-119)/(- c[4]+c[3]), a[5,1] = 119/23287176*(14161-35343*c[3]-35343*c[4]+117612* c[3]*c[4])/c[3]/c[4], a[10,6] = -3128630583935097105138851480293877190 9/719245288891454780892327614375584661*a[7,6]-155083243453526191142918 638/436648564509816360719190147, a[9,1] = -1/1227711242132317326460196 2757870082463528673812500*(-377703726980475413796706562180879018627023 174826850*c[3]*c[4]*a[7,6]+1227711242132317326460196275787008246352867 3812500*c[4]*c[3]*a[9,5]+785729023461548939392610738355402277018230008 78612*c[3]*c[4]+337945439929899054449684818793418069297862840634550*c[ 4]*a[7,6]-7378668576451806153977947314073433399797536281250*c[4]*a[9,5 ]-12337373352653755290350251243362336336300635662500*c[4]-307942097135 463653781173864546828123354179864300-302372235726751785560244311552005 640950719383725650*a[7,6]+44346543464533582440574531837108008816964990 78125*a[9,5]-12337373352653755290350251243362336336300635662500*c[3]+3 37945439929899054449684818793418069297862840634550*c[3]*a[7,6]-7378668 576451806153977947314073433399797536281250*c[3]*a[9,5])/c[3]/c[4], a[7 ,5] = -1120754/44835598304260228348742964527608887165776157*(-13566462 1810277368249307514139473152087334*c[3]*c[4]+6739654999185334925094756 2228237064029265*c[4]+50978612968056746751103402240186914082500*c[4]*c [3]*a[9,5]+102854252801751655288293471093801982275186*c[4]*c[3]*a[6,1] -30638661329286630623137903366576983716250*c[4]*a[9,5]-403199644490487 53784927550234633814504258+18414144940328833556330356063750813445625*a [9,5]+67396549991853349250947562228237064029265*c[3]-30638661329286630 623137903366576983716250*c[3]*a[9,5])/(198*c[4]-119)/(198*c[3]-119), a [9,7] = 231443608884459407960125470613488480291827060789/3141890410475 8208921880920386941521649080865000, a[10,5] = 4266/2787227535839991960 200407597792117*(-13245561823820159816242829034835842*c[3]*c[4]+606702 5440649022202726188676113099*c[4]+36216308109672357804844464890632500* c[4]*c[3]*a[9,5]+13775416847423105693009686374547206*c[4]*c[3]*a[6,1]- 21766366995207124135234804656491250*c[4]*a[9,5]+1308180642641236248531 7887647083125*a[9,5]-3621426585954523137922816260598870-21766366995207 124135234804656491250*c[3]*a[9,5]+6067025440649022202726188676113099*c [3])/(198*c[4]-119)/(198*c[3]-119), b[10] = 1754022361907/143029462080 0, b[9] = -68588512868559375/32259978418667428, c[7] = 9526409/4724514 9, a[4,1] = 1/2*c[4]*(2*c[3]-c[4])/c[3], c[10] = 1, b[2] = 0, b[3] = 0 , b[4] = 0, b[6] = 383280144717771439/119491812371674080, b[8] = -1607 0516558763250309/7373332958527296000, a[6,4] = 17/13718*(96026*c[3]*a[ 6,1]-21964*c[3]+289)/(-c[4]+c[3])/(198*c[4]-119), a[4,2] = 0, a[7,4] = 17/135973341712971148966814362385116446705728342937666*(-218686771559 62767709779961238434716170479157533444+2155285219451317190166750737625 3657491254393170804*c[3]+365744009260634133353653663066789100516239537 24287*c[4]+11175328953062462775024782022951218221875177676875*a[9,5]-7 61978034031414499611815665111054215195812846988638*a[7,6]-185942448126 58551508024427231465052167489791428750*c[4]*a[9,5]-5013859552019655247 7358677496946871905756678714332*c[3]*c[4]+1267828997800168663219659678 083938946292192804233196*c[4]*a[7,6]+851622508623345617213205743359413 534630614358399066*c[3]*a[7,6]+309383233017343966267969461498326078080 92258007500*c[4]*c[3]*a[9,5]+62421041704937023884333536978575385715399 923247126*c[4]*c[3]*a[6,1]-1416985350482541447127854934329108234091274 310613572*c[3]*c[4]*a[7,6]-1859424481265855150802442723146505216748979 1428750*c[3]*a[9,5])/c[4]/(198*c[4]-119)/(-c[4]+c[3]), a[10,9] = -5450 09894469844730599500000/14146180885244972240022216299, a[9,6] = -13805 29138549729517193195043/3699785893166328082036595625-47476728723127829 5040534584194848281561/15432125613116704608858544552002116250*a[7,6], \+ a[9,3] = 1/28887323344289819446122265312635488149479232500*(2902911377 0950012447882944102029026673648554500*c[4]-795165741011527186940434867 749218986583206683846*c[4]*a[7,6]+173615731210630733034775230919374903 52464791250*c[4]*a[9,5]+724569640318738008896879681286654407892187916+ 711464084062945377788810144828248566942869138178*a[7,6]-10434480815184 372338958713373437178545168233125*a[9,5])/(-c[4]+c[3])/c[3], a[8,6] = \+ 206363383747027392130307880/22333209930559273876754828633+445235814428 99150891801946540764448504675/7835662846814939203499311087069447624027 *a[7,6], a[8,7] = -28380858623627672736703320753126600/189707578384001 00725353564886292693, b[7] = 32739939833109197146986610130768938398484 878542073/102638217510541641713801705364029167418501950476800, c[9] = \+ 44/45, c[8] = 25/29, c[5] = 119/198, c[6] = 17/19, c[2] = 1/200, a[5,4 ] = 14161/23287176*(-119+297*c[3])/c[4]/(-c[4]+c[3]), a[8,5] = -270/20 256737767530230495996689572230881*(-2788163251002065970453483468443498 658*c[3]*c[4]+3466813239934814932740771070927713510*c[4]*c[3]*a[6,1]+5 43244621645085367072666973359487500*c[4]*c[3]*a[9,5]-32649550492810686 2028522069847368750*c[4]*a[9,5]+1199135373180534716914300620861363579* c[4]+196227096396185437279768314706246875*a[9,5]-326495504928106862028 522069847368750*c[3]*a[9,5]-714421696708217437748405223634765142+11991 35373180534716914300620861363579*c[3])/(198*c[4]-119)/(198*c[3]-119), \+ a[6,3] = -17/13718*(96026*c[4]*a[6,1]+289-21964*c[4])/(198*c[3]^2-198* c[3]*c[4]+119*c[4]-119*c[3]), a[8,3] = -25/152493946991955808325043914 146431107840376136480318*(-2651463152486899046477041544436651746062348 5057383*c[3]-186751126846457585584215693614464764754553196386484*c[4]+ 48426425463270672025107388766121945628125769933125*a[9,5]-330190481413 6129498317867882147901599181855670284098*a[7,6]-8057506085485372320143 9184669681892725789096191250*c[4]*a[9,5]+11503523052289958820534618177 0481776521353360585532*c[3]*c[4]+3690364204034497674590558221224125316 732662219729286*c[4]*a[7,6]+549392565713406420728519193836373543393283 5485010516*c[3]*a[7,6]+17483111256930402758680366094087418655584747596 612+134066067640849052049453433315941300501733118032500*c[4]*c[3]*a[9, 5]+855566718572954599295247580449479525498181798689114*c[4]*c[3]*a[6,1 ]-6140269852091012937554038048759469014395522012658812*c[3]*c[4]*a[7,6 ]-80575060854853723201439184669681892725789096191250*c[3]*a[9,5])/(198 *c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3])/c[3], a[9,8] = 41013323458190 49706502423/10667097363909719917125000, a[3,1] = -100*c[3]^2+c[3], a[7 ,1] = 1/951813391990798042767700536695815126940098400563662*(-21868677 155962767709779961238434716170479157533444+215528521945131719016675073 76253657491254393170804*c[3]+21552852194513171901667507376253657491254 393170804*c[4]+11175328953062462775024782022951218221875177676875*a[9, 5]-761978034031414499611815665111054215195812846988638*a[7,6]-18594244 812658551508024427231465052167489791428750*c[4]*a[9,5]+109588309951475 368543506898405335773301457428765348*c[3]*c[4]+85162250862334561721320 5743359413534630614358399066*c[4]*a[7,6]+85162250862334561721320574335 9413534630614358399066*c[3]*a[7,6]+30938323301734396626796946149832607 808092258007500*c[4]*c[3]*a[9,5]+6242104170493702388433353697857538571 5399923247126*c[4]*c[3]*a[6,1]-951813391990798042767700536695815126940 098400563662*c[3]*c[4]*a[7,6]-1859424481265855150802442723146505216748 9791428750*c[3]*a[9,5])/c[3]/c[4]\}:" }}}{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 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[7,4]=subs(e11,a[7,4]);\n``;\na[8,5]=sub s(e11,a[8,7]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%, $*&#\"#<\"Tmw$HMGdqYk6&QiV\"o'*[6(HrTL(f8\"\"\"**,<\"SWM`d\"z/<;ZVQ7'* z(4xwif:x'o=#!\"\"*&\"S/3<$Ra7\\dODwt]n;!><8X>_Gb@F.&%\"cG6#\"\"$F.F.* &\"S(GCP&Ri^+\"*ymIm`ONLTjg#4Sul$F.&F66#F(F.F.*&\"Svonx^(=A=7&H-#yC]xi C1`*G`<6F.&F%6$\"\"*\"\"&F.F.*&\"TQ'))p%G\"e>:U06^m:=h*\\99.M!y>wF.&F% 6$F'\"\"'F.F2*(\"S](G9z*[n@0l9BFW-3:beE\"[C%f=F.F;F.F?F.F2*(\"SKVrymv0 >(o%p\\x'etZ_l>?bfQ,&F.F5F.F;F.F2*(\"U'>LU!G>#HY*QR3y'f'>Kmo,!y**GyE\" F.F;F.FEF.F.*(\"Tm!*ReVhIY`8%fLu0K@F.F ;F.F.\"$>\"F2F2,&F;F2F5F.F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!D+m7`2K.ntswi B'e3QG\"D$p#H')[c``s+,SQyvq*=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 117 "We can find some simple order conditions that are not yet satisfied and determine which paramers are related b y them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "for ii from 64 by -1 to 43 do\n eq := simplify(sub s(e11,SO7_10[ii]));\n if lhs(eq)<>rhs(eq) then\n print(ii,indet s(lhs(eq)))\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" #b<'&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$\"\"(\"\"'&F-6$F4\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[<'&%\"cG6#\"\"$&F&6#\"\"%& %\"aG6$\"\"*\"\"&&F-6$\"\"(\"\"'&F-6$F4\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#X<'&%\"cG6#\"\"$&F&6#\"\"%&%\"aG6$\"\"*\"\"&&F-6$\" \"(\"\"'&F-6$F4\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We construct a system of equations that consists of the simple order comditions given in abreviated form as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SO7 := SimpleOrderConditions(7):\n[seq([i,SO7[i]],i=[45,55])]:\nl inalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim]( %))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7$7%\"#X%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F-- F06#*&)F.\"\"#F-F3F-F-F-F-#F-\"$?%7%\"#bF)/*(F,F-F.F--F06#*&F3F--F06#* &)F.\"\"$F-F3F-F-F-#F-\"$S\"Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "We can solve this system of two eq uations to obtain numerical values for the remaining parameters " } {XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[9,5]" "6#&%\"aG6$\"\"*\"\"&" }{TEXT -1 103 " and sub stitute these values back into the expressions obtained previously for the other coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "cdcns5 := [seq(SO7_10[i],i=[45,55] )]:\neqns5 := simplify(subs(e11,cdcns5)):\nnops(%);\nindets(eqns5);\nn ops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%\"cG6#\"\"$&F%6#\"\"%&%\"aG6$\"\"*\"\"&&F,6$\"\"(\" \"'&F,6$F3\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "info level[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "e12 \+ := solve(\{op(eqns5)\},\{a[7,6],a[9,5]\}):\ninfolevel[solve] := 0:\ne1 3 := `union`(map(u_->lhs(u_)=simplify(subs(e12,rhs(u_))),e11),e12):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[7,6]=subs(e12,a[7,6]);\n``;\na[9,5]=subs(e12,a[9,5]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!>me@B4^*45\"[,S!> f\"A&f5b^`X\"yNBuz)R%*>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,$*&#\"(/<(R\"Av$f6 VUKh#=T&H,pKB$\"\"\"*&,,*(\"?uVcy#oPYEd3Q$z7yF.&%\"cG6#\"\"$F.&F46#\" \"%F.F.*&\"?X2_6eJ&Q\")QR)H5!o&F.F3F.!\"\"*&\"?X2_6eJ&Q\")QR)H5!o&F.F7 F.F<\"?elc]I!ywXU#zOvEMF.**\"?%*[.lIBUEzn&px=;(F.F7F.F3F.&F%6$\"\"'F.F .F.F.,**(\"&/#RF.F3F.F7F.F.*&\"&iN#F.F3F.F<\"&hT\"F.*&FIF.F7F.F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5624 "e13 := \{a[10,8] = 141940909070976808611971 717/336796637787572286621096705, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a [9,2] = 0, a[10,2] = 0, a[7,5] = -1103782008619774/3109658947480129803 773133650895*(15299655948479104986*c[4]*c[3]*a[6,1]-765438476664843294 *c[3]*c[4]-1643190182023988955*c[3]-1643190182023988955*c[4]+101524792 9358640502)/(198*c[3]-119)/(198*c[4]-119), a[7,1] = 9526409/7556471242 37671542316871477167485*(15835893059176968332916324*c[3]*c[4]+10987822 760971006983768057*c[4]*c[3]*a[6,1]-3142846173723401860595496*c[3]-314 2846173723401860595496*c[4]+1259254864219140812831192)/c[3]/c[4], a[5, 2] = 0, a[8,4] = 55/143140684477154*(493403551930954*c[4]*c[3]*a[6,1]+ 202038989500468*c[3]*c[4]-189255102985872*c[3]-124685120554567*c[4]+75 829485708144)/c[4]/(-c[4]+c[3])/(198*c[4]-119), b[1] = 18530703372187/ 317986769215500, a[7,4] = 161948953/215899178353620440661963279190710* (5432063705211955751723268*c[3]*c[4]+21975645521942013967536114*c[4]*c [3]*a[6,1]-6285692347446803721190992*c[3]-4124323627782509674991907*c[ 4]+2518509728438281625662384)/c[4]/(-c[4]+c[3])/(198*c[4]-119), b[5] = 397406378366413672608/803680368165950784145, a[6,5] = -6534/6859*(164 73*c[3]+41154*c[4]*c[3]*a[6,1]-36822*c[3]*c[4]-9826+16473*c[4])/(198*c [3]-119)/(198*c[4]-119), a[5,3] = -14161/23287176*(-119+297*c[4])/c[3] /(-c[4]+c[3]), a[4,3] = 1/2*c[4]^2/c[3], a[3,2] = 100*c[3]^2, a[9,1] = -836/601433880219870821100495984375*(1614407497354698195959839785*c[4 ]*c[3]*a[6,1]+6474598996087652994106311564*c[3]*c[4]-21010650964296865 88593398930*c[3]-2101065096429686588593398930*c[4]+8418408972226690371 80520110)/c[3]/c[4], a[10,7] = 769638228283495127710326245120974996100 54141220500895/7457737217451650694295122868750841403431006754989491, a [10,4] = -1/27134725938701290*(11981916947658886802*c[4]*c[3]*a[6,1]+1 9074515377304801924*c[3]*c[4]-13111117015324060656*c[3]-87046838808704 45651*c[4]+5253275841156778512)/c[4]/(-c[4]+c[3])/(198*c[4]-119), a[2, 1] = 1/200, a[10,6] = -2147337295919765809044534716/152826997578435726 25171655145, a[10,5] = 1548558/124907764090140484772105334087995*(1173 4004292029675197733259804862*c[4]*c[3]*a[6,1]+931000944737349003532882 5553942*c[3]*c[4]-7208470047052258693872492624505*c[3]-720847004705225 8693872492624505*c[4]+4353587789116130351845125371794)/(198*c[3]-119)/ (198*c[4]-119), a[10,3] = 1/27134725938701290*(11981916947658886802*c[ 4]*c[3]*a[6,1]+19074515377304801924*c[3]*c[4]-8704683880870445651*c[3] -13111117015324060656*c[4]+5253275841156778512)/(-c[4]+c[3])/c[3]/(198 *c[3]-119), a[5,1] = 119/23287176*(14161-35343*c[3]-35343*c[4]+117612* c[3]*c[4])/c[3]/c[4], a[8,5] = -538695630/1178464813011121732104281*(1 36633897826020641606*c[4]*c[3]*a[6,1]-42522110254249620546*c[3]*c[4]+6 773310030692389611*c[3]+6773310030692389611*c[4]-3823684222714205206)/ (198*c[3]-119)/(198*c[4]-119), a[9,7] = 231443608884459407960125470613 488480291827060789/31418904104758208921880920386941521649080865000, b[ 10] = 1754022361907/1430294620800, b[9] = -68588512868559375/322599784 18667428, c[7] = 9526409/47245149, a[4,1] = 1/2*c[4]*(2*c[3]-c[4])/c[3 ], c[10] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[6] = 383280144717771439/ 119491812371674080, b[8] = -16070516558763250309/7373332958527296000, \+ a[6,4] = 17/13718*(96026*c[3]*a[6,1]-21964*c[3]+289)/(-c[4]+c[3])/(198 *c[4]-119), a[4,2] = 0, a[9,4] = -418/691575989934375*(855667482534996 *c[3]*c[4]-575000845115004*c[3]+441816713302398*c[4]*c[3]*a[6,1]-38200 4204470989*c[4]+230387544002308)/c[4]/(-c[4]+c[3])/(198*c[4]-119), a[9 ,5] = 3971704/32332690129541182613242431159375*(7812793380857264637682 78564374*c[3]*c[4]-568010298393881385315811520745*c[3]-568010298393881 385315811520745*c[4]+342675367924245767803050566558+716187769567792642 233065034894*c[4]*c[3]*a[6,1])/(39204*c[3]*c[4]-23562*c[3]+14161-23562 *c[4]), a[10,9] = -545009894469844730599500000/14146180885244972240022 216299, a[9,6] = -372567259603706115805768838168/168340258139067927732 6651009375, a[8,7] = -28380858623627672736703320753126600/189707578384 00100725353564886292693, b[7] = 32739939833109197146986610130768938398 484878542073/102638217510541641713801705364029167418501950476800, c[9] = 44/45, c[8] = 25/29, c[5] = 119/198, c[6] = 17/19, c[2] = 1/200, a[ 5,4] = 14161/23287176*(-119+297*c[3])/c[4]/(-c[4]+c[3]), a[10,1] = -1/ 169185957545012719667100745*(627795527960865814883047099*c[4]*c[3]*a[6 ,1]+2115931162241451889938035756*c[3]*c[4]-686960247241599481424253672 *c[3]-686960247241599481424253672*c[4]+275246698389731778752478744)/c[ 3]/c[4], a[7,3] = -161948953/215899178353620440661963279190710*(219756 45521942013967536114*c[4]*c[3]*a[6,1]+5432063705211955751723268*c[3]*c [4]-4124323627782509674991907*c[3]-6285692347446803721190992*c[4]+2518 509728438281625662384)/(-c[4]+c[3])/c[3]/(198*c[3]-119), a[8,3] = -55/ 143140684477154*(493403551930954*c[4]*c[3]*a[6,1]+202038989500468*c[3] *c[4]-124685120554567*c[3]-189255102985872*c[4]+75829485708144)/c[3]/( 198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3]), a[6,3] = -17/13718*(96026 *c[4]*a[6,1]+289-21964*c[4])/(198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[ 3]), a[9,3] = 418/691575989934375*(855667482534996*c[3]*c[4]+441816713 302398*c[4]*c[3]*a[6,1]-575000845115004*c[4]-382004204470989*c[3]+2303 87544002308)/c[3]/(-c[4]+c[3])/(198*c[3]-119), a[8,6] = -7115819429493 526581990/378493240381452882505729, a[9,8] = 4101332345819049706502423 /10667097363909719917125000, a[8,1] = 5/81135193939724549519167*(25852 002207608541603023*c[4]*c[3]*a[6,1]+30524614968278821279356*c[3]*c[4]- 9916068340092958415064*c[3]-9916068340092958415064*c[4]+39731048231348 89061928)/c[3]/c[4], a[3,1] = -100*c[3]^2+c[3], a[7,6] = -591904001481 10099510923215866/11994398797423357814553515510595\}:" }}}{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 0 443 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(5-i)],i=2..5) ]));print(``);\nfor ii from 6 to 10 do\n print(c[ii]=subs(e13,c[ii]) );print(``); \n for jj to ii-1 do\n if ii=6 and jj=1 then\n \+ print(a[ii,jj]*` is a parameter`);\n else\n print(a [ii,jj]=subs(e13,a[ii,jj]));\n end if;\n end do:\n print(`___ ______________________________`);\nend do:print(``);\nfor ii to 10 do \n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"\"\"$+#F(%!GF+F+7'&%\"cG6#\"\"$,&* &\"$+\"F))F-\"\"#F)!\"\"F-F),$*&F3F)F4F)F)F+F+7'&F.6#\"\"%,$*&#F)F5F)* (F:F),&*&F5F)F-F)F)F:F6F)F-F6F)F)\"\"!,$*&F?F)*&F:F5F-F6F)F)F+7'#\"$> \"\"$)>,$*&#FI\")wrGBF)*(,*\"&hT\"F)*&\"&V`$F)F-F)F6*&FSF)F:F)F6*(\"'7 w6F)F-F)F:F)F)F)F-F6F:F6F)F)FC,$*&#FQFNF)*(,&FIF6*&\"$(HF)F:F)F)F)F-F6 ,&F:F6F-F)F6F)F6,$*&#FQFNF)*(,&FIF6*&FgnF)F-F)F)F)F:F6FhnF6F)F)Q(pprin t46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#<\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"'\"\"\"F(%0 ~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\" #<\"&=P\"\"\"\"*&,(*(\"&Eg*F.&%\"cG6#\"\"%F.&F%6$F'F.F.F.\"$*GF.*&\"&k >#F.F3F.!\"\"F.,**&\"$)>F.)&F46#F(\"\"#F.F.*(F?F.FAF.F3F.F<*&\"$>\"F.F 3F.F.*&FFF.FAF.F#F.F3F.!\"\"\"$*GF.F.,&&F46#F(F;F3F.F;,&*&\"$)>F.F>F.F.\"$>\" F;F;F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,$*&#\" %Ml\"%fo\"\"\"*(,,*&\"&tk\"F.&%\"cG6#\"\"$F.F.**\"&a6%F.&F46#\"\"%F.F3 F.&F%6$F'F.F.F.*(\"&Ao$F.F3F.F9F.!\"\"\"%E)*F@*&F2F.F9F.F.F.,&*&\"$)>F .F3F.F.\"$>\"F@F@,&*&FEF.F9F.F.FFF@F@F.F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"(4k_*\")\\^CZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"\",$*&#\"(4k_*\"B&[nrZroJU:nPU7ZcvF(*(,,*(\";Cj\"HLopJG\" 39>U'[Df7F(F(F2F@F6F@F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"(\"\"#\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$ ,$*&#\"*`*[>;\"B52>zK'>mS/i`$y\"**e@\"\"\"**,,**\";9h`nR,U>_Xc(>#F.&% 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FGF?F?F.F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#!7!* >eEN\\H%>e6(\"9Hd]#)GX\"QSK\\y$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"(#!D+m7`2K.ntswiB'e3QG\"D$p#H')[c``s+,SQyvq*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#W\"#X" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"\",$*&#\"$O)\"?vV)f\\+6#3()>-)QV,'F(*(,,**\"=&yR)ff>)pat\\2Wh \"F(&%\"cG6#\"\"%F(&F36#\"\"$F(&F%6$\"\"'F(F(F(*(\"=k:J1T*Hl(3'**)fukF (F6F(F2F(F(*&\"=I*)R$f)e'oHk4l55#F(F6F(!\"\"*&\"=I*)R$f)e'oHk4l55#F(F2 F(F@\"<5,_!=P!pEA(*3%=%)F(F(F6F@F2F@F(F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"$,$*&#\"$=%\"0vV$**)fd\"p\"\"\"**,,*(\"0'*\\`#[nc&)F.&% \"cG6#F(F.&F46#\"\"%F.F.**\"0)R-Lr;=WF.F6F.F3F.&F%6$\"\"'F.F.F.*&\"0/] 6X3+v&F.F6F.!\"\"*&\"0*)4Z/U+#QF.F3F.F@\"03B+WvQI#F.F.F3F@,&F6F@F3F.F@ 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5\"\"'#!=;Z`W!4ew>fHPt9#\">X^lr^isN%yv*p#G:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"V&*3]?79a+h*\\(47XiK5x7&\\$GG#Q'p( \"U\"\\*)\\v15V.9%3voG7&H%p];Xh3o(4244%>9\"<0n4@mGsvyPmzO$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!<++]*fIZ%)pW*)4]a\">*H;A-SA( \\C&)3=YT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_____________________ ____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"/(=sLqI&=\"0+b@pn)zJ" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&# \"63En8kOyjS(R\"6XTy]f;o.o.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"'#\"3R9x\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"St?ay[[)RQ*o285m)p9(>4J$)R*RF$\"T+oZ]>]=u;HSO0OASv\"\".+3i%HI9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "subs(e13,OrderConditions(7,10,'expanded')):\nmap(u->s implify(lhs(u)-rhs(u)),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We obtain valu es for the remaining parameters by requiring that certain principal er ror terms are approximately zero." }}{PARA 0 "" 0 "" {TEXT -1 105 "We \+ shall make use of equations given by the order 8 order conditions give n in abreviated form as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "SO8 := SimpleOrderCondition s(8):\n[seq([i,SO8[i]],i=[122,102,65])]:\nlinalg[augment](linalg[delco ls](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"$A\"%#~~G/*( %\"bG\"\"\")%\"cG\"\"$F--%!G6#*&F.F-%\"aGF-F-#F-\"#K7%\"$-\"F)/*(F,F-F /F--F26#*&F5F--F26#*&F5F-F1F-F-F-#F-\"$g*7%\"#lF)/*&F,F--F26#*&F5F--F2 6#*&F5F--F26#*&F5F--F26#*&F5F--F26#*&F5F--F26#*&F5F-F/F-F-F-F-F-F-F-#F -\"&?.%Q(pprint86\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "In detai l, these order condition are as follows." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^3*Sum(a[i,j]*c[j]^3,j=2..i-1),i =3..10)=1/32" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"$F,-F% 6$*&&%\"aG6$F+%\"jGF,*$&F/6#F8F1F,/F8;\"\"#,&F+F,F,!\"\"F,/F+;F1\"#5*& F,F,\"#KF@" }{TEXT -1 16 " ------- (A), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[ i]*Sum(a[i,j]*Sum(a[j,k]*Sum(a[k,l]*c[l]^3,l=2..k-1),k=3..j-1),j=4..i- 1),i=5..10)=1/960" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$ *&&%\"aG6$F+%\"jGF,-F%6$*&&F46$F6%\"kGF,-F%6$*&&F46$F<%\"lGF,*$&F.6#FB \"\"$F,/FB;\"\"#,&F " 0 "" {MPLTEXT 1 0 358 "cdnA := add(b[i]*c[i]^3*add(a[i,j]*c[j]^3,j=2..i-1), i=3..10)=1/32:##122\ncdnB := add(b[i]*c[i]*add(a[i,j]*add(a[j,k]*add(a [k,l]*c[l]^3,l=2..k-1),k=3..j-1),j=4..i-1),i=5..10)=1/960:##102\ncdnC \+ := add(b[i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*add(a[m,n]*add (a[n,p]*c[p],p=2..n-1),n=3..m-1),\n m=4..l-1),l=5..k-1),k=6..j-1),j= 7..i-1),i=8..10)=1/40320:##65" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "The condition (A) gives the following equ ation relating the parameters " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\" $" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqA := simplify(subs(e13,cdnA)):\neqA;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,,*&#\"0fJ:vt%zG\"2+q@/'o#4z#\"\"\"*(&%\"cG6#\" \"%F)&F,6#\"\"$F)&%\"aG6$\"\"'F)F)F)F)*&#\"1JDB[rj_5\"3+:+%y]Zjm\"F)*& F/F)F+F)F)F)*&#\"1\">b?T%f'H#\"3+g+OJ+RlmF)F/F)!\"\"*&#\"1\">b?T%f'H#F >F)F+F)F?#\"1([OK4*ot#)\"3]A+wh7_*\\#F)#F)\"#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "This equation can readily be solve to give an expression for " }{XPPEDIT 18 0 "a[6,1]" "6#&%\" aG6$\"\"'\"\"\"" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]" "6#&%\"c G6#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 126 "We can then \+ use this formula to obtain expressions for all the coefficients of the scheme in terms of the two parameters of " }{XPPEDIT 18 0 "c[3]" "6 #&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6 #\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "e14 := solve(\{eqA\},a[6,1]):\n``; a[6,1]=subs(e14,a[6,1]);\ne15 := `union`(map(u_->lhs(u_)=simplify(subs (e14,rhs(u_))),e13),e14):" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",$*&#\"#<\"3C`4te!)R9qF(*(,*\"1nEy.C'HS(F(*( \"2W2ed:Hj_#F(&%\"cG6#\"\"$F(&F46#\"\"%F(F(*&\"2Y6LskczP\"F(F3F(!\"\"* &\"2Y6LskczP\"F(F7F(F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6104 "e15 := \{a[10,8] = 14194090907097680861 1971717/336796637787572286621096705, a[6,2] = 0, a[7,2] = 0, a[8,2] = \+ 0, a[9,2] = 0, a[10,2] = 0, a[7,3] = 161948953/80875680439391845273705 04819766857156350572*(300546041691835589454489053453725872*c[3]*c[4]-1 20420804583597424730923223437688144*c[3]-39456084283940173580069893913 372022*c[4]+53354136681849938298638853777262789)/(-c[4]+c[3])/c[3]/(19 8*c[3]-119), a[5,2] = 0, a[6,5] = 1072180395/58453317156091277*(-15693 06510152*c[3]+3220988141788*c[3]*c[4]-1569306510152*c[4]+893786548367) /(198*c[3]-119)/(198*c[4]-119), b[1] = 18530703372187/317986769215500, b[5] = 397406378366413672608/803680368165950784145, a[8,3] = 275/1631 45232598539920920044*(22808995504638776970768*c[3]*c[4]-91389577947879 27473136*c[3]+5579827211888797308510*c[4]+2893972069535750992175)/c[3] /(198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3]), a[10,3] = 1/17937616343 9730898164081852*(77596623028414695026496706416*c[3]*c[4]-310909028295 66830667182182032*c[3]-60219963203995199151682831326*c[4]+205160142290 82050232195900337)/(-c[4]+c[3])/c[3]/(198*c[3]-119), a[5,3] = -14161/2 3287176*(-119+297*c[4])/c[3]/(-c[4]+c[3]), a[4,3] = 1/2*c[4]^2/c[3], a [3,2] = 100*c[3]^2, a[9,5] = 21844372/39363359728650421823508775635782 654429559375*(75874337028012676913408397999209635166508*c[3]*c[4]-7278 8419305318649575279683406534860595384*c[3]-727884193053186495752796834 06534860595384*c[4]+47409399319113007313725730921183211729347)/(39204* c[3]*c[4]-23562*c[3]+14161-23562*c[4]), a[10,7] = 76963822828349512771 032624512097499610054141220500895/745773721745165069429512286875084140 3431006754989491, a[10,5] = 2839023/1520686720183952416004481542147637 76596375355*(1411491303298256991538765158900482668870212*c[3]*c[4]-218 4622786119901932091744866594702414833976*c[3]-218462278611990193209174 4866594702414833976*c[4]+1493015243138557574260808421118939375264633)/ (198*c[3]-119)/(198*c[4]-119), a[7,5] = 309610853417846607/42064968678 4674092358662516345898491785495*(4886706729696625856963075572*c[3]*c[4 ]-1671623176347940903094618456*c[3]-1671623176347940903094618456*c[4]+ 834188418761395887687138373)/(198*c[3]-119)/(198*c[4]-119), a[2,1] = 1 /200, a[9,3] = 209/761952587259725198544375*(1289399333769453717407520 *c[3]*c[4]-516628015887424216739040*c[3]-941900991848018035648734*c[4] +332992397941149639984773)/(-c[4]+c[3])/c[3]/(198*c[3]-119), a[10,6] = -2147337295919765809044534716/15282699757843572625171655145, a[8,4] = -275/163145232598539920920044*(22808995504638776970768*c[3]*c[4]+5579 827211888797308510*c[3]-9138957794787927473136*c[4]+289397206953575099 2175)/c[4]/(-c[4]+c[3])/(198*c[4]-119), a[8,5] = 808043445/49473079110 207149855788210173455981*(3531437994096984392504735971572*c[3]*c[4]-14 66633405799026829907997028696*c[3]-1466633405799026829907997028696*c[4 ]+793108542146990042475213402373)/(198*c[3]-119)/(198*c[4]-119), a[5,1 ] = 119/23287176*(14161-35343*c[3]-35343*c[4]+117612*c[3]*c[4])/c[3]/c [4], a[9,1] = -209/56324177184241321179716163206621740921875*(20551016 67896990032796778119601370978008*c[3]*c[4]-585093402145618422191774718 708826133850*c[3]-585093402145618422191774718708826133850*c[4]+2068493 99975419331942859646950608166575)/c[3]/c[4], a[9,7] = 2314436088844594 07960125470613488480291827060789/3141890410475820892188092038694152164 9080865000, b[10] = 1754022361907/1430294620800, b[9] = -6858851286855 9375/32259978418667428, a[6,3] = 289/100205686553299332*(3470094624681 1152*c[3]*c[4]-13903746139294704*c[3]-13779566472331146*c[4]+740296240 3782667)/c[3]/(198*c[3]^2-198*c[3]*c[4]+119*c[4]-119*c[3]), c[7] = 952 6409/47245149, a[4,1] = 1/2*c[4]*(2*c[3]-c[4])/c[3], c[10] = 1, b[2] = 0, b[3] = 0, b[4] = 0, a[6,1] = -17/701439805873095324*(2526329155758 0744*c[3]*c[4]-13779566472331146*c[3]-13779566472331146*c[4]+740296240 3782667)/c[3]/c[4], b[6] = 383280144717771439/119491812371674080, b[8] = -16070516558763250309/7373332958527296000, a[8,1] = 25/314412275314 6726133082766779157908*(113898933790023718129197512024088*c[3]*c[4]-99 40118923749298641966537390330*c[3]-9940118923749298641966537390330*c[4 ]-5155433213398138761765976870525)/c[3]/c[4], a[4,2] = 0, a[10,9] = -5 45009894469844730599500000/14146180885244972240022216299, a[9,6] = -37 2567259603706115805768838168/1683402581390679277326651009375, a[8,7] = -28380858623627672736703320753126600/18970757838400100725353564886292 693, b[7] = 32739939833109197146986610130768938398484878542073/1026382 17510541641713801705364029167418501950476800, a[6,4] = -289/1002056865 53299332*(34700946246811152*c[3]*c[4]-13779566472331146*c[3]-139037461 39294704*c[4]+7402962403782667)/c[4]/(-c[4]+c[3])/(198*c[4]-119), c[9] = 44/45, c[8] = 25/29, c[5] = 119/198, c[6] = 17/19, a[10,4] = -1/179 376163439730898164081852*(77596623028414695026496706416*c[3]*c[4]-6021 9963203995199151682831326*c[3]-31090902829566830667182182032*c[4]+2051 6014229082050232195900337)/c[4]/(-c[4]+c[3])/(198*c[4]-119), c[2] = 1/ 200, a[5,4] = 14161/23287176*(-119+297*c[3])/c[4]/(-c[4]+c[3]), a[10,1 ] = -1/190130822288238392526421428411252757020*(1945907693011548630166 061968258204155768*c[3]*c[4]-536390799482205135580283749672671592290*c [3]-536390799482205135580283749672671592290*c[4]+182740086327311710240 864312275822343855)/c[3]/c[4], a[7,4] = -161948953/8087568043939184527 370504819766857156350572*(300546041691835589454489053453725872*c[3]*c[ 4]-39456084283940173580069893913372022*c[3]-12042080458359742473092322 3437688144*c[4]+53354136681849938298638853777262789)/c[4]/(-c[4]+c[3]) /(198*c[4]-119), a[8,6] = -7115819429493526581990/37849324038145288250 5729, a[9,8] = 4101332345819049706502423/10667097363909719917125000, a [9,4] = -209/761952587259725198544375*(1289399333769453717407520*c[3]* c[4]-941900991848018035648734*c[3]-516628015887424216739040*c[4]+33299 2397941149639984773)/c[4]/(-c[4]+c[3])/(198*c[4]-119), a[7,1] = 952640 9/56612976307574291691593533738368000094454004*(6823921153157600815725 81336140387544*c[3]*c[4]+39456084283940173580069893913372022*c[3]+3945 6084283940173580069893913372022*c[4]-533541366818499382986388537772627 89)/c[3]/c[4], a[3,1] = -100*c[3]^2+c[3], a[7,6] = -591904001481100995 10923215866/11994398797423357814553515510595\}:" }}}{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[9,5]=subs(e15,a[9,5]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"&,$*&#\")sV%=#\"Mv$f&HWl#yNcx3N#=U]'G(fLOR\"\"\"*&,**(\"J3l;N '4#**zR3M\"pn7!GqLue(F.&%\"cG6#\"\"$F.&F46#\"\"%F.F.*&\"J%Q&fg[`1Moz_d \\'=`I>%)ysF.F3F.!\"\"*&\"J%Q&fg[`1Moz_d\\'=`I>%)ysF.F7F.F<\"JZ$H<@$=@ 4tDPJ2I6>$*R4u%F.F.,**(\"&/#RF.F3F.F7F.F.*&\"&iN#F.F3F.F<\"&hT\"F.*&FD F.F7F.F\"F1F,F1" } {TEXT -1 15 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "17699330840439779068196736* c[3]^5-34862770172180534108917392*c[3]^4+25522289439066332758667248*c[ 3]^3-``" "6#,**&\";On>o!z(R/%3L*p<\"\"\"*$&%\"cG6#\"\"$\"\"&F&F&*&\";# R<*3T`!=s,xi[$F&*$&F)6#F+\"\"%F&!\"\"*&\";[smeFLm!R%*GAb#F&*$&F)6#F+F+ F&F&%!GF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "8810156669345501948169468*c[3]^2+1458689872322045643053 436*c[3]-91065981268024619458329" "6#,(*&\":o%p\"[>]X$pm:5))\"\"\"*$&% \"cG6#\"\"$\"\"#F&F&*&\":OM0Vc/AB()*oe9F&&F)6#F+F&F&\"8H$e%>Y-o7)f1\"* !\"\"" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "sol := eliminate(\{op(eqn sBC)\},c[4]):\nop(sol[1]);\n`if`(coeff(op(sol[2]),c[3]^5)<0,-1,1)*op(s ol[2])=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&#\"\"\" \"%kmF+*&,&*&\"';IzF+&F%6#\"\"$F+F+\"'\"et$!\"\"F+,&\"$>\"F5*&\"$(HF+F 1F+F+F5F+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*&\":OM0Vc/AB()*oe9 \"\"\"&%\"cG6#\"\"$F'F'*&\";[smeFLm!R%*GAb#F')F(F+F'F'*&\";#R<*3T`!=s, xi[$F')F(\"\"%F'!\"\"*&\":o%p\"[>]X$pm:5))F')F(\"\"#F'F3\"8H$e%>Y-o7)f 1\"*F3*&\";On>o!z(R/%3L*p " 0 "" {MPLTEXT 1 0 97 "phi := unapply(su bs(c[3]=z,`if`(coeff(op(sol[2]),c[3]^5)<0,-1,1)*op(sol[2])),z):\n'phi' (z)=phi(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"zG,.*&\":O M0Vc/AB()*oe9\"\"\"F'F+F+*&\";[smeFLm!R%*GAb#F+)F'\"\"$F+F+*&\";#R<*3T `!=s,xi[$F+)F'\"\"%F+!\"\"*&\":o%p\"[>]X$pm:5))F+)F'\"\"#F+F4\"8H$e%>Y -o7)f1\"*F4*&\";On>o!z(R/%3L*p " 0 "" {MPLTEXT 1 0 82 "plot(phi(z)*10^(-23),z=0..1, -0.5..0.5,color=COLOR(RGB,.5,0,1),font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 497 347 347 {PLOTDATA 2 "6'-%'CURVESG6#7X7$$\"\"!F)$!31 iC!o7)f1\"*!#=7$$\"3emmm;arz@!#>$!3V><>Q3+?jF,7$$\"3[LL$e9ui2%F0$!3I= \"Q-'\\.hWF,7$$\"3nmmm\"z_\"4iF0$!3#excly?_)GF,7$$\"3[mmmT&phN)F0$!3)3 9TVj')Gu\"F,7$$\"3CLLe*=)H\\5F,$!3]!f\"e\")3>B&*F07$$\"3gmm\"z/3uC\"F, $!3=]j\\J!RRc%F07$$\"3%)***\\7LRDX\"F,$!3MjUb1a%fA\"F07$$\"3]mm\"zR'ok ;F,$\"3Liir_jKG%)!#?7$$\"3w***\\i5`h(=F,$\"3GPr@]H#4'>F07$$\"3WLLL3En$ 4#F,$\"3^J#[(*G!)*4DF07$$\"3qmm;/RE&G#F,$\"3O0dD51eDFF07$$\"3\")***** \\K]4]#F,$\"3[pU^*QE)fGF07$$\"3$******\\PAvr#F,$\"3f'\\3<%zp6IF07$$\"3 )******\\nHi#HF,$\"3z9wY$)GMVKF07$$\"3jmm\"z*ev:JF,$\"3JYiA(\\D1a$F07$ $\"3?LLL347TLF,$\"3#*=G[3fsvRF07$$\"3,LLLLY.KNF,$\"3]7!z83ARO%F07$$\"3 w***\\7o7Tv$F,$\"3wl^0o*R;v%F07$$\"3'GLLLQ*o]RF,$\"3c)G\"RGmf[\\F07$$ \"3A++D\"=lj;%F,$\"3]_ZKVA%))*[F07$$\"31++vV&R4h#FV7$$\"3&em;zRQb@&F,$!3#GV/9,Qr[#F07 $$\"3\\***\\(=>Y2aF,$!3O/d4F2![S&F07$$\"39mm;zXu9cF,$!3Q*>h1l\\R.*F07$ $\"3l******\\y))GeF,$!3-V=pexu;8F,7$$\"3'*)***\\i_QQgF,$!3q2I#pD.#Q<8!*RVk\"F,7$$\"3R***\\i?=bq(F,$!3+E'o _j$))RPF07$$\"3\"HLL$3s?6zF,$\"3;(oK3EqXY\"F,7$$\"3a***\\7`Wl7)F,$\"3y SEZb'y?;%F,7$$\"3#pmmm'*RRL)F,$\"3-pcl-+.fwF,7$$\"3Qmm;a<.Y&)F,$\"3\") *on#e&\\?B\"!#<7$$\"3=LLe9tOc()F,$\"3u]5'Q3:<#=Ffx7$$\"3u******\\Qk\\* )F,$\"3%*Gic#o+L\\#Ffx7$$\"31nmT5ASg!*F,$\"3QX,$4bc6%HFfx7$$\"3CLL$3dg 6<*F,$\"3P7J1=/IRMFfx7$$\"3y***\\(oTAq#*F,$\"3)H[*ewqZIRFfx7$$\"3Immmm xGp$*F,$\"3kWz[PJhnWFfx7$$\"3sK$eRA5\\Z*F,$\"3WdY()oe<%4&Ffx7$$\"3A++D \"oK0e*F,$\"3+\\e*ynz)zdFfx7$$\"3C+++]oi\"o*F,$\"3uX/'>]d]\\'Ffx7$$\"3 A++v=5s#y*F,$\"3]?m\"3M,8F(Ffx7$$\"35+]P40O\"*)*F,$\"3#*4AKdcix\")Ffx7 $$\"\"\"F)$\"3AN5M!HtJ;*Ffx-%&COLORG6&%$RGBG$\"\"&!\"\"F(F_\\l-%+AXESL ABELSG6$Q\"z6\"Q!F^]l-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(F_\\l;$! \"&Fi\\lFg\\l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The polynomial has three zeros between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "zrs := \+ [fsolve(phi(z),z=0..1)]:\nzrs;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$ \"+\"eZMc\"!#5$\"+bld=]F&$\"+Ptz`xF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "We choose the largest of the three ze ros and obtain an accurate value." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "c[3]=evalf[840](fsolve(phi(z),z=0..0.2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$$\"c_n3C+]$)y&z([/!4umJfk'o,1N zc'yp__99&4[ls'Q.$*Gs\\\">5(3#ye\\'Q(QiZfk!>3]h^D))Gr-8\"RoWV`7j\\C>qL 0<3L-$[O3wfxA#HM.u(*eP!HKYm2Wu!H@DzxQGc(z_*[my%G#ea*=.!o6dQ?6wQL[?Oa%e ?:Z_uY\"Q>N>KQ\"yp'ef`phMG#fXoiulr!)f#**Gc^8@*G%f;f#z[PdB,er^P(Ga!))3EsG0c d2yJEqA)Rlu<-K\\B#3)4CF?Kco^.tdFiJM]&H\"Rkz7,N#[CAgdsz8h@5U#R^'GHcC GeaIC(oBfl()pYQ.@.n4!*4i!=QbX;uC.QW>=5s,S)Gz!))*[*4i\"e9Q(GhV1Qu)p=\") [qF?M_Q')\\p#eIzXdKA97wiQ5*)R94eZMc\"!$S)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4] = (793016*c[3]-373581)/(6664*(2 97*c[3]-119));" "6#/&%\"cG6#\"\"%*&,&*&\"';Iz\"\"\"&F%6#\"\"$F,F,\"'\" et$!\"\"F,*&\"%kmF,,&*&\"$(HF,&F%6#F/F,F,\"$>\"F1F,F1" }{TEXT -1 15 " \+ ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 79 "can be used to express \+ all the coefficients of the order 7 scheme in terms of " }{XPPEDIT 18 0 "c[3] = r;" "6#/&%\"cG6#\"\"$%\"rG" }{TEXT -1 8 ", where " } {TEXT 269 1 "r" }{TEXT -1 62 " is the zero of the degree 5 polynomial \+ calculated previously." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "e16 := \{c[4]=(793016*c[3]-373581) /(6664*(297*c[3]-119))\}:\ne17 := `union`(map(u_->lhs(u_)=subs(c[3]=r, simplify(subs(e16,rhs(u_)))),e15),subs(c[3]=r,e16),\{c[3]=r\}):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4888 "e17 := \{b [1] = 18530703372187/317986769215500, b[5] = 397406378366413672608/803 680368165950784145, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[10,8] = 1419 40909070976808611971717/336796637787572286621096705, a[8,2] = 0, a[9,2 ] = 0, a[10,2] = 0, b[7] = 3273993983310919714698661013076893839848487 8542073/102638217510541641713801705364029167418501950476800, c[3] = r, a[10,4] = -5284658624/5262067085721*(-119+297*r)^3*(23926333761739086 *r-4821099352520075)/(39254292*r-10199933)/(-1586032*r+373581+1979208* r^2)/(793016*r-373581), a[8,1] = 25/37447012587565176701154*(841454330 944525618880626584*r^2-628310520362659740784506864*r+92920473849806665 985042065)/r/(793016*r-373581), a[6,1] = 17/288078*(2972790408*r^2-214 1412984*r+296884787)/r/(793016*r-373581), a[7,6] = -591904001481100995 10923215866/11994398797423357814553515510595, a[8,6] = -71158194294935 26581990/378493240381452882505729, a[9,8] = 4101332345819049706502423/ 10667097363909719917125000, c[2] = 1/200, c[6] = 17/19, c[9] = 44/45, \+ c[8] = 25/29, c[5] = 119/198, a[8,7] = -283808586236276727367033207531 26600/18970757838400100725353564886292693, a[10,9] = -5450098944698447 30599500000/14146180885244972240022216299, a[9,6] = -37256725960370611 5805768838168/1683402581390679277326651009375, a[4,2] = 0, b[6] = 3832 80144717771439/119491812371674080, b[8] = -16070516558763250309/737333 2958527296000, b[4] = 0, c[10] = 1, b[2] = 0, b[3] = 0, c[7] = 9526409 /47245149, a[9,7] = 231443608884459407960125470613488480291827060789/3 1418904104758208921880920386941521649080865000, b[10] = 1754022361907/ 1430294620800, b[9] = -68588512868559375/32259978418667428, a[10,6] = \+ -2147337295919765809044534716/15282699757843572625171655145, a[2,1] = \+ 1/200, a[10,7] = 76963822828349512771032624512097499610054141220500895 /7457737217451650694295122868750841403431006754989491, c[4] = (793016* r-373581)/(1979208*r-793016), a[6,5] = 2144360790/48013*(-116163*r+113 288*r^2+25160)/(39254292*r-10199933)/(198*r-119), a[4,1] = 3/88817792* (793016*r-373581)*(-793016*r+1319472*r^2+124527)/r/(-119+297*r)^2, a[6 ,3] = 289/41154*(733856844*r-296884787)/(-549560088*r^2+391883184*r^3+ 262706846*r-44456139)/r, a[9,1] = -418/46264144632297755469268921875*( 193727811082097025418053331802616*r^2-14717221300597552649246949966345 6*r+22401329358655482023238796021925)/r/(793016*r-373581), a[3,1] = -1 00*r^2+r, a[7,1] = 9526409/23250680745774508994365276220538*(254318791 511859585322981007160*r^2-148067000515155483853845258672*r+11323131287 229298506926196749)/r/(793016*r-373581), a[9,4] = -75105568364288/8940 8660625*(-119+297*r)^3*(20547952158*r-4329921281)/(39254292*r-10199933 )/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a[3,2] = 100*r^2, a[9,5] = 43688744/32332690129541182613242431159375*(17227404817373070 525201921438731784*r^2-13447774980756307315160789735713383*r+213644528 7741558614336689130224628)/(7772349816*r^2-6690847482*r+1213792027), a [4,3] = 1/88817792*(793016*r-373581)^2/r/(-119+297*r)^2, a[5,3] = 2348 55271133/23287176/r/(-1586032*r+373581+1979208*r^2), a[10,3] = -17/736 68939200094*(277411973149903835964*r-150447493203123226199)/(198*r-119 )/r/(-1586032*r+373581+1979208*r^2), a[7,4] = 146774983899712/69169633 9197194888866700667*(-119+297*r)^3*(1130562099516707654934906*r-471239 62106382485602745)/(39254292*r-10199933)/(-1586032*r+373581+1979208*r^ 2)/(793016*r-373581), a[10,1] = -1/78085826559236639846354190*(1977526 29356057708154596842884696*r^2-150016093703264621818653995024496*r+227 81101095348595330071055369945)/r/(793016*r-373581), a[5,4] = 786092970 32/2910897*(-119+297*r)^3/(-1586032*r+373581+1979208*r^2)/(793016*r-37 3581), a[6,4] = 89839196608/20577*(-119+297*r)^3*(12138*r+83383)/(3925 4292*r-10199933)/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a[ 8,3] = 4675/1943086214622*(6220605891953292*r-3068256512149915)/(-5495 60088*r^2+391883184*r^3+262706846*r-44456139)/r, a[7,3] = 161948953/33 21525820824929856337896602934*(23626050326990952091794755604*r-1132313 1287229298506926196749)/(198*r-119)/r/(-1586032*r+373581+1979208*r^2), a[5,1] = 119/23287176*(1973573707-15910044084*r+23317049448*r^2)/r/(7 93016*r-373581), a[8,5] = 1616086890/1178464813011121732104281*(609119 872272403028410104*r^2-1491441952850471754741213*r+4826080536650382947 63048)/(39254292*r-10199933)/(198*r-119), a[8,4] = 1453281121600/13879 1872473*(-119+297*r)^3*(350606239446*r-33617364751)/(39254292*r-101999 33)/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a[9,3] = -7106/ 625860624375*(3862519994901540*r-2121313830282151)/(198*r-119)/r/(-158 6032*r+373581+1979208*r^2), a[7,5] = -619221706835693214/3455176608311 25533752570405655*(116380082119314635944*r^2-35843098777289346303*r-76 05687041606942152)/(39254292*r-10199933)/(198*r-119), a[10,5] = 567804 6/124907764090140484772105334087995*(658033948305627158049889527017365 176*r^2-498518611969357062066905434423260837*r+75537414535272270827832 800085651592)/(39254292*r-10199933)/(198*r-119)\}:" }}}{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 9 "Examples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a[9,4]=subs(e17,a[9,4]); \n``;\na[10,4]=subs(e17,a[10,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\"*\"\"%,$*0\"/)Gk$ob5v\"\"\"\",D1m3%*)!\"\",&\"$>\"F.*&\"$( HF,%\"rGF,F,\"\"$,&*&\",e@&za?F,F3F,F,\"+\"G@*HVF.F,,&*&\")#Ha#RF,F3F, F,\")L**>5F.F.,(*&\"(Kge\"F,F3F,F.\"'\"et$F,*&\"(3#z>F,)F3\"\"#F,F,F., &*&\"';IzF,F3F,F,F@F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%,$*0\"+C'eYG&\"\"\" \".@d3n?E&!\"\",&\"$>\"F.*&\"$(HF,%\"rGF,F,\"\"$,&*&\"2'3R5F.F.,(*&\"(Kge\"F,F3F,F .\"'\"et$F,*&\"(3#z>F,)F3\"\"#F,F,F.,&*&\"';IzF,F3F,F,F@F.F.F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "RK7_10 := OrderConditions(7 ,10,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "simplify(subs(e17,RK7_10)):\nmap(u->lhs(u)-rh s(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}} {PARA 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 0 265 "for ii from 2 to 10 do\n print(c[ii]=subs(e17,c[ii]));print(``) ; \n for jj to ii-1 do\n print(a[ii,jj]=subs(e17,a[ii,jj])); \n end do:\n print(`_________________________________`);\nend do:p rint(``);\nfor ii to 10 do\n print(b[ii]=subs(e17,b[ii]));\nend do: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"$+#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"#\"\"\"#F(\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B___ ______________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"$%\"rG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",&*&\"$+\"F()%\"rG\"\"#F(!\" \"F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,$*&\"$+ \"\"\"\")%\"rGF(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B____________ _____________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"%*&,&*&\"';Iz\"\"\"%\"rGF,F,\"'\"et$!\"\"F,,&*&\"(3#z>F,F-F,F,F+F/F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",$*.\"\"$F(\")#z<)))!\"\",&*&\"';IzF(%\"rGF( F(\"'\"et$F-F(,(*&F0F(F1F(F-*&\"(s%>8F()F1\"\"#F(F(\"'FX7F(F(F1F-,&\"$ >\"F-*&\"$(HF(F1F(F(!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\" $,$**\")#z<)))!\"\",&*&\"';Iz\"\"\"%\"rGF0F0\"'\"et$F,\"\"#F1F,,&\"$> \"F,*&\"$(HF0F1F0F0!\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_______ __________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG 6#\"\"&#\"$>\"\"$)>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*,\"$>\"F(\")wrGB!\"\" ,(\"+2Pdt>F(*&\",%3W+\"f\"F(%\"rGF(F-*&\",[%\\qJBF()F2\"\"#F(F(F(F2F-, &*&\"';IzF(F2F(F(\"'\"et$F-F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" &\"\"$,$**\"-L6Fb[B\"\"\"\")wrGB!\"\"%\"rGF.,(*&\"(Kge\"F,F/F,F.\"'\"e t$F,*&\"(3#z>F,)F/\"\"#F,F,F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"&\"\"%,$*,\",KqH4'y\"\"\"\"((*3\"H!\"\",&\"$>\"F.*&\"$(HF,% \"rGF,F,\"\"$,(*&\"(Kge\"F,F3F,F.\"'\"et$F,*&\"(3#z>F,)F3\"\"#F,F,F.,& *&\"';IzF,F3F,F,F8F.F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B________ _________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"'#\"#<\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",$*,\"#R )*)\"\"\"\"&x0#!\"\",&\"$>\"F.*&\"$(HF,%\"rGF,F,\"\"$,&*&\"&Q@\"F,F3F, F,\"&$Q$)F,F,,&*&\")#Ha#RF,F3F,F,\")L**>5F.F.,(*&\"(Kge\"F,F3F,F.\"'\" et$F,*&\"(3#z>F,)F3\"\"#F,F,F.,&*&\"';IzF,F3F,F,F@F.F.F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,$*,\"+!zgV9#\"\"\"\"&8![!\" \",(*&\"'jh6F,%\"rGF,F.*&\"')G8\"F,)F2\"\"#F,F,\"&g^#F,F,,&*&\")#Ha#RF ,F2F,F,\")L**>5F.F.,&*&\"$)>F,F2F,F,\"$>\"F.F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"(4k_*\")\\^CZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"\",$*,\"(4k_*F(\"AQ0Aw_O%**3XxX2o]K#!\"\",(*&\"?gr+\")HK&ef=^\"z=VDF ()%\"rG\"\"#F(F(*&\"?s'e_%Q&Q[b^^+q1[\"F(F2F(F-\">\\n>Ep])HHsGJJK6F(F( F2F-,&*&\"';IzF(F2F(F(\"'\"et$F-F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"(\"\"$,$*.\"*`*[>;\"\"\"\"@MHg'*yLc)H\\#3#e_@L!\"\",&*&\">/cv%z\" 4_4*pK]giBF,%\"rGF,F,\">\\n>Ep])HHsGJJK6F.F,,&*&\"$)>F,F2F,F,\"$>\"F.F .F2F.,(*&\"(Kge\"F,F2F,F.\"'\"et$F,*&\"(3#z>F,)F2\"\"#F,F,F.F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,$*0\"07(**Q)\\xY\" \"\"\"\"(>Rjp\"p!\"\",&\"$>\"F.*&\"$(HF,%\"rGF,F,\"\"$,&*&\" :1\\$\\l2n^*4i08\"F,F3F,F,\"8XFg&[#Q1@'R7ZF.F,,&*&\")#Ha#RF,F3F,F,\")L **>5F.F.,(*&\"(Kge\"F,F3F,F.\"'\"et$F,*&\"(3#z>F,)F3\"\"#F,F,F.,&*&\"' ;IzF,F3F,F,F@F.F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"( \"\"&,$*,\"39KpNoq@#>'\"\"\"\"?bcSqDvLb7J3m@3!Q; \"F,)%\"rG\"\"#F,F,*&\"5.jM*Gx()4Ve$F,F3F,F.\"4_@%pgTqo0wF.F,,&*&\")#H a#RF,F3F,F,\")L**>5F.F.,&*&\"$)>F,F3F,F,\"$>\"F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#!>me@B4^*45\"[,S!>f\"A&f5b^`X\"y NBuz)R%*>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_____________________ ____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#D\" #H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",$*,\"#DF(\"8a6qw^c(e7qWP!\"\",(*& \"<%eE1))=c_W4La9%)F()%\"rG\"\"#F(F(*&\"!\"\",&*&\" 1#H`>*eg?iF,%\"rGF,F,\"1:*\\@^c#oIF.F,,**&\"*)3g&\\&F,)F2\"\"#F,F.*&\" *%=$)=RF,)F2F(F,F,*&\"*Yoqi#F,F2F,F,\")RhXWF.F.F2F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,$*0\".+;7\"G`9\"\"\"\"-tC(=zQ \"!\"\",&\"$>\"F.*&\"$(HF,%\"rGF,F,\"\"$,&*&\"-Y%Rig]$F,F3F,F,\",^ZO5F.F.,(*&\"(Kge\"F,F3F,F.\"'\"et$F,*& \"(3#z>F,)F3\"\"#F,F,F.,&*&\"';IzF,F3F,F,F@F.F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,$*,\"+!*o3;;\"\"\"\":\"G/@t@6,8[Yy6 !\"\",(*&\"9/,TGISsA()>\"4'F,)%\"rG\"\"#F,F,*&\":87uaW\"\\\"F,F3 F,F.\"9[Iw%HQ]m`!3E[F,F,,&*&\")#Ha#RF,F3F,F,\")L**>5F.F.,&*&\"$)>F,F3F ,F,\"$>\"F.F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"' #!7!*>eEN\\H%>e6(\"9Hd]#)GX\"QSK\\y$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\")\"\"(#!D+m7`2K.ntswiB'e3QG\"D$p#H')[c``s+,SQyvq*=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#W\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"\",$*,\"$=%F(\">v=#*o#pav(HKY9ki%!\"\",(*&\"B;E!=L`!=a-(4#36 ys$>F()%\"rG\"\"#F(F(*&\"BcMm*\\pC\\Eb(f+8A-'zQK-#[b' e$H8SAF(F(F2F-,&*&\"';IzF(F2F(F(\"'\"et$F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$,$*.\"%1r\"\"\"\"-vVigei!\"\",&*&\"1S:!\\**>D 'QF,%\"rGF,F,\"1^@GIQJ@@F.F,,&*&\"$)>F,F2F,F,\"$>\"F.F.F2F.,(*&\"(Kge \"F,F2F,F.\"'\"et$F,*&\"(3#z>F,)F2\"\"#F,F,F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%,$*0\"/)Gk$ob5v\"\"\"\",D1m3%*)!\"\" ,&\"$>\"F.*&\"$(HF,%\"rGF,F,\"\"$,&*&\",e@&za?F,F3F,F,\"+\"G@*HVF.F,,& *&\")#Ha#RF,F3F,F,\")L**>5F.F.,(*&\"(Kge\"F,F3F,F.\"'\"et$F,*&\"(3#z>F ,)F3\"\"#F,F,F.,&*&\"';IzF,F3F,F,F@F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&,$**\")W()oV\"\"\"\"Av$f6VUKh#=T&H,p KB$!\"\",(*&\"D%yJ(Q9#>?D02tt\"[SFs\"F,)%\"rG\"\"#F,F,*&\"D$Q8dt*yg^J2 jv!)\\xZM\"F,F3F,F.\"CGYAI\"*oOVhe:u(GXk8#F,F,,(*&\"+;)\\Bx(F,F2F,F,*& \"+#[Z3p'F,F3F,F.\"+F?z87F,F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"'#!?o\"Q)od!e61Pgfscs$\"@v$45lEtFz1R\"e-Mo\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"Q*ygq#=H![)[81ZD,'zSfW ))3OWJ#\"P+]'33\\;_TpQ?4)=#*3#eZ5/*=9$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\":BC]1(\\!>eMK85%\";+]7<*>(4RO(4n1\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"\",$**\";!>aj%)RmBfl#e3y!\"\",(*&\"B'p%)G%ofa\"3x0c$HEv(>F()%\"r G\"\"#F(F(*&\"B'\\C]*Rl==ikKq$4;+:F(F1F(F,\"AX*p`0r+L&f[`4,6yAF(F(F1F, ,&*&\"';IzF(F1F(F(\"'\"et$F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"$,$*.\"#<\"\"\"\"/%4+#R*oO(!\"\",&*&\"6kf$Q!*\\J(>Tx#F,%\"rGF,F,\" 6*>EK7.K\\Z/:F.F,,&*&\"$)>F,F2F,F,\"$>\"F.F.F2F.,(*&\"(Kge\"F,F2F,F.\" '\"et$F,*&\"(3#z>F,)F2\"\"#F,F,F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#5\"\"%,$*0\"+C'eYG&\"\"\"\".@d3n?E&!\"\",&\"$>\"F.*&\"$(H F,%\"rGF,F,\"\"$,&*&\"2'3R5F.F.,(*&\"(Kge\"F,F3F,F.\"'\"et$F,*&\"(3#z>F,)F3\"\"#F, F,F.,&*&\"';IzF,F3F,F,F@F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#5\"\"&,$*,\"(Y!yc\"\"\"\"B&*z3M`5sZ[S,4kx!\\7!\"\",(*&\"Ew^O< q_*))\\!eri0$[R.e'F,)%\"rG\"\"#F,F,*&\"EP3EBWV0p1iqNp>h=&)\\F,F3F,F.\" D#f^c3+G$y#3Fs_`9u`vF,F,,&*&\")#Ha#RF,F3F,F,\")L**>5F.F.,&*&\"$)>F,F3F ,F,\"$>\"F.F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'# !=;Z`W!4ew>fHPt9#\">X^lr^isN%yv*p#G:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#5\"\"(#\"V&*3]?79a+h*\\(47XiK5x7&\\$GG#Q'p(\"U\"\\*)\\v1 5V.9%3voG7&H%p];Xh3o(4244%>9\"<0n4@mGsvyPmzO$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!<++]*fIZ%)pW*)4]a\">*H;A-SA(\\C&)3= YT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B____________________________ _____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"/(=sLqI&=\"0+b@pn)zJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&#\"63En8kO yjS(R\"6XTy]f;o.o.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'# \"3R9x\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"(#\"St?ay[[)RQ*o285m)p9(>4J$)R*RF$\"T+oZ]>]=u;HSO0O ASv\"\".+3i%HI9" }}}{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 35 "We delay substituting a value for " }{XPPEDIT 18 0 "r=c[3]" "6#/%\"rG&%\"cG6#\"\"$" }{TEXT -1 39 " until the end of the next subsection." }}{PARA 0 "" 0 "" {TEXT -1 76 "#--------------- ------------------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#------------------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the embedded \+ order 6 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4888 "e17 := \{b[1] = 18530703372187/317986769215500, b[5] = 397406378 366413672608/803680368165950784145, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0 , a[10,8] = 141940909070976808611971717/336796637787572286621096705, a [8,2] = 0, a[9,2] = 0, a[10,2] = 0, b[7] = 327399398331091971469866101 30768938398484878542073/1026382175105416417138017053640291674185019504 76800, c[3] = r, a[10,4] = -5284658624/5262067085721*(-119+297*r)^3*(2 3926333761739086*r-4821099352520075)/(39254292*r-10199933)/(-1586032*r +373581+1979208*r^2)/(793016*r-373581), a[8,1] = 25/374470125875651767 01154*(841454330944525618880626584*r^2-628310520362659740784506864*r+9 2920473849806665985042065)/r/(793016*r-373581), a[6,1] = 17/288078*(29 72790408*r^2-2141412984*r+296884787)/r/(793016*r-373581), a[7,6] = -59 190400148110099510923215866/11994398797423357814553515510595, a[8,6] = -7115819429493526581990/378493240381452882505729, a[9,8] = 4101332345 819049706502423/10667097363909719917125000, c[2] = 1/200, c[6] = 17/19 , c[9] = 44/45, c[8] = 25/29, c[5] = 119/198, a[8,7] = -28380858623627 672736703320753126600/18970757838400100725353564886292693, a[10,9] = - 545009894469844730599500000/14146180885244972240022216299, a[9,6] = -3 72567259603706115805768838168/1683402581390679277326651009375, a[4,2] \+ = 0, b[6] = 383280144717771439/119491812371674080, b[8] = -16070516558 763250309/7373332958527296000, b[4] = 0, c[10] = 1, b[2] = 0, b[3] = 0 , c[7] = 9526409/47245149, a[9,7] = 2314436088844594079601254706134884 80291827060789/31418904104758208921880920386941521649080865000, b[10] \+ = 1754022361907/1430294620800, b[9] = -68588512868559375/3225997841866 7428, a[10,6] = -2147337295919765809044534716/152826997578435726251716 55145, a[2,1] = 1/200, a[10,7] = 7696382282834951277103262451209749961 0054141220500895/7457737217451650694295122868750841403431006754989491, c[4] = (793016*r-373581)/(1979208*r-793016), a[6,5] = 2144360790/4801 3*(-116163*r+113288*r^2+25160)/(39254292*r-10199933)/(198*r-119), a[4, 1] = 3/88817792*(793016*r-373581)*(-793016*r+1319472*r^2+124527)/r/(-1 19+297*r)^2, a[6,3] = 289/41154*(733856844*r-296884787)/(-549560088*r^ 2+391883184*r^3+262706846*r-44456139)/r, a[9,1] = -418/462641446322977 55469268921875*(193727811082097025418053331802616*r^2-1471722130059755 26492469499663456*r+22401329358655482023238796021925)/r/(793016*r-3735 81), a[3,1] = -100*r^2+r, a[7,1] = 9526409/232506807457745089943652762 20538*(254318791511859585322981007160*r^2-1480670005151554838538452586 72*r+11323131287229298506926196749)/r/(793016*r-373581), a[9,4] = -751 05568364288/89408660625*(-119+297*r)^3*(20547952158*r-4329921281)/(392 54292*r-10199933)/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a [3,2] = 100*r^2, a[9,5] = 43688744/32332690129541182613242431159375*(1 7227404817373070525201921438731784*r^2-1344777498075630731516078973571 3383*r+2136445287741558614336689130224628)/(7772349816*r^2-6690847482* r+1213792027), a[4,3] = 1/88817792*(793016*r-373581)^2/r/(-119+297*r)^ 2, a[5,3] = 234855271133/23287176/r/(-1586032*r+373581+1979208*r^2), a [10,3] = -17/73668939200094*(277411973149903835964*r-15044749320312322 6199)/(198*r-119)/r/(-1586032*r+373581+1979208*r^2), a[7,4] = 14677498 3899712/691696339197194888866700667*(-119+297*r)^3*(113056209951670765 4934906*r-47123962106382485602745)/(39254292*r-10199933)/(-1586032*r+3 73581+1979208*r^2)/(793016*r-373581), a[10,1] = -1/7808582655923663984 6354190*(197752629356057708154596842884696*r^2-15001609370326462181865 3995024496*r+22781101095348595330071055369945)/r/(793016*r-373581), a[ 5,4] = 78609297032/2910897*(-119+297*r)^3/(-1586032*r+373581+1979208*r ^2)/(793016*r-373581), a[6,4] = 89839196608/20577*(-119+297*r)^3*(1213 8*r+83383)/(39254292*r-10199933)/(-1586032*r+373581+1979208*r^2)/(7930 16*r-373581), a[8,3] = 4675/1943086214622*(6220605891953292*r-30682565 12149915)/(-549560088*r^2+391883184*r^3+262706846*r-44456139)/r, a[7,3 ] = 161948953/3321525820824929856337896602934*(23626050326990952091794 755604*r-11323131287229298506926196749)/(198*r-119)/r/(-1586032*r+3735 81+1979208*r^2), a[5,1] = 119/23287176*(1973573707-15910044084*r+23317 049448*r^2)/r/(793016*r-373581), a[8,5] = 1616086890/11784648130111217 32104281*(609119872272403028410104*r^2-1491441952850471754741213*r+482 608053665038294763048)/(39254292*r-10199933)/(198*r-119), a[8,4] = 145 3281121600/138791872473*(-119+297*r)^3*(350606239446*r-33617364751)/(3 9254292*r-10199933)/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a[9,3] = -7106/625860624375*(3862519994901540*r-2121313830282151)/(19 8*r-119)/r/(-1586032*r+373581+1979208*r^2), a[7,5] = -6192217068356932 14/345517660831125533752570405655*(116380082119314635944*r^2-358430987 77289346303*r-7605687041606942152)/(39254292*r-10199933)/(198*r-119), \+ a[10,5] = 5678046/124907764090140484772105334087995*(65803394830562715 8049889527017365176*r^2-498518611969357062066905434423260837*r+7553741 4535272270827832800085651592)/(39254292*r-10199933)/(198*r-119)\}:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We now turn our attention to the embedded order 6 scheme and intr oduce a new row corresponding to the node " }{XPPEDIT 18 0 "c[11] = 1 ;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The linking coefficients and weights can be chosen so as to form a n 11 stage order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We use the order 6 quadrature conditions which ar e given in abreviated form as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "subs(b=`b*`,QuadratureConditions(6)):\nListTools[Enu merate](%):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(l inalg[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(F57%\"\"%F)/*&F,F()F2F5F( #F(F;7%\"\"&F)/*&F,F()F2F;F(#F(FA7%\"\"'F)/*&F,F()F2FAF(#F(FGQ(pprint0 6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We incorporate the row \+ sum condition for the new tenth row" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(a[11,j],j = 1 .. 10) = c[11];" "6#/-%$SumG6$&%\" aG6$\"#6%\"jG/F+;\"\"\"\"#5&%\"cG6#F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "together with the stage-order equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j],j = 2 .. 1 0) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"#6%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"# \"#5*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^2;" "6#*$&%\" cG6#\"#6\"\"#" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(a[11,j]*c[j] ^2,j = 2 .. 10) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"#6%\"jG\"\"\"*$&%\"cG 6#F,\"\"#F-/F,;F2\"#5*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]^3;" "6#*$&%\"cG6#\"#6\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 50 "which ensure that the tenth row has stage-order 3." }} {PARA 0 "" 0 "" {TEXT -1 53 "We also incorporate the column simplifyin g conditions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(` b*`[i]*a[i,1],i=2..11)=`b*`[1]" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\" aG6$F+F,F,/F+;\"\"#\"#6&F)6#F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Sum(` b*`[i]*a[i,j],i = j+1 .. 11) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b* G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\" cG6#F0!\"\"F," }{TEXT -1 6 " , " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG \"\"&" }{TEXT -1 10 ", 6, 7, 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "`Qeqs*` := subs(b=`b*`,Quad ratureConditions(6,11,'expanded')):\nSO_eqs2 := [add(a[11,j],j=1..10)= c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n add(a[11,j]*c[j]^2,j= 2..10)=1/3*c[11]^3]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*` [1],seq(add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[$5..8])]:\n` cdns*` := [op(SO_eqs2),op(`Qeqs*`),op(`simp_eqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We specify that " } {XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[11,2] = 0;" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[11,10] = 0;" "6#/&%\"aG6$\"#6\"#5\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[10]=0" "6#/&%#b*G6#\"#5\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[11] = 3/2;" "6#/&%#b*G6#\"#6 *&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We also set " }{XPPEDIT 18 0 "b[11] = 0;" "6#/&%\"bG6#\"#6\"\"!" }{TEXT -1 66 ", so that the order 7 scheme can be regarded as a 11 sta ge scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 14 equations for the 14 unknown coefficients." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "e18 := \{c[11]=1,b[11]=0,a[ 11,2]=0,a[11,10]=0,`b*`[2]=0,`b*`[3]=0,`b*`[4]=0,`b*`[10]=0,`b*`[11]=3 /2\}:\ne19 := `union`(e17,e18):\n`eqns*` := simplify(subs(e19,`cdns*`) ):\nnops(%);\nindets(`eqns*`) minus \{r\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<0&%#b*G6#\" \"*&F%6#\"\"\"&%\"aG6$\"#6F'&F%6#\"\"'&F%6#\"\"&&F,6$F.F*&F,6$F.\"\"$& F%6#\"\"(&F,6$F.F<&F,6$F.F4&F%6#\"\")&F,6$F.\"\"%&F,6$F.F1&F,6$F.FC" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "e20 := solve(\{op(`eqns* `)\},indets(`eqns*`) minus \{r\}):\ninfolevel[solve] := 0:\ne21 := `un ion`(e19,e20):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e21" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6369 "e21 := \{`b*`[11] = 3/2, `b*`[1] = 10644890436863/181706725266 000, `b*`[7] = 432325592089717918474482148525568808656853/136057325232 1546818820057421907905293476160, a[11,4] = -2642329312/8830485*(210861 8078646025386*r^4-2960191138243599513*r^3+1527114722155249689*r^2-3406 05691688788827*r+27375531037328201)/(-1586032*r+373581+1979208*r^2)/(3 1129281624672*r^2-22753367727580*r+3810501170073), c[6] = 17/19, c[9] \+ = 44/45, c[8] = 25/29, a[9,8] = 4101332345819049706502423/106670973639 09719917125000, c[2] = 1/200, c[3] = r, a[10,4] = -5284658624/52620670 85721*(-119+297*r)^3*(23926333761739086*r-4821099352520075)/(39254292* r-10199933)/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a[5,4] \+ = 78609297032/2910897*(-119+297*r)^3/(-1586032*r+373581+1979208*r^2)/( 793016*r-373581), a[3,1] = -100*r^2+r, a[10,2] = 0, b[7] = 32739939833 109197146986610130768938398484878542073/102638217510541641713801705364 029167418501950476800, a[7,6] = -59190400148110099510923215866/1199439 8797423357814553515510595, a[5,2] = 0, a[6,2] = 0, b[1] = 185307033721 87/317986769215500, b[5] = 397406378366413672608/803680368165950784145 , a[7,2] = 0, a[10,8] = 141940909070976808611971717/336796637787572286 621096705, a[8,6] = -7115819429493526581990/378493240381452882505729, \+ a[8,2] = 0, a[9,2] = 0, a[6,1] = 17/288078*(2972790408*r^2-2141412984* r+296884787)/r/(793016*r-373581), a[8,1] = 25/37447012587565176701154* (841454330944525618880626584*r^2-628310520362659740784506864*r+9292047 3849806665985042065)/r/(793016*r-373581), a[10,6] = -21473372959197658 09044534716/15282699757843572625171655145, a[2,1] = 1/200, a[10,7] = 7 6963822828349512771032624512097499610054141220500895/74577372174516506 94295122868750841403431006754989491, a[10,9] = -5450098944698447305995 00000/14146180885244972240022216299, a[9,6] = -37256725960370611580576 8838168/1683402581390679277326651009375, a[9,4] = -75105568364288/8940 8660625*(-119+297*r)^3*(20547952158*r-4329921281)/(39254292*r-10199933 )/(-1586032*r+373581+1979208*r^2)/(793016*r-373581), a[11,3] = -17/247 253580*(931632861925500*r-505470810747877)/r/(-549560088*r^2+391883184 *r^3+262706846*r-44456139), `b*`[5] = 3667053975210116568/726654944092 1797325, `b*`[8] = -7030001847946787323/2764999859447736000, `b*`[6] = 839799471859174949/224047148196888900, a[10,5] = 5678046/124907764090 140484772105334087995*(658033948305627158049889527017365176*r^2-498518 611969357062066905434423260837*r+75537414535272270827832800085651592)/ (39254292*r-10199933)/(198*r-119), a[7,5] = -619221706835693214/345517 660831125533752570405655*(116380082119314635944*r^2-358430987772893463 03*r-7605687041606942152)/(39254292*r-10199933)/(198*r-119), a[8,4] = \+ 1453281121600/138791872473*(-119+297*r)^3*(350606239446*r-33617364751) /(39254292*r-10199933)/(-1586032*r+373581+1979208*r^2)/(793016*r-37358 1), a[9,3] = -7106/625860624375*(3862519994901540*r-2121313830282151)/ (198*r-119)/r/(-1586032*r+373581+1979208*r^2), a[5,1] = 119/23287176*( 1973573707-15910044084*r+23317049448*r^2)/r/(793016*r-373581), a[8,5] \+ = 1616086890/1178464813011121732104281*(609119872272403028410104*r^2-1 491441952850471754741213*r+482608053665038294763048)/(39254292*r-10199 933)/(198*r-119), a[7,3] = 161948953/3321525820824929856337896602934*( 23626050326990952091794755604*r-11323131287229298506926196749)/(198*r- 119)/r/(-1586032*r+373581+1979208*r^2), a[6,4] = 89839196608/20577*(-1 19+297*r)^3*(12138*r+83383)/(39254292*r-10199933)/(-1586032*r+373581+1 979208*r^2)/(793016*r-373581), a[8,3] = 4675/1943086214622*(6220605891 953292*r-3068256512149915)/(-549560088*r^2+391883184*r^3+262706846*r-4 4456139)/r, a[10,1] = -1/78085826559236639846354190*(19775262935605770 8154596842884696*r^2-150016093703264621818653995024496*r+2278110109534 8595330071055369945)/r/(793016*r-373581), a[7,4] = 146774983899712/691 696339197194888866700667*(-119+297*r)^3*(1130562099516707654934906*r-4 7123962106382485602745)/(39254292*r-10199933)/(-1586032*r+373581+19792 08*r^2)/(793016*r-373581), a[10,3] = -17/73668939200094*(2774119731499 03835964*r-150447493203123226199)/(198*r-119)/r/(-1586032*r+373581+197 9208*r^2), a[5,3] = 234855271133/23287176/r/(-1586032*r+373581+1979208 *r^2), a[9,5] = 43688744/32332690129541182613242431159375*(17227404817 373070525201921438731784*r^2-13447774980756307315160789735713383*r+213 6445287741558614336689130224628)/(7772349816*r^2-6690847482*r+12137920 27), a[4,3] = 1/88817792*(793016*r-373581)^2/r/(-119+297*r)^2, a[3,2] \+ = 100*r^2, a[7,1] = 9526409/23250680745774508994365276220538*(25431879 1511859585322981007160*r^2-148067000515155483853845258672*r+1132313128 7229298506926196749)/r/(793016*r-373581), a[9,1] = -418/46264144632297 755469268921875*(193727811082097025418053331802616*r^2-147172213005975 526492469499663456*r+22401329358655482023238796021925)/r/(793016*r-373 581), a[4,1] = 3/88817792*(793016*r-373581)*(-793016*r+1319472*r^2+124 527)/r/(-119+297*r)^2, a[6,3] = 289/41154*(733856844*r-296884787)/(-54 9560088*r^2+391883184*r^3+262706846*r-44456139)/r, a[6,5] = 2144360790 /48013*(-116163*r+113288*r^2+25160)/(39254292*r-10199933)/(198*r-119), c[4] = (793016*r-373581)/(1979208*r-793016), c[11] = 1, `b*`[10] = 0, `b*`[4] = 0, a[11,2] = 0, b[11] = 0, `b*`[2] = 0, `b*`[3] = 0, a[11,1 0] = 0, b[10] = 1754022361907/1430294620800, b[9] = -68588512868559375 /32259978418667428, b[2] = 0, b[3] = 0, c[7] = 9526409/47245149, a[9,7 ] = 231443608884459407960125470613488480291827060789/31418904104758208 921880920386941521649080865000, b[8] = -16070516558763250309/737333295 8527296000, b[4] = 0, c[10] = 1, a[4,2] = 0, b[6] = 383280144717771439 /119491812371674080, c[5] = 119/198, a[8,7] = -28380858623627672736703 320753126600/18970757838400100725353564886292693, a[11,6] = -871221978 04121104293267367/582943250254476957489125300, a[11,5] = 1331/35733610 12835218108070941280725*(-61078988567250554437463207205411923013*r+926 8380465737691551922715015613193908+80551640562060170770680173200952894 424*r^2)/(7772349816*r^2-6690847482*r+1213792027), a[11,8] = 607016351 56866657458077/141408544478495930424000, a[11,9] = -2472599248597125/6 4519956837334856, `b*`[9] = -333800898560611875/129039913674669712, a[ 11,1] = -13/402098244705402714202041000*(-5949614416817261359060274443 0491*r+78437349263344717172597486523576*r^2+90332551649538555589454389 65650)/r/(793016*r-373581), a[11,7] = 14114528561974414913592938450722 5418152792014862255637/13654459409305766148231745889098379435265792575 731520\}:" }}}{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 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a [11,5]=subs(e21,a[11,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"&,$**\"%J8\"\"\"\"@D2GT423\"=_$G,hLd$!\"\",(*&\"G8I#>T0s?juVa0 Dn&))*y5'F,%\"rGF,F.\"F3R>8c,:F#>b\"pPdY!Qo#*F,*&\"GCW*G&4?t,oq2 " 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 0 379 "for ii from 2 to 11 do\n print(c[ii]=subs(e21,c[ii ]));print(``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e21,a [ii,jj]));\n end do:\n print(`_________________________________`); \nend do:print(``);\nfor ii to 11 do\n print(b[ii]=subs(e21,b[ii])); \nend do:\nprint(`_________________________________`);print(``);\nfor \+ ii to 11 do\n print(`b*`[ii]=subs(e21,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" #\"\"\"#F(\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_________________ ________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$%\" rG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",&*&\"$+\"F()%\"rG\"\"#F(!\"\"F-F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,$*&\"$+\"\"\" \")%\"rGF(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B__________________ _______________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%*&,& *&\"';Iz\"\"\"%\"rGF,F,\"'\"et$!\"\"F,,&*&\"(3#z>F,F-F,F,F+F/F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"%\"\"\",$*.\"\"$F(\")#z<)))!\"\",&*&\"';IzF(%\"rGF(F(\"' \"et$F-F(,(*&F0F(F1F(F-*&\"(s%>8F()F1\"\"#F(F(\"'FX7F(F(F1F-,&\"$>\"F- *&\"$(HF(F1F(F(!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" %\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"St?ay[[)RQ*o285m)p9(>4J$)R*RF$\"T+oZ]>]=u;HSO0OASv\"\".+3i%HI9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B_______________________ __________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"/joV!*[k5\"0+gEDnq\"=" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&# \"4ol65_(R0nO\"4Dtz@4W\\lE(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G 6#\"\"'#\"3\\\\[r/C#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"K`ol3)ob_[@[u%=zr*3#fDBV\"LghZ$H0z!>Ud+#)=oa@BDt0O \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#!4BtyYz%=+Iq\"4+g tZ%f)**\\w#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#!3v=hg&) *3!QL\"37(pYn8*R!H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"\"$\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK7_11eqs := [op(RowSumConditions(11,'expanded')),op (OrderConditions(7,11,'expanded'))]:\n`RK6_11eqs*` := subs(b=`b*`,Orde rConditions(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "simplify(subs(e21,RK7_11eqs)):\nmap(u->lhs(u)-rhs(u) ,%);\nnops(%);\nsimplify(subs(e21,`RK6_11eqs*`)):\nmap(u->lhs(u)-rhs(u ),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 14 "We can obtain " }{TEXT 260 25 "accurate numerical values" }{TEXT -1 70 " for the coefficients by subsituting an accurat e numerical value for " }{XPPEDIT 18 0 "r=c[3]" "6#/%\"rG&%\"cG6#\"\" $" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 901 "eR := \{r=.156344758091439891038627612142232 5745793058269498638523420277048811869874380643612873814581620994898807 9288400172101819443803247416455538180620990096703210338466987655923687 2430545828245619017292865139242102161137972576022244823501127964391295 5034316227577303516856322027240980822349320217746539822702631780757560 5287226088805428737511977597140220186123221771580123573748792591659428 9211351562899259807165746219032260828274758621708659313547834882401406 2618985976250416329550953771006040384756486391221768455922834616953595 8669781383219351938146745247152058454362048333876112038571168003189545 8228478664895279756283877792521290744407664632290375897740334292227759 7608364830233081705337019244963125343446839113027128882551615008190645 9476238738649587820871019149722893033867265480951414525269786567935060 1686645931667409004487795788350002408\}:\nevalf[840](subs(eR,e21)):\ne 825 := evalf[825](%):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "e825: coefficients correct to 825 digits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57232 "e825 := \+ \{b[9] = -2.1261177542781685465260756865807396820266772341987714614017 0063027162393905539129792495753605834112376756933205602551670760113406 3315391964264690245896454843640739503752588918253671755555226805965226 6715706840214367474318916177811507251742874409569103042124983503911797 5181142341211246390285482188019253658578646776612427271809810945138572 2095838028600413895548046394062598733735569105188864375986762838610647 5433089600875834159210150893746536771452112149261861905206016496856174 6426378615772145708855583287508631331954931663320246176297782752442445 3399243078065119677380844096011851770442402711856555961903960706098522 0809719041270978287352014604302802531300794161372553140015663590233020 1515914094070694814685619183082988712242271377013838314074423505143274 137351260605826482262353212297074702904191757701691748960074436941, a[ 8,4] = -.4807142659618367597822391540871314829221696726643599551071504 9784051357013621638616962222878603149999551954570344041590139219985528 1776225706067447506381560064697228143690257642868675154646572981936578 8802949803859278590653704002083117021641880249548975767292722148246773 8842326711129171033615346832133548017485302855759144368130737651282108 7469660935008071217642469790060888474666274510870615191279474982063166 9089482407429683613579682718333161371423359500264298718137114086643032 7902220986604718519110490752010126970680555902494973664732117363265485 2829169773889459020448095303509716185211059033805832505140533853976794 1009976082963987452321694187132087079001550159173442952020628934006773 6748623135571716456542216884432218586269458270437563130105646663713023 9290967664978005557544387288850486857384278392359328270173995578, a[7, 5] = .5193029634013270828988884720187179868848783957920475461773356539 6001365293647071211098643145086772097617446873698132691032558979232921 0762934774186347328638331487147616555743535546448477691501538246681097 9720143469960126114581237740557970621599008244175307601922442688191319 8919905972475134240064946404534972443943109856992921766767230294525681 8494203019361108216487697629495874729717517507389538367060904336864763 3483801950490440103947506684154142231647765398430428836763574977828864 7765171845069121991117794917872730031925674331569923913609203358866855 1758382306440638705112736278459262902403454978409497258858485587110744 8553390071932696818474324212888142137762994480246344282263721338935297 3684959892627933315647688204292601385168593282518085604977754467118371 1080504621927212935435952022350634430412626019071317330969992e-1, a[8, 1] = -.260994278112986324034047099898264146640514636468446500245951734 5829644252145903489969246911493910398783852368118354949059101008953110 4394419393707571528684812379340402667425164884499528431353592178079072 9389406429849911404733405161311473530320212445956423542481277911288316 2823701089817240526183161621111454755328830267355481692957824433085900 8584589933697945363366851370396910694607450410760264308414424923822853 5733135937405507436041536587276302155899604016639190317014174182451649 3174857943940542172435684233252147785264559961641183312439580275471862 5361080641291845014326922797441096465324631444702507032845926932826110 5623582809150501264611938126058534635682104240891076839968772859137717 2087193773665063666263123022856373725433954541400672130471219255880378 17162068823282771966991775158697853848694838315622100650574454, c[6] = .89473684210526315789473684210526315789473684210526315789473684210526 3157894736842105263157894736842105263157894736842105263157894736842105 2631578947368421052631578947368421052631578947368421052631578947368421 0526315789473684210526315789473684210526315789473684210526315789473684 2105263157894736842105263157894736842105263157894736842105263157894736 8421052631578947368421052631578947368421052631578947368421052631578947 3684210526315789473684210526315789473684210526315789473684210526315789 4736842105263157894736842105263157894736842105263157894736842105263157 8947368421052631578947368421052631578947368421052631578947368421052631 5789473684210526315789473684210526315789473684210526315789473684210526 3157894736842105263157894736842105263157894736842105263157894736842105 263157894736842105263157894736842105263157894736842105263, c[9] = .977 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777777777777777777777777 7777777777777777777777777777777777777777777777777778, a[10,1] = 1.3654 2734165224359169272208404842791794965285905826293393408470483248180139 7164709438067499289676825954248416999812560284946100375175480394298141 1319674989814137518493968374501121266886876352415390688875274980150139 1706334761065457057592896602678856415429424181342216867164284001966805 6605003925000643075781686680039228189846003213827469299727675531202059 5758587624898225392240701520487241625210404771388914800433008257799684 5993283317805338844093332097879630829413948096746507183132965721138903 0775386527602533856338350775038364445871581634719142994546109194743986 4828756530910312530099512879858606152021712447101578323036877852691171 1200190340838781346881303955356920677085912878934443786815573871855442 6916554512335696540708027014433225367355462762318040756995473636731903 12713071781153648901377785681220413617690708302781, a[8,3] = 2.1052621 3803539884982239390471321887480873980383581555623192474381678519776772 3519212762262660582171956819180113961568922884201216556151460320529142 7305147369322113496460318201663690800919006434610100319777980806666881 8972830775400143318091024970451341712639847684928308000668911070192812 2130777310208019065111822587421864215349338661247504938148009612348623 8120106032595735889741046850418190054957545135975334168578638340134745 9286128081025755033276799490909549911694356456300733851924845978337633 6279752481979330021491992329807367790085698005090564049720454103285169 6992548786313484968453193919389706879304905702921708793182264110153912 4884168798774772963284467766391503605694388187711872314583946825500323 9071984802742278939752573904477579187608278902020072999461321318897550 19296543496491912196194240003013513052212913555, a[7,3] = .15531540346 2653352899504463709704545171027769160714222894848594380217748724373030 7450265418444982874019647289879682554904860312513530446782911506365700 1810773595914777591322422104869162991181637240796293447367655497413473 7004083735354389513106818333048359171820694341284780669391796777125069 4305787546604802448914946346421993086085604801562712790779898757764652 1162224405691513496103853227544836462224058909350489945148528833177073 2089657514597693131212018024341130185674651463863179535541581430440277 2679493751978240354476885766544033100924394003790992848127982839646645 3819046560054043974472930990763532619807745285296499796664800822133158 2745829081338567406725937778750207627959832353766772426846793745044195 1223671506304941425853921454988466572075487351433836746310065141014611 78227459471048063503065721900891259554637958, a[7,1] = .58901215642276 4352272020349179558611168210489118580756894311082952122123926873486426 7518500741636032915396067142654059226633390358414456598590159797239763 4172668866368377336755021488475890941184292418635416608963662838462887 0551642696117063634309031678528404628140600046596900802578689275032736 6282033694744491077849164761433289070215166850752076927325063230633920 9848212760521653569497711101233061437554706980721163398588246265080367 8616654533909888517627728313335215583485748706027394302432434588110428 7192872607524929592038808393790492341600631918753073884480961301114664 0552686998402292336047014159606958768315885430290505813421128670108951 7482516864757857726994211098006581473048714394782874971643608743026708 7676612446753692791135564484095300352945114844521041677818405251273066 32272015382614662175804904170769490143022e-1, a[10,3] = -10.3167528254 5292163394755035756687456081457944872052196026886811554378684298364983 3306140819827120392098337459455930601463968707449989578782841435533298 3403014511262989754811616165492456267710463833590551276893786078420329 0626712792402477363739364848557778806374378161583534766834227593136959 8276050826837702127246633028180180105713234082749349296431152257040346 2219526473674078458318292394762984201608984155677645370256328699989278 2778399270484837872645185691167331845501354484112781745709523690079012 1878083454892310720228498523031733777868987356074142175198690973703983 0075384751243910677846242518223273793245958453091160704705780130860495 4889239182963091134556433581520813891362609792720048810352052957394069 6053674103678359273512007045026323573215797022996675205311648083664497 2267347235932293759032269697091045447044262, a[6,3] = .534076298945585 3802077633913678076550345046963931714018067813698321333152122935083800 5710897294383888887441574043524993618990470905217532264176038985999709 3354725608443227278560911707049852729525640462257882638234259496546353 6492397981540882632841371124212341672114846863688184797266334658418422 5358728941924885976600632681611692254426258459116583670181437103259453 0367805184640944314277047825605076086339457581039193792965106409353026 1568949861749409817926065883204854497144672902529436191070040895509199 0682095564207300275084941164900198512320609271268333284867507360346924 3612421711669092550054161697144815767185757934116768613196862434269511 1081087049817161176965947773270263204698999492965416019883811652471431 9694296381326732428928758343736673377084072371792328384961293688933184 7553376954093954478827265891798467324998, a[9,1] = .955561779929348360 0208548823007076277061903733301676149288546251823077129509222064179265 8466382989564962800727785971204738963336426658149129877809249031251676 9783263274791134496612919808800160950826655513152677195417514057418641 4225532979426708497502949469030486192481378906084051684003710289687562 8561177159904545913842184028120307750184674760439135332560000753689918 1757328595950177433433917448938599587130534978762881312346945354005597 6418049941950964090796124319201576674387841560450098159581916238184918 9511877999449896997120191035586172501801318419839863717147328696053530 8820504855210980057993854779230431288545673949582049157923138996949998 2753966153877627302253810683392862192308094680949008724854885397388644 4714827001239226626865826315321637999863513632566106704830644559277342 2569441566677930423305171625308118791, a[10,4] = -2.349778040727092612 0107097224666549560843188344244116660222882815467060284935986031716423 2448343144138815250982015681678370280210406241578349736300114550380138 4593096752878464539110091570436077550788904976760244181696770329939541 2931902117597034961688083614851541775129752707097955106100661872193163 7445282855688380714053686348582269169663752456273346736128170088549943 4360899045277056896142609068553957974031944886646519687672155956468061 3335593008162261134323553214271844152811189358645969912143314885266109 9248440971516771796693632889624143723440823424496712486774764954634128 5528471324057887364997735342418023271086150194672074121394865744555255 9752058829668627038427004249439865192816920867761406842850463094329922 8010850392401124759358522198547077926742358412327804592754975604477933 131804278221426431141604216513714128, a[5,4] = .2376128287153792882814 8496608348868267319765940833343694170507992095457510931280188629401778 6813508950008761865101531999575177603341174263501117214082369237964584 1330204105589152901818402883289568297521506946021369173814115940450995 7392113781025895872225426476968694132775745959059002970776241122171213 8617838294297089871339617160003646680579015557521757930169609639112436 1288231087518206166606897635601089519784630643198532460797497110900990 5555362816386649588706549216858500570367230630439865629415367925429320 5961076175328082011393602158162002605797964087233838750345962512383582 6002664942205331828229574160568713056024189743306794221091360450364282 8858436063299147475945193179536245342454710806810663390083910395944096 1754736605214472722801171082321654372078499708292777942792232677905305 673216304943758162794781713995786, a[6,1] = -.525528479836865959304222 0802836927533529760394456099529786529575067646255979073597320179850800 2150416721822235335867942644580627401924778495625160670516694329587170 9604960210579278022848468467147167597185148515700833749256344706567204 7829673521994313186555066476084917226179298116423269115976302560591402 3964263873333619900674921966015232801681860907853061231720451965784353 7452368752417255557408406858138793997042627599683636508851533098694426 3633329653842433826446145693247830943048136343775522753470414256157784 0108079660425469315004756802076202918792399598297096970745132366305141 7156838437591480682942604495229563390130242412706797099744813852937534 4799797237223522383295091093910991494883166962095919763046461567216258 2422285177073934271982736533669633423464368128889965100330536720465294 7190153834594579803980720310464e-1, a[7,6] = -.49348367640423468407142 6288869737389281835749886142153303063507737995170710395074462037014075 9441238060657091364579586142991052648889658359052115275639703482705814 0423895537182412017136102780956705619436816734675731576190761944516127 9950582644464516040648301682414426887120409417419561531979091411984040 4100415568369690432278853272461845534616076592465785044449700398886408 2511484991006671647686770257308029871322028409910480574286684550480169 3485065318882774964236247818157207822649235981759501452572799054254982 3944723943473815948824789978595120568153146338144431856095821058568188 0174044995107837344607571427290584109115871804066536613067484433858976 4454425222953657222512112634075320058342619824392967849585697650926478 1453179144734766793970940199339847290044184936246726935245883523622694 55200609117726195263219130039142e-2, a[10,6] = -.140507719836456467981 7135328927924806945492038984288960370504750791822976502728975025294044 3884194882357608188435650128961732687247155069946612865986334453669262 3619289316421575511593294753140011581732728615142768336749402599856587 7202498146375999853169183512945726047188870223016297322226149410153014 7232758449799094006134176587695780108311155844229325953204951734374080 5483054478706934234353654348942045005590228752880326818852072398755312 1522829962230897337785213356310650098498452743202688461563031215644391 5213797058257434569346693078172124703257545741146863367459625083574495 5078421454730953055464733248640428138707143517281706229351992587119166 4395436649835638972227592274237192360871074949737035195684003612887078 5819876211468295986997479797116839288084085727343463307974615330459449 1127243649484107277751152086958017, a[2,1] = .500000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000e-2, a[3,1] = -2.2880235801756460369936 1828967241914480881230276069905046757413920423056580945190880292352015 2785787850560326916464486743256656542637018174461605968488668541977130 2647325218845722109336658033634721375738311601183922583647941451570109 2217656286737668185616930011397185641015382013473089235769151896044128 6868272204955138396342883086061804566849279074940801670869020250990670 5299317715655742344915498465482802208946974268731511028621155024668286 6085467366424413659428649728713303827438651226225546305291288104592253 7875703135892984354713099804490127833404735220866827281033357117564787 6174216176053259974111956919033310669019504644812236189285546230512255 2868786019104638329103136929313820329966219309391109515925058839354208 2632500921844752885069033153770430473174425773618024351089097773154947 51721191686765958193442088683528, b[6] = 3.207584997753613744999728890 6445837838684186590775496301999177361299480448770732506133147138624596 1333908867893842372410543063564004072090456418800695178696904488295753 9003995093920125733324084788189884044168063855096242923120544862878644 4295518230152807572160065546586355433565767700270895302082281904154619 3054021449763806304642623332415054883744672797619738020573808700484084 8773101055542103055850889142524946904061104056102529319446792132682592 7213653720329542664044126584344123699123044687967787774451737406872723 5879475556313037848357872970462679509340911317261348248659081930400590 8779233873697675007962725038234933596295838510973896802278025560432316 0285646562779881324200066246537115378912024998742354895877167609119174 9261851602765138639338247211043256746520269594148305301813945573016998 419764814412325246075674821, c[5] = .601010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101, a[4,1] = -.33584189932752773989147788071991456 8362232844059976143705289766054505550649540946700424646747200151586051 1330066644396514328151809097221251160897508857106618902114093435116378 7710157337687065115609620926274968119242328348991663599247384191570819 0293325682964773215732978496406449502157300294784271873725213672951346 7632258585707119118985931359087371090342202303335014488890557746867653 1342592178670452882725362430040720281799099377981792376484344444789613 5983258542053199083834301089571948089228198311484939924000865144097619 2194466653071697774914763707617979170463327969194689171671857524148327 7718195190097550406934948369809959937900381847420356769256788769468643 2127593525960896608899492644945637523814081925221964079228762043687904 6092558909470513494585339217900531646876102676358219382831649627237412 70108205093704591345, a[9,4] = -2.032774485543327751845846899356653000 6448605716761158456678345869598358466346447510704119855850817275797429 6904226644745435427116106626190321160141681958430962893563156276424831 3010226733933451657251103021404506818205751273998336610743986131252594 7999577407023568065421549790267096433598342323586394304058672190501554 2407286493128433617178216042909051701394416359071646136556975668370389 5498874294339136358792641382614426009494449226477095743416624401913168 0281004760521056326570395173569573664326323953824511669266393521858214 1854823564047744079375194373214406732389620607286737527032447741751389 7106201176525919503508700821184771601386844085217843012150269903911789 1065525777160779235416648698632734608552364901827117927567116813430471 9845955899102877151382760268985487005083262513040419101595995027001263 173018792822203628, a[8,6] = -.188003870883455269502755430221338389585 4033546940015699959883559517807852823214785941676093863929986188401889 6836901399370726633272958934488538057517745133342990457252156648933148 2501333774012583531861880226204138031079875568717643067525516746753696 6248457336238816890451713686268729410709408255121640482721965258463857 9747648589826586938224349683250236820787467423639175232381155781845857 9430032744454872835659490795856106917425990853496370693397960539837233 3539044299454206502021422016050044503248376465726773558080587701046470 3898741544667807270151843455742356807396657618817973257030224388683135 6579195215444938960327210346365362865903217580764268138405943081260699 8327295004447781462707567425087039364659707970428381005039175439056653 4519104410527248295397978161404729818096039297470352313343263257883975 3384252601219360e-1, a[10,8] = .42144396097119943803723889298139233220 3586527305376678827675134038936708907476159253946194123939393712319178 9794077985259855585085508937746321225152973722043612766887895622813006 8377189923620613253978099740118856446934950914110300239479694751690717 9465903918070942054066278794063591402842649164840080390051549217310486 2174889756957515056181164601531987204038389844488208692792946011081752 4431863815810840367922931174770062110411620329658075138633621761029957 5169407940552210771606143358178816355120878504795652905180695621287903 7409767482523322987817907302787348201529745795195009988836607933279015 1137797929385958869949451133012626665084033886129783134148311016451795 8733358331057151270395730656665477515918118798152819491873022295376806 6758903317340396097574649848638377536589967505105300748048906039592194 79365513355336270, b[1] = .5827507672065664142994104818193773375769737 8834074114766885248196399734523784092444816558006355389427006044954279 8345561122879711948076122862802494537331087593789635390579202285395031 3495508070783026858410758945799035579308357426214324579809209146846972 8338114513007100375446658502609726798688293458155401301516136413597990 3077968413480429454267537315696760038677421234670980049262376068810454 0512229524617782249439654909810899606756126179401051772500236772512377 6314214589866431693841274163130999383956348299376591173465914244117796 8646633054586779290607997161271752824866550552111991665979869105753070 5238563347335654706593048249530149797604480577040091990896200388645317 5548537809349200004246866633261713604606299572820724632279067692102185 9005460033415174462177166570728672688919239389919094122662805560209221 792416503668e-1, b[7] = .318983908988351079518342976376755971224239126 8473214591672025464654613689221098730459407742377722645282550673870602 7740635086800101491841543209573086030562807438639704657838941883539351 8664704868023257421306419885262859194775713611182801819016335049886405 1598088654518124832717445589497294022755128650455613659762965153342544 8096464828656565166745715352568411174118020337510339355236513627709093 3337858464011612240300753692798636399656564338605475436934291727996416 2784371836502876534884353558656520695985008454644029139858715725962519 9503340800393257143815974398114122392506183099053422693502910653967234 6316957386473786889102690519233828105974263521567734133484809952569303 6901982292504698079978249656205659795574859538427967742368521420629115 0546899382920021763152761442381090765122250238677669939560639573699739 0858691761, a[8,5] = 1.01334764442344312437224222812708275074946436608 0576212911294590337370862205604518830487771637623751499773836350701624 6252453380379657127823761642803179158566618919568095969281534614064459 6997383539904548151228299367843099614770240308671048682129433304740801 1527270393373475877912977601709610724615079533353365706669942021791716 7611586582174184868109460873929827172364310791527659598450790657124712 7303601876528556143420775016116931522070527805460389224188508606381425 5186355779585349973156899900124492346311047217211400806870973388340737 4323222109776317020453479194303982742417489259526511702241042912278947 5894075147556181449634446231447190610620604216395740463503645688188453 8773547197473978903426073379409205488346770630469363541425396157489075 5027520995422200535354033735872885744185541269568844667098282652220211 4320631, a[5,1] = -.73432196848555860257549147449805476572882939389609 0862971379188016273202704616761562632095142628732907638665813512853148 7099597747558254353858112118200097366942262663799259022517656159662455 8337780414138542593276701180674521508184846319881608367147253897908083 0270911726426921319006715125566553506414502246468671260567401710780379 2643764802600023389228190981533819760410021780836964808353773855887813 0305473809316025228652151245338509800610603778565692196041070023146265 7440457085316600707147814699483992245166397699823724116267661255895485 1121889906537917405083983702055177959236962076268660866372305641236460 6688864738529431855640001260273608786175498038341958754605544123059042 5524059740048398294652105711890619560968697830009354525344390030975894 3834473551822640888920721384713743323972062387824496797698700925129480 46728e-2, a[9,3] = -7.193562790025188642746037560530784653026836953325 5601965447116323935206412318815684388687399089146727765817108557513839 9519131719300335024729879088343942531122641617026899279527962685989223 1655288900404039749189303786106021570621435409680228373214295819797334 6793968545103975372123940171280664733317251369700832467820731715146393 5244133242109589376103760911681677099180270506673182794446177833416505 1379249744275465043469665409400276856180980828795058795262166975692908 1998429019501364765179655425798226877811507784316984794125754912347809 6999761493216227098070270409886061895158281635675659673728974808485160 1672896190422765234708811519078343211487246147457807554803816518770725 2360750506726192676134538887899029080289959855881720285345058717336495 2589214189083721296112195400223570042862047587417232637089744038246496 424413, c[8] = .862068965517241379310344827586206896551724137931034482 7586206896551724137931034482758620689655172413793103448275862068965517 2413793103448275862068965517241379310344827586206896551724137931034482 7586206896551724137931034482758620689655172413793103448275862068965517 2413793103448275862068965517241379310344827586206896551724137931034482 7586206896551724137931034482758620689655172413793103448275862068965517 2413793103448275862068965517241379310344827586206896551724137931034482 7586206896551724137931034482758620689655172413793103448275862068965517 2413793103448275862068965517241379310344827586206896551724137931034482 7586206896551724137931034482758620689655172413793103448275862068965517 2413793103448275862068965517241379310344827586206896551724137931034482 7586206896551724137931034482758620689655172413793103448275862068965517 2, b[5] = .49448312302728018185851013296614710886261150097525455834036 3832058208210786285199063539305222643229502495278389686520645570191429 7984881367503830507538215000895554894340508250463199112508108506905834 1308247953598318090238913909331319346540663690610603377852376127214062 3642026653018723920296822447776245585284339972353249406235602659509590 1185902006239779303213463516804461759684230816430413724405172097731509 8067541465339046230944741628537967923312144304141435027720290768834915 6429617312998686734631795852244388717568481322096099262762339881243476 2596224559476236476064339257698119330835377416306486709971388054150187 9288156141592025741621208395000803731573992762804495687176022882995032 1517926027381454350869427868670665951475110446636358425928621341600156 380282107550047258882098800448374132690309084531114151813702778983, c[ 2] = .5000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000e-2, a[8, 7] = -1.49603188577843198411772950824656526048525538738315067403189757 6480327572301185706741424284458626842341493870224964174444904787732725 0506555495301526538446382695187687572455211294513245862227623067523147 6505163229285470778403615858093479073709498262767218749592272153020395 0379696185755393706923451656666482848870071485806853773803386086792385 7641132324748349911566915408161000531584682255657985086998297252999479 2208880361050118526616497816759832997974035986343434069399631111652803 7006588108597085431709797746419650360210864950110790896123976109931559 4617803699247255751077232377136170405388530463561589509486285988591463 0851520369793884944649613983415733470537435069542343442161376300037565 4028865138698434196620826151457052730502271493514045272164090032739475 46304834236072071116159077778802906777494770926730512649799472, c[7] = .20163782317630112670403473592601009682496715165402483967189943670195 6427314897451164774609981651237886878079271164961295814730100650121772 2903149273589972168359549464009521908799567972576401441764952418712871 4526860736538263431024421152740993577986175892894315985753373325164029 0096238240247691884726620292805087777371598510568778182919901469672579 5065224579988095709043059637720689588681369170832755760808374210016778 6538253906237019169946950532423974364013541369083204711662566669013997 6063997596874972285514434508397888638259983051381634969549995492658939 4394755745187722870764996423230668613194552524323714165871294003115536 7929943453030489966282040935038642803306642127427728082728662788215568 9677261892009272740361132102684235369857760423191807480594462724628088 272089056169555100778706402217082646940112306556594836858, c[4] = .516 1473767552496839183849376838036967595743406212719948888628161544885588 1173722689052529412561970256681432501928077307242229218856053296337716 5039344102981840609567903558311382607369274795772758874788833202817207 6470060347458258571010171267981455139159180073478874736074389657583051 9843013087214242600763962251778506302419430873365471126585157565839156 8011660474644836422364143842691595152584776795254091856248834176478704 9470915097597344880724490509507801789450922537877783811556347710578112 4887098804535910788505481328318372097270588120323591254068552945492069 6632734220419113484664070628531886324883665443870983749319839307992397 1381989249533810055208609649966888726022973672058601958689089384615890 8343612976948972580518325686849741381281385066345650448874543262289182 0406382784645726417430695778865402066734240492999790, c[3] = .15634475 8091439891038627612142232574579305826949863852342027704881186987438064 3612873814581620994898807928840017210181944380324741645553818062099009 6703210338466987655923687243054582824561901729286513924210216113797257 6022244823501127964391295503431622757730351685632202724098082234932021 7746539822702631780757560528722608880542873751197759714022018612322177 1580123573748792591659428921135156289925980716574621903226082827475862 1708659313547834882401406261898597625041632955095377100604038475648639 1221768455922834616953595866978138321935193814674524715205845436204833 3876112038571168003189545822847866489527975628387779252129074440766463 2290375897740334292227759760836483023308170533701924496312534344683911 3027128882551615008190645947623873864958782087101914972289303386726548 09514145252697865679350601686645931667409004488, c[10] = 1., a[4,2] = \+ 0., b[4] = 0., b[2] = 0., b[3] = 0., c[11] = 1., `b*`[10] = 0., a[9,2] = 0., a[8,2] = 0., a[7,2] = 0., a[5,2] = 0., a[6,2] = 0., a[10,2] = 0 ., a[11,3] = -10.32936288816363880657000532897001857771766638003174253 2359288499003249931471517346574470046615858779370007317325040377862981 0188897672920575025034841189979328992553874184092797075547728452977842 1773069134004791457640991465027040949241172730512819355400603219641827 1232820605420512764858956081948594430229238303410289265359295207279968 0419182596734213495268252316047215935073790900200560965274413437534602 7183247038684222611872323897412595400907323532989898648692212836974315 5638288129576803972404826668033510031648595194665162657178496464697067 9657715119810608431053362681193249249108489829752764255934989078945746 6175462790647656509171504175150782890359773208381263710183640079600432 5942803965141565175046137211008049142997464048594062099943020125028789 8943695012124740926547174360188424531335156174225689951202138665367342 , a[6,5] = 1.219592837482178054040188049503337679370663063169973478214 9742277319195979426654704762082534170377182991183979320421534065211088 7069893441872309556441899886804628460134013795416995461956471080926303 3641250134430236171221984659601901925149453760319129668596574713335272 2848174597355005493944485107920921950006512739541137416865621566508476 3424174814123416949449820112879139851275468129091846093761692842860582 3165159841306238022577224924048738851864584064488006295589252180481165 0428902650712731486372055245035066503998994171543098982898596835179342 2630619646766734557115548352175483057643219110733875398438770077670225 6249490161359367387502462206412690525052846073818846040060085859689954 6592814717400546755898296307833147521214924999877704411157859346213573 6441451886612129564542978416448368361167655042136496665351547428854, a [9,8] = .3844843827614434989460427447205101477825659907283179228627418 5953754421851256115535923509269417572127497068293812165244607483000374 1923482942848426423342900765714126176571718059531447770686686805084113 4218433447431941821168402683339120764360094407809519143475952193013742 1383299205289097148377206968267569327199423475144570916563752887570783 2454458611866503175018977483580524288248674195234339167409387308755380 9249441298155761494739174645861401229405471664424038425683452529721644 5865469085891766076147259879331001874842375850354856671891528359588442 0075228683686861767290589346579030941187605626799789870832410014418551 7646445742892958176304273883668158736367024640548492350010004556176206 9164901564081686842213256450725015604235098154728534346430537713607301 5134443353521675859264513770466348500439569126446975615574302947, b[10 ] = 1.2263364039822305119306227904678140840841229849083132341456681230 3570400171919600636171252249458225746827894383422685665462302772019206 6319767284690049503400887012550806063969782078061773271265315521488675 9364368295329605143684533949412725079305562889242741952427819687986901 7822429329799238520634727119012877434153879466229759311418085702416703 0972028948289253050094390734633740992672549663832169255404361792031707 8214099929585640234269697394641841052500447186188564598704250401946278 5076007467565804048083014366322365462677967445516662884173240959727169 5199540528118862376413686082989706787548592310275994851871290768403342 9317362080650286117611049327663107995099298915016936068728589040639143 8202352218480775817513303200419894916240462448923725966934713930792964 078523646224147220116567469090211654944091641793861104340105, a[9,7] = 7.3663807022921792355102348489593224329933669195994530575659268287597 6558419027898146158625130750396961595236074752407116093024373799416073 1288776413154558113120305756145939944574582243798623510126616488331233 6583242505289648856507144219615658281348758251485210072936505263507272 1915183524277262540417464452679795852693598200788553078439342883303907 4361497946728054201992018212714914622090715528307293596200522684568224 4013510056860020059789506613223473601998071401741867813275844642768765 3321658138537477457695308075624909719149721404653045524371577012221480 8689958625181937047767884484897429166769963619813727372130981559304706 2401742263033475953933980530261822836810570259657201612259103497820487 0037480285474419449025417924760233894774538709117392117263659651389743 707574855536902713565028246426563281217597708669765216058, b[8] = -2.1 7954575619396361321107015205649899977041241644374153521833685561409742 1773057123604366337765471630497554681447651696660485876698925609424535 5414821726957632144768546121661813160212398557991804049216402373140245 7431588892286959829361425567378675501829073093458359604721892285644503 1144555558378949368515552870246392450209293565795111939651335714801665 5838815221647124224291182210983528025410680173208183770098507130343826 4701511424241849873076485789878019143285307205819285972301851092392933 9698734347443830187266888120038009947937093756629532395857427666945996 3130190768645511495488916502348282572156394946952865053303828039180365 5600293250668171951295146653103289079815517545948532880575162330300429 5447526845162673160358466900693516984281006243453106587873167724733695 68283896032912904942686186080048352649577285485890097, a[10,5] = 1.738 6976687436041439411254138134336505875227098585607474938658278588349272 9169284067616498410404049451324018887089336473592320764295696277217253 0606377727081624632398215180226293794572661026516773039609351943488737 0575027844151695375293360716262270010140715736686977307905983082307141 4739683557326764988362825231834884147498849197487807165749696821737913 6359423810144204868704937966025831820840960230927210958639603428916960 7030440454360911161000425255687483974024438201007599039160079154251036 6451804949776362874424858375501459753210492753064476964421011342451442 4971567231502312141870823604203155634649326979437850678256152912011616 3910370414725359965689777919243158786950919589869126993349432820279736 0922149124220003334461872654001895306952127793516464951364288853353828 988006559320955078849673832189495278341739891648332, a[6,4] = -.806379 4463388136804227923907375129011751333135133207268291534597081132927004 3140077780040598724256466600783354240469329480327505761234285760607289 8871637161808265144086027106409780629592874586810799449052660111637189 4488685366752218449305084462780604912575213736806446410297483919576870 1123585317796830723804113300468118544410942038698972287333738018602554 1141180470351314080492493993111677944323269092289296083336877602392660 0725349021589222901400338550053464403821182785263119184495694322055553 9395858237302918609660118234999699851859375394432727399482424302298670 5682944475106210304280336655739030771132992936470321002488642573398833 3846519729776107397444589559853108567120560060628499567870764092642028 2060420410142765861549543383682619554432782659537055997213567657886072 2481943370053437376314647411469616670179027477233, a[9,6] = -.22131798 0453566720023209658017078291608001960859333767158302077477465613076201 6168441731824200481860234181809504988128546279297936925404746602326374 1817896031006269197616637152726783150659144623191814837435747081852085 7698629600648650519157091862355828154084290137486711564025328369636528 2243941468392556812311544927324697138242757155610584896320952985692855 2652499200036381712798093495046004722246968681368303934861555739891706 3154898398513561769600735588576074596098316334081575979176254435224165 4038962782336256824671707019571657070424602530110546110852427901227759 1942288256537151640962603613546614201439286786379676535845635558599004 6512475137434318970861006079985406614736108881104676398594279182708155 0506009744300701793927223892961644024218956923388030281751160457581005 51599081894839584577777059197048403681682587093, a[10,9] = -.385269988 3389103753806753315050015610693796012770762791634318668812919297701321 4755379717199902576597161659323245197727507695011959277471988516675306 1025864035705318010881274859706524507850740895775876629219628210421001 8033530977607174153024889190185065757682760305952509224701631314641697 7701796048306901212062920031633402013989533572373559784768046480556300 6647048981244379030648584602332265247329540042046148263687233267225810 5375909568817904524894320200264616012209674446120184056505208573421826 9247846747001792409189131117283063978921190573943986334417793864220163 5280140411542580255244786494991018905652731360775047749441389568845084 7004750793835404817992930810199600012045075557308596107083418496831537 9842245346768794474577906385933213167464093656887020938128169260817029 0268911538989214991712615424999483498761167901e-1, a[10,7] = 10.319996 6134833145778069547552335682529596233509488697899889243921275509245082 0083936751358843163964472741992563357539371794166491179948579669064961 5970692958532252610417184217510536549694847373590659002619144036063473 7371779749084705069147568137137969194557434670728395260777051417558132 3088947507017772029799247565258452946390051796010626820409847317944207 1275732417872381019545933268798858807028930577932692002262791358660945 4238284991797828116957236864177995873300983053338459449582302492736450 8186824394381802488318501658108722564032639564020003839125909708510773 8974868616702368434403136204301613399868038525083401423831576665797440 8964723400532682511122156975086129280137866536836104668137680570967948 1462113751188903665665836601499930196645169031604016001777695718995844 48353901781859490054941555705545038863924497949, a[9,5] = 1.7190061688 1688979791573941970175351457535397998084899179110276112898236306674337 0892483757026312777616969587662788986427556588819537264697417999783386 7528282873891259883983957669560845324239730733201759470592644554282676 3283177600169462734998702382231146012745298646111626319841181166468301 5637665629644304642230987181642313840318880072056110634800304280074374 0543993848561657951568873661312485359963971035891335021198183726699096 0608041183697852735621802121977209416111533942293051921127365351991763 9678620687598976303593793757018431636453216867848768377709327111588527 7043068779519547275903126453409413565080246073278817037207501552284867 4370946543012896114350157534899947189475275671601172371032698712474751 5403573788012583366789553216942775048785204000994580498168319222175516 32326964079094214540090877826751791411206297, a[5,3] = .37074049197957 7307845280049671592875085100735540728481789018812969309167018743466739 4333132656228793891686348940436975420129320954164850009537509990459318 6950336823065348971020823757532947513697795839027326066729176283766595 9115819920811860451587857013236536139625868799607819723610138453594264 4143621074147274025165259036875916538900071852336854759684803618220251 9074898399396782922308874721417622526302260550347863921311297914579424 0900063014925238506363209230037832558999499877281869803682922862942170 7691685590260230720140907662290578067901293851186748402722553343661936 7567666446281987545043419958369392141389859097244406986498558811790141 6852290271320483663894086805061384209902888182954386015849988166380812 3371788262336644984317908221092396867231902024834239426971528796025820 75519676490181620563140546007153516494782, a[7,4] = -.5957425550471902 2871846347014824734258551148498890791997083196292094947388706048689405 3898745907407037974001705633878918272029808710988899211626105425116854 2311596846260439073142682598232148543835924586471866821715605145932644 6581205254009924014811738629662126051150169230714007690184672139461507 6749412657567428704948578637156345744803723001930405844401018391132095 5884025322087012181293320669956778242259376396306901552249582041463683 9729370366451983429487993434801075337590767154290401114926356809601663 0418574396721629529052709109325045164649410928586417037042153646872341 5684755442785594487218064898975417230110929513981713562398113704823421 4368846929683196004541668972330522606398767138106140996806975925979174 7345623701434839575018989885330683760064489325838826268245635904014499 795457246193235864476189562794506120101e-1, a[4,3] = .8519892760827774 2380986281840371826512180718468124813859415258220899410946127817359094 9940872819854152865458025945212723855107369470255088493254790229813643 7308209772470699492597089426516664239149709980959524984000702895246624 6184957485904250633580724160097212110320658593537220780735573042565641 4299732853295469131826250052879445566609858987484395500602231890808146 2853114199185294569085785065634997829191098731345836815229380852895776 5825291589349884691616220363430698686221545672434300590047706929953029 9031885919957380802940419371188373013404017056291252837742965531139888 0306518498230433646635068355664193750530988035374075917761420173991569 4869808874865968312100019582332810246909536155388409911289751805499145 3805033945300556402023068704797714805963022667222676056508016650420467 664700595658502606672174939334197591135, a[3,2] = 2.444368338267085928 0322459018146517193881181297105629028096018440854175532475162700903049 7831488527773135321091818550493769468901680157355626781586945570064536 1800141291758757002756761911422380765002713073262279530230940816389980 5120501409541628801134789270304656574886128779182169658243794661729427 1155004634796100801065723093737343692434282068127680203388858482148672 8158455848365854785586054449106355445967113729687701425861483824161780 3764957370750705060351919115389200881336514871163007311116920350578915 1482133739849547968216854529332387159528587625580541044776437211823795 0499297453513063554476076090971659614944694471755225631283560844998948 9658629800879508072197740546774636719052241585056282557790705533012817 9724413331998643951527245552903197914066467165470395669700589861191840 764538289126746934622786608829583977, a[11,10] = 0., `b*`[4] = 0., a[1 1,2] = 0., b[11] = 0., `b*`[2] = 0., `b*`[3] = 0., `b*`[5] = .50464859 6286841315687488770805059342595535715715047363706082329756698279919844 2699562484966093827703717102064291835707978587402970433374587093907010 5458275109293289843100463757689691409008316645967890478289214876823301 5903341980284333812227347668397860460635063840102810936216378533855402 5782978320954666146634102760575812764514280980928670440241520681427964 5060706454283904428761563907698663827857589692172604737800795794821276 2801298023882832616502093018262822113113449680070046377251281102798610 8667578528839276265612048155750637061964017874931806570992748402113873 4492025154526625773391097263464298450259190620128696534362295122413433 3799150136335312172190040715753429330080688680204600540147101616554727 0567193274175274861558946477280861630995017144595888981907535377682734 17148995423913808379978353302826130687903219962, `b*`[6] = 3.748315828 2433176463580078968824134239437325558568807811075112885462899115347738 3357194561637194576889997103065837951992946315272247929828564051030798 3720149932121080925473801209790366539214797331395585047565915880921185 1458733894257500158702865867335956057484283091135722624318734116592152 2482031299734230884428651548016204465266356886000703680940835064919496 4043308052481763906165514923025793431978285479226767698507230528484030 2417076990337026607262290176643514675044436198298063163905627897250007 4828050611995236939413890659118911986354775869808408225070909154419748 6262416763935007340541004044510114936972339550539365649357919479520147 8190148868490696424516067130401938927466310290653547528373549062275111 5583610186722707723587588230676052611967396009377510861460607397461576 329380133168409302396557967796020942584843848, `b*`[8] = -2.5424962767 8133643261752931286697685707700835362674374291191465247868245231538170 3435421498019916579012944763618181494929562631195135025053617922959392 5724847038435838085299118128830441305809113074353415707910637931284944 2771701075117910032208434710744455150567195970589612223525284954268666 0280641860775077361703406134741637884060130080586182626000150163394708 7902698900879328111718320759802861869255851502054945338742574957148579 8976259996369701534319468952215211758248039009469278966084845255263731 1815462331478184772505365781538557954203878393693824887649334820491734 9418252945249379522411694925932174291891286498342553061366394327369857 5871937248599115499651541364743345354186968703385291341029715110785732 4035969288331682064913814521458081687887870643609980867457484237914406 28686598258041494259019249203673741467103610, a[11,8] = .4292642667437 7930618348068185765518295479618618432881508519633196884860299427035983 2458297788304715481731243763760209149376512003689453794975855812620983 7527515577641543087914518295254185903608327743423113369557252647526507 5742846658264112777312481117767520676585841065794440168467483484328586 1424069238054566634310889946178248982805250226050486846776641540510006 1051737859716803878644429992437384030370489104443885247394730373932504 8685216124363733019406863236474467391813537036590469632147966255237320 4907679642670056954675010626145113225270229971630948982092609213623762 6475615413682450663532891687735309861827565654214473004090187669714181 4385086968042979806058375251205858269190001132636343071332091433216971 0238513295948737733248380441007577880081670158740052795436481278618771 079310836796492993028455690489625215795372, a[11,7] = 10.3369369221275 0886714624218035272659530604981805785027414761193919405267505535441461 4327603935246889158920479865284765111139658220286153395751243636372863 3186777285827488007184790180115971818424155892418599554138589213233153 6887762269545574191317255654237929853839606850692476441733251282574469 1549441426622798079495971826818000444109600258556932039157156916612111 1601300214177452547361684856390258700332633542688514754111242528175405 8737975620606451223527254671269674553286417144077161928826563949807780 3863156186720370808354356072148356846515708608735806777917940619255616 1045725319065225852212854175754217561593821940925195281676313075230146 8036537607357874451393996543656444751579283222628168920358303167751985 5871622136980771601943599073397429765464571207041895993549358356698731 1055360560596194611163889094072966328044, a[11,6] = -.1494522799021842 0543328425452226290681870706237755163150989283812007172247271884532510 2358504243644744188593083141659826228699893381701151979956106191217849 4739383243809355995057249135837664824546773478759827174173484477331161 4562931686820404531440218441734330007200717428248696001834403142892113 0209150305995470708486227522416057368802836039493753064928361518654540 2519505793336302387233976658701348830847802379305444677648999403941202 3348786676829111923149127982603022911131983467249604587694782831873752 7217809586749921457254686778012937583524635186035872280706055154433851 7218673943249209439873626516803020735566349524908198427742073655506659 8452729812206746527313378589844561153888119431500141258785313191336940 4989780268831028949838239030540625703910419905724958016386655983277061 433882770740696059436682199046455718511, `b*`[11] = 1.5000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000, `b*`[1] = .5858280931143287472 2489524390568299066029793054913487915172166657916506221738404811476208 8202697421012251946155217878274522004504660720739753843910980606357189 9097789995609616876688751672789171974994762877661204198920649101386353 9669829492815003709473102953565172749394190273701457044010960113298418 7011384450712937213030275579145167554736267633683633940571838338993182 5683821742800677390531472151724865481127271332528533688268334806654090 2183450392489337946065761222357111520517517089927961962217755661217695 6803117383191903194947538403677435629428696833163256023565302262377078 2188039391156169492089293421056639208036653407639426628639707841203112 9515980872742872273166532363277706927842819010654724766878293646040749 9510717642014345408647831395891796787888736943674462092597800567749955 589032336658521573424611892977836877e-1, `b*`[7] = .317752529201967292 1936599753752468409987480596559912154070055538771595521503235340446470 3385323509281377410974511338651416859709401969432025159066722477658660 4607432204240948248219996307919776286450176989550646452688476668970635 5897071451584037210759803529936067257262148305428588475534705665303201 9620876167514331674110149185479445749421449178757339752098244378972702 5954054744147303936108387460613894753590913674474073069945538643119411 6369949665725530997912770113843367285466640457926767446778595604916986 6380194694808459827407009692807693789627959910577605322511260242438761 4524888413921453540570626927759334667778967521652124044427093598167817 6891598241515846246666257757360419940894771562976929679974557214718472 9115055025450422749405132851666609091352561879655863905242917714954536 0351329460949311705428819578149185234, a[11,4] = -2.372679083479449557 6799852659373678503979989362654608798604232361722826814481930405355451 1053999627385083123302400900755075724556060744837673313960817235277317 3185676345453868889983926710260218117124495245697470456460283824952223 7400192662820581005203008339145473400684258063188325621461898933914477 8816415424596484392044202669227569221785773212816098241616565206623195 6063652056464259626102014248250597494189999523323548616212725300254667 4589620554629670728312535850804093643488371354905781792351775126670192 1998218525359351131112539795406180638219755861729067138087627875115116 3117389644690014283668525109013352010018215041416535644818166774016405 5597645837737409367490456539192973073285073248365551787120334729783071 1543744564680512262824415499009030839497248874351916861890009498185769 899721983621390794664673291675197176, a[11,5] = 1.75772889076033824844 5308849863698456667742463696256070727224709027475477444976539245172243 3471868892615947429891597050118044309628131656282328920715239326523006 4528169200098699052615354095302277440345129370068031308021997087157681 6992905814969507399152566324450166340120383735418993076206200325153122 3452045522778474854974983150463707318121437839708469156927918584949492 6861189787278265066286204095359455751215074447743070882601493202848144 3787251115516034749265849009055342345458579844768066849380080432896754 5551156813027452124087190421631529887347138409871643090251204759025767 0992867580001017435031158137704841544696765150748763275201948572635305 8475788080455769449761562638313540957892580059815010742985799093228368 3749343390226154504277962206821574663181508426699050165021461183447151 0489594121158452042328685043326225, a[11,1] = 1.3658871865249383360022 3005371981074518396224807937305569881391438462582874984715339425412403 5803415843132317967023651451192734406876903983329073649934693537574148 6307445348522395312725133157731443927474838164489877775856755110123715 9955099437978345494406959969633045929982795040744715596013885684019226 6316572538750226468071271624788089529739600011979557848952549492497326 2575765130175932726484747753832500502809959428877031687183090193130834 3572359048687464618431180212872275113677720843688527508421420936388693 5625087177203236154461680560635230575003937497760715077290025843000401 6605599501139835702581537920173506731865573038829254197044026178052348 3162596375832436637972419438852684370367394620853898806584800956864125 9529542449029026088518785751069966182681681030582548827365195932040740 60923927375374455960477548519341, `b*`[9] = -2.58680348626222269634411 6854586311049527037770656089105223856686359381797511298338948895857634 9167951737357778300167701393800593688733770849589531079685676386448088 0565214557139079112032336472048983927287799425935070338978148285808828 1702426768632809464970824992087986096097229559400569412669481121846943 7462662079958654062026327405111219760167585020295436363334837290494437 6834418463263404584770575974065914950693044028938781015704442907549201 6678850169784142364235642106749872341466863600146611051619375813031884 0698692421449782791188741903492567933832514269757625622288914683370230 9593004382242205140187234501816185140974257274958163098653602549951706 6464669459512085982077384769723253422354882507064438714903637348037524 5033455623775483047191875430745020685184288282127666637313622511289277 1915596890435152088346652759127, a[11,9] = -.3832301461129218809398691 6364241645178178337343053171929242321279398248852019234651094753446443 2117803516411530372854835463712499092352160734659719699047057577008711 9484660105764134993071651437762724848708139643482523540417011641612269 1887803979745325406975185067970164386625623102230657965473794397732499 9446845629017097215204850446092144595075333634006464641997589488806484 1991384640939327181786310726902443713971022650944903936362117148877061 7464446959765072062750253433331442095805386668838682246212975007879763 9980628465922708017610991162852858279000211403816675885761699012892310 5081546403588224299070140767647187273692700369750564424497815554840098 4661769770549421956701996588492643294146407512065758739313145896852214 8643787018896045143583339714741047187915381957446913145387000167248550 986069467311336427180226309818e-1\}:" }}}{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 53 "The Butcher tableau in ap proximate form is as folows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "subs(e825,matrix([seq([c[i] ,seq(a[i,j],j=1..i-1),``$(12-i)],i=2..11),\n[`b`,seq(b[j],j=1..11)],[` b*`,seq(`b*`[j],j=1..11)]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'++]!\")F(%!GF+F+F+F+F+F+F+F+F+7.$\" 'Xj:!\"'$!'-)G#!\"&$\"'PWCF2F+F+F+F+F+F+F+F+F+7.$\"'Zh^F/$!'UeLF/$\"\" !F;$\"'*)>&)F/F+F+F+F+F+F+F+F+7.$\"'55gF/$!'AVtF*F:$\"'S2PF/$\"'8wBF/F +F+F+F+F+F+F+7.$\"'PZ*)F/$!'Gb_!\"(F:$\"'wS`F/$!'zj!)F/$\"'f>7F2F+F+F+ F+F+F+7.$\"'Q;?F/$\"'7!*eFLF:$\"':`:F/$!'VdfFL$\"'.$>&FL$!'%[$\\F*F+F+ F+F+F+7.$\"'p?')F/$!'%*4EF/F:$\"'E0@F2$!'92[F/$\"'N85F2$!'/!)=FL$!'.' \\\"F2F+F+F+F+7.$\"'yx(*F/$\"'ib&*F/F:$!'c$>(F2$!'xK?F2$\"',> " 0 "" {MPLTEXT 1 0 68 "eS := convert(eR,rational,24):\nop(%);\ne22 := simpli fy(subs(eS,e21)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"rG#\"-Km;)yD \"\"-:=McX!)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "e22: r ational values for coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6239 "e22 := \{`b*`[11] = 3/2, ` b*`[1] = 10644890436863/181706725266000, `b*`[7] = 4323255920897179184 74482148525568808656853/1360573252321546818820057421907905293476160, c [6] = 17/19, c[9] = 44/45, c[8] = 25/29, a[9,8] = 41013323458190497065 02423/10667097363909719917125000, c[2] = 1/200, a[10,2] = 0, b[7] = 32 739939833109197146986610130768938398484878542073/102638217510541641713 801705364029167418501950476800, a[7,6] = -5919040014811009951092321586 6/11994398797423357814553515510595, a[5,2] = 0, a[6,2] = 0, b[1] = 185 30703372187/317986769215500, b[5] = 397406378366413672608/803680368165 950784145, a[7,2] = 0, a[10,8] = 141940909070976808611971717/336796637 787572286621096705, a[8,6] = -7115819429493526581990/37849324038145288 2505729, a[8,2] = 0, a[9,2] = 0, a[3,1] = -296212523854984450605064/12 9462181430947023498845, a[10,6] = -2147337295919765809044534716/152826 99757843572625171655145, a[2,1] = 1/200, a[10,7] = 7696382282834951277 1032624512097499610054141220500895/74577372174516506942951228687508414 03431006754989491, a[3,2] = 63290651458559192893696/258924362861894046 99769, a[10,9] = -545009894469844730599500000/141461808852449722400222 16299, a[9,6] = -372567259603706115805768838168/1683402581390679277326 651009375, c[3] = 125788166632/804556341815, `b*`[5] = 366705397521011 6568/7266549440921797325, a[6,5] = 10429409903250357502298256364223097 60/855155063453934295147878200275545499, `b*`[8] = -703000184794678732 3/2764999859447736000, `b*`[6] = 839799471859174949/224047148196888900 , c[11] = 1, `b*`[10] = 0, `b*`[4] = 0, a[11,2] = 0, b[11] = 0, `b*`[2 ] = 0, `b*`[3] = 0, a[11,10] = 0, b[10] = 1754022361907/1430294620800, b[9] = -68588512868559375/32259978418667428, b[2] = 0, b[3] = 0, c[7] = 9526409/47245149, a[9,7] = 2314436088844594079601254706134884802918 27060789/31418904104758208921880920386941521649080865000, b[8] = -1607 0516558763250309/7373332958527296000, b[4] = 0, c[10] = 1, a[4,2] = 0, b[6] = 383280144717771439/119491812371674080, c[5] = 119/198, a[8,7] \+ = -28380858623627672736703320753126600/1897075783840010072535356488629 2693, a[11,6] = -87122197804121104293267367/58294325025447695748912530 0, a[11,8] = 60701635156866657458077/141408544478495930424000, a[11,5] = 111870303021148790792075935935918275878271070980030897149566532/636 44799609999705548422499697144046294650653180108535971446675, a[11,1] = 6936698309764946835725063360649553583306425655914739761/5078529455579 087652393405412856849782233648892058268000, a[10,4] = -914106378133783 649744437882531239843522925103950674595588223056094266864010432/389018 1805643784996468101015923056880817477048476711228054318720508705563933 41, a[5,4] = 15643567285020817980224090362461345189277853300810312/658 36374953303610877005093358513531407567316792726287, a[6,1] = -38242136 2069232952645558482090899/7276891295937830443110867514470288, a[8,1] = -246878890533156807885166723922846376763102826028025/9459168695885399 37065443533422779502568160417427184, a[9,4] = -13436380927846244044757 803583952427765782251762574076119478709593044856969472/660987287247208 6679585908821819361338763002158561414498614431157761209623125, a[10,5] = 3868116403444609936063780569978932749234663970855789282731320288/22 24720532488974689873485313867737218754855271707412737609361885, a[7,5] = 319578151553445472127225629704831153236060812834175249153952/615398 2820746383354666623304676483805766376504613845749252065, a[8,4] = -493 2471259935775148927656253720019927767958725726761483449498264713532779 200/102607132951768009631258187707110446245083894204871897020455714319 72578553533, a[9,3] = -36145100904073145340688210364717549936347886287 66244060214613687/5024645222280263344619889845850599097795949523127865 29837471612, a[5,1] = -4319556318016342264550098836819533/588237381338 985769780687033796930496, a[8,5] = 21269687576229892809323240545664650 122099097000742563680/209895268354144631405886982147081123632443195967 05442863, a[7,3] = 517714047024312279768356989660598751616539775823173 072267317364897221236919340917125/333330780773975293533161697727890456 2594877430003089707149292892535414217827153748784, a[6,4] = -943608126 87526097028123187068745127462524044420500458282895614759616/1170178792 57650174135972597811683016497852743167990737825503364250809, a[8,3] = \+ 4105216230872532336132293706540464473954640713998671441385245340625/19 49978654298824108996488510322665339088184896936844826392190722672, a[1 1,9] = -2472599248597125/64519956837334856, a[10,1] = 2693249580763133 758720448819763251094287099002263195049/197245909658155142588365716192 3834583484247606165156240, a[7,4] = -304640485365550663069595193447523 8619460577978474785952003633121978548393408440668463488064/51136263942 3029538585081763862319819637306053965614118086930675717683221504825261 66450770407, a[10,3] = -7627201649863027483554299716391667556100940646 85904534324565414570375/7393025478952658115641133413312191063510030066 9232471904805722424944, a[5,3] = 1223121779188860605856654584132649406 75777568875/329913188780101678448178322632546427763796661824, a[9,5] = 989931864897123139010896933221647717910221123481001424643841888/57587 4527302276156101678113881665298242482451696692517167915625, a[4,3] = 3 2445052113731709472131307285859833545001662335/38081526404775332901721 474741064649138530038784, a[7,1] = 34593597270203433663140902653521768 969908784170157140274555/587315506021132570902446233005723262039375581 289686893222448, a[9,1] = 55835332432633377407304490629070855275648472 3746581752261/58431944020157116455094579324063186414904198900102231250 0, a[4,1] = -12789372157071146745045111775696471200626449407/380815264 04775332901721474741064649138530038784, a[6,3] = 220573403925972020047 99056057490702918544557070119184193875/4129997986457085733092356492524 8608645282955350725839567504, c[4] = 200814933981747403/38906510625737 6584, `b*`[9] = -333800898560611875/129039913674669712, a[11,4] = -154 8948850268913302819182056302536132743256445416938614472267815479730912 /652826950367613499218361365117961613704987867512238468998956163729502 185, a[11,7] = 141145285619744149135929384507225418152792014862255637/ 13654459409305766148231745889098379435265792575731520, a[11,3] = -5126 06260594671504932297273531042063679229867086229919385168625/4962612565 2150749082665578724450361459713081709904842453083616\}:" }}}{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 9 "Examples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "c[4]=subs(e21,c[4]);\n`` ;\na[11,4]=subs(e21,a[11,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\"%*&,&*&\"';Iz\"\"\"%\"rGF,F,\"'\"et$!\"\"F,,&*&\"(3#z>F,F-F,F, F+F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%,$*,\"+7$HBk#\"\"\"\"(&[I))!\"\",,*& \"4'QDgky!='3@F,)%\"rGF(F,F,*&\"48&*fV#Q6>gHF,)F3\"\"$F,F.*&\"4*o\\_:A Z6F:F,)F3\"\"#F,F,*&\"3F))y)o\"p01MF,F3F,F.\"2,#Gt.JbPFF,F,,(*&\"(Kge \"F,F3F,F.\"'\"et$F,*&\"(3#z>F,F:F,F,F.,(*&\"/sYi\"GH6$F,F:F,F,*&\"/!e FxO`F#F,F3F,F.\".t+<,0\"QF,F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "Substituting " }{XPPEDIT 18 0 "r = 12578 8166632/804556341815" "6#/%\"rG*&\"-Km;)yD\"\"\"\"\"-:=McX!)!\"\"" } {TEXT -1 14 " gives . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "c[4]=subs(e22,c[4]);\n``;\na[11,4]=subs(e22,a[11,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"3.uu\")R$\\\"3?\"3%ewtD1^1*Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!do74tza\"yEsWhQpTXkDVF8ODIc?=>GI8*o-&)[*[:\"co& =-&HP;c*)**o%QA^ny)\\q8;'z6l8O=#*\\8wO]p#Gl" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Evaluating the preceding two va lues to 24 digits gives . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "c[4]=evalf[24](subs(e22,c[4]));\n``;\na[11,4]=evalf[24](subs(e22 ,a[11,4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%$\"9&Q=Ro \\_vwt9;&!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%$!9,!od&\\%zM3zEP#!#B" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Rounding the 825 d igit values to 24 digits gives . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "c[4]=evalf[24](subs(e825,c[4]));\n``;\na[11,4]=evalf[ 24](subs(e825,a[11,4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"%$\"9&Q=Ro\\_vwt9;&!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%$!9**zwb\\%zM3zEP#!# B" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#-- -------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 62 "#--------- ----------------------------------------------------" }}{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 6302 "ee := \{c[2]=1/200,\nc[3]=125788166632/8045563418 15,\nc[4]=200814933981747403/389065106257376584,\nc[5]=119/198,\nc[6]= 17/19,\nc[7]=9526409/47245149,\nc[8]=25/29,\nc[9]=44/45,\nc[10]=1,\nc[ 11]=1,\n\na[2,1]=1/200,\na[3,1]=-296212523854984450605064/129462181430 947023498845,\na[3,2]=63290651458559192893696/25892436286189404699769, \na[4,1]=-12789372157071146745045111775696471200626449407/380815264047 75332901721474741064649138530038784,\na[4,2]=0,\na[4,3]=32445052113731 709472131307285859833545001662335/380815264047753329017214747410646491 38530038784,\na[5,1]=-4319556318016342264550098836819533/5882373813389 85769780687033796930496,\na[5,2]=0,\na[5,3]=12231217791888606058566545 8413264940675777568875/32991318878010167844817832263254642776379666182 4,\na[5,4]=15643567285020817980224090362461345189277853300810312/\n \+ 65836374953303610877005093358513531407567316792726287,\na[6,1]=-38 2421362069232952645558482090899/7276891295937830443110867514470288,\na [6,2]=0,\na[6,3]=22057340392597202004799056057490702918544557070119184 193875/\n 412999798645708573309235649252486086452829553507258395 67504,\na[6,4]=-943608126875260970281231870687451274625240444205004582 82895614759616/\n 117017879257650174135972597811683016497852743 167990737825503364250809,\na[6,5]=104294099032503575022982563642230976 0/855155063453934295147878200275545499,\na[7,1]=3459359727020343366314 0902653521768969908784170157140274555/\n 58731550602113257090244 6233005723262039375581289686893222448,\na[7,2]=0,\na[7,3]=517714047024 3122797683569896605987516165397758231730722673173648972212369193409171 25/\n 3333307807739752935331616977278904562594877430003089707149 292892535414217827153748784,\na[7,4]=-30464048536555066306959519344752 38619460577978474785952003633121978548393408440668463488064/\n \+ 5113626394230295385850817638623198196373060539656141180869306757176832 2150482526166450770407,\na[7,5]=31957815155344547212722562970483115323 6060812834175249153952/\n 61539828207463833546666233046764838057 66376504613845749252065,\na[7,6]=-59190400148110099510923215866/119943 98797423357814553515510595,\na[8,1]=-246878890533156807885166723922846 376763102826028025/\n 94591686958853993706544353342277950256816 0417427184,\na[8,2]=0,\na[8,3]=410521623087253233613229370654046447395 4640713998671441385245340625/\n 19499786542988241089964885103226 65339088184896936844826392190722672,\na[8,4]=-493247125993577514892765 6253720019927767958725726761483449498264713532779200/\n 1026071 3295176800963125818770711044624508389420487189702045571431972578553533 ,\na[8,5]=21269687576229892809323240545664650122099097000742563680/\n \+ 20989526835414463140588698214708112363244319596705442863,\na[8,6 ]=-7115819429493526581990/378493240381452882505729,\na[8,7]=-283808586 23627672736703320753126600/18970757838400100725353564886292693,\na[9,1 ]=558353324326333774073044906290708552756484723746581752261/\n 5 84319440201571164550945793240631864149041989001022312500,\na[9,2]=0,\n a[9,3]=-36145100904073145340688210364717549936347886287662440602146136 87/\n 502464522228026334461988984585059909779594952312786529837 471612,\na[9,4]=-13436380927846244044757803583952427765782251762574076 119478709593044856969472/\n 66098728724720866795859088218193613 38763002158561414498614431157761209623125,\na[9,5]=9899318648971231390 10896933221647717910221123481001424643841888/\n 5758745273022761 56101678113881665298242482451696692517167915625,\na[9,6]=-372567259603 706115805768838168/1683402581390679277326651009375,\na[9,7]=2314436088 84459407960125470613488480291827060789/3141890410475820892188092038694 1521649080865000,\na[9,8]=4101332345819049706502423/106670973639097199 17125000,\na[10,1]=269324958076313375872044881976325109428709900226319 5049/\n 1972459096581551425883657161923834583484247606165156240 ,\na[10,2]=0,\na[10,3]=-7627201649863027483554299716391667556100940646 85904534324565414570375/\n 73930254789526581156411334133121910 635100300669232471904805722424944,\na[10,4]=-9141063781337836497444378 82531239843522925103950674595588223056094266864010432/\n 38901 8180564378499646810101592305688081747704847671122805431872050870556393 341,\na[10,5]=38681164034446099360637805699789327492346639708557892827 31320288/\n 222472053248897468987348531386773721875485527170741 2737609361885,\na[10,6]=-2147337295919765809044534716/1528269975784357 2625171655145,\na[10,7]=7696382282834951277103262451209749961005414122 0500895/\n 7457737217451650694295122868750841403431006754989491 ,\na[10,8]=141940909070976808611971717/336796637787572286621096705,\na [10,9]=-545009894469844730599500000/14146180885244972240022216299,\na[ 11,1]=6936698309764946835725063360649553583306425655914739761/\n \+ 5078529455579087652393405412856849782233648892058268000,\na[11,2]=0, \na[11,3]=-51260626059467150493229727353104206367922986708622991938516 8625/\n 496261256521507490826655787244503614597130817099048424 53083616,\na[11,4]=-15489488502689133028191820563025361327432564454169 38614472267815479730912/\n 65282695036761349921836136511796161 3704987867512238468998956163729502185,\na[11,5]=1118703030211487907920 75935935918275878271070980030897149566532/\n 636447996099997055 48422499697144046294650653180108535971446675,\na[11,6]=-87122197804121 104293267367/582943250254476957489125300,\na[11,7]=1411452856197441491 35929384507225418152792014862255637/\n 136544594093057661482317 45889098379435265792575731520,\na[11,8]=60701635156866657458077/141408 544478495930424000,\na[11,9]=-2472599248597125/64519956837334856,\na[1 1,10]=0,\n\nb[1]=18530703372187/317986769215500,\nb[2]=0,\nb[3]=0,\nb[ 4]=0,\nb[5]=397406378366413672608/803680368165950784145,\nb[6]=3832801 44717771439/119491812371674080,\nb[7]=32739939833109197146986610130768 938398484878542073/\n 10263821751054164171380170536402916741850195 0476800,\nb[8]=-16070516558763250309/7373332958527296000,\nb[9]=-68588 512868559375/32259978418667428,\nb[10]=1754022361907/1430294620800,\nb [11]=0,\n\n`b*`[1]=10644890436863/181706725266000,\n`b*`[2]=0,\n`b*`[3 ]=0,\n`b*`[4]=0,\n`b*`[5]=3667053975210116568/7266549440921797325,\n`b *`[6]=839799471859174949/224047148196888900,\n`b*`[7]=4323255920897179 18474482148525568808656853/\n 136057325232154681882005742190790 5293476160,\n`b*`[8]=-7030001847946787323/2764999859447736000,\n`b*`[9 ]=-333800898560611875/129039913674669712,\n`b*`[10]=0,\n`b*`[11]=3/2\} :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[7,11];" "6#&%\"TG6$\"\"(\"#6" } {TEXT -1 128 " denote the vector whose components are the principal e rror terms of the 11 stage, order 7 scheme (the error terms of order 8 )." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[6,11];" "6#&%#T*G6$\"\"'\"#6" }{TEXT -1 146 " denote the vector whose compone nts are the principal error terms of the embedded 11 stage, order 6 sc heme (the error terms of order 7) and let " }{XPPEDIT 18 0 "`T*`[7,11 ];" "6#&%#T*G6$\"\"(\"#6" }{TEXT -1 99 " denote the vector whose comp onents are the error terms of order 8 of the embedded order 6 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denot e the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[7,11])); " "6#-%$absG6#-F$6#&%\"TG6$\"\"(\"#6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[6,11]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[7,11]));" "6#-%$absG6#-F$6 #&%#T*G6$\"\"(\"#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[8] = abs(abs(T[7,11]));" "6#/&% \"AG6#\"\")-%$absG6#-F)6#&%\"TG6$\"\"(\"#6" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "B[8] = abs(abs(`T*`[7,11]))/abs(abs(`T*`[6,11]));" "6#/ &%\"BG6#\"\")*&-%$absG6#-F*6#&%#T*G6$\"\"(\"#6\"\"\"-F*6#-F*6#&F/6$\" \"'F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[8] = abs(abs(`T*`[7 ,11]-T[7,11]))/abs(abs(`T*`[6,11]));" "6#/&%\"CG6#\"\")*&-%$absG6#-F*6 #,&&%#T*G6$\"\"(\"#6\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"'F3F8 " }{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[8];" "6#&%\"AG6#\"\")" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[8];" "6#&%\"BG6#\"\")" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[8];" "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 0 192 "errterms7_1 1 := PrincipalErrorTerms(7,11,'expanded'):\n`errterms7_11*` :=subs(b=` b*`,PrincipalErrorTerms(7,11,'expanded')):\n`errterms6_11*` := subs(b= `b*`,PrincipalErrorTerms(6,11,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add( evalf(subs(ee,`errterms7_11*`[i]))^2,i=1..nops(`errterms7_11*`))):\nsd nB := sqrt(add(evalf(subs(ee,`errterms6_11*`[i]))^2,i=1..nops(`errterm s6_11*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms7_11*`[i])-subs( ee,errterms7_11[i])))^2,i=1..nops(errterms7_11))):\n'B[8]'= evalf[8](s nmB/sdnB);\n'C[8]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\")$\")!o[f\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"CG6#\"\")$\")7w^@!\"(" }}}{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 6302 "ee := \{c[2]=1/200,\nc[3]=125788166632/804556341815,\nc[4]=20081 4933981747403/389065106257376584,\nc[5]=119/198,\nc[6]=17/19,\nc[7]=95 26409/47245149,\nc[8]=25/29,\nc[9]=44/45,\nc[10]=1,\nc[11]=1,\n\na[2,1 ]=1/200,\na[3,1]=-296212523854984450605064/129462181430947023498845,\n a[3,2]=63290651458559192893696/25892436286189404699769,\na[4,1]=-12789 372157071146745045111775696471200626449407/380815264047753329017214747 41064649138530038784,\na[4,2]=0,\na[4,3]=32445052113731709472131307285 859833545001662335/38081526404775332901721474741064649138530038784,\na [5,1]=-4319556318016342264550098836819533/5882373813389857697806870337 96930496,\na[5,2]=0,\na[5,3]=12231217791888606058566545841326494067577 7568875/329913188780101678448178322632546427763796661824,\na[5,4]=1564 3567285020817980224090362461345189277853300810312/\n 65836374953 303610877005093358513531407567316792726287,\na[6,1]=-38242136206923295 2645558482090899/7276891295937830443110867514470288,\na[6,2]=0,\na[6,3 ]=22057340392597202004799056057490702918544557070119184193875/\n \+ 41299979864570857330923564925248608645282955350725839567504,\na[6,4]= -94360812687526097028123187068745127462524044420500458282895614759616/ \n 117017879257650174135972597811683016497852743167990737825503 364250809,\na[6,5]=1042940990325035750229825636422309760/8551550634539 34295147878200275545499,\na[7,1]=3459359727020343366314090265352176896 9908784170157140274555/\n 58731550602113257090244623300572326203 9375581289686893222448,\na[7,2]=0,\na[7,3]=517714047024312279768356989 660598751616539775823173072267317364897221236919340917125/\n 333 3307807739752935331616977278904562594877430003089707149292892535414217 827153748784,\na[7,4]=-30464048536555066306959519344752386194605779784 74785952003633121978548393408440668463488064/\n 511362639423029 5385850817638623198196373060539656141180869306757176832215048252616645 0770407,\na[7,5]=31957815155344547212722562970483115323606081283417524 9153952/\n 61539828207463833546666233046764838057663765046138457 49252065,\na[7,6]=-59190400148110099510923215866/119943987974233578145 53515510595,\na[8,1]=-246878890533156807885166723922846376763102826028 025/\n 945916869588539937065443533422779502568160417427184,\na[ 8,2]=0,\na[8,3]=410521623087253233613229370654046447395464071399867144 1385245340625/\n 19499786542988241089964885103226653390881848969 36844826392190722672,\na[8,4]=-493247125993577514892765625372001992776 7958725726761483449498264713532779200/\n 1026071329517680096312 5818770711044624508389420487189702045571431972578553533,\na[8,5]=21269 687576229892809323240545664650122099097000742563680/\n 209895268 35414463140588698214708112363244319596705442863,\na[8,6]=-711581942949 3526581990/378493240381452882505729,\na[8,7]=-283808586236276727367033 20753126600/18970757838400100725353564886292693,\na[9,1]=5583533243263 33774073044906290708552756484723746581752261/\n 5843194402015711 64550945793240631864149041989001022312500,\na[9,2]=0,\na[9,3]=-3614510 090407314534068821036471754993634788628766244060214613687/\n 50 2464522228026334461988984585059909779594952312786529837471612,\na[9,4] =-13436380927846244044757803583952427765782251762574076119478709593044 856969472/\n 66098728724720866795859088218193613387630021585614 14498614431157761209623125,\na[9,5]=9899318648971231390108969332216477 17910221123481001424643841888/\n 5758745273022761561016781138816 65298242482451696692517167915625,\na[9,6]=-372567259603706115805768838 168/1683402581390679277326651009375,\na[9,7]=2314436088844594079601254 70613488480291827060789/3141890410475820892188092038694152164908086500 0,\na[9,8]=4101332345819049706502423/10667097363909719917125000,\na[10 ,1]=2693249580763133758720448819763251094287099002263195049/\n \+ 1972459096581551425883657161923834583484247606165156240,\na[10,2]=0,\n a[10,3]=-7627201649863027483554299716391667556100940646859045343245654 14570375/\n 73930254789526581156411334133121910635100300669232 471904805722424944,\na[10,4]=-9141063781337836497444378825312398435229 25103950674595588223056094266864010432/\n 38901818056437849964 6810101592305688081747704847671122805431872050870556393341,\na[10,5]=3 868116403444609936063780569978932749234663970855789282731320288/\n \+ 2224720532488974689873485313867737218754855271707412737609361885, \na[10,6]=-2147337295919765809044534716/15282699757843572625171655145, \na[10,7]=76963822828349512771032624512097499610054141220500895/\n \+ 7457737217451650694295122868750841403431006754989491,\na[10,8]=141 940909070976808611971717/336796637787572286621096705,\na[10,9]=-545009 894469844730599500000/14146180885244972240022216299,\na[11,1]=69366983 09764946835725063360649553583306425655914739761/\n 507852945557 9087652393405412856849782233648892058268000,\na[11,2]=0,\na[11,3]=-512 606260594671504932297273531042063679229867086229919385168625/\n \+ 49626125652150749082665578724450361459713081709904842453083616,\na[1 1,4]=-1548948850268913302819182056302536132743256445416938614472267815 479730912/\n 6528269503676134992183613651179616137049878675122 38468998956163729502185,\na[11,5]=111870303021148790792075935935918275 878271070980030897149566532/\n 63644799609999705548422499697144 046294650653180108535971446675,\na[11,6]=-87122197804121104293267367/5 82943250254476957489125300,\na[11,7]=141145285619744149135929384507225 418152792014862255637/\n 13654459409305766148231745889098379435 265792575731520,\na[11,8]=60701635156866657458077/14140854447849593042 4000,\na[11,9]=-2472599248597125/64519956837334856,\na[11,10]=0,\n\nb[ 1]=18530703372187/317986769215500,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=39 7406378366413672608/803680368165950784145,\nb[6]=383280144717771439/11 9491812371674080,\nb[7]=3273993983310919714698661013076893839848487854 2073/\n 102638217510541641713801705364029167418501950476800,\nb[8] =-16070516558763250309/7373332958527296000,\nb[9]=-68588512868559375/3 2259978418667428,\nb[10]=1754022361907/1430294620800,\nb[11]=0,\n\n`b* `[1]=10644890436863/181706725266000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]= 0,\n`b*`[5]=3667053975210116568/7266549440921797325,\n`b*`[6]=83979947 1859174949/224047148196888900,\n`b*`[7]=432325592089717918474482148525 568808656853/\n 1360573252321546818820057421907905293476160,\n` b*`[8]=-7030001847946787323/2764999859447736000,\n`b*`[9]=-33380089856 0611875/129039913674669712,\n`b*`[10]=0,\n`b*`[11]=3/2\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The sta bility function R for the 11 stage, order 7 scheme is given (approxima tely) as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "subs(e e,StabilityFunction(7,11,'expanded')):\nconvert(evalf[28](%),rational, 24):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F )*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F )F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)* $)F'\"\")F)F)F)*&#\"*z\"Q\"3%\"0PV^t;xG#F)*$)F'\"\"*F)F)!\"\"*&#\"*_1B !=\"0*y!HMYNX%F)*$)F'\"#5F)F)FY" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z ) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+&oYDI%!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "z0 := new ton(R(z)=1,z=-4.3):\np1 := plot([R(z),1],z=-4.79..0.49,color=[red,blue ]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamond] ,color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB ,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.79..0.49,-0.07..1.47],f ont=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 399 251 251 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3/++++++!z%!#<$\"3W-\\IdW(3D#F*7$$!3n *****H^bCt%F*$\"35uyp`8#p0#F*7$$!3=+++E5\"\\n%F*$\"3mq1a$fKo(=F*7$$!3[ ++bv=%[i%F*$\"3a(\\q>`t3t\"F*7$$!3))****4DFxuXF*$\"3In`(zBEWf\"F*7$$!3 h++&)GZY=XF*$\"3'[!)=wJ$y^9F*7$$!3Y++gKn:iWF*$\"3*37mG:z*>8F*7$$!3k++S !\\vaS%F*$\"3=5Wph2j(>\"F*7$$!3%*****>[Uz[VF*$\"3WwJ&e6U]3\"F*7$$!3K++ 0-CQ#H%F*$\"3&yUa\\Er4#)*!#=7$$!3$)*****ebqfB%F*$\"3![@9B1)yw))FZ7$$!3 U++q]&o88%F*$\"3X1_y6*)[LtFZ7$$!3p****4L#fI-%F*$\"3qM(Hb^h/*fFZ7$$!3&) *****=eX5\"RF*$\"3[j8.f)\\&R[FZ7$$!3')*****)f6R*z$F*$\"3\"=L>]oy/!RFZ7 $$!3/++!G'3a%o$F*$\"3]J%z4#R4?JFZ7$$!3K++ge1Q$e$F*$\"3MY%QsDLfc#FZ7$$! 3+++SG#)\\pMF*$\"3->$Rt2l(o?FZ7$$!33+++'=[^N$F*$\"3;UQ1d3a$o\"FZ7$$!3# *****fJ2&\\C$F*$\"3gMQdZZi-9FZ7$$!3:++!f)3)[9$F*$\"3B+KqAM!>@\"FZ7$$!3 3++Sg\"))e-$F*$\"3S#\\BN4^;0\"FZ7$$!3p****f8d3DHF*$\"3v6f\"*QAuK'*!#>7 $$!3v****H/&Gy!GF*$\"3[`2?**=Kf!*Ffr7$$!3N++g0g./FF*$\"3k`\"p;\\Ed*))F fr7$$!3-++I%=f,f#F*$\"3&*Q>t9dvI!*Ffr7$$!3))*****3_@<[#F*$\"3*yGNxmCPU *Ffr7$$!3++++&>y&oBF*$\"3d$yiL7)f25FZ7$$!3)*****H-tnkAF*$\"3I%)fBC=K(3 \"FZ7$$!3C+++`hg_@F*$\"3C887r\\I$>\"FZ7$$!3')*****es&>O?F*$\"3m\"=Q6xd ZK\"FZ7$$!3*******o5g[$>F*$\"3MuqF3D+d9FZ7$$!3r****>i[TD=F*$\"3x[DPBB4 >;FZ7$$!3L++?:sM7S*3b6F*$\"31TEyWAV]JFZ 7$$!3G++]2=y^5F*$\"3M3#fjY3K\\$FZ7$$!3c&****f>$R)Q*FZ$\"3B\\jb[r$3\"RF Z7$$!3N-++.CpD$)FZ$\"3A9)>tv6$\\VFZ7$$!3#o****4(Q'[@(FZ$\"3')3yei]Hg[F Z7$$!3D-++Sf#)GhFZ$\"3Qsz4[\"pyT&FZ7$$!3'******\\([%=*\\FZ$\"3?zL()G]D qgFZ7$$!3'4++g(pz'*QFZ$\"3K.,hIxtsnFZ7$$!3u-++QB&px#FZ$\"3o[+0_lFvvFZ7 $$!3/$*****y0Qm;FZ$\"33zhN?%f]Y)FZ7$$!3%)******>Z!)ekFfr$\"3c$=iYqNXP* FZ7$$\"3:!*****R\"ysB&Ffr$\"3os\")f%\\oP0\"F*7$$\"3++++3%R)p:FZ$\"3]]z &[#o(*p6F*7$$\"3;0++dD@&o#FZ$\"3ZQ\"o(z(G!38F*7$$\"3+/++zpw_PFZ$\"3G&> #oiSRb9F*7$$\"3!***************[FZ$\"34U(4?@;Bj\"F*-%'COLOURG6&%$RGBG$ \"*++++\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$ FGF_]l7$FQF_]l7$FfnF_]l7$F[oF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$ FdpF_]l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FhrF_]l 7$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$$!3E+++&oYDI%F*F_]l-%'SYMBOLG6#%'CIR CLEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&C ROSSGF]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#\"\"$-%%FONTG6$%*HE LVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%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 1368 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5 040*z^7+\n 1/40320*z^8-408138179/228771673514337*z^9-180230652/445 354634290789*z^10:\npts := []: z0 := 0:\nfor ct from 0 to 360 do\n z z := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.15,.23 ,.5)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.15,0]],i=2. .nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.3,.45,1)): \npts := []: z0 := 2.16+4.3*I:\nfor ct from 0 to 60 do\n zz := newto n(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,.15,.23,.5)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.1,4.29]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,.3,.45,1)):\npts := \+ []: z0 := 2.16-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=ex p(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)] ]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.15,.23,.5)):\np6 := plots [polygonplot]([seq([pts[i-1],pts[i],[2.1,-4.29]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.3,.45,1)):\np7 := plot([[[- 4.89,0],[2.49,0]],[[0,-4.59],[0,4.59]]],color=black,linestyle=3):\nplo ts[display]([p||(1..7)],view=[-4.89..2.49,-4.59..4.59],font=[HELVETICA ,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constr ained);" }}{PARA 13 "" 1 "" {GLPLOT2D 484 535 535 {PLOTDATA 2 "6/-%'CU RVESG6$7eal7$$\"\"!F)F(7$F($\"31+++^v>Z5!#=7$F($\"35+++-^R%4#F-7$$\"3' ******z7aQd$!#G$\"31+++bEfTJF-7$$\"3]+++!4*p!H'!#F$\"31+++A-z)=%F-7$$ \"3)******R4#Q#z&!#E$\"37+++xy)fB&F-7$$\"3\")*****4lOm`$!#D$\"3]+++0f= $G'F-7$$\"38+++/'z[i\"!#C$\"39+++=^QItF-7$$\"3Q+++R;$y0'FL$\"3#******* =uex$)F-7$$\"3#******p%*zR#>!#B$\"3l*****fd'zC%*F-7$$\"3%)*****\\e\"= \"Q&FW$\"3/+++**=?Z5!#<7$$\"3)******>E%3d8!#A$\"3++++(RE>:\"Fin7$$\"3! )*****Hs!4SJF]o$\"33+++YUlc7Fin7$$\"3e******)pKXv'F]o$\"3!******>E'Qh8 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ONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F__\\l6#%(DEFA ULTG-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$*[Fe fp$\"$\\#Fefp;$!$f%Fefp$\"$f%Fefp" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute \+ stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " } {XPPEDIT 18 0 "[-4.3025, 0];" "6#7$,$-%&FloatG6$\"&DI%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the largest interval on the nonnegative imaginary axis th at contains the origin and lies inside the stability region is " } {XPPEDIT 18 0 "[0, 3.45];" "6#7$\"\"!-%&FloatG6$\"$X$!\"#" }{TEXT -1 18 " approximately. 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0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The relevant int ersection point of the boundary curve with the imaginary axis can be d etermined as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look fo r points on the boundary curve either side of the intersection point. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Digits := 15:\nz0 := 3.45*I:\nfor ct from 120 to 123 do\n n ewton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0S**=l(G>T!#<$\"0G`Ia8=W$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Nqs:)\\/G!#=$\"0%e#peC\"eM!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0BD\">;VTP!#<$\"0pKe_=TZ$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0_bU+#pTz!#<$\"0!3)zL*z*[$!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we \+ apply the bisection method to calculate the parameter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi *I),z=3.45*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.20..1.23); \nnewton(R(z)=exp(u0*Pi*I),z=3.45*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0 " 0 "" {MPLTEXT 1 0 133 "subs(ee,subs(b=`b*`,StabilityFunction(6,11,'exp anded'))):\nconvert(evalf[28](%),rational,24):\n`R*` := unapply(%,z): \n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG ,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&# F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F' F1F)F)F)*&#\"+dKsGh\"/g0**fN^HF)*$)F'\"\"(F)F)F)*&#\"*JAMb&\"/7fsV\"zx \"F)*$)F'\"\")F)F)F)*&#\"*_;TY#\"0$)4fG^IM\"F)*$)F'\"\"*F)F)!\"\"*&#\" *s8B@$\"0lMx4yS_'F)*$)F'\"#5F)F)Fen" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the bounda ry of the stability region intersects the negative real axis by solvin g the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-4.1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Y;8UT!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := ne wton(`R*`(z)=1,z=-4.1):\np_1 := plot([`R*`(z),1],z=-4.69..0.49,color=[ red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,cros s,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]],linestyle=3,co lor=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.69..0.4 9,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 463 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3Q++++++!p%!#<$\"3wQR0f?crDF* 7$$!3F*7$$!3!Q$eC***[)yWF*$\"3BKauB.[<=F*7$$!3. ^(=BW2OU%F*$\"3]jv^HC!Rl\"F*7$$!3On;R&)eOoVF*$\"3U.C,&\\YG]\"F*7$$!3Jn mc^\"eFJ%F*$\"3YL7rR)3FO\"F*7$$!3En;uL\"F^o7$$!3QLL[p$*H fHF*$\"3W>M=Am@Z6F^o7$$!3kLL$*fgSgGF*$\"3we*=aXeR/\"F^o7$$!3A+D6J'p`u# F*$\"3'zEKLR;ou*!#>7$$!3lLLV**GaVEF*$\"3[c`Eu_y:&*F[s7$$!39+D6OG#=`#F* 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0:\nfor ct to 361 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_1 := plot(pts,color=COLOR(R GB,0,.13,.4)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.1, 0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,. 25,.8)):\npts := []: z0 := 2.3+4.2*I:\nfor ct from 0 to 60 do\n zz : = newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,0,.13, .4)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.24,4.20]],i= 2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.25,.8) ):\npts := []: z0 := 2.3-4.2*I:\nfor ct from 0 to 60 do\n zz := newt on(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[R e(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,0,.13,.4)): \np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.24,-4.20]],i=2..n ops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.25,.8)):\n p_7 := 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99859447736000,\n`b*`[9]=-333800898560611875/129039913674669712,\n`b*` [10]=0,\n`b*`[11]=3/2\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..11);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6,/&%\"cG6#\"\"##\"\"\"\"$+#/&F%6#\"\"$ #\"-Km;)yD\"\"-:=McX!)/&F%6#\"\"%#\"3.uu\")R$\\\"3?\"3%ewtD1^1*Q/&F%6# \"\"&#\"$>\"\"$)>/&F%6#\"\"'#\"#<\"#>/&F%6#\"\"(#\"(4k_*\")\\^CZ/&F%6# \"\")#\"#D\"#H/&F%6#\"\"*#\"#W\"#X/&F%6#\"#5F)/&F%6#\"#6F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coeffi cients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "se q(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6Y/&%\"aG6$\"\"#\"\"\"#F(\"$+#/&F%6$\"\"$F(#!9k]g]W)\\&Q_ 7iH\"9X))\\Bq%4V\"=i%H\"/&F%6$F.F'#\"8'p$*G>f&e9l!Hj\"8p(*p/%*='GOC*e# 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yoC\"T%=FuTg\"oD]zFULNWlq$*R&)epo\"f%*/&F%6$FirF'FB/&F%6$FirF.#\"^oD1M X_QT9n)*RrSY&RZk/a1PHKhLKD(3B;_5%\"^osEs!>#RE[%o$p*[=)3R`mA.^)['**3T#) )Ha'y*\\>/&F%6$FirF;#!go+#zF`8ZE)\\\\M[hnsD(ezwF*>+s`ilF*[^xN*f7ZK\\\" hoLNbyD(>Vrb/-(*=([?%*Q3XiW5rq(=e7j4!o<&H82E5/&F%6$FirFK#\"Y!ojDu+q4*4 A,lkcaSKK4G*)Hid(op7#\"YjGW0nf>VCjB63Z@)p)eSJY9a$o_*)4#/&F%6$FirF[o#!7 !*>eEN\\H%>e6(\"9Hd]#)GX\"QSK\\y$/&F%6$FirFgp#!D+m7`2K.ntswiB'e3QG\"D$ p#H')[c``s+,SQyvq*=/&F%6$\"\"*F(#\"ZhAv\"eYPs%[cFb32H1\\/tSxLjKCLNe&\" Z+DJA5+*)>/\\T'=jSKzX4bk6d,-W>Ve/&F%6$FauF'FB/&F%6$FauF.#![o(o8Y@gSCm( G')yMO*\\vrk.@)oS`9tS!45Xh$\"jn7;ZP)Hly7B&\\fz(4*f]e%)*))>YMj-GA_kC]/& F%6$FauF;#!hos%pp&[/$f4(y%>h2uDw^AylxU_Re.yvWSCYy#4QOM\"\"goDJi47wd6V9 ')\\99ce@+j(Q8O>=#)3fezm3sC(G()4m/&F%6$FauFK#\"jn))=%QkC9+\"[B6A5zrZ;A Lp*3,RJ7(*['=$**)*\"jnDc\"z;eMK85%\";+]7<*>(4RO(4n1\"/&F%6$\"#5F(#\"X\\]>jA+*4(G%4^Kw>)[/seP8j2e \\Kp#\"XSi:lhgZU[$eMQ#>;dO)eU^:e'4fC(>/&F%6$F_xF'FB/&F%6$F_xF.#!`ov.d9 acCV`/fokS45cvm\"R;(*HaN[FI')\\;?Fw\"_oW\\UAd![!>ZK#p1I+^j5>7LTL6k:\"e E&*ya-$R(/&F%6$F_xF;#!ioK/,koE%4cIA)e&fu1&R5DH_V)R7`#)yVW(\\OyL\"yj59* \"ioTLRc0(30s=V0G7rw%[qZ<3)o0Bf,,\"ok*\\yVc!==!*Q/&F%6$F_xFK#\"[o)G?8t #G*yb3(RmM#\\F$*y*p0yjg$*4YW.k6oQ\"[o&)=O4wt7uqr_&[v=stnQJ&[t)*ou*)[K0 sCA/&F%6$F_xF[o#!=;Z`W!4ew>fHPt9#\">X^lr^isN%yv*p#G:/&F%6$F_xFgp#\"V&* 3]?79a+h*\\(47XiK5x7&\\$GG#Q'p(\"U\"\\*)\\v15V.9%3voG7&H%p];Xh3o(4244%>9\"<0n4@mGsvyPmzO$/&F%6$F_xFau#!<++]*fIZ%)pW *)4]a\">*H;A-SA(\\C&)3=YT\"/&F%6$\"#6F(#\"Xh(RZ\"flDkI$e`&\\1Oj]sNo%\\ w4$)pOp\"X+!o#e?*)[OB#y\\o>0MR_w3zbXH&y]/&F%6$Fc[lF'FB/&F%6$Fc[lF.#! jnD'o^Q>*Hi3n)H#zO1U5`tsHK\\]rYfgig7&\"in;O3`C%[!*4<38(f9O]Wsybm#3\\2: _c7E'\\/&F%6$Fc[lF;#!do74tza\"yEsWhQpTXkDVF8ODIc?=>GI8*o-&)[*[:\"co&=- &HP;c*)**o%QA^ny)\\q8;'z6l8O=#*\\8wO]p#Gl/&F%6$Fc[lFK#\"jnKlc\\r*3.!)4 2r#yeF=f$f$f2#z!z[6-..(=6\"invmWrf`3,=`1l%HYS9(p*\\A%[bq***4'*zWO'/&F% 6$Fc[lF[o#!;ntE$H/67/y>Ar)\"<+`7*[dpZa-DVHe/&F%6$Fc[lFgp#\"WPcDi[,#z_ \"=aA2XQHf8\\Tu>cGX69\"V?:tvDzl_Vz$)4*)euJ#[hw0$4%fWl8/&F%6$Fc[lFir#\" 8x!eulmo:N;qg\"9+SUIf\\yWa399/&F%6$Fc[lFau#!1Drf[#*fsC\"2c[LPo&*>X'/&F %6$Fc[lF_xFB" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 11 stage order 7 scheme" }{TEXT -1 1 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i =1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"bG6#\"\"\"#\"/(=sLqI& =\"0+b@pn)zJ/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\"63 En8kOyjS(R\"6XTy]f;o.o.)/&F%6#\"\"'#\"3R9x\"/&F%6# \"\"(#\"St?ay[[)RQ*o285m)p9(>4J$)R*RF$\"T+oZ]>]=u;HSO0OASv\"\".+3i%HI9/&F%6#\"#6F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 11 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%#b*G6#\"\"\"#\"/joV!*[k5\"0+gEDnq\"=/&F%6#\"\"#\"\"! /&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&#\"4ol65_(R0nO\"4Dtz@4W\\lE(/&F%6 #\"\"'#\"3\\\\[r/C#/&F%6#\"\"(#\"K`ol3)ob_[@[u%=zr*3# fDBV\"LghZ$H0z!>Ud+#)=oa@BDt0O\"/&F%6#\"\")#!4BtyYz%=+Iq\"4+gtZ%f)**\\ w#/&F%6#\"\"*#!3v=hg&)*3!QL\"37(pYn8*R!H\"/&F%6#\"#5F//&F%6#\"#6#F3F. " }}}{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 284 30 "_________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "#======================== =============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Minimization of \+ the principal error norm " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 903 "SO7_10 := SimpleOrderCondi tions(7,10,'expanded'):\nQeqs := [seq(SO7_10[i],i=[1,2,4,8,16,32,64])] :\nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[ 6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-\n 7*c[5]^2*c[6]-c[6]-c[5]-1 2*c[7]*c[5]+14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-\n 12* c[6]*c[7]+14*c[7]*c[5]^2=0:\ncdns1 := [op(Qeqs),node_eq]:\nSO7_10 := S impleOrderConditions(7,10,'expanded'):\nSO_eqs := [op(RowSumConditions (10,'expanded')),op(StageOrderConditions(2,3..10,'expanded')),\n op(S tageOrderConditions(3,5..10,'expanded'))]:\nord_cdns := [seq(SO7_10[i] ,i=[45,50,51,54,55,59,61])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..10)=b[ 1],seq(add(b[i]*a[i,j],i=j+1..10)=b[j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 : = [add(b[i]*c[i]*a[i,3],i=5..10)=0,add(b[i]*c[i]^2*a[i,3],i=5..10)=0]: \ncdns2 := [op(simp_eqs),op(simp_eqs2),op(SO_eqs),op(ord_cdns)]:\n\ner rterms7_10 := PrincipalErrorTerms(7,10,'expanded'):" }}}{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 24 "The follo wing procedure " }{TEXT 0 13 "prin_err_norm" }{TEXT -1 84 " constructs an 11 stage order 7 scheme and then calculates the principal error no rm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "prin_err_norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 884 "prin_err_norm := proc(c2,c3 ,c4,c5,c6,c8,c9,a1)\n local eqns1,eqns2,sm,ct,saveDigits;\n global e1,e2,e3,e4,e5,e6,e7;\n\n e1 := \{c[2]=convert(c2,rational,Digits+2 ),c[3]=convert(c3,rational,Digits+2),\n c[4]=convert(c4,rational,Di gits+2),c[5]=convert(c5,rational,Digits+2),\n c[6]=convert(c6,ratio nal,Digits+2),c[8]=convert(c8,rational,Digits+2),\n c[9]=convert(c9 ,rational,Digits+2),c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\n eqns1 := subs( e1,cdns1):\n e2 := solve(\{op(eqns1)\}):\n e3 := `union`(e1,e2):\n e4 := \{seq(a[i,2]=0,i=4..10),a[6,1]=convert(a1,rational,Digits+2) \}:\n e5 := `union`(e3,e4):\n eqns2 := subs(e5,cdns2):\n e6 := s olve(\{op(eqns2)\}):\n e7 := `union`(e5,e6):\n saveDigits := Digit s;\n Digits := max(Digits,14);\n sm := 0:\n for ct to nops(errte rms7_10) do\n sm := sm+evalf(subs(e7,errterms7_10[ct]))^2;\n en d do:\n evalf[saveDigits](sqrt(sm));\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 38 "#============== =======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "scheme with a small principal error norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "We can use the procedure to calculate the principal error norm for the scheme of Sharp and Smart." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "Digits := 10:\nc_2 := 1/50: c_3 := 27/125: c_4 := 41/100: c_5 := 57/100: c_6 := 43/50: c_8 := 18/25: c_9 := 5/6:\na6_1 := -31/200:\nprin_err_norm(c_2,c_3,c_4,c_5,c_6,c_8,c_9, a6_1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lDou7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "We can use the speci al procedure " }{TEXT 0 7 "findmin" }{TEXT -1 55 " to minimize the pri ncipal error norm with respect to " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6 #\"\"*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "Digits := 10:\nc_2 := 1/50: c_3 := 27/125: c_4 := 41/100: c_5 := 57/100: c_6 := 43/50: c_8 := 18/25: c_9 := 5/6:\na6_1 := -31/200:\nfindmin('prin_err_norm'(c_2,c_3,c_4,c_5,c_ 6,c_8,c[9],a6_1),c[9]=\{c_9-.3e-1,c_9,c_9+.3e-1\},accuracy=0.5);\nc_9 \+ := op(1,%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+T)3VU)!#5$\"+d(=< F\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "We minimize the principal error with respect the parameters in suc cessively." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 853 "findmin('prin _err_norm'(c_2,c_3,c_4,c_5,c_6,c[8],c_9,a6_1),c[8]=\{c_8-.3e-1,c_8,c_8 +.3e-1\},accuracy=0.5);\nc_8 := op(1,%):\nfindmin('prin_err_norm'(c_2, c_3,c_4,c_5,c[6],c_8,c_9,a6_1),c[6]=\{c_6-.3e-1,c_6,c_6+.3e-1\},accura cy=0.5);\nc_6 := op(1,%):\nfindmin('prin_err_norm'(c_2,c_3,c_4,c[5],c_ 6,c_8,c_9,a6_1),c[5]=\{c_5-.3e-1,c_5,c_5+.3e-1\},accuracy=0.5);\nc_5 : = op(1,%):\nfindmin('prin_err_norm'(c_2,c_3,c[4],c_5,c_6,c_8,c_9,a6_1) ,c[4]=\{c_4-.3e-1,c_4,c_4+.3e-1\},accuracy=0.5);\nc_4 := op(1,%):\nfin dmin('prin_err_norm'(c_2,c[3],c_4,c_5,c_6,c_8,c_9,a6_1),c[3]=\{c_3-.3e -1,c_3,c_3+.3e-1\},accuracy=0.5);\nc_3 := op(1,%):\nfindmin('prin_err_ norm'(c[2],c_3,c_4,c_5,c_6,c_8,c_9,a6_1),c[2]=\{c_2-.3e-1,c_2,c_2+.3e- 1\},accuracy=0.5);\nc_2 := op(1,%):\nfindmin('prin_err_norm'(c_2,c_3,c _4,c_5,c_6,c_8,c_9,a[6,1]),a[6,1]=\{a6_1-.3e-1,a6_1,a6_1+.3e-1\},accur acy=0.5);\na6_1 := op(1,%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+d +3mt!#5$\"+_y1d7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+Dt7*e)!#5 $\"+Wv8X7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+V`Avc!#5$\"+MhX, 7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+'ybp4%!#5$\"+cd<*>\"!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+:ncn@!#5$\"+3YR#>\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+c\")yv5!#6$\"+#\\<88\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+7=8z:!#5$\"+QlC56!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We may repeat thi s process." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 846 "findmin('prin _err_norm'(c_2,c_3,c_4,c_5,c_6,c_8,c[9],a6_1),c[9]=\{c_9-.3e-1,c_9,c_9 +.3e-1\},accuracy=0.5);\nc_9 := op(1,%):\nfindmin('prin_err_norm'(c_2, c_3,c_4,c_5,c_6,c[8],c_9,a6_1),c[8]=\{c_8-.3e-1,c_8,c_8+.3e-1\},accura cy=0.5);\nc_8 := op(1,%):\nfindmin('prin_err_norm'(c_2,c_3,c_4,c_5,c[6 ],c_8,c_9,a6_1),c[6]=\{c_6-.3e-1,c_6,c_6+.3e-1\},accuracy=0.5);\nc_6 : = op(1,%):\nfindmin('prin_err_norm'(c_2,c_3,c_4,c[5],c_6,c_8,c_9,a6_1) ,c[5]=\{c_5-.3e-1,c_5,c_5+.3e-1\},accuracy=0.5);\nc_5 := op(1,%):\nfin dmin('prin_err_norm'(c_2,c_3,c[4],c_5,c_6,c_8,c_9,a6_1),c[4]=\{c_4-.3e -1,c_4,c_4+.3e-1\},accuracy=0.5);\nc_4 := op(1,%):\nfindmin('prin_err_ norm'(c_2,c[3],c_4,c_5,c_6,c_8,c_9,a6_1),c[3]=\{c_3-.3e-1,c_3,c_3+.3e- 1\},accuracy=0.5);\nc_3 := op(1,%):\nfindmin('prin_err_norm'(c[2],c_3, c_4,c_5,c_6,c_8,c_9,a6_1),c[2]=\{c_2-.3e-1,c_2,c_2+.3e-1\},accuracy=0. 5);\nc_2 := op(1,%):" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$$\"+MCxh&)!#5$\"+H%eJ5\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$ \"+h#*oVv!#5$\"+@)yc3\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+LG ,%e)!#5$\"+G]J#3\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+68@Uc!# 5$\"+cmaI)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+<2l$4%!#5$\"+n VN\"z*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+O#3o;#!#5$\"+%o!G!z *!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+?4)=\\&!#7$\"+[,99&*!#: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1498 "Digits := 10:\nc_2 := .5491880920e-2: c_3 := .21668 08236: c_4 := .4093650717: c_5 := .5642211311:\nc_6 := .8584012833: c_ 8 := .7543689261: c_9 := .8561772434: a6_1 := -.1579131812:\nfor ct to 16 do\n c_9 := op(1,findmin('prin_err_norm'(c_2,c_3,c_4,c_5,c_6,c_8 ,c[9],a6_1),\n c[9]=\{c_9-.3e-1,c_9,c_9+.3e-1\},acc uracy=0.5));\n c_8 := op(1,findmin('prin_err_norm'(c_2,c_3,c_4,c_5,c _6,c[8],c_9,a6_1),\n c[8]=\{c_8-.3e-1,c_8,c_8+.3e-1 \},accuracy=0.5));\n c_6 := op(1,findmin('prin_err_norm'(c_2,c_3,c_4 ,c_5,c[6],c_8,c_9,a6_1),\n c[6]=\{c_6-.5e-2,c_6,c_6 +.5e-2\},accuracy=0.5));\n c_5 := op(1,findmin('prin_err_norm'(c_2,c _3,c_4,c[5],c_6,c_8,c_9,a6_1),\n c[5]=\{c_5-.3e-1,c _5,c_5+.3e-1\},accuracy=0.5));\n c_4 := op(1,findmin('prin_err_norm' (c_2,c_3,c[4],c_5,c_6,c_8,c_9,a6_1),\n c[4]=\{c_4-. 3e-2,c_4,c_4+.3e-2\},accuracy=0.5));\n c_3 := op(1,findmin('prin_err _norm'(c_2,c[3],c_4,c_5,c_6,c_8,c_9,a6_1),\n c[3]= \{c_3-.3e-2,c_3,c_3+.3e-2\},accuracy=0.5));\n c_2 := op(1,findmin('p rin_err_norm'(c[2],c_3,c_4,c_5,c_6,c_8,c_9,a6_1),\n \+ c[2]=\{c_2-.3e-3,c_2,c_2+.3e-3\},accuracy=0.5));\n mn := findmin('p rin_err_norm'(c_2,c_3,c_4,c_5,c_6,c_8,c_9,a[6,1]),\n \+ a[6,1]=\{a6_1-.1e-1,a6_1,a6_1+.1e-1\},accuracy=0.5);\n a6_1 := op( 1,mn);\n print(c[2]=c_2,c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_6);\n pr int(c[8]=c_8,c[9]=c_9,a[6,1]=a6_1);\n print(`principal error norm`=o p(2,mn));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"cG6#\"\"#$ \"+JR'>>&!#7/&F%6#\"\"$$\"+X\"zr;#!#5/&F%6#\"\"%$\"+G@G\"4%F1/&F%6#\" \"&$\"+P.H>cF1/&F%6#\"\"'$\"+0p\"Rd)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+y!p!*o(!#5/&F%6#\"\"*$\"+L)p\"y')F*/&%\"aG6$\" \"'\"\"\"$!+qH\")z:F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~ error~normG$\"+Xe01&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"cG6# \"\"#$\"+Tz/#*[!#7/&F%6#\"\"$$\"+7x')p@!#5/&F%6#\"\"%$\"+<7u*3%F1/&F%6 #\"\"&$\"+rzq-cF1/&F%6#\"\"'$\"+#z>dc)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+6;e%*y!#5/&F%6#\"\"*$\"+g%)\\k))F*/&%\"aG6$\" \"'\"\"\"$!+#R5\\e\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal ~error~normG$\"+<)4'yu!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"cG6# \"\"#$\"+^>8#f%!#7/&F%6#\"\"$$\"+'e[D<#!#5/&F%6#\"\"%$\"+j(z&)3%F1/&F% 6#\"\"&$\"+9'e$)e&F1/&F%6#\"\"'$\"+#pU2c)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+r.M*3)!#5/&F%6#\"\"*$\"+#f\"y&**)F*/& %\"aG6$\"\"'\"\"\"$!+]1Y!f\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5p rincipal~error~normG$\"+wTMEm!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/& %\"cG6#\"\"#$\"+hf@#H%!#7/&F%6#\"\"$$\"+A())[<#!#5/&F%6#\"\"%$\"+6))p( 3%F1/&F%6#\"\"&$\"+/F>wbF1/&F%6#\"\"'$\"+^LOd&)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+)3gGE)!#5/&F%6#\"\"*$\"+RO=1\"*F*/&% \"aG6$\"\"'\"\"\"$!+UPu&f\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5pr incipal~error~normG$\"+&p.Z!f!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/& %\"cG6#\"\"#$\"+[!fA*R!#7/&F%6#\"\"$$\"+%Q?o<#!#5/&F%6#\"\"%$\"+=l,(3% F1/&F%6#\"\"&$\"+aQ(fc&F1/&F%6#\"\"'$\"+*eJZb)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+QZ!zT)!#5/&F%6#\"\"*$\"+6?r.#*F*/&%\" aG6$\"\"'\"\"\"$!+ViZ+;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5princi pal~error~normG$\"+F1<'G&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"c G6#\"\"#$\"+N^I#p$!#7/&F%6#\"\"$$\"+3`Vy@!#5/&F%6#\"\"%$\"+rmX'3%F1/&F %6#\"\"&$\"+cvCdbF1/&F%6#\"\"'$\"+S3g_&)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+Tvhd&)!#5/&F%6#\"\"*$\"+re!)*G*F*/&% \"aG6$\"\"'\"\"\"$!+s,q/;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5prin cipal~error~normG$\"+Y**HZZ!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&% \"cG6#\"\"#$\"+@UN#R$!#7/&F%6#\"\"$$\"+!Hc(z@!#5/&F%6#\"\"%$\"+!=6g3%F 1/&F%6#\"\"&$\"+vwz\\bF1/&F%6#\"\"'$\"+.1\"3b)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+j&ySo)!#5/&F%6#\"\"*$\"+.o#fO*F*/&%\" aG6$\"\"'\"\"\"$!+RxT3;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5princi pal~error~normG$\"+;[UtU!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"cG 6#\"\"#$\"+UhS#4$!#7/&F%6#\"\"$$\"+ru&3=#!#5/&F%6#\"\"%$\"+fpi&3%F1/&F %6#\"\"&$\"+xZJVbF1/&F%6#\"\"'$\"+6$z#\\&)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+E*=!*z)!#5/&F%6#\"\"*$\"+pz=L%*F*/&% \"aG6$\"\"'\"\"\"$!+emr6;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5prin cipal~error~normG$\"+.F3^Q!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\" cG6#\"\"#$\"+j!eCz#!#7/&F%6#\"\"$$\"+3qv\"=#!#5/&F%6#\"\"%$\"+faJ&3%F1 /&F%6#\"\"&$\"+wKrPbF1/&F%6#\"\"'$\"+8S$za)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+vgv.*)!#5/&F%6#\"\"*$\"+3V%G\\*F*/&% \"aG6$\"\"'\"\"\"$!+\\qi9;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5pri ncipal~error~normG$\"+O3OsM!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&% \"cG6#\"\"#$\"+fS[#\\#!#7/&F%6#\"\"$$\"+=H^#=#!#5/&F%6#\"\"%$\"+3R.&3% F1/&F%6#\"\"&$\"+rVvKbF1/&F%6#\"\"'$\"+0ivY&)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+xa?***)!#5/&F%6#\"\"*$\"+p$Qca*F*/&% \"aG6$\"\"'\"\"\"$!+eKA<;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5prin cipal~error~normG$\"+\"R4*GJ!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&% \"cG6#\"\"#$\"+bI^#>#!#7/&F%6#\"\"$$\"+jy7$=#!#5/&F%6#\"\"%$\"+o&)z%3% F1/&F%6#\"\"&$\"+))\\TGbF1/&F%6#\"\"'$\"+ampX&)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+z/K'3*!#5/&F%6#\"\"*$\"+,.l#f*F*/&%\" aG6$\"\"'\"\"\"$!+k`_>;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5princi pal~error~normG$\"+^r[;G!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"cG 6#\"\"#$\"+]]a#*=!#7/&F%6#\"\"$$\"+AQj$=#!#5/&F%6#\"\"%$\"+HAf%3%F1/&F %6#\"\"&$\"+#omX_&F1/&F%6#\"\"'$\"+R0vW&)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+@&>f;*!#5/&F%6#\"\"*$\"+v4CM'*F*/&%\" aG6$\"\"'\"\"\"$!+*G$e@;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5princ ipal~error~normG$\"+enQID!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"c G6#\"\"#$\"+Rrd#f\"!#7/&F%6#\"\"$$\"+=n/%=#!#5/&F%6#\"\"%$\"+#o4W3%F1/ &F%6#\"\"&$\"+)fR6_&F1/&F%6#\"\"'$\"+!)o*Qa)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+B*e&Q#*!#5/&F%6#\"\"*$\"+=:Gr'*F*/&% \"aG6$\"\"'\"\"\"$!+`mUB;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5prin cipal~error~normG$\"+\"G7zE#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&% \"cG6#\"\"#$\"+-`f#H\"!#7/&F%6#\"\"$$\"+\"zzV=#!#5/&F%6#\"\"%$\"+>oC%3 %F1/&F%6#\"\"&$\"+(4v!=bF1/&F%6#\"\"'$\"+*H@Ja)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+ " 0 "" {MPLTEXT 1 0 38 ".6926270398e-3;\nconv ert(%,rational,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)Rqi#p!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"%V9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1338 "Digits := \+ 10:\nc_2 := 1/1440: c_3 := .2184846320: c_4 := .4083975237: c_5 := .55 12863360:\nc_6 := .8541789472: c_8 := .9420576636: c_9 := .9759620500: a6_1 := -.1627931179:\nfor ct to 4 do\n c_9 := op(1,findmin('prin_e rr_norm'(c_2,c_3,c_4,c_5,c_6,c_8,c[9],a6_1),\n c[9] =\{c_9-.3e-1,c_9,c_9+.3e-1\},accuracy=0.5));\n c_8 := op(1,findmin(' prin_err_norm'(c_2,c_3,c_4,c_5,c_6,c[8],c_9,a6_1),\n \+ c[8]=\{c_8-.3e-1,c_8,c_8+.3e-1\},accuracy=0.5));\n c_6 := op(1,fin dmin('prin_err_norm'(c_2,c_3,c_4,c_5,c[6],c_8,c_9,a6_1),\n \+ c[6]=\{c_6-.5e-2,c_6,c_6+.5e-2\},accuracy=0.5));\n c_5 := op (1,findmin('prin_err_norm'(c_2,c_3,c_4,c[5],c_6,c_8,c_9,a6_1),\n \+ c[5]=\{c_5-.3e-1,c_5,c_5+.3e-1\},accuracy=0.5));\n c_4 := op(1,findmin('prin_err_norm'(c_2,c_3,c[4],c_5,c_6,c_8,c_9,a6_1),\n c[4]=\{c_4-.3e-2,c_4,c_4+.3e-2\},accuracy=0.5));\n c_3 := op(1,findmin('prin_err_norm'(c_2,c[3],c_4,c_5,c_6,c_8,c_9,a6 _1),\n c[3]=\{c_3-.3e-2,c_3,c_3+.3e-2\},accuracy=0. 5));\n mn := findmin('prin_err_norm'(c_2,c_3,c_4,c_5,c_6,c_8,c_9,a[6 ,1]),\n a[6,1]=\{a6_1-.1e-1,a6_1,a6_1+.1e-1\},accur acy=0.5);\n a6_1 := op(1,mn);\n print(c[3]=c_3,c[4]=c_4,c[5]=c_5,c [6]=c_6);\n print(c[8]=c_8,c[9]=c_9,a[6,1]=a6_1);\n print(`princip al error norm`=op(2,mn));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6& /&%\"cG6#\"\"$$\"+>!)*\\=#!#5/&F%6#\"\"%$\"++.'Q3%F*/&F%6#\"\"&$\"+2`k 5bF*/&F%6#\"\"'$\"+s0?T&)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG 6#\"\")$\"+Zw!4Z*!#5/&F%6#\"\"*$\"+`0A$y*F*/&%\"aG6$\"\"'\"\"\"$!+>0UG ;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+cQl j9!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+AaG&=#!#5/& F%6#\"\"%$\"+nev$3%F*/&F%6#\"\"&$\"+:F))3bF*/&F%6#\"\"'$\"+-=bS&)F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+wu%*>&*!#5/&F%6#\" \"*$\"+\\0j4)*F*/&%\"aG6$\"\"'\"\"\"$!+wNIH;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+`#y8L\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+`\")e&=#!#5/&F%6#\"\"%$\"+3,l$3% F*/&F%6#\"\"&$\"+h\"Rs]&F*/&F%6#\"\"'$\"+064S&)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\")$\"+\">ncc*!#5/&F%6#\"\"*$\"+Fu7I)*F*/&% \"aG6$\"\"'\"\"\"$!+\"G:-j\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5p rincipal~error~normG$\"+dQC67!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/& %\"cG6#\"\"$$\"+yF'e=#!#5/&F%6#\"\"%$\"+`db$3%F*/&F%6#\"\"&$\"++0y0bF* /&F%6#\"\"'$\"+5FsR&)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\" \")$\"+ " 0 "" {MPLTEXT 1 0 180 "[c[3]=.2185862778,c[4]=.4083555753,c[5]=.5505780500, c[6]=.8539722710,\nc[8]=.9607147517,c[9]=.9847541245,a[6,1]=-.16310777 13]:\nzip((u_,v_)->convert(u_,rational,v_),%,[5$3,6$2,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)/&%\"cG6#\"\"$#\"#S\"$$=/&F&6#\"\"%#\"#\\ \"$?\"/&F&6#\"\"&#F1\"#*)/&F&6#\"\"'#\"$p#\"$:$/&F&6#\"\")#F>\"$!G/&F& 6#\"\"*#\"#l\"#m/&%\"aG6$F<\"\"\"#!#%)\"$:&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 13 "ca lc_RKcoeffs" }{TEXT -1 72 " from a later section gives the characteris tics of the resulting scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "c_2 := 1/1440: c_3 := 40/18 3: c_4 := 49/120: c_5 := 49/89: c_6 := 269/315: c_8 := 269/280: c_9 := 65/66:\na6_1 := -84/515: bs_11 := 5/2:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\"\"\"\"%S9/&F&6#\"\"$# \"#S\"$$=/&F&6#\"\"%#\"#\\\"$?\"/&F&6#\"\"&#F8\"#*)/&F&6#\"\"'#\"$p#\" $:$/&F&6#\"\"(#\")p#43\"\")BV1d/&F&6#\"\")#FE\"$!G/&F&6#\"\"*#\"#l\"#m " }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\" $\"'@\"[&!\"(/&F&6#\"\"&$\"'ZgR!\"'/&F&6#\"\"'$!'EZGF+/&F&6#\"\"($\"'/ *)HF2/&F&6#\"\")$\"'_BC!\"&/&F&6#\"\"*$!'vlVFE/&F&6#\"#5$\"'%4A#FE" }} {PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"' q\"\\&!\"(/&F&6#\"\"&$\"'xyR!\"'/&F&6#\"\"'$!'%GO&F+/&F&6#\"\"($\"'8&) HF2/&F&6#\"\")$\"'\"3r#!\"&/&F&6#\"\"*$!'\\3\\FE/&F&6#\"#5$\"\"!FQ/&F& 6#\"#6$\"'++DFE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\" \"%8-norm~of~linking~coeffsGF&$\"+z7'o9&!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~schemeG$\"+W\"3 U5\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~terms~ of~order~7~schemeG$\"+@9mF\")!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% 5ratio~of~error~normsG$\"'igt!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %L2-norm~of~principal~error~of~order~6~schemeG$\"+r8=L5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)%Hkw$!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~interval G7$$!)*z2o$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~pri ncipal~error~ratio~of~order~6~schemeG$\"%fK!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")g5bC!\"(/&%\"CGF&$\")?7*[#F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e11,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(12-i)],i=2..1 1),\n[`b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11)]])):\nevalf[6] (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7.$\"'WWp!\"*F(% !GF+F+F+F+F+F+F+F+F+7.$\"'z&=#!\"'$!'3=M!\"%$\"'%*RMF2F+F+F+F+F+F+F+F+ F+7.$\"'L$3%F/$\"'X#p#!\"($\"\"!F<$\"'49QF/F+F+F+F+F+F+F+F+7.$\"'i0bF/ $\"'z#4\"F/F;$\"'')3:F/$\"''R!HF/F+F+F+F+F+F+F+7.$\"'oR&)F/$!'2J;F/F;$ \"'nu8!\"&$!'KN=FO$\"'tx9FOF+F+F+F+F+F+7.$\"'B%*=F/$\"'F0WF:F;$\"'&H:$ F/$!''yK$F/$\"'H&y\"F/$!'$oc\"F:F+F+F+F+F+7.$\"'92'*F/$\"'bpNF/F;$!'Y> JF2$\"'qq " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 44 " #===========================================" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 50 "scheme with a moderately large stability interval " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The follo wing scheme was obtained by \"trial and error\"." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 29 ": The code for the procedure " } {TEXT 0 13 "calc_RKcoeffs" }{TEXT -1 29 " ia given in a later section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "c_2 := 1/180: c_3 := 9/46: c_4 := 4513/10000: c_5 := 118/197: c_6 := 89/100: c_8 := 36/47: c_9 := 56/65: \na6_1 := -9/58: bs_11 := \+ 13/25:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/ &%\"cG6#\"\"##\"\"\"\"$!=/&F&6#\"\"$#\"\"*\"#Y/&F&6#\"\"%#\"%8X\"&++\" /&F&6#\"\"&#\"$=\"\"$(>/&F&6#\"\"'#\"#*)\"$+\"/&F&6#\"\"(#\"*JnyD\"\"* B3-H'/&F&6#\"\")#\"#O\"#Z/&F&6#F1#\"#c\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'f4e!\"(/&F&6#\"\" &$\"'O2c!\"'/&F&6#\"\"'$!'pg6!\"&/&F&6#\"\"($\"'jZJF2/&F&6#\"\")$!'I-o F2/&F&6#\"\"*$\"'s?=F9/&F&6#\"#5$\"'3g')F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'99l!\"(/&F&6#\"\" &$\"'%3m*!\"'/&F&6#\"\"'$!')>I)!\"&/&F&6#\"\"($\"'6ZGF2/&F&6#\"\")$!'& y9$F9/&F&6#\"\"*$\"'Rh5!\"%/&F&6#\"#5$\"\"!FR/&F&6#\"#6$\"'++_F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+ha(4-\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm ~of~principal~error~of~order~7~schemeG$\"+>\">%\\[!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$ \"'2LG!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~e rror~of~order~6~schemeG$\"+n[>R!)!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%;order~7~stability~intervalG7$$!)/\"zi%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)\"e0k$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~ order~6~schemeG$\"%TQ!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6 #\"\")$\"),!z$>!\"(/&%\"CGF&$\")2))Q>F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We minimize the principal error no rm with respect to " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 51 "We only perform a few iterations because allowing " } {XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 30 " to vary causes \+ the weights " }{XPPEDIT 18 0 "b[8]" "6#&%\"bG6#\"\")" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "b[9]" "6#&%\"bG6#\"\"*" }{TEXT -1 31 " to beco me large in magnitude." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 828 "D igits := 10:\nc_2 := 1/180: c_3 := 9/46: c_4 := 4513/10000: c_5 := 118 /197:\nc_6 := 89/100: c_8 := 36/47: c_9 := 56/65:\na6_1 := -9/58:\nfor ct to 20 do\n c_9 := op(1,findmin('prin_err_norm'(c_2,c_3,c_4,c_5,c _6,c_8,c[9],a6_1),\n c[9]=\{c_9-.3e-1,c_9,c_9+.3e-1 \},accuracy=0.5));\n c_8 := op(1,findmin('prin_err_norm'(c_2,c_3,c_4 ,c_5,c_6,c[8],c_9,a6_1),\n c[8]=\{c_8-.3e-1,c_8,c_8 +.3e-1\},accuracy=0.5));\n c_5 := op(1,findmin('prin_err_norm'(c_2,c _3,c_4,c[5],c_6,c_8,c_9,a6_1),\n c[5]=\{c_5-.3e-1,c _5,c_5+.3e-1\},accuracy=0.5));\n mn := findmin('prin_err_norm'(c_2,c _3,c[4],c_5,c_6,c_8,c_9,a6_1),\n c[4]=\{c_4-.3e-2,c _4,c_4+.3e-2\},accuracy=0.5);\n c_4 := op(1,mn);\n print(c[4]=c_4, c[5]=c_5,c[8]=c_8,c[9]=c_9);\n print(`principal error norm`=op(2,mn) );\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"%$\"+A^AGX !#5/&F%6#\"\"&$\"+d$>C)fF*/&F%6#\"\")$\"+2e&4r(F*/&F%6#\"\"*$\"+ac^)o) F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+shb[ ;!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"%$\"+<)yAa%!#5/&F %6#\"\"&$\"+9u7xfF*/&F%6#\"\")$\"+KV(o%zF*/&F%6#\"\"*$\"+v'Qy(*)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+8y)[]\"!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"%$\"+dv)\\b%!#5/&F%6# \"\"&$\"+v$[T(fF*/&F%6#\"\")$\"+Z]!\\>)F*/&F%6#\"\"*$\"+Fb*=<*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+`:E$Q\"!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"%$\"+UJOmX!#5/&F%6#\" \"&$\"+z[!H(fF*/&F%6#\"\")$\"+b0WB%)F*/&F%6#\"\"*$\"+)H4*=$*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+[X2!G\"!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We u se rational approximations for the values obtained for " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6# \"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "[c[4]=.456 6363142,c[5]=.5972904879,c[8]=.8423440555,c[9]=.9318909298]:\nzip((u_, v_)->convert(u_,rational,v_),%,[5$2,6$2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"cG6#\"\"%#\"#z\"$t\"/&F&6#\"\"&#\"#*)\"$\\\"/&F& 6#\"\")#\"$(=\"$A#/&F&6#\"\"*#\"$g#\"$z#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We obtain a scheme with the follow ing characteristics." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "c_2 := 1/180: c_3 := 9/46: c_4 := 79/173: c_5 := 89/149: c_6 := 89/100: c _8 := 187/222: c_9 := 260/279: \na6_1 := -9/58: bs_11 := 13/25:\ncalc_ RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"# #\"\"\"\"$!=/&F&6#\"\"$#\"\"*\"#Y/&F&6#\"\"%#\"#z\"$t\"/&F&6#\"\"&#\"# *)\"$\\\"/&F&6#\"\"'#F?\"$+\"/&F&6#\"\"(#\")*ohv%\"*VCNP#/&F&6#\"\")# \"$(=\"$A#/&F&6#F1#\"$g#\"$z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1ord er~7~weights:G/&%\"bG6#\"\"\"$\"'A*z&!\"(/&F&6#\"\"&$\"'e)*\\!\"'/&F&6 #\"\"'$\"':#f$!\"&/&F&6#\"\"($\"'\"f;$F2/&F&6#\"\")$!'\"*)o\"F9/&F&6# \"\"*$!'\"[2#F9/&F&6#\"#5$\"'DrHF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+ %1order~6~weights:G/&%#b*G6#\"\"\"$\"'e+f!\"(/&F&6#\"\"&$\"'bo`!\"'/&F &6#\"\"'$\"'9Sd!\"&/&F&6#\"\"($\"'wCJF2/&F&6#\"\")$!'`REF9/&F&6#\"\"*$ !'&*GNF9/&F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++_F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+'Gr ')3\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~er ror~of~order~7~schemeG$\"+V2hx7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%J2-norm~of~order~9~terms~of~order~7~schemeG$\"+s42`f!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'_fY!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~sche meG$\"+u5#H1\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stabil ity~intervalG7$$!) " 0 "" {MPLTEXT 1 0 535 "Digits := 10:\nc_2 := 1/180: c_3 := 9/46 : c_4 := .4566363142: c_5 := .5972904879:\nc_6 := 89/100: c_8 := 187/2 22: c_9 := 260/279:\na6_1 := -9/58:\nfor ct to 20 do\n c_5 := op(1,f indmin('prin_err_norm'(c_2,c_3,c_4,c[5],c_6,c_8,c_9,a6_1),\n \+ c[5]=\{c_5-.3e-1,c_5,c_5+.3e-1\},accuracy=0.5));\n mn := f indmin('prin_err_norm'(c_2,c_3,c[4],c_5,c_6,c_8,c_9,a6_1),\n \+ c[4]=\{c_4-.3e-2,c_4,c_4+.3e-2\},accuracy=0.5);\n c_4 := o p(1,mn);\n print(c[4]=c_4,c[5]=c_5);\n print(`principal error norm `=op(2,mn));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\" %$\"+%)3%fd%!#5/&F%6#\"\"&$\"+i#y7)fF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+I.\"4?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+8:+%e%!#5/&F%6#\"\"&$\"+\"4.$))fF*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+kRTT6!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+(>,3f%!#5/&F%6# \"\"&$\"+k\"*=%*fF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~er ror~normG$\"+G[F(4\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\" \"%$\"+!H*\\'f%!#5/&F%6#\"\"&$\"+hg8**fF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+q)\\Z1\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+8QK,Y!#5/&F%6#\"\"&$\"+0)oK+'F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+[P5T5! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+V4T0Y!#5/&F%6# \"\"&$\"+p!fn+'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~erro r~normG$\"+SO#Q-\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\" %$\"+Fg()3Y!#5/&F%6#\"\"&$\"+,'4(4gF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+@^C65!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+1K!=h%!#5/&F%6#\"\"&$\"+`i>7gF*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+I\"[@+\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+J#*H9Y!#5/&F%6#\"\"& $\"+(y7V,'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~nor mG$\"+N@Ab**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+s xT;Y!#5/&F%6#\"\"&$\"+7k5;gF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5pr incipal~error~normG$\"+$G&H2**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ &%\"cG6#\"\"%$\"+lh@=Y!#5/&F%6#\"\"&$\"+wqiY!#5/&F%6#\"\"&$\"+&p;*=gF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+k2oZ)*!# :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+U!R5i%!#5/&F%6# \"\"&$\"+'f5+-'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~erro r~normG$\"+$)*e'H)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\" \"%$\"+<%R@i%!#5/&F%6#\"\"&$\"+7'Q4-'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+73n;)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+sN2BY!#5/&F%6#\"\"&$\"+tfs@gF*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+^bJ2)*!#: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+*emQi%!#5/&F%6# \"\"&$\"+TSRAgF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error ~normG$\"+p.e+)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$ \"+o(RXi%!#5/&F%6#\"\"&$\"+94'H-'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%5principal~error~normG$\"+4Pt&z*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+%>6^i%!#5/&F%6#\"\"&$\"+L>WBgF*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+2wC#z*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+JifDY!#5/&F%6#\"\"&$\"+4,& Q-'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+q 7u*y*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"+Bz+EY!#5 /&F%6#\"\"&$\"+vk>CgF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal ~error~normG$\"+@-%zy*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We use rational approximations for the values obt ained for " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "[c[4]=.4626007923,c[5]=.6024196475] :\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG 6#\"\"%#\"#o\"$Z\"/&F&6#\"\"&#\"#]\"#$)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We obtain a scheme with the follow ing characteristics." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "c_2 := 1/180: c_3 := 9/46: c_4 := 68/147: c_5 := 50/83: c_6 := 89/100: c_ 8 := 187/222: c_9 := 260/279: \na6_1 := -9/58: bs_11 := 13/25:\ncalc_ RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"# #\"\"\"\"$!=/&F&6#\"\"$#\"\"*\"#Y/&F&6#\"\"%#\"#o\"$Z\"/&F&6#\"\"&#\"# ]\"#$)/&F&6#\"\"'#\"#*)\"$+\"/&F&6#\"\"(#\"(DXd(\")hU.Q/&F&6#\"\")#\"$ (=\"$A#/&F&6#F1#\"$g#\"$z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~ 7~weights:G/&%\"bG6#\"\"\"$\"'+Rd!\"(/&F&6#\"\"&$\"'FB`!\"'/&F&6#\"\"' $\"'@6W!\"&/&F&6#\"\"($\"'%3;$F2/&F&6#\"\")$!'42@F9/&F&6#\"\"*$!'!Gd#F 9/&F&6#\"#5$\"'vGOF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weig hts:G/&%#b*G6#\"\"\"$\"']4e!\"(/&F&6#\"\"&$\"'M#f&!\"'/&F&6#\"\"'$\"'[ Kf!\"&/&F&6#\"\"($\"'jKJF2/&F&6#\"\")$!'e#y#F9/&F&6#\"\"*$!'\\+OF9/&F& 6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++_F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+:9Ij))!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 7~schemeG$\"+ceV)y*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of ~order~9~terms~of~order~7~schemeG$\"+'z)GZg!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'*z<'!\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+&) >k^l!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~interv alG7$$!)=B/Z!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~st ability~intervalG7$$!)n/lW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%Pmax-min~principal~error~ratio~of~order~6~schemeG$\"%))H!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")%>\"4@!\"(/&%\"CGF&$ \")MeS@F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 171 ": This scheme is taken as the starting poi nt for a later investigation that involves minimizing the principal er ror norm subject to the stability function remaining fixed." }}{PARA 0 "" 0 "" {TEXT -1 40 "#---------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 44 "#===========================================" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 48 "#======================================== =======" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Constructing a scheme \+ with a given stability function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 878 "Qeqs := Quadratur eConditions(7,10,'expanded'):\nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2 *c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c [6]-c[6]-c[5]-12*c[7]*c[5]+14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-1 2*c[6]*c[7]+14*c[7]*c[5]^2=0:\ncdns1 := [op(Qeqs),node_eq]:\nSO_eqs := [op(RowSumConditions(10,'expanded')),op(StageOrderConditions(2,3..10, 'expanded')),\n op(StageOrderConditions(3,5..10,'expanded'))]:\nsimp_ eqs := [add(b[i]*a[i,1],i=2..10)=b[1],seq(add(b[i]*a[i,j],i=j+1..10)=b [j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5..10)=0, add(b[i]*c[i]^2*a[i,3],i=5..10)=0]:\ncdcns2 := [op(simp_eqs),op(simp_e qs2),op(SO_eqs)]:\nSO7_10 := SimpleOrderConditions(7,10,'expanded'):\n cdcns3 := [seq(SO7_10[i],i=[54,59,61])]:\ncdcns4 := [seq(SO7_10[i],i=[ 50,51])]:\ncdcns5 := [seq(SO7_10[i],i=[45,55])]:\nerrterms7_10 := Prin cipalErrorTerms(7,10,'expanded'):" }}}{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 24 "The following procedure " } {TEXT 0 14 "prin_err_normB" }{TEXT -1 54 " constructs a 10 stage order 7 scheme when the nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 58 " are specified along with a \+ specified stability function." }}{PARA 0 "" 0 "" {TEXT -1 69 "The proc edure then calculates the principal error norm of the scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 0 14 "prin_err_normB" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1790 "prin_err_normB := proc(c2, c5,c6,c8,c9)\n local eqns1,eqns2,eqns3,eqns4,eqns5,Rz,eqA,eqB,eqC,so l,sm,ct,saveDigits;\n global e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12, e13,e14,e15;\n\n e1 := \{c[2]=convert(c2,rational,Digits+2),c[5]=con vert(c5,rational,Digits+2),\n c[6]=convert(c6,rational,Digits+2),c[ 8]=convert(c8,rational,Digits+2),\n c[9]=convert(c9,rational,Digits +2),c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\n eqns1 := subs(e1,cdns1);\n e 2 := solve(\{op(eqns1)\});\n e3 := `union`(e1,e2);\n e4 := \{seq(a [i,2]=0,i=4..10)\};\n e5 := `union`(e3,e4);\n eqns2 := subs(e5,cdc ns2);\n e6 := solve(\{op(eqns2)\},indets(eqns2) minus \n \{c[3] ,c[4],a[6,1],a[7,6],a[8,6],a[8,7],a[9,5],a[9,6],a[9,7],a[9,8]\});\n \+ e7 := `union`(e5,e6);\n eqns3 := simplify(subs(e7,cdcns3));\n e8 : = solve(\{op(eqns3)\},\{a[8,7],a[9,7],a[9,8]\});\n e9 := `union`(map (u_->lhs(u_)=simplify(subs(e8,rhs(u_))),e7),e8);\n eqns4 := simplify (subs(e9,cdcns4));\n e10 := solve(\{op(eqns4)\},\{a[8,6],a[9,6]\}); \n e11 := `union`(map(u_->lhs(u_)=simplify(subs(e10,rhs(u_))),e9),e1 0);\n eqns5 := simplify(subs(e11,cdcns5));\n e12 := solve(\{op(eqn s5)\},\{a[7,6],a[9,5]\});\n e13 := `union`(map(u_->lhs(u_)=simplify( subs(e12,rhs(u_))),e11),e12);\n Rz := StabilityFunction(7,10,'expand ed');\n eqA := simplify(subs(e13,coeff(Rz,z^10)))=Gam10/10!;\n eqB := simplify(subs(e13,coeff(Rz,z^9)))=Gam9/9!;\n eqC := simplify(sub s(e13,coeff(Rz,z^8)))=Gam8/8!;\n sol := solve(\{eqA,eqB,eqC\},\{c[3] ,c[4],a[6,1]\});\n e14 := convert(evalf[14](sol),rational,rat_digits );\n e15 := `union`(map(u_->lhs(u_)=simplify(subs(e14,rhs(u_))),e13) ,e14);\n saveDigits := Digits;\n Digits := max(Digits,14);\n sm \+ := 0:\n for ct to nops(errterms7_10) do\n sm := sm+evalf(subs(e 15,errterms7_10[ct]))^2;\n end do:\n evalf[saveDigits](sqrt(sm)); \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 79 "#--------------------------------------------------- ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "sc heme with a moderately large stability region (continued)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "#----------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 58 "The scheme of Sha rp and Smart has the stability function: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z)=1+z+z^2/2 !+z^3/3!+z^4/4!+z^5/5!+z^6/6!+z^7/7!+``" "6#/-%\"RG6#%\"zG,4\"\"\"F)F' F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F 5F/F)*&F'\"\"&-F-6#F9F/F)*&F'\"\"'-F-6#F=F/F)*&F'\"\"(-F-6#FAF/F)%!GF) " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "10676759633808473 539/516965630708022048000000*z^8+9588168759125163797/86160938451337008 00000000*z^9-5750490893904312513/4786718802852056000000000*z^10" "6#,( *(\"5RNZ3Qjfnn5\"\"\"\"9+++[?-32jlp^!\"\"%\"zG\"\")F&*(\"4(zj^7f(o\")e *F&\":++++3qL^%Q4;')F(F)\"\"*F&*(\"48DJ/R*3\\]dF&\":++++g0_G!)=ny%F(F) \"#5F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "['gamma'(8)=10676759633808473539/5 16965630708022048000000*8!,\n'gamma'(9)=9588168759125163797/8616093845 133700800000000*9!,\n'gamma'(10)=-5750490893904312513/4786718802852056 000000000*10!]:\nmap(u_->lhs(u_)=convert(evalf[10](rhs(u_)),rational,6 ),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%&gammaG6#\"\")#\"$C#\"$p #/-F&6#\"\"*#\"$'H\"$L(/-F&6#\"#5#!$Y*\"$<#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "This stability function is appr oximately: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2!+z^3/3!+z^4/4!+z^5/5!+z^6/6!+z^7/7!+gamma(8);" "6#/-%\"RG6# %\"zG,4\"\"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/ F)*&F'\"\"%-F-6#F5F/F)*&F'\"\"&-F-6#F9F/F)*&F'\"\"'-F-6#F=F/F)*&F'\"\" (-F-6#FAF/F)-%&gammaG6#\"\")F)" }{TEXT -1 1 " " }{TEXT 273 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "z^8/8!+gamma(9);" "6#,&*&%\"zG\"\")-%*f actorialG6#F&!\"\"\"\"\"-%&gammaG6#\"\"*F+" }{TEXT -1 1 " " }{TEXT 274 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^9/9!+gamma(10);" "6#,&*&% \"zG\"\"*-%*factorialG6#F&!\"\"\"\"\"-%&gammaG6#\"#5F+" }{TEXT -1 1 " \+ " }{TEXT 275 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^10/10!" "6#*&%\"z G\"#5-%*factorialG6#F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "gamma(8)=224/269" "6#/-%&gammaG6#\"\")* &\"$C#\"\"\"\"$p#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "gamma(9)=296/ 733" "6#/-%&gammaG6#\"\"**&\"$'H\"\"\"\"$L(!\"\"" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "gamma(10)=-946/217" "6#/-%&gammaG6#\"#5,$*&\"$Y*\"\" \"\"$<#!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "These three values are supplied to the pr ocedure " }{TEXT 0 14 "prin_err_normB" }{TEXT -1 20 " via the variable s " }{TEXT 262 4 "Gam8" }{TEXT -1 2 ", " }{TEXT 262 4 "Gam9" }{TEXT -1 5 " and " }{TEXT 262 5 "Gam10" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "Digits := 1 0:\nc_2 := 1/50: c_5 := 57/100: c_6 := 43/50: c_8 := 18/25: c_9 := 5/6 :\nGam8 := 224/269: Gam9 := 296/733: Gam10 := -946/217: rat_digits := \+ 8:\nprin_err_normB(c_2,c_5,c_6,c_8,c_9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+F3mu7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 119 "The following scheme (obtained in a previous section) \+ has a larger stability region than the scheme of Sharp and Smart." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "c_2 := 1/180: c_3 := 9/46: \+ c_4 := 68/147: c_5 := 50/83: c_6 := 89/100: c_8 := 187/222: c_9 := 260 /279: \na6_1 := -9/58: bs_11 := 13/25:\ncalc_RKcoeffs():" }}{PARA 11 " " 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\"\"\"\"$!=/&F&6#\"\"$# \"\"*\"#Y/&F&6#\"\"%#\"#o\"$Z\"/&F&6#\"\"&#\"#]\"#$)/&F&6#\"\"'#\"#*) \"$+\"/&F&6#\"\"(#\"(DXd(\")hU.Q/&F&6#\"\")#\"$(=\"$A#/&F&6#F1#\"$g#\" $z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\" \"\"$\"'+Rd!\"(/&F&6#\"\"&$\"'FB`!\"'/&F&6#\"\"'$\"'@6W!\"&/&F&6#\"\"( $\"'%3;$F2/&F&6#\"\")$!'42@F9/&F&6#\"\"*$!'!Gd#F9/&F&6#\"#5$\"'vGOF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$ \"']4e!\"(/&F&6#\"\"&$\"'M#f&!\"'/&F&6#\"\"'$\"'[Kf!\"&/&F&6#\"\"($\"' jKJF2/&F&6#\"\")$!'e#y#F9/&F&6#\"\"*$!'\\+OF9/&F&6#\"#5$\"\"!FQ/&F&6# \"#6$\"'++_F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"% 8-norm~of~linking~coeffsGF&$\"+:9Ij))!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~schemeG$\"+ceV) y*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~terms~of ~order~7~schemeG$\"+'z)GZg!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ra tio~of~error~normsG$\"'*z<'!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L 2-norm~of~principal~error~of~order~6~schemeG$\"+&)>k^l!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)=B/Z!\"(\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$ !)n/lW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principa l~error~ratio~of~order~6~schemeG$\"%))H!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")%>\"4@!\"(/&%\"CGF&$\")MeS@F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The prece ding scheme has the following stability function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Rz := subs(e11,StabilityFunction(7,11,'expand ed'));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#RzG,8\"\"\"F&%\"zGF&*&#F& \"\"#F&*$)F'F*F&F&F&*&#F&\"\"'F&*$)F'\"\"$F&F&F&*&#F&\"#CF&*$)F'\"\"%F &F&F&*&#F&\"$?\"F&*$)F'\"\"&F&F&F&*&#F&\"$?(F&*$)F'F/F&F&F&*&#F&\"%S]F &*$)F'\"\"(F&F&F&*&#\"N#QC4:y5F&*$)F'\" \")F&F&F&*&#\"9\"fUy@oFtCEN6(\"@+_X!GmtJ3:y=ok?YF&*$)F'\"\"*F&F&!\"\"* &#\"6vr>kR4o,@'=\"<_In.#)*ef!\\,!=nOF&*$)F'\"#5F&F&FX" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We shall attempt to construct a scheme with essentially the same stability function but w ith a smaller principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 43 "We start by obtaining suitable values for " }{XPPEDIT 18 0 "gamma(8)" " 6#-%&gammaG6#\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "gamma(9)" "6#-%& gammaG6#\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "gamma(10)" "6#-%& gammaG6#\"#5" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "['gamma'(8)=241295183969821660143965779/1078150924382351940718798 7728800*8!,\n'gamma'(9)=-711352624732768217842591/46206468187815083173 66280455200*9!,\n'gamma'(10)=-186210168093964197175/366718001490595898 203673052*10!]:\nmap(u_->lhs(u_)=convert(evalf[10](rhs(u_)),rational,4 ),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%&gammaG6#\"\")#\"#P\"#T/ -F&6#\"\"*#!#5\"$z\"/-F&6#\"#5#!#N\"#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We make perturbations in the values \+ for " }{XPPEDIT 18 0 "gamma(8" "6#-%&gammaG6#\"\")" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "gamma(9)" "6#-%&gammaG6#\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "gamma(10)" "6#-%&gammaG6#\"#5" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "['gamma'(8)=37/41+.6e-4,'ga mma'(9)=-10/179-.1e-5,'gamma'(10)=-35/19-.24e-2];\nmap(u_->lhs(u_)=con vert(evalf[10](rhs(u_)),rational,4),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%&gammaG6#\"\")$\"+W-*\\-*!#5/-F&6#\"\"*$!+z@p'e&!#6/-F&6#\" #5$!+j_]W=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%&gammaG6#\"\")# \"#G\"#J/-F&6#\"\"*#!\"*\"$h\"/-F&6#\"#5#!#C\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "Digits := 1 0:\nc_2 := 1/180: c_5 := 50/83: c_6 := 89/100: c_8 := 187/222: c_9 := \+ 260/279:\nGam8 := 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits : = 8:\nprin_err_normB(c_2,c_5,c_6,c_8,c_9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[0;b%*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "We now attempt to minimize the principal \+ error norm with respect to the nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"c G6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 46 " while retaining the same stability function." }}{PARA 0 "" 0 "" {TEXT -1 47 "We may s ome changes by \"trial and error\" first." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "Digits := 10:\nc_2 := .005556: c_5 := .6024: c_6 := \+ .89: c_8 := .8423: c_9 := .9319:\nGam8 := 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits := 8:\nprin_err_normB(c_2,c_5,c_6,c_8,c_9);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+we*=Y*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "Digits := 1 0:\nc_2 := .005: c_5 := .60242: c_6 := .89: c_8 := .85: c_9 := .936:\n Gam8 := 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits := 8:\nprin _err_normB(c_2,c_5,c_6,c_8,c_9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"++%=D+*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 579 "Digits := 10:\nc_2 := .005: c_5 := .60242: c_6 \+ := .89: c_8 := .85: c_9 := .936:\nGam8 := 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits := 8:\nfindmin('prin_err_normB'(c_2,c_5,c[6],c_ 8,c_9),c[6]=\{c_6-.3e-1,c_6,c_6+.3e-1\},accuracy=0.5);\nc_6 := op(1,%) :\nfindmin('prin_err_normB'(c_2,c_5,c_6,c_8,c[9]),c[9]=\{c_9-.3e-1,c_9 ,c_9+.3e-1\},accuracy=0.5);\nc_9 := op(1,%):\nfindmin('prin_err_normB' (c_2,c[5],c_6,c_8,c_9),c[5]=\{c_5-.3e-4,c_5,c_5+.3e-4\},accuracy=0.5); \nc_5 := op(1,%):\nfindmin('prin_err_normB'(c_2,c_5,c_6,c_8,c[9]),c[9] =\{c_9-.3e-2,c_9,c_9+.3e-2\},accuracy=0.5);\nc_9 := op(1,%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+I*\\=#*)!#5$\"+\"G**yi)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+(ps#Q%*!#5$\"+1H\"yc)!#:" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#7$$\"+!f9S-'!#5$\"+y5sh%)!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+9'\\,U*!#5$\"+p\\We%)!#:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 687 "Digits := 10:\nc_2 := .005: c_5 := .6024014590: c_6 := .8921849930: c_8 := .85: c_9 := .942014961 4:\nGam8 := 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits := 10: \nfor ct to 8 do\n print(ct);\n c_9 := op(1,findmin('prin_err_norm B'(c_2,c_5,c_6,c_8,c[9]),\n c[9]=\{c_9-.15e-1,c_9,c _9+.15e-1\},accuracy=0.5));\n c_6 := op(1,findmin('prin_err_normB'(c _2,c_5,c[6],c_8,c_9),\n c[6]=\{c_6-.1e-1,c_6,c_6+.1 e-1\},accuracy=0.5));\n mn := findmin('prin_err_normB'(c_2,c[5],c_6, c_8,c_9),\n c[5]=\{c_5-.3e-4,c_5,c_5+.3e-4\},accura cy=0.5);\n c_5 := op(1,mn);\n print(c[5]=c_5,c[6]=c_6,c[9]=c_9);\n print(`principal error norm`=op(2,mn));\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\" \"&$\"+!=HQ-'!#5/&F%6#\"\"'$\"+r?3H*)F*/&F%6#\"\"*$\"+9'\\,U*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+6'QBM)!#: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"&$\"+!f9P-'!#5/&F%6#\"\"'$\"+t.$R$*)F*/&F% 6#\"\"*$\"+L\"Q\"G%*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal ~error~normG$\"+^?W.$)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"&$\"+!f9P-'!#5/&F%6#\"\"' $\"+XY/P*)F*/&F%6#\"\"*$\"+wM:M%*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%5principal~error~normG$\"+3 " 0 "" {MPLTEXT 1 0 268 "Digits := 10:\nc_2 := 1/200: c_5 := .6023714590: c_6 := .8937652358: c_8 := \+ 17/20: c_9 := .9447494746:\nGam8 := 28/31: Gam9 := -9/161: Gam10 := -2 4/13: rat_digits := 6:\nprin_err_normB(c_2,c_5,c_6,c_8,c_9);\nc[3]=sub s(e15,c[3]),c[4]=subs(e15,c[4]),a[6,1]=subs(e15,a[6,1]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+4?;$H)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %/&%\"cG6#\"\"$#\"$6\"\"$s&/&F%6#\"\"%#\"$1$\"$r'/&%\"aG6$\"\"'\"\"\"# !$2$\"%V;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "[c[5]=.6023714590,c[6]=.8937652358,c[9]=.9447494746 ]:\nconvert(%,rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"c G6#\"\"&#\"$.#\"$P$/&F&6#\"\"'#\"$W#\"$t#/&F&6#\"\"*#\"$r\"\"$\"=" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "We fix th e values " }{XPPEDIT 18 0 "c[6] = 244/273" "6#/&%\"cG6#\"\"'*&\"$W#\" \"\"\"$t#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9] = 171/181" "6#/& %\"cG6#\"\"**&\"$r\"\"\"\"\"$\"=!\"\"" }{TEXT -1 86 " (and the stabil ity function) and minimize the principal error norm with respect to \+ " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "Digits := 10:\nc_2 := 1/200: c_5 : = .6023714590: c_6 := 244/273: c_8 := 17/20: c_9 := 171/181:\nGam8 := \+ 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits := 6:\nfindmin('pri n_err_normB'(c_2,c[5],c_6,c_8,c_9),c[5]=\{c_5-.3e-4,c_5,c_5+.3e-4\},ac curacy=0.65,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+eMnBg !#5$\"+r'=-H)!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "c[5]=.6023673458;\nconvert(%,rational,7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&$\"+eMnBg!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$4&\"$X)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We can obtain the corre sponding values for " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "Digits := 10:\nc_2 := 1/200 : c_5 := 509/845: c_6 := 244/273: c_8 := 17/20: c_9 := 171/181:\nGam8 \+ := 28/31: Gam9 := -9/161: Gam10 := -24/13: rat_digits := 6:\nprin_err_ normB(c_2,c_5,c_6,c_8,c_9);\nc[3]=subs(e15,c[3]),c[4]=subs(e15,c[4]),a [6,1]=subs(e15,a[6,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V(47H)! #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"$#\"#$*\"$z%/&F%6# \"\"%#\"$5\"\"$T#/&%\"aG6$\"\"'\"\"\"#!$G#\"%J7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can use the procedure \+ " }{TEXT 0 13 "calc_RKcoeffs" }{TEXT -1 98 " from a later section to c heck the characteristics of the scheme and choose a suitable value for " }{XPPEDIT 18 0 "`b*`[11]" "6#&%#b*G6#\"#6" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "c_2 := 1/200: c_3 := 93/479: c_4 := 110/241: c_5 := 509/845: c_6 \+ := 244/273: c_8 := 17/20: c_9 := 171/181:\na6_1 := -228/1231: bs_11 := 16/25:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G /&%\"cG6#\"\"##\"\"\"\"$+#/&F&6#\"\"$#\"#$*\"$z%/&F&6#\"\"%#\"$5\"\"$T #/&F&6#\"\"&#\"$4&\"$X)/&F&6#\"\"'#\"$W#\"$t#/&F&6#\"\"(#\"*\\eyA#\"+@ _946/&F&6#\"\")#\"#<\"#?/&F&6#\"\"*#\"$r\"\"$\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'s,e!\"(/&F&6#\"\" &$\"'\"y8&!\"'/&F&6#\"\"'$\"'-0Q!\"&/&F&6#\"\"($\"'6!=$F2/&F&6#\"\")$! 'f>?F9/&F&6#\"\"*$!'\\I@F9/&F&6#\"#5$\"'Z_XF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'Bhe!\"(/&F&6#\"\" &$\"'Lb`!\"'/&F&6#\"\"'$\"';8]!\"&/&F&6#\"\"($\"'9cJF2/&F&6#\"\")$!'eF EF9/&F&6#\"\"*$!'MNHF9/&F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++kF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeff sGF&$\"+Q#Rj[*!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~pri ncipal~error~of~order~7~schemeG$\"+V(47H)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~terms~of~order~7~schemeG$\"+H.Jtc! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'cUo! \"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of ~order~6~schemeG$\"+(3D-9'!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;or der~7~stability~intervalG7$$!)\"[Tq%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!),#fZ%!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~or der~6~schemeG$\"%zK!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\")$\")6D$4#!\"(/&%\"CGF&$\"))=J8#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "#----------------------------------- ----------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 79 "#------------------------------------ ------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "scheme with a larger stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We attempt to construct a scheme with the following stability function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Rz := add(z^ i/i!,i=0..8)+82/91*z^9/9!+71/82*z^10/10!:\nR := unapply(%,z):\n'R(z)'= R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,8\"\"\"F)F'F)* &#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\" \"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\" %S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#\"#T\")S5^;F)*$)F '\"\"*F)F)F)*&#\"#r\"*+;c(HF)*$)F'\"#5F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Trial and error leads to \+ a scheme with " }{XPPEDIT 18 0 "c[2]= 1/24" "6#/&%\"cG6#\"\"#*&\"\"\" F)\"#C!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=23/125" "6#/&%\"cG 6#\"\"&*&\"#B\"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] =879/1000" "6#/&%\"cG6#\"\"'*&\"$z)\"\"\"\"%+5!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[8]=104/105" "6#/&%\"cG6#\"\")*&\"$/\"\"\"\"\"$0\"! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9]=20/21" "6#/&%\"cG6#\"\"** &\"#?\"\"\"\"#@!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "Digits := 10:\nc_2 := 1/24: c_5 := 23/125: c_6 := 87 9/1000: c_8 := 104/105: c_9 := 20/21:\nGam8 := 1: Gam9 := 82/91: Gam10 := 71/82: rat_digits := 6:\nprin_err_normB(c_2,c_5,c_6,c_8,c_9);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ba*[v$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can minimize the princ ipal error norm with respect to " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\" \"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 525 "Digits := \+ 10:\nc_2 := 1/24: c_5 := .184: c_6 := .879: c_8 := 104/105: c_9 := 20/ 21:\nGam8 := 1: Gam9 := 82/91: Gam10 := 71/82: rat_digits := 10:\nfor \+ ct to 15 do\n c_5 := op(1,findmin('prin_err_normB'(c_2,c[5],c_6,c_8, c_9),\n c[5]=\{c_5-.5e-2,c_5,c_5+.5e-2\},accuracy=0 .5));\n mn := findmin('prin_err_normB'(c_2,c_5,c[6],c_8,c_9),\n \+ c[6]=\{c_6-.5e-2,c_6,c_6+.5e-2\},accuracy=0.5);\n c_6 := op(1,mn);\n print(ct);\n print(c[5]=c_5,c[6]=c_6);\n print(` principal error norm`=op(2,mn));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\" &$\"+\\s9R=!#5/&F%6#\"\"'$\"+bRz$z)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+04m`P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+!oD>% =!#5/&F%6#\"\"'$\"+IhG'z)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5prin cipal~error~normG$\"+tQO_P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+U#HP%=!#5/&F%6# \"\"'$\"+@!zzz)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~erro r~normG$\"+7Sx^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+@G(\\%=!#5/&F%6#\"\"'$\"+v 8=*z)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\" +US[^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+)eQe%=!#5/&F%6#\"\"'$\"+-c.+))F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+&yQ8v$!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+M:XY=!#5/&F%6#\"\"'$\"+g$\\1!))F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+7UE^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+!p!*o%=!#5/&F%6#\"\"'$\"+CH4,))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+n_A^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&% \"cG6#\"\"&$\"+0hAZ=!#5/&F%6#\"\"'$\"+*eK9!))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+\"f.7v$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6# \"\"&$\"+0,XZ=!#5/&F%6#\"\"'$\"+*zh;!))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+fI>^P!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\" &$\"+mFhZ=!#5/&F%6#\"\"'$\"+:2%=!))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+&Q(=^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+],uZ=! #5/&F%6#\"\"'$\"+av'>!))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5princ ipal~error~normG$\"+vT=^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+&zIy%=!#5/&F%6#\" \"'$\"+0Q1-))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~ normG$\"+mC=^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+H#**y%=!#5/&F%6#\"\"'$\"+O=:- ))F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+X: =^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+6;'z%=!#5/&F%6#\"\"'$\"+O=:-))F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+c5=^P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&$\"+6;'z%=!#5/&F%6#\"\"'$\"+O=:-))F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"+c5=^P!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We may use rati onal approximations for the final values." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "[c[5]=.1847961611,c[6]=.8802151836]:\nconvert(%,ratio nal,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"&#\"$T\"\"$j (/&F&6#\"\"'#\"$p\"\"$#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The corresponding values for " }{XPPEDIT 18 0 "c[ 3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"c G6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\" \"'\"\"\"" }{TEXT -1 17 " are as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 249 "Digits := 10:\nc_2 := 1/24: c_5 := 141/763: c_6 := 169/192: c_8 := 104/105: c_9 := 20/21:\nGam8 := 1: Gam9 := 82/91: Gam 10 := 71/82: rat_digits := 6:\nprin_err_normB(c_2,c_5,c_6,c_8,c_9);\nc [3]=subs(e15,c[3]),c[4]=subs(e15,c[4]),a[6,1]=subs(e15,a[6,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U(f6v$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"$#\"$(*)\"%s5/&F%6#\"\"%#\"$g\"\"$*\\/&%\" aG6$\"\"'\"\"\"#\"$u\"\"$2%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can use the procedure " }{TEXT 0 13 "calc_RK coeffs" }{TEXT -1 98 " from a later section to check the characteristi cs of the scheme and choose a suitable value for " }{XPPEDIT 18 0 "`b *`[11]" "6#&%#b*G6#\"#6" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "c_2 := 1/24: c_3 := 89 7/1072: c_4 := 160/499: c_5 := 141/763: c_6 := 169/192:\nc_8 := 104/10 5: c_9 := 20/21: a6_1 := 174/407: bs_11 := -38/5:\ncalc_RKcoeffs():" } }{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#C/&F &6#\"\"$#\"$(*)\"%s5/&F&6#\"\"%#\"$g\"\"$*\\/&F&6#\"\"&#\"$T\"\"$j(/&F &6#\"\"'#\"$p\"\"$#>/&F&6#\"\"(#\"*REi)**\"+&fkw+#/&F&6#\"\")#\"$/\"\" $0\"/&F&6#\"\"*#\"#?\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~ weights:G/&%\"bG6#\"\"\"$\"'Hab!\"(/&F&6#\"\"&$\"'o'z#!\"'/&F&6#\"\"'$ \"'>i7!\"&/&F&6#\"\"($\"'9XKF2/&F&6#\"\")$\"'FB'*F9/&F&6#\"\"*$!'0zRF9 /&F&6#\"#5$!'9mlF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weight s:G/&%#b*G6#\"\"\"$\"'9Cb!\"(/&F&6#\"\"&$\"'/3G!\"'/&F&6#\"\"'$\"'@l8! \"&/&F&6#\"\"($\"''3@$F2/&F&6#\"\")$\"'_36!\"%/&F&6#\"\"*$!'_2XF9/&F&6 #\"#5$\"\"!FR/&F&6#\"#6$!'++wF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*& %)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+$o\")=S)!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 7~schemeG$\"+U(f6v$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of ~order~9~terms~of~order~7~schemeG$\"+53ax=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'C0]!\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+W@ 8[e!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~interva lG7$$!)*GE)\\!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~s tability~intervalG7$$!)+.p^!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%Pmax-min~principal~error~ratio~of~order~6~schemeG$\"%,$)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")@)[>#!\"(/&%\"CGF&$ \"))y()H#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e11,matrix([seq([c[i],seq(a[i,j],j=1..i-1),` `$(12-i)],i=2..11),\n[`b`,seq(b[j],j=1..11)],[`b*`,seq(`b*`[j],j=1..11 )]])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7. 7.$\"'nmT!\"(F(%!GF+F+F+F+F+F+F+F+F+7.$\"'an$)!\"'$!'8lv!\"&$\"')=S)F2 F+F+F+F+F+F+F+F+F+7.$\"'T1KF/$\"'2#f#F/$\"\"!F;$\"'VVhF*F+F+F+F+F+F+F+ F+7.$\"'(z%=F/$\"'z*=\"F/F:$!'c1y!\")$\"'ZitF*F+F+F+F+F+F+F+7.$\"'3-)) F/$\"'=vUF/F:$\"'$Rt\"F/$\"'p.9F2$!'RC6F2F+F+F+F+F+F+7.$\"'0u\\F/$!'dM ()F*F:$\"'YVhF*$!'&R1)F*$\"'+LiF/$!'YM>F*F+F+F+F+F+7.$\"'w/**F/$\"'[G_ F/F:$\"'Qc9F2$\"'djjF2$!'HtMF2$!'F_^F/$!'zjLF2F+F+F+F+7.$\"'\"Q_*F/$\" 'r^^F/F:$\"')H'))F/$\"'KIF/$!'+X=F2$\"''p3\"F*F+F+F+ 7.$\"\"\"F;$\"'i&G&F/F:$\"'RL;F2$\"'oRqF2$!'8xPF2$!'%4'fF/$!'sOQF2$!'] a?F*$\"'p&)GF*F+F+7.F[q$\"'-t_F/F:$\"'.K;F2$\"'$>.(F2$!'7rPF2$!'GZfF/$ !'KLQF2$!'zL?F*$\"'ECGF*F:F+7.%\"bG$\"'HabF*F:F:F:$\"'o'z#F/$\"'>i7F2$ \"'9XKF/$\"'FB'*F2$!'0zRF2$!'9mlF2F:7.%#b*G$\"'9CbF*F:F:F:$\"'/3GF/$\" '@l8F2$\"''3@$F/$\"'_36!\"%$!'_2XF2F:$!'++wF2Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#-------------- -----------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 50 "#---------------------------- ---------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 52 "#===================================== ==============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Constructing a \+ scheme that satisfies some error conditions" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 55 "#---------------- --------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set up orde r conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1291 "Qeqs := QuadratureConditions(7,10,'expanded '):\nnode_eq := 52*c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]* c[6]^2-70*c[7]*c[5]^2*c[6]+3*c[7]-7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5] +14*c[6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-12*c[6]*c[7]+14*c[7]*c[5]^2 =0:\ncdns1 := [op(Qeqs),node_eq]:\nSO_eqs := [op(RowSumConditions(10,' expanded')),op(StageOrderConditions(2,3..10,'expanded')),\n op(StageO rderConditions(3,5..10,'expanded'))]:\nsimp_eqs := [add(b[i]*a[i,1],i= 2..10)=b[1],seq(add(b[i]*a[i,j],i=j+1..10)=b[j]*(1-c[j]),j=6..9)]:\nsi mp_eqs2 := [add(b[i]*c[i]*a[i,3],i=5..10)=0,add(b[i]*c[i]^2*a[i,3],i=5 ..10)=0]:\ncdcns2 := [op(simp_eqs),op(simp_eqs2),op(SO_eqs)]:\nSO7_10 \+ := SimpleOrderConditions(7,10,'expanded'):\ncdcns3 := [seq(SO7_10[i],i =[54,59,61])]:\ncdcns4 := [seq(SO7_10[i],i=[50,51])]:\ncdcns5 := [seq( SO7_10[i],i=[45,55])]:\ncdnA := add(b[i]*c[i]^3*add(a[i,j]*c[j]^3,j=2. .i-1),i=3..10)=1/32:##122\ncdnB := add(b[i]*c[i]*add(a[i,j]*add(a[j,k] *add(a[k,l]*c[l]^3,l=2..k-1),k=3..j-1),j=4..i-1),i=5..10)=1/960:##102 \ncdnC := add(b[i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*add(a[m ,n]*add(a[n,p]*c[p],p=2..n-1),n=3..m-1),\n m=4..l-1),l=5..k-1),k=6.. j-1),j=7..i-1),i=8..10)=1/40320:##65\nerrterms7_10 := PrincipalErrorTe rms(7,10,'expanded'):\nerrterms8_10 := PrincipalErrorTerms(8,10,'expan ded'):" }}}{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 14 "The procedure " }{TEXT 0 17 "combined_err_norm" }{TEXT -1 227 " constructs a 10 stage order 7 scheme subject to the condition that certain principal error terms are approximately zero and calcula tes a weighted combination of the principal error norm and the norm of the order 9 error terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 17 "combined_err_norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2739 "combined_err_norm := proc(c2,c5,c 6,c8,c9,alph)\n local eqns1,eqns2,eqns3,eqns4,eqns5,eqnA,eqnsB,sol,P ,Q,R,sm,ct,ii,\n nrms,mn,saveDigits;\n global e1,e2,e3,e4,e5 ,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17;\n\n e1 := \{c[2]=conve rt(c2,rational,Digits+2),c[5]=convert(c5,rational,Digits+2),\n c[6] =convert(c6,rational,Digits+2),c[8]=convert(c8,rational,Digits+2),\n \+ c[9]=convert(c9,rational,Digits+2),c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\n eqns1 := subs(e1,cdns1);\n e2 := solve(\{op(eqns1)\});\n e3 := \+ `union`(e1,e2);\n e4 := \{seq(a[i,2]=0,i=4..10)\};\n e5 := `union` (e3,e4);\n eqns2 := subs(e5,cdcns2);\n e6 := solve(\{op(eqns2)\},i ndets(eqns2) minus \n \{c[3],c[4],a[6,1],a[7,6],a[8,6],a[8,7],a[9 ,5],a[9,6],a[9,7],a[9,8]\});\n e7 := `union`(e5,e6);\n eqns3 := si mplify(subs(e7,cdcns3));\n e8 := solve(\{op(eqns3)\},\{a[8,7],a[9,7] ,a[9,8]\});\n e9 := `union`(map(u_->lhs(u_)=simplify(subs(e8,rhs(u_) )),e7),e8);\n eqns4 := simplify(subs(e9,cdcns4));\n e10 := solve( \{op(eqns4)\},\{a[8,6],a[9,6]\});\n e11 := `union`(map(u_->lhs(u_)=s implify(subs(e10,rhs(u_))),e9),e10);\n eqns5 := simplify(subs(e11,cd cns5));\n e12 := solve(\{op(eqns5)\},\{a[7,6],a[9,5]\});\n e13 := \+ `union`(map(u_->lhs(u_)=simplify(subs(e12,rhs(u_))),e11),e12);\n eqn A := simplify(subs(e13,cdnA));\n e14 := solve(\{eqnA\},a[6,1]);\n \+ e15 := `union`(map(u_->lhs(u_)=simplify(subs(e14,rhs(u_))),e13),e14): \n eqnsB := simplify(subs(e15,[cdnB,cdnC]));\n sol := eliminate(\{ op(eqnsB)\},c[4]):\n if sol=NULL then\n error \"could not obtai n a scheme satisfying the additional order conditions\"\n end if;\n \+ saveDigits := Digits;\n Digits := max(Digits,14);\n P := convert ([fsolve(op(sol[2]),c[3]=0..1)],rational,rat_digits);\n if P=[] then \n error \"could not obtain a scheme satisfying the additional or der conditions\"\n end if;\n Q := map(u_->subs(c[3]=u_,rhs(op(sol[ 1]))),P);\n R := [];\n for ii to nops(P) do\n if Q[ii]>0 then \n R := [op(R),\{c[3]=convert(P[ii],rational,rat_digits),c[4]= convert(Q[ii],rational,rat_digits)\}]\n end if;\n end do;\n n rms := []:\n for ii to nops(P) do\n e16 := R[ii];\n e17 := `union`(map(u_->lhs(u_)=simplify(subs(e16,rhs(u_))),e15),e16);\n \+ nrms := [op(nrms),sqrt(add(evalf(subs(e17,errterms7_10[i]))^2,i=1..no ps(errterms7_10)))];\n end do;\n mn := min(op(nrms));\n for ii t o nops(nrms) do\n if nrms[ii]=mn then break end if;\n end do;\n e16 := R[ii];\n e17 := `union`(map(u_->lhs(u_)=simplify(subs(e16, rhs(u_))),e15),e16);\n sm := 0.;\n for ct to nops(errterms7_10) do \n sm := sm+evalf(subs(e17,errterms7_10[ct]))^2;\n end do;\n \+ for ct to nops(errterms8_10) do\n sm := sm+(alph*evalf(subs(e17,e rrterms8_10[ct])))^2;\n end do;\n evalf[saveDigits](sqrt(sm));\nen d 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T [7, 10];" "6#&%\"TG6$\"\"(\"#5" }{TEXT -1 132 " denote the vector who se components are the 115 principal error terms of the 10 stage, order 7 scheme (the error terms of order 8)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[8, 10];" "6#&%\"TG6$\"\")\"#5" }{TEXT -1 104 " denote the vector whose components are the 286 error terms of o rder 9 of the 10 stage, order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We consider the weighted combination " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(Sum(T[7, 10][i]^2,i = 1 .. 115)+alpha^2*Sum(T[ 8,10][i]^2,i = 1 .. 286));" "6#-%%sqrtG6#,&-%$SumG6$*$&&%\"TG6$\"\"(\" #56#%\"iG\"\"#/F2;\"\"\"\"$:\"F6*&%&alphaGF3-F(6$*$&&F-6$\"\")F06#F2F3 /F2;F6\"$'GF6F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "for va rious values of the parameter " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "alpha=0" "6#/%&alphaG\"\"!" } {TEXT -1 55 " this gives the principal error norm, that is, 2-norm " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(abs(T[6, 9])); " "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\"*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "of the vector " }{XPPEDIT 18 0 "T[6,9];" "6#&%\"TG6$ \"\"'\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "When " } {XPPEDIT 18 0 "alpha = 1;" "6#/%&alphaG\"\"\"" }{TEXT -1 37 " the com bined norm gives the 2-norm " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {TEXT 276 2 "||" }{TEXT -1 1 " " }{XPPEDIT 18 0 "T[7, 10];" "6#&%\"TG6 $\"\"(\"#5" }{TEXT -1 2 " " }{TEXT 278 1 "v" }{TEXT -1 2 " " } {XPPEDIT 18 0 "T[8,10];" "6#&%\"TG6$\"\")\"#5" }{TEXT -1 1 " " }{TEXT 277 2 "||" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "of the \"conjoined\" vector " }{XPPEDIT 18 0 " T[7,10];" "6#&%\"TG6$\"\"(\"#5" }{TEXT -1 2 " " }{TEXT 279 1 "v" } {TEXT -1 2 " " }{XPPEDIT 18 0 "T[8,10];" "6#&%\"TG6$\"\")\"#5" } {TEXT -1 94 " with the 401 compnents given by the order 8 error terms followed by the order 9 error terms." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------------ ------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "minimizatio n of a combined error norm " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The following coefficients were obtained by tri al and error. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "Digits := 10:\nc_2 := 1/200: c_5 := 119/198: c_6 := 117/131: c_8 := 63/73: c_9 \+ := 39/40:\nalpha := 1/4: rat_digits := 6:\ncombined_err_norm(c_2,c_5,c _6,c_8,c_9,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+AX$z7$!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The cor responding values for " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 17 " are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "c[3]=s ubs(e17,c[3]),c[4]=subs(e17,c[4]),a[6,1]=subs(e17,a[6,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"$#\"$)Q\"%$[#/&F%6#\"\"%#\"*:9 \"*>'\"+Cg7,7/&%\"aG6$\"\"'\"\"\"#!4xX*=pkc#zz'\"6g^zp[W$>#pG\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We may us e simple rational approximations for " }{XPPEDIT 18 0 "c[4]" "6#&%\"c G6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\" \"'\"\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "[c[4]=619911415/1201126024,a[6,1]=-6797925664691894577/1286921934 44869795160]:\nconvert(evalf[10](%),rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"%#\"$,)\"%_:/&%\"aG6$\"\"'\"\"\"#!#H\"$ \\&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "W e may use the procedure " }{TEXT 0 13 "calc_RKcoeffs" }{TEXT -1 112 " \+ from a later section to construct a scheme directly using the values o btained along with a suitable value for " }{XPPEDIT 18 0 "`b*`[11]" " 6#&%#b*G6#\"#6" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "c_2 := 1/200: c_3 := 388/24 83: c_4 := 801/1552: c_5 := 119/198: c_6 := 117/131: c_8 := 63/73: c_9 := 39/40:\na6_1 := -29/549: bs_11 := 7/5:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\"\"\"\"$+#/&F&6#\" \"$#\"$)Q\"%$[#/&F&6#\"\"%#\"$,)\"%_:/&F&6#\"\"&#\"$>\"\"$)>/&F&6#\"\" '#\"$<\"\"$J\"/&F&6#\"\"(#\"*d4r[%\"+t,QLA/&F&6#\"\")#\"#j\"#t/&F&6#\" \"*#\"#R\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&% \"bG6#\"\"\"$\"'B*z&!\"(/&F&6#\"\"&$\"'Z(*\\!\"'/&F&6#\"\"'$\"'9XO!\"& /&F&6#\"\"($\"'b#=$F2/&F&6#\"\")$!'cWDF9/&F&6#\"\"*$!'Z2@F9/&F&6#\"#5$ \"'!48\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b *G6#\"\"\"$\"'%Q$e!\"(/&F&6#\"\"&$\"'\"=6&!\"'/&F&6#\"\"'$\"'*\\I%!\"& /&F&6#\"\"($\"'xoJF2/&F&6#\"\")$!'c)*HF9/&F&6#\"\"*$!'#Gf#F9/&F&6#\"#5 $\"\"!FQ/&F&6#\"#6$\"'++9F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)in finityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+fj3-5!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~schemeG$\" +=Z%o<#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~ter ms~of~order~7~schemeG$\"+>n:%)*)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%5ratio~of~error~normsG$\"':FT!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+[$y&\\M!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)PJ, V!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~int ervalG7$$!)N\\KT!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-mi n~principal~error~ratio~of~order~6~schemeG$\"%&y'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")kle;!\"(/&%\"CGF&$\")Gij@F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We make s ome adjustments to the parameter coefficients of the previous scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "c_2 := 1/200: c_5 := 11 9/198: c_6 := .896: c_8 := .862: c_9 := 44/45:\nalpha := 1/4: rat_digi ts := 8:\ncombined_err_norm(c_2,c_5,c_6,c_8,c_9,alpha);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+]q;CJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 52 "Minimizing the combined error norm with \+ respect to " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 54 " \+ gives a slight reduction of the combined error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "c_2 := 1/200: c_5 := 119/198: c_6 := 896 /1000: c_8 := 862/1000: c_9 := 44/45:\nalpha := 1/4: rat_digits := 6: \nfindmin('combined_err_norm'(c_2,c_5,c[6],c_8,c_9,alpha),c[6]=\{.895, .896,.897\},accuracy=0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+u` )p&*)!#5$\"+:PPAJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Minimizing the combined error norm with respect to " }{XPPEDIT 18 0 "c[8];" "6#&%\"cG6#\"\")" }{TEXT -1 62 " gives a furth er slight reduction of the combined error norm." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 204 "c_2 := 1/200: c_5 := 119/198: c_6 := .8956985 374: c_8 := 862/1000: c_9 := 44/45:\nalpha := 1/4: rat_digits := 8:\nf indmin('combined_err_norm'(c_2,c_5,c_6,c[8],c_9,alpha),c[8]=\{.861,.86 2,.863\},accuracy=0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+K1V?' )!#5$\"+X+6AJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We continue minimizing the combined error norm with respe ct to " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 51 " alternately \+ until we obtain no further reduction." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "c_2 := 1/200: c_5 := 119/198: c_6 := .8956985374: c_ 8 := .8620430632: c_9 := 44/45:\nalpha := 1/4: rat_digits := 8:\nfindm in('combined_err_norm'(c_2,c_5,c[6],c_8,c_9,alpha),c[6]=\{c_6-.2e-2,c_ 6,c_6+.2e-2\},accuracy=0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+ Wv:Z*)!#5$\"+)fT)>J!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "c_2 := 1/200: c_5 := 119/198: c_6 \+ := .8947157544: c_8 := .8620430632: c_9 := 44/45:\nalpha := 1/4: rat_d igits := 8:\nfindmin('combined_err_norm'(c_2,c_5,c_6,c[8],c_9,alpha),c [8]=\{c_8-.2e-2,c_8,c_8+.2e-2\},accuracy=0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+dj&3i)!#5$\"+y=U>J!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "c_2 := 1/200: c_5 := 119/198: c_6 := .8947157544: c_8 := .8620856357: c_9 := 44/45:\nal pha := 1/4: rat_digits := 8:\nfindmin('combined_err_norm'(c_2,c_5,c[6] ,c_8,c_9,alpha),c[6]=\{c_6-.2e-2,c_6,c_6+.2e-2\},accuracy=0.5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+i!oq%*)!#5$\"+q3U>J!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "Digits := 10:\nc_2 := 1/200: c_5 := 119/198: c_6 := .8947068062: \+ c_8 := .8620856357: c_9 := 44/45:\nalpha := 1/4: rat_digits := 8:\ncom bined_err_norm(c_2,c_5,c_6,c_8,c_9,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r3U>J!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We may use rational approximations for the valu es obtained for " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "[c[6]=.8947068062,c[8]=.8620856357];\nconvert(%,rational,5);\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"'$\"+i!oq%*)!# 5/&F&6#\"\")$\"+dj&3i)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6 #\"\"'#\"#<\"#>/&F&6#\"\")#\"#D\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$/&%\"cG6#\"\"'$\"+@%ot%*)!#5/&F&6#\"\")$\"+b'*o?')F+" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "Digi ts := 10:\nc_2 := 1/200: c_5 := 119/198: c_6 := 17/19: c_8 := 25/29: c _9 := 44/45: rat_digits := 8:\nalpha := 1/4:\ncombined_err_norm(c_2,c_ 5,c_6,c_8,c_9,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:ie>J!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ corresponding values for " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 17 " are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "c[3]=s ubs(e17,c[3]),c[4]=subs(e17,c[4]),a[6,1]=subs(e17,a[6,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"$#\"%UI\"&d%>/&F%6#\"\"%#\"+X3T c[\"+w:'*3%*/&%\"aG6$\"\"'\"\"\"#!3V'=MH,ilB#\"4?#y2SDW$eD%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We take simpler rational approximations for these values." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 139 "[c[3]=3042/19457,c[4]=4856410845/9408961576,a[6,1] =-223656201293418643/4255834425400778220]:\nevalf[10](%);\nconvert(%,r ational,5);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6# \"\"$$\"+-wWj:!#5/&F&6#\"\"%$\"+yPZh^F+/&%\"aG6$\"\"'\"\"\"$!+oVGb_!#6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"$#\"#[\"$2$/&F&6#\" \"%#\"#;\"#J/&%\"aG6$\"\"'\"\"\"#!#N\"$m'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"$$\"+:z^j:!#5/&F&6#\"\"%$\"+B.Hh^F+/&%\" aG6$\"\"'\"\"\"$!+b_Db_!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "We can construct a combined scheme based on the values obtained for the parameters." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "c_2 := 1/200: c_3 := 48/307: c_4 := 16/31: c_5 := 11 9/198: c_6 := 17/19: c_8 := 25/29: c_9 := 44/45:\na6_1 := -35/666: bs_ 11 := 3/2:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'node s:G/&%\"cG6#\"\"##\"\"\"\"$+#/&F&6#\"\"$#\"#[\"$2$/&F&6#\"\"%#\"#;\"#J /&F&6#\"\"&#\"$>\"\"$)>/&F&6#\"\"'#\"#<\"#>/&F&6#\"\"(#\"(4k_*\")\\^CZ /&F&6#\"\")#\"#D\"#H/&F&6#\"\"*#\"#W\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'^Fe!\"(/&F&6#\"\"&$\"'$[% \\!\"'/&F&6#\"\"'$\"'e2K!\"&/&F&6#\"\"($\"'%)*=$F2/&F&6#\"\")$!'bz@F9/ &F&6#\"\"*$!'7E@F9/&F&6#\"#5$\"'ME7F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'Gee!\"(/&F&6#\"\"&$\"'\\Y]!\" '/&F&6#\"\"'$\"'K[P!\"&/&F&6#\"\"($\"'`xJF2/&F&6#\"\")$!']UDF9/&F&6#\" \"*$!'!oe#F9/&F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++:F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+#p$ pL5!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~erro r~of~order~7~schemeG$\"+^!>>;#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %J2-norm~of~order~9~terms~of~order~7~schemeG$\"+ZEOj*)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'-YT!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~sche meG$\"+:Ne=K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stabilit y~intervalG7$$!)F)HI%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;o rder~6~stability~intervalG7$$!)IdUT!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~order~6~schemeG$\"% A(*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")q`'f\"! \"(/&%\"CGF&$\")ve^@F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#-------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 283 39 "_______________________________________" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "#======== ==============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Abreviated calc ulations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set up orde r conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1429 "SO7_10 := SimpleOrderConditions(7,10,'expan ded'):\nQeqs := [seq(SO7_10[i],i=[1,2,4,8,16,32,64])]:\nnode_eq := 52* c[7]*c[5]*c[6]+105*c[5]^2*c[6]^2*c[7]-70*c[7]*c[5]*c[6]^2-70*c[7]*c[5] ^2*c[6]+3*c[7]-\n 7*c[5]^2*c[6]-c[6]-c[5]-12*c[7]*c[5]+14*c[ 6]^2*c[7]-7*c[5]*c[6]^2+12*c[5]*c[6]-\n 12*c[6]*c[7]+14*c[7] *c[5]^2=0:\ncdns1 := [op(Qeqs),node_eq]:\nSO7_10 := SimpleOrderConditi ons(7,10,'expanded'):\nSO_eqs := [op(RowSumConditions(10,'expanded')), op(StageOrderConditions(2,3..10,'expanded')),\n op(StageOrderConditio ns(3,5..10,'expanded'))]:\nord_cdns := [seq(SO7_10[i],i=[45,50,51,54,5 5,59,61])]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..10)=b[1],seq(add(b[i]*a [i,j],i=j+1..10)=b[j]*(1-c[j]),j=6..9)]:\nsimp_eqs2 := [add(b[i]*c[i]* a[i,3],i=5..10)=0,add(b[i]*c[i]^2*a[i,3],i=5..10)=0]:\ncdns2 := [op(si mp_eqs),op(simp_eqs2),op(SO_eqs),op(ord_cdns)]:\n`Qeqs*` := subs(b=`b* `,QuadratureConditions(6,11,'expanded')):\nSO_eqs2 := [add(a[11,j],j=1 ..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n add(a[11,j]*c[j ]^2,j=2..10)=1/3*c[11]^3]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11 )=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[$5..8] )]:\n`cdns*` := [op(SO_eqs2),op(`Qeqs*`),op(`simp_eqs*`)]:\n\nerrterms 7_11 := PrincipalErrorTerms(7,11,'expanded'):\nerrterms8_11 := Princip alErrorTerms(8,11,'expanded'):\n`errterms7_11*` := subs(b=`b*`,errterm s7_11):\n`errterms6_11*` := subs(b=`b*`,PrincipalErrorTerms(6,11,'expa nded')):" }}}{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 "ca lc_RKcoeffs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 2895 "calc_RKcoeffs := proc()\n local eqns1,eqns2,`eq ns*`,sm,ct,ssm,sm2,ssm2,rt,Rz,stb7,stb6,\n nmB,snmB,dnB,abserr s,errtrm,sdnB,nmC,snmC,B_8,C_8,nrm;\n global e1,e2,e3,e4,e5,e6,e7,e8 ,e9,e10,e11;\n\n e1 := \{c[2]=c_2,c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_ 6,c[8]=c_8,c[9]=c_9,c[10]=1,b[2]=0,b[3]=0,b[4]=0\}:\n eqns1 := subs( e1,cdns1):\n e2 := solve(\{op(eqns1)\}):\n e3 := `union`(e1,e2):\n e4 := \{seq(a[i,2]=0,i=4..10),a[6,1]=a6_1\}:\n e5 := `union`(e3,e 4):\n eqns2 := subs(e5,cdns2):\n e6 := solve(\{op(eqns2)\}):\n e 7 := `union`(e5,e6):\n e8 := \{b[11]=0,c[11]=1,a[11,2]=0,a[11,10]=0, `b*`[2]=0,`b*`[3]=0,\n `b*`[4]=0,`b*`[10]=0,`b*`[11]=bs_11 \}:\n e9 := `union`(e7,e8):\n `eqns*` := subs(e9,`cdns*`):\n e10 := solve(\{op(`eqns*`)\}):\n e11 := `union`(e9,e10):\n Digits := \+ 14;\n sm := 0:\n for ct to nops(errterms7_11) do\n sm := sm+s ubs(e11,errterms7_11[ct])^2;\n end do:\n ssm := sqrt(sm);\n sm2 \+ := 0;\n for ct to nops(errterms8_11) do\n sm2 := sm2+(evalf(sub s(e11,errterms8_11[ct])))^2;\n end do;\n ssm2 := sqrt(sm2);\n rt := ssm2/ssm;\n Rz := subs(e11,StabilityFunction(7,11,'expanded')); \n stb7 := max(fsolve(Rz=1,z=-10..-1e-7),fsolve(Rz=-1,z=-10..-1e-7)) ;\n stb7 := evalf[8](stb7);\n Rz := subs(e11,subs(b=`b*`,Stability Function(6,11,'expanded')));\n stb6 := max(fsolve(Rz=1,z=-10..-1e-7) ,fsolve(Rz=-1,z=-10..-1e-7));\n stb6 := evalf[8](stb6);\n nmB := 0 ;\n for ct to nops(`errterms7_11*`) do\n nmB := nmB+subs(e11,`e rrterms7_11*`[ct])^2;\n end do:\n snmB := sqrt(nmB);\n dnB := 0; \n abserrs := [];\n for ct to nops(`errterms6_11*`) do\n errt rm := evalf(subs(e11,`errterms6_11*`[ct]));\n dnB := dnB+errtrm^2 ;\n abserrs := [op(abserrs),abs(errtrm)];\n end do;\n sdnB := sqrt(dnB);\n nmC := 0;\n for ct to nops(errterms7_11) do\n n mC := nmC+(subs(e11,`errterms7_11*`[ct])-subs(e11,errterms7_11[ct]))^2 ;\n end do;\n snmC := sqrt(nmC);\n B_8 := evalf[8](snmB/sdnB);\n C_8 := evalf[8](snmC/sdnB);\n print(`nodes:`,c[2]=c_2,c[3]=c_3,c[ 4]=c_4,c[5]=c_5,c[6]=c_6,\n c[7]=subs(e11,c[7]),c[8]=c_8,c[9 ]=c_9);\n print(`order 7 weights:`,seq(b[i]=evalf[6](subs(e11,b[i])) ,i=[1,$5..10]));\n print(`order 6 weights:`,seq(`b*`[i]=evalf[6](sub s(e11,`b*`[i])),i=[1,$5..11]));\n nrm := max(seq(seq(subs(e11,abs(a[ i,j])),j=1..i-1),i=2..11));\n print(infinity*`-norm of linking coeff s`=evalf[10](nrm));\n print(`2-norm of principal error of order 7 sc heme` = evalf[10](ssm));\n print(`2-norm of order 9 terms of order 7 scheme` = evalf[10](ssm2));\n print(`ratio of error norms` = evalf[ 6](rt));\n print(`2-norm of principal error of order 6 scheme` = eva lf[10](sdnB));\n print(`order 7 stability interval` = [stb7,0]);\n \+ print(`order 6 stability interval` = [stb6,0]);\n print(`max-min pr incipal error ratio of order 6 scheme` = evalf[4](max(op(abserrs))/min (op(abserrs))));\n print('B[8]'=B_8,'C[8]'=C_8);\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#--------------------------------------------------" }}{PARA 0 "" 0 " " {TEXT -1 25 "Scheme of Sharp and Smart" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "c_2 := 1/50: c_3 := 27/125: c_4 := 41/100: c_5 := 57 /100: c_6 := 43/50: c_8 := 18/25: c_9 := 5/6:\na6_1 := -31/200: bs_11 \+ := 16/243:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'node s:G/&%\"cG6#\"\"##\"\"\"\"#]/&F&6#\"\"$#\"#F\"$D\"/&F&6#\"\"%#\"#T\"$+ \"/&F&6#\"\"&#\"#dF9/&F&6#\"\"'#\"#VF+/&F&6#\"\"(#\"(5DF#\")@t(>\"/&F& 6#\"\")#\"#=\"#D/&F&6#\"\"*#F=FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1 order~7~weights:G/&%\"bG6#\"\"\"$\"'+(\\&!\"(/&F&6#\"\"&$\"'=q_!\"'/&F &6#\"\"'$!'+kdF2/&F&6#\"\"($\"'u%*HF2/&F&6#\"\")$!'9^VF2/&F&6#\"\"*$\" 'sp5!\"&/&F&6#\"#5$\"'4LgF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order ~6~weights:G/&%#b*G6#\"\"\"$\"'\\>b!\"(/&F&6#\"\"&$\"'r(Q&!\"'/&F&6#\" \"'$!'C+sF2/&F&6#\"\"($\"'N&)HF2/&F&6#\"\")$!'NW[F2/&F&6#\"\"*$\"'6Y7! \"&/&F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'O%e'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+eg* p+\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~err or~of~order~7~schemeG$\"+lDou7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %J2-norm~of~order~9~terms~of~order~7~schemeG$\"+!R!eIO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'B[G!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~sche meG$\"+a,:=>!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stabilit y~intervalG7$$!)NX**Q!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;o rder~6~stability~intervalG7$$!)W2'y$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~order~6~schemeG$\"% :M!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")ma;>!\"(/ &%\"CGF&$\")/I;>F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 35 "#----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 41 "Scheme with a small principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "c_2 := 1/1440: c_3 := 40/183: c_4 := 49/120: c_5 := 49/89: c_6 := 269/315: c _8 := 269/280: c_9 := 65/66:\na6_1 := -84/515: bs_11 := 5/2:\ncalc_RKc oeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\" \"\"\"%S9/&F&6#\"\"$#\"#S\"$$=/&F&6#\"\"%#\"#\\\"$?\"/&F&6#\"\"&#F8\"# *)/&F&6#\"\"'#\"$p#\"$:$/&F&6#\"\"(#\")p#43\"\")BV1d/&F&6#\"\")#FE\"$! G/&F&6#\"\"*#\"#l\"#m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~wei ghts:G/&%\"bG6#\"\"\"$\"'@\"[&!\"(/&F&6#\"\"&$\"'ZgR!\"'/&F&6#\"\"'$!' EZGF+/&F&6#\"\"($\"'/*)HF2/&F&6#\"\")$\"'_BC!\"&/&F&6#\"\"*$!'vlVFE/&F &6#\"#5$\"'%4A#FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights :G/&%#b*G6#\"\"\"$\"'q\"\\&!\"(/&F&6#\"\"&$\"'xyR!\"'/&F&6#\"\"'$!'%GO &F+/&F&6#\"\"($\"'8&)HF2/&F&6#\"\")$\"'\"3r#!\"&/&F&6#\"\"*$!'\\3\\FE/ &F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++DFE" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+z7'o9&!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 7~schemeG$\"+W\"3U5\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~ of~order~9~terms~of~order~7~schemeG$\"+@9mF\")!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'igt!\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+r8 =L5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~interva lG7$$!)%Hkw$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~st ability~intervalG7$$!)*z2o$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%Pmax-min~principal~error~ratio~of~order~6~schemeG$\"%fK!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")g5bC!\"(/&%\"CGF&$\" )?7*[#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#-----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 54 "Fir st scheme with a moderately large stability region." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "c_2 := 1/180: c_3 := 9/46: c_4 := 68/147: \+ c_5 := 50/83: c_6 := 89/100: c_8 := 187/222: c_9 := 260/279: \na6_1 := -9/58: bs_11 := 13/25:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\"\"\"\"$!=/&F&6#\"\"$#\"\"*\"# Y/&F&6#\"\"%#\"#o\"$Z\"/&F&6#\"\"&#\"#]\"#$)/&F&6#\"\"'#\"#*)\"$+\"/&F &6#\"\"(#\"(DXd(\")hU.Q/&F&6#\"\")#\"$(=\"$A#/&F&6#F1#\"$g#\"$z#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"' +Rd!\"(/&F&6#\"\"&$\"'FB`!\"'/&F&6#\"\"'$\"'@6W!\"&/&F&6#\"\"($\"'%3;$ F2/&F&6#\"\")$!'42@F9/&F&6#\"\"*$!'!Gd#F9/&F&6#\"#5$\"'vGOF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"']4e!\" (/&F&6#\"\"&$\"'M#f&!\"'/&F&6#\"\"'$\"'[Kf!\"&/&F&6#\"\"($\"'jKJF2/&F& 6#\"\")$!'e#y#F9/&F&6#\"\"*$!'\\+OF9/&F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++ _F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~ linking~coeffsGF&$\"+:9Ij))!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L 2-norm~of~principal~error~of~order~7~schemeG$\"+ceV)y*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~terms~of~order~7~schemeG$ \"+'z)GZg!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~norm sG$\"'*z<'!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~princip al~error~of~order~6~schemeG$\"+&)>k^l!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)=B/Z!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)n/lW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~ order~6~schemeG$\"%))H!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG 6#\"\")$\")%>\"4@!\"(/&%\"CGF&$\")MeS@F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#-----------------------------" }} {PARA 0 "" 0 "" {TEXT -1 55 "Second scheme with a moderately large sta bility region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "c_2 := 1/200: c_3 := 93/479: c_4 := 110/241: c_ 5 := 509/845: c_6 := 244/273: c_8 := 17/20: c_9 := 171/181:\na6_1 := - 228/1231: bs_11 := 16/25:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG6#\"\"##\"\"\"\"$+#/&F&6#\"\"$#\"#$*\"$ z%/&F&6#\"\"%#\"$5\"\"$T#/&F&6#\"\"&#\"$4&\"$X)/&F&6#\"\"'#\"$W#\"$t#/ &F&6#\"\"(#\"*\\eyA#\"+@_946/&F&6#\"\")#\"#<\"#?/&F&6#\"\"*#\"$r\"\"$ \"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\" \"\"$\"'s,e!\"(/&F&6#\"\"&$\"'\"y8&!\"'/&F&6#\"\"'$\"'-0Q!\"&/&F&6#\" \"($\"'6!=$F2/&F&6#\"\")$!'f>?F9/&F&6#\"\"*$!'\\I@F9/&F&6#\"#5$\"'Z_XF 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\" \"$\"'Bhe!\"(/&F&6#\"\"&$\"'Lb`!\"'/&F&6#\"\"'$\"';8]!\"&/&F&6#\"\"($ \"'9cJF2/&F&6#\"\")$!'eFEF9/&F&6#\"\"*$!'MNHF9/&F&6#\"#5$\"\"!FQ/&F&6# \"#6$\"'++kF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"% 8-norm~of~linking~coeffsGF&$\"+Q#Rj[*!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~schemeG$\"+V(47 H)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~terms~of ~order~7~schemeG$\"+H.Jtc!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5rat io~of~error~normsG$\"'cUo!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2- norm~of~principal~error~of~order~6~schemeG$\"+(3D-9'!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)\"[Tq%!\"(\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7 $$!),#fZ%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~princ ipal~error~ratio~of~order~6~schemeG$\"%zK!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")6D$4#!\"(/&%\"CGF&$\"))=J8#F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#-------- ---------------------" }}{PARA 0 "" 0 "" {TEXT -1 76 "Scheme with a la rgrer stability region and a large imaginary axis inclusion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "c _2 := 1/24: c_3 := 897/1072: c_4 := 160/499: c_5 := 141/763: c_6 := 16 9/192:\nc_8 := 104/105: c_9 := 20/21: a6_1 := 174/407: bs_11 := -38/5: \ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'nodes:G/&%\"cG 6#\"\"##\"\"\"\"#C/&F&6#\"\"$#\"$(*)\"%s5/&F&6#\"\"%#\"$g\"\"$*\\/&F&6 #\"\"&#\"$T\"\"$j(/&F&6#\"\"'#\"$p\"\"$#>/&F&6#\"\"(#\"*REi)**\"+&fkw+ #/&F&6#\"\")#\"$/\"\"$0\"/&F&6#\"\"*#\"#?\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'Hab!\"(/&F&6#\"\" &$\"'o'z#!\"'/&F&6#\"\"'$\"'>i7!\"&/&F&6#\"\"($\"'9XKF2/&F&6#\"\")$\"' FB'*F9/&F&6#\"\"*$!'0zRF9/&F&6#\"#5$!'9mlF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'9Cb!\"(/&F&6#\"\" &$\"'/3G!\"'/&F&6#\"\"'$\"'@l8!\"&/&F&6#\"\"($\"''3@$F2/&F&6#\"\")$\"' _36!\"%/&F&6#\"\"*$!'_2XF9/&F&6#\"#5$\"\"!FR/&F&6#\"#6$!'++wF9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+$o\")=S)!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-nor m~of~principal~error~of~order~7~schemeG$\"+U(f6v$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%J2-norm~of~order~9~terms~of~order~7~schemeG$\"+53a x=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'C 0]!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~6~schemeG$\"+W@8[e!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%; order~7~stability~intervalG7$$!)*GE)\\!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)+.p^!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~or der~6~schemeG$\"%,$)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\")$\")@)[>#!\"(/&%\"CGF&$\"))y()H#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#-----------------------------" }} {PARA 0 "" 0 "" {TEXT -1 44 "Scheme that satisfies some error conditio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "c_2 := 1/200: c_3 := 48/307: c_4 := 16/31: c_5 := 11 9/198: c_6 := 17/19: c_8 := 25/29: c_9 := 44/45:\na6_1 := -35/666: bs_ 11 := 3/2:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%'node s:G/&%\"cG6#\"\"##\"\"\"\"$+#/&F&6#\"\"$#\"#[\"$2$/&F&6#\"\"%#\"#;\"#J /&F&6#\"\"&#\"$>\"\"$)>/&F&6#\"\"'#\"#<\"#>/&F&6#\"\"(#\"(4k_*\")\\^CZ /&F&6#\"\")#\"#D\"#H/&F&6#\"\"*#\"#W\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%\"bG6#\"\"\"$\"'^Fe!\"(/&F&6#\"\"&$\"'$[% \\!\"'/&F&6#\"\"'$\"'e2K!\"&/&F&6#\"\"($\"'%)*=$F2/&F&6#\"\")$!'bz@F9/ &F&6#\"\"*$!'7E@F9/&F&6#\"#5$\"'ME7F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~6~weights:G/&%#b*G6#\"\"\"$\"'Gee!\"(/&F&6#\"\"&$\"'\\Y]!\" '/&F&6#\"\"'$\"'K[P!\"&/&F&6#\"\"($\"'`xJF2/&F&6#\"\")$!']UDF9/&F&6#\" \"*$!'!oe#F9/&F&6#\"#5$\"\"!FQ/&F&6#\"#6$\"'++:F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+#p$ pL5!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~erro r~of~order~7~schemeG$\"+^!>>;#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %J2-norm~of~order~9~terms~of~order~7~schemeG$\"+ZEOj*)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5ratio~of~error~normsG$\"'-YT!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~sche meG$\"+:Ne=K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stabilit y~intervalG7$$!)F)HI%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;o rder~6~stability~intervalG7$$!)IdUT!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~order~6~schemeG$\"% A(*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\")$\")q`'f\"! \"(/&%\"CGF&$\")ve^@F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------------------- -----" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "#=================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the exa mples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "RK7_10step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3569 "rk7_10step := proc(x_rk7st ep::realcons)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a42 ,a43,a51,a52,a53,\n a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76, a81,a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,a A2,aA3,aA4,aA5,aA6,aA7,\n aA8,aA9,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,b1,b 2,b3,b4,b5,b6,b7,b8,b9,bA,\n xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,save Digits;\n options `Copyright 2012 by Peter Stone`;\n \n data := \+ SOLN_;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digi ts)),Digits+5);\n\n # procedure to evaluate the slope field\n fn : = proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_)); \n if val=lasterror or not type(val,numeric) then\n error \"evaluation of slope field failed at %1\",evalf([X_,Y_],saveDigits); \n end if;\n val;\n end proc;\n\n xx := evalf(x_rk7st ep);\n n := nops(data);\n\n if (data[1,1]data[n ,1] or xxdata[1,1])) then\n error \"independent variable is outside the interpolation interval: %1\",evalf(data[1,1])..evalf(data[n,1]); \n end if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; c5 := c5_; c6 := c 6_; c7 := c7_; c8 := c8_;\n c9 := c9_; cA := cA_;\n a21 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a43_;\n a51 \+ := a51_; a52 := a52_; a53 := a53_; a54 := a54_;\n a61 := a61_; a62 : = a62_; a63 := a63_; a64 := a64_; a65 := a65_;\n a71 := a71_; a72 := a72_; a73 := a73_; a74 := a74_; a75 := a75_; a76 := a76_;\n a81 := \+ a81_; a82 := a82_; a83 := a83_; a84 := a84_; a85 := a85_; a86 := a86_; \n a87 := a87_;\n a91 := a91_; a92 := a92_; a93 := a93_; a94 := a9 4_; a95 := a95_; a96 := a96_; \n a97 := a97_; a98 := a98_; \n aA 1 := aA1_; aA2 := aA2_; aA3 := aA3_; aA4 := aA4_; aA5 := aA5_; aA6 := \+ aA6_; \n aA7 := aA7_; aA8 := aA8_; aA9 := aA9_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_; b5 := b5_; b6 := b6_; b7 := b7_; \n b8 : = b8_; b9 := b9_; bA := bA_;\n # Perform a binary search for the int erval 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 jF := n-1; jS := n e nd if;\n else\n while jS-jF> 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 jF := n-1; jS := n end if;\n end i f;\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 + a65*f5;\n f6 := f n(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a7 5*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82 *f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8 *h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a9 6*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1 *f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + a A9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA; \n \+ ys := yk + t*h;\n\n evalf[saveDigits](ys);\nend proc: # of rk7_10s tep" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "RK7_1 sche me of Sharp and Smart" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5923 "RK7_1 := proc(fxy,x,y,xx,yy,h,stps,bb) \n local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a42,a43,a51,a52, a53,\n a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a 84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA 5,aA6,aA7,\n aA8,aA9,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,b1,b2,b3,b4,b5,b6 ,b7,b8,b9,bA,\n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigit s := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n \+ fn := unapply(fxy,x,y);\n\n A := matrix([[1/50,1/50,0,0,0,0,0,0,0,0, 0],\n[27/125,-594/625,729/625,0,0,0,0,0,0,0,0],\n[41/100,451/21600,0,1 681/4320,0,0,0,0,0,0,0],\n[57/100,19/160,0,361/3104,3249/9700,0,0,0,0, 0,0],\n[43/50,-31/200,0,520921/412056,-17371/11640,132023/106200,0,0,0 ,0,0],\n[2272510/11977321,25959766877768976976598957736980/48759451412 9628295945513157189933,0,347890318302644246405985993187156250/13218174 02067092875750818220388519949,-1717046972617147709491116450178750/7467 894926932728111586543618014237,29780304732725103577764751746216250/258 912687002832625147067486467854423,-302662548054389051180423185000/2566 2869164717278733974376694207,0,0,0,0],\n[18/25,42409705291266846/41646 2256407406875,0,3247095172038/883201854817,-518509279926/374238074075, 435669225629732566638/393965828849029186615,-6468694559114760/61945939 006089637,-8593750881095206170491007194502/321350454354555815090388058 5625,0,0,0],\n[5/6,-1401024812030113404025/19887564677841032175639,0,1 3281373111234375/5150833217292744,-50491693720625/29100752640072,89097 76468783164583973193125/6271093223575470807674793192,-4792324941735635 008750/159776107397443897190271,-1532806290465891141166096531902118541 769245/1203242011387872547807852011647420329982736,-7500029126894375/1 32689679447323376,0,0],\n[1,36393032615434450612/324390586094889663425 ,0,-1462401427649331250/154787214582248211,4135780451822750/8745040371 87843,-2349378733647002895234008950/1090914599757106529355865311,-7868 6605908422443750/52446632451499515953,23150798134912045244350678993658 85119542372444358703/3161690420395271575952352315737883080312607605842 00,-33473047374792524975/32907430028856870472,5594658687556280397846/1 893189870520997940175,0],\n[0,771570009067/14036203465200,0,0,0,283047 79228000000/53707434325074117,-296881060859375/515060733835389,7448583 03758379680905615939985761920312207508379/2487223884477764590764433396 524922145673887618400,-5118512171875/11763620626464,136801854099/12788 5521925,103626500437/1717635089268]]);\n\n c2 := evalf(A[1,1]);\n \+ c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]); \n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7 ,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n 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 := ev alf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a 85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8] );\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := eval f(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]); \n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf( A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 : = evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n b1 := evalf(A[10,2]);\n b2 := evalf(A[ 10,3]);\n b3 := evalf(A[10,4]);\n b4 := evalf(A[10,5]);\n b5 := \+ evalf(A[10,6]);\n b6 := evalf(A[10,7]);\n b7 := evalf(A[10,8]);\n \+ b8 := evalf(A[10,9]);\n b9 := evalf(A[10,10]);\n bA := evalf(A[1 0,11]);\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 + a4 3*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 := a 61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,y k + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76 *f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 \+ + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8* h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + \+ a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n \+ t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA 8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n \n t := b 1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA;\n yk := yk + t*h;\n xk := xk + h:\n soln := sol n,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[sol n],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7 _=c7,c8_=c8,c9_=c9,cA_=cA,\n a31_=a31,a32_=a32,a41_=a41,a42_=a 42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a6 2_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a7 3,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84 _=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93 ,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_ =aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8, aA9_=aA9,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b 8_=b8,b9_=b9,bA_=bA\};\n return subs(eqns,eval(rk7_10step)); \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 46 "RK7_2 sche me with a small principal error norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6738 "RK7_2 := proc(fxy,x,y,xx ,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a4 2,a43,a51,a52,a53,\n a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76 ,a81,a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1, aA2,aA3,aA4,aA5,aA6,aA7,\n aA8,aA9,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,b1, b2,b3,b4,b5,b6,b7,b8,b9,bA,\n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n \n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Dig its+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/1440,1/144 0,0,0,0,0,0,0,0,0,0],\n[40/183,-381560/11163,128000/3721,0,0,0,0,0,0,0 ,0],\n[49/120,10339/384000,0,146461/384000,0,0,0,0,0,0,0],\n[49/89,616 3073/56397520,0,3939947361/26112051760,94785600/326400647,0,0,0,0,0,0] ,\n[269/315,-84/515,0,4431643925510749/3223786245394050,-1662186069392 32/90566512053885,5209733339403491/3525508500182550,0,0,0,0,0],\n[1080 9269/57064323,8990648437774041845694826394521064409811/204088432323318 903399600031639662643622130,0,5479303396912256095018225030133602480458 68971/1737834520761638771583667016353839075551118195,-1083137587315071 8357864250769796161780597600/32547569657073291338827325045823088043427 021,1221450428943196975560168771911668179455657701/6841738152147548926 226151700669066667202112842,-2346622810543340886158743517271000/149768 789910631840990687560369315577,0,0,0,0],\n[269/280,5627145254214745541 87476828603/1576430076296951412495515648000,0,-23243407331995374498867 4545591/7451087574129649589321728000,2895719528678335306735611/1635351 45776061958545920,-3877135960439326710451592840782413717/4003204766164 17483052489638612992000,775651514059488014593497/587040197938164259487 744,188187525132648765731051207041043965977201/83805834990257366980828 07492981506048000,0,0,0],\n[65/66,310494343462355365184359482864331617 0473/5493198556137918605753979473772031976448,0,-803790884920078309156 69557451/1850963041401006078449519616,8519139775589074767837125/338538 398802108406103202,-247048879110939373809210763186554743125624445065/1 7667627894259931943374574719133235934016725376,17478588815033183487684 605738930625/9470320455773783765475210993539584,5680115883329072612844 775910212680982448198390650839995/184203510561647602575374593408831494 422662678761353728,-373168103571100370594887280000/2087620503761272199 2947009240147,0,0],\n[1,1169280695264980001334049232853059/16268206757 59937201701331837062960,0,-1468513512866349156603823997571/28584615782 549766069481480240,97509332134466915877268800/323803905433914032135332 1,-5076559736617024493905835117377217502141590/30139735361305596755059 4290701709567863619,18947515655188164002766115308564375/86660946015541 76945344432830508061,4572756892714768555146592045820970400789327611776 7183866475/1262464382386583873617557530342278467896533166668617282491, 653479766201588970500527390720000/84524048323760095514312876121590233, -638152078395004810819432233984/21426232114086837877217243378141,0],\n [0,157555978800757/2874472964350575,0,0,0,2105087478471774562042359/53 15246555379439663510400,-11645559732863512771875/409009954655655860540 672,89543313432661343622256182806845122951561468606052473/299571815814 765694033635559633841149884537293619715840,54185546099841228800000/223 58202267632352356631,-18745074956372100336/4293668048721948475,1247548 24260719/56172137577600]]);\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 c 9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\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 := e valf(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 := eva lf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a7 6 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]) ;\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf (A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 \+ := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]); \n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf( A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 : = evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[ 9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := \+ evalf(A[9,10]);\n b1 := evalf(A[10,2]);\n b2 := evalf(A[10,3]);\n \+ b3 := evalf(A[10,4]);\n b4 := evalf(A[10,5]);\n b5 := evalf(A[10 ,6]);\n b6 := evalf(A[10,7]);\n b7 := evalf(A[10,8]);\n b8 := ev alf(A[10,9]);\n b9 := evalf(A[10,10]);\n bA := evalf(A[10,11]);\n \+ xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for \+ k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 \+ := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62 *f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n \+ f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + \+ a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h) ;\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a9 7*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9* f9;\n fA := fn(xk + cA*h,yk + t*h);\n \n t := b1*f1 + b2*f 2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA;\n \+ yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk,yk]; \n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fx y,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c 8,c9_=c9,cA_=cA,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a4 3,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63 _=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74 ,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_ =a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94, a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_= aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,b 1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_= b9,bA_=bA\};\n return subs(eqns,eval(rk7_10step)); \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 53 "RK7_3 scheme with a moderatel y large stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7199 "RK7_3 := proc(fxy,x,y,xx,yy,h,stp s,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a42,a43,a51 ,a52,a53,\n a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82, a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,a A4,aA5,aA6,aA7,\n aA8,aA9,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,b1,b2,b3,b4, b5,b6,b7,b8,b9,bA,\n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n save Digits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n \n fn := unapply(fxy,x,y);\n\n A := matrix([[1/200,1/200,0,0,0,0,0 ,0,0,0,0],\n[93/479,-820353/229441,864900/229441,0,0,0,0,0,0,0,0],\n[1 10/241,-432520/5401533,0,2897950/5401533,0,0,0,0,0,0,0],\n[509/845,311 712996187/3366699277500,0,996040375149076/5096677701243375,34454681049 6217/1096059720697500,0,0,0,0,0,0],\n[244/273,-228/1231,0,150895012526 16323812/14457231345452578137,-6255616188978805360/5200809295819122531 ,103959712518200500/83969153915827131,0,0,0,0,0],\n[222785849/11091452 21,20784021508544698439004341302945788046687497048823/3338090077390109 23825478541340748044990946025336204,0,59087076776381536334498651011356 34541623011524075277164/3154555408778125545630482740453787587569225386 8011534003,-48597609826899455036511090495924959702383070843517685/4880 91226723681375828622960423688756917868034622078292,4083724255290790494 0455804285850036846494300434360648500/73437256266214599098958725391464 2887902719996408150162943,-2267236012274582026887011527795707923286/47 7373186945391376596029499088911158349111,0,0,0,0],\n[17/20,-1485037557 2183825804482268314325533/131869870061161226458875010296840000,0,25892 910773607576734414005208753/10529291320508273201747374752000,-61352285 2101531567576566644397/814576707522236069963552640000,8122715759499790 031095209746537365648167969/836296543559300372283670042627232294078259 2,-42937178562595251680361031143/2273295839722617724238904704000,-7424 205995886059956357359508852528825590423/437819729105165125748625856395 7705093888000,0,0,0],\n[171/181,-1538316898001585922524874095420554624 65307763/517713869734060023599533102763919424014608068,0,-133498421650 9015564184248020843701584/667067642557939663406130018977478253,-576334 97085298645064371869898744017/30963779845911010423286154735455876,4891 77972756706421411944299196898256388191463578294580375/3192912011009866 22665402847780351230046376607122311276442,-118567930035837775623348960 6885369481439862/18438601757968638166315261704657797344684897,11944482 045828658346298806908238121417524848379605193796762013/341237631731129 8392350722812851096867170775526794634954003706,10397310879870999283792 9472000/762442644104164301247937014633,0,0],\n[1,-40734473647257020156 088313054846628592379/112331307637222010640347265020564130793284,0,-11 11042984674821389643230968704444252/1475199643357862920568353175937980 19,-5213063483277854003640774248205317/2282514999055829721733809658986 516,4344125602637174263526484423723747582563035478367000/2819697174719 008494099755685231013933352915992886809,142812438169955950969368011788 034519784/281984196771808918111529577339005012981,35011829759852921774 894466285700128707491070752921128030483969446296/371804493842727990442 8695460050071040231420419464687750808112491477,-2632810508797581531988 7872000/966150763091150023719018517747,-119857578740612499110097443516 9141280/4635648233480921649154730958571956313,0],\n[0,1749959215584179 3/301627899585331605,0,0,0,89929293580433773826711625625/1750342069550 63764470139052544,43965679317071687650068867/1155464162092299275205473 0,639711292154532559948032664549872463482448777204372657657897547/2011 601425501588622427604443016911932371008211435797382991660160,-43614217 354789120000/21595629876813903267,-210996982649291857501475159/9903665 7939412172256631200, 7077278529846113/15546034297382400]]);\n\n c2 : = evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n \+ c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]) ;\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[ 9,1]);\n 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 := eva lf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a5 3 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]) ;\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf (A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 \+ := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]); \n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf( A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 : = evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[ 8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := \+ evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n \+ a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9, 3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := ev alf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n a A8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n b1 := evalf(A[10,2 ]);\n b2 := evalf(A[10,3]);\n b3 := evalf(A[10,4]);\n b4 := eval f(A[10,5]);\n b5 := evalf(A[10,6]);\n b6 := evalf(A[10,7]);\n b7 := evalf(A[10,8]);\n b8 := evalf(A[10,9]);\n b9 := evalf(A[10,10] );\n bA := evalf(A[10,11]);\n xk := evalf(xx);\n yk := evalf(yy) ;\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk ,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n \+ t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n \+ t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk \+ + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n \+ f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + \+ a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n \+ f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9 *h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h );\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7 *f7 + b8*f8 + b9*f9 + bA*fA;\n yk := yk + t*h;\n xk := xk + \+ h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n \+ eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4 _=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,\n a31_=a31,a32 _=a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53 ,a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_ =a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n \+ a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n \+ a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n \+ aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7 ,\n aA8_=aA8,aA9_=aA9,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b 6,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA\};\n return subs(eqns,ev al(rk7_10step)); \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 43 "RK7_4 scheme with a larger stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7195 "RK7_4 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21, a31,a32,a41,a42,a43,a51,a52,a53,\n a54,a61,a62,a63,a64,a65,a71,a72,a 73,a74,a75,a76,a81,a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a9 6,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,aA7,\n aA8,aA9,f1,f2,f3,f4,f5,f6,f 7,f8,f9,fA,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,\n t,k,fn,xk,yk,soln,eqns,A ,saveDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalh f(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix( [[1/24,1/24,0,0,0,0,0,0,0,0,0],\n[897/1072,-2173431/287296,2413827/287 296,0,0,0,0,0,0,0,0],\n[160/499,57894880/223353897,0,13721600/22335389 7,0,0,0,0,0,0,0],\n[141/763,5056664940861/42500572528960,0,-2862491496 86016/36667767392227699,2889259817673483/39243095536832320,0,0,0,0,0,0 ],\n[169/192,174/407,0,861765466269918263/4970009789268581376,10826192 777269994995/7712691245665173504,-16750236560040896731/148971940357507 15392,0,0,0,0,0],\n[998622639/2007664595,-3439778790612913527970381936 467065474966136664036779/393812093894245617780441062177821043882900286 25000000,0,2631414486943314085034368959766533330714638432739014413056/ 42832809612378095870766530128571820142439373914153488671875,-322681781 3200049695169462352322166717564113587934800370011/40015346463976160761 929086193516183832403838724337375000000,786190577550706940215874040419 538587608361105183350570182/126133509654299409689905111692458979939826 4209359351953125,-5411006332774526240948446773664264724938752/27971706 8193641240266628706086067991948046875,0,0,0,0],\n[104/105,153220176326 90457879761200795014874391/29304923702961208406725757182921875000,0,33 46824815910846126629531019335170048/2298048540174087438681694595766796 875,189748010561998005455394521375650921/29817877073894091150172096921 875000,-382845331583835193195089397869250404738354182/1102254632855885 17977748940118009136362890625,-120391896911431755652090822656/23366770 3961223196028992578125,-79695128060562505037238941490242415433886906/2 3692035580167289837348760854832233004866581,0,0,0],\n[20/21,1032897956 298204654305923433045759171807509/200496113676988554606751063366686154 3410000,0,8472900696201518374537737273138176/9559881927124203744915849 518389875,1046253623595072271778442420598129/2480847372547988383694318 46390000,-2255808953526127614059281040689393645809880661461199/8915192 31565352783145488093784087901162293552521625,-178869878874495788389143 3528082038784/5923143676314255446643283231146667875,-57480046875499710 3944423838666252602754508075193305936911875/31154518469390721570659706 3235165448890282539301416442871559,88873364366227887565000/81762900459 32637014386689,0,0],\n[1,86280208734474390054682197174277603231/163235 672733343926284895054973116600000,0,8711914259012320702383973719347192 32/533362446801744790377544382297593125,896418341744389143174385771404 6506701/1273379282362081167514274901917400000,-25204784038408156321958 3820196041648831131662918986674/66729954558238813530334990145293701430 108220259249375,-5290899117464226058433745200964698112/887595109564567 0561315492876310816875,-3069893906680376471707933893233719720682072201 84067211641263783356000/8001341807972529284403437443422408231395852556 0207876667654196707053,-7376114064684719178734872354875/35902295584038 3035894248406127836,255482840375593769438967652731/8853429096639550069 183960833070,0],\n[0,164949402359214517/2969763122318428800,0,0,0,1661 64647343343629239462104853/594150218553182839761586950000,883515736589 9998531682304/6999851448505791234978125,110567843372870117205263375074 449005180061856568412069340896875/340717754649392931183331876846338804 668211857149398834541328544,758177409913942921875/78785859799058312576 ,-2973333145929506703/747247079460651200,-1895689785077201/28870708445 0720]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n \+ c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]); \n cA := evalf(A[9,1]);\n 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 a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[ 8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := \+ evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n \+ aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9, 5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := ev alf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n \+ b1 := evalf(A[10,2]);\n b2 := evalf(A[10,3]);\n b3 := evalf(A[10,4 ]);\n b4 := evalf(A[10,5]);\n b5 := evalf(A[10,6]);\n b6 := eval f(A[10,7]);\n b7 := evalf(A[10,8]);\n b8 := evalf(A[10,9]);\n b9 := evalf(A[10,10]);\n bA := evalf(A[10,11]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do \n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h ,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk \+ + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4* h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 \+ := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64* f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h ,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a 86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f 1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n \+ f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 \+ + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn (xk + cA*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA;\n yk := yk + t*h; \n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if \+ bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c 3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,\n \+ a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52, \n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_= a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a 81_=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a 87,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a9 7_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA 5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,b1_=b1,b2_=b2,b3_=b3, b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA\};\n \+ return subs(eqns,eval(rk7_10step)); \n else\n return evalf[save Digits]([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 49 "RK7_5 scheme that satisfies some error conditio ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8405 "RK7_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a42,a43,a51,a52,a53,\n a54, a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,a85,\n a 86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,aA7,\n \+ aA8,aA9,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA, \n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits; \n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unappl y(fxy,x,y);\n\n A := matrix([[1/200,1/200,0,0,0,0,0,0,0,0,0],\n[1257 88166632/804556341815,-296212523854984450605064/1294621814309470234988 45,63290651458559192893696/25892436286189404699769,0,0,0,0,0,0,0,0],\n [200814933981747403/389065106257376584,-127893721570711467450451117756 96471200626449407/38081526404775332901721474741064649138530038784,0,32 445052113731709472131307285859833545001662335/380815264047753329017214 74741064649138530038784,0,0,0,0,0,0,0],\n[119/198,-4319556318016342264 550098836819533/588237381338985769780687033796930496,0,122312177918886 060585665458413264940675777568875/329913188780101678448178322632546427 763796661824,15643567285020817980224090362461345189277853300810312/658 36374953303610877005093358513531407567316792726287,0,0,0,0,0,0],\n[17/ 19,-382421362069232952645558482090899/72768912959378304431108675144702 88,0,22057340392597202004799056057490702918544557070119184193875/41299 979864570857330923564925248608645282955350725839567504,-94360812687526 097028123187068745127462524044420500458282895614759616/117017879257650 174135972597811683016497852743167990737825503364250809,104294099032503 5750229825636422309760/855155063453934295147878200275545499,0,0,0,0,0] ,\n[9526409/47245149,3459359727020343366314090265352176896990878417015 7140274555/58731550602113257090244623300572326203937558128968689322244 8,0,517714047024312279768356989660598751616539775823173072267317364897 221236919340917125/333330780773975293533161697727890456259487743000308 9707149292892535414217827153748784,-3046404853655506630695951934475238 619460577978474785952003633121978548393408440668463488064/511362639423 0295385850817638623198196373060539656141180869306757176832215048252616 6450770407,31957815155344547212722562970483115323606081283417524915395 2/6153982820746383354666623304676483805766376504613845749252065,-59190 400148110099510923215866/11994398797423357814553515510595,0,0,0,0],\n[ 25/29,-246878890533156807885166723922846376763102826028025/94591686958 8539937065443533422779502568160417427184,0,410521623087253233613229370 6540464473954640713998671441385245340625/19499786542988241089964885103 22665339088184896936844826392190722672,-493247125993577514892765625372 0019927767958725726761483449498264713532779200/10260713295176800963125 818770711044624508389420487189702045571431972578553533,212696875762298 92809323240545664650122099097000742563680/2098952683541446314058869821 4708112363244319596705442863,-7115819429493526581990/37849324038145288 2505729,-28380858623627672736703320753126600/1897075783840010072535356 4886292693,0,0,0],\n[44/45,5583533243263337740730449062907085527564847 23746581752261/5843194402015711645509457932406318641490419890010223125 00,0,-3614510090407314534068821036471754993634788628766244060214613687 /502464522228026334461988984585059909779594952312786529837471612,-1343 6380927846244044757803583952427765782251762574076119478709593044856969 472/660987287247208667958590882181936133876300215856141449861443115776 1209623125,98993186489712313901089693322164771791022112348100142464384 1888/575874527302276156101678113881665298242482451696692517167915625,- 372567259603706115805768838168/1683402581390679277326651009375,2314436 08884459407960125470613488480291827060789/3141890410475820892188092038 6941521649080865000,4101332345819049706502423/106670973639097199171250 00,0,0],\n[1,2693249580763133758720448819763251094287099002263195049/1 972459096581551425883657161923834583484247606165156240,0,-762720164986 302748355429971639166755610094064685904534324565414570375/739302547895 26581156411334133121910635100300669232471904805722424944,-914106378133 783649744437882531239843522925103950674595588223056094266864010432/389 0181805643784996468101015923056880817477048476711228054318720508705563 93341,3868116403444609936063780569978932749234663970855789282731320288 /2224720532488974689873485313867737218754855271707412737609361885,-214 7337295919765809044534716/15282699757843572625171655145,76963822828349 512771032624512097499610054141220500895/745773721745165069429512286875 0841403431006754989491,141940909070976808611971717/3367966377875722866 21096705,-545009894469844730599500000/14146180885244972240022216299,0] ,\n[0,18530703372187/317986769215500,0,0,0,397406378366413672608/80368 0368165950784145,383280144717771439/119491812371674080,327399398331091 97146986610130768938398484878542073/1026382175105416417138017053640291 67418501950476800,-16070516558763250309/7373332958527296000,-685885128 68559375/32259978418667428,1754022361907/1430294620800]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n \+ c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]); \n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9 ,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3] );\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := eval f(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]); \n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf( A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 : = evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[ 7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := \+ evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n \+ a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8, 3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := ev alf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a 98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3] );\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := eval f(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n b1 := evalf(A[10,2]) ;\n b2 := evalf(A[10,3]);\n b3 := evalf(A[10,4]);\n b4 := evalf( A[10,5]);\n b5 := evalf(A[10,6]);\n b6 := evalf(A[10,7]);\n b7 : = evalf(A[10,8]);\n b8 := evalf(A[10,9]);\n b9 := evalf(A[10,10]); \n bA := evalf(A[10,11]);\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 t := a91*f1 + a92*f2 + a93*f3 \+ + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9* h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + \+ aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h) ;\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7* f7 + b8*f8 + b9*f9 + bA*fA;\n yk := yk + t*h;\n xk := xk + h :\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n \+ eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_ =c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,\n a31_=a31,a32_ =a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53, a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_= a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a 82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n \+ a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n \+ aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7, \n aA8_=aA8,aA9_=aA9,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6 ,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA\};\n return subs(eqns,eva l(rk7_10step)); \n else\n return evalf[saveDigits]([soln]);\n \+ end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {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 49 "Test 1 of 10 stage, order 7 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%# dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F +F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6 #\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x* cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/ 289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ \"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX$\\'FJ$ 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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$%:scheme~of~Sharp~and~SmartG$\"+!*Q8$y\"!#C7$%Ischeme ~with~a~small~principal~error~normG$\"+X&4I$>!#D7$%Pscheme~with~a~mode rately~large~stability~regionG$\"+$e:gb)F07$%Fscheme~with~a~larger~sta bility~regionG$\"+#GI9F$F07$%Lscheme~that~satisfies~some~error~conditi onsG$\"+YEtKWF0Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"! \"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme with a small princi pal error norm`,`scheme with a moderately large stability region`,`sch eme with a larger stability region`,`scheme that satisfies some error \+ conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NC int((f(x)-'fn_RK7_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor =200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlin alg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+\")45G;!#C7$%Ischeme ~with~a~small~principal~error~normG$\"+lE_q6*\\F07$%Lscheme~that~satisfies~some~error~conditi onsG$\"+rl.OXF0Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed usin g the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 515 "evalf[25](plot(['fn_RK7_1'(x)-f(x),'fn_RK7_2'(x )-f(x),'fn_RK7_3'(x)-f(x),'fn_RK7_4'(x)-f(x),\n'fn_RK7_5'(x)-f(x)],x=0 ..5,-1.3e-15..3.59e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),C OLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of S harp and Smart`,`scheme with a small principal error norm`,`scheme wit h a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`],\ntitle=`error curves for 10 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 915 582 582 {PLOTDATA 2 "6+-%'CURVESG6%7cr7$$\"\"!F)F(7$$ \":qmmmmmm;a)G\\a!#E$!(.7N(!#C7$$\":MLLLLLL$3x&)*3\"!#D$!)C>w8F07$$\": nmmmm;z%\\v#pK\"F4$!)oL.7F07$$\":++++++D1R(*Rc\"F4$!(cS*fF07$$\":MLLLL $3xJs1,=F4$\")GH97F07$$\":nmmmmm;H2P\"Q?F4$\")i$e/$F07$$\":++++++vVtc8 d#F4$\")'o:j)F07$$\":MLLLLLLeRwX5$F4$\"*^qI?\"F07$$\":MLLLLL3F%\\wQKF4 $\"*(*eV>\"F07$$\":MLLLLL$e*[`HP$F4$\"*jbr:\"F07$$\":MLLLLLek.Ur]$F4$ \"*,%\\B5F07$$\":MLLLLLLLeI8k$F4$\")VB$)F 4$\"+&G+:7'F47$$\":qmmmmm\"H#o)fb%)F4$\"+OXcigF47$$\":0++++++]Plxe)F4$ 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" }}{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 0 771 "G := \+ (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([ [`slope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of S harp and Smart`,`scheme with a small principal error norm`,`scheme wit h a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`]:\nerrs := []: \nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK7_||c t := RK7_||ct(G(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK7 _||ct[ii,2]-g(Gn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs) ,sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,e valf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slo pe~field:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7 $%/step~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q(pprint36\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+#>OI+$!#D7$%Ischeme~w ith~a~small~principal~error~normG$\"+vg9VY!#E7$%Pscheme~with~a~moderat ely~large~stability~regionG$\"+J2$f+)F07$%Fscheme~with~a~larger~stabil ity~regionG$\"+`vULlF+7$%Lscheme~that~satisfies~some~error~conditionsG $\"+wm&)4nF0Q(pprint46\"" }}}{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 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "G := (x,y) -> x/y: hh : = 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: ` ,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. o f steps: `,numsteps]]);``;\nmthds := [`scheme of Sharp and Smart`,`s cheme with a small principal error norm`,`scheme with a moderately lar ge stability region`,`scheme with a larger stability region`,`scheme t hat satisfies some error conditions`]:\nerrs := []:\nDigits := 30:\nfo r ct to 5 do\n gn_RK7_||ct := RK7_||ct(G(x,y),x,y,x0,y0,hh,numsteps, true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := evalf(g(xx )):\nfor ct to 5 do\n errs := [op(errs),abs(gn_RK7_||ct(xx)-gxx)];\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(pprint56\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%:scheme~of~Sharp~and~SmartG$\"+w*H]I)!#E7$%Ischeme~with~a~small~pri ncipal~error~normG$\"+yH,6\"*!#F7$%Pscheme~with~a~moderately~large~sta bility~regionG$\"+wLh#*=F+7$%Fscheme~with~a~larger~stability~regionG$ \"+!>#*yv\"!#D7$%Lscheme~that~satisfies~some~error~conditionsG$\"+^TF_ bF0Q(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "mthds := [`scheme of Sharp and Smart`,`scheme with a small principal error \+ norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions `]:\nerrs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 d o\n sm := NCint((g(x)-'gn_RK7_||ct'(x))^2,x=0..10,adaptive=false,num points=7,factor=100);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDi gits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+K$f^ *H!#D7$%Ischeme~with~a~small~principal~error~normG$\"+&[(*3j%!#E7$%Psc heme~with~a~moderately~large~stability~regionG$\"+Gm&4*zF07$%Fscheme~w ith~a~larger~stability~regionG$\"+\\u?9lF+7$%Lscheme~that~satisfies~so me~error~conditionsG$\"+8u<%o'F0Q(pprint76\"" }}}{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 0 516 "evalf[20](plot([g(x)-'gn_R K7_1'(x),g(x)-'gn_RK7_2'(x),g(x)-'gn_RK7_3'(x),g(x)-'gn_RK7_4'(x),\ng( x)-'gn_RK7_5'(x)],x=0..10,-9.5e-16..1.79e-15,font=[HELVETICA,9],\ncolo r=[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)], \nlegend=[`scheme of Sharp and Smart`,`scheme with a small principal e rror norm`,`scheme with a moderately large stability region`,`scheme w ith a larger stability region`,`scheme that satisfies some error condi tions`],\ntitle=`error curves for 10 stage order 7 Runge-Kutta methods `));" }}{PARA 13 "" 1 "" {GLPLOT2D 935 587 587 {PLOTDATA 2 "6+-%'CURVE SG6%7hp7$$\"\"!F)F(7$$\"5mmmmTN@Ki8!#@F(7$$\"5KLLL$3FWYs#F-$!#=!#>7$$ \"5lmm;aQ`!eS$F-$!$3\"F37$$\"5)******\\iSmp3%F-$!$j%F37$$\"5KLL$eRZF\" oZF-$!%)e\"F37$$\"5lmmmmT&)G\\aF-$!%?BF37$$\"5+++]P4'\\/8'F-FF7$$\"5IL LL3x1h6oF-FF7$$\"5lmm;zWF_o$!%6lF37$$\"5mmmTN'4n`(>F_o$!%hrF37$$\"5+++]7.K[V?F_o$!%4wF37$$\" 5LLLe*)4$*f6@F_o$!%)f(F37$$\"5mmmmm;arz@F_o$!%)e(F37$$\"5LLL$e*)4bQl#F _o$!%;%)F37$$\"5++++D\"y%*z7$F_o$!%\"y)F37$$\"5mm;H#=qHlC$F_o$!%_()F37 $$\"5LLLeRAY1lLF_o$!%J()F37$$\"5mm\"H#o#3KVU$F_o$!%R()F37$$\"5++](oHa* f$[$F_oF^u7$$\"5LL3_D.q'Ga$F_o$!%!z)F37$$\"5mmm;ajW8-OF_o$!%t()F37$$\" 5+++vo/V?RQF_o$!%,()F37$$\"5LLLL$e9ui2%F_o$!%1&)F37$$\"5mmm;H2Q\\4YF_o $!%r!)F37$$\"5++++voMrU^F_o$!%!e(F37$$\"5NLL$3-8Lfn&F_o$!%,rF37$$\"5mm mmm\"z_\"4iF_o$!%tmF37$$\"5lmmmmm6m#G(F_o$!%DgF37$$\"5lmmmmT&phN)F_o$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h of the methods and gives the " }{TEXT 260 22 "root mean square error " }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 771 "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 Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error con ditions`]:\nerrs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n Hn_RK7_||ct := RK7_||ct(H(x,y),x,y,x0,y0,hh,numsteps,fals e);\n sm := 0: numpts := nops(Hn_RK7_||ct):\n for ii to numpts do \n sm := sm+(Hn_RK7_||ct[ii,2]-h(Hn_RK7_||ct[ii,1]))^2;\n end 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,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0in itial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~ ~G\"$+\"Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG $\"+S\"*eGM!#@7$%Ischeme~with~a~small~principal~error~normG$\"+z-@2N!# A7$%Pscheme~with~a~moderately~large~stability~regionG$\"+;3w\"4\"F+7$% Fscheme~with~a~larger~stability~regionG$\"+nL:I:F+7$%Lscheme~that~sati sfies~some~error~conditionsG$\"+yHkV " 0 "" {MPLTEXT 1 0 701 "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 Sharp and Smart`,`scheme with a small principal error norm` ,`scheme with a moderately large stability region`,`scheme with a larg er stability region`,`scheme that satisfies some error conditions`]:\n errs := []:\nDigits := 20:\nfor ct to 5 do\n hn_RK7_||ct := RK7_||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(err s),abs(hn_RK7_||ct(xx)-hxx)];\nend do:\nDigits := 10:\nlinalg[transpos e]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~ G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)ppr int106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+L;$45#! #N7$%Ischeme~with~a~small~principal~error~normG$\"+sq@l')!#O7$%Pscheme ~with~a~moderately~large~stability~regionG$\"+iMTP>F07$%Fscheme~with~a ~larger~stability~regionG$\"+@&*GVW!#P7$%Lscheme~that~satisfies~some~e rror~conditionsG$\"+)=)[>?F+Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 110 " over the interval [0, 0.5] of each Rung e-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 467 "mthds := [`scheme of Sharp \+ and Smart`,`scheme with a small principal error norm`,`scheme with a m oderately large stability region`,`scheme with a larger stability regi on`,`scheme that satisfies some error conditions`]:\nerrs := []:\nDigi ts := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n sm := NCint((h(x )-'hn_RK7_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+59muJ!#@7$%Ischeme~with~a~s mall~principal~error~normG$\"+G1vNK!#A7$%Pscheme~with~a~moderately~lar ge~stability~regionG$\"+\\l<55F+7$%Fscheme~with~a~larger~stability~reg ionG$\"++&[QV\"F+7$%Lscheme~that~satisfies~some~error~conditionsG$\"+T #\\Bh\"F+Q)pprint126\"" }}}{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 0 510 "evalf[20](plot([h(x)-'hn_RK7_1'(x),h(x)-'hn_RK7_2'(x ),h(x)-'hn_RK7_3'(x),h(x)-'hn_RK7_4'(x),\nh(x)-'hn_RK7_5'(x)],x=0..5,n umpoints=100,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,. 9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of Sharp and S mart`,`scheme with a small principal error norm`,`scheme with a modera tely large stability region`,`scheme with a larger stability region`,` scheme that satisfies some error conditions`],\ntitle=`error curves fo r 10 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 835 507 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0 "" {TEXT -1 48 "Test 4 of 10 stage, ord er 7 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)):\n y(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$-%\"yG6#%\"xGF,,$**\"#K\"\"\"F,F0F)F0- %#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6# !\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\" \"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$\"3+++++++]7!#=7$$ !3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$! 3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F= 7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!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> 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$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H (\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$\"3ylm;HCjV9F-$\"3 )e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn =F-$\"31))p(30[[c&Fap7$$\"3#z****\\_qn2#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:$Fa p7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap 7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31 MiF2c]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;HYt 7vF-$\"3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6 KU$)F-$\"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 778 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: \+ x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small pri ncipal error norm`,`scheme with a moderately large stability region`,` scheme with a larger stability region`,`scheme that satisfies some err or conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n Kn_RK 7_||ct := RK7_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,false); \n sm := 0: numpts := nops(Kn_RK7_||ct):\n for ii to numpts do\n \+ sm := sm+(Kn_RK7_||ct[ii,2]-k(Kn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#l nG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~ G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %:scheme~of~Sharp~and~SmartG$\"+\"*HX5y!#;7$%Ischeme~with~a~small~prin cipal~error~normG$\"+8&*zMl!#<7$%Pscheme~with~a~moderately~large~stabi lity~regionG$\"+d+0#>&F+7$%Fscheme~with~a~larger~stability~regionG$\"+ 9T[T;F07$%Lscheme~that~satisfies~some~error~conditionsG$\"+KZH.6F+Q)pp rint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{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 0 709 "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 st eps: `,numsteps]]);``;\nmthds := [`scheme of Sharp and Smart`,`schem e with a small principal error norm`,`scheme with a moderately large s tability region`,`scheme with a larger stability region`,`scheme that \+ satisfies some error conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n kn_RK7_||ct := RK7_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh ,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)):\nfor ct t o 5 do\n errs := [op(errs),abs(kn_RK7_||ct(xx)-kxx)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xG F,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\") 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint156\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+2s2Xn!#?7$%Ischeme~wi th~a~small~principal~error~normG$\"*sT3y&F+7$%Pscheme~with~a~moderatel y~large~stability~regionG$\"+N)fEp$F+7$%Fscheme~with~a~larger~stabilit y~regionG$\"*%pKwOF+7$%Lscheme~that~satisfies~some~error~conditionsG$ \"*vpa!yF+Q)pprint166\"" }}}{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 using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes met hod over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "mthds := [`scheme of Sharp and Smart`,`scheme with a small pr incipal error norm`,`scheme with a moderately large stability region`, `scheme with a larger stability region`,`scheme that satisfies some er ror conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm : = NCint((k(x)-'kn_RK7_||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7,f actor=100);\n errs := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+*)3&*Hy !#;7$%Ischeme~with~a~small~principal~error~normG$\"+]T6^l!#<7$%Pscheme ~with~a~moderately~large~stability~regionG$\"+&R8]?&F+7$%Fscheme~with~ a~larger~stability~regionG$\"+-3_X;F07$%Lscheme~that~satisfies~some~er ror~conditionsG$\"+Uq/16F+Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are con structed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "evalf[20](plot([k(x)-'kn_RK7_1'(x) ,k(x)-'kn_RK7_2'(x),k(x)-'kn_RK7_3'(x),k(x)-'kn_RK7_4'(x),\nk(x)-'kn_R K7_5'(x)],x=-1..1,color=[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brow n,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of Sharp and Smart`,`scheme wi th a small principal error norm`,`scheme with a moderately large stabi lity region`,`scheme with a larger stability region`,`scheme that sati sfies some error conditions`],\nfont=[HELVETICA,9],title=`error curves for 10 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 840 605 605 {PLOTDATA 2 "6+-%'CURVESG6%7eo7$$!\"\"\"\"!$F*F* 7$$!5nmmmm;p0k&*!#?$\"+*)\\/,AF/7$$!5LLLL$37$$!5nmmmm\"4m(G$)F/$\"+MmIqlF<7$$!5LLLL$3i .9!zF/$\",epw'o:F<7$$!5mmmm;/R=0vF/$\"+u/MNK!#=7$$!5++++]P8#\\4(F/$\"+ 'pT>]'FL7$$!5mmmm;/siqmF/$\",(z]&fE\"FL7$$!5++++](y$pZiF/$\"+pN]YB!#<7 $$!5LLLLL$yaE\"eF/$\"+Ql%yA%Ffn7$$!5mmmmm\">s%HaF/$\"+c=rRoFfn7$$!5+++ ++]$*4)*\\F/$\",\\t8*G6Ffn7$$!5+++++]_&\\c%F/$\",!pjk)y\"Ffn7$$!5+++++ ]1aZTF/$\"+$[O#yE!#;7$$!5mmmm;/#)[oPF/$\"+jLEOPF`p7$$!5LLLLL$=exJ$F/$ \"+SitC`F`p7$$!5LLLLLeW%o7$F/$\"+?%fK5'F`p7$$!5LLLLLL2$f$HF/$\"+88DRpF `p7$$!5mmmmT&o_Qr#F/$\"+#Gm_(zF`p7$$!5********\\PYx\"\\#F/$\"+/CAm!*F` p7$$!5mmmmTNz>&H#F/$\",Xa^j+\"F`p7$$!5LLLLLL7i)4#F/$\",Q'y^26F`p7$$!5m 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`y$\"+'\\BBW#F`p7$Fey$\"+LYroBF`p7$Fjy$\"+\"pv2F#F`p7$F_z$\"+;rCc@F`p7 $Fdz$\"+4^kB?F`p7$Fiz$\"+L0**z=F`p7$F^[l$\"+dw?KyG\"F`p7$Fb\\l$\"+!R&[J6F`p7$Fg\\l $\"*=Y+$)*F`p7$F\\]l$\"*ve.e)F`p7$Fa]l$\"*/8>U(F`p7$Ff]l$\"*`:>G&F`p7$ F[^l$\"*OaRu$F`p7$F`^l$\"+'H.0Z#Ffn7$Fe^l$\"+_j#zg\"Ffn7$Fj^l$\"*\"=YZ )*Ffn7$F__l$\"*(pjYeFfn7$Fd_l$\"*zd7B$Ffn7$Fi_l$\"+Gl`K " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 5 of 10 stage, order 7 Rung e-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"#;F&*&,&*&F*F&%\" xGF&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "y(0)= 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-%%sqrtG6#,&*&\"\"#\"\"\"-%#lnG6#,& *&\"#;F+%\"xGF+F+F+F+F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff( y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,font=[HELVETICA,9],l abels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diff G6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6#,&*& \"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a! #?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A!pa&\\6F27$$\"3-+]i!R (*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3]_vibZz]7F27$$\"3MLL$ eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3CLL$3x%3yTF6$\"31#\\\\E7=EU\"F27$$\"3=mm \"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$$\"3)HL$eR-/PiF6$\"3.fPtw=4W:F27$$\"3A* **\\il'pisF6$\"3/07@a`R%f\"F27$$\"3`KLe*)>VB$)F6$\"3K!\\`od36k\"F27$$ 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co$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF27$$\"3))*****\\#>#[ Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G!e&e%Fco$\"3)zsS%e\"3%*G#F27$$\"3;LLL $)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h.zwhI#F27$$\"35+] P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>())[`fABF2-%'COLOUR G6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"x G%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F($\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following 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 0 761 "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 Sharp and Smart`,`scheme with a small princi pal error norm`,`scheme with a moderately large stability region`,`sch eme with a larger stability region`,`scheme that satisfies some error \+ conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n Sn_RK7_| |ct := RK7_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: nump ts := nops(Sn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Sn_R K7_||ct[ii,2]-s(Sn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(err s),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0in itial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~ste ps:~~~G\"$+\"Q)pprint186\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~ SmartG$\"+>N*z]\"!#@7$%Ischeme~with~a~small~principal~error~normG$\"+` $Goj&!#A7$%Pscheme~with~a~moderately~large~stability~regionG$\"+wpxX5F +7$%Fscheme~with~a~larger~stability~regionG$\"+!e(z$Q&F07$%Lscheme~tha t~satisfies~some~error~conditionsG$\"+goy6:F+Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following c ode constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " f or 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 metho ds at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`initi al point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,num steps]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a sma ll principal error norm`,`scheme with a moderately large stability reg ion`,`scheme with a larger stability region`,`scheme that satisfies so me error conditions`]:\nerrs := []:\nDigits := 25:\nfor ct to 5 do\n \+ sn_RK7_||ct := RK7_||ct(S(x,y),x,y,x0,y0,hh,numsteps,true);\nend do: \nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct to 5 do\n errs := [op(e rrs),abs(sn_RK7_||ct(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"y GF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$% 1no.~of~steps:~~~G\"$+\"Q)pprint206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~ Sharp~and~SmartG$\"+Hn@R7!#@7$%Ischeme~with~a~small~principal~error~no rmG$\"+_@YJY!#A7$%Pscheme~with~a~moderately~large~stability~regionG$\" +V.)))f)F07$%Fscheme~with~a~larger~stability~regionG$\"+EnuIWF07$%Lsch eme~that~satisfies~some~error~conditionsG$\"+TR]T7F+Q)pprint216\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interva l [0, 0.5] of each Runge-Kutta method is estimated as follows using \+ the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equa l subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 446 "mthds := [`scheme of Sharp and Smart`,`scheme with a small principal error nor m`,`scheme with a moderately large stability region`,`scheme with a la rger stability region`,`scheme that satisfies some error conditions`]: \nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((s(x)-'sn _RK7_||ct'(x))^2,x=0..0.5,adaptive=false,numpoints=7,factor=50);\n e rrs := [op(errs),sqrt(sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+)y[u]\"!#@7$%Ischeme~with~a~sm all~principal~error~normG$\"+&pJYj&!#A7$%Pscheme~with~a~moderately~lar ge~stability~regionG$\"+I&*QX5F+7$%Fscheme~with~a~larger~stability~reg ionG$\"+:`)>Q&F07$%Lscheme~that~satisfies~some~error~conditionsG$\"+AF D6:F+Q)pprint226\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "The following error graphs are constructed using the nu merical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "evalf[20](plot(['sn_RK7_1'(x)-s(x),'sn_RK7_2'(x)-s(x ),'sn_RK7_3'(x)-s(x),'sn_RK7_4'(x)-s(x),\n'sn_RK7_5'(x)-s(x)],x=0..0.5 ,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red, brown,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of Sharp and Smart`,`schem e with a small principal error norm`,`scheme with a moderately large s tability region`,`scheme with a larger stability region`,`scheme that \+ satisfies some error conditions`],\ntitle=`error curves for 10 stage o rder 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 891 475 475 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al$\")PW\"H\"F07$Fbal$\")\\Y&G\"F07$Fgal$\")-\"*z7F07$F\\bl$\")UUu7F07 $Fabl$\")m;p7F07$Ffbl$\")r[k7F07$F[cl$\")JHf7F07$F`cl$\")Cza7F07$Fecl$ \")l8]7F07$Fjcl$\"):\"eC\"F07$F_dl$\")$*HT7F0-Fedl6&FgdlF(FadmF_dm-F[e l6#%Lscheme~that~satisfies~some~error~conditionsG-%%FONTG6$%*HELVETICA GF`dm-%+AXESLABELSG6$Q\"x6\"Q!F[cp-%&TITLEG6#%Verror~curves~for~10~sta ge~order~7~Runge-Kutta~methodsG-%%VIEWG6$;F(F_dl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sharp and S mart" "scheme with a small principal error norm" "scheme with a modera tely large stability region" "scheme with a larger stability region" " scheme that satisfies some error conditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 6 of 10 stage, order 7 Runge -Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy /dx = (1+2*(x+1)*sin(3*x))*exp(-y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F& F&*(\"\"#F&,&%\"xGF&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F&F&-%$expG6#,$%\"yG F(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\" !F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x+2/9*sin(3*x)- 2/3*x*cos(3*x)-2/3*cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\"xG\"\"\"*(\"\" #F*\"\"*!\"\"-%$sinG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$cosG6#*&F3F*F) F*F*F.*(F,F*F3F.-F66#*&F3F*F)F*F*F.*&\"\"&F*F3F.F*" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff(y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y(x));\nic := y(0) =0;\ndsolve(\{de,ic\},y(x));\nu := unapply(rhs(%),x):\nplot(u(x),x=0.. 5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/*(\"\"#F/,& F,F/F/F/F/-%$sinG6#,$*&\"\"$F/F,F/F/F/F/F/-%$expG6#,$F)!\"\"F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,,F'\"\"\"*&#\"\"#\"\"*F,-%$ sinG6#,$*&\"\"$F,F'F,F,F,F,*&#F/F6F,*&F'F,-%$cosGF3F,F,!\"\"*&#F/F6F,F :F,F<#\"\"&F6F," }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\"3QWK+t!=.P\"F-7$ $\"3umm\"H2P\"Q?F-$\"3pUCE&GmM$HF-7$$\"3MLL$eRwX5$F-$\"3l!G\"yWq,6\\F- 7$$\"33ML$3x%3yTF-$\"3dz%)zauhMpF-7$$\"3emm\"z%4\\Y_F-$\"3,G5kQO>C))F- 7$$\"3`LLeR-/PiF-$\"36YrjIBvP5!#<7$$\"3]***\\il'pisF-$\"3wGtPF*HL<\"FI 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/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59%FI$\"3mH1#H$\\ k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp;%FI$\"3A7j1wip y')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>%FI$\"3Gri,**=4 g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**>UFI$\"3Z**e,OD #4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,tUFI$\"3[2R[)*e VA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z-lU%FI$\"3)4DVD mlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-G n\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_ eaMDR_K " 0 "" {MPLTEXT 1 0 773 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`scheme of Sharp and Smart`,`schem e with a small principal error norm`,`scheme with a moderately large s tability region`,`scheme with a larger stability region`,`scheme that \+ satisfies some error conditions`]:\nerrs := []:\nDigits := 25:\nfor ct to 5 do\n Un_RK7_||ct := RK7_||ct(U(x,y),x,y,x0,y0,hh,numsteps,fals e);\n sm := 0: numpts := nops(Un_RK7_||ct):\n for ii to numpts do \n sm := sm+(Un_RK7_||ct[ii,2]-u(Un_RK7_||ct[ii,1]))^2;\n end d o:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nli nalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"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\" $+&Q)pprint236\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+ P*=6q%!#E7$%Ischeme~with~a~small~principal~error~normG$\"+OVJJO!#F7$%P scheme~with~a~moderately~large~stability~regionG$\"+bv1&G\"F+7$%Fschem e~with~a~larger~stability~regionG$\"+\"3lp7#!#D7$%Lscheme~that~satisfi es~some~error~conditionsG$\"+dv\">I%F+Q)pprint246\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 704 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`scheme of Sharp and Smart`,`schem e with a small principal error norm`,`scheme with a moderately large s tability region`,`scheme with a larger stability region`,`scheme that \+ satisfies some error conditions`]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n un_RK7_||ct := RK7_||ct(U(x,y),x,y,x0,y0,hh,numsteps,true );\nend do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 5 do\n err s := [op(errs),abs(un_RK7_||ct(xx)-uxx)];\nend do:\nDigits := 10:\nlin alg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+ F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~p oint:~G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$ +&Q)pprint256\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+Qc &)zM!#E7$%Ischeme~with~a~small~principal~error~normG$\"+C(>e)[!#G7$%Ps cheme~with~a~moderately~large~stability~regionG$\"+d<;*4(!#F7$%Fscheme ~with~a~larger~stability~regionG$\"+EZ>c?!#D7$%Lscheme~that~satisfies~ some~error~conditionsG$\"+azTMWF+Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme w ith a small principal error norm`,`scheme with a moderately large stab ility region`,`scheme with a larger stability region`,`scheme that sat isfies some error conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((u(x)-'un_RK7_||ct'(x))^2,x=0..5,adaptive=false, numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\n Digits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+i( Ggn%!#E7$%Ischeme~with~a~small~principal~error~normG$\"+4$R#>O!#F7$%Ps cheme~with~a~moderately~large~stability~regionG$\"+.+()z7F+7$%Fscheme~ with~a~larger~stability~regionG$\"+s2o;@!#D7$%Lscheme~that~satisfies~s ome~error~conditionsG$\"+6dg\"G%F+Q)pprint276\"" }}}{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 0 528 "evalf[20](plot([u(x)-'un_R K7_1'(x),u(x)-'un_RK7_2'(x),u(x)-'un_RK7_3'(x),u(x)-'un_RK7_4'(x),\nu( x)-'un_RK7_5'(x)],x=0..5,-1.9e-16..6.8e-16,numpoints=100,font=[HELVETI CA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RG B,0,.3,.9)],\nlegend=[`scheme of Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stability regio n`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`],\ntitle=`error curves for 10 stage order 7 Runge-K utta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 965 498 498 {PLOTDATA 2 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e~order~7~Runge-Kutta~methodsG-%%VIEWG6$;F(Fhgn;$!#zFa\\m$FjhlFeip" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sharp and Smart" "scheme with a small principal error norm" "scheme with a \+ moderately large stability region" "scheme with a larger stability reg ion" "scheme that satisfies some error conditions" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 7 of 10 stage, order 7 Runge -Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d y/dx=-(1+4*cos(3*x))*(y-1/3)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&*& \"\"%F&-%$cosG6#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF&*&F&F&F2F(F(F&F(" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F &\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin (3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sin G6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+ F*F7F,F1F*F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3 *sin(3*x)-x)" "6#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"x GF)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*co s(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/-%%dif fG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0 F0F0,&F)F0#F0F8!\"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-% \"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&* &#\"\"\"\"\"$F+-%$expG6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*& #\"\")F,F+*&-F56#,$*(F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F .6#,&F'F9*&#F3F,F+F4F+F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!# >$\"3W+7cSy5h&*!#=7$$\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3 %F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\ 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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 "d iscrete solution" }{TEXT -1 44 " based on each of the methods and give s the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each \+ solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 768 "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 Sharp and Smart`,`scheme with a small principal error norm`, `scheme with a moderately large stability region`,`scheme with a large r stability region`,`scheme that satisfies some error conditions`]:\ne rrs := []:\nDigits := 30:\nfor ct to 5 do\n Vn_RK7_||ct := RK7_||ct( V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Vn_RK 7_||ct):\n for ii to numpts do\n sm := sm+(Vn_RK7_||ct[ii,2]-v( Vn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpt s)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F, F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#! \"#7$%1no.~of~steps:~~~G\"$]#Q)pprint286\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %:scheme~of~Sharp~and~SmartG$\"+=I-OA!#B7$%Ischeme~with~a~small~princi pal~error~normG$\"+#Qxt5$!#C7$%Pscheme~with~a~moderately~large~stabili ty~regionG$\"+vEgi7F+7$%Fscheme~with~a~larger~stability~regionG$\"+wd( fy)F+7$%Lscheme~that~satisfies~some~error~conditionsG$\"+Mh7,mF+Q)ppri nt296\"" }}}{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 0 699 "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 Sha rp and Smart`,`scheme with a small principal error norm`,`scheme with \+ a moderately large stability region`,`scheme with a larger stability r egion`,`scheme that satisfies some error conditions`]:\nerrs := []:\nD igits := 30:\nfor ct to 5 do\n vn_RK7_||ct := RK7_||ct(V(x,y),x,y,x0 ,y0,hh,numsteps,true);\nend do:\nxx := 4.999: vxx := evalf(v(xx)):\nfo r ct to 5 do\n errs := [op(errs),abs(vn_RK7_||ct(xx)-vxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F, *&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$% 0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~ steps:~~~G\"$]#Q)pprint306\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and ~SmartG$\"+L5t.^!#D7$%Ischeme~with~a~small~principal~error~normG$\"+&= *Gm8F+7$%Pscheme~with~a~moderately~large~stability~regionG$\"+p!*yU9F+ 7$%Fscheme~with~a~larger~stability~regionG$\"+E.T9M!#C7$%Lscheme~that~ satisfies~some~error~conditionsG$\"+jrEeAF8Q)pprint316\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge -Kutta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 100 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp \+ and Smart`,`scheme with a small principal error norm`,`scheme with a m oderately large stability region`,`scheme with a larger stability regi on`,`scheme that satisfies some error conditions`]:\nerrs := []:\nDigi ts := 20:\nfor ct to 5 do\n sm := NCint((v(x)-'vn_RK7_||ct'(x))^2,x= 0..5,adaptive=false,numpoints=7,factor=100);\n errs := [op(errs),sqr t(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sh arp~and~SmartG$\"+a&)[JA!#B7$%Ischeme~with~a~small~principal~error~nor mG$\"+[O9'4$!#C7$%Pscheme~with~a~moderately~large~stability~regionG$\" +-RRg7F+7$%Fscheme~with~a~larger~stability~regionG$\"+C`3#z)F+7$%Lsche me~that~satisfies~some~error~conditionsG$\"+\"*Q)Hf'F+Q)pprint326\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The fol lowing error graphs are constructed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "evalf[20 ](plot([v(x)-'vn_RK7_1'(x),v(x)-'vn_RK7_2'(x),v(x)-'vn_RK7_3'(x),v(x)- 'vn_RK7_4'(x),\nv(x)-'vn_RK7_5'(x)],x=0..5,color=[COLOR(RGB,.5,0,1),CO LOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of Sh arp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stability region`,`scheme with a larger stability \+ region`,`scheme that satisfies some error conditions`],\nfont=[HELVETI CA,9],title=`error curves for 10 stage order 7 Runge-Kutta methods`)); " }}{PARA 13 "" 1 "" {GLPLOT2D 929 671 671 {PLOTDATA 2 "6+-%'CURVESG6% 7^u7$$\"\"!F)F(7$$\"5qmm;aQ`!eS$!#A$\"\"\"!#?7$$\"5SLLL3x1h6oF-$\"$p\" F07$$\"5,++Dc,;u@5!#@$\"%#G%F07$$\"5ommmTN@Ki8F9$\"&MC%F07$$\"5NLL3FpE !Hq\"F9$\"'u1DF07$$\"5-++]7.K[V?F9$\"'Hx*)F07$$\"5omm\"zptjSQ#F9$\"')f #))F07$$\"5NLLL$3FWYs#F9$\"'4!o)F07$$\"5-++vo/[AlIF9$\"'P\"f)F07$$\"5o 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SY!GF0$\"(\\Yg'F07$F^cn$\"(\"yClF07$$\"5+]7`%Rz#HrGF0$\"(3mW'F07$$\"5M LekyA!>Y!HF0$\"(D9F07$Fcil$\") Nm=7F07$F]jl$\")b\"=/\"F07$Fgjl$\"(=mF*F07$Fa[m$\"(M)=!)F07$Fi]m$\"(c^ w&F07$Fc^m$\"(Lq-%F07$Fa`m$\"(%R#*GF07$F[am$\"(`rC#F07$Feam$\"()\\`J(F07$Ffem$\"(v)ovF07$Ff]p$\"(1$3wF07$F[fm$\"(X^i(F07$F^^p$\"(qL i(F07$F`fm$\"(;fh(F07$Fjfm$\"(cZ_(F07$Fdgm$\"(d6Q(F07$Figm$\"(td$pF07$ F^hm$\"(\\EI'F07$Fhhm$\"(GaF07$Fdx$\"(2NI &F07$F^y$\"(\"[,TF07$Fcy$\"(p&\\OF07$Fhy$\"(x\"eLF07$F]z$\"(@8G$F07$Fb z$\"(xIB$F07$Fgz$\"(;=@$F07$F\\[l$\"(&H;KF07$Fa[l$\"(D\\C$F07$Ff[l$\"( PvH$F07$F[\\l$\"(^MP$F07$F`\\l$\"(FQZ$F07$Fe\\l$\"($HmPF07$Fj\\l$\"(gJ >%F07$F_]l$\"(8zy%F07$Fd]l$\"(^.c&F07$Fi]l$\"(?$\\lF07$F^^l$\"(F07$F^]n$\"',:=F07$Fc]n$\"'H4>F07$Fh ]n$\"'/jAF0-F]^n6&F_^nF(FabpF_bp-Fd^n6#%Lscheme~that~satisfies~some~er ror~conditionsG-%%FONTG6$%*HELVETICAGF`bp-%+AXESLABELSG6$Q\"x6\"Q!F_\\ u-%&TITLEG6#%Verror~curves~for~10~stage~order~7~Runge-Kutta~methodsG-% %VIEWG6$;F(Fh]n%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sharp and Smart" "scheme with a small princip al error norm" "scheme with a moderately large stability region" "sche me with a larger stability region" "scheme that satisfies some error c onditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 8 o f 10 stage, order 7 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x*(9-x^2)/(1+y^2)" "6#/*&%#dyG\"\"\"%#d xG!\"\"*(%\"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+sq rt(64+9*x^8-324*x^6+2916*x^4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*&\"#a\"\" \"*$F'\"\"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\"\"*F,*$F '\"\")F,F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,F0F3" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := diff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply(rhs(%),x):\nplot(w(x), x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F, \"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\"\"$\"\"\")F'\"\"%F/F+ *&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/*&\"$C$F/)F'\"\"'F/F+* &\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7$$\"\"!F)F(7$$\"3 ()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\"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\"=lB$F>$\"3L\\!fpl0?S%F>7$$ 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Fho$\"3uI!*3pTpuMFho7$$\"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)$\"3CxC=rRoRDFho-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\" xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point : `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]) ;``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small princ ipal error norm`,`scheme with a moderately large stability region`,`sc heme with a larger stability region`,`scheme that satisfies some error conditions`]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n Wn_RK7_ ||ct := RK7_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: num pts := nops(Wn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Wn_ RK7_||ct[ii,2]-w(Wn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(er rs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0 slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"y GF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$ %1no.~of~steps:~~~G\"$+%Q)pprint336\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~ Sharp~and~SmartG$\"+v$39D\"!#E7$%Ischeme~with~a~small~principal~error~ normG$\"+u66&y&!#F7$%Pscheme~with~a~moderately~large~stability~regionG $\"+>6IV&*F07$%Fscheme~with~a~larger~stability~regionG$\"+=*ovK&F+7$%L scheme~that~satisfies~some~error~conditionsG$\"+%QnBw#F+Q)pprint346\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ following code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 3.499;" "6 #/%\"xG-%&FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "W := (x,y) -> x*(9-x^2)/(1+ y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope \+ field: `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh] ,\n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a mode rately large stability region`,`scheme with a larger stability region` ,`scheme that satisfies some error conditions`]:\nerrs := []:\nDigits \+ := 30:\nfor ct to 5 do\n wn_RK7_||ct := RK7_||ct(W(x,y),x,y,x0,y0,hh ,numsteps,true);\nend do:\nxx := 3.499: wxx := evalf(w(xx)):\nfor ct t o 5 do\n errs := [op(errs),abs(wn_RK7_||ct(xx)-wxx)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\" *F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G6$ \"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint356 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+[sQ4N!# F7$%Ischeme~with~a~small~principal~error~normG$\"+'>&p*R\"F+7$%Pscheme ~with~a~moderately~large~stability~regionG$\"+h!fAx#F+7$%Fscheme~with~ a~larger~stability~regionG$\"+.:bB>!#E7$%Lscheme~that~satisfies~some~e rror~conditionsG$\"+f6(R-\"F8Q)pprint366\"" }}}{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 "[0,4] " "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfie s some error conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do \n sm := NCint((w(x)-'wn_RK7_||ct'(x))^2,x=0..4,adaptive=false,numpo ints=7,factor=200);\n errs := [op(errs),sqrt(sm/4)];\nend do:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+G.*GA\" !#E7$%Ischeme~with~a~small~principal~error~normG$\"+\"=xz_&!#F7$%Psche me~with~a~moderately~large~stability~regionG$\"+()Rp.\"*F07$%Fscheme~w ith~a~larger~stability~regionG$\"+%QhT>&F+7$%Lscheme~that~satisfies~so me~error~conditionsG$\"+db4VFF+Q)pprint376\"" }}}{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 0 515 "evalf[20](plot([w(x)-'wn_R K7_1'(x),w(x)-'wn_RK7_2'(x),w(x)-'wn_RK7_3'(x),w(x)-'wn_RK7_4'(x),\nw( x)-'wn_RK7_5'(x)],x=0..4,-6.5e-17..2.43e-16,font=[HELVETICA,9],\ncolor =[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)], \nlegend=[`scheme of Sharp and Smart`,`scheme with a small principal e rror norm`,`scheme with a moderately large stability region`,`scheme w ith a larger stability region`,`scheme that satisfies some error condi tions`],\ntitle=`error curves for 10 stage order 7 Runge-Kutta methods `));" }}{PARA 13 "" 1 "" {GLPLOT2D 1000 633 633 {PLOTDATA 2 "6+-%'CURV ESG6%7_s7$$\"\"!F)F(7$$\"5LLLLL3x&)*3\"!#@$\"&I0\"!#B7$$\"5mmmmm;arz@F -$\"%*[\"!#A7$$\"5+++++DJdpKF-$\"%CJF67$$\"5LLLLLL3VfVF-$\"%gmF67$$\"5 +++++]i9RlF-$\"%`=F-7$$\"5mmmmmm;')=()F-$\"%ZWF-7$$\"5MLLLeR?ah5!#?$\" %W\")F-7$$\"5++++]7z>^7FN$\"&-I\"F-7$$\"5mmmmT&y`3W\"FN$\"&*3=F-7$$\"5 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!$J#FN7$Fjy$\"$6$FN7$Fdz$\"$S)FN7$F^[l$\"%W8FN7$Fh[l$\"%npF67$FB$\"%H:F-7$FG$\"%,JF-7$FR$\"%FvF-7$Ffn$\"%,7FN7$ Fdp$\"%S7FN7$F\\s$\"%q7FN7$F`t$\"%p@FN7$Fjt$\"%^PFN7$Fhv$\"%EfFN7$Fbw$ \"%=%)FN7$Ffx$\"&\\3\"FN7$Fdz$\"&()G\"FN7$Fh[l$\"&'Q9FN7$Fb\\l$\"&^`\" FN7$Faio$\"&7a\"FN7$Fg\\l$\"&!R:FN7$Fiio$\"&%G:FN7$F\\]l$\"&=^\"FN7$F` ^l$\"&OY\"FN7$Fd_l$\"&jS\"FN7$Fi_l$\"&&y7FN7$F^`l$\"%f6Fb`l7$$\"5NLLLL e9XMiFN$\"%]5Fb`l7$Fd`l$\"$e*Fb`l7$Fi`l$\"$4)Fb`l7$F^al$\"$#pFb`l7$Fca l$\"$6'Fb`l7$Fhal$\"$P&Fb`l7$F]bl$\"$z%Fb`l7$Fbbl$\"$L%Fb`l7$Fgbl$\"$) RFb`l7$F\\cl$\"$j$Fb`l7$Facl$\"$P$Fb`l7$Ffcl$\"$7$Fb`l7$F[dl$\"$$HFb`l 7$F`dl$\"$v#Fb`l7$Fedl$\"$h#Fb`l7$Fjdl$\"$Y#Fb`l7$F^el$\"$O#Fb`l7$Fcel $\"$E#Fb`l7$Fhel$\"$:#Fb`l7$F]fl$\"$3#Fb`l7$Fbfl$\"$,#Fb`l7$Fgfl$\"$&> Fb`l7$F\\gl$\"$!>Fb`l7$Fagl$\"$&=Fb`l7$Ffgl$\"$#=Fb`l7$Figl$\"$x\"Fb`l 7$F^hl$\"$v\"Fb`l7$FahlFb\\q7$Fdhl$\"$t\"Fb`l7$FghlFf\\q7$F\\ilFf\\q7$ F_il$\"$w\"Fb`l7$FdilF[]q7$Fgil$\"$!=Fb`l7$FjilFi[q7$F]jl$\"$%>Fb`l7$F `jl$\"$.#Fb`l7$F^\\m$\"$?#Fb`l7$Fa\\m$\"$I#Fb`l7$Fd\\m$\"$U#Fb`l7$Fg\\ m$\"$p#Fb`l7$F\\]m$\"$9$Fb`l7$Ff]m$\"$!RFb`l7$F`^m$\"$Q&Fb`l7$Fe^m$\"$ w'Fb`l7$Fj^m$\"$:*Fb`l7$$\"5++]7Gyh(>'RFb`l$\"%q5Fb`l7$F__m$\"%,6Fb`l7 $$\"5++]PMF,%G(RFb`l$\"%,8Fb`l7$Fd_m$\"%z8Fb`l7$Fi_m$\"%'f\"Fb`l7$F^`m $\"%'y\"Fb`l7$Fc`m$\"%&)>Fb`l7$Fh`m$\"%4?Fb`l7$F]am$\"%Q?Fb`l7$Fbam$\" %;DFb`l-F^bp6&Fiam$\")#)eqkFbbp$\"))eqk\"FbbpFiaq-F`bm6#%Fscheme~with~ a~larger~stability~regionG-F$6%7hpF'7$F2$\"$7\"F67$F=$\"%UBF67$$FhbmF- $\"$o%F-7$FB$\"$<)F-7$$\"5NLLLLeR+HwF-$\"%\\8F-7$FG$\"%XBF-7$FR$\"%\\y F-7$Ffn$\"%=:FN7$F`o$\"%\\;FN7$Fjo$\"%]Fb`l7$Facl$\"$\"=Fb`l7$Ffcl$\"$o\"Fb`l7$F[dl$\"$e\"Fb`l7 $F`dl$\"$[\"Fb`l7$Fedl$\"$T\"Fb`l7$Fjdl$\"$K\"Fb`l7$F^el$\"$G\"Fb`l7$F cel$\"$A\"Fb`l7$FhelFeal7$F]fl$FcbqFb`l7$Fbfl$\"$3\"Fb`l7$Fgfl$\"$0\"F b`l7$F\\glFa[p7$Fagl$\"$+\"Fb`l7$Ffgl$\"#)*Fb`l7$Figl$FidoFb`l7$F^hl$ \"#%*Fb`l7$FahlFh]r7$Fdhl$\"#$*Fb`l7$FghlF\\^r7$F\\il$\"##*Fb`l7$F_ilF f]r7$FdilFh]r7$Fgil$\"#(*Fb`l7$Fjil$\"#**Fb`l7$F]jl$\"$/\"Fb`l7$F`jlFi \\r7$F^\\m$\"$8\"Fb`l7$Fd\\m$\"$9\"Fb`l7$Fg\\mFeal7$F\\]m$\"$=\"Fb`l7$ Fa]mFg\\r7$Ff]m$\"$4\"Fb`l7$F[^mFf]r7$F`^m$\"#oFb`l7$Fe^mF_^p7$Fj^m$!# TFb`l7$F__mFg^m7$Fd_m$!$`\"Fb`l7$Fi_m$!$F#Fb`l7$F^`m$!$z#Fb`l7$Fc`mF[` m7$Fh`m$!$H$Fb`l7$F]am$!$h$Fb`l7$Fbam$!$\"[Fb`l-Fgam6&FiamF(F\\aoFj`o- F`bm6#%Lscheme~that~satisfies~some~error~conditionsG-%%FONTG6$%*HELVET ICAGF[ao-%+AXESLABELSG6$Q\"x6\"Q!Febr-%&TITLEG6#%Verror~curves~for~10~ stage~order~7~Runge-Kutta~methodsG-%%VIEWG6$;F(Fbam;$!#lFj\\n$\"$V#Fj \\n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme o f Sharp and Smart" "scheme with a small principal error norm" "scheme \+ with a moderately large stability region" "scheme with a larger stabil ity region" "scheme that satisfies some error conditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 9 of 10 stage, order 7 Run ge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x))*y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-%$co sG6#*&\"\"#F&%\"xGF&F&F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" "6#/%\"yG *&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\"xGF&F&*&F/F&F0F&F&*&F&F&F/! \"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x)=-(1+cos(2*x))*y (x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x));\nm := unapply(rhs (%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9],labels=[`x`,`y(x) `]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\"\"$F0!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F)*$,(*(\"\"#F )-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!F)$\"3 :&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H8F,7$$\"3')*****\\7t& pKF0$\"3!G<)\\ef9f7F,7$$\"3z****\\(ofV!\\F0$\"3oe\"F,7$$\"3s*** ***\\i9RlF0$\"3kESFh\"zh9\"F,7$$\"33++vVV)RQ*F0$\"3'f)*)e-w\\p5F,7$$\" 3/++vVA)GA\"!#=$\"3V)o6<$fq15F,7$$\"3;+]iSS\"Ga\"FJ$\"3IyW%eHk>[*FJ7$$ \"3+++]Peui=FJ$\"3#4`!o2+#G**)FJ7$$\"37+++]$)z%=#FJ$\"3OGH4wwYu&)FJ7$$ \"3A++]i3&o]#FJ$\"3=1g%=M2W@)FJ7$$\"3%)***\\(oX*y9$FJ$\"31u2v$Q9&GwFJ7 $$\"3z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\P*zhdVFJ$\"3P$G(zQ8#4%o FJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY%G?7 \"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"33++]7[X$ocbFJ7$$\"3))***\\PpnsM*FJ$\" 3!\\;$Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aFJ7$$\"3-+++!R5'f5F,$\"3 q`:6QhHm`FJ7$$\"3)***\\P/QBE6F,$\"3@Igj*yDKK&FJ7$$\"3!******\\\"o?&=\" F,$\"3i/K.-M\\%H&FJ7$$\"31+]Pa&4*\\7F,$\"3OjcS#)ygr_FJ7$$\"33+]7j=_68F ,$\"3'e4m\")R`oD&FJ7$$\"33++vVy!eP\"F,$\"3a@U-1/NZ_FJ7$$\"34+](=WU[V\" F,$\"3Nrr*HO\"oU_FJ7$$\"3)****\\7B>&)\\\"F,$\"3'HX%)zwR1C&FJ7$$\"3)*** \\P>:mk:F,$\"3<^\"Q\"4\"y-C&FJ7$$\"3'***\\iv&QAi\"F,$\"3:*4?^OZ,C&FJ7$ $\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"3!******\\Nm'[F,$\"3[h+0^h(R>&FJ7$$\"3z*****\\@80+#F,$\" 3!zBIi>A%o^FJ7$$\"31++]7,Hl?F,$\"3<)30`]&>L^FJ7$$\"3()**\\P4w)R7#F,$\" 3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#)G/&FJ7$$\"3!)**\\P/-a[AF ,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>;?IA\\FJ7$$\"3')****\\i@O tBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8tZFJ7$$\"3>+++!*> =+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1YFJ7$$\"3=+]P%>5p i#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%FJ7$$\"33++Dr\"[8v #F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ7$$\"31+]P/)fT(GF ,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4%FJ7$$\"\"$F)$\"3n tdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l;F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 775 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numst eps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y) ],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps : `,numsteps]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme w ith a small principal error norm`,`scheme with a moderately large stab ility region`,`scheme with a larger stability region`,`scheme that sat isfies some error conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n Mn_RK7_||ct := RK7_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps, false);\n sm := 0: numpts := nops(Mn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Mn_RK7_||ct[ii,2]-m(Mn_RK7_||ct[ii,1]))^2;\n en d do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG 6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$ \"\"!*$F2#F,F27$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pp rint386\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+V[+fT!# C7$%Ischeme~with~a~small~principal~error~normG$\"+$['[vN!#B7$%Pscheme~ with~a~moderately~large~stability~regionG$\"+jy7=EF07$%Fscheme~with~a~ larger~stability~regionG$\"+#>BKK#F07$%Lscheme~that~satisfies~some~err or~conditionsG$\"+,wzKMF0Q)pprint396\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 706 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numsteps := 300: x 0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y)],[`initial p oint: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numstep s]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small p rincipal error norm`,`scheme with a moderately large stability region` ,`scheme with a larger stability region`,`scheme that satisfies some e rror conditions`]:\nerrs := []:\nDigits := 25:\nfor ct to 5 do\n mn_ RK7_||ct := RK7_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend d o:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 5 do\n errs := [op( errs),abs(mn_RK7_||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F, F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step ~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint406\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+))z[Ww!#D7$%Ischeme~with~a~sma ll~principal~error~normG$\"+\\FE&\\'!#C7$%Pscheme~with~a~moderately~la rge~stability~regionG$\"+WI_!y%F07$%Fscheme~with~a~larger~stability~re gionG$\"+'HzNE%F07$%Lscheme~that~satisfies~some~error~conditionsG$\"+$ 3ceD'F0Q)pprint416\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows 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 150 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme with a small principal error \+ norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions `]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((m(x)- 'mn_RK7_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7,factor=150);\n \+ errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+\"pZ'*4%!#C7$%Ischeme~with~a~s mall~principal~error~normG$\"+iNMCN!#B7$%Pscheme~with~a~moderately~lar ge~stability~regionG$\"+*\\v3e#F07$%Fscheme~with~a~larger~stability~re gionG$\"+7xV!H#F07$%Lscheme~that~satisfies~some~error~conditionsG$\"+( G\")QQ$F0Q)pprint426\"" }}}{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 0 497 "evalf[20](plot(['mn_RK7_1'(x)-m(x),'mn_RK7_2'(x)-m(x ),'mn_RK7_3'(x)-m(x),'mn_RK7_4'(x)-m(x),\n'mn_RK7_5'(x)-m(x)],x=0..0.5 ,color=[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3, .9)],\nlegend=[`scheme of Sharp and Smart`,`scheme with a small princi pal error norm`,`scheme with a moderately large stability region`,`sch eme with a larger stability region`,`scheme that satisfies some error \+ conditions`],\nfont=[HELVETICA,9],title=`error curves for 10 stage ord er 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 904 545 545 {PLOTDATA 2 "6+-%'CURVESG6%7jp7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#A$!\" \"!#>7$$\"5qmmmmT&)G\\aF-$\"$/$F07$$\"5SLLL3x1h6oF-$\"%'3$F07$$\"50+++ ]7G$R<)F-$\"&#Q=F07$$\"5qmmTNYL^9&)F-$\"&wr#F07$$\"5SLL$3-)Q4b))F-$\"& r%RF07$$\"50++D19Wn&>*F-$\"&6k&F07$$\"5qmmm\"z%\\DO&*F-$\"&S%zF07$$\"5 NLL3x\"[No()*F-$\"'k.6F07$$\"5+++Dc,;u@5!#@$\"'5O7F07$$\"5nm;z%\\l*zb5 Ffn$\"'XJ7F07$$\"5MLLLL3x&)*3\"Ffn$\"'$oA\"F07$$\"5nmm\"z%\\v#pK\"Ffn$ \"'T&>\"F07$$\"5+++]i!R(*Rc\"Ffn$\"'Pn6F07$$\"5MLe9\"4&[EB;Ffn$\"'\"R; \"F07$$\"5nm;z>6B`#o\"Ffn$\"'ql6F07$$\"5++vV[r(*zT >Ffn$\"'(\\R\"F07$$\"5n;zp[#)eB\\>Ffn$\"''o[\"F07$$\"5ML3-j7'p)y>Ffn$ 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n$\"'!of'F07$F_o$\"'.slF07$Fdo$\"'4/kF07$Fio$\"'usiF07$F^p$\"'QsiF07$F cp$\"'%4J'F07$Fhp$\"'Y!)RFh\\l7$Fh`l$\"(Ls#QFh\\l7$F]al$\"(F Pn$Fh\\l7$Fbal$\"(ra_$Fh\\l7$Fgal$\"(F\\S$Fh\\l7$F\\bl$\"(oGG$Fh\\l7$F abl$\"(6\\;$Fh\\l7$Ffbl$\"(Ro0$Fh\\l7$F[cl$\"(5'eHFh\\l7$F`cl$\"(Ji&GF h\\l7$Fecl$\"(y(pFFh\\l7$Fjcl$\"(&y#o#Fh\\l7$F_dl$\"(a$3EFh\\l7$Fddl$ \"(.9`#Fh\\l7$Fidl$\"(CHY#Fh\\l7$F^el$\"(9^R#Fh\\l7$Fcel$\"(!HKBFh\\l7 $Fhel$\"([*pAFh\\l7$F]fl$\"(#*H@#Fh\\l7$Fbfl$\"(sw:#Fh\\l7$Fgfl$\"(Ub5 #Fh\\l7$F\\gl$\"(*))f?Fh\\l7$Fagl$\"(^+,#Fh\\l7$Ffgl$\"($en>Fh\\l7$F[h l$\"(uV#>Fh\\l7$F`hl$\"(!*[)=Fh\\l7$Fehl$\"(&RW=Fh\\l-F`]o6&F\\il$\")# )eqkFd]o$\"))eqk\"Fd]oFc_p-F`il6#%Fscheme~with~a~larger~stability~regi onG-F$6%7jpF'Ffil7$F+$\"#8F07$F]jl$\"$R%F07$F2$\"%a_F07$F7$\"&\"HOF07$ F<$\"'NkWP'F h\\l7$F^`l$\"(6C5'Fh\\l7$Fc`l$\"(+(ReFh\\l7$Fh`l$\"(o_h&Fh\\l7$F]al$\" (d+R&Fh\\l7$Fbal$\"(aD<&Fh\\l7$Fgal$\"(7d*\\Fh\\l7$F\\bl$\"(\\m\"[Fh\\ l7$Fabl$\"(.Ok%Fh\\l7$Ffbl$\"(i][%Fh\\l7$F[cl$\"(g4M%Fh\\l7$F`cl$\"(l2 >%Fh\\l7$Fecl$\"(OR1%Fh\\l7$Fjcl$\"(7j$RFh\\l7$F_dl$\"(>r#QFh\\l7$Fddl $\"(DUr$Fh\\l7$Fidl$\"(gPh$Fh\\l7$F^el$\"(wU^$Fh\\l7$Fcel$\"(2@U$Fh\\l 7$Fhel$\"(T1L$Fh\\l7$F]fl$\"(zqC$Fh\\l7$Fbfl$\"(;f;$Fh\\l7$Fgfl$\"(L%* 3$Fh\\l7$F\\gl$\"(_C-$Fh\\l7$Fagl$\"(H$\\HFh\\l7$Ffgl$\"(?q)GFh\\l7$F[ hl$\"(AO#GFh\\l7$F`hl$\"(#plFFh\\l7$Fehl$\"(xiq#Fh\\l-Fjhl6&F\\ilF(Fc[ nFa[n-F`il6#%Lscheme~that~satisfies~some~error~conditionsG-%%FONTG6$%* HELVETICAGFb[n-%+AXESLABELSG6$Q\"x6\"Q!F\\bq-%&TITLEG6#%Verror~curves~ for~10~stage~order~7~Runge-Kutta~methodsG-%%VIEWG6$;F(Fehl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sha rp and Smart" "scheme with a small principal error norm" "scheme with \+ a moderately large stability region" "scheme with a larger stability r egion" "scheme that satisfies some error conditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 10 of 10 stage, order 7 Rung e-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx = -(2*sin(5*x)+3*cos(7*x))*sinh(y);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$ *&,&*&\"\"#F&-%$sinG6#*&\"\"&F&%\"xGF&F&F&*&\"\"$F&-%$cosG6#*&\"\"(F&F 3F&F&F&F&-%%sinhG6#%\"yGF&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y(0) =sqrt(5)/2" "6#/-%\"yG6#\"\"!*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de : = diff(y(x),x)=-(2*sin(5*x)+3*cos(7*x))*sinh(y(x));\nic := y(0)=sqrt(5 )/2;\ndsolve(\{de,ic\},y(x));\nsimplify(convert(%,exp));\np := unapply (rhs(%),x):\nplot(p(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$ *&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F2F,F2F2F2F2*&\"\"$F2-%$cosG6#,$*& \"\"(F2F,F2F2F2F2F2-%%sinhG6#F)F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!,$*&\"\"#!\"\"\"\"&#\"\"\"F,F0" }}{PARA 11 " " 1 "" {XPPMATH 20 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F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F- F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint476\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+>l6`G!#B7$%Ischeme~w ith~a~small~principal~error~normG$\"+&[\\Qn\"F+7$%Pscheme~with~a~moder ately~large~stability~regionG$\"+b:X/=F+7$%Fscheme~with~a~larger~stabi lity~regionG$\"+)oFqN#!#A7$%Lscheme~that~satisfies~some~error~conditio nsG$\"+W$*49>F+Q)pprint486\"" }}}{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 0 723 " 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 Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stabili ty region`,`scheme with a larger stability region`,`scheme that satisf ies some error conditions`]:\nerrs := []:\nDigits := 30:\nfor ct to 5 \+ do\n pn_RK7_||ct := RK7_||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,tru e);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 5 do\n er rs := [op(errs),abs(pn_RK7_||ct(xx)-pxx)];\nend do:\nDigits := 10:\nli nalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*& \"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6 #%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~ width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint456\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%:scheme~of~Sharp~and~SmartG$\"+A!3.j#!#B7$%Ischeme~with~a~smal l~principal~error~normG$\"+U\"zrV%!#C7$%Pscheme~with~a~moderately~larg e~stability~regionG$\"+G7=naF+7$%Fscheme~with~a~larger~stability~regio nG$\"+-11!Q'!#A7$%Lscheme~that~satisfies~some~error~conditionsG$\"+7R( p_&F+Q)pprint466\"" }}}{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 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme with a small principal error \+ norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions `]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((p(x)- 'pn_RK7_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+o$o%eG!#B7$%Ischeme~with~a~sma ll~principal~error~normG$\"+wg-q;F+7$%Pscheme~with~a~moderately~large~ stability~regionG$\"+$=q.z\"F+7$%Fscheme~with~a~larger~stability~regio nG$\"+7!GvK#!#A7$%Lscheme~that~satisfies~some~error~conditionsG$\"+hXG ,>F+Q)pprint496\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the nume rical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 497 "evalf[20](plot([p(x)-'pn_RK7_1'(x),p(x)-'pn_RK7_2'(x ),p(x)-'pn_RK7_3'(x),p(x)-'pn_RK7_4'(x),\np(x)-'pn_RK7_5'(x)],x=0..2.2 ,color=[COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3, .9)],\nlegend=[`scheme of Sharp and Smart`,`scheme with a small princi pal error norm`,`scheme with a moderately large stability region`,`sch eme with a larger stability region`,`scheme that satisfies some error \+ conditions`],\nfont=[HELVETICA,9],title=`error curves for 10 stage ord er 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 860 651 651 {PLOTDATA 2 "6+-%'CURVESG6%7js7$$\"\"!F)F(7$$\"5LLLLLepo(R#!#@$!%: ()!#>7$$\"5mmmmm;RP&z%F-$!'mY8!#?7$$\"5NLL$3x@\"[QeF-$!'T$[\"F67$$\"5+ +++v=&)e\")oF-$!'\\u:F67$$\"5ILL3Fp@9.uF-$!'1;;F67$$\"5lmm;z>epCzF-$!' #4j\"F67$$\"5+++DJq%\\iW)F-$!'qQ;F67$$\"5LLLL$37.y'*)F-$!'+g;F67$$\"5& ****\\P%[ZMa&*F-$!'(fj\"F67$$\"5mmmTgP')395F6$!'D`;F67$$\"5LL$ek.!Gus5 F6$!';;;F67$$\"5+++]7jpRJ6F6$!'c<;F67$$\"5MLLek)G0([7F6$!'+u:F67$$\"5n 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MF17$Fg^m$!'FMMF17$Fa_m$!'FQMF17$F]s$!'%HV$F17$$\"5ML$e9JPhy\\$F1$!'q9 MF17$F^`m$!'4)Q$F17$$\"5mm;/EC:lANF1$!'#pM$F17$Fbs$!'h'H$F17$Fgs$!(,<3 $F-7$F\\t$!(8Ez#F-7$Fat$!(t\\a#F-7$Fft$!(#p:BF-7$F[u$!(cm:#F-7$Fibm$!( e%*4#F-7$F`u$!(T$f?F-7$Ffcm$!(Or.#F-7$Feu$!()pG?F-7$Fhdm$!(LT.#F-7$Fju $!(gP0#F-7$Feem$!(T#)3#F-7$F_v$!(lk8#F-7$Fdv$!(Y)[AF-7$Fiv$!(k/S#F-7$F ^w$!(!HlDF-7$Fcw$!(/Yj#F-7$Fhw$!(7**o#F-7$F\\gm$!(y/r#F-7$F]x$!(=`s#F- 7$Figm$!(`St#F-7$Fbx$!(-pt#F-7$F[im$!(XFt#F-7$Fgx$!(V5s#F-7$Fhim$!(3Aq #F-7$F\\y$!(#HvEF-7$F`jm$!(l'4EF-7$Fay$!(cY_#F-7$Ffy$!(19I#F-7$F[z$!(c #z?F-7$F`z$!(kz%=F-7$Fez$!(l,l\"F-7$Fjz$!(]eZ\"F-7$F_[l$!(`&[8F-7$F^]n $!(SAI\"F-7$Fd[l$!(*fq7F-7$Ff]n$!(Z/E\"F-7$F[^n$!(\"*QD\"F-7$F`^n$!(=; D\"F-7$Fi[l$!((H`7F-7$Fh^n$!(\"Gf7F-7$F]_n$!(#zp7F-7$Fb_n$!(G_G\"F-7$F ^\\l$!(HmI\"F-7$Fj_n$!(pMO\"F-7$Fc\\l$!(ZDW\"F-7$Fh\\l$!(5-l\"F-7$F]]l $!(I*))>F-7$Fban$!(2(y@F-7$Fb]l$!(FmS#F-7$Fg]l$!(#z6FF-7$F\\^l$!(kL4$F -7$Fa^l$!(93J$F-7$Ff^l$!',eNF17$F[_l$!'nFQF17$F`_l$!'/ATF17$Fj_l$!'UjW F17$Fd`l$!'XF[F17$Fhal$!'A-_F17$Ffcl$!'!)>bF1-F[dl6&F]dlF`dlF_jnF]jn-F edl6#%Lscheme~that~satisfies~some~error~conditionsG-%%FONTG6$%*HELVETI CAGF^jn-%+AXESLABELSG6$Q\"x6\"Q!Fa]q-%&TITLEG6#%Verror~curves~for~10~s tage~order~7~Runge-Kutta~methodsG-%%VIEWG6$;F(Ffcl;$!$j&!#;F`dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sharp a nd Smart" "scheme with a small principal error norm" "scheme with a mo derately large stability region" "scheme that satisfies some error con ditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 11 of 10 stage, order 7 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that appears in an article by F. G. Lether: Mathematics of Computation, V ol. 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(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x) G-%%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 0 815 "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 Sharp and Smart`,`scheme with a small princi pal error norm`,`scheme with a moderately large stability region`,`sch eme with a larger stability region`,`scheme that satisfies some error \+ conditions`]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n Qn_RK7_| |ct := RK7_||ct(Q(x,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,fal se);\n sm := 0: numpts := nops(Qn_RK7_||ct):\n for ii to numpts do \n sm := sm+(Qn_RK7_||ct[ii,2]-q(Qn_RK7_||ct[ii,1]))^2;\n end 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,&*(-%$expG6#,$%\"xG!\"\"\"\"\",& F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\" \"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F 07$%1no.~of~steps:~~~GFFQ)pprint506\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~ Sharp~and~SmartG$\"+I@2XZ!#@7$%Ischeme~with~a~small~principal~error~no rmG$\"+/A&py\"F+7$%Pscheme~with~a~moderately~large~stability~regionG$ \"+2'R2X%F+7$%Fscheme~with~a~larger~stability~regionG$\"+jzPOE!#?7$%Ls cheme~that~satisfies~some~error~conditionsG$\"+S>V%p%F+Q)pprint516\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fo llowing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }} {PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each o f the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG \"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 739 "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 S harp and Smart`,`scheme with a small principal error norm`,`scheme wit h a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`]:\nerrs := []: \nDigits := 30:\nfor ct to 5 do\n qn_RK7_||ct := RK7_||ct(Q(x,y),x,y ,x0,evalf(y0),evalf(hh),numsteps,true);\nend do:\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(qn_RK7_||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)pprint5 26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+ht\"G(p !#?7$%Ischeme~with~a~small~principal~error~normG$\"+%Rn9y#F+7$%Pscheme ~with~a~moderately~large~stability~regionG$\"+oi4vuF+7$%Fscheme~with~a ~larger~stability~regionG$\"+@?P-I!#>7$%Lscheme~that~satisfies~some~er ror~conditionsG$\"+(y+L)yF+Q)pprint536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re 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 num erical integration by the 7 point Newton-Cotes method over 200 equal s ubintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "mthds := [` scheme of Sharp and Smart`,`scheme with a small principal error norm`, `scheme with a moderately large stability region`,`scheme with a large r stability region`,`scheme that satisfies some error conditions`]:\ne rrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((q(x)-'qn_RK 7_||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$%:scheme~of~Sharp~and~SmartG$\"+9(QV\\$ !#@7$%Ischeme~with~a~small~principal~error~normG$\"+d;`K5F+7$%Pscheme~ with~a~moderately~large~stability~regionG$\"+;RFnEF+7$%Fscheme~with~a~ larger~stability~regionG$\"+2mF'G#!#?7$%Lscheme~that~satisfies~some~er ror~conditionsG$\"+^r*R\"GF+Q)pprint546\"" }}}{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 0 514 "evalf[25](plot(['qn_RK7_1'( x)-q(x),'qn_RK7_2'(x)-q(x),'qn_RK7_3'(x)-q(x),'qn_RK7_4'(x)-q(x),\n'qn _RK7_5'(x)-q(x)],x=0..0.5,-7.5e-22..5e-22,font=[HELVETICA,9],\ncolor=[ COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nl egend=[`scheme of Sharp and Smart`,`scheme with a small principal erro r norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditio ns`],\ntitle=`error curves for 10 stage order 7 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 783 536 536 {PLOTDATA 2 "6+-%'CURVESG6 %7hn7$$\"\"!F)F(7$$\":MLLLLLL$3x&)*3\"!#E$\"\"#!#D7$$\":nmmmmm;H2P\"Q? 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{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/ *&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" } {TEXT -1 2 " " }{XPPEDIT 18 0 "5*y*sin^7*7*x;" "6#*,\"\"&\"\"\"%\"yGF %%$sinG\"\"(F(F%%\"xGF%" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0) = 1; " "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(16/49+5/3136*cos*49*x-cos*35*x/64+5/64*cos*21*x-25/64*cos*7*x);" "6#/%\"yG-%$expG6#,,*&\"#;\"\"\"\"#\\!\"\"F+*,\"\"&F+\"%OJF-%$cosGF+F, F+%\"xGF+F+**F1F+\"#NF+F2F+\"#kF-F-*,F/F+F5F-F1F+\"#@F+F2F+F+*,\"#DF+F 5F-F1F+\"\"(F+F2F+F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "de := diff(y(x),x)=5*y( x)*sin(7*x)^7;\nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=combine(( numer(rhs(%))/convert(denom(rhs(%)),exp)));\nr := unapply(rhs(%),x):\n plot(r(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 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\\=;'G%F2$\"32(*p'*eu>U7F27$$\"3G#[Z%F2$\"3S)\\h/++++\"F27$$\"3 TM$3_5,-`%F2$\"3y/++\"F27$$\"3SnmT&G!e&e%F2$\"3$eA?a3RF+\"F27$$\" 3fLe*[=Y.h%F2$\"3M:+/vy285F27$$\"3m+]P%37^j%F2$\"356M^f6but%F2$\"3%\\.QNQyVc\"F 27$$\"3ID19>zl]ZF2$\"3S[]>-K'z\"F27$$\"37+]iSjE!z%F2$\"3&)4L&Q*oyW=F27$ $\"3y*\\7G))Rb\"[F2$\"3)fCVDB`!)*=F27$$\"3L+++DM\"3%[F2$\"3$Gxb\\1an\" >F27$$\"3i]P4'>]M&[F2$\"3p-qe54u>>F27$$\"3)3](=np3m[F2$\"3zS9J5F#4#>F2 7$$\"3G]7GQPsy[F2$\"3j:7^7UI@>F27$$\"3a+]P40O\"*[F2$\"3L:$**HR(R@>F27$ $\"3s+voa-oX\\F2$\"3%e\"e*Rr89#>F27$$\"\"&F)$\"3n\\kX'z.7#>F2-%'COLOUR G6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"x G%%y(x)G-%%VIEWG6$;F(F_[q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete 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 "R := \+ (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1 :\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0)],[` step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [` scheme of Sharp and Smart`,`scheme with a small principal error norm`, `scheme with a moderately large stability region`,`scheme with a large r stability region`,`scheme that satisfies some error conditions`]:\ne rrs := []:\nDigits := 30:\nfor ct to 5 do\n Rn_RK7_||ct := RK7_||ct( R(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Rn_RK 7_||ct):\n for ii to numpts do\n sm := sm+(Rn_RK7_||ct[ii,2]-r( Rn_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpt s)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~ point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\" $+&Q)pprint556\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+ >[F,7!#@7$%Ischeme~with~a~small~principal~error~normG$\"+(3@r^)!#A7$%P scheme~with~a~moderately~large~stability~regionG$\"+z7'f%HF07$%Fscheme ~with~a~larger~stability~regionG$\"+F?e#H\"!#?7$%Lscheme~that~satisfie s~some~error~conditionsG$\"+)Q\">egF+Q)pprint566\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 690 "R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numsteps : = 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numst eps]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stability regio n`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`]:\nerrs := []:\nDigits := 25:\nfor ct to 5 do\n r n_RK7_||ct := RK7_||ct(R(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nx x := 4.999: rxx := evalf(r(xx)):\nfor ct to 5 do\n errs := [op(errs) ,abs(rn_RK7_||ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7&7$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xG F,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7 $%1no.~of~steps:~~~G\"$+&Q)pprint576\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~ of~Sharp~and~SmartG$\"+Z`WKE!#@7$%Ischeme~with~a~small~principal~error ~normG$\"+bF%=c\"F+7$%Pscheme~with~a~moderately~large~stability~region G$\"+)\\I7.'!#A7$%Fscheme~with~a~larger~stability~regionG$\"+Y8^$y#!#? 7$%Lscheme~that~satisfies~some~error~conditionsG$\"+Q%3RI\"F9Q)pprint5 86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the in terval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special pro cedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical inte gration by the 7 point Newton-Cotes method over 200 equal subintervals ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of S harp and Smart`,`scheme with a small principal error norm`,`scheme wit h a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`]:\nerrs := []: \nDigits := 20:\nfor ct to 5 do\n sm := NCint((r(x)-'rn_RK7_||ct'(x) )^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(err s),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme ~of~Sharp~and~SmartG$\"+`T#y>\"!#@7$%Ischeme~with~a~small~principal~er ror~normG$\"+&fDE])!#A7$%Pscheme~with~a~moderately~large~stability~reg ionG$\"+8>UNHF07$%Fscheme~with~a~larger~stability~regionG$\"+8Ig*G\"!# ?7$%Lscheme~that~satisfies~some~error~conditionsG$\"+'y0M/'F+Q)pprint5 96\"" }}}{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 0 495 "e valf[20](plot([r(x)-'rn_RK7_1'(x),r(x)-'rn_RK7_2'(x),r(x)-'rn_RK7_3'(x ),r(x)-'rn_RK7_4'(x),\nr(x)-'rn_RK7_5'(x)],x=0..5,color=[COLOR(RGB,.5, 0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`schem e of Sharp and Smart`,`scheme with a small principal error norm`,`sche me with a moderately large stability region`,`scheme with a larger sta bility region`,`scheme that satisfies some error conditions`],\nfont=[ HELVETICA,9],title=`error curves for 10 stage order 7 Runge-Kutta meth ods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 959 559 559 {PLOTDATA 2 "6+-%'CU RVESG6%7icl7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$\"%AE!#>7$$\"5MLLLL3x&)*3 \"!#?$!&5&eF07$$\"5nm\"z>'o^7\\6F4$\"&YU#F07$$\"5++]i!*GER37F4$\"'oy:F 07$$\"5ML3F>*3gwE\"F4$\"'>>;F07$$\"5nmm\"z%\\v#pK\"F4$\"'eOAF07$$\"5++ Dcw4]>'Q\"F4$\"'>o@F07$$\"5ML$3_+ZiaW\"F4$\"'Yd=F07$$\"5nmT&Q.$*HZ]\"F 4$\"&'**eF07$$\"5+++]i!R(*Rc\"F4$\"&xu&F07$$\"5n;H#o27JOf\"F4$!&2o\"F0 7$$\"5MLe9\"4&[EB;F4$!&Z=)F07$$\"5+](oa5e)*Gl\"F4$!&\\B)F07$$\"5nm;z>6 B`#o\"F4$!&W%))F07$$\"5++vV[r(*zT>F4$\"'&4*=F07$$\"5nmmm\"H2P\"Q?F4$\"'6TKF07$$\"5MLLek. pu/BF4$!((z!=#F07$$\"5+++]PMnNrDF4$!(PLR%F07$$\"5nmmT5ll'z$GF4$!(rGh\" F07$$\"5MLLL$eRwX5$F4$!'(eD\"F07$$\"5MLLLe*[`HP$F4$!($[?9F07$$\"5MLLLL 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(`1!GF07$$\"5NL$e9T.&\\ZYF0$!(id&HF07$Fghx$!(q#*3$F07$$\"5-+]PfL4EsYF0 $!(cU.$F07$F[\\r$!([l\"HF07$$\"5o\"zpB^eWzo%F0$!(Hs$HF07$F`\\r$!(^X%GF 07$$\"5N3FWq))fa%p%F0$!(M\"*)GF07$Fe\\r$!(!eeHF07$Fj\\r$!(&*Q<$F07$F_] r$!(:mA$F07$Fd]r$!(fKN$F07$Fi]r$!(H7)RF07$$\"5Ne9mW*>]Vr%F0$!(r$[SF07$ Feix$!(Y7@%F07$$\"5-vVt-.;&4s%F0$!(HK5&F07$F^^r$!(rs=&F07$F]jx$!(>WM&F 07$Fbjx$!(k'>hF07$Fgjx$!(VZ@'F07$Fc^r$!(FrK'F07$F_[y$!(8`f'F07$Fd[y$!( E))o'F07$Fi[y$!(UAu'F07$Fh^r$!(W#ojF07$$\"5Ne*)4$4ieRv%F0$!(=zW'F07$Fa \\y$!(,oX'F07$$\"5-v=<^C+cgZF0$!(omk&F07$F]_r$!(*)oq&F07$Fi\\y$!(%*)*p &F07$F^]y$!(M([[F07$Fc]y$!(q7*[F07$Fb_r$!(![#*[F07$F[^y$!(&zsVF07$Fg_r $!(-[G%F07$F[ar$!(^Uj%F07$Fear$!('*)*>&F07$Fjar$!(e;(eF07$F_br$!(^*\\l F07$$\"5-]PMx " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 13 of 10 stage, order 7 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!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\" &!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5;" "6#/%\"yG* &\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6#,&* &%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation " } {XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponenti al term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5 " "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this t erm disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF& F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a nume rical solution from the correct solution tends to become rapidly magni fied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5; \ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%),x):\nplot(e(x),x=0..8,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$ cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3E LLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLL LLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm ^&Q%R)F,$!39J7$$\"3))** ***\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LLL=\"\\g**FK7 $$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g (y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@E VOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n#=i\"RF,7$$\"3)*******RP)4M#FO$ \"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3:@(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO $\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+<'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$ \"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\" 3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3![MOl&\\jZVF,7$$\"3!)*****\\9!H. 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37+2hu:afHF,7$$\"35LLL.a#o$oFO$!3;\"e/Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4` 1I$pa!o\"F,7$$\"3y******z]rfrFO$!3pfL'*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3? j;%3XMsQ#FK7$$\"3/LLL8-V&\\(FO$\"3qi(R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3Z X^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,- %'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABEL SG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 765 "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 Sharp and Smart`,`scheme with a small principal error norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error con ditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n En_RK7_||ct := RK7_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: \+ numpts := nops(En_RK7_||ct):\n for ii to numpts do\n sm := sm+( En_RK7_||ct[ii,2]-e(En_RK7_||ct[ii,1]))^2;\n end do:\n errs := [op (errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,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)pprint606\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~ SmartG$\"+T[%yB\"!#?7$%Ischeme~with~a~small~principal~error~normG$\"+R <#**4#!#A7$%Pscheme~with~a~moderately~large~stability~regionG$\"+#H8;W \"F+7$%Fscheme~with~a~larger~stability~regionG$\"+\"Hdac%F+7$%Lscheme~ that~satisfies~some~error~conditionsG$\"+%\\'R#\\$!#@Q)pprint616\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The foll owing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }} {PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each o f the methods at the point where " }{XPPEDIT 18 0 "x = 7.999;" "6#/% \"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 696 "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 Sharp and Smart`,`sc heme with a small principal error norm`,`scheme with a moderately larg e stability region`,`scheme with a larger stability region`,`scheme th at satisfies some error conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n en_RK7_||ct := RK7_||ct(E(x,y),x,y,x0,evalf(y0),hh,num steps,true);\nend do:\nxx := 7.999: exx := evalf(e(xx)):\nfor ct to 5 \+ do\n errs := [op(errs),abs(en_RK7_||ct(xx)-exx)];\nend do:\nDigits : = 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\" \"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~ width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint626\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%:scheme~of~Sharp~and~SmartG$\"+b'4%fo!#?7$%Ischeme~with~a~small~ principal~error~normG$\"*uaO;\"F+7$%Pscheme~with~a~moderately~large~st ability~regionG$\"+Uvd))zF+7$%Fscheme~with~a~larger~stability~regionG$ \"+3!4*HD!#>7$%Lscheme~that~satisfies~some~error~conditionsG$\"+q9GN>F +Q)pprint636\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme with a small principal error \+ norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions `]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((e(x)- 'en_RK7_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+(e6]@\"!#?7$%Ischeme~with~a~sm all~principal~error~normG$\"+!['=h?!#A7$%Pscheme~with~a~moderately~lar ge~stability~regionG$\"+*H@]T\"F+7$%Fscheme~with~a~larger~stability~re gionG$\"+EHC\"[%F+7$%Lscheme~that~satisfies~some~error~conditionsG$\"+ 1e(zU$!#@Q)pprint646\"" }}}{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 0 515 "evalf[20](plot([e(x)-'en_RK7_1'(x),e(x)-'en_RK7_2'(x ),e(x)-'en_RK7_3'(x),e(x)-'en_RK7_4'(x),\ne(x)-'en_RK7_5'(x)],x=0..2,- 5.1e-16..1.63e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR( RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of Sharp \+ and Smart`,`scheme with a small principal error norm`,`scheme with a m oderately large stability region`,`scheme with a larger stability regi on`,`scheme that satisfies some error conditions`],\ntitle=`error curv es for 10 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 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y$\",(*4W\")\\&Fdq7$Fcy$\"+3,pziF^q7$Fhy$\"+o1LsrF^q7$F]z$\"+0$)px!)F^ q7$Fbz$\"*/^t4*F-7$Fgz$\"+btmK5F-7$F\\[l$\"+19@s6F-7$Fa[l$\"+T6SB8F-7$ Ff[l$\"+b54%\\\"F-7$F[\\l$\"+`cs%f\"F-7$F`\\l$\"+2&Q@q\"F-7$Fe\\l$\"+c hy;=F-7$Fj\\l$\"+.f:R>F--F_]l6&Fa]lFe]lF[ilFihl-Fi]l6#%Lscheme~that~sa tisfies~some~error~conditionsG-%%FONTG6$%*HELVETICAGFjhl-%+AXESLABELSG 6$Q\"x6\"Q!F`[o-%&TITLEG6#%Verror~curves~for~10~stage~order~7~Runge-Ku tta~methodsG-%%VIEWG6$;F(Fj\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sharp and Smart" "scheme w ith a small principal error norm" "scheme with a moderately large stab ility region" "scheme with a larger stability region" "scheme that sat isfies some error conditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 49 "Test 14 of 10 stage, order 7 Runge-Kutta methods " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx = 10*x*cos*x-10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",& **\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F&%\"yGF&F(" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "y(0) = sqrt(5);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101" "6#/%\"yG*&\"$+\"\" \"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*cos*x-990/10201" "6 #,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\"$!**F&\"&,-\"!\"\"F+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "cos*x+10/101" "6#,&*&%$cosG\"\"\"%\"xGF&F&*&\"#5F&\" $,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x-200/10201" "6#,&* (%\"xG\"\"\"%$sinGF&F%F&F&*&\"$+#F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sin*x+(990/10201+sqrt(5))*exp(-10*x)" "6#,&*&%$sinG\"\" \"%\"xGF&F&*&,&*&\"$!**F&\"&,-\"!\"\"F&-%%sqrtG6#\"\"&F&F&-%$expG6#,$* &\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos (x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapp ly(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF, ,&*(\"#5\"\"\"F,F0-%$cosGF+F0F0*&F/F0F)F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&#\"$+\"\"$,\"\"\"\"*&F'F--%$cosG F&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&-%'CURVESG 6$7hn7$$\"\"!F)$\"3\")*y*\\xz1OA!#<7$$\"3ALL$3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$\"3m****\\7G$R<)F0$\"3<_u( oLbK,\"F,7$$\"3GLLL3x&)*3\"!#=$\"3(**[ro!GyVzF@7$$\"3))**\\i!R(*Rc\"F@ $\"3A'ysO]2xW&F@7$$\"3umm\"H2P\"Q?F@$\"3/$)oqvSKmSF@7$$\"3YLek.pu/BF@$ \"3$Qjx*Gs<7OF@7$$\"3!***\\PMnNrDF@$\"3M:4%*3rt@LF@7$$\"3MmT5ll'z$GF@$ 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\"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$$\"37+]iSjE!z%F,$!3+r[gfMO'=*F07$$\"3a+]P40O\"*[ F,$\"3+2*eSHde(QF@7$$\"\"&F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;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 "d iscrete solution" }{TEXT -1 44 " based on each of the methods and give s the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each \+ solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 781 "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)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`s cheme of Sharp and Smart`,`scheme with a small principal error norm`,` scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditions`]:\ner rs := []:\nDigits := 20:\nfor ct to 5 do\n Bn_RK7_||ct := RK7_||ct(B (x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops (Bn_RK7_||ct):\n for ii to numpts do\n sm := sm+(Bn_RK7_||ct[ii ,2]-evalf(b(Bn_RK7_||ct[ii,1])))^2;\n end do:\n errs := [op(errs), sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,ev alf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slop e~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0 initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$ %1no.~of~steps:~~~G\"$+&Q)pprint676\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~ Sharp~and~SmartG$\"+4%*3*H(!#B7$%Ischeme~with~a~small~principal~error~ normG$\"+%p%=]^!#C7$%Pscheme~with~a~moderately~large~stability~regionG $\"+5!>f'QF+7$%Fscheme~with~a~larger~stability~regionG$\"+VbK=^F07$%Ls cheme~that~satisfies~some~error~conditionsG$\"+([EM@)F0Q)pprint686\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fo llowing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }} {PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each o f the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/% \"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 705 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`slope fi eld: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`scheme of Sharp and \+ Smart`,`scheme with a small principal error norm`,`scheme with a moder ately large stability region`,`scheme with a larger stability region`, `scheme that satisfies some error conditions`]:\nerrs := []:\nDigits : = 25:\nfor ct to 5 do\n bn_RK7_||ct := RK7_||ct(B(x,y),x,y,x0,evalf( y0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfo r ct to 5 do\n errs := [op(errs),abs(bn_RK7_||ct(xx)-bxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([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) pprint696\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+$[]US $!#C7$%Ischeme~with~a~small~principal~error~normG$\"+#[,3v\"!#D7$%Psch eme~with~a~moderately~large~stability~regionG$\"+#H\")R7\"F+7$%Fscheme ~with~a~larger~stability~regionG$\"+L%f[0#F+7$%Lscheme~that~satisfies~ some~error~conditionsG$\"*j_9<&F0Q)pprint706\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`scheme of Sharp and Smart`,`scheme w ith a small principal error norm`,`scheme with a moderately large stab ility region`,`scheme with a larger stability region`,`scheme that sat isfies some error conditions`]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((b(x)-'bn_RK7_||ct'(x))^2,x=0..5,adaptive=false, numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\n Digits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+MG GIq!#B7$%Ischeme~with~a~small~principal~error~normG$\"+eK=g\\!#C7$%Psc heme~with~a~moderately~large~stability~regionG$\"+NL%Rs$F+7$%Fscheme~w ith~a~larger~stability~regionG$\"+jv1I\\F07$%Lscheme~that~satisfies~so me~error~conditionsG$\"+0zH-zF0Q)pprint716\"" }}}{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 0 513 "evalf[20](plot([b(x)-'bn_R K7_1'(x),b(x)-'bn_RK7_2'(x),b(x)-'bn_RK7_3'(x),b(x)-'bn_RK7_4'(x),\nb( x)-'bn_RK7_5'(x)],x=0..0.65,numpoints=100,font=[HELVETICA,9],\ncolor=[ COLOR(RGB,.5,0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nl egend=[`scheme of Sharp and Smart`,`scheme with a small principal erro r norm`,`scheme with a moderately large stability region`,`scheme with a larger stability region`,`scheme that satisfies some error conditio ns`],\ntitle=`error curves for 10 stage order 7 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 1041 574 574 {PLOTDATA 2 "6+-%'CURVESG 6%7^\\l7$$\"\"!F)F(7$$\"5SSSSSS:N<7 $$\"5+,,,,^)yLH%F-$\"%+6F37$$\"5?@@@@@Y0_^F-$\"%:ZF37$$\"5STTTT\"RI2,' F-$\"&Oh\"F37$$\"5ihhhhhhSpoF-$\"&:o%F37$$\"5XXXX&zNNlh(F-$\"'Xm5F37$$ \"5IHHHHaXmj$)F-$\"'j[AF37$$\"5?@@@Y_\"Hst)F-$\"'[&=$F37$$\"55888j]Pz5 \"*F-$\"'7ZWF37$$\"5+000!)[$eV[*F-$\"'XDhF37$$\"5&ppppp%H#z&)*F-$\"'7L $)F37$$\"5HHHagC0Z/5!#@$\"'T*)F37$$\"5mmmmmTRt?;Fco$\"'+m*)F37$$\"5____xLkv/Fco$\"(rKW\"F37$$\"5GGGG.s(oz(>Fco$\"($ea:F37$$\"5YYYY'Rj8\"** >Fco$\"(tUo\"F37$$\"5kkkk*e\\e--#Fco$\"(Eno\"F37$$\"5$GGGGyN.9/#Fco$\" 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'F1-F[`o6&F]`oF``oFhhqFfhq-Fd`o6#%Lscheme~that~satisfies~some~error~co nditionsG-%%FONTG6$%*HELVETICAGFghq-%+AXESLABELSG6$Q\"x6\"Q!Fj_x-%&TIT LEG6#%Verror~curves~for~10~stage~order~7~Runge-Kutta~methodsG-%%VIEWG6 $;F(Fe_o%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme of Sharp and Smart" "scheme with a small princip al error norm" "scheme with a moderately large stability region" "sche me with a larger stability region" "scheme that satisfies some error c onditions" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 15 \+ of 10 stage, order 7 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods \+ for Ordinary Differential Equations, Hull, Enright, Fellen and Sedgwic k,\n Siam Journal on Numerical Analysis, Vol. 9, No. 4 (Dec. 19 72), page 617, Example A5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = (y-x)/(y+x);" "6# /*&%#dyG\"\"\"%#dxG!\"\"*&,&%\"yGF&%\"xGF(F&,&F+F&F,F&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/(x^2))+4*arctan(y/x)+4*l n*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"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$-%\"yG6#%\"xGF,*&,&F) \"\"\"F,!\"\"F/,&F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/ -%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'R ootOfG6#,,*&\"\"#\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F. !\"\"*&\"\"%F.-%'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can b e given more simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+ Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6 #*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 18 "The solution (for " }{TEXT 282 1 "x" } {TEXT -1 47 " increasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thet aGF2F*" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#Pi G\"\"\"\"\"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG \"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff (%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"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,linestyle=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symb ol=[circle$2,diamond,cross],\n symbolsize=[12,10$3],c olor=[black,green$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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G6#%&POINTG-F$6&Fh\\n-Fjz6&F\\[lF`[lF][lF`[l-Fb]n6$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_nFc_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The following procedure uses " }{TEXT 0 6 "fsolve" } {TEXT -1 23 " to solve the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6 #,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,, &*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{TEXT 280 1 "y" }{TEXT -1 25 " numerically in terms of " }{TEXT 281 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 := evalf(exp(Pi/2)):\nplot('phi'(x),x=1..uu,numpoint s=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$$\"+4:b2 8F-$\"+-Eit(*F37$$\"+s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig9X'*F37 $$\"+jM?A9F-$\"+N0$)y&*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?]\"F-$\"+ 2v=4%*F37$$\"+d@iS:F-$\"+c3U<$*F37$$\"+1sjv:F-$\"+R4cG#*F37$$\"+'[tsh \"F-$\"+gf2;\"*F37$$\"+x[a_;F-$\"+=V+:!*F37$$\"+*)Qd$p\"F-$\"+F-$\"+rA&\\1)F37$$\"+-Hdj>F-$\"+oiz%*yF37$$\"+`3.**>F-$\"+4o(3u(F37$$ \"+6kKP?F-$\"+KCbovF37$$\"+q%*)o2#F-$\"+LT&QQ(F37$$\"+0Uf:@F-$\"+Okc'> (F37$$\"+EI/`@F-$\"+7w54qF37$$\"+cNi%>#F-$\"+Snr$z'F37$$\"+8b)>B#F-$\" 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$\"+C/;ERF-$!+B0!=5\"F-7$$\"+$oA@'RF-$!+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$$\"+L76SXF-$!+'GZ)QE F-7$$\"+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))>qZF-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q .z%F-$!+!p\\]>%F-7$$\"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 826 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small pr incipal error norm`,`scheme with a moderately large stability region`, `scheme with a larger stability region`,`scheme that satisfies some er ror conditions`]:\nerrs := []: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn_RK7_||ct := RK7_||ct(C(x,y),x,y,x0,y0,hh,numsteps,false); \n sm := 0: numpts := nops(Cn_RK7_||ct):\n for ii to numpts do\n \+ if ct=1 then vals := [op(vals),phi(Cn_RK7_||ct[ii,1])] end if;\n \+ sm := sm+(Cn_RK7_||ct[ii,2]-vals[ii])^2;\n end do:\n errs := [ op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]( [mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 &7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initia l~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ v$Q)pprint726\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+Ag ^l6!#B7$%Ischeme~with~a~small~principal~error~normG$\"+#z-/l%!#C7$%Psc heme~with~a~moderately~large~stability~regionG$\"+N1/C8F+7$%Fscheme~wi th~a~larger~stability~regionG$\"+.'=2>(F07$%Lscheme~that~satisfies~som e~error~conditionsG$\"+l7(Qb\"F+Q)pprint736\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6#/%\"xG-%&FloatG6$\" %\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 689 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`scheme of Sharp and Smart`,`scheme with a small pr incipal error norm`,`scheme with a moderately large stability region`, `scheme with a larger stability region`,`scheme that satisfies some er ror conditions`]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n cn_R K7_||ct := RK7_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx : = 4.749: cxx := evalf(phi(xx)):\nfor ct to 5 do\n errs := [op(errs), abs(cn_RK7_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]( [mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 &7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initia l~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ v$Q)pprint746\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:scheme~of~Sharp~and~SmartG$\"+9f $oO\"!#A7$%Ischeme~with~a~small~principal~error~normG$\"+UeXRa!#B7$%Ps cheme~with~a~moderately~large~stability~regionG$\"+J**fN:F+7$%Fscheme~ with~a~larger~stability~regionG$\"+Guv$)yF07$%Lscheme~that~satisfies~s ome~error~conditionsG$\"+(pOx#=F+Q)pprint756\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "mthds := [`scheme of Sharp \+ and Smart`,`scheme with a small principal error norm`,`scheme with a m oderately large stability region`,`scheme with a larger stability regi on`,`scheme that satisfies some error conditions`]:\nerrs := []:\nDigi ts := 20:\nfor ct to 5 do\n sm := NCint(('phi'(x)-'cn_RK7_||ct'(x))^ 2,x=1..4.75,adaptive=false,numpoints=7,factor=200);\n errs := [op(er rs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%:schem e~of~Sharp~and~SmartG$\"+?EP\">&!#C7$%Ischeme~with~a~small~principal~e rror~normG$\"+w'*)e9#F+7$%Pscheme~with~a~moderately~large~stability~re gionG$\"+ic;`eF+7$%Fscheme~with~a~larger~stability~regionG$\"+N'GJ/$F+ 7$%Lscheme~that~satisfies~some~error~conditionsG$\"+/sk4qF+Q)pprint766 \"" }}}{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 0 530 "eva lf[30](plot(['cn_RK7_1'(x)-'phi'(x),'cn_RK7_2'(x)-'phi'(x),'cn_RK7_3'( x)-'phi'(x),'cn_RK7_4'(x)-'phi'(x),\n'cn_RK7_5'(x)-'phi'(x)],x=1..3.75 ,0..4.5e-20,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5,0,1),COLOR(RGB,.9 ,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`scheme of Sharp and Sm art`,`scheme with a small principal error norm`,`scheme with a moderat ely large stability region`,`scheme with a larger stability region`,`s cheme that satisfies some error conditions`],\ntitle=`error curves for 10 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 887 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45.000000 0 1 "sche me of Sharp and Smart" "scheme with a small principal error norm" "sch eme with a moderately large stability region" "scheme with a larger st ability region" "scheme that satisfies some error conditions" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "evalf[30](plot(['cn_RK7_1'(x)-'phi'(x),'cn_RK7_2'(x)-'phi'(x),'cn _RK7_3'(x)-'phi'(x),'cn_RK7_4'(x)-'phi'(x),\n'cn_RK7_5'(x)-'phi'(x)],x =3.75..4.6,-4.8e-17..9.8e-17,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.5, 0,1),COLOR(RGB,.9,.3,0),red,brown,COLOR(RGB,0,.3,.9)],\nlegend=[`schem e of Sharp and Smart`,`scheme with a small principal error norm`,`sche me with a moderately large stability region`,`scheme with a larger sta bility region`,`scheme that satisfies some error conditions`],\ntitle= `error curves for 10 stage order 7 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 900 623 623 {PLOTDATA 2 "6+-%'CURVESG6%7co7$$\"$v 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